poisson di geometric price replace brownian motion geometric financial market link process context di market market intensity model preserve property move velocity g account observe de propose implicit study type asymptotically asymptotically increment switch instant interest likelihood cs unfortunately treat square use context author chen al mean mesh play role organize observation consistency change change point price box trajectory shift occur govern poisson formula introduction explicit method follow square type estimator view et increment role study asymptotic estimator proof along require technical crucial increment indicate value depend increment consistent gaussian change represent value eq indicate sum residual follow square lead formula concern hypothesis permit brownian introduce conclude q brownian inequality yield tends let n b ny c c nm negligible study add consistency either sided motion manner brownian present convergence side brownian proof sketch k note write eq invariance explicit limiting expression prove show term multiply invariance principle analogously theorem transform define since converge zero k limit convergence set stock price asymptotic nevertheless finding confirm analysis evidence occur week value return process frequency asymptotic realize first rescale increment construct estimator construct confirm velocity part point confirm intuition graphical inspection period well change around two plot stock price set box discover fit difference et fit ar sequential couple discover first confirm finding et right series convert proposition di motion velocity usually real
accordingly vector indeed variable understand coefficient indeed make algebraic correspondence equation filter kf light smoothing secondly kf devise specific dedicated deriving along line variable linear innovation series mutually individually uncorrelated uncorrelated uncorrelated uncorrelated uncorrelated attain distribution moment specify kalman kf derive contain kf make rely assumption believe derivation simple purpose denote forecast forecast also kf recursive updating equation suppose wish write prove write b minimize minimize eq equation denote solve sense covariance kf traditionally kf derive follow distributional likelihood available note correspondence take general possible arrive rearrange kf thus tx ty x tx tx start tx tx tx r recursion kf smoothing parameter covariance reduce usual element correspondence kf apparent inversion matrix necessary clear kf multiplication application arrive although quite principle merely solve kf step linear step optimize know give kf regression coefficient perform error kf specify assume error develop statistical index minimum index highly asset stream p cover year stream stream price list poor web whereas stream yahoo add basis characteristic index stock namely historical datum period evolve stream comprise asset explanatory raw price process artificial commonly rule risk stream exhibit spread well assumption formal trading rule stream take associate pattern trading decision daily order trading return realize report result obtain base trading system financial goodness fit financial indicator excess return consider satisfactory financial movement peak cumulative mean rr rr rr rr gain loss ann mse incremental summary daily daily loss maximum percentage percentage win return mse multiply return three svd system without asset management index steady existence transaction restrictive transaction daily initial testing period economic soon figure regression figure change year smoothly jump fairly give vary context argue vary connection cost minimize show algorithmic trading aspect discussion point price index explain asset explanatory perhaps even dynamically stream asset investigation relate stream issue temporal mining core approach use arrive streaming pattern find trend stream adopt explanatory target easy task indeed slide window euclidean similarity stream among available stream grid measure dynamic suggest time extraction explanatory reduction incremental component well simulation show assume transaction negligible trading mean pattern data stream transaction trading spread great order rather asset portfolio may optimize capture financial capability potentially application predict evolve delay outli acknowledgement would david comment study stream infinite flow data stream task depend accounting dependence pattern probabilistic argue flexible penalize ordinary regression motivate application financial stream know kalman equivalence understand efficient promising trading report temporal mining flexible trading temporal mining develop concerned processing analyze high volume speed stream stream univariate stream structure collect sequential mining tool increasingly finance sensor security management many stream explore purpose trading core application lie need aware instant exploit another data decade trend trading trading variety resort serve specific strategy algorithmic automate trading enter intervention decide aspect recognition moreover automate system trade small financial stock exchange year trading statistical develop well many extension variation insight case face develop algorithmic trading firstly development storage generate massive datum require stream become frequency quickly information almost instantaneous decision secondly exploratory detect little require specification intervention identify vary evolve stream regression specify extent large explanatory algorithmic trading intra day stream stream model evolve smoothly organize briefly number trading arise background motivation flexible methodology exploratory fit impose specification know light modification original efficient numerically experimental trading complement extraction discussion relate direction popular trading market price market asset go go e price trend attempt capture market trend commonly relate serial trend asset price move serial trend attempt price trade direction occur vary strategy widely expect simple strategy historical stock tie together take equilibrium trading present figure two around trading implement two exploit instance great go make back term relationship quite may spread show capture implement refined trading simple upon extension look stock comprise index stock contract market exploit adopt simple somewhat relate trading pair asset asset represent stream possibly stream stream behind period underlie unobserved systematic component systematic include economic marker relate ultimately good asset asset estimate explanatory evolve stream asset interpret asset condition asset market factor exploit purpose analogy absolute soon correct market crucially rely accurately dynamically artificial asset involve predictor variable predictor parameter least ol evolve stream change evolve dynamically flexible generalization coefficient coefficient allow evolve probabilistic residual favorable aspect application temporal
condition simplify worth extreme exact value moreover condition multivariate couple year many stationary often aforementioned trend parametric structure tail residual parametric main direct dependence dominate describe serial asymptotic maxima last statistical method statistic maxima reflect dependence structure behavior naturally analyze de statistic diverse review short decide know largely matter discuss extreme markov chain time also discuss de contribution extreme continuous assume copy observe extension theory multivariate thus discuss article belong domain extreme section tail marginal end serial must take estimator interval different approach construct observation behavior classical tail originally propose directly want interval asymptotic mild assumption serial fashion serial dependence try tail behavior time seem well suited heavy tail appropriate consist clearly extreme event nice set wave analyze de co start wave wave period water hour point de de unfortunately application either one several often difficult trial sequel nonparametric compare robustness result hill serial study manuscript normality estimator strong independently prove normality hill comparable serial hill examine paper value extreme quantile estimate en de examine type eq suitable function tail sum develop publish de normality tail array sum en weak convergence regularity series verify en series tail asymptotic general estimator estimator quantile cf seem promise precisely move assume balanced tail prove strong time value suggest hill extreme result residual approximately moreover quantile turn work hill residual denote small statistic weakly absolute tend q result good rate hill asymptotic variance coefficient sequence number hill apply hill consider asymptotic equal claim first conclusion justify use hill occur analog directly hill unclear number hill low hill former estimator conversely apply hill inefficient series know behavior however move average move average relationship cd differ expansion small tail behave shift pareto second tail general result model plausible expect multiply g however even serious drawback mention mainly quantity may fulfil remark trivial nevertheless aware deviation see subsection satisfy estimate read eq favor equal variation equivalent may latter hill pareto variation imply u uk side pareto double pareto particularly favorable hill hill estimator residual unbiased contrast number statistics hill sensitive priori behavior aforementioned inspection proof straightforward calculations hill hill equals hence expect hill superiority carry quantile estimator quantile approximate simulated figure display square rmse solid resp dot dot versus use base outperform smaller sensitive one large residual direct quickly conversely somewhat rmse effect see summarize minimal error optimal choice lead rmse rmse minimal cc error k estimation k yield performance nonparametric next interpretation indeed quite nonlinear time eq perturb logarithmic relationship likely increase shift pareto series autoregressive fit turn difference residual capable deviation nominal size apply maximal power rejection alarm suitable sum accord dash indicate fit autoregressive figure rmse quantile simulate accord model analogously table base nonlinear extent cause optimal sharp contrast direct rely specific precise consequently minimal rmse rmse large rmse plot model cc rmse rmse error bias direct ex estimator plot solid quantile display mode vertical dotted model skew large spread contrast symmetric value indicate vertical dotted give even deviation assume time detect statistical estimator care analysis justify extreme estimator application interest financial period behavior consecutive call role denote sequence prove condition u converge type independence nx standardize typically reciprocal although know since maxima consecutive observation dependence application value interest field threshold maxima series consider recurrence denote relationship describe stationary standardize maxima fr de et al convention random drift tends th value limit compound process determined large statistic et consecutive recurrence precisely prove exist j limit express equal say dt ex w p w maximum instead sequence hill discuss arise naturally hill observation recurrence equation asymptotically analogous walk many index size statistical broad basis aspect different interesting cluster introduce far roughly short vector contain theory functional type series direct model residual analyze point justify extremely moderate deviation difficult direct analysis paper de seem preferable analysis problematic stochastic recurrence tail whole since extreme index suitably residual estimate sensitive
exploit quantify complicated mostly unknown observer plot experimental appear unknown observer relation cloud reveal unknown interaction among trading representation spectral laplacian essential object subset exploit fuzzy membership character subset biological incomplete fundamental biological sequence use fuzzy reduce proper subspace content modify gain observation back hilbert fuzzy content quantum mechanic books ba equivalent way row b set human real intensity dna array ultimately represent book library element paper element student method formalism abstract various note assignment point view set real free serve index orthonormal therefore ba b represent element orthonormal free dirac notation mechanic document retrieval information various context attribute convenient device function vector equip scalar equip scalar vector hilbert scalar induce hilbert normalise particular vector norm symbol ray structure allow pseudo db precise pseudo compatible stick publication pseudo mechanic hilbert unified probabilistic hold system act feature experimental encode adjoint trace act document possess probability algebraic description incorporate probability careful reader certainly encode belong initially disjoint pass represent attribute vector introduce thus include pseudo extend sake reader ba result however similarity irrelevant abstract foundation example explanation significance publication construct edge graph similarity exposition sake reader suppose pair level reduction datum dimensionality graph low span laplacian author definition reader like exposition always suppose non balanced construct define necessary specify recursively non trivial good semantic fuzzy vertex set hilbert subspace two letter alphabet root root stop precisely word letter coincide concatenation word letter start sequence far fuzzy centroid fine cluster fuzzy therefore explore branch singleton nn nu ordering n set totally attribute act specificity induce specificity frank role similarly procedure producing hilbert onto construction start c weight hilbert matrix laplacian c eigenvector new lead subset new cluster dna publish attribute heart brain step eigenvector representation gene cell associate small turn intermediate present type complete specificity degree dimension provide material home page author represent level cluster singleton context solely horizontal gene fuzzy membership vertical axis experimentally measure marginally specification degeneracy finally provide database gene specific line annotation induce specificity symbol mm gene annotate determine specific check experimental work matter context algorithmic experimental similar yet annotate exist database investigation character irrelevant use let concern linguistic genetic concern complexity dominant come dense real space low time step method multiplicative experimental semantic graph fuzzy analyse inspire application analyse certainly component seek direction correspond combination underlying vector major drawback assumption vector principal approximate variability representation review retrieve digital library google review genome latent indexing implementation dataset interaction among reproduce method seem formulation hilbert feature microarray graph incorporate propose graph intrinsic absence non interaction walk although always explicitly article walk unified formalism worth many context range biological array fuzzy cluster objective express term enyi abstract semantic strongly measurement computer share quantum mechanic worth underlie logic logic introduce represent biological traditional
perfect mcmc practice mean proportion phenomenon mx dx ax mx ds note stochastic analytic available fine evaluate approximation moment dominate condition diffusion however valid measure furthermore volatility therefore inference dominate next specify appropriate dominating transformation induce property address sde therefore one reason either euler formula definite whereas assumption make require full convenient algorithm involve unit naturally cholesky decomposition xx diag diffusion cholesky c cholesky structure eliminate always cholesky q coordinate restriction compatibility cholesky decomposition translate cholesky establishe entire highlighted provide scalar transformation unit volatility exist diffusion dispersion restrict dimensional diffusion sde xx diag x triangular diffusion x identity explicitly alternative proposition proposition suppose derivative diffusion unit transformation specify appropriate efficiency augmentation augmentation necessary long invertible whereas volatility sde weak diag diagonal ease vector observation datum define successive property consecutive applying likelihood problematic dominating likelihood dominate step require dimensional identity exist sde coordinate transform diffusion manner wiener transform jacobian dominating distribution brownian depend second eq bridge finish preserve volatility sde write independent brownian start finish likelihood irreducible degenerate analytically evaluate fine partition diffusion path transformation base posterior assume diffusion sde satisfy transformation outside broad application example volatility volatility model general dimensional correlate usually log price volatility provide generally transform unit volatility require nevertheless datum note need volatility couple cholesky model dx correlate top equation bivariate may x may dispersion triangular cholesky see contain successive component bi covariance augment accurate remain modify volatility case volatility entirely unobserved partially formulation observation use diffusion rather bivariate noisy transformation replace I relative remove volatility model satisfy sde transformation combine cholesky specification irreducible mcmc scheme path generally drift execute walk conditional tractable gibbs subsection regard update path option augment divide connect lebesgue dominating likelihood brownian proposal independence bridge substitute accept split path dominate alternative proposal adapt option propose local move spirit walk metropolis volatility therefore may choose diffusion bridge drift propose bridge substitute th dimension accept occur bridge small strategy detail use stochastic diffusion error observe early trivial full posterior closed form step step ensure preserve reasonably acceptance rate propose may implement use symmetric wishart high may replace update correlation restriction imply diffusion manner positivity diagonal hence random metropolis link see draw augmentation diffusion x correlation may substantial notice framework allow formulation main mainly correlation dispersion cholesky non simulate time run autocorrelation augmentation sampler concern regard figure plot draw chain figure depict look plausible contain summary good agreement lag lag lag lag simulate cccc posterior exchange rate rd nd month imply imply adjustment table imply iv iv ccccc mean st median note take provide model one line table redundant like correlation word exist assign informative business year autocorrelation draw reveal sign lag lag lag regard diffusion path provide approximate augmentation scheme exchange dataset summary draw correlation appear estimate variation process amenable mean median provide point pricing alternatively posterior pricing useful sd introduce handling correlation framework diffusion preserve substantial diffusion cholesky connection augmentation apply partially generalise include provide stand augmentation mcmc become arbitrarily augmentation nonetheless couple approximate analytic expansion appeal diffusion difficulty apart diffusion differ augmentation scheme sampler path bridge alternatively target matrix amount visit scheme base need eq hold diffusion matrix lemma q independent substitute also gx xx become q page pt proposition lemma corollary department process economic business university department likelihood diffusion markov chain carlo mcmc ensure update positive definite constraint observe point generally overcome augmentation volatility methodology daily rate keyword chain volatility cholesky phenomena evolve appeal term specifically diffusion differential sde drive motion term notation applicable weak translate linear chapter address modelling allow correlation increment quite common series cause methodology example example pricing often exhibit inter
optimal upper recovery extensively relaxation test finding directly optimize important extension expansion use condition derive matrix around denote u u f imply expansion u cauchy matrix curve complex eigenvalue interior easily see introduction complex formula specific expansion expansion around exclude eigenvalue apply f nx n x xt xt cc x cc cc proposition eigenvalue eigenvalue one eigenvalue remaining eigenvalue mt part aa part sum linear combination variable engineering formulate semidefinite derive greedy compute good target number coefficient application subset example principal visualization wide range science start multivariate datum combination direction maximize variance pca linear factor loading mean pca visualization use often hard application axis direct asset trade component easier interpret transaction finance fidelity aim efficiently explain amount fidelity obtain maximum positive covariance sparsity zero numerically require compute hard fact ordinary hard post interpretable subspace simple loading value systematic recent propose principal loading pca type optimization penalization simple thresholding recently semidefinite complexity use branch complexity derive total eigenvalue derive perform pattern optimality conference provide condition application certainly maximum la constrain lasso call isometry constant guarantee decode thus prove lasso recovery solution sufficient necessary hardness eigenvalue observe duality paper organize complexity convex relaxation use tractable optimality recovery finally test complement variable control generality semidefinite variance note permutation square root practice instead provably globally begin relatively maximization solution rewrite b finally nonconvex select greedy preprocesse recall simple first contribution derive simple algorithm variance diagonal rough theorem state sort quick solution eigenvector zero coefficient magnitude solution cardinality update sequence find index contribution algorithm sort diagonal initialization compute k pattern q form zero submatrix matrix gram eigenvalue eigenvector transform costly subgradient eigenvector increase least variable add provide preprocesse sort cholesky kx j go back output ex ex point form submatrix find test large value greedy use classic eigenvalue approximate maximum gram advantageous root complexity get product original pca appear solely equivalent rewrite maximize constraint hard however element also equal relaxation semidefinite variable root nonnegative semi one eigenvalue equal ia x x function concave symmetric affine relax optimization drop nonnegative ia desire semidefinite one optimality solution relaxation obtain relaxation kkt let kkt pair eq maximize primal kkt pair sdp kkt problem candidate optimality optimality I check span check optimality dual semidefinite write kkt condition I sufficiently feasible proportional kkt summarize provide optimality eigenvector define optimal optimal variable gap convex hence iteration large eigenvector define duality hence I interval efficient pattern basic note interval minimize zero problem efficiently form outer subgradient finding target precision derive explicit plug solution consistency strictly tight technique application pca selection datum predict estimate selection sparse many zero consider sparsity eq pattern correspond pattern pattern optimal less sparse condition derive unlike selection eq v optimality instance condition backward eigenvalue relaxation help check posteriori corrupt measurement code unknown error find classic trick problem substitute combinatorial equivalent integer submatrix form isometry word provide error restrict isometry compute sparse tf ic tu tf isometry failure condition pca finite isometry provide slightly weak restrict orthogonality condition extend sparse perfect relaxation tight upper sparse semidefinite significant artificial author generate vector form test signal ratio give plot full greedy produce almost answer roc curve dominate greedy rate greedy solution tradeoff various error duality gap solution code fail globally optimal note relaxation come search interval eq cardinality tradeoff curve greedy bound dot line compute cardinality cardinality tradeoff line dual section plot line dual dotted compute dash point duality bold present experiment optimality selection generating set certain satisfy greedy backward two provably simulation frequency optimal lasso greedy
indicate instrumental peak display blue limit target correspond peak background white noise produce dft target rule peak instrumental tell examine relate target area contaminate light available statement peak peak entire successfully et conditional assess peak derive comparison time series mean frequency dataset guide star star suffer contamination instrumental trend present single equally affected instrumental dft spectrum reference object star additive intensity approximation mean cope situation star hybrid star al example variability band affect four star able sect deviation fold peak spurious peak spectrum graph represent spectra peak black composition light long trend well signal light daily light frequency visible spectra time therefore regard bar peak detect five series individually find bar compose sect introduce origin clean description quantitative dataset return conditional identify target unique target compose compose signal strength run e g experience outline sect reliably identify instrumental fairly sophisticated distinguish intrinsic instrumental signal correct achieve evaluate example determine peak dft spectra kind acknowledgement pr receive chen project p thank huber g thank university careful valuable comment presentation least square reduction fouri increasingly object observe pose clean datum analysis frequency relate comparison signal strength instrumental intrinsic unbiased statistical background individual alarm probability deduce probability star alternatively frequencie dft simultaneously lead composed star observe significance measure none peak dft reach beyond quantitative datum mode al reveal instrumental sect ray detector stability problem bad periodic call see sect observation require enhance automatic without combine discrete fouri dft standard frequency clean quantity significance white basic domain unbiased comparison different necessarily measurement physical valuable beyond scope space primary scientific fail star detector turn remarkable quality al lead al successfully volume comparable environmental e light requirement series thousand frame sub consist present optical single identical pointing approximately achieve imaging image outer diameter pixel pixel outside image star et huber guide star et et pixels light technique simultaneous star variable star illustrate et raw noise et instrumental peak dot green line dotted black red frequency dot dft amplitude dominant peak amplitude significant ratio function frequency examine white dft result account peak amplitude avoid spectral hereafter logarithm false alarm uncorrelated datum set pure appear phase peak consideration refer white frequency phase spectra compatibility whether deterministic comparison star star star subsequently star keep mind everything star obviously compare circumstance extension handle comparison multi occur return statistically value conditional probability peak peak resolution dedicated testing whether peak dft spectra sense compose peak acceptable dft spectra definition eq denote width employ amplitude obtain realistic peak frequency eq numerical excellent compatibility quantity subsequently term frequency error enhance flexibility exponent attain resolution present make comparable introduce conditional peak counterpart multiple assuming additive term readily intensitie magnitude variation appear scale magnitude instrumental effect intensity convert magnitude eq variation strict transform magnitude magnitude light amplitude magnitude light denote intensity comparison amplitude correspond intensity reason variation scale distortion towards intensity propagation producing confirm induced term intensity star artificial variation comparison star reasonable application contaminate measurement demand special calibration magnitude example sect dft spectrum frequency resolution phase generally match perfectly dft deviation transform amplitude frequency angle fourier target peak however calculation perform fit satisfactory extent status omit instrumental environmental responsible target expect deviation target comparison point al light detector phase measure position lag reduction procedure sense omit technique phase phase interesting peak amplitude transform peak amplitude sect comparison noise include dataset variance light measure amplitude evaluate transform alarm produce amplitude fraction alarm probability process define logarithm false alarm correspond peak amplitude consideration comparison transformation peak comparison analysis target dataset consideration average reasonable trust absolute dataset peak sect none two star statistically individual peak probability complementary peak real joint introduce alarm application problem along forward implementation namely calculate evaluate differ series examine reliability dft spectra may evident peak compose evaluate evaluate clear peak consistently decrease dependence dataset potentially numerical interpret thus intuitive scaling mean become peak unique individual peak reproduce quantity consideration peak therein peak false decision whether peak noise rejection may write accept peak reliable peak accepted peak noise transform trust relate compose compose examine trust compose combination figure three reasonable significant peak represent search peak raise comparison spectra accord level expect
contrast interior expense nesterov smooth continuous write parameter complexity apply constant theorem know always prox choose frobenius prox non replace penalize involve prox approximation everywhere continuous gradient specific corresponding center satisfie choice proceed k q approximation gradient frobenius produce q ready smooth computed essentially amount project step advance graphical binary version formulate approximate multivariate maximum sparse use datum wish structure maximum penalty know difficulty partition outer problem next follow suppose sample moment follow optimize form rewrite eq upper approximate relate variable relaxation simple point degree since variance binary approximate sparse present sparse matrix end randomly choose positive give number add multiple identity need multiple necessary inversion approach threshold however even easily level synthetic size density drawing sort absolute right amount observe satisfy ht un thresholded test ability sparse size display use thresholded pick covariance matrix repeat use expected ability inverse matrix sample two correspond underlie blue line correspond vertical mark range blue correspond mark sparsity experiment illustrate instead empirical randomly generate value randomly misclassifie zero nonzero average percentage misclassifie misclassifie divide bar deviation show sense performance nesterov coordinate descent matrix range sample choose cpu duality gap ghz gb typically rest result penalty gene gene gene gene associate fusion analyze gene drug sample order variable large neighbor gaussian one key list tb gene bf interact bf est bf co ai ai co nm nm perhaps surprising directly either ai nm provide sparse maximum us record th put vote record many vote solely necessary experiment vote choose accord significance model node correspond relaxation color color medium make separate ct observation match medium make thus picture largely figure show low likewise primal gap last block coordinate quadratic column diagonal rise descent prior first initial complement even great prove second moment constraint time column always coordinate shall function variable adaptation consider drop determinant set subgradient ready sort together correct diagonal eq block inside constrain block suppose q since remain correlation student freedom student degree pa put fix satisfied choose proof nan freedom cdf square distribution inequality q desire bind choosing connect maximum determinant relaxation prove conjugate normalization low conjugate eq spin mean write use express help formulate obtain replace collect unconstrained symmetric sparse problem add term certain dual problem equivalent formulation primal negative graphical problem formulate method prohibitive node solve norm nesterov complexity dependence determinant log show maximum problem synthetic model estimation gaussian undirected offer among principle simplest explain way norm idea involve find zero matrix correspond conditional among traditionally forward backward infeasible datum moderate one nonzero thousand variable use penalty sparse list neighbor show graph formulation simple computation smooth result number point store prohibitive higher specialized consider gaussian heavily norm provably coordinate descent recursive penalize nesterov recent rigorous point develop solve problem binary determinant relaxation voting set maximum property suppose independently variate covariance estimator sum absolute value element technique classical invertible case invertible matter ratio even trade examine write denote bad perturbation second robustness make estimation machine dual analytically objective note determinant act barrier create thing dual add penalty maximum eigenvalue symmetric low hundred solve software point however infeasible large maximum likelihood know accomplish specify hope structure moment graphical denote
replace whose compute explicitly solve smooth solution computing exponential comprehensive detail different implement decomposition comprise eigenvalue diagonal relatively inefficient exponential rational e see control precision due issue scale first scale e inversion scale choice computing see expect problem eigenvalue problem classic method exponential towards produce solution algorithm compute precision costly achieve gradient f u parameter control eigenvalue compute become eigenvalue close phenomenon seem appear detail result clearly dominate give example necessity size eigenvalue cancer gene follow times interior solver achieve comparable percentage reference lead principal decrease htp dim var var analyze set gene expression feature recent factor projection intensity normalize sample deviation experimental effect semidefinite large preprocessing increasingly eigenvalue increase decomposition substantial require necessity condition depict test runtime dimension plot dash plus view proceed duality gap eigenvalue get optimal htp compare performance pca data cl top plot use cancer represent great predicting second good far cancer gene htp b clustering analyze quality derive numerically rand cluster two cluster rand pca index similarity pair error pair total plot mark rand gene derive derive however get htp cluster impact sparsity cluster vary factor separation drop cl begin sharp mostly cl get htp analysis derive relation machine feature compare software svm use contain predictive show gene appear include compute use cancer gene identify remove true factor cl factor appear maintain svm produce nonconvex convergence l description na htp l na na gene relaxation allow apply pca two classic gene expression case relevant original grateful acknowledge nsf dms bm application pca seek coefficient sparse cluster reduce pca detail et selection arise biology component feature focus area motivation visualization interpretable analysis give little efficiency pca tool multivariate maximum amount numerically eigenvector variable pca construct pca coordinate axis physical interpretation finance biology axis correspond financial asset greatly relevance interpretability pca seek goal expressive variance interpretability sure factor involve axis pca clustering allow loading absolute thresholded penalization seek globally use greedy approximate one introduction describe implement expensive algorithm gradient key contribution decomposition current sufficiently gradient drastically improve computational datum simple gene recursive rank organized motivation implementation toolbox available introduce
test validity give n size validity possess sort minimax penalty schwarz penalty minimum description penalty development penalization build consistent drive choice penalty penalization penalty paper illustration see example concern transform partition well loose solution adaptive automatically propose include design sequence construction space index index center minimizer countable family family build penalty risk model selection paper square square case maximum optimizing condition therefore testing range setup lebesgue measurable mathematical expectation fix advance space example natural hypothesis use test substitution covariance semi complicated could simpler getting avoid course consistency condition regardless choice serious restriction formula likelihood multidimensional statistic additionally general reason statistic special class special finding form inverse deconvolution appear signal physics imaging block datum drive formulate respect kf f define formula continuous nan score test among thus hypothesis nontrivial number rule drive statistic drive alternative impose section identically independent consist new obtain transformation obeys transformation transform choose empty distribution put serious choice uniquely nonetheless distribute counterpart allow kp k generality assumption include possibility handle analogously short l avoid possible confusion modify bit sometimes need stress dimension accordingly order k simplify hold q infinity possible grow control satisfied uniformity weak law number illustration bound expect rate problem drive nan eigenvalue penalty real number every every eq notational convenience eq formulate prove rather object let penalty monotonically monotonically b definition equivalently statistic consistency statistic require inequality deviation model regularity start easy sometimes desirable cost restrictive regularity arise regularity advance inequality drive use typical statistical basic sequence proper statistic need sure affect dimension practically establish prove happen nan hypothese inconsistent formally statistic weight I euclidean corollary drive simple hypothesis uniformity precise tend crucial continuous concern quadratic independent let satisfy b proper suppose condition type definition inequality variation sufficient property many approximate gaussian condition give second form random hope existence sometimes eigenvalue notion composite always concept need applicable composite measurable distribute let measurable take w distribution symmetric positive definite matrix let q call know explicitly reasonably definition generalize score establish parametric finding exist construct estimate often see general statistic put composite density drive score construction set eq identity likelihood q regular enough situation practically regularity consider problem follow block statistic deconvolution let lebesgue real set eq estimator regularity quantity studied construct statistic fact example difference integral process nonparametric ratio modification tool perform whole proof consistency sample splitting complicated involve opinion convenient tool prove drive drive consistency follow theorem meaningful use introduce auxiliary interest due definition definition serve purpose technical correctness proper nan alternative show make possible prove alternative sufficiently good possess assume satisfied satisfy relaxation much strong purpose growth important consistency testing tend infinity tend alternative kind minimax problem measure rate rate example minimax estimator optimal convergence like thank helpful discussion literature reference research determine constant prove lemma case case proceed analogously cm rewrite notation cm rewrite show complete follow equal assume eq statistic therefore applicable guarantee exponentially matter simple calculation prove variable applicable since nan statement theorem proof theorems theorem lemma proposition financial smooth test score consistency test class composite incorporate rule rule modify change propose powerful class construct good statistical essential statistic construct test speak von kolmogorov graphical usually type capable asymptotically hypothesis increase construct test construct way test asymptotically optimal sense construct example development approach adaptive situation score test infinite alternative performance concern minimax alternative discuss test exist classical substantially test likelihood notion test generalize concept smooth drive score main proof establish derive consistency inequality classical
recover since depend enjoy baseline mixture covariance adjustment require dimension emission computational resource adjustment unlike em computation adjustment point calculate simulate producing computation stochastically course investigation viterbi state question formal irreducible initial emission function respect dominate lebesgue measure count denote realization sequence know realization chain emission distribution completely determined emission moreover process give depend also ergodic throughout short central viterbi alignment maximize viterbi maximum give viterbi alignment refer alignment eq map element require hmm parameter emission emission parametrization straightforward free unknown unless classical mle computationally viterbi viterbi replace expensive alignment computationally practice hmm give alignment eq partition subsample sub mle subsample empirical take viterbi suppose use parameter base even n l moreover converge measure depend choice alignment fact mild restriction eliminate alignment writing whenever consistency assumption class viterbi attempt reduce propose viterbi nonetheless define follow correction adjust viterbi follow choose alignment empirical note sufficiently adjust fix recall density satisfy measure alignment via partition alignment wise lx strong density respective special adjusted viterbi easy largely support estimation accuracy adjustment compute extra affect adjusted viterbi generalize concept hmm one begin gradually build notion refer notion infinite viterbi develop notion section emission parameter respectively object observation constrain trivial recursion well viterbi alignment name viterbi programming find non uniqueness viterbi require oppose case identify subscript stand l let property maximize hypothesis imply latter yield aim alignment infinite objective e g proposition g barrier theory start pt lb lb lb lb lb lb lb lb lb lb lb lb lb lb lb lb lb lb lb lb lb lb lb lb lb lb lb lb illustrate newly introduce notion whether statement immediately also follow side alignment possible since hold finally give enforce always alignment alignment observation general imply structure significance remark decompose proceeding e ix kx element unique otherwise I ix ix well technical nonetheless go require selection lx u impose formally admissible lx state consistently proper lx scheme discussion valid provide base reverse neither equal immediate going refer scheme concern initial alignment purely presentation initial component lx ss www final every alignment alignment tie break consistently actually imply existence arbitrary emission positivity eliminate aforementioned positivity desirable notice recursion generalize define coincide implie rewrite generalize concept let font lb lb lb lb lb lb lb order node knowledge finally immediately give successively ti lx uv ti u lx ii u unlike remain valid u r lr lx lx suppose order repeat similarly yield result thus establish existence u u k k ni u u ir alignment alignment order selection concern proper definition piecewise order formally vx kx u u kx vx define node aim extend like hide immediately require restriction introduction large restriction virtue say prevent moreover would order hand order barrier barrier th technical denote q define satisfied function certainly hence subset state correspond necessarily emission state reveal one moreover exist finite integer th zero eq element barrier ergodicity e realization infinitely realization node need obviously technical reason alignment define statement infinitely let lemma barrier presence order say separate roughly barrier separate barrier suppose barrier general however suppose contain separate word tuple might unnecessary cluster fulfil enforce condition gain satisfied hence note alignment support hence stay constant see exactly belong cluster cluster apply e recursively let let emission observation unlike concrete barrier way modify example emission support hold barrier barrier construction observation subset buffer state arise possible realization node way lemma separate barrier clearly distribute provide correspond barrier barrier extend say correspond delay time eq clearly proper vx lx lx lx lx l lx infinite proper alignment namely lx lx infinitely lemma adopt u lx lx lx u realization define measure central define alignment allow define alignment course sensible infinitely many shall define let q every definition process thus hmm differ subsequent however still hmm rely essentially initial law would recall yield proposition appendix lemma measure satisfie involved facilitate divide short maximal construction subset auxiliary sequence cycle inside prototype barrier barrier commonly likelihood bound ratio implication auxiliary kf ii cc assumption statement would first second definition cluster imply sufficiently z k c consider sc cf auxiliary zero e find u notation u next exhibit auxiliary characterized b b particular transition positive iii ij ci j maximum path every sufficiently eq introduce concatenation cycle positive consecutive segment maximal beginning path path exhibit viterbi backtrack guarantee begin node prove long maximum need resp resp resp likelihood formally frequently score construction implication follow last path thus since claim construct note indeed therefore leave end state definition eq eq recall every put definition argument behind apply bind px lx pp px px px pp true positive necessary imply let clearly positivity assumption p p p j combine finally q integer otherwise expand li sp p x li replace state repeat arrive path sufficiently large observation observe recall introduce imply q inequality kp x il contradict generalization start return substitute apply appropriate time strict since go imply path observation go formally node let case go fundamental virtue kl j kl kl path satisfy argue likelihood right equal virtue recall similarly must construction positive barrier complete nothing case extension ensure separately z x every barrier already state sequence consecutive verify exist shifted verification consistent note admissible separate separate one indeed position right make impossible since disjoint value stop clearly finite irreducible hmm occur interval belong irreducible positive statement hold initial verify statement e r b simple I recall accord underlie
way use initialize map fix dissimilarity analysis projection factorial china clusters north west china warm correspond china warm china longitudinal eq represent cluster htbp datum type distortion euclidean hausdorff distance deduce hausdorff adaptation show apply symbolic pt journal symbolic research institute control de le france fr new symbolic tree query symbolic self symbolic symbolic dissimilarity visualization algorithm map enable input preserve operate som without symbolic internal datum etc datum trees document center solve operator result som complex som alternative modify som implementation dissimilarity raw end dissimilarity necessary process som quantization preserve spatial ordering prototype som map neurons topology map short impose neuron batch algorithm note present representation space batch define c cardinality belong neuron representation sharp smooth parameterized control neighborhood example representation neuron represent term cost induced partition assignment carry indeed classical minimization immediate average weight method adaptation som dissimilarity kind structure associate dissimilarity concern maximal china temperature record daily mean daily maximal
applicability real reason translate fast still trade speed valuable literature simplification inferential volatility rw often adopt literature work reference therein approximate inferential method extend filter mean approximation evolution volatility suggest rw work elaborate mention use convolution wishart multivariate beta first rw volatility adopt rw estimator base distribution notice incorrectly derive give volatility realistic beta volatility weight square return obtain volatility guarantee appropriate analyze volatility define section assessment namely mean forecast comprise exchange time integer drift volatility variate root comprise sensible time unchanged column stack random zero evolution multiplicative law wishart discount factor stand exponent trace gamma also innovation uncorrelate turn order completely specify show order guarantee expectation invariance rw specify q posterior derive paragraph evolution pn pn ks p eq wishart rw evolution verify proceed posterior likelihood observation distribution py together constitute usual univariate expand exclude influence prior increase follow walk suitable estimator volatility wishart posterior mean admissible unstable capture spike recommend explore range discussion performance discount factor standardized forecast provide forecast expectation I write standardize give datum yield sequential two compete first bayes factor student become denote suggest sense superior forecast suggest preferred model situation decision case one threshold choose exchange frank period correspond daily frequency data york eight exchange volatility return collect vector study rate adopt evolution specify suitable follow suggestion consider summarize table log function evaluate bayes forecast bayes current ideal suggest forecast covariance get close forecast attain diagnostic produce figure factor superiority sequential sign particularly advantageous ccc point small indicate period versus vary window platform take complete pc processor mb ram evaluation factor describe approach methodology rely closely notably volatility type instead walk evolution volatility view suggest attractive crucial computational finance trading research effort direct towards financial special portfolio derive density diagonal matrix detail jacobian transformation transformation equation logarithm develop procedure volatility multiplicative wishart distribution volatility flexible sequential volatility likelihood criterion step error bayes comprise eight exchange bayesian kalman wishart two decade effort vary recognize implication stock exchange rate examine volatility derivative
maintain highlight count cost variation satisfied guarantee action act rely heavily go algorithm exist current context context visit least iii act active discount first act respect respect action hypothesis demonstrate conclusion immediate eq estimate probability define fact solve bellman active eq fact discount tx reach hold requirement lemma action count context certain exposition event least one hold possesse step inaccurate probability else context q inaccurate visit sufficiently estimate analysis active algorithm broad one occur vanish fraction show fact suffice vanishing choose action optimal vanish control exploration attain briefly universal prediction alphabet probability assignment cost assignment occurrence take sequential probability probability assignment notice estimate employ active algorithm return essentially lemma combinatorial hoeffding part et arbitrary constant inaccurate go via give rate turn exploration decay ensure inaccurate go vanish inaccurate variation visit past probability realize correspond small inaccurate probability sufficiently slowly impossible inaccurate inaccurate go precise prove wherein assumption satisfied almost consider apply next event inaccurate occur event dt tt take go zero assume action cost structure somewhat separate active sub go rate provide converge fraction go surprising effect tree tree weight et plausible approach yield significantly compression optimality inspire dynamic programming model distribution reinforcement next markovian past variation use context converge consideration great model consider observable development management mit electrical engineering stanford student stanford fellowship student mit business receive degree electrical engineering science institute mathematics electrical engineering stanford university member benchmark stanford van van management science engineering electrical engineering computer stanford hold finance receive computer science engineering sm electrical computer mit event system mathematics research research mit newton laboratory award mit master thesis award mit award stanford award david department electrical engineering stanford university department electrical since interest signal involve member technical recent nsf award fellowship technology research school stanford com good device lemma combinatorial without observe assignment equivalent occur context occur distinct applying follow remainder kullback hoeffde inequality arbitrary define negativity kullback leibler inequality visit event yield result remark conjecture van mit edu stanford edu stanford edu reinforcement interact agent incur cost goal active optimal control compression integer conditionally past tree reinforcement programming agent action space incur expectation randomness tx prior strategy guarantee attain assumption integer situation neither agent follow example fall formalism paper person sum rich history reinforcement opponent incur player opponent action action space integer cost opponent mix opponent structure player game play opponent fall joint code fix channel alphabet transmission across encoding symbol corruption channel channel times l decoder time symbol distortion find minimize distortion order source unknown optimal within therein knowledge average either via dynamic programming straightforward optimality develop broad first efficient process store available action compression programming scheme probabilistic select suitably long knowledge task building kernel create exploitation action build selection balance two average knowledge kernel reinforcement decision context asymptotic guarantee control markov set theoretical difficult present constant require equilibrium challenge de select thought compete know function active extend however take active effect action exception cache formulation extension around observation structure structure belief represent experience engineering compression denoise understand value reinforcement paper formulate section state asymptotically endowed action history observation action observe action evolve accord stationary action manner observation time policy evolve state k policy underlie finitely policy average policy infimum policy enable average cost independent positive generally structural average difficult cost assumption recurrent optimal initial play critical note subsequent valid recurrent controller thereby dynamic achieve would discount bellman discount discount alpha solution bellman optimal problem discount minimizer acting action immediate plus quantify capture average sometimes game opponent memory transition bellman cost go entire history play recent game play optimally account immediate action future direct bellman kernel instead course active variable context dynamically scheme universal function bellman equation fashion solve select exploration acting minimize cost incur exploitation active algorithm discount probability proceed th phrase observation occur phrase discount factor exploration pick probability context nx nx phrase define precise action x next initialize uniform empirical visit empirical multinomial similar time subsequently initialize iterate realize trajectory exploit former action possibility fully explore ensure impact possible future act relative take versus exploit nx time maintain probability separately store readily count phrase interval leaf leaf algorithm cost go path storage active average cost however consider algorithm player term know
value suppose distribution another easy may moment equivalent use function estimator generate mostly paper concept outside paper deal comment would quantity even though metric zero occur equation equivalent arrive multiplied term whole show property likelihood state mis express used likelihood ratio instead way less concerned parameter population original discussion log generate less one distribution always considerable expand
dy dy ds dy dy dy v e dy j ds e dy v exist depend transformation dy dy dy dy dy equality page q proof theorem lipschitz mixed variate converge set uniformly uniformly independent previously permutation leave symmetry exchangeable random vector possess j b b b I generate conditional exchangeability e v see appendix e v v dt dt v v v dt proposition uniformly tend converge prove like thank write spend time grateful smooth suitably induction x dt dt dt x x x k dt ft k dt x mix j dx I finally observe side j u x j x u u u ds ds u dx u j j u j b q u u l u j j u q c c u c j c j j l u symmetry q omit use convenient subset u j c u l l j l l c l l l l l j l hand follow e q q q j l j l j l l write cardinality finite l l j j c l l j j j l l l u j j u use observe j k b j j j l j j l j l b conclude let uniformly write e immediate consequence q q e e u e j j j u e q k u q u q u j b j j k j j u j u u u q e u b k I I u j u j e conclude lemma uniformly e proof first u I I hence partition denote summation hand convenient subset j u l j l l ed ed u u u j j ed u u u j u k I I k u j k u u ed k I j j j u q equality etc introduce factor ed u u u u u u u j I u e e equality consequently next w b u q u u k k j k k j say p u k j j j u u j j k q u j k j u j k e e j u j j u k j j u u j k u u j q k j u consequently w notation e u u j u k j j u u j u u u r r r u u k j j k j u j conclude notation e w u j u j j u w u j u j j u u u e ed q u u j u k I k r r ed u j I u r ed u u j k j j ed u u k j w w proof q j b j k u ed u u j j I k u j ed u k k u u u r ed u k ed u u u k u I ed u ed j u ed u j u ed u u prof observe j u j j j k j k j u u j j j u u k j I conclude v I e observe e depend fashion dt I e j e j I I k j j u j l I u e consequently j proof e e v e I v e j j j lemma I q j u u u j w ed j r ed u u j j ed u u u k j ed u j u u ed j u j I ed ed k j j ed u last next w q u j u u j q k k I j j u r r e u j j k k j j u u j u q j u q u u k I I j I k k u u j j q u u u k k u j j j u last equality use let exposition subsequent suppose claim false exist converge subsequence say subsequence l theorem standard converge prove proposition law p pz denote prove converge law normal corollary array design computer experiment classification secondary f limit hypercube randomize stein method national integrable objective evaluate finite design point popular article multivariate central orthogonal design hypercube objective many stein estimate evaluation ultimately dominant high dimensionality integer orthogonal array strength row symbol submatrix occur account array book independently array design design contrast margin let th element permutation uniformly possible permutation uniform apply column satisfie array variate margin margin margin call b net net variate integer let array replace position entry permutation permutation da q version page denote randomization randomize permutation element estimator see article well argument orthogonal array important continuous dx imply always analogous ii dramatically additive lipschitz continuous mix partial j fx j l j
ordinary square fan li panel compare situation compare least square bad scad ordinary scad panel far make location suggest n relate scad bad component parameter either scad perform poorly poorly situation statistically decide poor diameter go slow finding give brief summarize proceed detail slice surface ii setup additional qualitatively scad fan li scad cross strong leading favorable bad value performance iv choose iv get similar less iv setup result figure setup setup setup vi enforce discuss ii setup carlo setup qualitatively conservative selection aic value true vi show undesirable risk property easily much model really worth recall oracle uniformity play remarkable year later newly propose estimator surprising pointwise estimator present show distributional post bad type estimator include relative beneficial achieve shrinkage least estimator certainly simply acknowledgement version grateful helpful f hazard fan li fan fan li selection cox li modeling datum penalize frank discussion v york information criteria e north fu asymptotic type wang censor supplement text statistic fact b estimator risk consistency von wishart strength share corollary hour height ne ne ne pt ne compatibility head part head head compatibility conjecture true mu size false ex mu th mu h mu th align align end end environment style use environment construct allow type construct allow false tag tag relate concept oracle fan li property sparsity estimator maximal supremum ease beyond study assess smoothly deviation fan perform poorly sample primary secondary f penalize penalize square scad bridge estimator hard maximal limit recent year see increase penalize penalize bridge frank include type fu smoothly deviation scad estimator fan li fan fan fan fan li scad possess parameter zero sample component remain correct zero restriction impose course scad point asymptotic scad scad parsimonious coincide fan li regularization also several fu suitably thresholde maximum estimator consistent procedure suitably choose sparsity use procedure property estimator property know asymptotically zero restriction price simple thresholding arithmetic exceed threshold example feature picture finite property oracle theory predict g although pointwise well estimator detail infinity along alternative perfectly normal fu certainly belief good infeasible zero reasoning oracle property bad sparsity entail finite estimator scale mean square size increase constant quadratic risk similar result seem suggest phenomenon simplicity presentation cite inspection proof show extend framework relate traditional post study demonstrate fan li scad monte finite especially sparsity fan li paper mention regressor error variance obviously continue hold take possesse absolutely continuous derivative guarantee sample base sample say sparsity guarantee scad member choice consistent selection estimator also establish subsequent result quite purpose note square bound increase tn use instead wang result continue matrix type condition scale square generally risk bad scale quadratic maximal infinity sample suffice arbitrary contiguous far inspection supremum ball center origin long bad part gain sparsity ball radius center limit inferior local result continue ball supremum center arbitrary possess component satisfy g represent inspection quadratic quadratic weighted quadratic correspond generally nonnegative loss inspection nu nu n nu nu discuss partial oracle property partition converge post selection procedure minor maximal extend theorem variation square maximal square base regression regressor obvious stochastic regressor nonlinear model whenever replicate monte carlo li demonstrate sparsity estimator estimator something fan li conduct happen favorable regressor regressor regressor component fan li namely describe estimator large close scad minimize penalty function fan
analogous appealing methodology type minor cope need task uniform bandwidth shall reader attention see also relevant reference therein bandwidth limit law turn particular allow establish estimator law conditional distribution function estimator hazard function especially point law construction band band simulate display devote triple em dd throughout replica directly correspond em c compact usual assume resp function lebesgue assumption denote f yy I measurable assume mostly regression censor p first build known naturally censor estimate define measurable em x unless let vary positive either fulfil cm denominator surely positive generally adopt nx z x give notation instance properly order functional accordingly hold exist h function hand conditional function weaker restrict compare hypothesis establish function sup fulfil pp enable easily py attention l measure pp set assumption em r kk easily check fulfil polynomial admit auxiliary probability convergence center center factor motivate stress attention bias em introduce main positive automatically fulfil complement consequence theorem estimate conditional hazard corollary ft center two hypothesis assumption establish corollary derivative denote additional estimator constant far hold hazard ft ft establish enable band paragraph impose regularity time cm ensure du remark discussion quantity along choice consider sequence give c constant assume bandwidth probability line consequence argument treat especially n n h constant however regularity impose version derive become alternatively generalization functional reader functional law relevant therein example simultaneous band almost fulfil simultaneous consistent theorem asymptotic simultaneous band precise confidence work size stand select simulate integer follow regard censor variable generate yield band appear adequate contain expect pt empirically involve q simulate sample estimate various bandwidth crucial since pair highlight build procedure select formal involve could achieve study example establish treat censor strong consistency cope censor identically replica q setting f sup variance center mention successive sequence constant condition direct capture naturally follow replace estimate sequence hypothesis recall definition combine property ensure cover pointwise em almost sup apply probability h allow hold theorem conclude line chen direct consequence capture multivariate index function define process base eq n nh close enough continuity imply constant involve moreover check pointwise class cover fact n conclude statement complete statement general upper part precede paragraph straightforward w show couple keep mind uniformly first function endow uniformly totally moreover relative existence enough select triangle complete hold assumption absolute surely fact function far I h introduce class uniformity end nonparametric functional first introduce nh nh n h h nh h identical omit sake corollary consequence follow show nh em deterministic ft h tf corollary h f df f
chebyshev follow cumulative norm classical find probability theory pp perfectly scalar conservative chebyshev denote matrix v x symmetry ik ki tr x indicate deduce show well inequality establish engineering element separate separate banach less conservative chebyshev inequality let inequality ellipsoid chebyshev inequality ratio e definition apply b arithmetic diagonal note matrix definite hence hadamard follow complete illustrative variable variance respectively ratio volume monotonically ellipsoid construct sample ellipsoid include ellipsoid indicate less classical chebyshev
scaling average sample draw function choose exactly equal true slowly sample probably acceptable low consequence small visually cdf straight plot identify beyond relatively sensitive tail reference principle desirable minimizing distance put represent model simply task fit fit directly count parameter list parameter kind increase flexible would achieved maximize marginal likelihood analytically expand order result yield determinant schwarz show simplify conventional maximum also justify problem circumstance difficulty need distribution law nonetheless represent bic tends subsequently calculate scale unclear bic represent empirical discrete behind high conversely small fundamental describe lie quantify probability kolmogorov ks fit law note skew sensitive deviation dynamic cdf estimate examine test datum form distribution follow pass point make calculation law value estimate ks text true show ks ks estimate slightly yield display tendency recommend ks synthetic estimate extract k achieve part form design determination choose law would would achieve true power power nonetheless case return ks yield consistent appear work rigorous variation ks goodness statistic circumstance ks relatively insensitive necessarily tend zero reweighte avoid across goodness fit ks conservative application give estimate magnitude many degree acceptable greatly statistical also law estimate scale estimate set original describe take estimate large repetition fail mention power law within finance perhaps brief summary thorough explanation family eq limit calculation dominate span law decay call hill appear technique yield quantitative finance large observation technique computational foundation analyse value perhaps importantly ks remove portion agreement section allow fit estimate whether power regardless fit law way match distribute hypothesis qualitative claim law look straight plot inspection follow albeit scale wrong roughly straight plot unfortunately draw extremely follow power form distinguish approach synthetic far power measurement typical law plausible effectiveness principle another goodness fit happen fail odd reason straight logarithmic scale law power increase power distribute threshold identify case power law exponential law hypothesis plausible give kind question goodness test plausibility distance distribution synthetic set synthetic distance fluctuation plausible quantifying distance two ks encounter truly expression example ref unfortunately case fit vary next recommend empirical model calculate ks scale parameter fit individually law calculate one crucially synthetic statistic fit set way perform wish estimate synthetic accurate law suppose total element repeat synthetic decide synthetic bad good wish wish accurate digit choose range depend value conversely plausible probability would agree poorly context author would candidate really particular circumstance normally since unlikely value hypothesis discussion well test alternative closely method reflect hard datum reason correctly law behavior continuous law log normal average calculate law distinguish go size become law power poor remain since law small fall law phenomenon fig reliable way well log eliminate possibility calculate believe discover reasonably law compete possible k calculate exponential combine good case value compete small competition although absolutely favor course fit compete fitting fit family therefore use prior knowledge constitute hypothesis decide place previous candidate law alternative neither well practical situation fit already perform goodness fit fail move thing another exist considerably easy ks ratio length basic idea behind compete likelihood likelihood logarithm depend event sign alone subject fluctuation could change order could close observe I estimate method give whether statistically fit favor hypothese insufficient favor goodness insufficient smoothing extent fluctuation capable detail case nest mean power cutoff nest nest small member small member modify properly distinguish describe appendix draw either continuous law average replication size gray method ratio synthetic constrain produce value time assess sampling ratio normalize way convenient sense unnecessary actually normalize contain behavior increase positive grow power repetition normal grey show normal law interval ignore simply classify synthetic set raw log say reach wrong fluctuation misclassifie test number decrease moderate account law log normal fraction part per quite indicate distinction utility variety world follow indeed case possible branch physics sciences sciences engineering sciences unique protein partially degree group ip address internet handle rather mechanism receive customer service states intensity per combine attack measure web http request laboratory hour roughly speak represent size web file internet datum compose specie context thousand american survey united number book unite period human census book size peak intensity intensity occur california number publish web site com occurrence name census aggregate united states publication scientific paper publish list citation list american database receive web customer internet service day number million page entity entire known interaction internet bias network computer internet f intensity attack receive web request computer laboratory period united customer affect electrical united books united power use variety plausible datum fitting scale parameter considerably method interaction report take quite similarly compatible value seven enough http web web get chance optimistic behavior http law imply characterize likelihood law column value test summarize law quantity degree call receive attack http species sale book population email book intensity paper paper site site cc cc law law book sale moderate cut cut cut give power law alternative statistically significant value alternative nest exponential table list statistical law moderate indicate law plausible alternative plausible none frequency word cut mean cutoff pure power law normal distribution cc cc cc exp power lr lr lr lr internet moderate cut none moderate protein specie cut moderate frequencie english text appear excellent none carry fit email address book power poisson normal exponential case value sufficiently large book protein plausible law plausible normal motivate distributional reasonable argument law discuss ratio sale call citation forest log difficult log closely equal apart set forest web email book power cut power cut pure form power include physics biology economic political statistic analyze law yet rigorously possibility power law behavior case argue practice identify quantify doubly straight mean sufficient law principle validation quantification law properly evidence distribution principle could although large set various law statistically description compatible compatible exponential remain power hypothesis power law plausible assume exponential cut extreme measure answer question power perfectly enough merely tail distribution internet quantity file size http connection node degree heavy tail visually careful prove strong typically compete largely scientific goal quantify heavy tail concern instance rare event fundamentally tail tailed difficulty hand infer plausible mechanism might formation internet matter comment year face power law addition validate law behavior hope hold fulfil acknowledgment comment wiener sharing support ac foundation common observe imply histogram alternatively construct rank order datum interpret package exist perform kind calculate line take procedure estimate subject systematic serious hard formula construct histogram introduce free power law account even fit power favor law fit requirement distribution little formula slope variable histogram though frequency gaussian fluctuation would fit cdf value independent logarithm fail adjacent cdf correlate empirically valid improvement pdf correlation truly law straight low reject law unfortunately nothing closely order high approximate power significant fraction practice rarely see regardless tell terminology value law cdf ordinary however incorporate consideration pdf method significant extent literature give likelihood estimator derive know hill power law power hold set know draw proportional call model scaling parameter maximize function commonly logarithm mle result motivate regularity identically maximum power verify rate strong law surely expectation mle version fisher continuous power asymptotically gaussian calculation interval around er rao elementary mle since attain become large correction estimate result coupling correction lead correction fluctuation hard analytically bootstrappe hill start parameter deviation arise repetition decay repetition sample small law estimator argument give power log straightforward efficiency law theorem reader somewhat moderately reasonable convenient formula section differentiable integer constant expression also power ratio function comparison identical continuous denominator comparison eq see fig datum set likelihood think likelihood single measurement hypothesis central usual value fact fluctuation give available scientific library program value small say fit model discriminate rigorous proof particular deal introduce likelihood treat care provide family done maximize additional hold fit family large converge consequently expression kind version rule square become likelihood take family value special wish give distribution transformation continuous discrete variant probability random typically large variety generator density interval integrate respect get q indicate cumulative eq computer language use though case inverse list equivalent unfortunately closed write direct binary equation step assignment english repeatedly meet fall search continue discard law generalize evaluate eq slow speed wish many array ahead worth rarely name law cutoff expression generate two log normally distribute uniform cutoff exponentially random accept reject repeat accept give need possible previous section result approximate law continuous near definition law eq reasonably value small generator large approach round ccc c generate set random number transformation technique show cdf power law give cumulative integer set number difference somewhat generator enough application law distribution situation scientific consequence natural make phenomena characterization law difficulty identify range law behavior commonly analyze law least fitting substantially inaccurate return answer give law principled statistical quantifying combine goodness test kolmogorov statistic ratio synthetic give also law consistent pareto tailed law empirical air york thing vary place make representative observation american size deviation either distribution problematic reason observation underlie attract attention property consequence appearance range make intensity power instance census city
wavelet henceforth implicit similarly note modify wavelet obtain expression curve deviation neighborhood curve find within amplitude wish solve amplitude analytic wavelet transform note term real part frequency take apply scale ridge lead obtain amplitude residual eq instantaneous include order assess note derivative truncation level writing rearrange ridge find anonymous constructive feedback theorem axiom conjecture criterion note summary condition j analytic available author web site create collective work list work exact general analytic wavelet analytic locally instantaneous expand time instantaneous vary increasingly high frequency bandwidth wavelet instantaneous wavelet induce hierarchy transform away suggest match wavelet variability expression simplify quantify vary estimation ridge analysis wavelet minimize complex hilbert ridge frequency derive family wavelet transform value property together may continuous complex wavelet signal include signal series quadrature flow attention discrete filter transform important addition discuss regard relevant broad recover signal parametric observation amplitude contaminate direct analytic signal amplitude phase reflect interest localize analytic analysis term underlie analytic accurately exhibit contamination due localization analytic absence unlike direct analytic e analytic negligible strong since substantial paper derive analytic signal error moderate provide guide wavelet explicitly background introduce representation signal pure ridge conclude associate appendix review amplitude phase analytic signal wavelet analytic amplitude amplitude phase define analytic signal domain analytic permit amplitude signal recover amplitude phase amplitude amplitude phase fundamental instantaneous quantity connection domain spectrum g therein frequently arise property present series discrete powerful subsequently method analytic procedure source error purpose henceforth vanish perfect wavelet analyze function simultaneously zero energy additionally satisfy wavelet transform projection rescale index scale normalization convenient signal vanish frequency wavelet define represent require wavelet maximum amplitude domain peak wavelet convenient version frequency domain wavelet peak real also real value within central measure deviation p description wavelet offer next interpret quantifying frequency wavelet performance wavelet calculate generalized wavelet wavelet define normalizing unit wavelet control role wavelet examine valid wavelet wavelet broad exactly analytic replace wavelet peak e wavelet third peak find wavelet broad wavelet show wavelet popular analytic wavelet analytic gaussian wavelet wavelet example increase vanish time frequency domain negative right fix window long wavelet change peak shift peak illustrate effect separately vary third wavelet equal wavelet special call know curve ridge constitute aggregation ridge amplitude ridge scale fix time scale magnitude phase satisfy condition phase match interpretable associate see identify amplitude minima introduce point group call ridge curve amplitude wavelet denote ridge henceforth map contiguous ridge curve condition latter scale also estimate signal evaluate wavelet along signal superposition ridge amplitude transform curve show negligible tend choice yet velocity record experiment center reflect property infer record regard example noise transform heavy frequency dash signal quantitie fourth contribution square dotted plotted wavelet transform wavelet show amplitude result signal present fig record wavelet clarity behavior wavelet space frequency amplitude exceed cycle fig c ridge nearly record appear drastically reveal variability residual increase increase major low compare signal decide result subsequent essential ridge ask vary bias term ridge amplitude superior performance wavelet minimize bias address question goal attention wavelet datum example analytic expansion quantify increasingly high amplitude frequency express analytic series uniform hand interpret global derivative equal signal instantaneous time power important position theorem real value e differentiable truncation interval th taylor expansion time employ lagrange remainder signal expansion version taylor expansion pure th instantaneous give deviation pure instantaneous interpretable fundamental uniform frequency instantaneous vanish everywhere instantaneous instantaneous find group instantaneous single value quantity part occur simplify expression make amplitude instantaneous may q complex instantaneous may upon exponent side expression relationship instantaneous right instantaneous leave derive expression turn complete argument define coefficient appear first power leave side recursion relation compare expression term order obtain first instantaneous function instantaneous thus combine power power instantaneous power interpret rate general value consider complex instantaneous q constant complex instantaneous frequency local instantaneous instantaneous frequency become matter take fluctuation instantaneous bandwidth cause instantaneous return panel present instantaneous bandwidth reveal nature magnitude three signal degree correspond long duration wavelet column right estimate amplitude generally small instantaneous writing panel g compare instantaneous frequency imply instantaneous quantify stability truncation interval stability small clear recursive polynomial power instantaneous frequency furthermore imply determined variability signal level describe square analytic th expression ridge section signal ridge accurate rational wavelet constrain appropriate criterion signal local stability give signal frequency derivative satisfy criterion frequency grow power low implie mean localized place tight condition odd odd quantify degree wavelet symmetry function vanish definition low order third quantity wavelet choose stability level curve along ask along criterion fig plot n quantity unity wavelet criterion gray dot gray begin term vanish wavelet difference behavior fig wavelet derivative always less rapidly value unity plot decay increase slow thus truncation choose satisfy choose application later goal wavelet transform make analytic wavelet support need ratio window width energy q width energy thus wavelet localize analytic consist series represent place constraint measure duration low vanish peak choice wavelet interpret appear high order involve sufficiently amplitude localize analytic keep low expansion point necessarily canonical analytic deviation one amplitude proportional wavelet deviation amplitude maxima minima intuitive localize true localize raise issue address section explicitly signal property wavelet wavelet minimize precede localize simply curve know frequency ridge section instantaneous estimate find ridge either amplitude ridge respectively ridge exist bias ridge instantaneous frequency employ taylor express ridge wavelet transform frequency curve power quantity refer instantaneous frequency obtain expansion representation theorem assumption cast involve triple summation wavelet derivative fix power relate signal instantaneous curve scale signal instantaneous frequency scale plane wavelet instantaneous final frequency instantaneous neighborhood time plane deviation th plane permit instantaneous frequency curve choose hold constrain imply write serve separate perturbation perturbation form early find expansion ridge order important term high stability way set derivative analytic signal involve derivative signal signal wavelet local stability involve neighborhood simplification instantaneous neighborhood ridge close ridge henceforth assume family analytic wavelet appendix amplitude ridge unique instantaneous amplitude explicit term truncation form term complicate firstly choice wavelet wavelet criterion thus curve omit deviation instantaneous form analytic substituting find expression analytic error due independent along instantaneous frequency instantaneous curve write eq recover difference add perturbation prefer amplitude practice amplitude superior indeed break isolated algorithm example identical amplitude show experience favor amplitude due aspect numerical similarly estimate estimate write analytic estimate amplitude localize analytic derivation ridge representation neighborhood instantaneous frequency solution equation find state ridge lie within neighborhood instantaneous find direct instantaneous scale however instantaneous estimate way reflect discrete level likewise amplitude phase analytic experience rather instantaneous bandwidth along reverse form instantaneous appendix residual quantity instantaneous plot lead instantaneous frequency perturb estimation recover instantaneous frequency great fidelity instantaneous desirable amplitude curve phase unlike direct instantaneous frequency either ridge lead estimate instantaneous phase ridge curve instantaneous bandwidth localize phase amplitude instantaneous identical low perturbation localize reflect amplitude occur evaluate proportional find attribute change motion across instantaneous amplitude instantaneous low order localize analytic instantaneous defining proceeding term order twice residual panel version three wavelet development show compare unity choose suitable wavelet ridge base analytic signal instantaneous instantaneous true course stability know wavelet insight consideration whether sufficiently wavelet simplicity record truncation j horizontal order line wavelet mind signal somewhat inspection line l difference three difficult visually level ratio median quantity unity wavelet column duration poor exceed signal extensive outside dotted analyze produce estimate result correspond value respectively variability column require column rough rapid fig lead contaminate assume ask iterate estimate play iterate middle recover produce great fidelity signal unknown say quantitative preferred wavelet criterion lowest satisfied choose fix wavelet g respectively reveal ridge reflect successfully perturbation develop paper appropriate signal wavelet reference address wavelet condition wavelet contribution signal variability expansion contrast analytic choose wavelet criterion understand consider role control meanwhile control clear analysis close frequency decrease hold distant role wavelet suggest wavelet short minimized discuss presence fluctuation impact scope paper small attractive spread clearly spread sense useful value small satisfying wavelet within duration suppose analysis problematic early analytic wavelet negligible quantify involve interaction increasingly order instantaneous increasingly order domain wavelet interval locally simplify substantially frequency wavelet taylor scale instantaneous frequency wavelet ridge extend bound globally amplitude amplitude due analytic signal instantaneous curve
something quantitie piece information relationship neither attention focus select maximize constraint important traditionally consider piece new I require lagrange multiplier impose usual normalization eq multiplier determine substitute familiar p unchanged accordance general wish something example constraint moment lagrange multiplier determine first yield additionally rewrite p side resemble put name piece normalization reference box know box inform company box get box assume suggest like idea perhaps perhaps customer kind ball another count box mathematical outcome average us sample outcome get implement usual relative entropy multinomial information fair bias completely status incorporate come constant numerator flat example moment reflect special replace discrete relate relationship something side divide complicated reduce final n denominator factor solution side calculate k count parameter series lagrange come sum evaluate involve lastly converge kind next relationship fig purpose enforce moment go large suggest problem would write etc produce mean I treat frequency unfortunately fluctuation I course asymptotic fluctuation equivalent update show detail template world problem method compare employ I superior fluctuation asymptotic I case I robust traditional recover would method implement finally I method ill acknowledge discussion probability present rd method example
simulate simplify build simplify different make composite align parallel section take produce composite pattern centre sphere example standard problem classical available whether problem sign pattern way pattern test distribution could distribution statistic tend find realization hard core explicitly basis pair correlation provide correlation simulation yet present likelihood ratio base detect nan datum truly differ poisson cell unless I occur nan parameter datum assess fail non difficulty single identify difference often difference test try determine evidence formal come look every realization process indistinguishable call try quantum atom sphere pack electrical process complicated material concrete engineering comprehensive electrical necessary require base identically realization process use exploratory tool difference seek statistic constitute continue evidence fit difference evidence fit effort manner try establish fit inference idea establish fit always risk sample realization chance eliminate give almost reduce chance use way develop powerful elegant describe sphere find experimental useful finding model formally try access library point distinguish require pattern develop context inference poisson process intensity estimate poisson number moment last realization come long realization transform separated radius would although random statistic moment union stationary reduce origin estimate find moment third translate window intersect third fraction set grid histogram measurement statistic smoothly define scale within assume proportional circuit weight loop path circuit sphere electrical path smoothly summarize pattern describe realization interpret apply process average great smoothly statistic build process need find hyperplane separate impossible visualize point separate provide smoothly arise sphere discrimination effective smoothly average law quite would unclear circumstance cart cart separate realization know process process core process dl process process unit portion realization point allow close unit coverage htbp realization kind testing realization realization distance realization fit fit realization fit intercept polynomial realization statistic statistic artificial vertex triangle find maxima statistic mr mr ci mr ci mr dl discriminant distinguish hard core well rule make classification training statistically indistinguishable cc mr mr ci mr ci mr dl classification rule division table suggest classification analogous htbp nearest use unchanged investigate realization pool realization take spaced material set eigenvalue statistic realization suggest pool separation clearly visible pool mr mr ci ci discriminant misclassification discrimination well cart attribute small depend heavily case need effective rule realization powerful near case intensity close show solely determined enough statistic
small cost update u li nb l iv li li uk li c l ik lc l lc lc li hybrid choose dynamically change threshold threshold fine algorithm implement program equip ghz iv operating server virtual start account implement time completion run ten subsequent otherwise ratio report deviation time compare deviation operating algorithm decide test combination always method force stopping optimize evaluate euclidean report performance force scheme test five test avoid value report p c model compatible model model square divide running dominate small complexity computer memory cache matrix cache rely cache model bad reference correspond table improvement force suffer force something htbp p datum big show simulation cost predict application early partial performance ratio time partial early algorithm scheme improvement appear stop representation term phase table roughly decrease extreme efficient happen stop report experiment htbp c early stop contrary reduce pre show update fine threshold time choose universal running time vary second lead rough summarize stop reduce efficiency two reason phase modification p early stop expectation improvement much especially early c c time compatible world usage grid second hardware force hour result world choose benchmark english word list list small list correspond english form word already least implementation reduce string series transformation transform allow replace experiment drawback length distance version string string divide use grid big grid lead bad quantization epoch l htbp force obtain artificial acceptable force small propose acceptable time mostly reduce order algorithm especially result identical self epoch proportional epoch induce run actual associate validate verify theoretical describing show overhead favorable reduction induce modification permit current computer minute run algorithm one basic acknowledgement thank anonymous valuable real accurately possible rely dissimilarity enable sensible comparison som adapt data dissimilarity unfortunately suffer quite algorithm important reduction dissimilarity som outcome time world time implementation fast projection dissimilarity proximity pairwise vast fix finite unfortunately quite vary instance strongly structure represent done adapt structured tree processing strong datum design compare meaningful solely dissimilarity rely dissimilarity solely dissimilarity adapt recently anneal dissimilarity formulation som dissimilarity anneal som focus adaptation visualization string call dissimilarity som also median som som variant drawback run som som linearly number behave see propose avoid essentially dissimilarity nevertheless proportional produce modification implementation reduce burden important property modification algorithm som adaptation theoretical new running practice method evaluation conduct list recall propose dissimilarity positive som neuron arrange structure prototype denote som arbitrary undirected length short relationship model standard som kernel possible test possibility nh lc obvious therefore calculation cost representation total one epoch therefore dominates optimize term time limitation datum produce different som optimize dissimilarity result try solve contrary modify without review use optimization problem complexity epoch calculated calculate calculate cost total cost k one calculate representation phase need therefore representation force situation efficient force simple early stop loop idea loop sure overhead loop induce early favor early stop inner outer fast
q lagrange multiplier become remain lagrange multiplier implicit moment bay bayes I allow constraint answer nature ask handle common updating infer process compatible rule implement comes experiment repeat common value experiment practice composite relevant px outcome general receive constraint maximize entropy subject second essentially different way process current maximize constraint maximize constraint simultaneously equivalent update p decide I method treat I principle distribution aspect nothing state indeed receive must allow update retain alternatively decide new posterior reflect new constraint process arrive summarize appropriate become appropriate old remain valid therefore inference process failure error example updating make poor twice likely come face mathematical follow deal model multinomial sided yield instance infer moment note refer general form particular lead write q accordingly I update figure multipli rewrite find particular time datum constraint q infer whole refer actual relevant use I update joint eq reader recognize precisely familiar reproduce solution simultaneous want available collect production whole randomly yield constraint refer produce old posterior normalization look crucial update intermediate satisfie satisfie update multiplier applie produce simultaneous realization I rule case allow bayes rule datum simultaneously put I additional information bayes raise I might sequence different inference way I constraint regard rather capable moment purely might life early example think fairly application really type average large method become ensemble away entropy favor probability favor claim purely artificial construction handle incomplete expect without make reality acknowledge valuable theorem axiom case conjecture criterion exercise summary em department physics usa maximum moment problem obtained process sequentially illustration multinomial solve detail maximum entropy assign probability update I special significant I prove complete compatibility bayesian entropy second could information although bayes rule quite impose constraint prior propagate process provide lie reach bayes concern infer something
satisfy quite theorem well subsequent b b randomize b g cdf correspond infeasible procedure cdf interest omit per g subset selection pp analyze cf briefly probability converge limit convergence n probability possible theorem cdf characteristic like square uniformity phenomenon theorem cause appropriate crucial property correspond sense entail converge sense b tp b formalize refined existence uniformly estimator technical process result strength section derive formalize general abstract derive closeness distance convert cf aforementione paper gaussian work verification property inter property property typically regularity g locally asymptotically rely property finite post regular semi distribution selection establish establishe result class parametric model e convergence additional consideration employed discuss next regular hence parametric ease gaussian matter rather necessity uniformity theorem conservative family theorem conservative enable reduction apparent reason precede paragraph extend semi estimating value view uniformly make formal p key step g find g differ certainly provide answer immediately note simple argument solve g convex combination dimension exploit establish without purpose general g continuous allow paragraph extend include discussion follow verification g b alternative sense c van locally formalize cdf regular consistently relative weak despite relative ease satisfactory asymptotically estimator pose title predictor stress procedure consistent finite post basically bootstrap consistent selection section bootstrap subsampling estimator however randomize post estimator suffer uniformity hold regression model else obtain general allow nonlinearity dependent present derive large include aic typical hypothesis test consistent selection procedure like bic testing suitably always cf asymptotic picture detail elsewhere estimate distribution se valid topic certainly challenge task independent independence reduce hence converse entail shorthand let expression display contradiction negative identically cdf line hold cf component upon least negative attain immediately fx assign g possesse set consist hold trivial otherwise consist index clearly row moreover let denote vector remain fx f z origin true measure let row establish tt necessity converging point find subsequence along subsequence along component either converge subsequence sequel converge distribution kn converge n absolutely lebesgue happen suppose asymptotically satisfy choose critical correlate satisfy every lemma pz rw mean variance show p ap pz b pz r r r r follow take follow observe rr rr cdf write agree sign coincide z note remark c suffice return p q q display normal concern ab note w w identically row z third q indeed write w z w write cx q c zero hold every give contradiction inspection simplify claim b exist show hold mean pp z loss discussion paragraph lb see differ ii become arbitrarily sufficiently shorthand prove eq q observe indeed non discuss q obvious p p q nk proof cf fx desire property ii continuously th satisfy observe obviously absolutely let obtain ph consequently entirely zero conditional lebesgue desire uniformly r form correlate constant critical asymptotically uncorrelated satisfy converge constant b radius u fact total variation show cdf g distribution base overall remark establishes describe case reduce recall note map tt suffice hence return value turn light establish proceed tp tv tv b tv n assumption convention second supremum extend let b proof convenient show notation variate zero p p p elementary converge suppose pz pz pr pz pz pz pz nr pz pz proposition remark error remainder furthermore subsequence along b surely support involve along clearly loss generality hold least q p variation trivially total carry index p pz except possibly zero concentrated pz pz pz q show index term total variation along subsequence establish tv x p vice versa b g hold mutual p inferior apply proposition view use tv paragraph establish similar lemma I use suffice show q satisfying b establish p rearrange regressor correspondingly rearrange row generality procedure selection coordinate hold every since sequence prior follow condition continue replace sequence b exist b square preliminary stein type procedure ed bootstrappe improve ignore pre testing drive regressor bootstrap pe variable fitting equation specification bootstrap work economics university model incorporate lag uncertainty accounting order advance h predictor unconditional sample estimator uniform versus b fact estimate working department statistics university shrinkage result distribution selection nd texts thesis department university b inference comment estimator result rao procedure k property preliminary van university theorem theorem corollary remark hour ne ne ne ne chapter chapter head head head head head ex compatibility lemma conjecture em em ex ex false true em mu mu ex mu th mu mu mu align align end end style define environment allow type allow tag post selection combine procedure result criterion g impossible estimate even measure exceed low sample situation estimator selection estimator regressor thresholde consistency risk support national science foundation grant mat material paper many inference stage specification model regressor lag etc contrast assume know prior analysis except consequence actual statistical estimator inference traditional differ substantially theory wu section ignore theory drive relatively recent unconditional post framework compete asymptotic post study set non model compete p simulation extension finite unconditional choose possibly incorrect framework study asymptotic sense predictor study pe latter simple inference serve paper sample distribution typically complicated purpose e confidence suitably replace sample distribution plug finite large limit uniformity unknown replace motivation investigate distribution necessarily plug limit ask post estimator suffer phenomenon answer post estimator negative result idea regression non regressor singular matrix parameter regressor avoid misspecification suppose necessity check basis selection retain unconditional scale center cdf function true particular estimator satisfy estimator estimator necessarily depend converge every show even consideration distribution reliably assess precision apart also provide necessarily consistent function example imply infimum section ball replace ball origin shrink rate uniformity phenomenon uniformity phenomenon typically unconditional function resample resample approximate distributional see general large procedure aic procedure base fact uniformity phenomenon specific procedure cf show result limit unconditional post selection conditional inference argue step object unconditional similar report treatment estimator general selection introduce notation select unconditional cdf model contain provide detailed encounter section extend large aic nested remark collect discuss scope paper auxiliary collect finally notation euclidean eigenvalue relation usual stochastic shall p pm give nest regressor parsimonious let converge c p np pp p q post exception g selection post estimator particular manner shall consistent estimator despite typically uniformly subset infinity asymptotic variate p probability estimate obtain critical go infinity little cdf variation formula expect poorly cf since case auxiliary decision correct decision sample next poor construct consideration cdf exceed remain attention subset parameter rate nature low cdf asymptotic pa vector equal matrix reduce regressor column asymptotically regressor discussion follow shall argument estimator reason instead estimator evaluate cdf priori notational convention subsequent uniformity phenomenon e satisfy let denote large hold every satisfie may critical moreover far left imply b variation sup satisfied range uniformly condition proposition large suggest estimator asymptotically uncorrelated satisfying inspection proof continue estimator never view proposition square well show cover highly impossible perform even uniformity shrink sake completeness radius uniformity p supremum side conclude section represent correspond f
graph connect similarity edge reformulate similarity mean group point within basic go undirected two adjacency graph undirecte require ij note sum adjacent diagonal vertex vector f nf otherwise convenience shorthand way intuitively edge vertex connect path intermediate connect nonempty partition construction similarity distance construct neighborhood datum connect distance distance connect roughly incorporate graph consider near neighbor connect vertex neighbor relationship symmetric making ignore neighbor among neighbor call connect call near case connect edge connect similarity sx I width neighborhood play graph influence exist behavior refer tool spectral exist e want carefully literature convention author laplacian lot literature matrix necessarily eigenvector order increasingly eigenvector unnormalize w small eigenvalue real f f f part consequence note laplacian adjacency coincide position unnormalize self change unnormalized graph spectral spectrum eigenvalue equal number span eigenvector eq ij f term vanish equal path whole graph obviously indicator loss accord belong note laplacian spectra connect component matrix w l random summarize normalize eigenvector constant eigenvector non prove see immediately multiply eigenvalue leave directly multiply obvious statement statement graph laplacian connected spectra equal span span proof proposition spectral section arbitrary similarity unnormalize mm similarity cluster way unnormalized cm eigenvector cm version use name mm unnormalized eigenvector matrix contain column th row cm point cluster algorithm use eigenvector work eigenvector hence normalize laplacian normalization need accord mm way laplacian contain column cm cm state graph three algorithm trick abstract property representation section cluster trivially new simple clustering difficulty detect reader familiar numerous bt corner histogram graph fourth row eigenvalue eigenvector base spectral would place quantity sample accord gaussians show axis set fully near ignore shape mean discuss eigenvector see vector reason part show eigenvector information unnormalized case eigenvector cluster thresholde third eigenvector separate cluster fourth eigenvector separate eigenvector carry four intuition separate accord similarity similarity weight point within group spectral adjacency simple define please recall complement subset simply choose notational otherwise problem course want explicitly request reasonably encode normalized subset graph vertex k ia ia small coincide coincide try balance condition np hard discussion cluster lead unnormalized rewrite problem vector def v rewrite unnormalized graph calculation w ij satisfie equivalently rewrite relaxation set discard optimization f lf eigenvector approximate real relax discrete indicator simple spectral coordinate point algorithm carry I unnormalize minimization similar principle define indicator eq set orthonormal check l fact get eq trace minimize rewrite h relax arbitrary value relax h standard form problem g tell solution eigenvector spectral partition mean lead unnormalized spectral cluster def eq f df problem f df substitution eigenvector give eigenvector eigenvector indicator matrix observe h substitute relaxed norm minimization eigenvector derivation guarantee relaxed compare exact minimize unnormalized simple graph vertical v v eigenvector unnormalized laplacian graph author unnormalize set cut clustering construct unnormalized spectral balanced cut exist contrary hard relaxation unique semi might many relaxation relaxation particularly solution popularity due solve another argument cluster graph jump vertex spectral interpret try walk stay jump intuitively explanation balanced cut random reading transition walk connect possess tight eigenvalue correspond random walk come small describe property walk let bi random subset pa first observe p ij nice normalize spectral tell versa vertex back nice property particularly appeal oppose short vertex decrease get vertex short distance look path lie path graph seem remarkably generalize inverse decompose inverse diagonal entry undirected vertex ij l jj l publish prove walk see help laplacian matrix proposition map vertex induce inner product subspace construction unnormalize spectral vertex embed compare additionally embed column justify construct cluster euclidean building cluster differ considerably example consist indicator consist column completely ignore column eigenvalue ignore eigenvalue multiplied situation spectral embed thing make assumption loose relation possible similarity strictly however perturbation study eigenvector perturbation perturb perturbation state certain eigenvalue usually eigenvalue spectrum statement justification spectral cluster consider ideal cluster similarity cluster point belong coincide trivially place distinct perturb version one theory tell eigenvector indicator completely mean spectral argument cluster problematic eigenvector particularly low give really summarize unnormalized cluster spectral justified theory justify treat issue actually implement problem occur various construct trivial implication construction think construct similarity go construct make sure need sure also similarity text make sense document belong range behavior really connect case live euclidean reasonable sx I depend type bt choice concern want neighbor neighborhood illustrate different toy choose cluster choose large one upper panel sample neighborhood difficult data point neighbor scale near neighbor near break several reasonably far away connect connect region nearest graph neighborhood graph act mix scale hence mutual neighbor particularly suit detect cluster use connection sx neighborhood point neighborhood far negligible neighbor simple adjacency experience less decided known guide task ask spectral trivially perfectly correct sure consist isolated achieve result hold example neighbor g limit really work connectivity want medium graph connectivity mutual admit lost advantage mutual cluster induce separate area cluster near neighbor neighbor choose significantly standard graph neighbor connect meaningful part unfortunately general choose parameter achieve determine small fully connect graph latter minimal span heuristic choose similarity function similar need one neighbor similarly span rule ad give inter experience show spectral sensitive similarity unfortunately study similarity justify rule recommendation future research implement spectral cluster practice graph laplace neighbor neighborhood sparse eigenvector one power method eigenvalue eigenvalue unfortunately necessarily usually exactly converge bad note vector cluster return encode reconstruct method justify criterion criterion usually treat large variety index pick example hoc similarity information theoretic use tool heuristic relatively ideal completely spectral many geometric help closely topic refer like heuristic introduce difficulty cluster histogram nearest plot unnormalize separate cluster gap eigenvalue relatively behavior connect graph plot see well however case approximately overlap difficulty make human obvious number illustrate choose work contain bt text neighborhood affect connectivity neighborhood component soon neighborhood graph interact choice problem nothing spectral algorithm extract nothing means seen express separate cluster belong cluster map exactly unit extract cluster clustering least euclidean meaningful quantity euclidean relate author diffusion use technique solution use hyperplane advanced eigenvector span eigenvector distance spectral cluster eigenvector degree regular vertex well however distribute considerably opinion rather unnormalized normalize eigenvector rather objective satisfy partitioning point simplicity objective graph mean cluster maximize similarity implement explicitly incorporate objective differently note cluster similarity maximize minimize implement explicitly aa different exactly normalized eigenvector normalize consider maximize instead think weighted minimize maximize relaxation unnormalized cluster keep cluster implement cluster objective unnormalize spectral implement objective bt red text completely superiority normalize come algorithm assume point distribution underlie data space fundamental datum spectral normalize spectral mathematically prove eigenvalue converge statement cluster partition diffusion space statement normalize unfortunately go beyond convergence statement spectral unnormalized spectral fail converge single space mathematically property limit prevent spectral possible occur make sure corresponding eigenvector unnormalize significantly mathematical reason dirac function eigenvector eigenvector want refer statement illustration toy eigenvector unnormalized laplacian fully see row star plot dash eigenvector dash line significantly already move case eigenvalue figure see soon dirac course eigenvector dirac function finite size toy concern preferable avoid unnormalized spectral normalized laplacian algorithm cluster favor reason indicator eigenvector computational go back suggest partition suggest eigenvector discover discover nice overview spectral cluster work spectral example co additional learning environment new insight link principal also large gram interpretation matrix interpret interpretation extension concern case cluster paper publish various scientific area encourage reader
correlation volatility dimension construction appropriate augmentation scale diffusion path run ease illustration assume constant volatility extension volatility model intermediate point time induce x diffusion volatility quantity eq pair consecutive deviation volatility occur therefore path u exploit transformation scale occur path sampler brownian motion current time motion need transform accept closed metropolis step propose point need accurate metropolis reject able store infinitely thin course possible assumption partition non record nothing irrelevant record alternatively motion scale compatibility record neighboring brownian suppose situation represent store triangle new require draw former via update pair successive appropriately bit accept fy fy transform diffusion therefore similar discuss issue convergence mcmc fact augmentation increase also volatility experiment check scheme augmentation estimation retrieve correct despite partially finite stochastic reflect increment effect euler approximate record transformation irreducible list consecutive observation transform remove proceed flat restrict point overlap need update acceptance acceptance rate sign autocorrelation plot value plot indicate sufficiently good agreement confirm lag lag post post post volatility month plot analysis slight deviation model sde brownian proceed parameter exhibit variance volatility become consecutive q transform eq algorithm time measure year successive assign restrict positive identifiability eliminate possibility run efficiency acceptance autocorrelation plot show reveal issue provide volatility evidence support lag lag lag lag ccccc post post median constitute tool base inference diffusion appeal elimination close expression nevertheless augmentation construction degeneracy operate accommodate transformation efficient rapidly expensive also additional include moreover state jump fundamental inherent volatility non construction unique choice adopt investigate grant gr thank helpful pt pt address carlo degeneracy issue transformation operate time mcmc present stochastic volatility illustrate consist rate keyword volatility diffusion natural phenomenon extensively diverse finance biology stochastic differential sde standard brownian motion drift reflect standard deviation existence weak solution translate condition see chapter particularly receive recent see review finite development approach indirect function transition observation property write approximation may analytical succeed sense allow monte carlo dependence markov regime different observe stochastic volatility use extensively model price volatility stock interest future stock price history alternative utilize carlo mcmc bayesian treat observation introduce however simulation degenerate overcome dimensional framework multidimensional diffusion volatility alternative operate diffusion path irreducible scheme accommodate diffusion every stochastic volatility organize problematic algorithm introduce whereas propose test conclude augmentation problematic introduce latent simplifie involve posterior observation price evaluation entirely fine partition augment specifie augmentation euler converge relate quadratic new define ensure value time change sde q another brownian law brownian motion scale wiener measure new brownian bridge condition applie define transformation give q sde drive brownian motion operation essentially volatility dominate despite fact integral prove law motion respect dominate depend parameter brownian extension wiener bridge transformation transformation brownian dominate sde correspond unconditional sde law general observation divide part remain diffusion overcome arise latter likelihood let g path volatility view similar introduce brownian time sde respect
perspective smoothly parameterization unknown location flexibility approach structural tree computational base gp alternative enforce effort extend partition simple fit stationary trend instead tree condition average get full accounting recursively space region contain consist covariate input plus intercept gp inverse gamma wishart respectively treat known specifie trend correlation coefficient common region ensure two depict smoothness frequently problem transition response boundary distinct regime vs isotropic family family range parameter generate family well correlation function preference global structure equal surface quite smooth take alternatively could encode specification implication ability interface limiting model sensible prior similar term conditional denote level hierarchical distribution first draw draw step draw show use joint drawing conjugacy available linear conditional n inverse q matrix eq improve chain obtain parameter metropolis hasting mh ratio see prior family draw integrated gibbs tree change propose move accomplished cause mh acceptance reduce split hold swap parent child input node pick internal node cause child parent become tree always accept rotation adjust configuration height g ht proposal rotation well thereby successful rotation swap empty location partition leave remain unchanged ratio part mh acceptance ratio minimal calculate ratio tree rotation depth rotation decrease q ratio leave grow operation change either select grow add new must discard mh model integrate newly child parent ergodic draw prior unity proposal elsewhere operation child draw parent create split grow randomly parameterization parent analogous require numerator denominator model straightforward distribute eqs across boundary aggregate tend smooth boundary grow translate high uncertainty boundary gp consistent continuous datum fit almost indistinguishable code use gp code interface either platform library link automatically execution improve interface http www project web package package translate unit cube make prior parameter condition proposal take uniform slide window around accept parameter function probability mh acceptance ratio straightforward allow processor speed helpful machine posterior classic acceleration head attempt predict suggest structure k thing family function combine dataset leave credible interval illustrate partition completely unable capture central end flat left reflect uncertainty fit segment smooth mcmc yield partition particular use gp dirichlet mixture report fit considerable always region rarely speed gp also winner discard take hour ghz allow mcmc round discard minute ghz essentially continuous fit average behave case suffice case parsimonious much computationally prior practical look flat rather gp clearly looks mostly give posterior mix gp fit nearly flat large nearly greatly computational gaussian process limit package advantage analysis herein gp ten mcmc first discard burn treat posterior single ghz processor algebra library invert single stationary take inverse need per round count like factorize multiplication posterior posterior lift response six level angle mean lift speed angle attack plot predictive gp work reflect increase variability rapidly high issue convergence large region increase attack datum level lift response figure mcmc notice near large angle address outline credible interval slice predict lift plot slice item essentially figure example gp structure typical quite even individual goodness qualitative example trace mix slice projection quantitative assessment cross validation predictive location hold hold response find proportion thus fit wider necessary process computer wide use model fully treat parameterization modular easily correlation limit gp estimation prediction result uniquely contain large contribution design computer much linearity simulator provide full particularly towards active exploitation characteristic exploration parameter space lee computer explore partition simple method deal development efficient detail analysis simulator demonstrate effective simulator recursive partitioning nonparametric modeling fast particularly early stage proceed sophisticated create heavily wind still fully moment wind use computer involve amount require differential equation simulator require five thus interested create computer approach literature gp prove able issue gps conceptually accommodate expect wide first require calculate covariance second structure strong estimate oppose predictive observe location nearby global use regard stationarity section particular real desirable instead gp schmidt defer partitioning region fit separate straightforward model explicit beyond constraint implementation project package index average smooth indistinguishable datum motivate detail material combine gps gps implement return propose return represent direction look somewhat primarily model euler mesh euler solver hour computer simulator theoretically solver start always automate mark accept inferior automate solver randomly arbitrarily estimate stop neither simple error term adequate impact inaccurate simulator force degree lift force roll focused lift force input simulator entry measure angle attack goal lift force six level range angle attack alpha vary turn surface discover determine trajectory contingency arise fail enter angle adjust necessary interested associate uncertainty surface modeling gps partition bayesian hierarchical reader gp specify inputs explanatory zero k prior sometimes simple specify often formally index correlation kronecker refer always serve mechanism introduce process correlation govern help become numerically singular notational convenience conceptual motivate though force wherein depict specification typically specify guarantee symmetric method clearly mat ern class function simple isotropic parameterization range parameter determine
evy characteristic evy measure measure distance whole rate depend function deconvolution know estimator numerical proof assume dimensional evy process triplet characteristic triplet drift volatility jump function reason explain zero finite borel evaluate characteristic b finite borel follow sequel evy characteristic increment increment distribute finite borel basic arise asymptotically efficient since sure infimum sequence implication norm convergence distance integrate characteristic mode convergence weak result evy distance satisfy property strongly consistent probability b uniformly characteristic standard law function hence since volatility restrict l evy intensity practical method raise certainly optimisation example consider exist obtain b analogy probability existence moment mild devise attain henceforth evy evy convenient kolmogorov evy instead measure define express characteristic nonparametric estimator start decide appropriate lie borel naturally strong usually rather finance option price cf measure estimator class test uniformly function respect convergence instance us fourier function equality imply use distinguish complex cf indicate strongly estimate empirical th derivative c proof logarithmic decay accordance hold uniformly choose nd give depend smoothness theory stronger require finite smoothness decay difficult consider bound dominate fx decay decay exponentially inverse distribution characteristic typical example positive risk bound continuously rate yield convergence loss b higher achieve prove logarithmic naturally cover characteristic nonparametric obtain low rate understand deconvolution decay characteristic rather due noise characteristic contrast exponent attractive govern nu nu point abstract hilbert ill pose read regularity regularity degree ill regularity criterion criterion regularity logarithmic obtain logarithmic mainly eq supremum certainly remarkable parameter involve procedure imply consistency already well natural yield hx f error function fu theorem logarithmic provide conclude allow obtain smoothness coherent rate compare continuous classical focus example optimisation simulate anneal look preliminary minimize around turn stable global optimisation identification plug n careful fu might estimate fourier occurrence denominator might effect particularly problem estimate certainly good noise level rely nu nu prove nu nu positive expect preliminary q positive nu fu pointwise serve routine pilot certain drawback usually necessarily work reasonably simulate superposition brownian motion convolution histogram increment increment true characteristic gaussian absolute empirical pilot estimate jump haar zero pilot discretized pilot evy measure display characteristic pilot error fitting pilot give good characteristic rescale l evy pilot mass final pilot estimator exclude gaussian deconvolution quite hard rough functional jump value estimate increment one yield rarely jump hence indeed estimator proof definition two satisfy associate function corollary c nu envelope right specify analogously remain set grid lipschitz wu wu wu z wu u consequently small integer generalize g use u nu triangle obtain proof follow r e analyse occur obtain nu c fu fu wu wu monotonicity supremum supremum arrive condition yield c improve upon evy derivative necessarily evy well moment imply eq latter obviously finite u sequel establish wu hand side look local evy triplet eq infinitely eq sufficiently also infinitely calculation fu dx dx dx e continue u du du du last distribution closeness u polynomial decay minimax never fast exactly infinitely characteristic fast exponential decay stable law sufficiently large requirement exactly e b distinguish n u nu follow say nu nu nu u nu therefore obtain nu nu nu n conjunction finally hoeffding nu nu u nu n yield nu equation fu e nu fu useful discussion bb decrease evy spectral evy ct p jump l probability ed cs characteristic uniformly real fan
scope product might relevance behaviour c prove ergodic ergodic induce suggest augmentation geometrically ergodic present relevant step appropriate metropolis hasting step worth establish ergodicity obtain computable advance direction see one interesting behave asymptotically tail auto case demonstrate algorithm walk correspond furthermore identical probability scale become tail ignore component phenomenon subject investigation denote lebesgue marginal chain analogous marginal ergodicity dx everywhere positive positive property sufficiently e e prove fashion td pick sufficiently e inequality analogously notice bound integration prove establish notice generate neighbourhood sufficiently exist e argument compact geometrically second result prove almost possess finite generate neighbourhood lemma prove theorem reversible respect transition tail sense continuous symmetric heavy tail eq geometrically use integrate iterate statement argument argument reversible chain imply geometric fail denote restrict normalise imply sufficiently large rip marginal gibbs walk geometrically u throughout shall denote x prove statement triangular verify write distribution statement value dx x dx prove stability consider separately easy demonstrate ergodicity easily see geometrically ergodic analogous ergodic prove fashion ergodic review symmetry weakly point result neither possible establish ergodicity property hold instead drift ergodicity since identically rip geometrically notice ratio function integrable proper correspond converge lx rip geometrically ergodic result surprisingly first lx lx x fail geometrically uniformly sampler geometric ergodicity proposition definition gr gibbs joint miss model symmetric behaviour distribution choose sampler gaussian theoretical introduce widely adopt construct method specify simple exchangeability flexibility extend simplify use powerful markov chain monte develop decade area longitudinal disease hierarchical specify distribution dimension datum application cite impose bayesian posterior intractable relatively use sampler mr furthermore compare crucial approximately behaviour specifie model concern discuss motivate example paper stability sampler gaussian three main stability base provide characterization broad augmentation component collapse component counterpart extend hierarchical latent section discuss extension proof establish drift argument follow iid iid symmetric distribution bold letter capital letter second several application clearly wish presence reference therein tail tail influence prior model improper although intractable use term terminology refer sampler collect invertible rest paper refer improve update h natural expansion apparent theoretical model exception result wrong linear observe sample one start mode mix negligible lag diagnostic assess chain converge nevertheless never event start around contour plot contour spherical near become concentrated mode look tail lx gibbs geometrically x section gibbs htb cc b contour joint sampler iteratively iteration generate coincide two sequel probability g start function l regularity total say geometrically rate starting term ergodic ergodicity ensure burn arbitrarily bad certain ergodicity qualitative property ergodic high fail ergodic lead undesirable break poorly real proved generate function neighbourhood geometrically ergodic neighbourhood geometrically ergodic establish geometric drift requirement section gibbs model specification proof broader see page code cauchy mm double power double exponential special tail give double two refer geometric sub geometric ergodicity stability cccc stability c u u u n c e g specification remark effect z z identically exist since prove although remark obtain symmetric possess everywhere positive proper impose convergence theorem necessarily context stability ratio geometric heuristic tail situation reverse algorithm augmentation adopt convenience mixture belong gaussian notice implement group block gibbs convergence collapse gibbs integrate scheme norm operator group collapse sampler enough collapse concrete answer special cauchy proposition geometrically distribution consider establish geometric ergodicity variance effect larger big one readily give conclusion practical certainly model address identically identically geometrically uniformly ergodic generalised student sampler numerical ar observe error simulate
dynamic iii comparable domain update square iii localize half signal envelope amplitude I filter cutoff frequency hz signal visualize compare height generate mixture see fig separate satisfactory confirm separate time factor execute technique processing spend sort domain fix source besides efficient sort frequency optimize time correlation weight square filter frequency domain mixture satisfactory separation record synthetic concern computational partially grant dc research grant mi pilot award center lemma study dynamic blind source sound base frequency minimize weight adjacent frame correlation coefficient lag method adaptive record music excellent blind bss aim extract source signal mixture signal mix environment instantaneous mixture realistic medium channel transmission signal delay paper study sound decompose instantaneous matrix method second fourth sound matrix joint orthogonal lead matrix source remain permutation estimate source large delay process utilize dynamical information propose dynamically receive frame statistic permutation domain optimize channel channel cross coefficient lag cross similarity propose allow accurately reliably freedom mix weight iterative dynamic bss adapt acoustic environment encourage satisfactory organize update result capability separate speech music room let component process processing divide partially frame mixed impulse response source mixture additive may add sound interested music shall number dft jt j pa dft dft frame approximation independent convert blind instantaneous reasonable approximate eigen matrix approach g iterative theoretical function direct reduce random dependence assume proper scaling follow independence conjugate transpose identity hermitian uniqueness order phase matrix call multiply statistic determine mean eq mutually independent group last except imply diag mu w multipli modulus row joint call equal maximize quantity b b minimize rotation cost stationarity robust stationarity satisfied music speech music signal reality short scale scale depend production speech become envelope time permit meaningful spectra equation ms unless acoustic synthetic potential real time able capture consist find initialization frame separation measure sign though absolute lag inside envelope reduce amount yield capture degree two production viewpoint drastically motion signal may music maximize function norm multiple lag sound maximize help reflect iii scaling de multiply reconstruct shall part part want multiply least exponential mathematically mix matrix impose automatically fix freedom choose scale nontrivial square chapter variable make half separate iv frame arrive generate tn tn job consistent order n signal extend frame newly continuity maintain time arrival correlation equation however moment fourth follow symmetry among conjugate compute th statistical quantity suggest update moment adjustment contribution early sample fourth moment move reference therein due different occur compute information transform back low experiment separation reliable reasonable carry production find frequency ii often frequency search sort permutation orient method among reduction consideration matlab source http mit paris maintain separation dynamic type mixture room speech music synthetic speech noise three list value share size overlap percentage frame limit computation namely frequency report
rao field decade e review due realization complex handle even computer filter incorporate consideration exhibit subsequently filter filter simplification efficient system converge filter instead move slowly expectation multiscale markov illustrate method however variety markov fast multiscale reference therein filter trajectory differential reconstruction pure chemical place speak estimation accomplish step evolution second likelihood allow fast mode rao standard filter phenomenon wide area exchange suggest slow scale example scale include chemical reaction several reaction similar problem see molecular see fast scale motion weather daily weather exhibit extremely large formalism separation worker multiscale phenomenon vast spend evolve may interested filtering structure address incorporation common technique control particle filter markov factor slow conditional conditional rao monte carlo separation scale make distribution quite example closely recently directly allow utility propose improve particle accuracy moderate observe significant analytical continuous paper organize general discuss presence separation lead allow filter construction analytically rao step main calculation analytically numerical differential type markov multiscale chemical filter dimensional markov transition discrete observation spaced variable variable admit density expectation family constitute call conditional filter write integral compute analytically particle simple form filter consist follow filter evolve accord get calculate rise challenge example integrate fast atomic order long past system scale framework average utilize dynamic substantial simplification system impossible reduce form motivated integration presentation concrete system equation chapter introduction idea apply multiscale markov independent brownian evolve macro evolve time scale micro I admit sx dx evolve step costly scale separation inherent multiscale phenomenon suffer difficulty sample high system show average rao assume rao reduce unfortunately rarely next multiscale integration rao q consequence enough dy central tool multiscale particle feature multiscale approximate particle eq particle filter particle evolve calculate measure end iteration reason indeed equality filter particle integrate kernel particle vector set quantity variable stop particle order illustrate asymptotic difference describe independently satisfy theorems application delta jensen assumption multiscale respect ergodic measure assume easily verify approximation course euler exactly removal impossible integration overcome multiscale discuss multiscale process therefore require different multiscale integration numerical recall simplicity inspire limit brownian refer macro denote transition sx k ergodic imply ensemble average rapid analytically short short run interval long average limit average yet solution couple coarse associate coarse micro simple euler eq write however imply invariant result estimate trajectory sequel symbol eq henceforth notation integration filtering application trajectory single marginally simplification dependence average integral particle rao correspond evolve particle instead update instead average particle I vector equation sx l evolve parameter sample particle practice require resample calculating average require practice initialize multiscale form markov process system simple numerical example demonstrate system parameter ergodic measure gaussian every unit particle filter run system discretize euler multiscale integration parameter filter definition particle filter reconstruction symmetry observation true method effective sample weight produce roughly gives produce unweighted empirical draw measure plotted figure plot particle filter indicate improvement quality comparable multiscale next correspondingly large integration motivate chemical stochastic behavior well molecular specie interact reaction channel accurately chemical master equation master simulation system specie chemical often take simulate numerous reaction event species molecular population reaction channel reaction produce reaction law master integer dependent intensity reaction evolution fast reversible reaction reaction specify use
bit upper r circle probability recover poor decrease transition increasingly point transition capture predictive model size know causal two recurrent future pick assign zero finite measure consequence resolution discuss length demonstrate become prediction error capture inherent recover partition however error substantially periodic predictive bit bit highly essentially causal available historical require keep historical information maximally state h finite length use nonetheless order different distortion much consider fluctuation indistinguishable naturally shape distortion depend reflect organization circle state box color green full dash dash causal box use successive random symbol equal symbol exclusive hide state total causal state complicated analyst plane prediction effective occurrence integer history process shown allow substantial show partition full even capture future probability fig pair eight state rate distortion curve decrease transition real joint length slide mutual x evaluating estimate variation may large causal true ref subtract control approximate calculate rather true approximate low temperature total number mutual correction generalize trade sample fluctuation r correct plot vertical quantity retain line know length length denote state ht x correct correct size indicate compute length use even process sec correct compression figure mutual estimate mutual calculate subtract low see peak thereby state sec process complexity sec historical probability future square gm spread input correct future probability causal filtering ii finite first principle process causal minimize predictive architecture stand approach hide assume state find minimum mention complexity source happen even fair find small partition previously mechanic store information give specify future ib unique minimal compression allow go plausibility theoretic causal minimal generate physics balance state computational mechanic state assignment mechanic partition principled ideal causal architecture approximate substantially application show correct fluctuation fit tendency hide application acknowledgment institute dynamics physical intelligence support department education fellowship ss thank l notation introduce infer extend principle learning causal causal correspond ideal principled desire complexity constraint relaxed filter causal system historical causal find hierarchy approximation causal change organization model allow correct fluctuation estimate thereby fit organization observer reflect vary measured complexity capture causal architecture capability investigate find cost one system lead new inherent nonlinearity many phenomenon highly correlate linearity novel discover structure successful attempt model mechanic minimal directly calculate introduce ref state variable predict series attention discover variable method ib complexity ref relationship mechanic ib predictive modeling make scenario distortion theory use balance implication automate building restrict causal architecture previously mechanic capture organization result mean store fluctuation datum fluctuation probability estimate take demonstrate stochastic process nontrivial bi denote measure ref whole stationary assume reader information theory ref mechanic past store information freedom demand infer many question review equivalence relation rise future partition specify history equivalence lead p give information future share denote information variable quantity name amongst ref therein distinguished rise furthermore eq consequence state variable predictive point past future past give p call process past store present causal state series broadly consider compression distortion principled find information distortion determine relevant keep irrelevant discard universal distortion application bottleneck mutual derive relevant modeling notion relevance low generalization future past directly specification reconstructing variable moment future specify maximally predictive historical establishes temperature sense value establish suboptimal give eq complexity constraint maximize predictive suffice recover causal solution visualize trade curve solution feasible infeasible give analogy distortion amount conversely predictive section solving apply anneal anneal time great rate distortion place look kl give length x exactly period plane vertical line entropy statistical various guide length plot exactly limit cycle fall deterministic reversible curve work fig dot horizontal cycle bit complexity vertical curve analog horizontal axis distortion axis ref x distortion optimal eq plot versus code evaluate value anneal curve compression predictive capture single fair distinct curve half cost find four capture number first one predictive curve straight trade future capture odd odd capture odd never vice versa overall long approximate compressed trade mean correct find full block h upper bit upper label annealing
eq acknowledgment thank suggest interesting proposition corollary remark paper metropolis spin system ise slowly problem keep usual metropolis metropolis decrease polynomially energy jump strictly connect world chain metropolis monte chain statistic want approximate proposal chain stationary achieve construct reversible initial important markov well slowly stationary variance inefficient metropolis slow peak physics energy temperature metropolis metropolis efficient local separate design move across barrier construct auxiliary see g essentially means change stochastically remarkable variant method energy level rely chain energy one energy efficient nothing formally prove finally mention world combine chain modification proposal mechanism fast field field ise field usual low slowly metropolis slight completely usual metropolis jump level metropolis slowly mix show modify intend well mathematical understanding sampling consideration concern review treat deal proof defer shall act gx reversible proposal metropolis chain slowly detail mix exploit form usually whenever add jump chain sampling actually improve performance metropolis chain collect fact concern reversible ergodic chain distribution adjoint important quantity relate eigenvalue spectral gap measure understand form realization chapter relate example total classical valid roughly speak chain define gap decrease exponentially mix occur move phenomenon bottleneck tool detect presence bottleneck inequality define reversible spectral chain chain reversible movement markov eq eq variant publish decomposition shall mix suggest fast set give big group act stationary bind way generate ball box proportional content distribution kernel q irreducible chain shall theorem describe every every chain stationary chain belong chain chain reversible distribution chain bound gap birth death chain eq birth chain prove derive independently variance metropolis proposal grow choice metropolis mix cost metropolis proposal usual cost need one number flip proposal thing slight begin decide flip random last expensive cost lattice spin statistical mechanic diverse mixture system simplify model even q normalization denote convention metropolis straightforward metropolis chain stationary kk k ising pass slowly proposal understand kind act odd metropolis kernel belong eq irreducible reversible spectral gap couple note reversible stationary define belong way respectively every chain set yield bind preliminary result unfortunately analogous additional prove nk least confirm nk eps eps eps eps death chain birth death ni nj eigenvalue follow variant path set ip ns q theorem appendix eq first every whenever integer q x note derivative q rearrange eq conclude proposition direct hence q whenever enough eq
analogous theorem exchangeable model prediction rather specify example fully exchangeability use backward look use justify region argument extend compression fact develop compression within formal look suffice bring line compression summarize bag contain look probability simple equal result successively replacement picture distinct say accomplish example contain form bag contain panel depict build step update bag different bag order bag account possibility repetition bag element call step bag appear appear remove backward backward obtain combine readily term outcome draw mean important look back way look moreover one line compression mm fall need exchangeability exchangeability consist number column context th full vector probability write intersection empty case tangent surface sphere plane sphere sphere put sphere kernel define compression summary full kernel property updating update summary term sum conditioning conditioning uniform distribution sphere surface hyperplane equation uniform surface sphere indeed model closely coefficient independent normally zero common freedom formula assumption variance first fully theoretical literature classical model conditional sphere uniformly surface freedom infinite integer prediction interval statistic consider give monotonically distribution proposition imply exactly simply classical length roughly make make exchangeability assume exchangeable normally article theory would thank wu many corollary cm edu cs ac edu ac uk use make prediction typically etc successively valuable successive sample successive even though dataset successive example independently widely present prediction numerical example treatment provide confident close think question rough past produce prediction typically limited number consist ideal prediction decision boost prediction look turn region contain probability least low make predict label explain set object successively predict one prediction use object precede example reader interested implement wish elementary need explain picture validity confidence valid probability contain law apply picture repeatedly independent valid prediction prediction line sense probability distribution weak compression widely validity prediction validity efficiency informative like prediction valid exchangeability prediction measure think may measure point predictor give great strong long give detail connection randomness article line mean exchangeability line compression leave aside important topic extension confidence laplace emphasis prediction important prediction successive interpret normally confidence informative always preferable even probability usually predict reader take explain find useful repetition picture successive prediction know prediction validity picture consider make draw fisher explain sense illustrate explain prediction weak interesting modern predict old example general favorable complicated old example simplicity predict help answer addition freedom predict fact freedom regardless illustrate binomial therefore approximately account picture entirely conduct separate consist draw st using might involve provide event approximately seem complicated experiment predict entirely experiment involve involve second master analytical geometry notice think actually overlap law applie know approximately happen generalize independent general even regression appear independence fisher accumulate return compression weak normally event uncertain quantity turn interval would like able seem assumption make insufficient enable numerical date involve mean normal well understand theoretic view think offer offer odd valid offer put disadvantage equally reasonable offer opponent multiply capital risk line model valid different general predictor precede new say predict safe odd observe calculate offer instance arise classification finitely problem possibility nx x case happen though uninformative certain correct fourth prediction clearly arise know whether define seem opponent know turn use fisher interval apply widely use probability fisher work english work mathematical extend influential subsequent movement advance extend equally exchangeability next relationship look exchangeability understand theoretically conclude sequence right exchangeability give clear intuitive make distribution might list avoid permutation permutation exchangeability exchangeable independent exchangeable suppose distribution independent assign z exchangeable exchangeability z property independent identically h middle obtain average large exchangeable exchangeability exchangeable average distinct joint independent averaging preserve accord exchangeable joint distribution show picture define bag list five conditional five one observe put bag possibly know know order probability successively without replacement exchangeability formalize notion bag sometimes list bag list exchangeability bag successive replacement remain bag distinct respective leave reader second emphasize condition exchangeability permutation value side explore cc cc cc cc draw leave bag draw reader familiar net recognize diagram example framework result probability theory capital factor express odd multiply capital risk idea sequence probability think game move total capital risk capital want possible odd bag brevity bag capital give odd n capital matter move exchangeability multiply capital large exchangeability equally determine exchangeability note study fisher number proportion high region predictor usual exchangeability exchangeable space adopt determined random bag say never exceed error predictor rare suppose event large fraction formalize way theoretically number tell mutually exactly event happen mutually unconditional mean happen happen game theoretic game number require successive rate look protocol rare protocol allow rate game theoretic pp apply theoretic well prove number exchangeability valid nest prediction distinguish case repeatedly predict observe predict symbol observing predict observe far seem new example old alone advantageous old prediction illustrate new old produce good nest exchangeability reader simplicity algorithm scope discussion encourage reader largely contain account apply dataset call real bag choose value prediction bag naturally add measure produce transform replace consequently relatively predictor concrete number number take predictor distance difference median already new change monotonic transformation numerical measure predict following suppose neighbor natural wrong prediction wrong old old way fitting pair coefficient affect bag example measure alternatively square new coefficient simplify explanation region exchangeable region simplicity state symbol assume measure significance implement force calculate measure region consist fraction bag example widely pearson interval false calculate reject probability reject make hypothesis say bag value q bag successive overlap observation error rare event event among strictly exchangeable proposition least correct discuss fisher interval valid normally take integer integer exchangeability exchangeability alone decide include value deviation average number chance large number large account lose normally distribute old normality may bag back reduce reach write ab define score change might convenient form turn example give special state symbol equal significance decide z x differ old alone produce prediction suffice algorithm alone change frequency old produce least rule time equivalent distinguish use species width plot classification list plot basis would classify confident sample answer obvious clustered measurement width separate specie perfectly wide c score species v used length specie second third specie measure calculate hypothesis fraction length long degree separation use evidence relatively precision learning calculation column label near give score obtain case numerator happen th us region confidence want uninformative false length specie reveal length great produce confident object wise report call near consider nearby length full advantage long expect efficient region two measure define specie bag bag consist old calculate bag bag bag calculate score region want prediction uninformative case near neighbor explain pp hyperplane mistake wrong side mind look dimensional hyperplane separate separate band wrong side band hyperplane possible hyperplane hyperplane obvious separate band interval calculate plot figure group mistake minimize short length check separation may interval minimize count bag list implement thousand reason quadratic separate mistake widely multiplier old present difficulty overcome randomization svm singleton uncertain singleton empty sample odd except whose specie try get picture apply different correct visible specie great produce tie informative oppose region specie empty table uncertain empty uninformative wrong turn predict number column task th length width predict actual conventional calculate least th predict normally pp take place prediction fisher bag review exchangeability interval comparable without measure one neighbor line new know neighbor obvious find close width several bag length equally fourth square measure become evaluate produce table always multiply calculation prediction interval least interval
margin parameter desirable small size make associated theorem bx x z nb virtue q eq check satisfied proportion allow size recent exact minimum parameter many evaluation error proof preliminary let n n consecutive z consecutive integer n lem continuous fix sn exist sn sn sn sn sn sn argument write c sn sn sn g g h g decrease increase monotonically decrease monotonically c coverage finite proof eq multivariate simplicity drop preliminary plus n consecutive element consecutive integer apply lem c n h n eq sn sn h sn observe h g independent since continuous sn sn sn h h c consecutive b c probability calculate determination estimation parameter prescribe margin confidence reduce evaluation coverage reduction coverage interval discrete introduction numerous field sciences engineering poisson define frequent base x nice estimate maximum likely possesse among question mean sn I proceed margin interval introduce determination advantage behavior characterize nb
higher contain occur consequently predictor group variable train compressed original prior easily stable distribution carlo compress split make prediction test handle problem high interaction amount organize term logistic parameter conclusion interface method divide sum training regression coefficient predictor value parameter specific note regression coefficient occur case distribution extra depend give relevant define prior easily assign stable additive symmetric index index symmetric stable cauchy location parameter gaussian standard prior cauchy parameter common moment denote individually treat unknown denote sample sample probably sample split splitting distribution depend distribution function omit evaluation markov evaluate sampling procedure collective interaction chain prediction actually sample huge next correctness original contain posterior parameter invertible original original map symbol use make original formula transform posterior jacobian mapping additive symmetric integrate result distribution clear ss equation discuss splitting prediction test case store huge huge therefore depend indexing variable extra suppose divided predictive need prediction test write prediction test sample easily analogously first contain away symmetric since gaussian use cauchy normalize degree width cumulative equation cdf inversion sample compute fairly prediction huge still split gaussian split useful sampling original however save sequence describe hidden markov create english need training base precede e use pattern ignore define intercept define equal write value sequence response x modelling pattern range pattern express x indicator intercept term assign value response use use model binary summation cauchy width denote treat hyperparameter assign inverse distribution shape leave summary inverse inverse gamma around need cauchy heavy side absolute variable mean cauchy width prior interaction may order cauchy therefore gaussian belief response redundant difference bayesian could say symmetric justify inclusion appropriate belief distribution linear share share incorporate strength sequence similar sequence replication make conclusion small replication cauchy model gamma eq multiply gibbs compression slice region plane show marginal sample distribution slice infeasible draw scheme particularly shrinkage procedure point expand end reach guarantee correctness cut part two gaussian case bayesian logistic compressed original iteration update update slice discard hyperparameter divide sum probability divide group function interaction pattern pattern appear therefore one group pattern express display shape pattern equal leaf pattern leaf expression expression grouping procedure continue take pattern pattern split leaf group must easily translate computer algorithm algorithm grouping like language index express meaning pattern leave show list le storing assume prediction give infinite length expression accordingly split tree say expression advance remove expression introduce therefore case increased train original regression pattern training coefficient grow x x carlo chain posterior express divide sum need split associate compressed parameter splitting identify pattern apply sequence compression set demonstrate length compressed converge predictive performance amount long hmm apply analysis et simple model observable markov whose dominate markov move probability number exhibit transition observable rectangle rectangle likely next ht figure length use vary state compare parameter train clear parameter decrease reach big original grow finite section compressed hyperparameter grow split variable express grow figure compression time time compress method include identify pattern less compression need repeatedly read huge disk improve hyperparameter sequence clear autocorrelation lag compress direction large likelihood reduction consideration lag autocorrelation accord time reduction markov chain error prediction minus observe response case practice chain cauchy slightly model well overfitting apply online website create encode character letter letter character special collapse multiple datum similar prior conclusion draw difference summary compression improve splitting consider order useful character example cauchy prior chain slightly worse identical cauchy well rate prior plot chain parameter model figure right scale rectangle show model trace compress plot predict character plot middle symbol median compressed sense character cc ccc technique cauchy move back character thing different surprising two stand letter rarely gaussian favor markov word repeatedly article cauchy move indicate posterior investigation cauchy useful keep region around word powerful information high interaction sided tail reduce predictor variable greatly training compress
proportional alternatively rejection various q proposal local mode beta wide dominate available software density tx r r column statistically probability orthonormal pz ta z n nz cx iteration generate reversible irreducible section column orthogonal eq linear first simply sampling conditional gibbs pz pz pz pz n introduction protein originally protein pairwise measurement among interaction consist introduction essentially j I think dimensional decomposition relation latent identifiability issue attempt complicated mean covariance symmetric pz replace symmetric observed element markov chain uniform prior mean z u z normal posterior distribution eigenvalue independent normal fitting eigenvalue binary decomposition measure fit protein two sampler length transformation rank tie randomly plot eigenvalue chain eigenvalue drop retain chain percent large thought point positive eigenvalue plot panel interact protein plot name eigenvalue direction contribute tendency node many model make magnitude panel figure display identify protein large positive negative value member group primarily interact member opposite bipartite biological reader plot circle statistic theory von fisher value von sampling mf provide study complicated additionally member arise posterior probit ordinal non function scheme outline website http www edu http edu role reduce value von fisher relational rejection distribution illustrate interaction inference decomposition network normal vector orthonormal role manifold primarily literature sphere commonly use family term eq von langevin density spatial describe simulating method version feasible dimensional manifold describe use heterogeneity represent orthonormal model framework variate arise multivariate represent variate measurement within row parameterization orthonormal respectively ordinal desirable error matrix make identically imply tu tv fisher normal writing py yy np uniform matrix von von posterior consist measurement link indicator diagonal factor probit z take think u parameter article describe von gibbs rejection infeasible generate von fisher converge implement algorithm interaction ability inference von mf sphere modify relatively straightforward implement ability vector mf rejection approach envelope sampling mf sample mf orthogonal concentration decomposition orthogonal rejection rejection sample mf reject indicate mf broad generally latter rarely ccc ccc ccc scheme chain sequence column express product tx column probability rewrite tn tx chapter give pz mf sampler markov proceed new mf column multiplication situation irreducible chain sampling detail von finally orthogonality autocorrelation sampler undesirable autocorrelation perform chain carlo derive involve resample generate stationary converge
strong law subsection several process satisfy chain two uncorrelated uncorrelated satisfy probability assume follow equivalent consider clear remain q measurable weak introduce discuss hold process strong z b z kolmogorov obvious process show kolmogorov martingale measurable z notion measurable define ergodic invariant ergodic follow mainly ergodic space conversely stationary recall space satisfie help know z function dynamical finally interesting ergodic process notion ergodicity say weakly mix weak ergodicity converse implication g introduce stationary ergodic space value stochastic mix e invariant definition coincide mix recall weakly ergodicity product lead invariant dynamical system probability variable subsection law number markov chain fix function homogeneous chain initial determine projection canonical chain homogeneous markov chain step ahead iteratively transition nb number finite automatically density hold theorem simple stationary homogeneous transition homogeneous identically generalization detail finally mention countable homogeneous markov notion risk satisfy number follow measurable mention subset equip borel equip corresponding stand integrable measurable convex risk l pf define continuous ensure sophisticated property loss exist constant integrable satisfy follow basically locally restriction lipschitz moreover locally locally lipschitz integrable moreover algorithms function margin margin loss hinge many locally lipschitz integrable loss estimate margin derive characterization integrable call base huber insensitive logistic loss insensitive usually continuous lipschitz say analogously say growth recall loss function obvious continuous converse implication trivial convex growth order also integrable realization number respect actually future loss bound actually almost finally integrable help reasonable ability provide set way whether measurable stochastic process strongly surely svms whenever recall reproduce hilbert exist nan svms I risk approximated less find rbf rich integrable countable space present main separable topological dense whose topology metric open consistent bound space rkh kernel exist strictly real satisfy consistent growth satisfy finally rkh sequence positive skeleton skeleton use reasonably omit let moment loss rkhs gaussian rbf number suitable sequence sequence unfortunately ergodic neither possible consistent ergodic moreover method consistent ergodic denote classification ty roughly speak universal speed stochastic satisfy law determine sequence suitably large however exist consequently stationary process number svms version number process law establish law svms much fail independent recall subsection standard mix basic thorough treatment hilbert integrable convention b ib j h p p definition equal symmetric b addition satisfy reference therein b b equivalent coefficient note constant yield give view estimate mainly early g learn mix stochastic probability bi eq algebra mix mix process respect mix tend mix tend weakly bi mix notion immediately trivial observation since typically stationary homogeneous nn stationary show ki respectively discuss stationary process mix particular mixing process z nn stationary gaussian n z result mix brief survey reference markov chain satisfie exponentially fast explicit suffice exponential mix contrast mix coefficient e variant see moreover ergodic chain mix irreducible stationary markov process satisfy information mix find let invariant dynamical system hence obtain z aa consequently weakly bi mix trivial strongly mix sequence invariant system independence information mix law number simple asymptotically weakly bi statement refer process actually show identically satisfy whenever consistency generate upper bi case process law explicit bound kernel supremum deal lipschitz closed subset convex continuous rich rkh nan problem hinge eq obviously continuous width consequently exponent g hence classification consistent generalize result generate svms loss rich svm robust assumption quantitative l pf employ markov detail like consistency sense satisfy obviously z consequently consistency result mix process lower bind establish consistency stationary lower weak theorem assumption polynomial svm svm mix deal part behaviour practical relevance interesting consistency mix necessarily stability markov type inequality skeleton consideration show strong establishe result distance growth separable rich rkhs moreover let space stochastic positive loss insensitive huber loss obviously lipschitz continuous remark hinge regression svms loss square reduce however weakly bi theorem know svms unbounded svms justification svms like replace describe complicated hence algebra write measure moreover satisfy show eq existence obviously sure convergence assertion law fix measurable consequently f z p p obtain almost convergence lebesgue supremum adjust convergence theorem n z b assertion define measurable yx z projection shift e weakly theorem conclude moreover z e follow elementary q locally since bound assertion assertion infinite integrable rkh bound exactly describe empirical essentially continuous furthermore define measurable denote expectation operator associate recall convex base continuous loss result let canonical map depend measurable growth additionally constant obtain measurable suitable easily assertion yield obtain p p pf l f p depend law th z almost subset property consequently image q consequently well n h z h n show case assertion loss fix measure regular hence integrable vanish outside consequently h n n h p satisfie obtain assertion satisfy number instead technical let exist eq without generality eq exist exist yield also estimate assertion since locally assume loss generality lead pf h pf pf r f respect set x g p moreover yield exist r pf consideration l pf assertion assertion obviously may generality satisfy guarantee almost l r pf pf pf p proof write z I assertion algebra together r pf p q imply b pf pf l h n b n theorem universal constant g n k estimate assertion without generality obviously eq together yield dt dt dt end obviously l pf p pf pf show pf pf p p p p h c n h p constant function parameter see estimate define find constant markov inequality h n g n z j p depend eq c conclude r pf r pf h p pn assertion let measurable hilbert z assertion trivial trivial bb ia already converse implication theorem ergodic theorem end stationarity define obviously b bs ba ergodicity ergodicity therefore theorem yield obviously preliminary pt theorem conjecture remark pt ex ex pt pt ms national
hmm shall state automatically satisfied strong occurrence infinitely node go cause technical defining infinite viterbi alignment treat fortunately two almost every infinitely many node resolution uniqueness state hmms follow three possibility aa ba bb aa ba bb aa bb ab q equivalent viterbi satisfy similarly mutually exclusive case transition satisfie condition case fulfil possibility change satisfie one case correspond see examine satisfie hold subsequently follow guarantee without automatically hold guarantee hold main lb lb lb lb mention condition nonetheless almost hmm imply exist large satisfie observation sequence u k bx contain barrier none none among strong eq contradict must ergodicity hmm realization infinitely many next almost every realization strong since however neither argument barrier ergodic hmms white one fail bx ax imply almost infinitely strong strong furthermore every realization occur observation complement inspection actually weak e occur imply observation infinitely strong stay state mention viterbi tie possible case break favor constant alignment generalization state assumption realization infinitely stop alone ensure existence notion generalize prove become similar often realization infinitely next observation hence contradict refine hence multiply right left whereas thus similarly apply bb bb bb ab node p ba ba node guarantee thus infinitely strong almost realization contain lemma node clearly discussion establish proof theorem simultaneous existence infinitely ab ba bb aa strong condition meet result infinitely realization theorem suggest analogous mainly state proof case due typical help analogy two proof could every counterpart theorem realization strong infinitely asymptotic viterbi hmms every realization almost realization chain otherwise infinitely strong definition example section corollary cm viterbi hide university uk email ac abstract early day digital hmms speech language image bioinformatic hmm distribution solely application hmm viterbi find viterbi attempt viterbi hmms indeed case viterbi alignment attempt rather assumption existence viterbi posterior viterbi viterbi viterbi hide irreducible markov positive stationary technical every correspond emission generate resp independently everything else application digital reference hmms define speech recently computational biology label code alphabet observation parameter hmms typically posteriori recognize path viterbi force name viterbi state hmms restrictive hmms existence infinite viterbi recursion viterbi namely maximum likelihood mix hmm special alignment recursion pointwise follow generalize ax bx generally viterbi alignment namely truncation coincide viterbi memory replace hold realization
inter connectivity present infer model module community receive much physics literature wherein module node popular sbm less study question inherent exist regardless wherein limit detect module modular develop relie modular give interpretable generalize module specify edge module use sbm module assignment bernoulli module pa roll assignment flip module determine extension direct graph write p k ij contain ij community ij community n constant hamiltonian potential previous assume sized group previous require user specifie average avoid functional preserve integrate update hyperparameter act framework module state probable module infer coupling constant chemical potential module assignment absence belief number module refer jeffreys fidelity determine context intuitive spin calculate spin integral eqn assignment accommodate application learn vb proceed beta function value partition dirichlet q optimum vb good multiple global number empty module evidence vb module probable module bayesian value specifically suggest module indistinguishable modularity vb consistently identify correct module runtime matlab ghz minute degree correspond module variational module identify assign modularity failure incorrectly group bottom range module clique method community range modularity initially find fail analytically furthermore em mode determine module module vb em control addition synthetic network vb american schedule team play schedule play make module belong misclassifie emphasize unlike probable automatically module detection latent principled procedure optimization modular avoiding computer cast united advantage design model parametric family world network acknowledge carlo methods model david
technique sense bit almost report table pca cholesky evident decomposition total variance cccc cholesky lt decompose matrix optimal show second cccc cccc cholesky pca lt cholesky times despite particular qr approach lt allow justified decomposition possibility depend order reduce computational pca qr study dependent fundamental cholesky couple qr mc order fair price option replication standard generator version simulation mc dimension propose super cube briefly grouping rearrange random consist fix one even computational insight really select sense reduce generator coherent suboptimal remain cc cc mc rmse cholesky rmse price price rmse cholesky rmse rmse cholesky lt lt cc cc mc rmse cholesky lt price cholesky pca lt lt rmse rmse cholesky pca table value expect cholesky sensitive used generation bad lt improvement far sensitive use lt lt reduce superposition sum accomplish lt evident result simulation decomposition superior extreme provide price option decomposition almost equally well optimally reduce view investigate view implement procedure qr factorization lt decomposition extensively investigate improvement option set extreme discuss cite reference sequence remain result generators lt cholesky considerable improvement carry accuracy notably attain fast qr decomposition implement compare slow burden time suboptimal matrix introduce qr computationally qr triangular qr provide vector approach calculate qr transformation van fundamental former correction identity rotation plane orthogonal order zero indeed th follow scheme qr factorization rotation g ta highlight define rotation rotation qr decomposition column form zero make upper triangular di dm pricing asset path extensive simulation monte improve efficiency nominal method investigate detail indeed rely ad considerably burden set transformation convenient combine option publish sequence transformation hypercube set accuracy give select also standard cholesky pseudo generator time quasi carlo mc computational intensive problem dimension several financial option pricing convergence intrinsic probabilistic reduction reduce quasi enhance rate mean sequence previous sequence purely mean estimation introduce discrepancy give extend superiority estimation notion truncation superposition truncation reflect really superposition take action general construction superposition author offer considerable respect pca high low property method lt maintain efficiency lt mc discrepancy consider simulate hypercube suppose target optimally intend lt generation superposition mc standard pca decomposition organize financial pricing notion effective describe lt several present step qr decomposition use estimate contract standard financial market free asset price drive theorem find unique neutral geometric brownian motion denote asset price represent instantaneous volatility w mt motion satisfy quadratic instantaneous risk neutral pricing contract neutral measure measurable determine contract explicitly entire restrict write security hereafter consider european market tackle financial portfolio payoff european options payoff weight average payoff terminal price option option section account drive geometric motion total volatility asset motion r ds dimensional geometric motion form solution pricing option sampling constant mt nm depend four index arithmetic option index denote great integer calculation price integral way purpose mc numerically estimate hypercube mc draw calculated vector variability similar principal sense decomposition good orthogonal eigenvector diagonal decrease linear combination normal lt minimize truncation variable trivial combination superposition sense lt procedure decomposed express normal variance minimize truncation equivalent maximize iterate impose must orthogonality lt involve combination normal lt constant payoff option price contract understand computational perform combination lt previous example show nominal dimension surprising product still normal variate highlight decomposition former normal variability focus combination payoff european neither normal geometric option european derivative contract subsection consider column complete expand subject provide step procedure column gram schmidt numerically return step sign adjustment affect solution stress qr indeed column reduce burden k one choice orthogonal matrix moreover equation provide lt contribution obtain matrix apply general expand column nm I order tc pp eq already must exp tc eigenvector impose constant combination generation technique generator far path concern standard cholesky lt decomposition option subsection rely kronecker decomposition iterative calculation orthogonal attain implement qr factorization require
liu discuss limit model canonical exponential gradually refer line single jump et consider generalized problem model change phase present contribution function multi lastly general study multi introduce huber property huber generalize among result ml l important linear respect thus modify regime middle regime completely give notation prove converge weakly small compound break purpose parameter point let variable absolutely lebesgue absolutely continuous everywhere case consider f l bx degree replace consider function xx jump finite euclidean convention construct estimator maximize square assumption satisfy ds continuous obtaining obtain convergence notice b consider obviously e c metric regression estimator consecutive order process possible value order percentile take end include likelihood deviation include normal double multi phase impose convergence absolutely mention identifiable multi estimator van obtain strong convergence derive ml ml model point simplify convergence difference process vary around vary change relation process decomposition assumption I theorem huber ii strongly consistent exist great b n study n positive gx k kn gx k arbitrarily side sum mean give estimator standardize decomposition random process regard let independence variance asymptotic similar break true point compound jump proof n n obtain two process standardize relation respect relation imply law account converge gaussian jointly et nx nx integrable z k kt kn coincide minimizer compact first prove bt b h ki e q cauchy bt change kt k result compound asymptotic ls ml consequence regression influence differently design van random two limit brownian motion change asymptotically valid uniform theorem change possible case generality schwarz p kx k k u present nu k x u lemma give v et existence point linearity assumption positive exist let exposition
confidence localize shape regularization adapt certain degree sign residual explicitly require distribute median consequence confidence sensitive section region possibility function monotone concerned monotone monotone make monotone mainly concerned determining point point adequate investigate detect basis size general conservative reflect real procedure shall strong convergence neighbourhood sense shape automatically smoothness restriction inequality determine mathematic simple involve taylor expansion intrinsic simple shape minimize local wish determine minimum problem explicitly analyse ability peak generating size interval interval right similar local independently identically extreme one local rapidly minimum power peak extreme local must local wish point short calculation small small study use string result minimize extreme upper constant proof local small tend local maximum local attain monotonicity argument alone investigate function extreme value follow inequality monotonicity depend small neighbourhood asymptotically degenerate local argument show apart shape concavity decrease tend derivative every value tend maximum tend tend tend furthermore similarly local behaviour large value large infinity non degenerate point convex repeat argument correspond finally f course result regularization th derivative evaluate eq similarly supremum lead programming restrict minimize tb noisy piecewise number peak reconstruction low spline figure use use determine concavity minimizing function smoothness obvious shape constraint figure minimize derivative minimization string shape generate taylor tend derive construct region asymptotic region f universal tight impose shape quantitative smoothness qualitative replace bound function bound give respectively solve impossible sample software scheme handle fast deduce latter fast necessarily make put lb lb lb nt panel datum replace bound dyadic second time dominate show term tb low convexity concavity treat similarly consistent linear programming upper programming solve value note gives somewhat consider point derive algorithmic complexity algorithmic panel upper panel dyadic corresponding upper calculation bound take hour second well almost indistinguishable piecewise monotone position string methodology position bound location extreme take confidence maximum fast bound finally mid string default extreme figure indicate combine concavity repeat idea determine concavity interval total derivative concavity figure impose convexity case sign residual band shape restrict smoothness minimum result restriction coincide figure panel compare degenerate centre panel panel algorithmic restrict form small lie small panel fast acknowledge smoothness acknowledge financial support science structure helpful comment make hand calculation ft put sufficiently maximum note tend imply automatically optimal interval tend adapt kf f c possible kf f ni deduce taylor points calculate subsequently describe construct interpolation explicit describe dyadic index consist calculate string possibly repeat eventually remark section university offer unify confidence involve centre simple shape interval concavity regularization decide region conceptually simple specify function representation design letter generic letter specific
censor convergent location operation may problem root seven cover possible shape slope censor pareto show neither subset horizontal pareto censor slope form far censor typical also explore result cubic investigate dispersion cf therein special appear case generalize minima monotone hazard slope function parameter define slope proportional summarize fr introduce example thus dispersion distribution satisfie characterize dispersion cf p positive extreme turn calculate slope side satisfy functional equation use depend generalized choice scale censor agreement value share due variance slope slope turn convergence convergence variance exponential family function say compact exponential family variance result appendix domain two follow exist hazard family uniformly substitute remark similar determining censor consider every censor formula integer min hand represent center scale convergence present form dispersion slope asymptotic asymptotic leave hazard power asymptotic involve q behave satisfied result albeit strong density hazard whereas case center location appear keep case slope function exist finite turn account continuity turn besides two extreme slope function extreme generate survival function right censor suitable operation slope extreme similar exponential fix slope exponential asymptotic shift dispersion model shift slope integral behave like condition main case may write follow side distribution distribution much infinitely note sense proceed theorem simplification idea hazard limit turn sequence fix inverse hazard nh etc parametrization n n nk nh monotone ny ny ny extend invoke condition integrate may connection support result proof arbitrarily choose small close complete proof acknowledgement support survival bar natural ann n ed distribution model application new york pp model extreme de weak ann von ann fisher frequency member du maximum ann heterogeneous population derive dispersion dispersion behaviour variance decision ed college cubic variance ann nd ed convergence exponential ann positive california rand ann life advance reliability ed hill york j family exponential family ed statistics international conference individual theorem axiom example exercise section mm mail dispersion introduce slope analogue power characterize family pareto slope classical value asymptotic extreme dispersion location family natural exponential generalize hazard slope secondary seminal ask binomial make wide function natural exponential exponential reference particular author bar investigate variance function correspond call dispersion exponential stable relevance may pareto make context variance perhaps relate fr paper extreme dispersion spirit lead constructive question material exponential family extreme dispersion exponential dispersion model manner variance extreme model convergence lead classical extreme dispersion finally introduce rate slope survival function hazard convenient use maxima et assume survival twice density let positive survival understand right except finite connection min way moment generate mt analogue moment generating analogy slope analogously name cf p letting analogous like scale slope satisfy translation integrate hazard identically behave combine invariant number involve denote shift include notation variable imply slope measure deviation hence play setup surprising law suggest fact min survival converging conditioning event correspond rescale obtain asymptotic distribution remark show characterize slope recall point exponential inverse lebesgue mean analogy imply analogous make need monotone survival monotone necessary order survival say hazard hence hazard though monotone hazard restriction survival analogue variance give open map analogously find hazard slope family among inversion family slope independent slope property c gamma pareto binomial cosine hazard familiar except family six exponential slope like transformation censor truncation rise hazard operation right censor replace modelling introduce consideration truncation restrict censor correspond restrict truncation censor give hazard concentrate consistent survival restrict change without extreme dispersion consist family proportional latter model function form index parameter index proportional moment hazard straightforward see rescale hazard function rate much correspond hazard location slope hence rescale survival analogous dispersion property preserved index parameter proportional generalization case parameter shift power become shift pareto logistic exponential pass hazard exponential reveal let parameter moment generate survival function et special follow survival hazard slope hence pareto gamma slope illustrate domain proper improper compound cf improper point exponential dispersion exponential asymptotically rate survival function give expansion letting obtain follow slope transformation formal sake brevity certain detail happen horizontal
mutually exclusive sum respective frequency safe axiom include discuss property hold monotonically lattice search fact query valid er query free database instance without empty table l reference domain domain next proposition assign properly valid relational exactly x tuple f ff er note two basic fact valid conjunction conjunction er er atomic valid tuple tuple f c f observation least hypothesis eq establish inductive safe query valid contain free inductive q formula safe er since semantic clearly inductive inductive third mutually exclusive event frequency fail straightforwardly language safe query would require fr fr fr ff another difficulty note theory requirement complement event safe safe safe safe free safe query safe tuple fr frequencie tuple always particular domain domain entity query potential answer entity satisfy answer unless entity probability measure boolean outcome point frequency definition express axiom logical receive view safe conceptually difficulty frequency definition single possible ask dynamically query express database decrease respect algorithm query avoid exhaustive result less refer valid valid query whose clearly previous mining relational potentially search atom iterative repeatedly apply single table mining consider query query mine conjunction twice table property approach application mining student receive student also table query respect frequent respect query student call rule entity relationship relational flexible expressive allow nest quantification conceptual definition query begin atom dynamically base domain individual er frequency customer prove axiom conjunction great frequency mining language difficulty search pattern language rule require language explore interesting entity grant author engineering axiom theorem conclusion conjecture corollary exercise theorem theorem proposition remark theorem em cs mining task search relationship concept association represent broad association entity association domain relational calculus entity relationship prove axiom property goal interesting always logical form association implication hold sufficiently sufficiently often hold confidence traditional concept capture essentially rule boolean involve student take database science well course association express quantification relate rule concept association query intuitively entity dependency entity entity safe correspond expressive order logic nest boolean quantification relationship extend notion implication er refer frequency er immediately namely generalize define discusse motivate usefulness entity statement quantification concept rule quantification contribution format characteristic target define set tuple support contrast set support think dynamically entity query respect entity main contribution support extend format organize follow database concept schema domain relational calculus entity query define class query entity entity entity frequent frequency show conjunction great present background database concept entity field subsection define safe domain relational calculus entity query table attribute possibly notation either name table display tv schema string string area string la l daily area global table schema entity relational schema entity er relationship relation type survey two tv represent tv table introduce assumption concern facilitate entity assume key unary single field relational schema entity tv schema entity table field tv program name tv entity generality form single key field table two composite key second assumption entity every denote entity assumption recognize entity ai similar refer name table entity distinguish occurrence transaction transaction entry table indexing would convention refer entity table another social security refer person contrast transaction key transaction transaction transaction name key constraint tv field tv tv tv instance relation unary assumption valid definition field database key field entity entity appear tv survey database table entity tv name tv week tv tv entity review relational calculus logical language give schema relational calculus formula presentation standard calculus table schema exactly language field entity unary key table convention list comment constant symbol exactly logical operator predicate tv tv sn sn must entity intuitively definition database instance formula quantify part constant argument candidate entity variable candidate database instance safe variable tv survey formula formula safe er query safe domain formula assignment formula safe specify tuple draw restrict table cf issue concern intersection database schema entity customer entity customer quite customer symmetric theoretic operation union intersection base define association therefore world entity query member type potential answer query close assumption basis safe query customer entity query base mention positively customer base domain make result assign domain entity free tackle one base instance think instance formula tuple f relation triple think name name refer relational single atomic x f g tv survey table table example sn sn f predicate specify direct bound domain affect attribute query query q several frequency query tv survey table query pt formula sn sn sn sn every tuple entity denote form combine entity consider rule logical implication say yx I composite entity include relation idea treat tuple familiar chemical treat entity compose element schema far constraint limit rather safe er query pair tuple rule atomic er maximal conjunction contain conjunction contain er free free entity valid er free context query basic case atomic formula safe think formula tuple query free contain name query return think column name name tuple single need compound statement database variable tuple atomic atomic conjunction x c ff x g x gx example find
follow complete square expectation differentiable notation claim invariance prove claim full strictly product abc real define quadratic lead composition let b nm p ip p b directly derive write law random k gamma gamma substituting result prove family parameter shannon degrees fix entropy come claim conclusion conjecture exercise remark use numerous field finance operation preference exact intractable inference prohibitive become variational alternative bayes logit solve extensive fraction thus method analyze modification history varied product development portfolio health service decision select alternative either repeatedly make number heterogeneous preference regression formulation encoding attribute draw preference model us distribution fully integrate create similar marginal heterogeneous include empirical asymptotic agent fast square root usual fully model markov carlo provide approximate draw joint relate integral output need collect store interest repeat draw variational offer maximize fully result variational technique approximate advantage variational versus variational far generate adequate mcmc draw biased evidence bias assess contrast mcmc number draw exceed gb discard draw burn many million agent preference mcmc chain require hundred thousand difficulty rarely apply indeed address individual valuable inferential bias derive variational algorithms discrete model multinomial logit mml study theory multinomial conceptually yet organize present mml procedure suitable mml novel delta moment mcmc mml simulate direction derivation agent index outcome set item accord th agent store covariate event variable use pair infer agent item event utility utility attribute agent preference loading unobserved select maximize utility multinomial logit denote logistic logistic often function research assume vector matrix bayes specify ht attribute preference hyperparameter wishart advance call approach mml bayes mixed multinomial logit turn procedure variable name inside distinguish density pdfs respectively empirical version mml preference density bayes numerator density density hierarchy integral close inference intractable variational inference deterministic mcmc usually family approximation true member calculation use plug idea place posterior substitute distribution mcmc give bayes mml family factor factored find kl h express distribution context formulate maximization q equivalence approximation posterior rather need inference coordinate solve appendix gradient update finish explain notice variational variational constitute step em current ascent new h bound function adjust surrogate maximize detail initialization multinomial logit calculations posterior sense simple extend procedure report idea factorize continue factorize variational eq factor factored posterior variate treat good approximate analog use agent choice randomly draw carlo estimate estimate variability significant handle three procedure posterior predictive posterior variational approximated usual handle integral exhaustive distance choice tv call equal attribute draw compare yield median error sense representative typical example procedure conclusion draw difference among vb tv item procedure exhibit make carry replication put table tv every significant suitable accurate conclusion attribute em agent vb mcmc vb low em low na na rl r attribute mcmc na accuracy three plot triangular figure proximity item qualitatively contour vb neighborhood simulate procedure turn ghz intel processor gb memory package package draw memory decision allow accurately display criterion panel axis fast mcmc magnitude mcmc computation agent heterogeneity versus hour compare hour hour vb scenario times minute mcmc variational discrete bayesian wide problem resource mml heterogeneous appear option one mml examine mml utility heterogeneou variational great subsampling consider covariate instrumental weakly draw covariance heterogeneous mcmc variational mixture normal mcmc favor method contrary use possible analog variational decade e discard fit resource mcmc previously intractable make appendix variational procedure logit semidefinite determinant function mml shannon unnormalized miss normalization constant q straightforward require attention multinomial logit mass function eq variational therefore new alternative first delta method call respectively use express parameter approximation appendix notice simulation result accurate variational derivation bayes distribution use block ascent convex update solve unconstrained requirement common concavity follow log sum exp standard unconstrained th j consist hand sum avoid make triangular matrix cholesky factor unconstraine triangular compare function appendix lead cauchy invariance
infection individual infect individual contact rate rate individual time contact brownian equation proportion individual write brownian initial assume sensible interpretation average number class actual assume suitably choose integrate sir present prior parameter prior brownian filter estimate filter conditional week quite would proportion decrease value absolute depend prior filter sir differ bit still look filter make estimate actually classical filter value bit range value surprising sir effect htb useful whose sir epidemic decrease zero figure go value could tell epidemic predict ahead future epidemic reach filter filter actually week filter value correct interesting accurate week far actual week epidemic accord likely cause epidemic filter estimate show begin maximum reach correct value long reach predict extra cause importance example simple well alternative particle filtering case law smoothing filter solution method exactly process discrete dispersion bit treat room possibility equation modify scaling introduce discretization due usage discretization exist without discretization use eliminate evy compound allow model possible importance framework filtering problem extend filter could form importance actual result properly consider select process would filter dispersion unlike base sampling implementation detail apply individual u extend kalman process u marginalization case eq brownian motion diffusion brownian example sde solution let brownian motion define solution solve measure brownian matrix rearrange sde assume invertible motion define weak measure weak helpful comment manuscript definition corollary application particle continuous filtering differential discrete time ratio need model drive law continuous rao static model consider particle differential brownian solution equation dimensionality dimensionality brownian motion singular state posterior engineering system phenomenon instance sensor infer transformation u sequential importance extend mathematical computing stochastic measure respect brownian represent exponential martingale base approach particularly successful discrete sampling idea several consider simulation diffusion show discussion filter multidimensional methodology base exact simulation discretization parameter stochastic perfectly approximate transition path methodology parameter drive filter unlike transformation methodology restrict sde dispersion higher drive motion matrix sde efficient apply dispersion diffusion drive error model model concentrate dispersion invertible drive restrict embed inside process drive motion deterministic plain integral kind kind handle sample inner integrate filter form approximate static contain static form certain particle stage brownian definite initial condition construct sde rough filtering time usage great importance shall see later already likely degenerate filtering absolutely respect drive brownian ratio importance derivation ratio sir recursion start sample initial carlo samples discrete sir cd sir sequential importance process q brownian ratio normalize unity extract inner dynamic differential consider section inner filter rao approximated distribution dynamic dynamic conditionally state application rao singular rao easily kind brownian give condition conditionally brownian driven process apply part process equation sample probability dirac delta function also measurement sir single gaussian sir realization simulate brownian realization q kalman unity particle low perform form rao procedure kalman context article rao static depend kt measurement assume static kt efficiently assume marginal condition meet algorithm sir weight sir static simulate scale likelihood new weight unity functional whole continuous importance spread application find differential equation angular random white angular velocity linearize notation brownian motion per gaussian suitable static
se v se v e theorem necessary possess top exponent associate te suitable periodic periodic fortunately easy multiplicative difficult model strict stationarity coincide periodic asymptotic turn periodic log moment periodic al suppose compact maximizer strong degenerate see assumption existence prove normality irrespective identifiability condition consider establishe periodic order may chose normality validate necessary establishe normality apply periodic stationarity moreover case reduce necessity establish admit iterate give negativity ib want observe jj imply top lyapunov k lyapunov exponent lyapunov sequence matrix say thus equivalent conclusion corollary ne integer multiplicative sa n n sa sa imply b ab b argument normality standard similarity spirit refer detail criterion se sn ns ns follow toeplitz sl backward shift invertible time vary st st degenerate therefore absence root prove se se v se minimize open n ns ns applying sequence sl ns sl ns lemma recover union neighborhood neighborhood sub complete theorem ns coordinate proof lemma aim derivative limit criterion third difference together ergodic suffice replace stationarity periodic stationarity periodic omit remark de universit mail yahoo com universit de mail establish consistency normality quasi give necessary sufficient strictly periodic prove moment underlie periodic strict periodic periodic strong consistency primary secondary periodic periodic proved encounter reference therein process noise periodic dynamic square imply structure underlie periodic write correspondence fact process model definition may trivially theory constitute introduction fairly consider order periodic stationarity study general periodic important existence ergodicity mention work concern considerable execute lee et al aim establish asymptotic process weak undesirable consider periodic constrain objective study structure indeed write cast stochastic recurrence equation coefficient h nm expectation periodic recurrence existence henceforth solution recall period furthermore borel set ergodic periodic ergodicity periodic analog stationary sequence e eq periodic translate stationarity process transformation seminal know
amount small ij p deduce imply finish result master thesis count bound submatrix uniquely must ij nm imply sum thus index inclusion department university corollary thm thm thm thm normal miss normal nine case equation exponentially though common problem arise longitudinal biological participant drop nearly replicate involve usually cause censor false conclusion technique useful reference estimation bivariate datum censor maximization replicate parametric covariate miss miss depend observe variable function observe censor wish maximize focus data bivariate assume dimensional case goal maximum likelihood algebraic connection algebraic geometry study critical solution fix intrinsic compute degree bivariate multinomial outline normal nine simulation bivariate normal censor mechanism real maximum also maxima possible maxima important likelihood section jointly combinatorial ml j constant convenient computation substitution identity bar log ml bivariate miss nine critical solution real coefficient choice parameter value random ideal polynomial field gr denominator encounter algorithm vanish ideal respectively complex ideal solution polynomial ideal degree quick nine complex solution ml bivariate miss nine equation complex conjugate least local denote definite similarly parameter tend maximum nine seven nine various singular case list distribution miss mechanism completely consistently maximum gaussian gaussian uniform test scenario sample run statistically significant local randomly regard run case real summary computation data covariate suggest one pay multinomial count record vector multinomial raw maximize multinomial region hyperplane p every real nonnegative maximize standard formula region probability linear convex exactly calculate ml need count region hyperplane remainder devote count proceed integer q multinomial ml monotone ml bivariate hyperplane hyperplane determine ml specify possibly amount hyperplane characterize nonempty prove classify appear nonempty unbounded position zero elsewhere suppose il il arbitrarily large unbounded ij ij sequence zero nonempty row column rectangular submatrix number row element form entry row contain contain lemma argument row put nonempty bound nonempty unbounded show exist coordinate absolute make derive contradiction boundary strict weak consider case suppose row column column map suppose belong
formula depict figure limit pearson numerical investigation coverage confidence show excellent htbp htbp normal rigorous c htbp normal htbp proposition definition construct formula guarantee coverage nature thus error application moreover formula excellent coverage communication area science pearson rigorous construct computational complexity involve event rate system probability instability uncertain recently prove normal approximation poor specify even situation quickly confidence goal rigorous algebra let bernoulli fix trial trial pearson respectively n j define u l coverage pr easy see hard prohibitive widely formula example therein central n n n n confidence limit problem trivially successful trial obviously complete lemma monotonically x j x n consider random sampling k x x x argument lemma consider I j x x x argument lemma completed obviously consider jx lemma monotonically prove easily computation show trivially lemma u finally complete implication interval pearson perspective interval make figure substantially low level bad confidence figure
cd cd cells baseline performance amplitude potential click e click amplitude click individual stress several include pair click day rt rt rest n normal day normal normal normal rt moderate minor baseline impact rest n figure run head keyword universit phone email abstract ratio quantitie specify limit method appropriate bootstrap use method deviation discuss use measurement resort specific repeat aware spurious correlation situation researcher researcher drug drug whenever calculate relative prediction calibration procedure situation arise ratio investigation speed discrimination biological basis stress specify confidence limit ratio classic also g health economic surprisingly issue cognitive cite call numerator denominator meet small coverage study ad hoc even problematic determine process mass index body divide height quite spurious correlation discuss detail I problem ratio denominator serious denominator normally denominator far behavior ratio cauchy distribute denominator numerator look like tail neither expect even identically e behavior expected mean random decrease allow behavior surprising deal ratio discuss case discuss confidence limit ratio numerator denominator normally simple geometric description alternative method development area allow relax normality show use fail numerator supplementary article short summary recommendation part ratio special case intercept ratio correspond slope variability numerator use justify form numerator denominator fourth spurious discuss old spurious ratio appear method three discuss context central justify intercept slope zero summary give allow quickly situation assumption pair assume discuss method restrict pair measurement restrict independent variance cv individual sample cv distribute normal often normally distribute relax assumption example point ratio conjunction behavior neither expect exist specify pseudo denominator bias second taylor certain propose practical situation equation problem size cv distribute normally distribute divide approximately student degree case correspond following meet b constant chi cf limit quantile follow three denominator denominator need discriminate set include exclusive exclude unbounded behavior possible fashion equivalent coordinate origin slope correspond intersection vertical determine confidence limit gray onto appropriate ratio projection determine covariance method qualitative confidence significantly axis confidence denominator unbounded get exclude unbounded exclusive b exclude value unbounde unbounded confidence interval denominator equivalent denominator significance significantly interval consequence arbitrarily unbounded unbounded exclusive nothing unbounde force fact method generate unbounded confidence limit alternative bootstrap away contribute certain unbounded count ratio level measure bound effectively conditional arbitrarily conditional never limit also unbounde alternative base notion go article bayesian ratio taylor limit bivariate limit symmetric mathematically handle fail problematic denominator close never denominator provide serious simulation taylor method cf bootstrap bootstrap determine confidence way complicated use population pair measurement draw original calculate distribution calculation confidence perform certain correction widely correct provide case normally bootstrap bootstrap limit b bootstrap apply intend confidence ratio see simulation denominator overcome bootstrap bootstrap determine quantile proceed result confidence therefore limited method perform enough show normal distribution skewness bootstrap superior method first converge second correct see correctness bootstrap clear method always pair subject individually ratio assume distribute limit calculate index use often almost example method justify value denominator unlikely specific happen method normal mean ratio show bias study g variability value calculate procedure numerator standard estimate denominator justify approach measure variability denominator reason call variance variability test systematically provide bootstrap supplementary material article raw ratio reconstruct figure difference estimate occur denominator significantly denominator zero construction unbounded discrepancy suggest small intend simulation course always bad result alternative much intend carlo data percentage limit contain value determine level limit contain simulation run e percentage significant pair explore across typical range use plot detail supplementary article correlation choice normally numerator denominator implement calculate method determine percentile similar always perform bootstrap equation detail pair bootstrap bootstrap bootstrap bootstrap calculate empirical proceed pair reflect estimate additional calculation perform deviation cf discussion method close bootstrap equally accurate cv denominator zero cv large numerator method denominator typically zero vertical infer soon denominator significantly accurate increase size size leave line accordingly area accurate index however area loose study locate figure band denominator leave band lead deviation show plot simulation perform band band b c confidence limit taylor surprising denominator typically result deviation figure method individual basis estimator ratio problematic around method reduce tendency ratio systematic note bias eliminate deviation systematically percentage extreme statistic limit panel summary fail denominator taylor fail denominator fail denominator large variability numerator never fail recommendation issue denominator approximately normally bootstrap bootstrap deviation normality denominator clearly limit intend taylor small method intend sake brevity index problematic necessarily wrong area figure intend confidence index denominator ever limit cv numerator exceed denominator use view slope therefore question arise limit would flexible sometimes careful assumption question whether error regressor denominator ratio depend model part describe situation measurement standard measurement regressor cf true pair call structural functional cf error correspond error additive uncorrelated equation functional measurement regressor similar classic measurement error typically repeat account variance additive third variety sketch classic interesting typical issue well alternative control regression classic measurement error uncorrelate create ignore estimate often cf see argument semantic give regression procedure measurement error standard error unique additional ratio measurement repeat intercept even enable fix predefine mean measurement classic value uncorrelated perfectly subject summary use regression accurately want certain correspond control ratio correspond ratio situation want ratio group ratio interest error allow ratio sometimes need want prediction slope calculate interested ratio indicate drug drug ratio limit combination ratio parameter nonlinear g effect deal general ratio addition handle datum perform approximation point discuss denominator elegant beneficial linear error index model specify specification confidence pose additional ratio attain close without justify I negligible need case error lead serious deviation g sometimes turn meet notably normally determine index estimate linear regressor ratio variable denominator see justify interest typically note justify error term parameter eq meet notably normally standard method limit incorporate give km body mass fit use turn plausible make correction division denominator fan fashion spurious spurious correlation spurious numerator denominator ratio intercept typically effect third relate number want assume random measure think step linearly determine much include cf inspection almost addition improve assume correction index equation problematic ratio test comparison full restrict equation equation equation mean intercept spurious correction properly see birth significantly correlate problematic argue indeed reason assume zero obviously easily non day short distance linearity need amount wrong ratio short distance car use long distance course serious example likely problematic rely strong error test also section closely spurious ratio ratio medical normal serious bias volume heart relate weight intercept person automatically automatically give illustrative conclude classified disease summary spurious use ratio zero intercept literature restrict index indice error scale conclusion measure deal ratio whether justify function denominator intercept otherwise intercept spurious standard simplify method measure complicated compare ratio denominator model indice study specific likely problematic assumption deviation confidence meet closely systematic variable estimate structure assume simple might denominator away taylor describe also alternative denominator bound author helpful comment manuscript correspondence mail universit phone behavior limit
attain lk uk interval without confidence application discuss exact infimum negative integer infimum k cb negative binomial variable negative variable binomial attain lk uk negative monotone infimum equal cb u c p next previous generalize intersection denote intersection u u cb c lk uk lk coverage theorem unimodal set unimodal note special unimodal specify infimum unimodal theory uk lk uk lk uk binomial variable theorem infimum l cb u cb consider infimum contain case treat follow I bernoulli uv lk uk lk uk lk lk uk lk uk lk uk lk lk lk uk l reveal infimum open equal infimum discovery confirmed investigate direct infimum probability random interval surprising local coverage binomial mild nonnegative minima argument prove theorems sequel sample bernoulli random variable lk u uk lk lk lk uk uk uk poisson mean function let lk uk u uv lk uk uk lk uk uk side poisson function nonnegative lk uk lk lk lk uk uk negative let binomial variable nonnegative l lk l lk uk lk uk lk uk lk uk p lk lk lk lk lk uk uk u uk far variable interval discrete certain unit unit number distribution notational know take suppose lk k uk interval interval preliminary lk uk lk lk lk lk lk lk lk uk uk uk take intersection event make position shall regard minimum consecutive lk uk lk uk lk uk lk virtue continuity lk uk combine lk uk show infimum lk lk uk lk uk lk uk show get great eq consequence virtue lk uk lk k l u k lead cb c cb u prove third statement already justify prove statement conclude lk lk lk lk lk uk lk uk uv lk uk v unimodal virtue lk uk lk lk u lk uk uv lk uk integer lk uk lk lk uk w lk lk uk lk lk uk lk lead conclusion lk lk lk lk lk uk notion intersection denote prove lk uk v lk uk p l observe lk p lk lk uk l p q u p p lk uk sufficient observe consequence notation exhaustive mutually exclusive case ii case write lk lk uk lk uk lk p p lk p uk minima lk p uk lk uk thus lk uk lk uk p great lk pp lk lk uk lk uk lk uk lk p uk lk l monotone unimodal conclude local proof complete non negative arbitrary n induction integer suppose k n k l k n vi l lem lm non n lm note nk nk nk nk k function decrease statement lem integer unimodal suffice case iii iv k l five vi nm l decrease n k statement increase consequently n exist conclude true k nn trivially suffice nn k g nn lk uk lk uk fact lk lk lk lk lk lk lk uk uk uk uk uk make lk lk lk lk lk lk lk lk lk lk lk lk show lk lk lk lk lk lk lk lk lk lk uk uk uk uk uk uk uk uk uk uk uk lk lk uk lk
notion relevant algebra affine factor code integer situation unit computable distinct notion geometry algebra corner hilbert states ideal finitely generator generate indicate conversely ideal algebraic generator set exist algebraic f contain see algebraic design ideal see gr else generator consider issue briefly illustrate occur outside scope general membership sum handle indicator factorial indicator fractional factorial design presentation design replicate orthogonal polynomial integer factorial treat level code root unity coefficient coefficient interesting orthogonality fraction coefficient ratio full interaction corresponding factor orthogonal exponent sum strength fraction cubic root unity indicator fact orthogonal two fact coefficient interaction fraction factorial design strength indicator interpolation code function factorial computed point factorial indicator factorial design provide level code rd root unity indicator mutually orthogonal indicator work choose order compatible ordering term term term confusion arise x ideal sometimes basis generate gr note gr basis generator ideal order gr gr basis gr generator ideal finite basis reduce gr polynomial point one gr ordering recognize sum mixture ideal every unique combination gr product give matrix invertible vector basis gr design gr example lead gr basis four invertible build column identify column list ideal response design gr basis relation fraction relation set present fraction model whether indicator gr base informative refer summary experiment lt procedure return slack intercept factor cone pass lt specialized exploit homogeneous polynomial cone homogeneous gr lead table generalize x small generator cone xy substitute requirement gr basis lose order dd jx simplex lattice interest gr vice see example see switch gr basis vice versa indicator gr consist union polynomial equivalently gr representation often terminate item easily adapt homogeneous ideal polynomial ideal switch gr design indicator belong design ideal moreover gr hence algebra gr basis choose resp resp lt f f j basis sufficient linearly ground say g gr different iterate indicator f implement item adapt polynomial degree computation vector degree polynomial base piece appendix f x eq var x x basis z z centroid use centroid design define definition integer double fraction typical simplex fraction include corner simplex default order mc discussion compute last combination indicator gr basis require think fraction gr ordering indicator informative work advantage complex response full complex response physical moreover factorial relation impose finite infinite response space hierarchical gr basis use light interest return satisfactory seem convenient lt regression gr significant tool relevant working solve address gr seem complexity algorithm normal gr last mention complete perform diagram indicator pa h end end return tuple algorithm section simplex lattice g sf sf code indicator affine computation var basis gr list polynomial f var tt cc minus tt end tt end end gr homogeneous ideal gr homogeneous ideal var list minimal ratio polynomial give power tt f e minus tt l tt end l
efficiency ac n p margin absolute many desirable small make possible compute minimum let n r n r see since observation relative error previously effectiveness binomial c characteristic poisson recent determining issue inaccurate exact demand sample permit new practitioner file size margin upon request useful old estimation binomial note actually simplicity notation drop preliminary plus gp consecutive b pp consecutive apply gp observe gp n p since sn h p result sn cp observe n element calculate z nb conclude clearly c bp na nb na nb make statement ii iii sn bn bn r sn taylor expansion sn sn bn bn bn bn bn bn bn bn n bn bn bn bn decrease lead observe actually throughout proof theorem preliminary minus p p n gp distinct element b integer lemma p lem observe p n n small sn gp p cp sn sn sn cp np gp n n sn exist sn sn sn h cp follow consecutive distinct n ready number minimum size prescribe margin old important demonstrate develop computing require approach bernoulli chernoff two bound attain discrete binomial infinite coverage recursive bounding technique introduction binomial fundamental application various specifically frequent estimate identical nice property possess unbiased crucial prescribe confidence high prescribe g reference show power modern computer contribution exact exist aim avoid unnecessary technique develop margin error absolute use bind margin computing section throughout notation integer I represent integer binomial assume summation tend mean notation size elaborate difficulty probability confidence desirable refer interval available classical size associate extensively point page impossible due intuitive suitably choose q make direct almost see page line side determine whether coverage motivate prohibitive bound size therein drawback coverage significantly prescribe severe binomial issue sample application report statistical eliminate resort inequality chernoff bind upon sample bind bernoulli problem size substantially sample since fundamental goal provide conservative quantification statistical persistent practitioner determine thorough discover determination tractable thm identical independent p b z obvious minimum determine start one gradually check enough n b c na nb na bn k b nb nb bn bc n define bn bn bn bn bn bn bn bn bn bn true na nb purpose shall complementary determine enough usually computation recursively similarly computation sample large bounding note bound conservative
place learner cope fundamental limitation distortion constraint learner noiseless channel bit arise location sensor measurement sensor unknown I assume sensor region sensor follow array channel measurement sensor compress sensor measurement feed function theoretic achievable error relate agnostic rate distortion concerned source code encoder decoder show input side code operating member scheme agnostic partial operate triple nr nr often denote learner nx f ny nx generalization keep notation encoder interest expect achievable operating derive sec sufficient achievable set outline estimation compress paper underlie parametric draw rate nonparametric namely good author nonparametric compressed observation state measurable theoretic bit otherwise learn algorithm generalize optimally absence rate constraint even ingredient proof learn minimization erm pac assume generalized lipschitz property concave pose measurable define minimal call kolmogorov entropy entropy every example set family compact euclidean absolutely continuous shall conditional distortion real number jointly expectation joint w r mutual information operational bit need describe expect distortion perfect two lead nr nr n substitution prove achievable correspond low usual strictly obvious nontrivial leave technical bound lead superior get finite replace requirement form construct theorem every argument rest well concavity compact regression draw variance independent square underlie absolutely density volume suppose detailed exposition integrable uniformly density divergence let r prove lemma say eq unconditional distortion distortion w assumption triple operate hard substitute information achievable compressed side code side encoder major difference technique long underlie family existence source code theory nonparametric incur decay exponentially prove theorem adopt erm algorithm code impose separation source code modular clearly gain attain design decoder learner justify source decide another performance complete may costly modular merely necessary sensor network direction work interest derive information bound algorithm could asymptotic excess infimum learner operate secondly learner analogy mind sensor acknowledgment support fellowship singleton finite tuple iy x w nr follow information theoretic therefore pr l theoretic weak l specify sample part rate suitable regularity admissible predictor probability information characterization achievable term distortion idea illustrate jointly distribute input construct basis copy prior theoretic simplify
easy q infinity tend indeed combine distinguish drawing trivially draw make ir get write q convergence make size index seek consider k I na ii verify condition turn use exist j ik important proposition proposition l km tend appear h easily deduce converge allocation may step force firstly let point iii case addition force thus lead check deduce also eq force indeed kn kn kn I kn deduce choose quantile convention denote law thank integration part convention optimal adaptive compute benchmark numerical choose plot evolution convergence run estimate word adaptive allocation horizontal convergence fast later one adaptive estimator allocation one would use introduction run manner additional computation non time allocation say introduction ensure take efficient proportional ip allocation optimal sequel wish optimal allocation analytical exactly adaptive useful test total importance plus proportional procedure plus precisely choose call variance ccccc divide improvement indeed often price variance ratio explain plot conditional expectation put option put case ii case option estimate value parameter conditional option conditional variance correspond thus proportional zero really function notation denote value quantity zero size change minimize minimize component constraint bind zero tend index see value last thus truly dm worth proportional variance analysis avoid automatically make computation nearly unlike toy negligible comparison justify minimize index index procedure work seek lagrangian minimization ix mh I look function deduce reach p proof plus proposition corollary modify allocation reduction asymptotic minimal confirm value interested remove positive denote distribute indeed distribute denote cumulative simulate option price q sequel force monte carlo estimator I variance proportion properly allocation equal force estimator eq q attain allocation small proportion monte suggest different importance expectation run successive procedure get first compute estimator could allocate proportion goal section allocation minimize theorem asymptotic confirm experiment toy price arithmetic proportion expectation estimate recently probabilitie estimator asymptotically asymptotic well advantage adaptively work step conditional compute total make end convention draw step convention increment begin contain step variance least draw ensure see seek convention second systematic sampling ensure systematic procedure asymptotically never b may think allocation find appendix index denote k km quantity decrease I ki use deduce estimator necessary strongly thank large follow procedure one consequence follow
divide ultimately development tool quantitative datum bridge biology computation recently center modern computational biology ability probe biological play important partly united institute medical sciences national foundation grant health gm thanks david manuscript conjecture assumption science university probabilistic popular tool domain use discover extent formulate behind statistical approach pattern biological mathematical express intuitive mathematical scientific ultimately development tool quantitative appear year resolution biological science dense introduction probabilistic successful biology exposition essential concept involve stage discusse contribute start specific give abundance across stage reveal one develop begin identify biological appear development illustrative process context reasonable typically probe abundance serial hard set definition create membership observable probe end conceptual development abundance membership translate biological specify fine tune variability carry information latent gene scale absolute abundance believe assign specification development fully address briefly enable biology assign quantity likely overview publish probabilistic graphical graph correspond conditional node express observe gene measure may unobserve arc specify distribution complete constant paradigm hyper see discussion distinction practice example gene considerable class choose graph encode measurement bayesian latent active observe certain express summarize explain structural encode graph model follow constant end refer quantity treatment process optimization estimation inference identify successfully summarize variability frequentist statistical probabilistic graphical infer consider strategy many strategy often inform integral alternative aim approximate end idea share approach likelihood make jensen em iteratively maximize equation analytic iteration think approximate estimating constant underlie exist choose instance likelihood empirical alternatively surrogate alternative model package available mcmc see variational inference bring biological intuition let continue encode membership gene observable may nonzero denote value constant biological probabilistic number specify across microarray specification exist fit inference summarie expression trend collection assign information make fine grain infer sure capturing set insight assessment relevance qualitative visual inspection focus biological gene pathway bioinformatic carry arguably interpretable functional family phenomenon investigation technique move model fit well biological biology goodness fit validation goal infer review underlie initial hypothesis analyse sense iterative statistical biological outline problem biological science infer measurement infer mutation longitudinal infer recent basis neighbor establish recently expression investigate predict clinical
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otherwise even depend rate local global satisfied necessarily leave side display display hold view give remark obviously replace continue supremum supremum center find zero probability zero every procedure subject set nuisance apply remark immediately arbitrary row rank brevity detail easily outline confidence depend partially full partition n na q inside inner proof arrive linearly range space consequently complement space arbitrary upon generalization inspection may replace kp furthermore remark apply condition example satisfied however already cover desire sort case behave however additional also necessarily result discuss statistical partially converge matrix partitioned let set sense c inside replace exploit arbitrarily large far hand arbitrarily cover except assumption post procedure consistently usual regularity restrict likelihood satisfy converge alternative n coincide show assumption estimator share hard estimator denote arithmetic nothing else post versus alternative satisfie selection standard converge zero instance reference selection possess oracle interval interval naive coverage converging interval n c nc feasible n c detail confidence interval thus symmetry restriction result assumption every coverage satisfy na n denote cumulative calculation normally give n normally distribute denote albeit coverage coverage show case infimum small side right limit coverage two jump one except trivial two merge consequence strong coverage probability positive inferior coverage n na prescribe short satisfy follow word asymptotic equal strong example illustrate discussion axiom exercise lemma theorem phone mail ac department sparse demonstrate substantial term parametric primary c post estimator sparse increase attention recent true parameter ii penalize scad variant suitable asymptotic estimator coincide infeasible restriction p fan li scad oracle fan li receive attention paper oracle fan li wang wang li wang li zhang closely seem superior show translate good necessarily coverage substantial reveal pointwise special sample go infinity see confidence asymptotic center procedure coverage nominal level parameter mention sense infimum probability paragraph problematic much optimistic actual problematic oracle discuss risk view problematic fact finite estimator limit assume subset procedure schwarz minimum regressor hard square zero threshold hold mention square sparsity return every belong inside inner condition trivially usual euclidean arbitrary element extend suppose satisfie sequence confidence coverage inside measurable probability assumption consequently obvious inclusion suppose every particular follow upon observe ie vector regression diameter
find ml degree matrix arise unable concrete biology dna sequence consist alignment match occur matrix hold resolution conjecture range constraint especially symmetry exposition aim include primarily binary gaussian work algebraic problem definite general behind realize gaussian geometry variable art mat mat solve information contain symmetric principal almost minor index example correspond minor symmetric equal follow gaussian random symmetric model algebraic subset polynomial equation algebraic geometry study algebraic equation particularly closure reasoning independence examine polynomial intersection dimension symmetric union precisely irreducible component variety five plus extra satisfy inequality multiply side respectively find equation intersection contain space algebraic axiom statement five statement question almost simultaneously vanish corollary next discuss sufficient axiom necessity axiom check calculation involve definite symmetric axiom arbitrary principal satisfie axiom apply axiom j k axiom axiom vanish complete conceptual framework introduce cone inequality cone space cone dimensional definite principal matrix equality subset cone image face map cone highlight variable geometry entropy map map various face map algebraic characterization relation term logarithmic variety variety seek section ci gaussians concern return consider random state collection ci specify call variety ci variety know strict ci hold ci strict ci open ci variety strict ci intersection strict consist ci statement valid table ci offer arithmetic problem might ci rational rational always appear large nothing discuss numerous aside list might rank nine suffice theoretically ideal principal statistical article present open mathematical emphasis hide gaussian present program geometry algebraic statistic module multivariate biology hold present contribution algebraic geometry statistic play role broad research algebraic concern probabilistic book subsequently reader statistic reader various cite list concern contingency table dna table polynomial nine vanish variety suffice variety serve broad direction algebra model hide familiar represent constraint geometry mostly point negative c parametric representation vanish ideal polynomial q principle compute generators gr basis parametrization software singular gr appear slow size real may language pure mathematic name table rank model model four fourth p markov superposition pure state literature geometric object language encounter generators generator degree nine table four generator author reader ask replace generator nine slice fix precise row row equal check matrix denominator tree conference problem proposition language theory define book module description insight independence current model conditional independence statement shorthand see absence implicitly conditional independence characterize content graphical complicated group greatly enhance calculus widely graphical illustrate hidden discuss algebraic problem publish projective equal one statistical algebraic statistical projective nice instance field ml proportional systematic wish feature relate correspond relevant definition fix projective coordinate event negative real non th observe define
follow function may vector k arbitrary latter function infinitely infinitely differentiable eq entail property maximize need taylor second fact function strict concavity formulae maximizer integer density maximize easily explicit expression first derivative moreover fx x x continuous concave q mc assumption subset simplicity essential assumption affine algorithm compute aa vector perform arbitrary goal vector aa l replace concavity hence repeat precede finally subset new l aa latter violate b l strictly basic advance algorithm start indicate indicate latter numerically optimum square regression estimation turn reliable start aa table mark correspond end basic stage moreover large maximum finitely set terminate finitely virtue implement vector specific application avoid loop ib b b end l cm algorithm cm cm b independent make relaxed particular previous consideration drop cone generator every case l basic procedure perform component note convenient corresponding change instance constraint knot throughout uniquely formulae si x ix ki optimization correspond function check check directional show variable smooth illustrate density picture together candidate picture start linear additional new plot result replace final fourth distribution density concave happen study viewpoint whereas censor contain purely censor mi candidate interval datum rather likelihood trivial treat elsewhere simplicity simplify arbitrary upper iteratively follow replace target function depend borel assume true measure support concave active replace auxiliary utilize q derivative one utilize calculation reveal k ab ty dt dt numerical j j minimum component moreover rv secondly dx dx dx follow concavity equal function maximize correspond function equivalent kx dx du dx du du dr yield characterization maximizer yield assertion correspond dx l v obviously equivalent partially foundation grateful draw attention active discussion matlab www em lemma fast indicate censor concave model unimodal family thorough treat aspect maximum related concave identically section latter tool cf key terminate
two probability model environmental modelling response vector logistic kind work theoretical practical case predictor cover neighbourhood date environment variable call notice estimate perform sample write penalize q penalization nk k n estimate maximization example predictor really penalty difference class side reasonable expect affect estimation numerical finally newton problem especially overcome instance resp perceptron property neural network take since optimal minimized usual training choose categorical gradient problem overcome mm compare methodology comparison statistical stage train neighbourhood network concern consider choose neighbourhood pixel cover pixel call pixel consider map perceptron set estimation randomly pixel map fact less pixel lead construct purpose compare significant bias parameter neighbourhood penalization neighbourhood penalization give parameter q predict date probable program core team request perceptron neural calibrate type neighbourhood variable estimation stage neuron neighbourhood train procedure stop calculate start repeat various neighbourhood validation neighbourhood moreover train various set well choose among make use toolbox request mm step lead neighbourhood ml perceptron neighbourhood perceptron build predict datum performance summarize frequency cover focus frequent cover see map c c frequency area forest forest type area real cover perceptron error map coherent smooth reality approach data approach help cover environmental statistical fast take need modelling perceptron approach lead significant difference train attractive think point great could usefulness predict cover parametric advantage automatic fact spatio aspect work step predict temporal spatial environmental spatially partially performance lead predict map perceptron modelling approach much low error look rate hard grow forest tend add kind know forest conclusion great replace big perform totally work form understand grateful support de grateful detailed constructive suggestion manuscript h network toolbox guide network pattern r evolution sense journal capability feedforward logistic new c regression journal american association stochastic journal american development core team environment foundation computing possibility compare france international journal l mod l du le une spatio I mm fr universit le france universit france france mail mm modelling perceptron abstract approach future landscape future old propose classical modelling perceptron map country rather precise square side whose pre determined list forest forest cover map old environmental variable point develop area idea ability date categorical variable cover date neighbourhood pixel environmental variable aspect proximity qualitative quantitative approach method type perceptron approach approach es south west france survey various map section finally map step et detail set spatio nature two evolution generalize approach likelihood recently idea linear parametric space suffer existence
cluster biological datum review experimental alternative find maximal connection alternative reduction monotonicity order assume matrix partitioning optimal way column cluster denote submatrix row submatrix context row partition transpose instead fix norm formula dependent norm row cluster analogously one eq minimize clustering notice way clustering one way row row partition exist approximation first increase cluster equal row bound sum equation prove lemma value distance deal real ht clarity row continuous suffice concentrate many median take towards upper equation row respectively change change figure minimize mention iii denote give equation iii ii exactly half theorem rely generalize matrix ratio column third row one way row optimal row want ratio hold matrix achieve good multiply achieve show standard column read manuscript give comment theorem lemma originally publish cs ds institute information university box column induced column way cluster value norm value matrix vector formulation final cluster another median distance function cluster several intensive research focus homogeneous matrix goal problem give identify reduction np fixing cluster efficiently approximate requirement simultaneous column independently word induce find standard guarantee
pyramid find agent pyramid boltzmann normalization th coordinate pyramid whose volume pyramid normalize whose boltzmann boltzmann also agent real depend physical situation interpretation temperature entropy boltzmann gibbs integral expression calculation temperature image canonical ensemble confirm statistical ensemble canonical usual literature energy heat reservoir time system repeat huge finish cover hyperplane limit system ref temperature obtain eq ensemble v statistical maintain limit ensemble argument gibbs formalism factor canonical clearly statistical present modeling isolate model contact heat reservoir energy maintain give interpret phase ensemble imply kind heat reservoir order visit reason equivalent discuss suppose evolve consequence mean visit energy limit picture say think heat reservoir remove number freedom suppose observe ensemble entropy implicit reservoir identical infinitely almost locate thin coincide ensemble canonical behaved derive volume insight origin volume pyramid finish volume family canonical ensemble derivation ideal particle momentum energy root reservoir sphere volume radius finding coordinate proportional volume form equal particle remain particle sphere easily normalize satisfy whose final calculation q call per factor energy index velocity particle
calculation avoid filter system hand auxiliary filter discrete explore alternative algorithm random weight particle investigation effectiveness auxiliary particle limit type expectation weight variance differ cause equation comment ready particle intermediate high iii particle see order error one intermediate time fix decrease depend decrease slow already need w brownian bridge methodology deriving estimator regard beyond particle assume homogeneity brownian bridge methodology extend limit unbiased estimator empty main pe negative variance finite introduce unbiased estimator pe consider depend discrete introduction let satisfy conditional dominate positivity specify generalise eq follow give conditional generalise give moment right guide poisson way conservative take call estimator moment mild guarantee variance alternative distribution negative estimator whenever pe efficiency gain achieve ad hoc jensen exchange subsequently w simulation work order magnitude pe usually easily presentation pe w arbitrary bridge exact simulate implement wider construct periodic minima maxima pe argue choice experiment quite choice dispersion table simulate value ccccc pe pe also report increment estimate see significantly pe particular pe give var couple pe efficiency construction term investigate pe vary time increment suggest variation error pe pe differentiable unbounded case significantly pe mention monte exist yield bias base say construct transition density diffusion process filtering filter two section first top black unit time circle diffusion particle circle filter mean circle credible clarity show time simulate use bootstrap simulate sampling filter pe simulate simulate propose efficiently exploit model drift implementation hard favorable acceptance speed simplification every choose proposal distribution obtain uninformative behave still diffusion ess high optimal value roughly bridge reasonably robust choice uninformative observation ess use approximate diffusion introduce uninformative time interval cox example absolute black arrival green filter red circle credible clarity bottom ess increase apply example cox impossible propose particle simulate calculate choose time avoid brownian bridge simulate inter weight likelihood observation ess pf suggest choose particle ess particle filter decay bottom ess exact simulation diffusion functional particle particle implement allocate methodology auxiliary particle filter apply appropriately expand involve focus filter current extension merely require ability simulate history approximation store diffusion path inference particle conditionally current firstly introduce particle bridge start simulate diffusion bridge brownian regime density directly inference finite skeleton brownian create sample subset x define determine determine bridge construction simplified diffusion bridge avoid since w ts w w w w l g w dominate valid constraint appendix brownian bridge w w w w growth furthermore w hand side expression formula brownian motion like conclude follow procedure particle accept proposal accept expectation law brownian return weight proceed step propose density proportional sde simplicity sde expansion sde calculate qx j appendix central consider sampling pf split simulate particle propagation particle iid limit residual observation easily let pf particle similarly variance function filter shorthand variance recursively measurable x I h convergence infinity comment refer due weight stage respective represent randomness weight particle filter bound weight proof weight adapt identical v take auxiliary combining give cm acknowledge comment filter class observation include diffusion arrival whose intensity cox unlike currently require transition build recent diffusion density generalise estimator particle filtering exact central theorem cox considerable interest process phenomena directly hierarchical focus estimate path diffusion filter class partially discrete filter monte partially one observation increment piece appropriate gaussian converge true filtering increase exact unbiased diffusion transition density way filter filter contribution simply use propagate particle entirely simulation pair algorithm filtering sampling particle go consistent filtering associate assign positive unbiased replacement unbiased estimator general one section convenient one contribution interest poisson generalise guarantee unlike efficiency term variance generalise investigate theoretically filter implement flexible adapt context show considerably central limit theorem estimation extension particle filtering limitation methodology equation specify diffusion diffusion matrix drift although volatility noisy observation component arrival know introduce underlying diffusion scheme test section require construct generalise simulation assess investigation implementation issue extension model diffusion drift know summarize paragraph continuously differentiable argument ii exist exist among iii ii correspond reversible sde coefficient extension replace appropriately transition expression available expression distribution evaluate expectation brownian bridge formula evaluate regime signal allow evolve time application fit partial observe invertible component cox consist poisson finance analyse single significant regime notation define density integrate conditionally know brownian bridge expectation law brownian bridge bayesian lie calculation filtering density density intractable scheme recursively flexible solve sde contrast coefficient dependent provide solve type drift formula condition already become general transform drift transform process satisfy condition transformation impossible transformation imply drift nevertheless physical successfully transform simulated consist top brownian illustrative code request consist arrival whose intensity q gaussian model although transition density density intractable regime handle include section density write filter convention simplify subscript particle calculation yield recursion particle whose substitute continuous approximation aim far discrete via give
accord fact element poses challenge dirichlet truncation adopt e show stick break construct effectively distribution iteratively update mechanism conditional configuration point allocate call allocate v give design sampler iteratively hasting invariant update chain full distribution let conditionally conditionally independent consist conditionally proposition stick break see conditional simulation dealing model family hasting carry sample stick break z fy jj model conditionally easily break independent however conditional mass stem need stage resort sum wish simulation nontrivial chain simulation simple avoid computation replace direct configuration eq maximal element given conditionally respect hasting produce move mass ik parameter propose proposal point allocate probability conditional allocation new way accord mass propose mass careful accept proceed composition update simulation satisfy start check value already previous simulate carlo step simulate conditional step calculate simulate leave step update distribution adapt early start identify recommend systematically proposal algorithm construct variable precise complementary need prior completion explore several modification chain monte carlo relate propose q desire proportion propose recommend guarantee propose prior independence perspective independence geometrically tail ensure tail tail simulation study discover interesting possibility allocation algorithm leave theoretically computational storing verify mass integer slightly gain modification switching move allocation infinite weakly identifiable exhibit highlight phenomenon consider simplified scenario illustration posterior process hierarchical size observation mass observe allocation common still regard dirichlet generate denote sampler exploring value give model let sample markov assume stationarity e representation posterior equality dirichlet process functional illustrative example correspond achieve obtain markov carlo therefore satisfied simulate compute figure provides need assume appropriate choose aim hyperparameter monte carlo distribution conditionally upon however amount information therefore successively full infer review upon update v technique hyperparameter recommend gap comparison markov chain algorithm implementation simulation later z fy proportional model allow excellent implementation gap algorithm gap receive chain suggest marginal one support independence structure sampler update decide specification draw unimodal mixture consist dataset suggest datum commonly mixture move every rest correspond functional update functional case well calculate follow output output marginal choose efficiency sampler report lag autocorrelation square integrable ergodic integrate autocorrelation specific require roughly achieve carlo follow estimate estimate lag size formula prior specification assess compete algorithm update allocation three difference among algorithm moderate implementation however optimize computational gap monte intensive compute careful inspection outperform explore mode measure ambiguity marginal approach explore multimodal indicate marginal achieve integrate time approach switch move include substantially approach infer latent flexibility extend stick break dirichlet latter obvious extension adapt crucial independence context work already sampler allocation unbounde permit simple construction behind elsewhere methodology involve stochastic application completion process grateful several constructive comment like anonymous suggestion solid bottom ccccc gap gap state allocate ccccc gap gap allocate cccc gap dataset across omit proposition cv al uk summary perform roughly marginal former integrate marginal sampler sampler imputation dimensional approximation approximation markov posterior sampling also functional dirichlet careful study conjugate specification exact switching stick prior hierarchical model standard include estimation analysis parametric cluster modelling hierarchical parametric distribution therefore dirac clustering last hierarchy process dirichlet property dirichlet dirichlet last rise rich stick measure general live flexibility component allocation allocate weight impose weak identifiability sense cluster allocate concentrate variable dirichlet gibbs thesis alternative carlo include sampler sampler paper reversible jump update broadly speak two augmentation scheme marginal exploit use integrating induce among make label analytic considerably complicated implementation conditional method augmentation scheme imputation subsequent gibb create imputation introduce develop considerable advantage flexible current stick covariate advantage principle pose interesting challenge imputation vector approximate determine desirable avoid implementation avoid propose monte carlo readily extend stick break introduce involve carlo weak impose component multimodal sampler different eliminate secondary mode become energy mode big area design tailor switch markov chain chain state art conjugate particular gap model method slightly outperform idea introduce hierarchical simulation diffusion simulate dirichlet conditionally marginal inference joint scheme allocation variable independent eq q denote associate proportional associated component note totally although label otherwise simulate independently draw least index arbitrary
observation account one observation component label empty component particular assign component portion page normal density variance natural place ga independent prior sample shape half degree predictive relatively thick order discuss computable normalizing expression exactly estimate substitute context component allocation play discussion everything sum numerator empty denominator equal use equation eq rearrange model side denominator probability mcmc sample computable readily rescale somewhat employ quantity posterior probability component empty show replace probability mcmc rescale normalize formulae invariant hyperparameter non one replace set estimate illustrate galaxy velocity km galaxy region appearance study likelihood model normal weight hyperparameter choice discuss allocation parameter instance component burn run allocation marginal likelihood normalize display contain uniform prior sampler run million burn keep draw allocation dot likelihood use text value agreement inspection suggest six display range probability believe next yield group invariant permutation label marginal hold eq understanding figure nest structure denote allocation vector digits digits allocation digit box box denote serve formula extent cluster nine separate extent obtain table n discrete put posterior display uniform discussion overall large component formula return combinatorial would f account ten way nine one empty likelihood many way empty feature suggest attention number non distribution try combinatorial likelihood write explicitly normalize constant binomial keep contribution relative require verify distribution satisfie prior discrete weak ht c illustration nine well rather uniform artificial concern normal natural conjugate prior adapt component continue galaxy mark specification experience demonstrate pattern page remark galaxy section computation figure dependent give value move apart yield corner plot considerable prior likely component accounting observation hyperparameter considerable extent affect long tail change adopt bayes mix prior proportion within component expectation metropolis hasting display logarithmic scale pattern consist draw discard rough occur posterior component median small past section substitute sensitivity formula formulae produce ratio rearrange consequence rearrange rearrange rearrange produce q side respect nk distribution satisfying equation truncate proof proceed derive j follow equation restrict mathematical york inferential mixture journal american association likelihood journal american journal american association journal fr carlo estimation journal american normalizing constant bridge decomposition journal association department www ac uk distribution bayesian allocation department university jump carlo monte practice mixture statistical density confidence galaxy american bayesian model reversible deal
aic wrong include aic separate I include label circle label triangle gap ready explain work superior evolve universe thing member universe aic universe subset start converge minimum work algorithm universe parallel universe get solution spurious variable parallel important variable majority vote correct already selection wrong criterion easy importantly problem regardless whether hazard selection meaningful logistic except label svm give constrain straight tucker also hand solution satisfie onto adaboost stagewise well derive boost stagewise course make statistical office valuable comment participant member thank provide I national xu provide I office college respectively article little partially engineer research complex mathematic like acknowledge library library http r project thank anonymous I support theorem lemma definition adaboost wave kernel method article idea behind one kernel ensemble rather influence shape research unbalanced rare parallel ensemble variable adaboost advance machine adaboost two fundamental behind classical highly nonlinear function build fine carefully fine present main idea four machine section rare idea mostly although adaboost detail affect main idea omit old wave area seek hyperplane hyperplane hyperplane clearly g hyperplane hyperplane satisfy canonical separate hyperplane class perfectly separable hyperplane figure compete hyperplane situation well separate hyperplane two away notion formalize margin hyperplane margin hyperplane sign hyperplane illustration margin elaborate justify together margin equal eq large margin solve problem extra relax separability general class act tune brief try well therefore classifier usual make svm logistic regression exercise familiar indeed familiar ridge especially optimal success derivation like strictly solution hyperplane inner product predictor lot simply inner hyper empirically become eq original predictor pick pick make general know create perhaps somewhat less familiar actually really either familiar student idea fairly old idea stack observation see depend matrix easily replace product kernel principal successful classic algorithm focus center order eigenvector principal represent let plug multiply j show matrix detail onto immediately pairwise product old find project onto principal toy spherical spherical component principal meaningful far away successfully classical reader notice far really discuss claim modularity problem function therefore well carry practice must hyper parameter sensitive considerable analytic experience knowledge regard great difficult expect great set use appropriately poor piece operate lie try spam available total predictor refer aforementioned site remain training apply test test plot figure penalty figure sensitive give performance svm stable serious carefully misclassification two tuning mainly predictor construct collection make ensemble alone make contain adaboost plain english start assign weight sequentially fit use calculate wrong incorrectly weight next cast weighted wrong people adaboost certainly perhaps extent wrong classifier clear incorrectly classify observation ratio vote individual member logarithm adaboost update tree draw randomly subset call good rather discussion lead I ask big company business business accounting resource I line operation various produce great difficult division high specialized market end build next camera parallelism statistical division division specialize special characteristic division particular view mind end describe learn product new division mass division class rare observation belong majority ahead course capable probability classification observation effective decision diagonal determinant radial speak specify parameter front kernel square set treat determine np combinatorial problem view simply solve program division construct parameter global distance near e neighbor g compare also construct like specify rare kernel spherical center near priori simple fast support special nature rare class use significant highly svm theoretical justified suppose density factor e monotonic belong class nontrivial calculation special chinese label b figure go b close imagine function nearby long nearby equation basic spherical often limited amount like often general specialized algorithm rare detection simplicity carefully empirical procedure validation repeatedly minute difference still purpose fold run apart important principle construction exploit special nature rare exploit fact quick predict criterion fair include criterion
stimulus create brain analyze fmri quantify experience competition offer collect natural environment fmri three subject segment reality quantitative series total range cognitive experience subject search task environment take people pick consist competitive natural represent external presence virtual entire fmri activation diagram brain occur level various phenomenon fmri fmri scale neuron activity couple result configuration activate entire fmri dataset voxel free study cognitive dimensional volume fmri compose voxel stack spatio fmri matrix scan feature predict describe subject dynamical change measure formally knowledge fmri times replace fmri small feature brain easier construct voxel low xt vary set brain configuration map parametrization brain provide set parameter different brain note play neighboring equip parametrization learn coordinate implicit brain state certain location map globally compute parametrization x xt build stage proxy entire xt eigenfunction represent vertex distance measure change head remove principal pca remove small fmri volume voxel white gray inside network connect brain state create distinguish fmri data cognitive weakly fmri euclidean locally circuit standard alternative geodesic short quantifie path principal principal volume expand component divide use predict predict sample quantify real average b method low reach coordinate coefficient nonlinear perform global global predictive across cognitive within whether experience divide region neuron area characterize distinct functional role role provide though partition open competition area correspond voxel thousand voxel virtual reality episode voxel series simple stimulus decode model mode rank index area region minimize mode prediction stimulus average virtual reality episode make could prediction sample power remarkably mode predict face stable subject interpretability subject top area area human language face stimulus visual area velocity visual mode self report interestingly cognitive htb usefulness cognitive likely simple global likely include collection area rank stimulus area area also interaction predictor generally significantly prediction ease laplacian principal nonlinear global knowledge region fmri build low surprisingly competition cognitive interpretation area prediction virtual presence part area cognitive brain appear global inspire manifold technique complex dynamic fmri towards broad brain construction decode experience relative simplicity competition acknowledgment support national mh foundation partially mind grateful member discussion brain program mathematics center department physics author work image dynamical brain sequential connection fmri cognitive voxel activity
compression stationary source regularity wish rate decoder reliably reconstruct reliably asymptotically manner term measurable dominate write radius ensure class sufficiently coefficient call parametrization variational condition meet asymptotic fisher density sequence space collection measurable integer deviation probability vc universal expectation consist see therein require satisfied every memory length description encoder code perform well design knowledge decoder active asymptotically recall discussion immediately weakly notion code every c suffice code n throughout operate database associate string suppose lagrangian optimum encoder block encoder compute convention infimum specify later encoder concatenation string I whether finite ii string equal string receive determine produce n shall happen eventually variational radius comprise parameter si encoder encoder composition decoder decoder decoder define nc n x ss assess function normalize first instantaneous lagrangian stage prove show database ensure ok x op surely divide length block act copy mix q n approximate probability expectation suitably I process proceed everywhere density accord use minimum block set empirical induced key regardless I np show nn kp choose sequence second lagrangian measure bound one argument lemma code achieve satisfie straightforward coefficient yield gx n thus lagrangian approximated follow triangle eventually surely first encoder handle estimate stage l ok realization straightforward example family scheme measurable collection process source trivially equal source hard condition check condition n autoregressive ar real variance root plane show exist meet verify dimensional nr nx meet process reference ergodic unique corresponding channel alphabet alphabet density lebesgue assume transition density know process exponentially establish technical ia v author like thank discussion fellowship joint treat code recently generalize fix source ergodic distortion parametrize source universal scheme source also example source code scheme intuition suggest operate code intuition rigorous comprise suitably optimize redundancy extend code parametrize continuous regularity exist code distortion redundancy fidelity block operate code match parameter moreover certain source alphabet practically class autoregressive markov space member index nonempty measure dimensional denote wish process code decoder
j value penalize minimize penalize iteratively weight g write distribution weight use estimate evaluate z initialize adjustment divergence rare mle parameter minimize make directly variance thereby orient smoothing parameter less cycle failure frequent well avoid fundamental issue smoothness evaluate convergence effective degree freedom seek otherwise w ad hoc sometimes usual choose automatically fold validation see exposition derive canonical relaxed provide justification next idea kullback depend version leave criterion replace contribution predictive predictor minimization attractive choosing seek minimize kl attempt minimize predictor omit work final taylor expansion note employ yield score p pearson although motivation approach cite accommodate link final little summation aic tr view tr tr course discuss respect basically respect parameter smooth imply estimate find possibly obtain respect smoothing stable give criterion minimize newton newton method al depend conceptually scheme update derivative expression derivative poor prototype sort build instead scheme fast solve fix expression smoothing convergence evaluate number throughout decomposition calculation avoid calculation derivative thereby system accumulation purely practically derivative add simplify subsection explain initialization derivative r initially iterate derivative step derivative derivative second derivative calculation routine expression provide appendix derivative smoothness update need initially matrix derivative note g direct naive expression form unstable address conditioning problem develop way derivative maximally keep computational cost lead find actually available linearity rank factor triangular triangular fairly straightforward detect left row drop course give remain row find root decomposition decomposition qr column triangular must subsequently routine way redundant drop row remainder explicitly either decomposition take irrespective qr prefer avoid ill explicit formation method advantage singular van possible part matrix exception mention column derivative inverse establish inspection precede multiplication lead computing pearson section degree evaluate trace calculation I ii hc calculate iv column section appendix lead evaluation finite derivative require derivative component quantity whole omit criterion newton parts term readily identifiable eigen hessian newton replacement guarantee descent eigen decomposition burden smoothing converge optimize drop underlie newton enter quasi newton build associated speed expect first obtain produce degradation pure newton method always reliable effect illustrate problematic display adapt replicate covariate simulate covariate produce form x take probability iii iv replicate term model logit represent thin regression spline iv routine fast purpose c gradient optimizer orient e mixed estimating via mixed aic binary try bad mse new concerned error minus use truth predictive incorrect covariate replicate fact predictive use eps cm new method derivative derivative pi orient estimate mixed method reliable text fail replicate gamma fail replicate orient reflect design successfully failure seem failure course exclude cpu second replicate ghz processor operate low apart orient error skew summarize model indicate pair sign indicate oriented iteration rather failure replicate produce mse new appear greatly reliability group replicate linear predictor I otherwise smooth penalty coefficient smoothing control eps cpu model quasi little predict cpu second need replicate fail gamma fail mse performance test pair fail mse quasi penalize quasi difference seem speed reliability substantial use severe predictor obvious advantage associate fitting right fail provide satisfactory fit new method substantially effective reduced estimate orient convergent working iterate regularization narrow window clearly eps obtain hand way fail replicate simulation sort method alternative replicate replicate replicate alternative final example west france abundance infer produce datum collect water surface figure record depth proxy water water depth net al presence introduction successfully aic aic definite hessian optimum make modelling absence survey area give distribution eps central show root temperature orient example turn density reasonable quasi thin spline see basis performance orient regularization cycle ever routine linear fail fit overfitte significance measure freedom reason effective degree e show orient likely relate location addition count assumption underlie somewhat linearize problem orient unlikely orient substantial smoothness reliable smoothing fitting suffer iteration optimize aic relate fit rather work approximation addition competitive oriented median cost another obvious procedure p early fitting binary data convergence easy orient smoothing change step possible decrease disadvantage associate disadvantage implement hard imagine circumstance orient relative score method much difficult modelling situation enhance reliability approximation lead iterative adaptively cope ill unless finite apply get level truncation cope ill derivative code criterion auto operation count finite make storage impractical go towards state introduction make routine aim orient iteration guarantee force direct method well optima issue replace optimum introduce succeed derivative smoothness criterion relative orient far substantial regard highlight stability model address rank qr basic fitting stable determination gold short achieve state seem implement grateful deal start core thank helpful suggestion update refer converge I I derivative evaluate eq pearson define il derivative noting expression derivative derivative evaluation write decomposition van step term rhs etc store diag diag tr tr equality advance term tr km tr km tr km diag diag k tr tr tr tr k p diag easily require diag diag k tr tr g tr smoothing point still considerable first total z user principle international improve production journal science techniques linear journal american association structure base spline statistic www w journal analysis estimate smoothing nonlinear equation nj guide journal graphical structure bootstrap air journal health optimization h van rd university green w nonparametric generalized parameter scientific statistical journal c york smoothing spline generalize journal science journal smoothing spline scalable via journal statistical journal lin use smoothing spline journal comment additive statistic generalize nd ed journal perspective ill pose statistical generalize journal american new york language statistical foundation explore additive air environmental health automatic marginal asymptotic model validation spline york modelling penalty journal thin journal statistical b efficient additive american n g cross smoothing spline cm mm iterative working fail particularly frequent perform attempt computationally inefficient convergence computationally direct computation scheme work offer reliable fitting generalize additive keyword penalize additive aic smoothness additive spline also apply work weight initially term approach orient extend penalize regression spline develop smoothness approach treat generalize model become variance estimate iterative work mixed iteratively
element derivative let matrix p ph ph p p p h n ps ne h km ps I pn pc n ns concern generalization fan multivariate far even pd proceed prove main theorem ix ix n define ix ni I write surely partition j n partition repeat recursively partition sequence l l denote round choose point ni l ni ni n ni together quantify let b nh r n r lm cm l therefore lm n nn check enough n lm almost surely borel nr ni r b nh dm n nh nb nh n n nj select quantification involve lemma let ik ik ix dt k np p nj n borel surely point ki ik ik ik second ik imply quantify nh nh nn c w om ni ni I lemma ni dm dm nh c nn exist om om ik np I k n n loss ct ci ik independent l obviously nh first ni dm n nu ni ni nh hand ni ni ni choice ni ik h n nb n nb nb mx ie nh ni ni ni h I center write ni ni nz ni nh conclude complete proof impose additive note ignore strongly mix nm pn pn expectation term h w standard nh ns x np therefore lead variance variance nh np kx yield f next generalization nb z cn nb consecutive block write nz stationarity independent supremum definition markov independence bind notice j c first f nj nj univariate distinct x lx l integral therefore h h f l h nj cb summation ni h h u dt dt h ni conclusion n h ni dm ni cm rest complete uniformly cauchy nh proof meet nh g p fall therefore inner nh complete nh ps impose validate line nh n I desire w k r quantile ann process van fan n generalized fan integration multivariate lee regression huber j ann quantile ann partial curse comparison estimate van ann polynomial strong consistency least h optimal rate nonparametric regression ann principle ann wu sample cm mail mail lemma theorem em fit strongly mix representation inference plug estimator functional apply local fitting mix many context want estimator useful plausible assumption estimator parametric context estimator asymptotically property case need explicit expansion expansion widely agreement expansion one derive purpose mind implicit nonparametric chen van model additive model van suppose estimator nk xy suitable bx x mx r nx surely lead expansion central suppose parameter expect asymptotically base first mx mx mx dx widely mx mx dx chen van verification term term variable hx nx I mx obey central possible derive useful expansion uniformly remainder uniform mx mx hx mx nk hx follow therefore normality term complicate bound sum independent random theorem argument expansion probabilistic integral easy argument however addition financial concern tail outlier robust perhaps combine local examine location functional objective derivative series multivariate strongly uniform expansion almost surely almost lead functional restriction relation smoothness nonparametric strongly dramatically curse reduce curse assume normalize nonparametric function additive lee additive volatility superior property arise suggest moderately often heavy tailed expansion notable recent wu extend class provide review close local regression pointwise univariate limit applicability specifically application process dependent coefficient algebra variable strongly strong goal partial derivative base quantile yet another huber huber pm give r r density bandwidth observation quantity main define term expansion distinct tuple position pn pn rewrite minimizer p matrix diagonal element define piecewise derivative p aforementioned order quantity lead lead pf appendix ps ps sup accord point remainder quantile result function nevertheless see continuous uniform measure mix weak accordance result prove estimator practical interest determine hold impose algebraic calculation would positive denominator tradeoff mixing decay slowly trivial generalize functional pg dc weakly dependent suppose regression term component know direction partition x thus note
fully bayesian approach instead tractable bayesian serious challenge bayesian call square explore within reflect portfolio get distribution portfolio weight extension normality marginal make promising apply pair copula extend beyond become enforce restrictive way dimensional may support marginal various form asset argue thing quantitative allocation covariance vary monotone pattern convenient asset normally total asset count obtain repeat ordinary ol regression one asset historical ol unstable matrix explore partial apply large offer accuracy interpretation extend method show factor incorporate assumption historical contain code implement estimator herein word financial miss principal least factor eliminate estimate observation style financial return length historical stock price deal utilize portion across approach imputation little third aside datum typically spike historical group pattern differ start different reason multivariate focus sensible portfolio balance mind restrict share current handle immediately useful portfolio balancing handle monotone approximately augmentation beyond arbitrary specialized slow lead likelihood asset normally asset ml estimator ordinary ol finance attribute text see historical ol regression unstable sometimes big attention bioinformatic applicability financial historical explore setting short involve replace parsimonious regression partial shrinkage enable covariance essentially even situation parsimonious parsimonious motivate exploit book market traditionally restrictive decide independence accomplish even condition return factor remainder follow derive factorize regression analytically ml method deal big context transform highlight benefit accuracy interpretability parsimonious regression generate essentially return show datum large highly estimation portfolio balance efficiency discussion inherent maximum completely missing represent historical return asset monotone arrange ht property collect column monotone miss completely neither depend note asset back index asset counter illustrated generally factorize exploit auxiliary parameterization mapping conditioning assume q follow may financial simplifying assumption tractable compare j design intercept ht intercept involve monotone diagram design intercept one straightforward calculation q comprising describe correct denominator typical obtain unbiased covariance several length typical replace importantly asset history shorter greater full sometimes whenever nearly large overcome shrinkage principal subproblem intercept response ols mle two reason parsimonious provide ols lead fit qualitative interpretability assume cause latter design singular even say far return asset history method coefficient probably aim way minimize searching become greedy forward selection start nan intercept sequentially add backward discard predictor produce shrinkage forward subsection shrinkage ridge lasso family direction principal component partial square parsimonious choose package available comprehensive http provide implementation nice monotone typical input outline scale ridge penalty ridge parameter shrinkage intercept interpretation posteriori impose convex minimize ol implementation ridge mass library r call lm ridge difference ridge coefficient towards effect importantly many something choose way implement subset selection decrease though monotonic lar package lasso two stagewise special least lar possible effort magnitude ol select final e rule cv within conclusion equally benchmark odd lar lar lasso final decomposition principal combination partial decomposition combination value decomposition basis diagonal value order call loading pc extraction sum univariate regression importantly write column obtain ol choose component proportion variation less hoc reliable intensive even involve predictive incorporate loading proceed initialize loading optimally capture obtain similarly correlate advantage less record operational behavior ridge ridge coefficient principal diagonal whereas pls provide unified estimate cv algorithm find maximize outline iterate onto column predictor small full increasingly approach double precision computer implement package therein support choose number parsimonious regression fold loo parsimonious describe section validate involve towards balance mean set analysis thousand package generate wishart impose monotone base presence tail comparison relative strength variation method calibration tool illustrate leverage simple completely put another way I provide incomplete maximization consequently also sort two software package available package matlab toolbox nearly run time fast solely stop improvement sensitive fail issue numerical due representation e handle expect kl comparison pdfs parameterization data integral expression sample large rank estimator distance comparison parsimonious regression repeat repeat trial parsimonious regression fold table winner nearly complete financial return tail monotone pattern freedom roughly good estimator improve ridge line exploit distributional completely evidence sensible ingredient despite underlie violate scenario estimator financial asset determine parsimonious varied stochastically uniform ridge proportion well trial loo objective optimal repeat trial loo cv observe ridge thing prefer control experiment fix one describe give fail method converge well fail increase example fails examine characteristic historical variance portfolio balance stock require stock construct keep return short portfolio must typical weight bold forecast typically poor fully quality unnecessary outline early variation historical return portfolio stock parsimonious regression incorporate value portfolio parsimonious method unable handle example converge thousand slow second ghz assess return rate deviation track portfolio return return market p stock closely setup stock hold subsample size thus also serve enable bootstrap previous stock replacement sense outline least differ estimator historical choose highlight benefit incorporate portfolio via excess last return return sd te com factor return track stock summarize return average year break weighted base historical return whereas com complete history completely return remove portfolio ratio six return year tracking model upon na I inspection part improve ratio deviation expense indicate low market ratio estimator complete inclusion value far add yield unchanged one high low low placing ridge similar lasso obtain lar market factor via seem lar base good result top appropriate lar largely parsimonious tracking bottom obtain year horizontal bar correspond vertical ratio random year approximation characteristic various superiority small resample lar variability ratio amongst estimator good recover mean variance ratio statistic transformation portfolio weight via statistic say high section complete shall demonstrate take thousand financial www com stock united universe approximately return completeness stationarity methodology exclude artificial serial stock price
apart preliminary present suggest kernel may walk raise issue complexity graph concern parameterization issue concern length walk consider walk precise optimally cross validation focus walk precise safe default limit walks walk walk penalize account limit walk finite flexibility control offer advantage algorithm kernel lead filter walk introduce obtain nd improve virtual reduce computational filtering phenomenon helpful structure biological include drug kernel rely spatial positive negative responsible biological activity drug follow compose three form apply different base slight abuse atom arrange configuration precisely atom number atom ix take notation hand follow way precise atom equivalently atomic distance consider bin discretization define space correspond triplet atom alphabet triplet coordinate extract discretized version mean triplet atom triplet bin discretization specify resolution constitute bin prevent bin lead match practice consider bin within range reach atomic dimension suggest explicitly limited bin thereby molecular rely hash onto limited induce representation highlight line discuss enable pair index million course represent similarity know discussion nevertheless kernel without compute store drawback choice precision match prevent side bin match close actually issue check pair let lead introduce kernel intuitively quantify triplet compatible implement atomic compare elementary compare distance resp resp distance define couple compare atom kernel intuitively elementary notion simply label inter atomic rbf parameterization continuous discretized counter share feature triplet atom two lie strength inter atomic correspond allow account spatial configuration differ kernel inner parameterization know valid function discuss consideration without kernel compute computation equivalently see label undirected vertex atomic distance equation interpret computing walk graph product operation cubic product compare prohibitive discretized implementation derive string algorithm refer reader implementation discretize bin inter distance note precision identical priori optimize validation procedure study optimize inter atomic range match fine grained impact issue parameterization definition inter distance study comparison select validation discrete coincide formulation lead study mechanism virtual screening involve property drug target interaction molecular responsible bind drug usually atom particular feature interest similarly discuss base construction external use atom compose partial positively neutral atom label alternative consider base sp atom atom feature study influence subsequent relationship enable drastically cost world raise issue alternate spatial approach sample admissible operation instance draw considerable since kernel particularly extension kernel kernel structure possible adopt evaluation construction virtual notably target art intrinsic modularity extent unify virtual screening molecular descriptor focus variety virtual screening concern practical dataset kernel demand interest community virtual screening database certainly importantly representation molecular application bind prediction nevertheless efficient model show outperform study extension benefit scheme thorough chemical could improve expressive kernel molecular structure exist merge consider atom improve reduce issue opinion worth relate precisely space kernel tend outperform counterpart believe handle high great virtual extension would global integrate structure would propose combination semi structure virtual benefit introduction molecular virtual experience development indeed year count exhaustive short vertex assign sum kernel assignment atom big unfortunately definite computational geometry walk extend graph molecular surface together give presentation list constitute molecular virtual screening source implementation toolbox publicly hope motivate molecular acknowledgment pm student paris france let writing function year directly structure extraction molecular descriptor prediction relationship introduction computational play drug accounting biological help time cost new infer biological chemical target throughput subsequent availability characterize machine relationship characterize decade machine provide classical artificial strength issue one want concern way represent represent vector use descriptor molecular descriptor often significant need choose molecular descriptor property molecular descriptor keep task machine svm regression difficulty representation dimension measure computational trick number demand small inner combine give vector dimension leverage new imagine molecular descriptor svm popularity various include bioinformatic activity drug brain barrier name molecular descriptor line seek beyond molecular thank kernel trick simultaneously independently infinite show svm later refined attempt flexibility molecular promising direction model unique possibility kernel offer art description development community possibility purpose quick introduction illustrate trick representation structure discuss issue conclude suggestion future originally develop early extension multiclass interested know know problem object belong set rule object class beyond situation object represent various recognition binary label object class use produce predict dimensional svm sign geometric interpretation hyperplane separate side hyperplane svm solves hinge good prediction loss prediction sum candidate make good small slope often especially large well rational behind find reach goodness fit smoothness quantify control trade balancing point hyperplane allow separate hyperplane misclassifie hyperplane overfitte interesting way theory problem dual instead structure notice dot point product training dual moreover svm importantly easier characterize space class kernel satisfy positive svm kernel offer formulation enable extension svm decision use nonlinear keeping e etc x result form eq nonlinear second possibility kernel label assign node typically relationship graph practice alternatively decide combine walk inner walk walk length small need counter increase product grow walk weight contribution walk walk factor adjacency rewrite explicitly cost inverting provide good approximation complete different weighting walk define markov occurrence walk walk weight along walk kernel restriction walk definition walk behaviour lead walk alternate two connect expressive walk prevent terminology define path albeit unfortunately alternative enumeration illustrate walk back vertex notion path strong concept stem transformation add additional vertex transform graph find make walk walk limited subsection subgraph kernel raise hard alternative label information walk approach take vertex graph environment atom atom procedure initially define implement computed read unity adjacency index topological include walk particular topological configuration practice advantage topological compare second make illustrated pair automatically reduce surprisingly note systematically walk branch reflect atom subsection walk walk indistinguishable kernel walk issue graph feature hard know must computational towards walk introduce
multiply absolutely part density truncate typical shape interpretation scad seven intersection f x n formulae mixture normal absolutely continuous complicated six piece truncation multimodal choose fan li choice scad estimator coincide atomic large region sample estimator multimodal also lead formulae restrict coincide conditional select conditional identical complicated class asymptotic distributional moving sample consider asymptotic quite picture description accumulation estimator remark consistent selection characterize behavior tune true weakly p include brevity cf total atomic converge absolutely lebesgue everywhere absolutely mass absolutely continuous mass absolute absolutely continuous letting thresholding coincide replace limit pointwise distribution statistically contrast well coincide except fact limit soft true note completely thresholding arise case relate fu fix parameter distribution pointwise agreement equal contrast well coincide scad converge weakly proof hard soft estimator discuss scad reflect case pointwise asymptotic agreement especially statistically contrast much scad well variation distance mass atomic converge third theorem see theorem select subsequence subsequence along subsequence limit finite particularly estimator distribution cf pointwise coincide maximum thresholding oracle fan li soft somewhat behavior able consistency actually raise framework finite move hard converge weakly mean mass convex absolutely whose kind combination absolutely converge total distance atomic converge condition atomic locate view immediately first show hold imply atomic part continuous n nn show display absolute proof part treat absolutely continuous boundary interval discuss recover asymptotic hard thresholding complicate predict show hard estimator uniformly stochastically bound case sense appear thing finite non normal finite theorem also property pointwise asymptotic highly picture estimator certain nx entail distribution thresholding set total soft act selector pointwise certainly oracle contradict incorrect claim yu selector estimator pointwise cf appendix scad scad n scad scad nx scad convention weakly total variation cdf contribution atomic contribution break contribution atomic part hand contribution respectively argument nx mass atomic assume atomic precede formula evident integral nx require subsequence limit space scad nx nx nx rf nx nx converge display almost mass compute furthermore check display lemma total cdf formula density dominate role handle reduce establish observe scad scad n n scad indicate argument scad oracle property clearly like thresholding discussion different see essential statistical observation consistent question estimator degenerate parameter asymptotic nevertheless stochastically theorem hence precise estimator theorem turn asymptotic locate soft even pointwise stochastically unbounded easily thresholding show stochastically bound limiting note stochastically bound distribution move asymptotic oracle property scad estimator picture finite put responsible problem eliminate distinguish statistically reason sensible quite perturbation reasoning actually consistently scad n scad desire oracle na nb contain element order converge unity scad n top highly parameter estimation use analogous statement hold set contain I zero tend unity thresholding procedure adopt value substantially difficult distinguish support adopt theorem actually limit finite sequence theorem subsequence subsequence etc converge interest restrict phenomenon theorem tie rely usual local asymptotic possible consistent selection pointwise asymptotic capture distribution well discussion estimator center scale estimator depend parameter manner interest estimation uniformly follow present post thresholding phenomenon tune conservative soft scad consistently tune uniformity phenomenon large conservative choice straight popular possibility pointwise limit pre functional depend pre test limit estimate relate scenario fu intrinsic feature let tuning arbitrary estimator probability contiguous verify write shorthand jump absolutely continuous formulae dominate supremum n supremum bind stress apply cdf randomize estimator speak cdf interest suffer govern I equal phenomenon tune conservative trivial surprising uniformly asymptotically cf uniformity somewhat en extend prove let variation e p easy computation get close already see f observe precede display bind f apart right insight phenomena inference shrinkage estimator recently attract attention cdf thresholding note consistent nevertheless show phenomenon cdf tuning result range n sense limit theorem appendix thresholding scad large absolutely multimodal large tuning essence case asymptotic hold size reflect move asymptotic sample picture see scad thresholde normal irrespective particular statistically interesting case sense also phenomenon occur view scad fan li oracle asymptotic seem suggest perform asymptotic whole sample phenomena move asymptotic addition phenomena actually hard irrespective tuning sensible choice act conservative selector long hold act selector scad hard estimator uniformly scale scale hard cdf pointwise consistent construct ease inconsistent cdf phenomenon distributional consideration estimator facilitate risk focus error loss n compare base scad compare situation hard soft cf formulae go case distinguish perform conservative estimator remain size increase perform scad thresholde oracle size fact estimator asymptotic risk favorable pointwise behavior property estimator bad risk uniformly tune conservative stress per distributional suggestion greatly estimator tune consistent interesting always hold pass stand g next asymptotic note remark theorem hard converge z p proposition h n view consider easy scad estimator assume converge scad scad identical proof hard rescale next atomic cdf give proof nx scad limit subsequence easily scad n x rx n na scad nr complete see ng x furthermore indicator converge weakly inspection nn nx ng scad subsequence next inspection immediately weakly prove completely analogous estimation finite distribution cdf infimum extend consistent n trivially prove n c n relation trivial success b fan penalize oracle frank view effect region fu asymptotic estimation design e bootstrap estimator g north post approximation selection inference shrinkage type p distribution post selection oracle property b unconditional texts york b averaging estimator regard note k asymptotic regression yu selection lasso theorem axiom case corollary theorem exercise notation theorem summary university statistic university scad derive selection perform fu fan li show highly tune sample uniform rate tune regard primary scad thresholding post oracle property penalize maximum last prominent example least lasso estimator study frank regression lar smoothly deviation scad fan thresholding likelihood estimator limit probably contribution fu study distribution bridge lasso select fu alternative setup tune act fan li scad perform particular picture estimator especially model close actual finite hard facilitate sample strength readily yet consider rich enough phenomenon perform conservative estimator normal result parameter asymptotic reliable estimator also estimator move framework capture turn case move asymptotic framework asymptotic necessary exhibit full distribution convergence estimator estimator character show
membership model appear include variational deterministic alternative mcmc idea behind free parameter fit close review paper marginal jensen introduce specify factorize multinomial variational kl mixture conjugate ascent multinomial variational dirichlet analytical appendix ascent l f q inner detail panel step respectively variational parameter update convergence relational introduce variational schedule update na I scheme dirichlet p qp pp algorithm I fail converge dependence satisfied I algorithm happen purely perspective na I variational process I variational maintain dependence mainly schedule various maintain optimize update thus provide maintain dependence keep value nest us deal variational cycle need scalar increase offset convergence rate well term rate I inference present parameter variational hyper computation however satisfactory cause concern sensitive choice hyper hyper surrogate fitting latter sufficient turn approximate newton mle parameter prior interaction matrix latter estimator membership provide quick computational burden exploratory strategy available model translate determination plausible hold approximation bic datum social interaction recover nest na I implementation peak likelihood real latent connectivity application publish latent interpretable context analyse develop correspond hope recover measure respectively use simulate setting membership node cluster run na I variational enhance inference cluster hold likelihood cluster identify cluster successfully recover model membership adjacency interaction nine panel grid panel whereas column value panel column reveal block interaction likely consequence phenomenon interaction evident na I variational em na I implementation axis likelihood deviation nest algorithm furthermore nest panel perform network hold log fold peak identify optimal em peak hold national explore social family friend neighborhood health risk outcome analyze among student school sample student pick collect student friend analyze asymmetric student raw right panel right bic selection group follow nest fairly connectivity mix mixed signal situation signal repeat cluster measure c c cluster cluster attempt student mix stochastic blockmodel simple conclude complementary source one hand connectivity unconstraine hyper membership collection fashion prediction basis pathway biological throughput interact protein wide hybrid protein throughput weakly associate detection encode process precision biological signal throughput protein role analyze global via composition suitable operation carry function stable protein situation interact intuition e interact protein category interaction energy sub protein institute sequencing database technique throughput association collection protein amongst annotation annotation organize annotation leaf annotation protein functional protein category display correspond functional annotation display protein obtain figure protein order presence annotation display axis usefulness latent analyze protein datum testing assess biological identify protein member interact one identifiable category cell protein synthesis activity infer protein absence annotation logistic enough predict annotation category bayes estimate hyper conclusion consistent comprise analysis broad protein priori identifiable resolve mapping latent frequency membership fraction mapping versus annotation mapping membership broad category along category membership predict mixed annotation category gene source functional annotation truth finer grain produce sequence large extent reduce protein analysis nest validation determine parsimonious model provide good protein interaction qualitatively quality parsimonious group interact biological aggregation high biological group interaction predict dark term rna go go dna go complex go modification go rna go go go go modification go pathway go go go go extensive measure source membership discover hierarchical mixed membership model issue choice school computer university gene expression manuscript j gene biology incomplete score graphical volume page status engine mit allocation r house r network discrete taylor appear g fuzzy network manuscript maximum incomplete via social thesis e w page west frank organization systematic analysis scientific topic network p national health technical report university north hill k systematic space social network z technical report ai mit n system concept infinite relational j yu j zhang b wu j thompson st landscape protein li category c structural wang computer science et annotation protein whole generative aspect functional structured experimental social thesis mark stochastic latent structure b link process collapse dirichlet allocation national study health health ii wave iii report university north hill censor survival model graphical technical statistic university berkeley p logit regression social term four variable measure pair node node sub population node level membership population node membership membership across latent realization replication measure node observe whole give parameter q full specification adapt kind datum type semi specification detail em present available extension inference relation response minor modification derivation follow general em variational possible make jensen maximize particular unfortunately define parametric involve entail low kullback optimal posterior accord field intractable fully factor n define minimize z z working expectation derivative evaluate natural index exponential parameter natural statistic property gamma function ascent natural mp underlie group indicator step carry value find lower tractable derive bayes hyper maximize contain unfortunately form likelihood newton method linear contain whose index pair control relative importance I estimation may sparsity fix estimate sparsity attribute information non mass mixed quick burden exploratory analysis fact university edu david university cs edu university edu arise gene network collection email analyze exchangeability long describe variable model extend membership thus dimensional fast posterior protein network field interaction relational information pairwise relation modern analysis machine scientific connect citation web connect page connect physical interaction pairwise property example web page protein assess datum collect individual independence assumption make fact develop purpose devoted group pattern interaction immediately applicable relational assume independent assignment blockmodel model dyadic relationship govern via one role blockmodel object advance recent extension relax cardinality assumption latent cluster via hierarchical formalism dirichlet blockmodel suffer limitation word latent protein social actor social actor may act role possible play relax actor membership flexible heterogeneity apply survey process mix single cluster vector capture aspect topic membership formalism particularly relational object membership influence relational us object play govern set manually collection interaction problem generate problem find compute amount small develop fast many world mean connect absence underlie latent group interaction
slightly estimate going apply finally want practical aspect system sometimes singular terminate fail phenomenon become sample large table initial replace fail consequently cf small begin newton update already incorporate implementation language run cpu ram via since cv analyse data consist measurement line infect study measurement measurement median year cover variability cd useful process use fan zhang wu functional component cubic spline equally knot use select show give eigenvalue eigenfunction hand much converge experience indicator reliable next eigenfunction panel together bandwidth disease cell tend decrease trajectory eigenfunction panel capture cell shape function panel shape eigenfunction fact relatively seem study cell count fig fourth magnitude eigenfunction shape stage disease measure likely elaborate incorporate eigenfunction individual method utilize obtain problem eigenfunction comparative cross validation approach real capture variability go work relate study asymptotic result good local polynomial closely eigenvalue know mle counterpart pca estimator pca intrinsic geometry propose relatively work extend kullback essential analysis role suffice spatio temporal covariate natural identifiability constraint long compute closely relate alternative eigenfunction add loss eigenfunction cf green orthonormal function penalty algebra obtain modification problem relate incorporation covariate longitudinal example covariate xt propose estimate eigenfunction modification eigenvalue parametric function eigenfunction depend assume capture eigenfunction curve covariate express maximization eigenfunction represent basis function acknowledgement g code thank simultaneous unbalanced component university model smooth component new york data university geometry orthogonality fan zhang estimation functional linear longitudinal p green l rate wang principal component component model j organization participant lee wang wang wang nonparametric principal smooth ph thesis wu effect noisy subspace intrinsic er wu coefficient longitudinal g wang data longitudinal f wang l longitudinal riemannian geometric concept manifold riemannian denote tangent give reference lee smooth smooth xy yx bi bi linear national sometimes write calculated curve dt x follow since geodesic satisfy understand bi riemannian metric fact I basic fact necessary implement description sd reduction mse sd sd mse sd sd cccc cccc model converge fail converge cccc converge replicate select cccc deviation estimate ccccc figure estimate eigenfunction wise estimate eigenfunction eigenfunction average eigenfunction eigenfunction noise eigenfunction eigenfunction eigenfunction eigenfunction panel panel estimate panel eigenfunction material table spline initial replacing replace fails converge block iv replicate converge replicate integrated eigenfunction sd sd normalize time noise ccccc c converge replicate c square eigenfunction sd sd normalize reduction reduction number ccccc c I converge replicate sd sd sd iv v converge replicate integrate square eigenfunction sd sd sd iii normalize reduction normalize estimate ccccc converge replicate estimated eigenfunction sd sd c mean estimate iv normalize estimate ccccc converge eigenfunction sd sd sd sd iii square reduction exp ccccc converge replicate eigenfunction sd sd sd sd iv error variance ccccc converge eigenfunction sd sd normalize error ccccc replicate eigenfunction sd sd reduction sd estimate reduction basis ccccc error estimate eigenfunction sd sd sd sd sd reduction reduction iv normalize select converge replicate eigenfunction sd sd error iv normalize distribution ccccc converge replicate ii estimate eigenfunction sd sd sd sd sd reduction iii mean error estimate iv select ccccc converge eigenfunction sd sd sd sd iii eigenvalue estimate ccccc converge mean integrated eigenfunction sd sd sd sd eigenvalue reduction square exp ccccc integrate sd sd reduction sd normalize square estimate eigenvalue square ccccc converge replicate error eigenfunction sd sd sd squared variance select converge mean sd reduction sd sd sd iii reduction c converge replicate ii integrated square error eigenfunction sd sd sd estimate eigenvalue reduction iv ccccc replicate integrate square error eigenfunction sd reduction normalize reduction iv square cccc cccc replicate select cccc cccc converge replicate ii frequency noise cccc converge replicate frequency converge ii select noise integrate sd reduction sd sd sd ii normalize reduction reduction reduction mean estimate c square eigenfunction sd sd sd sd mean square eigenvalue iii mean noise cccc deviation obtaining model eigenvalue eigenfunction longitudinal exploit eigenfunction restrict use eigenfunction address dimension order leave set empirical em newton year work noisy observation regard measurement classified functional viewpoint become increasingly popular think scenario situation g spectra chemical second longitudinal curve measurement subject setting compression studying effect extract mean variability first scenario e regular grid long level technique require treatment main goal paper estimation functional eigenfunction nice related building functional survey kernel space operator nonlinear statistical also argument favor utilize geometry obtain er intrinsic hessian log bring viewpoint motivation intrinsic geometry shall geometry parameter space maximum shall outline realization observed model variance semi term eigenfunction kernel eigenvalue orthonormal eigenfunction variance sample base rank eigenfunction result orthogonality describe specifically lie orthonormal procedure work normality base utilize intrinsic propose author include wu utilize likelihood result handle implementation challenge current measurement accurate study geometric viewpoint longitudinal prohibitive utilize computationally contribution finally hessian asymptotic likelihood necessarily understand present serve section overview exist idea important em approach algorithm estimator correct eigen nevertheless inefficient secondly utilize redundancy also though attain address find cv rely gradient algorithm propose basis another th smoothed weighted kernel center number measurement bias demonstrate wang smoothing empirical observe pair optimality process choice separate eigenfunction latter nonparametric somewhat inefficient secondly negative parameter cross far discover mention indicate significant satisfactory model derive want estimation paper usefulness geometric viewpoint though paper describe develop easily extend situation organize maximum newton manifold leave devoted comparison discussion section application cd count possible technical first weak e eigenfunction various smoothness eigenfunction stable basis basis eigenfunction model b tm dt know loss generality hereafter covariance motivate underlie infinite expansion form imply eigenfunction imply process covariance kernel rhs r motivate furthermore help eigenfunction lot get approximation eigenfunction situation decay modeling instability eigenfunction estimation grow gap successive appropriate develop advantage wu note parametrization highly likelihood parametrization rough maxima restrict view likelihood give I immediately difficulty lie orthonormal non optimization minimize newton develop estimator r rr newton involve hessian objective solve shall important assume cv assume throughout eigenvalue kernel eigenvalue eigenfunction point em algorithm restriction seminal newton conjugate counterpart euclidean space set function geodesic update tangent loss notational simplicity drop irrelevant since estimate parameter manifold orthonormal break part update update initial iteratively orthonormal basis function dimension convenient treat newton update straightforward rest newton treat field act manifold hessian bilinear act tangent essential implement appendix notation use denote step current satisfy act space intrinsic tangent skew orthonormal decomposition repeat mean sup arise inversion formulae reduce handle relatively measurement large quantity need propose considerable inversion basis eigenfunction cubic basis equally space flexible wide smooth certainly implement structure besides basis basis number problem basis fix project estimate one key question selection select number eigenvalue eigenfunction scheme second select basis criterion aic leave curve choose curve exclude curve cf proportional negative prohibitive efficient fitting generalize satisfie product space canonical refer view vector manifold curve estimate expand side shall notation first mt g b j bb approximate approximation considerably involve keep ib taylor expansion approximation two former treatment riemannian concept hessian respect newton aim whenever additional negligible huge computational curve discuss setting indeed first cv conduct estimation accuracy method henceforth polynomial henceforth em henceforth aim usefulness generate principal I cubic equally knot value natural spline distinction em spline correspond select space detail supplementary material three replicate parameter scheme table variance respectively eigenfunction represent cubic equally eigenfunction represent cubic spline name spline spike rely eigenfunction bandwidth truth fit result projection converge combination primarily cause therefore fair converge replicate estimation eigenfunction square deviation integrate square error mean square mse order magnitude ease reflect lack illustration figure eigenfunction converge quantile see close meaning accurately bias spline quantile fairly narrow variation variance
mechanism l derive estimator base demonstrate stein type blind compare blind ls regularization stein rule vector denote letter indicate unique semidefinite matrix qp ta deterministic know simplicity definite regression popularity ls estimator achieve novel l dominate blind minimax moment case estimator mse close derive show estimator lower long bound technique outperform l stage minimax design use variety shrinkage subsequently section general case bb l blind minimax radius measurement minimax ls estimate close small would exclude unknown alternative propose thompson condition thompson strictly dominate define shall dominate l blind minimax approach use generalization stein center origin weight may effect measurement shall outperform estimator reduce completely around constant vector lie particular give q continue merely sake notational demonstrate outperform l term mse large q dominate estimator dimension roughly effective vector theorem requirement requirement result l parameter estimate hundred measurement requirement variety prove strictly proof due stein q q q distribute normally substitute expectation strictly strictly dominate mse ls provide blind minimax estimation shrinkage ls multiply square estimate equally scalar shrinkage factor seem researcher shrink propose one close estimate iteratively solve equation furthermore whether dominate provide whenever substantial guarantee unless variance shrinkage easily adapt consider first narrow axis ellipsoid large eigenvalue fig wide axis l method unbounded shrinkage component illustrate want variance form highly ellipsoid result shrinkage result identical exist motivation application cosine dct corrupt high frequency contain dct number dct estimate choose l estimate merely multiply appropriately scalar reduce significant specifically result coefficient low l preliminary demonstrate improvement ls technique scalar dominate third define implicitly value sufficiently ig yield substitute dominate far dominate l suitable thompson technique dominate stein improvement dominate l indicate arguably readily hence opt eq nonzero substitute minimax blind approach estimator well unless balanced stein dominate l suitable balanced strictly dominate ls substitute proposition well drawback cause shrinkage shrinkage perspective shrinkage replace value minimax thus may word specifically stein dominate present balanced cause shrinkage improve performance degradation snr particular yield reduce mse high section identical positive db low snr preferable fact extend blind independently development researcher aware ls ill condition dominate intend quality specific approach common ridge intend ill nearly invertible definite depend cause severe effect ill pose snr ls attempt improve know normally minimum mse wiener choose empirically nothing possibility estimate optimally like imply analogy derivation substitute dominate dominate perform illustrate perform ls estimate magnitude varied obtain snr paper snr mse calculate snr snr consistently snr db candidate technique estimator effective dimension h c condition test stein consist shrinkage nj gauss pour des des j em minus ed plus new york pp admissible mean vector pp compare admissible dominate may unify admissible multivariate pp minimax admissible multivariate normal improvement ph stein thompson mean pp minimax stein bayes minimax arbitrary quadratic pp j v h poor filtering pp sep ed plus em new york possibly relate dominate least iv pa blind square france optimal filter square integrable gaussian pp robust square presence uncertainty pp family normal department stanford stanford em ridge biased pp plus minus new york pp theorem unknown parameter corrupt colored blind parameter minimax parameter estimate assumption propose linear dominate error stein estimator within minimax readily extend wide stein white previous extension stein stein estimation application typically set deterministic far minimal square gauss least l line reasoning l measurement h l maximum neither criterion mse l minimal yet remove yield mse appeal primary hand achieve indeed example deterministic word another trivial achieve mse nonetheless estimation follow strictly dominate mse never strictly least dominate estimator say admissible ls turn class estimator dominate l sometimes admissible dominate admissible give dominate considerably restriction g
unit root behaviour approximate approximation bt weak strong interval stationary bt c weak root analogue area reject efficient testing martingale different grateful david partially support foundation cm remark theorem cs uk preliminary theoretic calculate interval parameter game theoretic theoretic scalar intercept interested computing fix let procedure guarantee probability intersection empty guarantee theorem probability involve confidence satisfy accordingly interval produce refer confidence probability least precisely conservative definition goal construct random variable probability starting assume due e indeed true probability reject increment parameter integrate gives find confidence interval interval size worse usual iterate logarithm agree stationary concentrated around proof law iterate logarithm chapter interval familiar mainly interval centre give statistic
analytically periodic generally carry li matrix start analytically form appropriate adopting show due note calculate approach explicitly whose analytical exhibit circular naturally model order ns h normalize cross coefficient causal process see stand need start recursively lag main formulate necessary start sp h k sp unique whenever infinite circular property indeed substitute correspond conclude remark despite drawback require operation costly periodic separately suitable
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solution five root unfortunately argument infinitely many yet finitely solution al polynomial variable integer number complex coefficient generic point need integer degree furthermore sense equation formula word complex solution al provide generic argument present positive exist hence prove give claim check kk kk solution generic form generic hold map contradiction theorem fisher clear minimize critical theorem degree coefficient expansion expand rational equal elementary calculation equal gr root degree real real derivation variance population basis sample wish parameter section rational ml q writing form odd always calculation combinatorial follow summation problem yield recognize formula equation natural solution surprisingly determine system solution present empirical rarely generate triangular set follow constitute population sample equation compute software package implement homotopy summary solution exception indeed bivariate fisher equation real five real multivariate distribution instance test small bivariate simulation outcome simulation result case pt pt result case mean covariance randomly generate seem chance randomly fisher three solution high chance population root likelihood equation without argue generate module polynomial cubic ig imply term anti page cox cox et al module hence continuity comment manuscript part nsf dms introduction york maximum equation symbolic cox using york solution equation http gaussian journal likelihood bivariate regression ann behavioral yu elimination nuisance parameter berkeley statistic pp berkeley york l computational advanced polynomial american mathematical york ny purpose solver systems homotopy problem york percentage h solution department mathematics mail edu mathematics university mail edu department pa apply triangle nc mail edu lead eq identity u simplify ensure solve fisher problem initial calculate repeat process algorithm challenge property generic problem affine leave hand almost solution affine variety case choice affine subspace ml intersection subspace dimension affine subspace empty fisher pc lemma thm corollary ten st plus minus ten st plus minus abstract population fisher likelihood equation shall utilize simulation phrase fisher plus minus definite symmetric normal give fisher mean likelihood ff importantly early result restrict fail nuisance construction literature discussion many effort solution three unknown variance case system cubic surely sum cf side
minus mu mu mu corollary theorem proposition ex mm http www mm institute converge rapidly computable universal unknown despite nearby literature strong result sequence open interesting randomness show universal partial answer provide positive define measure hellinger closeness randomness prove complicated enough mistake hence contradiction truth law attack general less keep consistent formalize principle universal priori assign low simple environment formally mixture characterization sense dominate central observe past observation computable converge rapidly inform distribution convergence say individual pass randomness g law iterate natural ask individually clearly fail sequence randomness attempt paper sequence answer open probably particularly additional property l contribution construction universal converge computable convergence measure predictive play basic number asymptotic concept kolmogorov complexity concept universal convergence hellinger improve expect convergence hold construct universal give constructive constructive proof virtue sum sequence double universal converge summarize I alphabet concatenation string specific infinite sequence say sequence string number mu binary logarithm say without imply mu mu plus low mu mu f ff computable computable n fx length short universal machine object standard code define need co bound mu mu kx mu increase mu plus universal string mu equality start probability dominate mu mu plus mu x class constructive large constructive universal predictor universal monotone coin definition mixture repetition multiplicative assign drop generalize class contain exploit result computable n n subtle converge stay whether matter kn mu mu mu denote expectation measure e probability hellinger hellinger hold plus mu mu nx nx time marginally exceed precisely h mu plus w mu plus mu take jensen exploit multiply second inequality plus plus mu mu plus n plus mu plus mu sum tn one essentially computable sense tell close say nothing randomnes important default sequence closely kolmogorov universal give definition random iff one randomness test iterate logarithm etc sequence question whether converge fail randomness regard slightly indicate hard subtle detail answer universal mu mu since whether hold concept alphabet mx mx imply lemma c r vanishing define equivalently know hold infinitely refine fraction otherwise since hence infinitely formally infinitely r n mu plus mu mu plus plus infinitely r n plus plus show infinitely define complete proof finally could small show true constructive define similarly else else odd define even possibility sequence rx rx plus mu plus mu plus mu mu mu mu mu mu mu mu plus mu mu mu formal odd satisfy rx rx xx n p p triangle inequality summation apply triple h k convert sort converse imply instance hellinger sum h imply integral test increase universal hence desire mu plus mu plus mu plus f mu mu plus mu plus mu mu mu plus mu mu plus mu computable mu mu plus mu ex nf plus ex plus n nf mu mu plus measure mu mu mu mu mu mu mu limit mu computable decrease subtle constructive object computable computable mu plus ix follow proposition sequence convergence computable index computable constructive ex ii plus lemma h db db sequence ratio bound instead idea convert computable k tb mu mu plus mu mu plus k kk mu plus plus mu mu mu computable imply last mu mu specify mu put everything mu mu plus mu plus k xx mu plus mu mu applie show mu plus mu k k kk mu mu mu plus mu mu mu mu mu mu plus mu plus k mu mu k approximations measure close measure include require string xx string measure definition one enumeration convert correspondence convert directly obvious mu tt nx enumeration enumeration enumeration enumeration property define mu plus plus mu hellinger rl mu plus b locally quadratic scale closeness imply deviation extend define exploit mu mu mu mu mu mu mu mu yy mu plus mu mu mu additionally mu plus mu plus mu mu mu plus sum prove mu mu w iii mu plus mu da plus plus mu convergence contribution long hence convergence assume e mu mu mx plus mu c hold computable ix dx mu mu plus mu dx plus mu plus mu mu theorem proposition tm tx respectively speed finally dominate measure dominate measure proportional dominating computable dominate dominate computable dominate conjecture computable lie normalize measure converge intermediate quantity seem sentence exist sentence follow zero string computable dx mu sentence computable dominate computable dominate extend let sequence compute accuracy
fig curve fig curve special random knowledge threshold general let bernoulli define p p furthermore eq know desirable computational may small essential enough seem evaluate fortunately evaluation sequence z b obvious coverage p p convenient computation way reduce proportion infinite population application technology bit length receiver recover bit bit assume possible bit identically continue take explicit method bernoulli random assume conclude remark numerous history lack rigorous progress researcher fold discover critical mistake exist determine reliability efficiency second developed formula computational threshold efficient precision suffice exclude otherwise need true derivative f x u z completes similarly establish property similarly monotone monotone decrease tend z z z monotone decrease prove estimator n integer shall case case definition positive z lemma bound note actually preliminary p two consecutive element z pp consecutive set lem cp gp p cp observe small since gp exist cp gp cp unimodal let incomplete suffice increase k p b ks case must exist decrease lemma must consecutive z z cp drop argument tc respectively readily deduce statement immediately coverage sequence variable al propose rule section page reliability point al represent al lem et page let size recall page suffice l n use inequality complete instead give claim point al argument al prove spirit rely rely l z demonstrate al reasoning efficiency stop replace page page ensure page ct I variable common n first paragraph page event equality mistake definition conditioning claim equation reason validity sample justify development rigorously demonstrate prescribed precision take variable develop explicit search determination knowledge special random variable numerical bind thus detect introduction field science frequent problem bernoulli extremely universal formulated estimating network reliability uncertain approximate probabilistic cast application need estimate quantity bound operation typical design use average outcome refer monte carlo method tackle multidimensional integration volume count find enumeration value mechanic error rate sufficiently error know chernoff bind poor relative seek error estimator want usually easy reasonably tight scheme loose lead prescribed force size scheme rich modern estimation brief seminal book exposition sequential precision mean estimate guarantee tend drawback inherent see researcher area uncertainty namely error reference therein resort monte rigorous et develop sequential bound guarantee propose one sampling value obviously inverse binomial determination et ensure improve threshold paper sequential discover incomplete gap argument reliability estimator incorrect importantly threshold make substantially explicit claim prove special apply develop computational threshold bernoulli remainder follow general theory inverse binomial application conclusion proof examine notation denote denote denote less limit notation notation clear define scheme continue inverse consider bernoulli variable respectively estimator binomial estimator considerable small coefficient highly desirable minimum
arise deal inferential arise nonetheless deal parameter parameter evidence concern call significance inference nuisance significance level challenge channel ten channel broad agreement central bayesian use give widely prior one argue hold lack uniqueness inference discuss event regard tend principle toward used take publish account modern outline wherein reference find depend wish central nuisance include realistic represent background signal measurement let satisfie lie space open subset take limit summary profile log generate j corner preferable interval quantity significance cumulative monotonic decrease significance inference equation contain obtain interval approximation significance quantity preferable set subset invariant confidence parametrization body asymptotic bayesian converge key formulae derivation formulae density conditional section account perhaps improved inference likelihood determine exponential approximation canonical partial determinant whose sided significance sample depend discrete approximation reduce slightly take independent contribution scalar product function need numerically invariant addition affine quantity leave unchanged use mathematic intend size parameter increase approximation outline little available three component variable kk ny background positive principle nuisance parameter positive mathematically value purpose restrict meaningful inference form pt maximize nuisance argument exposition model log canonical parameter give summarize evidence profile log inference equal regard adjust log figure profile significance explain analogous estimate obtain leave right preferable parametrization bound interval significance one minus significance side give respectively surprising upper indicate bound get negative admissible fact low coherent case although suited signal limit confidence give confidence see perfectly sensible answer example alternative would strongly suggest positive cast model support observe challenge leave tail interval test regard inference limit simulated coverage minor simulated nuisance highly display worst apart issue well boundary extend channel nuisance channel profile simply sum profile channel maximize remain ingredient root v eq give modify root interval bound simulated dataset nuisance confidence nominal analytical posterior inference choose prior derivative lead marginal use discuss inference pt model show element mild condition jeffreys side posterior contain sense unfortunately require express model parametrization impossible arbitrary parametrization model write element matrix relate cccc orthogonal express next section channel loss inference frequentist inference lead constant u fisher I orthogonal hence use readily algebra turn available yield apart additive prior quantity depend heuristic channel nuisance parameter consider significance constant typically yield upper parameter small respectively sided positive coverage solution quite show significance approximate ordinary result quite good modify modern detecting signal use comprehensive observed sided limit frequentist appear essentially inference signal worse slightly equation analogous serious coverage quite arise weak although challenge concern general
view bethe loop nontrivial due parameterization marginal approximation argue meaning end lead viewpoint start may basic view correction apply continuous case loop scheme sense apply variable compare expectation mind firstly scheme generalization initially interact denote probability local pairwise manner interaction j loop discrete continuous expression model interaction neighbor cavity way write normalization equation lead repeat moment true cavity function interest clear cavity restrict able perform insufficient choose notice belief recover choose approximate cavity distribution include correlation cavity loop correction parameterization correction cavity loop cavity specify average covariance belief yield investigate exact propagation exact parameterization cavity dimension cavity bethe cavity consistency equation vector cavity set cavity cavity second subsequently average follow obstacle exact covariance cavity recover gaussian response possible covariance binary response propagation yield identity write define entry run variance average bp equation follow ik I equation allow meaning message average variance cavity come via belief message via ordinary next together argument ordinary form correct impose bp compare equation cavity covariance interpretation average cavity may run variable bp original appendix entire suggest invert useful subsequently bp run grow along run fact multiplication invert loop correction total bp local marginal general loop correction able increase bethe formalism seem tractable alternative expectation propagation special generalization loop correction equation function form ep bp family ep fully bp base large neighborhood derive general e nonlinear correction ep target distribution approximate contribution remain relate approximation potential follow notation intractable proceed remove defining minimize update furthermore deduce parameter contribute scalar thus marginalization inversion equation prohibitive cavity implicitly kl generalization correction potential start loop generalize formalism still parameterization approximation easy equation ep subsection since limit somewhat derive equation slight fact cavity absence target kl approximate cavity vanish gaussian integral may perform equivalent ordinary approximation algorithm optimal since moment cavity distribution integral calculate observation approximation neighboring cavity reduce ep fully moment interaction ep optimize way albeit correct update correction current approximation sensible formalism cavity variable follow derivative correspond cavity covariance reference propagation equation value variance obtain run ep fast response involve cavity determine update central response obtain estimate costly necessary update might loop correction beneficial far derive loop belief work tractable derive exact message correlation cavity part moreover various bp lead involve order relation propagation loop correction strategy potential loop like expectation grow costly heavily stage cavity
need relate function unknown subject intensive testing recent testing use use however lower base result possibility membership complexity complement establish low testing oracle call oracle finally query access section query allow membership query complexity substantially eliminate consumption example another classical giving nearly section give testing give keeping standard stand concept concept omit subscript class depend variable agree fraction study see version boolean class behave concept least notion test al separation particular note know identity primary call algorithm interested concept test source type classical boolean oracle assign order pair simulate oracle query consider example label example label weather financial market efficiently pac learnable limited efficient quantum generalization respectively membership query act basis state oracle act basis query invoke query pac quantum turn work use value function real endow q induce norm well function value product consequently every uniquely alternatively function relate value value f eq boolean take value random random take different probability chernoff sum let boolean return variable probability section oracle distribution set describe root spectrum boolean possible simulate describe classical testing give membership give several efficient test membership subsequently al far emphasize concern classical query call new use oracle example algorithm call response output output suffice successive call oracle immediately st new add obtain oracle far know variable fact contradict least consequently uniform query query since membership first follow must query set boolean variable choose among yield low possible draw easy random spectrum boolean sum square fourier boolean imply scenario query make check p scenario thus algorithm oracle access must oracle integer address address element depict decision address formally address address useful us spectrum spread difficult address shall far replace place see draw range permutation clear support fix variable fraction input precisely nonzero correspond leaf take value input rhs term consequently give rise address equation consequently easily rearrange replace choose order value return function rr make tree place randomly decision tree place index j location range range depend spectrum leave tree equation I rhs sum give rise address moreover draw independent uniformly ready prove every consider oracle return tn j nf response probability similarly fix draw form outcome occur crucial symmetry consider next draw alternatively one since successive determine odd sequence call call distinguish sequence call random subset independent oracle start corresponding wherea distinguish low accuracy membership query exact query suppose know case learner least value chernoff unlikely learner uniform result uniform computational problem fast et multiplication exponent give oracle formula use variable since pac try optimize obtain concept learning query prove consequently query input quantum query set make dim query vc dim since vc dimension boolean oracle sufficient high query concept whose single oracle query bit concept class motivate reduce sample drastically quantum answer use classical p output list query output response occur oracle call yield learn oracle claim satisfy requirement construct influence variable encounter boolean depend oracle reveal successful group description encounter relevant depend notational e assignment assignment assignment every stage hypothesis produce observe linearity imply fraction unseen assignment p stage terminate p terminate desire verify let string let total denote indicator function value hold false start give string clearly nf ff expect incur list less subset assignment obtain describe agree fraction back generation hypothesis drawing coordinate draw
prove give condition take eqn iid drop index factor first factor per convergence lemma convergence depend however value convergence consider issue state converge iff convergence part constructive show convergence irreducible irreducible realization network irreducible pattern replace non let eqn eqn prove part vector corollary necessary convergence laplacian connectivity formation necessary sufficient proceeding convergence variable variable space converge definition variable chebyshev also imply subsequence formalize sequence convergence part like constructive eqn eqn hence lemma converge always assignment scheme weight particular argue separate two convergence weight assignment weight link converge always weight metric sequel convergence per iteration convergence particular actual play significant maximize formation perform minimization achievable eqn depend laplacian minimum perform range since fast optimal follow analyze consensus topology consensus presence inter weight throughout entry communication incur iteration sensor connectivity fast cost structure network topology deterministic fix equal cost topology fast easy translate link solution essentially class topology cost cost may seek topology convergence combinatorial problem cost look random fast convergence rate communication network make constrain cost convergence summarize problem concerned design formation fast entry later solution medium constraint force satisfy analogy assumption reference optimize probability topology protocol probability impose contrast fast give reliable communication ratio snr often enforce select snr sensor cost communication incur single formation entry access distribution laplacian total incur diagonal entry incur random incur distribute small per step inequality constraint inequality follow difficult formulate cost successively convex solve good topology iii analyze convex lemma constraint convex maximize optimization semidefinite problem numerically see reference solve laplacian discuss topology good original topology eqn subsection establish constraint consensus stem fact joint plausible justify plausible first justify step eqn bind lead suggest quantity order element provide monte carlo see replace justify suggests successively verify numerical sense os enyi network formation collect eqn display result remarkably similar local fig confirm order evaluate sense lead general study topology topology dependence matrix tu since establish concavity derive recall graph set edge edge associate total cost quantity link formation satisfy proof concavity l prove need interesting state topology may cost expect study next solve semidefinite solve snr fraction time active compare topology connectivity display fig geometric propagation euclidean topology topology incur cost step gain optimal topology topology topology much significant medium topology two sharp increase horizontal meet times example show achieve use top bottom sensor communication sensor fail communication network term algebraic define topology optimal topology subject communication topology specifie topology consensus design topology sensor mm algorithm axiom case claim conclusion theorem criterion summary htb among sensor operate resource datum communication main determine communication failure proxy snr operate design e assign reliable communication sensor failure preliminary sufficient consensus particular topology design subject random link failure communication convex semidefinite programming consensus decision design I communication maximize distribute thesis recently research receive literature topology question constraint reduce realistic fail entail constrain field algorithm network link probability formation link link formation across iteration design fix know communication sensor budget preliminary result paper ergodicity randomize relate single node consensus link topology simple probability communication entail cost communication recent evolve topology switching delay identical link outline summarize concept laplacian distribute average consensus failure state term average bound address distribute constraint problem programming topology numerically sdp show design factor compare sensor sensor radius performance topology paragraph consensus topology communication whenever paragraph recall concept make sense sensor become drop topology paragraph topology undirecte sensor set refer motivated topology sensor set vertex integer sensor call loop self edge connect vertex term require edge vertex consensus random topology update accord iterative collect edge network connectivity iterate choice weight laplacian eigenvalue eigenvector still reference nonzero weight adjacency provide link see optimality consensus algorithm iterative weight probability formation likewise matrix also iid drop matrix iterate lead eq since influence property iterate subsection describe formation probability iid mean need problem mean laplacian connectivity algebraic laplacian follow link property laplacian lemma mean eigenvalue arrange normalize eqn negative eqn weight call irreducible laplacian easy compute function eq follow jensen eigenvector result laplacian spectral study
neighbourhood vertex degeneracy condition main focus reconstruction sparse mrfs two differ degeneracy degeneracy condition observe fully observe algorithm problem perturb conversely relatively weak compare couple theorem observation subset node available provide reconstruction generate high computational decay realize prove liu random underlie tree computation maximum span mutual markov whose work biology devote reconstruct tree sample degeneracy condition recent method allow clique interaction line requirement interaction weak strong refer hardness verification refer generating return refer alg x x reconstruct graph complexity reconstruct close kullback leibl application often reconstruct furthermore differ generate large polynomial generate sampling kl present ise efficient graph ise clique allow condition require thing however ise lattice regular e subsequent work consider produce nearly asymptotic temperature difference ise I potential take limited pairwise interaction mrfs markov disjoint path pass factorize normalize constant reconstruct graph degree map consideration interested relationship node successful approach theorem necessary argument theoretic compare accord probability need applie identifiable minimized maximum posteriori error deterministic max following give adequate graph max satisfie explicit graph vertex maximum degree graph distinct way degree neighbor edge edge label vertex vertex reconstruct label vertex hold markov say potential degenerate recover condition constraint come fast versus assignment replace exist x correct reconstruction estimator computable assume neighborhood subset compute select true otherwise exactly choose hold thus hence reject every determine run previous essentially proposition soft graphical degree exist neighborhood u exposition ise external coupling constant normalize constant couple ise model parameter condition eq eq eq moreover reconstruction algorithm polynomial correlation denote correlation q weak condition decay correlation exist runtime high probability correlation cv du vertex modify neighborhood neighborhood correctly vertex correlation neighborhood choice neighborhood run reconstruction operation calculate large dominate reconstruct field observation instead amount condition state eq reconstruction assumption correspond thus reconstruction impossible let ise xu xu xu perfectly vertex extra spin observation distance graph structure identifiable shall symmetry ise external px px px continuity jacobian lebesgue neighborhood h region identifiable missing fit vertex vertex mild restriction eq correctly vertex algorithm clique maximal clique size miss vertex simply vertice clique least necessary simplify clique algorithm reconstruct vertex show condition recovery satisfy graph degree satisfied property first e helpful hide definition fellowship support fellowship mathematics nsf
function table size problem yes correspond benchmark processing method use directly performance assess procedure choose penalty term choose provide pattern rely calculation coordinate allow prevent point safe small validation use example c c htbp c linear direct yes correspond analysis perform projection subspace plain directly apply project obvious plain performance bad rule affect dominate ridge adapt datum kernel reach seem functional perform train mid gaussian package whereas minute report sensitive conduct value yes set high leave give select validation set search extend error performance grid raw projection gaussian derivative difficult one nevertheless appear improve derivative significantly linear test performance plain feature linear expert kernel show use svms plain svms directly show benefit kernel kernel provide consistent allow account expert discriminant satisfactory result show type functional acknowledgement anonymous suggestion improve simplify obvious project l q inequality union capability linear observation precisely us yx kx kx subset point separate classifier space kx l lf f equality oracle ml direct consequence fulfil valid kernel satisfy compact value request infimum define number requirement therefore compatible take analysis motivate traditional adapted investigate discrimination svms tool base implicit mapping space thank classification conduct datum emphasize benefit functional functional machine consistency discretize rather application mapping implicit response study correspond study example problem series complete face difficulty discretize represent high coordinate highly consequence problem theoretical work functional space infinite practical discretized functional functional nature comprehensive method linear extensively paper adapt machine svms extend nature take construction svms functional datum ill pose problem provide generalization present illustrate set article focus value finite positive borel hilbert integrable value additional g existence derivative need development structure space product basis course apply view arbitrary neighbor method calculate obviously use layer almost arbitrary space infinite fact simple ill pose dimension view instance target covariance matrix technique g ill pose schmidt direct problematic infinite far principle finite near instance space consist function curvature example regularization include parametric discrimination find lot successfully study case machine filter brief presentation svms reader comprehensive arbitrary presentation introduction belong rather make svm arbitrary difficult discuss problem functional classify predefine realization variable pair affine observation margin problem request error satisfactory separable partly problem separable secondly prevent overfitte discussion version error thank slack margin cost dominate contrary close note satisfactory non transform function construct linear mapping problem note feature define space solve problem seem infinite dual precisely optimization original map map solve first rather dimension link write optimization problem classification use transform kx x definite nx j map kx short introduction define functional hilbert software possible rely numerical integral effect hilbert margin problem general dimensional solution avoid soft margin regularization space section performance know optimization regularization behave function svm expect seem dimension behave lead approximate dot product see interesting functional point g classifier also penalization might operator hilbert therefore hilbert kernel kernel obviously difficulty implement appear plain kernel discuss kernel provide take advantage hilbert kernel easy transformation domain transformation center normalization enough restrict transformation hilbert function derivative allow focus transformation kernel standard dot product detail field directed expert nature outline two b spline separable hilbert system consideration wavelet stationary wavelet fourier explain projection give projection interesting practical spline spline regularity enforce knowledge spectra smooth physical light transmission spline replace unconstrained observation combine operation perfectly therefore kernel situation discretization consist counterpart regular standard svms vector sampling regular integral thank quadrature position situation sample location discretization point depend possible value spline detail tool operation instance spline implementation sample whereas propose request unique projection spline convenient function world spline reduction functional goal asymptotic usual classifier bayes achieve error course admissible note data svms belong choose specific type map associate dense continuous consider compact subset compact detail propose turn consistent hilbert adapt svm precisely choose addition universal kernel etc kernel denote list example validation classifier set fix list calculate ax kp dx please everything one select optimal estimation generalization
show n continuous ds n proceed term addition rewrite definite q state latter show proof recursive give h kullback leibler matrix g eigenvalue g strictly assertion directly author feedback exp lemma theorem contribution generic sometimes usual leibl distribution regression online approximation regression maximum posteriori estimation common sense broad demonstrate strength yield include censor etc em appeal generally conditional moreover naturally sense converge data stream impractical strong possible store dominant online algorithm update new incomplete different propose online algorithm decompose first stochastic incorporate newly algorithm consequence view long secondly rely previous provide follow fit relevant conditional propose finally limit normalise criterion kullback leibler paper section review algorithm discuss simple mixture regression property non observation deterministic value latent variable provide situation include miss censor datum induce latent stress restrict notation set em optimisation normalise possibly complicated traditionally essence force log step regularity recursion locally equivalent adjust use search condition adjustment may avoid problem well replace fisher information mild guarantee recursion fisher except particular complete canonical naturally family thus depend behaviour instead able estimation datum difficulty consider sequel iteration index identical describe datum recursion size version newton correspond although recursively however robust title paper note stochastic little algorithm consider unchanged precisely maximum feasible automatically explicitly require inversion practical rate algorithm belong assumption iteration correspond reflect application em care issue comment may also exponential much low involve history sequentially update step learn see relate incremental em derive incremental process sequentially several incremental define k k ki pseudo see mostly differ traditional expectation sufficient online coincide online instance call normalise gaussian extend canonical family sketch scheme mixture call correspond miss rather simply expectation really complete kronecker delta rewrite case conditional th generic evaluate complete matrix deal inverting update online latter intensity poisson em ensure constraint happen negative lyapunov argument analog monotonicity kullback continuously stationary divergence instance stay assumption stability expand sequence see construction exposition online gradient surprising counterpart asymptotic express online em assumption recursion gradient remarkable em online gradient without approximation particular coincide em lead guarantee weak minimum leibl real upper converge lyapunov q specify I lyapunov specify hence lyapunov solve explicitly coincide constraint hand step difficulty recommend post step follow asymptotic estimate equal inverse propose method complete determination weight efficient averaging regression variable explanatory finite distribute regression model specify datum distribution regressor triple couple specify determined therein model regression part delta put one q mixture indicator check function example role scalar definite matrix invertible unless take care first word build great simple q variance regressor parameter first component differently illustration purpose despite consider regression explanatory since previous theory apply straightforwardly use regressor specify online recursion weight correlate orthogonality regressor th correspond value bold use average start iteration relation model near empirical em computation actual parameter comparison check linearity approximation million associate correlate unweighted stochastic length algorithmic use em em step average start note whereas em costly require many figure compatible iteration batch long observation claim approach bias increase problem avoid asymptotically speed slow illustrated figure appear vanish online parameter batch easy
attain similarity let margin absolute frequently find sample integer l u mn mn kn kn r k z less show characteristic coverage population bernoulli population computation detail method proportion efficient efficiency characteristic theorem simplicity notation drop equation integer therefore consider case case sn sn sn n sn v sn sn sn sn sn sn sn l sn sn sn sn sn sn lem lm hence lm prove part deduce nk n nk non show eq statement direct computation lem show suffice case iv case decrease respect see vi n sn n sn nk nm l l non decrease lead k thus k nm therefore integer l lemma integer notational kn kn kn n kn r r k kn mn mn k complete integer notational r kn mn kn kn mn mn ng sn sn g n sn h sn h sn nn h n sn sn sn sn g sn nn nn consecutive kn k kn obviously focus suffice kn sn g sn h kn eq sn follow sn gm iii kn sn cm sn gm c kn kn kn consider case focus vi include focus comparison belong vi prove clearly coverage compute kn xy fact inequality number kn kn u l remark proportion finite error complexity regardless introduction basic problem among attribute frequent sampling replacement carry attribute prescribed margin prescribe exact require coverage probability
mention let rewrite spherical segment mention spherical I normalization I canonical analogue canonical measure eq rewrite introduce coordinate second agreement eq produce another possible rewrite yet produce integration representation eq decompose spatial axis point sphere orthogonal represent twice opposite undirected line expression integration length line canonical uniform isotropic multipli invariance rotation let introduce volume respect produce normalization formal integral body cauchy r use optical variable rewrite radius sx dx analogue axis intersection line represent integration fig analogue optical body integral complete analogue calculation matter difficulty multipli analogue formula surface integral represented introduce finally present proof average path isotropic depict path symmetry respect path coincide get average straight proof path minus ex plus minus discuss sign sign definition via autocorrelation point body work constructive interpretation theory dirac length three way straight intersect nonconvex body line respectively proportional autocorrelation generalize calculation possibility negativity nonconvex sign distinguish body possibility sign probability physical negative give assume need review physical mathematical measure purpose use call sign extension decomposition measure sign obvious simple object sign appearance negativity natural decomposition nonconvex fig represent convex body intuitive possibility use hull express derive sec six sign negative sign nonconvex body sec arbitrary briefly mention extension trajectory affect appendix length segment length precisely ambiguity widely distance inside fig function body point isotropic b length distribution body convenient autocorrelation distance eq autocorrelation together remarkable correspondence express via volume surface body derive later dirac also source integral function widely convenient completeness nonconvex usual generalize integrable generalize functional definition ensure arbitrary derivative dirac six integral body expression q distance volume surface volume ref particular result follow relation due side rewrite autocorrelation integral dimensional appendix accordance appendix express integration point finally rewrite body justify associate functional relation integral consider transformation functional necessity quite integral rewrite derivative eq correspond integral quite dirac expression integration part eq derivative ensure formal generalized length density nonconvex body distance autocorrelation nonconvex body write coincide convex produce interpretation sign analogue body ray body point ray surface denote maximal distribution positive respectively nonzero eq nonnegative nonconvex stochastic body kind primary ray also kind secondary event positive integral via average expectation etc radius event event e introduce q ray r k analogue integration similar intersection coordinate new whole inside body point interval inside minus term j integration triangle analogue nonconvex due q term positive negative integral arrange negative depict draw line decompose normalize multipli express write analogue show early body quantity proportional derivative sec formal equation q describe formal body yet relation stochastic useful uniform isotropic body secondary event segment interval contribution effect work difficulty express sec e coincide due eq finally ref convex concern nonconvex prove yet area due clear equation represent body stochastic consider length length segment represent proper sign eq proper normalization total number note page definition interpretation discuss definition line uniform case integral analogue length use complete formal may analogue certainly nonnegative density formal consideration body complement hull optical length mention nonconvex body already path concern derivation uniform concern trajectory important sometimes integral particle justify formal integration necessary condition uniformity discuss line justify compare make nonconvex sec due spherical symmetry basic model completeness always describe essential construction much complicated describe sec example application represent propagation trajectory intersect body straight use segment inside body intersect environment medium integral boundary intersect depend concrete
overall rate complexity reconstruct three simplex x state box future future distribution distribution causal process due extreme randomness introduction biased coin completely rate x vanish p p opt extreme periodic distortion straight x pf x r rate e excess distortion causal accuracy bit distortion lie exact randomness show curve serve mechanic give distinction must turn connect machine avoid fluctuation find vary degree abstraction principle model distortion employ iterative distortion compute anneal calculated curve demonstrate reveal provide automate making structure practically speak admit parsimonious capture behavior point find relaxed show automatically abstraction focusing limitation finite error compact understanding point size impose level previous fellowship dynamic intel automate theory building naturally distinguish randomness model construct hierarchy complexity minimal maximal intrinsic organization extract causal reflect optimal balancing often discovery novel physics mind principle last decade collect truly vast example automate translation social exceed analyze present automate discovery understanding make provide develop automate procedure theoretic criterion construct degree abstraction importantly show appropriate recover organization without approach ad hoc character store information building challenge extract structure physical randomness phenomenon structure irrelevant irrelevant structure distinction constitute theory assess importance prediction typically give small however prediction find distinction causal occurrence merely implement attempt nonlinear system linearity require address notion goal principle definition organization discover x communication storing equivalent future define causal constitute maximally capture past causal future call markovian past partition partition state history partitioning therefore causal equivalent causal capture information unique capture efficiency state amount store excess future causal serve alternative compare approximate causal way quantify minimized assignment building result accurate prediction take condition excess vanish partition distortion eq conditional past lagrange control trade balance excess entropy dependent therefore maximize future past ib ib give cf eqs consistently eq specify family parametrize gibb within analogy mechanic correspond optimal compute carry concern gaussian dash line past horizontal excess infinite sequence solid conditional six six circle anneal retrieve causal optimal deterministic p zero condition probability condition past conditional equivalence eq finds argue goal ad hoc partition code causal partition capture less distortion less optimal original result shape curve characterize vanishe curve fix achieve vice versa causal distortion determine trade specify nontrivial statistical causal whose entry
base space tangent eq would tangent sphere continuous modeling really topological us carlo tangent sphere bundle state analogue build continuous sphere circle direction north south start bundle way carlo modeling tangent generate rotation one correspondence random dash specify generate monte projective produce similarly rotation unitary treatment point satisfy property describe four construction bundle principal bundle random tangent vector plane describe tangent question monte nontrivial basic example straight principle present apply nontrivial come total bundle less consideration skew bundle adequate description construction action structure uniform space necessary possibility generation space principal bundle homogeneous group density use definition associate simplify isotropic inside sphere cosine angular yet parametrization line associate line surface ex ex plus question nontrivial example line interaction solid nontrivial bundle equivalent sphere carlo trial variable wider wide due growth capability computer comparison traditional always justified example geometry theory simplest monte carlo integral integral compact support distribute multidimensional rectangular area law quite euclidean base produce initial apply monte way always classical associated name quite discussion state big produce problem example physical problem carlo trajectory necessary basic necessity consideration ref approach choose co object unique way next measure invariant parametrization parametrization plane straight plain origin co plane straight htb space line represent bundle definition straight line draw origin space base bundle direct sphere definition unit origin plane draw direct pass definition due direction plane uniform tangent straight bundle generation difficulty really could sphere plane sufficient particle volume meet problem nontrivial bundle distribution bundle sphere illustrative frame may tangent smoothly case isotropic carlo way symmetry matter plane plane normalize advantage monte even without analytical necessity difficult indirect procedure e construction
linear intensity domain boundary field two temperature ambient temperature everywhere background refer mapping compactly call write zero neutral element operation meaningful development enkf large complicated zero possible derivative jacobian matrix q inverse however value etc boundary intermediate imply original recover choose residual argument way formula easier require inverse like computation find track residual spurious formula amplitude array associate rectangular grid call array mapping array grid grid denote array bilinear interpolation assume value mapping bilinear compose calculate straightforward manner bilinear interpolation grid bilinear interpolation spaced node approximate grid accomplish matlab space grid mapping optimization pixel wish find norm good many minima match solve minimization use modification description completeness automatic invertible speed chance minimum proceed hierarchy mesh build grid approximated approximated scale sum respective grid guess grid none notation array restrict grid like mesh iteration index coarse correction adjust node time number residual coarse mesh first capture coarse grid capture fine track large scale perturbation grid thin maintain small grid smooth resolution tune grid replace determine also already coarse grid initial correction grid guess value j node refine several optimization local value use coordinate alternate matlab cf boundary move inside difference describe refinement position function guess move multiplication operation multiplication implement pixel fortunately objective unnecessary change grid objective function jk grid grid grid large node grid consist array simplicity array array discuss ensemble consist array member concept hierarchy grid automatic array fix automatic ensemble determined optimization cycle good order magnitude enkf make combination enkf enkf ensemble w z representation forecast member ensemble similarly state array impose potential correction limitation seem elsewhere temperature transform adjusted temperature reading ensemble member high member ensemble completely difficulty feature array dense measurement transform like become array ensemble need enkf fraction temperature ambient temperature start member mapping error enkf apply I level stop relative improvement last infinity parameter simplicity include assimilation spatially ensemble member guarantee test one element represent ensemble state advance advanced cycle previous enkf ensemble analysis indicate fig indicate error much close fig residual bandwidth time non fig normality result plot highly highly normality visualize closeness normality distribution see around prototype enkf highly feature ensemble add penalty spread variance limitation automatically specify difference ensemble divergence essential limitation sufficiently similar guess limitation ensemble go number grow almost degree method assimilation move paper report practical science problem residual force small domain shift rotation weather application solid test could function force local stay reciprocal norm multiply function would care bilinear grid grid order able relatively coarse interpolation spline update often initial regardless significantly ensemble track perhaps ensemble member analysis member residual member much affect enkf might currently test significant characterize generalized use grid perhaps use different mapping state different position equal thin horizontal connect mapping ccccc em em vary location locally alternate direction library bilinear thus within thick dot line side put point segment line reaction transformation cccc combine enkf solution move thin composition plus residual automatic identify enkf transform state intermediate instead linear research motivate data assimilation present challenge assimilation center thin reaction sharp effort build drive assimilation network enkf fail highly coherent degree penalization state localization enkf employ observation function problem enkf work increment location enkf even
node end delay node form remarkably traffic internal loss network study algebraic inefficient estimate complexity pseudo likelihood measurement mle study discrete bin link bin width link ease however heterogeneous scale slow vary structured delay side round bin appropriate bin width force bin suffer delay link delay component link delay issue instance mild identifiable propose function delay develop fast gmm allow delay heterogeneous delay extensive suggest yield estimate yet remain address identifiability describe delay develop fast section extensive evaluating conclude identifiability mild tool characteristic basic review eq convention characteristic manner consider uniquely specify versa independent function characteristic function characteristic easy convolution eq mutually general establish identifiability otherwise distribution identifiability discuss issue condition namely characteristic analytic first issue simple tt argument difference function order side leaf tree model root leave four leaf purpose denote link delay delay end end delay node partial sum parameter induction distribution determine ambiguity argument identifiable shift ambiguity provide identifiability traffic demand topology identifiable shift ambiguity ambiguity constraint e convenience shift ambiguity element wise notice denote common identifiable neighborhood uniquely decide discuss ambiguity ambiguity recognize avoid ambiguity poisson model start delay message despite distributional shape determine order moment uniquely identify bring achievable example delay relationship demand estimation distribution analytic link delay characteristic tailed distribution distribution despite theoretical counter identify tree appendix full condition context traffic demand gaussian condition realistic topology distribution conjecture variance ambiguity leave delay describe class flexible mixture link delay well know parametric delay define flexible delay link delay follow delay mixture mixture characteristic function distribution appropriately characteristic function delay mass link delay queue steady distribution probability queue body piecewise flexibility tail traffic long model reduce bin advance interest denote space delay vary bin strategy heterogeneous link delay bin width grain bandwidth delay grain characteristic along bandwidth density vary lot area important place bin area use researcher also link top bottom bin scale bin many fast link fit vary clear bin slow use bin width bin bins quantile delay reality quantile initial link delay process iterate less last tail simplification tail mixing coefficient point parameter delay model identifiable delay complexity mle present easily good motivation characteristic though moment formally formal discuss previous function joint eq likelihood kullback minimize characteristic use size integral monte carlo draw sample rewrite conjugate base cf residual evaluate covariance motivate identity inversion practice either estimator iterate process base estimator generalized gmm considerable body sample choose efficiently follow simulation space characteristic close characteristic efficiency sampling dim dimensional implie view pseudo network coefficient develop characteristic function obtain parameter optimization quadratic optimal ta ta ta ta dim column programming quickly iterative repeat nice property function care serve good iteration due become segment delay solid link tree piecewise tail delay solid mean mixture piecewise bin line line fast flexible delay delay continuous delay distribution piece estimate compare mle delay equally space comparable mle continuous spatial hold demonstrate adapt scale delay link satisfactory drive realistic scenario assumption trace drive simulation match delay spaced bin four link discrete link delay seven mle repeat seed cf use also cf total randomly dim subspace delay normalize develop much seven delay observe compare link truth experiment figure quantile represent median median high mle comparable subsection delay one delay delay heterogeneous network challenge homogeneous delay represent link large delay addition exist mle rely discretization internet leaf heterogeneous environment mass treat comprehensive evaluation different due report representative modal ii multi modal delay four respectively assign resemble delay delay delay estimate link delay distribution cf four mixture link delay mass lie bin equally bin link obtain cf bin distribution delay along ground figure identical estimate equal give satisfactory complex estimate cumulative view absolute quantile two distribution normalized deviation repeat simulation distance link clear bin improve cf suggest bin distribution improve network dependency arrive link simulate use trace collect internet arrival behavior trace web header trace ten link internet differ bandwidth traffic hour trace different simulate network tool trace delay delay whereas delay link due bandwidth average delay symmetric leaf tree tree core trace trace edge vary along across link
max message edge factor thus real number simplify I w j j I k I edge belief b ij define find first belief product max output computation interpretation depth computation root recursively child leave initial full product every update tree time alternatively max execute message time four computation full root bold copy copy match maximal course say possibly depth ready main result product lp lp tight program value say max converge belief node asynchronous converge say product answer convergence uniqueness recognize optima characterize hard unique program lp relaxation answer lp tight say uniqueness graph tight edge satisfie match convergent suppose matching copy maximal matching alternate path path alternate copy lemma suppose eq leaf last inequality mean mm tt contradiction establish relaxation loose correct answer loose max product correspond match optimality possible max product incorrect either show correct loose step combinatorial characterization lp loose node matching exist edge match odd arbitrary base matching bad satisfy odd cycle bad bold edge last base h end pair one provide characterization loose relaxation loose bad bad match bad subgraph converge answer max converge look neighborhood step full depth matching root copy root result loose belief converge belief incorrect converge max matching lp loose prop bad bad suppose maximal path remain edge membership match easy bad max converge projection root start copy alternate form alternate path respect end thus path gain augmentation match contradict product incorrect see max converge walk live walk map path strictly lead general similar paper lp outline direct operational link program bipartite operational algorithm form message update message vice versa idea connection programming author acknowledge max non graph responsible pointing author loose bad optimal fractional augmentation cycle alternate path augmentation every alternate end match path exist increase exceed contradict augmentation beyond one follow leave loose decrease increase remain leaf leaf extend edge loose argument lp loose exist remain increase decrease new optimal strictly thus pair bad successfully variety relatively convergence correctness provide exact paper use maximum arbitrary edge whose mode correspond optimal run max relaxation tight correct relaxation loose converge provide dependent characterization precise connection tight graph product bipartite message pass belief propagation variant generalization solve instance intensive problem field originally calculation max marginal structure probability application involve update graph however correctness general characterize pass area correctness graph find mean tight bipartite bipartite ensure lp lp edge dual problem minimize
confirm calculation processor server collection independent free example demonstrate software axiom theorem definition exercise arise population population formula population still population complexity multiple able arise population imply statement provide extend computable occur hypothesis appear book provide thorough function fast variable compute joint function formula addition grow approximate algorithm complexity show population exponential improvement joint complexity sign column exactly determinant cost evaluating algorithm theorem explicit sort joint write denote index remain k summation loop loop implementation complete set order generic value sample member cumulative distribution function index block block row subscript take k expand group repetition sake ground define denote k jx ji kn independent compare simplify becomes identically carry arbitrary population omit etc cover population exactly population theorem section element take number term induction assumption identity twice number call growth distribution function eq require term compute bound computation exponential fortunately possible case population fact complexity joint statistic random variable bound polynomial term index evaluate product sum bound indistinguishable bin give twice inequality function exponential polynomial population consider depend number population general fix formula improvement fig confirm illustrate result take fig theorem implement
brownian bridge write chebyshev inequality ny derive possible obtain argument convergence change particularly fast get first consistency q iterate imply additional able use framework omit consistent pr pr
make relate stock financial observation price asset observe price asset convenient representation explanatory column analogously log might suffer moreover trading application trading signal update quickly acquire much explanatory stream remarkable speed address extraction dimensionality explanatory stream explanatory stream extract principal explanatory stream effectively incremental brief outline procedure suggest sequel first call characteristic eq incremental observe set q influence old computation find stable dominant eigenvector exclude experimental term give contract value stock allow trade price contract daily ratio contract call explanatory stream either stream method explanatory asset
use dependent stationary ann product field tail inference theory additive process within hill tail j dependent heavy tailed ann tail order behaviour extreme iii publication hill autoregressive en tail sequence manuscript en weak tail en tail quantile strongly stationary department north en h tail array sum stationary ann functional l equation infinitely yu index functional cm aspect extreme statistic serial time series emphasis important contribution tail second dependence recurrence primary secondary key extreme index condition deviation publication ph thesis one drive force development extreme statistic decade
formalism purpose abstract problem measurement fuzzy inspire analyse array cluster accord specificity quantum dna microarray logic sequence genome analytic biology biology among induce differential lead functional specificity coherent cell however yet explanatory matter establish precise formulate novel method term dna degree specificity list order specificity context improve diagnosis tailor etc idea exist algorithm various situation think worth
section pair correlate due section datum multivariate inclusion affect characteristic price together denote instantaneous sde q motion drift term dispersion xx unique definite diffusion entirely diffusion term exact ease exposition initially partial regime necessary relaxed denote inference trivial receive remarkable literature review problem generally available except approximation refinement technique succeed inference monte bayesian carlo augmentation treat suitably note theoretically algorithm initial implementation degenerate increase scalar principle apply mcmc scheme issue generally
relaxation condition complement equivalent become I ix eq eigenvector show semidefinite solve semidefinite feasibility rank one semidefinite nonconvex provide combinatorial optimality condition result section rather lead refined fully denote eigenvector eq I ia necessary condition xx precisely thus expansion local maximum equivalent optimal dual technique check consequence conclude highly degenerate optimality specific b xy still require program candidate trace feasibility proposition solve eq give tx mean problem calculation tx
threshold define denoting represent arbitrarily use put rely peak mean conditional exceed fig agreement point contrast datum removal outlier reject peak nothing addition peak reject correspond due light periodic al sect compose allow sect instance various characteristic employ compose star frequency consistently detect peak produce likely run correspondingly decrease increase reference dataset thus split grid ten fine resolution bin consecutive bin spectra peak bin contribute peak bin peak compose background observe signal influence analyze individually spectra induce
ability inverse lasso lasso obtain way either lasso quantity note choose formula follow size matrix run trial number estimate correctly identify nonzero set true set plot correctly predictive correct yield sparsity term predictive offer lowest tie seem conservative lasso achieve choice lasso pattern ht parallelization attractive estimate sparsity pattern test algorithm three set voting record graphical towards obtaining replace result
minus deviation htp partial time method dominate percentage test gain limitation become gb ram also compute e perform eigenvalue numerical cost multiplication computational algorithm mean satisfy prove convergence duality vary number eigenvalue require eigenvalue dimension vary degree ie note
tool pass automatically additionally lot penalty regularity applicable dependent conjecture support likelihood process could test determine describe closed alternative describe statistic definition main study behaviour statistic section investigate statistic section direct application concern introduce notion hypothesis measurable suppose suppose family arbitrary e correspondingly identically infinite allow stochastic impose would moment observable condition statistic score depend test score score construction truncate partial likelihood theory generalize score test estimation generalize
calculate clearly l il theory establish support foundation grant later work convention section theorem cm cm technical school sciences uk statistics ng rd uk ac uk emission motivation wherein replace training fast less need viterbi adjust viterbi viterbi accurate estimator elsewhere viterbi viterbi training model estimation viterbi procedure estimate markov markov state transition emission parametrization independent emission forward computationally seek alternative alternative use recognition bioinformatic motivate constrain quantization replace expectation maximization simple speech recognition describe quantization viterbi especially distribution describe al context name constrain quantization emission hide follow maximize viterbi every viterbi alignment subsample regard find converge finitely em significant inconsistent ascent objective despite speech significant degradation phenomenon curse speech speech regard merely special em
temperature secondly describe use hausdorff show map dissimilarity htbp compare metric square distance vertex symbolic compare different individual
suggest verify correct evolution preserve tp claim assume preserve expectation update degree freedom wishart suggest beta integer integer resolve beta comparison via closed adopt order theorem possibility choose maximize likelihood log find model bayesian inference compare root nuisance discount possibility
inaccurate context visit least non eq I large possibility vanish fraction us constant hoeffding combining follow stationary theorem active arbitrary optimal transition generality depend fix time k k tx couple accord stationary match otherwise evolve evolve independently accord transition average policy therefore pick optimal tn follow sufficiently discount average presentation give assumption guarantee discount optimal hoc sufficiently yield nonetheless trick conjunction attain optimality subroutine non negative sequence sufficiently slowly cost optimality rigorous would argument sketch epoch remain th epoch less step borel may finite subsequent epoch time step suffice provide cost discussion
extend alternative parametric model distribution alternative cumulative
fy article defer mixed order law remainder expansion main base expansion haar decomposition proxy suffice stein normal dependent see stein shall stein limit latter well article add asymptotic condition seem indicator denote either depend expansion mutually permutation randomized orthogonal array j page hypercube j square haar analysis integer q zero k page rx j c equality change set lebesgue j j u mutually u u u integer denote integer write constant repeatedly sequel similar manner brevity sequel j denote cardinality end lipschitz la j j u j u l j j u e j u u e u u j u
report brevity setup carlo scenario ii results setup legend description graphic consideration figure consideration component quite large first generalize e iv setup summarize choose scad tune clearly estimator monte iv cc figure setup result scad properties vi identical setup I precisely fan li result condition fan result finding gain square factor considerable range study fan surprisingly bad boundedness bad post
dt db f state pointwise exist c choose arbitrarily exist satisfie k ng k l h h h h n l n k h I h h h z de et universit paris du paris france de universit paris paris france bandwidth law censor c estimator density hazard main law yield simultaneous band example band obtain
htbp corollary remark new conservative classical chebyshev fundamental relationship
least broad outline propose law law great reject hypothesis zero power statistically principled fully validation power brief quantity power basic quantity interested continuous describe density normalization hold power law straightforward normalizing find paper integer calculate normalize q table useful continuous continuous discrete continuous approximate behavior counterpart law result reliable integer power law approach many round probability generation integer continuous poor law follow good scaling observational appendix cc continuous law cutoff discrete law discrete turn correct fitting study empirical law give scale often task histogram logarithm side law obey imply straight doubly way probe law therefore plot histogram axis approximately fall bold distribution scaling give straight line logarithm histogram pareto variation generate systematic error power distribution equally really follow law parameter estimate let know unknown fitting provably accurate parameter estimate size distribution derive scale derivation appendix use elsewhere symbol estimate symbol equation hill positive cutoff distribution case discrete ref recently respect us maximization function simple discrete take justify large example evaluate reasonably although discrete mention true power distribute integer detail result give systematic employ author fact calculate approach easier implement reason circumstance decay fast becomes compare
long wavelet specify q generalize wavelet wavelet fix decay fraction eq imply residual provide residual paragraph take decay time mean non analytic assume everywhere vanish wavelet eq analytic anti portion former eq derivative write substitute obtain infinite thus arise taylor signal involve form frequency wavelet appendix arise decay comment precede view fundamental frequency wavelet certain quantity instantaneous frequency derivative wavelet role frequency set transform explicitly permit wavelet derivative assume instantaneous hierarchy much broad variety behavior generalization everything term assume wavelet envelope multiply exponential et al representation examine instantaneous frequency quantity ridge set ridge localize analytic signal evaluate along instantaneous frequency curve wavelet representation immediately recall residual transform evaluate instantaneous curve fact localize analytic signal filter wavelet sequence local projection signal rescale wavelet proportional instantaneous period localize signal analyze wavelet instantaneous analyze clear localize time frequency since merely function analytic localization fourier note multiplication become fourier product support energy manner wavelet localize analytic stability match wavelet localize analytic instantaneous order localize analytic interaction invoke introduce bias localize characterized truncation true analytic truncation level
constraint updating method I method traditionally process either compatibility I update different resemble use effort name piece call moment ill behave detail solution produce illustrate
smoothed analogue estimating form moment order emphasize use realization regular skeleton near point link point calculate directly include mean volume cell aspect realization sparse non iff matrix characteristic characteristic triangle count adjust regular composite material material electrical yield material context statistic use make different list
modify broad organization dissimilarity som l compute compute article simple epoch prototype therefore som mention epoch note point induce certain type dissimilarity solution use build affinity breaking instance bad complexity paper provide mostly partial scheme fill epoch follow phase case complexity good solution observation major drawback formula optimization equation force use use consist
posterior although bayes capable enforce brief process datum canonical distribution simultaneous updating bayes modify exponential although handle information constraint must question process sequentially correspond lead inference multinomial solve two problem appear fact process sequentially familiar derive
pm pp row concatenation denote denote column similarly consist write abuse notation respectively always contain interpret pp interpret relative sequence parameter model stress distinguish nothing else select select list lead model hypothesis procedure aic decrease start process reject rejection model choice formally ensure remark statistic convention root element hypothesis statistic distribute freedom conservative selecting probability contrast limit example estimator define event least denote event summarize subsequent rank consider cdf several cover cdf ap pi distribution selection access need expression equal restrict square np restrict center cdf gaussian variance k lebesgue depend dependency explicitly basis generalize invariant choice inverse column also precisely furthermore uncorrelated pz univariate gaussian equal indicator present explicit formula display normal square distribute degree post figure exception normal uncorrelated b pg n pp far necessary partition eq cdf variate variance pp pe pp pz next cdf alternative take variation supremum total distance uniformly
perturbation perturbation completeness background read refer section angle also define principal angle subspace matrix column orthonormal value angle angle angle diagonal matrix norm respectively perturb set eigenvalue contain quantity unnormalize work analogously laplacian ideal connect perturb long connect perturb laplacian choose eigenvalue first contain easy perturbation perturb distance eigenvector ideal piecewise perturb perturbation coincide spectral gap close perturb also different namely cluster statement weak eigenvector become bit argument justify base property eigenvector zero outside individual block argument similarity adjacency discover ideal separate eigenvector sure eigenvalue eigenvector case know connect possesse exactly hence component eigenvector laplacian know eigenvector matrix similarity come block unless take reason second bound connect ideal fact cluster point belong case situation perturb usually tell original unclear interpret situation either indicate belong think class classify point normalize ideal eigenvector lot low small ideal normalization vector multiply run describe cluster even belong
process specifically determine diffusion perfect implication mcmc translate scalar confirm experiment deterministic analogue problematic augment exceed amount diffusion long provide stochastic transformation paper target diffusion appeal volatility ease illustration provide time relevant nevertheless volatility sde generality possible every successive link spirit parameter free dominate theorem get diffusion wiener denote lebesgue dominating depend reflect brownian bridge introduce
degree simulator location grid relatively equally two successively fine grid primarily generally sound barrier simulator completely regime close show interpolation simulator lift surface angle attack angle ridge appear angle attack transition sharp surface part around modeling process work yet produce feasible feature appear corner speed spike look place believe false simulator surrogate contrast modeling want deterministic simulator smoothing simulator half surface attack angle region instability thus across level simulator slice one surface instead clean ridge level angle attack simulator really likely want concern deviation high attack deviation numerical simulator priori become appropriate single uncertainty uniform stationarity
formula un di vi kt minimax new york mr origin evy economics nonparametric homogeneity process increment appear van kernel bernoulli van university r canonical infinitely convergence characteristic mm pt thm criterion thm remark evy process universit mail cm universit mail evy minimum evy characteristic tend infinity keep fix rate deconvolution problem characteristic subject secondary keyword phrase evy characteristic deconvolution title evy evy process fundamental building trend evy finance biology evy problem evy discrete
sampler geometrically respectively ergodic geometrically ergodic geometrically appropriately large generalization find geometrically ergodic converge rapidly precise rapidly stability valid page scalar symmetric everywhere include cauchy double gibbs deal dependent defer concept interpretation stability show partially tight geometrically probability exponentially real index scalar strong concept say rip robust parameter q accord guess conversely influence rip model everywhere linear everywhere positive geometrically rip ergodic
collect frequency properly component frequency amplitude phase instantaneous become permutation across frequency step sort consistent order domain processing propose require sort envelope average number beyond stationarity envelope component compute measure separate measure sim involve ensemble drop similarity sort denote sort frequency frequency repeat sort use segment entire compute delay processing correlation un lag step degree
one actual trajectory evidence well gain efficiency empirical sample produce plot averaging time great particle indicate quality generate significant due separation filter stage filter create separation large encouraging separation increase minor particle slightly multiscale law evolution system generate trajectory laboratory particle also apply importance multiscale markov grateful like dr anonymous helpful suggestion manuscript support national dms office contract de sf filter construction exhibit separation two principle particle
summary contain historical capturing predictive intuitively want vice versa write formally constrain solve control complexity represent hoc justification greatly successful solution parametrize multipli assignment distribution energy effectively different true distinct gain probabilistic future substantially self iteratively mechanic pseudo control randomness ref guide intuition ready assignment become write assume assignment property connect solution causal within mechanic transform causal call assignment x subspace maximize objective function also prop recover state find causal partition temperature therefore recall state
fast metropolis decompose eq appendix hence suitable fast every slowly mix let every q call ise particle spin probably study spin usual metropolis use whenever eigenvalue derive yield fast aim avoid symmetric order q union even irreducible chain shall realization respectively note
cc compression new old bag mm cc cc diagram generalize net diagram represent diagram whose summary summary also general exchangeability summary step use full kernel compression model initially term kernel usually easy compression successively update word early kernel one step drawing produce kernel relation compression large different summarize write step alone significance na n see reduce give recall model fraction argument readily event obvious way rare equal probability successively rate less exact proposition suppose compression law apply happen feature generalize level decide na n validity old example exchangeability line model model exchangeability label linear already exchangeability assumption hold hold hold closely prediction line add article right exchangeability way label exchangeable appear open possibility probability th consist list nz easily probabilitie bag position equal update kernel summary update kernel step exchangeability object exchangeability label rate exchangeability particularly digit panel exchangeability near neighbor stay uncertain error exchangeability track exchangeability much exchangeability label keep prediction region contain digit uncertain exchangeability overlap exchangeability
belong interval guarantee prescribe margin prescribe contribution answer develop absolute sample method margin computing criterion notation represent represent integer
easily compress therefore small compressed may possible simulate huge reduce compression high interaction feature variable explanatory variable human disease interaction gene environmental character english text precede character many interaction introduce indicator interaction pattern occur measurement variable distinguish interaction moderate prohibitive real primarily say omit high predictor computational unless training signal case prior bayesian express prior belief parsimonious approximate interaction irrelevant response assign distribution appropriate interaction additionally gaussian favor coefficient interaction incorporate huge bayesian markov long iteration require converge
value whereby uv interpret modal orientation interpret mode one generate mf envelope maximize uniform procedure extremely much approximation von close von fisher orthogonal column consider sample orthonormal basis calculation density express matrix mf bind ratio rejection may generally orthogonal make ratio
assumption violate consistency generalize I relaxation machine learn pac investigate regularize process namely necessarily kernel estimator process exponentially decay mix coefficient consistent step ahead prediction polynomially decay result refer relaxation common knowledge deal reason literature fact identically observation define resemble empirical law show law large number reasonable limit law law definition recall algorithm process law svms establish consistency sequence consistent whenever generate type introduce law number subsection important notion risk consistency number finally svms mainly law general necessarily concept present already elsewhere cover part section notion give function set measurable small algebra probability measurable map consequently furthermore distribute wide probability I know converse implication later interested generating call introduce process say weak constant strong law almost obvious converse implication moreover must
likelihood viterbi maximization core extraction estimate typically differ path run obvious propose hybrid computational important viterbi principle viterbi positively general hmms infinite viterbi generalizing extend piecewise separate detailed formal hmms rather long technical hmms existence viterbi need special result complete generalize lebesgue paper hmm emission also positivity natural
gibbs multiply divide bind log define eqn energy variational mechanic next q subsequently take summarize provably approximation evidence posterior expect constant chemical potential spin normalize iii iv optimize
replicate arithmetic state normal sample root mean numerical accuracy arise quasi mc rely deterministic discrepancy hypercube star measure uniformity df orthogonal f variance subset enjoy nice line definition follow dimension superposition sense effective truncation small might superposition computation path generation minimize sense problem accurate many pricing attention payoff pca multi dimensional dimensional good linear investigate lt high simulation european option pricing cite random start lt cholesky generate optimally choose maximize explanatory variability consist find orthogonal iteratively optimization subject
multiple change multi jump parameter consistent change rate compound large keyword random intrinsic economic physical vast amount specifically design break maximum likelihood least estimator wide statistical influence continuity character variable exhaustive paper area impossible recent paper ls refer
simple white robust central play confidence specify sense within region regularization smoothness asymptotic often express li van emphasis ball concept shape monotonicity confidence shape justification yy limit hold coverage exceed finally universal construct point interval generate restriction universal exact confidence mention appropriate observation belong evaluation inequality sense nothing consequence shape
von sufficient convergence min rather max condition express slope contrary conventional theory separate proposition illustrate exponential truncation domain uniform power asymptotic line outside extreme survival function show slope behave near let extreme dispersion yield fr convergence worth pp extensively next survival application like apply monotone suitable censor hazard long hazard monotone slope exponential exponential variance distribution transformation left removal left truncation right censor maintain give slope extreme characterize slope slope positive unit slope calculate slope side obtain continuity
la notion form formula form formula domain symbol atomic formula atomic formula formula form application valid type tv quantify sn sn v sn sn quantify ground formula satisfied usual logic free occurrence term hold iff tuple iff iff query variable set tuple make formally write constant constant table involve query formula sn sn sn v restrict set formula serve result tuple bound independent adopt safe behind safe database query key idea safe query restriction form database expressive safe query formally replace connect consider consist conjunction formula must limit comparison query safe f sn sn complete come restriction query basic
ascent coordinate coordinate form close discrete parameter control preference population agent attribute consist agent choice event set choice simulate item agent low evenly spaced value heterogeneity heterogeneity collection vb scenario soon step step change relative mean criterion vb instead change variational approximation namely mcmc set burn technique default setting lead large burn iteration trace sample indicate burn unnecessary manually investigate autocorrelation datum partial autocorrelation iteration burn small reasonably fair inference informally forecast item unobserved decision maker
realization brownian importance normalize particle kalman article process stochastic differential euler euler differential equation restrict law drive brownian kind common mathematical treatment differential find mathematical finance similar sir continuous embed kind part plain integral operator outer ratio stochastic importance kind process process consider function brownian invertible brownian joint brownian turn eq form importance importance sir start measurement sir u sequential resample simulate realization
spirit goal consistency normality first strict periodic stationarity necessary positive consistency normality irrespective requirement rest give necessary stationarity process consequence find appendix stand almost sure order e modulus square transpose element study period identically periodic
consistently critical region natural censor seven statistically statistically relevant local maxima statistic randomly regard nine could possible datum strategy censor censor obvious randomly censor cell mechanism whenever run relevant scenario censor mechanism data strongly censor
asymptotic asymptotic enough event desirable formula construction construct limit n pearson limit compute pearson need preliminary lemma course
click click versus describe density describe relationship analyze predictor heart event brain potential click e conditioning amplitude extensively examine potential pair stimulus conditioning paradigm excellent detail condition p simplicity case change investigate relative contribution cell response ratio light nr two ratio variance sometimes estimate population calculation light record dark slice transfer nr reaction contribution na ratio change post na day relax day pre rt rt represent volume pressure well disease currently experimental clinical ratio flow local pressure extraction fraction five level line minor five group state present force load force examine force hold load compare load force object hold low lf frequency band heart period lf transform stress subject heart variability associate physical affect device hour analyze band lf cells cd specify examine gender difference stress record passive stress versus rest condition week week free week week week week pre stress stress post baseline speech p
interval show coverage unimodal rigorous sec bernoulli many confidence notation drop assume throughout paper reader distinguish notation conditional clearly construction quantity bad case purpose suppose monotone discrete lk uk b like emphasis assumption confidence interval generalize version respect purpose infimum cb l c l either increase respect interval interval without restrict interval specialized sec poisson let independent frequent interval l drop argument bad case coverage
x x x sx I xy homogeneous ratio degree well line function first polynomial every represent cone design design simple lattice design polynomial furthermore algebraic design polynomial fs ds fs fs pf f low exist way ready representation design gr respect gr gr gr admit loose generate I generator homogeneous polynomial degree generator also polynomial
q enough sn sn sn gp sn h unimodal ready suffice sn h sn iv decrease vi sn sn g n n h h p p see case true monotonically monotonically number monotonically lemma vi let consecutive element distinct element cp drop c c immediately
strictly everywhere distortion denote shall straightforward extension argument encoder decoder satisfy entropy bound letter nr nr n iy operating rate risk minimum emphasize role encoder learner loss concavity follow eq x n f f suppose expectation concavity jensen continuity
cost balance wish black present underlie asset price neutral rate asset volatility wiener option asset payoff arithmetic price option simulate thus q author importance vector density law gx dx indicate payoff density define aim standard show gx indeed detail inverse law way law proportional
close available belong biology however probabilistic specification solve close resort three strategy monte maximization em chain gibbs metropolis joint likelihood intractable complete likelihood joint block conditional sample instance conditional without hasting compute desire proposal accept reject use depend particle
ce mod le en est mod du est de du instant des plus un de en une les et dans la est la en pour est p un il pour les pour la le p en ce une de la possible pr il pr de pr pour la r de de par mod du la date h les r de la du en mod mod le lin de surface du mod il la pr cat du le compare les cat en surface r de cat les le tend le ce est la mod est cat du de le mod le un pr pour cat les spatio les les es pr les de des tr mod le lin g en de mod le lin r par cat mod les mod pr des spatio dans ce des des mod la mod de l une la le une le mod par pr dans ce
reveal actual distribution analysis asymptotic capture view favorable suggest result confidence estimator necessarily partially consider sparse sequence live size sequence contiguous datum example certainly satisfied experiment presentation essential measurable estimator sequence respectively find focus furthermore estimator however sensible typically consistent estimator follow consider linear standard simplicity
adjoint lie entry term involve entry look cca p ideal reveal consider format format fix module module irreducible module nine three slice denominator rational one numerator homogeneous nine remain homogeneous nine generate module appearance algebraic statistic conceptual play algebraic leave branching show root construct generator nine therefore build arbitrary
set model briefly distribution censor frequently time person certain cause quality failure object survey ask fall absolutely al latter censor unimodal log concavity treat censor aforementioned building consider consistent estimator information relate
compare obtain analyze advantage description study stand es drift recovery field forest none area explain much slow less change contrary consider cover quantitative qualitative area big know categorical cover slow change opinion categorical choice use make c several environmental slope categorical management environmental environmental environmental presentation cover precise consist
behavior across solution require dimension simultaneously name submatrix lie partition optimal change affect row column cluster twice cluster vector clustering original matrix adjacent np directly
image recover ensemble confirm ensemble problem positive adequate state volume average limit answer probably calculate temperature respect boltzmann obtain
diffusion appendix result particle particle proposal particle brownian simulation reject iii algorithm comparison simulated particle weight comparative cost four acceptance diffusion stay spend time region respectively cpu cost filter distribution one comparative cost filter line green red blue algorithm affect measurement similar decrease increase performance robust implement pf use every th th filter different scenario ess ess calculate mean run particle run ess filter ess compare give ess drop inefficient performance diffusion monte try particle bridge numerically filter ignore see ess suggest monte variability small poor try integer contribution
simulation carlo density take mode mixture weakly secondary prominent gap mode unimodal monte multimodal add switching jump particularly important mode area inspection propose change label choose move exchange accept proposal label hand proposal reject attempt allocate move complementary move integrate definition address metropolis directly posterior distribution general whose measure formulate simulate random probability independent stick break rule although general incorporate allocation scheme q provide appropriately construct generally tackle almost application correspond case lk lk construction mathematically report elsewhere
journal email uk number value normal prior keyword likelihood markov normal distribution tool parametric conceptual parametric flexibility one bayesian mixture new computation posterior exploit specific hyperparameter finite mixture normal galaxy datum illustrative remainder brief introduction mixture representation marginal normal deal section determination normal conjugate variance distribution underlie
evolutionary example apply bag principle algorithm choose ensemble vote probability bootstrappe create ensemble thing aic section evolution well mathematical simple evolution e incorrect evolution search true actually ensemble incorrect search criterion learn kernel many hyperplane nonlinear algorithm e framework practice carefully fundamental tune single careful usually fine amount fine ensemble easy kernel argue expert tend prefer easy expert tend prefer flexibility hence researcher advance methodology fast target evolution ensemble list sign hyperplane equivalence straight forward
random degree vertex eigenfunction embed embed use indicate good algorithm construction embed pt number construct eigenfunction ki fig minimizing inaccurate describe face velocity response face dimension eventually drop describe replace voxel appropriately coordinate formulate choice need nonlinear embedding refer
justify relax certain model lagrangian redundancy good length fidelity converge collection marginal source key universal learn probabilistic moreover clearly rich turn affect performance source learn past encode block digital control constraint ergodic source alphabet space equip adopt apart
whole derivative careful possibility approximately optimum setup package base package problematic ep form model fail use use thin fail fail purpose derivative substantially failure occur whether help exact note impact difficulty discuss variance reliably presence ill correction sort might counter cause direct approach drive free convergence failure problematic eliminate method simply glm without need way practical reliable estimation aic type smoothness optimum criterion newton utilize derivative deal degeneracy severe heavy modelling situation method meet consideration give regard none difficult achieve allow count cubic task produce answer hard detect convergence become simulation reduce variability increase burn serious problem computation art use spline feasibility matrix choice spline cube sparse multiply covariance multiply use inference try difficulty kind ensure
ni nh dm quantification much involved let x ix om om n k k n ik k ni ik easy check lemma dm h cr indeed know nh nh nk nh k suppose minimum np eigenvalue last note c nh h equivalent nh ni nk write nj ik ix ik c ik ik nh nu follow lie lipschitz iterative disjoint continuity note ni ni n I ph nj p nr ni nh nh dm
rank simultaneously instead perform regression outline proceed estimation parsimonious aid yield lar intercept nonzero parsimonious section alternatively parsimonious regardless one rule asset involve behind derive formalize excess model independent residual instance index book factor therefore estimate square observe main mutually considerably parsimonious result encode result suffer assumption adequate integrate factor framework full term family apply minor asset become usual give asset indicate importantly parsimonious regression ol onto set regression zero otherwise observe possible historical instead factor method classical mix generalize optimal shrinkage standard portion available optimal factor estimator completely observe joint return distinct publish success combine estimate traditional estimator involve combine also historical regression shrink possibly definite towards apply advantage finally r package make
must count walk relevance walk length adjacent definition edge clearly walk graph walk edge walk moreover sequence alternate label label extract walk label count word equal coordinate formally approach linear require explicit dataset problematic keep atom type label reach walk walks walk suggest walk approach useful solution hash limit call molecular obvious drawback solution indeed indeed inner us inner eq inner pair introduce vertex connect pair word conversely walk pair pair walk walk pair walk length label counting walk turn count walk easily number walk vertex follow therefore number walk sum observe adjacency inner represent compute I adjacency product reach exponentially pattern ever store vector many flexibility review walk
distributional study organize probability consistency uniform derive whereas asymptotic section concern finite sample convergence orthogonal normal variance orthogonal square result component mutually least square location simple sequel without generality apart likelihood parameter thresholding view square hard thresholding arise penalize coincide fu tune latter fan context small large piecewise alternatively penalize square li scad induce restrict hard furthermore else choose estimator likelihood hypothesis probability I select suffice probability coincide stand cdf generic impose consideration incorrectly restrict vanishe vanish estimator seem interest shall basic hard thresholding soft thresholding hold case case thresholde soft scad act case ii act vanish inspection fact precede paragraph asymptotic nature infinity pointwise essential aspect especially finite next present vary convergence neighborhood let conservative suppose parameter corresponding n part immediate second
interaction govern exist membership appropriate latent vector relational relationship like ensemble membership object modern mixed membership model mix relational fast demonstrate application scale protein interaction role exhibit interaction role observe mixed membership reliably section application student block test dimensionality protein model membership observe datum value think among repeatedly like trust asymmetric four asymmetric relation collapse binary direct analyze determine social latent actor membership strength draw belong degree strength bernoulli represent link group denote denote node belong put everything blockmodel n draw indicator z membership indicator receiver rp node context interact interaction q group membership interaction asymmetric interaction factor actor generate useful analyze relational measurement positive influence collection datum example interaction affinity return probabilistic thus
b b mb riemannian euclidean hessian derive geodesic bb f skew b h bb exponential skew let svd show purely k tx tx td td mean kronecker product permutation gradient cf repeatedly I I I eq simplify notation drop subscript thus use f bb bb bt tr tr bb gradient bb x bb bt tr tr appendix c expand get inverse together b follow hessian inverse linearity substitute ignore brief last definition much argument also combine q derivative follow q drop treat latter also use noise mse sd reduction mse sd mse sd mse sd mse sd reduction sd sd sd sd sd sd mse sd sd mse sd sd mse sd reduction mse sd mse sd mse mse sd sd mse sd sd mse sd sd noise sd reduction mse sd sd mse sd reduction sd mse sd reduction sd mse sd sd mse mse mse mse sd mse sd sd reduction sd sd mse sd sd sd
shrinkage guarantee improve obtain expression minimax estimator close eigenvalue minimax value seek l appear computational complexity estimator major calculation dominate l ls note substituting l condition condition situation much high l admissible modification stein whenever stein technique dominate stein technique resemble modification since shrinkage dd dominate diagonal matrix element pt n bi I option result hence mse never must negative end choose since complete prove part suffice
behind magnitude take table result drastically neutral go would set bt capital coefficient bt
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observe real univariate real solution tend prove theory algebraic generalizing result almost surely fisher sample calculation likelihood solve calculate corresponding value formula sequence point implie saddle local multimodal recently model necessarily information initial equation comprise inspection reveal completely eliminate system
mu xy rx rx n mu nr r tr r hence satisfied mu mu mu mu mu non constructive constructive dominate nm define random contain infinitely vanish mu n n nm mx mu universal mu plus mu mu mu mu ex n mu mu mu mu mu plus mu plus mu mu plus plus mu plus mu small constructive contamination converse exist give predictive alphabet universal measure prediction construct sense dominate unlike normalize whose prediction prove intermediate computable definition mixture immediately proposition I p plus plus mu plus
definition complete statement case virtue eq yield complete monotone decrease monotone q q light contradict upper q making z n preliminary lem variable k p tt ss definition follow k p p p p p lemma p combine tail property q finally p cp p cp
number signal derive signal realistic frequentist interval bayesian yield estimate argue root comprehensive presentation particle physics essentially recent therein limit observation channel channel particle suppose positive device frequentist subject property decide energy physic hold might arise large participant attempt limit underlying channel challenge realization poisson know unknown formulation
derive loop propagation variable use average covariance propagation cavity perturb nonlinear pass approximate scaling polynomially discovery year error vision probably prominent imply belief negligible necessary treatment loop motivate research apply involve regular ep latter correct loop framework strategy basic underlie recent analysis show order cavity cavity neighbor central remove
classical quantum must additionally query additional query query quantum classical goal quantum call another future explore query theorem definition article quantum test unknown efficient algorithm whose domain function example quantum quantum possibly many classical relative quantum subroutine enable subroutine early accuracy quantum example classical bind quantum relate low field algorithm prominent membership query black box membership query oracle receive value query receive model consider year quantum variant e model membership
every eq structure degree set symmetric semidefinite laplacian connect component connect graph connectivity normalize eigenvalue normalize one concept sensor failure drop online fail transmission successful one topology paragraph channel may directly course message path topology network link failure fail bernoulli failure assume bernoulli statistically independent model describe edge identically iid zero stand tuple collect edge formation probability edge formation element loop structure row notation refer topology consensus compute average sensor node neighbor edge vector one
label vertex follow take logarithm enough proof reconstruct markov vertex every reconstruct size theorem reconstruct network differ non degeneracy degeneracy fast run graphical model correct accurately calculate empirical measure collection choose remainder neighborhood imply property conversely equation neighborhood correctly determine straightforward neighborhood check correlation wide spin marginal would incorrectly empty
locality limitation apply speech ms duration length restrict difficult sub test example aa training split parameter set use report represent decide hierarchical penalty functional svm svm appear apply wavelet coefficient mean evaluate optimal previous additional really spectrum feed spectra consist nm classification separate content htbp maxima decide spectra I figure difference derivative spectra spectra compare kernel use procedure subspace calculate split sample sample spectra validation subspace obtain leave procedure repeat splitting fast give error test mean test linear linear derivative
intensity however behave similarly simple proposition operate set em let follow comes treat nuisance latent version regression expert framework consist parametric pdfs expectation specify thus algorithm directly set algorithm straightforwardly extend consider expectation form instead notational straightforward case covariate set likelihood somewhat broad specify require twice define continuously compact step stochastic call variable field solve preliminary root mean belong kullback leibler divergence root function define existence
large contribution computation evaluation coverage regardless demonstrate integer organize technique developed take absolute method conclusion throughout shall less large great integer notation desirable minimum interval whether
cauchy formula area may rewrite density introduce lemma lemma step shorter uniform lemma correspondence integration describe distribution isotropic randomness direction sometimes interior line uniform multiplier precise randomness yet another pair inside randomness normalize multiplier precise body q distribution necessity dp lp dl produce equality eq normalize part derivation case preferable nontrivial volume express agreement q body lose generality hull zero complement consider case choice analogue
systematically sense complexity amount importantly appeal frame limited capacity rate principled compression minimal fidelity general map deterministic state induce measure complexity turn h equal h illustrate assignment p indistinguishable x x due past reflect vanishe reflect code away toward less controlling code model distinguish however theory
product action example associate bundle sphere use describe description bundle necessary space equivalence relation embed rotation axis plane act rotation origin quite structure invariant early four property application five dimensional satisfying discuss
concentrate ambient temperature need adjust additive include position efficient movement spatial correction transformation polynomial alignment step preprocesse move intermediate essence combination ensemble filter intermediate additive position single transform consist position component analysis convert advanced article assimilation enkf formulation enkf detail refer literature automatic formulation enkf numerical conclusion extension purpose assimilation discrete consider work call modified accounting bayes forecast datum datum datum find assume
link complexity apply estimation acknowledgment discussion conference main appear counter identifiable aa ia check condition characteristic ct group leaf tree correspond link delay open delay shape identifiable even plot function proposition mutually measurement represent topology consideration often ill paper study statistical address identifiability identifiable mild characteristic extensive trace favorable infer delay heterogeneous identifiability characteristic mixture monitor diagnosis decentralize nature service collect link statistic tool whereas end unfortunately global view internet degradation internet cause several system service internet may end characteristic directly network address issue form
upon asymptotic correctness decode paper arbitrary edge formulate lp relaxation lp relaxation lp loose max product show converge answer recently bipartite graph tight well weight sufficient instance characterization decide lp relaxation well broad algorithm similarity comparison program interesting investigate channel code lp match set negative total weight ip lp constraint
distribute classical theorem reduce distribution sample population indicate eq expand special joint order range random interval term list illustration order type permutation multinomial element
constant velocity govern underlie mean square consistency convergence obtain volatility regime switch process describe motion velocity alternative define particle initially locate move alternatively velocity change govern homogeneous poisson process q
unable precisely assume violate illustrate minimal assumption penalize ol scalar determine way point fashion sequel small cost recursively scalar eq start identity apply available sequentially subsequently rely dynamic define act smoothness vary vector away prefer alternative parameterization control scalar vary close total weight static away dynamic note point situation sequentially dimensional stream recently acquire give eq initially start vary setting stream example generate use explanatory stream evolve complex flexibility gaussian deviation distribute example feature non error coefficient encourage jump dynamic rely quite coefficient
deviation usually tail tail restrictive quite nonlinear nice example arise observation estimator dependence optimistic start author grateful estimate ph calculus remainder expansion stop topic nd ed heavy I move vary probability l extreme estimation ann weight approximation tail mix ann empirical bernoulli extreme tail portfolio l correction mathematics de de solution application wind regular seminal contribution extreme univariate also influence value contribution second principle aspect de aim great generality article univariate generalize extreme extreme general consequence deviation ideal
preliminary far seem confirm direction progress logic fuzzy currently work generalise fuzzy logic semantic context degree freedom multi connection semantic semantic contextual specificity mining database etc developed computer ir analyse collection raw physical powerful classifying finally quantum logical aspect method rely store database help absence act significant hence stage contain available information represent concept mechanic
thus rise correlation among time therefore crucial move increase introduce preserve diffusion draw matrix mcmc usually appropriate decomposition contribution introduce diffusion environment cholesky explicitly appropriately posterior transform cholesky approximation perform dimensional stand alone cholesky augmentation several multivariate diffusion volatility augmentation justify convenient property highlight potential algorithm problem paper require cholesky methodology simulated daily pair link data augmentation simulate directly posterior observed introduce step problem price constitute fine multivariate control augmentation base approximated euler relate variation determine hence
backward provable method posteriori backward even sufficient cardinality replication dot exhaustive satisfied consistency backward enough even inconsistent intensity condition consistency consistency sparse produce pick exhaustive feasible plot exhaustive dot greedy dash square explicitly star uniformly algorithm provably rank biological gaussian random hard duality gap semidefinite relaxation bound solution mean relaxation tight large value remain max plus algorithm plot tradeoff curve dual case table compare important pca cancer select software observe gene p duality condition candidate
white significant detect default sect peak dft amplitude link peak dataset resolution consider background constant intrinsic instrumental environmental peak counterpart consider intrinsic frequency observation characteristic decision shall typical handle peak series time answer old rely substantial advantage unbiased compare multi three build handle outline series mention sect achieve pairwise vs arithmetic provide good peak target spectrum estimate peak pixel bar bar represent concern measurement star datum reduce accord obtain restrictive bad comparison intensity appropriate pixel one resolution apply output
penalty estimate graphical model connectivity correspond penalty point student empirical fix problem penalty maximum likelihood turn solve descent remove column remove plan one repeatedly column convergence iterate quadratic replace descent suboptimal column fix exclude meet implie pairwise independent show find sufficient achieve strong minor sample covariance consequence solution except never minor interpret recent obtain rigorous interior obtain another fairly seek nesterov formalism estimate framework
partial sufficiently precise give symmetric dimension explain maximum amount control cardinality numerically hard compute solution ax x semidefinite bind approximate eigenvector instance solve efficiently interior semidefinite solver inefficient instance interior towards large precision smoothing technique much solve write dual fu
statistic hypothesis problem simple generalize marginal likelihood numerous cox variable transform likelihood sequence substantially interest nuisance specific assume statistic section method make find asymptotic alternative idea quadratic random moment eigenvalue detail extensively efficient test statistic refinement dimension make I automatically offer lot penalty theoretical possibility test test restrict number grow important observation information component rate denote satisfying assumption parameter belong nested require require mean
chain behavior fact establish always nod definition segment formally stand parameter asymptotically algorithm parameter much arbitrarily attempt make measure use true asymptotically adjusted must alignment sufficiently lp desire generally improve aforementioned sense need measure every limit continuity viterbi briefly idea paper stand hmm core straightforward handle notion predefine follow contiguous outside observation indicate underlie state viterbi realization special could generalization concept barrier contain term suppose barrier node existence alignment go barrier state certain existence positive barrier ergodicity infinitely many next know mc go proposition infinitely every barrier generic let note also special block infinite alignment hence alignment process empirical hard measure emission easier describe simulation indeed
month temperature year period year htbp c dissimilarity choice factorial dissimilarity visualize choose work hausdorff distance item combine hausdorff
price portfolio selection management enable forecasting although volatility describe model correlation framework link together influence unobserved consequence market exchange many effort fall auto paper capability brief parameter volatility model applicability volatility stochastically review need resort simulation heavily intensive make front iterative
opponent memory armed opponent strategy long opponent active achieve begin opponent select play make decision proceed opponent current game recent make opponent play go refined play opponent true decision cost go proceed algorithm importance decision opponent opponent opponent play opponent play easy opponent play game repeat strategy incur opponent call predictive predictor opponent likely history response offer strong practical would would detect opponent optimize cost opponent improvement well active improvement convergence appear substantial xlabel width legend west particular upon employ active start phrase current phrase current active
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e u u j j u j j l q observe l u u j q j q j case namely partial give I observe w u u q u u u u chebyshev probability chebyshev probability q probability latter statement prove conclude tend univariate normal suffice tends prove converge weakly variate tend infinity stein stein measurable dy close supremum exist indicator g constant v h r h dy dy differential tv dy write l v v l l l dy dy dy dy dy dy dy e dy v dy since v ds prove
li equal denote fan fan li scad sparsity go infinity provide li implement scad author request measure fan mean square I regressor define square I I ls denote least overall median l identical scad cf study fan li unbiased replicate li size hence view fan li choice setup vi median relative carlo replications fan li replication report line figure monte estimate parameter estimate median scad scad maximal gray indicate
df consequence first ft ft ft ft ft corollary ft em evaluate quantity simulate censor law logarithm inverse censor estimate pt classification nonparametric law function provide efficient give interest explanatory work deal yet situation arise reliability censor censor transformation fan especially transformation paper make transformation estimator recently gain popularity censor literature particularly property therein behavior easily
describe variance random extensive devote generalization vector reference
united purpose significant fraction york population order review therein quantity obey distribution scaling typically although phenomenon power tail follow power law article issue scientific literature question recognize power rarely ever hypothesis case describe reach conclusion calculate law bring together give box software implement also also apply set describe phenomenon claim process law hypothesis appear ability know calculation wrong law form method latter section solid represent good fit text ccccc est est method ls pdf cdf l ls fit logarithm two bin cumulative cdf discrete mle measurement bold generate discrete uncertainty digit calculate agree scaling parameter along use estimate plot function cumulative robust fluctuation size tail give scale straight line fit slope transform histogram slope bin width tail slope constant bin slope bins rank frequency regression perhaps fit produce biased estimate bias nothing substantially incorrect true synthetic observation logarithmic bin bin cdf law omit small symbol accurate limit finite choice bias significant ignore statistical decay reasonable extract fig accurate set note important treat namely truly conversely good draw power normally power calculate therefore need discard sample point power wish need get biased scale non power demonstrate
object instantaneous type lead amplitude fidelity signal appropriately enable characterization add value decomposition mention property approximate wavelet decomposition scale one tool signal wavelet maintain control redundancy base observe usually phase property understand transform order although wavelet base signal signal software matlab toolbox module numerous analytic construct instantaneous wavelet peak frequency derivative spread wavelet implement build automate generate notation rescale wavelet obtain instantaneous signal anti vanish account wavelet implicitly define wavelet expansion wavelet integrable simplify substitute bound split wavelet inner wavelet define write q outer give integral inner integral eq outer also th summation interval outside residual summation find magnitude inner wavelet schwarz denote norm wavelet negligible choose unity line algebra inequality three note normalize taylor expansion yield power write lead instantaneous denote high note fact vanish expansion residual additional factor denominator convenient assume st derivative transform recall unity quantity derivative leave obtain precede expression follow write transform wavelet fourier transform incorporate wavelet remain function fouri wavelet satisfie constitute analogously original integral imply side truncation
template numerical I design probability entropy family specific I moment case ill behave theoretic rely I need I solution discuss compare attain I prior posterior conduct infer
mind build correctly see build classification pt pt fitting model poisson process use statistical absence realization statistic currently concept physical law understand provide poisson among set define sensible either little exist potential difficulty process regular distance could model probabilistic type point cell classification find identify formal poisson realization
good prototype loop early compute induce overhead order inner loop outer loop organization prototype prior prototype epoch candidate epoch evaluation scheme construct scheme calculation k put position line next prototype position prototype special proceeds equal reduce whether modify overhead add additional pre additional coarse observation therefore fine consider epoch formulae induce counter observation move old perform extreme case total calculation
value piece information prior prior relationship maximize subject constraint p information least process new posterior reflect fact constraint lagrange multipli maximizing plus yield posterior joint posterior familiar summarize accordance minimal section emphasize impose differ
consequently apply suffer uniformity phenomenon asymptotically precisely regressor asymptotically surprising estimator enforce orthogonality column vice helpful case avoid consideration selection coincide column long coincide estimator always post vector uniformity phenomenon estimate theorem uniformity arise uniformity degenerate proposition asymptotically obtain carry class procedure use aic consider denote restrict user subset select equivalently nest candidate inclusion priori obviously upon suitable arbitrary procedure throughout section finite possibly denote symmetric usual hypothesis index coordinate zero non coordinate procedure probability unity decide natural asymptotically show aic matrix rank post model q square convention correspond index n may selection employ certainly asymptotically model employ ratio test versus use applie discuss test hypothesis overall residual sum square employ ic aic ic precisely select incorrect converge elementary post estimator aic like procedure obtain verification enable post selection aic procedure certainly require post special case aic like element solely yy depend selection size satisfy allow depend sample
literature get spectral mainly make result set convex cluster long make sure linear get minima initialization mention unstable serve automatically correct machine learn cluster overview application encode close label I function small quadratic regularizer smoothness encode little lie dense I connect allow call request lie region like laplace transform similarity observe like look quadratic dx make graph laplacian construct similarity laplace deal operator graph laplace operator generalize pointwise convergence similarity manifold general manifold distribution uniform manifold study smoothness functional partitioning problem draw connection topology property mention understand reader explore huge literature proposition remark corollary recent year spectral become algorithm implement software algorithm first cluster obvious really give intuition describe graph spectral keyword spectral cluster exploratory statistic science biology science scientific dealing get datum try reader traditional algorithm linkage spectral often spectral implement derive cluster algebra mathematical attempt impossible devoted spectral next algorithm work explanation describe section walk study divide similarity nice way
brownian introduce dominating reflect motion condition remove introduce change sde write hard standard therefore dominate measure multiply increment word assume presence leverage effect sde drive brownian regard treat drift ensure sde issue proceed define sde manner respectively dependent volatility conditional sde transformation may define transformation translate dependent dominate measure act enter transformation multidimensional version allow triangular cholesky eq treat transformation two
smoothness process point euclidean isotropic power range isotropic spatial stationary process partition different partition model typically divide boundary axis partition example partition partition divide first split split split leaf cart example cart fit leaf become clear interpretation ability cart meaningful process enforce parameter start region input split detail available paper default well although splitting involve split uniformly accomplish via reversible mcmc describe al generalize cart create lm propose fit gps leave gps toward distinct partitioning add interpret align suffice align nice review partition modeling include behind map dispersion wherein stationary thin scaling construct explore similar
fan deconvolution characteristic estimation l evy process kolmogorov limit sum variable ed read adaptive functional variance ann nd van van drive harmonic nd new york kolmogorov n observation evy since evy jump want face nonparametric l evy increment jump occur insight analogue nonparametric process high extent increment jump brownian evy way increment form shall behaviour estimator I question evy aware work implement fix cut pilot special jump reference assumption law evy evy idea minimum
give however highly efficient every result affect depict finding behaviour rapidly area instability tail context I remain ergodicity mix gibbs explore arise hierarchical show structure adopt improve gibb certainly use clearly might address extend conclusion table case tail distribution replace concept rip already state translate natural lyapunov family investigate extension stochastic volatility finance go stability relatively light non table expect possess property mcmc competition desirable tail explore hybrid iteration sampler provide find outside
motivate favorable precisely maximal lag mixture separate exception lack source computation list table mixture large dynamically separate signal value batch signal ratio closeness signal list separate batch denote c c separation source x yx mixture separate signal first way x batch method signal relative closeness ratio large close piece source one count office room fig second result
shorthand slow external couple variable reaction eq reaction trajectory evolve roughly evolve population observation take time trajectory multiscale compare handle paragraph implementation multiscale scheme please see particle filter correspond roughly multiscale thereby proceed latter accomplished unit parameter proceed except time set step severe average filter test trajectory estimate trajectory connect dash plot hide reconstruct upon factorization rao base multiscale differential multiscale filter variance
gm order produce string never gm process chain ref demonstrate show behavior plane high left corner compression dominate prediction result causal increasingly information horizontal axis effective state curve denote occurrence increase dominate find albeit statistical retain complexity maximum remainder filter circle conditional causal box find partition hide markov process stochastic process allow symbolic string block observe sequence never word proper system irreducible infinite even since irreducible markovian source matter even choose fair generate ht estimate vertical dash entropy
thesis stage yield q note well note combine lemma prove result satisfy deviation q consequence e symmetry bind get suitable constant thesis lemma walk computation q obtain reversible chain death odd odd let lemma integer eq finally equality easy birth death analogously lemma combining
satisfying right exceed cccc square square quite square actual invariant respect old nest optimal precise function old integer always bb element consider unique mistake mistake element em measure large hold obtain agree rank respect produce tight confidence strictly z z nz sensible want exchangeability practical predictor want rely exchangeability case exchangeability weak match exactly conclusion wrong illustrate machine example gray scale matrix use hundred book article illustrate example dataset perfectly example treat systematically remain test predict exchangeability satisfy approximately test total singleton uncertain singleton uncertain empty prediction contain one exchangeability affect record measure distinction predict example produce necessarily exchangeable steady example mistake move mistake jump exchangeability statistically conventional game reasonable exchangeability circumstance acceptable exchangeability compression exchangeability probability compression done add line compression drastically compression study start statistical mechanic start randomness study predict observe probability summary specify variable kernel interested could interest widely use seem less uniform seem something summary adopt possibility surface sphere radius pp assume
attain determined computer whether sample one gradually check enough interesting point reduce coverage reduction accomplish poisson n n nb
efficiently demonstrate theoretically parameter interaction predictive order interaction algorithm prediction expression difficult li also empirically side capture belief parameter group may much absolute think appropriate implement difficulty compression method use distribution response continuous discrete however probably transform research c j li bayesian ph university http j dimensionality among breast page markov ph thesis r pt logistic abstract interaction chain monte great whose reduce method compress value original local mode apply reduce bayesian interaction use interaction pattern predictor pattern occur
method iteratively uniform q symmetric decomposition orthonormal py sphere uniform proportional density lebesgue gibbs conditional straightforward markov full conditional non ensure keep mind density p computing rejection equal sign affect randomly obtain monte iterate von add way sign distribute value
usa establish far beyond show vector essentially satisfy law explicitly allow unbounded support stationary recent support machine become theoretical term I often justify example learning application diagnosis inherently unlikely svms justification goal establish svms somewhat show correspond universal stationary ergodic sequence generate adaptively mix polynomially decay definition suitable addition svms common measurable exercise every measurable process goal remain satisfy ib z asymptotically measurable system grey introduce idea simple formula process obviously value weak event converse implication measurable value satisfy stationary measurable moreover almost describe high loss subsection definition natural approximate make rigorous large begin imply law large number satisfying let hold surely usually restrict lemma definition serve space stochastic process let large number
strong infinitely time u viterbi infinite infinitely time verify fortunately precede ignore specifically easy meet rather ergodic ergodicity every realization infinitely element lemma observation infinitely term essentially hmms noise emission gaussian support hence infinitely remove observation barrier generally barrier existence infinitely class hmms condition
variational manuscript support pn nsf infer assignment module module variant resolution module variational past decade technique synthetic outline among application full group module density intra
computational cost lt cholesky cholesky lt mc generator property remain investigate improvement method couple indeed prove sum dimensional lt capture superposition combination consequence run lt lt suboptimal dimension introduce among central rmse rely repeat time batches burden transform cholesky decomposition generator adopt version property fair option paper nominal problem correlation cholesky pca correlation lt describe concern kronecker burden fast qr decomposition simulation run intel processor gb show percentage positive correlation ratio equation cccc correlation cholesky lt notice
u k ks nu n kn right hand divide strictly nu nu kn kn b nu nu nb nx kb big nn assumption generality obtained note n k k ci q write eq finally eq conclusion relation section two
stochastic model estimator bandwidth mention spline green often bayesian non bayesian version shall theoretical interval whether exist allow follow wavelet calculation recently default see positively bias coverage decrease universal confidence separate specify investigate always theoretical interpret inversion multiscale test residual interval idea region estimator location hypothesis region use kernel character
support slope censor hazard right censor survival survival hazard right censor survival function see hazard ga gb censor location quadratic slope vertical family c reflect extend correspond slope slope domain slope survival hazard hazard component uniform distribution function remark vertical exponential exponential encounter family slope leave restrict slope corresponding yield type transformation slope censor transformation hazard family slope remove transformation transformation map q operation us classification leave censor hazard next identity substitution useful checking serve slope convergent
calculus formulate sn sn database formula sn sn f table query program er illustrate various query formula v f query variable program notion er rule concept far entity database implication contain free er query er association usual indicate logical implication true survey let formula rule complete goal entity give brief comparison rule rule language approach suppose transaction transaction item appear item query transaction involve frequency transaction transaction survey find frequent negative correlation query specify variable query key query quantify customer formula translate domain calculus query target child table assign agree definition definition well probability fact involve division entail
rule gradient triangular triangular term triangle diagonal form practice optimize convex maximize identify recognize easily concave fully mml corresponding middle term change treat formula entropy entropy eq wishart normalization work change hard concave ascent update inspection need note similarity conjugate update copy variational weighted combination variational update derive compute leave unchanged variational delta q hessian hadamard product argue leave hessian order denote definite equation concave concavity term univariate see minus dimensional function definition arrive
importance process continuous kalman approximation form fortunately rough practice case approximation immediately measurement dirac delta taylor series expansion get time state mean covariance approximate q recursively limit covariance result measurement process predict distribution state posterior mean sampling process write set particle variance u particle proposal particle process angular velocity signal true unknown classic disease sir sir variable reason denote initial
n periodic translate usual product correspond ergodic process ergodic substitute sequel strict study stationarity top lyapunov exponent find arbitrary lyapunov exponent sequence se top lyapunov property lyapunov follow ns apply ergodic note property exhaustive periodic pair
nonempty must case row letting tend unbounded contradiction exist entry equal let tend unbounded nonempty row entry requirement belong boundary tends infinity none might follow p n ij equation involve occur coefficient deduce strictly thus nonempty bound
frequency lead light normal approximation rigorous probability conservative actual around tune formula formula meet performance n n
thick vertical line correspond interval determined covariance significance confidence zero right indicate unbounded exclusive exclude vertical middle panel unbounded exclude right limit datum calculate study different denote upper beyond lead limit would unbounde exclusive limit confidence level code gray zero cv ex numerator report point see figure cv run order practically identical show limit expect indicate marks b panel large large scale scope indicate cccc low taylor calculate limit quantile student supplementary material htbp restriction adequate bivariate bivariate necessarily bootstrap normal restriction htbp birth rate material cite supplement short show figure numerator variability confidence limit p degradation activation contribute patient receive vs treatment record object reach ratio size effect method certain size movement movement intermediate table stress human induce production imbalance cause imbalance predict production response stress pre post stress divide status ratio cd cd cd explore human assess response dependent
uk lk uk make lk lk lk lk uk lk uk lk lk lk uk uk uk uk uk uk unimodal respect iii making fact lk uk lk uk lk uk lk uk lk lk k lk lk k lk unimodal make lk uk lk lk uk lk uk mm recall unimodal position prove lk lk uk mm consecutive distinct proof theory coverage belong thm
end k end polynomial context experimental finite mostly gr basis include full factorial design p algebraic factorial gr literature study gr complete algorithm switch diagram relative practical advantage algebraic four coordinate diagram horizontal set equation solve point analogue implementation algorithm mathematic design experiment chemical study dedicated symbolic main point note point sum begin
evaluation early determination whether computational trick motivated experience respect situation feasibility method computer file combination confidence high formula margin confidence determine sample section essential infinite evaluation evaluation accomplish x p p b nb see convenience com bn bn bc b n p b actually type technique improve
go risk predictor infimum predictor hypothesis class I accord get well learn agnostic free typically assumption causal capability capture classification regression algorithm short nz nz pz quantity interest p nz random interested excess loss suitable generalize every g assume available separate
proposition computation subsection main proof proposition recall integrable martingale suppose deterministic satisfied write could think fx indeed j kn computation thank verify involve handle go introduce build sequence new martingale little check sequence subsequence define quantity inductive among give several great great index I k kn n
status breast gene profile infer location array comparative extension comparative genomic profile individual recover pattern reconstruct organization nest perturbation effect phenotype infer protein among pattern family go far specify exhaustive specify subtle graphical offer common conceptual architecture common formalism effective communication across mathematical
op une analyse les date pr est une en de le se dans es les de cat du de une cat du la par ne des de l la probable une des un un de est les et la la date projection en une probable la mod mod par des de adaptation tr de pr aspects les es se dans le pr tr dans la pr le dans un spatio des le plus de les ci un issue des mod des base le si est de es pour plus tail l est une es es pour un une architecture est architecture un une un le de ce est des mod le est par un pour une des de la les des de la une multiplication par la des de mod la les w une de ce es les une les la est le des du mod le de la la de de est les ce de
ellipsoid stochastically stochastically necessarily equal nuisance sequence nuisance provide consequence n analogous extend confidence satisfy aforementione interested generality cover estimator confidence coverage every infinity slow cf inspection stronger large predict theorem extension apply illustrate say euclidean essential see equally euclidean
specify ignore rational likelihood point vanish intersection critical zero generic characterization derive normal restrictive almost statistical require illustrate curve zero critical restriction equivalent curve considerably curve instance general plane six special special equal arise statistic rank model I example q represent rational equal u mixture variable rank cubic show remain problem
finitely principle adapt yield iterative spirit vertex reduction describe censor elsewhere contain density give happen rounding observe place element x I j appropriate q point e want function family constraint constant family precede function concavity pointwise equality equivalent maximization rewrite
contrary moreover easy make toolbox matlab describe set variable cover pixel cover date neighbourhood choice simple neighbourhood shape neighbourhood sophisticated slope influence neighbourhood influence pixel respect weight influence pixel decrease figure h l neighbourhood date type try estimate perceptron cover cover many observation account modelling perceptron cover case
case ratio value value twice multiply approximation ratio clarity lift restriction directly basic algorithmic initially name direct refer clustering
temperature velocity image ensemble recover obtain confirm ensemble boltzmann distribution represent total system evolve pyramid role reservoir play bank formula pyramid
auxiliary filter proposal choice particle proportional particle sample approximation sampling sampling theoretical condition particle z auxiliary simulate accord distribution replace unbiased weight particle filter pf simulate otherwise set I notice algorithm particle decision ess interpret acceptable liu chen resample condition resample occur set pf optimally pf sample scheme sequential monte carlo approximation unbiased ordinary auxiliary particle richer equivalent obvious establishe construction conditionally positivity x lebesgue alternative homogeneous density density respect observation density I densitie auxiliary particle filter eq particle tractable particle time firstly make inefficient realization estimator instead illustrate combine bootstrap easy simulate probability distribution density
alternatively two initially allocation however entail generation avoid simulate simulate uniform simulation simulate first simulate decision order back pair proceed process repeat simulate true keep track infinite visit simulation easily implement scheme behind impossible avoid approximation whether summary dimension toy example formulate simulate avoid error scheme simulation simulate conditionally summary elaborate illustration gave extend however simulation far fit quantity variable classify hyperparameter predictive datum distribution kind suggestion exact posterior distribution functional feasible exploration augmentation parameter gibbs sampler
model weight sum component belong parametric proceeding number rewrite introduce component analysis typically assume assume priori conditionally green identifiability always contribution fr likelihood model eq make fix note estimation posterior inference consequence see reference therein likelihood rewrite set entry notably likelihood representation partition allocation allocation
aic optimize challenge predictor answer year serious shall moderately exhaustive impossible challenge substantial especially good evaluation criterion aic bic aic subset bic favor classic bic aic use therefore appear must logic certainly easy pointed bic aic construct new criterion strength criterion behave division method create parallel universe run evolutionary aic evolutionary give universe universe universe select rest answer simple toy idea generate eq predictor contain aic figure size possible number characteristic observation make subset
combine internal seek fmri simplicity prediction use subject virtual local remarkably low model cognitive area predictive area body temporal predict base dimensionality reduction interpretability relative approach conceptual sophisticated offer window natural functional imaging change concentration image hardware handle nature commonly detection activate voxel simple cognitive
countable string nk fx nk nk ji ff nn letter ix x md x n variable convenient distortion lagrangian trade lagrangian achievable zero block length give order operational rate code nonzero improve use neighbor block lagrangian mn mc r rp
select smoothing go al performance orient extra inefficient datum exist problem sub common might absence abundance interest covariate edge surface temperature depth presence assume tensor rank spline flexible term represent regression fit converge treat penalize model direct optimization invariance smooth interaction g quasi mcmc hand bayesian mcmc usually essential smoothness criterion comparative method include generalize model partly compose basic variance relationship use quasi smooth link strictly model component covariate constraint sometimes multiply yield coefficient handle recognize early arrive lin zhang link penalize mixed method e associate function one know evaluate something univariate spline thin spline may smooth penalty j term section intermediate sort discuss variety problem smooth term glm column column covariate contain smooth glm use possible mean several way behave basis reasonably avoid mi certainly
h bandwidth asymptotically suggest et et provide study sample include asymptotic support density function condition hold replace notation nh ps stand convergence k regression f normality develop like generalize smooth marginal useful context quite general functional regression show define generic constant appearance absolutely j u bound first derivative derivative bandwidth satisfy exist f l satisfy almost regression regression huber ti specification conditional neighborhood assumption nonparametric weakly cause dependent strong course loss ps ni ni n therefore ni kf kf
employ exclude exhibit stationarity six lag stock step return asset proportion miss always lar indicate marginally uncorrelated indicate yield produce less qualitative summary asset return result definite pearson simple enough reject yield entry everywhere position asset uncorrelate investigate zero conditionally show histogram illustrate beyond scope paper market index contiguous residual return series run experiment discover asset marginally uncorrelated market histogram insight ol parsimonious derive demonstrate handle thousand argue even ol suffice parsimonious advantageous selection parsimonious show standard say due use moment future employ assumption condition historical therefore irrelevant without modification need estimate vector portfolio bayesian posterior ideal nice closed situation improper calculation notation matrix analytic section quantify wise relate argument preference
compare common figure promise since capture principled feature typically branching pattern mean recursively detect build size typically control play molecular graph degree vertex relevance well kernel issue mention elegant compute walk lie product formalism initially basic construction walk walks walk graph actually close computation kernel infinite choose walk weight even dimensionality kernel kernel product graph practice product time consume relatively small virtual screening dataset walk application fast search procedure moreover implementation drastically recently drawback structured kernel object tend high degree serious tackle issue normalization set diagonal kernel object self similarity amount vector norm molecular widely assess molecular ratio point operation kernel define kernel alternative normalize function variation allow generalize way come parameterization subsequent kernel largely open
post estimator handle completely describe without manner subsequence subsequence quantity conservative behavior detect positive deviation detect consistent deviation procedure zero deviation would pick consequence later detailed phenomena p model part govern exponential govern convergence speed scad consistency fu fan scad condition show stand uniformly consistent converge fast furthermore nonnegative q consistent prove normally far r zero exponentially get relation consequently equal establish carry scad observe two entail repeat argument see mn large term note uniform follow estimator tune conservative precede guarantee consistency actually sense scad purpose obvious correspond model easily write multiply respective event relation well post selection estimator recognize hard singular coincide restrict maximum absolutely continuous represent absolutely shape c right hand segment location weight mass equal picture obtain finite relation correspond
membership interaction useful application introduce importance interaction group membership directly specify inference value cause interaction determine turn detail estimation ask spend join rooted strongly suggest whose first side event take place support member difference label outline group estimate mixed membership b bic select hyper number relation note group identify fit map name via bayes project interesting statistic illustrate mean dominate corner central play exhibit relation three group uncertain later six encode relation analysis allow mix lose datum mix six negative express finding support lose graph collapse thus prefer extend mixed membership blockmodel multivariate tackle problem membership bernoulli employ scheme latent membership normalize constant
perform eigenfunction adequate reduction vary greater large evident big exception less moreover eigenvalue sometimes term adequate sometimes easy case replicate simulation around replicate negative mse suffer lack high however converge much tend eigenfunction well compare eigenfunction represent cubic natural among converge replicate comparable main converge replicate converge replicate converge result eigenfunction cubic study impact distribution see material table quite improvement alternative average standard deviation measure good convergence long difference eigenfunction way mention shall select procedure fix value long converge almost moreover unless converge selection simultaneously simulation adequate eigenfunction cubic spline equally spaced knot datum sample gaussian idea mean time adequate select project spurious even large people drop converge replicate big adequate latter large go nearly apply whenever factor find prefer select mainly due therefore focus hereafter cv small selection criterion frequently become preferred dominant negligible result select cv indeed effective select correct basis eigenfunction
focus find nonlinear dominate l l dominate among stein approach include stein stein later alternative ls reason estimator justify reason result shrinkage estimator gain estimate technique certainly mse suboptimal image reconstruction multiply generate complexity dominate construct principle minimax generating tailor blind stein continue dominate solution analytically appropriate construct shrinkage considerably outperform minimax method design set parameter blind first minimax use independent construction process ls actually lie estimate provide whereby estimator generate
q fixed extensively follow asymptotically white show brownian motion replace generate model behaviour
causality li periodic autoregressive evaluation q periodic periodic
f une une quantification des est affect un prototype ce une des es prototype est de la des affect ce auto plus prototype observation une des en un plan une projection lin car class les des les auto une de lin les lin la es les sp des auto es pour les es es de les es les es dans des auto dans comment adapt de par de par une un I analyse usage site pr send dans analyse du web national des auto som pour est de lin de il un de ensemble de une de en prototype des une des est est la des non lin des es la est la est un la est prototype du une ensemble connect dans la pour couple la distance est du et le de l ensemble les un prototype un de les une et pour fa les il une la correspond aspect de som de il est si distance les de I pour par dans tr dans la des es des et la notion des auto en tail pour des es est un dans est ensemble dans les es la notion de de la une dans de par est analogue de une
simplify v v follow equation degree namely proof theorem hold respectively likelihood q equation multiply deduce positive determining equation since ideal define ideal zero need cubic root polynomial root ideal solution hence exactly solution four root similarly nine nine
n x computable measure concentrate get mu plus mu mu mu mu mu mu plus mu plus plus km dx plus mu plus mu plus mu plus mu mu mu sum w k mu plus mu plus mu nk mx td immediate drop define non already investigate theorem latter state universal individually universal fail property measure restrict show exploit excellent double exponentially bad minor question unlikely finally dominate universal show tt prediction insight favorable showing mixture possibly distribution dense promise sequence mu plus mu
variable integer nn follow hold readily monotone iii application threshold bind exp nn n x provide sampling ensure reliability actually see theorem implicit efficient explicit implicit ensure eq suffice claim satisfie justification fundamentally statement v theorem explicit et ensuring fail claim reliability useful average sample make thus average sample implicit effort htbp htbp
theory precision order break inference replace circumstance modification yield model take might main test correction correspond default jeffreys coverage property optimistic effect may interval include frequentist perspective concentration acknowledgment national foundation education part discussion david cox thank particularly computation particle physics experiment huge uncertainty suppose concern suggest presence parameter extent
bethe approximation recover cavity correction one estimation run bp order approach tree nontrivial cavity correction applicable sense loop relatively interaction principal requirement cavity third well outside approach discrete graphical loop belief propagation tractable message pass extra loop correct belief model like linear pass interact belief
hypothesis moderate bound classical oracle classical query computationally quantum mean almost query convert context access oracle quantum mention classical harmonic formula exponential classical quantum information theoretic quantum membership query pac quantum article computer remain challenge implementation quantum desirable oppose query quantum algorithm motivate interested design classical source minimize subroutine subset subset spectrum boolean oracle implement pac pac query quantum focus oracle abstract away useful primarily theoretic
eqn eigenvector lemma one eigenvalue spectral eqn note respect symmetric equal norm lemma distribution eqn lebesgue integration jensen consensus topology sense sure subsection state vector optimize recall eqn iid recall eq average convergence factor mean factor call convergence fast convergence topology factor follow note choose mean sufficient condition contradiction eqn eqn converge generalize convergence eqn quantity eqn lead fast let hence vector minimum generalize matrix derive guarantee subsection study converge sense q first lemma follow repeat
award grant dms grant n nsf grant dms dms field high apply devoted reconstruction dependency structure field analyze reconstruct define markov observation mild degeneracy multiplicative constant depend interaction guarantee generating provide case time effective noise reconstruct structure field mrf high attract biology sharp need infer argument require degree propose reconstruction
selection propose consistent dimension projection specify thank parameter regularization constant hilbert ball whole g svm induce closely norm ball cover kernel induce suppose finally functional svm optimal proof follow grid search countable behavior see point hypothesis universal course include search depend could could single gaussian flexibility functional real application functional consider discretization point deal functional efficiency permit derivative among value general grow kernel family candidate value illustrate learn procedure give procedure consist classify speech sample
implement combination typical generalised another model gradient whether apply value matrix transpose root function stationary eq note thus imply assertion conversely establish assertion use eq plug relation definition relation alternatively invertible score equation proof note state compact k il martingale angle chebyshev martingale proof estimate continuity compact let compact decompose imply taylor remainder
coverage population ensure coverage total discover probability significantly reduce coverage property integer kn appendix symmetry mn assume less situation control error find size purpose reduce evaluation coverage n mn
even b illustrative uniformly inside body isotropic treat origin trajectory integral volume six multipli treat via integration define body definition functional show distribution equal unit otherwise less possible dx non lebesgue regular case look functional straightforward relation produce integral eq body six euclidean measure independent uniform use alternative ref point segment consider second direction uniform multipli alternative density proportional autocorrelation autocorrelation function rewrite
per bit historical curve anneal slowly iterate continue temperature anneal procedure state effectively equal allow observe zero temperature causal future string recover upper comparison show analytically drop rapidly away finite effective predictive successively increase gain compression specify four distortion capture excess distortion bit bit
bundle parametrization line uniformity convenient tangent bundle projective bundle bundle convenient principal definition bundle lie act bundle frame bundle bundle already mention represent case principal space space bundle group base action absence
state well non arise system evolution approximately density near reaction measurement state assimilation kalman linear kalman advance enkf measure member simulation dirac member advance independently enkf approximate ensemble enkf ensemble algebra column implementation enkf variant well survey enkf build tool image use enkf later find spatial manually map feature pixel use mapping convenient notation read convenient b b z ia say scalar mapping
vary factor easy extremely link link average distance link leaf trace give satisfactory delay exclude tail surprising delay queue fractional motion traffic consistent identifiability general gmm choice delay individual study characteristic measurement demand traffic traffic internal end user therein traffic loss delay consider root leave subject delay loss receiver active measurement send send copy probe base correlation less overhead scalability internet available dynamic measurement represent topology network denote transpose network scenario assume delay component internal delay example tree send root leaf delay denote delay
place cover q tight satisfy mp node pass message maintain belief max marginals product involve general answer correct answer mention understand mp insight max product give correspond root computation describe computation max beliefs product assignment induce cycle find suitably fact continue associate variable neighborhood follow indicate assign constitute correspond max matching factor
automatic partial calculating grow polynomially consequently complexity exponential actual calculate formula time versus log attribute speedup partial predict experiment formula fig table formula population
stop problem precede sequential procedure article carefully valuable suggestion comment mm mm test mm mm optimal multiple stochastic process setting keyword analysis bayes subject introduction whose consider classical article characterize test product respective randomize suppose eq suppose measurable component q value interpret proceed make stage experiment continue stage rule apply etc eventually stop suppose stop make accept experiment stop observe use observation cause adopt another important sequential minimize latter problem account minimax procedure problem follow easily latter unconstraine minimization multipli infimum stop class stopping obtain ii minimization problem ii unconstrained lagrange proceed sequential subject lagrange multipli function constant multiplier suppose test lagrange sequential test inequality strict least strict let last take account get multipli sequential hypothesis lagrange multiplier implicitly proof method lagrange multiplier extend complement similar exist strict strict rule theorem problem minimize correspond attain indicator attain measurable negative measurable eq equality suffice integral non negative leave apply almost easy truncated minimize section step problem truncate measurable almost start measure eq immediately q equality prove hand q rule eq lemma hand dx equality get series start etc right side condition let stop follow true recursively equal among truncate make sequential without inequality trivial test consider observation difficult experiment receive essentially treatment stop recursively loss additionally essentially ii broad easy test example characterize q cf lagrange multiplier truncate precede apply inequality bound least stop suppose let order go far due imply series convergent chebyshev complete imply let hand weight infimum rule behaviour statistical sample bayesian fulfil plan pass behaviour induction complete everything pass part justify monotone reason pass stop eq follow achieve virtue optimal optimal test bind theorem
private exposition lie mean version resp always maximal mab mab algorithms reward mab allocation rule social benchmark bad instance round iid expectation instance difference social select separate obtain well well arm arrange v mab namely regret separate allocation gap bad regret even sublinear gap regret separate mab agent let allocation gap significant mab mab mechanism exploration separate mab mechanism gap agent immediately mab require separate characterization bind whether weakly separate allocation assume exploration separate mechanism agent allocation allocation satisfie imply gap exploration translate gap good agent immediately bound satisfy strong condition allocation weakly separate separate require regret expect separate mechanism click time counter agent require show say contradiction mab two agent mab agent mechanism add receive click run pick pick agent guarantee separate characterization picture separate mab allocation rule deterministic normalize factor separate phase exploration follow exploitation well exploration crucially exploration mechanism assume particular two bind sense weak theorem randomize mab realization internal mechanism rule randomize mab restrictive click realization seed monotone exploration turn mechanism mab design click observe literature mechanism mab design notion even click randomness deterministic define ask whether structure seek similar likely surprisingly monotone allocation rule mab expectation minor increase theoretical mab mechanism must directly require variability optimal mab rise mab allocation randomness third weak mechanism click click provide algorithmic mechanism design problem click fourth expectation click kind obstacle design mechanism observable information obstacle appear still mechanism set perhaps importantly conjecture extend mechanism interact provide conjecture follow section current snapshot open question mechanism compare algorithm mechanism gap due combinatorial combinatorial public showing gap mechanism mechanism area include online dynamic pricing offline interested topic beyond claim satisfy claim confirm search et author lie agent nash equilibrium may highly hand suffice whenever reasonable lie unless prove bounding provide open still rational agent outcome might far seem yet notice claim price bid work respect consider mechanism mechanism mechanism match almost identical mechanism two agent bid demand private information reveal need agent reveal stay exist agent hand maximize feasible mechanism mechanism exactly expect expectation prior notion paper framework economic refer adversarial vast amount lower typically mab various payoff dependencie g etc pay click ad mab focus notion bound regret recent consider like understand gap prove characterize bound achievable follow gap prove computationally combinatorial combinatorial public project constraint schedule aware arm conference publication several open snapshot direction concern weakly mab mechanism informally main significantly deterministic counterpart resolve weakly mechanism regret mab henceforth design weakly mab design allocation monotonicity allocation click monotone perhaps terminology notion monotonicity terminology mab arbitrary background click mab vast mab suffer explicit tradeoff recently case bad bad optimality parameter development concern rule mab monotonicity transform result reduction expectation new deterministic mab allocation regret conjunction mab mechanism intuition crucial deterministic mechanism obstacle insufficient rigorous setting mab mechanism obstacle arise offline pay ad et scheduling potentially observable arrival click event information arrival miss pay ad ad slot pay click ad side slot multi mechanism mab mechanism development mab snapshot open characterization develop prove bound match randomized adversarial click click agent round click information agent round tuple round bit click select horizon realization rule payment bid click choose receive allocation round click observe know realization round realization round payment realization derive clicks click realization exist agent payment per click click realization bid profile click receive get mechanism click realization v ib ib mechanism agent value bid profile iv mab mechanism design specify round click independently thus rather private similarly define take supremum instance allocation follow bid profile click realization round allocation degenerate degenerate tuple round allocation degenerate contain degenerate hold interval weakly present characterization describe background click agent bid everything else domain relate monotonicity result characterization mab click realization click realization use information every contribution mab payment click actual payment click computation literature characterization mechanism parameter context mab design payment click click decrease payment rule mab mechanism payment restrict notation give click realization click first notable version monotonicity rule click bid mab normalize mechanism degenerate pointwise click bid agent round bid word b generality click round round round affect round allocation depend click click realization differ bit get contradiction claim generality otherwise ix click click realization agent get since degenerate degenerate interval bid ix click fashion click bid round click realization click influential bid allocation bid round agent round bid realization round allocation future round round call agent round call influence realization influential mab allocation click round influential bid allocation round click bid vector round influential influence agent separate separate follow realization bid influential influence need influential thus r allocation separate bid allocation round separate mab allocation rule straightforward definition click realization influential bid influential bid influential influence since influence round let structural characterization general theorem structural implication imply structural condition rule separate weakly separate idea behind consider mab degenerate scale allocation separate exploration click realization round influential want round influential click bid profile round click realization click value ease exposition also click know get round bid profile click mechanism denoting pointwise monotonicity lemma exist bid price click realization therefore violate round round round mechanism violate click realization click contradiction degenerate exist degenerate distinguish bid agent mechanism price contradiction characterization mab degenerate allocation payment rule pointwise monotone pointwise weakly separate prove proposition albeit weakly click bid round round influential influence agent hold generality bid need bid click realization I tb difference allocation round influence round I round fact monotonicity b I I pointwise monotonicity violate contradiction degeneracy degenerate interval eq however distinguish bid see contradiction direction allocation rule pointwise payment normalize monotone I increase bid click decrease mechanism normalize show click bit reveal want payment bid click simulate execution round pointwise agent bid round bit bit irrelevant bit arbitrarily bit execution prove round get realization lx ij lx lx ix l lb ib however change click realization concern round allocation bid profile formally use claim prove change allocation bid claim round lx round prove follow get round raise contradiction begin round influential influence influence increase influence derive contradiction show must case get click must get click contradiction minimal must agent hard see weakly separated argue degeneracy normalize pointwise monotone yet weakly separate round allocate agent allocate show allocation payment consider alternate allocation rule select except mechanism payment rule even change easy allocation degenerate round influential mab weakly non allocation pointwise exploration separate weakly separate sketch preserve place agent right cause transfer label denote cause click realization bid sketch proof lemma focus weakly contradiction realization round bid round bid influence bid r exist bid round bid get round mechanism far prove gets bid bid bid round either agent agent bid symmetry easy increase round round get value prove prove use value dependent belong degeneracy infinitely allocation argument exploration rt let expectation clicks given let bid separate cause bid click allocation distinguish allocation click agent crucial proved technique fix agent instance agent round algorithm incur regret exist require bid instance expectation bid round contribute agent round suppose good round agent prove agent therefore hand side side round algorithm incur instance rt constant claim instance rt rt contradiction claim prove let bid instance gap agent incur entropy summarize define probability measure function property fp fp e fp p fp conditional fair fact specify qx yx px qx fp commonly denote relative claim round click click bid exploration separate bid history determine round independent support bit universe abuse treat fp fp tt fp interested fp bid fair coin bid bid claim fix bid history treat theorem bind allocation satisfie mechanism degenerate suppose satisfie rt sketch fix influential bid round particular click realization round bid round influential agent define click round agent say click realization click property click influential round fix click round influential bid influential influence definition without bid bid suppose bid profile bid bid transform adjust bid agent initially last transformation adjust bid bid adjust happen bid adjust happen pointwise carry minimal modification horizon influential round click let influential round w r click induce click influential vector expectation constant later agent allocation incur rt similar paragraph randomized mechanism realization seed mechanism randomize mechanism relatively bound regret mab mechanisms mab mechanism randomize mab mechanism deterministic support problem expectation mechanism let mechanism normalize degenerate extend agent bound theorem assume exploration separate exploration deterministic mechanism mechanism separate rule suffice need mab mechanism separate separate deterministic agent follow regret theorem tight logarithmic factor naive mab mechanism simple rt describe match agent horizon bid phase agent round click exploration choose every click exploitation every agent mechanism mab problem naive bad regret rt price exploration exploration round influential click phase price click round exploitation irrespective bid less click weakly regret chernoff agent lie specify run clean clean exploration v rv exploitation regret claim section discuss apply design click adversary specify click optimize term allocation realization profile increase bid decrease allocate round click depend click round discard round randomly run pointwise monotone choice click mechanism click realization seed randomize click bid bid decrease allocate round coincide click round discard report allocation round pointwise click randomize normalize weakly mab let separate allocation payment rule result weakly mab allocation mab design rt separate weakly mechanism adversarial mab adversary whose k adversarial mab regret achieve bind immediately mab allocation open improved horizon bid agent agent vary bid agent stay click bid payment take internal randomness recall select look round monotone round round agent round value click exploration mechanism payment payment assign allocation pointwise therefore follow normalize seed payment click allocate agent bid click round separate random seed take interpret allocation rule separate sake present bid divide horizon consecutive round round round randomly arm denote exploration phase exploration call agent according equivalently observe click update q b discard k clear choose round begin look allocation pointwise monotone monotone bid agent pick round assign term relaxed vector click internal randomness mechanism deterministic similarly allocation click mechanism expectation bid receive expectation click mab allocation monotone convert mechanism expectation minor increase introduction result mab monotone expectation mab rather well form mab include give rise monotone mab trivial bound trivial structural least show monotone time expectation allocation allocation moreover click absolute payment per click bid consider monotone approximation mechanism bid polynomial argue payment payment rule number claim good stochastic mechanism allocation monotone rt gap initially active round sample time bid active complete perhaps mab regret bound along crucial observation time time bid maximal allocation expectation bid cause agent activate let random stochastic design respectively click payment treat payment bid algorithm let polynomial degree history include round integral separately fix run allocation click history payment ie proof possibly amount degree induce payment order time time round choose outcome click b despite mab mechanism understand snapshot open current write deterministic mab agent weakly mab conjecture insufficient observable class mechanism allocation environment mention follow work suggest setting mab obstacle conclude obstacle prominent payment mechanism powerful surprisingly still understand limitation expectation accord regret rule extend mab analyze slightly mab decide skip model could trivially extend bound regret case negative agent extend immediately follow recall exhibit variance high crucial bad optimality monotone mab simultaneously reduction tradeoff monotone allocation rule tradeoff regret result simply mab mechanism click mab mab achieve tight discuss pay many version mab mechanism various could weakly mab mechanism well reduce design weakly mab new angle mab account pay click ad unknown seem extend precise remain would probably multi slot click correlate ad multi mab mechanism independently remains see obtain slot mab even mab understand acknowledgement thank helpful direction bid click round influential allocation weakly influence tuple bid w click realization occur separate bid property b prove degeneracy tuple degenerate tuple influence round page figure assumption bid lemma make deterministic allocation bid case change sake agent agent arbitrarily bid tm tm k jk let mab allocation rule pointwise monotone round bid bid high enough note pointwise ti lemma deduce bid bid profile bid notion allocation rule monotone click satisfy possibly bid see define b I satisfied say existence existence work existence existence bid j prove contradiction assume b yy yx yx b yx x contradiction b follow bid j agree contradiction agent arbitrarily consider bid vector tm bm bm bm tm I b contradiction define bid positive assume
reasonable good change recovered cover lie stable indicate range quantitative adopt fitness module fitness simplicity cover recognize equal histogram cover stable peak result fitness cover rank combine seem existence apparent hierarchical partition overlap partially partition hard depend extent specific time backtrack rough module cover moment square loop check fitness worst comparable run os enyi complexity htb computational complexity algorithm run graph os enyi range quadratic community linear network require resolve cover display large run cover resolve hierarchy complete quickly note iteration calculation trivially value computer run cover similar initial run complete repeat way conclude several complexity fair besides optimization fitness considerably complexity seem promise direction future extensively method artificial adopt simple arrange group order every link group node link level consist fig top example mixing parameter tune tune community prefer micro community fuzzy mixed hard htb accuracy algorithm high four include find mixed well inside micro community line graph configuration link connect macro community check build parameter build realization modular adopt normalize information prove overlap community several extension mutual hierarchical four macro community identify start link outside macro community mix performance remarkable modular structure htb histograms american college leave peak cover coincide cover except correspond modularity bottom right fitness os enyi american test cover reasonable fortunately cover fitness histogram top right lie whereas community perform interaction network graph unstable remarkable well cover study overlap systematic pattern conclude www corresponding subset analyze graph link hour skewed tail exponent analysis concern www stress processor carry necessary distribution necessarily correspond cover representative cover turn fair actual htb community domain www skewed agreement graph tail fit power law exponent dash simultaneously overlap network find maxima fitness enable include module lead natural description overlap tuning probe explore application network excellent like large expression fitness meaningful fitness setup reliable exclude framework flexible community design fitness accounting module structure carry computer systematically size million aspect organization graph possibility quantify community value fitness extend network thresholding fitness strength easily plausible node subgraph subgraph exceed produce link thank suggestion read manuscript jk thanks seed start may affect cover principle fitness histogram seed seed cover seed seed ranking cover additional seed scan cover reliable seed peak computational run discuss cover yet issue entropy equal another normalize interpret infer vice normalization helpful follow suppose belong membership entry partition say cluster regard array whose distribution hold possible define cover amount order infer among candidate turn normalize divide normalize appeal property vice happen case belong vice versa similar close add far vanish play quantify use encode express half cluster complementary none take note accord repeat reference organization form connect module link field community embed community find fitness fitness tune enable hierarchical give excellent complexity successful understand natural system complex I existence node module community reflect topological relationship element underlying entity group relate page deal pathway central importance organization level usually highly non trivial reason module community nest etc could argue life map network hierarchical organization module care presence hierarchy community rich detect modular hierarchical well finance community accord call dendrogram know partition identify meaningful belong one overlap community belong community depend friend belong detect often overlap community clique clique reach mostly modular overlap often exhaustive modular overlap local procedure visit many matter assign community way overlap recovered size explore htb structure internal overlap community nod basic module extend neighborhood certainly plausible network node graph www even form base social community subgraph fitness try option fitness external degree module real control community module double module external subgraph inclusion new node elimination node subgraph community maxima fitness large possibly attain idea detect metric apply helpful introduce fitness fitness fitness fitness subgraph natural identify covered subgraph stop examine sort fitness look high instance
discriminant background testing event introduce classification discriminant signal histogram event analytical calculation show signal input distinguished separation histogram calculate increase roc analytical area number cell cell cell theoretical area background rejection cell cell classification algorithm core binary contain event build discard memory consumption overhead within inactive empty consumption geometry store information correspond number division division visible inspection main default optimisation classification representative example use training large statistical box sensitivity fluctuation train large density box influence time case cpu large reduce volume contain collect box roc uncorrelated gauss shift b compare original fig signal background target cut minimum cell number size box reach drop precise box fluctuation additional stage inside cell reach box wider stable original box size careful optimisation box towards slow accuracy increased provide large small training fluctuation space drop increase need cell figure five moderately gaussian signal sample training consist signal large contain restriction event cell sample exceed training training range cell reach drop cell assume statistical accuracy density distribution cell guarantee phase effect event contain inside consider cell splitting cell stop target reach requirement affect cell fluctuation sample improve drastically event event cell suffer decrease identical drop point merge drop cell study number cell avoid htp per division affect phase build enough statistical increase large cell improve number build approximately default reduce choose without histogram evaluate cell division histogram cell division step study dependence number bin small sufficient training sample suffer fluctuation cell classification reduce gauss length box reconstruct event background reconstruct width visible case procedure separation signal htp event gauss shift minimum exceed one method train min gaussian htp background two gauss shift kernel classification phase limitation original slowly result typically need time training event cpu consumption example moderately study background event respectively display roc rejection single sampling size event box original approximately normalised training event average volume allow probe volume cost method event cell behave cut number event large cell fine size reach size event classify method reach rejection background event example tree sample dominate density cell cut background depend mostly almost independent variation cell slight geometry original binary background minute less recursive geometry long htp quantity value implement method value dimension target multi build phase box sum box build density target target additional build density build store projection centre cell axis form geometry target active cell simulate accuracy relative reconstruct target display per event add htp multi variate adaptive far generation search build result give optimisation show exceed original furthermore consumption limitation event implement discussion grateful discussion implement main help grateful lot universit mm pde discrimination technique signal density mc simulation multi modification pde use adapt dimensional hyper size predefine cell phase background mc package present example toy discuss improved capability classification pde search mm mail f variate discrimination physics distinguish signal characteristic individual discriminant cut background introduction discrimination e pde use event classify background density probe volume search pde dimensional observable space correlation mc signal search dimensional box involve uncertainty box free parameter limitation densely furthermore computer scale convolution box variate geometry therefore optimally case density vary original fluctuation sample self adapt phase box efficient event base pde event probability background cut lemma estimate event type event either simulation discriminant densely sampling assign discriminate background framework physics background approximate pose challenge search classified background volume around event background event discriminant obtain propagation uncertainty event contain build volume entire create build cell cell division histogram gauss build start correspond contain mc coordinate normalise correspond linearly coordinate tail distribution remove base cell side one exclude start along hyperplane implementation identical code cell predefine uniformly cell box consider event contain box divide sampling beyond cell boundary axis project predefine split
half plan property include half plan property rectangle call infinite orthogonal property completeness recall proof suffice property use concern disjoint origin height angular centre rectangle height disjoint angular centre height lower v angular portion origin height together eq u imply deduce end concave measure concern property consider gaussian measure lebesgue strictly radial proposition result subsection real write identically result surprising point q calculate explicitly exponent believe step imply assume loss generality claim inequality q end distinguish several disjoint demonstrate begin step desire conclusion lead desire lead desire conclusion decompose step deduce eq cumulative gaussian variable z analytic domain precede equation hence p q acknowledgement la proposition remark analysis analysis problem design dimensional straightforward give section banach binary aim unknown seek rule give incorrect underlying make measure label weight probability eq label want set mind detection class frequently necessarily medical procedure counterpart want later simple formulate interested rise use sequel set case logarithm I derivative real life thing substitute classifier simple risk log perturbation word simple real value contrary investigate general case affine correspond usually linear discriminant affine discriminant correspond also plug study different context therein differ address consist find good substitute give natural order get satisfactory classification procedure dimension justify extremely study performance discriminant build responsible sequel theoretical asymptotic overcome poor propose rely fan fan select multiple study procedure constitute classification act high procedure theoretical two priori get reflect nothing relatively clear like justify thresholding regression framework see introduction technique answer infinite functional far context classification therein rather expect apply hence norm depend abstract space notation article covariance cumulative real gaussian plan application extended infinite framework symmetric schmidt organized result lead problem procedure section relate lead light curve introduce geometric link excess devoted proof symmetric restrict affine substitute decide define angle play role solution respectively sub simple area rotation ex ex ex make wrong gx ex motivate comment eq also perturbation affine perturbation affine yield answer sequel optimal whenever quantity separation ex p r distinguish perfectly separate q note orthogonal p mutually absolutely finite link excess believe tend link indeed think hard risk q sequel see behave like excess use elaborate quantify intuition lead estimator excess distribution seem behaviour procedure plug rule straightforward procedure analysis excess particular intrinsic procedure establish thresholde independence keeping mean lead ultimately constant part inequality give satisfy first compose observation uniquely rest learn p precede probability proof theorem precede sharp everywhere ex ex arbitrarily believe estimate direction link small ex invariance unknown seem quite use let illustrate link estimation classification problem exactly noisy want grow infinity sufficiently set zero coefficient thresholding estimation shall precede thank dimension act equivalent direction intra big maximize use problem give try matrix possible precise arise error already detail easy exercise know measure point angle get good risk close shall almost refer discussion recall definite matrix generalise generalise arise equal draw generalise comment comment proposition fisher already form comparison bayes alternative base covariance operator together sophisticated aggregation procedure done therein link stationarity toeplitz circular convolution operator fouri type harmonic combine quasi search huge harmonic stationarity stationary process process see idea variable form vector distribution degree fashion intersection unitary sphere surely symmetry angle inequality desire probability covariance associate mean ex comment precede proposition excess converge tend estimator estimation dimensional restrictive suppose basis precede also define cauchy consequence calculation l well rapidly well separate tend theorem separate rule f enough degree give advantage play lead symmetric problem degree suppose polynomial fact dimension infinite could infinite introduce subject highlight contain remark framework separable banach case class separate trivial sufficient measure reproduce operator equivalent define finite counterpart understand integrable chapter transpose q straightforward perturbation follow easy symmetric exist precede less procedure might concern explain quantify observation base quantity upper consequence excess risk concern procedure term relation lead equivalence allow conjecture recall parallel require correspond distinguished hypothesis natural link error proof obvious excess distinguished reflect know base norm separable hilbert stay decrease long rule ie il pe plug subspace span eigenvalue equivalent choose enough suppose rule figure case give practical gaussian dimension treat procedure recall wavelet variable unknown partition construct function presentation suppose covariance section mean separation two vector tends note definite matrix eq plug procedure discovery fdr hypothesis decrease quantile standardized depend covariance successively article et inequality remark close wavelet attractive transform wide operator speed universal aware indeed fan fan high lower bound unknown hypothesis gaussian hilbert operator gaussian measure support control rest empirical going unbiased choose come belong calculate term equal substitute standardized precede practice keep mind divide view constitute keep application associate direction extend estimation vertical reveal diagonal one use hypothesis order decrease eq constitute direction rule need finally give use support svm kernel recall procedure solution q label set notably et al record curve sampling compose aa compose aa aa label curve almost medical spectra characterize area spectra associate tumor spectra spectra retain spectra second type spectra consider leave lead case htp type figure
matrix let least square estimator true computation estimator optimality associate good multidimensional model cm universit paris paris concern multidimensional model perceptron mlp logarithm determinant optimal couple mlp minimize choice euclidean widely suboptimal estimator covariance square approximation noise devote true mlp sequel positive minimizing cost asymptotically square true compute example find partial derivative depend parameter get inverse get eq get give line easy moment order estimator consistent mlp exist
clinical library wavelet dirichlet survival hazard illustrate survival aim show posterior meaningful significance available represent posterior inclusion probability depend large lm age diagnosis lm td gender lm distant run iteration chain root consistently analysis use swap top visit chain leave sampling area leave bottom show probability covariate accord diameter large follow age tumor status estimate inclusion survival modal posterior inference full posterior maintain tree km survival cluster value table low survival low survival leave high difference main latter define interaction predictor plot figure approximated model leave laplace schwarz increase fix leave leave employ schwarz unbalanced group survival penalty laplace involve survival censor indicator increase unbalanced laplace marginal inference survival separate size without section convergence fluctuation inherently metric criterion assess chain although increase selection development appropriate important field future research taylor real parametrization numerically integrate conditional k j jt penalty arise integrated jt acknowledgement thank helpful early manuscript motivation implement employ request introduce simultaneously prove metropolis hasting selection namely cluster selection selection chain chain bayesian selection regression survival let markov chain monte generate time mcmc mechanic adopt approximate e integrable respect integrate analytically thorough publish reader novel find useful highly multimodal parallel prominent role one carlo sample run parallel fact connection couple monte sampler respect sampler propose accept ensure mix review emphasis metropolis hastings mh joint relationship section illustrate inference mh model survival section discuss current let density lebesgue draw markov condition construct let transition value probability kernel irreducible state irreducible kernel detail reversible irreducible transition strong number hold condition theorem asymptotic deviation transition metropolis implement transition kernel reversible distribution draw accept assign current mh ratio mh transition practical hinge check furthermore equation evaluation typically multiplicative extension mh mh distribution multimodal mechanic stochastic interact spin ise occur informative prior highly integrated provide bayesian structure replica carlo metropolis exchange carlo liu level index temperature act smooth temperature tail less mode proceed alternate step update sampler use mh choose proposal sampler ss I swap index pt define probability iteration swap current index current index swap order couple accept respect independent chain correlation mix successful analogously mh pt mainly proposal cross chain finally equation suitable sampler key difficulty dependence mechanic latter physical model energy temperature simulate possess alternative solution multiple equilibrium specifically multiple independent swap move draw swap chain marginal emphasize analogy label sampler prominent carry analogously joint use mh irreducible chain kernel reversible marginal joint distribution density probability update chain carry write swap mh former proposal swap acceptance rewrite respect side take respect side equal right mh knowledge suitable multiplicative mixture joint probability analogy carlo estimate two transition step jump unnecessary competition local global mix marginal conceptual conceptual neither analogously index practice determine sensible trial error pursuit swap unless family implement temperature latter treat carry simplify hasting acceptance concentrated mode practically nature temperature explain posterior see metropolis swap accept mark evident sample augmentation sampler general variable typically sample algorithm augmentation chain iteration chain chain generalization proposal iteration sampler attain balance acceptance ratio involve several current computed multiple generalize chain proposal main update retain use first retain individually update proposal update mh mixture mh walk proposal distribution current update within swap example nine proposal spread mh start initial uniformly standard deviation finally interval normalised mixture deviation result mode compare mh sampler three start visit bayesian address among dimensional gaussian element otherwise include latter potential derive inference order inference employ mh marginal right hand unstable compute cholesky numerically estimate mh simulate dependent second strong let z j mh chain dataset estimate inclusion draw predictor chain additive transition nine target spaced sampler chain chain carry component scan metropolis propose current chain proposal uniform also proposal batch liu illustrate plot without whereas plot report inference compare row appears estimate inclusion generally true value sampler dataset shift high swap marked circle result predictor regression coefficient algorithm suggest comparable swap pt algorithm wrong estimate inclusion cart disjoint set leave leave node root partition tree within leaf survival framework bayesian cart appear paper tree sampler tree censor survival propose adopt split within strength survival quickly mode bayesian marginal space tree structure upon distribution combination define visit tree survival leaf write number th unit include covariate leaf uniform multiplicative array leaf place uninformative prior uninformative matching parameter parametrization leaf
go address denote compact simplex another k observation covariance point complete thus parametrization amplitude controlling call interested reconstruct pmf covariance tensor happen tensor unknown closeness use formulate pmf pmf recover solution ji ji ji square geodesic say field bound continuous p k kn default euclidean default full manifold curvature plane curvature confirm need pmf choice much fact e solve correspond obvious straightforward convex show one let prove consistency matrix true pmf random triangular q practice non difficult fact try convexity eq eq need put geodesic lipschitz necessary lipschitz domain open whether ex map geodesic compact default rank sequence alone exist geodesic diameter uniform define choice operator criterion w separate minimum therefore consistency constructive choose criterion requirement theorem basically domain lipschitz criterion relaxed criterion look back h mf recover default field fail euclidean curvature generally condition infinitely infinitely converge condition might relax problem field amplitude introduce tensor operator operator tangent field recover representation reconstruction possible concept euclidean manifold riemannian manifold metric specifie operator operator operator circumstance field problem recover field surprisingly euclidean space euclidean covariance field true subject riemannian clarity recall comprehensive n manifold parametrization coordinate local coordinate chart tangent tangent mapping call mapping let pg ng coordinate j tangent metric change eq endow volume coordinate change change manifold dimension equation equation chart say satisfying depend differentially geodesic differentiable q exponential map maximal neighborhood call neighborhood inverse map indeed gauss differentiable differentiable sense differentiable local coordinate jacobian adopt brevity exponential analogy co component variant expression similarly bi expression variant tensor contraction coordinate non coordinate two matrix change tensor manifold co variant globally co differentiable function differentiable write p like symmetric negative variant tensor moreover correspond coordinate back manifold set linear summarize differentiable differentiable riemannian open set algebra denote volume space x f additive say except measure distribution absolute mass function pmf def definite field q claim differentiable continuous field definite except hyperplane correspondingly tensor positive space symmetric positive definite tensor linear e let q moreover eq mean metric geodesic distance q give let distribution show applicable say geodesic let distribution proof lose dominate exactly claim variable bounded finite dominate give proposition I field space field define proposition emphasize continuity condition field prop let canonical tangent ix xx xu continuous course research useful field whole variant field particular distribution continuous thus control purpose amplitude circle uniform sample sample generic apply calculate clearly confirm benefit amplitude typical eq geodesic member family recall geodesic radius space e
example remark stage affect depend classical alternative datum control start assign response analysis etc suppose experiment stop article mathematics subject c keyword sequential stop sequential stage use unknown goal obtain start observe analyze obtain suppose eventually stop final aim characterize hypothesis alternative etc let triplet stopping rule suppose suppose start first decision respective continue next apply stop accept reject interpret reject hypothesis vector recursively cause sequential error like characteristic testing goal minimize sequential procedure essentially sequential hypothesis control variable lagrange multiplier truncate stop characterize class truncate section obtain minimize procedure proceed function sequential testing relation non test q testing hold strict least theorem minimize multipli rule infimum derivative distribution respect stage observation probability control calculate easy q us step minimization us find simple measurable function measurable everywhere form see immediately minimum stopping step truncate take natural everywhere almost everywhere function right hand side transform apply see hand side equal everywhere everywhere stop truncate attain rule attain everywhere suppose side converge convergent series remain series convergent chebyshev immediately behaviour inequality satisfied lemma fix non everything pass hold definition lebesgue monotone possible q leave additionally hand prove coincide let eq obviously first contrary lemma contradict follow structure control almost almost everywhere fulfilled strategy hold right hand first successively apply almost everywhere everywhere prove satisfie apply inequality mean follow leave hand prove treat least observation see give particular characterize hypothesis identically observation rule statistic likelihood general strategy define ratio eq functions measurable function see lemma argument us expression almost sure sure sure satisfie see q rule stopping occur interval problem help essential construction control apparent stop drop accept continue sure ever unable speak fulfil may expectation hypothesis fact even control hypothesis optimization take problem
well establish propose value generate copula method mutual method competitive mi negative method mutual estimation entropy call base mutual measurement copula margin form separate margin random mutual density entropy marginal function copula copula mutual contain corollary instant result mutual hence information propose
projection cluster indicator combine metric separable cluster tensor clustering combine manner analogous combine simplicity proof collection index run index respectively clustering relation part objective clustering last approximation clustered term sum complete bregman divergence theorem bregman euclidean good tensor mean scalar strictly divergence familiar turn tensor extend bregman divergence upper entry divergence away representative f express clustering full cluster tensor curvature seem necessary bregman divergence approximation intuition avoid theoretic obtain tensor metric euclidean result cluster metric arise conditionally function integer choice element result connect metric metric embed hilbert hilbert construct metric behave squared euclidean hilbert cx cx distance base follow square product independent dimensionality run kernel lead guarantee apply slightly bind j name cluster thm tensor mc j thm bregman bregman cluster metric j underlie study close wherein mind impact tensor build empirical usually small factor objective greedy final approximation depend assess three microarray describe microarray preprocesse microarray matrix whose refer breast datum selection remain microarray include true merely agree label clustering data repeat display simple improvement method encode stand say improvement via algorithm depend improvement lower generally combine seem table variant distance gain uniform become mean clustering outperform variant turn kl divergence impact improvement kl divergence bit besides initialization way iteration take decrease even dimensional clustering co simultaneous gray ii lc cx r pt cx pt htbp c r pt lc breast cx pt pt lc c breast kl cx pt r pt htbp r ii l lc cx pt lc kl cx c r lc cx r pt cx pt pt c htbp gray l pt htbp paper present approximation bregman factor order bregman divergence slightly metric clustering suitable initialization latter approximation clustering algorithm interesting direction specific subroutine conjecture section gray theorem generalization make bregman co simultaneous input suffer hardness researcher k bregman algorithms co tensor beyond divergence prove tensor cluster separable evaluate characteristic also practical impact fundamentally euclidean sum corresponding centroid rapidly minima applicability paper pt euclidean minimize divergence detail theoretic co divergence divergence cluster heuristic well local minimize algorithm approximate reader numerous reference therein approximation tensor cluster attempt cluster namely follow additional matrix factor bregman divergence separable generalization version cluster know gain tensor achieve factor result broad norm divergence corollary pose euclidean clustering could could insight amount inherent consideration several dimension theoretical traditionally center seek partition matrix cluster k eq measure bregman co extend seek center column minimize denote center column easily show section formally present tensor tensor well multilinear algebra mining machine part section notation turn particularly tensor use letter multiply value multilinear tensor multilinear generalization familiar multiplication three act multiplication matrix dimension p compactly multilinear tensor order multilinear multiplication property multilinear generalized tensor arbitrary eq dimension vector norm vector extend tensor write frobenius property inner product verify familiar q product tensor divergence scalar divergence arise wish coherent sub tensor reader familiar recognize th problem name outline pt obtain cluster call tensor illustration tensor divergence tensor e treat vector apart dimensional dimensional algorithm outline method cluster approach discover bregman factor combine seem analysis wise suffice strong guarantee cluster amount lie simultaneous follow remainder order th guarantee dimensional eq theorem might great euclidean begin proof technique require notational exposition optimal strong generality integer
cost function b term take index latter term k number converge almost term term tend curve stems otherwise expand yield k k sum third term equal detail unique actual nk behave denote noise k minimizer mean zero shift big appendix note yield bind motivation contribution curve tend may curve order align proposition word shift state k bound probability converge constant inequality proposition proposition yield ab hand converge still replace prove let need standard rate due give suggest plug shift estimate weak estimator law surely show pointwise convolution estimation error latter distribution would real data biological simulation case compare method practitioner measure fit square shifted square shift automatically parameter suggest build tend infinity fix block align create equally shift provide present alignment curve alignment comparison b well estimate display notice free value whereas thus trivial refer depth bandwidth rule retrieve particular situation block c k align signal maxima heart cycle compare alignment compare curve take figure outperform one moreover average heart cycle separate heart wave wave visible signal ht fit reasonably well fit data heart perturbation type kind model hz interference amplitude cause effort motion contact produce movement choose perturbation namely effect frequency take ht observe regard kind perturbation observe curve shape naturally baseline preliminary baseline baseline interference order simulate interference perturbation average interference illustrate possible amplitude frequency interference kind distortion retrieve signal alignment note segmentation amplitude describe display segmentation display align obtain signal representative illustration proposition align block shift take method information noisy surprisingly block remain small however come large shift propose simulation outperform method interest estimate emphasis like thank anonymous thorough improve quality grateful international partially write grant equation expand collect easily come vanish lead term gamma distribute moment latter random hereafter latter hermitian product markov similar identically variance consequently schwarz obtain since eq expansion rhs order eventually term follow whose alignment error far curve curve equation assume artificial curve eq therefore cm proposition theorem alignment application shift biological application possibly noise ratio shape lead first set shift shift eventually estimator mild weakly simulation alignment real estimation shift nonlinear inverse problem investigate specific inverse value represent parameter standard process estimate either model commonly numerous align vary shift extract curve alignment observation variation variability individual curve paper focus biology alignment encounter growth traffic many preliminary shift derivative discussion find identification estimate jointly parametric since interest nuisance shift one shift proceed jt remain invariant fit estimation shift fix curve paper estimator optimization practical deal curve shift infinite penalize approach curve shift analyze signal aim heart electrical enough shape preliminary take complex since heart measurement encounter align yield lose efficient correct consequently shift correct convolution shape individual method dataset signal align reduce shift jointly also therefore vector shift minimize cost continuous invariant signal translate introduce importance signal function th integral latter let nonnegative propose aforementione replace c cm invariant add curve reference shift reference plug estimate shift nonnegative bound derivative mild shall theorem converge fourier dft dft close
minus mu mu mu h mu minus agent achieve high reward agent maximize formally concern ai narrow ai consider either opponent perfect belong environment environment assumption minimal possess structure experience need determine summary game static opponent well summary physics location consecutive time bandit n planning explicitly function history discretized version belief generalize sufficient look robot equip camera pixel usually extract feature neither precise context minor car huge copy essentially importance properly formalize mean information code know compressed seek approximately goal relevant primary ask find compare model state action review code adapt consider px shannon mu plus plus mu mu plus minus else empty category code rigorously principle mml combinatorial incremental ignore code code similarly mdp may recall mu mu plus minus plus minus mu ps plus ps plus plus mu mu plus minus mu minus mu exposition identify far state repeat shorthand state ss ar ss ar ar consider subsequence reach mdp occurring times code join total code mu plus minus eq plus minus mu code reward standard simple reward time code minus mu mu minus mu minus plus mu minus bit length non transition depend ultimately reward want well two state short need code structure detail hence frequency code mu plus minus minus mu mu mu mu minus mu discussion keep predict reward regard reward look minimal mu mu plus mu plus minus mu mdp sequences sequence closeness simplicity simplicity example insight reader observation coin e deterministic various regard relevant summary confirm state large ct mdp cc ct ct cc ct ct ct mdp rt r cc lc probability lead learn nonzero generate previous bernoulli subsequence observation occur consist equal code mu plus minus mu plus mu mu plus mu sequence get minus plus mu mu plus mu mu mu plus minus mu term reflect code learn mdp state realize bit need away leave figure regard allow state allow subsequence reward minus mu mu mu plus mu minus plus h h minus mu minus shortest code realistic short code minimization reinforcement learn briefly explain gain representation neighborhood concrete search context summary generic ill reinforcement employ challenging come propose blind inform adaptive heuristic structure directly simple case minimum genetic major algorithm tree list grids powerful one logical recursively specification relation minimal refinement mu minus mu mu plus mu h mu mu mu minus among splitting rule randomly space similarly mu minus plus mu minus mu minus mu plus h mu minus minus regard neighbor search high highly effective despite idea choose neighbor well small version likely occur section observation generalize set tree string g iff regard relevant work binary code tree optimizer ex improve ex ex ex ex number split else merge agent transition probability routine probability exploration problem promise solution suggest section I present mdp know finite achievable discount satisfie bellman mu mu plus mu minus plus mu discount typically equation solve polynomial process observe plus minus mu plus mu mu plus minus mu plus mu mu transition plus mu mu plus mu minus mu minus see shannon ts n relevant frequency estimate attribute code estimate mu minus plus minus mu minus mu simply lead part never explore stay try action explore cause agent stay without try trading versus exploitation optimally bandit polynomially agent add never plus minus minus plus mu mu mu minus mu mu mu minus exploration polynomially compute agent action policy try state region sub action mdp follow ex mdp ex n mu output action ultimately care result reward spurious likely space proper integrate code reward reward mdp probability mu mu mu plus minus n mu plus minus plus minus mu minus mu mu minus mu define ar ss u ar ss ss ar ss ss mm u mu minus mu mu mu plus plus minus transition formal present learn flow h accelerate produce suggest value primary explore impractical mdps powerful dynamic
matrix norm formulation previous polytope advantage model considerable model mention illustrate power refer detail robust broad uncertainty yield problem see cardinality unknown take robustness uncertainty set column resemble net regularization ability recover treat g instead lead remainder receive exist perturbation irrelevant hold loss far among tool result satisfy angular separation linearly zero substantial regard sparsity lasso literature particular establish tool investigate robustness robustness sparsity wise way index subset equal elsewhere write norm identical perturbation make eq q let establish help generative setup random feature belong assign perturb assign translate condition irrelevant condition namely orthogonality property satisfied line receive nearly e perturbation relevant j ij jj increase j j regression statistical perspective consistency optimization formulation error include subsection present intermediate utility w definition kernel consistency lasso insight robust establish equivalence robust utility give equal hold eq explain corollary equality generate probabilistic operation take ns na nc partition denote lemma bound stand distribution n nb db slow theorem know encourage sparsity interest desirable namely property encourage algorithm certain sparsity stability space l regularize establish sense regularize get trivial bad arbitrary set label function repeatedly yield uniform stability observation stability observation jointly eq remove full sample robust robust consider perturbation show regularize regression among robustness perspective formulate lasso wider scheme investigate free theorem algorithmic show stability property scheme perspective extend result regularization norm broad regularization new offer solid scheme design new parameterize define rf rf ip ellipsoid take indicator assume empty relative notice function duality show eq equation back take minimum side give borel set establish leave mass since lead construct hold lead q combine lemma optimal arbitrarily eq generality denote arbitrarily hand formulation equal furthermore eq prove assumption remark email regularize extensively remarkable addition problem consequence connection regularizer property principled selection regularizer optimization secondly explore robustness explain specific standard estimation give stable statistical robustness minimize regard element observe correspond approximate target least square sensitive among cite minimize norm regularity tendency recently attract attention ability reconstruct pattern exactly observation subject therein solution lasso robust square line reduce sensitivity robust minimize residual observation consider either row norm none robust property previously investigate wise process magnitude couple couple uncertainty intuitively wise couple variation satisfy two consequence principle regularizer particular uncertainty generalization lasso convex secondly investigate solution explain solution sparsity intuition ultimately additional incur error exploit solution perturbation essentially nonzero feature perturbation addition lasso list organization robust wise square provide perspective regression arbitrary norm consider section new explanation obtain beyond lasso estimation setup also state encourage stable notation capital letter letter vector use evaluate denote recover guide formulation standard know coincide regularization flexible powerful corrupted find bad formulate min max q set admissible section uncertainty place requirement uncertainty contrast couple uncertainty feature discuss significance show uncertainty hand write observe notice combine inequality proves set robust fundamental chance constraint constraint know g first particularly important large optimization q
w di demonstrate classify rare survey pure modify output accommodate discrete classifier supervise machine low resolution optical spectra look classification necessary threshold completeness contamination limit contamination completeness magnitude object contamination account serious poor sample completeness discuss classification influence tune training survey find rare two reason look prior rare survey obviously modify incorrectly classify object goal satisfactory maximize positive positive implication illustrate problem optical star million study star galaxy large number whole reference frame clean low contamination intrinsic classifier report simulate build pure acceptable relate comprehensive colour identify contamination completeness al et classifier density identify uv completeness contamination et decision tree object object classification tree confirm object proportion contamination rate percent class start present prior application notation class relative number modify appropriate refer true subscript output effective mod rare modify name e denote names star assign truly refer g notation assign stand completeness contamination assign context probability prior always whether explicit many star posterior probability thus prior something appropriate star every likewise exclusive exhaustive unity classifier assign output probable confident pure sample contamination completeness label build select user completeness divide set object truly object class object could precision completeness contamination believe prior include prior explicit know reason something discuss calculate hoc output quantity reflect architecture write model object deal updating reflect new probability one survey look fact survey make possess depend train whole set train learn star change training classification influence different us refer imbalance set support tree affect classification might probability depend equal prior via marginalization hence calculate think leave absence equally perfect would perfect train class help reliably boundary nominal want class rare poor issue ideally replace instead modify inspection nominal nominal equation equation calculate modify yet posterior modify posterior nominal modify define normalization impact discussion section necessarily expect good proxy star I suppose object therefore want star nine common normalization inference think change change training completeness contamination actual incorrectly classify modify modify c create set reflect effectively quantity sample actual test unchanged yet completeness generally change probability set calculate completeness contamination already modify calculation must effective modify class equation use class ten time modify ten could posterior check really classifier algorithm develop work classify primarily call bp spectra proper variability three class star galaxy algorithm system operate library train test lin optimally separate svms object close vector trick implicitly input minimize regularizer control svms fundamentally probabilistic probability function actually train use method wu et produce probability advantage however cost radial tune hyperparameter concept present dependent perfect least modelling mean nonetheless alternative rbf library spectra library library plus star library span temperature galaxy library spectra star formation history library slope emission equivalent flat flat exponentially influence contamination little library purpose article datum rather purely galaxy tune discuss set simulator et library simulate rp motion source time number stack call cycle normalize case spectra two mirror red removal region range rp nm separate input strongly blue nm pixel instrumental spread significantly broad pixel measure spectra source approximately square time small snr library simulator fig original library resolution nm line confusion pixel spectral bp rp expectation help zero cause processing assessment purpose simulate apparent standard proper motion twice classifier improve noisy single band draw replacement unless object clear object emission training leave test unchanged nominal galaxy list case accommodate time expect suffer galaxy reality fraction modification confusion assign table correct classification misclassification value histogram histogram unit actual adjustment case set threshold order third completeness contamination report motivate removal low galaxy star prior confusion table histogram classification translate trade panel show star rather galaxy curve fig contamination impossible bottom explain effective star recall would class inspection highly mostly dominate peak tendency significant fig plot value emission line broad bp rp sharp dispersion emission smooth spectra readily star class class model modify em datum star first nominal row object equal prior give class proxy give prior rather base next motivated datum tune svm test unchanged galaxy compare confusion classify misclassifie star loss balanced star misclassifie star modify training galaxy misclassifie star change previous many galaxy always galaxy act mostly classification hard focus confident positive central show assign probability assign still compare remove reflect completeness sample correspondingly see contamination star contamination completeness contamination pure object galaxy prior modify prior time histogram posterior case difference plot peak barrier classifying probability whereby impact ignore provide nominal show c nominal contamination drop completeness translate nominal drop bottom actual thin contain significance explain evaluate galaxy galaxy star confusion table threshold class sum use threshold galaxy respectively completeness star galaxy curve completeness contamination think star heavily yet rare fraction express object star contamination galaxy look like relatively classified correspond star modify low dash model prediction modify set building really test retain modified variance time fig measure prediction nominal theoretical argument aside nominal classify plot dash line agree c effective datum I use line agree nominal modify superior obtain pure I experiment agree nominal star galaxy galaxy modify maximum star correctly classified reflect low confidence classification fig shift
big score pr follow specific history lk kn eventually probability section dedicate another product distinct imply node happen space omit record draw history record satisfy n eq simultaneously product solely node compose distinct probability node reveal lk bit go balanced recall happen follow balanced cut view event space put leave random pr expand subspace leave correspond expand subspace cut history expectation outcome bit pr v u h l deviation individual bit l balance corollary immediately pattern balanced cut cut cut bit actual deviation cut entire record history convenient introduce reveal random bit another prove ready history record balance h lk side e l lemma remain full completeness balanced show remain stay subspace node omit n independence event bound l acknowledgment research office agreement grant thank bit ht examine bad event history lt lt l expand subspace k block represent every point along map say fix fixed let map know corresponding know thus upper regard induce deviation nn lemma take constant inequality appear l thus optimize optimal n k inequality due l kn l l n take summary requirement implicit case k section boolean distance distribute demonstrate balanced input instance form hamming corresponding bit vector edge edge cut result nice one trade certain like cluster cut classification arise context small g dna markers snps nucleotide frequency represent reasonably draw independently objective consider minimize feature classify individual origin throughout previously mixture unlabele point general reveal balanced type balanced cut cut weight hamming two base allow construction dimension bit draw across dimension always classify unbalanced base paper new idea goal bit idea appear handle dimension contain nonetheless complete find hill strong technique reproduce achieve balanced case require balanced show enough explore interest single draw vertical dashed line give curve generate term distance seminal present inference membership individual spectral partitioning original individual feature analyze examine cluster spectral draw aim classify accord extended distribution population requirement sample motivated state dimension correctly classify every universal focus case center separation requirement requirement principle distribution concentration distribution center fix large discard individual correctly classify second kullback product domain minimal distance distribution aim point boolean boolean cube refer discrete particular boolean resolve random bit attribute vector center vector definition complete node point realization cut term high object represent let u lx partition I balanced theorem say probability kn n pairwise ts iy u nod red dotted green solid belong instead value difference unique partition stay cut edge edge differ exactly node need bit present score although g significantly pair figure prevent much exclude event individual refer z pr assume n pr pr n pr n htb fix random conditional cut measure score event always positive individual nice guarantee pair score come perfect minimum cut score across comes expect k proposition proceed hoeffding bound I x k independent let kx hoeffding e combine expectation focus bound event idea score simultaneously enough idea three contain completeness presentation regard probability possible bit event subset individual formally underlie bit field event figure high balanced high cut less initially subspace exclude
symbolic relation symbolic analogy lexical resource lexical analogy question college demonstrate symbolic analogy symbolic system hand code update examine symbolic research concern symbolic couple collection find school system extract conventional keyword source laboratory finance discover lexical test module use symbolic relational module relational much represent fix code automatically pattern experiment analysis examine choice proportional analogy college use simplified simplification beyond analogy either computation concept specific influence extensive evidence system analogy make able analogy derive merely involve rather involve analogy problem large beyond analogy daily especially solution ten claim ten semantic discuss seem candidate compare section human human matter find right mapping future relax requirement add leave develop take analogy input automatically expand corpus new add analogy step atomic rest idea couple seem analogy people start search mapping lead ideal statement heat water air rough jj nn nn jj jj sound jj jj sound reflect light jj dim jj agreement analogy htbp nn trajectory nn elliptical jj air nn agreement competition adaptation artificial jj natural popularity fitness jj ball nn jj fast agreement read mistake average slot machine win mutation average agreement chapter argument vb argument acceptance attack logic agreement nn belief valuable jj support solid reason jj agreement person nn vb schedule effective quick jj expensive common derive htbp seed plant inspire jj seed grow vb vb agreement mind turn jj turn jj break intelligence average nn vb object heavy agreement understand vb nn path understand jj complicate straight jj understand light nn confusion interpretation jj average agreement common mapping intend mapping participant intend participant map speech pos tag source tag assign manually mapping tag term tag pos tag pos show participant intelligence pt pt pt ai researcher cognitive argue analogy core influential computational mapping theory engine code relation engine combine idea relational code build mapping list use corpus automatically discover mapping scientific ten common achieve human compare approach able try past current analogy situation current situation survey theory engine influential analogy target familiar concrete whereas target unknown abstract transfer source kind distinction attribute relation argument large engine mapping mapping representation source model atom target map map similarity attribute small likewise little around child rely similarity gradually relational mostly analogy refer mostly relational similarity focus look attribute thing atom go beyond input requirement hand code representation argue creates code entity name name mass form name name name name force type atom expression name name mass form name name opposite sign cause form name representation system qualitative physics text interface person draw label knowledge code far require hand qualitative physics learning use code present code idea engine relation two element corpus raw text derive code list source list term construct list table analogy although effort considerably effort figure system may seem believe analogy daily application identify role attribute appropriate semantic labeling frame contain role topic identify sentence role topic proposal medium phone sentence test unlabeled sentence view labeling create mapping sentence source test sentence help map company mapping transfer knowledge source company environment semantic labeling mapping briefly behind define specific mapping build relational application mapping analogy ten table science problem table validate intend give ask mapping present result problem agreement algorithm human variety analogy good achieve code speech question precede limitation consider conclude list relation among concept involve involve atom term significant tend co occur window e corpus relation cause co occurrence occurrence reliable relation word word put constrain possible put sentence constrain tuple problematic perhaps apart relation semantic co example cause statistical signature hypothesis generate mapping simplicity restrict term map term word speech map independent generate unique ordering give possible mapping show consensus mapping stand permutation follow generating mapping relational depend correspondence correspondence relational word relational similarity cat relatively degree mapping mapping term break randomly mapping seek map relational degree cat relational similarity cat assume always less define define term well mapping discuss mostly analogy abstract analogy appearance normalize combine simplified analysis calculate describe generate proportional analogy evaluate similarity quality analogy give sim sim proportional extend proportional describe ten proportional theory classify semantic word information extraction answer automatic identifying evaluate classifying experimentally compare task solve college corpus question human score hand well human apply task relation noun noun phrase head noun semantic noun hand noun different accuracy variation different language test achieve analogy question distinguish parameter extension potential application handle might section evaluate science support benefit superior semantic semantic relation connection link table six binary agent affect use perform action htbp agent agent action discount affect tool agent affect six relation manually dataset expand binary relation underlie facilitate automatic relation result benefit handle evaluate mapping problem consist intend validate information technology experiment web server institute people way participant map participant separately agreement intend agreement linguistic annotation agreement system atom water heat science artificial mind slot mutation item belief reason difficulty seed mind mapping mapping source mapping target third participant agreement figure average gives view agreement mapping mapping majority participant intend agreement select majority participant get score precisely try participant agree mapping problem intend apply similarity mostly similarity accord analogy intend mapping appearance relational hypothesis perform mapping mapping problem appearance relational mapping equally similarity relational primary analogy hypothesis measurable therefore relational relational similarity distinction cognitive relational capture distinction hypothesis engine seek maximize relational evaluate possibility break use calculate build pair row correspond correspond example might correspond element term center system center thus element truncate pair term cosine angle two problem process list contain mapping add pair ij order system token phrase corpus contain pair corpus phrase find phrase member template experiment word gb plain web phrase engine retrieval http www find phrase phrase pair next list phrase phrase replace leave replace replace word replace pattern add token illustrate center generate eight pattern illustrate pattern replace yield center illustrate another eight system illustrate make pair correspond row phrase find either empty remove pair frequency stage million many computer share pattern phrase say sort set top pattern column let center center corpus create keep check record generate transform raw measurement suggest kind tf inverse frequency element statistically surprising achieved call therefore frequency calculate probability mutual information e pattern center estimate pair center estimate probability thus relation aspect semantic relation expect hence positive may design otherwise zero indicate nothing semantic term low truncate svd use svd svd product column form singular matrix produce select column sense minimize approximation error f frobenius think compress original two simplify calculation drop result term suppose th pair th either maximize similarity eq form way form term raw frequency use slightly section two test change motivate increase efficiency call search corpus package permutation experiment dual computer bit time search hour minute phrase take minute take minute disk access baseline configuration agreement table difference participant statistically confidence htbp system water flow heat sound artificial natural mind slot mutation difficulty seed idea machine mind object configuration performance label human people whereas compare score score give person seven map incorrectly mapping incorrect look column incorrect mapping label people wrong various mapping participant participant mistake table human yet perspective htbp way examine sensitivity eight dimensionality truncate value decomposition eight row show effect column number multiply decomposition set row final row sensitive variation significantly differently pair confidence would need show dropping sensitivity modification seek maximize search possibility tie break experiment lc lin implement similarity build treat increase try word valuable noun lc lin search similarity speech select similarity word lin could find noun also evaluate similarity behind company keep statistical corpus ir pointwise analysis online option read st college factor space ir speech similarity measure intend mapping speech speech tag speech tag part speech automatic intend manually speech tag intend seven measure create seven st measure combine simply return pos tag match weight manual tag mapping present label label lin pos high lin pos everything else pair significantly lin pos pair confidence level summary human perform st lc lin ir st pos pos lin pos lin pos ir mapping examine precede science analogy common combine advantage approach suggest science support participant intend vary science great performance difference problem pair test level c font score science might contribute difficulty agreement people find easy science whereas difference pair confidence perform science htb lc lin ir pos pos lin pos ir pos deviation science table human font score case problem evidence significantly human definition normalize negative combine score add multiply lin pos mapping manual tag try multiply combine lin pos significant probability lin pos pos multiply show significant problem significant advantage combination elementary benefit kind mapping involve coherence swap term mapping similarity
amount component portfolio justify matrix normal wishart disadvantage introduce restriction rp fine decomposition introduce note specify latent latent severe portfolio mechanism difficulty paper coherent involve confusion turn little specification want component remainder informative likely especially survey source expect ask rate grateful statistical office economic inequality author thank remark indice economic various aspect inequality scalar index input population population country survey sometimes subsample obtain point account uncertainty inequality nonlinear base per datum match censor monte monte carlo markov subsample center estimate linearization take adequate covariate methodology statistic carlo approximately six office several aspect focus type concern nature difficult due difficulty measurement selection sensitive probability relate bias piece unless assess say imputation practice institute bound among predefined replace miss bias censor article focus population specific survey everything assess measure value collect ask home visit collect total unbounded sampling increase final aim index censor contribute index sensitive misspecification though usual specify logarithm normal residual assumption distributional residual absence pareto literature contribution differ censor high possible capture ray possible misspecification example residual censor specification indice component total obtain rely point rule inference rely take censored assumption censor conditional maker instrumental produce predictive though simultaneously numerical heterogeneity al american sampler chain account rectangular censor detail gibbs discuss result compose initial particular compose census self employ people rich selection adequate ignore selection mechanism little target interest inter recall rank denote base well use linearized estimate justify rigorously enter start q gaussian censor able tool rely model adopt point hierarchy conditional macro remainder private certain parametric class prior density adequate heterogeneity portfolio remain unobserved heterogeneity covariate time cause residual product good model collect possess variable matrix context limit et informative indeed observe wishart already question answer form collect financial include collect per se information match condition pay information always pay bound information interval censor total censor domain note external specify motivation sample collect assume able draw would might probably sample total adopt reason model censor observation simulate basic introduction variant em several try also normal simulator rectangular require importance hierarchical modeling feasible accept instrumental unconditional know especially gibbs procedure intensive viewpoint modify account gibbs multinomial choice augmentation gibbs state state gibbs rely exhaustive block initial markov decompose block simulate iteratively update block block stage update take finish model updating truncation truncate univariate normal update previously component unbounded uniform ergodicity law marginal chain predictive probability fast main markov chain useful theorem maker want single ask answer say quadratic posterior covariate give posterior could path drop accord replace state take large optimality among quantity
number give make mistake mistake mistake note trade diameter cluster number diameter structure size simple predict adjacent account vertex vertex label mistake case loose nice literature g use mistake note mistake account wish adversarial number mistake label optimal factor would lemma research online labeling specifically guarantee mistake cut induced partitioning know problem graph high graph problem problem space laplacian rely perceptron algorithm proof purely directly generalize label connect algorithm adversary label answer minimize mistake round presentation clean mistake online perceptron mistake cut induced vertex effective exploit label graph mistake combinatorial predict vertex know label neighbor mistake label adjacent loose count mistake obtain follow connection problem connect receive unit know guarantee include mistake first shall mistake predict mistake thus use cut edge mistake mistake cut mistake fact label well obtain theorem appeared call offline discover setting set path connect path bind span use follow mistake label edge way optimality adversary mistake mistake present algorithm perceptron
straightforward adaptation rough tail often heavy increase elegant proof adaptive reader condition easy understand methodology scheme reader preliminary approximation weight tail distribution time laplace preliminary weight equal weight speed could far careful adaptive range work preliminary harmonic accept draw definite space big period acceptance mh acceptance low probability poor fit sensible chance cover area unclear require ergodicity limit update preliminary period update interval determine require acceptance period strict adaptation proposal draw probability however accept nearly normal problematic finding though cluster algorithm sampler divide approximately normal skew normal draw normal matrix build mixture mean straightforward component block eq come normal attractive arbitrarily evaluate estimating normal already identically distribute go concrete possibility maximization author experience reliable reject run rise small covariance expensive converge update attention estimate mixture normal quickly without even cluster reliability decrease optimal fit harmonic outline harmonic algorithm harmonic less sensitive unsupervised start perform low computational computationally normal line recursively cost parameter require batch efficiency severe early phase chain direct phase mh produce limitation gradient descent maximize case mixture estimate sensitive draw later draw verify line rapidly representative exploratory phase though acceptance high costly implementation example section constant whereas advantageous outline appendix propose mixture proposal density report performance proposal strength limitation adaptive wish density accurate explore wish possible fail explore ability region want quickly map example target little approximate ability rarely explore slowly several enhance explore frequent fit area useful improve finally ideally long proposal update accommodate change proposal poorly course cover region explore next show often poor initial univariate mixture large support quickly fast valuable consider normal symmetric skewed distribution initialize tail laplace minus log equal acceptance learning learn low mixture skewed acceptance successfully ideal initial often integrate therefore possible parameter block exploit obtain initialization proposal gibbs inefficient check sampler adaptation strict adaptation number iterate sampler draw sampling define scheme obtain compute time iteration walk let matrix dimension matrix distribution iterate thus follow autoregressive straightforward conjugate inverse parameter report iterate inefficient gibbs draw posterior also reveal full extent period observation item gamma common mode likelihood ar ols parameter therefore posterior suggest nearly burn recursive distribution mode suggest persistent little reasonable tell proposal normal z posterior soon distribution highly around mode amount lower run outline start acceptance obtain posterior particular find table sampler relative efficient sampler additive error eq q regressor enter enter quadratic spline eq otherwise knot h knot spline basis basis intercept simplicity include transform model parametrize diagonal convenient normal nonparametric parameter independent j simply set different prior log gamma shape imply prior inverse imply residual metropolis approach update prior normal use approximation fast analytical derivative correlation lead rejection hasting block sampler integrate make model median value covariate web linear follow use five distance center status seven initialize laplace laplace extra need mh ratio result acceptance improve seven update jointly distribute except connect benefit add smoothing updating proposal estimate adaptively would nearly report gibbs prior sampler relative relative nearly seven time sampler average look give ratio adaptive walk metropolis datum initialize acceptance start factor sampler acceptance conjecture poor behavior sampler posterior may poorly less htb l l mean ig gibbs gibbs simple correct data log chi indicator one block distribution recommend draw integrate accomplish metropolis kalman filter posterior available less code sampler explore begin analyze daily return difference nominal exchange integrate center prior truncate acceptance rate show take around satisfactory adaptive walk sampler acceptance around chain sampler metropolis htb build hasting adapt arise current practice inefficient start early article provide fast reliable markov metropolis helpful suggestion improve support grant axiom claim conjecture criterion example exercise proposition remark proof hasting sampler information tune automatically repeatedly carefully hasting sampler normal take potential normal frequently start early build speed reliability sampling real article give readily proof iterate sampling monte markov chain simulation greatly influence year implement mh proposal move efficiently across state crucially proposal provide good approximation distribution play experience preliminary tune adaptation adaptation ensure iterate correct target mh obtain adequate proposal start point strict adaptation sampling functional quickly literature stem article scheme important proof strict mh adaptive sampling body justification adaptive mh sampler efficient reliable sampler apply example partial use limit period reversible call adaptation mh discrete inefficient code effort generate within particularly building increasingly proposal building proposal thousand draw
experiment initial initial per corner detector fast detector corner per varied detect corner robustness image increase camera noise aggregate see test random add detailed show good feature frame arbitrarily close frame use dataset frame varied key fast despite generally er presence despite detector modern hardware er low hardware detector test learn turn segment detector despite speed result excellent generalize idea corner detector still er improvement noisy detector variation corner detector corner much intuition want good detector experiment detector measure fast tree http project http mi uk corner useful real world scene yield location detector operate frame rate detection detector use available operate frame detector sift second generalize allow optimize loss third carry rigorous corner despite detector significantly detector demonstrate learn produce significant detector corner feature step vision track simultaneous corner massive computing power corner detector live video stream frame rate feature detector leave corner match database world detect amount hold application review place advance literature point corner corner object patch texture capable kind point often design detector corner maxima change corner technique detect often high corner curve curve concentrate effectively slope curve pair curve method curve link corner minima corner fix compute curve determine isolated pair central peak look angle corner instead fix maxima length corner must corner centre curvature region centre free slope angle rate slope point curvature rapidly minima angle maxima corner stable decrease form branch consider stable corner smoothing wavelet transform modulus curvature scale define corners maxima curvature maxima closest minima locally curvature propose instead direct cubic corner detect derivative extend curve saddle minima maxima nearby edge thereby slope curvature find identify histogram chain candidate corner local minima slope smoothed corner angle rely segmentation point corner absence curvature corner maxima curvature window corner find edge generalize replace segment corner line intersect manner segment corner strength change edge fail corner point nearby edge contour rapid measuring gradient direction along bivariate fit multiply direction along corner gradient fit image surface corner polynomial total image corner along texture detector directly require edge detector define corner along scale corner take maxima laplacian maxima corner also maxima connect nearby curvature gradient direction consider gradient elementary magnitude compute contour corner strength patch patch shift version build derivative shift eq perform area claim negative derivative autocorrelation equal term f explain detector affine motion corner suggestion corner channel explain form surface second general appearance pointwise behave consider generalization maximally appearance give matching transform illumination scalar take laplacian greatly filter gaussian determine location maxima log different scale particularly scale corner extract image per select maxima space approximation much especially intermediate reject eigenvalue hessian image keep similar satisfactory significant speed filter convolution detect pyramid detect pyramid maxima scale transformation space detector corner detect feature pyramid major detector examine patch look corner detector describe paper corner appearance intensity intensity instance corner model family orientation curvature convolution detect use corner base moment detect corner direct contiguous arc centre pixel corner intersection angle belong line angle corner use derive shaped corner strength angle corner self corner pixel centre corner minima pass rule qualitatively bad centre orient dominant average pair orient corner dominant self instead center pixel either end diameter line orient orientation corner since stop soon encounter detector step pixel point maximal large use corner project candidate corner location intersection centre window median percentile oppose patch quadrature filter give orient energy corner maxima total orient energy direction radial transform along radius detect corner apply train corner apply processing use instead image test consistently track recognize train behave detector maximize state detect point define point detect corner detector optimize style detector er detector detector detector detector broad category corner evaluate term positive corner location existence false negative image performance often track evaluate corner change detect corner however obtain detector result generalize well system counter though necessarily system corner view corner detector detect identify corner provide usefulness corner instance pixel useful corner detector angle corner corner adjacency corner additive noise detect corner position varied consist shape corner pattern decrease add pattern na corner localization positive detector generate use scene plot positive varied add corner angle criterion firstly detector detect corner vary consistency corner transform detect corner measure number corner truth corner corner truth corner image subject corner method corner corner agree keep corner rely remarkably otherwise consistency axis roc second category define strong detect frame algorithm divide corner corner frame successfully detect optical match corner corner interest descriptor closely relate detector computing perform matching type generalize detector vary descriptor third category propose reliability detect view detect one nearby position scene interest processing descriptor patch corner detection patch highlight square corner centre candidate corner arc pass contiguous test criterion operate consider circle corner detector exist contiguous pixel circle intensity pixel plus originally admit exclude corner corner corner segment criterion remain candidate examine exhibit reject candidate corner pixel assume dark detector distribution corner unlikely feature another expand learning stage build corner detector target label segment convenient circle pixel relative denote partition pixel point centre corner false yield pixel total arbitrary select yield recursively select terminate mean subset either occur exact summary procedure create decision classify corner fast corner detector case separate convert create else speed allow branch equal pixel second batch test perform pixel reject lead significant increase corner precisely detector case straightforward include weight ensure detector exactly compute corner non result increase detect corner decrease produce corner strength corner determine pixel corner corner corner alternatively iteration scheme centre pixel classify corner detector pass detector take iterate fail processor non mask extract potentially detect nearby world image average pair frame vary aggregate test checking feature view measurement pixel surface scene detect feature detector corner also model plane test affine margin alignment camera radial perfect furthermore detector find different corner likely corner marker align simulated annealing sum square frame filter frequency system location try capture eps eps eps eps eps eps eps eps eps eps dataset consist change radial distortion corner detector texture eps eps eps eps eps eps eps eps eps eps eps eps take prop reality consist projective scale eps eps eps eps eps eps eps eps dataset appearance affine segment decision detector generalize detector decision detector convex configuration perfect tree grow bind quite capable one repeat useful result define corner frame frame decision cost threshold oppose corner inversion prevent rotation combination intensity inversion detector corner tree corner offset centre pixel refer corner corner apart branch outcome branch corner generally increase simulated annealing optimizer first
denote short program generate string short order keep notation simple concatenation equation sense side depend string machine likewise context node constant string shortest necessary distinguish trivial compute shortest general symbol program follow useful measure algorithmic algorithmic string mutual bit description short use ensure symmetric symmetric string non vanishing take indicator relation information causal mutual measure versus object briefly correspond string short equality strength causal much common though relate two string amount li al et intuitive obvious complex string share et evolutionary letter kind statistical causal adjacent instead relation distance algorithmic less conditional analog three string constant algorithmic conditional mutual string call conditionally word additional compression kolmogorov develop describe law string think sequence statement independence however string represent heavily symbol cutoff motivate statistical analog mathematical relationship statistical showing complexity kolmogorov string symbol set shannon entropy expect algorithmic statistical draw joint writing mutual entropy refer limit focus mainly limit call input string pair short program intuition various mutual long assume necessarily far know could description save knowledge allow product measure generalize throughout string define label string definition assigning provide hence algorithmic mutual construction arise statistical algorithmic causal markov individual algorithmic causal algorithmic condition description observation whose formalize acyclic graph concatenation concatenation except definition specify justify compression parent string turn nice equivalence condition analogy string length issue apply causal principle two object significantly past influence cause type causal vice cause include three read similarity text author influence influence construct causality makes discuss significant similarity genome similarity evolution usually instance common history identify common algorithmic low write stre binary imagine word similarity observe similarity understand algorithmic causal condition implication justification lemma string direct acyclic recursive form compression condition parent q separate distinguish version algorithmic intuitive string string cause modularity description later mutual every algorithmic analog string string string eq need star operation clearly string subtracting invert state generalized processing inequality string name processing justify arise scenario inequality obviously pair equality use equivalence kx kx ix kx kx ix kx z ix third fourth assumption tuple string tuple string contrast interpret conditioning string concatenation inequality expression e remarkably kolmogorov complexity multiplicative equality apply complexity since string perform kx combine n n px imply conclude subset string induction kolmogorov complexities complexity eqs kx px separate recursion ks ps r pt ks iii suffice separate analogy terminal assume mean string string twice step length step nd eqs show derive functional model causality dag causal among parent additional input formally compute input additional input jointly sense program contain drop assumption program assume parent causal mechanism universal spirit interpretation thesis formulation way influence signal computation process message bit string term communication scenario yet quantum theory also rule classical variable break classical mathematic string instance define believe intend motivation imply algorithmic respect proof w observe via second computed w parent parent trivial program mutually nod satisfy compute empty input last sentence mechanism generate independent general mutual independence mechanism formalize object observe parent complex another explain sensible define algorithmic existence genetic significant human however human genetic code particularly genome expect kolmogorov genome minimal make relation go human unconditional mutual evolution causal property individual specie relevant causal background similaritie conditional algorithmic go beyond evolutionary every causal specification information causality relative concept ask causality version causality relevance evident statistical ask height without specify people age translate relevant object binary implication algorithmic statistical inference generate accord eq conclude causal depend send causal design causal prefer hypothesis causal connection provide prefer markov short description formulate concatenation short causal reject selection cyclic causal side causal every causal minimal short total short description infer modification go bayesian discovery model provide rule give show causal causal go know focus example hypothesis reject gaussians elementary calculation observe occur string count reject need look converse simpler reject algorithmic would plausible sigmoid independent fact gaussian six example obtain independence relate stability occur naturally free causal length describe world notice binary crucial independence reject evidence case ii underlie joint could instance pair product hence would infer connection causal hand explanation switch ii I link connection fact two machine string detailed causal subsection example large precede subsection serious define kolmogorov occur finite computable coin produce head quite formalism avoid however root cause link total complexity show factor conditional compute construct attain construct conditional conditional define introduce double index stochastic describe transition two string q define matrix joint distribution string probability string define kolmogorov canonical conditional joint component string complex generic compare generic sense marginal conditional complexity vanish string digits digit conditional formally depend depend contribute q conditional general consist element class quantify complex detecting cause dependent follow cause joint kolmogorov small mass let singleton thus causal hypothesis complexity become relevant moderate sample skip count position guess frequency decrease exponentially copy digit want sample grow way conditional increase logarithm identically random independence assumption prior object dag algorithmic markov trivial implication coin time certainly justify believe coin influence relation coin coin dag relevant coin common fig relevant head conditional string consider node arbitrary cause common subsection discuss generate symbol important mind format read concatenation string consider partition relation string always causal condition causal relation instance source determine ensemble individual resolution get interesting consider scenario generating generate cause effect analogy causal causal access analogy set condition easily separate separate exhaustive triple subset condition remarkable asymmetric role violate reason example show provide string remove randomly string likely last digit vice process depict partition greater miss complex q correctly prefer causal direction even explicitly contain complexity markov closely relate algorithmic independence generate algorithmic would separate observable close checking kernel kolmogorov true use finite approach promise theoretical kolmogorov complexity minimum describe cover sample computable converge density reject causal observe mutually true argue mechanism justification irrelevant complexity estimator coincide complexity true attain general datum plus complexity correspondingly prefer causal direction algorithmic condition construct inference rule use justification idea full estimator subsample long carry significant independent markov causal hypothesis causal fold leave correspond still program hence subgraph robustness program likewise program selection rule use additional length subsample define also eq string program length full computable description interesting frequency description derive conditional instead conditional subsample simplified merging fig mutual low density desire property empirical process formally global eq behind generate subsample try subsample algorithmic apply conditional mutual reject show data map stre variable string know distinguish likewise subsample sufficiently large choose ordering contain description subsection already aspect string characterize algorithmic computable remarkable strategy provide consider example string string generate digits probability string observation subsample first information choice causal series useful ground give unknown give observation structure infinity direction joint inversion provide graph simple seem plausible apply recall total markov remark algorithmic fail coincide causal constant time justification complexity argument principle unique describe unique kolmogorov latter exceed first walk describe step conditional read first distribution bernoulli number right e elementary px j priori object mutually j x initial alone already share algorithmic process surprising represent instance process statistical generate independence essential cause effect describe causal machine generating value series machine resolution entail constraint direction recall two dag skeleton undirected structure structure skeleton inversion agreement sample rule independent condition replace computable notion subsection practical develop kolmogorov complexity kolmogorov complexity simplicity make develop far subsection describe kolmogorov conditional seem already lead lead furthermore identification direction often easy one point consider marginal sharp peak height position center great applying define right peak peak ask map also map hypothesis eq column already kolmogorov choice position equality location lead conversely uniquely however prove high value kolmogorov family necessarily denote shannon need stochastic achieve exceed coincide processing apply different never I derive require set matrix choose appropriately exceed attain ix j ix r r iy iy follow since second mutual inequality combine last least exponential complete apply peak mix generate assume doubly mix yield distribution average kolmogorov require double walk two peak position kolmogorov ask peak peak necessarily make datum choose distribution centralize peak random position seem random reality expect like peak one smoothed gaussian dash rather cause lead opposite operation whole reconstruct peak opposite discuss behind way reason another kolmogorov complexity probability translation real apart continuously differentiable probability fisher see show map monotonicity let quantifie degree process able convolution gaussian hence translation invariant map backward easily argument quantity general group denote random correspond easy map map concatenation map invariant haar information theoretic measure degree respect reference haar context occur physical reference temporal quantity interpret mutual introduce increase conditional markov kernel first additive group act lead stochastic extend act string form distribution would asymmetric way map absence mutual kolmogorov real value variable sample statistically correlation previously justify detect function cf sure pair complex string program yield correlation statistical generate possible kolmogorov empirically develop inference rule compression intend algorithmic chen al kolmogorov genetic sequence evaluate extent subject theory resource description define step advantage resource disadvantage statistical algorithmic counterpart analogy statistical algorithmic mutual unbounded symmetry resource version nevertheless infer resource future reason take provide additional causal argue logical follow describe object shortest logical device logical depth indicate object process many causal information follow shortest description resource role causal compute hard computation cause goal describe cause complexity ignore algorithmic observation causal way causal condition link causal
lack integration agree consider suggest three co formal comparison update calibration ii ii I complete invert note pair trade rely construction degree unobserve spread exploit excess return building previous modeling spread first quick estimation trading use monitor device uncertainty experimental result historical integration exchange trading know exhibit certain year considerable amount attention financial study run stock price attention stock price mean series evolve tendency constant historical observation price able forecast past seminal stock horizon several market e asset potentially historical study examine implication allocation asset management work asset fall simple portfolio trading go asset another asset net broad often illustrate implement recent instance first find financial term tie common necessarily spread instance price spread quantify asset relative exist trade open equilibrium would spread extended asset price exploit notion price relevant statistical involve exchange aspect investigate explicitly result spread stochastic evolve analyst precisely salient mean trading specific adopt spread arise trading observe noisy realization spread true market unobserve spread discovery suggest use tracking process unknown exposition review build methodology introduce dependency parameter gain quickly change generate motivate discuss advantage satisfied parameter estimation convergence well suggestion prior also produce measure cost discuss important realistic approach illustrative monte carlo estimate particularly advantageous track change monitoring may stop rule trade historical stock pair remark find argument appendix candidate financial price point denote let usually ordinary ol historical large capture movement mention penalize ol use recursive spread may return initial requirement spread spread state real restriction impose otherwise innovation series take variance state rewrite expand expectation variance regardless run otherwise unbounded analogously regardless conversely unbounded geometric mean adopting rewrite white uncorrelated equation define develop involve unobserved state variable reader refer length unknown estimate datum parameter kalman order fully specify mle demonstrate recursive may base back size run window ahead implicitly analyst effective ensure vary place instance although value guarantee notable special market price structural price change question much trading strategy accommodate strategy domain notably automatically modification cope limitation three contribution release vary create price persistence secondly efficiently burden frequent calibration window generating intervention line monitoring characterize continuous feature enable decision recursive computed informative quantifying assess estimation error potentially exploit trade practical propose space force process autoregressive ar substitution obtain ar time evolve stationary case via ar series vary ar accordingly spread unit form govern gaussian assume innovation series uncorrelated also initial inclusion component mean property true spread condition apply circle mean ar change enable detail involve change corollary give important weakly ar benefit gain reducing far spread autoregressive vary ar order ar ar lag mutually distribution readily see space diagonal element adopt inside solution unit effectively consider adopt naturally section root initially follow bivariate comment prior place elaborate posterior tt writing recurrence refer ahead recursive need bayes distribution easily interval degree freedom interval recurrence series frequentist perspective ar discuss work recursive time vary autoregressive nonparametric gaussian particle bayesian early widely via package http www west update posterior initial calibration quantity extract bound recursively extract assess possibility mean example specification responsible evolution sequential discount discount move specify ta section evaluate exposition diagnostic standardized forecast likelihood factor standardized standardized residual distribution freedom therefore tool write define good py denote carry follow bic particular maximize discussion hyperparameter application diagnostic criterion historical update adaptive ratio briefly may value respective tn sequentially extract empirical quite kalman exactly result kalman value important implication stability instance covariance make system vary aspect instability result instability stable provide already reduce time ar satisfied equilibrium special easily convergence appropriately series eq asymptotically start component brief section provide course specific preferred analyst want spread g nevertheless sensitivity calibration specification negligible streaming phenomenon detail study prior specification ar expectation availability historical example set follow place report crucial estimation forecasting proportional reflect informative prior zero provide place observational sensible proceed specification maximize give condition alternatively corollary present simulation latter analyze optimize factor two discount factor multiple assumption spread process involve eqs describe observational sufficiently trade hide state window true sensible position later spread conversely decide portfolio spread strategy ask additional question transaction word trade entry threshold guarantee cost become procedure may alternative way point perhaps extreme theoretical rate execute trade single forecast ahead spread dynamic spread trade short stop implement sure strategy surely signal quickly appear estimation deviation soon integration suitable universe search extremely build co integration alternative include turning factor final note may spread note recursive use coefficient capture integration asset price necessary historical model vary penalize heavily solution originally q initially start arbitrarily kalman develop monte demonstrate fast describe simulation simplicity quickly achieve initial explore situation vary spread equilibrium jump clearly mean sub process mean different discount monitoring result throughout contrary vary track almost piece able track
devoted trend mean confusion link instance stress choose consider general unit heart series except build add precisely consequence parameter g k k l jj phase example certain record distinguished reason detect phase parameter several estimator consist logarithmic representation wavelet noise study choose wavelet estimator performance exhibit persistence correlation decay characterize long memory record phase lead check dependent try aggregate present close fractional simulated parameter fractional gaussian series figure detect fractional sequel testing similarity graph let stationary sequel stationary say eq lag write slowly I self exponent aggregate causal distribution self process taylor zero corresponding fractional brownian motion increment increment trajectory suitably estimating estimator else unchanged frequently al aggregated variance long briefly aggregate window window deviation error power choose fig suppose behave stationary add previously second stage suppose trend finally fourth aggregated method apply optimal reach likelihood estimator extension frame minimax al type trend add competition encounter polynomial great trend trend improve polynomial law wavelet modulus wavelet define wavelet eq continuous provide wavelet coefficient suppose function satisfying vanish moment wavelet apply self similarity base increment increment prove therefore coefficient power regression discretize graph priori general establish et al converge criterion window behavior processes degree trend vanish chi square goodness test path therefore aggregated analysis square line scale behave appear estimation simulation concrete analysis wavelet fast wavelet path wavelet mean level wavelet estimation wavelet estimation essentially hand seem slightly trend estimator apply series follow figure example estimation phase obtain change estimation phase table series middle different estimation process assumption sequel still satisfied imply wavelet method accept seem behave small frequency frequency conclusion remark clear characterize signal wavelet multiscale self quite similarity self similarity frequency self similarity frequency signal center pressure force platform measure frequency fit aggregated behave self differently frequencie brownian motion mean er existence stationary increment like band q density wavelet convergent estimation st dt property moreover limit convergence na h central establish base wavelet generalize regression remark problem band assume consist large band scale compute estimator band use consider secondly wavelet apply wavelet large frequency band frequency band signal h h signal wavelet reject comparison series estimation reflect modeling series sample obtain close characteristic distinguished indeed correspond p middle stage relate phase relatively tendency like begin clearly trend datum log plot straight prove long case wavelet remove goodness accept wavelet study use series distinguishing indicate time concern hour exercise heart stage observe appear sample phase value indicate value c fig difference behavior begin important middle start indicate heart law obtain reflect time series estimator prove stationary case wavelet goodness show classical exactly suitable model graph locally fractional gaussian could chi confirm goodness wavelet frequency band obtain increase appear detect validate estimate end bring interpret behavior effort detect change several recent long dependence main firstly clearly smooth secondly give article wavelet context smoothly fractional introduce characterize large process phase evolution local obtain al regard time exercise observe heart parameter
come close consideration bias pl rather separately parametric supervise formula dy py x give square decompose etc usual direct concern distribution random covariance impose strong consideration outline weak explore pl important intuition many address explore pl include stack bag essentially apply pl justify exploit besides pl ce use ce problem multivariate importance find convert stochastic problem eq dirac delta center permit parametrization degenerate sampling degenerate distribution homotopy one initialize iteration sample correspond percentile kl divergence suitably give computing problem minimize parametrize track properly constant enabling importance parametrize ce however ignore importance unity sample notation sec ce actually perform generate estimate integral optimize simply extensive experience pl perform suffer overfitte prevent overfitting recommend dynamic prevent shrink percentile large slow algorithm point extremely slow towards percentile thought hyperparameter affect pl hyperparameter technique work give partition two hold generate parametrize integral choose optimize hold performance way validation fold leave cross randomly partition divide test use sense case fold trade dynamically pick percentile set dynamic distribution come specify exactly instead well em precisely entropy broadly partition probability mixture specify pick hyperparameter select adaptively equivalently parametrize optimize ce share optimum ce ce generate cross minimize sample minimize subscript keep track hold ce algorithm adaptively validation ce pl ce present conventional method problem unconstraine dimensional problem varied dimensionality make optima minima minima gradient multi ce dimensional function detail ce algorithm number component xx population dynamically value precede version dynamically validation rather learn practice validation pick optimize hold ce gaussian mixture trial initial algorithm performance difference algorithm reflect algorithm varied trial experiment performance true plot comprehensive summary plot log optimization evolutionary visually one want find number scale reason interval logarithmic nature plot besides minus log positivity skew median dominate show performance run claim fig dimensional plot part reason show see performance magnitude gaussian gaussian surprising multimodal problem optimum median single gaussian median gaussian mean mixture gaussian ccc mixture single performance mixture mean gaussian mixture performance h dimensional unimodal flat gain ce ce converge mixture improve compare ce mean indicate ce ce minima mixture multimodal nevertheless certain distribution see trial variance conclusion final gain fig early final variant eventually optimum perform poorly bad scaling seem arguably slightly careful reveal evaluation arrive optimum inefficient picture relation pl follow pl addition largely sample pl address parametrize integral publish moment variable coupling consideration ignore seem pl tradeoff alone capture improve ce use ce early moreover gain without technique iterate closely investigation median gain use possibility overfitte lack would tune hyperparameter exception probably require use mixture perhaps consequently extremely ignore iterative seem subtle likely believe interesting insight discovery examine ce broad monte carlo pl know pl bias tradeoff pl range estimation integral estimation pl similarity variance pl enhance tradeoff significantly improve ce pl technique replace marginal mse term variance control depend quantify stochastic estimate way mse trade variance parametric exploit tradeoff extension parametric machine community describe bias tradeoff concrete conventional bias couple technique integral unbiased tradeoff monte optimization illustrate machine learning ignore apply bias variance nonetheless simplest accurately predict consider number input element target next single response guess define suppose identify square make value general statistically nothing know establish write application rule conditioning algorithm ignore expect provide insight value apply simple law distribution depend also moment moment arise consideration provide therefore addition moment coupling ignore coupling ignore high moment ignore
sound reasonable initial collect solve set available randomize obvious aim combination ignore set previous keep unknown problem guide selection exploitation online obstacle bandit problem advance introduce loss expect online provable paper organize algorithm bandit introduction introduce solver along regret selection computational contain multiple copy randomize seed feature vs available attain measure focus one normally decision improve optimisation vs among entire distinction allocation selection execution problem technique knowledge subsequent distinguished technique every instance solve seminal selection offline instance optimisation terminology field deal focus solution quality runtime condition numerous instance per learn offline focus area static allocation resource portfolio paradigm recently candidate algorithm predefine amount horizon sensitive policy solver case reinforcement recursive form dynamically sequential set static method offline find accounting usage minimize offline use branch tree allocation present generate square unfortunately grow process knowledge indicator good contract share adopt state reach estimate window problem aim bandit variant attribute round optimisation run allocate reference armed choose loss incur loss optimistic aim bandit solver loss choice pick original assumption allow loss bandit consider history game adversarial probabilistic strategy full game consider instance multi set arm instance view generate mechanism problem available unless decide bad receive bandit typically minimize regret single arm implement present unfortunately per dominate problem instance well situation instance great nice property allow among algorithm selection mapping algorithm resource allocate correspond spend portfolio runtime solution runtime example assign single vary heuristic sound parametric use value among working case pick algorithms bandit initialize il non selection combine exploration represent whose meet interesting relax require solve practice allow combination incomplete sect determine good time say r allocation eventually inspire play distribution expert history reward level game play expert arm distribution expert distribution low arm expert suggest per arm game exp exp straightforward convex moreover exp loss plain assume use tune solver poor set particularly deal exhibit among instance even interesting regard loss consider full information game algorithm adapt sign reward work grow instance meet constant suited game loss arm quantity relate incur solver indicate trial quantity consist trial logarithm exp otherwise process extract actual value epoch update estimate cumulative starts solver bandit losse arm trial ji ji np ii ei ji n ji assume first know exp loss light expect appendix variation loss subroutine use unknown new current problem loss initialize exp light ei l l iii arm unknown obtain situation relatively algorithm exhibit variation eventually light use follow include nine optimize quantile whole portfolio share evaluate quantile time allocate share compare minimize advantage improper solver obtain conditioning spend far fix nine range uniform scenario complete rand benchmark solver guarantee terminate overhead algorithm fast allocation share alone solve instance cumulative overhead set upper confidence bound random
select type high n mn expanded st cf provide justification application st identify projection ip nn q towards determined divergence ball metric center observe type say type high value physics form expand parametrized type suppose px jx j j parametric solve equivalently obtain rx jx realistic analogue result grant concern maximum parametric extension latter entropy view certain probabilistic justification number posteriori state sampling sampling objective mass simplicity presentation restrict treat pmf endowed topology pmf sampling prior summarize nothing posteriori qx ie number counterpart deviation sect introduce respect divergence nr set formal posterior sampling consistency note supremum asymptotically condition supremum permit view another maximum q trivial transformation specify posteriori probable say qr turn pmf ed distribution characterize jx put induce probabilistic justification also rx know one
combinatorial optimization follow decoder ii index stand submatrix compose column index np decoder solve one relaxation comparable omp sublinear relaxation offer equivalent precisely viewpoint great I produce isometry matrix goal aim solve generalize original well performance measure success recall sparse solve subject play construct quadratic use binary symmetry suffice relaxation turn exact relaxed problem lagrange function easily concatenation resp lagrange constraint component zero dual eq l positive eq semi definite try definite situation non singleton contain rank equal subspace coordinate last exist vector easily check value thus iterate dimension non uniqueness despite sdp relaxation drawback imply sdp naturally great try nice candidate moreover problem overcome enough hoc constraint trivial sdp gap propose primal seem semi great naive relaxation overcome drawback scheme formulation could nonconvex merge unique choosing implicit lagrangian vector dual main problem dual performance sdp due dual relaxation interest theoretical suboptimal alternate restrict one lagrangian optimize lagrangian suboptimal ax lx lx z ax lx l lx know indexed thresholding solution carlo alternate reweighted reweighte square support support nonzero normalize lagrange multipli equivalence alternate nothing successive algorithm decide choose alternate nothing plain decoder reasonable choose success method outperform iteratively reweighte least reweighted method reweighte success reweighte l hard ahead alternate lagrange multipli monte decide base intuitive criterion suggest well plain interesting duality bit perform heuristic alternate function pt pt pt thm definition thm email fr construct allow often
partition gp burn begin limit lm partition value gp initialization segment reduce datum lm calculate cv test random logarithm tune lm iteration gp fair response calibration st median rd notice extremely local quadratic gp herein think resource work contribution domain computer experiment computer linear linear dimension entirely largely ignore bayesian gp particularly towards learn input domain characteristic framework experiment parameter space gaussian retain exploit perspective encode mind prior show beyond conceptual simplicity linearity extract combined yield highly word surface extension variety environmental requirement complexity produce e lot computational lm fit well see limiting case gp numerous issue instability desirable choose adaptively lm goal gps linear major flexibility greatly situation gps remainder paper organize review gp argue intervention limit feasible thorough exploratory reveal broad gp behave tend motivate parametric lie versus synthetic modern discussion linear dimension gamma wishart g random process separable family generalization section defer central allow model highly metropolis hasting mh conditional give posterior mh draw normally distribute posterior cart classification leave place usual regression requires place reversible jump simultaneous boundary posterior boundary tree gps yield extremely gps provide divide model special g etc extension discuss paper package http www web package index implement default specification describe throughout unless see special limit lm replace hierarchical parameterization lm gp flexible gp consideration also numerically operation computing grow cube sample numerically diagonal scale input lie cube range mostly surface gp limit come stability exploit equivalence great prefer key idea indeed intervention range model krige krige predict determine krige couple range towards exploit parameterization impossible regard krige neighborhood reveal shall conduct exploratory platform jump study posterior gps evenly space interesting dash line dash gp maximize solid dotted ml indicate ml conditioning variance solving give form dd column bar dot dash column gp surface likelihood sample fit predictive surface gp set parameterization likelihood linear likelihood surface correspond show mle horizontal surface look drastically top dominate contrast surface look likelihood axis parameterization sample variability datum gp comparable lr surface sample lm simulation surface corresponding likelihood like upper would sample small size less sample min st mean rd gp limit clarity show ml parameterization ml lm evenly space random lm quantile histogram show gp well lm sample favor ratio covariance numerically decompose illustrate phenomenon model stability avoid ern lm suppose examine gp ml eq specification specify dropping term solely specification carry encode prior gp zero problematic certainly far treat stochastic gp fit truly linear task put rescale nontrivial response large interval linear roughly population surface smooth depict histogram usually alternatively encode specification however extreme intuitive linearity large away serve platform evaluation parameter fix ht fit middle row bottom row integrate posterior axis sample one per column row surface parameterization gp fit show top bottom row influence posterior would surface interesting representative dominate still cumulative posterior look surface limit parameterization map gp sample small extremely low contrast bottom panel show top right right integrated posterior range line look gps parameterization actually fit likelihood posterior column sample quadratic successive increasingly middle show row integrate range axis line four sample column sample less axis influence become although surface remarkably due likelihood compute ht augmentation latent ib I range operation matlab preference likely jump prior preference determine whether gp minimum take surface primary unless intermediate describe lie gp extension full covariate spatial gp act traditionally priori implement subset prior still covariate increase scope high dataset high dimension curse dimension dimension reduce gp product drop detect marginally lose prediction parameterization gp know know term zero helpful formula simplification look familiar writing obtain prefer invert zero proceed usual gp fit nonlinear surface partitioning dramatically increase parameterization predictive jump hereafter illustrate synthetic extend gp whereby recursively thus experiment partition linearity extract spatial separable consistency case clearly isotropic proposal accept rejected construct otherwise right true gp histogram partitioning gp split partition action piece gp piece show surface wherein average obtain ten ghz histogram histogram show three fast classic acceleration head moment fit mean predictive bar typical noise analyze dirichlet gaussian process one hour typical inference effort contrast fit comparable one ghz identify spatial partition datum
assignment consist np hard eq constraint constraint case obviously depend attribute functional fix precisely restriction relax paper parametrize vector propose another solution way maximize produce maximize compatibility function supervise comprise typical vector element small pair function w minimize assume plus order avoid minimize expression predictor predict ability practice define parametrize address specify parametrize class discriminant consist optimal estimate discriminant maximal discriminant try property far determine joint encode formulation interpret interpret parameter encode compatibility function choose parametrization compatibility parametrization compatibility linearly map involve map attribute stress necessarily nod instance see section encode node perspective constant naturally experiment computer edge match follow section define loss incur match correct situation context match graph map simply euclidean scale simply distant regularizer combine order procedure incorporating consist non although regularization empirical number class loss certainly type approach upper exploit learn easy linear q feasible n hold claim constraint margin I gap exceed induced estimating intuitive reflect want mis prediction care mis enforce violate objective many finding bad matching grow solve exactly use known instead directly violate optimization eq q term bundle regularize minimization solver merely violated define input graph matching compute nn argument become w jj maximization carry assignment throughout cubic solve efficient solver house assignment c implementation assignment discuss independent note case unable find violate duality properly minimize full linear actually experiment difference g ir I decay result coordinate attribute standard graph g ii tuning feature use encode instance node perspective respect graph angular bin angle radial radius scale average distance scene similar set use graph edge scalar frame match red match correct infer match house label explore baseline separation frame separate exactly adjacency feature top previously normalise hamming proportion incorrectly match validation assignment beneficial maintain assignment outperform likely relative linear learning note similarly result quadratic assignment assignment computationally centre assignment frame feature point correspond indicate first radial bin radial important last bottom show frame assignment show noted effect run strong assignment well still either significantly experiment human video sequence corner detector detector identify within radius tune matching human identify figure advance scene find target scene correct scene hamming wish incorrect match distance match match interested graph figure show increase monotonic move throughout difficulty house also baseline figure centre heavily angular bin final image contain image shape robust rotation image reflect consistent orientation corner detector scene frame varied scenario error selection training matching loss testing training pair graph match margin structure efficiently despite constraint experimental reveal matching improve art speed major matching quadratic assignment matching camera reasonable camera surveillance change summarize substantially mm definition plus height depth em matching vision biology matching pattern model correspondence formulation assignment encode compatibility encode edge compatibility research approximately np turn attention compatibility function match human pair label reveal substantially improve find scheme art quadratic algorithm commonly structure include dna text image vision formulate attribute matching relational attribute vector find correspondence node maximize research focus np compatibility attribute semidefinite relaxation
quantum predictive rely preserve efficiency give quantum quasi predictive observation predictive quasi admit hypothesis quasi efficiently equivalence question follow new quantum learnable one learner query acknowledgment grateful receive helpful anonymous check proof q accept query call hypothesis testing check agree notion efficiency learn say quasi learner query testing ask call answer efficiently learnable version testing query phase check agree notion answer query learnable learnable version polynomial query notational predictive query would phase produce learn remain imply quasi reasonable classical turn achieve produce consist description answer subroutine stre answer subroutine predictive mode equivalent definition give rise learnable concept situation let nx hx x hx hx relation every final approximate class final hypothesis exponential outline imply learnable exist approximate least particular neither classical quantum class learn concept quantum quantum learn efficient quantum concept impossible relational concept mode summary cf theorem bar prove new receive surprisingly communication quantum trivial relational involved element integer accordingly r string stand kullback leibler divergence need x rd completeness x x rd rd kl x logarithm x x rd kl c cd classical sketch behind mostly reader limited analyze mechanic start quantum computation quantum computer efficiently membership teacher information theoretic consequence quantum classical hand principle quantum observer computational student quantum deal feed reasonable function quantum denote basis example correspond uniform measure computational relation quantum superposition naturally time obtain superposition satisfy quantum teacher model predictive ignore define follow end algorithm receive least learn learnable learn pair projective would follow projective space outcome assume outcome last state former state orthogonal learner answer proof use approximate simultaneously approximate answer c qx q ne forest consist span tree component e view element string distribution uniformly c c entropy mutually independent c q c q e choice separation demonstrate modify usual relational essential learn functional concept concept quasi set establish connection communication proof communication independent quantum complexity follow relational party begin player receive goal player message input length length latter possible law mechanic quantum consist protocol strategy share randomness allow protocol behave solve answer produce least message protocol solve similarly cost way classical relational theorem fy ff present new way communication call mode receive denote communication receive base alone trivial may appear side single answer theorem surprisingly generally communication quantum classical way pp apply case hold happen protocol random answer upon use share randomness point element cost answer sample z z z x negativity follow error require
shorthand string start symbol indicate symbol let denote ta sided interaction string finite string agent formalize system distribution conditional semantic couple interaction develop limited output policy diagram effect policy c area choose environment amounts partially control state state transition g chain environmental thought environment policy represent curve policy diagram analyze hypothesis environment agent endow operation see associate simplify interpretation diagram mapping state define markov space transition central whether control correct pa pa operation mode general amount posterior subset essentially understand asymptotic probability posterior q denominator reference indeed figure illustrate simultaneous controller speaking process measure whole growth use happen stay policy stochastically operation mode variable realization draw pm pm pm htbp heterogeneous temporal belong policy tm q divergence sub divergence form partition show decomposition htbp sub divergence eq gibbs particular control stationarity complexity prediction rate divergence realization vary hence requirement ergodicity class divergence process analytically q illustrate boundedness go result important input output divergence wants vanish mode policy share wrong accumulate apply controller eventually enter evidence long execute short period controller risk region strategy policy time motivate follow contain mode core iff tm expect demand execution agent finite probability operation mode vanish operation divergence process pm almost core indistinguishable happen figure hypothesis differ policy operation mode core unclear whether ever undesirable law clear need restriction mapping follow observation dynamic know us drop policy region say policy show consistency sufficient right control operation agent section operation hold boundedness impose divergence partial policy contain operation mode ergodicity prediction relevant formalize potential arise context controller determine away detail one think partition operation region operation mode neighborhood reward reward objective reward parameterized indicate difficulty arise bandit balance knowledge acquire new knowledge exploration versus exploitation tradeoff bandit high operation avg l l control greedy index begin curve strategy select action empirically horizon average suboptimal improved value decay significantly outperform least worth make complexity horizon several hour machine contrast control apply pre computation issue action bayesian action mode convergence imply bottleneck thus affect define tuple action state rx ta rx tt assign action mdps stationary mdps reward fix assumption chain ergodic mdps ergodic policy policy average reward non equation qx ax bayesian control mdp characterized rewrite instantaneous instantaneous reward plus reward mdp reasonably immediate reward determine intervention simply give apply posterior fortunately simplicity devise mode action intervention algorithm bayesian differ action test agent intuition achieve r learn variant exploration correspond fix probability maximize try high enforce grid especially useful test rl purpose easy contain controller move state another trial form enter bottom play agent move square default reward reach simulation follow carry learn sufficient control learn policy initially controller latter attain mode initially exploratory behavior lc reward key underlie adaptive control work idea compression experience minimize amount write require action code action inference decoder one lead turn recent researcher focus issue contribute body provide evidence treat calculus dependency agent stochastic completely general show control treat conditional stream never causality setup maker expect utility outcome kronecker delta distribution coincide hence compatible decision setup fact generalize bayesian thereby paradigm define environment construct capture knowledge intractable operation pre compute policy give environment environment define operation characterize probabilistic mdp section approach adaptive control treat bayesian associate bayesian computational nature solution intensive probabilistic realistic utility bayes optimal controller mdp consecutive action reach lead back environment controller either consecutive control environment prior mode stay uniform choose exponentially allow control operation step however break mdp bayes controller step boundedness idea underlie publish reinforcement learn important amount relate compression intelligence compression intelligence basis passive mix extensively bayes mixture universal expert previously universal environment study selection approach exploration amongst particular discuss armed accord likely strategy converge thus criterion derive law kl show class solve reformulate control approach equivalence return applicable duality continuous special treatment calculus decomposition adaptive action furthermore turn obtain causal calculus agent construct rule converge operation mode ergodicity demand indistinguishable minimum regard principled novel optimality heuristic take bayes translate stochastically posterior problem statement formulate usual relationship problem statement investigation crucial generality equation difference two one depend varied pm p pm cross minimize choosing obtain pa note weight mutually hence tm marginalization equality probability follow causal factorization point realization divergence sub divergence inequality hold right exponential rearrange divide side normalize valid pm process divergence see probability stochastically since gm upper bind yield mt pa tw mt pm
study usual voting particular trade work detail refer provably bring light variable lasso easily procedure suggest bag may computational eventually performance extended various focus fix allow variable grow setting norm regularization block loss finally note carefully logarithmic interesting uniform bound straightforwardly order simplify support extend account generate minor theoretical support concentration bootstrap replication exactly tight variable replication two p covariance pn compact enough pn e closed form desire mm norm problem asymptotic analysis decay correct show tend positive sample support lasso bootstrap consistent variable compare synthetic uci repository attract lot recent year much effort dedicate efficiently particular algorithm regularization path value cost single inversion justification lead vector know actually grow generate matrix verify setting however lasso light data weight add allow situation sparsity focus proportional number lasso enter variable tend exponentially several latter suggest consider intersection would always irrelevant variable enter support eliminate resample resample actually lasso refer enhance finally support lasso consensus keep regressor agree also get hyperparameter organize describe section illustrate synthetic datum uci repository sign vector sign index similarly denote submatrix compose row describe regard capability lasso predict covariate assumption finite joint invertible tail matrix simple assumption study grow scope consider p tend exponentially term norm behavior I consistency consider mutually exclusive explain fine present tend tend sign minimum v pattern equal satisfied tend pattern agree tend sign obtain pattern tend particular tend tend sign variable may regimes one induce tend zero tend infinite hope consistency lead regularize section paper derive result relevant tend asymptotically positive probability many tend potentially exactly multiple copy tend zero sign p pattern proposition relevant tend tend actual I replication n ii sample suggest support bootstrap intersect least fit sample grow slow always cm k simultaneously lar simulation regularization lasso cost appendix correct model eq constant tend slow infinity estimate correct well improve comparison synthetic obtain synthetic dataset uci repository zero independent diagonal first variable non loading sample magnitude e distribute probability odd black satisfied satisfied cm ratio variable probability vs consistency satisfy right empirical regularization asymptotic detailed right lead exactly range fix replication consistency inconsistent right region select correct pattern bootstrap replication see inconsistent right increase look always seem asymptotic analysis consistency fact relevant replication cm replication finally various linear
accurate term although estimate magnitude wrong loop calculus approach significantly especially uncertainty problem see algorithm mechanic state convex functional bethe free energy bp equation vector allow exactly belief relation probability ib I sum serve loop bi graph belief normalization belief bethe minima beliefs unity attain interior domain encounter e dominant present minima corresponding follow equation multiplier chemical eqs bethe energy interior might configuration discard reduce programming yield accordance remark modify minus sign latter saddle approximation numerically iterative eq indicate chosen ensure unity b numerical bethe loop ls explicitly adapt lead bethe accordance stand define subgraph bi connectivity loop odd give contribution therefore definite fact low exact hold finally generalized individual exponential derivative approximated integral representation order analytic analytic part eqs contour contours origin notice real contour contour component numerically concavity shall various maxima index account integral correction maxima correction lead sp reason fourth order give saddle saddle exact ratio zero weak suffice typical therefore saddle next section empirically dominant term indeed correspond sum maxima simplify g sp compare simulation fully polynomial scheme idea applicable problem assess order slow study different search respect parameter bp calculate accord saddle eqs covariance saddle accordance finally fourth respective contribution saddle choice particle particle find contribution sign separate linearly allow saddle contribution exhaustive general contribution contribution limited sign randomly saddle hold sufficiently typically saddle point particle plot result find scenario decrease thus sufficient characterize panel show use show black value peak although low right value green curve correction section use correction plot use scheme mcmc right panel velocity axis actual velocity gradient use particle similar trend correction around peak well extremely run time per point tool passive dynamic experiment self biological tool excellent bp order magnitude important loop perfect loop calculus model perfect matching loop rewrite bp proceed belief analogous desire loop seek must focus problem multiplier take derive product maximize pair hand pair whether loop university ny paris france proposition statistical moving consist passive flow consecutive graphical namely propagation providing want learn l match ls cauchy accurately point numerical experiment comparable one polynomial randomize carlo substantially scheme statistical science range machine bioinformatic physics error application evaluation sum configuration track random particle tracking frame particle acquisition increase trajectory statistically acquire uncertainty environment possible particle sufficiently move move shall passive particle mechanic indistinguishable probable state search maximum weighted particle turn complete belief bp calculate weight arise bp minimum bethe suitable graphical contain reason flow particle furthermore effective particle acquisition keep modern thousand frame per huge dimensional slice cell extremely impossible unless previous motivate development element environment flow modeling particle label typical neighboring
economic keep application situation book introduce predict sale expect book online side contribution sale tail side sale notation jump item item item potential sale around head ranking accord item tail theorem section rank potential sale negligible limit formula theorem imply sufficiently long since system ranking reach hold denote total sale rank sale pareto gamma q sale per sale convergent integral correspond tail sale rate tail calculus head great note b find represent sale situation great dominate sale contribute trivially contribution tail dominate intuitively explain difference result limit separately note dominate great former latter total sale change contribution total item sale unit store sale sale unit expect list sale top ranking ranking chance business extreme sale less extreme concern ranking amazon com value exponent range adopt calculation find price index use introduction value experiment less consider million web item business sense decrease online let return stochastic control online store item store sale past record online store decrease sale store sale express introduction item sale record fluctuation determination regime difficult deep regime rank sale rate title sometimes sale book inherent sale title observe top store sale sale show ratio calculate value adopted turn insensitive range approach ratio approach sensitive sale average sale sale explicitly cause item evolution item detail nothing divergence sale rate theoretically reflect ranking occur intuitively speak infinitely copy per top book follow sale side proportional word formula contain great total sale per total sale potential b tail side sr particular sale cause select instead top item sale measure ratio show ratio adopt insensitive near approach return would introduce sale suggest pareto exponent pareto pareto economic reverse one probably close theoretical line end realistic situation head lose head sale pareto extend pareto basically assume left side reproduce find side place place evolution pareto equal actually begin effect position rank amazon hour book per hour book intuitively clear cause rank value total sale online store rank estimating store return parameter ratio contrast discuss section ratio significantly contribution sale long tail long business also calculation amazon support amazon aware business sale economic impact example phrase store mass medium internet business site online business sale advance paper mathematical new sale sale ranking explicit formula example obtain amazon co particle stochastic process theoretically accurate suitable study long expand advance computer side theory serve store business method long tail sale book amazon open sale amazon sale ranking increase tail business business significance pos system page ranking web page web web obtain value pareto exponent page order principle drawback fluctuation social general free work grant aid education technology support aid scientific education technology institute school email new estimate sale rate book online evolution store method mathematical stochastic rank suitable quantitative study tail online give amazon co pareto slope sale store internet pareto school science drastically product variety transaction capacity possibility claim huge poorly product internet contribution sale long business advance online cost handling item drastically item item sale long tail business idea deal item sale record various market place try collect kind product thing advance result activity study possibility need precise quick long online million book collect sale record end list ten result book well sale ability book quantitative sale dominate want fluctuation item extensive law number hope sale store thank law number tail store observer purpose detailed item would specifically sale potential store ratio sale previous paragraph sale long observe sale sale back statistical page sale ranking book amazon com web page see title description book range million book sale store study ranking fluctuation sale observe rank single sale potential calculation online fact amazon com involve ranking number definition serve efficient sale purpose provide store business follow applicability practical situation formula summarize ranking amazon implication obtain amazon summarize simple sale ranking book rank range copy item jump rank sale well poorly motion item rank sale cause sale numerous assumption item ranking sale deterministic trajectory observe development book sale amazon website fit sale rate mathematically notation distinguish item sale rank initial satisfy sale non namely various sale ranking accord sale sale item sale occur sale j eq property distribution sale ranking increase sale rank I sale ranking sale time sale ranking ct ax ct queue mark item high see trajectory sale start trajectory n trajectory follow sale determine sale sale weakly q intuitively process sale determine sale towards tail observation book reflect jump order sale deterministic imply large trajectory sale count sale avoid precise time sale especially month ensure observe sale ranking reproduce sale distribution uniqueness accord laplace determine near item large sale rate regime make rigorous sale scale sale ranking x dy r regard sale x com focus fact evolution amazon amazon com involve process secondly amazon co home country book update title smoothly jump copy book check order copy amazon website hour observe million book amazon co book less per hour qualitative motion percent book amazon note stochastic correspondence amazon rank entirely I sale sale record far small though chance model usual point sale ranking number amazon co give evolution book formula start sale section measure function book sale book copy book sale pareto sale book book never omit theory actually sale sale reproduce exponent slope impact business exponent economic intuitive mean exponent roughly say great dominate sale contribution sale dominate possibility implication fluctuation strongly datum rank book sale tail observe hence number deterministic appear particle substitute z dx formula satisfactory integration come slightly different correction evidence follow formula time short title ranking fluctuation amazon huge fluctuation use unit elementary calculus underlie sale rate rank fine uniqueness inverse determination pareto fine smoothing laplace original amazon see update per hour amazon assume pareto amazon company plan control evaluate sale tail method practical preferable update rank accurate amazon title algorithm evolution sale amazon situation total book determine amazon website amazon website however amazon book describe pareto discard analysis
period typically distribution service impossible state scheme conclude remark article sample instead section inversion monotone preserve uniformly offer combine technique moment match mc financial engineering field high application intractable brownian break moderate emphasize usage nonparametric nonparametric whenever sampling require briefly investigate establish w c v mh analogously u w dl notational convenience generality unnormalize histogram taylor histogram bin addition q mi analogously eq analogously multivariate remain assumption line I negligible expression counterpart let calculation I begin analogously taylor summing bins dimensional yield treat analogously end term former proof omit together asymptotically negligible analogously crucial prove remainder lemma zhang l l main dependency weight j j j analogously negligible bin marginal begin analysis mid step describe histogram recursively mid relation search mid associate discuss dd marginal marginal bin mid carry small generating evaluation negligible generating apply e importance relationships american association estimator transaction model asymptotically pricing option mathematical finance monte carlo financial engineering york management science lee event conference c cox r computer pseudo generator transaction simulation improve practice york sampling multimodal importance journal association deterministic population model american york carlo new york curse likelihood ratio journal area communications american association mixtures uk journal zhang importance journal american statistical association efficient bayesian network figure lc l ccc ms ms ms lc cc method cc cc cc method cv mc cc cv server dot line server thm thm pc nonparametric mm importance choice nonparametric importance sampling parametric normalize unnormalized importance burden solve utilize recommendation application importance attain heavily outperform criterion compatible inversion method generation evidence usefulness benchmark queue length spam carlo multivariate rare option pricing introduction sampling apply demand intractable limited unless massive carefully change expectation rewrite importance sampling know derivative choose include impose constraint importance estimate draw proposal self sis strong large implie converge surely set error desirable central limit central limit theorem variance estimator w proposal median merely conceptual integration find easy sample approximate proposal traditionally proposal parametric density assumption limit support decay density investigate family parametrize proposal adaptively al expectation optimal proposal consequence new order suitable alternative rely investigation nonparametric nonparametric base proposal west restrictive estimator achieve mc proposal include importance aspect effect treat heavily nonparametric multivariate computationally superior technique sis loose explore finding result suggestion variance different provide technique suggest promising section sis mse discuss implementation issue toy sampling density estimator approximate analytically proposal empirical usefulness establish essentially restrictive compact support theoretical weak practical demanding purpose suffice employ nonparametric interest also well arbitrary computationally use histogram zhang drawback slow paper usage frequency cite construct interpolation histogram mid computationally slightly histogram consider multivariate histogram bin height bin consist trial step subject increase compact zhang step v omit integrable derivative furthermore bin mc h rewrite variance attain bin choose optimally hold h implication proportion strong result estimator zhang move achieve substitute distribution write p om suppose bin eq f consequence improvement bin theorem due theoretically asymptotic stem non usage proposal lead approximate proposal surprising mc reasonable positive algorithm separately achieve done carry denote simulation study section nonparametric function integral self solving problem choose know proportional self analogy step j j proposal q sis bias asymptotically easy asymptotically self frequency fp draw use cumulative distribution fp convenient linear univariate q distribution piecewise provide underlie bin inverse see linear intercept follow recursion let uniform distribution compute calculate mid remark sample request remark complexity detail order mc prove estimate regularization whitening see instance al induce severe besides arithmetic operation parametric nonparametric require call function example parametric first evaluate third reduction efficiency mse interest simulation precision intel cpu ghz code pseudo report define dd unit cube separately bin plug spread show mc report least computation significant improvement also favorable order investigate efficiency plot mse small dimension computationally strongly magnitude whereas convergence variance indicate rapid concerned pricing call volatility evolution stock differential sde brownian motion solution sde rt st pricing integration payoff standard pricing shift standard proposal drift technique optimal drift simulation drift simulation option price option say latter affect relative whereas coincide convergence variance confirm applicable proposal employ reasonably explain satisfy report however significantly payoff imply multimodal proposal advantage expensive mc massive burden roughly remarkably dimensional investigate strong dependency become vanish favor relationship sis proposal trial west envelope density sis far initial guess show respect sis application investigate spam system active readily basic briefly single server single room capacity service distribute respectively theoretically restrictive world arrive
delay belief error datum simulate accept define maximize acceptance summary live sense relate innovation inference give exact inference accept posterior give note accept algorithm toy approximation mixture normal q assumption possible approximation standard cumulative suppose error truncate onto acceptance mean variable approximation limit understand posterior implicitly error approximate rejection say give cut euclidean poor measurement outside way choose either error measurement straight measurement replace distribution belief specify zero unknown care constant theory infer include measurement build noise rather datum give analytically abc population occurrence structure structure cause known algorithm without algorithm give illustrate describe record date branching depth time interest branching parametrize treat prior specie record model preserve cumulative branch epoch find process explicitly base posterior unknown abc use epoch give perform analytically exact accept need measurement parameter make normalizing normalizing slow practical measurement acceptance carefully error straight wrong hard model calibration assimilation provide purpose carefully model represent reality argue deterministic model break part physical process model process etc specification use simple economic cell underlie differential may measurement error separately aware far make likely multidimensional approximate interpretation measurement easy break error wrong assume prediction wrong reason helpful acceptance principal small summary meaningful interpretation general summary suggest select summary summary require summary informative increase exist use think choose abc study use mutation eight world summarize number summary mutation literature effective population growth model mutation summary measure triplet summary result summary summary cut quantile close meaning posterior measurement conclusion preferable surprising light value assume measurement use perhaps surprising constrain prior distribution rejection inefficient high likely application behind chain chain successive spend mcmc hasting assume give version hasting approximate metric equivalent tolerance assume arbitrary space stationary construct obeys move probability q set introduce auxiliary chain space simulate q chain distribution sufficient detailed balance equation stationary equation algorithm detailed balance equation stationary equation detailed balance mcmc ratio rate instead advantage normalizing expectation respect simple prior simulator expectation instrumental acceptance reduce proposal accept reject version sequential algorithm consider algorithm slowly variant metropolis move generalise kernel move cut introduce due store particle partial rejection control introduce abc reject keep particle theoretical acceptance rejection evidence evidence selection perform although unstable vary eq stable acceptance tend equation abc summarie bayes summary statistic general represent simulator acceptance careful simulator simulator simulator far similarly summary simulator careful simulator reproduce automate summary coincide expectation simulator reproduce construct summary strongly produce sensitive simulator capture phase insensitive part summary appearance unclear learn sound physical view choice see difficulty abc clear paper consider part towards understand paper summary reduce whether sufficient summary layer assume summary nothing inference simulator discrepancy term want move simulator reality currently model term represent inherent physical acknowledgement interpretation thank dr early manuscript theorem abc likelihood free posterior making depend error abc make paper guide choice abc replace acceptance distance acceptance term enable apply chain value light belief error
involve cell cluster cluster consist need z tc verify posterior easily tw article cluster functional approach together multivariate function flexibility jump point branch employ far reproduce parsimonious general heterogeneity datum fit rejection control parameter validation real open code available functional spline title penalize rapid throughput repeat take scientific temporal study sequentially biological time expression thousand simultaneously gene gene gene similar profile co participant biological thus gene homogeneous mechanism account temporal dependency repeat hierarchical datum order observation yield unbalanced application multivariate multivariate cluster measurement replicate point study treatment factorial incorporation covariate add I subject active functional approach datum factor accommodate knot cluster must drastically lead classification motivated gene propose aforementione link nonparametric effect great jump branch employ hilbert notion parsimonious effect correlation structure mixed effect nest heterogeneity functional characterize design rejection eliminate decompose simultaneous maximization penalize track functional fluctuation method subject confidence extremely powerful article simulation remark theorem assume homogeneous cluster present mixed functional mixed population assume generic order sampling ib random reference specie functional decomposition multivariate main identifiability term specification associate accommodate e correlation across let x profile profile b estimate eq fidelity quadratic quantifie parameter control smoothness since two function therein use extensively hilbert nice hilbert loose evaluation functional simple hilbert integrable well define consequently constrain evaluation reproduce rkh suggest hilbert derivative inner product functional exist negative function call scope reader reproduce space semi norm function space decomposition subspace function form subspace reproduce coefficient distinct combination consider take fix level decompose satisfying satisfie variation contrast use force additive yield list td ts r ir mt il z derivative one follow fit cc array cross treat parameter extra adopt one may standard function tuning newton minimizer distinguish generalize cross justify theoretic quadratic track asymptotically score approximately minimize rigorously nonparametric exception interval nice variance impose decompose prior space eq prior specify generic expression solution tb percentile letting interval confidence pointwise unclear coverage base homogeneous heterogeneous assume model cluster ib kp membership belong cluster probability smoothing derive e ib reproduce hilbert td k tc tb ts I ir mt il z nk sr b use play optimal parameter smooth observation probability th setting give observation adopt estimate ik thousand consideration result huge small involve expensive unstable algorithm em rejection control em control ik w ik right em reduce variant accurate greatly reduce avoid optima run early gradually original arise algorithm stop stop rejection control stop gibb high model impose total parameter scale logarithm balance issue determine effective bic define number simulate simulation function replicate indicator otherwise distribution matrix analyze simulated individual random feature either enforce additive mixture specify implement software give algorithm let true eight plot agreement membership popular rand rand value cluster moreover range rand method cluster priori development biological feature specie common level gene measure stage day report life cycle include stage genomic connection across expression result q effect model branching spline analytic form branch branch branch cubic spline cluster biological gene use cluster discover cluster mean bayesian interval consist gene peak cluster formation transition development body cluster expression reach peak expression gene involve
slightly well alarm interpolation increase equivalently plan divide measurement investigate autocorrelation possible part pc detect framework predict series equation compute recent occur algorithm change likelihood scenario decrease useful parameter like used resource scheduling transfer choose characteristic time internet describe stationary simple characterize stationarity transition cause throughput explore assumption certain interval process discard old indicate likely extensive variation elementary detection control application location interval series test conventional interval formulation occur location tracking algorithm monitor offline set real constraint online keep incremental update base previously incremental embed generally algorithm online goal soon identify sensitivity specificity avoid positive window fast enough satisfy requirement nevertheless application evaluate online predictive imagine observe want predict absence observe next irrelevant usually know infer whether occur exactly common functional likely previously threshold discard predictive assume assume weight intermediate goal recent intend seminal tracking estimator normally generate predictive current extensive recent work autoregressive process detect change mean receive relatively estimation et service detect database david detect stream current window past reference present summary technique bayesian know iterative section include synthetic introduction interpretation bayes relevant body evidence evidence sometimes normalize interested process prior computation reflect belief evidence formulate mutually exclusive compute drop compute suppose random selection proportional use event al shorthand write bayes want distribution produce generalize yield arrival random meaningful bayesian interpretation reflect belief use unknown compute scale inverse chi posterior derivation et page imagine simplify version start occur interval interval normal mean variance know know hypothesis normalize equation easy follow section simplified relax hypothesis datum propose equal drop exclusive complete normalize generalize technique add exclusive normalize drop know occur evaluate propose q subscript like probability last second know exactly compute store estimate time step start approximation accumulate total roughly partial function partial sum likelihood proportional algorithm online real show series shift bottom cumulative appropriately mode recent mode another near chance probability add chance tb feature case unknown two plug approach alternative equation choice detection seminal dataset record figure three salient three near conclusion near last evidence time quick quick reason uncertain change likewise indicate generalized work effect much estimate accurate algorithm since two evaluation could probability reach simple implement criterion goal online alarm occur frequency false propose compare geometrically alarm alarm alarm gold standard online sum delay parameter parameter distribution unknown use construct alarm indicate
via eq value width interval yield set denote begin henceforth definition us component growth express sample induce order lexical ordering first order version short q brevity sometimes write denote union order collection let non contain union verify mutually exclusive equal also let remove complement condition exist argument still however combine distinct reason point differ e I follow prove element maximally overlap collection follow class binary let proceed class define set interval form least hence distinct interval subset set comprise subset say h cardinality large lemma cardinality theorem mm plus mm minus plus minus plus minus plus minus em minus em cm letter letter letter letter letter letter minus package ps university ac finite subset hx set obtain growth trace class sample let function limit necessarily behavior label learner obtain smoothness notion q call width denote resemble notion sample sample width description know efficiency implicit see logical denote statement integer refer convenient function refer brevity cardinality wide complexity correspond hence cardinality complexity follow subset subset g obtain trace trace collection convenient union value growth let binary growth follow generalize write indicator statement thresholding choose
kn bayesian procedure truncate truncate decision eq everywhere everywhere additionally control rule rule additionally completely analogous eq satisfie particular fulfil e equality without draw consequence mean flow incorporate experiment value sequential two component case notation section ny nf nr nr precede section function transform stop equality everywhere multiple simple distribute see suppose sequential thank read manuscript carefully comment suggestion author city support cb receive mm control cm v x x mm control affect distribution hypothesis allow control follow sequential start assign control observe variable choose observe suppose eventually stop moment favor characterize procedure type keyword mm sequential sequential hypothesis control sequential procedure test set depend usual value observe control observe analyze supposed experiment stop favor article structure multiple hypothesis dependent variable vary sense start randomization control experiment yet use subsequent use variable every related control subsequently sequentially fit mainly cost follow case hypothesis hypothesis briefly randomize hypothesis triplet suppose control suppose measurable value suppose measurable interpretation experiment start apply determine use probability continue next define control way experiment suppose stop decision accept interpret accept stop stage control policy variable throughout paper eq cause follow sign hypothesis sequential article minimize procedure q sequential reduce minimize constraint unconstraine new lagrange multipli finding problem stop final control unconstraine lagrange reduce problem respective unconstrained idea multiplier define multiplier exist testing least strict lagrange multiplier testing least minimize minimize problem q infimum finite stage give control apply q attain everywhere everywhere eq low among truncate applie nothing easily see mean good think obvious rule stop control policy precede apply idea bound obviously need practically coincide except completely lemma instead increase q passing limit side limit lebesgue monotone virtue stop low hold see hand coincide characterize almost almost satisfie everywhere satisfy fulfil hold coincide theorem substitute theorem treat optimality take remark easy structure optimal strategy ratio process distribution absolutely respect precisely us ratio definition recursively well measurable see see argument use start expression almost sure sure minimize testing error probability recall sequential hold well thing prove assertion obviously strict analogously hold inequality remark extend see hold obviously extension moreover theorem remain strategy prefer
interest mixture relate predictor enkf move towards region pf adjust character estimation pf enkf pf fail enkf formulate background common construct equip column function mesh domain smooth fast converge define gaussian eigenvalue equal fourier cosine vector another norm original ensemble member advance consist forward assume link would absence state forecast density bayes density weight step forecast q ensemble enkf sample distribution forecast enkf weight use thought sample ideally sis giving distance th member norm cc likelihood density enkf sis enkf sis enkf sis able sis enkf alone enkf non gaussian sis enkf assign ensemble enkf sis enkf sis alone equilibria unstable equilibrium stochastic make flip equilibrium ensemble determine track model one fig enkf describe track unlikely ensemble member close sis tracking enkf enkf evolution numerically likelihood scale quadrature point take one exhibit switch explicit euler perturbation add right step simulation assimilation cc filtering function fig fourier generate initial norm enkf handle resample forecast ensemble sis sample demonstrate potential filter presence question apply bayesian mode model national foundation grant com center predictor assimilation enkf large particle pf nonparametric advantage enkf pf assimilation kalman non dynamic drive application aim integrate acquisition dynamic assimilation modify component generally million time change filter evolve
l precede section objective express coverage unnecessary sampling operational effort emphasis stop stop deal process introduce refer function desire plan terminate I accomplish event stage observational accordingly I si si precede tuning weighting coefficient random u equivalent requirement coefficient meet requirement satisfied tuning apply search tuning sufficient satisfactory properly limitation testing plan effective plan parametric hence weight formulate minimax reject plan propose determine coverage minimize task propose maximum follow use search k return weighting coefficient approximately minimize b minimax optimization begin risk guarantee indicate behind reject ib I argument parametrization stop connection consequently believe u I reject reject time inclusion plan framework stage stage n x notion ii hypothesis consideration sample scheme condition number th estimator consequently analysis significantly simplify parameterize great associated likelihood joint r note mle may sampling define nn n unimodal likelihood th principle propose prohibitive burden design global construct structure risk parameter risk parameter idea tune acceptable virtue risk prescribe level tune moreover accomplish furthermore function possess check multivariate z n z z f inclusion nn ig n vi I reject reject I accept accept unbiased unimodal reject hold reject reject reject reject see apply simplify rule x apply stop simplify n z otherwise modification remain unchanged approximation simplifying rule g accordingly rule n n w rule remain unchanged guarantee rule binomial population simplify rule proceed n moreover g c n scheme x maximum less minimum ii ig I precede simplify stop technique likelihood ratio construct purpose let let function parameterize f hold nonempty virtue construct stop base inclusion principle shall tuning parameterized assume objective requirement statement vi reject imply small every determine consequently specification risk wrong constraint requirement choose large reduce subroutine risk infinite parametric essential develop risk vi check h virtue statement purpose determine reject h vi checking algorithm therefore efficient subroutine determine value number subroutine requirement apply interval risk risk assume extended control accept accept accept j j accept statement iv spirit virtue reject incorporated idea technique exhaustive computation monotonicity appropriate form theorem apply propose terminology follow intuition reject wrong affected apply theorem clearly weight coefficient significantly test make testing plan efficient parametric determining formulate consider testing moreover restriction associate plan parametric propose minimization minimize actually minimax minimize maximum iterative state iteration value weight let I exist return coefficient efficiency minimization reasonably return weight minimize r coefficient risk technique computational situation probability incorrectly hypothesis attain computation like n perform virtue recursive parameter test population unit certain course population unit attribute formulae without proved virtue note domain truncation significantly reduce computation scheme frequent problem subset computational associate type summation truncation recently considerably result thm shall discuss application previous risk control prescribe I follow ig reject h hypothesis chernoff sufficiently large suppose less ig unimodal corollary reveal choose sufficiently address sequel risk necessary specifically weight apply corollary weighting consider see requirement satisfy associate determine minimize since notational describing throughout resolve minimax intuition reject b weighting ii value tuning parameter weight return weight tuning order less sided versus problem cast general formulation make decision priori inequality type ii error impose theorem side reject h sided chernoff let minimum unbiased unimodal space shall z n f n g moreover make monotone likelihood say monotonically increase x continuous ready side ratio monotonically increase respect maximum minimum accept tuning weighting testing define testing requirement risk weighting coefficient constraint accomplish minimax develop adapt weighting let determine value denote q b sec equal frequent two side wrong priori error impose refer hypothesis reject h accept reject accept chernoff unimodal corollary testing sided requirement need procedure specifically coefficient due suffice ensure family plan risk requirement propose weight minimized intuition reject reject h solve special q b b b return desire b sided formulation decision typically prescribe number hypothese h triple statement hold true h reject triple chernoff large suppose less unbiased corollary triple requirement perform b weighting coefficient suffice value test b efficient satisfying determine minimize reject reject reject virtue present special set weighting base b b return tuning weight risk prescribe impose refer test decide fall call applying reject accept g accept accept reject reject reject result suppose suppose unimodal estimator conclusion different hypothesis requirement weight satisfied associate determine tuning coefficient truth follow intuition reject reject reject adapt present adapt maximum iteration value q b b b use b determine satisfy many shall demonstrate important iv monotonically variable px x n z k ni p g ni f n ni p p n ni p p z possess begin section plan frequent unit certain attribute replacement remain equal characteristic th sample sense sample attribute nature sampling treat bernoulli x recently x np testing outline choice n nn p nn n nn nn verify previous stop boundary n g p n otherwise n moreover p c hypothesis testing replace finite let j f ig ii sm reject p accept ip p vi vi reject reject reject reject p reject h b reject p reject requirement risk control accordingly parameter argument chen inequalitie simple bound less shall n n z n n z eq possess beginning imply test test know n testing section variable possess section applicable situation x observation section definition readily z lemma thus plan arbitrarily specifically parameter show accept accept reject h reject definition size variance odd unknown variance rewrite unity establish chen integer I unity q follow statement true I value f I z j j many n possesse begin method applicable accept reject rule unity probability density function form shape unbiased unimodal likelihood integer follow like shall length test variable refer refer failure reliability engineering issue failure practice efficiency replacement fail hypothesis failure accumulate accumulate run main life accumulate whenever failure test sequential probability drawback hypothesis test narrow accumulate time may specify may truncation status drawback tackle life testing mutually exclusive wrong decision pre specify accept accept accept address general principle describe positive connection length attempt life testing accordingly possess describe testing section applicable unit convert preferred derive let direction divide stage test time stage estimator h multivariate function k function time ii f ig z z c assumption sf ig I established clearly limit test plan depend evaluate plan satisfied change correspond satisfactory hypothesis readily triple procedure variance sample sampling hypothesis regard ratio standard exclusive exhaustive composite wrong decision typically number accept h variance integer reject I sufficiently probability reject carlo method risk risk propose testing requirement tune weight specific testing follow devote concerned gaussian formulate hypothesis testing situation risk require prescribed accept large n true accept sufficiently develop testing plan satisfy risk need tune accomplished estimate risk risk iterative frequent problem versus risk require prescribed number accept h accept hypothesis h reject accept follow size sample hold I accept h reject sufficiently virtue monte risk appropriate weighting satisfied three hypothesis control risk prescribe accept accept accept normal large n n statement accept h accept small apply carlo method risk idea risk iterative determine typically prescribe reject since probability accept size sample I reject make carlo estimating risk idea describe optimization hypothesis purpose prescribe size I ig I reject h use carlo risk tune minimax section risk weight variable exclusive exhaustive typically pre accept I shall sample mutually freedom square u v know ratio respectively u mean sample eq I u sufficiently appropriate coefficient requirement satisfied make risk tune minimax problem general special sided hypothesis concrete procedure work present advantage exist consider family xx xx k k use stop parametric x k n k virtue exponential simplify reject k number result termination statement ii true eq v q notation statement risk average precede boundary incorrect hypothesis address boundary variable consecutive mutually define mutually frequent g sg yy main recursively denote readily equivalently formula boundary fail rigorously control mainly finite domain limitation establish recursive differentiable sm accomplish eq precision control desire interval virtue suppose I du positive proof bc di f b dx I c f b u u c satisfied accomplished virtue theorem simplicity illustration address repeatedly method sequel clearly bind similarly exist il u ia bound calculate program j u I construct upper ensure requirement express general coverage construction hence purpose globally fast f w w j w namely apply bound compute bound form w summation associate repeatedly split low gap decrease pre see description recursive computing purpose reduce complexity technique integration illustrate context letting establish b thm truncation I reduce conclusion test arbitrary mutually exclusive exhaustive develop several advantage test always prescribe requirement power absolutely preliminary preliminary basic dependent increase x nx nx z great increase z x z e z z z nz nz ap parametric probability z use definition plan statement virtue lemma notice reject definition test plan statement virtue f z clear I j rr I g j z j j evident I prove make established reject establish iii statement virtue accept I consequence vi increase respect n I note reject reject making proves statement definition plan tuple number reject reject h reject reject reject reject b reject show virtue reject plan finally j sr sf sg g j notation publish x prove x n n x x f x n f x n show x f among number attribute accumulate stage probability I recursive shall rigorous justification notion space k k suffice unit exclusive attribute without attribute permutation character stre attribute otherwise need permutation possible hence nk ik bp among I connect string obtain step character v purpose simplify sample sample establish point k k permutation recursive relationship proof ap parametric plan accept n f f testing plan ratio accept establish conclude normal simplicity notation unit chi freedom possess accordingly variance chebyshev n show suffice fix monotonically gaussian unity variance v without introduce confusion virtue know z z z
one prove penalization procedure include rademacher nest datum even demonstrate margin selection motivation supervise introduction introduce variable prediction error bayes regression sx x build excess expect prediction distant minimax sense situation smaller introduce eq bayes predictor vc lc situation prove consider decrease risk lead risk strong margin loose back use contrast reference therein whole select lead close assume aforementioned minimax small may complexity e vc appear define prediction error margin ed prove upper bound several state adaptive model call emphasize procedure sake minimizer include contrast consider provide correspondence expect assume margin decrease margin margin condition stay dimension constant constant small constant method penalization strong nest satisfied complexity margin local respect inclusion ordering exist hold penalty completely reasonably corollary would emphasize valid definition soon right large remainder hence constant function minimax estimation know remainder risk bind even belong know local complexity theorem contrary satisfied belong one whether method bias trade theorem penalization focus base complexity precisely complexitie mainly minimal modulus eq choose later local small positive fix point precisely follow theorem enough assumption local rademacher state need empirical diameter modulus numerical constant margin rademacher complexity assume probability nest hold k make oracle inequality close whether change penalization collection improvement margin case reasonable happen situation suggest assumption might corollary complexity replace numerical margin example vc class margin condition holds depend n provide optimal nested aim model selection select u c selection loss indeed selection corollary hold corollary randomize aggregate conclusion theorem note penalization coincide minimization instance complexity ideal penalty focus whereas simplicity hope strong meaning well highly non know penalty deviation inequality positive value large margin main favorable situation condition least indeed fix minimizer right nest ensure margin condition example challenging situation one depend situation straightforward explanation surprise sufficient give generally small excess challenging margin depend positive depend iii pf pf margin tight pf margin k pf k n strong margin form right proved selection problem among focused penalty define complexity strong drive whether penalty natural think resample penalty kind resample scheme rp generalizing computed complexity point resample fold penalty define fold validation depend single front heuristic contrary local rademacher complexity certainly corollary rp would rp assumption large conjecture rp proof seem theoretical seem less may rp prove result provide penalization reasonably whether nest look happen nest selection easy term remainder replace margin merely satisfied hand compare challenge contrary large term induce complexity detect make nest require nest nest least order corollary finally definition n rewrite p f hold bernstein proposition intersection derive f f f result follow theorem theorem probability least right use convex use side small let nest occur decrease n eq proof lemma know exist constant hold addition inequality yx eq deduce follow accord every b q imply main fact binomial variable crucial remark replace ordering probability refer assign stay prove generalization point small
noisy allow simulation study goodness approximation phrase potential random polynomial measure turn know even noisy complex moment complex bold character problem central appear context constant analysis instrumental one white process become complex interpolation existence fact retrieve generalize e g right identity prove measure density generalize dirac moreover generalize q te nz consistent provide device solve original approximated distribution associate root heuristic algorithm peak idea rigorous consider eigenvalue computable approximate expression identifiability give give section statistical study estimate describe experimental make notice uniformly unit assume relate spaced f n n pf process loss signal result use extensively map realization thesis base would notice generality assume I define perturbation choose analytic circle depend continuously admit analytic component function tv h bt bt I pg admit taylor independence finally start eigenvalue qualitative statement already solution exponential interpolation transform circle analytic get location close polynomial saddle asymptotically satisfy algebraic equation equilibrium presence external corresponding therefore attract n high zero close zero numerator root polynomial small therefore attract attract complex close sum zero moreover point gap expect relate expect attract likely far picture behavior qualitative result discuss quantify qualitative statement eigenvalue count plane snr qualitative statement location positive measure pz q nz n z n weakly let bound pz p z p p k analytic neighbor dominate use argument lemma process q e limit stacking element taylor e f h f rewritten take straightforward pure odd similar drop taylor expansion depend power density unfortunately follow respectively moreover nz let r ie nz nz z j n pz nz j last thesis weakly nonnegative test denote nz nz nz nz exploit location prove relate nz check enough information meet know noiseless open amount ability devise identifiability identifiable nz pn generalized eigenvalue peak converge must enough check identifiability impossible many instance possible good mask signal properly choose identifiable correspond admit consume approximation quickly hermitian eigenvalue exponential identifiable eq distinct sort q thesis value independent replication define interpolation therefore nz e q solution condition computable variance estimator respect despite theorem fact verify suitably prove square squared sum equal define transform lattice cope appear convenient expression former dirac easier cope cx c usually automatic get let drop simplicity residual use identify sort maxima maxima close predefined percentage notice spurious cluster close prescribe candidate spurious correspond relatively evidence claim section component give component involve quality close identifiable close along restriction radius circular generate circle estimate reliable make identifiable simulation mse
extension section allow challenging split part important p wise make rejection dimensionality reduction part disjoint active dimensionality potential parameter idea break fraction select predictor determine ordinary model testing adjust p value cutoff inclusion positive easy control weak split split drastically split differently divide split remainder give answer splitting control quantile empirically split split choose split method original disjoint group set ordinary square calculate remain aggregated finally procedure predictor statistic quantile empirical quantile function predictor quantity twice median proper guarantee search instead select quantile datum correction section choice fact comment relation propose adjustment false family procedure correction procedure hypothesis empirical take test single hypothesis histogram variable split pick value multi distribution break line give indeed histogram section pick randomly pick p split broken variable equivalently quantile turn bind break fdr value number variable besides well multi powerful selection control rate consider conservative instead fdr false total discovery denominator false fdr control order variable value reject rejection empty fdr conservative wider however assume dimensional dependency level fdr work raw assume division correction split producing p division correct order small fdr fdr replace completely analogous control multi split procedure later control make crucial requirement procedure variable retain irrelevant impossible classical test retain appropriate condition include assume condition boost sure use scenario satisfied repeat work assumption selection adjust choice adjust provide control assume whenever rate level valid pre pose propose adjust value adaptively wise control appendix brief asymptotic control truly important non analogous adjusted use fdr section select variable fdr control false control could value experience work require otherwise let level vanish property turn split choose split ensure various variable reverse necessarily consistent split necessary select approach hand need quantile detail analogue raw improving datum thorough pick default everywhere use distribution artificial response gene gene logarithm production rate know simulation new uniform sampling choose component remain strength component error adjust ratio maintain size prevent datum calculation situation split simulation compute qualitatively average positive family wise snr either screen choose reasonably hand easily slightly arbitrary avoid parameter fold correspond coefficient adaptive parameter fold regression single typically increase include split method control split nominal asymptotic apart nearly setting sometimes substantially error seem multi though split conservative substantially general control wise controlling look twice median value split suggest recommend adaptive choice average number positive error indicate broken vertical line solid coefficient sampling break multi selector employ adaptive fold penalty adaptive cv asymptotic offer error way multi match show simulation select truly false positive ccc cc cc c positive multi adaptive snr split yes yes yes yes yes yes yes lasso make cv price pay wise multi average truly relevant split select general correct correct wrong depend study prefer important hence beneficial make validate medical place available likely multi split dna bind intensity protein score represent dna bp candidate bind protein bind site variable show bind intensity model split split asymptotic evidence bind application mention could measure fdr control split dark bar fdr bar setting snr height correspond fdr attractive correct fdr turn simulation correlation truly shown already split preferable interested traditional fdr controlling procedure obtain fdr dimensional empirical fdr regard track control fact say explanation estimate increase substantially ol ability truly small half sample repeat split even low think dimensional generic situation asymptotically graphical form recently rely e likelihood select methodology regression conservative fdr adjust false reject adjust modify procedure control reject positive set assigning hypothesis problem predictor large extension discovery combine split split method split share split argue false positive split positive area application dimensional variable exceed fdr fdr
decrease eventually steady fix steady furthermore behavior become well reach error infinity behavior ensemble monotonic steady independent quite minimum mobile ensemble line good state us perceptron show development fig steady case behavior value maximum exceeds certainly move close teacher move teacher ensemble generalization error perceptron student ensemble expect behavior fig steady essential behavior ensemble moving interval mean play student h teacher steady curve represent differential line direction teacher symbol teacher ensemble steady symbol line j teacher ensemble sharp ensemble step precise moment perceptron contrast find steady state steady coincide mobile ensemble movement reach fundamental minimum value mobile ensemble depend mobile teacher movement drawback present minimum stop point perceptron interval fig convenient explicit practical way acknowledgment grateful reading manuscript aid scientific area expansion education technology parameter rule standard calculus ordinal equation correlate limit step give limit find order index calculate average derive calculus equation q condition eqs eqs omit subscript perceptron perceptron delta equations perceptron one consist contrary example repeatedly batch learn extensively extension make generalization analyze teacher go teacher student teacher teacher learn teacher output ref monotonic perceptron teacher perceptron infer teacher generalization error teacher go around fix turn relatively student teacher move teacher model ensemble regard student mechanism ref true teacher student model teacher rule perceptron learn monotonically asymptotic reduce ensemble perceptron monotonic exhibit decrease rate total ref student teacher ensemble teacher mobile teacher trivial effective student study teacher teacher move student perceptron generalize adopt perceptron learn rule go around teacher analyze mobile perceptron ensemble steady movement improve performance student process sec ensemble go teacher sec ordinal formula generalization order sec numerical generalization perceptron conclusion differential sec teacher student dimensional simplicity component draw respectively assume independently learn asymptotic find norm move value introduce extensive quantity assume teacher perceptron monotonic fix threshold ensemble move simply sign true teacher student function purpose product ax bx one define generalization error obtain gaussian function note irrespective matter learn given let update x output teacher function choose function ensemble analysis perceptron steady focus dynamical ensemble effect student learn steady student update recursion formula student random move particularly student learning perceptron constant previous ensemble term evolve limit learn extensive call ref take move assume self average evolution dynamic rule find closed equation omit subscript subscript differential easily depend threshold true teacher take steady student begin dynamical therefore solution student dynamic ordinal equation parameter perceptron refer derivation equation calculation differential perceptron equation generalization error time error solve equation
algorithm slope section provide justification heuristic provide rely necessary concentrate around expectation prove quite describe although prove understand piecewise constant empirical computation easy uniquely fix family classical assumption merely moreover I bias decrease section concern existence nd reason prove dimension belong may model select penalization suppose satisfied exist constant heuristic prove minimal small select estimator coupling penalty theorem nevertheless understand theoretical penalization procedure generalize existence penalty even restrict moreover framework situation dimension therein much moreover relaxed speak boundedness long mild assumption proof upper medium otherwise would allow regular r measure state general density phenomenon soon choose usual estimate penalty formal penalty lead one satisfied bias exist eq tend probability moreover oracle depend smaller make small theorem twice point almost consequence theorem close shape estimate state instead constant constant comment additional classical proving make w r w complete right hand restrict equivalent small loss situation like appropriate show bic minimal penalty slope heuristic correctness follow combination sake simplicity read picture quite accord penalization associate inequality soon imply soon multiply since soon front generally theorem slope heuristic validity slope heuristic extend theorem existence crucial dimension jump complexity jump look proof theorem constant probability dimension jump definition output tend heuristic index c replace problem selection grouping sufficient calibration rely since devoted prove prove devoted main technical denote occurrence depend parameter include negative part take convention consequence infimum attain soon every construction hence terminate convention clear either every k gm use combine difference hence limit tend case last statement statement definition sum gm give statement take state corollary moreover q depend make price penalty motivate ideal shape fold penalty tend theorem show oracle lead tend mp md mp depend assumption condition give sake completeness p sm concentration inequality nm hold consequence n q fact deduce every control l oracle proof theorem q model mm nm nd q argument show theorem dimension choose low imply soon lemma l dd remove condition quite first model large piecewise constant general moreover eq derive piecewise constant associate ap lx finally recall particular partition crucial theorem prove piecewise partition finally technical negative real give since fact everything fulfil assume general result homogeneity belong continuous function positive absolute constant real one piecewise statement classical lemma result look every first triangular us inequality triangular eq schwarz inequality every side supremum du jensen concave obtain upper simplify term notice center deduce q already notice multiply bind multiply penalization procedure multiply whose either unknown assume particular shape rely penalize square regression side evaluate drive stochastic purpose paper state general design drive penalty regression least selection penalization decade receive commonly penalization choose minimize empirical fit datum see penalization among complexity penalty fold goal asymptotically quadratic asymptotically consistent asymptotically deal procedure assume true huge prove suitable moment moment lead moment error tend infinity practical aic serious drawback hand optimal correct noise involve independently difficult error dependent rule plug efficiency improve strong drawback gap inequality prove global rademacher multiply factor necessary calibration penalty local complexity large multiplying address popular certainly cross validation general rely widely however high instance validation entire calibrate penalty proportional model calibration procedure completely penalty extend wider briefly recall main penalty calibrate estimator quadratic efficient word minimal relationship characterize slope crucial minimal penalize huge successfully understand existence minimal address variance consider contribute calibration slope section shape shape penalty validation point allow usual model belong allows describe penalty regression prove theorem theorem prove feature restriction heuristic provide evidence prove inequality restriction framework mixture spatial heuristic formally validate framework article step prove ideal penalty follow framework heuristic section theoretical state proof framework heuristic observe center noise term variance conditionally predictor bayes therefore good q empirical counterpart q exist unique empirical minimizer square contrast family empirical looking instance prove oracle inequality event let dependent define depend natural idea choose penalty every hence therefore oracle sufficient show close side imply oracle contrary side oracle shall penalty minimal complex contrary complexity large medium support framework q p mp penalty penalty twice minimal heuristic formal validity heuristic knowledge slope heuristic heuristic penalty contrary much small lead calibration penalization define generalize shape n penalty estimate section penalty complexity measure slope heuristic provide heuristic huge reasonably efficient huge reasonably proposition advantage require compute every computationally notation choose order decrease always define reason efficiently piecewise increase summarize jump jump non sequence eq fm gm gm correctness terminate n algorithm k detection complexity ni subsection heuristic describe jump whereas medium complexity select space natural choice stop soon reach another match jump jump kk simulate set partition maximal jump jump give jump corresponding value definition select distant value strongly select make appear
assume lebesgue exist variate lebesgue subspace let order log log conditional log concern density b multidimensional compare present share property log infinite density log density handle concentrate degenerate random nx exist distinct hull dimensional convex exist concave informally complete show give class member seek iterative algorithm precisely subset event iii theorem coincide tuple distinct j j follow three eq inner product need find fortunately algorithm converge theorem try principle difficulty intensive another stem set attain general possible could function modify objective convex minimum say informally generic notation section whose simplex far long version may zero equal complicated function vast technique cf include newton informally study support height height therefore exist differentiable constitute lebesgue ignore optimisation differentiable differentiable fundamental introduction subgradient differentiable euclidean subgradient sequence formula property either index slow somewhat never compare towards adjust size selection produce subgradient direction linear analogy newton smooth inverse nonsmooth exist include direction hope improve bad towards variant formal claim accurate terminate original termination practice denote vector terminate two termination criterion knowledge optimum throughout take time iteration ordinary computer ghz gb ram increase relatively long terminate recommend active set time concave integrate error error square error follow location normal location size square carlo iterations density normal example bandwidth possible take mean formulae knowledge density bandwidth selector computed package option estimate large sample remarkably also outperform choose bandwidth size log concave concave deal true decay boundary log concavity assumption violate optimally bandwidth approach concavity divergence fx interesting purpose underlie still sensible paper combine univariate concave em form mixture proportion sum em estimate obtain carry assign observation component considerably performance true density lack multidimensional model multivariate marginal assume dependence model multidimensional carry simply find nice introduction density set posterior belong th component incorporation converge happen sequence component concentrated observation arise fitting density high likelihood cancer uci website panel solid open give contour misclassifie instance plot log create mass instance two radius distance texture grey datum patient mean different image reasonably means normally take illustrate particular fitting study unsupervised performance assessment skewness suggest may contour plot misclassifie em plot reduce algorithm example estimate approach need interested density example fx fx x informative restriction fx functional correspond plug plug offer empirical plug potentially functional possible large strong law use procedure must sample fortunately rejection th eq selecting eq repeat ii functional section estimate compare kernel matrix knowledge square differential entropy plug high region density region approximate section log concave plug kernel moderate illustrate point kernel guarantee error estimate r estimating high monte unable assess uncertainty take repeat true likelihood methodology remark plug extend methodology nonparametric condition concave include functional area currently rapid cite paragraph development refinement display theoretical challenge include attention univariate datum set diagnostic assess shape constraint assess density band datum distribution number constrain say call identically concave affine subspace empty convex interior regard subset hull closure interior affine closure function function every finitely hull call pl half contain hyperplane ps face polytope support contain close half opposite modify dimensional l ny ny element limit log concave functions dimensional subspace whose complement writing value density note index jensen prove strictly large lebesgue suppose write concave middle equality close supremum attain I may write may iv q follow jensen denote lebesgue measure prove complete support normalise geometric mean cauchy schwarz equality uniqueness tf tf ty ty ty ty n ty ty must sigma function subgradient independent set supporting zero lebesgue measure subgradient suffice derivative statement th vector hand denote one j te ta j final substitution exist coordinate subgradient enough subgradient may use formula mm thm thm proposition laboratory centre mathematical sciences road cb mail j uk laboratory mathematics university let identically lebesgue prove maximum estimator attractive unlike fully smoothing constructive computing optimisation geometry converge moderate maximum likelihood improvement performance bandwidth present real clustering use conjunction version package concave estimation keyword concavity differentiable optimisation modern introduction work distribute appeal lead asymptotically unfortunately density second considerable focused find automatic cf chapter reference therein result relative namely specification symmetric bandwidth involved mean attention identity issue automatic bandwidth remain automatic choose density economic reliability see section discussion application property show identically distribute density eq worth shape constraint density unbounded define successively mixture spike nx lf cf unbounded restrict attention possible hence maximum monte bootstrap assess functional validity contour contour exploit elliptical case concave address detect presence issue pressure pressure pressure
f use von fisher distribution center leaf example define q unfortunately von gamma explicit second make tensor namely measure dispersion want obtain approximate covariance tensor moment normal coordinate volume tensor approximation volume density addition derive variance sphere hyperplane formula coordinate eigenvalue decomposition determinant maximal positive tv satisfy definite kronecker element ready straightforward derivation observe eventually integral assume proceed volume derive element integral since integral q plug unique jacobian unchanged quantity follow normal coordinate co ji nr q claim q stand normal euclidean q well q taylor utilizing show density conclusion confirm blue green curves correspond estimate coordinate covariance green curve see stay close predict riemannian endow exponential map whole tangent empty normal plane curvature dimensional q approximation volume expression unit cc decrease curve correspond expect confirm experimentally stays predict precise request matlab program experiment try manifold manifold parametrization manifold attribute treat carefully variety view coordinate purpose center euclidean like normal make relate covariance variable euclidean counterpart interesting formally regard confirm unit directional statistic normal plane lack interesting distribution obtain projection construction popular easy utilize tensor relation distribution particular relate control concentration approximate manifold curvature confirm distribution plane property complete riemannian manifold primary limited direction rotation von shape bioinformatics mining field directional von unit q normalize studying field one circular another three center distribution going consider include nature obtain tangent however aspect rigorously one care manner issue tangent central point define log eventually current star definition cover maximal meet closed volume non neighborhood p open w cover fact log define entire
order consequence study clear impact coherent shall equally likely occurrence essence de coherent low subject concern event possible exchangeability indeed conservative coherent exchangeable dominate conclusion old exchangeable go approach first version precise counterpart lower call low envelope decide lower base bernstein case elegant contain certainly attention essence theorem coherent countable coherent polynomial nothing language gamble rather low emphasis keep probabilistic merely matter shall language gamble expressive event expressive power plan introduce understand rest establish replacement exchangeability countable develop limit deal exchangeable bernstein polynomial provide coherent section follow subject uncertain actual actual coin coin really distinguish subject outcome hide subject partial lack go subject uncertain transaction belief transaction capture mathematical map transaction assume receive possibly gamble subject belief certain unique price price belief almost every make price subject price lower acceptable acceptable hence price price price price accept concentrate focus mainly make supremum low gamble negative number negative combination acceptable transaction transaction subject matter lower gamble negative coherence acceptable price domain acceptable transaction gamble avoid sure upper coherent low gamble coherence requirement gamble negative p sure gain homogeneity super moreover domain coherent coherent indicator coherent event gamble avoid sure small conservative coherent dominate call extension supremum acceptable gamble coherence coincide wise small coherent extend subject precise fair specify conjugacy f equivalent invariant associate coherent follow gamble non real number gamble gamble indicator event finitely namely finitely consider model event expressive gamble indicator coherent link linear convex dominate coherent event I infinity coherent extend gamble gamble event precise main section formulate term gamble let list consequence use far already coherent whenever gamble f low gamble gamble uniquely coherent closure immediate coherent uniquely gamble e gamble end introduce notion familiar theoretic use shall model belief variable variable lower define consider gamble coherent say define always exchangeability theory variable necessarily result decide subject jointly coherent define gamble permutation associate procedure map exchange account gamble permutation coherent process exchangeability gamble equivalent see exchangeability follow immediate strong de exchangeability exchangeable envelope exchangeable exchangeable converse break coherent low evidence exchangeability special evidence invariance coincide exchangeable exchangeable moreover nf nf nz nz place shall try reason random variable look exchangeability assessment coin might event exchangeable consequence tail success failure often failure random element mass coherent low envelope interestingly exchangeable coherent representation without give come permutation permutation denote constitute define way q tuple element negative whose map onto atom invariant element shall denote atom way joint random available information give gamble conversely exchangeable coherent call establishes exchangeability replacement ball element set count ball subsequently select ball replace denote variable ball select outcome precisely atom selection outcome possible outcome zero mean expectation notation marginal moreover count shall know tuple exchangeable tuple happen definition countable sequence empty call exchangeable require variable sequence random exchangeable distribution nn nf gamble extension collection exchangeable coherent conversely exchangeable coherent coherent marginal exchangeable nf detail follow consequence marginal family count nf nf consistency equivalent prove multinomial one collection result collection sequence multinomial limit multiple one another arguably even work exchangeability context indicate representation uniquely prove multinomial observe multinomial count ng namely count polynomial list interesting shall soon bernstein form multivariate hence bernstein equation equation ny vectors ny ny simplex every element count multinomial count possible sn univariate bernstein basis polynomial degree linearly follow linear combination polynomial simplex coherent coherent define exchangeable corresponding show converse exchangeable coherent coherent low count coherent family course coherent consistent find hold exchangeable automatically np nb thing polynomial unique check corresponding apparent polynomial degree b assume bernstein decomposition consistency coherent coherence consider must p coherence tell real equality coherence homogeneity count tell low homogeneity super low tell coherent sequence coherent unique coherent linear gamble belief countable completely space polynomial gamble case exchangeable basis polynomial coherence uniquely gamble sequence gamble coherence gamble determine gamble simplex equation study gamble whose agree polynomial gamble gamble represent bit sequence assume coherent low model n take account equality see bernstein polynomial uniquely uniformly find provide see limit frequency converge back exchangeable variable form gamble number count binomial polynomial extend low gamble go infinity representation linear equivalently completely completely basis determined know finitely determine except bring back une pour il une exchangeable exchangeable taking eq sequence define f simplex mean eq bernstein gamble approximate bernstein coherent uniformly tell mean throughout set gamble impossible exchangeable specify infinity supremum acceptable keep exchangeability coherence subject specify mean infinity number gamble specify number extremely realistic exchangeable coherent show subject exchangeable terminology specify acceptable price gamble necessarily gamble furthermore conservative look conservative small exchangeable coherent coherent ng ng finite exchangeable gamble model sample without ball completely composition prove elsewhere avoid replace infimum operator coherent lower become immediately apparent lower dominate coherent n combine equation technical prove exchangeable extension exchangeable dominate n elegant manner combine exchangeability coherent low consider indeed see gamble k ball replacement composition formula expansion bernstein unique tell gamble coherent coherent exchangeable avoid sure simplify g applicable special gamble wise small coherent coherent count know linear side drawing ball envelope exchangeable envelope exchangeability place coherent bernstein gamble low case result coherent exchangeable fairly converge matter strong completely square strong non convergence consider
head suggest different volume compound distance could go cdr cdr analyze test necessary convert cdr cdr subsequently cdr subject follow convert cross predict convert distance way baseline convert variation volume cox proportional surface volume leave select predictor surface observe subject later convert cdr pattern discriminate subject suggest early find cdr cdr leave cdr right cdr cdr different cdr cdr follow concerned local change might overall might cdr cdr agree dominate dominate shape influence scaling normalize difference difference change use diagnostic discrimination discrimination technique linear discrimination metric together qualitative logistic discrimination furthermore optimize cost entire cross set subject e distinguish ad able diagnostic great extent component volume consider loading conclude volume partly metric distance mostly partly longitudinal distance volume metric increase volume decrease summary neither metric baseline lb lf cdr cdr volume reduction distance cdr neither cdr subject diagnosis measure obtain volume classification result compare result volume distance distance change year volume early distance cdr discrimination classifier good constitute present detailed change diagnosis group great interest distance depend template address issue template journal mild disease thompson disease detect disease population brain subject study disease relate et specificity distinguish temporal volume memory journal associated memory quantitative mr formation year age american journal outcomes trial disease modify health p al volume index study n c disease voxel compression template al sciences united states base mapping brain thompson mapping disease brain development I computational apply mathematics w visualization brain pattern wang et volume serial journal quantification ad wang mr volumes brain structures ii study disease image five change cognitive performance project segmentation serial p longitudinal brain mild cognitive p et modify disease evidence cognitive al mapping geodesic flow international journal vision metric euler engineering p collaborative perfect human multidimensional analysis wang et surface mild application et parallel clinical version agreement scale measurement al clinical rating j et clinical training reliability disease disease marker p c disease concept cognitive course outcome vs wang et momentum discrimination image al high journal computer vision g series forecasting control ed day modern nd k p york york lee york rd ed york california du et high j analysis du rate shape volume p et brain sciences united j figures c gender age scan education gender sd cdr cdr na na baseline follow min median max iii sd volumes cdr cdr volume lb lf cdr cdr c mean lf rf cdr cdr summary subject baseline sd stand standard deviation stand mean volume diagnosis diagnosis iv distance diagnosis rank na applicable volume lb lf baseline rf htbp c c pc pc pc var prop pc loading principal volume principal component prop explain principal prop baseline volume volume htbp component pc pc prop var prop loading loading pc pc pc loading volume baseline eigenvalue brain volume volume c pc pc pc prop prop loading variable loading pc pc pc pc pc pc loading distance volume volume compound symmetry autoregressive var metric distance common repeat factor order c c var un vs criterion compound autoregressive heterogeneous var criterion bic criterion likelihood htbp c distance side lb cdr lb lf cdr lf cdr cdr cdr rf cdr rf cdr comparison pair sided lb lf cdr cdr cdr lf cdr cdr rf cdr metric pair leave metric normality homogeneity variance level mark vs distance group lb cdr lf cdr cdr rf cdr lb cdr lf cdr cdr rf cdr coefficient alternatives pearson coefficient correlation significant mark htbp c vs group lb cdr cdr lf cdr cdr lb cdr cdr lf cdr rf cdr correlation side zero correlation value mark htbp c cdf group lb cdr cdr cdr cdr lf cdr cdr rf cdr von g stand sided alternative cdf cdf great second alternative er er von htbp c cdr cdr cdr cdr c c predict cdr cdr cdr c truth cm cm predict cdr cdr cdr c cm predict cdr cdr cdr matrix metric distance classification subject equation classify cdr cdr classification cdr subject use follow specificity mark use optimum optimum c specificity metric threshold classification c sided lb cdr lb cdr lf cdr cdr cdr cdr rf cdr value pair test pair group side sided cdr lf cdr cdr rf cdr cdr lf cdr cdr rf cdr lb cdr cdr lf cdr cdr lf cdr rf cdr rank pair volume bottom mark htbp c optimum correct sensitivity specificity metric probability bottom mark c optimum cost rate specificity classification use metric volume probability optimum middle optimum cost optimum classification rate specificity volume bottom performance mark optimum c classification specificity probability threshold base threshold base well mark use optimum cost rate base volume value middle optimum cost performance mark optimum cost cost function correct rate sensitivity procedure volume optimum middle cost mark flow template target htbp generation distance baseline plot volume metric distance number stand scatter distance plot diagnosis level level side slope change cdr subject right diagnosis ignore differ htbp plot diagnosis level right metric left difference vs fit cdr proportion eps bb ps theorem conjecture remark school school st university school center university school st school institute md mathematics mail edu phone predict change metric article analyze shape subject type control year difference metric large metric calculate correspondence field metric image quantization term give demonstrate metric longitudinal comparison brain repeat interaction diagnosis explain difference parametric parametric analysis distance baseline metric subject subject subject significantly follow subject diagnostic discrimination compare volume metric metric template tool detect difference longitudinal numerous prominent within marker become disease accumulation characteristic death gray matter loss currently mr imaging volume mild moderate mr show ad human brain distribution ad course disease region affect ad disease process prefer distinguish mild ad normal enable quantification brain volume shape individual disease group year principle general pattern mr brain manifold sensitivity shape associate ad assessment volume optimally distinguish subject mild distinct change volume shape within longitudinal ad important task consist template union transformation template geodesic point evolution template connect template template geodesic induce metric distance template velocity find enforce length minimal transformation connect distance shape optimizer construction allow quantify similarity thompson year distance evolution template notion distance distance relative another allow sophisticated base could powerful subtle time considerably load change shape mild subject shape residual surface individual distinguish subject metric baseline follow track change template could difference find correspondence difficulty template longitudinal computation statistical group change develop concept describe computation finding include subject mild distance discriminative power volume volume final baseline mild clinical scale cdr match cdr approximately apart clinical rating cdr scale subject conduct structured subject assess obtain cdr overall cdr cdr indicate severe cdr inter reliability weight confirm subject clinical cdr e individual ad cdr subtle cognitive progress severe stage I cdr sign ad cdr diagnostic summary sp head sequence min slice template use e cdr age detailed template subject template subject surface scan create manual leave surface subject surface convert outside surface align template surface scale rotation image map enhanced resolution voxel voxel surface structural voxel resolution versus mapping convert voxel smoothing voxel voxel template pair compute metric controlling big tend big inherently correct brain prototype shape gender age education use affect subject evenly variable brain volume volume right template image analysis mapping via variational velocity take eq optimizer generate upon dt enforce velocity field solution v space velocity type lf determinant jacobian self adjoint operator uniquely field holds first investigate measure volume compound baseline volume characterize trait quantity measure statistic sd third repeat distance effect repeat diagnosis group subject leave baseline follow compete structure compound I covariance autoregressive covariance var covariance criterion aic competing fit hoc cdr vs cdr cdr baseline follow right test underlie normality homogeneity independent employ median homogeneity deviation distance group normally distribute population distance lb cdr lf cdr lb cdr cdr follow cdr cdr cdr distance dependent come subject nan metric cdr cdr cdr baseline cdr calculate correlation pearson correlation cumulative cdf distance kolmogorov k er er von hypothesis diagnosis cdr cdr leave follow calculation critical test discrimination since diagnosis level cdr cdr word cdr follow parameter logistic discrimination see cdr side etc consideration full py cdr choose reduce backward procedure stop elimination significant predictor logistic odd classification cdr cdr probability large probability cdr cdr decision optimize incorporate apply distance differential volume rate volume distance logistic discrimination incorporate together volume summary measure gender education scan age diagnostic brain volume volume volume cdr cdr hand increase cdr diagnostic assume test diagnostic age education brain distance distance significantly diagnostic plot scatter plot pair continuous correlation present age education discard analysis discrimination observe volume volume distance moderately volume brain volume hold distance extent maximum present seem distribution location distance distance likewise leave metric seem right follow distance leave distance tail reveal level subject small follow leave follow tend baseline template baseline scan age point baseline would would test significant reveal change template large left distance provide cdr distances cdr follow cdr surprising consider cdr cdr cdr subject variability template cdr statistical provide scatter metric distance group axis avoid plot volume distance measure relate correlated figure principal represent aspect volume observe principal component variation loading pc volume metric distance volume dominate pc remove eigenvalue score component loading volume baseline present pc almost variation compare loading head brain volume right table loading pca follow suggest brain volume mostly head partly measure volume mostly measure partly shape head volume metric volume mostly distance emphasize one due subject dependence main repeat group metric cdr cdr label lf cdr cdr individuals lb lf cdr similar labeling metric hence set repeat compound convenience cdr diagnosis effect diagnosis interaction I effect right correlation marked follow overall cdr correlate lb cdr cdr level pearson cdr cdr level pearson cdr together follow cdr right analysis distance overall baseline distance level test lb cdr significant level correlation suggest difference template difference template tend increase slightly lb cdr vs lb cdr cdr cdr lf cdr lf cdr rf cdr cdr comparison provide value side cdf er value er cdf rf cdr cdf rf cdr rf cdr stochastically large rf cdr cdr shape template rf cdr shape lf cdr lf cdr er lf cdr stochastically metric distance agreement subject cdr baseline follow logistic subject cdr possible predictor diagnosis cdr cdr intercept line however proportion cdr subject group distance relationship analysis indicate quadratic logistic classify top image cdr cdr subject classify discrimination follow clinical cdr suffice classify cdr cdr rule notice cdr subject classify correctly cdr classified section diagnosis matrix notice cdr subject would classified cdr logistic classifier get notice cdr subject correctly cdr correctly validation discrimination calculate sensitivity specificity sensitivity subject cdr cdr subject cdr cdr number cdr cdr data specificity proportion subject classify cdr cdr cdr cdr correctly classify cdr cdr specificity classification specificity classification observe cdr subject cdr classification procedure specificity rate large sensitivity probability equation correct sensitivity specificity good cdr get decrease specificity tend decrease optimize threshold minimize misclassification appropriately odd cdr cdr cdr minimize maximize classification misclassification rate threshold optimal specificity sensitivity rate respectively obviously clinical view cdr cdr classify subject might desirable label cdr subject cdr modify reflect practical cdr cdr high sensitivity alternatively maximize specificity sensitivity specificity increase decrease good sensitivity classification specificity give lb cdr subject mm lb cdr average volume cdr lf cdr show ns cdr subject stand lf cdr subject volume reduction rf cdr repeat significant volume group total volume take account visit scan covariate volume comparison interaction interval repeat volume model volume use effect compound symmetry var volume volume diagnosis level diagnosis effect diagnosis side leave ignore within group group change time volume repeat compound var volume repeat subject side leave effect interaction diagnosis cdr ignore main significant conclude line join volume plot meaningful right diagnosis diagnosis interaction selection criterion aic volume diagnosis diagnosis diagnosis interaction var effect diagnosis I instead compare group main side baseline volume volume year volume volume significant reduce cdr cdr cdr cdr follow volume cdr cdr cdr leave cdr group reduction cdr cdr volume different cdr lb cdr cdr cdr rf cdr rf cdr lf cdr cdr cdr volume small volume leave cdr volume stochastically cdr volume follow discrimination method logistic predictor variable elimination subject diagnosis cdr cdr baseline right intercept slope fit group interaction diagnosis group rate classifier good sensitivity increase correct specificity good classifier model classifier sensitivity increase specificity decrease volume hoc volume distance difference volume metric distance distance volume contain volume performance volume well cost performance table logistic performance logistic discrimination side interaction predictor elimination subject cdr cdr side leave intercept coefficient volume coefficient volume distance eq good q observe consider volume together discrimination classification compare metric metric longitudinal measure need adjust variability differential
procedure right everywhere sign almost equality almost everywhere decision eq lemma right measure definition fulfil aim section optimal stop rule truncate stop everywhere suppose definition introduce recursively please implicitly unitary cost characterize almost everywhere follow less routine despite optimal stopping hold decision something stop time non randomize describe rule purely class procedure may optimal discussion therein like sequential crucial pearson non stop stop risk truncate particular lower bind pass thus exist monotone nx nx fr easy close almost everywhere hand well close omit sufficient hypothesis consider article stop stop stop fulfil eq integral equal suppose side tend former sufficient hand tend I fulfil fulfil fulfil problem function case corollary equivalent vanish risk stop everywhere observation problem pose bayesian respect rule everywhere rule eq everywhere procedure strict everywhere almost problem theorem reformulate equivalent way e stand dimensional meaning nx x nx q nx bayesian rule surely general case due incorrect observation structure sequential testing theory sequential acknowledgement anonymous valuable comment suggestion author city grant cb mm corollary mm mm mm r u e x p optimal rule mm article consider minimize average incorrect exceed procedure bayesian due incorrect process time existence uniqueness stochastic process eq derivative respective usual sequential pair stop rule decision element interpret proceed stage continue observation rule way etc stop stop decision stop rule distribute independent q stand I discrete article develop extend testing variable bayes minimize procedure stop rule particular minimize procedure satisfy problem sequential usual general define multipli multiplier class application problem
transaction give transaction item contain tp transaction number transaction give interval transaction fix last substitution unconditional short vice relationship vector alternatively use observe item rough exist sophisticated example item database function alternatively draw suitable flexible fit independence mixture many conjunction rule model use context database independence model parsimonious quality assumption violate significantly transaction former paper customer outli customer rank later frequent entropy learn something item therefore usefulness independence wise even explicitly independent item month typical category transaction article call estimate transaction per day simulate comparable use parameter transaction experiment use relative estimate information transaction transaction association database expect association use datum set use exhibit world association concentrate co item rule restrict association easily item frequent frequent front axis plot analyze b naturally frequent item frequent front plot support similar simulate occur event occur transaction present distribution figure world confidence indicate side rule hand increase b dominate problematic selecting rank rule filter order lift lift lift co occur database great application generally argue complementary b simulated lift plot figure extremely lift occur co chance avoid association mining minimum plot show high lift datum find rule lift great indicate lift poorly filter transaction rare plot lift tendency produce always refer treatment lift discover tendency towards less lift variation observe occurrence item transaction contingency table model marginal count arise contain trial without replacement hyper applicable occurrence follow way transaction therefore ball transaction without item randomly transaction without transaction transaction assign occurrence reasoning hyper transaction marginal count simplify omit rest develop measure lift hyper deviation use assess cluster hyper geometric parameter ball occurrence count two transaction transaction equation relationship count lift item high occurrence lift majority transaction database use expect example two support database however hyper chance lift item especially problematic database transaction sufficient database usually collect period contain change may deviation occurrence count divide hyper geometric lift hyper lift inequality result call lift lift lift time high count expect hyper lift rule item exceed compare hyper lift value item lift evenly lift figure rule hyper lift rule indicate lift filter lift lift great rule lift hyper lift occurrence rule frequent item intermediate close lift lift problem section lift present quantile hyper lift small calculate use equation indicate confidence interest confidence accept rule count chance pure formally use threshold rule interpret sided test contingency depict positively relate value fisher exact contingency evaluate odd ratio association randomize low lead rejection would reject nan use hyper exact tp c confidence special sided contingency table directly geometric computationally negligible counting table test dependency author test side accepted rule count contingency test test suffer exact drawback sided application rule positively correlate one side figure confidence simulate vary intensity dot dot hyper threshold remove threshold rule pass support result able reject nan figures hyper proportional randomly rule test exist chance accept rough proportion spurious rule rule spurious conduct simultaneously generate set rule individual accept spurious rule correction correct significance use test alpha correct depict figure simulate correction spurious accepted still rule represent association simulate b lift require lift hyper need follow conversely complete item adapt substitution co together independence hyper confidence side lift quantile however figure show item simulate find low triangle database contain visible figure substitution rule spurious rule lift publicly available procedure evaluate third database provide contain click stream line news database characteristic data source item database market artificial click stream avg size distinct item association free paper generate specify support lift lift hyper present find minimum lift hyper lift database set rule support assumption contain least potentially useful association performance lift hyper table trade rule database simulate rule rule three never spurious reduce rule especially result rule lie lift rule tp lr rr rr sim sim sim support c find analyze trade proceed lift lift assess accept repeat hyper lift lift confidence minimum accept database accept simulated lift confidence lift classifier similarly corner positive database false spurious rule accept simulated datum leave hand database perform lie outside left side plot lift visible four lift omit inspection range spurious accept simulated right plot hyper lift lift lift lift look generation market click click web page page page restriction incorporate hyper lift thus produce still consistent previous accept rule database produces well lift great see figure set generation process rule spurious report generate represent spurious contain default generator produce produce detect use corruption basic hard spurious set item transaction use generation set support lift omit lift hyper count many accept represent rule plot plot figure call axis pn roc corner coverage use evaluation association rule generate frequent natural averaged pn hyper lift dominate lift large support world database statistic lift provide inferior individual datum use generation produce dominate world compare comparison find lift point argue isolate well picture strongly influence
general article explore several som som batch som version former converge state som maps original hilbert som apply map carry som specify associate kernel hilbert describe batch som map prior structure algorithm concentrate iteration neuron prototype book vector map suggest li nx neuron update prototype accord trick solely q assignment give lx step directly algorithm rewrite observation close prototype coordinate batch som depend present structure bi temperature parameter anneal keep decrease process repeat ensure final organization behavior som test original initialization method kernel discover build coordinate vertex initial done choose parameter som importantly grid latter specific rkh use directly quantization decrease assess som literature topology adapt som criterion path prior start match matching quantization whereas term point map contiguous matching unit statement equation unit term decrease quantization close favor map addition modularity q fraction graph connect vertex edge connect vertex subgraph edge used g constitute majority position mainly concerned population consequence anonymous master approach location several presentation locate km locate south france rectangle side decrease share property sale concern transaction relate neighbor transaction additional still record whole corpus keep du france interesting especially automatic synthetic corpus database relational person name analyze graph obvious central individual mask organization link appear contract year de account rule link specify analysis less simple direct drawing frequently social network connectivity diameter short connectivity average neighbor graph obey decay decrease exponentially relationship number numerous emphasize dense discriminate neighbor perfect extract high vertex edge connect vertex map dark color divide dense right right dense dense bottom cluster vertex represent connect large som subgraph induce vertex methodology except map sparse main vertex whole subgraph reflect figure show high degree cluster map large degree subgraph law subgraph one initial graph phenomenon degree subgraph distribution date depict cluster high map ht three part homogeneous top middle leave recent cluster organization som therefore year individual little dominant live family close community provide belong always cluster map perfect community reverse arbitrary assign perfect community community two som perfect belong cluster self make link community link color often belong nearby som som link respectively similarity approach consequence realistic organization nevertheless separate community arguably cluster som depict three cluster som majority name find none time cluster none cluster cluster advantage som clearly correspond perfect community seem perfect community community bottom part link represent cluster separate rich contain perfect community explain group link remark show element organization advantage perfect induce way represent definition partly question drawing bias som alternative proximity even communities kernel som clusters som g two outside kernel som probably graph perfect restrictive social group group easy social historical automate sense social som instance give broad extract aspect go som chosen spirit deep conduct drive cluster conversely som help perfect create draw distance som currently thank universit work interesting database explain historical thank le france anonymous detailed constructive natural mathematical field mine biological etc application synthetic discover relation goal conjunction spectral explore structure social network non trivial arise naturally numerous name web complex grow pathway article individual complex common property mathematical description often understand complex begin number search subgraph highly connect connect recently several go model network account community note deal million vertex open explore way vertex laplacian building eigen similarity measure dissimilarity vertice community graph cluster quality measure measure generally np laplacian relaxation community classical recently som combination tool complementary view spectral community interpretation bias cover one som solution cluster link analysis classification get limit case community organization two intend come field extract homogeneous paper organize follow alternative complementary derive laplacian implement som less map dimensional structure dedicate application propose social network historical section historical network challenge densely outside consensus restrictive type addition precise view community outside perfect van weight assignment follow appear come social community perfect community instance together nod community simple unique connection community summary graph one set contain whole vertex belong perfect network characterize vertex measure high degree role link subgraph diameter two edge possible construction start totally decrease process stop diameter reach practice choose diameter rich vertex diameter satisfactory subgraph diameter high density share people social role know belong perfect community rich another vertex path vertex occur vertex essential graph vertex sort subgraph first high measure sharp drop flat region see example vertex reduce logical lie drop actual remain matter node clutter visualization leave call rich perfect coverage maintain first central explain perfect community therefore perfect community central vertex vertex add another simplification rich vertex summarize link perfect visualize edge positive summarize many structural topological laplacian semi connected way spectral allow perfect community vertex call contain eigenvector vanish eigenvector consequence look nan simple efficient way perfect moreover perfect constant write eigenvector define orthogonal eigenvector orthogonal complement span co belong perfect presence moreover relevant perfect might community complement help numerous method community dissimilarity introduce explain build able separate community idea map section emphasize link smooth spectral clustering minimize key point discrete partition eigenvector eigenvalue column solution matrix convert usual eigenvalue map belong perfect community consequence vertex perfect spectral perfect equal laplacian small corresponding eigenvalue entire provide laplacian laplacian section author notion regularization operator function give kernel diffusion kernel eq eigen orthonormal eigenvector l past year limit value energy tend energy vertex edge case see definite reproduce space feature previous way work map avoid calculate mapping induce product embed equation use therefore loose perfect indistinguishable contrary eigen decomposition local permit heavily totally whereas common neighbor attractive tool popular computational extract pathway gene datum gene vertex cluster
segregation denote rl seem estimate independence remove test qr adjustment henceforth design independence rl distinction rl context hypothesis demonstrate independence class g specie age rl individual specie nn structure spatially major bivariate clustering pattern association segregation likely individual member detail alternative pattern qr adjustment test segregation extensive carlo article adopt convention capital fix denote low letter test illustrative conclusion extension multi case record nn relationship cell class constitute base category frequency cell obtain nn also lack segregation row sum overall sum cc class pt base segregation large expect expect independence rl independence detect rl rl e test segregation test overall depend sum serve sum segregation cell count rl write join derive covariance cell count frequency expect size column specific suggest pick triplet give point twice side side use normal give test equation cell deviation respective cell four combine segregation nan test rl specific test except variance difference status rl independence expectation expression rl variance unconditional variance replace expectation independence covariance covariance value pattern alternatively empirically follow iid unit times carlo record plot ratio homogeneous poisson replace qr adjusted covariance empirically expectation segregation hypothesis cell count n ii rl suggest test r n independence covariance however variance independence fix rl independence empirical qr correct segregation cluster cell statistic combine four cell let correct cell invertible use segregation independence variance covariance hence conditional sum estimate still conditional cell eq vector wise count test distribution segregation segregation independence variance covariance covariance replace obtain qr version column test version incorporate class sum serve sum count propose segregation discussion rl empirical qr adjust denote segregation cell normality cell asymptotic rigorously prove yet normality hold version ii would independence distribution qr adjust estimate nan case class replicate data point iid size combination choose influence abundance corresponding statistics monte respectively qr qr smaller large mark indicate conservative normal determining significance empirical estimate test size conservative trend qr version qr version significantly different test ht significance test unconditional I significance version version carlo significance version qr stand qr adjust version conservative qr adjust significantly small qr alternative square least seem desire nominal necessary rl pattern avoid alternative rl I fix segregation segregation considered pattern appropriate imply nn pair constitute observe increase segregation homogeneity respect square homogeneity segregation segregation pattern symmetric equally realization triangle present qr adjust indicate get large power estimate segregation get strong performance segregation notice adjust indistinguishable monte replication segregation alternative number horizontal axis combination alternative pattern jj jx choice likely consider constitute three get occur homogeneous exhibit region pattern homogeneity ht realizations solid square power alternative size large get strong power large qr version sample qr slightly estimate qr size combination independence cell different simulation study sample cell appropriate carlo randomization count test furthermore version adjustment qr adjustment improve independence empirical adjust furthermore qr segregation alternative qr adjust I illustrate artificial consider spatial specie rectangular plot test live diameter record realization pattern contain specie stem stem plot water stem specie live frequent tree specie water black conduct segregation less specie perform base sum example water base tree figure segregation specie large ht scatter water circle triangle ht cc cc species specie size water tree specie appropriate nan observe size associate test qr adjustment however substantial conclusion segregation specie consider c associate statistic tree set stand stand qr although version conclusion evidence segregation specie question interest spatial plot location table observe mild segregation scatter location circle point triangle cc class marginal nan observe size statistic associate decrease adjustment conclusion find spatial significantly adjustment difference independence segregation segregation adjustment get ht c associate value artificial test segregation near overall segregation adjustment perform test e nan case test denote
smooth describe spline li verify smooth oracle pm ds k splines pilot simulation start pilot smooth criterion stop stop start pilot stop remarkable h pilot despite pilot smooth succeed signal large weak learner correction desirable accord show biased estimator estimator explain pilot enough almost residual stop weak scheme investigate present select driven stop examine pilot smooth regression ratio performance replication random standard range mx error gaussian student pilot smoothing near smooth smoothing ideal mx since focus first density ratio smooth smoother evaluate stop cv cv splitting ds aic smooth datum small positive value essentially unchanged small selection big small present integrate error clearly estimate denote stop space smooth datum big big act modify small ideal fold cross intensive validate conclusion kernel near pilot smoother enough enough small discriminate bias conclusion datum cross big recommend finite sample stop smooth report median initial smooth smooth small correction table gives smoother select use square improvement present spline gaussian student student spline gaussian b gaussian student student bandwidth modify report pilot iteration square smooth figure boost thereby provide new interpretation seminal case kernel interpretation surprising smoother stable iterate convergence well scheme rule optimally one smoothing finally iterative let argument si last spectrum I simplify exposition term order eigen symmetric let zero except expand x I last larger suitably nn versa k lead k belong nn potentially nn ij term able relative position l x l away without interested entry recall entry quadratic quantity grow evaluate x fourier inverse x dx density know lebesgue symmetric obviously virtue version identity lead kx dx dy consider follow form u hx fs h h ds ds nh ds b h du latter arbitrarily enough dominate kt less statistic proof proposition recall semidefinite principal minor produce minor determinant end column hx quantity kernel large readily calculate principal conclude bivariate determinant positive correction schema show interpretation depend spectrum smoother combine stop practical smooth study simulation relationship traditional parametric variable minimizing predict smoothly observe covariate nonparametric smoother predict variable past numerous smooth bin smooth spline smoothing spline smoother run mention depth require parametric specification function pair smooth mean variance discussion compactly form smooth smoothing matrix contraction tune smoothness goodness smooth matrix produce want kernel smooth span bin smoother smoother much select smoothing ideally want without underlie compute stein risk paper take tuning reasonably ensure result smooth variance substantial bias smooth estimating residual enable subtract estimate residual correct pilot go back introduce misspecification multivariate repeat correction iterative correction iterative bias boost boost learner boost seminal method view boost adaboost variant boost context nonparametric show boost reduce reduction boost apply effect reduction decrease come increase correct regression iterative boost provide explain eventually variability commonly use smoothing spline good behavior spline positive behave boost bias sequence modification smooth sequence stop boost iterative propose several aic aic fold error splitting using stop smooth either desirable smooth pilot pilot conclusion study present datum desire prediction boost stop boost reduction simulation compare optimum optimum correct cross conclude correct smooth finally proof iteratively highlight ease vector error covariate multivariate typical smooth call shrinkage smooth smooth question x I sm estimate residual bias spline projection improve smoothing indeed express si recognize smooth say plug bias subtract estimate smooth si smooth smooth repeat discussion smoother iterate correction construct correct iteratively bias correct smooth nice representation smoothing write smooth throughout section boost various common take function solid gaussian produce pilot heavily pilot smooth plain iterative correct iteration smoother start go peak increase start boost projection residual boost smooth spline smooth consider neighbor smooth matrix near smoothing enjoy desirable suited boost design soon big value big proposition neighbor smooth produce hence confirm datum pilot smooth pilot plot plain smooth nearly bias correct qualitatively high type symmetric general algebraic symmetric orthonormal smooth I p eigen iterative unstable density function spectrum smooth inverse fourier positive conversely suppose away stochastic conclude concern remark converse size lead negative singular inverse fourier equivalent definite detailed theorem kernel produce behavior smooth state spectrum smoother state large smoothing matrix regression smooth value example smooth pilot smooth bandwidth equal pilot smooth plain dotted line pilot smoother since third contrast spline smooth smoothing denote basis u eigenvector write iterate iterative smooth figure pilot smooth spline equal pilot plain dot line pilot pilot fast smooth show operate correction bias resolve problem neighbor compact kernel previous resolve propose suitably modify smoother equivalent boost iteration estimate b si partially correct smooth algebraic right b si boost rewrite smooth combination smoother smooth understand produce analogy depend conversely inspection reveal
lem random lem z n n z establish g virtue conclude establish z exist n transform quadratic obtain explicit follow hand b remark approach distribution engineering frequent specifically desirable probability real express summation situation large require advantage purpose obviously show ic I eq hence prescribe suffice eq prescribe follow statement ii iv exist jensen hence z chernoff bound complete statement show let e decrease q x statement q result q iv q monotonically determine set reduce truncation paper chebyshev inequality truncate domain though function close form hoeffding bound truncation tight use bound regard
moreover impose shape avoid smoothness procedure start statistical qualitative al functional assume give close instance cone minimize standard problem pool set extend numerous author see soon differ integer matrix let correspond ij recognize cite therein exploit structure functional paper special give rise introduction dynamic instance good fashion algorithm finitely least strictly adapt denoise method indicate yet observe regard always mean plausible would define rectangular contain least different stand describe introduction index differ fail nevertheless minimize later uniquely min max sense minimize suitably penalize strictly w prefer show quadratic strictly complete smooth nontrivial appropriately term condition hessian light regularization irrelevant illustrate difference simple interpolation two observation leave panel fit gray column remove randomly result grey panel simple note difference occur h may quantify turn well return begin cone explicit coincide characterization infinitely criterion involve finitely component easily verify thus check check cone r may correspondence end eq hence exponentially functional possible obviously deduce sa sl r recursion ij se cm k cm exposition minimization feasible definite solve arbitrarily type describe alternate procedure procedure procedure check two already e determine know cone replace optimal idea finite family subspace replace constraint see potentially correspond partition subspace correspond index procedure finite linear contain start suppose space also get subspace q finitely show reach optimum finitely mention applicable need unique cone error latter signal u respectively later basis r transformation obtain coefficient spline degree k ex b unit special equation determine decomposition namely correspond write moderately expect trend motivate spline z u benefit candidate eventually shrinkage denote frobenius measure risk estimator ij distribution risk ij high frequency later factor rather poor improve give cf strategy utilize restrict contain shrinkage show cf ij particular give denote idea propose inspection almost increase left fluctuation shall estimator beneficial quadratic loss candidate normalize risk denote generic constant variance consistent differ particular generate row turn figure gray signal effect vary replace heavy c nd row depict transform left panel right panel ij average stable severe beneficial conduct matrix shrinkage matrix fine turn yield although signal monte loss f deviation latter loss thresholding year upper character year bring year year summarize effect covariate summarize vary pattern preliminary run triplet reveal view measurement transformation basis plausible interpolation element bring
straightforward index shift verification simply consistent consistent case I yy clearly one shift impossible set disjoint us study viterbi process reveal underlie specifically viterbi estimate path incorrect conclusion certainly inference simply suggest aforementione one viterbi hmm viterbi use g segmentation sequence code dna segmentation dna g code code region state hidden path block clear due vanish asymptotically probability often one emission overall link find might find acknowledgment author support science grant author research hide author significance topic viterbi process day digital model language bioinformatic hmm assume independent explanatory moreover solely hmm viterbi posteriori viterbi attempt study behavior viterbi case limit viterbi alignment assumption involve prove infinite constructive manner class hmms posteriori viterbi viterbi extraction training et markov irreducible unique stationary exist emission distribution euclidean space suitable commonly lebesgue continuously measurable I emission emission also ergodic fixed treating estimate py viterbi viterbi think maximize map path besides significance viterbi path central application hmms emission viterbi also crucial question ad trivial change alignment fully fortunately hmms extend contiguous viterbi call barrier observation go principle node time viterbi alignment construct let alignment let alignment observation segment infinitely infinitely piecewise prove infinitely shall call viterbi alignment ensure implementation memory store compute viterbi although viterbi good white states special strong node prevent independently unknown viterbi extraction compete procedure provide computationally appealing hand bias viterbi existence appear scope paper present whereas formulate hmms detail special generalizing specifically prove infinitely infinite viterbi irreducible state hmm generalize turn generalization advance furthermore hmm infinitely nod infinite node markov zero exclude case state sufficient existence viterbi communication correct positivity assume accommodate matrix notion effectively issue uniqueness viterbi construction viterbi process adjusted viterbi art outline construction appear alignment necessity technical brief consider help viterbi tu next realize path connect rp q p give observation say say say separate alignment end note path maximize unconstrained restrictive infinite backtrack tie break tie break unless break result neither break favor rare lx empty lx lx lx l mx lx li notation piecewise path piecewise follow immediately alignment u must break want infinite alignment impose rule kx l kx understand break favor subsequently special involve sequence guarantee separation adjust order impose often achieve simple separate incorporate satisfied barrier px unnecessary give hmm meet occur q satisfied correspond assumption change regard block alternate stay node let transition let emission satisfie previously guarantee barrier another change emission namely support hold lemma facilitate exposition divide assumption disjoint kf lemma existence imply existence everywhere make guarantee next eq clearly sc follow exist p simplify assume state clearly next going exhibit b r b ii ci define hold introduce fix state sequence additionally property consecutive transition maximal transition satisfy property path ready complete backtracking begin path q barrier observation contain tail definition likelihood notation appropriate q score construction hence general implication every v v last inequality maximum claim make become true imply iy ij hand become last imply end become recall get put path fact bind side true belong prove get since imply positivity every integer expand recursively li generic state path repeat argument cycle prove relation would imply last kp lf contradict satisfy exist argument replace follow recursively start return stop substitute appropriate hand side strict virtue
apply achieve indeed behave like aic aic bic whenever well empirically seem model meta yu procedure design method provably consistent somewhat aic show form validation aic aic prediction bic aic paradigm individual equivalently sequence score heavily idea suggest base term size relative especially nonparametric consideration switch change literature algorithm intend sequentially generate switch especially I source lead cause design may thought meta expert predictor meta change even though np strategy make remain switch probability switch outcome consistent expert design surprisingly exist kx density reflect belief model represent reflect belief clear maker like prediction phenomenon depict unlikely see version competitive optimality exponentially small plug bit accord phenomenon gain either wrong gray randomly markov prior try argue phenomenon occur switch distribution model want one model agree come strongly force wrong time never prefer regard strategy raise bayes prior compare switch substantially outperform probability first predict large sample well yet happen bayesian switch would standard hierarchical first discrete index try nonparametric agree nonparametric recent often work surprisingly performance strongly often inconsistent advantage switch decision convergence approximate model fact identify think prior agreement nonparametric argue irrelevant never consideration aic leave optimality reason selection depend distribution certainly completely irrelevant primary whether would aic sometimes think matter regard cross switch satisfy principle switch predictor prediction actually loo non represent weather future weather day air pressure temperature day way phenomenon selection switch broad achieve thus aic bic approach test initial report elsewhere remarkably typically moderate comparable leave loo experiment behave loo interesting open whether analogue lemma averaging switch achieve minimax switch well posteriori switch switch outperform predict predict however always analogous bound switch hellinger kl risk define yet highly important seem phenomenon situation efficient acknowledgement es point serious grateful helpful european publication view incorrect probability false parameter bad observe sequence switch predictor outcome imply I rx mx dd mutual b mixture mutually mutually say mutually ratio select incorrect relative suppose mutually absolutely contradiction exactly proof bayesian denominator x take equipped strategy correspond measurable probability measurable strategy mutually mutually interested proof random variable take respective separable integer mutually space separable lemma countable open particular subsequence p b exist iv let enumeration f pa k f sum ip switch multiplicative select proof exhibit approximate logarithmic use approximately bin outcome integral summation allow ip kp precede switch oracle p n leave remain follow imply logarithmic switch q verify select bin sense apply satisfied apply proposition actually general let strategy nonparametric achieve ip proposition argument suffice nonparametric ni integral parameterized kp py error normally outer final equality orthogonality parameterization rewrite sample entry event imply remain prove adjust equivalent gauss qp imply vector become j theorem additional respectively whether switch outcome switch nm nk value new prediction turn determine proceed property k efficiency kn km k proceed nm compute conditional value imply also imply probability theorem nx q go hold probability k kk k invariant subsequent kx k k k nx z z nx z nx end loop hold start next report proposition definition gb bic sometimes slow aic leave slow switch modification switch thereby aic compression yet usually factor consider countable set focus model averaging task goal explain weight good attractive property criterion go selection minimum selection widely selection aic validation loo inconsistent especially nonparametric many rate loo improve bayesian model therein marginal obtain average model posteriori bayes kx index model average x kx decrease call allow code code refer think accumulate log incur sequentially predict posterior x kx I achieve code sensible satisfied length possible well common say typically predict well need predict illustrate accumulate picture gray marginal distribution markov book outcome dirichlet model phenomenon common jeffreys outcome ideally predict start behave like code drop behave outperform serve see theorem nonparametric elsewhere phenomenon occur realistic well world take lead achieve combine figure behave start start differ prior prior avoid implicit assumption one size give perform switch related paper convenience section basic switch allow switch finite base switch show model contrast average switch nonparametric include case switch achieve section put work broader explain recent describe implication proof theorem result variable take nx nx mx mx predict outcome prediction issue sequential sometimes think strategy simplicity relative lebesgue measure measure latter case probability natural define joint ensure strategy fix countable explanation observe infinite consider define index element small freedom commonly view explain think strategy density bin minus selection number k example aic estimator map represent guess kx selection define joint density mixture thus approach strategy strategy prediction distribute variable ml use n ix smoothed laplace estimator uniform prior case predictor coincide general parametric kx nx general bayes list strategy infinitely development modification adjust family prediction switch prediction k I extra switch represent use switch take place prediction strategy define element switch probability q although combine correspond parametric may formally unique switch current last switch x sx switch strategy non turn posterior whether select strong standard model follow bayes mutually example nested b measure hold consistency mutual mutual condition require nx countable mutual imply mutual denote extension switch bayesian switch c e nx extend turn meta hold must predictive verify exponential family smoothed hold well general strategie theorem p nk mutually switch switch relative investigate switch converge kullback section central notion minimax switch proof concept present establish contrast switch nonparametric setting tool basic setup abuse want restriction restriction formulate assume restrict throughout lebesgue count uniformly derivative thus deal emphasize large predict divergence equal definition divergence theoretic redundancy switch convergence equivalent mean connection convergence investigate detail convergence asymptotic order nonnegative n denote absence subscript way write statement never n ip multiplication convergence bound cumulative always risk always show must indicate notion standard ordinary risk switch infimum define multiplicative nx section oracle n nx np nx nx n distinct let denote exist segment oracle select complex switch maximum additional depend maximum segment comparison establish oracle switch sequence big depend implicit claim prove never index constant imply bit encode achieve never select well additional switching idea rather lemma cumulative cumulative oracle switch condition let n om en note parametric base apply minimax risk oracle achieve iii present rather weak standard non minimax useful compare infimum lie risk infimum nonparametric e depend standard risk analogously histogram present base keep achieve supremum fortunately n theoretic variety nonparametric mean standard nonparametric call relative count minimax rate h neither develop nonparametric exist lemma convenient achieve rate nonparametric lag lag mean early constant lag may model switch distribution satisfie achieve multiplicative switch switch exponentially x apply remain iii satisfied p satisfy n n hold condition switch point jt imply situation achieve convergence select sample exponential assumption show achieve minimax rate space continuous finite k take independence infinite linearly independent three knot allow polynomial associate kx kx kx rather constitute polynomial spline satisfy spline arbitrary kn kn r lemma minimax establishes hold show thesis consistency consistency rate nonparametric family variation lemma within assume I kl dp dp kp k dp kn pz np furthermore kk depend datum lag behind increase lemma error rewrite see np coincide need look immediately verify exist minimax achieve lemma linear normally equivalently section achieve rate nonparametric distribution therefore formal restricted condition verify fix function span family give kp normally variance x kp n kk k nj zero q immediate almost surely hold lebesgue achieve minimax verify whether np k nk np omit hold include space full strong beta state c replace random proof ultimately require extend automatically achieve minimax
average dominate large correction var ml consider generalised tail order keep finite benchmark historical daily figure cumulative bootstrap copy series detail solid dash connect boundary credible bootstrap well normal var var day mainly day computation indeed risk market capital var normalize equation ahead return penalty range traffic rule daily non day day forecast generalise recommend square day however already strongly assumption normally I estimate parametric statistically law approach well return although value previous section represent prior problem illustrative first var state day nan occur day upon statistic reject window return var compare realize stage ex estimate return check ex exception occur exception obtain statistic original var year mi var reasonably respect test produce low reason locate series study frequency volatility decrease volatility regime volatility regime conservative series present novel methodology advantage allow remain identify parametric literature cluster agreement level apply variance work extension approach portfolio translate augment replace explore several filter e acknowledgement helpful de acknowledge value computation product market risk asset purpose popularity partly due understand remain presence closed compare different product partition mean methodology market clearly quantify exposure full approach methodology rich cluster point chain detection product value financial intensive easily understand var financial meet capital requirement var potential loss asset time horizon obviously financial market cause capital effect leave free var level capital g web novel parametric product likelihood ml pay attention good management pointwise precise normally assumption fail effective market resort identically return paper latter normal structure interest assign identify mean belong hypothesis identical normal setting anomalous identification common impose variance common unknown prior reduce sensitivity fixing hyperparameter experience behaviour drawback effect variance monte mcmc physic generalise extensively resort financial organized briefly introduce var var form expression extend exploit cluster analysis remark var refer typically var potential var asset normalise price inverse shall refer quantity express apply section parametric presentation generic asset return jointly vector element observation period partition sense assign partition weight formation opinion cluster yield formation partition reduce large subset contiguous interesting parametric marginal specify problem contiguous block connection daily asset normally impose partition structure partition variance vector vector former latter correspond ny product equation partition variance allow return alternative account return var deal identify describe model var approach impose mean accommodate use hierarchical ig complete model nonparametric provide iteratively joint model describe sampling remove entry dirac delta distribution point proceeding partition sampling value already belong represent replace old newly increase generate high cluster parameter gamma hierarchical reduce distribution translate minor gamma split ty iii relax identical return assumption variance aim necessarily contiguous sharing value ia factor joint gibbs conditional dy mixture remove dirac correspond gamma euler order fact volatility series extensively aspect shall result label cp consequently output describe focus attention respectively cluster share value characterize different order combine arithmetic var compute analogous arithmetic average perform cluster value var quantity var cp similar outlier identification follow propose outlier identification shift extent induce return aim select separate correspond fix norm subscript subscript conditionally estimate describe evaluation gibbs simple version sample distribution perform iii exhaustive infeasible fact partition equal extremely moderate need search partition minimum exhaustive estimate partition ji cardinality representative return right correspond element explore trivial alternative necessary exhaustive cardinality outlier index set previous test
lattice lattice satisfie condition I local limit lattice extend vector covariance straightforward page tx central nc hold equality unchanged discussion outcome initially believe reason uniform constraint chernoff hoeffding jx j nc nc original express moment indicator distribution vast satisfying denote event accord chernoff satisfy constraint look obviously continuous technical report must require go cause turn adapt continuous pay conditioning continuous write n continuous version implicitly understand lattice range theorem phenomenon regular lattice weak well proof involve much technical original closely approximation item tend somewhat interesting study theorem weak explicit bind tend infinity extend considerably e g rather explicit simplified version prove connection deviation reference weak limit consequence function code briefly countable code length actually implication code concentration precise interpret assign sequence constraint qx qx theorems close assign zero length satisfy constraint problematic approximately modification length satisfying actual turn exceed guess advance follow theorem countable lattice infimum need show equation reach result qx nx corollary equality per achieved ask distribution whether achieve small satisfy surprisingly answer number give constraint lattice exist enough exist know guarantee justification suppose satisfy follow rather achieve infimum c mass function slightly x increase decrease probability total way whereas x mp nn nx c nj jx induction function j take universal integer induce actually contradiction mention theorem n nj jx I specify value start theory infinitely constraint infinitely decide encode value well contradict competitive shannon consequence mass px qx theorem example page th result appear conference technical publish part european network publication reflect view entropy entropy provide theorem concentration phenomenon van cover theorems characterize exactly condition constraint constraint game theoretic characterization entropy entropy principle inductive range protein stock would phenomenon includes actually operate maximum observe hand conditional section limit use justification frequently cite principle kullback formulation exist stick maximize entropy tell absence guess problem usually make prediction come moment one justification finite absence prior one pick phenomenon apply one indicator constraint close uniform illustration closely several consider say eq theorem say set concentration phenomenon n concern outcome might conjecture wider namely one hold approximate phenomenon go broadly speak theorem say sequence measurable q give concentration compression phenomenon good prediction characterize non minimax surprisingly answer crucially dimensionality good sense consistently outperform walk symbol formally borel interested whenever confusion joint otherwise fold distribution also average
work universal forecasting checking rule non learnable sequence generator one forecasting system outcome base check infinite subsequence characteristic one overall notion theory systematically treat effective enumeration triple integer identify upper low recursion universal upper function every represent maximal triple distinct analogous sequence number exist minimal otherwise interval computable operation recursively finite sequence satisfy binary computable operation sense pair measure probabilistic forecasting computable distribution pr f outcome check rule select subsequence et case class construct binary partial forecasting define selection characteristic associate easy verify call probabilistic hold infinite network start vertex terminal vertex edge take rational minimal recursive elementary set extra edge delay dx qx qx construction work computable computable exist infinitely many program visit infinitely step bit sequence call program binary finite bit forecast less large sequence total sequence label portion exceed define lx n string specify induction sequence elementary extra define split edge happen finitely construction goal precede put task define case extra edge network task task edge unit delay flow set task exist finite opposite assertion edge value besides easy see topology extra contradiction existence true delay edge true sequence close extra th proof sequence maximal exist sequence extra construction hold sufficient definition extra prove np pe pe qr network flow delay qr ns flow delay combine case combine computable partial computable forecasting define compute rational arbitrary measure decrease length bit forecasting define rule extra let definition forecast initial real infinitely j obtain visit infinitely obtain analogously say lemma deterministic forecasting output every forecasting thank anonymous pointing work foundation fundamental remarkable paper outcome forecast dependent checking outcome checking rule violate close consider partial outcome forecast call calibration informally forecaster say calibrate sequence concatenation theoretic assign word probability eq specify reality recognize forecast fall whole principle say forecaster probability forecast outcome forecast conditional enter framework tends forecast et checking rule forecast base
prior posterior line though distribution tail credible informative prior credible cover present inferential section discover interesting provide motivate literature provide aspect review goal scientific discover mainly include unless keep science hypothesis rejection hypothesis rejection nan discovery false multiple false discovery rate false introduce fdr nominal selective inference valid selection rule long nominal criterion construct cover method criterion construct multiple testing vs show express procedure ensure adjust ci hypothesis cover respective apply select control directional proportion parameter wrong procedure directional simulated directional fdr side credible example cover green need correct widely recognize genome wide hundred marker throughout genome express odd disease risk allele finding occurrence false positive odd analyze report et association odd remain estimate absence log odd normally simulate normal bayesian treatment include analysis find microarray prior fold discovery gene active action small expect loss verify decision et select fdr nominal rule fdr procedure statistic fdr rule selection quantity choose review consider effect mean class compound iid posterior distribution overall provide binomial observe experiment framework regard discuss way regard distinct relationship within regard draw box distinction classification laboratory different chemical batch distinction carry mean mean call sample likelihood actually selective inference analysis microarray gene variance expectation level log fold expression gene differentially express frequentist mechanism provide selective example cover adjust selective suffer limitation impossible incorporate adjusted selection adjustment criterion selection adjustment need toward adjustment selective however selective inference actually event selective truncation problem suggest selective call selection predict school college bayesian selective mechanism predict student discuss selection definition truncate derive case informative prior effect bayes bayesian selective relation fdr specify rule make algorithm correspond box bayesian fdr exist fdr microarray express control directional fdr provide inference change fail frequentist however frequentist selective selection statistic yield fdr statistic yield selective differentially gene paper conceptual contribution specify truncated distribution selective use selective action risk incur selective example selection unlike fix describe parameter type derive truncated conditional selection apply give unknown remain unchanged incur selective fy truncate truncate provide inference eq fy divide mix distribution incur selective inference fy change integrate truncate reveal truncate dependent basis college student school expression reveal fix school student parameter college student example compound selective max value remain unchanged parameter batch batch compound compound conditional mixed selection adjust adjust marginal act informative use conditional inference also informative argue selective opposite decision informative treat fix adjusted informative updating accord adjust selection adjust selection distribution conditioning make redundant conditioning condition mixed selection conditioning apply remark kind likely mixed compute adjust mean joint density selection adjustment stochastically terminology box call random mixed selective incorporate yield adjust integrate integrate adjust incorporate integrate box effect exchangeable distribution independent marginal joint q scatter plot display model important joint observe conduct another simulation fix realization repeatedly truncate repeatedly component model display scatter plot truncation leave joint display joint left panel model value towards keep first panel reveal shrink express average incur provide selective thus rule selective adjust selective adjust credible h serve estimator generate assume random flat informative flat prior non informative distribution pr flat mode credible spike mode credible flat prior mode credible much negligible correspond shrink mode credible unlikely adjust selection adjustment adjust improper include selection adjusted likelihood parameter illustrate unique selective paper large selection towards alternative adjust stochastically small much adjusted adjusted interval frequentist adjust rule flat prior credible interval frequentist frequentist hypothesis reject nan quantile large set rule confidence iv non interval false coverage refer frequentist paper indicator event integrate confidence construct give incur selective iy f selection adjust credible posterior effect mutually value numerator denominator adjust interval selection random serve conservative dispersion specify prior credible interval informative coverage informative yield proposition marginal credible yield display dash curve credible flat curve credible interval flat prior light adjust credible interval interval credible frequentist coverage proportion explain offer explanation informative credible adjusted rather adjusted posterior credible interval specify effect exchangeable effect seek control proportion rejection specify nan hypothese specify discovery false select provide select regard analysis discovery discovery number false result conditional selection false ensure fdr pearson type option rule define risk adjust follow subsection exchangeable effect select possible false section derive assume I q false risk non exchangeable specify control exchangeable bayes estimate unknown marginal fdr approximate among specify fdr controlling rule compute specify fdr form fdr control directional selection use ensure risk set criterion risk directional fdr notice example criterion correspond effect directional e directional fdr model iid region hypothesis conditional reject fdr previous generate distribution random loss adjust discovery special adjust expect fdr special lastly fdr valid fix informative posterior selection include compare mutation rna gene expect log change variance assume variance expect scale prior apply package variance flat informative fdr procedure specify rule q good expression independent marginal specify directional select less gene discover gene directional fdr rule performance directional controlling implement nan hypothesis differential expression observe ratio deviation observation directional fdr notice error less directional conservative directional frequentist differential expression degree statistic since
follow estimator behave always approximate assumption hold let path sample limit interpret almost surely integral construction integral zero deduce follow parameter mle recover answer compatible model appropriate asymptotic hold technique ergodicity small would also every hold path dy dx absolutely continuous sde l straightforward see complete replace assumption continuity dy function happen mle equation multiscale differ average subsection correct unless something however path limit interpret almost identical q lemma eq deduce q put previous limit behave weakly come equation consider follow subsample behave bias time subsampling overcome provide appropriately expectation measure recall choose initial apply likelihood principle euler statistical obtain likelihood noting depend prove provide subsample appropriate prove rely proposition prove assumption hold limit follow present sufficiently increment denote martingale deduce fx n n fx n correct variation since c assumption prove zero square integrable path apply eq identity log function form version apply fact ergodicity limit already see maximizer equivalent consistency ergodicity first assumption continuous respect prove countable dense ergodicity provide q ball need old numerical find experiment framework illustrate problem construct identify responsible langevin smooth depend temperature brownian fast generator square integrable equation grain want estimator asymptotically biased correct responsible assume potential eqn sde process write stand inner smooth notation py dy appropriate simplicity sde formula eq dy eq q v vx dx dy vx py dy vx dx z p vx vx derivation formula cauchy laplace show fast admit description slow estimator asymptotically drift coarse grain model slow mle asymptotic likelihood infinite parameter subsample appropriate rate several decade application dynamic chemical mathematical finance believe great care take use maximum likelihood infer various interest plan list bayesian technique estimation multiscale coarse grain slow work context involve reduction investigate fast ap partially international ct partially sde derivative generator invariant constant isometry invoke boundedness average almost surely apply concern follow boundedness together sure clearly assumption u probability proposition reader convenience assumption increment write martingale term assumption hold q respect condition need lemma rough increment assume hold eq formula vs ds x j inequality imply j c j define q bound hx ds ds x assumption equation together old inequality imply hx ds hx old hx hx p hx p ds ds estimate ds n hx ds hx therein gx ds gx n ds p gx remain law prove zero require note respect stationarity dy dirichlet assumption choose converge remark corollary pt statistic al mathematics institute university cv al uk maximum fast slow differential aim light scale fast grain rigorously refer ask correctly coarse grain slow maximum unless subsample explicit formula function simple dynamic keyword likelihood subsample differential often useful extract simple capture important aspect compatible small technique scale phenomenon appear frequency molecular essence coarse grain context parameter incorrect create framework issue order gain investigate multiscale grain freedom rigorously coarse grain description dynamic couple multiscale fast equation separation give average model slow extensively decade reference therein numerically limit fast slow system approach infer coarse grain multiscale mainly multiscale know merely multiscale fit coarse equation slow understanding problem move multiscale diffusion system drift estimate use fast case asymptotically unbiased correctly slow hand rigorously term appear become subsample fast slow refer equation coarse average slow compatible possible state informally mle averaging mle e formula error obtain precise theorem failure due identify subsample asymptotically subsample rate roughly obtain q technique process similarly eq let denote compactly martingale pose continuous deduce slight theorem chapter deduce produce sensible necessary impose process solution assumption define well cholesky solve sde brownian poisson unique side zero density construction argument proof apply eq poisson technique theorem imagine subset suppose actual ask whether find mle average previous weakly equation model ergodic invariant measure preliminary effect multiscale property asymptotically path order irrelevant estimation estimation drift path
bernstein cover variable proof lemma numerical showing may possibly q small low prove bc ff eq strictly prove similar q comment lagrange infinitely differentiable since f divide exchangeable eq deduce recall split center appear exchangeable depend combine resp expression term convention detailed order deviation upper definition remark validation improve penalization remark expectation reference former latter additional sect section eight compare procedure sect benchmark independent differ fig jump histogram fig sample notation regular eq regular bin size dyadic regular dyadic contrary bin interest bin size experiment b eight experiment another kind kind bin experiment sect possible estimate uncertainty estimate report uncertainty deviation divide regular loo uncertainty divide experiment regular regular dyadic dyadic bin size loo completeness similar one uncertainty deviation divide bin size dyadic regular dyadic bin size loo f uncertainty standard divide size regular bin size loo f f uncertainty
yet determine clearly boundary apply general disadvantage dependency also interestingly find model encode type directly compound exclusive invariance reveal principled encodes variation substantial maintain run mm lemma infer match however shape even transformation actually subject limited variation appearance involve structured prediction outcome appearance reveal substantial upon successful time shape image alignment typically correspondence distance great devote shape feature encode related optimisation transformation perspective impose convenient algebraic correspondence efficient graphical small clique substantially advantage namely flexibility encoding framework correspondence explicitly match np hard match quadratic improve power use geometry computational efficiency combine shape isometry scale feature encode relative scale regard know appearance term speed remainder brief structure learn thing depend study identify template scene template scene whereas query scene correspond common extract local generic match correspond feature assignment cubic unary pairwise second location unary feature generalise quadratic near shape matching matching setting assignment encode depict represent point represent optimally energy bottom graphical single propagation sufficient form single convergence bottom number convergence unary term c order cost vector depend specific shape match various matching optimisation mapping third convention achieve simply cost choose assignment function avoid sufficiently smooth choice training specifically set pair u control importance risk feasible parameter advance structure consist relaxation go consistent incur solve optimally clique graphical look instance depict seem capture distant unlikely contribute mind new feature brevity shorthand scale width scene scale average angle unary pointwise square transformation rotation invariant dependency include clique ensure include remain capture adjacent clique scale distance angle scale distance angle detector identify clearly impractical target scene unary feature single weight vector simply determine requirement consequence tune perform work regularity order experiment performance house sequence frames house unary assignment present pairwise specifically pair template scene template experiment however form adjacency leave rd frame correct match red match report match quadratic bottom bar error fix baseline frame normalise increase top without exhibit substantial unary likely unary author learn author unary pointwise exponent linearly work well fact achieve large increasingly violate see assignment improve run time scene stage run assignment even approximately error baseline note identical iteration figure stage angle see explanation learn shape separate radial angular angular bin important distant second briefly run dependency perform benefit exceed capture dependency indeed play significant experiment randomly exhibit experiment point randomly randomly range experiment aim examine robustness different note zero outlier intractable experiment adjacency outlier perform worse hope get normalise hamming loss reward effect range pixel bottom outlier show right bar standard error plot identical observe improvement high
q approximate popular replace singular singular equal sum equal singular strictly alternatively name include fan introduction concern solution coincide efficiently via include semidefinite survey whenever set solution criterion nuclear heuristic minimum nan rank great minimizer hold include constraint reformulate secondly equality nuclear heuristic show obeys space mean gaussian space statement translate rectangular zeros random ensemble ensemble result low sample nuclear norm recover random surely norm heuristic succeed recover strong rank eq satisfy norm rank depend solution hand nuclear heuristic succeed bind test experimentally show correspond experimental datum weak recovery nonetheless surprising bind versus rectangular transpose vector stack ever define call schmidt frobenius norm large nuclear relate associate follow readily dual frobenius trivially norm duality several time analysis sufficient success lemma verify eq allow project onto column space generality full rank since rank lemma converse violate nuclear project equal appendix x x non imply interested reader argument find proof imply nan banach space sufficient condition meet explore imply necessary sufficient projection subspace rapidly differentiable theorem deviation tail normally weak rise find complete prove strong hold net projection net examine change fourth projection operator rearrange probability onto proceed need project onto subspace endow net metric cardinality net eq show upper eq prove define constant use solve bind eq net comparison sufficient supremum infimum great random comparison theorems generality lemma interesting right norm let dual definition sample sample variance variable observe fashion dual let functional matrix difference greater equal q complete follow plug expect nuclear eq secondly reveal plug appropriate value variety size number measurement fix construct recovery low matrix vary determined rank cutoff rank triple time rank choose factor entry sample ensemble column nuclear semidefinite ghz could solve less minute recover display reflect rate scale remarkable failure region within sufficient hence trace affine argument full equivalent rank last note imply lemma bound maximal convex subgradient bound q last expression argument complete proof continuous exist net complete lemma claim em pt xu arise machine np hard yield solution popular replace decision quantifie successfully find probability affine success finally evidence heuristic convex compressed sense rank arise controller reduction rank embedding metric space euclidean instance singular decomposition general good involve elimination dimension heuristic solve minimization problem control heuristic minimize positive semidefinite introduce sum symmetric value optimize efficiently trace heuristic nuclear produce solution could elementary algebra success nuclear norm
upper adjacent result line process element r graphical construction color chain eliminate adjacent local stack begin b red indicate stack element stack big restriction lower adjacent update restriction adjacent element restriction adjacent cost element stack adjacent big update set element element grey stack eliminate new interval turn black grey l u h element stack restriction turn mathematical module iteration calculation maximal element present implement build begin prevent select resulted remove component lr minimal calculate u stop next prove correctness return minimal minimal maximal element begin complement element search space thus new add upper big equal establishe permit restriction construct version decomposable shape curve big equal chain result big element describe already cover I contain stop line cardinality restriction cardinality restriction r lr ar l restriction list low element contain computation heavy visit single minimum recursively recursion adjacent big element set later explore computing cost whole process process huge heuristic stop procedure another shape u algorithm minimum local element one shaped contain reach element reach minimum permit relax curve curve window genetic set uci machine repository attribute criterion avoid moment reach perform feature inclusion desire default program web page information employ feature randomness eq convention probability give possible yield large gain practice feature insufficient e rarely instance penalization curve part curve uci alternative chain process processing process test quantitative gb follow summarize table column present result window test reach remain process frequently take winner classifier experiment test reach process improve process happen experiment involve computational spend process overhead responsible consume htp winner test feature two reach reach vote na na na na take boolean u optimization present branch bind decomposable shape chain permit problem context constitute explore boolean nature domain permit restriction restriction explore construct adjacent chain adjacent node option randomly domain make minimum cut execute adjacency adopt diagram connectivity partial order visit cost decomposable shape support formal u constitute optimization restriction boolean lattice property constitute new structure combinatorial find u involve operator six uci repository equal obtain precision many case curve several position local efficient minima bad one minima result encourage version minima lattice optimization subject future begin reach early version la return process step construction line result result initial element lk l element version curve true cc l decomposable shape author grateful tw health usa helpful biological helpful comparison generate sc computer science topic bioinformatics pattern parallelism currently ph student electrical engineering apply ph electrical engineering lattice lattice bioinformatics bioinformatic biology david receive sc computer science de sc vision bioinformatics gene currently ph student computer research laboratory year branch shape boolean present combinatorial boolean lattice curve lattice apply heuristic equivalent branch explore shape contribution paper architecture exploration prove several public superiority give good time lattice branch algorithm shape combinatorial search space cost simple object huge minimum heuristic kind heuristic formal find branch mathematical guarantee study combinatorial optimization space finite search object organize space apply recognition window mathematical minimum subset sufficient finite shape form classifier join increase induce classifier class become cover heuristic well good branch base cost branch real joint large practice estimate monotonic curse know curve problem phenomena branch differ explore shape window architecture repository set encouraging set result cover pattern lattice search particularly interval cut cut search property adopt represent cut perform branch mathematical experimental conclusion discuss step inclusion set cardinality compose boolean lattice partially lattice small large element product complement denote zero one belong mean abuse language km shape maximal u shape cb cx decomposable present maximal chain local minimum maximal lx element boolean moderately shape maximal key branch deal combinatorial subset give ab leave characterize decomposition boolean right interval interval collection collection eliminate search explore explore minimum minimum extend set region
markovian I guess contribute significantly decrease multipli particle r n expensive involve sum introduce suggested target set position instrumental assign discard present sequel adjustment multipli weight expect adjustment multipli model suggest x ix x adjustment multipli application point lead even worse quite varie prior study adjustment multipli single criterion whereas criterion kernel adapt clear behave differently behaviour particle infinity expression limit derive adjustment proposal measure absence arbitrarily consist coincide develop turn absence nice interpretation within limit time joint couple conditionally moreover limit filtering reweighte new markovian dynamic completely instrumental quantity reflect model adjustment still marginal identical choice proposal kernel weight family proposal optimisation optimisation nice explicit definition surprisingly function sophisticated adaptive elementary scheme burden quickly limit smc simplify presentation sequel proposal jointly adjustment weight proposal clarity prefer presentation technique work optimisation adjustment weight rather complex adjustment close model formulae find gain extra cost sort necessarily extend model precise algorithm implementation issue result precisely result random induce transform use hold linear space sometimes weight wish sample new measure marginal simulating position instrumental difference hereafter take ht I n I follow function I assumption define weight n describe pass somewhat general l direct consequence two absolutely respectively eq measure kernel assume converge l satisfactory q illustration return ce adaptation particle type mutually distribute wish expression optimal weight respectively express weight adjustment function adjustment mode variance hessian mode let x respect expression proposal obvious may parameter instrumental adjustment instrumental adjustment straightforward optimisation close form n rich family property toy conduct plain filter adjustment numerical algorithm adaptive ce base reference filter weight experiment yield adaptation procedure run burn iteration reach stationarity state equal due poor particle mode transition adjust automatically variance proposal auxiliary filter observation display filter particle filter ce inside particle iteration study indicate l step figure plain choose prior contrary success change regime step filter need several adaptive reduce observation ran particle scheme particle ce expand plain matlab plain bootstrap ce figure equal runtime adaptive bootstrap three particle mse particle except reference dash identity hold outer product derivative use establish assertion show assertion enough function belong theorem uniformly detail definition f select thus suffice follow tight satisfied proper dominate follow light establish hence consistency n q imply moreover recall consistency prof complete belong continuous q belong apply q entropy follow rhs depend adjustment multiplier establish rhs associated density multipli establishes assertion grateful useful anonymous comment suggestion section fr se strategy particle filter rely particle proposal major show weight instrumental auxiliary analysis adjustment multipli type minimal leibler involve distribution design illustrate role user tune key smc notably particle adjust adaptively adaptation particle reach threshold avoid situation particle region adjust cloud resampling suggest increase ingredient mention design achieve efficient sampling require sample specify importance mean zero variance estimate ensure grows exponentially adapt adjust filtering important propose state region bootstrap usually inefficient iterate map decompose correction amount compute appealing update approximation variance importance equal enjoy optimality condition consume auxiliary accept reject difficulty several approach try behavior sometimes prohibitive sampling therein specific optimisation particle consume approximate tool therein technique conditional next mode carry jacobian carefully computation rather involved overhead technique suggest build smc observation comprise stage stage particle modify order region multiply depend well possibly adjustment position necessarily form proposal multiplier weight accuracy filter choose way mixed leading computationally particle see none suboptimal sensible criterion practical choice correct plain bootstrap see particle filter drive different try guess proposal seem sensible theoretically step construction sensible straightforward smc interest approach lead reason expression time depend explicitly hence recursive satisfactory sampling appropriate sequel risk criterion target coincide estimate I equivalent system practice estimate particle empirical converge tend particle result come approach currently attractive empirical particle still proposal particle appropriate proposal use detect resample adapt formulation filter multipli call parameter together mixture instrumental expression limit auxiliary target auxiliary smc adjustment proposal optimisation adapt proposal drive objective coherence detect see use algorithm improve make observation state rigorously informally finding develop briefly adaptation expectation refer implicitly dominate ii px self normalise nf application availability provide integrable tend infinity function estimator asymptotically choice asymptotic variance respect proposal distribution optimisation close negative fx impractical reject choose optimal therein discuss point sequel type impractical expression asymptotic variance recursive proposal looking often express I unnormalize q variation accuracy examine importance detect variation ess overall small ess possible fit proposal proposal negative shannon entropy maximal minimal rare proposal ce methodology think element subset family mixture multi student recently issue context efficiency course evaluation quantity involve evaluation integral detailed fact p substituting expression optimisation formally share chi square likelihood important definition constitute estimation approach could keep inefficient optimisation proposal principle outline optimisation recursively define iteration number update either quantity sufficiently use conjunction search direction issue dependent particle obtain particle index optimisation even suffice simulation increase generate another consist stepsize tend zero expectation approach assume appropriate regularity q eq immediately compute derivative share may uniform f regularity differential quantity quantity w xx realization coincide keep use outline solve follow optimisation
logistic idea fold datum testing note vector score score pass evaluation tool website summarize td table clear win fold mark map box model least art rank algorithm intercept ir could pair different benchmark limitation implement conjunction et dependent rank category determine neighbor immediate idea within york ny contain retrieval benchmark collection ir ndcg precision map idea free intercept query relevance nuisance test report idea http microsoft com contain benchmark algorithm challenge require result ir measure ndcg precision relative problem record record ir irrelevant irrelevant rely record comparable record compute query nonlinear follow naive compare relevance train relevance score response check hypothesis query result determine cost idea relatively nuisance extremely logistic test ranking arguably simplest test base challenge complicate well give query set k result pair explain label td logit simplicity global vector aside query dependent call intuition query query pass logit thought result obtain coefficient query number query eq ranking query likelihood
course free soon leave response hamiltonian assume permit q integrate perturbation sum treat ideal effect mostly dependent contamination use calculation assume symmetric index symmetric coefficient integration assume impose continue permutation convenient shift background read since correlation latter need calculate connect moment connect diagram line vertex end line end interpret end correspond external label since depend diagram field correlation depend calculate open end line internal coordinate represent represent internal locality integrate external diagram divide permutation leave field derivative remove vertex diagram diagram second repeat index assume purely q expression diagram simplify considerably fourth line connect represent term diagram calculation space scalar assign inverse line momentum real momentum internal line integrate momentum integration term front expression divide source momentum vertex open end get sign delta momentum permit calculation hamiltonian fully characterize space volume signal describe instrumental convolution locality interaction simplify interaction hamiltonian km sphere need provide properly require reason energy energy theory logarithm signal latter hoc ill issue however try expansion loop diagram investigate representative adopt describe diagram notational convenience volume diagram convergent due unbounded filter field dark matter fluctuation sufficiently present since freedom chose behave ensure although theory beyond formalism sensible price evaluation hamiltonian provide try simplicity moment case diagram diagram diagram non expansion around classical power line diagram correction around correction quantum correction quantum uncertainty correction due truncation scheme classical completely different purpose truncation incorporate systematic carry measurable adopt boltzmann volume available uncertainty latter temperature calculation relation internal energy boltzmann temperature normalize choose arbitrarily calculate consistently energy generator probability logarithm possible diagram internal end generate interaction diagram perturb interaction proportional vertex internal vertex thus zero notational convenience power diagram diagram diagram loop diagram affect x x permutation obtain information identically gain response measurement fidelity linearly level increase theory gain actual typical variance mx general specify prior probability distribution field prior relate physical taylor freedom diagram quantity consist line end log evidence diagram end diagram end around connect diagram end read diagram specify sect use algebra package vertex sect implement matrix especially filter verify monte simulation use likelihood establish hamiltonian prevent diagram summation saddle classical posteriori detailed along sect poisson noise dependent therefore well non galaxy scale expectation dark matter initial field contaminate galaxy datum currently galaxy survey chart distribution three improve galaxy important reconstruct affect signal crucial problem image detector treat discuss problem galaxy superiority galaxy formation reference sect treat cell volume galaxy cell galaxy dark assume negative statistic density galaxy everywhere reality exhibit variation spatially observational linear non depend log exhibit negative definition purpose exchangeable propose investigate seem reproduce observational galaxy underlie density field likelihood actual galaxy cell scalar read see read weighted decrease galaxy bias formalism information response bias galaxy signal density galaxy exactly vanish analyst volume individual size grain however series vertex correspondingly patch exactly small counting provide among diagram cast class frequency universe go formulae also galaxie galaxy variable galaxy multi galaxy type spatial encode datum hamiltonian interpret integrated type read live solely matter reconstruction problem affect book keep galaxy marginalization various since know galaxy type marginalization justify explain reconstruction simplification order magnitude experiment exhibit dependence bias fluctuation unity scale galaxy therefore galaxy bias matter galaxy bias galaxy shot order interaction comparison galaxy numerous order source reduction dominate therefore accurate reconstruction galaxy improvement describe compact linear formula correction correction linear symmetric thereby signal display fig sx sx datum process reconstruction exhibit naive high diagram display correction mostly obvious correction technique summation contain diagram close b resolution galaxy galaxy response response wiener sigma next classical accord wiener reconstruct correction signal deviation wiener length strongly response unit galaxy mask mask signal interval mask linear wiener reconstruction well correct partly display reconstruction estimate galaxy signal location signal field last galaxy form especially suitable dispersion reconstruction probe wide hamiltonian reconstruction realization curve wiener uncertainty average uncertainty see remain uncertainty flow read top un uncertainty around clearly see uncertainty galaxy estimator visualize effect height propagation information source structure mask location block become visible mask response thank rich additionally region high count low galaxy region galaxy galaxy per galaxy observational setup sect want amount strongly depend actual either factor control unity present strongly bottom observational strong one gain expand order realization fluctuation fluctuation bias gaussian approximation scheme due density gain high galaxy trace show case fig approximate adequate benefit eqs uncertainty galaxy non inverse maximally reconstruction fig fig uniform observation advantageous location show demonstrate observational gap high galaxy large asymmetric shape gain galaxy density advantageous around galaxy real observational location achieve response consideration interact temperature fluctuation currently scientific availability high measurement constrain physical early epoch universe reference sect temperature fluctuation intensity due e mode fluctuation follow gaussian secondary universe integrate signature secondary temperature fluctuation initially write potential observe induced intensity foreground response translate publicly code sect matter development concept deviation instrumental foreground wave contribution non fluctuation consequence non gaussian p realization via respectively read subscript covariance usual notational convenience aim principle produce compare foreground galaxy nonlinearity observable able least briefly convenient generating permit spectrum read transform power potential possibility spatially vary parameter track coordinate spectrum calculate spectrum denote case noise bi dependent calibration add expression usually formulae apply field marginalization hamiltonian tensor read convention permutation local response diagonal therefore due different versus frequentist angular scale wolfe dominate fluctuation space surface permit simplify coefficient stand term expansion series usage wiener evidence interested reconstruct fluctuation extract quadratic provide diagram time twice diagram feasible calculate expression proportional quadratic low therefore sphere finite sort matrix coefficient compare angular foreground galaxy level contamination removal assess reference sect therefore suppose significant bi angular scale since insensitive sign actual usage subtle proportional seem power combination bi spectrum estimator filter exposure chance coupling unbiased wiener filter normalization fix realization vary regard spatially linearity experiment measure temperature response spatially homogeneous therefore universe homogeneous diagram diagram yield used order combine diagram result diagram result loop diagram homogeneous exposure realistic vanish see encode already point coverage second base correlation exposure estimator estimator order exclude traditional estimator l since signal realization bring third normalization diagram expression diagram traditional suppose probability traditional sense posterior perform encodes reason normalization dependent understand suit even concrete prominent sect fundamental noise physical reality universe consideration reformulate language identify spatially technical field ensemble field datum prior information conceptual hamiltonian theory wiener present non interact particular latter field spherical space argue usually well normalize wiener algorithm exist make term weight tool hand convert well boltzmann shannon free mechanic motivation think inference reconstruct spatially matter galaxy galaxy survey reconstruction imaging temperature fluctuation serve signal signal inference involve field coefficient may incorporate realistic need understand galaxy formation underlie dark term statistical hamiltonian dark detailed numerical semi description benefit inclusion foreground solid fundamental fluctuation like non heuristic prove serve circumstance implicitly suit well meet whether suitable problem field acknowledgement people scientific various perturbation theory structure science isotropic thank connection acknowledge helpful comment manuscript constructive briefly summarize position usually principle regard infinite dimensional volume finite domain volume origin delta transformation operator two denote complex spectrum denote function also permit compact q component tensor vector diagonal whose component eq diagonal multivariate tensor power spectrum fourier representation trace fourier determinant hermitian dependency e instead write work provide flat scalar sphere isotropic depend orientation spherical orthogonality therefore assign fourier proper angular read external momentum line connect ml c dl dl angular angular line vertex quantum momentum internal angular summation front get symmetry interaction complicate orthogonality power spherical harmonic function compare spherical structure orthogonality relation successively due probably calculate propagation space term individual diagram xy cm em lin lin develop spatially field reality derive hamiltonian field wiener interact expand provide rule position fourier field reconstruction matter galaxy incomplete galaxy survey galaxy formation universe response galaxy formation reconstruct equation filter predict universe linearity even spatially temperature fluctuation signal relate scientific design possess many conceptual information theoretical theory distribute field contrast mostly full interact thus theory flow optimal uncertainty fluctuation quantum perspective technical permit experimental deep insight mechanism accumulation dependence pure empirical hoc alone hope work interest reader apply practical spatially especially theoretically serve understand many extraction expect reader familiar mathematical everything interest structure detail brief fall two abstract application distinction physical formalism introduce summarize knowledge concept read sec sec new present interact shannon well field define provide section reader specific inference address serve matter galaxy survey analyze generalize non find work try information conceptual theory image reader expert might decide skip bayes inference problem belief knowledge information work shannon information negative entropy mechanic uncertainty numerical bayesian integral curse high massive method idea already mind hamiltonian partly relevance dimensional get recent review list exist reconstruction incomplete important distant object weather mainly observer well optimal prominent base implementation assume wiener image wiener regard full signal noise sect formalism wiener expectation data hamiltonian decompose information step build wiener reconstruct truncation iterative instability miss implicitly flat prior gaussian entropy partly partly outside reconstruction discover prominent direction path exist regard linear mechanic course early usage moment field already simultaneously argue model theory transformation summarize nuisance field direct aim visual approach ad hoc enforce posteriori smoothness controlling topic also follow publication recognize tight statistical prefer field put emphasis whereas obvious concentrate reconstruction probability potential posteriori book essential insight theory follow probably work publication remarkable circumstance rigorously mathematical require come vast make book first towards improve galaxy survey large matter overview work universe galaxy initial density strength self instability partly universe fluctuation produce universe carry valuable function observational formation perturbation however later galaxy formation description observational complicated sensitive velocity cause partially path integral integral paper extensive body simulation formation probably provide property year recognize address method virtue correlation future reconstruct fluctuation observational recognize fluctuation reconstruct galaxy development reconstruction implementation wiener filter principle least approach various wiener apply galaxy survey partly matter behind disk close address sect sect extract galaxy power power normalization overview method statistical universe small epoch neutral streaming form dark matter physical temperature velocity fluctuation mapping fluctuation permit precisely simultaneously like dark matter pressure sophisticated extract noise foreground emission accuracy b h w wiener treatment temperature fluctuation spectrum calculate boltzmann dynamical active specie well publicly temperature prediction recognize happen fluctuation fluctuation initially fluctuation early universe fluctuation discriminate observational h sensitive enough constrain detection sect make side current status attempt infer universe incomplete quantify uncertainty result galaxy survey map experiment theory interpretation uncertainty offer ideal permit describe relevant involve universe system consideration denote universe datum universe completely universe spatially function coordinate likelihood physical condition outcome functional integral possible sect universe aspect physical reality interest capacity device use physical influence receive conditional signal derive span proportional need divide datum ps pd mathematical eqs normalization term evidence formalism combine class underlie physical may correlation express choose denote dependence response noise reaction signal might whereas exploit content denote complex definition usual language processing connection need trivial since variable aspect noise signal transformation coordinate transform exhibit theory suitable signal response signal recover may aim may require definite signal state negative signal non operation construct optimal fidelity signal uncertainty q realization trace since value signal posterior estimate latter quantity characterize overall estimator inference illustrative copy statistic signal bad reveal signal however perfect zero complicated noise definition practical reason usually process well steady analytic reality degree freedom sense correlation exist degree insensitive add budget avoid lead mathematical convenience example present guess example assume solely characterize matter observable phenomena galaxy refer epoch universe later galaxy galaxy trivial mode position reconstruct sensible pass attempt uncertainty reconstruction define low filter datum manifold sphere take appropriate expectation fairly fortunately exactly formulae moment often huge decade couple
penalty exclude predictor big see close ccc tf simulation try network topology copy number positively correlation much thus block section block define ij il j magnitude matrix simulate give material predictor table nine edge master predictor cross number much count slightly count finding suggest control ccc fp tf tf breast mention early genome significant rna genome finding result may cast dna copy number rna level tumor expression microarray array describe step map proper dna copy array output copy interval basic unit genome sample employ order estimate copy expression map human gene array also sample median median breast cancer publish breast cancer copy affect rna gene possibilitie rna copy gene affect figure illustration study find relationship type direct interaction sparse partial estimation pair hereafter rna significantly expression level consider breast cancer existence distinct tumor distinct association gene expression divide distinct correspond suggested breast breast cancer set five variable iii select coefficient include copy often infer rna note slight modification idea five indicator force gene achieve update fitting parameter model fold search positive away fold contiguous relationship annotation ccccc begin nucleotide bp array pearson dna copy expression level expect high correlation potential association copy unlikely tumor influence gene symbol bm q bm q table list table list gene probe annotate identify span hereafter breast cancer growth protein tumor breast normalize ratio tumor six result literature importantly suggest influence imply relate nr co influence serve drug c end want besides rna model focus I predictor response investigate multiple master play role tackle challenge utilize penalty consist network impose sparsity detection take account interpretability computational penalty impose help information across regression incorporate type joint efficiency also detection master identification make cross treat validation consistently select majority fold suggest false negative broad apply breast cancer goal influence dna base breast cancer tumor suggest region influence rna gene breast cancer finding additional investigation way verify common region rna biological great interest understand nucleotide rna levels rna level well disease utilize select group encourage group also apply public available grateful valuable h fp truncate fp bar represent bar prove theorem show solution give abuse denote xy q xy similarly xy xy xy abuse obvious xy xy xy expression denote show exist equation plug achieve minima bic section bic selecting assume th df overall criterion derive also column equation prove explain definition current scale future therefore un bias freedom criterion apply stein identity normality residual unbiased estimator p x x last rule eq regression penalty estimation simply induce reflect preprocesse preprocesse array standardize median smoothed gain lose region genome region copy equal smoothed accord noise define hierarchical genome cut height copy number fall agglomerative leave fix genome array adjacent tree bottom refer array divide non overlap array addition breast cancer combine seven breast gene intrinsic gene gene signature list recurrence list index breast relate set away gene intrinsic gene derive breast rna apply sparse infer rna identifying assume overall partial employ fitting indicate many tuning choose result include maintain stability associate marker breast encode gene breast cancer encode player breast cancer breast cancer annotation gene list refer rna interaction investigate rna symbol like tumor transform bind protein member p could detection expression expression could correlation expression across correlation rather weak indicator predictor label pattern work expression total across cluster divide patient five five suggest b breast illustrate five information tumor influence rna report drug target genetic associate breast possibly encode tumor gene tumor cell dna induce death tumor play breast serve remark rgb california usa statistics university ann mi usa medical research university division public health identify master fit low sample motivate investigate relationship among biological base genomic study copy rna rna dna regression utilize select discuss simulation breast genome wide rna dna copy region rna gene lead sparse dna rna breast microarray genomic experiment primary breast tumor cancer center rna level dna copy experiment tumor molecular marker clinical reveal array alone array alone characterization rna gene primary secondary dna give loss cancer dna rna help subtle genetic relationship widely variation copy number play important development expression cancer dna influence rna rna level example copy expression genome hard assessment understand genome event analytical tool reveal subtle complicated dna copy number rna level copy number regression dna copy dimensionality response identify iii complicated relationship response unlikely satisfactory method variability fit help accuracy incorporate relationship link response help hold large need regression extend multivariate recently utilize within region propose propose union multivariate bayesian response way reduce analysis include compared enter reasonable affect moreover predictor term build scientific widely exist role mention account response influence none response boost aim select novel multivariate master predictor penalty coefficient impose essence discussion section put regression illustrate study breast cancer mention early rna finding analyze array array reveal portion relationship gene however identify penalty penalty encourage effect correlate induce penalty illustrate penalty method utilize regression master perform suggest enhance exist cost coefficient help information regression correspond response thus incorporate variable degree hand control individual subject variability fitting see greatly efficiency section iterative solving problem zero exist row q supplementary say specify correspond univariate ordinary square current lasso univariate soft ol conduct matrix orthonormal ol update adopt define fix repeat achieve final computational tuning v fold perform cross subset complement estimate I ordinary least square calculating assume note use fitting result result number finding aic tuning result section limitation idea treat
describe measure approximate human texture specification representation haar drastically come haar coefficient store color descriptor element across descriptor spatial extent use image process rest area bin find low capturing inspire specification interpolation give representative color cosine dct result frequency first v wavelet compare pyramid resolution simultaneous autoregressive mr base texture descriptor wavelet gaussians complete imply expand orthonormal redundant present set redundancy gaussian peak meet like u gaussian gaussian domain width related gaussian word wide gaussian bandwidth eq center frequency scale q orientation orientation generate filter bank see filter bank kernel rotation invariance descriptor direction dominant feature article feature texture human introduce brief measure column gray size pixel without center pixel th entry sum pixel pixel gray different rectangular region oppose replace rough surface particle compose weighted center pixel cope neighbor f q gray reflect scale gray variation intensity texture spatial numerator measure denominator summation formula slightly article complexity could patches different elaborate texture build easily attractive texture numerator tend denominator coarse texture high dimensional discrete q circular convolution fourier ft ft multiply wise filter origin figure case important dominate dm width image lie cell infect indicate classify correct cell typical name variant meta broken cell cell descriptor set possible within boost library output reduce time differ gradient ascent variant stochastic ascent svm implement tucker kkt select first update one rapid kkt progress make updating coefficient simple histogram bin see release receive represent sum histogram color calculate descriptor expensive descriptor increase discrete cosine dct use library transform west wide band v extract frame simplify square next else else else x length invariance descriptor achieve dominant project image filter bank gray divide subproblem center write calculate position sure accumulate thereby convolution calculation pixel convolution image much efficient fourier dft image power fourier transform west size still great speed classifier view hold abstraction name realize abstract classifier view contain represent boost array convenient view validation view view convenient represent multiclass join group new removed achieve two part classify simplify merge cross subset act test merge expert often opinion skew merged remove remove class break cell e since white cell classify remove rare heterogeneous rl variant successful slightly well occur class band total misclassifie compare simplified confusion percentage confusion misclassifie svm c result rbf implement r rbf rbf rbf c polynomial primary truth uncertain dm bad mean something mean use descriptor texture combination investigation investigate whether set variant perform stress svm artificial classifier find impossible know company field expert cell five certain cell truth machine primary state truth human cell dm class indicate several ground truth probably svm wrong truly situation simplify confusion occur human thesis cell discrepancy even simplify diagnosis involve count white determine infect interesting cell cache descriptor e area quickly cache view library boost virtual convolution soon realize big filter pixel big calculation section improve wavelet many however improve gradient ascent replace improve divide problem kkt optimize heuristic call literature gram keep something purpose decomposition optimize decomposition thereby kkt true variant beyond modeling approach interesting first visual simple elaborate analytical article cauchy wavelet suggest show software http nu test software write ms window g boost library fast scientific library brief overview program mostly dataset load save file main help train save load load fold fold dataset integer gap list save create validation pass number kkt double precision allow default gap set mask control I stop feasibility pass primary bit kkt satisfied terminate default generate example feature truth cell file file main help db generation image generate generate program call l f extract image extract class dm program help main db integer db extract cell program main help integer extract cell db class save feature format program feature format chapter chapter r thesis department science pt classify diagnostic increasingly svm implement descriptor scalable descriptor homogeneous texture descriptor property visually human image implement classify cell come dm micro machine simplify classify differ common white cell encourage achieve rate ground investigation machine machines ef de descriptors color descriptor descriptor texture descriptor tt som fr en tt en tr fr som fr fr en dm som som som om svm thesis dr practical detail thesis descriptor field machine thesis subset cell specifically field disease classify kind cell diagnosis certain frequency make diagnosis rl abundance classifying cell classify determine amount typical cell find colored cell flow complicated intensive cell expert expert able even regular frequent impossible develop sample send vision help less preliminary result result thesis try lot support machine introduction broad field bioinformatic engineering instance quantity specify hyperplane instance call measure euclidean point define margin hyperplane linearly separable margin classifier margin least margin optimal distance hyperplane still close hyperplane easy theory lagrange multiplier lagrangian
path clearly infer autocorrelation model plot give method analysis network social actor pair actor actor tie refer various build co worker place example political piece house study business describe business tie relatively family tie relationship family popular random sample statistic capture aspect uniform posterior particle variance use previous turn target strongly relate walk metropolis definite reach acceptance choose adaptively chain figure table quantile estimate significant covariance quantile plot plot adaptive plot mcmc constant far literature vanish exception propose strong limit promise remain method scale respect involve scheme imagine would allow properly research la paris mc study condition e ni multiplicative instead equivalent eq combine part deduce introduce well nh h ng nn ng n nd k martingale increment deduce ng aa exist poisson geometrically transition kernel q eq lebesgue dominate algorithm section section example deal model constant presence function sample think mle monte illustrate segmentation model behavior law intractable normalizing pose major problem include point protein problem describe statistical ex easily problematic carlo use hasting acceptance normalize early pseudo approximate perform poorly inference mle develop simulation study asymptotically exact bayesian exist mcmc algorithm develop use variable intractable normalize possible expensive problem bayesian sample generate stochastic intractable normalizing constant computing take another entire carlo perhaps certainly behind mle estimate poor get principle possible building idea marginal limiting include detail illustrate example ise theoretical aspect throughout fix sample space similarity decomposition decomposition sample estimate build wang algorithm fed carry respect adaptive monte carlo sampler sample reader think adaptively adjust approximately possibly propose markovian n ex ex give convention finally transition invariant possibly number general idea particle possibility sample distribution approach possibility grid particle otherwise imply low recursion write approximation size step heuristic approach typically larger easier large keep measure approximately point switch deterministic form let arbitrary vi specification initial run switch law condition triplet use result compact equip borel algebra lebesgue example check notation operate kernel recursively measure pp irreducible clearly symmetric away ball imply ex p kernel geometrically ise rectangular lattice simulate give generate perfect generate independently flat proposal
obtain integer j integer integer four moment non jj jj jj give representation parameter function skewness increase decrease come expansion shannon define distribution ba unit vanish b ii nu nu obtain numerically define matrix lr test upper model strength cm measure national laboratory maximize lr hypothesis therefore reject nan favor plot estimate well beta ne name exponential introduce mathematical distribution moment ne sum moment discuss hope generalization wide de universit pe com pe yahoo com introduce beta provide treatment derive generalize expression statistic quite quite analyze positive place keyword two gamma distribution name exponential investigate et al four type ex generalize gamma fr also mathematical vary significantly shape function non exponential therefore use might random distribution tail ba ba bx cdf define beta bn take first bn cdf distribution extreme maximum work exponential distribution four extreme likelihood result li eq ba beta parameter factorial beta establish density tractable pdf except special pdf cdf hazard involve complicated pdf generalize flexible plot value shape monotonically basically organized derive expression cdf pdfs order moment expression inference information conclusion formulae integer integer integer integer ba ba reveal distribution sum distribution fx jj expansion property cdf expression cdf distribution addition cdf exponential integer density distribution ba jj jx integer ba x jx beta distribution I I ai x bn bi ba n real j q eq sum tuple non
iii iii iii category table small bias superior mle advantageous normal mixture situation flexible finance inconsistent recommend mix likelihood know bind also convenient extensive conduct practical likelihood provide mixture decade include variety al review human normal mixture normal draw substantial attention recently system equation parameter crucial fit ill placing help result place variance well result shrink though apply ray univariate mixture hence difficult penalize estimating mixing estimation method know likelihood estimator conduct work removal maxima univariate advantageous follow strong present defer likelihood density assigning unbounded arbitrarily penalize attain maximum penalize choose ng pp ng ng pg c n dc condition flexible satisfy condition c simplify numerical condition limit effect degenerate well view via density ng np g defer consistent surely score let matrix second derivative log positive definite use classical omit may upper deal mixture mix almost defer recommend code local maximum risk maxima recommend penalty th equal em cg ng p n p ng e maximize suggest penalty eq function maximize q view put wishart prior strong likelihood non ordinary maxima poor simulation maxima attain large among remain identify mle solid support mixture guarantee superiority practice thorough study context multivariate standard mle clarity result subsection parameter multivariate use aspect two model component bivariate model form mean practical situation meaningful eigenvalue angle configuration bivariate ex specify mix due invariance distribution configuration thus three situation location eq matrix effect size ratio angle orientation choice component proportion triangle three representative six triplet covariance three fall normal covariance triplet mixture mixture reasonable mix normal category configuration configuration need although penalty two correspond mle ten initial true mix group base vector proportion mix distribution degenerate quality algorithm large computing mle degenerate record ten value degenerate outcome component bivariate immediately clear degenerate outcome eigenvector configuration counter intuitive heavily sensible component vector distant outcome mostly due category phenomenon degeneracy addition category quality degeneracy long degenerate
kk end moreover k kk r summation correspond eigenvalue hermitian depend condition check x assume quadratic thesis appear previous thesis use linear distribution series generalize polynomial convergent specifically distribution term expansion moment free parameter series determine quadratic form check r converge close expand uniformly convergent moreover kk truncate r kk condition distribute central true conditional density kf w r convergent represent determine moment uniformly convergent expansion kf ki convergent integrate get kk eq constants ny dy yy yy practice moment corollary expansion terminate happen expansion support factorization density expansion e correspond consequence unstable expect error finite define realization proof secondary one matrix block tr dependence integration tr tr tr h tr parameter derive expression respectively fix matrix positive nonnegative quantity nonnegative q thesis k quadratic deterministic nonzero orthogonal nonzero estimate correspond eq view increase mode density relate likely smoothed however lattice must lattice order unitary reduce upper triangular unitary transformation rotation lattice notice n h ft jt k n ki il eq k hz local along arcs arc close branch elsewhere branch correspond maxima sufficiently maxima rough discrete median error top original plot top rough part plot notice procedure outline identifiable cluster reconstruction plot perfect well bad comparable abstract joint modulus determinant computed get eigenvalue implication numerical logarithmic potential introduction us quantitie bold character generalize root equivalently borel denote important hausdorff uniformly make combination noisy sample want build start information location complex plane approximation explicit coefficient order essential average case dirac distribution center marginal asymptotically dirac maxima neighbor estimate process stochastic computation eigenvalue many measure burden however give reduce
bring factor slope less unitary slope graph ratio bf report table exact consequence poor acceptance tolerance version recommend sequence proposal quantile comparison scale tolerance bf logarithmic use quantile simulate proposal sub sub decision accord strong c c weak weak weak comparison quantile bf numerous genome sequence available provide amount unknown call fold provide method ray description consume structure homology protein say control study hereafter compare query sequence often assess homology sequence fold onto find compatible fitting structure correspond say happen base protein homology information help make necessary contact complementary observe protein formal perspective represent protein indicate contact label allocate associate classified field ise graph structure algorithm select study treat query purpose evaluate help dedicated structure pick protein bank http home home sequence optimisation score alignment score less reach additionally query similarity tm structure library candidate cover whole spectrum good candidate make tm score consider alignment onto good similarity candidate query alignment structure share structural figure superposition grey structure green dt superposition grey green dt seq tm na seq percentage tm score quality alignment tm score summary protein seq percentage tm candidate mc bayes factor predict ise simulate obtain gibbs pick euclidean bayes indicate alternative classify evidence structure alternative structure tolerance factor tolerance quantile overcome closed form computation bayes include require advanced technique usual avoid availability toy improve abc simulation application implement abc ranking acknowledgement universit paris author partly de paris project grant de grateful encourage comment universit paris france universit france et mod universit france en france analyse spatially challenge perspective competition special free technique computation towards random demonstrating exist approximation far test bernoulli order structure approximate gibbs field configuration said denote word element clique undirecte u mrf everywhere mrf consider ps ps cliques neighbourhood constant depend consider term constant require versus statistic take rely face depend unknown constant free abc abc tune purpose dependency spatially correlate datum summation identical difficulty increase section propose aim model accuracy toy iid sequence trivial neighbourhood aim select available free overcome difficulty rejection briefly describe generate parameter read generate accept truly impractical impossible introduce sense open distance I thus associate simulation accept rejection rejection practical crucial pick quantile marginal datum many development allow simulation perfect discussion gibbs update one clique empirical frequency denote associate evidence relative estimate approximate estimate substitute indeed assume binomial rv thus go ratio distribution since sufficient unstable suffer difficulty drive increase exhibit run extreme ever unlikely bring poor approximation conclusion choice model approximation bayesian model first example compare bernoulli special neighbourhood structure neighbourhood constant probability x
paper arbitrary high significance provide advantage significance class detection realization terminology clear disadvantage probable vector complicate dimensional space reason rarely would detection numerous group roughly normality combination classifier outli normality give work base rely point choose automatically upon set near course necessary reasoning th near simplify publish apply per point cost furthermore minor outlier avoid estimation detect apply outli score estimation regression away belong category idea classifier classifier sample outlier available create cloud outlier apply classifier apply outli example simple category find generalize traditional concept essential know future feature outlier significance typical mean region vector fall integration solve serve question cumulative assertion realization marginalization convert probability comparison show write significance provide probability single provide valuable assessment allow decide probable standard distribution cauchy significance close symmetric unimodal level interval identically distribution usually valid significance reasonable please restrict th contrary integration appropriately knowledge correspondingly random cumulative guarantee unnecessary computation sort possible square error make desire significance speed neither density calculate dataset unbiased furthermore k yield finally generate unknown suitable upper generate interesting calculate level classify reason propose detection contrast compare simple distribution close significance distribution form vary average difference form estimation summarize generalization matter generate dimensional multimodal density integration transform prediction easily accomplish
residual new penalization framework implement via thresholde efficient lead principal addition grouping highly nearly multivariate construct small throughput profile identify molecular signature profile analysis early drug efficacy entail massive possible biological identify structure noisy make task step unstable modify ridge penalty lasso gain popularity ability select exhibit stability ridge interpretation laplace tend grouping play role cluster predictor molecular signature group biological exclude hand highly correlate predictor variant therefore group implicitly elastic net en norm penalty another encourage similarity propose group reduce eigen solution curse group predictor dimension method many pls paper regression penalization candidate idea penalization section adaptively predictor computationally summarize simulation study analysis illustrate behind dimensional column non moment satisfactory first highly predictor classical component construct eigenvector lead univariate component remove lead jj uncorrelated constitute component lead regression furthermore eigenvector appeal furthermore correlate solution calculation fast uncorrelated implement subject finding construct involve covariance observe estimate partial subsequently construct combination especially number contribute square regression besides loading eq implement sparse problem construct eigenvector pt eigenvector e large version introduce benefit covariance element I find follow tr x false discovery fdr fdr ten find en pls since neither pls select fdr report report large loss predictor interesting pls loss pls lasso loss en correlation mild loss dramatically increase loss response loss indeed comparable loss c c case en pls en lasso pls ridge predictor losse cluster predictor build correlation perform fdr except include correlate lasso present surprisingly fdr effect lasso although except fdr c en experiment population array phenotype also measure quantitative rt expression http www remove gender phenotype phenotype gene phenotype investigate rank randomly include build training square test en result number report report phenotype number en generate component phenotype summary c en h fit phenotype generate report e build similar reduction successful unsupervise exclude construct false pls nature signature predictor measure functional year search clinical signature build orthogonal along
robust summary statistic concentrate statistic variability run use replicate rather variability varied display estimate quantile interval wider produce algorithm distribution close reduce posterior feed datum long record frequentist power potential open evolution social science statistical abc increase r code perform available author website acknowledge institute grateful anonymous helpful bayesian centre national de la france author la france mail fr likelihood curse dimensionality feed abstract well suited likelihood mathematically rejection suffer curse dimensionality summary statistic two fit conditional summary statistic approximate bayesian reduction burden implicit regression indirect making inference neither analytically computationally find seminal implicit statistical term generate grow implicit model population genetic carlo tree many propose process et name computation abc et al year rely simulation large parameter summary generate estimation originally classify broad category al rejection describe paragraph markov monte carlo embed account statistic metropolis hasting third class share monte carlo samplers liu combine idea sequential sampling al severe rejection dimensionality statistic increase attempt observe suffer curse acceptance percent value adjustment curse adopt construct functional summary assume ideally utilize produce exploit stage relationship nonlinear flexible regression like exploit within adaptive iteratively discrepancy posterior historical evidence reduce burden abc adjustment abc multidimensional interest generate draw sample datum simulate generative parameter denote euclidean norm retain span scale norm rescale place euclidean distance statistic accept approximate posterior sufficient addition approximate large note interpretations standard abc weight estimator smooth subject curse estimator decrease dramatically increase see et avoid curse abc describe draw empirical adjust posterior bias increase reduce thank increase linearity modification previously adjustment abc linearity location g fan nonlinear fitting flexible variance estimate residual q mean feed network reduce summary internal hide neural reduce easily within transform equation logistic general weight represent parameter call decay parameter network literature perform correspond equal similarly adjust k adjust fall distribution original parameter lie logit transformation cox method estimation improve liu logic stage algorithm acceptance rate adaptively build well pool estimate fall implement step number simulate new form principle mean suppose approximate support estimate support two one indirect empirical local use programming neural implement network approach unit weight increase decay increase cross alternative previous well example implement base public library lin biology decade fu li al inference describe implicit usually dna sequence concern effective mutation occur dna affect mutation say summary statistic fu sum independent exponential distribution simulation tree times al accord description mutation site dna take site million sample posterior total quantile posterior quantile empirical quantile exact sample quantile assess median discrepancy replicate tolerance standard algorithm find distribution model approximate tolerance increase tolerance close triangle achieve tolerance range triangle dot allow eliminate rejection approximate bayesian algorithm benefit method adaptive stem give region tolerance additional rate turn could population von size start grow year reach present individual perform nuisance individual marker al characterize four repeat record individual implicit grow tree branch mutation simulate walk equally likely intensity genetic imbalance al al statistic peak value expansion peak shift old expansion average take uniform distribution range interval distribution test date population population study summary replicate implicit record abc median quantile run median marginal tolerance green curve indicate summary performance decrease tolerance substantial give estimation variant study tolerance quantile
estimate poisson deterministic stand sample stop k index double variable partition sampling scheme theorem regard scheme develop theorem poisson distribution prescribed confidence need classical random readily ps p b kn p p integer n bernoulli random integer z lemma ss pp hence good k pp p n b hence position direct well sampling prove stop stage scheme p b see p conclude proof chen lem b k lemma argument argument theorem well stop lemma stop p introduce preliminary result chen lem kn kn ix possible event k completed complete chen lem lemma r well must index scheme lemma stop r conclude proof remark definition rigorously prescribe confidence binomial poisson extremely numerous engineering arise context occurrence characteristic utilize inference year phenomena deal count event reference therein persistent statistic despite devoted issue drawback drawback conservative sampling scheme limitation binomial parameter rigorously guarantee prescribe remainder section present sampling binomial different proof integer less represent function integer z z x drop introduce virtue satisfy coverage stop sampling would like note define truncation partition construct estimator criterion prescribe stop p sampling estimating relative obtain one view desirable satisfy
handwritten database handwritten digit apply subset dataset reduce dimension image assessment near neighbor digit nonlinear principal point transformation smoothed prior demonstrate discover dimension investigate automatic dimension putting seem compare adopt still possible graph connect nearby support novel nonlinear motivated component specify location smoothing field feasible advance von principal component pca old technique reduction exploratory analysis pca aim variable visualization sometimes preprocesse pca typically center definition weight subspace optimal linear minimize associate large eigenvalue principal similarly define principal component total onto span several nonlinear statistical curve curve center point onto visually curve conceptually neural kernel generative propose absence probabilistic approach adopt advantage provide degree naturally incomplete latent priori center change q put covariance gaussian likelihood probabilistic author notice replace kernel nonlinearity conceptually regard cubic approximation contribution put observation correspond part field space make explore transformation burden square next discuss familiar von background material simulate manifold handwritten digits datum matrix play model definition matrix common von respect exponential omit maximize give regard concentration parameter closeness distribution von detail simple proposal uniform
confirm often improve acknowledgment wish di provide discussion life reduce solution complex pose formulate moment plane tool useful nuclear moment problem logarithmic potential many processing formulate complex complex equivalently problem generalize relate eigenvector matrix equivalent compact dirac harmonic exponential interpolation consider define white solve problem consist equivalent I series eigenvalue problem would ill pose known see optimally fact reduce orthogonal well difficult relatively recently approximation exploit eigenvalue assume make tool logarithmic potential polynomial numerical address tool develop test follow solve nuclear interpolation moment provide start realization us u polynomial support prove generalized circle concentrate true therefore evident order solve signal hope step provide tool base eigenvalue nz k pn relation unknown measure identifiable tend continuously support small identifiable program experiment get build step course assume use hard acting eigenvalue fact mask therefore identifiability properly real measure measure candidate measure identifiable random extracting discrete stochastic k propose cope dirac appear alternative expression replication define condition solution computable order nz one look discrete transform lattice take discretization rise cx iy j disjoint center belong associate transform eq inverse discuss main address method suggest algorithm computation transform approach context unknown well eigenvalue several advanced account factorization scale achieve square classical split least toeplitz fast use burden propose algorithm quantification nuclear spectra usually hoc competition artificial european participant third interpolation acoustic sound shape plane moment specific synthetic address hyperparameter order select good hyperparameter value top part spectrum induction physical combination positive weight spectrum fourier transform turn function estimate related argument model experimental ideal peak peak interactive choice model ill snr interest analysis mark theoretical million interactive procedure peak band filter middle filter peak cluster cluster separate frequency filter mark plot sum line spectrum segment fix easy show eigenvalue eigenvector apply pool eigenvalue modify matrix computing gap observation segment eq time time want interpolation synthetic truth argue discrepancy fit smoothing cubic spline improve cubic residual subtracting smoothing leave plot true reconstruct one example illustrate hz plot apply miss show part middle right spectral characteristic signal bottom spectrum complete plotted signal miss value show sound produce signal star e shape e correspond degenerate vertex direction index
compute either simulation provide approximation somewhat present simulate begin along conjugate typical relevance diagonal identically gamma improper derive marginal likelihood integration maximize tend model validity improper prior contrast integrate modal e g machine adjust integration prior distribution reveal density univariate equivalent elliptical contour instead produce along axis choose maximize ever contour lasso etc encourage hence place regression coefficient conjugate direct prior objective integrate joint see analogously normalizing prior density normal kernel distribution multivariate conditional regression inverse prior parameter follow distribution deriving use frequently simulate allow compute joint facilitate maximize iteratively density readily appeal mode equation maximization modal ridge iterate penalize square substitute obtain essentially procedure problem weight final understand ols I demonstrate proceed contribution response underlie iteratively marginal logarithm expectation side em work consist iterative guess repeat hierarchical become unlike let setup notice form exclude expectation maximization sequence minimization eq mode extremely similar adopt mention univariate degree far would superior mode integrate slight coefficient independent think try marginal actually create yet independent convention form model thing explicit statement regression could variance would yet become write apart tuning quantity plug procedure fundamentally identical discuss choice eq plug remain convex initial modal would thus ol point e ols covariance plug likelihood correct estimator hypothesis statistic quantity piece inverse square sequence problem predictor absolute testing intuitively propose estimation maximization joint hierarchical simulation base computed laplacian design explanatory relate irrelevant toward solution thus mode contribute irrelevant claim removal irrelevant variable removal approximation perform certain section analytically posterior condition integration obtain region root inverse adjust width report number computationally elastic library model adopt lasso iid pg scenario lasso selection simulate combination table lasso adaptive result perform lasso ordinary term selection medium significantly consistency iid adjust noise conduct report prediction observation case value stand predictor calculate find median calculate mse net ridge ordinary estimate determine lasso method laplace marginal also integration carry sd cm mse utilize empirical bayes correct especially surprising mse prediction accurately lasso case decrease variant bayes step poorly situation superior model term outperform version laplace detect term term almost accuracy show estimator perform clear winner ridge estimator variable make efficient optimization tailor marginal spirit normal laplace coefficient superior prediction selection improvement increase inversion obvious regression computational iteration eliminate dramatically offer single ol inferior model computationally number predictor offer open integration obviously particularly stable suggestion significantly integration upon suffice integrable respect dependent take respect finite statistic operation management tn statistic management science tn introduce shrinkage linear variable extend proper ordinary
simulation run execute select key unbiased estimator threshold table variance sampling time try analytical discussion exhaustive proceed cover permutation bank simulate count gain biological protein year deterministic module get complex adopt acknowledgment national grant statement compete financial interest exist mass function computable normalize constant backward recursive q let probability markovian sequentially via transition generate xx cx proceed let score quick relation compute desire letter sequentially markovian relation unbiased j indeed use independent importance first whenever ensure unbiased carlo since monte asymptotically answer section involve include subscript define quantity highlight similar implicitly b fix let v xy lemma asymptotic check occurrence family bound replace w normalize argument reference zhang r spatially gene expression deviation rare algorithm large deviation york distribution application carlo simulation pt h dynamic ann importance moderate application determination significance identification j z recursive detection control chen clusters yu scan dna sequence origin nonlinear pt within region statistical significance compound poisson occurrence sequences j approach occurrence statistical overview j sequential test ann g zhang b comprehensive microarray molecular biology zhang multivariate data ann de discovery module hierarchical zhang analytical mc monte run direct mc importance rr carlo execute display search root need apply probability suggest count tendency correction special general table analytical direct monte importance reveal maintain implement encourage basis leave desire score generate dna sequence sequence simulation direct retain time quite substantial co locate signal co bind factor important use successfully understand cf al simple occur length calculate prescribed limit respectively family occur u create repeat independently respective select markov store bank store co operate region et al probability appear probability direct underlie co occur consider al essentially separate
bound bound matrix something p panel one advantage propose capability forecast three forecasting range forecast day forecast forecast day forecast correspond interesting throughout covariance portfolio allocation frank us daily exchange return exchange return figure paper autoregressive model bayesian prior accurate eliminate include make useful volatility use multivariate flexible multiplicative evolutionary volatility propose forecasting forecasting volatility bivariate share fx provide tv var work var vary research likely direction acknowledgement like comment early purpose vary autoregressive model tv var model volatility covariance model via wishart allow conjugate standardized forecast error deviation error order formula consist bivariate datum share exchange fx forecasting detail discuss portfolio set empirical finding suggest var var forecasting multivariate volatility exchange portfolio allocation past l multivariate enable optimal portfolio allocation trading exchange comprehensive advance series development time vary develop wu study refer vary autoregressive tv var include al capture behaviour desirable volatility use vector var incorporate relatively need resort simulation technique carlo simulation direction west describe filter allocation forecasting forecasting resort analytic fast stable counterpart suppose serie roughly integer variate var g express possess compact model hard multi step forecasting forecast include degenerate singular wishart difficulty difficulty ar order large employ reduce form lose aim suggest state overcome difficulty propose put volatility evolutionary wishart forecast forecast autoregressive de al sequential comparison goodness standardized error mean forecast factor share exchange illustrate multi forecasting strategy find invariant var superior var forecasting discuss brief conclude evolutionary markovian attention write denote transpose walk follow column stack operator denote discuss specification discount independent evolution notation remain evolution evolutionary law convolution beta triangular decomposition independently write reflect fact singular determinant matrix detail refer discount resemble expectation volatility remain unchanged volatility prior wishart suggest responsible define discount tr discount regard goodness model volatility var invariant model volatility assume behaviour environmental generalization briefly suppose posterior pm pn iw denote wishart wishart distribution n dp pm iw previous observe information posterior dp pm iw pn tm te tf tf te play role kalman filter step ahead forecast start give conditional kalman define element volatility log function f n posterior sequence property forecast totally unstable writing bound scalar series bound write adopt since bound forecast performance univariate west compare contrast ar var lag order differ forecast distribution student pn forecast section indicate preference forecast function bayes differ discount factor sequentially indicate preference scheme ar vary forecasting positive forecast horizon forecast include denote replace n dp pm h th tt distribution forecast vector matrix exact I th p p quantile credible standardized measure standardized forecast n v h mean forecast absolute th e indicate absolute third bias forecast volatility bayes volatility evolution find evolution singular beta
consist convergence hx argue major concern achieve convergence sampling empirical average element rapidly correlate draw say possess mix mixing explore region research develop one sampler mode convergence mix exploit facilitate efficient comprehensive implementation mathematic introduction discussion issue researcher practitioner develop reliable problem speed relatively smooth dimensional might prove reliable mcmc incorrectly explore lie assessment balance provide qualitative mcmc notion chain section derive possible implementation one asymptotic qualitative annealing find estimator tool metropolis hasting slice computation remark transition proof irreducible limit ergodic detailed balance irreducible chain initial prove irreducible reversible need introduce decrease positive geometrically uniformly ergodic hold polynomially order ergodic stationary let borel function polynomially ergodic order ergodic geometrically ergodic geometrically hx geometrically ergodic uniformly initial distribution hx convergence irreducible reach equilibrium principle mechanic mechanic concern study physical particle approach equilibrium principle balance system equilibrium kt energy assume satisfy detailed balance check see discussion metropolis hasting implement temperature sequence decrease probability state working function develop method sequence use output number simulation function proceed reach stationarity mm ccccc j notice cccc z z cccc asymptotic stationarity w z begin point irreducible ergodic x device mm x n markov detail reversible infinite finite v n n z n variable show p j b cc I irreducible markov burn draw discard assessment criterion hypothesis chain stationarity lyapunov central limit remains follow satisfied lyapunov limit follow result enough resort summation assessment via test confidence irreducible satisfie balance discard first statistic f z n stationarity level stop else continue iteration return replace implement qualitative tool assessment specific depend multi modal correctly lack unimodal qualitative tool define algorithm kn kn kn take patient possibly variable distribution number relative versus slice application decrease trend difference increase initialize temperature value kn n energy value differ unit systematic average average anneal proposal hasting approximately slice almost proposal advantage hasting two alternate horizontal slice produce adaptive sometimes outperform metropolis hasting gibbs sampler approximate grid width implement metropolis single variable apply proposal center truncate number close grid slice sampling update initial iteration column display histogram value relative difference autocorrelation application slice hasting hasting stable slice compare true histogram chain respectively metropolis negative value slice sampling tail positive display behaviour relative variance function method end fails reflect incorrect tail hasting would assess plot autocorrelation slice hasting continue even indicate metropolis hasting sample diagnostic http www hasting autocorrelation chain start follow quantile whether chain chain draw hasting versus slice trace indicate positive algorithm figure pooling slice pose concern stationarity frequent whereas metropolis behaviour whereas behaviour metropolis allow exploration support year development popularity particular statistical simulate output question arise diagnostic decade general develop tool assessment continue challenge method irreducible chain space mcmc sampler balance statistic whose behaviour analyze theoretically experimentally present implementation assessment qualitative tool high insight lack stationarity analyse behaviour possibly monitor behaviour stationarity extent explore space particularly shape distribution analyze value chain diagnostic employ exist assessment weight convergence weight estimate converge replication chain criterion criterion px px px px x j obtained sample close result posterior compute different drawback conditional exist however state method disadvantage computationally expensive ergodic average various marginal probability method replicate incorrect sampler fail reason recommend replicate start space criterion min space come finally criterion intuitive representation min function regardless underlie assessment approximate discrete state chain time subsample homogeneous would explore extend chain theorem remark markov employ form simulate
allow large set highlight utilize identical classical geodesic identify geometry visualize calculate parameterization density rely nonparametric fisher implement nn applicable describe nonparametric fine combine method find realization underlie pdf distribution lie parameterization actual note set density space benefit geometry enable embed multiple single dimensional reduce manifold gaussian distribution covariance lead manifold significantly dimension manifold x focus relationship xx wish preserve geometry manifold pdfs interested reduction individually low preserve fisher information estimate pdfs refer fisher calculate interested projection projection like identity constraint orthonormal keep matrix different function benefit sense locality fisher pdfs differ prevent away provide operate kullback leibler strictly preserve minimization proof bound tight negligible dependent projection similarity maximally preserve dimensional allow comparative solution problem surface direction great iterate minimum value objective fast calculate eq ensure converge local guarantee absolute minimum dimensional direction take give projection eq find iterate collection step calculate fisher distance initialize orthonormal fisher matrix ia ji note often initialize bias carry beneficial available stress descent future loading projection appropriately well preserve example pdfs dimension contribution entirely differ distance go weight loading give discriminative variable exploratory detail fisher pdfs realization pdfs goal approximation well direction respect hellinger limited value single density ann school edu concern manifold considerable reduction purpose visualization primarily riemannian sufficient straightforward meaningful case may represented distribution lie manifold manifold pdfs dimensionality reduction medium storage capability amount information datum flow micro array vast substantial often representation redundant entirely correlate retrieve naturally actually constrained allow datum exhibit intrinsic data domain one dimensionality value vector often hoc highly suboptimal geometric framework realizations generative pdf dealing manifold construct offer form euclidean reconstruct scaling use pdfs offer pdfs useful visualization variable high dimensional field construct manifold analyze tool geometry theory statistical inference neural background method thorough geometry manifold lie point variety construct coordinate image regardless one mapping onto mapping infinitely differentiable manifold surface sphere roll system metric approximation divergence divergence real evaluate divergence kullback divergence hellinger kullback kl equal divergence important theory kullback leibler pdfs parameterization note kl symmetry symmetry divergence triangle fisher symmetric symmetric kl call closely hellinger axiom hellinger hellinger relate kullback leibler multinomial sphere throughout hellinger great continuous pdfs show ensure distribution multinomial divide hellinger monotonic transformation divergence provide probabilistic fisher refer reader application manifold promise flow recognition texture cluster alternative use euclidean datum outside address problem
similarly associate terminal entail draw conditionally sampler additive generalization refer inverse gamma routine challenging implement draw equivalent play constant draw elaborate metropolis propose tree tree move probability terminal pruning change terminal integrate avoid reversible continuous space draw draw enable fortunately reversible jump initialize simple single satisfactory increase terminal abundance incremental add subtracting compare mcmc dramatically single tree algorithm quickly neighborhood sharp contrast remarkably run multiple modification provide benefit converge induced function run algorithm burn sequence regard approximate inferential sample need inference application experience estimate choice median variation interval obtain quantile see uncertainty behave estimate functional interest partial summarize effect precisely partition eq observe appear fit tree less effective redundancy mix irrelevant predictor decrease redundancy heavily relevant accomplished component usage sequence mcmc sample number proportion use use th small favor inclusion prediction seem create force predict might useful measure present thereby initial split interest however extend probit normal use aggregate majority vote see implement bayesian fortunately minor section oppose implicitly proceed eq tree interval practical relevance motivation recommend shrink toward zero automatic shrink toward shrink toward interest replace offset modification augmentation introduce sampler successive successive sum eq q converge distribution suitable burn regard posterior demonstrate example different real next illustrate capability simulate finally apply probit classification library predictive datum subset datum forest unable predictor split correspond categorical categorical convert sample apply square make half reduce transform lc lc lc lc budget edu college cpu diabetes set create split randomly training predict predictive consider version cv prior operational default cv default specification least burn inspection typically mean linear black boost implement r implement black box predictor cart predictive interpretability operational choose fold particular cv conservative prior hyperparameter four value moderate heavy hyperparameter potential cv prior range tree net text decay sample grow node wide range candidate problem neural operational unit predictor directly unit candidate value comparison relative rmse divide obtain train obtain oppose meaningful invariance response split quantile box figure table visually comparison lc lasso forest default although relative performance clear rmse notable default arguably second good since net forest boost control avoid hyperparameter specification default cv selection hyperparameter follow fitting application want winner various simulate give irrelevant find spline illustrate selection partial irrelevant detect dimensional dramatically begin basic simplicity apply default use burn iteration begin posterior averaging plot use generate interval select interval tend degradation wider great frequentist interest figure b value replicate bootstrap similar frequentist however may toward frequency low entire plus horizontal line draw chain although autocorrelation around sequence draw tree substantially move beyond inference value estimate dependence reveal individual figure partial function mcmc nonzero effect zero form identify promise sum illustrate record simulate dramatically tree increasingly make use assumption actual form exactly variable appeal feature number illustrate figure display rmse various plot text default feature plot obtain need fit slight degradation comparison sample comparison suggests fit remarkably sample fit great correlation fit reasonable replicate seed fit stability use long mcmc require start variable draw five embed dimensional subsection detect observation extent purpose display default exception naive sample anchor prior naive also use display interval remarkably plot reliable indeed especially less make toward compare remarkable reliable figure sequence draw solid estimate however get increasingly tend back toward uncertainty note draw systematically high draw due part rather use rise attractive feature avoid simulate run clearly become uninformative interval evidence forest net drop hope nonlinear modification spirit estimate cv burn draw data validation simulate method fx rmse parameter sample change neural net unit decay neural net increase order difficulty force neural nets rmse note clearly cv little default cv performance poor prior cross anchor quantile five adequate fit bayesian perspective express express forest neural net boost default probit drug discovery activity compound molecular structure compound activity ability outcome molecular classified topological aspect molecular set cancer institute activity compound probit mean molecular promising split probit probit obtain essentially result plot provide posterior interval identification drug fact rate interval wider compare forest support machine use penalized exclude due randomly inactive allocate set choose comparative experiment replication default default namely cross cross select predictor lead tune utilize unit decay unit replicate test receiver operate roc must predict activity predict activity though large auc superior order choose forest auc competitive cv support default score difference auc significant effect replicate avoid validate section modify hold build iteration number stability percent usage probit consider figure percent case e trees percent percentile percentile horizontal execution various light number execution forest dependence varied run version burn burn iteration replicate run replicate variation calculation residual metropolis proposal iterate observation observation contain execution time linearity indeed short long dependence become quadratic nonlinear nature iteration tend toward fine tree base terminal terminal whereas terminal average keep tree execution time execution version quadratic execution time predictor execution display figure reveal execution independent especially compare underlie long lead execution forest boost net execution version default burn indistinguishable hold typical display execution comparable scale fashion tuning typically validation competitive need illustrate compare stand alone incorporate component term instance might eq vector also multivariate covariance generalization modularity random simple fit variable selection illustrate promising identification important modification enhance instance put split put tree small split tendency spurious split spurious predictive thereby difficult distinguish selection part advance technical report add component problem conventional census base tf dna include regression network machine conclude quantification interpretability inclusion successfully predictor outperform cart support machine independently discover probit extension logistic forest support machine cart neural evidence potential oppose bayesian approach explain variation account overall accomplish simpler formulate rapidly highly immediate tailor algorithm effectively inferential quantity sample application data set demonstrate feature rmse compare boost net forest default perform reliable true regression marginal remarkably hyperparameter regression ever effective free competitive tool discover promise molecular structure report support nsf dms institute prior inference accomplish via generate nonparametric use element motivated method model enable interval potential keep predictor variable experiment drug fundamental vector sum regression tree sum tree fundamentally generalized model component naturally combine set ensemble attract bag forest boost fit tree early bagging forest randomization average prediction across yet combination apply tree average paper propose approach bayesian essential elaborate tree impose fit keeping effect adaptive explain portion equivalent fit tree tailor bayesian construct fit successive spirit boost approach differ individual instead tree conceptually rich influential inference successive sample estimate successive sum draw pointwise interval easily quantile interval functional partial function reveal effect component parametric open implement stand available library remainder organize consist regularization mcmc describe simulated potential execution extension variety development early discussion detail elaborate sum mod establish single let terminal binary split predictor form single vs terminal rule top sum observe
simplicity let v v ns b b b ns sample n n n n k induction first k mp set size application size p n non assumption induction n k consequently n making v v v v n v n really p remains show observe n k il l p kk n evident remain show n n l p complete lemma add mutually independent integer u induction true l kn lemma establish shall poisson negative integer u poisson iv min max min volume two volume assume sample size mutually independent generate p ns gs gs lemma establish follow step gs noting v z v z min min min max v min min v min max v min h min v min complete mean k I k theorem e e get dense de dd corollary remark proof approach mild computable technique complexity many situation vector expectation consideration reasonable great context robustness system lack knowledge due special maximum equal obviously intuitive relation monte unique powerful remainder lower base iii monte carlo bounds introduce method computational carlo lower q purpose fundamental slight principle propose reveal computation ny cumulative imply virtue simulation huge complexity mn investigate basic arbitrary nest poisson v volume consecutive tends volume refer lebesgue virtue derive kk original
iteration describe rule proportional estimate clique step update ar mle clique suffice submatrix mp compute complexity theoretic step require use rewrite efficient compute obtain extension htbp extension perfect clique k follow return theorem paper since k iterate accordance perfect adjust add fill induce perfect elimination accordance elimination describe k propose algorithm calculation multiplication range measure unit multiplication division cost computational cost amount cycle time localize cycle mle direct estimate computation intel core ghz cpu cpu iterative performance expensive use machine compute small cpu almost proportion theoretical propose slow cost sense consider direct cpu way decomposition subgraph localize iterative graphical cost confirm mention require enumeration clique graph enumeration propose characteristic sequence order obtain author grateful anonymous constructive suggestion lead improvement presentation theorem remark b iterative graphical computational reduce localization update iterative decomposable multivariate gaussian investigate graphical cox year effort graphical call decomposable decomposable general iterative proportional introduce contingency table certain marginal convergence study many framework formulate proof expensive large table computational localize decomposable contain technique numerical computation linear localize extension improve model variable require iterative technique decomposable model mle definite extension relate problem point al enumeration maximal exponential large however relatively simple clique organization follow section notation pair clique say mp mp subgraph induce subgraph denote clique satisfy clique minimal figure minimal eq order set clique vertex hence sequence clique perfect sequence maximal clique contain vertex elimination vertex g possess perfect elimination maximal elimination perfect induce perfect basic
jt n chi square poisson remarkable pre poisson arrival rate accuracy quantify relative obtain inter arrival time estimate service observation service moreover chi poisson thus complete see information show obtain relative error level come htbp design communication service estimating process service require satisfy pre specify remarkable priori parameter post evaluation rigorously traffic communication develop assume service server traffic service interest service sound theoretical service accurately load device service service purpose evaluation accuracy deviation construction post accuracy conventional specify absolute confidence arrival determine method determine absolute
self contain tool exploit geometry da algorithm like euclidean ar mat I broad context intuition precisely reverse I projection density omit decrease markov prove let prove without odd even last cover omit induction verify hypothesis eq odd induction q exist imply decrease space text since odd conditional everywhere finish corollary hold relative entropy dd ar iii consist part show ar projection theoretic ar part weak possess conditional yu generalization gibbs tailor issue investigate da let reversible theory reversible chain adapt albeit derivation author thank li grateful associate anonymous valuable augmentation da powerful computing da kullback leibler entropy reverse projection variation like simulation impractical tractable appeal replace description da powerful widely da convergence chain say roughly relative entropy leibler inequality density dp analyze relative result measurable x density
dot figure illustrate candidate notice location circle dot row row snapshot first input final bottom alm gps much mse serial fashion despite occur first least crucially transition partition take stationary gp region outperform alm ht alm alm alm alm alm alm alm rmse alm I I alm I I I alm I alm I I alm I highlight sequential deep comparison involve combination five cart generate candidate maximum entropy heuristic rmse dense grid repeat run initial configuration choose calculation grid predictive location regardless fourth rmse surprisingly cart really well alm though alm would concentrate high region candidate one maximum design notable cart stationary non maximum entropy design study perhaps linear dimensional active varie lift response beta alpha upper correspond sample map separate concentrated region heavily alpha surface remaining show beta slice slice look side roll slice slice six response characteristic counterpart occur near angle attack alpha concentrate reduced burden gp environment asynchronous base configuration alm criterion optimally spaced candidate next determine classic methodology bayesian partition alm entropy design implement code asynchronous available request apply herein broad array find contour contour learn computer fidelity code cost pair physical acknowledgment association university center national national science foundation grant dms thank project help two helpful comment suggestion lead improved derive hierarchical term n partition rather yield simplification equation give reduction turn back limit straightforward expand ac uk california experiment allow costly simulator expensive advance insufficient particularly explore simultaneously surface guide newly surrogate model explicit uncertainty cope asynchronous agent approach motivate computational dynamic active many phenomenon instead become alternative phenomenon towards fidelity simulation continue fast computer environment dynamic flow phenomenon excellent flow surface model accurately resource explore problem surface computer adaptively bayesian suited combine model design learning result flexible interface sequential design deterministic response analytic input configuration order build outcome even extremely environment expense approach experiment dense configuration orthogonal maximum offer version however approach identically design call maximum linear strategy model dynamic concentrate complicated region develop back much characteristic lift roll function input angle side triplet simulation ht lift angle attack alpha side beta characteristic flow indicate surface even differentiable continuous equation demand euler solver adaptively uncertainty effort design need behavior physical commonly conceptually predictive firstly computation poorly cube importantly throughout entire computational limitation predict gp response design limitation ideally able combine gp design classic suit bayesian inherently serial control isolated node agent serve diverse strategy soon available resource devoted process interface asynchronous execute design monte design make contribution sequential asynchronous put classic sequential efficient flexibility remainder review active gp average sequential illustrative synthetic section motivate dynamic describe design classic active modeling review simulation output particular configuration stationary process canonical I normal matrix element associate nearby stationarity contribute influence follow lee kronecker delta refer positive introduce valid power mat ern reference separable modeling computer fix omit importantly simulator theoretically necessarily solver start force indicate sometimes behavior arbitrarily fail potentially smooth secondary numerical matrix improve stationarity surrogate process gp extend trend place gps multivariate inverse wishart separable prior depend sample chain step predict normally surface smooth near boundary swap tree gps probability translate boundary indicate capability model short jump linear detection linearity region range determine model adaptively model fast parsimonious stable suggest computer response gp particularly extremely gps divide spatial suit align find compare appropriate structure design high implement stationary obtain http www package package implement specification paper unless otherwise statistic experiment sequential analysis experiment computer depend whether goal prediction design derive correlation matrix equivalent maximize call excellent design intensive stationary repeat maxima stochastic find estimate parameter treat whereas assume parametric priori design distance computationally intensive whereas relative sampling degenerate less advantageous sampling mechanism ensure configuration relative previously sample location design demand convert fix location obtain optimize machine would fall research active literature query selective refer situation perhaps limited input train essentially try gain select location great alm show maximum information design similar maximize reduction notation gps average location integral replace grid location parameterization gps alm approach surrogate stationary gps aggregate language aid active mining experiment surrogate distribute architecture work yield computer start model configuration initial run region run concerned fidelity agent processor time agent allow begin execution therefore start finish perhaps master program simulation configuration queue get queue interact design queue design classic traditional approach primary optimally boundary region measurement severe difficult checking put truth boundary highly full favor partition drawback difficulty inherent surrogate asynchronous response node become sequential design sample spaced relative sample location stage gp surrogate monte alm queue optimally spaced candidate become available input section learn posterior alm great deviation output convenient width predictive maximize square reduction notation component column equation integrate reduction calculate average candidate alm heavily concentrate boundary partition ranking location alm comparison alm comparison alm gps full specify dense grid advance sequential already sequentially application priori instead mcmc sample possibly sample candidate come subsection sequentially candidate space configuration algorithm model computationally intensive full variance linear proceed variance base preferred replace matrix densely expensive alm asynchronous reason candidate alm agent go run would want pick point pick sequentially candidate order reality asynchronous environment fit run thus spaced candidate entropy seem reasonable encourage entropy parameterization model thus suit mcmc wherein closeness entropy may candidate continue understand change another disadvantage require repeat decomposition large use sequential entropy separate obtain depict posteriori candidate proportional create candidate neutral use infer community stability adaptively choose employ search unnecessary spaced simple propose swap accept upon possibly maxima search run parallel typically much sample location dot design show entropy design configuration sample design dot candidate sample circle space exist possibly near roughly sample circle point trial trial trial accordance alm conditional follow candidate queue sort list run input configuration add estimate surrogate response run configuration available candidate number develop simulate asynchronous response finish interface section artificial experiment use real treat gp surrogate surrogate dimensional alm pool develop treat physical parallel highly advantage multi processor becoming couple speedup processor multivariate scope create surrogate response recommend chain chain trial sampling individual trial find mode posterior aggregate explore random pruning represent tree shorter burn begin tree initialize run lm burn chain burn alm need burden response incorporate round trial response affect candidate compare slow yield optimal offer improvement I synthetic input second example provide improper fixing section
random mild regularity assumption path normality suitable trend function space approximate band inf une des un du une e les non de la dans des pour introduction traffic monitor medical allowed collect collection curve dimensional term comprehensive typical david david des sciences fix trend nonparametric infinity square rate spline universal normality task building nonparametric band diagnostic receive considerable case process suitable estimator space simultaneous band asymptotic square path surely variation variance x c form triangular mm error mutually mean order quasi write j p space fulfil space regular suitable nonparametric estimator denote correction recall compactly say equip sup norm process index weak convergence continuity consider c rely follow theorem noise iv particular normality feature monotonicity like make proof straightforward hand classical kernel motivate popularity drop noise besides carry autoregressive lt lt ph define normalize consistent find simultaneous confidence band simultaneous confidence either e converge weakly denote correspond function suffice result get derive
wasserstein property see e law stand measurable function conditioning inequality jensen inequality mean interesting functional statistical problem deal functional think eq weakly satisfying compact continuous possess law every resort kk kk whereas uniqueness converge law dx random common euler kind beta setting markov hence attention wasserstein order metric dual give random variable e body distribute p p pn notation denote dual n thesis integration subsection shall choice variation finite topology binomial n k observe need suitable recall tractable eq prove eq use jensen inequality martingale markov jensen bayesian view treat prior guess normalize increment probably example nonparametric dirichlet increment study measure increment recall increment collection variable stochastically characterize give follow normalize independent increment put assumption see classical dx dx di g sequence exchangeable specie sequence non atomic measure among task condition observation coincide law sequence stick break prior surely almost surely identically take common either independent let nonnegative say exist see sequence drive task moment acknowledgment I grateful behind want virtual work corollary partially universit main object statistical inference determination sometimes law empirical serious vanishe framework exchangeability fairly tractable posteriori application aspect reason quantitative posterior appear assumption infinite exchangeability rise posteriori attractive infinite independent give parameter fact acknowledge theoretical experimentally bayesian procedure inference hypothesis point state complete mention e probability traditional role modeling entity hypothesis drawback vanish consider horizon always least ideally observable take preserve hypothesis avoid assess particular distribution give conditional usual nonparametric view empirical concern entity whereas technical generally deal define hence infinite exchangeable thought law procedure follow overview section quantify law remark total exchangeable say represent exchangeable define complete separable endow measure parameter space endow view attention common parameter decision formulation statistical assume decision action consider real represents suppose finite realization bayes common determine connection value lx hypothesis bayes tp usual estimator number loss classical bayes eq ny estimation mean usual bayes bayes plug turn endow metric induce borel
log therefore sufficient statistic notation simple statistic show distinct adjacent distinct proceed induction linearly minor index say element notice linear linearly false adjacent adjacent cardinality adjacent explicit done simply algebra investigate structure contingency define vertex cell define binomial vertex cell involve binomial equivalently appear cell connect consecutive form connect add definition example contingency define figure cell generate dimension independence table adjacent complete present cell minor involve require repeating analyze complement graph corner notice corner define adjacent orthogonal adjacent minor adjacent straightforward straightforward use lemma statistic enough adjacent remove basis sufficient statistic model introduce column sub linear strictly positive representation sufficient statistic negative determine statistic formulae apart normalize eq unique way switch implicit parametrization matrix vanish direct substitution converse intuitive obvious large summarize model independence sufficient noticed nevertheless union suitable conclude statistical contingency table figure h markov symbolic nevertheless independence theoretically follow determine representation markov generator ideal I ideal therefore computation computation generator ideal associate order generator ideal use symbolic algebra operation present collect student evaluate scale final ccccc analyze adjacent use package mcmc consist goodness show table phase table use intra category category homogeneity datum subject ordinal high present effect exact independence total independence accord free cell markov second fourth easy direct substitution belong local maximum contain maxima suffice binomial new independence log positive importance wide first find plan analyze interested model independence selection must study investigate biology like explore structure adjacent thm proposition thm remark thm study contingency example range contingency apply categorical concern application independence detect complex introduce identify contingency log wide spectrum biology book great deal example datum set area contingency table use algebraic geometry structure name algebraic relate paper focus attention geometry polynomial algebra probability particularly log graphical exposition biology find counterpart symbolic traditionally package specifically design ti consider contingency probability representation independence independence vanish vanish call binomial hold pattern contingency property normalize constant formulae table vanish zero zero representation phenomenon adjacent probability point view independence variety simplex vanish choice model course mean statement locally pattern first application contingency present
entire history play binary consist payoff first payoff instance enumeration ball j nb k ib input instance construction reference construction payoff rise precede shorthand define count round analogous eq q may reasoning sample construct play side inequality complete I bound wrong concrete root call exactly binary encodes diameter contain subtree root whereas min covering subtree cover dimension open dimension metric locate active strategy locate near contain achieve impose active strategy outside cover intuitively strategy impose lie cover eventually stay start becomes cover ever extend fix close di either tree decomposition depth metric decomposition separate ball report ball return strategy cover ball space require lipschitz mab compact decomposition remark relax theorem instead compact define leaf outside require modification per guarantee problem instance upper extension analyze phase round cover version rule initially net large contain point rest partition pool denote round routine cover pick add strategy radius never repeat horizon lemma phase fix proving algorithm phase simply run near end phase remainder mab decomposition strategy cover run active well cover run write eq small set claim irrelevant eventually away always throughout step run algorithm provide function continuous strictly beginning strategy activate suppose covered claim cover clean run cover clean regret cover clean run room let set claim sufficiently cover set moreover ta tr tu tu tu general extend idea generalize depth metric denote ordinal subset close note depth definition definition achieve dimension max dimension decomposition max decomposition decomposition imply decomposition depth ordinal whose cardinality exceed open complement ordinal ordinal induction observe otherwise assumption suppose every vx intersection open cover satisfy cover consequently mab problem exist theorem satisfy require oracle cover ball return ordinal closure exist open ordinal report cover phase round phase ordinal precede initially play compute per simply per phase find initialization time play phase enumeration ball extend direction follow analysis work confidence essentially history playing give strategy hold lead play shape concrete provide improve radius maximal chernoff example dimension unknown metric relax lipschitz function triangle lipschitz hold one mab extend distribution gaussians moment rely throughout phase round play time reward radius history strategy within phase history tv cn tv tv property property need line carry instance lipschitz radius become regret one vanish set plus lipschitz every strategy reveal take advantage shape interestingly slight abuse notation corollary noise lipschitz mab problem side multiply multiply radius corollary noise distribution satisfie bound establish stochastically derivation needs modify slightly omit version concentrate noisy mab least radius give regret via p get confidence tv consider locate estimate chernoff moreover chernoff lipschitz mab increase peak piecewise least constant admit radius regret follow separate length sub use reward tv bx fx probability use sample approximate tv tv interval reward set elaborate lipschitz mab constant dimension ab radius standard mab maximal ingredient refine chernoff appear literature average e na n minus reward strategy check property ii radius maximal reward tv imply else simply claim lipschitz mab reward reveal quasi distance mab strategy decrease mab problem mab function well payoff target mab maximal reward take dimension cover small covered refine space target mab dimension cover prove dimension word suffice cover cover diameter diameter cover remark need target mab extend guarantee cover mab never essentially one let properly relax refer replace guarantee decrease eq quasi refine example goal clean illustrative concrete theorem follow finite cover pair prove diameter proof diameter remark metric shape suppose shape constraint guarantee restrict moreover similar neighborhood reward strategy suffice random mean absolute trial mab trial mean furthermore algorithm nice survey inequality failure take bind accordingly refined parameterize event clean tail violate optimal nsf author associate award armed choose sequence strategy performance small finite bandit problem set investigation motivate broad strategy enable set multi armed metric satisfie condition metric multi armed lipschitz arbitrarily technique online algorithm must trial payoff choose tool inherent three decade increasingly visible computer science diverse armed bandit often gap bandit strategy infinitely investigation strategy payoff large multi broad natural induced metric strategy bandit study interval case despite extremely motivate website thousand ad display click ad display ad web infeasible highly inefficient ad website generalize inference similar ad satisfy predefine constraint expectation moment think reveal abstract state refer triple lipschitz treat mab space armed space experimental bandit tree describe bandit r author follow theorem paper multi arm payoff always play mab r bind extremely na arm suitable regret mab odd set mesh refine useful choose really close raise payoff nearly proximity information payoff lipschitz mab problem tune strong answer maxima modify na I play phenomenon general whose outperform instance paper mab possible solution describe every metric space achieve satisfactory combine bandit apparent explore mesh ingredient design moreover perform instance call problem require contribution early bandit step maxima mesh strategy region ingredient algorithm stronger achieve state sketch arise naturally try cover diameter dimension space max dimension whose local cover number mab compact bandit particular achieve na I bound cover homogeneous equal generalization I early lie x strictly design suggest dimension far important treat generality mab problem metric space expect level hierarchy early subset notion capture connect lower highly near set optimal quantify cover diameter cover payoff fall short maximum instance suffer min poorly point cover cover space part root branch infinite weight exponentially shortest infinite away root lie bound cube covering point dimension state theorem specify issue infinite number interpret theorem ignore interpret abstract possibly randomize mapping history theorem valid precise algorithmic require take ball either output standard mab oracle round ball metric complicated definition tailor mab metric obtain sharp use precise extend notion dimension extend notion feasible leave extension elaborate example play plus shape algorithm satisfie guarantee enjoy maximal reward reveal fourth relax analyze known extend distribution unbounded moment round select select place lipschitz condition give define na I adversarial aspect create considerable technical future hope refined style guarantee goal point hope characterization per outperform instance perform similar prove topic metric min paper characterize open question publication conference version metric mab depend open ball radius notation otherwise metric diameter payoff reveal strategy observe random running payoff dr diameter discuss optimality adaptive phase round time let reward chernoff clean algorithm play cover active return else cover run round active cover active play active maximal break tie section mab yet guarantee cover activate cover entire metric assume strategy focus phase clean strategy lipschitz cover phase play trivial play time definition r tv clean assume activate cover diameter one strategy tv leave essentially current remain contribution past phase let leave hand claim ask algorithm instance metric everywhere else focus exponent large motivated shape theorem let mab take metric mab I extend I phase choose regret mab cover dimension use phase I algorithm phase assume dimension constant suffice suffice accumulate round phase choose net diameter expect arm round maximal plugging obtain claim ask achieve perhaps sophisticated indeed case
life underlie relationship context systematic seminal topological diverse expect gene microarray gene expression measurement expression level stage point stage focus known range range visualization two evolve cluster expression microarray rough microarray microarray measurement state reason learn binary mrfs post activity score group annotation study global pattern evolve figure gene function statistic edge cluster wave peak mid begin similar pattern gene contrast drop mid stage stay begin development gene localize function mid role rapid turn gene mid mid time respectively tight interact visible mid mid evolution coefficient figure group gene relate development connection figure plot three normalize nice suppose increase gene stage activity stage development becomes increasingly recover list occur whether actual interaction cell interaction dl development compound chi development gene yet guide cell color right factor stage representative relation cascade cascade important role development box point involve coherence reveal dynamic nature relation persistent across stage instance five factor dpp across across development proceed furthermore connect cascade gene right side network produce gene different accord large topological interaction many huge r gene margin surface gene actually active interact gene abundance due development hold td role propose aspect genome reverse task gene microarray hundred algorithm correctly function graph degree intuition positively freedom possible give identifiable intuitively distinguish dependency strong identifiable intuitively covariate look similar relevant hard covariate common away require nonzero distinguish vector adjacent point observation smooth sufficiently achieve kernel symmetric support assumption reason statement parameter furthermore condition hold g asymptotically appropriate dimension recovery surprising tv difficulty tv multiple bias toward segment structure segment partition first stage segment restrict fused technique analyze method tv besides assumption appear may important vary static limited connection persistent describe dynamic interaction although network pattern discover come extra parameter throughout independent future direction opinion straightforward practical work datum extend directly still principle select tuning procedure static vary challenging incorporation topology develop tv end smoothly tailor toward estimation structural bring together future estimate vary acknowledgment grateful anonymous valuable improve support ray support nsf fellowship plausible representation information observation estimate varying network series present vary network logistic formalism optimization solve solver scalable large network vary reverse political record evolve underlie life cycle course necessary complex dependency gene genome activity community understand information assess impact natural build network dynamic noisy incomplete even add complexity network methodology dynamic reverse network attribute rich toward dynamic time various complex problem development exist dynamically context dependent follow microarray time stage reverse finance stock suppose measure value stock like stock insight structure change us depth investigation mechanism impossible determine specific two system measurement microarray stock price status time base measurement estimation assume relational suitable dynamic network entity field mrf entity edge represent entity stock edge distribution index mrf assume pe independence assumption conditionally rest paper kind mrf binary potential type ise express family x recent assume emphasis node concern structure mrfs set distribute sample mrfs much mrf estimation domain actor relational seem define assume change segment body invariant covariance graph structure zero dimension research handle sample microarray measurement assumption employ successfully recovery use pseudo across consistently suitable approach consider different selection procedure neighborhood estimate empirically improve neighborhood procedure suitable solver maximization simultaneously efficiency penalize solve program author work solver exploit seem graphical author penalty try introduce difficult since due partition node consistently aforementioned time structure model guide topological toward vary topology observe attribute form evolve call adjacent edge stability density transition capture topological incorporate hidden topological change specific time unfortunately expressive method change smoothly counterpart covariance concentration concentration structure address consistency follow describe structure obtain demonstrate theoretical detail obtain step simplicity come index assume value function independence subset address insight change one need example every vector index pair node vector rest nonzero time observe recover information alone nonzero pattern pattern use combine combine use operation appear estimator edge include conditional variable u objective structure adjacent composite fuse penalty g create signal solver package computationally expensive know algorithm efficiently much solver large u n block u procedure convergence descent optimization experiment hour hundred bottleneck linearly overall collection optimization equation atomic covariate instance solve equation regularization plane example repeat regard identically incorporate time procedure sample assign small change denote require tuning way time tune smooth structure tune freedom network control small bias penalty change tv function onto classification number technique set use employ estimate equal score next address bandwidth tuning parameter sample risk miss change parameter make adopt heuristic scale distance form entry simulation find guess quite exhaustive method smooth bic equation conduct generate recover evaluation scenario discuss piecewise wise combine equation min operation take maximum generate one pair degree less add edge remove take obtain graph prototype prototype parameter graph nonzero independently generate accord smooth I piecewise prototype independent estimate tv term precision harmonic f maximize bic score parameter penalty penalty log tv network experiment reader illustrative purpose recover smoothly lower combine neighborhood increase dot smoothly see static benefit surprising generating underlie see min experience max min edge seem piecewise method tv worth note tv marginally inspection point change piecewise upper
idea coverage adjust unobserved happen place first value place value unobserve value thorough explanation imply prove response region bootstrappe bit word bit solid confidence interval obtain bootstrappe times bit curve berkeley california berkeley information response stimulus instead entropy ignore stimulus mutual trial plug probability entropy average conditional entropy time average entropy response joint stationarity ergodicity stimulus quantity mutual stimulus mutual long across spike stimulus estimate mutual stimulus response stimulus response vary stimulus response average ergodic usually make explicit consideration lead time conditional distribution stationarity estimate mutual interpretation vary specialized direct entropy concern stationarity unlikely stationarity appear violate second explicitly dynamic nature begin brief plug actually measure variability observation trial formalize describe limit example graphical direct stimulus choose repeatedly stimulus two response potentially relate stimulus trial figure panel neuron song panel audio natural stimulus response divide bin spike form bin spike bin become letter correspond belong spike theoretically unbounded figure bin vertical bin different entropy response stimulus response trial noise associate length total rate direct strong prescribed necessarily stimulus response stationarity potentially difficult primarily non reader discussion notational dependence plug distribution estimate use information estimate entropy divergence trial average kullback divergence response response stimulus repeatedly present trial stimulus trial identically furthermore law number increase response view eq response quantity plug leibler write formalize section summarize average analogous specific analogous stimulus function decomposition stimulus explore section direction amount observe increase length necessarily occur serial autocorrelation difficult stimulus stimulus guarantee suppose repeat trial q statement entropy ergodicity stationary stimulus potentially depend average ergodic pointwise moreover likewise entropy primary ergodicity necessarily mutual statement quantity appropriate justify assertion plug average interpretation time distribution vary familiar consider taylor thus variability course characteristic due stimulus measure stimulus eliminate aside stimulus may across time picture stimulus examine graphical turn conditionally coverage adjustment describe appendix contain figure response neuron panel stimulus stimulus synthetic stimulus song show adjust divergence confidence bootstrapping included exclude going represent per nearly plot stimulus appear time fluctuation roughly fair second stimulus song isolate vary flat vary time plot stimulus location isolate divergence stimulus coincide stimulus offset stimulus confirm energy stimulus range match correlation plot isolate structure divergence plug measure stimulus stimulus jointly deterministic long think interpretation stationary stochastic process assumption meet mathematical formalism abstraction impose hope variability display relevant assumption highly stationary make property stationarity speak concrete natural song display look stimulus amplitude stimulus period stimulus look good stochastic long indeed difficult
node tw child terminal child terminal tw edge terminal tw tw node reality reality ct reality infinity move element reality situation reality game drive account uncertain driving reality event subset terminology explanation generally thing move partial call course situation situation natural partial set path situation restrict terminal cut assume associate consider variable notation associate event identify terminal second coin yet terminal tail distance mm corner fill black corner draw rectangle corner draw distance mm dot root inner child child child child east h north h north tt real return terminal get variable whose stop soon give measurable consider c return outcome outcome second coin flip element sample coin measurable soon reach determined outcome coin flip see fig element outcome coin flip coin flip stop new equal turn player terminal situation move terminal associate make move reality represent capital make introduce terminology game play terminal situation move correspond whose capital accumulate far start zero capital play recursion start capital accumulate capital situation accumulate capital variable non terminal capital also tell capital start capital situation strategy outline sufficient terminal ft tf ft protocol move gain terminal move ta tt call terminal terminal reality playing increase capital might protocol price associate measurable f coherent protocol terminal situation tf tf tf tf tf tf tf tf tf iterate specific instance coherent law iterate logarithm measure shall come pay attention might reality might school probability belief incorporate belief theoretic book theory consider explain coherent really gamble non empty alternative actually obtains determine actually interpret uncertain linear utility may gamble assume gamble boundedness transaction obtain belief gamble gamble call follow avoid belong belong belong belong number argue desirable axiom consequence axiom infinite partition may conditioning avoid detail belief really require actually coherent gamble upper bf bf supremum event occurrence b stress conditional acceptable price occurrence mean gamble unless occur conditional occurrence supremum else accept interpretation conditional stand acceptable occur seem consider third interpretation update briefly come distinction partition bb lower coherent desirable gamble give potentially unbounded gamble property coherent really gamble assume mean bf bf bf c bf property analogy proposition equality identify correspondence rather situation extend fine partition partition deal gamble fine partition blue colour colour gamble requirement coherent really gamble ff colour ball subject g r g g infimum supremum price gamble inequality lead degree f bf subject extreme bf conditional call model consider equality indicator finitely additive connection theoretic enter world another forecaster move reality specifically non terminal situation get impose requirement desirable interpretation sense reality specification forecaster stress interpret dynamically I gamble forecaster shall generally local belief really desirable gamble perhaps equivalently ask conditional prediction gamble path reality reality situation combine reality investigate shall piece really desirable gamble coherent mean gamble coherence terminal situation associate tt tu far process terminal black rectangle draw circle draw black minimum distance segment amplitude grow tw child terminal child terminal parent child tw edge child node right tw tw forecaster value indicate represent end call reality move path forecaster conditional reality get translate gamble forecaster situation thing leave coherent desirable gamble indeed coherent coherence really desirable gamble cut partition consequence attention space cut contain cut make cut trivial requirement cut eventually gamble forecaster reality get relate conditional collection event constitute sum gamble convex cone sum terminal cut belong protocol discuss reality union terminal cut sum situation terminal call fig relation q initial value terminal letting argue gamble terminal capital mm circle fill rectangle distance distance segment amplitude mm root grow sep node terminal parent child terminal edge edge parent terminal tw child tw terminal situation really forecaster step term set really gamble coherent much de theorem gamble include extension eq coherent really desirable gamble define explain shall situation f terminal besides proposition low upper satisfy additional predictive gamble terminal ff tf tf tf tf monotonicity go important proposition two gamble f assume cut gamble u u appear close correspondence coherent protocol coherent protocol immediate model lead price predictive mark correspondence approach theory coherent protocol immediate match conversely immediate model protocol forecaster immediate forecaster correspondence local belief gamble derive something question entail gamble associate see reward depend move reality make get forecaster price reality get derive forecaster forecaster acceptable price reality global directly infer concatenation cut tf f gamble situation element cut readily infer stop stop reality reach also restrict call proposition terminal cut measurable tf f consequently child cut terminal interpret gamble immediate proposition completely local modal recursion illustrate back begin coin tail stop otherwise flip tail flip coin word depict circle draw black dash grow child label leave left leave terminal label parent solid child tn parent parent dot child node terminal child child right h tn tn cut introduce situation terminal forecaster belief reality situation th flip belief term really gamble move identify child enough forecaster assume fair sense flip lie assessment lead k situation terminal situation clear equality child g n n repeating course find generally illustrate recursion appear call r consist non terminal leave forecaster make tell theoretic framework forecaster move gain forecaster reality get present extend idea explain belief move reality terminal belief really gamble bring forecaster specifie situation reality get situation combination gamble accord combination axiom price forecaster conditional reality get situation forecaster terminal non negative combination gamble offer forecaster accept reality move situation change capital negative amount move essence tell lead protocol correspond price coincide forecaster game theoretic subject lot price certain decide upper update game framework law large law interpretation example law consider terminal cut tree simply word protocol minimum mean forecaster reality common upper thing forecaster provide gamble forecaster accept price segment tree start average gain forecaster associate reality move know write forecaster belief initial reality get occurrence word want forecaster reality get law increase version weak see hoeffding event look reality path forecaster hand forecaster upper occurrence coherence forecaster occur coherence forecaster predictive force choose event forecaster plot exp node exp function exp construct try calculate forecaster upper forecaster situation reality never shall upper never situation value indeed thing ask coincide gain event upper forecaster get ss predictive instance theorem enable probability probability chapter proposition tree proposition tree arguably discover discuss draw tree calculation treatment vi allow reduction expectation seem phenomenon computational tool sequence alignment algorithm illustration look forecaster time coherent also system jump system go depend state go model situation kk reveal mm rectangle circle black mm mm dash draw grow inner sep child leave child terminal child terminal child leave terminal right terminal leave child label child right terminal label child leave child label node terminal label ht th reveal model notational convenience generally forecaster belief instant want belief time actually want calculate f kf cx equality apply cx x tf therefore proceed n nk calculate number calculate time set really desirable gamble call language gamble extreme set bayesian terminal tree minimum expectation mass see typical step lead efficient algorithm involve network another phenomenon precise event bound game infinite become immediate coherence axiom lead dominate correspond matching argue requirement rational topic low predictive happen tf ff immediate arguably introduce guarantee possible I correspond possible situation lead want model much say envelope theorems set type ultimately concatenation specific general inequality generally speak result forecaster specifie model relate belief terminal acknowledgement discuss discussion take place year read grateful three discuss significance prove gamble possibly unbounded gamble proposition two generality second inequality f eqs real I bf sum term define without generality great taking supremum real lead first get statement immediate consequence observe therefore bf bf bf deduce bf bf bf inequality trivially fix ci ci bf bf g f sum side already argue really desirable gamble gamble coherent prove extension set suffice selection hold non conclude disjoint union f u invoke second follow non terminal situation non situation moreover call immediate obviously suffice hand eqs terminal situation ss complement ss without empty mean minimal bring continue situation situation tf clearly suffice necessity assume mean ts otherwise lemma cut terminal apply lemma contradict terminal I find conjugacy f observe tf last statement first ti tell g proof statement base I consider
markov th assertion remark study asymptotic behavior growth eigenvalue part quantity polynomially process real starting hold part let invariant law logarithm follow notation I eigenvalue real part follow lemma almost surely jointly f definite surely dt dt consider number possibly uniformly large since eigenvalue half e mean uniformly notation hence tr p possibly depend tr e large tr clear argument corollary case eigenvalue part eigenvalue negative part combine discuss decompose rational root space define define identity obtain tu dt therefore hence term go increase c complete remain spirit discrete decompose rational characteristic root define I c one obtain mm observe ec ec dt clear grow like integration part martingale respect martingale since use tu tr tc u apply obtains define monotone arbitrarily mm tu proof cf corollary theorem observe theorem section show efficient process matrix see already therein process type deduce f ft dt ec te tr ec f eigenvalue ft etc tc ft e dt e ec pe ec purely purely negative possibly eigenvalue half part stable depend large proof theorem es value bound away argue depend uniformly ec get therefore possibly depend value similar ec ec tr ec decompose symmetric invertible partition matrix invertible dt converge surely converge almost surely obtain c ec ec tr ec conclude remark state autoregressive process dx multidimensional hold asymptotic efficiency procedure may develop whether univariate need stationary whether real often assumption arise normality approximate far investigate mixed normality may consider drift depend markov help switch find consistent important ask setup application sample ask focus asymptotic process purely purely rational canonical moreover lemma similarly j term hence martingale u aa theorem eq study zero eigenvalue summarize zero rational since scale theorem generators example lee paper consistency efficiency drift case half part process part real part stationary grow eigenvalue polynomially consider separately combine show coefficient estimation use modelling phenomenon paper dt td note follow come show strong square estimator discrete general estimator drift multidimensional follow sde process dimensional usually task generally dy dy dt dt dt invertible unbiased term consistent irrespective condition automatically autoregressive positive relax possible discussion remark drift parameter extensively example rao therefore asymptotic know multidimensional mixed showing consistency multidimensional rao methodology deal section basic part discuss fact eigenvalue appendix consistency concluding matrix rational canonical matrix root half root lie distinct irreducible minimal similarly rational canonical rational canonical form covariance rao assumption high definite theorem proof respectively dy follow tr f ec real gaussian real part derive part moreover positive consequently throughout shall I particular hypothesis ft f ft let almost sequence almost surely ft z definite suppose almost f k nonzero f contradiction
outcome analyze small amplitude usual greatly decrease reflect cover percentage adopting confidence interval amplitude decrease r cc cc n c usual mse cc usefulness chemical laboratory estimate chemical element concentration correspond intensity uncertainty guide intensity present intensity refer calibration observe increase concentration multiply program likelihood nx nx ny estimator expand propose estimate usual usual element observe obtain element considerably adopt r intensity intensity c c element element e arise read consider error read uncertainty great future g consider skew particular drawback usual concentrate one acknowledgment grateful dr carefully read manuscript grant stage respectively interval order standard pt pt b calibration analytical g de chemical one chemical tackle linear assume control chemical attain value observable calibration keyword chemical usual chemical specie assume solution due propagation law estimator chemical analyst process attain variable unobserve control formulate framework usual calibration independent certain study uncertainty uncertainty depend motivate calibration call variable measurable property variable intensity concentration second relate measurement concentration analyst produce unobserved attain calibration define calibration model follow variance suppose control usual calibration logarithm give respectively logarithm likelihood making obtain solve
wang even mix experience similar argument mix run eventually publish sampler incomplete list include et variable spatial longitudinal study though development rejection thompson second problem deep though handle model mostly continue meaningful realization possibility iterate monte molecular avoid due rejection technique occur physics early occurrence particle back connection formally connect past de particle filter correction volatility filter produce approximation incoming tendency degenerate close recover variety modern develop liu liu chen point resample sequential carlo recent closely mcmc sampling perfect place method discovery large book review material quickly major sampler close impossible impractical advance implementation fill activity stochastic geometry particular mainly apply process birth death bayesian space allow bayesian subsequently exist solution later spirit reversible jump completion cross give setup explore look european conference stochastic organize aim direct implementation green technique version early jump variable algorithm like central mcmc take sufficiently recall state fairly normality strongly lack markov require hold ergodicity chain half estimate variance na usual standard dependent batch stationary allow estimate early apply yu calculation derive adaptive work interesting development however al estimator valid al thorough cumulative mean change represent could solve te change emphasis close expand improve algorithm lead world truly evolution current deep application continue expand influence big something von continue advance future solve appendix university college theoretical aspect university university sciences california berkeley university college institute mark david college van sampling university university linear model university university software green newton thompson panel bayesian acknowledgment grateful suggestion david green liu ne work partly vi grateful de paris grant dms version chapter li modern paris university paris paris distinguished statistics university usa attempt early lead use importantly development methodology monte technique though impact truly early except specialized appear physics treatment mathematical chain monte generation mid reversible jump sampling conclude compute mention markov chain could variety situation despite advanced compute computer chain trust researcher demonstrate series application method practical lee al rapid dedicated bayesian argument adopt development carlo early close statistical reasonably metropolis publish metropolis research work physics atomic back regular von computation win game idea von carlo suggest early car brief history closely appearance computer life von early von inversion accept reject technique simulate distribution without computer car rand publish font metropolis et font algorithm se computer direction I member team include well publish primary al particle parameterize boltzmann normalization factor dimension standard technique approximate since realization improve monte walk particle configuration previous compute accept counter move walk move particle together algorithm appear primitive et validity establish second part discretization prove full minus discretization iteration paper step burn subsequent variation develop anneal temperature schedule anneal show fact homogeneous chain font font generalize statistical simulation tool curse meet et fall sampler ten cox define methodology reversible chain treat discretization analogy acceptance state state proposal form et metropolis mention opposite exploration example mix rather strategy transition stationary hasting sampler pre metropolis within except even though remainder paper deal connection pt somewhat physics responsible name name gibbs implementation sampler apply influence green within point actually quite field metropolis issue normalizing york activity spatial community metropolis et still attribute lack treat even hundred process however mid piece year highlight usefulness image statistic influence write paper start intensive process compute metropolis algorithm introduction title give presentation interestingly early essentially namely fact sufficient important enough paper approximate z simulation posterior estimation connect original datum point discussion costly take end sense imputation start new another reason may functional analysis uniformly author focus green ten hierarchical lot compute yet generic method development sampling result seminal paper subject later mcmc conference university conference like impact conference research papers liu conference year later chain follow discussion appear clearly evident even perfectly understand look summarize advance mid influential mcmc property ergodic average relax condition central limit new condition play important role research pair influential liu
sup associate rademacher bernstein exponential supremum rademacher process version knowledge context adaptive instead estimation sup mention interested adapting model technique direct construct hard wavelet introduce satisfie functional adaptive sup h proof constant become quite present moment measurable lebesgue norm space time function integer r fx fact section z hx jx jj subsection onto certain series jx decay pointwise onto main case article l k kf convergence pointwise replace fact wavelet variable law denote nx compactly support wavelet coefficient first compactly wavelet wavelet may sum consist family class compactly decay wavelet term spline generate spline suitably translate spline exact derivation detailed definition sum extend projection outside spline wavelet find main follow wavelet compactly support wavelet estimator nj ep ny ep ny bind ny pp central banach usual brownian emphasize optimal sup loss n p drive choice resolution state generate compactly support rademacher need grid indexing choose integer pt estimator resolution offer est remark minimum alternative equal briefly procedure drive resolution rademacher type usual threshold start bias stop start bind correct lack monotonicity resolution somewhat refined bias domain exceed main result whose contain compactly support bound variation wavelet sup minimax old ball rate possess compactly wavelet sequence constant law furthermore continuity continuity density relax modify definition candidate ball automatically even functional control prove theorem constant drive haar e require operator obtain bind obtain haar spline available bind see therein note section show replace respective definition inequality constant bernstein author let countable case conjunction describe class satisfy entropy borel function radius early due type historical hand far well prefer constant especially tail supremum similar countable absolute set imply take pf v large consideration adaptive mind inspire nice context namely consist supremum supremum rademacher page rademacher independent note bind take variance estimation mind small quantity valid enough range convenient well know analog variable countable absolute nx pf nx pf nx nx apply sum well nx combine prove version q ga vc exponent potentially let countable uniformly value proposition nx together review wavelet represent shall need wavelet ij z equally spaced knot spline degree differentiable spline less equal coincide page basis spline page eq spline cubic spline element basis derive show j rx kl j rx jx spline projection note argument operator see page section exist decay kernel describe projection simple lemma spline since haar basis page operator therefore q combination product imply almost everywhere proof decomposition kx kx rx rx rx since since index function support variation however apply wavelet toeplitz band limited spline still prove type extend borel haar result toeplitz change compact toeplitz function page limit write q last l last fix extend set function give dominate hold r h cover assume bound scaling give see supremum remark second compactly wavelet lemma follow proof differentiable quantity interest est precisely process prove functional rather quantity interest bound cr p rp rf f decay bound first note also wavelets decay lt inequality bias fluctuation shrink consist compact finite wavelet consider measure analogy proof absolutely interval equal contain decompose collection linearly inclusion vc envelope independent prove classical compactly ss ny jt precede lemma inequality lemma prove compactly wavelet I section wavelet wavelet uniformity control respective derivation preliminary independent every bias continuous still uniformly define mp enough p ep nj nj ep nj step furthermore iii j see obtain furthermore rademacher expectation r satisfy first p j ny nk convergence nk j l use uniformly nj fy
poor predict detector detector sensitive near hold user db vary plot discrete continue near far additional discrete degradation due detection reveal discrete either iteration extension acceleration powerful interference channel section sequential detector outline scheme see outline completely insight sequential schedule implementation standard sequential discrete sequential scheduling consider joint variance amplitude plus em simplify differ aspect update user bit estimate instead initialization year impact code rarely investigate variational gaussian detector variational solution receiver exact detector inference perfect parameter however one say term auxiliary say carry iteration compute probability step yield jointly offer joint wireless concrete assume assume remain block bit code estimate write channel denote dirac delta property free omit constitute knowledge minimization free prove special intractable simple arrive initialization th value rest section implement resolve receiver noisy receiver attempt adaptively estimate noise amplitude jointly index free gauss model parameter prior channel however receiver channel equivalently estimation gauss need square writing side equation gauss produce substitute decrease free coupling acceptable em decode table l kb k extension algorithm sequential straightforward outer update free output making derive complete utilize ignore independent give em k kb k end backward clear converge last term section complete grant instance investigate detector employ suffer poor convergence non convexity consider scenario benchmark four assume code generator flat channel convolutional generator also receiver inaccurate channel amplitude introduction result mean elaborate complete near present omit poor whose paper variation predict pass rule complete employ scenario record perform hybrid originally outside variational inference small scheduling sound sequential schedule schedule inaccurate estimate schedule ease need arise scenario situation receiver inaccurate actual assume inaccurate channel noisy estimate compare receiver consistent curve plot outer see em detector error reach actual amplitude degradation start replace outer keep iteration I every outer update inner prescribed add step em seem perfectly certain attribute leave work despite inferior discrete detector produce excellent even unknown variance exist detector would fail setting initialization far inaccurate channel addition channel channel depict discrete channel iteratively discrete concept reach formulation belief sum product center minimization would channel point scheduling scheduling good conventional view focus detector alone paradigm detector detector show composition subsequently refine systematically effort obtain good energy scheme insight variational efficiently estimate conjunction propose framework derive study soft soft interference concept inference framework interference inter symbol interference generality attention facilitate avoid closely viewpoint provide convenient decoding utilize error control code detection inaccurate channel receiver may systematically base arbitrary square rigorous variational setting detection code interference treat control code interference channel dramatically conventional interference follow decode channel inter symbol interference channel vector common modification restrict scenario understand generalized evolution detector ic decade grow interest detector detector practical detector simplicity detector importance design challenge lie differ must symbol access symbol unfortunately unlike decoder generator code complexity infeasible design success detector generalize method adopting include special extension focus treat inference engine channel symbol paper recent attempt provide detector generalize contain elegant detector bit paper regard detector detector arise approximate distribution optimize highlight implication significant way introduce formulation minimize term energy minimization detector mmse interference successive interference detector derive naturally produce detector scenario receiver motivate joint unknown solution iteratively minimize em replace inference parameter demonstrate estimate inaccurate amplitude refine conjunction bandwidth channel carry framework hoc optimal modify scheme iterative detection technique channel separate detector discuss decode scheduling study constraint detector section example inference justify simulation case face letter column stand element diagonal stand variance gaussian wireless link user match filter represent signal amplitude distribution match match filter normalize signature whiten triangular cholesky computational selective asynchronous form reader refer insight jointly detector output detector maximize individually prohibitive exponential individually detector optimal detector minimize detector derive produce complexity find optimize cost energy detector accept decoder call send rigorously message algorithm graph detector good exact sum product individually optimal detector addition statistical contain cycle valid sequential wang hybrid give message scheduling iterative code scheduling code channel run propagation perform decode inference process see view approximate sum way pass example message b scheduling pass prior error ignore even user detector linear mmse sequential schedule substantially generator ignore detection schedule resemble message pass protocol product restrictive introduce detection must overall linearly simplification schedule bit k unlike scheduling prior schedule round decode write hence schedule reasoning generate divide parallel fold pass shot generate shot detector implement challenge approximate hybrid schedule without use sequential computed schedule remove issue scheduling use justification schedule schedule implementation demonstrate prior approximated scheduling scheme poor detector treat detector model estimator generalize arrive detector mmse detector detector match filter attain important nonlinear detector freedom approximate technique variational treatment statistical find observation understood suppose computationally intractable rule discrete assume write omit possible excellent measure use equal denote explicitly specify rest drop accordance decoder detector detector density prior detector detector capable posterior bit ground break wang poor two first bit k come decoder mmse filter zero except k b pz k b essence target approximate derive procedure without heuristic schedule ignore modify denote piece mmse detector prior ignore lemma k derive wang poor remarkable viewpoint assumption mmse filter account wang algorithm systematically possible scheduling scheme end k kb k end k kb l summarize three standard associated introduce schedule section elegant variational obtain detector coincide soft interference mmse filtering store user process parallel schedule pass decoder decoder immediately user parallel since ignore become gauss marginal schedule extract finally obtain k k k common wang poor scheme differ schedule generate store ready pass decode hybrid bring compare optimal parallel decoding assume detector include wang poor framework generalize even albeit choice symbol assumption detector routine salient distribution posterior indeed distinction jointly detector assumption general closely tie replica spread field derive detector code near grant interference scheme eq receive combine model gaussian approximated ratio fed channel iteration proceed link effective scheme describe represent provide decoder derivation traditional viewpoint interference stage ic recursive energy let denote mean utilize b rearrange descent minimize eq define ratio I iteration schedule iteration detector pass decoder remove serial updating robust channel plus iteratively em multi update demonstrate solid theoretical foundation grant reveal suboptimal en discrete scheduling pt b l lm l k end kb k end l l l lm end b summarize three version standard discrete highlight major characteristic sequential schedule obtain serial update inner set scheduling send prior schedule inefficient serial need scheme propose simplification sequential discrete inner iteration serial
implement sampling intensive involve auxiliary approach recent novel implementation possibly base small restrict easily extension laplace compute al routine correct laplace approximation laplace copula generalize laplace approximate density estimation gaussian sensible flexible contain attractive generalization copula replace notable exception et discuss marginal likelihood laplace sampling estimator copula framework consider method consider probit datum section conclude describe across whole example simulate continuous give distribution uniform transform variable due correlation background copula song obtain obtain ordinary laplace inverse approximation copula approximation extension gaussian copula write partial derivative decompose c correlation posterior zero one fix modal note modal value unconditional maintain change unconditional operation correct approximation ai correlation ensure compute copula dimensional suppose want expectation approximation numerical exact posterior independent seem preferable normal base sample mcmc estimate kernel et al multivariate estimation clearly fairly situation posterior component estimate correlation matrix van similar di al implementation speak gaussian copula approximation approximation previously degree density scale freedom j k copula simulation output formula freedom let identically r assume numerically grid et find laplace bridge improve bridge single want likelihood yy normalize choice choice adjustment account also involve implement iterative bridge successively bridge estimator approximation determine approximation copula gaussian copula density estimate simulate copula normalize easily assess skew distribution convenient use accommodate skewness heavy tail simulate calculate consider dimensional multivariate degree skewness obviously choose way skew control skewness give skewness give skewness follow vector give simulate heavy consider laplace volume van simulation inverse hessian simulation copula cl tc bridge copula replication volume ellipsoid al bridge copula estimate marginal package core development team simulation value replicate deviation replicate single replicate de estimate value true conclusion copula respective simulation higher perhaps intuitively increase skewness bridge well bridge laplace small work well perhaps surprising freedom choose include integer later example consider likelihood binary consider variant bridge variability laplace copula choose maximum choose nearly generally inferior copula concern state present scenario indicator probit et analyze approach notation index index consider arise variable formulation simple probit effect covariate covariate often intercept w table follow total q ig update metropolis update gibbs step et interest et al reference latent effectively variable apply integrate yy simulate density accurate context use term different block discussion handle conditional see accurate run similar require code cl marginal approximation effort computational negligible whereas first table replicate try inferior copula conjunction situation could copula scheme grouping copula block potentially attractive great allow become increasingly common fast like laplace approximation whole method usually improve variant believe method conjunction journal american association cm multivariate elliptical bayesian chain true pt constant compute health policy quantitative chapter pt true cm true asymptotic l simulate normalizing constant hill use hierarchical default technical available http www edu pdf cm reversible jump markov computation determination probit available http com pt diagnostic improve parametrization bayesian true apply logistic york cm marginal formula cm multivariate concept cm laplacian bayesian true simulation laplace american simulating via identity exploration pt cm approximate high integration advanced pt development team r environment statistical computing http project markov chain pp newton integrate j pp university cm technique computationally demand van pt bayesian p gaussian copula cm l posterior moment pt york l cl cl tc integrate report ccc cl tc cl tc lb cl cl lb apply integrate multivariate degree extreme skewness c cl tc lb cl cl tc lb l cl cl tc lb skew density freedom zero moderate extreme skewness log l cl tc lb cl tc lb birth predictor age year weight indicator indicator white history visit approximation likelihood c cl tc lb cl gs log likelihood interaction additive covariate subset ccc logistic cl tc cl cl tc lb gs rl gp patient gp log likelihood cl copula david mm activity conventional problem calculation examine specific fit difficult laplace relate copula bridge effect accuracy well capture
dynamic model kalman wishart variate mutually uncorrelated variate denote typically return exchange financial interest non definite density volatility volatility evolution al allow fast forecasting stem work experience use west therein propose aim high necessary dimensional yet need law signal rather estimate resort g volatility paper adopt estimation desirable analysis facilitate fast indeed volatility set restrictive correlation matrix later paper explain clearly flexible covariance stochastic discount decomposition beta appendix ia tt evolution accommodate accommodate convolution beta conjugate generalization pm know compare facilitate resort remain organize algorithm volatility volatility appendix proof section wishart freedom notation exponent trace generalize wishart square scalar define propose wishart clearly generalization wishart wishart write next expectation property reflect motivate estimator equal wishart estimator desire estimator reduce expectation wishart wishart eq see density generalize wishart wishart freedom reduce wishart terminology cause confusion generalization wishart generalization wishart integer denote freedom ia integer let decomposition evolution gaussian generalize wishart limit covariance value limit definite prior know generalize relevant model page degree chapter next approximate distribution forecast pm forecast matrix infinity p wishart densitie ks argument evolution reduce dimensionality procedure consuming mean grid application suffice exchange vs vs vs daily frequency van return return use criterion differ set large range return log likelihood posterior estimate mse I log acceptable except even indicate exchange rate figure confirm correlation time observe correlate time theorem implement r procedure likelihood take minute run pc generalization wishart wishart walk plus correlation methodology easily log volatility important forward volatility various model availability efficient comparison procedure require case volatility volatility follow wishart combine volatility develop attractive computational procedure aim address volatility determinant jacobian jx x x sx pa iw pn iw pn iw pn x since proof proceed note range sure factor normalize role sx ax pn somewhat pm pg sc nz jacobian h h immediate beta matrix definite satisfy lemma give convergent suffice monotonic tr prove pr p monotonicity constant tp pp proof root inductive evolution py py proceeding theorem lemma matrix
eq q rx rx px px rx px modify behave like old pair visit unchanged result execute state visit yield construction tx tx tx exploit k get slightly fail fail action function execute mdp dominate l h transformation tx imply hold imply inequality similarly dp update carry dp initially tx tx iy tx assumption value ks mdp denote infinite trajectory step truncation reward fix along agent nonnegative r q hold trajectory expect follow tell modify count estimate consequently prof statement sl action encounter trajectory occur pair trajectory mdp discount total receive agent step h converge start hold mdp requirement unknown mdp approximate model visit least close mdp identical identical unknown lemma eq pair unknown along apply ensure behave probability furthermore preserve follow sl step suppose happen become r q x truncation proof shall follow sl proceed series lemma pair visit estimate rx px px statement already let visit times martingale exclude side minor ks sl lemma parameter mdps mdps px rx px x v x rx rx px rx px old perform visit modification sl modify stop update execute mdp identical modify preserve modified execute mdp q probability tx dominate omit equivalent tx imply assumption term lead r dp carry proceed dp initially x q tx r iy tx tx tx q r tx apply x truncation furthermore discount trajectory h fix agent reward v r assumption r small inequality relation hold expect mdp identical state lemma least mdp state mdp identical state encounter start let surely requirement value denote mdp let pair consider visit time lemma mdp identical action pair unknown encounter along get original behave modify probability conclude line sl high happen time become near x r truncation decrease value abstract loop phase thm thm definition remark part example requirement consequence exploitation reinforcement face role advance integrate evidence robust rl art long rl exploration central problem process mdp greedy boltzmann poor formal review method combine source advanced include proof find polynomial comparison benchmark review process basis numerous extension partially observable factored state rx discount shall nonnegative bound policy agent mdp maximize expect value discount total state action short greedy value action optimal bellman reinforcement model mdp collect information interact little spend without know reward balance acquire agent concentrate rl exploration mdp markovian armed greedy action work approximation select action nonzero visit suitable path q learn nonzero function schedule boltzmann exploration follow action greedy unfortunately greedy may scale exponentially state boost trick visited often estimate become low thus try rarely visit still explore optimistic value sometimes e g exploration give justification optimistic value apparent disadvantage estimation long may parameterized collect successive calculate principle approximate computation exploration infeasible demand simplify interval algorithm assume state interval agent choose high initially interval gradually calculate avoid confusion refer give direct exploitation usually immediate reward list agent act intrinsic force use effectively need advantage switch set furthermore converge estimation greedy bellman polynomial policy estimation real probability max maintain assume state update make iteration naive combination save boost high propagate move correspond may lower boost high boost identical max reward tell model reward precision employ experience soon accurate extra reward understood exploration agent get b tx like tx form method ad hoc estimation valuable slight modification x near several benchmark benchmark change setting format benchmark agent always succeed chance yield reward consist six payoff play armed bandit payoff choose win reward parameter four search complete record confidence interval table method max another start position goal corner move get reach get learn run algorithm collect reward meet challenge rule summarize table task goal many run l greedy k next mdps gets ahead loop arrange shape complete loop combination action consist goal whenever reach
qualitative rd ib acquire ib modification area analysis text mining show concern rd paper extend analogous ib focus attention statistical generalize shannon g physics find http cat aspect utilize rd statistic rd employ originally statistic possess form theory briefly introduce ib alphabet rd encounter rd statistic utilize compression rd eq denote problem engineer partition induce expectation rd specify nature compressed rd specify ib introduce need specifically ib follow quantification respect ib ib condition discuss un regime qx qx qx entropy entropy versa interest desirable express divergence possess form parameterization derive ib seminal minimization simultaneous minimization describe l q rd ib extensive statistic inherently entropy k thus logarithm exponential define govern analogy geometric salient employ consequence logarithm eq dual joint obeys eq relate parameterization parameterization upon total markov yield q rule relation term express generalized formulate theory employ yield p introduce subtract x p x numerator tradeoff evaluate alphabet x statistic valid free total probability generalize k effective distortion relate add qx x positivity positivity employ multiplying x f set minima iteration proof present herein constraint q x x iii b paper minimization define f qx employ show minimizer
thus sample satisfie assume independent predictor covariate exclusive derivative twice logit probit denote among special attention rich chapter regression possible regular regularity cox hold obtain log dx r regularity imply ik h u obtain maximization al diag diag diag fisher orthogonality estimator information q rr obtain bias expression correct follow likelihood adopt moment log likelihood derivative derivative derivative typical bias mle bias cox write need obtain algebra arrive k form th element diagonal element one order bias write bias combine previous become bias coefficient weight express quantity model generalize et ii due nonlinearity vanish linear bias correct estimator normality k aa diagonal fisher evaluate modify original score function beta substitution yield modify asymptotically aa diagonal inverse matrix estimation scheme random index write application bootstrap consist obtain number extract classified perform bootstrap shall whereas version bias respect replace true parametric nonparametric eq bootstrap replication bb possible b version arrive bias deal regression need minor modification sample save bias call resample index describe bootstrap replication coefficient bias correct estimate method resample result valuable namely rely sharp estimate response true parameter moreover et expansion derivative side expression q page asymptotic page matrix bias estimator formulae lastly bias bias last analogously able define estimator bias vector bootstrap bootstrap bias correct link derivative may interested normal htb probit link probit htb cccc commonly arise beta model regression beta dispersion covariate far detail score matrix bias beta row define fisher calculation agree expression score fisher move vanish thus agree find al equation vanish easy al bias parameter vary row covariate fisher vector correction score estimator bias page define consider turn e page define matrix bias give q q page generalize let regression generalize beta remain model give case vector write modify q bias follow page section carlo correct version logit nonlinear value x normal hold throughout replication perform software cox variance mse variance cp mse true cp mse mse mse cox bias mse bias mse bias mse cp mse mse mse mse cp mse mse mse cp mse true mse mle cox mse mse bias bias cp mse mse mse cp cp mse mse cp mse mse cp mse mse correct version cox version bootstrap replication bootstrap size present want taylor expansion one present simulation sample begin bias initially bias bias big original moreover bias mse phenomenon occur et problem small correction correct though bias mse intensive lastly mse mse show satisfactory implement assumption size cox correction poor correction scheme one mse mle distant satisfactory parametric correction base bias method maximum estimator bias general good mse satisfactory overall bootstrap also correction scheme work scheme bias mse mle mse observe performance bootstrap satisfactory nonparametric bootstrap size initially mse performance also bootstrap similar performance regard table bias estimator parametric mse method small bias mse summarize bootstrap nonparametric behavior sample good mse size nonparametric bad performance maximum likelihood table bad bias consider mse consistently mse bootstrap considerably mse considerably mle sample improve second set goal first performance estimator behavior another usage logit link nonlinear identity value take ix nonlinear replication replication linearize mle linearize analogously model mle cox mse mse mse cox p variance variance mse mse mse model parameter bias mse mse variance bias mse present respect mse parametric bootstrap consider bootstrap nonlinear compare parametric estimator except considerably nonlinear model hence I bootstrap linearize mse present mse achieve consider nonparametric well mle worse good mle similar compare bootstrap nonparametric modelling modelling linearize mse achieve bootstrap estimator mle probably size bootstrap good follow nonlinear model prefer linearize present linear dispersion source want proportion convert fractional two explanatory proportion second temperature al illustration beta correction seen intercept different situation measure temperature include intercept ccc value st htb cox np htb logit relate relate newton bfgs instance correct section write look see normality significance value statistic quantile conclude nan reject present correct version adjust version correct cox correct correct np estimate correct maximum version difference correct estimate nevertheless correct one beta use develop cox bias formulae cox formulae estimating bias simulation beta regression model nonlinear dispersion superiority bootstrap regard far advantage intensive method check nonlinear linearize preferred present illustrate usefulness dispersion grateful support author derivative page give contain equation check appendix incorrect put minor
express component shape hypercube elliptical form elliptical unimodal uniformity definition give third theorem enhance flow result symmetry visualize mathematically symmetry depict elliptical unimodal role definition unimodal elliptical general researcher unimodal active research alternative elliptical elliptical unimodal density moment conversely decomposable elliptical unimodal subsection represent density summarize dimensional decomposable volume elliptical unimodal via weighted gaussian component create parametric analysis need know underlie require elliptical unimodal robust furthermore design automatically improve kernel cluster relate give therefore area component improvement density mcmc particle filtering example particle filter particle fit particle fit mixture density weight determine proposal mcmc proposal density mix acceptance step name denote rotation play lemma acknowledgement ph mathematic would express kind helpful research elliptical unimodal estimation probability generalize elliptical unimodal density derive unimodal density demonstrate develop theory paragraph express form say decomposable intuitively peak decomposable unimodal symmetric unimodal second word moment weight component density logistic author possibility either multimodal contribute front generalize space equivalent elliptical unimodal multivariate laplace via gaussian density application decade cluster research likely widely survey date status class parametric analytical mixture parametrize density detail reversible jump markov approach parametric popular tool algorithm spherical elliptical assign sample distance variation drawback elliptical cluster near cluster evaluate suitable analysis unnecessary assumption require cluster elliptical unimodal limitation probably perform ideally shape cluster elliptical unimodal however approximate elliptical unimodal clustering via volume instead allocation exist cluster recent volume originally outlier intensive often achieve algorithmic outline heuristic via volume cluster cluster determinant minimize volume another possible estimation estimation derivation bandwidth become
sequence usage markovian traffic author assume traffic statistic priori secondary mac transition secondary primary channel make receiver sequel optimize capability protocol channel synchronization information superiority protocol receiver access channel upon second scenario assume receiver slot problem optimal algorithm balance exploitation remain inspire recent construct view phase protocol transition primary consist channel slot structure channel refer slot index statistic state channel channel diagram single markov figure markov chain unknown priori secondary secondary channel channel slot secondary choose free two special case version secondary receiver channel capable channel assumption decode receiver behind sense receiver diversity secondary conceptually cognitive mac protocol decompose stage secondary channel receiver decide primary channel estimate receiver synchronization synchronization require dedicated channel throughput synchronization overhead successful send stage empty channel spectrum false alarm characterize alarm detector result interference detection paper state secondary receiver access channel free primary section secondary time slot receiver include receiver channel slot sense history channel receiver access receiver decide access secondary channel free channel explain receiver include transmission slot fail receive back receive forward receiver fa j md fa md fa I share perfect sensing belief observation differ transmission succeed slot failure receiver compute channel secondary priori secondary channel hmm describe probability hmm primary channel begin slot keep track metric free transition ij secondary send receiver update successful transmission order channel access begin slot propose use sense exploitation observation hand absence belief greedy conjecture optimality work journal analytical throughput scenario delay receiver decide channel access slot steady first summation one represent high throughput final order assume channel free become strategy assume secondary channel free compare make probability secondary user access scenario observable instantaneous reward already information new strategy strategy maximize per slot throughput already know problem medium relax probability secondary receiver add another blind appropriate inspire et armed strategy begin primary channel continuously period get assign slot channel continuously period transition probability estimate secondary decide secondary receive back receiver eq successfully receiver update channel detailed channel state armed scenario lack dedicate cognitive follow apply initial avoid ensure begin slot secondary receiver decide access j jx channel choose access channel receiver receiver successfully receive number channel spectrum usage primary assume remain unchanged randomly plot discount index report perfect sense throughput slot blind achievable offline throughput mac strategy throughput capability apparent high throughput offline describe high steady illustrate throughput probability channel secondary achievable throughput converge asymptotically achievable expense strategy upper known tradeoff learn overhead final achievable throughput clearly intuitive block steady achievable throughput
sample monte generally forward simulator formulation rich repeat call forward solver constitute impractical distribution multi modal give apparent computationally critical efficiently across attempt parallel present paper nonparametric independent solver parametric parametric superposition various cardinality treat unknown infer rise variability sampler random name particle simple importance resample algorithm directly rest solver operate successively fine solver operate update fine posterior update extremely possess purpose heat interval spatially field temperature profile impose temperature measurement noise identify observation location pde hold relate harmonic variability latter constant within continuity necessarily consider consist seem plausible readily imagine parametric polynomial high continuity uniqueness measurement though solution possibility detect occur length much generally analyst negligible comparable long identify unknown noise engineering field variability greatly multiscale property research devote scalable black capturing fluctuation assume scale various general assumption limit applicability framework since impossible prescribed input scale variation problem example apart estimate measurement response etc limit introduce lot uncertainty variation field propose paradigm solver spatial accounting process produce emphasis specification implication efficient determination discuss prediction finally capabilitie numerical central build utilize noisy order spatial arise point furthermore important confidence adopt formulation differ denote basic likelihood likelihood formulation commonly issue formulation manner analyst frequent inherently context mathematically analyst insight physical prior likelihood summarize bayesian possibility whose plausibility quantify uncertainty solver box solver solver field sensor without discretization domain resolution operate upon fine solver view etc medium fine variability operate grid fine forward come expense hour fidelity solver identification call analysis perform choice piece however message expense quantify decision project propose framework forward solver operate solver couple analog introduce early propose use solver piece expensive fine solver time fine sub propose emphasis sub determination mapping discretize differential evaluation field resolution treat box experimentally deviation salient physics error quantify solver frequently correspond phenomenon couple solver rise simple dependence sensor readily variance conjugate adopt variance equation simplify note explicit make datum mean equally plausible observation likely imply bias perhaps sensor collect suited quantify posteriori measure plausibility restrict variability critical involve discretize forward solver use element offer pose difficulty problematic several way variability large solver level amount lead produce value nevertheless perfectly predictive variability small solver discover order convolution stationary noise determine since version expressive ability improvement process correspond view radial overcomplete representation great robustness presence flexibility matching structure parameterization scale variability material isotropic imply concentrated fluctuation large slow center kernel form wavelet wavelet wavelet multiscale cosine wide context offer allow principle assumption hence I interpretation adopt approximation equation resolution forward illustrate section need scalar coupling elastic modulus precision kernel scale accordance paradigm quantify knowledge analyst process specific reflect exclude model exploit structural hierarchical robustness unknown priori absence sparse advantageous also truncate poisson computer define allow representation assess support detail hyper great flexibility e integrate perhaps control scale variability prior absence exist ad hoc make framework solution plausibility quantify unify involve adopt select special invariant rescaling furthermore offer small order gamma express lead prior multivariate control spread integrated kernel location I volume incorporate complete assume example equation operate posterior correspond posterior posterior numerical term equation quantify validation absence observation use conjugate adopt gamma context aforementioned support k density solver appear imply mapping solver fine fluctuation spatially mode small posterior increasingly solver refinement exploit accuracy posterior determine dimensional black box solver involve consume determine also possible call expensive fine solver analytically reason monte infer traditionally technique carry build asymptotically target appropriately transition rate lot call solver target might constitute impractical infeasible operate dimensional perhaps mcmc hierarchical resolution several mcmc solver operate need fine significantly mcmc would hence way inference distribution mcmc distribution intractable utilize commonly resample mechanism particle think state proportional respective particle sequentially particle location hence weight dirac furthermore integrable discuss worth multiscale scheme inference make refine elaborate forward solver use possible forward solver directly utilize inference solver reduce coarse fine scales inference solver increasingly fine notation artificial coincide forward demonstrate process forward successive inferential auxiliary reciprocal temperature trivially recover bridge gap provide update solver start trivially draw goal gradually update importance location order various computationally section smc htp p effective size population prescribe threshold current I sense stop distribution employ aim small multiplying weight size ess table degeneracy quantify degeneracy exceeds specify resample resample even multinomial resample often problem examine critical component perturbation kernel determine although freedom distinguish posterior intermediate equation live dimension must involve dimensional proposal reversible add expansion invariance constraint mix fast require solver desirable minimize implementation closely spaced understood computational efficiency desirable intermediate significantly automatic determination evolution particle readily several processor distribution readily update explain solver fine computationally solver go respective solver engine advantageous make readily quantify credible interval posterior unknown field exceed approximation credible guide refinement traditionally error norm spatially variance serve select make resource easily understand significantly need versa impossible priori implementation extend automatically number need ess table favorable next hand smc htp adaptive smc w w ess ess stop propose early critical smc enter equation exploration employ move dimension proposal representation unchanged dimensional proposal ratio purpose proposal cardinality reversible mcmc formulation death proposal dimension paragraph user reverse dim dim proposal u similarly reverse dimension decrease provide reversible death simplify result propose birth death c constant equal death move birth move amplitude equal iteration equation draw consist obvious jacobian split move correspond splitting merge birth death select obvious move kernel ensure acceptance merge move require meet uniformly pair aforementioned kernel remove new associate ensure split ensure kernel equation way achieve scalar draw uniform compatibility specify x xx well scalar new kernel determine ensure compatibility well dimension algebra jacobian transformation remain proposal scale location equation candidate move birth p merge perturb perturb positivity perturb mcmc formula variance proposal adaptively respective markov past property issue framework additional mathematical simulation physical serve engineering ultimately presence source uncertainty infer accordance predict specify py forward predictive mixture likelihood weight simulation readily mechanic von stress modulus ratio field interest examine yield spatially determine elastic material square two stress solver order operate consider grid yield stress alone deviation detail spatially inaccurate solver lead role approximation fine efficiently expensive scheme adaptive particle used section equation stress vary nonlinear uniform mesh follow boundary along coordinate record result record contaminate order record stress course inconsistent geometry variability play critical property understand construction resolution solve operate solely adaptive smc result sequence auxiliary construct exhibit similarity difference
polynomial sum aim express polynomial power sum equivalence fast algorithm recover proof theorem satisfy generalize call simply uncorrelated singleton usually term sum compressed order term implement present compressed formula involve increase rapidly formula construct generalize deal whose moment therefore satisfy integer q q result follow order partition polynomial straightforward us product uncorrelated polynomial sufficient express term summarize polynomial correspond replace occurrence step I build fast notion moment multivariate tool basic notation generalize multivariate length element write length non support therefore n n partition partition empty element build partition replace example partition type give summary notation gm gm gm multiplicative indeed generic equivalence q r joint compound length partition replace work uncorrelated let gm gm equivalence observe since replace recall give block cardinality block n n term replace gm gm correspond partition recall statistic equivalent term sum hand satisfying generalize uncorrelated uncorrelated implement result generalize generalization union disjoint union union respectively n behave filter distinct equivalence refinement lattice partition sum hand split immediately different rewrite assume calculation express uncorrelated respectively respectively respectively obtain replace product rewrite compare equivalence express follow equivalence evaluate sum polynomial mean generalization equivalence replace occurrence table comparison computational time different package refer procedure http fisher david nb package devote multivariate third package name name platform good r statistic multivariate statistic depend parameter whole well multivariate mean mean procedure c evident realize package available web fast table release mac os mac release theorem theorem generate statistic exist symbolic classical rule number polynomial connection variable compound poisson symbolic area symbolic computational paper statistic rewrite estimator go treat language main reference accurate reference investigate allow multivariate price cost generate improvement ask amount datum gaussian population challenge procedure involve algebraic symbolic language application generate compute classical calculus probabilistic aspect polynomial symbol operator symbolic light factorial come symmetric unbiased express sample recall notion suitable r via compound poisson speed recall symbolic device largely multivariate compound poisson summarize execute ghz ram symbolic aim recall terminology useful handle detail formally consist alphabet whose name integral zero evaluation polynomial take classical choose linear functional integer partition integer property extensively interesting agreement product feature calculus new auxiliary symbolic replace prove contrary happen dot product symbolic via law dot product parallelism law dot composition power compositional inverse satisfy fundamental equivalence go power inverse singleton virtue equivalence power factorial account well expression therefore factorial factorial
experiment expect quantum action greedy select algorithm inspire measurement denote fs cell action eigen leave right initialize uniformly plot performance td td horizontal axis learning require correspondingly effective computer quantum design quantum computer explore td begin fast balance exploration easy tune parallelism theoretical accord theoretical great potential quantum quantum exist rl rate illustrate learn alpha give figure give learn explore converge step goal explore learn slowly compare td action determine update reward exploration exploitation relate theoretical side machine intelligence development subject essence computation likely evolve influence applicability near technical simulated demonstrate superiority unknown environment progress technology via cavity physical need hadamard easy computation system simultaneously effectively robot task quantum rapidly birth computation influence appropriately way quantum artificial intelligence advantage probably angle intelligence thank anonymous dr associate constructive comment suggestion manuscript greatly improve dr helpful environment novel quantum reinforcement combine quantum reinforcement learn rl superposition principle parallelism introduce eigen quantum superposition eigen eigen quantum quantum measurement determine amplitude update characteristic exploitation exploration exploitation amplitude parallelism several superiority intelligence quantum reinforcement superposition amplitude classify supervise rl feedback input pair map input unsupervise rl use name evaluate mapping interaction trial intelligence powerful difficult one exploration contribute lot balance trying previously advantage complex curse dimensionality action huge parameter exponentially abstraction decomposition explore rl programming speed different rl rl som adaptation q fuzzy large action space continuous implement practice attempt satisfactory explore difficulty rl quantum theory reinforcement learn processing develop field computation efficiently speedup use speedup classical database search implementation quantum explore quantum quantum artificial pure implementation parallelization fuzzy logic control fuzzy quantum evolutionary evolutionary combinatorial asymmetric recently dynamic taking inspire quantum characteristic algorithm traditional computer development relate research area quantum computer superposition quantum parallelism formal quantum reinforcement framework obtain exploration rl relate contain description quantum iii quantum reinforcement introduce systematically quantum exploration achieve algorithm iv balance experiment superiority vi relate future conclude briefly review reinforcement reinforcement base finite state mdp accord mdp know history compose successive decision history strategy rl divide value observe receive reflect action change choose learn say discount maximize discount reward state bellman action bellman stand action rate update widely rl rl bit concept basic denote bit besides superposition complex phenomenon difference computation quantum atom spin physical basis indicate choose two observable quantum system directly superposition state determine amplitude represent satisfy quantum fundamental superposition act superposition basis simultaneously parallelism powerful ability parallelism output superposition eq joint first second evaluate value simultaneously oracle superposition states quantum learn speedup product complex coefficient represent occurrence superposition superposition integer unitary simultaneously state superposition circuit execute parallelism make tradeoff need parallelism circuit exploit quantum space effectively classical quantum gate gate gate quantum complete simple gate quantum gate introduce quantum logic reinforcement discussion ref hadamard gate represent hadamard gate equally superposition similarly state e amplitude probabilistic algorithm phase analog mechanic quantum gate gate operation important element carry decision quantum implement quantum also policy reward reinforcement remarkably traditional rl intrinsic representation parallelism eigen state eigen observable eigenvector orthonormal basis hilbert denote orthogonal basis eigen eigen remark observable orthogonal eigen eigen eigen state mechanic general system dirac representation hilbert inner quantum superposition quantum quantum reinforcement learn lie state correspond lie superposition linear superposition worth note goal quantum characteristic learn fact action state action convenience practical eigen eigen action reinforcement arbitrary action expand eigen state traditional rl traditional sum action still eigen state exclusive eigen analysis build concept ii convenience processing express number represent eigen action word eigen inequality rl representation eigen eigen action amplitude satisfy q discount mapping probability action action q accord get reward method action theoretically fundamental measure result balance exploitation natural action detail point every expand orthogonal complete accord parallelism unitary operation simultaneously state td update rule mean traditional rl value rl state function simulate computer quantum execute measure relate probability superposition eigen interact quantum probability amplitude iteration equally superposition eigen easily hadamard sequence initial represent eigen action irrespective construct iteration combine unitary external obviously orthogonal act trivially act vector hyperplane orthogonal quantum effectively justify action similarly preserve unitary consider act span initial express iteration fig plane axis reflect fig respective reward stepsize action execute amplitude time value open ref time obviously action far select amplitude amplitude subroutine want emphasize database search aim appropriately eigen essential demonstrate algorithm observe eigen execute give reward td time device firstly eigen state eigen respective eigen action store memory loss since finally simple classical may etc superposition principle quantum parallelism eigen quantum occurrence probability every eigen reward whole action superposition tradeoff exploitation algorithm effective algorithm novel computation method parallelism quantum computer major proposition balance follow obvious show search become difference td update td prove converge converge function follow hold verify iterative many prove main exploration quantum measurement carry mean update learn rl affect quantum correct decision repeat algorithm acquire system policy implement quantum give quantum make repeat computation several suppose converge repeat repeating time quantum due powerful computing capability system current focus mention develop simulate theory argue complete dynamic compute policy bellman optimality policy rl lie probabilistic amplitude still simulate quantum parallelism
dominate represent posterior conditional entropy approximately imply bayes say satisfying versus cardinality characterize size advance gain whether complement unbalanced property prove complement entropy binomial parameter take include define cardinality accord kolmogorov efficiency sum vanish display follow may let grow efficient increase rate respect remain small efficiency property vc display efficiency linearly introduce fundamental concept combinatorial space finite domain quantify compute standard boolean many interesting examine search instance finite hypothesis e accounting algorithm include acquire information algorithm area retrieval object application depend remain consist estimate number technique element corresponding noting notion use count shannon play proceed next probability generate probabilistic call function extensively graph coin success application distribution equal q probability collection another construct matrix coin denote probability process column element less element denote whose column claim conditional knowing probability enable computation henceforth resort independently draw use accord distribution correspond clear lemma independently trial ks get function check decrease right give negative guarantee tend infinity eq kb kb statement theorems section therefore f f kp p possess behavior cardinality draw respectively brevity suffice state condition class union cardinality equal n disjoint hand equal tend clear critical tend probability probable continue theorem xt ratio class increase satisfie ratio denominator theorem probability function numerator approximate integral side cumulative ix description cardinality equal eq let nn k letting n op substitute integer cardinality uniform proceed represent dd dd event submatrix whose row define consider event submatrix row index least fulfil continue theorem value since exponentially take approximated eq notation binomial distribution approximate integral factor equal numerator nn denominator tend q substitute back approximated identical theorem numerator n rs denominator tend yield statement cardinality satisfy property condition class lemma suffice equal hence binomial become denominator tends therefore tend yield first q probability denominator multiply factor hence author thank dr remark mm minus plus minus plus minus plus plus em minus minus em use letter letter letter letter letter letter letter article package ps proposition corollary mm z cm center ac kolmogorov argue shannon entropy paper pose follow description cost information question kolmogorov concept term input evaluate application measure algorithmic shannon setting object represent variable necessary object string realization often use english text finite universal representation kolmogorov mean book probability hand rapidly distance page lead kolmogorov alternate algorithmic notion binary string later kolmogorov complexity combinatorial kolmogorov take object entropy cardinality know entropy eliminate bit projection restriction kolmogorov eq view pair eq kolmogorov clearly instance vector make information recognition base contain side implicit family set repeat collection I subset property target partial effectively search quantify see reference possible object represent know contain input kolmogorov target restrict collection useful collection ignore f g additional equally know implicit collection structural kolmogorov combinatorial build static object shannon principle concern underlying object base approach pattern recognition contribution introduction framework secondly application framework class maximally informative complexity compute use information value different measure definition serve universal reference type kind function finite via class relate term dimension interest investigate information property stem binary theory deal computing sometimes inductive bias unknown vc dimension sample learn target infinite quantify side answer treat problem computing value uniform source generality take object applicable classifier tree boolean formulae applicable biology dna rna interested amount specify structure sequence rank example consider dna rna protein sequence sort activity active top task affinity bind chemical specify require specify activity applicable level complexity later information consider compare consider property source dimension efficiency efficiency additional measure set may leave stochastic description complexity denote define description positive real bit take cardinality great certain property element knowledge element bit describe string precisely side alternatively gain property complexity increase set complement description x c clearly follow proportion element plus instance opposite may grow either question whether gain hold scenario take set amount singleton impose description equal distinct force e intersection may empty leave great price introduce represent description bit scenario kolmogorov scenario follow unknown contain differ side equal satisfied entropy still leave great less input strictly average possible set fact scenario description side specific long mention know kind continue introduce concept measure information cost unknown property amongst notion formalize next resemble width state free choose structure limited description time consider input width width eq compute particular empty low rl equal form replace notion width approximation rich simple kolmogorov nf perhaps basic quantity consider vc define vc width correspond h notion cost possible description efficiency consider apply denote space set consist class possible property rate write set statement string set contain describe property aim information efficiency previous satisfy sometimes write section
table carlo repetition row square proportion scad lasso number add criterion evaluate median test differs fit compare van sis sis var sis variable oppose report three sis encourage regularization make mean error aic sis good sis value use var include consider var van sis var sis var sis prop median final aic iterate version van sis var sis van stage criterion fast choose scad non iterate sis iterate method van extremely ht van sis van var sis lasso prop model train aic test van sis van sis prop prop model aic response conditional extra cross penalty magnitude predictor sis sis van extremely always method continue nb diagnosis microarray reliably predict disease comprehensive gene responsible microarray quality free survival patient diagnosis exclude positive negative sis reduce competitive whenever fold randomly subject positive negative test result report design gender goal design performance classifier non outlier array train test sis half method sis var sis year gender testing compare lasso especially year giving indicate parsimonious var probe sis sis var lasso x x g x g probe sis var sis lasso x x child cancer report artificial develop blue one nb profile diagnosis diagnostic category category filter profile online http microarray supplement include apply linear reduce lasso directly appropriate tuning error set var new pt wu proposition remark example variable selection characterize problem scientific discovery simple correlation possess sure screening condition sure independent needed extend explicit residual pseudo include new improve high use rate screen two mm mathematic subject classification phrase technology collect video frequency financial demand progress recent year great common covariate modern problem paragraph dimensionality mathematically infinity rapidly scientific example microarray order hundred number thousand study protein predictor million phenomenon accumulation long reference therein demonstrate popular square include scad selector attract algorithmic reference therein involve challenge algorithmic take demonstrate supervised limited screening moderate sophisticated smoothly absolute deviation scad final utility correspond correlation response regularity surprisingly method independence screen technique sure screen sis iterate independence cover fail instance predictor marginally uncorrelated correlated predictor jointly high important base variable work crucial work residual obvious improved independence feature bioinformatic expression treatment differ statistically select frequently screen justified indicate illustrated marginally jointly test disease molecular mechanism disease feature procedure sis make issue aim find covariate important application outline section logistic fit framework particular compete common hinge I conventional huber loss fit screen default discovery fdr select variant sis methodology reduce fdr partition group incorrectly select model sure screen sis technique let column identically population design matrix relationship parameter fitting pseudo regard marginal feature eq minimize sis framework compute parameter quickly even dimensional problem select sis typically take utilize possess sure method sis sis reduce screen statistic likelihood sis simultaneously refine penalize consideration loss variable sis seek choose five fold example zero commonly smoothly scad concavity mcp scad penalty tail fundamental bias penalization solution scad penalize minimize iteratively quadratic whereas local optimization suggest minimize penalty scad mc motivated differently adaptive take maximum choose approach iteratively never function include oracle non extend cover adaptive lasso possess study stage procedure describe sis lasso scad sis sis break predictor jointly uncorrelate former rank seek overcome joint covariate sis sis employ mcp subset index bivariate attractive feature dimensional statistical computing optimization easily note subtracting difference contribution element approach substitute fit bivariate quadratic fitting conditional relevant variable show encourage solution zero estimate active step least square case allow previously index iteratively index reach prescribe choose take iterated version sis terminate another possibility example estimate explicit residual improvement square show sis difficult correlate low residual chance density write dispersion generalize give vector dispersion conditional response relate dispersion dispersion parameter simplicity throughout dispersion immediate generalize fit fact canonical elegant classification logistic regression particular eq independence quick irrelevant conservative many outline sis theoretical particularly procedure convenient true model corresponding set active inactive respectively assume two sis yield sure appear tend one possess sure screen fewer inactive indeed original sis likelihood scad variable proceed penalization false theoretical sis make condition natural exchangeability inactive likely sis give inactive require version exchangeability satisfy exchangeability level sis prescribe let q splitting last inactive tuple tuple equally likely follow exchangeability condition satisfy comparison report also true dimensionality probability false positive decrease set index true increase natural variant sis apply sis intersection carry first outline separately set penalize true second partitioning sized index require element penalize pseudo selection case variant
significant distribution multinomial hellinger desire monotonic probabilistic reader distance calculate variety closely together manifold another consider achieve densely end connected segment sum approximate along specifically parameterize p fisher divergence hellinger immediate concern available collection intuitively shortest manner approximate manifold compare actual univariate densely mean distance dissimilarity low multi generally purpose find classical take dissimilarity point euclidean centering dissimilarity decomposition unsupervise permit coordinate reveal separation dissimilarity approximate double subtract add back multiplying origin mathematically solve eq vector take eigenvalue consist refer continue distance embed geodesic exact rectangular rectangular effect change laplacian lem linear reduction principal analysis construct give compute assign eq diagonal weight collection small detail lem dimensionality reduction fashion discriminant constrain neighbourhood highlight identical geodesic visualize low dissimilarity family approximate fisher family practical interest parameterization instead scale estimate density mixture normalize density unlike mixture comprise normalize point within estimate entire probable sum large density kde pdf smoothing estimator kernel priori original kernel secondly parameter yield peak generate smooth kde slow calculate dimension area future datum desire density geodesic parametric embed fine realization manifold view embed manifold note matrix use present section consist set density synthetic visualization application density assume manifold parameterization method reconstruct statistical roll manifold know parameterization pdfs utilize manner strictly set roll reconstruct construct euclidean statistical manifold clinical flow presence surface scatter cell pass cell characteristic dimensional marker distinct disease entity clinical multi characteristic thousand analyze clinical interpret clinical two scatter parameter routine clinical flow additional parameter gate exclude cell scatter characteristic clinical flow histogram multidimensional nature may recent show dimensionality document visualization document represent pdfs dimensionality fine document classify different computer expect see group discuss see count article computer pdfs let multinomial utilize frequency multinomial tie kde density priori unnecessary multinomial pdfs commonly use method separate choose domain number wish classify high datum first natural dissimilarity calculate hellinger natural geometric expect embed optimality euclidean separation compare fine document classification e intrinsic estimation sample document dimensional embedding vs potential class accord computational sample pca jointly fine outperform embed ease concern lem multinomial calculate embed lem dissimilarity metric kernel svm comparable embed essentially lem additional embed separation even dimension kernel frequency utilize stress difference l r std std setting I fine maximize linear set validation highlight fine significant diffusion kernel increase however diffusion fine decrease kernel eventually modify assign rate classify divide total sample structure set size fine compare yet fine focus jointly unlabeled use test sample decrease fact already exist embed fine reduce sample analysis one fine diffusion coarse easy highly lead dimension dimension document draw could bring class close entire multinomial dimensionality kernel approximate pdf representative dimension anomalous gain elsewhere perform lead utilize diffusion size region fine illustrated fig performance fine radial function weight visualization purpose e fine kernel lie manifold ease due euclidean many practical ability find separation reconstruct use approximate distance geodesic leibler hellinger information fine tie although method obtain pdfs secondary concern primary dissimilarity pdfs similarly illustrate visualization seem nothing around set utilize fine utilize nn maximize include kernel plan lastly continue fine internet anomaly would offer university analysis thank university help classification implementation partially science university ann mi school department university ann mi edu edu visualization high dimensional straightforward typically task reduce high many many manifold justify geometry similarity use entirely parametric parameterization pdfs low euclidean refer non embed medical knowledge model make observe association cluster predict unlabele belong classification base machine parametric aim govern constraint effectively dimension intrinsic aim introduce perspective problem three field exhibit intrinsic dimension manifold straightforward strategy processing ad hoc dependent good absence suboptimal obtain optimally extract parameter characterize datum interest parametric statistical problem many represent flow cell cell realization parameter marker purpose visualization dimensionality collection form cluster similarity form geometry statistical manifold case face segmentation shape unknown handle focus statistical manifold fine set geodesic approximation fisher purpose classification visualization manifold parametric setting discuss derive statistical manifold information propose alternative euclidean cluster address low euclidean focus necessity perform finish enable linear geometry explicit realization work similar demonstrate originally enough work lee framework segmentation sphere exploit property manifold cosine fisher reduction work differ manifold information account later lie entire geodesic utilize rather consist problem interest describe geometry statistical manifold wish algorithm set draw possibility future geometry tool method theory largely network thorough smooth curve lie manifold coordinate system construct domain situation differentiable coordinate differentiable entire infinitely global coordinate surface roll manifold local system fortunately contain property coordinate level solely coordinate manifold whose q value pdfs set satisfying mapping parameterization parameterization rest manifold manifold distance manifold measure geodesic analogous fisher amount information parameter case fx dx meet whose element fisher information single obtain geodesic univariate density fisher metric like illustrate deriving compute geodesic consider family q fisher omit derivation straight two inner product parameterization define minimum short length arc radius univariate straight hold change geodesic calculus variation univariate distribution calculate determine eq visualization grid lk figure
way appendix early readily theorem employ hoeffding via analogy argument formula regard define eq integer thm conclusion rigorous variable base nominal lemma classical hoeffding inequality lem bound n z z lemma establish chen lem lem integer x I lem number positive integer hz z z imply monotonically increase lem lem lem exist therefore write hence prove lem case set h q show inequality inclusion prove limit define low limit uniqueness upper obvious result inclusion relationship relationship justified follow lem satisfy remainder theorem base prescribed threshold estimate scaling translation typical
penalty term via lasso carry inverse structure penalize cholesky metric roc simulation though acknowledgment author acknowledge support nsf dms nf grant yu thanks california berkeley fellowship comment perhaps simple global consistently comparison inconsistent vanish small occurrence create sensible estimate alternative improve covariance consider eigenvalue shrinkage covariance discriminant analysis filter covariance estimate inverse fall impose precision stem parametrization covariance ensure spectral cholesky translate sparsity sensitive random vector paper constitute attempt aic estimate computationally tractable alternative perform computational fit aic stem entire modern day penalization selection estimate cholesky term validation computationally tractable homotopy regression detail suffer cholesky alternatively precision case impose directly precision penalization computational definite present impose sparsity moreover entire homotopy graphical separate regression time merging regression variance pseudo symmetry approximate quadratic pseudo main advantage parametrization amenable homotopy avoid criterion select regression bic regularization estimate penalize likelihood cholesky sample set simulation commonly star topology spectral precision deviation remarkably comparison cholesky incur positive positive comparison careful alternative remainder follow section likelihood surrogate homotopy short establish coefficient regression emphasize differ one use alternative parametrization extend precision result pseudo non function still yield propose well homotopy counterpart suggest vector dimensional invert matrix method regression j j parameter correspond coefficient expect along diagonal hold pattern estimate graphical obtain example could define rule estimate symmetry must surrogate consistently minimize pseudo advantage precision use weight jk adjust accommodate remainder advantage define fix minimizer solution penalize fix homotopy lar estimate value minimizer alternate advantage drawback precision continuity semi norm replace positive penalty semi definite addition present penalize estimate convenient symmetry symmetry constraint define symmetric explored enforce symmetry within trace estimate write definite satisfied small estimate minimizer diagonal pseudo term parametrization function secondly surrogate matrix vanish weighting equal identity term case approximation grow infinity clearly unconstraine impose symmetry conjunction use estimate point coincide likelihood likelihood penalize well simultaneously interesting quality pseudo set grow efficiency exact likelihood well penalize pseudo grow suffer drawback enforce costly computational slow path unconstraine estimate semi path nearly penalization definite penalize pseudo likelihood positive semi definite entire precision precision useful induce subscript net penalty section semi hence leave future research path estimate advantage reduce optimizer close propose estimate path regularization choice path jj path subtle detail correction make jk jk jk jk throughout current execute remainder homotopy lar enforce regularization issue relate enforce homotopy lar constrain penalize modified force create matrix corresponding element vector row constrain emphasize element thus homotopy trace regularization homotopy lar regressor homotopy denote lar constant plausible regularize many incorporate greatly estimate entire complexity path order entire order solution desire pick criterion freedom path kn kn freedom plus zero upper triangular finish loop look diagonal k exceeds fix result cause solely issue warm simulation matrix become stable reach term covariance selection namely cholesky penalize previously compare different aic bic pick exact likelihood entire roc see semi estimation covariance imply model relatively estimate distribution replication simulate normalize edge show great dependent cholesky entry first meanwhile added ar panel random panel observation auto process tend give order dependency structure family design f randomly matrix appendix sure environment evaluate precision metric precision precision cholesky exact aic bic cholesky respective regularization cholesky exact minimize criterion space absolute exact likelihood matlab criterion sample figure case sample suffer later cholesky comparison reveal loss achieve loss attribute similarity loss pseudo norm precision perform large size perform couple size size term exact somewhat insensitive selection many cause grid miss penalize rapidly case affect seem yield ease reference result table recommend aic smaller penalize aic suitable covariance structure show combination base precision metric ar like metric generate estimate case method operate curve horizontal axis positive incur vertical identify roc roc negative fix vertical false positive curve upper roc false positive horizontal axis vertical solid line corner well roc cholesky selection within panel operate positive horizontal vertical axis solid dash well left corner plot result suggest pick covariance cholesky seem roc cholesky sizes roc use include grid would illustrate path grid path follow thorough mean roc performance selection cholesky method positive false positive incur exception chance contain cholesky attention star cholesky correct positive replication cholesky path select positive wide margin cholesky suffer degradation variable roc curve star positive indicate positive think horizontal insensitive cholesky direct order note guarantee estimate positive somewhat behave
gaussian rbf decomposition covariance distribution kernel sum kernel dimensional last infinite exponential one kernel operator simulation try obtain variable decomposition cm structured feature technique paper assume dimensional sum view define sum bring many say early hull e complement hull iv look v positive impose vector minimization norm q dimensional choice norm path dag dag dag moreover hull finally hilbert absolutely continuous selecting select must optimization simulation essential cauchy schwarz v v general appendix know cm variable supervise moreover x associate entirely appendix optimal ia learn give maximize moreover duality gap two gap flat regular dag know indeed although exponential section complement namely optimal primal condition respect problem value cm extreme point cm fairly constitute contribution large optimization exponentially consider great bring specific grid also factorize sum w save run kernel space primal generic g exist supervise code use preferable v learning kernel ridge add differentiable moreover differentiable b project gradient proportional matrix plus solve ready search kernel need problem condition matrix gap cm set solution obtain satisfied optimal stop gap reach depth dag duality gap check fact know optimal complexity select kernel complexity assume kernel conservative solve kernel compute quadratic sufficient kernel mkl say sparsity thus simplicity hold infinity decrease joint invertible almost hull consistently proposition consequence result mkl overlap hull satisfied correlation correct worth slowly consistently depth currently investigate parametric pattern label sparse number compare hierarchical greedy strategy similar polynomial full replication rotation generate sparse outperform situation perform e sparse problem really help help sparsity cm ccccc ccccc bank bank bank fm bank fm bank nh bank nh rbf bank nm nm rbf rbf fm fm rbf nh rbf nm cm gaussian rbf dimension early greedy context strategy mkl hold report good dataset bank nm bank nm nh dataset dedicate dag number ccccc ccc greedy rbf rbf rbf cm dataset loss greedy generating know slightly bad perform kernel variable selection particular try penalty advantageous inside pyramid match string cm relate obtain relationship cauchy inequality dag parent parent root parent situation v convex dag optimality condition loss convex conjugate example ib learn dual auxiliary lagrangian use simple subject derive duality minimize strict duality gap variational decompose duality pair f I gap entire remove necessary dual problem candidate lead turn relax sufficient goal candidate maxima lead simply derive optimality directional thus v assume finite loss row assumption fast probability generate optimization indeed precise previous selection dual dual close equal bound lead propositions v c w c lead project team sup paris unsupervise large space number penalization euclidean explore induce norm large basis acyclic polynomial variable selection synthetic repository efficiently explore state appropriate regularization enable large space work implicit observation numerous work algorithms many induce lot year work focus efficient solve property case paper bridge try norm indeed space require exactly
equation taylor center x yy come hence continuity fourth derivative implie imply hence conclude form derivative pdf student variable degree beta imply obey student possibility unit pdf rx dm dy sg sx sf sa x r gamma q unit furth theorem reciprocal result distribution obey dispersion introduce dy distribution theorem density smooth circular serial distribution symmetric beta family connection motion dy unit dy pdf inverse gaussian third kind j reduce fx approximately dispersion asymptotic extra wide class distribution fashion derivative unit vanish time detail normal gamma inverse von dispersion j ba ty function uniformly extend et dispersion define model preserve could dispersion dispersion position dispersion function pdfs lebesgue counting measure denote symbol dispersion b density decompose dispersion take dispersion model model gamma reciprocal inverse distribution von dispersion circular dispersion moreover dispersion depend statistic definition satisfie construct log partial analogously twice differentiable satisfie easy approximation dispersion notation mean model even normal simplex dispersion dispersion asymptotic equivalent whose dispersion convergent exponential know student degree say standard normal
dx u many case expansion estimate recommend take approximation derivative making use taylor approximation rule follow truth h fu v u u may example elliptical application concavity concavity quadratic distribution beta decompose integer modify integration b u vs st fu v execution algorithm establish fu fu fu u v r r upper achieve k gap point quadratic binomial binomial poisson hyper obtain subset concavity zero analytic zero determine case small extremely e proceed choose step u st u estimate believe small case adapt adaptive finding maximum prescribe upper apply course conclusion develop scheme absolutely rigorously prescribed involve weight evaluation readily accomplish tight adaptive integration summation x u variance possess chi possess moreover n I unit independent unit n n x nn n mn mn v position lemma algebraic operation b sa reject n n accept reject evident u n z follow accept reject accept generality du r theorem generality coordinate r dr dr dr horizontal axis vertical axis axis iv vi visible boundary express g k theorem make integration locate k visible variable g htbp visible part boundary express make integration side locate boundary completely integration see visible v htbp shall proceed axis write denote arc end refer proof domain proof accomplish lemma sequel satisfy one condition h k tuple g discriminant divide condition root g k b u u k divide quadratic root observe k divided iv imply attempt root branch specific sequel lem visible need lem r r r negative number complete lemma divide visible refer vertical line h p investigate case situation branch express region convex domain tangent visible yield situation arc precede tangent line critical v htbp situation show must arc arc lemma parts np htbp situation htbp situation l np v investigate lemma lemma visible htbp g r p observe arc visible u htbp figure critical arc visible determine l u pp visible determine htbp u ga v figure since completely visible I htbp situation part boundary determine case u km figure make part determined respectively b r htbp km boundary situation consequence slope line slope line note visible boundary respectively r b situation show critical arc part r htbp parts b htbp investigate branch therefore htbp situation must must visible visible v I figure visible boundary visible b v proof pt prescribe previous test average operation truncation testing argument operate function relevant adaptive present area introduction application determine prescribe setting testing specify great extensively study sequential probability suffer several drawback sampling testing deterministic force g prescribed may truncation sample instead third optimal extremely inefficient say third construct testing framework construction weight efficiency overcome limitation test normal feature reduce sampling absolutely without truncation iii prescribe rigorously iv testing consist th stage notation accept stage risk incremental requirement power appropriate purpose readily operate present testing method variance summation area conclusion proof throughout notation denote cosine indicate random parameterize drop introduce zero variance plan distinct z z guarantee accept bound n h accept result determine domain large probability v computation base theorems h u k b b eq np np np n I n I appendix notation respectively last frequently integral form integral exist approximation integral integration decision hypothesis quantification develop method moreover optimization hypothesis test numerical rule quadrature rule integral
drawback point latter expression bayes rhs construct likelihood rao mcmc sampler case mixture approximation later formal level converge parametric virtue converge unfortunately discuss previously mixture describe via occur fix explore label switching posteriori replace permutation component apply justification modification stem rao exploited galaxy estimate marginal simulation permutation already approximation gibbs sampler check simulate average fourth digit agreement point difference unlikely less mode permutation correction control unnecessary galaxy estimation approximation base permutation select subsample keep compute reasonable level keep identity permutation see original permutation mix property convergence indicator compare class marginal preference acknowledgement grateful careful reading suggestion universit paris technology universit paris survey exhibit lastly light elementary offer much wide possibility component challenge inferential many advance study review solely book depth short perspective paradigm probability statement unknown parameter opinion include analysis naturally variable survey construction within paradigm technique sampler carlo setting benchmark introduce include appear component precise derivation multinomial fundamental object prior modelling include distribution interest mixture dominate dominate measure simplex multinomial denote contingency situation contingency homogeneous simulate contingency give histogram contingency independent histogram another mixture distribution observe take modality th speak model medical associate study influential lastly category modality belong suggest sale dominate count integer may distribution integer well normal unknown degree dataset present multimodal feature understand formation environmental modelling particle size concentration particle particle figure day diameter red blue elementary estimator take explicit posterior likelihood use fundamental difficulty deal example latent datum subsample priori inferential behind modeling different exploit device facilitate always associate second identify estimate keep availability variable help draw technique augmentation start involve take operation version analytic consider distribution compute rely variable define allocation simplex subset size instance order partition r derivation write lot inferential bayes allocation construct allocation even consider posterior exact derivation surprising phenomenon take place discrete distribution essence belong family consider likelihood easily dirichlet simplex conjugate eq family observe complete likelihood configuration sum poisson sum view truly multinomial mixture note former previously component uniform eq jj important feature involve simply act statistic require distinct complete distinct propose recurrent efficient component equal one recursively allow straightforward representation constant eq computed product irrelevant computation factor considerable posterior pressure large row table simulate poisson mostly make take impact statistic easily assess happen correspond explain ccc due multinomial conjugate prior observation allocate q since correspond weight partition q allocate applie namely occurrence statistic book apply close extremely modality modality product identifiable beyond switching issue advantage posterior partition make involve pick eliminate statistic middle partition weight likely partition posterior probability eliminate partition apparent face inferential arbitrary truly assess exist nonetheless improper demonstrate ad perspective quite easy argue mixture exchangeable identical posterior component prior must prior explicit first common reference location original term reference express representation use prior component start metropolis diversity carlo drawback identifiability constraint drawback constraint sampler extreme prevent sampler choose fx mixture geometrically ergodic theoretical severe augmentation stage induce nature lead concentrated iteration allocate concentrated move mixture reproduce permutation visit replication mean conjugate prior straightforward implement dirichlet conditionally simulate implementation galaxy radial obvious first histogram one label sampler three isolated leave right component gibbs sampler galaxy dataset evolution note mode contrary perspective target explore fail restrict single intuition derive mixture surface modal check argument extent invariance make argument state perspective article simplicity assignment expect hasting identifiable component normal exist mode posterior value mode nonetheless attractive gibbs visit mode incoherent illustration multimodal identifiable approach mcmc recommend visit likelihood lack mix difficulty individual histogram mode helpful identifiable moreover interpret continuous variance scale gibbs q include simulation conjugate conditional dirichlet inverse illustrate sampler explore variable modality involve two completion beta distribution gibbs sampler figure exhibit predict theory separate simple histogram iteration local behave posterior identifiable difficulty completion chain increase metropolis computable initialization generate gibb priori mixed metropolis unconstraine mean proposal constrain model remove permutation help correspond move realistic freedom implement far alternative introduce difficulty hyperparameter density therefore resort walk hyperparameter performance variance unknown indicate weakly informative gibbs density adequate recover cm green blue particle describe average illustrate mixture c student possible direct
theoretical tree maximize illustration node node appear figure edge tree set red path tree become rise usefulness demonstrate tree describe exploratory brain tree tree component correspondence color gold reveal symmetry h correspondence correspondence type concept support tree correspondence type different color figure tree correspondence bottom correspondence indicate correspondence result population likely pca representation already aspect tree child population show start tree human brain back gold split side back expect mirror image contain main branch consequently imagine vertical axis see tree right brain mirror unlike branch reason symmetry later suggest early relatively remain however consider pc right third structural visible note pc side strong insight correspondence graphic line analog familiar commonly high visualization sometimes score indicate datum eigen pairwise view correspond unlike pc age versus age plot hypothesis pc component correspondence des thick pc result significance add interpret leave child split right split significant close root however look deep tree th split seem contain population appear split back sub population none web also parallel result web give significant result correspondence h correspondence variation function left brain location tree sub correspondence total pc much population correspondence important consequence branch reveal symmetry superior correspondence future population symmetric room improvement rich branching last explore devoted node begin tree common point l brevity use symmetry imply contain maximize whose sum weight subtree q main using since disjoint l follow statement general line claim path nsf grant es nsf dms partially grant partially support grant es e functional recently orient consider population particularly extension idea population structure analog principal algorithm brain research area see introduction recent viewpoint analysis statistical wang extend oriented atom example population movie functional image population standard functional particular whose solution result spirit pca challenge tree work wang appear develop toy tree manual illustrate main idea interesting discover size production structure illustrate human brain collect choose variation tree I branch structure ignore curvature still need branch put later orientation study wang main analytic concept notion devote analysis carefully correspondence component claim brain human range slice image white dimension tree slice colored point gold blue therefore biological simplicity choose sub color indicate brain right front store tree enable colored segment branch consist sequence white slice sphere center indicate radius single colored connectivity tree tree green segment child branch connect parent top initial gold near bottom thin show back patient branch retain branch ignore thin blue binary tree make put way ambiguity terminology image put median sequence fit leave child put compare type correspondence also attractive suggest discussion child perhaps leave would address idea wang idea extend idea originally tree child single let tree toy tree use wang index remain node right child index enable binary basis appropriate distance tree use common notion hamming distance purpose tree
b imply coverage b coverage less sufficiently b n thus establish uniqueness construction subsequence theorem accumulation point prove prove claim choose locate second inspection note elementary n p sa n nh absolute difference n bound follow observe constant elementary inequality note nc central theorem delta real second immediately symmetry theorem theorem claim conjecture theorem corollary criterion exercise lemma theorem theorem summary phone mail ac phone mail department statistics institute mathematical penalize maximum determine symmetric short interval short base length estimator tune possess sparsity interval estimator magnitude sparse smoothly absolute deviation know case show secondary penalize adaptive thresholding soft confidence coverage year increase maximum likelihood square estimator variant estimator fan li regressor hard soft reformulate thresholding distributional likelihood square estimator study literature mostly fu fan li fu bridge estimator tune perform conservative model selection fan scad concentrate case possess possess oracle asymptotic scad estimator general obtain estimator oracle typically picture performance fan li literature establish oracle estimator asymptotic p discussion base square context formulae width thresholde lasso among interval short interval long maximum tuned property formulae interval show estimator magnitude tune case simple penalize show asymptotic plan variance case treat regressor penalize maximum likelihood least identically entail unknown would replace residual regression set thresholding denote maximum ny infeasible use value lasso thresholding shorthand adaptive give infeasible counterpart simple assume whereas natural take account scale let cumulative function obvious operate linear regression thresholding coincide interval form nb c probability due note obvious true interval follow n n assume loss generality shall thresholde denote determine infimum coverage piecewise thresholding let infimum coverage p c repeat convenience every interval point interest coverage infimum attain adaptive interval infimum coverage interval every infimum attain refinement open interval n n n coverage half open interval ii open h n infimum probability interval attain iii probability continuous everywhere trivial coincide coverage infimum minimum furthermore interval soft thresholding simple reasoning two side prescribe coverage probability short interval long standard unique short interval n solution coverage equal coverage h unique equal short n short thresholding symmetric seem short interval phenomenon gain error mirror another coverage soft thresholding short estimator short interval hard long show short interval thresholde soft various interval hard except large base p half base regime parameter distinguished regime tuning perform discussion regime hence large interval order situation similarly note derivation length length infinity estimator tune possess interval result arbitrary tune hard thresholding call light property sometimes argue literature obtain interval converge zero valid interval inf rely estimator smoothly possess consistency tune furthermore limit appropriate moving always concentrate interval stand let interval distributional case finite coverage one construct half h arbitrarily sound compare interval asymptotic phenomenon component bad component show proposition hold smoothly absolute deviation tune possess sparsity argument model coverage limit zero correct unknown interested coverage n n n brevity symmetric generality infimum respect coverage nh nh root freedom divide obtain finite probability determine coverage symmetry standard normal cdf freedom relation open interval follow appendix coverage length n hold theorem remark hold open open thresholding adaptive analogously sa sa sa sa sa n sa h sa sa sa n sa n sa n n sa n sa n sa sa sa sa sa sa n sa n sa n sa interesting inconsistent sequence h correspond subsection q appendix nonnegative c analogous theorem result carry converge respectively short regime hard theorem soft immediately carry unknown expression sample note proposition see coverage consequence lower derivative dp b elementary calculation give imply eq last display together dp less remain rearrange elementary equivalently write follow use proof remain unchanged replace
discretization saddle rare massive computationally strategy miss value indicate sometimes fail explain result proposal skewness optimal proposal pure execution algorithm need approximately execution ten exist pay arithmetic average exceed option narrow two one respectively trial stage miss table improve finally option option although option ed mc ed multi asset option multi asset deal keep set ed first price compute option final qualitatively option ed option table price difficult ed nominal massive gain sample applicability ed size case r r ed ed cox interest rate square root diffusion discretization rate subject discount value report ed explain outperform r ed ed conclusion algorithm choose mse property asymptotic asymptotically efficiency mc usefulness option price advantageous ed finance situation dependency multi optimal method fail efficiency gain description analytical restrict kind diffusion apply occur finance evaluation risk necessary compact continuous integrable derivative assumption trial choose width om theorem quantify bin width determination ed wang variance ed reference continuity topic gaussian california institute statistic sample multimodal function pricing simulation conference w brownian dimension journal importance security pricing finance cox interest density journal american monte financial engineering york p pricing finance finance west lee rare conference p efficiency conference j uniform transaction nonparametric sampling journal american h quasi mathematic real pricing derivative finance pricing journal economic array visualization net sequence quasi p york financial management h p nd ed methods york york cube integral mathematics mathematical fu pricing conference importance american multilinear spline technical sciences information university pricing simulation conference wang carlo journal zhang journal american statistical thm thm thm proposition importance simulation pricing propose proposal contrast nonparametric allow close proposal nonparametric importance inefficient issue low subspace dimension square property optimality easy require demonstrate path dependent asset option pricing lead significant gain dimension pricing option carlo reduction decade complexity pricing product distinct increase pure mc stem independent inefficient power increase mc mc space ce estimator realization definition observe improvement reduce include reduction moment technique aim give behind force sample domain intuitively rare obvious dependency would rarely lead rare technique usage difficult additional pricing gaussian proposal shift obtain adaptive fu square drift utilize approach complex approximate proposal reasonably derivative pricing idea dimensional zhang lee limitation pricing suggest restrict coordinate effective ed identify principal component pca proposal converge sense algorithm computationally benchmark option pricing ed mse accurate computationally addition easy implement implementation property sequence quasi mc next general mc pricing review convergence investigate concept ed review section base introduction section implementation simulation conclusion section pricing importance sampling describe asset differential brownian motion bm free interest european option compute neutral case kind euler discretization yield space discretization price kp density multivariate identity trajectory keep discretization approximate integral mc basic idea define eq confidence rate remarkably optimal proposal denominator nonparametric zhang stage trial nonparametric proposal subspace low subproblem curse decompose cover burden estimate choice nonparametric inefficient article multivariate nonparametric attain rate estimator evaluation require construction mid height bin ht k h estimator show select bin width trial obtain j partial sampling mse obtain assumption hold interpret impractical rewrite discretize bm reduce ed construction discretize bm path bm expression approximately respectively number discretization claim suffice matter long nice roughly path space figure common method ed pca certain situation bridge technique superior may brownian bridge integration mc integration use pseudo random construct fill evenly discrepancy reader merely theoretical benefit involve infeasible low integration problem engineer stem early finance rather ed compare property integration drawback lack randomness computation mse assess deterministic low different discrepancy digit shift simulation shift shift quasi sequence mixture apply parameter define suggest tail gaussian proposal tail aim minimize discuss variant algorithm directly solve optimal adjust grow becomes suggest ed convergence computing weight need evaluation cost compare combination carry parametric coordinate propose overview ed compute trial follow marginal distribution coordinate heavy discuss alternatively tailor good choice reference subsection case optimal detail package request contrast trial error selection practice trial distribution ensure uniform small
useful odd implicitly example implicitly basis interpretation simply trick relation conditional ultimately moreover base unchanged projective sum one projective define projective space homogeneous projective conditional I ei ideal algebra eq ideal point projective conditional probability distribution basis ideal equal form product ideal ideal purpose necessary independence py z without main gr gr gr basis algorithmic ideal cox overview binomial relation ideal generate bayes name contain probability ii p ki kb j p j ei j ji ei graph one vertex edge label cycle undirected follow edge direct cycle direct cycle edge correspond outer universal gr induce cycle gr need recall matrix characterize follow fan normalize volume ideal ideal gr special matrix come bipartite vertex incidence label column cycle binomial define relatively irreducible minimal support ideal bipartite circuit cycle universal gr incidence circuit gr incidence circuit universal gr fact ideal cycle opposite associate cycle polynomial contain however necessarily gr htb outer along induce cycle g probability coincide ei ei bayes binomial associate inclusion preserve q reverse bayes p cg ef multiplication mod I j k mod j p j break long cycle polytope point factor among sense result thus interior nonzero coordinate another restrict hessian connect tu tu change lie boundary zero outside argument extend zero assume tu b side contradict arise htb dot simplex choose maximize htb general high upon triangle homogeneous coordinate give coordinate hull permutation six htb mat way project understand regular merely singleton product may assume state also singleton correspond cube convenient say disease relation relation explain version outer cycle n cycle outer cycle figure htb bayes projective eq copy version represent projective I sum familiar bayes collect primarily integer matrix ideal minimal generate say minimal define polynomial initial gr gr basis multiple lead lead algorithm ideal variety column cone span correspond coordinate x embed closure parameterization exponential family polytope projective variety closure ideal cone one add unless homogeneity span restriction require instead obtain projective variety onto polytope nonnegative projective projective variety j standard name define n x empty define corresponding polytope composition map ambiguity correspondingly map point maximum entropy feasible equivalently minimize divergence uniform otherwise hessian interior segment connect minimum lie vanish lie sign lie boundary zero remain go arise cm thm describe gr probability set event specialize purely special generalized describe discrete conditional order observable singleton event assign chance become relation must relation infinitely ideal nice ideal quick language gr et cox universal gr basis ideal type mat state event map generalize upon generalized polytope provide probability analog family focus event correspond cast compatibility family full conditional relate organize necessary condition condition come gr computation specialized simply see role geometric result case generalize accomplish geometry theorem partially variable relation finally relationship construction fact polynomial elementary denote explain index
monotonicity regression series method year spline cubic spline employ four four net figure conditional capture growth age monotonic curve high age perform poorly age many curve upon original monotonicity requirement quantify improvement next multivariate quantile quantile plot age fourier severe non monotonicity figure mit paper value problem process monotone estimate simultaneous conditional fourier series original repetition error rearrange end simultaneous quantile confidence correct original integrate confidence quantile height fourier bootstrap repetition dark band intervals monte experiment estimation relative compare describe detail closely expectation monotone report error measure ratio estimate spline norm curve target accurately original curve average outperform local spline bad norm report find bad mit spline intensive monotone spline spline il average monotone monte carlo x estimate quantile process multivariate estimate index argument estimate obtain multivariate estimate sequentially apply possible age consider multivariate curve target accurately winner average local il original exist mass sort sort point sorting complete gain sort dx dx extend proof proposition induction argument weakly variable dominate quantile jx jx jx virtue conclude weakly argument integrate recursively inequality imply state weak claim proof describe rearrange sequentially state proposition homogeneity proof monotonicity rest proof rely preserve operator property u claim claim multivariate preserve preserve case dominate variable quantile must quantile induction true gx j x mx x x mx j jx j jx univariate univariate hold fix preserve part weak inequality completeness proof proof start vector sort sort operator sort element th element leave element simply proposition weakly reduce become weakly close inequality pass multivariate dimension fact submodular v strict proposition location function regressor transformation regressor slope conditional function ordinary chart specification dependent normal whose regressor observe age nonparametric replication software package axiom conclusion conjecture corollary exercise notation solution summary mail edu target monotonic weakly univariate metric simultaneous confidence cover short norm cover probability demonstrate utility improve point estimate height chart univariate growth chart quantile regression secondary monotonic example age demand quantile distribution function non variational use distribution g mit without generalization estimate produce monotonicity original improvement univariate multivariate improvement show length rearrange low long history mostly relation provide extensive review study estimator smoothing show procedure project basic statistic unknown suppose original monotonic tractable indeed improve answer yes transform estimate monotonic global local spline produce conditional give absolute deviation produce example play need monotone monotonic improve upon estimate related estimator attractive tractable interval take measurable function random simply transform sort operation fine sort function measurable weakly gain quality strict namely proposition strictly applicable property independent estimation error strict subset reduction estimation inequality become quantile function direct consequence classical correspond qx dx f almost pair measure submodular pair hold submodular weak monotonicity dependence exclude monotonicity hold apply every permutation integer weakly two resolve among let collection average possible smoothed show estimation weakly individual describe property weakly measurable weakly moreover increase produce weak reduction error subset measure iii v ff satisfy error small average estimation eq demonstrate several multivariate see function monotonic argument smooth conditional monotonic bivariate improve property function argument average dotted line light informally begin reduced step function finite case function point property panel illustrate decrease monotonicity requirement co bring set original fig projection original set weakly value monotonicity leave unchanged flat pool adjacent domain point monotonicity violate definition upon e therefore hull upon obvious homogeneity property sequential hull multivariate class original estimate good distance reduce approximate illustrate panel fig consider pass middle target neither flat outperform combination flat particular apply univariate multivariate simultaneous proposal coverage simultaneous interval low point confidence asymptotic q contain probability hold finite sense finitely interval specify estimate standard value attain confidence excellent critical problem monotonic indeed monotonic confidence accordingly intersect initial may contain due misspecification say incorrectly cover weakly monotone incorrect center non center confidence nonparametric estimation correct centering application regressor researcher rather development justification smooth thus robust equivalently reason interval instead hence improve improve entire simultaneous low end increase multivariate symbol multivariate emphasize describe interval function confidence increase empty sense end imply correct specification increase cover correct equal cover weakly
multinomial virtue idea seek simultaneous p coverage coverage tuning word denote simplicity probability tuning branch bind whether p simultaneous iv tuning determine actually construct requirement ii therein four freedom confidence upper class binomial chen al propose define simultaneous confidence simultaneous interval guarantee confidence control coverage tuning ensure simultaneous confidence computable establish sequel integer l z p p reason p impossible complementary coverage direction c c c lb lb I lb n n b I I I truncation propose bounding truncation truncation tolerance choose reduction note method negligible far bound virtue theorem recursive b b ib ip ip I ia employ adapt branch technique hypercube branching process hypercube eliminate consideration give check coverage simultaneous less focus proportion pre specify requirement reliability general formulate pre specify reduce error specify parameter margin hence p find small coverage probability denote interval define simultaneous tend tend bound establish upper hypercube determine truth pn pn focus idea generalize virtue absolute theorem hoeffde sm appendix sample approach stop termination general criterion sample purpose develop computable coverage interval define decision stage assume complementary coverage complementary coverage accomplish symmetry coverage virtue chernoff bound b otherwise otherwise assumption follow chernoff hoeffding n upper complementary coverage complementary coverage complementary coverage great fixed virtue propose possible optimize choosing estimate b sequence theorem complete construction theorem computation sequel propose computational sn propose stage coverage section computable complementary contain assume rule continue propose coverage probability complementary coverage accomplish virtue chernoff hoeffding stop c virtue chernoff stop rule x immediately argument right apply complementary coverage interval bound complementary coverage probability virtue tuning technique optimize sample sample inverse scheme scheme infinitely stage stop different rule otherwise rule chernoff hoeffding cdf rule derive cdf describe establish statement n accomplish technique similar chernoff variable mixed criterion construct scheme x scheme sn l mix sn proof bound margin relative ab ab hz ab z ab b hz virtue establish scheme mix w efficiency size scheme integer mix n note maximum computed branch bind need one ab hz ab hz hz narrow hz w hz hz hz subset hz z point section valid sample relax interval surely f f f chen discover inherent variable theorem mean binomial binomial sequence random sequence eq estimating binomial define th structure scheme estimate wish see stage deterministic sample confidence stage virtue pos decision statement hold true moreover b l v size n l pos chernoff virtue chernoff propose scheme follow pos chernoff variable following provide ii l b number subsection chernoff regard pos statement hold n e appendix sampling criterion wish scheme stage th virtue cdf sampling scheme pos g statement provide n vi sizes e chernoff cdf propose scheme chernoff assume provide p vi sequence see shall focus asymptotic scheme rule cdf distinct p p pos n statement r z proof design mixed wish scheme associate sequel stage chernoff bound precision scheme rule stop f otherwise stop virtue virtue chernoff cdf scheme pos mix cdf u u element integer p size chernoff useful pos statement z r z z z variation simplify expansion size ii establishe shall asymptotic throughout rule derive chernoff regard double pos mix regard performance tend pos mix sampling n j hold appendix proportion population scheme describe sequel k replacement sampling interval mixed precision estimate u z nu z nz p l nu z nz therefore absolute cast constructing immediately suppose kn kn p p p kn u kn l kn kn criterion section size choose distinct sampling kn kn kn k otherwise p p p l corollary chen tail hoeffding sampling replacement size estimator formula absolute scheme margin absolute error size element integer stop eq sufficiently rule sampling satisfied introduce finite restriction w n integer p virtue tuning stop population small coverage propose stop continue shape stop maximum th define probability stop sufficiently n p p r sufficiently small reduce propose continue w minimum choose suggest stop conclude section stop rule sampling boundary stop rule ensure p parameter stopping ensure simultaneous estimating category category proportion construct simultaneous interval sample replacement simultaneous confidence th unit draw replacement follow n probability multivariate simultaneous proportion virtue tuning idea seek simultaneous control coverage lower function adapt complementary p interval tuning determine order construct typical later n n pi n p confidence determine control subroutine tuning determine via branch sufficiently coverage simultaneous associate subroutine computable hypercube p sequel vector I u u u complementary b lb b lb l lb lb I I n lb lb I theorem estimate multinomial proportion bound complexity recursive computing ib sa ix I I ia nn n derive recursive multinomial readily I ip p adapt branch coverage less determination size estimate confidence margin p consequence true u coverage interval apply bound hypercube p obtain pn large motivate construct absolute criterion criterion directly work unbounded stage main convert substantially small give frequent construct therefore convert scheme mixed section absolutely small maximum sample size number desirable construct scheme arbitrary scheme criterion construct sampling ensure precede section conclude emphasis direct use convert always maximum frequently practical situation sampling eq stage size choose odd sampling scheme stein stage improvement see therein analytic statement n z c appendix ensure coverage tune much coverage sequel approach determination integer sequence exponential variable common unity statement b assume ii j j j j b note generalization formulae appropriate tuning parameter useful odd c mn I n h p n appendix coverage respect finite sample number suppose tune proof frequent define let stand n deterministic stand th gamma section shall discuss say gamma scale parameter sample let eq notation x nk k sequential prescribed interval poisson bound en construction bound interval binomial paper width base ratio unfortunately width sample pre length size overall level specifically construct width confidence interval nu nu extremely useful constructing n u u n stage l n n coverage tuning bound confidence adjust confidence interval stop continue confidence termination desire confidence coverage desire coverage limit bounded I bernoulli sample deterministic use lower adjust coverage subsection scheme pearson establish l sd otherwise suppose n u l sampling p p u l sure event chernoff hoeffding establish fw let th sl otherwise suppose l p u l p l p criteria sample scheme interval chen otherwise rule u l u criterion element c describe section virtue bound ci u kn n p p nu kn nu criterion size choose max boundaries ci define n z sample small kn sp stop p u virtue corollary primary category typical interval cumulative function g interval pearson ci termination nu u l problem interval estimation computational common finite general describe ci finite termination sampling limit nu p suffice observation ii ensure n describe suffer drawback conservative interval interval tuning coverage n section virtue bound maximum checking value approximate limit form level bernoulli sample size use limit note limit confidence rigorously guarantee virtue tuning describe beginning poisson variable parameter overcome difficulty design probability guarantee describe describe section n u eliminate necessity probability course pos test upper hold test variance describe appropriate size h index stage complete accordingly sample complete termination realization certain odd unity determination purpose describe number completed accordingly complete confidence n u complete confidence realization low z random unity rewrite determination interval exact side sequence classical researcher shall approach parameterize th stage random tuple construct interval lower assume simplicity mention interval possible coverage coverage b bound identical discrete parameterized l define k statement hold consecutive achieve similarly maximum achieve iii b u b k establish summation independently recursive use sampling limit limit desire rigorously virtue tuning population describe bound u n l u sa b mn la bl si l u pp consecutive consecutive distinct maximum iii suppose u establish tune recursive checking section poisson large coverage confidence coverage tune x l n u eliminate necessity interval infinitely wide coverage pos hold see coverage generality suffice complementary z r z z bound consecutive specifically x n z therein significance shall construct confidence sequence odd number define note eq z unity coverage virtue shall construction freedom common unity coverage computed theory technique precede may stage bind sequence treat construction finite illustration formally let interested determine stage provide well know ensure fulfil involve computational modern computer statement obtain recursively computable adapt branch quickly confidence level rigorously guarantee virtue coverage another construction sample variable generality interest determine martingale well suffice ensure rigorously virtue statistical investigate modeling application almost sciences management life biological name consider random major regression various estimation error iii purpose observe stop n purpose criterion integer suppose observe n stage stop see quantile importance uncertain desirable percentage estimating formulate quantile prescribed control uncertainty order statistic positive follow quantile integer k n p x n stage sampling appendix margin integer n p n p appendix margin procedure follow quantile mix integer p p x p x x r rigorously preliminary preliminary true contain r r r complete remain r r write r lead contradiction either r probability generalization inequality establish literature variable event z z z basic basic depend non random denote e x x mx assumption mm mx ii modifying replace x conclude proof monotonicity u inequality inclusion make b stop b b measure complete chernoff independence function write z z inequality confusion assumption chain n tn n virtue respect equivalently tt rt z z z inequality virtue chernoff theorem case f g z impossible f g n n l l l occur distinct establish occur n n occur occur occur argument consecutive distinct statement u n e occur u e consecutive establish regard occur occur e occur n occur hold distinct element statement regard occur occur e l l b e occur occur occur occur occur occur b b statement iv lemma z apply z z l b show theorem ap great kn kn kn kn nk kn n nh mn kn investigate kn n kn np kn kn less define sl f assumption exist number e e complete shall statement chebyshev f k c proof thm statement n n n assumption b k virtue establish fa fa fa observe fa fa fa bound j continuous theorem ii gx fa fa gx fx dx fa tt fa f b cdf need readily hoeffde lemma kn k n kn z z expansion formula b see z proof sure n p p p immediately lem z z lemma trivially remain lemma accomplish note p sure suffice z n p accomplished b b z z z proof p suffice b p p p sure p suffice need show clearly must imply establish virtue shall establish hold virtue complete prove sure describe immediately follow chernoff result virtue fact lemma scheme requirement describe corollary immediately inequality sure event virtue respectively lemma requirement immediately chernoff sequel lem suffice bp p bp claim case lemma lemma establish prove consider ii bp p second lem bp p z z z hand exist intermediate z z n b bp proof chernoff q preliminary lem bivariate enough e dx proof lem b taylor since exist b z z z z z conclude z show z z intermediate unique complete proof lemma lem take fix iv z ii monotonically respect z iii claim enough true denote enough apply base enough b prove iii proof iv z small many small monotonicity z contradict statement iv b use z prove z exist b virtue consequently b z z z p unique z b b virtue prove lem main show definition make chernoff similarly definition use chernoff lemma consequence make statement z sufficiently p p virtue statement p chernoff z small enough p pp statement sufficiently statement chernoff small enough virtue statement chernoff z enough observe lemma suffice show result prove lem readily show p c p p mean u symmetry notation p iv p u u shall p independent gaussian mean function u u u u u arbitrarily small p z z statement therefore virtue p claim l p u complete independent unity u v u u v u include domain boundary visible shall statement purpose n c statement sampling n p p z apply p statement z sufficiently clearly p z sufficiently n p p p law p n p manner show conclude three ii lemma p l n n statement scheme lemma p p l n e n p j p establish make notation base p p p p p l p p p l p u iii shall x denote p p derive p e p e p z see virtue p z p q yield lemma complete position sure requirement follow result theorem I bernoulli sequel shall focus associate lem note z apply b pm z pm z p z z pm follow result state p p rewrite let follow p notation virtue h prove statement suffice inequality check hand statement prove ii sufficient iii poisson random ny virtue chernoff x z r e e infimum e lemma e e unique make lemma fact z z z z p direct show definition size z z z unique p thus prove small p lemma note definition apply note z pz combine p p theorem inherent show z z n p u virtue fact requirement corollary immediately p complete inverse lem simplicity p suffice z z z complete virtue similar lead identity stop l chernoff need lemma exist pm prove get z lemma z contradiction claim multiplicative chernoff n complete proof prove hence choose l n n r imply p b b p p p course prove statement z hand p l b iv p b p p statement iv preliminary b z pm z get since z n eq multiplicative n n n lem small statement unique restriction statement statement since monotonically sufficiently small establishe statement notation small claim contradiction exist fact lemma small enough lemma restriction yield iv get contradiction claim true prove claim enough virtue z z statement p claim statement shall statement small z sufficiently consequence statement small virtue independent lemma enough virtue definition complete employ lem I p p p p mean unit l p u v p p p p l p iv p pp v central p z p z p p therefore complete shall statement purpose c note p note p pz p law number p statement note first three pz claim claim justify investigate trivially p claim law true also shall evident sampling l lemma statement shall u variable see p ii employ similar ii preliminary lem statement I exist unique iv intermediate simplicity p definition sample eq suppose z prove b b b z iii iv z claim statement lem show definition make statement last chernoff p enough similarly see definition lemma last statement independent pr make statement z chernoff bind distinct c b p b n n assumption complete p r p p p j let random unit u lemma similar purpose show c p pp z r virtue next shall statement hold statement n statement complete step ii e l n n e making n therefore lemma n p n e therefore n l pp rp p shall evident l l p rp true apply l p p limit standard variable mix monotonically provide p pg g g monotonically p p establish z z pm negative p proof fact p p notation order show lemma suffice p assumption z side impossible consequently note side consequently establish position rule bound sure event lemma recall sampling theorem scheme follow rule cdf l p b b thus event mix preliminary lem p b z z z establish lem z z r intermediate theorem r lem b p z p intermediate z n b complete lem b z exist unique z r b z z theorem b show statement p r n equality due statement ii mix especially monotonically lemma check lem z trivially accomplish z z lem lemma eq negative decrease monotonically fix monotonically monotonically lemma check partial z good u p sn sn virtue virtue making b complete proof simplicity p combine yield event note hand side impossible complete position event satisfy requirement immediately mix need lem sufficiently true unique z z limit vi sample sufficiently note b intermediate statement p p note b p b z intermediate unique n statement sufficiently z z iii statement sn suppose infinitely b make definition prove claim set infinitely imply rp r statement claim contradiction infinitely note contradict x z sufficiently small get infinitely contradict prove small enough imply p vi b b r claim sufficiently b noting lemma z ap z p p z establish r p r b r lead contradiction prove r b p moreover r p simplicity step shall hold make use four statement b p chernoff b independent see statement statement definition p direct definition statement p small enough result make four statement chernoff independent make four enough chernoff great use statement p chernoff rp complete argument mix definition prove theorem lem p p p next shall virtue r b p r p p p independent unit variance l l v u shall p shall show statement follow n lemma note sn pz r establish n number rp statement z pz show enough investigate sufficiently law p j similar manner show statement ii iii lemma course order prove follow central u p simplicity complementary u induction limit g lb I p e p b b p I c c complete hoeffding preliminary I simplicity n n n x notation x x b x n theorem lemma b sure bound need preliminary let I variable x n n let x n x n n thus theorem similar similar sn sn follow lemma p b lemma b similar sure argument establish pos chernoff first show statement number n get contradiction contradiction therefore k n chernoff bound proof statement statement use iv iv statement follow x pos lem sufficiently statement p take eq first b c j statement b make statement apply c statement pos distinct z z k assumption z pos chernoff statement statement argument iv argument z pos theorem pos lem statement unique take iv lemma define notation sufficiently last z provide lemma statement lemma therefore b make statement size distinct c p pos mix cdf lem p kn n complete z z z z z g p notation rl u suffice show note pz recall event pz recall monotonically hand side impossible since second derive requirement derive cdf thus requirement theorem pos mix result sufficiently statement z constraint fix vi statement I size note intermediate z statement ii note intermediate theorem imply statement monotonically respect z z p z establish iv define check sample claim enough contradiction suppose claim infinitely small enough z check enough contradiction z interval imply proof shall bound series statement deduce consequently z r taylor formula definition deduce consequently statement vi p claim p n n z z prove z p establishes r p lead r r throughout restrict enough p proof shall c z p similarly first four statement statement virtue great shall use four statement statement virtue sufficiently view make statement last statement statement proof employ argument pos mix preliminary lem n definition p c shall consider p r r zero l z employ enough statement make observation central analytic thm shall sure note u x v freedom eq combine q imply prove shall statement ii virtue argument ii iii iv making proof clearly gaussian zero unity orthogonal unity x sampling x statement follow stop similarly q virtue n h nn note chi freedom observe increase respect monotonically b chernoff bound eq notation follow statement normal cut combine monotonically decrease increase unique hand right side monotonically exist stop virtue identity show argument proof theorem stop r virtue identity therefore ci virtue z z z sure event imply inequality lemma recall l n inequality virtue pos test l u n n immediately pos n immediately proof variable mb tn freedom stop recall virtue stop sampling I tn degrees scheme quantile let x kf definition hand j kf x nk eq last property argument similar stop finite mix definition stop identity p j n p r p r n p p j r argument statement adapt branch discuss branch improve branch prescribe immediate application interval quick determination coverage exceed branch general finding consist systematic enumeration solution candidate discard quantity optimize discrete reference contain hypercube mean multidimensional region n converge let specify typical describe b follow eliminate k k k return whether computational adapt algorithm lb u nonempty great follow pick remove l lb otherwise drawback portion effort lead propose algorithm l lb follow hypercube split one set splitting eliminate hypercube lb let portion branching lead improve check great b nonempty follow hypercube procedure process u remark therein chen unify demonstrate problem sample confidence interval sequence cast construct prescribed coverage sequential interval principle adjust obtain effort introduction fundamental area enjoy various field science parameter unknown application confidence persistent reference despite devoted drawback either drawback frequently use designing method seek bad assumption tight wide value include second employ deviation theory nonlinear reference therein solution asymptotic tends unfortunately scheme approximate motivated limitation establish sample sequential efficient precision get available prescribe rigorously new develop spirit important branch accomplish sampling quantify word many researcher offer inferential prescribe differ value less number addition close problem point precision requirement construction interval interval construct prescribed estimation word size stop parameterize coverage control adjust sequential interval whose control sequence stop sequence termination publish version present methodology coverage probability interval principle coverage sequential coverage limit course bound approximation confidence rule coverage control calculation complicated stop possible coverage parameter coverage sequential unnecessary obtain accuracy increasingly level coverage high subroutine desire subroutine complementary coverage probability consume second infinity therefore avoid compute parametric overcome adapt branch global see algorithm checking interval bound bind probability sequential coverage idea complementary virtue q bound conservative tight complementary probability exploit sequential unimodal estimator iv coverage tuning interval first coverage subroutine integer start possible complementary sequential interval pre specify paper theory propose principle construction necessity design scheme powerful coverage consecutive bound recursive computation maximum checking domain truncation triangular crucial scheme scheme binomial proportion section multinomial information unknown bound confidence base construction sequence investigate quantile conclusion proof various branch throughout follow integer denote row respectively small integer denote sign assume value gamma integer takes denote denote parameterize introduce confusion drop respectively student freedom denote percentile chi square presentation scheme ni n k z z z z z z define size use confidence tuning later section estimation stage termination sampling decision notion process also remainder index denote equal scheme version fix stage important estimation sampling inverse binomial special version inverse therein criterion minimum minimum impossible ii guarantee purpose size stage arithmetic stage rule sample sample prescribe sample reach scheme inverse rx sequence positive integer base sequential limit sample x n u nu drop simplification sequel focus n framework wide spectrum obvious cast control point estimation framework pose way priori construct margin problem respectively appendix put implie concerned construction limit nu obviously problem cast interval level sampling impossible extremely parameter present binomial proportion population parametric overcome difficulty associate number value coverage sequential coverage desire play crucial mle numerous monotone develop shall concept unimodal sequence event time say great non increase likelihood mf emphasize mle contain clearly x n stage illustration binomial generally belong discuss sampling shall critical problem l distribution cdf cumulative distribution eq assume structure result random requirement n address pose tell rule function bound coverage prescribe refer illustrate intuition behind rule stop test u u h l rejected reject g otherwise consequently highly likely interval hypothesis stop rule interpret appendix I lower limit coverage probability confidence monotonicity control sequential random constitute convention purpose interval call confusion interval virtue concept sequential method interpret l sequential control relationship impose call methodology sequence rule desire sequential interval inclusion inclusion termination consideration stop impose interval familiar coincide control something requirement pre requirement random except inclusion process sequential interval consequence inclusion principle coverage sequential control provide coverage control precisely connection see ci st scheme follow l th l note remain valid sequential complex point probabilistic inclusion nb region dimensional assume inclusion scheme sequential control generalize confidence extensively inclusion derive stop rule first publish simplification stop stop control obvious careful reading proof version publish derive connection sequence readily see version six systematic stopping rule random version confidence imply coverage limit control sequence replace estimate approximation obtain assume identical establish propose replace suggest modify accuracy approximation low n performance class level binomial poisson explicit interval solve equation illustrate derive check take et n apply confidence n explicit q n example construct replacement attribute size let attribute random n p n n z introduce approximation virtue although tuning determine appropriate desire excellent trying rule confidence eliminate order stop simplify stop describe interpret g e stopping performance apply simplify stop coverage lead simple applie etc bernoulli x procedure size I stage termination prescribe interval let procedure p bound continue coverage appropriate prescribe bound p continue z virtue technique appropriate p prescribe purpose stop random u respectively violate coverage prescribe coverage stop proportion adaptation variable draw rule adaptation derive level relative precision derive inverse I common mean integer purpose bernoulli sample I scheme stage termination value stop rule identify l stop rule continue virtue coverage technique less prescribed observe actually version rule termination prescribe margin design sequential follow choose positive number sum margin apply rule r margin error rule derive continue either violate prescribed coverage tuning technique extensively approximation simplify stop addition normal cdf size x virtue chernoff sampling n l u sf event n n imply assumption chernoff z define stop inclusion principle confidence limit bound cdf see prescribe assume value sd virtue cdf rule virtue cdf integer p sure chernoff choose size choose distinct stop choose p chernoff size need express accomplished chernoff solution z k taylor formula sampling c eq simplify chernoff claim rule rule size central limit eq stop algebraic value exist stop put stop scheme stop general n proof restriction sample rule satisfied stage w stage satisfied integer coverage tuning optimize stop price effort extra introduce conclude subsection satisfied otherwise boundary stage sure first small coverage focus asymptotic sampling scheme chernoff assume distinct p sampling scheme p statement n shall binomial relative wish construct scheme integer refer threshold sample k tends appendix note inherent connection inverse scheme size variable let iy geometric stage termination p construct ip apply coverage great cdf stop rule virtue cdf scheme cdf decision otherwise threshold sum th stage p p z p appendix threshold distinct maximum scheme otherwise equal p unique threshold distinct define readily compute search monotonicity virtue inequality inverse p p appendix criterion threshold choose I z truncation computation view occurrence hence triangular reduce computing probability v subset natural number compute recursive n p taylor expansion formula inverse scheme sequence distinct otherwise make small coverage value satisfied assume conclude subsection modify stop assume value shape boundary threshold associate spirit specifically sample event first stage small propose sampling binomial high
location primary lack channel unknown hypothesis likelihood parameterized distribution cdf condition monotonically furthermore optimal access eq bad solution parameterize problem know structure channel belief policy belief channel describe appear value snr belong choose interference constraint channel assume reward channel alternative posteriori appendix simulation necessary give plot greedy reward instead aware spectral improvement observe simulation hence primary spectrum thus significant throughput cognitive show channel posteriori almost satisfy distinct posteriori channel posteriori channel converge algorithm posteriori chain without generality sense slot obvious hold chain hold version slot observation channel posteriori condition take eq hence would proof thm remark pt conference conference signal computer author science laboratory electrical computer engineering il usa email cognitive markov cognitive user try channel follow scenario cognitive perfect knowledge maximize instantaneous performance scheme exist scenario unknown parametric true constraint naive bad cognitive spectrum access channel partially decision learn cognitive ever band place band refer primary system efficient channel cognitive also select sensing policy jointly primary usage user slot propose study secondary aware exact signal receive primary develop unknown performance assume primary spectrum access secondary user obtain analytical optimal suboptimal solution show statistic signal learn reward constraint interference literature spectrum cognitive secondary primary secondary receive signal secondary user channel different user secondary primary channel present use maintain synchronization receiver difficulty dedicate channel secondary unknown statistic interference sense problem describe elaborate simulation secondary channel channel secondary channel slot access channel slot secondary strict potential slot receive access secondary select slot expect maximized primary channel secondary explicit channel constrain decision term primary stationary slot represent free channel primary user secondary system include center center dedicated channel dedicate secondary receiver receiver tune correct channel decision make center slot channel slot represent output wireless cognitive channel slot density slot collection slot channel slot channel slot denote slot decision channel slot denote channel slot whenever secondary slot bandwidth channel satisfy follow constraint primary slot simplify slot observation slot slot channel channel furthermore state channel establish access access whenever exceed joint slot express represent indicator advantage access policy obtain secondary user threshold meet probability primary rely objective make constraint discount horizon program discount seek channel observation state identical channel channel identical reward reward slot maximize know access decision access user channel threshold unconstraine state slot markovian decide slot function past decision slot slot slot dynamic slot past k channel conditional equivalently channel condition slot probability slot slot value distribution state channel state slot slot use bayes belief slot channel select slot otherwise sufficient statistic instead access decision depend conclude scalar observation scalar secondary scalar slot denote slot decision threshold evaluation evaluation function return give slot express dynamic bellman reward vector channel slot perform bellman adopt suboptimal obtain rest transition regularity irreducible recurrent ensure channel free slot free slot current switch become slot relaxed hold channel e policy instantaneous slot reward give slot choose channel slot greedy channel condition past policy fact condition greedy policy slot channel argue policy reality future make reward initial probability belief stationary upper reward assumption give represent step assumption state become know slot call function want evaluate binary string represent channel slot priori slot determine function become slot sense action slot affect expect instantaneous achieve add would free slot time slot channel slot reward slot precede slot write reach matrix joint channel state element system slot system slot binary slight abuse element vector slot element bit use value analytical reward access scheme cognitive also employ user spectrum access describe reveal channel depend channel optimal similar scheme channel track use receiver slot receiver transmission secondary receiver bit information provide fact secondary know advance expect slot synchronization receiver presence receiver free regard practical aside dedicated channel low reliably observation primary channel utilize transmission another independent general channel propose handle add statistic would posteriori access threshold would depend avoid keep secondary primary cognitive rely coherent detection sense even coherent primary hence aware path secondary reasonable secondary observation hypothesis assume observation primary model parametric present denote unknown channel log section two approach restrict ease illustration comprise user knowledge access meet interference constraint interference optimal access decision would concern need address perform satisfy condition hold parameterize practical discuss suboptimal solution run update hold place observation belief update give channel slot monotonically solely clearly belief guarantee greedy channel perform likelihood access single likelihood test bad access decision conclusion lemma sense policy min max average reward true bad lemma reasoning approach severe relative learn reliable channel parameter across channel convergence cardinality parameter secondary user belief long posteriori keep belief state belief product slot represent posteriori slot posteriori distribution state slot condition slot belief slot appendix posteriori mass slot frequently realization update belief knowledge use constraint use design mind choose slot access spectrum channel slot order posteriori group partition low posteriori upper partitioning constraint posteriori partition ignore design policy posteriori probability interference access meet interference constraint condition partition posteriori condition slot arrange c ik partition early channel slot slot evaluate threshold complex conditioning interference meet average posteriori converge delta function true almost asymptotically meet u policy problem allow access slot belief
hausdorff simplify replace hausdorff hausdorff interval hausdorff read hausdorff distance central interval hausdorff read minimization problem combination hausdorff distance low resp upper length read combination distance half length central combination hausdorff hausdorff distance positive rewrite use resp rectangle simplify quadratic whose resolution describe eq proportional ex set kk view resp resp language use central hypercube prototype define many less hausdorff two see hausdorff simplify dimension involve euclidean metric exist compute hausdorff know compute explicit solution original still subject investigate another make explicit definition easy wise pd central section concern central apply directly solution determination existence useful clustering see criterion coordinate prototype dynamical interval hausdorff low de determination combination hausdorff hausdorff concern datum scale already globally instance distance natural wise could dispersion central seem aggregate central exponent exponent extension tendency point investigation acknowledgment resolution associate comment problem resolution respectively axis parallel resolution tucker system eliminate end case minimizer edge solution simplicity half length distinct compute convention index notice index q minimizer reasoning line unconstraine minimizer line go least general unique admit rectangle small minimum point skip problem say along edge edge corner say g leave corner component parallel universit la france mail universit iv datum rather fall propose determination tendency dispersion keyword multidimensional summary involve location tendency arithmetic focus basic central tendency dispersion measure interval value meet interval measure
example mean coincide mode mode discuss weight tail denote unless derivative similarly requirement center eq smooth conceptually tail coincide usual distribution side evident cr corresponding weighted critical define cr therefore test center conversely choose independent requirement exponential standard tail whenever leave test test indicator cr generality statistic derivative equal bias base test variance bias lemma hold unbalanced size figure sample leave plot finally p mode unimodal unimodal pair distribution region minimum inverting binomial relate interval standard ratio variance population chi conditional comparison example triangular unimodal linearity symmetric somewhat whole deal unimodal exactly respective value leave truncate normal zero reach derivative reach plot plot main tail rather hand zero seem leave tail perhaps definition tail depend value circumstance arise square sided value sided mean value opposite side plot plot three test right plot figure biased agree slightly right side exp plot difficulty equivalent test distinct sided test side generally region advantage give function increase right thin sided binomial test reject thing happen observe plot tail sided cp cp c binomial tail thin tail tail symmetric c shall exact margin define margin importance odd ratio estimate association denote side sided test base column equivalent statistic function fisher randomization mode exact side association negative association side distribution pn nn ij ij table standard homogeneity odd chi likelihood statistic far order table margin example value total table right look increase statistic follow due monotonicity side differ therefore test fisher chi practical linkage table sided conditional second table margin thin right tail probability value table margin side symmetric software package even implement general test importance conceptually computationally simple sided evident sided introduce discrete p transform test resolve side association option symmetric equal small considerably unbalanced test bias equal tail test variance power binomial test asymptotically require density statement matter open question also sided p recommend report nan spirit side value p introduce equivalent transform sided variance exact two sided test exact variance side widely accept journal journal cancer institute journal clinical statistical well statistic symmetric though discussion numerous year discussion recent proposal far fisher side p motivation invariance chi evident drawback rule use majority statistical primary article introduction denote intuitive demonstrate sided value definite currently side discrete popular sided discrete implement package multinomial principle likelihood fisher j shape unimodal justify event necessarily unfortunately demonstrate feature p side tail opposite may low side p side base perfectly normal sided test symmetric tail dotted line
p function conditional replace way model comparison useful performance monitoring compare sequential standardized west chapter quantitative value discount factor forecast standardized forecast propose literature test carlo simulation work volatility process propose procedure modify exponentially move chart univariate indicate preference base apply sequential west chapter alternative make adopt exchange exchange primary web site http www co uk al review four collect price per daily price remain collect trading figure exclude week end bank trading day site http www uk compound return focus volatility purpose eq compound volatility volatility subject might choose might add delay covariance matrix discount accord decompose discount discount responsible evolution responsible volatility discount factor compound specify likelihood pick smooth high smooth I range apparent level likewise lead evolve reflect performance measure forecast maximize return discount small I produce table also reveal poor drive discount factor due infinity likelihood due appear application model dynamic correlation aim flexibility drawback introduce drive correlation overcome discount rate series compound exchange volatility recognize volatility invariant al level assume simple evolution autoregressive estimation separately ar forecasting allow complicated evolution structural characteristic apply superposition model west multivariate volatility alone build multivariate inferential consider topic model important invariant study combination portfolio risk point write explicit precision transformation readily see ta pt west bayesian carlo estimation reasonably large computationally expensive estimate come computationally suitable volatility dimensional acknowledgement grateful comment suggestion paper wish anonymous detailed lead considerably paper detail precision pn pp consequence assumption maximization likelihood require notation likelihood employ procedure var posterior forecast distribution tf r take eq derivative respect second stacking portion one diagonal diagonal write definite chain definite prove multivariate eigenvalue orthogonal column density non ia jacobian multivariate k p derive py py n q kalman tp c p n theorem corollary forecasting series foundation matrix dynamic distribution diagonal discount discount allow flexible variance diagnostic series comprise price lead exchange finding suggest effectively financial volatility dynamic exchange decade emphasis west chapter de vary volatility model develop essence correlation change univariate forecasting volatility widely et serial volatility family model multivariate volatility family multivariate conditional correlation factor model li model lot see al et west procedures yu computationally commonly use follow al et aim application space dimensionality volatility return desirable construct rely carlo volatility propose consider volatility wishart distribution update forecast discount change slow fast ahead forecast entire ahead forecast volatility methodology illustrate consider price exchange price lead factor diagnostic include standardized forecast diagnostic test exchange market detail proof variate dynamic west develop et west paper value al develop variate volatility covariance innovation denote stack kronecker innovation mutually uncorrelated wishart degree freedom function multivariate gamma denote determinant follow wishart comprise specify discount imply information equation calculate generalize west order form update discount capturing characteristic volatility useful consideration financial exhibit short assume pn ns decomposition triangular law represent beta p refer ia evolution reduce singular beta west west pp retain shrinkage generate perform discount evolution discount close reference practically discount unity typically agreement section discount evolution discount single equation introduce diagonal discount factor eq clarity presentation write joint f f wishart follow forecast see result derivation note likelihood single py g tm f g step forecast write kalman filter lemma denote requirement beta wishart result close error q tf equation forecast employ discount volatility discount volatility variance discount rate lead volatility impractical wishart relate estimator first maximum forecast error estimator wishart volatility
interval l ip ip contradict complete hold virtue moreover p show p ip suffice two follow l p p p p l p shall l p p p p p b intersection l p p complete follow consecutive p p position cp cp cp distinct p p statement regard conclude theorem p z pz pp pz z p z pz z p p pp ph pz p theorem lem n z z n pz hoeffding mn n mn z lemma introduce r j u l rl rl r lr lr lr lrr mr lr l ml lr lr r gm mr u r r gm u gm n u virtue monotonicity r b u r b l l r range virtue monotonicity characterize maximizer r b p establish conclude establish lemma x hence pp z g pz l lemma pp pp p ph pz p h p immediately follow notice since I e sampling attribute classical note identical dependent bernoulli apply pe ne establish lemma g z g g z z theorem z lem r r z thus lem lem suppose z z hence w z z z p lemma lemma accomplish proof show inequality theorem z n z z g z z z h z pt establish estimate binomial derive scheme prescribe level moreover develop branch operation research biology medical science computer engineering class theoretical put random familiar example include binomial population mean studied estimating binomial studied chen mean estimate theoretically practical quite resource specify mean frequently inverse scheme sense prescribe sample maximum size ideal inverse draw truly truncate inverse scheme shall truncate inverse equally central regard pre planning determination threshold sample level estimator truncate organized estimating proportion investigate notation integer represent integer denote integer drop whenever introduce make space shall continue stop random e regard random estimation estimation binomial bernoulli p integer k kn determination threshold maximum make margin r explicit formula determine direction monotone support I theorem variable refer virtue margin relative prescribed level guarantee determine ni p ni note interval last binomial population unit attribute replacement replacement draw without remain every population define attribute otherwise moreover tend tend I continue unit attribute number let variable sample see reference therein goal inverse section prescribe margin error regard nm establish truncated variable binomial hyper variable rigorous method establish lemma k k lemma remain z definition scheme n complete n z z assumption z z z z I nm z sampling n n k k k z n z complete proof definition positive corresponding chebyshev ik ik k I number less classical inverse apply x e e ne n e pe complete preliminary slight modification hoeffding lem common z z z z hoeffding z z b z xx xx I prove statement similarly prove lem
equation shrink process mild regularity condition depend put differentiable exist adjoint functional calculus correspondence term operator notation relation notice al et derive estimating eigenvalue coefficient suffice notice eigenvalue moment rate use non diffusion al indeed form law sx l following write yy properly obtain plug expression give term transition scalar proxy perfectly identify process generator identification guarantee nevertheless help process markov markov inference process four synthetic financial review basis spline support support distance calculate piecewise read essentially discover similarity regard shift distance euclidean distance literature ahead calculate sequence analysis wang allows necessarily weight shift distance distance implement r drive stochastic although value distance correctly similar put together separate aggregated trajectory really drift diffusion stochastic capture although aggregate diffusion occur separate real outli daily financial microsoft plots micro device intel software co bank db bank group yahoo com hardware software financial company path visual volatility trend volatility look financial db present cyclic drift seem volatility inspection alone try report four seem separate collect parent metric separate singleton well close appear give sharp dendrogram separate cluster hardware final financial hardware company produce game present multidimensional group identify cluster cluster contain two ellipsoid financial implication nevertheless conclusion discriminate sharp cut dendrogram group convert diffusion process department economic business dynamic differential equation mesh quadratic correspond real stock capable diffusion contrary metric keyword diffusion financial lot mining particular financial datum study dissimilarity see stochastic property model sequence information process see al black modern pricing assume diffusion paper dissimilarity diffusion diffusion process observation shrink parametric datum true operator generator drift
positive distribution know overcome cost converge asymptotic compute sequel asymptotic sequel q remark inverse note strong law identity true deduce consistency compact uniformly set finally parameter notation depend resp get use get analytic derivative equation asymptotic lemma w li easily q q trace circular integrable w consequence replace around chapter mlp traditional match twice concentrated nice covariance matrix good good derivative easy theorem multidimensional concern testing perceptron mlp purpose model transformation unit framework asymptotic determinant couple variable assume one mlp mlp note agree identifiability restrictive
efficient broadly narrow ensemble monte stochastically nonlinear numerical available alternative dependent process general statistical mechanic assimilation weather financial quantitative significantly impact mc slow assume path improve upon reduce discussion technique sample set ability user exploit lack applicability technique strategy database devise bring relie exploit add estimation nominal achieve efficiency neighboring generality applicability easy code computational exploration phase extensive costly setup justification justify project estimation parameter project cost enough subsequent justified project setup line passive cost statistical project setup cost implementation technique technique cv utilize utilize control neighborhood cv estimate argue elsewhere broad choice cv effective share histogram chain carlo broad rely structure give weather source biological remainder detail numerical well method consider difference curve context discuss numerical conclude numerical model note purpose spatial dimension g denote transpose zero covariance low use forward euler point independent space time apply well lattice depend representative q represent complete path parameter know noise quantity give brief classical control reduction see correlated assume estimator use degree deviation correct adjust bring alternatively unbiased variance reduction ratio statistic achievable highly cv technique potentially effective order practice effective square precisely ratio converge separate pilot estimating consider task cv find satisfy requirement need need barrier find second namely modification cv call control variate reduce allow analytically resample beyond calculate control mean may prefer elaborate seed number generator store precisely simulated implementation complete store e vector path estimate location actual property generate much explore important question investigation involve study qualitative study implementation utilize discussion implementation database discussion input generate uniformly variance database cv scalar unbiased optimal prescribe classical cv define measure variance control estimator bias bias oppose probabilistic assessment database specifically approximate confidence distribution large early correspond initial cost involve assumption generate obtain average control approximately ratio control approximately ratio computational cost serve setup cost approach justify estimation instance fix set cost total estimation application setup cost view enable gain merely result intend give qualitative illustration efficiency specifically regular interest specific choice numerical qualitatively dynamic lattice evolve exhibit behavior temperature critical simulate path time total interest control estimator cv control correspond respectively choose anchor nominal opposite side third use control simultaneously micro macro approach micro variance macro simulation overall ratio control result total p quite exclude reduction cv scale control nominal substantial variance moderately close problem add control substantial compare control incorporate side temperature coverage estimator cv
singleton point form locate singleton cluster begin call join singleton way whenever hand move fall way originally come back cluster want cm pt circle circle circle circle cm right pt right circle circle call yshift circle circle circle circle circle node yshift cm cm cm cm node node leave right circle circle right circle large I ip move back join new center position come end say fall day singleton center configuration join back singleton give distance intra distance define positive value describe start location unitary radius constant later align vertical equal formally coordinate uniquely impose cluster solution uniquely impose mean term root positive radius center outer radius location intra distance intra distance observe c stage since cluster leaf join already next iteration claim join join x I verify c e ic w w dc c iw hold cluster stable stage cluster call join stable proved analyze iteration ii join proven call use stage singleton want iteration join proved point dp ip stage analyze wants consider stage want verify stable q cluster proof previous assume set center briefly simplicity begin want center towards move step affect iteration spread distance modify construction dimension mean conjecture hold construct require bind result iteration hold twice would see point next note would simplify optimal upper construction lie smoothed natural ask ordinary acknowledgement david confirm recently improve conjecture present plane widely propose local cluster close center center assigning repeat despite far cluster algorithm survey mine usage artificial intelligence biology graphic name particularly popular simplicity say pattern sample practice recognize theoretical trivially occur twice improve prove case show spread point polynomial upper improve since decade suggest truth show polynomially open mean dimension low spread aforementioned conjecture aim gap make use smoothed interested polynomial plane exponentially lie center run prove conjecture optimal rewrite lower easily translate analogously subset center among run construction spread iteration imply smoothed run center name position compute center assign definition close mass repeat center change set might situation center remove less cluster degeneracy close center break arbitrarily stress fall situation take actually cluster replace point unitary center mass close mean affect plane high behind construction formal couple extension center spread address py simple phrase day fall time time auto node thick black pt style draw center node style draw thick split text em rectangle split part center fall fall construction ii center clustering point indicate locate center move step fact note center build instance imply tuple set center center denote radius later similarly leaf compose weight union center center total center mention center coincide stage clustering go scale circle circle node circle node right circle node right node node xshift cm cm right circle circle pt right circle
confidence reveal signal n nj r nj p condition cover instance wavelet estimate high wavelet divide situation distribute variance define satisfie inequality strongly nest approximate setting albeit strong long rely also remark candidate estimator risk denote cardinality aim simultaneous interval end utilize nj nc oracle arbitrary upper entail maximal risk exceed factor close minimal substantially nest suboptimal leading term carefully special verify general estimator advanced multiscale theory latter lie bound eq conclusion hold naive naive w naive naive near precisely confidence contain entail entail ingredient extend inequality quadratic form furthermore z calculation one obtain exponential arbitrary particular nn apply consideration reveal great q precede bind second throughout assume loss let n place necessary pointwise capacity number cf entail utilize theorem subsequent remark start remark imply hand ready recall standard exceed consideration show prove assertion refer dual topology weak stochastically independent assertion immediate fact result process stochastically ne k moreover process increment write nj j n arbitrary n j virtue gaussian step ii eq suffice moreover accord claim ingredient transform index entail equality illustrate time dot line versus stochastically variable j kn kn z random proof two inequality constant eq entail strong part lead q entail definition entail bc together consequence oracle throughout shorthand furthermore generic variable neither index precisely consider rewrite combine yield n elementary apply certain nr nj jj nj nk attention q deduce simple statement need reasonable minimal risk equation nj nj nj nj nj k employ nj p n uniformly hand nj p latter remain loss denote generic first equal nj p n nj pr pr nj inequality shown restrict index maximal p n nj fix obtain finally arbitrary nc c c moreover large k n p p nc nd eq assertion risk arbitrary utilize tend hence assume fix conclude l nd r nc nd entail empirical process metric space capacity separable bound equip supremum norm path outer probability expectation modulus continuity denote process iii assumption x xx n n arbitrarily sufficiently assumption elementary consideration reveal eq converge inequality sum variable eq bound sample note set continuous path pseudo without let stochastically arbitrary follow fy fx fy fy fy fy fy latter functional constructive comment university national possibility statement connect numerous cf li light set approximate term construction multiscale coupling argument utilize loss within natural arise approximated focus contain characterized model require equal sufficiently modulus perturbation parameter vector adequate alternatively optimal risk appear class compete concerned approximating suppose together deviation represent orthonormal vast estimator eq euclidean know set oracle inequality read family candidate cx assume constant arbitrary framework gaussian latter particular fact partly setting et mention severe limitation li euclidean let use distribute versus alternative reject hypothesis pearson c chi square degree throughout paper asymptotic statement previous entail order space long set present similar possible consist instance minimizer naive introduction observation eq risk aim depend procedure minimize risk estimator subsequent mean I asymptotic statement unless unknown mx analysis nj nj nk nj nk nj k nj depend freedom identically distribute
theorem asymptotically observation achieve deviation cumulative incur symbol symbol loss hence symbol define p eq continuity norm eq family symbol symbol loss loss exponentially block behind assess cumulative minimum mapping induce bayes envelope clean signal give true true conditional track distribution underlie clean closeness clean empirical sure marginal mapping empirical subsequently induce density satisfy loss bind correspond eventually deviation cumulative propose formalize converge distribution use lemma analogous spirit output set subtle deviation theorem example growth growth subscript symbol universal give theorem establish universal optimality symbol sequence symbol section slide window denoise scheme slide window symbol sliding know slide window slide scheme length clean sequence necessity symbol channel process tuple symbol length successive super subsequence count symbol symbol fix subsequence symbol idea symbol slide window order optimality scheme symbol scheme estimate set contain hypercube slide window symbol slide window individual sequence monotonicity loss know clean well slide arbitrary extend loss subsequence empirical kk empirical symbol slide window order I kn slide operate cumulative incur sequence cumulative incur propose subsequence minimum slide window incur slide window entire wherein step width density estimate satisfy incur slide growth constraint specify bind deviation deviation incur length slide super channel eq k verify n particularly gaussian channel variance absolute growth direct lemma go cumulative incur bound corollary theorem similarly computational already fast kernel inversion polynomial slide scheme fast continue length contexts channel inversion scheme become implement lead symbol performance optimality clean symbol access underlie clean wherein establish incur scheme achieve ergodic analogous subtle difference due continuous output statement completeness compare underlie clean noisy illustrate apply gray efficacy additive noise gray underlie technique order accordance present heuristic modification practical aspect greater denoise white clean denoise context range context around value evident report square quality achieve improve denoise finally square translate length reliability see primarily aim demonstrate fully scheme channel benefit highlight channel h rmse rmse rmse right image simulate function clean gray location x location symbol favor efficacy scheme discussion estimate histogram clean smooth clean produce inspection evident able reasonably clean correspondingly image denoise multiplicative noise case qualitatively propose validate efficacy family salient loss draw enhance performance pass whereby neighboring context addition applicability scheme value suffer statistic alphabet size address underlie value alphabet simultaneously provide framework value function instead tractable requirement demonstrate experimental seem promise exploration aspect propose future currently direction study applicability design recursive scheme multidimensional video theoretical understand implication recursive optimality round satisfie lipschitz additionally densitie uniformly derivative universal infinitely mild technical enable marginal estimate true impose propose channel density differentiable family derivative alternative conditional channel modulus q modulus continuity member comprise proof lemma uniformly alternate formulation notion proof lx lx figure symmetric furthermore absolutely taylor expansion q kernel kernel absolutely derivative uniformly simplify limit side get extend derivative early outside continuous derivative smoothing family bound show limit theorem integrate x integration justify obtain pp necessary multinomial inequality possible cardinality expect empirical measure give outside new later call partition equal lebesgue third side fact partition infinite cut tail finite intersection care argue let stand small give nonnegative force symbol symbol case law law true uniformly bl fy channel b b dominate distribution density output density get continuity p use fy quantization statement lipschitz channel deviation function marginal channel correspond signal channel bound law law induced density corresponding discuss follow measurable continuous proof discuss measurable thus expression inside bound magnitude allow hoeffding leading theorem need two df definition df df df see theorem family inequality fact n opt x let close clear follow ax bound measurable also q ax ax lipschitz eq proof q equality eq eq fa fa equation f present proof symbol symbol merely leave ax iy measurable measurable bound law associate incur propose slide subsequence lemma theorem apply window possible claim necessary claim follow decrease increase integer claim nevertheless completeness bayes envelope concave show concave definition conditional measure absolutely permit e l conclude item martingale random k nf support probability function impose continuous mapping f n f bound end yield x kk lemma f side double imply convergence assume establish f combine imply hand lemma output alphabet level alphabet result output correspond mass tuple conditional density correspondingly form denote uniform quantization symbol q cubic histogram define n nj borel set rich borel algebra class cubic partition sequence exponentially sequence channel translate smooth number integer quantization simultaneously satisfy whose super symbol map equation lift get minimizer symbol rule sequence stanford edu consider reconstruct continuous corrupted mild regularity underlie clean limit sequence universal slide denoise optimize clean initially noiseless individual source randomness show fully noiseless stationary scheme attain practically gray white noise less conventional estimation slide window denoise clean mechanism channel find range bioinformatic significant problem notably additive various asymptotic optimality criterion scope beyond additive optimality establish sense design corruption mechanism restriction make provide know author corruption mechanism leave room distribution corruption cf provably compression improvement however much recent progress amplitude particularly microarray image comparative medical imaging parametric model nature universal channel therein attractive theoretically restrict problem collect noisy mapping estimate channel moderately alphabet leave challenge compression room improvement denoise discrete input propose quantization output similar show denoise underlying theoretical issue scope channel mild continuous propose two count statistic reference therein point reliable specify slide universal increase length large sequence development universal denoise image smoothing assumption inherent redundancy consistency denoise expect clean noisy symbol potential improve result incorporate pixel heart universal generality signal arbitrary regularity condition ii notation description technical iv sequence bound choose full clean extension slide window knowledge vi implication guarantee semi set clean process rather modify propose scheme clean alphabet extension set value clean take preliminary apply gray proposition throughout maintain flow state lemma symbol think symbol subsection empirical empirical component member form channel induce feasible density symbol correspond clean noisy achievable density could lie unobserve member close exactly minimizer minimized hence channel output additionally minimizer uniquely achieve minimizer distribution quantization level carry fig quantization step symbol manner else pa pa indicate distribution quantization length high end pa borel sigma quantization underlie clean correspond q carry denote quantization primarily motivate argue precision representation clean symbol value asymptotic result clean symbol attained formalize discuss envelope construct mass
proposition q asymptotic nz nm pool trend clearly limit due x bar covariance define side side depend estimate quantity even allow correlation nuisance independence suppose hold cc large estimator consistent pool inconsistent extra bias term constructing denote correction iteration n ik suppose zero example bias estimator normal inference coincide asymptotic estimation asymptotic correct rewrite estimator asymptotic subtracting subsection different bi replace fm correction serial correlation obtain iteratively solve correction serial make assumption asymptotic differently estimator correction iteration correct usual chi alternative augment regressor proxy slope fully estimator state rotation rate similarly loading rate ng bound integer criterion factor ignore lead however cross independence error assumption suggest ng precede common relaxed brownian process limit presence I regressor convergence rate I regressor normality distribution discuss independence trend intercept matrix estimation version identical fm estimator distribution brownian replace asymptotically estimate residual procedure prescribe replace brownian ik f tt whether underlying brownian motion include whether trend walk deterministic trend consider interested reader refer panel regressor stationary suggest robust common sketch argument observe argument l f ls mixed assumption regressor common factor f observe submatrix bi bi factor appropriate choice consistently fm treat construction correction degenerate choice bandwidth suppose regressor regressor consider inclusion loss mean q rule abuse square approach add exactly know situation asymptotically analyze correction asymptotically intercept fm construction residual regressor regressor intercept practice proceed integration major separate derive hypothesis testing scaling end one regressor long covariance fm bandwidth reader regressor factor practice whether I since conduct assess estimator compare stage generate design f generate gauss split series factor control control importance trend covariance estimate quadratic report deviation estimator replication less standard deviation consider perform statistic package clear statistic approximate standard interesting final stage table increase associate bias statistic except large deviation poor increase trend standard inconsistent consistent consistent estimator regressor trend exist property appendix follow generate uncorrelated relaxed lemma assumption across invariant conditioning number independent sequential limit prove rewrite ci martingale across ne I save schwarz prove central eq ne seq rewrite part proof proposition observable proposition supplementary et distribution hereafter suppose hold nm ik n seq uncorrelated q ik ik b ib f f f nm ik ik hence q notice nz nm x n uncorrelated ni na ij na b proper limit dependent also derive limit imply correlate bi u bi tx bi bi cc part n ik ik ia ik ik ik nb ia ni appear nb ik n ik ni nz ik mn ik prove directly ne ni examine limit fm modify variable f remove serial correction term bi bi infeasible fm f limit hold w bi bi define f let mn n modify cc cc u u ni prove un un bi un un variance bar version basically demonstrate focus without bar first argument result un un q un prove establish n u bi next f I I nx nx supplementary replace nx nx bi bi nx bi bi bi bi bi c theorems bi bi bi bi bi bi bi cauchy bi ii c nx bi bi bi bi nx p nx nx bi bi bi use lemma get nx p n collect since step basically nx tc tc un un I f u un un side corollary ng study cross trend standard induce trend procedure jointly slope trend bias correct update limit conduct statistic stationary cross pt concerned panel focus stationary common source stochastic estimator inconsistent slow trend hereafter slope jointly asymptotically standard test valid factor stationary regressor stationary bring dimension analysis inefficient impossible development meaning size series dimension large span horizon many researcher exploit rich panel panel issue tackle instead strongly persistent stationary explanatory put stationarity small root dimension treat analysis say literature panel serious drawback section cross potentially hypothesis panel grow panel test cross consider trend adopt component stationary component panel ty suggest analyze factor evidence factor correlation naturally panel capital production latent component trend spurious stationary treat jointly use estimator bias account bias serial correlation limit around denote asymptotic denote distribution nuisance continuously update coefficient use robust well regressor argue interesting rest basic panel trend modify f slope parameter common loading joint limit assumption loading I nf b definite w iw correlation ts ij ts ts u l panel assumption cross term principle sum process partition take together bm standardize brownian motion assume relationship f finally need
recently approximate moment small many frequency bound moment counter prove achieve optimal necessarily base successful algorithm large approximate stream conceptual various none study capture simple counter suffice low continuously particularly practically cc count account decaying estimation moment th moment lemma prove dependency say statistical estimation I estimator convenient analysis geometric harmonic mean ok harmonic gm achieve term variance q unbiased considerably accurate begin analyze skewed harmonic devote harmonic logarithmic stable simplified appendix yield variance decrease decrease small brevity simply rest rewrite concern euler constant denominator depend convenience equivalent lemma tail plot tail infer monotonicity tail q leave gm understand behavior precise value along gm extremely bind use estimator gm harmonic mean except harmonic nice tail property harmonic estimator j bias correct harmonic eq small ac logarithmic xx logarithmic norm estimation element stream practically fact eq show expansion logarithmic statistical soon count private communication develop algorithm stream suppose q delta hx monotonically stream compress counting moment consider different let estimate another suppose care balance choose precise compressed counting tail parametrization note advantage function task useful thing euler use exchange integration rest matter infinite cosine rewrite eq monotonically euler ok term monotonically e take suffice increase tw q tw show suffice decrease monotonically monotonicity rewrite show suffice decrease equivalent monotonically markov logarithm I derivative expansion rewrite eq prove let solution alternatively asymptotically expression minimize guess know express right attain skip proof leave eq series representation rewrite show must word euler series logarithm c rate bind lemma rewrite consider sure provide eq series finally consider number use treat positive q combine estimator bias order bias recommend version obtain taylor expansion correct bias tail instead q tail eq leave tail count fundamental counting frequency area theoretical computer mining accomplish counter trivial space memory moment stream strict data stream restriction minor technique skewed random projection capture continuously understand good entropy stream small th example might decay interest rate thus ideal decay approximate logarithmic norm estimation logarithmic practice heavy tail among fundamental almost simple counting time counting moment total zero actually sum denote example stream stream practically important stream challenge problem research time I count system answer ip account approximate moment stream skew sequentially describe meaning increment either positive either strict describe phenomenon example check allow compressed count applicable time strict cc stream believe suffice stream restriction frequency moment trivial counter increment count non counting system counter small physical may propose compressed counting count skewed projection introduction skew distribution skew fourier fourier unbounded inverse count
ratio long metropolis numerical coupling variate call ratio observable amount estimator estimate fig along ratio expect distribution datum rejection metropolis decrease fig pair distance nearest coupling control correlation show circle square near product circle fig number location nearly vanish interior couple site boundary near couple explanation couple site number agree couple interior system number neighboring site agree spatial variate accuracy mcmc coupling variate less twice copy site variate offer simply copy perform solid ratio correlation estimate time reverse direct computed unit formula tell ratio write sect reflect gain factor situation plot coincide plot instead integrated autocorrelation reduce variance harmonic lattice nonnegative unlike system site rate energy pool I energy jump new density dynamic site interior provide heat equilibrium product equal temperature profile reflect correlation similar limit difficulty estimate attain profile simple coupling two copy copy heat use randomness nearby site font lb lb lb lb lb lb sect slight modification jump simply replace density singular consequence nonetheless check ratio rl interaction nz apply sect coupling preserve cc square mean cc solid open square reverse numerical ratio site couple amount reduce range fig ratio product distance ratio tend ratio control variate despite coupling variate estimator even improve conclusion effectively accuracy calculation situation case one coupling variate result suggest various observation coupling control variate large correlation original try trade idea like site heat partial well variance net issue dependence desirable error interest mention related coupling depend observable use curse dynamic impractical distribution common path likelihood ratio score ratio generalization thank point reference support department contract er kl support science fellowship pt lin calculation situation steady illustrate monte markov leave carlo crucial explicit steady markov improve factor situation approximate steady technique coupling control build early assume explicit steady value basic second invariant closely correct simulate couple expect steady approximately local equilibrium equilibrium detailed balance boltzmann chain equilibrium interest theory heat consist site couple heat chemical etc steady boltzmann heat lattice steady locally equilibrium heat govern distribution temperature sect show equilibrium idea work cite another molecular markov analysis exact monte recall mc variable want estimate reduce variate expect estimate use variate estimator eq variate var initial monte carlo independent successive let homogeneous continuous finite completely specify jump x rx unique steady x x converge surely integrate observable mc typically aim reduce either autocorrelation notion look notion precise process transition couple markov process project component project onto stationary computed coupling control variate estimator variate simplicity variate compute state know available rate transition rate markov process range model case couple markov lyapunov exponent sect factor scale variate integrate autocorrelation time respect keep reject process slow correlation time amount may competition overhead run complexity coupling coupling computationally expensive effort easy process lattice reservoir place maintain net particle hold particle thought follow carry particle jump site equal probability leave rate rate reservoir act site analogous manner see particle result unique convenient control motivation understand quantity interest great distant particle density easy equilibrium become iid nonzero distribution product nonzero dynamic detail balance correlation well weak correlation number entail
indicate plotted spectrum symmetric show constant increase completely weight frequency process bias correction estimator frequency kk kk figure calculate display length intensity correspond trading day notice high frequency moderate intensity market ratio fourier integrate volatility multiscale ratio specify plot kk right two correct figure plot ease comparison single estimate white multiscale realistic aggregated process display multiscale ratio plot suggest frequency summation across frequency make integrated path matlab kk kk consistently parameter limit information multiscale decay high spectrum appear worth note multiscale process ie equal volatility realize volatility observable multiscale bias variance calculate estimator denote perform estimator realize volatility high estimator competitive first approach volatility parameter marginally remarkable correction globally weight realize integrate produce study henceforth lead order deviation estimator remarkably rmse expect become order rmse close integrated volatility estimator new volatility l correlate noise model corresponding multiscale multiscale estimate white good despite nuisance structure multiscale white remove j lt lt ds du k lt ds tt tt since combine calculation q acknowledge drift establish x compare contribution well variance observe converge deduce eq find normality via see length time inverse signal power use deduce firstly estimator transfer therefore therefore f x x jj j jj kk jj kk jj l j jj k jj note kk term identically principle z z z x realistic decay decay rapidly integrate need integrate note df df smooth smoothing delta non axiom claim conjecture corollary theorem theorem remark multiscale observation multiscale represent quantify realize integrated volatility due multiscale estimate bias procedure extend correlate imply integrated performance simulation bias correction market volatility decade available diverse area molecular essential physics efficient make inference set science inherently sense abundance temporal quite simplified coarse grain describe essential reduce subtle simplified compatible clear contaminate multiscale contaminate share wish compatible misspecification system usage quite extensively last year first presence high frequency observation noise various similar context e sde quadratic variation estimate model use bias available quadratic integrate volatility aggregate necessary integrate integrate suggest contaminate high frequency secondly fast rigorously show paper make inference coarse grain sde generate slow system examine estimator biased diffusion coefficient grain likelihood subsampling rate characteristic slow variable explicit scale separation paper inference multiscale high property frequency domain technique enhance study multiplication domain estimate e process domain bias integrate volatility observe frequency contaminate high de accuracy increment domain process additive upon point process sde brownian motion see example drive three correlate observation relate eq spaced noise estimate volatility integrate volatility realize integrated volatility market estimation necessary roughly dft sample volatility write dft variance coefficient heavily variance frequency unknown integrate via reduce show competitive noise formulate readily generalize correlated property analysis state domain result carlo various include give simplest integrate volatility ignore integrate realize integrate inconsistent biased comparative define realize integrate volatility derive firstly define increment series u jj u examine time series inefficient k j ds approximate formally need determine complex value h n n approximation coefficient strong integrated volatility relationship uniformity dt dt modulus bound transform decay also precisely sample white naive k frequency specification naive sampling integrate combine notice inconsistent equivalent estimator domain contribution biased usage multiscale optimal correction quantity knowledge simplify k continuity n x consequently lead order naive estimator multiscale shall develop observation frequency estimation see satisfied approximate improve possible stationary meet kk kk produce energy likelihood stress merely device multiscale estimate multiscale give appendix estimator integrate volatility define naive moment thus remove decrease variance purpose multiscale construct integrated take property unbiased imply inverse signal square root period compare variance shall relative argue less intuitive frequency spectral estimator smoothed view transform smoothed q continuous utilize integration plane decrease sequence lf rapidly
maximize subset leave tree leave distinguish call exist standard weight binary classification build multiclass starting bound offer guarantee regret previously classifier classifier classical cl due explicit build classifier error root give error copy worst optimal regret measurement prediction choose upper good depend fidelity state form training similarity fidelity symmetry property fidelity efficient require state bernstein pure perform state close euclidean give acting probability predict class two future testing different section affect algorithms ml experimentally compare numerous publicly repository university california character experimentally quantum represent realistic situation likely access description want simulator possible moreover situation quantum information processing maximize discussion fr ed proposition conjecture learn reduction et op universit centre quantum classification unknown ensemble copy state discrimination within machine notion come variant multiclass pure state predict quantum study back seminal field course ensemble pure hilbert quantum state live copy receive take idea insight help characterization task amount instance copy perform originally take give goal fidelity put put complementary quantum receive copy quantum training state produce generalize accuracy allow relate task task discussion field task predict observation goal discover find meaningful represent empirically ml consider machine assume computer classical logical circuit point use classification classification person classify belong detection music classification etc main latter number quantum dataset pure quantum pure state live hilbert form interested binary dy work quantum remain generalization object superposition allow mixed intrinsic define quantum dataset contain copy explicit latter powerful description produce quantum copy formalize notion concept dataset goal class subscript ml quantum everything classical exception something specific usual sense learn another computer learn quantum serve concerned description e challenge universe atom individually memory bit atom superposition atom suffice store quantum copy make potentially extract impossible hilbert copy quantum see instance quantum learn copy ml copy particular obviously define class task class high hierarchy belong since correspond quantum form operator hierarchy relation form hierarchy copy tend infinity reconstruct description precision cl copy potentially information integer copy copy must allow interact potentially measurement plus interesting question strict hierarchy cl reason answer positive use constructive manner task notion formalize box oracle learn task model although desirable differ sense characterize computational learning progress moreover primitive task box solve problem instance classification generally relate subproblem global predict quantum state characterize precisely back essence discussion error regret range meaningful context raw measure inherent difficulty situation possible even set high regret quantum distinguish class level associate indeed copy perform count copy copy act classify reduction equal call multiply copy task copy description complete knowledge quantum copy state classifier class give case available define manner classification act quantum copy construct minimize training error n question ask hope error occur generally devise copy even advance state reveal minimize error course quantum negative positive total negative complementary class true mixture manner n draw idea class inside single choose class determine choose description state return box identify minimal mixture kind quantum reach distinguish class bind binary follow directly measurement case measurement mean measurement quantum set constructive come close difficult characterize relationship copy state contrary classical ml always generalization bring near quantum express unless impossible draw unless state ensemble include estimation copy classification copy classify quantum come strategy correspond state specific measurement sometimes confident known know choose minimize answer confidence perfect discrimination exception maximize measurement act classifier classification construction take copy quantum train form circuit oracle cost copy implementation measurement learning situation quantum possible efficiently circuit description correspond realize quantum could circuit number assume burden act require design binary task practice fundamental implement implement explicitly copy focus oracle albeit almost access number argument number copy spend quantum world measurement situation generally mean outcome datum associate correctly state wrong class classification w dy act class quantum minimize rate directly incorporate description reduction solve binary classification oracle description convert eq equal complementary demonstrate incorporate weighted suffice call copy unknown quantum minimize measurement measurement minimize discrimination et correspond matrix minimize weight bin flip keep copy put aside new distribution tt oracle classifier finite copy quantum efficient enable reduce weighted classification proceed mechanism algorithm algorithm generate binary choose final simply classifier form classifier clear standard training call oracle multiplied require reduction demonstrate average minimize error global eq version rejection additional copy quantum generation state weight high probability keep aside state among classifier copy class multiclass dy act multiclass unknown quantum state multiclass classifier minimize move binary far thing know root classification copy unknown quantum logarithmic section way class state recover unless state identical label copy available state require strategy analogue training search quantum identical classification formalize fidelity test fidelity maximal state identification quantum state copy copy global cost identical near unchanged facilitate near retrieve copy na copy consider know description state negligible kt bin empty bin j binary adapt act density compose click predict state click reduce k class random binary call call copy binary lead cost reduction situation happen class click negative uniformly false classifier click error binary classifier rate classifier simplify rate
rigorously mathematical evidence light wavelet coefficient localization localization value location sphere non plan section isotropic field sphere construction establish provide necessary statistical application concern field hold square triangular orthogonal random angular power spectrum call spherical I lm lm lm spherical well spherical complete many establish usually argument instance spherical harmonic mean field angular must slightly condition sequence fully region motivate wavelet construction detail completeness function immediate verify corresponding ensure main show localize motivate spherical immediate jk jk jk lp jk last well know property random condition inequality jk jk jk jk main compactly tight enjoy firstly cover manifold moreover nice benefit certainly valuable practitioner report spherical wavelet exploit paper sound vanish eigenvalue obtain function frame tight make rigorous formulae exactly infinitely entail minor practical possible spherical coefficient lm jk dx pe lm jk consider construction shape allow us asset practitioner shall drop confusion mention suggest widely wavelet stochastic property tangent point jk write develop argue normalization asymptotic theory validity implication currently investigate introduction development spherical field circumstance extend construction full characterization positive write provide coefficient statement method entirely believe independent constant idea notational l denote use integral follow summation recall transform computation b integer noticed argument differ side j p modification case pc numerator correlation inequality q let back cb similar omit sequel pe pe without argument pe j argument cb p explicit show circumstance angular decay slowly enough extra log technical difficulty deal integral transformation would asymptotic indeed compactly support various automatically observe field correlate establish indeed correlate angular memory different uncorrelated introduction asymptotic behaviour asymptotic crucial play constant across different usual drop component angular decay fast prevent angular spectrum condition exist jk jk divide variance coefficient b j b pe j show easily pe j b j similarly second c pe j pe l p dx e j pe pe denominator summation numerator consequence jk b j pe lp j pe pe p jk jk l pe jk pe pe pe complete result illustrate property spherical choosing introduce high improve localization localization frequency lead pixel uncertainty see instance issue evidence phenomenon decide assumption highlight hold fast polynomial simplify version easy decay dominate component correlation application polynomial function coefficient polynomial cardinality mesh statistic exist condition example polynomial angular power spectrum jk b view presence miss observation noise consistency asymptotic observe consistency parameter unity implement skewness wavelet instance jk jk jk jk jk isotropic q polynomial establish moment omit brevity noted consistently subsample argument extension circumstance rely inequality wavelet statistical functional much challenging relate investigation theorem axiom conjecture criterion definition example exercise summary structure wavelet sphere label recently uncorrelated frequency domain statistical also spherical wavelet asymptotic primary rapidly grow wavelet system reference motivate interesting application concern emphasis devote background universe
per dimension reach indicate recovery model quite many setting experimental benefit analysis behavioral question mind multivariate estimation recover behavioral brain activity rather know ground truth x impose couple distinct goal state couple side determined site result present brain take additional simulate hmc chain take couple state result depict comparison correlation brain state depict successfully recover depict second metric couple show third range interaction might many instance correlation example dense spurious sample state display indicate angle bi effort statistical circular break effort analyze major previous correlation true probabilistic characterize univariate bivariate see notably capture hyper unimodal difference capture instantaneous blind analysis slice capable direct example relate modeling interaction causality distribution instrumental attempt infer causality technique recover observation also neural acknowledgment would comment manuscript nsf grant grant figure true thank thank title pt c author ex plus minus minus ex plus ex ex plus ex ex minus ex ex pt minus plus minus pt pc em minus plus minus minus minus pt california berkeley berkeley ca berkeley edu circular phase orientation considerable attention prominent analytic phase little phase capture estimate parameter distribution coupling area different behavioral broadly circular circular orientation represent physical signal system couple range couple couple describe coupled nature chemical reaction coupling see phase fourier analysis scale receive bind establish surface site band hz phase phase highly relationship relationship phase figure section common task analytical statistical multidimensional may provide adequate probabilistic binary variable lack effort turn offset conclusion effort restrict offer paper provide solution motivate examine empirically phase grid band introduce capable capturing empirically provide recover demonstrate weakly couple investigate dimensionality could behavioral state dependency phase variety examine phase band human experimental ref human patient phenomenon capture distribution datum complex dimensional hermitian positive definite normalization real computation contain depend set function follow j expectation quadratic jacobian compute derivative produces far demonstrate recover simulate weakly couple diagonal hamiltonian e th equation second integrating obtain simulate couple angular model system couple couple coupling third directional simulate term third variable concentration e correlation depict figure ht distribution wise phase distribution recover technique sample recover pair marginal able equation estimate depict importantly indicate indicate compare produce empirical carlo use carlo sample hmc closely match couple flat close ability
universit paris paris universit paris fr application number component mixture proper alternative density truly label switching issue solve conjugate rao markov choice mixture modelling nonparametric unknown uncertain inferential goal true especially weakly informative provide always generate component component standard consider choice perspective define corresponding solution chen derive reversible jump fundamental reversible jump mcmc distribution rather inherent randomness seem finite thing propose reversible jump accept scheme introduction rely conditional true likelihood nature force probable probable lack adjacent space course use reversible probabilistic less large intermediate answer proper simulation additional jump explore separately produce therefore correctly approximate posterior approximate discuss instance stress correction due proposal recall mixture present correction demonstrate correction recover eq rhs constant approximation mixture approximation rao latent variable set variable indicator ii latent prior theory approximation demonstrate via significantly constrain constraint constraint mode distribution demonstrate component posterior exchangeable set permutation accord word distribution permutation bring order integrate factor randomness recover switch symmetry posteriori rao argument replace take predict theory principle state show next recover miss lose exploration mode point start explore mix lack exchangeability may call additional simulation start mode distribution may secondary standard occurrence approximation use device simulation population produce sample product benchmark galaxy represent radial galaxy variance switch mostly occur marginal simulation introduce lead note gibbs mode exactly check force likelihood fourth digit similar
proof omit describe probability accumulation selection apply every subsequence subsequence subsequence case local alternative describe sense deviation detect one picture zero asymptotically namely note deviation would pick consequence shall later fact post selection limit govern slow ii govern condition discuss precede certain every furthermore nonnegative tuned selection precede guarantee estimator equivalent fact estimator normal finite sample n p x expression n n x follow quadratic equal precisely equation give n x treat similar fashion obtain finite atomic absolutely shape highly plot mass locate asymptotic move asymptotic asymptotic depend note early picture complete accumulation remark surprisingly result tune perform follow theorem converge normal distribution obtain coincide however clearly disagreement zero capture also coincide fact limit value e cdf mass follow x mass coincide maximum provide shift infinitely get additionally precisely impose oracle translate theorem estimator satisfy property maximum case oracle guarantee carry mean validity oracle property limiting framework essential feature particular precede consistent sample stochastically property picture face singular b n reasoning property think highly adopt diameter order classify converge procedure zero choice selection naturally parameter statistical condition favorable mention particular inspection reveal stochastic phenomenon boundedness hence uniform see precise next trivial reduce hold next scaling limit degenerate something tune selection scale I unbounded asymptotic stand finite f move asymptotic limit weakly converge cdf weakly cdf apply subsequence subsequence subsequence comment apply consist already arise sequence phenomena theorem local nature tie way parameter tune limit small rely asymptotic reveal limit classical asymptotic reveal reason pointwise distribution convergence former underlie discussion post restrict orthogonal regressor correlate regressor phenomena normality uniformity section correspond correlate regressor result sparsity cdf cdf consistent straightforward estimator distribution choice subsampling bootstrap possibility hypothesis different test limit ensure large sample formula cdf behave case sense eq bootstrap subsample arbitrary theorem follow speak estimator suffer error phenomenon tune less error condition trivial conclusion uniformity phenomenon estimator occur estimator less gives result cdf n infimum extend clearly tune large sense cdf arbitrary n extend consistent compact origin n trivially consistent trivially estimator sample case regressor finding show marginal center exhibit derive orthogonal repetition block cholesky regressor value monte study apparent reason neighboring case equal zero lar choose way consistent selection require select minimize fold cross lar package figure center estimator represent dot draw height histogram form value smooth absolutely finite naturally rescale appropriate relative zero distribution practically outcome line oracle property predict fact atomic absolutely part shift origin absolutely perfectly demonstrate oracle give estimator n picture except component less finding validate regardless value consider tune find act conservative rather theoretical agreement result absolutely normal cross depend situation speak validation value estimator penalize least square sample regression find mixture absolutely estimator tuning distinguish tuning model selector adaptive uniformly asymptotic reflect asymptotic size picture moving irrespective close finite phenomenon observe occur sample case tune asymptotic phenomenon occur move asymptotic even sample find actually consistent property property give result finding section adaptive orthogonality restriction remove analysis simulation confirm finally scale cdf ease would stress result read lasso quite prove observe n n let z prove z last zero z nx nx n follow second claim analogously result converge go zero limit prove atomic furthermore atomic prof replace n assess indicator function limit elementary consequence n prove assumption prove observe part observe w xx limit exist elementary result proper follow continuity consider first converge short elementary particular imply supremum f rest argument similar analogous except place axiom condition conjecture corollary theorem example exercise notation solution proof wang wang case finite well typically highly regardless slow tune particular question statistical theoretical orthogonal study primary f j distribution uniform consistency penalize study prominent related bridge lar smoothly absolute scad fit hard soft thresholding many penalize understand distributional distribution
several interest conclusion draw mdp set action agent agent give start discount agent tell choose agent mdp determine process agent policy maximize discount total easy policy transition distribution instant bellman eq upon bellman mdp mdps bellman equation iterative assignment start arbitrary sake compactly maximum map value well norm contraction contraction banach exact compactly initial bellman equation furthermore require shall mdps hold effectively compact polynomially apply factorize markovian combination r kk r substitute right h assignment general image put project right hand start expansion converge compactly contraction banach matrix examine choose projection shall fit value usually projection square well book reason expansion case enforce q define projection minimize max computation define turn contrast programming et al minimize norm bellman constraint proof compatible require solution program step hand know operator compatible normalization element therefore absolute increase element nonnegative sum assume least normalization role note otherwise projection value hold h h statement transformation triangle expansion error proportional represent get optimum check basis argument association lp base mdps attractive sequential exponentially large mdp description factor process product space notational full space derivation require depend factor formal variable z denote index notation specify full vector value shorthand index function small local function different action take probability factor scope factor markov characterize j I ir appear description value represent efficiently unfortunately case scope depend basis function select apply overview method concentrate factored form transition reward yield k rearrange operation exploit occur local scope compactly notation eq contain basis index notation scope mean computation unfortunately many equation subset consequently scale sufficiently necessary determined later reduce original sake simplicity assume projection correspond update true let unique solution constant polynomially proof closely although norm instead factored structure I kn solution factor mdps infeasible mdp factored first algorithm hybrid factor partially mdps mdps alternative mdps first mdps major approximate decision list way policy compactly et present unfortunately grow representation bellman equation transform program sense optimum programming linear form exploit appear decision formulate lp van maximum scope function compute serial program accord et cost exponential network undirecte variable appear course elimination order problem order hard order yield width approximate long approximate lp dual program report policy bellman policy evaluation costly explicit decision tree representation grow artificial test york drive lp task consider model successfully generalize task far algorithm one factored exist structure bellman minimize add piecewise splitting sampling validity de van thorough overview technique use specific programming linear polynomially reach error however probabilitie new factored markov decision approximate iteration projection max preserve secondly uniformly become size convergent projection combine provably polynomial time avoid programming acknowledgement grateful attention article al research ec grant reference manuscript ec project wish interested implication rigorous claim subspace figure radius sphere ensure project fall region calculation let sphere radius absolute trivial statement note polytope consider cone vertex edge vector pointing consequently contain image strict contradict complete function sum identical entry without loss stochastic auxiliary mdp consider observation factor identical item order transition probability observe step infer state agent mdp probability path mdp use uniform reward correspond first
maximization em em maximize define informative distribution hierarchy fully adopt apply semi hyperspectral paper reconstruction principle image problem iterative thresholding penalize require maximization often maxima present quickly produce estimate entire indeed hierarchical bayesian generate image hyperparameter response extend blind deconvolution outside scope organized deconvolution bayesian accord extensive reconstruction report recover projection eq stand function device additive noise convolution version describe convolution spread address section consist bilinear dimensional observation q denote norm section prior consider eq lead penalize positivity constraint lasso component priori allow one x coupling atom encourage instance spike train deconvolution order increase distribution similar atom introduce example account positivity pixel imaging component independent introduce conjugate inverse gamma choose choose inverse gamma hyperparameter aforementione accuracy hyperparameter sometimes knowledge pixel case parameter manually value situation information conjugate gamma scale intensity similarly informative jeffreys gamma density noise q prior yield informative jeffreys informative hierarchical prior yield complex example pair gaussian spatially vary noise ratio zero image sparsity factor hyperparameter represent acyclic dag hyperparameter obtain stand beta uv gamma present image distribute describe accord posterior chain maximize subsection sparse hyperparameter pdf imply simulate inverse prior pixel condition intensity computation appendix stand stand truncate distribution use variable generate l description name external field value moment gradient depict use fig derivation model define convolution leave size fig panel different variance ratio process propose gibbs complete intensity depict dot red line propose hierarchical em unknown hyperparameter therein base perform bayesian map mmse bayes mmse averaging sample sampler recover point human observer visually non intensity abuse notation since reconstruct computed paragraph snr measure bayesian propose map outperform mmse image sparse sense pixel pixel balance course reconstruction low square error error mmse map propose mmse prototype projection space spatial sample subsection adapt concrete image pixel depict recover right result image image image assume observe application two operation hierarchical bayesian sparse burn propose illustrate real projection collection constitute whole follow scan depict nm nm nm nm slice bayesian scan experimental e resolution fine slice rate define unknown initialize iteration reconstruction horizontal slice figure mmse estimate reconstruction produce project spread visually evaluate goodness fit raw fig figure clearly agreement reconstruction represent figure propose significantly hierarchical algorithm corrupt truncate exponential propose image mmse gibbs approximation estimator outperform reconstruction reconstruct method uncertain spread investigation molecular prior comparison link q identity decompose whose element rewrite compute similar truncate hide expensive w hide hide bernoulli bilinear appendix recursive accelerate sampling updating consist draw gaussian simulate simulate evaluate bilinear simulating moreover exploit property consequently stage carlo similarly q therefore bilinear gibbs evaluate simulation f I iw accord variable distribute bernoulli truncate pdf draw indicator take pixel simulation truncate positive truncate appropriate accept instrumental distribution acknowledgement would technology provide point university suggestion grateful university france ann mi usa fr edu hierarchical image observation obtain corrupted account positivity appropriate hyperparameter tune marginalization propose sample use e posterior fully posterior information propose sparse reconstruction sparse illustrate collect prototype deconvolution image sparse decade deconvolution deconvolution reconstruct optical device
interest bandit minimize minimized equivalence cumulative one interpretation simple tool prove result armed interval topological call mapping latter bandit figure environment want environment round forecaster action strategy cumulative regret I expectation classical armed forecaster choose forecaster recommendation environment next allocation strategy recommendation armed family environment environment respectively exist respect auxiliary course requirement see recommendation give notion henceforth concentrate exhibit environment find rely designing strategy exploration exploitation exploration fed environment section environment result armed bandit form generalize example let family technical ingredient give set exist report environment main respect distribution dependent compact space proportional link article statement family allocation strategy environment satisfy cumulative regret informally round get reward play get recommendation play distribution form integer regime regime use ss keep obtain regime eq explore come exploit rewrite conclude cumulative ingredient characterization separability rely application subset separability characterize existence full direct exist converse implication separable dense sequence contain least implication separable open ball positive ball impossible many converse provide rely somewhat depend set regime begin draw formally tie break play phase already denote regret decomposed approximation vanish open provide tie break number quantity hoeffde twice regime q substituting sum superior limit true lemma denote let ball consider e aa associate environment get reward forecaster expectation forecaster vanish outside eq existence indicate thus fix behavior forecaster yield payoff round forecaster pick payoff payoff pick integer measurable countable positive mass empty forecaster get nan consequence behave measurable recommendation adopt la application define particular separable countable thus immediately uniform bound individual environment first sublinear continuous environment infinitely disjoint however specific instance totally form acknowledge research grant application ec abstract statement since suboptimal quantity function variable difference range quantity plus quantity sum classical use moment generate hoeffding lemma maxima subgaussian variable claim put thing sufficiently precise result part ucb arm large g apply choice suboptimal optimal suboptimal allocation recommendation arm moderately large allocation much bad whereas ucb strategy focus well corollary paris france paris armed bandit possibility term regret capture round necessarily advance cumulative resource discuss cumulative regret result forecaster cumulative former mind end devoted regret able separable distribution payoff bandit armed regret exploration get environment action exploitation bandit stage forecaster give arm get assessment criterion forecaster cumulative reward obtain exploitation known ask recommend arm average armed evaluation occurrence armed problem give medical severe disease ill include pick treatment minimize phase coincide product phase minimize perform pure address cpu making occur preliminary term resource come budget motivate recent cpu play search hierarchy instance strategy right cost explore option one actually final maximization whenever costly cost issue possible order good address exactly strategy make precise resource turn restriction forecaster ahead amount pure exploration present armed bandit open another notion related exploration number round aim article possibility achieve probability indicate identification arm assessment criterion forecaster either wrong closeness recommend suit free present simple rely exploration exploitation simple cumulative uniform arm benchmark exploration round regret uniform exponentially fast somewhat lower indicate small simple theoretical arm topological simple regret tool exactly sublinear cumulative family function contribution decision armed parameterize fix reward independently previous arm round associate forecaster simultaneously secondary one secondary exploration forecaster forecaster refer strategy shoot environment exploration refer forecaster summarize forecaster randomization definition reward forecaster choose environment draw notation introduce output take next start pure armed recommendation pure bandit forecaster denote reward sequel gap equal recommend quantity interest regret popular treatment armed bandit ensure see g borel quantity differ indicate arm arm immediate recommendation simple regret randomization regret lead smaller guarantee upper interpret exploration exploitation recommendation exploration design arm exploration past payoff exploitation corollary quantify need exploitation asymptotic regret forecaster logarithmic suboptimal arm strong low forecaster bernoulli reward allocation recommendation strategy bernoulli reward constant reward collection put differently merely therein tuple might get fraction order tuple sometimes tuple equivalently bernoulli distribution reward prove far keep mind growth ucb suffer distinct put piece n implication pointing moderate phase small natural allocation strategy mostly link cumulative really sophisticated upper regret allocation second ucb exploration parameter exploration suboptimal logarithmic round play let broken arm allocation recommendation recommend play arm play formally history play action reward group accord arm compute form recommendation tie play tie break summarize distribution two family constant pr ucb ucb bind upper allocation row recommendation strategy universal whereas within show two hand together play perhaps expect bound decrease simple similar arise versus free table cumulative prove consider use exactly stochastic indicate reference happen base round consider explain combination allocation indicate briefly decrease exponential dependent distribution recommendation empirical denote prefer simple round prefer recommendation instead dependent strategy ensure arm inequality random quantity bound q inequality concentration find free bind corollary yield allocation recommendation empirical arm round supremum simple take whenever study recommendation distribution round statement interest second state theorem comparable ucb ucb recommendation play arm kk bind corollary surprisingly dependent large small play distribution free ucb allocation ucb recommendation give play ensure supremum tuple distribution easily arm allocation strategy associate ensure arm one suboptimal use summation second suboptimal get q exist suboptimal arm arm thus extract arm integer part third inequality recalling write conclusion recommendation choice
consider mcmc burn summary consists perform previously detection consider material display tolerance obtain example simulate sir consist abc tolerance large tail tolerance scenario abc target prior typically detection time summary statistic partial estimate detection summary type contact number positive type summary mean detect compute weight follow spherical x k span scale summary case convergent tolerance increase keep variance bias may phenomenon curse paragraph present curse principle general setting within obtaining couple accord sample draw correction possibly depend uniform formula adjustment inversion practice constitute sample describe use local regression approximate weight squared consistency correction nonlinear regression relax linearity denote adjust feed neural adjustment neural regression expectation approach regression whose transform logit non negative logarithm transformation synthetic epidemic population group year four estimate summary statistic rejection correction summary final detect estimation posterior perform simulation sir mass prior order magnitude degenerate half life detection contact quantile except summary mode less biased low four possible ci variant tolerance slightly increase tolerance bias rejection increase remain considerably variable method adjustment comparison rescale mean rescale q point obtained find small obtain tolerance rate always statistic four additionally rescale ci ci display rejection increase obtain intermediate motivated sir contact path statistic abc simulation different rate contact tolerance first epidemic tolerance rate error epidemic rate sir simulating path sir distribution posteriori statistic posterior predictive obtain summary consider detect individual contain explanation age take infection order non markovian effect infection maintain infection pressure constant model action rate figure supplementary material mass contact linearly detect rapid comparison extremely contact suggest summary contain provide consider dependence trajectory compute ci ci ci infection yet dynamic predict sir individual contact individual interestingly really predict individual detect random period follow less slight discrepancy year real predict contact reveal contact mode contact new name test contact compute epidemic display sir predict coverage since sir model predict coverage still contact individual per screen contact equal ct ct order epidemic contact almost simulation sir evolution sir infected epidemic individual proportion proportion contact display sir contact individual contact infected period time sir screen individual contact context temporal infection rate abc computationally intensive population population individual whereas database individual dimension infection imputation mcmc demand day experience algorithm method monitor abc evaluate favorable situation practice estimation sir mode provide tolerance generate tail abc application restrict moderate whereas involve constitute way abc set small error rejection however method contrast method particularly less abc however adjustment develop sir posterior model day rate contact infected detection probable contact contribute largely incomplete percentage individual acknowledgment grateful pr h university la dr disease thank anonymous valuable suggestion material http www cm cm cm theorem corollary remark p universit france miss infection partially observe monte propose rely numerical simulation distance weight propose abc summary model recover sir distribution sir contact comparison evolution percentage sir population simulation base understanding predict spread disease evaluate health although describe importance realistic quantify confidence standard recover sir sir model recurrent issue infect population infection miss involve integration mcmc treat mcmc computationally prohibitive missing proposal sir miss abc originally make simulation motivated contain individual database individual detect contact contact detect individual infection time introduce equation size individual contact material apart inherent epidemic particularly small sir deterministic simulation model consider formalism process abc alternative density
actual train truly fire modified spike receive elsewhere efficient discover train pattern also significance pattern present efficiency simulator spike network neuron model whose rate firing update ms real number period poisson time fire neuron total input neuron see weight take spike neuron time delay unit fire determine background rate fire fix background fire specify neuron low method specify background fire keep conditional instantaneous firing specify reach instantaneous spike background fire rate fix neuron fire neuron neuron absence strong firing neuron specify expect hence also simplicity specify stated paragraph connection fire get weight mean fluctuation nonlinear determine operate sigmoid fluctuation direction neuron significance still effective simulator run neuron connect neuron uniform delay background firing assign connection choose interval probability background fire hz fire correspond probability decrease direction around fix incorporate simulation experiment probability spike train period neuron neuron neuron neuron fire randomly effective conditional probability range hz firing resolution neuron connection vary factor compare incorporate connection among episode episode episode episode strength connection strength node episode episode span probability connection delay ms simulator spike train sec duration spike obtain occurrence episode present repetition datum sequence duration mine dual core couple minute simulator update ms spike receive neuron especially independent process show replication neuron connection neuron calculate non count give accurate episode episode value easily count panel panel node pattern count show replication value range standard indicate well capture non capture count explain significance strength observe episode strength connection count per test confidence unlikely episode strength inferring actual strength connection simulation infer strength actual replication infer actual strength significance sequential occurrence emphasize count connection strength observe episode particular call infer strength cc value replication cloud fit line result episode quite infer base count finally test correctly sequential episode episode calculate significance nan hypothesis varied episode test episode strength higher significant mining count loose relative different sequential address detect statistically spike datum efficient frequently occur pattern prescribe inter delay frequent occurrence occurrence make contribution statistically different pattern assess mechanism one neuron specify term user delay compound include many dependent influence weak probability reject conclude pattern represent significant connectivity interestingly term strength influence conditional independent usual much high decide interaction method specify hypothesis computationally count use frequency statistical occurrence possible probability bind present occurrence count occurrence count relation variable chebyshev loose effective appropriately illustration extension analyze pattern calculate suppose hypothesis probability assess episode use applicable without course specify distinguish accommodate idea fire pattern number non occurrence fire pattern possibility fire nan appropriate expression could analyze occurrence fire sequence suppose significance length occurrence presented assess instant use assess issue mine discover sequential pattern computationally issue spike sec duration background work spike neuron fig certainly strength differ notice plus sigma count fig reliably strength say good level obviously strength suffice issue conclusion mining approach problem discover fire pattern temporal mining literature partially use discovery serial episode episode would discover occur mining technique spike train present lot specialized handle many train explore issue acknowledgment help simulator well analyze stream mr simulator mr report support project science statistically pattern spike train characterize spike neuron delay spike previously frequent count overlap occurrence propose repeating include neuron neuron dependency strength certain put construct model capture process allow spike train homogeneous spike train hundred neuron significant micro record neuron spike neural slice even human would mixture activity external input discover spike train neuron manner analyze infer connectivity spike brain embed train lead brain ultimately systematically question pattern fire neuron lag delay successive neuron denote event spike connectivity trace activation find neuron pattern fire fairly delay successive detecting assess significance reference fire sequential pattern pattern order fire delay unit time fire pattern spike spike delay pattern delay interval time measurement pattern involve neuron pattern pattern due slide spike another specific delay detect pattern train statistic expensive detect great order firing order different delay consecutive pattern technique recently detect pattern whose frequency specify occurrence essence algorithm pattern occurrence pattern delay detect choose set detail main assess significance sequential detect pattern repeat threshold automatically tackle framework assess significance fire current analytical generally employ train process assume one repetition pattern able assess many stream experimentally spike assess significance whether preserved possibility perturbation impose empirical implicit main follow pattern enough influence represent detect strength influence among strength among neuron employ specify whether able rank neuron probability bind parameter independent neuron neuron pair wise influence neuron hypothesis significant interaction among neuron sense extend currently derive probability counting repetition fire pattern mention early non occurrence repetition count non occurrence pattern discover neuron effectiveness simulation experiment spike non poisson process rest organize overview explain detect sequential elsewhere provide spike train effectiveness capable efficacy present generalize handle pattern briefly temporal analyze symbolic discover episode framework analyze spike train many explain order efficiently technique present address question detect significant frequent framework discover frequently sequential episode temporal analyze occurrence follow event event multi event neuron generate spike event neuron ensemble potential spike spike order datum frequent episode time temporal wish discover episode order type episode totally order tuple type example serial episode episode event prescribe constitute serial episode occurrence event type occurrence spike episode spike frequent discovery episode episode exceed specify threshold episode intend event choose episode discovery computationally episode occur episode occurrence episode episode occurrence event say pair occurrence occurrence distinct episode occurrence occurrence episode occurrence non occurrence count interesting addition occurrence look happen spike useful counting possibility thus frequent episode datum available section issue significance pattern discover discover episode choose frequent episode statistically answer question classical testing intuitively want pattern strong influence system neuron generate early allow neuron conditional essentially conditional unconditional fire interval interval fire hz main firing within window datum recent train assumption assume fire vary matter mechanism conditional spike affect fire duration neuron fire essentially spike delay analyze intuitive behind nan functional repeat pattern delay spike neuron repetition define specific delay formalize probability formulate composite nan composite model interact neuron delay interval model neuron would nan spike fire result less interact interaction neuron independent fire hz choose agree call influence reject reasonable episode discover strong appropriate interaction method occurrence episode nan occurrence episode inter constraint iid variable variable depend important show variable represent occurrence episode span sum delay duration nan occurrence instant counting occurrence counting capture evolution counting consider episode inter counting counting process explain view discretized axis occurrence episode instant discretized correspond interval occurrence episode represent episode unit occurrence episode instant advance unit axis look occurrence occurrence look occurrence independent occurrence complete occurrence capture end capture value last take reach occurrence episode occurrence complete occurrence value number occurrence episode length episode b sa bb fire objective reasonable discuss episode conditional probability fire prescribe successive delay unconditional neuron fire instant episode recurrence calculate episode episode let recurrence word occurrence happen occurrence recurrence recurrence idea recurrence q solving variance occurrence beyond something chebyshev positive explain
e c calculation cs c c ce co c l c c ce ce calculation cs c co c c co c ce ce co c characterize theory network model thick ann prediction ref prescribe ann overall c c c ann mode c c c calculation c c c c c pn calculation c c c c l c l measure characterize ann present operating et quality index eqs percentage within prescribed second certain first experimental ccccc prediction c c c c c c c calculation c c l c calculation l l subsection prominent adopt quality introduce compare global phase et efficacy ann micro statistical theory implement table ann specific odd odd thick table tend shorter odd ann tend even parent measure global decay separate odd odd membership result update micro combine decay first ff ann table ann determine independently odd odd range quality ann close agreement ann quality eqs include along comparison index ann response go mode shorter regard date strength fit turning calculation evident ann exhibit network model reproduce know shorter note whole analysis clear validation make model many ann construct al quality index eqs percentage range column calculate tolerance factor column prediction ref c c c l ann ref ce co c ce co e co ce root ann svm construct number ann c c c svms calculation c c ann exploratory application neural network report study connect feedforward ann effort binary encoding backpropagation incorporate momentum enhance difference early ann analog represent table operating concentrate since relate network display result evaluate base performance network layer ann odd network superiority ann reflect advantage current ann early one number degree freedom bias analog secondly relative latter interest need theoretical introduction nuclear carry svms class kernel rigorous theory vc notably architecture difference tradeoff generalization framework automate process determine explicit pattern remain methodology originally develop problem svms atomic decay considerable approach separate study property divide odd odd odd odd distinguished ann well base even somewhat leave validation construction regard class four svm execute spurious fluctuation chain output note li experimental ann desirable experimentally know know nuclear landscape achieve imply especially poor lack ability away exist accordingly important challenge global learn nuclear decide respect adequate nearby ann develop fig estimate chain estimate include calculation et mass criterion rich short tendency odd ann produce probably similar behavior though appear calculation fig chain decay process ann improvements quantitative study process delay emission lie r decay upon global need dynamical calculation calculation fig significant various ann consistent al decay path compare ann global propose mode artificial feedforward reproduce experimentally large network intrinsic outside develop kind demonstrate term problem specific theoretical purely good process match newly create experience gain previously modeling suggest significant sophisticated strategy study feedforward also explore support machine program substitute pursuit traditional thick global provide great underlie physics responsible nuclear hybrid ref ann measure excess ref thereby enable away remark range property beyond decay acknowledgement research support part u national science foundation university wish g team communication grateful university da well nuclear establish approach theory previous machine implement advance generalization reproduce feedforward develop use optimization predictive extensive system early neural discuss place prediction far stability match accordingly valuable additional exploring expand nuclear landscape number thing devote artificial network decay work among nuclear reliable estimate decay future establish nuclear totally area intrinsic subsequent core wave form element notably element scale sensitive decay rich chart decay process decay version recently scene approach emphasize self comprehensive ref beta theory decay final state subsequently various modification introduce parent due limit effort particle model include treat refined decay decay environment configuration beyond start calculation co worker nuclear range addition residual effect theory calculation consistency calculation semi plus integral force prediction density spin theory operate develop ground property stability line apply near close east ground dependent effective rich momentum energy calculation decay rich despite improvement power conventional rather decay far considerable poorly environment technique statistical notably interesting potentially decay approach prove physics modeling implement learn thin drive observable view atomic attempt body attempt question extent answer surely database refine physics interface feed device code ii iii observable say beta adequate weight case generate example serve predictor nuclear atomic energy branch probability machine steady improvement performance thick model reliability new physical insight create physical model nuclear mode develop describe take improve set scheme assess review sec iii large sect drive ann assessment prediction line focus particular sect summarize conclusion architecture feedforward hidden layer contain five whose dynamical unit neuron arrange analogy biological determine report focus feedforward intermediate layer connect unit nonlinear whereas linearly indicate notation input complete bias network modeling neuron carry input neuron neuron together neuron fed activation characterize neuron neuron activity neuron view activation bias connection neuron virtual always scalar notably global nuclear present compute response neuron parallel update successively proceed beta apply weight subsection follow learn predict form atomic number experimental course vary magnitude machine learn neural physical may encode indicate introduction investigation degree determine exact functional decay perspective mathematical noise influence along model include experimental constitute training pattern exp often measure practice modify perform input outside mean predictive supervised network epoch suitable improve involve tradeoff closely interpolation present goal network model would would neuron obviously point construct inherent complicated employ algorithm backpropagation descent bias backpropagation output proceed toward input backpropagation fast require accomplished update q form matrix gradient epoch cost hessian mass restrict case ground parent decay channel consist sort range na cd uniform predictive capability partitioning reflect view diagram great histogram uniform validation exclude vertical fig deal accordingly focus odd character whether similarly operation learning set code prediction employ analog point code activity variable interval range tangent activation lie interval base calculate single analog unit indicate target exp scale study beta properly essence formula drop bind physical quantity plus term form nuclear nuclear landscape input ideal determine physical body experimental input however property decay present clear limitation associate atomic classic correction different surface odd odd effect character alone analog develop yield represent naturally curve result within effect structure fluctuation validation prediction fit modification suggest represent number analogous constant namely even odd odd odd conceptual thin purely physical intuition accept knowledge integer replace ann behavior allow transition ann nuclear prove advantageous encode odd htb plot decay chain dots ann mode proper initialization parameter bias highly nontrivial error eqs close possible global minimum lie evenly neuron clear advantage naive initialization operating range input receive neuron network accelerate measure develop assess provide produce understand detail experimental close unity slope p indicate quality model several index analyze lead measure wherein attain assessment also prescribe range prescribe specifically quantity give point set deviation datum question take namely mean range q total near unity capability index within prescribed already accordingly presentation well develop use b database compare detail cite
share share construct way b b u bx bx bx b present section preserve basically study merge process consider building protocol little construct applicable tree thing party responsible precisely party possesse attribute limited party know partition construct classify new unseen root know node root node identification party classifying pass party party depend visit party tuple site know nothing site start party return local node value node decision attribute tuple introduce grid partition privacy two basic attribute label explain preserve horizontal fact determine empty determine frequent use value party tuple site site attribute union protocol adopt party analogously union party party possibility remain class symbol provide mean still protocol stop attribute accurately calculate tuple provide party party multiply protocol circuit tuple attribute attribute distribute protocol calculate class sa informally site know limited horizontal distribution denote recall attribute case first continue horizontal merge leave vertical vertical distribution attribute compute precisely figure attribute possess input sum protocol pass circuit frequent determined follow tuple reach tuple merge construct union vertical protocol locate layer vertical horizontal merge vertical entire database continue tuple tuple reach use frequent intersection imply certain leaf leaf attribute transaction analogously frequent class party intersection intersection might reach tree tuple intersection tuple leaf vertical know class attribute specific node node classify determine tuple class lie circuit except construct known know compute classify attribute transaction tuple number gain target class proportion tuple datum tuple reach current tuple merge possess merge horizontal intersection tuple consideration protocol tuple node formula done attribute horizontal tuple attribute use protocol horizontal sum call random multiply protocol circuit share use circuit output tuple construct classify among attribute distribute hold circuit whether empty attribute intersection protocol return intersection circuit class return tuple part return root branch analyse previous first merge develop next play attribute attribute maximal maximal length task attribute discuss computation sense consider message protocol never pass never assume union protocol intersection protocol computation cost set make length party value communication size protocol far develop contain circuit create input party determine size circuit party circuit circuit explain depend consideration attribute party possess attribute party attribute attribute attribute attribute remark protocol make protocol protocol calculate intersection protocol agree per attribute transaction reach transaction reach node class transaction reach current transaction reach node execute horizontal call horizontal group protocol horizontal protocol follow protocol rl v rl look logical merge test former case protocol give real small circuit preferable advantageous indeed protocol union protocol execute extend current preserve motivate great world define introduce three mining start e preserve decision two contribution induce partitioned solution merge complexity merge develop definition distribute problem situation privacy preserve partition involve partition discuss evaluation preserve namely develop first merge develop privacy preserve mining partition contribution show privacy preserve active possibility mining combine internet attract inspire area bioinformatics economic relatively field naturally lead security mining clear discover database raise approach allow discover individually guarantee privacy centralize mining neither guarantee datum derive mining try hard discover pattern rather individual mining recognize back access database unable attain database grow naturally carry mining bad typical identification public use anonymous demonstrate combination near mean individual past year state preserve partition datum mean tuple yield essentially site collect instance chain company branch distribute site instance financial collect customer call kind significant financial account partition vice versa financial branch bank end new preserve privacy partition involve closely party technique classification partition protocol partition datum also life datum mining consist partition motivating discuss partition method privacy namely develop merge develop show former protocol everything model end introduction motivate mining sketch definition introduce discuss preserve service possibility stock interested customer bad one loss behavior financial transaction identify huge loss precisely datum risk case transaction branch e set occur illustrate table nr stock yes would knowledge lead precision behavior simplify mean transaction rule bank bank tuple rule combine site discover globally discover individually imply great accuracy identify behavior save great would transaction obvious room separate neither exchange datum customer high word people usually involve kind stock imply group partition show appear distribute contain could service people illustrate need privacy preserve technique significant distribute party induce decision tree originally thorough algorithm interested reader tuple finite attribute tree manner start choose attribute separate create branch attribute repeat determine measure reduction measure homogeneity tuple tuple ht tuple return leaf frequent specific value attribute part value branch label formal grid projection database set consist suppose contain build attribute join attribute party holding contain attribute include tuple ij tuple attribute contain tuple call database c party party party party party party party party party party result party sure process else polynomial amount precisely great partition mining lot prevent party involve security protocol necessary circuit intersection context two already mention semi strictly everything model goal party way party party add party party value add party continue party party add receive party party receive party party reveal safe protocol polynomial simulator protocol domain computation break party party party party value sum calculate avoid party party discover party computation path discover party security concentrate function circuit form receive
prescribe immediately recall sequential hold strict strict well thing last eq strict analogously combine problem sequential inequality strict satisfie observation numerical relate identically obviously sequential test study acknowledgement city cb anonymous reading
mainly online armed mab click mechanism algorithm round pick click round display feedback click round try maximize derive click minus payment initially likelihood bayesian prior design mechanism strategie utility click realization click never social maximize derive click total time click receive call behavior repeatedly choose click mab tradeoff information current choose mab past decade understand choose good notion naturally extend payoff exactly equal invariant benchmark thus mab mab mechanism broadly ask mab affect impose structure online negative believe fundamental good attempt incorporate pay per click reveal context rich produce mechanism separate exploitation extend click attractive agent neutral randomness click belief change agent weak paper dominant emphasize regardless formally mab problem essentially mab get input determine quantity click click click ad display allocation click event distinction allocation payment essential particular completely payment opposite sign amount mechanism mab mechanism well understand type mechanism allocation bid agent cause click payment rule way main agent choose click observe mechanism mechanism know agent click round payment click restriction payment require simulate allocation unobserved restriction implication notable mab realization click specifie whether click observe information round rule click bid profile round select bid everything else mab mechanism strict exploitation crucial exploration click realization influence bid change click realization affect allocation round show influential round influential essentially exploitation allocation realization allocation influential round click per click mechanism call mechanism normalize never allocation rule multiply unit easily convert rule independent heuristic decision use convert mab rule assign click approach respectively ready present main agent allocation bid profile degenerate interval structural bid exposition regret clarity degeneracy throughout rule mechanism payment
american bottom social member network whether interaction david american college play game conference frequent community correspond spike show structure stress cover modularity indicate meaningful split give merge main interestingly share high community bottom leave htb level correspond two misclassifie overlap reflect illustrate circle b find separation circle figure share group htb node extent american college network clique method information find american college prove hand many clique represent build free association norm category method category usually
uncertainty number good probe number mc discriminant background value peak value efficiency background sample relation signal background performance perfect discrimination roc measure high often search rare rejection relevant roc area often background rejection fig htp calculate suffer memory consumption accuracy bin increase bin dimension fine change region overlap phase effectively adaptive method project contain cell generator hyper rectangular cell dense slowly iteratively produce act build context pde splitting distribution sample usage optimisation discuss outlier exclude fraction volume
four rely member f first end procedure error correct v begin precede good plug procedure deduce relation true quadratic seem less put quantity measurable thought next paragraph introduce banach space banach measurable subject hilbert affine denote nan integral measurable classical decomposition dimension reproduce space schmidt associate measurable hilbert schmidt operator gaussian schmidt operators v equation depend depend present methodology section essentially near inequality lemma exist h measure effect perturbation variable permit recover event eq perturbation equation fix rely principle si sc end standard p us covariance associate result help assumption entirely banach measure assumption conclude use element precede satisfied theorem
estimator model minimization ordinary square sub optimal overcome
survival leaf number diameter number size present markov label comparison parallel metropolis high acceptance usually move exhibit serial compare pt swap proposal perform sample temperature advantage proposal within chain index even point attractive sampler marginal chain chain swap transition note pt transition largely property direct analytical establish ordering criterion fall beyond scope regard computable criterion three paper mcmc probability variety provide numerically inference metropolis hasting section inference behind choice require several processor comparable chain whereas course observe appear explore configuration effectively mechanism observe swap rate lead precision advantage application deriving
triangular strong actually former mean triangular criterion prove th criterion pmf sequence choice simplify random matrix one follow corollary invariant satisfie consistency satisfy give constructive recover covariance specify field general representation correspond fact seem first notable make field moment recover start make consistency invariant criterion q ex lf k f
immediately rule everywhere satisfied everywhere reason see trivial test bad take obvious structure sequential testing minimize lagrange multipli component precede low question let strategy easy equivalent let proof difference go hand
respectively letter variable clarity please
objective simultaneous divergence euclidean draw initialization refer clustering distance assign point close initialize bregman bregman sum square generalization bregman additionally outcome variant version version guarantee euclidean square tensor generalize another initialize initialize initialize monotonically tensor mean euclidean distance vary divergence repeat time tensor estimate factor achieve true underlie qualitatively tensor decrease factor estimate hand clustering yield add k improve well though help
average alignment organize describe derive estimator comparison spectra theoretical efficiency density estimate performance methodology alignment show propose outperform standard appendix assume typical express q process assume standard gaussian whole formalize bound support respectively periodic associated period generality far simplify l imply identical curve guarantee study curve appear boundedness easily discretization later part also curve make appear shift identically without intuitive sake argument shift
minimize ex minus plus minus mu mu plus mu mm feature decision height act learn cycle complex non develop markov perform suitable mdps part realistic bayesian keyword evolutionary selection efficiency intelligence ai approach arguably rl rl consider receive agent collect possible formulate environmental observable mdps reasonably understand extension approximation mdps algorithm still namely observation potentially find clear one better already perfect mobile robot equip camera unknown imagine image potentially one search mdp experience contribution article criterion reality uncertain
side arbitrary notice right depend sample represent na see proposition hence definition reference detail generate property converge discussion equal robustness generalization uniform recall lasso tradeoff however include g stability range neighbor infinite vc dimension stable interpretation allow consistency almost surely step kernel sample denote belong n convergence kernel bound universal side follow notice nc nf remove result slowly need belong hence radius distribution tc
threshold higher want test number rare still correctly second multi believe modification rare contamination reliably demonstrate course conjunction classifier select object vary completeness contamination pure rare object take target act modify star pure zero yet sample magnitude limit contamination adequate observe build sample complete galaxy star contamination achieve width emission somewhat arbitrarily include establish contamination set low well star either mixture change majority actually star extend include full slightly explicit influence simple method calculate prior nominal model recommend train object properly class determine replace appropriate target rare produce rare approach appropriate prior change acknowledgement make use national galaxy star library team
n node shall thing pr kn bit node independently rely history kn mm kn kn increasingly partition extension martingale variable refer single individual follow bit htb h immediately plug inequality bit reveal definition similarly k ready difference bit k k p k lemma exclude exclude cut event bit event whose let q space pr I n next deviation addition event variable
structure ultimately computer corpus exhaustive search possible acceptable small problematic search exhaustive hour process hardware software even great continuously database pair ready create soon request vision svd algorithm propose nonnegative factorization probabilistic semantic scaling interesting substantially truncate scale efficient benefit table many information phrase automatic identify label role analogous cognitive semantic memory memory fact past information like semantic extract sentence memory experience semantic past accumulate suggest extraction extraction analogy core understand past predict situation analogy understand human work computer analogy put burden analogy burden computer remain principle problem human try far reach support relation attribute mapping human user computer surprising thank make available I thank team thank team thank anonymous suggestion give display mapping problem column summarize mapping atom give give htbp ten scientific look l drop right car analogy car car car car movement car generate product right side item mark error helpful unclear please exercise output computer proceed confident participant atom atom average water heat flow pressure temperature nn water nn
checking form remainder therefore group portfolio remain capital random letter bold character follow concern model pattern restriction rp last inference index unknown equation survey finite super population quantitie may kind age age square life education social receive location history parent square covariate coverage certain stage vector justification justify law unconditional selection unknown mechanism
request incur algorithm algorithm incur great prediction mistake pair label mistake bind prediction bound approach map label mistake theorem
space problem outlier dynamic preliminary strict mh continuous state increase goal current walk sampling satisfactory equally adaptive expand handle sampler adaptive conjugate prior make adaptive build four main preliminary strict mixture gaussian history draw part adaptation fourth theoretical ergodic behavior sampler strict adaptation need fast reliable good goal carefully select need study sampler commonly inefficient metropolis sampler build provide sampling convergence rate successful metropolis target suggest build sufficiently answer rather thorough walk metropolis switch sampler proposal tune first stage early reversible problem safe efficiency cover reduce risk may large phase consume sampler
ability good er er fast slightly perform corner density er greatly detector detector towards corner consequently frame threshold effect reduce number corner per detector effect er corner carefully detector fall detector derive affine perform affine hold detector outperform detector detector outperform corner density corner detector correct point typical probably somewhat point non projective sift good start corner detector well hence drop quite start soon level detector remarkably presence convolution convolution kernel symmetric convolution match filtering shape robustness match filter additive gaussian perform ghz representative modern computer set resolution pixel source train speed detector frame video fast detector original detector implementation sift detector process fast use yield detector pixel original er selection video sequence percentage video widely approximately frame take roughly twice handwritten addition efficient reliable fast detector er
involve however local imply must markovian obvious dag say node contain separate block undirected imply condition respect statement recursive shorthand conditional parent condition conditionally parent markov three product density causal important kernel markovian markov free parameter causal causal markovian prefer impose markov restrict attention natural parameterization markovian form surely choose independently accord lebesgue close spirit markov specify probability dag procedure sampling condition choose bayesian justification justify markov end functional model direct acyclic q jointly joint model satisfy idea extend graph joint induce eq completely check lemma globally parent non local formalize idea value stem hide mechanic functional helpful causality law phenomena micro causal implement causal limitation equivalence end markov graph reason desirable whole ability repeatedly oppose distribution causal observation theory similarity shortest description joint description analogy due algorithmic theory inference new describe causal problem causal among individual object straightforward novel rule statistical algorithmic occur obtain algorithmic causal describe subject represent statistical string string conditional px x n pa kx pa nd jx pa separation functional section imply statistical causal artificial read graph markovian prefer yield simple idea plausible kernel option define illustrate idea value determine switching prefer former explain way model conditional section also paper kolmogorov complexity principle
order characterize suggest vary formal around paper bayesian autoregressive express recursive algorithm discount factor track spread see efficient studied analytically monitor financial strategy trade related trading integration benefit finance believe broad within management science community arise domain aspect methodology work first well understand methodology relate procedure hypothesis stationarity effort direct test walk complement procedure plan tune keep technique artificial vary factor scope improvement acknowledgement thank helpful paper appendix trivial complete proceeding ar model write ty ta ia convergent write bound convergent variance q series give proportional follow convergent series follow e obvious show convergent mean structure
precisely data evolution paris period ms successive record focus estimation relevant characterize instantaneous heart signal exponent indicate path dependency parameter range process behavior recently memory detect detection signal unknown characteristic middle end estimate fit estimating method wavelet version trend aggregate study property noise extend long trend wavelet et optimal convergence minimax goodness deduce popularity numerous paper
guess solely exact approximate monte carlo choice difficult integral region yield q degenerate dirac variance mc importance sampling result importance contain try successively importance generate importance unbiased estimator tradeoff bias variance minimize parametrize integral minimize problem technique integral adaptive precede mc dirac distribution correspond accordingly introduce variable component degenerate dirac call call variable analyze random minimize propose difference value integral adopt eq discussion additive constant loss additive affect
original obtain value play loss alg multiply side rearrange write loss u optimal last epoch indicate loss trial end worst increase consecutive term epoch side bound jensen depend note bind note large finite imply eq becomes rearrange foundation grant sequence performance use guide selection trade exist arbitrary regret propose bind adapt result preliminary experiment solver decade
straightforward see satisfie indeed maximization maximize divergence view type value view self problem instance obviously name
introduce alternate recovery recovery plain relaxation reweighte compressed sense cs impact application code acquisition already remarkable date website http com problem one reconstruct frequency appear see survey reference problem state
calibrate upon readily lm rmse almost ones rmse times low recently cv randomly partition time set size un transformed ridge relevance gps ard covariance repeat gp give ard algorithm far ard gp bad comparison ard perhaps surprisingly rd gp hand tail take response otherwise predict flexible term benefit straightforward combine lm tractable tool focus function relevant yield limit scaled covariance characterize straightforward mat ern unlike mat ern methodology similarly allow mean polynomial interaction term straightforwardly within allow reduce clear limit case well lose move
match uniquely acquire surveillance camera assume database optimal match situation account match match limitation extract obviously impractical purely perspective extract acquire want solve word surveillance camera optimal map assignment amount loose language adjust compatibility match quadratic readily column evidence dramatically outperform assignment improve approximation assignment problem include study spectra adjacency graph relaxation probabilistic discrete variant belief
view quantum generalization relational introduction quantum get thing accuracy phase want able cope simplifie make strong concept communication consider efficiently learnable prove string range bit measure something perfect become quantum fit show amount query correct exactly learner keep copy rest give c ready assume ask n
process approximate sampler stand remove respectively matrix collect value shorthand substitute equation definition conditioning q distribution fix amount treat gibbs constitute operation basis cycle adaptive carry mixing perform reasonably proceed entry occur similarly q visit state expert environment formalize minimize entropy agent suitable environment observer bayesian however agent need causal solution problem controller implement adaptive behavior mixture mild assumption expert artificial intelligence entropy control operation similarly desire know weather control however unknown face design far counterpart good versus action environment agent even toy thus
theoretical boost measure prediction error multiply cm lasso cm bagging cm ridge lasso cm bag cm namely pattern solution close rewrite sign pattern focus accord proposition limit lemma condition sign lemma relate derive concentration proof obtain argument constant normal error give large j p p distribution covariance
connect bipartite particle edge weight enforce per node active bp give maximum maximum loop characterize model bp understand parametrization graphical furthermore bp loop series weight particle loop term explicit reduce demonstrate bp saddle polynomial randomized scheme improve achieve comparable significant gain speed experiment thousand modern mechanic optical particle density negligible flow particle snapshot
dy absolutely regard lebesgue ratio ranking increase inverse shift converge jump say item regard continuous notation assumption explicit theorem observe rank sale sale potential list mathematically nontrivial dependent variable also measurable subsequent section ranking find regard model title title order accord definition book order rank title guess naive popularity book book item dominant rank book rarely though book stochastic jump book top position side sale count jump intuition rigorous form distribution e theoretical sale amazon co amazon com brief explanation sale find page book amazon actual sale book web amazon irrespective web book title title author represent sale book notice serve quantitative economic impact reflect sale book internet sale publicly refer structure web page long number amazon com sale book amazon
normalize unless impossible sis unnecessary inter however similarly attain certain suppose width replace first therefore asymptotically efficient reason normalization constant discuss require implement one need care selection implementation practical view trial tail proposal obviously choice assumption expectation close proposal suffice ii bin incorporate dependent proposal quantity plug method zhang derivative unknown plug apply reference case investigate corollary proportion depend clear mse know suggest iv application restriction compact omit bias make small particularly precision practice implement account histogram implementation see emphasize suffice store underlie involve author discuss apply method crucial
beyond give throughout paper simulator simulator give distinguish let approach denote clear distribution probability usual normalize intractable develop doubly normalizing intractable standard monte carlo technique normalize sometimes enable inference simulation basic simulator accept tolerance determine algorithm accept write draw draw smaller well consequently tolerance control several extension make project onto lower change accept choose unknown use sufficient add top use tolerance
cell interestingly mean c go rapidly condition estimate value expression partly explain increase continue covariate nonparametric mixed nested model penalize employ smoothing generalize cross use automatically capture control suggest outperform propose accommodate genomic study although temporal involve wave report open r sequel work arise temporal development appendix covariance kf ks lf process follow normal mean gaussian tf bn eq therefore
show alarm alarm consistently good wide scenario advantage increase bayesian oppose prior probability tracking apply problem compare scenario characteristic log transform alternatively handle convenient think promise novel statistic evaluate location evaluate capability rigorously problem drawback step necessary raise possibility actually contain correlation appear future work
possibly leave consist non become map collection collection order let procedure consists overlap distinct element satisfy two index singleton element satisfy
I measure easy eq minimization otherwise equality reduce find eq satisfy solve rest article respectively optimal control stop truncate rule solve truncated rule I class way truncate recursively everywhere everywhere suppose follow measurable optimal immediately truncate rule
degree pose space infinitely usually gaussian assimilation attempt kalman enkf sampling sis pf enkf carlo kalman kf kf covariance enkf advanced independently
reason decide accept consecutive hypothesis tend narrow consequence make practically lie decade well sequential special hypothesis drawback useful plan force termination g reference therein prescribed power number batch sample one inefficient say suffer problem boundary variation feature finite stage cost sampling ii absolutely truncation power rigorously guarantee efficient parametric organize unify general framework sequential present theory testing apply important binomial normal exponential life dedicate variance normal exact test recursive evaluate risk incorrect decision conclusion proof shall notation empty integer denote small integer great gamma integer take otherwise variable denote probability parameterize drop without introduce confusion denote critical let percentile chi square denote percentile support notation proceed demonstrate cast interval probability sequel fast desirable observational st pre determine rule observational terminate insufficient making proceed accept practical stopping surely finite stop decision rule sequential interval l index termination th requirement requirement l I I l termination process ensure sequential probability general shall stop stage observational stop late parametrization decision rule control sequential term number random u say sided sequence confidence coverage weight credible similarly credible stop sequential observe turn observe u termination outcome interval interval interval base rule infer location sequential nature comparison note parameter simplification u confidence sequence
finite illustrate difference model infinite infinity prove margin build complexity therefore adaptive nest look carefully main one small much look general strong slightly different framework margin classifier selection function different focus cardinality probably prove explain satisfy countable event q comment first kind oracle look close sake every right inequality penalty general main difficulty prove strong lie ideal
generalize right tv notice zero complex unit nz ne ne ne z ne p get change p dy p hz hz hz h hz hz snr dominate jacobian exponential maxima admit closed expression however jacobian variable h notice z z important role easily computable approximation building u drop simplicity hence eq
deduce average bootstrap definition event imply bind proof fdr controlling adjust let reject reject equation begin argument eq rewritten analogy proof split consistent must hold split hence converge adjust lemma b significance high dimensional efficient inclusion valid exception proposal split size split classical remain variable second minimal involve arbitrary reproduce split control inclusion family fdr power selection comparison receive attention decade extension sparse interpretable result variable problem prediction result low
symbol teacher mobile symbol plot h true teacher ensemble j teacher mobile plot perceptron teacher symbol teacher mobile ensemble line analyze move framework adopt perceptron teacher student perceptron calculate parameter ensemble student teacher use perceptron begin learn separate ensemble close teacher ensemble continue unity learn even minimum reach fundamental regardless
infinity drawback cross large computation empirical perform fold validation set contrary framework prove quadratic factor third know unstable final strongly depend conclude split quadratic preferred aggregation model aggregate algorithm stable paper drive penalty remain relate slope criterion unbiased risk come selection empirical criterion likelihood criterion nm assume minimize penalize amount bias concentration mind equivalent regression constant stand far use minimal explain connect heuristic second develop use jump way minimal approximately slope shape penaltie slope heuristic take approach framework heuristic nevertheless believe heuristic prove help mention contrary assume small like power collection penalty nevertheless slope heuristics slope
unimodal density likelihood successively spike log maximum estimate plot dot figure diagram easily imagine plane certain logarithm maximum estimator function constitute uniqueness logarithm challenge correspond key function apply powerful subgradient call estimates standard bivariate package interactive likelihood optimal bandwidth available maximum rather integrated square large size suggest adapt smoothness automatically nonparametric fundamental exploratory include certainly procedure problem population interest population occur huge diagnosis density population pz kernel estimator automatic classification notation proportion positive density show methodology density breast cancer aim observation population true density concave maximum functional include possible concave analytically
parametrization another specific comparison study aim improvement exist free large complete manifold upon generalize precise something miss illustrate development riemannian manifold exponential manifold origin invertible inverse borel matrix lebesgue co call reason density
sequence draw without therefore q hyper permutation linear establish strong result sequence coherent exchangeable f permutation nf nf nf nf nf statement fairly immediate turn n super homogeneity exchangeability similar coherence super nf g sure gain nf nf statement imply exchangeable q exchangeable imply equivalently exchangeable draw replacement ball unknown low reasoning see interpret unknown course essentially back de proof lower conceptually though special case general essence special case linear capture exchangeable sequence probability back index iff failure identify ss shall element depicted color gray cm pos coordinate atom atom atom exchangeability force atom equally mass freedom exchangeability leave lie essence tell exchangeable composition guarantee random variable count selection amongst exchangeable
diagnosis var structure ignore time figure diagnosis time line segment distance baseline year cdr group compound side diagnosis cdr cdr ignore df interaction significant df consequently far apart meaningful line parallel slope right eventually slope estimate look group would know cdr subject cdr type question model interaction diagnosis var subject subject compound try autoregressive var diagonal heterogeneity covariance seem suggest fit criterion likelihood promise aic favor structure besides small log ar var diagnosis cdr cdr mean effect diagnosis effect diagnosis diagnosis diagnosis interaction term var three interaction group interaction effect interaction main diagnosis close main various post see significantly change baseline distance regard mark normality value assume distribution lb cdr lb cdr cdr cdr cdr significantly lf cdr rf cdr cdr comparison comparison rank test cdr mean significantly cdr lf cdr distances lf level lb cdr lb cdr likewise cdr cdr cdr cdr respect template indicate significant shape however necessarily shape implication leave cdr subject tend significantly differ template compare cdr significance right tend cdr pair pair assume pair difference lb cdr significantly lf cdr likewise cdr vs rf cdr cdr tend template distance perhaps reduction cdr large cdr lb cdr cdr rf cdr usual pair cdr significantly less cdr evidence shape mild detect significantly cdr cdr significantly cdr cdr baseline cdr influence vs comparison lb cdr cdr significantly lf cdr rf cdr distance lb cdr cdr cdr vs rf cdr cdr template cdr small cdr right follow template cdr cdr leave template cdr right leave leave leave right template otherwise side metric moment correlation hypothesis
duality throughout probability average procedure weight give goal minimize sequential decision generally speak measure procedure sometimes put let restrict testing simple value q eq restriction classical sequential ratio see hand see consider
frequency item lift poorly transaction measure lift filter new lift spurious mining datum association important meaningful transaction database association transaction transaction item significance transaction transaction transaction association rule frequent combination frequent database search bad item decade research center problem variety introduce exploit various property currently fast frequent satisfy measure rule suggest propose filter lift instead take mining visually lift simulated contain random simulate use bernoulli trial base probabilistic suited measure suffer confidence lift follow introduce simulate free lift simulated measure develop lift conclude finding measure available extension comprehensive r tp transaction transaction transaction transaction record fix interval transaction
subgraph ht ccc cm choose good derive need provide community social rich central representation community surface disk size inside gray disk encode member community white name add come single effort perfect community nearby position community form outside contrary outside link community community seven community connect rich representation colored location course interpretation vertex respective relevant link nevertheless understand important fact graph shape rich perfect link community way main great community share name community even community homogeneous link appear high degree share perfect community link central person rich som varied instability sense positive diffusion map test pca initial several lead single note choice initial lead modularity cluster almost second modularity decide seem non give graphical proportional width square proportional weight
depend difficulty calculate pattern henceforth present carlo observed unconditional empirically qr adjusted independence segregation association adjustment significantly segregation alternative illustrative conditional keyword association near contingency labeling spatial segregation author mail tr spatial implication biology study univariate I e label stand segregation class label point pattern one class point distribute region hypothesis two class latter henceforth note generate uniformly fix allocation aggregate clustered type nan rl pattern test spatial segregation near near contingency table
smooth smooth bias correct smooth ti ti smooth decompose lemma get si tm si magnitude eigenvalue increase iteration drive correction step precede correct tie iterate satisfy conversely value iterate reach iteration suitably stop improve upon smooth bias correction reason hence numerically behavior bias correction imply shrinkage smooth shrinkage numerically shrinkage correction fail boost community boost interpret step set pilot hx smoother one smooth smoothing b boost modify f great deal select use boost selection pilot smooth spline bias neighbor smooth fix convergence impact modification devote iterative schema singular
follow apply lemma I h virtue conclude n prove
modify basic subspace contain active x determine qx new qx active many repetition provide cm qx qx cm determination handle subspace involve possibility avoid replace respectively x v work precisely basic replace basic procedure yield vector pi pi x functional modification equality define whole validity
decode tie break favor reverse tie within interval adjacent fail infinite infinitely strong definition realization depend th relative existence persistent call barrier order call condition satisfy maximal intersection emission distinct single latter otherwise observation let th every barrier length imply every barrier ergodicity almost realization infinitely hence every realization infinitely hmms strict positivity condition verify f hmms positivity fact turn sufficient infinitely thus irreducible infinitely many strong
look bad irrespective achieve switch factored decrease exponentially polynomially decrease polynomially tt k predictive switch switch respect estimator introduction one almost include overall share circumstance switch bin select base bin suppose bound exist minimax convergence use histogram bin equal width histogram within bin degree less model q outcome bin estimator interpret lead estimator rate statement irrespective histogram model bin predict switch histogram switch bin switch minimax model select factor although inconsistent still correctly bin disadvantage let switch experiment consistency might construct achieve rate division bayesian model expect minimax rate formal bayesian experiment significantly switch aforementioned elsewhere cause bayesian occur cause make suboptimal prediction switch explanation support switch phenomenon mind achieve develop result result define convenient oracle switch oracle result restrict usually nonparametric amount impose order least nonparametric formally relate rate main nonparametric achieve concrete nonparametric setting finally parametric switch look proof particular therefore keep technical cumulative switch know switch distribution formulate lemma serve basis
mi result remarkably mb mi var exhibit slightly decrease gamma convenient present figure hand around stability within credible identical stable confirm figure play effect test able reduction represent triangle circle loss approach var could interesting investigate reason responsible fluctuation price summarize quite expected ii year coverage var unconditional test one focus number binomial assess generalise ratio observe quantity asymptotically freedom reject serial var combine occur day exception conditional
lattice span accord encode length qx need number bit establish theory work chapter focus theoretic consequence corollary assume countable algebra implicitly understand distribution product algebra distribution regularity q member p know nature constraint want worst context turn minimax q strategy decision justification seem deal empirical version nature
number probability generate forecasting system individual individual calibrate impossible unfortunately mild forecast generation non present assumption paper deterministic forecasting correspond probability show everywhere calibrate well forecast forecast randomize randomized forecasting distribution usual omit generate randomized forecasting event
selection affect carry important implication explicitly determine even specify provide fully bayesian notable exception datum specify selection distribution correspond consists estimate fdr nest rejection region rule express fdr problem fdr fdr frequentist fdr controlling may fdr discovery simple control directional fdr fdr hypothesis selection reduction selective suggest tradeoff take account eps bb thm corollary remark example pt provide inference select base provide inference show informative adjust contribution introduction discovery control exist fdr mixture simulate datum provide apply former student high school student ability student school student college wish student ability observe student college high student college college student college student school
sampling come scale character simplify model multiscale subtle overcome rest paper appropriate average introduce average section present framework slow system variable need justify sde stochastic generator alone generate brownian either hilbert inner function respect uniformly equation compact diffusion pde without uniform assumption ergodic generator process understand denote brownian resp dynamic resp course evolve slowly intuition understanding arise long invariant eliminate equation complicated idea result subsection via cholesky hold solve sde motion use denote brownian poisson equation unique
report comparison also report appear always difficult contrary always improve eight uncertainty report divide p dyadic dyadic f
limitation reveal feature great present low either scene concern assignment effect link issue detector candidate important future another encounter suppose angle difference scale like feature good bad force address fundamental issue apply heterogeneous aware principled detect scene detector possible part location learn clique matching performing
column respectively every space decomposition eq partition dual description nuclear similarly supremum follow trivially generality may coordinate project onto space span lemma note theorem key proving provide identically nothing space however subspace gaussian clear homogeneity attention crucial characterize onto dimensional section plug divide right powerful concentration smoothly mean smoothly small
consist highest fully determined adopt become eq sample number observation formula assume logarithm normalizing exhibit shape number increase consequently include w design look similar training obtain compose file class biological time microarray experiment one present paper predictor distinct compose file uci repository subtract respective value account file respectively nan compose vote feature compose compose binary feature binary feature cost chain theoretical minima lose however experimentally observe could element examine operator window almost biological
l adjustment weight choose minimize rhs hold adjustment weight proposal kernel weight possible relate asymptotic variance n iff n nf sequence produce show asymptotically normal function recall extended decrease variable keep noise variable varie via f suggest particle n burden adaptation mechanism threshold possible central similarly class function sense omit brevity select proposal instrumental thus instrumental straightforward r intractable key iteratively use approximation successfully ce method split adjustment weight keep constant necessary program eq happen example admit simple expression straightforward optimisation em context model occur within successive trade typically recommend step implementation show fix iteration particle
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iteratively ensure form field naive inversion q regime contain galaxy inversion classical solution apply divergence one turn system trivial successively trick original example successively allow equation take derivative starting appeal like wiener incoming stream filter accumulate imply system bayesian illustrate dataset successively fuse single knowledge higher possible investigate fig solution problem especially classical solution region fig well exist diagram miss correction correction derive normalization hamiltonian reflect galaxy replacement hamiltonian define sort counter non effective term arise relevant part background approach desirable diagram draw diagram want simultaneously exist diagram loop loop diagram interaction vertex via summation effectively reflect field fluctuation shape exponential hamiltonian fluctuation fluctuation supplement miss loop correction equation field rigorous calculation within loop correct dark logarithm relation dark estimator minimal deviation contain loop correction account grey datum display variance analyze bottom top corner vanish periodic put corner diagram quite something trick keep either diagram number term concentrate hamiltonian data proportional full regard proportional immediately drop analysis guess characterize hamiltonian form probability guess restrict want forget adopt measure much full regard pseudo next state guess correspond proportional kind source fed trick shape dispersion field acquire update next hamiltonian express expansion simplicity save even diagonal guess approximate hamiltonian free shift connected diagram value shift net force represent mirror two order yield differential previously eqs loop enhance classical unify classical convenience begin calculation temperature regime fluctuation energy minimum reconstruction fig fig seem reconstruction difference surprising latter solution one replace
mechanism master example dna one dna copy copy number master paper specifically refer dna detect array array examine extensive breast conclude discussion primary paper sample p pn shall focus breast cancer gene couple hundred screen ordinary least suitable heavily year constraint dimension regularization genetic pathway genetic widely sparsity treat predictor life network network specifically indicate impose denote stand product wise multiplication indicator base knowledge advance affect coefficient coefficient matrix induce overall matrix product row affect relatively variable combine master multivariate identify master select variable model response et al constraint loading regression constraint union case group additive penalty shrinkage penalty thus penalty together select also limit predictor necessarily predictor norm
recognition support article artificial ann much long measure image descriptor value describe essence thesis texture minor guide audio mp video consist broad deal digital information identify descriptor color texture search classification successful retrieval system wavelet bank briefly find recommend read book taylor thorough book sometimes additional I learn feed tie machine belong recognize member class call notation would previously train observe q produce imagine classify hyperplane affine separate hyperplane linearly separable imagine dimensional place straight place instance separable straight hyperplane dimension hyperplane offset consider function system hyperplane example set networks ann guarantee datum criterion
estimate mcmc analog mle indeed respect poor smoothing variance overall trial error implementation dimensional space aware fact slow take case n discuss category early essentially wang difference improve integrate stress markovian process chain chain marginal conditional give invariant precise strong law
fulfil space multivariate construct interval hazard normality goodness distribution compare well equivalent test lr statistic parameter likelihood construct lr lr check superior versus statistic lr testing hypothesis dimension
deviation mix model iii vector gain mean nevertheless report simulation three iv iv low deviation component proportion small comparison superiority superior mixed parameter good estimator model estimator job mix e fisher dimension model degeneracy good deviation unnecessary whether safe
region suppose size properly iy laplacian operator approximate mesh taylor eq fix unbounded choose look mesh balanced get error remark computing start instead follow hold g qr however convenient computational exploit contribution allow
propose approximation level impact since rarely dimension available implement occur set field beneficial show read follow otherwise competition sufficient likelihood neighbourhood probability namely include target e statistic statistic obviously gibbs sg cardinality set constant concatenation parameter obviously specific family rarely model different give one algorithm produce tolerance generate otherwise simulate
cumulative monotonic could value interval value integration interval level transform level threshold classify vector significance outli condition fulfil function significance level expression apply summarize random provide
state derive specify problem approach approximate minimize partially account dispersion penalization parameter dispersion loss hereafter consideration sparsity employ point mass e dirac since marginal calculate maximize estimate show thresholding sense iii construct independent estimate optimal study suffice far accordingly proceed
value approximate wide rejection abc approximate accurately expense heavy method et cpu linear example network generation cost work subspace low reduce summary statistic estimator distribution predictive averaging replicate many equivalent popularity modify change justification feed neural ability implement allow unified choice consider parameter infer candidate weighted procedure extension logistic two equivalent linear neural multinomial logistic abc population recently context compositional
stop well stop b k conclude integer rs I n state chen z z z making z complete b similar position
maximize converge marginal observation come marginalization unit distribution maximization perform use missing reconstruct store responsible observation piece information
problem structure generalize solve select eigenvalue large bind initial scheme derivative implement fast transform compute form speed usually iteration therefore cost step transform discrete expensive summation lattice need however maxima likely monotonic increase e algorithm initialize step mesh centroid th select problem case
underlie usually result arrive easy interpret attain well extensively case suffer fundamental limitation need search direct improve shrinkage increase penalize criterion nonnegative lasso elastic net selector propose penalty coefficient lasso variant demonstrate polytope unlike contour axis lead independent hierarchical create create parsimonious upon continuous concern monte incomplete bayesian hierarchical major development model hyper unlike regression
word section asymptotically hence r case q subsequence follow word family check u u mc entry deviation theory see hold simplify analysis root statement assumption iv chebyshev follow co
west aim allocation include return forecast assume return several portfolio allocation evaluation forecast covariance allocation unconstraine portfolio weight portfolio cp allocation naive allocate portfolio discuss west realize tv single discount historical west var tv var also model cp model rr rr cp cp percentage realize cumulative return portfolio cp
covariate patient parameter maximum simulate joint log likelihood mass place outside grid introduce optimization improvement analogy equilibrium low simulate global temperature equilibrium temperature schedule schedule global equivalently low state schedule slow implement schedule burn simulated annealing schedule discussion aim sampler propose move method move univariate briefly temperature metropolis compute every obtain low conclude mle figure sharp drop indicate thus increase variance versus proposal
system act neighborhood coordinate manifold fortunately local coordinate system global coordinate notion manifold whose pdf eq value family pdf parameterize know equation mapping parameterization statistical term amount contain reference fx dx meet parameter fisher specify family distribution pdfs parameter manifold parameterization pdfs shortest geodesic regardless parameterization enable manifold
bottom value terminal assign tree give equal terminal node assign express terminal equal terminal unlike effect variable I incorporate may size vary order terminal depend single component sum tree reduce gain representation predictive capability tree determine bottom individual tree component regard overcomplete choice discuss specification fit keep influential rich thereby representation facilitate recommend automatic default specification appear remarkably demonstrated basically proceed reduce hyperparameter recommend reasonable external consideration specify range plausible value value computationally demand coherence prior concern sure severe data specification prior prior node independence restriction simplify form subsection cart benefit control interpretable hyperparameter calibrate effective default specification choice hyperparameter form recommend specify give ii splitting iii splitting ii iii namely uniform ii splitting iii splitting variable size alpha depth p single node complicated tree model want regularization individual component example terminal receive prior respectively put tree terminal considerable set benefit
reduce accomplished method follow mm mutually extremely large section idea sample result later conventional hence effort distribute number element x regard appendix use describe mx nb
give brief gaussian implementation illustrate propose procedure summarize fact decomposition need induce subgraph clique clique connect consider say say clique clique adjacent inclusion minimal vertex non adjacent
connect terminal size approximate poisson length accurate estimate guarantee long observation
markov irreducible possess stationary converge iterate da total variation regularity condition discrepancy discrepancy measure natural yet rarely
threshold gp usually typically put non second variable adaptively vary artificial one describe five node evaluate response finish second repeat start initial configuration maximum design lm burn stationary exclude comparison variation order come burn come statistically experiment motivate simulator vary output lift roll experiment interface plot result lift alpha angle attack beta preliminary datum appear output live sequential environment notice structure choose statistic guide illustrate fast architecture extremely design interface cart queue location sample configuration candidate maximum hierarchical prior default pool vast response surface homogeneous fix beta alpha angle configuration sample panel show angle project side beta project alpha beta recommend treat beta set locate alpha sample uniform across beta bottom plot speed alpha angle attack beta angle fix sample gp upper plot slice
appear literature require indicate correction drive bandwidth selection apply construct test linearity currently corollary definition remark proposition inference trend functional
fx fy topology consequence hold pn place sake variable surely get identity dp thesis proof martingale proposition worth condition define banach cf moment order hold true positive together theorem jensen yield dx pn play everything impossible check
belong linear basis symbolic recently biology statistic use algebraic inference algebraic statistic way restrict concerned binary extension example prove belong class consequence basis independence model particular devote relationship highlight development contingency sample contingency simplex contingency table raw positivity probability set
belong perform oracle always play index remain explain enumeration ball I since variant q prove round play prove play instead together fact clean adding ordinal clean iw iw ordinal strategy activate let time deduce well tv clean finish radius increase time ball contain closure may hold clean cover throughout phase ball consecutive clean calls ordinal I ball also radius activate cover definition contain strategy reach include cover throughout explain differ decomposition look optimal show consecutive clean phase keep ingredient neighborhood clean play play together clean follow add use fact ordinal strategy phase compact clean claim iv iw activate bound eq provide clean index deduce well tv phase finish closure ball conclude ordinal desire clean throughout ball center consecutive call cover proper ordinal optimal ball note active never radius strategy activated cover net strategy never reach phase conclude
v although scenario example study graph ise pseudo distribution dot simplicity view response generalize vary play role node pattern maximize across depend primary problem ill deal graph nonzero node number genetic parameter impossible different smoothly exist collect high temporal resolution interval structure adjacent differ less formally partition u b b partition point become become change segment estimate subsection propose suitable smooth nonzero pattern wise parameter estimate lot control also define control around well interior method subgradient fast specialized order coordinate initial u covariate property estimation collection neighborhood number contain plane hour nonzero piecewise piece wise oppose define equation estimate
define fraction refer observe adjust obtain plug kullback weighting accord estimate correct miss detail trial replicate lebesgue dominate sequence integrable equip pointwise limit suppose g inferior right integral imply statement proposition trial stimulus statement law tr nr q eq sequence eq use assumption statement expansion eq eq
behaviour influence system principle interpret historical practical strongly connect develop comment end imply convert shall coherent prediction interpretation suggest way satisfie believe important relevant ai intend even action theory reasoning decision uncertainty precise property bayesian choice alternative view always explicitly discussion ai deal incomplete partial instance grow literature net associate keep become model believe report complexity indeed connect develop tree transition child inference tree tree relate bring start process call recursion dynamic complexity inference book first solution game theoretic probability game reality certain obtain devoted mean begin reality play reality possible move previous move previous tree event restrict reality end exclude reality come circle draw rectangle draw fill mm dash black grow right inner sep mm child label terminal node terminal child terminal right terminal child label child terminal child terminal cut cut circle circle observe cut reality tree tree represent move reality begin game path game connect starts root identify reality situation initial segment example notion alternatively path elsewhere event reality event call event may occur reality
necessarily multi matrix value thing properly value pp aa dy aa dt df dy however efficient f ec tr ec ec tr tr ec prove eigenvalue half te te f finite almost te te ft ec te ft ec e tt ft ft ec ft dt dt ft fs z moment ft ft z dt dominate convergence
variance formulate control calibration test consider seem adequate conclude present appendix table chemical chemical compound give tackle calibration consider calibration
remark matrix gibb publish probability discrete understand variance direct consequence establish negative algorithm conclude pt early seed particular produce constructive argument conditional besides mention contain seed gibbs work sufficient graph mean absence must product depend component historical explain never positivity joint density recover conditional joint support conditional published put full stop certainly discovery sampler express many binary situation practical optimistic early monte topic sampling monte surprisingly nevertheless carlo em connection sampling instance try start replace current complete unfortunately however decrease
grid tn l bound uniformly cover lx l p n fx I pf fx pf n q admissible est reduce case nj helpful conclusion conjecture lemma give sup convergence old ball sup close threshold projection onto wavelet spline threshold rademacher wavelet process classical mathematical optimality distribution function sup minimax instance estimator sup purely efficient prescribed density goal article adapt whereas existence practical among threshold spline
write place nothing gain closed method perform inference often omit distribution physics detail outline derive detector minimize iteratively optimization physics surprising inference root physics detector variational free minimization adjust mmse detector somewhat sophisticated routine calculate decision maximize recognize identical output exact tractable also bayes detector subsequently utilize second rise familiar detector routine yield mmse detector mmse detector produce conventional signal technique something detector use valuable section iterative detector convergence gauss iteration solve linear analyze investigation variational vi view coordinate minimization except describe rearrange define express familiar interference unbounded minimize mmse detector hand lead coordinate detector finite mmse solution gain invoke ng optimal nonempty open strictly iterate accord gauss converge least code conclude guarantee converge additionally relax cyclic relaxed detector within variational provide decode unique detector determine biased message schedule detector obtain routine
low birth desire study carry binary response indicator transform ten continuous eight predictor covariate link default et link hill pp consider logit use inclusion interaction final except well interaction term consider direct include logit link comparison easily via traditional examine linear expand additive covariate smooth adequate check two link calculate marginal using consider comparison sample show affect generating mcmc iterate metropolis scheme normal walk covariance hessian comparison table gold gs laplace bridge picture simulation study remarkably set approximation laplace
motivates increase discuss n l n np diagonal element eigenvalue maximize optimize forecast mse standardize step forecast I prior historical case initial setting critical specify measure step forecast modulus standardize preferred account covariance look mse I wish biased propose follow distinct element relatively
maximum near yet know agent optimistic collect new become realistic computed asynchronous choose exploration induce unknown reward algorithmic detail similarly separate value summarize external assume extend exceed rx environment build time external reward exploration reward model equal action value tx ty tx ty tx task usual way external reward space limit neighborhood state programming method summarize state update r ty q e similarity emphasis sketch proof near uncertainty point exploration reward reach combine concerned asynchronous however optimistic
range alphabet ib lagrangian self consistently enforce free generalize distortion rd theory branch compression rd address minimal alphabet codebook approximately reconstruct receiver expect distortion bottleneck
estimator extend model regression family nonlinear generalize regression precision extension linear dispersion g linear precision restriction dispersion sense covariate section numerical real usually dispersion covariate model variance largely literature method detect develop diagnostic variance nonlinear whereas compare move dispersion expression dispersion student discuss diagnostic aspect biased bias problematic done bias serious explored investigate magnitude unconditional estimate linear nonlinear parameter curvature model et explanatory position give formulae index distribution result hold model structure obtain expression class cox bias modify perform adjustment estimate resample goal close second bias response precision bias adjustment adjustment rest introduce class derive expression bias
goodness estimate guarantee kullback leibler note analytical demonstrate theorems scope elliptical unimodal decomposable belong elliptical unimodal density second go decomposable well via density define closely result practitioner density unimodal
avoid interference primary primary traffic priori secondary investigate scenario secondary channel whereas channel protocol efficiently simulation blind protocol achieve throughput primary resource dynamic secondary user cognitive long primary achieve secondary user primary traffic exploit mac sense spectrum detect primary spectrum channel interference cognitive protocol decision several cognitive mac protocol
solver axis divide spatially parameter utilize parsimonious determined appropriate forward solver coarse salient field subsequently solver fine computational novel automatically propose methodology markov solver multi formulation approximation represent readily update analyst want employ operate even fine inference estimate contain important quantitative measure datum efficiency move relation possibility refinement solver accuracy resolution unknown field consequence credible interval datum resource available hierarchical resolution model spatially response pde quantity pressure field solid mechanic heat field advantage bayesian formulation ability inference provide uncertainty contrast exist utilize parametric formulation operate coarse solver salient subsequently operate fine significant computationally demand adaptive mixing encounter carlo scheme capabilitie mechanic attention contaminate noise small advance computational physical model identification frequently forward burden phenomena extensive pose challenge accuracy various mechanic material guide inform assessment system reliability identify functional material detection reveal appearance disease also assess
ai k integer call augmentation element unity polynomial occur uncorrelated polynomial element moment moment say similar addition infinitely sequence factorial alphabet singleton moment except factorial moment eq whose factorial moment parameter thank extend alphabet auxiliary obtain among calculus construct treat alphabet uncorrelated symbol dot power equivalent polynomial
fig result advantageous small alpha policy show process learn rate big hard us episode effective three aspect tradeoff learn search action well superposition quantum mechanic parallelism much prominent practical come use computer ref simulate learn quantum mechanic reinforcement discuss regard important pointed way implement quantum physical system version traditional prove effective rl follow environment solve much easy environment accurately simply consider accord discrete action state generalization rl artificial intelligence issue task aspect update etc lot especially representation computation theoretical need kind environment believe promise learn environment new computation problem rl tradeoff learning etc superposition update rl upon state eigen superposition state eigen action simulate quantum probability eigen eigen
henceforth let denote power represent collection subset set uniquely representation index exist representation kolmogorov representation therefore represent set representation subset hence previous section view hold value wish information represent side intersect subset informative fit variable definition may switch informative exist correspond definition q fix measure bit henceforth convenient refer acquire define representation theoretic expression restriction singleton implicitly extend kolmogorov general definition neither I necessarily dimensional pair view relationship know choose start know intermediate medium general see primarily refer conditional combinatorial entropy conditional convenient express entropy combinatorial notion description
suffer ht van sis var sis prop prop van var prop model prop model bic test van prop final final training aic three manner example result version effective select error hand lasso much omit detail important proposal difficult one independent repetition include repetition new corresponding repetition important include repetition improvement final four problem configuration case response generate accord py case standard case variable marginally challenge standard marginal hinge estimate f pick scad error iterate methodology predictor van sis van var sis lasso prop prop final modal error prop final modal response second hinge method
clinical scatter illustrate two disease scatter contour cell patient disease axis represent marker disease patient significant scatter contour plot overlap dimensional scatter plot primitive potentially clinical dimension cell routine clinical characterization able fine purpose patient respective marker flow analysis compare illustrate express surface generally distinct expression common b cd distinct cd typically positive negative difference disease let set analyze scatter marker cd datum cell give comprise patient patient wish analyze fine scatter fine consist patient patient order clinical diagnosis patient department leibler individual reduction embed unsupervised display class note natural separation conclude experience priori important cluster ability visualize comparison fig scatter patient lie within embed create give quick determining set
run independently time outcome estimation scheme describe probability follow sum reduce scheme
maximum likelihood provide mid pick reveal thorough exploration penalize pre entire singular problem case pseudo rewrite p jk p p penalize tucker sufficient matrix anti anti symmetry positive conclude follow lemma definite positive rewrite pa particular every path precision cholesky coordinate spirit diagonal matrix triangular diagonal j break separate j j subproblem homotopy lar scale subproblem next understand broken piece stem program show program homotopy lar lasso merging simulation therein sample precision entry random sample geometric parameter condition form diagonal select pick definite add observation inverse e
situation prove advantageous natural try answer feasible input selection precisely definite exactly feature kernel one small apply decomposition extra dag norm dag restrict able dag kernel allow non variable compare regular show performance uci repository finally know kernel capability estimate variable hence restrict power gain predict variable general observation function I hinge logistic machine loss setting assume express space feature dot precise
distribution parameter inverse unit unit x pdf respectively hence f xx satisfy pdf theorem density fx pdf unit yield far lead
student satisfy plan minimum element n b small tight possess normal chi square sp see b make number demonstrate describe domain dimensional convex express two domain visible observer origin precise definition boundary domain otherwise unit origin visible point suppose set theorem know thm variance g k see see evaluation design z number max min max z z z I I z ip z eq refine
subsample empty permutation unless likelihood first exchangeability mixture mode via technique exchangeable severe latent modality subset extract sum weight q denote occurrence reflect prior prior posterior beta exchangeability issue call properly converge mcmc produce provide assessment mixture point device consequence result surface reduce mode neighbourhood constrain include modal result posterior region constraint sample purpose simulation estimation completely distribution solution modal region term region base view fact observation direct important drawback improper since improper little pose improper issue particular proper matter variance
age et extend red age interest circle sign visualization would reveal respect age range generate wide instead onto pc shorthand show pc follow pc limit possible serious limitation tree analyze unfortunately readily visible structure however particular subtle view slope suggest old people tend small people slope calculate standard value nan hypothesis slope consistent et trend
every formulae sa n n side property n subsequence argument remain away finite positive say bound say give sa n h n sa h n sa sa sa h sa sa h sa n n differ n n c p sa c c e h sa n n sa sa sa nh n n prove closed remark every later reasoning side n n corresponding proof theorem c n sa sa n sa
component expression quantify word optimal efficiency grow fast massive mc zhang section also integration justify ed integration financial engineering term ed rigorous ed pd p recursively variance explain lower define ed due truncation ed coordinate require variance mc ed purpose integration distribution inefficient result curse unless ed typical financial integration unless path sample huge event definite v without loss correspond sort low approximation first fraction option pricing lead discretized bm path sample easily random walk component count restriction
pass induce prove contain enable pure exclude universal gr intersect gr correspond remove gr generalize use geometry main expand upon event project mat dimensional restriction projective distribution consider variety theorem projective variety projective lie p develop applicable projective hereafter column notion play convex q j w j polytope simplex define submodular sum set condition think redundant additional ambiguity
multivariate compute average du report integrate length function measure original original repetition critical band entire case coverage length particular local global problematic estimate coverage l spline like thank many participant comment considerably acknowledge partly et assume step associate vector whose denote vice versa sort act integer exist th element lf k lf lf fx use sort time repeat argument reduce function quantile integer almost limit p subsequence show paragraph use expansion haar approximated function subsequence necessarily extract subsequence run converge everywhere replace quantile retain define take
reduced virtue tuning note exponential probability know let z sample simplify theorem coverage tuning coverage sequential l u stage requirement hope mild u regard stage sample size sx u l pose regard identical discrete index function la bl interval distinct moreover occur occur occur element occur u occur n occur e occur occur e occur occur u occur u l univariate u sure u u see appendix establish theorem publish concept support say support term random rigorous sure propose begin subsection consequence l condition probability u exhaustive u n statement adaptive checking determine coverage stage deterministic u c l proportion discussion proportion dependent unit mn proportion replacement unit equal assume otherwise unbiased mle however random variable describe stage random stage estimator stage n sampling termination scheme interval follow direct consequence theorems main bivariate l nn bi consecutive consecutive distinct l n express respectively assumption stage finite parametric function p pi find important familiar example structure I number termination justify estimator superior relevant result unbiased finite variance infinity appendix prohibitive burden computational complexity shall focus adjust coverage tuning idea tune meet sequel construct scheme ensure confidence tuning scheme accomplish search whether probability subsection problem follow subroutine small prescribe difficult since interval infinitely extremely adapt branch multidimensional reduce checking assumption bc cb backward proceed mean intermediate reach backward attempt attempt repeatedly width interval width prescribe tolerance relevant situation make rare interval practical limit l u cc unnecessary desirable accomplish essential define complementary coverage discrete care need first small prescribe note small prescribe multidimensional limit application purpose exact coverage confidence extremely computation coverage significantly evaluate complementary coverage virtue statement interval narrow satisfied n determine u suggest alternative construct prescribe level idea construct scheme note precision use probabilistic n subset calculation computation k immediate eq case design hypothesis test drug screening increment unity recursive page scheme deterministic number conditional attribute simple replacement population unit note idea domain truncation reduce design low integration still summation integration truncation recently power reduce small truncation assume probabilistic termination evaluate describe bernoulli sp termination sampling k kn design analysis scheme summation coverage probability clearly complexity prohibitive burden computer break curse tight regard real r j w r j see sum consecutive particular interesting method bound brevity probabilistic involve refer e bound increase price reason method allow powerful dimension estimation two n double computation probability r reduce computational reduce decision employ sequel class scheme conservative upper decision relaxed computation since single illustrate complementary coverage probability associate reduce shall event w term l compute meaningful identify upper coverage rely course expense apply sum upper computational replace term hoeffding illustrate consider size deterministic number ce c chernoff chernoff hoeffding par design scheme form b v f domain note develop systematic purpose split let uv separate last side triangle triangle triangle discrete variable value integer triangular save probability rectangular side parent partition probability include orthogonal rectangular triangle readily rectangle trick repeatedly save since crucial scheme coverage prescribed level probability cover triangular go triangle become small eq low triangular triangle become accordingly avoid triangle large triangular sampling application routine variable possess concavity convexity readily compute use virtue concavity calculate low bound variable upper upper low repeatedly partition gap follow low discrete integer r fa fa b fa r ba minimum gap ba random apply bound interval statement concave fa fx fa fa fx fa fa tt fa frequent routine factorial integer store make recursive computer easily table double sized proof page shall throughout sn stage size clearly index mention finite infinity let variable classical result establish hoeffding inequality paper show e z chernoff want chernoff construct
bad performance snr minimal accurate belief update design snr level case learn snr value gap posteriori time conservative use initial drop infinite reward program discount reward assume cardinality compact example illustrate done assume describe suppose uniformly distribute interval compute probability db db db expect see posteriori probability safe threshold snr db db posteriori probability converge sum require order parameterize scheme estimating channel gain justify dedicated channel synchronization receive signal unknown design unknown parameter significant drop drop primary require reliable probabilistic ensure belief parameter access consider bad mean primary hypothesis reasoning behind receiver locate interference power primary region bad secondary user force access decision secondary snr receive
symbolic comprise categorical quantitative empirical summary calculation variance determination summary variance absolute deviation approach set vector norm central dispersion minimization value dispersion r r mx consistent coherent
dx ap increase proof ensure equivalent test transform sided sided test true tail side value discrete also tail tail add tail continuity correction formal discrete weight coincide symmetric value distribution tail mode mode integer odd integer tail even mean version value modify package side binomial www com see modify
suggest volatility constant cccc cccc show var confidence use refer invariant adopt type et generate dependent parameter var large result suggest produce relatively forecast diagonal respective forecast appear look confirm modulus integrate lead term linear locally sense period time stationary integrate reflect choice discount drive volatility west volatility model wishart beta result volatility element employ several discount allow discount decade discussion advantage multivariate review generalization univariate model correlation correlation even restrict
reach sampling refer denote implie put scheme tend infinity proportion scheme estimating take n prescribe precision let margin explicit thm function support u virtue respectively absolute error q r p condition smaller clearly precision confidence integer un regard truncate restrict take kn size line construction confidence purpose nk nk I method method
package combination drift diffusion coefficient present c cccc x hence simulate e time lag length around x final report cluster complete linkage rescaling gives really figure quite apart agrees unfortunately correctly classify path
parameter unit sequence variable invertible law leave permutation finite mlp space
requirement choice implicitly assume task statistical obtain good barrier r completely control dramatically expand start observation include correlate input close page identify potentially control sufficient need bring detail setup phase involve random set evaluate value find eq pt select sample simulate sample database control estimate select database simulate control estimator database
inter consecutive q radius arbitrary note mean center addition order establish reflect exactly cluster describe rational behavior change enough multiply integer another point cluster cluster resp day go sake convenience radius q note always need easy w w point stable dp dp n n nx ny db n prove understand stage day
constraint consider substantially set region perform finitely many via split class treat model broad question bridge point possibility candidate construct confidence region inference setting estimator compete entail particularly situation several look explore candidate cover theory point whole remainder convenience develop estimator correspond discuss rich family estimator defer section observe brownian motion interested construct confidence f naive denote refine
channel set clean noisy sequence continuous amplitude analog semi stochastic channel symbol symbol probability family channel uniformly sense condition guarantee track density regardless nonparametric density absolutely density crucial guarantee conditional equivalent channel correspond channel distribution precise reasonably behave l induce distribution metric analytical describe arise practice address channel additive channel easy additive gaussian channel channel clean take c satisfied fact additive absolutely density mean discuss satisfy requirement measurable channel impose function e x formally eq incur estimate clean also easily verify satisfy aforementione contain denote envelope assumption achievable symbol loss map denote symbol symbol clean per optimal symbol assess universal symbol different symbol noisy realization irrelevant current probability empirical clean expectation probability channel q denote envelope denote concavity attain know observer noisy towards channel track evolution average accord distribute input
trend mixed regressor issue testing present result brief contain material regressor across least estimator normal shift away zero asymptotic exception panel modify along normality restrictive justify investigation economic economic dependence impose vector loading stationary stationary consistent ignore like pool ol obtain fully fm essentially serial covariance treat developed q regression treat actually treat estimate correlate rate regressor explore primary focus difference estimate versus pool long valid single correlate imply pool ol convergence walk problem panel create spurious loading show assume potentially stochastic trend hereafter integral ambiguity bm brownian convergence generic number project
approximate logarithmic small course strictly approximate logarithmic massive heavy static take pairwise account weight logarithmic replace interested format fundamental operation datum stream counting frequency propose compressed counting take stream stream compress successfully capture intuition suffice small practically decay useful study compress counting sense geometric harmonic tail complexity bound complexity neighborhood actually require various
steady steady constraint coupling coupling realization see sect basic idea metropolis component respect discrete chain g precisely total jump state algorithm generate couple joint probability reject process always whether coupling coupling expectation continuous bit approximate discuss sect physical situation relevant master give note coupling estimator know approximation couple choose value compute necessary coupling variate estimator close expect close coupling variate may variance theoretical tool study
histogram bias plot along histogram observe first estimator suggest see new magnitude also root vary great path consistent narrow roughly unbiased k brownian keep simulation simulate marginally estimator simulation table brownian c shorter reduce subsampling affect multiscale ratio cf except simulation length model still approximate process accurately multiscale white magnitude greatly rather multiscale ratio small kk right bottom bias correct estimator bottom right scale figure estimator fail sample always reasonably effective average calculate sub bias new
copy quantum training dataset en fidelity si j similarity precision state training precision entry therefore state copy measurement copy possible bind make fidelity entry upper similarity bound eigenvalue good state formula estimate intuitively fidelity distinguish fidelity low discriminate time efficiently quantum dataset want build explicitly seem know exponential copy unknown research oracle information necessary learn good measurement copy learn circuit implement good follow table see multiclass classification via oracle cost oracle multiclass identification test binary reduction unknown copy dataset act classification error trivial even almost paper likely design minimum copy essence ml come hope situation paper state act generalize recognize close training without closeness fidelity distance pure
fix angular distance frequency grow property carefully give specific inference isotropic sphere radius available grow experiment increase array spherical random introduction procedure non estimate angular technique datum asymptotic datum array extent construction actually share focus reason make interesting wavelet procedure literature fill gap aim investigate coefficient wavelet extensively start seminal paper stress explain early concerned circumstance make high frequency spherical rely harmonic provide
examine filter typical frequency cognitive useful use hilbert extract filter figure simultaneous band filter phase record examine dependency display joint distribution site site phase condition highly compactly von variable distribution write q offset solid variation importantly among many neighboring site strong dependency behavioral offset phase variable specify present
ii estimate obviously family conjugate prior efficient require mixture computable approximation level virtue stationarity rarely converge essence lack label phenomenon lack permutation multimodal case prior exchangeable fail reproduce predict
scad lasso established estimator suitably well asymptotic picture actual behavior e take close distributional lasso present distribution tune selection often multimodal give analysis provide reliable assessment reason capture actual estimator interestingly turn selection usual asymptotic exhibit lasso estimator tune selection mention establish e organize estimator section theoretically linear convergence asymptotic carlo imposing finally summarize finding technical denote convex hence minimizer convenience adopt convention measurable tuning parameter consider version point section simplify assume assume I latter remove regressor occur include
tv os mail com inf correspondence propose factored iteration factor traditional way projection operator modify max preserve convergence polynomially exponentially prove convergent derive solution analyze projection combine approximate mdps useful solve sequential problem mdps subject curse mdp grow even many problem polynomial size description factor offer assume dependency factor component mdps parameter iteration programming book excellent overview program variant direct traditional furthermore programming
matlab implementation ghz intel core iteration depict actual right panel low snr resp reconstruct db distribution estimate truth db db application easily measure event know value appendix distribute require indicator reflect presence absence value gibbs sample addition use non pixel inside case probability db presence though db detect confident red rectangular panel fig resp area posterior pixel dot red pixel
regret slight ucb rely achieve regret large dependent ucb forecaster distribution intuitive basic consider happen forecaster basically pick regret would arm formally fix pair thank obtain reward distinct distribution satisfy theorem actually order introduces depend permutation first notation expectation tuple arm expectation respect tuple form respectively dirac permutation symbol tuple form th locate tuple dirac worst locate low arm ensure arm reward get repeat argument deal see conditioning jensen times arm latter nc theorem value statement third side arm payoff interested realization form index jensen argument right depend common distribution vary tuple permutation tuple k k j tuple
sir sir compare detection path value introduce statistic detection section database address concern epidemic epidemic contact program restrict epidemic see modelling case sir divide remove figure population individual becoming infect size individual yet disease individual infection individual total detect contact detection contact remove two total contact detection remove detect determine remove contact detect principle contribution detect correspond parameter contact show sir model
occurrence time specify give require successive occurrence serial serial episode serial episode inter time call temporal inter delay neuron connection delay occurrence inter sequential inter serial inter event spike around variation duration occurrence spike occurrence sequential event span episode occurrence event time interval discover frequent serial episode discover episode occurrence conceptually operate time event spike record time length episode frequency want following suppose episode instant check occurrence instant instant within window increment counter occurrence next time instant hand either instant follow instant search inefficient describe candidate frequency output frequency threshold issue obtain candidate candidate tackle combinatorial possible serial episode popular understand frequent unless node namely frequent occurrence occurrence node episode frequent episode candidate node episode episode episode quite combinatorial come drastically size increase many frequent episode efficiently stage frequency set candidate pattern whose occurrence stream care occurrence
nonlinear jacobian matrix compose derivative matrix bias network pattern adopt gauss newton appear control size near minimum coincide newton execute hessian enough algorithm newton error prefer toward successful cost iteration begin small value cost repeat multiplied decrease produce newton advantageous implement fast key jacobian matrix backpropagation derivative bias jacobian generalization avoid overfitte table fail network seek avoid overfitte combination bayesian cross validation divide three second outside guide model validation network begin set continues specify stop bias validation use assess model validation produce bayesian standard sum square write explicitly decay backpropagation minimize error square multiplier number bias characterize hessian evaluated regularize gauss approximation bias detailed discussion ref training backpropagation execute pattern mode record jacobian pattern mode jacobian updating present report batch choice basis substantial carry strategy htb eps partitioning set validation atomic also htb lie vertical gray develop ann property atomic
transaction tuple proportion value number tuple reach current tuple attribute gain v attribute gain need first tuple reach current I tuple merge union vertical union party attribute attribute number tuple attribute intersection merge gain attribute repeat far vertical party possess classify construct decision possess attribute description attribute among attribute hold attribute circuit empty case attribute protocol merge development intersection protocol circuit node return circuit true leaf return classify tuple union protocol value root branch branch contain attribute need class merge party tuple reach attribute possess merge let horizontal many protocol manner horizontal group vertical number class attribute intersection tuple class protocol pass circuit frequent leaf leaf know attribute determine
immediately sufficiently lemma well contradict lebesgue convergence remark suppose follow structure stop stop rule inequality inequality successively part apply thus hand virtue prove rule difficult minimize q repeating stop recursively start remark
bid bid bid decrease get round let bid bid particular round see click information round round pointwise difference influential influential influence contradiction contradiction l l l side since bid lose bid l lx l pointwise monotonicity note us equation since bid bid let prove differ click round click b agent lb minus plus ex minus ex plus ex ex plus minus minus apart presentation snapshot open microsoft microsoft com usa com microsoft view usa microsoft com set motivated pay click select derive click click initially click dominant mechanism bid true view characterize good investigate armed affect certain strong exploitation multi armed bandit essentially categorie subject descriptor computer behavioral keyword multi bandit recent year understand implication behavior input among motivated internet interaction goal mechanism mechanism setting background recent book much market numerous every mostly focus equilibria price google rely know click click consider observe user behavior influence focus method implication game two agent profile agent allocation bid mab realization bid profile round influential realization bid every influence weakly separate weak separate agent influential influence fix allocation condition equivalent consider degenerate payment monotone separate allocation free agent rule imply investigate mab regret adapt notion social call short allocation click bid bid condition trivially degenerate allocation pointwise hold exploration limitation agent mab yet pointwise later satisfy suffer design perform pointwise challenge allocation paper leave open mab allocation monotone change short click bid profile round bid transfer prove degenerate deterministic pointwise monotone hold rule separate weakly separate mechanism consider term click payoff choose
fitness neighboring fitness high short cluster node possible cover discover straightforward repeat however computationally expensive many coincide spend module economic stop htb blue member fitness red respect justified argument community start outside hand non recover module without cover community module extensive test one proceed suggest community choice seed resolution fixing scale look value yield community module end degree correspond weak definition cover find specific modularity know cover different vary explore hierarchy graph single way
division bin split cell remain mark inactive elsewhere cell region gradient cell create distribute angle gauss radius cell cell belong volume consist centre corner plane radial centre radial densely example rectangular geometry cell angular symmetry implementation option cell cut decision tree correspond cell store binary represent optimisation boost implement separate signal number contain cell contain cell volume discriminant analogy discriminant distribution calculate discriminant background event contain discriminant analogy store cell uncertainty retrieve implementation find similar figure
reduction priori would tumor predict tumor variable easily spectra spectra exist angle theoretical sufficiently choose ask manually spectra keep real spectra phase optimal create inside incorporation classification however phase translate interestingly already word classify lead mostly section theoretical sensitive rule eq article theorem reverse rule binary classification measurable function let constant result comment tend excess risk proximity error sensitive ex ex ex gives good accord proof lemma et partition concern prove subsection fix value equation lemma lead q u end precede precede precede tend enough obtain observe depend limit end require four angle follow invariance call angular straight include
moment order justification let li circular hessian
proper point leaf joint leaf integrate analytical index possible employ tree integration need visit computational search laplace leaf equation form appendix cart section construct proposal lack model issue note reversible specification chain distribution generalize besides five select new splitting child split splitting random splitting splitting rule close root specification chain transition move improve splitting type tree distinct current splitting rule adopt type chain chain second chain include swap iteration proceed chain chain chain transition mh cause death cause death survival tumor along profile international cancer table report nine available
family minimal need point geodesic spherical define coordinate spherical say exist spherical rescaling define lemma geodesic radius volume coordinate moreover distribution member recover possible structure distribution representation circumstance answer symmetric riemannian important apply non negative similarity invariant distance satisfy unique apply continuous field view representation would recover
remark bayesian solution bayesian use modify etc obtain precede minimize sequential prescribe level theorem immediately satisfy test strict least strict theorem thing eq obviously strict well author greatly city national grant cb c
copula I empirical entropy entropy
empty cluster entry tensor normalize let projection matrix rewrite immediately zero use recursively initial dimension go level set dimension represent range dimension recursively divide middle individual collection complete cluster recursive clustering product clustering collection relate sub clusterings li li bind clustering eq always map first prove induction decompose l l combine induction term property guarantee cluster dimension bind objective clustering follow factor look permutation clustering
actual discretization domain middle obtain shift property dft stem lagrange sequence ok result purely contribution deterministic function induce bin might indeed simplicity random long central appearing understand homogeneity need dependency mix b sampling observation concentrate signal beyond scope shift alignment block assume odd align define integer l energy spectral side rhs approximately harmonic later note obtain get q hence minimize band weight establish normality tend
stochastic environment experience expensive want need exploit past optimally virtual need option parametric far rl ultimately depend indeed derive length partly detail later mdp scale way side open extend state aggregation planning simulation rl code generalize yet develop string else generally omit separate confusion arise string xx state reward realization never lead confusion number state time several place special describe formal framework reward mdp environment camera image real reward win game time cycle action receive reward loss may history plus plus
decision uncertainty quickly I set evaluate bad distribution inequality theorem second definite semi replace inequality hold even still bind reason connection like generalization robust counterpart theorem formulation way couple uncertainty norm regression remove feature wise bad eq function hence formulation significant equivalently formulation empty interior robust follow regularize z z convex efficiently corollary direct theorem regularize regularizers robust regression perspective straightforward
histogram modify stand clean contamination completeness compare completeness contamination translate completeness acceptable contamination notice contamination drop expect proper discriminant object remove surprising majority consistent training distant star build predict mixture class probability svm flexibility object learn classify well object minus contamination fraction solution eq logarithm vary sum contamination panel plot contamination agree approximately intuition contamination posterior apply achieve completeness know would contamination calculation fig tell consistent strictly could also despite inference ultimately modify contamination regardless use accommodate modify nominal yet poor completeness contamination sample dash line fig demonstrate application rare give contamination modify
event nice union bad subspace event occur probability examine record record node history balanced let outcome history occur concerned view particular n l l point difference history history bit fix expand expand bit unconstrained bad node apply clean history space h advantageous occur point exclude history individually expand pr definition require reveal bit martingale amenable concentration tight need exclude regard k event lemma definition exclude cause difference
impact swap limited part relational swap limited way affect kind global similarity similarity somewhat experiment isolate problem intend mapping map two use generate internal coherence case internal must constraint internal coherence internal calculate mapping coherence explore involve coherence internal coherence external select mapping ten e whereas average coherence difference significant pair benefit connection attribute part optimize independently cover relational mapping decompose relation connect part relational mapping inherent internal total coherence use new accuracy internal coherence whereas total statistically pair reason tie cause variation suggest analogy easy logic analogy overlap course difference attribute experimentally give analogy cognitive emphasis computational human make symbolic ai cognitive call conceptual space influential latent analogy appear around survey take spatial analogy make analogy make date analogy proportional analogy geometric figure could word code rule limit code theory symbolic influential date analogy range child development explicitly shift emphasis major principle match originally identical predicate map relation prefer mapping structural intend domain principle relational coherent system prefer mapping individual corpus quite symbolic hand code symbolic handle argue process memory conventional unclear conventional dictionary semantic relations english
unobserve nonlinearity unobserve maker symmetric could high region region minimal boundary obtain central limit invert approximated use burn little sampling formula combine tool quantity interest interval sampling total parameter parametric due uncertainty measurement methodology try convergence convergence seem convergence empirical collect quantity finally histogram similar specify et still introduce much issue covariance block unique
number mistake large bad small diameter give mistake exploit cluster graph prove bind mistake vertex induce laplacian improve
initial possible outline intensive advantage proposal sufficiently flexible frequent chain reduce increase good performance proposal approximate density present try proposal reject entirely accept proposal initially proposal draw fitting tail proposal course tail reduce local mode mh update proposal history proposal iteration section framework give proof result extend proof methodology measure mh converge version laplace generate take take density density constant covariance relation note adaptive condition ergodicity result measurable integrable eq describe omit show nz nz normal harmonic normal mean normal obtain nz tail nz nz nz nz appropriate
select node leaf subtree flip leaf node replace leaf constraint remove replace copy branch branch randomly subtree consist offset exception constrain leaf leave subtree modification allow grow mutation shrink modification motivate identical exhibit modification accord boltzmann acceptance iteration q cost temperature iteration algorithm initialize randomly depth fix perform single optimizer seed detector every code execute result especially generate result leaf value box give entire consist ghz high optima expensive technique simulate anneal optimizer recall training image box dataset parameter distribution median value box determine effect corner affect corner detector sensitivity detector three different value combination detector compute dataset optimization produce score quite vary demonstrate fast er detector variety detector speed eight sequence six viewpoint approximately er detector pair pair use mask control thresholding laplace detector detector note detector case fast er detector
algorithmic implication causal assumption observation sample joint replace causal causal infer description complex causal follow causal dependent since algorithmic kolmogorov complexity motivate theorem corollary de max institute describe explain algorithmic complexity imply density make causal kolmogorov algorithmic analog empirically question independence test attract decade kind causal explicitly direction upon crucial connect causality introduce correspond direct representing indicate causal direct flow variable via next infer sense causal causal joint satisfies markov coincide follow version refer direct acyclic graph algorithmic version occur property like semi axiom causal also abstract notion abstract condition local cause observable something formalize variable quantity string object accordingly statistical like direct cause condition rather intuitive effect causal obtain product plausible prefer plausible follow causal inference prefer complexity refer one observe additive require direction prefer justify belief conditional causal tend causal conditional principle often benefit variable learn life apart former object necessarily sample comparison text similarity author appear later statement text similar statistical occurrence letter sequence direction infer relation though common object text come text idea real attack efficient decide key cca detect simple counting similaritie plain infer relation whether object object kolmogorov start notation string string description convert binary string length
mean accurately cm historical spread compute flexible equivalent ordinary least square ol finding formal availability regression several develop include maximum ol procedure test daily price stock period report confidence band clearly posterior stay spread figure observe confidence band likelihood historical reject hypothesis root agreement reveal period pp unit root significance unit root agree monitor device illustrate co exist operating market financial last stock index index stock major stock trading collect historical gold share reflect price gold whereas try replicate possible yield gold achieve proportion historical available period compare spread band co integrate maximize historical indicate integration suggest
observe may associate appear last article motivate general signal record instance interesting view estimator firstly exceed large secondly stage always reject section stage gaussian locally fractional kind band relevance confirm obtain regard al evolve change differently transition step begin reach arrival phase imply automatic cutting begin middle characteristic mean phenomenon begin component series observe exist define j k estimator principle
moment couple due quite acceptable covariance large covariance incorrectly preserve order whole ignore insight impose requirement plus variance single precisely scale opposite moment naive ignore quite way address previous density estimation together machine pl attempt input correspond output stochastic try problem instance quadratic loss notation express associate precisely training mention algorithm poorly pl poor call formal approach pl try interestingly none consideration mathematic bias pl
performance share rand good evolution overhead oracle dot bound bandit loss selection avoid regret converge show degradation compare present advance runtime light adaptive trial replace current efficient variation light exp light offline thank exp light remark grant trivially field intelligence alternative many kind display variability trial error still method follow availability learn pair approach
dual objective result see bayesian qx unlike sample pmf opposite sampling objective summarize nothing else
equation relaxation hard combinatorial consist norm short idea lagrangian nonzero simply give eq compare challenge construction problem indice nonzero sense equivalent problem call moreover
matrix jump well picture use multivariate spline datum response observe combine nonlinear linear irrelevant comparison several lm parameterization greedy root rmse simulation al nice number experiment stationary gp model essentially gp single give range min rd linear report al report mean comparison gp winner fit three covariate table double bar accurately correctly irrelevant gp bag svm cross cv randomize mse uniformly bag repeat bagging commonly show lm gave compare popular list gp lm lm computationally intensive lm initialize chain break large
draw equilibrium equation graph assignment combine series assignment method vision graph focus try address compatibility principled author relaxation problem mean nevertheless initial task closely use estimation alignment performance appropriate compatibility involve generic generic index instance match predictor discriminant map instance quadratic problem violate constraint compatibility compatibility joint slack instance monitoring threshold follow pair matching denote empty edge binary absence match compatibility compatibility
imply predictive teacher know domain able least party receive teacher know nevertheless able regard ask take relational single input x turn protocol protocol readily approximate therefore require simplicity proof learner testing distributional version prove problem demonstrate efficient quantum predictive relational class limitation predictive proof argue extract
q side let choosing hold simultaneously proceed member hence inequality precision place front inequality simplify bound combine precision rule obtain contain simplified inspection independent immediate index reward reward transition give hyperparameter mdp fully carry posterior action tractable recently minimization reformulate deviation control approach give minimum particular agent minimize observation variational treat distinction control translate problem show follow introduce set control relative consideration motivate section illustrate usage bandit relate work conclude agent environment formalize symbol signal probabilistic contrast theoretic proof letter thing string length sequence alphabet tuple write follow
select select e pattern outperform selection particular compare bag loading way extra purpose bag estimate additional thus test satisfied replication selection compute square distance show perform regression compare alternative namely ridge instead replication randomly know loading table repository replication one free generate sparse table sometimes competitive
particle evolve straightforward measure reference velocity field langevin describe molecular particle indistinguishable frame particle subsequent frame parameter acquisition small identify statistically particle weighting boolean particle match particle simplicity rescale function goal frame probable probability provide reliable algorithm conversely exponentially complex systematically approach observation fully bi loop b flow scenario conversely
book website receive book number make jump tail side website strictly formulation number search result fit strictly observation plot fig evolution sale mid book curve square fit square randomness sale perform amazon website observation determination mechanism obtain value book copy say book less economic impact variance roughly expectation would suggest deviation small jump control book amazon updating external month suggest constant point solid horizontal axis hour concern series record least eq total book shorter change amazon control book low sale series explanation change another reason fine fit point section change fine week determination long require daily trend title overcome automated acquisition programming look proportional graph axis adopt notation
spam filter mail spam consist service record pm business request grateful provide produce single queue server spam filter empirical display former maximum contrary service parametric service require bin standard interested queue al probability period reach begin mail empty end empty queue reach queue period service server case represent service reach else th period queue queue hence importance work follow simulate sampling service obtain time estimate exponential ki equal estimate service system know proposal achieve exponential parametric benchmark use exponential proposal mc cv former cv table give find become server estimate queue level
keyword calibration implicit computation abc need evaluation algorithm implicit likelihood become popularity primarily difficult compute despite popularity effect choice basic rejection inference presence word metric tolerance future work algorithm completely error account posterior abc rejection monte carlo approximate extend carlo calculating enable weighting understand carefully weight closely measurement belief wrong box wrong reality error deterministic stochastic datum measurement consider measurement
value development growth c contain gene development peak expression early representation relate function growth growth determination production growth gene type gradually minute hour concentration normal period ten minute refer microarray conduct condition pool normalize expression gene expression filter differentially thus differentially express gene cluster effect use penalty discover represent biological function interval three consist gene furthermore shift accordingly gene
occur threshold alarm straightforward unknown occur adapt minor minor change signal effect vary actual parameter alarm alarm false alarm delay trial false alarm discard small reason insensitive delay right wrong reason generate false occur arbitrarily long go plot delay versus alarm short mean delay show generator wide
rich enough binary set interval hence form uniquely unless explicitly root outside continue take unique set repetition choice definition generalize sequence correspond
definition minimize problem average eq arbitrary q theorem respectively apply concrete experiment normal simple fulfil constant likelihood time start terminate stage accept change characterize bayesian respectively due incorrect testing unitary observation cf call minimize show characterize procedure control notation therein
ensemble simulation assign member weight step weight kf enkf covariance tendency enkf unimodal enkf make enkf pf update move ensemble combine interest
notation n dx n student degree freedom respect xt n q tend follow virtue virtue recalling hand q let small enough last combine right tend conclude let z therefore small guarantee enough ensure lem loss great show obviously shall iii noting x n clearly note eq virtue n n shall suffice suffice ii iii n n n enough three finally complete position plan integer great use proof scheme relies state mn uv gr chernoff observe respective freedom establish complete theorem make lemma n ii ii u ratio bind establish note un un un un u n provide u u v consequently recursive virtue lead h l u sufficiently complete definition reference therein chen apply exclusive exhaustive rigorously risk error average operation sample absolutely introduction inference population view rx engineering sciences formulate testing mutually exclusive exhaustive I probability wrong specify I continuous process requirement impose controlling decision I concept confidence sequence desire variable l u ii propose exist index control confidence termination process decision include index controlling include interval interval specify inclusion termination simplify method construct rule inclusion stop value inclusion let let li
large show otherwise key low measure formula n soon taylor h assume infimum soon see decrease result proposition element denote every later k statement imply every margin depend hence ii k side strong adaptive keep value small global hold accord l author acknowledge dms dms author mostly carry paris de project anonymous presentation proposition condition tackle selection consider weak easy
square estimate unknown obtain less step fourth mse discard look residual amplitude contribution spurious cluster consider one low independent order order plot generate zero add I obtain average original produce result five necessarily simulation pt conjecture remark abstract make dirac point plane complex moment additive approximate approximate
control reasonably generic broad control multiple include linear adjusted otherwise assumption fix j show superior definition notation simplify considerably quantity fraction bootstrap yield equal event equivalent hence eq furthermore base widely area ad hoc assigning relevance predictor often progress build extension clean selection moreover family false fdr focus assign significance extend procedure greatly exceed show competitive indeed highly variable article split choice splitting dependence fdr method numerically real false design
dynamical case steady keep learn steady input teacher steady update finite perform figure avoid simulation agree theoretical one confirm average stop steady rate get learn teacher go approach monotonically steady increase steady value great ref hand continue state begin monotonic unity fig ensemble efficient even teacher dynamical student monotonically
inequality definition classical overall conclusion experiment competitive evidence definition value model difference definition automatically look confident use shape framework first square paper power collection elaborate asymptotically detail advantage without know advance estimation quadratic far instance multiplicative performance whereas increase general satisfie large plug lead really general shape penalty instance computation assume eq proportional suggest fold penalty cost factor automatically penaltie penalty asymptotically square naturally extend lead contrast shape rademacher however whether slope extend several concentration slope heuristic square article include come closeness conjecture end successfully different shape interesting open problem
maximum methodology immediately dimension univariate density straightforward discuss automatic valuable include iterative three vertex describe increase support decrease brief concluding suggest future research appear reference proof defer appendix give idea introduce commonly encounter gamma distribution example density unimodal fairly tail may logarithm half thus cauchy pareto log mixture density instance assumption concavity economic concavity prefer mean competition concavity equilibrium produce concavity economic concave sampling density density density densitie pointwise density increase hazard reliability theory unimodal unimodal dimension mention concern multidimensional density log concavity one relate notion insight issue section although design
eq read refer basically determine system benefit clear eq integral multi log point point radius w qp qp tangent direction sense extension generalization concept line
bernstein polynomial elegant acknowledgement acknowledge financial project ph institute innovation like thank literature multivariate polynomial bernstein polynomial linear whose property b polynomial expansion polynomial tell combination bernstein imply increase polynomial uniformly instance deduce theorem theorem notion countable sequence subject belief model use coherent replacement problem conservative coherent exchangeable dominate deal belief random finite exchangeable exchangeability terminology often sequence exchangeable independent iid de amongst going consider refer modern exchangeability exchangeability important virtue de exchangeable claim eliminate restrict exchangeable exchangeability term fair price able specify fair transaction require able decide real price explicitly allow distinguish price strictly specify subject coherence precise de probability treatment try realistic brief overview e
self volume measurement volume change percentage percentage change repeat compound side diagnosis mean diagnosis cdr cdr diagnosis side diagnosis effect neither diagnosis side line join apart main diagnosis meaningful hoc cdr volume significantly cdr logistic discrimination apply procedure get subject good fit rate model compare rate sensitivity specificity volume volume unlike well classifier compare distance volume time notice distance oppose volume definition modeling measure var repeat diagnosis diagnosis effect diagnosis side consequently join interaction far apart diagnosis comparison meaningful cdr distance significantly absolute discrimination side get predictor fit relatively performance table observe sensitivity expense substantial classification specificity cross logistic distance volume logistic side elimination leave right predictor logistic cost correct specificity likewise cost rate specificity well optimal rate specificity classifier correct specificity compare classifier table logistic volume classification performance well separately discrimination apply logistic discrimination apply elimination logistic rate cost distance volume algorithm generate group subject label cdr cdr baseline follow diagnosis comparison field diagnosis metric measurement metric whereas momentum vector also change momentum template field template previously volume loss subject distinguish control largely distinguished subject distinct loss wang analyze leave right superior cdr left subject subject subject odd proximity mild cdr subject momentum metric difference baseline subject cdr al velocity show similar pattern reason metric compound baseline cdr cdr table cdr subjects cdr find diagnosis detect could volume velocity field baseline significantly cdr cdr quantify shape cdr cdr subjects baseline cdr cdr subject within cdr subject change baseline follow largely cdr change involve body
let exist procedure strict satisfying account q equal whenever treat minimize easy sequential decision cost sequential additionally discuss bayesian g hand thus make risk essence rule let sequential
item vary dominate bad lift tendency support lift unstable small change rank high lift hyper quantify nan occurrence independent database hyper lift lift quantile geometric skewed lift order lift setup side fisher measure problematic lift outperform lift statistic datum construct superior complicated incorporate domain stream page page artificial dependency item could effectiveness datum mining rule university economic business discover transaction different fail property start present simple use association present database lift measure confidence systematically influence occur transaction occur homogeneous transaction poisson intensity transaction observe occur item contain transaction transaction bernoulli database incidence column transaction indicate correspond
clear hand understand community potentially complex structure leverage self map unsupervise call arranged partition way prior structure som relaxed arrange member turn som preserve som euclidean directly adapt variant som lot past ten framework adapt whole focus node possible variant dataset dissimilarity propose som som generalization center generalize median variant som field introduce rely som natural well laplacian heat kernel dissimilarity object som type som map version link outline propose paragraph reflect implicit call trick apply topology map exactly spectral rather algorithm via map som present poorly organization map unbalanced quite surprising help distinguish restrict rely visualization batch som dissimilarity som embed via numerous som variant
allocation test bias result difficulty carlo substitute expect qr size test simulation base carlo find qr adjustment moreover adjustment segregation qr theorem proposition theorem theorem theorem conjecture university spatial spatial segregation test contingency frequency type nearest neighbor nn pair test nan pattern randomness class label rl rl point distribution share contingency point provide spatial adjust underlie segregation two multi design pearson ease segregation frequently rl however show rl distribution join count also appropriate base nn segregation test new
near smooth kernel smoother singular large hence resolve difficulty arise resolve potential issue suitably modify underlie smooth singular follow eigenvalue smoother well behave smooth produces fail boost pilot smooth raw iterate desirable boost increase iteration pilot bring decide stop iterative question stop aic error specify numerically estimate mean tendency smoother typically correct version simplify unbiased three fold cross evaluate design end smooth bias correct rewrite size give size predict advantageous vector weight boost add fit parameter basis function tn note value parameter property smooth test smooth selector simply validation generally insight property stop
establish binomial integer even need preliminary z sign z partial lemma variation
trend pattern year estimate occur grow change additive include v j ij plot pattern nonlinear interaction reveal simple work national foundation grateful constructive comment instance regression square smooth absolute shrinkage ordinal represent basis numerical programming pool adjacent regularization c monotonicity qualitative role nonparametric often justify theoretically
p k kl order node let state observation kl kl mp mp p mp py consider separately right hand express virtue kl kl j j p kl argue obtain product equal kl p kl kl kl virtue implies finally recall positive path path construction eq observe required construction choose barrier consider situation already nothing situation complementary extension ensure separation x barrier make arbitrary q consecutive
special hold convergence small size achieve happen sequence small soon put already show switch predictive predictive complexity extend always achieve additionally minimax rate although switch define estimator x ig n inequality consequence part switch compare reasonable restriction achieve proportional risk predictor require example kk resembles implicitly impose geometric allow prior prior crucial loss strategy time grow run switch switch though may relative switch suffice oracle switch sample allow switch section q strategy outcome online switch w nk w n w nk report x section selection aic bic run switch report posterior marginal question whether receive lot apply literature well aic prediction vs truth combine type parametric setting consistent average minimax rate seem selection criterion consistent minimax result prove variation regression design setup notion combination somewhat formally imagine hold result allow respect minimax consistent keep exist consistent game allow explain procedure essential whenever minimax rate minimax reason switch achieve vary whenever fix rate proof find intuitively slow proposition
fail analysis mb mi mi return daily return hyperparameter horizon typically week gamma vector concern impose cp use suggestion conjugate occur instant interested repository propose web site mcmc assess numerical ghz processor ram os minute ergodic compute cluster frequency compute sort program use sort preserve var credible interval agreement former natural extension identification general variance empirically justify daily contribution volatility ten depict posterior var moreover present arithmetic
either countable lebesgue measure case mass countable lattice form b regular continuous admit density lebesgue logarithm distribution q constraint exist express tx qx shall condition family lie boundary restrict x satisfy invertible hold infimum uniquely attain unique determine phenomenon give lattice span q concentration phenomenon
check checking rule forecasting realize forecast rule characteristic et universal forecasting checking rule event tend computable partial recursive slightly randomized forecasting forecasting past realize forecast take place define modify past forecast define recursively eliminate condition require forecast computing forecasting fed finish
tail thus directional fdr derive random directional error use numerically compute directional fdr curve dash curve rule curve solid yield dash curve correspond log mutation gene mark frequentist adjust freedom confidence marginal posterior black curve corresponds adjust adjusted posterior credible interval posterior gene curve shrink towards weak mode credible probability select joint density student school teacher predict student predict college ability predict draw event predict close conditional decrease stochastically student selection large inference discussion microarray fold gene expression datum necessary bayesian iid laplace hereafter control credible interesting yield construct credible model later credible construct per draw inference respective credible assume unknown informative
limit limit provide drift eq eq assumption ergodicity lemma assumption invertible necessary drift order prove analogue drift parameter system identifiable integrable assumption compact straightforward likelihood address make natural invariant multiscale system slow resp absolutely continuous respect sde absolutely lebesgue density dy dy fast invariant resp dx dx dy satisfy every dx dy uniformly cc independent straightforward periodic need essentially require prove generator fast gap structure smooth criterion facilitate determination whether reference ask happen multiscale equation
divide experiment dyadic dyadic bin size loo f worth sect come jensen absolute proof satisfy
weight although observe poor dataset logical fairly align detect frame include background use target error initially sift describe error reveal performance higher reveal benefit sift feature rotation second affine invariant sift testing validation training sift long transformation method identify invariant sift well show match unary high right interestingly weight stage candidate unable
heuristic focus seek low restrict isometry define nuclear solution ensemble affine build seminal development determine condition strong parallelism diagonal non exploit developed heuristic guarantee compress sense heuristic coincide affine characterize affine map define constraint nan hold dimension equality constraint appropriate dimension small nuclear heuristic observe probabilistic accurately succeed variable generality assume mb
list restriction execute u eliminate exploration direction method new direction element element element low restriction calculate minimal say restriction cover restriction calculate cover discard restriction iteration begin exploration flow big previous point explore adjacent ca representation contain line I list list save cost prevent element recursive procedure procedure perform line select upper element big definition u adjacent cost big l u lr r restriction update update recursive visit element upper restriction adjacent
see advantage fix appropriate condition tend theoretical method approximate pair address select common recursively I evaluate choose successive closely em incomplete happen ef see convergence estimator simulate density sequel exist generic equation generate state mutually random precise form imply markovian past past depend capture assume initial dynamic frequently tracking vision finance monitor communication audio engineering list general system involve apparent case finite alphabet vast analytic solution posterior obtain predictive distribution state implementation smc density fix work analogously indeed since let qx dirac singular respect handle theoretic section adapt consider density time drop notation irrelevant nf yield simulate particle application infeasible computationally auxiliary reject overcome simulate instrumental
conjunction state label choose benchmark result query statistically compare benchmark benchmark statistically benchmark irrelevant inspired method benchmark
make develop tackle sphere research one reconstruction sect purpose show expand permit map code computation perturbation series diagram information former thereby become moreover diagram encode skeleton however practice digital permit choose discretized distribute happen spherical computationally force implicit step significantly read scalar product component dimensional concrete discrete pixel volume attribute product discretize vector matter often signal product integration count within tensor product analogously path realization discretize integral normalize integrate probability distribution white stress formally take integral detail sec argument could way sometimes avoid signal existence pose reconstruction differ space scale small expect identical vanish many physical theory define variation incomplete give observational function approach could incomplete sampling cost volume space nuisance realization function q integral delta also express q contain information argue contain fr configuration permit q hamiltonian moment definition permit fr connect function moment dispersion express counterpart correction gaussian connect vanish four define hamiltonian taylor fr permit think taylor role compare linear information define permit partially reconstruct take finally create mode free harmonic interaction investigate simple signal assume device accord response linear contain window transformation space signal hamiltonian response fashion read discrete space constant field gaussian yield eq permit signal generate generalize describe field contain autocorrelation connect correlation zero around test latter free via permit approximation exponential principle signal filter exhaustive however field fluctuation probe maximum hamiltonian become theory get classical mapping field eq calculate hessian maximum residual pure gaussian
consistently cross fold appear fold specify integer increase negative bic require bic supplementary derive degree column material show study tend model perform lasso tuning strategy nine tuning validation result bic material select simulate generate predictor x x normal specify set copy number simulate regression setting residual ratio equal pre specify level response predictor response deviation average datum simulation setting pre specify indicator false negative selection say predictor incorrectly simulation I various setting fix adjacency predictor response total clear fp positive relation perform reasonably considerably compare incorporate response master predictor false positive impact parameter play role figure tend select negative figure orthogonality degree freedom consequently small seem though master predictor especially
eq know generally constraint quickly rewrite lagrangian negative arrive dual maximum maximize maximal function inner avoid curse dimensionality cause trick implicitly ann dual separable another non feature know easy implicitly inner hilbert space could linear hilbert one could string symbol soon definite element support job kernel show book guarantee hyperplane rest classifier create classifier predicting give high binary create class direct acyclic one mechanism classifier place root class mean move leaf useful texture use color pixel matter scalable color descriptor descriptor color descriptor descriptor description descriptor similar texture descriptor descriptor name visual
first burn converge infer seem partition ise model plot histogram autocorrelation ise methodology image segmentation noisy lattice represent color white noisy unknown parameter though provide good noise ise proportional chain adaptive metropolis metropolis boundary generate inverse gamma xu pixel neighbor sample
negative easily non mt ba b jj e change mt ba jj u du analogously assume binomial expansion take q agree agree appendix expression obtain integer respectively
outcome variance large degenerate maxima regard wise simulate present representative request representative I standard model present low result method three estimating orientation mean apart bias standard high compare notice dominate line triangle configuration bias although help deviation category
burden show noise smoothed extent simply act closed general estimate exploit issue case let part drop simplify notation hypothesis full point consider upper triangular generalize eigenvalue interest factorization gram schmidt l k
posterior model rational fraction remain eq z prior model choose avoid consist value impossible distinguish case therefore position approximations posterior simulate exact bayes mc loop number accept bad figure quantile euclidean slight difference evaluate bayes clearly fit fit occur limit thus difference occurrence observe factor belong category set close category
integration dimensional function evaluation furthermore density enable plausibility realization realization transformation negligible please class quality apply quality contrary would like multi topic omit experimental recognition one classification
response residual sequentially orthogonal component penalize principal training code take minute computer intel ghz core implement penalization via thresholde construct sparsity automatically clear predictor utility molecular indicate protein penalization noisy comment prove uncorrelated orthogonal conclude denote tr large
phenomenon infinitely simulation step tolerance estimation adapt reduction upper quantile totally context indirect indirect proceed step auxiliary usually simplify version true obtain summary statistic maximize build euclidean means informative queue service uniformly arrival arrival customer inter generative algorithm inference inter arrival unobserved inference involve dimensional generate successive inter observation uniform sensitivity summary abc summary include inter quantile inter use replicate distribution mode close ground
relative prescribe r k k p p rule b p p poisson variable base regard virtue relative prescribe k p positive stand stage
randomly orthonormal span derivation omit iterate ix jx x decomposition inference burn period bayesian reduction nonlinearity compare histogram distance reconstruction visualization
indexing therefore identify complex equivalent compare involve replication moment good rmse average obtain use setup assume transform cluster slow improvement notice estimate plotted coincide full fast report column fig star shape e report third ill exploit need average respect provide properly select retrieve
ridge difference explore keyword elastic net shrinkage lar penalize familiar regression pn dimensional noise normally arbitrary regressor contribute minimize incorrectly use conditional elegant effect lead variance zero posterior mean justify exploit adaptive ridge selector shrinkage irrelevant maximized select impose converge rapid selection hierarchical motivate interest explain step regression focus derive marginal likelihood
word word follow manner mutation otherwise basis randomly generate sequentially markov occurrence consider dna markov transition estimate analytical al p value numerically via recursive display illustrate example combine eight generally p algorithm
freedom respectively wishart consist return share p daily chapter tv forecast series volatility log evaluate square standardized one absolute var discount indicate produce also tv vary inferior invariant var inferior tv model var factor previous show factor superiority large one forecast forecast share forecast indicate share correlate index correlation dynamic bound stability forecast
available diagnostic assessment tool principle balance provide qualitative derive criterion anneal estimator via slice metropolis base versus slice central limit interest lie monte importance resample sir closely resemble although possible method greatly integral arise mechanic metropolis algorithm equilibrium property interact behaviour property construct chain iteration tend conditional regardless start hasting generalize propose metropolis transform reversible normalizing introduce full conditional metropolis hasting variant gibbs sample augmentation substitution algorithm
dimensionality visualization necessity potentially good dimensionality datum approximation fisher distance similarity reduction low visualization drive geometry natural parameterization us approximation information distance variety closely approximation good manifold densely specifically give pdfs parameterize pdfs pdfs intuitively shortest connect approximate distance riemannian dissimilarity entity classify supervise purpose manifold matrix fisher
essence substantial choose use aspect try ensure prior right substantial plausible avoid goal meet robust change inform prior reliable convenience implement rescaling dependent center choose suitable shrink limit keep tree increase prior become tight shrinkage counterpart choice find recommend default choice alternatively cross range choice transformation tree split invariant simplicity appeal contrast like neural combination conjugate chi square guide specification hyperparameter inform prior region plausible avoid essentially rough natural naive sample deviation specification residual deviation shape value quantile consider illustrate prior three rough refer conservative default mode toward value increase recommend choose seem much three similar automatic default avoid validation reasonable treat put proceed fully validation reasonable strategy requirement cost strategy make experience predictive begin large avoid see yield excellent finally shall consideration choose play variable bayesian setup distribution eq determine sum exhaustive level set
proportional dimension define e upper unimodal achieve large density estimate random completely establish poisson method denote volume r fall k linear subject analysis achieve summation tend j n
section decompose block display element positive denote gaussian graphical correspond q base clique mp subgraphs vertex decomposable write hand commonly use note hand calculate successively adjust
relative arrival integer inter arrival time I random characteristic inter arrival denote
density tx dy da entropy generalize gibbs sampler yu weak liu qualitative comment hilbert space square
synthetic data eq three maximum candidate estimate predictive solid alm show alm ten thing adaptive ht middle partition alm solid dash snapshot figure learn probably one near alm side right alm relative boundary alm quantiles alm select middle snapshot sample heavily deal alm agreement far concerned uncertainty near snapshot cosine focus alm agree near sample marginal final bottom panel truth left panel increase axis sample decrease despite alm difference alm negligible likely quality candidate prevent behavior alm illustration something square evolve dataset allow sample mse add add design alm implement et powerful alm even impact sampling input wherein candidate sample figure snapshot sample room evenly split region observe plot consider split along mix reversible jump space ht middle left surface criterion map sample figure situation improve high
simultaneous confidence band normality new tool simultaneous extend multivariate inference derivative band david universit paris du paris france present
field necessary order contain variable need random stand distribution distribution say compare posterior see appear natural statistical sake measurable law analogously f nc pf nf independent moment surely almost surely nf p surely proof apply conditioning order quantitative version resort wasserstein wasserstein
choice instant agreement later section classical notice introduction row function yield indicator extend ij prove cell familiar field convenient indicator write table probability mean I review relationship minor define follow ij p sequence orthogonal symbol matrix sub space abuse statistical statistic detail
arbitrarily rest regret dimension cover develop regret fact evenly space precise prove match low I version na I idea armed evenly spaced phase achievable na I cover dimension regret space well know construction mab rely create payoff remain open slightly contain infinitely need ensure open continue recursion infimum open infimum may arise ball sufficiently small large covering explain supremum metric contain ball every radius cover great disjoint ball center otherwise disjoint element ball cover hence desire min let lemma finitely disjoint ball single radius radius center point collection denote ib ib bx function expectation claim base half pick defer positive right dominate finitely strong surely seem well obviously hard subsection proof consider randomize construction fix bandit conditional first strategy pick inside us notation let fix arbitrary lipschitz mab algorithm conditioning bit remove average bit value
consist consist operation combine neighborhood precision f plot solid dash dot decide os enyi world likely property degree perspective regardless degree vary efficient direction datum consist vote voting record vote record yes time discover cast map st smooth tuning select parameter square blue represent learn network political division estimate either connect member party member party similar party need evolve political division invariant evolve pattern place capture interestingly interact event discover political evolve discover category consider neighbor conservative share view toward although political view increasingly neighbor time point actually view environmental reflect emphasize static network
stationarity long interval valid problematic bit second long stimulus varie extent response factor deterministic ergodicity trial stimulus affect determine invariance response part intuition setting mutual may intuitive effect acknowledgment california berkeley discussion also thank anonymous comment greatly fellowship dc b nsf grant dms dms nf fellowship work institute california berkeley
immediate define requirement would coherence lead selection situation strategie capital immediate conversely protocol move space set gamble p terminal gamble predictive cone gamble gamble cone set gamble obtain terminal situation first show really desirable gamble e observe contain gamble contradict reality make possible terminal tt ss therefore prove f possibility terminal situation hand since cut unique u f tt terminal follow converse f invoke notation find terminal eq u yield together tf tf sufficient necessary suppose elsewhere situation equality terminal situation supremum eq follow proof build situation form st elementary measurable ease find belong take side q indeed prove theorem low theoretic correspondence ii allow result process framework theory indicate deal prove interesting law number recent year uncertainty probability value theoretic outline seem certainly influence rational subject belief situation alternatively cut corresponding partition event cut interpret stop situation say element terminal situation immediately terminal terminal situation reality situation arc connect child mean concatenation draw draw fill grow right
arrive measure prove continue theorem lemma hand hence proof notation outside ft ft ft ft almost square integrable cf te te bt part relax notice minimal polynomial estimate characteristic l ix give root b rectangular dimension diagonal block block dimension repeat characteristic polynomial characteristic repeat exactly time polynomial degree minimal characteristic sde separately sde
ccccc n reduce reduce know variance present study control model consider generate propose inferior near central superior parameter n theoretical expression relevant observe
analyze establish validity rao justify exact paper mcmc distribution numerous even cite explicit acceptance rate random get death book gibbs tendency specify almost positivity unfortunately specify perfectly conditional correspond joint convergent arise improper prior much continue grow put chain follow give liu wu develop augmentation construction theorems original real simple variance frequentist prefer bayesian integral usual conditional sample gibbs early researcher hasting almost apply previously model good example building quickly getting
ii case modification omit define consider balance rate case first iii even nj first nj ep nj n l pt ei pt enough complete verification verify pick denote underlying projection onto span wavelet wavelet central derivation article wavelet compactly support wavelet readily projection onto space candidate estimator density project onto study know theory spline knot orthonormal wavelet spline wavelet wavelet exponentially decay wavelet wavelet spline compactly new asymptotic spline wavelet inequality typically bound
update free energy monotonically detector parallel immediately send user facilitate parallel decoding detector interference detector far decision subsequent insight include early distribution identical replace statistic therefore directly make derivation arrive detector scenario reduce monotonically take step detector k f f k difficult detector intuitive perfect interference derive stage simply f k k detector obtain routine replace energy minimization effect transform first act match extract indicate interference define simplification first imply need free specifie discrete execution coordinate enable feedback ask robust free energy solve problem entail inversion convex become local minima detection detection discrete work system channel free able minima bring minima due substitution refine iteratively exact discrete detector obtain soft routine algorithm detector utilize implement subsequent low discrete improve original cross snr small user implemented follow discrete offer identical suffer
keyword selection copula inference essential calculate model bayes bayesian respective prior denote respective pm p yy side bayes compare suggestion marginal method give show commonly marginal bayesian laplace calculation application many suggest newton approach use harmonic identity suggest possibility inefficient suggest method routine green algorithm large compare move calculate likelihood identity calculate marginal attribute employ gibb consist density estimate multiple hasting hasting block bridge also discuss bridge approach extend approach integration discuss physics
singular multivariate element orthogonal non positive first ia jacobian px n qp log logarithm lemma corollary procedure autoregressive return employ generalize innovation vector extend beta multiplicative beta flexible sequential volatility update volatility suggest vary correlation efficiently volatility multivariate forecasting
base exploration variant boltzmann dp source parameter optimally three phase measure greedy boltzmann optimal boltzmann policy phase greedy boltzmann advanced exploration component optimistic accord agent explore help make follow extent optimistic initialization find mdps acknowledgment grateful support ec grant manuscript ec ks sl lemma reward small form martingale ks learn visit become mdp visit px rx rx px visit step right similarly q martingale state exclude minor modification ks sl mdps consider reward px rx
rd assess demonstrate compression counterpart b observation curve tend ib divergence contract bottleneck ib formulate within relate entropy ib tradeoff compression regard ib represent
issue formulae text initially derivative derivative dd dr nd dr u dd rs dd rs r r b r define th th element l elsewhere q rs r k r abuse I page analogously define precision response structure regressor formulae bias bias formulae supplementary compare correct different analytical empirical nonlinear bias open line turn interesting proportion instance extension well reduce auxiliary linear bias section detail original finally final say distribution admit lebesgue let parameterization denote parameterization parameterization far play role sense new parameterization beta
become multimodal everywhere unimodal density aspect practitioner space reader application rest flow space involve dimension distance length high henceforth shall volume author two paper show relax limitation
p blind cognitive mac protocol primary traffic scenario cognitive sense capability utilize learn primary ensure secondary receiver dedicate second focus complexity slot mac protocol allow transition traffic numerical result blind protocol traffic receiver design mac protocol enable secondary maximize datum maintain synchronization secondary mac protocol equip control dedicate tune channel hand period
suggest probably mode expect due nonlinear solver space could recover increase effort impractical call expensive require solver obtain close solver correctly underlie greatly improve solver finer depict respective equation determine smc particle intermediate decrease solver consequence scheme result effective stage call depict infer see quantify correctly contamination depict marginal sparse traditional usually e mesh operation carry effort resolution exploration etc phenomena describe couple domain spatially advance mathematics solver primary potentially yx index mechanic heat diffusion fall physical associate problem exhibit characteristic approach address identification one hand deterministic technique attempt minimize prediction technique task augment issue ill inverse square rigorously system uncertainty provide quantification uncertainty stochastic counterpart involve frequentist inverse statistical approach bayesian attempt posterior formulation unified framework deal uncertainty noisy measurement assess inferential uncertainty medical well biological system spatially vary parameter furthermore large dimensional
equivalence may field characteristic rewrite integer scalar polynomial moment nonnegative factorial auxiliary symbol auxiliary indeed eq say multiplicative recall deal calculus multiplicative inverse multiplicative symbol previous simplify read symmetric statistical virtue term shall compound r remark state polynomial sum uncorrelated recover suitable replacement usually statistic power express
execute physical system much well fall physical realization scheme reduce exploitation method effective drawback explore difference proper offer balance action choose exploitation adjust similar simulate sa select strategy action select accomplish fundamental phenomenon superposition measure effectively superposition parallel updating observe eigen kind greatly exclude possible result yield specific quantum exploration balancing exploitation simulate phenomenon physical make quantum quantum physical realization equally weighted superposition quantum update amplitude combination hadamard physical implementation demonstrate feasibility cell individual eigen environment primary action eigen left action cell goal point minimize reward episode time state find step episode episode policy moving step state goal cell agent receive reward end episode reward
basic function trace relate chapter fundamental different graph member term allow restriction integer take dimension denote trace clarity henceforth sequence denote normal element know ix complexity estimate let element know large imply decrease vc widely study field statistical property convergence average property class element know complexity cost information approximate relate remark proportion class decrease actual binary class property cardinality complement description class vc label express state restriction property ix description dependence description complexity rapidly minor effectively almost complement space class condition later show respect dominate approximately entropy condition first property hold
sis possess screen property allow converge maximal allow infinity technical ensure non small dimensionality discussion screen iterate version sis enough equal sized index set second active continue index applicability variant study context machine marginally distinct jointly marginally predictor straightforward correlation decay marginally situation challenge especially iterate version sis sis yy n less value explain choose sis outline van sis second var sis variant since far choice step mean apply near sis reason tune set lack particularly overfitte logistic bayes randomly independent marginally response another coefficient make ideal pick risk large true may bayes close independence technique
contour density property information metric parameterization remains calculate instead lie manifold parameterization pdf embed density circumstance much determine dissimilarity available approximation problem use visualization effect perform lie information regardless parameterization parameterization manifold approximation divergence divergence important metric hellinger leibler divergence alpha kl kl theory entropy pdf leibler fisher eq information pdfs need return illustration univariate normal distribution manifold available closed kl divergence fp fig contour hold varie early b reference note star fp j star note divergence distance symmetry inequality property distance metric obtain divergence symmetric relate divergence approximate fisher divergence evaluation axiom hellinger hellinger kullback leibler divergence kullback leibler hellinger distance great mean pdfs region
approach experiment complete desirable confidence mean term z confidence readily case confidence
either bic positive simulate tell case provide evidence experiment replication keep near singular replication sample ij sample close wishart distribution design happen form align form divide varie value eigenvalue index simulate replication occur near overview min path eigenvalue vertical scale panel normalize panel definite pseudo inverse pseudo precision form coefficient variance advantage linear regression efficiently homotopy lar variance residual close suggest computational comparison pseudo argument favor stop move penalize expression precision matrix case namely spectral metric exception metric false positive cholesky estimate simulated grid size regularization
omit sum correspond situation map predictor look jointly direct graph refer dag define convention restrict parent belong subset hull cm goal selection kernel namely instead consider hull relevant section specific sparsity focus product sum grid applicable many kernel namely assume input kx p ix ij efficiently define connect correspond select dag nonlinear context select interaction interaction kernel kx whose degree kernel degree
namely note dispersion convergent assume unit fx uniformly convergent time furth theorem convergent derivative k shall entirely look function
domain quadrature iii quadrature rule rectangle method quadrature quadrature rule newton formula frequently split recognize assessment composite absolute cause term derivative fourth bound conservative rule drawback requirement rigorously guarantee salient adaptively force meet certain requirement start great form take determine difference v b integration nature formally assume test u v proceeds initial b v u u I u readily execution generate issue u general address respectively
distance reversible death process depend perspective testing reversible mcmc survey include birth process much except essential description reversible care therefore description advance approach kullback divergence mixture spirit give calibration kullback interpretation cover proposal factor orient direct exploitation major difference factor reversible jump proposal advance instead concentrate simulation effort mcmc often model competition estimating visit whole range exhaustive j solution integral use bridge anneal switching rely integration iterative attention different density produce carlo sampler implement auxiliary select costly another variable simulating require base alternative mixture uncertainty quality approximation
difference thin blue study figure main idea child tree start label insight onto wang follow principal component toy tree distance analog restrict component restriction analogous union onto line member q orthogonality tree much q concept pc tree line useful importance need fit data direction furthermore pc line aim capture first make
claim side precede display q immediately distinguish observe elementary suppose positive solution solution strictly coverage probability continuous n obviously since belong length rise short interval coverage must hold suppose generality decrease continuity coverage probability must thus uniqueness claim increase short within may hold entail n continuity coverage long close interval shorter n last view see proof lead rise short assume increase hold interval b
intractable plug density covariance dominant role distribution explain precede subsection base weighted european pricing scenario combination mc mc ratio ce ce mc cost increase asset option within time simulation pricing show effectiveness root mc description empirically bin estimate al carry cpu ghz code sequence pseudo quasi randomize random technique number normal example option world pricing integration modal optimal proposal inefficient modal approximate bin bad table massive efficiency adjust gain see factor unimodal integration suit ed option price approximately represent rare event option pricing show
event contain extend set definition mat p k I j j probability theory notion simplex definition disjoint atomic represent simplex describe term prefer lie projective vector span singleton represent two match simplex projective identify simplex part figure htb q alternatively equivalent special comprised unit simplex projective moment identity simplex projective circle view hilbert modify classical event suggest circumstance allow computation
short univariate v r j jx jx jx x x u weakly short original also interval univariate strict region moreover univariate demonstrate coverage confidence quantify strict improvement coverage idea preserve operator interval apply instance another order operator property age nonparametric function introduction growth assess health status evolution measure height weight body index classical perspective reference make height increase height survey subsample white giving let th quantile index target interest function index entire age monotonicity requirement target age quantile use regression satisfy
involve fact sampling sampling scheme section infinitely stage confidence virtue scheme cdf decision I provide p p p p p size n chernoff virtue chernoff scheme follow chernoff assume statement true provide p z z moreover p sample sequence appendix virtue cdf assume statement true provide p chernoff appendix scheme threshold p proof recall regard performance f j p statement hold shall asymptotic cdf size choose distinct b asymptotic x pr pr p c p statement p p p design construct p equivalent p sequel finite mix absolute relative error prescribed confidence develop assume p l p otherwise virtue cdf derive chernoff cdf scheme describe mix cdf size p associate need event mix follow l z cdf mixed criterion mix p p size r choice size decision simplify sufficiently establish p modify rule follow value shape modify variable assume parameter stop stop rule associate suggest sample last sure event stage small p p small subsection asymptotic inverse sampling chernoff distinct regard double decision variable mix number limit proof regard mix analysis p p pr rp statement proof proportion important comparative randomize clinical bernoulli refer proportion proportion exact sample notation tuple px ip propose virtue p parameter word coverage p ii coverage tuning adapt branch complementary apply determine many interval see reference therein illustration investigate root quadratic equation p l x x n n p n substituting root c p p c coverage follow let interval x yu p similarly long confidence interval control determine whether give complementary branch bind algorithm need computable domain p p gp p p apply define x lb lb lb multivariate function virtue term integer integer uk follow k p respectively k p respective positive uk p bound adapt branch technique coverage interval discussion pre specify reliability follow gp r error absolute relative margin p p virtue identity express u p large precede technique section shall interval second prescribe sampling width size sample p sure limit stage sampling propose estimation main small termination gp iii tuning apply illustration estimation binomial proportion confidence u complementary coverage lp p construct third reliability cast interval lp apply inclusion principle propose seek class sampling requirement notation x lp stage termination iv sampling guarantee th stage scheme section requirement lp n gp lp lp stage termination guarantee sampling determine possible size function give determine sequential sampling guarantee seek sequential small tuning coverage lp l subroutine appendix coverage random interval bound probability rectangular p sequel stop sampling stop rule complexity bounding apply establish let truncation define x x x x x lb lb lb n define virtue inequality see bound recursive k k sample p x p n k k k sequel shall apply develop computable purpose x p k k x k p p k p satisfy x k k like pp virtue bound computed employ branch hypercube coverage interval lp readily population population multinomial proportion multinomial binomial number trial success trial analog trial probability random number outcome multinomial multinomial give classical statistical volume confidence difficult category get hand region simultaneous interval offer straightforward intuitive assessment reliability interval p ep wang statement page method curse dimensionality parameter check grow exponentially respect justification statement argument strictly gx intersection page line bottom strictly page intersection unfortunately incorrect x gx x readily intersection wang wang conclusion intersection incorrect certainly incorrect affect wang determine simultaneous proportion construct confidence interval
long obvious slot access justify practice past suggest achieve thus access greedy channel expect instantaneous posteriori parameter make slight channel channel enforce modify order consider propose employ consider channel belief scheme scheme make channel belief vector secondary receiver third use well interference keep index represent discount spectrum user simulation scalar hypothesis choose decision belief channel slot give variance db sense center interference signal demonstrate rely amount learn observe upper incorporate signal especially interference explanation receive signal successfully get across receiver primary channel interference channel value low carry state signal good signal performance spectrum access distribution discuss section version section user secondary employ lack
central tendency interval x b interval interval correspond dispersion brief recall section bound already resolve hausdorff result conclude one account symbolic instance chapter simple commonly distance l set hausdorff
reject value line dot test dash horizontal line tail c level plot bias define critical tail generalize bias line coincide normal p plot formal side discussion important side sided strictly continuous observed side left tail generic location separate tail mean mode median separate tail depend exponential regardless detail
close sequential bayesian procedure think advantageous preferable opposed likelihood log dependent next section exist standardized denote tu u goodness square forecast error produce fit e n ts u definition unchanged mae I q another word amount asset way calculate term var portfolio portfolio percentage level portfolio define portfolio volatility tight var amount probable ensure clearly cover proportion probable var chapter goodness design discount give element
verify lem prove statement lem b b statement second lem show b lem r z monotonically lemma assumption apply first statement complete lem prove case case case lemma increase respect summary shall case follow p thus lem case iii I obvious second statement iii lemma summary p prove suppose interval set empty p p
two process couple identification care present ergodicity stationarity relaxed mention section assumption markov section study behaviour analyze group result comparison dissimilarity dynamic reference homogeneous diffusion expression standard drift coefficient assume ergodic process
emphasize multidimensional output mlp satisfactory improve let test classic denote statistic show
suggest relevant problem expect path new carlo specific dependent nonlinear show gain hoc understand option computational involve end focus derivation specific effective conjunction reduction application system paper improve carlo core extract nominal gain neighboring idea basis strategy improve monte describe control extract incur computational setup justify project estimation monte reduction
xshift cm cm circle circle circle yshift xshift circle right circle circle right circle circle pt circle pt right circle yshift cm xshift circle circle pt node circle right pt cm xshift circle node circle pt right circle circle actual big size big l fall join join instead join end ready fall join cluster join join note
construction powerful multiscale approach one refined increment process subgaussian tail correction j j nc bound weakly eq standard brownian limit additive correction term definition crucial confidence confidence mean auxiliary depend know imply common since rescale numerator center toy easy location risk situation index statistic numerator stochastic also coupling mixture random observation couple whole coupling arbitrary index denote simplicity case case
alphabet step extend accommodate symbol nature channel symbol value density estimate borel function positive borel measurable author coordinate always proceed function kernel I kernel correspond surely c integrable implication project approximate density member achievable additionally true f n discuss discuss law dp metric q particularly alphabet early clean pmf quantization next mechanic estimation core previous discretization evaluate dimensional channel inversion discretize estimate force greatly use method recently fast gauss base complexity factor estimation choice bandwidth recent dual reduce channel clearly convex subspace simplex algorithm broad n quantization naturally build search component x yy yy I inversion heart formulation particularly elegant symbol validation evaluation use barrier method quantization map get x g theorem asymptotic symbol respect symbol symbol technical theorem clean deterministic benchmark symbol symbol
derive need consistency consider two procedure subsection model f q estimator ne ib b bi estimator strictly bias individual correlation error nuisance use kernel u bias assumption kk correct correct modify spirit reason panel allow long covariance cc arise correlation summarize consistent stationary regression context converge true limit fast deal subsection linear infeasible inconsistent estimating minimizing function subject least give q I rewrite function tr w w concentrated term depend respect tr f f replace unobserved leave side substitute update solution equation inside arrange decrease note solving estimator though iteration obtain triplet jointly minimize objective
two independent property transform eq sign stable unless compressed need meaning represent stable scale skewed stable dynamic stream every maintain remain convenient three flexible moment count moment may stream instead parameter fractional value estimator reasonable mainly accuracy cause choice pareto closely logarithmic machine learn tail either dynamic distance entropy important summary
subtract much coupling coupling choice notion contact different approximate equilibrium bernoulli variable profile tell attain equilibrium local stationary site property sect variate copy move whenever precisely let mean site site change site rate jump reservoir etc go whether move move move copy hasting sect coupling variate metropolis site z z ratio involve rejection probability metropolis
zero find sequence delta us plot snr coincide almost agreement function however tail extremely variance try correlation due peak choose value snr perfectly mirror gaussian estimate despite represent white variable dft eq define new multiscale minimize fitting accounting penalty retrieve multiplier smooth fourier domain window noise period detail describe fouri square various multiscale estimator brownian path shorter another greatly error capture integrated multiscale eq brownian parameter coefficient simulate average realization
reduction offer difficulty situation nothing guarantee reduction instance average important case happen matrix state classifier fine reduction call offer exploit false negative detect classifier wrong mean reduce multiclass standard binary main advantage reduction global generate reduction copy use oracle description reduction reduction version use divide subset class classify go predict leaf global classifier possible construct recursively leave bin leaf subset among accord matrix oracle binary binary correspond kf traversal root copy tree label multiclass binary access dataset binary oracle call split sum size copy oracle training depth tree directly proportional probability go leaf upper simplification node imply quantum quantum non trivial copy simply class apply situation subset
immediate adjacency big object last decade among spherical devoted functional show capital necessary uncorrelated depend behaviour angular isotropic random field contrary happen indeed coefficient angular power decay decay angular sense heuristic basically realization compact develop drop dominant introduce version well localize less frequency component localization property domain principle impossible transform view suggest localization formulate
phase entry sample ij ij dense coupling column display sample model clear visually quantify metric calculate mse ij ij third display wise metric indicate ref u ij uk maximum example figure amplitude angle alpha scale estimate third wise scaling display column per dimension trial plot recovery
also unlikely get surface part average galaxy current prior estimation likelihood select grow reasonable keep identity discrepancy permutation convergence acknowledgement de paris france
still respect asymptotic bridge parameter influence term lasso selection asymptotically consider act contribution asymptotic way wavelet variance separately component section see observation assumption define nonnegative coincide spirit consist distribution respectively denote incorrectly select restrict vanishe conversely vanish asymptotically hence seem distinguished always pass adaptive lasso estimator conservative select even act procedure vanish limit see inspection impose paragraph pointwise sense vary well reveal proposition model selection asymptotic analysis uniform fail conservative satisfie consistent satisfie rr
random select matrix factor sample similar integer uniformly discrete pi hoeffding equivalently row q sum km scope scope scope let selection previous compatible identical local scope property column keep class lemma small get multiply eq gain compare exponential scope selection n individual independence sum hand statement complete greedy rewrite c scope informally tell desire require furthermore bad mean polynomially equation high lemma decision
include numerous medical technology force improve image nm resolution recently prototype image atomic result pixel observe signal deconvolution motivate many scientific application deconvolution hierarchical bayesian select compose weighted mixture standard exponential center prior deconvolution recently previous determine mixture weighting vs likelihood hierarchical approach difficulty positivity conjugate derive monte instability bayes unbiased sure however instability especially high ratio snr hyperparameter
empirical play ucb indicate result notice begin form ucb less something eq yield summation concavity ucb form recommendation accord provide picking follow depend leading constant useful free get bind provide analysis ucb state argument dependent constant gap situation moderate typically latter focus ucb uniform furth figure distinguish regime round regime involve small moderate combination moderate ucb preferable say perhaps magnitude illustrate theoretical plot carlo method accurate experiment interesting behavior make arbitrarily second plot numerical unclear picture become moderate favor allocation appear recommendation recommendation remark probably bernoulli parameter axis round axis mostly illustrate small computer yet ucb strategy however bind theorem
focus posterior summary infinite course choose density involve draw generate simulated tolerance threshold possibly n approximate target ci marginal material time consist individual last total database detect screen contact statistic begin epidemic consist simply view partial product k r tolerance give percentage simulation replicate particularly favorable achieve size might serious standard sir contact consist time infection time mcmc algorithm
consider type say node episode nan episode episode assess consider serial episode episode prescribe confidence fix inter event time episode span unit pair neuron high increase monotonically upper probability episode model though recurrence value calculate fire prior assess episode fix node deduce analyze test allow type assess sequential bind total firing neuron datum exceeds emphasize episode interaction chebyshev often loose example result usual hypothesis discover strength deviation good correspondingly discover connection little discover pattern threshold look episode connection strength connection strength much useful analyze experiment assume delay episode delay interval calculate adequate care delay resolution reasonable delay specify integer delay iid random early first take variation recurrence recurrence also easily derive recurrence little difference significance recurrence care distribution modify recurrence suitably significance stochastic separate conditional probability properly strength matter justification conditional denote strength train ground strength statistical theory use
provide machine svms large architecture code activation ann preferred network sigmoid activation neuron layer improve ii cross epoch bias subsection ann experimental overall error mode ann decay performance set ann specific hand application fig experimental validation white versus total ratio atomic dash line absolute b versus bar indicate gray square value validation test set full database plot model fig deviation line exp figure imply direction fig stability infer absolute calculate experimental indicate balanced behavior network region fig accurate stability finally fit serve slight calculate imply smoothly response regression database represent desirable exp good respective coefficient give calculate ann mode quality odd character parent lie prescribe ann overall mode cs c ce c c ce co c e co ce ce c ann cs c c c ce co ce quality et calculation calculation limit c ce ce ce c ce
party input party party want party wants suppose protocol learn party let execute polynomial time party protocol fact learn polynomial therefore give refer type circuit even set problem union belong union four phase sketch security generate prevent determination cardinality size party party cyclic party party next party party item party long item party odd party party avoid party get receive party everything party remove union party fully union send item belong security indeed one party interested intersection propose computation union repeat refer protocol use circuit early circuit suitable protocol suitable assume call protocol respectively input return share taylor receive share protocol share multiplication protocol execute give
converge average compression geometry say finite net net n exist achievable scheme nz nj element true encoder operate indeed expect true net z b encoder remain step various definition triangle property two distance measurable number entropy instance euclidean space dominate constant f scheme encoder distribution finite description encoder unless reliably identify enough ability underlie encoder unknown away encoder description agent essentially le existence operational distortion infimum n define limit operational distortion state type ii operational f achievable encoder achieve infimum enumeration construct encoder learner nj x excess f nj p x nj p nj take achieve limit would express purely theoretic straightforward upon work communication immediate eq requirement list define distortion xy general omit code rp pp ix np ny p f rp triple random p rp pn result would constructing achieve rich slowly grow combine union code rate overhead devise difficulty distortion average f find infimum rate noisy sequence I unknown wish reconstruct distortion page within dirac concentrated leave general nature find bound variable goal performance setting description performance presence proof separation tailor joint rate constrain accurate predictor variable input focus access description context repeat agent agent inference rate channel trade present kind training limited noiseless digital channel part part scheme impose separation compression restriction encoder reliably tailor learn optimum encoder construct part namely design happen pair result theoretic constrain communication classifier also paper arise limitation structure motivation complementary regression allow knowledge predictor constrain one learner map z nz nz pz main depend training show probably pac excess achievable excess channel training learn set encoder fig observe noiseless digital bit per operating encoder f n nj nz nz type ii encoder output training fig perfect digital n nj ny ny shall
amongst posterior probability correct winning affect plausibility argument thing introduce large drastically particular thing however reason complicated sure thing reduce introduce take account datum take bayes updating subscript product usual marginal describe related data plan concerned knowledge distribution twice posterior thus stage however prior stage knowledge update think prefer update rather write stage probability use update behind split probability I lot relationship principle understand rule proposition build product whereas I proposition space stage joint calibrate typically often therefore however usually parameter constraint distribution distribution must satisfy constraint distribution lagrange multiplier marginal multiplied correspond I subsequently illustrate logic selection paragraph prior rescale measure closeness may break soon knowledge get specify specify yet must assign joint hypothesis incorporate zero reweighte exponential original entropy odd evidence win sharp example confident prior nature dark think responsible accelerated expansion universe survey dark energy exactly equal minus former indicate play really sure evolve least literature answer present contribution four equation varied scale factor constant evolve complex ia probe test forecast assign automatically evenly amongst predictive distribution prior solely confident course explicitly assign key check careful realistic predictive energy address physics california address ts maximum ease usual simple thing hypothesis situation alternative sure thing may number alternative formalism modify explicit enumeration outline resolve amongst dark way decide datum compete hypothesis possibly wish plausibility drop hereafter provide mean side posterior encode justify conclusion base regard primarily difficult hoc calculate carry comparison equation publish author community whole plausibility vice versa sure thing suppose
diabetes body diabetes indicator bernoulli random variable standard probit parameter illustration opt metropolis hasting posterior dimensional walk scale normalization body significant variate indicate end previous simple regression scale start diabetes row value additional control variate bring acknowledgment author grateful mcmc work team lead improved presentation support present rao accept reject metropolis high adopt completely universal scheme illustrate toy probit accept reject hasting generation acceptance precisely dominate metropolis hasting respect dominate measure hasting value uniformity directly take flow auxiliary preserving integrating size rao reject candidate part due rao argument variance result derivation asymptotic improvement toy metropolis approximation metropolis hasting construction reference markov distribute transition dy transition density integrate geometric time p balance reversible estimator accept weight estimate geometric obvious hasting small quantity estimator conditional pair expectation u therefore surely involve variability iteration require intermediate fix fortunately unbiased dy proposal computational cost additional compare metropolis weight get variable algebra rewrite depend expectation version bring estimation follow result denote reference behave following q pt directly denote I first imply q array show c g c ip f lebesgue without generality q denominator thus central numerator since prove n n eq fact side bound result arbitrarily go term hand side n f r q p ph go go let proof note universal control variate metropolis hasting unbiased estimator estimate setting exploit series toy target acceptance target estimate walk proposal rao repeating represent gain rao overlap source randomness addition gain illustrate replication proposal start function gain fact acceptance probability rejection improvement variance variability estimate additional previous pt additional require additional despite occurrence difficulty unconstraine metropolis hasting target proposal quite produce slightly indicate clearly gain widely target proposal proposition rao version importance illustrate table rao variance importance extreme ideal bring considerable replication
example lemma lx fy fx fx l lx fy fx lx fx lx fy dy dy dy let lx lx lx r lx r lx r lx lx e together derive bound function straightforward function depend poisson q eq inequality adapt proposition omit evaluate nf l use poisson lf lf u straightforward uniform drift moreover converge n view px show finite lemma eq exist finite martingale lf px cn rely measure measurable value measurable chapter proposition chapter measurable increasingly e vx kp ne let eq convergence let notation use next path go conclude q lx dy lx n lx lx lf theorem exist invariant follow exist check lemma e sample bind see g process covariance central limit uniformly ergodic chain check corollary suffice I trivially write h x u u nx u x nx p x additional straightforward simple nk n nk n sample define back partial k p negligible gx nk k gx nk nx nk converge almost follow nx arrive g nk k v almost every path use lemma n arrive converge xx central one xu p p u nk define x I j n nk nk x difference nk j nk nk dx ergodic term author grateful helpful discussion remark section chain method transition distribution design sampler adaptive mcmc paper chain substantial chain carlo carlo efficiency seminal let adaptively marginal behave way chain mention assumption invariant interest equal design limit otherwise resample central limit kernel variance always limit substantial illustrate literature law large number study also numerically equation develop interact ir mcmc prove ingredient martingale example section metropolis space equip borel measure measurable function l monte family order far measurable measurable kernel real nh throughout initial simplicity expectation k follow k l heuristic invariant lx way choose choice ergodic limit choice monte simulation define importance invariant ir probability resample lx lx form q impossible try limit hasting lx lx dy distribution always independently follow resample accept lx x sampler simplify version e ix ix partition get sampler limit metropolis hasting lx dy l partitioning pt lx dy l work practice allow jump state accept theoretical partitioning remain restrict ce moreover drift drift check metropolis langevin energy enough quantify kernel ergodic measurable immediate irreducible invariant distribution admit invariant distribution natural central center denote give surely central sampler restrict case compact equip precisely subset constant endow define eq simplify notation dependence sampler denote imply lemma comment let fs n introduce dy nk j nx field behavior eq kernel g see dx show sampler asymptotic chain arrive interact also estimate tend particularly initial enjoy whether unfortunately answer hold present often check p hasting metropolis hasting target resp acceptance resp ax ax dy away transform measurable function hold rely stochastic use long result still use compare walk rwm algorithm sampler limit ir algorithm
denote support sublinear visualize hull dual vector angle original visualize surface hull deduce property subdifferential hence functional differentiable iff subdifferential support achieve examine roughly serve hyperplane find differentiable subdifferential singleton criterion state picture implication easy verify strict convexity uniqueness minimizer hence curvature lead decay occur e suggest strongly norm state convex norm q suppose functional show minimizer proof fix let achieve theorem expression get subdifferential order expansion arrive result dual convexity follow assumption concluding intuition bound functional differentiable state term rademacher complexity bound prove actually strategy typically provide construction throughout refer rate slow rate notion rate minimizer arise slow differentiable derivative grow fast rate upon show regret always arise satisfie flat norm immediately strongly tight quadratic attain difference concavity write vanish proceed indeed eq bound average one notion rademacher rademacher uniform omit dependence sake average guarantee minimization rademacher average key sample rademacher averages bound average show eq minimizer particular suboptimal supremum q even whole indeed exchange go supremum tangent fix eq worst average rademacher average lipschitz complexity multiplied lipschitz loss hull finite vc dimension give minimax remark bound depend bound examine game norm dual boundary achieve take round distribution easy zero arise away adversary play maximization optimum develop describe upper quadratic later converse infinite curvature bound viewpoint point suggest minimizer translate flat singleton face face support hyperplane origin support normalize singleton face equivalent distinct discuss support singleton face convex minimizer sp low arise fluctuation refer suppose singleton I contain shift class center constant recall f pf gaussian absolute rearrange least remark case reduce discuss low bound bound enough describe primal dual average act primal game dual optimum true suppose put intersection problem conclude non lower far put round asymptotically surface ball regret grow adversary sequence distribution come within game case visualize low look expert suffer loss per action contain simplex shape pyramid I game times deviation minus ball random deviation let turn hull support uniform process verify index structure lemma adversary n maximum result therefore present address low whereas hope easily require I construct convenience shorthand choose expectation identical statement prove induction lead define induction follow since base conditional unnecessary conditional expression done follow nonempty respective function take subset nonempty eliminate let adjoint nonempty invertible transformation contain face transformation face e away regret modify condition transformation mapping last clear point last understand sum quantify variable adopt proposition swap expectation maximize replace supremum inside square distribution invoke q ranging depend maximize compare concavity easy distribution concavity first clearly linear concave strongly convex expectation rearrange lipschitz establish result curvature subdifferential q evident substituting choice thus verify corollary computer science division berkeley study von regret adversarial behavior minimization process distribution adversary difference sum loss geometric interpretation concave minimizer optimal strategy within theory topic gain popularity past adversarial manner work statistical find root theory argue online early recently argue attractive indeed problem middle adversarial quite analysis security spam largely responsible interest learn recent book similarity online algorithm link phenomenon bad strategy paper attempt build bridge von regret difference loss loss stochastic lead similarity game round round player predict convex determine point emphasize adversary note differ instance constitute adversary draw loss sequence joint face conditional pt sake clarity proof consider quantity clear optimize eq compact supremum observe appeal depend token influence see prove divergence bregman differentiable subgradient notion infinite instance define put subgradient subgradient fact generalize divergence focus omit definition immediately eq q since expectation useful word expectation obtain round roughly speak lemma say jensen inside suppose joint marginal easy
respectively odd together use observe follow gaussian z chi bound suffice condition together bind final expand bs b drop subscript get formula follow lead version appear bound except small whenever need know incoherent basis satisfy equal hand maximally coherence mi member mi centre f mail fr theorem proposition treat via coefficient nonconvex coefficient ensure identifiability derive dictionary assume bernoulli basis identifiable grow logarithmic I require compressed matrix identification optimisation component separation blind source efficiently know huge inverse blind separation representation exactly dictionary success heavily fit class dictionary dictionary wavelet stem tool harmonic mathematical require type dictionary meet treat know certain surprisingly work dictionary dictionary recently people start learn component ica identifiability available well geometric identifiability overcomplete condition size grow exponentially provably good identification identifiability outlier hand concerned relatively e availability resource provably combinatorial dictionary learn availability inspire give dictionary investigate detail minimum possibility replace learn efficient investigate minima surprising result sample draw generate incoherent matrix locally identifiable distribution consider basis essentially real identification identifiability initial optimisation perform optimisation matrix experiment indicate natural turn dimension gaussian imply vector complete matrix representation consist find sparse significantly select ideal amount pseudo number entry nonconvex nonsmooth hard solve turn strategy principle news admit relaxed related finding sense representation sparsity collect column fit total zero prescribe admissible consider representation give dictionary np hard try fortunately since stable respect suited goal criterion might success pursuit norm several unlike problem change still convex programming identify transmission observation dictionary channel channel dictionary convex seem behave criterion make amenable numerical need define admissible dictionary family library orthonormal basis cosine fast selection possible search parametric learn jointly common domain concentrate sparse assume reader inequality see constraint let aspect treat adopt ability dictionary ideal determine extent objective spirit dictionary uniqueness overcomplete dictionary specify advance optimisation want condition successfully ambiguity face consist usual ambiguity avoid ambiguity dp sign relax requirement ask condition indicate mean permutation like incoherent minima even find optimisation spurious minima independently raise complementary identifiability guarantee match permutation change concentrate minima carry certainly serve question contrast identification assumption ideally bernoulli sufficiently incoherent basis illustrate section make atom observe perfectly proportion htbp ny cloud figure angle dictionary minima locate sign ambiguity restrict moreover despite spurious minima none htbp infinity varied sample repeat display associate sparse black spurious outlier seem grey observe fully transition several possible orthogonality difficulty globally optima solution section provide tool progress even bernoulli model well account believe warm tool understand robust outlier b dictionary identification number fortunately mathematically initial reader local draw describe correspond relatively necessary satisfied consider check satisfied index indeed consider sufficient htbp bernoulli many sparse observe belong inside polytope radius necessarily orthogonal incoherent lie polytope conclude condition true word coherence polytope state radius large include provide dictionary incoherent sense incoherent admit strict compare assumption dictionary considerably quantity quantity specific value accord involve dictionary example substantial htbp draw shape coefficient highly outlier polytope seem shape cube constant function however heavily nature determine size bernoulli large shape essentially choice behaviour htbp draw gaussian almost shape ball derive need sufficiently constitute draw perspective would laplacian draw prior locate nonzero observe set compress ill solve allow previously take follow role strictly assumption gaussian mainly simplicity reason theory believe many certain concentration determine image large small probability ball e significantly exponentially use yield coherence constraint type eq wish basis identifiable answer question bernoulli describe section assume incoherent identifiable except failure rapidly soon scale need example basis resp maximally incoherent check bernoulli necessary algebraic local learn incoherent random sparse long signal dictionary learning question case assume local minima corresponding desirable converse direction basis generate investigate resemble current ideally could coherence extremely unlikely minimum hard overcomplete noisy overcomplete prevent intrinsic condition lemma theorem formulation contaminate dictionary close need introduce follow product operator convention proof eq similar relation follow decomposition matrix zero consider gram matrix nan dictionary subspace make abuse cover unit point eq point minima x yy equation notion square admissible ccc admissible completeness write admissible cc matrix pair tangent arbitrary c complete define local inequality local b may f x x x tangent yx yield admissible consist admissible cx therefore k equivalent row entry row denominator decompose decompose numerator sum observe match q numerator straightforward proof global diagonal nan x thank understand mean matrix characterization
incomplete robustness coefficient variation cluster largely context genomic rather mixture produce interpretable cluster immediately significant fluctuation order challenge simplified component reduce factorization elementary product probability limit development produce cluster anonymous note tend easily proportional provide advantageous gene weak assign gene make small feature gamma tune determined assignment cluster category substantial simpler analyze gamma small nothing temporal dependence seem involved line microarray impose complicated identify gene act act different another confident fit though cluster cluster seem associate correlate rich model dependent care need parameter intrinsic mechanism mixture support expression trait factor order probability develop ranking enable implementation flexibility shape method property explore distribution pe line move factor far mind evaluate z poisson probability indicate equivalence stem relationship process accumulate hence value basically eq shape proceed z p z probability highlight right precisely function gamma distribute gamma conjugacy page integral evaluate contribution possibly conclude token work force row table eliminate force variable great say second force seven repeat table repeat domain know term eliminate contribution degree complete replicate three balanced across replicate component observation simple exercise strict concavity likelihood invoke lagrange derivative calculus ij I constant determine quadratic form f ix jx ax regardless nonnegative concavity know force linear independence complete distribution imply every put linear verify linearly minus pick one nonzero impose drop drop let complete pt cd na parameter gamma gamma recall hence round calculation five order structure reduce set model one gene great mapping structure approximation ideal eliminate gene cluster affect cluster examine quantile relate sample coefficient note largely em cycle cycle estimate shape change receive computation cpu second per second affect gene analyze size acknowledge find report r associate comment improve development support grant gm mixture although challenge scope structure computation depend gamma variable attain binomial find dynamic programming accord among concavity beneficial method promise gene cell exhibit sort examine control post amount pattern investigate relatively gene identify hundred thousand pattern substantial multi many different ever gene share recent perspective method partition approach informative satisfactory like select subtle anonymous external pattern gene approximate contribute technical problem minimize narrow biological treat mixture assign probable procedure mixture popular page considerable cluster technical affect modal establish constraint identifiability switching somewhat group possibly call place gene anonymous component linearly computational characteristic structure record pattern specification density embed gamma extend poisson response characteristic gene measure next sequencing relatively vector represent cell chemical stage along course record say indicate group analyst intensity microarray adjust systematic relate advance allow explicit abundance mixture gamma gene datum treat finite discrete structure order latent specifically proportion modeling partition carry ordering subset structure denote group subset partition expect group structure number grow think structure correspondence allow tie pt sp number association example entail group single mean without amount order subset subset constitute replicate induce gene replicate differentially equal subset low e absence expression generally union replicate language assumption necessarily regardless probability replicate determine bayes alternatively though gene go gene characteristic discrete calculation integrate transformation rate reflect identically gamma ordering parameterization center nan specification assume measurement independently empirically stochastic population consideration move decrease represent depend choose normalization joint gamma integral variable precede arrange product multiply factor involve simplification follow mutually shape structure entail single exchangeable multivariate compound gamma variation scale inspection shape parameter parameter special report density evaluate contour display contour three mass one way constraint structure restrict way share something wherein model solve order event mutually gamma possibly special variable compute numerical beta distribution clearly indicate assume formula develop monte certainly fast accurate preferable efficiency target shape setting numerical computing shape positive collection process denote distribute gamma gap form marginally process dependent overlap point next equal variable appendix sum construct inner indeed recursion although version viterbi density seem linearly identifiability strict concavity establish identifiability key step denote real number structure finite special balanced replicate density proceed multivariate degree lead center rational factor rational rational require parameter different treat linearly independent structured admit unique maximizer sure expectation apply strict concavity multiple insensitive starting position page confirm implementation recover draw show mix proportion share prototype share simple mixture proportion gamma www mixing structure effect identify appendix assumption quantile coefficient observation component check comparison infer cluster pattern parameter guide reduce issue represent biology though examine example summarize identify large code infer gene standardized display raw cycle rule heart stress three replicate measure five hour stress several consider old process produce temporal fitting mixture structure work mean aspect diagnosis appendix possibility cluster gamma contain stress baseline significant example give different find choose www adjust rand measure dissimilarity gamma rank gamma worth investigate involve activity activity molecular complete picture biology apparent set rank datum repository set relevant exhibit case f gene gamma order cluster gene fact gamma smaller wide low number significantly way rank em gr km cc al al pilot al et empirical ranking often seem frequently category functional gene category gene biological calculation cluster across go proportion small comparison set rank biological green plotted proportion
initialization local optimum pca dim train input dim dim mnist letter breast less diabetes peak converge mnist letter breast cancer diabetes peak diabete parameter simply wise adaboost criterion constraint adaboost affected value uci mahalanobis computational running algorithm task ram operation corresponding time dimension artificial dimension keep triplet interior sdp solver scale instead combine sdp solver back cone need fig input comparable dimension become significantly involved randomly pair test local represent histogram visual subset image visual histogram subset classification carefully rate respectively slightly svm classifier right triplet trend triplet lead face google two object retrieval problem target class subset use face calculate retrieve image precision subset subset report precision codebook consistently attain high advantage triplet codebook test triplet face face new generalize sense algorithm show art exist currently handle theorem conjecture updating eq eq exactly adaboost solve z denote base optimization summarize triplet ss parameter si u ss v ss tasks right pca recover datum preserve visualization artificial toy dataset circle dataset circle eight noise fail first informative dimension lda center overlap informative indicate successfully eliminate speed handwritten face uci last randomly use
cca maximal correlation cca address cca semantic indexing cca provide capture space bag word sparse cca subset language interpretability identify document inner bag require document immediately improve retrieval computation bag word illustrate achieve retrieval specifically english model feature feature associate vocabulary indicate collect vector collect english feature vector matrix collect document respectively english document product datum cca perform across english efficient dc cca successive sparse stack subsequent component reader english convert project english one loading differently say onto span associated language select projection distance neighbor use sentence smaller align english rare english term computing generate english appropriate query retrieval test document canonical percentage zero loading dc cca dc cca sparse cca cca curve query rank project vector euclidean query performance example return suppose approximation sparse apply method program iterative computationally use instead convex sdp algorithm formulation paragraph support vector machine minimize loss support formulation well study show study sparse propose tight cardinality formulate optimization program behavior dc cca dc cca demonstrate pca cca application case sparse experimentally benchmark real life vary dc pca explain variance perform pca scalability relevance cca language vocabulary music retrieval priori guarantee level similar sdp well shoot although original propose solve quadratic use version acknowledge national foundation california micro appendix derivation idea deriving program derive bi lagrangian dual dual program convex consider relaxed lagrangian eq q lemma bi obtain result derivation alg alg differently apply idea program indicator eq check derivation alg alg maximizer lie boundary lagrangian give q equivalently alg wherein goal obtain sparse principal pca canonical correlation cca cardinality previous method context sparse tight log problem solve convex program minimization result exhibit initialization subsequence iterate point program performance demonstrate gene gene cca vocabulary selection retrieval component fisher minimization music language eigenvalue find identity fundamental area multivariate prominent deal high visualization positive problem maximum matrix pair call know widely specific pca classic analysis compression visualization maximal variance use wherein ambient dimensional significant variational covariance semidefinite multivariate canonical cca dimensionality however pca deal space multivariate space information reflect correlation maximally correlate represent rewrite write xy yx xx yy fisher discriminant find projection onto lead covariance variational give rewrite formulation multi lead discriminant simplicity popularity lack suffer disadvantage vector interpret different pca cca application coordinate axis biology might correspond specific loading moreover asset trading technique solution consequence imply transaction cca copy corpus write english extract multiple dimensional variation document language aid translation interpret well music annotation description review acoustic content music retrieval sparse summarize generally desirable aid understanding reduce economic denote cardinality problem cardinality one usual approximate early cardinality constraint solve iterative pca theory iterative subsequence iterate c like mention cca sparse scalable dc possible show dc cca deal document vocabulary music annotation document retrieval application different language say query string language language experimentally cca loading canonical cca select pruning vocabulary underlie audio generalize programming computationally intensive briefly section cca instance algorithm section algorithm respectively mean definite semidefinite absolute value denote zero x I principal generalize solve former let consider p concave constraint non computationally intractable quadratic objective homogeneous quadratic two reason objective solve briefly discuss convex could version program objective except follow sdp tractable large sdp expensive wherein instead cardinality constraint minimization formulation concave convex relaxation simplify cardinality constraint scalability explore oppose e regularize version penalization parameter equivalent limit equivalent q combinatorial program selection machine factorial bayesian improper show demonstrate validity expansion approximation machine tight end know would know maximization easy formulation derive sparse value two function I I g formulate program trivially choose introduce difference convex optimization branch cutting solve c program nonlinear solve quadratic present generalization know maximization em behind algorithm numerical appear place robust correspondence recovery mm algorithm reference general idea mm idea construct function update already stop jensen quadratic upper equality follow sign change min max refers call put thing perspective jensen construction refer refer negative name study sm stand surrogate stand call surrogate example construct derive let optimization convex differentiable solve note concave suppose strict achieve g helpful return program eq write idea derive satisfie result hand check sequence constrain quadratic program clear alg irrespective unique lie boundary alg also appendix detail define diagonal alg similar constraint interpretation l l compute approximation norm ellipsoid iterative therefore small weighting therefore toward discussion clear problem alg l choose setup unsupervise cca free tune desire noted sparsity reduce lead searching value say cardinality check present obtain remove entry pattern nevertheless replace p solution termination certainly discard loading obtain loading surely improve mention iterative converge converge address question like initialization behavior mention front convergence theory iterative show globally convergent sequence converge constrain tucker kkt condition kkt obtain apply alg carry globally convergent corollary problem derivation mention set assign power map set say tucker kkt case sense fact term global result provide convergence ix jx uv continuous dc solve point generate dc alg nonempty suitable dc l vx vx f lx l f vx alg set alg obtain apply uv alg correspond alg nx follow convergence algorithm since whether follow corollary algorithm match eigenvector converge algorithm solve l variable n imply multiply side complementary give eigen suppose result algorithm generalize follow converge alg constraint solve sparse consider address pca special dc covariance reduce correspond sparse involve complexity dc pca exhibit dc pca exhibit property suppose interest dc converge dc reduce power method suppose proceed far briefly discuss rotation component thresholding principal true framework lasso enforce pca bound non pca angle cardinality lead sdp complexity scale benchmark set scalable high even nesterov propose combinatorial method lead target sparsity numerical sparse pca mention knowledge art compare approximate let version program maximization apply solve dc lx guarantee dc pca ensure irrelevant cardinality dc reduce zero svd principal unit loading semidefinite regression interpret circular mean laplacian maximum posteriori penalization aforementione define improper prior replace dc solution note like unlike pose approximate program dc obtained obtain program ellipsoid result gradient scheme confirm dc performance mention unlike eigenvalue setting cca effectiveness small dc dc except scalability wherein greedy algorithm comparison comparable carry ghz ram motivated discussion pc dc become benchmark pc explanatory pca eigenvector mention table pc loading dc pcs capture cardinality loading pc non loading capture pattern non loading total loading capture cumulative cardinality explain loading addition c show dc similar variation pc compute dc plot summarize compute dc pca setup assume curve variance cardinality vs cardinality computational size explain various induce regularization case pca increase vector decrease display computation see dc pca perform perform well
equilibrium population consist therefore variant player one player player bit choice quantity length bit feasible interval quantity ensure nash equilibrium prove induction equilibrium quantity always value ensure homogeneity population choice make choice accordance size etc run game social number time simulation equilibrium game estimate lead estimate give ne strategy one trajectory market quantity calculate figure value respectively ne unbiased deviation evolution see estimator player c c c vi player individual parameter run parameter testing player correct play hypothesis accept quantity polynomial model see player quantity value individual player cm p cm individual co alg co establish ne play nash ne quantity subset play strategy co co mutation population ne within mutation population require ne state course population differ less bit nash matter high calculate unbiased ad expect state arrival ne two heuristic ne potential equilibrium cs state end identical equilibrium qualitatively potential game since ne attention game play order check quantity one quantity heuristic depend complexity investigate player quantity lead nash equilibrium quantity mutation number inside prove statistical heuristic equilibrium ne belong finally social version algorithm allow multi environment simulate bit length population bit encode feasible value therefore model especially ne finally could theoretic ct like thank la h evolutionary stability behavior iv rao statistical process economic potential stable genetic optimization read van company tc co evolutionary market behavioral stability genetic evolutionary games equilibrium optima http www individual learn co evolutionary genetic learn version co establish nash contrast version see player strategy nash outcome player canonical evolutionary methodology general statistical find social ne play chain individual case ne simulations ne ne large indeed nash equilibrium game theory nash equilibrium two quantity turn price quantity market evolutionary study classical genetic optimization evolutionary version genetic evolutionary player determine player choice evolve goal dynamic system consideration player represent population algorithm strategy fitness establish play define active quantity player market quantity total quantity lead player fitness dependent previous co establish fitness proportional player produce price determine quantity demand algorithm update game converge nash agent determine tend nash strategy co evolutionary price quantity market see nash multi article fitness calculate game population current population pick give round update usual genetic operator mutation regard equilibrium quantity market prove finally population single order current fitness choose mutation lead market ne identical function game study et al decrease game pseudo potential game genetic therefore discrete principle course dense strategy ne game ne coincide investigate nash pure strategy case give player linear cost q decrease finally demand represent alternative choice genetic algorithm individual lead ne consideration introduce characterized opponent generation population create algorithms version player take generation form pool usual genetic operator mutation generation aggregate finally player population idea player pseudo represent player ga mutation player create realize fitness correspond evolutionary programming population population update take pseudo follow player decide fitness repeat strategy assign apply selection mutation keep implementation use selection proportional fitness mutation mutation rate bit classify ensure nash equilibrium introduce social algorithm population transform randomly draw player ga single mutation consist union population copy correspond population social evolutionary initialize player strategy assign decide value strategy evolutionary operator mutation union player copy new generation repeat difference social aggregate generation form economic choice update strategy learn since economic genetic allow player opponent every population assume k ij nj condition selection fitness scale solely
various justify reversible chain subsample asymptotic efficiency behaviour central chain f construct mean argument interpolation precision decrease estimator median space estimation uniformly ergodic chain bound hoeffding available lead bound functional consider technique result rigorous result cl truncation directly bound analogous sequential identification time difficult implement approach know towards small integral unbounded section parameter drift parameter design total elementary median multiple run illustrative toy emphasis unbounde ergodic practically drift apply particular effect paper interest measurable state homogeneous kernel interest mcmc walk transition qx g e function define evaluate variation precisely norm sequel geometrically ergodic chain markov say chain ergodicity drift define drift towards small satisfying sequel drift type condition assume geometric ergodicity definition ergodicity allow relevant devote c total sequel make ergodicity convergence drift explicit convergence unique establish appendix explicit improve ergodicity establish nx inequality hence theorem define bind without effort error essential confidence main drift constant explicitly corollary particular motivated quickly inequality confidence lead trajectory take chebyshev term roughly autocorrelation bottleneck somewhat improvement e burn approach measure appropriate precision typical bound imply therefore chebyshev get define good minimize calculation complete trick complexity need general odd random p k algorithm average markov chain base run estimate addition cf illustrate chain parameter drift k dy indicate reversible reversible relationship point reversible formula confidence possibly unbounded mcmc example starting compute lemma c total one walk e e e p uniform ergodicity optimize one walk find total reason bottleneck short run significantly effort long mathematically tractable reality estimator phenomenon infer asymptotic function compute result available http www ac uk message chain reach relevance unbounded function priori often practice visual look bound burn use possible derive conservative total drift remain universal tool obtain markov bound tight difficult convenience reader repeat sequel term refer respectively eq unique lr markov reversible dx reversible chain say reversible thm proposition thm thm remark partially education uk drift
would inclusion order covering interval interval partial let show quantile partial acyclic direct direct illustrate acyclic quantile comparison relation relation arc surface statement section dimension partial index vc whose e condition drop finally uniform van theorem display index estimate quantile look quantile fig show light quantile slow bottom right corner correspond point comparison fall diagonal quantile evenly space instead near minimum grow large estimate quantile monotonicity expand estimate unit square computing interval quantile rate illustrate quantile fig violate coincide except modify eliminate monotonicity dispersion quantile region boundary dispersion region extend concept quantile involve detailed graphical quantitative summary information regard c collect department periodic survey use formulate education consumption define run daily et propose approach quantile focus separately two protein gram partial situation usual sense comparable recognize extremely undesirable yet partial positive correlation alignment dependence important designing moreover quantile preserve transformation multidimensional protein subset survey use diagram fig align scatter diagram monotonically increase expect comparable people emphasize nonetheless interpret univariate quantile derive policy program maker quantile reasonable comparison quantile quantile partial quantile quantile reflect intuitive interpret quantile multidimensional quantile table quantile whereas generate quantile c ptc ptc index level give estimate color partial quality comparative quantile surface level hand light think well surface comparative need stay quantile estimate partial protein boundary rough symmetry surface location partial quantile figure figure dispersion def end finance literature central return return asset asset arise exposure intercept adjust yield return zero finance see reference market return break negative capture note well capture partial fig partial previous align b ptc quantile comparison surface narrow probability quite everywhere quantile monotonic quantile monotonic note fall support datum partial quantile strong true interpret result choice dominate slight performance dominate datum explain target ideal trade near datum extent comparable project effect school social media intervention pt yes yes partial maker pass fail regardless cost political consideration social cc media intervention cc cc tv acyclic direct probability pass partial value quantile similar traditional quantile outcome tv make partial propose quantile multivariate order important partial might space include robustness preserve partial regard discuss particular important order order additional partial quantile crucially depend heavily concept link application achieve quantile desire partial probability partial partial etc type application concept tie partial order quantile instance partial possibility partial surface covariate surface concept censor wide attract quantile suitable apply censor multidimensional motivation connection axiom preference allow identification decision partial quantile strategy valuable although pursuit scope believe future event proposition mapping mx mx x mx px x partial quantile px px px px lemma establish compact probability associate measurable measure vc vc therefore measure support dx e van singleton argument vc ensure van building upon p fx pointing let k fx k cone interior vector positively therefore continuously differentiable implicit twice differentiable compact condition continuity continuous follow union condition cardinality take note trivially follow piecewise constant mapping jump include jump p h eq theorem proof proceed quantile derive step pt x x definition feasible identification optimality x p dx x nu yield leave hand side use ii relation dx corollary complete satisfied convenience w p x n p x x p p w get multidimensional central simplification x follow corollary build space p g pg px proof satisfy p n x x p note proof since every thus dx dx x combine differentiable since strict minimum interior every p pt proof transform indicator see proceed modification fw iw iw restriction analytic zero open nonzero borel contradiction vanish nonempty open bx convolution everywhere turn us contradiction proper cone proof generality proceed connect separately general already px yy terminate individually strategy previously similar x du du u u du proof theorem point jx j jx jx df jx j j optimal optimality feasibility j take note variance around z f p q vary nonempty interior strict trivially assume point concavity pr show show def constraints optimal contradict lemma x arbitrary integration log walk construct membership control oracle construct oracle thank comment thorough version grateful suggestion type style e e proposition focus generalize preserve outlier characterize distribution partial sufficiently rich generalize concept partial perspective partial quantile furthermore procedure might establish complexity concentration natural order finally discuss several impact policy concept quantile prove notion robustness quantile role interest counterpart attract survey recent focus develop measure usually suitable nest partial instead incorporation work focus fundamental difficulty reach agreement suitable quantile arguably lack multidimensional various quantile character quantile acquire loose usage page simple multivariate quantile quantile fail multivariate feature attempt feature influence exploit metric multivariate quantile notion gradient univariate quantile variable away quantile relate notion assume minimum key insight rely family induce lack distinguish partial analysis different detailed discussion section definition framework generalization quantile study quantile mapping instrumental case applicability relation convergence infinitely many quantile accommodate restrict identification difficulty convergence partial index subset index probability study probability distribution quantile due quantile could curves context partial lattice new point monotone upon estimation mild possible curse dispersion partial moreover interesting primitive finally illustrate application evaluate within quantile imply value probability arbitrary set point compare comparable define x xx derive relation order relation exposition imply iii binary partial comparison draw point comparable usefulness fact xx xx sensible involve probability draw precede order index partial quantile define associate quantile univariate simply quantile would univariate quantile representative quantile quantile use maximize comparable partial quantile partial complete exploit quantile surface geometric multivariate well quantile occur partial balance correct comparable good approximate maximizer probability consequently interpretation quantile allow partial traditional characterize overall minimum quantile partial quantile considerably note def write saddle move imply definition notable interesting mapping preserving imply invariance preserve value quantity quantile surface comparison quantile preserve transform partial common invariant translation require symmetry distribution every partial quantile automatically quantile relation interest explore quantile view counterpart empirical let na nb carry notation depend notation level imply primitive discuss impose regularity induce eq nx px condition partial behave requirement condition imply primitive several primitive technical three considerably weak however lead sharp treatment derive mind partial quantile contain positive quantile nr identification partially identify spirit quantile continuous singleton convex criterion estimator van functional mild index weakly directly mild verify main example cone order nonempty case setup dx probability map acyclic ex partial describe direct direct acyclic sampling hold hold example estimation partial quantile order estimator univariate lack restrict achieve result quantile drawing comparable quantile interest whole framework huber partial reasonable highlight difficulty quantile rare might completely eq create ambiguity choice quantile probability comparison quantile surface partial quantile uniformly uniform intuitively ensure likely point estimator slack go comment aim ensure nonempty associated pt x cone partial describe sufficiently general suffice simplify affect could quantile practical case underlie bring measure estimate might application metric rely avoid application moreover need need account underlie motivated develop monotonicity curve quantile greatest refer join meet close construction base partial monotone scheme cone x monotone previously derive improvement conditional quantile always monotone assumption side therefore corollary lack partial point quantile quantile reflect independence carry random variable independent let partial value variable quantile norm eq partial close univariate quantile quantile phenomenon decrease contrast extreme comparable advantage soon concentration curse dimensionality comparison positively less align order perfect correlation transformation positively correlate trivial univariate quantile surprising concentration measure exhibit concentration quantile value order quantile logistic zero mean extreme measure grow likely extreme quantile surface close connection mass corner corner notion quantile surface connect order definition surface allow generalize efficient random realization interpret quantile surface partially parametrize probability draw comparable interest partially efficient close partially efficient might quite appeal support univariate dispersion measure quantile traditionally dispersion median expand quantile interval eq dispersion variable I shift interval region moreover quantile specify quantile comparison quantile typical partial quantile quantile help characterize dispersion quantile contain surface constrain unbounded region complete coverage apply go whether computation efficiently literature mr tie regularity relevant object object could report alternatively distribution quantile evaluate might problematic cardinality moreover emphasize regard discretization suffer curse requirement large surprising case cube unknown px x exponentially ever cube extreme arguably explore regularity representation probability allow draw relevant regularity let concave every every cone nonempty interior particular concavity convex concave achieve representation carlo chains survey assume evaluate concave cover many case interest illustrate condition partial order practical equal cone equivalence considerable condition equivalent problem due concavity change convex programming membership available maximizer efficient anneal power objective rl step p iv x independent random run empirical maximizer membership approximate walk simulate use construct follow evaluate
consequence stage identically variable suggest epidemic branch al non negligible major number later limit epidemic individual individual refer arbitrary infection let period intensity contact contact occur define variable follow activate without activate period correspond contact period stop recover former happen person person integer imply person get infected period start contact activate already infect nothing happen contact activate contact process period select individual already infect go contact period stop finite probability straightforward check desire period individual rate homogeneous branching process constant birth variable whole branching agree point detailed branch branch individual apply epidemic infect except contact epidemic already infect consequence epidemic branching agree contact infect epidemic contact infect contact equal contact contact branching process epidemic agree contact contact epidemic branching agree precise ball epidemic branch approximation branching study e instance length birth quantity previously branch total ever branch branching grow beyond well branching epidemic coincide space branching go arbitrary special opposite branching individual go happen go denote initial birth event satisfy birth life length course duration poisson process must hold third use transform e initial epidemic approximate homogeneous birth life epidemic whereas approximation epidemic branch correspond epidemic epidemic initial treat period parameter e epidemic small mean major life total rely individual infect branching approximation show epidemic approximated branching individual goes never happen large enough branching grow beyond limit implicitly question course happen epidemic something outline elegant interested infect initially call infection pressure contact multiply period community individual infection pressure become infection uniformly thus increase accumulated infection pressure epidemic stop distribution lie infection accumulate embed process straight process make obey perhaps something expression final infected probability infection period equal approximation approximation fraction infect equation negligible minor unique plot else rigorously summarize minor therein von consider epidemic initial sample space square variation illustrate epidemic start initially look subsection conclude equal equal conclusion simulation minor see distinction minor major theoretical look course hard seem agree previous question occur interest epidemic eventually sketch minor focus major hence study time depend see epidemic branch approximately individual infect branching infect epidemic counter arbitrary fraction initially grow decrease epidemic already infect epidemic behave average follow branching duration plausible duration epidemic result technical many modelling disease model attention last decade attack way come severe measure restriction contact e particular disease somewhat change reduce make suppose available fraction individual initially contact equals want hence instead epidemic approximated branching birth mean life denote conclude major approximately unique central limit major total infect normally state apply point impossible fraction surely prevent example prevent whole far give parameter epidemic stage stage mean individual infect vast quantity define epidemic enable error illustrate epidemic infect individual let fraction satisfy normally distribute around standard deviation asymptotically normally deviation delta e cox asymptotic square root replace quantity variation impossible infer period proceed available example critical natural normally variance delta community infect upper estimator normally distribute make number epidemic continuously precise refer standard stochastic epidemic stochastic epidemic finite randomly time intensity choose despite model contain simplify behaviour spread quantitative epidemic individual period reality population heterogeneous social infect quite nearly age gender experience define epidemic individual individual close give population epidemic branching attention reason imply average individual individual secondly simple whereas occur threshold limit state number infect epidemic take involved proof desire refer perhaps term uniform epidemic previous include uniform sense assumption contact specific individual situation tend mix epidemic behaviour epidemic common epidemic individual group assume contact pair individual contact individual individual period rigorously et al infect infected treat approximated branching process refer biased infect major happen et number equation additional derive central another epidemic contact call specifie upon epidemic assume frequent cycle stochastic epidemic open problem solve influential disease highly disease edge correspond focus sir epidemic allow enter behaviour sensible approximation disease g certain disease country another potential disease brief outline refer reader study sir epidemic model dynamic individual rate exponentially state size markovian epidemic individual contact infect immediately recover become life individual irrespective infection status epidemic jump enough individual play roll question property large correspond get equilibrium show disease equilibrium basic number individual typical stage leave recover disease stable equilibrium call stochastic reach state fact state stationary individual epidemic question size go population influential important present epidemic reality complicate affect spread list effect play roll dynamic reproduce high weather perhaps social school start event effect al transmission school transmission school present spatial increase dramatically last spatial component modelling study disease always take et al individual decay main epidemic growth assume infection think product contact transmission contact period first infect often eventually activity start drop dynamic long epidemic growth period equal infection infection process activity length complete reality rarely response et et reduce compare effect infection higher critical true call rest something efficacy course enough make often clinical usually hard reduction since actual rarely book discussion trying include realistic model completely predict happen situation nearly people adapt behaviour disease say health measure reduce disease epidemic minor modification area spread g recently wide epidemic even use terminology disease spread give introduction big mathematical disease spread probably spread come epidemic classic book cover acknowledgement grateful financial cm plus minus survey paper model simple stochastic epidemic property model critical towards epidemic disease coverage epidemic epidemic epidemic threshold early disease aim evaluate model rigorous make contribution consider early question big fraction community epidemic fraction arrival model generalise way individual equally example spatial deterministic epidemic preferable study community aim epidemic intervention stochastic advantageous contact contain graph network say epidemic role present epidemic epidemic model close stage behaviour epidemic epidemic duration epidemic main summarize assume early epidemic branch birth branch epidemic branching grow beyond limit happen determine final infect divide three beginning fraction place end infected order answer study describe many epidemic realistic complete guide contribution epidemic define epidemic property insufficient end call sir epidemic approximation rely community generalization epidemic model one deterministic epidemic g individual individual get infected individual remainder assume period community consequence assumption move I say sir epidemic individual sis model infect becoming call call allow refer sir close recover effect receive contact infect rule individual remain become distribute also contact agree epidemic start epidemic evolve new individual infect community imply epidemic
multiscale interaction borel regular function measurable integrable area simply weakly road regular close also easy restriction configuration weak union weakly thus measurable argument weakly clearly integrable note x integrable derivative dominate integrable integrable strong generalise absolutely continuous respect poisson perfect feasible neither purely multiscale model correctness monte rarely simulation reach solve introduction perfect spatial example attractive rigorously check burn error estimate identically reduce unfortunately response require evaluation literature theory exception simple define process interaction mention capable modelling cluster regular suited whose vary multiscale area either demonstrate sample simulation coupling move dominate justify perfect area describe perfect simulate infer datum discuss suppose desirable finite go run return chain chain let minus infinity question two chain couple stochastically whenever stochastically maximal minimal chain bottom state ingredient coupling although continuous state space notably section still truly general high moderate dominate coupling coupling past soon type space poisson impose constraint sample poisson evolve birth death configuration step mark refer initial process evolve birth death happen accept point however happen remove event mark keeping involve expensive costly version calculate dominate partially unique partial replace preserve equation wish density monotonic respect induce intensity poisson dominate process mention modify step requirement point parameter compact reduce homogeneous randomness cluster unfortunately allow place sort distribution nuclear force particle law model follow type radius markov range scale scale van standard multiscale write functions intensity substitute find dominating process simulate evolve time point whenever dark either amount accept whereas right little add maximum intensity pseudo arbitrary subset observe integral side note quadrature weight use extend likelihood approximate q pattern log independent generalised order procedure must nuisance fit value narrow different moderate large brief look set consider scale seem scale rather dotted give envelope simulation model solid dash line simulation choose remarkably necessary small logarithmic carlo function fit thing firstly fit reasonably choose slightly would secondly seem scale
preserve neighbourhood essentially well singular say somewhat degenerate bilinear index dimension resp product euclidean compact maximal n k kk geometry special algebra n k complement n intersection know decomposition via element except possibly wish matrix project define totally real distinguished property totally plane plane well hamiltonian mechanic subspace subspace conjugate transpose euclidean transpose unitary act action real observe I n ng see admit factorization totally conversely totally admit degenerate hence r symmetric admit maps n c nk simultaneously unitary transformation validity position n h transpose homogeneous addition lagrangian natural embedding obtain eq define totally real arise mechanic compute neighbourhood diag k consequently sum one geometrically amount totally isotropic uniquely degenerate neighbourhood nj orientation preserve conjugate structure complex structure conjugacy ng nx thus neighbourhood unitary homogeneous space compact tr ad let lie one know neighbourhood formulate term ng one obtain decomposition map neighbourhood deal despite minimax geometry construction similar regressor manifold embed density unknown unknown say discrepancy function smooth iff hessian euclidean intrinsic riemannian one parameterize dimensional highlight result differential topology fr manifold fr manifold hausdorff topological coordinate fr canonical define compact manifold lie functional determine geodesic assume second way positive definite definite along subspace vanish riemannian smooth minimum orientation angle compute onto formally regressor regressor canonical particular vanishing know differentiable translate minimum satisfie differentiable set iff resp triangle sr euler regressor euler angle draw increment computation solution histogram normalise euler angle regressor report kolmogorov normality angles regressor euler ccc see text information htb normality introduction section fy quantitie q dependence omit notational therefore measurable smooth r let fix bayesian estimator satisfy proposition observe hand integrate value single embed norm q bayesian due embed leave prove euclidean space inspection hand restriction geometry semi definite let defined transformation thing prove non degenerate span scalar multiple identity matrix projection dim riemannian manifolds tangent shortest smooth bad lie neighbourhood uniquely map integrable define riemannian measure curve cc ts clear figure discussion exist proposition examine application unit orientation preserve normalise haar logarithm part integrate q equal investigate theorem joint vanishing introduce orthonormal one rotation plane vanish vanish multi euler angles maxima compute vanish euler tr mat mat else mat mat euler j theorem comment thm thm example question thm thm compact manifold euclidean space riemannian manifold white smooth smooth bayesian technique variety second equality problem orient manifold boundary map observe via white estimator map second asymptotic technique variety geometric tool early applicable one observe plus geometry state estimator example geometric naturally extend regression observe state map observe output attempt infer belong transpose regard observe state may evaluation commonly geometry situation real dimensional inner resp riemannian riemannian manifold embed inclusion infinitely summarize diagram open neighbourhood orthogonal projection basic geometry compact smooth map space vector transpose row e minimax map riemannian minimax one approach determine view risk state riemannian permit derivative hessian denote curvature manifold random assume conditional point interested admissible parameterize regression regression assume volume functional regressor examine prove special map determine structure determine riemannian induce special sphere group preserve linear detail section dimensional denote e es linearly basis orthogonal latter euclidean unit transformation space inner denote naturally pass tangent typically call lie algebra equip lie denote element observe g call adjoint know ad trace adjoint representation act subspace denote complement similarly bundle bundle g fact neighbourhood open neighbourhood see neighbourhood analytic whose suffice g g g neighbourhood intersect figure value neighbourhood globally lemma consequence neighbourhood many linear decomposition neighbourhood introduction transpose dual induced euclidean inclusion ef f v derivative impose act attain haar haar measure act isometry note independent product flat invariant flat dim us dimensional sphere orientation pt orthogonal v plane alpha alpha alpha alpha alpha curvature unit
dash generate segmentation affinity dash segmentation algorithm combine classifier pixel weight tend belong compute affinity edge patch remove connected affinity segment image segmentation algorithm misclassifie dramatically segmentation split merging optimize rather affinity sophisticated spectral cut simplicity direct supervise optimize segmentation graph partition possible sophisticated still prefer great learnable rather hand design clean affinity graph spirit large assumption sophisticated partitioning segmentation way classifier segmentation special cluster similarity clustering recently rand define segmentation assignment pixel belong pixel fraction rand fraction rand similarity dissimilarity rand segmentation truth function rand sensible incur huge truth classified lead pixel segmentation affinity rand index affinity rand index affinity pair binary correspond belong rand index incur pixel must connect vice segment incur penalty penalize work thresholded affinity let train classifier relate indicator classifier characterize whether two connected thresholded affinity graph affinity pair affinity affinity let graph path affinity affinity affinity path important pair pixel thresholde affinity exceed pair thresholded graph path affinity minimal affinity affinity consequence affinity connect connectivity q efficiently span maximum sign maximum span tree neighbor affinity affinity number grow share image perform time edge classifier rand q replace continuous cost suitable operation differentiable continuously gradient edge choose ji affinity neighbor pair near function pixel adjacent affinity speak cost similar affinity cause incorrectly classify gradient often affinity pair neighbor pixel randomly w pair near draw w w w learning pair pixel image gradient weight pick train affinity edge pick train affinity integration neighbor affinity graph connectivity decision distance train train decision truly learn superiority affinity compute local affinity train computational brain identify diagram brain piece brain cubic brain image advance making collect image remains require accuracy serial block volume voxel train affinity convolutional restrict convolutional network previously standard second map sigmoid lead affinity cubic patch classify function lx affinity slow affinity predict proxy pick overlap sub original significantly image train less pixel total iteration measure correctly pixel curve recall quantification classification segmentation classifier pixel connectivity rand classifiers spectrum lead threshold rand image pair connect pixel reflect rand index imbalance affinity comparison instead precision quantification imbalance observe affinity affinity performance connect component standard learn affinity poor segmentation mistake merge segment contrary properly thresholded follow connected image segment miss segment merge merge cross neighbor boundary affinity affinity graph result partitioned graph thresholde key segmentation cost function affinity segmentation fast contrast base segmentation dominate simple proportional graph number segment size time partitioning connection linkage span ultrametric partitioning minimum resemble segmentation part ultrametric map algorithm generate hierarchical identical vary threshold graph incorporate improved affinity acknowledgement medical foundation max medical research max h predict affinity reflect image partition segmentation learning affinity affinity relate ultimately present affinity graph produce directly rand well rand quantify pixel segmentation use graph component
density conjugate dirichlet bernoulli corresponding policy mdp dynamic first function mdp follow policy hoeffding bounding estimate iterate follow probability implicitly belief root whose high expand either branch take let discount cumulative reward finally leaf specific branch bound simply heuristic algorithm shall employ upper low bound expansion perform start node among sample leaf child serial nearly balanced leaf expand expand lead unbalanced tree lower expand expand currently high expand ci less branch long agreement appear aim interesting rl node heuristic expansion result difference complexity expansion try exploration apart perform sophisticated depth simply optimality importantly observation space continuous reward also include stochastic one mp upper another interesting perhaps line successful problem acknowledgment denote mdps specifically state pair dirichlet count transition simply product transition mdp pair transition transition statistic way exposition mdps produce employ optimality increase planning programming infinite state employ regret present strategy explore compare ucb recent probabilistic mdps observable decision exploration firstly compact current secondly augment mdp child possible subsequent belief large fast problematic grow concentrate expand tree whole trade optimal computationally recently extend propose special structure belief order node look method tree branch help idea multi armed optimal algorithm already introduce mdp formalism discuss tree expansion introduce evaluate development interest agent seek expected simply discount arise markov decision process define tuple comprise transition satisfy state shall define value infinite mdp uncertainty f essentially maintain order optimally select suggest bellman name mdp comprise solve shall call mdps analogously bayes adaptive density I belief mdp augment mdp measure reward jointly mdp transition component hyper abuse shall horizon future action eq hyper step clear continue expand belief observe bayesian utility current future expand tree exception wang sampling use thompson thompson expand expansion closely therein branch bind also upper propose structure belief upper value expansion property propose combine bound leaf experimental bandit believe important towards applicability look ahead exploration current suppose observation together mdp transition recursively belief unbalanced hope fully expand tree especially true continuous even far computationally horizon expansion largely use reduce approach search leaf leaf strategy child formally wish expand space reward deterministic deterministic enumeration reward
intersection correct invariance problem deterministic broad class program generator polytope combine construction generator explicit form intersection regular constant depend constant remark gaussian counting quasi algorithm counting program let program exist deterministic run estimate program program contingency table regular program within hypercube quasi polynomial obtain integer solution work count solution program algorithm give task integer multidimensional run difference give strong far state invariance whose bound hyperplane regular however polytope hyperplane symmetric space universal additionally invariance sphere modify intersection intersection spherical explicit uniform cut similar hyperplane randomize box specific study play reference generic tight recent develop lee theoretic notion cover connection also give generator oppose good give outline invariance principle proof proceed two first replacement hybrid literature intermediate prove invariance prove normal coefficient smooth fourth derivative notice polynomial univariate wise construct framework bound hybrid bad form intersection regular hybrid group block irrelevant order suffice hybrid argument intuition randomly proceed partitioning smoothing well roughly speak contrast argument small turn analysis block construction translate closeness smooth closeness smoothness become test multivariate version test et rather invariance hybrid fourth p q small fortunately construct final closeness closeness differ end surface well intersection proof invariance obtain fourth suffice smooth construct replacement sequence principle low polynomial al proof involve principle smooth cdf behave involve prove smooth supremum regularity influence hybrid argument expand error fourth step principle smooth approximation cdf smooth approximate fourth uniformly smooth cdf everywhere small obey anti state variable low polynomial obey anti complete proof try invariance characteristic polytope equivalently logical instead wise polynomially face ball lie polytope grow combine fashion derive dependence exponentially step ask dependence improve lead read principle reveal obtain invariance variable obtain derivative approximation quantity outline ip column define face polytope uniformly small polytope invariance might face polytope true improve w w ip back outline invariance principle one step irrelevant suffice replace choose block replacement block invariance ip ess quantitative central give invariance principle single ess multidimensional ess identity matrix deal implication history approximately count solution especially regard contingency however much term algorithm deterministic covering problem cover kind regard contingency give run quasi relative error approximation state explicitly machine contingency table time algorithm contingency case box contingency table intersection al give al generators intersection seed bound generators seed intersection least yield set bound regularity lemma regular use handle strong difficulty reduction case use face unless section principle functions smooth show closeness smooth anti variable imply closeness lemma anti concentration variable finally anti concentration gaussian concentration let eq follow function proceed therefore straightforwardly surface denote element polytope face tx prove explicit hash family set however wise analysis complicated family construct form lemma regular wise indicator moreover inequality moment equation therefore equation careful outline detail hyperplane orient surface hybrid hyperplane disjoint hyperplane conjunction series reader find smooth p taylor first function hybrid choose similarly view replace generality variable form lastly taylor q equation follow sum essentially regular rr fact regular note factor intersection w kf first bound volume neighborhood invariance bound boolean boundary theorem agnostic constant subgaussian constant follow say regular perturbation perturbation k w ip union eq get sufficiently perturbation random flip bit see anti regular follow directly imply k applying immediately result namely main generator result construction polynomial et reveal construct principle observation ask invariance regular different careful application begin used family hash family know avoid technical easily generate construction generator generator consider generator argue generator regular argument indeed independent rely hash independent block word generate involve consequence independent move closeness cdf argument equation enumeration possible seed immediately time small integer turn regular program broad dense cover contingency quasi polynomial time algorithm contingency nontrivial algorithm program class integer aware approximately program notable counting count contingency count dense count contingency table contingency r c c note integer program proper result bound generator appropriately generator ct ks string get deterministic run dense cover get count table cover program negative important integer program combinatorial standard set universe give dense least linear appear dense constraint continue regular seed generator dense set problem constraint universe approximate additive time long elaborate approximately contingency contingency positive integer wish solution whose matrix appropriately correspond lie notion dense instance count number contingency contingency discrete cover instance contingency contingency c c ks moreover proper contingency quasi polynomial set prove invariance unitary rotation spherical hyperplane prove rotation bound hyperplane tail require apply let normalize hadamard entry probability observation diagonal wise fix wise later l dx observe degree multilinear markov distribution constant generator recall invariance unitary rotation thus regular invariance principle use area polytope faces distribute fix later distribute apply distribute two follow prove later sufficiently follow equation claim generality odd acknowledgment thank integral david section corollary conjecture spherical gaussian possibly intersection e dependence least prove theorem elsewhere important application invariance boolean noise sensitivity intersection regular give agnostic learning intersection seed length hypercube construction obtain algorithm program dense cover contingency table computer science problem analytic spectra sensitivity fundamental tool prove hardness proving conjecture hardness problem notably max principle relate uniform gaussians invariance multilinear multivariate parameter depend small say cdf polynomial cdf polynomial coefficient invariance generalization powerful hardness hardness choice boolean principle widely understand cumulative invariance principle possibly unbounded intersection support refer regular regular invariance regularity threshold main invariance invariance principle generally
maximum extract instance build reader familiar security concept depth converge dominate attack rational select valuable optimal path construct eq algorithm construct budget single cut cut minimum small attack surface start vertex boundary organization internet interior many security depth depth model system attack version center network depict figure receive server receive front web server front server large attack surface database server server interface application whereas database budget try sensitive database e rational budget unity budget right database attack end achieve unbounded attack analyze security playing alternate select select vertex yet bound competitive good via literature literature include play boost machine heuristic although extension apply online management formalize game round choose attack budget sense security principle make accurately allocation exist knowledge black reveal attack edge post function sequence edge notice beyond round column attack algorithm edge path attack already reveal reveal edge notice vertex begin knowledge graph update point update attack ever sum unity round allocation move attack small budget appropriate strategy compare online relate ta surface theorem establish well carefully construct attack principled strategy attack appeal measure cost result competitive converge cumulative abuse slightly singleton e know entire remove know vertex competitive ratio optimal proof well perform every inaccurate far optimal reason instead learn observe reward without knowledge payoff difficulty limit star system equal attack surface reward little equally indistinguishable rational number leaf course ratio bad vertex rational learns attack another way mistake matter rational actual whereas variation system figure budget rational budget edge view edge near unity rational perfectly rational room situation attack might knowledge attack software security equilibrium al discuss pareto improve security accord al optimal security division loss due lack generalize modeling et highlight theoretic theoretic due adversarial setting applicable nash equilibrium security scenario arguably abstraction detailed adversary readily equilibria many security expert ignore principled adopt management competitive attack never actually occur although abstraction support make security employ monitoring tool analyze attack tool effort attack security build roll quickly detect exploit determine security recent attack discount attack exponentially security security ill suited attack round learn attack conclusion security always reader appropriate part strategy significantly like acknowledge support nsf dms program establish know second hypothesis know advance full specifically surface reward round make output ta strategie lemma regret run mp lemma matrix add entry game sequence obtain respectively produce allocation sequence game game thus b yield ta ta bt online assume row advance relax perhaps allocation subgraph induce reveal visible letting actually round output identical round unity correspondence ability ta ta consider edge budget allocate time budget allocate thus instantaneous definition optimize reveal ta te receive immediately enjoy complexity constant bound quantity large technical ta ta b feasible eq b prove allocation attack reduce attack es de maximization allocation ready main equivalence convert ta ta imply ta b ta ta ta ta yield ta ta b ta ta briefly graph e converge attack path past vertex vertex edge attack ratio attack select attack iid round attack every entire cost coin attack allocation well frequently well probabilistic must attack lemma ignore try easy algebra conclude generalize graph small thm thm science superior security security security learn past attack security bad act competitive ratio inspire unlike robust security security risk security last typically perform benefit identify risk constitute strategy conventional look risk examine exploit security security attack study efficacy economic security benefit trade economic act strategy security make knowledge require make conservative knowledge assume know perform single meaning attack consistent say business meaning without security team rule improve access control single act attack attack make choose study patch security might portion budget web token make patch interaction place patch viewpoint evaluate cumulative study metric seek attack instead receive performing say business show competitive prove theorem technique learner know let situation result clause allow although likely strategy strategy gradually edge construct effective implement capability less budget attack actually encourage management inherently arise naturally bound security security generalize clause relate related work conclude theoretic attack capture security sense situation include internet direct graph define graph represent represent induce another might machine two system second select begin result general clause discuss generalization think attack drive edge include server connectivity back form make spam reward derive drive
choose please bias column fix contrast choose unconstrained initialize draw htbp cc high htbp glm show simulated dm snr glm dm show data bar unit deviation attempt optimize emphasis relationship importance bias contrast leave choose potential dm augment dm unbiased sign change unbiased pure solution cc unconstraine dm dm snr nan dm dm unconstrained shape curve fractional bias gauss markov bias glm analysis optimize simulated dm glm dm simulation generate dm bar represent deviation variance block application consist shift response maximum user want bias entire paradigm weight dm design shift figure v htbp run optimize dm c example dm snr glm dm also glm snr illustration purpose plot bar standard deviation deviation quantify response impulse response voxel illustrate enable simultaneous capture plausible shape plausible shape shape half parameterization use parameterization give plausible shape generate uniform automatic initialization describe result capture glm analysis capturing enter dm glm capture underlie optimally unconstraine dotted optimize solid fractional r gauss variance optimization glm analysis set optimize dm example fit dm plot dm error bar represent quantify detail scan half mean hour terminal half al scan baseline minute ml rate control automatic collect parameter tr te flip angle slice slice volume acquisition acquire rapid gradient tr te ti fa slice slice library volume mr instability volume raw fmri correction mm automate perform map minimum identify pattern co probabilistic pursuit test analyze dimension reduction latent via decompose base core use constrain demonstrate subject htbp produce spatial drift find figure response covariate drift contrast multiple profile baseline could general response exact easily model expect brain structure seem delay response subsequent inspection stage capture represent pure drift purpose optimization potential dms capturing delay let dms dms dms dm process drift pure negative linear drift absence response drift conservative snr fix response htbp constrain leave automatic dotted bias example htbp glm optimize dm c roc curve dm bar quantify computation dm glm analysis represent derivative figure result validation maximum optimization size dm equal potential dms value unconstraine initialize maintain around contrast gauss monotonically imply gauss estimator model maintain curve automatically optimization column example compare contrast bias increase maximum monotonically imply maintain column unconstrained initialize uniform maintain decrease monotonically becomes indicate gauss maintain use unconstrained dms augment explicitly enable signal presence nan decrease monotonically four variance bias illustrate user want bias indicate primary primary reduce around increase gauss maintain capture enable explicitly indicate nan signal optimal dm find optimal dm unconstraine find contrast bias maintain mean maintain shape maintain signal fmri reveal profile take baseline initialize section column column estimate dms dms dms presence drift signal dm absolute model roc generate drift choose extreme produce absolute contrast rule roc cutoff residual dm specificity snr specificity real approximately primary response specificity detection correspond fmri illustration purpose sample detail fmri partial obtain illustration full partial fit temporal derivative covariate delayed response delay point delay find dm delay unbiased produce dms snr match size specificity sensitivity real fmri snr cutoff opt cc e figure dm dm cc st show partial fit figure dm dm htbp temporal derivative full interest figure optimal dm objective theoretical enable design fmri analysis framework allow capture potential specify weight occurrence various design associate validate numerical sophisticated test column design selection algorithm bias illustrate study control solution algorithmic fmri come model shape maintain bias variance capture amplitude optimize temporal derivative bias reduction optimize comparison temporal good next examine property glm design find variation basis recommend fmri shape relate notable shape allow presence include amplitude analysis amplitude never approach control capture example shape amplitude objective instead propose potential explicitly objective result capture amplitude pe glm optimize dm column rd column plot dm glm use analysis simultaneously signal glm optimize develop motivation fmri general variety quantity denote identity eq consideration small combine apply repeatedly term simplify r combine note compute problem constraint slack replace constraint slack equality solve lagrangian define eq subproblem tucker kkt optimality penalty update base feasibility monitor sufficient accuracy subproblem point choose try px k cx iteration augment sub use gradient quasi update recommend recommend convex switch newton framework trust update progress monitor b xx bfgs sr limited bfgs limited sr newton non projection vector elsewhere truncate newton cg cg inexact cholesky bfgs sr bfgs sr quasi newton even find describe plausible shape half david glm tool analyze functional fmri fmri univariate fashion analyze voxel limitation vary nature well potentially main develop set design validate numerical fmri implication ability match signal magnitude also size thereby specificity enable capture multiple profile interest oppose optimize enable pass estimate variance group fmri capture fmri fmri fmri quite wide applicability fmri analyse design dm glm stimulus paradigm explanatory variable dm pe fit define meaningful brain paradigm fmri voxel brain glm fmri univariate mean dm voxel pe glm gauss exact mechanism fmri fmri signal possible effect signal interest region profile correctly dm voxel fmri glm handle bias pe mis specify dm extremely ignore implication gauss markov theorem mis specify bias pe generalizing group subject correct add dm mostly heuristic still result pe approach flexibility response shape regressor fit use difficult amplitude pe bias trivial wish derive theoretical enable dm glm analysis practitioner develop simultaneous dm contrast dm controlling bias optimization optimize detect second specific task fmri experiment design fmri fmri response start generate section whiten whiten misspecification use analyze eq glm contrast interest show eq recover glm quantity define follow pe becomes normalize normalized change ideally tradeoff capture simultaneously interest residual function enable optimal ideally pe well residual compare performance capture enable optimal dm candidate dms expect contain regressor noisy contrast dm frequency matrix htbp cc first contrast unconstraine contrast find initial acknowledge initialization one initialization heuristic try find primary column singular leave singular matrix singular optimize column primary primary modify strategy many account via inverse framework allow inclusion datum simply noise maximize available level high number variable algorithmic user choose cutoff default cutoff dm z bc c column compute j est algorithm algorithm maximum reduction value please choose example variance curve local correct essentially initialization problem algorithmic initial choose cutoff default cutoff output number dm find cutoff slow tail sensible design motivate write
concern screen dimensionality empirical likelihood propose allow marginal moderate enhance multi procedure sis sis select variable simultaneously nevertheless marginal challenge jointly nonlinear sure natural effect family improvement sometimes class ordinary enter nonparametric closely signal converge extension adaptive high challenge penalize modeling fit nonparametric response covariate model goodness preserve non minimum regard nontrivial sis error modeling nonparametric depend function bring challenge extent iterative screening reduce computation two spline basis selection additive group variable page wavelet selection implication basis whose select simultaneously additive present demonstrate effectiveness conditional zero important curse dimensionality marginal nonparametric regression denote joint integrable onto rank utility select group marginal spline polynomial spline degree sup smoothness nonparametric projection approximate version express respect smooth rapidly np correspondingly define square regression nj j denote screen rank importance strength view rank nonparametric sense descent residual regression u residual sum fit screen possibly much whether active procedure sure property sure screen property rate screen additive assume admit identifiability whenever theoretical sure screening vanish follow simplicity support marginal belong rt n positive lemma nj consistency partial signal active separation sufficiently large set sup follow sure sure screening convergence exist addition condition f hold n nd eigenvalue design upper whereas f spline compare univariate marginal independence function small signal converge long lemma taking nonparametric property control selection rate basically false negative ideal nj active inactive variable nonparametric zero tend nj c ii achieve consistency select set show relate exist nd size whereas original np converge fast size fan case covariate diagonal hence naturally select additive model example additive active enhance employ term independence permutation inactive enter nan permutation nan permutation nj row quantile nj nj ny nj numerical use maximum far inside cross validation minimize component marginally pick determination except apply spam adaptive apply enhance positive stop variable study simplicity method selection show sure major difference unlike square remove make flexible particularly effective correlate tend high choose positive due screening improve chance select variable performance numerical experience outperform high positive rate illustrate datum analysis setting sure screen effectiveness also advantage method choose parameter example example I normally linear model vanish normally distribute ex sis summarize respectively screen fail contrast screening worth note sis particularly normally select whereas sis select nonlinear behave sis underlie newly propose simulation fold notation additive accord effect follow pn though model aim sparse show row computation report tp pe g greedy much false positive whereas scad important nonlinear example reason look signal contribution signal significantly introduce additive snr varie lot merely snr play measure difficulty scad nonlinear bad scad quite selection prediction subsection conduct estimator snr follow covariate take different make table positive poorly true active inactive one signal look table correlation competitive performance snr snr achieve sure screening current analyze illustrate week old microarray analyze rna contain probe genome intensity use probe interested heterogeneous system probe represent genome array whole follow probe express marginal implement analysis probe follow follow select pe evaluate performance validation compare prediction set observation observation select prediction set repeat deviation replication far fewer small biological probe selection additive use marginal marginal important correlation propose apply preserve sure screening selection modification propose deal marginally uncorrelated response appropriate additive approximated series expansion paper readily smoothing wavelet approximation smoothing spline proof property ef nj nj decomposition desire result condition together bernstein need reproduce bernstein exist c nd denote iy bernstein lemma tail inequality utilize bernstein ij lemma probability b j nd euclidean
analytic open definition cover grid covering analytic neighborhood cover next ease notation kb great cardinality easy bound convenient choice one subset let equal radius radius regression precision impossible assume calculation fix rd x pr r eq proposition trivially v closed suffice proposition xu u consequently probability q imply due x since complete main proposition immediately theorem shall facilitate subsequent discussion real function different suppose analytically xu v ki n formula side equal let xu thus therefore due fix p notation eq provide bind end cover k u entire connect finite tu w let therein eq series define let apply dominate cover satisfied compact covering grid lemma hence brevity first proof compact pz jj sl j de se mb sd fx mi db u mb sg sg complete proposition ft radius convergence assumption fix cauchy contour supremum get grid cover f z cx z contour regularize sparse underlie linear chi department statistics university ct email ex nonzero coordinate number parameter sufficient establish base error ls regression analytic rely expansion require take regression selection nonlinearity power analytic primary secondary support dms large machine ls error mean nonzero coordinate much nonlinear structure henceforth clearly criterion precision article sparse matrix shall despite conceptually besides regularization take advantage linear fail model prototype mind instead basic regularize yield estimation reduce result also collect establish model analytic analytic exponential much handle due explicit mle discussion establish series example corrupt vector impose vector remove unnormalized collection exactly observe denote seem eq general form regularize search mle minus likelihood typically position value mle ls precision proceed easy condition ease notation statement constant check mle form random letting meaningful need make sure try play role study main proposition condition due great establish rise satisfy fix later mild reasonable condition recall shall example suppose since aa rise say wide way correspond family counting hold result somewhat simplify k define find constant contain easy length mild proposition estimate reasonable order behave extra impose section devote establish outline easy conjugate relation u nonlinear exploit expansion expansion row transformation desirable bind work generally fall power series expansion may fall deal approach line connect account treatment whether use bound answer seem polynomial finite expansion general get guarantee bound hold simultaneously neighborhood unique open contain henceforth
also localization choose formulation locality constraint representation remove simplicity case origin appropriately shift invariance requirement put try optimize work exist machine nonlinear manifold laplacian method pre affinity graph compact handle unseen importantly coordinate direct sound unsupervised pre facilitate set model fix local kernel smoothing regard include widely machine non kernel kernel suffer high hand order overfitte dimensional learn locality compare coordinate balance balance coordinate code quantization widely process acoustic zero method relationship regard signal knowledge answer question space reveal important good code sparse code code locality property method learn th low order order yes code order various point claim particular code locality robustness first base roll primary goal demonstrate simple representation traditional code newly code formulate mainly point basis learn evaluate square rmse coding basis nonlinear behave look datum representation basis figure sparse basis encode nonzero coefficient code nearby basis get coefficient sparse data locality fail facilitate interesting coordinate encourage datum linearly enforce negativity remain interesting initialization unlabele base basis depict indicate coding remains code near point low coordinate code approach increase unitary variance smoothing consistent suffer cccc rmse rmse rmse rmse rmse b digit gray anchor code formulation anchor nonlinear enforce locality representation call svms code processing change sign anchor manifold anchor point obtain optimize basis compare smooth neighbor obtain various auto encoder manifold testing comparison code code raw local coordinate code across various basis check locality find unlike roll portion nonzero mostly distance work remove far locality table belief back network classification cc svm sparse code svm coordinate l rate raw smoothing code linear belief svm third classification patch background visual degree variation rotation code entire approach pool representation state method extraction sift grid sift descriptor pooling code classification examine replace code simple setup repeat sift extracted center sift descriptor partition block scale pool block average pooling try code block pool code local much accuracy rely nonlinear coordinate code locality ensure good code average average pooling svm code pooling pool max pooling nonlinear distribute see coordinate unsupervise local unseen also scheme locality coding depend handwritten digit object finding manifold valid manifold manifold coding theory apply mean worth many locality code local explain effectiveness code origin shift invariance bind definition requirement projection span orthogonality imply eq imply th follow stability lemma terminology hold obtain derivation convexity imply loss sum qx derivation third respect measurable independent definition ex introduce dimensional unsupervised basis learn basis provide anchor manifold approximate nearby anchor coordinate approximate global quality code nonlinear global learn drawback inspire use since sufficiently locality possibly suboptimal empirically verify synthetic handwritten digit unknown underlie distribution dimensionality traditional predict curse dimensionality argument become large distance large real dimensionality although new learn high idea point manifold respect observation show effectively localization dimensional extensively dimensional method interested function define let although restrict specific often norm curse sample require observe intrinsic depend intrinsic manifold use typical take manifold intrinsic dimensionality q statistical dimensionality involve cover manifold intrinsic coordinate code set anchor lipschitz accuracy manifold code data point anchor
convenience horizon eq modify discount agent assume know predictive improve gain experience take environment assign rule posterior experience implicit within predict setup agent definition identity section next allow environment model whenever environment step context describe mixture environment environment predict next equation model prediction posterior model give equation maintain maintain likelihood good adaptation agent entropy difference n arbitrary supremum finite fast predict rapid weak statement tell good motivate agent class observe action include computable environment computable function seek computationally replacement ideally bias place suitable candidate limit resource compute action computational use constitute approximation na I take tree use horizon mdps extend algorithm domain deal problem space pair produce state construct node give time converge agent history next reflect environment directly agent planning process true ignore uncertainty recommend carlo search technique construct search tree compose decision chance node history chance end reward contain child conceptual phase iteration phase root exist chance expansion decision child environment root finally trajectory four conceptual limit select child exploration gradually estimate reward cause towards high predict reward future choose heuristic horizon focus exploration practice allow build chance width search sparse stochastic branching chance search sized chance node star circle line star node indicate expand policy proceed flow back detail search always child represent history pose classic exploration exploitation child node like armed action logarithmic carry ucb domain playing ensure node gets select sample visit take horizon instantaneous history action pick ucb positive exploitation choose ucb reward follow immediately decision decision also reward action see decision tree remain require future reward sampling repeat natural baseline choose step tend structure search long limit full determine overall performance unless state complete root tree leaf maintain history follow q increment visit counter chance receive history construct tree estimate available retrieve importantly computational resource horizon simplicity exposition search carry obtain end experience keep subtree root tree use routine routine routine pick accord policy horizon accumulate reward trajectory value estimate update per history argument tree node create reward reward reward th action choose ucb policy child explore add search parameter value create deep selective high short tree ucb automatically focus explore alternate action eventually exploration exploitation uniformly node remain search generate estimate state general horizon mdp main consistency pick monte carlo tree search routine easily main invoke routine providing mechanism interior node scope note applicable program environment problem experience slow mixture environment construct context weight weight efficient summation tree huge covering I computation require outline way generalise compute action brief kt bernoulli one kt via put uninformative jeffreys beta probability string one q next form markov model work tree right identify node child set string stre pair binary call accordance terminology map intend example learn model tree learn learn kt depth aside bit see repeat long need describe history sequence binary predict bit furthermore achieve without kt idea convenience loss l reward symbol symbol action leaf node aside initial cycle variable long mm mm bit bit prediction action reward reward grouping node observe deal action specify prior code work give depth pre traversal perform encountered depth leaf otherwise nothing model length code tree describe impose like penalty structure ingredient depth internal estimate context tree history observe bit update maintain probability understand depth reason binary node probability binary node empty history aside middle processing tree fourth bit practice course count instead complete henceforth tree important depth tree treat leaf block node law highly context induction true leaf depth statement consider depth tree leave right assign simply aside sufficiently mm mm select next bit probability new drawback action potential history string may predict domain choose infer tree remove restriction arbitrary one would multi alphabet consist exploit space note difference former latter work symbol property helpful deal large fortunately describe technique incorporate level action precisely history bit string bit bit factor assume induce computable lemma modify predict follow scheme maintain mm create receive bit history factor conditional well environment markov environment reinforcement learn recent environment markov say ax markov represent converge model update kt see updating produce markov meaningful environment bound environment mixture imply cumulative difference also square zero rapid environment way computable environment first present improve automatically extensive clarity also environment move relationship equation optimal agent contrast reproduce kolmogorov expression share describe scale factored prediction tree model gain property environment result combine lemma adaptation negative depth life planning maximum planning instantaneous policy define j ax max apply absolute distance divergence give x precede b b w b km drop fix square difference average sufficiently importantly horizon environment ergodic intuitively mistake long thus environment mistake average markov say ergodic every occur infinitely agent sequence say self policy long environment policy exist restrict ergodic environment ergodicity early agent stationarity behave learn stationary ergodic environment model ergodic mdp ergodic mdp function action arbitrary apply countable set ergodic environment sequence produce self sequence policy choose sufficiently agent environment agent usage kt class number conclusion entirely justification mixture mixture form node interpretation kt would feasible result grow dynamically considerably tree process piece reasonable cycle furthermore exact suffer issue early additional needed set history length context tree length context recover reverse perform operation bit write phase context discard belief uncertainty lead high e situation complicate issue recommend horizon computationally force exploration intuition domain good achieve use exploratory action within planning amount exploration help avoid set belief underlie u routine exploitation softmax environment pair agent incorporate cycle agent environment across perspective noise partial test domain domain uninformative yes yes yes yes biased yes yes partially observable agent begin location action agent effect reach third receive reward reward problem regardless agent location inside piece agent choose four action move receive find movement depict equivalent bit observation receive exhibit observation familiar environment start action agent suffer receive begin stand right successfully open two stand action agent plan solution slightly simple agent gold grid corner receive reward remain bottom receive attempt remain large scale part correct step move domain opponent randomly agent reward reward top move end reward repeatedly play opponent opponent play cycle play pick uniformly opponent receive domain involve opponent play nash zero slow playing remain strong round pass player player put put opponent pass subsequently otherwise occur high round opponent pass either pass pass immediately opponent agent winner receive number remove play another begin play nash second player partially classic game agent exact bit indicating except indicate within location another indicate single indicate whether effect power every empty receive movement action collect e episode representation domain domain fundamentally put compare compete reinforcement literature page find slightly previous agent environment cycle begin agent bit encode table symbol interpret integer negative reward handle non remove grid observable bias partial agent break learn exploratory various time agent report reward cycle per reward receive cycle phase reduce early exploratory performance core intel ghz parameter phase recent capability expense model phase explore recorded amount experience active discount exploration exploitation discount greedy exploration small gave slightly reproduce experimental result report phase complicated tree general completely specify due absence publicly reference splitting criterion apply exploitation policy design decision list mm split step resource choice allow cycle domain split try recent distant exploration tune separately domain help fair tune exploration extend implementation agent performance exceed domain active experience learn overhead split limited period tree advantage cycle symbol require enumeration cycle large domain illustrate u observation vary domain learn cycle experience except cycle normalise domain significant planning extended effort good performance c grid biased agent domain resource near report day induce benefit dramatically context reason exhibit slow reason ensure dominate structure decision provide search action although practice least per cycle compare tree act motivate active guarantee u tree heuristic criterion never configuration dramatically somewhat applicability learn suggest framework along adaptation history action return sample remain complete unlike build tree highly stochastic planning improve exist reward differ total horizon overhead account simulate trajectory evaluate planning combination grid versus plan importantly performance bold perform multi plan particular region set match believe important challenge environment plan difficult choose action explore reward per cycle greedy selection time point vary show cycle effort use affected behavior inferior learn visual inspection agent play perfectly concept know know seek limited provide sensor know away learn red become nearby visible reasonably exhibit optimistic several attempt study fast asymptotically parallel pick fast program enumeration run cycle pick well provable agent g b domain universal feasible force expert game exhibit monte algorithm universal early influential partition leaf tree partition tree incremental fashion leaf begin leaf split history fall leaf show exhibit deal effectively environment algorithm attempt discover raw stream tree discriminate allows effectively ignore irrelevant observation represent sequence experience around kolmogorov heuristic try place learn state action selection combine scheme produce optimal build distinct accumulate statistic estimate refined time symbol algorithm probable prediction much distinction usage prior predictive representation maintain probability experience experience experience markov observable environment complete dynamic unfortunately impractical form improve algorithm currently area representation approximate abstract prediction future prediction network give current update promise recent network give parameter maintain contrast bayesian variable share similarity estimate ucb limitation perform environment bound depth prohibitive amount experience need cope limitation unless planning solution exploitation intractable need augment
bind satisfie second rhs rhs proceed consider sequence introduce partial proof deal omit let inequality eq inequality proposition proposition b ax ax check x hence martingale get u bound interval mm boundedness get mean covariance grateful r n last ac h n see stationary show lyapunov trace handle nonempty empty interior include ac assume tailed class compact convex measurable finite weakly characteristic variance take ac study related adaptive mala mala drift truncate cm remark fr central drive geometrically stochastic asymptotic adjust heavy tailed subject include theorem drive geometrically ergodic sub many markov kernel geometrically example target interest tail langevin mala kernel drive enjoy uniformly central stability chain irreducible v martingale proof proof study law adaptive paper mention review development organize section adaptive drive approximation section illustrate theory adaptive langevin tailed transition measurable stand dirac act function nf dy dx nx space norm resp endow open borel markov kernel measurable invariant nonempty compact subspace measurable ax practice dx x main paper order paper without far chain chain describe n initial state arbitrary systematically write instead control vary compact develop take aa commonly chain easy markov eq expectation natural convenience notation compact convention strategy strategy former main projection law large hold measurable nf probability section strong number hold imply law measurable assumption notation usual define ax show admit martingale let take let martingale hold free adaptive array value random refer path random weakly characteristic establish weak law restrictive recurrent markov suppose satisfie let simple checking condition exist hold take ii eq establish drift uniformly polynomial v v b geometrically uniformly exist measure explicit ergodic hold also find get assume whether check drive condition indeed difficult typically cx check drive let convenience write eq h ny random continuous continuously w dy x x magnitude approximation notice projection key sa framework proof line enough hold exist assume b suppose langevin mala density lebesgue mala metropolis whose langevin dimensional brownian mala work give mala practice depend regularity fine properly transition mala obviously many mala target acceptance generate x make cosine frobenius compact smoothness bound section variable characteristic x recurrent satisfying consequence hold imply consequence easy conclude corollary asymptotic proof follow weak law theorem basic serve allow markov constant compact notation keep give family depend notation resp approximate well satisfy eq xx dx aa let ii direct write q integral sign assume rhs rhs p x f v q yield part ii write gx ax ax ax ii conclusion compact integer x dx expectation consequence exist next proposition general bind inequality martingale fix initial kernel p measurable nf nf n x p g p notice eq consider k martingale array choice converge r k q rhs since p ax ax central theorem take ax kn n kn sequence x x k assume k n n p n deduce k obtain na n converge n g ax p v probability give non markov allow transfer limit adaptive equality follow one w sequence variable nc jt equality law number triangular array non x kn idea proposition write write k ns ks n w ns get
derivative relax shall apply several rate normality throughout value choose constant namely theorem type student pearson corollary specific random quite explicit finite bound depend long explicit denote application complicated state reader constant subscript pre bind remain moment pre self normalize pearson correlation constant simple bound corollary corollary place appendix hilbert let satisfy counterpart number hold exist constant depend bound explicit expression uniform essential appendix satisfy enough mention remain case small bind probability bind conduct constant theorem complicate corollary simple explicit constant size especially unbounded improved factor able obtain theorem natural number continuously twice smoothness denote upon specify r theorem explicit first involve particularly nonlinear statistic investigate f c f coincide quadratic statistic respectively note without generality adjust use z cf conjunction l point work well nonlinearity method tailor specific specific commonly student ty ia let generality let easy condition inequality satisfy distribution bernoulli concern appear show distribution degree concern degeneracy equivalent appear cf imposed achieve expansion distribution order central nan discuss remark end hold therein remark self close self nz cf uniform form depend namely show great necessarily distribute obtain also I type bound structure bound state number mistake kind triple several choose range parameter thus rather slightly great also advance concern central make normal wang wang probability moderate wang concern logarithm wang bound concern central normalize somewhat early central exist significantly stochastic central heuristic reflect well triple appear competitive remark detail corollary involve theorem triple triple try weight constant bad assumption eq eq triple either ht statistic bind reflect small maximum attain whereas rather put perspective recall even simpler identically explicit similarly proof corollary demonstrate obtain variety constant numerous accurately idea hand complicated suppose somewhat constant proof ht well moment order contrast high moment instance degree freedom symmetric vary one tail absolute bind triple check monotonicity namely rx reasoning dt dd nontrivial standardize skew enough nontrivial note mean h blue student freedom dot ht red green center pareto potential cause eliminate truncation truncation truncation moment kind look comparison truncation sensitive sense knowledge needed compute another corollary eliminate product entail accuracy accurate complicated appearance z either triple proof degree center pareto shape numerically also minimize significant tail heavy e find truncate truncate small small follow large consideration somewhat surprising bound develop self normalize compare tailor sum also want case statistic behavior particular take continuity expression view standardized freedom standardize skewed enough py pp get use directly short dotted precede imply e picture ratio approximately hand contrast paper certain various choose asymptotic behavior bind provide though still assume stand coincide wider taken grow slowly hand moment specific pearson correlation coefficient eq denominator invariant transformation r standardize x smoothness x immediately satisfy exist point union straight origin understand axis equation root vice versa lie union line origin r cx cx c p standardize uniform third rely let value vector let accordance type derive cf statistic linear particularly satisfie condition proof lemma defer subsection q p complete recall condition place provide hold consequence write satisfied definition let expression positive q recall view obtains replace inequality display combine accordance type respectively prove easily verify inequality inequality respectively first old inequality definition reason q well truncation definition xy xy satisfied assume definition follow since take recall accordance notation use eq remark concern ic replace one q z concern pre constant choose n correspond theorem normalize q accordingly place allow recall specifically ab triple rational expression ccccc lemma expression tail need wherein expression several q q restrict real abuse symbol represent expression take express use also indeed result establish immediately theorem constraint remark improve pre inspection pre complicated appendix constant assume theorem hold take condition g xt cf lemma k w jj satisfied definition combine definition lemma also accordance z use use definition imply trivial convention together definition upon case remain yield well prove bind bind hence factor triple denote inequality exist depend chebyshev chebyshev hold satisfy smoothness modification fail extra contradiction triple rest far w q enough contradict statement prove modification argument easy verify previously corollary computation arise proof course calculation could principle aid computer r v arbitrary far recall smoothness whenever norm increase specific rational number whenever happen necessarily satisfied positive q take u try weight expression substitute parameter give interpret exact rational triple ccc adopt notation use corollary recall satisfying shall specify proof eq cf triple shall specific triple use decrease inequality inequality inequality fail eq definition definition upon list accordance definition w restriction recall definition one concern table necessity perform round intermediate exact rational within prescribed precision take expression algebraic calculate triple calculation replace absolute triple list contain general number use attain paragraph formula mention root value list place one prescribe remark value table c cc cc cc cc whenever supremum attain finite positive function define x hx expression table lemma trivial verify statement suffice statement iii iv decrease since change hence statement easily finish make hence statement positive critical rational attain checking checking complete case case old q take condition cf discussion line use instance expression desire expression listed verify recall identity denote open origin u namely far bound bound q combine program algebraic number rational particular statistic pearson define rr population normality rapidly zero come normal david suggest close normality approximate distribution close normality closeness variance therefore whether nice transform moderate non population carlo skewness tail normal expression kf rf asymptotic bivariate easily consideration briefly statistic correspond aside application namely upon let g moreover view whether population despite appear g particular worse view pearson yield bound thank remark part nsf grant dms uniform order closeness normality abstract statistic bound delta pearson student kind study central student previously method stein chen er transform constant pearson independent identically property relative efficiency pearson correlation well g test close normality neighborhood closeness bound correlation statistic pearson surprising consider bind somewhat perhaps somewhat simple student identically distribute I student statistic normal establish e method linearization chebyshev inequalities form statistic function delta origin since linear demonstrate main sum assumption moment norm obtain comparable factor infinity extra logarithmic factor far interested soon remarkable chen make offer relatively refer chen pearson deal define variable take multiplicative allow moment chen er yet modification level bind group closeness abstract may upper tw present fs z delta effort finally delta method statistic one bound appear know central student obtain turn one delta section know identically distribute general replace necessarily identically organize tw mention state vector delta bound fs ls z commonly namely student pearson proof section defer proof statement appearance bind prove even relaxed practical make computer preferable pearson statistic value measurable borel brevity stand borel assume q note hand surely independent let r q simplicity replace validity replacement show theorem upper behave chen modification make define simply paper would place instead allow possibly happen third improve constant application state counterpart minimum let p even state independent mean obtain g center g shall upper application real bound possess monotonicity clearly follow relaxed condition true e consider paper expression cause moment bb root exact low bl c j independent borel
figure case loop head relation polynomial chebyshev modification assignment graph concern cycle belief assignment theorem hand positively proportional comment sum equal proportional assignment regard bethe loop series expansion run loop sum loop interest bivariate integer cycle q assume side bind loop generalize loop generalize degree equality factor model pairwise straightforwardly hypergraph hypergraph type example hypergraph mrf node satisfy message rule bethe partition convention connect modify sketch choose loop sum call bethe topology strength formula technique polynomial since contraction contraction relation polynomial edge loop end formula essential proof contraction loop edge classify include interesting divide loop loop connect omit theorem match matching match introduce index word denote restriction principal minor summation run cycle regular connect denote omit polynomial assertion acknowledgment grant aid aid scientific field joint set variable dependence joint often give form normalization mrf mrf computationally require approximation attract computer equivalent successfully image algorithm compute surprisingly cycle behavior marginal assignment know hand many little topological structure underlie give loop expansion term series bethe loops bethe correct expansion highlight pass diagram expand easy derivation bivariate ratio bethe approximated investigation understanding graph pass message initialize message approximated formula bethe partition ambiguity update specify depend choice normalize b ix x marginal prove play polynomial transformation chebyshev polynomial induced edge hand bethe rewrite q bethe
consequence shall far respectively proceed subject follow multipli direct application lagrange multipli conditional test let exist satisfy inequality quite straightforward eq mm test test type probability obvious assertion hold inequality stop rule let fulfil almost everywhere condition proof sequential test original characterize natural truncate fulfilled stop stop rule eq rule coincide eq finally attain everywhere proof characterize minimize stop truncate result precede apply particular basically proof lemma exist lebesgue monotone pass side leave need stop stop need side stop rule conduct follow fulfilled rule condition satisfy satisfie find generally violate eq suppose normally mean also stop take eq remarkable finite process definition justify fulfil lagrange truncate may optimal truncate stop additionally g condition formalize let call test condition fulfil everywhere immediate theorem satisfy regular locally test vs strict least locally see remark similarly locally irrespective note regular need additional fulfilled arbitrary regular problem vs locally powerful vs strict strict analogously fulfil strict inequality show much also sequential exist another test kk number happen thus follow vs interesting example powerful average irrespective family I symmetric theorem test class I sample exceed theorem space integrable equality everywhere lemma side substitute right hand equal respective hand everywhere proof lemma major let integrable equality everywhere k x easily eq thus obviously equality inequality happen fix limit virtue particular equality integral finite fulfil almost part everywhere us b l r check infimum family side tail converge series second tends third start everywhere schwarz let virtue everywhere kx k almost everywhere everywhere almost everywhere theorem eq follow eq let let therefore analogy show satisfie essential non decrease obvious claim side virtue lebesgue monotone go non go possess thing property wise similarly pass thus lemma tend jensen addition non tend reach easy convex prove theorem generate see follow author work partially cb c subset testing hypothesis versus goal article characterize powerful sequential rule maximize sequential suppose structure discrete optimal test sequential testing locally let real testing fixed structure powerful sequential definition well interpretation characteristic see also hypothesis suppose measurable value probability make experiment observation stop suppose stop stage interpret reject hypothesis generate paper process
cpu three ht edge mb efficient perform study variable span forest complex forest complexity cpu grow linearly vertex cpu grow ht curve reflect short much start figure consume large iteration analyse package adjacency matrix list model likelihood aic find vertex cycle graph return connection return perfect free correspond neighbourhood find find perfect residual subgraph base list subgraph give ci degree degree vertex neighbourhood pos label false vertex col c label vertice col restrict neighbourhood subgraph neighbourhood radius neighbourhood actually general close high degree aic homogeneous heterogeneous could information false clique return list strongly decomposable list return short edge return inf far vertex distance diameter plot plot default circle continuous grey circle define shape place iterative force place consume clear iteration plot allow appearance plot label vertex shape black triangle vs r c r vs pos symbol code isolate displayed figure r r col ed col col ht vertex package package package way class forest model convert package limitation numerical acknowledgement stage research european support project life science innovation david package high package decomposable criterion aic bic graph package graphical involve useful wide application g application science internet model whose independence encode vertex correspond conditionally represent indicate independence two contingency linear variable modern account graphical model variable difficulty model package model central search forest decomposable typically term package distinct basic definition notation present cover discrete show profile site non moderate sample total site profile evaluate genome array probe independent process use compose characterize gene expression patient international project genetic single nucleotide four different basis g occur position dna individual base occur allele snps select snps snps snps structure minor frequency website www allele individual allele dna reference allele copy allele decomposable representation datum know record consider different give sketch complete vertex connect conditionally independent give graph present conditionally conditionally independent ht relationship conditionally give b add non loop allow conditionally independent subgraph every edge subgraph maximally addition would subgraph incomplete clique graph say distinct vertex vertex example great since separate path connect adjacent discrete describe graphical search http r package selection discrete row vertex use lr aic lr aic bic name name name original datum vector homogeneous case first last edge index besides true na core package aic bic forest tree search minimize criterion specify calculation use format storage type discrete discrete first represent refer index representation format index r frame level factor indicate type name length length attribute object refer column number dataset variable connect width edge attribute consist aic use repeatedly add optimize edge preserve cycle continue edge measure computed step reduce bic exist bic bic addition edge return add edge return decomposable forest summary span find
specie classify specie apply example class normalize mean near leave validation I mistake test error mistake order explanation result window classifier choose one cross explanation vector live input visualize initially show dot explanation class group dot dot feature roughly different dimension part distinguish scatter explanation class blue versus digit follow digit kernel width training approximate window example part example display explanation digit end explanation interpret look top example eight part vector dark add digit classify eight lack digit classified explanation red adding white two mistake explanation part change classify probably inside cloud explanation correctly classify focus explanation make digit weight classification similarly explanation mainly suggest remove dark part add light part look overall finding vector example digit label reason classify problem fed determine predict chemical cause requirement investigation drug discovery chemical modeling machine class randomly compound class split investigate individually confirm result example compound represent molecular software manual normalize mean rbf confirm figure remainder section evaluation feature histogram local indicate examine cause set certain seem mostly model outcome modify predict probability indeed conclusion explanation agree knowledge exist could explanation question rank trend exception rank consequently random use establish methodology example compound global explanation need identify relevant display kolmogorov ks relative frequency leibler prevent zero lead add bin generally figure almost positive gradient global chemical explanation model influence seem knowledge yet discuss literature conclusion learn explanation make potentially feature overall notion explanation error label decide way sensitivity area science outli sensitivity case remove estimate remove line influential point actually change regressor whose input explanation vector view thus influential vector sensitive influential decade explanation expert topic ai especially context sensitivity use remove explanation variable affect target connect explanation value decision train knn svm omit difference odd difference probability prediction without save combinatorial consider calculate difference layer measure importance change variable input turn interaction principal framework framework start binary discretization ii combination approach feature estimating discuss situation panel middle e project representative slice explanation middle maximal finding influential g x py derivative expansion start quadratic explanation second interesting direction eigenvalue instead meaningful mention practically estimator coarse structure far g extension demand useful classifier gradient explanation precisely classify explain hand estimate correct agree exactly boundary train area boundary space explanation away data area side corner away vector pose locally influential investigate high respective explanation gradient datum local gradient predictive model assume stationarity explanation reflect g explanation assume deal set taken stationary shift book paper propose box explain decision arbitrary possibly classification characterize point change predict information approximate validate conclusion various fisher different identify digit distinguish challenge drug agree exist available chemical space discover tool practitioner would biology decision make experiment promise approach prediction acknowledgment fp european mu thank follow illustrative explain respective examine exhibit kernel introduce locally ht effect toy gradient explain situation fail misspecification reflect rational quadratic kernel able linear separation illustrative purpose obtain local perturbation trend class clear previously observe feature interact triangle class ht two rbf kernel capture affect depend parameter item local gradient explain explanation extract local gradient outli near outlier reflect explanation feature reality nevertheless model place histogram figure distribution affect single outlier local region slide thus gradient hypercube center appropriate averaged appropriately window locally boundary point accurately follow circle shape range towards instance introduce small reflect gradient elaborate explanation fit window derivation estimation gradient window calculate approximation svm use rbf window minimize absolute predict local boundary resemble accurately choose width pointing width practically useful region width fail obtain gradient leave blue bottom tu tu de tu tu modern unseen answer question answer influential currently base explain decision automatic powerful tool learning classify unseen label nevertheless explain decision essential give answer typically jointly relevance determination salient coarse still provide instance conclusion equally classification view combination influential see answer corner contribute jointly solely ensemble pruning provide individual tree propose framework explanation vector order result explanation organize explanation gradient apply learn distinguish estimate explanation vector classic classifier explain digit distinguished digit discuss world scenario capability human decide capable modification calculate gp posterior analytically model use
normalize numerator side complete define notion hypercube prove thm area noise show sensitivity noise quantity asymptotic surface furthermore asymptotically tight distinct homogeneous area agnostic polynomial threshold learnable along prove sensitivity relate essentially also boolean interface origin recently multilinear polynomial point uniformly hypercube always note similar thought first prove conjecture related surface area set volume metric open area furthermore surface probability two define degree follow theorem q threshold threshold devote begin let gaussian define eq gaussians integer function equation sign interval note periodic sign change sign limit threshold show sign polynomial note sign happen ignore noted root way change sign circle number sign spaced distinct homogeneous occur asymptotically need slight sensitivity threshold begin prove follow variable q lie dimensional change bind probability least one change suffice correct degree distribution q
natural latent consider e ip probit give bernoulli rv distribution probit suggest ng corresponding complete latent deterministic call conditional gibb complete conditional indeed posterior intractable implement conditional new estimate probit population test diabetes health organization collect national diabete disease supervise tolerance pressure mm diabetes accord criteria goal diabetes explanatory bp analysis median max std dispersion degree aic scoring relevance reproduce perspective bayes probit covariate probit testing nest probit covariate intercept already factor hypothesis nuisance improper obviously variance elementary approximation ratio standard correspond prior prior eq estimator force distribution prior monte evaluation evidence inefficient produce integrable infinite require effort efficiency monte usually figure survey obviously integrable simulation simulation define support two offer importance function possible approximation sample candidate approximation hard use importance distribution distribution estimate provide general specific probit since replication methodology figure table require compute simulation sampling simulation prior simulation bridge factor rely representation bayes sample distribution perspective bridge parameter common poor possibly variance nothing importance function integral bridge representation posterior apply long exist choice poor performance connection harmonic quasi optimum se approximate base posterior approximation alternative difficulty difficulty derivation restrict embed correspond advanced bridge appear space joint distribution clearly approximation two generate depend completion form consider posterior bridge sampling handle implementation pseudo technical device bring bridge close cross alternative complete globally link posterior green jump mcmc cross model seem randomness pick step useful infinity average form asymptotic ml quasi optimal average gaussian suboptimal result replication methodology hand bridge variation excellent sampling considerable upon initial monte side superior accounting example iteration leave right generic harmonic matter representation remarkable direct processing mcmc variability estimator importance tail instance harmonic approximation infinite discuss opposite support like hull simulation correspond region importance approximation mean ml estimate replication methodology simulation importance fast compute due gibbs necessarily account dataset harmonic versus estimate approximate bayes q rhs say select harmonic unlikely framework use preliminary special model distribution probit approximation particularly attractive sampler rao available probit replication simulation approximation reliable dominate asymptotic particular case dominate harmonic estimate harmonic importance distribution bridge harmonic median estimation sampling
contain previously estimator differential ordinary widely physics biology usually involve parameter dynamical seek concerned variable ode use complicated thus analytical besides estimate covariate vary assume know generality investigate likelihood optimization require eul principle exploration since require solution seem ignore nonparametric smooth spline smoother find side estimate covariate simple implement take recent besides spline method work ode coefficient minimize functional reflect satisfy ode numerical ode take asymptotic vary tx dt extension multiple straightforward non covariate regard simple case involve dimensional functional coefficient associate error author analyze difficulty problem extra dl lt lt simplicity derivative bandwidth localization local result dl though ht h differential try around approximate together derivative identity note order even kernel avoid discussion issue complicated seem choice affect except multiplicative always fix go infinity support well derivative zero measurement finite denote independent distribute support main result concern lt p diag I I I lt lt know state completeness q eq general appearance probably entry r iii dl rx nh ft lt dl ta vi dl ta rx nh expression dense time consider besides appear calculation involve slightly rw ft nt uk uk equation bias component ta l dl ta expansion conditional ta r ii rd dl ta ta dl ta
object use database instead link member subsample object since always empirical matrix link maximum leave present application sensible basically query dependent since useful measure extended relationship relationship measure occurrence frequency word metric continuous however substitute regression web page collection relation web come recover page relationship asymmetric web imply page page simplify I web page unique web page later bayesian standard bernoulli merge link page evaluating serve purpose treat probability methodology object indicate word document avoid introduce extra approximation original obtaining measure b I intercept logistic cosine measure practical advantage linearly adopt feature make comparison suitable relationship web page symmetric relationship reflect pair subsampling computer evaluation item measure define setup query web page label fourth page class return pair criterion reasonable objective page possibility hard particular demand way omit page return however page page query four four standard binary standard original cosine page combine cosine document relationship demonstrate superior equal precision perform ask retrieve fall detect link close evident adopt pair retrieve notice hard instance group mostly short basically member web page recall intersect summarize performance bold indicate pt model molecular biology protein interact protein physical carry cell resource collect protein bind protein role source protein protein localize degree hybrid collection total analyze aspect large interaction network include de prediction consider categorization protein evaluation individual annotation protein sequence gene collection protein encode collection bind experimentally protein collection mode interaction bind place context pathway functional annotation go combine functional annotation say ii random replacement iv interact comparison purpose approach process large pair query vi calculate comparative summary genetic localization bind protein generate attribute process attribute correspond expression different attribute experimental measure perform well metric batch protein interaction select query link pair network satisfied category rank pair connect protein path reason filter pair undesirable filtering correct match trivial query pair interaction generate ranking respective auc replication last smoothed version replication obtain replication top give cb b c c provide include indicator however notice considerably reason degradation change method increase slightly method analyze criterion random add tie smoothed count method winner obtain relational bayesian winner rest pairwise comparison often method another besides proportion among categorization categorization explain pair top rank intend well link database link protein link categorization instead perform query gene category select category query replication challenge scenario optimize able pairwise top filter link protein coverage category pair categorization include neighborhood proportion histogram search coverage categorization perfect valid pair gain positive rank considerable across particularly experimental setup protein categorization select filter candidate pair protein link protein query include category replication category link summarize evident pairwise automate criterion believe much reflect high coverage top rank table success room artificial intelligence cluster reduction classical planning exploit derive would graphical incorporate existence relational use step probabilistic motivate retrieve query reasoning discuss interpret link choice future extension consist discover latent formal discover latent relationship inductive programming inverse overview relational particularly active lie analysis discover gene retrieval text index method answer idea give treatment simplicity task conditional compare predictive framework similarity remain appear international conference artificial intelligence formulation calculating compare relational size graph kernel provide metric property membership object cluster role framework indicator conditionally available relational reasoning application exploratory anonymous several suggestion presentation additional reference reasoning formulation support nsf grant gm foundation fundamentally develop relation question retrieval analogous object way object combine require specify relationship illustrate text application discover protein work even american assessment record include reasoning reasoning implicit although relation implicit nontrivial measuring similarity way discover extensively discuss cognitive analogy isolate atomic discover relationship paper concern implicitly tool exploratory protein protein cell cycle functional interact protein molecular experimentally protein bind clear interaction molecular protein particular generate list prediction expression encode protein localization relationship protein biological know interaction role aim detailed protein pair correspond analogy example section present role protein possible pair want match possible metric protein interaction section perform query ranking molecular general exploratory link relational database link way retrieval text illustrative web page page link relate search analogous analogous wikipedia evaluation criterion review tailor analyze base corpus characterize relevant set similarity unlike feature need define similarity avoid mistake compare relation similarity paper initially group use multidimensional pair represent connect similarity pair compare operate object instead distance solely complex approach logical reasoning seen reason latent relationship relational approach discuss however reasoning tackle planning create exploratory text way exactly rank respect query query retrieve exploit one retrieval query mass rank equivalent expression collect advance query give estimate general query instead define use class datum art ranking inspire event model model hand provide interpretation compare compare define hyperparameter query generate generate next biological modification analogy object aspect link feature object query pair illustration pair good query reasonable would match nevertheless infer similarity feature kind world hypothesis need object relational reasoning similarity meaningful feature represent say interaction b similarity query correspond pair directly object similarity capture classify want quantify link unobserved integrate bayes explain latent dimensional logistic parameter particular pt mapping attribute potentially predict link model bayesian methodology explain object parameter compare link indicate link relevance set feature integrate prior bayes give compare hyperparameter generative point represent rectangle conditioning set
score scale lack small turn rademacher ensemble hadamard rademacher hadamard develop fit display rademacher typical already hypothesis mostly shift meaning realization rademacher ensemble typical less hadamard success ensemble mostly slope score sign success treat random varie realization expectation fit without display output symbol denote interaction median coefficient std multiple adjust df score toward residual score close hinge term increase significance scale call residual min coefficient pr intercept de degree freedom square f residual ensemble particular analysis rigorous quite belong particularly unclear hadamard display score fit show describe ensemble transition develop entirely display present early describe ensemble comparison demonstrate rademacher validation ensemble early explain gain develop ensemble mean decay ordinary ensemble existence seem confirm extensive experiment success programming recover million cover ensemble limit diagram success success tends measure sparsity ensemble location closely match behave survival width transition unlikely tend broadly ensemble locate advanced maintain conclusion use ensemble counterpart level score support fitting statistically nonzero fraction mean exhibit trend ensemble weak finite accounting mean score ensemble phase match case far current author software great deal describe resource compute dms jt partially support transition occur dimensional processing sense reconstruction model threshold combinatorial geometry transition vary location appropriate subject area model hard throughput hard limit degree break compressed sensing define sparsity tradeoff theorem derivation transition combinatorial assume underlie identically distribute required experiment inferential analysis underlie ensemble run million span several ensemble phase transition agree asymptotic careful explain decay experimental large ensemble reject throughput measurement dataset sense transition phase amount rapid shift critical threshold concrete example surprising processing cloud intuition surface hull intuition interior segment intersect expectation even intersect interior human hard visualize phenomenon phase appear continue bit small threshold bit intuition work tuple indeed intersect phenomenon one consequence field select model sample design imaging device consequence nothing really hour medical necessary help reader transition occur choice observe phase transition evidence million random transition iid formally geometric open contain standard ensemble class broad view attractive feature tendency ever observe pixel patient technology put throughput measurement ever detailed protein expression whole fluctuation activity ever st mark observational hundred throughput measurement device fundamental obtain observational face observe distant galaxy many field observational perhaps hundred set abundance characteristic modern develop new tool comprise activity institute gauss variable measurement expect measurement observational datum allow set gauss throughput batch potential automatically know useful project throughput popular believe fraction among unfortunately throughput everything together batch stage expand conduct stop choose denote predictor regression record square observation world observational unit fall slightly increase position curve derive combinatorial notion fractional axis vertical observation curve combinatorial rapid fall match curve field geometric think unlike normal variable contain know design partial hadamard column hadamard matrix design choose generator either large negative well relationship try standard outlier quantify create range digit agree record outlier six digit attribute six digit panel axis vertical leave panel use expert agreement change behaviour fraction observational unit fit large contamination break n derive curve coincide axis look interpretation fit design experiment robust certain critical outlier calibrate match critical fraction match geometric analysis decide acquire measurement imaging signal actually kind observe although degree freedom attempt measurement experiment choose horizontal axis measure vertical axis contour success roughly fraction image accurate digits horizontal axis fraction axis fraction interpretation usual limit twice already arise recall geometric draw denote hull random polytope suppose figure simplex interpretation tuple vertex line polytope dimensional second black see curve define dimensional hull origin typical every segment polytope dimensional face involve positivity recover describe derive lack formal connection establish ball observe frequency visible accurately situation present believe kind limit formalize geometry polytope hull polytope point project face projection family call regular simplex analogue polytope analogue hypercube polytope available reader object think however face count reveal system frequently responsible three lp course fraction lp solution face simplex tell lp correctly consider polytope optimization instance equation infinite general position sparse solution underlie solution q face count polytope cross polytope tell short simplex cross rather tool polytope theory rigorously existence threshold face count iid triple count display curve polytope year run experiment generate million various specific whether polytope accurately matrix involve gaussian ensemble detail material supplement l name bernoulli iid equally likely fourier row dct fourier dct element equally likely iid equally iid iid iid hadamard row hadamard matrix hadamard special binary rademacher equally likely equally likely specify pattern ensemble column fraction varied sparsity range lp solve exact reconstruction correspond polytope face face success frequency success systematically sample success show success ensemble nonnegative nine curve present solve note use sign excellent agreement ensemble qualitative et al show qualitative agreement transition theory surprising merely prove polytope accurately ensemble ensemble accurately suggest phase transition behave prove behave sized asymptotic writing ensemble know know discover phenomenon identify stage identify comparable identify call phase transition phase diagram ensemble physics mean something different instead phase underlie structural strong transition conduct like instance specify factor coefficient ensemble coefficient indicate draw uniformly sign ensemble indicate visit triple triple give run observable take value within accuracy otherwise aim rather exploratory inferential carefully explain apparent attention scale separate span situation many available require cpu different table exploratory study day comparison calibration vast majority experimental run compare popular consistent rather inferential comparison say ensemble may gaussian everything else ensemble realization ensemble success ensemble hypothesis really assertion equality standard compare proportion comparison experiment non gaussian standard merely consideration score formalize nan asymptotic seem priori consider universal correct behaviour size asymmetric occur analogously systematic score arise binomial proportion verify show describe exhibit ensemble obey slight shift away negligible shift cause lack small ensemble ensemble complete use validation validity lack good hypothesis device list table vector either underlie process optimizer underlying sign process solve optimizer optimizer desire solution digits replication coefficient replication specifie rademacher type type presentation coordinate shape success dataset slice hold vary success fraction monotone consider compare baseline varied systematically grid range trial region report happen analysis conduct hadamard rademacher ensemble rademacher hadamard restrict question transition profile function reduce manner empirical point appropriate place complete rigorous asymptotic exist either width rigorous rigorously scale rigorous phase constrain death probability problem slice gaussian monotone increase failure function success three probit response complementary distribution present checking identify choose match display curve residual column datum model column generalize linear model binomial response take eq success obeys call logit probit links fit call fact achieve among probit residual probit zero thin deviation logistic link good residual probit adequate ordinary statistically sensitive residual fit inferential test failure estimate way find large solve small spline difference estimate limit l phase transition quantify show approach limit roughly transition response present nonparametric analog base spline location normalize distance probit nm methodology score ground difference asymptotic independent compare setting sure procedure compare score present present plot versus fraction score exactly plot quantitative ensemble either sign score absolute line assume distribution comparison value much line thus behave observation need check score construct know normal figure score compare ensemble ensemble fit trend show obtain case table l methodology detect difference compare reflect curve reflect plot identity also display truly normal every deviation apparent dependency force lose polytope decrease effect conclusion main strict asymptotic report focus exclude discuss fit consider first report result later restriction display restrict symbol symbol denote list en sum de en z median max std e na na de de de de residual freedom df value model significant roughly proper reduction sum square nan residuals min coefficient estimate std pr intercept e de de e de de de de de de code degree square fit associated contrast mostly worse adjusted restrict df df pr exceed significance word vary sample root function motivation fix success ensemble asymptotic transition c theory width describe success indicator perhaps interpretation rigorous provide surprising underlie indicator variable correspond one common score gaussian ensemble obtain collection intercept fit table l l l intercept slope exponent case easy find software exponent fit two ensemble include term vanish joint eq term exception effect significant fit give include although tendency vanish joint intercept term exception treat two give degree tendency vanish exclude figure panel score versus differ score overall get evident plot dramatically apparent mean adequate provide case residual versus mild nonlinearity model model preferred scale q de residual median std pr na na de code degree square square statistic substantial residual dramatically give residual min std error pr e na code freedom multiple square adjust square f e individual coefficient compare statistic df df df sum pr fit term fit score contain immediately residual pr intercept de e de de residual degree statistic df error allow extra explanatory drop scale model adjust bad scaling remain cause remain error everything satisfactory way summarize residual group summary deviation median absolute deviation fairly residual deviation
dag matrix every connection element denote dag later early influence influence imply influence define dag identifiable strength discovery algorithm define reject evaluate reliable return bootstrappe reliability th create replacement bootstrap eq q ij sign important way divide would make boundary region close solely multiscale multiscale mb mb replicate generate bootstrap replicate denote hypothesis compute multiscale specifically multiscale bootstrap ica point change estimate ica step practice maxima test equal imply accept model violate could might give order case tell order create two boundary create laplace size mb select replicate bp mb mb mb bootstrap bp bottom multiscale bootstrap mb give line bootstrap bootstrap six compute bootstrap histograms multiscale imply rejection probability ordinary bootstrap versus level six case equal level plot bootstrap bootstrap give reject nominal level multiscale much show multiscale bootstrap provide propose statistical value variable estimate tell investigate sensitive smoothness although might problematic support continuous recently gaussian call model various direction affect random conduct ordinary hypothesis propose procedure advance call multiscale multiscale asymptotic artificial utility widely causal recently call variable datum alone order causal relation variable infinitely bioinformatics tree replicate bayesian probability
summarize split right stand perform get conclusion clearly small show cloud mask analyze multivariate estimation dimensionality independent factor latent mixing use mirror average aggregation density classifier achieve excess support sense outperform several intensive efficient algorithm develop mirror note imply transform dp x x p jk bias h variance w suppose canonical denote al diagonal convolution kernel brevity term statistic far since uniformly inequality hoeffde straightforward imply nh h nh standard lemma apply inequality l imply j b get constant follow corollary expectation aggregate estimator first subsample suppose take inequality respect subsample outside hand entire recalling obtain rank brevity denote j g j proof proposition ni ni g systematically k ai f fm condition ng cf complete classifier r bt bt j I I f plug brevity j prove statistical berkeley berkeley california xu comparative model journal l minimax ph thesis plug asymptotically estimation infinitely censor institute mathematics integral representation theorem g machine reduction regression probabilistic recognition york fan inequality completely continuous k n n density statistic negativity estimator journal multivariate scoring association journal american new york new york mirror average mirror average new component mathematic family use principal discussion journal american association statistic example projection discriminant science estimator advance modeling inference world fourier framework introduction york nonparametric adaptation freedom cc aggregate ks ratio size noisy black area dark gray clear white classified figure figure cm cm multivariate form noisy model either component estimate mirror aggregation achieve estimator density construct classifier logarithmic dimension experiment promise factor aggregation complex lie multidimensional occurrence science source kind include biology genetic network gene molecular imaging internet phone management important challenge visualize well even moderately motivate de li al paper generalize ordinary ica recover hide ordinary assume source unknown ica equal mixture without source situation involve source factor ica ica concentrate source xu xu mixture model present serve reduction suppose noiseless estimate know section recently supervise plug classifier label achieve excess condition classifier penalize empirical risk difficulty plug moderately region purpose gr suggest overcome outperform quadratic poor early discrimination orient projection pursuit produce quite high procedure appear model source distribute report promise result section give excess plug plug achieve logarithmic bayes describe implement report application consider deterministic loading zero mean normal mean covariance identity ica widely blind source separation source basic ica assume et unlike literature goal serve different gaussian technique column know propose factor extra arbitrary rotation factor allow throughout assume orthonormal convolution convolution matter whether mix kernel hard rate give moderately example show source orthogonality eliminate dimension specify none quantity orthonormal substitution expression q q denote th density estimate density convolution smoothness property estimator bandwidth use la integrable several follow suggest outside nonnegative density q discussion give know estimator integrate note distribution identifiability factor allow since section still outline strict factor example noise imply ii sample arrange decrease consistency root consistency matrix fan slow conditional estimator bind plug nearly excess size density union sample one problem consist predict assign membership explanatory associate denote population borel misclassification exist sample misclassification goal construct classifier possible estimator relate excess plug follow noisy let mirror denote excess numerical aspect dimensional bandwidth singular let svd center sort use cf form density feasible fast huge procedure control implementation average computation integral realize mean form node associate integral involve multidimensional domain prohibitive exponentially realistic alternative monte integration sample monte carlo several consider fast one realization density array dimension end estimate kernel estimator algorithm estimate independent compute kernel iy diagonal output predefine generate cumulative pre linear noisy extensive source include distribution well unimodal multimodal dimensional run generate signal ratio snr process q density estimation ks implement ks package implement matlab request ks effectively contrast integral compute necessarily lattice impose conduct numerical mix matrix gram schmidt monte replication correspond noisy brevity representative figure display present snr chi superiority aggregated ks show analogous case factor keep
selection base cluster necessarily fall impose fig cluster plot red bar member bar field galaxy red symbol weight weight square color error galaxy comparison width ordinary gmm panel show measure ordinary strong nearly clearly power without contamination location mean scatter strong scatter gmm mostly member point relation galaxy member generally respect lack galaxy galaxy extend us slope cluster measurement red follow process removal determination member galaxy measure slope slightly method directly likelihood red galaxy section assign membership difference galaxy within fit member galaxy color cluster background likelihood identification galaxy straight measurement call tb tb point dot bar every strong despite slope see mean slope bin mean place measurement deviation apparent observe slope statistically slope evolution slope elsewhere slope red cluster strong measurement determination galaxy red galaxy cluster contaminate foreground reject via galaxy address foreground result preserve galaxy galaxy choose galaxy extension slope field fair bin galaxy bin velocity slice km galaxy separate galaxy sequence galaxy big galaxy sample fashion one galaxy galaxy band come color inverse error record bin illustrate slope slope panel high environment correction perform scatter increase magnitude scatter evolve red frame galaxy find intrinsic rest increase frame reveal scatter show statistically significant iv qualitative agreement trend slope kind scatter trend slope cluster lack color fraction galaxy star formation increase contribution come choice importance use place galaxy motivate choice derive band indicate intrinsic precede intrinsic evolution color toward member paper purely sample variation location scatter width correct measure improved cluster galaxy apply method recover namely variation slope scatter sequence slope trend observational correct k corrections measurement individual attention formation measurement check acknowledgment lee helpful tm acknowledge grant de er would thank physics nsf theoretical national science u energy site http www institute group university national institute university gmm introduce brevity represent mixture color galaxy color member galaxy sample component density color color extend parameter could relate eq iteratively parameter maximize likelihood lagrange multipli multipli q arrive similarly within analytic iterative major iteration relation fine eq back arrive relation round ignore measurement relation easily simply datum variance repeat upon request p david center laboratory il department ann mi department university il department university ann mi california b center university department il institute particle physics stanford stanford university stanford department physics il laboratory red cluster optical measurement scatter red sequence red sequence galaxy new correct gaussian mixture galaxy use technique remove effect intrinsic select galaxy sequence measurement red galaxy find scatter increase slope observe slope galaxy observational check galaxy trend scatter intrinsic evolution present base galaxy large universe abundance growth expansion history universe feasibility parameter demonstrate author galaxy recently red evolve core varied least mean cluster part red galaxy population galaxy dominate color old star observe color smoothly optical cluster application yield proxy precision measurement extent exploit accurately characteristic measure cluster red play important complex physical galaxy include galaxy red environment relation identify galaxy high environment galaxy red sequence galaxy population rich cluster scenario summarize fill formation formation magnitude space slope scatter red scatter sequence effect age sequence measure color allow galaxy well refined cluster measurement image red measure various individual turn considerable digital survey sequence elliptical galaxy various environment study identify red galaxy constrain galaxy relevant galaxy sample robust red using scatter systematic slope scatter handling error correct gaussian reliably recover property measurement relevance cluster trend red emission early galaxy dominate old rise remarkably galaxy color addition galaxy color filter band long therefore informative color detect member galaxy field galaxy galaxy member whose color cluster narrow width galaxy color broader separate component represent galaxy double adequate model gmm suit gmm negligible measurement galaxy measure intrinsic scatter cluster contamination galaxy without account scatter large intrinsic color scatter break toward band intrinsic measurement error maximization em fit clearly good sense decide free bayesian bic component corresponding small intrinsic scatter measurement method gmm fit subscript cycle cycle denote location width point gaussian brevity parameter give q maximize expectation way introduce tell case negligible arrive lead maximum component fail initial suppose point repeat process resample set datum get estimate beyond good result take real monte whether reliably identify cluster component reliably input parameter respect cluster member color cl galaxy galaxy color cl generate allow cl vary keep choose color member background galaxy field note color generate noise noise add plot cluster
gradient descent work magnitude distance lipschitz continuous denote first reduce claim follow opposite arrive parametrize curve equal distance path note derivative path expect assume loss vanish fast exclusive integrate yield q side switch prove note proof monotonic information part generate consider path optimal particular reach define eventually restrict always align derivative choose loss generality weakly monotonically increase point definition integrable integrate multiply side become expect vanish smoothly vanish control theorem assume gradient risk bound quite gradient optimality observe move make oppose mean reasoning extract bound l delay expectation feasible independent upper bind appeal monotonically stepsize small wish rate risk delay small last bound bound likewise divergence lastly integral term period cover segment guarantee plug collect yield divide govern regime initially quite increasingly delay essentially dramatically affect parallelism step average dominant parallelization set conclude set strongly smooth occur logistic surprising ratio eigenvalue theorem rate provide second inequality rate decrease combine obtain also simplify rhs divide dependency factor fully make generalize bregman begin convexity moreover convex whenever finally function scalar delay convex obtain algorithm unnormalize delay identical strongly update bind key constitute yield exploit continuity transform obtain tight easy example considerably scenario feature bins bin come canonical distortion hash dimensionality pick try delay goodness check system delay secondly whether scale well delay update upon hash regularization e code machine gb core parallelization divide piece give piece piece master piece master master add piece together proportion magnitude quickly multiple dot dot maximum would simply first run delay observe delay example bad run delay delay try turn handle slightly one find parallelization dramatically show parallelization trying delay intuitively delay theoretically example effect small three simulate delay secondly hard problem small prevent prove delay parallel paradigm choice large framework stochastic online excellent tool address current algorithms process receive instance make update word entirely process modern machine graphic core core disk processor speed typical network interface throughput mb disk array reach size whenever amount cpu distribute bottleneck propose evidence work guarantee variant core gradient sharing update accelerate intensive problem whenever gradient computation subsequently update occur delay core available parallelization synchronization consequently comprehensive cloud home core computer processor execute piece code processor easy exploit affinity core share architecture processing graphic tend element execute piece synchronization kernel process synchronization mechanism undesirable come expense significant memory resource availability graphic mb high speed ram per communication nontrivial bandwidth computer communication communication equivalent cycle tend slow server configuration communication unable directly disk network storage transfer moreover typically second reduce processing stage play critical analysis exclude parallel suit problem banach family category vector variant game communication within team adversary response whenever induce loss goal cumulative loss minimize abuse loss achievable radius anneal schedule compute update gradient anneal entirely delay current gradient previously extend extend implicit update divergence section lead parallel modify bound plan delay function measure bregman define need auxiliary lemma instantaneous divergence decompose expand product delay gradient distinguish difference protocol yield plug show project decomposition key characterize successive gradient bad cost constant prove briefly identity inequality lipschitz gradient via tackle term diameter last sum discard contribution negative become lipschitz property decrease hence gradient q plugging rhs processor claim converge bad adversary may algorithm fast result practice bad assume online regard least old construction function chance every consequently instance guarantee delay could set strongly difference correlation eq monotonically increase pay delay version minimize objective typically small function combination successive stochastic gradient descent
collaborative reconstruct considerable collaborative ranking subset movie rating factor contribute user preference infer dimensional incomplete distance wireless recent algorithm low guarantee successful high recovered rank entry np adapt compress sense incoherence correctly recover recently bound sense without impossible fix due entry per row value suboptimal appear manuscript guarantee relaxation incoherence original relaxation recover non rank completion unique introduce randomize complete specific furthermore minimum theoretical focus prove completion low approximate provide rmse plan generalization relaxation rmse analogous side directly solve grow proportional year solve thresholde atomic trying solve describe solve problem minimize singular matching problem nuclear pursuit iterative approach hard thresholding procedure estimation incremental broad performance guarantee estimate original reveal turn broad show underlie condition carry reconstruction algorithm applicable organization relaxation mostly introduce efficient modification discuss numerical assume dimension entry let matrix original matrix rank solve recover operator matrix match observe notice doubly especially matrix sense convex relaxation equivalently recovery completion nuclear nuclear I singular lagrangian rank namely performance compete minimizing provide excellent high initialization add description represent estimate rank input initial condition represent twice entry row analogously column input represent grow spurious dominate respectively singular provide unobserved entry degree throughout singular following base minimizer idea reveal clear separation reveal matrix reconstruct spurious one describe guarantee reconstruct appendix bound rank probability consist perform rescale singular appropriately svd projection notice require available form involve define estimate compare eq allow suitable proving throughout paper fx p complementary set paper explicitly x definition generate interpretation justify manifold descent r manifold k left matrix start numerical good stop p f basic criterion also author provide initial tolerance iteration size w em condition novel case reconstruct ill far discrepancy start first singular next incremental output projection e x yx k em demonstrate incremental bring gain implement test computer gb use modification simulation section different scenario completion generate sample independently reveal reveal notable use stop criterion generate corrupt additive identically follow subsection reveal independently probability entry stop matrix illustrate convergence matrix decay iteration close validity criterion next reconstruct reconstruction fraction curve prove plot plot rate extra come surprisingly low one solution lower display rank proved figure plot use plot rank rate sharp threshold location surprisingly close bind admit compete algorithm rank plot relaxation solve algorithm low rate algorithm consistent various present correspond hence outperform algorithm error time per c time entry per column high novel robustness incremental exact completion generate simulation ill condition let orthonormal respectively diagonal entry linearly form criterion incremental improve different mean square error define comparison start take entry entry standard relaxation performance relaxation performance oracle comparison root one small square error become indistinguishable low compare average error root illustrate different rank c time observation corrupt ratio row column illustrate performance change entry gaussian distribution unless independently add noise generate matrix observation missing estimate variance entry depend completion scenario reason error suit ensure matrix almost evenly distribute performance change reveal per row scenario distribute equal accuracy measure rmse result rmse show rank bad gaussian coincide rmse close oracle implement performance observation curve reason rmse decrease return return performance bind factor projection reason rmse snr less good rather get rmse close correctly localization sensor observation assume formulae choose note case entry rmse multiplicative rmse correspond display figure respect noise difficult distinguish motion position capture location failure corrupt outlier rl accord target independent entry noise outlier value affect figure clearly noise norm error however standard quantization regular near choose carefully quantization bad multiplicative entry whereas show
pass form considerably consume expansion phase slow indirect diagram align force see eq series drive force implementation start amplitude unit svd transfer successive assume series drive vary smoothly considerably slow closely work drive allow absence driving consider delay define scalar odd even index center alignment driving visually drive bring drive alignment offset sign define align constant indicator low value slow signal signal dimension unit fulfil e extension result embed logistic completely corrupted number expand eigenvalue svd mean order magnitude drive fast eigenvalue blue dot red break whenever negative complex eigenvalue experiment implementation fail expand become indicate expand space routine generalize eigenvalue singular fact eigenvalue occur matrix svd matrix regularity time happen short logistic move singular observe low high natural noise unlikely singular perhaps reason algorithm circumstance frequently application happen svd occur r r signal amplitude case solution rank might correct exception r r e way deal rank embed thus svd extent try parallel way signal constraint cutoff algorithm one rely eigenvalue secondly usually condition test usual result phase dependency large least show eigenvalue circumstance wrong term slow circumstance svd closely approach stable matrix implementation available algorithm original execution since consume part expand datum rank span number dimension expand cutoff threshold insensitive span decade reach noise amount noise dimension drawback carefully tune new drive force plan always look covariance sometimes worth modify accordingly review regard give define covariance diagonal dependency dimension carry amount whiten search zero transformation eigenvalue contain eigenvector row eigenvalue usually singular easily verify course run close become infinity noise error multiply trick svd deal singular replace effectively remove become row invertible subspace zero eigenvalue eigenvalue investigation numerically contrast eigenvalue eigenvalue package feature maintain namely algorithm look main modification v svd interface line execute code signal new pattern goal minimize pairwise difference new diagnostic drive force experiment line handwritten benchmark uci repository full modification available package grateful work university science grant cm cm cm http implementation http www de analysis extract slowly vary multidimensional signal easily circumstance expand decomposition free handling multidimensional signal modify datum tune slow ok approach ok ok eigenvalue ok svd ok improve ok broken slow ok svd ok discussion ok remark ok ok zero ok dependent cutoff ok conclusion ok appendix ok ok deal ok ok ok ok slow processing slowly multidimensional series already successfully numerous reproduce complex cell primary formation handwritten digit extract drive force role datum understand various temperature drift vary heart parameter refer drive force dynamic smoothly slow rarely e eeg electrical particularly drive force observe aspect clear convenient sphere expand transform basis accordingly signal singular however eigenvalue pairwise eigenvector define desire obvious direct fourth eigenvalue sequence signal signal zero eigenvalue equation q sec multiply line four expand possibly nonlinear ii sphere expand obtain component zero mean derivative expand matrix
full stop count wrong approach conservative expect batch draw batch batch tend ip batch ip batch number calculate analogously substitute voting possibility batch batch contrast bound ip batch bit draw strong single correct conversely five zero confirm low counting draw expect number test three independently simultaneous procedure chance hand incorrect outcome incorrect maintain keep split across chance least count count count incorrect outcome total column design control draws expect distinct batch vote draw distinct distinct vote row batch far work vote independent control expect batch analogously expect batch number independent control effort vote associate batch count work batch high independent control far low pairwise batch incorrect outcome compare independently batch vote en n en na na nb nb nc tc nc nc nc en na nc nb single random relative margin limit entire build sampling scheme control chance fail full count per batch keyword size sequential simultaneous post full count pilot california risk conduct collection arbitrarily count appear build state california sample vote pairwise margin extension margin winner margin candidate error batch summarize pairwise margin exist incorrect instance limit chance go set application use chance hand wrong wrong margin winner suppose together represent batch every candidate total take candidate candidate lose apparent vote candidate batch report apparent apparent actual vote candidate vote would include actual vote apparent must relative pairwise margin maximum pairwise margin apparent outcome hand think incorrect family strong apparent outcome hypothesis batch rp batch margin batch need otherwise cause wrong must spread batch even batch proportional error observe compound apparent outcome differ outcome calculation procedure outcome test hypothesis outcome keep incorrectly conclude outcome batch convenient batch draw apparent winner apparent b involve half apparent winner winner batch either mail batch illustration batch margin batch rr ip batch winner winner winner entire include include b batch cast ip cast mail
take program reference list least sum compute program uniformly requirement upper version depend every list neither logarithmic additive logarithmic string denote word xy x ne yx list element additive constant short short program program mapping element program addition nothing string show origin fundamental string theorem list string length extend finite list triangle property divide improper normalization list reduce restricted lead proposal improper symmetry violate equal hold kx inequality divide divide equal triangle inequality divide divide element correspond list change equal th dividing divide inequality contain useful kolmogorov call symmetry hold logarithmic short universal function additive kolmogorov string logarithm result short code great section proposition remark centre science address email support parameter free extend pair study kolmogorov purpose approximate version use real world pattern kolmogorov mining information information objective object object object multiple object classical kolmogorov objective information pair practical focused arise normalize kolmogorov real alignment measure impact decade conclusion classification represent file weather forecast music bioinformatics internet abstract information engine produce aggregate phrase relative semantic run semantic question answer reference many interested object example customer review article occurrence comprehensive specialized thus go object extract essence example list internet review tv list binary string string order list express string universal machine convenience string distance string term state interpret length comprehensive interpret specialized object similar information list practically theoretically promising case imply constant overlap take correspond state argue prove minimum normalize list element distance v necessity idea general case pairwise kolmogorov information subject informally kolmogorov complexity string universal constitute program technical reason choose read right without computation take upon initial call set program free reference definition kolmogorov shall simply kolmogorov formally kolmogorov short reference input output unconditional kolmogorov consist nonempty string present appendix list list string may order abuse conditional kolmogorov list length short machine list kolmogorov put law term list x overlap go single string length everything additive logarithmic length bit possibly suffice element finite length string k put edge next care trivially color length color appear since color always know reconstruct know appropriate length length suffice select element take encode vice versa string length possibly program encode string side shortest assume interpret kolmogorov comprehensive fact choose go list program overlap quantity short list string order increase list finite distance list order length distance metric symmetry permutation triangle inequality
sr r prior actually loading ibp prior prior synthetic dataset propose nonparametric gene connectivity synthetic sample gene underlie ground factor efficacy factor loading bind site also expression breast gene prominent figure actual infer factor recover loading bind ground approach loading loading permutation follow spurious factor hierarchy fast configuration never std mse response compare variant approach fit separate discover see phenotype binary figure real hold predict reconstruction random initialization variance suggest fairly w initialization nonparametric factor ibp power ease integration specific gene improve output factor interesting open ibp model cs edu account relationship factor variant couple base apply data task solely achieve benefit discover underlie predictive compact motivated feature greatly potentially overfitte approach stem reconstruct gene expression datum pathway contribution parallel need gene pathway couple predictive instead model fundamentally treat relationship propose variant ibp design account ibp explain pathway fundamentally involved synthesis nonparametric ibp distribution infinite motivation bioinformatic alternative sample movie versus action movie spurious process e pathway relationship pathway matrix originally motivated observation analogy customer enter restaurant infinite customer incoming select select customer select easily precisely customer select stochastic thus infinite binary turn stochastic limit exchangeable kp km z ibp nice possible ibp second parameter control factor parent easily individual limit exactly continuous process singleton evolve leave event pair binary topology infinitely exchangeable therefore limit markov evolve brownian dimension covariance non node gaussian pass al propose agglomerative approximately maximize propagation associate message current child recall factor consist feature factor variation treat factor purpose simple begin factor ibp ibp hierarchical infer presentation mechanism ibp factor ibp apply nonparametric past ibp places ibp feature gene factor factor loading context gene usually small factor ibp prior number model unbounde expression binary factor loading instead use hadamard analysis I ibp prior hence inverse gamma thousand pathway factor analogy enter restaurant spurious effectively ibp sparsity fundamentally conventional ibp ibp ibp rich get rich get truly whether likelihood correspond ibp ibp exception customer gene bernoulli basic fact hierarchy ibp describe exchangeable mean efficient ht consist b depicted aspect ibp prior follow example component nonparametric factor analysis response treat factor binary extra probit predict binary response summarize propose set pz ik pz ik ik simultaneously proposal acceptance fast show figure propose new leaf new find tree node tree accord prior proposal uniform predictive newly pass pass variable indicate
minimum arbitrary direction often example strictly hessian low family behave quantifie almost strong let analytic standardize eq q mle phase initially somewhat newton quantify enter arbitrarily burn central theorem key idea central shorthand moment eq hold key characterize expansion attempt use moment expansion argument e comparable selection small subset away whose support mostly see fisher eigenvalue complement quantify substantially weak subset previous different need replace small regularize optimization expectation reduce set regularization noise state deterministic e free appropriate quantify satisfie analytic optimization fisher bound intuitively expect think dimension hence favorable condition quantify distributional assumption actually relaxed sub eq bound sub gaussian unbounded long estimate linear sparsity general characterization level solution two support first proof generate analytic well expansion proof specify analytic prove core theorem furthermore convexity prove consider jensen know fourth standardized proceed analytic moment second claim max argument claim follow lemma case ready claim us solve prove prove claim assumption second use triangle proof support divide add ready prove theorem theorem note satisfie observe eq use restrict use conclude showed care standardize specify clear exposition bound leave coordinate plug choose obtain theorem applicable complete second case q us simplify conclude q q simplifying claim bernoulli thank email existence exist classical follow easily completeness work polynomial root claim root root gauss root derivative hull real root extra claim express effective dimension sparsity characterize convexity ability quantify show exponential discrete optimization generalization issue ambient much large size special high body characterize rate tackle challenging grow model family hold though modern problem rapidly asymptotically case quantify relevant aspect family throughout agnostic necessarily generate analyze log nature convex asymptotic limit log gaussian information quantifying occur particular rather natural standardized standardized recall standardize moment th grow similar study tail growth rate characterize rate exponential draw newton burn behave locally quantify strong eigenvalue design show family enjoy rate condition optimal incoherence condition provide essentially family relate final selection low drawback mutual incoherence permit perfect feature price sparse eigenvalue multiplicative level recover exponential family merely nearly mild risk result rather mild favorable lie statistic finite exponential general though keep mind variable covariate point loss eq natural space later eq consider expect fisher minimizer main quantify family family also property regularization find standardize satisfied family prediction behave quadratic analogous exponential tail standardize th power normalization deviation standardize moment use term reflect analytic standardized denominator analytic respect subspace direction univariate mild use obtain sharp bernstein th analytic tail standardized generating th neither analogously th standardized deviation quantity use certain setting natural standardize univariate bound hold denominator analytic standardized subspace bind interior analytic standardized finite analytic going issue mind quantify
decrease eq pareto admit arise mix derivative q whenever derive tend tend toward apply easy whenever law although may theorem explanation build process see model prove right think scatter characteristic link multiplicative scale invariance real set generalize version replace indeed idea regular mod preserve scatter scatter closely u henceforth theorem continuous law thank implication lead digit general apply six mathematical sequence slowly perfectly last experiment l kolmogorov term exception display kolmogorov arrange speed fast go fast understand integer kolmogorov odd perfect six uniformity allowing theorems v except everywhere absolute distribution digit uniform logarithmic function tend toward tend uniformity root concave exp j j dt expression toward complete mean l yes yes exp last kolmogorov apply read noticed root rather exactly rare roughly approach law directly relate multiplicative formalize general regular intuitive idea regularity actually thus explanation course argue fact study ii apply may argue mixture density regular multiplication lead density explanation simpler arguably good argument favor invariance relate easily historical implication digit proof remark mod approximate version phenomenon receive depend assumption often link author implicitly characteristic variable come regularity part prove intuition regular simple corollary result law link log view law scatter digit code law sequence number mod random uniform stand logarithm recently many vast fit law pass seminal paper test population half law r binomial array tend toward law law detect anomaly pricing report finance indeed one put forward appearance particular variable rule special show invariance imply limit multiplicative datum look truly explanation notice likely law order cover magnitude normal invariance assumption view mathematical random law link circular express transform signal expert would smooth implication explicitly actually explanation relate r formalize probability henceforth example scatter idea follow scatter imply irrespective surprising explanation several far simple property normality admit r uniform law special law datum various explanation uniform mod soon regular precisely regularity scatter
preliminary remark well approximated eqs analogously use inequality course imply f b v point bound bound proof eqs cn notice begin four form b bound c r c c us thesis b three one n observe f small hypothesis us u ad u degree eqs simplicity case treat manner side ib desire bound j j ji analysis exist generalization strong matrix discrepancy hold bipartite show subset discrepancy assume thesis remark let principal angle span thesis inequality x write x tu u f u tu therefore thesis dx perform explicitly hand f whereby thesis plus pt minus pt minus proposition corollary remark sufficiently guarantee reconstruction complexity massive proving statement spectrum matrix imagine customer available dataset customer movie pair rating rating miss suggestion make community predict rating error know one problem solve particularly important massive actual mathematical rating rank assign movie dimension dimension retrieval refer large limit away factor notion formalize es incoherence satisfy uniformly alternatively incoherence bound rating reveal decomposition iy singular rescale matrix rescale poorly contain column singular concentrated respectively singular operation column reveal diag entry per heavy column amount rank apparent important reveal heavy following let project residual eliminate small particularly consists locally minimize quadratic low furth column manifold well understand definite descent search practice loop cost instance collaborative sparse matrix achieve small entry frobenius matrix correspond usual section top value efficiently iteration iteration operation prove one exponentially calculation mention systematically assume procedure prove basic close approximated quadratic observation due necessary contain one entry consequence bound suboptimal collaborative use machine processing satisfy set heart compress collaborative matrix match entry entry thus incoherent prove correctly purely consideration reconstruct point counting treat realistic semidefinite program pose important provide convex science important fast rank hold short international completion prove substantially different point simulation completion fast degree indicate characterize point theoretical recent one study collaborative far uniqueness solution completion fast rank incoherent order formalize incoherence factor shall say satisfy apart difference normalization coincide assumption entry whenever depend reveal entry reveal probability guarantee performance well vanish notice transpose denote denote vector matrix sometimes first integer explain crucial consider matrix let rescale understood eigenvalue spectrum rank probability theorem inequality q use lemma want show basic belonging apply discretize I heavy discrepancy challenge bad enough definition light heavy z notice relate original discretize x max would apply contribution j subsection prove remark prove thesis subset column j entry rough size column event belong want proceed bound tail estimate l enough pz lm bound proof whose potential rl ij max mn z e follow thesis follow pz lm bound analogously finish contribution bind mh matrix entry satisfy mn iy notice bipartite set result mn ij iy n result adjacency bipartite hold inequality heavy recall singular understand max analogously minimize naturally view manifold important fact geometry calculation defer section recall numerical etc denote matrix orthogonal manifold subspace easy depend matrix mm gm n manifold reconstruct give correspond usual scalar w distance geodesic arc arc principal angle column useful fact propose admit computable form arc length express singular decomposition time vector curve one cost condition fully factor regularize row force remain incoherent take appendix compute kt kt kt x kt x kt kt way consequence c theorem discuss gradient pose indeed need repeat hessian lemma function numerical happen c stationary define lemma claim c gx triangular x claim observation gradient descent exact unique stationary point time write recall degree order term numerical thesis contain set neighborhood obviously equivalence canonical remark
indicate ref expect resample replica sophisticated resample section backward unable discrepancy run back particle c component focus particle thus cost complexity path main gaussian condition publish search support office science u department contract de ac national foundation dms tu department mathematics university california berkeley berkeley laboratory berkeley particle filter datum assimilation sequence function pdfs significant number maintain offer detail keyword particle normalization many must consist sde brownian scalar diagonal identity motion equation assume random simplicity evolve state nonlinear independent linear solution kalman distribution approximate filter posterior information past avoid identical particle expensive backward markov monte see alternative approach sample bayes backward monte carlo density parametrize construction idea sequentially nest conditionally subset sde explain already interpolation mesh use f nx nx eq function see solution mcmc increment know explicitly hand pdf pdfs check get pick obtain similarly fix normalization repeat sample previous obtain want pick draw remain iteration guess increment long case increment repeat start iterate vector leave become different also use strategy readily work satisfied iteration establish suitable course choose result variable context problem discuss demonstrate capability filter set point plane variable previous observer make measurement quantity scalar denote letter reconstruct position follow particle section show result datum discuss section increment known sde come normalization increment position reference particle pick independent reference present calculation unchanged increment normalize call say increment assume value brevity start explain observation like evaluate taylor series around define dy j dx dy normalization variable variance orthogonal connect find begin complete phase converge time short iteration create particle sample gaussian vary one number distribution density bayesian get probability sample respect phase cumulative weight exceed summation resample terminology divide size create great diversity strategy step create accuracy refinement future past tag probable observation make observation sampling often solely diversity particle boost problem sake completeness set step partial history particle occur history share among particle know last member project quantity remain deal slight increment increment slightly usual go intermediate correction observation half way reach connect replace accordingly look dy dy gaussian equality wish define forward impose observation time equality parameter increment separate equation motion converge approximate direction approximate multiply pdfs variance time create dx dy phase iteration phase forward step one
mixture copula universal approximation contrary dataset copula poorly model normal component conversely normal normal weak finding normal density factor marginal approximated show normal marginally adapt mixture normal expect mixture normal approximated adaptation concept outside normal mixture normal simple behave fast approximation copula multivariate copula cube marginal correspond transform popular copula implicitly transformation multivariate cdf marginal cdf transformation imply copula implicitly take normal challenging fit separate treat copula follow distribution marginal demanding give estimate function suit incorporate outline copula define normal copula mixture normal number choose normal implicitly copula component pose evaluate account distribution normal implicitly mixture normal section empirically copula normal skew estimator main I normal copula perform copula moderate need hold mixture normal copula bad normal depend simulation marginal mixture normal normal copula stand alone never simulation estimator appendix divergence estimate compare performance estimator kullback loss q similarly simulation number nc freedom tc normal frank copulas normal mn skew st aim capture describe briefly copula normal comprehensive set extended simulation normal report divergence multiply percentage increase copula ratio logarithm ratio ratio loss median report mixture copula generate confirm simulation normal copula perform similarly criterion almost always select component normal copula normal copula copula negligible copula despite loss normal estimator skew substantial simulation datum mixture normal perform copula generate one normal outperform depend density two reason normal fit well bad normal direct marginal indirect joint transform normal difficult normal conceptually report indirect density joint misspecification mild misspecification parameterization normal define impose normal parameter normal copula normal univariate density normal normal become highly parametrize get large normal copula marginal component medium large normal copula confirm analysis ability normal process bad rather likely group normal poor criterion generate perform multivariate easy cluster evident bivariate normal variance show marginal distribution copula normal component confirm need separate copula datum normal normal poorly poor normal copula improve copula still loss compare normal section highlight place strong conversely focus hard effectively idea fit datum fit marginally close make precise dimensional marginal I f iy h iy iy give sufficient condition define suppose multivariate density marginal lemma everywhere note give know suppose know note condition verify know density estimator marginal necessarily note complex multivariate distribution degree normal nonparametric density complex assume marginal specify normal normal therefore straightforward marginally adjust normal normal poor normal general normalize slightly prevent bad ratio iterate adaptation need example component normal considerably experience help tail capture marginal marginally metropolis proposal efficiency marginally normal normal marginally mixture normal estimate generate mixture normal mixture normal normal normal multivariate estimation estimator datum multivariate normal normal copula mixture well component three fourth component relate efficiency loss normal marginally adjust mixture normal marginally adjust normal mixture normal normals copula pure regression simple regressor unknown straightforward financial display moreover portfolio use practitioner excess asset health website http page ten validation copula skew mixture normal choose ten subset mixture normal three factor assume insufficient representation well model volatility finance decade construction treatment volatility distributional dynamic find exhibit long memory skewed logarithm realize display approximate study economic realize volatility report pay capture gain volatility volatility relative return daily realize period year daily return day bivariate vector lag specification capture cross validate sp explain normal apparent marginally normal copula disease cause concern molecular great develop anti expression level gene hour cycle expression process cluster number multivariate degree freedom estimate result normal bic ten component model none eight cross subsample marginally normal average copula mixture normal multivariate identify modification simultaneously well marginal challenge able perform thank pt pt axiom conjecture
sampler metropolis hasting yield algorithm include ease mcmc implementation empirical full associated wise markov particular connection strategy ensure existence mcmc confident identically sample hierarchical one maximum estimation mix fundamental carlo metropolis update proposal acceptance create state chain proposal particularly lead optimal kernel update variable sub block choice unclear advantageous component correlate target distribute version optimal hand also show metropolis langevin updating use consideration rate borel x chain value quantity value whose simulation markov approximate average n gx justify ergodic along condition existence ensure central limit existence sufficient batch mean method strongly ensure monte least tool confident independent much establish ergodicity metropolis full dimensional study establish yield geometrically hasting chain include well none full ergodic many updating example ergodic com establish ergodicity scan metropolis walk fix walk step perform gibbs sampler geometrically scan sequence sampler probability ensure uniform ergodicity strategy despite none metropolis hasting sampler especially two fix state particular connect scan sampler scan random scan develop condition ergodicity component metropolis practically relevant gibbs likelihood empirical sampler counterpart component wise sampler support wise practical technical fundamental combine kernel mix markov mix eq kernel preserve invariance ib b dx iy satisfy conditional invariant correspond elementary correspond hasting dirac delta wise irreducible combine scan eq admit q another wise update easy order composition use combine order clearly special employ satisfy com mcmc focus deterministic scan sampler goal rate sampler mix brief description establish existence chapter borel p nx j j ergodicity small ergodicity begin sampler component ergodic establish ergodicity ergodic ergodic wise ergodicity follow case ergodic ergodic probability component chain random walk chain block expect produce markov hand full hasting ergodic component I algorithm independent proposal case sampler uniformly typical accept reject truly think extend proposal update existence ergodicity argument case able give pair together ergodicity describe position markov additionally nan result define suppose exist eq ergodic ergodic selection follow corollary indicate verify proposal ix ix dp condition amount require jointly almost observable th k normally analytically challenge monte carlo monte maximum monte unknown require chain independence sampler proposal full proposal marginal invariant ergodic performance geometrically ergodic walk sampler concrete suppose geometrically nonnegative call metropolis gibbs markovian invariant se markovian let nx ny geometrically ergodic geometrically connect straightforward ergodic geometrically sampler condition yy lx effect know know nx p satisfy density hyperparameter posterior observe generic conditional report gamma gibbs sample gibbs sampler update conditional markov say theorem establish ergodicity row n chain gs gs ergodic geometrically ergodic gs consider performance full efficiency chain geometrically ergodic central limit valid confidence interval quantile chain sample integrate autocorrelation carlo random provide size quality estimate assess mean replication quantity efficiency estimating chain move square jump square successive denote jump chain summary graphical summary trace take picture examine consider subject subject full coefficient q next definite matrix model gs geometrically gs comparison focus rw use normal equal rw acceptance rate know rw geometrically simulate value markov prior k kk geometric gs variance asymptotic run rw substantially sampler rw rw gs mixed middle show half width integrated act relative gs rw gs four simulation reflect half act size rw half act rw effective carlo replication rw gs take obtain rw ratio gs give standard notice ratio great rw sampler rw gs consistent discussion sampler explore sampler gs mixed independently distribute introduce current condition four sampler ergodic rw sampler normally distribute proposal metropolis sampler proposal ergodic compare implement carlo call q likelihood effect set table chain density define entire markov asymptotic denote consistently mean generate rw logit section implement skip report implementation simulate chain metropolis jump proposal determine minimize autocorrelation result yield plot update panel analogous appear panel four result rw significant autocorrelation panel mix rw panel appear sampler suggest conclusion entirely closeness sufficiently conditional worth recall theorem chain illustrate implementation time act panel estimate replication standard
approach wide sm sm c sm outperform et al et relatively presentation label solve box constrained mean method iterate regularize constrained programming appear problem apply instead matlab code terminate use absolute approximate end cycle consist coordinate descent code terminate accuracy see pp experience criterion terminate relatively set obviously thus fair termination detail duality word cycle consist terminate gap since inverse need reasonable duality gap iteration iteration intel ghz machine instance present table sample four eight cpu column substantially also outperform small termination hour code number five maximization hour except scale one hour well objective instance scale increase digit rr rr rr time c smooth concave maximization approach variant show substantially latter study et et subsection impose associate view iterate would nesterov technique compare though outperform optimization preliminary occur highly analyze behavior sequence hence k completely open smooth heuristic nesterov general max written matlab www code suitably plan extend eigenvalue like nesterov scheme author two greatly improve pt theorem remark author support discovery grant strictly concave maximization nesterov iteration primal sparse approximately solve outperform compare approach namely nesterov block method sparse covariance smooth outperform method variant c k nesterov convex problem close method final proximal optimization concave admit smooth convex dual counterpart result find dual problem one give impose estimation apply determine simultaneously discover despite popularity numerous real reference therein combinatorial technique lin show solve norm penalize author efficient order nesterov approximation scheme show iterate quadratic theoretically release paper variant coordinate square programming appear paper attractive optimal substantially outperform compare randomly instance show nesterov substantially mention smooth maximization interested propose smooth briefly solve penalize maximum estimation optimization compare smooth optimization randomly finally conclude remark matrix semidefinite write semidefinite otherwise frobenius identity entry determinant eigenvalue symmetric denote space endow denote functional endow dual operator concave non maximization strictly every conclude convex give endow arbitrary norm continuous assumption suitably solve prox strongly modulus generality nesterov approach smooth concave sd fu u du fu fu kk algorithm nesterov sd k minimization ready maximization special smooth algorithm termination apply minimization algorithm exceed q function xu kk imply fu fu u xu fu fu fu u gx conclusion mention enjoy addition propose give nesterov smooth except former prox subproblem every prox subproblem cost smooth subsection denote clearly know prove strictly indeed since continuous therefore strictly saddle together strictly unique sequence minimization statement compact suffice convergent subsequence convergent subsequence otherwise one convergent u f desire immediately covariance selection subsection solve relation together et bound derive handle eq expression derivation scalar one show discussion observe rewrite define therefore complexity interior method al nesterov scheme lie endow norm concave modulus conclude q denote decomposition show problem therefore suitably give proximal modulus solve ease aforementione smooth minimization algorithm problem smooth xu sd fu fu fu u kk fu gx complexity algorithm solve hold easily follow fu case iteration algorithm smooth accuracy non smooth approximated smooth continuous nesterov apply perturb problem problem iteration problem et block coordinate iterate box mention rate global theoretically moreover reformulate suitably ip nesterov bad initial newton optimal eigenvalue multiplication cost find superior ip small subsection nice theoretical ip nesterov block performance attractive section computational concern indeed computation new termination use know use termination moreover termination one despite advantage shall mention complexity termination useful termination criterion usually fairly know iteration solve complexity drawback dynamically update easily observe unique problem ideally unknown generate asymptotically view know asymptotically approach generate introduce notation active inactive let give fix iterate find inactive kx kx kp k recursively generate inactive fact imply termination replace accordingly termination criterion aforementione convenience presentation omit subscript em covariance choose
series different curse consider determinant cover determinant correlation amount uncertainty curse matrix cholesky eq innovation uncorrelated turn former strong location nonzero recall triangular great eq recall mild average say easier uncorrelated apply hc power apply argument fix positive integer nk plan polynomial interesting phenomenon hold sequence tend sequence apply standard hc yield result learn apply hc yield hc case higher powerful begin fix diagonal zero elsewhere vector coordinate elsewhere compare nonzero coordinate entry coordinate approximately function singleton hc approach strong hc sensitive signal expect great write normalize call hc comment calculation nb n n nb mean stronger stronger hard selecting must mention case decay diagonal nonzero large strength significantly strong preferred modified turn behavior hypothesis large bandwidth suppose reject cut value fast sec suggest select summary lower upper reasonably investigate range dependence strong effect estimate relate still could comment represent series characteristic datum long period record use deduce evidence early detection communication maximum great constant multiple uniformly equal p conventional converge zero consider signal amount acquire increase would involve genomic difficulty information dependence genomic quite argue correlation decay base et figure upper tailed genomic possible effectively expression genomic case signal readily independent randomly distribute note follow problem among location integer distribute distribute among integer pre dependence place assume negligible toeplitz truncate toeplitz k nf first value definite putting see toeplitz matrix convenient asymptotic suppose constant known inverse asymptotically toeplitz generate known result compare combine direct thm toeplitz symmetric satisfie hypothesis asymptotically error infinity bandwidth converge power curve plane uncorrelated region current view corresponding region uncorrelate region rectangular region see enough stand correspond uncorrelated cluster signal generate randomly probability section investigate appear whose location strength right follow shift position backward add signal comprise cluster consecutive g toeplitz density consider eq equivalently view entry expect symmetric hypothesis merge nan converge zero weakly investigate variate equal specify location without display slowly decay calibrate place first boundary turn definition assumption secondly significantly see hc open detection adapt hc optimal key idea correlation three relatively next toeplitz lower diagonal elsewhere additionally entry let equivalently asymptotically toeplitz detail introduce converge coordinate therefore expect consider model display strength merge error converge bandwidth power scale compare hc bandwidth denote hc hc hc hc correspondingly include fix generate randomly otherwise explore parameter setting describe take detection cf datum apply hc hc score nan hc way report type ii error possible cut second percentile hc score hc correspond exceed threshold display cutoff save threshold report power cut asymptotic moderately recommend instead leave three blue display hc hc hc outperform outperform hc large mean strong correlation detection increasingly increasingly hc score small hc hc mainly definition hc hc bottom axis dash display hc hc hc bottom cut percentile hc correspond panel dash green display hc hc hc choice cut hc consistently outperform hc hc consistently outperform hc cut percentile percentage bottom display display correspond hc hc hc toeplitz matrix generate experiment sum report hc outperforms hc hc hc investigate hc hc hc take type hc hc hc improve investigation case omit discussion extend hc hc play curve correlation toeplitz upper wiener interpolation therefore plane hc full fall interior neither call optimally hc performance hc explore hc paper relate mixture work relate low focused proportion hc situation relatively feature useful contribute weakly relate hc classification work focus unknown available even stay current progress show matrix estimate situation approach perform well error experiment hc hc interesting explore correlation challenge improve recent aforementioned b derive way incorporate raise leave proof theorem omit subscript confusion independently show monotonicity hellinger function short hellinger write old integrable dx hellinger combine theorem hellinger distance infinity equivalently eq compare model establish model x tends prove note define observe generic constant c symmetric great norm view greater write converge inter nonzero disjoint simplify hellinger summarize lemma lb state theorem put reduce proof location draw replacement way hc establish hc short n distribution monotone family fix x x consequently tends zero infinity put let survival r n q use proof ns n follow remain detailed state inspection matrix theorem diagonal entry decay condition cholesky factorization cholesky triangular entry lemma diagonal decay decay remain proof hellinger tends infinity small g eigenvalue hellinger suffices hellinger infinity g hellinger uncorrelated converge remove negligible hellinger distance combine hellinger denote divide sec cluster nonzero equal recall cf r nb matrix bandwidth show negligible except lemma last claim cholesky constant continue hold side proof take operation fixing define constant form row direct calculation basic algebra combine give k follow algebra give sequence infinity location signal arrange asymptotically vanish tend fast event define uk calculus last uniformly infinity since negligible inter less decay small therefore calculation n ks small moreover inequality independent sufficiently derive constant page use deduce bandwidth except probability note assume wise bandwidth k thm jt argument algebra independent therefore statement infinity establish note sub close th b norm norm tend zero paragraph proof theorem lb verify location triangular u note hellinger law introduce index distinct np apply restrict old inequality desired consider combine claim follow fix next distance write k k great follow hellinger bivariate satisfy dr proceed derivation inter q combine shall side near lemma first near two set index first one candidate outside generality independence pair q third equality independence use cardinality elementary show q recall deduce last exceed dr n cdf survival claim show write unit give old recalling direct calculation c view dr statement right side converge infinity need tend infinity model nonzero strength randomly draw replacement write direct q chebyshev identity deduce compare statement sufficient definite autoregressive structure exist clearly show equal j kt kt know suffice j jt er k jt j prove discuss separately j k k ts tt zero claim follow directly derive kf q er x x strictly combine acknowledgment would thank extensive van help proof also thank david gr theorem section nsf award dms high detecting signal noise reasonable setting nature exploit correlation indeed accurate level advantage decay rate toeplitz class introduction hc could hc capable optimally signal weak estimate signal al relate white make
row x iy iy apart normalization well regularization introduce technique one regard justified differentiable riemannian mm gm gradient descent generality shall maximum far uniformly subset correspond revealed column work model reveal independently since model allow vanish shift universal numerical projection map subspace frobenius norm sometimes indicate integer guarantee appropriate incoherence present provide reader interested go generality rank define represent rank reveal let max cc number entry degree contribution effect miss stress crucial achieving guarantee key satisfie matrix generality jk r incoherent rank hand discuss rather couple matrix variable zero gaussian latter basic main norm model far elementary column represent high regime independent cr c minimization theoretic analogous apply require point mainly indeed rather figure average rank rank letting noise take take relaxation bind iteration small root indistinguishable rank theoretic root manifold easily generate trace rmse evolution many error two rmse later error decay exponentially converge perfect reconstruction still complete include real next main guarantee stress strong incoherence concerned semidefinite front small improve several observe entry norm frobenius uniformly random entry provide subspace correct lie precisely dimension constraint project least error estimator show side gaussian oracle semidefinite finally deduce e optimal study review recent application collaborative study introduce restrict boltzmann machines rbm learn intractable performance approximate use argue collaborative consider descent residual justify lead factor gradient also record line quadratic factor basic sum square residual stochastic obtain convex provide nuclear singular regularization quadratic factor review norm regard convex surrogate prove strong implication immediate present correct quadratic rank try establish implication promise counter intuitive seem describe fact actually full exploit treat column probably row stick reveal non binomial different column large positive made want index test computing conclude singular crucial prove max n value matrix quite large singular summarize bad normalize phenomenon explain crucial consider decomposition apart trivial rescale matrix great max understand exist large cm e max max proves function riemannian controlling point reconstruct two manifold tx manifold optimization du f minimum two bound constant happen c hypothese e far follow enough verified sequence generate appear make modify constant appear statement corollary eqs eqs x x together x max term upper bound get claim constraint rescaling remark reader next k non increase thesis n x u enough f x twice converge x eqs imply z corollary noiseless reader convenience simply triangular proceed get cn cn replace e cn get desire corollary putting c take bx triangular instance u analogous bind trivial velocity tangent x expression obtained x get absence left upper use proceed analogously u inequality claim achieve e maximize correspond analogously generality observation differ side eq imply also tight bound norm random I function lipschitz inequality square function zero gaussian tail z e possibly variable jensen inequality matrix thus enough symmetric variable I column positive bind sub result independent variable choice tail apply chernoff analogous substituting z get
realization plain evidence exponent available provide consist hypothesis composite sum composite nuisance integrate away nothing uninformative uniform change condition hypothesis eq calculate analytical limiting exactly time distribution gradually transform limit illustrate find kullback leibler attain sum order divergence scale sum transform simple result upper log n log sum amount always log likelihood hypothesis favor latter nan desirable sum value sufficient exponent simple way value assume uninformative give expression maxima maximum assume minus reciprocal evaluate find likelihood permit quality state nothing bad bad detect fs middle focus include rescaled range local respective variant ssc mild accurate term decrease estimator online quality generate middle question arise ever fs fs corrupt importance short give fs mask away contamination contaminate autoregressive move average drive sample plot series presence go plain leave different proportion justify claim series fully adjust main summarize stationary anomalous evy type additionally distribute estimator enable portion two make valid interest motion preprocesse scale universit exponent assessment analytical drawing inference fs technique exploit result compute accurate characterization support scale regime close outperform time scale fs fs nature conclusion domain assess exhibit fs address exponent often verification look establish fs fair see heuristic estimate constitute assessment contrast discussion distinguish hypothesis rely series think fluctuation perspective generate stationary process normally series one walk associate exponent normal gaussian noise sample order reference diffusion scaling priori
ask nh nonnegative node single ensemble constraint optimization small node pick select ensemble receive zero sparsity zero ignore root always include equal set solution nonempty solution solution minimal amount add form simply weighted point fall fall new denominator define contain demand root nodes choose converge nonempty region unlikely node node root response reasonable observation fall select node nh initial turn insensitive long order node follow essential maximal interaction minimal advantageous competitive predictive fact maximal show empirically node solve get clearly costly hundred example calculate practically overfitte first attempt maximal node keep interpretable maximal factor choose assign impose could choice yet result across remarkably generate generate alternatively fit seem insensitive choice require node predictive initially forest rf ensemble fit rather order minimal add node choose one state qp qp solver efficient suppose root clearly fulfil actual nontrivial node satisfy many nontrivial basis arbitrarily choose orthonormal guarantee uniqueness term constraint price pay computation svd however remain qp implement language explicit take second ghz processor ram nh adaptive smoothing doubly symmetric value doubly fit training g g hence put define ij remain sum follow g q complete ensure irrespective size mean fit node weight hold convex real n minimal node size follow ij j sum nonnegative less positive strictly strict lemma last obtain positive replace pair observation member nh node might greatly exceed substantial forest build idea interpret forest ensemble nh since explicitly average bag possible possibly hundred consist turn variable involve measure ensemble despite computational nh interesting stacking weight classifier minimize weight stack tree nh trees spirit algorithm interpretability indicator variable prediction combination exactly define style put coefficient variable nh nh essential nh inherent nh select weight whereas particular node magnitude response nh breast patient fall nh response easy relate fall group group power ensemble seem well nh experience high nh cope complementary ensemble make rule currently build either node exist tree ensemble forest nod various center observation improve nh outline nh imputation sample fraction number sample row equality stem within surprisingly inverse thus fraction least extreme constraint effect fraction vector node beneficial unless look predictive accuracy nh sensitivity size fit validate choice penalty fit two poor structure default contour contour line fit tree forest fit predictive contour noisy variability default throughout node pick forest fit clean contour plot forest contrast tree across shaped fall broadly speak uncertainty environment precise behavior thousand perturbation well sample relevant section parameter gaussian note outcome hence explanation response temperature change period scenario area node weight node mean observation axis subset node fall annotate simply x annotate type contain receive axis training simply fall area sample happen fall mean pt four coefficient parameter belong node impose ignore new figure simply annotate nh though nh least tree interaction nh get interaction breast clinical variable tumor apply nh rf people tumor position sample patient fall one patient average patient patient assessment example characteristic nh breast select patient within patient tumor horizontal patient vertical plot nh belong annotated node main factor interaction select patient tumor tumor patient weight across people group train nh compare validate seem low label forest drop nh maintain completely variable concentration know diabetes median house price census measurement uci machine radial velocity galaxy contain gene product measure production gene essentially genome gene relevant gene nh could deal time part half nh employ cart ensemble nh select forest without tune node cv tuning remark nh forest fine tune know nearly nh rf boost tree depth weak optimize test datum record split available diabetes galaxy regularize estimator fraction well ptc diabetes machine galaxy solution table unconstrained estimator always solution average node apply improve advantage unconstraine regularization nh data estimator good additional desire extremely aim nh ensemble forest nh share ease interpretability simplicity node response nh overlap observation member predictive often nh interaction tree lack tuning thus nh forest seem comparable ratio seem nh drop nh simplicity arguably interpretability nh mix monotone nh deal without imputation censor forest forest associate stein helpful comment suitable predictor classical classification tree understand tree yet aim interpretability combine extremely initial generate randomly response observation identical new fall role nod weight optimal handle interpret predictive observation covariate trying predict covariate tree attractive understand partition simplicity notion tree subspace identical node rectangular eq subset support leaf intersection leaf correspond tree vector prediction observation error loss try partitioning equivalently minimize empirical complexity typically tree improve bagging popular ensemble average allow observation forest tree ensemble
satisfy property test datum gaussian mean alternate correlation simulation dataset perform error depict roc curves vary correlation calculate roc curve correlation datum experiment power method mining row randomize margin dataset threshold use list mean deviation dataset mean test combination dataset value large store property depict pattern control level calculation limit intuitive conclude reasonable topology randomization extend graph individually dataset calculate depicted ccc number frequent similar value plot level dotted line randomization especially test test generic mining dependency mining mining scenario make adversarial toy consistently argue serious limitation significance study scenario control fdr exploratory looking would detailed test fdr real also powerful control desire avoid negative randomization hypothesis hypothesis simultaneously mining assess frequent entail type generic algorithm control performance show control maintain power keyword test mining address statistical significance produce mining method whether nan randomization sample specify original dataset discard recently randomization mining margin matrix preserve matrix traditional interpret discover frequent define fraction dataset significance know advance hypothesis frequent false positive call significance choice collection frequency user specify apply significance pattern though evaluation testing simultaneously inference exist statistic tackle correction assumption dependency structure within common test likely algorithm mining exponential attribute pattern separate hypothesis naive would pattern due overcome example one frequent another specific association rule split fold half half significance pattern hypothesis work partial feasible find frequent subgraph completely association bootstrap limitation association bootstrapping dependency course sense mining contexts contribution randomize multiple testing suitable verification validity power provide section contribution proof summary method read paper end pattern set universe still define randomization one nan hypothesis intuition extreme dataset definition define datum mining pattern apply frequent mining subgraph mining general unlike restrict frequent dataset could could could frequency margin nan sample swap randomization decide frequent output statistically test shape follow within discuss detailed derivation experimental first base value randomize equally nan pattern return sample pool p dataset let return define pool eq nan extreme become equally control conversely treat denote sort value I error formal obtain adjust read section check calculated case property satisfied testing correctness result remainder ignore pattern theory testing reference test simultaneously know advance respectively unknown statistic statistic level statistical hypothesis nan hypothesis reject sample call negative call acceptable multiple hypothesis hypothesis significant incorrectly count correspond error positive significant significant hypothesis multiple observe count I way acceptable rate even type fdr control fdr control depend application would example fdr hypothesis multiple method adjust test value control testing control understand implement adjusted hypothesis use powerful slightly neither dependency control absence false nan hypothesis eq property test different encounter pattern risk pattern extreme statistic value possibly pattern might satisfy property vary visually plot never diagonal admit property mining constant pattern value prove property identically h fy return consider fy fx simple output number property restrictive mining pattern assume sample number uniform interval elsewhere occur output pattern probability denote happen output property assess randomization analytically property violate illustrate plot show empirical nan different however method may ignore theorem testing test arguably multiple testing mining output pattern pattern control significant small satisfie since q framework contribution broad zhang attribute calculate transaction attribute dataset break store dataset finally value show calculate use check significant rule et rule type error conversely strong rule might association variance association test random original dataset pattern dataset sort error level generalization control al define bootstrappe bootstrappe multiple directly assumption original statistic replacement difference sort decrease store contrast reasonable rule group transaction group disjoint calculate respective contingency contingency adequate
collect take used view member approximation simply contribute pl propagate gp classification essentially aggregate incorporate hide analogy px I necessary propagation upon information sampling may via setup prefer local pl propagate follow use mh approach class underlie prior drop index extend may accept ratio mh acceptance comprise acceptance acceptance nothing way loop obtain illustration real concavity third sign gps pair initialize round minute take hour less last minute pl exponential three gray open circle location circle posterior predictive surface test leave ix input circle misclassifie solid red one near efficiency pl less multimodal logit probit subtle versus computational offer comparison obtain every pair pl pl class pair run ht right progress pl section initialize size track additional point optima alternative right track maximum ei decrease except magnitude spike good diagnostic pl search take minute identical figure useful heuristic boundary exploration irrelevant influence base undesirable way versus heuristic mean pl class al entropy design figure design pl particle pre misclassifie point compare static section run long e candidate greedy near explore decrease nearby location result show smc pl contexts argue also well suit arrive component significant aspect subsequently context mcmc ill suited online acquisition arrive pl propagate maintain another relevant independently contrast mcmc inferential proceed serial get smc pl approach careful asynchronous implementation propagate particle resample evaluate package parallelism loop calculate propagate promising business method exploit sequential monte produce relative alternative ideal iterate new point attractive smc approach optimize boundary online key monte nonparametric sequential highly flexible nonlinear regression classification gps build point gp fit goal keep save gp via design utility design criterion alm gps task ei stationary mcmc determine circuit device explore label information new drawback tailor nature guide early iteration may carlo pl analytically rao quick fit heuristic calculate particle approximation smc pl gps establish right together sequential design remainder paper review pl strength pl pl fast al software implement specific illustrative covariance cx yx yx separate work parameter inferential thus problem likelihood response vector row covariance clutter drop subscript context mle via profile infer numerically proceed specify mix hasting mh multimodal hard crucially pl response degree classification use collection latent particular variable determine independence assumption softmax add degree expand proper linear take gp via scheme predictive pl algorithm irrespective detail classification hard practice fix say introduce undesirable simplify notation shall throughout implementation monte smc design inference smc sufficient information uncertainty time approximate sufficient smc update particle prefer gp pl derive decomposition suggest particle index pz ps ps core pl propagation resample propagation statistic ahead pl accumulation however setup extent firstly design keep possible gps poorly regardless smc mcmc etc drastically recommend secondly analytically integrate pl class smc note initialize explanation implement pl sufficient particle propagate classification smc use number maintain quantity store sufficient comprise define necessary prefer presentation efficient implementation initialization initialization initialize particle sample take proper word must start later work I much large calculate thereby obtain improper parameter sensible exist weight dynamic conditional probability z I propagate propagate update correlation pl directly counterpart propagate prefer propagate propagate liu gp represent equilibrium sensible tune mh proposal likely acceptance far decrease mh propagate whereas resample predictive global method follow delta introduce smooth encoding process noise parameter upon input one plausible greatly restrict exp mh proposal uniform slide center around work baseline may locality q capture low fidelity noise cosine enough fidelity pl particle improper mh round take second implementation pl mh take minute take take second fast round update exploit drastically line average right particle term credible student surface show fidelity surface find
exactly tp method base small storage comparison isometry iteration solution noiseless noisy case since method tune method large involve coefficient partial fourier art algorithm regime algorithm regime significantly outperform furthermore snr reweighte ii element converge fast present optimistic since algorithm portion support contain element also quantity snr mean criterion square absolute coefficient raw coefficient measurement use zero converge run noise measurement noise run run ii package package use fourier randomly l describe output square pursuit iteration iteration solution converge result quality db lasso pursuit knowledge outperform value quality produce reweighte knowledge also term visual keep thus actually perform mean observe higher choose fast much keep performance sparse affect test construction algorithm improve shannon remain unchanged set general construct small storage member frame reconstruction matrix decode art interest outperform fouri various research area pass schedule message pass amenable another may degree adaptive decode area finally tree decode may change accordingly improve schedule pass check code schedule neighboring node pass neighbor free posteriori schedule serial node message neighbor propose schedule base intuition message vice versa schedule mark edge connect inactive receive mark edge message information active index income message node active valuable reliable calculate edge schedule tend perform thm j school sciences edu compressive signal matrix decode presence additive noise frame matrix storage frame demonstrate significantly outperform art db range low frame sum expectation mixture said equation interested come estimate observe perfectly recover linearly uniquely np criterion noiseless various belong contaminate hamming theory exploit ensemble decode minimization complexity regularization satisfy property isometry program eq parameter selector limitation improve expansion represent estimate work matching pursuit variant subspace pursuit measurement rip noiseless also offer similar algorithm multiplication ensemble signal noise yet direction compressive apply compressive performance study dimension decode generalization various idea outline next introduce property introduce concept decode criterion finally conclusion future frame form frame formally non element redundancy restrict one dimension consider code bipartite sense graph show variable factorization undirected factor factor depict sum product exact computational increase issue one treat look subtree connect represent subtree associate summation yield message node node connect include go node factor start leaf whereas leaf description parent expression way calculate write structure factor modify every twice many calculation calculate marginal case could posterior marginal code row relationship code bipartite disjoint vertex contain node satisfy node zero bipartite graph graphical call connected sum purpose decode represent node code without tree product prior frequently ignore create interval compare accord jeffreys peak even demonstrate highly origin coordinate axis far enhance sparsity coordinate commonly expectation em find hidden parametrize em initial distinguish argument maximization respect random maximize em algorithm posteriori factor decode vector I use algorithm em em stage q px underlie density maximize summarize pass schedule really behavior alternative pass schedule maximum pass schedule attain schedule indicate fix note exactly construct soft stop schedule initial index message come check message incoming k criterion node enforce ii algorithm message likely zero denote develop compressive pursuit measurement stop criterion message schedule large index step absolute indexing vertex message decide decide make decision vertex calculate index index intuitively reliability maintain list large merge element decision update force e keep
differentiable map real number nonnegative nonsmooth subdifferential alternate penalize situation subspace alternate kullback proximal proof adopt space precisely rr set next write q alternate lasso proximal follow cluster kullback interior subset tucker point hold assumption belong without mixture check likelihood fact inside orthogonal verify restrict iv follow p ik probability let clearly differentiable around satisfie follow set number tucker optimality result tucker satisfy cluster et definition pt pt many problem set self regression penalization component mixture alternate penalize prove present obtain see maximum keyword finite regression widely great field pattern recognition biology finance mixture model comprehensive application mixture model traditionally associate notion mixture model independent identically multinomial random index latent estimating maximize log eq semidefinite case many practitioner notice maximizer log function viewpoint procedure mixture mixture available package instance question right local optimizer sufficiently many model biological propose estimation size small aim certain robustness spirit idea express simply restrict span obtain impose simplify instance likelihood condition whole formally em step encourage monte showing assume bayesian study chain I mixture approach regression simple easy strategy give involve stay small formalize subsection simple idea variance ahead notion follow instead stay impose combination sparse difficulty approach vector seem hard rely concern simple regression formalism estimate enforce unobserved cluster proportion maximize eq parameter obtain need accelerate methodology average iterate algorithm trajectory follow restrict multiple matrix detail method summarize convergence analysis alternate input problem cyclic iteration formula index address question testing simulate alternate em simulate datum generate permutation index recovery present carlo scheme expectation uniformly run code going obtain correctly obtain alternate uniformly cube example close correctly obtain cube box vector uniformly c cube cube correctly recover index similar initial choose standard gaussian mixture c recovered monte show increase space correctly recover standard size recover index output classification expectation increase recovery good dimension quite likelihood gaussian especially classify penalize compare investigation detail context goal propose base regression satisfy
exploit estimation follow prediction idea provide account similarity describe illustrate ridge along contribution adapt lasso penalize least study aspect aim lasso probability estimator important task contrary ridge pick provide way choose penalty selection organize procedure explicit form predictor present discuss generalization procedure illustrate performance sparse briefly book prediction predict label pair observation describe augment sample relative notion quantity lie associated explicit score adapt connection testing consider hypothesis permit py recall concept search ny new py yx new yx confident prediction reason measure procedure validity issue power simple eq note perspective asymptotic validity cumulative rather cumulative provide accuracy adapt shall point case sparse estimator namely nonzero originally linear label observation estimation estimation produce large sparse equal naturally define set lasso modification lar approximation lasso transition write sparsity obviously estimator cardinality sign evaluate finally variable invertible characteristic detail e begin end ordinary least square ol mention remain interval highlight lasso piecewise lar help regularization linearity lar variable index linearity lasso interesting encourage use sequel lar mind analogy lar tuning decrease reflect lar algorithm select yy estimator augment k n k sign equal purpose observation correspond let real k k make loose generality multiply whole collection propose choose particular study result augment satisfie actually exchangeability fulfil element depend believe predictor well present propose driven predictor explore methodology drive base technique attractive practical discuss validity select small measure validity validity validity union bind consider suggest validity construct modification lar kk associate thank use predictor fact piecewise form consist interval belong lar lasso predictor I kb nb I n km u nj b predictor w lebesgue say lebesgue construct valid selection construct value bring illustrated predictor true suitable predictor small predictor criterion add illustrated hard lar tuning aspect considerably reduce version could lasso lar th value correspond vertical line separation unstable generalize selection net linearity response score piecewise parameter computational effort lar estimator vector let sign base dataset soon construct estimator consider obviously elastic net predictor net modification lar predictor correspond elastic net lar k replace component make obtain dependency note lar lar computation least one test way grid window lar permit considerably tune point lar construction predictor treat turn sparse predictor reduce present performance accuracy observe predictor embed varie see validity elastic select introduce section net predictor last ridge select modification lar predictor level consider specify correlate moreover describe example sparse highly zero separately accuracy validity log top iteration blue predictor mark bottom show iteration modification lar illustrate predictor follow iteration appear predictor drastically even length predictor time unstable happen iteration let aspect valid variation predictor observe lc example stop validity cf show table observe variation validity good expect point part gap way early soon predictor least ii previous enforce early stop rule small consider
strong experience process sde wireless communications rao implementation consist static length series case mcmc proposal tune accord pre chain keep adaptive proposal adaptive metropolis issue relevant pg potentially space function elsewhere pg sample strong linearity particle likely preserve smc leave ti adaptive comment relate particle equation rely nx approximate mode path space variance study involve state degeneracy filter mutation kernel regard rao acceptance use mixture expert adapt distinct region separate softmax partition htbp static g htbp r particle sir rao effect particularly visible mix high total proposal note vertical unlikely prevent mmse converge iteration begin concentrate path around close mode satisfactory iteration intractable smc abc algorithm tolerance simulate observation degeneracy path control see due restriction htbp require replace smc compute incremental weight online sample k ks incremental q n comment approximate abc sake brevity set
literature look distribution minimal model investigate early possible adjacency matrix reference lead yet calculation hasting conditional census degree mutual paper explore conditioning likelihood exponential largely mechanism avoid unconditional something dynamic minimal offer suggest proper generate exact distribution matter discrete family utilize appropriate basis explicitly unclear proposal literature reach associate generate graph explicitly make early paper notion focus characteristic ensemble estimation assessment search optimal node block term stochastic go equivalence rise see g determine idea random could predefine discussion blockmodel involve discovery attempt framework generalization focus issue restrict blockmodel comprehensive treatment analyze interaction far traditional detail science statistical blockmodel basic algorithmic detection heavily display connection block maximize statistical link maximize hoc related link block blockmodel rely intuitive equivalence equivalence imagine super mind adjacency belong connectivity run index run suitable rely pre blockmodel social stable protein block leverage equivalence consider green definition although among tight direct connectivity blockmodel diagonal block equal technical blockmodel blockmodel block connection direct block node summarize specify mapping array specify interaction interaction nod blockmodel explain asymmetric block pattern mix explain connectivity pattern blockmodel identifiability beyond characterize concrete example stochastic blockmodel process membership context dependent different membership interact statistically p membership equal characterize network may symmetric interaction blockmodel integral evaluate analytically simplicity exact option thing nod social nest variational variable contribution bring approach discovery blockmodel community modularity biased modularity discovery incorrect favorable community substantial compose likelihood correct exchangeability develop pair intuition low adjacency distance measure individually first treatment cluster heterogeneity node probability adjacency position low pair relevant representation pair covariate odd general formalism generate quantitative edge weight example link negative integer node inverse may explicit distance separate position reference suggest latent likely distance euclidean distance carry scalability address network analyze latent project invert practice often interest identify group protein identify cluster position infer allow joint cluster introduce gaussians approach come blockmodel membership node binary relationship per condition blockmodel allow hierarchical distribution belong position group uncertainty cluster membership mixed membership carry latent proportion former infer variability membership posterior matrix million node interesting computational covariate argue blockmodel mixed membership concept extract ultimately new hypothesis mix blockmodel bic fit major benefit mixed could formation confirm longitudinal certain specification refer specification latent latent nice singular connectivity latent number eigenvector interpretation among capture connectivity eigenvector interpret latent interpret tight micro community interpret enyi way phrase os r enyi branch intuitively branch start node branching keep grow intersect node formal proof pick node work pick node place list take list neighbor already consider distribution add chernoff connect belong detail please p chernoff binomial line chernoff process carry analysis transition mathematically model static translate birth death old node drop due link partly partly study gain popularity begin increase dataset long span rich mind chapter begin os r generalization time dynamic recently os r static model static snapshot network record different step process link though view pseudo dynamic discuss enyi view os process start convention extend assume get rich description discrete branching process particular representation explore enyi issue dynamic major center produce network result claim exhibit subsequent generalization os enyi construct degree dynamic pa design generate scale subsequent node pick accord multinomial undirected much early intend grow get page likely page oppose little result law empirically whereas os enyi allow flexible modification dependence create decaying lead law appear mat forest name empirical describe infect main network certain network network often point contrary lot attention distribution company plot log fit straight visually careful case suggests usually justify law ordinary degree except cutoff degree adjustment search cutoff effort law g example careful often linearity give metric whether graph link description generative fall study fitting mcmc framework notable kronecker multiplication started turn analyze fitting real principled thought sense world model os enyi produce previously edge node drastically begin probability construction move toward os r small network dynamic evolve world variation finite grid connect depend greedy path number work attempt perform greedy within search reach walk show converge size different end goal amenable randomly perform probability range result chain stationarity span range link optimum search series study discuss link typical small involve aggregate formal examine small assess fit originally world model snapshot web static direct however dynamic demonstrate newly add web web web page direct edge web hyper page regardless match content web link web page closely match content specification et prototype among link th choose follow prototype possible since generate particular derive prototype node goal remain appear model protein interaction mixture assess evolutionary dynamic protein routine posterior statistic review dynamic evolution protein interaction recursive enable principled markov process network first shall become markov tie family evolution variant specification change party change modification begin provide quick notation outcome conditioning conditioning determine future distribution know network outcome possible configuration network configuration take node flip opposite specifie arc employ simple model derive close maximum model reciprocal q currently edge one direct exist reciprocal add direct edge exist either complicated popularity change edge node dynamic orient orient intensity factor component control specify two edge orient interpret configuration difference orient statistic oriented moreover choice flip two formulation orient write general orient edge edge combination statistic look familiar indeed orient equivalent update arcs number arc ji arcs target ji arcs jk statistic assume correspond undirected undirected reciprocal oriented suffer degeneracy triplets networks degeneracy longitudinal distant defines intensity q oppose edge seek neighborhood would suggest potential oriented modify flip configuration allow treat detail please proposal dynamic network operating domain markov factor simple version unlike potential consecutive configuration q list ij ij ij ij may attribute distribution may snapshot dependency pair extended estimate hessian covariance pair sampling well behave static degeneracy recall link dynamic position distribute observation euclidean influence node likely edge citation author co ensure radius one noise radius follow position author propose latent position base multidimensional transform observe ambiguity align position evolution rich previous ability word author mode kalman dynamic procedure line offer explanation step enable state network dynamic collection explicit behind network another citation physics community citation acyclic node show completely unsupervised manner opinion reference model reveal something new modularity variable modularity validate deterministic centrality discover reveal deterministic show several significant drop age mean gradually complement contextual represent aspect life meet interact context time strength social interaction people interact choose accord node appear update weight pair meet coin individual weight update birth death dynamic capture basic formalize context context hyperparameter idea weight show various relation brief relationship capable term past cost right represent line quadratic dataset weight contain email aggregate simulate relationship cf per represent strength take article drawback lack form frequently come weight life rich mechanism step realistic ultimately especially additional context individual model analysis section quality ease inference rise social visualization automate degree popular package longitudinal platform technique effectively combine visualization kind review want tool network estimation computation network g need newly package program longitudinal package capable learn truth really million estimate drawback sensitivity point really take come size dense contain assumption important focus asymptotic network serious problem confidence estimate behave asymptotic problem growth comment briefly asymptotic lack assess address assess especially involve could form assessment effect associate problem asymptotic exploit useful broad number entire subgraph even random condition bring parameter cf bias early random subgraph exploit statistical frank focus binomial size question hoc relevance sampling network address adapt account sampling design date work sensitivity sampling share divergence topological property expect relevance consequence parameter estimate along question non survey exclude consider directly empirical survey implication subject justify assumption interesting open datum mechanism specification inclusion actor response censor vertex scientific treat task treat estimate prediction next work evaluate dynamic paper relational develop prediction www literature predict biological work discover link evaluation usually validation know link interest distinguish utilize dynamic fit within paradigm paper distant future dynamic model implication epoch time oppose refer cumulative circumstance care actual realization markov process social process finance address identifiability refer lead procedure mixture solution various different g solution blockmodel pre identify reference especially arise context link example mail cascade full author attempt model message text membership membership way kind combine evolve diverse social biology computer science economic subject trend model proposal point visual diagram influence pointing influence pseudo dynamic green dynamic indicate influence motivation primarily literature science concern addition main lack model statistical lack degeneracy early broad dynamic clearly static snapshot network continuous hand clearly dynamic seek evolve os ultimately snapshot usually either pseudo within category main direction network equivalence existence understanding model evolve accord markov intensity dynamic observe various time network latent dynamic advance modeling decade issue model create model realistic mechanism great acknowledgment united national institute medical grant gm foundation grant health gm national grant grant dms office contract computer science institute thank anonymous valuable comment helpful correction citation list thank correction wish give original life major interest model date back social early active substantial effort literature past decade literature physics computer online community facebook specialized community begin overview historical example discussion focus prominent static emphasize description interpretation description challenge scientific field network interaction pattern much book decade facebook work selective statistical social computer biology book conference publish survey would impossible far attempt chart year major gap effort overview deduce promise complementary overview exist organize axis article static concentrate snapshot dynamic concerned mechanism change network early single static static year recent interest brief overview give comparative select approach statistically literature derive seminal last study mathematics os enyi random paper one impact network develop mathematical incidence structure small world phenomenon connection shortest people complete majority experiment provide title play movie degree ignore due censor formal chain like analysis early description variation mail chain network build upon effort os enyi os enyi along papers os work number fix choose description might sequentially version enyi associate value connect trees component literature extend os popularity effect allow estimate contingency formulation generalization multidimensional paper demonstrate lead spread structure cumulative link full maximum approach appear example social network point report network could truly evolution relatively reflect computation assess fit form statistic area increase section truly evolve ask continuous many work network model macro description physics think os enyi probability intend give law rich rich back world model back distinct adjacent world utilize law include statistical physics detect phenomenon counterpart social link pick idea epidemic variation network book length process description dynamic exploit extensive literature already existence complementary property problem diverse notable assess physics consequently attention noisy network often extraction graphical law centrality either sufficient descriptor dynamic free probability send delay mail nature activity demonstrate solely bayes representative mail poor fit law show parameter key structural comprehensive primary example author input mutation model seven mutation well several decade grid degree degree requirement treat model combine blockmodel idea membership generative model way sized mixed membership resemble dirichlet allocation offer kind briefly model truly extensive theorem proof largely present literature mathematic network substantial science deal short diameter connectivity centrality cluster volume issue strategy modify beneficial search rare population detail neural connection recently computational tool pattern model relatively area study economic book excellent technical concept structure relational area probabilistic uncertainty net etc different meaning review give arise property representative uncertainty feature good exception population suggest bipartite old agent attempt simultaneous effort create complex often agent idea recent advance high simulation social become research strong interest security biological well counterpart interface artificial intelligence social science analyze behind origin start example dataset interested reader may wish often interpretation social arise something mean characterize orient dedicated mechanism testing tend interested parsimonious formation common explain dynamic several find thought well biological similarity subgraph among understand find subgraph also help genetic interaction heterogeneity lot network purpose de degree structure prior graph biological analyze latent cell relate discover business organization machine know science statistic relate question member dynamic exist connectivity machine network often information likely movie box office application crucial business pattern state predict preference preference research name medium attention competition movie company netflix company million team able customer rating high house information propagation many domain infection work finding assume focus disease spread quick dataset ground break construction result length complete chain phrase separation study interaction relationship elementary student lot focus study gene mail web citation citation history collection subsequently title average author website raw along author connectivity static analyze two dynamic reproduce illustration ht network static nature activity statistic summarize os enyi generalization statistical physics generalization law free exchangeable graph edge ultimately structure connectivity strategy science community response social enyi model modeling model generative evolutionary set dynamic interpretation generating process generative node edge node often usually instance concern absence edge relation adjacency henceforth graph mostly set node contain direct correspond interaction os enyi nod science actor tie largely science terminology random equal edge model network simple usually back examine os enyi undirecte involve interpretation graph likely equivalently induce specify every edge expect edge modern simplify analysis binomial os enyi change remain mostly component contain node contain node use branching chapter concept useful os enyi mathematical study actual essence approximately network poor fit provide kind lead focused random second identify select physics several describe exchangeable simple extension introduce weak exchangeability attribute help connectivity source follow generate observation bit string edge node pair generation weakly conditionally string perspective exchangeable simple step generation string probable edge graph explore mathematic arguably interesting specification node dependent family hypercube dependent probability node bit binary string node direct node variability exchangeable direct argument string bit exchangeable main probability edge edge corner hypercube fit definition homogeneous leverage branching process intersect high form soon string match graphical illustration matrix correspond connect panel graph component bridge three bit string bit unlikely infer string set fitting assess observe leverage connectivity quantify retain bit plot correspond histogram exchangeable algorithmic model summarize illustration graph alternative independently likelihood need give fit sample profile histogram sample model support graph bit string bit integer distribution bridge member match practice specify correspond instance protein absence
jeffreys numerically chain case estimator error copula increase environmental study also currently finance precisely copula copula risk measurement compute simulate asset model copula provide book methodology copula within bivariate function uniform margin paper denote copulas lipschitz constant bound fr hoeffding copulas cumulative representation unique couple cumulative consider sample treat present function setup parametric margin rescale show estimator comparison likelihood rank depend describe latter exploit equation ci k describe v use copula call henceforth margin identical since inherent copula moreover estimator review optimal small think practitioner aware study rescale call bayes sample purely bayesian case invariant monotone transformation construct sup copula exist copula parametrize doubly mean numerical evaluation topic estimator use metropolis much problem copula write mixture doubly specify prior specify weight information reason candidate jeffreys main contribution derivation jeffreys generally difficult mixture literature face column well nothing kernel estimator every construction use basis unity partition unity nonnegative indicator function bernstein li partitions unity doubly lemma straightforward doubly absolutely continuous unity uniformly spaced restriction indicator evaluation copula constraint first know cx v give upper old mu copula appear extensively function equation copula generate doubly stochastic doubly matrix indicator multinomial experiment cell count define estimator empirical copula consider matrix copula q concentrated doubly matrix adopt objective jeffreys discuss doubly specification polytope polytope computing mathematics follow let function pt copula w ij h equality fact model vc w mm result matrix reduction greatly enable paper w w jeffreys proper specify consider b v dimensional hilbert subset positive prior uniform sampler distribution polytope another decomposition von doubly decompose combination matrix extreme furthermore doubly combination permutation see permutation exist weight lie simplex dirichlet realization couple structure know transform follow consider pseudo describe numerically bayesian estimator jeffreys call metropolis within approximated chain repeat select direction every take draw uniform accept eq expression prior previous draw stochastic prior eq jeffreys probability contain radius doubly meanwhile cc bm approximation jeffreys probability radius doubly doubly polytope show pt figure jeffreys estimator bivariate evidence important jeffreys prior six copula u u bivariate correlation univariate normal cumulative function u frank family popular family describe cross margin copula copula highlight cross dependence illustration extensive carry part family space determine value first four family respectively corresponding part experiment margin situation equally space range seven square margin sample size estimator jeffreys priors uniform g maximize expression estimator numerically estimator order doubly value latter commonly figure cc family family family pt c family cc outperform kernel estimator near frank family increase fr hoeffding bind call copula almost dependence bayes one remarkable result margin result margin latter estimator mention result margin margin case unknown margin worth case especially go stay boundary select life probable phenomenon marginal empirical procedure plug outlier consequently invariant increase transformation margin
previous especially prediction close principled balance seem automate hoc could compete base methodology use home familiar experience adapt interest rate walk tool predict outcome simultaneously several outcome single etc observation year outcome year player year outcome rate position past could extended experience prediction home consider multiple argue possibility player population refine home rate sophisticated model player population suggest dominate player mixture non limit shrinkage towards population home run prediction bayesian investigate status directly primary home prediction also estimate secondary trajectory position addition evaluate position home run trajectory note represent major represent conditional conditional prediction however interested unconditional trajectory sophisticated censor focus home make assumption throughout predict know maintain fair current modeling like discussion cm major bayesian paradigm principled balance player share player improve method held limitation performance award top assumption past course balanced age project observe player truly par consistent answer prediction within current include sophisticated player set player similar player historical past player effect effect recent major recent heavily methodology overall method prediction multiple publicly database day fit model player within follow datum home run age home position home run exclude leave nine position third ss lf center cf rf home use major vast majority player give player home home run walk effort home binomial justify coefficient age trajectory spline position nine position team share player home coefficient since play particular team home run home separate team effect would examine aspect performance capture outline firstly age treat home identically create home position share home run represent place mixture term force status model age extra odd home year player move status maintain year year address past performance trajectory would small year player favor prevent player build past status indicator player specifically status indicator player status player year markovian induce dependence home player share player position differ position initialize start status desire consequence player must must need model age share coefficient age intercept normal identity bivariate indicator non variance hyperparameter specific use prior parameter entire gibbs standard transition conditional posterior imply represent year observe involve sampling status use sum hide k hasting step player position variance variance procedure predict home player external validation include home performance several internal choice via gibbs distribution home home wide possibility home evaluate prediction focus external home run ab ab home run home fair external advantage full produce comparable term mae compare prediction overall percentage home home total among table give measure separate age year versus player ccc players mae mae competitive examine player absolute superior scale examine player rmse mae superior performance sophisticated position well past method root old player player position beyond shrink position fit validation encouraging method player principle past performance advantageous among examine year decide home run question indicator examine attention subset home datum figure player year player home fact player year home ij player player examine balance age question old player age examine versus entirely year contribution prediction home find good suggest information evenly balanced prediction age imply home run age ball status year position differ position status consequence age ss year vs arbitrary contain curve graph imply value gibbs output examine curve sample see ss shape home fact home position vs status substantial magnitude home overall restrict player equation difference part range surprisingly grow home across position examine non position distribution intercept imply home run age dataset
reference satisfy diagram look set observe standard gaussian white space assumption interested confidence quantity play value value integral weak strong separability follow auxiliary sensitivity prior arbitrary pp b w ii schwarz pp eq fix complete inequality nothing choose b easily apply estimate situation use w hand choice follow consider next assume utilize finally choose continuity pp pp b pp hold v q r r k r index convenient lipschitz observe observation r obvious accord immediately pp r contain norm give couple depend obtain use weight normalizing equal jensen cauchy direct apply place constant independently r pp proof low lemma recall thank lemma section satisfie pp pp pp h special inequality scalar assumption previous measurement possible choose condition yield special force limit present section indice pp define familiar truncation appendix proper variable follow norm formulate quantitative assumption pp r pp pp pp example difficulty state work circle identify measurement measurement finite function limit independent variable white indicator jx hoc random weakly take converge notice measurement behave proper condition violate proper white motivated white noise actually hold see smooth weakly space h j measurement white function dual verify complicated construct scope wavelet detail comment elsewhere discretization smoothness jx jt projection subspace duality value n jx q ni green differential imply lipschitz smooth almost surely old fix e every strongly identity infer define variation let consider e end vanishing orthonormal random jt variation gaussian stay linear example let dimensional disjoint triangle union function triangle b weakly variable zero smoothness dt white measurement well define verify proper measurement formally speak green study field formula dimensional meet difficulty therefore measurable measurement almost surely quantity could subspace surely value change subspace unfortunately intersection realization cm lemma proposition inversion space prior department mathematics statistics university measurement realization linear device inversion discretized formula kn nu kn nu choose achieve infinite case discretization priori wavelet invariant inversion norm wavelet coefficient inversion inversion wavelet smooth white object apply physics interested find measurement domain euclidean device realization variable operator relate device realization specific space brain imaging equation describe surface produce external model inner linearly device measure surface determine unit orthonormal model bayesian inversion quantity know value choose projection dimensional range finite source brain finite virtual sense appear particular relate random bayes formula exponential white statistic form information problem mean mean differ involve confidence approximately chain carlo software ray electrical imaging general reference inversion require practical implementation imaging correspond field discrete equation effect possibility lead gaussian fall discussion model measurement noisy measurement fully priori posterior work construct bayesian inversion discretization consequently sequence increase motivated phenomenon computational resource necessarily reconstruction perform lead reconstruction discretization converge mistake different discretization finite represent use surely state come realization prove strategy involve measurement dimensional white eq coincide km general analyze confidence infinite process achieve discretization invariance inversion total preserve reconstruction posteriori total inversion discretize particular estimate regularization band filter derivative processing prior discretization discretization proof quantitative convergence inversion use wavelet two know estimate norm wavelet us review literature inversion dimensional space discretization study kind statistical inversion hilbert space specialized work entirely within computational point inversion early invariance however appropriate realization emphasize inversion discretization organize invariant bayesian invariance value convergence generalized banach denote inner hilbert stand banach specific space denote realization acknowledgement discussion anonymous improve paper ml ss support national contract group project section prior inverse give result later discuss deconvolution smoothness prior regularization derivative ignore boundary periodic generality periodic cover compactly case dimensional space eq space inner generalized variable gaussian value assume hilbert space smoothness also stands take covariance parameter operator prior prior almost surely let differentiable index white gaussian operator function value hilbert space zero space negative inversion place measurement mean location thus omit formal rigorously practical range pixel truncate enable device us turn need series expansion dimensional converge estimate correspond smoothness discretization subsection emphasize practical use compare close discuss connection minimize functional eq define form convergence smoothness represent vary however sometimes priori prior gaussian geometric prior analysis discretization invariant inversion replace discretization dependent space prior band use edge inversion compactly measurement convolution wavelet normalization arise naturally scale function fine refer wavelet range choice efficient term probability mean approximately use chain carlo sampler hasting e modify involve fast transform converge either involve prior distribution define converge distribution phenomenon rigorous space immediate recall banach banach dual algebra topology separability borel algebra measurable separable banach space diagram separable measurement gaussian separable operator adjoint unique adjoint power henceforth space well call white domain consider triple tw z hold complete discuss tend weak value let nested variable variable u example mainly conditional pd pd vector fact value expectation ready variable deterministic measurement deterministic coincide thus formula realization measurement datum measurement essential model representation generalize value existence conditional expectation establish specific situation follow usual conditional present proof completeness measurable stand formula define equality integration thus integrable integrable sure pointwise since motivate would write derivative define particular formula see gm continue fm dm especially hold distribution dm respect well dm p sure formula assertion look everywhere correspondingly satisfy formula see appendix discussion respect point view practical inversion quantitative
min pls ridge hour adaptive lasso major rely cross scheme regression pls scale lasso adaptive adaptive nest partial parallel alternatively might consider construct network without cross method refine run propose partial application simulation assess reverse engineering investigate estimation perform fdr significance test fdr graph even sample dense network performance five world sparse except pls close mean replication yield structure suit compare stability regression stable difference non seem unstable constitute severe decrease topology cross pls slow fairly allow representation I partial correlation pls package repository assign nk study initial version manuscript package analysis concept manuscript acknowledgment support grant european network financial product constructive mr axiom axiom axiom axiom lemma axiom axiom axiom cm technology clinical popular undirected association issue greatly exceed sample partial correlation inverse poor scenario adequate need biased estimate scheme least investigate regularize regression respectively extensively repository investigate decide difference confirm know tendency towards select lasso provide reconstruct network six clearly obtain method automatically assumption uncorrelated replication lasso yield complex structure indicate suited approach auto model microarray article undirecte graph edge correlate correlation indirect variable correlation strength primarily article genetic microarray numerous study nonetheless reconstruct microarray remain covariance estimation observation array gene alternative regularize high dimensional focus comparative use various high dimensional extensively real truth true focus difference investigate examine result graph moreover dimensional represent whose dependency relation node assume quantify variable finite respect involve exist network plug partial formulate outline notation rest array unbiased q inversion estimate invertible partial denote least stand include intercept least column gene moreover sense microarray firstly estimate eqs ill condition even large criterion pose hence regularize regularize review secondly approach set hypothesis base sparse correlation separate testing discovery multiple review estimate estimation regularize ridge propose shrinkage construct control shrinkage assume covariance correlation whereas shrinkage diagonal entry variance shrinkage target intensity analytically shrinkage favorable subsequently gene association multiple bootstrap aggregating bootstrap help bag inferior shrinkage expensive augment closely shrinkage shrinkage finally recent base graphical corresponding sparsity conduct parameter challenge least square formula estimator define partial ensure estimate always interval method pls estimation two attractive least al pls pls method seed scientific field bioinformatic idea pls pls assumption mutually pls hence pls orthogonal component rescale vector lead formulation observation pls regression criterion freedom gene network empirically experiment well validation pls correlation testing use alternatively pls correlation mention estimate use pls speak root et recommend ridge popular regularize perform ridge term penalty avoid fold scale pls apply coefficient minimize penalize form coefficient mind lasso advantage yield assign gene regression successively scheme control connect e tailor however graph prediction determine stage adaptive require necessary among show procedure rather consider dependent manner specifically follow estimator pick estimator penalization lasso adaptive lasso use gene connect correlation selection determine validation split fit computational cost reconstruct network propose technique pls lasso covariance follow overview five characteristic package performance assess simulation set sample range investigate consider vary topology network topology scenario partial replication vary density draw rescaling ensure partial package correlation depend absolute file histogram partial correlation network high degree select correct effect eliminate simulation partial trivial lasso shrinkage estimator successively fold component penalty two follow parametrization ratio lasso estimate avoid phenomenon ridge range parameter range pls component estimation partial correlation ii recovery topology difference square panel figure mse display estimate base lasso mse sparse likely non significant ultimately vanish degree notable exception network conjecture pls number four reconstruction underlie panel number network regularization pls contrast conservative promising difference density difficult explain high degree panel rate panel come low decrease density argue report fewer less change relative method independently particular fdr detect non partial specificity fdr display roc supplementary pls slightly covariance lasso threshold depict roc finally runtime figure display lasso compute load lasso high pls ridge fast consume validation computation different regression discrepancy become apparent study topology consider topology simulate topology display show different simulation cluster shape cluster star star mse discovery scenario probably insufficient towards perform reconstruct network different diverse real set availability consider shrinkage ridge pls selection procedure approach simulation use cross datum ever performance power possible compare estimate percentage proportion six adaptive seem behave differently partly different select set except datum non datum likely explicitly standard regression independent implicit use fdr assessment estimate conjecture pls validation reliable method network strategy incorporation pls among fdr assessment pls conservative pls identify refinement present simulation
uncorrelated individual load correspond component uncorrelated explain achieve solve control however look plausible modify modification consideration component connection formulation pca technical integer define solution satisfie orthonormal r address problem orthonormal first whose consist orthonormal feasible hold also part show define matrix whose column consist orthonormal diag tv tv rp r therefore p tu u tu iy tu moreover together right hence diagonal let first respectively obtain identity v consist orthonormal correspond em orthonormal eigenvalue loading provide pca suitable nonsmooth constrain nonlinear programming problem pca subsection subsection minimize nonsmooth close suitably subproblem global nonsmooth constrain programming nonlinear necessarily presentation denote establish active jacobian expression demonstrate sake completeness cone cone tangent imply gradient condition satisfied identity definition hold order optimality let lagrange denote eq arbitrarily sequence equality second fx fx hold simplicity next contradiction nonempty contradict closed know convex unbounded px subsequence necessary xx closed side limit subsequence necessary contradict convex mild accumulation lagrangian method program feasible drawback novel lagrangian subsection make follow feasible moreover denote augment penalty roughly speak subproblem update multiplier penalty know assumption framework novel lagrangian sequence choose find lagrange multipliers method lagrangian augment step ii penalty lagrangian multiplier augment lagrangian address approximate subproblem lagrangian converge feasible subsequence accumulation point lagrange second gx k hx side inequality follow show statement q contradiction subsequence necessary x convergent identity fact imply cone contradict identity subsequence bound see accumulation see every accumulation fact first optimality discuss satisfy lagrangian method particular apply subsection able approximate second subproblem approximate solution th subproblem satisfy additionally propose subsection possess value iterate q minimize nonsmooth suitably subproblem augment lagrangian mention method relate known project method study propose subsection smooth lemma stationarity observe convex immediately change fast theorem conclusion immediately lemma indicator statement q eq definition px fx td fx fx td l together immediately statement set choose ij go differ distinct indeed lipschitz continuously method namely local convergence theorem two technical lemma first show hence exist whenever claim limit side contradict imply use relation provide conclusion objective converge observe integer eq bound limit hold definition together lk lk q second hold argument lead together use argument hence definition lk lemma ready show convergent uniformly accumulation method accumulation subsequence exist without simplicity since exist particular specify pass subsequence upon k relation boundedness k k sufficiently inequality contradict finally limit side h x begin proof convergence make establish inspire similar local nonsmooth bound gradient provide sufficiently constant kx use result lemma lie lk lk lk inequality lemma h ik limit side rearrange term q next follow choose integer h hx accord integer define relate proof global hold view smooth generally global proceeding exist satisfies lemma together lemma scale direction zero also converge suppose uniformly continuous sequence sequence together see limit hold replace immediately easily show use lk assumption lemma follow ready globally convergent level accumulation gradient stationary suppose contradiction accumulation subsequence subsequence assume limit side lead lemma stepsize monotonicity sufficiently fx px px k fx px px fx fx fx pt fx contradict hold establish rate suppose bound continuous set ii satisfy constant ix argument index ik lying follow lk lk lk lemma constant proof detail augment lagrangian pca reformulate subsection sufficient augment lagrangian include explicitly hold accumulation point trivially satisfied accumulation condition feasible due condition proceeding subsequently feasible v I ji fact orthogonal remain lemma reduce diagonal unique let assume condition b change thus suffice p conclusion ready show condition hold feasible feasible hold v ji j active inactive inequality inactive first diagonal addition equality one directly hold solution accumulation point augment lagrangian fail augment equivalently discuss outer inner lagrangian inequality constraint multiplier equality constraint convenience presentation matrix whose entry resp th resp stacking rewrite clearly complexity evaluating assume sample covariance efficient store compute initialization termination pcs eigenvectors lagrangian multiplier terminate lagrangian sufficiently prescribe equality augment second one method first choose subsection subproblem become choose scheme study initially termination criterion solution prescribe shall sake numerical lagrangian pp adjust lagrangian multiplier update speak decrease minimization lagrangian multiplier constraint decrease recommend conduct augment lagrangian method detailed subsection equivalently synthetic real term explain pc et specifically via penalty block via penalty thresholding pcs pca uncorrelated total pc variance pc pc pc correlate much explain overlap total total variance pc dramatically pc explain variance drawback account introduce pc pc uncorrelated usual pc adjust explain explain pc shall stress pc three pca nevertheless shall solve datum al test effectiveness sparse synthetic actual find pc independent pc table fact pc explain second sparse pc pc uncorrelated termination loading pc column pc interestingly one quite subsection rr pca pc pc pc first center subsection set choose termination criterion pc loading method properly method randomly instance column one loading average two three average orthogonality loading vector angle form loading column orthogonality pc present loading three pc almost uncorrelated loading method pc c c method pc average loading around sparsity present substantially outperform pc loading pc increase accordingly pc non sparse pc surprising orthogonality orthogonality pc data result introduce classic illustrate pc several method six pc ease pc pc shall pc sparsity one table pc large sparsity one sparse orthogonality pc obtain standard five measure third report measure absolute angle load fourth present maximum correlation pc pca nice table except give pc apply method test termination r r pc knot r variable pc knot pc knot length clear experiment uncorrelated pc sparse pca pc present sparsity orthogonality seven yet explain r knot l r pc knot pc orthogonality experiment choose pc pc pc orthogonality dramatically combine deduce seem possible six g nearly uncorrelated pc exist knot correlation control performance pc non orthogonality pc table pc explain variance correlate surprising drawback observe sparse pc still experiment six nearly orthogonal uncorrelated pc acceptable pc one r r pc knot c correlation formulation pca principal pc globally convergent lagrangian class nonsmooth suited gradient method lagrangian subproblem local finally sparse pca exist pc produce substantially method total pc orthogonality loading effective finding desire sparse set sparse uncorrelated pc loading word actual associate orthonormal since practice approximated sample question exist pc show also accumulation however open hold recently augment method nonlinear nlp program sdp sdp relaxation hard combinatorial augment generally converge due theoretically approach sdp augment mild nlp code available www extensive exploring application acknowledge comment west
correspond associate player instead balance game resource singleton resource machine assume profile strategy give scheduling policy player example schedule order clearly balance allocation game load balance load balance whose unitary introduce exist strategy pure I strategy basically intended initially select player lead random player player f consider validity probability say preserve function formally size support problematic please game stay allocation decision decentralize instant interest nash equilibria general consider weakly sense problem differential ode informally replace bp behave ordinary ode eq dynamic unity dynamic stay also mean field dynamic convergence limit nash equilibria convergence equilibria theorem ordinal game game admit lyapunov lyapunov take call lyapunov game balance game lyapunov martingale lyapunov lyapunov equilibria happen exact game potential load balance particular actually iff prove allocation game potential game value decrease player response correspond player good response move game ordinal potential game terminology ordinal pure nash take strict response must pure equilibrium turn player low load idea response investigate play result investigate require move choose version good response dynamic terminate uniform task machine expect time game pure nash equilibria game pls complete dynamic nash equilibria investigate nash equilibria occur extend different asymmetric game elementary partially game potential game prove equation follow incorrect happen towards nash unstable stationary super martingale rely theory evolutionary game set construction time discussion game consider lyapunov particular game nash equilibrium modify cost lyapunov player team define assume hypothesis convergence stochastic define limit approximation formalize piecewise interpolation bt k interested limit family variable converge weakly unique value presentation presentation mean law instantaneous variance covariance differential particular diffusion continuous consider markov clearly stay keep mean sup theorem correspond differential turn ordinary solution unique classical restrict like follow dynamic like rewrite equation lead ordinary rescale q limit point equilibria theorem evolutionary corner ever irreducible ordinary whose nash equilibria ordinary equation equilibria game interior mixed equilibria method problematic nash equilibrium set belong interior dynamic game dynamic generally provably convergent lyapunov argument terminology lyapunov say lyapunov game hence potential lyapunov degree game lyapunov game lyapunov function ordinal linearity clearly linearity I dynamic side game ordinal q ie ie lyapunov function ordinal take lyapunov function dynamic class continuous potential lyapunov definition lyapunov clear derivative ie define continuous potential conversely strategy equal cost case characterization continuous differentiable connect path pure part proposition part q proof lyapunov lyapunov dynamic ordinal game lyapunov differ ordinal lyapunov function accumulation trajectory lyapunov game dynamic entirely dynamic lemma slight lyapunov theorem ordinary stationary ordinary near prof stationary limit hence lyapunov game respect full lyapunov dynamic condition field nash equilibria field nc property nash equilibria unstable stationary fortunately previous continuous balancing game lyapunov possible avoid differential equation double proof second term lyapunov q ft definition hence variable ti observe indeed one move consider elementary lyapunov dynamic hand super martingale reach corner super eq martingale get stability subset enough underlying chain ergodic visit say closure positive neighborhood proposition one nash equilibria probability default require neighborhood lyapunov game point compact correspond nash equilibria fortunately nash iff ie q nash improve time current equilibrium u ie e perturb dynamic form b q u iteration near I opposite lyapunov lyapunov ordinal hence potential take surely nash equilibrium furthermore ie ie side equation already nothing otherwise sigma algebra generate whenever n proposition ordinal gain gain variation preserve course game game game potential give rt following say satisfy satisfy whenever nash player cost adopt would gain least believe perturb polynomially many game lemma fr france fr algorithms learn equilibria sense weak limit ordinary ode case stochastic dynamic ode turn dynamic fact convergence discuss dynamic game convergence ode equilibrium lyapunov lyapunov game stochastic dynamic prove ordinal game potential lyapunov lyapunov lyapunov game lyapunov function lyapunov super martingale way time martingale game consider include load balance choose portfolio interested converge rational equilibria game theory dynamic fully game description several game mainly deterministic good description market variation avoid consider
source negligible second calculate simulation nan complex assumption commonly numerically also simulate happen apply commonly remove contamination simulate filter filter proportion tool pp achieve comparable outline approach replace pp necessarily section describe see image pixel vary spatially clearly violate like background problem derivative use region stand behind multi examine derivative filter work toy location remain stay roughly reach big source point smoothed bandwidth occur tell alternatively source value smooth increase bandwidth get nan smoothed tp compare smooth source stand large smoothing quickly perform smooth filter origin bandwidth image scale image create pixel derivative high scale derivative cause source background smoothly vary enhance image flat background figure show look datum ground detect galaxy cluster detect leave strong source wiener combine three see middle detect galaxy wiener filter vary filter right highlight filter alternatively view incorporate peak create expect create design enhance make stand describe previously run derivative image confidence procedure source detect follow make detect object tp without image make underlying explicit pure design patch detection wide verify initially ability keep detection without benefit relax proportion grow publish follow optical observation detection contour ray background spurious ray count visually false follow object confident previous rigorous error find recover outside optical unable whether spurious still accept expect proportion false source determine wide set collect follow scenario control reach aim analysis beyond imaging place ask carefully arrange behavioral dimensional brain acquire regular perform subtle flow response brain location compute statistic change image area see detail process ask visually visual image acquire fmri resolution surface david use primary responsible processing call fmri roughly visual first phase location goal region appear series visual band problem region prefer inferential location fisher stimulus phase stimulus type location want location response phase location stimulus want test phase false cluster adapt definition deal multiple oppose pixel grid classify cluster belong define test two location less large location confidence determine percentile uniform total number information previous connect location great analogous limit band surface false proportion situation control region detect show control object regular variety source guarantee false check science behind multi type enhance false cluster follow detection whereas run false verify control comparable detect control without critical generation provide manually generalize type plane grid object proportion false future apply proportion technique wide clustering try false token token token token token token token interesting product important input detect error control several rigorous control detect aggregated pixel technique rigorous statistical ray source source detect detect previous paper extend blind detection david student department statistic pa mail edu department pa mail work support national nsf grant dms national health ns space grant center like thank ray david fmri helpful discussion record light section various detector count detector exposure record solely interest reduce seek coordinate source survey input scientific early star object produce direct visual improve much require year collect comprise change digital imaging design computer available power storage relative could wider deeply fast ever search automatically collect previously digital hundred million survey collect comprise past decade poor lie challenge lie answer challenge lie rule often come generation operation entire prohibitive yes several new cut massive controlling source incorrectly reduce object comprehensive human scientific likely automatic misclassifie method control advantage especially large survey although context wide similar give later array value pixel arise include object anomaly essentially poisson base denote source respectively disjoint poisson random apply good report base observation image reasonable approximation gaussian utilize generalize problem pixel contain pixel hypothesis coarse want characterize pixel accurate construct criterion coherent localize wise operate pixel false proportion control effective control false image show procedure power technique wide source take procedure powerful technique provide excellent ray even source maintain although setting concept band activity multiple give rate underlie center problem functional response field region contain fmri dimension general detection processing operation effect well detection typically operational planning typically simulate real detect source depend simulate qualitatively effort construct simulation run fail detection use fall peak consist cutoff wise statistic classify fast computed popular software apply raw signal match filter simulate create ray image simple thresholding simulate image pixel false south small million second observe exposure mean able resolve distant interference galaxy approximately angular size also location complementary figure scientific source think galaxy center ray make rate publish combine source modify version background source region pixel look real select include several detect refined observation effort replicate design patch follow back reject spurious exchange fail large make impractical follow automatically reliably control one pass conduct pixel appropriate pixel apply pixel unit inference pixel compose collection therefore alternative pixel wise proportion introduce false random treat field derive location unknown confidence pixel decompose component pixel pixel measure tolerance come proportion detect sufficiently cluster envelope false proportion wish search proportion detect equation calculate look q pixel intensity p pass value quadratic homogeneous gaussian field locally field satisfy equation incorporate appropriate detect cutoff less henceforth see plug software software way active galaxy detect make spurious classifying really ray detector model background pp assume deal smooth apply pixel smoothing filter count structure poisson make close zero square background root normalize root low rate normal different rate normalize transformation pp rate source conservative pixel select power nothing know nature assume check pp calculate publish since many nine tp boost
combination classifier interpret hyper plane space select decision weight find maximize separability criterion criterion technique asymmetric address cascade unlike discriminant analysis criterion wu optimal lda technique neural feed forward class lda consider neural increase superior adaboost key contribution introduce detector algorithm combine detection performance objective word two separated detector flexibility well beneficial consider highly detector criterion incorporate hence well adaboost exponential loss detector detection incorporate classifier propose experimental result show section technique adaboost concept lda handle asymmetric also sample select analyze training discriminant lda cast eigenvalue decomposition pair matrix covariance separability maximal lda q count nonzero integer become np nevertheless infeasible extend efficient add number select predefine meet eigenvalue case column vector determine inverse greedy adopt suboptimal simple rank computing reduce computation mainly forward index calculate q drawback control number control tune decide predefine sparse explain classical explanation cascade reader refer detector operate initialize train weak classifier focus haar later haar examine continue predefine algorithm set haar like rectangle feature minimum acceptable rate cascade cascade false train parameterize threshold error add weak classifier add misclassifie cascade cascade cascade would prefer high rate false positive bernoulli easily sense linear classifier equal covariance linear matrix normal input image feature minimum example central limit close asymmetric cascade trade rejection eq lda minimize total covariance select asymmetric easily adaboost detection little positive think negative prior number lose contrast sample hence sample artificial consist show weak adaboost select small weak introduce since classify adaboost weak classifier try introduce positive adaboost weak classifier yield classifier less false positive adaboost distribution adaboost experimentally significant boost round term cause gradually pay attention boost separability boost boost popular originally design problem combine accuracy least minimize hinge property majority vote boost adaboost additive classifier example final decide label receive determine significance weak boost computed adaboost select update final rule weight coefficient learner weak classifier dimensionality linearly project direction introduce concept domain learn save small error remain however decision achieve decision boost training receive weight decision u ix margin minimize subject weight select weight update base detection replace line normalize weak classifier optimal add classifier output yield weight manner remove correctly classifier misclassifie boost cascade layer need classifier classifier calculate complexity total dominate spend weak classifier organize experiment efficiency haar rectangle know haar adaboost face fast adaboost haar feature technique classification consist consist face rescale image pixel face face patch non obtain image split train experiment classifier selecting set remain measure test alarm set mean dual forward fig haar like comparable adaboost fig slightly haar adaboost slightly haar classifier confidence error indicate adaboost curve result figure adaboost experiment layer positive previous cascade bootstrapping cascade terminate negative bootstrap fair train technique cascade backward dual cascade detector face detector low resolution face mit test set contain face merge overlap ground truth overlap bound ground box exceed face fig receiver curve produce classifier weak cascade classifier meet roc curve adjust threshold adaboost instead weight adaboost weak perform unlike adaboost train algorithm sequentially whose process continue meet classifier experiment svm lda detector believe another provide finding open object rectangle feature cascade select haar like cover compare classifier classifier average number haar evaluate detection window adaboost number classifier classifier adaboost decision nevertheless classifier indicate classifier class suitable domain skewed separation compare different decision scheme correspond skewed asymmetric boost apply figure classifier adaboost surprising adaboost adaboost low haar rectangle feature alarm positive evaluation curve adaboost train cascade roc curve roc classifiers cascade predefine meet evaluate performance asymmetric asymmetric multiplier every show classifier roc curve train false rate might suitable domain example gain small adaboost lda problem reduce table indicate perform comparable slightly cascade intel cpu ram total hyperplane class weighting conduct scheme remain previous roc configuration outperform adaboost classifier high positive perform positive bt stage total haar adaboost adaboost scheme apply difficult face dataset scale set contains analyze two six roc conduct three haar scheme experimental fig roc curve face conduct manually time sample preserve contour extract haar poorly haar feature however applicable dimensional overcome dimensional brief stack covariance project onto space decision discriminant drawback slow assign classifier store project technique project space replace lda feature train classifier threshold rectangular filter subsample approximately stage objective meet detection rate cascade
tree hypergraph density arbitrary measure distribution implicitly take depend disjoint union vertex determine density factor way factor set denote respectively factor even decomposable eq may specify give determine law parameter hyper inverse wishart useful factor far hyper previously little typically take decomposable perhaps os r enyi encourage express arise model posterior graph factor independence entail selection graphical either little represent discrete space obvious geometry case challenge flexible metropolis mcmc parametrization intersection region proximity form less ball radius intersect diameter range intersection convex consider compute concept finite collection distinct nonempty hypergraph three paper complex denote set alpha denote construct family example complex diagram vertex plus illustrate alpha display table alpha complex illustrate index intersect produce isolated vertex index intersect triangle vertex index intersect face include known abstract complex subset collection set hypergraph arise skeleton skeleton complex cardinality skeleton uniquely skeleton obtain nonempty intersection obtain skeleton induce alpha exceed although still disjoint index dr parametrization space set determine implicit whenever obvious generic element alpha induce parametrization two advantage need hypergraph vertex hypergraph mcmc induce distribution integer radius intersection cube ball widely clear choice write pt height pt set os edge model exhibit class construct describe asymptotic count alpha complex regard distribution area recent survey discuss count joint vertex suggest result sometimes multivariate sometimes require approach infer specify law inverse wishart normal principal law strong build construct random dr construction transformation angle simultaneous scale change restrict distribution reduce feasible represent ball r triangle inequality must great fit ball center segment separate diameter translation rescaling may diameter complete proof fix simplify writing understand induce configuration uniform edge lead take enough empty feasible rich graph converse prop dr r slightly strong attain dd tuple distribution wish draw distribution vector fx metropolis hasting begin diffusion walk parametrize spherical radius euler angle informally radial walk brownian radius angular brownian radius fix walk begin reflect expression dd random walk generate local simplify ergodicity option global move uniform one hybrid walk ht draw graph begin hasting distribution invariant accepted multivariate law close efficient variable parameter depend configuration reversible mcmc use auxiliary variable keeping need nest auxiliary auxiliary instrumental move metropolis last metropolis ratio non move graph proposal ratio denote p feasible statement recurrence markov section strictly global move replace draw p markovian nevertheless prop encourage specify factorization point vertex display htb complex use list skeleton complex alpha factorization factorization factor similarly figure fx particular graph set contrast os enyi edge clique plane mat ern core radius asymptotic ern iii comparison make os enyi os r enyi inclusion joint ern draw os enyi joint radius enyi inclusion width depth clique mat ern er mat ern mat ern mat ern mat ern er er c complex sample draw core construct change hypergraph maximal represent need normalize monte store I give point radius triangle hypergraph encoding point unlikely lie radius hasting proposal vertex mixture walk pick probability random pick prior proposal burn posterior bold prior vertex lebesgue simulation result concentrate section set investigate methodology class use model consider class conditional marginal class metropolis hasting random enforce uniform draw summarize table mode alpha alpha obtain alpha feasible alpha complex marginal observation class summarize observe alpha mode section sec mode match l alpha alpha propose describe subgraph graph triangle cycle cycle star estimate network compute count subgraph network os regime assume vertex prior vertex radius move move subgraph convention induce procedure lasso decomposable graph display decomposable c incur exception star also outperform regime exception fit highlight green another simulation random package formula triangle specification encourage triangle produce triangle os enyi experiment triangle surprising triangle os model tend decomposable proportion decomposable table graph decomposable compete incur produce error table runtime discussion term clique distinguish estimating regard hypergraph method estimate scale multiplication lasso require scale regard method complexity compute skeleton scale lead compute alpha large scale run employ normalize decomposable ghz gb ram h variable width length datum geometric graph assume walk proposal markov specification law hyper adopt prior center deal normalize constant decomposable graph run burn display posterior concentrate coincide model attribute leave missing report appear conduct assess gaussian via miss model incomplete set summarize simulation average average predict vector surprising since contrast decomposable daily exchange rate consist model interesting trivial miss yahoo finance r data assume vertex unit intersection different law keep simulate burn assume miss posterior decomposable parametrization perspective support prior also useful characterize intersection set particularly suited strategy generalize subtle proposal lead move graph hypergraph proposal perturbation consist resample produce model law hyper law encourage feature think lot possibility specification worth couple representation restrictive gaussian graphical model general convenience graphical develop relation hypergraph state decomposable hyper markov put mass conjecture characterize feasible consider graph embed propose ii segment straight every complex geometrically realize dimension iii ball different idea deep subspace span design prior specific graph tree width acyclic insight core incorporate tool prior future development involve process parametrization structure dag obtain strong topology convex subset abstract theory perspective construction modify communication accord grid may improve behaviour infer among easily direct hypergraph think encode individual picture people picture people arguably properly complex topology efficient computing alpha r decomposable decomposable adaptation construction example central idea decomposable complex decomposable decomposable formalize I c produce decomposable initialize decomposable empty graph test finitely decomposable compute use decomposable simple show figure table induce complex decomposable present clique edge reject ht edge accord note reject quite point unit algorithm skeleton restriction radius appear complex preserve htb complex display b decomposable complex ht pt section parametrization geometry idea induce place prior configuration retrieve alone hasting monte move great comparative evaluation abstract complex copula model inference dependence random observation dominant formalism graphical formalism specify carry detail distribution undirected problem among distribution random stage distribution summarize form index edge say markov pair simultaneous decomposable proposal promise extension take parallel computing batch incorporate local move
signal classical cholesky lar lasso single gram small first iteration step lead large prevent phenomenon implementation use dictionary never happen practice solve practice problem case difficult highly choose strategy gradually method algorithm reference improve inverse hessian matrix efficiently yield remove optimization hessian reason share illustrate major modification formulation unconstraine course original formulation nonetheless recursive formula algorithm sequence fast implementation reference therein tool process prove reasonable support impose convex semi definite great consequence invertible strictly verified experimentally consist dictionary dct wavelet enforce replace definite omit penalization sufficient uniqueness coding present briefly optimality denote necessary unique eq correspond eigenvalue equal eq build invertible linear lar aim elastic net replace improve homotopy objective paper optimization neither one good practical proposition almost surely meaning prove variation prop assumption surrogate short calculation growth f prove smooth support feasible continuously x continuously f therefore one restrict unique appendix statement since continuously second immediately simplicity slightly condition optimize unique imply matrix convex show moreover e x smooth show almost surrogate surrogate surely process quasi theorem variation obtain fact tf past obtain bind u f f tt strong corollary easy hypothesis verify differentiable exists bound applie exist u prove surely u sure tt appendix almost converge note addition prove prove final result asymptotically stationary assumption almost matrix compact possible extract converge moment converge case converge u limit converge taylor follow optimality condition cone close accumulation close normal address optimization matrix code prior coefficient regularizer verify positivity regularization homotopy handle regularization elastic regularizer case note sparse induce regularizer subsection claim unit step solve column keep solve project u j j classical amount ball convex long union negative elastic j induce regularizer analogy constraint elastic constraint control negativity constraint problem factorization slightly look piecewise consecutive prove array replace projection ball onto constraint converge optimization onto additional thresholding onto set compute onto net constraint extend ball homotopy solve lasso piecewise part solution fuse problem present efficiently numerous application scope allow fast solving fuse could use complex constraint test address regularizer within formulate dictionary regularizers nmf follow x matrix force negative lead solution face localize one address project research objective control sparsity vector negativity vector work tool datum interpret direction maximize variance matrix propose different formulation sparse analysis extend pca maximize formulation enforce orthogonality sparse formulate matrix factorization regularization net low operation well optimize matlab lars matlab measure performance test method objective act surrogate column setting full version version subset set systematically outperform batch counterpart desire similar speed choose try try power objective datum plot optimal though plot curve contrary set curve set large datum one bring tune hand observe big slow cycle obtain mini give sampling among give strategy except mini compare sgd improve use new much learning form instead tune high value lead bad asymptotic try power couple refined trying select show curve curve see curve still asymptotic set different initialization initialization lead similar variance computation classical negative code face size pixel mit database face image extend f compose patch pixel database even though implementation advantage speed matlab spend multiplication well optimize ghz minibatch original experiment choose face objective matrix run run sparse vector report initialization great curve average initialization term test even nmf bit implementation cm nmf method report computation logarithmic present address various type face qualitatively nmf learn principal component analysis dictionary vector unit figure e dictionary set result different level scalar indicate compose represent value white systematically face database neither nmf localize feature patch hand sparsity among demonstrate technique analyze genomic expression measurement dna copy comparative genomic try analyze determine gain therein measurement order analyze correlation set suggest correlation obtain recursively orthogonality v p u center good r furthermore regularizer gene experiment gene non coefficient select divide repeat split factor tr te te experiment result curve report purpose genomic compare carefully substantial conclusion need lasso average denote demonstrate remove use implementation patch minute ghz eight core text remove thorough comparison art instead wish indeed trivial dictionary use mode cm stochastic online dictionary adapt significantly batch alternative million training stochastic gradient moreover extended matrix factorization negative solve already course need bioinformatic plan propose computationally demand video extend framework address sensitive paper f pf uv differentiable x stochastic measurable realization stochastic f converge index x suppose element borel measurable e f additional address project extend formulation project efficiently lemma solve elastic solve define lagrangian minimize calculation b condition l denote solution complementary imply q close necessary short optimality b j k modification similar convergence solution exactly u u b handle replace scalar problem term homotopy solve constrain u u solution natural lasso propose use lar exploit formula require operation admits close c c p allow homotopy cholesky factorization operation adapt require path stop whenever still note lasso solve eq improve modelling linear focus factorization basis adapt variation signal negative propose base stochastic approximation scale set million sample naturally formulation wide image genomic demonstrate art optimization large online principal analysis atom predefine wavelet recently lead state art task image denoise texture synthesis classification learn natural signal decomposition base principal basis allow datum machine slightly factorization paper algorithm address prove art computational challenge include million address design generic capable various topic approximation atom show modelling code many processing natural predefine dictionary wavelet dictionary instead although look tuned algorithm batch iteration minimize training change time video address mini batch particularly common dictionary several million patch per frame set online stochastic attractive descent sometimes low consumption low batch experiment scale large million training problem convergence problem generalize dictionary learning devote demonstrate suit make smooth desire solve quadratic cost constraint stationary experimentally fast approach dictionary small task dictionary show matrix procedure project onto define p sake n tu space norm denote product kronecker dictionary represent good signal sparse usually process say overcomplete dictionary basis pursuit norm q admit direct analytic sparsity prevent constrain k minimize cost respect rewrite k decomposition rewrite x f alternate variable minimize also prove behave general pseudo dictionary computation coefficient dominate accurately suppose unknown effort inaccurate optimize algorithm whose fact certain theoretically empirically reach solution cost batch set overfitte may become speed memory project iteration orthogonal obtain train speed way fall process one mini batch structure allow without rate section kalman compose distribution loop iteration dictionary minimize
measurement polynomially bound information polynomially observation conjecture infeasible presence classification generator generator specialize binary quantum define quantum formally quantum generator generator tuple state unitary transformation finite set quantum nx symbol basis degenerate generality hardness hold highly link measurement quantum system certainly formal identify concept exist stress model relevant generator distribution assume take arguably hardness practice turn symbol quantum copy hardness appendix improper learning notion divergence learnable kl output satisfy appropriate polynomial improper learn efficient definition boolean efficiently learnable input output representation polynomial learnable quantum generator pac learnable kl gate cc k gate unitary gate particular quantum representation net distance generator size distribution converge calculate former give divergence infinite string support prevent minimum symbol accordingly kl divergence symbol string perturb divergence net generator n hardness distribution generator formally noisy polynomial circle state belong come output quantum generator divergence acknowledgement thank question relevance thank quantum convenience quantum quantum generator unitary starting gate kk k perturb similarly whenever put np px claim hoeffding range perturb therefore take circuit evaluate gate merely involve follow rate construction generator think represent explicitly generator describe transformation entrie nonzero entry appear together zero unitary unitary index preserve application decompose entry real suppose two entry column latter contribution q sum similarly entry weight sum unitary generator basis outcome satisfy previous quantum generator verify induction inspection measurement generator basis claim induction state case inspection support sx b symbol output generator precisely accord suppose could quantum particular learn circuit function easy verify calculus contribute
similarly exploration search embedding e non operation focus split interested split base split specificity sensitivity trade think bayes b estimator tree estimate ht nj neighbor tree sample estimator ht assign left figure b b h conjecture thm thm thm problem thm remark bayes bayes estimator reconstruction p w li r biology institute building pa department work department phone measure address sample maximize expect closeness distance unify focus especially distance euclidean notable hill likelihood consensus metric reconstruct reconstruct wrong slightly refer reconstruct issue help cope bootstrappe bootstrapping occur almost highly support regard similarly common close way closeness rf know reflect common likely reconstruction least tree likelihood accurate representative yet ml goal closeness true optimize likelihood approach bayesian view accord tree many distance easily express vector euclidean statistical understand square minimize distance rule estimator closeness closely distance hard compute hill popular heuristics hill compute hill comparable hill difference hill bayes hill tree compare ml encourage pilot study bayes conclude discuss improvement direction develop sequence specie many evolutionary exist express give observe could topology method create bootstrappe tree tree nj entirely obtain bootstrap notation whether bootstrap tree expectation distribute expect regard bayes close common decision theory give dt estimator simply say tree call recall vector popular embed distance tree branch length branch denote split half size realize squared euclidean maps vector correspond partition induce size symmetric realize euclidean figure square map distance define study branch length topological distance topology length dissimilarity distance analog count leave figure leaf combinatorial distance interpret mean near depend project split onto v consensus consensus tree bayes project nearby analog tree reconstruction onto base see cm bayes minimize frequency since weight set weight find compatible maximal weight traditional split frequency tree sample frequency apply though hence hard considerable toward see special collection tree easier least square ols evolution square dissimilarity bayes estimator reconstruction ol I ol I first length topology length comprise ol I however difference I square map ols I dissimilarity sharp contrast summarize govern underlying sequence bayes treat perturb tree tree exponentially hill ml work hill minima hill move topology combinatorial move subtree move composition move tree move detail quickly ml choose hill move hill must definition situation much express additive constant tree begin hill need expense calculate bayes choice distance consensus extensively lie foundation dissimilarity choose length prevent short topological estimator believe property suggest importance conclusion state path evolutionary tree study tree choose square think believe bayes connection outline deferred tree vector compute simulate briefly review tree process k seq program datum available website sample burn run total hill software compute nj pairwise house software euclidean hill hill along various choice trees nj tree five ml tree sample call briefly hill list hill input compute pt neighbor end practice allow hill might several statement loop study always code write available compare reconstruct ideally would obviously unless particularly set frequency v pt attention topology probable tree topology fairly probable topology compute three record tie topology proxy set hill nj start start bayes five plot nj ml figure report interpret difference typical plot pairwise pairwise also analogous plot empirical estimator true might global true plot table hill nj ml hill
local super propose xt xt minimize likelihood connect source due chose candidate function lag structure sequence cause reliably maximum adopt regularization suggest shrink zero functional brain connectivity fmri property shrink point coefficient appeal connection might also appropriate area eeg datum penalty hoc adopt lasso correlate assumption instantaneous hence order apply split original since correspond source propose put e sum group p call connected extend causal discovery eeg connectivity correlate source observation mix volume effect compare fit sensor space instantaneous e source effect ideally turn ica employ usually g explain arise model nevertheless regard maximum perform instantaneous source possible connectivity ica connectivity model possible filter even connectivity invert generally equivalent source sparse carry sensor able comparable ica allow cross latter see traditional ica measure mutual htb step compute ica ar fitting second statistic cause drop compare obtain vector l bfgs optimizer converge sign difficulty compare retain df become l bfgs special care gradient bfgs df uniquely subdifferential straightforward subdifferential sign inversion care must exactly corresponding norm minimum subgradient attain minimum norm care practice outline procedure sparse shorter obtain heuristic pruning connection might composite alternative alternate justified expectation call unconstrained importantly parameter convexity concavity convex great unique guarantee bfgs eq regularizer method bfgs unlikely solution however joint problem lagrangian introduce minimize additional loss conjugate hessian source gradient conjugate evaluate eq conjugate give turn nonconvex exactly convergence part autocorrelation however increase especially use augment function minimize assess simulate seven accord seven model draw mixed source eeg channel spread seven place spread realistic forward build image head see generation never noise none explicitly evaluate robustness additional variant pseudo eeg variant differ degree temporal correlation variant variant source share mix xt variant simulate cover brain distinguish noise n distribution instant variant temporal ar note since delay model causal snr snr norm eeg source dataset construct htb six structure create correlation introduce distinguish noise source coincide contribute sensor sensor n ica reconstruct seven source instantaneous ica fundamentally source fulfil dependent connectivity analyze variant lag temporal disadvantage temporal filter reconstruct source computation lag provide author information selecting lag constant either jointly bfgs variant since relevant source unfortunately generally reason perform similarity fit achieve another goodness scale possible coefficient goodness fit whole matrix quality decomposition match pattern location typical example estimate reconstruct finally source ridge coefficient area auc correctly discover significance true connectivity order equally necessary connectivity however could examine coefficient ridge testing non actual reason preferred regression different noiseless six variant plot outlier red remove result simulation follow difference box correct affect localization good achieve fig occur estimate fig sometimes superior amount criterion another might instability role current performance solely regard noise say relative degradation source seem problematic uncorrelated sensor spatial mix temporal affect however error de quite connectivity processing finish short em medium still room htb make source inconsistent studying require minimum brain hence make sense innovation ar pay price effectively stable stability connection observable eeg presence background emphasize assume innovation non innovation brain complicated question correct decomposition scope address channel significance decrease channel brain connectivity hard problem volume measurement rise spurious novel overcome elegant numerically detail eeg model source jointly drive super gaussian spatially use group achieve interpolation two ica cross extract connectivity usefulness excellent work study stationary analysis novel assessment addition aim connectivity enhance interpretability e european fp technique brain connectivity eeg signal volume follow eeg autoregressive source parameter overfitte avoid extract appropriate manner functional usefulness compare exist excellent two decade become possible thank progress make field mathematical imaging modality allow spatial temporal simultaneously record series brain functional relate connection sometimes region significant correspond quantify spectrum coherence g causality place outside head arise rather measure site sensor superposition brain instantaneous detect spurious recently eeg analysis account volume cross instantaneous effect couple robust volume conclude
tendency class grow vice versa notation element alternatively member tendency member around see alternative parametrize one generality dx fp segregation pattern association segregation alternative family association alternative segregation restricted lie remain denote vertex b j segregation occur association occur around alternative family parametrization triangle triangle segregation association alternative triangle segregation parallel segregation standard arbitrary away vertex segment segregation available association alternative segregation depend explicit finite asymptotic association alternative asymptotic much segregation uniform furthermore number follow mh om nh nr nr r mm segregation alternative proof r hold limit hold association notice segregation degenerate cr cr triangle segregation define one triangle number segregation association furthermore order extend manner translate segregation alternative extreme segregation association expect segregation association triangle segregation alternative extreme segregation small standardized test statistic critical sided segregation normal distribution segregation association segregation yield binomial base carlo power investigation realization segregation leave middle association number segregation test segregation test association give mf p n mf r mf normality consistency prove similarly hold segregation multiple triangle case triangle j gr gr mh h mn gr gr gr gr segregation sense theorem efficiency investigation involve detailed discussion segregation alternative applicable association direction hand segregation alternative association sensitive segregation st nr nr mp around choice pattern seem uniformly distribute support intersect intersect clear violate detect test statistic tend segregation support intersection point intersection support intersection calculate record per triangle replicate repeat monte carlo times critical base empirical conservative determine deviation significance level size conservative upper figure level value side tend small conservative association power restrictive sense realistic crucial lose substantial proportion might outside extensive simulation outside affect empirical monte carlo respectively set respectively form deviation generation record x expect area realization independence support segregation point away square restrict segregation overlap segregation segregation furthermore large segregation outside segregation finally associate level association e generate combination combination proportion point outside denote mean value ht fit value various proportion point outside relationship base solid fit carlo propose adjust proportion namely outside suggest per triangle suggest eq hull affect segregation consider large adjustment seem correct increase equation hold might large guide sample correction extensive suggest statistic adjust mn mn ht j mb extension proximity let polytope vertex faces form opposite vertex fall vertex face opposite hyperplane rx polytope rx rx xx r j x r nr pe nr pe ir nr pe conjecture iid simplex intensive calculation article bivariate segregation knowledge theoretic cover unlike mathematically tractable computable numerical dominating proportional minimum set nice triangle base region obvious particular sort dissimilarity find empirically work perhaps complicated edge geometry triangle class vertice construction imbalance abundance imbalance interaction use provide tend remove hull nan pattern since point circular result alternative parametrization alternative invariant likely might parametrization segregation point pattern segregation parametrize small parametrization construction convex hull hence correction pattern might pattern inference ii triangle recommend binomial approximation simulation randomization finite sample section correction segregation recommend power edge building manner acknowledgment air force scientific contract grant project theorem remark identity mail edu tr parametrize family multivariate spatial relative position extend proportional edge segregation spatial randomness binomial size class whose point constitute infinity normality size infinity evaluate carlo prove consistency restriction small find optimal segregation association discuss article keyword map interaction implication specie relative allocation method graph approach test spatial association respect spatial interaction give together tend occur convenience call characteristic observation example segregation investigate specie spatial neighbor contingency table nn pattern rl nn test design spatial mostly pattern rl realization arbitrary bivariate interaction graph gain popularity analysis move metric landscape suit concerned connectivity conventional explicitly reference reduce utility geometric system path location preserve spatial datum usually lose spatial spatial require adjacency allow meaningful edge interaction reflect patch relationship quantify patch propose association interaction class nonlinear correlation theoretic spatial arcs relation pair order arc vertex place arc vertex proximity near set arc iff near first triangle trivial proof omit short proof give article graph vertex correspond arc define bivariate datum introduce arc utilize radius dx ix r ir involve multiple dimension one prove uniform rather appeal dimension find minimum dominate np tractable maps alternative introduce type call parametrized family parameter number one sufficiently arc design distributional mathematical new family applicable classification proximity parametrize apply spatial testing spatial segregation relative purpose proportional extensive treatment article investigate proportional spatial segregation furthermore extend range expansion arcs small probabilistic behavior segregation association applicability pattern new choose section present one triangle section segregation association present suggest adjustment outside hull sample extension proportional dimension proximity associate point think close name x proximity map building survey map vertex arc define iff call arcs author vertex dominate minimum dominating dominate set proximity set proximity refer iid square triangles iid hull ht circle point distribution unit square set arc relationship symmetric rather iid proximity dominate point random depend explicitly implicitly n example briefly define high exact set size interval proximity map associate rx arc iff open pure contain element interior sphere natural proximity rx proximity extension higher spherical introduce whose view dt iid illustrative purpose point triangle preserve transform scale vertex vx xx r line interior triangle lie e mass figure region line edge possibility vertex region assign arbitrarily edge pass distance let triangle orientation opposite hence ht proportional vertex cm I line vertex n pe pe xx pe rx proportional bottom vertex arc iff arcs n pe likely small imply probabilistic spatial segregation association transformation also similarity rx rx r z rx rx additional degenerate f special falls occur probability star shape necessarily ht ne I f rx n nr pe ir ir rr basic triangle disjoint figure triangle region orthogonal region orthogonal important role proportional uniform triangle complete spatial randomness desire variable nr pe pe r vertex geometry segment edge line invariance us special iid variable distribution geometry note geometry theorem pe vertex notice geometry projection hence geometry triangle projection map orthogonal
test see interesting degenerate alternative realize gaussian noise alternative alternative give statistic limit hypothesis test regularity function continuous derivative converge process type use equation wiener let fulfil write explicitly follow calculus done write q wiener put wiener q eq mention slightly simplify solve equation introduce limit random come observed wiener function usual allow write kalman theory value innovation elementary hence correspond alternative equation eq q function condition write hypothesis show consistency function uniformly kolmogorov test hypothesis test hx correspond hypothesis start behavior fix write direct yield equation put inverse yield limit test statistic put wu wiener signal white case power condition fulfil alternative condition degenerate power already different put continuous hence put n test lead think relation eq replace forget wiener formally let formally modify fourier coefficient write integral mathematical meaning starting introduce statistic hypothesis introduce quantile lead valid important observe condition fulfil q fix contiguous sx put alternative course minimax alternative weight special choice white gaussian poisson process representation time start play ergodic asymptotically proposition moreover convergence weakly let c statistic eq constant asymptotic limit construction ordinary differential weakly local wiener equality q integrating put hence outside suggest q free case local time limit close white suppose hypothesis impossible property coefficient solution test take smooth consistent normal fisher regular smooth limit coincide integral another consistent structure see et calculus eq empirical integral test way integral formula contain integral I converge uniformly process statistic statistic limit kullback
pick edge theorem obtain formula correction miss tree totally backtrack error bind length shortest derive chain inequality k intuition accurate interaction omit equivalence product compute aid z correction express whereas correction still include serves totally backtrack non basic idea underlie depicted cycle base cycle copy tree root illustrate tree undirecte understand allow computation follow omit embed equivalence cyclic walk trace cyclic computation reduce infinitely tree walk totally backtrack segment tree walk edge direction step break walk embed cyclic walk calculation let weight single loop graph product intersect subgraph determinant corresponding matrix equivalent tree loop diagonal consequence impractical however walk correction block k lb w maximal block negative submatrix approximation elsewhere however insight gaussian w definition weight error result scheme converge correct decay exponentially estimate cover estimate cover cd z z exact hard distinguish bad correction need need cover graph grid four may shift increment see block loop short intersection block add weight complexity compute determinant block correspond numerically periodic quality approximation rapidly correction determinant interpreted totally backtrack show estimate way involve cycle particular product grid method address system leave plan model explore factorization r k kk direction model walk corollary propagation determinant determinant backtracking group equivalence furthermore multiplicative correction efficient correction length belief propagation numerous application sufficient also also insight sum within graph estimation determinant inspire recently study new perspective tie determinant cyclic walk determinant backtracking computation universal cover group interpret determinant graph edge solution truncate specify grid estimate differ heavily multiplicative expansion additive expansion loop calculus walk truncate let treat undirected edge direct adjacent direct end visit cross precede vertex begin end multiple shorter primitive close primitive walk walks shift cyclic walk follow classification walk play role said backtrack unique walk denote walk totally totally walk irreducible elsewhere set disjoint equivalence class denote backtrack irreducible totally backtrack neither backtrack sparse symmetric partition j h dx normalize vector px x dense use gaussian graph complexity growing distribute parameterized dx reduce follow iteratively message convergence marginal I method terminate elimination cover computation converge correct variance covariance jj concerned link partition motivation graph unstable solution interpret walk describe covariance determinant independent message correctly mainly concern correspond covariance approach walk diagonal diagonal walk sum eigenvalue interpret walk count occur walk regardless add walk require absolutely equivalent spectral wise sufficient variance walk sum reweighte walk interpret recursively sum imply walk model walk message interpretation compute correct mean walk variance totally backtrack totally backtrack walk estimate walk call w w primitive primitive cyclic totally backtrack totally walk play role walk interpretation estimate analogous z totally backtracking seem intuitive prior proof previously prove theorem summarize useful use identity
fix verify decrease iii original theorem proposition specify choose nontrivial fortunately necessary achieve substantial improvement already considerably set constraint upper leave remark iii equivalent ii surprising enter picture study algorithm introduce reduce result without ii limited availability expect little iii simply channel nevertheless convergence proposition v proposition follow entry also obtain symmetric eigenvalue proposition radius eigenvector unchanged particular restrict precede eigenvalue stochastic frobenius theorem prove involved place proof prove reduce special omit q final output eigenvalue right calculation eq deduce proposition capacity preserve provably alternate capacity calculation channel appeal geometric possesse monotonic extension study variant aim speed focus channel capacity calculation convergent typically valuable insight slow construction usual measure global substantial approach differ acceleration proximal iii broadly term overlap monotonic iv theoretical discrete specifie letter mathematically e nonnegative logarithm loss identically zero algorithm eq convergence original generalization constraint guarantee iv reflect intuitive explain end iii illustration fig iteration appear start give comparison algorithm henceforth define entry nonnegative write scalar satisfy doubly upon stop convenient restriction discussion reduce ii slightly implement determine time simple minor substantial consider inspection reveal regardless easier verify ii iii lead fast possible restriction recommendation conduct illustration example channel accord satisfy algorithm ii record use evident display bivariate sometimes hundred throughout reduction summarize acceleration ratio acceleration around ii support preference still implement satisfy guarantee intuitive setting maintain write study property calculate break restriction equivalent indeed algebra specify reduce useful proposition iv reduces conversely satisfy deduce equivalent iii converge monotonically bad slow channel matrix try overlap iii simply call proportional iii call work iii long I inspection regardless mention vector obvious relation key iii slightly straightforward follow maximize maximize jointly maximize maximize tucker condition iii never decrease alternate monotonic sequence generate iii exist solve throughout general convergence theorem comparison theorems convergence iteration mapping emphasize assume lie interior enough hence eventually fast notion global version see formula
I complement spectra ideal spectrum perform turn strict f orthonormal accord fix th matter eventually fast da per iteration lebesgue count k mf obvious minor symmetric component let q density datum version highly intractable moreover explain every interesting equal furthermore vary exact mode separate area despite year careful description facilitate fs algorithm density integrate pt pair joint give integer density datum density xy section conditional first follow eq formula reveal two fact independent conditional draw sequential example define know available concern section deal suggest chain converge move mode section describe alternative chain move iteration encourage transition symmetric mode move proceed choose permutation get choose permutation call fs explore effectively chain establish fs chain fs respect develop formula permutation represent switch fs choose simply observation cluster choose suppose satisfy give depend hence argument show indeed establish couple demonstrate balance little thought thing happen either r uniformly obviously step uniformly I fourth fs indeed theorem applicable operator compact spectrum pt eigenvalue fs fs actually chain I move exact recall graphical latter marginally fs target density fs spectrum compare spectra fs chain mixture quite flip fair coin take tail fs r four eigenvalue along early easy see nontrivial eigen switch replace theorem order case add replace replace affect fs evidence fs substantially small base appear surprising minor huge function form conditionally q proportional bernoulli complicate complete via routine eq analogous form imply chain consist probability entire idea use approximate eigenvalue fs section express da respect write operator fs dp carlo idea spectra must furthermore heavily generate random mixture result observation contain third set ten observation use classical carlo row carlo da fs row calculate eigenvalue record large dominant close dominant eigenvalue fs eigenvalue increase clear f may eventually would begin fs estimate monte carlo random seed eigenvalue correct place element must estimate thus simulated mixture process purpose result analogue nearly irreducible section show reversible respect eigen equal row could imply determined eigenvector eigenvalue element eigenvalue element yield two root correspond eigenvector eigenvalue row acknowledgment visit author wants acknowledge first support nsf grant third author la paris thank universit paris paris author thank project visit anonymous suggestion theorem paris paris paris reversible chain augmentation da self adjoint encode convergence generally quite handle spectra augmentation finite operator da compact dominate spectrum operator former less study density associate bayesian particular compare da fr random label switch intractable monte resort monte augmentation da algorithm liu build da must density say p algorithm free satisfie obviously densitie da goal joint da good formalize unfortunately ideal da fast function da draw mind inherent tool reason da possess two variable explore da simulate reversible adjoint whose encode property chain zero define p gx dx da chain formal value operator definition da operator imply eq invertible fast unfortunately I even get currently yy yy fx consist element directly call reversible chain yx small prove associate alternative chain close ideal draw call draw surprising van decade great deal modify da liu wu liu van yu alternative auxiliary reversible new chain move routine show chain reversible alternative name yu base despite perturbation negligible relative drawing concrete liu wu van operator condition close spectrum consist small addition eigenvalue dominate uniformly da gold standard monte hope strong quantify auxiliary degenerate start illustrate huge gain new involve sample take j negative course mixture thus make dirichlet proper let denote observe result posterior highly modal matrix note maximum else da introduce augment level prior eq consist use conditional call da chain simplex move mode lee switch movement applicable spectrum operator fs chain dominate illustrate extent switch speed study fs chain toy get frequently eigenvalue monte carlo conclusions converge affected sample organize brief operator use analyze reversible chain da appear review proof fs fs chain example eigen special equip irreducible hilbert x xx dx dx act follow show adjoint exist ig il eigenvalue eigen eigen define application jensen negative univariate define p extend quantity call define liu particular geometrically ergodic driven geometrically central theorem asymptotically standard estimate ergodicity reversible operator rather call whose probably satisfy eigen solution less da suppose integral intractable direct simulation indirect liu important spectrum da dx dx dx yy dy kf quite compact operator function ce particularly compact iii accumulation eigenvalue compact along eigenvalue define conjugate da chain geometrically ergodic finite indeed eigen chain start stationary liu conjugate calculate integrable function abuse double inner norm exploit
factorization joint probability depend q therefore presence arcs variability whole indicator goodness bayesian criterion develop call summarize interest difficult value marginal success distribution collection simultaneous subset vector useful depend approximation multivariate bernoulli vector consider link correlation univariate bernoulli bernoulli eq complete variable independent pair conversely equal pair eq independence sigma sigma set correspondence analogous variable apply disjoint subset bernoulli bernoulli l bernoulli variable bernoulli dependency uniquely identify probability bernoulli covariance matrix two zero imply several moment attain minimum eigenvalue show multivariate let negative prof hold eq simplex preserve whose variate moment underlie unique direct arcs state naturally bernoulli include bernoulli via dag undirected empirical use obtain variability network structure undirecte simplify bootstrap variability structure learn sample outcome frequency moment bernoulli structure bad outcome independently half univariate spread normality multivariate associate unstable network structure due easy equation hadamard determinant non respective maxima generalize maximum reach equal rank instead norm structure rewrite multiplier method coincide associate network high distance case second moment generalized variance frobenius matrix three matrix limit statistic interested relate total achieve hypothesis significance correction inequality distribution respective correction one compute correction bold associate frobenius spectral decomposition see significance value unlike display sample raw asymptotic statistic compute bernoulli specify diagonal matrix completely marginal normalize compute monte eq evaluate remove distortion cause lack problematic dimension low bootstrap carlo small network pt multivariate bernoulli significance various size bootstrap use derive property along arcs ph school article give useful suggestion thank pure university help respectively frobenius property maxima depend multiplier multivariate bernoulli prevent direct interpretation quantity matrix statistic k line global correspond
result actor ii likely connect actor mixed membership case actor mix role within affinity actor obvious discrepancy truth percent actor percent still retain employ different scheme mixed possess membership could accuracy mixed vertex difference display ground estimate metric result ten time error bar perform slightly significant difference model compute experiment question simple log derive via table goodness fit model dominate dirichlet normal assess fitness consist time point remain role furthermore network adjacent point certain similarity compatibility display figure performance average membership vector value percent suggest indeed integrate test confirm membership grey learn role compatibility entry arcs value outside example record social ranking toward study major member interesting look separation undirected vice static point researcher study static fit role select estimation label correspond mark member place place demonstrate estimate role compatibility appear intra pure role boundary leave bic score suggest role compatibility network vary role infer illustrate big change mixed membership time occur overall dominant early mixed membership grouping static role dynamic roughly role later besides support member change time membership trend member isolate finally lead email communication email process email network email pattern property gene engineering fit coarse quality dl life among high dynamic role evolve plot trajectory wide simplex dl diverse pattern formation axis specification system compound development relate heart role invariant diverse gene cluster cluster combination across evolve biological role consist functional role group different gene functional group heart cell development fourth role development study level ensemble path node mixed blockmodel propose relation dynamic social biological role independent structure actor network role third membership actor vary provide extra expressive network rich temporal practice proper development fulfil move cycle evolve change drastically stage process specification posterior axis may dominant many gene interact various aspect hence lead understanding ingredient prior membership context diagonal dependency structure role clearly readily couple tracking role drawback therefore learn develop necessity role indicator possible role focus develop efficient thousand rather million appropriate web offer want economic extend explicitly clique state enforce mixed membership interesting derivation simplicity drop subscript eq q derivative ks kx exponent x tx tx ts x second derivative k jensen apply approximation specifically analytical set derivative mle approximation likelihood tractable bind log grant nsf cs ii award fellowship dynamic social biological environment actor systematic evolve offer infer actor underlie topology build mixed membership blockmodel many actor account interaction actor allow actor behave differently carry interact reality approximate communication gene interaction network full case pattern dynamic role actor fundamental form science entity also actor communication study reveal position pattern biological evolve investigate statistical inference evolve vocabulary imaging network internal network gene network relationship vertice event evolve terminate stochastically semantic static stand role biological entity affinity compatibility algorithm infer dynamically concern evolve episode major therefore behind change email communication network company record perhaps behavioral trend business time span development capture inference function system social company capture role individual interaction among community behavioral process biology translate latent genetic interact protein topology molecular advance understanding mechanism biological broadly lead consequence diffusion hierarchy organization formation appropriately simulate mechanism discover change actor network network various trend network include free formal characterization characterize detect characterize additionally progress traditionally toward semantic review major limitation modeling actor biological label etc interact actor realistic role multiple influence play actor role actor response exist relationship stochastic molecular systematic distribution understand biological process phenomenon capture actor role evolve modify enable link specific connection separately fractional role capture thereby actor statistically infer embed membership characteristic membership intuitive community model latent infer actor biological entity function observe actor position dimensional simplex role actor role functional actor among actor reflect distance actor dynamic process drive evolution furthermore tracking position actor blockmodel emission shall short allow infer role result membership wise microarray actor remain briefly study simulation model algebraic derivation vast body network traditionally focus exponential generative cause actor position latent base latent explore membership blockmodel idea model node belong multiple block e fractional population model analysis membership project aspect space normalize reflect weight aspect role etc serve surrogate develop early role network use aforementioned actor multinomial actor role sample interact actor interaction may contexts interaction protein context space tracking trajectory infer entity termination underlie adopt extract text author network network invariant j possibly link vertex class relationship point vertex role realize predefine latent single membership paper different membership stochastically role role role compatibility realize role interaction mechanism link pair actor actor interact interact specifically different role pair role actor unique actor role combination basic mixed membership blockmodel static mixed interact vertex draw indicator role j j ji j unit draw specifically generative define among vertex reflect latent e identity link actor actor existence link package send person vertex label undirecte ignore semantic capture membership unique interaction strength interaction compatibility pair membership actor actor expect role role actor actor different interact neighbor role affinity role dominate actor role connect differential preference role rich pattern role compatibility complex role represent capture probability actor actor actor membership membership employ multinomial nontrivial correlation among within role role cell employ logistic simplex result normal logistic membership break draw simplex follow constant constrain parameter freedom need draw leave description model role every vertex compatibility evolve conditioning sequence trajectory function state static logistic random mixed compatibility relationship behavior generative transformation mix adjacent membership represent transition shape trajectory model emission define dynamic propose dynamical change network entity sense termination function topology generative graphical membership k k compatibility coefficient compatibility subsequent compatibility probability via point dimensional tb link assume actor mean evolve accord topic unlike outline mixed directly kalman smoother intermediate principle membership capture vertice dynamic simple walk membership compatibility semantic membership unlikely time expect membership difficulty base prior vector intractable additional logistic normal direct infeasible mean approximate b factor show expectation variational approximate marginal apparent description subsequent variational simple building simplicity inference role observation role indicator latent marginalization hide intractable approximation update continue formula multinomial multinomial therefore laplace base possible previous number repeat multiple pick simplest via em style current maximize log posterior intractable use update formula em derivation
plain see graphical refer problem restriction generality correspond independence graph follow edge require dependent detail copy analyze infer regression require global put later penalty likelihood obtain consistent edge glasso mutual neighborhood fisher matrix counterpart generalizing via infer graphical general condition closely relate pursuit regression conjecture use glasso study section also glasso limited restriction via consider many regression recognize hand behave problem glasso study adaptive strong every pair irrelevant recover correct selection assume mutual incoherence pairwise eigenvalue examine property procedure relaxed design design parameter know framework multi study probability rather analyze ensemble satisfies incoherence open lower sample assume adaptive general property present restrict section estimator design design consequence distinguish fix design proposal use lasso variable make assumption relate denote eigenvalue throughout consequence bound constant describe assume allow use notation fix row index active behave throughout hold imply also relevant property initial true hold optimal argue set hence multiple unique emphasize require random require upper use nice estimator section selector could restrict design moreover hence proof analyze later section adaptive standard lasso specify sensible view build work assumption formalize therein selector condition introduce integer vector index value outside integer hold often fix condition adaptive lasso possible eigenvalue result argument know size positive definite cover sparsity corollary condition follow restriction sparsity weak assumption derive discussion arguably upper thresholding follow hold set large restrict every regression pursuit assumption model assumption among derive adaptive correctly relevant ordinary stability sufficient easy stability eigenvalue high relation former require assumption restrictive stability appear trivial condition certainly derive additional thorough relation direction frequent appearance dimensional reasoning check analyze glasso algorithm clarity denote parameter weight solve distinction assume former pre specify latter slightly strong support sign pattern weight design matrix ordinary lasso coincide impose lasso incoherence let nonempty j general recover sign furthermore statement define q variable n ny c bound throughout rest follow plug fact kx large union bind unique c p crucially loss show bound lasso triangle fix satisfy strong hold hence ks ks x conclude finish check invoke finish regard dominate k design shorthand hand coordinate assumption ks ks design subset perturbation design set define hold subset ks ks ks sparsity hold proposition invoke finish sufficient event let submatrix j positive invertible weighted hold weight I solution j subgradient simultaneously subgradient p cs w cs c direction reverse guarantee q simultaneously hence similar detail illustrative adaptive program w eq sparse standard uniqueness definite elsewhere define relevant j x x x condition event prove least straight hence finish check incoherence condition invoke exclude event incoherence satisfied need concerned sn shorthand event design term section thm thm thm pt van f two procedure dimensional model accomplish gain stream feasibility provable penalization scenario easy convex oracle require condition design refer coherence compatibility restrictive restriction severe penalization shrink also correspond become infeasible alternative multi procedure example adaptive loose penalization analyze gaussian framework
diameter horizon ball arbitrary x play bx total tb plus term reflect location metric uncertainty insufficient active distance follow confidence reward pre ucb payoff center ucb obtain observation refine use observation ball well ucb expect reward ball ensure property ball activate allow accumulate activate ball dominate active relevant context ball round oracle call current ball let running time oracle implementation run similarity upper ball prove third must played ball put tr rr jj start ensure several maintain confidence time active domain ball cover similarity space ball radius immediate ball cover ball radius activate center notation active ball center set x increment hoeffding n tn tb n ucb clean run heart clean active b activation put piece together tb clean activate activation lemma claim simply ball ready ball activate throughout parent ball activate round active ball activate select moreover corollary activate center lie distance another fix round select regret contextual provide difficulty correspond definition carry without add fold matter crucial ball accomplish center form packing make efficient radius active ball full child give bound constant similarity active radius potentially much radius ball full center pack additional assign arrival call packing packing fall region tailor application informally context optimal suboptimal radius ball center select round contribute contextual therefore consistent r rr consider payoff contextual contextual absolute dimension cover modify activate ball radius become ball full become ball child factor give upper pick upper uniformly must possible namely pack mab stochastic payoff contextual bandit theorem extension context free payoff payoff parameterize pack collection pack number packing note r n metric point context similarly arm sequence permutation repeat problem instance payoff define completes summarize regret choose lower contribute contextual regret proceed lemma derive arbitrary horizon increase function write tn mab condition brevity write loss exist p rp simplify define whenever obtain contextual random contextual consider separately form free bandit round payoff instance pick informally know full reveal contextual context independently expectation respective remain handle separately draw use divergence arm exactly instance stand alone theorem fix ensemble describe mab slow change stochastically evolve bandit discuss recent learning publish free mab payoff specifically payoff round priori known algorithm adapt change converge arm mab problem maximize payoff context expect payoff benchmark see dynamic regret sublinear time term performance quantify dynamic goal limit bind payoff mab payoff arrival payoff specialized arm constraint q mab long performance contextual suitably horizon every round version prevent strong provable provable provide theorem tractable mab contextual period whenever period analyze period take obtain ok theorem packing number time modify section omit corollary contextual stochastic payoff combine temporal time across arm mab literature payoff variation analog regret depend cover arm mab problem volatility arm average ok corollary periodic contextual first bind covering cover arm work former take bind easily arm generalize mab walk payoff evolve uniform analysis particular name mab defer version call mab marginal assume payoff evolve stochastic mutually marginal arm dynamic q strong uniform period ok interestingly whereas seem mab walk walk stay boundary assume require accord walk contextual space q corollary dynamic arm parameterize round arm bandit payoff result mab arrival every arm pair lipschitz set high index payoff easy mab payoff arm payoff e arm context arm follow sum contextual mab problem extend bandit incorporate contextual similarity preliminary publication apply bandit interestingly contexts motivated web round user rank document click document namely relevant specify vector document minimize click extension user style connect expect click separate contextual document sequentially slot round document relevant treat algorithm condition expect click suffice guarantee lipschitz corollary contextual stochastically evolve payoff corollary dynamic periodic non periodic mab show benchmark payoff lipschitz likewise let let result provide completeness conditional hold let obtain corollary implicit rt namely sake r see definition fix pack rt rt six fix time marginal least claim fix exist claim follow show winner winner proof eq q take possible simple namely ok tt suffice periodic payoff satisfy constraint consider failure implicit corollary plug section maintain partition thus advantage algorithm bandit call reveal pick observe payoff section arm payoff round specific fix adversary advance generalize regret free mab mab arm tie break define metric arm loss metric reduce stochastic setting set context mab randomize statement achieve cover arm space flexible leverage distribution take bandit line bandit payoff payoff achieve notion quantify guarantee refine outlier cover cover slack mab instance fix bandit multipli slack remark version raw number page contextual parameterize bandit use subroutine finite collection ball ball radius stay instance create parameterized radius proceed call report time round ball break activate specifically see space diameter set data ball counter loop b b else x bb bx specifically activate activation active contradict continue satisfy basic ball active radius separation immediate note specification round activate child ball hand else activate contradiction child active ball within fix horizon contextual ball ball select regret claim definition collection full ball radius radius say full heavy exist heavy ball round specification claim q ball q plug derive payoff contribution maintain adaptively partition time refer resolve arm pair either slowly nearly invariant payoff special change adversarial payoff maintain partition context adaptively advantage payoff non choice tailor concern requirement quality provable would weak version obtain several result make meaningful would suffice setting upper difference payoff arm pair want provable contextual publish publication question payoff desirable payoff context knowledge author armed manuscript comment anonymous improve presentation analyze construction kl divergence technique implicit stand alone contain along relevant definition section minor feasible payoff function function rely subset child feasible payoff exist coincide payoff lie play arm round incur subset child ball payoff end subtree root feasible mab payoff bandit algorithm regret payoff assign payoff arm preliminary manuscript early address issue point journal mab round receive associate case strategy focus exponentially recent literature similarity extension round payoff round contextual motivated place crucial particularly similarity information bandit arm payoff prior bandit similarity approximate payoff context advantage central fine partition several mab bandit descriptor abstract bandit henceforth armed bandit mab present round past payoff operation economic clean exploitation trade crucial sequential uncertainty regret optimal mab arm understand arm infinitely unless assumption bandit need find investigation discussion work assume certain assumption efficient work start arm mab give contextual bandit directly place crucial cast ad payoff click page perhaps follow bandit simple bind contextual arm round follow choose payoff fix expectation subsequent definition set whereas payoff mab set demand arm free space call lipschitz word lipschitz without generality generality context metric mab similarity context suggest choose create bandit adjust horizon box mab pick run point horizon regret idea adversarial payoff cover space arm potentially payoff adaptive partition adjust frequently occur context instance payoff two main payoff payoff payoff region correspond frequently occur region prior space maintain one context arm develop provable payoff match uniform obtain matching bound divergence fix upper bind mab contextual ad scenario per se incorporate spatial constraint contextual meaningful recover one context contextual recover publication version contextual bandit constraint arm context study mab payoff random payoff case problem choose mapping context case arm applicable contextual mab essentially algorithm match use partition expert bind distinct translate context reduce mab define mab payoff notation sequence context arm break way ensure attain assume similarity hold space cover related paper diameter less minimal covering similarly relate packing point maximal packing packing cover namely small follow know introduce know dd euclidean radius
display likelihood simulation technique public explain illustrate need analysis tool practical side grow complexity available approach handle identifiable infer graphical estimation dense time markov bayesian perspective support presentation comment acknowledgement relevance unity neutral agnostic manner bayesian handling model increase proportion past irrelevant keyword choice informative computational statement believe valuable bayesian toolbox elaborate argument science paradigm perspective I toolbox set theoretical ensure coherence procedure proper encourage historical people jeffreys answer greatly towards bayesian procedure popularity even trick bring keep mostly usual already text selective run support advance practical past anti rational asymptotic normal conjugate first statistical much book bayes incorporate seem class parametric mostly prior parametric much acceptable work common parametric nonetheless approximation choice obviously state handle model really give reasonable answer minimal type go beyond bayesian question draw run obviously opponent inference impossible immediately discussion paradigm operate truth true production parameter since even inferential optimality alternative model agreement demonstrate coherent statistical relative support drawback paradigm contrary strength thing true reference besides belief test analysis check little reason besides name state theorem toy position wide jeffreys stein optimality bayes surprisingly bayes express formalism economic constant ignore inference validate measure conditioning approach give mean properly datum report engine raise experience inference view decision automatically derive automate derivation optimality explicitly search square minimax sometimes also economic utility bayesian whole update recurrent bayesian perspective whole inferential upon straightforward come decision answer take impact would plus bayesian allow infinite range item advantage incorporate focus prior e surprisingly completely parameter thing informative expect moderately remain probability make particular generalised manner space mathematically mathematical fail criterion family like haar mention prior rely classical difficulty natural factor rich bic bayesian law test time specific contrary strongly bayes factor contrary maintain specify theoretic rarely improper prior mostly set lack proper constant solution domain ad hypothesis whether inferential question justify assign type hypothesis define factor direct pearson fisher namely coverage percent interval live coverage thing question frequentist jeffreys occur theoretic relate classical consequence nan must subsequent towards decision theoretic perspective posterior bayesian bayesian se acceptable cover seem I fundamentally sound since statistical limitation fisher paradigm impose perspective measurable hypothesis sum measurable mathematically probability co classical belief dimensionality nest contain vice versa bayes properly applicable interpret point apparent absence standard output illustrate default bayes bayes standard lm obviously answer else frequentist strength region bf intercept strong substantial ten bayesian provide carlo computer recently extreme elaborate less mind reason simulation past decade detail computational seem infinite inferential assessment simulation discussion allow inferential question nature namely generic principle simulation abc confusion reliability development allow mcmc essence method output convergence apply mcmc sample help stationarity therefore happen similarly software fail important numerical little often detect together
infect individual ultimately infect probability avoid infection simplicity community infection dependency inherent avoid infection argument hold z infection new final describe model natural parameter original new size size hasting algorithm three give center size period epidemic outside quite result final outcome set example median l simulate giving decrease community dataset c true mean l table result point table size alone sufficient effective precise sense former low latter reduce even dataset threshold similar kind finding sir affect extent individually dataset conversely four experience hard infection drive epidemic reflect complete conversely correlation distinguish community infection estimation interesting note likelihood poorly consider final estimate nothing either true average seem likely occur surface need really level model mix ignore e group zero final hope considerable second consider divide belong two kind model population secondary contact go school go individual child allocation allocate school child school school allocate go respectively go obviously numerous possible allocation seem impact model approximate branching epidemic epidemic group community transmission ignore recursive formulae e g p initially become infected course epidemic far mean initially ever infect early epidemic receive external contact average total individual infect infected therefore type individual matrix denote number individual community contact child half period infect equal school child infect equal child plus child infect school infect school probability number school school similarly large work place school ever infect ever infect individual ever school child respective infection child infection transmission within exactly infected explicitly number infect school place community child ultimately infect number child infect infection event second epidemic child behave individual infect initially individual adjust type child infect eventually infect amount final treat complete final size perform analysis key finding use follow simulate epidemic typical parameter infect comprise child difference infect purely random identical dataset except result epidemic far infect comprise child c mean mle complete final size pt median median mle remark parameter analyse table broadly deviation especially suggest comes associate less form infection picture transmission occur infection within contact school community datum great data utility datum affect former attack infect information deviation datum large explanation arise detailed consideration dataset particular child individual whose infect contact infect individual know conversely possible infection argument apply analysis already far source infection would precision heterogeneity estimation heterogeneity compare size purely question study try question one consider intermediate final kind enable considerable precision distinguish infection temporal epidemic distinguish community occur secondary group structure homogeneous factor generally threshold change ignore turn level would affected level mix unity could natural two individual similar seem broad qualitative finding unlikely basic acknowledgement partly uk engineering sciences ep pt minus pt pt epidemic secondary typically school place accord infection parameter understanding infer precision thing considerable inferential heterogeneity considerable keyword number inference epidemic disease classical mathematical epidemic mix individual disease e rarely reflect disease propagation therein broadly speak g contact structure child heterogeneity chapter cause assume mix within g considerable essential population mathematical address issue various disease propagation example age structure school location national characteristic period possible kind study intensive simulation identify disease realistic assign plausible inform study mix population large precisely enough relative transmission transmission measure school restriction aim precisely reduce transmission al recent transmission longitudinal report school transmission child intermediate main aim establish procedure longitudinal use assess simulate epidemic incorporate assume two kind epidemic time kind maximal actual mix child former go belong belong paper epidemic derive epidemic devote community finish community group school sequel terminology belong consist applie label individual thought type potential difference similar equally apart behaviour individual individual remove contract individual capable disease remove individual matrix sized group transmission contain ever infect ball final infect initial recall threshold section complicated turn statistical kind I individual throughout disease start epidemic knowledge initially motivation extreme scenario gain extreme evaluate collection social know secondary grouping method extend take account likelihood leave limit th infection infection infect period contact straightforward potentially censor last carry period transmission product contact respect parameter ji jt jt j j ds ji jt I cm ds extend equally plus number number define hasting obtain
although formulate operation global optimum still need initialization multiple neighboring class state e c gradually increase well mechanism explain organized notation motivate section explain derive present latent lda conclude assign latent th denote element available assignment assignment datum indicator vector product product kronecker matrix b lb available class state quantum use extension conventional call classical statistic example well indicator indicate occurrence indicate diagonal probability introduce quantum quantum real however trace finite distribution specifie unit quantum correspondence generalize distribution mixture third employ phenomenon involve vector diagonal vector therefore quantum uncertainty expect useful vb variant proportion maintain term mixture via uncertainty introduce describe enhance generalization explain apply define indicator density log posterior conditional introducing posteriori variational free maximize temperature follow identity classical explain approximated posteriori distribution maximize assignment state approximately see j boundary accurately take completely sum free energy approach index interaction initialization label update indicate class explain b quantum denote inner update temperature decrease field estimate projection implicit precise extract solve whose latent allocation probabilistic corpus document vocabulary schedule temperature process schedule temperature simulation obtain mathematically rigorous consider schedule paper schedule schedule low inverse temperature markov varied fig run five average amount five repetition batch fair term execution time average execution outer iteration iteration try lda novel increase interaction limit work think well eq accurately quantum variational bayes vb simulate annealing vb density generalize add look demonstrate allocation actually typical poor complexity projection class work construction quantum suitable schedule mixture gaussian aid scientific work research area physics material grant generation super project program thank institute physics university google institute physics university technology university present anneal variational easy implement find free latent allocation relate machine quantum mechanic conduct density adjoint one connect mechanic learning maximization concept quantum use quantum mechanic generalize density bayes
fan chen li sir popular regression chen j model ann quasi unknown j regression graphic idea study regression graphic new york li ann comment dimension smooth noisy spline functional measure explanatory fan local regression pursuit ann p index partially york demand air environmental economics management spline coefficient ann ann li projective resample reduction li data visualization stein li reduction response li asymptotics ann york york convergence partially p york university ann adapt link ann h li l estimation reduction r distribution yu spline linear l asymptotics ng check confidence partially index x discretization manuscript university simulation angles eps bandwidth bandwidth dot htbp scale pt plus true linear model wang california china china associate model link function index parameter model establish estimate convergence error variance result facilitate region study application illustrate extension index primary g bandwidth attract combine nonparametric recent comprehensive book curse dimensionality accommodate covariate reduction assume collapse single nonparametric partial index specifically response index component metric index inverse sir li li discrete fan assume yu confirm also yu difficulty employ link fall dimensional flexible smooth h mild suffice often reduction justification suffice summarize predict reality select analyze census area build eight describe neighborhood describe center air covariate specify house binary demonstrate chen li variable interpretable drop consider datum part reduction structure note component detailed procedure et estimate sequentially simple optimally approach plug lead difficulty come remove impose identifiability positive approach sir resample li estimate smooth residual get estimation accomplish sir proceed via residual since plug employ square result employ obtain estimate conclude role estimate specifically regression indicate efficient importantly equation normality equation perform well procedure directly target iteration convergence asymptotic normality attractive method limit variance compare h reduce function h small limit provide asymptotic elaborate present asymptotic report proof independent identically error general explore stage elaborate dimension versus estimator smooth perform versus initial dimension find use profile new residual update step complete theoretically practical iterate simulation report benefit iterate elaborate case index sir response need recommend reduction li already choose smooth fan link bandwidth sequence minimize estimating calculation smooth q estimate replace smooth estimator correspond x smooth bandwidth minimize respect linear smooth need later achieve rate relate specify asymptotic normality estimator fix estimator nonparametric reality possibility term partial spline correlate profile smoother short obtain desirable smooth express estimate advantage current estimator g regression constraint hence include et et model worth variance estimator asymptotic estimating asymptotically efficiency attribute make least possible solution equation true tp remove yu obtain motivate estimate least estimation theorem al final n final theorem follow additional need avoid curse high smooth index moreover asymptotic straightforward beyond scope assume extend great smooth remain unchanged except change sir sir save employ sir perhaps sir major intensive sir difficulty covariate dimensional dimension may hold show li sir save li correction contrast sir direction sir employ conceptually however per wang sir obtain asymptotic regard two theorem estimator normality estimator condition hold r convergence rate corollary give optimal convergence estimator obtain result quantile plug need follow x ig theorem q theorem let quantile far great smooth multivariate estimator sir sample sir use slice slice scalar uniform probability case one choose check scenario orthogonal throughout bivariate save computing pilot reveal residual estimate frequently select generalize utilize estimator bandwidth bandwidth minimize bandwidth calculate bandwidth correct magnitude ng note satisfy ii sir notation sir sir slice estimate iterate sd mse last serve gold column iterate iterate column table table pursuit sir direction sir might iterated typically estimation angle lead far sir fix replicate plot seem parametric chen partially model investigation try et procedure use unable meaningful seven difficulty analyze mention determine variable describe census median census variable covariate final age coefficient hypothesis versus coefficient determination estimate chen li examine drop fit sir inverse meet single choice well thus transformation describe sir met either try suggest satisfy sir sir well new handle slice sir lead slice sample much sir slice lead slice chen li point sir sensitive slice try slice estimate small bandwidth choose sir approach chen li high sir statistic reduction estimate yield significant inclusion minor show estimate axis effective variate variate make transformation use smoothing theorem proof divide appearance denote show central easy divide step existence prove normality condition c expression z obtain minimize much separately existence enough omit ii eq follow equation r r definition recall j decomposition page find complement far cc cc cc cc r cc cc r asymptotic give definite inequality easy proof vector theorem imply complete corollary
product q identity counterpart motivate penalty propose drive far call orthonormal coincide component form jx programming desirable theoretical property problematic thorough overview combinatorial estimation estimation machine estimate penalty let index indicator inequality user hold least first inequality coherence number satisfy oracle coherence require bind correlation cf almost suggest maximal local satisfy f deal gram entry clearly obtain immediate gram sparsity assertion condition valid gram simple frame consider necessarily vice minimal eigenvalue choice without modification useful wavelet drive corollary choice remain valid replace deterministic inequality replace event n j triangle hoeffding sum independent view lemma leave f f f apply inequality easily proof theorem analogue j j theorem q side replace modification rest identical cauchy schwarz j yield xy proof sum jx f ef j xy conclude exponential inequality independent j l n f r x r x nr represent mixture density identification convenient normalize write form q unknown clarity exposition simplify weight cf clarity exposition upper recall true paper general sparsity dimension strong mind dimension model valid equal require correct selection typically mixture density specifically use estimate mixture substantially regard deal identify close let component investigate need ensure quantify mixture quantify follow I restrict nonnegative inspection result section indeed introduce burden section replace suppose belong probability lemma f kx j k hoeffding recall f consider event index elsewhere pt recall position reasoning sum apply rl lk collect bound conclude density mean q identify density square euclidean large eq apply density grid approximation unknown target identify correctly index constant compactly support wavelet basis compactly assume orthonormal n nm nj k converge follow probability constant allow attain minimax logarithmic simultaneously c various old appropriately basis reference condition inequality find probability expression rate logarithmic factor satisfy construction logarithmic pointed logarithmic sometimes asymptotic constant cite therein class density uniformly fix transformation drawback rate class densitie q wavelet assumption depend minimax obtain minimization study differentiable adopt descent instead descent technique start step choose order find direction optimum observation norm optimum derivative th coordinate become iterative coordinate minimization direction minimization procedure apply quantity detail drive choose candidate validation describe principle construction avoid preliminary give parameter tuning proceed queue general queue queue back queue queue method experimentally discussion times accuracy dimension whole partition candidate determine index criterion jj ji pl final estimator take need slightly approximation density good practical validation goal validate bic bic array numerical context beyond scope research simulation investigate ability tuning parameter choose ii identify component conduct density random gaussian choice true weight consider bar begin evaluate accuracy estimate investigate relative mixture panel panel consider three size vary increase significantly increase large size investigate ability mixture figure percentage time mixture find versus consider correct accordance recall indeed simulation need identification l finally percentage small result requirement small happen another present sufficient see error crucial suggest percentage time correctly decrease function dimensional left center least density choose obtain candidate density circle exactly finite component natural exactly approximate isotropic many application reflect sufficient constitute crucial step analysis offer approximation application demand constraint determine weight display right figure successfully approximate number minimizer minimizer nf kx rv k v k minimizer nf kx f f lr nf kx k lr coordinate fact therefore event prove assertion component minimizer minimizer recall distinct point minima constant jx derivative mx jx mx continuity j lemma newton institute
delay error sf denote time sf case analysis bias ci range mse avg ci ci avg ci ci range mse avg sf pair test sign test sign test sf sf sf sf sf sf sf minimum apply version sf sf contrary scale noise short delay day significance ci sf statistic therefore accurate method sf measurement hence practice wiener conclusion outperform method bad stress optical novel automatic delay successful drive result promise approach involve one way procedure see fitness technique may fitness costly moreover approach presence uncertainty test huge next scale monitoring project dedicate http www http edu produce multiply available effort currently one would automate cope gap develop estimate represent delay context distant mix within evolutionary artificial detailed delay day readily optical monitoring involve regression statistical evolutionary algorithm delay delay arrival path observer time optical distant light nearby fact come get pass massive galaxy observer receive various direction phenomenon massive object source sequence path delay depend direct method universe often dark scenario underlie pattern time intensity gets delay corrupt observational sample possibly gap availability weather system currently long period delay claim multiply discover claim generate currently delay common employ optical multiply observation inherent modification largely optical well novel evolutionary delay fitness mse novel procedure decomposition integer direction evolutionary introduce form novel principle automatic propose drive iv study evolutionary optimisation devoted type come optimisation series instrumental fail etc sample availability influence collect way assess dispersion spectra b refer delay generate employ artificial wiener outline justify also significant method observation observe several describe outline conclusion future optical monitoring usa measure filter table sample day bc observe image b measure filter measurement deviation std bar weather scheduling monitoring monitoring peak light curve day correspond c time error day day day delay real attempt generate synthetic performance method kind large wiener delay offset optical et five set gap l h without gap bar correspond ds five bar set function variance represent sample periodic gap monitor eight series delay delay shift day shift model noise represent true day fig noise shift bar previous introduce approach paper detail derivation repeat detail section come monitor either multiplication offset image optical option time model zero underlie light curve whereas eq delay image generalise regression superposition kernel width imply width delay light curve image specific goodness square involved eqs variable may system time involve inversion decomposition svd tell may artificial optical singular pattern find fall change estimate fall range furthermore fit inside outside none aim fall review c axis versus singular good evaluate trial range noise gap increment concern pattern true actual optical unitary increment table formulation optimisation optimisation discrete variable force validation apart deal consume manner section estimate delay landscape unitary increment optical band explain landscape set hill search also local show combination shift follow three ii width value fitness loss follow avoid minima refer square validation might apply artificial genetic mutation population generation well accord fitness block include validation compute mse artificial global evolution evaluate generation link individual correspond use employ represent population fitness population sub population index individual population individually mutation link fitness repeat procedure mutation double mutation mutation population individual unless evolutionary good refer hereafter unless mixed type kind population fitness real artificial dispersion spectra width observational large spectra involve nearby correlation real observational synthetic wiener observational optical outline evolve integer fitness ten run table good solution individual accord column generation day approach es gray neighbourhood mean parent select produce evolutionary es choose es fitness function fitness superior benchmark problem world medical variable allowing reach fitness fitness variable es mse c run day fall pattern crucial estimation omit yield estimate regardless es iteration evaluation fitness tend iteration across every correspond fitness es artificial since fitness similar objective small measure day day table however day q report predict therefore delay increment ratio true value light curve besides validate range highlight delay quantity delay mean estimator delay estimate square delay absolute eq ci l r ci ci range also hypothesis estimate group underlie gap student zero significant show significance level value dot group highlight test significance nonparametric ht k table within ci group illustrate ci e c detail locate delay result group previous give grouping table observational explore various noise group level regardless well highlight bold table ht mse l mse performance pair delay pair bar represent delay
control subsequently identify approach outline note nan properly cat score centroid pool mean reduce cat cat extensive group comparison ranking refer cat centroid group gene respective score table discriminant function form pool centroid reduce definition weight splitting product page however selection simplify interpretation selection level center predictor interpretation involve feature typically substantial nan overall variable group cat feature greatly grouping contain cat lose usefulness lead instead exclude discriminant prediction rule fast inverse root stein estimator detail normalize nan summary employ fdr software apply cube transformation normalize shrinkage selection cat reference compare feature et investigate set cancer patient analyze summarize correspond plot gene fdr plot generalization box median prediction error balanced fold repetition control local small gene nan gene need include fdr cutoff gene figure note recommend feature differentially prediction balanced repetition split feature ranking estimate avoid select feature lda hc fdr based hc hc statistic maximum hc top feature hc limitation fdr shrinkage rule hc hc obtain fitting mixture nan cat employ remarkably hc imply framework hc choose large fdr hc threshold buffer dimensional discriminant analysis prediction contain stein predictor rank employ stein shrinkage estimator well perspective shrinkage improve cat rank among predictor cat lda dimensional score predictor note induced score differ procedure interesting propose efficient term high variable fdr lead inferior dimensional difficult discriminant analysis procedure hence shrinkage lda cat score computationally shrinkage accuracy typically intensive large lda cat rely stein shrink shrink empirical toward estimate propose shrinkage link present detail gene correlation soft selection bayes false discovery thus provide correction error modify lda stein shrinkage pool matrix employ greedy algorithm optimal criterion regularize ridge expense problem find unique discriminant selection public later critical comment helpful grateful von ki ki signal problem lda first pool centroid predictor give adjusted cat thresholde cat control third stein analytically overall effective shrinkage procedure implement package profile genomic study specific challenge see large prediction sophisticated proposal forest conceptually effective prediction applicable recognize essential specifically datum particular care need take covariance yet highly dimension zero employ discriminant classification high dimensional advantage straightforward selection relevant group multiclass mean centroid commonly software regularize multiclass gene procedure see essential ignore include imaging correlation spatial dependency pathway suggestion include approach version offer automatic correlation lack elegant feature optima paper framework high analysis base employ stein shrinkage train analytic fashion expensive resample second correlation score cat lda equation presence correlation third thresholding detail discriminant subsequently genomic conclude lda start multivariate centroid weight application bayes k lda evaluating choose maximize variable way prediction pool mean eq centroid group observation discriminant center interpret ratio completely equivalent careful simplify eq mahalanobis transform predictor make variance remarkable discriminant test much variable contribute group lda function centroid rely different stein shrinkage rule ridge estimator variance proportion frequency stein shrink analytically precise advantage stein rule large stein lda stein shrinkage compete approach vector lda adjust cat scale version pool adjust minus
graph remove associated uv uv argument follow kullback distinct st uv leibler st uv uv exactly uv uv st uv use base bound st cliques separation conclude uv claim apply require uv least construction conclude prove third ensemble second bind binary stay claim ensemble markov contain previously valid family large note certainly remove set z distinct uv pair distribution uv uv uv uv x uv expand recall st st x st establishes claim kullback divergence st uv st uv uv x uv x symmetry term decomposition x decomposition expectation st uv uv x x uv uv inequality apply bind shorthand denominator uv combine b correctness claim theorem maximum likelihood ml graph model describe provide deviation performance remainder term edge structural collection rescale likelihood give set uniquely choose attain conservative failure consequently decoder vanish deviation previously notation apply chernoff thereby obtain associate exploit deviation claim derive low divergence divergence graph make recall theoretic terminology match distinct matching pair comment great auxiliary elementary hoeffding vertex st sample hoeffding yield st st edge vertex u distribution unnormalized obtain sum fix ising define distribution manner lemma hence inequality yield analogous exploit bind quantity quantity involve bound obtain tv tv definition neighborhood uv combine bind theoretic prove four main necessary demonstrate succeed succeed characterization constant remain minor gap open immediate summarize gap binary appealing computational theoretically binary technique construct variant deviation apply would interesting extension direction acknowledgement nsf grant recall divergence uv definition uv via proof contradiction contradiction unnormalize little shorthand notation xx st appendix characterize change configuration agree definition observe original assumption sum yield root denote achieve quadratic formula since use uv equation u contradiction root contradict remain exploit lemma appendix distinct sx st ss set definition namely notation proceed proof contrary stay fix note add together yield equality equation note ix ij c inequalities st st thereby complete cm proposition corollary example ex ex em p decoder succeed family similarly exhibit fail succeed nsf dms preliminary result international ccc j department berkeley berkeley structure social associated graph independence recover domain attract search vertex reduce problem hand theoretic constraint base relaxation analyze selection penalize form particular increase address problem allow vertex sample line consistency method variant pc graphical practically appeal interest graphical selection identically sample size index triplet goal address triplet correct conversely triplet although applicable issue analysis paper field physics phenomenon model analysis ise recover approximate class perspective view observation channel parametric accordingly develop relate kullback leibler divergence statistical capacity practically way computationally theoretic optimal constant reveal date motivate indeed type sufficient edge vertex model theorem indirect precise consequence devoted proof whereas devoted condition discussion reader throughout standard constant notation mean background field formulation degree denote edge variable vertex specify graph ise form factor nature tx little calculation conditional x physics physical phenomenon modeling network represent vote negative edge impose collection structural depend property obviously mask three ccccc field parameter arbitrarily separate ise model identifiable nonetheless finite identical distinguish require exponentially markov field parameterize low maximum neighborhood weight minimum satisfy analogous belong class markov observe definition precisely consider refer loss risk probability incorrect scaling sample specifically size maximum decoder variant edge decoder know decoder graph know unknown variant low discussion follow sufficient begin state upper weight let comment regard interpretation consequence weight compactly observation scaling make although dependence refined increase grow case grow wish impose family method recover correct roughly speak fisher distribution recover graph factor theoretic analogous upper decoder comment consequence sequence compactly weight subtle number construct form subgraph require least correct subtle compare graph total edge development sufficient class true small degree necessary case adversarial choice completely condition match discuss bound size complementary condition discuss exist give edge weight decoder worst graph statement sample compare theoretic scale require guarantee careful grow degree like scale exponentially wish growth follow family succeed probability use theorem provide bound within condition et guarantee restrictive incoherence sufficient case satisfy decoder unknown weight exist decoder condition condition imply exponentially parameter stay control discussion case scale minimum scale know decoder succeed estimator half section introduce sufficient state definition model quantify kullback leibler special ise kullback leibler kullback leibler two divergence symmetric natural secondly via note symmetric calculation divergence family st give straightforward leibl denote divergence summarize bound b turning ensures claim group vertex discard property component edge permutation use permutation vertex permutation
transition system paper one lr decaying law std annealing temperature complete dynamic way compare result organize description brief subsequently conclusion state ise hamiltonian spin site assume summation perform length take periodic lr hamiltonian lr interaction I order apply briefly discuss update metropolis carry choose possible dependence temperature critical evaluate fluctuation l spin autocorrelation critical uncorrelated ground state average start analyse expect short start uncorrelated al static exponent account large critical setting become q hold short length lattice law initial follow law critical exponent depend exponent equation hand randomly generate configuration initial exponent avoid correlation determination ct r std measurement started fully order one validity mean std temperature validity g ground configuration pointing anneal q scale law valid take logarithmic reduce evaluate critical get exponent slightly furthermore worth std free size effect finite truncate short regime investigate worth know influence critical time size find search equation bar assess close deviation fit number configuration detail finite size carry several see range distinguish source cause finite often obtain interaction physical depend effective temperature cause range simulation system interaction spin hamiltonian periodic condition carry relaxation effective law behaviour short figure sum overlap suitable temperature meanwhile analogy behind range scaling eq aid get value fit report obtain calculation transfer method already size enough critical within time statement complete show second fit power list exponent significantly principle expect depend short ise nevertheless discrepancy attribute ising perform adjacent effective critical derivative temperature obtain observable exhibit behaviour table exponent correspond one equilibrium solid fit aid correspond effective indicate htbp worth evaluate source insufficient statistic interval power estimation former observable fit measurement bar account account major error report bar value estimate std std exhibit temperature simulation use obtain evolution suitable power behaviour function figure bar estimate way htbp c contrast measurement perform measure size get exponent std value obtain std calculation asymptotic expansion yield relationship excellent std estimation ht average indicate configuration indicate configuration autocorrelation increase exponent figure fluctuation calculation function consequently bar time close obtain agreement estimation average indicate obtain exponent scale spin range panel hold result bar collapse form show space excellent agreement self result different cr tx rr panel configuration discuss extensive lr ise interaction decay regime affect effective temperature law contrast study range finite analysis order temperature agreement std estimation exponent measurement agreement two may monte carlo dynamics present report dynamic relevant critical long range evaluation dynamic critical work
algorithm design condition rip sparse good application requirement super overcomplete fine thus highly correlate relaxation inconsistent see necessity objective practically parsimonious model representation stable parsimonious account datum never encourage penalty convex norm maintain penalty statistical setup group group desire family penalty allow regularization scad addition penalty family application briefly summarize important absolutely setup soft thresholded solve coordinate view recently linear approximate least penalty guarantee may another popular dc solve nonconvex represent difference similarly neither apply penalty group address grouping concern assumption group penalty group penalize algorithm design attain trick penalty include scad predictor group less impose penalty contain conclusion apply mild organize introduce thresholde rigorous present concrete discuss high study ridge section propose selective super resolution reconstruction example methodology leave setup go assume observation fy il fy canonical link fisher group r want keep predictor whole super size singleton reduce associated function nonconvex zero great exist nuisance feature directly use build parsimonious nonconvex class thresholding rule solve somewhat interestingly convenient tackle viewpoint tool define rigorously real value odd unbounded shrinkage version monotonically increase introduce thresholding define group satisfy canonical avoid influence ambiguity thresholded correspond assumption rarely application easy norm sparsity refer turn solve problem dimensional k kt corresponding limit theorem threshold cover essentially penalty practical indicate matter predictor group arbitrarily simple guarantee appropriately glm glm rule glm classification except multiplication predictor identical approximate original penalize logistic guarantee hand experience small computation concrete bind seem implementation give example estimation figure illustration attain discrete penalty group glm scale comparison predictor net elastic net q solve penalty justify group predictor solution attain scad shrinkage mcp scad focus minimum pg g numerically give property function odd equal offer coefficient fy normal introduce outli intercept intercept vanish center involve inversion improve dimensional computation incorporate accelerate asynchronous recently update penalty convex original reduce dramatically must take account run correlate avoid greedy preliminary b perform iterative feature step threshold nonzero similar long reasonably maintain set safe much fast quantile screening independence base correlation apply penalize estimation penalty lasso scad penalty multi similar calibrate restrict predictor weight construct ml behave apply neither introduce hard penalty thresholding grid search look good validation seek performance understand ideal situation allow parameter setup norm penalization lie group prominent predictor predictor vary generate well tune combination measure performance simulate sde n fy run successful joint probability probability simulation much serious dimension intercept transform suffer power bp resolve cosine atom ignore resolution high similar high corrupted frequency performance additional effective error report goodness fit frequency joint bp hard ridge regression compute mm tune c large large ideal experiment validation observation tune hard penalty spectrum reconstruction ridge validation bic show minimum necessary start good regularization maximum see solve time acceptable tradeoff resolution cancer real cell patient normal probe label iterative quantile ridge small ridge penalize tune aic bic correction test near centroid penalize refer tune get optimistic cross outer evaluation classifier hard ridge behave give parsimonious htp c error mean median median aic identify parameter tune plot replication give select gene frequently appear visit triple bootstrapping annotation show associate solve arbitrarily group sparsity require group treatment condition
q measure tight boundedness w notion apply system irrespective boundedness trade uniformly boundedness b critical behavior critical mark transition boundedness operate infimum eqn obtain unstable operator computable bound critical relate probability unstable invertible clear upper proposition system operating boundedness unstable boundedness necessary operate whereas need unstable important boundedness operate strictly next main paper operate unique initial sequence convergence stochastically theorem system converge invariant irrespective operate distribution operating may guarantee invariant discuss implication system boundedness unstable note existence uniqueness boundedness verify operate invariant case proposition boundedness invariant measure denote closure dirac concentrated eqn natural dense unbounded study example show positive exhibit next devoted rely dynamical establishe complete proof prove subsection sided space dynamical st mapping xt xt iterate guarantee iterate non negative sided transformation purely see later iterate sense suitably algebra define sample sided sequence projection random family path side equip shift assumption dynamical iterate eqn follow transformation jointly virtue construction pair initial distribution iterate construct indeed iterate map construction probability investigate distributional distributional carry paper sequel generic one generic empty real banach solid banach order hold sequel space preserve sublinear eq strongly sublinear solution eqn almost equilibrium transformation preserve eq eqn iterate object technical convenience asymptotic analyzing lead understand asymptotic sequence possess u boundedness sequel boundedness back eq compact back family compact topological property conditionally result sublinear preserve banach cone strongly sublinear order preserve ergodic one equilibrium follow space preserve sublinear establish order eqn consider eqn fix preserve point obtain eqn definition topological clearly tt sx sp continuity permit sp tt tf f sp xx q eqn reverse closure eqn establish inclusion inductive unstable hold follow preserve property eqn eqn thus since interval detail structure scalar characterize support study scalar show dense interval highly subset self explain proposition prove interpret reflect proper scale restriction restriction alternatively restriction placing show dense interval yy yy ni start every use eqn q show inclusion reverse scale rigorous self explain describe nature large iterate system countable explain contain thorough studying pattern consecutive proposition visualization component separate one look see top version consecutive study ac assumption plot cumulative eigenvalue approach dirac entire concentrated cdf vary assertion ergodic measure attractive transition process example consequence moment probability obtain complementary say moment condition instance process long belong consequence moment need costly simulation generate empirically generation suffice positive iterate possess support open set empty measurable existence uniqueness situation possible unbounded functional sequence important operate invoke mean present study kalman lose analog channel sensor novel analysis steady filter critical steady arrival steady show characteristic latter combine ergodicity provide numerically steady amenable address general provide theoretical evolution paper control channel interaction via random proposition consequence also thus verify map eqn property weak every verify eqn linearity chebyshev part obvious ii trivial unconditional reach suboptimal estimate point ii unstable since invertible substitute eqn eq tx large eigenvalue zero indeed imply inequality possibly loose purpose follow mm mm theorem axiom problem summary htb paper equation arise arrival sublinear boundedness limit preserve strongly sublinear asymptotic random matrix converge exhibit boundedness arrival weak convergence operate arrival rate possess moment weak distribution closure countable general characterization sure ergodicity distribution non self name first great kalman filter linear kalman powerful steady consequence steady implementation time complicate naturally identify arise study consider sufficient filter conditionally finite instant need identify recursively recently wang interest system concern control namely sensor purpose area network characteristic channel additional source analog channel drop limitation limit quantization address fundamental common quantization kalman suffer delay sensor arrival observation model bernoulli process receive kalman optimal differently asymptotic depend critical arrival time provide close special characterize critical relationship spectral widely adopt extended author although present chain observation establish stability I boundedness information grow characterize behavior distribution asymptotic bernoulli boundedness see subsection provide question existence invariant weak irrespective consider boundedness operate arrival boundedness operating arrival lead whereas critical boundedness instability mean ensure finite preserving explain later limit preserve strongly sublinear invariant distribution take transform compute bernoulli process numerically sound assume infinity asymptotically first weak theory iterate e invertible overlap satisfied satisfie contraction existence invariant use unique stability show operating stability enables characterize countable functional algebraic general dense highly self finally explicit identification ergodicity enable easily moment context complete analytic result example address follow characterize invariant arrival organized subsection preliminary present formulation establish proof present scalar subsection conclude euclidean natural subset indicator otherwise partially order banach banach space field banach space cone induce namely solid non order cone monotone various ensure banach compatible ordering induce supremum focus banach equip norm close solid interior matrix partial induced notation denote denote semidefinite theoretic metric
fig interaction initial static interaction radius apart scale avoid boundary robot use infeasible apply user use possible every hence box user interestingly robot minimize perform hamming rest centre marginally bad hamming e conclusion might sensitive location stroke graph show summarize centre robot user strategy feasible since centre nearly ham always centre call robot misclassifie hamming result hamming variation weight boundary differently investigate hamming evaluation giving ham particular fair quality user want form fact setting inspection image visually error visually incorrectly visually discriminate large due runtime robot colour want small confirm considerably fix interaction different free predefine minimize loo estimator sensible important thing big suffer fit train addition value rough perform define perform static interactive static start reweighte error system iterate optimisation dynamically learn strength course interaction summary illustrate plot differently lead study also interactive static look close learn system dynamic change introduction participant infeasible robot user run user good smoothing comparison segmentation system start contain number max exposition reference adjust segmentation image simplicity involve operation respect vector balance trade symmetric degree fit segmentation give margin rescale read rewrite quadratic exponentially segmentation fit cut kn graph cut connectivity approximately train empirically interaction binary one partial fed unlabeled stay optimisation choose maximal reweighte pixel formulation interaction prove less convenient compatible cutting difficult basically geometrically possible remove closely regard selection kernel special kind selection explore discrete try variable short guarantee conceptually optimisation user removal k k user human relax strategy k concatenation properly proxy forward optimisation unary gmm potential ise contrast robot begin look little final include unary potential also potential unary unary behavior without either learn optimisation evolution user interactive system demonstrate approach show grid parameter segmentation approach infeasible max framework show solve optimisation art segmentation include enforce unary potential handle future tackle challenge enable interactive learning grid parameter learn interactive segmentation microsoft microsoft microsoft cb microsoft com successful vision parameter traditionally interactive manner interaction propose bring user loop human art interactive segmentation popular propose computer hard automatic show challenging input condition past sure laboratory decade research primarily interactive system help interactive interactive popular area interest interactive lead vision graphic interface interactive crucial automatic counterpart come surprisingly little devoted learn interactive system try bridge gap interactive segmentation efficacy interactive interactive segmentation aim separate rest treat assign label foreground interaction come pixel mark user help belong question interactive segmentation system interactive system imagine generate possible change segmentation evaluate one efficacy system interactive go margin automate summarize study evaluate interactive thorough segmentation algorithm margin user discuss give segmentation explain na I structure interaction conclusion problem development world choice set test measure traditional vision learn misclassified interactive choice hard presence behave differently prefer interaction learn interactive intuitive want achieve quickly literature interaction mostly choose researcher encode assignment give evaluation consider user current interaction refer static user prefer user good compete system prefer segmentation cut ground result performance scheme static tool newly propose involve group use use correct segmentation example full advanced job static evaluation participant significance participant car fast normal car experience car evaluate independently participant infeasible try segmentation thousand million crowdsourcing attract lot community primarily scheme collect datum crowdsource excellent platform interactive vision could imagine user system require suffer study light new fix human current ground output code middle label alternatively interaction interactive market similarity agent reinforcement one learn task summarize effort price user model yes yes yes crowd yes slow bit user yes yes infeasible static far low paper ground truth code consider kind input blue ignore database image ground comparison keep ratio create computed ground truth segmentation indicate foreground user interactive system gmm make short ratio get system cut undirected node correspond segmentation color background foreground unary learn color foreground computed channel pixel concept practice term ise
recent rely ultimately density algorithm ergodic use argument mild additional bound away imply evolve matrix position covariance x k nx dp density stand characteristic determine portion proposal template appear decay verify setting correspond apply instead original essentially fit differently covariance symmetric metropolis increment proposal deal constant improper target distribution recursion template weight result suppose adaptive walk hold speed original setting value scale smooth proposal behave almost grow reach decay slowly however covariance figure therefore use significance successful burn may ensure space borel algebra lebesgue n k n n unbounded follow walk vector almost surely dimensional uniformly adaptive small enough n fx converge probably target compact support extension however handle convention compute independent mean determined recursion analysis first show increase estimate imply also substitute algebraic equivalent strictly suppose conversely consequently geometrically imply contradiction strictly ultimately additional sequence growth respectively assumption index lemma another q show sequence clear z hold z g similarly decrease sufficiently small sequence suppose decrease imply establish obtain contraction consequently triangle converge constant latter satisfy lemma first us eq imply combine point sufficiently n hold k x section define stand behaviour recursion express simplify first follow symmetric degenerate identically distribute real measurable nr n n invariance notice particularly unity behave quantify behaviour random non degenerate kolmogorov set thus n conclude technical mention require zero assume degenerate variable measurable adaptation estimate j j nz sufficiently hold whenever use write choose sufficiently I kn sufficiently right first estimate conclude adaptation adaptive walk satisfy process surely proof estimate apply martingale convention assumption martingale converge limit satisfie due implie converge hold simple result similarly process adaptive small walk increment walk sx sx sx n select n appendix b j construct apply fix also jt ib surely sufficiently assumption therefore whenever hand consequently establish surely converge martingale difference trivial infinitely indice one stay index infinite inequality sufficiently must whenever infinitely index infinitely exist whenever thereby hold trivially conclude proof strong number run ingredient check simultaneous ergodicity next suppose everywhere non increase x condition measurable vx sx lc adaptation target satisfy event eq us truncation construct truncate start truncation function n coincide law select law number deal component proposal mix fix adapt result ergodicity result key speaking regardless adaptation compactly measure absolutely measure constant measurable dx ds da interested ball fulfil eigenvalue relatively use adaptation stand sphere independent auxiliary n use variable measurable p nx w write lemma hereafter denote define n martingale ny w write w w bn b cover suppose measurable satisfy w w almost absolutely measure du measurable bound stay tail contour eq process weight moreover adaptation satisfy sufficient fact show cone ax bx r hold fulfil density fairly easy verify practice hold exclude unbounded contour theorem corollary hold proposal require use record ergodicity large differentiable stay super tail contour use mixture stand weight neighbourhood origin adaptation surely imply compact set hold vx dd put auxiliary truncated process ensure law large letting
education internal web page site external user visit period consideration record list page website visit internal part context attribute external user site target incidence pair visit site analogously page target pair visit preprocessing receive visit visit total visit page site describe term site tackle dimensionality case reduce size datum follow observation give information interest target site group site page merge domain bank page page page take month reduction size large context size give concept interesting group employ index concept construct stability extent context factor show much stable motivation index formal obviously stability concept indicate particular extent index close several several stop site part concept stability lattice large threshold look correlate set group fig part lattice site www external internet visit user www month concept many concept correspond political read fig present order several survey stability web site site paper group decrease amount arise employ group formal proceeding terminology attribute
choose translate mild shape bind question establish principled either dependent dependency mix process specific ec fp give let binomial upon classification km obviously jensen directly markov suffice see q p f version block measurable bayes compound need differentiable concentration respect second stand eq thank eq give q lemma step inequality q let increase function take nonnegative end mp corollary replace thank equation correspond appropriate effective identically independently make follow concentration random range fractional make distribute inequality rise generalize bind applie allow function draw z bound corollary theorem range identically allow retrieve tight given establish pac bayesian mixing follow stationary denote order bound mix suffice follow concentration countable z suppose function bayes mix mix sample corollary align ne iid particularly classifier even guide classifier situation data dependency iid state framework generalization call dependency encodes dependencie datum graph fractional fractional sufficient result train way note bayes stationary mixing rank much progress bound bound bayes bound bayes refine appeal advance concern bound classifier directly bayes show bayes risk also variety outcome explain able learn classifier real e bipartite first constitute iid principled establish bound setting establish convexity exploit contribution cope dependency subset bayes bound non distribute mention dealing variable establish concentration fractional cover derive inequality extend function provide bound complexity subset dependent beyond inequality call bayes illustrate ranking ranking function performance area exhibit interesting strong skew imbalance negative besides rest vc dimension rank algorithmic stability qualitative somewhat bayes uniform bound already observation carry bound consider result deal use new process recently bayes price dependency graph straightforward shall calculation lead sake provide bind process together tool allow derive iid new fractional provide iid mixing give rise generalization stationary mix possible product space bind iid probability predict draw iid mh throughout generalization bound value appendix kullback distribution success kullback posterior absolutely prior straightforward tt apply argument deterministic margin focus generalize dependency identically independently result rely build accord dependency state bound role dependency graph dependency edge independent connected cover fractional fractional proper fractional cover cover cover colored color way problem fractional number graph come precisely graph order clique split independent dependency fractional bound draw assume dependency distribution equal proper exact fractional cover follow stand q n defer technique proper cover proper fractional reason cover weighted classifier distribution ne q z classifier n z j third iid prior enter scheme cover factorize look factorize minimize iid accord respect readily choose cover least mc side q z km comment say iid iid amount bayes valid prior posterior depend get give side decrease reader check indeed suffice nf vs induced subgraph draw amount subgraph induce situation empirical error I see fractional must color fractional subgraph remove twice big differ carry circle inner sep fill gray origin size gray might comment comment well subgraph compute subgraph bind subgraph let least random h simply subgraphs probability form small induce node fractional graph subgraph size replace keep still iid fractional obtain bipartite rank negative fractional dependency graph figure totally see plug give fact distribution look rank rule hx yy allow pair high rule predict conversely make measure natural question rank difficulty sum distribution henceforth clearly simply need upper bind xx bind rank claim bind fractional theory u rewrite permutation iid random index cover decomposition appear permutation permutation take fractional cover definition knowledge ranking argument analysis easily fractional even require practical situation value limited pair rank minimizer able appropriately moment bayes course base scoring state similarly important usual tight clique clique make equal odd bipartite respect e one interested sequel whenever consideration opposite sign situation straightforward iid q estimate fraction pair incorrectly incorrectly express scoring notation minus name consequence bayes provide evaluate score negative early share dependent depend figure read resp entail positive building bayes step part necessary deal depend sequence check cf say early identically exist determine p z q hz q propose call event directly unconditional p fractional part finish fractional clique define construct addition check cover mean minimal proper fractional cover note skew express therefore score classifier acting carry assume gaussian posterior parameterize bayes follow straightforward bind parametrization done choose minimization selection iid empirical turn proper fractional cover moment empirical error convexity linearity z q z side decomposition dependency tackle order datum another graph depict easy clique besides proper implicit reasonable easy bayes sake theorem datum mix stationarity stationary identically denote algebra integer coefficient process say note detail mix process dependence difference datum rademacher bounds mix integer p z block even odd drop block variable within stationarity block dependency variable connection block theorem qp
pls algorithm svd extract factor pls selection stream pls pls assume discrete streaming datum introduce challenge offer update bridge pls simplify bridge pls latent inversion loading reduce division pls stream efficient svd weight unable least square least current datum essentially adaptive individual covariance eq exponentially contribution past current bridge pls construct sum pls point eigenvector perform iteration sim column orthogonal power column eigenvector schmidt follow modify iterative bridge pls algorithm estimate simplify pls latent vector since effectively loading sparse regression detail full initialize c ty tc tm tu tw ts initial mutually orthogonal also initialize choose place penalization make penalization design line govern ar autoregressive coefficient factor means give time stream hide create stream bridge pls accurately select line algorithm lead convergence take place point create response coefficient center variable inactive ease visualization interpretation define univariate pls response multivariate result sparse pls simulated coefficient line correctly factor pls pls factor area pls quickly line suggest period pls algorithm correct furthermore pls use stream univariate strongly regression select center analogously second mean assign period third pick automatically rapid switch keep switch gain show pls able accurately mostly component neither inactive suggest change fast control report solid percentage second carlo data pls little percentage decrease quickly select time result place adapt change change quickly factor cause become see large monte carlo case pls select capital return track algorithm portfolio tracking propose lasso latent tracking suggest component reduction latent regularize application publish index motivate present enhanced perform index tracking target asset return plus pls algorithm figure enhance track static stock static portfolio period poor financial suggest portfolio test pls p benchmark stock stock assess track randomly sure fair portfolio weight vary square make ability portfolio composition really advantageous tracking see pls consistently portfolio achieve exactly target portfolio index result suggest importance stock detect certainly advantageous drive advantageous market heavily portfolio entire period stock whereas pick drop period suggest adapt track every transaction cost change place return trade tracking algorithm streaming aware show able factor stationary pls able accurately pls specify pls per towards development tune several method literature automatically update adapt achieve evaluate use minimize aic incorporate pls selecting retain reconstruct ensures retain lower retain retain new likewise retain high important incremental pls pls method number pls initially projection recursively keep function component pls pls increase add pls per choose change achieve covariance maximize explain threshold quantify however remain pls stream fashion technique literature concern neural network plan relate application trading application mining efficient multivariate problem stream recursively latent select factor singular line fashion simulation artificial dynamic observe stream time report track financial data stream consist line asset allocation benchmark streaming arise web monitor asset management contexts quantity collected analyze often incoming stream stream refer may multivariate common fundamental regression penalize coefficient problem arise firstly select truly important component correlate arise ill pose special care difficulty take approach relationship change quite development adaptive method dependency datum generate development methodology unify vary stream numerous analysis amongst track recursive yield regression propose lar know penalize solve algorithm domain finally aim propose incremental sparse square pls track stream pls linear assume existence latent property deal format review pls emphasis development pls propose pls allow bridge thresholding singular although second contribution incremental pls incremental pls pls real time simultaneous iteration method bridge pls ensure full pls component eigenvalue possibility pls ridge notice covariance covariance regular pls ridge far notice small prevent becoming follow pls eq eigenvectors loading loading final pls pls direction may extract computing relate necessity computation pls single svd review bridge pls perform pls find pls computation pls svd efficient bridge pls svd recently device loading briefly achieve pca low approximation
replace simulation smc sensitivity smc bayesian molecular dataset smc apply type deterministic stochastic computationally efficient allow discrimination candidate selection give sensitivity review abc application dynamical formal bayesian model aim computationally intractable costly exploit computational modern simulation vector abc generic distribution tolerance tolerance algorithm approximation replace summary dynamical without statistic simple simulate accept reject disadvantage sampler acceptance rate introduce proceed simulate go go abc correlate couple acceptance may get low probability period avoid abc monte carlo smc method smc sequential sis sample represent sample gradually evolve target principle get area proceed population indicator indicator else perturbation return candidate return particle go normalize go particle denote perturbation kernel walk smc abc selection employ concept selection find let model define prior uniform simplify bayes factor weak strong compare hypothesis testing firstly need secondly equally traditional insufficient explanatory reject translate direct evidence true additional specific particle specific estimation particle proceed ms initialize ms indicator ms candidate calculate q every normalize output happen factor directly use however explanatory extent inference typically great particle thereby ensure estimate poorly posterior belong wish independently abc smc selection parameter perturb particle ode solver code scientific library implement ode solver code delay part deterministic dataset typically world molecular highlight abc smc dynamical demonstrate describe interaction specie add simulated sum conventional noisy low reach tolerance accordingly rejection sampler approach order particle approximately need infer distribution apply outline comparable abc rejection gaussian distribution next apply distribution summarize range median obtain substantial need reach result need time abc abc mcmc ccccc step inner histogram result try describe equation population fix resource inference master exact simulate point smc population laboratory setting replicate also experimental three run purpose system dynamic one repetition rate generation prior parameter magnitude large accept result summarize gene gene loop protein gene loop gene six parameter display cycle behaviour available point function inter infer histogram four infer posterior large credible interval recover change change change hence sensitivity analysis intermediate construct intermediate visualize distribution project intermediate accepted pca scaling pc explain variance parameter sensitivity problem increasingly difficult visualize behaviour secondly determine varied individual pca output abc go sensitivity particle rank parameter eigenvector describe orthogonal pc define ellipsoid accepted particle eigenvalue specify pc variance proportion sensitive pca account posterior summarize pc first interest lie pc extend across region distribution pc large pc last mainly combination change two third composition pc outcome agree obtain range smc stochastic transform correspondingly process condition correspond include protein choose particle average summarize get hard infer infer deterministic figure analyze compare deterministic stochastic parameter sensitivity link system insensitive impossible vary stochastic scenario variation one case infer stochastic disease sir describe spread infect recover r individual simple delay get infected ability individual individual death irrespective infection rate recovery realistic adding gets infected time incorporate delay include infection state infect equation rate latent allow individual become rate obviously basic four impossible inspection alone sophisticated need select experimental group gaussian deviation similarity intermediate inference simultaneously bayes calculate basic time population bayes factor weak compare true rt da isolate approximately observe break arrival use day stochastic abc infect recover individual extra epidemic previous expect period particle use prior perturbation kernel uniform interesting population probable local model pass selection describe marginally draw infer wish posterior estimate compare whole general epidemic broadly establish abc select plausible realistic suggest smc tool reliable dynamical generality abc unlike deterministic time obtain merely intuitive meaning frequentist advantage abc shape cost information different parameter simulation sensitivity infer narrow one infer population localize change population resemble distribution correspond straightforward number care selection domain care prevent reject population e observe bayes change tolerance perturbation variance care need stress way posterior range test goodness successive intermediate crucial signature proposal impose limit procedure different abc systems particle develop sequential carlo abc compete approach apply chemical science reaction sensitivity abc smc also critical identify quickly relatively insensitive financial division molecular college sciences trust helpful discussion david manuscript especially grateful helpful manuscript abc sis improve rejection impossible sample weight dirac delta proposal reach series importance sis sis draw set stop sis intermediate take sis base define abc smc sis proposal intermediate distribution q dataset indicator tolerance particle allow dd bx equal proposal perturb accepted kernel walk calculate define approximation particle intermediate weight calculate b dd bx bb tb tb abc write indicator weight particle simulate particle go weight go simulate base smc sis disadvantage smc sampler optimal good backward suggest simplify prior result approximation weight affect credible toy example distribution three population abc poorly figure approximated population particle run schedule show variance green weighted particle blue line result smc variance abc rejection use population
case matrix overcome follow rectangular define shift k entry loop construction confirm lemma add make sign gaussian notation enough model rescale hermitian use loop parameter input k f kf channel decision rule matrix approximate complete adjust accomplished linear channel appear extensively storage characterize noise ts definite often channel medium bank match physical assume ambient noise error mmse detection suboptimal mechanism linear equation scheme maximum introduce algorithm mmse detector work apply sequence implementation typically convergence spread since indeed axis plot color depict well next loading loading axis show residual option loading first force dominant radius dd radius iteration tradeoff amount show outer loop weight dominant iteration loop increase average loop iteration per tend number inner global h present iterative propagation essentially involve add diagonal become walk add feedback cause loading believe numerous detection give art linear iterative fail convergence compute mmse direction importantly overall double trade beyond converge make outer slowly balance compete choosing loop develop lastly class beyond loading computer science support foundation thm thm assumption condition thm remark inference converge vector compute converge compute constrain linear construction link engineering discuss force convergence originally fail consequence able pass gaussian link fundamental science processing rank social machine recently exist instance densitie condition graph numerous novel convergence definite application discuss practical original section introduction describe extend system positive information potential solve optimization rescale variable zero diagonal correspond pattern reflect markov distribution edge radius condition imply definite key idea loading walk perturb convergent solve walk equation propagation obtain walk normalize diagonal walk dominant hence satisfying condition explore solve iterative effect iterative follow objective basically version newton minimize size typically ensure hessian optimize
inactive fixing fuse inactive set instead path choose nature algorithm define set determine fuse becoming fuse everything go fuse set specify define base active correspond correspond inactive fuse correct also fuse finite short optimality locally w f get active inactive coefficient inactive remain subgradient unconstraine get determine setup find set maximal flow let associate define assume source fuse lasso k kl define fused status time define activate activation activation active putting length remain variable split inactive maximal come source capacity two iterate step nothing inactive active fuse result proposition assume set give denote calculate use rule fuse valid accord necessary outline extension inactive set particularly fuse additional q function regular subgradient start see kk np ii ff ii time activate ij h f ia gain fuse update fuse hand active composition include inactive treat calculation necessary possible old fused computationally lot apart moment delay program lasso later date subgradient equation value eq show node see come capacity flow come flow go every maximum imply eq proposition first continuous actual proof j unless condition f jt kl lk fail restriction existence satisfy construction graph node merge immediately otherwise occur loss generality therefore continuity fuse fuse necessary flow come use grouping optimal solution space linear flow flow derivative force break thing remain sure set piecewise easy continuity linearity inactive active l g active thus inactive definition stanford university supplement article path fuse give complexity alternative article increase fuse become proof result fuse lasso predictor article fuse order however calculate derivative identify fused small require operation use save fuse stay derivative per decrease therefore entire possibility store retrieve store whole coefficient fuse linearly impossible efficiently memory requirement store node fuse coefficient parent store end example retrieve start next interpolation store node look vector value therefore get complexity speed article want result assume flow fuse sufficient edge node node essentially situation f rsc rl ks inner flow whereas maximum flow go residual also graph node know definition merge penalty turn fuse immediately split lead fuse split want expand result fused predictor want difference get calculate whole fix additional lar see inactive fuse restrict development jump
dimension h compose proportional f f corrupt additive snr observation burn number iteration chain precisely output example depict middle converge reconstruction iteration sufficient ensure ghz course challenge mcmc gibbs blue reconstruction source result active detect amplitude implicitly indicate absence paragraph indicator regard active estimate consider histogram fig component line mmse active mmse estimating depict range paragraph sparsity identify follow atom mix patch source illustration dictionary depict complete blind separation problem orthogonal prior source hyperparameter prior unsupervise collapse sampler study generate noise approximate generate sample estimation simulation conduct svd favor investigate unsupervise jump extension currently investigation domain would thank university suggestion relate grateful fr address identify formulate mixed unknown orthogonal formulate model process prior inference gibbs sampler joint matrix synthetic illustrate representation chain carlo year representation consist identify decomposition signal among main alternative pose recently compressive sense reconstruct projection reconstruction sparse formulate penalize numerical unfortunately np problem reconstruction solution well pursuit mp orthogonal omp norm exploit property devote constrain generally estimation mp design course completeness redundancy atom recover activity signal machine community procedure problem assume recently formulate strategy negativity orthogonality demonstrate gene datum analysis blind sparsity signal image recently molecular source unknown source assume therefore source mixture zero problem among hyperparameter stein instability noise manner couple monte strategy consist level assign informative hyperparameter parameter bayesian paper hyperspectral image standard noticed standard process lead demonstrate deconvolution result partially collapse sampler van strategy efficiently dictionary ensure source orthogonality main impose recover property preliminary address source base mix knowledge level precisely unobserve noisy bayesian description idea decompose free orthogonal column conduct assign couple von fisher strategy assign mix generate sample develop formulate blind source separation problem derive quantity algorithm accord posterior illustrate application natural processing future consider unobserved stand notation q accord center unknown source consequently index sparse propose variance unknown mix section likelihood observation independent vector stand full numerous work gamma choose noise main simplify column introduction regard mix gamma eq generating highlight require orthogonal accord column firstly nan sample sphere set unit set unit sampling detailed achieve strategy matrix accord distribution element therefore prior recommend coupling classical locate deconvolution frequency sparse approximation molecular take dirac center hyperparameter degree assume strategy adopt would source sparsity level times explain introduce subset active jeffreys prior hyperparameter distribution hyperparameter variance eq assume hyperparameter hyperparameter q define graphical acyclic joint nuisance lead infer matrix generate bayesian estimator mcmc method allow one generate collection asymptotically reader consist highlight detailed lead mix weak alternative deconvolution recently study rely presence collapse van drawback inherent cite sampler consist replace conditional summarize step subsection h von hyperparameter independence vector source consequently achieve successively f derivation needs explore mainly gibbs sampler local introduce overcome introduce active conditionally indicator rewrite active govern hyperparameter sampling successively notice efficiently initially introduce adapt account orthogonality rely cholesky decomposition inversion avoid calculate compute intensive determinant sampler u
gps include mat ern entirely herein generally describe classification case generative variable via generative draw trial require class without generality prior sample hybrid monte carlo hierarchical almost suggest sampling prefer hasting mh draw carefully consider variable notable drawback popular typically stationary stationarity categorical operation necessary modeling computationally mention natural suggest sampling model partition categorical separate input gp predictor treat cart fashion cart cart branch predictor split assign branch assign recursive arbitrary partition cart standard real value output cart theoretic cross may cart partitioning hereafter model otherwise identical model proceed open detail computing default begin start queue root recursively queue accord child define explanatory uniform distinct informative leave sensible requirement take automatically overlap region partition region define extension intercept wishart hyperparameter aim tree set hyperparameter treat monte carlo gps gibbs exception integration gp contrast difficulty change leaf simple mh instead reversible jump markov transition model parameter leaf node proposal mh ignore leaf maintain collection allow jacobian leaf gp set jacobian factor ratio unity leave describe proposal make move call move acceptance largely exception move solve predictive distribution conditional normally eq mean r boundary aggregate near partition boundary grow swap integrate tree gps posterior uncertainty gp flexibility datum gp fit almost indistinguishable cope quite ordinal categorical split make marginally input code binary treat handle real input perhaps standard wu wu input scale apart parameterized scale parameter output tend order function binary even variable cause unique consequently tree handle input present indicator ignore come leave exclude ever boolean indicator upon lm zero beyond produce speak boolean gp parsimonious accounting relationship value include model mix separable presence indicator necessarily setting allow whether particular categorical input visit tree split binary indicator technique require tree relationship remain value chain propose operation benefit context possible drawback allow tree relationship predictor gp relate real input separately region mechanism directly handle input predictor imagine correlation input part set partitioning categorical ability learn real value account correlation different categorical predictor argue possibility influence take back speed interpretability benefit section benefit classification single softmax case imagine possibility recall partition fit model consist output gps variable region entire might tree could tree call note latter tree imagine perfect copy partition differently partition region across expect may model imagine datum set occur tree capture split partition greater likely split immediately relevant class gps enough move directly proposal grow change grow take advantage structure essentially hierarchical classification formulation code modification package henceforth tree region set hyperparameter model identical prefer mean incorporate freedom less generative incorporate softmax summarize diagram extent sequentially region three modify extension latent along would along newly newly propose f grow move play direct indeed define try determine variable grow benchmark input real value set handle categorical tree inference identification categorical last would difficult consider modification covariate categorical normal noise q original reverse irrespective response linear irrelevant response note encoding split indicator note previously context cart categorical predictor split categorical encoding generalize arbitrary predictor split predictor range achievable cm median training random compare summary root square rmse sample behavior method I gp ignore categorical account rmse test clearly indicator include cart appropriate input constant leave indeed fit unfortunately improper indicator partition proposal leave respectively parsimonious smoothly vary cart improvement real balance cost categorical translate bayesian cart proper drawback boolean come leave boolean partition lm full implement scheme consider linear map value input lower rmse treat value fitting leave direct linear model boolean process smoothly indicator represent categorical boolean partition predictor matrix cart lm partition indicator indicator leave indicator become function well parsimonious monte carlo value input analysis trivial consider generative assign great distribute evenly depict order expect cart one partition another perfectly class set divide likewise capture mechanism represent htp latent follow vary smoothly enter determine class correspond latent mark distinction amongst latent class likewise negative region remain modulus reduce almost smooth limiting exhibit divide divide remain latent constant similarly latent variable divide separate latent variable care separate difference arise separate two would parsimonious class tree place true capable nonetheless much likely partition partitioning fundamentally predictive interpretation tree model meaning finally see run benefit take second roughly difficult trivial candidate amongst easy model hessian synthetic partition illustrate amount sd add convert assign base indicate direction exponential concavity change quickly similarly require fit hessian right multiple maximum posteriori encounter separate regression able upon relative grid misclassification location rate relative range well represent softmax partitioning suffice even take hour execute whereas fold thing context strength compare gp mind shall categorical uci database group class aspect six input distinct category set predictor treat continuous form attribute neural network implement library test library hide ten training evaluate via validation select parameter setting keep partition rp select base default avg rp fold slight misclassification standard misclassification fold slightly cpu use hour per fold hour appropriate worth map tree principal split binary therefore predict success extract devise single figure show chain log figure tree illustrate many benefit separately success process latent softmax powerful simple drawback implicit assumption stationarity handling categorical evaluation due repeat decomposition divide gp suggest partition input advantage divide handle categorical gps leaf drawback behave inefficient involve basically integrate direct preferred however categorical value appropriate categorical gp real one dimension result interpretable highly acknowledgment sciences tb research support thank anonymous comment statistics california berkeley uk laboratory uk interpretable regression gaussian process gps application output utilize classification formulate gp sampling expand helpful develop handling categorical collection synthetic inference produces interpret keyword gp function ease typical prior stationarity
shall underlie currently requirement adopt practical experimental verification algorithm usually observe metric consider reasonable verification main contribution establish assess experimental global instability drop instability multi verification reinforcement agent reinforcement algorithm learn receive theoretical prove action arguably simple system domain consequence verification agent researcher stability evolution global surprising since optimize global allocation eventually usually reasonable verification argue end global challenge widely establish framework assess global instability system drop alternative local entropy experimentally propose traditional metric instability multi system organize throughout review use experimental initial global lead conclusion algorithm use instability conclude simplify task allocation assign agent service minimize scenario figure execute option forward reach although experience task well without know appear world yet capture analyze delay effect action appear message time delay consequence delay message link unit agent queue simulator make agent state feedback get agent agent make good unobserved agent gradient ascent neighbor practically load learn incoming request neighbor receive later indicate load ascent adjust successfully domain load balance concern main stochastic expect benchmark game equation intuition similarity neither analysis nevertheless mention elsewhere update ensure policy update adjacent arrive x sub center time unit execute task unit arrival service illustrate htbp algorithm look plot safe actually spurious ideally would evolution learn parameter include small aggregated summarize agent neighbor case ascent policy learn learner effective policy count neighbor deviation agent agent policy
penalty example long hence fashion guarantee detail argument statistical paper concrete norm regularization give decomposition thresholding create grid initialize initialize compute assign norm shrink retain validation shrinkage choose solution svd iv iv meet positivity negative sign singular vector correspond minima warm reasonable process version appear speed demand calculate matrix assumption explicit p rewrite factorization svd computational heavily upon matrix indirect iterative calculate lot algebra sophisticated requirement effective count compute absolute maximum error algorithm rank bi manifold approximation svd use suitable output algorithm uv normal iid completion meaningful standardized error nuclear panel exclude snr percentage indicate unique correspondence nuclear plot rank nuclear true minimizing summarize perform simulation size iteration convergence time version acknowledgement support dms national health miss entry miss entry true noise middle get type b post process poorly predict apart type get however snr time lemma relaxation matrix regularizer simple nuclear norm algorithm replace element svd warm efficiently solution measure represent entry netflix competition primary datum recommender correspond rating movie potential rate movie movie yet rate lie much represent recommender structure suggest movie small score movie affinity recently theoretically assumption unobserved recover within observed contaminate entry constraint make singular svd modification make convex nuclear norm norm explore programming efficiently modern software algorithm second method propose large sized criterion equivalence rank problem al singular scalable comment suggest prohibitive problem force minimization lie path grid value warm inspire svd different step prohibitive exploit structure require large iterative partial adopt notation onto observe complementary form basic ingredient svd refer proof follow look minimize lagrangian control norm minimizer solution warm z initialize grid iterate assign repeatedly replace guess figure curve see test smooth objective fix satisfie nuclear norm sake brevity expand singular product arbitrary lemma obtain svd minimize tuple pass subsequence k w pz pass
formula quadratic give minimization amount unconstraine aside interesting note thing low correspond norm one big little say nevertheless might useful construction finite library construct potential among recognition tracking physical extract image want library dimensional contour transformation projective object might collection estimator x selection sequel recognize actually priori solution respect basis profile obtain example know class discretization vector class bound lower typically one proceed construct r fm linear suggest devise select among collection estimator rank inferior identifiable identifiability statistical possible identifiability hypothesis body know priori noiseless uniquely noiseless formula shall impose type detail please p recover subspace though generally little norm write obey derive expression recovery least norm say state start derive expression one one equality power equality derive say estimator quadratic equivalently eq ideally minimize hence refer oracle serve benchmark assessing add challenging indeed inferior quite derive satisfactory drive identifiability one write new formula hence formula remainder possible useful procedure propose estimator penalize criterion q arbitrary penalty minimize risk shall penalize help estimator assume fix eigen positive eq defer consider inverse problem argue give might help motivation like emphasize begin identifiability might might enhance risk performance linear estimator select risk introduce denote large positive eigen value matrix large value number number function select simple fact quadratic risk amount consider result issue suggest reasonable bind penalize accordingly assume remainder obeys identifiability choose penalty put penalize formula performance compare predictive oracle course hence constant positive take satisfie shall choice assume positive eq obtain upper put find equality universal say model model start obtain collection performance bind situation q sense measure write start intuition subsequent assume moment discrete integer l sense since among difference I achieve low term risk take make actual instead hope expression reasonable way achieve quantity diffusion gaussian bandwidth expression proper bring enough constant formula penalize risk reasonable decrease versa nevertheless exist intuitive fact take one p hence amount energy concentrate interval regard role prevent upper risk model find instance trade constant reasonable remain parameter optimal speaking suggest reasonably knowledge examine penalize firstly far unless happen increase decrease dominant hence great adequate default reasonable penalty constitute course turn enforce exclude optimize penalty study attempt involve penalty application dedicated reader optimize whenever estimation concern inferior identifiability address selection put forward noiseless x real satisfy unless belong respect identifiable recover estimation highly hypothesis call identifiability know write equivalently happen practical engineering field one like one respective basis zero mild happen realistic numerous practical application basis capable extract vector construct enforce two cx hold one generally cx might priori know solution identity sense mention illustrative assume multiply smoothing kernel increase kernel correspond cx one meet identifiability eq actually idea semi definite symmetric formula formula might ellipsoid know semi know construct follow decay law appropriately like add investigate identifiability scheme scope please penalty directly concern concern statistical ill linear regularity signal mail request stand constant take filter signal summarize condition matrix check ill conditioning ratio find stand third discrete follow stand three stand differential achieve degree recover triplet filter repeat experience instance penalize risk experiment need first reason treat since large upper penalize strong oracle recover perfectly error recover almost perfectly oracle latter relative error order assessment please direct inversion practical noisy green drive smoothed signal visualization green drive red estimate shall framework address estimation penalization work non mainly show good difficult application penalty constant simulation well propose note paper present plan mainly process constitute attempt satisfactory inverse problem course challenge address along near final answer could though make lemma nevertheless penalty small penalty however bind behavior penalty mainly open question solve near spirit finding identifiability really rapid question would yes assumption go direction recover interest among infinite impose nevertheless exclude generic belong devise identifiability comparable result noise modify simultaneous believe future answer question currently amount try question practical point sometimes fact find fix subtract quantity latter equality find put derive find need control put q eigen value value put number derive simultaneously large derive sharp let put integration concentration real square vector eigen symmetric prove derive stand respective eigen eigen stand two follow inequality hold true lemma consider rewrite let compute put derive technical detail k finally technical proof derive f proceed inequality inequality inequality derive r concentration variable acknowledgement author france paris university discussion video scientific stay en remark grant propose address estimation statistical depend inverse problem drive mild important due e sharp paper estimator propose approach find code name recover interest trade fidelity term regularity selection achieve summary notation shall sequel problem lie hilbert endowed euclidean follow shall euclidean matrix identity mean diagonal element write one solution case countable framework section collection quadratic risk extend quadratic mahalanobis risk rank interest linearly identifiable know priori adopt via penalization estimator stand upon propose remainder great inequality q constant collection generally collection choice sharp universal stand correlate dimensional estimate resp vector adopt via penalization regard deterministic matrix general ill great therefore seek paper allow number viewpoint refine point consider example go shall latter coefficient orthonormal otherwise property indeed recover library thorough discussion estimation certainly continue researcher either work appear indeed early model concentration line consideration collection oracle inequalities propose extension aforementione work mild model selection quadratic risk extend simultaneous probably present subtle author subspace orthonormal basis estimator construct error admit constitute contribution author might suffer result especially family basis play relaxed estimator construct construction encounter computer highlight proceed unconstraine matrix generally compete h fidelity illustrative quantity regularization operator pass band filter enforce smoothness recovered solution generally regularization however want mahalanobi optimally application collection putting obtain finite linear estimator low find selection representation family orthonormal complexity g wavelet correspond proceed mp solution put collection estimator among
value ridge fold multidimensional computed roc construct varying significance range sparse illustrate correspond situation underlie box reconstruction base causality ridge regression white noise approach discovery subsequent testing var show outperform research problem give split subproblem solver regime estimation flow fmri accuracy support european discussion technology institute technology institute interaction autoregressive vanish belong respective parsimonious causality assume discovery consist absence causal lag zero achieved regularize recently outperform recover causality use causality widely accept assumption effect cause series cause knowledge improve series spurious causal relation common factor detailed form elegant handle causal vector var graph ol fitting var model coefficient enforce estimation lasso account absence ar belong pair furthermore detail give extensive briefly relate autoregressive causal strategy autoregressive tp gaussian noise cause causality vector autoregressive causal coefficient interaction conduct matrix set z p var latter transform partial correlation copy coefficient time copy estimate correlation propose case control high sparsity alternative white ar square straightforward way ridge however fully dependency graph ridge improper estimate bootstrappe explicitly estimation k z kt furthermore x hypothesis multivariate normal sparse causal discovery ridge suffer assumption direct sparse via desirable would causal lag suggest overcome coefficient vector correspond incorporate belief influence restrict estimation positive modeling lead eeg problem vector first univariate coefficient hence overfitte avoid scenario eqs solve order carry efficiently coefficient solver lead considerable reduction memory combination solver conduct experiment recover compare lasso ridge causality four case reconstruction cross multivariate time generate var accord choose distribution var randomly causal randomly test look e eigenvalue accept var datum noise strength across
decrease perspective open bring low radius achievable single datum rich geometric streaming provably store empirical classification theoretically many ball improve upper upper identical alternatively analyze stream set specific place around appear streaming mean probability toward stream could ellipsoid ellipsoid upon inclusion unnecessary ellipsoid axis scale variation allow ellipsoid expand addition approach along line gaussian space weight incoming maintain uncertainty ellipsoid covariance exist streaming however conservative stream pass svm streaming prove despite conservative experimentally competitive technique learn much believe careful classification alternative cs edu streaming svm leverage connection streaming impose allow formulation minimum ball idea learn require multiple streaming weight way use stochastic perform computation storage even efficiently accuracie comparable solver online give extension common traffic fashion streaming apply disk store propose allow pass data storage severe stream streaming successfully employ domain geometry adapt streaming setting formulation streaming naturally motivate efficient technique approximate stream context encourage high latter geometry admit stream adapt provide outline experiment synthetic machine kernel provably generalization formulate quadratic typically solver require requirement svms decomposition method divide small scaling approach rigorous guarantee due success stochastic stochastic base quite requirement recent considerably single pass easy train approach suffice require several converge reasonable define w n ny difference form nonlinear map far assume property fix isotropic dot kernel criterion label unsupervise one second account misclassification vector except entry instance metric product margin induce ball extensively geometry inner programming become dimensionality cardinality svm turn radius solution extract small subset originally core easy equation correspond center dimensional space kernel pass mutually never store vector compare perceptron radius store exposition kernel kernel store weight vector initialize kx kx kx weight update lagrange dy input example slack rw dy nx ht example slack support vector initialize nx x sm l sm use storage obtain conservative classification well end therefore weight ball choose ball ball merge ball end ball merge variant zero amount store radius income ball store buffer whenever buffer closed rather take size size whenever buffer take number practice less synthetic art evaluation pass accuracy compare online iterative batch pass accuracy solver table accuracy pass dataset suggest pass use perform single solver htbp cc cc dim c
snps discount option namely multiple snps even computation computationally c snps uniformly use snps show reasonable nominal one amongst true reduce snps considerably bring close improve nominal false maintain misclassification still sensitivity overlap distribution sample apply apply column snps another correct purely collect set nan see return bar quantile blue situation individual publicly available nan threshold incorrectly half line illustrate choose sample resemble choice sect similar european descent converse well match illustrated incorrectly classify majority outside population sample nan empirically sample poorly match separate threshold false negative power specificity practice remain unknown population produce false greatly error less nominal positive nominal false rate sample accurately importantly threshold specificity assess match specificity describe considerable quantifie post prior theorem posterior prior write plot specificity accurately need exceed post probability subject specificity assume sensitivity belief difficulty assess specificity practice great positive fact result specificity sensitivity versus nominal yield difficult match depend strongly underlying meet consider table severe resolve membership individual positive sect frequently effect family child parent another child snp test child value child large yield center amongst reflect see resolve member group become big homogeneous move close note know true positive know positive distinguish positive similarity ideal situation snps independent ideal hold carry sect population control sect simulation sample snps identical rate fraction varied show identity find half time identity note fractional identity percent simulating varie percentage fig rate exceed yield misclassification eqs together true member eqs member possible variation cause turn green show distribution close close genetic inferential significant responsible misclassification example far characterize genetic quantifie allele frequency classifier positive classification sensitivity quite classifying specificity distribution eqs utility test circumstance dominate see different amount nan assumption positive sect high false result nan I amount snps sect meet despite sample adjust deviation nan obtain sect additionally conclusion neither positive assumption meet classified despite assumption sect amongst even threshold adjust produce classification considerably expect nominal nominal report table practice result difficulty yield probability probability high specificity test sect correctly sample sect finding privacy become topic positively identify mixed pool unknown scenario evidence need population obtain sample composition discount value sensitivity assumption sensitivity difficulty threshold rate party identify relative concern one access subject careful need member well show classified value sample narrow sensitivity need probability truly status eqs discover despite limitation eqs quite sample likely neither also little worth investigate column fig lie sample fig nan control control shift snps statistically close control statistically control disease individual conversely find subset snps snps use snps comparison location successful positively presence dna finite genetic useful appendix nan underlie writing two trial eq per person invoke binomial value notation consequence write analogously absolute value consider separately treat expand polynomial simplify notation rewrite perform expand well indirect indirect become dependence uniform sample nan hypothesis sample range varied circle plot uniform obtain closely national health md fellowship national cancer institute national md individuals population control control control case child child specificity specificity nominal snps sensitivity specificity specificity specificity nominal rate apply metric quantify genetic suggest infer allele limitation characterize strength limitation reveal individual member several circumstance crucial yet explore method improve additionally specificity may ideal circumstance despite identify disease strength limitation situation implication finding individual amount dna highly complex mixture snp author may material minor allele idea genetic sample subtle systematic introduce dna comprise binomial question contribute variation contain intuition statistic measure distance underlying identically reference detect versus hypothesis major minor q assume central large modern normally denote average snps exploit individual neither article propose composition e appropriate reference positive individual intuitively author construct near rate presence specific genomic specific summary many publicly available study method concern participant false assess rather nominal quantile expand investigate robustness inherent I hence difference independent central g via enough assumption begin attempt address analytically empirically explore conclude finding well individual dna reveal sensitive small nan frequently finding false individual likely exactly although finding may reliably detect valuable find extended genetic explore analytically empirically attempt publicly available source retrieve genomic breast sample describe comprise breast cancer case comparable match participant participant american european sample array snps share snps additionally european individual project individual member comprise parent snps frequency control three pair group even reasonably resemble allele difference minor allele sometimes allele frequency magnitude lead non size small representative sect derivation fail separate case analogous center situation separation leave control misclassifie achieve use nominal accuracy control classification sect three frequently nominal simulated identical genetic method individual possess eqs quantify furthermore positive sect sensitivity appropriate pool comprise individual case snps use positive sample fairly correctly classify sample misclassifie group positive nominal sample test participant classify subset yield misclassifie rate amongst false positive rate amongst amongst specificity reason
test negligible therefore bp justify dependence mention situation causal nonlinear demonstrate noise verify lead et tu ab ir tu wu valid normalize w provide basically argument worth asymptotic case box test powerful box test power lag whereas detect lag goodness problem long memory commonly long long true value lie test goodness attract lot construct roughly distribution usually trivial power alternative type asymptotic parameter chen al smoothing alternative neighborhood model disadvantage kind asymptotic martingale chen utilize practically far chen et justify process conditionally martingale exclude interesting invertible observation follow similar nk present literature even pseudo likelihood memory series reformulate equivalence series chen test prove case allow process iid partially solve conjecture even limitation assume know turn modification main reason series population slowly rate become adjustment point chen statistic mean invariant mean adjustment possible extend directly beyond scope short affect hold adjust residual seem natural central might nan transformation use et al care unknown fourth innovation happen error feasible base show large sample white residual still aspect although chen fit series error find skewed adopt chen domain along correction device yet certainly university j goodness series dependence journal series autoregressive integrate series autoregressive integrate move average journal american j day em chen goodness chen induce methodology variance ratio em j bootstrap test frequency distribution fit statistic r test martingale martingale journal em daily flow science estimation plan recent model j diagnostic uncorrelated journal american introduction bilinear time w j model economic relationship york university series york drive specification p theory em bootstrap stationary journal em serial lee serial unknown bootstrappe box robust l test parametric mean alternative long dependence statistical statistic wu stationary j journal american li li autoregressive average li integrate autoregressive conditional journal american association forecasting fractional journal zero theory r ng testing time series journal new york p journal american statistical association extend theory em wu asymptotic theory wu national science wu principle wu dependent stochastic process em wu limit functions journal wu em wu approximations uncorrelated ergodic martingale f x f kk wu accord wu cr contribution achieve approximate double well study wu wu wu among directly major set martingale bound martingale presence separate along lemma respectively ii wu basically wu omit eq part covariance martingale u u u u u k k j u c j l jt jt jt tu j tu u tu tu u tu u tu u u kt v j k j schwarz tu tu c j apply f f l l proof since nu nu pn nk nj nu nk nj suffice show cauchy schwarz nz jk nz jk side view hereafter term nz n j cm nj nk nj j nk imply nj j u u nk nj absolute wu n nj nk nj k write martingale difference nj nd om regard k nj k nj n n r nk nj om martingale nj r martingale difference constant nk n nk nk nj nk sequence difference martingale central suffice verify nk n nk nj cm nk nj nj write argument follow uniformly td j conclusion nj nj td pm proof j td pm view notational n nk nj nj n nk nj nj nj nj since dependent get cauchy absolutely wu derivation z r nh j h r u cm regard j table cc th establish assumption nk nj nj td r pm write nj nj j n n nj n nk nj nj nj nj r j r n om nj nj nj j l nj r dependent cn nk nj rd rd om pm appendix ny kt n nk nj nk nj j nj pm let follow show tu n absolute since mean get k kn kn e k k k ny p ng ng ng n part nj pn cauchy proof theorem need nj pm schwarz end nk nj nu nu nu nu n proceed u u k u om wu bound om n show k u u u k u u cf wu u k fourth term n derivation conclusion assumption n kn kn n j k j e l suffices k n e k j h h pn k u u k h h h j g p partition k h h h contribute shall partition nonzero j similarly involve restriction fix typical fix schwarz fashion similarly argument assumption n om kn absolutely sequence u exist let q noise statistic box lag truncation nan distribute assumption popularity conditional imply autocorrelation understand asymptotic nan box dependence lag diagnostic checking address go early finding series white lack serial correlation process variance covariance function respectively alternative statistical common checking systematic fit statistic domain time probably admit form eq call truncation empirical nu u reduce frequency rigorous contribution derive nan work lack stress statistic presence dependence show bp lead inference uncorrelated demonstrate test uncorrelated modification dependence statistic asymptotic shall seminal distance density lag normalize density nonnegative function bandwidth sample quadratic distance equivalently nk worth regard case iid establish nk j stand normal lee establish martingale general white establish replace residual uncorrelated distributional theoretical asymptotic fix negligible change contrast fourth appear negligible result restrictive
equation corollary kl get problem constant ignore margin linearity discriminant pac bayes bind generalize multi base wrong independently draw definition pac bayes easy reading copy pac bayes continuous margin threshold respectively independently e discriminant random draw define distribution f hoeffding achieve mh mf result fact two last hoeffding substitute mf expectation side cm exactly statement get f h h hold finish remove mf since pac bayes true possible cm maximum prediction lack probabilistic dual probabilistic model bayesian allow hide combine extend generalize markov style entropy discrimination discrimination paradigm know mn model similar mn simultaneously type plug combine mn pac generalization bind combine learn generative model structure maximum discriminative outperform compete structure extraction maximum entropy discrimination network margin infer modal g web intelligence scientific genome discriminative graphical structured dependency field network specialized graphical model explore learn remarkable success structure output flexible capture impose desirable bias reduce structured maximum margin machine concentrate output due dual depend optimality arguably desirable paradigm prediction context lack unable domain sparse irrelevant extensively likelihood estimation learn linear model svm univariate output turn extremely later guarantee estimation discard generalization discrimination simply formalism offer formal paradigm generative discriminative principle bayesian regularization structured spirit discriminative structured prediction mn maximum discrimination margin offer advantage mn learn primal label formalism extremely contribution concentrated define solve generalize arbitrary insight general solution reduce prior achieve effect laplace primal mn convex thorough compete include mn mn synthetic mostly superior comparable formalism general discrimination follow laplace offer sparsity laplace discuss web extraction language parse annotation decode map sentence parsing scene consist denote combinatorial structured interpretation object input entity structure must arrive satisfactory represent pair function without class optimization maximize input structure finite discriminant predictive possible f column structure likelihood formalism classical label discriminative boltzmann mn feasible discriminative principle elegant generative regularization find advance structure crf method utilize discriminant optimum frequentist furthermore concentrate directly map elegant probabilistic cause obvious easy derive sparse paper bayesian obtain infinitely propose formalism objective mn special achieve resemble knowledge base principle substantial principle yet explore important exposition approach margin solve follow cm slack adopt iy j equal one otherwise constraint explore label reflect specific pair wise duality cutting pass optimum directly employ mn learn max margin employ question average devise constraint spirit scheme optimum sequel regularize formalism offer simultaneous primal generalization traditional bayesian style use entropy principle additional predictive basic discrimination learn binary general ratio px bb find solve follow optimization slack keep optimization input feasible mn feasible subspace expect entropy discrimination principle optimum correspond relative entropy measure minimize kl entropy theoretic favor among feasible appropriate accommodate separable prediction usual kl optimize entropy close put everything formalism discrimination discrimination nf relative cm optimization function expectation problem nice calculus variation p brief insight underlie optimum markov lagrangian multiplier solve slack program lagrange function saddle dual dual defer appendix program lagrange kkt kkt condition show enjoy lagrangian multiplier correspond active equality enjoy mn due constraint learn duality conjugate slack easy c equal hold infinity training perfectly ignore multiplier problem c uncertainty slack expectation side successful rely estimator optimum special generality offer important mn admit primal enjoy generalization third incorporate generative predictive train partially analyze partially combine possibly latent discriminative structured choice parameter subsection standard mn somewhat surprising reduction offer totally predictive counterpart respectively underlie mn make claim explicit assume p posterior ip mn state primal assumption detail corollary slack shall exist mn applicable solve trend markov extension implement sparse graphical context margin gap merely due discuss mn adopt strategy primal regularize mn directly dual problem constraint non trivial due complicated solve mn preliminary expensive real another possible method interpretation resemble regularization effect parameter reveal mn normal weight standard regularization model around dimension standard infeasible mn dimension employ learn apply regularizer mn variational express e heavy tail encode around nice concave negative convex exploit estimation laplace mn obtain f k intermediate note laplace hierarchical mean admit hyper straightforwardly new multivariate vector p k multivariate substitute hierarchical normalization column due derivation avoid infinity convex conjugate arrive function depend logarithm problem laplace qp mn variational laplace cm solve develop corollary posterior primal subgradient cut optimize respect take set since univariate gaussian dimension representation get calculate expectation convexity optimum apply posterior distribution shrinkage laplacian irrelevant zero qp qp hyper pac provide theoretical average classic pac rule prove follow mild boundedness mh definition pac pac bind bound margin cm event true spirit bind extension structured margin detail present stochastic structured average predictor pac dependence pac classifiers posterior generally evaluation laplace mn mn regularize regularize newton method mn use correspond good mn mn open thorough beyond scope appear elsewhere experiment implement equivalent mn slow apply ball mn evaluate structure e synthetic pre field dimensional feature vector correlate sample sequence draw define crf gibbs synthetic randomly transition capture pairwise independently sampler generate iteration draw rest mn validation sample setting outperform mn encourage outperform un case derive note convex try optimize mn perform mn algorithm consistently performance reality correlate evaluate model linear generate contain input relevant partition group get feature mean method solve qp variational fold cv datum like part k search get state generate compete average correspond average mn large mn exhibit shrinkage plot generate assign label predict fact learn generate variance since variance two variance relevant group variance whereas figure sparse structure partition fold cv sample fold put web cutting method slow warm simplex example stop iteration hour finish cv thousand performance approximate project sub combining sub cut plane synthetic figure instance small variance consistently outperform mn regularize bit reasonable examine accuracy obvious fitting contrast robust unlike usually small ability lead instability regularization put weight regularize fitting suggest later mn stable log likelihood right question magnitude regularization regularization fix sensitive regularization mn mn stability instead thresholde zero max margin regularize primal dual less outlier last conduct problem regard web extraction web page record web page characteristic type structural tag well flat construct construct vision accordingly product image hierarchical long dependency level level item web attribute image price generate template page evaluate record record accuracy extract attribute record page testing record criteria comprehensive measure four price identify percent datum model especially mn mn regularization outperform regularize max outperform third mn enjoy primal especially number training mn suggest margin mn mn scope defer later motivate discrimination method propose average maximum essentially structure version mn structure svm lead powerful new paradigm structure prediction enjoy primal raise algorithmic inference propose example along relevance machine unlike optimize margin function logistic sigmoid although sparsity due justify unlike style learn margin margin also explore interpretation hyper advantage free hierarchical laplace encourage two replace weight explicitly add applied maximum discrimination straightforward fundamentally state interesting online laplace conference input formalism offer formal paradigm integrate discriminative principle bayesian technique vector machine entropy margin
covariance satisfy approximately briefly let good desirable inverse covariance concentrate find pair positive approach inverse covariance give make complete choose suitable subsection update perform instead vary amount incur ij discuss convergence nonlinear nlp nlp associate problem follow penalty establish penalty nlp assume exist cf nlp bound finite optimal value easily fact first know due together ready generate optimal update inequality maximization value observation first see proposition respectively exist saddle immediately desire definition diag derive give immediately far rewrite concave conclude convex solution smooth let similar solve simultaneously solution next nesterov method equivalently subsection adaptive project gradient solve equivalently spectral al smooth close integrate al classical namely respectively optimal view moreover close set suitable ease subsequent describe detail also h problem integer g k k k establish generate suppose interpretation observe observe q q follow nearly optimal expect nesterov study show u iteration nesterov method unique nesterov slowly propose adaptive nesterov subsection continuous general replace corresponding constant ease detail method assume give definition subsequently inactive ready inactive kx u u f sd g u sd kk end subsection convenience sparse adaptive project method nesterov solve instance describe al generate generate contain value uniform positive randomly q instance experiment purpose code matlab method addition initial terminate find ghz table iteration evaluation cpu rr rr rr cn rr nf cn c rr rr rr rr nf cn table able within time spectral generally outperform nesterov carry entry nonzero sample pattern covariance approximate solution observe capable recover independence know formulate norm penalize nesterov demonstrate able half within amount outperform online www namely completely straightforwardly lemma author research grant inverse conditional formulate method nesterov smooth instance able solve least constraint nearly reasonable amount inverse nesterov undirecte capable describe among zero zero year matrix notation norm estimation control trade nesterov approximation scheme coordinate lin propose suitably et al capable discover graphical nesterov substantially outperform exist addition maximum estimation conditional formulate pair method practice graphical partially know sparse inverse covariance partially completely covariance constrain penalize pair control sparsity worth observe ii view easy reformulate convex homogeneous self barrier suitably interior al bad iteration ip ip solve dense arithmetic ip prohibitive et convert large apply slight square method dual ill surprising slowly square often fail huge consist found project adaptive nesterov smooth paper rest introduce first generate conclude remark assume write cone matrix resp operator unless explicitly norm one dimension clear context absolute determinant
material balance pseudo piece square preference single single preference opponent master piece multiply rank factor expansion relate four complex preference opponent c relate preference isolate isolate non isolate open file isolated feature view limitation move game avoid early move normally program search depth check account evaluation carry carry game train random validation record train natural temporal learning position game black loss generality game white black actually play white mae feature divide position without white game classify player I identify playing play actually still valid game black perspective replace define opponent play cardinality cf white summation black perspective e white perspective note discriminate player game examine examine perspective obviously undesirable know white black algorithm game subsection first mae player differ much bias mae selection evaluation may useful choose train move difference position detect example nevertheless reflect structure point consideration opponent situation break move varied method note mae counting game despite moderate success replicate low value enough weight feature evidence difference difference mae mae converge increase adequate feature partly limitation next subsection introduction player player normally position abstraction player feature deep blue increase play blue rest subsequently lose well capture level playing versus algorithmic indicate range many readily solve aid program exist often involve rather solution plan readily aid computer technology put emphasis force planning possibility leave direction human knowledge discriminate player record play use engine yield success believe methodology present fundamental need far would capture concept classify player match since domain game try game go conclude profile agent record ac describe style record base individual temporal aid engine architecture encourage attempt learn discriminate try white playing discuss limitation possible present applicable domain keyword record player game player player affect position generally tackle computer discriminate player review difference machine occur difference actual outcome feasibility write master strength human world less technical description machine program demonstrate learn enable self effort chinese self consume game record strong player evaluation temporal minimax record game pattern move prediction record game principle interact record exist generally accomplish fast learn include meta play result game module play look player reformulate play black piece high human already program computer already human play mainly increase rely far classification suggest temporal cite attempt computer program scenario attempt learn evaluation function course consideration likely improve method could weak correspond prediction td decay factor force accelerate momentum describe section world learn weight guess encourage fundamental player point feature probably strong seek maintain piece player concept difficult formulate translate algorithmic conclude remark component playing program example component play minimax game play world go level employ advance promise improve concentrate essence player weight record useful discuss incorporate relate structure addition record know player novel subsection accelerate training momentum generality evaluation define game sum value n measure unit material balance evaluation move play play version program tuning convert apply sigmoid choose adjust assume initially white move white perspective position white move white move game record word result record follow small rate logistic set recommend literature experiment choose rate subsection update normalise preferred fix material balance logic position advantage material position factor measure material dominate interesting note rule td opponent move make move possibility player program choose self become program move make player
current define possibly randomize measurable function observation history play feedback full double definition abstract potentially represent infinite metric metric call algorithmic represent result metric oracle access suffice space classic notion guarantee whether admit whereas arm instance instance whereas instance need guarantee apart several study background bayesian formulation payoff maximize payoff expectation mdp play difficult formulation passive science include formulation offline mdp action formulation probabilistic payoff adversary minimize strategy set lie subset payoff question idea lipschitz mab adversarial version mab well naive stochastic invariant view problem lipschitz payoff estimate payoff distant arm whereas linear mab arm crucial mab mainly fix payoff advance make present joint prove algorithmic complementary infinite space result oracle space reduce low topological self particular use covered course consider payoff payoff arm independent expectation bandit reward collect algorithm round round notation denote expected regret analogously open radius ball contain finitely every limit cauchy dx I cauchy identify formally subspace cover complete vice let family intersection topology element throughout refer small topology open ball topology singleton topological empty least order statement equivalent axiom use concept extend beyond infinity require notion namely von definition necessary material found prove lipschitz tractable much exist distribution problem expert first existence perfect useful metric binary center parent correspond ball parent could necessarily center radius perfect construct pick root construct perfect point define payoff fix let large enough tree nod child one node complete leaf child belong ball probability measure function first sign choose sign function child independently associate payoff hold I generalize lipschitz problem metric implicitly payoff metric give triple apply technique exposition one idea bandit similar formulate formulate expert payoff indistinguishable learn consider lipschitz along borel measure feasible kk x tuple ensemble borel algebra mutually expert mab bandit payoff remark theorem small among node probability measure induce complete node three let follow exist event random many event happen metric lipschitz mab double feedback exposition expert version lipschitz expert metric topological function segment order denote rely strategy since value compact open set contain call maximal segment open complement therefore attain towards play small ball significantly large strategy well via follow lie within output ball subroutine input call receive cover consist point let collection ball call take round sufficiently problem optimal consider function large least return us notation run rt chernoff let clean ball show imply suffice rt sx kt cover claim consider lipschitz non use proceed doubly length length tractable exploration subroutine subroutine play end fix total reward phase share exist let duration part exploration return phase proceed phase round run exploration complete incur round infinite compact diameter payoff baseline payoff baseline r mab constant intuitively ability payoff ball formalize ball round induce event divergence technique detail proof claim x ir r via intuitive less space oracle need metric covering space point space point six say finite finite metric suppose topological order iii arbitrary initial contain true isolate initial segment exist xx reveal mab expert cover provide tractable every even consider metric cover n section consider subroutine oracle receive play rx winner winner exist else point complete sufficiently strategy subroutine notation algorithm rt chernoff bounds clean payoff optimal rank isolate indeed open set optimal pick enough contain indeed dominate claim strategy complete cover compact metric cover radius run armed center ball mab sense look tune tune correspond fairly natural metric ball sufficiently rational denominator radius cover true support contain close contain near move let space hamming cube distance pair probability radius unit move distance least obtain follow existence asymptotically binary hamming lemma easy bind imply cardinality point least element least subset cardinality pair point imply cover expert call mab problem use space finite parameterized run phase round play phase pick break tie complete term completeness explain regret lipschitz feedback let phase bound event guess total accumulate phase claim feedback problem function version guarantee via involved analysis space include upper bind involved expert expect payoff phase let set choice hold feedback guess proof use separately essentially union many efficient bind chernoff slack chernoff follow root empty internal level singleton child diameter definition cover tree cover break algorithm say clean consider chernoff lb c lb sufficiently turn determine slack interestingly right ignore incurred phase clean phase base phase diameter induction prove pick break fix covering clean root arbitrary refined tree metric feedback tractable tractable suitably idea naive see lipschitz theorem g uniform plug theorem characterization regret except lipschitz full many upper bind proceed efficient repeatedly pack nonempty exist contain disjoint positive covering let maximal center disjoint every ball every ball cover packing recursively construct set consist finitely open ball equal positive ball ball disjoint ball radius ball let ib ib ball mapping absolutely infinite one ensure one problem expert distribution payoff function sampling distribution lipschitz subset sampling sampling random independently analogy notion complete infinite nested ball specify expectation payoff finish lower I bt low q tractable incorporate via lipschitz sufficiently feedback point sample average break tie arbitrarily lipschitz expert log rather fix denote arbitrary ordinal depth contain existence suitable decomposition every equal infimum let depth maximal ordinal ordinal via union ball let closure report cover return arbitrary covering net successive construct net depth union successive net b phase round phase strategy call throughout phase good guess previous end feedback arbitrarily phase cover construct roughly construct net point let feedback oracle specify chernoff lipschitz expert whose depth use covering point roughly version regret phase clean high clean depth argument show sufficiently large depth phase clean good guess within optimum reason depth ta final section lipschitz metric mab expert metric copy abuse expert lipschitz property never select metric payoff mab mab algorithm conversely algorithm tractable lipschitz mab playing time draw report let instance mab payoff expectation algorithm behavior identical payoff apart query infinitely success theoretically impossible output receive call kl omit let ft tractable metric completion remark lemma bound algorithmic direction lemma directly type less elegant payoff denote tb tn pick belong value set relation expression equal denote random count account least algorithm finitely kl x choice payoff give play payoff define two imply non integer obtain q last time select strategy play playing role sake convenience equivalence implication arbitrary countable iii space circular perfect perfect subspace ball leaf path root nest ball empty intersection call distinct leave correspond distinct tree disjoint thus point ball ordinal cardinality point recursion isolate isolate nonempty empty exceed index ordering open construct order ii imply order contain order know open imply point separate arbitrary implication fact induce open initial open topological order property perfect element well initial segment open topology perfect infinite complete direction nsf air office scientific microsoft research fellowship foundation fellowship arm bandit mab classical arm priori payoff certain imply logarithmic lipschitz mab finite bandit problem generalization infinite regret lipschitz mab either bound perhaps coincide compact connect technique online notion perfect set lipschitz mab term exhibit give nearly regret space form metric characterize tailor expert show sense completion compact category descriptor algorithm abstract general keyword armed expert metric multi henceforth year exploration sequential k payoff mab commonly evaluate payoff play one strategy range experimental armed growth time decade begin seminal subsequent application mab set bind apply making assumption payoff bandit trivial mab payoff become subject intensive mab information consist upper situation maker access similarity similar payoff information model define imply payoff mab et feedback essentially space precede real interval regret dimensionality metric upper exponent depend regret tight infinite isometry factor picture regret metric exist work provide space finite metric metric study ask metric achieve metric issue concrete
definite fact define zero actually include would subsection transform distribution contribute reduction depicted request statistical decision estimator define square x poisson density parameter suppose estimator call tp ik poisson gamma obtain apply view server analysis software numerator time increase range weighting regard effect www real www traffic access www request minute except period server date date describe likelihood px x px px previously follow lk analytical maximum however sub plot show mle date server real www processed evaluate request model point mle request b table show interval horizontal axis solid solid histogram estimate request respectively dot propose stationary plot server server model server table request third expect table result server h server max limit accord mle become show cause maximum non stationarity www aic server aic stationary exception aic small server traffic server drastically decrease request day traffic time server traffic strong www traffic data viewpoint model request closely stationary special stationarity stationarity www traffic forecasting planning www extent strongly mle differ server square error large cause small figure square around table estimate server server suggest length maximum estimation sufficient request concern request server derive expected value reduce squared point effect would strong model superior thus advantage h c aic stationary c server stationary show forecasting www traffic steady term theory calculate server stationarity express vary constant pointed consider traffic stationary real www traffic real also prove www traffic forecasting strong validity classical stationary poisson value point interval mean traffic advantage support grant aid scientific research project remark research institute engineering university department traffic forecasting past calculation focus wide www traffic deal traffic time viewpoint forecast arithmetic calculation www traffic theoretical empirical wide web www traffic engineering environment problem stable operation look analysis software analog web www server count file page intuition case forecasting point researcher traffic suggest lot wide suitable traffic structure express forecast assumption wide statistical substituting suitable assumption future bayesian alternative bayesian new parameter forecast posterior forecasting problem bioinformatics traffic distribution poisson estimator variation inferential depict assume time cc px vary cc conditionally regard kind definition x vary variance distribution vary model vary degree include stationary another
colour sharp expect solid colour region point interested demonstrate smoothing parameter estimate make wish simple residual behave residual propose I image surface gaussian response space simulate left note estimate top show vertex connect spaced covariate sake choose global bandwidth exhibit signal practically flat bottom choose identify location show easily pose constrain I subject j tucker require existence lagrange multiplier j otherwise negativity active acyclic system j lagrange multiplier system minimize event subsection kf reach l requirement region give region sense remove happen suppose swap f f km j c f f j change clearly since c c f l f equality merge therefore active without increase never twice terminate york regularization taylor pp application run string numerically multiscale multidimensional smoothing van locally regression spline e nonlinear removal r lasso extend take vertex familiar article discuss penalize total result challenge new algorithm include graphical discrete variation penalize regression fully automatic smoothing statistical contain sort graphical mapping variation focus penalize thought consider explanatory often sort graphical rise covariate graph term assume realization value complexity regression response grey pixel connect pixel image display image graph regression neighbourhood noiseless shown discuss penalize datum rough observation value vertex measure difference estimate measurement therefore penalize f plus vertex different vertex edge usual denote treat order pair direct completely vertex make split edge motivating nonparametric continuous natural first adjacent second observation convenient shorthand variation extended dimension analysis think pixel pixel neighbourhood pixel suggest picture variation plus van de smoothing allow smoothing procedure minimization complexity norm shrinkage lasso graph apply regularization every spaced create square et algorithm idea active method give minimize convex function minimum global minimum minima convex strictly unique share consist acyclic active optimization entire contain vertex denote subset hold example join together vertex hold since acyclic remove edge associate region j edge acyclic appendix string describe van consider algorithm value define f proceed reduce change active occur change f k reach target region might meet would decrease change merge join share acyclic minimizer region meet vertex event break therefore k edge join ensure acyclic occur must whether remove region take swap order edge possible event occur remove l f complexity analysis simplicity image vertex generic mainly combination stop edge active set add change active many repeat decrease monotonically remove active active active without region also check condition acyclic calculation work sub gradually small describe grow simplicity integer adapt satisfied stage p p consider grow look follow look vertex follow square continue connected implementation allow therefore connect rectangle vertex furthermore edge active total computational must constraint check perform calculation datum flat
learning half learn subset recursively contain one label make classification root leave transform label draw multiclass leaf node subtree give chain start root follow gives even classifier yield multiclass binary tree multiclass find correspond let fix chance give example predictor choose multiclass predictor choose label suffer error rate imply classifier node predict filter algorithm illustrate elimination label structure round pair round round win round pair predict winner classifier train stage subtree root example form round filter example reach second tree identify node accord training match multiclass binary label set yy ns nf nf f root idea multiclass associate upon learning importance underlie sensitive simple classifier analyze complexity reduction oracle call search time back toward read bit computation sensitive cost testing require per specify transform sensitive multiclass example importance weighted label implicitly cost weighted analysis transformation allow add single classifier cost sensitive importance weight instead pick random create weight accord use quantify thus multiple deal classifier leave close problematic iterative technique cyclic classifier node regard imply induce algorithm transform sensitive multiclass classification rejection binary time relevant quantity importance weight form two label binary regret core theorem relate result sensitive path analysis boost distribution call oracle round original term cost multiclass form choose prove corollary multiclass classifier multiclass depth importance multiclass induced therefore cost bad cost tight use weight x draw sensitive predict accord induce sensitive induction weighted classifier subtree subtree class importance regret c whether without generality predictor subtree root inductive hypothesis trivially satisfied tree label subtree subtree possibility leave second prove induction induction inductive paper definition theorem make use filter tree evaluate vector node winner immediate claim provide respective use hypothesis l r c li li rs r proof take mistake recall denote small importance weight since expect importance hold respective input even odd split quantity let subtree choose inequality last proof three case rewrite follow fourth side inductive hand side third three term lc lc lc yield ts l complete tight tree binary pair except multiclass ratio variant filter significant practice essentially label compute test pair give node namely achievable use often maximize classifier tree publicly multiclass split isolated letter speech handwritten digit handwritten result split split pair pair filter different multiclass dominate relatively tree filter computation tree learner row bold although tree build section multiclass previous operate phase consist elimination pair label per round elimination binary tree elimination substantial first pair mechanism lose eliminate root final elimination phase select winner elimination subtree leave subtree phase leave elimination importance depend play winner versa label set preference amongst choice elimination blue one red elimination node importance depth game importance match elimination play bind computational elimination simplification weight control computation importance importance training unlabeled multiclass induce small power distribution classifier case achievable second case good label regret use subtree part proof tree invariant subtree winner phase elimination time w ar inductive desire finally maximum apply depth depth bound case error fourth depth relaxation relaxation allow elimination broken version important elimination single elimination depth relaxation elimination odd label odd play round thus compare label remain chernoff coin kk verify computationally factor formula compare elimination phase importance I put everything elimination depth add phase eliminate low hold somewhat powerful adversary weight equivalently constraint transform elimination repeat comparison majority vote depth adversary regret say adversary round make force bad outcome denote multiclass classifier induce use deterministic binary outcome pick algorithm query adversary assign adversary nothing suffer yield generality appear round adversary win label multiply number round adversary adversary regret roughly maximal adversary choice k winner outcome adversary incur answer win round break arbitrarily ask label total must exist round answer consistently choose winner unbounded reference abstract classification select powerful adversary new depth small classification multiclass instance goal accord approach multiclass reduction way key multiclass binary informally good refined technique analyze multiclass formally achievable loss suboptimal create class classifier predict binary classifier label answer classifier reduction
posterior incorporate likely return argument prior reflect mostly instead assessment band rejection basis acknowledgment partially la paris project technology choice discuss approach include assessment mean error model calibrate balance inference make manner original modification valuable distribution translate poisson iid integer value feature converge even proper necessary necessarily location therefore mostly conversely concentrated support use obviously remainder paper assess validity marginal likelihood model still ad hoc relate argue thing difference solely prior abc bayes assessment bayesian aspect consequence inferential machine current gain compare directly derive simulated poisson abc produce evaluation parametric equally version eqn counterpart exploit use device poor produce every consuming obviously moderate argue former rejection subsection seem rejection bind additional analyse product
quadratic countable regression inverse application give correlate let let put concentration technical defer hold proof rewrite follow symmetric derive respective orthonormal eigen stand p lemma consequence notation project bernstein serve sharp selection criterion penalization exclude broad bernstein variable concentration stochastic key importance statistical getting penalization sharp proving bound control uniformly statistic sharp concentration considerable interest analogue bernstein overview advance discover refined topic
short memory converge towards range estimator fast bring parameter fast completely dependent estimator method ml ls correlate change estimator let generality argument define huber solution n way property follow n moreover expansion imply assumption v huber strongly convergence prove solution without et al consequence theorem may neighbourhood consider account relation readily argument r I n term straightforwardly term take behaviour pn pn relation convergence neighbourhood generality relation eq pn yield function making imply hand imply bound positive var var obtain arbitrarily bound probability contain belong compact since show exist p p strong q triangular eq schwarz account writing two solution hand argument obtain account obtain relation imply proposition fr consider regressor stationary estimate property classical include convergence rate point asymptotic classification secondary compact denote regressor variance consider study statistic l wide distinguish error account condition error relate parametric vast development consider square process mean coefficient condition ml case paper limit change restrictive numerous sequence tx rr consider contrary memory error regressor parametric point vast error long x functional fractional brownian ls estimator concern cause suggest spurious point none absolute regime change define minimizer respect introduce point determine method past estimator behaviour change et et regressor gaussian change difficulty especially function convergence totally obtain consider notice change towards value classic make long science exchange memory another sp daily stock although contain serial absolute serial lag need throughout symmetric continuously differentiable cl vector slowly sequence long range lx value measurable vary infinity slowly vary reader refer long process process slowly tv residual consider notation parameter previous least zero strong exist arbitrarily solution obviously equation define equation twice paper go depend estimator purpose calculate partial choosing suitably close play moreover consistency differentiable assumption remain probability consequence solution system intervention realize accord fix neighbourhood order supplementary
chain define estimator clear enough therefore estimator surely e small markov chain recall denote markov aic aic bic clearly contain much concern consideration great simulation generation matrix advance hundred markov suppose concern converge right depend complexity order case keep example behave well tool create schwarz number exist et chain make behave aic inconsistent establish bic relatively sample corollary mathematic mathematics department mathematics review relevant information definition explore markov contain estimator already establish aic stochastic space transition hold I integer n decade great order chain j schwarz information aic impact statistical evaluation problem autoregressive process move process chain derive kullback leibler discrepancy tool model maximum later matter aic present mainly bic consistent bic estimator sample natural admit chain order identification though behaviour variable multinomial distribution sample finally propose establish aic bic outcome review estimator elaborate law iterate logarithm situation chain sample describe exploratory simulation discrepancy intuitively odd study sometimes density support ft u triangular sense eq formally fp p test relate square thorough multiple stationary unknown value v q homogeneous derive process irreducible mc see irreducible consequently ergodic exist satisfy q likewise sake simplicity return positive random extract mc sample divergence iterate convergence ergodic markov chain ng ng q almost well begin complementary exist k definition limit j ni accordance previous n
e subsection shall generative make normalize max alternative perform discrete py parameter via extend glm take corpus posterior distribution factorize problem extension class cm develop co descent tb input distribution step solve plug dual optimize optimize row vector derivative discrimination one label datum model px bb optimization keep slack want proof ignore corollary conclude I normal algebra another optimum solve primal state corollary supervised utilize document discover document supervise topic model categorical response variable discrimination utilize margin model arguably max estimation direct supervise supervised demonstrate advantage topic movie topic maximum discrimination max topic latent allocation gain collection discovering capture semantic collection latent topic represent vocabulary document variable represent task tool otherwise lda unsupervise incorporate corpus online user usually post rating score rating image may discover secondary dominant user goal unlabele unsupervise prominent perhaps orthogonal goal latent serious alternative well corpus incorporate gain major supervise report classification lda document review rating score variant supervise design different model predict aspect associate paper loss supervision arbitrary model level differ well exist train maximize lda use fed classifier document rating movie discover sub prediction develop supervise lda report task later later discriminative share goal differ train train joint response likelihood learn latent contrast stage procedure first discover discrimination dirichlet margin vector optimize objective special partially observe discrimination structure hide undirected discover latent classification learned sense discovery latent max yield representation suitable apply topic g lda correlated topic principle lda underlying implement variational comparable lda property stems directly optimize suffer normalization make learning model introduce basic efficient variational discuss generalization present classification present conclude direction conditional paper lda discrimination couple max define use lda separate lda building variational allocation proportion document document draw proportion topic response vocabulary problem follow draw response proportion document define pz document posterior exact hide intractable variational solution variational e assumption like efficiently detail pz approximate inference approximate independence assumption inference efficiently change type model lda delta I conditional although mle great success max arguably empirically svms recognition allow deal integrate advantage procedure latent discover task real machine margin supervise unsupervised lda model exposition basic vector upon machine publish brief regression find deviation response datum flat choice find function problem slack trade norm amount empirical loss insensitive qp solve formulation lagrange support package routine bayesian possible represent prediction constraint margin sequel dirichlet extension elegant max variable extension latent discover semantic document collection principle assignment unsupervise accordingly integration parameter sample directly optimize intractable optimize upper random define bind kullback kl divergence integrate problem cm multiplier slack version topic constraint insensitive loss want hand correctly sufficient explain well e minimize variational margin estimation topic discovery couple expectation latent representation suitable constrain intractable obtain mean variational form factorize free variational parameter small topic topic iteratively solve step unknown e step constraint topic since alg last lagrangian argument infer infer specifically rule tb solve dual update time document element essential lie firstly expect version topic co involve secondly max margin lie around document last affect representation document therefore representation row vector diag lagrange multiplier plug partial derivative get e diag optimum k co also normal cholesky laplace effect prior programming qp solver may recent development corollary achieve cholesky decomposition prior prior similarly prior qp qp solver efficient leverage development corollary formulate exist specifically cholesky matrix let u primal problem formulate unknown equation discover representation apply lda naive unsupervise g use unsupervised latent topic representation dimensional representation couple optimal g movie discover low representation sub present unsupervise inter play discriminative apply learn integrated learning cm bias implicitly independence assumption lda setting suggest less affected margin choose prior depend latent representation independent model less affected topic rule coupling topic representation coupling lead inferior see qp reformulate leverage recent either primal un normalize partition involve reasonable margin specify lda show use normalize beneficial appear discrete brevity multi easily similarly class stacking equivalently write fy derive model q distribution develop lda discover latent topic however impossible dual prediction latent easily state supervise intractable except normal variational method high expansion max loss response instead I corpu similar regression integrate variational variational upper dy dy fundamentally expectation latent well kl term regularizer l last iteratively optimize rule omit explain insight since factorize perform unsupervised reflect discover topic example boundary e lagrange multiplier act bias model discover latent representation tend example word document latent discriminative classification optimum optimum regularization effect prior normal shift dual solve svm present margin lda supervise apply formally discrimination topic lda g likelihood slack general em underlie topic bayesian contain recent discrimination extend multiple mutual exploit latent extension promise use lda unsupervised lda topic empirically compare integration statistic collection marginal likelihood slack regression classification implement compare section qualitative quantitative text model regression c class c per class graphic image db file file mail software file window mail format bit format file graphic power water rs air circuit nuclear loop circuit mail neutral signal email temperature people people attack article center people price mail mb disk offer package mail controller mb condition disk video mail controller email email remove relate topic explore unsupervised pair grouping separation document embed category mix embed examine discover figure top topic moreover distribution average latent yield decay discrimination fact effect seem discover topic detail regard discrimination topic per topic graphic document salient lda distribution case discover provide evaluation lda document build baseline evaluate relative ratio build gibbs binary svm threshold whether max margin utilize build via run experiment final ratio number per category set align since margin tune regularization change perform multi category package solve sub learn align close discover topic max margin believe slight movie approximately unsupervise implement denote document input evaluation criterion pr response slightly topic representation alone highly separable integration max margin discover discriminative representation topic document stay eq decrease behavior suggest discover separable obvious improvement rule fourth term discovery get pr bad r classification classification svm partial lda find vision etc problem result classification important issue method high statistic concept maximize margin generative handle spirit supervise fully generative minimize field differ leverage supervise include review label lda discrimination provide max style structured network application principle dirichlet version discover topic document collection prediction variant predict mutual dependency globally consistent prediction scenario annotation neighboring tend smooth machine token align discrimination use supervise principle optimize integration predictive suitable variational
q spectral associate optimal define stationary eq section bound fx uniformly ergodic simple chain start point lead effort improve decay valid appear computable ergodicity one literature specifically appear function drift simplify statement entail q exploit explicit simplify write block proposition modify make reasonably appendix mention c instead cite easy variance satisfie indeed result put everything theorem need bind imply analogously final vx iii compute vx vx vx f vx obtain effort bind one experiment describe prove actual error normal play role unknown parameter uninformative improper determine location scale gibbs whose since notation symbol gibbs draw conditional small transition let rhs regard first student degree therefore sampler rescale student degree freedom drift let quantity analogously give take assumption student attain respect eq interesting parameter interest bayes zero mean q report repetition experiment name henceforth formula analytical table inequality quite sharp mse primary large satisfactory proceed drift corollary computable drift examine influence proposition depend correspond grey assume equal actual figure analogous bound black grey good contrast inequality depend c value bottleneck approach drift theorem specific derive norm present drift computable chain alternative might work well successfully space reference aim obtain concern square general application relate vast setting give finite relate exponential inequality explicit derive effort constant give turn come phenomenon dependent inequality martingale use motivated valid metric expression ix iy apply well functional uniformly ergodic chain chain detail refer reversible bound bound available result apply integral regularity stationary turn exponentially seem dimension inequality functional technique apply e geometrically possible inequality geometrically cl truncation constant rate chain one assume drift need focus either total translate burn stationarity asymptotic always example section avoid could chain often validate g lead introduce bias design confidence chain often trade rigorous heuristic refer mcmc argue essentially asymptotic difficulty estimation markov chain chain context heavily lot markov chain consistent geometrically ergodic markov satisfy condition condition non overlap batch markov chain condition geometrically markov condition ergodicity drift drift require check typically ensure clearly algorithmic require whereas variance estimator conclude upper acknowledgement discussion early grateful anonymous constructive comment eq decompose shorter visit notation let easy distribution vx consequently begin eq partially support science high education et department statistic university cv also continuous block markov trajectory consider term ergodic geometrically possibly high respect monte idea simulate markov converge ergodic average reliable long must run prescribe use distribute block chain starting propose sequential length trajectory promise total trajectory cycle provide identify introduce simulation tool quantitative bound mcmc aim end split independent inequality proof classical result process identity median trick lead far express computable uniformly ergodic asymptotic chain turn small motivate towards unknown drift parameter drift imply ergodicity purpose build derive variance uniformly assumption identical moreover quantity practitioner connection one benchmark technology hierarchical bayesian variance area simple regard analytic assess full also construct goal estimator combine chebyshev small trick concern far independent boost independent chain estimate chebyshev odd deviation require bound moment precisely choose describe conjunction scheme behave like reasonably less choose big enough make run chain absolute constant tight familiar asymptotic
observable variable trace protocol limited security protocol identify protocol intend crowd anonymous david start project fall suggestion use fact make privacy mutually vote retain hence vote protocol periodic random vote protocol contract message protocol achieve relation observable datum protocol make observable correctness protocol verification protocol study protocol formal exploration verify express logic security protocol model mdps check temporal logic logic issue randomize biased reveal information crowd might adversary guess message project protocol quantify security protocol create environment pseudo generator implementation detect random protocol implementation attack randomization source randomization check hold protocol black reason protocol could hardware core able correctness observe input output inspire reverse genetic network molecular process form feedback architecture discover among randomized trace work observable trace randomized protocol measure channel technique probabilistic example available www contribution statistical probabilistic security protocol protocol mdp trace aware exist security protocol probabilistic organize mention limitation work protocol mutual well several effort make towards security protocol summarize provide protocol measure lack notion assume model channel nature security protocol worst different quantitative flow space guarantee guess posterior probabilistic relation observable guess guess knowledge flow kullback leibler di al attack see conduct validate similar approach al error protocol find approach protocol work pi calculus check probabilistic checking check extension check formal finding circuit exhibit specify give technique protocol abstract implementation code black analysis protocol rest summarize measure channel mutual trace observable basically bayesian describe trace estimate learn htp protocol analyze channel number process observable protocol ensure across maximum across channel thus channel capacity take maximize algebraic decrease freedom always use expression still channel randomized characterize need protocol choose estimate knowledge except would accomplish maximize information marginal conditional analytically maximum attain conditional bayesian dependency variable graphical way join maintain give observable depend variable bayesian protocol need network trace previously genetic adapt available knowledge dependency variable reduce possible structure begin identify observable mutual identifying node edge proceed node similar start maximum receive algorithm size record increment empty learn learn maximum unknown distribution maximum need special case approach probabilistic checking code infer code h ps ss need capacity mutual information detail co maximum ab technique terminate answer correspond maximum channel capacity protocol validate statistical technique compare checking protocol example choose mdps available pay coin say exclusive exclusive channel maintain guarantee rely try conduct protocol experiment vary increment capacity protocol probabilistic checking infer plot estimate channel channel fair coin protocol fair coin channel attain close coin
assume also regard model well separate attain value beta level fit good classification obtain mean straight along beta whereas produces take figure beta identify histogram concern beta heterogeneous respect independent see heterogeneity secondly exhibit negative confirm incidence ht cccc elliptical competitive generalization model remark flexible powerful useful fitting respect inference valid well suited student context procedure parameter recent robust research another issue em behaviour condition mixture regression point initialization simulation either preliminary numerical pointed reduce covariance implement g provide decision surface general illustrate point consider surface follow case surface circular circular surface eq figure equation remark surface elliptical acknowledgement comment valuable suggestion thank proposition prop lemma prop cluster modeling regard joint property case mixture mixture regression far base student distribution long tail observation simulate mixture model flexible approach wide phenomenon unobserved heterogeneity work sample arise identify conditional density variable refer unconditional otherwise refer g know mixture expert machine biology explanatory multinomial logistic focus approach response variable explanatory model medium technology digital refer moreover propose market segment derivation literature quite indirect statistical view assumption special secondly student long normal theoretical illustrate numerical base simulated organize framework sec discuss simulated conclusion surface cluster modeling introduce response weight hence broad sense generalize idea give type characterize relation parameter type form indirect mathematical tool concentrate th depend conditional since density assume gaussian g gp g g mapping denote particular relationship assumption mixture pg finally random value value assume gp g yy g g write argument lead linear include quite group secondly th mix consider probability result mixture regression augment multinomial satisfied multivariate gaussian assumption gaussian every follow multinomial logistic p density complete immediately posterior get complete joint side side neural network diagnostic list table relationship assumption none g traditional mixture py eq student provide robust normal tail observation asset pricing variate random parameter definite denote mahalanobis gamma chi degree freedom throughout location g g q refer moreover vector random g g classification plot classify accord cc classification classify illustrate different situation datum parameter b gb unit properly model attain plot classify accord symbol classify group denote classify third distribution group cccc identify matter reason classification classify true classification classify present concern noisy datum fit base unit mark secondly reduce whole group step follow strategy group mahalanobis distance maximum likelihood outli classify classified group otherwise denote chi distribution forward outli minimum determinant scope concern parameter reduce student classify group concern unit list different gb distribution variance augment rectangle noise tt tt outperform tt recognize large small small true outli cccc cccc outli error misclassification student gaussian student misclassification correspondence gaussian set generate gb divide group previous generate rectangle unit outperform misclassification recognize misclassification fit matter identify cluster tt point misclassification ccc student outli outli cccc outli case student cccc outli outli confusion square misclassification student student small misclassification bivariate simulate concern accord sample rectangle add unit summarize outperform tt misclassification attain small square student gaussian ccc outli cccc outli student
use encourage estimate simultaneously consistency consider type approximate mention large relaxation rank subject trace tight manifold norm goal explore method namely nesterov outperform code beta require less latter one organize subsection provide simplification programming saddle point nesterov aforementione second interior aforementioned cone variant nesterov smooth finally conclude remark notation symbol euclidean denote dimension diagonal whose diagonal let semidefinite interior let denote minimal eigenvalue z n denote standard norm norm xu u tx qr x z consist functional endow denote vector define lipschitz respect differentiable subsection eigenvalue presentation identity symmetric scalar statement I equality due fact possible follow hold identity scalar identity note eliminate conclude dual strictly conclude identity view duality w moreover fact perform variable last arrange integer prove page immediate immediately lemma integer eq identity every scalar identity subsection subsection restrict reduce row subsection observe matrix appear many similar type clearly assume orthonormal diagonal f ta tb tu optimal relaxation optimal hence formulation subsection fact quadratic optimal f quadratic strongly convex strongly conclude unique cone respectively reformulate nn r l easily proposition reformulate nn gx define smooth saddle suitably variant nesterov subsection introduce notation optimal immediately relation addition max possible develop solve nesterov scheme min first computationally superior report paragraph problem lipschitz continuous subsection nesterov solve yield nearly report paragraph subsection saddle suitably nesterov subsection result let scalar penalize coincide saddle point suppose let objective assumption together immediately saddle point min namely scheme base lipschitz subsection detail subsection nesterov yield hence section review nesterov smooth solve convex minimization subsection detail variant nesterov formulation specifically problem review nesterov smooth cp respective cp nonempty make follow regard function every strictly b differentiable well imply differentiable solution proposition dual namely optimal finally recently nesterov smooth notice type subproblem variant follow need prox subproblem modulus immediately note complexity corollary discussion arithmetic operation iteration variant bound expensive variant method nesterov solve maximization subproblem fact see scheme iteration nesterov subproblem subproblem real scalar solving value decomposition eigenvalue h nf nf tf zero easy see solution problem efficiently terminate variant nesterov objective nesterov subsection solving address prox problem formulation discussion also scalar specify aim lipschitz help form fix easy convex modulus respect proposition assumption b specify prox set variant nesterov let fix minimum achieve last equality identity allow due identity modulus view nesterov dual result smooth prox find exceed precede variant nesterov iteration relate original smooth prox function apply exceed view find arithmetic per nesterov consist overall arithmetic require nesterov actual maximization order detail subsection paragraph nesterov efficiently together optimal solving terminate variant nesterov properly dual dual computed q report compare nesterov smooth method version beta instance use experiment first generate value cpu gb variant nesterov discuss subsection interior version cone write worth interface call task suitably program represent product nonnegative terminate terminate performance randomly table number seven amount column computational instance variant nesterov smooth method less instance memory memory rr rr c optimization trace multivariate explore variant smooth computing penalize least generate version former memory efficient drawback handle turn exist alternative problem e dimension qr triangular handle subsection show nesterov lagrangian subproblem replace upper triangular matrix later hard subproblem presentation discuss constrain problem unconstraine penalty exact penalty strong consequence make function value nonempty region statement every solution gx imply statement let statement inequality every hold conclude inequality statement moreover gx fx f hold threshold compute instead yield view slight h page yield optimal solution version solution feasible h ix x fact feasible every feasible accomplish lagrangian subproblem different lagrange idea acknowledgement thank anonymous numerous comment pt pt support grant author support part grant grant author part nsf grant dms estimation nice finite compute however variant nesterov
region covariate nan hypothesis curve plot empirical horizontal intercept empirical simulate intercept case label case restriction precede case figure intercept power reach also error power identical scale balance case identical symmetric normality test case w violate violate asymmetric test sensitive power estimate value covariate adjusted residual overall regression line treatment specific interval might treatment prefer adjusted conclusion use significance treatment reject usual k reject test treatment reject tool covariate similar residual underlie among iv range treatment residual extremely covariate corollary main theorem conjecture remark university mail phone adjust control covariate particular homogeneity variance adjust residual adjust line entire ignore level compare appropriate adjust residual appropriate remove covariate specific power extensive normality size covariate adjusted treatment treatment show sample covariate mean power nonparametric covariate distribution variance heterogeneous different functional power test covariate residual cluster relative factor covariate cluster exhibit suggest covariate treatment factor homogeneity isometry may factor often strong external remove influence quantitative effect adopt covariate article affect relationship factor covariate experimental choose technique instead covariate available assign treatment covariate etc several influence covariate statistical biological remove covariate compare ratio site technique ratio residual iii analysis ratio g effect recommend still ratio covariate ratio remove covariate covariate relationship relationship isometry response covariate nonlinear intercept ratio introduce heterogeneity assumption homogeneity ratio give spurious recommend remove covariate remove research set residual refer henceforth residual remove covariate recommend ratio covariate adjust widely show superior ratio residual argue residual totally overall pool residual analysis wrong iii well recommend tool rather assumption treatment group covariate demonstrate balance sample design otherwise especially covariate inference common group hence recommendation favor article wrong residual also covariate adjusted empirical power monte nonparametric test covariate adjust residual entirely sense fully covariate continuous categorical see covariate additionally fact unbalanced heterogeneity group alternative outperform influence present covariate covariate adjusted residual detailed nan hypothese condition simulation use power discussion provide model adjust residual provide single suppose factor replicate value level line covariate intercept slope error term regression analysis stand identically treatment factor ij ij nan eq reject covariate depend reach parallel parallel otherwise article parallel slope parallel line q slope treatment equality treatment different intercept overall level residual value equal treatment assumption adjust residual r residual residual identically parameterize covariate treatment parallel treatment covariate residual difference error homogeneity necessarily assume extension group k treatment notice parametric section distributional equality test adjust compare treatment parallel hypothesis test without covariate adjust respectively parallelism sufficient hypothesis test respectively stand equal estimate residual rewrite average treatment covariate treatment n r assumption take expectation rewrite equation pair condition imply equivalence hypothesis slope covariate overall response slope ir iy slope ij xx side iff hold equivalent treatment hypothesis test statistic square treatment mean error degree df since treatment covariate adjust square covariate adjust residual calculate ir mse r n ti test r adjustment source inference df literature become statistic hypothesis f f ff ff level score similar reach likewise see large increase regression entire line adjust residual covariate mean get parallel base df calculate approximation k normality loss slope arbitrarily intercept response generate error distribution error introduce generate covariate treatment sample size replicate summarize slope apart power increase choice increment simulation treatment specific line expect occur approximately pilot size error replicate replicate covariate variance interval replicate variance error iid double parameter parameter pdf x x df log location parameter pdf fx heterogeneity variance variance dependent variance iid treatment iid treatment case treatment choice variance case roughly difference consumption consumption non covariate consumption covariate consumption heterogeneity adjust residual assign restriction different generation symmetric around notice case scale variance non normality case distributional vs asymmetric term investigate generality replication heterogeneity normality variance might naturally distinct partially covariate covariate mild overlap covariate covariate uniformly generate treatment treatment difference figure cluster first covariate covariate note different treatment covariate treatment choice research consumption randomly select treatment hence treatment middle treatment mild comparison present simulation size relationship treatment difference detect record difference record nominal base mc detect
present subsection nn conditional probability p median well associate divide recorded panel panel record associate sis record sis scad lc c pt demonstrate difficulty minimum regression oracle know statistically difficulty significant oracle setting take see magnitude get ratio oracle difficulty evident value font indicate lasso scad estimate lc pt table lc e c pt le correlation correlation comparable increase size sis scad scenario setting sis scad fail correlation large set interestingly help increase signal notable lasso scad third sis reasonably well normally scenario scenario size correspondingly reflect linear lasso scad logistic magnitude job screen irrelevant scad regression l pt c pt lc l rank generalize show possess marginal screening surrogate screen utility screening option variable signal estimation example meanwhile sure vanish false method current cover link leave broadly applicable outcome covariate arbitrary therefore besides rich regression transformation censor pursuit interesting topic extension number covariate marginal procedure highly jointly marginally weak screen lead interesting future choose important sis preference fdr also employ final choose interesting scope current follow contraction van value rademacher sequence element value rademacher q positive font mean table e define tail idea involve bound application n contraction side cauchy schwarz jensen expectation bound e n since combine p nk k small utilize definition minimizer regard set leave n na bound triangular theorem b ft contradiction constant lipschitz continuity x b score conclusion sufficiently interval function deduce n nj condition b ga r condition consequently strictly increase obvious generality jointly normally distribute independent first side chebyshev satisfied tail j j mi n term tail schwarz take bind complete key proof exceed n b expansion eq interior x bound sufficiently tail eq b b put side conclude positive entail q bound prove case x j c n bound hand equal hence negative continue cauchy schwarz prove signal taylor nm nk nm choose tend one prove obvious consequently tend minimum x k k k g theorem easily see exception tend exception set exception exception negligible taylor last consequently exception tail q exception negligible probability eq conclude contraction negligible acknowledgment conduct song constructive presentation paper corollary remark part nsf grant dms dms play increasingly important research fan marginal correlation possess response propose rank maximum linear fan special possess screen property vanishing possesse sure screen surprisingly applicability spectrum quantify extent screening depend covariate true parameter study addition useful learn scientific research brain study phenotype height million snps disease classification profile grow rapidly demand lot challenge statistical also dimensionality accumulation computer comprehensive reference therein small predictor contribute lead prominent role statistical propose bridge scad penalty method concentrate property learn problem simultaneous challenge computational algorithmic screening sis select deal aforementioned challenge relate show possess independence screen sis screening technique sis ordinary argument heavily easily even limit significantly categorical ordinary heavily explicit expression call research sis model less ranking regression intercept preserve marginal preserve marginal well former surprisingly marginal marginal utility interesting method assess select likelihood ordinary covariate impose aspect sis normality set obvious generalize current framework sis third generalize ranking ranking base fitting type misspecification drop establish tail traditional asymptotic misspecification interest practical variable marginally jointly marginal develop iteratively screen selection procedure former focus linear applicability sis surrogate propose sis procedure sort ranking whereas sure key maximum noise technical difficulty monotonic transform generalize sis exponential likelihood sis present formulate sis summary section scalar family dispersion easily derivative consider dimensional intercept copy covariate whenever note take standardize rank base standardized denote model covariate regression robust instability np version regression denote predefine threshold learn rank coefficient independence learn dramatically possibly hundred although implication pass sure device mle use self response depend sufficiently convex fisher positive moreover control noise relative uniform convergence rate sure screening method former control select concavity minimum b update condition parameter j l nm hold linear linear regression poisson second condition exponentially lemma exponential family bound give convergence sure property screening depend use positive p nm nk fan bernoulli constant bound ordinary weak fan permit whereas handle covariate sure screen property number reduce independence answer simple see one probability tend ii consistency q regression deal euclidean nonnegative differ condition exist nk nm explain interestingly screen property depend correlate indeed percent discover negligible order fan condition result fan condition screen generalize suggest sort view build increment screen equivalent possess screening formulate screening sort independence rank marginal magnitude screening common computation procedure optimization two parameter computation traditional utilize magnitude incorporate whole increment magnitude current comparable level implication convexity otherwise still need discuss beyond scope sure hold screening increment increment follow purpose selection consistency hold minimum
pa pa pa pa pa pa pa pa pa pa pa pa pa pa pa remark hence pa pa pa note p property random variable side stochastically independence exist k j pa pa pa conditional almost sure problem pa iv follow triangular influence influence I suffice kk nr function random result definition acyclic commonly among arise biological moreover may order reduce estimate network paper propose adjacency lasso penalty grow size achieve simulated datum example efficient study compact joint variable conceptual type direct causal relationship relate direct also bayesian base acyclic edge direct direct cycle belief important application study cell pathway play np early greedy search hill super large impractical intensive particularly skeleton causal relationship setting order gene expression present graph conditional undirected graph zero concentration penalization use node explore estimating concentration lasso consistency frobenius norm regularize result estimation matrix precision covariance penalization cholesky covariance matrix order interpretation cholesky estimating skeleton natural theoretic adjacency offer considerable improvement variable selection consistent adaptive consistently estimate usual assumption theoretical evidence mechanism network method gaussian simulation order sensitive permutation association amongst direct parent undirecte undirected use adjacency whose entry specifically regardless infer illustration result new original graph change probability start skeleton true cm eps equivalence challenge estimate variable conditional remove graph parent node reveal graph parent illustrate suppose normally covariance connect variable direct acyclic graph parent variation association simplification let represent eq coefficient normal simple simple latent variance x establish influence adjacency establish relationship skeleton latent give entry depend regardless joint result datum non without scale one precision control penalty involve therefore solution edge lasso penalty find latent graph formulate order triangular estimate solution problem lasso obtained facilitate modification original lasso estimate adaptive penalty differentiable reformulate use number triangular prevent nonzero optimization non negativity solve algorithm scale applicable hundred penalty denote see solve optimization row I equivalent fact n reformulate regularize square project section connection neighborhood set order w estimate suffice solve estimation square range r package summarize comparison computational complexity introduction exponential node surprising pc restriction although considerable improvement estimating gene pathway exhibit iterative require lasso solve comprise covariate coordinate matrix graph calculate include overlap number estimation adaptive lasso regular compare cpu well complexity pc neighborhood pc adaptive penalty accord cpu graph plot demonstrate high pc repetition gb ram adjacency asymptotic property type estimate researcher estimate section matrix overlap property estimator focus asymptotic estimating nonzero adjacency establish structure price main lasso require adaptive estimation lasso adjacency lasso exist iv well adjacency depend choice tuning include validation bayesian choice propose tuning state control probability distinct set next every consist node tune respectively general tune false probability graph lasso goal easy requirement recommend prevent generate low triangular generator control well size edge difference performance represent graph equal drawback dependency node goodness coefficient commonly classification false respectively fit worst compare establish report pc value investigate lambda image gray obtain calculate specific offset effect present observation gray precision observation base base false positive although computationally use gene network direct h along gray edge simulation ham lasso parameter lasso give propose likelihood outperform size may undirected network skeleton pc ordering determine estimation partially complete accord additional simulation considerable hand magnitude decrease comparison remain observe addition significantly power true create consideration similar result observe exclude performance tuning confirm focus aspect estimation distinction plot suggest vary variation positive deviation base time large estimation normal correctly study simulation distribution freedom performance similar penalize additional simplifie disadvantage complex expect ordering variable play significant well unknown generate three dense pc cause recognize correspondingly degree structure crucial lasso adaptive carry flow human system pathway establish perturbation cell molecular know include protein analyze moderate false negative rate lasso give include undirected edge enforce see close play control expression e provide whole datum application goal connection whole genome gene present use method relatively attribute successfully reveal combine consider pc true connection lasso drop adaptive penalty small true edge positive network positive structure lasso penalty estimation adjacency
extreme distribution algorithm algorithm derivative sometimes derivative censor complicated avoid disadvantage visualization graphic graphical reduce iteration complete procedure censor type extreme estimation transformation illustrate use type call equation eq fact point unlike extreme technique monotone decrease prove ng extreme value solution obtain deduce mle consider maxima km software book file fit iteration extreme decrease existence r I guarantee existence uniqueness observation observation sample meanwhile relation q uniqueness mle q censor replace r censor testing fail lead solution reduce censor I censor time fix iteration mle q argument precede decrease point mle type sample exist unique fix type ii censor distribution newton compare iteration newton fix newton take iteration take take converge axiom theorem conclusion theorem theorem exercise theorem likelihood estimation parameter extreme paper newton require unlike
represent option predictive often need common study efficacy medical pearson enable conservative hypothesis coherent decision would probabilistic estimator give negligible concern level terminology establish confidence consistently indicator indicator usual statistical associated significance x side next proposition represent function asymptotically xx interior converge x yield prove proposition consistency smooth suitably transform likelihood ratio statistic consistent estimator regularity indicator side discussion frequentist bring consistency frequentist coherence establish ability hypothesis frequentist flexible distribution require need confidence posterior require set interest dimensional composite neither consistent specifically convergence interior manuscript lead clarity thank discussion coherence provide university thm remark thm example department road odd accord measure call loss frequentist posterior frequentist posterior confidence automatic reduction apply case result frequentist coherent axiom theoretic theoretic cite support unlike interval hypothesis truth keyword attain coherence confidence utility region test utility motivation interpretation observe confidence almost confidence repeat result probability lie coverage rate report match addition matching enable leverage inference lie basis coverage rate predictive probability model location model location scale yield asymptotic model g hierarchical mixture model achieve necessarily asymptotically suggest function integrate nuisance attain bayesian rate advance vision objective universal angle matching prior inference raise goal motivate formula distribution confidence benefit inference interval conservative appropriate value either largely effect various shrinkage likewise know tend avoid interval valid interval parameter confidence contain true give support coherent inference available fair odd condition conservative fail reflect rely extent maker conservative control pre post confidence fisher approximate fisher particular instead careful actually observe frequentist formalize extend level lie confidence rate repeat understand achieve often substantially true lie confidence reasoning frequency level notable inductive logic often effective decision know hour whether particular accurately absence relevant reading car indicate hold equally level confidence hypothesis employ framework applicability inexact third recent arbitrary general motivate extend include theory additive odd depend theory scenario price hypothesis confidence level price probability posterior summary concept frequentist posterior decision state probability completely idea foundation decision truth framework frequentist reporting interval hypothesis determine level interval situation circumstance frequentist posterior reduce exact automatic confidence decision loss measure coherence axiom compatible case frequentist framework example example side assign region assess practical scientific pearson symbol angular tuple pair index parameter quantity sample generality unless nuisance except otherwise note kolmogorov measure measure measurable borel complement field unnecessary value usage latter call space triple estimator nominal coefficient set xx measure turn nest likewise provide sum level equation confidence estimator eq kp x mutual consider hypothesis analogously call accordingly hypothesis confidence strongly ensure confidence measure field framework choice estimator induce extend structure credible improper taking advantage advantage likelihood function distribute accord nest confidence upper tailed yield equality student nest special upper cumulative probability c estimator valid set induce equation p drop difference confidence hypothesis observation case confidence level correct confidence integer equal binomial reasoning process ideal field low odd determine confidence coherence odd upper low probability probability decision interpretation would pay small price express function call family specify evidence satisfy axiom avoid make framework coherent price gamble extend measure induce turn upper initial price agent loss generalize make decision unbounde behavior amount average preferred decision restrict dominate multi statistic use decision preferred yield member member member multi alternative practical broken additional consideration argue dominate rather need x undesirable eliminate replace implication application assessment science arguably valuable apply report inferential role play many value become identity origin radius outside radius chi cumulative cdf justify approximation among conclusion hypothesis implication work confidence scalar enable approximation relate justification level branch maxima estimate characteristic bootstrap introduction fundamental bootstrapping coverage rate justification continue merely neither latter test available approximation acceptable lie confidence value consider minimal scientific significance application researcher datum increasingly minimal biological significance gene effect confidence hypothesis confidence hypothesis practical application minimize confidence estimation bayesian bayesian frequentist confidence posterior posterior expect minimize p prove frequentist mean prove frequentist mode maximize analog xx like distribution confidence appropriate prediction bootstrap value bag prediction bag study uncertainty method determine rely versus illustrate use compatible update confidence theory differ fundamentally dominant statistic broadly prior invariance nonetheless theory make demonstrate framework frequentist require distribution correspond observe kolmogorov suppose decision must assumption vector mapping quantity write respectively successive name conditional odd prior odd determine odd actual observation current learn point bayes sampling upon whenever prediction frequentist check poor reflect initial decision avoid reduce expect loss single confidence measure expect coherence concept place law probability none define kolmogorov replace theorem statement illustration agent whose self agent version unless none ever finitely distribution agree probabilistic logic conditional either parameter update statistical support understand mostly david transformation book know use odd principle occur consider agent odd book argument consider coherence argument requirement distinguished game rule incoherent book coherence bayesian temporal theorems coherent theory thus represent value quantity ground minimal concerned conditioning dynamic coherence maximization case light distinction coherence confidence hypothesis reasonable odd distribution case place contrary conditional hold incorrect work consider conditional bayesian temporal non light subsequent bayesian posterior uniqueness inference coherent give inference frequentist selection prior intend criterion scalar tail confidence prove consistency property condition level composite truth confidence decision cdf value likewise distribute nan side test value pair central name scientific order asymptotic prefer avoid distinguish kolmogorov measure incomplete theory incoherence distinguish value hypothesis
envelope accept probability algorithm independently compute rejection area trial accept assume notational convenience scaling unnormalized constant clearly envelope mixture inversion probability compute envelope c independently return acceptance coincide acceptance use every reject low c ce xx optimize acceptance change frequently fortunately insensitive various note apply avoid author associate valuable comment bivariate conditional bivariate conditional work simulate conditional simulation exponential conditional specification conditional receive attention concern actually suggest convenient configuration acceptance efficient approach g readily
long convex work j series inequality accurate accurate solution accurate furthermore identical minimize change application inequality plug accurate setting k claim argument problem svms discuss algorithm improve subroutine problem wherein bound briefly maximum solve training hyperplane use notation identify empty suffice solution svm produce let margin substituting size point notable vector guarantee define inside convex origin equivalently look small finding polytope similar polytope exhaustive list start version problem rewrite dual read polytope problem minor modification since whose geometric purely optimization viewpoint algorithm treatment appendix competitive version set polytope rotation multiplicative please ask stem variable derivative substitute kkt root note monotonically monotonically increase nonsmooth adjacent sort sign root time sort make fs loop terminate iteration linear total sum aggregation slope offset sm fu support svms recent interest geometric idea iy hyperplane margin separation class write yield qp j one dual feature boundary boundary space live associated svms context become ic therefore iteration compute name core vector rate convergence hope convergence aim efficacy performance algorithm random average guarantee multiplicative choose reproduce report fair comparison time relative refer cccc cccc bc bc bc comparable furthermore usually take believe benefit mm theorem lemma corollary conjecture theorem theorem assumption zhang minimum smallest contain produce radius yield greedy polytope problem polytope polytope present face polytope heavily duality argument n ball small radius diverse statistic graphic wide dependence svms million dimension significant interest art extensively concept small ball expand build greedy every build current stop away current include continue know problem algorithm achieve denote value approximation simple show scale approximation effort minimum convex polytope set find polytope shape set rotation polytope apply two machine find hyperplane computing distance polytope algorithm one study iteration follow section notation duality future find appendix bold bold letter denote entry euclidean dot n definition f q accurate standard concept convex use sequel strongly brevity gradient brevity lipschitz gradient duality section respectively cast convex nesterov subset map function standard certain respect necessarily smooth aim c prox verify smooth constraint use transpose maintain figure conjunction derive follow duality eq word duality gap idea answer question find point maintain desirable allowing update efficiently cast convex answer htbp plot node right right right black thick give set nd cast problem reformulate n I set nesterov minimize efficient towards cauchy satisfy turn prox notational convenience place algorithm recursive plug immediately gap lie ball radius conservative plug accurate ensure require map conjunction find accurate time computation cast qp linear appendix simple algebraic cast quadratic qp constraint bottleneck apply exist point radius give
compatible constraint bayes note practice space px side implement constrained information outcome thus impose imply bp p receive form maximize lead receive second piece constraint update process current new updating lead might exist possibility simultaneous maximize simultaneously fortunately check consistency b update posterior decide clear treat constraint distribution however state expect indeed really retain posterior decide old validity refinement family correctly reflect restrictive processed updating remain comment understand process failure lead reflect background kind assign box ball box amount kind box inform company know kind ball information getting box allow randomly ball box get perhaps open box look format outcome represent sample total let original outcome would get start eq normalization yield normalization lagrange multiplier determine mathematical wish select ball multinomial distribution use model use flat integral form sake joint rewrite wish piece information kronecker delta infer particular box constraint apply whole refer actual box take expect become process familiar bayes recognize use familiar derive reader reproduce standard contrast process different general inform company know information get old ball know getting allow select ball box since piece processed simultaneously maximize normalization simultaneously piece look like sequential crucial updating multiplier multiplier satisfie purpose section second wish current theory pose problem one summarize different different company know expected type ball time many ball box select ball box let ball problem determine get description intend key see identically distribution type addition closure interior become asymptotic essentially say true entropy lagrange multipli sample avg expect use think kind method version label information take box arrive maximize proper multipli plot represent compute ball produce close however drawback represent expectation multinomial produce one perhaps almost indicate underlying probability asymptotic use incorporate special allow go would emphasize method constraint inference argue contrary constraint regard case template currently need assumption fluctuation analogous moment recover rule acknowledge discussion c create detail dropping differ force calculation nest sum take symbol equal technical worth sum second involve lastly sum first newton currently less feasible nest numerically good physics apply economic situation entropy economic information relative example detail template deviation advantage inference much idea physics economic paper economic purpose shannon entropy statistical assigning probability regardless idea assigning assign allow method expect apply moment address large justify use justify simplify bit issue data maximum entropy update form moment resemble produce produce compatibility far address individually one datum comment discuss whether process sequentially conclusion accordingly lead inference potential economic similar two ill behave q dirac delta require lagrange impose usual normalization include function gx constraint emphasize satisfied posterior bayesian proceed want close vary derivative lagrange constraint constraint
complete elsewhere statement first mat ern universal quadratic toeplitz apply analysis maximum check integral say value root small equation two derivative ml maximizer interval interior regularity condition well fulfil mat ern know result regularity series argument b asymptotic square follow surprising small nearly full reach close common limit mse estimation delta law full efficiency large mean error v b efficiency error similarity statement zhang asymptotic discuss introduction adjust discuss wang reference therein recently develop also well estimate generally equivalence ml estimate theorem concern know variance rather restrictive law denote obtain consistent say full parameter restrictive study refer believe reason least good estimation mat ern reference herein equivalently law already directly ml result large particular case implication possible extension grid mind decrease datum likely accurately log determinant worth similarly obtain true square error one line candidate say generality firstly check collect interest ern lemma denote g g equation fx dx fx dx part boundedness enough let filter integral claim proof observe g dx dominant j w j take un lemma prove un lemma claim h h w g algebraic schwarz paragraph g w sign w w easily obtain close remain g g g g boundedness scale ann correlation toeplitz system long zhang likelihood ann spatial statistic series near randomized base noisy single http mat ern correlation prediction http explicit estimate c k mit texts stein data theory krige spline observational conference fix journal spectral fields optimization zhang inconsistent interpolation model zhang toward framework spatial zhang hybrid mathematical call energy ml multidimensional gaussian belong mat ern family regularity white noise square range analog grid toward close analog benefit implementation discuss obtain observe dense regular mat ern commonly use stein correspond known autocorrelation stationary ern process autocorrelation c interpret range correlation independently drop concerned dense grid two successive ideal without correlate follow less analysis useful insight approximation article setting measurement equivalently call actually restrictive since course digit include equal due ill may possibly rescale observation vector whose k k td stein expression effectively parameter especially costly eliminate search maximizer zhang also recently costly likelihood range maximize respect say explicit score idea even wrong remain du wang stein zhang obtain firstly role correct otherwise secondly propose maximization estimating case equation r n energy underlie sample gibbs candidate equation estimate equation new parameter derivative r w r base first plausible idea proposal instead fix coincide thus second known case estimate justification isotropic field time lebesgue list article context much et series exist implementation chen al rather strong insight capability complex limited lattice grid use mat ern randomize principle mat ern family restrict spherical mat ern notation adopt thought framework wrong quite behave toward monotonic plugging quantify provide indeed asymptotic reach efficiency large
proof adopt analytic notation function log denote notation paragraph entropy express cc else continuity know variational alternate introduce support outline later support inequality lead prove support general omit fix supremum full achieve since shift even scalar add result assume differentiable class form denote treatment test potential testing include channel code detection normal represent unknown hoeffding alphabet support sequence hoeffding cardinality weak result hand chi degree report weak elaborate section decay irrespective argue bias address observation draw divergence decay address later guide constraint application alarm probability hoeffding variable moderate false alarm predict simulation hoeffding alphabet show compare quantity alarm simulation central see note give difficult instance nc integral divergence ball hoeffding address issue universal partitioning alphabet extend hoeffding sense apply conclude yy instead xx n trivial suppose class yy partition incorporate regard alternate ratio alternate may anomalous desire finite alphabet letter shannon expect optimal letter bound away sequence distribution symbol length symbol else sigma algebra symbol satisfie else establish cardinality alphabet conclude hoeffding reduce paper paper consider approximation problem xx known express eq onto exponential di divergence express project recursion current scheme robust section test ball analogously q next establish basic ball collection g intersection zero supremum support hyperplane obtain h f follow exponent constant satisfy q adopt criterion evaluation hoeffde notably evolve type limit exponent exponent alarm exponent pearson supremum exponent hoeffding describe knowledge yet optimality choice proposition support eq likelihood test equality exponent clear follow achieve h proposition separate mean disjoint theorem prove approximation alternate several equivalent relate likelihood representation infimum rhs interpretation identity supremum infimum minimizer consequently identity conclusion optimizer know correspondence generalized analyze let ml estimate n value assume attain solve reverse define divergence maxima identity test use restricted denote optimization use divergence evaluate divergence reverse establish characterization projection achieve must vanish first equivalent r r convexity follow minimize establish reverse infimum achieve onto hyperplane reverse onto family geometry linear suppose supremum achieve r linear states support furthermore minimize test asymptotically finite consider solve composite exponent achieve discussion follow proposition moreover f dl f r conclusion solves establish easily composite corollary alternate refer detail prior class coincide strictly log class choose suppose use consistently follow identity statistic establish involve specialized establish specific support open supremum achieve independence satisfy basis lemma alternate q denote identify asymptotic bias alternate depend cardinality far f ii ii sufficient follow function class function part linear part ii linear suppose linear first f r second limit impose assumption ensure assumption whenever furthermore eq theorem derive interpretation call threshold alarm hoeffding threshold section divergence hoeffding uniqueness contain span constant assumption alphabet coincide thus application prove follow appendix theorem part denote divergence define suppose f f apply observation assume decomposition central variance dominate bias dominate term xx contain interior suppose together compact contain interior hessian asymptotic condition directional h decay eq gradient directional dimensional chi square variable freedom denote define condition optimality value obtain generality assume observation marginal law concern appear z nz concatenation requirement satisfied follow lemma neighborhood function coincide function satisfied substitute definite function open open uniquely respect consequently results derivative derivative uniformly u representation random give g gd derive limiting term apply g conclude since decomposition limit cauchy together prove apply third decomposition first generalization characterize variance statement define analogous verify rest hold establish note supremum linear ii establish follow r hoeffding much theorem informally approximation denote distributional approximation large chi square variance interpret result degradation parameterize alternate increase intuitive reasoning hoeffding alphabet choose chose basis test alarm misclassification correct roc curve vary hoeffding increase test suggest divergence hoeffding class match summarize suggest hoeffding exponent priori alternate belong testing incorporate reduce variance hoeffding dimensionality phase ensure alternate corollary make approach universal testing test applicable satisfy central research synthesis adaptation extension markovian extension detection synthesis report address computational report exact optimization tractable building surveillance acknowledgement thank ar behavior continuous version appear identity follow follow simple place
dms grant mh eq q respectively henceforth nonzero regularize estimator model learn much reference note concerned case set interest purpose similar proposition often certain weighted type estimator norm type estimator attain counterpart amenable unable attain precision omit focus require idea vector vector respectively regularize satisfy inequality maximum least illustrate focus use two bound select notational ease verify case always let eq also moment coherence next comment tail error typically value condition domain restrictive typically restriction proceed constant restriction cf need see probability ease follow derive low j b nx put inequality group far choice continue put suppose later indeed make union large outside lemma note assumption combine get complete e right great contradict assumption suitable mle respectively condition bound regularize almost brevity omit get condition let family borel let condition gx inequality interval length continuously extend open contain x get case state regularize include purely second necessarily contain analytic become hard k proposition trivial constraint order nc treatment analytic identically reflect nonlinear unknown become
point large indirect measure indirect might figure map normalize distance viterbi noise intensity intensity behave differently small map superior ml range intensity complicated although perform similarly intensity theoretically examine map estimation consider intensity estimation agree vanish agreement upon intensity operational regime define like interesting slightly linearly stable intensity less stability concern range application carefully development analytical average figure furthermore think semi estimation knowledge remarkably modify interesting beyond process map like behavior observe binary attribute hide dynamical exact acknowledgement thank sciences institute support u nf institute mail sciences institute usa edu maximum posteriori hide markov energy spin focus finite value exponentially reflect zero entropy finally noise intensity reduce phase ising order various bioinformatics economic many hmms amount state corrupted posteriori approach find maximize map readily viterbi extensive estimation surprisingly clear state insufficient understanding infer complete picture know method sequence simple employ comparison average pr hmm relation computer science physics way average cost pr pr logarithm respectively temperature symmetric hmm rich sequence linearly intermediate many solution problem reflect entropy ise intensity poor furthermore regime phase ise phase transition general section also discuss concrete binary paper discussion respectively write probability influence describe far assume generate offer method generate p equally sense typical set overall converge nearly delta strong dominate prior distribution put source viterbi possible repeat ergodicity argument solution observe ise due employ see moment another quantity observe overlap discrete parameterize distribution unbiased error refer refer depend act realization composite likewise combine spin model external govern spin spin interaction constant uniquely determined probability constant situation spin spin align note situation field uncorrelated markovian xt partition terminology statistical physics temperature fix pick state minimize configuration equal ising express free define map account overlap sequence hamming limit equal eq spin act spin temperature limit repeat express recursion govern depend value take assume infinite check positive integer never limit add mix original identical merely make realization composite take depending recursion turn task characteristic study next return free write take entropy kronecker probability typical physical extensive act spin spin amount temperature note check recursion binary fig respectively reader easily generalize graph cc cc cc cc transition add bar compose hence actual element serve indicate general read see fig matrix tensor also block zero augment markov go value amount change matrix sparse treat quantity process process moment overlap former free redundant probability symmetry lower relevant indicate normalize half energy one confirm note depend formalism require moment overlap see inequality realization assume transition regime contribute write latter lead continuity energy stationary probability maximum ml entropy uniquely stress
hx n bb invertible central wishart density eq polynomial polynomial hermitian integer write negative integer eigenvalue polynomial notational simplicity symmetric degree vary coefficient expansion replace differential hermitian verify coincide polynomial hermitian following give symmetric k equation hermitian case case three hermitian let coefficient j include one partition polynomial c yy table symmetric cc cc cc ccccc definite put differential invariant invertible notational convenience real symmetric differential invertible consider differential hand put call plane prove fy assume fy md upper z dx analytic get theorem absolutely laplace absolutely I mx ba yy k hermitian summation special absolutely terminate hermitian matrix suppose assume lemma transform x ng z q z hand analytic wishart distribution generalize case aa w dd differential form q give density eigenvalue present denote summation partition function corollary put corollary large resp eq resp resp resp take grateful comment point lemma cm polynomial keyword wishart matrix mathematics china china deduce formulae matrix distribution small wishart polynomial real early introduce density wishart tool appear polynomial argument definition appear involve partial easy polynomial division division argument argument formula wishart derive polynomial function eigenvalue denote complex division matrix put hermitian eigenvalue say aa na na ij ia determinant volume dx dx dx ss constant invertible define invertible follow come paper element eq dt ss td st
asymptotic fulfil easy fulfil ergodic fulfil strictly function correspond limit process therefore composite statistic consistent estimator obvious contribution modify statistic equality x contradict alternative equation consistent easily alternative eq kullback alternative chi square test find interesting test ergodic de universit du france goodness fit diffusion hypothesis suppose trend coefficient asymptotic von test particularly transformation modification composite basic play special role bridge real I hypothesis function diversity test come er von family er von hx f metric kolmogorov statistic allow statistic tend similar ergodic diffusion suppose hypothesis concern mean basic hypothesis trajectory differential trend hypothesis satisfy bound solution condition fulfil q ergodic law simplify exposition stationary asymptotic interested asymptotic moreover test test q hypothesis constant test belong moreover test special propose similar goodness observation study chen reference therein goal test direct kolmogorov hypothesis er von test estimator kolmogorov test hx relation statistic invariant asymptotically equivalent estimator course choice make therefore q process work certain let median testing trend fix side alternative value first von normality class density mild condition normality therefore test simplified use avoid q wiener introduce condition fulfil hypothesis time density suppose stationary eq law central limit distribution multidimensional law last double sided wiener formally follow distribution wiener property ergodic integrable estimate q strictly monotone decrease note first term alternative course correspond integral similar goal choose q vanishing
close lasso within lasso multiply run active likely support associate assume cn polynomially thus grow pattern union procedure homotopy method development method solution compute computing quantity path unique variable become bootstrappe pair different replication bootstrappe run avoid lasso objective penalty quadratic path within follow homotopy make per homotopy complexity put bootstrappe residual replication bootstrappe even require active propose cm cm cm white correspond equal rest consistency bottom consistency similar know bootstrappe design normal covariance satisfy one lead lasso figure replication variable value regularization noise bootstrappe bootstrappe residual report replication exactly present illustration tend fast pattern relevant proposition top plot satisfied path since bootstrap lead behavior resample knowledge need bootstrappe slightly lasso cm consistency bottom marginal probability select way obtain design middle right top satisfied consistency plot create consistency region bootstrappe early regularization I value bootstrappe residual replication increase essentially inconsistent variable leave intersection replication eq zero black resample know generating bootstrapping except middle relevant design design design simulation bootstrappe consistency multiply marginal study general scheme see bootstrappe behave differently resample noise residual resample behave correctly compare top currently various replication bottom plot analysis lasso general high bootstrap linear bootstrap residual work bring variable enhance thank also bag random forest minor soon union bind z optimality condition noiseless sufficient eq need sign j j j j c satisfied occur great p follow p long p probability covariance complement rank strictly n result inequality pattern absence zeros eq q j j thus allow well weak consider term upper bound use upper well infinity elementary standard constraint notation eq optimal full w c soft triangle shape bound moreover design constant normalize q apply j q asymptotically normal get apply c select variable type upper c hc use appendix ac concentration appendix variable hoeffde distribution normal use nz follow upper c first select thus lead allow lead desire j follow reasoning include inequality upper j j appendix eq exclude bootstrap satisfied give great u regard c combine outline j consider eq lead constraint x n follow non iy iy k k n q thus eq extra moreover hoeffding need n variable set relevant variable lemma select q use appendix q assume p n approach proposition proportional great full analysis outline together note lemma j c j j z reasoning c z large rest proof along line affect pattern q sign pattern lead desire union thank work support france project cm lasso lasso decay regularization correct specific decay enter select run replication support lasso selection novel setting provably attract lot machine processing refer effort dedicate efficiently homotopy regularization know loading selection I load certain correlation variable irrelevant light lasso design dependent step procedure main contribution propose approach resample focus detailed asymptotic pattern estimation extend number variable much select enter variable tend one latter would estimate irrelevant enter support sufficiently eliminate resample dedicated availability bootstrap support consistency regular support bootstrap rather consensus scheme agree case provably consistent also potential additional bootstrap bootstrapping bootstrapping type bootstrap moreover empirical setting bootstrappe lead consistent bootstrappe residual currently unable bootstrappe residual high prove residual run order new low fine new regularization properly version bootstrappe run homotopy bootstrappe residual design time lar section example follow extend denote denote matrix singular element eigenvalue subset vector element index include index consider form setting depend ratio potentially large high fix design identical distribution invertible normalize constant loading setting section extension lasso population e regular include moreover sign equivalent consistency tend infinity derive non bound compare due review power exclusive regularization path many finer particular asymptotic regularization illustrate zero generate sharp situation notably simply exactly correct tend see assume homotopy enough provide noiseless global minimum desirable regular exponentially pattern limit unstable limit exactly hinge sign away hinge path tend slow sign sign noiseless sign first away hinge regularization path eq equal occur consistency sign tend statement probability select correct pattern regime regime tend rate agree probability tend pattern pattern tend consistent probability appendix note early design ps fs assume ps universal inequality detail behavior small one tend previous zero otherwise next section tend tend stability relevant variable non trivial could fast tend sign norm obtain n simply fast constant appendix sign give currently explore possibility share aspect consistent induce tend zero tend infinity hope satisfy least way resample next finer allow presence without consider consistent upper certain relevant j n probability selection selection claim tend zero tend irrelevant unstable several copy piece two bootstrap distribution replication set give original restrict estimate active probability incorrect pattern denote generic p include irrelevant original replication term natural replication feasible us copy design active proportional remove irrelevant variable scale incorrect selection exponentially fast multiple copy split enough piece happen main goal show use bootstrap copy cut piece small matrix strictly fc piece partition present two section together replication consider I n nk select twice note replication show running procedure q large enough bind incorrect b b design constant scale polynomially currently try bind extra replication remove variable overall incorrect tend exponential copy minimax improve research finally explore way intersection appear replication important loading however scope alternative resample residual adapt replication consistency resample scheme differ slightly
two specie restrict common far large population interestingly study consistent gene flow nonetheless support population applicability model single model estimate attribute origin exclude check mutation pattern match use remove software total simulation per report acceptance variation acceptance simulation glm smoothing reject favor among discussion still computation tackle innovation algorithm adjustment term frequently relationship summary summary advantage tolerance value distribution glm spirit advantage glm likelihood glm glm approach framework glm type sampler mcmc glm great glm set simplicity implementation standard package show factor population structure among strongly significantly think exactly never summary namely attribute previously arise almost impossible simulate value fall model acknowledgement author david recently realistic population genetic calculate analytically change abc key innovation adjustment allow large computation realistic magnitude adjustment allow integration factor methodology population ever powerful computer refinement gibb become tool scientific past bayesian distribution turn alternative hard core discussion issue create dna determine quantity bayes often fact realistic population genetic stochastic sampling simulate summary number set capture rejection question scale mutation extend multiple summary von vector statistic space tolerance zero around truncate accept b n f dirac center distribution process truncate hard q give exactly truncated make guess parametric posterior full localization exhibit simpler easy sample reasonable time respective save whose lie list retain yield tolerance small unknown formula complex curse raise acceptance rate approximation partially process et variant metropolis hasting term directly sequential monte iteration method rough within ball retain basically summary statistic local implicitly vector suggest many density posterior density closely center true empirical statistic adjustment prior distribution posterior posterior prior actually vanish moreover clear abc glm literature unfortunately share statistic truncate constant account local linearity statistic summary sm empirical put distribution sharp get curve might several explain introduction normal covariance normally multivariate already square quite insensitive influence estimate dropping exhaustive treatment book observe truncate perform proof multiplicative exceed impractical calculate marginal integration univariate component normalize analytically numerical speaking posterior element selection principal method nearly impossible curse logit model hand readily bayes factor determine marginal density check rejection estimate aid statistic count fraction fall center glm run q probability favor model comparison obtain rejection glm analytically posterior infer mutation site sequence bp tolerance level estimation always report replication assess infer analytical calculation measure curve equal vanish uniform figure abc glm rejection algorithm tolerance value rejection posterior observe tolerance rejection abc
development circuit evolve fire relatively minute day node episode occurrence ratio dramatically development medium provide despite large firing day day fire rate episode connection include increase methodology track development show dramatically medium h display overlap occurrence episode discover functional connectivity spike connection frequent episode inter event discover serial connectivity method allow episode strong interaction among delay cause propagation delay chemical building number occurrence serial episode show occurrence episode directly among demonstrate spike discovery frequent pattern discovery fix consideration strength context symbolic time episode motivation dependency among symbolic source conditional event characterization strength present application delay general delay record connection delay event currently episode episode connectivity address future dr simulator counting report general center university ann structure activity multi challenge repeat activity group neuron neural temporal framework counting occurrence propose statistically significant counting strength connection resolution previously pattern strength present pattern discover neuron train array episode occurrence statistical interact time characteristic electrical spike sequence spike refer multi spike train contain fire spike individual activity group datum identify network neuron underlie array tool lead collection multi spike neuron critical understand region act currently amount temporal efficiently analyze appropriate technique intensity stimulus aim pairwise small neuron frequent episode discovery temporal firing spike train temporal mining deal symbolic stream specifically sequence event analysis multi event spike neuron spike interested collection event prescribe firing follow fire episode constraint delay compute frequent episode precisely pattern approach major implement algorithm episode statistically threshold random connectivity episode overlap episode implementation mining work recurrence relation result characterization distribution moment section discuss connectivity strength simulate demonstrate usefulness employ delay neuron fire pattern serial extended encode spike fire rate suggest produce spatio temporal discover frequent episode piece connect subsequent episode discover reconstruction popular detecting repeat shift spike train early currently neuron temporal circuit involve suggest base ability simultaneously neuron infer firing computing involve frequent view time order denote denote type finite denote event type collect spike pattern episode general partially order consider order sequence event type serial episode order serial episode event appropriate episode neuron neuron occurrence episode serial datum specific firing processing discover frequent episode tractable episode focus episode episode non time period occurrence illustration episode say every counting algorithm style wise discovery datum episode pass count episode count frequent episode use count candidate episode efficient count occurrence apart consideration computational efficiency frequency spike serial episode would episode base count occurrence occur repeatedly frequent serial episode structure frequent episode time discuss set episode frequent count neuron want count indicate restrict difficult threshold episode limit assumption interesting notion look threshold indicate probability introduce refer time bin delay neuron delay conditional weak know hypothesis use detect episode simple episode occurrence observe time period neuron interest time word connection overlap fire furthermore identically success hence occurrence binomial trial occurrence bin occurrence occurrence interval binomial success probability readily take side independence decrease expect vary parameter span episode episode occur characterization mass moment function use exist inverse time w moment numerically mass cumulative good normal range easy note except obtain variance replication independent hz firing rate quantile plot consider approximation reasonable refinement compute threshold count skewness fire hz suppose conduct episode neuron independently fire use interval test hypothesis demonstrate variety strength occurrence map true confidence replication connection coverage mp arbitrary episode expectation expression depend neuron episode become dependent demonstrate combine statistical efficiently discover strength complicated firing neuron mining episode occurrence episode event constraint map count episode occur episode fire episode would respective count divide datum functional delay strength ratio mining interested connection episode ratio frequent episode candidate count much node permutation node episode frequent true connection embed node rank connection count node eliminate
describe bayesian imply thing rule stop irrelevant entirely collect analyze contribution hypothesis ratio use successive ratio martingale martingale martingale reasonable independently red realization martingale false dotted dotted line evidence martingale nan hypothesis martingale compute role easily martingale show figure test martingale even though b logarithmic figure unbounded trial reader notice martingale around reject calibration conditional increment variance increment iterate tend almost surely eventually obtain hypothesis want conclusion conclusion correct choose measure conclusion conclusion nan test leave corner observe figure figure generate sequence slowly converge show enough reject two false hypothesis identically independently martingale test moderately conservative test stop whether positive precede paragraph test matter rule select nevertheless report value follow follow matter know advance asymptotically tend reach conventional size define q take value stop chance stop interpret report guarantee advance reject choose close possible significance satisfie define require wider determined much wide question know advance interval slowly sequence weak detail stand thus posterior always observation ratio spirit interpret reject rejection stop really correct hypothesis fully mind resource decide sequential take shall seem stop statement consider irrelevant treat statistical advance come e discussion report continue naive hypothesis law happen term pointed happen something without know frequentist mainly notion adequate purpose hypothesis fix consider hypothesis martingale accord decomposition discard compare essential hypothesis modification introduce principal often say process member process exists stopping stop popular standard vi expectation adapt strictly decomposition test therefore purpose test martingale dominate martingale local continuous essential admit brownian motion vi local martingale martingale vector harmonic nevertheless fail detailed calculation martingale replace purpose martingale represent martingale belong dl dl times integrable integrable belong dl martingale sense martingale let martingale arbitrarily dl vi stopping dl appendix first observe iterate version euler theorem pt iterate entropy equality rewrite far rewrite finally write log kind hypothesis alternative finally detail roughly size tendency article shape discussion gr lead correction improvement section grateful development type de france department mathematical logic mechanic department computer science university unite inverse value interpret bayes factor large attain way eliminate inverse characterization increase function eliminate regard simple well relationship nonnegative visual initially evidence value begin martingale mean risk cumulative lot look make look well relate familiar support alternative say value stopping leave great value far understand complementary answer martingale testing every perhaps shrink shall say exist martingale randomness treat martingale dynamic measure establish interesting reader familiar mathematical notion answer converse exist scale parallel supremum martingale bayes factor bottom probably people picture bring algorithmic randomness reader familiar field historical discussion formalism theory reader mathematical section coin fair depict prove devoted mathematical introduce wide conservative test explain explain maximal current tool carry thesis start capital capital begin strategy tends infinity zero zero martingale idea predict event subsequently role randomness historical perspective behavior tool mathematic subsequently important technical tool mathematical time survival mathematical slow concept test nothing reject establish pearson pearson hypothesis likelihood reciprocal ratio density say continue reciprocal martingale directly goal nothing reciprocal role become increase importance bayesian start often call bayes ratio multiply ratio factor informally statistical probability justify rejection differently report hypothesis reject pearson pearson probability advance merely report significance adopt point define bayes value version narrow wider equality version attain conservative version conservative conservative narrow recall triplet value measurable random take tt interested formalize convenient specify martingale order index whenever adapt measurable resp martingale algebra almost surely automatically limit surely see vi satisfie namely contain subset space complete usual se vi early define value prove test page vi call relate factor discuss informally derivative satisfy bb conversely whenever nonnegative measurable relate concept consistent call usually measurable set satisfy also satisfy treat tie carefully satisfying generation precise statement assertion former assertion whose reciprocal give factor include completeness practical arbitrarily correspond statement almost bayes bayes martingale moreover measurable test convergence necessarily expectation right evy page martingale imply ta final stop vi supremum true supremum point test inverse test artificial construction directly practical help give explanation discrete level merely test equal whether whether martingale start capital last amount precise martingale supremum construct capital capital everything check time nontrivial integrable algebra complement algebra generate either borel subset borel check suffices borel subset equality rewrite strictly increase characterize increase say strictly dominate admissible admissible prove give perhaps borel algebra bayes check hold c fp formula random fp fp inequality stochastically inequality part equation give produce admissible primarily interested behavior take significantly etc example let increase martingale independently demonstrate convention say martingale martingale admissible continuous start suppose increase bayes fx ty fx ty statement argument show check martingale modification precise equip complete
series devise markovian scheme thus usual sampling simulating conditioning truly move derive ar good calibration particle figure evolution particle observation autocorrelation graph configuration around particle particle around observation grows show particle evolution simulation plot index could possibly occur large simulation higher demand iteration particle hour severe rate value row write http simply calculation gibbs advantage comment loop paper mention well deep extent finite horizon nature noise mechanism rao denominator eqn past obvious information provide fortunately value care selecting evaluate bias stochastic volatility example et al sufficient reasonable unable likely explanation good implementation may optimal nest perform hour report meaningful paper b theoretical paper complicated evaluation setting offer state comprise trajectory enough remove limitation smc come complete smc ask resample rao proposal distribution technical
shift pool especially yield research effort towards multiple comparison problem kind fitting package implement suggestion multiple comparison arise frequently study participant interest etc arguably yield simplest comparison correction pose burden type correction reference american control practical powerful multiple discovery testing report department stanford analysis stein generalizations american rates bayesian journal biology hill use hierarchical american rates american operating characteristic extension false discovery rate grant liu identify microarray bioinformatics sometimes journal american comparison health trial health journal american stein fourth berkeley ed california thresholding possibly sparse parent biology teacher effectiveness york city economics education nan journal american association estimating journal american association center education c office impact student american economic review may estimation randomize experiment journal statistic potential assessment education journal behavioral w inference gibbs j multiple encounter try state journal validity study journal resample testing adjustment york v v pt www hill http www edu pt apply find inference setting seem challenge paradigm correction moreover multiple entirely hierarchical building partial pooling toward typically keep center adjust multiple make interval wider equivalently adjust width yield efficient low comparison concern bayesian comparison statistical nearly social physical find simultaneously evaluate question compare point compare several different compare indicator rate state examine effect examine impact program outcome comparison conclude set test even nothing go additional serious propose review relate comparison concern test may identify statistically fact real paper perspective rarely believe multiple insufficient modeling correspond within appropriately shift toward often shrinkage classical adjust wide adjust interval appropriately sense interval say make likely adjustment detect difference many problem examine many comparison significance paradigm question simply put problem put burden goal procedure realistic comparison illustrative solution outline correction several relatively use classical health development intervention birth service home intensive care program randomization take within site birth experimental slightly complicate purpose randomize block eight overall effect give composition site program implementation varied site like know statistically concern make false risk sometimes arise interested whether effect site might look like j program way represent site allow significance site hypothesis incorrectly test least raise perform independent eight site significance would chance reject popular evaluate test specifically work calculate divide assume example overall significance translate test wide confidence estimate along correct nominal significance reject intervention site adjust reject nan hypothesis expense type reject claim average false reduce researcher goal dependence variety correction bootstrapping method test example focus reduce rate instead false rate control rather rate powerful paradigm rate independence test test fdr make field would expect vast quantity effect examine al science application likely thousand hypothesis less truly zero distinction formally control rate already perform variable group statistically significant difference perspective significance yield correction probably helpful would one significance alternatively sometimes specify perform expectation similar comparison tie hypothesis moreover fail test goal propose circumstance present perspective implication argue perspective simply perspective primarily establish site effect concern accept fact ever importance test similarly occur accept truly group shall discuss statistically serious concern might fact phenomenon treatment new york fact reverse analysis concern compare different none might want effect actually near likely instance statistically actually plot estimator distribution great produce less uncertainty deviation examine automatically increase something go help view within error substantially simple score parameter allow intercept across seem think realization addition specify kind could real power notably intercept model pooling treatment site site often figure plot next interval correction horizontal dash line display horizontal solid quickly statistically significant effect site estimate really reflect shifted estimate predictor partially pool fit group surface mean pool sense intuitive uncertainty estimate population treatment site true certainly really inference essence use effect estimate site actually ignore site allow site site sense ignore find site site level great site get less uncertainty site trust illustrate run decrease site sample result display increased bootstrapping result right way reject hypothesis procedure ignore comparison raw graph actually though clearly nothing piece default graphical obtain average mathematic student state average student take average distribution parameter case ambiguity claim confidence make little paradigm examine parameter fit simulate effect state purpose classical cutoff state simulation claim plot significantly one blue classical summary confidence central comparison classical set extreme true evidence correction evidence comparison comparison correction ccccc treatment raw e one additional correct deviation treatment evidence little pool toward none comparison close statistically discuss chapter meta new notable analysis effect estimate effect get standard eight order sort situation might multiple actual raw happen statistically study bayesian estimate likelihood mode zero bayes pool toward insight deviation plausible simulate eight value eight separate distribution group analysis compute count statistically estimate time error number correct simulation statistically sign correct avoid statement bayesian get sign correct essentially multiple comparison way estimate statistically comparison bayesian occur school classical inference inference correspondingly statistically happens repeat simulation treatment statistically statistically whether classical price pay claim huge child york city assess factor determine effectiveness finding teacher effect moderately deviation level system researcher use approximate model variation teacher effect teacher effect persistent background decade teacher broadly school like award leave problematic aside analysis rarely ever address involve comparison try get compare thousand distinguish teacher individual therefore concerned type fact analysis health find likely rate attractive participant survey statistically significant versus plausible comparison physical survey use measure five point scale attractive vs category possibility compare people compare compare comparison summary wave wave wave statistically study multiple comparison bad idea percentage correction would change significant properly multiple either proportion measure grain pattern beyond importantly sample simply literature analysis people risk type reasonably uninformative prior heavy tail effect reveal information effect well percentage program multi site expand birth weight birth group differently intervention weight additionally treatment across group effect site child birth weight low birth allow intercept site hyperparameter plot set serve group arise site birth child stable treatment correct birth child sample site conclusion analyse none close zero adjusted width four include close researcher attempt impact many eventually positive significant chance require big conceptual natural describe effect think draw knowledge big trivial
computer university correspondence edu cm edu distribution class necessarily follow normal positive independent chi square variable chi square especially two genome estimation particularly second em infer em need substantial burden eliminate mathematical practical extensive study two statistic find typical gene weight square association association complex disease evidence single interact create super allele trait et al design statistic goodness quadratic sample statistics chi nan chi distribution permutation power al lin attempt specific advanced similarity normal normal inaccurate sample statistic assume normality test write follow discuss systematically central alternative statistic square comparison procedure permutation permutation increase test significance permutation procedure intensive estimating typical power significance power must calculate apply association genome many order account comparison additional arise permutation current generation study un trait association estimate computational arise category rare phase intensive distribution solve et rare merge common thereby reduce efficient fast permutation moreover prune rare situation consideration apparent way forward generalize paper asymptotic value power need permutation assess robustness use illustrative display positive case definite assume symmetry study plot distribution distribution likewise assess performance distance al need study unknown population count notation k ks hypothesis focus multinomial therefore variance asymptotically positive orthogonal tr z r w w asymptotic assumption hypothesis let represent let semi chi chi approximate adjust statistic chi freedom base approximation chi chi tr quantile reject level formula indicate coefficient chi approximation calculate major inaccurate high dimensionality ii chi square positive definite similarity similarity consecutive subsequence similarity formula simple carlo simulation random use base formula procedure much simple fast alternatively eigenvalue positive negative estimate chi difference variable may monte carlo technique describe study find asymptotic md b kk discussion case singular statistic appendix multivariate positive definite definite central shift square singular easy verify propose shift chi quadratic idea liu al able fit list formula refer square liu df df df critical note find value rejection usually correspond replace estimate alternatively q quantile automatically conclusion et al eigenvalue approximation power singular diagonal aa sd distribution singular dimensionality convert non illustrative statistic al actually long necessarily z weight moreover therefore freedom singular conclusion show chi chi square chi illustrate deviation claim hypothesis normal previous discussion follow chi large different normal inaccurate difference rate approximation demonstrate propose yx calculation central write x x software integrate source file approximate quantile size specific power file http edu software set et ii table complete length counting measure proportion seven simulation approximation nan hypothesis examine plot million simulation axis small million significance million combination level chi chi procedure proportion true one hour ghz ram estimate however second procedure stay summarize table approximation prefer simple high list approximation evident provide probability estimate value large mean million permutation give conclusion base date examine examine purpose simulation note good approximation hypothesis figure examine kolmogorov true effect nan moreover situation usually formula case value examine approximation tail parameter fairly moderate power claim normal rate substantially show variance suitable figure chi small size increase also become acceptable figure far compare normal square kolmogorov von combinations second illustration observe chi distance especially candidate distribution around phase information unknown distinct category matching length test indicate significant measure error unknown counting measure additional require significance approximation describe easily calculate required population separately em package start use em refine calculation minute ghz gb ram finish single calculation procedure discussion analytic quadratic statistic use test well efficient way calculate specifically show quadratic combination chi square distribution situation square distribution liu degenerate generally speak approximation deal nevertheless latter moderate probability matrix computationally less intensive similarity perform dimension decompose appear well property hard accurately population recognize lead error testing et several develop structure al focus population square would similarity conduct association genome association fast estimate genome region manually limitation length define method explore acknowledgement support grant institute medical center ny suggestion student statistical association trait correction test statistic principal hessian letters b genomic association mf improve relate chi american estimation molecular frequency population frequency available via http project package liu h zhang chi negative quadratic form central statistic lin base association test effect population genetic association nature nj I na wide association jk na genetic marker american journal association american human de population ga trait linkage complex trait human chen zhang http weighted journal american association similarity goodness fit zhang variance kl trace application independent random hypothesis start chi approximation shift see diag ta liu liu al formula quadratic form degenerate multivariate c kolmogorov piecewise let keep generate kolmogorov check von formula
qp challenge accordingly reliably precise solution fairly accuracy future rule method interesting efficient deal gradient descent hard coordinate might decompose iteratively hinge leave thank constructive comment soon helpful rewrite mkl follow turn know lagrangian express lagrangian sufficiently neighborhood proof assertion remark mm ac new mkl type iteratively solve qp proximal cost minimization roughly proportional active aim scale mkl include net efficient exist powerful rkh choice mkl combination assume function source combine weight set example heterogeneous numerical text link principled manner challenge kernel give put evaluate mkl various programming sdp order solve sdp heavy propose kernel update nice tuned solver program approach utilize cut plane weight instability solution sequence mkl drawback call improvement rather scale show behavior solve linear qp newton solve qp qp grow article mkl norm formulation introduce also formulation sparse estimation efficient view efficient scale call thousand original presentation primal mkl problem believe compare interested svm lp qp minimize cost kernel thousand train less second recognize initially think point often outperform kernel accordingly instead well mkl uniform combination elastic smoothly connect paper norm allow mkl review slightly kernel combination general organize mkl weight section extension algorithm primal application proximal carry efficiently exploit mkl section mkl elastic net mkl formulation special elastic mkl efficient fast block summarize super appendix matlab software regularization block mkl direct extension formulation square version later belong output usual setting gram fix consider rkh kernel sec also rkh accordingly hinge loss mkl learn objective without get serious way prevent increase explicitly follow inequality arithmetic take formulation motivate squared block constraint instead solution map let minimizer formulation minimize square formulation regularization subdifferential convert mkl notation attain problem nm nm simplicity rewrite k eq section extension proximal twice differentiable discuss minimization algorithm iteratively minimize proximity proximity adaptively objective last proximity try keep solution proximal seem mkl primal proximal mkl minimize original mkl necessarily interpret gradient subgradient equation zero q subgradient learn unique scalar solution q converge optimal linearly proof mkl dual efficiently variable result apply mkl threshold proximal mkl minimizer update tb mkl augment prox prox repeat lagrangian mkl lagrangian multiplier maximize mkl lagrangian respect problem lagrangian minimization lagrangian see note minimization respect conjugate second conjugate hinge loss illustration regularizer envelope soft operation algebra convex positive semidefinite generalization eqs proximal ignore problem eq dimension conjugate whereas smooth every iteration follow hessian efficient require correspond sparsity make formulation dual region prox point interior intermediate hessian objective minimization call mkl update correspond lagrangian technique search minimization inner use unbounded attain boundary situation separately line last net mkl uniform combination formulation separable handle question technique conjugate immediately formulation equivalent mkl mkl mkl table discussion relation rkh obtain optimizer formulation correspond multiplication regularizer exponent exponent mkl mkl mkl mkl original iteration need proximity operator regularizer envelope proximity follow note along due elastic mkl regularizer write proximity obtain therefore conjugate q regularizer convex obtain eq envelope conjugate regularizer cauchy schwarz envelope modification case envelope become manner generalization straightforward need prox matlab general loss block dual primal dual proximity convex regularization smooth outer dual show elastic primal q expression rewrite follow differentiable regularization penalty function primal compute primal q kkt therefore primal differentiable e loss resolve elastic net block norm elastic net mkl cx net kernel solve norm calculation conjugate formulae exist basically formulation say constraint give rkhs corresponding consider instead average scheme convert one inner easily publicly efficient updating update differ totally update descent envelope envelope fx fy fx xx proximal update envelope increment next fortunately envelope smooth optimization discussion envelope optimization norm mkl regularization norm mkl formulation reason utilize bfgs special case set mkl regularization confirm binary uci repository logistic loss report elastic describe optimization regularization candidate gaussian e normalize experimental choose repeat run convert duality tolerance compute dual multipli project domain dual objective mx mx j dual kernel publicly available code solver program inside performance summarize addition standard show tend fast factor factor increase active dataset iterative procedure accuracy regularization mkl hinge seem perform elastic varie logistic hinge strong hinge rapidly accuracy nearly identical hinge regularization kernel net much block decrease regularization uci objective current refer outer line search gradient obviously outer outer iteration kernel become intrinsic small drastically light per require qp heavy relative duality gap cpu dataset see drop fast support decrease duality gap order duality gap gradually drop early
jk jk jk jk jk q feasible recall magnitude birth make modify appropriately live location calculate nearby location assume generate deal issue monotonically indeed dominate location gain dominate value solution unnecessary thus large estimate present study investigate four namely block arise function simulate spaced signal add take replication posterior distribution add value parameter deviation trial establish wavelet estimator reconstruct signal cross false asymmetric wavelet block haar wavelet discrete cross area interaction ss bt false estimator noise replicate rr rr bt fdr ex fdr ex bt fdr goodness measure replication clear moderate procedure naturally cluster perform noise reasonably level future number point location modify tail behaviour behind behaviour adjust could also speed inclusion minor software include would develop method software implement describe request thank discussion reconstruct noisy rather make possibility correlate scale frequency lead structure analytically past approach regression eq spaced interval noise normally zero approach transform large behave transform distribute distribute combine remove transform small coefficient perform large evenly discard discard threshold wavelet propose median wavelet coefficient give zero coefficient capture wavelet allow prior extension area basic disadvantage long possible chain sure chain simulation stationary introduce exact section discuss coupling past compare conclusion discuss area produce cluster moderately order pattern introduce left interaction process neighbourhood set neighbourhood follow compact metric configuration suppose finite borel regular class compact case rest technical trivially subject true wavelet zero wavelet transform constant formulation prior discrete location variance wavelet nonzero view replace allow integer assume lattice concentrate zero observe pure support wavelet choose periodic equally simulation begin past follow stationary space stationary return I practice show goal go back finite space couple wish spatial point q respect rate value monotonic process evolve birth death birth death configuration constrain birth birth let birth death start death started accept evolve follow also equivalently sample require back time keep rejection lattice modify normal possible integrate lattice performing rewrite jk jk dd dd jk jk simulate process mark lattice process amenable use slight abuse third subset dominate intensity
track expert forecaster sequence switch limited formulate tracking hard maker tuple measure switch interval tuple average meta reference moreover generation tuple achieve version decision maker observe outcome remains consider bandit problem become always imply exponentially forecaster satisfie average implement compute essentially invert detail study multi maker round computed manner incur norm constraint first forecaster full maker loss correspond task extra maker model characterize task among course hard could example paragraph inspire loss incur may example loss whenever performance make markovian define big round action make single replace supremum loss need explain modify unnecessary multiply loss care include min addition measure incur round additional hard constraint design task forecaster implementation discretization take back previously index decision maker give opponent choose maker maker hard one eq must contain shift definition shift contain denote shift take aim maker minimize cumulative eq pick denote uniformly ideal forecaster distribution draw application give simultaneous round forecaster continuity forecaster regret bound inequality take convention bind obtain inequality appear achieve infimum regret exist take element whose cumulative infimum order action differ length action define close simplex take uniform neighbor distribution take value put together n n simulation propose although would typically markov pair form time convention chain uniform index take simulate simulate exact state programming discuss take approximate space efficient resort sequential monte broadly know ideally suit approximate formulae expectation function law large particle importance particle interact particle property population approximation target constant go carefully design importance step characteristic section approximate approximate version forecaster describe partitioning except last fix action aggregate super task able precisely restrict element start simultaneous action compatible partition super task loss satisfy forecaster run shift ensure choice complexity yield comparable moderate easily bandit super task unable unable computable position constraint put action position keep track row diagonal another trick berkeley use issue trick task half view es sup paris france paris en discuss maker simultaneously task related impose maker restriction tractable introduce efficient select short discuss bandit additive loss task considerable attention multi simultaneously relationship maker choose simultaneously repeat task put simultaneous motivating consider company several customer customer order company loss receive offer consideration suggest offer customer age company select batch offer budget company offer send customer response send simultaneously action budget constraint maker accumulate good problem play repeatedly game consider nash equilibria address feasibility game play reference parallel enforce constraint loss game loss difficulty requirement maker choose restriction maker survey show graph tracking decision maker restrict set limited bandit decision maker instead observe game learn finally infinitely maker function natural finitely discuss closely relate concentrate average feedback complexity order maker deal simultaneously index finite action space n action integer real associate task repeatedly round maker choose choose index assume among task outcome maker outcome vector completely arbitrary maker compare maker important maker restriction simultaneous action forecaster allow vector maker aim regret difference cumulative cumulative action allow basic maker vector outcome maker reduce expert history maker treat task maintain forecast least bandit solve restriction need satisfy structural satisfied well meaningful like basic maker freedom maker maker budget round make thing one integer value become value meta problem exhibit forecaster implementation proportional cardinality precisely denote simultaneous instance cumulative loss put mass eq tune direct application denote cardinality complexity prohibitive cardinality example shift lower exactly shift finally draw action accord reduce online short e maker assign round loss path exponentially dynamic edge graph draw online short example order short state hide mean satisfied thing four simple action space sequence one define underlie shift action spend view example markovian follow x put differently indicates impose already stay thing concrete rely transition transition transition eq one ready graph multi online short sequel round acyclic correspond task two associated round
total divergence minimize eq conditional rewrite pm ta ta hypothesis new agent weight update bayes suggest system arise fact system one illustrate statistical ask likelihood give pdf q likelihood informative change thus belief would mathematically imply independence result produce next obtain datum look almost exception posterior eq q obtain simulated datum order separate converge differ design toy mixture bayesian rule ta bias towards observe act furthermore uniform pm interaction instantaneous instantaneous value instantaneous deviation result imply either result whereas correct propose causal calculus agent bayesian mixture I integrate agent heart adaptive agent na lead output observation crucially vanish intervention present unique agent environment propose representation ai superposition previously expert like architecture stochastic find amongst usage principle ai main derivation inference rule divergence potential application would take similar bayes control translate stochastically formulate bayes relationship investigation engineering cb pz uk intelligence formalize sequence output possible actually compatible world obtain leibl world uncertain pure stream agent intervention calculus calculus model adaptive behavior stream allow approach control behavior intervention calculus bayesian kullback environment consider intelligence environment system exchange symbol symbol environment sequence perfectly tailor environment face robot endow behavioral problem act primitive distribution interaction uniquely determine define coupling system rise describe stream specify valid model true coupling stream produce observation history stream role observation sequence output stream stream
filter alternatively estimate systematic performance currently explore problem compress gaussian describe measure analogously consider minimum ht ht significantly transform sparse fewer suitable satisfy isometry recover optimization q effort cs generalization include also specifically causal discrete system impulse response signal neither recover uniquely turn belong dimensional ar isometry orthonormal build intuition practically relevant piecewise integral operator act sparse long orthonormal usually variation compress sense develop alternative filter impulse mathematically familiar toeplitz consequently toeplitz rip tool indeed property toeplitz idea stable idea deal turn every finite system order filter employ convolution leave pose duality speak indeed measurement rip projection provably toeplitz scenario random construction organize mathematical section describe filter state reader understand main proof address blind deconvolution section technique namely decode causal reconstruct autoregressive non autoregressive known driving assume vector task compress ar driving measurement ar standard setup ar efficiently variable toeplitz preserve speak assume shift version effect shift particularly suitable notational purpose submatrix compose rearrange equation use simplify project reason choose iid rip rip constant toeplitz construction projection projection toeplitz entry th order confusion true train refer possible decode similarly omp directly apply regard original signal still mind contaminate noise algorithm take equation follow minimization find derive recover invertible paper solve minimization equation propagation state isometry quantity obey cardinality direct sake completion unique unknown need process e impulse ar spike define jk spike amplitude satisfy drive equation drive spike necessary spike consecutive solve random benefit random naturally toeplitz exploit consider blind deconvolution reconstruct coefficient problem blind deconvolution simplify compressed identity well completely version ar matrix solve problem show state technical denote noiseless define comprise comprise u condition small tx cp practice generally persistent converge compressed say energy relaxed simplify condition let scenario scenario bernoulli sum hand spike align signal phenomenon illustrate experiment ar blue curve see blue spike db sign spike sign spike condition satisfied also assumption snr small spike ensure spike zero lasso estimator hard analyze clear kkt choice theorem provide ar autoregressive contain transform process past depend input equation toeplitz triangular side assume decoder follow process note toeplitz toeplitz kkt condition unique applying condition convert toeplitz equivalent invertible last prove situation decoding lie neither spike train combination iterative comprise step iteration use require minima switch rewrite eq final iteration round round step update equation fast rate hand small practical implementation early stage stage figure illustrate un un round round update choose image pixel neighboring consider causal ar causal ar impulse impulse process discriminate process subtle difference deal condition boundary case follow random since boundary submatrix comprise multiply side follow p simplify p p minimization lasso p p u complicated way view problem perturbation noisy p unfortunately consider namely sense eq u write formulation proof sign ensure primal pair construct obeys give choose value magnitude ar contain place simple contain idea case note assumption automatically drive loss assume root function ar tell furthermore summation j j ty convexity primal dual ensure give violate spike next break whole satisfied check exist condition construct theorem inequality rr verify clearly follow denote begins already impulse cause almost finally correspond reasonably prove ty ty ty exist construct fix I give know last satisfie sign p jx u magnitude determined sign general choose r similar kkt respect need check inequality ty ty ty ty ty next multiplying compute yy ty py pe pe pe ip ie show correct I ip ip ie I follow I hold probability equality proof omit simplify formula submatrix comprise comprise row index simplify matrix comprise row index index comprise index comprise row permutation rewrite check I p note ty p ty matrix ty ty get ty ty ty ty ty ty ty ty ty ty I simplification ty ty ty
generalize nice feature update well result generalizing prediction nash nash algorithm profile guarantee agent strategy representation anonymous game still fundamental know agent must choose agent something unlikely action another guarantee choice payoff action would round payoff equilibrium slow furthermore guarantee distribution equilibrium strategy converge learner nash game use stage rather equilibrium search strategy use move move equilibrium give require history examine procedure frequency guarantee game game tend observe agent payoff payoff base learn fast body empirical pricing game grid converge would explanation game rapidly equilibrium practice anonymous game straightforward tuple infinite choose simplicity interpretation agent mixed action second interpretation notion say agent action mix profile choose action want fraction end action action profile process profile limit something payoff payoff result perform payoff rather agent payoff change agent perform action agent follow p sa ensure anonymous require notion continuity distance difference agent utility anonymous round agent player game opponent game payoff choose action payoff play utility game action maximize good amount difficulty determine allow action denote well well try agent start action action agent apply say probability note notion merely sequence say approximate dynamic every determine learner anonymous converge stage need game well short sequence action distinction irrelevant small show property degenerate strategy agent want know payoff payoff divide game stage almost play explore choose next agent maintain stable environment explore originally learn reasonable show agent learn stage round need select mixed action agent strategy play explore uniformly previous know payoff distribution agent know action I I convenience agent stage ai ts stage learner observe statement run triple agent choose receive notion tuple g stage learn anonymous profile error learner fraction stage action stage play expectation average exponentially close true expect via standard hoeffding variable correctly anonymous game converge game learner agent rate exploration close change strategy fraction fraction action symmetric pure incorporate explore play explore distribution exist play width depth close example reverse closeness notion specify distinction close calculate game agent actually follow lemma sufficiently lipschitz give requirement acceptable lemma acceptable well least learn well nash profile specifically nash actually nash equilibrium three practical converge quickly system robust know payoff requirement guarantee stage argue case robustness fraction assume error due noise agent leave issue increase equilibrium population learner allow within round apply slowly learn implement regret converge nash equilibrium game converge closely stage make two tell get random payoff match observation distribution prediction lead particularly game payoff due mistake decrease agent experimental illustrate behavior learner include describe payoff payoff match payoff adjust game symmetric report contribution strategy contribute collective agent contribute contribute game strategy agent second implementation learner describe stage learner result strategy distance average payoff nearby close notion play distance action length mistake presentation graph similar population agent population agent mistake mistake cause successfully although mistake strategy agent converge equilibrium mistake rare equilibrium due exploration also fraction action action cause asymptotic equilibrium allow far depend tight require converge equilibrium nash equilibrium agent determine stage convergence tight ten result payoff determine strategy short stage stage convergence mistake agent successfully regret learner payoff game payoff end random take approximately design result system collect ensure enough expected payoff learn game statistical typical result show natural learn interesting issue exploration agent stage average payoff receive certainly guarantee arbitrarily give stationary strict exponentially recent estimate value new base stage spirit play game possibility tolerance mistake equilibrium notably utility add mistake time fraction agent play could leave system reasonable newly agent follow treat mistake tell newly quickly agent include agent utility function hold possible type infinite agent interval care define agent pay service internal seek currently extend game stage agent control next stage slowly purpose typically need hundred low exploration mean need explore pair learning problem guarantee another agent explore action payoff agent game well behave concern well game state game stage action converge agent equilibrium strategy sufficiently equilibrium dynamic near learn play try agent simply well utility sufficiently strategy hold agent make mistake treat entire stage cause mistake scalable
sample facebook sample specifically use pick error discard repeat process require sequentially evenly allocate explicitly facebook allocate sequential nevertheless rejection allocate proposition yield sample allocate social allocate allocate consecutive pick element valid w w easy indeed accept pr completeness interpret sample uniform allocate space implement meet measurement facebook assign simply knowledge actual select exist support facebook time furthermore acceptance experiment long facebook emphasize compare ground appendix consider practically static facebook duration day thank confirm facebook take privacy bit node setting comparison sample across day facebook grow growth day growth fact analysis therefore growth facebook negligible evaluate facebook run rw collect next day visit change day collection approximately space define degree summary day day absolute additionally day importantly way growth day former node magnitude variance essentially interaction latter consider become important however problematic vs experimental facebook far take broad internet topology case conclusion topology life internet topology table library long require sample achieve facebook measurement life internet topology play major design degree counter intuitively visit rw instead visit leave therefore round stay bias rw measure node fix number random introduce variance end rw simplicity ignore clearly rw choose uniformly contrast stay round drastically limit count consecutive independent calculation description email email large www graph sample length estimate median triangle green circle actual repetition node kolmogorov ks statistic vertical bottom corner part whose connection therefore exploit test toy graph clique rw error deviation bar comparable size rw topology carefully carry strongly estimate process switch clique clique say inside behave enter typically stay good se eventually rw chance return significantly leave systematically outperform rw intuition confirm fig case fig outperform life topology require translate bandwidth gain recommend internet topology com uci edu edu develop obtain property candidate weight walk walk rw substantially biased result offline assessment process show adequate facebook collect knowledge user publicly service facebook internet internet facebook popular million member membership twitter context population user population user medium site per month account spend online spend email website facebook visit website internet google minute day google top top regard traffic ranking facebook internet example collect study strategy engineering perspective enable design social user activity improve user storage may understand traffic activity traffic engineering serve potential influential user social trust collaborative spam enable online effectively party service well become interest rise number characterization attempt understanding study complete view collect social university company privacy envelope need social million encode bit friend topological node attribute amount least access service requirement view limit impossible fully case necessary overhead instead desirable property precise population inference ability property list individual sample make principled difficult primitive obtaining systematically reweighte uniformity user social appropriately use implementation contribution contribution consider widely bias towards highly characterize consider random rw sampling lead bias quantify correct appropriate hasting walk stationary user technique past facebook collect uniform facebook select facebook bit serve method collect rw analysis former ready latter practical second markov carlo adequate safe stop sample third compare aforementioned technique facebook highly non trivial various access limitation implementation representative publicly available request facebook privacy property describe methodology candidate include high facebook evaluate efficiency quality facebook section conclude elaborate point obtain rejection temporal facebook sampling vs work investigate quality efficiency ii characterization relate work walk traversal sample replacement visit visit method visit node depth ff basic technique research popularity incomplete node however towards artificial topology confirm bias sample online social statistical estimate suggest possible compute random distribution walk graph well excellent sampling wide www towards high bias analyze correct classical chain bias metropolis describe apply select representative alternatively collect result recently improvement walk jump walk weight walk unbiased rw weight baseline complement formal diagnostic test use several knowledge close property evolve state implement multiple chain recently use demonstrate walk analyze corrected practice bias walk agree term network study context namely facebook publicly available twitter main throughout treatment unbiased temporal include evolution yahoo social social evaluate dynamic sampler assume social b facebook context asset study rejection truth uniform allocation star induce evaluate paper summary challenge researcher face collect datum analyze online small study collect weakly use study facebook collect graph graph facebook region collect user profile friend large appropriate facebook wide percentage user privacy conduct york difference example value coefficient follow also dataset facebook property collect social work examine setting facebook privacy common additional privacy neighborhood demonstrate accurately large body service mention community show network present drive content video popularity distribution collect range usage file characteristic fm music site feature conference appear paper material sample truth section empirical assumption social graph static appendix purpose characterization facebook additional comprehensive model discuss extent undirected facebook direct collect publicly ignore isolated facebook thank contrast fm approach remain duration b facebook support mean identity neighbor mechanism uniform generally user social graph frequency attribute setting user us property degree cluster section last base node therefore random towards characterize node edge possibly representative entire facebook global attribute combine process social iteratively visit node discover neighbor proceed choose visit implement new iteration visit select node distance start incomplete densely graph walk move rw inherently bias connect probability node twice visit rw property correlate bias weighting later visit unique belong discrete interest node transition metropolis chain carlo mcmc directly achieve exactly distribution paper initial stopping meet uniformly stay move accept towards reject move towards high truth assess graph measure system truth typically fortunately facebook exception time bit interest random allocate user regardless actual allocate evenly across space completeness derive appendix refer although node facebook allocate facebook user user retrieve per attempt consist string infeasible bit interestingly complete facebook change bit information usage facebook obtain uniform baseline truth primitive believe design multiple walk convergence walk explore graph convergence walk reduce accurate advantage multiple view valid inference convergence diagnostic develop answer least question sample discard start collect approach long discard initial burn burn measurement day sampling decide target request account server activate type friend continue address backtrack exist degree upon discard consistent assumption exclude publicly isolate want rw node collect sample independent repetition returning visit rw ran collect collect collect sample burn rw collect total rw repetition partially especially collect show rw observe still overlap rw probably uniformly user friend isolate consistent statement result characteristic collect unique k neighbor period cluster avg million unique facebook evaluate efficiency quality walk interest period exclude independent x sample exclude burn estimation sample summarize chain apply yet inspection trace base discard seed formal resource section property burn walk sample convergence diagnostic membership space iteration z see diagnostic diagnostic walk walk membership new follow user metric score considerably pick spike new york membership likely walk york particularly iteration drop test walk discard total evaluation remain rw make exclude burn length way analyze confidence analysis formal convergence assess approximate equilibrium top property bottom discard burn convergence attain determined burn facebook connect random seed visual inspection reach drastically fig walk individually combine degree walk iteration likely get average seem walk additionally average rw stability iteration clear interest number inspection need sample per walk walk resource walk another allow correlation consecutive sample process examine estimation percentage rather average entire iteration even equilibrium reach sometimes break bottom iteration perform well despite membership york generate visit walk mcmc advantage space collect bottleneck perform rather storage post effect collect indeed collect information visit friend due correlation consecutive essentially sub node compare technique iteration estimate truth kl kullback capture accounting calculate kolmogorov ks vertical distribution respect usage ks online importantly grind cc cc cc represent real bandwidth mcmc use choose metric principle metric want estimate need mcmc converge respect slow several metric membership uncorrelated use easily metric relevant estimation network burn cover metric interest degree result metric three node degree strongly converge uniform walk stay poorly take long reach network collect representative fig confirm expectation perform example york ny error I np kn give walk deviation walk population present estimate namely degree size essentially baseline c rw chain identical rw degree represent order magnitude rw rw much notice rw shape lead wrong perform true metric network result present poorly coverage rw bias possibly degree well albeit high degree c cdf look distribution topology facebook facebook assign bit happen user profile add friend expect rw usage big difference usage rw cover uniformly rw clearly shift origin shift probably sharp facebook first bit old degree correlate degree indeed show facebook user history website phase initially auto assign stanford assign introduce degree rw explain shift finding recommendation comparison demonstrate facebook property simple rw rw directly uncorrelate node degree move generation traversal community principle correct rw chain time appropriately remarkably ingredient employ formal diagnostic several show reasonable believe implementation walk sample adequate subsequent another ingredient start use single improve duration efficient less finding partly due partly slow avoid present simulation yield subsequent heavily biased weight correct intend facebook ready intend people ease desire target distribution
space generator sort speech recognition engine speech equivalently trajectorie invertible order recognize recover bss coordinate train engine identify bss global permutation global determined data match produce bss finding pair human procedure separate source describe separate statistically independent component component separability source coordinate system product correlation range follow vanish velocity correlation form consequently system diagonal respectively order prove block transform correspond prove belong block vanish correlation factor block scalar must alone although derive imply permutation coordinate must index respectively function alone consequence bss compute velocity algebra find satisfie choice range partition datum index group contain choice choice step subspace space plotted subspace compute function map embed therefore related coordinate system source separability must invertible also manner must separability source perform exist linearly bss source choice lie denote subspace another linearly relate source follow determine source linearly mention span derivative set pdf occur series objective bss comprise statistically measure function equal product bss permutation component wise transformation separability certain locally invariant derive local velocity constraint constraint illustrate record device often evolve simultaneously situation necessary separate signal knowledge considerable blind source variable linearly although bss human usual objective bss mixture find transform datum define special system total locate location usual bss statistically product bss permutation wise transformation however weak suffer problem solution mix source reference issue uniqueness fraction velocity trajectory within location early bss pdf component strong separability note recover side former one fact guarantee bss virtue physical interact generators interest pdf induce geometry second velocity bss metric compute first respect mathematically bss suffer space deal require densely calculate derivative accurately current correlation advantageous require speech separation experiment iii record single minute differential method differ independence one exploit usual time unlike mix derive constructive parametric employ without neural differentiable unlike class describe source variable simultaneous speech record implication discuss multidimensional source bss procedure section scalar datum combination velocity scalar invariant space impose scalar necessary simply satisfy source coordinate explicitly transform factorization construct velocity trajectory segment neighborhood denote possible factor velocity moment formal computed average velocity subscript vanish identically next let use corresponding factor definite always diagonal therefore coordinate scalar must equal alone derive coordinate separability true coordinate system coordinate function likewise function separability bss compute velocity correlation find continuous satisfie compute triplet vary plot lie single require separability manifold coordinate point six coordinate system statement understand manner invertible relate manner separable describe mix variable bss invertible transformation coordinate x proportional denote proportional source rescale bss derivative determine relate partial coordinate linearly source varied manifold speech record single human human impulse extract unknown bss differential simulate simulate cavity represent series spike separate hz hz impulse amplitude impulse slowly vary smoothly latter statistically energie db pre short use energy hz frame nonlinear function two datum determine redundant component inspection within ambient produce hide dynamical degree freedom redundant eliminate dimensional neighborhood establish sound coordinate bss ii statistically step bss procedure compute function
minus kind include dependency arise examine dependent crp connection derive gibbs setting study corpus show exchangeability distance sequential traditional crp original dirichlet process flexible text vision biology advance scalable approximate dp valuable modern mixture describe chinese crp partition distribution structure describe sequence customer chinese customer concentration crp customer belong flexible crp exchangeable permutation essential connect crp mixture dirichlet process distribution dp mixture parameter exchangeable crp exchangeable equivalent dp assumption cluster time collection news article article nearby tend non exchangeability crp assignment version base dependent dependent arrange recover crp dependent crp infinite allow crp represent partition table assignment crp connect table distance crp connect customer assignment arise model customer representation gibbs tool cluster datum traditional crp appear research model crp partition crp customer partition dependent crp customer crp prevent reverse note present crp employ nonparametric include place collection draw dirichlet covariate induce customer value equal draw respective exchangeable condition covariate distance dependent alternative modeling exchangeability difference affect property class crp distribution include special distance crp crp assign product review process far crp section develop distance crp fully observe customer assignment algorithm dependent crp assumption dependent well also alternative formulation crp fast sample one original customer choose connect another link customer assignment htp cc chinese restaurant crp chinese restaurant table enter restaurant randomly configuration table traditional crp customer table customer assignment customer table customer sequentially customer table exchangeable invariant plan term customer customer table product customer customer table illustrate figure assignment index customer customer measurement customer crp draw customer distance notice customer assignment customer assignment customer customer customer restriction sequential mention customer table customer customer see might finally customer assignment cycle customer customer cycle assign customer assignment expressive determine measurement customer time encourage customer customer encourage customer proximity present eq normalization requirement present version crp write crp many set distance exchangeable appropriate exchangeability decay customer result decay take satisfies example consider customer decay customer distance window partition crp previous requirement guarantee customer customer define window decay logistic decay recover examine detail express traditional specifically crp recover marginal customer assignment proportional already customer precisely crp derive draw decay include crp setting crp customer nearby emphasize lead partition crp particular customer customer property capture way precise characterization marginally might goal modeling preference reflect likely share marginally model common preference unobserved might prefer marginally preference regardless whether discover city status city disease marginally require computed ratio contrast factorization easy computation marginal invariance marginally invariant choose computational crp illustrate crp data terminology collection word fix vocabulary document language modeling document crp iid base place table customer assignment exhibit share dirichlet smoothed language alternative also set crp assume occur document formally decay distance draw assignment assign customer assignment induce assignment indicate draw traditional crp emphasize crp customer successively previous dp nearby endow draw set opposed assignment document sequential crp structure cluster lag external date distance covariate mixture process sequentially long generally dependent crp mixture provide dirichlet mixture kind setting integrate parameter proportion distance dependent crp amenable sampling formalism regression mixture distance unlike gibb integrate different crp nearby nearby crp consider node distance far pair group impossible group mixture crp nonparametric interpretable draw crp mixture generally mixture invariance distance generally capture capture assumption appropriate model posterior exploratory datum intractable compute crp place combinatorial number customer strategy approximate markov chain mcmc observe setting section distance distance fully chain crp consider assignment figure denote hyperparameter factor let draw variable markov crp gibbs iteratively draw crp observation remove link customer examine describe remove affect partition assignment gibbs sampler scenario highlight way condition table join customer link two split happen table remove customer customer point case remain case effect link customer another change customer self change term partition factor table gibbs correspond partition link self join table might link finally join gibbs sampler term type type four could table simply representative customer indicator ensure observation index customer table mixture probability table parameter draw collapse sampler conjugate pair integral setting compute new rely posterior customer inner customer assignment sequential arbitrary sequential future point case customer thus unchanged use approximate distance occur middle discovery change assignment thus new know advance sampler leave customer unchanged unobserved simply ignore crp marginally distance crp necessarily crp distance crp marginally invariant marginally crp prior partition measure dependent suppose covariate latent draw parent dependence formally measure measure conditionally implicitly point equal cluster marginally crp dependent crp crp crp situation invariance dependent obtain marginal modeling mixture setting text set explore decay distance connect crp well fit text fully observe gibbs dp mixture crp dp mixture customer fast sampler bayes dependent versus document denote traditional crp denote crp distance crp collection journal article model assess sampler visually autocorrelation chain factor distance crp traditional crp indicate dependent crp crp bayes document various function crp decay function hierarchical base shape curve setting tb corpus bar across sample dependent crp outperform traditional crp decay decay traditional crp test corpora month article analyze contain corpora article datum three news paper th year retain hold article likelihood article well early hold corpus crp best corpus logistic decay crp exchangeable crp mixture well exchangeable crp model dependent crp emphasize gibbs sequential collection paper connection window decay enforce customer another customer refer immediately treat undirected subset abstract citation color graph assign connect color repeat figure note group possible connect crp crp emphasize crp crp simple cluster emphasize concept flexibility crp choice explore long window treat modeling image distance express flexible crp sampler inference dp collapse dp e algorithm applicable conjugate iteratively customer assignment collapse crp sampler customer cluster assignment customer posterior sampler two likelihood either crp traditional add set amount constant lie computation sampler example traditional sampler sampler local optima tb illustrate identity decay red sampler dependent crp identity crp log crp representation leave panel corpus show corpus indicate well local crp fast sampler document collection crp uniform dirichlet illustrate traditional crp gibbs iteration proportional posterior posterior likelihood constant traditional crp sampler local optima corpus chinese restaurant partition crp derive gibbs examine text dependent development image dirichlet fix correspond variational worth explore approximate acknowledgment david nsf foundation google fa thank anonymous comment
mu mu repeatedly chain start remain mode less original gibbs random walk logit hastings acceptance logit exchangeable necessarily parameter exchangeable give imply exchangeable applie modification achieve share get neighbourhood converge go main second simulate convergent slowly iteration application mixture book prior complete I trick gibbs sampler generate ij difficulty move around compare mode guarantee simulation reproduce appear show normal likelihood eq partition single allocate particular reduce go bound therefore unbounded code behaviour mu seq seq mu surface like col color exhibit illustration unbounded mixture go average behavior walk reduce target avoid pick rather ratio need choice power computation sample computation impossible explain need constant twice numerator denominator acceptance twice power vanish reason derive mcmc detailed balance balance generalise represent distribution q marginal similarly finite respective generic importance approximate posterior set need distribution support least correspond compute importance obviously efficiency importance impossible reversible acceptance move balance proposal relate acceptance markov kernel reverse reverse move exercise rigorous derivation perfectly kernel forward backward marginal exercise distribute identically truly process sequence random variance stationary bt process moreover tw bt bt necessary moreover I student series evaluate replace book degree new proportional indeed indicate proportional eq integrate jacobian expand unconstrained next band around causal process autoregressive polynomial causal root circle plane symmetry root causality empirical sphere change plot root outside root n col program look triangular shape analytical look either root root eq together region amount equivalently causality q region triangle q therefore pm eq well book value posterior integral integrable parameter bound remain integrable derive root relation set sentence first book expand root recurrence process proposal prior prior acceptance book hasting ratio extend reversible jump modification consider death move root new program acceptance eq convention conclude distribution normal distribution covariance proportional distribution proportional costly require derive recursive construct single step whole argument compute give conditional distribution exercise formal construct account obviously horizon correlation ability far horizon marginal distribution deduce identifiability eq reason switch write therefore double close integral prediction formula book obvious development ji fx r lead px r fx near illustration relation sufficient new close decrease point near neighbor h five experiment evaluate carlo produce neighborhood diag n neighborhood sum summarize size figure neighborhood joint discuss general fy fy fy fy fy k extension solve compatible e joint distribution conditional joint despite formal agreement conditional joint mass compatible distribution make use size joint pseudo define satisfie therefore unfortunately get differ distribution since line conditional support marginal marginal exercise replace exercise conditional never product support marginal support conditional clique neighborhood structure regular clique draw member square clique neighborhood structure conditional deduce ise mrf development exercise exactly neighborhood array determine normalize array summation summation indicator I array array neighbor structure array exponential summation neighborhood deduce index odd wang exercise replace former exercise array four near neighbor deduce update whole simulate pixel even pixel index simple case wang obvious book graph node index node update node odd powerful stage gibbs computational array color exercise normalizing summing term exponential involve sum even neighborhood normalize face establish associate conditional deduce exercise initial conditional multinomial proposal proportional another possibility select propose multinomial q efficient purely show wang exercise obvious modify number sub interpolation pair program txt correction boundary estimator minimize lead solution mode similar completely basically look permutation class arbitrary therefore obvious pick minimize allocate choose scheme reach configuration converge produce experimental checking risk representation optimization run obvious since integral basis detail two distribution cdf integrable lebesgue negative condition derive positive semi test seq help try seq lm simulate generate lm lead residual std pr intercept degree freedom r df therefore simulate function meet check write mean nan na complete else complete pairwise complete else deduce knowledge write four density exploit x z z z think could classify reason propose illustration find claim minor high claim tail normal extreme http www reproduce histogram conduct report relative file book create file histogram inference follow region whether strongly differ define sample histogram density ad smooth col col lead roughly normal different two joint conditional show quadratic solution eq q normal geometric family distribution book fit representation fit failure also fit fit exponential component define show update imply therefore iid show depend statistic see conjugate update e obviously statistic give posterior varie q model gamma family jacobian check exponential impossible iid derive distribution prior jacobian student marginal q distribution eq get prior show model jeffreys prior location pz therefore constant jeffreys long space change location therefore jacobian negative result jeffreys q get fisher density integral devise parameterize improper matter sufficient go fast go cauchy go exponential prior associate posterior expect decision x go x go go recall denominator use simplify without term appear numerator monte mean appropriate compare precision carlo density like nu nu pi nu nu nu sigma mu b n exp comparison seq b precision converge h comparison bayes happen importance evaluate integral eq importance variance exponential polynomial q integral less exercise dominate matter importance like nu nu df col output distribution large high dispersion log importance weight student density density factor deduce miss marginal q ratio integral expectation respectively posterior bridge infinity converge dirac regularity identifiability constraint converge distribute cdf uniform property purpose available rank transpose r tx deduce happen dimension x xx x linearly dependent decompose expression eq check via solve linear lm note x identity x xx establish correct virtue form coverage probability mean regression explanatory x deduce mn posterior show exercise x student degree freedom location equal exercise restrict represent matrix hypothesis k satisfy constraint mean combination expectation actually write format notational conjugate apply sense factor matter differ nc c student difference prior consequence exercise predictive distribution derive integrate produce student n nc xx x exercise nc coverage exercise distribution eigenvector deduce determinant obvious I z generate vector whole x x n marginal distribution n exercise predictive exercise indeed jeffreys nothing prior exercise mn file txt provide suffice instance ty cc ty cc ty converge obviously vector explanatory predictive student exercise distribute produce irreducible work axis center necessarily chain depend disk bt sampler iteration jump positive disk density conditional exist density distribution irrelevant eq role check value gibbs mu program lack influence start need loop mu seq mu col triple machine show c series lead posterior take fix prior equivalent since bank jeffreys corresponding file txt bank dataset available bank bank book call residual median std pr intercept code adjust statistic exercise jeffreys sufficient give statistic sense update rather due fact link last covariate figure book bank via prior right auto variable ny I check mu flat call good compare bank difference unobserve py pz simple inverse exercise cdf define irrelevant back flat give kx introduction gibbs code file function smooth converge fast modification move smooth transition compare bank dataset txt bank probit intercept flat right auto distribution proper create enough control nonetheless traditional control define bank bayes hypothesis bayes k simulation multivariate normal suggest direct txt bf probit multivariate contain bf probit divide approximate factor hypotheses jacobian deduce transform density jacobian determinant make multiply square logit kp exercise jacobian examine sufficient exercise probit logit little asymptotic density bank logit bayes compare exercise exercise estimate file txt prior full sample log bf logit bf logit strongly probit exercise except twice factor support contingency probability dirichlet deduce associate q variable apply multinomial contingency comes replace restrict contingency four matter since build variable zero since pick factor exclude model exercise term major control regressor rank go least whole exponential term go quantity show improper equal relevance converge normalise available close equal median rather intuitive late number binomial irrelevant code posterior post equal median would produce stage distribution size deduce expectation expectation eq capture day keep track number say give expectation derivation estimator proportional n quantity increase statistic extend capture episode individual number case capture episode converge proportional extension capture consider capture future extend capture different episode observe likelihood therefore number individual another stage mark mark recover mark give contain lose mark complete second partition tag loose obviously possible summation acceptable term must keep simplified form reproduce switch exercise modify file book posterior change direct prefer metropolis step modify conditional proposal thus simulate modify nc p nc prop n p n prop nc book constraint r integrate cost full conditional exercise sum remain gibbs full conditional simulate distribution simulation standard appear consume former write complexity elementary individual life adopt history constraint likelihood case accounting constraint computation replace marginal deduce normalize constant surface therefore marginal give simulation output matter normalise gx
subgraph temperature introduce individual factor edge leave subgraph world section field eliminate without loss subgraph unlike ise random like subgraph state index edge edge maximally configuration path random independently assign node cluster behind different component mixture edge indicate might well index mixture develop constant relate easy calculate go proof insight remarkable relationship correspondence configuration physical section proof subgraph world importantly graph could idea fully hence subgraph draw mixture distribution ise use parameter value many proceed formulate variable random variable reverse give begin choose choose desire draw directly draw possible proceeding stage stage provide choose choice normalize constant binary probability divide yield side multiply term right side turning complete easily suffice utilize fix start edge nod leaf exist maintain degree requirement preserve receive choice minus degree yield equivalently therefore e complete random wang devise chain fast index generate draw ability move random subgraph wang subgraphs subgraphs cluster show subgraph f j f I return subgraph draw section maximal forest edge weight let forest connect let configuration configuration along node onto condition equal condition word draw plus combine case value line correct choose way task remove nonempty leave leave forest either first search either forest leave subgraph provide obtain spin let consistent pe pe pe mean e combine z e desire coupling past algorithm perfectly immediately simulation world prove study theorem huber computer college edu via family say simulation utilize extra randomness whose runtime flip come draw simulation input size simulation wang approach ise edge call wang draw turn view ise third assign call subgraph remainder organize discuss show subgraph direction subgraph draw convert similarly convert single draw wang approximately ise answer direct relationship discuss early create reduction create series drawback multiple configuration draw vice perfect ising initially propose extensively phase transition application either refer spin spin configuration weight function strength ise factor directly control say
make partly choice lead third qr analyze post post ordinary quantile penalize qr qr true perform well qr qr true subset qr design interest post qr international economic growth contribute penalize technique comparison implicitly index notation f n f n na na ba formulate primitive regression underlie quantile eq compact quantile index quantile function population minimize asymmetric absolute minimizer analog high setting ordinary inconsistent motivate remove least zero penalization lead penalize quantile estimator penalty quantile depend penalize asymptotic problem programming define optimal solve polynomial avoid curse dimensionality selection estimator post qr ordinary quantile quantile specifically penalize remove model work estimator oracle unlikely post well choice introduce independently distribute conditional quantile quantity recommend c recommend account sample recommendation parameter contract implementation practical variance cross also choose allow behind lead precisely slightly qr recommend select dominate noise suitably rescale criterion function value general subgradient regression index omit indexing ease exposition continuously constant conditional quantile away impose regressor quantile condition jacobian u condition impose assumption vector empty restrict eigenvalue condition primitive condition bound impact appear concept restrict set eq invoke post analyze lasso se highlight usefulness condition brevity correlate normal estimating one eigenvalue away constant design satisfy eq normality smooth obeys state general primitive specified remark coefficient hold n chernoff small design hold concave design regressor consider estimate location differentiable uniformly bound zero design linear coefficient un hoeffding inequality approach small population design away nonlinear impact replace primitive form suffice minimum eigenvalue restrict condition quantile analog rate design prove least control modulus conditional evaluate overall penalty restrict identifiability rate follow lastly require fast mild going condition appear precisely turn behaved design interest concave hand hold bind invoke se analyze se less penalize dimension unnecessary outside characterize control quantile quadratic function neighborhood covariate se b unnecessary component support design assumption straightforwardly compare qr choice oracle polynomially result restriction polynomial model penalize dimension select estimator correctly minimal provide thresholded penalize apply ordinary obeys component obey growth eq singleton qr indeed post qr qr relatively permit qr well due qr select obeys select contain converge post qr case perfect selection rate post qr singleton drop necessarily happen coefficient separate additional regressor coherence regressor dimension polynomial order well single penalize quantile penalize paper post regression apart perfect oracle regression qr analogous qr different fundamental excess loss function logistic principle regression problem assume penalize hold design regressor imply qr sparsity property drive qr start characterization determine via equation universal establish rate qr preliminary penalty specify belong least inspire analogous relation condition derive show control norm restrict control quality quadratic preliminary derive bound error q combination argument contraction fundamentally principle alone neighborhood define intrinsic uniformity armed qr assume condition level least provide obeys condition derive penalize intrinsic norm uniformly range recommend rate qr depend significant strength summarize quantile extreme quantile slow convergence obtain role rely proof quadratic nature control quality refer discussion design proof shape geometry restrict classical argument insight follow event uniformly event preserve n u f p upper growth bind obtain quantile fundamentally sparsity link quantile regression gradient highly piece wise control sparsity rely empirical process argument gradient crucially exploit result eigenvalue pre probability initial restrict pn design interest k main eq obeys eq states initial control crucial penalize estimator corollary theory supplementary possibly need qr growth virtue lemma hold upper bind u turn selection qr thresholded estimator analogous regression say separate support one sided support unnecessary coefficient hard eliminate unnecessary hard characterize zero quite unlikely practice certainly example motivate post allow selection unnecessary post estimator rely crucially identifiability u u obey dimensional theorem establish post selection error post qr condition assume hold bound hold obeys growth describe qr inspection proof reveal qr fast qr fail contain rate qr hold recall design theory material appendix calculation overview intuition qr component allow qr still case succeed u unnecessary component obey post fast qr extreme high post qr refer reader note last eq imply optimality u u identifiability growth access practical propose monte international economic sample conduct monte penalize post oracle post canonical quantile select quantile course outside carlo focus penalize risk error identically distribute equal penalize estimator panel select much design penalize always miss regressor coefficient regressor notably despite partial failure post perform report right penalize rarely support penalize estimator select theoretical theorem stochastic agree summarize recover penalize quantile zero drastically improve penalize quantile reduce notably estimator never always regressor post risk perform identically ideal expect penalize well make estimator find estimation performance post mean qr post qr oracle design qr post penalize qr qr penalize international economic growth primarily lee consist panel national period analysis total observation goal covariate central growth initial per growth rate per growth hypothesis typically fast rich thus hypothesis point reject positive characteristic characteristic include education science market predict initial covariate analysis severe rely hoc case previous finding simple resolve lower large million specifically median resolve important turn perform covariate drive penalization lead separate slowly decrease order market exchange characterize political instability additional several report reader discussion ordinary median confidence estimate confidence interval work work additional select covariate coefficient interval conclusion quantile middle range believe empirical finding growth inferential finding agree report parameter per black market political instability political restriction consumption education exchange school population school population black market political instability ratio real consumption net education school school ratio education education year secondary population instability restriction education exchange rate high school secondary school complete high education education proportion year secondary growth population school population nominal nominal k rademacher independent condition j envelope obeys ball empirical hoeffding I choose make side display condition therefore diag probability least furthermore u u u n event lemma probability lemma population claim proceed large eq proceed radius criterion quadratic scalar expectation mean proof error divide main argument tail note lemma greater far p nc e p q ix ib k k k p k p n ij hold last inequality law iterate hold inequality intermediate x ix elementary hold p inequality inequality characterize sparsity eq variable uncorrelated play sparsity sign property conditional uniformly complementary problem show part row I x linearly independent one basic solution wu un number quantile equality establish conditional feasible polytope moreover imply therefore dual generate finite intersection row event absolutely basic degenerate empirical bind cauchy probability supremum side establish note p contradiction convenient rank score dual solve ij u j non provide next step inequality triangle probability c linearization control linearization definition cauchy schwarz next control define shall preliminary number function class q bound universal constant bound proof argument supplementary material supplementary brevity restriction universal since restriction total cover statement control uniform sparse derivation early semi definite positive proof inclusion converse u thresholded inclusion u x c e nf z z nf c nf n n nz ny n n n proof interest condition mild reader
least side consider lead least least sufficient statistic count marginal respectively prove marginal statistic likelihood mixture try describe model study light classic fact basic variety describe describe vary describe non zero coordinate set vary describe equation repeat fixing describe simplex base join diagonal model form element scalar order second gives add lp lr mc ic mf g g vanish entry diagonal term variable appear appear I ip kp k ir add variable kp ir ir ir kp p proof proceed I p k ip kp kp kp k computation find polynomial model true di department definition theorem contingency encode explore element non number simplex contingency space j geometric structure algebraic polynomial p vanish negativity complexity algebraic geometry must intersect negativity algebraic study involve one take probability contingency log algebra algebra introduce geometric model structural exact basis negativity issue basis lattice basis effect special contingency table agreement medical science result discuss due statistical mathematical definition describe object especially show vanishing differ boundary simplex basic emphasis independence model effect model show special diagonal effect model common behavior diagonal cell diagonal contribution present diagonal study geometry model contingency categorical therefore product x statistical define algebraic cell define form power encode simplex log probability know obtain ideal polynomial ideal pure binomial I I define integer move pure binomial part finite connect contingency table path count notion move basis generate ideal deduce generator ideal contrary next implication independence role px px py independence suitable non negativity reflect negativity namely equation negative rank probability must formulae easy write implicit markov basis independence polynomial ideal say corresponding implicit simplex notation q define define sub space description issue find two field vector literature see relatively easy parameter obtain result model distinct q minimal form move move distinct generator define intersect hyperplane framework definition diagonal model matrix normalization geometry model give write polynomial easy check equation polynomial detail connection investigate negativity condition impose example respectively construction belong
minimize leibler eq uniquely strictly partition pseudo estimator root give objective scad pn large satisfie pn large enough completely center large minimizer helpful fit concern nuisance parameter property derive standard bic consider lemma argument weak kullback I maximum likelihood true different former maximum likelihood penalize fit fan wu pn combine complete yield identify establishe criterion penalize minimize criterion bic adaptive consistent adaptive lasso tuning sequence accord fan wu tend edge estimator nonzero partial similar derive scad let defined penalty pn enough term last imply choose radius light study fit work minimize adaptive study conduct likelihood consistency result bic cross commonly fold validation disjoint index subject cross validation score calculate optimum graphical structure ar model sparse graphical employ point connect distance entry covariance ensure penalize scad lasso penalties validation criterion specificity define true false positive negative true false positive classifier deviation different inverse covariance sample exceed setting reveal different assess method size tend glasso fan wu method initial obtain examine tuning via table specificity coefficient simulate set standard graphical adaptive consistently well outperform increase advantage scad scad bic yield specificity ar specificity sensitivity scad specificity sensitivity adaptive lasso per specificity confirm tend infinity specificity penalize sensitivity sensitivity structure compare fold edge per versus consistently specificity validation simulation overall bic exhibit large cross time intensive compute cm conclusion investigate tuning selection graphical establish true scad bic condition research science grant hold wu bic cv bic bic cv bic scad penalty cv cv scad york fan variable fan wu exploration scad covariance li network shrinkage tune smoothly white lin li discussion cm mathematics york mail mathematics york mail pt wu department york pt mail mathematics york pt mail cm pc pc cm pc thm axiom section graphical plus minus minus abstract graphical independence maximum smoothly absolute fan li penalty article establish bayesian criterion bic penalize penalty lead empirical performance bic demonstrate advantageous tuning selection study phrase oracle introduction relationship zero among covariance exactly vector multivariate denote covariance indicate represent vertex corresponding coordinate represent dependency relationship identify simultaneously address likelihood penalize unstable another standard approach perform forward elimination however hard furthermore computational search computationally propose quadratic lin implement inherent wise box quadratic solve show graphical lasso fast convenient tackle wu propose deviation scad approximation penalty result method lead penalize penalty lin coordinate bias estimation bias fan scad usually fan li hinge scad quadratic knot origin ensure penalty heavily penalty scad method select correct produces know true namely adaptive impose regard penalty efficiently implement lasso scad fan wu li iterative optimize denote
asymptotic efficiency segregation sr r ar r investigation underlie order analysis investigation asymptotic critical segregation sr ar investigate article implement monte nan monte investigation segregation calculate mc nr stand estimate imply carlo replicate mc th use segregation alternative percentile alternative underlying case monte yield mc notice also skewed monte carlo segregation depict density estimate may consider power test proximity present carlo monte carlo underlying severe segregation case compare carlo critical segregation case top two case replicate levels monte carlo sample analyzing factor base critical monte investigation critical corresponding table empirical level moderate appropriate significance severe segregation high furthermore segregation alternative size circle line triangle critical alternative underlie bottom underlie significance investigation estimate carlo mc nr mc nr stand similar imply small power monte carlo empirical power mc mc case separation much carlo replicate estimate monte mc density skewness experiment bottom right dash value power estimate case severe suggest ht monte carlo critical leave c underlie c carlo critical critical value critical power close yield severe association version level solid triangle critical association underlie power finite assumed let triangle triangle wish h segregation alternative relative underlie underlie construct use version density nr ac asymptotic rr jensen hold segregation alternative vertex allow mn nr nr n nr nr rw nr denominator complete maximum suggest density nr nr nr I give rr nan iff segregation alternative underlie case version nr nr equivalent limit however finite infinite nr nr nr j r nr multiple triangle asymptotic normality nr segregation association figure accord segregation association ht segregation realization segregation leave association great except segregation realization except underlie association realization case nan segregation association segregation consider value consider underlie segregation segregation repeat realization segregation association figure result indicate point per enough estimate relative segregation large per triangle test suggest choice moderate alternative well segregation circle line triangle top circle line triangle dot critical underlying case bottom multiple right triangle size triangle necessarily update conditional unconditional triangle form simplex vertex edge q boundary assign arbitrarily opposite contain let euclidean distance vx vx polytope vertex pe rx rx rx rx rx ir pe rx transform polytope face preserve uniformity become boundary sphere particular simplex regular face underlie parametrize proportional proximity segregation association theoretic test spatial proportional property triangle respectively point density compare imply assume fix abundance imbalance perform randomization conditioning employ relative arc edge testing bivariate spatial consider graph demonstrate statistic employ asymptotic normality statistic test segregation nan two class triangle co type parametrization geometry independent triangle e likely parametrization segregation association alternative tend point pattern segregation expect large association parametrization reveal power segregation hand well performance association underlying monte randomization otherwise recommend furthermore testing segregation recommend association acknowledgment partially advanced project air office scientific contract f office grant proposition example mail edu tr introduce direct various proximity graph determine family parameterize statistic provide alternative arc employ relative analytic asymptotic statistic illustrate bivariate spatial segregation parameter efficiency asymptotic alternative dimension keyword efficiency randomness segregation statistic classification base testing bivariate spatial extensively population pattern point one investigate two class implication especially specie see article derive family spatial segregation randomness roughly segregation association frequently generality characteristic observation segregation specie class mathematical popularity tool provide move movement although landscape suited application movement conventional maintain reduce graph integrate geometric tie patch spatial dimension preserve relevant spatial graph usually lose see concept depend adjacency express allow spatial edge domain network model spatial interaction intra inter relationship quantify patch potential patch theoretic measure design spatial interaction instead generalized coefficient point arc bivariate neighbor place arc vertex give demonstrate good dominating find minimum dominating e multiple distribution number prove extend non parametrize arc pattern family calculate arc proportional two datum obtain arc arcs arc edge without underlie tool article scale demonstrate underlying graph central analytically difficulty encounter edge edge describe asymptotic power segregation association triangle provide extension proof defer appendix arc arc pair edge replace arc underlie graph refer uv uv ng represent number order proximity region z set arc ix x x jx x x n proximity comes represent denote x ix nx nx symmetric arcs random px x px px furthermore ji ij x nx nx h find joint nx h h density ij ix jx nx finite brevity similar underlie nx nx x note ij mass note h nx h h triangle define proportional triangle nr pe nr n pe nr rotation v bt preserve uniformity furthermore scale map triangle boundary edge distribution collection region preserve uniformity preserve edge proximity uniform henceforth proximity map recall px pe rx rx px rx occur two underlying section equation value r r r limit equivalently underlie natural relative density recall r n pe rx pe rx r pe rx rx pe rx pe rx rx see distribution relative edge small st rr rr indicate skewness underlie figure skewness skewness may derive asymptotic successively depict histogram vertical ht depict histogram base replicate line normal axis scale depict histogram carlo replicate indicate severe extreme ht depict histogram replicate indicate severe small skewness extreme vertical axis proportional underlie graph proportional edge nr nr nr pe pe rx pe rx x pe rx pe rx pe nr ii nr pe rx pe rx ip nr nr nr nr nr nr nr order segregation class tendency fall tendency near segregation let
accuracy beyond paper super fista fista clearly approach synthetic section benchmark detail measure server ram regression reason measure conjugate minimum duality algorithm compute criterion implement absolute duality duality iteration construction finally criterion fista additional whereas vector non g matlab inner solve matlab choose initial proximity conservative since appear soft multiplied seem intuitive formal argument detail variant equation l solve dual implement notice happen poor order undesirable proximity constraint modification without specifically use proximity al function rewrite initialize conservative set proximity condition counter increase factor note section e simply conservative equality computation duality vector fista matlab instead unnecessary gradient implement matlab implementation optimize algebra algorithm implement matlab code logistic implement regularize logistic regularizer diag newton method matlab file library bias include subsection first convergence size proximity sample label sign mm repeat ten confirm fista minimizer eq run correct obtain minimum trajectory multiply initial residual estimate tb theorems residual vs cpu residual v show run describe keep mean bound top leave residual theorem result result difference optimistic realistic analysis order reach quality fista iteration step panel fista value fast fista needs every spend much accurate second obtain computation precision higher clearly bias term panel residual cpu spend parameter variant increase factor increase residual primal linearly roughly report slightly concave super convergence probably information spend roughly achieve residual tb summarize plot cpu spend reach fast roughly scale parameter computational show cpu error stop iteration run converge run except solve less advantage demand advantage large without subsection run factor inner spend conservative set cpu show stack bar segment bar correspond outer one use roughly outer half hand slightly therefore outer note half iteration spend iteration fast conservative makes recommend figure total spend variant conservative proximity conservative previous except problem clearly outperform benchmark five bioinformatic provide validation combine split dataset regularize set well cpu format dense category graphic example contain goal treatment multiple patient dataset denote gene gene subject subject setting begin polynomial order iii triplet obtain standardized mean dense even fista keep design deviation standard deviation zero place cpu whole order separate regularization constant warm strategy regularization solution conservative initialize summarize spend second show bold see fast number tend accuracy fista typical contrast except grow reduce seem almost r dense sparse time cm tb normalize efficient tb regularization regularize minimization minimization generalize super augment lagrangian importantly check assume convex assume loss check regularizer check look projection onto result inner approximately compare need convexity obviously many argument primal lagrangian logistic rapid confirm simulated regularize logistic regularize fista synthetic dataset fast large number observation dense relationship inner outer computation improvement change make condition basically light fista category convergence another small iteration prominent member empirically show shrinkage first class effectively use analytically sparsity shown compute efficiently sparse include primal lagrangian split advanced thank helpful discussion center development mathematic proximal convex proximal right proximal operator convex follow elegant convex prox prox similarly give sum side eqs operation onto convex take ball radius regularizer soft operator therefore special ball attain envelope envelope envelope pair prox conjugate line envelope conjugate tb threshold regularizer indicator envelope consider inf quadratic envelope envelope differentiable completeness subgradient envelope prox prox envelope projection figure line step generalize allow minimizer bind close namely minimizer follow arithmetic geometric expression accordingly substitute complete follow term let prox ready analogue decompose residual follow b reduce follow arithmetic applying expression depend front bound inequality term hand expression line true last line minimizer show x f third attain l bind value cl mc denote delta primal gradient hessian diag diag z ic diag logit ik ik ik ik ic ic mc ik ik mc envelope function element wise p c proximity operator envelope regularizer n prox n j norm tr prox en prox en convergence recently estimation minimization theoretically super due model analysis lagrangian algorithm interpretation generalize wide extensively efficiency propose lagrangian super linearly estimation become common application bioinformatics rapid development tailor machine sparse minimization plus paper loss term regularizer differentiable various factor tool machine diversity arguably square signal reconstruction estimation variety wide logistic loss function square loss matrix design stack input g minimize design compare regularize apply context denoise sparse reference therein contrast focus design recently shrinkage see iterative lagrangian version algorithm proximal rigorously converge super linearly mild grow framework wide practically regularizers improve convergence augment structure sparse instead consider al efficiently exploit intermediate solution al play important role analysis primal section recently dual lagrangian paper review derive minimization special discuss section theoretically behaviour contrast simulated moreover compare recently regularize dataset variety finally given formulate regularization close proper sequel function close proper see continuous see twice equivalent hessian uniformly quadratic loss smooth hinge exclude quantify examine nf ff differentiable strict important study line regularizer notational convenience close information see regularizer w theorem dual indicator radius lagrangian primal variable multipli vector al note ordinary sequence primal respect carry involve separate follow vector outside domain obtain onto ball th soft way substitute soft soft processing minimizer q call slight abuse terminology turn minimization contribution review framework rigorous section minimization sequence number repeat e duality term proximity term next even original although carry e decrease obvious differentiable decompose small minimization short substitute constant omit right coupling right contain equation know shrinkage section bind precision use parametrize adjust wise maximum substitute eq maximization saddle concave denote maximizer general different max min naive final derive compute maximizer slightly derivation write turn envelope iteration minimize minimizer like inner section derive review proximity envelope specific function al function envelope prox regularizer choose similar slope optimize minimization next point highlight use adjust become tb mm derivation part handle term rest term discuss special qualitatively efficiency minimize simple discuss regularizers equation proximity operator regularizer conjugate regularizer envelope see converge linearly asymptotic note increase exponentially generate eqs continuous modulus compute inner weak stopping rather compute accordingly stop side increase approximate analogue objective theorem moreover eq super linearly theorem inequality see obtain weak obtain perform minimization let set assumption inner precision early safe unfortunately exchange criterion practice practical subsection discuss assumption term proximity may restrictive setting translate residual residual think locally strongly within bound quadratic q constant depend bound bound sure increase minimization contain use strong convexity around rapidly objective eigenvalue hessian term hold globally exist convex conjugate unique positive constant continuity imply minimizer guarantee become weak valid weak hold point constant asymptotic require proceed predict close super convergence complementary super convergence assumption number justify section study category comprise try overcome term category overcome pose separability efficiently minimize three constrained iii subgradient constrain rewrite auxiliary cone challenge auxiliary pg method compute project pg linearly pg overcome scale bfgs constraint lasso constrain ip basically generate call connect center ip well convergence upper non arithmetic geometric iteratively solve regularize weight technique study variational generalize jensen context kernel context challenge framework remain arbitrarily
logit boost logit theory include selection survey idea output small confusion subset minimizer empirical empirical estimator choice regular estimator piecewise span vector fourier basis algorithm smoothing square ball datum radius amount perform terminology choice convex mention cv instance local estimator focus length solve choose difficulty algorithm compute selection contrast function contrast poor statistical except target think generally dimension depend use contrast call reader find much deep insight give selection procedure main identification goal target selection build measure excess possible estimation call almost depend optimality procedure assess way framework efficient asymptotically optimal sometimes weak hold inequality remainder tend tend build adaptive belong procedure minimax model aim among typical selection build identification bic quality recover model algorithm true replace b identification consistency strong efficiency define framework exist former case bic aic aic bic prove model minimax rate sometimes share recent sketch particularly cv work procedure select dependent penalization exhaustive completely approach reader procedure book five come framework model procedure form asymptotically unbiased heuristic explain start theoretical inequality directly imply prove selection principle cross include section penalization approach principle classical penalization principle aic square bar contrast penalty prove several framework constant reference therein drawback penalty aic depend suboptimal overcome large computational possibly simple framework paper depend multiply level drive multiply several weight bias slightly penalization example procedure leave well noise grow unbiased risk principle penalization penalty oracle hold choose procedure take proved inequality typically identification bic framework soon factor infinity part picture since coincide show penalization also soon idea bootstrap tend procedure also category context context statistical risk roughly penalty complexity penalty call localization exhaustive cv procedure successively formally loo loo loo name n successively define loo computationally cv introduce alternative loo preliminary partitioning approximately successively formally much less loo loo size indeed incomplete design idea point idea index almost belong appear j bb choose randomly coincide split several time fix observe drawback risk closely penalization penalization introduce loo regression estimate risk estimator independent close see approximation estimator among use replace several apply loo loo bootstrap give bootstrap estimator modify theoretical behaviour area assess quality first split distinguished division primitive loo procedure evaluate rule primitive loo loo independently loo discuss loo selection understand cv purpose cv estimator cv cv make two evaluate biased imply risk particular risk rigorously decrease function near tend decrease rigorously precisely cv make grow list statistical behaviour cv confirm several estimator projection estimator design asymptotic expansion yield square spline expectation loo calculation picture expansion loo loo tail problem shift loo loo shift loo stay realistic biological compare bias loo bootstrap bias decrease loo fold bias nearly minimal bootstrap moderate sample ratio otherwise minimize nd report loo nd n call double cv small cv asymptotically regression equal penalize prove unbiased cv behave differently variance b several proportional around strongly unstable conversely trend stable notice bound regression upper deviation bootstrap loo tend variability cv maximal hold minimal loo complex vary quantification split complex sum split link furthermore variance cv behave differently optimal simulation detection confirm contrary usual performance kind knot prove correct asymptotically statistical regression oracle inequality estimator among square small constant performance cv bandwidth estimator contrast efficiency loo multivariate inequality efficiency kullback suffer target condition cv efficient framework classifier oracle compare penalty contrary still enjoy good nearly procedure st cv procedure naturally propertie statistical loo prove inequality describe loo several base particular classification paper identify procedure identification bic may cv loo efficient identification goal cv confirm result prove cv method inconsistent loo even asymptotically select tend context order selection compute cv numerically procedure specific result slightly hence loo variability consistency confirm put somewhat validation cv understand consider goal fast two inconsistent depend convergence framework measure cv see consistent selection prove rate consistency soon proportional negligible front gap condition good cv consider candidate simplify cv parametric strong imply cv loo condition experiment average cv second originally principle cv procedure dependent cv smoothed smoothed excellent smoothness density globally presence cv modify efficiency consistent change achieve choose robust regressor robust classical cv cv huber one model cv part play rest independent cv model selection procedure framework independent positively correlate use issue choose change bandwidth almost enough short dependence enjoy optimality long method seem several alternative propose cv modify term correct short framework partition cv bandwidth cv combination parametric dependency time datum dependent particular predict observation cv similar specific stationary cv difficult small therefore cv directly provide chapter add penalty prove oracle lead change selection first meta polynomially criterion trick cardinality contrary procedure cv consume naive usually intractable nevertheless cv framework greatly decrease cost cv density formula loo histogram recently result projection formula hold estimator risk polynomial square spline lead relate naive close projection closed formula partition write sum dynamic minimize risk piece detection cv form efficient avoid j formula discriminant loo expensive empirical formula loo user appropriately cv impossible framework cv induce behavior criterion choose cv particular model compare variance estimation snr well otherwise suboptimal snr keeping often take see identification model cv risk decrease split cv cv close link variability cv quantify minimal variability seem framework unless formula split hold loo optimal estimator final trade point account final user preference allow computational reduce aforementioned term hold question split choose proportion well define every scheme practitioner split every validation sample make compare splitting improvement take split like seem intuition explain similarly cv every point nevertheless cv procedure distinguish additional variability quantify empirically theoretically mild remain appropriate strategy uniquely split phenomena bias decrease training whereas loo high framework estimation minimal framework like report literature remain claim problem conclusion statistical instance depend loo perform risk asymptotically optimal whereas contrary snr small loo thank bias bias independently mainly goal selection drawback often consistently order provide contrary procedure candidate perhaps provide quantitative split split result direction prove cv method greatly cv generally make distinction order realistic show
mixture assign j positive specific cluster cluster model nonzero effect sparsity marginalization extend able zero induction zero discovery choice set nj dp prior prior concentration mass follow emphasize structure component dp measure entirely call dp mind nest dp group patient significance presentation empty base mean indicate similarly augmentation iterate step probabilitie j nu c p j j pc use ik draw distribution nonzero associate basically make algorithm completeness singleton b cn I accept singleton set pc c nx pc nx c used partially collapse integrate eq index draw approach obtain calculation step gibbs sample conditional step singleton prior high mass dirichlet typically extremely surprising draw implement accept solve successfully sequential pc nx condition k kp expression propose previously perform change prior draw sequential describe inference need label switching refer reader although distinguished variance extra indicate group variance sparsity many choose cause well omit shift microarray researcher shift distinguish study different gray distinguish attribute distinguish figure format posterior identify markov strong support cluster simulation burn inference show figure c attribute cluster fail identify single th discrimination encourage clearly signal attribute simple least exactly identify attribute finally simulate dp consistent example vector example simulation example increase attribute informative four cluster sample belong cluster cluster selection different error overlap true attribute joint besides correct table outperformed consider square perhaps favorable intend situation work situation square tend discovery distinguish subset tend note attribute subset select utility contain divide microarray within interval whose finally select gene end mcmc concern variable diagnostic run burn start markov another assign quantity agreement indicate dataset put conditional gene demonstrate cluster structure condition help deal switch sample allocate posterior misclassifie sample misclassifie know consist discover gene gene relevant cluster distinguish separate dirichlet shrinkage mass sequential mean study sample effective challenge choose approach modify sequential sampling plotted estimate procedure microarray heterogeneity genetic existence signal dirichlet simultaneous variable also distinguish formulation double usage dp make mcmc update model study gene expression markov monte microarray discovery decade simultaneous investigation potentially characterize distinguish known cancer obviously possess power different attract attention recently class come principled inference compare procedure largely heuristic important inference cluster proposal next formulate prior computation mcmc sampling
remain create bregman derive introduce term handle coincide unknown develop interior point truncate inner scale area compute soft every extremely light gradient solution naive size recently author propose augment dual minimization addition minimization converge inner precision method primal explicitly update lagrangian multiplier soft every iteration apply coefficient ip exploit organize algorithm present sec experimentally compare brief direction base surrogate surrogate lagrangian dual duality dual indicator duality hold maximum minimizer maximizer dual respectively multiplier associated constraint coefficient primal barrier barrier reduce ordinary lagrangian lagrangian duality f eliminate depend radius method barrier increase super see l maximizer kf strict accordingly dual lagrangian function everywhere except soft hessian hessian complexity active second derivative complementary multipli regular point newton factorization second conjugate truncate newton element backtrack objective test various state art algorithm interior algorithm size experiment mean increase increase keep choice singular value replace series ratio largest additionally approximately correspond experiment increase target experiment regularization decrease equal approximately computation element bottom ghz processors gb memory criterion variable primal term define definition objective eq dual tolerance inner choose large require barrier parameter affect behavior large gap manually choose problem guarantee super condition conjugate less large grow hessian datum poorly condition propose fast constant fig increase decrease robust efficiency keep barrier parameter well matrix law horizontal variable optimization framework dual sparse coefficient explicitly algorithm favorable condition solve million minute improve
leave unique h prove continuity external expectation converge local precisely ball radius try probability go rt g pp expectation take ss lk g rt cs il rt rt apply expectation reduce calculation expectation latter arise calculation long exercise outline acknowledgment work partially support fellowship nsf dms proposition corollary theorem conjecture remark electrical stanford university department electrical department stanford problem ise several propose limitation remain analyze complexity systematically precisely coincide graph random binary mechanic graphical vision spatial introduce statistical mechanic convention ise structural sake simplicity know double unbounded resource question precisely parameter denote probability sample change unbounded resource general graph pattern long range correlation complexity strongly graph correspond beyond critical low complexity appears phase coincide illustrate strength thresholding thresholding correlation denote dominate computation correlation straightforward exist choice graph bound range correlation graphical decay vertex happen graph family degree range characterize advanced range limitation result encode fix vertex neighborhood conditional rx change possible change fixing marginal allow set motivate local threshold candidate minimum empirical calculate neighbor minima maxima contribute consequence imply impractical set neighbor vertex vertex degree regularize logistic logistic fail indeed graph degree show necessary reconstruct notice incoherence picture difficult evaluate family restriction expand second term surprising relevant practical extensive simulation good ising model generate bias sign conservative temperature indeed graph model easy figure remove independently success vertex average empirically phenomenon threshold poorly irrespective ise grid evolution sufficient threshold prove auxiliary convenient notation submatrix index neighborhood since hereafter shorthand r x minimum omit context quantity throughout ss min min sense graph one edge node finally ss high rely ising prove uniformly sufficiently c min ss omit
j si jt false negative report happen follow chernoff bound e former solve especially tight binomial precise fig show exception slightly close report close hand would probability binomial mean chernoff illustrate time item algorithm trade negative alternative procedure efficiently eliminate accurately value measure threshold filter away technique wise hash sketch similarity decide likely filter could built maintain use hash lift overlap set hash large cost extra pass usage representation counting introduction subsequent time cosine information retrieval item weight extend instead user may informally go achieve maintain rate rate movie actor actor work rate movie relation actor actor cosine random confirm concentrated around case around possible behaviour due choose bring close surprising reporting report scenario omit dominated phase count mean fluctuation effect possibly threshold significantly change time see speedup come large transaction analysis suggest transaction speedup item transaction moderate give support usage range though locality hashing appear distinct item lsh comparison hash signature actor signature hash table range necessarily indicate lsh signature count connect actor base appear association mining system outline way thank software pointing observation lemma full version http www people paper mine strong mistake report base find association low item rely show variety high theoretically average transaction experiment mine speedup order significant mining association market setup sequence item customer canonical define association indeed capture exist lift cosine confidence closely overlap refer measure take aspect rule previous rely mean occurrence compute signature item similarity partially compare signature approach offer flexibility sense go directly item negative rigorously understand efficiency main focus many association paper usage recognize believe come carefully consider transaction pair small pair read ram modern able per second require occur frequently initially frequently enough potentially report support pruning focus mining elaborate contribution capacity fast device gb ram sequentially speed bit fusion offer read around million word even massive read challenging keep must million million per ghz cycle cycle item processing item transaction likely cpu rather I hash table cache mb core hash operation item cpu rather conclusion believe time carefully optimize count os pass cpu remain bottleneck efficient frequent item set refine transaction however similarity measure value degenerate count number usage pass least occur transaction item equivalent multiplying incidence vector transaction transaction appear algorithm sparse transaction get huge factor interest transaction frequent random transaction considerably count occurrence miss support association transaction relevant locality hashing occurrence signature occurrence sample possibly false negative sum plus initially read result form support similarity hash locality sensitive method describe show signature angle incidence positive negative signature item transaction show handle use locality deterministic signature database community threshold sometimes refer join described locality avoid employ signature serve signature take priori method join exhibit incidence transaction present measure lift find occur transaction probability support significantly I transaction transaction time sample infer false negative nearly input similarity item close hope similarity factor transaction mining negative give speedup magnitude present set work locality sensitive hashing transaction occurrence capture handle n special increase decrease similarity increase computable time fig measure lift cosine overlap find randomized probability return similarity factor reduce eliminate basic occur transaction pair expect strictly function indeed except occurrence define xx xx sort j occurrence transaction occurrence iterate transaction sort build either time relationship memory sufficient sort count hash consider ram table internal constant concrete c item transaction accord occurrence item transaction occur occur occur occur would output add particular binomial trial look si measure exactly sample standard ram latter external implementation transaction denote pair run report dominate analyze complexity sort transaction sort step spend loop assume j f consider spend proportional sample otherwise case transaction similarity expect well input happen dominate run scheme count large transaction require thorough report highly great similarity frequently could imagine greater choose small term find item
distinguish degeneracy map treat covariance agree reproduce shape degeneracy true estimate equation relation implication report single report ridge solution use ridge grid fit study initialization usually adequate good usually identifiable value degeneracy somewhat snr high degeneracy relevance contour degeneracy small error table degeneracy apparent band magnitude directly need provide degeneracy built estimate accurately magnitude prior galaxy outperform assess detail explore help sensitivity possibility model good metric combine suitable analytic many carlo thus probabilistic smoothly galaxy et parameter et forward principle use advantage hence many would like discussion simulated effort people respect whose effort would grateful team university purpose estimate modelling nonlinear interpolation template avoid use parameter treat weak approach uncertainty predict parameter goodness fit provide outlier parameter simulation ap machine zero cover surface temperature spectra error h accuracy star depend magnitude prior vary range still strong degeneracy parameter magnitude advance probability survey help reduce surveys star infer datum task galaxy physical evolution star population require via parameter temperature numerous parameter signal snr phenomena spectral type l band index generally narrow reasonably method pattern recognition use generally feature determination ap mapping ap spectra spectra variety name machine al galaxy company classification tree al linear function projection estimation example volume line index really place enable relationship label template star produce despite nonetheless try fit cause severe independent ap contrast generative transfer ap affect yet already try overcome et ball plus extension distance likelihood create solution way create label template close smoothly good within error covariance grid grow parametrization might ap template metric use mahalanobis scatter add mix impact mahalanobis loose sensitivity interpolation template template far consume also unnecessary generative forward template generative shall estimate assessment solution ap base idea interpolation spectra g star magnitude galaxy star priori span wide accurately want intrinsic integral processing comprise section latter report plot discussion http www outline terminology multivariate table summarize notation band refer general band pixel band spectrum counter band counter ap band sensitivity model band true band generative transfer explicit unknown ap template spectra generative generative forward nonlinear grid forward band demand continuous also calculate ap fit grid keep predict train basic newton forward fit detail fig remain square residual local ap calculate discrepancy predict offset n taylor reciprocal make toward estimate ap iterate vi stop basically iteration vi way spectrum stop fix ap large move function opposite limit likewise initialize true solution band several sensitivity multiply eq ap v matter model provide function derivative arbitrary work I find strong ap explain weak relative term explain fit weak minimize function reproduce weak ap little optimization ap separately strong case strong ap generalize ap ap value strong ap fit residual increment add illustrate strong bottom weak solid precise follow let ap ap band ap point strong red residual illustrate bottom panel semi regular grid weak ap easily fulfil grid weak value strong dimensionality apply evaluate near grid identify close component increment change component specify increment component component smooth combine axis arbitrary forward carry ap axis practical working mean describe purpose paper progress irrelevant constant logarithmic bring I band ap variance spectra covariance strong forward smoothing e conventional cubic spline drawback control knot spline apply smoothing control specify via result fit h unique maximum smooth fit however many overfitte component practical known author inverse model well similar datum something analytical derivative calculate select priori ap resolution choose ap update depend upon limit standardize rarely apply impact ap ap thus ap update noisy spectrum valid update standardize low limit arbitrarily ap undesirable code upper limit step offset incorrect limit rarely expect forward model good prediction ap grid ap estimate ap apply important assess via ap evaluation error include systematic mostly former distribution outlier vector transformation algebra x apply equation assume update estimate ap calculate goodness reduce q forward diag name large refer fit model naturally provide usually resort intensive sampling measure update take band snr band ap measurement proportional measurement performance improvement g illustrate thereby observe dispersion blue red vary nm nm spectra line broader remove snr model spectra retain pixel bp cover nm pixel cover nm pixel spectra experiment extensive library bp rp simulated generator library library former value k uniform unique grid incomplete reason fig range star combination star simulate ten parameter al band library step combine show five three weak estimate ignore contribute scatter case low curve spectrum highest high present limitation library offset clarity dash band calibrate publish classification currently spectra demonstrate spectra break nm dispersion rp dispersion plot varied hold variation weak little spectra course snr spectrum law end observation observation magnitude simulator background well spectra instead zero pixel g magnitude band define mirror rp sigma number spectra specific four distinct try determine vary case scatter forward represent split near half indicate g band rp adopt rise normalization offset spectra magnitude bp rp primarily area divide band procedure evaluate global ap range present measure grid temperature estimation ap star evaluate full evaluation tm strong ap either tm star tm star tm full grid evaluation tag ap star random evaluation star star evaluation ap training mean systematic twice statistically marginally magnitude standardized multiply fractional evaluation f tm g tm tm tm l tm tm tag tag problem band central nm point star ap point plot standardized unit prediction forward forward band fig weak describe smoothing fig forward band fit plus robust fig standardize varie compare weak ap band library ex leave top plot reduce goodness equation ap final adopt ap horizontal look sometimes rapid star long sometimes star depend noisy near away something adaptive encourage property cycle problem specific ap estimation ap residual minus statistic accurately significant systematic scatter spectra obviously acceptable accurately reliably distinguish subject inter table magnitude result nn limited density template grid report ap template grid noisy exact template weak star grid noise spectra leave time confirm grid limitation star evaluate full error average g report entirely model star priori variance limited line dominate ap gaussian fact clear tm grid tm tm assume already rough spread act scatter set level magnitude summary show star even full range precision twice g compare star within full star bottom different systematic star vanish turn performance quite lot little h see panel fig plot exceed limit grid section suggest include strong systematic see essentially star update predict corresponding grid ap report logical desirable report average ap act result implicitly star relaxed test train tm h precision galaxy identify poor star tm uncertainty residual lower positive residual black plot tm element uncertainty panel compare residual low uncertainty important forward comprise forward almost forward unique training grid point vary strong component build residual respect strong combine nothing else change two different problem weak ap important parametrization accommodate fig forward spline et fit band black plot standardize full tag cut show accurately variation summary applying small limit change star fortunately many star swap train broadly summary list near bottom significantly tm acceptable additional ap systematic trend correction make g star star star tm problem uncertainty h weak uncertainty g evaluation error tag star present analyse star accurately star surprising spectra simple signature
cb subgaussian theorem satisfie selector cb research foundation grant author grateful reading manuscript constructive significant presentation reference throughout present immediate implication shall admissible exist index conversely hold decompose correspond large large absolute fact hold assumption index large absolute integer factor similarly q obvious direction location large clear admissible immediately admissible hold ks ks inequality argument denote q location absolute argument observe obeys admissible k imply ks x definition preliminary also provide respect metric ball cover cover net main exploiting carry onto include completeness state proposition selector c design j nc apply union obtain immediately proposition condition optimality imply theorem hold part condition thus hold immediately crucially exploit condition selector apply proposition selector true feasible x optimality obey cone desire optimal hold see front let constraint crucially exploit random fundamental apply method tight compose gaussian similar small gaussian come tighter developed set extension follow derive cf therein lemma immediately plug lemma concentration measure normal call canonical vector np fa b gauss fa conjecture definition pt pt pt pt department kind restrict associate necessarily condition bind isometry define hence broad condition functional intrinsic low complexity implication keyword sparsity selector isometry subgaussian typical appear graphical modeling approximation recover noise throughout norm random shall section impose gram guarantee property selector nonparametric elaborate definition put model penalization pursuit factor convenience selector refer non lasso selector appropriately section column represent ready introduce restricted formalize selector purpose completeness say therein see condition tailor particular check absolute guarantee large body work principle condition strong restricted isometry constant submatrix extract integer isometry small quantity hold subject subgaussian compose column order extend family subgaussian ensemble hence behave certain introduce covariance column theorem constant eigenvalue specify section believe case random randomly orthonormal selector matrix entirely self exploit thresholding adjust select rely selector condition select significant conduct propose matrix eigenvalue impose need let vector subgaussian random vector basis copy vector note random copy strong context suppose integer number case check admissible coefficient integer admissible equivalently sense definition sufficient admissible necessarily paper I matrix normal hull cardinality denote throughout understand understand main result subgaussian ensemble class subset subject cone constraint even broad set coefficient contribution element need rip isotropic independent copy row least absolute immediate consequence euclidean guarantee small follow bind vector concentration around proposition see admissible location coefficient long clear generalize restrict property rip particular sparse lemma theorem identify canonical understand appear appear elsewhere
potential statistical significance partial stream future describe list episode block frequent episode recall identical node partially output purpose illustration type edge edge node type per per neither sure exist type contradiction r z z opposite contradict partial order x z contradiction arise proof generate among closed possibility prove drop belong z l ix z l ix l need close perform list lemma exist prove reverse indicate possible hypothesis exist lemma prove exist type closed hypothesis never present demand node scenario closed iff statement z z exist hypothesis l type prove hypothesis b hypothesis existence type z l close node analogous add I I e l l remark claim frequent episode framework event episode node node separate episode serial episode trivial parallel episode discover episode partial order specialize serial episode flexible specialized mining frequent alone sufficient propose partial episode effectiveness episode discover temporal pattern symbolic several domain www biology text mining etc stream event partially collection order episode event constitute occurrence episode call serial episode whose exceeds currently exist discover frequent serial episode stream available episode relate sequential pattern order collection sequence contrast frequent episode discovery look pattern repeat make quite different discover episode restrict attention pattern episode episode restriction order algorithm handle constraint episode occurrence occurrence specialized discover frequent serial episode episode method discovery certain partial maximal serial episode episode example order inherent frequent episode episode occur often episode frequent episode discovery episode serial alone insufficient measure episode order tackle evidence event occurrence require extensive organize sec frequent episode formalism episode describe track occurrence episode episode sec candidate generation sec conclude event tuple occurrence sequence episode tuple node type alphabet serial episode empty episode episode neither serial episode imply follow episode occurrence constitute valid occur integer subsequence fourth event occurrence event occurrence arrange accord order occurrence episode follow episode say g w must hold episode whose exceed frequency episode stream frequency episode occurrence frequency episode informally occurrence episode occurrence appear episode occurrence episode stream stream episode say cardinality occurrence frequency paper consider episode call episode say episode episode episode alphabet subset denote partial induce denote simple episode episode episode episode depend occurrence come episode represent indistinguishable occurrence ambiguity obtain alphabet order note order episode early example episode ax cr v cx cr v v v cx b er v episode cf associate relate episode episode give track episode order manner serial episode node cm auto edge swap loop node node swap loop illustrate episode track episode subset namely already accept I ready initially accept first continue ready accept encounter instead encounter rather thus either move recognize occurrence episode episode parent see initially element immediately subset episode track event event ready initial namely pair ie e tuple constitute could accept accepted list property state exactly element e state make matter state represent pair contain empty type accept reach thus definition parent accept e complete episode episode per e since added e conversely consider consider true event show started accept set e e accept eventually state accept event proof sake completeness argument property belong ie l increment frequent episode extract episode frequent serial style episode order generation candidate combine frequent episode candidate exploit frequent frequency frequency episode frequent episode detailed explanation candidate generation episode give size serial episode episode generalize episode partial order start state transition prescribe stream state soon stream final increment start track candidate episode data stream look appropriate transition algorithm count non occurrence pass count episode track stream temporal occurrence episode often application occurrence span difference episode occurrence span user window implement pattern separate count occurrence track span soon event appear occurrence allow consider start episode initialize first accept accept would second second accept move initialize occurrence final since increment episode episode begin track episode method reach whenever exist event I make stream counting like reach drop track amongst end together check span increment episode track reach use non occurrence episode constraint input episode set type episode array store episode notation iff array element store list possible state pair c tuple list tuple next event transition know next episode store explain list list event ensure transition list array piece store episode frequency episode track episode keep track transition initialize state accept start properly also encode event episode initialize work line initialize episode main loop stream list affect processing line k state set bit line recall active episode record initialize memory initialize process tuple tuple initialize line become current transition list line complete computation accept hence contain type ready accept compute list also list event state process list current old list make first state remove element appropriate list line reach constraint span accept increment remove episode start new complete pass loop algorithm handle type event share process unconditional state transition slight compare episode accept state set event time st definition event strategy stream move parse accept time add thing element start process initially inactive parse next instant perform transition event time instant essentially check removal follow increment parse type store lt false e j start zero currently process event associate current stream transition j I transition exist g increment k bag add start start bag central currently entry recall currently accepted event transition start state accept affect transition hence time entry take find add list information make transition maintain ready accept episode employ procedure level involve two candidate frequency count candidate candidate frequent episode generation exploit construct episode cf simple episode associate episode represent order array contain sort per episode principal generate node episode explain episode combine explain episode frequent obtain respective episode ba combine obtain drop ba ba combine obtain drop last node dl candidate episode formalize episode combine episode distinct event index combine episode episode pick order construct potential episode share drop maximum potential anti closure episode event type get partial episode combine er candidate valid candidate combine episode mention attempt valid partial order candidate verify check share dropping last closure r close closure check subset l style font pt b plus b leave right leave xshift scale construct drop line note dropping already frequent find frequent important partial whether frequent episode generation detail easy generate episode hence different episode exactly candidate pair vary depend combination pair case come form candidate ii x lx l lx also candidate combine x r lx lx x distinct argument every candidate order uniquely episode would candidate episode episode contain frequent episode suppose episode node episode particular frequent note drop last episode hence combine frequent combination episode valid generate generation whether maximal induction list candidate thus output valid candidate episode serial episode episode combination singleton combination candidate empty level would generate episode thus order serial episode partial explain partial every partial maximal lie easily specialize refer maximal serial episode episode property order retain potential candidate belong mine serial check generate class way satisfy maximal maximal bound user episode b parallel non episode maximal corresponding partial contain parallel episode serial episode easy belong increase order episode end class number maximal threshold exactly serial episode maximal serial episode episode episode maximal belong check use candidate order upper constraint make process efficient compare partial order wish count candidate episode node map lie chain associate interesting serial episode contain special episode keep episode episode total suppose map remain map far along special ambiguity possible episode redundancy non episode either accept choose per count device track state interestingly episode e even try counting even though convert state equivalent note count straight episode argue general require procedure episode episode tuple repeat interestingly proper element parent constructive disjoint two element episode deal node proper transition ensure counting algorithm episode elaborate candidate generation combine episode g v v vary iii suppose generate iff episode thing need ask type partial episode ultimately episode frequency alone episode episode every constitute occurrence episode episode mining restrict serial episode serial episode consider general exponentially episode da c thus inherent episode partial evidence episode frequency meaningful tackle episode frequent set episode say episode episode episode episode episode mining frequent episode size episode frequent episode order episode partial episode frequent specificity satisfactory frequent episode output suppose actually partial episode occurrence episode serial episode great specificity filter episode preference serial episode satisfactory depend count episode instead datum decide order fit datum would episode see roughly often follow well dependency episode episode addition nice partial demand episode event type either formalize give episode j occurrence occurrence partial episode measure try magnitude symmetric entropy term tie subset threshold episode threshold ii count episode maintain end candidate episode initialize count initialize zero store already increment reach increment relevant output size may reduce output efficient well wise efficiency unlike frequency threshold anti monotonicity main set episode subset case embed pattern pattern evidence embed since maximal embed often mining simulation evidence maximal almost maximal frequent sub episode threshold non level specific pattern pattern point happen threshold generate partial order episode vary generator episode episode occurrence episode stream involve episode stream stream string stream order datum consist generation three explain episode stream generate occurrence generate event need difference occurrence successive event geometric successive occurrence episode geometric denote event embed episode noise event stream occurrence successive geometric similarly type stream occurrence successive embed order stream merge stream way multiple instant stream merge episode stream final order stream improve efficiency mining embed order candidate frequent find frequent episode table ii frequency threshold iii threshold frequency level threshold time give th c post run embed three threshold embed report candidate remain frequent episode drastically table marginally overhead calculate improve considerably reduction candidate whether essentially embed along threshold frequency episode non max mix super c mix super c max max super indicate candidate episode column table episode frequent pattern various episode indicate non episode event embed order episode episode contain event two embed column two episode either category column necessarily embed say episode belong category maximal category table episode frequent table episode maximal maximal super episode contain threshold episode frequent one maximal episode use threshold episode report non maximal never eliminate frequency frequently inherent mining point evidence report actual partial group maximal category event episode occurrence episode verify pattern table frequent episode use post provide substantial improvement efficiency sized pattern run threshold episode embed second count mainly inherent partial pattern max level contribute huge frequent high huge candidate
comparison figure compute batch em depend implementation long estimate fair online would em hide markov fix appear case correspond recursive smoothing obviously explain rather whole use numerical online em recursive em batch individual entry performance estimate standard applying averaging sequence range comparable unitary also display post process average start center mild averaging recover suggest equivalent picture estimate effect index start variance early guess parameter figure thousand important optimally limiting provide demonstrate hmms rely recursive computation functional proposition require quantity course encourage theoretical analysis miss although idea analyze become point view many generally carlo computation report direction theorem side sided bound allow transition q variation obviously note x x probability lemma familiar normalization factor second determine side bound g backward function pseudo easily check indicator interior family finite first markov borel quantity latter converge prove check imply deal normalize log suppose drop law proceeding vanish corollary one simple lemma prove equal increase decrease lemma concentrate apply equal define light analyze prove exp algorithm remark call fix time series combine root methodology problem consist exploit purely hmms recursion algorithm resemble sufficiently convergence potential recursion quantity involve propose comparable hide markov maximization concept range practical impact year markov classical variable value rise procedure allow modelling situation ever em dedicate routine maximize preferred alternative ease hmms store hmms challenge trivial smoothing computation approximation log principle comprehensive recent review advanced require method em algorithm hmms take ingredient recursion allow smoothing functional algorithm specific general address purpose case independent proposal hmms possibly continuous key observation recursion smoothing scheme introduce currently provide algorithm coincide interpret em recursion infinite generalize argument provide large organize brief computation hmms propose online em devote previous numerical proof finally online estimation observe gaussian hide state observation parameterize convention pdf arbitrary respectively state initial pdf hmm start parameter consistently trajectory discussion density function characterize recursion belong sx non necessarily invertible natural parameterization maximum assumption ii maximum belong exponential family algorithm stick representation section detail hmms usual familiar avoid unnecessary log likelihood briefly ingredient recursively recursion idea study largely exploit discussion chapter addition usual q quantity obviously quantity allow interest decomposition update recursively available observation check claim proposition constitute carry analogy observation choose decrease stochastic approximation require performing update compute role guarantee behave degenerate hence properly sufficient first early intend dependent reduce simple recursion analyze approximation random perturbation key hmms maintain filter become acceptable proxy auxiliary although recursive auxiliary put em argument maintain filter backward constitute usual complete mathematical though recursive author particle filter markov acknowledge constrained instance fail principle algorithm analysis algorithm time important observation tends interestingly result limit auxiliary instrumental section word relate limited important mle adapt space inspection limit suitable converge limit normalize n various influence vanish ignore result combine family vanishing rewrite follow define hmms compact interior continuously interior fix stationary contrast interpretation em sequence hmms investigate convergence kullback leibler divergence property filter case use find identity conditioning find infinite future recursively indeed requirement estimation hmms show converge constant h theorem quantity obtain imply ii stable limit parameter highlight contrast past verify sake case take state conditional case consider matrix respective constraint transition mean statistic separate form nature state example slight incorporate h early modify term iff notation refer application step g n nature update component approximate covariance derivation straightforward scalar additive gaussian model several continuous channel comprise batch may maximization directly easily constraint avoid complete example observe chain k q ni ni ni ni trajectory q identification separation noise actual reflect systematically start initial illustrate consequence em variability estimate bold observation slow iteration obvious similar picture limit guarantee theorem range thousand plot large rather computational
facilitate object live metric oracle retrieve sort list distance respect relationship new question rank relationship object rank r introduce notion rigorous approximate triangle rank capture inequality relationship notion rank investigate randomized scheme exist near neighbor oracle average ask partial develop decompose likely get object tendency stay generalize build similarly locality hashing retrieve near query hash depend distortion criterion combinatorial rank distortion space less homogeneous asymmetric every capture neighbor variation extensively see survey space intrinsic net structure search application net author growth metric restrict growth guarantee randomly object neighbor underlie necessarily prior generalization rank search oracle study algorithm question database author combinatorial near neighbor define approximate analogous bound depend crucially combinatorial database notion triangle spirit net show complexity complexity retrieve near neighbor phase builds exponentially improve polynomial accept list framework access distance force number question ask particular infer rank also sample top believe search ask address show build decomposition diameter set tree intrinsic ask framework access exist hierarchical decomposition constant approximate neighbor query query time remarkable survey lsh instead hash table case several choose object near neighbor lower bind hash space distortion depend knowledge study first moreover hierarchical demonstrate efficient formally distance object directly access return rank set equal near simplify notation indicate unclear ambiguity note rank object object r question create rank add ask precisely ask object select question ask sort object characterization space form approximate first define relationship triangle imply rank object small triangle rank around rank symmetric column give distortion monotonically exist linear four triangle inequality imply first inequality nearest problem neighbor problem object hash sensitive probability search rank whether problematic sense tell close point candidate hide extent violate see triangle provide randomization lower randomize require neighbor exploit fact choose database densely ultimately sample observation close close sample conceptually randomize reduce consumption performance small retrieve near bit average search retrieve near query randomize develop indeed constant search answer oracle randomize question ask neighbor expect consequently within object database limitation might considerably inaccurate come point extend tree analogous result new hash application sensitive hash sensitive hashing depend rank capture rank pick sort object exploit far distortion special locality sensitive one consequence retrieve point question rank retrieve section assumption consequently rapidly exclude candidate object algorithm neighbor build succeed query neighbor w build top database bin close find r oracle comparison bin one chose hierarchy top answer question ask level question need question even sample close query low level near repeat database retrieve neighbor leave union ds ia nj ni ji ji union probability higher succeed h immediate easily modify neighbor hierarchy p interested hierarchy desire section configuration guarantee neighbor query expect question ask query database show star branch weight database object connect edge range object branch star edge connect neighbor object answer question ask need near neighbor graph short distance see randomized example direct find neighbor direct neighbor fewer expect sure retrieve ask find query know exclude know building decomposition separate root leaf close show neighbor lie centered object lie object r w narrow close appendix query retrieve neighbor question requires expect sort sample retrieve nearest happen scheme fact illustrate exclude object nearest vice versa width exclude ask characteristic word decompose separate know characterization build search recursively decompose database binary tree clearly illustrate rank notion diameter set proof diameter symmetric distance decrease diameter hence diameter equal diameter choosing take assume want appendix cut diameter reduce diameter constant general roughly good cut divide h interesting fall outlier likely bin bin away ever set node rank object end object notion median randomly select hash also separate computed time randomly sort object popularity next exploit randomly cut stay together sufficient search object rank rank distortion say single object input rank prove retrieve neighbor distortion distortion intuitively situation roughly constant r box distance address object database exist whether efficiently object need object assign meaningful numerical similarity raise interesting good efficient right characterization work capture one distortion relate present combinatorial bind search characterization sensitive hashing depend rank manner locality sensitive hashing search comparison bridge search situation technical lemma ball uniformly ball bin choose less let fall bin else see denote appropriately object query visualize distance htbp locate property tell rank set object less w close proof expect close less half identical property choosing make five true object level level object object let close property triangle section eq object object bin less tell close sample fall bin close level less inequality property summarize need question total level fail probability fall level upper question immediate store close requirement exceed bit property proof appendix external query find object ask question particular object near neighbor triple let two inside star contain case could ball even place end branch contain node xy htbp branch star edge line graph path connect database choice neighbor query weight assign direct direct neighbor weight path principle expect run solid star object know answer question position point fig query branch object call edge equally word near neighbor must neighbor equal question except question
svm parameter gaussian denote cardinality model hold set follow q finding scenario confidence prediction make unbiased label argument give non outline model strategy training randomly train training point label proceed split validation calculate validation proposition require specifie function would test use empirically determine would estimate uniform cumulative difference bounding difference therefore measurable vc apply estimation accord unknown generation minimum misclassification satisfy choose function true opposite bound proposition apply unable cv traditional svm model cv make vote acquire uci repository sample unknown list attribute gaussian negative sample vote carry list fold routine split fold testing procedure train validation training carry procedure validation exclude training sample cv strategy result standard validation selection fast cv exclude set exclude selection cv less percent cv inferior number approach select margin testing despite improvement close cv rate overall bad cv exclude rate except standard vote since number traditional possible show bind value subsequently small figure bind early monotonically considerable partly quite vc correlate encourage achieve validation generally gold strategy margin perform error htbp give novel encourage shift selection competitive relation present restrict svms measure choose base margin applicability future research near measure acknowledgement would acknowledge project ep project fp cs ac selection measure batch help leave selection bind comparable method theoretical success demonstrate choose analysis fit analysis popular practitioner loo concentrate vector svm loo number alternative literature instance explore model span scale application drug automate methodology classification svms selection use put employ predefine propose pac bayes measure performance classifier estimating svms use optimisation recently achieve maximum obtain select svm cv involve cv keep model cv applicable margin idea label measure observe unknown map sample svm svm optimisation slack primal vector denote dual optimisation nonlinear optimisation problem x lagrangian discuss validation whole fraction z measure validation functional large
enter distinct forecast distinct functional multiple function evaluation functional participant allow possibly distinct forecast issue relate huber huber bregman characterize probable arithmetic perspective apply conditional quantile traditional quantile bregman original form could employ generalize least distinction example census census measure use census estimate include squared mean percentage census impossible designing estimate aim se consistent distinct statistical functional census way point open nonnegative nonnegative part strictly consistent equivalently part fy dominating hold unless relative consistent hence sketch statement immediate argument general necessity prove let probability f exist x strictly remain necessity principle x pairwise partial yield score validity scoring function sketch yet part x fy fy fy fy nonnegative positive strictly prove necessity principle usual integration convex measure point functional relative class focus center absolutely compact appendix forecast scoring forecast rule take normal forecast robust forecast international journal stein loss berkeley mathematical ed university california note consistency guide evaluate quantile j banach norm ed north average law lead journal law refinement quantile median message toward fr mean journal la et de quantiles quantile quantiles journal distribution conference f performance volatility mathematical point nd competition implication international journal k competition study international journal forecasting newton j forecasting competition journal forecast national evolution sale forecasting management forecasting journal forecast j k usage application deterministic forecasting technology forecast science r pp r forecast verification weather journal american association asymmetric van p proper h ph thesis california berkeley k schemes journal public economics l error volatility forecast comparison volatility volatility correlation forecast time pp machine journal van design apply statistical economic analysis university journal journal building minimize transaction journal american approach focus statistical activity news c survey production possibility inform journal statistical functional west asymptotic scoring conjecture example universit single forecast continue science forecasting assess depend forecast observation error inference forecasting point scoring function forecaster functional score forecaster forecast rule loss forecaster receive functional scoring rule forecast link bayes scoring scoring expectation ratio quantile bregman quantile piecewise ratio expectation weight score consistent functional instance quantitative finance phrase rule median point forecast aspect human activity major uncertain forecast nature still reason report requirement communication type situation assess average thus criterion take realization list commonly score generally score orient forecast absolute discusse point forecast proxy se mm percentage mm table public table survey volume review forecasting group application area iv article forecasting paper contain predictive forecaster forecasting score monotone square surprisingly forecasting scoring function square popular particularly group absolute percentage survey conduct summarize al square business demand sale monotone transformation employ percentage scoring sum column exceed column simultaneous scoring article estimation study fp se mm international journal forecasting journal forecasting statistic mm apply journal american journal statistical iii journal business journal mm american mm weather journal weather mm se option consideration practice standard theoretical arguably theoretically principled year note verification measure effort concept principle forecast verification nothing change ask deterministic forecasting verification al states requirement still circle day ahead forecast blue focus seek forecast asset realization series issue forecast asset actual true forecast forecast along asset successive trading day little performance forecast score list low absolute percentage scoring perform yet forecast scoring se se mm value growth case report probability point prediction square specifically means ask ask provide distributional point may report mode apply function may similarly receive concern example help variable square minimize refer rule practice argue complementary way ex forecaster functional forecaster scoring permit mr issue point forecast scoring forecast absolute bayes median percentage scoring density optimal forecast median random whose call arise summarize discussion forecast rule distribution generate mr forecast forecaster predictive distribution study understand follow fractional exist readily see forecast small thus forecaster predictor score mm rule forecast se mm correspond mr mr bayes forecast mm bayes alternative request forecaster quantile scoring roughly quantity quantity distribution concentrate mapping ft consistent equality strictly consistent remainder notion comprehensive addition scoring finding forecast ratio expectation quantile subject weak regularity bregman subgradient apply quantile piecewise functional notably functional popularity application practice forecast forecast either expectation quantile scoring develop forecast theoretic whose comprise outcome probability domain equip constitute probability distribution observation decision maker represent cost maker act rule maker uncertain future represent loss bayes act bayes domain action simplicity common equip borel algebra furthermore score theoretic observation score pe forecast optimal forecast act interval case partial derivative subsequent impose score share multiply forecast pose predictor concern argue homogeneity desirable scoring domain bc x decision problem scoring probability distribution instance prediction b decision resemble al much work assume domain b favor forecast focus forecast distributional forecaster functional huber current point presentation definition score class scoring relative note opposed scoring line probability finite moment parametric property forecast decision dual connect forecast evaluate scoring give forecast state differently consistent identical function despite immediate define appear widely result scoring suggest score satisfie score relative proper distinction useful fail make forecast mu discuss score consistent framework assume expectation domain penalty arise forecaster forecaster loss belief encourage probabilistic scoring act domain score consistent scoring induce natural score functional proper general theoretic proper describe evaluation notion back whenever feasible definition consistent relative relative domain map consistent scoring strictly strictly consistent concern scoring function admit dominating domain weight w integral proportional strictly relative strictly weight score predictive probability density proportional density result forecast forecast function result score median functional recover correspond mae prop forecast permit table notably relative quantity represent freedom limit multiply carlo derive see appendix b optimal original scoring consistent functional functional become forecast positive squared percentage optimal point percentage scoring derive situation weight forecast routine show consistent eq weight scoring consistent characterize class scoring general functional al condition functional sum quantile question include converse generally characterization practical way describe characterize scoring function consistent useful functional include point identification available argument partial derivative example expectation derive square fourth identification mm mm forecast probability argument respect yy consistent principle subsequent principle example expectation quantile ratio expectation technical rely property refer know squared scoring functional probability moment expectation turn subsequent identify bregman result score compactly subgradient score mean functional measure bregman bregman representation score homogeneous arise multiplicative constant unique symmetric introduce rich family homogeneous bregman function namely homogeneous scoring restriction worth forecast bregman event compare mr mr forecast bregman forecaster mr bayes functional measurable f function satisfie compactly subgradient arise section relative consistent subgradient scoring respectively cumulative finance var evaluation forecast generally relative measure heart quantile score function consistent equivalence historical quantile functional relative class suppose satisfie class compactly quantile form generalize piecewise order transformation functional monotone mapping homogeneous mae log mae sd score mr green simulation score mr bayes introduce functional measure rule point piecewise score similarly rule asymmetric piecewise surprisingly quantile class scoring interesting combine key characteristic bregman family measure interval moment scoring satisfy compactly probability strictly finite expectation distribution convenient quantile huber popular finance vary elegant appealing sense consider relative distribution mixture continuous challenge use measure evaluation forecast lee consistent scoring remain unclear might forecasts measure mode state informally mode optimal forecast rigorous forecast rule scoring modal length explore differently know member lebesgue sense represented become available put scoring attain origin lebesgue continuous unimodal median coincide theorem convexity survival adequate people accept reasonably accurate death forecast apply et restrict attention forecast domain wind forecast volatility theoretic forecast target take discuss assume wise forecast bregman denote product sufficiently smooth score generalization representation
lie cube u contradict combine contingency integer wish estimate number constraint program begin expectation determine solution integer satisfy approximated see discussion deviation sum independent geometric characteristic assess refer characteristic time separately associate theorem parameter require absolute zero drop away hold matrix satisfy since hold possible non increase maximum r uniqueness achieve determined row maximize tu st r tu constrain lie add fix maximization equality follow concave fix equality strictly strictly strict concavity tu say paragraph unless hold turn maximization fix accomplish choice lie compact maximize maximal point contradict fixed choice row entry conclude occur extreme take away require occur nm mn quadratic dt jk jk jk jk dm form establishes allow independent relationship covariance diagonal note w ie ir ir ir reduce dimensionality reverse hold v concluding proposition prove q corollary bound taylor eq jk jk jk jk ga ex u u count direct construct progress small index index pass value remain variable I tt ct condition require prove analytic prove evaluate rest behave integer characteristic individual characteristic function value happen handle transformation constant prove q proof integration denote condition iv nm enough v v ir characteristic function constant validity sum cm sum equal calculation convenient dropping exponential c estimate contingency constant column exact approximation long accurate cm approximation joint correction produce integer row sum undirected degree begin consist q provided solve uniform degree eq lattice possible degree near produce total twice formula near regular graph graph whose asymptotic depend characteristic expectation approximation expectation similar contingency case binomial degree let expectation invert variance fourth formula estimate degree identical formula improve give note symmetric since degree graph maximize far assign degree get degree exactly correction constant table error vertex formula work near half number degree bad even consider degree entropy edge maximum variance p j lattice possible degree compute p l gauss accurate formula degree I approximate correct entropy satisfy integer satisfy satisfy entropy density first moment validity correct demonstrate contingency graph subject polytope volume ann mi usa mail address new ct nsf grant dms dms united count integer c distribute central apply sum suitable moment select propose maximum gaussian volume kt entropy discrete entropy distribution cube integer satisfy density form density may constraint uniformly integer suggest mean say expectation ba maxima j far entropy sum infinity variance sum sum approximate accurate correction produce proportional derive cauchy integral coefficient generate maximum widely specification clique cm extend integral asymptotic enumeration contingency table integer sum know sum maximum formulae vary formulae integer integer advance unify use example moment sum variable determine maximum de square statistic uniform
collect undesirable time concern heterogeneity difference category model likelihood bias relatively variance category na estimator need increase variability bias trade recommendation actual study motivated reduce particular focus estimate genome wide control disease status study adopt model significance methodology discussion specification assess performance specific utility association candidate three conclude log odd interest follow asymptotically snp code copy allele loss minor allele vs case essentially familiar significance conduct vs statistic standard calculate subsequently without adjust nan hypothesis critical rate simplify although conditional unless na I bias demonstrate genome wide linkage study correct q standard conceptual connect logistic control correspond correspond statistic correspond bayesian focus factor influence association disease population additive dominant allele snp power association simulation study practically range significance sample turn normal moreover conceptual cover association analyse quantitative conceptual coefficient show build conceptual normal publish association coefficient threshold unbiased mean conditionally development evaluation performance author mle correct reduce simulation tend large estimator treatment conditioning author unbiased population completeness family estimator conditional statistical n b dirac define equation completeness n estimator unbiased information genome diverse genome linkage common apparent snp positive performance bayesian paradigm belief significance mathematically discrete probability hyperparameter prior history shrinkage theoretical discusse spike frequentist procedure reflect belief imply favor could belief extreme outcome value small correspond snp beta evaluating propose robustness prior simulation indicate preserve shape inverse shaped density inference exist although variance precision subject hyperparameter inverse produce component represent however estimator snp difficult influence problem influence sample actual reflect disease trait snp know perhaps genetic date show long largely association detailed equation provide use characteristic n traditional chain technique mixture augmentation extremely alternatively carry inverse metropolis iteration discard first burn discard sample conceptually method take account uncertainty lack information test confident detect reflect robustness paradigm inferential say eq set model discovery belief specify utilize decrease q view normalize constant therefore two normalizing use bridge sampling propose iterative st compute begin easy ij obtain carry simulation examine performance bayesian compared develop second nine examine na I unconditional mle estimator likelihood recommend estimator prior high power discovery estimator negative truncate zero flip occur snp population snp associate disease allele allele allele decrease factorial design factor investigate reflect genome association rate typical snp throughput depend snps statistic population uniquely determine detail simulation simulation generate statistic nine genetic apparent relatively consider h inferior implication put put little I estimate behind bayesian bias well compare perform power bias big standard deviation root mean square half match pair base run factor influence bias qualitatively sample small significance stay show table show slight variance difference drastically association study snp keep comparable increase comparable range simulate circle bar long horizontal deviation sd root rmse conduct range generating summary size apply let detect snp log qualitatively genetic scale study practically significance na I discovery report twice association power size na I never sufficient snp na I sample center believe practically useful sampling examine level equal report apparent perform snp power association circle bar average sd rmse rate calculate na I simulate significant circle horizontal bar long horizontal require genome wide study outcome specifically association diabetes previously via genetic estimate report easily study original genetic na I five estimator literature design power knowledge snp rs snps five complete serve variation original ci interpretations ci different although sample way credible normal averaging estimator equation ci ci specifically ci quantiles conditional note propose compete estimate mle wang et candidate study control significant I rs independent considerably rs I association ci ci ci b pt follow quality allele test focus snps analyze snp propose use discovery actual control control discovery ci ci ci b estimate mle ci pt ci l follow snp roughly value apply report table extreme correction estimator little change publish less general produce longitudinal phenotype diabetes control trial identify conventional value association c vs snp perform pass effect pt ci ci ci follow follow robust reflect belief iteration odd addition conceptual normal
share probably rare position challenge enable simultaneously improvement sequencing machine include enable convolution additive dominant averaging e replace build add column proposition conjecture assumption variant rapid sequencing enable prohibitive improve trade pooling design individual relatively low individual pool design sense general simple computer enable rare allele especially high individual technique enhance genomic sequencing recover identity rare genome association successfully association phenotype numerous find various trait limited bias although find association trait far explain trait disease interest however study large infeasible recently currently sequence sequence throughput cost hardware generation sequence utilize read genomic order higher sequence high rapid sequencing address question infeasible sequence possibility genomic sequence individual extensive amount genetic ability enable rare population thus fill us discover rare predefine seek variation nucleotide minor large population certain sake discovery rare variant generation sequence million typically dna pre genomic sequencing hybrid dna rna region interest region throughput make task naive costly option hundred thousand since whole genome throughput much capacity naive inefficient pool feasible pool several individual together sequence pool allele population measurement allele allele traditional pool sequencing infer rare recover rare rare sequencing pool tackle identify individual design e mid biology streaming communication see comprehensive survey work try rare allele pe er design code enable allele construct provide signature enable allele observe contain signature offer resource recovery single rare allele detecting albeit al enable identification sequence accord ideally could mix use therefore suggest pool read obtain identity chinese accurate recovery allele pool keep approach recover individual variant testing yet study allele identification identity allele analyze deal cs enable testing thus identify snps cs paradigm reads effectively sense active development research daily basis cs many field pixel zero reconstruct term dot measurement reconstruct testing set allele interested indeed measurement dna pool individual take rare single whether consecutive theory theoretical robustness noise numerous technique benefit development fast improve suitable identify rare allele direction extensive aim benefit apply cs identify rare scenario sequence find benefit apply large improvement per organize rare reconstruction pool simulation efficiency approach offer conclusion provide cs description identify mathematical sequencing utilize standard cs reconstruct k solving equation measurement sense vector case scalar recover uniquely uniquely specifically solution find number zero entry somewhat stem contain information require isometry rip briefly state matrix almost matrix invertible vector able follow computationally relax close still solution problem reformulate follow problem convex realistic problem measurement norm therefore reformulate maximal level obtain add term cause available enable practical thousand choose wish reconstruct individual reference allele alternative allele represent allele allele count individual interested rare minor zero cs real restriction expect reconstruction enable fast namely sense matrix individual random individual sense sense measurement perform pool ensure affect individual equally dna read genome perform allele together cover number measurement represent measurement introduce various section measurement measurement present illustrate formulation people snp green allele assign pool probability example first pool dna individual pool pool sense incorporate constraint original rare range aim sequence apply generation clarity experimental factor add establish allele vector rare allele perfect pool likely read sequence read pool segment read allele probability vector allele eq q dna pool practice position cover read read generally read cover specific main variation bias content experimental adopt rare allele fluctuation sample c assume noise exist besides limited read scenario sequence read sequence reflect read sequencing may sample due basis read mis wrong genome base dna read different result three different allele allele read read produce reduce observe read vary sequence technology library quality control alignment algorithm realistic value know correct r unknown run one single incorporate framework pool factor far error resemble one propose model namely dp hard exactly amount amount result measurement actual obtain dp error entry mixture dp noise unknown unchanged effect modify actual rare allele oppose classic cs know dp region consideration give size snps study small region result treat isolate snps indeed snps read cover snp contiguous interpret target region read cover multiply length genomic read successfully sequence technology million sequence machine greatly influence perfect might desire varie read alignment million fix rather modern genome report simulation adapting need per snp denote merely section interpret read person pool approximately individual pool therefore average coverage pool visualize noise read error specific scenario instance rare identify level sampling noise number snp correspond zero separate read right panel dp actual error reconstructed obtain clearly visible infinite expect deviation size moderate error panel rough quantization vanish reconstruction aggregate information read level irrespective error reconstruct leave probably sample per sequencing lead cc dash expect frequency pool read dp pool dominant observe gradient trade fit specify maximal allow often desirable measurement adopt throughout although different entry integer post processing zero post large value ss equal describe cs combine strategy improve obtain dna dna additional identification mix together attribute sample pool dna pool dna specific apply still pool utilize increase effective number read decrease usage solve total read easily cs vary accord technology present trade run extensive simulation evaluate parameter range simulate instance apply order accuracy input reconstruct sequence reconstruct due coverage error term reconstruct yield false false show restrictive reconstruction reconstruction reconstruction occur problem read etc individual correctly rare allele false discovery discover allele measure reconstruction efficiently various simulation experimental simulation follow individual rare allele vector allele frequency allele low allele varied region length read length different coverage kept fix read estimate cs relevant error present different finally display behavior dramatically rare fig present number snps allele vertical display correspond demonstrate naive sample available merely evident panel decrease insufficient cause almost cc treat simultaneously mean certain parameter simulation yield reconstruction simply demonstrate right axis display panel region present scenario unit individual correspond case appear cs efficiency present simply I individual cs divide naive cs black line scenario snps score high number approach resource provide hand read per coverage question read successful small snps yet mostly noise coverage coverage individual code percentage reconstruction white line accuracy transition sharp overcome reconstruction criterion lower naive difference efficiency considerable naive allele cs simulate individual addition higher encounter practice take reconstruction fig cs way allele rare allele much naive one rare allele frequency rare allele noise specific reference appear dp figure compare performance read read person insufficient reduce coverage snps read read read significant performance see study dp pooling protocol overcome cs read dp solid read dp
stein shrink shrinkage estimator introduce empirical entropy loss class prior address invertible case minimize mse show advantage improve develop rao theorem idea mse matrix first haar obtain solution provably mse shrinkage try begin naive replace obtain remarkably expression refer limit oracle estimator intuitive compute provably dominate size estimation autoregressive show implement author substantially introduce represent simulation adaptive summarize conclusion letter letter transpose conjugate transpose frobenius positive definite independent identical distribute assume mse constraint specific shrinkage solution necessarily mse variance usually ill pose hand well reduce reasonable towards follow shrinkage matrix generalize restrict attention shrinkage shrinkage next unknown approximate optimal shrinkage begin derive oracle error follow subsection show optimal let equation gaussian evaluation expectation identity eq specify distribution depends prove show shrinkage rao estimator describe provably improve sufficient estimator rao rao expectation never rao covariance expectation satisfie q large asymptotically achieve optimality point indicate contradict reasonable parsimonious formula assumption oracle iterative procedure initialize initial iteratively refine covariance non negative yielding generate limit eq latter force meaningful plug simplify q easy shrinkage iterative plot estimator gradually towards htbp htbp htbp increment fractional long often internet traffic phenomena mse fig fig experiment evident perform closely small improve obvious figure converge increase decrease regard characteristic explain well correspond preferable covariance time mse mse mse always dominate occur shrinkage c coefficient next narrow band sensor complex signal interference vector assume linearly combine arrival obtain vector estimator improve method estimator yield value hermitian complex part covariance represent extend estimator element half mutually independent one angular db sensor implemented measure varie gain fig performance several even well improve great paper shrinkage covariance art virtue rao oracle estimate analytically provably dominate conduct estimator structure coefficient significant target scale theory identity target target theorem require treatment haar singular wishart state lemma mean singular decomposition comprise orthogonal joint jacobian denote diagonal substituting factorize similarly column treat separately haar matrix unitary independent complex fourth since eq take identically substitute element identically note change variable obtain eq jacobian similarly eq scale eq calculate go prove th column expand coefficient
usually bad case instance square instead eq work term essentially close close bias estimate crucial linear theorem non oracle section hold numerical constant consequence yield asymptotic leading imply optimality front mean actually observable check experiment prove select near close selection restrictive relaxed sufficient family estimator ingredient uniform probability event define prove uniform result refer proof instance ridge regression certainly additional deviation core least theorem would assume gaussian asymptotic oracle heavy tail clear scope comparison estimator close term negligible oracle asymptotic meaning implicitly assume problematic may grow difference since like statistical non selection therein inequality procedure mostly ridge assumption sure asymptotic optimality ridge whereas non exist bind weak compare e exist regression require assumption near regression case method computational price increase study comparison mild lead penalty deal put aside nevertheless complex dependent since explicitly appear mention special feature minimal may appear dimensionality jump occur theoretical risk prove interesting correct minimize proportional simulation illustrate take use spaced goal regularization learn number neighbor bandwidth size jump penalty variable get jump penalty leave amplitude jump particular due replication knowledge minimal consider find freedom integer jump second degree jump heuristic jump occur penalty outperform outperform multiply penalty factor useful minimal proved extent slope heuristic modify light penalty prove theorem assume nn extension example regression problem simultaneously covariance regularization parameter penalty successfully mild regression e base certain generalize probably modification rely case slope still still linear general appear naturally conjecture valid dimension contrary modify x repeatedly every every define main every actually consider separately ridge parameter prove useful elementary specific true symmetric convention define hold equal result q lemma schwarz every concentration main extend form gaussian vector ridge set come result standard proof constant new framework event proposition concentration real extend optimal good assumption satisfy case combine prove prove lower similar slightly constant consequence proposition prove separately apply ia orthogonal ib corollary let particular least n q proof choose become eq soon derive imply deduce event jump use controlling consider showing imply distinguish eq take compare choose hold soon inequality hold soon satisfied section yield take use n compare hence side since soon q comparing get eq hold hand soon choose lemma proposition hold true define replace proof start eq become algebraic bound use definition bound remainder every inequality eq imply handle implie since reasoning union theorem us hold get deduce noting remark use end inequality upper bind assumption continuous common well vector therefore corollary similarly every taking possibly zero term limit ss integrable eq q hence result first root well rational fraction every q either polynomial degree general orthonormal hence every actually prove since concentration similarly exist every realize let q every similarly eq follow apply every finite lemma proposition acknowledgment improve acknowledge reference detect european grant independent post cm cm cm cm project team cs paris france l cs paris france select several linear non include kernel ridge spline context plug prove simulation multiple significantly calibration validation smoothing spline kernel classification issue framework practitioner use cross drive procedure rarely choice square select among quadratic include choice spline neighbor study signal constant minimax rich harmonic function main present class drive selection satisfy section parameterized ridge spline one parameter inequality context heuristic curve improve procedure kernel near neighbor locally solve example estimator let assume observe design set quadratic norm linear rest vector family deterministic parameterized combination result apply mind input projection projection selection wants parameterize column matrix assume reproduce rkh spline estimator regularization unique equal parameterize case positive framework back b norm f pf detail identity lead smoothing combination depend group single parameterization parameterization particular way alternative oracle selection procedure particular upon value large drawback estimate drawback give directly drawback minimal study risk proportional although several paper projection hold intuitive behave completely rigorous conclusion bias well classical selection empirical opposite suggest sense behave distinguish huge increase penalization follow first analyze paper indeed suggest around jump select degree describe generalize previous assume projection
follow note initially begin deviation weaker large weak sufficient error rapidly careful dynamic proceed need lemma weak need say bad hoeffding bernstein permutation fix price period linear let objective least b ij nb satisfy similar appear next relate value offline ready long violate achieve bound fact condition competitive item unless vector demand demand demand pair complement vector consist least item achieve boolean string represent item illustrative string p cm vector c c c c complement demand pick bit string vector share boolean string demand demand coefficient input demand demand demand accept construct hold eq proof offline sum input demand w type item unit item remain e prove future research x proof bad bad p inequality lemma sum price inequality ease omit subscript primal bn bad permutation term define easy z h lemma p second define tn p price distinct take union prove proof similar p pp price learn optimal dual kkt x nb probability permutation h nb z nb next distinct value customer bid bid conditional bid value customer bid bid constant similar necessity end bid potential mistake instance w bid mistake call mistake size mistake total mistake claim potential therefore consist v common complete probability central constant probability equal lagrangian optimal pg pg pg return differ value expression expression distinct online wide attention management reveal come online management arrive period stay request decision bid capacity receive sequence request user intend usage utility objective appear match online management resource recent development reader problem formulate online linear integer discussion focus linear programming integer online linear constraint reveal corresponding coefficient function far immediate without observe precise offline restrictive constraint meet q linear objective maximize propose algorithm programming need make assumption adopt column coefficient arrive pick start permutation number priori adopt comprehensive review intermediate case bad uncertainty evaluate bad input offline hand priori input great extent choice critical suffer weak draw one drawing algorithm history price relax knowledge multiplicative error let model offline solution permutation arrive near program also multi decision use offline program decision choose satisfy use objective maximize entire special online programming usually refer maker face fashion maximize application arise context worker schedule permutation widely paper constant competitive sized near sized paper higher near computer edge request request offline decision integer discussion problem name online packing problem main yahoo etc ad attract past decade majority daily budget relevance bid display maker engine dimensional allocate corresponding offline special case programming vector except entry attract interest recently competitive comprehensive section paper competitive dual act pa work threshold initial price decision price initially step precisely state program way competitive ratio condition far demand large prove necessary counterpart online lp theorem show appear algorithm problem permutation competitive random first depend side model quite uncertainty implement dependence dependence strict sized however contrary input engine search category million reasonably large input furthermore result might interest finish section programming entry condition q science operation management community recently random model attract grow popularity adversarial still capture matching online packing result obtain category near dimensional prof competitive achieve look multi need technique apply randomized maintain programming schedule prove necessity dimension achievable clearly high programming online utilize decision author develop update price carefully achieve competitive check problem although idea nature require answer price significant general packing vary extension propose achieve side dependence remove also improve dependence improvement dynamic recently study competitive way root cubic competitive expense contrast cubic dependence table om opt besides allocation call adversarial generalize permutation situation input model develop achieve competitive dependence make comparable technique operation research price various management resource problem arrival availability poisson bid price investigate author bid asymptotically arrival idea discuss arrival make contribution scope many dynamic learn knowledge arrival instead reveal design answer update often quantify update trivial bind add difficulty apply concentration cover dynamic share idea trick trick unknown horizon price horizon careful design section ratio simpler regard necessity condition extension notation integer price state initialize p dual price vector decide allocation execute constraint attractive solve small program side guarantee trick subsection learning rely program observe unless ratio first optimal dual price sufficient column reveal online fashion algorithm substitute substitute constraint optimal value simplify discussion necessarily point add uniform p simultaneously pa effect perturbation program differ offline program q py dual p pa condition therefore value pa value optimal dual close offline however price p discussion attempt price p random input replacement construct feasible dual linear program precisely follow price p say bad bad every union price
affine subspace avoid sphere incorporate necessary algebraic improve problematic minimum local minima affine initialization explore number develop would semi set substantially stream algorithm minimum assume currently develop robustness instance identify careful initialization thank anonymous helpful comment chen constructive manuscript hybrid question thank benchmark motion segmentation support nsf grant partially nsf zhang school mathematics school california se mn edu mn zhang edu linear e cluster minimize implementation amount storage modeling subspace require evaluate http www edu model vector video lie affine face condition face give rise hybrid linear mixture suggest utilize generalize algebraic agglomerative spectral curvature use multiscale hand e separation subspace probably kf aim partition formally parameter try minimize function example following assign newly cluster significantly worse global recent subspace affine replace function replace goal point beneficial moderate stream accurate order address algorithm name see approximate strategy approximation implementation apply affine superior particular outli relatively intrinsic motion intrinsic dimension well minimum carefully hybrid segmentation video conclude brief possibility introduce storage discuss implementation algorithm convention subspace I identify approximate subspace correspond try partition subspace normalize element lie sphere use gradient energy need derivative orthogonal calculation near subspace summarize partition update step convergence assign storage algorithm find near subspace operation computing update cost cost typically usually increase become complex etc stop ratio previous computation datum online replace functional square angle often ki large find near little empirically greatly obtain initialization initialization subspace initialize guess pick tuple hand kf guess respectively kf kf kf restrict represent various dimension equal follow fix instance linear http edu randomly distribution cross cube subspace center center corresponding far corrupt uniformly outlier cube maximal instance misclassification table time various hybrid linear advantage outlier reduce kf conclude algorithm ambient intrinsic run table indicate usually apply kf would deviation misclassification minima available www coordinate extract frame accord object background video formally video frame sequence move camera background point j segmentation camera correspond move live four kf htbp percentage misclassifie database kf median kf kf kf htbp misclassifie three kf r median median kf kf kf compare connect improve segmentation kf implement subspace subspace consensus kf http www vision edu base rate misclassification kf segmentation result randomness kf time misclassification kf ambient kf initialization
standard argument identify similarly rule fix interpret stopping criterion control cg particular consider ill filter know iteration polynomial unlike additional technique establish consistency attain square regression learn reproduce stand minimization procedure linear solution nest exploit conjugate algorithm empirical estimation iteration step partial second condition provide pls technique construct predictor covariance dimensional representation fitting regression latent component principal component trick pls nonlinear dimensionality pls consistency e ridge component pls define sense depend linearly instead pls minimize nested subset define latent estimator pls study consistency pls target know pls early configuration scenario dimensionality grow regularization universal pls infinite derivation pls pls combination early stop equivalence version pls step population pls converge ii population pls ensure pls empirical condition estimator provide universal emphasize stop rule base joint convention perfect knowledge bar represent technique true mapping kx boundedness surely dense briefly interpretation inverse denote operator adjoint usual operator g reproducing coincide gx finally variable learning cast formally problem multiplication even formal motivation regularization come inverse order approximation belong right interpret perturb version wherein empirical counterpart note usual wherein map operator x operator space dimensional correspond perturb right predictor ill pose regularization ridge correspond regularization boost pls describe greedy iterative produce pls response project component th step conjugate cg normal detailed overview establish pls e use pls exact clarity remainder cg subspace range real polynomial degree step cg minimize e mapping equivalently represent vector extremely important scalar multiplication iteratively construct space pairwise equivalently uncorrelated algorithm convenient cg population version iteration replace operator component different geometry norm respect projection onto closure u asymptotic simplicity I infinitely decomposition eigenfunction version theoretically run population stop step modification projection closure conjugate population write projection onto first operator eigenfunction degree convenient property eigenfunction principal converge principal principal fact pls empirical token less biased boost correspond however finding suggest suitable control since ingredient stopping quantity version quantitie positive x justified banach banach satisfy compatibility eq bind monitoring note monitor initialization observable quantity u monitor procedure subscript estimate rule b u use stop step almost
functional study view statistical besides parallelism state utilize scad besides superior practice advantage penalty extra tune think formally little simplify nonconvex solution guarantee might think point computational scad visual arguably less follow briefly review tv point trivial hope reader follow development adapt denoise problem discuss minimization review regularization computational superiority denoise emphasize encounter paper tv model decade desirable smoothing carry replacement tv near weight avoid tuning amount variation partial tv derivative choose sufficiently basically artificial edge numerically unstable image difficult lead increase computation require mention introduction almost use different context identically distribute sometimes reason zero formulate encourage propose possesse lasso prove practice reasonable estimate rigorous see parallel processing desirable difference argument lead form tv respectively statistical study mention adaptive appearance address variable smoothly desirable possess oracle property behave coefficient scad processing get parameter superior linear penalty amount fig derivative even odd penalty penalty image formally write functional eq reader solution seem note encourage reader change expression continuous prefer nuisance compare tv consider minimizer compare scad penalty penalty defer appendix tv I simple pixel shrink penalty desire shrinkage tv already context difference big enough complicated functional evolution euler lagrange equation gradient potentially also tv scad taylor estimate scad getting actually minimize penalty weight involve extra inner evolutionary derive euler analogy bound stability extra tuning unnecessary scad produce monotonically iteration take thus comparable largely quality mse calculate knowledge technique carlo require denoise formulation noisy image consider image input prove divergence mapping regularization filter operation carlo add recover carlo pilot experiment empirically pixel four neighbor normal black white regularization compare whole deviation tv weight function good different intensity value choice see outperform show evolution mse regularization figure clearly scad significantly tv get histogram histogram tv tv bias colored pixel value generally shift colored intensity shift previously address scad besides histogram result result fig mse big well different wrong make sure burden even good scad mse fig former white square image structure large feature deviation regularization carlo sure briefly previously tv scad denoise sure accurately table situation mse require choice priori conclusion scad slightly complicated test performance library record scientific show fig transform piece wise visually standard add result scad image since visually difficult matlab format penalization functional denoise know scad originally statistical I pixel scad solve inherent spatially adaptive tv newly method stability achieve general denoise wavelet become popular exist pde extra recover also current proposition b make small derivative constrain easy quadratic form minimizer meanwhile imply partial add subtract partial q combine prove solution argument contradiction functional tv cccc tv e cccc tv image numerous improve different context motivate efficiently realize spatially theoretically penalty inherent variation
hour c chain c chain hyper well value around truth confirm estimation increase parameter reach orthonormal wavelet basis resolution image noisy image denoise noisy b reference image separable shift filter denoise mmse frame frame recover image wavelet frame depict purpose bayesian experiment hyper denoise ball interest deal wavelet literature assume frame generalize another propose framework situation explain sampling ball independently along appendix focus difficult norm pdf k unit sphere ball derive pl distribution ball ball radius straightforwardly ph paris est la france mail paris est fr university france mail fr ph bm point france mail fr sup des communication sup com en et communication mail tn example assumption study frame become necessary characterize frame coefficient difficult general synthesis observable introduce frame hyper markov subsequently posterior experiment propose accurate hyper problem image denoise impact bayesian frame generalize sparsity sense wavelet crucial operation processing include signal transform representation domain fouri good frequency localization expense spatial localization localization wavelet tool wavelet mention basis redundant become many decade sake clarity point frame understand sense advantage frame process use frame frame synthesis operator determination frame frame conceptually frame focus gaussian restrict attention concave function provide take current development carlo instance separation brain investigate impose reconstruction assess redundant address mcmc move accord deal denoise framework contrast overcomplete representation basis hyper estimation organize brief overview hierarchical representation introduce draw digital endow vector frame adjoint synthesis whereas redundant orthonormal tight identity signal write frame fr model impose belong error nothing measure adopt assume realization characterize parametric great image denoise actually denoise domain wavelet investigate seminal description signal relate estimate estimation perform inverting deduce hyper since present need coefficient representation pdf frame pdf close convex random hyper parameter frame assumption use lead follow prior shape coefficient modelling signal laplace recovery rewrite distribution define split group hyper vector multiscale frame frame resolution belong hierarchical complete improper kind summarize cover encounter reflect prior parameter posteriori square close study sampling generating distribute accord posterior generate sample unknown sample posterior provide sampler generate distribute precisely iteratively accord straightforward accord detailed method frame norm inefficient especially design distribution union orthonormal euclidean analysis adjoint f f kk ml x sampler generation coefficient accord f nn n mh support generation pdf define close acceptance mh preferable choose ball region associate propose n b n b adjust exploration enough sampler successively see strategy consume component dimensional follow parameterization truncate achieve use mh move note sampler I accept intuitive implement point restrictive consider frame orthonormal basis assumption hold propose hyper exploit algebraic sample impossible replace move mh globally accept reject acceptance efficiency strongly depend candidate stems fact yield algebraic frame frame decompose x realization value f sample h perform x u account ff f interesting generation technique proceed generate u ff explain simulate x reason draw ff easy pdf semi rule transform due q express remain yield finally simplify hyper sampler initialize u b ff u ff ratio accept accept result application carry frame basis filter wavelet j basis stand horizontal vertical resolution accordance model form uniform hyper distribution frame suppose principle reference value square hyper parameter belong monte
iv equal partial differentiable therefore assumption together characterization subdifferential hold addition hold proximal assume active I follow subset q resemble traditional tucker optimality since necessarily subsequence converge every iv iteration n proof indice linearly space gradient linear small continuity linearly large rewrite convergent subsequence subdifferential exist subsequence equation I family ij claim finish theorem ij pass use outer semi subdifferential sum moreover active index imply tucker type condition addition either r ij ij ij ij optimality hold penalization function aic penalty approach lack variable increase method type selector subset variable enumeration optimization bundle however purpose difficult situation induce certain em chen presence finite mixture realization example chen chen generalization smoothly deviation scad fan li propose spirit convergence mm purpose cluster theoretical scad chen penalty eq define label component complete case invertible choice chen jointly optimize variable consist successively alternatively respect differentiable non differentiable option turn conditional kullback like subset successively trivially iv imply ik real http www report bic criterion algorithm plain em obtain resp plot optimal start similar plain close fact notice plain chen expectation algorithm kullback stationarity cluster show nonsmooth tucker regression differentiable require reference locally admit directional directional subdifferential singleton subdifferential relate map value I crucial subdifferential outer say locally lipschitz optimality tucker particular main fact theorem section definition section france email fr methodology penalize provable convergence likelihood relaxed penalty smooth paper alternate penalize kullback proximal extension em nonsmooth stationary lie sparse em methodology extensively study generalization wu propose attention variable I many several approach contribution penalty contribution among alternative selector attempt penalization logistic chen present non use kullback interpretation em prove nonsmooth coordinate extension component version acceleration speed apply al cluster alternate nonsmooth tucker penalize kullback point penalty present implementation regression fan li far study chen ml observe random sample parametrize em maximization density alternate parametrize consist maximizer accept current iterate maximize concave point iterative write penalty control convergence relationship algorithm discover detail analogy motivate alternate generalization sense absolutely projection distance differentiable kl st coordinate vi kullback positive strong continuously differentiable definition real number convex iv assumption notice know divergence satisfied gaussian check make assumption iteration define maximizer em order prove technical assumption important theory likelihood important establish tucker often satisfy fact regular scad assumption need simplify analysis iterate lead always onto impose ensure grow assumption tucker behave simplification requirement basic proximal
note table word semi bold detail substitution many choose word bold candidate neither ccc anti sup semi sup oracle word language consideration alternative remain strategy perform anti sup sup comparison complete great note variety possible take last word comparison anti remain two candidate anti oracle set entirely relative way comparison sup sup set contain summarize validate trend sup quality result affect censor ratio speech article select evidence distinct speech processing depend upon detection phone rate consistent limitation neither candidate ever recognize candidate approach interest requirement notice finally recognition optimize development set leverage amongst generalize expert censor likelihood likely perspective much speech utilize acoustic main appealing direction future forward acoustic ccc error anti oracle sup semi oracle acknowledge speech team speech university plan co acknowledge wolfe manuscript manuscript conventional pattern selection accomplish minimize ground mean label development costly train error automatic amenable robust minimax censor limit validate experimentally candidate use utility suggest application sign category powerful speech speech engineering community likelihood long play prominent role aid system development log ratio turn ever area likelihood many processing compete serve regression likelihood observe speech assume know datum come acquisition limit scalability sample speech engineering likelihood evaluate compete serve truth yield sound algorithmic datum selection strategy speech standard metric serve benefit unlabele construct permit algorithm derive consider label incorrect assignment influence incorrect labeling limit well technical show optimality apply semi maximal induced labeling minimize relative compete significance select close kullback divergence notion fundamentally model predict possible comprise instance acoustic traditional devise fidelity effective unseen comprise classical optimize calculate held label truth manner validation fitting goal building training testing subsequently practical assume training draw speech engineering benefit ever great amount build amount training datum whose use semi paradigm application supervise limited augmentation self involve speech system desirable even certain instance new engineering approach acoustic acquire digital amount employ unlabeled approach understand system indicate bring likelihood error automatic mean supervise label unlabeled speech select compete performance section art speech recognition detection force conclude implication improve speech processing processing manner speech recognition acoustic likelihood typically model parameter simultaneously stage maximization amount match validation adapt aside purpose recall represent continuous function density evaluate acoustic pair training proceed absence direct knowledge speech likelihood seek overall use choose amongst compete compete approach leibler distribute empirical likelihood respect sample arithmetic average amongst multi clarity exposition admit outcome represent natural describe fit follow careful expectation work potentially likelihood possess technical assume give force appropriately standardize straightforward proceed appropriate statistic asymptotic deviation evaluation root fails statistic surely directional fixing normalize evaluate select model evaluate decide favor model conclude insufficient evidence conclude distinguish compete already wish leverage model class seek fit replace ratio ratio course label decode incur error section compete model procedure suffer misclassification error reasonable marginal tend course recover label encountered true principled adapt p misclassification recover nonparametric simply actual statistical enhanced robustness error automatically limit reasonable provably misclassification automatic consequence inexact procedure distribution respective contaminated determine exist contamination incorrect amongst whenever ratio monotone enough ensure contaminate disjoint comparison include possible comparison currently machine tailor select amongst compete prototype selection processing task amongst compete two speech system differ conventional audio crucial processing speech recognition synthesis form term comprise mapping er create however expensive inconsistent individual lack broad interest approach put must word maintain speech processing system create vocabulary must extended name usage rare significantly dynamically adjust generation thereby effective automatically amongst date focus modeling letter sound previous building large phone additional speech work variation concern compete scenario word current however end consider speech condition say subsequently supervise hence oppose conventional show ccccc ax ax ax ax ax er z n n contain word force acoustic assign viterbi g cast follow word comprise q decide likelihood force alignment conventional method production difficult consume external potentially need identify speech segment instance use candidate news item rich serve correspond example occur know weather record episode give word know many word segment choose example output alignment select candidate speech recognition candidate datum acoustic evaluate entire likelihood semi analogy sequence likelihood use decide ratio indicate section consist scale speech processing force output candidate hour correspond recognize speech total recognize speech evaluate decision trade curve phone speech experiment conduct state recognition retrieval supervise selection variety g retrieval synthesis variety word place english english consideration often require selection word acoustic interest verify system contain consideration acoustic candidate consider letter sound word letter sound produce subsequent sign log ratio correspond reflect priori equally likely enforce candidate selection vocabulary build recognition train datum hour detection train word news rt hour test employ system use lattice detection task detection index refer procedure letter word purpose make either supervise
estimation equivalent add constraint problem add constraint redundant claim equivalence lagrangian function add spherical constraint optimize sphere lagrangian eq use variational formulation eigenvalue symmetric notice parametrize linear lagrangian dual weak lagrange duality use provide primal weak duality duality bind precisely value relaxation section weak duality eigenvalue relaxation quite tight nesterov problem well function admit make prove fact exist say go denote subdifferential subdifferential affine adjoint whose subdifferential prove nothing define product e know set subdifferential perhaps duality prove denote closure hull true constrain technical property prove duality apply relaxation norm consequence specialize short subdifferential differentiable must prove nice relaxation exact combinatorial relaxation optimum one help answer two recover binary problem max graph nesterov allow precise view practical favorable good average eigenvalue natural approach sdp thus eigenvector solution simplify presentation establish nesterov obtain sdp problem equivalence relate matrix last q nonconvex sdp relaxation give program definite follow subdifferential formula minimizer hand candidate initially prove give relaxation preferred argument recall get use obtain well eigenvalue equal relaxation sdp relaxation use fact eigenvector great obtain sdp find work specify obtain practice eigenvector square subdifferential subgradient obtain bundle desire lie feature bundle sufficient opt eq eq subdifferential multiply cauchy sdp desire common sdp reader solution recover quite subgradient optimum answer question although part section exactly necessary condition strong max sense lebesgue eigenvalue fix diagonal frobenius deduce subscript decomposition non say eigenvector associate differentiable eigenvector associate subgradient n strategy whose expression greatest round coordinate nearest give higher feasible round near binary sum cut objective randomized imply cholesky factorization clear prove cut exactly eigenvalue explanation take section image devote problem simply consider column penalization require sense index like quantity property index south east west main associate eigenvalue eigenvalue optimization outside diagonal prove duality hold perturb problem perturb eq image denote denoise perturb binary denoise cost page minimize notice confirm relaxation least hand eigenvalue complicate quadratic constraint experiment report noisy additive independent identically apply influence smoothing display vs recover result encourage suggested word compare letter seem posteriori consist satisfactory experiment numerous confirm interval identify study approach square k ds signature th take equal user overall filter bank match filter whose give equal approach compare extension shift symbol issue former primal greatly increase overcome semidefinite duality prove eigenvalue equally point analysis prove correlation outside strong signature research theory frame interesting viewpoint finding negativity htb signature duality indeed situation heuristic indeed compare value primal verify monte carlo axis software sdp solver possibly cost time vs user dash style sdp curve plain eigenvalue complexity reader complexity routine solve bundle software eq study polynomial property part physics sketch order easily case zero formula deduce root interval two discuss main square relaxation solution problem address sufficient allow solution nesterov binary detection image denoise strong hold signature thm thm proposition subsection france email fr survey property eigenvalue program obtain lagrangian dual implicit compact relaxation definite programming tool handle image several engineering corrupt noise address square relax recognition size bundle quadratic superiority bundle semi one appear favor eigenvalue relaxation user prefer sdp primal recovered motivation recover geometric penalize least eq parameter priori term binary vector know construct programming relaxation semi definite programming sdp play role inside signal problem nice survey lagrange general bound
estimation belief bp red dash expression bias toward look observe solve bp observe isolated one obviously nontrivial isolated ml calculate count straightforwardly derive recursion permutation b return case phenomenon split nonzero temperature one temperature nontrivial bp local dominate positive solution good matching exist constructive weakly ml configuration without generality observe nontrivial derive j v bp constraints translate u equation temperature straightforward verify nontrivial perfect matching conjecture solution bp extend solution smoothly another plausible minimum lie doubly stochastic polytope bp generic gm ls term adapt algebra expression accordance subgraph bipartite e loop even formula degree formula determinant derive evaluating observe eq ls doubly belief derive utilize version prove z ls x van minimum doubly attain conjecture open finally surprisingly short elegant allow van conjecture call van simplify van theorem non matrix respect bp transformation transform side one naturally bound van namely match without generality positivity entry sake completeness review specialize independent square hadamard consider study detailed size temperature provably nontrivial bp van deterministic provable calculus realistic course mathematic support student visit advanced study grateful national nuclear security department energy national laboratory contract na via nsf collaborative communication reference proposition theorem claim computation equivalently likely function calculus representation bethe functional doubly belief also pass propagation type z expression state bethe alternative multiplicative calculating context physics intrinsic particle solve unlikely naturally look randomize algorithmic problem approximated polynomial relative complexity impractical realistic motivate find continue belief propagation heuristic absolute bp originally code artificial intelligence state loop evaluate partition maximum gm normally expect result surprising existence ml realize bp raise question understand performance heuristic capable handle gm calculus gm bp call subgraph series doubly matrix marginal perfect describe minimum bethe energy understand first result correction recover expression pf key gap estimate pf exact technical bethe energy bp conduct evaluate bp entire ls collapse term state ls lower state corollary van lower derive original lower discuss text sufficiently dominate iv discuss transform application hadamard discuss negative j parameterize via pm complete binary interpretation allow represent perfect perfect matching temperature degenerate gm un variational kullback leibler statistics functional find condition belief understand proxy probability unity achieve minimum approximation underlie gm bp gm relaxation gibbs paragraph gm bp states perfect accord belief belief belief associate variable substituting bound absolute minimum simplify express solely satisfy
cell set consider cell contingency frequency vector denote denote I configuration denote exist contingency common sufficient element negative total frequency degree move write move primitive primitive basis correspond specify unique minimal basis coincide move reduce element distance markov reduce usual call basis connect reduce zero table reduce zero table finitely difference element large move connect investigation connectivity one table basic fact basis primitive sign side entry sign add therefore strongly reduce large connect entry condition strong frequency vector distinct move form markov existence two basis imply discuss end condition table entry zero remain one least reduce effective connect several generalization cell move condition distance reduce entry suggest connectivity table pattern b distinct one combine follow induction primitive suffice reduction check move primitive recursively reduction proposition cell remove zero last discuss weak basis basis similar weak cell move equivalent distance reduce reduce move reduce index see hold investigate common contingency long question table contingency entry express difficulty set sum statistic extensively study practically evaluate develop e sample practice case easily show exact need connect every zero row sum show connect table row sum goodness model imply move show three independence htbp table complete q move independence degenerate define move complete classify I rr rr I I move move require move connect see square free move even seem difficult ti practical test limit connect implement zero include social statistic move quasi denote cell zero table social quasi implement sequential two way way structural provide independence way contingency complete description basis quasi loop section df support df exactly element array df loop basic df contingency show connect zero df loop connect every table quasi square diagonal move df result table incidence matrix square symbol row symbol square axis array square sum square give may minimal connect case first table move interaction need degree move form interaction line slice loop slice loop degree generality slice loop pattern move contradict move see move move prove move degree connectivity program check basic degree hence require connect chapter connect generate array representation view move move square basic move level correspond degree move connect table general find basis table basis contingency result contingency table entry zero zero particular contingency table enough proposition suggest basis table basis behave cell consider challenge author anonymous constructive true mm mm section section discuss connect entry correspond minimal markov basis since basis tend interest particular minimal connect table table minimal table one table basis methodology test exponential family sharing call markov basis markov tend scale researcher interested subset markov restriction count one case interpret logistic logit trial covariate one element frequency occurrence non occurrence record
select open specific spectral already grow improvement translate directly compression size offer improvement wolfe latter approach practitioner dependent offer liu rigorously date bound quantitative guarantee quantitative nystr approximation appropriately definite symmetric definite kernel spectral coordinate matrix sort non approximate kernel follow notion unitary invariance unitary invariance term matrix depend singular coordinate evaluate quality task ordering however cost costly method rely dimension dimension impose computational burden illustrate former category datum via nystr om modern preserve nystr om cardinality submatrix partition follow positive principal rectangular dimension eigenvector eigenvalue typically complexity reconstruction nystr om serve complete row cardinality norm lower small eigenvalue general complement correspondingly general definition selection cardinality minimize remain open whether threshold spectral force enumeration divide minimize liu zhang quality kernel typically wolfe sampling currently stochastic determinant detail nystr om adopt norm trace norm arbitrary denote singular semi definite decomposition frobenius notion function e trace amongst invariant norm adopt minimax unitary result end definite om complement per obtain trace nystr om error norm complement norm induce nystr om approximation express denote index select om accord complement always definite norm subset term latter zhang trace regression partition accordance om obtain project admit decomposition loss generality choose therefore diagonal nystr om square spanned column characterization provide comparison semi fix exponent distribution principal determinant computation amount course mass concentrated practitioner et al al trivially recover associated wolfe error uniform sampling nystr om tight average contrast principal eigenvalue incur fail place wolfe kernel subset via om completion require reconstruction nystr om suppose nystr rank admit determinant rank square imply determinant know q selection gaussian minimax dimensional upon correspond represent first covariance maximize entropy maximal relative upon extend follow maximization nystr om j result wolfe bounding case improve let om extension large eigenvalue tx x al possible note present combinatorial wolfe easily cover approximation computation light selection draw field computer manifold stream particularly aspect impact efficacy diverse segmentation et al spectral et appearance manifold lee spectral world require indeed aforementioned task share feature approximation select serve video database lee al video dataset often dynamical evolve line rotation extract low object recognition appearance manifold lee recognition pose lee ingredient video stream obtain definite number frame video stream prohibitive entail begin test efficacy couple procedure exact embed dimension diffusion al database lee head front motion position straight camera circular low front right normalize nystr om video selection overall average map average trial monte carlo wolfe determinant subset determinant choice sampling determinant range observe maximal analysis tend concentrate front yield locally region properly avoid subsequent collect video data author slowly front camera embed diffusion straight evaluate visual quality display recover uniformly curve centre show trial yield good sampling uniformly analysis practical implication subsample determinant sampling centre uniform map embed video movement speed pixel recover almost scheme lose curve fold material base project grant national health grant national science grant dms perform mathematical science high dimensional average error nystr om reconstruction map sampling consistently outperform kernel nystr om extension year spectral appropriately define extract dimensional currently practitioner kernel follow spectral analysis term nystr provide algorithmic performance optimize selection implication bound real world draw vision whereby low video year capability development decade new treat structure cluster laplacian different computation principal eigenvalue semi definite spectral long hold central matrix variety principal subspace discriminant determine separate data embedding kernel scale cube wolfe accordingly pose severe modern construct kernel partial analyse select summary burden enable practitioner apply aforementioned original solely subsequently upon adaptive manner begin review dimension formally clear conclude demonstrate statistical landscape indeed pca introduce enjoy cost process normalize transition step simply eigenvector eigenvector correspond unity eigenvector eigenvalue nonlinear embedding embed show overlap colour indicate panel respectively obtain map
extension mechanic work general definition contribution cluster anneal experimentally optimize well sa also sa implement carlo sampler sampler mcmc thus approximate st derive tractable sampler look interaction annealing find optima continue stronger close pick st quantum work globally suppose equal fig local interaction search assignment quantum metric interaction chance go close finds often cluster majority cluster assignment pick locate st one th indicator denote assignment denote indicator length assignment product matrix b point cluster assign second matrix assignment store use anneal like sa th aa c optimum optimum briefly anneal give energy search next function sa almost choose high hill function summarize inverse sa update energy many intractable sample focus mcmc draw denominator tractable quantum anneal fold form section schedule mechanic equation hamiltonian physics example matrix exponential e e equal calculate easy equal sample equation effect matrix one depend work try couple explain bad later sa mcmc method mcmc intractable evaluate unlike formula product product follow completely different similarity finite show exact pass anneal schedule dominate anneal schedule residual annealing continue effect different anneal temperature assignment increase inverse inverse anneal address propose observation pilot pilot observe st suboptimal assignments st assignment far optima necessarily well compare become become close strong begin strong interaction close search middle anneal step eq th sampling f difficulty sa choose anneal schedule schedule next choose anneal st reduce schedule sa mnist lda b sa sa fig three schedule st sa schedule show st sa schedule improve sa gaussians inverse wishart latent model sa posteriori corpus point apply reduce corpus word vocabulary word vocabulary corpus st st sa st keep st similar sa term anneal vary become st st fig plot st sa interested st achieve sa column mean result schedule schedule sa third st still work experimental consistent work lda lda optima show st achieve sa slow schedule st still sa accelerate study permutation augment minima techniques exchange carlo fit cluster schedule simulate sa randomly sa run sa actually sa time fast necessarily sa thus experimentally try develop algorithm network quantum look like genetic multiple interesting acknowledgement research physics new material grant super
alternative conventional probabilistic incorporate center realize drop interpretation replace amplitude quantum mechanic associate center find quantum state point operator gaussian wave trace represent minima quantum begin focus attention wave construction expectation coordinate dynamical state mass moving choose evolve evolution accord expectation wave towards potential relation minima trajectory clearly locate near local come move apart dynamic visually trace associate one successfully move seem replace conceptually implement gradient solve complicated difficulty solution simplify considerably allow translate form capture less advantage formula analytic evaluate thus multiplication simultaneously multi processor time produce linearly display introduce employ introduce minima also connect nearby degenerate minima reduce calculation final worth wave strategy handle set discuss work brief outline assume datum associate wave center coordinate scalar product matrix gaussian orthonormal basis solution desire stop obvious restrict feature derive analytic operator computation experience difficulty apply text book five belong reader example wish simplest capture essential reasonably reduction unitary occur diagonal entry consist call component think assign five full within principal apply consist pc compose three use five quantum color place quantum potential good job capture cluster roll near produce separation three first column guarantee normalize unity conventional project onto sphere study temporal henceforth show unit sphere quantum visually separation colored class however incorporate unsupervised begin specie red green fairly problematic middle plot quantum stop cluster occur quantum evolution enhance separate accomplish e point band difficult tight toward cluster arrive minima pattern stop evolution configuration evolution hand happen end matter quite evident agree color group together color blind full proceed lead insight dynamic point lie due assignment true proximity difference extreme phenotype absence discriminative important conceptual message classification message include geometry measure euclidean influence analysis may replace manner define dynamic point euclidean clearly geometric distance reduce investigate evolve semi close example intermediate point evolve thus interesting exist reason scientific large dealing point problem pc intensive simply lie map cube hilbert tuple way svd filter simple modification dataset eigenvalue define entropy define quantity remove filter technique remove raw datum pc represented follow apply evolution latter cluster blue filter blue separate cluster separate point identify stage iterate stage however begin svd base filter trying enhance red start blue distinguish blue remove higher important biological clustering reduce go six feature begin among eliminate responsible separation track robust repeat filter blue sort green fact figure remove accord svd happen remove comparison time svd entropy five filtering separate complete svd filter stage cluster accomplished evolution distinct merge evolution certainly say reality cell look cluster give fp stand correspondingly high summary dynamical explore start show dynamical visual state derive analytic calculation temporal evolution treat quite put system sure produce gaussians associate center evolve wave gain full wave expand dynamic notion particularly core manifold methodology potential minima svd value reduction dimensional handle difficulty computational point define associate related computing multiplying give operator clearly number feature potential small feature construct experience stage evolution occur structure see plot readily construct use reduce addition employ contribution nature especially evolution note contain advantage carry see reduction assign datum remarkably explain unnecessary allow set contain large number turn hilbert point naturally hilbert display employed datum wish usually learn e appropriate stage dimensional filter lie visually subsequent datum accomplish point appropriate construct include system intermediate give full fact old colored accord original possible set plus datum happen original fail properly feature filtering change sort identification existence identify cluster quantum potential hamiltonian already evolve guarantee cluster accord cluster conventional quantum wave dirac employ operator relation hermitian operator identity calculate order meaning prove easily construct gaussian center wave coherent gaussian gaussian matrix orthogonal shift derive appendix q I generalize derivation expand point identity speed series original include computation purpose need eq behind remain mechanic evolution hamiltonian shift computed hamiltonian computing truncate operator matrix set find away compute whose exponential operator simply eigenvalue one basis compute q put construct pt cluster stanford stanford usa school university whose determine use hamiltonian calculation wave around original allow exploration dynamical point convergence formalism decomposition ill define nonetheless give point looks sort sense together investigate
represent may generalization corollary bayes appear risk unbiased priori may flexible shrinkage unknown notational eq operator adjust naturally unbiased risk enable haar haar wavelet shrinkage neither haar direct application haar soft iy rather haar empirical via coefficient simple effective empirical estimating rule substitute bayesian matrix poisson intensity estimate risk unimodal estimator available imply ix coefficient variance solve numerically yield consider early simulation overall performance efficacy consider bayes numerically laplacian soft laplacian linear ss soft thresholding adjust shrinkage haar moment shrinkage rule relative fig mse inference min std min max std median std reconstruction test bit image frequently literature house interest level reconstruction report snr compete conjunction implementation baseline comprise haar wavelet priori amongst propose offer alternative term quality visual appropriately locally yield dark fig importance incorporate coefficient process variance completely resolve smoothed texture entirely lose exception typically estimator widely error db readily confirm compete remain competitive great adjust snr save intensity way near frequentist haar along low wavelet confirm offer appealing method literature indeed haar yield exact along substantial application amongst haar gain presence additive proposition remark material base national foundation grant publish th imaging science international conference obtain independently et international imaging information laboratory usa mail integrating imply variety world model poisson turn proportional underlie article arise haar measurement type difference near provide unbiased frequentist new haar unbiased risk complexity demonstrate efficacy shrinkage estimator univariate wavelet test image substantial class device subject loss resolution e quantization inherent degradation prominent variety digital communication image popularity diverse wavelet transform spectral scenario readily transform measurement range effect across transform instance accumulate element detector model poisson random density ii signal sensor grow linearly strength imply inefficient detector back summary seek invertible typically compressive nonlinearity familiar white estimate directly poisson leverage independence strength upon maximum approach dependency haar describe well repeat directly haar date wavelet describe domain mean canonical tail prior analytical practical variability variant stein parametric estimator transform discussion subspace axiom require family kk moreover scale wavelet form wavelet family together expand time q scaling term analogous transform haar take scale mirror turn recursive canonical type frequency digital processing decompose component wavelet band recursive decomposition hadamard haar requirement transform make attractive haar orthogonality computational simplicity haar wavelet transform axiom analysis serve admit efficient sequel coefficient aggregated scale notational finite subscript generic turning transform domain estimate stein sure differentiable risk unbiased transform nonlinear thresholding example poisson denoise straightforward case haar wavelet close form difference poisson count end unnormalized haar transform wavelet element observe haar empirical coefficient effectively poisson coefficient characterize generate fix difference poisson verification summation observation taylor q poisson clarity variate limit difference integer hand skewness skewness random variable proportional poisson rate proportion distribution proportional fig text skewness fig haar transform depend similarly haar wavelet let definition haar transform empirical consequence eq verify entry leverage haar wavelet intensity conclusion variability variability haar haar first toward achieve turn univariate mean frequentist generic scalar quantity haar coefficient haar coefficient latent coefficient directly estimation assumption plug estimator haar scale constitute poisson context signal ratio argument moreover admit recurrence probabilistic derivation begin derivative derivative admit comprise operator act linearity similarly derivative property slope curvature identity imply likelihood recursion admit recurrence fix eq calculation likelihood initialize combine equation linear equation consider underlie prior distribution area range generalize mixture attempt estimation univariate however determination parameter infeasible end approximate obtain closed shrinkage random bayes score approximation equation expectation schwarz latter term control condition bind equivalence analogous derivative go average high shrinkage rule prior expectation formulation amenable former tail go derivative generalize parameter gamma unimodal expression admit truncate laplace prior px px
robustness direct consequence assumption central application central definition law consistently one since proof acknowledgment thank english responsible mistake lemma axiom exercise package interval central team laboratory france france abstract describe package implement confidence central mean context test express robustness normality keyword parametric test interval central package test chi variance however use even fail chi hypothesis level sensitivity normality may implement sample variance well thing much study test direct remark valid ratio variance user tool different community basic e wish hypothesis limit call already find chi square testing normality contrary test g procedure implemented available package etc never paper unified sample mean variance derivation test package normality large alternative fisher purpose term present test framework test unify present give concept explanation chi finally general derive asymptotic variance package notably finally section devote discussion proof sample identically version denote compare gaussian framework resp resp mean resp resp quantity variable chi ratio statistic measure gaussian normality theoretical explain section q obtain differ variance gap framework govern indeed asymptotic assess large c widely confidence thank limit express term central theorem parameter limit law definition standard nd alternative illustrate measurement specie frame tt length width normality width specie frame specie mean specie var var mean width specie par ref test width specie alternative less percent interval inf sample width frame tt n width specie difference width specie e percent inf difference width specie frame tt specie width test species inf ratio difference frame number width specie e alternative weighted percent inf sample simulation reliability sample perform datum chi test versus alternative chi versus false fisher asymptotic type error worse suit asymptotic direct summarize introduction order illustrate output resp sample empirical propose seem fit chi apparent normality may prefer tt par l ref statistic variance percent inf variance decision inferior case kind test tt leave variance ratio percent apparent prefer frame number two hypothesis interval difference variance think might test various write
possibility latent consider approximate simulate converge sequence therein amount everywhere right hand side uniquely depend version simulation joint converge approximation stress artificial sample convergent estimator biased availability ratio computational first alternative require possibility representation bridge formalism approximate factor sample z follow converge eq gibbs distribution normal compute estimator compare variation approximation mle distribution bridge complete mle replicate simulation sampler comparable section address http www fr evaluate impact diabetes occurrence diabetes probit mr acknowledgements grateful discussion comment team improve exposition author point bridge sampling university support de la paris project big mr framework know hypothesis allow demonstrate fundamentally rely theoretic version density involve impose mathematically completely measure bridge approach versus bayesian core choice indeed marginal mathematically uniquely exist differ literature chen improvement numerical precision bayes relate complex method representation nuisance decompose plug obvious notation marginal reduce justification illustrate representation artificial computing hypothesis embed model since representation address aspect give rarely face challenge deep nature consider consider uniquely alternative hypothesis posterior though second section difficulty examine axiom state mathematical difficulty proceed identity exploit early probit measure rigorously probability q measurable property conditional distribution test advance completely manner reason theory satisfied conversely assumption prior contrary state mathematically artificial arbitrary agreement instead disagreement valid specific version density derivation depend last rely namely rigorous availability already test prior experiment thus modify completely rigorous representation hold density verify stress mathematically prior choice mean choice establish approximation bayes obviously
continuity moment v k co closed complement co co relative assume surely therefore moment computable joint computable topology right random case moment computable moment give work regardless location consider randomized number computable absolutely everywhere real denote generate open continuity compute measure extension uniformly relative k measure multiplication convergence c moment aa rational interval whose equal thus mixed ia de relative immediately lemma computable relative joint determine moreover respect ij aa k note continuity union intersection ij supremum computable almost surely surely relative expectation respect relative make complete proof explicitly construction algorithm alternatively sketch computable theory prove computable could densely interval continuity set metric approach complete probability boundary less boundary set abuse define c ik ik arbitrarily polynomial pointwise polynomial define n n cv cn real co final follow order product call ready computable exchangeable computable de give show computable proof ij v cv uniformly supremum continuity range cv preserve dimension open topology eq recall polynomial v k p p continuity p c c cv cv cv cv continuity v cv q v continuity continuity low note dominate convergence cv q real supremum c uniformly implication semantic programming language eliminate code automatically context background functional language code transformation computable early partial exchangeability recent application programming language choice operator language researcher number language operator semantic program theoretical science randomize checking area somewhat different language type inductive evidence think infer program datum implement programming al describe extend mit perform execution monte think walk execution describe language implement approximate sampling describe calculus expectation g tree naturally helpful work express nonparametric model produce program entropy beta bernoulli describe language de exchangeable computable furthermore representation automatically make computable probabilistic language extend scheme binary value return false move definition call bernoulli semantics associate evaluation expression binomial evaluation procedure behave denote dirac concentrate real equal probability always modify use via non local may implement thereby mutation manner associate probability expression could keep counter variable repeat translate mutation mutation perform transformation throughout program represent pass operation original counter accept return transformation particular remove require rest section concrete mathematical beta bernoulli describe recall directly bernoulli possible mutation require keep track value instead introduce operator track black ball compactly scheme description beta fix beta coin coin code code environment create argument environment coin coin argument return procedure coin coin coin coin repeat beta coin coin otherwise coin flip sample sequence ten return hide beta coin coin beta coin beta coin weight independent application coin ball ball return application coin exchangeable therefore draw know beta coin bernoulli informally think beta coin bernoulli process although coin important coin coin coin change coin non sequence execution computable sequence measure transform mutation return distribution general program generate exchangeable computable produce mutation procedure random randomness transform code sequence produce procedure return coin semantic mutation reason example overhead necessary exploit independence exchangeability probabilistic language execution sequence exchangeable chinese turn distribution sequence exchangeable process stick break produce certainly form exchangeable object combinatorial write analogous p restaurant dirichlet detail result encoding subset countable support discrete measure computable complicated breaking process depend arise evy method construct process de random extension suffice type exchangeability variable exchangeable permutation nearly year show array separate exchangeability conditionally adjacency exchangeable beta bernoulli g inherent parallelism computable de exchangeable could representation wide range include increasingly exchangeable block process explicit nsf dms fellowship conference abstract would thank extended present comment computer artificial intelligence laboratory institute technology usa exchangeable exchangeable language automatically modify local computable de theorem computable language mutation classical theorem sequence almost exchangeable sequence computable de distribution computable measure computable readily de term exchangeable along prove computable independent interest proof moment random moment computable suffice compute computation general survey representation computable real automate domain study semantic programming language survey statistical probability measure coincide generate computer numerically science machine recursion probabilistic language directly relevant computer exchangeable use communication access modify program indirect classical description conditionally thereby implementation classical finding moreover construct desire describe would perform local induce beyond language semantic desirable term de measure example machine computable assume theoretic formulation theory let nonnegative logic basic borel bx value index real conditionally b bx measure expectation side characterization perspective interpretation exchangeable arise latent implication de prove mixture repeatedly coin draw extend value exchangeable hausdorff give conditionally history development book comprehensive role measure borel dirac uniform almost surely sequence example dirac dirac marginally distribute compose marginal give x n verify kolmogorov marginal clearly process exchangeable sequential turn q repeat ball ball return additional variable independent x n finally de measure measure notion precise programming language ask computable exchangeable sequence computable de computable exchangeable sequence converse computable exchangeable exchangeable computable computable limit computable rate limit able reconstruct measure borel open denote rational value borel uniquely place moment variable I de marginals exchangeable open lattice open note place boundary I rational lower property imply e moment suffice characterization computable sequence order computable topological enumeration set topological discrete topology enumeration precisely consider computable representation topology effective enumeration straightforward let enumeration function km tn computable recover definition computable measure topological unit interval admissible borel measure equivalence computable program randomness name topological algebra finite intersection second countable space topological computable closure intersection enumeration derive set enumeration fashion enumeration enumeration space measure computable note computable uniformly index co open co e close singleton space strong open strong arbitrary computable measure concrete notion topology computable sequence computable topological characterize sequence real enumeration simple computable side precisely corollary one characterization embed measure play distribution joint computable product right ik ij one note standard characterization integral respect topology topology variable
far construct w hull first svm convex reduce construct formally enable optimality solution value large writing vector let horizontal word formally boundedness finally know define choose choice polytope ray illustrated prove indicate three em change go discrete make complexity guarantee exponentially none hull attain equality solution program many subset solution general result program ignore svm solve program track path bad path solution prescribe quality constant bad path project pa work anonymous comment suggestion discussion proposition france cl france variety machine entire develop method support path assume exponential instance svm parameter become tool biology vision regularization contain special regularization tradeoff term tradeoff generalization performance programming extensively algorithm whole varie function prove complexity svms program worst exponentially distinct occur regularization change exponentially valid number space eq describe vary always semi gram include regression multiple square angle lar also pursuit denoise compressed parametric program control predictive geometry moving point mean portfolio variate solve parameter majority prominent g svm probably example path parameter piece path turn alternatively number change svm distinct number show svm exponential number path linear avoid confusion construction report exponential exponentially many exist conceptually construction motivated finding instance imply probably parameterize restrict svms exponential geometric path relate homotopy long particular particular entire parameterize compute exact solution machine technique similar program solution path svm give lasso multi svms svm point separate regularization path interesting quadratic path mention usually require invertible always later point svm indeed already arbitrary matrix invertible matrix recently g instead optimization would cube support simplex cube objective parameter compute method maintain linear vertex exist program variable many solution vary contribution adapt overcome parameterization regularization class svm different secondly continuous path solution parameter carefully transform objective turn geometry svm svm parameterize eq point solution appear coefficient distance reduce point note discover equivalence note slightly commonly svm variant geometric distance regularization parameter geometric explain path svm convex role first construct two plane class regularization class svm path piecewise linear path roughly align circle align class vertex depict polytope formulation end walk nearly boundary face path inner claim formal main guide surprisingly lower contain class parameter dimensional plane crucial construction hull vertex moreover walk fraction depict figure tool slightly cube vertex visible cube intersect plane cube keep let fact way hull polytope interior suffice face full vertex hull imply strong every statement equivalent use actually cube variant cube inequality standard cube transformation cube program rule property polytope interior hold inequality intersection exactly vertex index set vertex denote vertex v differ expression thus property cube hence variant cube coordinate dimensional precisely vertex tell cube appear polytope project interior transformation map strictly satisfy call hull set polytope finitely inequality boundedness transform interesting polytope origin interior polytope vertex define simple consider dual cube dual cube therefore perturb version polytope see outline intersect already property polytope cube summing mean cube geometric duality one correspondence q accord lie slightly plane ideally cube sure walk intersect accord achieve except omit version define face define point behind transform enough onto key define fix sufficiently exponentially property unique cube ray em dl define choose cube vertex inequality onto vertical chosen show eq eq see also projection item outline begin remain statement eq hence prove second part write problem minimize quadratic equivalent optimality pair satisfied pair prove unique relaxed dropping tucker relaxed problem determine solution observe tucker complementary turn hence actually easy
notice aggregate good performance stage diagnosis combination aa bad even choose particular aggregate mix stage coefficient aa result amount allow carry follow calculate approach peak intensity carry violate unlikely calculate nan nan calculate describe apply calculate event triplet categorical aa triplet aa value calculate state diagnosis trial assign label e present include value include frequent choose combination peak algorithm choose peak peak manner good time period error index loss algorithm value good number error contain correspondingly l l l l practice often choose suitable level advance time peak combination aa prediction disease predict suffer less intensity another investigate stage diagnosis aggregate mix predict check peak give time diagnosis get value reject statistic never bad deal cancer assumption direction future consider disease patient put triplet like thank discussion fp grant method application diagnosis development machine complex grant ep learning management optical network apply method level conjunction mass collect period year gives convert strict make peak contain diagnosis previously set reliable cancer appear stage disease lead difficulty patient improve diagnosis reliable early stage investigate quality detection demonstrate contain diagnosis stage peak intensity form set case choose age storage condition triplet sample individual cancer information peak precise reliable diagnosis use rule process combine prediction decision rule algorithm triplet convert label weight label normalize reliable prediction vast majority thorough experiment perform peak information stage choose organize describe data work description stage diagnosis section predict step make suffer goal prediction combine expert close impose loss probability measure loss function probability concentrate game repeatedly expert protocol learner reality cumulative lot probably weight aggregate aggregate behind algorithm assign correspondence expert adapt outcome strong aggregate algorithm exclude case lack triplet control sample sample individual triplet diagnosis triplet calculate triplet outcome represent apply construct predictor patient peak expert cancer diagnosis purpose sort date measurement sample person triplet choose theoretically aggregate evolution expert minus triplet clearly line expert aa expert line high bad axis present triplet aggregate triplet aa expert group cluster graph separate mistake stage aa prediction moreover loss aggregate loss say fair expert strict flexible find expert convert triplet present way strict aa prediction convert calculate see aa
framework analysis inequality analyze family template problem recent particular simple mistake batch multi ii categorization norm interpret decide multi perceptron previously similarly task complexity issue bound simplify previous recently framework previously task furthermore previous learning discuss convexity growing study organized class categorization online pca lasso e significantly derivation improve adequate tool context learn technique e set generative specifie task correctly identify relevant contrast focus agnostic understanding sample identify discuss convexity smoothness strong attribute relatively recently machine derive online generalization batch duality convexity strong derive online along regret algorithms case directly describe related involve bound detailed survey application derive duality complexity rademacher duality characterization immediate corollary need strong strong second different literature banach seek understand concentration smooth say recently derive concentration matrix compose rest organize section general present strong inequality show rademacher draw attention strongly strongly convex strongly number corollary recent next systematically adequate prior I turn applicability approach categorization section naturally unified enable simplify understand background new believe presentation notion reader familiar object short f euclidean equip fx deal space nm increase arrange f f infinite restrict proper define smooth state strong convexity property close smooth close strongly strongly dual proper important setting always domain smooth finite everywhere differentiable p smoothness find reverse implication easily people direct prove generalization convex denote st nd smoothness online round player prediction rule mapping assess risk generalization w excess online convex optimization bind family strongly lipschitz r dual enjoy regret plug rearrange tw tu tw w receive let I I know lipschitz bound consist predictor strong argument corollary proof essentially original highlight importance fw n get side last become divide throughout combine contraction w fw satisfies expectation choice easy bind expectation recall strong norm fw mainly respect slightly mean function shall set slightly become fw family strongly counterpart define n q corollary dual weak easy calculus group absolutely suppose smooth condition norm convexity duality combine corollary bind return learning corollary easy derive nx exist online w nx form problem online subset dimension learn argument absolute hinge batch set online encode particular shall eq analogously section bound simplicity ignore class property well know ignore organized refer usual suppose distribution still inequality course question dense bound away g k guess small tackle apply vector column vector us column column close dense small preferable class course problem might dense sparse similarly sense imply sparsity zero dense group sparsity use dense employ difference instead consider consider demonstrate general methodology order bound solve prediction example define mix different task heterogeneous multi task learn recent natural regularizer task together similarity consideration common regularizer comparison class consideration lie linear norm regularizer comparison describe learner use predict j dl j j also w x regret implication bound represent assume well scenario happen reasonably predict value let group roughly evenly evenly column well adequate prior belief similarly maximal singular assume spread draw k rademacher bind strong rademacher argument k tw line proceeding proof theorem k tw group norm row corollary corresponding excess risk table multi categorization online round receive predict number prediction zero verify r zero note two fact let w let learn bound give c implication happen inferior well space previously suggest observe class share much share demonstrate expert expert predict label e represent expert ignore logarithmic term utilize derive perceptron discussion multi perceptron perceptron class categorization online mirror procedure conservative update ignore round mistake recall di dt receive ty ty mistake mistake bound conclude mistake make sequence bound briefly family kernel consider k unconstrained class class margin jk class j lasso mild kernel large kernel discuss upon logarithmic inferior bind rademacher however result generalization bound devote dedicated effort candidate proof derive strong duality helpful discussion key excellent reference function vector equip inner fx deal dual property dual conjugate conjugate f subdifferential fx vector l n product inner singular eigen value von equality orthogonal g gx permutation define start ng allow immediately conjugate absolutely symmetric singular eigenvalue entirely fan inequality instead norm lf n ng corollary absolutely convex norm ng another allow subdifferential ng prove case
clearly mathematical iteration dependent predict paper thresholding inspire message pass algorithm associate encode relevant measurement rule mp lp paper albeit mp thresholding estimate significantly easy exclude eqn depend weakly dependence careful analysis lead correction correction capture hand side eqn amp statistical reaction reconstruction iteration curve success amp se describe property properly version mp powerful reconstruction conduct extensive amp mp intensive detail complementary show model map accord se mse recursion although map lead threshold notice sparsity function concave indeed claim derivative omit remark due vanish combination concave concave sufficient explicit please information soft define optimal se optimal se analogous minimax constrained support nonlinearity amp se formalism mse hope design monotone amp inaccurate offer room nonlinearity support exploit eq offer essentially se phase transition experiment little threshold sensitive detailed support ensemble exhibit phase often large apply operator example fourier find case acknowledgement award dms nsf theorem compress certain accurately reconstruct exploit characteristic achieve reconstruct iterative alternative optimization algorithm substantially bad convex modification thresholding make iterative agree calculation formalism derive tradeoff agreement formalism sense refer body traditional make content sampling apply formula scheme use elegant promising recover signal expensive reconstruction scheme image acquire involve thousand million despite advance lp dramatically would achieve sense lp reconstruction dramatically fast iteratively threshold transpose finally iterative popular researcher year focus scheme scheme per attack application lp solver attack fall sparsity reconstruction iterative scheme base ta z include model numerical carlo amp lp sparsity limit phase mp reconstruct success failure lp partitions sparsity identical reconstruction region strong formalism accurately predict dynamical numerous formalism square variable change formalism predict amp recover mse sparsity ratio develop find coincide precisely ie fail amp within statistical precision fast perform programming base random simulation formalism remarkably success phase lp give boundary principle entry measurement normal canonical nonlinear linear reconstruct nonzero cast entry despite condition procedure perfectly recover take tradeoff sparsity limit control fraction domain phase reconstruction formally whose domain two region reconstruction tend zero exponentially succeed exponentially definition green analytical prediction geometry line datum amp random dash present estimate percentile percentile iteration obey reconstruction green ordering say sparse sparse curve behave surprisingly amount sparsity green really help quite agree respectively principal paper combinatorial geometry need modern problem image demand reconstruction thousand million practical system describe behavior convex run slowly minute hour require operation operator vector number really rather section inspire use extremely stop iteration scheme depend iteration paper j empirical sign vector drop depend mean mse answer orthogonal stop invertible correctly really sketch truly understand course immediately behave kind random suppose accurately model entry version sparse recover noisy heavily understand error sparsity consequently level accurate valuable digital communication interpretation channel interference interference coordinate weakly thresholding interference detect many zero remaining cause reconstruct weakly interact fraction significantly reduce interference actual behavior sample reconstruction track iteration fix less stable point unstable formal mse form independent random soft thresholding sparse u sort familiar soft supplement recursive system stay
sum degree polynomial independent hold multilinear polynomial use notion straightforwardly structural restriction degree critical al index multilinear polynomial regular multilinear px x x multilinear degree kp lx structural lemma variable qx qx qx qx use show use large apply get degree apply regular kp lemma suppose kp flip sign suitably constant definition lp combine markov bad l lx lx sx bind anti polynomial multilinear polynomial et sensitivity degree polynomial degree anti qx pz last anti apply eq setting tail early error claim work multilinear degree polynomial I ix x l ix note value fx fy fx eq sensitivity boolean hypercube bound sensitivity boolean immediately sensitivity boolean nf observe n eq sensitivity degree q ig ig induction cauchy schwarz z f ix I term expression average vanish notation x ix constant go acknowledgment thank early version appendix polynomial denote sequence cardinality derivative calculate eq derivative since depend without multivariate polynomial basis polynomial sx call coefficient polynomial work p claim recall exist since calculate assume loss bivariate degree expand suffice constant sx sx constant theorem corollary question cs edu give first nontrivial sensitivity sensitivity boolean equal total sensitivity combinatorial case sensitivity application new resolve result al due iii structural theorem structural interest template transform problem threshold hypercube polynomial let real say boolean play computational circuit quantum interesting property spectra sensitivity characterize average sensitivity conjecture nontrivial bound average sensitivity sensitivity average sensitivity sensitivity fundamental arise boolean roughly speak sensitivity randomly input sensitivity definition average application hardness circuit theory social quantum focus sensitivity boolean function agnostic hypercube also efficient et learning begin boolean sensitivity boolean boolean boolean hypercube perturbation measure change order analyze sensitivity noise similarly let univariate boolean gaussian define boolean degree gaussian boolean handle boolean boolean degree gaussian noise multilinear ellipsoid hold total sensitivity random hypercube sum influence sensitivity function clearly know sensitivity monotone tight sensitivity progress upper use sensitivity boolean give elementary argument show sensitivity degree elsewhere depend coordinate would important ingredient sensitivity bound structural restriction result paradigm play fundamental theory least restriction restrict regular influence bias motivated restriction degree reasonably generic one regular precisely reason author generator nearly outline obtain bound sensitivity move sensitivity lemma al sensitivity argument sensitivity regard theorem principle majority proof majority main anti concentration bound ball probability bound change either perturb slightly noise involve large second easily distribute opposed hypercube anti concentration boolean invariance invariance polynomial e sensitivity result structural biased former resort noise sensitivity latter merely bound extend issue exist variable restriction restrict polynomial polynomial biased resort bind merely sensitivity easily turn broad also polynomial anti invariance boolean sensitivity sensitivity boolean boolean however bound sensitivity challenge agnostic follow error gx mapping marginal cn sensitivity polynomial degree sensitivity replace multivariate existence function imply agnostic precise relationship sensitivity essentially follow bound noise sensitivity concept class appropriate use sensitivity hypercube concept class degree learnable respect concept learnable respect first hypercube degree learnable spherical gaussian sensitivity spherical distribution open implicit boolean trivial broad continuous believe obtain bound sensitivity
post burn sample trace knot convergence appear around show posterior modal single around parameter estimate finally continuity pose discussion posterior knot modelling level daily theory upon approximately follow generalised pareto dependent determine daily record international almost previously figure model shape point model crucial making prediction concern event reversible merge model mix inference curve specifically day variation temporal fluctuation expect scale simultaneously express coefficient adjacent year curve impose indexing clarity unable analytically integrate mcmc update generalised optimisation posteriori sampler effort factor prior specification spaced interval display plot pointwise level year year return conversely change tail fit pareto density variation year return correspond indicate early article focus approach allow via metropolis adopt sophisticated extend method depend interval knot accurate particularly instance map find example overlap interval interval reversible jump gibbs relax need coefficient detection converted point complex implementation jump involve split birth death move case analyse reversible sampler handle complex non standard found implement software efficiently material program please relevant david discussion research scheme dp de universit france school mathematics method regression unknown allow non provide sampling make material include computing keyword gibbs markov carlo splines article curve independent curve wish interval article powerful curve function represent location spline well curve knot coefficient exposition regression model spline introduce number subset select potential knot bayesian g carry require sampler curve fit straightforward easy knot reversible avoid square mcmc nevertheless define candidate knot limitation sort distinct knot recommend place sort cubic spline clearly problematic spaced treatment conjugate reversible jump run number article auxiliary variable introduce context knot knot lie expect curve knot give general modelling carry extend auxiliary gaussian auxiliary variable beneficial conclude discussion adopt indicator unknown absence range adopt denote product value give denote prior parameter prior relate prior conjugacy integrate correspond default work recommend range uninformative lead posterior chapter knot use knot allow quantity configuration equal probability gaussian easily posterior distribution sampler successive update involve two add swap propose involve swap two exchange move accept usual hasting uniform posterior metropolis sampler circumstance produce draw conditional commonly use use average configuration value mcmc carry discuss selection cubic sample add rescaled unit gaussian evaluate grid compare use mean difference method order mse sampler perform mainly attribute allow selection finding estimate difficult visit time quickly relatively short component randomly secondly quick since metropolis hasting jump regard mcmc long particularly find result suggest issue htb example map x ex ex ex l l ex consider denote nuisance methodology employ integrate step update correspond propose parameter precisely update update add swap otherwise chain update accept metropolis remain step facilitate note application computational mle estimate maximum posteriori posterior case recommend acceptance probability greatly poor choice delay accept reject decision acceptance beneficial move make distribution facilitate mode refer mixing become poisson sample fit consecutive limit avoid numerical sometimes calculation deviation run perform mix use update map differ mle plug smooth updating slightly expense include hasting exist alternative propose importance
section user adversary adversary user belief adversary play role probability define admissible posterior schwarz norm appendix follow since w form possibility user collect term people user rule kx decide set adversary belief adversary user world form procedure dirichlet distribution estimate dirichlet multinomial concrete observation multinomial sequence observation c function parameter dirichlet parameter use follow htb propose synthetic multinomial estimate via compare oracle differently biased model adversary second experiment access discretize model course user role correspond figure enjoy perfect concern distribution world use last similarly bias observation case percentile multiple per result since equip degree model framework run select user error record call appropriate baseline model world use bias bias approach examine oracle half single world bad biased model run though biased performance wise note ratio false positive negative world partially model htb datum prior adapt label upper attack examine essential naive dimensionality outperform partially possible approach sensitive case probability make tune achieve desire automatic useful comparison complex give promising lack adversarial marginal x prove statement sufficient obtain induce schwarz inequality organization detection project support economic grant depend prior variable tend irrespective empirical approach standard review technique deal statistical decision toy validate indicate outperform useful make scenario lack classical example maker whose accuracy measure decision base decision typically decision remain decision training decision rule detection spam normal relatively wish generate ask decide whether belong person set overall experience conceptual adversary distinction shall example instance user must separate specific adversary people use adversary attack decision making employ current observation upper good bad approach approach world model result validate detect building discuss framework introduce sec present main disadvantage label attack learn label attack train simple concern datum base nevertheless many detection predict expect application place individual attack severe cost alarm cost decision unknown expect rather body sensitive statistical decision sensitive assume availability label dataset attack cluster able detect attack disadvantage adversarial view main detection efficiency decrease extensively considerable call examine paper since seminal model consideration adversary essence frequently et adversarial adversary adversary adversary investigate reverse allow retrieve attack consider adversary lot adversary main attack without adversary current observation create bad conditioning adversary prior population user world essentially soft bad adversary construct relate nan appear explore sophisticated problem case maximally similar spirit contribution experimental model consider assume set observation ambiguity decide whether adversary complementary adversary user
near elimination bad ii quick convergence former per simple implement speedup modification brief exchange strategy formally algorithm algorithm monotonically multiplicative case continuous space design denote closure represent proportion assign design rounding usually convert assign model parameter interest response independent matrix eq equivalently design determinant unbiased widely criterion shall design alternatively e criterion motivate state weight crucial numerical approach refer well multiplicative choose let mapping highlight normalize al paper concern improve multiplicative principle exchange point exclude modification large step iteration maintain monotonic yu another yu conditional concerned theoretical ingredient direction define follow set maximize directional great ascent abuse shall closely exchange k maximizing perform optimal exchange carry denominator cauchy one numerator constant say rarely exchange method point ii resp minimize maximize optimal transfer nonzero choice shall key ingredient near multiplicative difficulty mass adjacent design together measure metric proportion put support would significantly mass add neighbor easy example eq single quantitative close whenever consider perform fractional intermediate output step nearest neighbor support exclude I intuitively appeal depend natural sometimes encode factor neighborhood interesting approach dynamically determine specifically much experience number minimized turn exclude shall composite mapping rather adopt two exchange serious stand alone outside support assign put previously define iteration neighbor fractional intermediate help keep iteration cost effective seem extend easily alternative consider monotonic convergence property increase establish see al yu monotonic convergence theoretical concern wu example immediate neighbor monotonic iii consequence rank iii effectiveness model exclude code request author line purpose conjugate cg quasi newton bfgs powerful dimensional effective consider system one count favor inspection iteration spend receive e algorithm reader start set randomly point intend ensure keep small design remove iteration relatively insensitive initial multiplicative always start exclude design priori linearization design grid evenly spaced parameter include analytic consider surface nonlinear stop criterion meet exceed large experiment compare omit similar evident improvement algorithm vary qualitative remain table median count multiplicative clear improve upon often tend fine exist seem insensitive concern newton via r function quasi popular bfgs conjugate test design iii substitution al derivative bfgs general purpose stop bfgs moderate gradient table bfgs quantitative affect nevertheless limit confident global conjugate cg bfgs cg bfgs cg bfgs design linear either multiplicative algorithm optimality yu vertex strategy close
theoretical mathematical model biology complicate become primary tool analysis large complex different probabilistic exist standard frequentist intractable assessment observe know integrate thereby treat nuisance likelihood abc calculation replace simulate desire agreement approximation material parameter whenever plausible candidate model closely shift onto posterior framework conceptual non bayesian practical consider show make expensive evaluation model develop abc selection formalism smc sampler abc employ whole bayesian formalism new model reaction dynamic real describe explain joint indicator likelihood posterior model scheme idea smc approach powerful reader material derivation well smc marginal basic consist candidate otherwise particles smc particle pm intermediate distribution gradually present smc indicator calculate return calculate tm b km kp I weight q set go particle previous denote perturbation allow perturbation replicate fix particle particle marginalization straightforwardly distance informative sense reaction rate informative preference particular inform find try arrive truncate adapt specify result special algorithm illustrate reaction abc gibbs select publish selection approach pathway stochastic reaction reaction occur protein synthetic measure use abc smc correct confidence tm n b tolerance schedule gibbs random become application biology bioinformatic gibbs form smc computational rejection collection iid ising eq sufficient respectively simulate parameter abc estimate computational speed smc compare n km tm kp exclude correctly analysis rejection abc approximately fold speed average abc smc occur infection infection address molecular spread infect distinguish spread inside infect community become infect infer separate question consider four characteristic hypothesis share function denote strongly appear four overlapping distribution show posterior posterior median run support mean table confirm marginal three share week model genetic difference heterogeneity infection among combine well suggest shape molecular well population parameter c abc smc agree conventional abc rejection turn previously perspective though crucial survival create site bind activate become act hypothesis originally get histogram population distribution schedule perturbation dd interval distance function root sum square ambiguity development mathematical action activate basic leave add appropriate developed leave original analysis evaluation course physical act equation propose abc delay without delay numerically equation add time noise identically population marginal bayes population accord evidence positive appear receive without delay allow nest methodology monte illustrate usefulness wide applicability smc even experimental unknown dynamical apply modelling
efficiency however effective prior knowledge light construct proposal proposal standard deviation proposal mcmc datum light symmetric simulation set simulation baseline count center concentrated infer event posterior center near relate simulation curve run property bayesian coverage interval summarize appear center calibrate interval overall result exercise simulate serve increase confidence validity apply discard burn parallel approximately cpu second hasting event event figure appearance characterize change appear minima evenly interval zero notable tendency median classical median likely p deviation symmetry relative event conduct count even split median equally fall zero calculate certain deviation extent reflect instrumental maxima curvature deviation support detect far possibly instrumental instrumental close nm result dominate phenomena effect count symmetric light expect small translate study curve shape differ signature study range clear object correspond finding time origin present herein minor examine symmetry parameterization flexible inference make quantity width half maxima extend apply angle incidence account uncertainty light offer construct location bit tractable multiple c acknowledge would like thank participant feedback particularly run computing group plot event omit anomalous omit due anomalous department computing university light curve ray center parameterization event characterization event key carlo carry novel curve light provide detect survey vast database series become increasingly examine traditionally examine skewness stack satisfactory ray light number particularly nonparametric skewness significantly parameterize skewness estimator false design publication claim detection diameter dynamical early population observation ground star present observational finding year group claim stack profile asymmetric however entirely satisfactory heavily transform take uncertainty wherein claim x instrumental al present evidence event event detector unfortunately beyond stack validity conclusion remain uncertain present ks year paper parameterization unimodal light profile describe wide naturally parameterization poisson event profile chain introduction approach proposal center event datum identify instrumental suggest describe apply discuss curve conclude assume intensity event occur sake discussion event extend trivially character intensity count sign deviation characterize intensity characterize pattern throughout index parameter characterize event baseline peak put constant make characterize location deviation still event proper balance parameterization give enough wide range event shape introduce event characterize addition first two deviation refer life approximately correction numerator event characterize symmetry approximate width maximum characterize rate characterize rapidly source second characterize rapidly intensity curvature event profile change intensity event see figure priori center interior contain light curve complete analysis improve become near end light curve restriction chain metropolis hasting symmetry excellent please quite result analytically tractable simulate metropolis draw metropolis hasting hold draw choose accept calculate holding etc new proposal analogous shape metropolis version signal center move analysis window ten point narrow noise capture result value light curve narrow curve initialization far proposal initialization first robust initialization brief duration smoothing smoothed light contrast simply overall contaminated initialization baseline assume symmetric acceptable
process combination point string function collect process state translation language string act sketch cone generate closure closure routine exercise check cone ideal computing however act might state mean parameterization rise hide process determined string length thing aware dimension obtain generator onto class unconstrained hold theoretic equivalent would ideal unconstrained string maximum degree increase obtain equation induce positive summing case entry row section accordance law string string matrix associate iff straightforwardly exploit unconstrained image usual string obvious lie finite mean induce translate theorem assertion necessary rise apply involve connection theory introduction see trace shall try general dimensional language trace trace therefore string application function dimension hold assertion application string function large stre kind relation trace relationship trace hide characterize chain correspond characterization state thm thm conjecture thm thm definition thm remark thm problem outline novel arise string function theorem outline model identifiability early view around value stay discrete accordance contribution example random process walks mostly quantum computer introduce string formally string function far provide helpful sec generate ideal algebraic obtain conjecture list hidden model detail sec obtain base precede trace relatively detailed proof finite letter concatenation operation direct string discrete view rise associated function discrete iff condition hold string call iff apply accordance terminology string function notation prediction case binary string string string also characterization source exist theorem prominent markov state alphabet emission probability initial define eq representation characterization string equivalent exist transpose follow actual dt dx matrix contrary resp row vector string function note govern condition generic need entire string practical resp span resp q resp row eq need theorem driving theorem finite minor column string resp string matrix size naive exponential runtime determine string basis row resp resp string imply let prove string parameterization case parametrization fact parameterization accord idea string rank eq done string matrix necessity claim give rise desire parametrization term elementary contain statement induction induction p span suffice induction q let two fully condition translate state column condition induce therein linearly dependent closure irreducible eq choice handle certain care rise unconstrained process string big give rise string ideal would consideration mild lift set generator generators string give great string define string member let unconstrained process step nh n nx n na generators generator generators ideal generator uniquely string least
disk pc addition inverse distance relatively uncertainty uncertainty dominant count uncertainty developed star star correlation negligible want describe star frame around system axis point rotation toward measure projection velocity individual star observation object project sphere coordinate term observe frame angular position angular frame depend coordinate transformation matrix epoch position coordinate north north respectively epoch deconvolution radial star use component remove restrict study velocity star uncertainty star star van reconstruction star gaussian variance restrict parameter rather method survey aa velocity one fit velocity good fit velocity hyperparameter km compare reconstruction star neighborhood base star penalize star available aa aa aa agree peak consistent move therefore well complicated velocity turn cluster probably cause dynamical bar way show merge plot actual high number give axis step implement programming language depend code reduce ensure full optimize increase describe maxima extremely merge select gaussian merge merge gaussian merge gaussians split split equally gaussian initialize random vector dm split merge gaussian gaussians hold affect gaussians sum gaussians convergence follow parameter great triplet split merge candidate question remain triplet quickly triplet define merge criterion observation equal probability gaussian merge therefore merge probability th gaussian kullback leibler distance local complete il il th density current leibler two nonzero finite local split determine problematic kullback model gaussian candidate merge merge pair rank decrease summarize step em specify correspond merge split sort candidate triplet sort list merge equation run affected gaussian begin triplet none merge list first stop go split merge constraint find maxima thank fr ed comment anonymous associate valuable support grant nsf grant perform w von expectation carry unique missing distribution even individual uncertainty project conjugate merge procedure avoid maxima infer velocity measurement infer find significant commonly many science case property different though uncertainty source well variation uncertainty significantly vary apparent interested really description uncertainty never know uncertainty property underlie distribution incomplete pose similar miss exist approach approach technique e zhang uncertainty recently attract none approach use account noiseless mixture generalize incomplete give model objective multiplying point obtain expectation algorithm optimize much way degree projection estimate incomplete current limit reduce show report merge search also incorporate merge issue datum purpose correctly velocity measurement component velocity measurement cover nevertheless good velocity describe application sample uncertainty mixture maximum likelihood seed goal fit true value datum matrix matrix infinite latter good infer velocity star know star exceed speed light exceed galaxy point projection greatly reduce place probability mix q unity observation ip gaussians parameter model choose explicit justified log eq q optimize optimizer reach maximum parameter must nonnegative add surface follow monotonically restriction mixture precise indicator extra handling datum would use prove update also likelihood observation consist likelihood respect model log show component current draw I denote expand eq covariance eq q thus ij ij straightforward full monotonically em wu maxima capability compute unknown encounter iteratively maximum equation become update might get monotonic increase avoid merge problem singular covariance produce rather different penalize maximum briefly regularization introduce conjugate mixture density normalize optimizing replace amplitude issue improper choice switch bayesian issue maximization split start algorithm variance reach gaussian gaussians merge alternative birth reversible mcmc variational approach move component suit detail merge complete specify set regularization assume basically function reliably could hand small believe get objective mention validation gaussian could well significant overlap overlap available test explore star km sigma plane proportional logarithm amplitude panel contour contain percent percent dark km move correspond often fully uncertainty reversible infinite infinite variational extend heterogeneous incomplete beyond extension straightforward method true value literature gibbs infer neighborhood angular
leaf allocate leaf categorical leaf response count z leaf expect count default leaf trivial leaf covariate allocate leaf lead representative particle include encode upon arrival sample version particle learn pl sequential carlo original pl pl mixture dynamic pl recursive due particle resample proportional wide question pl address make quick specific crucially resample introduce reduce dependence particle find filter remain track analytical greatly monte rao evaluation l marginalization move identify particle update low detail approximation depend set leaf section restriction plausible move accumulate update leave consist upon observe covariate follow residual resample new ty via stay move propose grow draw non u calculate posterior probability root sample leaf finish update particle approximation division incorporate global tree provide particle practice careful assess mention gain consist split rule sequentially discard except predict pl marginal available tree overcome datum aside model parameter condition large proper predictive distribution root factor pl constant pl marginal specification marginal likelihood ratio bf comparison leave leaf favor leave leave general covariate regressor bf assumption datum see consistent serve effective present series consider constant mean pl compare compete section focus describe application multinomial tree classification throughout dynamic tree parametrization inference parametrization four datum available mass acceleration simulate impact two row show filter fit pl row interval variation pl datum model appear adapt leaf lead high height opposite leaf evidence favor leaf estimate bayes present filter leave calculate first random ordering comparable preferable although evidence favor majority bf run increase agree visually leave particle interval cm filter log factor leaf mean compare tree regression bayesian constant make partially problem mcmc previously take broad partition relatively complicate response surface model practically leaf global additive tc gp forest hide respective size uniform root new candidate predict adaptation lead global summary finding leave relative fitting rapid interestingly routine flexibility gp augment candidate range hard standard addition repeat application focus response informative location efficient automatic covariate prediction error search draw I mt heuristic statistic optimum solely understand specify input gain heuristic alm choose maximize average upon maximum design alm heuristic alm extensive alm constant active inherently dynamic tree remainder illustrate alm scheme build particle alternative leaf functional evaluate candidate particle hence heuristic tree update alm seek maximize simple allocate mean equation leaf unconditional straightforward maximize give constant approximated candidate maximize alm leaf illustrate two heuristic column alm look much alm different exhibit additive ccc surface p cauchy rmse alm alm alm I tc tc alm alm alm design alm alm alm tc alm tc alm gp alm I gp design progress tree model leave heuristic hypercube statistic rapidly compare static start follow candidate round square predictive mean I repeat expensive dominate alm static fit mcmc amongst bad neither evenly gp alm offline learn tree clearly dominate second leaf I gp interest space tree categorical input high particle surface available mass library book response multinomial leaf dynamic gp outline fit gp latent p outcome tree particle gp max result bottom chain column figure surface fit decay cm class result soft classifier bottom black plot label mean surface illustrate problem top offer gp cluster status categorical encode aside note encoding incorporate categorical datum linear leaf adaptation exclude binary multinomial repetition sample soft classifier I gp correlation sophisticated leaf winner order gp particle feasible application dynamic max fold cm reformulate tree dynamic entirely class automatic specification avoid sampling tree point tree space posterior search essential model improper constant leave look find mechanism lead cm dynamic option surface accumulation learn efficient line filter state major partition division potential local newly capture along propose default specification integrate conditional illustrate nonparametric regression classification detail methodology motivate application commonly fraction partition cart school business edu laboratory university production article research partially sequential detail example experiment design characterization dynamic model grow practical effect algorithm example demonstrate propose generic formulation probability simple powerful situation priori hyper simple rectangle state induce restrictive formulation conditional simple tree depend approximate model updating sampling alone thereby specification alternative dynamic herein appeal e expense correlation specify realization easily finally perhaps problem serve engineering computer prior additive gp decompose build around sequential new datum intensive nature problem search calculation explicit expensive serial sequential gp fall design search due specification may area interest along response optima stationarity modeling scheme counterpart contrast fundamental consideration may flexible function set simple posterior suited class model specification automatic inference sequentially filter state prediction surface complicate review core outline characterization evolution define leaf prediction detail provide application alternative literature sequential experiment provide tree relationship relevant base approach partition regression cart force transformation use conceptual rectangular partition scheme recursive shall outline detail covariate correspond tree consist hierarchy subset split also terminal node new tree include sub tree subset diagram form recursive partitioning child parent contain equivalent node otherwise leaf internal leave parent child illustrate split set tree complete simple leaf covariate response regression tree allow assessment partition predictive band generative specifie place rule leaf depth implicit partition create contain ignore prior probability seminal develop partition tree stochastically propose incremental change swap move accept although provide chain move probability represent drastically batch inefficient characterization lead particle framework able difficulty mcmc trees model introduce recursive rule associate observe time define function newly covariate allow evolve small neighborhood specify section likelihood leaf marginal likelihood evolution detail describe multinomial outline partitioning statistic update combine resample account finally section evolution keep possible localize evolution equally move node include parent node uniform dimension
analyse independent dependence efficient approximation distribution study negligible piecewise competitive infer uniquely commonly allow change application posterior liu producing draw efficiently sc base assume note parameter segment give segment segment within start conjugacy side update piecewise polynomial segment ccc x x independently distribution geometric segment underlie refer segment note determine segment consistent dependence conjugate prior gaussian km km previous start observation end curve calculate interact multiple let step piecewise tm match moment number simulate give final segment simulate simulate model repeat segment conditional c pc pc tc segment simplify notation mp get need simulate smoothing algorithm give assume simulate mc simulate simulate smoothing position accurate obtain simulate simulate segment tractable piecewise polynomial give calculation become pdf inverse multivariate evaluate come independence simulate multivariate normal multivariate normal hold give multivariate segment filtering smoothing point linearly expense approximation particle resample liu chen discrete resample result bound constant investigate substantial obtain negligible evaluate model first look filter smoothing simulate accuracy curve underlie implement use filter rejection method resample filter observation variance segment segment smooth roughly second sample run value underlie could draw true quantile demonstrate deviation equivalent performance datum set equally spaced posterior plot dash dot dot line simulation black dash plot quantile case close extra quantile interval generally plot suggest approximation posterior look regression new fitting firstly implement parameter uninformative estimate hyper whereby default choice simulation study choose see repeat effect quantify set curve square coverage credible wavelet detail ccc c mse credible simulate mse estimate substantially two wavelet mse method coverage likely use bayes effect repeat default scale increase default increase mean error repeat last case repeat time lead table advantage study cubic expect modulus efficient increase underlie new coverage interval curve method still substantially wavelet curve ccc error coverage credible piecewise cubic simulated comparison various henceforth dms datum block signal denote dms reversible piecewise continuity approximation approximate segment magnitude analyse ccc snr dms dms set set give point observation mse dms compare method considerably estimate curve underlie dms set peak rapidly use polynomial cubic fit curve suggest dms method accurate wavelet investigate calculate error obtain use wavelet implement table integer power apply wavelet dms block detect curve curve due curve dms point
hoc task distinction sentence job create query applicable multiple document formulate term document relevant document correspond search query build concept behind ir represent query document probabilistic bag look rank make researcher build build kl divergence strictly assume distribution assign probability exactly smoothed final query focus sentence sentence eq system sentence document relevant query background information contain english appear relevant query word english level query background english general english capture english background document language query language specify exactly word rest sentence layer also degree lie dimensional simplex english variable language document corpus parameterize multinomial restriction continuous normalization term generative define document generate document select generate word relevance document query document wrong english depict know square circle relevance unobserved circle indicator degree contain document observe give expression datum accomplish prior sum value final word condition select intractable coupling integral give rise technique monte mcmc saddle approximation effective deal expectation propagation propagation roughly long propagation generalization belief propagation assume filter thesis superiority variational product integral give inclusion leave expression fix method ensure integral well approximation ep reliable ep omit experiment document document return search engine short conference data relevance require query typically break title sentence sentence concept keyword model train query relevant amount roughly median document relevant query remove seven manually extraction ask select document need overlap evaluation agreement annotation doubly annotate data inter agreement keep sentence precision recall criteria reciprocal calculate sentence order average reciprocal rank first precision relevant sentence human baseline four information retrieval rank fourth interpretation ir query rank sentence fourth cosine compute sentence document smooth collection retrieval blind feedback expansion first return relevance retrieve top expand sentence method interpolation parameter oracle optimistic relevance however access model evaluation corpus system ep hour query field evaluation description consider concept alone baseline position bit turn p maximize baseline either experimental show system thing standard without perform position model title relevance feedback might initially seem actually available several sentence however blind map kl kl kl input kl description title make metric add improve performance improve substantially model give collection violate access collection relevant deal collection irrelevant dotted kl star title red circle title indicate ir evaluation field engine linearly relevance interpolation obtain ir engine improve relevance six interpolation observe dot ir little difference obtain roughly dominate ir engine look perhaps surprising difference ir title believe title field contain nevertheless trend ir conference evaluation competition fortunately account redundancy document technique select sentence central well accord pyramid automatic element evaluation confidence significantly understand conference competition focus nearly mse additional compression component summary query system score competition linguistic evaluation worse likely sentence top automatic perform third never significantly bad describe generate focus bayesian focus state retrieval force relevance alone achieve score primary operate purely question arise outperform formalism relevance access
standardized compute bind pt x assume choice skewness equal skewness formally proper example positively skew symmetric skewness initial introduce skewness inverse positively due rarely symmetric algorithm search skewness skewness guarantee make impossible see return either upper positively skewed rarely shift transformation initial affect triple alg pass update input accurate level skewness k return triple great skewness student uniformly symmetric disadvantage probabilistic skewness remain must separate slightly second jointly underlie algorithm check reason optimistic true ii ignore branch leave considerably available kolmogorov input package vector specification skewness rv simulate though simplification would guarantee indeed general obtain although nontrivial expression alg sim finite input particular mle gaussian skew package whereas accuracy skew versus rv additional value return introduce reveal branch mle skew replication input rv relation set euclidean gaussian skew standard variable thus comparison imply estimate x impose cause loss finite sample symmetric distribution ml rmse unbiased whereas moment explain effect truly distribution estimate give confirm normality skewed gamma asset exhibit slightly skewness c ml skew present ignore mle mle rmse present rmse small reason though extremely skewed gamma skew mle skewness lie theoretically branch mle skew mle fail accurate heavily skewed practically rmse skew increase sample rmse ignore branch surprisingly size c c normal ml pt whereas extent skew inferior compare restrict extend class mle little normal precise location skew skewness absolute skew normal include bottom iteration right input degree freedom table increase away origin close bottom size find total number number bottom approximately time show rv degree freedom take input result return start process property fairly assumption quick heavily skewed demonstrate skewed transform triangle pt top normality test min mean skewness distribution appropriate well standardize auto risk body index dot several fairly skewness normality assume give yield positively skew datum reject w estimate significant evaluation triangle adequate lie half open lie local closeness density improper approximation health explain skewness support sense natural albeit financial introduce excess typically generalize auto volatility theoretical however usefulness distribution direction note resemble closely future news return interpretation stock input latent news reject consequence location table coefficient particular increase solely address negative daily leave transform data mle table kolmogorov student adequate unconditional vertical horizontal news scatter observe return outcome bad worth location powerful market show bad financial estimate potential asset fix period percentage expect confidence period statistically quantile distribution way empirical quantile comparative c pt quantile capability excess degree freedom high skewness datum skew tail capture news winner skewed skew one median true concentration return financial call return mle parametric model successful uncorrelated unit detail student available find standardized residual fit return package return mle standardize residual still give estimate study combine possibility skew suggest area research finding skewness asset skew lead suit asymmetric price financial return recover f rather skew control tail idea skewness specific get skewness heavy tail reveal important play role mathematic physics field introduce financial great flexibility respect rv wide transformation acknowledgment grateful give I cat de anonymous comment suggestion manuscript notation type point generalize parametric skew particular nonzero skewed variant counterpart maximum estimator show affect finance relevance particularly useful skewed package author publicly exploratory look gaussian asset tail skewed sense fairly generalize prominent generalization skew include skew rv density skew skewness lead skew cauchy skewness skew skewness seem put cart skewed variable start skewness propose novel naturally model observable rv input either chemical physical biological kind simply represent restriction analyze detail methodology instance stock asset asset percentage price skewness excess call left daily return percent series excess skewness excess kolmogorov skewness asymmetric xy skewed choice regression quantile estimation convert back skew world news w asset return perfectly empirical evidence student skew fundamental consider result bad market return good news positive return empirical news skew really thing happen really thing return get news get news drastically negative news distribute act skew rv way exploit skewed procedure skew skew transform result skew truth take skewness consideration ignore summary test kolmogorov ks pt st skewness define counterpart parameter sec show skewness affect estimate useful skewness set return particular output detailed quantile estimate essential appropriate plot evidence link square rv distribution figure simulation realize source statistic package notion translate terminology rv continuous rv parameter family rv value possesse continuity expect also rv factor positively exist rewrite exist typical scaling signal affect system suffice become useful transformation branch branch unique two real stable principal branch similar input extreme denote cause evidence student ignore root matter much branch obtain generate stand estimate branch alg w cdf likely ease yy u monotonicity split separate derive f scale analogously branch coincide xu scale rv equal analogously depend restrict c c importance value rv equal scale rv nonnegative rv inverse f f follow note rv f rv flexibility expression researcher easily create variant four coincide become skewed although particular sometimes aspect might return whereas solely generalize analyze concentrate inspection directly rv median transformation pass furthermore input median corollary interpretation general u analogously explicitly obtain quantile rv well insight gaussian moment particular
easily v consequence nd h converge surely lemma together imply combine surely respectively n ks nm x nd converge surely pt square integrable martingale equation x ng dx g say k furthermore martingale g conclude consequence dx x proposition check n combine dominated theorem constant g nj hx c nj j n ng eq inequality inequality v give acknowledgment grateful helpful discussion point reference helpful comment technical argument supplement section algorithm nsf dms asymptotic variance chain almost result chain literature weak result adaptive algorithm flexible framework sampler see interest mcmc markov hx kn role assess performance variance sampler markov chain markov precise constitute framework analyze mcmc well method notable mostly stationarity broadly contribute variance ergodic general example chain mixture variable geometric stability markov bandwidth almost coincide converge deterministic random derive bandwidth mcmc carlo describe method overlap batch consistency chain assumption ergodicity moment variance time model ordinary square condition converge estimation version call various hold martingale approximation adapt treat law number array differ sure take class section adaptive markov understand behavior result also supplementary logistic act measurable fy qx metropolis throughout impose ergodicity exist constant q ergodicity assumption moment probably redundant geometric ergodicity imply dr either drift large establish adaptive short proof state theorem assess g omit notational convenience nonnegative eq easier check positive metropolis independence similar metropolis langevin reflect fluctuation metropolis adaptation side surely good kernel often say value h justify follow hold section eq twice continuously kernel impose strong replace restriction continuously supplementary article instance fail choice satisfy markov chain transition kernel satisfie holds apply ball continuously imply vx xx assume thus small focus conclusion derive theorem vx hx h hold choose almost surely similarly deduce term give eq take choice issue take th autocorrelation choose high bias autocorrelation process decay issue one hand asymptotic square fluctuation assessment adaptive adaptation chain weakly metropolis subset typically adaptation h multiple clearly even confidence interval become asymptotic consequence valid run chain advantage adaptive monte assessment important adaptive adaptation mechanism define case rwm mala langevin illustrate markov follow I assume process chain take define introduce hold kernel choose bandwidth follow approach outline run discard burn sample path plot
density update procedure slightly density serve illustrate initial importance multivariate consist nine place centre second degree circle indicate mean proportional weight start shape become separate tail target importance target density normalise normalise ess first iteration simulation shown normalise rapidly importance approximately importance normalise increase importance normalise start indicate need density general need observe horizontal axis vertical axis second indicate dot every rd importance plot run plot thick contain outside circle start algorithm poor initial importance function narrow may part tail match importance inform guess possibly play small adapt may provide sample consideration density take large increase issue datum sect simulate compare consider encounter density normal change co unchanged density center uncorrelated jacobian shape two density interest tail target curvature slightly typically highlight difficulty cover guide choice mcmc absence except choose student distribution randomly away centre variable mean component degree adequate coverage albeit region distribution preferable variable importance dimension show iteration panel pilot importance density fitting require seven coverage issue relatively long unable sufficiently problem occur dimension curse prevent numerical updating less density adaptive fast use either independent walk proposal concern common gaussian stationary product target despite however effect recursive sample suitably condition empirical pilot run schedule quantity pilot value sensitive choice position counterpart find schedule every assess update simulation ensure burn period sect acceptance proposal update chain stop burn outline appear th follow final point point burn successive interval provide performance approach simulation interested mean also tail target indicator respectively result show deviation estimate calculate fold reduction compare close look reveal see quite reduce quite majority run panel difference explain visit positively second despite estimate overall variability cccc mean htp variability plot measure density skewed bottom nevertheless significantly point panel panel highlight variance evaluation simulate represent challenge importance take properly see significant alternative structure observe adapt true change proposal covariance precise comparison mcmc apply importance parameter constant dark energy parameter test ia release release analysis publicly public tt te bb theoretical return part compute associate covariance angular scale tt te bb angular compute pseudo te v te tt uncertainty ignore correction due impose large constant alone degeneracy acoustic peak spectrum therefore angular diameter distance energy weakly amplitude determine normalization peak dark matter density optical wolfe probe describe use curve fit frame band colour ia colour respectively parameter specific distance regard year release mass dispersion observable likelihood correlation angular theoretical dispersion scale galaxy rescale distribution galaxy histogram introduce parameter model multivariate uncorrelate include independent weak angular amount universe probe degenerate survey include latter determination weakly constrain hypercube exist implicit represent numerical formulae break exclude non allow rarely solution exclude rare description matter dark l normalization optical magnitude colour sect important function rely estimate hessian initial proposal use find fisher mixture consist student freedom turn bad shifted scale fisher shift shift result shift component stay near tail sample randomly typical case fisher element adequate use reliable explore parameter effective quickly reach although posterior satisfactory yield consistent mean efficiency sampling tb mixture close consecutive tb contour movement plot circle iteration mark circle panel position importance method order consume parallel obtain core readily acceptance normalise large final sample posterior calculation make computing mcmc issue mcmc algorithm avoid adaptive result reach acceptance rate well find excellent marginal superior follow inherent sample way mcmc chain visible second exhibit anomaly chain converge feature importance uncorrelated issue nearly unconstrained illustrate choose weak fisher stay small direction flat jump flat proposal acceptance rate modify initial increase sect explore parameter step adaptation modification low acceptance fine initial recover tb tb mcmc dash green dotted alone respectively interval less percent region agree mcmc carlo aim overcome achieve towards alternative lie massive parallel posterior essence form produce exploit regular sampling output previous combine approximation storage sample improve posterior may successive importance importance tail importance procedure usually involve matrix case alternatively iteration quickly poor ess normalise outline significantly inform iteration iteration counter potentially parameter absence importance available reasonably variance reasonably sect information example point covariance point approximation singular along interest component k reasonably successful examine place support reduce iteration difficult density main worth point ability increasingly useful availability multi cpu computer cluster computer software implement message parse interface sect computer reduce target significant use fold reduction mcmc time requirement total variability valuable adapt importance target combine across iteration absence desirable attribute reduce sample sect sect simple assessment potential associate diagnostic ability evidence naturally far explore acknowledge lambda lambda provide office computational support contract base mixture principle error function increase iteratively equivalently formulate maximization bayes posterior belong evaluate intermediate quantity concavity easily check em maximal requirement maximization solution whenever depend routine calculation denominator integral propose normalise weighted n empirical eqs sect density except equal shape factor tail polynomially decrease tail whose finite family mixture student mixture student sake formula student density multivariate easily derivation chi advantage straightforward opt call adaptive importance population considerably benefit assess performance problem actual type ia provide art markov mcmc type parameter hour cluster recent advance observational availability quality testing high thank technique monte produce markov chain burn regard sample approximately design mass region visit spend grid model mcmc particular metropolis thank package form hybrid hamiltonian interesting usage estimation spectrum reference advantage sampling approach suffer correct convergence presence greatly third need computation slow speed computer posterior public code precision second explore flat course require order magnitude efficiency apart improve likelihood code availability computer speed want big complex effort devote recently code interpolation network look improvement algorithm former provide pre step latter requirement availability computation derivative non markovian monte apply present improvement availability computer computer partly way lead speed improvement iterative mcmc running chain end big chain explore converge absence bias sample determine chain support expectation integral link q provide converge context right hand normalise importance ratio normalise converge approximation normalization
significance triangle c significance empirical even power alternative implement monte carlo monte critical level segregation ten skewed almost symmetric occurring estimate skewness get kernel skewed skewness get skewed segregation alternative observe get estimate symmetric symmetry occur skewed skewness large estimate solid dash power mc alternative imply power two alternative density nan dash estimate segregation ten skewed right occur skewed leave solid maximum carlo estimate association base monte estimate value leave middle estimate asymptotic alternative value asymptotic critical nr nr power critical association alternative circle significance triangle figure monte significance nr z appropriate approximation severe association power significance plot curve power consideration asymptotic efficacy local involve limit well test sequence neighborhood nan q pc equivalent pc pc satisfy pc pc testing region n pc need calculate asymptotic efficacy segregation consider test small suppose equation h h q derivative detailed yield thus second pc pc sr give segregation sr efficacy segregation segregation large test segregation small moderate skewness density moderate around convex behaviour efficacy section sufficiently whose r equation segregation get pc r numerator pc hold association alternatives r ar suggest small association appropriate skewness moderate alternative unlike involve require asymptotic variance see rr r sr degenerate sr plot asymptotic association alternative finite denote triangle triangle convex wish segregation alternative realization realization segregation association realization segregation association segregation association use relative base yield triangle corollary corollary respectively appendix similarly segregation triangle replace case h segregation association segregation e realization figure figure realization value segregation realization implement carlo empirical power table alternative alternative realization small empirical segregation use segregation small notice also increase estimate get alternative significance realization function right realization circle empirical significance triangle leave circle represent empirical significance triangle conditional unconditional size triangle poisson point derive arc triangle rw r j r adjust nr triangle size triangle optimal efficacy segregation association alternative asymptotic efficacy segregation association right realization vertical axis differently segregation association notice efficacy moderate segregation segregation sr efficacy segregation alternative conditional severe segregation ar sr efficacy association j r r severe association straightforward detail invariance asymptotic normality statistic proximity similar factor map literature slice proximity central proximity proximity advantage tractable dominating set geometry triangle mean high segregation association survey relevant independence triangle fix nan label complete randomness interest number triangle segregation alternative asymptotic efficacy suggest efficacy unbounded case arc moderate efficacy preferable preferable work project air force office scientific contract direct position employ parameterize proximity map relative arc summary provide alternative employ relative arc properly analytic study central segregation efficacy efficacy classification receive considerable year position cover give apply employ involve dominate set prototype find minimum dominating particular tractable proximity respectively triangle proximity advantage dominate tractable latter segregation arc properly rescale hypothesis segregation association alternative central efficacy relate segregation association test detail visualization describe dimensional nx xx use represent briefly proximity triangle interior form map segment edge partition fall fall adjacent opposite line euclidean orientation edge proximity notice tx rx set set occur direct graph base vertex arc since need define functional represent number arcs arc density henceforth proximity statistic brevity arcs variance simplify central statistic depend segregation involve tend away test segregation generally spatial randomness thus sample alternative segregation association let segregation occur near association association definition invariance follow triangle vertex parallel triangle segregation geometry triangle simplify subsequent theorem transform triangle v uniformity transformation boundary median parallel edge line joint intersection bound line content preserve uniformity preserve uniform assume triangle proximity nan rx occur geometric calculation limit establishe summarize q degenerate form increase sn sn r rx degenerate continuous observe skewness analytically much asymptotic variance r successively calculation value approximation small indicate figure severe skewness depict histogram replicate severe skewness extreme recurrence note normality hypothesis segregation segregation alternative covariance r normality obtain likewise segregation association h asymptotic universal asymptotic relative proximity appropriate segregation association use segregation standard since degenerate great mean alternative follow detailed segregation likewise association follow alternative segregation likewise indicate segregation alternative implement monte degenerate degenerate large empirical relative value empirical n h segregation alternative eight skewed skewness get symmetry occur estimate skew
post manual explanation calculate limit manual provide publicly make voting include device bring together expert follow computer program fair discrepancy ability strategy complete whether outcome expand sample algorithm development candidate select look college methodology principle resource routine limit outcome need go development two acknowledgment thank david article helpful suggestion political science suggest describe author field post thank develop weighted limiting size sample many I make mathematic valuable suggestion dedicate motivated derivation vote small margin report win small win candidate vote margin number could give cast report vote winner report vote close estimate minimum cause outcome margin sized unit could cause winner win vote unit could margin sufficient cause outcome eq reduce eq increase win winning unit vote candidate margin need incorrect outcome estimate detect actual win pair margin win plotted vote candidate excellent calculate win pair bound may chance circumstance win candidate produce win margin post vote initial express q denominator get cast candidate apparent two apparent initially receive vote error win vote initial cast way way vote situation vote margin occur candidate initially incorrectly incorrectly possible margin vote vote candidate really vote vote really candidate vote really vote pair vote candidate candidate vote margin error add scenario scenario occur margin improve four limit paper b c c c c c confidence bound small unit maximum occur confidence improve size conservative unit article advance improve post post extensive sampling various focusing discuss risk post exist unit winner tool four calculate show apply margin efficacy exist discuss mistake reduce adequate post sample three article two article discuss algorithm purpose detect incorrect act computer article define post check report manually count randomly vote checking record risk limit post minimum detect correct control worth million minimum outcome incorrect winner ahead post amount cause incorrect vote incorrect largely political recommend ad systematically recommend post national institute technology us development recommend voting voting discover ed modify consider computer provide calculation vote count vary rely small report vote count detect vote propose individual voting system design report count thus make sized develop detect outcome sample size win weight fair selection prefer ten sided generator publicly voting cast live voting voting device count batch unit automatic associate number maintain unit individual produce public report vote preserve privacy size manually count vote cast vote vote candidate security procedure substitution record effective manual therefore minimum count manual count ensure voting machine percentage publicly report inaccurate outcome outcome unit need detect outcome incorrect vote detect incorrect outcome post appropriate post purpose checking outcome achieve risk limit post desire sample unit minimum outcome voting machine accurately tolerance typically margin small every vote compare effectiveness cast margin calculate uniform estimate could cause outcome flat inaccurate high detecting could incorrect outcome red roughly vote detect correct regardless win size limit win margin decrease could eliminate automatically manual necessary ensure outcome bar ensure wide house vote roughly rate limiting way approach detect incorrect outcome category flat detect incorrect risk limit inaccurate separate limit risk incorrect post author continue detect risk limit post various limiting conduct california united limiting state ensure small incorrect outcome post sampling weight risk sample size depend outcome margin article calculate post detailed datum draw quick planning precise achieve desire minimum incorrect c cast vote win total vote win vote total vote vote vote candidate vote number vote nr vote win candidate winner margin vote win vote margin margin divide cast vote margin margin upper unit nj formula depend reverse total unit assume confidence initial unit select number size manually count might small cause vote count incorrect maximum thus margin look cause assume overall winner immediate minimum could cause incorrect outcome unit fewer incorrect large unit unit still bound would crucial consequence size candidate additional unit multiply could exist without see author margin cast inaccurate assumed level necessity additional one maximum article level ratio measure herein margin replacement course notice fact vote cast candidate expect target vote notice unit upper calculation method determine precise win pair calculation recommend method multiply time vote cast maximum derive vote apply place recommendation number cast incorrectly take normalized margin win pair risk limit large author continue recommend less margin insufficient use favor candidate expression margin vote win win impossible amount share margin available contribute incorrect case candidate vote account cause vote winner versa vote win count vote win vote vote win precisely cast assumption proportion vote switch winner vote vote cast vote could vote within margin size detect vote incorrect outcome limit win pair compare actual bound upper margin win candidate win pair sign difference margin manual win margin margin author agree occur pair unit winner individual upper margin winning reduce margin bind win cast expression q error winner vote vote full manual initial really vote winner vote cast margin vote number vote margin error thus incorrectly report winner percentage vote find actual minus margin winner b c bound win initial incorrectly record vote vote margin individual candidate contribute vote incorrect vote margin contribute win unit vote vote cast vote vote win winner maximum margin become winner win perhaps simple win pair formula include vote win candidate vote vote win one contribute could cause outcome contribute margin incorrect cm c c win accurate incorrect win detect incorrect win win candidate bound reason conservative margin small ratio vote margin margin win unit margin win win candidate vote winner vote vote vote win analyze limited win win margin win candidate margin affect pair overall unit win pair margin win consideration analyze error consider separately win pair unit analyze unit wide upper margin error win candidate pair win give cause incorrect limit presence sample sized probability large size could incorrect outcome calculation cast accurate detailed variation minimum initial cause order order bind win candidate margin win unit take margin follow pseudo create array r order cumulative margin vote count vote count etc compare margin minimum vote count incorrect calculate calculation vote win pair within calculate large sample prior random algebraic rough planning purpose initial detailed vote count vote estimate size need c c bound calculate est margin occur summarize three mechanism risk limit obtain rough overall total estimating suggest estimate quick vote cast cause incorrect margin error bind eq incorrect unit error cast large cast unit derivation equation simply margin cast ratio far close value detecting object sometimes detect unit unit eq eq substituting formula combine get substitute expression combine derivation sample size upper formula follow desire detecting cause incorrect notice cause incorrect outcome avg calculate c avg column risk sample various formula calculate need provide vote proportional probability margin post size necessary desire unit randomly select propose sized calculate probability probably good avoid initial win incorrect contribute margin error win develop ensure approach adequate sample margin total margin win unit describe error recommendation calculate certainly weight select unit bound win vote cast receive treatment avoid margin win candidate avoid assumption incorrectly report know win provide vote initial show order vote need margin win candidate increase detect error candidate vote vote unit winner initial win candidate f margin unit benefit winner multi suggest winner winner candidate win consistent amount contribute incorrect win candidate pair incorrect proportional bound look many pattern cause sampling develop et size unit improve possibly cause take reasonable winning without immediate unit size sum contribute outcome probability bind develop recommend win margin calculate overall replacement vote win sum margin win number cast initial vote win candidate vote report desire report outcome incorrect total vote win total sample another sum win win candidate vote unit calculate win multiply error time evident easily add plus total winning minus vote margin error win candidate summing times total vote win calculate win candidate initial derivation describe previous calculate calculate upper candidate htbp example probability size conservative sample size vote count win stop j find method unit vote approximately overcome select kb win pair ai ij ai win candidate pair round vote count incorrect outcome avg kb w r two c desire select unit supplement necessary size margin investigate necessity crucial would candidate pay separately confidence detect essence occur within unit include part manual fail sample separately programming system sample providing chance unit variation vote small error available sufficient incorrect especially require wide manually count risk
laplace measurement give h define control resolution tuning ill pose section notice deconvolution thresholding alternatively give choosing perform density wavelet confirm superiority base fouri decomposition nd instability interested small study reasonable estimator compare divide oracle independent average model selection repetition thresholding model selection expect uniform see also ill pose inverse direct introduce level quality combine remark reasonable satisfactory estimator solid jk em jk jk j jk jk inequality obtain j complete exist inequality constant c cc n c p q q remark although always various uniformly study behavior one j ps ps ps remain term dense decompose j r pt jk jk j em jk jk p v jk em jk j dense imply integer write pf j j r combine n n complete proof cm assumption lemma section aim usefulness wavelet deconvolution use wavelet computation rely fouri wavelet band compute avoid wavelet dyadic main drawback classical wavelet threshold performance logarithmic performance direct deconvolution deconvolution threshold inequality adaptive minimax secondary de universit acknowledgement acknowledge provide compute density identically iid representing size wavelet know advantage spatially use estimate function therefore receive attention last decade detailed subject wavelet usual coarse resolution frequency dyadic various instance sufficiently scheme wavelet limited wavelet transform approach fast transform exist wavelet band deconvolution moreover wavelet fast contaminate q iid represent additive fundamental importance communication theory e density additive relate indirect observation instance problem deconvolution wavelet thresholding wavelet fouri numerical scheme contribution threshold density threshold empirical wavelet threshold poisson estimation haar compute wavelet use threshold exploit variance wavelet compute compute threshold attain performance logarithmic depend quantity wavelet inequality gain popularity procedure performance recover inequality currently introduce wavelet problem white intensity poisson name background wavelet correspond direct explain performance compare oracle oracle study depend large propose estimator compare proof function compact support hold mainly mathematical outside center fall apply transformation wavelet respectively construct smooth scale wavelet space wavelet fu du fu u wavelet transform indeed estimator fast show wavelet band avoid scheme dyadic analogously c deconvolution e unbiased difficulty deconvolution quantify error ill fourier depend decay estimation ordinary fourier decay mean rate wavelet depend smoothness base wavelet wavelet form achieve optimal smoothness quadratic risk however nonlinear estimator estimator hard thresholding positive direct take level deconvolution deconvolution however threshold practice coefficient depend deconvolution let definition jk algebra deconvolution v make direct also write n calculation obtain dyadic point upper close oracle derive coefficient suggest threshold form tune universal conservative paper small moreover tuning depend finally propose threshold instead context nonparametric regression exploit wavelet oracle ideal estimator practice depend unknown shall assess quality quadratic equation retrieve classical risk oracle coefficients basis j addition obviously integrable supremum choice tune oracle define satisfie oracle performance moreover performance classical universal one set additive risk belong choose possible fundamental constant one level price pay term obtain derive rely wavelet basis haar deconvolution ordinary assumption j constant behave tend comment make choose additive great deconvolution degree ill deconvolution cut usually ill typical various literature see small pose inverse choice understand influence hyperparameter validate result characterize basis ball respective replace ensure subspace shall allow local smoothness typical old piecewise piece locally element despite possibility one piece function local let theorem deconvolution near presentation hyperparameter theorem condition assume minimax study converge usually pay deconvolution smooth deconvolution q rate respectively spread function vanish coefficient effect detail white show reference therein wavelet toolbox matlab fast wavelet density distribution em e pt test figure scale compute coefficient theorem hyperparameter control frequency cut threshold equation rate additive inequality take
previously location explicitly paper node mobile point process channel receive act obtain success accounting interference use tool analyze scheme analytical introduce inclusion spatial mathematical wireless service consist area call bs mobile cell boundary distance inter cell interference base increase expensive paradigm mobile boundary significant architecture effectively benefit communication two may significant scheduling bs receive information mobile cell act choose subset fashion interference simple interference geometry analyze provide asymptotic analyze complicated emphasis methodology rather communication specific although extension bs bs consider act locate circle bs power simulation distribute distribute code tight precise choose maximize diversity coefficient average node location incorporate start spatial emphasis introduce metric section probability connection bs section employ analyze scheme direct assume bss arrange square lattice deterministic bs bs poisson would see observe necessary near bs outside cell assumption ms bs serve assumption location mobile base cell probability h cell bold dot bss dot space consist frequency choose node hence increase center around one form non singular path loss treating interference imply transmission locate receiver bss additional mobile bs want receive never able connect set bs connect bs connect bss origin bs cell subset potential intend connect belong connect receiver potentially ms intermediate reference selection compare gain characterize high q diversity gain transmission transmission interference correspond limited even x curve scale receiver b bs bss interference bs origin hence position intensity intensity eq able connect follow connect section respective ms channel path receive channel node fair direct success inter interference precisely reason cell b slot fair condition one begin first contact observe contact rf interference cause cell though contact lie pdf condition event yield calculate asymptotic bs easy average potential scale f fr gr fr gr follow asymptotic basic expansion dash derive interference fr remark higher limit may happen scheme monte obtain theoretically include selection orthogonal choose channel alternatively channel use well distribute fashion previous success cell empty early shall unconditional multiply mathematically selection interference aim behaviour denote eq comparison transmission easy interference cause cell intensity obtain expansion mx let difficult calculate reason asymptotic since unconditional gain low interference conditioning require similarly observe upper basic exhibit q cell maximum interference limit direct transmission monte purpose bs threshold
centre unit truncate agree confidence double restrict realize capability kernel benchmark leave tail outcome confirm realize consistently suffer small realize precision neighbourhood quantile show matter concern neighbourhood vertical set involve iterate fast double range low produce essentially transformation generate double precision approach branching evaluate iterate suffer degradation production suitable approximation except relative relative precision change leave region construct quantile double b realize precision interesting feature reciprocal logarithm map sample map ode intermediate composition cdf probit rational difficult serve fast branch rational satisfie furthermore give divergence therefore back branch computation computed precision indistinguishable implement branch implement central central fall back vote modern gpu seven algorithm four code double dp double precision pure exponential hybrid understanding characteristic capability gpu iterate iterate b hybrid appendix clear provide double range monte argue preferred advantage relatively logarithm double precision note branch optimize tail behind hybrid robustness respect increase per briefly polynomial representation central avoid divide supremum finance normal base side direct give shall explore marginal couple simplify computation base tail elegant q translate evident q characterize base solve result boundary get correct clearly choose trivial make differential right initial condition method parameter identity model return asymptotic asymptotic match exponential integral give easily first identical remain straightforward ode step singular otherwise neighbourhood origin elsewhere risk depend tailed asset return allow traditional distribution ode transform couple tail complicate handle essentially quantile interest distribution translation solve relevant develop copula hypercube course rely explicitly characteristic elsewhere concern good quantile implementation formulae approach speed develop offer precision double gpu preserving enough monte carlo acknowledgment knowledge transfer thank theory gpu produce windows architecture members numerical group system grateful allow financial modelling understand helpful manner noise induce distribution monte form monte carlo may traditional fisher expansion distributional assess free regard student variance change employ allow place single rational wide range branching statement avoid quantile offer environment divergence comparison old argue mode fast precision offer quantile yet monte library function keyword carlo gamma finance mechanic quantile inverse probit distribution solution make sample distribution characterize density uniform base transform marginal generator effort leverage manner principle associate form see direct way develop form composite mapping transformation way skew base control introduction traditional gram simplify brief review insight quantile expansion already series explore normal student case explicitly become object offer performance environment branching algorithm approach costly branching avoid environment paper computer mathematic approximation learn reveal type norm construction term precision function differential characterize one probability another give focus transformation student traditional expansion brief introduction might part quantile distribution idea quantile precision implement change argue offer characteristic section make double computation introduce generate two double fast gpu preserve full range enough ode rule rational pearson allow analytical order quantile suppose algebra quantile map ode ode arrive equation two rather suggestion eq ode arrive ode eq positive line ode eq brevity encode exact composition cdf distribution relationship information term know expansion fisher consider notable hill consider differential interesting illustrate know asymptotic series explore develop explore purely student case freedom integer ode look solution behaviour rather different always purely matter large value far go wrong tail gaussian centre wish apply ode derivative centre treat solution student indicate whether convergent algebra tail assume ode change cdf determine determine simple property tail behaviour cdf deduce step calculation ode ordinary quantile find literature reasonably student degree freedom cf transform expand power incomplete every matter observe series constitute summation coefficient series assume valid domain turn assess precisely normal student cdf formula student beta simple form available page interesting daily shall turn central within treat tail tail goal excellent course efficiency arise numerical make issue transform student transform form useful explore elsewhere consider building quantile live distribution quantile distribution know probit interesting see reference general modelling precision rigorously achieve intermediate distribution gpu four explore issue give quantile mechanic diverse think appropriate consider difference precision seek minimize seek figure mind goal typically rational abundance second actual answer significantly detail mathematically model newton quantile relative matter error accept normal quantile option distribution interest side exponential degree chi square give student freedom pdf cdf quantile function quantile chi square random quantile also look quantile formulae side essentially base transform gamma normal tail double tail computer impact mathematical try cpu normally sophisticated management branching extent branch efficiently branch divergence group execute branch execute branch problematic evaluation evaluation change tail algorithm apply lower value effort essentially region final precision vote whole architecture multiply operation hardware create performance support cubic support polynomial rational device reduce factorization speed reduce polynomial specific trick degree exercise loss several iterate rational evaluated architecture quadratic comment cost work polynomial various mode expensive main quantile helpful inefficient note reciprocal root suitable scale filtering explore precision exist double consider modification iterate construction implementation employ arithmetic way essential precision region outside interval precision majority theoretical relative rather supremum norm region plot close machine growth concern early design precision precise room optimize speed goal approach exponential sample exponential sample detailed mapping region odd symmetry cdf solve however neighbourhood coefficient far enough different need retain equation point statement computer branch rational positive quantile region quantile iterate break computation thing break sensible break split slowly vary central fig much aim rational pick fig show character function precision relative rational target explore arithmetic quantile tail create small neighbourhood near employ desire fig polynomial respectively nest form produce v denominator completeness normal sample scale simplify unity evaluate rational pair employ entire range appendix next precision realize formula illustrate fact main tail fig precise tail carlo obtain request reality pass c see computation reveal scheme realize precision reality utilize
heavy detailed collection tuple consist string eq large class vc call inequality relate vc dimension maximal event frequency respect empirical distribution dirac measure inequality measure tuple q expectation refine empirical well come expense large let measurable polynomial vc concern universal source universal parameter euclidean nonempty interior reliably reconstruct regularity implicit depend condition approximate invoke exploit proof back bernstein yu laws stationary nonparametric strong finite move integer constant show absolutely continuous root polynomial circle complex plane decay open ball radius center weak source indeed p function lie open around attain evaluate taylor expansion regularity fisher normalize entropy recover easy check fact form satisfied independently density idea class distinct note consist q pass hypothesis draw independently classifier nz yield classification q relation vc cardinality sample imply finite hypothesis test allow weakly list regularity must result source every effective notation description encoder decoder active variational one say code finite memory dimensional compression finite memory code additional allow decoder identify asymptotically infimum immediate improve lagrangian code code fx expectation l c n shannon gray universal also sense terminology minimax universal behind notation suffice universal variable exist code prove throughout code operate decoder access bit later exist achieve come lagrangian optimum encode encoder convention infimum empty encoder look database find first encoder vector string string iii receive determine happen ball around estimate slightly furthermore almost lagrangian optimum comprise si encoder refer encoder parameter decoder decoder stage define code encoder n nc n assess gp contribution x random expectation term optimum motivate section decoder fidelity inequality assume unless use exponent divide block parameter although act use distribute marginal denote copy distance increasingly mix equality yu shall heavily hide database codebook construct absolutely everywhere stage codebook infinite decoder order store difficulty generate encoding generator encoder entry suffice describe define th density call key md estimator regardless construct encoder decoder pair parameter encoder look codebook find encoder hold every codebook performance bind need expectation zhang x everywhere almost eventually surely surely respect source codebook geometric parameter q x lemma almost codebook asymptotic bind n event triangle write nf n follow parameter follow sufficiently sphere imply accord invoke sec imply lemma side take expectation q n codebook realization codebook turn code boundedness distortion invoke finite codebook bit marginal together condition furthermore n dp boundedness lagrangian nx nx lagrangian expectation term due examine approximate expectation nz nz nn term encoder eventually expectation estimate obtain put everything together almost codebook np wish surely cf recalling imply follow inequality choose condition borel process class source remain order entropy source np algebra fix around degree polynomial six vc therefore autoregressive source autoregressive filter exist filter coefficient unit set root lie outside unit absolutely invoke process geometrically mix sufficiently condition asymptotic fisher information recall discussion conclude condition verify set form x r entry real variable bivariate process chain conditionally markov observable interest reference therein finite homogeneous chain sequence matrix ia ergodic irreducible exist unique initialize sufficiently far away past side alphabet specify density lebesgue channel source fix channel fix parametric though proceed verify meet ij n exponentially mix measurable fs random uniformly independently bivariate invariant mixing bound mix well condition condition examine fisher transition density invoke finally write satisfied mix universal scheme joint compression identification lagrangian redundancy variational estimate converging block quantify marginal generalize previous outline research paper priori would interest parameter hierarchical could technique structural g chapter adaptively distance play especially gray require select code base scheme toward compression would devise objective issue optimality interest say conceptually indicate link source exploit present universal coding therein treat statistical problem distribution close spirit minimum source complementary perspective code detail lagrange optimal exposition modification elsewhere alphabet alphabet induce measure marginal define np p n see gray eq infimum side eq set finite concentrate countable ci satisfies minimum expected see probability eq intuitive lagrange multipli encode point deterministic lagrangian block variable operate distribution py q np I c associate q np nx n x c prove give lagrangian close np q np np dp use triangle show distortion lagrange positive rate let encoder encoder pc p zero encoder decoder pc pc author anonymous useful suggestion joint rate compression performance source source mix smoothness universal joint compression lagrangian infinity source several parametric source minimum universal quantization source code complementary objective capture provide optimal precisely within class coding show modeling accomplish jointly regular alphabet source maximum likelihood stationary ergodic alphabet source amount construct address statistical modeling universal coding long knowledge statistic certainly helpful design apart hamming source error measure extract reliable rate distortion code distribution realization rate distortion emphasis compression instance chapter control observation controller modify discrete govern finite controller digital capacity bit
loss generality normal degenerate stand kernel hoeffding statistic symmetric kernel result h also eq order z z independent circular symmetry length arc circle point asymptotic variance z normal volume volume vertex spherical derivation variance correlation matrix formula symmetry derivation general geometric reasoning use correlation q note x expression spherical sake brevity omit reader david correspond concern omit integral express one though variable appendix label e nh h paragraph close strictly enough introduce find z inequality vanish tend well fact rs theorem v factor l nn h w w w w w complement event last expression suppose arbitrary adopt mean mean string alternate suppose r alternatively frequent throughout special rule refined rule along monotonicity et variant wherein use express pairwise justify proof six statement rt accordingly ensure algebraic interval algorithmic monotonicity choice lemma one phase second rule monotonicity throughout shall unless treat rather calculation perform software detail output make request rt ts format label stand phase dedicated lemma rt third rt prove increase perhaps even quadratic true yet proof eventually adapt r polynomial degree arbitrary upon recall visual q rational implication root shall simply similarly denote simply root concern us root respective rt argument accordance imply note general rule existence imply note finally hence continuity notation repeat rule next root show sign next whereas root imply next similarly imply lastly ts appendix ts adopt imply refined rule special case rule next root refine imply r g l establish program evaluate refine also rule similarly existence rule continuous root two root imply imply imply existence root rule note continuity lastly rule continuity rule remark precede hence recall reasoning proof stand regardless interpretation ts appendix follow adopt notation repeat imply case rule imply refine general existence imply imply yield z z f f g r g f g r r see lemma accordance rule single imply imply distinct root similarly root show general rule four interval find existence well via rule hence continuity imply rule continuity limit rs argument lemma repeat rule limit existence imply show rule rule imply exist special case show continuity find imply rule lastly imply place place begin six corollary replacing recall even desire proof sl r sign monotonicity b refine rule thm thm thm thm pearson common correlation setting nonparametric show normal efficiency monotonically coefficient increase monotonicity ta r another immediate corollary proof monotonicity pattern pearson correlation bivariate reduce correlation asymptotic could form show strictly stay additionally quadratic shown corollary theorem statistical certain sequence condition properly parameter neighborhood continuously differentiable express asymptotic ess bound mean increase increase
threshold set computational frequentist sparse network address yet give log likelihood marginal know density tractable value involve p likelihood variational maximize maximization minimization kl divergence approximate analytically believe kl depend nevertheless criterion rely algorithm bind criterion call estimate vertex practice result carry throughout experiment indeed contrary algorithm analyze core recall sbm criterion detection data sbm retrieve concentrate criterion experiment generative otherwise complex structure model connectivity matrix probability proportion class apply network various note l q simulated different initialization estimation henceforth keep tendency context behave consistently true whereas context graph make community class methodology represent chemical compound part vice section prior beta il repeat initialization indeed initialization class axis axis correspond learn frequentist dot representation bayesian maximum posteriori frequentist eight six probability connectivity compound responsible clique single clique sub clique class since structure retrieve topology class merge block sbm full bayesian informative approximate posterior non sbm focus density relevant number illustrate capacity sbm retrieve network look computation seem sbm likelihood accept acquire cluster vertex connection paper sbm sbm work variational expectation criterion estimate component criterion sbm study tend case propose criterion variational em random variational em bayes em date date field biology social science edge object internet lot attention develop group model mix vertex mostly cluster depend propose asymptotic approach map quickly position cluster conversely position simultaneously well consume r particularly suitable bipartite numerous encode group inter network author cite paper detail description mixing look efficient parameter belong otherwise introduce non paper focus sbm originally develop science network vertex belong describe intra inter proportion make structure account sbm characterize network locally dense extent many exist many sbm difficulty model variable due propose approach introduce posterior software package give posteriori handle vertex em sbm strategy lack bayesian criterion aic intractable tackle use criterion integrate easily compute sbm originally separate case fail interesting structure network sbm class deal emphasize sbm call integrate view detailed overview sbm tractable asymptotic obtain informative model implement work web http undirecte relation symmetric sbm vertex vertex vector x iid edge suppose sbm describe general allow concentrate without loop q z graph account loop sbm block membership stochastic block capture partial class sbm single identifiability identifiable except two sbm bayesian consider otherwise see constrain element sbm frequentist informative model distribution mix informative jeffreys distribution dimensional simplex fix informative jeffreys direct long variable hyperparameter distribution algorithm sbm full rely section asymptotic approximation term quickly become tackle well em stage estimation describe value old old
natural test vector generate sampling overlap face vector training search solver experiment dictionary weak order respectively threshold negative classifier order classifier w ix opt h ix jointly expect order value frame ix synthetic datum display comprehensive test minor modification find drop round eqn employ ensure refer investigation scaling suggest reduction classifier chose determine validation stop minimal result dimensional quantum instance variable result obtain coincide determined training overlap represent fit infer minimum quantum simulator initial hamiltonian quantum hamiltonian gap ground hamiltonian notational seminal need gap collection typical note extract minimum special resort consist unfortunately number variable currently attempt simulator minimum estimating derivative state relate gap ds correspond interested assume derivative polynomial exponential whether synthetic set encourage quantum wave quadratic unconstrained loss version traditionally learn preliminary usually attractive application dataset depict eqn employ adaboost outer square dimension large objective action unfortunately expense minimize small minimize training classify replace value purpose ever becoming need hardness objective could search conduct analysis resource lead small classify bit versa classify correctly objective contain need variable number weak classifier parameter process outside spirit thus far formulation quadratic amount variable live stay wave processor format elegant form sum introduce encourage weak b look like special context dependent exhibit execution importantly example incorporate principle allow incorporate priori principle example train detector impose image nearby symmetry continuity weight optimize formally thank zhang discussion boost david quantum initial g wave system com discrete classifier thresholded sum classifier motivation cast format amenable yield superior heuristic solver solver advantage communication training candidate weak weak learner use classifier exceed handle effectively piecewise optimization numerical loss add superior version boost carlo quantum detector strong choosing simultaneously set choose regularization complex regularization encourage strong build weak classifier maintain training accomplish solve follow number boost optimization bit depth represent small deal binary consist blue training color datum light color parallel hyperplane classifier situation employ four negative area adaboost subsequent round contain negative become severe greedy configuration adaboost see handle adaboost exponential employ quadratic show lead bind classifier vc dimension classifier bind compact achieve weak come look weak e merely demand reduction switching associate incorrectly eliminate expense vc lower equal adaboost classifier use rich need illustrate practice weak regime determine regularization strength perform trade increase gain baseline system namely employ adaboost rather exponential essential function employ perspective quadratic increase I possible contain global large classifier fulfil weak handle global look art solver wave train often sift moreover dictionary learner dependent mean cardinality typical classifier thousand rather strong solver problem hope reasonable quality inner loop algorithm handle learner need construct classifier small weighted opt hx unweighted yield small inner
share issue far later time contrast situation happen co next add statistic say likely formally circle support relationship dot statistic emphasis plug formulation still trade compete complementary retain flexibility model plug actor phenomenon would ability box acknowledgment participant comment fu cs evolution extension graph model well evolve network communication field social population communication relationship actor structure behavior global increase demand tool subject modeling depth network single flexible include classic extension model cluster model role relevance capture signature connectivity pattern statistic social probability intractable represent model graphical regard clique potential representation vary possibly include type asymmetric actor relation actor actor evolution sequential observation wish network community trend evolution rise time model dynamic example event represent actor link local neighborhood dynamic view exploration model beyond work quality series previously though recently explore issue flexibility parametrization model specify compare elegant indicator section refer capable evolution flexibility general furthermore formalism exist past decade readily temporal maximum likelihood fit capture signature dynamic hypothesis application begin way simplify evolve representation single make put property give generalize admit representation specify k understand clique specify accomplish simplicity presentation detail special form function possess framework weight adjacency matrix govern tie tendency control tendency link result tendency simple however actor possibility deal type task model network sensible normalize often mcmc study modification expectation network gibbs conditional unconstrained optimization newton direction relate family tailor model initialize convergence sample I I affect accurate iteration need however early stage sufficiently precision resource remain general procedure given computationally perform newton examine loss pmf z initialize range initialize density represent average convergence distance exact newton rather sample much approximation perform return almost identical indistinguishable concern express degeneracy issue arise model explore space mass complete several expect degenerate distribution intuitively degenerate slight variation become degenerate namely degeneracy generating degeneracy distribution converge aforementioned require fail question whether issue also affect temporal extension factor edge problem initial phenomenon example entry minimize empty maximize empty entropy quantity thus reasonably get intuitive entry entry plot example option fix yield plot plot bernoulli briefly graph calculate class analytically show class accord since edge exchangeable purely situation equivalence purely distinct significant calculation entropy computationally tractable small magnitude generalize discussion satisfy eq note number eq entropy imply entropy upper entry take factor expect marginal aforementioned hypothesis see generality pay broad scientific hypothesis hypothesis write significance represent serve plug potential compute united actor proposal resolution serve proposal possibly record create slide consecutive window direct relation window toward evenly spaced slide window proposal proposal window proposal series first test inherently formalize previously membership political otherwise potential represent alternative hypothesis write ratio compute estimator ratio mle value hypothesis mle alternative composite value bit seem tractable analytic unconstrained optimization population sequence nan mle calculate calculate empirical likelihood genetic decrease brevity introduction approximate form likelihood derivative analytically particular likelihood optimization directly without however might likelihood possibility likelihood divide value ratio dividing alternative even encode hypothesis well write specify alternative densely transition clique densely structure additionally actor allow random subset population know label evolve accurately infer actor label party modify party multinomial know party randomly leave posterior unknown infer assume label sequence fully observe since likelihood straightforward model infer party algorithm use observable unknown
select compatible assume compatible termination prescribe algorithm denote exist constant diameter confidence may constant either c usually hoeffde inequality bad rate region diameter ensure crucially depend construction regularity continuity approach assumption duration phase expectation regret obtain constant duration exploration terminate reach diameter maximal terminate balance terminate policy regret may instead bound terminate region rather assume actual formalize alternative construct exist constant policy bound decrease instance access interesting illustration different optimal long compute play role stochastically identical channel consider single channel bandwidth channel receive decide observe channel policy bring light define secondary user channel consists observe correspond policy represent straightforward policy infinity aggregate observation horizon confidence fully consists sense resp resp visit markov ensure confidence region regret algorithm assumption secondly half center large finally reward lipschitz exploration observe since channel lowest free positively positively long policy exactly compute secondary value negative knowing region similar verify rectangle whose include distance less illustrate run replication process horizon take equal empirical distribution may observe represent otherwise indeed actual policy exploration phase correspond inside exploration long policy capture exploration phase important exploration close resp really resp later central limit transition order hold observe transition difficult however long particularly corner indeed channel imply transition strongly positively decide reinforcement algorithm applicable channel balance monitoring phase pre specify furthermore guarantee single stochastically channel indeed exploration stochastically rapidly potential wireless communication concern able general stochastically channel adapt principle phase parameter possibility either region sum duration exploration violate f additional fact true denote use assumption minimize nc distance soon policy equal therefore c appendix prove event c c remark last hold nc nn conclude paris france fr consider task channel primary independent secondary statistical system problem cast markov decision aim exploitation requirement provide finite horizon regret performance channel stochastically identical channel cognitive focus effort make use portion band primary secondary cognitive user secondary user carefully identify resource primary access potential wireless previously channel secondary search channel availability evolve markovian long channel planning class partially decision assume primary traffic secondary user statistical traffic must secondary user select close reinforcement learning carry policy reach issue consider asymptotic rule exploration consider user simple source reinforcement none allocation contribution propose strategy adaptively exploration phase determine come form horizon sake clarity abstract parametric remark correspond transition availability channel rely parameter value planning problem case allocation detailed use stochastically identical channel consider article organize follow access detail stochastically consist channel channel primary channel either secondary channel primary sense channel secondary channel maximize transmission introduce channel channel channel resp channel additionally denote channel channel channel gain available unobserved channel reward depend observe may channel receive channel penalty channel receive channel exploit dimensional internal ti ti state enable secondary channel sense internal recursion denote last ki ti equation may channel interpret introduce planning bandit achievable number channel become important nevertheless recent near call reduced cost consist separate channel interestingly determine planning value explicit expression index sense learn act optimally learn high chance channel learn coincide cost question secondary apply well exploitation adaptively monitor function condition restrictive case mdps case like channel also q f unknown probability choose
modify denote run monte simulation fashion follow run simulation probability w p computed exact sec equal probability fact element threshold increase c one select maximum map model signal mutually mutually I train compute close thus step algorithm feed back appendix solution summarize discussion perfectly outside causal simulate camera static model acquisition orthonormal video denote norm mask contain randomly row entry take cs ls cs minimization table big store need solver include ii multiplication big like dft time solver use modification code interior sampling mask bit storage fast multiplication link image fig set table even size reconstruction measurement approximately result cs cs mod cs mod cs instant showing greatly work either tv former parallel modified question recursive noisy stability currently open dependence recursive time approximately anonymous contain pixel spatial magnitude nonzero minimize pixel subject designing homotopy cs sequentially measurement application contradiction nonzero full rank happen constraint since solution define iteration apply j follow apply c disjoint equation simplify absolutely convergent define two j r rt belong thus give summation u become dirac delta markov statement laplace estimate solution causal time normalize constant since I support institute ph university college electrical currently transaction interest current reconstruction sequential large tracking receive electrical engineering department china ph electrical research focus interest include theorem corollary definition edu part appear support nsf material purpose request projection although knowledge spatial support instant weak compressive error compare support important extension signal show noiseless measurement although part example mr discrete wavelet basis know reconstruct spatial support previous application dynamic camera video compression image refer energy transform number notice change sense cs conditions reconstruction cs residual ls problem know replace observation l residual greatly reconstruction signal require sparsity cs able reconstruction fewer noiseless measurement need cs reconstruction need modify cs relaxation exact reconstruction cs estimate weak cs support change slowly measurement true practice develop extension regularize modify reconstruction shorter appear work probabilistic include sequence use except reconstruction even demonstrate reconstruct difference approach reconstruct cs gradually become sparsity result achieve whenever cs actually point anonymous bp anonymous multiscale cs compression improve organize modify sec condition exact modify cs cs modify cs sequence sec sec count element norm contain belong notation complement w operation denote restrict isometry small cardinality orthogonality respectively notation vector part unknown thus denote denote denote size know denote property rip rip describe introduction unknown coefficient certain threshold tell intensity mostly index nonzero detail series instant occur case nonzero newly reconstruction current similarly case small may element word like empty know I threshold cs give result sec complete argument give disjoint reconstruct minimizer equivalent compare version true relaxation sufficient slightly strong next subsection whose unique condition cs simplify u k reconstruction consider reconstruct usually consider measurement give large ensure hand hold third term third term value small example differentiable multipli lagrange subgradient using replace satisfy subsection construct disjoint prove apply lemma iteratively construct satisfies condition theorem disjoint proof subsection outline prove theorem proof rs j fashion define argue satisfie entire appendix previous subsection motivated least minimizer need hold equal minimizer inequality less xy ax jj x full thus minimizer even thus set disjoint rhs rhs since ta definite denote matrix denote decrease take ks thus hold height eps nm nu small simulation exact modified cs sec compare cs cs express serve decide need meaningful cs corollary cs two obvious weak cs reconstruction either compare scalar e condition clearly alternatively probability binary reconstruction random h reconstruction random fig three different choice see allow measurement reconstruction cs maximum allow reason significant sufficient cs modify cs generate gaussian entry rip rip affect performance way follow time generate random generate uniformly element call output cs divide various value result cs cs hand cs work cs work reliably require say modify cs estimate b modify anonymous value reconstruction occur pursuit cs handle modify se c c e cs e cs
seed leverage implementation appendix metropolis sampling scheme seven scheme rr rr rr ac min ir ir median ir ir median ir rwm rwm likelihood leverage outlier allow outlier run small highest computed independent metropolis bridge cc particle pt also describe adaptive sampler processor furth table metropolis adaptive hasting parallelization rr rr rr rr ac min median ir ir median ir ir median ir rwm mn consider poisson gamma mcmc binomial model mean shape success marginal example poisson walk figure possible series priori poisson present monte replication seed simulation implementation walk metropolis seven hasting rr rr rr rate max median ir median ir ir ir rwm mn rwm mn show distribution summary result present mean standard particle binomial run use sampler eight processor nearly adaptive l rr rr min ir ir ir median ir ir rwm consider dynamic priori j poisson pattern model representation differ also explanatory multiplicative analysis capture change ht presents monte study seed adaptive walk metropolis seven hasting rr rr rr rr min median max median median ir median ir ir rwm c mn rwm mn eight use five show intervention consistent report intervention ht cc particle filter intervention intervention level intervention intervention run second iteration metropolis hasting run eight processor implementation summarize median ir ir rwm auxiliary particle rwm table contain trend ht cc cc research partially arc discovery dp datum resample sir fix notational suppose particle take filter density distribute filter dirac delta sample estimate denominator rao form efficient weight discrete univariate multinomial bootstrap sample associate method proceed time replace multinomial resample bootstrap strategy hold auxiliary density py z proof appendix paper walk proposal multivariate density covariance iteration laplace scalar sampler locally walk proposal multivariate adaptive simplifie refine take random tailed leave proposal adaptive parameter scheme stage throughout two term heavy third fourth tail preliminary run walk mean normal normal covariance begin component schedule depend ratio stage construct tail local mode vector explicitly bridge iterate q adaptive proposal positive reasonable py simulation alternative eq tailed importance ratio code matlab code write file use file resample step algorithm library cluster compute intel gb run processor give implementation simulation simulation processor particle iteration sampler metropolis hasting sampler iteration metropolis hasting draw eight processor update completion block standard particle detail sampling simulation processor particle filter metropolis hasting sampler perform particle hasting simulation sampler metropolis eight update occur particle eight process sampler corollary south edu se economics university south edu economics university ac uk feasible model adaptive hasting approximated filter base construct adaptive independent metropolis hasting proposal attractive parallel processor marginal obtain efficient bridge feasible exact state adaptive sampling evaluate analytically transition justified work likelihood uniform metropolis base time construct efficient adaptive proposal evaluate q likelihood compute mcmc parameter integrate kalman general auxiliary sample review carlo integral computationally standard particle approximate become particle tends infinity standard particle particle likelihood particle filter suppose wish hasting give initial proposal otherwise depend regularity detail iterate regularity iterate draw application available auxiliary particle provide unbiased particle filter conditional iterate use auxiliary augment u adaptive hasting scheme follow space suppose ii proposal metropolis scheme converge sense set integrable respect tail binary binomial similar ensure outline theorem theorem particle parallel processor available apply estimate processor filter use single processor make second apply hasting sampling
position generalize test memory link small integer total sum see define case decrease tm relie expression backward refer normalization algorithm correspond order interest introduce subsequent introduce two order either eqs iterative functions eqs function right normalization since step subsequent z boundary approximation constant eqs benefit tm another couple wish tm tm mc random sampling compute know technique set sample indeed happen become perform initial purpose approximation marginal e chain value tm gaussian random possible independently draw derivation iterate normalization constant expression replace return final tm introduces factorize introduce expression drawback variable connect arbitrary successive relationship probability interaction close neighbor consider variance expression couple write appendix behavior approximate impossible tm approximation average number perform start study channel discuss channel reduce eq probability maximized distribution occur expression event general finally expression channel consider write parameter notation error truncate x tt hand define gaussian impractical numerically take I use previous write expectation eq expression separate accordingly consider fact function expression distribution study subsection behavior section tm discard vary eight figure couple fig fig value memory thus tm vary dependent couple average error vary h bottom one top mc tm mc tm tm mc use tm mc mc tm mc expect trend tm notably tm bit decode perform tm par specifically tm g perform well tm tm mc also tm tm tm mc similarly outperform tm though link explain mc use poorly tm mc fig per sample different exactly combination tm tm case four combination tm mc tm tm tm function two tm mc tm combination algorithm relatively though g outperform grow study total length p f tm mc probability correctly decode total length big rapidly decrease explain graph fig long tm mc sample tm show thing tm mc slightly tm tm latter behave variation fig never vanish use notable instead error various incorporation inspire test sequence system obtain template range test genome gs machine separate sequencing model additional approximation estimate gs introduce deviation base compose alphabet dna read repetition sequence see consideration thing non incorporation rate positive read length datum estimate value small slight issue communication present mc plot correctly would extract tm mc gauss within total gs capable correctly read template basis template gs least indeed value since encounter decode tm mc concern indeed correctly chain length also decode sequence great error decode chain tm mc emphasize direct algorithm matrix tm tm gauss tm final tm g gauss perform memory small tm memory large besides yet test variation need quite future preliminary assume x gaussian variance value eq normalization keep recover iteration keep close reconstruct one marginal marginal need latter procedure calculate compute expression write need situation decompose conditional probability expectation eqs follow necessary distribution overcome regarded distance minimize wish constraint multiplier furthermore eqs constraint hand q right equal prevent appearance static choose place wish thank proposition conjecture infer markov physics language dimensional external accomplish become intractable several field transfer realistic model dna markov model application range speech sequence whereby state chain conditionally formulae fundamental algorithmic hmm infer thought state symbol reduce physics boltzmann dimensional temperature sequence act analogy marginal sequence state precisely multiply time become memory underlie multiple state standard hmm augmentation exponential lead severe length propose basic intuition length get transfer mean concrete infer dna carry trace position scale thus plain impractical dna motivate complexity describe analytical result collect positive integer e observe non recursive depend observation memory also side effectively sequence describe include except exactly noisy regard state precede high model posteriori boundary condition probability write construct use symbol decode map decode would computation entail summation grow section four limitation give subsection take success construct integer bernoulli correspond generate directly use decay rapidly still distribution distribution c enable decoding case otherwise derivation equal decide subscript original synthesis review concrete history one sequence dna chemical test become standard low efficiency sequencing
requirement practical believe allow assume truth concern corollary remark selection different concept relate eigenvalue slightly allow hence coherence restrict isometry keyword phrase compatibility lasso eigenvalue isometry examine relation oracle among selector property relation fairly study noiseless space dictionary consider fixed inequality oracle eigenvalue present condition tailor difference overview compare literature explicit display enable indicate implication section rigorously deal compatibility condition many compatibility approximation compatibility compatibility improvement study additional implication depend discuss invoke gram product form play small positive singular gram matrix compatibility name compatible definition compatibility condition restrict compatibility successively strong version size invoke simplicity define set q necessarily eigenvalue complement restrict eigenvalue call eigenvalue introduce restrict adaptive restrict restrict depend gram section compatibility condition ss l mainly situation study definition orthogonality orthogonality constant moreover isometry isometry eigenvalue eigenvalue restrict isometry weak isometry restrict isometry rip isometry rip modify coefficient condition condition strong definition condition meet spirit tight coherence compatibility imply derive later compatibility go lemma assertion lasso note inequality hold inequality implication lemma assertion statistical case condition essentially weak oracle noiseless isometry uniformity rip rip clear restrict isometry constant demand rip oracle inequality selector show weak isometry property restrict restrict n restrict condition compatibility easy calibrate prove compatibility might pay oracle regression end give put combining give essentially summarize sufficient compatibility somewhat elementary projection moreover adaptive word regression restrict oracle refer condition restrict definition hold imply conjugate quantity eq replace might eq take apply lemma give eq lemma vector use orthogonality constant q anti define moreover eq hence subset eq eq small positive since q kkt suppose weak know arbitrary must arbitrary weak condition equivalently restrict corollary q rip enough constant condition picture programming minimizer programming condition exact say restrictive compatibility prediction also multiply eq condition q moreover span uniform eq mu verification lasso type therefore look rather trivial non restrict population restrictive compatibility gram relevance replace design variable often population even design compatibility supremum generally may eigenvalue eigenvalue corollary r mu q result show find well eigenvalue consider column column denote population row one eigenvalue extend tail empirical section implication eigenvalue matrix vector restrict hold condition meet hence toeplitz sense spectral unique toeplitz small bind block drop minimal satisfy eigenvalue restrictive small matrix behave example compatibility calculate first satisfying small easily moreover uniform hold compatibility follow get behave noisy noisy design abuse notation define behave case suppose distribute noisy take define hence thus eq may case compatibility involve compatibility eigenvalue situation kkt noisy matrix repetition define generalization anti space span kkt
learn expression various agglomerative regularization hierarchical regularizer induce structured effect bias covariate strength coefficient regularization lasso snps association strength treat select separately lasso penalty union support capture penalty let define overlap response covariate snps properly calibrate consistent group organization although previously use penalty take advantage structural response due weight different group arbitrarily lead contrast systematic scheme apply balanced penalization coefficient lasso lasso weight adopt loss structured induce group computational typical time number node advantage graph fusion introduce fusion penalty fuse weak association false could regularization selection relate response different select response agglomerative widely preprocesse classification subtree height regression extract cluster gene form average regularization incorporate present datum detect signal successfully remainder paper organize brief discussion previous estimation section introduce experimental collect trait snp minor response center column intercept column th number covariate effective identify element matrix tuning control provide mechanism enforce literature group adopt multivariate nonzero share community estimate coefficient eq denote coefficient response covariate norm encourage coefficient encourage th covariate would vary different response realistic expression level snps snps gene pathway snps relax add penalty regression within zero share limitation regularize regression grouping structure response multiple considerably add advantage correlation hierarchical structured sparsity norm microarray measure thousand gene correlate sample imply share common pathway variation snps module relate act module derive run agglomerative expression natural extension incorporate output hierarchical cluster genetic variation influence module build lasso leverage statistical expression gene primarily mapping generally applicable multivariate dependency object speech valuable enhance relationship vertex illustrate node response subtree root internal bottom tight response root weak subtree knowledge resource gene database hierarchical agglomerative cluster associated subtree root represent correlate normalize tree expand overlap group overlap tree follow group whose member response variable subtree root regression tree bound weight near leaf common covariate amount penalization group internal relevant separately response child root tree term equation net slight difference elastic weighted nonzero response covariate heavily equivalent multi response share response parent penalty penalty tree single node leaf root node operation norm penalty norm encourage joint balance norm lasso penalty reduce penalty contour surface penalty show pt cc penalty ensure overall amount group article belong multiple different internal appear always weight scheme guarantee variation lasso overlap group previously overlap arbitrarily result unbalanced show weighting scheme inconsistent construct tree response tree recursively root leaf th covariate w w g introduce structure branch forest tree tree leaf individual main nonsmooth coordinate algorithm apply nonsmooth group overlap penalty prevent closed optimization tree formulate cone interior approach gene general sparsity induce penalty function share handle group fuse special overlap method introduce approximation nonsmooth smooth fista accelerate objective accelerate descent rewrite split two correspond internal leaf overlap weighted weight response contain overlap auxiliary covariate j v j j c matrix row column note tree penalty dual optimize overcome challenge smoothness nonsmooth g ty smooth gradient continuous v penalty fista adopt lasso give proximal penalty obtain close thresholding w give role determine compute back tracking iteration store tree lasso complexity quadratic cubic demonstrate compare regression assume response parameter model range error validation give low prediction evaluate criterion sensitivity specificity specificity sensitivity rate square base scenario generate correspond predefine group response share covariate cluster figure three avoid expression study divide set b hierarchical response avoid clutter row response column covariate method lasso regularize multi nonzero lasso strength positive coefficient across combine get response vertical hierarchical visually clear positive response true relevant covariate use average simulated datum systematically performance generate receiver operate regression coefficient average nonzero figure outperform regularize regression especially knowledge correlation prediction error average simulated figure show figure include error low error addition true hierarchical agglomerative along internal node since obtain manner represent discard root thresholded tree even available benefit account response error experiment range value cluster gene snps expression miss know module hierarchical module correlate agglomerative internal goal variation gene response incorporate able activity co express pt ptc ptc correlation agglomerative cluster coefficient panel accord agglomerative estimate row represent gene choose validation lasso extremely reveal snp weak unable detect signal strength gene correlate expression regularize task across tend vertical expression perform pt go category nonzero coefficient group module go interested identify snps gene module encourage hierarchical reveal element go gene nonzero snp absolute value estimate figure biological biological snps go regardless threshold select association estimate generally find lack figure finding provide snps influence gene focus affect gene biological molecular bp biological process molecular cc bp mf activity bp mf activity bp mf mf activity activity pt family stimulus response bp translation cc mf activity mechanism cc generation cc mf mf list group snp lasso strength category genomic location
assume follow symmetric select secondary shall decide channel bid bid single channel channel cognitive cognitive utility finally bid stay allocation channel allocation equal bid winner pay presentation assume mechanism write observe current rate implement observe shot entry well typically strategy history far participant knowledge player assess accumulate observed cost payment access history channel accumulate utility accumulate cost action utility intuitively beneficial stay incur accumulation thus avoid optimize even centralize manner space individually bid straightforwardly dominate strategy bid price consist game differ channel detail subsequent single drop stay reduce first thresholding identically cdf cdf state low evaluation difficult accumulate reward individually associate payment end simple approximate side maintain private update tc obtain associate payment choose stay payment winner payment amount payment payment amount estimate move old summarize output ta bm I som mt mt mt p mt access problem regret explore exhibit set action proportional play action two playing denote payoff bid view average channel k p channel prove access action decide though control channel I sufficiently investigate randomly place fix away center set transmission bandwidth assume length hz propose compare htb understand propose figure snapshot change monitoring converge quickly furthermore always bid decide bid past active pay monitoring decrease effect simulation user htb monitoring cost well see entry increase utility decrease gap selective likely high high show adopt resource allocation channel repeatedly entry htb htb next average bid bid generally agree high bid average bid also stay treat bid slot cost win htb vary respectively decrease dominate performance utility gain channel experiment performance convergence user ga regret users channel ga channel instead channel ga upper bind practical bad bid perform problem channel adopt access history always show greedy plan convergence may inaccurate acknowledgment air force office scientific foundation grant access cognitive model subject entry cost cost incur primary activity secondary user successful channel regard activity game propose outcome balance recent study despite actual spectrum period cr exploit spectral secondary wireless device whenever challenge research cognitive spectral efficiency tradeoff previous aspect spectrum access sense term throughput share instantaneous channel study sense propose improve detect aspect spectrum also receive attention allow maximize access account penalty therein sense cr network affect author access find correlate propose change arrival arise desirable access availability decrease transmission channel transmission spectrum explore transmission across heterogeneous common control channel information operator access allocate area intelligence outcome contribution paper spectrum access problem cognitive single rest paper organize terminology mechanism give cognitive channel reasonably estimate channel information primary
thus homology coefficient homology degree later topology form differential analogue adjoint definite operator theorem space identify kernel harmonic compute definite harmonic almost orthogonal class former representative space alternate due preserve proposition formula alternate define relate definition inclusion induce variant simplify relevant fact hilbert adjoint furthermore condition q former unique representative latter condition prove eq orthogonal hold imply v f pf pg pf w hilbert adjoint close include completeness q open suffice establish imply act trivially subspace map complement complement indeed leave zero leave side argument difficulty close sufficient first suppose bound banach finite without mapping close already finish lemma homology trivial check gx x proof guarantee k follow theorem theorem alternate trivial geometry borel measure non also throughout goal section alternate thing check bound proposition imply borel application theorem prove chain subspace constant assume function immediately alternate suffice think regularity pde continuous unique since bound identical proposition easily section chain construct whole trivial group theory operator throughout product observe closed diameter appear remark sense probability take alternate theory space respect f nonempty essentially note depend subsequent analysis q want emphasize dependence harmonic function eq component h furthermore equivalence inclusion consider map large rich condition metric hold immediate course formula harmonic minimize f ff see define section equal compact scale harmonic slice b xx x maximum component harmonic continuity fact slice locally harmonic form assumption unable form poisson regularity give show need regularity solution let lebesgue atom real number limit outside poisson regularity borel function measurable right suffice function continuous characteristic function measurable dominate continuity form partly theory put back symmetric extension define compact hilbert schmidt adjoint difficult section reproduce kernel poisson normalization kernel operator correspond operator eigenvalue complete eigenfunction reproduce next paragraph q reproduce finite limit theorem harmonic compact connect orient riemannian induce volume form ball convex geodesic lie ball point simplex face totally geodesic dimensional face geodesic geodesic segment construction vertex dimensional complex consider important visible determine tend homology orientation orientation orientation sign volume orient geodesic degeneracy open volume simplex vary continuously harmonic harmonic generate generate top constant curvature orient curvature harmonic scaling uniqueness degenerate x therefore orientation orientation geodesic segment curvature isometry onto onto easy define tt side establish let generality orientation orientation face orientation orientation therefore equal since case face depend face simplify opposite orientation right side equal geodesic term case orient surface totally generally geodesic triangle define case show orient manifold tuple iteratively combination assign orient evaluate generator space development relate homology theory metric metric section complex x complex necessarily complex open empty course complex chain check dim trivial intermediate complex rational coordinate intermediate case restriction b dx limit h xx construction see thus defined modification definition scale equip extent via continuous theory question extent map inclusion induce compact metric space equip borel topological reason see function space denote denote harmonic denote closed space analogous inner direct decomposition analytical analogous regularity equation regularity poisson answer problem continuous question show recall decomposition regularity show complete namely regularity harmonic function harmonic imply inclusion function induce regularity every degree representative suffice establish imply different class riemannian coefficient introduce similar sufficiently scale complex give detailed argument rather neighborhood diagonal cause difficulty compact metric alternate value discuss precede function manifold space ph h dimensionality involve fact collect rectangular space one check thus couple space operator respectively map linear map contraction follow fact algebra prove dimensionality leave complex augment chain map induce chain row augment homology need row de fact column lemma linear eq column follow complex also subspace corollary banach topological topology induce borel corresponding follow act horizontal explicitly tuple tuple chain horizontal map vertical bottom assign vertical map riemannian everywhere complex respectively cover complex complex condition borel open smooth riemannian induce natural inclusion map function arbitrary value suffice indeed vanish cover ic th measurable hc h th prove partition unity choose continuous riemannian manifold contraction replace ff restriction bottom augment argument row induce finite manifold ball complex take column trivially fix check define define contraction fix contraction slice x x empty finitely open borel measure hypothesis riemannian dimensional ball hold let covering since correspond show chain contraction column borel positive playing term chain contraction replace take give notion b rx rp x r proving assertion suppose imply x x iw u kb radius hypothesis open ball radius close closure let qx b b iw p thus continuity riemannian manifold riemannian metric strongly geodesic geometry hold hold proposition course second intersection radius proposition claim strongly minimize geodesic geodesic equal assume without geodesic sphere curvature imply sphere inside convexity claim single interior interior w hold denote exist establish single take subsequence take subsequence easy contradiction interior subsequence denote eq another subsequence large interior point interior contradiction proof formula arc point curvature geodesic comparison check sphere convex imply proposition convenience metric space metric converge respectively disjoint require require cluster diameter banach space canonical diagonal contain belong add tuple belong intersection center borel zero consider ignore tuple point characteristic scale look banach projection two time kernel complex vanishing sequence kf kf long etc obtain come six permutation degenerate simplex contribute without determined vanishe vanish value addition constant without constant observe function coincide way closure close ball coincide volume except ball ball point short inside metric write suitable similarly pair point diagonal neighborhood quite explicitly complex namely neighborhood projection map neighborhood compatible projection onto reverse map sided inverse function tuple cauchy bound projection consequently induce zero norm zero hausdorff inspire space infinite dimensional homology homology separable metric scale associate x dx point homology homology complex exploration compact metric homology miss infinite homology homology promise result homology derive fail attempt homology difficulty perturb equality homology high homology homology cycle equivalent high homology group scale homology length equal exist simplex substitute equivalence rely infinite dimensional homology group countable easy embed show sensitivity infinite consideration necessary homology dimensional homology infinite discussion diagram rgb diagram b degenerate cycle act homology last highlight must r shown suppose include eliminate boundary term new eliminate impossible return eliminate homology suppose need eliminate elimination either case boundary generator homology homology homology tailor nature homology change degenerate homology group homology reduce decrease enhanced version scale homology homology first case let homology satisfy value original exist cycle prove close
know unify confirm differentiable invertible matrix nonnegative nonnegative definite criterion hand side restriction concavity usually concave criterion illustration monotonicity assume iteration q word well denominator criterion simplicity formulae hold eq monotonic read consider monotonic read take ii monotonic criterion note point suggest desirable recommendation another henceforth vector zero denote identity design intercept nevertheless measure concerned coincide considerable numerical accumulate argument p use conjecture consequence g transform problem minimize jointly van ar yu although mathematic intuitively interpretation lead monotonicity monotonic establish regression key formulate specifically denote square obvious explain variance unbiased rank assumption square linear estimator sense definite matrix minimize estimator reduce upon immediately subproblem ii minimize formulate joint minimize equality problem upon natural minimize part closed form point maxima boundary therefore optimal present yu yu comment ensure highlight basic start assign weight point assigning tend valuable say check see satisfied coordinate nonzero equality check technical concern simplify positive maintain limit condition mention satisfied fail often condition illustrate converge remark iteration x z z tw rare practical think reason wide em monotonically although slow upon g monotonicity hold mathematical contribution way space possibility construct desirable monotonic logistic compute design prior guess space criterion weight design iteration locally verify simple serve monotonicity apply design space concavity long monotonic evident plot theorem monotonicity calculation monotonic evidence insight resolve alternative seek expected notation moment would section plan extension future work like david van grateful comment multiplicative introduce computing design conjecture confirm illustration approximate theory general focus finite space probability entry proportional closure nonnegative eq extend case allow use although positive typical criterion th criterion often combination naturally may special optimality seek variance unbiased coordinate blue interpretation argument apply guess chernoff optimality ignore prior dependence li henceforth assume dependence extension design early wu iterate
reconstruct motivated deduce small number application estimation wireless communication usually extra facilitate instance structural prescribed toeplitz etc complete completion maximize realization matter efficiently give arise context localization complete partially euclidean motion fundamental computer vision reconstruct analyze structure find matrix presence failure account difficulty complete customer complicate track customer preference system year maintain preference column correspond specify preference record entry consider preference item completion maximize force information specify reconstruction ill pose may criterion shall theoretic aspect rank wish reconstruct simplicity suppose initially available allow reconstruct natural ask entry manner theoretical problem network localization aforementione distance guarantee successful turn number pairwise distance reconstruct network perform solve semidefinite sdp get determine freedom singular decomposition svd orthonormal degree specify observe thus degree th freedom specify degree freedom specify freedom need entry infinitely many exactly whether reconstruct crucial reconstruction may zero reconstruct depend motivate reconstruction entry tradeoff entry reconstruct propose idea compressed optimization define notion coherence measure similar notion informally case reconstruct entry set singular whenever randomly formulate polynomial limit practice specialized solve sdp associate least subsequent improvement sampling runtime bound reconstruction certain coherence entry uniformly iterate probability refined analysis show input away reconstruction yet complexity first introduce notion speak sub intuitively stability large crucial present many small subset column amenable turn notion separable theory moreover analytic nature coherence nevertheless notion compare pursuit fashion note strategy rank namely strategy assume involve produce polynomial discussion et exact high particular low reconstruct furthermore runtime sampling complexity yield moreover essentially extra attribute phenomenon discussion derive coherence define study construction fact pursuit analyze sampling complexity possible direction section ability depend paper stable rank rank matrix unchanged removal nk nk row rank column may shall column stable sequel stability nice generalize notion stability I give let construction equal rank one whose equal easy verify stable two distinct singular coherence raise coherence formalize statement recall let orthogonal basis let orthonormal view span column coherence column establish arbitrary matrix let column iff collection complete ready negative suffice follow non measure linearly probability determinant turn statement haar conclusion reconstruction low matrix introduction reconstruction since priori information guess currently entry uniform strategy certainly reconstruction illustration form hope reconstruct entry randomly entry bound eq uses reconstruct large entry entry wise may critical localize matrix reconstruct first row general largely motivate follow reconstruction knowledge column examine draw column span eq proceed column linearly let entry express basis column terminate matrix illustrate flow rank row identify sub reconstruction entry low rank pursuit matrix proceed let remark speed lie index remove facilitate section analyze goal let terminate reconstruction simple theorem compute algorithm randomization complete terminate suffice b execute throughout towards end divide epoch begin end column iteration th basis column epoch execute parameter execute course execute r quantity distinct examine theorem corollary significantly orthogonal model least terminate reconstruction entry computational complexity initialization operation pursuit newly span index achieve via gram schmidt set orthonormal th orthonormal vector span index suppose select proceed select new column add basis pursuit since conclude pursuit bound row gram carry step use operation need reconstruct column follow total step summarize suppose let total perform suitably add counter time still however idea develop knowledge matrix reader bind compare sdp polynomial approximate polynomial reader issue early assume rank however priori raise question modify mention end last section keep pre specify pursuit proceed sufficiently probably find input exactly formalize propose initialize correspond recovered basis next column hence otherwise proceed b drawn belong increment span index belong new index find examine entry express th column combination basis coefficient turn rf least rf terminate exact total since total entry rank generic reconstruct note reconstruction much flexible need reconstruction rf algorithm find proceed step proceed facilitate proof divide epoch st epoch define exactly iteration st epoch hold rf find proceed rf proceeding probability inductive observe terminate
generate frequency word word database merge highly correlate coverage rectangle rectangle rectangle bf c bf level rectangle rectangle rectangle bf bf level word exclude right long ssc hierarchy acquire concrete age whereas continue visualize figure across entire connect across within triangle coordinate mark coordinate coordinate bf plot mark coordinate coordinate coordinate coordinate mark bf hierarchy write frequency stay factor analysis within exclude correlate hence effect analysis hierarchy across dictionary show c hierarchy hoc see test show triangle plot plot mark bf plot coordinate mark mark mark right bf right external initial age acquisition level level hierarchy whole abstract less writing far refine outer rest space level increase bottom level source whether effect distance hierarchy induce entire connect within alone turn successive level induce hierarchy beyond dictionary turn frequency continue frequency likewise continue decrease age acquisition level outer small acquire early corpus frequently ht corner fill edge style arc cm cm arc arc cm cm arc arc arc cm arc cm cm cm cm cm cm cm cm cm cm cm cm cm category abstract concrete distinguish thing thing category category reflect amenable ever category define something category constraint abstract category increasingly category abstract mathematic constraint though still law increasingly mean word turn counterpart cognitive des universit du universit du p et en analyse des dictionary need one reduce turn early concrete rest turn strongly core distance age write category feedback semantic symbol category thing trait state thing right thing specie trial category almost name category definition category acquire know define already symbol induction allow answer reach definition reduce dictionary word age variance correlate age remove residual abstract cause ground reach feedback vertex np hope able special case meanwhile dictionaries international english english begin analyze play important word acquire concrete rest tend acquisition abstract rest whole make comparison length distance strongly connect age acquisition object reader theory mathematic couple call graph edge graph vertex vertex nan integer starting cycle subgraph ii use couple word define dictionary graph loop toy color dark dark light good dark red dark color red light style edge style bad dark good good light light red bad color dark edge dark light dark edge bad good light light edge dark red core minimum contain good dark direct acyclic cover cycle find np unlikely find hope exploit around difficulty report effort extract operator define easily acyclic must stop step also include hence linguistic cognitive strongly subsection relation vertex path therefore construct exist fact graph acyclic acyclic induce vertex particular minimum belong dictionary turn refine division vertex eq categorization function hierarchy connect vertex q acyclic
n increment identical original way estimate approximate likelihood maximize approximation integration expectation break class version coincide improve rarely iteration robust indeed redundant reliable batch online gibbs joint variable sequence simulation simulate node j context version old complete il il simulated vertex online describe consequently label adapt original likelihood node notice network large associate expect log online equation gibbs modify already exist estimate propose use update convenient bernoulli poisson visit improved likelihood approximate variational newly new optimize leibl probability choose multinomial natural conditional expectation variational factorize hidden suppose entropy independent variable additive aim update step online new node necessary know lagrangian maximize solve give obtain thank successive statistic g update hidden term online use estimation give propose effect framework real growing assess implementation http software http project web package public online visually explore take advantage reveal first simulation simulate connection model free control modular structure enyi module intra module strong module module illustrate kind allow generate size graph simulate times r enyi criteria comparison reflect estimator estimate belonging partition lie partition adjust rand variational propose sequel comparison initialization start integrate consecutive step estimator mse online online precision batch behave well limitation burden impact online variational average precision rmse pt partition estimation reveal index partition illustration rand expect increase c pt c execution speed web web individual discuss page respectively represent edge like web focus study form community concerned community opinion existence page explore actually web comparison community detection consist compare political agreement aim find module intra tend another political link manual naturally module explain module value modularity community compare link community apply variational manual partition necessarily module modularity optimal definition complementary give global detect dense structure community rather cluster confirm political highlight role political com com com com central confirm political already political connectivity pattern affinity medium thought toward pt mean probability c c c c c c c c c c c c c determine characteristic conservative site political determinant com com know core intra clique position like website spread hierarchy intermediate interestingly c constitute core conservative compare community show expect model highlight political small way site explain tendency ignore internet interestingly core tight observation web interpretation short geodesic path vertex core political look similar random set method constitute potential amount estimation even online precise may size exist simulation study remain method political political classical module highlight strategy online flexible could membership online adapt context assume might intensity traffic mining describe article commonly publish author network n l n z z l n z z n n n n n page simulate political political trade speed estimating become essential include grid focus political new facebook show political far medium political political one day snapshot link manually classify citation strategy study distinction make model summarize connection consider spread among connectivity unknown cluster strategy mixture belong alphabet propose variable et structure computational strategy diverse constitute core put challenge development speed execution network extent bayesian strategy heuristic satisfactory statistical strategy limitation assess network time online alternative batch grow application study many algorithm model traffic difficulty graph conditionally factorize base simulation approximation describe principal modeling data framework estimate parameter variational design finally us algorithm extraction company day http com www manual classification automatic seed site web link visit still belong web create node page intra account political identify political corpus check validity seed confirm site http fr sg set vertex model connection suppose node among belong proportion formula loop introduce software edge supposed conditionally suppose belong vector normalize function consequently conditional graph exponential mainly nod membership whereas membership blockmodel play multiple role stand group whereas consider per node could dirac zero difference optimization strategy bayesian choose author allow integration structure contrary approach rely strategy multimodal approach lead switch converge local framework order political describe figure first provide category class inter connection intra
design minimize k interpret nuisance deviation e order particular variability local attempt understanding behave nuisance essentially roll believe mostly taylor expansion motivation set form lagrange multiplier compute inversion h top simulation realization random field know diagram corner scale near corner interpolation bottom bias job recover function region bias originally generate generate isotropic mat ern spatially smoothness proved convolution thesis let definite q positive definite equal g positive since combination limit definite positive definite g pd kind definite claim conjecture smoothly away work sampling trade bias reduce exposition technique estimate positive stationary field local na I likelihood stationary neighborhood difficult parameter control distant present estimating smoothness local mat ern random field observe sampling play role apply unfortunately stationarity real datum visible challenge spatial stationarity sort field stationarity statistical field lack due add assumption field local stationarity example enough scale dependency field approximate random definition stationarity literature enter discussion estimate observe realization dense possibly goal mention decompose log sum weight spatial covariate independence field neighborhood stationary random range undesirable present exposition local likelihood paper devote construct local likelihood distant weighted apply convenient local one prior fractional local ern realization observation present advantageous domain clear real distinction versus model local approximation random index argument return call determine stationary informally h l law remainder function concrete field fix denote could model tradeoff variability estimate increase local likelihood balance compete term smoothly single realization field spatial location observation parameter define first full likelihood incremental change likelihood field decomposition typically additive bandwidth typically start estimate function observe gaussian mat ern concrete estimation close solution need notation neighborhood response eq maximum weighted mean mat ern middle increment evenly space true hard thresholding I estimate line diagram I right hand simulate evenly mat ern use parameterization page unknown even variance change throughout observation region difficult compare however look increment middle diagram visible plot dash also section minimize global truth yield consuming make inverse formula may order study spatial parameter situation expansion high prior risk goal attempt understand aa tt thresholding I estimate local investigate estimation thresholding heat map percent threshold use hard thresholding choose leibler heat correspond column mat ern location evenly notice random field smoother possible explanation smooth neighborhood attain neighborhood high bias mention right diagram reach almost figure bias kernel hard mat random even location polynomial last paragraph likelihood vary depend estimate comprise realization datum highly leave problematic construct present present numerical claim justification heuristic bandwidth say one bandwidth variability first variability true try come realization field small bandwidth spatial variation random realization interpretation example might simulate realization stationary due field selector similar variation replace recognize statistic spatial stationary behavior quantity fit simulation expect stationary green dotted mat ern location plot criterion profile maximize dot blue green exception section evenly spaced sampling location interval mean zero stationary mat ern leave dash estimate green dotted smoothness true observation location kullback leibler right profile maximize green diagram standardize exception instead dash green dotted standardize good bandwidth mode drive criterion unclear moment two regardless think mode mode result truth fitting observation letting gets take smoothness observe realization suppose locally vary spatially smoothness spatially smoothness
calculate stage c make efficacy simulate datum student reach ap bp bp bp bf bp bf ap bf ap bf ap student simulate situation progress student event half half pose convenience hard distribution mark student pass distinction path illustration purpose combine factor edge merge point reach prior homogeneity strength exploit discover qualitative fashion homogeneity theory currently rich even category dynamic chain number case parameter big map class clearly paper student another arise similar search one describe case order reformulate great finding ki unit paths independent gamma root node leaf lemma theorem axiom theorem condition exercise proposition summary class advantage propagation naturally asymmetric event sample homogeneity technique event model finite discrete graphical however long scenario depend variable zhang create context however place class seek context belief lead encode common single graph end tree represent unit extend possible atom event space encode leaf exactly atomic argue expressive framework accurately belief particularly explain contain redundancy describe unable express deal combine topology independence statement possible follow run successful component allocate module student allocate module pass fail automatic module proceed fail student proceed module fail distinction round component fail depict ap bf bf r ad ap bf bf take edge edge pass node specify unfold student natural situation situation figure consider second pass component mark pass hypothesis leaf situation partition hypothesis consist stage p atom tree probability stage encode tree combine student record want take model posteriori common et specifically structure definition develop section analogous prior homogeneity condition algorithm use discover explanatory student finish event discussion overview direct v st possible event induce conditional reach v mutually independent event calculate primitive together primitive colour bernoulli pass module indicator variable subsequent module primitive multiplying primitive v starting define v e v child subtree represent random one eliminate two say despite conditioning situation concept stage write partition situation colour leave must colour situation similar stage eq path sequence map identical denote obvious variable position subsequent fine partition chain event tree position colour edge colour construct probability graph position use mixed vertex undirecte w w v construct closely analogous fashion conjugacy lead closed candidate make space demonstrate conjugate proceed stage edge vector q student stage assume analogous random separate component model local posteriori write posterior marginal likelihood function calculated theorem give prior score try posteriori intuitive c problem describe section tree partition tree stage situation except node edge search score high stage mcmc agglomerative local begin henceforth seek yield combined simplify algorithm assume stage form absolutely differ stage unchanged logarithm equation two proposal show calculation form two practical give value task way informative exploratory advantage potential cluster priori accord exploit modular state completely modular simple stage bayes obvious search exploit assumption likelihood cd marginal I set assumption stage equivalent vector c reduce set trivial prior cluster stage surprising global previous paragraph second form prior assume priori rate root leaf path independent entirely root leaf path assess uncertainty way dirichlet prior space kn integer I theorem leaf path equivalent least prior edge construct tree leaf path node extend possible describe atomic event variable event tree event leaf lemma trivial dirichlet show stage prior inductive identically compose identical
obstacle filtering sample arrive maintain gaussian kalman fix approximated tractable mixture approximation scheme apply nonlinear introduce representation linear call model behavior many boost machine unify explicit prior knowledge implicitly embed knowledge encode learning algorithm aspect require come bayesian inference posterior update instead issue incorporate update supervise sense scheme representation rule guarantee supervise statistical literature apply theoretical attack problem predicting depend therefore posterior parameterize updating equivalent sufficient augment incorporate statistic show fig leave state eq event rule update completely key observation capture general flexible posterior require approach approximation ensemble replace stochastic traditional explicitly problem explicit complexity incorporate learn advantage sophisticated supervise functional moreover derive rule hand sufficiently powerful believe approach model mis specification commonly occur change probabilistic dynamic goal remove refer dm map pre characterize vector essential learn middle panel evolution prediction panel time predict jt approach implicit evolution operator various natural part architecture essentially system solve kind graph understand circle make prediction predict conventional quantity integrate integrate evolve intermediate ambiguity problem middle panel use though source one solved integrate learn choice obtain rule without want computationally specialized provide straightforward multiple deal multiple correctness everything else difficult thing multiple choose take information near initialization improve train change future observation negative stochastic mixing apply improvement alone consistent gain execute imply system improve describe imagine done evolve state break state state transform depend example extra generally show dm limit algorithm limitation non agnostic dm exact agnostic dm correct technique insight agnostic analysis constraint dm vector generate nontrivial dm function understand dynamic show markov third set hide express state state invertible hide markov induce imply specification impossible invertible limited dms still nontrivial invertible long dependency accord accord observation state converge exponentially fast bayes chernoff concept look give class state short induce invertible efficiently different dynamic identical short known effect imply significant limitation efficient next recover consistency supervise consistency dms solve perfectly projection hold perfectly define c inductive imply exist consequently prove inductive case result dataset involve high structured come graphic nine leave average horizon model condition nonlinear use right panel state hmm predictor vector zero introduce use logistic dataset initialization previous gradient backpropagation rate robust conditioning step sequence angle plus body orientation invariant contain split datum sequence datum sequence dimensionality leave compare error autoregressive model nonlinear use compare show range autoregressive operate space linear prediction step fit prediction five step parameter autoregressive grow input operator quite term probably perform compare linear long range prediction panel perform obvious model unable cope unable outperform hmm long considerably video sequence
penalty b initialization observe start respect maximize algorithm imputation brief comment initialization estimating center instead iterate row ignore convergence computation matrix mle miss set imputation yet r way multivariate formula kronecker value covariance medium sized expensive inverting amount miss relatively inversion cost calculation sample bayesian imputation prohibitive thus cost multivariate imputation imputation regularize covariance iterate take expectation cm respect intensive iterate step iterate maximum starting often approximation solution seek start recall marginal variate one among column penalize imputation instead marginal start two missing applying row set complete apply em algorithm good start em expectation take imputation part however correction unnecessary goal miss imputation initial imputation miss miss mle discuss calculation whether good miss imputation detail final computed inverting avoid exploit property variate namely k j j l index row column respectively vector column I k k l j l j theorem row calculate take computation matrix conditional interest row splitting manner avoid invert kronecker conditional element value alternate expectation mi r j repeat alternate show invert kronecker reduce around complement matrix operation dimensional variate computationally model observe imputation example autoregressive certain report one slightly imputation could give missing estimate converge stationary point apply also apply beyond step denote log figure marginal indeed higher observe also iterate step comparable feasible dimensional datum data set variate normal begin star comparison follow imputation regularize variety situation exist assess one simulation suggest rating netflix movie rating distribute assess performance commonly imputation method svd reduce validation parameter effect pairwise correlation close missing compare cross validation array step take mean quantile close microarray value use penalty expensive compete imputation value cross choose indicate array independent often microarray mse error make investigate issue assess pattern miss array display absolute imputation quantile assess two method utility imputation microarray compare exist netflix rating framework suit movie customer movie customer movie customer netflix simply customer rate interest may customer set movie remove begin thus exhibit due miss value link covariance use rating find netflix movie assess currently number rating around rating customer subset movie movie customer one rating leave rating compare dense movie rating movie customer value compare absolute dense pattern customer rank movie entry leave miss svd discuss error netflix leave potentially thousand correlate customer greatly method predictive fact entire netflix rmse thus conjecture rmse use entire set small imputation penalty notably subset rmse average potentially great imputation large percentage miss model choose cross indicate movie power cross choose indicate well missing result follow svd right large absolute penalty lead error lead imputation netflix ensemble would individual formulate parametric advance miss set row step imputation approach one column vice roughly cost miss respectively application value imputation single extend imputation repeat imputation form foundation also address kronecker covariance covariance flexibility recall distribution restrict variate parameter say location within often reasonable either familiar flexibility numerous advantage potential mathematical practical interest covariance essential add restriction observe introduction make application many classification mining thank helpful improvement numerical covariance discussion property alternate expectation bayesian step gibbs penalty proposition challenge arrange column present modification separate place penalty inverse row maximum covariance imputation exploit allow imputation dimensional netflix imputation technique outperform great degree become common miss netflix movie around customer rate percentage predict rating movie recommend movie customer movie however relationship customer rate rating movie neighbor method movie customer rating movie customer movie rating linear rating fail sophisticated customer imputation netflix call movie customer column treat row interest base variate movie incorporate long matrix customer movie customer rating movie model interaction customer practice largely rarely real matrix variate restriction employ decomposition graphical foundation computationally expectation em imputation contribution single reasonable computational user regularize type imputation netflix section conclude section result enable give miss feature log regularize absolute square penalize graphical use likelihood penalty undirected graph analytical singular svd nonzero model alternative underlie imputation penalize likelihood covariance via except addition method fit penalize enable call discuss integral present previously model row column variate interpretable distribution regularize variate singular graphical variate normal restriction mean variate mean variate mostly sample formulation appropriate estimate marginal improve interpretability restrict variate n np mean model compose row column imply element column belong point random jj effect covariance share matrix variate error random product illustrate marginally jx jx distribution restrict variate statement two row ij n jj j j thus general multivariate completeness p np formulation variate mean give desirable term marginal section reformulate covariance estimate imputation seek two separate penalty inverse column penalize absolute square element penalty note penalty equivalent place kronecker great flexibility covariance penalty strategy counterpart place concentration graphical especially penalty nonzero conditional row correlate link form conversely imply regularize row graphical covariance hence mean column th row center additive result thus covariance difficult penalize concave coordinate coordinate wise block reach global due maximization monotonically log maximization accomplish set list penalty change penalty coefficient term parameter penalty eigenvalue coordinate maximization iterative maximum penalty global penalty penalty eigenvalue analytical optimal svd td pt ip penalty singular reveal equation eigenvalue quadratic give eigenvalue singular covariance globally say penalty covariance penalty decomposition commonly employ write reduce
datum constrained scheme perform presence element operational criterion design learning involve converse theorem formalism theoretic distribution assume infimum joint incur particular framework cover function indicator pg f function measurable problem paper basis
mostly probability often scientific job assign modelling though influence sampling fact entire rather look posterior thing discuss principle eliminate inference must belief carefully use evidence pd pd
term equation use triangular nonnegative u h ph p use sufficiently indeed definition lebesgue prove main theorem asymptotically vanish extend central markov accept additional assumption maximal exist induce markov start moreover measurable generality proof resp start let go infinity
fix actually extend classical extend take arbitrary continuity continuous dense lx l lx n lx lx lx lx lx lx lx lx lx lx successive therefore uniformly conclude end proof strong law number conclude finish since martingale write l lemma lf hold martingale eq f lf fundamental nd np k l combine dominate virtue follow strong nd l
go allow case original infimum convexity von various upper low maximize define joint array analytical try value joint put interval force decision root appear nature I maximize follow inequality I expect study follow distribution gap hierarchy analyze minimax introduce help expression derive inner adversary conditional deviation next functional concavity already concavity section show description interpret provide yet divergence begin simplify notice immediate application similarly interpret jensen albeit precise
resp vice overcomplete original coefficient pair global minimum representation entry whenever entry standard numerical opposite develop replacement admits side directional appendix indexing row index kn appendix provide sharp explicit decomposition kx k indexing nonzero entry index resp keep index resp also th equip notation state condition consider notation matter local restrict e strict open whether overcomplete dictionary side additional relating collection next discuss belong convex polytope project hypercube cf column resp difference column reference orthonormal necessary simply read polytope figure live bernoulli htbp
hence prove furthermore chance gamma distribute result assume k true evaluate establishe theorem gain k k equivalence event range summation implication mutually though factorization mutually force evaluation fact surprising dependent balance replicate form vector density mixture table structure q replicate take normalize constant stand x z z z z x z n n z leave side denominator factor control inspection suggest polynomial indeed degree reduce construction multipli assertion table indeed contribute high equal indicate contribution term zero zero equal reduce consideration able focus bivariate focus degree instance coefficient equal require
implementation expect artificial need need compare four class recognition face google identify content characterize sift descriptor object categorization detection face background google
relevant query result roc area percentile basically measure language rank result query average chance cca regular cca cca loading canonical approximately cca able good cca result section achieve cca loading cca sparse narrow effective dimensionality reduction technique short summary result cca application involve meaningful retrieve database semantic probabilistic acoustic signal tag word content decrease computational away noisy detail scope song represent song semantic tag coefficient song allow small span semantic highly audio sparse cca generate contain horizontal axis train area receiver operate vertical independent initially clearly cca dc remove noisy figure system base vocabulary heuristic amongst subject collect improvement initially summary selection cca improve retrieval computer effectively remove agreement special special primarily interested advantage formulation become clear sparse introduce relaxation gives specifically give regularization parameter perform write simplify solve version sparsity constraint constraint result convex convex optimum occur unlike discussion clear significantly
ne ne nd simulation reach need nash percentage ne table cm p cm p ne time ne games linear co co see lead nash equilibrium contrary version frequent nash games ne players ne version well social strategy update new previous strategy converge ne course function symmetry cost cost realize update learning individually note value display case hold state expect inter arrival time estimate arrival yield nash nash quite reach
small big may reasonable often obvious calculation term know directly markov chain see corollary easy autocorrelation fact yield loss generality obviously q bind side note state bind cover setting indicate set need apply perfect deterministic without burn burn corollary chebyshev
def define lack allow usual convergence give van condition space rate quantile surface unknown replace uniform partial quantile compact quantile hold metric typical rr logarithmic recovering binary presence issue identification criterion since partial quantile quantile space perfectly quantile point index consequence lemma quantile moreover characterize partial index make quantile index partial quantile assume quantile quantile point turn aim characterize quantity mild notation asymptotic quantile quantile hold assume continuously gaussian limit set gaussian limit generalization quantile relevant study monotonicity quantile counterpart characterize underlying establish independence measure quantile dispersion quantile detailed process develop quantile index let quantity quantile comparison contaminate result whose direct function probability comparison depend outli probability point nonetheless influential associate influence notion outside scope partial univariate rely partial reference well classic determine class probability identity index characterize determine determine order describe partial quantile characterize proper example determine acyclic follow necessary characterize otherwise partial order univariate order monotonicity partial surface partial quantile proposition deal assume monotonicity case monotone true quantile quantile partial monotone estimate quantile monotonicity show monotonicity condition quantile monotonicity quantile conditional
assumption together differential fraction indicate notation initially individual ignore choose interpretation still contact infect mix mass action linear worth pointing since track two differential equation show monotonically monotonically initially increase decrease start little happen eventually get infected fraction major occur first minor occur small separate different figure configuration right fundamental importance number refer individual epidemic epidemic balance divide equation epidemic recover balance determine epidemic infect interpret infect must infected infection pressure cause infect fraction infect useful related epidemic previous epidemic major fraction community mix homogeneous mixing accept model may example epidemic day care seem uncertainty infect possible chance epidemic motivate epidemic reason motivate stochastic epidemic disease equip standard error study epidemic disease general homogeneous uniformly mix recover time contact constant contact mutually contact infection individual contact
expensive calculate calculation algorithm polynomial practice trivial look step alternative evolve time way death happen happen birth mark happen next obeys dominate note property property monotonicity locally step dominate involve calculation process multiscale require configuration element construct thus refine
sphere radius derivative smooth curve say follow say resp smooth manifold matrix square view column view row especially corollary group simplify eq look spherical least observe singular decomposition rewrite obtain decomposition formula spherical act linearly one simplify inclusion drop simplify transform rearrange function smooth open geodesic cut subset short one cut complement tc ty since geodesic geodesic geodesic many intrinsic lead appear dimensional riemannian smooth structure hessian calculus
ground truth human berkeley parametrize map image quantify segmentation relative objective parametrize predict neighbor affinity affinity find connected component measure segmentation adjust rand affinity key edge graph opposite sign machine graph focus incorrectly split merge segment learn segmentation performance affinity previously number misclassifie train affinity partition cut optimize rand close however minimize rand classifier
sample calculation iteration two retain regret bernoulli benchmark ucb secondly bayesian simple optimistic htb select accumulate e suffer high determine perform cumulative average run compare ucb ucb bayesian baseline approach cumulative serial upper bound node expansion evident improve due algorithm serial expansion seem slightly serial consistently expansion
otherwise polytope often transpose denote denote identity covariance call function mind constant polytope regular require regular principle converge highly non main applicable proper constant proper hypercube spherical multilinear hypercube multilinear lipschitz function constant q pac sensitivity translate define noise define independently denote learn focus agnostic label example agnostic concept fit draw opt correspond agnostic see product multilinear estimating fouri concentration one function know work let submodular output example due polynomial polynomial output opt principle invariance regular polytope divide principle extension devote prove random closeness anti imply closeness surface bind ok result surface state invariance involve make hybrid introduction clarity next
five severe security resource might fixing exploit complete computer resource customer management attack database observe attack learn tune receive result extend model clause clause previous level attack might clause hypergraph model consist clause attack reward budget attack rule list set proposition proposition generalize directly valid row become clause matter obvious substitution become clause security game practice security objective show mathematically equivalent attack per attack edge interpretation attack root hold appropriate modification simple budget qualitatively informally design security account various attack patch benefit security economic shot security game identify
objective leave q quantity composite write put fix certain impose eq composite bias dms rank attain bias decomposition modify introduce equally weight bias local weighting definition easily extend basically gradient descent htbp locally post step generate user inspection size dm orthogonal find j gradient multiple solution solver produce solver optimization core use inspire solve handle transform via slack result constrain optimization problem trust use subproblem conjugate cg accurate handle hessian approximation solve region follow newton symmetric sr quasi newton approximation hessian use suggest reject step option bfgs option implement limited variant sr simultaneously test performance provide technical detail illustration purpose base profile response propose choose reflect choose choose vector zero elsewhere two column contrast htbp iteration htbp vector everything else convergence optimal contrast determine attain shift unconstraine htbp two shift
j e lemma n n nd fact condition conclude follow positive c nj nj nn nd first note event pa nd f j nj nj nj marginal let joint score snr fp pe snr tp fp pe remark correlation fan dimensionality true regression highly nonlinear correlation marginal nonparametric nonparametric member several screening general nonparametric mild independence screening extent quantify drive iterative nonparametric moderate dimension perform keyword model regression screening
regression assumption tail mild hoeffding hold coherence property u particular inequality mention early need domain eq mapping constraint property condition roughly speak concern one function indicate constraint well portion need condition constraint mild small comment constraint aa let nonempty counting measure extend rest give I see
dimensionality paper problem localize localize anchor intrinsic manifold code code anchor point map induce q moreover code theory condition shift origin coordinate system shift invariance requirement become represent code concept code localize illustrate linearization linearization code leave side linear coefficient right first second localize quality convenience introduce locality code localization tuning quality depend complexity coordinate depend manifold dimensionality manifold dimensionality call intrinsic constant
ex university national ex com ex national south david com technology introduce principled scalable learn direct approximation notion general reinforcement unclear motivate open provide first approximation monte agent context encourage propose future ex reinforcement confidence tree observable prediction tree reinforcement popular influential paradigm experience reinforcement agent environment introduce evaluate reinforcement agent inspire exist environment cycle history agent time environment achieve computable observation reward action compute machine mm horizon sequential universal horizon pick environment eq consider step ahead pick expect reward kolmogorov rigorously agent accurate environment act optimally description agent computable algorithmic bayesian making environment ai way principle view machine principle optimal behaviour complexity issue agent theory plan mixture reward past experience part many generalised generalised weighting idea contribution paper paper notation environment accumulate include familiar reward describe bayesian present carlo operation context generalise set put idea form related limitation highlight area investigation terminology agent experience true underlie environment string denote denote empty concatenation string space denote joint observation agent call drop way mean describe underlie may agent environment typically learn approximation environment agent agent previous definition environment wide variety environment reinforcement mdps familiar notion much seek horizon instantaneous reward
helpful measure offer optimize drive recursion assume euclidean measurable number function initial transition conditionally hold old illustrate metropolis langevin mala euclidean mala effective metropolis hasting proposal langevin diffusion symmetric definite brownian mean mala propose accept probability define mala practice computation run mala mcmc definite c mala proposal let family nonempty convex obviously choice initialization acceptance x x transpose q n define rhs expression recursion
require statistic student asymptotic bound closeness consideration test concern deviation expansion follow simplification uniform pearson statistic obtain variety value list try minimize however appendix choose let eq statistic q generalize though rather matrix us eq k let couple find closed ball smoothness condition let inf immediately finite degeneracy applicable yy degeneracy recall hyperplane appear central expansion normal proof corollary corollary assertion cf modify proof recall one prove remark note valid distinction definition schwarz yield remainder proof provide ease use extension obtain er absolutely continuous exponential introduce r definition concern assertion comparative truncation especially regard er transformation q also follow definition must recall restriction third two establish second case independence argument mt dt cf last recall definition easily identity next cf q chen absolute er random vector necessarily moreover replace define display use thus recall follow view replace establish
topology review mrf belief propagation pairwise undirected edge associate make direct denote graph compatibility normalization probability distribution undirected graphical generality compatibility function compatibility operation bethe bethe partition
theorem consider obviously imply fulfil thus problem generate see easily matter nx I side denote define recursively easy induction bf bf bf far pass equivalent sequential everywhere almost generate plan rest equivalent almost everywhere test everywhere everywhere fulfil test locally strict strict inequality proof
subset path addition triple disjoint definition decomposable exist decomposable subgraph decomposable example clique sequence perfect name residual graph clique residual property decomposable graph factorize eq occur forest graph cycle several tree acyclic edge forest forest bic minus ht spanning assume model homogeneous heterogeneous decomposable path occur path non adjacent pass show page represent circle dot
automatically chemical domain functional characteristic property limitation section classification start bayes specialized binary predict label unlabele model theory bayes bayes derivative formally differentiable order support class neighboring pair bayes highlight positive sign explanation form continuously change orientation understand remark become fit region issue estimator appropriate classification explanation learn explanation gradient dimensional vector sign decrease increase entry amount ascent test negative need predict model learn gaussian use interested classify drug discovery area process e gp natural datum worth insight predict capability complex definition directly start gradient
immediately note bound gaussian function distribution hypercube aspect perhaps symmetry original sensitivity sensitivity surface equivalent
mcmc include monte harmonic functional equality survey efficiently significance probit performance dataset keyword model choice factor functional probit contribution fundamental establish difficulty value bayes difficulty contribution area obviously present factor procedure stand hypothesis crucial framework give hypothesis compatible odd prior odd namely paradigm g comparison
variance prove normality strategy calculate assumption differentiable assumption order investigate ode complement open drive bandwidth extension link asymptotic differential equation current associate without measurement
protein datum result approximately positively data protein protein procedure correlation relational remain majority remove link total extra missing value normalize vector euclidean initially fit pt choice matrix rescale norm protein link mle computation measure rank protein pair gene page categorization retrieval receive concern link follow protein measure algorithm precision calculate area auc proportion vice versa belong multiple sense measure measure go categorization pair analogous agree believe categorization system organization protein lead category protein interaction correspond auc compare different variant metric retrieval cosine near score measure euclidean query given initially setup obtain fit use generate give analogous pl ij pl integrate parameter neither take account interaction theoretically require
identify trend score weak score e variance tail already table probability success curve figure curve strong agreement formal sensitive visual display score ensemble odd figure present odd eight panel difference ensemble problem along horizontal combine score consistent conduct test fourier hadamard panel dot red color panel hadamard dyadic blue dot fourier f hadamard rademacher rate panel e blue dot linear fit score match color hadamard dyadic dot mark display figure trend visually evident model drift otherwise truly difference probability score exact agreement success term refine tend early line fit within close plausible inspire score large expect find score reject weak model dependent record scale verify width drop verify success rate probit ensemble figure ensemble offer relatively poor ensemble size curve success shift shift character evident appendix row hadamard air present classical cyclic show notably special forward conduct extensive observe transition observe compete yet derivation phase dimensional informally approximately ensemble instance range empirical ensemble band phase band consistent prove behaviour ensemble independent diagram tend tool counterpart fit statistically significant trend problem fit trend finite dimensional class ensemble task geometry ensemble quantify drop verify probit logit logit well probability
size finite induce thus reliability ordering discuss reliability statistical compute hypothesis great bootstrappe computational confidence random sample create bootstrapping selection molecular
rank model develop mirror whose lemma define aggregate estimator subsample denote base subsample collection k consider function second subsample cardinality proceed eq goal collection mirror average recursive weight denote subsample aggregated density give norm restrict accordingly restrict l l mirror average restrict euclidean remark inspection one lebesgue restrict norm mild assumption inspection valid tend origin factor densitie integral ball value main learning label difficulty moderately dimension tail e classification purpose excess misclassification plug
change line sample fig trend need range panel statistically confirm environment affect frame cluster observer galaxy formation galaxy light particular optical effort red sequence herein bootstrappe cluster extent agree simulation future work illustrative purpose list make context literature five place comparison summarize galaxy impose color actual result relation correct red iv slope negative negative increase red slope fact rest frame difference come red iteratively sigma blue cloud present automatically initial fit account slope cut magnitude ideally fit line unfortunately indicator ii slope environment basic agreement
see core different sharing parameter allow delay core preferable consume synchronization core read write atomic synchronization memory architecture key multi date gradient computation retrieve copy message exceed bandwidth variant considerably less term template solely purpose occur g dimensional decompose partial value partial update combine partial cause stage already quite vector delay much processor communication channel access entirely delay adversary aware together time delayed rate political treat iid pay modification since exceed game analysis bound minimizer convert randomize
aspect draw uniformly commonly incoherence netflix figure decomposition say incoherent satisfie movie rating run netflix cumulative norm leave permutation notation define comparison generate independently plot straight plot among randomly movie rating matrix say movie user hence challenge reconstructed factorize diagonal singular start numerical understand provide rr max calculus establish prove thank discussion helpful comment work fellowship award grant dms pt minus pt plus pt corollary conjecture singular
drive extraction time experiment drive force regularity fail slow signal analyze expand present paper review well call sec drive force experiment demonstrate discuss originally review approach objective slowly multidimensional raw input indicate temporal proportional large place preprocesse input component also formalize generate output variation slowly measure derivative trivial meet calculus constrain basis contain yield expand reason become
incorrect chance outcome chance wrong bit rather small sample size keep total size would stop full count keep comparison error batch vote batch vote comparison control chance full hand count incorrect outcome error chance fail incorrect wrong chance chance
encode string substitute theorem straightforwardly two string code determine string mutual four quantitative difference provide admissible list distance account dominant admissible absolute want express string think string information bit universal distance list maximally maximally normalize universal list information say version metric alternative normalize divide either
discover factor loading spurious see variable effectively spurious gene without factor evolutionary search false loading show gene htbp result synthetic datum gene factor interpret hierarchy column matrix prominent tree column correspond prominent factor locate appropriate breast factor discover interested agglomerative usually contrast factor degradation likelihood result
merely trading consider say threshold find estimate restrict regularization parameter standardize solution threshold coordinate use assumption gaussian condition either analytic standardized significantly see appendix central function notice dt dt dt exponential family namely interior see chain central analytic interior generate namely particular hold analytic standardized moment assertion notice argument generate also desire abuse
aside reveal completely determine fast decide speed number concave converge toward look converge uniformity uniformity integer limit distribution law kolmogorov discrepancy uniformity rule kolmogorov uniformity month third country area square population million display area confirm rule absolute country fast
prove desire calculate f f term theorem combine estimate completely expression r tu eqs tu l x acknowledgement discussion support nsf generalization prove inequality chernoff p analogously proof independently reveal z j remark denote indicator tm ii max second x j x I max definition sum average adjacency denote later discrepancy property probability partition row j ib ib ib value group summation uv e definition leave bounded discrepancy show summation
back particle reconstruction reconstruction obtain reconstruction perturb initially calculation otherwise large reconstruction filter little perturbation individual run estimate filter function produce g independence observation detect pick trajectory particle increment value early trajectory single depend iii computation find correct value run c c run run filter design
much normal reliable datum normal nonparametric article concern density construct estimator reliable generating process function density reliably density relatively skew symmetric available disadvantage model bivariate copula normal flexibility density separately gaussian copula nonparametric copula copula attractive copula originally addition transform standard transform likely suppose joint hope capture copula allow one extra coefficient capture dependence contain remain multivariate overcome encounter large grow one survey univariate aspect choice bandwidth second normal example discussion univariate analysis
observation geometrically demonstrate establish geometric ergodicity uniformly ergodic uniformly also geometrically ergodic irreducible analogue albeit additional markov reversible geometrically geometrically geometrically ergodic geometrically relate result apply reversible yy application attention replace markov form usual gs update kernel easy qualitative geometrically exploit sampler practically model easy put say geometrically connect scan suppose yy clear conclusion condition instead gibbs easy claim gibbs sampler practically statistical geometrically reference ergodicity random scan sampler prove ergodic substitute I hasting hybrid composition
penalize estimation subsection smooth propose subsection covariance reference covariance certain zero since covariance sparse underlie graphical several dependency conditional identifying model well available give formulate control apply gene np hard g see relax follow penalize recently lin propose covariance suitably also selection et heuristic newton coordinate partially al lin effectively paper reformulate subsection notation subsection show q yield eq concavity xt xt xt xt xt two invertible upon side let obtain
error xu wu differ author advantage dependence exploit simple improvement inferior design work advantage available organized review boundary introduce hc boundary toeplitz low discuss dependence proof theorem lemmas hc unit sparse formulae quantity represent strength decrease increase increase ease would connect relationship adjust treat entry proportion much location without magnitude throughout sec variable uncorrelated identity variation equal upper result strength test two hypothesis uncorrelated case also detail identity record closely space test whether closely least clinical outcome et profile gene another expression quantitative trait genetic marker chen observable gene associate clinical outcome contribute clinical boundary partition plane region successfully nan error infinity drawback need detailed hc uncorrelated tackle aforementione uncorrelated apply th value sort high statistic version hc hypothesis infinity fall interior region type associate satisfy hc detection theorem use
ij adjacency bipartite true thesis hypothesis n conjecture claim stanford edu stanford electrical stanford stanford usa rank reconstruct subset collaborative netflix complexity combination optimal circumstance statistic process significant low rank implementation practical sparse discover unless reconstruct first analyze convex introduce tight carry iterative scheme appear optimization complexity complex operation theoretic reconstruct performance crucially
maximum sum order I sample interval plot nan hypothesis figure show limit plot vertical clearly fs support strongly present state know notice scaling choose well theory resolution one
surrogate predicting reduce nonconvex misclassification replace apply mean among nh prediction surrogate appropriate nh beneficial since interpretation weight node interesting nh cope missing imputation surrogate split fit replace value technique estimator calculate prediction member question whether node group simple turn say node membership sufficient node membership sense value usually interaction node membership observation still member involve variable th observation member root node variable member root refined amount drop stop split node across occur nh tree nh tuning procedure apart choice insensitive choose numerical necessary predictive improve constrain node minimal fraction
frequent statistic lift relative frequency association mining small frequent individual column randomization margin name randomization randomization additionally maintain row margin mining method list frequent randomized expect deviation difference dataset frequent sufficient depict test ht randomization plotted control dotted line satisfied depict find level randomization reason rule swap randomization less randomization frequent subgraph frequent transaction graph frequent pattern subgraph graph use mining readily laboratory transaction statistic subgraph large subgraph interesting select switch edge swap create
cv covariate pair final pl fold give misclassification average hold take hour average smc issue pl pl suit design al probably alm gp instead sequential design context sequential optimize black essence distribution ei obtain integrate maximize ei iterative global optimize bayesian inference account order surface particle posterior ei use student predictive equation let applicable opposite embed heuristic perform random candidate ei direct optimizer heuristic augment include predictive map smc pl ps I initialize
decode code graph product distribution fact variant algorithm long try complexity may grow grow exponentially however prior jeffreys informative provide sense noiseless simulation indicate algorithms noise snr sum compressive set interest nonetheless employ naturally pdf constitute message set node product scale pdf check pdfs normalization note preserve gaussian laplacian straightforward implementation family random symmetric tail successfully process order completely choose jeffreys prior improper improper
like eq complete latent variable density parametrize traditionally denote penalize precisely penalize likelihood like eq notation I order practical optimize manner gauss fact separability subproblem subproblem alternate option helpful keep step level recently kullback successively reasonably optimize
superiority elastic remark group variable lar use permutation house choose consist observation without variable consider variable section well irrelevant concern business build notice predictor present remark length whereas predictor elastic net version illustrate validity adopt sparse predictor case early stop neighbor construction comment definition remark et paris de predictor introduce build exploit previously novel construct multivariate emphasis model even number
sampler mutation avoid computation normalize constant auxiliary smc concept recover set allow threshold merge read journal
start seminal network social researcher network simulate network especially network economic trading network resource mobile public package lot well contain domain domain protein network direct communication internet stanford network package six study describe reasonable include network classic derive survey publish relation dataset spend join analysis root existence member member accept main event take stay member difference year leave ask term relation influence span stay well social response survey subtle lead order past decade researcher truth email widely energy company company united cast management top subsequently account investigation publish online cognitive learn project correct email message contain user datum mail top contain give mail threshold message respectively research activity document classification visualization paper working corpus become molecular biology protein way instance protein physical recent year amount resource collect experimental protein bind effort infer protein role song interaction protein target protein localize interaction iii pca interaction iv collection protein estimate produce part ht method large include identification bind national longitudinal health health united middle focused patterns date survey year wave survey occur country complete background school life activity health status complete school service student home topic select two health home survey student construction valuable resource wave collection occur wave home cover school collect wave I construct amongst student two span oppose previously core core wave iii topic include disease original wave home wave iii participant among wave iii wave wave locate comprehensive survey aspect health measurement access public child complete height period derive appear heart tie member series statistical analysis claim examine world depict network individual explore via longitudinal publish similar focus dynamic behavior time come plausible alternative methodology social answer question network subject width body mass color inside status color tie relationship denote appear processing system conference volume
jj transformation absolute mm decompose integral q parameterize find w b pt b pt q justification j b b random distribute w eq namely j w variable therefore np n acknowledgement grateful comment suggestion lead wish support science et universit bivariate margin decompose copula function finite
player issue home overall home use true value fair particularly set actual know ahead report prediction player rmse n predictive interval width interval full outline couple simpler examine full substantially truly last home year home ie point comparison solely rmse full rather sophisticated external ht ccc interval indicator na na transition consider extended transition parameter markov motivation allow transition display past however suggest prediction somewhat even player long transition extreme force sample
lb pp lb pp quantitative r b pp resp obtain notation apart uniform bound neighbourhood norm topology first claim definition last statement follow immediately cover theory take projection truncation analogously average grid filter analogous explicit speed assumption hold term wavelet mu b random variable assume satisfy infinitely smoothing depend projection satisfy compactly wavelet analysis smoothness construct wavelet periodic period jx jx constant function successive wavelet periodic word sequence one convention range function constitute orthonormal function correspond remain index exact ordering choose characterize namely basis space especially relevant b function understand follow simplicity basic embed embed stand duality respect duality operator nf
conservative remarkable behavior four replication adaptive seem account decrease effective way investigate optimize influence introduce adaptive world study validation adaptive lasso tuning parameter attention future value without proper cross validation yield nice incorrect biased favor strength regularization estimate analytic formula thus make consume unnecessary emphasize depend false theoretic reconstruction gene comparative explore research method potentially use causal presence longitudinal al identify variable investigate partial correlation shift lasso regression promise alternative finding simulation investigate sparse discovery density decrease topology pls high mse real opinion pls suit pls
via however practice detail record generality center commonly pc loading worth pc uncorrelated pc see sequentially minimal possible pc different pc pc direction loading practice typically pc great dimensionality success nice pca drawback pc combination loading often especially indeed variable physical meaning biology gene pc would compose small original loading develop find pc sparse loading nice sparse active research topic decade approach ad hoc post pc pc simple thresholding pc small sparse achieve loading much possible algorithm call find loading explain loading pca penalty pc program relaxation pca recently induce formulate sparse penalty maximization propose stationary method pc aforementioned property pc lose pc pc quite explain attempt optimistic overlap among loading pc method lack orthogonality formulation account nice total pc loading vector connection lagrangian class nonsmooth constrain problem suit method method
perturb dynamic affine introduce perturb like field correspond nash equilibria game equilibria stationary unstable include nash equilibria definition strategy profile theorem evolutionary game completeness go follow corollary clearly equilibrium vanish right stable indeed continuity strictly say neighborhood great fast fast corner nan hence strategie dynamic corner generally nash strategy perform unstable field leave perform never nash unstable interior patch imply stationarity nash pay purely dealing avoid nash randomize already guarantee unstable leave almost surely purely unstable like
peak local image source large wavelet base technique detect ray provide wavelet tool tailor wavelet determine interest bayesian algorithm often gaussian separate source propose way slow typically make priori computationally peak detect error rate deal error focus detect aggregate rigorous controlling error rate source demonstrate ray powerful detect ray source give good error source new multi design enhance integrate procedure maintain evident use competitive detect ray description application resolution datum overview direction study ray powerful ray third extend observe without ray million galaxy matter circle black space answer question several ray infer galaxy universe x ray provide evidence dark matter interact visible matter account universe deep
exist seem poorly nonetheless world often day pc context visual detection discriminant aim maximize skew boost scheme algorithm classifier fig ability research publication dr c laboratory bag act e mail com zhang laboratory south mail zhang com video surveillance face detection effort spend boost used training detector discriminant conceptual simplicity efficiency well term train detection cascade weight property boost separability domain demonstrate outperform significant argue adaboost approach achieve classification detection object adaboost analysis classifier face numerous video surveillance base retrieval detector challenge visual pose illumination condition furthermore typical background pattern
os er probability control uniform edge see clique control os enyi edge obvious section decomposable edge decomposable develop four graph markov structure span use fit model third skeleton way pairwise fourth decomposable marginal copula graph necessary standard marginal independent prior vertex plane walk algorithm employ metropolis approach random proposal sample burn period high display compute six figure compute appear iteration actual vary ht construct decomposable marginal procedure gaussian graphical encode zero wishart mass denote form graph approximate via recommend identity precision independence figure employ hybrid move walk section replace vertex independent draw sample summarize c distribution entirely bivariate marginal edge appear dependency familiar joint complete continuous strictly q illustration q z early example graphical structure associate graphical alpha maximal recover show proposal
dictionary orthogonal constraint update j solve linear present addition fuse problem encounter loss sense equivalent instance using proven experimentally adapt denoise carry sparse code computing solving since quadratic surrogate cost situation I structured group simultaneous code name sparsity group suppose wants obtain active way impose consider norm jointly decomposition define use active group n optimization formulation relatively compute matrix keep variant evaluate matlab interface available project natural genomic efficiency factorization principal component randomly patch database compose varied testing increase represent various typical image color patch norm use regularization experiment normalization factor datum experiment matlab interface lead implementation experiment single core ghz signal try power variable show good low empirically tune since performance intend regular order stochastic gradient share set meaningful algorithm alternative would batch
learnable restrict algorithm noise well beyond require recently show problem learnable reduce establish central moreover related decode code bad learn serious barrier represent simply generator size produce exactly label construction hide markov construction
make ml smoothed accurate ml close move approximate quite guess tree hill like thank third fourth research grant dms program r gm p removed sample burn give hill initial tree summarize column percentage hill tree geometric hill hill drop distance distance map tree nj remove sample burn two summarize percentage hill hill hill avg ml nj ht pt true pt nj neighbor package
process divide source eeg inverse ii example find inversion physical source de mix transformation formulate briefly possibilitie principal component ica prominent unfortunately even eeg successfully sophisticated way meaningful decomposition interesting ica assume trace argue instantaneous exist ica propose mix coefficient model temporal combination instantaneous ica furthermore approach integrate connectivity brain source lasso avoid overfitte interpretable brain remark sensor source connect analysis ica follow connected relation exist detail finally implement bfgs numerically realistic eeg plausibility source model future research direction context
large desire size segregation association alternative present carlo recommend segregation side right sided deviation level fact binomial hold instead recommend asymptotic size leave segregation association middle pattern base circle solid line horizontal line nominal association iid generate replicate observe even segregation considerable separation kernel segregation moderate value observe segregation association segregation association empirical critical empirical ht monte segregation depict replicate right nan solid segregation dash observe even mild moderate value suggest critical power alternative depict density replicate leave replicate solid association alternative segregation carlo power value replicate binomial plot power estimate recommend around segregation segregation middle plot plot triangle alternative three power test replicate base approximation poor empirical association power association circle base plot line table ht power empirical stand point relative segregation size relative
et universit du france testing diffusion coefficient always von kolmogorov chi special alternative discuss goodness fit asymptotically free g goodness test dynamical I initial concern trend basic consideration always simple eq correspond trend coefficient
weight incoming subtree elimination illustrate formula execute summing multiply edge preserve value determinant base straight direct hence ij r correction graph element construction take power define weight weight modify weight b walk respect follow rate gauss h clear capture walk depth computation include walk model hold error conclude construction preserving equivalent
modification situation alphabet order overlap distribution adjacent ii update hold time report open fast lead implement suggest david introduce computing field step sort say calculate convention overall minimization measure observe fix respectively slight abuse notation use calculation format
formula model frequently classical monte bernoulli know ever success show posterior mass r denote bernoulli trial never explore fs base really need form development ir z furthermore I km r toy analyze probability eigen r r r j write order row transition strictly positive ergodic satisfie solution follow eigen finally g fs quite poorly numerical datum exactly chain slowly due move high point follow conditional leave chance back stay translate get bad increase size maintain dominant size size get fs bernoulli
follow sigma sigma gb ram update b tail adjust tail lower b true frobenius norm false true year successfully apply biology rapid pc grow main development algorithm reduction comparison positive false negative grow learn distance
actual build email month dynamic latent trajectory company role first vertex interact role role actor likely role another clique role appear information mean clique yet discover red within many people green clique role reflect management role receive report role among behave possibly dominate role clique connection position source find role stand receiver publicly analyst lead massive internal systematic temporal vector actor people increase weight visualize membership vector entity track mixed evolve role base provide explore behind actor simplex vertex represent cross role active role trajectory mixed membership gene estimate various life cycle underlie determine stochastic point invariant evolve experimentally topology technology evolve reverse microarray point across analyze gene know mixed role gene cycle gene development many gene exhibit transition underlie accommodate new stage role interact role reveal visualization compatibility examine whether
proof b immediate substitute consider model define follow substitute design assumption adaptive satisfie formula assumption essentially sublinear fix norm suppose question adaptive lasso consistency assumption random integer set admissible realization whose row q suppose set define probability estimator section regression denoting imply ij last symmetry corresponding regression vanish theoretical sparsity linear regression satisfy equivalently element
analysis extend moreover side pair mab adversary constrain amount naturally contextual arrival correspond expect arm time metric provide change interestingly across good knowledge mab rare combine payoff bound aggregate temporal notable case per behavior random provable guarantee mab round arm round precisely context subset arm pair feasible distance contextual bandit contextual exploit absence essentially reduce carry mab regret play round interest typically current arm motion boundary treat speed brownian analysis contextual obtain superior provide nearly maintain take context develop provable contextual cover adversarial call subroutine flexible depend payoff plug bandit algorithm mab regret bandit overall payoff payoff introduce explore specifically contextual adaptive analysis clean contextual bound result alternative maintain context lead present adversarial payoff priori priori ability adapt expected payoff present adversarial treat discussion bandit beyond scope bandit background bandit arm bandit similarity another payoff payoff allow away arm assumption distinction stochastic adversarial payoff payoff whereas similarity payoff notable adversarial payoff lipschitz expect payoff realize payoff trivial distinction payoff whereas one payoff
attempt produce meaningful prior opinion device kullback divergence see thing always solution require think reference upon could back information confusion informative mid nature prior often serve informative often probabilistic easily variety topological optimality another much eq regular inferential jeffreys synthesis synthesis hypothesis test rather give proper probability space advance jeffreys
infect latent specify negative follow individual distribute mutually function period may mutually independent place result infection member times homogeneous process individual belong individual accord contact rate finally poisson initially population individual simply ignore epidemic individual community else infect individual epidemic typically behaviour mean quickly else negligible behaviour epidemic quickly fraction infect key mixing describe later unchanged become note various instance fix increase distinct value assume key give contact outside tend branch individual infection non infection branching constitute parameter epidemic branching reflect well process describe threshold
et unified calculus density rule translation conditional quantum mechanic retain conventional theory difficult framework framework bayes use framework idea quantum mechanic probabilistic assign latent latent optimization energy cost suffer practice deterministic em vb sa overcome issue local optima vb learning
would curse extend index result assumption hold parametric assumption list distribution positive derivative lipschitz order component satisfying lipschitz nh nh e derivative function away ensure bound necessary asymptotic analyzing estimator slow deal estimator motivate another variability construct limit behavior estimator asymptotic efficiency linear consistent obtain numerous example exist show pursuit elliptical ng sir latter include correction variance estimation save hold li variance regard consistency need far equal therefore sir r ex j et correspond infinitely inverse jacobian theorem small v notation negative generalize inverse r asymptotically efficient et addition iterate estimate
belong estimate importance represent known representation sparse nonzero index assume orthonormal series achieve cf method cover general orthonormal necessarily small achieve adaptively would construct among general notion reference mind suitably choose suggest also define introduce widely statistical refer regression orthogonal equivalent soft type paper include regression prediction use particular refer vast literature component example literature mixture develop aspect component mixture consistent specialized suggest rely well various mixture complexity
column avg average use minimum sf sf table pair difference statistically bold sf accurate datum come bayesian test detail datum aim c statistic pair bayesian highlight bold pair sign well tie datum positively bias table pair estimate significant medium converge properly biased estimate assess dataset v employ log delay fitness compare square observational criterion k log lead accurate fitness instead fitness log consume try evolve allow poor evolve test inferior evolutionary approach perhaps optical suggest plausible distant regard match low art optical monitoring acquire measurement sf sf
split determine cutoff include feature whereas naive fdr cutoff predictor yield massive program inclusion rate datum set approach gene prediction correlation cat score interestingly differentially gene gene buffer become cutoff predictor pt gene brain datum set thus summary difference fail feature contrast express correlation ignore yield failed feature yield predictor clearly outperform fdr happen predictor sufficiently call pt hc hc hc hc
shorthand particular context divergence edge match write vertex note combination kullback divergence usual kl appendix minimum weight pair class edge graph necessarily union deviation obtain consider number graph notion namely set involve maximal vertex matching specify edge belong cover iii vertex yield possibility claim consequently union deviation bind condition theorem condition assume decoder ml decoder since assume knowledge would ratio maximize begin describe set x exponential st graph minimum conservative failure tie therefore apply prove suffice upper lemma begin deviation bind concern x u lemma lemma complete bind
state also infinite finite range estimate agreement previously obtain equilibrium dynamic exponent fact specific insufficient behaviour long interaction still topic field physics state towards regime pose additional simple lr understanding attract great attention decade g std formulate dynamic predict existence exponent initial parameter subsequently validate obtain variety study generalize
rule p hybrid ridge thresholding obtain penalty design group design pursuit either implement range threshold assume normalize interval extensive performance iteration involve matrix multiplication tradeoff omit iteration allowed reach start theory try start nice need achieve gain find simply start good choice empirically find close building parsimonious course initialization e penalty differentiable warm start poor optima warm start finally necessarily
hence preserve eqn order preserve eqn order preserve concavity q eqn strong strong definition induction assume imply hold preserve strong eq q strongly sublinear preserve conditionally compact cone hypothesis theorem precisely assertion hold contrary word sequence probability eq thus since every imply contradict eqn existence equilibrium distributional equivalence forward uniquely induced eqn must hold extend condition equilibrium f eqn proof step invoke weak convergence would I lemma define denote weak convergence distribution eqn eqn complete ii result eqn establish invariant locally separable metric define attractive proof attractive follow
surprising value lie loose interestingly learn two study might static give good choice system reach minimum summarize conclusion set interactive system solely static c ccc static sake conclusion thing notice choose lot far automatically remove pairwise concentrate sure isolate pairwise step dynamic static interesting slightly surprising error actual collect table dynamically well term instantaneous term cumulative c dynamic er er b er value get picture weighting figure compare interactive decide advance user study select advanced
author thank discussion comment theorem adaptive covariance adaptive covariance sample history plus identity induce theoretically convenient practically good easy choose eigenvalue study eigenvalue tend stay practically restrictive super decay regular contour markov attract increase last original advance review propose date seminal symmetric walk distribution definite accept let ks metropolis prove fx compactly support support super decay regular
attribute ambiguity instead application closure extensive therefore categorical attribute concept relation concept order form lattice relation web
pick obtain subgraph converge towards contrary towards still soon one imagine way relate idea fractional fractional comparable assume deal cast within study fractional cover deriving amount hand obviously use cover argument directly benefit instance cover fractional hand fractional allow iid pac bayes summarize elegant deal dependency occur probably step follow line iid stand mh sample nh z markov inequality least suffice jensen h j h z q z n j lemma z j q lemma instance easily generalization problem learning process iid setting scale mm z circle inner pt z control control z control cm control z
sequential introduce behavior change artificial pls model output residual find covariance input extract latent factor sequentially matrix subtract current extract latent recent pls variant usually differ computation focus pls iteratively hide rewrite covariance stream weight find solve equivalent solve normalize large alternatively load regression extraction first factor subsequent must subtract current give towards pls iterative extract limited case perform main limit pls algorithm remove propose avoid require
population computationally smc backward number cost vast computational spend simulate update population bit order correctly backward square dynamical minimize solution straightforwardly time abc ode closely evaluate likelihood normally supplementary material video video particle distribution intermediate c approximate package statistical environment simulate material graphic computation inference david h centre bioinformatics division molecular college uk institute mathematical college uk department health college uk college department college uk ac bayesian apply bayesian method base sequential monte smc smc perform abc credible interval infer abc range description dynamical engineering delay differential equation vast system particularly biological reliable structure underlie moreover datum incomplete surface quantitative
characterization positive algorithm message pass interested map specify potential compatibility potential px nx j x j x marginal mean I ij else exact variance estimate correct variance condition guarantee generalize dominant current
always f split case assume fused break apart want one indeed show equation know know come go either source capacity flow therefore flow kl equality propose solve want fusion finish ef group valid end trivially set hold fusion converge merged conclude possible infinite cycle split converge
agreement dot propose learn refer widely image promise matrix randomly orthogonal column decomposition svd dictionary set corrupt additive burn estimate matrix iteration performance express reconstruction root actual noise estimate vector zero non zero monte trial depict bayesian analysis k reconstruction combination mix provide source source svd implicitly fix match pursuit omp unsupervise framework propose complete fraction pixel depict decompose patch reconstruct
acceptance however trial suggest intuition seem incorporate proposal acceptance proposal certainly leaf correspondence accept move instead proposal parsimonious variable flexibility capture input label suggest reasonable mh gp region pz x rx distribution set eq slight distribution generalize definition role distribution formula r sample mh begin parameter acceptance z rx rx rx region employ scheme whereby keep propose mutually exclusive block iterated conditioning rejected process yet iteration trade small acceptance poor propose individually change region sensible block leaf likely class predictor small softmax generative label recall zero class record predict label round monte take vote upon summarize posterior proportion class obtain full accounting fully fashion real categorical context
request agent execute task add queue execute queue delay euclidean unit interact request request send indicate agent take step main service average request receive task request network request
multiplication require computation svd outer truncate svd upon multiplication although multiplication also cost multiplication cost meaningful add multiplication dominant calculate truncate svd calculate svd cost rank cost truncate large package request singular value fall current number
solution way instance fidelity consequently might limit deal construction estimator extent goal interesting bias spirit interestingly framework would overview probably point explore many research information code sparse compressive sense latter try admit parsimonious programming author oracle inequality describe view regard estimator application sake make fact briefly always concern problem concern estimator quadratic risk presence state shall remainder namely however identifiability inverse shall idea help overcome many engineering application risk reconstruction scan brain patient matrix image device image vector reconstruct diagnosis interest illustrative problem recognition library interest curve etc stand depend application eigen eigen shape wavelet stand combination vector predictive good prefer subsection risk say organized term paper help penalty practical devote numerical application extension paper typically collection estimator parameterize select
fold display influence exhibit come tendency select influence causal dependency ccccc true glasso short outperform significantly regression noiseless yet advantage influence roc curve roc curve dense interestingly model cccc white glasso
maximum margin hyperplane rely core adapt streaming pass train pass stream first storage make current ball require approximation adapt margin classifier stream suitably center core input result verify center old radius center result ib close construct compute bind bad upper bind adapt update select point belong
common study locate keep exclude participant total snps meet criterion minor allele compute allele retrieve project difference fig explore sample misclassifie create independently pair bernoulli allele control allele allele frequency explore create begin draw simulate draw simulate construct percent snps choose identical individual fraction increment identity individual eqs step create sample individual simulate percent increment generate create sample snps sample summarize introduction sect sect snps allele additionally use sect analytical follow sect realistic computation various distribution figure yield nominal yield vast majority test neither misclassifie member group rejection also threshold sensitivity sect finding sect characteristic choice control minor allele
noise dependence consistent empirical goodness long memory mention discrete nan series et martingale use domain approximate article distributional heavily rely conditionally martingale decade error al model crucial specify misspecification misspecification diagnostic model unknown li li li bp type test assume knowledge diagnostic methodology justify long article unobserved error notation vector p tc denote denote follow distribution nan discuss observable noise proof assumption certainly need causal measurable function contraction wu wu wu nu exponentially bilinear wu min wu besides model difference martingale satisfy bilinear q iid wu z wu nonlinear move average p differentiable number long technical chen decide retain hold pointed moment parameter assumption unable relax view asymptotic statistic lag
shall see complicated theorem posterior convenient popular kl laplace sequel closely relate laplace result effect law mn enjoy dual dual strong shrinkage result regularization look effect regularizer mn equivalent prediction examine integration show laplace exhibit shrinkage laplace k integration k z eqs k eq plot reveal red curve shrinkage toward obtain solve optimum mean point since gaussian mn sparsity kkt imply uniquely shrinkage sparse comparison posterior estimate versus laplace prior shrinkage shape smoothly mn relate mn go norm mean laplace nearly mn explicit compare mn region regularizer bb turn point kl smoothly intuitive explanation kl figure move close norm discriminant function smooth mn offer close lagrangian estimate easily laplace corollary difficult optimize laplace distribution layer exponential admit hyper p straightforwardly follow eq jensen upper kl necessarily kl alternate algorithm detail iterative optimizing outline detailed tb input respect get ia analogous dual
solution suppose terminate satisfy accumulation accumulation observation bound replace x k contradiction conclusion discussion solution problem equivalently converge however slowly large lipschitz frobenius respectively thus converge slowly directly aforementione next equivalently adaptive method terminate go establish solve solution equivalently first clearly terminate total terminate clearly method
player difference great value table preference saddle opponent white h black difference learn game decide close highly tendency white attack normally black opt white opposite pair factor h white side preference central preference side preference indicate preference keep safe white feature iv difference indicate preference absence subsection indicate preference absence c difference preferred move average sufficient capture may high apart balance fix value value normally nevertheless feature
take account tune essentially number regret accumulate define tune fine tune cover specifically complete mab metric induction hold prove general ball support ball constant function ball randomly add subtracting assume ball chance ever ball statistically fraction ordinary ball fix cover ball infinite ball mutually disjoint construct ball pick sequence interval follow support sign define payoff note instance subset uniformly consider value function element eliminate filter whenever play probability jj sampling tn pick strategy bound condition eq imply bound equal select strategy round demonstrate account index contain expect bound probability finitely r jt k tractable full lipschitz expert notion describe space notion relevant notion restrict obtain regret analysis cover minimal cover diameter sufficient cover essential expert metric broad trivial trivial root node child infinitely deep common uniform branching cover finite diameter wasserstein infimum wasserstein distance define probability widely context retrieval log cover diameter whose wasserstein
etc framework take factor traffic stationary vary poisson model forecasting define definite would transformation www traffic effectiveness www traffic time vary role value arithmetic calculation combination solve calculation www tool example www traffic validate discussion discrete paper depend hereafter discrete www request focus
f minimizer move one stop edge occur state terminate minimizer smoothing proof contain start start tell penalty order place constraint whole region preserve close close active happen step change enough subsection discuss change possible happen merge region splitting associate also active algorithm appendix contain complete stop split place
multiclass regret may label balanced leaf reduction binary probability label hard solve operate loss case concern question multiclass binary class basic leaf predict noise root preference utilize multiclass multiclass bound binary bottom single elimination player elimination control somewhat surprisingly single player player play twice simultaneous elimination follow carefully elimination tree time multiclass label construction robust lie set previous work fail adversary number budget error always comparison repeat game analysis error player severe bias elimination elimination player player win play
poisson quantitative assessment slow decrease mention monotonicity comparison decrease blue black derive provide
form random square gaussian mean eigen put concentration
p ps relation application al notational stage going write another evidence account convergence estimator gaussian express nh analyse yield I tn last hand side term inequality ergodic imply relation relation obtain q convergence obtain study order analyse three
show prove chain spread exponentially empirical seem retrieve chain keep mind numerical apply estimator size c c c c c consistent regard secondly size c c constant order
update get formula separately n due factorize perform document separately derivative cm label difficult compute second order taylor expansion py py tr var derivative evaluate cm taylor expansion choose introduce effect shift nice normal extend derivation variational normal see step log partition function involve reasonable max margin however problem use beneficial high order factorize optimization svm lagrangian cm last fully factorize perform document sub plug get dy shift mean nice shrinkage derivation variational problem program exist solve estimate step see second learn lda py py point estimate seek high cm level z approximate factorize define follow dy develop co follow normalization step get update optimize fully factorize separately cm derivative second var py py tr var var derivative evaluate derivative get
irreducible split construction borel depend residual eq unless symbol block simulate differently actual avoid denote practice indicator eq initial part trajectory scheme absolutely borel quantity assume sum estimator fix define basic trajectory excess square error compute q q asymptotically assumption necessary fact prove appear excess k identity claim time classical theory symbol start invoke elegant conclusion quantity
observable characterize system rely tool reveal computation important notion random denote base bit express intuitively variable assume content notion distance log px qx p difference bit code kl expect length code wrong code relate px summarize maximum remain observe deterministic uncertainty provide quantity call little algebra interest randomize protocol tuple mutual
variate vector degree location product hence prove q coincide conclusion define far author numerical classification either real dataset approach implement avoid optima organized subsection case medical area carry see aim unit sample variable obtain accord lambda use variance within scatter index misclassification percentage classify wrong density aim gb g figure summarize cccc lead separate classified model contrary bad misclassification rate
respect subtract may generality bregman similarly subsequently scalar variant nesterov smooth pt pt go regard nesterov smooth dual variant nesterov typical sequence theorem variant nesterov variant smooth exceed state satisfy find differentiable strongly modulus u function nesterov solving address describe context prox subproblem nesterov algorithm multiply result u p norm nesterov saddle done indeed form f frobenius modulus proposition prox nesterov differentiable modulus easy nesterov dual exceed apply dual iteration exceed use relation conclusion
everything estimate large size case small intercept e intercept case term remove covariate difference treatment level method usual analysis adjust homogeneity adjust fit ignore level adjust influence overall parallel test mean value covariate extend specific simulation indicate similar sensitive difference hypothesis test treatment line range covariate adjust give symmetric distribution give situation heterogeneous different covariate range residual treatment covariate similar covariate line covariate adjust extremely whereas nominal level covariate residual covariate get close difference treatment exercise ignore grouping overlap covariate advantage spurious conclusion
lipschitz supremum constant constant condition satisfy analogous assumption ensure satisfied establish connect tail utilize hold sure divide fitting regression type misspecification preserve specifically interested regression summarize sure screen marginally consistency jointly marginally answer reveal easily orthogonality condition orthogonality essentially linear sure important hold basis independence screen hand partial orthogonality marginally select version signal strong follow b bound constant poisson light theorem effort assumption implication many final model covariate jointly normally distribute simplify jointly monotonic addition covariance part proposition monotonic regard direct positive positive word suffice establish sis property
grow network sparse assumption lasso present study parameter appendix denote simplify diagonal value outside pa bn remain effect receive parent restrictive neighborhood neighboring indicate parent node node propose violate show weak refer initial assumption neighborhood well require variable selection consistency relaxation lasso special section adaptive precision well random additional mild variable constant pa control error pa estimate argument present minor modification replace conditional separation next property lasso without consistency sparse cholesky completeness simplify remark element adjacency ps estimate replace pa bn
mle case ii eq variable great location value shape ii censor censor type I great
mention principle minimize likelihood likelihood play role decision consequently always sampling encode coherent method give hypothesis estimate principle coherence constrain two inferential value coherence coverage tend order simultaneous frequentist controlling report normalize uniform measure frequentist situation reasonably confidence measure degeneracy availability achieve either transform confidence approximate remark concern rely solely inference making enjoy utility hypothesis whether reduce measure argue minimize expected respect within outside dominate x xx dp rule convex automatic reduce also recommendation family entropy kullback binomial gray indicate confidence notable irrelevant situation half correct confidence also level record agent payoff difference situation involve hypothesis agent x odd act accept odd would benefit gain unless equal accept function minimization degeneracy confidence
random otherwise go go return parameter desire b easily amount
convex word find extension even every crucially half applicable algorithm problem ball make clearly convexity try hope efficient rotation algorithm naturally set rotation orthogonal insight gain solve optimization begin explore believe appendix derive result begin show satisfie induction update imply update convexity eq put bound obtain plugging qp assume also otherwise solve nonempty
mention moment process sequentially traditionally principle prior bayes alone simultaneous actually question solve solve solution rely realistic method need assumption recover deviation concern posterior conduct wish something piece know relationship since concern product attention must focus maximize prior call explicit important prior traditionally think prior information represent already piece express posterior posterior fact
indeed widely study zhang ml demonstrate well near performance range mention noise improvement numerical multidimensional trace field refer reader say measurement stein simple squared multidimensional appear wiener denote signal ratio whole apply replace standardize interesting whether similar near efficiency result still elaborate estimate base hybrid zhang likelihood easily solver
distribution unknown hoeffding universal universal leibler hoeffding k divergence result test alternate lie test certain alternate shown exponential hoeffding test size involve alphabet advantage due statistic nan test kullback evolve wish say characterize observation favor marginal probability relative entropy leibler observation universal testing information available regard propose testing q alternate suppose alternate lie parametric specific paper hypothesis interpret similar refer divergence terminology within framework view hypothesis identical consequence also establish advantage finite size hypothesis show divergence alphabet
within analytic set thus detail k cauchy contour dominate r fy I hoeffde proposition regularize ex chi department model bound
separated phase regime map ml agree become influence performance characteristic transition avoid machine translation one sufficiently motivated map exponential number whenever ml might implication instance application translation top solution accord heuristic need careful might nearly solution choose directly notion intuitively define ability accurately fact al weak entry emission characterization equal space sequence quantity entropy x intensity
term latent root condition differential root hermitian hermitian verify polynomial invariance put analytic play j k ij I matrix assertion real high let eigenvalue
regularity fulfil dot derivation substitute rewrite formula mean derivation w shift parameter converge limit negligible suppose note mle situation kullback regularity f alternative suppose interior
key cm cm cm cm plain work extend various bootstrapping residual give consistent apply regression norm hierarchical norm generalize locally locally w w multiple moreover extension connection resample boost use estimate interest finally application resample signal processing remain explore pursuit pursuit greedy selection distribute moment derive another lipschitz lipschitz lipschitz constant note improvement improvement selecting give among use leave like appendix design function real subgaussian subgaussian exists normally hoeffding bounded variable amount subgaussian also quadratic form independent subgaussian eq positive r lemma sign j j jj directional derivative along sign far j optimality relate appendix continuity short calculation invertible set result p
strong summary assumption unbounde around fall practice disadvantage live outline tend vanish refer abc glm glm summarize glm give value retain aid abc distribution tolerance glm estimate truncate formula look two review computational genetic tend correlated function reduce analysis mahalanobis identical advantage reduce summary retain sophisticated statistic pls fix simulate summary
episode neuron influence neuron influence episode however neuron delay motivate characterize give brief explanation communication thin neuron electrical impulse across side spike randomness furthermore refer delay ten propagation potential spike spike frequent episode neuron occurrence episode event represent episode event episode spike serial episode inter event constraint fire neuron delay neuron call fire sequence detect sequence spike general delay neuron specify inter assume delay delay frequent episode discovery generalize serial inter specify user specifie
sequence main ratio namely side respect probability something logarithm good whose standard interval grow carefully half logarithm simplify trial test measure obtain previous figure vertical logarithmic red would conclusion unknown compare composite hypothesis nan uniform uniform context martingale turn law iterate almost upper confirm intuition side long enough increasingly walk universe depict value de none reasonably typical setup evidence hypothesis point seem likelihood avoid observation likelihood ratio ratio affect plan plan probability coin stop identical odd evidence decide calibrate wrong acknowledge model wrong may provide evidence anomalous phenomenon continue probability want hypothesis special g make odd turn observe
implication towards space monte paper marginal quantity approach seem naturally provide close denominator estimate output et
leave observation shrinkage site change drastically scenario drastically plot even though size keep level heterogeneity overall pooling tend comparison consider pool toward common posterior variance toward mean decrease statistically bayesian become likely score estimate pool zero adjustment posterior deviation go frame improve ensemble conventional bayes provide precise barrier sure fitting model way software hill fit package package treatment site treatment effect hill fit available well appendix hill site site site illustrate arise comparison national education statistic national assessment mathematic make mathematics fdr many comparison statistically theory average score display well concern multiple fdr appropriate ahead hypothesis difference reason motivation state regard model average state test score year type error rate concern although exactly nan weak classical
control observe around straight conclusion approximation rare moderate tail straight tail tail straight p table increase approximation size assess kolmogorov von function von cm empirical total size million computationally intensive especially replace approximation simulate sample compare performance size table replace permutation slow deviation distance result table converge approximate component contribute increase approximate decrease distance stay appear one preferred procedure comparable reasonably hundred conclusion similar table parameter approximation table well size around preferred counting empirical check around von distance indicate predefine moderate moderate know significance ideally true
differentiable point interior domain augment lagrangian hinge discuss apply hinge slack variable equation rewrite non q exchange explicitly maximize follow similar algorithm analogous broad function auxiliary loss regression define write example h h formulation optimize objective add term eq solve exchange remove eq nb similar thus use minimization lagrangian prox criterion newton computational norm c necessary active kernel c reduce considerably fortunately th files matlab improvement overall parallelization motivate sparse formulation mkl mkl elastic choice previous formulation furthermore efficient elastic mkl consider generalized block formulation nonnegative function example norm mkl give elastic
shrinkage extend consider expect sparse transform make cluster wavelet interaction eq cluster cluster density discuss past help specification lattice coefficient decide two adjacent make total nine coefficient illustrate time frequency examine periodic boundary black frequency counting unless row immediate variation
formally maker determine action vector opponent decision maker variable state illustrate full observe outcome propose state sum tracking instead compete good switch restriction loss task infinitely interval natural connection describe handle propose set task linearly consecutive task similarity action consecutive game far away interpret internal coherence maker assume set logic maximal action task order action higher favorable maker maker high rank need receive within simultaneous action must order maker maker stick action shift budget associate cost
natural measure true environment term theoretic kl divergence environment extend output extension special calculus exposition discrete interaction construct call string length denote string agent formalize
rip stability hold spike separate iid toeplitz know ensure toeplitz sense rip toeplitz construction simulate depict figure reconstruction spike well filter process fix fix minimization show toeplitz bernoulli preferable toeplitz govern equation closely interested note bernoulli experiment ar fix nonzero component ar process sense matrix bernoulli sense toeplitz outperform toeplitz article corollary thm thm definition electrical computer edu realize drive impulse operator compress reconstruct suitable random projection turns indeed reconstruct lp reliably blind de convolution filter naturally
behavior highlight factor aid learner improve learner payoff mistake information publicly learn acknowledgement ef ik grant grant fa ef part nsf proposition non section systems agent act effectively find game converge efficiently nash equilibria anonymous game good dynamic two convergence beneficial many provide agent distribute unable behavior user preference agent strategy however large system agent choose fast user period thousand million convergence sublinear agent provide polynomial exponential number mistake agent payoff agent effect message delay agent learn need find characteristic make involve agent seem hard interaction advantage typically depend agent less useful
convergence walk mcmc describe section property include membership diagnostic markov consider begin last compute z move metric identically converge number fall interval monitoring sequence node stay long enough reason intuitively speak diagnostic chain empirical sequence convergence bit explanation propose view friend message friend page basic uniquely change bit name core network facebook always accept side thus undirected network user join one allow twice network email domain join uci edu email many privacy setting restrict information reveal possibility four refer result setting facebook set friend web page user network membership collect friend additional profile potentially profile display friend collect profile information available method enable statistically elaborate topological measure purely view decide collect interested node rnn extend collect whole population k randomly select l rw total valid k neighbor avg degree median degree rw rw uniform uniform life explore systematic mechanism automate mining per ip facebook traffic call restrictive force modern site enable asynchronous loading web require sophisticated satisfy overview limit memory ram disk collection parallelism machine run share server control number connection track already iii maintain queue account manually
q construct rotation rescaling transform rotation fourth rotation rescale matrix long neighborhood unique order velocity transform scalar imagine quantity system prove substitute transform tensor kk kk kl satisfy solution permutation reasoning equal element permutation permutation necessary scalar separability involve measurement describe methodology multidimensional separability imply transformation velocity correlation product correlation independent source number vanish velocity correlation source
necessity choice multiply concentration concentration choice proposition generality distance dependent marginally sequential invariance dependent crp distance identically marginally sequential decay marginally sequential decay another invariance table absence presence customer customer way table customer customer customer customer customer way link occur first customer customer link occur table marginal consider may satisfied root possibility describe describe result one customer distribution marginally quickly corollary decay distance crp marginally distance sequential necessity probability induce alone marginally enhance concentration accordingly crp customer note customer assignment prior generative z generally sequential follow special window sequential case operator identity traditional np posterior gamma distribute exactly gibbs entail transform variable hyperparameter window rate function section use hyperparameter decay since conditionally case depend function simplify decay normalize continuous might approximately
canonical model poisson indeed canonical link function property contingency margin cell margin cell boltzmann therefore acceptance property hold positive normal write implement proposal independence hasting follow exp rate rate col code figure histogram mix behaviour random density maxima histogram mcmc mean direct metropolis hasting distribution metropolis proposal independence hasting via code sampler evaluate output average lead something range mcmc mcmc valid type remove carlo replication run output hasting biased difficulty converge stationary sense range sampler mean iid walk cauchy prior density check walk metropolis code sd df df df check sigma n sigma rt mcmc valid show acceptable small sense leave window medium scale induce acceptable mean gold use cauchy noise leave right mixture five walk variance file txt sigma pick walk normal center figure book histogram explain nothing top every middle density bottom function condition pair distribution proper proper therefore inner set eq whole follow go infinity include intercept probit version discuss intercept column matrix code I type main lag autocorrelation ci bank intercept book intercept posterior covariate bank magnitude covariate
take parameterized times proportional come draw e index either force precisely pe particular chance must could chance choice agree chance agree subgraph unless degree z z even degree find even degree everywhere edge part value ht
bind analysis upper rate dependence k matrix collect vector supremum proceed supremum relevant packing generalize low concentration step show fixing pack disjoint radius pack agree set cardinality consist claim step convenient every element k k ts satisfy van process n prove claim next combine control correlate intercept covariate b nj p kk covariate contain intercept design lemma allow uncorrelated design lee thorough read paper detailed comment would thank anonymous would thank participant quantile regression conference university mit american american business school conference college theorem conjecture comment etc j bt version acknowledge national science foundation regression high model regressor size regressor grow slowly quantile regression qr post penalize qr qr apply ordinary qr condition qr uniformly compact index drive penalty post qr near uniformly qr rate close condition model select select uniformly quantile response capture regressor exhibit outlier excellent property asymptotic
closure variety need consider segment join construct get effect definition kind first note effect invariant follow list binomial term case theoretically markov basis show markov basis effect cccc cccc move cccc move ccccc analog save complicated fact partial result define
possess oracle practice likelihood proper choice selection estimation wang select penalize scad scad able consistently set lin tune consistency investigate scad select yield identical tend cross validation demonstrate include greatly comparable computationally intensive validation organize penalize likelihood section tune parameter bic graphical selection scad adaptive tuning bic cross penalize depend maximum xx I center
x pe see use segment edge vx xx fall boundary region mass arbitrarily edge opposite line vx vx xt rx orientation proportional rx z rx rx ir pe rx pe xx pe rx r rx j dimensional implie fall boundary mass note convex ht xx xx arc pe rx pe pe rx pe pe pe rx j pe rx r nr pe pe pe simplify central thus order paragraph nr pe nr pe vx pe pe pe px pe therefore lebesgue pe pe px pe px pe r arc h hx ix ix hx pe pe rx ii h explicit form mean hypothesis form spatial thus size random geometry invariance result
also q see design sequence proximity initial tolerance augment lagrangian eqs stop tt final newton al prox correspond zero equal general expression note eq proportional element thus let disjoint proximity regularizer obtain moreover analogous soft operation prox operation compute hessian regularization convex norm support group generalize define indicator projection onto lemma finally compute envelope let prox sup prox sup prox regularizer section group regularizer component set thresholding operation regularizer soft identity support envelope prox regularize word update write minimizer show proximal minimization impractical minimization tolerance present rate improve inspire partly however objective check exactly early minimize nx j line arithmetic eqs minimization generate eqs let objective proximity parameter increase converge denote solution c substituting sum side proximal minimization claim convert result norm residual generalize minimizer us denote objective minimum distance minimizer w inner minimization solved exactly follow
framework value despite risk assess framework linear yield expansion decrease variance imply asymptotically asymptotic closed variance quantify explicitly simple population shift loo large make computation empirically minimal loo stability show simulation distribution estimator test bias histogram assess cv procedure unbiased risk conclude cv procedure bias asymptotically good instance model estimator loo cv typically grow belong second explain risk beneficial third small probability near specific cv model efficiency framework directly cv make good cv cv depend asymptotic cv aic multiply loo loo loo bootstrap density show multiply risk framework cv generalize selection estimation model rigorously every empirically regression relate candidate loo framework cv confirm naturally select among despite constant penalization coincide section loo suggest
interpretation consideration much select propose perform principal covariate mixture model top number approach include difficulty argue cluster adapt originally jump discriminate model dirichlet dp global attributes simultaneous mean shift improve significant signal noise lead reduced dirichlet dp mean gamma usage shrinkage inefficient utilize embed step
exploit show favorable poorly interest natural inverse problem eeg localization deconvolution identify much valuable explain predict neural machine consider arise sparse matrix element pursuit denoise brain imaging community efficiently
vertex intuition mention graph approach empirical likelihood control fluctuation cause limited term add low r extension estimate select first show indeed sufficiently exist constant exist exponentially success converse reasonable old graph compose
queue reflect pair line sample pair solve remove pair maximum queue item pair pair queue stop increase search least lift fully implementation detail architecture focus table operation hash table space distinct pair similarly base count input since occur transaction good update require hundred unless cache fraction spend well update approach lie hash use hash table essentially speedup applicable filter reduce artificial mining repository three internet avg pos actor connect uci click stream pos contain see market store traffic dataset generate generator actor set rate
weak prior unconditional rp band star common star star likely highly likewise star latter prior much well estimate want ap detail give star around ap estimate measure triplet nm independently additional finally survey incorporate ad hoc system address goal survey desirable maximize scientific replace less bp rp ultimately forward adapt ap weak h simultaneously practical assume low fit forward section bottom h considerably correspond tm star expect snr model extra variance three extra noise ap bring performance determine two worth point degeneracy mapping degeneracy g distinction approach forward modelling weak smooth unlikely adequate fit lot structure dimensional e newton intend overcome non direct finite issue datum much automatically division independently smooth spline forward naturally actually goodness model low resolution rp summarize limit well entirely accuracy model h star estimate course study average star systematic correlation strong wide range accuracy estimate remarkably variance also affect statistical show intrinsic rp addition report
new variable dependent random prove measure exist apply consist analogous ball support well standard example exist ms cover set obtain gaussian need reasonable bound hold v equality ex h lemma bound kk large understand use k ks u ks prove gaussian h optimal selector satisfy cone hold selector show
c xshift yshift right right c e background width r center width cm text come style circle font plus xshift c xshift c center width cm text text center candidate font inner size e leave b scale leave b c c e xshift yshift leave b leave width center width text come false g rl rl rl e k interestingly number episode enough check node episode sub episode serial episode episode general episode example denote node obtain restriction node maximal obtain obtain dropping node maximal node frequent frequent describe frequent level ensure order set type denote episode frequent episode frequent episode share episode drop episode episode serial episode episode candidate combine candidate note share dropping event three drop event drop candidate episode block respect event block episode combine store episode episode place share event candidate episode episode give rise block episode common episode drop naturally construction episode block sort array type array type episode event type block episode appear since belong may appear later candidate list episode episode organize episode appear array store episode episode episode line episode start try use procedure identical return function candidate equation three possibility closure validity separately possibility save explain decide generate
case difference point particular value hmms easily check parameterization sx recursion recursively coincide eqs practically equivalent indicate limit behavior recursion filter use step potentially limit may sufficient consistency option use work batch mention avoid hmms interval average see regard complexity sx bring operation x j interestingly observation iteration comparison directly meaningful several respect constitute model say gradient equivalent main burden come necessity via recursion state value transition low dimensional cost
case reduce configuration locate ask expect entropy inequality phase r object want find sequentially store see far every oracle close current minimum minimum ask learn know object center htbp object retrieve neighbor times mr ask oracle close hence distance away vice versa need constant object w u r reduce reduce number level time expect chernoff take union hash object analogous locality sensitive scheme neighbor evaluation hash value near search address nearest neighbor near object contrast access object live oracle give object attempt human user capable statement similarity assigning object hope sort object call object difficulty oracle depend inequality rank speak define inequality violate build
evaluate different available notion formalism repeat high observation testing order restrict margin broad margin derive learning theory bind learning algorithm definition algorithmic contribution novel finally mainly output space pair measure set must prediction measure outline step predict set paper confident
score consistent score proportional degree freedom forecast whose compute numerically degree forecast carlo continue multiply acknowledgement author huber discussion reference financial von national science dms university tucker survey express author mm spatial quantile statistic letter coherent risk finance taylor measures international journal forecasting evaluate forecasting principle forecast ed error measure generalize forecast international journal finance wang conditional expectation predictor predictive market reality check forecast accurate finance journal loss classification paper evaluation survey journal scoring pp geometric quantile american journal institute mathematics coherent dispersion business forecast management international overview journal compare prediction probability business economic united york r forecasting international journal management forecast operational journal operational journal business economic journal statistical optimal forecast report quantile international journal forecast proper scoring journal american association hold assess forecast application ensemble prediction surface operational w h economic measure forecast forecasting decision interface college pp quantile j wind journal american scoring c functional incomplete huber huber
constrain number formula covariance maximal order occur pair partition frequently within one tree order tree size maximal product tree small covariance thus tree set covariance tree size denote tree ex u u pair way partition variable one corresponding contribution definition ex covariance sum u k product tree hold partition show n
significance threshold incorporate threshold another source bias snps effect rank estimate biased complex correlation due linkage rank effect allow snp correction however limit I provide fall within clear frequentist significance analysis estimation consider note current association prior move part computational incur inference full acknowledgment three constructive suggestion substantially improve dr association study type institute grant engineer genetic ed follow sample significance threshold propose spike prior possibility chance method testing average procedure bayesian yield likelihood outperform four odd ratio genetic snp
base careful typically testing decide whether reconstruction enable adaptation sparsity check stop thus unnecessary beneficial pooling randomize statistical start drawback limitation sequence technology first heart cs problematic application expect certain long beneficial cs preferable frequency take pool reconstruction pool target efficiently genomic efficiency could maintain large coverage increase future sequencing technology pool difficulty relate randomness discard instance sense randomness view enable pooling scheme carefully design drawback mention contain half may problematic might slow minimize pool pool advantageous assign pool pool design approach experimentally purpose adapt simulate experiment beneficial high coverage thus design suitable benefit cs variant amenable detect next generation sequence technology population interest copy important studying serve reconstruct copy level rather another give present evidence read head map genomic tail map two pair end read genomic rare discover extension read introduce procedure
therefore naturally ad hoc turn comparison distinct share express parameterized function statistic arise scale locally proportional shrinkage test compare improved performance covariance different shape mse simulation repeat shrinkage plotted coefficient small marker figure covariance structure ar entry fig estimator fig
setting find nature combinatorial amenable gradient procedure optima measure neighbor zero otherwise know window function positive evaluation normalize sum sum smoothing ridge kernel except hold mention follow result matrix ordinary regression ridge instance integer eq example ia prove alternative classical spectral inverse cut reference estimator review selection usually select choose family goal therefore drive satisfy constant negligible selection unbiased estimation minimize eq cross heuristic penalization consist form estimation heuristic deterministic covariance imply drop note dimensionality classical penalty lead
obviously contribute remain consider item common available accept vs ir b ib mean get brief proof demand order get demand total number input select cause happen decision claim get complete general program decision section tie arbitrarily use assume prove tie argue discussion proof provide distinct distinct lemma exactly integer competitive compare online competitive online program hold linear program apart offline large program rigorous achieve approach reduce randomly competitive problem random arrival objective distribution input dynamic dynamically update threshold price vector interval current online problem research hand tight actual
method time affine add subspace become work clean outlier work affine make mixed dimension algorithm artificial hybrid rate misclassifie eq pt pt c kf pt kf kf mixture kf local spectral curvature voting matlab kf http http www algorithm http www vision db paper default
assume bound prove explicit hence deviation appear summarize deviation control find whenever universal least lemma almost surely satisfied mn fix almost surely population surely stop consistent covariance cross difference solution discrepancy covariance closed knowledge heavy pls early connect context stands study
inferior fine extension pde wavelet correlate hard denote noiseless denote traditional filtering think pde square standard tv image denoise estimate minimization eq function control tradeoff fidelity image somewhat variation adopt study theoretically pde base image belong space existence establish tv discretization implement continuous formulation tv competitive denoise ideal piece point partial first difference inherent tv
denoise ref also evaluate denoise lie wiener snr different display wiener mcmc snr quantitative mmse well wiener variational approach latter hyper constitute bring interested image observation distribute translation invariant transform decomposition resolution fig noisy image radius fig illustrate denoise mmse sample ht snr db propose kind representation variational approach c mmse ht mmse mmse estimator mmse signal signal perturbation introduce signal prior hyper hyper
kullback leibler divergence domain use distance proximal proximal penalize log likelihood follow implementation r r r penalize point let r th rp nonnegative nonsmooth lipschitz bound subdifferential set standard mapping allow mapping use operate bivariate analysis assumption locally subdifferential lipschitz differentiable locally domain iii convergent nonnegative mapping
huber minimize contamination vice versa huber partitioning range yield contaminate test evaluation censor huber occur set disjoint begin unity unity log large test trial identically distribute directional semi supervise fix significance nan side tail side alternate p n appeal investigate efficiency reader summarize model selection well likelihood minimax efficacy testing suitably standardized statistic hypothesis limit size test efficiency relative twice large computation test strictly label give appropriately standardize efficiency square evaluates imply slightly scale asymptotic relation exponent gaussian distribution confirm density would twice large limit demonstrate guarantee amongst optimality long necessarily retain several
rewrite set projection r u subdifferential case equation equation notice open induced set set lebesgue almost negative diagonal property diagonal nothing imply eigenvalue optimum duality negative outside nesterov cholesky factorization sdp allow write transpose cholesky random cut sign coordinate sample good probability make nesterov remarkable fact future research hope nesterov main drawback former presentation uniform motivate eigenvalue column may require optimum reduce unique binary turn unit
positive negative part subsection give derivation formula ls concept remark even though discussion manuscript count report generalization perfect graph solution derive q combine formula consider expand rh matrix expansion ls expansion l step consequently relate determinant l determinant obtain undirected edge matrix bipartite x l ls diag
provide implement test sequential sampling regard distinct move loop write know complete independence contingency move condition part part loop connect table fix sum extension extensively rate aspect article aspect table v model coincide model sufficient case test connect basic move connect model write sufficient independence free move basis connect table square three via ti two include
storage landscape massive amongst become ever new computational become necessary part treat unique pose modern us notation issue illustrative comprise contain temperature particular period day would component dimension amongst intuitively hard imagine say speed decrease level wind small know value mean contain nonnegative eigenvalue eigenvector associate yield variability analysis panel wind reduction refer relationship sample three appropriately semi describe serve laplacian cardinality positive integer form definite entry matrix evolution diffusion
derive give interaction unlikely construct base fashion tractable mechanic anneal fashion gibbs computational sampler optima example change mechanic cluster assignment draw label label complexity sampler summation choice mechanic replace define omit require sa different expensive less label significantly affect good
point wish state evolve observe lie state express compute overlap gaussians determine maximally point point note evolution quality time whole analyze variation density apparent inspection plot combination need visual select gaussian essentially linearly expand display stage different examine colored plot structure belong follow distance exploration involve perform svd coordinate feature within also employ subset effectiveness use filter visual easy expert method select feature conjunction happen apply et cell patient type expert datum divide experiment expression
interest laplacian visualization reveal soft frequentist ss turn offer improvement thresholde gaussian cb std mse truncate laplacian cb median truncate fix univariate wavelet angle thorough comparative evaluation several repeat estimator outline well review technique experimental level invariant decomposition implementation employ retain propose alternative diversity possibility wavelet multiscale comparable prototype correspond corrupted version scale hard soft universal cs denote correct indicate multiscale tn multiscale multiplicative approach ss db std std median max ss median
e j n I htbp cm reader illustrate asymptotic allow interval compare ratio reference weight cc cc cc n package consist six design except want confidence object vector alternative hypothesis side default reference default contribution
q solution unbiased approximation sequence differ involve simulation although superiority embed briefly model generalise diabetes study probit diabetes predictor concentration pressure diabetes assess diabetes e coefficient associate rely probit probit latent
general prevent computing inspire randomize avoid recover notion definition notion though sometimes representation real borel papers notion computable recall less call great e co space topology suppose countable topological every yx xx therefore topological space computable topological count topological space countable enumeration possibly sm sn topological derive often enumeration b sx notice open space set correspond observation topology computable instance section name computable topological enumeration sn sn computable name similar name usage able computable functional enumeration implication computable partial computable computable increase domain eventually every informally use enumeration name enumeration computable situation establish may refer implicitly relate topology computable topology context order computable enumeration topological standard enumeration computable topological enumeration enumeration give
q reduce segment q reduction th calculation slightly simplify q u claim point remain lie right hand u du value solution theorem also distance shortest pair solution svm vector point uniquely every possible previous word subset set artificial problem must optimal hull tell uniqueness part occur also low path let bend straight point relative confirm finding cube cube h dimensional em vertex directly exact arithmetic programming solver dl vary bend
aa sufficiently loss probably aggregate one forecast case outcome aa theoretical aggregate learn prove cumulative expert expert equal ht read prediction g w work description goal description extract procedure match control ca diagnosis intensity peak diagnosis date take date percentage align peak peak purpose
interior domain interior strictly ok twice fu fu fu fu r norm let prove smoothness plugging prove I r k claim school university university body appropriately sophisticated describe construct statistical decide methodology duality conjugate smooth respect dual analyze potential framework method incorporate recent describe systematic method construct strongly function already derive utilize generalization bound online matrix derive property problem help decide adequate framework derive novel learn tackle learn problem efficiently sophisticated form knowledge share feature across task central understand restriction impose nature impose
formalism accurate se actual formal well true false alarm ti x ti experiment agreement actual dimension se numerous property false measure evolve match formalism success predict indeed empirically mp tune reconstruction provide natural tune mp amp choose achieve eqn refer sometimes mp amp short phase transition tune mp se optimally tune mp evolve se formal evolve property show supremum lie mse
sake completeness simplify presentation sensitivity sensitivity section prove certain anti concentration bound invariance anti hypercube prove anti concentration bound distribution anti concentration distribution multilinear polynomial multilinear work hypercube first bound sensitivity anti bound noise sensitivity follow negative bind expression I follow concentration regular interval length let interval sensitivity regular degree f applying exist kp bias towards np sx ns lp give x
mse three posteriori equation hereafter concern specification use datum poisson example largely insensitive knot allow prior location choose range sort value respectively spline basis formulate instability g trade quickly interval computation prior burn follow update effective update increase number iteration much calculation perhaps quickly fit map use burn edu bar size competitive particularly marginally example generally
bootstrap assess significance create drawing independently distribution prior distribution draw distribution probable record experiment increase experiment actual estimation user subsequently user posterior user prior adversary show biased outperform biased model encourage access use organization organization scientific half month include initial datum include three date access discretized hour per hour
wang associate valuable comment tend monotonicity subsequence assume moreover assume perform infinitely continuous letting monotonic inspection unless theorem global maximizer fast new numerical share simplicity dramatically improve speed improve strategy act global effect multiplicative extension maximum keyword hybrid design approximate encode explanatory extension strategy classical
room likelihood information qualitative inferential reliable selection study sound available close many biology abc employ grateful comment member biology group version grant article bm graphic division molecular institute mathematical sciences college uk sciences important hypothesis exist reality require suitable allow g signal particularly biology
observe nan positive negative equally precede technique deviation see deviation symmetric see clear structure inspection interval include sign quite unlikely occur posterior evenly consistent decrease deviation distribution case apparent characterize rapid count typical right see interesting examine joint immediately apparent origin indicate event asymmetric behind outcome half maxima deviation base short event
present generate system interest markov quantum walk consequence string uniquely determined sketch idea string coincide string vector determined string subsequently string coincide due see therein determined word whose uniquely string string necessarily counterpart crucial map q comprise generalize obviously irreducible derive set strong view inspection let case put
code com p extreme share library incorporate code everything quick include split slow split step split easily turn set specify split merge heterogeneous incomplete integrate true incomplete result marginalization derive em monotonically find noisy incorporate prior without analytical accommodate merge deal maxima suffer incorporate nonparametric modeling infinite dirichlet advance model apply science likelihood jensen inequality continuous case integrable normalize observation entropy distribution take q reduce calculate posterior drop play ij
model particle carlo repeat data dynamic tree outperform counterpart dynamic linear nearly offer mean method sd rf search scheme rank location maximize expect gp leaf repeat mcmc hybrid setting true impossible promise local appropriately iterate towards regression flexible attractive tool hybrid usefulness deterministic stochastic due outline tree search space optimum default input expectation statistic optimum optimum improvement minimize hybrid incorporate idea maximize precision favor scope close give tree ct cumulative freedom n ar ar detail generic candidate location draw sample posterior fit tree leave precision quickly wide leaf seem cm iteration tree posterior grey green blue red cm tree cm exponential solution number dimensional initial location routine routine dynamic tree except
infer appear nearly model use red blue dot dash middle bottom segment approximation segment resample storage linear quantify use importance fitting quadratic allow possibility study contain also well procedure uncertainty advantage underlie appendix detail recent time mean gr like helpful liu mathematics
random method connectivity various metric properly hoc nearly impossible interesting bag move bag bag gram interesting gram move tree similar fashion relax relevance attempt ir anonymous comment great benefit partially grant present extraction focus common reinforcement query short datum furthermore understand justify document algorithm function document formulation multiple question limitation access know ir engine retrieve multiple relevant document
rv rv multiply skewness allow comparable system characterization scalable scale rv rv family deviation rv pt invariant scaling system accordingly exponentially introduce skewness symmetric student input transformation invariant unit shift shift scale rv rv deviation unit variance version rv parameter closeness skew parametrization ignore parametrization case vice versa student input degree parametrization distribution rv simplicity definition scale rv transformation example rv refer family rv look interesting transformation essential f analyze minimum equal physics commonly since define outcome domain exist real branch value include functional identity reference skewness axis map asymmetric output axis event event inverse curvature curvature skewness verify work
analysis advantage positive number estimator assume process pt theorem condition b imply try bandwidth choice investigate carefully admissible term enjoy martingale conjecture case limit carlo improve impose weak moment almost convergence desirable setting typically assumption impose either restrictive growth translate impose condition type surely remove
inherently difficult largely improvement parallelization study statistic solve mcmc course analyse advantage outline brief hope reader discuss challenge sect assess density obtain sect actual consist type ia conclude sect inference expression regard combine information incorporate experiment consist absence information restriction entirely probability normalization constant normalize constant available analytical approximation relate denote function convergent integral mean grow covariance domain indicator otherwise option necessary complex chain production markov chain target many markovian metropolis algorithm give current call denote acceptance q arbitrary applie proposal properly tune proposal small take step support extreme fail converge hand scale exhibit fail sufficiently require multiple number aspect perspective mcmc recommendation schedule proposal propose carlo version importance population sequential manner construct improve estimation fundamental hold include
r sr segregation severe choose large reduce ever complete efficacy segregation variance plot graph segregation agreement order r increase efficacy give pick recall degenerate degenerate get close get ar local supremum l ar ar ar suggest efficacy r segregation plot association small get agreement small decrease increase asymptotic normality segregation large reject percentile specific segregation alternative z plot attain supremum attain power segregation function observe segregation alternative leave plot observe segregation function association percentile e sample level r plot power association
real always use large help estimate formula margin derive simplify conservative nb large cast large margin margin unit uniform mathematical fact natural side estimate distinct sample replacement n count ask give replacement draw vote count chance chance replacement draw drawing drawing count vote chance select count replacement estimate select c begin formula distinct sample set replace q provide detecting size total count count slightly exact calculate replacement employ vote count method appendix post see vote incorrect drawing without vote formula would fix appendix weight work new approach post improvement recommend incorrect confidence enough occur report vote candidate decrease win vote win low contain vote win let high receive vote candidate et continue change error incorrect vote sentence precede paragraph incorrect vote vote number incorrect vote vote win produce report produce sensible recommend ok often vote cast weight desirable weighting within vote quantity vote rather cast put focus contain vote incorrect outcome show margin maximum margin increase quantity size impossible vote contribute margin total
dotted selection divide risk htbp j represent divided argument technical suppose wavelet limit cardinality f k k assumption hold j fu dy periodic dy argument bind p z j hence z jk w n jk z proposition jk proof jk k k j f bernstein e obtain one check jk follow let set integer j proposition jk j jk inspire see jk obtain inequality obtain follow lemma cn pt j
snr transmission direct transmission allocate across capture performance network diversity definition diversity diversity single definition receive time analyze scheme observe tend interference bs density evident interference define translate interference performance system plan transmission bss interference may cause transmission direct connection success probability definition follow x x limit observe definition
work double error peak theoretical precision bit double bit precision matter variation circumstance perfectly acceptable norm gpu useful realize somewhat matter different choice possible along matter web possibility get close unity arithmetic symmetry take inverse double large symmetric sensible seem go case unit zero view quantile precision small precision double precision diverse view line entirely appropriate monte argue simulation independent distribute work elementary eq small invert quantile approximately eq carlo simulation chance produce might influence expectation one imagine dominate copula lot correlation put quantile argue end truncation point high accurately sample special require ignore high mind algorithms region double seem machine already small super carlo go alone small double monte unlikely make poor characteristic error
problem alphabet process distribution completely wish answer question letter distortion code code give per letter scheme code construct reconstruct process fashion address I continuous alphabet source regularity code identification whose gap theoretical shannon distortion fidelity tend code operate code code match source precede rate rate block compression identification vc limitation restrictive g autoregressive source parameter sense diameter priori relax study identification satisfy belong parametric class open subset code infinite assess composite lagrangian regularity distortion code identification length lagrangian achievable code block fidelity converge dimension certain class region dimensional result compression identification capture rich hard affect compression performance decide
verify chen convenient ess pearson statistic formula bivariate spherical significant paper david rule interval knowledge sign monotonicity pattern see application state theorems independent bivariate note correlation q commonly moment rx I moment rank surely pair two sided alternative express odd strictly plot well monotonicity immediate hypothesis
finally learn maximize criterion approximate know conversely focus estimation capacity retrieve class well cluster interest outperform generate indeed class contain retrieve distinguish community appear community explain community frequentist sbm handle topology might structure observe lead estimate class thus class network percentage h c display structure presence central complex
linearly polynomially statement datum relate search often depend dictionary fact take cause scale exponential building train binary classifier enable handle training production generalization execution employ accomplish quickly since boost find gain varied global amenable heuristic show map negative cost improvement increase advantage conventional greedy hardware quantum establish depend propose attractive employ quadratic map naturally
since contain irrelevant estimate parameter transition entire relatively markov transition estimate cross blue line line time change note capture sharp instance somewhat seem sharp might assumption hand quite room improvement design model evolve eps eps eps rise one observe markov traditionally single light degeneracy find distribution investigate modeling variable seem difficult study stationary knowledge directly specify move hope beyond incorporate evolve phenomena existence form merge split flexibility temporal
addition depend far close empirical phase remark different far exploration range independently represent parameter precise therefore phase nan exploration regret exploration phase algorithm smaller far follow exploitation phase full allocation channel bandwidth stochastically usage channel channel reward ki channel result positively detail explain important positively case function positively optimal observe equal ki
sequence pixel imaging l cs nonzero exact cs error exact implementation conclusion sequence fig measurement entire partial cs use set use exp opt give fig exp opt sequence cs rmse stable large enough cs frame keep slowly rmse ls plot l error l number much l inaccurate notice fig significantly outperform cs outperform always fig fig poorly error large cs bad though slowly study reconstruct partly contain know cs relaxation outside constraint sufficient exact reconstruction modify size known extension
result metropolis time adaptive algorithm rr rr rr ac min median max median ir ir median ir ir median ir rwm daily area figure poisson peak term school break occur include harmonic effect peak explanatory variable start term beta parameter explanatory preliminary include eight correspond daily intervention take present monte replication different seed trend explanatory variable adaptive random walk least twice adaptive hasting rr rr rr c median max median ir ir
perform tm limit length time pick therefore set fig besides tm behave trend limit error function tm tm tm versus incorporation tm mc finally distribution tm tm tm tm tm indeed keep tm tm computer fig show contour line decode incorporation detail case smaller slight two tm confirm f tm average accord versus difference huge subsection never compare cutoff enable one relatively number little difference subsection tm quite tm small though tm tm tm though slight tm tm much
consequence spirit coherence coherence similar cumulative local coherence condition recall large eigenvalue weak isometry coherence theorem observe schwarz inequality factor furthermore find important tucker kkt condition context subdifferential calculus kkt q span anti lemma kkt must leave multiply equality q define hold condition moreover prove ns n kkt kkt min ss
learn multivariate throughput technology expression genetic genome provide complex disease diabetes know phenotype aim identify marker nucleotide snps rise clinical phenotype disease across individual deep disease directly express pathway use analysis examine many often co co incorporate association gene exist gene analysis obviously exploit source genetic expression gene jointly trait individual gene module set snp module process single incorporate gene snps search leverage module discover cluster module gene gene snps module detailed module coefficient white gray correlation response highly cluster tree likely influence group tree lasso combine across related leveraging cluster gene gene
beneficial achieve long design fair sharing cognitive repeat monitoring repeatedly begin spectrum access bid history regard secondary activity limited game high bid irrespective entry incur regard allocation make monitoring incur propose outcome transaction term scheme outperform multi comment feasibility exploitation different interest architecture cognitive currently pilot gain communication resource management
recognize surface hand top visible space map remark question direct boundary harmonic regularity question harmonic regularity suppose imply proposition conjecture convention question space g www support high universit university city usa city china setting manifold space image mathematical pattern recognition framework develop theory constitute towards geometry vision appendix metric homology theory geometry extend theory laplacian domain space manifold deep vision lead development develop mathematics vision e space occur manifold paper example recently decade laplacian classical fan extension separable endowed measure tuple define boundary operator geometric special homology interpret finite sample equation finite kernel gaussian reproduce substantial second adjacency threshold picture regularity regularity form solve harmonic progress coincide classical answer theory case riemannian manifold riemannian manifold upper de general topology image lee investigate real basis find homology class evidence homology surface attempt could
exact minimization express involve iteration q iteration become eq produces choose exact result desire iteration concern assume handle restrict ii possibility handle limit note monotonicity play role continuous positive tend increase maxima monotonically proof subtle point
entry rank removal rank stable rank matrix removal generality column write row observe contradict argument one consequence coherence interest arise converse hold row define one entry symmetry stable hand proposition sensitive attribute coherence stability combinatorial notion stability regard generalization construction high stable nice construction quite full clear attain turn typical integer nn consider rely well find suppose equivalently determinant zero square polynomial lebesgue whose column proposition theorem stable consider construct via orthogonal haar orthogonal arbitrary orthogonal haar
within alone induce age acquisition flat age acquisition outside inside light learn early concrete shift within concrete learn whereas acquisition age finding exclude eliminate correlation subset differ substantially dictionary work early dictionary age refine small early rest concrete outer layer acquisition outer remain concrete outer abstract relate outer
execution show speed execution among online method estimation result case make variational consequently spectral search whereas connectivity low connectivity five rand index c thus display blockmodel online variational model fairly bias spectral accurate well adapt particularly partition set densely template realistic consist political day snapshot extract analyze web page represent different identical orient realistic group six community network centre moderate party economic package present organization connectivity correspond dot generate realistic network simulate network accord sequentially minute enhance recent
order kernel polynomial bandwidth random field p bias mean become convenient c c p k expect nuisance parameter decompose note risk come mild get dense bound particular use small asymptotic stay domain improvement bias threshold use bandwidth selector weight generate kernel heat value column mat ern bayes leave plot hard ern exposition class kernel define course estimate result thresholde weight w b tt
leave apart apply algorithm iterate p three map correct one example real dataset efficacy life issue appropriately mark year period core module illustrative simplification investigate trivial stage stage table htb student situation location student student appear detail stage place student situation et situation stage comment membership identify situation discover student situation getting get probability achieve low situation qualitatively student four situation low storing event natural expert
learning perform rather standard hmms present framework entirely device remove traditional human allow well establish supervise automatically acknowledgment visit yahoo partially nsf dms dynamic lead capable otherwise long structure motion high alternative hmms kalman unobserve transition event maintain distribution state condition range forecast finance control video speech detail dynamic application capture small predict
neighbor test simulate variate autoregressive ii compare performance distribution variate simulation either missing give miss random test normally dimension random square poisson chi distribution poisson three set simulation imputation table penalty near imputation variate c c pt c c c pt parameter fold cross validation c imputation perform lc c chi poisson competitive imputation method svd near imputation penalty fact globally procedure reach permit flexibility see cross type prefer marginal multivariate penalty covariance allow observed variate svd imputation full equal base imputation normality outlier imputation compare usually gene array variate normal indeed microarray span marginal array microarray truly accommodate array appropriately correlation last nature information microarray cancer tumor datum miss
often abuse main object error independent equal I ii value asymptotic expect excess say achievable prove type setting idea provide estimate classification function assume throughout sign associate norm tuple dirac mass concentrate I every
person fair win produce winner explain evidence read propose fair winner evidence equally plausible something conclusion sure thing implicitly state get would sure involve contain sure thing hypothesis
cauchy start hx replication set exponential optimal gain importance weight sampling estimator fourth final toy example geometric target one random compute q gain rao probit diabetes year old test health organization criteria collect national institute diabetes disease available
ir rwm dimensional identity adaptive rwm check five sampler compare sampler mean time expect rwm ir rwm ir rwm plain rwm conclusion replication iteration rwm ratio pt ir ratios ir mse ratios mse ratio kp l drift actually depend imply vx vx lx vx lx lx vx r lx lx lx vx vx lx vx line deduce distribution imply exist
light regret product imply I transition product product arbitrary appeal make I low regret obtain cost curvature loss curvature lead example slow rate convex minimizer expect unique regret curvature play role speak exp ensure grow fast g linear grow achieve curvature divergence suggest determine regret shall curvature curvature curvature directly rate first let rooted mirror care notion introduce chapter infinite dimensional space analysis compact present exposition geometric interpretation proof long vector basic analysis take hull end
equivalent strong check since large reader check include equivalent conclude minimum typical follow q follow immediately union row x start number estimate splitting separately q large denote need begin sphere r I case derive sphere distribute mp mp ks ks let sketch inequality special version hoeffding independent surely mean valid coefficient zero indicator zero eq use converse chi distribution moment especially recurrence eq suffice eq part I ij ij identically drop subscript directly little trick always moment formula recurrence formula gamma treat even
versus horizontal computed database band indicate set true evident gamma find different relate anonymous gene profile set gene gamma utilize grouping label information various develop construction ranking hypothesis microarray technology continuous gene essentially count copy gamma ranking extend readily gene tag th microarray replicate library state library say adopt notation library put order latent expectation latent conjugate gamma conditioning gamma shape analogously conditionally poisson response gamma distribute normalize q notice refer sequence arise whereas inverse computation key thing monotone transformation gamma multinomial utility investigation within reason rank section model calculation pattern biological gene expression part round structure comprehensive beyond towards control list
generate triplet briefly train near find triplet target except dataset accord table code code conclude use dataset mahalanobis classification outperform dataset statistically
average cardinality pca well grow extract cardinality b dc really affect cardinality magnitude demonstrate greedy computational pc fix average instance induce figure dc pca scale discard deal dc experiment examine fix ratio parameter solution algorithm cardinality display table algorithm see dc pca ccc dna gene hundred thousand experiment enhance interpretation extract involve usually gene expression say compute expression cccc dataset sample gene microarray first principal consist training sample set sample use widely feature normalize value extract microarray gene first principal total high candidate investigate interpretable scalable study dc cardinality sparse pc algorithm time range result handle dc scalability explain vs mention sparse cm fix cccc cancer dc cca cca dc matrices cca relate task retrieval deal retrieval sparse relaxation cca cca scale document collection document string retrieve semantic language translate space
since fitness ga state also population mutation population actually markov imply mean adequate course genetic inter limit observation chain state observation player make estimation limit impossible limiting without limit equilibrium symmetric call state one union entire although expect still definition average population denote nash quantity hamming bit average hamming nash eq player state nash maximum maximum hamming maximum value bit nash population consist correspond nash
estimator take along monte width unbounded precisely derive burn ensure drift analyse estimator multiple short run allow bound require practically bound many computation integral know normalize feasible approach ensure approximately introduce et seminal underlie discard call burn validity law number
partial quantile polynomially existence efficient partial relation author broad generalized process one incorporation raise issue interested index quantile process quantile quantile build instead probability conditioning automatically condition strictly regard mapping counterpart however interest dependent fix able weak seem even process non quantile property set indice potentially interesting apply generalize nest unlike unknown priori thereby characteristic provide intuition variety interaction key role characterize quantile maximize figure partial quantile b shape partial surface infer band band interval quantile generalize ordering partial example population lead convex collection quantile square partial see representation cc potential partial arise order align quantile example point quantile population well partial align distribution partial px x partial quantile x impact quantile alignment quantile member diagonal draw lead quantile consistent draw quantile seem square alignment lack try separate extreme order illustrate simplex case might suggest deal univariate nonetheless try compare pareto point dominate allow efficient describe order fail nonetheless order binary relation represent value behave therefore consider encounter example consequence multidimensional might payoff emphasize standard univariate
stop epidemic describe infect recall infect consist infect infect approximate distribution period reason first intensity epidemic markovian intensity epidemic period non random version end mathematical contact imply individual explain stochastic sir epidemic section devote large matter possible derive exact simple close expression dynamic epidemic possible recursive final epidemic explain infected infection way e g outline detail formula idea identity final total infection pressure express getting initially get start write number infect exclude possible infect label infection pressure infection pressure sometimes also cost epidemic exactly infected dependence initially infect initially product infect subset infect total start identity transform period step identity period mutually period contact dividing result condition q simplify return recursive formula use sequentially unstable large informative deriving early rely reason early unlikely happen individual conversely happen distinct individual
ignore case gain information seem order stable process main advantage allow instead step attractive purely dependence situation develop multiscale interaction method simulate process focus two extend multiscale might amenable perfect method fitting multiscale circumstance future author would thank helpful discussion prove generalise
risk q inspection right monotone yield minimax wide range space say neighbourhood neighbourhood description back riemannian map nice background aim ng without r g x g unit sphere tuple orthogonal orthonormal frame manifold naturally subset e ex kk ex v tx facts suffice map bundle projection neighbourhood positive definite g yield map bundle bundle np yield invertible see closely manifold characterize admit basis natural manifold normal eq recall g diag onto ny g construction may n n h transpose real hermitian structure
thresholded affinity able train classifier directly minimize rand result partitioning learning image serious ever train performance detecting belong boundary boundary detector affinity performance supervise image field label similar class segmentation require distinguish object probabilistic field parse reason tu module boost level module joint labeling never minimize rand weighted affinity near neighbor pixel group segmentation connect thresholded affinity affinity
state continuous examine expand utility perform htb v good leaf policy mdp mdp leaf node definition du trivial blind policy low analogue blind fix policy trivially greedy mean sampling express form integral approximate leaf expand perform iteration mdp mdp belief form easy mean transition state
first appropriately often dependence achieve nontrivial remark result single ess fundamental quantitative version therefore ess boolean give characteristic theory know intersection uniform approximately solution class elaborate main new structure integer topic sensitivity introduce seminal notion analysis boolean speak boolean randomly choose sign independently bound boolean hardness circuit complexity complexity theory sensitivity boolean yield algorithm agnostic invariance sensitivity intersection noise sensitivity intersection intersection boolean noise sensitivity noise current good sensitivity start noise important towards intersection necessarily believe intersection intersection even intersection remain open restriction see et imply intersection class intersection learnable learn adversarial precise definition intersection regular fall class easy pac time intersection preserve regular accomplish union result towards improve
server might database server easily number start action attack visit attack front server database server allocation edge decision resource might web application kind continuous denote budget difficult attack budget raise must pay attack server run attack surface server must pay edge surface wider large attack surface build height mapping ever edge abstraction justify rate piece software detail amount budget using believe capital decide result hold quantify simplicity purely rational attack maximize knowledge attack maximize evaluate management provide system fix main result term include minimize
fractional control multipli contribution fractional change r gauss markov multiplier suppose want emphasis term choice q denote emphasis eq interestingly choose e emphasis necessarily sensible pick one wants thus satisfy reasonable objective wants put time emphasis sensible eq side discard root result behavior equal emphasis speak want reasonable would make linear emphasis bias term dm contrast variance variance gaus illustration deal design fmri illustrate capture shape response fmri weighting variance choose matrix initialize draw imply equal dm imply weighting htbp dm simulate dm snr glm plot show generate bar unit quantify variance automatically initialize column result initial dm curve gauss imply likelihood observe dm bias htbp cc glm analysis set use dm figure simulate dm snr glm fit dm show dm bar simulation investigate effect bias curve emphasis reduce
take side equivalent switching role j side let hand bernstein inequality bind right jx j bernstein union probability j nd tb j ta c since b note conclude nj n j nj te j n j n n j variable rapid advance technology throughput frequently example genetic functional frequency grow specific np make possible contribute role statistical selection parametric concave selector elastic net mcp simultaneous challenge computational statistical algorithmic method meet
say convergent power norm radius q infimum contain expansion main conclusion f note reasonably mild combine comment estimator precision need analytic eq easy compact connect compact compact every unbounded closed main follow close conclusion c compact unlike regression small precision acceptable eq comment proposition see op
linearization depend although manifold code lie coordinate imply learn manifold nonlinearity coordinate code advantage problem unlabele datum significantly data prevent label coordinate merely significantly simplify consequently label g within interested coding consider linear function approximate estimate ridge lipschitz include svm generalization code expect optimize become optimize immediately consistency arbitrary manifold note convergence intrinsic manifold lie pick local coordinate coordinate code
heuristic prevent optimal performance set considerable progress particular pac free reinforcement remain whether approach history make optimistic important policy computer policy thus online conduct predict learn replace figure show learn work current first promising thorough investigation policy adversarial adapt would search bootstrapping look modify ucb technique computer go class convex combination principled expand power agent generalised define help predicate boolean node false associate leaf node represents reach predicate generalise context analogous retain provide powerful way extend notion example suitable possibility logical tree several extended depth without space overhead symbol case may significantly derivation investigate agent allow property parallel complete core confident improvement allow agent planning ability main single binary attempt attempt example together binary predictor apply technique bit predictor integer character convenient environment model compose possibility construct sophisticated mixture feasible directly approximate ideal well theoretically previously unclear theory agent answer environment monte carlo reinforcement powerful highlight interesting direction performance agent particular heuristic policy intelligence community reinforcement agent thank anonymous helpful feedback communication information technology centre program mm mm pt proposition conjecture height mm monte pt ex new south ng ng com national expect act history horizon act policy ax ax ta achievable reward agent history
inner since throughout take rhs nh n martingale array eq central theorem imply ft show ax x q argument almost term ax decomposition assumption study projection converge apply conclude usual define proposition rest almost surely zero finite compact na n h l since follow rhs define apply get c give q last term rhs
r v shall conjunction thus use allow count tail expression order need replace respectively modification satisfied accord expression since refer add assumption replace small obtain self sum chen turn precise paper allow order moment hand kind especially consider closeness closeness normal reader refer due imply independent analogous value exist shall equality whenever generalize banach use present every banach smooth next borel condition coincide enough statistic one corollarie magnitude numerical rather moderate satisfying term way chebyshev type even additional obtain hold q essence measure magnitude indeed banach satisfied say take enough explicit seem uniform whole traditional significant play exercise e space type type introduce remain mean correct magnitude depend space constant also usually closeness distribution obtain closeness motivation work deep closeness chen improve closeness banach g several infinite existence smoothness
see side proof approximated state property context procedure run subset contribution degree loop alternative fact normalization choose leave side prove give formula let show
suppose previous systematically throughout expectation main goal characterize test power exist locally candidate calculated pose employ throughout paper vector respect use calculate basic integrable hold nothing particular distribute everywhere everywhere condition power treat non various assumption property follow side
actually forest continuous object edge number statistic bic edge search decomposable measure decomposable determine add add preserve one stop component start forest isolate isolate change join true bic bic edge step example object number default model edge start empty decomposable figure forest tree graph object first consider isolate join evaluate cpu ghz gb ram running bit
probit error compute easily denote complementary function rbf derivative jj kx variance site show task point illustrate b together explanation vector corner triangle interact triangle edge dimension negative corner discussion issue often machine rbf tend thus derivative point axis htbp consequently define sophisticated try try explanation value thresholded resemble bayes logistic regression sigmoid vector density index k approximate want explain assign omit use ensure orientation assign summary practically classifier high care available may describe length certain
maximal multiply correct sharp approximately spaced factor polynomial gaussians equal next rapidly shortest close error ignore
harmonic counterpart probit sampling sampler recommendation approximation reference acknowledgement mod de universit paris centre de en paris survey cover importance sampling reversible jump survey fundamental dimensional model method bayes avoid importance approach survey single carlo assess recall probit
write n vector contain deviation detail omit notation finally lt lt lt pi nh finding bias error difficulty quadratic functional observation instead variable
repeat relation illustrate multiple point query share set score rewrite logarithm dropping constant detail dataset sample model assume link q pair decrease order integral present carlo compute compare query similarity sort due variational logistic impose initial approximated reasoning interaction function general decomposable present indicator give entry call decade structural mean node similar connectivity seminal introduce strategy role indicator block ij integrate computational exploratory expect relatively short alternative bayesian formulation merge binary relationship create pair marginal treat merge really relation fail dependency conditioning marginally nevertheless computationally evaluate section provide existence link useful generate automatically automate selection relational combination cell specie web page web relax requirement possibility treat formulation database
view structure scale conclusion ensemble agree ensemble however visible tendency variable largely behave turn indicate structure largely phenomenon size score much indicate structure transition width probit expect fit account substantially asymptotic eventually understand phenomenon success mean polytope projection face course precede rigorous receive pair site hand belong lose event sparse face ensemble quite small ensemble nan reader discuss nan denote ensemble find find sparse nan provide heuristic apply algorithm random sparsity record column sparsity level record sparse indicate common draw matrix contain zero case dramatically remarkable may explain bad intersect support become consequence increasingly common instance lp moderately yet asymptotic value although effect compare c ensemble indicate close constant tend zero rapidly value ensemble unable interpretation initially interpretation driving effect behind ensemble position chance rapidly report place happen two ensemble rademacher complete long analysis basic principle science validation building construction provide rademacher excellent reason point exponent study rigorous exponent lack fit argue lack mean long assume believe occurrence argue however instance validation actually validation hadamard special happen analysis
bias naive bootstrapping tend advanced accuracy multiscale bootstrapping implement follow section multiscale propose outputs multiscale bootstrap multiscale test conclusion gaussian call dag
difference ks former chi snr confirm finally aggregate ks single operating aggregate ks subsection apply performance context linear discriminant covariance class separate hyperplane density ica asymptotic bandwidth get nonparametric regression estimate european organization aim provide weather forecast channel km center degree every affected cloud processing thank band problem cloud infer presence image discriminant mask produce endow mod aim cloud mask pixel mod threshold mostly deal spectral band generation investigation statistical water perform mod water pixel pixel mod truth evaluate water divide randomly use estimation capability split pixel
location tb estimator gmm panel function estimator cluster calculate bias gmm clearly bottom panel removal bias measurement panel galaxy cluster degree camera enable acquisition observation calibration contain measure object galaxy magnitude star survey comprise galaxy dr description location selection publish elsewhere galaxy rich galaxy galaxy galaxy locate galaxy center completeness dispersion cluster spectrum count galaxy cluster varied therefore varied critical radius mass cluster weak scaling range next cluster color correspond measurement galaxy result red represent galaxy color component likelihood get new original continue analysis subsample component measurement
step expansion start contain large possible introduce exploit bind perform hilbert add kernel proceed device expense property scalar delay initialization beyond adversary act fashion case adversary arbitrary requirement term realize see copy product occur see past versa formalize martingale technique favor issue expectation key bound correlation case limitation improve assume lead guarantee increasingly via q xt hinge loss detection three loss huber except rescale value depend impose smoothness simple want show lipschitz piecewise occur delay
within performance shift oracle scenario quantization multiplicative ill condition standard scenario normalization case condition incremental rank context recommender base rating rating test mean absolute mae mae ij predict rating item rating incremental present four column rank use miss entry predict neighbor yield incremental c incremental last movie rating last incremental look user point approximately although make difficulty deal decomposition manifold low reveal enough give good local high
iv expand eigenvector output implement implement dimension expand j return signal jt j jt algorithm contain zero eigenvalue become long sec signal become wrong reformulate sort eigenvalue svd covariance find svd eq solve j line singular zero precisely row exclude keep zero exclude svd lead expand central proposition svd
vote situation batch vote ip vote eight case batch rr ip c b upper bound batch eight kind batch chance require count count different control risk incorrect count batch draw chance draw let batch stop
inequality remain triangle inequality nonempty string term xy xy reach respective element obvious reach drop moving argument observe reach additive asymmetric like exclude degenerate measure want important density distance computable broad total asymmetric element term admissible admissible list constant additive verify list element define
parameter turn get p p posterior ix jx lf l case v f ix v posterior add see correspond gamma tr ibp gene network account information topology mcmc evolutionary unfortunately converge integrate factor infinite component treat take account hierarchy standard ica propose fix assume mass
point require case free generalize spectral relaxed case consider scalar canonical bernoulli satisfy odd central standardized analytic moment burn begin look like quadratic evident standardized generating great mean multivariate matrix definite seek moment eq moment connection self response draw distribution exponential sufficient special setup family take q see either unit logistic least easy standardize analytic eq norm fisher
uniformity figure illustrate idea f rise uniformity restriction stack partly perfect second note enough formalize henceforth great uniformity variable satisfy condition prove put q integer prove fashion z still accept exceed previously assumption make regularity adjust r iff admit pareto
notice tu estimate constant bound incoherence denote minimum start prove together imply f lower prove us inequality ij ij remark follow n e call six calculation tu ns tu proceed condition contain diagonal reveal f cn tu f analogously combine na constant f leave prove thus fx ij ij bound u f bound velocity yx value respect geodesic parametrize th thesis together imply thesis cauchy schwarz let prove
interpolation many present accuracy expect single rely random variable position pick multiple orthogonal distance estimate deviation process generate trajectory display indicate indistinguishable zero estimate accuracy fair constraint table c component component want reliable filter reconstruct path reconstruction large reconstruction mean step table display x number particle
theorem exercise proposition remark theorem theorem division bank school business university new south south article estimate marginal correctly propose mixture normal copula avoid normal copula flexible copula marginally adapt normal improve normal use empirically copula behave much third dirichlet process normal constraint comparison mixture encounter mixture dimension imply model understand marginal mixture normal mixture normal moderate multivariate mixture propose normal marginal transform marginal normal marginal define copula normal perform copula copula normal
cd rw sample half width interval roughly rw half rw half half width time large rw act rw bad replication rw rw comparable comparable rw surprising similar chain component two gibbs sampler scan metropolis gibbs rate outline geometric enable identically distribute certainly study component enable uniform ergodicity true ergodicity ergodicity setting seem rate think especially study either geometrically ergodic mcmc often geometrically ergodic implementation section superior dimensional every case real literature practitioner acknowledgment health award city york conduct full
set inactive gx sd kk property terminate iterate update terminate optimal final iterate know gx u behave exactly termination statement compare minimization nesterov coordinate sparse invertible entry density prescribe generate finally small discuss smooth case nesterov however performance differ compare presentation code matlab code code accordance algorithm terminate duality gap sm randomly size sample covariance matrix give four objective cpu time give sm nesterov scheme ii namely variant minimization substantially experiment nesterov approximation equivalent
correlate ability hc appropriate accommodate method independence statistically speak nearby extreme perfectly remove standard hc utilize datum design exploit good wide setting context paper correlate brief detection describe strength boundary sparsity strength correlation able precisely construct lower special toeplitz asymptotically hc particularly extensive literature hypothesis dependence include control randomly despite appeal convenience uncorrelated correlate special exact detection complicate tight either subtracting amount difficulty measure etc adjust recent decay rate decay add inference hard specifically model noting see intuitively add hellinger hellinger distance diagonal review concern diagonal message cholesky positive polynomial decay rate writing correlation matrix uniformly matrix turn well sequence rate proof ready correlation boundary fix sequence error test key new higher bind hc really reason summarize structure hc coordinate build correlation new statistic establish connected innovation
make arbitrarily thesis sample tight second constant together desire thesis case z max application interest approach present similar obtain variety circumstance rank matrix entry perturb produce approximately entry reveal reveal position rl ij analogously respectively position convenience recall algorithm analyze normalize decomposition guess minimum standard descent description technical reason cost denote condition may reliably follow say reveal row column represent set
candidate task scale well amount determine prediction observation process scale window introduce redundancy impose rescale element combine scale evidence sum order investigate evidence across sum sum compute
leaf appear associate weight multiple appear every possibility grow tree obtain split pick tree node form weight loss weight quadratic practically adjustment prediction first latter hundred tree final measure calculate function fit tree cross interaction advantage interaction extension cover nh efficient basic nh space minimize empirical typical minimal maximal interaction depth partitioning would weight vector form empty optimization difficult solve correspond feasible partition also computationally latter demand partition sense suppose part exactly helpful observation belong receive empirical indicate whether part exactly demanding understand equality constraint yet still due relax take relax
away inequality define candidate specific generate mining frequent consider overcome problem cause support hypothesis may vary scenario part considered split spatial datum operate present contingency association find adjustment multipli pattern second adjust well resample set case raw pose explain test assess number gaussian value nan measure draw constitute dataset parameter amount covariance semi therefore proper simulate mining hoc third number start generate pattern value test dataset store method depict correspond ht negative method control
sample obtain clutter pattern take pl slow individual marginal mix show square numerical rmse predictive via pl repeat test sd rmse pl pair pl standard sided pl mcmc tune long narrow come expense prior comprise initialization fold gp initialization consequence sample identically neither require fold latent propagate helpful play hide dynamic treatment pl need total probability q second come conditional label quantity student integral monte collection student
modify enforce final input extra way determination large chapter sort relevant coefficient achieve undesirable propose modification compressive employ prior improve approximation motivate algorithms initialize recursively unchanged simulation algorithm cycle length snr pick generate normalize decoder output solve involve matrix distortion snr denote procedure performance decoder define convert db db performance simulation eq indicator true specify stop reach simulation sure algorithm stop pass schedule reweighte present
tucker optimality particular show robust increase understanding strength author discussion since small set maximum incorporation may rigorous space alternate proximal precise alternate proximal relationship discover analogy motivate density define kullback leibler maximize proximal wise implementation proximal rr proximal maximize log continuously
method selection lot ensure right stop pn observe remark pn also figure stop select elastic modification lar one mark green square selection link elastic lar fail true illustrate elastic turn lasso select often produce close one conclude superiority net elastic prediction penalize result provide drive choosing illustrate simulate lar confidence secondary f observation regression noise suppose collect label corresponding
ensure particle uniformly stage mutation degeneracy path introduce extension article impact scientific
domain distribution observe bind protein certain exchangeable control ultimately connectivity enable rooted notion information theory focus per se keep focus modeling establish issue list chapter detail graph model partially scalar difference exchangeable goal space position association community membership exchangeable graph binary string carry semantic meaning induce connectivity principled parametric fashion regard social conceptually separate science direct graph dyadic keep track link neither base rate popularity effect incoming add indicate model probability normalizing purpose one observe mutual enhanced factor problem representation identifiable interest os enyi direct appearance reciprocal focus solely version study likelihood dependent reciprocal edge parameter constant summing correspond subscript minimal set iterative major recognize lack asymptotic fit procedure likelihood example ad hoc deal e recently basis generator conceptually direct form incomplete contingency loop redundant standard show relation rational contingency analyze use analyze relation multiple generalization aggregation identify network bayesian typically refer set quantity partly individual refer variable quantity serve treat popularity fix think popularity effect develop paper reasonably take multivariate fix effect spirit extension involve effect distinction additional level high come approach work monte use version implementation network total dependent share undirected normalize triangle star star star triangle three generalization work three model maximize generalization frank maintain structure count pseudo bad sensitivity number edge change sensitivity variant major double counting overcome nearby estimation model describe associate use combination distinct mode configuration zero topic empirical root method relevance carefully construct package analyze package mcmc estimate current formulation formalism undirecte denote node clique parameter clique potential advantage em feasible expensive strategy monte lot methodology develop undirected primarily learn image os expect degree os enyi formation finite lattice os explore focus utilize physics style model fix sequence sample idea representation th
copula finite mixture margin latter mixture disjoint jeffreys proper jeffreys margin result proper author choose fall approach entire nonparametric copula take space copula bayesian adapt methodology inside mix weight pt lemma iw c w since elementary row get mi mi mi mi jeffreys proper follow partition mi parametrize doubly selection doubly polytope contribution jeffreys difficult
variability position position surprising corner home middle dot posterior prediction home mixture curve figure player full player past examine past player additional prediction home run home age plot middle poor home plot posterior draw gray dot mixture posterior indicate happen consistent player relatively see case substantially year player go non regardless year middle status sophisticated home major age home home information across player primary interest home hold
kn kn gm nu k assumption claim dominate convergence discretization inversion concern use consider inversion discuss projection follow coincide variable expectation extension p dimensional p p endow theorem k u assertion finally convergence addition k measurement satisfy limit hence assertion practical inversion datum provide device implementation monte algorithms let borel pe p give next substitute pe p algebra consider probability weakly theorem measure weakly measure compactly wavelet scale suitable expand variable prior next definition follow deduce ii independence p notation stand observe finally almost surely negative sure expectation hence discretization invariant sequence operator proper converge surely sure convergence weakly assertion embed realization accord quantitative result describe thank difficult together
estimate graph consider mind proportion select five method decrease considerably obviously proportion gene set validation split lasso connectivity graph six connect datum depict violate high lasso lasso figure five six vary also method assess reliability good subsample lead set exclude subsample five successively consider include repeat computationally expensive candidate time edge score degree agreement proportion assignment r agreement measure finally denote average denote agreement expect table absolute compare set display shrinkage probably rely subsample cross split approach less differ dramatically run approach shrinkage far one
usa proposition author discovery zhang principal pca widely dimension pc linear interpret drawback propose achieve standard lose pc orthogonality loading explain attempt optimistic paper uncorrelated pc loading novel augment lagrangian nonsmooth suit regularity subproblem pca several synthetic random respectively demonstrate pc correlation pc orthogonality pt pca lagrangian nonsmooth classification processing reduction numerous science biology face handwritten code gene analysis essence aim find combination orthogonal capturing augment function minimizer parameter detail stationary gradient nonsmooth close convex suitably subproblem lagrangian method random result pc obtain substantially outperform explain pc orthogonality pca formulation lagrangian nonsmooth problem nonsmooth close applicability lagrangian pca concluding remark vector assume symbol resp diagonal sign th entry nonnegative denote far clear semidefinite write whose euclidean associate explicitly state otherwise maximal denote denote real closed set formulation orthogonality loading formulation generality commonly load entry
ad hoc efficient mainly good dynamic expect convergence dynamic player game propose dynamic relate possibly perturb algorithmic evolutionary theory say see visit corner mix interior equilibrium player player correspond pure simplex pure strategy strategy component unity corner strategy specify mixed pure strategy correspond play classical write write denote game payoff random whenever profile know get play pure resource denote algorithmic game resource player singleton subset pure strategy space subset cost player pure load sum resource pure strategy cost player profile pure strategy
source help maintain control source varied mean pp simulate filter high maintain false derivation background approximation satisfy acceptable infeasible commonly enhance assumption pp scope develop simulation produce numerically assume rectangular pixel simulate maxima value pp plug value procedure equation wide pp image source pixel pixel want believe number computation increase build give area formula sized area remove square value account fall difference area get source default worth shape give tail benefit simulate value instead handle wider dotted line information solid dot mild gets small consider get run simplify since background thus computation simulate percentile nan ab maxima big image
adjust stage stage cascade collect positive previous cascade show use curve cascade classifier result seem report early perform want rate false specific cascade huge success time detector lot underlie boost limitation adaboost skewed li detection wu ignore scheme adaboost another train strong base divergence therefore haar publish preliminary face greedy sparse discriminative classifier manner face patch reject quickly reach patch face patch reject cascade detector lead fast cascade context soft cascade develop follow cascade contribute efficacy algorithm cascade adaboost forward choose proper coefficient
global stochastic search intend region still bayesian decomposable monte carlo specify law simulation construct prior think need offer exploration meaningful prior os undirected place prior decomposable specification allow vary feature solely place informative direct graph use agree feature potential markov tractable encode certain jointly copula jointly reduce easy case dependency distinguished dependence novel section configuration euclidean induce sampling modeling early subset benefit approach yield estimate graph undirected parametrization flexible family prior parametrization intersection system specifically approach induced graph graph hypergraph stochastic framework concern form theoretical concept concept distribution set resp say graph undirecte unless complete possible inclusion respectively collection clique two vertex incomplete disjoint nonempty path must nonempty collection subgraph complete
aspect almost surely act suitable warm computing initial iteration lar k warm learn th column update dictionary solve soft thresholding observe column empirically become much slow lead lar homotopy whole description implementation prove experimentally thresholding providing solution robust require stop update warm advantage learn store solve th keep one optimization block concentrated make coordinate correlate concentrate effective algorithm convergence invert impractical present building discuss simple enhance predefine case video stream situation examine simulate I set reference experimentally speed carry xt natural require store past block sized store impractical still exploit remove cycle auxiliary replace carry old information old weight aggregated decay forget date propose line algorithm practice apply iteration condition convergence even show understand become j c version could also speed draw
certainly encode find circuit iff correctly predict output theorem proposition observation conjecture mit mit complexity produce quantum show hidden generator find hard quantum generator quantum quantum repeatedly classical outcome perspective outcome work question learn
hill optimize tree nj improve hill hill particularly encourage core red hill minute depend nj long hill bayes tree close directly maximize contrast observe popular consensus minimize bayes estimator study exploratory hill give path metric hill close true tree nj cases hill initial tree hill encourage future choose type theoretical distance geodesic bayes distance connection hill try general interesting hill bayes
remark autoregressive ar frequently series original procedure past lead error information spurious causality include observable pairwise autoregressive causal coefficient g measure probably argument cause say influence series present alone th lag causality indirect cause eeg interact model eeg represent effect model innovation term accord innovation spatially uncorrelated assume source simplicity number invertible exist sensor pca innovation impulse coefficient thank innovation bss statistic would like preferable apply ica directly domain ica infer brain model mild spatially eeg super activity
triangle combination number stand stand r numerical line refer degenerate line degenerate limit nr replicate right number top mt rt rr monte carlo generate leave right r table example mt rt table case r discuss increase table expect c pe mr st nr stochastically hence desire multiple triangle triangle wish investigate segregation alternative realization identically accord correspond correspond triangle use define triangle sub number edge triangle point plot figure arcs iid construct become asymptotic binomial mr p ht arcs hull point circle leave right gr j mr r equation desire top plot give look since binomial hold increase curve become however order approximate normality triangle base point unit square present histogram triangle normal gets approximate normal top gr gr j carlo replicate curve depict j carlo replicate finite
find power alternative alternative discuss asymptotic trend unique solution generic deterministic present goodness von kolmogorov test limit moreover consistent alternative test classical test identically distribute simple von kolmogorov respectively bridge describe help
z final include subgraph submatrix lead quantity appear tree copy mark ij include marked edge let adjacency covariance node mark similar argument use correspondence totally backtrack expand express backtracking time appear numerator appear denominator totally backtrack see subtree tree power vertex node show embed could give intersect marked edge contribute power lastly power
interpret opposite overlap singular lead convergence quantitative maximizer algorithm eq proposition appendix proof eigenvalue deriving assume final see consequently remark write satisfied value since overlap sense conservative overlap recover theorem optimal depend satisfie minimize compare proof
h yy yy dy xy dy dx show adjoint inner product kx dx dx dy xy dy q conjugate markov reversible operator equivalent rewrite show eigen development course eigen eigen conversely eigen eigen se author play role extension chapter compact main relate da spectrum follow proposition also aside share eigenvalue dominate transition remain reversible draw draw draw finally draw product conditional explain da imply easy indeed start half never negative fortunately note marginal exists actually base therefore develop refinement spectrum chain space compact eq eigenvalue order characterization self e
covariance valid bernoulli probabilistic state configuration undirecte present simultaneously prove b else sim prop c create datum probability sufficiently sample presence composition fundamental level propose inference model random associated arc derivation variability arc bayesian
summarize initialize outer converge iteration inner loop ib worth role compatibility make compatibility mix evolve describe section need differently first appear equation replace kf initialize converge current estimate solve accord kf procedure smooth follow use logistic normal inference pose constant summarize initialize converge ib smooth notice marginal variational marginal optimum optimum validate algorithm well evaluate early investigate major laplace adequate estimate membership well fit role iii fit network role role role compatibility real life show three row cell estimate project represent truth truth estimation link grey color actor row display compatibility synthetic synthetic actor actor connect mostly actor mix actor simplex corner indicate dominate role actor corner close edge membership actor lie near simplex mixed role actor possess role synthetic membership compatibility diagonal
model situation plain fail restrict eigenvalue linear noisy noise design either throughout I object parameter infer pattern refer understand also follow problem choose convenience penalization attractive method provable large stability necessity lasso rather exclude case exhibit exploration computationally selection relax impose towards goal analyze lasso originally analyze progress high scenario paper lasso lasso initial estimator behavior submatrix whose index thus variable condition necessary
ball rather point lipschitz metric constant full access similarity information practice believe issue application via specific oracle answer query time horizon advance regret phase round typical point point space phase duration duration copy contextual mab payoff problem take adaptively maintain space region contextual complicated algorithm differently notion argument subsequent express contextual notion pack refined version take payoff guarantee follow form unless otherwise pack rt rr small guarantee suggest guarantee term include dependence bind eq dimension extend pack packing subset include optimal payoff pack number type call optimistic arm pair small packing likewise contextual optimistic contextual payoff contextual guarantee dimension theorem enjoy trade drastically dimension contrary prior essentially take account defer horizon round ball similarity add active call ball active activate initially active ball radius similarity select active arm accord arm activate forward fix number ball arm algorithm relevant definition defer subsection ready break tie arbitrarily activation rule rule select relevant let ball radius
numerical enough opinion indeed illustration probit demonstrate precision approximation discussion accuracy overcome difficulty bayes present single advance far impossible computation introduce model reasonable use simulation tool principle produce attain replace relaxed calibrate easily simulate method calibrate well choice graphical model illustration popularity opposite diversity promise researcher advance want try choice tool modify choice
posterior distribution detail contrast temporal epidemic inference final complete temporal impractical analytically likelihood overcome augmentation simple behave independently literature level discuss practice complete inference latent impossible period final invariant choice e ball greatly realistic choice affect infection see numerical sequel latent simply fix infection parameter latent derivation quantity start fairly allow explore interest mention threshold infect individual infect infected individual individual transmission assume occur contain individual summarize j infect adopt imputation describe approach nevertheless enable address question correspond
two space sa concept statistical mechanic temperature decrease temperature gradually landscape high temperature change see sa global practical temperature physics annealing attract alternative anneal analogous quantum fluctuation help optima low novel obtain generalization vb density motivation interestingly
yield note ne pn complete employ central relevant component useful nonparametric estimation simplify proof main proof lemma together lemma rate sequence r cn nx nc x put ff x hand condition hoeffding hoeffding together lemma assumption lemma c proof proof write e gx nx x use lemma gx eq side x side lemma note eq similarly prove assumption noting remain need cn verify calculate hence lemma obtain enough lemma obtain prove complete condition separate note pn proof suitable number obtain tr r sr lemma reference model analysis new york linear
mathematical sciences complex university university e paris paris france support nsf dms sparse penalization high recover density yield inequality offer drive construction substantial complement theoretical simulation employ dimensional mixture finding finite function estimate functional infeasible candidate large gap give intermediate identification component recover large since emphasis tuning penalty regression section inequality show adaptively factor class number wavelet together employ introduce computationally construct tune choose candidate validate study inner
plot k plus level large empty bar small plus symbol moreover appear case mean performing pair case pair pair sign comparison second number imagine column sign test rank good mse mse pair less c c fig find datum delay regardless gap row ci measure result follow time data delay bias place optical table fig group statistically see table show result well also fig suggest capable compare artificial generate methodology performance fix sf covariance sf sf sf several delay day unitary increment
sign regularize cat cat comparison square group cat maximum absolute value second constructing eliminate provide class conventional prediction validation primarily offer correction rate fdr emphasize construct control fdr contrast classifier aim identify confidence eliminate fdr see subtle important distinction differentially decide gene similar false confidence nan false feature retain discovery instead include argument fdr pt correlation
connect permutation connect permutation permutation state background variant need vector manner datum say summarize decoder widely statement family kullback kl form stay relatively edge control follow control leibler divergence pair apply follow begin denote denote subgraph edge remove uv follow lemma lower bound let convenient shorthand st begin imply remainder calculation low choosing shown always lie since st completes begin ensemble graphical family st st equal result field non kullback divergence mrfs uv st uv uv x st uv since st u uv v st kullback leibler probability bound order ensemble grouping base
support chen li fisher g mod e w da de g chen chen z li mat k mat mat mat l mat c mod b de dynamic critical behaviour dimensional long law system uncorrelated state infinite dynamic concept system al extend lr decaying ise relaxation preliminary numerical std dynamic
tight reduce iteration algorithm nonconvex attain exact discrete penalty satisfy p unique pp simply ambiguity first suffice satisfy q q detail though iterate characterize g pc expansion j j subsequence challenge model sparsity desired occur real suffer problem beyond meet challenge group rigorous allow group predictor nonconvex include discrete penalty examine estimation joint bioinformatics improvement estimation useful technique everywhere method contrast differentiable characteristic however dimensional component discard
borel banach space sup norm fx bb banach space norm radius center closed element bilinear operator operator markov exist linear markov space subset weak weak convergence metric markov uniquely probability review kalman observation gaussian vector matrix sequence uncorrelated signal kalman covariance pair observation drop observation randomness bernoulli arrival probability arrival protocol know signal kalman give recursive eqn matrix update accord unlike classical element emphasize arrival probability analyze governed eqn fix equip far property complete equivalent canonical eq denote path generate cb banach fix follow appendix dirac concentrated notion boundedness boundedness strong
ise expectation single system close simplify model unary front learn finally perform inspire foreground stroke foreground notion either operate super simply foreground robot image tight black segmentation comparison describe user specify stroke place next interaction history policy position label center second connect absolute current truth circular pixel away motivated user mark precise also take account analogous user learn interact mark stroke pixel low marginal hamming amount hamming pixel hamming expensive act advanced actually centre user
probability hold student cover contour grow tail article lower induce first propose asymptotically like eigenvalue bound verify modify define adapt covariance practice affect scheme particular large small eigenvalue covariance mix though projection avoid behaviour section first improper behave show set uniformly continuous laplace practical preserve belief ergodic result adaptive slightly parameter proposal fix advantage bad proportion take positive value
differently attribute site interested help optimize site example formal concept user economics www site site site correspond external group medium
stationarity dependent fractional number eq exact consequently q expression last appendix define measurable z z p q lead adjust end first fractional cover tool mix derive mix question decompose iid use encounter graphical consideration connection relaxation fractional cover besides remark multiple posterior empirically tight bipartite ranking good iid bayes rather contribution competitive likewise interesting bayes translate question remain kind learn perform obtain generalization connection make order thank plan investigate direction study note u work bound iid bound cover rademacher bayes possibly tail meta except randomization propose fractional bind subset
solve loading apply component variable pls framework calculate bridge pls whereby criterion unit may loading number see concentrate optimization problem fashion penalty component sparse iteratively procedure pls bridge pls proceed lead factor loading pls coefficient describe pls pls procedure u u rs control financial direct induce achieve exhaustive select correct select thresholding apply perform difference zero simple expensive portion binary search method penalization multiplied guess normally achieve less equal large number operation space expensive sort another pls
perturbation one therefore careful irrespective moment experience th toy axis population poor posterior update whole population weight abc perturbation equal whole parameter abc smc work big abc smc correct resample ess sis smc algorithms sample step perturbation require sis develop novel allow dynamical provide model allow describe decade conduct optimization develop extend parameterized bayesian simulate annealing find various system estimation extensively financial biological system maximum test nest investigate way bayes factor deterministic criterion selection couple multinomial logistic regression tool parameter estimation extent selection kind ordinary delay without estimate incomplete reliably paper sequential monte carlo abc
convex move slowly follow initialization diagonal possible non next th error h perform obtain loop perform inner loop single loop passing become necessary good loop efficient loop possibly column shift note pseudo construction drawback explicitly
enhance variable incorporate also inactive show solution piecewise slope change give solution possible therefore necessary keep currently associate coefficient inactive look subgradient general fuse fuse whole substitute precisely set keep condition ff f f lk j ik definition change reflect adjustment mention addition also formal definition
wu active source l avoid source describe denote eq detailed conditionally basis nan space distribution q fisher column iteratively conditionally draw sample von fisher look carefully hyperparameter probability paragraph accord achieve q ig source mix parameter detailed sample mmse estimator empirical average total mcmc chain collection easily hyperparameter ba consider comprehensive distribution probability source observation vector
stationarity exhibit stationarity stationary undesirable much unnecessary obtain advantage trend partition gps region leave interpretable inversion require separately gps already bayesian outline proposal tree carlo leading cart due may utilize class variable softmax two way gps gps region partition propose interest fast monte summarize full modification efficiently latent variable outline shall section begin partition cover prediction focus integrate structure categorical framework discussion categorical clearly interpretation feature section method flexible offer inference discussion nonlinear decade real input formally input gp define mean correlation explanatory define decompose underlie strictly identically predictor call term represent write gp function variable help remain frequent concern instability sampling necessary analysis popular parameter describe decay increase underlie isotropic vector differ result process still structure
still decrease adapt simulator simulator start decrease entropy stable depth understand system general beyond try metric verify stability reliable usefulness methodology performance underlie instability significant drop successfully instability paper
prove stationarity firstly unique accumulation point boundedness subsequence infinitely z contradict subsequence stationary point nuclear norm behave soft norm surrogate perform wide variety situation produce model error situation concave penalize regression popularity analogy problem penalty propose concavity penalty
want estimator framework come valid vector orthogonal replace please simpler positive look basis subspace respective vector basis matrix respective vector complement either orthonormal regular generally parametric regularization power depend smoothing operator nonetheless simplicity generality remark closeness estimator constrain appendix latter number put obtain collection want impose filter operate new operate reconstruct linear pass band band stop etc various application low g filtering processing white improve measured peak interpretation e image collection linear like image indeed image therefore goal low risk assume configuration kk say achieve low predictive insight risk influential explanatory minimize configuration linear remainder address application since efficient tool allow good recover start address issue ideal quadratic
instantaneous ar none possess measure receiver operating allow regime positive measure performance area auc curve auc value causality toolbox particular instead coefficient logarithm residual ar interaction exclude score divide score
mnist vs operate one vector interested know pass achieve accuracy streaming pass note pass kernel mnist vs turn hundred pass report htbp investigate varied stream deviation accuracy order permutation still perform single mnist limitation approach exact albeit thing secondly deviation
false grey line correct width along sample amongst inspection result distribution lie shift due draw false positive sect participant come population observe snp independently population false fig appropriately sensitivity specificity considerably improve know positive lie individual toward threshold analytical sect construct distribution pose substantial difficulty analytically need know width conduct deviation size sect amongst snps cf sect first population population also slight compare column table population correction simulation appropriately population move two unit misclassification nominal sound estimate alone hence upon assumption match sect correction sample size context unknown context size readily snps distribution nan either
look aware context density linear zero lack specification context smooth test optimal reader note fourth wu proposition work statistic white prove variance statistic parameter horizon process martingale additional impose moment white implication nonlinear uncorrelated martingale view lee chen nontrivial martingale discussion n critical speaking procedure asymptotically limit bp valid presence unknown estimate variable parametric model uncorrelated treat section interest cf reference goodness long memory time see autoregressive move backward z pz z ar polynomial respectively interior martingale iid pn tt initial q specify expect residual behave j follow simple omit detail smoothing type error impact limit relative reasonably phenomenon exactly critical critical
network important modeling lead max simultaneously dual structure structure output enable incorporation partially label detail two explore iterative scheme bayes procedure mn demonstrate sparse map real superior promise prediction benefit explore algorithm formulation tight adapt translation association genome trait framework interesting semi supervise hierarchical hierarchical structure plan relax since natural structure principle plan mn desirable perform suggest regularize markov stable desirable acknowledgement university national natural foundation china discussion discussion support nsf fellowship support national science foundation grant foundation project cb cb microsoft fellowship mn lagrange straightforward mn another formulation transformation derive problem dual variable lagrangian hand side zero true infinity I proof theorem corollary constraint another non dual normalization constraint rise lagrangian cm get dual ip mn solve appendix solution conclude ip rich normal mean optimum argument algebra mean optimum proof corollary present derivation divergence posterior divergence corollary solve quadratic
minimal resp real denote partial element integer general covariance first namely adaptive spectral smooth subsection provide framework inverse throughout follow addition however whenever establish problem unique optimal see sup fx x diag q along easily observe fs observe strict conclude present introduce terminology objective solution estimate
similar increase decrease specify w initially set gradient momentum w typically choose rule momentum give decay somewhat td subsection proof concept task learn result evaluation test try discriminate record game methodology discriminate player discuss subsection limitation progress carry feature matlab search however choose use matlab working challenge implement matlab carry windows ghz processor game notation file record module implement variation alpha prune minimax return game testing choose feature
mab bandit exist prove mab fix either tractable every occur bind countable furthermore show infinite metric tractable lipschitz mab exist payoff strategy reveal name strategy payoff could payoff reveal round select receive payoff payoff reveal strategy bandit version obvious specify triple space borel payoff borel induce product payoff lipschitz present metric algorithm measure expert formulate receive infinitely expert tractable even tractable full mab expert tractable tractable opposite ask space become completely characterization pre compact moreover upper even tractable compact consider view interested et dimension expert problem naive therefore natural ask lipschitz prove characterization low novel notion tailor expert space isometry tractable version feedback expert term feedback lipschitz expert feedback fix isometry tractable lipschitz tractable metric tractable feedback tractable min theorem point technique joint two identify topological entail topological topological isolate every call metric notion topology topological good perfect follow lemma tie topological compact countable iii theorem make mab complete space double feedback expert feedback expert lower possibly borel history past
update rule subsection divide theorem describe last mainly discuss already prior gamma observe theorem assume anomaly www traffic context follow cc update simply consider distribution assume prior poisson contribute constant newly remain conjugate discuss power model depending forecasting satisfy one mapping
spike spike leave show parameter satisfy interval run algorithm spike however baseline estimate solid variation penalty leave improve baseline right baseline index value influence new vertex connect idea baseline region baseline everywhere since contain observation baseline spike join region region knowledge spike encourage vertex join choose vertex influence cause baseline demonstrate algorithm suggest neighbourhood exhibit area
multiclass loss binary prediction multiclass scale regret classification composition reduction classification root undesirable regressor create multiclass bound dependence feature state raise winner repeatedly player loss start loss winner winner continue remain construction elimination see elimination good could single elimination imply concept introduce notation divide tree inconsistent motivating sensitive multiclass prove dependence return cost multiclass experimental filter multiclass present family integer control tradeoff minimize small setting multiclass early powerful adversary fail sort ratio large number call large label classifier define extend case multiclass binary
view speak proportional simulate entirely dominating suffer jacobian acknowledge turn thus turned translate compare
new inequality serve form gaussian happen useful controlling
see taylor al scale neighbourhood consistent h need estimator use standard expansion example rank symmetric without impose without point eq pn us rate q equation instance I pn l slowly example remarkable independence
otherwise assumption herein dependency level dependency level n local dependency level observe q sufficiently consequently pl finally let markov contain ts
method movie datum towards max present section several improvement extension horizon due nature optimization advance max margin variational incorporate field tight collapse variational gibbs moreover experimental incorporation expressive integrate utilize model advanced margin would efficient acknowledgement nsf support chinese nsf foundation project cb cb cb basic follow topic proportion word topic assignment n unknown manner take entropy discrimination principle call joint cm get cm evidence define intensive regression cm cm nonnegative multiplier fundamentally passive current tend measure closeness p infeasible compute apply develop alg term posterior solve optimize factorize cm form component lda formulation since thus expectation co variance affect step let vector factorize plug cm framework ki write normalization cholesky nice margin extend laplace derivation method normal prior quadratic programming via cm three see fix topic proportional normalize derivative cm tr
eq normal distribution account arrive conclusion big one task bounding appear geometrically chain unbounded defer next chain direct tight I definition uniform geometric ergodicity markov rest typical situation ergodic property acknowledge time problematic execute al intractable computing set clearly geometric bounding variance well reversible independence metropolis chain uniformly ergodic uniformly candidate translate spectrum contain lead spectral self adjoint see case
completely biased ensure message randomly select plot path two different first vary probability estimate check variation sampling uniform correspond crowd identity reveal identify technique symmetric protocol currently protocol assume source randomization security protocol arise due poor randomization identify provide wrong check formal correctness guarantee rely trace protocol promise technique extensively protocol project
outli outli outli confusion misclassification student small misclassification concern robust classification http www ac uk five measurement express rw mid line maximum body group class direct variable use reference add variate th table well read begin central initial without nan outperform grows slowly reach unchanged error finally remark equivalent suitable concentration last concern beta model compare relationship subset individual moreover schema consider remark bic
least compare use randomly instance variant nesterov efficient key programming smooth interior trace dimension multivariate relationship multiple response common observation response explanatory response explanatory coefficient independently classical square response widely response explanatory factor express column factor dimension decrease correction partial least formulate form linear factor regression proceed equivalently square able accurate estimate may relationship factor choose discrete nature unstable sense change determine loading advantage trace nuclear fan force definite fix idea
covariate adjusted residual proportion difference v n mc mc significant detect I I mc mc proportion interval whether nominal interval proportion either check significantly table conservative nominal see conservative nominal however replicate case level case replicate error symmetric tend normal desire term normally covariate mean term asymmetric tend covariate adjust residual specific mean covariate adjust stable desire nominal test desire error different error covariate extremely observe method w mean considerably similar case agreement estimate different usually case w acceptance rejection region k acceptance rejection
exceed condition x f g hold give utilize rank invariant strict monotonic able screening suppose nk nm size simplicity technical nk n nm sis generalized fan screen reduce size include sis procedure sis sis apply final avoid focus sis difficulty simulation generate standard variable double location proportion standardize correlation matrix variable simulation present normally setting lc median simulation size realization empirical zero covariance decrease behavior sis two sis mle sure screening choose difficulty computation burden addition sis worse complicated procedure like scad sis complicate screen scad scad regularization table logistic regression
choice long penalty biological order temporal causality correct order dimensional estimating entry diagonal also row tucker proof weight estimation iff j ij j th fix value pa pa j pa pa consider show similarly suffice follow show exist eq pa event lemma use pa pa pa ij pa pa ij ij pa pa pa without unique event problem j pa second lemma pa since imply verify suffice jx pa jx pa pa pa pa pa pa pa pa j
fix iteration propose literature point f extreme largely apply engineering environmental see parametric distribution widely approach
p p x correct involve parameter frequentist asymptotic asymptotic generalize quantity frequentist asymptotic prove result function bayesian interpretations pearson framework cdf confidence value thus interpret potential relationship framework aim frequentist report function component pearson alternative bayesian fail guarantee interestingly originally pearson main frequentist interpretation confidence proportional bayesian report interval disagreement imply implication coherence compatibility posterior coherence necessary dropping requirement bayesian enable coherence achieve pure toolbox statistical inference enable nuisance prior incorporate frequentist circumstance measure decision whenever information parameter observe independent confidence exactly function independent incorporate information decision apply combine cdf value bayesian level level function inference demonstrate hypothesis point mean give versus observe numerically probability value
therefore rejection inferior section new rejection evaluate envelope keep
inside ball produce center radius minimum first type guarantee seem scale follow solve obtain unknown center eq suffice multiplicative like scaling scale merely become ratio appears appropriately choose polytope problem translate rotation allow furthermore face intersection hyperplane polytope point want convex polytope polytope help equivalently radius center usually ball express maximization maximization rewrite mn identify set
detailed form lagrange expand rewrite inside eq multipliers substitute solution substitute divide substitute lagrange multiplier substitute integrate q rewrite side term likelihood exponential effort put name piece call give normalization notice bayes canonical section particularly process depend constraint ask constraint whether process simultaneously sequentially collect purpose infer x x process matter
full efficiency proof outline notation everywhere choose classical series assume everywhere without loss observe time well spectral g sum mat ern keep mind coincide constrain spectral simplify us b characteristic define sense square extract plus extract like important article w r w throughout denote integral index resp resp estimating identifiability encourage expect
follow bound loss generality law large boundedness n jx quantity result eq differentiable scalar decay exponentially combine using directional write n boundedness quantity moment apply lemma n know write hence taylor expansion directional also know large continuity together establish orthonormal span unitary proof chi square vector give freedom c pt mm corollary proposition pt pt pt nsf grant recommendation material author necessarily publish decide make impractical involve alphabet also divergence grow control remainder concept include ml formulae section term
impossible latter identical identically unknown know despite possible get tail present necessary contain important bound series analytic secondary research
agree presentation introduce notation strength skip fig via viterbi quantity regime jump remarkably indicate govern jump monotonically decrease decay jump dominate sequence virtue dot proper support prior increase towards maximal jump regime small leading dominate simulation obtain viterbi algorithm calculate directly continue value noise entropy naturally regime entropy usual transition decay point
st mh upper triangular h h h h svd dx I dx nu tv td nu tu td normalize arbitrary phase row let thus iid easy deduce density I
infinity already convergence well statistic put remark double wiener process function er von statistic statistic converge free limit form close condition fulfil tx eq mild regularity put strictly
bootstrappe pair empirically bootstrappe residual n ki iy bootstrappe strictly bootstrappe show behavior notably pair residual currently bootstrappe model replication large need regard various correlation sparse sign analysis carry tend show lasso asymptotically behave support index element optimality regularization path involve c solution condition note essentially hinge make quantity characterize noiseless speed relate q commonly simplify design consistent procedure satisfy weak lasso figure middle condition design observation relevant explore theoretical behavior extend current relevant ensemble weak various rate currently refine relax stability e probability average one local currently derive condition sign loading usually high regularization lasso zero hold asymptotic meaningful design would sign
glm perfect posterior depend smoothing may support fair adjust artificial gap quite distinct posterior frequently away true reason step abc outside support propose prior transformation form low upper prior complex abc introduce glm much less observe rejection abc glm posteriors posterior glm prior range uniform estimate maximal consistent human classify
rank w b p node table pattern find appear frequent neuron chain connection subtract count leave occurrence modify new occurrence middle calculate respective come conclude false apply procedure subtracting count strength great correctly true connection demonstrate high frequent episode spike upon future l technique place neuron connection neuron potentially connection produce spike methodology furthermore connection track circuit cognitive formation usefulness methodology analyze neuron record analyze al two repeat discover track
case martingale decrease admissible martingale analogous discrete greatly et al technique make mathematical finance article idea composite hypothesis possibility measure evidence composite measure nontrivial simultaneously martingale goal simultaneous produce demonstrate martingale simultaneous test simultaneous almost illustrate simple fair suppose random variable horizontal logarithmic power unbounded direction consider non frequentist require interpret probability merely principle section sampling r cox corresponding hypothesis hard sensible widely composite second composite follow hypothesis composite bayesian simple hypothesis example supremum ratio implication inverse supremum likelihood ratio value picture discuss composite composite represent former uniform latter calibration track supremum martingale relate reach high value conclusion sample evidence conventional finally back
smc impact single certainly demand e degeneracy particle contain partly de paris project mc author particle publish journal
hand within simple might fairly instance several phenomenon behavioral problem several might modeling trend outcome suitable outcome return treatment site eight type cognitive test stanford year primary year intelligence child allow site contribution treatment allow outcome index site site parameter contribution effect simply model variance test allow vary last piece plausibility exchangeability also display test score outcome different year age interval display intervention appear site outcome phenomenon outcome figure outcome correction linear example eight estimate towards adjust year type shrinkage cause conservative eight stanford score display hard arise model outcome exchangeable group model procedure comparison different comparison large variance procedure potential tuning multiple comparison false discovery hierarchical truly zero sense prefer issue term type social program equivalently interval wider prefer
project grid independent central chi cumulative dy dy z dx note transformation formula convert integrate nan consistent simplify jj important degree freedom sample plot blue base million row third use value count figure blue solid line leave middle matching square red n match l kolmogorov von nan gene na na na mean na na kolmogorov von nan matching counting count comparison use kolmogorov von hypothesis matching probability tail matching ii l distance chi chi match chi cm k chi counting chi require significance cm department university department university school il
step duality gap unstable drop qp increase proceed get hard fluctuation duality duality monotonically cut behavior cpu rapid decrease kernel huge computation per relation speed kernel spend dependency size repository duality tolerance kernel random report logistic hinge logistic increase median random train fast method however clearly fast roughly scale unstable programming iteration kernel good regularization increase efficiency block result converge small scale number good loss algorithm generates iteratively loop super linearly inner newton block elastic mkl mkl connection norm weight block norm give optimization elastic net numerical number scale
space context necessary dominate location minima rather global maxima location minima lower close dominate reduce dominate process maxima essential example dominate intensity location dominate eq start version accept process carry obvious naturally interest generate step give rule rule give ease notation pd jk
add direct loss equal finally vertex loss tuple action path generate tuple weight equivalent accord exponentially e connect consecutive task discuss exist consecutive short path tuple accord dirac mass proceed vector auxiliary maintain round state round action state show weight draw conditionally generate note equation put mass proportional action hide correspond previous task sm repeat draw space particular draw proportional realization store perform update number pair complexity negligible naive parameter action pair similar argument example transition problem
regime perfectly predict iff go countable going tailor design give interaction interaction realization convenient much kl construct minimize total expect kl divergence kl observation arbitrary divergence
ty justify z ty tx prove try component x te pa I rhs notice pa claim decompose term ty q call submatrix comprise index tight follow proof denote column cauchy distribution subsection iii satisfied least tail probability bound ty ty ii ty last pa x condition long arise system refer wavelet pass reflect coefficient reflect output refer trace record filter include nuclear train communication impulse output measure noise detect approach decomposition scheme
thousand implement minimal agent learning fail solution anonymous approximate equilibria game frequently design theorem system game confident learn become response agent guide beneficial second system something link depend send importantly distribute system agent play somewhat hard game efficiently game behave agent minimal converge algorithm round stage strategy agent choose reward current condition stage dynamic guarantee learner nash poorly number number agent many game converge dynamic game human weak common game interest dominant agent strategy agent result agent converge agent agent speed order magnitude play game
introduce weight impossible many purely procedure scale hierarchical simulated importance resample problem concern weight effective ultimately versus trade initial sampling practical case use trade produce nod facebook contrast bias facebook focus topological aspect one happen topological node facebook differently networks node law regime approximated power cover decade behavior formation degree degree cluster connect connect node degree vs study similar network summarize plot pearson coefficient possible explanation stop boundary high likely friend user friend phenomenon connection near node average facebook report clustering tend node fig privacy privacy bit privacy facebook setting aware level user privacy aware l l ab united co city bc city facebook user user extreme end facebook users middle seem concerned split english speak privacy facebook report finding present privacy facebook fig leave setting modify facebook modification friend facebook user factor friend classify b k fig degree find tend modify setting trend make concerned carefully facebook easy privacy c b average neighbor finally privacy privacy friend clear positive correlation framework recommendation contribution several rw bias unbiased efficient offline truth rejection implement finally facebook unbiased sample facebook characterize structural facebook publicly publication uniform
fig plot subspace bss procedure manifold separable must state invertible figs fig source version generate scatter plot nonlinear transformation nearly effect noise number information analogous multidimensional human speech content bss procedure capable recover speech content recover reformulate velocity space state space conventional attractive reformulate bss mix unique permutation independence velocity physical compose interact deterministic avoid difficulty bss previously bss ii recognition engine recognize state
suggestion marginally family subset entire construct contrast crp throughout relaxed version use notation collection customer number distinct customer iff crp customer among crp identical crp marginally invariant ij set take place customer assign value unique customer iff point partition crp decay note customer occur th distance crp customer divide unnormalized rewrite linkage crp traditional dependent crp decay produce traditional customer marginal crp separate customer fall within traditional crp group whose customer contain customer customer since marginally sufficient dependent distance equal customer may assignment suppose suppose crp collection distance marginally share customer customer customer directly distance crp marginally identical writing leave right possibility distance keep multiply either generality must claim proposition proposition decay proposition dependent crp result function marginally invariant
examine feasibility uniform even simulate simulating fall within accept reject step accept geometric conclude trial draw surface probability round first round accept algorithm normal bounding target exercise involve r n bound pi prop sd sd ratio prop sd mean sd rapidly moderately sampler conditional gibbs slice sampler chapter exercise distribution uniform conditional slice invert take exercise thus produce closed solution reproduce analysis exercise change deal simulation long beta interval necessarily confidence around establish exercise coverage devise rather conditional schwarz partly along time constitute captured observe past location conditioning make unobserve generate would bring choice proposal block thus far extend definition take thus consider mixture value define mixture distribution unless show deduce way trick ball case remove allocation ball represent ball case equivalently extreme speaking partition normal distribution unknown subsample weight therefore prior pair normal allocate via already find chapter exercise distribution exercise influence q lead tt sd step step sd sd em em em sd em sd em increase log along log start eq function give parameterization simulation conditional n I implementation simple mcmc allocation gibbs mu mu
field easy sense easily follow build create add edge choose field create merge merged node fashion seminal approximate instead ise edge indicate encode edge follow take subgraph value sum configuration equivalently normalize
convenience normal consider location q diagonal minimum uniformly coefficient constant hold l chernoff tail eigenvalue design maximum thus concave variable bind location regressor independent assume moreover satisfied since ff hoeffding inequality eigenvalue bound zero nonlinear impact k b f k class van lemma hand consider vc vc intersection basic class vc vc vc form van convenient positive matrix characterize behavior maximal regressor establish hold q converge uniformly suffice collect van n r nr p conclude converge conclude converge rely give asymptotic covering regressor negligible relative sample regressor number significant regressor quantile arise processing become widely investigate penalize acting demonstrate fundamental demonstrate fundamental forecasting thus excellent forecasting even growth total development review exist contribution develop result selection quantile since inconsistent quantile norm qr function near rate quantile cf quantile index cf hence complementary fundamental excess forecasting
either contradiction diagonal belong imply row contradiction column nevertheless open fact table element simplex inclusion proposition vector definition e divide j unit want find scalar study situation recall system prove chapter normalize generic equation quantity zero divide thus
bic selection lin fan wu regard selection tune drive edge edge true search upper search region find wang result model associate bic define suppose know correct two result construct working wu result identify coefficient theorem fan wu tune p ij ic penalize correct probability tend g fit essentially likelihood graphical denote ij theory model
vertex h segregation possibility definition association alternative definition alternative particular segregation triangle use segment segregation correspond away opposite edge argument segregation alternative normality relative mean asymptotic normality obtain degenerate r normality segregation universal normality base statistic segregation alternative value since segregation statistic underlie asymptotic critical sided segregation segregation test test nan might nan less mean alternative pe rx rx pe rx x rx pe rx rx pe rx p pe rx rx pe rx rx pe rx pe rx rx p x association consistency provide investigation asymptotic local segregation alternative graph likewise segregation provide work appendix segregation segregation arc detail figure segregation choose segregation segregation r sr suggest use association r sr imply
tb pg interior point ip iterative limited newton dual augmented discuss text row condition condition well parametrization see text mean extend beyond regularization exploit exploit sparsity intermediate solution efficient show bind subgradient optimization minimization method pg ip exploit efficient directly g stochastic subgradient problem strongly however since base fail problem poorly condition challenge information tackle poorly condition quasi quasi challenge differentiable regularizer deal fact easy many regularizer remain coupling show section consider strategy method manner operator forward need every current size since reduce ordinary gradient consequently loss term differentiable batch setting approach maintain throughout reduction compare interior sense difficulty size problematic poorly condition naive iteration need f minimal approximate curvature efficiency poorly condition denote guarantee per see member qualitatively low term rigorously study converge linearly course convergence cost nevertheless qualitatively mild intermediate effectively minimization optimize plus generalization exactly insight write problem else reliably employ duality stability optimal must trade
theory variance remark universit sciences et de france estimate simplicity exist performance relate emphasis distinguish statement rigorous conclusion accord likelihood least rely available call cv strategy generally selection cv risk error avoid overfitte popularity cv mostly come empirical cv entirely confirm prove fail depend identification question picture precisely aim question cv cv keep goal framework choose cv overview mind classical procedure cv risk cv framework discuss finally focus cv procedure conclude survey common observe throughout except purpose mean quality basically accordance accordance framework fit detailed statistical framework useful quantity relate purpose exhaustive aim excess kullback leibler aim predict predict multivariate sx minimal excess equivalent correspond finite discrete prediction contrast minimizer denote consider instead look classifier basically confident classical convex
update clarity exposition shift shift well brief cluster differ attribute model attribute standard deviation share sample process concentration paper rigorous notation since use least normality impose component discrete measure provide mechanism shrink towards derive specification mixture intend assign mixture follow
case include generalization loss inner david discussion ms support aid rt mathematical mail ac signal number unknown observation primal reweighte differentiable non series exist minimize component initial
choose regular ss without reasonable bind success il wise upper subgradient mean l n q n ss expression eq sample min q min denote expression ss bound line via defer version explicitly root
dominate roughly transaction item distinct transaction lift consider linearity expectation similarity indeed item efficient paper read measure sort carry track transaction belong compute sort list os pair transaction fit pair memory item sort trivial occurrence item os tuple transaction tuple write disk sort tuple transaction use os transaction sort support transaction fit inequality memory page phase os read total si os sort pair pair os os dominate cost previous expensive size imply transaction subset
effect residual h precisely star tm moreover suggest degenerate resolution rp investigate degeneracy estimation try try plus width degeneracy degeneracy amount observe would unable ap apply trivially star thus significantly function e g fails represent uncertainty accurately star panel dot connect show triangle red cross initialize converge run star near tag spectra various plane degeneracy star panel initialization quite offset due three wrong solution initialize poor practice reject turn k bad predict spectra indistinguishable star bottom panel band bp rp dash star convergence ridge solution initial final spectra agree spectrum degeneracy degeneracy reflect identification ap near star suggest degeneracy systematically spectra predict spectrum adopt original grid mahalanobis simplification inter degeneracy star arise nan hypothesis spectra degree freedom observe give less correspond word star degenerate square unnormalize one lost significance dd contour star grid get distance plot degeneracy star grid case word strong degeneracy plane noisy one indistinguishable lie contour
random relevant result map variable crucially exploit bind contribution let integer definition thus subset vector apply powerful context complexity measure extent uniformly isotropic constant independent copy immediately finish twice upper bind bad event constant condition proceed treat deterministic condition describe define formally design satisfy hold proposition constant theorem optimize theorem
lot reasonable serial mode mode general time c minor candidate generation serial episode episode satisfy maximal generate stream episode parallel episode run algorithm episode episode serial episode episode recover serial episode mode six episode embed result large similarly partial order embed episode episode see run threshold almost mining order burden candidate level level bound high font inner sep pt style c parallel episode xshift c cm b right g right font b xshift cm yshift c node node font c xshift iv c yshift cm node node font xshift node vi xshift yshift cm c font c g yshift node node font width cm episode xlabel ylabel frequent episode grid reduce xlabel ylabel episode xy rest run rest c embed fig vi iii mining robust frequency eliminate episode episode constant algorithm embed level generate event embed average number event datum stream signal refer come various episode stream episode verify ns tn l number simulation see say less signal candidate successive candidate level increase run go candidate episode variation observe keep run embed increase increase number paper discover frequent episode partial episode available discover serial episode present algorithm occurrence episode care time candidate interesting order serial episode serial episode episode order episode inherent episode one general discover partial order effectiveness extensive paper consider episode partial episode serial episode algorithm serial episode discovery extend present episode
grow grow likelihood converge rate towards behavior batch display leave parameter associate summarize box compare variability batch iteration range thousand independent em estimation small variance obtain use size online preferable despite slowly decrease batch provide choice guess size scheme poorly figure step pilot improve simple large size thousand figure cause slow reduction bias estimate contrast choice smooth reliable require observation forget l matlab batch forward
know neighbor direct respect construction query object oracle direct close query exclude neighbor learn near near deterministic must ask direct list direct neighbor ask compare every current candidate equally identify denote outcome answer ask probability tell answer branch star neighbor equivalent assume branch locate contain identify ask question direct neighbor object close question exclude direct ask direct know indeed choice inside direct way choose direct direct ask way neighbor consequently bit ask question store approximate phase answer question hash base application retrieve hash distortion underlie collection face angle distance similarity human human proximity respect human object make statement probably lead formally aim efficiently oracle object query ask phase answer
margin illustrate simple six order line value iff incorrectly picture label validation value use equation picture label conclude assign value p fix measure value label confidence q permutation value fraction permutation bound sequence proceed throughout prediction validation calculate
square norm error mahalanobis al rigorous depend restrictive continuously differentiable forecast al observation univariate setting et forecast stochastic definite respective hide apply forecast say resort score q wise derivative multiply one half whose element strictly convex standardized homogeneous extend general bregman representation rise scoring stein notion quantile score particular condition course could general type situation propose fr fr median traditional case ideally forecast quantity event al forecast evaluate name mean predictive consistent functional consistent scoring literature forecasting forecast develop theory notion consistency scoring expectation ratio expectation quantile forecast function mean consistent bregman form attractive homogeneous scoring evaluate volatility forecast recommend superior power west test ability depend direction desirable theoretically quantile forecast assess power appeal interestingly family square score absolute percentage median severe seem consider consequence score fair comparison compete employ score consistent scoring rule hence protocol point forecast predictive forecasting literature forecasting realization volatility notion forecast participant ex score employ target name name
table margin row count infinity bound number infinity expect valid asymptotic entropy variance integer sequence determinant lattice first tr dt matrix te zero multiply notation comment much prove particular third iv fourth require characteristic center cause determinant lattice possible let ik determinant large contrary reciprocal consist integer lattice reciprocal original reciprocal determinant reciprocal lattice
demonstrate via simulation connect robust iv phenotype status quantitative know effect size reasonably precise percentage phenotype explain snp expression snp phenotype unchanged table great estimation great association little I close choose big confirmed effect effect misspecification resample flat unlike mass via spike well balance setting clear meaningful standard base twice small study threshold snp nucleotide report discovery initially use selection threshold phenomenon winner curse winner curse gain major factor failure five three winner curse recent paper dedicate curse argue reliable
appears reduce effect simulation consider pool include approximately minimize individual possibility modify thus pool individual mark small together former preferable qualitative success matrix recover evidence set f preferable beneficial number panel case coverage read efficiency reduce dna pool oppose dna section price read black line incorporate cs without recover minimal prominent still number becomes keep beneficial number many drop hence add start enable number effective although read add high allele treat individual increase add naive black perform present identify allele deal rare display naive rare benefit put identification vast amount cs rapidly believe major advantage tailor may cs fashion use cs solver try keep optimize likely formulation incorporate allow sample example input signal consider post distribute frequency low deviation treat allele independently rare allele also
eq error specifically improve rao whose square iterative approximate iterative closed refer approximate although constant provably even well adaptive mmse fundamental accurately matrix year covariance matrix attract interest include microarray forecasting image brain activation fmri method motivation common unknown coincide maximum minimize mse
generalize heuristic projection propose estimator see reference apply selection drawback maximum criterion ridge support finally aggregate final present follow compute penalty optimal selecting algorithm practice grid embed thank possible case maximal jump entirely jump jump elsewhere state union obtain ridge regression parameterize assumption hold assumption happen true exist theorem assumption actually derive oracle section remark depend almost lead unknown look less possible ol estimator every applying inequality treatment compare generalization problematic great care deterministic oracle refer
probability learn price p sample program primal solution treat union distinct distinct price otherwise exactly price bind price show feasible primal optimal program high probability proof base two observation satisfy dual solution program ib hoeffding bernstein without replacement manner low observing hold solution feasible therefore objective online optimal decision period contribute relate optimal linear program program offline q solutions py dual weak duality p least follow event happen decision take uses fact horizon require claim propose improved price price history learn precise price rule dynamic set dynamic proving competitive intuition behind
purpose kf project onto span kf kf report table initialization despite probably even though low noise outlier superior kf kf kf two online good experimentally order typically exceed storage synthetic clear advantage study component outlier intrinsic much work extend
construction ensure n immediately lemma control error ensure compatibility control n violate borel inequality essence detailed continuity argument due limitation sketch iteration reach almost dependence obviously ensure almost control quantity iterate obviously initialization monitor locally alternate cg approach rewrite iterate cg
superior performance investigation total scad variation pde quite denoise modification success failure largely characteristic image scene color solid object exception tv denoise variation less near significant utilize spatially weight procedure obtain show computation tv piece development explain although priori numerous exactly imply shrink estimation piece wise shrink propose smoothly deviation scad become statistical image task tv
carlo run sampler section ht sampler two perform kind redundant frame conditional orthonormal histogram know reference wavelet pdfs tight frame translation filter norm frame coefficient value hyper suppose mmse estimator base frame report ht c automatically appropriately monitor base scale simulate chain use basis fig stop burn period computational use intel ghz architecture two propose sampler term simulation sampler fast sampler reduce
proof give generalization monotonicity sequence useful straightforward belong successive iterate decrease stop establish statement tend infinity subsection lie closure tucker type establish kullback distance notational alternate kullback lie particular first optimality assume subsequence small belong compact continuity proof make tend deduce claim decompose directional
method semi supervise denote select supervise force reference base automatic recognition accumulate list accordance censor additionally method oracle anti oracle hand oracle small anti oracle distance notion recall case g ax ax ax er experimental fig choose sup hour architecture choose use query indexing institute technology scoring evaluation trade demonstrate operating point quality rt rt rt sup sup supervise semi select table select observe error close change ax instance recognize incorrectly system recognize make consequently level prune word rt data rt phone error word way comparison may supervise candidate recognize contain word comparison standard rate reference necessarily well force alignment instance
relaxation problem propose max also advantage eigenvalue eigenvalue relaxation remarkable writing content follow section rapid lagrangian duality globally binary third section semi recover relaxation inexact deal binary primal solution give norm eigenvalue strong mean term maximal eigenvalue relaxation experiment method chain anneal metropolis gibb present recently relaxation far encourage duality denoise recover special pass second effort sdp inner real e hull closure denote diagonal relaxation quick duality collect role present
doubly entropy become bethe gm bethe free incorporate multiplier enforce look set quadratic observe contain among integer interior rely description set perfect refer bp approximation follow enforce belief enforce mf energy one example mf entropy explanation mf fact edge perfect properly solution bp homogeneous b show homogeneous blue green
table support methodology fit zero arise many practical study one incidence table viewpoint interest markov chain minimal may connect table cell contain square move connect table entry connect one find connect connectivity satisfied model markov restriction connect table organization rest preliminary fact theoretical table minimal connectivity one table contingency version discuss square conclude paper remark preliminary basis
recover original unfolding although symmetric eigenvector definite pd analysis necessarily case diffusion map neighbourhood vertex among point versa kernel map sum subject diagonal consider laplacian semi definite calculation constrain reformulate whose consist exclude diffusion map proportional laplacian comprise important point either map laplacian pca modern discriminant kernel multidimensional scaling extension embedding cloud quickly induce computational thereby motivate introduction increase use across field exploit redundancy often relative summarize subset actual turn accomplish via nystr om om upon
primal institute university cluster anneal schedule find assignment sa furthermore easy sa implement popular typically optimization relaxation np simulated sa promise able optimum slow schedule temperature schedule cluster sa find reasonably good solution schedule mechanic anneal novel add sa annealing see
property subset question association geometry analog heat operator evolution start hamiltonian method summarize quantum formalism vary detail evolution wave behavior quantum clustering begin state hilbert view datum move original distance change many center instant cluster association reason visually explore approach begin know window euclidean gaussians scale view observe maxima determine cluster maxima prominent location maxima clustering take hamiltonian potential function inversion harmonic center original expect minima maxima minima frequently turn minimum display maxima estimator associate potential simplify identify effectiveness two wave wave
combine proposition case recover moment normal likelihood tail implicit shrinkage illustration shrinkage show shrinkage zero asymptotic turn rule fit soft thresholding piecewise match odd symmetry functional x eq slope numerical indexed yield figure laplacian mmse compute numerically soft piecewise approximation compare numerically idea straightforwardly may unimodal mean shrinkage haar coefficient maximum identity multiple may detail equally set aggregation notation location haar result shrinkage operator furthermore exponential yield unbiased thresholding iy I result case white variance
form clearly quantity measure easy follow chi fisher formalism possibly equivalent eq r us concentrate equivalent lead give g eq name center chi center reduce distribution quantile build course would classical fisher formalism comment robustness express g kind
numerator posterior density alternative posterior define joint guarantee representation bayes marginal posterior impose uniquely impose bayes everywhere hold apply formal bridge conclusion whose uniquely define version therein satisfy differ density implication variable
topology uniformly ij consequence computable computable de computable vice joint generalize topology unit bound immediate integration continuous function borel relative access recover uniquely moment show mixed evy inversion characteristic function xt finite dimensional inversion direct obviously instead computable mixed moment mixed building polynomial pointwise polynomial approximation rational polynomial computable show uniformly x n f computable effective pour coefficient polynomial computable integration computable moment versa computable moment uniformly computable consider coordinate computable uniformly computable establish eq lemma computable converge pointwise indicator dominate convergence sup p linear moment supremum exchangeable sequence de distribution uniquely show distribution relative claim computable x let joint computable order furthermore preserve obviously lemma argument succeed result mix c mix uniformly co
theorem distinct contradict outlined problem express way either solving secondly appearing call polytope convert polytope convex polytope show dual cube distance point construct general particular exactly eq unique hull write combination definition cube hence convex actually system uniquely linearly know zero apply contradiction assumption construct maximum summarize short cube ray optimality suitable vary value cube dl point first section define p prove construct occur path correspond exponentially support
strict final categorical aa useful well combination follow aim investigate whether disease follow month stage triplet patient triplet interval length expert aggregate decrease accordance law peak sense peak sort peak frequent combination combination etc empirically coefficient aa error strict show aggregate good include combination
fan von strongly simplicity x square dual definition xu fx fu f xu xu u x fu f fu equation imply point modern impose restriction much understand nature implicit knowledge impose norm tailor margin suit sparsity deal systematic work regularization norm methodology property notion ability range batch impose complexity online central behavior lie tangent strictly characterize norm rather convexity understand duality strong smoothness approximate linearization function expectation sufficiently smoothness enable focus characterize derive rely previously specify systematically decide base underlie matrix much recent usefulness
evolution equivalently evolve partition se complement se recursion evolve sparsity guarantee mse subscript evolution depend essential simplification prescribe sparsity moment threshold proof along derivation explicit expression precision numerical evaluation canonical problem formal mse region optimization formal truly thresholding would imply allow solve hope last five possibility replicate behavior iterative property phase extensive iterative algorithm nonlinearity phase small lp base
necessary present structural boolean noise boolean bound sensitivity polynomial define regular oppose polynomial ia ii multi say multilinear work multilinear notation unless state degree polynomial p multilinear polynomial either px ss clarity exact extend determine play role intuitively regular variable high coordinate polynomial multilinear regular qx regular loss invariance anti concentration degree multilinear follow anti concave anti polynomial qx generalize
sampling mle considerably slow mse importance approximately find c map l equivalence spline one interpretation coefficient example model extreme modelling datum pair distance measure point formulate coefficient interpretation detection number location perspective carry birth proposal equivalently express framework retain interpretation high note current datum truncate poisson truncate change inspection sensible interval occurrence point mcmc iteration burn
rule estimate adversary observation explain discuss practical neither precisely obtain amount prior distribution possible adapt world employ point process decide discriminate employ rule compose shall omit discussion user discuss adversary model decision adversary shall rule perfect information adversary know priori adversary generate oracle adversary user concern adversary uncertainty
standard idea near may conceptual optimality closely problem censor wang yu support special california author like li david van introduce grateful problem vertex simple monotonically never algorithm slow strategy b propose vertex exchange effective consider work name combination include multiplicative ingredient contribute effectiveness neighbor strategy intend
able explanatory delay differential dynamical application availability simulation routine application system thousand parameter development believe exploit make improvement come adjustment iii develop kernel exploit inherent substantial molecular species uncertainty employ sequential range describe pathway simulation ability employ exact mathematical analyze complex describe model allow compare ultimately rely
starting give discussion initial event manually quite insensitive initialization yield similar keep mind discuss outline mt cumulative distribution cdf interpolation interpolation calculate quantile value analysis serve zero avoid issue vertical interpolation axis cdf polynomial interpolation calculate cdf would however spline inverse simply linear calculation interpolation quantile adapt presence potential curve beyond hasting would proposal maximize
show unique due string statement finite uniquely value term language introduce unconstrained translate contain yield yield ideal generator follow polynomial map constrain model string result attempt novel notice explicitly state adapt language function process view module language area amount column string function unconstraine hide
demonstrate detailed velocity interesting evolution disk deviation close gaussian velocity expect simple model disk dynamical bar shape region galaxy part disk rise velocity b similarly steady effect coherent infer signature directly observable need direction branch along line motion rise apparent background distance apparent shift star traditionally pc intrinsic star systematic background source angular motion proper velocity velocity star line around uncertainty star purely velocity measure line star distance reliably star diameter
regardless scale aside leaf would split prefer option posterior surface thing approach involve grow move proposal whereas sequential change swap present cost desirable global filter observation dynamic leaf combine l pass split leaf predictive covariate somewhat subtle point would predictive efficient impractical seem distinction dramatically concentrate option leaf reduce functional likelihood leaf regression ease evaluate prediction free fortunately subsection regression move example process gp leaf impose provide propose herein posterior inference lead reversible particle sequential thus dynamic tree nature expensive lead preferred choice application modeling leaf node mean eq leaf force leaf set framework scale leaf likelihood allocate leaf student leaf plus shift partition essential probability marginal marginal
produce look extent occur jump hard infer occur position number b plot black line green blue dash analyse log probe surface number change interest detect furthermore segment model thus idea change position remove analyse analysis possibility underlie allow model whenever plot variety way
relevance especially relevance feedback kl yet consistently one explanation separate word similar ad hoc expansion advantage work integrate coherent manner instance document explicitly redundancy could account preference similar walk advantage query underlie retrieval noisy leverage know know query successful ad hoc retrieval view perspective ir work interpret query expansion ir terminology importantly expansion hoc statistically language describe
pt central moment rv since central moment th central moment eq accurately skewness define dividing moment respect give rest expand expression skewness odd contrary taylor around indistinguishable first approximation skewness coefficient unbounded denominator numerator skewness dominate rational tend whereas one numerator determine behavior use skew skewness restrict sample commonly likelihood mle present input output function branch allow scale x w cdf pdf rewrite transform depends necessarily coordinate approximate mle know back compute uncertainty likelihood z w think transform dependent support location depend crucial asymptotic mle depend confirm gaussian behaved show unbiased however rarely kind estimating build output aspect skewness estimator see detailed discussion back iterate
kernel illustrate covariate transpose rwm posterior compare plain rwm rwm example easy b hold ny hold apply measurable example covariate presence absence disease variable chain discard burn contain section generic actual appearance toeplitz use follow martingale pt notice lemma imply similarly g ns
importance closely match estimate use importance equally suffer select accordance fitting carlo possible solution aim produce regular associate normalise sample approximate update term divergence kullback divergence relative select large density parameter specify usually latter preferred tail tail vast density mixture considerable flexibility wide posterior follow population monte carlo sample produce self normalise output improve increase density normalise importance induce extra variation residual systematic sampling generic gaussian arbitrarily parameter result proceed recursively iteration normalise counterpart function derivation expression simulation sect appendix kullback divergence time although target density improvement shannon normalise kullback good agreement sample non importance high weight ill fitting also estimate normalise tf fx importance formula derive normalise adaptation chain simulation consideration autocorrelation chain approach fig great depth next use student
increase density solid segregation dash empirical critical critical empirical segregation estimate nan solid monte investigation segregation alternative replicate mc separation nan density carlo replicate notice function skew segregation monte carlo segregation skew almost skewed segregation alternative empirical empirical empirical segregation carlo experiment six estimate nan case segregation symmetric occur skewed segregation dash level critical significance furthermore estimate increase decrease segregation alternative leave function power test standardize experiment value empirical power nr carlo investigation j r appropriate yield desire segregation carlo critical segregation function circle represent
limit roughly unit potential somewhat level manually unit uniform size number number need incorrect outcome voting report count store design make delay public release candidate mail publicly consume way mail batch sized unit article risk limit post insufficient initial inefficient insufficient likely detect well unit incorrect inefficient manually another constant pair insufficient win calculating produce insufficient proportion vote total upper sometimes actual insufficient poor insufficient sample state vote occur cause incorrect unless expand initial understand logical implication level paper say case manual calculate sample analyze win candidate even mix calculation precise methodology save computation resource calculation sum within margin win candidate separately win candidate pair separate multiple initially discrepancy conclusion less unnecessary failure manually count stack sample post confirm instead zero confirm win vote limit post risk wrong winner incorrectly desire summarize post detect incorrect outcome win article calculation size unit single winner treat candidate equally margin error ensure allow win candidate win candidate weight conservative method generalize sized improve material development count voting would vote specification provide count
combine argument suggest take quality typical example figure possibility choose already show take uniform goal choice compare hyperparameter usefulness take depend wavelet average repetition display wavelet deviation thresholding estimator keep wavelet behavior minimum laplace minimum small indicate take estimation level threshold perform generally display small oracle introduce resolution large confirm reasonable satisfactory j test nj htbp nj deconvolution performance wavelet deconvolution penalize collection contain integrable band adaptive band limit localize perform select make realization display
bs locate square specify figure well match perfectly diversity otherwise gain select channel selection l z gain equal throughput transmission throughput tend intuitive throughput increase use analyze consideration location two diversity alternatively noise interference limit otherwise system select exponentially conclude gain select good overhead compare help
capability build tail data gpu ran kernel formulae pass result preserve none formulae deal effectively value significantly generation extreme argument move compare quantile code quantile remain cpu improve scheme gpu gpu dramatically analyse rational gpu initially follow computation way gpu benefit improvement old form sde evaluate payoff make relative substantial increase gpu computation lack use provide understanding hardware etc optimization also would course base different distribution pre exponential normal evaluation benefit time work precision modern gpu need somewhat us satisfactory monte almost ok sure one repeat work generate concern associate unity issue concentration thus double unlikely candidate method feed high assess series formula suitable http en probit series code cross precision agree
objective encoder decoder controller reliable decoder encode redundancy decay scheme universal aim immediately ensure compression compression organize basic code list regularity result three suitable regularity direction contain source alphabet separable borel index marginal whenever carry subscript e process absolutely avoid clutter omit whenever measurable supremum partition gray density respectively dominate pf partition measure ball marginal coefficient observe split equivalently memory alphabet assume space letter distortion x countable binary string equivalently bit encoding block encoder precede composite denote compactly source process letter reproduce expect rate string distortion convenient distortion q lagrange multipli control distortion trade geometrically intercept line passing carry subscript
increase decrease end rt defer r nan hypothese corollary conjecture pairwise make piecewise illustrate approximation near various far tight course interval state another corollary similarity il corollary solution whose approximate dash immediately imply bound expression variance fact theorem aid pattern shall satisfy may open closure three transformation
hold model fix vanish old old hold maximized give increase sbm assume factorize node variational em variational look conversely variational sbm intractable full modelling decompose eq problem variable look tractable q factorize q n approximation equation bind initialize vertex euclidean distance assignment parameter variational cycle value small
tx sd think round weak produce validation possibly much empty solver draw dictionary round always find global equal objective low classifier algorithm rather try decrease may construct add classifier outer initialize current opt outer tx w hx tx outer outer outer classifier qp per adaptive weak yield accuracy apply
likelihood make prediction correct additionally interesting likelihood compare predictor predict split provide reasonable learn room specification graph sequence run graph type benefit evolution r transition dynamic outli initial behavior explain usual choose exclude fit type statistic transition magnitude validation start include plot magnitude play play involve two expand include explain section person support support person person person look link time effectiveness weight possibly co support could proposal happen
accord hence observe receive accumulate average long reward notation highlight average depend addition exist default correspond policy assume step make determine phase confidence fully turn sufficient algorithm suitable let instant terminate exploration phase policy three four policy follow finitely default select remain phase include otherwise confidence policy say empty exploration basically policy policy hence purpose stop even neighbor challenging course consideration suggest
c study robustness modify cs measurement case correct relax cs gaussian simulation change measurement varied normalize rmse expect cs cs regularize modify useful recursive reconstruction sequence clearly signal support contain application cs reconstruction estimate compute initialize I modify support wavelet index measurement e need measurement slightly ensure reduce increase increase enough alternatively support keep add element condition number l estimate l typically l h compute either empty compute cs tt knowledge reconstruction reconstruction something generate reduce regularize
iterate current density processor kp kk block processor carry draw chain draw marginal likelihood often marginal importance metropolis hasting scheme discuss use auxiliary particle comparison several datum illustrate flexibility applicability particle filtering acceptance compare define acceptance metropolis proposal sampling divide scheme generate autocorrelation lag lag acceptance factor measure effectiveness interpret sampler attain draw depend whether equation transition autoregressive outlier distribution truncate shape particle apply yahoo finance datum
directly connect detailed usage publication initial contain reaction light cycle nucleotide dna sequences complementary basis one side reaction produce repetition encounter sequence dna subsequent available reaction cycle test positive chemical reaction fraction copy call average incorporation mean copy next test response incorporation incorporation measure negative account usually description consideration differential incorporation effective description consider slight adaptation base sequence base explain usually cutoff keep definition rate incomplete plot direct look test fraction sequence test count instance position value direct path start
involve discuss subsection approximation condition conclude kkt condition exploit way albeit adjust restrictive relate illustrated figure condition compatibility look substantial room eigenvalue compatibility selection lasso condition zero perhaps strong compatibility verify compatibility approximate analogue realistic cover situation nonparametric slow contrast linear regime compatibility population regime general formulation compatibility condition well need
previously conventional association tree lasso mostly recover report indirect evidence grain module perturb article call jointly leverage response analysis snps activity co express smoothing originally induce lasso view reduce false part gm nsf research problem quantitative trait mapping discover genetic gene level investigate capture hierarchical response node response internal leverage recover relevant structure systematic coefficient manner membership employ induce penalty simulate show pattern run agglomerative expression scope compare hierarchical gene correlate leaf internal root internal represent subtree level common type effect strong correlate gene small height tree among correlated cluster tree great height grouping available
bernoulli generality secondary common power secondary assume bandwidth channel bandwidth exposition htb cost account spectrum monitoring sense control channel past access model access user always pay monitoring regardless decision begin whether stay spectrum option bid bid vector component q indicator function maximum bid spectrum incur accordingly spectrum access stay game transmission occur
significantly homology equivalence dimensional homology large close homology bind characteristic tie heavily homology throughout give promising area summarize point apply concrete particular al homology class systematic quite seem naive attempt argue rich object beyond real spirit different either via question decomposition arise failure extensively g property algebraic discuss generality complete calculation laplacian rarely possible rather homotopy invariance write essentially section theorem must point assertion moreover dimensionality theorem prove helpful measure result homology subsequent take homology complex homology endow product keep mind domain euclidean space lebesgue borel assume certain additional simplicity impose later section concentrate rich notion locality instance define complex map space construct confusion operator co formula sequel recover operator define set gradient linear integrate right give complete map side first term co define l inequality inequality side finish proposition theorem variable put
heuristic explanation observe value gain design adjust place design increase weight gain increase convenient small remarkable place literature algorithm attract attention appear case probabilistic inequality know assume see broad aim condition ensure certain
reconstruction proceed definition probability number bound complete finally rf threshold operation rf hold stable bind algorithm notion addition exist reconstruct specifically reconstruct know great setting interesting tradeoff direction sample corollary proposition conjecture axiom grant project motivate compress sense reconstruct attract theoretic variant main reconstruct number impossible reconstruct matrix number entry specifically stable matrix satisfy incoherence propose pursuit reconstruct factor away information limit compare sdp perhaps importantly sdp keyword randomize pursuit randomize subject arise frequently
particular belong next section hierarchy hierarchy order alone understand internal word small dictionary function ht good bad dark natural multiple variation word loop empty vertex subgraph strongly define represent c vertex derive natural model graph criterion world turn degree degree seem respectively connect remarkable source correspond strongly heart cyclic analyze dictionary relate age word word
connection like cluster assign green seem prefer assign cluster approach membership integration possibility relevance publish elsewhere provide cluster basis development online connectivity green red green gray w gray red green gray red gray belong family prove claim stand link connection aim estimate incremental recursively update current notation ij addition use addition new q n z update relate em purpose maximize difficulty factorize base strategy replace computing efficient alternative base monte approximate tractable node extent n
parameter mat ern present closed attain degree old smoothness behave ern prove covariance appear derive spatially vary second kind spatially vary mat parameter determine covariance problematic feature difficult separate scale local therefore desirable interpretation suppose geometric positively identity parameterization simplification smooth recognize mat parameterization page smoothness amount simplify prominent observation simulation automatically motivate theory equation local w therefore minimize exclude fix ignore form one
prior give stage stage dirichlet contain situation stage regular stage attribute unchanged stage dirichlet stage prior dirichlet prior child child dirichlet consider node child terminal child situation lastly child order stage root node leaf clear probability justification stage markov event analogous modularity property colour margin definition prior situation partition theorem trivial prior u v n leaf place terminal stage
nine place person action hence represent value test length normalize state nonlinear consistently autoregressive particularly long range dataset nonlinear model range contain history dependency incorporate k trick filter map reasonable classical recently time parameter estimate prediction system dynamic incorporate nonlinearity dynamic nonlinear sensible together observation purpose maintain formula maintain exponentially difficulty
difference leave incorporate svd basis incorporate right singular singular length svd covariance whereas row share differ imputation formulate computational limit variate application imputation maximize exploit maximize one coordinate considerable mathematically behind briefly discuss computational strategy consideration computationally suggest foundation calculate procedure example bayesian step address imputation method covariance multivariate notice penalty inverse covariance kronecker leading reason method repeatedly see imputation propose strategy present review part miss develop imputation mathematically begin seek indeed likelihood em e variate trace matrix let value step form pp variate structure imputation correction jj jj jj jj ii analogously form maximize along solver multivariate gradient penalty since differ term take eigenvalue decomposition eigenvalue lasso apply
initialize orthogonal previously normally convergence previously continue randomly vector differential entropy bss entropy parameterize flexible well desire component see sample per expectation average number via transformation projection differential entropy minimize realization include prior prevent make condition previous uncorrelated source drop term entropy much uncorrelated bss use algorithm show differential enforce bss replace work mix replace entropy partial dependence thus bss flexible partial order dependence source base bss create use mixture mix mix mixed picture source true source inspection http www project synthetic source use source fraction choose sum mean source eigen sample orthogonal matrix matrix construct sample blind mixing mix pm package odd coefficient compare run value non gaussian normality pass empirical cdf normally normal show run sample variance illustrate base bss real berkeley first standardized mix draw distribution noise image orthogonality odd orthogonality without orthogonality figure pdf htbp figure use middle row orthogonality constraint projection show estimate source projection htbp estimate use algorithm middle orthogonality constraint impose orthogonality constraint projection result show fail use upon converge fig detail default setting successful see visually fig intensity exactly well source compare orthogonality model extract source monotonic note distribution gaussian approximation adequate htbp pdf evolution log increase em estimate bottom project density jointly describe show algorithm distributional part distributional distributional root cubic objective instead property check equation next consider model bss bss sources bss difficult one et develop bss problem latent imply factorial latent derive em mix factorial provide exact bss exact intractable propose variational bss posterior bss couple assume bss model ml bss reduce orthogonal differential entropy approximation entropy et modeling factorial suffer work al flexible source al differential entropy distributional observe feasibility source arbitrarily densitie ml factorial source retain computational tractable source approximation variational beyond scope follow application result source simply bss generalize source bss alternative worth bss fig pursuit mix latent source mix transformation source uncorrelate become pp general contrast pp bss sources bss think preprocesse orthogonality solve maximize objective concave maximization problem maximization global method compare run mixture average significantly well would like mention true fact biased experiment illustrate true illustrative publicly visual inspection relatively compare orthogonal density multimodal fig model capture orthogonality perform orthogonal independence correlation dependence limitation bss slow present minute estimation versus ghz intel processor allow encourage explicitly become however dependence bss approximation average become square bss et follow application density illustrative furthermore non zero simply orthogonality requirement assumption unlikely bss optimization vector future acknowledgement acknowledge n medical db lb thm section convert gaussians project vector observe estimate blind bss differential minimize minimize entropy project space simultaneously projection flexible fitting near conventional bss feasibility mix distributional parameter maximization algorithm directly generate unknown projection project gaussian distributional particular derive em similar conventional entropy blind bss analysis ica mixing source signal mixture corrupt latent ica free bss bss reader bss entropy contrast density maximum importantly assume denote al expression flexible bss adequate true near al bss density model factorial advance density addition bss follow however algorithm become computationally moreover al develop latent apply source bss question show apply problem allow scalar denote font bold letter vector upper e p zero bold th symbol denote I multivariate denote ba natural density capital necessary indicate vector expect need estimating model projection need suppose vector form think project scalar scalar realization also matrix column unknown satisfie assume mixture gaussian q q equation project gaussian give project projection project model contrast done prevent onto simplify p write gaussian handle expectation allow along prevent onto gamma similarly vector improper prior choice posterior denominator density solve problem subsection algorithm maximize deal logarithm rather substitute get q density get introduce write write monotonically iteration converge local stage step outline alternate fashion ordinary optimize convex post detect error use subsection discuss detail lagrange tucker w q impose constraint noting give ignore optimization constraint always inactive upon get simplification satisfy condition lagrange multipli lagrange multipli derivative optimality upon simplification write summarize matrix diagonal distributional assumption require dependent maximally source bss difficult couple equation q would require tractable compute maximally correlation loose difficulty bss computation bss trivial fundamental follow detailed bss leibler divergence concept ng variance co invariance singular kl algebraic co eq length theoretic vector p kl simplify mutual linear satisfy q equation see use element second dependence see optimization minimize component write source variance variance constraint encourage nd order term minimize uncorrelated encourage source weight serve balance term note nd drop equivalently maximization minimize mutual satisfy maximize uncorrelated vector minimization bss estimate mixing try answer question approach uncorrelated et al problem second order allow seminal bss problem exact gaussians since flexible sufficient able complicated give component flexible factorial density al integral analytically also gaussian factorial derive obtain et note source describe component step intractable overcome et approximation em summary bss exact intractable
situation team overcome opponent pass team try move penalty area confirm conclusion relational domain whole multi system activity focus team base behaviour agent system team use symbolic sequence relational multi variate environment action characteristic team discover relation raw transform set symbolic b forward forward b forward right b behind c b front forward forward b forward front forward forward right behind university department science di electrical college uk ac research institute research multi agent whole team member systematic verify effective member team observation analyse recognize team action focus challenge team allow symbolic translate multi variate environment sequential team sequence express first logic atom relational meaningful pattern relational team reasoning field intelligence robot environment act task carry challenge always occur design environment create decrease task less realistic multiple capability perform goal possess detection internal planning execution recover engine must income world since outcome environment detect evolve plan either correct team execute task involve difficulty task decision limited robot result opponent agree decision complete robot robot achieve goal address collaborative behaviour environment aim effective team compare recognize behaviour direct team deal agent activity agent team precisely four robot base retrieval base message internal retrieve adaptation situation model action description perform retrieval action individual act allow challenge rich robot pass robot domain require action team implicit mechanism avoid ball towards direction force region etc result always try move alone towards try avoid turn pass avoid ball individually pass chance address represent perform relational enable human understand observed principle could qualitative team concept recognize proposal extract multi file relational describe discover relation interesting relational offer advantage frequent action team team distinguish pass adapt solve snapshot robot point robot robot internal belief correspond game contain retrieve action formally tuple relative ball reference robot position ball position ball scope within retrieve easily relate robot perform retrieval solution one retrieve evaluate adapt case feature index former robot move appropriate position latter goal modify idea modify adapt description indicate function similarity compute harmonic similarity compute cost modify adapt distance position adapt position obtaining adapt global execute manually create file system three real essential minimize speed index list store thus index similarity cost potential select case adapt robot retrieval order player maximized multi interact environment among exchange internal copy base responsible retrieval process distance robot high correspond begin part adapt reach execution rest next execute execution rest receive retrieve arrive rest stop relational mining extract frequent pattern define team language knowledge comprehensive introduction logic programming inductive logic represent atom consist empty symbol symbol argument atom formula symbol apply I symbol symbol apply x ml implication I I l clause term say contain argument ia ba bc indicate substitution subsequence least element subsequence distinguish atom dimensional atom characterize involve represent denote mining inductive programming discover relational mining take belong pattern evaluate support wise lattice order lattice generate lattice evaluate candidate discard frequency frequent pattern actually dataset base start add atom try potential discard store one user refined pattern pattern discard operator background clause contain description relational file able describe team frame successful overcome opponent player represent shape body orientation team compose team robot represent stream consecutive player stream recognize team basic use activity attempt team goal gain ball gain gain player move avoid opponent move move ball toward goal opponent process occur trial place ball recognize contribute describe team necessary take account perform agent environment qualitative world allow recognize event persistence take indicate subsequently opponent compatible previous therefore occur another opponent try ball try event event hold full move ball go recognize event contribute ball pass depend try case robot consider situation robot opponent take system consider event world event event system current represent ball describe coordinate identification relation specific viewpoint perform viewpoint robot opponent front back precise reflect generality abstract sequence symbolic abstraction raw relation opponent box use relation horizontal vertical ball player horizontal horizontal describe environment pass robot predicate trial ball opponent go pass field stop action environment direction intercept opponent opponent relation action playing point behaviour team work action team behaviour extract set able entire team allow strategy global well take thus try ball behaviour team team sequence extract team version simulator create team platform competition additional feature implement message simulator simplify noiseless ball position action perform degree randomness end ideal trajectory robot try simulate failure situation real ball start gradually decrease code automatically exploit default default home go beyond otherwise remain home region ball maintain view consist vs team account two configuration define configuration enter home overlap assign region player robot region player situation avoid define occur come position scenario ball middle field side figure remain home ball middle front decision make execute critical avoid ball set qualitatively situation respect perform since action near goal attack perform trial trial advance approach analyse file game implement able identify level construct sequence team record simulation atomic trial file configuration sequence ability reach goal reach reason end trial configuration trial trial sequence dimension one recognize predicate dimension relate analyse recognize understand team preliminary relate degree behaviour relational abstraction action recognize scenario recognize action indicate possible team table amount behaviour team high team play collaborative strategy try reach move adapt sequence action describe behaviour robot ball try area front opponent act pass recognize would opponent adopt behaviour succeed action recognize cc cc scenario scenario intercept cc cc cc scenario scenario
firstly integral via part secondly b unity ts ts ts k ts ts ts ts ts ts ts ts treat ts ts h k ts ts ts ts signal spread suffice highlight benefit walk reject repetition compare transformation yield version preferable transform statistic indicate intensive accurate weighted df chart version transform df control chart acknowledgement multivariate read minus corollary keyword robustness analyze difference stationary issue analysis often unit nan stationarity favor significant establish define existence equilibrium relationship favor stationarity equilibrium statistically justify test test residual monitoring procedure monitor horizon detect stationarity soon stationary method analyze detect trend usually stationarity see al b limit walk substantially detect change nan alternative apply control stop series horizon convention value design average value serious present characteristic focus article version strongly innovation limit white ensure stationarity modify df time cover local unity alternative df monitoring nuisance firstly nuisance secondly appropriate transformation long dependent avoid least propose root test type accuracy monitor procedure detail motivate carefully series type relate distribution hypothesis unity asymptotic drive brownian motion appear nonparametric assumption allow motivated approach series suppose ar error characteristic root polynomial root imply error calculation ar unit root motivate time sequel univariate satisfying concern impose assumption series functional central brownian wiener strong time coefficient equip satisfie regard nonparametric ensure converge orient definition stock scale notation uncorrelated process exist negative random put common assume common standard normal al unique addition decay coefficient control criterion detection estimator suppose uncorrelated rao standard brownian motion base nan unit quantile want signal root dependent alternative specify check deviation unit df use sophisticated weight current define detection df play smooth distant dominate kernel ensure decrease weight nuisance regularity scale procedure instance otherwise look ensure number get parameter appear limit choose relative therefore restrictive alternative stationary signal start monitor reasonable approach type ensure converge nuisance past datum west specify estimate west q lag truncation west study chart series point calculate estimate less transformation back weakly limit consequently denote ensure rule next asymptotically valid ar unit root statistic regression df df numerator rule know nuisance alternatively limit eq provide functional central theorem define walk nan define property hypothesis central provide weight df series asymptotic representation stochastic ts ts ts ts ts k ts ts ts dr bound variation cf hence eventually equivalent integral integration obviously ts tr ts b k ts ts tr k ts ts function ts functional weakly motion brevity integration ts ts k h ts tr tr ts dr u r dr uniform position numerator ts ts theorem entail dr follow detection require limit result proof brevity omit detail tt limit lag chart estimate e equivalence ts c ts continuous write pointwise nz ts immediately definition schwarz e pa mix decomposition ts tr r r secondly consequently triangle inequality ts ts r yield ts ts ts note c provide transform satisfie transform chart particularly asymptotic yield ts ts ts ts ts ts ks dr dr theorems weight process associate walk k depend parameter note ts ts ts ts ts ts ts ts ts ts ts q fact ts large stochastic yield stochastic
already appear physics aim go group symmetry regard viewpoint hope application let briefly mention predict object assume joint provide previously observe input random training construct unseen object binary pattern input usually describe call handle categorical encode signal etc simply system either quantum provide state respective learn could scenario quantum channel input need classify input alternatively binary coupling want quantum quantum measurement unknown learn training estimate apply intermediate error measurement excess converge size state estimation plug constant state perform minimax capture behaviour key theoretical tool quantum extension roughly say collective quantum system approximate gaussian classical derive strategy benchmark harmonic collective measurement one moreover collective optimal measurement showing optimally simultaneously close template framework template discrimination special size quantum special bayesian difference measurement special symmetry mix quantum propose series paper regard paper scope investigation give classical discusse emphasis problem problem variable subset first second stage present ask guess unseen depend overall measure distribution equal give know bayes classifier choose probable respect bayes fit naturally give optimal known ratio choose high likelihood verify density measure return unknown learning primarily find classifier parameter subset certain excess call q alternatively bayes classifier hand one replace equal excess fact away situation show behaviour bayes area curve area green deduce determine around call roughly speak satisfy parametric learning theory slow depend one principle learn solve procedure function plug classifier aim close paper front quantum well method phenomenon stem optimal measurement quantum counterpart quantum describe probability state denote counterpart state description think measure classical prior classifier measurement x copy together permutation copy operation unitary binary element random copy quantum state whose guess misclassification nothing optimally prior sign tr quantum arbitrary risk q vanish analogous set besides obvious coin similar rate discrete continuous explanation label outcome way conditional box additional appear distribution label resemble datum choose give formulation unknown problem belong available restrict sub define refined risk multiply expect non trivial limit risk well difficulty different capture behavior whole think training consist estimator local minimax sequence call local model le gaussian extend slowly describe version normality simple spin dimensional state unknown methodology concentrate structure convergence model devise initial high label ball local write u essentially unitary intuitively big sphere picture spin coherent spin state collective spin central represent uncertainty collective see law x suggest canonical quantum harmonic equilibrium basis rescale independent limit classical identify von perturb pick first remain unchanged precisely equilibrium bit quantum shift classical local shift define convergence natural quantum trace however quantum channel simply diagonal quantum definition one represent case sequence local spin state restriction parameter channel would comment intuitively provide rather weak rather operational meaning channel secondly devise strategy state asymptotically involve quantum serve transfer protocol local set quadratic form simple prior discriminate label training outcome projective randomness unknown mean optimal case measurement guess state measuring check whether excess eq satisfy state separately basis matrix average construct estimator concentration inequality exponentially plug zero projection projective aim plane expand expression back account rate excess coming term drop uniquely q associated risk need employ normality let local neighbourhood pt pt plane eq shift eq equilibrium state technical lemma normality argument since consider loss local risk aim optimally directly strategy optimally measurement come formulate asymptotically measurement excess previous shift like contain express frame plane angle z l contribution come write optimal jointly measure quantum component separately optimal risk combine additional square add contribution risk depend quantum minimax optimal construct rough system transfer optimal coherent combine classical ball red plane plug mixed set measurement step rough separate observe local classify state taylor contribution quadratic distribution write q component add canonical three contribution consider asymptotic maximum risk plug compare plug r point prior maximum opposite understand require measurement anti directly deal know statistic may measurement bring additional excess add local prior orthogonal training cast additional mean additional factor classify two statistical asymptotically reduce optimally quantum two state
match upper dimensional bound base bind exceed bind finally need depend subset arbitrary metric e distance measure assume dx ball radius dy give empty clustering set hierarchical collection clustering clustering diameter c large radius large radius finally radius discrete radius minimal radius finite find cluster minimal clustering give agglomerative repeatedly volume state distance lie finitely set dy define contradiction u p follow note ball deduce dp volume contradict agglomerative center introduction agglomerative algorithm start contain successively remain cluster minimize agglomerative radius diameter result agglomerative linkage clustering minimize analysis agglomerative center simple adapt center diameter case technique dependency range additive case center doubly cluster problem introduction cost solution three analyze agglomerative agglomerative algorithm minimize I compute clustering denote theorem partition cluster solution ball greedy frequently clustering radius always upper next proposition linear lemma center cluster radius cluster two cluster merge argument center simple analogously merge cluster satisfie divide step reduce increase phase additive term cluster fix center merge computation exist cluster discrete bound phase proof proposition algorithm get apply use lead two expansion q merge introduce term divide merge phase half merge volume lemma distance discrete optimal increase eq cluster exist loop merge use ball radius merging would radius prove consecutive phase ik k agglomerative problem rl minimize difference radius cost refer cost center analogous furthermore cluster always upper bound partition constant break analysis tie clustering compute uniquely dependency satisfie denote divide step phase phase one fourth discrete center intermediate need cover induce apply bind overlap ball radius twice ball contain existence pair bound optimal fix ii dc c radius ball radius one ball ball let merge algorithm merge ball cluster show consecutive let analogously I lemma deduce repeatedly get take exponent geometric convex function line put analysis merge center two dependency show sufficient analyze satisfy certain connectivity property able combinatorial rely merge define connectivity property relate set next input compute clustering input subset cluster another cluster compute input cluster algorithm know let union establishes follow hierarchical union cluster merge computation merge cluster inside absence cluster inside algorithm compute cluster e compute minimize observation k thus cluster ever cluster whose contradict e subset property unique intersect cluster cluster proposition use merge phase refer hold union observation compute clustering cluster lemma iteration loop pair set cluster assume connect use radius two connect bound either candidate radius connect remain either come n complete get nothing show hence cluster still prove deduce obtain repeatedly apply sum cost get u follow merge furthermore z every know cluster another intersect agglomerative I minimization analysis compute uniquely diameter case set cost equal diameter create diameter cost computed theorem partition q hide doubly proof theorem intermediate depend doubly satisfy denote solution two lie ball long diameter able additional proposition divide merge stage merge cluster first stage cluster call translate cluster near merge diameter essentially length translation first guarantee sufficiently large intermediate upper stage reduce stage long sufficiently merge second stage weak argument center leave sufficiently many intersect cluster plus diameter find cost main stage phase bind q fix get ball ball dy I dy ic configuration subset ball dy dy intersect configuration configuration gray gray gray gray gray gray cluster merge cluster exist lemma exist diameter u p mc cluster tc parameter side proof able upper increase number point input compute furthermore imply q expansion e geometric lead remain merge compute phase lemma let cluster exist merge point two k gray gray gray gray gray gray gray deduce apply inequality k u result immediately combine merge analogously problem section discuss cost merging cluster cf diameter cluster get replacement replace cluster upper case slightly lemma proposition become analogously section compute factor make omit present construction whenever choose merge step minimum show e factor least state upper metric base note exceed upper case moreover approximation already show set section approximation low approximation euclidean metric problem cost construction extend furthermore I gap cluster opt n bad possible distance assume figure compute follow lx vx vx merge diameter remain diameter gap choice merge merge cluster length gap create diameter figure go infinity approximation factor converge eight section always impact metric eight point maximum diameter however start merge merge fourth cluster diameter euclidean section low one suggest dimension easy euclidean compute cost construct eight x g diameter merging add one add result diameter fourth cluster remain four maximize cost optimal consider compute solution metric norm input compute problem optimal simplicity sequel unit large diameter diameter hamming start thereby next keep merge end cluster cluster first next diameter keep way round k compare solution diameter behavior power immediately corollary optimal
front allow simplify precision constitute base minimization causal controller convergence bayesian true analyze asymptotic controller consideration operation mode mode finite extra impose stability contain operation ergodicity hypothesis another hypothesis subset hypothesis share formalize proof develop appeal arise context controller approach infinite operation partitioning space operation essentially different operation mode mathematical control solid formulation control theorem corollary university cb pz approach control solution output boundedness ergodicity sure thing intervention calculus bayesian control kullback divergence control fully produce weather solve map appropriate measurement design robot exploration turn counterpart controller simple toy effort center tractable approximation new adaptive control relative attract problem solve statement reformulate dynamic control system causal controller solution rule particularly infer implicitly uncertainty dynamic exploration exploitation constitute promise adaptive guarantee policy develop limited operation illustrate exposition signal set action respectively shorthand like string controller proceed cycle controller stream fully represent action collect history characterized suitable controller stream maximize say many situation probability case prefer policy unknown face control randomly available mp tailor would minimize controller respect controller average incorrect deviation effect give cause minimize observed action give controller minimize q constitute causal calculus worth result define treat hypothesis context htbp diagram useful illustrate symbol highlight collect diagram dynamic state choose transition thereby choose subset direct illustrate diagram especially dynamic endow operation see give sake simplify interpretation existence policy happen stay determined stochastically variable realization draw pm pm pm deal introduce unique realization decompose sub form average divergence play realization inequality clearly divergence process interest go well increase complexity e specifically growth realization mode qualitatively realization hence impose stability requirement ergodicity possible tractable light insight say htbp boundedness key construct point realization divergence inequality previous bound well inequality rearrange identify side yield inequality eq pm want vanish characterize true hypothesis prediction region differ accumulate eventually enter htbp long enough mean execute risk right policy motivate operation contain sub strategy sub divergence grow demand execution guarantee controller operation vanish operation controller divergence apply stochastically include gm result final mode core indistinguishable
relate elementary way function list function l utilize theorem recall property therein thus index cdf define equal statement cdf reflect cdf weight indeed establish section therein decrease non function weight cdf consequently show submodular non note assume decomposition utilize fix existence decomposition bound h demonstrate equivalence decrease decomposition hold pair aforementione trivial otherwise define imply decrease depend position show submodular theorem next illustrative know mathematical convex shape usefulness example offer mathematical four function complex supplement mm literature reference therein known expectation weight w easy x h mathematical graph specifically xx xx lemma role establish strict monotonicity function section helpful determine certainly seven log prove equation x follow whenever follow eq right formulate prove establishe result x establish hand side complete easily check fact non follow follow p prove function increase monotonic somewhat surprising increase depending impose strict monotonicity require speak visible loss variable surely pair concern look seven imply satisfied thus make roughly speak closure line leave continuous even look somewhat nevertheless seven every continuous everywhere increase contradiction rewrite hand cdf measure right continuous almost right equation expectation coincide say almost contradict thus function increase side equation zero positive numerator leave equal contradict assumption conclude motivate end importance continuity naturally look seven continuous every w seven every connect function care random seven satisfy moment proof assumption satisfy x seven continuous every assumption fail singleton elementary omit satisfied continuous example crucial cdf order collect section shall omit elementary positive hold upon converge positivity verify check imply log contain interesting decrease namely hand increase check equation show show equation right equation see lemma conclude decrease calculation rewrite imply take denominator positive side conclude proof acknowledgment partially support science research reference risk apply york additive measure ia w transform random economic finance journal management submodular north weight calculation mathematic economic weight pricing functional application north american risk estimation mathematics economic u concept electrical north c f partial submodular lattice mm mm mm statement section definition loading wang motivated concern monotonicity loading loading appear literature increase interest weight loading answering establish loading consequently herein methodology loading monotonic c make conditional tail wang chebyshev uncertainty mm author mail triplet literature random know net negative note otherwise class net need accept risk remain order meet various financial large define consideration uv expectation well mathematical chebyshev integral reference utilize economic finance
r convergence tie sake attention investigate possess uniqueness strict convexity e conversely strictly strict uniqueness whenever tend sufficient contrary bound conversely state td ty show interior origin interior hull interior accord separate unit criterion constructive visually interior check solve attain condition accord happen hull existence origin segment attain reference converge unique minimum cover objective algorithmic isolated point program result summarize function minimize cluster constraint regard possesse solution tend dimensional experience al offer evidence algorithm work well high separation quasi newton surprisingly crucial treating programming extend even program inequality special constrain quadratic algorithm slow quasi newton acceleration dramatically acceleration matrix program sufficient convergence mm mm nature absence theoretical descent make technique diagnostic behavior programming fill stroke department box usa derive programming generic geometric inequality hyperplane simple converge boundary one interior occur easily quadratic simple branch optimization geometric programming chemical equilibrium mechanic circuit maximum process finite power may subject geometric type program geometric program computational hard program convex method contrast many recently unconstraine programming imagine geometric connection arithmetic derive new generic device mm possess advantage unify operate reduce minimization problem advantage ease compete geometric generalization introduction rest review mm derive program mm penalty section principle surrogate around iterate constitute second minimize denote minimizer fact f g remarkable stability strictly speak art mm choice fortunately invoke inequality coefficient geometric number mi scope mi handle support hyperplane z multiplication negative coefficient surrogate additive attain minimization surrogate function derivative equal x mi integer rational function solve accomplished root positivity constraint transform iy ix second mi convex possess argument continuously differentiable implicit stationary determine consequently worth mi support apply put back position already detail composition demonstrate iterate appear objective x x x x x intend initial value relative pre f mm mention furthermore constant slow way accelerate publish quasi newton acceleration reduce necessary iterate function acceleration fall convergence criterion condition require worth parallel parallel graphic processing hardware device separate min f p iterate leave mm condition low mm condition extend algorithm programming parameter develop couple x penalty depth relax positivity enforce achieve multiply side example constraint x become method constraint whenever toy choice x course power fractional achieve status minimize usually decrease iteration mm exposition far manner generalize result parameter positive converge summarize procedure traditionally minimize unconstrained unfortunately suffer error instability mm involved iterate property conditioning cause previously mention acceleration largely penaltie
indicator metropolis candidate allow near graphical sense advantage posterior indicator innovation time use ordinary model stage accomplish run processing usual way burn iteration store posterior distribution easily auxiliary k post compute way model posterior iterate value jacobian transformation rao method improvement need chain indicator form transition normalize chain discard burn code reversible relative likelihood assign prior model visit start discard burn estimate posterior bayes close agreement sample chain estimate steady base fitting logistic return length indicator five model il distribution logit standardized element htbp part choose leave specification update store fit particular prior prior coefficient case jacobian model repeat draw fit length discard burn indicator tune visit appear rapid half chain bf prior give lead steady simple jacobian identity matrix factor assign move p cc u appropriate gibb within full conjugacy constant indicator categorical chain rapid chain datum straight forward exact solution appeal bayes equivalently model barrier implementation post offer partial argue elsewhere choice efficient thought simplify must offer construct investigation benefit consider view determine representation efficient posterior direct distribution simplification move move pair involve space induce default construction lead inform weighting monte carlo ordinary implement use mix remove fitting model bayes example reversible equivalently involve model provide reversible jump chain output technique bayes factor generate joint indicator common instead criterion usually outline alternative alternate categorical relate careful recover invertible g k km km jk k accommodate
iteration square iteration row exploit highly lar implementation use qr efficiently row work recall strictly great qr compute dual solution qr factorization idea behind fairly straightforward complicated require least square point start fully regularize path toward moderately one compute path terminate could large leave operation approximation quite accurate time coordinate leave furthermore path lar lar discuss section find problem reason lasso critical piecewise preferable efficiency discrete example another interest simplicity separable therefore coordinate descent various sparse wise coordinate dual connection lar motivation lasso path via find lar strictly full state lar return continuously decrease correlation residual refer lar path lars path path elegant dual dual leaving drop precisely compare path show path familiar show lar diabetes color enter vertical ignore dual leave lar equivalence lar ignore primal lar notice rearrange get column residual correlation residual appropriately absolute current maximal among absolute match realize prove lar set lar recall freedom interest fit difficult degree fit arbitrary corollary freedom section freedom degree freedom interest formula come stein unbiased almost differentiable state easier use stein checking check almost locally affine essentially take divergence treat easier express fit onto polytope projection onto contraction lasso already filter derive transform rank recall coefficient concern second interpretation come possibly outlier reasonable simultaneous perform give x nonzero coordinate fuse fuse fuse fuse group order number outlier vector lasso outlier count appropriate difficult visually may knot important criterion like bic problem employ assess estimate choosing obtain one critical solution residual check use simultaneously norm seem fuse single group freedom end cubic degree cubic degree freedom knot ahead cubic trend knot remarkable property search fuse knot outlier shrink penalty back toward lagrange towards less greedy fitting group speak fact fuse degree right simply think replace give solve combinatorial property entry reduce constrain discussion fit freedom subset expectation problem fuse trend special case develop path instead lagrange solution original continuous piecewise linear respect original sign furthermore dual degree perspective fundamentally tie fused reveal freedom underlie graph corollary present direction website several describe possibility fuse project simple wise may therefore iteration lead connect leave boundary geometry investigate could path lar lar ignore dual leave boundary modify forward stagewise limit follow lagrange attention optimization problem complementary interpretable novel insight acknowledgment suggestion consideration thank lar help make wide choice solve dual lasso greatly lar derive unbiased freedom generalize turn introduction seem mathematic engineering regression predictor omit intercept solve lar solve also early piecewise path tuning case change slope large path efficiency aside lar characterize insight notably lar establish freedom lar exception enforce instead pure coefficient regression depend typically geometric fact choice know lasso entire also prove freedom fit note author propose relate annealing follow begin motivate penalty fit transform regular need section lagrange serve sake clarity build section fuse lasso case add one loop design familiar outline path case path lasso lar lar refer lar probably version lar lar lasso turn easy procedure exactly unbiased utilizes vary interpretable trend section save detail defer document application wide exhaustive illustrative example motivate place split main often signal soft coordinate equally exist gives highly observe signal row geometry solution fit adaptively structural begin address complex fuse lasso position straight adjacent fit setting coordinate genomic number copy order genome copy believe nearby copy copy valuable mean development tumor take natural extension neighboring pixel noisy arrange vertical difference pixel technique actually special carry field science electrical engineering toy color color intermediate datum solution piecewise idea define adjacency edge hence graph fuse fuse adjacent figure application underlie edge graph node include state edge take map reflect low measure log dark reflect high red noisy infection exhibit solve underlie connect state share west mid west south east north east color among region get live north east west south east certainly state region infection reader fuse penalty typically norm penalty identity actually carry yet trend call fused technique give setting unknown recursively define piecewise show cubic equal piecewise cubic respectively choose fit spline knot adaptively polynomial estimation nonzero jump piecewise fuse regression spline spline operate knot substantial literature place implement represent global smoothness local signal trend potential property localization field largely wavelet wavelet smooth popular compression wavelet perhaps formulation smooth orthogonality many desirable generalized long quite answer approach attention traditionally center synthesis suggest outperformed fused penalty outcome image penalty fuse explain contiguous region brain trend filter engine response air ratio engine interaction linear intercept slope subject intercept fit observation lie bin slope cubic course low trend last solve fit engine cubic filter line interval bootstrap unlike previously majority come point might set outlier exactly coordinate convex relaxation transform lagrange let rewrite coefficient matrix outli fit circle read natural generalize transform lasso discuss invertible row lasso lasso except clear onto back generalize lasso lar author establish solution fall case fuse lasso wavelet several satisfy admit fall summarize next dual nice arbitrary penalty essentially problem compose rewrite q constant immediately nice constraint problem original unique constraint duality see write respectively relate last equation tell dual look complicated moreover look restrict attention derive path give relationship focus simply though begin study case use build algorithm general strictly row efficiently path stage u box origin box fuse turn state fused coordinate q proof connection lemma lemma state say coordinate general fuse correspondence therefore lemma intend explain rigorous argument correctness section describe solution path apparent detail path piecewise slope boundary stay eliminate consideration know path move boundary maintain contain currently sign boundary interior pt start direction next complicated define index index use subscript index paragraph precede treat order list consistent unconstraine clearly sign lemma optimization every objective square estimate interior boundary solve call maximum respectively study property rigorous argument consider simple interpret path initially path primal picture path piecewise linear change slope piecewise obvious algorithm continuity also general primal coordinate path dual version primal path become fuse picture suggest primal coordinate coordinate fuse complicate dual prescribe primal prescribed lemma path check fuse dominant unfortunately fuse lasso boundary develop strategy path checking check boundary interior coordinate path boundary define leave path boundary idea call coordinate one leave maintain sign start compute next leave coordinate leave fuse detail technical view tucker optimality kkt subgradient function overview condition satisfy leaving happen interior coordinate continue interior propose boundary give time yield q boundary next leave boundary examine express leave th boundary coordinate first leave next iteration critical leaving happen verify visit kkt derivation leave property terminate return continuity hand subtle continuity appears recover transformation path continuous linear nontrivial nan solution slope help solution boundary coordinate know fit onto space several reason use along geometric argument prove primal slope turn slope achieve primal correspondence value coordinate various interpretation edge assume without otherwise recall set two
represent intensity pixel expensive one optimal case like plane still far perform bound behave metric learn build metric compare shape point set surface utilize key distance use interpret reproduce kernel hilbert rkhs moreover become tool compare point uncertain provide kernel distance similarity spatial uncertainty object claim kp q setting k algorithmic main fast set run take general reduce convenient representation point distance various surface tool map dimensional preserve believe map popular machine learning attention technical relate standard space unit associate curve tangent vector surface form identical measure replace kp kp type kp fp consider reproduce compactly map reproduce kp two kp p product function simplify approximate ignore far apart tail kp q xx kernel surface weight introduce error input result formal kernel curve surface correspond computation I measure imply algorithms approximation normalize additive normalize diameter orient surface point illustrate surface decompose orthogonal axis kp dp kp dp specify without unit present simple technique reduce continuous discrete grid shape point g correctness alternatively random weight weight work technique remainder assume weighted sum put observation yield ab b b pt point size ia ib ib ib weight contribution edge pair lemma pair w I ignore desired describe p shape reduce computation calculation immediately yield algorithm shape euclidean algorithm result allow kernel computation rkhs dimension unfortunately dimensionality approximate directly dimension directly projection shift fast gauss gaussian produce probability diameter ambient essentially implicit work shift fouri harmonic probability g hoeffding last follow union pair q manifold diameter building domain tail net adapt trick replace recall guarantee work operate domain fast improvement gauss brief full taylor sort come sum error degree dimension require map approximate advantage dependence lemma lemma problem apply euclidean combine feature near preprocessing set extract specifically binary denote total family subset p want notion value kp kp k vc notion fundamentally tie range directly combinatorial dimension subset every cardinality show give way relate space random heavily optimize construction complexity tie vc range realize cardinality show range space show construct thorough run exist exist construction algorithm create size recently provide algorithm method central prove algorithm claim approach context ball technique matching packing vc range size attain sample beyond show range align x th sample kernel instance variate kernel skew link variate gaussian super level set sp error approximate approximately time fail property scheme error sort super sort count simplicity assume differently negative value possibly sort sort place begin place since super empty construct partition reverse clean empty consecutive point sort point reverse empty place least third derive super level present negative p kp ks eq gaussian covariance query convenience weight deterministic weighted create unweighted one construct link kp sampling dependence appendix optimize subset algorithmic theorem link n notice link extra work gaussian trivial standard kernel transformation rotation vector r origin preserve translation rotation minimize rotation tr depend thus kp optimal translation adapt rotation parameter translation begin analytic pp kp kp n dp g g p imply guarantee translation lemma must close need kp kp pair know kp q w pair point produce runtime correct q find k describe translation rotation align origin perform rotation origin location choose rotation ignore extra modification ensure give pair q ss r p rotation reader possible construct
search yield constructive pattern number evaluation rand dt dt ds si ds ds ds constructive search iteratively add name respect randomly feasible computed control amount randomness value correspond greedy construction nonzero global optimum asymptotically begin iteration adopt feature selection save prove pattern great confidence level equal system validate particular achieve obtain accuracy conclude real lc system fisher class set label firstly method relational optimal lead adopting use search na I classifier performance real class relational discover label sequence firstly efficacy strongly sequence task solve adopt local embed I world fisher reasoning fundamental intelligence indeed may lot contexts day science sequential application video planning biology modelling speech recognition etc form structure different problem pattern mine firstly capture investigate assign however environment mine extended multi relational mining involve lead exploitation powerful formalism classify adopt adopt language markov hmms simple relational able description contrary paper tackle discriminant learn value form learner obtained adopt mining frequent pattern use boolean strongly represent carry task optimum however search perform obtain solution explore filter select select adopt preprocesse heuristic intrinsic characteristic ignore learner paper name public relational non I subset construct power compute na I hence probabilistic outline discuss present section follow experimentally mining lot mining effort design face restrict pattern involve predicate early domain highlight bioinformatics represent domain effort extend predicate involve category first logic frequent boolean tackle relational probabilistic combine author construction firstly construct adopt language discriminant feature logical combine search algorithm refinement logic framework operator sequence one indicate atom base language correlate tackle account combine approach combine relational description logical relational field design sequence sequence flat alphabet logical develop generative promise logical successively exploit handle flat symbol structured probabilistic finally extension alternative hmms dependency discriminative relational learn relational briefly report relational take account formalism mining implement probabilistic classifier construction capability selection approach logic review symbol symbol symbol atomic symbol say atomic x occur clause constant define substitution applicable obtain multi relational mining report briefly recall order atom separate consider atom sequence express dimensional relation dimension may atom dimension event represent dimension indicate closure step th direct dimensional prove operator appear multi relational clause ground concept iff identity object identity term represent object step system construction obtain mining frequent make lattice order search start level lattice lattice evaluate frequency generation phase monotonicity pattern pattern frequent none approach start step try discard frequent storing refined pattern discard phase basically operator pattern knowledge clause satisfied pattern particular type language predicate formulate bind variable must specify specify relational class whose label use pattern refinement pattern atom pattern add atom pattern dimensional add atom exist start length pattern equal atom refinement step final pattern classify unseen space relational single relational predict build component otherwise discriminant function discriminant involve discrete value conditionally binary define independent yes pattern expect th estimate count relevance determine assume conditional write
property appearance claim surely neither degenerate I q use z condition use nb n gaussians power gm h matrix use function get q assumption I notation ic q cauchy normal conditionally assume moment lemma bx nh albeit note induction w enough inverse eqs provide random var latter small eigenvalue hand induction r vector limit converge apply eigenvalue limit lemma definition tm lemma eq hand prove proof side combination converge hand notation induction hypothesis term h modify hypothesis r hold prove analogously last cr ts q deduce limit lemma r almost enough finally note definition contribution last lemma point since use induction hypothesis surely assumption corollary write distribution surely finite drop pseudo bound degenerate follow hold surely exist constant almost surely double grateful support fellowship nsf present heuristic amp standard message stress proof reader intuitive understand amp connection belief propagation rewrite great reader side message exclude guess na I correction produce vanish z I substitute negligible writing notice type expand term obtain analogously eliminate analogously q equation section theorem taylor next cn e generalize mean lead condition therefore ne ne always use truncation weakly bound since v v ki ki v bl functions l kb recall lipschitz full absolutely lebesgue nx pz compact thesis fix matrix complement formula vanish constant constant lebesgue vanishing law imply exist constant z thesis case constant independent gram schmidt construct orthonormal define schmidt thesis follow plus minus plus plus lemma claim rgb rgb rgb rgb rgb reconstruct incoherent linear numerical dynamic term rigorous foundation indeed hold asymptotically gaussian message class sparse fundamentally different standard short underlie context spin theory compress reconstruction vector small noise reconstruct start convention succeed cf normalized way nf ia finding class formalism evolution claim tradeoff amp se coincide appropriate stress e amp boundary determine polytope reference finding argument paper proving se hold system limit matrix prediction random prominent pass proceed argue approximate pass limit appendix argument amp rule derivation necessary help reader familiar pass develop intuition important analysis evolution sequence sparse graph locally graph algorithm indeed complete bipartite evolution regard sense analogue evolution dense sake graph instance study discuss proof theorem symmetric mention propagation bp gaussian refer generally none exist rigorous bp seem remarkable odd standard evolution work limitation standard literature begin miss show relaxed let everywhere differentiable sequence exist shall constant type explicitly constant properly adjust sense index weakly x nk kt surely finding eqs find finite principle theorem ix z n rely assumption particular hold instance broad heuristic argument clearly insensitive detail entry online supplement suggest state evolution even broad domain open description since involve bring play modification copy correspondingly vector eliminate follow dimension write bt recursion matrix iteration develop intuition central easy entry converge exercise eq entry together evolution draw independently across argument fall dependent dependency negligible system iteration several notably call object great standard optimization application study behavior dramatically system simply hand term lead consequence evolution fact keep evolution rely exploit density decode exact tool threshold construction threshold major locally like special case david study assignment speak increase neighborhood converge bipartite like nevertheless admit rather subgraph bipartite concrete derive case indeed eliminate argument outline argument sophisticated lemma tree impossible graph fact local limit bipartite graph vertex node root simple spin assignment prove context hundred publish replica user detection performance coincide rigorous foundation along concrete principle replica insight specific replica method compress rigorous limited statement replica provide sharp predict multiplicative determination constant practical application group base replica foundation work mind apply predict iterative algorithm generally evolution warm namely evolution scalar estimator lipschitz amp read theorem variance minimize square result recall mmse compute u random motivation mse predict case asymptotically mmse mmse law wishart calculation mmse variance asymptotic evolution result already derive e non vanishing assume converge weakly soft non nevertheless square choice nonlinearity indeed substantial g threshold sparse soft prove state evolution deduce yield sparse amp coincide popular argument optimality evolution way improve class idea choose construct mmse distribution mmse simply phenomenon model code division signature frequently theoretical giving signature I random independent justify signature far naturally square algorithm read state evolution recursion signature dense signature mmse expression method recursion deduce mmse receiver point instance apply conditioning technique develop spin relate idea simple recursion reference conditioning convenient somewhat namely I consider scalar evolution first argument minimum generally problem algebra technique avoid conditional instead ts conditioning event gm h gm equivalent linear play random linear projection formulae p appropriate gm g new I tm refer explicitly control law number array phenomenon vanish large term recursion claim describe amp regard sequence recall lipschitz everywhere differentiable derivative context abuse analogous sequence vector fix obtain f th derivative finite dimension via lipschitz function vector whose empirical bound nx order amp amp define equivalent mathematically recursion track current recursion coordinate similarly therefore I value calculate basic characterize conditional distribution write projection onto column define projection column r coefficient cf go transpose product measurable word trivial algebra surely consideration large represent subscript similarly property variance let define recursion column orthogonal eq lipschitz exist variable
category c ssc trivial effectiveness corrupt world analyze stock greatly company financial discuss lrr interesting albeit stock category stock exchange center world stock divide york stock exchange center category generally stock category basic consider stock category subspace stock stock categorization stock label historical price stock global exchange market divide exchange category stock within top rank one category price york price obtain unfortunately historical price stock interface stock price accumulate stock stock stock span trend price state previously sharp drop stock strategy stock price average stock interval plot normalization adopt pca reduce dimension stock pca pca market htp ccccc markets ssc lrr sc stock market method statistic sparse make perform well market raw uncertainty bottom sub stock category stock mark stock experimental sufficient class categorization impose datum way learn essential corrupted mm algorithm convert prove practical sc model give rule achieve motion promise stock contain many uncertainty phenomenon solve multiple bit consume lrr especially recommend learn rank direction method component principal sum low affinity consensus robust cluster subspace discussion property strictly hold claim infinity accordingly sequence every bound exist convergent subsequence also convergent rely decrease get since h theorem liu member recover play role various processing name sum recovery traditional directly utilize reasonable enhance intrinsic although propose effectively mm function iteratively surrogate fall algorithm successive apply typical robust principal rank lrr principal lrr apply motion segmentation stock rank log well sum heuristic compressive structure network efficient rank e typical approach handle preliminary sense nuclear lead structure community consider learn formulate world corruption recover ill pose address optimize optimization vector little use exact intractable approach make try minimize envelope via convex norm heuristic regard practical powerful learning complicate corrupt dense promise paper approximation heuristic enhance recovery mainly conduct main representation generalize advantage solve surrogate taylor expansion next replace upper paradigm reweighte accordingly possible prove framework finally converge point paradigm adapt application use compare principle pursuit handle many rank tolerance rate propose extend verify removal face video second task power sc goal performance video test besides highlight robustness stock historical record art improvement include three fold sparsity solve reweighte stationary point extend stock demonstrate structure powerful area vision remainder paper review introduce discuss address low subspace segmentation discuss conclude part work recovery lrr reweighte recovery consider low matrix real world device sensor due low topic simulation introduce reweighted improve performance corrupt low representation tool desire general original correlation residual lrr regard exist lrr much robust public inspire lrr recovery paradigm effective solve optimize therefore mm fall field mm tv compressive reweighted lead practical portfolio management processing reweighte prove work reweighte approach work reweighte semi could relative alternate due possible capability method heuristic theoretical prove approach state basic impossible solve usually intractable make alternative one try replace envelope nuclear norm via accordingly relaxation definite programming define summation essence verify point replace norm formulate therefore stand convex constraint envelope concave indicate norm ask whether might log sum signal prominent term indicate close approximation therefore encourage sparsity heuristic formulation obviously differ norm use log instead typical powerful enhance unfortunately cause convexity function concave case non define replace hard one maximization fashion repeat iterative construct mm algebra taylor e ij ij mm convert reweighted call denote update construct surrogate besides summation nuclear solve adapt two part corruption pursuit derivation matrix formulate reweighte weight place nuclear impossible nuclear inspire add equality solve single objective close proximal alternating since alm relax instead lagrange multiplier e accordingly problem alternate direction convergence work minimize minimization update rule minimization unique close eq value immediate solution summation address gradient discussion reweighte mn ij z k kt recover low precede comparison robust simulation low rank independent simplicity rank rate comparison lin inexact solver inexact solver indicate highlight effectiveness record cost second cccc frank times frank e obviously exactly high table provide htp basic low vary variable feasible feasible verification accuracy repeat time median recovery regard much get conclusion make fit boundary apparent could difficult task fails experimentally verify conduct respectively coordinate count report part occur assign converge need resource accelerate criterion apparent four step recovery recovery second phenomenon advantage reweighte powerful conduct real suggest stack face subject conduct extend respectively face accumulate rank htp original face b face faces fig dense face image effectiveness apparent leave dense highlight significant visual face recover face recovery correspond remove use evaluation video list fig htp normalize frame convert benchmark video use theoretical recovery limitation divide recover truth remove corrupt three video besides apparent keep dense sequence illumination make isolated noise part foreground keep many advanced concern without mixture comparison evaluation false foreground correspond machine
line outside normal decide available negligible computational hypothesis large subset accept likely subset visit processing model reasonable amount several integral joint probit probit represent independent iy latent volatility volatility variable thousand period considerable complexity simulate similarly nuisance inference evolutionary mechanism capture capture mark capture informative description chapter open episode associate involve unobserved indicator capture experiment mark subsequent second individual stage use improper prior n q summing create indeed summation dataset example chapter relate european cover year observation france capture completion require near class label large corresponding unobserved usual denote delta motivation full conditional constant closed call summation number respectively generic approach posterior procedure go name bayesian functions formal monte carlo effort posterior effort grow square assessment central stem integral integrating include effort independent importance n return sampling formal leading infinitely way surprisingly choice wide generality include fundamental poor choice lead use stress curse dimensionality contrary numerical carlo always parameter space derive satisfactory importance become difficult get satisfactory function dimension probit formally approach poor equal ml estimate ml estimate approximation specific probit posterior importance former weight much concentrated weight miss constant histogram simulation setup diabetes exclude integration constant compute replace normalize version respectively normalize converge constant normalised weight posterior need weight large close unlikely closeness relate importance give effective q completely importance weight evaluate iid direct comparison sampler probit size primarily resample sir derive distribution use weight importance technique careful break recent literature extension adaptively monte evolve either sequential importance population connection early literature principle criterion simulate behave population give technique dependence remain unbiased iteration recent discussion therein sampling importance resample major past simulate currently iteration decrease effort generate n choice open convenient proposal update abc produce approximation population purpose optimal multiple mixture already test bayes factor monte rapid decade prior nan hypothese proper nuisance carlo elementary approximation consist simulation indeed n consistent define importance support two importance estimate importance significant importance approach n come respectively apply upper lead performance method connection se since use approximation bridge posterior quite since common bridge thus embed bridge density equivalence obtain clearly obviously performance depend pseudo establish asymptotically marginal harmonic estimator generic harmonic markov generation core exploration weight lead representation include hence dimensional gibbs short introduction mcmc simulation monte alternative differ issue output distribute target distribution irrelevant quickly happen fail converge prescribe iteration result biased markov correlate iid metropolis truly offer universal simulate markov explore proposal q internal irreducible respect distribution associate visit whole distribution converge propose sometimes reject relate accept reject generate q simplify else sometimes accept diabetes walk mle iteration mle also chain figure behaviour algorithm describe rhs visible rhs half reject lead acceptance rate walk diabetes leave graph first component chain straightforward hasting recover standard strong limitation attractive algorithm present generally strength sampler break simulate successively dimensional systematic scan effort tt return hierarchical multidimensional note probit base variable sampler aim indeed conditional depend produce produces take estimate walk hasting figure mix superior metropolis hasting gibbs diabetes mix performance local classic hybrid replace non metropolis hasting hybrid solution probit depend proper alternatively simulation metropolis step move validate random find trial dataset contour last satisfactory surface posterior probit mcmc source mean good therefore difficulty hasting include proposal reasonable visit scale stay local several modal modal region jump sample mixture multimodal secondary spurious stem run metropolis prevent markov mode compare choice mode evolution walk hasting chain mixture hard determination curse dimensionality increasingly part intuition modal weak complex impossible proper hard increase unless difficulty tune I adaptive difficult convergence discrepancy chain line empirical acceptance reach like inherently modal lead scale scale produce iteration adaptive mcmc right preserve ergodicity implement g create independence preserve path proper ridge image valid basic ergodicity adaptive monte coupling construction weak large mcmc tune adjust logarithmic result well algorithm impossible density produce situation latent analytic impossible handle additional joint cause mcmc illustration inverse give equation case computation population wide rejection sampling effort I return free eq level p summary tolerance hasting sampler computing effort rejection get generate perform sequential alternative method abc carlo key issue simple subproblem algorithm begin simulate density leave abc mcmc black diabetes illustration probit describe abc density tune bound simulate euclidean predictive mle therefore avoid predictive good fit incomplete biased choice model interest rather design anneal parametric process choice available benefit conversely area method challenge partly la paris mc acknowledgement universit case associate paris email chapter chapter aspect approximate procedure recent carlo approximate considerably choice method algorithm computation choice coherence inferential etc perspective function prior construct usually involve optimisation implicit branch concern early laplace probability development year approximation chain sequential monte extent statistic make nature object involve challenge community opt solution numerical indeed specific bayes simulation essentially computer law large major build entirely upon approximation advance area chapter discuss connection
divide normalize check forecast insensitive testing evaluation figure original aggregated wind clear wind power change low variability spike wind power first short forecast hour ahead model often wind change throughout year day period give fit study cycle play cycle wind due air wind sometimes train help decide exclude cycle aggregated wind series wind nonnegative unconditional aggregated peak normal common transformation wind include root demonstrate wind datum logistic transformation wind approximate individual wind gaussian generate density forecast wind aim rely monte carlo approach iterate easily reason forecast transformation wind figure result approximately transform first difference transform volatility small autocorrelation qr past consider qr select minimizing criterion maximize likelihood ahead forecast obtain step forecast gaussian standard move density aggregate wind jacobian ty z h ahead conditional forecast second wind however ahead forecast handle ahead ahead ty conditional ty innovation dynamic evolve ahead density require stand function give conditional horizon approach stand function variance model additive may violate value final forecast reality ahead approximated function characterize scale parameter simply step forecast depend estimation conditional proxy model gaussian well good density normalize aggregated wind truncate successfully model nonnegative parameterize truncation true simultaneously method exponential successfully forecast volatility forecasting smoothing give framework ad forecasting forecast process simultaneous forecast corresponding process enable ahead forecast iterate exponential smoothing simple wind constant need estimate refer conditional parameter scale correspond smoothed give update accord step ahead iterate highly lag autocorrelation forecast forecast simple smoothing forecast case obviously conditional essentially distribution exponential trend trend stand ahead forecast identify forecast forecast forecast gaussian innovation ahead forecast ahead give estimate variance case explicit maximum nice property maximize ty continuous rank score much large amount slightly forecast minimize forecast efficient ahead density use next parameter scale smooth equip ahead forecast square wind smoothed series update parameter forecast due ahead forecast highly correlate include account forecast unfortunately extra negative approach smoothing negative allow logarithmic small equation forecast equation except take replace insensitive size forecast ahead forecast identify smoothing summarize smooth estimate logarithmic parameter describe smoothing apply write write update equation exponential variance unlike asset expect volatility distribution truncate due truncation distribution obtain transform forecast calculate forecast asymmetric wind always calculate forecast conditionally assume conditional expand taylor performance forecasting approach forecast test minute hour ahead horizon absolute mae rmse forecast mean ahead forecast forecast mae respectively ranking mae mae rmse forecast hour outperform forecast hour interestingly almost identically phenomenon smooth scale parameter logistic wind mae rmse explain investigate forecast capture change volatility forecast discuss aggregated wind forecast aggregated consider simple good power wind sense krige predictor spatial datum process context calculate spatial temporal covariance hour point forecast generate wind aggregate normalize divide aggregated aggregate performance course sophisticated interest one discuss persistence forecast forecast forecast continuous score show strictly score tool forecast forecast analyze forecast affect extreme order resolve calibration forecast density forecast ahead forecast cumulative indicator ahead forecast persistence forecast constant forecast forecast density forecast mean ranking mae rmse forecast forecast summarize contrast investigate conditional variance probability ahead coincide forecasting calculate th quantile deviation show indeed generate forecast calibrate indicate description volatility time note slope figure conservative large spread ahead conservative confident small calculate calibrate calibration reliable description forecast slope opposite valuable evaluate condition period capture propose confident density risk risk capture volatility variance essentially give realize variance large value times diagram show demonstrate step forecast model period two confident spread give moving realize early n right line deviation generate multi forecast wind generation normalize wind fit forecast show generate hour step truncate aggregated wind underlie exponential forecast forecast truncate although approach use truncate alternative forecast several first performance smoothing length set estimation model set take datum remain perform approach model forecast computationally forecast density transform forecasting problem choose forecast testing forecast summary general non particular forecast aggregate wind wind challenge reliable wind generation individual wind interesting nonlinear jump may long low wind individual power mass forecast wind power forecast portfolio wind location development include process truncate modify version site would importance system wind forecast aggregated power explore neighbor wind acknowledgment wind generation author taylor little associate suggestion quantitative finance engineering fellowship project ct multi forecast intensive wind intensive avoid transformation normalize approximately describe datum forecast iterate approach forecast govern forecast two underlie forecast truncate generate forecast power ahead generate forecast slightly approach computationally length attractive approach normalize wind forecast wind power wind wind store risk period wind wind wind drop power uncertainty wind addition wind accurate uncertainty wind generation penalty maximize wind speed forecasting increase research wind speed wind power forecast literature forecast year emphasis place quantify uncertainty forecast forecast performance forecast recursive quadrature significant nonparametric forecasts series monte carlo simulation approach intensive forecasting model wind wind power wind easily major power vary also difficult quantify power curve forecast wind power forecast ensemble forecast generate weather forecast day ahead approach wind forecasting focus direct modeling power difficulty lie wind
virtue ergodic think select ergodic process stationary ergodic weak ergodicity allow n n ergodic independently stationary ergodic put either tend respect sample length formulation close assume different gaussian hide markov one distribution fix model likelihood cluster clearly formulation problem homogeneity whether different cluster problem classification three three parametric I easy indeed ergodic sense stationary work asymptotically upper mixing cluster rather empirical distributional distributional real sigma probability space alphabet tuple could omit difference tuple triple etc formal definition useful tool various although summation minimal cluster assign cluster thus cluster point cluster unknown simply put together certain calculate base rate incorrect cluster quadratic general distributional replace distance theoretical preserve theoretical problem lead practical compression way combine consistency promise direction alphabet work consider extension multidimensional borel sigma volume b b kb kb kx nb tb ergodic occurrence distributional word volume count weight fine metric q three infinite distribution stationary ergodic idea analogously grow converge cumulative frequency converge lb jj lx l vx un prove statement symbol different unknown stationary ergodic partitioning partitioning target set number know asymptotically require sample ergodic perhaps assumption finitely cluster select statement loop calculate iteration need pairwise calculation computational apart computation precise algorithm grow infinity replace partition estimate sequence consistent integer go observe follow asymptotically consistent statement sequence number infinity computational l hand proposition burden precision cluster term however control far know advance cluster assumption stationary unknown number impossible two distribution impossible decide weak consistency come unknown make expectation rather model assumption unknown extension multidimensional range mix coefficient formulation informally stationary way assume stay non call many process irreducible markov finite coefficient reference therein input cluster far measure calculate pseudo obvious level select joint one parameter moreover bind provide consistent fix finite also joint partition output way finally mix coefficient process sum correct answer bind asymptotic term practically parameter multiplicative factor individual expectation take would frequencie expectation assumption namely follow speed decrease strong without define come advantage beyond stationarity ergodicity basic clustering initialization easy main establish mention suggest compression spirit direction concern optimal course rate length grow partially education research region national project theorem
result jt neuron represent change provide summarize specify relation encoding neural activity activity support extract activity bayesian fire profile evidence excellent neural calculate remain next focus highlight utilize pass restrict introduce assume linearly relate one another appropriate update minimize entropy respect weight implement specify without work capture evidence consistently pool support u national science ci e university conduct eqs fig fig neural network derive represent network dynamic require determine calculation source deal consistently inconsistent evidence neural distribute network neural code artificial ability quite numerous neural probabilistic either train interpretation explicitly strategy constructive explore representation neural encoding challenge alternate suitably broad end encode probabilistic allow calculation accurately usual activation formulation explore analysis original implement analog quantity investigate population extension develop interact neural population additionally stochastic boltzmann belief network machine neuron manner enable give connection neural architecture attempt encode readily approach match instead activation pass nonlinear depend encode impose neural network analog normally result value network usual form neural neural input transform activation summary procedure generate network procedure acyclic model arcs strength influence probability additionally take causality arc pointing cause causal belief arise cause effect opposite arc first upon graph separate node node comprise parent direct parent random decomposable denote decomposition neural network match desire organize bayesian facilitate probabilistic utilize naive wherein assume place network accurately encode rather take firing rate mean describe neuron random activation work temporal encode neural state determine interpret transfer function rule describe activation time fire recover relation decode minimize difference density
truly permutation partition observation handle limited conjugate distribution case normal pick mean scale possibly obviously open discussion without consequence prior fast acknowledgement chapter book follow take mathematical trust statistical support participant contribution technology exact analysis parametric mixture conjugate relevance limitation complete impossible conduct datum complex conjugate use keyword prior family poisson mixture reader want beginning mostly understand automatically datum structure crucial simulation advantage also former available relevant reference due build upon thus denote scalar product vector purpose chapter exponential multinomial representation natural function literature variable observation one end family extend prior prior conjugate complete amount conjugate prior different dirichlet exponential since fix simplex conjugate prior q family case poisson mixture conjugate gamma posterior gamma observation posterior normal gamma indeed allocate component square difference convention derivation correspond complete development obtain posterior consider expand complete configuration sum except process multinomial truly observe upon missing conjugate component compute availability estimate sense switch posterior since available obviously discriminative effort summation partition allow form distribution posterior equal hyperparameter spike identifiability proceed fairly purpose illustration sensible correspond important feature term act like rather nu ji jj nu ji ji update ii update correspond column add last plot shape seq le type seq type considerable posterior miss much much memory size pressure also observation show first table poisson make assess sample zero explain value pair dataset poisson miss storage comment distribution happen occurrence feature ccc sufficient posterior pair poisson term requirement turn dataset observation large since posterior marginal bottom marginal bottom remain compatible range completely component figure marginal remain nonetheless symmetric mixture prior dirichlet use note indeed consider also available constant allocated allocate sufficient poisson namely count particular book apply follow I
proportional conceptually relaxation far focus elegant devoted collect specify dynamic covariance consecutive ask condition support time recover make precise letting indeed keep indeed prove carefully control dependent time evolve discrete subsampling system model coincide uniformity reconstruction interpretation roughly regularize closely relate follow develop produce use corrupt successive configuration elementary sufficient significant gaussian reconstruct binary ref regularize logistic structural focus naive present guarantee discrete fix produce accurate scaling whereby interested slow degree freedom fast relevance address fix number dimension relate graphical paper mainly difficulty analogous high concrete check specific class vertex laplacian symmetric outside entry diagonal denote well negative semidefinite eigenvalue fit network connect strength result simple q least recover achieve take trajectory length polynomial logarithmic system determine section numerical synthetic confirm consider set draw independently random square observation row average leave depict success curve vs sample time trajectory equation success probability converge evolve portion row apart additive coincide indeed precise see model distribution determine lyapunov context refer notation shorthand discrete sign large logarithmic network particularly derive limit confirm intuition produce large fine limited correspond configuration namely finally equivalently x omit write hessian outline main dynamic state concentration lemma prove outline mention proof compact condition recover supplementary condition hold square sign continuous provide likelihood check configuration hold checking condition regard expectation respect stationary non proposition proposition supplementary material relate ij dd finally corollary follow subset proposition imply second guarantee obtaining impose proposition recall distinguish adopt couple supplementary large claim prof leave task continuous time couple slight abuse check initial continuous discrete define respective couple drive let follow standard motion fellowship nsf award nsf grant dms addition q relation take support kkt condition optimization proposition estimate introduce extra element solution easy lead represent row equal I represent zero everywhere one position represent one appear product eq n fact inequality proof bind prove let introduce block block way compute thing eq put together lead bind trajectory instant word mapping stationary vector symmetric bernstein denote continue inequality similar show respect instant assume consecutive state application us stationary statement write define eigenvalue x reasoning assume matrix ab b compute give statement theorem imply last eq probability want satisfied substituting must need far need q impose probability condition look hold notice allow simplify four restriction dominate actually let unit thus prove negative definite block b fact vertex visit generating function form walk move neighboring expression make theorem proposition electrical stanford stanford stanford stanford model dynamic associate freedom tackle learn regularize square set graph direct edge associate take evolve generality give current corrupt represent brownian motion entry pattern system stable
dynamic communication financial classic armed mab offer reward draw distribution unknown player choose total involve exploitation player face objective arm reward play reward statistic case know clear reward play mean essence arm policy sublinear growth achieve know slow effective minimum different reward finite liu logarithmic sublinear order heavy tailed reward evolve process successive play passive markovian reward liu ucb markovian armed classic mab state arm continue evolve play transition arm play accord arbitrary process centralize player decentralize multiple distribute centralized make player choose performance similarly loss compare markovian model arm evolution period markovian segment control bad experience switch potential steady frequency arm balance factor sequence exploitation structure propose exploitation carefully control epoch length player partition contiguous play arm player large e average reward play far grow geometrically tradeoff exploration balanced exploration cardinality epoch accurate rank achieve nontrivial system regret knowledge epoch order case propose reward classic mab weak classic mab markovian optimal stay term long large unfortunately markovian adopt weak regret know reward open discussion arm play exchange occur objective decentralize minimize growth arm perfectly among centralized scheduling perspective player arm evolve accord process play decentralized policy centralized scheduling regret among player global regret indeed know centralized scheduling inherent sharing unknown consider player liu adopt propose knowledge system available algorithm propose ucb arm determine obtain cycle cycle cycle classic mab policy deterministic epoch use offer discard cycle pilot arm reward choose pilot small pilot difficult pilot reward solve govern stochastically identical tractable semi exploit markovian show regret arbitrarily decentralize liu division fair share fair achieve reward decentralized player ucb basic idea sequence exploration exploitation carefully analysis also different furthermore decentralized require highly present develop non bayesian mab treat deterministic quantity value bayesian policy treat probabilistic knowledge prior past bellman mab solve policy classic mab mab markovian dynamic lagrangian certain index numerous example g armed bandit broad cognitive dynamic spectrum secondary search several primary user channel chain dynamic user channel sense subsequently maximize long without primary user secondary user paper apply environment specifically channel choose accordingly reward transmission throughput design policy dynamic problem capital select company year company evolve transition whether company vc long design know integer k rest organize establish logarithmic sec consider performance sec single centralized arm choose play arm amount reward define arm unknown markovian transition irreducible passive arm stationary arm permutation specifie play determined policy measure regret reward play stationary denote expectation order omit regret extension arm play learn markovian steady consecutive bad long enough balance factor illustrate policy partition exploitation epoch geometrically epoch epoch player far play epoch aim learn arm exploration make exploitation epoch accurate epoch every epoch play denote begin end epoch whether exploitation epoch determine fig epoch condition ensures spend exploration necessary frequent exploitation exploration epoch exploitation exploitation epoch statistic give h c exploration exploitation arm arm epoch htbp epoch complete decentralized fashion pre epoch player local arm fashion offset sharing player different rank arm play exploitation epoch detailed description decentralize policy divide fig player start exploration arm go spend epoch go step epoch large sample mean exploitation epoch length player play go epoch divide play th go decentralized logarithmic regret different arm mean policy sharing decentralize epoch achieve order require nontrivial achieve definition appendix detail show global pre agreement eliminate policy join system exploration exploitation epoch difference exploitation play arm player play exploitation random arm efficient sharing among player pre note epoch arm reward affect reward consequence logarithmic exploration epoch logarithmic establish logarithmic absence synchronization agreement exploitation epoch sharing decentralize without global agreement achieve logarithmic reward occur hold access communication condition transmission observe problem reflect reward corrupt rank achieve pre open compare context secondary search assume consist independent state evolve secondary select power channel channel use choose fig short period seem parameter sufficient logarithmic fig logarithmic regret monte run space consider state state arm reward arm make probabilitie arm find appendix avoid pilot state well fig use cycle learn pilot discard armed bandit centralize decentralize develop deterministic geometrically achieve order decentralize appendix rewrite definition show cause arm cause effect lemma irreducible space state time denote reward exist continue play cause switching goes cause logarithmic bind number number exploration epoch player exploration epoch spend epoch bound time spend exploitation epoch total spend spend spend epoch play arm exploration epoch spend epoch denote starting th epoch denote large sample arm arm mistake let condition ii contiguous growth length see loss generality derivation every realization notice occurrence state segment bind chernoff reversible space transition probability pa l arrive show regret cause arm exploitation epoch become arrive point markovian problem require evolve process condition consider multiple epoch segment detail carefully length exploitation epoch b theorem fix three part cause arm cause bad playing exploitation epoch upper bounded cause arm switching arm cause play bad arm cause play bad epoch order caused play bad epoch spend arm lt mt reasoning upper bind cause epoch appendix proof rewrite regret term cause cause cause epoch next spend arm time spend bad arm epoch play bad arm bind bad exploitation combine arrive bind bound follow constant logarithmic term exploitation epoch still consequently cause arm epoch play arm player play arm mistake upper cause exploitation epoch regret player player reward th player player
belong view prop er device center immediate consequence follow expansion claim easily follow tend event since assumption establish complete instrumental proving let let twice partially minimizer minimizer exist otherwise nothing prove twice partially taylor positive obtain eq note normality suppose minimizer belong I satisfied nonnegative specify coincide assumption current noting prove achieve apply neighbourhood positive definite appendix tend proposition subset go rest reasoning tend estimator respectively minimize eq note existence ii tending event coincide precede instrumental go expansion require property unable implicit consequence proof likely direct score normality weak density also alternative normality weak growth condition specify quite comparable difference iii generating mechanism iii p schwarz proof everywhere continuity prove iii trivial belong leave element part implicit suffice latter ii px prop p dp px interior radius interior depend p f nx ax hx derivative compact assumption consequently nx p h proposition follow statement pointwise iv vi vi arbitrary subsequence exist converge converge converge keep mind follow implie imply iv I f consequently follow choice bounded elementary follow central v l cover complete proposition generate close ball separable compact hence p k immediately parameterization view appendix norm empty borel every empty subset additionally empty bound v kx f df nf clearly borel follow make argument analogously sup view proposition appendix convergence parametrization continuity kf subsequent van empty function consider constant satisfied value tr every maximizer j sr k combine imply n k cr inequality outer depend present van constant assumption coordinate projection space proof unnecessary I denote empirical sup f real note integrable hypothesis l c p event entire hold consequence density sup dominate mod inclusion satisfied mod view remark sequence converge compact assumption imply complete proof nd p nd part mod density sup part ball inclusion imply fact sup hold event part mod twice continuously partially differentiable continuously p p note involve mod assumption formulae integral continuity dominate inclusion satisfied integral appear mod lemma large td follow continuity separability assertion precede true probability interpret thm axiom thm thm conjecture thm thm exercise thm thm thm summary em university indirect estimator minimum model parametric maximum density correctly furthermore asymptotic parameter estimator independent I random law correctly identifiable likelihood likelihood method expression density equation imply complicated dynamic concrete alternative well method also efficient simulate equation define simulate respectively specify example maximum estimate objective function auxiliary model model auxiliary simulated indirect estimator consistent asymptotically regularity condition indirect inference estimator efficient sense certainly long suggest asymptotically idea amount choose restrictive maximum likelihood span limit informative auxiliary present long indeed indirect asymptotically fisher information variance simulate show normality originally explicitly certainly possible comment estimator high admit spline asymptotic normality asymptotic efficiency correctly analyze simulated remark indirect topic proof use fisher define indirect q indirect simulated sense extension classical introduce additional proof establish careful weak uniform obtain establish result also derive rate outline existence derive zero estimator indirect estimator establish originally parametric efficient fisher proof technical result f banach real value equip sup empty subset banach space empty measurable associate borel lebesgue measure shall furthermore field banach value equip sup shall thus logarithm space measurable measurable mapping say integral furthermore say weakly borel measure denote say value real write converge outer space fold n ny p r ny n open integer integer uniformly continuous hilbert b b next proof multiplication continuously embed embed compactly compactly empty fx totally minimal ball entropy space norm abuse let non give open I law sample furthermore family simulate synthetic density argument consequently variable value indicate introduction form also arise way application indirect section shall estimator obtain projection measurable remark denote auxiliary repeatedly summarize subsequent proposition proof find statement empty equivalent element iii singleton affine hyperplane endowed topology coincide tt topology exclude singleton natural shall requirement coincide stress proposition norm sup apart maintain assumption frequent representative strict prop interior state interior strict assumption strict inequality interior happen g map mod equivalent e equivalent interior specify density assumption respective mod trivially hold already appendix equivalent ii behaviour imply simulation continuous mechanism satisfy mechanism relate could conversely mechanism principle prefer work assumption assumption introduce auxiliary indirect eq k p kp px view logarithm value give x satisfy similarly estimator element kt eq existence uniqueness uniform central go first lower equal extend consistency norm norm prove consistency estimator even extend appropriate stochastic cf k q kt satisfied view b non empty show radius singleton hilbert subset maintain continuity claim coincide property impossible view np x px p qx px gx px qx ix I apply make proposition potentially argument uniqueness fix lemma compact assumption property appendix mapping nx x v k turn theorem already contain less assumption apply however norm q define restriction value make maximizer c maximizer give theorem already let satisfied k restrict number real law b fix arbitrary event converge element definition maximizer lt go function pointwise integrable monotone conclude equal lp hence analogously part eq q argument assume k k dp k view proposition remark contradiction assumption maximizer monotonicity term k appendix recall h k p l c appendix h p conclude follow theorem tend satisfied tend iii assumption satisfy event idea van norm give analogous generating cf subsequent estimator convergence avoid restriction n trivial result choose remark estimator since kt lemma p ks instrumental assumption subsequent mod assumption mod inclusion establish borel verify k condition satisfy taylor td convexity verify empty denote logarithm imply sup consequently proposition b get display establish condition condition satisfied rate complete recall interpolation inequality mod strict iii coincide follow case assumption theoretic consequence let process result analogous hold h appendix n coincide assumption also k borel measurable kx ff space borel evaluation totally appendix f kx coincide apply establish limit process p part largely mean classical sum true fr turn empirical gaussian limit negligible term ingredient establish uniform previous apart classical major usual together eventually imply maximizer zero relative r conclude maximizer save norm either reasoning necessarily small provide lemma simplify lemma parallel empty h g may hold w differentiable prop conclude k repeatedly prop k conclude last prof contain note make suppose mod subset index bound path process separable range uniformly metric give let empty subtracting use prop lead k k subtract p gd gd every expression supremum display supremum display proposition expression observe p repeatedly prop put thing every real nonempty denote nonempty nonempty subset bound property every cauchy theorem next class view borel nonempty subset already l proof event tend estimator f weakly brownian measurable value function f index k latter property follow correspond bridge isometry view theorem proposition b obtain ff definition space measurable analogue lipschitz outer integral denote lipschitz hypothesis separable brownian bridge denote fix theorem separable constant h display brownian bridge index theorem describe weak discussion respective projection empirical canonical appendix study indirect auxiliary estimator define eq well value separability belong respectively jointly measurable measurable
knowledge indexing map likelihood note invertible equivalently remove correspondingly rewrite normalization calculate use law ps ps uniform distribution calculation agent know perform calculation calculate next estimator agent probability increasingly require place belief al neighboring belief estimator span previously see dimension possible occur converge belief slightly subtle argument convergence diameter current incorporate network option challenge calculation remain tractable leave standard martingale argument area study mechanism neighbor neighbor sufficient calculate even explore feasibility paper certainly convergence happen fast take propagate network star simulation lead believe sample diameter memory span conjecture converge minimal thank network thank al discussion follow helpful suggestion write result section corollary remark theorem open model social computationally rational calculation resource large result extend preserve computational feasibility agent social network member year measurement independently acquire piece deviation belief estimate update converge belief additionally iterative network efficient calculation carry mathematical behavioral economic human utility action world take may social take characterize rational intractable usually rational network clique field receive weak state aggregate majority high enough correct recovery go go known view attempt extension relationship connect goal finding de node perform computationally lead original leave desired maximize unclear hard rational average signal apparent proportional total connection network network matter large constant away setup chain individual set tie model somewhat limited realistic agent act prior network converge receive potentially unbounded furthermore fully rational carry agent begin agent completely aim utility round require know network study paradigm setup show belief agent converge possibility action generality model infinite belief private iteratively pick learn neighbor result prove rational calculation tractable optimal signal finally take place twice agent diameter thank attention briefly place appear main contrast computation theoretically dimension code agent requirement agent structure social justify large network calculation principle derive mean discussion follow variant preserve efficient conclusion part agent state world behavior model network tie social agent chain neighbor neighbor simple certain measurement normal deviation initially belief regard update posterior piece belief prior improper learn improper distribution pick assume current belief optimal want pick carry observe neighbor distribution base formally agent action next observe neighbor know agent involve rather memory computation agent calculation simple algebra involve matrix belief converge access private furthermore converge
gradient scalar de familiar calculus complex diagram vector calculus also combinatorial analog vector de complex triangle edge laplacian analog laplacian finite calculus tool computer graphic engineering mathematic differential form vector decompose potential vector although potential first free homology finite really four idea book subspace slightly prefer terminology algebra refer operator presence always dot identify vector away slightly use middle splitting two subspace orthogonal presence ingredient get splitting come subspace fine piece fundamental idea elementary unique orthogonal obvious complement b ta inner precise write equality identify dual space space early pairwise equation content drop theorems square minimize gradient equation converse sufficient book classical rank positive definite nontrivial kernel function minimize dot ax latter short square square ne tucker equation saddle calculus algebra analogous term purely harmonic existence state least normal tucker easy harmonic harmonic orthogonality also second formulation ease write square square normal notice contrast chain minimize use difference interesting rank fundamental chain relate datum vertex rank part loop clique cell loop length four capture harmonic square component value straight involve vertex orient inconsistent triangle choose part triangle also harmonic would square loop connect equation solver recent explore homology clique arise enyi homology homology extension homology homology harmonic pure globally bound edge enyi homology almost type omit phrase restrict homology relevant enyi homology homology homology et clique almost satisfy column serve new look numerically try mathematical mention apparent bound infinity homology region suppose trivial homology tool ask number measure measure harmonic stacking yield desire find value count number alternative evident experiment picture presence homology transition nonzero homology region homology region graph trend visible appear peak mark vertical line quantification homology investigate much contain homology nontrivial harmonic result column right experiment topology clique scale free graph develop graph plot enyi area heart usually different experiment accuracy system solver square graph experimental special star free perhaps study enyi probability phenomena cluster see shape distribution life graph reaction capture generative process accumulate experience study rank underlie come iterative solver worth variety iterative rectangular aggregation aggregation especially attractive ignore could kernel direct solver nontrivial handle vertex pressure remark rank normal symmetric solve method et optimality however reliable method et available need dominant graph clique six et solve due lack algebraic restriction mesh carry adjacency aggregating strength connection define way vertex belong aggregation center algebraic smoothed enyi random various solver table formulation sa aggregation indicate another variation table homology fail setup mark specify double symbol sparse partitioning list setup problem solution algebraic approach hybrid combine algebraic direct problem direct solve partition dense system hope small form use system approximate inverse guess ia I ix ir solve dense inverse successively refine equation description dense technique sort vertex scheme like reverse approximate order sort general sorting scheme likewise least none yield consider matrix feasible hence report rank enyi smoothed aggregation solver rank two graph result amongst conjugate well perform optimally certain differential would algebraic sometimes poorly algebraic exclude setup solving setup worse last could even reasonably table algebraic take second phase second algebraic experiment expert find contrast box serial currently mean million require computation open explore method result approximate marginally slow alone table system method size triangle detail storage requirement store storage figure visual representation displayed apparent little locality visible mesh er mesh locality whereas researcher formulate rank study present involve take elementary highlight square rank graph area calculus algebraic absence calculus geometry arise information vertex project aim student relate hard comparable elementary state formulate system involve involve decomposition harmonic manifold problem differential equation metric case require matrix star calculus discrete calculus analogous study require stability graph simple code use experimental formulate information field lead algebraic since successful equation poorly graph poisson equation involve operator lack partial equation solver normal low square triangle area small piece minimal one need method method problem create connection solver dominant reliable software implement comparison various surface exploit study surface assignment rank part dms rich wang h harmonic trend graph although comparison section figure gradient estimate laplacian ratio eigenvalue certain condition label enyi graph count obtain bind measure obtain label variety enyi graph cycle star graph graph marker plot next table r homology trivial absolute error enyi graph three experiment algebraic table homology fail phase mark symbol reach number mark double symbol mark symbol dense partition detail enyi graph graph setup phase algebraic table require phase enyi table show coarse grid algebraic useful fail phase enyi graph next nine pattern laplacian four top vertex number reverse label column show result visual figure example middle way ccc er er er er ba ccc ba ba ba ba scale ba ba graph edu http www cs edu rank comparison rank alternative value comparison come impossible residual really remarkable square graph reach area spectral graph laplacian clique connection explore paper easy fundamental elementary algebra aim basic connection topic numerical method work small algebraic generally count calculus theory chain item use lead square way precise usual laplacian square actor study study numerical analysis square find gradient however different field domain conceptual algorithmic implication give direction area seem rank ranking ranking use ranking opinion allow quantity rank formulation link link keyword focus rank independently interest comparison al calculus decomposition motivate rank different ease graph calculus al working calculus albeit never mind modern physics vector calculus manifold calculus make computer graphic partial equation computation try reduce simplify algebra suitable serial result result never appear many experiment need theorem yet numerical hope reader implication exposition aim idea aim mathematic community broad imply stem experience calculus skew tensor object associated orient object simple study discrete calculus display book chapter idea book solely numerical algebra aspect iterative far solver al ranking experiment find time write work performance quite problem reason square laplacian solver mention square rank graph type eigenvector manifold extensively geometry laplacian spectra well operator propose application understanding solver hope payoff go potential paragraph eigenvector detail numerical topology random clique clique augment subgraph consider establish connectivity naturally homology clique lead clique reproduce graph free something analyze elsewhere establish maintain list field development rank graph diverse development need easy old code differential equation away examine problem formulation rank real value pairwise represent comparison opposite skew al oriented precisely pairwise opposite orientation multiple version strength task translate find always exactly vertex impossible sign add procedure close possible whose difference reproduce part edge represent technical sense consistent loop sense rank still inconsistent square particular second equivalent formulate tensor follow introduce pose ranking method differential equation emphasize square graph inconsistent fit equation think al projection viewpoint address web graph theory alternative preference concerned quantification strength reasonable internet win high preferred pairwise exist winner hand square rank rank alternative vote scheme least rule edge weight arrive alternative acyclic graph edge weight originally acyclic produce graph square account remove satisfying requirement ranking future require accuracy rank treat random give mean variance strength variance rule rule account outcome alternative sort mean via interaction least globally optimal pairwise vote give connection ranking connection random propose ranking alternative rank proportional strength preference frobenius eigenvector subsequent rank strength amongst square ranking finds optimally require easy right side create preference preference analyze rank perturbation perturbation particular rank method ordinal perturbation rank page rank preferred ranking social concerned order alternative relative strength completion matrix expensive connection calculus quickly recall square problem describe rank basic idea basic terminology concept need calculus detail call simplex say refer dimension simplex refer abstract equivalence even permutation fall one orientation orientation orient complex orient weighted oriented abstract edge vertex graph augment clique abstract dimension clique yield clique triangle orient triangle refer clique loop valid complex reader may simplex change real orient change orientation algebraic topology integer chain since chain particular
polynomial statistical release release small lower bind corollary fact inefficient additive proceed multiplicative check update omit description remark proof query release call learn implication make prove concept preserve privacy release many point section present need concentration explicitly variable rx dx bounding function strong concentration independently dimension concentrate around crucial submodular satisfy self concentration generalize monotone submodular monotone satisfy string whenever ix roughly element weight rather probability hand string prove work keyword fact conjecture pt know answer query statistical ask query agnostic answer question concern small fraction answer submodular function graph interesting point main application efficiently differentially private answer progress problem preserve datum unconditional differentially potentially statistical query implement hence barrier differentially private element correspond privacy enable rich individual like rigorous privacy outcome nearly indistinguishable set differ sometimes contingency fraction individual open efficiently private datum encode answer output differentially private boolean query privacy imagine analyst wish cut subset cut small number constant complexity give fundamentally polynomial guarantee boolean permit implementation query statement preserve release theory put aside privacy pose question sufficient show operate conjunction apply submodular independent concern implication characterize compute privacy secondly unconditional query release permit regardless privacy knowledge class almost date include recently multiplicative mechanism statistical query make query yet implement statistical summarize develop query release expressive simple class conceptual result privacy theorem submodular submodular runtime fx give differentially private release boolean differential differentially private differentially private fraction may large fraction however strong work bound mechanism euclidean privacy preserve literature ability conjunction release agnostic learning query release exist learn release statistical moreover release make statistical agnostic preserve query complexity neither reduction preserve runtime equivalence characterization factor answer class ability make differentially thus answer simply vice versa release informally submodular submodular satisfie submodular recent concentration self imply query additive submodular correspond query database characterization release multiplicative recently maintain universe query observation datum set much possible query relative provide submodular style learn monotone continuous submodular distribution spirit inspire introduce potentially lipschitz constant submodular function studied introduce gave centralize pac value function class preserve class mechanism universe ask class give theoretic privacy agnostic characterized al conditional mechanism release al query inefficient run universe g universe latter mechanism set encoding mechanism answer mechanism query hardness small know computational set useful unconditional bind monotone interact release monotone lower depend representation almost private hardness private information query bound run differentially private particular average conjunction size recently arbitrarily conjunction query early show give interactive private release mechanism analyst slowly privacy ask make intuitively characterization release implement interactive mechanism statistical query analyst ask step induce agnostic answer preserve privacy database interested privacy privacy differential pair database rd rd query abuse two adjacent database laplacian distribution universe ask query oracle parameter differential distribution statistical early differentially private new example denote algorithm output privacy answer laplacian set predicate release datum interest release statistical query fs concept consider influence reject get claim demonstrate claim suppose must converse suppose sake minimal ps jx imply claim observe mapping property oracle submodular tolerance set function tolerance map property compute tolerance proof oracle always however use q differ establish detail call modification reader step proof size show every vb bs proof claim observe f vb u uv decide whether select inclusion analogous ps also manner place element establish product distribution subset universe product remain collection return bs approximate oracle submodular apply error exact additional lemma equivalent state corollary plugging vb claim follow monotone refined structure replace lemma strong guarantee monotone submodular submodular prove theorem oracle query collection return oracle mapping respect gb vb function guarantee vb c b cs compute item submodular b vb g cs directly bc g item analogous construct appropriate follow analogous already clear mapping tolerance tolerance invoke complete algorithm subset universe sample collection mapping vb tb cs notational throughout detail estimate estimate sufficiently number submodular product oracle tolerance let query tolerance oracle query apply submodular state plug cs combine submodular answer approximate submodular like appendix problem database release result previous monotone describe submodular indeed monotone section element submodular monotone universe element natural corollary differentially boolean completeness product though rely query answer collection mapping vb bs b corollary directly monotone boolean conjunction last release release replace monotone width distribution concentration statement denote uniform string follow lemma throughout less monotone conjunction width thus approximation monotone omit monotone presentation graph discussion multiple edge sensitivity observe cut count database submodular decomposition construct release preserve privacy modification lemma decomposition preserve show query agnostic specific release consider find concept p reduction query release agnostic weak take value agnostic concept satisfying p strong sense agnostic section agnostic sufficient release algorithm agnostic exist distribution query return query correspond proceed induce eq strategy uniform short distribution concept distinguish multiplicative
signal easily however suffer correlation correlation great strength priori assumption impose dataset restriction correlation causal result trivial I restriction correlation since correlate remove correlation find description toy singular assume consider observation time mean datum correlation information correlation spectrum statistically lie low spectrum represent reject exclude dimensionality spurious correlation however uncorrelated variable eigenvalue include find singular spectrum represent theoretical eigenvalue density theory q empirical suggest nontrivial correlation construction check whether describe index publish indicator variable economic indicator explanatory period I standardize carefully optimal internal correlation uncorrelated h cc uncorrelated create rectangular calculate compare result singular benchmark high energy exchange cc eps whether necessary proceed present path necessary conclusion future example correlation redundant significant svd truly correlation investigation stock
countable finitely lebesgue follow make finitely finite finitely vc figure process theory geometry probability inequality law large uniform insight immediate corollary vc vc approximation analogous finite extend vc join consist intersection empty vc dimension large boolean every dual vc introduce vc family use proof collection subset vc vc prove begin specifically boolean boolean pointwise subsequence finite boolean stage splitting mean weak incorporate subsequent stage boolean van private intersection construction critical space essential factor topological section devote family number deduce vc uniform family empirical process envelope f f countable function measurable particular say f need minimal element notion separability regard indicator vc routine argument countable generality cell remove denote follow relation c b c l fx suppose envelope integer finite level define f mx fx let family f uniform law large frequency limit probability give class establish vc class large uniform atomic probability vc appear number x stationary ergodic almost tend infinity corollary number f find complete law vc ergodic early version simple assumption argument follow outline improvement anonymous standard vc countable elementary every countable fail sequence stage wise splitting stage arbitrarily join collection vc boundary great set functional measurable I h mh adopt stage subsequent stage splitting wish nn nc kn kn kn j kn proceeding stage define note fix splitting positive suppose generality eq elementary claim induction note fix cell ensure cell imply cell join join necessary cell imply eq equal inequality hold every let join lemma vc dimension inspection measurable positive define clearly vc equal anonymous comment lead simple general acknowledge helpful van early paper present nsf set corollary family dimension number satisfy law corollary vc major vc space measurable combinatorial complexity say every
bc specify shall explain intuition detail process network actor actor actor community element e bc j z hierarchy likewise receiver use interaction scenario worth hierarchy compatibility hierarchy child position receiver appropriate receiver position place sale theory explicitly interaction community arbitrary severe parameterization decide thorough position share parent element infer path child hierarchy infinitely even guess appropriate bayesian chinese restaurant short extension chinese restaurant recursively integer shall level actor crp draw define hierarchy crp crp top unique top level associate child crp result crp ad newly prior tree crp priors crp share concentration finish describe generative actor display complete model side path imply infinite infinitely compatibility integrate path exact intractable integrate condition I z beta conjugacy value shorthand bc j iterate expectation conditioning actor stick break depth domain ij actor actor actor interact time explore compatibility element actor tend interact within element actor tend interact community element high noise actor actor interact analogously hierarchical denote spectral unclear cluster let pick likelihood calculate tp actor depth false count false total average show quantitative result illustrate result diagonal figure result edge hierarchy actor show illustrate result branch branch branch size correct cluster branch strongly diagonal little level branch imply actor interact community perform good figure figure essentially actor poorly model actor specific rich network web choice reflect two mode see web reflect social could hierarchy branch different organization chart subgraph actor parameter log log likelihood subgraph hold subgraph require tuning parameter experiment likelihood result either dataset model likelihood superior notably role extremely recover complexity rich blockmodel candidate author interpret world early community hierarchy actor via grid importance optimal parameter take lag time iteration sample posterior path generate pair actor share actor share position consensus simply post visual merge bottom actor sampler link augment interaction miss annotation shape specie contain outli community community web species community nd range large separate least six either specie community specie super level identify occur fine grain interaction specie community community level interaction interaction link g community subgraph physics citation construct subsampling paper involve parameter post require hour processor core run sampler burn take sample lag iteration complete hour processor infer community annotate paper frequent title actor learn solely citation sub citation imply paper topic keyword field physics sub community super focused keyword string concept reflect community dense separate specific researcher narrow topic community super surface deeply involve infer community level mixed actor bottom interact community edge head color assume interaction solid represent annotated membership stochastic blockmodel model multiple hierarchical membership actor interaction simultaneously recover mixed membership hierarchy discover role membership expressive non compatibility simulation dataset real recover intuitive membership organization collapse sampler efficiently scale dataset around actor complete processor support nsf gm fellowship technology school science le song school pa school computer science university pa actors realistic play one interact social community analyze interestingly organization mixed blockmodel generation social community membership actor automatically discover gibbs algorithm conduct synthetic real network citation network self organization arise information actor actor formation particularly interested network actor exist actor role single role interact actor social actor play influence actor interact mixed blockmodel specific nature actor structure social network formalism flat role actor structure actor beyond simple simultaneously finance department european american branch european might business orient membership european finance department normally business american finance european branch company american european resource european member specific american motivate yet infer visualize semantic actor link actor hierarchy send day hierarchical membership membership community actor form model structure implication link sequel begin specify branch next relax nest restaurant process direct graph actor adopt convention else edge actor receiver latent variable actor membership short record actor level depth hierarchy actor branch root
non component ac sa choice stochastic however gradient descent incorporate technique nesterov replace closed like fuse solve accelerate apply stochastic word deterministic gradient totally prove convergence combine smoothing gradient determine algorithm independently iteration choose ny hx aspect exact replace length maintain sequence point x ac sa unconstrained ac sa hard depend performance solve true convergence algorithm follow total fx gx technical lemma present algorithm inequality technical characterize q classical see subdifferential x z give follow third one imply updating equation convexity lemma l z fy fy hx z z z z z cauchy schwarz accord side come q imply n n n x convergence sa projection solve similar appear particular fuse lasso iteration could application modify utilize construct smooth control approximation well proved function eq therefore gx tv tv show constant convexity parameter smooth apply parameter get good modify iteration smooth lipschitz z ny tv step algorithm mapping non term close solve small return near fortunately appear dominate observation result find descent algorithm propose apply smoothing dominate obtain word price incorporate technique stochastic develop dominating lipschitz term incorporate strongly stochastic different regularize ac sa algorithm propose matlab implementation run cpu processor gb ram fit function occur equal chance square prove gradient iteration whole whose element interested input influence term force highly relevant regularize level choice eq q problem apply sg ac sa projection sg ac sa apply represent ta parameter generate e numerical represent cpu p n introduce al input subset set induce element group input inducing call apply algorithm sg ac case mapping sg long solution coordinate descent mapping adopt solve sg apply non formulation dual auxiliary associate let ball space sg ac mapping ac sa dataset iteration generate algorithms numerical result though sg ac sa performance sg ac sa reason ac sa length impact stochastic ac approximation sg influence nesterov reflect mapping close apply necessary green curve overlap sg sg sg mapping solution infinitely parameter numerical linear variable eq loss gradient discrete regularization stochastic set ac sa sg performance reflect figure converge fast ac belong minimize likelihood eq regularize choose norm lipschitz lipschitz regularize whole generate apply sg sa follow component probability three generate put decrease logistic ac sa solution projection sg ac rely subroutine smooth another gradient descent smoothing term convergence scalability proposition
finally expression limit variance formula limit way obtain adjacent form term finitely many acknowledgment author thank I helpful concern associate anonymous remark lead considerable improvement manuscript remark la procedure usually either combine approximation however establish posteriori due dependency weighted estimator equation maximization composite strongly convergent consequence overall simulate exist procedure economic analyze analysis datum scientific social science communication molecular biology protein vast book study graph bernoulli edge homogeneous capture feature network lack heterogeneity lead introduction primarily science group characterize relation refer adjacency matrix indicator respect belong exhibit block block intra group connection block behaviour diagonal call structure parsimonious lot model community structure inter intra traffic network consider adjacency financial last concern dense weighted network integrate primary problem vertex cluster widely community analyze graph existence however rely mixture increase stochastic network beyond scope approximations intractable core hide factorize form variational strategy behavior support indeed major procedure variational local maxima posteriori open simple variational focus treating main either moment composite likelihood consist marginal dependency cox weighted show estimate rely binary graph mixture identifiable identifiable moment composite induced establish normality rate distribute graph issue edge order convergence group proportion convergence also limit variance limit result fact variance organize notation limit introduce specific binary first equation proportion rely random model rely composite theoretical dedicated section illustrate datum finally proof notation v sake consider generalization graph without loop mixture first follow use heterogeneity index general independent worth adjacent precisely belong shall assumption particular observe degeneracy specific lemma section later thus proportion motivate single follow weighted assume absolutely namely vast continuous mixture maximum variable usefulness rely whether expectation answer yes good identifiable variate parameter tuple estimator core k kp eq consistency asymptotic estimator asymptotically normal g group proportion us comment expression useful interval parameter simple expression state convergence happen equal prove variance meaning converge limit shall consistency preserve one give insight rate mixture instance model random binary random indicate presence absence node conditional latent bernoulli group attention connect whether belong intra inter bernoulli respect simply display example equation parameter idea moment correspond triplet proportion give rise multiple solution indeed never uniqueness partly identifiability apply proportion know algorithmic procedure proportion use shall triplet finitely completely finite moment three moment induce triplet three characterize triplet namely equation examine proportion phenomenon take recover rational obtain estimate introduce estimator specific theorem able proportion suppose converge surely almost combine shall parameter composite moment method develop preliminary identically mixture component five appear follow q zero soon equation define one relation constrain bernoulli mixture shall already permutation recover mixture identify mixture indeed identical different one appear consider quantity derive obviously accord uniqueness proportion unique let criterion converge identifiability limit attain classical estimator normality define define comment mention model rely comprise graph value result assumption question slightly triplet distribution identify corresponding proportion already bernoulli unconstraine finite variate decade identifiable rigorous elaborate directly hard establish strongly believe simulation seem reasonable restrict parameter space greatly simplify restrictive generalization chapter parameter prove result consistent infinity moreover increase proportion comment result rate rate indicate might beyond group group solution consistent rate least always case approximate estimator assumption converge iterate grow infinity weighted observation either indicate specify parametric dirac dirac accounting sparsity parameter focus structure connectivity identifiability reason constrain parametric admit give parametric could density counting parameter namely poisson would sparsity density concern weight edge htbp weight binary value column truncate use group intra inter community fail find meaningful graph model never propose literature dirac enable proceed rely plug connectivity rely estimate rather section consistently estimate independent variable accord mixture p f directly soon identify mixture composite provide introduce need assumption identifiable range family next deal regularity quadratic twice continuously last compact express log stress quantity derive label issue estimate deal grow infinity least mainly consistency direct criterion assumption consistency note theorem seem indicate phenomenon degeneracy limit estimator composite procedure adapt thus provide implement issue recover graph follow range number triplet let st nu encoding hide triplet observation moreover state coordinate composite dimensional bernoulli distribution optimize iterative compute conditional use step maximizer triplet value constant observed aim recover procedure group plug previous section datum latent proportion simply remove datum simply call latter frequency introduce criterion estimator dependency section label estimator denote latent choose couple estimate weight propose use structure iterate procedure maxima value maxima estimate simply take frequency procedure summarize start z procedure analyze introduce value weight graph estimate section illustrate start triplet transform triplet estimate edge study examine bias proposal propose strategy implement assume distribute describe generate group number create different intra connectivity free proportion model picture three method estimate method triplet method propose result present simulation give density variance magnitude htbp computed group line true estimation three model produce enough asymptotically unbiased agreement bias among estimate number vertex equal group compare deviation simulation show vertex see deviation vertex relate similar converge deviation remain comparable estimation great dispersion rand evaluate agreement latent rand belong consider structure identical rand rand index three size allow well structure structure variational latent since analytical resolution structure miss distribute equal mahalanobis distance model different group proportion empirical deviation function average simulation c computation bias recover parameter bias c high deviation vertex vertice binary slope lying display perfect efficiently consider order economic non symmetric graph
q class reflect rewrite interpret give optimizer nonzero find propose singular globally convergent certain completion problem objective aware work generalize denote matrix satisfy frobenius define follow solved initialize arbitrary stop establish hyperplane w lemma satisfy e onto gradient local outline standard form prove nuclear problem incorporate form partial constructive pick define assume generality small outline kronecker product use replace basis main proof relate bind vc elementary two define subset basis define follow ir b obtain identity support grant fa nsf recommendation express material author view nsf assumption conjecture theorem remark edu alternate observation feature performance optimal hoeffding crucially contribution new extraction cast obtain optimization testing extraction decide favor either assumption uncertainty motivate behavior amount observation simplex hoeffding empirical leibl element kullback hoeffde nonnegative favor hoeffding characterized error exponent statistic summarize give hoeffding observation theorem derive imply drawback hoeffding exponent optimal size observation potentially address hoeffding name enjoy advantage design robust error knowledge knowledge heterogeneous incorrect optimization supremum ratio divergence restrict supremum special case hoeffde expect choose determine divergence exponent critical successful implementation us exponent come maximum different optimization provable convergence context framework paper family pca find estimator kl divergence improve upon motivated exploit limited possibility alternate kl divergence dimension might alternate divergence absolutely respect exist say given require basis fact distribution dimension family mutually absolutely continuous except obviously family characterize divergence neither absolutely continuous error ask pairwise need support parametrize call exponential
strong detection formation vector embed dimension wrong continue point vary strength direction grey te additive suffer obvious system large less measure regardless observe give discrimination perfect measure cause serious implication world presence investigate advantage te noise couple enhance add te allow level te two large sum make weakly drive first system delay absence regard dimension use environment matlab sample large strength future tend increase tend increase opposite direction balance take positive two coupling direction mask rank value direction couple system couple display free legend te value measure give fig b illustrate noise te variance discrimination coupling strength fig give note couple favor te turn sensitive sensitive presence rank te seem presence condition world result te rank useful coupling recently transfer entropy term symbolic te vector modification augment comprise reconstruct state consider show vector joint far suggest rank sample simulation couple assess receiver find couple varied three presence state system fix delay future summarize te mask direction less well te noisy dimension pass correctly justify work couple drive measure wrong measure te measure length strength provide term stable vector requirement future rank te embed length increase couple zero coupling measure couple maintain coupling lack te show large direction complex system complexity rank direction strength coupling discrimination higher particular te te estimate sum showed perform worse especially attribute estimate stable noisy te entropy vector journal system transfer te measure information perform setting define te reconstruct rank system regard version transfer rank propose ahead drive system assess te flow receiver operate roc form coupling realization couple state space observational show te particularly te ahead improve keyword couple causality transfer measure measure causality direction approach extend approach focus synchronization neighborhood reconstruct flow concentrate last measure entropy variant te operate symbolic entropy see comparative give te perform te application mask detail structure treat discretization te response scalar modify give te information mutual grain interact flow ahead regard financial index map well see interact correction measure interaction system reconstruction investigate one ahead measure briefly te compare te interact delay respectively te driving response system cause drive accounting response te shannon mass pmf cell variable appear entropy te estimation estimate directly occurrence dimensional reconstruction estimator demand estimator te near sum follow te discretization variable entropy approximate inter distance suitably estimate sum vector account euclidean symbolic transfer reconstruct rank assign small assign substituting rank vector combination cell combination vector occurrence scalar express rank setup entropy constitute sample horizon propose substitute transfer ahead entropy rank augment rank independently appear time capture time ahead reasoning use te direct te flow joint affect simulation favor te note lag rank vector distortion rank vector large g entropy term involve coupling measure use instead bias remove strength te simulated mean computation te point stable maintain neighborhood preserve distribution insufficient coupling measure bivariate time te good direction coupling coupling quantify discrimination information net assess discrimination set receiver operate characteristic roc coupling direction e g flow detect couple great fig te strength reach reach couple bivariate white te small white noise deviation small te strength
hold would time panel suppose within service panel job server service job service job leave change service job processor job start job leave impossible later job early conditional close unnormalized even compute require new service time affect computational conditional important consequence perspective natural arrival times times fact present severe fortunately though expect small long occur expect typical time therefore queue sufficient develop computationally efficient sampler note queue markovian service exponentially markovian queue chain except queue queue arrival incomplete section difficulty time service time inverting parameter either model em infeasible markov sampler sampling wish accomplish slice sampler leave invariant version iterate sampler sampling uniform slice update leave horizontal update spirit easy slice previous metropolis slice difficulty density normalize unnormalized job slice sampling refer times slice constant pointwise invert set service contain time number fortunately cost updating change factor product nu value mean value select sampler infeasible unnormalized compute change equation service job b e e old arrival job e b change immediately queue propagation service compute update service time change algorithm section relevant job set algorithm section propagation queue arrival queue main service depend previous processor time immediately previous job separate unnecessary queue job time update queue alg change hand none service recall arise job enter service queue penalty job queue become compute queue implement sort queue contain queue arrival compute dt finally time nu index propagation queue algorithm arrival algorithm service times ps function queue structure queue job ps function portion service search intersect implementation variation store initialize nonconvex latent service configuration make service belong cause value bad mix ed service finally service switching distribution time avoid generally processor subset unobserve initialization initialization proceed unobserve task path service time service service step service switching service prevent would service describe analyze propose experimental setup facebook develop web actual implement software profile twitter figure architecture common medium web service show incoming request first web server system image page request content comprise email change time user make request time computed request web server design run request copy store user ability external request purpose assign request copy copy obtain elsewhere run machine copy setup machine per load database involve machine series request generator total cause request range amazon ec utility request record handle request spend library time spend database cause database query bad high practical degradation variety evaluate prediction period average predict interval arrive queue ps networks service single queue arrival datum database baseline regression regression response maximum pt power law processor queue network ps reduction interestingly differently capacity important simple processor difference type poorly inspection suggest cause model outside datum g prediction poorly demonstrate reconstruct miss arrival database time spend measurement intrinsic distinction response add due intrinsic processing log minimize overhead service estimate well use estimate ps delay connection minimal service estimation miss observe axis time second display spend queue visually resemble increase quantitative measure bin bin system root mean square reconstruct service incomplete time infer subset baseline compose service denote job ms ps reconstruction baseline observe linear worse interestingly support limitation final although selection receive relatively attention context performance characteristic software often understand system build external software library whose internal fully code software hardware failure idea behavior pool behavior expect system bottleneck queue add couple consist arrival queue transition bottleneck queue unobserved basis goal miss queue choose determine lar lar aggregate covariate queue choose bottleneck assign second covariate coefficient bottleneck despite poorly relationship covariate response nested service time distribution gamma come arbitrarily close mass zero knowledge bottleneck queue choose augment whether service bottleneck queue statistic alternative selection likelihood demonstrate use bottleneck five ten alternative consider work bottleneck queue technique synthetic generate figure arrival homogeneous service bottleneck queue varied contain service distribution bayesian sampler iteration repeat independently c selection queue decision queue bottleneck queue queue lar well chance technique perfectly hard display roc generate r package curve lar perspective presence miss idea perspective demonstrate web fact network computer distribute application build perspective queue consider setting work focus single rather addition restrict restrict miss queue observe none arrival estimator inference algorithm markov theory algorithm indirect possibly situation single distribution example single processor queue place service arrival metric steady exist goal diagnostic serious distribution asymptotically unstable difficult modify incorporate sophisticated prior issue area network focus estimate delay network inferential link link include request current largely direction model service example run effect service imputation grain computer detailed web book google internet services yahoo amazon request day computer service operate strict great queue computer detailed request heavily bayesian arrival viewpoint viewpoint carlo benchmark web application model internet service google yahoo amazon serve facebook million user retrieve large scale web application thousand request request request request computation web operate strict minimize site web page return response delay cause business http com web concern response example ten web spike poor request handle slow request incorrectly distinguish performance hoc question regression response onto natural server single queue request allow incorporate two qualitative connectivity inherently incorporate maximum question way concern answer question request simply infer parameter arrival second request system much spend queue process request reach queue call request inferential service operate collect overhead overhead number request delay site receive million request per reason incomplete arrive request scheme minimize storage expense introduce inferential incomplete datum idea arrival arrival observation service enough advanced include service sharing arrival approximately monte arrival em present arrival complex arrival time queue necessarily arrival previous consider network model much require request share web service design architecture presentation generate web application storage often request typical service actually consist multiple request web internet randomly make request necessary generate turn request storage friend web page engine queue individual reflect system web service request request assign first necessary third response reality serve request ignore external request terminology request individual series cause request web service external request web typical would order probabilistic arrival job task queue task markov chain call reach potentially individual job time arrive queue job draw independently job queue amount queue job head processing service draw density jacobian vanish jacobian observe knot whenever job job knot write number queue job job precede jacobian triangular bring queue section network develop job queue job job queue job job task whole initial queue queue processor simply service queue notation define denote chain always specify base figure structure define queue structure arrival job service accord service parameter service service distribution solve system network system parameter service service job time use regime jacobian fortunately jacobian triangular time joint likelihood arise path queue likelihood work improper choice prior issue key deterministic service distinction service important statistically service I time dependency queue impose arrival assumption relax complex combination job nonempty queue job enter ps however distribution inferential also make posterior model
equality represent pair x f f argument g pt f f g f f g g vanish moreover f f conclude p x p x pt x x non expression set outcome iff condition need integrate acknowledgement like thank anonymous comment associate greatly improve von binary iv extension way association extension negative independence hold independence normalization take zero measure dependence well simple similarly simple ordinal monotone simulation study random order outcome continuous categorical continuous arise social science ordinal robustness cc b cccc popular variable replication range sign version ordinary expression interpretation firstly see observation see three specifically term reason score continuous aforementioned definition score e obtain implement carry modern computer ordinal test robust zero association alternative coefficient hoeffding function hoeffding henceforth see hoeffde severe drawback consistent page option suitable categorical square applicable suitable categorization chi account alternative test treat nominal rather ordinal although rather decide probabilistic notice complex simple natural call negative independence probability analogously simple independent coefficient main reason popularity pre computer value much hard google currently appear equally binary er von organization state interpretation theorem turn involved related hoeffding new give description independence permutation replication z absolute coefficient result paper distribution mass true equality proof give sign obtain show schwarz normalize exceed define analogously unfortunately possible set suppose distribute uniformly cumulative decrease chapter explore copula probability pair pair jointly pair jointly minus jointly separate pair ccccc convenient four two pair jointly four point notation point permutation logical shorthand straightforward z denote randomly sum two term equal last equal see four say neither tie likely variable monotone ordinal variable hoeffde probabilistic use leave ordinal coefficient simple drawback ordinal consistent characteristic correspond equality definition show hence definite negativity imply show kernel hilbert appear proof simple mathematical formulate square euclidean summation pearson chi square well onto contingency table suitable permutation test value permutation time consume moderately permutation compute th monte evaluation practically infeasible moderately approximated taking well permutation marginal distribution statistic independence usually conditional test b l independence simulation artificial table ordinal category visually monotonic yield chi give association hoeffding consistent ordinal possible like chi hoeffding use hoeffding compare chi yield use hoeffding independence expect alternative gain look value six box represent line maximize correlation function box represent square double respectively box minimize represent sense comparative furthermore choose hoeffding little poorly six marginal copula course give equally increment normal plot increment heavy automatically care use simulate normally distribute value increment increment normal average least hoeffding compare bad walk double box poor section lie zero see likely hoeffde necessarily positive tie bad walk increment significantly cross test monotone
order neighborhood pose stability indicator non cc behave point fold sparse result present problem particular important behave standard structural equation quadratic study clear relationship among difference variable cluster group relationship among clear residual advantage explanation adequate learn necessary nonparametric r lin gps lin fold fold fold fold subsample training pair sign rank test suggest behave mix dependent believe size sample also sparse alternative non decade review classic simple parametric construction interested relate factor misspecification conversely specify discuss factor non none nonparametric structure context graphical real problem often ignore bayesian mcmc work improve mix particularly relevant ordinal mix difficult bottleneck procedure pseudo least open determine graphical promising thank relevant concern detail draw interest number pseudo input latent hasting current variation non variable structural latent indicator mean mixture block latent mutually function pseudo function sample conditionally convention particular explicitly mention fix implicit respective implicit implementation use submatrix update library gamma marginalization done give posterior conditional gibbs parent correspond condition variance j dy j p coefficient constant analogous set conditional structural new value individually reject gaussian proposal center probability graph parent value shorthand point accord cost use submatrix cholesky latent take child parent include implicit drop factor mixture latent respective mean sampling latent gamma random b variable give b pz e ij observe choose aspect plausible child variable target differ application design explicitly account moreover implication distinguish among implication detailed discussion instead serve block spirit determine condition subproblem detect edge establish result exploit identifiability observable assume assume independent easy entail marginally marginal independence variable identifiable scale marginal without loss generality follow result independent estimate latent instrumental residual residual discard justification indeed error identifiable consequence however opposite connect path use technique error input input formulation formulation without pseudo input explicit dictionary space scalar country national product indicator freedom parent instance b dependence essential graphical structure independent acyclic dag simplify presentation likewise treat define cyclic exclude particular latent parent empty distribution parent terminology mixture gaussian measurement parent mutually independent coefficient example square observe circle emphasis direct model sparse study explore exception observe share share indicator less might lead many latent limited condition identifiability latent I identifiable structural identifiable deconvolution element term joint deconvolution extra describe identifiability joint regression regression equivalent x identifiability condition causal additional assumption discuss literature overcomplete brief appendix mcmc gaussian without discuss case imagine latent measure least equivalently mutually input mutual clique instead parent represent index represent variable fix value whole I n equation function sampling introduce representation adapt input reduce choose pseudo notation represent I I x I kk x kk k pseudo motivation alternative mean dd I dm k k conditionally imply context input model overfitte pseudo fact see gaussian rather motivation optimize pseudo variable next pseudo input avoid pseudo choice partially informative easily freedom choose fix structural imply parent cholesky pseudo may cause one way propose jointly cost pseudo metropolis probability latent variable mutually remain accept walk accept ix dx children graph correspond child c x function crucially cost briefly illustrate follow identifiability interpretable study predictive alternative independent gamma model five indicator fix edge intercept identifiable burn burn due prior coefficient inverse mixture gaussians significant intercept purpose observe value able reproduce linear functional relationship pseudo input cccc latent assume variable measurement represent however learn application allow functional analogous illustrate quadratic rotation enforce identifiability study student aim pay indicator variable simplicity latent dag correspond measurement identifiable fix intercept slope describe reference section identifiable dataset
affect affected atom empty slot attain eq write wish minimize yield substitute z j yield collect like j n j atom respect final solve applie notice nearest ise material mean material use determining acknowledge valuable material mm ise institute science technology materials university nj usa show I methodology reproduce solution drawback agent interact ise purpose I tool compare traditional field show case main regardless I approximation case otherwise least justify simple spin individual strong exchange effect material behave determine net difficult second atom net dimension dimension approximation computational difficulty shannon entropy information assigning method method entropy I probability family allow posterior drawback present way assign bayesian method deal expect reproduce purpose updating probability I I tool material lie traditional paper traditional third reproduce fourth traditional I mean atom act atom allow interact neighbor quantum suggest examine atomic spin spin suggest exchange exchange calculate interaction energy spin atom partition write easily strength net energy atom mean hamiltonian hamiltonian atom word average effective hamiltonian hamiltonian part use I justify material I use write canonical I moment must purpose arrive produce field act wish entropy canonical proceed free reduce q I good information provide entropy term I proceed I well find maximize constraint solution energy illustrative purpose atom pose continue similar except solution summation since z yield minimize respect well rewrite eq though atom behave like atom process ising spin examine possible atom
characteristic observer raw return place remove apply change use follow remain decompose table affect rotation random mark ten population guess accurate state everything perform transform component table normalize unity randomly result experiment time use separately length show experiment table test make sense keep mind observer object compare correctly classify difficult different vary proceeding chance albeit linearly observer unity dc band band train ground result identify time instead slightly moment transform mark x band band eight point repeat test show good signature follow transform preprocesse direct angle four moment skew add numerical percentage fold fold set scenario aspect angle object observer experiment enhance experiment result relate technique bayesian popular technique pattern conditional laboratory describe paper aspect increment overall performance bayesian approach process identification increment four vector give cross validation show statistically guess give feature easier transform bin nothing exclusive bayesian machine arrive accurate process result feature comprehensive classifying computation load extraction load main preprocesse extraction least classification operation especially applicable negative release result review proceed grateful thank mark identification signal space identification object datum good fusion band band svm number shape mean sized object little object great interest survival keep geometric shape object determine return cross section signature though geometry approach effective signature different geometric pattern recognition vast pattern signal signature widely recognition signature return representative generally one training separate class simulate shape accuracy percent precision standard run signature decision deal computational burden minimize maintain mention potentially fast enough approach advantage three require database label signature time train input signature label signature quickly weight poor testing evaluation sample pattern evaluation simply consist matrix multiplication input ten one multiplication ten element product ten element adequate application significant neural advantage preprocesse signature automate concerned recognition neural good reference network bayesian massive require technique spectral classification signature spectra computation often quite consume signature advantage nonlinear perhaps consequently global well secondly unlike accept database signature synthesis synthetic incorporate operating location upon intermediate versus object shape ghz band ghz band diameter height diameter edge feature except object sphere adapt optical eq q angle incidence solution derive singular superposition add circular correspond incidence aspect htb aspect b x classic formula utilize center key htb center geometry axis plane center origin axis phase set comprised object approximately broad observe half symmetry observer ghz simulating return object change perspective dynamic comprise object object observe static one five period class vary length result datum second set observer band ghz effectively simulate return ground observe object amount object length goal signature sensor fusion improve identification ready begin process support follow label signature label follow represent signature simplify signature else group depend signature output equation training relation hyperplane hyperplane separation call separate hyperplane problem basically subject constraint label give nc analogous bias compactly representation point weight programming perceptron know basically find support reference several svms basis rbf kernel tangent kernel calculation compute whether support train element phase support result support case object classify belong otherwise take one randomly remain test repeat statistical randomly remain weight train classifier sphere non time select object second test
differentiable gradient show efficient apply suffice proximal may alternatively adapt proximal polynomial slow purpose provably hard nevertheless dedicate cut plus modular min proximal done dedicate derive closed sort element beyond cut cardinality express easily maximize minimum orthogonal subspace greedy submodular synthetic cm vary unique go condition proximal obtain agglomerative example agglomerative cm path variable show prop submodular total thus additional extend extension proximal penalize fw soft minimizer proximal cm matlab replication proximal descent dedicated situation graph chain graph light red penalization level consider analysis design matrix already proximal make typically guarantee first face create see proof supplementary material constant absolutely respect lebesgue one minimizer set define partition lattice prop condition correct one recovery lattice constant minimizer probability well eq design meet interestingly validity imply proximal exactly prop non merging term cm situation increase show presence know issue limit face set intersection prop condition one start practice optimality lattice proximal lattice partition f belong face constant value imply soon extension optimum general maxima candidate prop simply sa fa ii thus satisfied lattice check bf merge lemma moreover proposition fc fc fc fc violate merging set satisfied condition condition equivalent true interval one eight depend possibility neighbor chain trivial fc fc v fc fc fc fc fc bind path agglomerative namely two neighbor get unique dual pointwise maxima distinct union proximal element show full fine grouping order correspond show distinct lebesgue lattice order relationship impose second meet j probability prop j lead bf lemma get fa si p uniquely symmetric fa k moreover j counter gray node complement show connect correct even cm true pt minus pt minus pt minus plus pt pt mm sparsity submodular focus function submodular extension envelope set index whose component lead knowledge level set support predictor unify proximal guarantee allow submodular function statistic cluster noisy cut detection variable solution zero either use match knowledge therein sparsity one lead solution image detection induce automatic design sparsity submodular predictor show similar parallel submodular contribution link extension see submodular specific depend exist norm cut application presence outlier unified operator efficient adapting slow unified predictor always subset indicator subset moreover define index small index denote relevant proof cm throughout power b fa b fa unless usual submodular negative section undirected extension function fw extension moreover indicator positively bf p sa one submodular analysis submodular greedy decrease fw bf g say know tight tight face intersection hyperplane prop say cut set connect cm submodular beyond convexity homogeneity submodular zero immediately invariant addition constant contrary decrease regularizer norm envelope certain combinatorial envelope extension envelope note envelope intersection level relaxation perhaps consider amenable effect next extreme induce cm note need variation fa fw extreme separable subset lattice always topological chain go construct example thus correspondence go back constant subset diagram inclusion order connect represent diagram immediate allow lattice variation ia ii v empty relative face modular I ii interpret certain submodular undirected interior face lattice belong cm section submodular extension cut variation cardinality base submodular ga mutual eigenvalue weight cut extension variation symmetric set justification behind variation allow prop lead one may application direct cut may increase jump along edge piecewise outlier node level good total variation base noisy set detection vs variable
non burn effective monte relative ratio px configuration number indicate rr r expand configuration rr estimate relative efficiency different summarize table sampler px usually inefficient roughly draw posterior table reflect introduce regression large predictor example top middle intuitively failure px offer presence conditionally direct fail conditional matter enter hierarchy autocorrelation conditionally independent indeed seem thing suppose restrict near behave large globally small may study expand big expand handle global parameter default difficult situation posterior mcmc precisely solve simply encourage well option heart conditionally handle slow autocorrelation either depend make poor marginalization family prior beta variance doubly nest sometimes remain area sigma sigma lambda sigma burn sigma save burn beta lambda beta sigma lambda sigma jeffreys lambda sigma sigma rate lambda comment shrinkage sigma lambda burn beta lambda sigma save sigma seed sigma sigma delta lambda p sigma burn matrix sigma save lambda beta sigma lambda sigma jeffreys lambda sigma sigma expansion g z sigma lambda lambda lambda shrinkage sigma delta lambda delta lambda burn burn burn lambda sigma save sigma consider mixture example prior usefulness global conclusion expansion type improvement less especially attempt component practice approach seem parameter matter perhaps view replacement expand keyword mcmc normal expansion simulation history autocorrelation expand expand sampler undirected graph right px indicate replicate many common equivalent parameter expand version identify improve mcmc convergence generalization expand generally equivalence hold cauchy excellent sake illustration despite fact individually identifiable model identical imply easy moreover translate yet mcmc drastically phenomenon widely exploit speed useful sampler particular simulate standard severe px code implement apparent arise fact independently posterior autocorrelation location exchangeable mean look interval exponential outside transform work slice step invertible invertible slice sampling expansion applicable conditionally conjugate prefer slice possible beta beta cauchy ratio two equivalently possible use conjugate note component omit compare px px sampler zero seem perform well default reasoning concentrate meanwhile half observation exhibit modeling goal goal logic order plot expand expand simulation autocorrelation expand
ask extra quasi error square simplicity consider ordinary ol term ol fit ol note choice use easy expectation replace version case close rate limit covariance limit population solution directional give may obtain subspace may lead variability difference latter q equivalently tx tx derivative try avoid estimated level level conditional self adjoint taylor expansion asymptotically tx classical equal level classical importantly tx tx j g tx us attain follow function nuisance approximation define estimator requirement efficient first side negligible certain positively correlate weighted purpose extra negligible tx tx e tx g negligible gain confusion appearance place asymptotically normal achieve fisher information g covariance nonparametric calculation justify interestingly word joint extra automatically extra estimation help achieve reduction compare side limit equal super efficiency root nuisance piece estimation estimate true regard expectation nuisance pseudo link estimation likelihood estimation motivate use nx density direction employ technical handle smoothing argument begin subsection intrinsic propose identifiable idea previous identifiable estimate quasi boundary surface directly parametrization popular parametrization remove yu wang parameter infinitely jacobian pe sp prove invertible inverse parametrization coordinate presentation otherwise th component jacobian matrix component replace brief conclusion clearly inverse see easy invertible whereas rest invertible either latter derivative involve estimator check sign another estimating define estimate solution step positive procedure estimate scalar apply appear limit tend go infinity achievable tend need identically smooth linear derivative sum fan bandwidth confusion classical quasi glm uncorrelated distribution phenomenon may benefit correlation predictor step towards glm exploration glm dispersion interest quasi replace initial estimator likelihood motivation easy similarly previous technical similar structure unknown nonparametric construct conditional estimator obtain classical know asymptotically efficient still nonparametric final equation simply derive case unknown compare simple parametrization variance attain minimum result parametrization issue glm nuisance regard link true automatically estimation positively fisher efficient regardless parametrization small property interest correlation small key extra show glm nuisance paper may extended handle advantage estimation cost expensive nonparametric possible instability quasi give technical compact derivative ii lipschitz condition vx bound vx nh vx derivative smoothing worth unlike use truncation second wang present difference one reference estimation sequence variable variable hold similar main conclusion replace respectively simplicity cauchy schwarz fact wang j nr condition lemma proof omit x I hx f I verify note n condition side imply zero cn separate write note pn pn pn tx tx expansion tx tx tx ie tx tx weak law number use denote derive vx vx iw nj tx g tx tx vx ni tx tx vx ni tx jx vx together imply conclude argument estimator estimating argument also proof theorem easy clear variance elementary presentation ti ta n proof result application distribution variance far j limit theorem imply p tv p last tv q tv derive covariance q vx vx write negative semidefinite definite thus v v v tv tv v v cm corollary china parameter introduce pseudo consider nuisance estimate importantly positively asymptotically fisher asymptotically small fisher fisher information necessarily stage scalar trace limit fisher quasi information extension parameter augmentation readily handle classification word investigate issue develop literature linear blue description gauss limit achievable generalized well asymptotically unbiased information measure variable carry unknown upon variable depend present criterion efficiency probable efficiency linear widely class regression analysis complete result glm dimension write unless suppose method weight square used method exponential variance covariance attain rao er effect linear estimating prove optimum property estimation matrix quasi benchmark comprehensive book rigorous optimality nevertheless seem parameter nuisance model perspective extra estimation large study stage base quasi pseudo nuisance parameter regard trace covariance small fisher particularly reduction relate correlation theorem section reference linear gauss provide distribute shall
particular case mle asymptotically asymptotically subsample problem suggest variation avoid subsampling step method multiscale subject classification accurate physical multiscale one apply coarse grain effectively efficient choose fitting particular coarse grain diffusion apart estimation multiscale coarse average multiscale estimation equation case grain fit grain equation combination example newton question dynamic grain drift process go step path system locally dynamic carefully simulation answer global dynamic possible statistical come full important available full issue horizon either fix time horizon issue separation dynamic scale brownian motion computation reach definite build intuition behavior general tackle goal dynamic address diffusion quadratic commonly go discuss rise review core coarse grain give free grained explicit computation outperform describe heuristic come multiscale describe limit multiscale allow discuss estimation multiscale affect algorithm limit equation completely come multiscale multiscale equation banach space call slow ergodic transpose set eq call average limit condition assume positive prove assumption basic type equation banach space call fast eq step x follow converge eq diffusion similar definite inner theorem replace interested slow dynamic reduce equation simulate much step dynamic simulate review multiscale slow small multiscale multiscale hold differential multiscale limit system case multiscale drift system might pose realistic coefficient precisely limiting depend linearly parameter extend normality come parameter interval relax explain multiscale slowly behave diffusion certain say diffusion demand certain simulate multiscale general answer coming evolve slowly see estimation interpolation concrete describe approximate local path multiscale another starting possibly lead multiscale see particular exactly prove would maximum equation since quadratic lead q fact reason choice square error small significantly investigate behavior intuition follow behave scale brownian length behave like process value go definition process finite total variation non process clearly total variation order behave behave limit real path go function maximize cardinality choose maximize maximize obvious proof satisfy z clearly zero I set equation fact supremum countable set countable give k e random finite section law grow prove estimate define properly study error need py expectation simplify already z z note explicitly p p dt theorem find behave thus finite need get know proceed write laplace behave laplace transform transform substitute back get inside operator put everything consequently variation error proceed moment pz w e come p p
control begin control intensive use euclidean hamming define distance minus mean case exhibit positive control exhibit snp quantify pathway derive pathway conceptually clear significant drawback snp analyze test drawback distance issue control employ allele frequency major minor frequency minor allele individual first term quantify different term quantify relative minor allele frequency centroid along close standardize across obtain quantify across consideration across provide mean whether consideration inspection numerator distance centroid sense centroid denominator distance across snps consistently close snps consideration assign control quantify relative distance systematically remove individual leave manner distribution hypothesis control close manner pathway select pathway remove needed control quantify control significance systematically step pathway identify pathway homogeneity control indicate play eqs magnitude influence difference distant centroid correlation penalization denominator application pathway yield influence pathway snps yet subtle pathway influence snps gene wish snp correlation gene rather end snp gene detect snp effect practice pathway pathway database contain annotation associate gene retrieve pathway exclude minor allele necessarily pathway gene snps select gene snp association snp significant snps significantly filter snp marker interest remove pathway snps yield another control case quantify computing rank amongst thus control close snps accumulate yield group contrast nan set control yield control figure snps snp pathway snps across minor snps relative snps none snps show association snp share pattern snps pathway comprise pathway snps snp neither snp directly truly evaluate snps gene pathway pathway distinction procedure informative snp pre pathway gene likely highly significant concern snp proxy toward pathway contain abundance gene account permutation describe select compute classification normalize distinction pathway effectively preserve correlation q permutation practice permutation snp snps pathway distinction score adjust quantify close high control pathway difference simply whether pathway permit distinction significance distinction pathway assess choose number present snp random pathway snp pathway snps pathway yield particular choose snps address pathway consideration permit separation control snp pathway expect chance author concern impact select snp proxy gene snps chance pathway appear snps kolmogorov assess whether pathway contain gene chance pathway conjecture pathway genomic biological retain pathway result report kolmogorov interpretability comprise breast match control participant participant european descent array snps genome smaller comprise sample snp array snp approximately one million result genomic marker comprised breast number control participant participant american array snps across comprise control population snp one million breast breast figure expect turn snp comprise snps rs rs rs distribution frequently control peak great obtain systematically pathway represent make use build pathway snps pathway snps snp pathway gene mean pathway distinction pathway pathway great score sample relative regard membership metric odd disease status odd adjust pathway adjustment significant ratio plot increase range odd allele snps closeness pathway carry first compute snp pathway remove pathway excess pathway greedy algorithm pathway order frequent output keep share pathway un supplementary pair pathway assess statistic one pathway correlation suggest biological playing process perturb novel cell connection element significant pathway breast cancer genetic difference mechanism disease pathway biological playing role process pathway observe cell majority pathway directly cell gap pathway activation addition associate breast cancer finding interestingly link breast breast sequence study breast risk functional could influence cancer modify production contrast wide variation contribute snps pathway four cancer contain gene associate significant randomly may reasonably ask pathway snp pathway resample base significant suggest difference kolmogorov whether pathway pathway large continue pathway pathway gene genomic relevant cancer pathway trivially cover snps unique keep snp interest pathway mean differential computed pathway pathway odd hypothesis fdr adjustment pathway breast cancer cover complete overlap pathway give examine remain pathway maximum pathway breast form various type include breast play protein several neural pathway drive cancer drive pathway genetic irrespective breast cancer many pathway cancer datum well know tumor associate far support mechanism contribute pathway breast cancer appearance interaction pathway suggest exist genetic irrespective disease site along breast pathway pathway notion contribute appearance pathway contain drive increase cancer driving pathway pathway overlap complement pathway il dependent il pathway nk interesting suggest activation play study research cell response connect human finding nuclear although human connection human appearance significant pathway suggest may variation contribute cancer pathway result observe even pathway odd unit typical approximately case pathway pathway overlap pathway snps take union sequentially pathway yield considerably combine significant illustration case comprise top pathway breast give finding spread mechanism rather single pathway distinction identify pathway closeness permit infer across snps snps gene test pathway breast cancer suggest disease dna find pathway link role complement snp unlike gene approach gene find significant subtle consistent marker permit pathway joint phenotype possible many pathway breast detect snp include highly pathway relevant role dna rna show pathway along statistically effectively closeness individual control conceptual relationship neighbor centroid classifier base derivative must goal genomic heterogeneous near centroid derive marker classify case control complementary application g f pathway significant snps drive near near neighbor pathway heterogeneity significance snps snp place snps likewise fashion importantly yield regard control snps benefit phenotype minor allele interact gene slight consistent eqs pathway geometrically situation centroid sample linearly separable exclusive two I one control plain yield pairwise permit interact snps package possible pathway identify cancer suggest generally study even phenotype different snp level pathway pathway examine pathway pathway set conduct run statistic analogous pathway cancer alternatively pathway pathway rich pathway one pathway potential use pathway snps case pathway advantage rely marker genomic cause disease would snp biological effect rich system level availability request future acknowledgment research institute md cancer fellowship cancer institute national md like helpful simulate snp status minor red yield fold snp pathway comprise snps pathway htb snps snp comprise snps close four snps value htb pathway breast cancer scatter plot pathway leave panel high distribution toward odd ratio fdr adjusted pathway scatter pathway sample case control shift toward see odd ratio fdr adjust union top pathway snps breast cancer cancer control shift toward pt wide study increasingly advance nucleotide disease typical marker complex diabetes amongst unlikely snp reveal case control snp pathways analysis pathway snp identify pathway ideally distinction technique upon pathway disease appear snps systematically apply potential hypothesis pathway snps exhibit great class importantly snp snp analysis require pathway independent permit pathway detail method cancer suggest exist pathway wide genomic contribute analytical complementary system novel snp genome study motivate snps associate case exhibit within across snps pathways pathway gene gene snp identify pathway distinction systematically pathway potential pathway snps control distinguished pathway disease demonstrate system genome wide association variant disease modern yield million nucleotide human genome yield insight basis disease diabetes list publish find national national genome institute typically produce analyze
extremely resource integral distinct discuss augmentation scheme td specialize td algorithms selection da td bernoulli interest status derive show additional effort care focus modeling seem fit individual one york compare three list death medical trend year multiple material class many order describe abundance accept loss change abundance outline simple alternative nsf grant rgb circle rgb cycle rgb rgb rgb rgb rgb rgb rgb cycle rgb model thin central credible vertical span symbol thin central correspond quantile model black gray rectangle rectangle rgb rgb rgb cycle et horizontal span central credible thick horizontal median area space ess ess per min model top six ess min model parameterize standard effect thereby alternative integrate monte model complete complete proceed monte eliminate sampler explicitly account move state product avoid move td algorithm subsequently augmentation specify refer approach complete phrase da refer miss auxiliary status consequence make modeling purpose fold firstly issue underlie various augment representation capture heterogeneous integrated likelihood super population within bayesian framework abundance implement generic extension software offer comment vary likelihood obvious difference approach ii absence iii presence absence conditional unless specify capture capture capture study align compatible individual history history integrate integrate individual parametric distribution hierarchical integrate across detail supplementary material term idea replace vector label exchangeable write probability observation subsequent two distinct step involve equivalent replace usual da specifie individual population integrate sum summing permutation multiply induce conditionally bernoulli random parameter specify binomial include material instead partition augment zero show material modeling eliminate provide integrate carlo fitting implement gibbs specify detail include difficult binomial induce unlikely single across wish specify outline fit note define overcome implement reversible procedure implement necessary upper value specification de specify likelihood purpose pilot explicit pilot de k induce realization integrate obtain satisfie requirement use unable specify zero material abundance year count count across closure ill manuscript suggest treat fit extension exchangeable logit specification
subsection estimator size combine spectral norm eigenvalue norm find family satisfy regularity ny log fitting maximize mle another n particular th result rewrite normal let eq nb follow establish hellinger exponential argument hellinger density exponential define satisfie chi inequality repeatedly normal basically accuracy terminology formally paper indicate usual change paragraph next constant bound approximation need quantity kk assume regularity gaussian proof initially emphasize serve essential estimate component estimate maximize try position invoke modify allow suitably regularity difficulty asymptotic true misspecification choose maximize cutoff differ argument three set x universal ensure ready truncate conditional sequence know x nc therefore total step lie invoke notation subsection respectively give q n independent cutoff define state conditional lie high st w iw taylor eq behavior l follow imply deduce q must unit establish bind mle decompose n cauchy cauchy schwarz n moreover may f r ensure x x dx matrix taylor invoke absolute value state generality virtue e preliminary lemma preliminary section eigenvalue definition proof sum function save lot argue pa use range contribute contribute covariance inequality prove covariance lemma rewrite uniformly c use iii iii lem adjoint cf crucial perturbation theoretic result extend q j k k prove six five x complicated expect appear kb p expect x assumption hold lemma equality together nc n f c n assertion eq form assertion hold inequality know bound section theorem definition nd triangular uniformly inequality plug n dms nsf grant dms functional rate parameter change measure eliminate cause extensive exploratory decade comprehensive discussion application among many problem functional regression receive minimax truncation show depend smoothness slope theory know device problem nonlinearity function analyze estimator establish estimator exponential parametrize change inspire le eliminate nonlinearity consist pair process index parameter exponential conditional measure fr section precise specification control decay eigenvalue characterize smoothness result bind assumption bind functional generalize provide optimality establish due issue overcome measure theoretical advance regression prediction prediction establish minimax consider generalize regression economic predict curve minimax result minimax proof family size aid reader stage kernel key idea invoke theoretic family pt otherwise yy quasi provide difficulty cause nonlinearity quasi specify achieving cover broadly regression unknown assume due model model become difficult interesting generalize eq exponential regularity set c observe pair like I satisfy decrease offer estimation procedure provide theory assumption justify regression widely application component zero curve functional activity analysis basis wavelet fourier basis regularity component regression applicable wavelet basis basis basis slope case theorem degree ill setting need low explain page eigenfunction lie denote depend regard strong assumption remove help regularity case exponentially decay logarithmic case eigenvalue slope decay exponential essentially much methodology rate state estimation trade mle misspecification
optimize optimize handling enable subsection ready optimize weight formulation regularization objective monotone function speaking correspond fit example severe h regularizer norm mkl consider consider sample task task dependent common propose optimization hyperparameter learn simultaneously norm regularizer generalize presentation early paper regularizer let variable kernel weight x h employ regularizer special mkl elastic mkl let x discuss early elastic net mkl block mkl combination connect base block norm base regularizer analytically eliminate weight regularizer origin satisfy monotonicity mh conjugate divide concavity straightforward composition convex line use monotonicity infimum exist formulation correspond original note convexity strong concavity regularizer separable regularizer suppose kernel regularizer separable function slight abuse follow convex regularizer separable straightforward omit regularizer summarize regularizer example mkl regularizer separable therefore accordingly formulation regularization norm regularizer example minimum minimization lagrangian taking set mx expression obtain structure task case mkl q jensen jensen note discuss joint different problem may task regularizer example convenience lagrangian back converse denote conjugate converse statement conjugate base elastic kernel elastic net regularizer function weight since concave square regularizer component take q substitute back equation therefore example norm mkl tb concave derivative q follow follow rapidly numerically compare mkl discuss take dataset sift pyramid kernel precisely combination four type sift sift sift sift px image choose local visual point center local histogram obtain partition partitioning whole level cell measure similarity histogram pyramid decay four pyramid see linearly space computed sift result block mkl elastic mkl mkl mkl loss mkl mkl logit include make comparison mkl possible difference square elastic net fix model logit implement toolbox update toolbox cross except norm tb tb mkl elastic mkl uniformly weight favorable mkl block mkl tend perform especially elastic net mkl mkl perform almost mkl component specifically vs dataset elastic mkl rbf agree vision literature elastic mkl band kernel never choose structure mkl elastic task see kernel regularization two formulation systematically map transformation likelihood employ furthermore derive block regularizer bayesian mkl test classification mkl categorization accuracy uniform roughly kernel usefulness mkl sparsity contrast mkl tend probably obtain extremely one avoid prior relax fine characterization regularizer bayesian theoretical concern sparse mkl mkl mkl non mkl thank helpful discussion partially point completely specify function mx mx n mx easy mix side form mkl receive attention paper understand weight systematically map separable naturally generative probabilistic numerical mkl comparable kernel might preferable mkl elastic net usefulness kernel elastic weight representation kernel represent web page bag might classify discuss also page useful classify whether page support b categorization descriptor classify car representation manner source framework kernel combine nonnegative weight base approach mkl recently mkl understand constraint regularization kernel weight meanwhile structured assumption employ regularization ask well base map manner conjugate far contribution
phase integral go configuration weight rewrite bayes language field theory partition actually negative logarithm joint permit transfer interest norm minimize often field pixel discretize signal fs sx integral multimodal density determine many functional integral path integral functional expand around integral reader theory treat analytically start depth wiener parameter use get interesting corresponding want reconstruct distribution covariance subscript value indicate read xy sx sx ps far signal process linear device response general depend theory case focus discussion concrete covariance eq function give source response fashion finally constant ignore depend quantity signal partition permit calculate call map apply filter signal work p identity connect correlation function obtain classical treatment hamiltonian classical field map signal characterize theoretical classical acceptable much easy approximation follow specify thereby abstract either determined determine joint give expression contain parameter fix permit construct partition parameter information estimator simply multiply additionally average compatible get recognize hamiltonian hamiltonian energy obey enter calculation analytical calculation hamiltonian theoretical representation hamiltonian around reference hamiltonian dp provide signal repeat thought interaction permutation permutation leave implicit tensor notation ax jx rank tensor uncertainty variance introduce signal x long definite relevant invariant fully characterize mean hilbert identity transform spectrum deal simply scalar product fourier adopt since applicable space like abstract basis also unknown spectrum combination support band positive band cover domain therefore variances verify distribution law amplitude cutoff jeffreys limit provide e reconstruct full field turn assume purely improper jeffreys interesting hamiltonian five cast determine equation discuss filter wiener spectrum assume iterate specific freedom follow assume filter decide principle wiener spectrum ask hyperplane pure approach first traditional map principle joint jeffreys filter jeffreys improper prior class would appropriate informative decide concrete inference individually subsection summarize fig display section joint filter pure filter sec critical filter line estimator spectra wiener realization gaussian verify j p p p rhs account lose wiener rich individual drop last jeffreys obviously information logarithmic critical scheme sec filter cast critical mark phase critical apply successfully sparse noisy transformation latter identical without normalization jeffreys hamiltonian symbol mark respectively stationary hamiltonian thin dotted hamiltonian hamiltonian eq spectrum hamiltonian sorting provide respect filter critical logarithmic map express hamiltonian calculate hamiltonian regard poor critical lose filter single signal hamiltonian dimensional show map provide acceptable skewed reconstruction goal calculate average effective optimize construct eqs expect case theory expansion truncate low order practical uncertainty localize hamiltonian repeatedly add uncertainty source accumulate thereby entropy parameter follow flow parameter decompose narrow mutually prior auxiliary onto mutual auxiliary mostly strict sum convenient value full effective increase accumulate uncertainty analytically hamiltonian assign measure accumulate far suitable accumulate dispersion auxiliary find time hamiltonian cast back spectrum hamiltonian construct posterior sense maximize let individual infinitely hamiltonian hamiltonian hamiltonian especially effective localization typically factor various coefficient effective interaction perform step uncertainty let specific hamiltonian virtue might correction hamiltonian eq vanishing therefore ignore introduce four vertex correction theory dominant correction account drop subscript rule inverse hamiltonian belong high uncertainty due contrast free uncertainty correction field lead free hamiltonian subscript uncertainty arbitrarily derive flow pseudo accumulate dispersion prior transform compact operator partial spectral sort closure hamiltonian original structure affect uncertainty ideally change map onto effective amount uncertainty distribution limit q apply problem sect need additive decompose moment concentrate parameter auxiliary convenience properly dispersion result normal take jeffreys logarithmic permit pure receive uncertainty rate center appendix expand free shifted flow dynamic flow latter differential equation pure quite expensive simplify filter equation original evolution equation need dt split evolution rhs signal bands evolution orthogonal contribute evolution obtain part evolution yield fast evolution strength whereas inverse evolve long evolve direction band get project reverse value accurate determine evolution evolution equation original pure uncertain project parametrization onto infinite result jeffreys prior trivial limit jeffreys arbitrary probable basically infinite however likelihood datum must finite amount stay stay either unlikely amplitude thus probable pure averaged imbalance dominate remove jeffreys imposing look alone take would remain small jeffreys spectrum formal future filter jeffrey project plane representation display jeffreys spectral coefficient filter band variance band low band map fluctuation appear calculate filter exhibit power thus investigate extreme algebra since mark vanishing filter jeffreys scheme filter exhibit vanishing lie threshold datum criterion position datum noise live spatial fully characterize spectral band band spectra instrumental kernel spectrum fully power spectrum band generic eqs separate independent individual fidelity far significantly narrow permit determine band coefficient drop normalize fidelity case jeffreys simplify formula solution three simultaneous solution take decision ignore trivial amplitude increase decrease branch root asymptotically datum work exhibit follow approach spectrum filter however filter bin filter threshold exhibit negligible spectral amplitude since might imply assumption significant filter spectrum measurement band filter regard response separate pass band critical power spectrum wiener map iteratively filter spectra filter power procedure name kl spectra filter exhibit mode expect exhibit mode application literature measurement usually kl iterate system initial correctly estimator reflect roughly grey guide double logarithmic spectra dataset top panel spectra display bottom many frequency trend spectra reflect spectral filter want filter performance test wiener basically small difference clearly adopt reason decade frequency mode convolution fourier coupling unknown spectral homogeneous observational signal response pixel display spectral band low since mode take negative one band identical band reconstruction five filter show roughly way threshold analysis filter band noise generality filter prior signal spectrum adopt member spectra exhibit incorporate want specify specific length double logarithmic spectrum limit additional smoothness logarithm spectrum derivative collect quadratic line actually combine log repeat use physical evolution force jeffreys specify reach neighboring band thereby produce smooth spectra without gap see fig regularize filter smoothness think fidelity pure derivation uncertainty lead trivial amount uncertainty spectral influence result desirable since guess might available abstract want include fair comparison filter inclusion non wiener correction filter seem well determined unchanged understand roughly poorly map one evolve similar speed evolution map large region uncertainty large thus observational remove fast pure filter aware assessment filter principle critical obviously fidelity smoothness regularize filter one probably lack gap comparable wiener power also wiener finally full fidelity comparable without spectral show deal uncertainty theory effective hamiltonian joint signal parameter beyond treatment uncertainty successively feed parameter uncertainty pure many calibration uncertainty pure provide full maximal consideration concrete noisy model uncertainty various four classical filter regard wiener operation spectra single filter prefer significantly informative jeffreys filter suffer threshold band power threshold filter four investigate threshold variance fourth filter try logarithmic mode noise soon successfully depth frequently estimate spectra galaxy threshold truncation full iterative scheme hide band signal adopt typically large exhibit large slightly asymmetric spectrum much bad know margin degree band datum determine result parameter pure filter principle tree poorly perform correction include pure essential improve threshold smoothness fourier may show pure smoothness approach wiener since identical critical already filter spectra keep spectrum critical logarithm spectral wiener correction evaluation flow equation filter conclude general uncertainty calibration implication assumption commonly improvement pseudo time flow amount dispersion feed knowledge circumstance real measurement device calibration noise evolution offer dispersion per equation permit control way pure calibration acknowledgment acknowledge comment manuscript signal covariance expect fluctuation free special invertible expansion logarithm general second normalization linear shift gaussian unity different combine
eigenvalue compatibility universal distortion explicit get prediction code deterministic construction construction sense knowledge use construction selector construction code code time unbalanced graph degree side chernoff bound inequality proposition unbalanced vertex signal selector polynomial red indicate entry target panel code draw bi design gaussian moreover gaussian performance latter establish dimensional design compare performance design figure simulation agree theoretical cm author would like anonymous adjacency unbalanced prove selector multiplicative estimate dimensional situation know explicit eigenvalue satisfy compatibility unbalanced deterministic time design processing soon choose find rely target recently emphasis put selector identically center random variable variance first impossible recover noisy ask tuning estimator suit sparse soon satisfy coherence property prove design challenge adjacency unbalanced great refer property selector design inspire frame article matrix I ball distortion design interestingly show design matrix compatibility universal property constant moreover four matrix unbalanced constant remove coordinate great optimal multiplicative less risk side bound show coordinate root code soft phenomenon coordinate error moreover ordinary support eventually sense value selector unbalanced constant left design construct follow positive deterministic euler number selector exist universal time sensible construct selector recover article fold part unbalanced third satisfy satisfy compatibility distortion explicit involve design unbalanced vertex denote regular vertex neighbor adjacency unbalanced unbalanced bipartite graph eq subsequently property small property art unbalanced theory let adjacency unbalanced degree adjacency matrix begin loss assume consist large coordinate neighbor finish ns ss ps give event choose finish unbalanced great lemma finish instance form begin restrict isometry property unbalanced graph short conversely binary satisfie proper one unbalanced satisfy property restrict isometry high vector property property deal context adjacency check unbalanced expansion q frame code eigenvalue cone compatibility analysis carry assumption
ks e notice continuous uniformly point weak continuous dominate case general decomposition j k large numerator equal converge theorem simplify eq e j virtue eq us equal vanish sum term vanish q term yield complete modification weight treat tu ks tr replace notice display virtue establish result suffice substitution detail treat brevity fx ts proof make lipschitz small arbitrary boundedness dominate ts open ball arrive finite coupling allow one paper value real coefficient notice otherwise assertion trivially cf consistency hand large finite sum fourth consequence dependent check moment recall bound estimating result consider moment ts us block schwarz coupling follow derivation consecutive partial sum sum partial sum odd independent putting conclude expression argument replace denote field put replace allow mix variable complete definition height cross sequential mind application aim prediction detection task law large consistency cross validate extension discuss error mix cross validate series illustrate class nonparametric regression attractive area science devote study selector subtle treatment prediction detect address setup classic especially choice thus classic next sequentially maximum design form result namely epoch issue order external time internet traffic sample application preferable issue matter nonlinear medical response explanatory pressure science study influence political opinion regressor practice ii process evidence mean differ usually nonparametric kernel smoothing spline wavelet amongst concern detect kind proposal chart type latter amongst frequently asymptotic classic invariance dependent asymptotic play well literature see separate change drawback course detector certain optimality restrictive sequentially prohibitive ease interpretation applicability base reasoning fit possess class smooth select smoothing extensively classic regression practitioner sequential validation treat yet bandwidth version arrive mind really quite time engineering consistency observe converge available aim present cross focus time point time grow also bandwidth consistency result epoch plan discuss assumption detail interest sequential error section extension data extension general ensure result appendix discuss study power module cell mathematical process distribution error would allow clearly validation mind possible aim detect large assume recall weakly dependent let semi specification sensitive mean function bandwidth sequential classic note classic control chart I kernel cover canonical sided detector monitor indeed stop use illustration assumption smooth kernel satisfy kernel number even weak discuss bandwidth concern choice kernel detector average result extent change cross criterion validate estimate bandwidth shall provide certain remark sequential classic estimation cv since mind past current present ts similar sided cross propose classic interested computational feasible exposition fix finite small appropriate would prefer large value nice relax allow increase devoted careful cross validate bandwidth consistency identify asymptotic q equivalent minimizing identify weak bound uniform usual integer number conduct uniform law put eq worth remain condition combine additionally q extend weak bandwidth weak mind parameter follow write abuse eq cross validate bandwidth minimizer sequential objective uniformly uniformly validation provide many functional functional continuous measurable formulate behavior validate sequential selector apply technique minimizer separate exactly validate large formula sufficient estimator cf perhaps suited criterion requirement separate check analytically numerically ensure sufficient criterion easy require differentiable derivative series communication collect e g longitudinal clinical trial social survey assume ask carry since subject scientific qualitative assumption let weakly discrete mix mix instance mixing provide cover exist eq dependence bridge generally definition approximate l coefficient satisfy
consider power subset modular norm cardinality indicator define resp give submodular section theory preserve separable optimization associate section submodular minimization detail result present subset cardinality value infinite submodular submodular subset simple I e multiplication scalar submodular discrete analog behave duality theory link extension equivalent difference submodular fa fa fc fc fc condition necessary opposite satisfied sum fa k difference submodular fa k condition weak simply inequality fa prop lead modular indicator submodular prove submodular pf sa fa bf sa pf empty unbounded empty interior let submodular interior show empty true constant pf cm function extension link submodular convex transfer extension prove actually definition unique ordering integration summation equivalence equal irrelevant integration fw w w w letting tend modular classical extension indeed set extension fw fw fw v v fa symmetric w f w fw regular cut co reference submodular def basis many result submodular show maximize extension otherwise programming require greedy submodular maximizer fw apply prop introduce multiplier constraint sa fa w fa w feasible I loss generality assume decompose interval sa fu fu f v sa non notably finally value draw link extension prop equivalent submodular convex equal fa fa fa fa proposition complete prop showing equivalent minimize minimize submodular submodular function submodular fa property fa p ia one consider large fw fa fa fa fa next proposition complete prop full support function behavior submodular let submodular exist proof greedy thus fw fw optimality prop algorithm characterize maximizer otherwise order unique convex combination solution exactly submodular w si fa ib optimization v I cm v objective equal equality result optimality condition fa solution let submodular take set sa last sufficient characterize face characterize interior submodular separable say let submodular base must intersection hyperplane interior factorize contraction sa bf bf ab fa fa fa separable detail base e contain hyperplane face intersection support hyperplane support hyperplane hyperplane face potentially empty interior sa together prop face interior face prop non empty interior prop submodular proposition conjugate note prop regular indicator conjugacy function conjugate submodular function extension fa sa submodular proposition desire assertion fact relevant submodular present minimizer submodular sufficient fa fa desire moreover strong equality one moreover w present several submodular exist one submodular affected operation simple restriction moreover extension moreover submodular projection component obtain notice conjugate contraction submodular function submodular contraction function extension pf pf sa tb pf pf bf sc fc fa fa pf build similarity submodular prop prop minimum submodular submodular extension fa ga b ff pf aa minimize get conjugate one sa ga fw q fw fw let fw v proposition give submodular set modular ga vb ga ga b ga ga ga bb pf pf sa let submodular ga ga ga ga ga ga obviously get take extreme definition inclusion consider separable penalize extension separable function problem reference therein simplify statement make equivalent dual continuously problem eq maximization prop w conjugate strictly separable optimal function also minimizer minimizer sufficiently contradiction optimality separable exchangeable q immediate prop relate tangent differentiable separable optimization define set check belong unique become prop another interpretation prop ps great maximize order derivative cast order base minimizer maximize vector pair prop moreover unchanged contradiction imply exchangeable imply prop base greater equal prop show one strict tb j base previous prop prop k ks prop leave zero exchangeable maximization prop equivalent submodular prop satisfie specific cut e maximize submodular general maximize cardinality review submodular first briefly consider function call present submodular know equivalently minimizer negative old norm base possible maximize function solution prop complexity essentially order know form value know prop identity combinatorial usually subset base tight output high complexity submodular f attain function minimize e subset regular include well sum particular cut general submodular direction submodular thus value soon assume fa affine obtain imply find minimum point minimization fw w fw base strategy prop large enough minimum recursively f procedure quadratic fa piecewise otherwise find f k adapt algorithm slightly decrease submodular minimizer minimize base associate associated contraction stop part let submodular aa aa tc th prop thus tight tight minimizer submodular ab fc fa tc decrease bf optimal exchangeable imply tight stem k kt come tight prop optimal prop show finally apply restrict g submodular decrease refer decrease truncate submodular decrease maximizer may fw prop constraint simply replace monotonicity also base include positive consequence prop include compact unbounded prop prop support prop face support take sa fa special non decrease submodular strict partition sa f f relative interior prop submodular positive base first non non decrease separable let primal pair maximizer statement consequence optimality correspond strictly denote prop hence set prop prop involve pair define view view apply prop positive submodular convex pair maximizer prop ks ks kf non separately non decrease fw w submodular associate submodular depend cardinality proposition concave submodular prop cardinality concave submodular result concave base extension extension function disjoint subset fa fa fa f fa da db c b da c fa denote fa b b fa fa fa f f b fa fa b weight submodular also symmetric prop cut extension play role image process compose total therein partial g cut weight fw fa submodular dedicated plus modular flow add node graph non weight cut lead minimizer proximal section define fw obtain use linearity subset always decrease induce norm fa k jj w replace cardinality prove type weight cover q show I
program dl boolean resp boolean correspond conjunction nm simple partition answering depend boolean semantic answering follow case query apply case database schema partly tuple view cast building stand name tuple fact belong integrated false induce hypothesis concept occur form mm mm contain database kb adapt concern build p predicate form usually assume guarantee target belong target kind tuple individual description theory incomplete inherent set appropriate answering reformulate kb rule satisfy axiom model consequently note precisely every thus force every force cover three support induction need equip subsection devoted hypothesis employ original problem induce hypothesis must issue language regarded clause logic program directly carry normal logic program treatment adapt result relie reasoning r boolean boolean difference nonetheless boolean answer aforementioned link answer check reformulate kb reformulate way solve apply consider mm example want immediately hold say r r r nothing else ground answer refinement operator refinement language hypothesis build rr refinement l since nm semantic add apply rule intuitively reduce rule mm rule mm mm mm discover induce nm kb main start contain element queue include deal induction database consider terminate specify redundant pruning post processing pass nm theory process queue become report satisfied framework far hybrid dl cl less expressive propose focus discriminant interpretation hypothesis recursive play coverage usual learning interpretation relation extension procedure provide pre processing enable represent clause framework mean induction generalize quasi check constrain setting interpretation query answer opposite partially implement support variant frequent pattern conceptual close problem nm illustrate case learn aim rule head head kind rule kb whereas extend principle powerful coverage generalize instead scope point notably relative interpretation deal formalism language hypothesis apply engineering task interactive rise call inductive functional dependency valid almost constraint decompose restrict framework reformulate nm notably evidence go beyond illustrate traditional nm dl hypothesis far accommodate viewpoint expressive treatment define generality refinement turn induction framework towards plan analyze produce adopt expressive may turn dl definition refinement operator ideal refinement operator theoretical inefficient constructive improper preferred refinement promise reason deal expressive mention rise currently expressive incomplete knowledge feature third deal rule rule group concern core likely inspire dl language use case final shift extension mining name onto relational systematic report relational integrate onto grateful valid universit di mail di community semantic web solve database well issue database ii schema reformulate expressive framework inductive database knowledge reasoning concern example classification purpose relation logic relational interesting extension attract little effort make able pose relational scalability respect far still remain prior conceptually rule organize conceptual seem ignore conceptual artificial intelligence refer precisely principle certain plus explicit regard intend word form logical vocabulary appear unary respectively concept formal explicit specification among thing semantic mean propose semantic dl language dl expressive semantic combine limited form framework argue difficulty address web help database understanding perspective consider view database schema reformulate problem benefit expressive mainly nm integration brief introduce reasoning induce view within survey conclude remark resp resp role n r relations atomic concept dl construct dl form atomic restriction limited restriction call concept restriction whereas dl add dl role restriction role expression contain inversion role r equivalence concept axiom role equivalence axiom role axiom assertion assertion assertion resp concept resp axiom also inclusion axiom role role axiom dl adopt nothing else possibly together dl kb directly theoretic show map default individual equality dl kb interpretation kb satisfy axiom kb coherent semantic dl kb least kb write reasoning dl kb try check decision mostly calculus another reasoning service check assertion implication dl sophisticated instance dl individual possibly expression individual entail intensive application retrieval aspect complex expression query relational express arbitrary cyclic possibility name name alphabet conjunction atom predicate whose tuple kb entail write check boolean follow boolean alphabet expressive finally ground clause aim induce require thorough discussion induction affect explain confirm explanatory setting observation clause hypothesis observation observation interpretation clause cover observation learn aim none example validity call drop due absence case interpretation traditionally partially order inductive theory single ii clause notion clause clause general define quasi clause clause substitution order clause substitution definite clause apart definite program say iff substitution appear substitution lattice partially order least infimum great clause yet generalize structure accord generality ii refinement operator generalization accord search kind refinement operator refinement desirable illustrate hold practical use condition application operator clause ever satisfy simultaneously refinement language implication language usually locally improper full retain possibility refinement operator couple bounding clause concern specify syntactic clause space connect definite link link term link link chain link length integration kb accord formula upon mutually name alphabet predicate call either alphabet expression tuple atom atom logic rp ir
course optimize simplicity type close likely mostly alternate likelihood hierarchical integral graph make bayesian integrate since posterior fitting goes determine although set tie type topology hide way active learn study type laboratory resource vertex vertice type vertice query mi type entropy average expected decrease result mutual good vertex entropy assignment accord gibbs heat vertex proportional assume type fix explore collect distribution offer theoretical time family exponentially r enyi vertex connect switch state type bottleneck state real world try far chain improve average mi algorithm say already vertex gibbs query large move next query another call give type agree correctly really maximize assignment draw gibbs project expect assignment draw condition imagine vertex independently give assignment agree agree thus three vertex vertex well sampler except assignment chain two independently initial denominator give keep track average draw increment numerator numerator denominator unchanged alternate estimate type vertex correct query large mutual mi vertex large agreement aa text label algorithm accuracy converge vertex test mi aa consist member split around around vertex form highly correctly identify vertex belong aa course community identify perfect review choose sort central lie boundary understand lie aa well experiment seem place examine take attribute attribute half vertex correctly attribute less although aa perform measure decrease easy vertex hard form fraction arrive time remain type attribute aa specie label correctly label word accuracy continue get end learning find extent attribute attribute species large include type attribute block half misclassifie block likely specie confidence know member friend close expect regard mistake cut aware belong confirm misclassifie modify artificial make block iterate follow species value equal accord block iteration change specie reach fig perfectly predict specie web difficult order suggest agree extent relative positively pearson begin type specie fill available mutual varie condition specie largely due topology adjust mutual perspective heuristic vertex high centrality heuristic popular centrality reflect heuristic mi aa although mi however situation resemble surprisingly mi early stage achieve poorly actually predict rest leave
weight random consistency open analyze base gradient optimization conjunction hour ahead wind problem classic management demand lose constraint constraint observable generate demand mixture variable include mixture generate bi width pdf width new pdf bandwidth accord package range kernel online period decision posterior gibbs collapse sampler run take iteration base dirichlet weight base sampling decision optimal dispersion measure sample use collapse method location wind ignore wind dp ignore display algorithm wind function type outperform ignore possible dirichlet still substantial wind simply free optimization optimization rely weight give kernel complex weight dirichlet show dirichlet super work statistic solution problem search space element aside decision observable machine tool avoid proposition conjecture david problem noisy search behavior variable shape currently density state outcome distribution query problem function examine two scheme weight method hour ahead wind result dirichlet substantial solution class uncertain hard compute instead problem decision classic stock uncertain demand distribution rich theory important variation search allow belief state belief decision challenge outcome something use current decision say conditioning often frequently hour ahead wind wind company hour future little difference lose depend market contain solve stochastic sharing use outcome observation decision weighted ensure summary mind formal search problem become q technique distribution treat state independently nonparametric affect similar new observable motivated realization depend reason subscript nonparametric approximate weight response state outcome gradient z solve wind entire resource allocation involve problem base restrict weight observation currently exist almost sure convergence global optimization popular variable explain extended try behavior original optimum previous weighted accord linear approximation around optimum separable heavily weight give easy implement good develop give unstable situation propose mixture dirichlet nonparametric distribution derive place current monte carlo clustering method kernel problem hour ahead wind synthetic weighting hour ahead synthetic wind north american assimilation computational solution scheme dirichlet produce contribute base dirichlet process dirichlet empirical promise paper review treatment variable propose incorporate state prove base state section weight scheme analysis synthetic hour ahead wind operation optimization study form search problem diverse within entire outline problem consider individually case construct wind select variable wind create require problem portfolio bandit literature theory community study portfolio handle treat separate bandit establish area sample study parametric study however mechanism parametric view problem sensitivity control community deterministic machine problem theory reinforcement state state arise way dynamic might state determine influence next visit choice propose problem example major close value learning deal challenge decision variable information portfolio reference reference close online player select convex propose decision value update give turn method use decision hour wind wind value sampling deterministic complex hold almost surely ratio problem study program average event gradient variable fairly straightforward create joint convex like respect observe change produce average close particularly solve deterministic solver decision would surely pointwise subset set compact subset surely convex every fix minimizer decision pointwise section discussion assumption decision per nx ns almost surely pointwise heavily every compact minimizer proof uniformly compact converge pointwise uniform satisfy turn corollary function however present optimization take stochastic use gradient general iterate back another construction piecewise function variable straightforward difficulty state base iterative state construct approximate give piecewise separable aim approximate well easy convex interpolation dimension component univariate piecewise convexity restriction ks weight outline place decision eq sufficient k minimization nx ns nx nx affect nx nx piecewise reconstruct fx v df kx ks step optimization perform set discuss extension subsection heavy part order number observation find easy format fine evenly parameter equation never grid second break converge optimum query care say sufficiently neighborhood accumulation however define assumption finite write separability convexity gradient parameter continuously observation unbiased equation cone every decision place net center partition convexity variable optimum eq often non phase infinitely subsequence every infinitely combine fact sufficiently therefore hold choice determine importantly well approximate discuss weighting procedure response calculate object density approximate either compose observational observational weighting kernel implement however dirichlet rich weighting material rely conditional common bandwidth weight bandwidth observation estimate q continuous every ease guarantee highly sensitive validation overcome propose dirichlet alternative weighting kernel non weight observation similarity dirichlet partition introduce gain popularity powerful computer computation partition model countable simple number dirichlet place lead trivial proportion dirichlet issue determine like derive cluster mixture query state average response state proportion produce method clustering discuss mixture satisfy mixture parameterize eq wish determine find review find good number dirichlet dp parameter proportion location dirichlet effectively place give probability call space prior place reason actually measure construct draw model condition mix give dirichlet base concentration produce dirichlet explain later dirichlet produces surely demonstrate dirac know also take control example approximation model conjugate normal dirichlet equation associate place place partition induce give partition wish place partition
size throughout definition transpose matrix way mode analogue fixing tensor rest robust gaussian perturbation conclude problem non convex minimize iterate always take respect converge stationary mm stationary jointly continuously full meet guarantee stationary point global simplification iterate ii unique minimize r fashion hold mode subproblem separable problem toolbox tensor draw dense variable generate tensor randomly draw shape parameter dense entry I add value initial mode toolbox goodness match factor perfect provide two recover score decrease increase exception find minima compare true median error factor capture solution base comparison lp found perform slightly fast similar factor show size tensor indeed consist independence parallelization simulation demonstrate suffer insensitive presence noise perturbation cause degradation conversely suit tensor finding tensor array cp typically propose perturbation alternate fitting cp cp factorization generalization decomposition fit cp tensor
employ leave analytic give become core suboptimal start theorem give partial yield translate argument bind yield claim formula optimize space fix build primal derivation kkt formulate macro mkl optimality compute accord solve subsequently compute analytical stop gap iteration disadvantage full propose thereby solution solely operate standard terminate fall precision normalize constraint mkl variable accord mi old old optimize accuracy subproblem exploit method useful gauss prop continuously define gauss recursively let proposition establish mkl hinge let positive globally p svm ignore speed hinge suffice show transform requirement prop fulfil expand thereby extend variable may loss prop fortunately prop substitute constraint maintain closeness gauss method solution block objective continuously differentiable latter minima uniquely attain solve svm definite analytical solution limit globally hilbert imply violate either due long strictly computed equation achieve one exist want improve little bit indicate cf sect lot optimization suboptimal overcome ensure svm try cf like primal svm section cut plane newton toolbox two class task decide svms algorithms optimization scheme outer offer list semi use violate thus single sequence constrain program alternatively analytic update perform modification core currently implement objective kernel step couple trivially typically perform svm positively access solver implement cache become mkl scale sense reach cache track partial objective individual cache size regularization optimal imply magnitude modify regularizer feature regularizers imply kernel sensible uninformative experiment fundamentally generalize spirit contrast nevertheless beneficial normalize formally positive rescaling rescale corresponding feature satisfy equation equivalently express function eq kernel empirical frequently kernel rescale unit also multiplicative spherical substitute point length whether far section show norm extension aim mkl mkl translate mixed formulation penalization equivalent use op prop yield close look parameter real range perspective extend sophisticated norm mixing explain straight q prop analytical update derive block mkl mkl unweighted study toy light sparse mkl localization find rna bind gene genomic study intuitively expect directly sparsity apart suboptimal set true go level sparsity scenario step study mkl illustration toy balanced covariance opposite mean encode true sparsity imply latter carry fraction set fix hence noise work classification norm mkl grid optimal attain interior grid relative gap optimize test pure I latter error reflect corresponding explain theory unstable result observe agreement test scenario kernel carry scenario perform mkl variant truth I select datum imply towards contrast unweighted kernel informative bayes block sect truly limitation shall keep artificial balanced scenario level mkl achieve less tuning sparsity mkl low increase nevertheless observe mkl prediction scenario variant scenario contrast reasoning discrepancy explain minimizer stable become test appropriately fix mkl summary size increase expect scenario mkl uniform unweighted svm well appropriately mkl rare norm mkl define capture protein mkl unweighted improve establish problem gram section investigate matrix set predefine split norm rest prediction r coefficient proper affect norm equally split yield optimize averaging split data split table non sparse regularizer hand mkl arbitrarily norm intermediate norm r std std std briefly mention mkl four kernel pick differ place gap part protein sequence consider vector carry overlap information gap principle high degree polynomial svms individually case similarly highly redundant exclude non norm model detect rna bind genomic site regard key determine start site start genomic detector appeal mkl contain gene utilize translate positive instance extract window around negative size represent angle energy describe instance use pool soft grid attain test size pool training bar indicate size contrary mkl high auc mkl suit norm mkl yield superior remarkable application domain unweighte recently confirm comparison program area roc test indicate error repetition mixture weight spectrum good explanation optimality sparse show moderately informative recall report auc individually energie drop remain discriminative investigate compute ai frobenius dot observe kernel show slight little auc carry complementary include mixture precisely mkl fig reason right energy expect auc performance auc auc surprisingly kernel auc originally database functional unknown employ experimental setup phrase edge capture localization profile additionally use integration kernel match unweighted sum validation fold however omit guarantee problem setting table slightly svms result observe unweighted svm perform approximated mkl improve result include maximal mkl unweighted kernel moderate show suggest hence orthogonal kernel experiment explain mkl experiment mkl mkl mkl mkl mkl mkl product mkl mkl exp one suggest beneficial non mkl task odd digit lead example analytical mkl experiment analytical convergence compare infinite program mkl svms unweighted mkl kernel train computation expect one counterpart optimize outer gap stop criterion duality gap norm counterpart trade method execution svm analytical training number kernel figure display result gaussian bar standard unweighted fast mkl slow compute matrix notably requirement gb double number time slow evaluation kernel optimization allow kernel gb mkl optimization naturally slightly slow taylor rank non figure vary sample unweighted taking time mkl increase fast mkl thereby somewhat fast sparse analytical might bad contrast conjecture part solution hardware storing gb allow effective cope gb considerably fast slow method increase memory propose analytic cut plane order limit impose store future improve mkl pay translate multiple minimization problem regularizer arbitrary penalty mix regularization dual kernel mkl analytic base allow application algorithmic approach norm implementation toolbox execution reveal commonly approach scalability world learn empirically validate mkl data computational control various mkl achieve scenario wise real bar either unweighted closely approximated mkl hence seem natural worse unweighted along suggest mkl bad despite appeal first understand appendix scenario mkl fast sparse well ground suggest mkl mkl inconsistent exist advantageous conjecture bound improvement serve conjecture bound slight advance cross case call non analysis world discriminative carry play distinction exploration beyond scope remark redundant uncorrelated variance sect speak observe redundant keep put zero strong preference scientific already connectivity causal flow social true zero interpretability correlate arbitrary predictive demonstrate wish discussion manuscript acknowledge contribution suggestion project fp european exchange service theoretical norm mkl rademacher rademacher generalization mkl mkl remarkably effort line mkl valid match want hypothesis risk class bound w rademacher eq rademacher value remove dependency random rademacher show technique rademacher main rademacher exponent old norm technique extend norm convert rademacher thing rademacher illustrate bounding let rademacher substantially improve loose favorable whenever plug obtain new rademacher exponent compare one p coincide differ polynomial asymptotic considerably infinity contrast approach precisely p om bound vc rademacher literature employ aim assume large w loss v hx iv functions loss lipschitz bound immediately bind mkl fix margin assume n w reason remark extend norm different example block sum norm norm seem beneficial somewhat contradict norm refined theoretical support intuitive claim uniformly priori promise beneficial block norm prop norm generalize introduce parametrization straight pc q exploit h illustration leave word bayes mean advantageous attain latter bind yield interestingly obtain norm mkl variant generalization theoretical priori optimal see fig simply precisely sparse mkl truth expect ground truth vice tight ingredient paper let convex f exist rise duality satisfied constraint vice versa op equivalent op vice suppose exist translate vice versa duality optimal point max removing show corollary multiple berkeley university california computer division berkeley usa yahoo com yahoo research laboratory com life learn combination appeal choice support scalability rarely outperform trivial practical mixture generalize mkl norm devise new mkl like demonstrate strategy compare control artificial mkl empirical biology sparse achieve accuracy kernel convex conjugate coordinate bioinformatics generalization allow find appropriate representation kernel immediately machine wide domain appropriate even good hand validation probably prominent right alignment aim vector support machine svms alternative select dc mixture problem relaxation obtain focus scenario guarantee seminal combination semi subject require combine learning mkl automatic version mkl namely sdp however sdp expensive application conceptual develop mkl utility simply constrain negative constraint restriction transform problem constrain drastically burden optimize dual formulation decompose problem mirror prox min independently mkl training generation amount repeatedly mixture coefficient immediately convenient algorithm benefit svm allow large scale training svms still prohibitive reason svm drastically scheme extension exist major coefficient regularize optimization control mention learn solution mix mixture first sparse combination irrelevant evaluate appear additional simplex simplify constrain beneficial mkl outperform svm unweighted consequently progress field mkl need useful model improve plain since compete formulation set achievable outline side cast loss function regularizer norm penalty mkl variant dual objective optimization state mkl well incorporation knowledge isotropic include sparse plain regard introduce form step infinite unnecessary utilize work svms mkl current mkl implementation employ enable ten thousand thousand completely small set machine toolbox claim experiment artificial world represent diverse application bioinformatics negativity unbounde partial allow optimality discard problem hinge supremum threshold recover solution apply kkt start composition side conjugate dual term dual recover regularizers point highlight arise theorem optimize employ depend dual term depend dual present duality limit strictly positive duality learner optimization unweighte recover special regularize risk minimization regularizer
tail attribute level transfer bank take equal bank word increase coverage comparative risk bank conversely figure bank attribute completely view bank similarly transfer condition bank risk receive bank understand must motivation behind price bank bank cross var mit bank consequently region region bank price risk deviation bank hence retain bank bank portion transaction region solid line exceed bank risk dot bank mit bank conversely var mit solid bank less risk measure optimum bank advantageous bank relationship distribution optimum point measure decrease portion entire exposure bank prefer criterion us bank give bank policy advanced tail index frequency set ht bank risk dot exposure receive bank dash dot unit unlike previous low frequency unit attribute infer coincide observe attribute coincide discount grow optimum infer discount apply bank light increase tailed linear cover limit linear bank bank exposure dash bank bank appear dot line discount event exposure demonstrate payment transfer perspective create line peak illustrate risk band moment less tailed shift bank appear risk band policy unlike dot consequently yield region bank fair range maximum uncertainty bank expect risk exposure light tail study context capital reduction multi period risk modelling investigate examine ii capital extensively demonstrate flexible capturing process process provide form claim lda cm remark operational advanced reduction capital policy capital extreme distributional lda modelling involve compound heavy comprise analysis capital question bank financial heavy tail question ii address quantification capital risk consider fundamental policy several two extensive simulation study policy var fair provide analytic solution loss multiple operational distributional capital reduction mathematic email mathematics north modelling impact risk business challenge operational management capital partially attribute limited impact fair policy ii advanced financial bank strong capital impact scenario allow accurate pricing study transfer bank aid model capital develop international ii specify resource hold capital phase distinct capital requirement level capital also naturally allow capital estimate measure differ particular unlike capital measure capital correction concern treat since contrary capital hard define management company explicit analog ii particular es claim product loss bank capital percentile claim allow level relate accounting business closed form analytic basic transfer capital policy capital lda claim understanding policy understand extent capital offset informed analysis scenario heavy rare extreme consequence introduce model claim family flexible incorporate tail infinite pose bank policy capital behave perspective claim build policy address structure cell business event model variable discrete year distribute bank year calculate requirement seven time eight business financial section limit select block structure linearly duration year second stochastic propose trade bank capital attribute loss risk begin individual follow policy applicable practice exploit risk see simple concept via year loss exceed bank incur black lda model accord process claim process lda provide loss complete limit bank loss bring bank bank loss bank simplify policy function claim period accord provide maximum basis maximum year illustrate claim exceed bank loss claim hence exposure incur bank express conversely period multi bi loss denote provide limit simplify result claim generate period express require modelling condition outline payment advance base particular aspect model policy year bank section model explicit arrival time consider scenario single discount sensitive factor simplicity year claim describe payment payment discuss generally bank financial loss realize perspective claim account uncertainty account severe loss likely payment delay severe discount payment arise counter party claim uncertainty stochastic across reflect coverage loss demonstrate influence measure loss equation result claim period quantify segmentation define identify band locate beta payment claim process unlikely band limit equal convert lose truncate model one family scale dt result chapter ht function lemma allow compound jt process mixture density comprise exact form lemma corollary compound mention compound jt express comprise poisson loss random furthermore loss analytic directly jt copula aggregated loss close capital var quantitie meaningful exercise report chapter furthermore measure var es jt poisson tail quantile allow unique compound maximum lda jt median poisson loss compound state theorem relate simulate loss loss lda fundamental extensive basic consideration meaningful fashion process consider quantification capital bank define var refer year expect loss exceed consider quantification policy deviation context correspond claim loss define demonstrate lda stable exact analytic purpose es potential second claim process ii jt loss accord closed form analytic derive process policy risk process quantify j claim finally es var depend whether risk example capital economic capital es bank infinite nc proof th second process es stable var nc numerical losses capital result capital capital capital var however although context represent report capital typically simulate require uncertainty capital estimate uncertainty sufficiently large restrictive report accurately estimate simulated loss experiment frequency poisson intensity rare modelling systematically simulate parameterized index stable well apply capture policy capital requirement exposure exposure capital generate level purpose multiply percentile purely individual range indicate provide study intel ghz simulated loss provide study capital measure ii risk measure detailed capital loss realize via allow question policy detail stable model case policy advance study statement type scenario comparative var exposure process comparative capital var bank question denote claim make quantify claim comparative present eq bank capital var reduction loss present single risk bi simulation study identify exposure
direct straight line elsewhere necessary unit decrease represent lie body pair body may group contribution bin decrease length equal straight straight affect different bin separate lie inside divided normalize sum sign equal quasi construct length produce limit line aggregate body consider measure straight body early express ratio cf straight cf devoted integral particle move along straight line medium fraction body straight trajectory possible describe amount intersection ray analogue angular limit direction projection body surface sphere yet calculation could general adapt develop integral b particle double integral eqs many body intersect modify due discuss separate calculation however may simpler disjoint almost straightforward advantage straight integral integral different due hundred carlo critical possibility take variation calculation boundary surface omit integration discuss distance expression write monte technical may ref analogue write express cauchy due ray q dirac finally application sign measure calculation integral may useful many area physics fairly technical theory elsewhere ray essential property extension necessity sometimes negativity certain delay application algorithm ray distribution quantum physics call function reasonable physics distribution conceptual challenge appearance illustrate ray outside body describe measure know pc pr pa pr overlap affect removal another bin effort simple distribution reason appearance ray complicated sign formula correspond set dirac quasi dirac suitable six integral distribution homogeneous body analogue dirac call function nonconvex relate interpretation sum integral two object dimensional calculation body apply calculation double pair surface integration due obvious length may analytical calculation six initially dirac ref analytical expression shape reasonable integral method dirac analogue get carlo neighbor even nonconvex straight intersect body widely construction nonconvex interval inside nonconvex multi distribution introduce divide justified body definition nonconvex range argument construction sum present analogue calculation ray dirac introduce version tool sec collect discussion body sec carlo mainly analog ray instead ray segment draw isotropic write expression intermediate ref explanation inside straight distance leave side six inside track ray proportional useful explanation appearance alternate nonconvex intersection ray nonconvex body possible introduce distance maximal intersection distribution rewrite produce analogue rigorous treatment sign quasi use ref description picture may consider essential condition possibility use integral depend merely absence inside medium indistinguishable ensure possibility generalization expression call factor isotropic distance introduce coordinate second consideration body precede together ray angle angle brevity ray direction early may take multiplier body area surface explain discussion emphasize ray particle trajectory formula disjoint nonconvex arbitrary isotropic medium inside environment term hand distance straight track inside fall outside boundaries monte calculation advantage integral many give calculate integral numerically analytically carlo visually maximal divide bin simply monte fairly useful application different create origin odd index bin interval index bin carlo generation reasonable relevant alternative definition via derivative relate ray length autocorrelation far convenient dd body length find widely autocorrelation define sphere radius multiplier comparison easily nonconvex system explanation relations q explain convenient hand multipli measure volume six space division correspondence mathematical expectation cumulative distribution function random distance dd rewrite agreement prove analogue detailed proof elsewhere briefly five product body use multiply surface area analogue ray equivalent derivative work integral space integrable function define consider body topology test simplify notation instead rewritten b b derivative body ensures rewrite derivative appropriate dd integration integral formal four straight rewrite produce straight body analogue derivation dirac ref double along integration source intersection
error measurement survey portion response space relevant solve play useful extend current section discuss condition identify sense possibly mis identification fourier transform fourier continuously grow fast topology function topology function class include lead discuss generalized density function derivative belong equation define ordinary positive eq g polynomial choice generalize discuss example regressor transform continuity characteristic characteristic hold set include interior necessary interior characteristic equal point shift continuously differentiable continuously differentiable continuous proof density proof univariate density though still b identification necessarily restrict include equivalently identification restrict factor exclude unbounded gaussian weak topology therefore b b n nx ne ne far lead function arbitrarily nonparametric support wider fact parametric pose al closeness support n function theorem converge solution pose exponent imply slight two gaussian differ gaussian belong class separation gaussian specification weight convolution super transform multiplication growth bound function cube wiener complex exponential long theorem axiom conjecture theorem theorem exercise proposition remark proof em la la la university abstract identification problem convolution nonparametric measurement panel examine issue receive attention well far give period let observe nonparametric period distributional
effect symbolic calculus analytic function program implement calculation gradient etc ad outside symbolic list ad possibility derive gradient use ad price code ad long might affect overall analytic ad iy j iy I j I iy j j j iy iy I iy model iy iy iy j c grant research science thank mathematical sciences research proposition physical phenomenon whose estimate sde intrinsic randomness identify drift interest model dynamic effect distinction source variability effect method like root automatic close transition equation cox class become theoretical make applicable field dynamical ode pde repeat take individual role form experimental subject parameter vary mixed model difficulty arise deal model distribute term extend kalman one dimensional growth curve reference demand dimension handle contaminate measurement likelihood normally sufficiently behave distribution develop density give integrate effect quadrature result quadrature application extend multidimensional automatic result obtain work methodology flexible accommodate complex normal employ satisfactory subject observation per often drawback measurement situation section introduce function consider method tool dataset summarize result limitation method bold character sde evolve different population unit pp dimensional fix effect effect unit dimensional standard mutually condition diffusion assume ensure state denote unit evolution unit realization brownian path random express sde among unit explanation variability unit distribution give r lebesgue unit time let unit di scheme goal datum parameter also estimate observable density density behave function effect result possible random integral computed effect case explicit estimating explicit integral analytically solve ii unknown evaluate numerically solve direct equation expansion mention comprehensive propose approximate show approximated transition homogeneous sde suggest review dimensional focus extension expansion reference drop sde observation infinitely drift growth expand approximate taylor exist transformation solution sde ease interpretation scalar sde namely transformation give reasonable reader mean many diffusion see software symbolic algebra homogeneous obtain obtaining obtain general integral numerical dimensional integral irrespective effect implement though grows suppose give approximated series q tf special laplace exact likelihood denote mle laplace inference still valid intra variation variation happen general either ii symbolic iii automatic recommend costly grow choice symbolic package become however package help specific situation see ad tool comprehensive example assume random name rand rand create automatically file name contain return precision hessian rand hessian rand b hessian gradient b rand returns rand trust derive strongly recommend ad tool speed estimation random symbolic toolbox hessian much effort remainder drop necessary follow internal sometimes symbolic calculus value limited precision gradient hessian trust external plugging especially internal variation external free namely internal estimate internal might however author latter inefficient consume iteration external efficacy e toy growth model allow growth rate growth growth datum growth tree effect balanced consist seven measurement tree reference age age mm day infinity logistic quadrature method solve consider ignore ode ij growth often observe proportional root state effect pdfs zero estimate dataset thus obtain trajectory scheme size extract interpolation trajectory linearly equally spaced sampling transition expansion technique close euler discretization symmetry report skewness mean ci ci skewness ci ci skewness ci skewness produce report datum simulate trajectory scheme band trajectory trajectory realization draw order panel solid dash estimate see figure report fit relation find histogram population fit density fit equal methodology expansion approximation euler exposition sde start sde approximate draw lead close euler simulate obtain approximation considerable frequency contain true result surprising sde model conduct although worth numerous physics engineering finance application anti density multiplication mutually effect external internal total internal external step strictly positive reduce superposition transition realization effect dimension parameter euler approximation transition use expansion skewness ci skewness ci skewness ci skewness ci skewness figure coordinate empirical smooth solid five report simulate euler scheme use band five trajectory draw estimate population estimate strong estimate sum equal sum determine occur individual deviation sum moderate return internal round plug explicit hessian plug pool equal distributional thus equal effect plug exact maximize converge computationally costly huge expression exact expansion polynomial root process ergodic many application finance interest integrate neuron emission electrical see reference call population p standard symmetric constant quantity determine moment determined plug variance p value observation interval setup respectively satisfactory drift parameter problem especially calculate order length interval observation
lee writing update drop index simplicity analogous lee case exactly eqn learn view complete hope encode guess guarantee encode remove eliminate remove particular generate term attribute generic imputation keep decomposition place finally estimate attribute decomposition done learn classification engine additional convert single test base multiple write obtain converge attribute error follow write write get increase eqn extension claim square q increase define analogous result dot close repository represent vector ar rest point random mark mark replace entire test comparison baseline experiment substitute substitute substitute dataset base accuracy random lemma nmf datum interpret factorization miss attribute provide joint miss classification miss factorization nmf recently environment recognition mining dna extensively lee field various extension recently nmf point generally basis researcher decomposition basic need limited dimensionality similarity introduce motivate classification would miss motivate due inherent optimal nature
computational goodness theoretical simulation snr use distance digital priori characteristic class base kolmogorov ks statistic classification accurate letter distance simplification classifier classifier sample symbol perturb receive loss generality propose method testing compute small cdf concatenation quadrature cdf give method obtain theoretically distribution respectively point order j equal evaluate get thresholding counting sort respectively although brevity address approach metric metric letter set point sort order notational partition conclude fall distribute pmf individual sample class classifier replace snr store snr value far store complexity sort sort complexity use cm short requirement application memory requirement smooth worth generate cdf observation classifier l multiply distinguish ml snr fix confirm small size region detection snr exceed converge snr ks perform constraint analytical match perfectly simulation result db set evaluate present observe cm classification upper perform size slightly ks third classification phase snr fig actual
last implication kkt strong rule j slope condition drop inactive bind kkt drop derive rule penalize logistic coordinate piecewise enter leave slope absolute change move start product plot predictor simplicity rise interval intuitively solve kkt exclude buffer account may move strong kkt rule kkt slope violate condition section counter slope rule believe counter yet somewhat exceed short took independently center standardized eliminate nd nd continuity interval break sequential rule random hence maximal solution line horizontal line plot red slope segment value slope red segment reference sufficient slope basic guarantee dominant result hence never general show problem dual slope lebesgue measure cone weak simple hold outside model suppose inverse one dominant haar haar triangular matrix arise dimensional transform design possess condition work express slope sample concept mutual nonzero condition extremely optimization put distribution design mutually incoherent subset condition unfortunately argument become complicated seem translate need hold tool check kkt condition fails generate sequential logistic global safe rule gaussian inner residual vast example regression discard predictor least association exceed million say solution snps check verify find k scalar suppose form variable satisfy norm globally sequentially easy check reduce rule covariance covariance penalize likelihood sum diagonal penalize would useful graphical optimize entire rule discard zero retain occur rule problem lasso group strong sequential q value predictor predictor kkt predictor predictor start active predictor maintain non coefficient solution kkt check repeat strategy ever ever non value current broken unit slope add plot contain ever active occur ever due sign also occur light advantageous value check kkt go warm start kkt predictor go step warm c fairly common step strategy table see offer case never seem slow sequential rule inner offline intensive subset paper strong sequential discard kkt condition offer speed yield plan version use discard predictor apply discard unable correctness time thank co share publication helpful work author support national grant national sparse seq strong seq seq discard safe univariate predictor outcome zero paper rule rarely fail practice tucker kkt convex rule optimization focus statistical fit start observation denote write omit solve tune considerable past derive fast give zero therefore perform kind solve problem substantial discard author derive safe penalize logistic set similar construction fit tucker kkt condition repeat screen related optimization screening rule author argue performance estimation propose rule discard predictor problem involve penalty discard safe rule exclude nonzero rely end effective sequentially generally rule stems fact discard rarely practice net penalize look counter example large counter although global extremely safe rule rarely mistake condition discard need check elastic net penalize logistic section general problem strong speed convex answer detail penalize linear safe discard I author lasso sketch find scalar represents bind nothing predictor certainly imply safe bind somewhat recursive safe safe advantage discard truly nonzero discard fewer strong global rule predictor hand always small safe weak illustrate safe basic rule safe keep order rule discard practice predictor useful much
term joint approximation expression eqs q integral eqs analytically conclude sufficient component computing eqs mean eqs analytic kalman predict kalman covariance many gp covariances eqs recover equation filter sufficient smoothing distribution measurement mean state terminal step equivalent distribution tt recursively integrate measurement multiply integrate tp integral p step rule conditional joint approximation joint close gaussian approximation first second filtering fully determine measurement cross covariance matrix filter computation rule obtain shorthand define invertible unnecessary obtain multiply gaussian state p tc integral eq integrate unnormalized constant constant eq smoothed conclude p tt approximation filter previous step smoothing recursion filter approximation consecutive smoothing mean covariance consecutive state joint smoothing implication filter distinguish covariance algorithm filtering smoothing implication covariance smooth kalman smooth x ig w x I overview cross notation kalman measurement respectively sigma use mapping desire computing computation slight modification coordinate none compute joint implicitly mean filter smoothing eqs paper recover presentation smooth gibbs infer gibbs infer mean filter sec alg priors time prior infer I filter fp wishart estimate covariance burn similarly generate map mapping conjugate inverse wishart prior gibbs posterior unbiased alg joint tb conjugate x generate datum iw parameter j posterior sample b estimate covariance mean prior parameter moment p require pass show per give rmse solely filter measure filter unlikely experiment choose gibbs system px unbiased filter error gibbs sampler tb cc filter filter nonlinear dynamic system exactly setup filter quadratic measurement nonlinear run start expect rmse c cccc filter filter rmse rmse system error coherent smooth I high consider consistently improve coherent gibbs distribution area confidence area area latent variance filter preserve lead differ infer infer system part unlike particle degeneracy due filter computationally efficient sufficiently approximation preserve infer infer covariance joint increase infer remove time due introduce relative avoid error gibbs able evaluation requirement elliptical slice potentially combine gps model publicly gaussian gp paper smoothing extension gp general perspective filtering probability filtering distinguished use determine filter kalman filtering smoothing compare filter robustness acknowledgement thank valuable suggestion support intel partially foundation research computer general filter smoothing allow show filtering distinguish solely approximate covariance novel filter straightforwardly insight smoother robust extracting respective model play signal decade context filter control smoothing efficiency filter appear filter smoothing implementation concept require get lose implementation derivation kalman nonlinear kalman kalman filter
update proportional magnitude motivated compressive sense author result framework algorithm cost sparse empirically superior performance rate steady state compare system furthermore penalty penalty addition time vary result regularize provably dominate conventional counterpart proper formula white signal introduce furthermore group adaptive useful enforce regularization propose close expression guarantee provable white demonstrate advantage simulation regularize mis base organize section iii filter identification iv simulation summarize proof denote upper letter transpose denote briefly derivation vector goal identify impulse input desire independent instantaneous error cost instantaneous size control steady yield scenario knowledge study subscript allow constraint time instantaneous coefficient counterpart term always regularization manner approach identically mutually vector filter bind regularize provably conventional upper bound true show regularize robust superiority simulation correlate provable suitably negligible application transmission acoustic divide sparse identification system location non bind impulse non know suited adopt surrogate regularization equation regularization alternative small yield wise many application often group zero originally whole index mixed encourage among coefficient inside reduce ensure denominator knowledge follow generalization q weighting zero coefficient assign plotted fig initially gaussian input variance zero filter implement filter refer project vector onto ball use tuning time estimate upon steady behavior well outperform operate memory begin perform well however storage computation memory computation investigate fig highly exhibit unstable converge behavior local minima indicate iteration phenomenon ball fashion projection mis specification rather correlate ar process system noise benchmark filter set employ calculate curve plot white input empirically confirm severe degradation signal independent frequent signal correlate b study initialize change active shift signal unknown repeat achieve conventional filter section system start group response input treat benchmark divide coefficient set number coefficient number average db steady white coefficient update next parameter filter value white average curve fig outperform improvement correlate filter system initialize response coefficient inside block iteration block active propose general filter penalty close provable filter sparse identification demonstrate simulation regularize simplicity memory requirement likely steady extension order filter extension filter study accord note
cluster scheme determine scheme r alternative way map require existence f use condition exist informally suggest finite define metric space finite metric space produce partition say whenever say metric space finitely existence generative let endowed fix write remark x bp depict diagram conversely z ki diagram refinement refine partition prove pick point f z regard bb endowed restrict hence bx bp finitely metric definition finitely represent fix metric px ii z z z xx iy k z p reflect follow assume recall uniqueness assign block piece piece proof r px xx xx claim indeed exist x conclude half block belong px qx x piece lie scale axiom cluster c assign metric block let metric consist piece piece piece fix finite write xx piece assign block space scale piece pick pick subset easy belong assign partition follow every partition proof find behavior also kk cx assign metric pick pick distinct block partition arbitrarily force produce partition piece prove partition rich cluster scheme include already scheme practical cluster meanwhile cluster metric admit interpretation produce degree point reflect sensitive point counterpart consideration cluster implement two together consecutive inter exceed proposal provide metric clique shorthand allow could densely connect cluster metric consist vertex together persistent equivalence relation object act regard easy implement hierarchical linkage dataset preserve cluster mean output name agglomerative hierarchical begin cloud construct dendrogram describe merge lack come correspond force order root output simply root tree amongst impose larger demand pass powerful unique satisfie natural condition complete linkage linkage agglomerative clustering explain metric metric length refine counter linkage htb uniqueness prove instead hierarchical xx persistent mean persistent two persistent element block partition point block block equal another characterization linkage book proof hx r effect lie lie skip follow lie claim lie lie lie persistence preserve hence block conclude skip b assume imply belong skip equivalence class equivalence equivalence distinct would imply contradiction write x block relation x rx x depict r conclude b observe condition point cloud multiply example rx trivially partition replace persistent exist block impose persistent dendrogram check persistent obtain evaluate indeed finitely belong minimal belong belong block let xx xx permit restrict positive define equivalence relation require cardinality class singleton persistent check xx rx regard lie cluster represent arise definition construction spirit thin chain point additional increase hierarchical clustering manner write paper graph fit case determine could produce classification desirable ingredient topology vary construct scheme diagram reflect stability independent believe conceptual cluster algorithm qualitative quite notion many argue valuable well claim open proposition stanford optimize partition framework happen impose refer set one analogous obtain uniqueness theorem vary notion space require rich family sensitivity technique hypothesis apply deal interest regard exploratory equip input might optimize choice clustering family define desirable property cluster practitioner notion satisfactory however one thing intuition fact incorporate linkage nice method average linkage sort sensitivity unstable theoretically support theory sound incorporate persistence little regarded consist collection ad method generate picture one demonstrate construct plausible map metric axiom cx xx cx upon reduce distance simultaneously consistency choosing hierarchical prove slightly present understanding theoretical develop rich version familiar include familiar regard recall path component topological continuous point path crucial one set key topological decade reader familiar map interpret notion symmetry one construct useful conceptually basis space value circle construct spherical harmonic function regard invertible rise understand organization topological algebraic homology powerful key homology determine homology group equation field rational observation associate symmetry theory day study transformation law modular form continue powerful present classify cluster density domain moreover construction similar set rule assign induce replacement topological space consider nest increase map point hierarchy give respect quite restrictive single linkage requirement restrictive enough permit specification isometry space implicitly property require existence mathematical powerful axiom odd bipartite generative interpretation parametrize connect criterion point permit respect find collection account include effect account whereas scheme find scheme construction rely study finally scheme admit generative composition linkage change input believe incorporate cluster operate space isolate space construction application topological method connectivity paper demonstrate notion induce operation category discuss construction output standard scheme whose set category finite partition b p fx constitute category scheme persistent set say persistence preserve refinement object persistent persistence preserve easily persistence preserve map persistence preserve persistence preserve three range value hence one proceed similarly collection object consist map easy composition clear therefore another map category consist isometry finite whose eq proper special finite space yx x next concept formal construction fx clarity refer pair letter diagram equip carry without multiplication respect inverse set x ff xy regard define framework study define object object hc input finite regard category object procedure assign follow case partition persistent concept refer cluster object happen let stand accord map input output diagram order view specify map object category diagram eq discuss section study standard idea order inclusion study category demand require algorithm category uniqueness category category order real
length useful understand imply almost surely mass sometimes purpose suffice moment case dirichlet suitably rescale pd sequence self average limit random extra assessment cluster ultimately govern whole random beta atomic converge detailed moment exchangeable partition self finite big big cluster average confirm size accordingly group cluster relatively dp mass beta autocorrelation priori decrease atom great consideration guide prior obtain weight latter specification lead characterization identity true evident determination problem sensitivity dirichlet process cluster example short asymptotic relationship choose priori low matter suggestion present consideration hand long base application cluster otherwise second moment autocorrelation possibility work end form initially suggest stick break characterization dirichlet stick breaking characterize dp sequence generate dp beta special matter stress stick analogy stick break new tag stick evident consider dp piece unitary stick define one prior straightforward dependency exchangeable series datum realization measure random beta define marginally center dp mixture model parameter interest hierarchical beta conclude section exchangeable conditionally identically distribute statement see entail effect assignment observation describe e indicator retrieve analyze refer th draw base datum assignment label turn scheme move space mix easy assign usual indicator set sequence block cluster analogously indicator rule appropriately adapt believe merge move problematic imply exchangeability cluster configuration fast integrate however interested possible unique gibbs observation conjugate otherwise hasting addition metropolis equation adaptation well beta thorough adequate elsewhere provide specification example develop popular simulation range detail inference beta generate point use assess typical analyze section variability model default well basic specification fitting gamma hyper mean large fit dp model concentration basis compatible parameter dp fit package framework dirichlet stress underlie exchangeability dependency report aim provide synthetic goodness fit cluster assignment together follow terminology metric correct assignment predictive bias output parameter cluster original nearly compare bias incorporate intrinsic generating process typical nonparametric ability ground relative magnitude hyper material specification choice remark beta mis specification exchangeable exchangeable dp sensible study generate normal component deviation either robustness level weight short priori rescaled choice gamma relatively simulation beta robust specification assignment correspondingly parameter close process accordance finding around simulation inference beta affect specification confirm represent guide choose estimate mixture material result specification overall remark beta specification beta cc truth assignment table compare specification generate process gaussian variability bias hide model specification generating level ccc c c yu study process markov process hide spend state markov refer duration variable duration generate dataset hide state binomial term negative poisson simulation present robustness beta beta hyper assume behavior beta hide state result report implement package alternative hmm markov affect hyper induce beta generate middle column allocation beta attain hmm avoid overall plot induce hmm fit seem shorter contiguous identical hand beta hmm fail fair representation practical experience default informative beta formulation accordance long contribute wide range generating mechanism result note beta classic link breast medical raw copy gain loss array comparative genomic precisely intensity cancer microarray presence dna repeat fit tumor region high contiguous typical replicate frequency genome copy gain loss genome location copy lowest accordingly low copy gain say minimum say current mean plus two consider level iteration report bottom expect tend localized genome increasingly reinforcement embed correspond live adjacent point shape different adapt compute rate fdr neutral value minimum fdr determine correspond copy describe paragraph region account string consecutive call constitute subsequent fdr alternatively specify information available literature optimality procedure discuss also value previous location platform kb fdr end rp rp rp rp rp rp rp rp rp rp rp rp apply methodology publish match location queue take hour also fit negative length parameter estimate technique length share lead deviation purpose genomic location breast cancer tumor file genome project consensus genomic size list list likely list particular reference comparative beta semi detecting consensus genomic give consensus genomic location span beta correctly location versus hide hidden course take single illustrative study suggest detect array account improved compete approach characterization specie beta random defines exchangeable discuss beta process specification latent beta use hierarchical finally study outline illustrate beta useful array complement tumor suggest clinical study currently speech flexible state model exchangeable formalism heterogeneity exchangeable sequentially order enable unknown single transition state persistence since beta convenient underlie g complicated issue framework dependence cluster arguably major obstacle wider rely hyper distribution suggestion case enough differently experience dp mixture suffice inferential latent conduct inference mcmc discuss specific beta limitation method inference scalable facilitate believe possibility cluster imply directly complex development substitute probit specification observation record covariate curve imagine sort paper model base call asset across market liu acknowledgement support national foundation dms national institute health grants research office research european community fp grant agreement author associate anonymous ex definition proposition example primary c
three example ex moreover simulate increasingly realistic scenario span kb project ex correlation useful column indicate pairwise follow firstly induce x x z jx j introduce relationship hereafter complex placing effect exception vary finally range difficulty six example dimension intercept intercept analyse idea create firstly involve strong covariate secondly since simulate interested select belong motivation ability combine version secondly side copy design covariate high complicated challenge covariate different challenging simulate contaminate sample size model respectively retain r know account without negligible easily jump mode simulated project population originally contain snps nucleotide redundant example linkage naturally force correlation snps test ability ess report ex replicate ex change across ess example visit burn tune temperature exception example fix value remain equal correspond uninformative section fact ess panel replicate ex overall ess variability albeit reach respect thank tune temperature distribution overlap allow exchange panel replicate chain mix gap tune drastically exchange information overall move implement acceptance delay stay stable deviation ex variability model across example average chain believe performance operator tune small simulate relate ability ess remarkable superiority find explanation ess indeed ess number actual drastically ess ex explore conclusion rich parallel ess space competitive art york genome selection use bayesian predictor implementation discussion ed uncertainty vs fully bayes prior bayesian em approach em j uk em liu evolutionary em green delay reversible jump search gene underlie nonparametric regression combination basis function reversible chain em variable application liu new york york j em proper drive wang em accurate assess regression north ratio de f example ess move panel mc use retain replicate mc move plot region uniform accumulate ess block top panel vertical correlation grey leave triangle ess vertical vertical dash replicate ess upper green triangle replicate probability inclusion right triangle ess ccccc gp min ess ess ess determination well visit unique visit chain specific dr exchange stand rejection exchange move cc ess ess ess mc ess l ess l ess ess adaptive ess ess ess ess ess ex ex mm mm swap exchange c c c p ex ex ex l stability l min mm cm theorem college uk model datum many field make operational upon monte design thus datum example demonstrate extensive simulation evolutionary fast scheme model selection discuss recent advance make different coefficient propose algorithm huge algorithm implement idea evolutionary monte difficulty sampler face high become covariate approach operational indicator illustrate prior far hierarchy present specification namely devote sampler portfolio include good algorithm variety algorithm contain conclude remark response dimension describe tp want variable selection interesting large predictor interpretability bayesian perspective place latent overall exponentially predict gaussian coefficient column correspond intercept contains recommend intercept separately assign gamma q value jeffreys specification joint far write main neutral lead prior scalar replicate likelihood rise first conditionally specification unified tx clear alternative prior binomial illustrate eq choice ep pn centre prior attractive firstly possess posterior standard recommend skewed measurement feature appeal constant become tx reason next qr regression despite prior complex comprehensive place back shrinkage z hereafter lead analyse prior point prior likelihood close something advantageous even though laplace derive appendix never define tx x demand extra determinant fix different predictor order place value illustrate place regardless specification use qr know difficulty multimodal unified mass space know tackle parallel monte hereafter green hereafter inspired extend algorithm finally neighbourhood predictor propose issue related dependence strategy hereafter add temperature swap every non try genetic scheme integrate conditional population temperature chain population retain population update move genetic ordinary metropolis hasting selection swap probabilistic measure operator different exchange move crucial allow propose several novel aspect two fast scan hasting particularly bold pattern predictor new adaptive proposal update sketch useful implement discuss furth new paradigm complex predictor space believe move use indexing indicate indicator population local fast sampler mc york add move require indicator select try rather affect hereafter affect chain update indicate chain l l main relate gibbs evaluate full cycle implement least consume rely however notice cycle much likely sample large thus mc scan hasting scheme hereafter pt demand idea index gibbs step sampler probability use base size temperature th chain therefore aim save simulated finding save computational consuming l p l variable rejection step temperature indicate chain l update approximately l l indicator higher become indicator give mc embed move select pair chain firstly boltzmann tf log assume use boltzmann chance select chain configuration probability refer suppose new latent chain chain different operator select random adaptive move uniform indicator chain essentially try swap variable correlate practice block calculate retain j consideration specific move exchange extreme operator receive chain exchange adjacent chain temperature limit capacity mix idea delay move exchange chain full detail bold well simulated temperature liu ingredient preserve accuracy population consideration limit recommend example relatively subsequently monitor acceptance delay rejection exchange operator strategy laplace quadrature interval strategy follow integrate implicitly coefficient situation arise product reach response chain marginal likelihood see chain unstable make hard reach increasingly place close temperature balance acceptance move instead lp whole condition allow fast come extra relation sample useful apply adaptive within asymptotic enforce go I benefit amongst gibb proper give upper exploration tail proposal evolutionary ess denote ess prior without generality independent matrix rescale conditional global move ess population state em point select sampling scheme independently moreover complete gibbs perform delay rejection exchange operator delay view update qr qr restrict help burn discover genetic variation quantitative datum type quantitative trait find parsimonious set covariate illustration report gene ess table proposal reach acceptance deviation quickly inside see mix panel ess case reach inspection reach automatic tuning distribution exchange chain confirm without reach exchange rejection exchange operator acceptance experience detail burn implementation ess size uncertainty support knowledge mean variable small ess couple visit ess chain visit drive discriminate compete relate snps select genome gene predict divide bin finding bin analysis whole bin mix note influence figure clearly control univariate example large around burn index believe visit big gain would ideally suit superiority ess scheme illustrate wide portfolio enable search complicate I ess mc move ess respectively comparison fair version ess burn finding ess reach visit ess model prior rank datum provide indirect great superiority explain compare figure exchange ess rather mc contrast ess retain chain mode see look right tail kernel record chain regard second comparison ess operator probability high acceptance already notice efficiency briefly comprehensive ess simulate fair assume prior simulation set appendix behind example g realistic include effective exchange show collection move implement remarkably specifically move temperature ess far swap equal overlap parallel hyperparameter size inclusion posterior overall ess algorithm ess explore find true goodness ess ability towards compete explanatory ess contaminate ess pick ess remarkable superiority especially estimate large design bayesian sampler competitive spirit example also version visit rank ess effective extra enable local mode balanced gain chain develop good chain bold conservative automatic burn ess range situation multimodal tackle important index update move scan metropolis local gibbs large simulation superiority move model variable update particular difficulty contain consume illustrate spirit present far response response identification regressor carry multidimensional acknowledgement adaptation grant detail omit related sampling indicator l j th normalise version gibbs binomial crucially l derive scan metropolis hasting evolutionary define l j moreover introduce metropolis gibbs similar proposition omit check first calculation marginal j l l l scan chain requirement acceptance since normalise work follow chain propose value bernoulli one similarly otherwise select covariate efficient scheme iteration gibbs low gibbs expensive become operation move exchange extreme second chain receive chain acceptance adjacent chain temperature capacity mix obtain implement related delay rejection green gibbs possible pair rejection exchange try swap usually apart temperature reject traditional chain drastically rate flexibility extra evaluation maintain balance report suppose swap since random difference move delay rejection acceptance pseudo detailed balance alternatively attempt exchange operator far
nonlinear get back concept nonsmooth nonconvex accord f v nonsmooth generalize point nonlinear condition useful euler positively homogeneous characterize critical condition necessary nonlinear critical sf sf prop continuously critical nonsmooth thus single value everywhere small eigenvalue building block iterative eigenvector transforming try solve fail problem otherwise unbounded convenience whereas yield rescale inner formulate completeness convergence suggest multiple eigenvector small htb initialization sf sf rf sf initialization g sf sf kf sg rf solve speed quickly suggest consider propose tailor case generalize omit constraint produce alg sequence terminate produce sense differentiable point throughout proof notation u objective problem lemma note homogeneous minimum attain set feasible otherwise rf sf sf ff k nonnegative limit every converge sequence subsequence towards definition subdifferential homogeneity contradiction converge fact everywhere minimizer argument convergent positively homogeneous homogeneity q imply generalize positively homogeneous functional subdifferential hold q subdifferential q sum yield limit become substitution f p rf necessary imply homogeneity eq note replace even lemma minimize eq attain p ff kt rearrange type proposition give minimizer problem terminate attain minimum sphere sequence bound convergent subsequence clearly ff ff nu ff pf contradict fact minimizer imply argument subsequence subsequence towards optimal inner ff k important convergence vector satisfy instead early one limit accurately initialization spectral graph method overview undirected graph find relaxation cut ratio normalize cut weighted vertex weight partition connect normalize motivation c compute cardinality subgradient condition inner invariant opposite initialization htb weight accuracy g produce terminate conclude terminate minimizer invariant verify divide side converge eigenvector f k satisfy negative terminate case use low converge limit contain sequence subtract proceed analogously though spectral lead spectral c else f f invariant generality assume c distinction sign follow fact q easily contradiction cc dc second f second eigenvector graph result cc follow solve nonsmooth subgradient exploit dual w bound note rewrite non empty standard min corollary unit give euclidean unit transform regard lipschitz primal solve use fista convergence fista lipschitz fista good descent make modify fast alg tb input initialization rs rs rs reduction subspace maximal compute covariance pca want like component human enforce component trade standard constraint cardinality simple thresholding component focus good candidate sufficient globally approach et al simultaneous component penalization enforce sparsity control recover whereas yield trivial easily fit function inner become closed analytical objective positively optimum iterate attain derive noting objective concave equality thus optimize get principal component subtle difference thresholde current whereas fix lead iteration htb control initialization total eigenvector laplacian construct cut error point initialization initialize unnormalized initialize laplacian propose tv yield clearly spectral avg avg unnormalize spectral resp point successively number reach minimized scheme initialize thresholded eigenvector laplacian normalize random initialization show eigenvector outperform effort run well however want bi cut least spectral thresholded gene compare power plot method coincide observe art b cs statistic problem linear critical nonlinear apply achieve quality beyond useful inverse eigenvalue semi machine view
monotonicity key n monotonic denote zero define full rank concave concave suppose hence definite ordering agree except relate follow algebra identity monotonicity strict monotonicity entail entail view calculation form contradict hold contradiction rank right hand positive hold first show limit inspection monotonic strictly monotonic strict monotonicity theorem let generate ii yu present yu example even inspection zero point eliminate yu omit analyze convergence let assume slightly weak starting pattern practical assumption tractable convergence section present notion algorithms context mapping convergence eq global rate spectral modulus eigenvalue restrict linear implication impose notion eigenvalue lead yu situation shannon interval surprising exceed nonnegative show precise iteration define speed appeal toward converge secondly slow design iteration reach increase converge note imply intuitively explicit q resp mapping corollary imply hence prove algorithm fast reasonably practical quite slow explain et version monotonic discuss iteration behave dynamic twice practical assumption early usually reasonable measure eq q similar equivalently guess mathematically employ dynamic design side first column interpret speed equal accurate ratio become concern dynamically agree ratio believe situation iteration design space dynamic acknowledgment author li david valuable pt lemma multiplicative compute strict monotonicity establish variant formula explain al keyword computational aspect approximate seek determinant closure linear et iterate iterate resemble closely read design intercept broadly applicable intercept refer ii interest example et al yu recently yu work aim extend monotonicity thereby monotonically
recursive dealing instability solution parameter tool g choice explanation portfolio zero portfolio optimization use statistical estimate reason become f later online moment reason find substantial discrepancy financial standard recursive could version et naive mean expect stable may update section core development allocation efficient nature algorithmic us notation lagrange often approximation qr q recursion q equivalence filter understand similar filter robust statistic choose datum window efficient compute recursive deviation consider exponentially median correction huber replace slide preliminary investigation term financial inherently ability accurately dependence pointed instability portfolio weight cause adopt low eliminate datum optimally lower find frobenius see denote return update less straight time objective n equal one eigen decomposition yu operation inversion specifically yu calculate eigen dimension q incremental free numerically initial perturbation portfolio datum initialize balance allocation return period generate large portfolio base recommendation find extremely scenario balance incur eigen procedure filtering criterion portfolio selection balance approach filter complementary firstly deal separate var secondly account outlier quantity embed account outlier var adaptive account need advance place every shift environment rank set advance give example subject cycle would significantly expect may expectation allocation link theory asset allocation version technique dataset rate perform exchange fx see american cover approximately year daily www com approximately year yahoo http uk yahoo com adjust financial daily market http edu pages ba daily yahoo data observe allocation financial trading subject account looking develop could distribution huber significantly naive compare idea way avoid look ahead bias var strategy portfolio balance instant go balance window balance datum balance period initialize day actual portfolio balance balance time trading transaction cost strategy use transaction cost assumption since transaction cost often substantially employ financial application day calculate daily gain win draw perhaps need explanation percentage constitute peak cumulative percentage term measurement frequency trade absolute portfolio balancing times fx equally spaced value exploratory depict contour ratio evident exhibit become evident structure suggest good performance basis plot approximation extent vice versa computational issue memory storage year substantially imply purpose decay great run balance day first third fourth var initial converge balance period speed batch code version dimension compare batch increase batch second strategy table clearly ratio volatility underlie maximum strategy volatility balancing period partly infer truncated portfolio contract var whose literature indicate imply naive return exhibit bad strategy growth necessarily price familiar pattern report display ratio perform marginally satisfactory financial performance tend increase success var link naive figure remark section secondly var method fair train year insufficient algorithm although case smoothness beneficial outcome well discussion final economic business wise many perform poorly financial trading return encourage extent suggest useful first procedure drift computation noise potentially derive table drawback potential naive naive assumption direction position ann var var naive var naive var var naive var var var var var r gain ann ann var var var detail method need technique benchmark measure et extend transaction bid ask portfolio well balance explicitly incorporate lead addition work make adaptive require statistic finance signal computer drawback allocation remove online portfolio require standard wise left exception zhang li little methodology currently g investigate portfolio algorithms portfolio management stream stock return technical fast recursive portfolio r update singular good dynamic portfolio inefficient v r portfolio finance appear optimization h portfolio program new line use multiplicative huber p york wrong dynamic portfolio algorithmic http www return portfolio li ng portfolio period portfolio capital market gaussian expect market exploratory datum mining trading recursive presence tail portfolio early history decomposition shrinkage via lasso stanford university zhang zhang online quadratic motivation context trade recursive portfolio exponentially regularize var simple statistic empirical financial allocation dataset benchmark allocation term demand portfolio management portfolio capital fraction capital asset fraction capital asset know weight portfolio portfolio maximize fix maximization small portfolio prefer portfolio small return portfolio small desirable capital allocation portfolio return capital little interest visit wish mean portfolio unstable difficulty substantial improving include al b et impose portfolio portfolio allocation concerned require historical oppose recursive mechanism procedure computationally efficient stream nature handle asset trading perspective regarding take automatically allocation statistic
small zero different hence overlap work image strategy calculate yu rewrite require q equivalent eq clear explicit component component make sum denote whether observation treat derive one treat indicator expectation check tucker condition maximize determined iteration em ensure efficiently yu less informative entire collection slight extension fix monotonic yu yu rate work within convergence comparison automatically combine comparison upper possible hence recommend helpful optimal choice convenience current eq uniquely suffice inspection show potentially transfer mass section neighbor em overlap fast scheme exist component severe apply example nonparametric assume draw normal mix distribution put denote mean increase adjacent density conventional em may nearby component hold somewhat pair element number support perform e use eq choose naturally implement tool composite near find add considerably speed usually start interior receive subsequent near bad support may eliminate one easily vertex direction derivative maximize choose choose add em near exchange convergent actually show add globally convergent near yu design report performance near output conventional iterate one overlap density room use combine hope near complement focus purely modification purely one conventional poor many guarantee start global maxima yu denote output step subsequence converge step index equivalence reveal uniquely say derive framework define observation interval inspection inspection failure censor may censor unit censor inspection variable failure observe leave censor right censor censor observe fall within random notational convenience open close though trivial set maximizer jump em type define maximize implementation em section censor take implementation substantial order bootstrap resample repeat briefly give em zhang doubly interval censor show admit implementation censor effectiveness algorithm doubly censor propose em decide evaluate include bivariate censor work progress simulation random censor degree censor start criterion experience benefit insensitive neighbor conventional em replication second censor heavy censor em mean count either situation reduce conventional large reduction significant number remarkable direct slow much bad moderately censor heavily censor somewhat moderately heavily censor time censor significantly higher another fewer censor moderately table superiority augmentation design type maximize likelihood advantage overlap conventional near exchange censor work conventional natural ordering censor bivariate censor variable censor present inferential challenge extend encouraging extension accommodate truncation censoring facilitate semi parametric adopt van type maximization one strategy former maximization investigate potential would helpful censor conventional em contain special fast zero zero note equivalently algorithm cumulative initialize obviously valid output output agree algorithm compute induction compute clearly conventional implementation apply track algorithm near exchange affect limited implement belong satisfy entire algorithm input sort slightly set censor em strategy efficient van proportion ease monotonic improve slow overlap component speed stability overlap censor improve adjacent consideration carefully improvement realistic augmentation doubly censor mixture density multinomial miss partially al example mix distribution support closely van among well censor al computing exchange wang em em widely use experimental yu emission shannon ar density mention motivation fast censor partly overlap em monotonic strategy overlap augmentation em observe mixture collection step dramatically expectation scheme van effectiveness wu liu van liu nearby component pair strategy maximize log overlap explain datum illustration censor effectiveness simulation bivariate censor implementation em type take
appendix add subtract break infimum property expectation nest expression part arrive loose indeed pick supremum due bad hand inside expectation condition identically independently draw identically history argument rademacher proceed identically proceed fashion way arrive b arrive pass supremum tree start immediate successive expansion coordinate zero argument assume q ball jensen inequality conclude lemma proposition jensen eq desire proposition specify verify smooth see definition tree martingale come smoothness tree tree far integrate trivial depth let path element sense pair value q successive let j trivially verify hold affine argument sequentially irrelevant specialized response put ensure claim smooth fix mapping fix cover q interval interval enough tree precisely ti I belong plain word partitioning define q simplex function homogeneous need prove close consist hence supremum value shorthand tc q pick appropriately sum integral fix later find player subgaussian tail augment restriction player belong pack choice observe simply equal response mass reasoning quantity supremum martingale choose third bound put upper get adversary almost sure similar divide episode episode length player play subgaussian episode step adversary infinite regret incur union episode choose ensure q prove section separable banach property value define sum follow embed everything work replace key define scalar function q derivative case g x g g plug immediate previous whenever first note second eq valid satisfy whenever control style union family divide optimize plug give expect recover choose slightly take outside proof game eq application minimax subtracting infimum nest expression three arrive mention replacement appear loose pick supremum random draw study wide notion well beyond framework simultaneously notion internal additive calibration perform know study focus algorithmic online external learning present extend wide appear framework give external internal recover extend improve quantity sequential rademacher play derivation decade reveal somewhat biased former focus primarily understand supervise learnable risk minimization serve learnable approach dominate decade base unified tool develop situation unify manner provably learnable feasible yet show scope precise characterize external within circle well know base rather banach ever tool show utilize successful situation section discuss result greater break contribution payoff formulation might appear show framework game upper various natural tool allow performance include smooth non payoff generalize cumulative consider abstract payoff rich whose complexity average number combinatorial usual regret regret see several equilibria infinite banach specifically space shoot calibrate forecasting improve upon rate calibration outcome e global al provide notion regret suited environment tool example great preserving study game unlike research online general problem proceed long study able even hope believe bound arise inefficient recover minimax often inherent describe organization serve whether progress paper whether specific decide specific risk potential first page avoid define framework appear establish general framework derive overview detail skip sake defer paper however basically notation omit abstract problem player adversary course generality say minimal assumption meaningful let move learner adversary generalize round end information set player adversary move separable banach player randomize learner minimize measure eq value payoff additive form cumulative sequence mapping adversary maximize game concern identifying general play payoff class formulation work likely one seem summary mainly payoff payoff transformation emphasize payoff mapping transformation payoff payoff mapping function act modify choice payoff say transformation write payoff transformation mapping shall abuse use mapping transformation metric probability adversary formally strategy mapping respectively adversary sequence mapping learnable game statement measure discuss sure might abstract mapping framework nothing illustrate external regret z mapping eq cover notion swap banach close z easy ensure vector vertex calibration detail global game let scenario consider capital ease already shorthand expectation distribution p f xy bx ba kk supremum infimum write quantify range understand denote tt notation payoff singleton sequence transformation banach let denote dual corresponding ball upper beyond section version rademacher progress additional assumption refine upper definition generalize complexity global payoff payoff function payoff mapping supremum take value depth I omit sequence transformation base write adversary appear definition rademacher external turn act player choice section notion study external sequential expand version online ability perform convexity condition third last twice first singleton consist let mention decompose game interpretable decomposition yield well constant nonetheless inequality essence treatment fact bind third inequality distinct achieve theorem quantity former choice draw mixed complexity draw crucially easy sequential oppose since random sign mathematical arise easy control martingale typically provide specific response adversary similar condition arguably set payoff assumption term proof might appear loose nevertheless substitute pass presentation decide give replace z z z z transformation instance conversely new x become exactly analysis detail smoothness cover existence banach crucial term increment yield informally promise appear consider say say exist finite smooth argument linearize term increment establish inequality argument true assumption lemma bind third lemma smooth bound complexity sum smooth q taking reduce gradient reader familiar notice resemble sequential rademacher studying ask finite question finite transformation finite sequential complexity gradient normalize account cover average power calibration basic power norm separately smooth unfortunately bind value smooth z concrete smooth calibration example statement assumption finite payoff z general case average turn get good payoff transformation z f result exponent previous smoothness smooth probably g f absolute payoff result expense amount resolution proceed however notion generalization notion introduce provide payoff whenever payoff invariant mapping complexity identical section detail comparable tree consider section particular assumption generalization eq particular drop soon repeat statement hold first number adapt coordinate sequential take familiar value tree simplification notion invariant transformation generalize dimension subject next section assume invariant definition dimension tree class exist dd define version generalize tree large tree invariant payoff base control instance proof tree role tree number bound value tree generality evident cover sequence transformation lead appropriately number bound transformation little hope non trivial good news variability cover external time obvious external decision appear dynamic path general one would assumption naturally capture notion change transformation cover comprehensive list payoff obtain piecewise change assumption proposition payoff norm transformation think segment accumulation point payoff transformation vary scale cardinality simplicity though tight payoff sequence payoff vary much sequence payoff transformation sequence claim cover tree increase payoff let large close clearly eq control solely move define notion repeat whenever proposition immediate strategy many since action strategy check face function low term convergence set minimizer expect singleton minimizer specific literature establish framework decide major unified simple inherent comprehensive randomized expectation randomization sufficient sure game randomize least randomization natural expectation measure exceed value non random inequality integrate well integrate respect fix existence player adversary exceed suffice want prove existence consistent strategy round formal round game calibration game existence strategy call trick rest devoted probability version range think conditional distribution decomposition bound roughly speak typically response adversary third third term apply bit mild draw tangent variable
image phase couple factorial sigmoid impose space phase couple generate unit hide unit square interaction restrict call build reference couple produce unit unit example unit generate dependency implicitly activation implicitly conversely implicitly unit ignore signal datum gradient ascent likelihood express energy visible bias possible contrast unit condition exact integrate hidden statistic update denote indicate expectation calculate straightforward however distribution equilibrium approximate summarize divergence approximate run run hybrid monte produce momentum run rejection weight function panel order model berkeley color patch preserve unit initialize random variance bias rate update length vector length vector grow match subspace set column normalize unit batch various part sequentially adapt parameter fix add hold adapt examine localize oriented band roughly quadrature fig learn pattern rigorous analysis need verify position weight hide learn hard visualize remove view filter high column eq coupling subspace matrix find sort pair filter map plot direction angle couple separate suggest explore couple hide varied matrix choice rather factorize factorize additional motivated model subspace structure phase like thank would thank document nsf grant nsf c minus ex minus plus minus ex ex plus minus minus ex plus plus minus minus plus minus minus minus pt minus pc pc plus plus minus plus minus pt minus pt pt pt pt ex center institute california berkeley berkeley describe capture amplitude phase factorize third boltzmann capture structure model dependency output subspace local norm subspace quadrature additional unit dependencie subspace phase form coupling difference spatial image recent year e decomposition utilize infinite mixture gaussian image focus radial step order phase across contour shape image machine phase restrict quadrature pair subspace couple condition phase model couple view within attempt orient filter art natural mathematical factor negative negative model learn dependency scalar way correlate noise angle pair filter coefficient high level high order filter related pairwise logic multivariate describe boltzmann local phase amplitude dependency variable dependencie combinatorial phase couple pair neighbor connect coupling unit cosine connect couple stroke bias omit review name covariance angle pair dependency phase coupling phase define energy term define entry bias filter sum hide however deterministic image unit combine produce covariance representation describe mean contribution connect unit form contribution rbm hide visible unit mean specify unit unit invariant local well model subspace
operator version measure result conditioning operation computable even set construct variable measure computable every individual conditional least construction encoding time joint computable central infinitely relative explanation phenomenon circumstance condition computable conditioning computable discrete corollary situation condition noisy capture computable absolutely corruption independent additive contribute give computable enable theory notion develop computable build recall potentially correspondence finitely say computer program output element eventually c precisely basic notion see sometimes real computable approximate e give report rational real lower rational low computable computable theory convenient largely use metric enumeration dense uniformly denote center bs enumeration metric characterize computable dense eventually string standard enumeration string metric enumeration denote borel algebra ball measurable computable sequence point computable space familiar computable computable every open set enumeration ideal note open intersection modulus subset domain computable enumeration computable sequence computable image set set continuity let computable program restriction equal consider canonical representative computable intuitively map input randomness induce independent coin randomness formalize infinite basic extend henceforth basic relation instead say drop hold font variable measurable measurable p neighborhood open ball computable also say characterize class computable measure term computable metric computable computable metric computable measure let real hand open computable empty whole computable borel open immediate almost computable ideal also ideal enumeration computable form almost union enumeration ideal ball computable e k enumeration almost construction almost characterization almost computable real let c theorem e compute sequence almost union computable real supremum desire computable computable computable computable computable informally event occur occur measurable write conditioning precisely insufficient define interested measurable set atom notation modern kolmogorov solution space function b uniquely set e surely refer generic equivalence version define conditional measurable variable natural proposition version natural want individual conditional order distribution detail g call probability measurable show measurable sx sx measure everywhere agree random variable version conditional map b g image conditional build elementary probability hand measurable b demonstrate almost computable measure corollary computable almost numerator computable real denominator likewise finally computable conditioning notion distribution metric probability give computable computable correspondence allow computable index ball make space computable rational subset topology metric let rational subset computable uniformly ball center write c suffice ideal characterize center constraint bt rational decide whether show uniformly computable computable uniformly interpret computable computable computable space uniformly rational c open weak open computable uniformly e computable function computable space density lebesgue measure therefore immediately lebesgue density rescaling admit lebesgue admit lebesgue measure positive version probability q x version unit eq result index I conditional let numerator k r computable open u rv q u u u exactly distinguish case decide equivalently uniformly computable follow computable computable sense although encode infinite computable compute set map computable obtain pair variable pair continuous nan conditional f computable relation let dense consist rational coefficient computable metric computable probability computable borel computable computable measurable equivalently discussion let consider derivative measurable computable conditional n express numerator one denominator along line proposition low decide rule ask conditional despite conditioning restrict begin address fundamental obstacle admit nonetheless computable possible think bit stage occur long simply stage variable rough cause set infinitely lebesgue everywhere dx n one lemma random surely variable py dx integral computable early uniformly k q kx k straightforward admit differentiable result result measurable let distribution disjoint open contain remainder similar replace replace k computable carry show simplicity computable value random value pair everywhere many setting understand situation inference possible begin common conditioning elementary event condition discrete variable computable metric countable let characterize almost numerator take computable note could proof sampling several formalize unbounded computable finite countable metric space characterize version conditional computable metric space exist computable computable result easy distribution discrete metric let computable discrete distribution distribution way calculate say conditional first definition integrable density complete dx parametric family gamma etc admit case rule give result bayes variable version satisfy dy dy denominator finite borel dy dx dy dy dy dx imply denominator set point denominator characterize dx definition bayes rule dy hypothesis ct respect probability low rx rx hypothesis integration continuous complete integration computable bayes rule computable density computable computable computable conditional computable immediate situation observe corrupted let computable computable computable variable absolutely computable pour show continuously computable computable derivative continuous computable computable corrupt computable even mean introduce theory capacity yet cause channel capacity drop prevent encode bit real much noise couple noise uniformly computable absolutely density magnitude nice reduce time despite raise question show classic exchangeable conditionally I sequence condition measure given version cover nonparametric statistic acknowledgment preliminary version appear partially support nsf dms dms publication grant publication reflect view foundation national mit newton international fellowship college thank comment regard computable comment em rgb claim proposition theorem font theory mathematics subject primary inductive inference success limit probabilistic probability fundamental notion bayesian inference inefficient computable variable unit first second encodes problem nevertheless broadly condition computable computable corrupt probability modern science inductive directly raise phenomenon exceed researcher propose language describe joint automate reasoning inference computable demonstrate generic exist possibility amenable automate challenge theory explain characterize circumstance broadly computable intelligence ai language describe answer language language abstraction modularity practitioner infinite produce effort logic probabilistic language research model formal language high order e structure essential modern statistic expressive language describe arbitrary traditionally contrast system introduce compute conditional toward hope eventually support computable progress towards random hoc explain probabilistic operation build classical notion arbitrary sufficiently programming basis precisely describe operation probabilistic programming capable perform describe computable computable study metric g distribution probability experiment outcome compute straightforward ratio modern probabilistic place high situation notion theoretic notion kolmogorov characterization conditional
tree inference inference go target application character recognition probability encourage show use real justify reasonably work coherent fairly coherence need call write special partial ts ts non ms real map complicate variable joint gamble formally singleton real number element denote similarly also mention tuple corresponding element tree independent meaning simplify device identify example sx device gamble x sa spirit gamble correspond gamble throughout conditional model may systematic gamble condition unconditional unconditional every subject supremum conjugate sc I gain super I homogeneity set follow intuitively similarly sc sc sc sc notation behaviour hereafter frequently thing interpret belief keep mind coherence uniquely consider low kk satisfy consistency coherent sense coherence concept describe unconditional easily recover sure conditioning disjoint subject cx ix subject separately irrelevant infer coherent sx sx ms unconditional uncertainty generic local conditioning show letter unconditional tree compatible ss reflect type list need explanation condition encode turn classical take parent irrelevant follow turn assessment child irrelevant tree infer empty interpretation focus help go tell irrelevant child child even tell non child irrelevant conditional child terminal ci irrelevant grow child child equivalently publish elsewhere refer detail result mention coherent low belief set construct low reflect structural disjoint proper word value affect belief generally speak unconditional disjoint proper assessment conditional independence family lead definition notion model coherent coincide domain low product necessary make distinction term implicitly version marginal small coherent coherent play tree separately proper non nf ng important separate right side equality explain coherent natural extension trivial empty independent coincide want separately joint proper subset jointly coherent low calculate consider eq meaning generally speak yield conditional n regular lie detail extension interestingly extension assessment independence shall course conditionally independent empty finite construct coincide marginal follow proper irrelevant variable belief speak conditional turn lower alone assessment conditional eq assessment family separately coherent domain family marginal conditionally coherent conditionally product denote conditionally extension n indeed independent imply global model conservative extend express encode justify crucial lie tree recursively leave root basic building block dot grow child construct recursively heuristic manner justification child already cx sx cc ci marginals cc blue child child c need combine sx sx see coincide conservative small conditional low blue level distance also account ss start recursion lead joint alternatively eq reference follow collection end discuss property global derive ss positive empty c ec ec singleton separate arguably tree family compatible requirement satisfy requirement coherence coherent reflect leave hand involve model one would model strong generally prefer condition positivity restriction uniquely reasonably coherent approach coincide uniquely coherence condition requirement coherent requirement inference model consideration encode tree dominate family turn model construct satisfy requirement ss eqs constitute low satisfy unique family finally empty derive kk coherent family inference discussion coherent inference go much detail property one net bayesian net tree message namely interested sign sign next thing tree tc indicate ht pi x x node node x node node x node node pi south constitute message node message reach independent arrive represent wise product eqs message child move stop low send leaf send remove overhead create x te left node leave left message along fig calculate recursion formula q eqs influence calculate find maximal principle concavity character drastically fig exp exp exp exp exp exp exp exp exp exp circle exp exp pt node yshift circle yshift exp exp circle exp exp exp node xshift exp pt ht ct sa pc mt intersection axis respectively next evaluation interval become become stop iterate soon interval exactly zero briefly complexity algorithm make iterate multiplying make employ fix tolerance linear represent base algorithm focus discuss step property strong tree observe markov chain thick dot blue grow child node x thing like mass restriction notational inference variable want section message message qx send transform lead obvious notation separate soon long additional observation probability include conservative computation unclear whether independence address availability fast preliminary different separation particularly leave root reason compare interval root binary interval chain interval former wide irrespective least ten chain seven extreme point strong similar independence summary difference inference notion outer safe light could still make tool right pt smooth paper expressive enough model interesting section character illustrate difference traditional notably arise come available reliably character one sequence hide generate refer processing speech text observable sequence hmms correspond tree informally every generative generative grow child child node child model precise version state bayesian technique learn multinomial identify might realistic unconditional sample positive inference local identification tree generally speak computing inference hmms likely observational sequence precise probable adopt overview state text observable text perfectly reliable process device count single transition character another generative two sequence model generative might character th albeit completely resort gram character quantification exponentially transition depict recognition might ed child grow observe child grow right grow observe child grow target child I percentage prediction alone hmm indicator contain percentage correct prediction return htp lrr precise accuracy output quantification character include character hmm take one report value accurate precise output happen rarely remarkable return accuracy display basically return character shape allow notion stochastic focus notion limited net develop efficient algorithm update belief precise distribute fashion pass message expectation remarkable feature fact never practical net moreover clear behave axiom separation net would result encourage open net requirement coherence establish requirement inference update tree able reliable counterpart go would extend expressive tree net graph might affect net infer could achieve global rather many tree lead assessment condition observation indicate lack certain might application way necessarily together separation necessarily require match net de project partially nsf p de comment anonymous net tree strong markov rgb circle draw fill cm width pt nd fill nd nd fill nd width draw line mark draw line mark lemma c focus net replace independence commonly net weak arguably suited theory focus combine local uncertainty justify compute update belief entirely coherent satisfy requirement character approach comment availability truly year intelligence address variety finance name still net create inference close convex notion net vast majority speak independent set stochastically strong precise net mathematical consist bayesian graph ideal net consideration net precisely consider net cause existence partial knowledge disagreement amongst another case
constraint propose know propose encourage combinatorial thus tractable group either entirely solve solve side note similar original understand whether discuss cast find direction assume orthogonal achieve take two problem consider row standard obtain optimal k far illustrate constraint impose toy plot plot x black square vertical matrix square dotted direction represent green dotted line differ correspond error example fact next precise pca additional pca red row sparsity constraint compare constraint dotted method produce v f high v directly implication set eq last strict cast propose produce row two observation choose base row express entirely recall view encourage matrix penalty lasso group entirely importantly convex solve follow form column row equal zero consist successful attain reconstruction correctly optimize compare reconstruction comparison case ii underlie I except always even tune iii notice performance expect vector term low rank pca type wise encourage entire microarray soft set observe x since measure precision plot randomize every practical light alternative plot compare specifically enforce gene exclude c gene microarray ps ps minimize consider explain unconstrained subgradient subgradient equation write claim explain subgradient equation b rearrange take norm side b b connection adopt highlight difference general modify intractable term encourage optimize operate randomness resource thereby understand concept acknowledgment would thank art support stanford fellowship stanford fellowship lemma definition department stanford stanford xu stanford stanford department mathematics stanford stanford provide reconstruction span appear take randomized whereas pca convex optimization viewpoint implicitly sparse pca observe attain possess method recently often svd decomposition interpretable motivation interpretable procedure recently machine learn start far approach purpose decomposition viewpoint connection put combinatorial optimize highlight interesting practical viewpoint second implicitly contribution formulate randomized problem relaxation original pca implicit objective third propose provide paper alternative help notation column contain respectively column brief background particular emphasis subsequent section seek hyperplane write datum column think factor equivalent nonzero provide number randomize variant variant nystr om variant consider set combination ask intractable randomness submatrix least randomly leverage sample singular norm relevant onto generalize interpretation attempt pca find
application mc real mc computational limit optimal decide neighbor differ dimension expect difference large spread dimension small spread separately skewed use result mc study need verify carry mini mc investigate question situation good kolmogorov generate integral hold case ks repeat time show nominal nominal slowly increase especially inferior encouraging size event event going indicate mc true type nominal agree statistical suggest equal preferable multivariate dimension go illustrate normal time figure reject ks fail routine carry available http edu approximation search sophisticated could combination code equality two dataset conceptually suffer curse study show high computational capable detect visible high hour ne ne ne ne compatibility abstract head chapter head head head head head head compatibility lemma corollary theorem ex ex false compatibility true em mu mu mu false mu mu mu mu mu align align end end align environment style style environment array allow tag tag pr box email usa could might dimension neighbor sciences rely monte simulation stage detector simulation mc experimental number higher belong consistent test concentrate simulate mc testing concentrate observation correct generate near neighbor bernoulli binomial slight ignore instead consider
institute statistic virtual center use patient visit retain per say visit standardized independently binomial particular enter spatial term adopt star link structured structured account particularly suit fix purely frequentist use north france public region disease model parameter latent prior inference mcmc various explain latent field strongly dependent development instance poor remain end marginal recently give accurate explain introduce justify assumption obtain comparison finish public north east france adjacent dense centroid km region record record unit unit median per population potential practitioner patient practitioner patient unit region statistical autocorrelation improve explanation reflect access influence specifically focused measure policy notion proximity show patient big specific age status herein patient prefer practitioner us star model adopt follow response binomial logit spatially component use proxy environmental autoregressive process assume adjacent normal distribution proportional unit area write adjacent adjacent unit accord common adjacent two share common ask spatial unit association residual heterogeneity traditionally disease mapping risk convolution autoregressive process heterogeneity gaussian hyperparameter structure prior posterior present approximate three approximate marginal approximation compute laplace laplace previous posterior approximate task sufficient integration mode quasi newton compute let gaussian assess compare penalize indicator measure fit measuring pc core processor compare hour former parameter indicate enough risk different lrr alone proximity density summarize indicate spatial effect prior contrary improve prior remain major patient live less greatly use covariate fix summarize figure adjust north south east h fix introduction mind nothing gain covariate remain furthermore powerful walk use combination b useful offer flexibility coefficient linear spline walk spline spaced walk prior flat distribution assume spline reformulate latent model spline knot example devote model prior rational round several spatial spline centroid population population population concern e population diabetes logical try adjust percentage example technique simplify laplace approximation integral use spline distance vary binomial logit assume distribution application manuscript enough smoothing walk retrieve easily aggregated observation application interested account patient number combination category poisson krige access time hence example proximity model walk third support university thank frank careful review recent spatio model environmental disease propose include potentially ratio patient population within response variable link relative additive account various spatial effect covariate close form integrate nest approximation recently model give fast model comparison assess lemma spatio temporal field environmental disease model potentially distribution patient unit framework response
thousand gene simultaneously decide differentially express gene response absence presence covariate whether nan hypothesis expression setup differentially test exposition thus concerned design mean wish depend standard rank decision whether practitioner software package concern global nan detection plain like know datum noise zero compare nest science nucleotide form position genome population reference allele common snps record copy allele quantitative trait decide trait scan need trait genetic thousand hundred thousand approach test hypothesis fit regression global adjusting see problem denote impulse user noise whether assume user detector max detectors concern detection white popular consist model sparse superposition element multi one would employ detect superposition time signal dictionary problem material arise compressive theory signal fix dictionary projection projection reconstruct interested place motivate study alternative absolute regression nonzero various detect sign need familiar alternative versus type alarm detection dimension asymptotically asymptotically understood bad risk define chi simple asymptotically powerful quantity fix alternative mild sparsity mean identity suboptimal require absolute sparsity max review literature mean asymptotically powerful see analyze procedure higher show within detection established explore matrix multiply triangular decay polynomially fast prove unchanged achieve theoretical literature locally setting would test resemble suggest level sparsity manuscript vector become comment paper signal salient end research focus entirely focused whether datum white paper match simulation match term sharp coin mean result cite applicable situation number greatly exceed simple hold assumption convenience since simplifies exposition essential majority condition mean unchanged interestingly number observation negligible sharp sparse soon threshold optimal max asymptotically powerful ratio good give design correlation role orthogonal low clinical trial compare balanced replicate treatment remove correspond model q block dimensional design even concatenation orthonormal basis result apply mutually incoherent incoherent basis e compressive communication application subsequently normalize column unit design fact know rademacher discussion simple straightforward bring moderately general match elaborate carry important moderately amplitude covariate correlate model put differently linear small may signal correspond way false yet full though design triangular rapidly decay result go much sense design far triangular include decay pattern simplify discuss whereas setting obviously involve nuisance nest nuisance balanced interaction nuisance represent clutter noise nuisance result concern apply provide far application space space also column low detect mechanism space obey organize design state assumption pair correlate contain design embed numerical section file provide brief summary subset bold resp resp letter bold represent column vector likewise denote sup distinguish f survival brevity say stochastically denote introduce column orthonormal viewpoint serve warm determine alternative resp early whenever content second orthogonal detection need define definition follow compare max exponent obey conversely sequence asymptotically pt absolutely clear statement understand simply identity high situation concern design clear asymptotically whereas begin matrix weakly correlate depend obey correlation relax main fix probability belong mean quantity computable alternative identical orthogonal design study lower understood sense generating suppose resp sequence asymptotically resp interpret least sharp increase exponent obey op resp asymptotically essentially multi linear reader appear nonempty instance bind aside special impose slightly weak lb turn bound follow random coefficient sign sign occur make potentially vanish study accordance low although obey asymptotically resp powerful design contrast say method moderately set lb say achieve however importantly negligible proposition turn ht multiply define obeys suppose op p op without conclusion state theorem say condition weak ever close related max fact discretization establish basis discretization leverage higher detailed relatively fully grid statistic strong adaptive exponent obey op cover combine correction operating mention high coefficient restriction dynamic amplitude large test assume design first similar equivalent proposition fourth moment reference consider apply design standardized entry effectively design weak implicit randomness carry discretization threshold high design course generate fashion turn orthogonal remain op asymptotically powerful max powerful assertion relationship response covariate mostly correlate whereas achieve maximal magnitude support use fine comment situation denote must accurate slight significance suppose apply methodology biased consider apply amplitude equal contribution complement argument mention case orthogonal design color unknown note treat matrix standard strategy construct possibility discuss signal near chi freedom say obeys draws apply obey valid obeys condition impose remark replace concentrated around correlation portion depend fine give apply impose impose balanced eq design exponent obey conclusion valid divide balanced way linear easily coordinate balanced replicate set balance
jacobian maximizing require gradient source u consider independent write jacobian mix write equation implicit ignore relation follow gradient entirely write must replace expression replace universit de france fr occur paper correct article maximum separating log mixed
satisfy satisfy k ix moment term moment therefore parameter invoke straight since c k minimum covariance cluster covariance take ij clustering recover leaf corollary university department computer university department distribute pattern several identify activity trace statistic wide aggregate consider arise activation contribution activation pattern method learn independent network activity weak corrupted node observation pattern strength gaussian baseline operating signal activation arbitrary activation physical structure signal arise practice consider pattern structure node exist pathway bandwidth embedding structure efficient social weak activation network trace internet chemical hazard differentially microarray analysis social detector aggregate location contain interested arbitrary detection global aggregation testing average node reliably signal strength location I drive signal strength reliably detect irrespective iid random probability limit activation detect measurement global aggregate adaptive fusion node aggregate fashion approach processing consider statistic pattern intractable weak study subtle test adaptive various range unknown investigate test propose level problem assume activation independent result strength must activation tend locate sensor environmental phenomenon aim detect weak leverage dependency activation closely offer also establish fundamental limit structure paper nature pattern paper pattern reflect present world lead method relatively add practical pattern support structure even orthonormal adapt transform basis coefficient canonical scale early method attain activation pattern exploit extremely activation three fold transform hierarchical structure propose motivate arbitrary pattern organize node focus exploit variable pattern effectively fusion summarize snr activation quantify relative method constructive procedure third necessarily priori dependency structure learn measurement rest organize structure characterize learn discuss introduction internet sensor structure exploit enable pattern activity node structure pattern agglomerative similarity hierarchical cluster group denote figure must satisfy agglomerative recover hierarchical several see agglomerative agglomerative produce binary case straightforward omit save collection similarity pair agglomerative figure cluster pattern cluster merge agglomerative basis denote project pattern onto compute result vector possess vanish moment haar wavelet activation merge cluster yield augment compute orthonormal unbalanced haar transform basis construction spirit lee al balanced haar wavelet dendrogram transform correlate result difficult analyze procedure unbalanced haar wavelet sub thus vector set similarity initialize merge unbalanced end difficult transform support contain multi activation constant basis ising structure since increase level pattern group high pattern approximately sparsity govern interaction mind situation activation network naive model scale pattern tree strength sufficiently zero canonical result determined canonical domain small strength propose sparsity pattern group enhance snr additive denote strength unknown activation project yield pattern energy concentrate thus pattern simple maximum bind max pattern draw tree ise model pattern ising strength activation alarm condition draw signal strength polynomial significant improvement canonical structure pattern detect signal pairwise similarity covariance hierarchical cluster constructing propose recovery multi estimate covariance similarity concentration measurement variable possess specifically satisfy condition let denote difference gap covariance leaf kronecker delta true variable affect behave purpose realization agglomerative network measurement need model add pattern imply signal strength vary compare domain global aggregate false false discovery method exploit algorithmic complexity detection unbalanced haar except summary vanish one contain activation agree parent tree edge basis node different value specify pd chernoff dl dl q dl dl sparsity sparsity inactive level level essentially argue canonical govern activate number condition root inactive consider parent active understanding expect canonical follow repeatedly apply similar fact hold enough proceed e l repeatedly apply similar get l l bound canonical invoke derive recursively binomial chernoff q pa chernoff p notice p n p conditioning probability pd dd canonical sparsity p l fact enough establish canonical upper recall chernoff q enough
average vc ergodic borel sigma collection vc denote small vc vc role machine let union intersect complement positive formally depend interest existence uniformly possess vc finite eq q corollary next immediate corollary least need note define minimal element countable remark routine countable weak function pointwise without positive remove clearly give ergodic ergodic ensure vc uniform show ergodic process set ergodic large number property argument uniform law major vc law dimension early work preserve mix mix countable vc countable mixing transformation nn nd approximation cell measurable satisfy follow triangle previous arbitrary countable vc mixing establish uniform vc sampling several follow definition exist countable elementary countable atom atomic countable atomic lebesgue interval equip borel existence preserve ensure generality may let countable finite proof assumption construct fashion sequence th stage sequential splitting set use arbitrarily collection join vc dimension proof contrary k nc great continue split set remark splitting stage let select join measure continue continuous define exist integer generality intersection lb lb lb induction let interval display subsequence set sufficiently lr cell contain definition boundary ii set hold join j let subsequence interval previous j l c inductive complete give dyadic disjoint intersect dyadic remainder contain acknowledgement author
denote instantaneous security simulate market immediately w p definition maker loss rewrite substitute bind market maker show converse market maker must quickly pricing function parameter use equation equivalent use majority mention apply weight state market market market crucial later finance originally different towards risk market return outcome preferred function property convex monotonicity negative translation financial interpretation important provide low allow result informally include completeness semi continuous quantity vector differentiable function decrease monotonicity translation theorem imply convex immediately concave constraint respect kkt necessary optimal q price precisely envelope ensure convex represent strictly differentiable price maximize ability represent form loss maker function correspond convex risk market maker n market scoring market maker scoring cost base regard show section market scoring differentiable satisfy mild market scoring score base market market guarantee receive every change quantity long achievable achievable correspondence certain market scoring function base market scoring market however provide guarantee circumstance satisfied market strong give function market score q class market proper market rule mapping equation market equation market class regular market rule differentiable scoring pair mapping hold market equivalent price market price price every price market strictly rule regular regret another market instead market price quantity share price share solve convex order share share share market price share function limit imply follow expert risk suggest penalty function function market maker bad market maker contrary regularizer necessary market scoring regularize discuss score equivalently majority expert relationship market write equivalently market market correspond descent setting give quadratic market scoring score market base payoff whenever price nonzero cost cost market uniform previously price nonzero maker market market regularizer price apply bn descent match demonstrate elegant connection market market think interpretation market mechanism function assumption way market scoring infinite accept permutation finish ahead either google run naive exploit majority price permutation connection market grow algorithm infinite set price type outcome space fix place integrate calculus dx I iy dx iy dy rearrange price play derivative n price differentiable iy dy dx iy dx dy absolute value every j b j p dropping equivalent kkt maximize price equal lagrange satisfy kkt condition price case eq price outcome outcome positive price outcome outcome denote bn em price price definition derivative price price derivative outcome change price derivative positive price everywhere rgb exactly choice market regularizer show equivalence market market cost function scoring follow commonly study market score aggregate imagine interested united states month fall hour news read informed could potentially save appeal design prediction market outcome event offer security state neutral p security collective accurate practice wide example rational expectation equilibrium offer insight market converge accurate say nothing market mechanism logarithmic scoring might accurate estimate practice insight mechanism deep prediction market come connection market market build proper loss bregman knowledge weight majority regret marker maker majority efficiently combinatorial space show converse maker however connection deep market interpret maker make observed think maker treat cost market time minimize combination cost market regularizer furthermore another market convex base market algorithm insight prediction market accurate describe review market regret variety mechanism market dynamic market broad scoring function market sequential mechanism section probabilistic forecast market rule encourage individual belief scoring rule mutually exclusive exhaustive rule score outcome take line intuitively reward forecaster receive predict relative scoring rule say outcome equality strictly score scoring rule rule extensive nice survey scoring many scoring market maintain enter may rule distribution allow pay infinite amount happen outcome turn possibly payoff score receive payoff payoff proper scoring belief market converge collective scoring essentially responsible thus maker responsible score bad market maker bad market uniform parameter logarithmic scoring affect payoff maker discuss mutually exclusive exhaustive outcome event market security outcome different mechanism specify market function quantity share security currently share must pay market instantaneous security share say price price security price guarantee second ensure price something respectively great quantity security ensure price valid outcome price market current occur function function elsewhere important completeness property monotonicity positive sufficient require monotonic require require equivalent price fix da da translation invariance invariance fix set translation invariance combine bad market maker maker might pay price formulation receive formulation briefly review loss goal almost cumulative expert assumption make adversary every receive loss cumulative maintain weight receive instantaneous receive always cumulative none say randomize weight know trial eq advance yield bind weight choose specifically minimize entropy broader name time algorithm simply place algorithm highly unstable dramatically change weight time next suffer alternate overcome instability suggest random perturbation perturb instead minimize regularize guarantee wide variety regularizer l quantify trade dramatically round range big choose minimizer unique compute efficiently strictly regret optimize appropriately regret foundation describe loss market maker expert market maker treat formally market
nominal understand effect calculation present fit calculate analytically statistic use goodness fit also bias observe value distribution histogram probability purpose illustration present bins histogram carlo true give histogram reconstruct reconstruct pdf histogram pdf divide take true pdf reconstruct pdf accord define simulation initial parametrization pdf reconstruct gx reconstruct monte carlo generate event bin histogram equal histogram reconstruct pdf weight equal gx px histogram convenience histogram px cc result reconstruct monte reconstruct pdf cc cm result monte reconstruct reconstruct pdf monte size calculation reconstruct carlo reconstruct present calculation part measure detector compare histogram unnormalize previous method new experimental permit decrease flexible restriction domain investigate goodness fit describe histogram various bin numerical goodness fitting model detector finite model find minimization experimental simulate example present validate monte experimental simulate homogeneity unfold density reconstruct detector resolution acceptance represent true acceptance characteristic experimental resolution variable pdf unfold problem true parametric reconstruct reconstruct possibility acceptance resolution must majority monte use unfold inverse ill pose solve statistic systematic priori influence preferable disadvantage approach matrix rather element case monte carlo author goodness fit distribution reconstructed present rather use illustration calculate author state degree impossible statistic alternative apply fitting acceptance fit goodness fit error practically run validate event correspond characteristic call weight oppose unnormalized carlo reconstruct weight avoid monte reconstruct pdfs weights histogram eq bin weight quantity estimator comparison reconstruct pdf carlo reconstruct histogram homogeneity represent distribution assume belong bin experiment monte equal histogram weight sum convenient normalization problematic statistic sum extend bin bin unknown find minimization substitution homogeneity valid substitute parametric formula weight dependent estimator find l fit example validate procedure take true reconstruct domain pdf describe
move limit different cumulative sum zero begin prove restriction except correction improvement point represent mass substitute special second consist point feature allow represent continue mean remainder notation appear identically similarly special positivity apply ergodicity positive case region neither recurrent split irreducible positive recurrent recurrent hence irreducible recurrent recurrent form recurrent irreducible recurrent argument proposition irreducible recurrent class period recurrent periodic contain recurrent thus period must reach odd even exception periodic evident give policy enough hide boundary unclear equivalent interval transformation write denominator exactly fractional transformation fractional interior inspection fix least three order fractional point contradiction thus must apply monotonically lemma exception threshold policy policy threshold since contradict information chain periodic period sensible depend appropriate policy without write closed limit give probability exist suffice ps obvious ps q mass transition multiply require exactly denominator exclude generalised case long nature policy limit good calculate limit expect hide natural place approach formula product infinite hence truncation specifically ic update limit entropy c c h c function maxima occur follow requirement vanish series series th combine inequality easily calculate calculate expect unbounded reach desire precision denominator replace calculate later simulation prescribe within computational arithmetic operation call call approach expect simulate drawback work state infinite possibility atom width variation regardless iteration greatly limit list contain location start list least either require calculation iteration computation correspond represent differ method grow indicate limit set threshold policy represent represent threshold simple collection threshold test threshold policy extremely inefficient countable move threshold change figure tend apart naive test accumulation space even subset next previous choose give policy policy equivalence avoid however depend iteration number determine point create circular depend policy tested adapt threshold policy prove order require bind exception policy ingredient bind depend uniform suffice estimate search space finite policy policy check simulation number policy feasible whether include four possibility inclusion round strategy simplify policy policy orientation lie mass policy thus past every threshold terminate finitely policy step require since policy policy generality contain threshold find small application respectively error empty right run next great encounter record minimum repeat primary content optimality analytically description valuable insight six orientation diagram figure demonstrate threshold qualitatively equivalence assign region accommodate datum partitioning set orientation inclusion define precisely depend space policy interval present extend interval policy equivalent illustrate precise definition six figure cm iii vi red accumulation equivalent mass region finitely belong blue equivalent h region vi strict non calculation generate significant extreme value occur half diagonal blue prove autocorrelation expect symmetric policy convex eq equality occur order prove optimal explicit lemma limit illustrate inequality concave symmetry inequality cm increase write identically vanish prove require fact equality attain empirical description provide perform notice unimodal circle prove simplification threshold test two policy policy entropy determine policy uniquely view hypercube infinite space finite hypercube justify decay geometrically tail contribute truncation policy similarly possible look sense change hope since hypercube connectivity region tend find locally truncate pick policy digit cycle give policy entropy leave previous otherwise find pick integer spaced figure hypercube locally policy indicate policy locally strongly globally policy infimum general locally globally new method find pick probability observation suggest slow significantly slow parameter entropy far uniform likely optimum unlikely boundary region deterministic simulation policy previous section consider criterion limit term attempt maximal gain strict strict current method newton allow computational result greedy idea appear greedy easy range discount suffice show close distribute greedy error tolerance average greedy maximum point greedy suboptimal policy h grey policy error boundary region make final globally optimal closeness greedy threshold figure exhibit behaviour suggest indeed qualitatively study thesis find strong policy aim able ergodicity positivity policy expect general policy find author aware paper date incorporate thesis algorithm policy algorithm computationally prescribe error thus room pt process requirement master department mathematics statistics department electrical university cm abstract consider hidden process policy deterministic focus derive calculation computationally almost thesis comprise original towards use thesis acknowledgement hour thesis together observation case model consider observation choose observation information sufficient past accord arise information state special definition limit entropy explicitly rational series limit optimal general finding also greedy reasonably suboptimal series paper name soon speech calculation influential consist observation observation vector simultaneous sometimes prevent observation evident operate mode must another example study must location limited device even simultaneous may restrict availability processor much sensor consist sensor insufficient analyse processor choose sensor receive multiple limited communication decide bandwidth virtual target move possible site special correspond special file channel give yet choose sensor calculation adapt replace multiple observation hide exist state underlie explicitly information choose measure additional choice infinite cost theory model observation markov exist sensor scheduling autoregressive horizon thereby write optimal suggest work dynamic programming markov dirac piecewise norm piecewise function could fine mean state information state observation path consider infinite horizon primary tradeoff use usage cost aim uncertainty measurement work function function prove policy restrictive assumption use monotonic choice observation still problem cost discount sum cost differ verify underlie chain oppose current simplification transform proceed far search move partially prescribe certain observation transformation appear optimal policy extensive state argument provide full however mistake recurrence formula correct precisely define convenience end list model notation consistently symbol uncertainty many choice good naturally real vector structure finite sensible choice natural countable measure accordance fact particular policy converge among policy limit interesting measure region observation observation set variable chain actual variable already ergodicity incorrectly en recurrence certain chain mostly elementary limiting begin find dirac th markov observation satisfie recurrence py definition simplification give satisfie recurrence observation ps nz ty ty ty ty give chain analyse ps ps recurrence z give criterion recurrent discrete irreducible positive limit state chain ergodicity observation process latter whenever observation reach probability recurrent transition many much recurrence anchor pair coefficient definition fix finitely exclude reach model measure outside bound rewrite definition notation integral integral measure integral union map q give since always large
function moreover know normal statement sequel property detail functional eq function cdf characteristic f f infinitely iff infinitely sense r relation allow obtain stable clearly f provide strictly usual sense moreover analogue analogue iid family f straightforwardly transfer paragraph stable distribution cm stability pz sense scheme direct exponential analogue strictly example branching scheme generate nz another family v nz n actually consider probably neither work familiar year aim involve summation purpose balance equality exactly pair approach however straightforward certain typical outline rational iteration fractional subset complex plane call invariant invariant conjugate conjugate ne see e number point condition union dense plane behave precisely cycle shape lie cycle neutral complement cardinality real left behave part generate exist eq polynomial power negative converge convergent non negative chebyshev function take directly property chebyshev odd consider let n generate clearly p chebyshev state eq q n plugging give well actually fact well clearly pdf form cdf needs apply relation represent somewhat property wiener consider see laplace equal coincide cdf let via reformulate kn attain constant therefore standard normal deviation certain analogy representation r scheme summation consider family e way turn iid pz z laplace analytic book example family laplace transform clearly monotone mention book page negative sufficient pair rational investigate family obviously geometric mention hence check generating pn pm omit notation already analogous applicable strictly follow give distribution normal similar transform standard wiener ex plus minus ex plus plus department email grant university school college research centre mathematic provide number add variable also call stable term chebyshev polynomial result class particular stability characteristic certain class useful
smoothed proximal fast shrinkage smoothed carefully summarize scalable qp thereby applicable convex structured regression enjoy convergence subgradient implement smoothing optimize function adopt mm quadratic surrogate fuse penalty detail connection organize formulation group lasso present along connection extend algorithm simulate overlap lasso penalty parallel outline high penalty feature denote lie univariate obtain solve g encourage regularization parameter lasso scenario many pairwise similarity employ induce penalty joint among specific structured kind penalty norm total variation penalty sparsity induce recently regularizer optimization encourage coefficient jointly composite power group group mix play widely hierarchical hierarchy achieve grouping effect comparison alone group precisely estimate use regularizer variable detail prior structural pairwise edge denote proper measure desirable encourage share similar magnitude fuse fusion effect feature monotonically increase correlation positively would tend whereas opposite effect calibrate graph fuse lasso encourage densely select q fuse globally difficulty arise nonsmooth although overlap lasso penalty type penalty nesterov induce penalty use technique nesterov easily calculate utilize induce reformulate problem norm write variable associate overlap single nonzero store memory first rewrite highly nonzero rewrite fuse auxiliary entry generalization penalty optimization norm make optimization use construct view one dd fuse verify know maximum control gap result continuously lipschitz smoothness nesterov pt plot onto suggest show nonsmooth dimensional project surface see compose sharp figure similarly axis sharp remove close equation group fuse proposition project fuse fuse lasso operator define simply approximation nonsmooth induce fast iterative shrinkage reformulate gradient smooth optimization q q large involve nonsmooth part penalty fista algorithm determine overlap penalty fuse penalty compute proximal q w closed soft operation coefficient soft thresholding objective w l step obtain zero e parameter could far accelerate backtracking could dynamically assign operator eq rate ensure iteration satisfy particular constant find predefine optimize rate next solution f dt upper pt theorem part involve bound balance three detail present much subgradient convergence term monotonically objective decrease f objective value estimator eventually depend convexity learning solution solution speed recognize open community main computational iteration calculate share iteration subgradient descent store computation x j per iteration order table time pre storage significantly store insight draw early composite widely problem loss simple nonsmooth achieve consider operator challenge smoothing work entire nonsmooth approach separate nonsmooth norm structure induce penalty enforce fuse lasso smooth complex leave benefit irrelevant feature avoid processing truncation condition predefine impractical optimal induce full rank obtain problem tree expense time boost idea construct smoothing adopt widely mm maximization maximization problem minimize replace difficult objective surrogate bind gap objective iteratively construct construct apply proximal optimize surrogate penalty mm matrix surrogate structure fuse gradient quasi newton smooth convergence seem derive convergence develop various overlap fuse lasso develop handle penalty focus group solve ascent quadratic min flow min flow min min flow subgradient summarize applicability give solver proximal sake computation proximal although propose worst importantly superior addition method active propose involve active alternating involve expensive global rate proximal guarantee proximal accumulate fuse ty coordinate proximal due compute two solve great advantage method update qr propose original version algorithm column add objective especially balance trade author extend deal extend require schmidt initialization extra avoid heavy pseudo practice column solution entire efficient require time consume entire column entire column parameter sparsity prior association goal genetic nucleotide input influence phenotype measurement phenotype naturally guide graph relevant phenotype regression penalty previous application briefly induce illustration matrix output b j coefficient task structure naturally wise sparsity induce multi coefficient output tree structure output output fuse task reformulate tc replace output auxiliary f constant per qp complexity evaluate scalability efficiency smoothing overlap lasso real genetic overlap lasso qp qp graph fuse qp proximal solve function g nonsmooth slow pc gb ram software matlab fair criterion dual gap stop criterion widely objective stop addition maximum purpose overlap lasso receive group incorporate view singleton ease singleton receive constrain smoothing reasonably univariate simulate univariate assume input order adjacent input q pt pt pt c pt report cpu parameter magnitude large unable collect scale reach iteration instead hundred pre terminate third lead moreover notice time show per iteration require method dimensional moderate computation newton simulate identify input phenotype simulate parent panel additional individual randomly individual like structure correlation pair nonzero size relevant input group subgraph sparsity pattern phenotype correlation show pixel regression lasso regularize fuse output distribute input information illustrative coefficient shown regularize multi lasso reveal clear fuse lasso relationship qp solve structured figure select regularization fuse coefficient fixing fix assume correlate input relevant select relevant every three consecutive group datum substantially moreover notice almost analysis overlap cancer demonstrate employ structure induce dimensional regression solver problem datum breast cancer research devote identify consist particular discover tumor growth group pathway gene provide pathway biological analysis consist pathway approach pathway significance two tumor gene canonical pathway molecular signature lasso overlap gene pathway pathway pathway gene gene analyze pathway logistic overlap tumor type gene expression solve adopt vary regularization cc vary select gene logistic regression different along group low overlap belong different pathway figure number pathway select belong b incorporated select overlap lasso pathway coherent easy interpretation gene pathway independently total compute whole second group perform functional analysis tool pathway breast tumor gene marker belong pathway pathway breast cancer protein pathway involve activity unnecessary protein chemical reaction one protein extensively select lasso marker breast previous pathway find breast relate dna protein relevant growth cell pathway notice pathway relate death occur widely know involve tumor pathway associate pathway case functional signal breast cancer among pathway involve pathway functional gene match biological insight
covariance approach estimation minute inverse twice scale general sparsity fuse lasso group contribution aspect reformulate regularize introduce lead term amenable bregman square root positive major consume quadratic eigenvalue conventional solver first compute eigen decomposition quadratic eigenvalue eigen consume solve eigen decomposition contribution crucial bregman definition problem covariance encourage result bregman method solve fast artificial importantly split bregman term provide efficient popular tool variety construct multivariate mean matrix reformulate aim nonzero entry covariance concentration idea behind explain select simple penalty result note entry learn model fitting model treat use require nonzero attractive account furthermore penalty regression able correctly skew overcome principled find maximize log encourage graph n empirical goal find definite tr trace denote concentration guarantee view approximation penalize formulation guarantee computationally challenge prohibitive coordinate descent coordinate descent lasso implement solve remarkably solve graphical specific limitation prevent penalty fuse paper penalize substantially graphical method broad penalty old gain efficiency recovery completion general optimization reformulate interact thereby enable split first generalize matrix estimation include update consume subsection formula step propose solve result quadratic section illustrate total later equivalent closely optimization multiplier method multiplier increasingly becoming regularize inverse regularization term proceed bregman reformulate general use norm net fuse lasso likelihood make auxiliary coupling amenable split demonstrate multiplier admm presentation bregman use lagrangian lagrangian corresponding constraint lagrangian augment lagrangian term augment lagrangian dual multiplier solve size ascent equation solved augment still function term multiplier minimization run step split bregman definite four eigen decomposition root equation convergence define moreover definite converge decomposition converge write govern initial matrix positive simplicity combine newton square htp newton f covariance penalty correspond special applied matrix satisfy solve htp use newton trial illustrate utility first demonstrate propose definite compare performance bregman inverse estimation compare conduct intel illustrate square root mostly motivate equation implement matlab sr upper triangular decomposition exactly list newton solve square root stop calculate precision demonstrate highly converge step mostly constant substantially decomposition example take newton newton next bregman method penalize graphical trial far solve scale estimation code code link language reasonable time trial despite language penalized terminate relative fall addition primal graphical guarantee matter value affect iteration involve empirically artificial gene expression artificial create generate parametrize inverse diagonal positive random approximately inverse covariance covariance graphical testing artificial less time take second solution second evaluate scale plot cpu
sample try new ibp ordering process exchangeable exchangeable customer consider customer exchangeability customer dimension bayesian finite version element model strongly e integrate sparsity prior motivated improve interpretability element increase reduce suggest uncertainty actually aid interpretation however generative adjust constant roughly source source active hyperparameter metropolis hasting marginal datum parameter diagonal upon current denote ibp matrix determine likelihood integrate th loading exchangeability ibp exclude expression column contribute likelihood hold account factor contain row prior mh integrate either new latent matrix high dimension proposal move note simple prior new slow modify poisson multiply accept take collapse sample likelihood term likelihood scheduling improve scheme describe ibp sample ibp conjugate act harmonic remain include completeness sampling column calculate invert calculate prior g variance allow wish share also put noise additive constrain isotropic b b gamma share ad hierarchical prior couple variance variant fa analysis sparse factor bayesian ibp propose nonparametric simply appropriate ibp toward derive literature binary variance calculate white noise give reconstruction permutation rotation rotation average ten sample infer spurious thresholding omit since consistent ten generate ten number refer use fix share dimension reconstruction different latent ten random average ten sample plain analysis bad factor four spurious noise ard improve long occur spurious feature improve perform well factor ard suggest default actually marginally well find underlie also see place ard control finally perform couple share variance reduce example check infer histogram number latent show significant standard deviation noise sampler infer show skew deviation suggest proposal sensible proposal slow sampler reach equilibrium add greatly increase spurious prior assess performance ground log ten percent remove large mcmc split test mcmc run various gene incorporate although ard sensible number predictive log missing indicate cpu second average across cpu pt algorithms breast cancer datum finite sampler measure used likelihood calculate log number ard prevent overfitte improve slightly model comparable sensitive sparse avoid bad order sampling encounter datum nonzero element sparsity model cpu number straight per increase suggest inversion precision ard negligible double finite cost decrease increase overfitte time decrease cpu somewhat feature birth marginally choose comparable sparse make normal considerable hierarchical dp datum sec either individual expensive gene split ten training test number gene slow large gene appropriate predictive performance note include breast capture variability surprisingly worse latent give log likelihood slow breast improve predictive ibp provide sparsity straightforward factor infer choose breast set ard like prior computationally birth death ibp manual finally feasible prior specific incorporate knowledge improve improve interpretability gene expression set share gene perform suffer slow acknowledgment would like anonymous comment algorithm describe consistent reconstruction microsoft support grant bayesian extension fa datum model superposition infinite process ibp use prior sparsity gene datum matrix datum set analysis pca independent ica superposition independent load usually assume gaussian inference probabilistic model matrix pca whereas diagonal ica latent assume tail integrate fully bayesian whose play role desirable irrelevant
involve whether call evidence alternative study expression effect representative observe evidence associate hypothesis hold interpret apart significance define interpretability frequentist observe favor nan hypothesis amount quantify p indicate evidence bayes limitation pose nuisance frequentist practical implication bayes specification distribution advantage coherence rarely choice improper algorithm apply selection extent divide generating proper expense presence result clear belief potential notably bayes satisfy interpretable criterion hypothesis distinguish pearson minimize provide review lead frequentist yield relevant statement actual example remain repeat cover confidence regard observe issue outline continue development approach inference enable minimax without repeat sampling origin source parametric herein reduce probability countable denote necessarily predictive averaging employ originally theory minimax optimality employ oppose accord length family algorithm efficiently density minimax involve value prove problem optimality nf contradiction assume since ratio n v contradict assumption call observed exceed regret worst optimal frequentist optimality average guarantee individual optimality select among terminology favor hypothesis nan interpretable evidence importantly information optimally quantify predict predictor average kullback leibler unknown logarithm choose interpretation logarithm yield broad enough refined scientific except distinction weak mirror originally cf accordingly fairly constitute quite strong c c negligible moderate heuristic evidence interval discrimination amount nan quantify discrimination impractical application typically nuisance regret relative ideal member hypothesis unless additional available biological normalizing denominator family distribution conceptual difficulty overcome replace datum largely depend nuisance compressed statistic compressed also include relevant weighted variance arbitrary efficient weight focus comparison sample equal share single approximate approximate except difference exact complexity vanish lemma since discrimination w favor generalizing except smoothness apply since I discrimination evidence discrimination observe nx xx I x weight satisfy constraint broadly regularity result extended space open bounded I inequality imply consider hold define together weight comparison equation pseudo statistic might expectation consider fisher weight entail smooth single pseudo assign comparison weight il ii weight case study unweighte since separate analysis comparison comparison population address problem biology illustrate weight reduce consist estimate eight site site nan hypothesis display discrimination discrimination weight assign site hypothesis affect feature measure primary question logarithm abundance treatment disease perturbation whether feature strategy prove direct correspondence proportion abundance ratio interest influence abundance level protein breast likewise breast mostly pr group thus compete th display approximate discrimination favor alternative hypothesis hypothesis protein protein denote right bit abundance level differ disease status versus protein giving abundance level protein visually indistinguishable nonetheless effect weighting size example fig display protein panel depend hypothesis abundance differ disease status use nan versus protein comparison little lose except reduce address infinite section interpretation comparison adjust extent tend researcher report medium large especially adjust evidence favor commonly often weak bayes favor hypothesis strongly favor insufficient nan important discrimination
investigation phenotype development biology wide large continue serve excellent landscape protein network responsible growth systematic mechanism operate protein network gene dynamic gene phenotype response activity response growth response root root velocity neighbor profile image mark back numerous algorithmic substitute often manual make develop technique utilize air structure follow throughout image originally production curvature need automate throughput system image software automate prototype hardware software product modular automate application article provide evidence throughput functional protocol biology quantify subtle trait gene quantify trait example development growth challenge insight image wang department usa database engineering department mathematics mathematics ny usa towards computation algorithm software nsf dms grateful helpful biology early root branching root com grid acknowledge genomic deal screen target genetic desirable biological consume article progress scale image gene protein growth trait implicit environmental condition common one typical scenario trait biology advanced number powerful method gene protein collective protein process trait systematic discovery major bottleneck progress trait method quantification systematic dna signature extract distinguish characteristic specie behavior course outline step identify quantify trait subject force orientation growth quantitative trait informative carry genomic signature vary great complexity simplicity visible start life simplify serve noise ignore root texture also part whose diversity growth dynamic become essential pathway article provide extraction root direct growth response water light temperature study area dark tend root propose quantitative proportional use root region root despite molecular sensing signal response molecular response response broad asymmetric formation include system use protein identify mechanism determine explore action reveal response band induce curvature reveal quantitative gene role describe give application protocol attempt quantify subtle gene protein design specifically focus hardware novel automate image component orient software accommodate essentially analysis ready protocol platform model suitable segmentation suitable application derive throughput simplify obtain resolution allow select delta function movement automate occur accumulation surface cover mean scale quality sharp system algorithms iterative recover fine idea et joint variable fixing overcome follow solve euler fix first switch role class convergence appropriate function turn movement slide sometimes occur automate mostly inside article development develop new object orient code hardware quantify feature continue discover serve trait carry distinguish depend argue generation article robust fast root expansion cell root way root growth direction horizontal region call vertical length calculate length make carry growth segment growth growth region geometric primary algorithm extract growth discussion biological significance mention list feature purpose dynamic growth growth acceleration root root htb follow feature representative feature growth vertical region horizontal region segment root length root growth velocity
mix effective three number require decomposition evaluation single intel ghz cpu hyperparameter update however latent algorithm hyperparameter satisfy datum ergodic eventually explore hyperparameter expensive computation covariance covariance hyperparameter sense every ten update elliptical hyperparameter apply elliptical simplicity surrogate hyperparameter six update representation whitening match site approximation moment match approximation use set expansion infinite base slight advantage notable mining poisson work poorly site lead large bin count mining dataset expand give bin zero count work come approximation site investigation discuss recently cox surrogate advanced applicable process fix practitioner poorly whiten posterior site advanced simple whitening serious performing inference hyperparameter acknowledgement anonymous comment publication reflect school computer science process gp popular specify probabilistic consider make carlo sampling careful slowly require incorporate explain generalize specify split class approximation monte easier develop priori observation update parameter possibility sampling slice use wide variety technical little generative distribute science consider summarize covariance focus popular square cm hyperparameter input covariance joint q new different covariance chain operator initial position operator leave invariant z posterior simulate transition leave conditional invariant fairly markov towards target distribution recent work transition variable covariance hyperparameter need invariant metropolis hasting possibility hamiltonian monte carlo cm use square fixing appeal markov slow joint covariance highly hyperparameter especially dramatically hyperparameter conditional quite constant sampling prior fix strongly couple strategy independent commonly normal draw square low cholesky decomposition would principal square square root behave like power instead link hyperparameter update actually change result trend change acceptance instead respect propose accept propose draw solve pt p h ideal weak restrictive strong likelihood ideal ignore update variable proposal observation plausible noise analytically hyperparameter accept posterior applie create auxiliary introduce surrogate guide proposal variable auxiliary true automatically auxiliary hyperparameter possible draw marginal auxiliary process spherical surrogate update plausible surrogate hyperparameter illustrate leave update hasting term far proposal must crucially careful scale distribution slice adaptive search slice surrogate proposal careful axis align hyperparameter move prop surrogate imply u h sec surrogate randomly h surrogate noise plausible region latent update determine algorithm current latent acceptance reduce uninformative current tend base preliminary likelihood individually fit fit one dimensional site noise site thresholde propagation expensive cm sampler move move narrow posterior jointly value hamiltonian hybrid monte tune hmc jointly hyperparameter target replace often taylor look like laplace likelihood approximate pt current pt fix hyperparameter hope likelihood independent chain mix expand log maximum probit likelihood expansion see give flat gaussian likelihood flat auxiliary covariance surrogate
whether wikipedia action abuse predicting time summary median min max set baseline predict majority logistic content provide information higher predict content c european relatively examine control semantic evolution abuse project wikipedia experiment wikipedia article formalism qualitatively predict segment detection document may reduce operation complete remain pixel successful control analogy likely example wavelet filtering field acknowledgement nsf grant static continuously author representation framework experiment concentrate modeling sequence version control lead sequence document consecutive difference consecutive version keep track control book project com wave control document index locally concentrate continuous generalize weighted sequence version document axis smoothing domain simplex vector mapping capture content smooth probability geometrically control document content mix time sharp fourth address field vocabulary integer word v component represent document vector potentially j control document array row version control information suggest smoothing obtain space vector trace surface normalize normalize vector measure around location thus difficulty length way resolve short version need refer word expand expand result display unit panel display non normalize representation fourth panel display field contour represent panel correspond due version track document term normalize length document make track version document dimensional proceeding describe framework low vocabulary word visualize component component redundant control contiguous segment bernoulli segment get segment length increase rate constant array distinguish display nice total version rate segment third apply four task remain union version document word sharp word three change first word neighboring document position within version shift high technical occur substantial change axis measure instantaneous alternatively provide describe describe total position region repeat substantial substantial integrate derivative curve measure dynamically two book space anchor shift version addition removal integrate axis parameterize anchor position measure anchor point synthetic expect segment magnitude display contour high magnitude segment boundary show maxima portion control wikipedia within boundary panel amount version word substantially relatively third maxima topic segment vertical horizontal line coherent segment streaming message email version control segment find edge consecutive study thing control document segment exist view segment boundary separate segment easily segment segment boundary partially predict edge consider segment task predict section edge segment thus characterize high particular vertical transition horizontal diagonal transition across version magnitude l c c language european amazon paragraph summary max question besides synthetic six article european union european version control document wave amazon control include tag stop number author google wave ground boundary separate display maxima control wikipedia articles maxima segment logistic regression min
amount independent bit capacity channel device problem also quantum physics define principle functional equip call preference order utility line separable measure completion probability dirac comprise affine functional make pre compatible axiom shall formalism functional linearity certain system focus problem preference utility quantum function hermitian hilbert space spectrum total pre eigen non operator early relate phenomena capacity often represent functional shannon leibler axiom axiom logarithm abundance metric variation proper interpretation information generalize resource use shall exposition lot normalization perform structure linear bilinear yx xx use fact space however rich algebraic multiplication hermitian element real every call hermitian hermitian form point convex cone dual yy x measure strictly respect action case act identically element primary algebra continuous locally topological hermitian banach sign non generalization example algebra contain contain unit coincide variational generalization represent outside banach unbounded example generally complex value evaluate hermitian accordingly maximization real normalize e measure set weakly simplex represent extreme point geometry topological different I consider resource hermitian element ht package color conjunction terminal explanation load graphic explanation terminal graphics macro ltb lt lt lt lt lt ltb lt lt lt lt lt lt close weak closed also effective domain shall also hermitian problem generalize value observe increase unbalanced unbounde function empty one unlike special follow eq maximization unconstraine problem therefore feasible special f solution fy show subdifferential transform otherwise convex algebraic interior fy element call subgradient subdifferential functional strictly map single subdifferential example inequality strict strictly monotone concave dual concave analogy concave lagrange achieve shall study monotonic sufficient set belong boundary belong boundary closure half therefore write lagrange multipli lagrange define f fy corresponding sufficient via value equation property clearly existence guarantee restrictive first banach desirable measure often banach complete chebyshev objective utility cost function unbounded polynomial generally unbalanced unbounded consideration motivate functional generalize measure value element call thus functional although address coincide support cx cx mean space element space define gradient minimized function q particular thus unbounded existence close subdifferential empty close empty geometrically f origin one dimensional equation close strictly set inclusion fy prove optimality equivalent property include monotone reasoning strictly fy x generating order dual cone x operator order generate cone non negative assume opposite empty positive measure maximize mutual formulate state operator unit algebra hilbert correspond dual space x restriction super act algebra modular generalization conditional expectation g clearly projection physical meaning always completely positive onto modular invariance refer localization absolute order generating point cone family close strictly contain correspond mutually resp resp f x let localization completely operator act convex case contain mutually absolutely mutually absolutely measure order inclusion resp singleton immediately strict convexity singleton chapter singleton optimal equality cone solution generalize distance resource absolutely belong measure belong case simplex functional cone boundary absolute prove fact image subdifferential absolutely constraint next optimality non deterministic transition convexity contain understand classical preference relation maximize problem may problems information kullback leibler positive equal norm element necessarily classical kl case algebra hermitian trace type convex functional clearly maximize mutually absolutely continuous maximize exist element consider module algebra unique additive multiplicative yy yy terminal option graphic terminal need graphic macro ltb lt lt lt lt lt lt ltb lt lt bp p p dash information dual total ray maximize maximize linear functional element distribution figure dual c mutually absolutely continuous figure family simplex understand utility preference relation include attain represent relation achieve maximization discuss axiom theoretic information strict weak axiom ensure generalize problem ensure directional derivative information appear requirement support utility support operator onto complement element restriction onto complement element measure localization measure treat algebra scalar vector subset every affine equivalence localization measure corollary language composite set transition element decision communication optimal exposition function understand classical however unnecessary shall kernel transition joint probability probability eq event statistically dependency correspond deterministic eq one composite joint define measurable understood channel give notion amount shannon marginal theory refer third distribution transition optimization channel kind kind allow joint kind area decision joint boundary interior transition case fp problem b eq deterministic deterministic transition interior minimize interior fy f fp transform equality f relation separate discuss simplex simplex particular one often choose include element fact mutual exponential example qualitative later transformation deterministic kernel section measurable measurable measurable measurable manifold invertible bp bb kernel mapping b deterministic transition supremum uniquely determine countable omit final show express indeed empty constant empty infinite infinite input measurable sequence mapping constant I without probability hold grow information deterministic transition image notion shannon maximize input maximize proposition entropy potentially loose deterministic kernel sense kernel important transition utility shannon belong condition correspond exponential depend exponential use fact exponential write normalizing integral depend energy db shannon observe utility derivative differential independent distribution product equivalent locally compact utility linear difference expression joint mutually utility constant dirac case constraint inequality qualitative rather quantitative illustration information transmission optimal respect measurable distribution denote conditional maximized hand deviation cauchy assume belong element large unbounded giving infinite image finite argument proposition however amount deterministic utility amount exponential mutual utility translation gaussian expectation derivative amount inverting depend differential entropy construct instance expect kernel finite information utility deterministic deterministic qualitatively markov give generalization function model output machine study system transform output domain consider word call algorithm randomize accord computational terminate reach answer terminate reach computation terminate kind positive size resource computation sequence boolean indicate terminate terminate utility computation boolean utility maximization terminate final deterministic input computation terminate onto final word main deterministic way markov transition sequence deterministic correspond markov map deterministic assign transition kernel deterministic deterministic context variational problem generalization utility theory polynomial error utility communication number query oracle class problem utility duality constraint resource relate study family problem formulae relate free channel recover simply define motivation generalization absolute continuity family establish family separability useful context reason absolute deterministic strictly optimal computational devoted procedure qualitative result suggest broad constraint exist show expect deterministic strict kernel well recently concern information hand optimality subject constraint asymptotic output loose qualitatively
extent understand assumption supervise study form requirement reach regularize scheme study availability condition impose c pf assumption supremum norm eigenfunction integral make explain introduction gradient approximate solution equation subspace closely link square pls kernel pls wide standard property consistency provide pls component assumption latent optimal slightly cg study difference risk derive convergence learn empirical operator empirical reproduce property check adjoint satisfy fact adjoint usual conjugate obtain multiplication advantage noiseless version algorithm wherein replace kernel integral latter hilbert differ inner kernel function inner well integrable follow proposition quantify covariance operator operator integral noiseless denote schmidt bernstein second true sharp establishing abstract context minimal number consider cg algorithm operator datum output discrepancy minor polynomial hold datum cg algorithm consider exactly identify provide satisfied imply ensure precisely simultaneously deviation strong principle rule obtaining result fundamental allow obtain ingredient concentration deviation quasi prove theorem unfortunately necessary present convergence take instead respectively deviation form introduce obtain regularization type inequality value apply outer introduce therefore fundamental idea introduce material derive early stop rule true depend situation adaptive rely importantly idea introduce cg cross model show cg ridge include note selection parametrization threshold rule cast hilbert display cg use norm technical justification focus kernel pls pls future present important effort derivation depend prevent go expectation desirable perhaps stop case choice regularization optimal rate theoretically property hold general strongly hold yield cg regard estimation study whole application proposition property I theorem theorem least square overfitte stop combine rate regularity intrinsic map depend factor true function reproduce fulfil rate rate art operator base least square conjugate goal sample unknown depend satisfie assume belong integrable reproduce hilbert ny response expansion adequate regression closeness target criterion square minimization empirical distribution generating give rise equation invertible perfectly poor error overfitte considerable overview support generalization nf regularize inspire inverse popular case component regression extensively cg learn cg subspace subspace euclidean rescale cg iteration correspond early conjugate gradient appealing construct forward multiplication cg response learn situation relatively depend pp assumption weak noise put measure coincide case kernel intrinsic rate determine rate consider discrepancy sequence threshold actually inner consider discrepancy stop sequence reason stop stop step obtain sc hold probability
backtrack proof store store compactly represent scalable proof set proof algorithm probability discuss storing proof like proof derivation tend share suggest try task try originally generalise recursive structure please refer try automated logic essential property store different path token term token common branch distinguish token storing token respectively internal transition may child sequentially become threshold dynamically index far reduce dynamically expand hash table require require reach leaf node storing list proof proof use fact element present store three proof root six token seven share save common store save common dash false represent set reduce view compact decision tree node label two leave assignment child take assign start obtain subgraph redundant reduction possible node subgraphs root path tool translate generation build execute via utility share file replace child ct ii n c I generation otherwise become extremely memory quickly build logical operation successively detail combine create child child node leave subtree translate result occurrence subtree translation graph result ce ed ac n generate calculate sum child node true false terminal terminal return return assign leaf label finally entire probability receive threshold determine end query stop section width execution quite two number program complete check accelerate process skip cache standard database database sample million proof often whether fact database sample suggest good take whether represent compactly three mean belong call fail directly database sampling fact biological extract around know result graph start three comparison calculate exactly use roughly acyclic long testing database store gb report respectively furthermore index database query explanation start measurement take spend proof spend execute whenever meaningful report exact interval include threshold shrink algorithm interval interval sample path k compare algorithm table result apart therefore obtain approximation proof query increase running path code run widely actually depend interface magnitude converge need remain exact implementation use total reach show fast first good confirm top probabilistic rr c c c c graph around edge inference long show probability node lead also bound reach loose bound formulae encounter process hour cause query backtrack heavily version obtain second require tight c c database cover million probability monte large possible imply success practically branch bind reasonably tight interval path c c give query unlikely consider restrict connect two none exact fast carlo reliably c bottom confirm implementation database original part thereby application language represent logic design scale success query example execute database address disjoint already scale implementation pd tight lead focus connectivity query work query database clean background engine interface promise closely pd semantic even though subtle semantic state art system without exclusive implementation calculate know meta implement sum symbolic build program mutually exclusive collect proof explanation mutually exclusive sum logic programming calculation receive increase logic primitive therefore relationship logic perspective hope general language efficient relational already cp elegant probabilistic representation closely probabilistic type estimation logical parameter probabilistic logical closely inductive limit relational learning inference repeat probability query implementation raise problem deal disjoint interesting transformation could optimize program take situation disjoint currently investigate efficiency incorporate like team logic foundation project ci university fc year relational develop extension mining biological fact indicate fact belong sample kind introduce top entity past database logic develop prominent include logic pd traditionally study focus impose first learnable expressive interesting inference expensive answer long inference inference even probabilistic hard probabilistic logic use context biological network edge label biological protein phenotype extract public database link various treat mutually whether belong randomly sample program success succeed semantic well time database however semantic infinite random variable semantic order inference additionally query represent network mutually exclusive implementation implementation contribute success explanation efficiently diagram adapt approximation low success contribute compute bind proof integration algorithm integration use effectively nod semantic algorithms approximate discuss report conclude upon fact clause mutually different program clause add shall use run encode subgraph coin edge set possible denote fact program since background fix definite interpretation define interpretation generalize restrict fact program ground ask node say randomly subgraph edge tree depict figure fact necessary proof probabilistic fact introduce indicate logic program represent time use goal contain b full program consider rewrite entail vice versa explanation contains outline initially neutral proof yet empty remain succeed clear monotonically never change contribute hence pp subset probabilistic conjunction represent construct know pp pp pp illustration additionally encode e formula full low order implement describe approximate query evaluate introduce sort lead success branch calculate explanation proof cf query represent correspond explanation overlap good proof account add edge cd bc cd propose carlo execute existence query provable compute give use stop carlo similar interval context biological represent monte carlo programs although neither differ mcmc method
well unseen kernel hilbert add sort exactly optimize instance regularization assumption capacity although obtain well sense modify optimization algorithmic tradeoff problematic interested scale well amongst practitioner heuristic speed computation sometimes regularization implicitly learn network iterative bin observe phenomenon application order meaningful domain motivate interested great extent formalize address nontrivial graph laplacian special interest vision vector characterization walk base approximation eigenvector one base computing nontrivial eigenvector graph procedure implicitly regularize optimization exactly interestingly identification relax program enter unit formulation distribution somewhat implication explore intuition matrix eigenvector power method start initial compute weak eigenvector expand eigenfunction iteration approximation eigen eigen draw intuition undirected walk associate usual matrix vector let normalized laplacian start optimization problem remainder subspace reference orthogonality statement impact argument check carry language description three walk matrix naturally heat alternatively write denote heat heat thus describe heat undirecte one walk r multiply connect undirected walk transition hold permit iterate walk except top use place nontrivial achieve initial random heat matrix operator spectral sdp sdp x operation program unit vector density let lx objective semidefinite invertible minimize second statement regard assumption statement kkt condition strong sdp formulation keep exposition decide present easy optimal diffusion generalize f hx establish scale heat tr lx conversely give parameter equation parameter parameter determinant function f f appropriately scale version vary ad conversely x hence appropriately thus establish solution give step depend x vary require ad conversely walk proof large body theoretical broadly inform geometrically learn explicitly regularization improve property act level regularization design ill novel zhang yu boost oppose convergence well describe aspect noise eigenvalue arise estimate walk improved partitioning heat call try community community empirically implicit quality heuristic interpret alternate perspective several small nontrivial adopt perhaps surprising associated statistical imagine one method nontrivial eigenvector adopting perhaps surprising identifying noisy motivated us ask recent characterize cluster large property formalize severe cluster phenomenon commonly observe machine phenomenon intractable eigenvector compute eigenvector rank understand concept extend problem though concept implicit extend obviously vector perhaps regularize intractable
map provide implicit key detailed understanding state notation determinant function quantum system formal treatment reference euler product periodic term power concentrate correction resemble surface case rigorous connection quantum nevertheless indicate beyond axis trace admit expression sum formal eq form formal clear depend hamiltonian maps physics physics devote boundary obstacle obstacle mathematical treatment case convex illustrate set three also near arc parametrize horizontal vertical angle velocity tangent circle green blue correspond region forward bound stable manifold latter unstable reduction quantum map physics study discrete construct transfer operator unitary replace unitary quantum map quantum toy system generalize unitary quantum map quantum physics atom dot map popularity quantum map stems much offer quantum classical level realistic flow quantum organization remainder classical particular introduce refer support flow consider tool need calculus operator scale strategy associate e complicated find formula problem pose serious difficulty suitably projection lead hamiltonian e map summarize acknowledgment thank national foundation dms complete author institute advance national science foundation la thank also fig apply give differential manifold reader interest recall physical operator assume x px pp hx uniformly respect compact compact self adjoint correspond assumption decay long admit extension assumption roughly flow energy encode e intersect abstract see classical symbol assume characteristic surface simple like ensure fix define finitely compact boundary eq disjoint uniquely remain neighbourhood neighbourhood smoothly eq always component component application change dx topological due flow eqs diameter partition realize add component make fig unstable dash analyze discuss quantization identify notation map arrival subset call obviously neighbourhood local neighbourhood neighbourhood totally mutually disjoint notice contain also define trajectory fig sketch definition trajectory boundary arrival contraction imply boundary group call view sometimes map support structural hold flow component quantization family mean thing r n symbol necessity come slightly fouri due exponential operator characterization single operator value quantization integral comment term simplify place confusion context involve function compact set norm possibly neighbourhood symbol make notation finally complex construction affect formulate present material reference proof start symbol quantization calculus class quantization exactly symbol symbol projection main symbol symbol admit power symbol operator principal symbol admits call notion symbol unless norm norm recall operator uniformly characterize symbol say q intersection space concern essential support function wave front replace always satisfie characterize individual family operator h exist specifie say asymptotically q l operator set w st fourier globally fourier global integral operator dependence example family unitary family dependent field real value symbol canonical transformation graph denote usual lagrangian unitary canonical graph locally unitary integral operator since canonical point integral class form operator characterization fouri near origin always possible family neighbourhood construction unitary cutoff operator associate neighbourhood unitary open define neighbourhood say unitary integral neighbourhood equivalent unitary integral construction instance contain self principal flow generate fix energy zero transformation r hamiltonian quantum operator certain space possibly associate neighbourhood fouri integral operator r compose briefly see reference independent operator totally follow coefficient analytically outside operator operator r r simplify r consider existence implicit z analytic region reason property come symbol identification principal symbol outside enough convenient explicitly long near infinity intermediate compactly infinity instance potential direction consider value function calculus functional self adjoint arbitrary write q put simply put equivalent govern property symbol show convenient notion give lead equivalence ball inside past future field need set interior time bound outside neighbourhood rx ik enough neighbourhood rx set notation eq complex neighbourhood drop transformation review result unitary compact direction fouri integral well unitary cutoff neighbourhood define map converse solution flow fouri integral segment relation associate operator smoothed region hermitian inner bring neighbourhood fourier operator symbol inside time terminology ignore symbol write fix writing define similar expression project well neighbourhood cut outside compatible extension solution fourier q property define operator integral l equip want solve solution show obtain integral operator operator correct ensure identity solve system finally solve full achieve notation see concentrated cutoff flow consider problem concentrate neighbourhood neighbourhood aim constraint precise slightly short intersect intersect fig neighbourhood near neighbourhood condition fulfil thank resp dash contour thick arrival connect dash two trajectory inside extend fourier cutoff set operator concentrate take set maximal e satisfie estimate cutoff near arrival due reads parametrize obviously define operator k formula solution k linearity problem let back operator relation identical composition relation associate graph coordinate jk unitary neighbourhood near forward consider property cutoff effectively unity ensure jk intersect concentrate notable state solve inside neighbourhood away procedure notice summarize concentrated neighbourhood datum full homogeneous solution concentrate course reflect neighbourhood unstable describe proposition problem care norm auxiliary difficulty remark end modify norm weight construction scale long q appear satisfy appear discuss still neighbourhood neighbourhood describe scaling weight near local g integral weight depend expansion invariance weight check solution analogue cutoff cutoff symbol calculus jump near return h k respectively argument carry estimate satisfy iteratively symbol change norm hold definition n estimate transform globally require transform acting take place auxiliary subspace compose near invertible construction appropriately build weight construction away modify take allow realization operator obtain modification describe place near let let neighbourhood eq coordinate last independence simplify make trajectory segment outside strict outside particle remain next function define auxiliary element hilbert q globally proposition construction nice space advantage operator suppose support small enough hilbert space satisfy set need adjust hold belong necessarily hand increase see inside layer neighbourhood arrival one appear set inside time outside symbol lead symbol choose know fix neighbourhood negative spectrum together diag g r reference calculus instance globally mod select realization restrict realization operator define put together short space prove suffice inverse rest devoted component operator act cutoff trajectory old project side negligible notice property split component arrival set rewrite data state arrival remark replace operator drastically modify hand space precisely bad case cutoff require ratio enough indeed small inside put crucial rank counterpart nz gx property jk z j trajectory connect jk disjoint support segment symbol multiply decompose near take eq solve calculus spatial cutoff q state adapt small enough cutoff outside norm eq remainder v h occur decompose cutoff q q remainder gm estimate
microarray gene investigate linear publicly microarray set assess randomization expression kf st decode human genome molecular genomic essentially within genome simultaneously provide far small take impossible g logistic regression microarray microarray important interpretation microarray microarray major topic induce classifier e diagnostic learning learned predict diagnostic unseen sample design decrease increase grow refer phenomenon classifier dimensional pool covariance discriminant lda hessian package fail call dimensionality microarray irrelevant gene gene maximal discrimination induce use procedure gene individually advantage low alternative dimensionality dimension extraction aim dimension reduction extraction microarray partial square pls inverse sir microarray investigate area decade paper address fan important future gap propose handle high microarray prediction conceptual modeling begin quite recently recognize variable relate set observable observable variable child intelligence variable assess intelligence ask wider include assessment originally probability correct trait ability intelligence participant response item gene row biological expression gene factor idea approach gene gene e factor account gene similar biological factor protein element factor gene measurement relate gene gene factor meta magnitude prediction carry main microarray gene expression validate parallel pca propose expression step response gene publicly microarray set contain sample gene sample gene tumor publicly filter intensity consist gene expression pi preprocesse datum procedure handle gene cpu consuming require many percentage expression gene two feature selection consist gene experimental set test measure significance change expression define statistic statistic briefly subsection short overview original expression datum trait item person item person outcome denote latent item describe trait produce specific response person score latent score score total person calculate calculation find terminology item gene see imply functional gene vice versa determine factor identify cluster partition gene partition profile mean discover describe need compare datum partitioning gene partition calculate score index let factor sample partition specific microarray dimensionality datum much variation original transform original variable sequentially covariance combination coordinate direction amount original component variability computation vector calculation decomposition eigenvector principal provide linear specify knowledge procedure rule reduce low dimension describe avoid use latent analysis pca factor constructed consider formally consist sample test class sample denote rule implement th sample profile k reduction step short predictor matrix lda assume discriminant assign class widely lda relie assume hypothesis satisfied set demonstrate performance complicate calculation reader factor meta give prediction sensible criterion purpose subset select one sample extraction fit use class subsection successively value run response class otherwise denote select experiment base class randomization evaluation create sample sample learn datum gene gene one method matrix subsection latent reduction loading factor l gene predict subsection repeat step whole predict indicator use measure probability bias skewness overcome receiver roc reader perform false specificity various result use varie class sample two proportion consider acceptable excellent computation carry statistical graphic generic gene function random splitting perform conduct function mass roc interest application microarray preprocesse apply gene subset randomly gene procedure set determine component performance prediction reduction randomization trial value parameter deviation roc see decrease increase size subset predictor include model building pca dimension excellent h legend lda base prediction pca factor present subset accord statistic gene extraction analysis describe roc curve compare method score excellent performance agreement hypothesis classification accuracy low legend mean component area roc data tumor random approach learn class method trial statistic pca prediction random roc pattern performance performance prediction generally lda legend lda error area roc yield figure method regard roc perform excellent legend lda
semidefinite keep considerably vast field optimization program familiar sdp constraint component format regard convert semidefinite constraint augment translate linear every cast finding cast program duality apply semidefinite decade thousand considerably sdp solver toolbox available translate level semidefinite program involve function mention semidefinite whenever union satisfy fisher introduction know space finitely state semidefinite compatible order design characterize optimal problem semidefinite programming support fix optimal assign allow problematic reason support design intractable support semidefinite optimal semidefinite semidefinite finite interval rational rational respect design subset zero polynomial degree whose aside matrix translate matrix semidefinite program example extension proof detailed demonstrate optimal translate investigate high degree rational source solver computer ghz core intercept consider illustrate approach verify semidefinite optimality scalar order hence entry constraint whereas correspondence summary drop subscript anti turned plug computation semidefinite solver explanatory toolbox trace represent high level translate solver leave work variable variable pi minimize pi pi pi toolbox index omit brevity optimal polynomial whose root eight e time polynomial model characterize design follow semidefinite semidefinite scalar first translate compare system equation along real polynomial high designing degree stability scalability high rarely ill polynomial fisher becomes ill condition considerably lead ill condition orthogonal basis look optimal polynomial ordinary basis st tp tn solve program computation second single root polynomial polynomial allow subject rational measuring measurement intensity obey square intensity source total estimate affected source detector measurable measurement detector semidefinite take yield point obtain zero optimal design solve useful zero polynomial might semidefinite simplify essentially calculation set w infinitely correspond design even trivial nonlinear rational model response variable rational function explanatory parameter consider class recently family rational consider satisfy many rational class fundamental non experiment optimization correspond immediate fisher whose estimation design guess useful sequential design concept nonlinear consider readily notation matrix write q finding equivalent linear rational applicable simplification polynomial find easy fisher put parameter nonlinear parameter find design associate derivative fix always one observation optimal chebyshev assumption drop criterion also consider rational rational express finitely constraint computationally work include truly generalize estimate number support root interior chebyshev optimal design linear rational space finite interval finite symbolic form limited interpretation rely effective method whose design applicable problem involve rational treat general generalize rational finitely interval generality far achieve symbolic solution generate program high trivial iterative base semidefinite theoretically guarantee low find globally considerable importance impact study insight demonstrate flexibility robust handle condition analogous generalize linear rather rational fourier polynomial may design nonlinear symbolic question remain extend criterion transformation find equivalent semidefinite representation chebyshev natural importantly idea identify would extremely multivariate location signal source close solution aside involve concrete applicability far study nonnegative know polynomial functional constructive claim apply let exist semidefinite satisfy polynomial semidefinite detail computation reveal characterize semidefinite semidefinite assign semidefinite notation may cone relaxation multiplier suppose admissible lagrangian dual attain finally optimization write eq aside constraint variable rational function variable multiplying denominator equivalent inequality give optimal optimum last way concentrate pt em theorem remark department engineering management sciences rd il linear polynomial rational rational treat zero optimal generalize previously result incorporate various establish previous optimization problem example consider polynomial discuss find locally optimal design rational extension corollary find considerable instance software paper optimal design polynomial rational error model rational rational error normally denominator estimate uncorrelated linearly characterization design translate numerically readily characterization every involve polynomial rational give rational handle run practical optimal design certain model find many component estimate entire identity design mean design show devote finite convex even difficulty arise characterized root support design discover ordinary constant zero degree design polynomial classic determine optimality criterion quadratic discuss design website comprehensive maintain considerable interaction give design odd term multivariate problem cube design incomplete polynomial even general weight consider rational union interval base orthogonal polynomial chebyshev rather scope proof numerically method compute design bottleneck carry finite support canonical carry relatively polynomial concerned univariate np restrict known difficult degree applicable attention turn design comprehensive oriented viewpoint numerical method variant variant classic gauss free maintain support another well current support idea serious drawback care abuse variant know iteration fact iteration coordinate problem iv necessarily converge satisfactory experiment report cf propose model rational optimization involve solve point exist numerical coordinate approach computing coefficient polynomial zero formally derive parameter illustrative case nonlinear involve mostly polynomial stand polynomial denominator function frobenius since decision operator unknown write avoid product adjoint identity cone positive semidefinite denote finitely support notation integral respect order end design call short compatible call small denote arbitrary extended value function finite difficulty design discuss definition summarize affine characterize inequality semidefinite set equivalently allow motivation
theorem eq write bound eq writing slowly slowly vary converge zero rapidly give rapidly generality clear long slowly x nx large eq n x l l writing since eventually decrease hence right asymptotically almost surely theorem write almost since expansion decrease decrease thank david lemma university college primary variation residual survival von limit extreme consistent robust new good see operating explore substantially set introduction tail play important finance pure tail ordering efficiency location value asymptotic maximum medium long tail fr standardize right classified classical categorization law medium may shorter thus distribution classification attempt order work middle ordering see paper propose pure tail pure distinguishing use popular one various behavior tail behavior base residual life hill argue qualitative hill include undesirable behavior hill plot concern restrict tail hill provide apply hill distinguish tail may sometimes hundred thousand large quantile tail actually exceed counterpart tail characteristic simulation work test hypothesis result focus tail consider take hypothesis medium tail distribution long represent statistic sample work section develop discuss relevant discuss alternative wise simulation result type good property comparison conclude residual good type error close gamma gamma medium good interest breaking base argue quantile eq leave inverse tail define short whether indicate write shape let left associated relationship approximate turn restrict monotone density discuss distribution quantile table interest major drawback classify issue estimate exponent use hill early u condition standardized classify come constant estimate analysis possess operate medium tailed sensitive medium separate definition rate refine rp classification q short medium medium medium rp normal short medium medium medium unfortunately rp distribution require value tail behavior es second data quantile interval define define ii iii zero mean connection rate exist pareto infinity failure go infinity converge positive value make rp es property slowly consist rp short medium category rp medium medium rp medium es short fast es medium es slow depend shape distribution medium tailed medium es distribution asymptotic extreme still precision provide sense include medium short long failure make precise es utilize medium tail result residual provide membership definition xt ht function always exist life example consequence corollary hereafter denote mean exp x class behavior es medium testing medium tail medium perhaps medium distribution underlie see long tail therefore pareto es long refine classification asymptotic tail several cccc exponential medium medium medium medium weakly short super cauchy moderately long super pareto long moderately long weakly medium medium medium short medium moderately classification extreme residual scale henceforth tail sense survival function tail behavior arise medium tailed exp unknown need asymptotic distribution medium function slowly nuisance limit estimator fy multiply divide parameter limit statistic es medium must verify medium tail turn fy medium rapidly whether instrumental prove tailed vary fx c cx eventually tail alternative short tail asymptotic simulation previous asymptotic property long short type distribution rejection gamma shape type es tail choose sampling table statistic n x desirable rather reflect es medium tail power statistic es tracking es short distribution serious misclassification serious misclassification classify versa short sample survival great expect l gives state es decrease w w value perfect detect power unfortunately extreme case percentage classification simulation reject classify tail lack table es tailed percentage desirable slightly sample excellent l introduce among es medium tail good distinguishing significantly show capability distinguish address differ find test power short normal approximately block give rise block subsample hypothesis medium sum hypothesis hypothesis es medium tail short tailed favor reject favor percentile size otherwise block large sample without error en stand exp stand table detect tail substantially ccccc exp exp block l l distribution show power short tail sample norm par tail classification classification par increase correct desirable extreme increase percentage similar x exponential selection drive trade power table tailed distribution consideration behavior small level level tail sometimes drawback error gamma medium parameter gamma favor type distribution long tail favor reason since center sequence achieve whether small manner shape quantile shape quantile implement reject close medium medium tail test statistic similarly medium high distribution quantile gamma shape observation gamma cccc cccc cccc devote claim european million adjust discuss exponential right great right tail confirm long behavior plot test tail long tail million shift subtract million claim hypothesis es medium tailed thus tail break strength appear plot tail fall exponential right tail give therefore hypothesis medium tail favor thousand datum fit histogram q suggest tail subtract observation test statistic yield medium f x f nx pz pz pz n pz c pz pf n n x f f ff
noise tend remove difficult actual bernstein establish integrated give mixture satisfy involve average consequence procedure study scheme f integral consume part form absolute taking absolute additional logarithmic relevant proposition ny nx differentiable elementary differentiable get nf jx side na r pf f display useful page let random eq lk lk nc lk display yield integrate get next jensen inequality basic eq give q display obtain obtain successively datum eq argument take j definition lemma http edu support pour difficulty consider achieve good performance minimization computable design establish rate achieve reasonably mild performance result measurable noise know expectation denote define pf p empirical integrate independent choose may treat question rp consider design rp construct index nonzero unknown performance consider penalize minimization pt eq penalization bic pt combinatorial consider computationally feasible example cf chen pursuit vector allow lar establish establish oracle inequality restriction recently propose minimization propose zhang alternative motivate performance either theoretical design van de survey simultaneously taylor framework adapt study condition high weight pac design establish excess risk study exponential first sub estimator initially set note estimator compute fraction consider aggregate scheme datum splitting splitting necessary least dimensional may deterministic pac develop framework inequality excess aspect monte reasonably introduction prior hasting organize procedure oracle design modification oracle excess noise n variable aggregate define suggest prior variable aggregate establish deterministic design display bad notion estimator display combine achieve ep section rate
find subspace generate discuss track motion condition extensive hybrid handwritten digit affine vision motion rise refer subspace tracking segmentation frame object affine camera lie equivalent cluster subspace prove object cone accurately approximate low linear subspace face subspace require partition datum I unknown dimension manifold kf direct affinity agglomerative spectral curvature appear recommend require initialization need collection lie underlie require propose geometric local model component appropriately size global hybrid suggest however neighborhood adaptive sized need recognize quantify study local fit flat art nearly deal obtain art motion particular outlier justify complete particular valid analyze ssc additional restricting modify rigorously quantify main precise local approximately find good neighborhood different extensive datum benchmark face handwritten database datum show particular synthetic extremely previously mention method face indicate fundamental suggest fit heuristic show heuristic quickly determine state find test compare hybrid segmentation video handwritten digits recognition fast discussion describe method capturing optimal neighborhood approximately find neighborhood justify capture fit optimal neighborhood radius neighborhood small around around hybrid consequently local flat neighborhood around underlie fit equivalent though common refer possible guess fix neighbor reasonably adapt look large neighborhood neighborhood dimension neighborhood become approximated sense belong enter neighborhood onto flat numerator flat utilize invariant optimal neighborhood precisely neighborhood near neighbor increase neighbor check neighborhood neighborhood search summarize nk kt kk try justify around r b underlie mixture arbitrary choose adapt probability appendix analog ball case l measure let minimum weak often sort computational preprocessing represent point complexity note limit term replace good candidate describe tuple near flat randomly list pass minimizer pick htbp local scale cluster pt generate contain random random flat power outlier square minimize indeed require evaluate configuration example strong robustness one follow related subspace datum choosing fit arbitrarily thus scale svd decomposition represent since cost complexity come complexity scale requirement organize size candidate storage candidate subspace find neighborhood fit neighborhood finally spectral matrix algorithm replace matrix multiplying corresponding root eigenvalue remark change use euclidean parameter default disjoint approximated pt em normalize accordingly step default input choose similar ssc affinity th entry affinity ideally affinity expect underlying subspace cluster suggest far choice close point segmentation embed small small norm row fit subspace neighborhood moreover force applicability base try enforce practice operate differently though ssc base calculation step limit replace storage exclude algorithm experiment notice without much noise allow allow technique tailor ms r r c kf oracle pt ssc r pt pt kf ms pt pt conduct artificial verify situation accurate mode discuss correct misclassifie exclude change trade accuracy large balance though perfect intel ghz section run intel ghz gb kf agglomerative matlab code kf http edu http www edu db http www edu http edu code motion http www edu http www ssc try modify tailor step refer ms motion segmentation mean ms matrix wise initialize guess membership initialize pick tuple follow kf guess since kf tend kf record run ssc algorithm algorithms oracle oracle noise construction speed database motion http www edu contain frame videos frames video formally frame due camera move distinguish feature background coordinate camera object across frame live affine subspace implement ms min min motion implement subspace linear manifold subspace learn early cluster segmentation copy www edu report ssc ssc ssc ms median misclassification two experiment misclassification record table demonstrate figure misclassification misclassification ssc twice explain ssc ms slightly well ssc besides ms ms run fast ssc ms ssc many energy true ms work good data information spectral combine adapt ms total two advance parameter paragraph ms ssc negligible randomness ms random comparable deviation test ms ms ssc face database http failure extend face database consist image vary condition image repeat randomly face variable lie dimensional nine database roughly slow well also provide example follow http co software table failure two factor sparse per dimensional relative consequently span cluster neighbor closer near distance whereas term consider subspace ssc find discriminate direction irrelevant classification dealing whiten remove exclude reduce greatly become whitening htbp whiten whiten ms voting ssc pt htbp whitening whiten ms ms ms work available http com work contain digit digit reduce dimension rich process record misclassification htbp c pt ms voting pt ssc htbp pt voting pt ms n pt ssc c ms ssc ms ms ssc misclassification misclassification ssc also good ms fast mnist explain use return algorithm point flat among determine add find logarithm e apply ms ms ms part artificial intel core ghz gb artificial user compare ssc artificial http edu sample unit cube corrupt deviation four restrict subspace separation let voting majority neighborhood heuristic true distortion find pca choose try pick obtain low algorithm part high ambient try idea nevertheless well report repeat speed find second htbp c c c minimum ms voting ssc good local flat heuristic distortion help reduce vote artificial option computation reasonable obvious face detect number ambient subspace affect experiment repeat whitening rate correct cluster record ms ssc perform ssc small difficulty indeed b ms vote perfect detection whiten ambient include intrinsic respectively tolerance repeat due error table htbp ms ms voting ssc c subset ms vote ms ssc large real datum except ms ssc demonstrate geometric fix neighborhood neighbor well flat flat continue stop parallel design favor neighborhood data three outlier favor sample rate neighborhood adapt neighborhood average run neighborhood wrong
take induced generalization orthogonal orthogonality follow coefficient function metric k dependence last implicitly prefer dictionary dictionary question arise probabilistic consist value outside dictionary function cover dictionary depend unit hilbert sphere sample dictionary sample length fast draw eq certain generalization representation give straight proposition draw result rate bind let least minimize right dictionary signal high infinite reconstruction treat magnitude feature element inner fulfil representation hilbert concrete word let count well bag imply reconstruct content could decide dictionary bag approximate dictionary suggest minimize eq act norm induce combined cover dictionary reasonable eq product describe remain cover order cardinality cover number generalization prove section restriction orthogonality dictionary rely lastly recall cover number definition wish introduction application generalization cover da sd note banach formalize cover cover lipschitz next clear metric function start define norm maximal bound certain induce qp I geometric representation dictionary induce representation error representation set another unit representation I number symmetric scale point sphere eigenvalue norm theorem eigenvalue particular inverse scale linearly eq norm lipschitz dictionary lemma whose existence conclude next lipschitz constant subset dictionary h least signal assign dictionary basis arbitrarily construct identical covering result need appendix let cover supremum least class function note cover metric minimal arbitrary member ef mf ef mf supremum choose inequality mf ef mf mb mb dc mb probability bound cover substitution metric quantify orthogonality dictionary uniqueness sparse algorithm dictionary proceed discuss elementary proportion dictionary orthogonality term product perhaps simple coherence product represent expressive power illustrate restrict exclude disjoint whose inner first extreme mean orthogonal sign well generic dictionary half proceed prove overcome maxima possibly unit uniformly light variable choose rotation maximum find eq q clear scale linearly triangle tight case estimate dictionary kernel show complexity property simple kernel present feature simple signal unit unit apply nothing representation corresponding representation near proposition euclidean norm dictionary dimension practical issue order address computational generalization bound consider signal represent metric denote show feature constraint pose difficulty beyond common depend dimension simple approach lead weak case space instead use representation dimension consideration map old hold relevance old map enough old old map dirac counter two map geometry sharp kernel older use old condition x root mapping define proposition give cover theorem representation old old cover dictionary sufficient substitution several implication language penalization sparse property dictionary low relaxation mistake g present exclude possibility upper large trivial future exploration view favorable geometry encourage acknowledgment thank partly european community fp agreement appendix justify growth rate instead combinatorial effort constant tight concept need body vector star shape closure sub root non decrease rademacher average integral xx xt te u du tail particular qx qx qx xx xx xx prove core decay complete lemma replace expectation upper entropy observation r applying since f dx department electrical institute electrical institute technology element application denoise separation previously unseen example magnitude assume distribution question generalization learn dictionary expect regularize dictionary level orthogonality assumption strong yield rate oppose localize complexity provide similar set smoothness requirement technique sparse representation combination detail measure often measure vector coefficient determine induced call representation generalization extent
minimization admissible path solution compute relate chain minimizer replace minimize maximization solution maximize bring newly elegant indeed recursion solve unconstrained optimization term risk notational hand simplicity consistency hand involve observe immediately generalize state solve recursion sufficiently small actual probability decode motivate impose positivity easily viterbi j x eq positive prior reduce back clearly start would arrive generally treat previous consider also equivalent newly introduce map viterbi decode combination maximum prior decode every admissible perhaps solution minimize generality assume exist admissible path risk suppose must imply hence imply viterbi decode almost surely take fairly nonetheless x tc tc admissible solution similarly almost surely computed solution find every next let follow score terminal path state completeness break slight abuse notation maximizer let yield every allow interpret recursion hence maximizer form recursion generalize far follow risk problem remark posterior viterbi decode surely decoder viterbi solve refer decoder distinguish generalized accuracy decoder characterize solution function fa ga fx x state completeness write b ga fa assume c c r lemma insight well drawback paragraph elegant ordinary concluding recall adjacent minimize derive loss usual maximization decode path natural think minimizer move towards viterbi indeed minimization viterbi decode experiment minimizer point monotonicity allow I use unlike grow block away replace eventually viterbi certainly code idea well long block result guarantee another risk positive integer generalization first since let q gives hence also divide complete result context path risk risk markov homogeneous priori apply risk decoder viterbi hybrid significance show solution compute fashion theorem risk marginal along inequality monotonically decoder embed risk family q viterbi clearly within case recall end reach sufficiently obvious recalling inequality corollary viterbi since would generally comment definition decoder inequality divide r vx result consider interpretation entire possible least viterbi inference risk approach mainly publication dedicate inference apply transformation clearly lead decode return break power transform nonetheless let function assume j propose return viterbi point motivation necessarily viterbi ignore viterbi symbol viterbi implement viterbi decode symbol sense emission emission sequence nine path viterbi path px symbol aware suboptimal actually viterbi absence symbol symbol uniqueness idea intermediate decoder time attempt viterbi follow map continuous verify function laplace calculus logarithmic identifying recursively backward respectively notation produce viterbi latter transform appear proof lemma notation match lemma leave implicit introduce state part start equation error contradict return essence transformation viterbi clear besides parameter transformation limit special reason really wish idea moreover explain hybrid also subsection become operational subsection operational modify decoder practice transform probability vanish see f moderate length soon exceed dynamic double precision bad computation chain indeed appear function use trick remark intermediate logarithm however transform I mean become transform generation store irrelevant trick require actual value course function operate resolve worth effort complexity issue hybrid jacobian refer via recursive generally albeit expensive commonly k mm irrelevant compute transform forward backward variable respectively claim operational decode trivially state symbolic toolbox machine hybrid decoder base already decoder base magnitude use practice example verify realistic practice symbol decoder vanish joint p ti I ti I py I px scale aforementione emphasize viterbi practice ordinary hmms secondly potentially principle redundant bank secondary structural atomic http ac uk hide represent realization position observation symbol distinguish four enumeration final seven class number five eight long specifically first long similarly last prediction protein divide class go likelihood transition compute situation parameter value repeat leave stationarity evidence appear short display position split piece truth row decoder gain viterbi subsection secondly accuracy transition state five respectively output member posterior viterbi legend monotonicity hybrid illustrate risk viterbi indistinguishable constrain monotonicity viterbi constrain row indistinguishable differ capture row probable neighbor likely illustration idea blind priori pointwise generalize matrix absence ht risk c additionally instead dataset initial emission realization simulate necessarily degenerate risk operate reasonably realization subsection stands give parameter underlie population could risk analytically simulation reality complicate specify cv output wise obviously reality cv rely likely report variability merely observational validation average sensible bootstrappe possibility simulate synthetic hmm distribution freedom margin viterbi round accurate less viterbi e obtain subsample consist realization appear notably viterbi decoder accurate decoder extreme difference short position shorter subsample confirm decoder decoder viterbi examine sensitivity histogram subsample consist subsample extreme histogram suggest replace viterbi decoder examine risk difference largely unchanged minor increase subsample position risk viterbi validation viterbi gain viterbi respective pointwise viterbi come accurate viterbi little accurate viterbi respectively replace risk compute viterbi six next log return avoid switch display return bar bar constrain new viterbi path constrain viterbi decoder return decoder realization monotonicity base notably admissible viterbi simulation compare measure posterior recall block parameterize give viterbi continuous parameterization case display performance member family identical remarkably blind viterbi circle classifier call evaluate sequence write besides interested risk viterbi viterbi path viterbi decode viterbi hmms ad viterbi possess ergodic viterbi see converge random risk imply risk appear actually proof risk help viterbi inference misclassification rate viterbi decode thus viterbi make error asymptotic risk theoretically reality however asymptotic expectation risk even viterbi decoder long run universal prove upon theorem risk risk decode possible viterbi nice prove completely probability prove minimize could prove risk combination three constant intermediate case together long vx predict outline say constrain decoder constrain viterbi decide like risk x tt active word computable latter measure x line risk member multiply prior risk vector transpose provide state undesirable transition recently loss consider provide generalize follow perturb version assess stage exploration want use output produce loss instead one risk ordinary risk combine family risk transformation minimization immediately member multiple readily corollary chain represent markov proof path extend annotation already mention namely assign problem solve already labeling priori admissible positivity class extend loss generalize incorporate since successful generalization offer possibility potentially arithmetic ts ts ts certainly risk dependent tuned cross generalization present extension base e could author grant nr sf visit ii author grateful thorough review work suggestion dr manuscript point subtle mistake mat simplify emission alphabet symbol idea relevant situation emission alphabet big emission continuous emission distribution unconstraine return decoder positivity return path despite prior probability zero posterior decoder constrain path estimate hmm dataset describe l decoder probability l decoder decoder pt hmms paper hide primarily posteriori viterbi path pd around dedicated application decade careful several problem issue propose practical furthermore simple hide show compute present interesting pd algorithmic interpretable bioinformatics task generalization admissible path hybrid interpolation accuracy symbol posterior decode viterbi besides application process communication reference become computational bioinformatics security hide field model influential spatial image segmentation hide idea year success distribution system possess albeit whereas marginal markovian posterior independence observe lead hide naturally special hmms particularly fact indexing realization relatively straightforward computational approximate technique configuration hide field posteriori efficiently dynamic name simulate anneal useful semi factorial applicable extension exposition ordinary adopt respectively convention commonly let include brevity cause ambiguity write matrix hide markov regime non emission write shall assume emission borel measure count integer often stand observe unobserved hmm notation addition largely function require argument shall expression stand stand unconditional call datum segment path view segmentation region label far solution digital literature problematic path viterbi viterbi viterbi commonly think viterbi posteriori viterbi path may reason sub expect viterbi path typical indeed might probability add representative general context viterbi path particularly parameter transition probability probability coin hope typical realization maximize individual posteriori well maximize expect characterization estimation mode biology know decode pd successful posterior viterbi wide biological application dimensional consensus absence centroid communication hmms largely influence term e decode viterbi decode natural question improve maintain computational viterbi inference negligible conclude remark make though decode optimal rate viterbi conclusion binary leave room inference differ gain clear viterbi fact viterbi writing application viterbi use return systematic viterbi apart aforementione soon version article matter example sense maximize time possibly viterbi easily subsection besides infinite subsection constrain decoder path specifically call admissible explain prior note form aggregation force path appear possible course understand chain refer decoder admissible constrain also early ignore distinction posteriori admissible protein rise latter distinction decoder distinguish posterior mode distinction decoder path guarantee decoder obtain pairwise use suggest description detailed context path pointwise nucleotide level answer respect map address move thus example aggregate individual small effect map state mapping annotation viterbi inferior map annotation practically hmms np chain unlike viterbi symbol annotation path extend admissible path also know satisfactory although viterbi decoder still popular biology demonstrate various example recently demonstrate surprisingly viterbi predict protein restrict decoder admissible strong map seminal sensible vanish albeit leave unclear viterbi could viterbi inference logic true possible produce prior path path aware moreover related concerned map decision theoretic extension partially asymmetric loss incorporate prominent secondary possibility viterbi inference interesting need intermediate mode publish propose interpolation algorithmic interpret general importantly claim explain apart trivial situation despite raise discuss consider new unlike analytic measure family datum family optimize performance measure advantageous aware emphasis automatic decode hard valuable state sense identify cluster posterior recently thorough geometric general appear beneficial family smoothly optimization criterion understanding gain path similarity score context relatively become inference mapping eq decode refer classification decoder cause naturally formulate whereby risk family viterbi furthermore
example evolution along branch distance date plain sufficient specify set marginal specifie covariance class turn offer equally evolutionary inference value trait separable exist function covariance space separable separable component specifie follow corollary wiener evolutionary evaluate follow isotropic meaning isotropic gaussian degenerate process mathematical find supplement part proposition help autocorrelation establish convenient range covariance inference reduction spatially spatially supplement detail illustrate way make functional trait modelling spatially simple space trait trait correspond markovian markovian gaussian kt evolutionary proposition obtain note isotropic end belief variation trait spatial stationary trait hyperparameter characteristic combine summation homogeneous covariance property lose covariance control delta functional sample regression datum regular lattice illustrative develop supplement construct univariate specify priori alternatively eq give representation functional function inference provide question straightforward trait principle evolutionary trait trait proposition theorem section section biological object feature correlate relationship flexible combine bayesian placing rate evolution brownian extend function relationship example functional unobserved functional root infer sense temperature therefore ambient temperature growth rate heart series process assumption effectively specify character evolution ii role evolutionary avoid confusion indexing model flexibility mathematical substantial use gaussian prior nonparametric view account functional relate spatial involve multidimensional index topology conditional independence particularly use evolutionary versus regression considerable recent cross unless specify mean common gaussian specifying encode depend equal argument covariance co might make random co co evaluate pair mean combination independent long model variable dimension evolution trait indexing point observation surface
assume number candidate expressive exhibit content cluster bit communication sensitive mutual approximation concept physics set ensemble mechanic logarithmic correction calculate factor know factor joint mutual relate boltzmann identification ensemble access rich physics analogy theory mechanic particle role free energy reflect code logarithm partition markov monte employ analytical technique deterministic function reliable capacity sensitive high data tradeoff rank describe paper explore set string symbol alphabet significant communication comparable respective reliably bit sensitive information fluctuation identify individual suggest solution set replace method question naturally choose capacity selection combinatorial depend noisy noise level estimate average ergodic exploit mechanic enables study properly valuable david partially support fp international theory mm validation suitable control order depend information thereby uncertainty clustering model high fluctuation superior selection generalize comprise cluster optimize directly employ centroid method normalize cut linkage inspire principle linkage base various clustering ask principle method source viewpoint conceptual root assumption ultimately complexity endow ability validate set select channel good robustness clustering employ code minimizer approximate clustering model yield highly although selection still measurement relation describe structure g three application use distinguish measurement object relation assign group parameter uniquely identify object measurement cluster require characterize product assignment exploratory selection assess criterion e mean measure prototype theoretic validation assume simplify notation explicitly hypothesis minimize cluster approximation reduce two statistic deviation subsequent respective object I furthermore uniquely sufficient set training noise erm unstable cluster cost training correspond measurement clustering training hypothesis reader might indice nearest cx cx cx enable basis training datum determine overlap mean empty intersection datum become measurement tradeoff control constraint describe communication receiver generator generator serve channel receiver take optimize cost calculate code channel characterize clustering determine function index receiver agree employ generate receiver problem calculate permutation define nr behind cover clustering respective nr communication succeed stable stochastic fluctuation criterion reliable ability receiver specific permutation codebook shannon setup receiver set available determine membership clustering receiver communication depict generator generate cluster worth conceptually restrict problem optimal optimization necessary sufficient reliably identify set reliable define protocol coding analyse probability capacity capacity determine precision scheme select receiver event introduce calculated receiver
minimizer visualize priori minimizer secondly minimizer either formula powerful application subgradient know belong strictly question arise naturally constrained devise study online task generalization tackle processing manuscript extension mapping let map dynamic priori study constrain convex continuous differentiable nu n cluster lie nu tackle task stand manuscript organize include special wide application online find section complexity increasingly system recovery task sequel denote n associate symbol stand real equip denote classical dot lt stand bx v v b v cx cx subdifferential subdifferential value continuous differentiable singleton nothing differential well subdifferential function close cx x dx c tv verify close ty call quasi call equivalent mapping mapping particular iff mapping share relaxed projection nonempty associate metric point quasi mapping follow symbol weak strong subgradient convex exist inner limit b symbol stand closure set fashion subset subsequence repeatedly sequel assumption nx mapping n subsequence contradiction lead follow hence dx establish choose arbitrarily subsequence clearly nx arbitrarily mapping later sequel another satisfie relate assume mapping existence sequence dx dx necessarily exist n sequence projection satisfie assumption assume point nonempty let hold true weakly sequential sequence nonempty nu nu assumption hold nu nu nu assumption hold u assumption nu nu nu nu nu combine easily verify recursion follow stand relaxed subgradient projection mapping verify nothing fix arbitrarily hence convergent establishes assume nu claim theorem quasi easily subgradient n merge cauchy definition side assume moreover side claim n easily obtain time map one word theorem side establish theorem u u nu nu verify inequality n become guarantee nn order tt tu verify using easily verify consequence assumption mapping stand time priori meet signal processing application whose impulse also sequence rich quasi priori mapping context incorporate devise ax monotone monotonicity ax induce definite monotone give stand nothing map minimizer proximity x inconsistent knowledge piece surely ideally quite case piece priori inconsistent tackle follow convex proximity everywhere fr differentiable differential rest mapping hx xt tx hx length convex identify convex learn instant introduce weight du il collection e q fix definition lemma true arbitrarily assume previous establishe arbitrarily calculus since form sequence nonempty set elementary algebra recursion ambiguity case define also u nd hold n hold nu assumption follow apply task assume equip hold n inequality claim deal observe establish impose however clearly suggest boundedness true previously already du nu n mind theorem become direct utilize regard assumption indeed notice suggest u nu v previously notice establishe guarantee consequence previously adaptive low basis retain exploitation recently interest previously importantly recovery realize efficient minimization recall integer stand define cs generate real stand stand norm vector norm k stress batch recent cs appropriate operation predefine cs recover update improve measurement feasible development importance engineering time vary storage resource deal available take study framework cs scenario major limited letting design additional task capability track variation place require real time batch develop cs e time become exhibit convergence counterpart operation devise albeit requirement demonstrate incorporate set algorithm close associate quantify closed l metric hyperplane scale linearly aware mapping need sort operation multiplication due relaxed subgradient reason introduce unweighted enhance different ball help incorporate radius suggest algorithm n w necessary condition radius ball follow invariant save couple extensive spirit reader methodology compare couple belong minimized accounting build classical employ weight norm term scoring solve lasso variant word respective sub account instant infeasible implementation benchmark nr total case invariant whose position nonzero zero equal one vector zero equal tag curve refer case lasso inherent tuned way produce low error iteration although parameter expense demonstrate propose projection lead performance propose drop exact mapping accounting sort operation necessary computation instant nr experiment similarly refer nonzero value coefficient change filter tracking performance practice use realize equal coefficient equal odd coefficient set change instead set tracking factor forget concentrate variation reduce factor expense choose value factor propose system would discussion dr information matlab manuscript theorem theorem problem wide non incorporate knowledge manuscript project subgradient priori mapping benefit class way cast priori introduce mapping nature information property special case potential propose scheme demonstrate system recovery task multidimensional method infer time vary mapping dimensional system contaminate noise show distinct difference counterpart follow batch scenario time instant newly incorporate process sequential prescribed efficiency online become tool case dynamic scenario change nature environment variation signal past put receive become clear tool need like aware
pm multiscale air fusion bias coverage output magnitude coefficient appear predictive include proxy covariate covariate focus attention proxy useful fusion proxy scale sufficiently distinguish proxy diagnostic discrepancy fusion united issue highlight implication spatial lack circumstance prediction proxy scientific goal spatio prediction fine pm effort smoothing use recent consider optical deterministic pm rather complexity temporal individual separately spatio little accounting focus average air relatively usefulness change correlation differ short period gold quality first variation month mid optical depth moderate proxy snapshot approximately value location vertical average base light available km analysis proxy spatial variation month km proxy regular km grid vertical level low pm ground average short term air quality pm day average second pm output report third day source process analysis review health effect fig show pm raw markov monte appendix basic spatial grid represent sub single simplicity pick element observation proxy account spatial proxy grid particularly relevant sensing grid relate grid sub heterogeneity term normally p remain simple smooth nonlinearity I thin spline cubic radial function basis control able small variation thin spline mid knot scale mrf mrf thin spline mrf weight generalize discrepancy represent smooth variation scale provide I analyse integrate spatially integrate instead focus understand particular car often weight realization asymptotically resolution de show approximate mat ern range go infinity differentiable path surprising realization car heterogeneous regardless value component penalty induce thin henceforth mrf mrf extend detail exact include fit posterior standard whose smooth car mrf fine albeit mrf give variation suggest alternative small scale spline reduce krige capture large discrepancy understand proxy interest implication proxy implication discrepancy application involve combination white proxy fine scale pixel noise scale discrepancy proxy reflect small scale spatially model without white gold variation proxy spatial model treat spatial ignore proxy without gold small scale proxy improve achievable variability proxy estimation tradeoff gap clear proxy help variation improve gold scale discrepancy proxy accurately reflect discrepancy correct proxy constrain large scale difficulty occur scale cause discrepancy tradeoff proxy prior remove scale proxy occur spline set uninformative weakly scale flexible discrepancy represent proxy largely proxy proxy sample scenario spatial scale discrepancy spatial diagnostic assess performance relative observation minus output observation diagnostic output sum ratio average analog quantity diagnostic use l scale proxy proxy quantify diagnostic appealing indicate variability proxy explain discrepancy ideal situation interest contain process trade addition dependence hyperparameter marginalization joint convergence high marginalization base detail mrfs effort analysis hour day respectively mrf informative simplify present scale discrepancy scenario construct mid gold proxy observe mat ern vary value simulate surface independently discrepancy operating scale cell differ spatial representation population road road class average square scale fitting road road first misspecification generation road analysis include capture generate near major road size fit replication surface component normally proxy white noise pixel variability covariate empirical true large correlation spatially proxy varied replication correlation exclude correlation replicate correlation occur residual spatially occur chance vary appendix mean square base replicate table variability difference error scenario replicate reasonably proxy include replication minus proxy spatial discrepancy full lack identifiability play role difficulty estimate quite sensitive inclusion proxy substantial discrepancy residual covariate posterior value one signal proxy term subtle make proxy resolve induced simulation covariate natural indicate discrepancy pick already need operate small proxy discrepancy scenario proxy scenario assume large fixing scenario model run add improve discrepancy vary scale fixing fix scenario surprisingly exclude unfortunately identifiability reasonable cross assess discrepancy greatly relative difference scenario remove entirely include proxy proxy small discrepancy scale discrepancy proxy covariate reduce model discrepancy scenario spatial proxy proxy covariate always small model reduce without pose take proxy ability range median difficulty exploit proxy correlation proxy high analysis proxy proxy recall error proxy still benefit proxy prediction complicate fashion signal proxy informative km grid large surface km mrf rather thin spline computationally mid represent small km posterior scale calculate covariate km cell fall value g km fall km grid covariate km weighted cell four km grid seven negative drive strong apparent ad hoc manner hoc formal orthogonality practice specification near would hope somewhat variability scale pm scale suggest resolve variability discrepancy surface fig scale real north proxy simulation sufficiently interest strength gold able exploit proxy pm extreme refinement might improve proxy term find validation without suggest term restrict observation pm prediction poor simulation adjust discrepancy dense proxy signal raise identifiability proxy fitting identify observation proxy penalization latent process tradeoff sensitive note trade occur analysis relative proxy data identifiability spatial difficulty exploit analysis include signal proxy small scale way discrepancy interpret relate proxy shrinkage proxy complicated proxy improve relative proxy information despite concern identifiability signal noise fundamental task know ideally literature lack identifiability scientific flexibility process model present assess proxy proxy spatial process result highlight difficulty actually improve prediction information concern proxy able information proxy variability well conclude variability gold note daily proxy value day observation local variation pm identify yet poor job capture variation proxy variable add useful absence part covariate local contribute proxy mask relationship proxy mrf spatial scale lack mrf recently mrf mat ern approach explicitly gap might limitation proxy scale think handle identifiability discrepancy impose identifiability signal discrepancy term seem unlikely substantially well identifiability treat proxy greatly complexity simulation proxy discrepancy discrepancy decrease proxy improved prediction scenario covariate purely understand proxy regression may good drawback sense cover sort one proxy subtle involve scale proxy adjust leverage proxy prediction phenomenon simulation concern proxy plausible large small spatial proxy interpret analyst proxy additive spatial proxy alternatively proxy measurement error coefficient limit gain scale signal carry proxy proxy discrepancy principle albeit practice explicitly decompose separate allow successively predictor suggest acknowledgement liu processing overall lead thank environmental power institute grant health effects institute spatial mid additive matrix representation explanatory proxy pm cloud site correlation site prior dimensional mrf fixing coefficient spatial implicitly give limit five knot knot knot covariate equally space spread covariate penalize exact informative component exchangeable amongst term follow I integrate normal q prior carry marginal likelihood result exponent calculation inversion combination form contribute infinite avoid determinant improper prior column collect sum block tune proposal determinant calculate operation simple proxy dense correspond basis computationally total must latter bb consider diagonal I equation recall quickly spam calculation remain normal matrix calculation describe draw indicate variance ii kn variance daily subsampling average daily month I simplify daily dl give kn month I advance locate enhance identifiability co locate integrate value add primary mcmc avoid calculation take vi involve diagonal analogous daily pm average pm work pm alternative modify log transformation scale single accomplish correlate residual tail approximately albeit skew outlier worth somewhat base pm several km class area national density value centroid fall grid population density road density extremely truncate location contribute mixed spline represent effect location emission strength source represent effect strength distance strength weight distance fine pm primary km source five proxy grid point grid box average integral emission hyperparameter informative variance low upper prevent part location distribution away toward smooth simple suffice even improve run burn th iteration storage cost reasonable size multiple month validation pair note justify follow computational represent grid parameter covariate km base pre advance average km pixel overlap knot include represent joint ac pp follow calculation integrate determine remain normal remain note grid cell km entirely water treat km grid include km cell intersect computational require year within km use rather run iteration subsequently cost reasonable environmental space gap observation proxy call discrepancy proxy complicated spatio dependency extend popular discrepancy employ little field thin spline capture scale discrepancy computationally conditional auto flexibility lack inherent context matter output estimate small discrepancy predictive improvement modeling observation proxy identifiability prevent prediction result
cdf pdf rr sd pdf pdf representations implementation regularize distinct comparison square rmse respective sampler use e round discard burn difference cdf pdf representation binomial term rmse cpu many rmse gain pdf draw proposal ratio even show slice never reject expense inner slice slow overall sampler hand maximum take time long base mh absence slice assess visually due autocorrelation nearly slice mh fully full map modern regularize logistic synthetic like vector zero unit metric approximate respectively prior precede burn mcmc round estimator find initialize value mean except mle package efficient omit cv penalty parameter cv give final reliably intensive pick use mle l avg mle irrelevant wherein apply dash red summarize low across repetition fully hand table employ penalty average increase hand region table due many case qualitatively estimator though former fix fully behavior choose offer stability setting bayesian preferable map hard increase uci contain classification treat comprise cv estimator bayes estimator burn round attribute expand map calculation avg section absence presence wherein estimator dash line summarize thing section good predictor relative regularization expand automated simulation logistic inferential goal development everything prior coefficient conjugacy gamma art context describe attractive implement generate pdf mh rate ess nine coefficient ess spam lead fast well movement lead around extension example handling convention simply likelihood independently trial logistic extend ordinal response probit logit cdf pdf applicable ordinal break regularization implement add sampler goal far approach regression via similar spike grant handling code associate valuable comment expression namely term quadratic collect pdf inverse kind inverse simulation school business usa paper logistic exploit normal carefully mixture implement mcmc say posteriori include flexibility computational applicability uncertainty assess optimal methodology modern word augmentation logistic numerous posteriori bayesian regularize bayesian collect include handle proceed employ advantage previously encode predictor I penalization amount regularization relative solve discuss lar subroutine scale inference covariate follow shrinkage finite intercept practice indexing ignore simplicity offer fully coincide help map simulation efficient logistic posterior write augmentation hereafter combine augmentation suggest finally recognize observe otherwise framework traditionally hierarchical include marginal expectation give context logistic outline scheme illustrate empirical section direction future package available mle logistic inspire g absolute tool calculate likelihood bayesian concentrate whereas mode motivate equivalently estimator solving recognize use computational response follow regard fix discussion defer likelihood yield bayes subsection augmentation primarily concentrate double possible briefly logistic represent latent product logistic write integral mixed queue primary interest include generalize rely pdf generative likelihood drop cumulative zero yx yx establish model summarize conditional z identical generative extract suggest alternative show represent inspection eliminate integral eq avoid analogy problematic fortunately mix long extra pose practice specification implement p regression heavy around mode origin towards provide discussion notable laplace typically proceed cv vary assess estimator pose difficulty power offer tractable uncertainty lead efficient option ig prior shape base second option ig identical identity tail purpose efficient draw unconditional yield overall simpler proposed reject mh cdf replace adaptation may auxiliary random although section scheme fast behave similarly cdf multivariate prior combine op represent significant formula pp instead pdf penalty likelihood observe enter representation adaptation result conditional reciprocal inverse ig conditionally outline multiplied ba b b r kb r p sr variation slice former replace draw approximate uncertainty include via anneal sa sa posterior increase schedule possible start systematically monte expectation determine threshold iteration initialize iteration chain chain procedure style importantly sa know optima certain optima although quick sa however burden schedule regularize short safe often perhaps em option beyond infer option classical annealing option experience former work dominate yield far less provide find predictor thousand require extension repeatedly contingency table typical un regularize way proceed describe section scheme require turn feature observe likelihood subject equivalently term
regular algebraic construction review handle fractional factorial design effect valuable characterize design effect carlo effect observation use gr basis order division fractional factorial code factor code complex unity factorial algebraic statistic set equation polynomial vanish factor factorial level factorial polynomial design ideal polynomial unique factorial two ideal basis theorem basis coincide fractional factorial calculate intersection consist write intersection gr gr respect elimination gr basis design resolution define relation q follow element contain reduce gr gr reverse reduce gr hereafter write level code indicate relation factorial fact design obtain eq ideal see design ideal fractional factorial design define addition non polynomial polynomial regular non design obvious regular gr fact hand gr argument gr gr basis next gr basis regular regular design express effect membership calculation gr basis design ideal give polynomial lt belong gr gr respect set standard lt theorem gr basis basis identifiable gr write x ideal identify effect effect high hierarchical resolution relation ideal gr whether define gr include gr follow level level th unity code satisfy representation correspondence algebraic fractional factorial design classify function concept design resolution orthogonality regular naturally see addition incorporate algebraic naturally function indicator equation characteristic indicator factorial function factorial design factorial example indicator exceed b e f generalize regular factorial design assume factorial fix hold work author design regular fractional factorial factorial full factorial design factorial conversely regular factorial factorial design fractional factorial satisfying property factorial simultaneous identifiability factorial main simultaneously full factorial full model fractional factor dimensional factorial mod determinant realistic situation unknown rely robustness interest require number active factor consider present evaluate fractional design minimize diagonal appropriate effect compare criterion dimensionality gr basis design experiment procedure construct give publish topic basis another branch investigate basis apply basis design section work integer regular fractional design simplicity run natural run variable mutually independently distribute expression treat nan write way nan statistic specify valuable fitting exact infeasible markov key conditional sufficient statistic eq basis calculate reversible conditional ideal illustrate fractional factorial fractional factorial relation several model effect seven may column interpret main effect also interaction effect two interaction among effect column note correspond additional two want relation effect membership factor complex specification sample space fractional design level code parameterize example constraint simple treat column covariate total code part conditional polynomial algebraic statistic work algebraic statistic close establish branch argument branch except design theory row often ideal therefore design factor code complex indicate end effect factor without factor argument reflect ideal algebraic design fractional factorial purpose design gr
text corpus scientific paper specialize motivate desirable datum able live node tree infinitely exchangeable breaking play prominent constructive dirichlet draw dp base eq stick length break call stick place break piece break right piece random view sequence include paper accord partition natural number idea distribution infinite partition possess topology sequence index partition pt construction string index node stick breaking size beta branching interval onto construct string eq decision branch determine child mass sum must satisfy mass depth throughout remainder induce partition sometimes chinese illustrate set order branch tree bias draw node child stay child later stay n previous stop child chinese restaurant customer stay child accord datum part popular without structure prior tree live arbitrary node infinitely exchangeable exchangeable chinese unbounded depth path distribution nest partition binary tree live marginal cluster exchangeability however topology hazard live internal seed gamma seed seed gamma pdf decay seed decay seed pdf gamma seed decay pdf stick break labeling mixture th continue direct requirement edge must tree give global diffusion transition coupling complete infinitely examine generalize describe transition fx nu weight challenge assignment frequently integrate infinity slice dirichlet path draw variate break imagine top need conditioning previous enforce tree draw slice th currently initialize determine currently must current determine correspond lexical prevent represent slice appropriately perform discard hull assignment use stick crucial include perform biased permutation set repeatedly draw imply weight keep invariant hasting proposal slice aforementioned hull slice far markov n inference cifar image dimensional factor bernoulli real component parent place diagonal monte hmc slice robustness stick break use modern mcmc slice approximately represent provide capture high low branch texture shape material bag word topic model thus multiple share node multinomial symmetric dirichlet kind chinese restaurant node topic path visualization datum document subtree histogram show year document node expand version supplementary material secondly assess predictive create ten partition perform inference predictive take pseudo improve lda number believe improvement constraint topic topic seem may term topic performance tree good lda fold figure fold procedure train follow first run tree word association association vary additional full posterior somewhat multinomial parameter
maximize improvement contour branch bind algorithm simulator run overview process branch improvement expect contour estimation present compare branch genetic test conclusion outline computer simulator I without input hypercube output gaussian spatial hyper estimate maximize close hyper maximize good predictor square power exponential mat ern correlation see choice correlation omit use near singular unstable overcome ill condition well section include gp expensive model simulator feature global maxima contour sequential budget simulator setting data process model new trial improvement estimate run simulator trial exceed limit develop new gain computer ei minimum simulator improvement interest criterion minimum maximizer denote cdf attractive ei exhibit second local performing attain reach simulator run become ei multimodal peak trial branch optimize ei review often global optimization present find real dimensional hypercube component branch pruning branch rectangle randomly bound gx edge guarantee pre specify tolerance nontrivial appropriate bound ei criteria specific feature pruning remove upper rectangle algorithm initialize counter list pick l ii min ix step branch bound pruning ei simulator function monotonic w take lb lb lb lb needed replace bounding ei bound easier estimate deterministic computer simulator next ei ei contour development maximize ei maintain versus simultaneous global specific scenario weather low maximal corresponding expect improvement ei ei criterion individually estimate maximum global subsequent neighbourhood dominate global otherwise explore overall ei global optimize upper accomplish partial equal straightforward show f monotonic monotonic w every bind monotonicity close bound equal ei contour simulator motivate involve bad performance server one server queue simulator estimate contour yx x sx neighbourhood around contour r normal pdf cdf respectively ei additional trial trade search gain computer motivated modify ei facilitate still uncertainty neighbourhood contour without criterion ei obtain drop change modification tradeoff section piecewise monotone bound let partial sign mean derivative cdf plot worth note height peak figure increase nature remain always may display maximum using contour maximize improvement outperform save propose genetic ga ei various approach ga design ga ei method long demonstrate compare true case use output ga implement maximize initial mutation augmentation x x augmentation cross ei retain ei produce ga magnitude mutation perturbed dimension information optimization ga trade search contour consist search balance search table denote proportion choose specify contour proportion new trial computer simulator output dimensional present average realization random hypercube choose ei ga contours interest respectively whether difference c cc cc contour add consider function add new notably contour implication integral contribute global desirable characteristic improvement criterion simulator local global reweighted reweighted ei enable branch ga ei start strategy fit surface fit ga ei estimate ei two ga estimate directly comparable though approximation estimating gp fit outline comparison present ei criterion ga budget budget entry table ei ga feature contour maximum function dimensional hypercube result gp fit complex see ga exclude ei row table ei ga ht cc ga ga contour ga ga table function maximize ga close initial maximum method indistinguishable ei ga ht cc contour ga ga analogous dimensional contour determination ga ei achieve ga ei criterion note design maximum fixing ei evaluation consequently iteration ga tolerance true design surface compare interest add design compare static approach hypercube serve ideally ga outperform sequential next eq global attain ga maximize simultaneous estimation global minimum hypercube design size trial sequentially maximize ei criterion realization bar denote clear ei criterion lead simulator red ga optimizer baseline hypercube estimate minimum seem though attain close ga turn static somewhat design sequential ei ga contour height measure estimate contour denote discretize static result average random design display contour ga similarly static contour quickly contour contour corner design hypercube design likely place contour static dimensional global ga ei criterion simultaneous size winner well much simulator ga static design size average hypercube design minimum blue lead contour ga static black design minimal improvement contour simulator careful ei save simulator result design hypercube sequentially highlight optimization evaluation ei figure display ga design simultaneously
person repeat game payoff average advance existence prove see notice existence strategy external use existence strategy also construction strong property call internal player regret external stage regret introduce shorter calibrate strategy complete survey sequential converse convex calibrate strategy statement prove construction strategy consistent construction game monitor I action receive idea follow asymptotically external well see strategy external prove explicitly strategy strategy signal depend play opponent restrict finite regularity assumption main give idea construction partial monitoring repeat game choose probability furthermore belong prediction call depend past h nh n stage calibrate eq word strategy calibrate close close go strategy expect general define calibration player stage choose partition go formally player every diameter player calibrate strategy calibrate definition calibrate diameter calibrate calibrate calibrate strategy respect grid grid yet algorithm polynomial calibration result condition replace payoff player player choose payoff behavioral h player payoff almost surely uniformly respect notice purely condition state define subset separate informally point matter outcome expect payoff moreover player player complete convex player particular player prove consists call proof person choose vector payoff internal average internal player strategy every increase payoff game state existence prove note auxiliary payoff strong converge rely result existence chain non therefore sum every player payoff stage respect player player useful strategy originally q norm get grid exist strategy player respect resp choose generate payoff auxiliary every grid close soon finite irrelevant proof calibrate game conversely existence calibrate give new mainly proof advance play payoff prediction since multilinear introduce auxiliary game action outcome player forecast let action game depict choice player close calibrate soon small probability therefore least inequality bound imply q sum cd exist play accordingly necessary convex proof work von allow projection onto gx z minimizer implicit construct sum program big elimination repeat polynomial fix compute obviously reduce solve solve polynomially elimination payoff one arbitrarily aim algorithm construct strategy approach stage without explain maximum external match dark coin choose coin otherwise coin payoff cc cc c payoff nature good payoff receive signal whose law evaluate payoff internal action stage partial response want framework family player assume player trivial guarantee converge behavioral stage player intrinsic player uniquely sequence I sl regularity bl ib regular ball good ball enough strategy part firstly main observe generate idea calibration transform implicit constructive corollary game predict grid choose sequence close play asymptotically action belong different calibrate slight time armed calibration regret chapter survey every define uniform every chen white calibrate know close soon big close therefore ii equation hoeffding every construct strategy external describe know compute regret accumulated block beginning decide next play follow fine external regret try payoff abstract abstract game monitor action almost calibration therefore internal quickly slowly zero depend drastically converge unobserved mapping star behavioral receive signal play nx equivalent nx receive stage signal mixture behavioral behavioral conclusion play strategy strategy player use I compute know prove continuous compact grid since define imply assumption application gx x I iw rr g require face opponent regular consistent strategy mention action denote mapping define choice receive law belong assume player player range polytope convex hull finite range sense p continuous bb good player playing need repeat game example player odd stage player must fact polytope definition continuous true strategy payoff strategy actual give easily show see every diameter every random history resp actual payoff rely g strategy player accordingly block accordingly h partial history long regret accumulate block stage trick prevent rate regret two restrict strategy actual payoff definition calibrate strategy monitor linearity payoff strategy property monitor regret payoff consistent respect actual payoff consistent
picture gibbs conditional distribution exactly hasting distribution simulate algorithm proposal surface see log center eq approximated h generate candidate acceptance mode full th give accept reversible mcmc autoregressive design proposal autoregressive good autoregressive refer reader work extend autoregressive process order autoregressive restriction nevertheless conditional autoregressive constraint deal stationarity suggest move innovation suggest parameter jump make rate move employ suitable order close order proposal newton approximation posterior let innovation lag innovation term root stationary innovation associate thank parameter prior reciprocal satisfied prior jump allow need order uniform process assume proposal truncate autoregressive autoregressive jacobian acceptance simplify identical identifiability parameter acceptance log one normalize log evaluate derivative u x convexity bar move control parameter vector propose jump propose vector assume unitary jacobian parametric choose approximately equal rate note gradient split nan define notice opposite mode dim variance simplex system hessian respect ta apply modal value approximation approximated recursion gradient respectively provide analytical choice period posterior mode reach acceptance rate move zero effect parameter jump gibbs simulate thus burden within mix well acceptance size convergence diagnostic first proposal hasting prior setting experiment random iterate typical present root inference choose regard correspond highlight precision frequent next higher consider exhibit fig precision dataset parameter setting efficiency mcmc truncate normal truncate typical raw chart gibbs mcmc parameter average observation simulate bar sample mcmc rmse average acc effective order rmse iteration discard number still work combine graphical inspection average kolmogorov statistic convergence rmse ess mcmc parameter rmse axis modify color circle display display precision rmse ess prior since fig show improvement rmse ess low circle depict order output necessary different chain estimate probability true order result associate ess markov mix aim dataset estimate modify gaussian observation chart subspace chain chart chart probabilitie ccccc probability bar order posterior mean size base specific last show secondly standard deviation decrease increase concentrated noticed bias base integer beta autoregressive high dispersion shape obtain section intercept certainly economic model challenging recent advance nonlinear characterize brief period rapid contraction long focus let say transformation usually recently beta consider mainly economic cycle observation source aggregate study dataset challenging amount represent issue european bank european statistic modify autoregressive autoregressive discard estimate model us acc acc rates capacity acc dimensional capacity rate production capacity production factor I e force stock capital economic capacity economic time series forecasting indicator issue role practice economic demand capacity level economic activity adjust inspection chart trend deterministic trend could naturally beta conditional imposing constraint intercept impose specification focus autoregressive parameter residual regression capacity beta beta fig rate autoregressive focused mean process allow easy order parameter deal parameter specification informative studied jointly estimate parameter different estimate true mixing reduce near boundary within informative truncate beta autoregressive allow evidence one united area paper autoregressive beta particular inclusion order beta conditional extension explanatory reciprocal jump autoregressive coefficient identification recursion constant belong simplex condition positivity satisfy appendix k pc correspond iii de de beta autoregressive restrict attention estimate due suitable specification moreover particular mcmc metropolis within gibbs solve reversible jump mcmc beta autoregressive reversible jump define interval proportion challenging issue year issue modelling transform real line standard example box cox early contribution follow contribution along series seminal beta consider instead employ mixture extend beta propose beta apply context beta recently markov switch autoregressive mean autoregressive parametrization autoregressive without attention appeal challenging nonlinear modification procedure consider autoregressive become autoregressive moreover reversible jump markov carlo beta autoregressive autoregressive jump iterate chain view kn x likelihood indicator distribution markov sampler state form accordingly markov probabilistic reversible jump see fan jump beta autoregressive process follow reverse jump space dimension jump acceptance move contribution propose combine find outline beta parametrization bar show beta autoregressive algebra denote beta refer parameter past transition density autoregressive process distribution fairly flexible unimodal family review identification issue part beta determine parametrization context variance quadratic location precision x k convexity path volatility chart chart conditional mean process anti unimodal switching type behavior fig anti unimodal switching bar
sake brevity complete result instance result quite conservative performance either method standardize compare show observe standard standardize especially high noise also slightly run comparison empirically yield odd ratio unclear whether rely simple avoid difficulty coefficient situation un square error odd sample estimate especially combine odd ratio need explain correlation closely consist regression return coefficient coefficient quite odd ratio model modification rely correlation regression l l association cause death possibly relevant death dataset cause death record death france year coverage death start cause death death death record analysis cause code international classification disease total category code analysis apply entity appendix therefore death record death five cause frequency report appendix cause death decrease heart failure heart disease disease diabetes cancer death clearly upon age gender vary age gender decide whole population gender sub consider age sparsity yield association good model different many association occur little association especially ratio ratio unbalanced compute take respectively analysis window machine computational computation un estimate derivation step reduce second final retain intersection derive apart obvious association association diabetes disease cancer disease disease negative association worth association cause death biological plausibility old save take second method estimate second detect agreement confirm agreement return ratio paper empirically compare approximate method association performance term f slight moreover method case could similarity term computational disadvantage especially low parameter method pseudo coefficient adaptive derive theoretical pseudo alternatively cost aforementioned lack cross start combine good time fast particularly truly might slow mention implement package think package glasso rely coordinate interestingly observe different especially ratio example decide retain intersection model lead conservative competitive example approach optimally approximate regression attain performance reach exact computational result gaussian ise absence justification recommend theoretical study enable retain cm cancer cancer cancer cancer cancer breast secondary c disease diabetes disease use f disease g disease g disease heart disease heart disease disease disease j status disease disease due disease system disease disease k disease disease disease finding elsewhere fall fall x association statistic due amount biology domain application recently gain popularity purpose literature exact inference slow approximate recently extensive study method rely approximation achieve association death death biology researcher involve study relationship example system biology underlie variable way study relationship total number cell contingency free greedy selection testing even contingency table plus hypothesis testing issue originally penalize enumeration profile plausible large alternative speaking instance tree become dense approximate inference notably recently rely two distinct penalize regression graphical model three derive maximum solution show extensive study solution either solution derive fast ever conduct thereby conduct outline principle slight modification fast application association death contain vertex independence binary focus q strict therefore independence consequently complexity need principle pt valid interaction also higher handle dramatically extensively hereafter terminology use estimate separately rigorous dimensional setting grow consistently graph sense solve separately asymmetric combine way possibility belong alternatively estimate belong comparison study recently maximization solve problem enforce symmetry differ neighborhood still represent set spin elsewhere resp th column implement maximize however pseudo exact less pseudo method refer interestingly point easy association minus twice compute obviously pseudo coincide interaction may sequel completeness shall maximize account spatial example genomic study describe ising suggest result put descent specific sample upper partition obtain likelihood solution solution handling compare original add glasso develop question work moreover related use put decide model pt computational precisely spaced scale small run window intel ghz gb intel ghz core gb ram mac windows cell draw multinomial distribution different normal distribution subsequently odd retain compute lead association potential greater retain lead pt graphical probability px panel graphical representation panel edge coefficient logit gibbs px evaluate rely poisson value tuning lead gaussian approximate likelihood log precisely quantity consider stand sparsity induce lastly define q k approximate likelihood base furthermore obtain covariance add term close exact undesirable effect obviously try conclusion consistently confirm finding corner exact black solid pseudo dash circle pseudo likelihood blue likelihood dot green circle solid solid performance oracle score slightly fourth simulation design focus save procedure although one insufficient c method acc score cm l cm l pre acc cm l cm suggest section half likelihood table grows achieve model un slow approach
shrinkage conditional mode motivate bayesian also average new penalty mixture iid interest make center omit important propose penalize residual sum solve minus mu mu mu plus mu minus mu control regression lar provide implementation lasso consistent choose yu show vanishing regardless choose due wang et consistent model inference select may bring undesirable introduce sub lasso interpret posterior use lin study lasso explore prior previously consider generalize way coefficient focus em algorithm somewhat broad mcmc prior coefficient choice explore model purpose generalization linear lasso originally property hoc thresholding rule difficulty bring recommend credible interval variable selection fail explore uncertainty space spike mixture lin increase plus mu minus mu minus mu mu mu plus mu minus different penalty naturally covariate wang preliminary treatment argument distribution draw al subsequently obtain array estimate gibbs permit unify flexible penalty least wang cox model outline novel structured penalty group lin variable selection yu rest organize tuning vector explanation discuss average analysis dataset present unified deal selection model software deal many penalty represent normal exponential motivate mu plus mu mu plus mu mu mu minus mu mu plus minus mu mu minus minus mu minus mu mu mu minus improper introduction use shrinkage motivate structure plus minus mu minus mu mu plus mu minus mu mu mu plus mu mu minus mu major difference allow intuitively penalty mode wang gibbs bayesian full multivariate conditionally conditionally inverse gibbs fast choose framework bayes prior shrinkage close deal approximated sampler rule mu mu mu mu mu plus mu minus mu minus mu th expensive obtain hyper parameter make hyper plus mu minus mu minus j mu mu plus minus mu minus mu gibbs size choose scope practice trial join minus mu minus mu mu plus mu advantage use algorithm conditional parameter rate parameter specification join although number increase use specify alternatively estimate rule mu minus mu minus mu e minus mu plus mu minus deep hierarchy allow demonstrate gibbs sample plot versus expect would put likely zero phenomenon demonstrate plot sample discard distribution central c trace plot likelihood versus decrease increase demonstrate signal may shrinkage allow minus mu plus mu mu plus minus plus mu become mu minus mu minus plus minus mu mu full conditional lead amount shrinkage coefficient choice choose demand lasso exploration model fail method hybrid coefficient hereafter suggest median posterior explore frequency choose consist less median mp surprising original lasso uncertainty make inference may set help improve bayesian framework average use therein refer formal treatment uncertainty use ensemble sparse mode observation predictive give plus mu mu mu plus minus mu mu mu minus mu mu performance scoring use predictive measure smoothing parameter mu mu plus dd mu nonnegative kullback leibler sense offer average sample example case prediction estimate gibb give conditional replace integral integrate accordingly fact predict draw select draw gain predictive condition fix compare lar example preliminary use marginally compare original measure fit replication summarize lasso mean example consistent seem go increase perform well even large area coefficient median lasso generally superior summarize example perform outperform variable selection ability experimentally conditioning suppose dataset square minus mu plus plus mu plus minus mu minus mu mean original similar average various size set experiment well well lasso summarize surprisingly due use outperform percentage important health require special percent percent body summarize carefully et al omit nd diagnostic percent age weight cm cm intercept lasso x correlation additionally remove help small bic ht proceed let model w shrinkage model mu mu mu mu pm dd mu plus usual formal mode uncertainty smoothing parameter gibbs straightforward high uncertainty high mostly account approach use predictive median predictive clinical measure weight age seminal score percentage response consider standardized intercept exclude smoothing coefficient smooth put predictor estimate ht obtain convergence mean give estimate correspond zero shrinkage shrinkage propose strategy include variable lasso choose present mostly select select presence account considerably different second examine form training rest prediction median well far regression complex generalize cox group lin composite yu unify broad context likelihood wang l mle approximately need keep specification discuss novel flexible frequentist mu plus minus mu mu plus minus mu plus mu mu plus mu minus mu minus mu mu mu plus mu mu mu mu mu conditional specify plus mu p j j mu minus mu plus group lin minimize plus mu minus plus plus minus plus mu th mu plus mu plus minus mu mu minus mu r minus mu mu mu size group identity conditional block size j j j mu mu minus mu consider natural ordering group extend yu vector plus
cutoff order spectral observe fig markov explore burn follow always initialization stationary run consider initialization length precisely concentrated small interval burn independent exploration burn period sample accordance width instantaneous burn chain burn period accordance rate hasting algorithm quite rotation hasting law simplicity counterpart rejection parameter accurate realistic image corruption white close visible give capability realistic coherent build hyperparameter mean minute assessment law addition unsupervise image frequency invariant encounter medical optical property methodology valid mean optimization penalization penalization penalization overcome difficulty development unsupervised deconvolution prior unknown parameter difficulty efficient unsupervised plan extend law penalization limitation function impossible resort idea overcome finally cutoff within hasting sampler laboratory france constructive suggestion author grateful laboratory vector covariance diagonal gamma write case hold mean precision pdf know student law conjugacy pdf give hasting law multiplicative law proposition follow law name metropolis law acceptance get simple ne le universit de universit paris sup place du france france pt national sup france phone r r pt de de france mail cm ref remark rgb paris de la france paper deconvolution framework infer law effectiveness parameter high spatial coherent deconvolution active field decade medical imaging generally problem induce observation precise importance critical development quality moreover poor limitation pose acquisition deconvolution class accordance information dedicate image numerous ill different bayesian information design build representative law maximizer optimization latter lead integration two addition firstly depend response devoted deconvolution contrary devote unknown know main extra practical model description usually relatively field addition deconvolution strategy parametric physical parametric practically parametric need interest refer difficulty blind ambiguity even resolve resolve moreover design nominal addition algorithmic respect contrary blind algorithmic linear unknown element despite difficulty format blind format moreover blind extensively case secondly law name hyperparameter variance law tune deconvolution hyperparameter approach devoted parameter recent address experiment laplacian address deconvolution hyperparameter image interest variable rule law noise uniform parameter regard attention parametrization facilitate conditioning degeneracy estimate choose simulation algorithm enable distribution despite firstly detailed law establish sec finally sec devoted consider square nx stand width convolution transpose convolution fourier image strict description spatial fourier domain coherent efficiency notational convenience x fouri component n coefficient account penalization differential operator law consider prior parametrization next parametrization facilitate law integration hyperparameter moreover result law parametrization sec parametrize read design diagonal domain sometimes refer law focus positive pixel penalty differential difference pixel scale build diagonal elementary parametrization equal corresponding absence frequency consequence vanish still despite degeneracy law format rule incomplete embed degeneracy format remain yield rely energy extra proper classical frequency approach degeneracy rely frequency determinant expression factorize control converge meet precision parameter degeneracy law chain provide compute law marginal representation analyse information uncertainty mark ambiguity sampler conditional law law burn complete joint next step posterior law eq q transpose regularize square solution wiener invertible describe sec finally diagonal matrix even law law update law c jeffreys k need law parameter direct metropolis algorithm sample propose law divide proposition sampling iteratively number mean approximate image spatial recursively single end describe obtain completely need entirely simulate study respective give protocol control entirely also different low signal broader different realistic show case image generate white laplacian image numerical profile fluctuation hyperparameter set non informative jeffreys law fourier normalize illustrated fig nominal rotation convolution add white smooth fluctuation corrupt level image nan coefficient h nuisance dirac sec fix fig line finally situation step sec line ignore obtain sufficient exploration difference successive sample compute approximately matlab law successive empirical give case respectively deconvolution notable profile order reconstruct profile match precisely fig pixel visible visualize circular average power spectrum hereafter radial frequency spectrum fig respectively spectrum image retrieve radial frequency dominant information lose produce correct properly wiener hyperparameter concern comparison difference fig visually indistinguishable especially euclidean image true report deconvolution
variance work former latter analytic sequential stationary variation design result design modularity sim easily package implement agnostic sim analytically ei generalization support far extend gp sim detail ei package ei classification model prior soft max logistic categorical imply sim sign easy say alternatively calculate matrix differ negative diagonal panel panel show sign b cluster relationship synthetic row panel sample imply color accord average posterior matrix rest identical c involve sign matrix set sign full sample index science sg sim parsimonious nonlinear univariate simple canonical use setting focus drastically simplify generalize sim favorable illustrate computer two package facilitate pursuit index sim parsimonious multivariate dimensional predictor variable response random predictor formulate apply section projection mx provide flexibility ad hoc step fit may author prefer sim interpretation sim computer run input configuration choice gaussian g prediction integrate base incorporate uncertainty fit calibration make one reason considerable insight nuisance quantity comprise projection index sim nice review use spline link sim illustrate empirically offer widely univariate limited suggest gp lead motivation model clear spline already naturally pose computer show gp sim error isotropic correlation sim complicated spline canonical unnecessary simple sim sim spline primary experiment formulation framework see multivariate computer package herein one suggest extension remainder organize follow hierarchical reformulate monte illustrative synthetic comparison section computer computational work sim experiment gp sim presentation notation version ig conditioning presence relevant finite shorthand fx nn correlation prior gaussian correlation prior interact identifiable choose flexible von sphere sign lead posterior setup carlo proceed chain metropolis gibbs iterate conditional actual expression conditional gibbs whereas metropolis mh proposal rw fashion von modal add lack identifiability sign switch offer post suggestion cluster mode posterior proceed krige equation discuss detail reformulate follow lack identifiability without unit ball gp constant quantity identifiable sign sign matter mcmc explore possibility direct variable selection sign pose primary identifiability heuristic allow explanatory effectively remove treated function none proposal interpret correlation gaussian inverse matrix sim odd equivalent combine iid q x correlation preferable equivalent I choose von recommend available sensible sensible design scale lie unit make sensible much easy benefit yet readily one fewer eliminate latent setup implement sim code trivial turn implementation consistency reformulate continuous uniform essentially prefer software interpretation simple inferential advantage augment analytically never predictive posterior stand imply jeffreys preferred may result require decompose newly avoid unnecessary eqs save slightly significantly carlo scheme comment start mh good rw uniform slide window reject accord implement metropolis draw similarly correlate update rw center choice know pilot crucial good hard von sign sample estimate pilot run sign compound product experience change mh acceptance ratio pilot run try always propose require probability easy value student degree freedom minor classic krige extremely sample predictive quantile summary basis path extension extremely original leveraging gp sim try forget root index collect location useful assess explanatory aspect identifiable sim real computer primarily gp sim isotropic separable separable scale isotropic see superior multivariate explicitly state software sign initialize allow compound involve sign diagnostic show predictor mahalanobis output predictive location estimate covariance location sim predictor carlo design conditionally form training row true make mahalanobi ht sim min rd time three mahalanobi summarize figure ranking mahalanobis isotropic observe sim distance cm cm display realization performance also fit index see b index prediction credible interval ci provide look advantage ideally well variability axis visualize mean horizontal dot prefer index though comment cause lack sign index correct sign mcmc heuristic index build true hard get nice plot index response adjust figure arise obtain ht negligible correlation obtain run new proposal seven possible von ht result normalize lie density sample length square histogram panel eight lie rectangular domain experimental obtain without highlight sim compare far outside sim class approximate eight comparison sim even see testing sim pilot determine mh proposal also indicate discard predictor ht b sep sim min st max competitive well worse separable project measure separate direction ht distribution describe contribution aspect sim show component sign rarely sim separable good gp sim arbitrarily sim experiment benefit estimation especially play predict try sim computer experiment computational back detail simulation relevant response speed angle attack shall roll detailed later portion like pose challenge experiment sim canonical experiment fold fold inside fold predictive fold response hold test code available aspect mahalanobi play major simply covariance explanation sep min median rd min max explanation improve visualization time fold generate mahalanobis sim summarize left part top version mahalanobis sim small indicate reliably project input correlation well axis align spatial ht aspect figure way towards sim mahalanobis model non estimate relationship suggest index explain deal exception relationship index cope accommodate section figure posterior index suggest fit mc mass origin suggest input mean lift response exhibit broadly summary mahalanobi nearly roll response relationship sample five response figure roll omit brief lift response probably roll roll seem input quite roll
input describe beneficial evaluation agreement ideal moderate draw arise evaluation random act representation produce version mutation efficiently mechanism mechanism mutation landscape ideal mutation jointly evolutionary perform class mechanism representation perform essentially pac provide target notation evolution randomize hypothesis mutation mutation class randomize representation neighborhood non negative mapping desire value predict context boolean consider function boolean disagreement loss early hypothesis range quadratic relative domain admissible explicit z use mutation beneficial neutral use beneficial candidate neutral relative neutral beneficial mutation tolerance candidate selection algorithm evolution output take concept mutation converge evolution probability evolve monotonically fr single evolution emphasize distribution require evolution refer learnable basic polynomial evolution require initialization performance process start reduction somewhat derive independent monotone base usual independent refer th point point point performance modification equal decrease combination modification converge loss modification improve type formalize nf definition case exist ni xx q bind low combine choice case eq compute convert evolution exactly done exist tn pn question whether characterization understanding learning address first low way distribution use specifically iteration turn boost agnostic particular distribution new agnostic believe insight learnable concept elaborate concept monotonically include addition independently boolean suggest need consider simple loss result subsequent work monotonicity monotonicity imply robustness evolution change function give along evolution monotone evolve les valuable comment I also grateful anonymous useful correction theorem fact definition bound characterization preserve characterization give characterization agnostic boost design demonstrate existence evolution loss phenomenon reason learn query restriction estimate unknown query oracle oracle query satisfy tolerance demonstrate convert robust small theoretic barrier query powerful algorithm fact introduction albeit know also extend noise scenario find preserve system crucial theoretic learning learn al prove give relatively simple statistical nearly uncorrelated gave characterize weak statistical query require bound query work notable bound dim upper easy exist strong uniform polynomial function require number query query dimension complexity specific invoke equivalence weak explicit respect recently derive result complexity agnostic informally characterization state learnable value close semi function x boolean idea say large characterization lead f correlation product function refer orthogonality efficiency learn efficient achieving error orthogonality easy analyze prove agnostic replace concept least model hard confirm dimension agnostic even generalization achieve task learn dimension simplified enyi elegant characterization maximum enyi proof complexity preserve efficiency computation proof accuracy preserve comparable direction distinguish give weakly approximate query execution imply desire property inner learn process closely area close closely core construction boost new type boost weak weak function namely explore learnable computable imply learnable learnable pool adjacent type acquisition particular converge rely decrease evolve monotone monotonicity allow adjust existence ability environmental monotonicity basic requirement recent show depend performance hypothesis namely uniform positive algorithm design specific use hypothesis equivalently agreement measure quadratic hypothesis function boolean loss vice versa fairly translate demonstrate concept monotonically quadratic robust mutation mutation variable correspond hypothesis exploit evolve list range range work scale quadratic hold define rely heavily particular fourier transform list uniform distribution formal model one preserve characterize proper emphasize characterization agnostic recent boost positive integer denote problem function product easy version dot semi g ff domain refer together represent concept representation representation complexity describe number drop length introduce distribution produce approximate oracle upon example independently output satisfy fx convenience hypothesis needs value think value hence fx expect polynomial pac pac learn respect weak learn disagreement target less precisely produce fix agnostic introduce situation agnostic class concept give hx agnostic access hypothesis pac run generally agnostic produce fix advance statistical query access oracle target concept respect distribution instead pair value fx convenience range extension say learn tolerance learn place oracle make tolerance evaluate minimum learn say extend agnostic analogously example denote agnostic learn statistical query alone relate al weakly use statistical exactly polynomial dim almost orthogonal use product gave characterize weak query weakly concept say set concept weakly number possible relate dim directly maximal almost approximate strong version let generalization orthogonality characterization query generalize characterization class simply use boolean define value exist either small give simple characterize query tolerance learn decompose value tolerance simulate every x x g provide simulator return therefore combine giving claim inequality standard confidence transformation unchanged establish direction every build step find point g every every tolerance g p desire query iteration establish every f xx fx x f cl iteration use query property give efficient convert algorithm statistical query vice note circuit efficiently size unlabele access learnable produce circuit run circuit give simulating would simulation easy replace easily access yield learn place estimate increase circuit generate learnable characterization orthogonality convert approximate every uncorrelated extend definition dim real say exist function set state boolean boolean easy generalization follow f f modification relation set minor need approximate every denote ensure f base obtain characterization concept learnable pn pn class require weak function function weakly agnostic imply learn formalize cb dc b word agnostic
huber asymptotically alternatively huber multiply case interest novel inconsistent meaningful globally convergent sensor sensor sparse outli km km km explicitly understand solver minimizer denote vector find everywhere subdifferential define differentiable subdifferential yield positively consider admit block minimizer huber evident rather surprisingly generalization huber capable l contribute specify vector sensor likely presence regard cutoff tend contain outlier suggest residual criterion sensor classify reliable heuristic rule practically select scalar mention sensor roughly know prior alternative solve prescribed note group lar descent warm grid value efficiency problem assume specification across case sensor across huber case contrary remain colored modify early solve interior solver exploit structure advantage successfully solver block variable block descent separately involve minimize keep fix step denote minimizer quadratic second minimizer sensor li jointly instead update residual thresholde offline demand product operation presence exploit save computation block coordinate complexity iteration readily small predefine termination vector affect robust huber appropriately argue non yield improved estimator appropriately initialize interest explore surrogate recall seek end convex follow solve initialize unity demonstrate solver lasso propose property validate weak weak refer occurrence single keep long large validate independently simply sm notice due normalization variance select identify uniquely solution empirically satisfy correspond intensity indicate east east fig integer small choice probable recovery black solid accord circle success probability validate moderate rs develop setup network collect observation previous sensor ls solver solver obtain ga ls ga implement serve simulation insensitive sensor successfully classify sensor residual reliable novel advantage even iteration evaluate solver reliable sensor mse empirically average comparison include ga ls conventional huber solver vi solver turn critical cutoff huber estimator worth coordinate algorithm snr db snr time time plot consistent sensor outperform task reasonable performance show serves good solver combine absence model sm solver modify remark mse curve snr db correlate setup superiority solver prominent classify sensor reliable classification solely sensors reliable ga attain correct sensor classification setup differ reliable db laplacian set solver iv residual huber scalar measurement sensor consider list reliable sensor yield sensor huber exploit scope exploit sensor outlier give rise compressive sense reveal sensor sensor show hold gaussian reformulate subsequently cost estimator derive solve efficient simulate design theorem recall imply minimizer reliable sensor minimizer vector indeed attain inequality necessity proving must cost attain attain conclude continuity hold inequality trivially appear last lipschitz continuous suffice continuous bound express supremum infinitely subgradient norm norm subgradient expect next introduce random entry functional triplet expectation exploit property readily proceed need I j c index set lemma verify deduce side establish exploit separability optimization yield hold n conclude ht line perfect circle fail ga ga ls statement edu sensor distinct sensor recover specify sense formulate finding feasible proved link establish signal sparse rise strength first relaxation measurement recover obtain concave fourth scheme tailor noisy cast combinatorial subsequently fall framework block capability verify sensor network robust convex compressive recent advance technology sophisticated task environmental surveillance medical imaging typical heterogeneous sensor signal term sensing entail operational measurement comprise observation snapshot view impulse response underlie multivariate costly delay stationarity constraint reduction cope curse large sensor observation sensor failure sense device sensor communication interference reliability sensor fusion aggregate instead sensor vector reliable sensor henceforth refer rs context establish sensor area even outli sensor network rs rs task one sub optimum rs section cone relaxation hard problem relaxation compressive cs cs vector satisfy equation pursuit bp block comprise variable non sparsity herein residual recover block signal develop solver generalize equivalence alternative concave constitute rs solver surrogate concave sequence weight third vector select sufficient establish whenever sensor per sensor world application contaminate quantization dynamic identifiability scheme signal ratio sparse cs noise lasso vector block sparse cs placing sense noise robust fourth initially interestingly cost huber function attractive block alternative non stand multivariate notation n equal collect vector possibly subset sensor compactly state I scene interest view possibly image system domain sensor environmental monitoring could chemical compound field green capture measure sensor sense due propagation effect failure even irrelevant rank either infeasible full th linear ignore admit check whether satisfie retain proceed check challenging admit infinitely easily handle focus note bandwidth delay stationarity row match henceforth proceed whereas subset aggregate matrix rigorously pose otherwise strictly positive minimize linear solve rs concave linearization utilize local let minimize cost minimize minimization drive equivalently weight error residual weighted indicate high reliable let r consistent total sensor total unique minimizer first thing consistent uniquely recover may minimizer assume uniqueness characterize vector satisfy every minimizer ss empty intersection belong minimizer avoid cardinality recover reliable second require stated next subsection partition hold exist exhaustive exclusive subset arrive contradict critical whether former hardness random assumption decay start equivalence condition provide follow equivalence theorem turn generalize minimizer worth range impossible useful establish characterization equivalence subsection another remark range valid constraint non lie ball though reduce set probabilistic bind subsection valid extra impose early practically infeasible sense prove condition decay exponentially entry summarize pf lt deviation result sufficiently multivariate generalize refine expression proof partly analyze unknown relate isometry sufficient relaxation independently minimizer low success suffice condition fail subset
empty network maximize maximize expand instance simplify diffusion inference interpret propagation tree note cascade contribution propagation tree take span tree network cascade remain stop cascade thus maximize maximize edge tree maximum weight direct edge observe forward node already infect cascade edge graph dag use edge acyclic direct span dag dag weight incoming maximum score sum node parent handle dag maximization create proposition span propagation income efficiently compute direct tree dag put together find likely cascade cascade cascade aim log observe monotonic add complete maximize monotonicity observe convert tree maximum span network usually edge infection cascade propagation maximization graph search would optimal np hard max collection element cover set index pick maximize produce cascade cg contain incoming max whenever solution max correspond efficient could decide whether value efficiently find add score add enable let cascade edge cascade cg cg submodular argue direct span dag obtain assign incoming incoming weight maximum proving maximize marginal stop return marginal gain edge heuristic think graph value adding change iteratively hardness however fundamental et monotonic return fraction moreover tight dependent decrease number edge gain great np hard algorithm network thousand node speed improvement cascade cascade infect cascade network may cascade cascade cascade cascade network localize update drastically behind evaluation sequence gain add graph whenever marginal gain greedy element gain insight exploit maintain queue respective iteration greedy high weight since decrease remain high algorithm decrease queue later run several implement formulation addition parallelization likelihood cascade edge big make structure scope empirical runtime cascade likely ci g g ci proceed experimental diffusion real show surprisingly outperform heuristic typical network experiment synthetic exponential performance simplification namely exactly likely network edge general proceed follow diffusion simulate cascade aim recover cascade node recover poorly error edge cascade parameter cascade direct generate namely forest produce kronecker core hierarchical structure generate network power law simulate cascade need choose parameter need fix cascade control control cascade cascade make fail get cascade cascade edge important cascade never trace infer choose amount principle need fraction pick start cascade cascade cascade cover edge look cascade forest node edge cascade edge take cascade cascade follow law median cascade edge forest flat cascade node exponential case set infer diffusion edge cascade propagate edge pick edge score baseline performance two optimize optimize exactly online fraction correctly log plot bind np red curve narrow return diffusion online optimal actually value generate present fraction edge appear recall cascade scenario node per cascade infect per cascade cascade mean generate cascade record infection time fraction cascade miss cascade cascade infection cascade percentage missing node naturally amount choice basically propagation likely thus give cascade chance propagate external source various percentage node infect external source influence external note appropriately diffusion due external strong infect million news article million web create link refer transmission cascade start recursively post trace chain reverse identify cascade cascade post short million distinct cumulative million phrase phrase cascade cascade phrase cluster website mention phrase cascade general methodology handle concept cascade cascade cascade truth create create post link construct take create receive connect post site link cascade right notice perform improvement medium perform prefer create site information ignore site satisfied life break point good result hard site mention particular infer underlie baseline fair synthetic realistic synthetic function bind tight synthetic least shape trace remainder medium site number document dataset cascade dataset diffusion edge track proportional strong strength cascade add influence medium rarely network bias notice main side medium devote distinguish media deal post play central role side site part dominate news establish rest figure connect network phrase track medium infer diffusion analyze information network first show node diffusion direction node shorter infect slightly sum node set reach node propagate influence diffusion network site split site medium link news medium say medium notice many medium later media medium link tend pick early seem later pick lastly capture medium mass show type site spread source distinguish medium medium site quick another even infect slow infect whether information come medium infer quantitative qualitative insight surprising use insight several diffusion receive considerable attention shape cascade infer diffusion formulate rich predict occurrence individual predict independently pick hyperplane predict vi link probabilistic infer cascade programming solve include transmission rate transmission rate computationally edge think degree probabilistic graphical fundamental difference graphical learning direct dag network learn direct first cycle edge direct reciprocal allow infer network direct acyclic dag undirected typically network bayesian heuristic network network framework however hill search offer performance work novel formulation relate static method unsupervised method relevance influence graphical relate visible although function maximization find work network diffusion formalize scalable network cascade good likelihood exploit objective infer exploiting localize able scale cascade accurately recover relatively number drastically outperform maximum evaluate real news exhibit medium site topic technology site capital site future work utilize etc accurately propagation network dynamic interesting assumption system protein protein infer connection result promise process base acknowledgment thank resource microsoft yahoo grant nsf nsf nsf fa foundation la infer infect di vi influence underlie network spread unobserved tackle path infer propagate node become infected explain observe infection time np million news article year flow media diffusion news media site tend media rest site circle influence medium acting mining behavior idea innovation disease setting spread news collective spread disease challenge address need actually moreover identify successfully task automatically large scale identify phrase place edge epidemic however place unobserved commonly infect observe infected propagation discover infect infected propagation people get without know infect people adopt behavior explicitly cause especially study network path diffusion information influence wrong inference complete identify propagate relatively web supervision global structure medium site interact site play diffusion influential question underlie influence propagate static node get infected piece product observe infect interested path take reconstruct propagate propagate infection recover b propose make perfectly recover edge disease work propagation network node infection unobserve figure create trace cascade cascade infect subsequent layer depict cascade create true connectivity infection cascade infer formulate spread node infect could network large surprisingly computation super exponential tree cubic tractable function exploit speed exploit function evaluation broad greedy hill contrast infer optimal synthetic reliably propagation influence network overall synthetic dataset outperform heuristic correctly apply information news article online news network news medium medium tend news keep discuss news medium infer understanding network gain various play diffusion process assess devoted statement optimization section optimization propose solution shown conclude discussion section propagate static direct create cascade infect unknown network adopt particular infer carry semantic influence give hide direct observe cascade triple cascade initially observe time reach node reach infected neighbor infect observe triple infect infection different recover network use cascade practice simply nod network cascade relatively node cascade observe cascade vector node infect infected infer describe cascade tree simply specify infected cascade occur show near maximum cascade occur probabilistic build cascade infect neighbor choose implicitly cascade infected neighbor neighbor infect cascade fully describe direct spread infect illustration cascade create implicitly cascade notion new gets infected currently neighbor cascade necessarily get neighbor yet infect cascade influence node observe pair node direct since propagate forward node property intuition decrease infection infect arbitrarily disease propagation scenario attribute could information status would strength type allow cascade infection depend property symbol description spread node infect cascade cascade node cascade cascade cascade propagation node e e purely intuitive transmission time distribution time argue particular multimodal interpret sense cascade cascade explicitly stop cascade stop never reach thus pass want cascade simply infected element spread cascade stop simplify simplification empirically work moreover rewrite edge fail observation come examine edge possibly probable product optimize specify cascade cascade gets influence parent likelihood aim cascade infection tree infect tree figure three cascade propagation tree combine propagation cascade consider propagation tree cascade spread cascade trees subgraph cascade even range span case inconsistent assume basically skeleton cascade propagate know tree cascade really propagation sum direct span cascade occur cascade occur independence cascade formulate transmission solve node infection cascade eq q edge equation constraint number seek sparse graph section intractable eq cascade
specify poisson follow theory rate many investigate expression count correct cluster signal switch induce ensemble equilibrium response neural furthermore periodic input profile derive periodic steady impact transmission interact population neuron effect short coarse dynamic abstract exhibit prototype phenomenon generally define rescale hazard drawback operational time inter event example constant period maintain dependent hazard function underlie approach regard choice hazard enable dynamic analytically investigate phenomenon pde dynamic analytical rate generalize compute hazard time neural activity ordinary equation like change rate transmission periodic input fix qualitatively dynamical class turn cover range generate hazard event instead history dependence hazard define hazard function component process density equation boundary rate fraction process age imply boundary interact population dependent solution uniquely lose trajectory determine evolution represent fig feasible hazard first obeys scale auto delay denote auto domain find constant due hold apply change use fig numerical display exhibit frequency ensemble gray mid dark gray periodic fourier steady active periodic ik obtain function mutually infinite ik spectrum different couple inverting give spectrum cosine obtain recurrence relation unique manner relation within require tolerance spectrum output display amplitude three low function emission rate maximize slightly hazard enhance emission interestingly second harmonic amplitude harmonic activity twice ensemble perform period maxima dominate maxima harmonic intensity process anti rate drive multiple distortion light mid dark amplitude phase dark gray mid light device neuron time introduction generation independent probability density duration remain event follow inactive see generalization normalization analogously obviously recover localize hazard generalize periodic random class density obvious hazard theory dependent hazard time expectation hazard function function hazard function fig special describe b time transform system equation enable replace differential equation constant tt temporal yield switch give simulation distribute function upon fig via qualitatively change spectra time easily localize pdf original hold coefficient emphasize validity output arbitrarily investigate pdf pdf laplace process call hazard process differentiable expression suitable sense translate hazard fulfil process inter pdf parameter choice illustrate know concatenation interval gamma frequently time cell phenomena hazard identification gamma entail generalize rate define inter event unit random number pdf parameter accord x study l line hazard solid upon simulation process hazard average bar denote deviation trial output point generation action potential release particle device ensemble produce delay delay describe property rate relation express dependent hazard result display stochastic might reliably rapid change demonstrate steady input adjacent couple rigorously solve term recurrence explain activity slow reliably fundamental stimulus frequency subject frequency contain stimulus stimulus
index step step subspace develop belong scope constraint form create associate mathematically efficiency converge direction em subspace former superior parent obtain mle acceleration minor situation illustrate example dynamically formulate generic solver linear system explore context em however later suffer hence inefficient common direction context solve analysis accelerate demonstrate dramatically popular perhaps effort gradient implement evaluation code monitor extra implement search em variant new efficiency cpu remain arrange follow motivating propose several conclude remark neighbourhood em eigenvalue determine show play illustrate acceleration determine magnitude dominate em growth assign group treatment weight mixed one fix effect effect stop example dramatically show figure take converge second converge em direction em entirely subspace fix since em induce implement step along large become eigenvalue observable nine loading call loading row em significant mixed convergence em explain fall entirely subspace add dm table eigenvalue remain unchanged eliminate dm eliminate fix subspace eigenvector eigenvalue span gain em choice converge depend potential acceleration dynamically information formulate generic iteration algorithm compute original b call large appendix obviously insufficient small neighbourhood fast direction lie iteration mle proof reveal illustrate relaxation factor generate nine example simulate em small neighbourhood mle mix one algorithm movement towards small line connect show two show implementation proceed repeat dash include iteration line search line connect numerical another way red dash first conduct along connect become clear generic cm search calculate demonstrate efficiency neighbourhood mle exact conjugate direction efficient near mle note much easier implement em algorithm typically code line search required effort note expensive optimisation one evaluation stand dimensionality descent view method system certain discuss bivariate adapt figure relatively fast second em second use second slightly em mixture popular machine pattern recognition extension dramatically fast class efficiency em ten norm cpu use em implementation dramatically iteration cpu overlap among first somewhat rare phenomenon may conduct suggest acceleration framework develop acceleration method em motivate development implementation nest equivalence conjugate justification also numerical simplicity attractive literature analyse optimum line however exact performance efficiency note acceleration work rate specific make work broadly leave future investigation chen david van zhang zhang zhang suggestion da establish mainly adapt neighbourhood may prove determined mentioned fraction name represent order matrix mle definite com com com com com com com pi com com ti com com eq equivalently determine eigenvalue simplicity eq make com immediately eq q eq two conclusion com dm obviously dm com I dm q hence prove statement conclusion dm statement conclusion induction proof generalised direction certainly true search em direction true hessian prove along com expand conjugate conduct achieve high routine may various line routine desirable trade propose transform constrain unconstrained implement search constrain along commonly software optimize narrow control optimize advantage force accurate mle magnitude constraint feasible find interval univariate several determine induced intersection follow vector represent direction cm degree boundary type e solution inequality handle way degree freedom linear mixed distribution current dimensional dimensional size covariance enforce spatial linear eigenvalue mix dm cc
tell singular simple approximation identification row orthonormal contain singular get eq row orthonormal orthonormal construction orthonormal mind let address restrict problem ga k da kf simple c c definite later simply choose comment condition showing actually formulate consider otherwise vanishe everywhere mutually finally distinguish like actually curse achieve sort decay consider discuss role contrary learn approximation require consequently sampling evaluation desire query necessary sake address simple g ridge mathematic ridge name projection pursuit recovery pursuit problem ridge reference therein notation assume column stand ingredient taylor giving access taylor expansion bound consider contain derivative matrix structure g g informally term approximation due compressed tell recover stable minimization precise get approximation set maximal derive z z final control satisfied shall hoeffding g formula reliable tractable approximation eq shall hold practice approximation application uniformly smooth additionally obtain evaluation compress sense matrix entry isometry least natural number part e reference part theorem immediately apply equation error recall copy lead assumption lemma relate normalize stability fix maximal ingredient bind j hoeffding recall reader convenience hoeffding assume surely exist scalar hoeffde inequality let hence c g violate hence ready ensure sign use ga c ax estimate g collect comment differ domain make heavy completely find address follow opposite arbitrary sampling hold success parameter choose say prescribe least posteriori sample properly calibrate otherwise beginning purely view arbitrarily derivative small extent appear ratio return detail stable reconstruction choice consider practical basically choice sphere haar rotation I borel rotation rotation ok identity natural become concentrated around sense fix measure dy r k informally matrix exploit ga ga long scale copy row via coincide span moreover eq right vector accord z singular contain define final kind accuracy compress span well singular equivalently separate related approximation use define r depend show theorem lead satisfie decompose may lead similarly put j z possible might norm singular tailor equivalent relate value necessary matrix singular decomposition understand size perturbation bind follow useful stability subspace eq say speak separation equivalent say away apply final ingredient value provide generalization hoeffde generalize semidefinite matrix improve technique chernoff dimension value sum matrix proof application thus get study random surely x g show recall lemma first singular vector define therefore priori follow posteriori ii far straightforwardly dimensional simple kb ga ga x calculation vanish due symmetry dy similarly expand taylor approximation computation deduce radial nearly radial respect able actually suppose neighborhood exactly discuss limitation firstly numerical secondly deal body sketch still rigorously interesting discussion evaluation precision difference measure lead sense substitute therefore estimate observe take effect noise unfortunately sketch natural random analogue identically variable mean variance regularization whose observe scale sparsity therefore noise error scope numerical minimization coordinate choose axis success white black successful mention free last picture increase decrease behavior careful inspection method modification replace point therefore prefer able even suppose actually x modification true circumstance easily enough consider convex need yy thus execute define xx yy yx hold drawback clearly understand subject I functional unit ball replace dual diameter always bound norm clearly investigation sketch special would get result depend acknowledgment like kind warm part acknowledge financial start award anonymous valuable mm mm theorem theorem conjecture notation remark continuous define ball budget point approximate algorithm point suitable approach compress sense chernoff semidefinite classical bound invariant decomposition subject key compressed semidefinite stability decomposition life capture approximate relatively smoothness regular accurately numerically complexity sampling measurement map notion performance grow exponentially result learning phenomenon curse dimensional problem exhibit eventually behave respect problem appear tractable approximation dimension behavior cf polynomially notion sort sum physics involve describe dynamic function tractable refer class consider q negative model various unconstraine address
volume optimal choice strategy function statistically asymptotic analysis ever portfolio market especially strategy increasingly research research adopt standardized variety influential financial financial mathematic economic market something statistically market et et et name develop add piece formulation treat market impose establish work research appear market within medium country economic medium portfolio package business economic modern stock access resource identify great deal situation competition choice worth portfolio market character share market comparative choose share package portfolio return request portfolio pass ahead familiar share share package characteristic valuable possible choice stop package portfolio choice base characteristic act fast successful go evaluate potential affected financial constraint essentially request request subject portfolio share consider return portfolio share stop share bank thereby bank compete bank medium relationship fairly typical modern valuable share employ shall minimax optimization rule major mathematical capable accurately market employ fairly mathematical prediction formulate strategy stock portfolio bank problem package parameterize assume customer financial sufficient several variant within portfolio analysis optimal ensure stability financial strategy behavior compete portfolio model assume package bank algebra markov suppose assume two stop interpret agent monitor concept p n borel moment lie mean monitor monitoring process refer value stop monitor convenience negative value include observer n definition material sequel hold q expression convergent mathematical expectation ij rewrite calculate let next small vx fx inequality infer calculate limit find excess function supremum q choice use act take every solution operation mean virtue problem validity vx fx n vx nonzero price analogously function price stop price zero observation solve stop markov observation fx fx formulate valid case find power probability ergodic asymptotic equality om ordinary scalar monitoring opposite operator exist strategy observation conclude proposition markov process state essential vector boundary quantity form least next consider portfolio share price constructive together constructive elementary event subset define space quantity form virtual possible process family important definition tt find fx subset map sx sx x usefulness choice database package bank portfolio supremum take markov stop stop choice valuable share bank portfolio stock market compete stock market bank package use stock valuable share portfolio make package package permutation number stock condition important thus round mathematical share package p desirable implicitly complicate package return portfolio prescribe induce event j taking involve valuable mathematical expectation price share bank share coefficient package stop moment detail choose valuable use choice valuable threshold stop markov first valuable share calculate probability associate conditional price x stop package choice candidate share figure main sequence chain additional add valuable package consequence follow relationship decomposition space subspace price set probability lk kolmogorov substitute get equation easy calculate valuable share stop moment find solve inequality verify induce decomposition inequality package obvious symmetry choice valuable simplify bank portfolio contain number share package stop moment satisfie take stop heavily bank assume low bank potential behavior bank portfolio share share package whose compare portfolio compete market stock appear bank share package notice somewhat simplified behavior priori information possess financial capital share bank portfolio financial constraint subject portfolio share package price prescribe strategy essentially bank portfolio process develop compete share package ranking usefulness characteristic package independently within portfolio permutation order number possible choice package n moment mathematical large price q bank transaction share basic
neuron create question parameter logical aspect formally database management importantly cognitive phenomenon simulated observer discuss updating involve formulation question realization consider period event clean observer record event algorithmic physical reaction observation possesse occur result add increase precision sensor language observer share question formal prescribe alphabet tag subset set endowed denote structure subset shall associate let happen toward observer underlie p correspond value gradually negligible limit face ignore treat irrelevant might boundary become transform nest complementary degenerate correspond face convenient corner corner full induce proper precede discussion pp mind distinct proper interval see probability probability diagram corner zero leading say denote q graph every relation every contain face main process introduction move removal redundant question question face infinite simplex regard observer leave abstract observer observer structure evolution observer study observer every face mean observer update either imagine alternate move state observer maintain whenever small grow become provide possible human issue study phenomenon bar change circumstance figure amount occur graph achieve update implementation achieve reconstruct complete picture reality price pay low benefit price turn update logical context elementary updating process vs natural aspect memory abstract observer observer consider negligible alphabet question group skeleton union irrelevant skeleton language role language formation formulation parent child question available begin environment lot parent serve child result parent memory sensor mean population common pattern memory agree meaning sound blind exist already decide population synchronization among set population describe child parent population subsection capability employ observer create question old main observer need capability exponential maintain efficiently prevent observer subtle gets incorporate discussion space impose algebraic boolean alphabet formulae alphabet together require admissible pa aa bc ac bc construction compare option call database arbitrary observe environment follow observer evolve observer interact observer assign observer goal capable dealing mean treat accord significance observer rather importance tune contribute accurate record part observer trait separation database physical observer vertex weight borel algebra represent observer picture mind life observer moment evolution actual physical physical observer relevant current observation division stage totally dependent physical stage algorithmic mention potential speak approach software hardware enable study question hardware support note observer choice boolean algebra sensor principle considerably fuzzy logic still one define throughout explore observer observe fuzzy phenomena record deterministic quantum temporal replace logic expect self structure could aware allow necessary role interaction database realization logic another logic question regard motivate propose enforce deterministic introduction strongly frequently observe thought include serve test title capable demonstrate aspect proper language develop summarize front put speak accordance update face database corner observer less implication unless force corner statement every act conclusion update lose manner least implementation imply propagation observation search reach either incoming unless mechanism value high ignore possibly context realistic implementation contain reason formally speak feed conclusion understand implication occur easy cascade propagation reach observer reaction wave reach like motivation network coherent family simultaneous observer relate introduce sensor correspond sensor automatically deal human difficulty able think write create loop reading arm update notion reach song stop material read work heavy explain precede paragraph stress biology physic totally stress peak become input periodic enough ignore stress capacity parallel input evaluate resource resource evaluation may search turn creating factor observer appeal serve mathematical precede computational propose biological force interact separately notion language level back stage evolutionary process language relate realization g underlie structure language language english understand natural language language acquisition individual restrict language capable produce thm proposition thm thm remark cm cm observer evolution state real process biological serve evaluation mechanism relevance observer observer assign recognize observer serve geometric produce serve transform model flexibility efficiency possibility neuron observable human thought reasoning loss acquisition dedicate memory complex extremely structure element comprise mention exclude possibility direct piece piece capability input desirable outcome specie exhibit remain evolution central ability roughly brain algorithmic underlie answer question goal intelligence whose na construct machine capable develop tool describe principle turn improve formation acquisition structure logical variety become desirable formulate answer main principle principle define answer invariance abstract mind realization us search exist discard tie physical incomplete put use inconsistent prediction database store maintain store memory retain handle necessity principle human shape management distribute among reasonable produce restrict unclear maintain principle responsible sentence contrast general capability human strongly say possess rather inherent example formula tune head failure phenomenon well difficulty else normally memory merge powerful tend management result evolutionary make desirable replicate undesirable effect daily occur principle resource indeed since constitute set admissible admissible situation regard storage ability explain functional occur frequently goal formal evolution environment environment state correspond observer additional information content updating involve content add receive evaluate mean procedure replace mathematically speak category belong database various possibility transform algebraic topological category correspond principle category consist every pair object composition define order triple bb c g say distinguished element triple compatible meaning axiom regard object compare improvement see difference two common may observer objective reality sense category model motivation source shannon introduce theoretic database observation space endow corresponding algebra observer sensor imagine explain notion entropy sensor absolutely preference know sensor know input correspond complementary producing become family power partition induce join sensor observer partition reason visible view possible computing observer approximate minimum expectation binary question ask order higher despite ability database put set edge construction space necessarily bipartite perhaps observe person observe space number precise north west imagine observer able ask point south west east us answer open number characterize quality well observer occur reader verify graph graph one well picture idea central geometric combinatorial content mark vertex visible state observer categorical follow sensor list list refinement hence onto induce every visible contain category modeling bipartite develop searching turn computationally observer come possess grow situation bad assume job interpretation business observer complementary subset probably never sensor go home fact observer maintain content without content vertex state make repeat observation probability vertex final track objective observer numerous time evolve add vertex think sample question result answer update quickly minor small cause practically big large intractable skeleton trait retain encode implication update structural e vertex visible interest detail close objective success paper underlie graph well study reader detail duality median treatment intersection precisely denote rectangular grid grid positive integer sub inclusion relevant consistent state respect realize inclusion vertex state one integer make straight segment figure demonstrate median three diagram interested set graph wu wu finite set coincide arise realization relate principle introduction restrict structure shall proceed idea median graph structure aspect duality translate median versa median graph relation impose prove restrictive purpose define property set embed proper element theoretic length moment henceforth coherent apply select precisely vertex visible denote family consistent lie illustrate example induce embedding belong box observe inconsistent vertex become reduce corner inconsistent look picture precision answer question correspond contribute circle obtain precise answer proof separate logical structure realization result distinction constitute introduction case representation realization realization seem figure ready observer seem memory environment classify individual environment may observe stress result context regard universe ideally ideally coherent note human hold coherent rarely hull observe family inclusion fact easy via intersection family relationship observer observer vertex observer event observer observer theoretic every observer state small observer aware source stress provide observer change stress try equilibrium regard plausible observer theoretic interpretation convenient describe phenomenon observer attract state observer choose result observer precise date correct probability state universe consider possible situation observer allow state universe ask observer view universe inconsistent state observer observer exist state universe impossible event probable situation motivate observer precede concerned observer representation would structural component rule bad common share logic inherent notion consider observer implication relation equivalence hold together automate reasoning environment action understand state matter logical effort understand operate combine inherent structure directly set cube example assign unit assign leave want disadvantage assume coincide inconsistent objective well way ask standard situation like observer needs identify answer come position would reasonable maintain relation increase keep ever encode record balancing exist implementation come possess observer observation search observer may update
become quite part suggest mcmc ps high true bad differ ps reliable relatively nested ps ps sc ps ps thing tend go wrong valid bayes factor validation explore change mcmc focus seem since probability one purpose bayes good discussion finding hope communication density feasible induce constant base first heavily good good moderately multimodal introduce bf sample implemented expand note often choose big correctly densitie density markov chain converge start state number chain converge similarly design converge iteration converge follow path different spread big contrast try directly marginal marginal thus fisher estimator bic ignore poor suggest simulate mcmc independent p variance convention apply need evaluate term care know approximation also usual compute information computation use neighborhood use see mcmc run sample laplace modification give behavior role selection ps big maximum likelihood regularity condition kullback projection easily expansion pn substantially unlike asymptotic mle still work pt pt pt pt ask manner procedure justify ensure role model joint prior usual density default choice jeffreys variance acceptable thought early regularity ingredient namely wrong bayes sampling factor true ps sc mcmc support conclusion computation bayes advance mcmc compute posterior major bayes complex phenomenon numerous paper bayesian book quick calculate posterior interval calculate select ratio marginal model probability ratio measure bayes factor modern recognize recently principle calculation bf calculation reversible jump lead different state chain connect space differ prevent mix another popular bf major explore nest dimensional last reference example geometric mean arithmetic path arithmetic path bridge modification usually cauchy recommend examine less p like popular recent popularity relative ease multivariate apply finance see ps implement integrate score discuss ps bridge summarize introduce path essentially except path need ps hold likelihood factor generalize prior behave mcmc dramatically properly big heuristic correct inaccurate wrong grid mcmc time start point ps work avoid propose ps well implementation point discuss validate heuristic projection likelihood gold sc draw burn burn suffice ps sc go show modify namely ps sc ps topics grid sc simulate ps sc choose conservative life ps relatively well section reduce sc say ps marginals preliminary screening comment briefly importance ps appendix one familiar asymptotic maximum course point asymptotic would point like ps sc validate partly partly subsection ps path selection model ps mostly ps review ps toy relate introduce introduce showing validity ingredient mcmc argument later section subsection among sampling p simple bs bs know difficulty suitable sampling try difficulty introduce act numerator extend connect bridge get numerator denominator generalized bridge ps idea estimate calculate construct bs path give unnormalized normalize constant taking derivative identity order integration eqn log normalize bf sample converge point mcmc bayes factor commonly scheme ps modification ps providing compare unfortunately example orthogonal bind one path convenient generally ps amp resp density factor later one common density parametric arithmetic path rao mean importantly mcmc amp ps regard degeneracy ps ps study discuss ps toy selection ps fail modification ps sc difficulties calculation bf later begin consider true wish independent wishart help bf conjugacy take wishart appendix arithmetic definite combination matrix path scheme describe early bf ps report define bf value bf bf mcmc true factor factor loading model imply outcome uncorrelated latent without rotation post triangular restrict parameter reciprocal entry may boundary equation easy calculate definite mle neighborhood example factor global give specification element normal prior diagonal low triangular conjugacy simplification idea loading scheme mix expansion induce class sign loading suitable prior whole range degree go use sampler stick convention factor define complex define factor depend say threshold path explore path tune grid sample arithmetic path path along construct gm prior estimate turn bf song along expansion respectively connect path score along path simulate sample distributional model point fy fy ps model normalize log mcmc factor parametric arithmetic assume r eqn path integral namely eqn notational write respectively show dt fy fy quadratic quadratic function apply moment dominate dct integral statement extend path remark like ps related theorem ps study estimate bf simulate report salient feature bf tend rather subsection happen simple prior scheme operate path path nest model small change contiguous scheme ps sc precision subsection ps sc ps sc demonstrate underlying issue load diagonal also mind remark tune namely grid size bf discussion remark path sampling converge finite ps cauchy cauchy moment integrable enough degree freedom f sensitivity bf change f table priors continuously show rd rd bf change prior report mcmc run mcmc standard deviation nd size mcmc size l pt major size fine grid deviation bf estimate differ order magnitude special bf correct focus prior grid size bayes estimate much value mcmc mcmc remain early ps explain bf low dim reduction model prior moreover cause relatively two ratio space properly divergence take likely make seem plausible stability lack bf show ps modifications idea ps remark identify cause poor subsection subsection ps try solve deviation ps sc ps pt factor factor give prior try compete keeping estimate step ps implement path reduce variance dimensional vector triangular correspondingly perform path computation I step score ph true close suggest fluctuation bf stability bf dimensional notice proportional computational grid size regard produce curve grid bf value set model standard estimate ps deviation ps argue ps sc bf keep loading diagonal table lie range respectively ps sc generated factor bayes ps sc parameter factor factor factor pt bf generally bf expect pattern estimate bf ps estimate bayes bayes usually choose true often ps ps bayes true ps sc well ps choose equally ps sc detail sc ps factor along remark study figure proxy fluctuation loading latter infer denote vector view sort mixture log sample nonzero come cluster present range appear representation vary moreover see fluctuation value represent near model ps tt use ps sc ps sc stand ard deviation avoid standard concentrate ps sc ps mode poor ps ps toward mode study nice mixing sc simulate use sc poor plot different lag sake ps except big ps sc figure ps slightly explanation mixing miss probably explain discrepancy notice bf ps ps sc look factor loading load top row show autocorrelation believe call autocorrelation loading dominate tend big simple cover model simple
infinitely element fmri environment improve resolution development genomic array increase distinct unit appear repetition copy briefly region genome span never contain genomic type genomic absence hence collapse per genomic discretized array number consist probe status hundred thousand region unnecessary appropriate mixed method point step resolution use reduction latter probe collective genomic neighborhood lose note high differ keep either number collapse mathematical correspond belong group similar secondly region use since permutation test independence valid permutation invariant wise error motivation spatial illustrate pair discretize correlation diagonal part notably wise structure span oppose spatial modeling describe apply purpose adapt take physical region practical concern leave region plot color correlation consideration spatial location th region first cluster besides class distinguish unconditional test usually margin margin permutation label permutation summary distribution testing association cluster permutation pearson statistic determine region motivate hierarchical cluster region procedure control firstly hundred fdr reflect highly share estimate fdr subset secondly cluster fdr usually subtle conclusion relevant plus permutation control feasible testing clinical permutation invariant testing invariant rejection combination improvement hierarchical context apply value threshold neighboring reject likewise connection cluster emphasize guarantee give rejection cluster cluster iii hierarchical reject j region region reject step cluster region less focus behavior standardize effect wise scenario cluster region scenario testing analyze region row er er negative row sample column involve concentrate per constrain sufficiently large decrease finally lie nominal correlation cluster validation precisely consecutive argument outline probability consecutive next need low use cluster region range confident create genomic dependency association label result cluster unbalanced hence peak differential proportion proportion small less region summarize identify cluster identify identifies region identify identify region hierarchical significant hierarchical identify cluster contain hierarchical comparable clear identify type type proportion differential scale axis dark grey significant mark bottom proportion gain group proportion c raw conceptual two separately array rigorous region versus unconditional available hypothese methodology apply unconditional cluster algorithm separate conditional unconditional likely conditional hypothesis cluster approach value testing acknowledge accordance hypothesis might prefer unconditional emphasize tumor importantly clinical good tumor large sample heterogeneous genomic location occur detect unconditional attractive external applie extent unconditional differ split part use test unconditional part significant reject confirm unconditional average region member reject reject conditional approach confirm unconditional cluster latter conclusion strength help region significance reflect wise adjusted wise extent distinguish genomic response neighbor far molecular level decide genomic dna really relate prefer discretize clinical information cluster useful would characteristic correspond copy rather gain consuming unlikely one digit digit loss normal would keep digit gain two gain double dna probe design strongly verify usually costly straightforward genomic cluster sample clinical outcome genomic individual small simple selection dedicate combination permutation multiple resolution clinical introduction resolution datum e massive parallel sequencing acknowledgement associate testing van partly software hierarchical testing package contain actual implementation software package site http www ii permutation test short prove distribution permutation permutation may conditionally sufficient allow principle improvement assume imply intersection relationship improvement dependency fully example x often wise random covariate dependency appendix multiple control consider case correction provide proof use also aim pre step hold condition region complementary sum appear hand c logical relation implie fact dna array tumor contiguous dna contiguous principle clustering pattern moreover really capture
full feasible speed carry point label classifier subset great magnitude practice fold variability minimize rate bx estimate reduce rate correct estimate x calculate validate addition original set feasible time note ten would estimate shall demonstrate application cancer bioinformatic al contain expression normal patient et al gene min max min minimum expression across sample gene level cell contain label breast cancer label sample separate give label et al gene small round blue cell b double firstly unit deviation row cross validate versus five machine multinomial plot list validate bias correct form guide level list centroid near unbiased estimate consider al question inspection heat illustrate heat sort visual separation negative top similarly separate pathway et present extremely dimensional need insight differentiable approximated ab reference lee parts object factorization nature van c university broad gene tumor arrays national sciences usa lee non advances system mit zhang localize base representation international conference pattern volume decomposition advance mit improve molecular class discovery factorization bioinformatics inferential microarray bioinformatic transaction intelligence component canonical matrix classification international conference bioinformatics chen california pp improve collaborative collaborative temporal conference discovery paris c g bias gene expression precision bioinformatic form plan r p j molecular discovery science discrimination method journal american yu early breast breast r diagnostic prediction gene expression nature r diagnosis cancer centroid national usa available representation molecular specie state sciences usa national zhang tumor use nonnegative transaction ccc svm square loss global iteration blue dot sort negative sort top iteration text latent factor ever demand differentiable bioinformatics factorization gene xx observe observation variable microarray gene might thousand let transpose overall zero reduction primary perform give form minimize frobenius square factorization replace b restrict nonnegative factorization nmf shall call approach factorization classic svd van exactly column matrix diagonal rank specify minimize al large essentially svd factorization applicable function square limit global iteration element factor fix loop minimize individually rather total second perform rank contrast factorization task minute demonstrate reduction supervise five know ab spirit variable interpretability consideration nonnegative b advantageous view interpretability lee lee improve interpretability whole nmf procedure lee parts subspace local localize et al et recently variation nmf datum mix combination al penalize decomposition describe function derivative member note limit loss since range algorithm follow input factor rate correction initial b generate randomly cycle step cycle step internal update ab loading derivative see total parameter loop fix role iterate difficulty convergence unconstraine iterate iterate switch partial process factor update recommender technique predict netflix note recommend maintain value correction rate illustrate figure
grid feature visual center rectangular partition whole partitioning cell cell compute measure local addition pyramid decay computed kernel level kernel kernel pyramid kernel band linearly spaced histogram combination sift result average active kernel mkl difference decrease explain result image classification generate normal increase band parameter weight sparse spectrum remarkably mkl true spectrum hand mkl row well often find consistent elastic band width weight dataset similar column medium mkl net regularization mkl change parameter mkl mkl two sample kernel mkl seem favorable dense kernel kernel neighbor intermediate favorable preferred result prefer ac trade uniformly learn mkl elastic depend weight spectrum also mkl often outperform weighted mkl however mkl helpful feature lot computation accuracy elastic regularization interpolation extend regularize mkl problem sparse mkl poor mkl small mkl among candidate reproduce hilbert functions mn nx iy minimization member rkh logistic regularization mixture determine trade rkhs non zero regularization square norm mkl correspond weighted mkl take th matrix nz weight define concave linearly concave substituting unchanged rewrite equivalent uniformly call elastic mkl mix hope mkl approach weight penalization al
event vector investigate jj second increase resolve step proceed iteratively loop step store orthogonal shall specify maximize separation wrong similar determine statistic compare k previously update remain consideration k kk substantial growth term send correct send hard weight false alarm indeed unlikely distributional property discuss define separation send eq specify matter grow near number normally induction high send send covariance joint constant density arise constant send exceed room stay case positivity separate gap choose consider quantity capacity capacity mean likewise play nearby compare growth case bound g lx analogous suitably alarm accumulation order source drop lemma non minimize evaluation favorable function make decrease skewed choice overall positive alarm satisfied produces indicate reliability successive normal bind deviation full manuscript pick satisfied generally incur error probability eq kullback leibler divergence bernoulli least investigate invoke preferred producing shown produce helpful simulation channel superposition code successive decode subset message index superposition code decode exponentially capacity presentation give decode vector partition power algebraic message string encode realize concatenation number bit specify power receive distribute map decoder produce estimate overall fraction section mistake reliability mistake high string supremum channel achieve arbitrary decode moderately decode moderately mistake adopt channel noise constraint code appropriate involve internet wireless phone space communication scheme practical scheme capacity equal sum simple allocate variable slight variant allocation power across agree setting find summarize finding capacity achievable albeit order gap capacity reduce factor mistake rate conference refine obtain exponentially equivalently journal measure gap capacity benchmark scheme include related superposition code theoretically optimal code gap capacity order initially compute receive inner comparison perform inner incorrect allow proportional constant variable mistake exceed result sign superposition improve subset without presentation prevent superposition code distance rs code overall interpret take either code fraction concatenation code mistake error exponentially small code good low iterative decoding mathematically capacity limit channel superposition convex arise statistical iterative preliminary iteration high inner residual communication reliable capacity express recovery combined signal convex accurate provide design establish complement typical recovery non lead minimum capacity superposition channel channel feasibility identifying
exist x measurable sigma x particular remove underlie loss generality ergodic eq f one point countable borel define elementary notion segment join family disjoint third tree establish segment gap subtree good binary rational integer define definition segment adjacent interval topological regularity family f countable countable family regular element ergodic join family disjoint join empty intersection lemma establish useful join segment suppose sub integer join cardinality remark intersection contain indexing without selection ensure tree binary tree locate refer distinct child exception two child internal leaf leave adjacent proceed depth short necessarily path depth node length show collection leave correspondingly nearby depth suppose leave exist child sum child section present arbitrarily tree segment segment root leaf call intersection principle ensure adjacent join final remove regularity satisfied internal child equal adjacent intersection empty interior pair non adjacent integer join adjacent cardinality every appear together establish order assumption regularity countable subset element join measurable borel lebesgue ii inverse borel measurable iv symmetric countable section multi identify join adjacent element differ least sample relative union expectation limit via stage important splitting identify stage split stage follow proposition treat value difference precede identical difference adjacent segment particular use hierarchical appear require involve binary keep difference substantially result detailed completeness countable borel measurable ergodic f dyadic consist countable lebesgue remove use splitting construct stage proceed f let join dyadic segment function pointwise sample join function fashion relation many average differ proof simplify apply follow state measure b tight subsequence weakly see absolutely stage appearing th produce define suppose splitting stage eq join fashion define lemma continuous set away integer intersection interior closure set proof condition internal set adjacent segment path empty interior intersection empty proposition let depth assign level begin child segment non empty construction lr subsequence set identical include exist assumption finite argue intersection adjacent segment measure exclude possibility segment adjacent segment intersect interior one segment segment ax ax k third yield suppose argument like g contradiction segment assignment set child depth property node appear select interval r subsequence inequality display sufficiently lemma l j proposition child follow fix proposition suppose interior node label segment adjacent pair non integer iii particular inequality node show integer argument integer subsequence adjacent embed subtree label child easy segment node appear root construction ensure member join empty tend infinity theorem statement f countable contain identity satisfie shrink diameter let map statement lemma process f define let f segment define sequence rational number include let fact interval family finite mod fix sense moreover rational family approximation argument proposition exist join obtain segment examine segment end h j kf argument zero join everywhere mod zero form join eq cell cardinality lemma completeness proof let ensure q follow display collection join include dyadic join diameter lebesgue bound two inequality sum join contain proof support nsf grant dms proposition family measurable combinatorial extent predefine convergence average ergodic sampling resolution ergodic average eventually within f gap finite average bound countable place place exist run title dimension number process complete separable f countable measurable sense every ergodic limiting difference expectation f discrepancy f theory inference consider identically substantial discrepancy mix focus ergodicity summarize ergodic call asymptotic discrepancy omit mention confusion say provide combinatorial quantity know scale definition family f gap arbitrarily mean et
h r together linear regularization model learn kernel mapping way regularize mkl weighted target one omit case optimize show equivalent strictly replace block criterion include net thus investigation criterion recover elastic easy section generalize begin expand value slack incorporate lagrangian multiplier incorporate equality lagrangian problem partial lagrangian kkt n rearrange lagrangian conjugate thereby remove dependency function follow inf convolution moreover conjugate dual generalize arbitrary regularizer loss solely correspond dual regularizer pair offer conceptual practical contain implicitly output optimizer induce form kernel need actual parameterization primal kkt optimality feature model focus case consider elastic net multiplicative elastic net regularizer optimality translate hand numerically optimal coefficient solely mean use identity efficient quasi newton hinge hinge optimization substitution dual inf convolution problem hence express k mx nm ms supremum call approximate give kernel formulation rademacher complexity let rademacher rademacher classifier literature far constant induced kernel q classifier normalize upper bound improve compare main rademacher regularizer term norm regularize weight influence regularizer recover approach term capacity elastic net regularizer decrease w b show first apply old dot c apply twice mean account factor analytically underlie mkl mkl elastic net mkl mkl unweighted balanced sparse intuitively non mkl robust mkl achieving less perform worst sparse mkl perform counterpart section prevent perform well block mkl error aim detect site rna bind genomic dna start regard key detector thereby rely task employ kernel represent shift st spectrum energy draw roc repetition experiment block norm elastic elastic elastic net net mkl norm block mkl norm block norm mkl vary model net approximate mkl norm classical outperform unweighted sum accordance highest surprisingly considerably recent norm mkl remarkable significance domain unweighte recently confirm state program detection experiment describe consist http traffic institute first http request randomly traffic traffic generate exploit buffer attack use virtual environment http gram avoid dependency http request length test example roc repetition auc elastic net elastic elastic net block norm mkl mkl mkl block mkl show net relatively typical detection net relatively reach mkl well sparse mkl mkl version bioinformatics mkl predictor present framework line mkl variant plug mkl variant term give concentration match previous bound mkl mkl depend scenario compare mkl bioinformatics network surprisingly mkl sparse practical future translate function hinge area roc curve ex e cs berkeley multiple lead kernel regularize minimization approach objective present show formulation special case analytically empirically select kernel task difficult view choose selection research allow predefine typically come criterion kernel leave variable base classifier approach take second optimize base norm trade contribution sensible encourage weight way extend
finding basis contribution improvement prevent local mode period jump secondly involve full two write application configuration point centroid configuration protein identify site atom atom sum match requirement match approach use shape et configuration green inference match treatment match configuration match regard multiplication rotation translation size translation chapter riemannian size rotation translation r dr td l ml translation q linear denote point parameter tx sx pm isotropic gaussian model constraint protein box obtain multiply direction protein likelihood variability lx v tv al assume use assume distribute possible gamma match metropolis position particular point match match probability select becomes match accept probability al metropolis hasting match require whole calculation alternative hasting match accept carry match ensure remove brevity shall refer model matching use computationally fast approximate get mode reach position satisfy certain propose much big move translation flip four reversible big jump effectively help subsequent match near match three translation flip step rotation near nx mx x nx distribution define phase jump phase interaction behind explore region big immediately else allow home big jump n p p match accept new match exactly configuration point turn green configuration body transformation co remove rotation matching translation perturbation uniformly volume concentrate rotation translation xx z axis euler angles euler angle haar perturbation underlie perturb mutually take haar density condition rotation angle hasting draw perturbation angle haar rotation group update lx match otherwise exactly mcmc update configuration similar appear note green matrix rotation update rotation angle green matching possibility configuration match situation al density rotation translation nuisance inference nuisance integrate marginal obtain optimize nuisance different density laplace method theory perspective rotation form explore performance bind site protein jump protein p criterion efficacy determine distinct configuration select random measure correct match figure configuration converge correct match reliably surprising configuration allow run continue million iteration monitoring match match reach allow continue big use initial histogram iteration success big configuration encourage big jump include success increase result within million run start point look run posterior go three thing look result compare result firstly effectively convergence method angle gibb make matrix convergence form proposal close formulate poisson much big start take lot converge jump jump let algorithm run period introduce jump converge big jump could increase despite take converge jump big jump always always since proposal accept half translation compare configuration scheme run start algorithm protein green accept match match principle one tend run run five proposal move status match value correspond fix effect protein independently run matrix match appear occur divide match match everywhere else match configuration change prior effect match model green similar configuration match readily relationship long suggest variability burn consider know algorithm without perform fix deviation point cube corner subject distance point uniformly cube corner start matrix match start record proportion matrix match record proportion matrix repeat match experiment choose four deviation iteration configuration estimate match reliably model match configuration give look increase perform model match reference illustrate study pose relationship simulation appear value two swap conclusion method jump however comparison converge reliably protein match optimisation
smoothness smoothness base since smoothness slight variation finite difference instance z x fx fx fx denote continuously imply induction generalize nonnegative z x continuous dimensional let countable product eq supremum partition il b I sf jj sum integral eq q finite coincide let integer ds k defer appendix lemma result let dd b ds lemma ds l dd ds b sd p sd sd p sd sd sd sd sd sd extend standard estimator numerical toy exhibit rate apart obtain ask use obtain question since variation describe alternative use permutation therefore another implement reduce still line simplify apply rmse mc integral approximated analytically digit mapping digit expansion form j b j get expansion digit expansion digit hence x dy countable exclude sa nj j b c j result b u u u c u u b u digits component b b b use schwarz b b last follow cauchy schwarz inequality dependent relate divide I since operator suffice b j jj kt l q ft u ft element case divide triangular supremum j inequality supremum take admissible choice b k sf j sd sf sd rd df technique integral statistic sampling instance root randomize monte achieve rmse strong achieve rmse derivative smoothness general additional smoothness achieve square integrable partial mixed rmse smoothness numerical rmse approximately accordance upper integral statistic instance estimation involve smooth standardized approximate unit cube transformation different carry bad power arbitrarily even mean satisfie smoothness yield improve instance algorithm achieve convergence smoothness know mc integral monte monte quasi uniformly criterion kolmogorov decay randomization achieve rmse bounded randomization method use apply digital net improvement digital net local therein rmse domain latter method mixed order coordinate smoothness use algorithm assume digital finite digital net base n dm dm concept classical digital net net suitable construction use expansion give function proof remain item dp digit also extend ds rmse introduce follow describe shall permutation digit permutation index mutually independent permutation permutation point drop subscript analyze generalize applie see
employ focus parametrization variability quantify bandwidth dimension approximation indicator note normalization function expectation respective derivative involve free leibler z z frequently negative pz constant ignore offer clear energy provide objective readily calculate well depend derivative equation equation proportional minimum bias information free visit neighborhood connect landscape unify criterion pose kernel equation representation discuss add detail carry expand nevertheless capture atomic end propose perform step equilibrium monte carlo scheme create advantage effort estimate readily change optimization aforementione algorithmic module contain propose scheme equation follow carlo available propose gradient estimate current employ early iteration gets discard gradually place recent p relate free energy cardinality expansion least way firstly salient ensemble secondly give vocabulary basis identify kernel surface divergence computational combination hierarchical add greedy procedure maximum generality isotropic kernel parametrize location propose selecting maximize expect value target current reason maximization readily carry overcomplete basis employ e wavelet propose efficiency free landscape naturally small successive fine detail capture e offer approximation successive equation kullback leibler assess general analytically whereas normalizing sampling augment efficient computation appear depend expectation analytically scheme degree force simulate suffer well know every change routine reason rely sequential sampler smc range path sampling work allow difficulty retain interact molecular dynamic dynamic hand change give approximate update mechanism proportional weight update particle dirac center particle expectation particular algorithm iteratively step descent successive z mp algorithm compute expectation carlo estimate mp building bridge base recover auxiliary intermediate affect overall smc automatically determine intermediate process ess intermediate priori reciprocal priori size I extreme arise particle whereas rest weight provide extreme informative weight ie dramatically accordingly p sense energy base produce high gradually towards energy temperature guess energy density employ well add convergence building distribution p purpose schedule employ adjust aforementioned difference eq demonstrate efficacy begin temperature remove straightforwardly general reaction coordinate reaction coordinate evident q integrate reaction see define straight forward pdf coincide derivation point adaptive smc mcmc mala sampler equation note technique finally parametrize free energy surface reaction artificial clearly recover value surface would complexity exploit parametrize rather construct gradually move large guess smc ensure level play inverse temperature wish free free upon simple used potential depict fix inverse region two apparent particle smc depict landscape capture less particle series pick three depict value select great majority free energy rest correction estimate kernel show number quantify equation offer great gain kl divergence periodic side interact pair give atom potential barrier separate two equilibria atom energy q ensemble volume probability atomic position temperature system assumption atom reaction norm box densitie high atom employ various box equilibrium move probable barrier slightly decrease equilibria move leave probable probable dimensional pairwise interaction play respectively potential system finally particle follow global truncate energy number minima parameter initially eq coordinate structure two separate dimensional reaction q domain temperature learn adaptive scheme automatically determine intermediate whole range aforementioned step mala adaptively adjust high approximately step nature smc scheme largely rest intermediate figure similarity energy surface neighboring optimization intermediate overall cost free energy surface four report study truncate correspond latter free landscape calculate previous assess result profile reaction perform numerical integration reaction coordinate depict agreement different e method leibler rigorous minimal adjustment representation free offer optimization conjugate employ wavelet sequential believe free system challenge subroutine uncertainty quantification propose free motivated molecular dynamic estimate energy markovian employ make use scheme sparse multidimensional case employ adaptive monte couple molecular dynamic sequential parametrize capability energy potential central concept physics rigorous consist probe experimentally global application force sample impractical infeasible overcome remain compute surface integration integration require long trajectory order user
friend walk union section walk different l friend event friend friend event group avg friend avg avg last fm identify fm store discover possible fact ground fm increase date assign exploration exception service sequentially rarely non conjecture non close account assign latter sample last fm user repeatedly million discard repeat uniformly space irrespective user refer examine population vs inactive vs solve fm valuable asset must efficient fm bit infeasible summary fm table week later capture fm estimate number fm internet use process substantial process fm evolve duration maximum fm period average therefore population day reveal distribution study fm network period ignore unlike graph collect fm stage discover graph user collect list friend neighbor list user list graph friend friend neighbor graph accordance stage event quite user group event thousand hand friend equivalent enumeration approach enumeration user uniformly event carefully implement action event study member group page return marked return cluster machine execute simultaneously walk walk randomly web site music country rely special purpose outcome seed seed let per online describe eventually collect sample user l type friend event group graph friend friend friend group friend summarize collect type observe repetition walk range event neighbor fraction event group neighbor relation besides event dominate combine group dominate friend occur many event hundred participant walk neighbor percentage user graph g event user play recorded expectation importantly table well type relation friend leads estimate exception percentage friend group thus weight emphasis compare table approximate percentage consider friend user fm plot number type ground friend event portion group average portion base however relation helps considerably utilize friend group close truth approximate fig close truth shape nevertheless probability distribution gap cause ii compare fm validate walk absence remain track popularity track type fm report automatically mention track tag track chart people plug service provide site fm stream validate track week week start track track popularity percentage top track available fm rank percentage track rank curve friend track quite actual additionally lie top last fm graph friend neighbor quite actual elsewhere get close truth early towards degree recent bias bias walk network graph walk fast mix case possible explore unknown graph sampling explore dependent improve goal instead achieve multiple relation sampling idea benefit technique couple improve across thompson condition cluster sampling broadly within design note interest confirm fm week apart consider study fm include track tag user user link explore fm crucial single part recommendation work etc sample multiple relation walk relation sampling connectivity relation fm approximations sampling majority user fail synthetic improve believe grow paper demonstrate utility sampling algorithm implement sampler performance expect correlate prove effective question helpful designing proposition corollary problem edu technique social rely connect time exist relation user induce perform perform select benefit fm internet music fast fail highly cluster service fm walks graph social decade popular present hundred continue study interaction behavior despite limitation service g limit treatment impossible obtain account estimation precise relatively ability draw know lack frame user directly principle especially focus limitation current scheme definition social graph need frame node initial explore produce poor cover body systematic walk social user posteriori know sampling ultimately dependent walk yield representative social speed walk target three group event union definition group union select pick relation connect user link social tie group either exploit graph compare typically graph graph naive collect relation exploit acceleration approach combine relation frequently connect graph graph require enumeration relation propose third stage walk select relation practice equilibrium distribution internet fm highly relation give structure methodology evaluate synthetic graph methodology fm practical recommendation discuss finally conclude different membership undirecte special relation several graph union union contain edge merging also implement sampling relation however union graph helpful conceptual tool seek sampling multiple naive way run random walk collect sample graph restrict never bias seed f graph allow quick walk practical walk union quite expensive require enumeration edge adjacent costly depend relation cost relation union graph employ vertex another edge enumeration node instead two f enumeration within neighboring edge save bandwidth neighboring relation high relation helpful offline application survey enumeration select select neighbor equilibrium process connect walk irreducible recurrent presence triangle issue need completeness briefly repeat parallel way walk traversal show walk sample weight weight hasting paper employ classic inherently degree per show apply throughput combination post hoc walk recommend obtain representative formal assess quality approximate hence stop use multiple walk critical scalar diagnostic walk third diagnostic across ensure walk demonstrate key benefit improve connectivity underlie graph cluster random determine quantify look component belong walk related node run walk axis characterize connectivity fraction belong er heavily connectivity relatively graph say approximately fast asymptotically union trivially exceed threshold characterize well relate associate top significantly drop grow new edge rare connection speed conduct experiment
chapter py augment define simulated weight region choice generally recover marginal integrate abc act practice discard realization dataset sampler simplify important way weight representation see simplification matrix replace statistic significantly sufficient statistic utilize kernel norm abc variate approximate computation use conjugate proposal reduce kernel sampler framework obtain variate proposal sample stable theorem non framework rejection proposal abc present day datum consideration comprise conjugate stable proposal propose variate proposal day assumption allow sample metropolis abc tolerance anneal tolerance utilize sample inversion proposal parameter propose sample propose unconstrained intra day vector perform involve numerical variate generate two heavy tail stable day versus approximation approximate ignore posterior assess bias inter significantly demonstrate day min line estimate posterior shift statistical variate utilize first impact bayesian framework appropriately shift next noise comprise stable innovation noise level shift illustrate deterministic time trade inter day market methodology extend modify apply conjugate bayesian transformation symmetric heavy tailed skewed non approximate mcmc utilize incorporate case asymmetric verify stable mmse matrix variate performance sampler gaussian financial demonstrating mark hence applicability trade framework perspective justify vary depend regime detail suitable setting fundamentally distinct describe property series deterministic series close market account fundamental appropriately underlie price model consideration mixture student innovation intra allow capture certain market still maintain conjugacy comment suggestion addition perform aspect cm proposition remark mathematics edu capital st electrical statistical auto extend incorporate price series shift accurately model develop variate comprise error intra day mixture normal stable inter day innovation inter boundary allow skewed current special price series shift either shift estimation shift market bayesian shift series inter fit variate inter day jump variate stable journal trading trade statistical pair trading consistently assess trading representation co integrate see overview address bayesian bayesian trading real price pair statistical fit portfolio shift jump series evident rank also shift economic social factor attempt economic occur close market trading time secondly appropriately shift co integration framework develop demonstrate robust statistical practically important observe series occur open market asset pair price shift solely market asset period overlap market open particularly accurately shift pair ignore shift consequence trade result affect carry effect design trading begin inter shift multiple contract segment span year demonstrate statistically significant level shift price flexible interest flexible term gaussian member fit stable price shift market trading demonstrate period imply fitting demonstrate distribution appropriate capturing shift typical statistical multi variate gaussian innovation fit basic utilize incorporate innovation thereby parameter period asset market include stochastic time point change switch trading tail simplification trading addition statistical generalize innovation symmetry realistic model novel selection generalize variate demonstrate data shift price reflect large model mean trading system arise day break frequency estimation several min simply discard shift occur significant trading trading close shift trade day discard period perspective shift integration issue address regular change portfolio transaction related trade volume paper price series make modeling price shift intra market paper variate distributional co normal family variate estimation newly aspect variate abc aspect involve standard variate model abc methodology aspect capture abc stable intra price shift contract segment price day model asset stable intra shift stable fit perform set frequentist due price series shift develop process simplify multivariate skewed conclude actual pair row denote random column successively kronecker product model furthermore integrate vector linear relationship assumed multivariate lag express format row dimension dimension represent trend autoregressive long multiplier long multiplier important root full univariate co occur co integration vector stationary univariate contain specify equilibrium variate gaussian capture capture stack variate model stable area data skewness heavy scale distribution consider shift include special sub purely incorporate composite process intra inter asset dependent denote respective market day univariate stable typically specify rate tail sign skewness scale term exponent tail analytically tractable member admit close evaluate gaussian member characteristic discussion intractable efficient observation abc develop day shift quantile stable http american stable corresponding paper market ignore roll effect historical behavior allow day shift stable fit sequentially window length fit historical assess evidence day shift substantial becomes contain reduce analyze inter shift extract open asset inter shift daily market jointly level cd note mini tu us year asset vary consecutive period segment end contract day market include shift contract day shift fit asset comprise inter asset table report parameterization propose fit stable model stable stable day shift series assessment stability constant period cccc result suggest shift distinct gaussian confidence even mini tail several demonstrate inter boundary day data pair series realization length error independently series asset estimate shift see series stable innovation raw versus stable price equivalent th sample stable fit asset h compare know procedure utilize posterior perform mmse specify assume knowledge nine stable hence assess impact noise display histogram bayesian mmse estimate group dash represent mmse set gaussian innovation inter day noise mmse top model mle mmse presence ignore vector presence day noise avoid appropriately inter day shift price us model introduce model assess admit ability apply long point wise would generalize simplification instead formulate bayesian specifically thus distribution intractable range context review review identical bayesian singular long run multiplier indistinguishable unique restriction identity sub skew bayesian prior distribution variate composite presence day conjugacy however conjugacy case via scale normal representation n variate mean positive definite matrix variate definite conjugate derive noise process framework specifically observation obtain matrix obtain covariance lemma variate likelihood factorization prove transform uniquely give transform transformed derive model meaningful allow variate important posterior instead significant equation strictly ie involve auxiliary framework mixed day intra innovation denote observation intra day observation vector include row subtract location parameter stable intra time exploit conjugacy beneficial decompose form specify theorem trace identity determinant identity x complete group inter day boundary grouping observation intra inter day variate likelihood price represent variate present combine variate statistic variate independent diagonal variate group preserve conjugacy result day prior reduction observation conjugacy remark gaussian model independence
mean equal slightly large detail almost estimator straightforwardly family nevertheless redundancy show conjecture propose modification ml put estimator contrast ml text bit outcome shorthand expectation shorthand finally discrete value write read mass however admit let countable x model family relative statistic countable sufficient remainder omit poisson multinomial gaussian variance suggest natural parameterization mean onto infinitely often differentiable define universal recursively note plug universal family common phrase plug universal constant eq plug plug plug model constant understand sequence plug introduce outcome outcome code ensure plug ml outcome practice hold redundancy define redundancy universal minimize follow definition kl exist unique redundancy major type code part specifically mild universal code depend log case major show model ml behave redundancy significantly examine expand taylor get exponential coincide another parameterization dm last step redundancy plug code family satisfy parameter exponential sufficient space widely satisfied exponential define bernoulli put easy x large variance take see family bernoulli long relevant statistic parameter estimator exist p set lebesgue immediate plug code plug behave family unless bernoulli fact exclude small modification put lead claim consider family equal estimator albeit lead conjecture something exponential modification ml plug estimator properly family I nm almost ml hard ml bayesian universal ml achieve redundancy satisfy statistic parameter denote q unbounded md dm validity whenever fourth rough express relative redundancy plug sum redundancy ordinary difference n ix v get expansion mm pt concern ml parameter establish plan analyze line conjecture analyse plug code sample redundancy behave inferior bayes slight almost resolve universal code code forecast code sequentially code previous outcome take equal ml maximum call code plug paper redundancy expect ml variety model family example paper plug redundancy behaviour way see plug calculate three code appear argument yet parameter model redundancy mp cite yet universal code typically behave differently substantially inferior plug selection
auc jensen mining difficult proportion actual link possible relationship observe although author conference another improve near comment manuscript laboratory research program laboratory company united energy national nuclear security contract ac social base link structure exploit consider link link structure predict link time evolve consider link year know moreover scalable truncate decomposition usefulness exploit natural temporal numerical tensor despite difficulty particularly periodic pattern numerical algebra author national nm national different web instance link exchange phone call g correspond phone call link exploit link handle task introduce predict link link relationship problem context movie interest user aspect predict picture link extend state temporal link step relationship periodic arise mail interaction pattern forecast make prediction far organize third array simple case tensor weight predict time analyze link publication link dot denote base slice approximation produce truncate singular consider bipartite proven method scalable approximate method cp higher prove successful network interpretable factorization temporal step periodic link web page web visit history place month computer network traffic computer science conference publication publication produce author conference pair prediction score impractical due requirement store score answer question conference year year practice problem predict link periodic example daily recognize use make heavily versus service summarize outperform summation bipartite use devise forecast cp expense periodic datum forecasting forward scalar letter vector letter e capital column matrix denote tensor euler letter slice entry element periodic conclusion different present unweighted weighted year score extend show approximation straightforward entry tensor motivated link backward call slice weight great link see demonstrate ct paper matrix specifically size column sum htbp score predict link calculate score low prove effective semantic indexing rank arguably one outperform undirected node link node node path term adjacency graph consideration adjacency first scalable inversion rank method applicable square represent situation rectangular represent q become generality diagonal matrix orthogonal incorporate rank rank replace diagonal base submatrix adjacency technique technique adjacency equivalent computation discuss specifically dense dense far consider bipartite represent adjacency eigenvalue sort magnitude matrix square matrix diagonal bipartite graph replace approximation result submatrix interesting except change singular rank via method require adjacency bipartite advance factorization factor amount storage rather operation explicitly model dimension one model cp also review symbol j k individual cp cp tensor svd tensor svd rank decomposition orthogonal svd orthogonality despite orthogonality cp permutation condition cp uniqueness cp tucker factor forecast easily depend rotation whether applicable topic make extract cp score outer quantify vector may different trend profile heuristic average choice component follow temporal heuristic alternative discuss temporal cp method periodic pattern require period method tool predict future period score compute vector step study correspond trend additive blue step red cp forecasting method analyze temporal work applicability leave cp predict cp link predictor matlab processors gb ram alternate tensor toolbox organize third let decrease large number slide seven contain correspond test year keep author least year author available train c new link exploratory temporal interpretability example component cp capture signature pattern publication author cp capture link factor mode third bottom list author conference combination early mid author conference discuss author trend mid relate primarily nice cp link predictor whether conference test year treat regardless indicate author predictor ct systematically use need determine observe entry magnitude slice form heuristic scoring first predict address well predict operate auc view presence imbalance link show predictor term predict link bar bar ct weight improve figure receiver operating clarity omit ct predict link method initially unchanged understand link predictor show cp achieve close less remove link test although accuracy seem order magnitude expect chance imbalance note ct bad predict link also observe among look even start give year ct ct link fast ct ct ct sec sec full evident reveal pattern section datum periodic pattern exploit set periodic play periodic predicting period length period ghz processors ram connection set entity year simulate day correspond roughly seven day might entity service service yahoo google service group people business service email service represent motivation temporal often service schedule may cache well business motivation traffic computer contact computer link could load balance predict link crucial matrix resp least note decide component entity choose strength pick length assume period train period use periodic pattern pattern correspond activity correspond different entity experiment show generate repeat adjust increase decrease neutral show column row train pattern tensor train make significantly train randomly swap select add percent entry instance cp large noise factorization predict generality e link zero positive trend approach nonlinear tolerance normalize gradient maximum iteration maximum technique obviously clear predict possibly week parsimonious prohibitive extremely difficult across comparison value train predict extremely randomly gaussian fit average recovery follow k assume vector ambiguity must condition permutation permutation well mention average yet cp ten temporal original blue pattern generally show line cp
show gain segment nearby lasso attractive ordering coefficient naturally order regularizer fuse solver difficulty optimization problem introduce auxiliary constraint medium derive bregman solve fuse fuse fuse fuse vector classifier preliminary experimental occur real iterative easy variety fuse minor modification aspect require equation elementary c involve nonnegative separately also theorem omit system fortunately solve efficiently ty x ty ty ty ty see ty matrix still solver energy minimizer subdifferential calculus differentiable subdifferential p contraction actually fact p get remark corollary fused exploit difference regularizer fuse solver deal medium fuse matrix split large fuse fuse lagrangian genomic array many solver efficient encourage increasingly classification procedure minimize usual procedure toward fast efficient solve million attractive large introduce fuse take placing coefficient assume standardize different lasso find encourage coefficient toward smoothness naturally real feature variation pattern order molecular order ratio dynamic inference gene distant exploit surprisingly application area wang fuse detect tumor array comparative genomic fuse tumor area fuse denoise social network quantitative trait strictly optimal solution computationally optimization one computationally demand solve lasso approach solve regularize problem include group elastic net lasso logistic coordinate fuse lasso loss regularization guarantee fuse name fuse notice et al pair observation develop fuse generalize fusion work general fuse lasso fuse paper propose bregman iteration solve fuse gain show compressed sense completion fuse reformulate split bregman apply organize lasso augment lagrangian fuse fuse lasso algorithm effectiveness section describe additional implement addition relax order along allow order arbitrarily g fuse error encourage sparsity variable specify toward fuse zero everywhere reformulate bregman convenient split bregman generalize lagrangian note lagrangian inner lagrangian finding saddle solution saddle iterative algorithm primal update base current lagrangian update relatively ascent step iterative problem solve objective induce involve term quadratic completely soft thresholde optimal efficiency iterative entirely minimization fuse analytically theory alternate minimization primal htp k convex split bregman hold furthermore regularization larger long choose fuse constitute special generalized fuse differentiable thus step quickly largely store minimal equation solve efficiently cholesky solve equation x z mention compute equation use step implementation introduce predictor identify primal stay precede matrix require solver split loss problem find optimal differentiable involve difficult hinge diagonal unconstrained reformulate function constraint derivation saddle iteratively update eq supplementary solve modification proposition prove q update yx get result proposition htp solve linear ty ty kb w k k yx property algorithm follow supplementary assume property furthermore whenever illustrate fuse lasso trial artificial application matlab windows platform core ghz fuse procedure frequently fuse design quadratic constraint package fuse additional objective solver implement implementation warm whenever evaluate compare path terminate stop convergence guarantee used choice choose parameter rate identify convergence select procedure certainly improve empirically work gaussian predictor outcome coefficient motivated coefficient change table cpu consistently test ten fold due second p plot cpu take solve figure cpu average run different parameter cpu success thresholding iteration figure solution thresholde htp htp equal use fusion solve art gaussian nonzero test p hour work work piece fuse fuse mass ms great identification protein application motivate illustrate fused problem raw mass site reconstruction equally spaced mass correction remove systematic
non dominate sense small parsimonious prior calibration elastic net selector associate predictor classical impose fundamental relevance keep within discuss application variable regressor microarray genetic deal poorly ill pose recently frequentist among paradigm demonstrate recently lasso literature induce question actor solution prior negligible effect jeffreys illustrate pay prior avoid problem consider variable purpose frequentist view regularization slightly great full comparison consider outcome dominate counterpart therein hierarchical frequentist real dataset conclude exclude vector correspond conditional variable symbol prior classical average observe regressor traditionally front fisher avoid would allow specification observational unit virtual pseudo fundamental feature prior improper probability uniquely improper framework allow improper simplicity reduce prior input often parameter center flat stress p rely namely model q p however impossible infinity eliminate influence jeffrey end information dependent amount observation schwarz criterion criterion g p connection criterion regression resort technique base particular nonetheless involve since author cauchy correspond augment form constraint must connection processing intercept measure specify used choice mostly jeffreys justify possible jeffreys prior indeed eq subspace span jeffreys leading jeffreys prior integrate detail model representation proceed close form p straightforward matrix explanatory predictor n predictor derivation estimate predictor exploit consistency factor prior depend despite argument location formulae jeffreys ensure invariance order would necessary center completely create location quite specific negligible situation location alternative consist exclude prior part center ensure present numerical popular method point selector elastic selection iy identify influential positive influential influential six dataset benchmark table approach variable naturally model averaging predictor l aic criterion prior g global base bayes take intrinsic hyper hyper exclude jeffreys selector net numerical parsimonious procedure overfitte fp solution method f scenario behave slightly slight tendency select somewhat seems tendency viewpoint model perform except note bic reject bad aic lead performance select achieve close systematically aic systematically notice mse compute well otherwise recommend one view ccc fp aic bic mse fp error htbp relative select aic bic fp htbp aic bic example ccc fp aic g lasso number htbp aic bic frequency bic fp relative frequency select htbp ccc fp aic lasso number aic lasso frequency oracle mse fp htbp aic relative comparison location impact last argument go relative distinguishing model example criterion summarize variable intercept otherwise criterion instead htbp ccc mean mse number htbp frequency replace moderate observation correspond aim percentage body regressor weight extend ten validate prediction parsimonious standard show mse stress compute bayesian highly remain open take daily eight variable concentration million response variable pressure height wind international percent air temperature inversion height inversion temperature pressure gradient mm original contain observation study observation bayesian opt five lack difference bayesian variable poorly selection parsimonious relevant perform point view
markov factorization respect factorize pa suggest across constraint respective subgraph pa gx implication independently individual valid give section parent follow clique sx vx gx sx pa gx corresponding function type graph parameterization complete parameter obtain I pa gx low introduce artificial direct operation add edge make child factor drop relation previous exploit probit gaussian generate singleton artificial figure illustrate describe section tie joint describe copula among dependence perspective put copula marginal joint distribution simply transform cdf incorporate encode return markov marginal dependence binary ordinal conditional function parent regressor fix basis among adopt copulas clique copula clique show copula plug become q cdf copula marginal maximize plug maximize via although likelihood consistent estimator substitute something feasibility hasting mh mh message pass scheme formalism compare fold validate probability dag ground truth non set uci repository ordinal parameterize frank copulas convenience average difference per r e driving alarm dag leave breast datum list introduce structure dag capture broad dependency broad parent small dag th predictive tell dag case log predictive significantly compare dag ordinal breast uci repository dag figure repeat procedure dag table perform use uci repository use fashion incorporate validation dag follow bi direct residual direct fix technique copula dag result validation fold bad computational efficiency alternate direct bi bi direct try fit fitting residual compare suggest dominate acyclic translate advance gaussian introduce family independence extend expect learn hypothesis prior knowledge structure play multivariate introduce inspire advanced structural procedure develop proposition factorize gx state except along parent let total say set x vx main result theorem induction probability accord must child acyclic except hence induction hypothesis marginal minor hold pa gx ix x order ordering accord respective subgraph fx pa gx valid non since vertex adjacent vertex pa gx ix side factor process repeat remain give cdf sx pa vx pa gx sx gx pa gx correspond cdf transform direct set marginally relation complete complete parameterization describe purely bi bi joint binary vector indicate summation contain equation interpret cdf transformation rewrite parameterization parameterization summation appear conditioning respective parent subset come necessity enforce independence path cdf clique accounting constraint hence construct different factor element enforce unnecessary understand coincide first figure specification marginal px parameterization come factorization fourth markov although level parameter coincide complicate extra complete reflect c cdf evaluate give generalization department statistical university college ac uk computational college ac uk computational college popular express direct mixed graph generalization much set conditional implicitly paper cumulative network copula propose construction encourage powerful framework encoding multivariate family acyclic dag undirected property dag monotonic sense know network allow flexible alternative direct acyclic mixed marginalization reading separation property practical bi direct model bi might much compare include obeys constraint exploit paper flexible approach copula literature review formalism copula summary graph clique px marginally relationship complete parameterization parameterization value cdf inclusion px x parameterization enforcing
dynamical give evolution relaxation regime angular momentum angular momentum different vary average change angular change slowly long lose intermediate regime enhance angular momentum evolution recognize relaxation angular radial take major uncertainty relaxation boundary especially intermediate coherent related relationship magnitude question dynamical momentum intermediate evolution angular suggest draw conclusion regard work angular evolution intermediate behaviour near angular momentum give brief review suitable angular evolution memory limitation object exhibit process define hausdorff hereafter exceed calculate dimension sketch suitable purpose self part self copy trivial correct segment copy rectangular cut identical give technique well consist measure detail length example measure give shall copy small copy general modify satisfied readily generalize self convenient curve check decrease total go broken going step term modify become determine simulate centre star semi major axis function final test star star star pick exact matter much somewhat resemble star centre uncertainty exist would around explain detail elsewhere field star potential include gr correction lead star individual star angular momentum consistent decrease significantly employ body star field adopt cluster gr parameter star axis gr behaviour nystr om discover part increase avoid scheme advance element correctly except truncation accumulate error star case treatment gr share last star curve record energy angular momentum star figure star plot curve fractional brownian angular momentum test measure star start feature long value gaussian change momentum unlike walk long decrease slope point determine accuracy star slope never evolution angular never slope reasonable vary star look evolution star expectation slowly star coherent short much star reach randomization dominate short randomization motion resolve intermediate angular enhance angular momentum regime rapid growth evolution brownian motion fractional brownian describe motion give generalization produce curve mathematically elegant simple easy physical brownian particle motion consist step duration choose increase decrease decide action sample step brownian motion walk small component brownian increment use repository average determine correlation strength length characteristic angular momentum long intermediate occur sampling match slope intermediate regime coherent slope walk walk start unit whenever take roughly walk show angular angular momentum near system test star cause back reaction cluster star mass spectrum apply possibly coherence potential angular evolve decrease randomization cluster relaxation gr comparison centre star mainly randomization would angular momentum body none mechanism angular momentum evolve evolution calculate dimension angular reliably regime key evolution angular result arise
share similarity occurrence multiple expansion functional whereas spline intensity link predictor internal covariate process develop package linear intensity expansion usage elsewhere focus theoretical likelihood dimensional setup treatment maximum show basis appear say penalize problem spline framework generalize banach general result similar result penalize maximum space solution infinite algorithm interpret reproduce term functional express integral functional consider trivially present involved property space reproducing filter addition homogeneous local vi take derivative martingale abstract always able decide likelihood martingale play count intensity theorem vi banach banach algebra linear functional observe belong measurable map limit hold continuity limit process process point intensity leave intensity require limit ensure wise interpret parametrize definition include continuous equipped uniform filter poisson present reproduce space give w filter analogy ordinary transform process terminology call inverse terminology evident negative strictly speak encode representation penalize play negative concave turn concrete give result suitably time log non second integral integral formula follow bilinear neither one deal derive derivative hilbert prove derivative log iterative formula put parameter stay formula derivative possible make continuously result formula play discretization eventually compute root smooth around root likelihood twice assume differentiable moreover root path second homogeneous move sample alone differentiable derivative differentiable martingale part equality side realization functional show counting case norm turn inner reproduce kernel moreover fix orthogonal onto hence give rise project penalize negative likelihood denote count minimizer span function practical estimation problem form q work trick outcome theorem q happen intensity however take explicit dimensional subspace log continuously differentiable explicit derivation consequence penalize solve minimizer minimizer must belong dimensional inspire gradient propose generic approximation minimize gradient differentiable need determine known literature fulfil choose h return continuously differentiable convergent strict convexity minimizer conclusion towards unique eq give brief treatment process intensity additive intensity function corollary linear functional equip v hilbert orthogonal wise minimizer negative likelihood span also generalize choose lead dimensional ordinary lasso implement model mark intensity likelihood parameter likelihood form case reduce separate example additive filter point example turn reproduce kernel hilbert q furthermore inner space reproducing reproduce onto order find integral description jump spline knot due see knot cubic spline mostly integrate enter r r function function polynomial motivated function specification structural algorithmic estimator result establish generality integral respect basis counting also expect integral negative observe reliable approximation penalize space integral establish functional jump trivially elementary reproduce requirement reproducing bound term term integration interval root derivative enough eq integrable useful straight norm though product kernel argument convenient continuous straight forward norm reproduce hilbert characterize reproduce evaluation cauchy schwarz since already integration linear continuous let functional precisely schwarz u h z z functional operator g g root continuous eq first consider z continuity follow embed proof continuous left limit strongly linear operator recall define outside mention gs gs g u gs denote translation act strongly unitary consider limit left limit let side prove sg sg sg I respectively hand side prove argument continuous continuous turn right limit requirement define moreover gs u gs z gs td functional represent hence combine conclude eq linear functional pg td pg td pg equality minimizer theorem result semi martingale give elementary gs regular weakly q verify check get right u g u u u gs u u continuous term conclusion lipschitz continuous since twice continuously sg u g
explicitly causal present finally amongst detect influence term contain measure interaction system prefer interaction causality equation causal little knowledge consideration fellowship dr foundation lead regression equation run invertible square invertible simplify extension trivial expand expansion repeat sum regression coefficient equivalent show partial may calculate matrix formula rearrange rhs simplify show prove rearrange multiply rearrange get follow immediately expand establish tr ga cd uk bn uk causality direct interaction causality univariate condition interaction take group ensemble establish causality interaction seminal causality support comprehensive theoretically causality individually specific illustrate multivariate motivate causal reveal type functional system mn keyword challenge many dynamical among acquire simultaneously increasingly aim inference system complement connectivity reveal dynamical analysis however interaction possibility reason usually incomplete operate typical fmri identify region fmri voxel voxel comprise change assess connectivity derive extract principal alternatively perform voxel approach voxel comprise range area economic biology principle assess causality causality causality particular g causality time autoregressive basic series causal incorporate inter manner predict future predict said predict importantly causality orthogonal causality among causality description generalize propose totally indeed recently appear numerically implication analysis section measure accord covariance matrix residual autoregressive total explore advantageous trace formulation transformation extend determinant important causality extend previously causal interaction show causality enhance g causality carry notion causal independence discussion summary use mathematical denote vector quantity matrix vector consider vector vector symbol transpose determinant square jointly multivariate covariance term random variable q identity useful g causality process focus use notation analysis concern thus contain comprise residual model uniquely specify zero regressor via obtain find stationary process z want autoregressive model process firstly lag lag lag useful version simply omit causality predictor variance regression notation multivariate write last equality formula variance know transformation asymptotically distribution central predictor long causality valid causality standard causality possibility multivariate square residual call multivariate causality note causality choice fit measure univariate residual uncorrelated minimize square nonetheless quantify reduce univariate autoregressive variance list causality generalize maximum likelihood z distribute large justify choice g causality henceforth appeal z unconditional q together cause cause identity subsection property causality simple factor transfer entropy quantify stem entropy proportional logarithm determinant conditional involve gaussian proportional logarithm determinant crucial equivalence justification measure causality entropy property causality development regression entropy extension causality covariance residual covariance linear account regressor formally ar general process behave conditional multivariate x pf partial covariance transform simple linear transformation matrix determinant find group causality measure causality unify rather arbitrarily define add component add difference think invariance transformation one angle preserve consequence broad transformation insensitive discussion one variable actually invariance mean compute observe principled component change value practical implication affect difference mechanism differently differentially context sensitivity connectivity content respective magnitude worth transfer symmetry group non transformation entropy transformation causality version transfer transformation predictor expansion depend decomposable product decomposition logarithm obvious expand combination expansion entirely sum present past entire variable past iv appendix univariate total multivariate help univariate predict help multivariate univariate support total univariate helps predict degree predict current plus help predict current whole predict implication high suffer stability univariate suggest expansion indicate extent present effect predictor enter equation stem determinant residual causality root lie outside lag exact exact lag lag since practice lag regression rely crucially trace density ii analogue analogue satisfactory trace causality remark g domain trace analogue form residual matrix simulate unstable process reject lag lag empirically achieve integral compute quadrature display confirm accuracy finite lag relative magnitude decrease sequence c display repeat confirm yield relative difference aside difference appear presence residual sensitivity group tr invariant restrict extend measure introduce exploit parallel order standard causality causality causality term correlation regression regression predictor causality covariance regression extend naturally multivariate rhs follow numerator denominator thus differ respective regression see always negative partial causality circumstance alternatively partial condition note section partial extent influence take explicit aim influence may influence strong relatively uniform measure partial affect predictor condition equal degree note express residual appear might understand proxy influence eq analogue trace partial covariance appropriately causality quantity develop trace refer problematic fail time f straightforward causal system previous define univariate condition dynamical system element element dynamic causal somewhat contribute globally potential various extension suggest various interaction principled scale size condition rest x r one predictor eq interestingly across currently explore could interesting predictor define average scale version measure progress density causality recently adapt operational system say g multivariate determination formalize along regression differ eqs multivariate situation element jointly self group activity group would g g activity macro extent micro extension consideration micro macro causal interaction ensemble perspective mechanic identification represent dynamical parsimonious extent dynamical extent candidate significant causal individually independence notion self independence useful description macro micro macro leave level description identify decompose causal within level finally iii characterize inter level standard causality measure originally totally residual term address residual covariance trace novel dynamical quantitative characterization novel independence analysis particularly biology may simple observed variable collection ensemble fmri voxel arbitrarily share similar eeg signal univariate meaningful fundamentally causality univariate important act influence cognitive behavioral since increasingly ensemble brain suit causal relationship onto principle brain operation important broad range decompose indeed acquire series approach multivariate propose measure residual variance provide determinant invariant linear transformation maximum distribute standard enhance fully
use say eigenvector e eigenvector cluster choice cluster bind distance incorrectly preserve span closeness canonical angle subspace span orthonormal eigenvector define angle affinity matrix ten eigenvector second affinity ten second canonical angle underlie fail coordinate justification figure coordinate eigenvector canonical maintain small underlie right affinity eigenvector eigenvector leave eigenvector difference angle still eigenvector local provide representation datum coordinate cluster compressed guarantee fraction entry measurement less ambient method distance compress completion perturbation eigenvector top eigenvector distance spectral define perturb first coordinate preserve original classification label membership match quantify misclassifie misclassification rate function handwritten project nd rd eigenvectors distance measurement often degree type perturb ambient dimension hide error matrix completion perturbation affinity membership unitary transformation compress sense measurement preserve perturbation sense measurement application collaborative computer wireless sensor network entry incoherence property reconstruct reconstruct distance preserve perturbation completion provide rigorous bound affect compressive generalize current perturb compressed measurement preserve span eigenvector close span show perturb unitary transform vi completion eigenvector cluster see previous perturbation eigenvector expand partitioning preserve small perturbation eigenvector column span perturb eigenvector eigenvalue similarly eigenvector write diagonal angle second require perturbation eigenvector orthogonal canonical angle space projection establish closeness span project eigenvector preserve form top unitary find unitary matrix find decomposition canonical space value cluster row corollary eigen perturbation thresholding incorporate eigenvector top develop perturbation coordinate knowledge amount error affinity many area wireless lose independent practice observe develop take inaccurate bound eq robustness perturbation let defined corollary probability give coordinate corollary technology lose corrupt costly incomplete fraction entry constraint solve nuclear convex task recover unknown know incoherence property obey resp resp sign rigorously property recovery completion completion local x kx obey high problem eq apply similarly multiply rewrite divide subtract simple ci proof combine corollary synthetic face handwritten digit distance frobenius datum synthetic dimensional sparse onto three nontrivial dimensional image compress sense measurement keyword keyword group wide balance feature create unitary transform level figure middle method compressive require perfect classification face pose view order experiment database head face fouri compression transform ideal capture desire difference transform level degree figure ambient measurement second capture underlie order often signal may measurement signal product fourier way image image face pose compressive spectral coordinate use range misclassification trial plot cluster image compress misclassification full dimensional standard spectral face addition grouping c image color classification label misclassification transform would level proportional show cluster handwritten compressive maintain make cluster first h handwritten digits data nd rd graph form distance random use varie stack row increase misclassification combination two completion compress measurement span completion traditional entry observe apply completion eigenvector cluster rank three measurement take spectral coordinate ph apply mathematics university california mining harmonic process diffusion mine ph mathematics spend university department california stanford technical institute interest harmonic digital signal theory book serve corollary spectral widely extract efficiently sparse partially combine cluster rigorous work spectral clustering incorporate eigenvector track compress sense affinity perturbation multi require affinity naturally number example mining become fast grow research topic mathematic spectral meaningful grouping similar eigenvector structure signal hyperspectral costly among work organization extract preserve compressed inaccurate measurement perturbation eigenvector compress affinity span eigenvector unitary assignment perturbation compress sense completion widely compressed noisy entry satisfy costly clustering compressed possible vector technique signal compress sense derive compressed turn lose numerous task portion understand impossible alone lack suppose stack contain use usually error propagate rigorous depict image cluster section h image range pose face
th trade suggest al european second structure comprise connect neighbor though area mid subsequent erm regard join common switch uk exclude uk regime lead party base erm adopt display market integrate possibility join european somewhat return exclusive regime use show minimum yield well return covariance assume per day regard show portfolio high changing thereby incorporate induce although mixture hide markov emission model extend nonparametric kernel improve rate high problem nonlinear generate outcome regression derive computing generalize regression implementation mixture model gibbs update group one computationally intensive slow plan explore near alternative particular move develop monte carlo algorithms feasible number however dimension experience walk inefficient base heuristic partition difference come graph predictive numerically recent development acknowledgment west provide ar support nsf dms brief auxiliary concentration parameter concentration auxiliary mixture l case lr full l introduce variable graphical independence lead turn tackle emission divide heterogeneous consider infinite mixture allow estimate illustration exchange fluctuation pre demonstrate provide extremely keyword markov inference size parameter challenge graphical deal type ill pose problem enforce sparsity graphical vertex node indicate covariance popular range finance multivariate conditional zero covariance relationship copula graphical model decompose joint pairwise copula however alternative copula countable mixture expectation consequence motivation investigate mixture often microarray encode conditional dependence information pathway implicit expression pathway individual different linearly study economic exchange economic block understand interact evolve membership mode interact mixture tool interpretable challenge implement determination estimation partition data grow exponentially mixture graphical mixture search specific determine number component address nonparametric mixture emission time employ since focus sampler integrate problem associate approach greatly avoid explicit representation state discuss outcome combination binary auxiliary begin bayesian inference graphical introduce graphical move mixture specie mixture markov emission illustrate section research multivariate mean decomposable graphical zero miss definite entry dependence relationship induce follow factorization clique correspond subgraph clique joint conditional mean prior precision g wishart lebesgue trace product clique wishart distribution normalize decomposable necessarily wishart factorize clique hence normalize subgraph associate explicitly prior variance specification identity result prior weight wishart likelihood k distribution jk become wishart distribution similar argument g assume decomposable calculate numerical approximation computation consider come model probability fully bayesian specification mixture heterogeneous involve practical practice know assign involve use reversible method inefficient dimensional alternative mixture q denote put define discrete say process dp distribute identically refer break dp appropriately useful dp prior integrate among sequence distribute distinct first model imply give kind grows increase cluster assignment also update follow neighborhood decomposable neighborhood connect graph graph differ candidate unchanged see improve cluster time cluster carry adequate mix interest n inference cluster predictive burden mean mixture easily sample conditional denote assign observation dp concentration parameter inference try datum auxiliary idea mixture gaussian graphical similar exploit analytically sampling model dirichlet combination extension first replace process species exchangeable follow specie distribute sample accord predictive weight satisfy species model include poisson normalize induce dirichlet imply give precision grow application consist exchangeable sampling baseline discount modify dirichlet modify weight reflect improve develop approach linear demonstrate assumption application markov emission correspond hidden hmms context hide trajectory trajectory evolve transition probability dependent infinite generalize hmms number process mixture model particular allow control evolve account nature transition state sub transition base dp precision update empty create trajectory kind auxiliary update simulation arise brevity result run representative run cluster connect cycle precision belong remainder compare dataset burn process run attention restrict cluster use full space decomposable full panel full observation incorrectly restrict graph properly dramatically correspond capability model translate edge exchange dataset consist daily observation five european european
slot arm index clarity algorithm logarithmic note feasible practically example polynomial computation traditionally armed bandit play expectation arm work turn bound regret consequently arm grow polynomial policy logarithmic like upper valid state chernoff hoeffding proof counter initialization period update follow slot period two happen play time pick increment arm pick element play arm equal indicator arbitrary eq indicator false arm pick get arm element h j jt chernoff similarly equation false l therefore policy consist binary reduce arm ucb generalize formation infinite set maximization least conclusion chernoff hoeffding need assumption optimization weight many associated matching constraint formulation weight learn regard propose storage regret logarithmic problem weight bipartite matching algorithm cognitive channel channel operation channel user due secondary potentially channel throughput spectrum channel period denote model mean linear ht minimization accordingly algorithm eq policy exponentially bad bellman short cost armed formulation path subgraph minimum cost constraint bellman time objective span section channel cognitive channel secondary throughput player show storage naive fig policy lot throughput I time high grow policy grow pt channel reward arm time slot total brevity policy problem modify pick play arm accordingly upper regret policy proof expect reward arm k expect arm play hold pick minimum arm note q arm play observe substituting regret consider reward unknown random policy ucb regret policy store alone storage combinatorial solve policy also computation short insight future question achievable conjecture intuitive rigorously unclear whether bind well context cognitive policy independently would policy open finally great tackle prove convex edu armed bandit dynamically operate yield reward accumulate player know mean arm accumulate policy policy regret grow multi bandit yield reward linear grow polynomially number exponentially policie storage generalization broadly applicable many formulate optimization problem span computation multi armed mab classic reward identically policy arm mab problem fundamental tradeoff hand explore order observation exploit gain immediate reward apply wide domain include network fundamentally combinatorial environment broad use armed argue barrier wider limitation basic formulation entity deal exponential setting exploit arm dependency handle provably well storage computation unknown action element obtain broad combinatorial multi armed bandit ucb yield regret storage approach focus maintain computing observation dependency base storage directly rather computation substantially selection policy linear storage essentially tight apply asymptotically time straightforward guarantee reward propose deterministic combinatorial np programming special interest thus property different policy readily extend regret well applicability bandit reward cognitive short minimum far work time allow random provably application combinatorial problem arise algorithmic economic finance operation engineer related armed bandit linear reward computation path span widely useful various combinatorial show extension large section contribution point paper infinite horizon arm bandit generate unknown linear logarithmic show allow compute policy logarithmic formulation work paper reward un restriction finite ucb regret rather exploit arm paper ucb therefore decentralized policy develop policy distribute player operate key focused arm treat dependency dependent case arm model assume present theoretical regret similarity regret difference reward show upper regret sum static across bind set work linear reward storage linearly regret regret bound among first combinatorial optimization grow polynomially restriction matching paper formulation system index evolve random restriction support normalize require unknown decision refer represent policy n arm component reveal instance discuss could eq reward arm indicate parameter arm
asymptotic reference think develop expansion sequence occur cf drive behaviour growth understand serial evolutionary describe overview phenotype stochastic model use feature age sequence dna relationship patient cell site patient examine problem part suppose record molecular member ask observer organization paper work homogeneous remainder asymptotic allele simulation asymptotic provide motivate sample dna patterns division switch site vice versa pattern obtain represent string binary patient cell two pattern neutral site measure cell cell pattern cell circle way simple consider information detailed body number allele sample third row number different sample cancer column distribution variation qualitatively cancer far cancer allocation among cancer cancer next section variation mm allele r allele identify mix collection typical pattern cell evolution mutation describe review provide population cell assume cell mutation assume cell division mutation sample neutral assumption express classical one neutral combine data allele count column depends denote write vector count cancer sample q cancer maximum solution give entire combine consistency see observer parameter interested assess goodness fit sample joint distribution count see observer chinese restaurant statistic relate multiple observer simulate sample cell allele count chinese restaurant crp individual individual type label copy individual assign individual low integer represent indeed subsample replacement sample count appropriate size sequentially remaining require say arrange crp run reject since allele frequency independent freedom determine observer sample observer course use statistic suggest see aim observer know answer aid understand evolution understanding variance number statistic average discrepancy use comparison combine th th crp simulation suggest anomaly underlie leave consistent investigate homogeneous crp homogeneous interesting suggest adequate attribute side percentile percentile share inconsistent homogeneity observer many start construct range empty proper rejection nan homogeneous reason mutation growth might apply mutation likely far functional homogeneity approximation variation begin nz sample label observer sample distribute irrespective let observer difference motivate approximation suitable poisson mean give combine sample observer observer number group allele observer observer present time sample component take approximate note allele black eq differ label independent interpret number assignment count distribute compare suffice prove poisson random approximation see proper th observer individual frequency combine observer individual joint need independent multinomial trial
without cover maximal cardinality code hx large n proof follow useful derive compare cardinality contain pm p support union see follow classic polytope every face simplex statement face minimal equal equal tight simplex hierarchical interaction fully accomplish believe provide well note set cover face correspond face see convenient idea remark cyclic polytope vertice distinct moment xt nk cyclic polytope k k cardinality satisfy rough bind set small ap p ap p p b parametrize r see span k vertex less simplex combinatorial cyclic polytope polytope cyclic tuple iff combinatorial cyclic polytope may criterion odd pair hence begin proof correspond r f f p let maximal bound cardinality ball radius binary hard sphere cardinality simplex x xy thus z grateful comment grateful valuable thank read grateful mis partly support fa question theorem mathematics university decomposition every exponential exponential face lattice write distribution combination number arise family measure parametrize partition family convex give express convex closure topology product random outer tensor record expressive understand whether count combination small connection treat mixture independent non identifiability largely variable equal soon negative sum generalize distribution large expect I particular model unit interaction smallest represent weak variable give bind small mixture distribution interaction set contain distribution aspect support family forward analysis special iff probability stand consider hull closure rise small set family convex support discuss treat interaction family call simplex variable define set vertex unit natural correspondence subscript paper hold call statistic distribution simplicity always sufficient depend depend row exactly strictly closure exponential distribution nest k kn choose hadamard realize sufficient image polytope hull point expectation detail polytope define affine combinatorial face together union face affine transformation preserve combinatorial type iff strictly statistic whereby line pair set except point probability two support red right support realize hull statistic example assess expressive different idea relate exponential minimal cardinality packing packing require follow whereby cover packing subset place exponential support within appendix lemma well implication distribution contain closure jx ks q x moment map direction px third item contain small simplex relate theoretical problem cover see polytope hull less simplex polytope face support interaction polytope dimension compute lattice g face face cover family call closure exponential context boundary point lemma
employ possibly parametric g gaussians researcher computer motion review geometric argument motion lie subspace algebraic method generalize principal sc affinity subsequent sec identify close classic local subspace recent affinity induce et al sparse subspace cluster ssc independently reconstruction sample rest coefficient regularize study theoretic ref liu et affinity employ nuclear minimization nuclear minimization surrogate semidefinite field effort sense generalize sparsity drive collaborative localization name cut solver accelerate lagrange multiplier alm see review learn symmetric reconstruction low without behavior learn affinity justify learn constrain affinity semidefinite lrr surprisingly connection canonical lrr formulation exactly characterize spectrum successfully uniqueness lrr lrr psd state lrr psd robust like lrr computational cost nontrivial elementary nuclear regularize semidefinite elegant aspect provide equivalence lrr lrr psd establish uniqueness spectrum lrr psd efficiently lrr notable difference throughout necessary framework affinity subspace next equivalence lrr psd lrr take briefly spectrum version lrr psd lrr proceed present tackle lrr psd robust lrr psd bold capital bold letter scalar g space norm induce value norm singular norm generalize vector concatenation sum besides inner induce alternative frobenius spectrum eigenvalue square collection real symmetric semidefinite cone say semidefinite simply requirement symmetry asymmetric matrix respectively notation choose subsection whole rank general invariant matrix unitary compatible value lie family norm apply norm singular fact place next duality nuclear characterization always achieve homogeneous formal dual nuclear duality take together characterization imply piece review np minimization envelope pointwise lower nonconvex convex envelope relate envelope pp sec envelope surrogate matrix prove mild optimization equivalent surrogate build lrr lrr tackle segmentation learn minimization surrogate incorporate semidefinite valid argue liu observation since affinity diagonal sort state favor segmentation reveal theoretic e exposition somewhat critical characterize lrr psd lrr obviously corollary immediately psd equivalence lrr lrr psd exactly minimizer semidefinite obey lrr psd lrr psd lrr psd lrr show objective classic nuclear get corrupt suffice spectrum property translate respective hence lrr psd however try employ alm see convert use generic form partial alm inexact alm routine fixing norm solution update update next show close basically asymmetric major cost decomposition counterpart robust partition hold similar solution take whereby ref remark theoretic proof derive convex unique minimizer remain admit spectrum cast unitary nuclear recover restrict since strict decrease reduce r programming obviously separable suggest form conclude proof translate since argue unique three ambiguity sign ambiguity freedom great sign cause value eigenvalue readily view I last problem repeat arrange act contribution rotation namely cause building devise real follow able thresholding unique take whereby element wise theorem ensure symmetry irrespective lrr psd either size svd convert eigen eigen eigen size include matlab evident eigen stick practice unstable problem solve systematically throughout experiment td I I dataset raw stack dimension heavily corrupt key equivalence lrr psd singular identical ref associate td hence simulate setting lrr psd lrr gradually increase regularization version intuitively tend clean present evident pass eigenvalue identically rest confirm empirically towards clean lrr psd please color psd lrr perturbation setting confirm although explanation produce thing evolve sense corrupt nuclear robust lrr psd spectra please argue noise essence objective version instead adopt totally random realistic variance collection percentage corruption total evolution corruption see well form speed psd conventional sc obviously c gauss sc ssc lrr lrr psd acc acc acc table present via affinity sc removal lrr psd lrr relatively virtue corruption removal affinity another denote lrr c lrr time lrr eigen place pursuit insight work affinity recently lrr psd towards understand behavior denoise scheme lrr lrr psd scale produce operational computational svd eigen large nuclear practical purpose work project office medium national helpful comment suggestion manuscript fact nuclear eq subset program objective constant formation solution readily definition interactive digital medium national computer national work employ high dimensional structural approximately lie affine affine subspace mention therein manifold considerably sc fail fundamental problem affinity advance one process affinity sc enforce semidefinite semidefinite
table regression use frequency ft motivation study reduction bandwidth cause increase surprising ar mainly absence short memory wider bandwidth quite control reduce computed contribution affect strongly increase spectral great extent bandwidth decrease bandwidth ft surprising nature ft correctly ft bias ar bias misspecification example misspecification end table estimate fractional correlation also calculate ft ft ft ft display show similarly ar counterpart correlation memory correlation increase coefficient ft method present superiority performance c mse ft ft ft ft conclusion performance c mse ft ft ft ft ft ft illustrative form observe box plot respectively table standardized claim fairly n mention property comparison would investigation model deal misspecification simulate part short misspecification relate specification memory ft estimate period misspecification ft ft estimate highly biased surprising bias non ar table misspecification non affect significantly ar stationary region part contrary contribution affect slowly decay lag ft method previously consider related coefficient c mean ht mean mse mse illustrate usefulness fractional third pm size two value display simple fractional estimate affect structure save fractional parameter artificial adjusted example analysis fit artificial request analyze short memory period present present request concentration daily pm concentration region greater comprise population million range throughout year average month period month month raw measure nd show autocorrelation autocorrelation figure plot series characteristic physical phenomenon expect datum since slow correlation lag lag multiple lag period process fractional parameter run period fractional estimate use tool section secondly obtain achieve ht pm display bandwidth use table estimate correspondingly nan reject table least fractional stable non large contribution fractional stationarity confirm max var var ar model ar infinite nearly scale autocorrelation adequate describe seem significant accordance memory identify suggest anomaly residual request correlation fall inside boundary request deal fractional parameter multilinear parametric estimator consider empirical general give estimate competitive presented implement sophisticated usefulness fractional daily pm acknowledge partial grant de suggestion part centre paris thank kind proposition ct department paper explore long property fractional one period counterpart stationarity series long accuracy investigate estimator pm estimation classification keyword fractional long memory common economic autoregressive phenomena period well know persistent long period become zero follow ar process also polynomial belong suitable fractional modeling short domain lag property frequency time non function spectral long memory model spectral memory ray usefulness model allow fractional paper gray france al flexible economic activity discuss test extend fit short contain introduce call publication property memory estimator multilinear guarantee carlo compare known method show multilinear regression estimation parameter however exhibit focus estimation one two period long ol fractional zero mean memory density generality even process next ordinary ol estimator example ols estimator normally distribute comparison estimator parametric approach misspecification problem ft estimator investigate final remark even difference binomial b ks long specific series fractional specific among fractional see ray return frequency filter odd term appear expression equation q fractional period process invertible spectral density spectral assume ij ij previously note filter order stationary process fractional satisfy property theorem prove deal simultaneous estimate memory et al reference well asymptotic unbiased inconsistent spectral series ol estimator slope otherwise integer ef ef k expression spectral variable center least run otherwise local center center noise constant procedure estimator introduction
respectively fr system bivariate case precisely reasoning multivariate fr hoeffding apparent obvious dimension presence variance bivariate construct first standard deviation correspond distribute random simplify notation construction present independently hx hx h quantity f ce also non decrease algorithm marginal correlation coupling note last paragraph work transformation preserve determination among know place perspective great entire distributional family remain marginal maximum derive follow dx distribute dx either double proof appendix valid choice marginal exponential worth many bivariate another recent concept constructive limit integer variable obtain output notice minimal correlation numerically correspond c density symmetry stand x minimum analytically minimal ii h minimum reader paper extension start identical pairwise correlation coefficient although applicable depend necessary matrix principal equal identically distribute correlation construction correlation choice dimensional determinant factorization requirement impose algorithm accommodate accommodate negative add restriction figure view plot distribution marginal region middle plot restrict give generally sign algorithm admissible number recommend reduce comparison concrete implementation algorithm thing chemical reaction beta describe exist compound copula correlate beta dimension form distribute illustrate fact cdf random scalar analogous quantity variable marginal bottom simulated note consider htp algorithm simple generation copula simpler require whole specification exact copula random generator desirable fast applicable accommodate range major determination range theoretical se emphasize low bound actually family commonly common distributional example straightforward ht independently stop semi one applicable produce w jx introduce huber li david discussion gm nsf dms exponential result section dx ij x xx I integer equal series converge add ii ii lemma section conjecture paper generation pre base fr simulation calculate example illustration detail implement dimensional positively beta copula algorithm beta simulation distribution specify margin back equilibrium fr evy fall hoeffde excellent overview al et al correlate pre specify play role stochastic encounter field finance environmental physics weather increasingly species dynamic far coupling copula become widely generation sample specify marginal dependency copula methodology rely characterized limitation surprisingly generate coefficient attention simulation notable work bivariate generation method present distribution mostly marginal common distributional gamma marginal
contribution monotonicity technical device idea yu maximization monotonically suppose increase objective choose call design general fisher provide assign design optimality seek represent choose log determinant view sample closure convert round chapter I use characterize maximizer generalize optimality mass example optimality general confirm satisfy mild parametric finite least generate assume limit maxima iteration establish yu omit question choice similar cost design yu carry treat iteration basic roughly basic slow al criterion theorem assign concentrated around adjacent design convergence slow potential convergence yu propose combine fast monotonic multiplicative et ingredient strategy exchange adjacent optimality investigate finding extend identity cauchy iw nonnegative coefficient eq write q multiply jensen apply completeness fix equality jensen proof finite finite nonnegative q integration yield form relate long hence relaxed inspection monotonicity finite remark yu layer prove partly grant university california author david computing multiplicative algorithm optimal design maximization principle multiplicative monotonic
project mmd mx xt dy sub moment probability class domain unlabele coefficient jensen one therefore k bx x allow keep separation invertible let vector onto space w tt pp pp lemma prove let ik x b set sub include index include b depend substitute assume line line g sub variable term k moment lemma l c l hold statement psd moment matrix orthogonal db q last equality fact ac il margin govern dimension conclude characterize rich tackle characterization rule learn source specifically tight characterization margin focus bound provide sample excess get dy understanding aspect rule compare upper bound often true sample tight lower bind exist source tight word concern g radius low support also precise characterization dimension determine actual complexity govern calculate rich light tailed distribution gaussian bound sample achieve bind concerned dependence desire obtain tight error contrast classical typically tight recently show learning example rigorously establish regularization discriminative novel tool believe work source least certain indicate sample complexity classical provide sufficient class distribution criterion see provide lower know learner specific learnable david set hypothesis line bad case vc balanced tight also distribution linear parametrize denote misclassification ds learning margin whose test concern size denote sample margin minimization characterize tail learnable restrict ensure tail direction sufficiently rich family distribution require restrictive namely coordinate far course multivariate sub extensive bound exist gaussian moment focus instance distribution bernoulli regard constant space mention introduction minimization term alternatively dimensionality tight rely respectively tight average dimensionality create converse arbitrarily high average option try trivially variance situation minimum dimension average integer dimension limit small example coordinate eigenvalue k k eigenvalue upper useful relate adapt margin establish follow prove proceed establish dimension term classic quantity prove upper minimization dimension column matrix space project project label similarly yy tm onto dimension ty ti tw yy w w j uniformly z I fashion corollary support sub variable decay appendix bind upper answer need complexity specific closely learn closely probability formulation precede dimension converse relate minimal complexity setting w dy involve existence equivalent iff margin margin margin correspond gram matrix condition denote th origin xx xx distribution sample apply independence hyperplane regularization homogeneous linearly require observation generalizing point thus use lower learn bind specify sample example distribute theorem let fourth asymptotic calculate provide low variance adapt dimension conclude lower bind quantity control conclusion derive however asymptotically highly separately family small provide survey sample distribution coordinate follow result also distribute proof find sub sub coordinate moment draw constant integer margin diagonal g whose small probability coordinate divide problem separately case assume draw theorem x x dm distribution draw hold depend conclusion influence conditional complexity highly easily interesting margin ng relevant irrelevant compare bound
marginal true probability compete section derive expansion marginal notational drop subscript glm vector response give regularity laplace taylor expanding exponent long bound meaningful derivation rescale design matrix n later determinant play simplicity take condition establishing asymptotic normality regression become align likelihood expansion satisfy r r prior n technical apply sensible normality regularity r bic compete bic misspecification second bic correctly specify expect introduce semi principle kl principle consider principle methodology compete fit glm index minimize divergence glm vector conditional principle proposition ig principle index divergence kl response true combine strength well drop subscript expectation nr expansion semi principle compete become become particular give negative penalty misspecification symmetric misspecification index natural interpretation regularity asymptotic glm glm approximate regularity n natural misspecification penalization misspecification expect ni implicitly refer cubic independent fit order comparison frequency regression aic bic outperform ccccc ccccc ccccc aic bic bic information simulate design independent simulate copy consider well apparent residual ccccc ccccc ccccc size criterion bic aic principle index well principle kl expansion bayesian principle family principle generalization natural penalty dimensionality misspecification respectively advantage correctly specify model impact analysis reveal covariance suggestion discuss take goodness misspecification possible introduce scope topic suffice da eq full proof easy uniqueness minimizer solve conclude q observe occur function maximum interior locate taylor segment n n nz schwarz lyapunov yield normality complete separate define n retain lagrange attain de denote kronecker delta idea expansion term imply order attain de along conclusion log likelihood easy concave attain value expand segment r entail key deriving list whose separately provide nj asymptotic expansion pick follow since fast rate claim q prove q give q inequality since remain observe n subscript follow proof entail complete definition variate distribution variate regression use formula least denote hadamard interesting true linear precisely involve due misspecification term maximum matrix quasi ignore attain condition remark university california university model classical principle kullback leibler lead criterion misspecification true true family principle combine strength expansion semi model new maximum penalty dimensionality misspecification directly demonstrate newly correctly specify principle model misspecification leibler principle principle model article explanatory desirable produce sparse involve subset one enhance interpretability fan variable problem compare predictor classical principle kullback leibler principle principle aic aic bic selection book account development aic performance study aic bic asymptotically true bic compare aic kl histogram asymptotic aic principle estimation measure absolute selection penalization wang li liu aic bic parametric priori fit maximum statistic wrong broad generalization aic bic model well among space set know predictor truly fan li fan family contain true neither aic bic misspecification discrepancy fit family true potentially helpful paper expansion several principle result semi sum maximum log misspecification independent observation extension classical rest organize normality estimator expansion principle principle present numerical illustrate propose methodology discussion implication necessary commonly foundation derive selection principle section white systematic treatment unknown function response entail ij generalize work z family contain glm set differentiable positive full two vector rewrite eq quasi concavity argument clearly maximum theory maximum white divergence misspecification observation kl density introduce upper kl divergence throughout paper specify regularity e follow divergence divergence unique solve entail f theory play quasi model estimator model include huber huber condition use misspecification extend maximum likelihood regularity condition n eigenvalue cn n small ny consistency intuitively neighborhood work far importance sampling target correctly outer product form normality row n norm specify normality conventional theory foundation propose selection methodology simplify technical presentation asymptotic analysis show asymptotic current g fan penalized grow polynomially glm estimate denote hadamard practice construct g bootstrappe treat square value study principle selection minimize kl show choosing maximize lead seminal compete connection
vector result experiment present middle panel already produce fairly vector believe remarkable another become panel cc satisfactory variance stable projection panel shannon size carefully choose good trade gm cc estimator projection right panel gm panel stable right panel stream strict technique entropy additive th accuracy study even entropy coefficient determine analytically know base skew recently th moment stream cc geometric geometric unfortunately still impractical prove cc estimator clean practical word improve stable geometric harmonic algorithm roughly extensive could accurate shannon even scale ten devoted stream tb database area g reach scale search typical source stream describe signal increment restrict strict suffice phenomenon relax strict moment stream q relax strict e hence strict e g web network shannon generalizations shannon enyi denote respectively enyi entropy converge shannon entropy enyi moment shannon entropy shannon entropy estimating numerically verify extremely let enyi clear frequency sufficiently perspective enyi entropies clear proportional variance course closely suppose complexity standard argument drawback provide shannon initially likely impractical know projection indeed exhibit traffic stream effective measurement network crucial anomaly diagnosis measurement metric shannon traffic goal describe histogram interested measure source service attack representative anomaly attempt computer computer resource service machine request attack typically site payment site attack statistical traffic certain since shannon entropy suit history anomaly entropy measurement detect attack low traffic time one traffic store stream recent devote shannon web search big analyze search use million triple particular representative stream history devote wide spread use shannon entropy entropy train approximate heavily computer science study algorithm space speedup processing note moment compute counter property maximally skewed random projection provide harmonic harmonic algorithm empirically another precise theoretical neighborhood neighborhood moment well fashion geometric adequate enyi entropy entropy contain harmonic provide geometric well note fix harmonic proportional harmonic adequate cdf increase g conjecture figure curve close approximate cdf dash curve lemma basically propose variance care random cumulative stable projection interested theory cdf see appendix compare mle addition considerably preferable prove asymptotically prove asymptotically use know complexity order normally tail least ideally really present bound tail estimator tail appendix complicated gain people even compute tail present follow formulate facilitate result bound write series rewrite necessarily word tail bind present together form lemma numerically k replaced prove largely overlap analytical expression actually analytical estimator actually fact intuitive small fact approach eq demonstrate sharp model stream stream shannon entropy application detect anomaly approximate moment even efficiently approximate frequency moment stream heavily science algorithm impractical shannon achieve bind base maximally skewed projection compress count impractical truly proportional previous algorithm entropy must note appropriate define j lemma algebra basically delta statistic careful make carry taylor take evaluate order moment eq property gamma eq monotonicity infer proof convexity convex suffice convex q eq eq tail minimize look tail minimize thus solution eq derivative prove prove propose compress geometric cc entropy algorithms estimation accuracy
train return reconstruct second spike fire hz firing course spike slowly presentation stimulus fire figure first panel firing depend spike history history select bit bit spike stay spike stimulus stimulus external immediately cause thus spike spike stimulus spike happen represent stimulus match stimulus history inaccurate panel fire stimulus presentation compare square discrepancy know external apply spike stimulus priori base something random firing panel panel wrong fact exceed indicate external stimulus quantify relative present post stimulus correctly internal stimulus strongly influence dynamic internal inaccurate us gain stimulus reconstruct second record iii discover state bit bit plot state randomness reduce resemble fire entropy low description suffice exhibit spike spike hz spike move subsequently spike neuron display spike triplet general spike move show neuron history predict spike spike statistic two reason neuron short due low firing second long present train emphasize parametric structure third respectively entropy spike entropy vary reconstruct spike train neuron bic select giving internal rate residual randomness bit fire history stimulus chain firing upon spike subsequently state represent external stimulus neuron spike experiment spike stimulus fire entropy stimulus average look however something complex place wrong stimulus entropy stimulus agrees drive complement different fire instantaneous external rate imposing want neuron encodes covariate encode identity existence uncertain pre cognitive state still use change population complexity examine macro entirely fire curve complexity array mutual apply calculate mutual spike train causal advantage causal represent behavioral pattern spike process spike calculate different neuron coherence reveal directly spike way spike train traditional analysis rigorously structure discover go beyond describe observe describe underlie spike also change thank valuable discussion valuable difficulty markov prediction time create filter strong dynamic linearly iid additive process amount maintain posterior update bayes call state whole unnecessary time transition remain update allow possibly chain circumstance go exponentially period useful place understand computational firing neuron considerable easily merely reconstruct cross part important check validity bootstrappe somewhat strong goodness test rescaling interval integrate describe rescale follow kolmogorov rescale kolmogorov ks cross neuron periodic firing spike set second stimulus fire bootstrappe stimulus fire largely fall stimulus panel rescale plot dash ks plot largely fall within bound stimulus stimulus show rescale stimulus worse surprising cause technique generalize reconstructing obtain perfect estimate inherent mathematically expand influence state mapping observable right statistic test distinguish set sequence idea state uniquely probabilitie produce input reconstruct future history spike g otherwise entirely computation statistical output train characterize spike train time quantify randomness show algorithmic content spike exactly describe minimal spike describe statistically randomness spike generate residual spike quantity regularity reconstruction analyse complexity spike train record device form one know activity inference spike train determine neuron channel capacity neuron give spike quantify randomness say produce throughout theoretically effective process reproduce train bit need reproduce describe noisy rigorous yet computational structure analysis spike train priori identify markov spike train train define computational letting quantify multiple group minimal conditional markovian spike train minimal capable call hmm splitting consistently paper use spike train history dependent familiar analysis analyse capable capture contain quantify spike relevant future reproduce effective statistically algorithmic information content average algorithmic content splitting complexity internal exact state residual randomness generative first quantify spike train precise functional version determine neuron require description method must spike quantify extent drive force simulate experimentally neuron train measure treat spike train binary divide equal step typically resolution structure present spike train program spike information need program quantify use minimal optimally predictive hmms reconstruct minimal computational predictor use available history limitation state group past activity spike train member predict construction ensure markovian spike train therefore like hmms graph node direct label symbol correspond spike average state markov chain idea figure simulate hz train second length iid train figure period spike period hz extra neuron hz state spike state spike equivalence past define period spike manner neuron proceed possible rest divide subsection theory consider understand spike train reconstruction spike priori discover spike notion namely statistical interpret reconstruct reconstruct response foundation causal sufficient one predict theoretic parametric predictive share optimality statistic actual statistic every statistic sufficient statistic summary retain basic statistic context spike train minimal sufficient statistic predictor minimal statistic always turn mean original homogeneous state causal stochastic minimal spike spike train statistically causal infer spike causal maximize spike train alone minimal empirically equivalence class distinct future sufficient statistic sufficient meaning minimal quantify observe causal recursively appendix find statistic spike hide state state cluster preserve length truncate infer building recursive long prediction rely sufficient future find save detail treat available program treat identically causal add state suppose alphabet statistic string map although contain multiple basis piece change conditional check kolmogorov ks fall right match wrong total l plausible irrelevant limit statistically reject sometimes complex nan start ii successive reach end ii sufficient state transition technical condition number discriminate traditional maximization bound use series knowledge structure reconstruct likelihood bic help causal grow increase datum aic bic markov chain class spike spike sequence spike state update recursively start state fix initial state use independent range grow fast next give symbol estimate limit less entropy output bin actual large see extended neural number hundred page inspection develop reduce spike probable remove appropriately transition probable train complexity sort find least probable state incoming edge remove keep keep state keep state stop transition merge reach repeat generate observe spike accept lowest iterate remove impossible chose gave choose show bic want check wise confidence checking coverage reconstruct check simulate spike train state probability forward inter spike pointwise pointwise often rapidly correct lie outside sort cross rescale algorithmic sequence comparison realization process algorithmic coincide entropy fact completely shannon statistic key determining algorithmic dropping term separate represent randomness spike quantify term intuitively hz figure zero state either probability contrast six need describe train period quantify high kind second quantify bit describe account spike capture train spike approximately two computationally represent generating process needs pick bit process always stay next symbol iid randomness total informative future bit randomness transition uniquely spike randomness stay spike need transition contribute bit reduce period spike less time fire spike iid bit rate update put spike fewer entirely quantify mean complexity complexity entropy quantity previous entire complexity entropy fire entropy variation stimulus fire probability vary spike stay invoke ergodic arbitrary integrable firing rate function randomness time entropy stimulus stimulus presentation analogously entropy calculation stimulus entropy time spike estimate appear definition interestingly outside average predict section external firing filter pass predict fire incorporate prediction simulation probability depend neuron rate generally external g currently formulate represent external precise spike train external
consider study communication sensor design sensor quantization rate scheduling good blue quantization decentralize multiple channel fc consider binary decentralize base communication communication receive considerable conclusion study terminal code popular scheme quadratic network access channel schedule transmission study show transmission efficient communication channel optimal orthogonal orthogonal law transmission result indicate source transmission channel access decentralize communication sensor code fc orthogonal quantization since quantization observation level general derive mle suboptimal local strategy represent various quantization transmission fc usually many receiver coefficient bandwidth fc symbol significance reduce provide summarize develop decentralize fc orthogonal serve low bound decentralize feasible estimator communication noiseless degenerate fusion maximal level redundancy redundancy sensor communication introduce various quantization decentralize transmission quantization quantization rest organize describe present introduce suboptimal estimator section analyze discuss codebook computational complexity asymptotic sensor fc deterministic inter communication among sensor sensor fc channel ideal orthogonal access protocol fc fc separate induce interference diagram decentralize system parameter transmission digital communication quantization communication digital analog sensor arrive fc channel use estimator digital communication extend analog parameter sensor independent identically transmission facilitate transmission quantization unknown write quantization nearest piecewise symbol much dynamic ignore transmission codebook quantization code decentralize estimator optimize codebook perfectly signal sensor channel channel period symbol receive I mean unit receiver transmission energy fc symbol special form receive statistically receiver fc px q shown give substitute omit likelihood simplicity digital communication constant symbol function mle maximize unknown fc function show received show readily substitute system symbol consist analyze define transmission upon signal symbol p regard minimum square error mmse gaussian mmse equivalent mmse ph upon mmse channel q l substituting mle fc model stage fc receive symbol receiver derive conditional function exactly unknown symbol implicitly provide pt mle show channel accounting channel channel contain correspond coherent coherent two simulation previous consider decentralized system prohibitive nevertheless practical derive low suboptimal estimator know unknown principle pmf lagrange interval satisfy interval expression pmf computing partial likelihood simplify necessary mle unfortunately explicit equation hand hand indicate regard receive stage complete estimator two stage view multiple necessary critical iterative good estimate stage mle iterative estimate criterion estimator probability uniform mmse first ip mmse unbiased inaccurate instead mmse hard system regard mmse variance mmse pt iterative follow substitute update repeat reach suboptimal differ applie criterion detect sensor true observation mle channel linear two stage iteratively accuracy perform simulation convergence study insight decentralize mle sensor communication perfect quantization resolution bit quantization derive mle pmf pt tell exploit signal enough perfect reconstruction communication level similar subspace follow ideal dirac delta term compute estimate mle obtain quantization detection perfect sensor receiver fc traditional technology communication meanwhile unnecessary substitute log recall normalize find quadratic max eigenvector fc estimator likelihood function observation mle communication pt eq mle apply centralize fc obtain raw sensor fc sensor error codebook show channel communication ht noiseless blue derive digital communication also transmission transmission transmission since rewrite rewrite normalization condition assume quadrature receive fc substitute log transmission derivation sufficient derive receiver part independent part h ignore real likelihood estimator transmission consider bandwidth contribution sensor local processor shift quantization equal uniform substitute cdf simplify channel channel digital communication transmission quantization scheme pdf show pdf pdfs correlation receive symbol optimal coherent rely real unknown ix ix symbols x ix digital communication transmission symbol transmission codebook ix ambiguity severe degradation decentralize auto codebook play especially unknown scheme code transmission codebook tn codebook cope phase ambiguity inherent codebook symbol transmission exploit symbol matrix enhance systematically codebook nonetheless preliminary result optimize transmission consider er rao fc second factor blue centralized statistically distribute among contribute equally fc signal long depend exploit infer know unknown bad mle take complexity mle mle searching order mse searching likelihood least fc term searching store conduct multiplication value get exhaustive mle estimator estimator consider simulation observation fairly energy scheme consider sensor low scheme cyclic code code codebook transmission also codebook denote code code transmission energy symbol ambiguity codebook symbol whenever unless codebook ts communication show mse quasi quantization blue bind practical comparison quantization estimator estimator examine sensor quantization rate bit rate codebook symbol simulation total identical use quantization total constraint sensor due example energy legend mark snr quantization bit quantization quantization blue low fig case extremely low medium snr level quantization inferior quantization bandwidth quantization observation consumption snr employ bit active similar conclusion draw consider communication consider communication channel convergence suboptimal estimator depict iteration unknown stand suboptimal communication mark depict transmission digital communication gain jointly traditional fc sensor combine final sensor receiver error discard error receive sensor symbol fc obtain quantization except use codebook codebook simulation codebook stand stand estimator stand bound communication suboptimal fusion quasi suboptimal snr mle transmission transmission channel transmission codebook exceed snr low snr cause transmission bad code reduce fusion base drop due exploit bad show impact observation stand symbol respectively denote mle apply true reference unknown symbol mle estimator codebook symbol evaluate symbol codebook communication ambiguity severe degradation symbol still bad
consistency compression promising alphabet set tuple borel sigma mean distribution dimensional define l b kb kb b kx nb bb b ergodic frequency occurrence generate priori ergodic distance set difference probability fine partition easy base empirical expression involve infinite consistent joint ergodic fall analogously grow whose already converge converge yet index moreover lb lb lx un thus partition cluster target target consistency sample partition asymptotically weakly consistent pf respect point next away assign second minimal already assign cluster cluster close assign iteration initialization design sample k jt distributional zero obvious complexity calculate enough last calculate statement let way stationary know consistent calculation requirement perhaps cluster finitely belong otherwise therefore c select consistency next pairwise distance second apart rest computation order calculate precise estimate want check sum replace partition subset increase q lemma integer infinity set increase enough estimate algorithm consistent cluster statement true condition cluster whose computational complexity think proposition possibility trade burden set take every unknown rate appear cluster know advance joint ergodic unknown impossible independently stationary ergodic asymptotic hold come make assumption expectation mix generate datum mixing without model assumption cluster bound multidimensional straightforward formulation informally process past one make assume stay sigma algebra generate strongly many process stationary irreducible markov mix underlie exponentially assign pseudo code distribution select weakly consistent let sample generate way joint satisfie satisfie depend parameter algorithm weakly process union sum I b eq every pair answer therefore together pair sample analogously theorem know bind multiplicative term make practically fact take individual frequency take union obtain realistic guarantee frequency cell partition coefficient speed strong without define cluster stationary process advantage framework make stationarity ergodicity simple check cluster initialization computationally implement distributional distance replace distance spirit direction concern rate finally consider length line problem length grow theorem claim net clustering parametric notion generalize statistical homogeneity consistency achieve stationary ergodic assumption neither mix give argument object object formalize work particular clear finding measure integral measure euclidean appear reason np hard concentrate certain numerous biological notion ergodicity define satisfied within assumption ergodic mean intuitively virtue ergodic select distribution observe time ergodicity countable sigma underlie alphabet tuple say sum
take ball true pt pt pt plus predictor variable support combinatorial turn replace cardinality envelope paper may knowledge many set envelope tool submodular norm algorithmic proximal operator support interpretation base group norm potentially overlap one factorial scientific signal process variable situation interpretable admit one look sparse low turn practical processing bioinformatic structure bayesian process prior dedicate focus mainly specific norm family instead follow support beyond cardinality limited pattern restrict penalty insight see function norm cardinality ensemble namely see submodular envelope extension algorithmic e proximal operator condition dimensional cm submodular recover give group potentially norm factorial experiment outperform relate absolute value vector submatrix define modular throughout fa cm refer also generality may monotonicity simply partition equal empty lead group norm extension extension note piecewise convex extension vector identify set sa fa submodular q component decrease order augment without strictly refer stable set cardinality set separable show submodular polytope soon strictly face p sa set play describe unit norm derive cardinality stable paragraph otherwise rely sa ga refer moreover look look lead bad fast practice equivalent algorithm solution eq compose absolute indeed envelope envelope strictly I norm envelope ball iii fa fa fa example submodular vice versa norm ball proof example point unit ball stable go possible sign example point concentration inequality supervise norm section novel submodular interesting example function function consider group norm overlap w soon submodular allow intersection strictly positive weight extend however zero go restrict sparsity give various topology group define acyclic element bioinformatics vision organize norm group vision group side figure empty gap already regularization order effect small norm come contiguous right side correspond equal plus contiguous limited fuse relaxation number jump extension semidefinite define nonnegative eigenvalue thus lead correspond entropy thus lead submodular see however dedicate submodular one applicable submodular minimum algorithm simple minimizer regularize induce pattern section lead induce norm rely submodular submodular restriction circumstance w w j set make assumption specification show one appendix simplicity situation assumption pattern assume lebesgue invertible minimizer unique one support assume support proposition decomposable property extensively j proposition proposition get similar jj consistency assume vector q concentration covariance set aa paper least square give unit gaussian signal function beyond compare proximal accelerated fista one fista fast n k three solve combinatorial w f approach induce norm forward ordinary method possible easy approach predictive greedy table factorial prior large support submodular multiply average replication deviation divide measure pair induce norm dedicate set synthetic norm worth current practice sparse enhance norm concept application link relaxation combinatorial study total variation cut cm partially project european author discussion positively homogeneous norm soon w p w w increase increase w w fa w desire face one happen get desire potential first prove subdifferential non complicated component magnitude e subdifferential subdifferential subdifferential stable contain subdifferential h w subdifferential norm zero unit norm desire nonnegative applicable z constrain nonnegative positivity equivalent constraint submodular apply approach regular solution negative submodular minimization unique zero may obtain minimum efficient regularize cost sa sa sa c fa fa result consider ii within interact include equal
subscript classifier change dataset minimize possible differential positive property perturbation additional cost compare perturb differential dataset p dataset previous due proof strictly convex objective function strong commonly machine technique huber adjacent perturb function convex minimize differentiable objective solution relation optimal perturb differentiable perturbation perturbation adjacent dataset obtain perturbation apply property calculate draw surface radius jacobian matrix mapping regularize rearrange definition square excess regularize classifier substituting prove regularize excess bind utility trade classifier parameter ellipsoid classifier intuition confirm proportional paper gaussian privacy technique add privacy learn directly trade extend work along technique theory arrive excess margin intuition large robust would insight design mechanism acknowledgement would anonymous within huber calculate huber instance eq algebra frobenius h frobenius huber one loss norm frobenius q lemma aggregate become develop mechanism process differential mechanism algorithm classifier perturb regularization excess perturbation year vast amount aggregate medical record database lead individual desire propose adversary instance instance instance observe new private able individual analyze propose modify add perturbation compare original preserve work binary situation multi class differential classifier classifier introduce modify year privacy privacy collection element randomize produce dataset say adjacent one substitution entry substitution modify dataset say satisfy execute adjacent query define differential dataset determine trade utility requirement think classification density algorithm differential dataset individual observe output algorithm already adversary differential oppose adversarial mining connection privacy study framework agnostic algorithm use create differentially classifier add estimate differentially private formulation modify laplace advantageous compute perturb introduce algorithm address problem expensive naturally class present differentially leaf theoretical analysis perturb investigate margin class training dimensional datum instance ellipsoid ellipsoid parametrize inverse offset decision mahalanobis scalar centroid simplify expand collect class also eq discriminative training involve semidefinite matrix rule training instance provably guarantee formally train centroid class least centroid one training penalty incorrect trace matrix label correctly ellipsoid prevent trade covariance upper identity replace surrogate hinge positive programming efficiently use interior
cause slow problematic carlo precision problem ht case q increasingly correlate arc eigenvalue generalize total square report carlo various bootstrap perform nan c correction bold test conditional test assess first sample generate alarm software package consider asymptotic fact log call mutual stein develop arcs acknowledgement ph school science article give many comment suggestion department pure constrain application frobenius matrix minimum maxima constrain test prevent interpretation statistic line strictly unique correspond covariance see valid use derive probabilistic direct let undirected configuration undirected graph present graph therefore also prove network uniquely assumption measure subset introduce monte test undirecte learn small bayesian bootstrap multivariate year bayesian successfully different include biology example rapid one grow score genetic hybrid one hill main need bayesian correctness assume size absence positive negative datum benchmark difference hamming evaluation real world either parametric apply large possible probability markov limit undirected graph underlie multivariate arc derivation exact variability network part network node variable article dependencie graphical dag distribution parent therefore measure specific presence arc particular variability goodness network bootstrap marginal arc direction original confidence evaluate dag confidence depend bernoulli marginal joint simultaneous every element result reduce parameter vector first result independence random completes extend ji turn sigma b c sigma two correspondence property normality second apply subset bernoulli variable also marginal bernoulli variable dependency uniquely identify numerical form element attain cauchy multivariate eigenvalue non hold complete sequence distribute preserve one univariate variate binomial closed guarantee therefore arc eq identify w parametric bootstrap several statistic variability direct graph simplify bootstrap case simulation two bernoulli several network frequency equal proof propose usually assumption multivariate normality three bernoulli call associate statistic structure bound generalized hadamard determinant negative definite reach reason convenient reduce instead frobenius network eigenvalue minimum behind instead correspond associate whose
complete hold prove regard plan q maximum submatrix follow simply generalization extreme singular permutation proposition value provide result well lemma along somewhat make develop union proposition regard lemma obey coherence therefore assumption elementary algebra hold difference martingale sequence inner independent gaussian let random draw every concerned complex random easy proposition let bound difference value eq real martingale simple union bounding follow inequality norm complex let independently distribute loss since follow rescale norm union event proposition definition remark linear structure graphical non model general regard generalize model coherence term bad coherence average coherence column design utilize measure coherence one term insight regard successfully carry fail coherence optimally signal average entry high key extend model use agnostic particular frame carry selection irrespective nonzero entry almost frame incoherence matching pursuit pattern recovery orthogonality process curse often break exploit live manifold vector p observe represent operate enable task computational fundamentally measurement compressed complementary nonetheless question need answer reliably reliably researcher year area known model denoise problem case enable objective problem among roughly case seem design coherence coherence introduce conference version spread within unit ball vector contribution area agnostic specifically selection term despite primitive optimally energy per nonzero notation equally nonzero objective regard contribution recent model recovery particular frame shift seed signal energy entry far generate coherence recover signal value entry nonzero entry away f solve rich context compress aic essentially attempt regularize square criterion seminal work well therein researcher year notable aic bic make procedure recent method lasso arguably become selection partly provide report correct certain result nonzero report asymptotic limitation verification selection result matrix plan correctly small nonzero entry singular bad coherence symmetric recent theoretical still agnostic propose thresholde use issue magnitude small nonzero know model plausible whether condition selection lasso beyond generic design arbitrary particular even tend demand much old complexity model present differ five completely agnostic order deterministic statistical nonzero linearly study design matrix influential report statistically around relate namely trivially achieve consistent model condition light rate consistent addition model selection also characterize partial regard probability cardinality hand study model matrix gaussian resp prior three result compress nevertheless conclusion long agnostic threshold enable threshold carry submatrix reduce third universal threshold report context recovery compressed setting exist problem bp selector variety ill suit complexity hand iterative match subspace pursuit hard pursuit fouri sampler show perform rip rip intractable provide design contrast sufficient condition entry characterize design highlight establish point signal bp nevertheless phase nonzero statistically uniformly lasso frames bp diagram arbitrary recover bp unlike letter scalar letter zero matrix use magnitude conjugate inner use denote index submatrix collect correspond agnostic extend result previous frame extension present use mathematically problem formulation begin unit white sake exposition perturbation norm make word intuitively speak incoherence quantify incoherence formulate incoherence coherence column coherence word roughly state somewhat superior incoherence key aspect simply exact proceeding selection word nonzero entry ratio small entry nonzero entry usual signal noise ratio worth point relationship see ready first concern selection selection coherence next quantity probability failure provide reduce measurement selection optimal opt notice quantify number need application fix specify successful selection suit satisfie p k failure complex model directly theorem important theorem easy threshold conservative reduce analytical tend might rely estimate magnitude entry characterize model design coherence depend p quantity therefore omit remark concern sort threshold cf preferred choice obtain next let complex result specifically far condition selection fix measurement circumstance still performance aspect even nonzero entry energy power average signal nonzero entry mathematically precise average nonzero ready performance coherence distribute k large integer failure respect rely great extent proof point never put study literature report devoted purpose assume selection literature average regard immediately bad gaussian reader bad case definition remain valid replace long appendix fix union fact establish design property long resp therefore correctly successful hand use maximum likelihood perform optimally matrix system high nonzero away average scale worth point also etc precede discussion regard gaussian selection thresholding prefer report bring lasso submatrix correspond away see require aforementioned part signal reconstruction whereas establish worth selection selection tight frame identify nonzero symmetric ii assume design satisfy early frame resp identify long suggest lasso succeed lasso nonzero entry away average equally attain nonzero certain perform optimally satisfy design devote construct goal first wavelet denoise design establish oracle like sense probability location noise thing present early specifically early guarantee location location nonzero entry energy nonzero recover require basis order agnostic signal recovery specify polynomial specify knowledge third impose entry limit exposition recovery sparse set noisy report study goal model intuitively note inherently model intuitive column strong illustrate coherence coherence property lemma design coherence suggest long regime gaussian satisfy coherence main satisfie nonzero significance consider approximately frame design satisfy recover sparse isometry rip guarantee sparse satisfy much weak scaling consequence see point order show slight variation sort difference theorem replace selection recovery three establish collection shift nonzero seed aforementioned geometry constitute design frame completely specify describe seed multiplication carry communication deterministic construction next finite hilbert frames constitute class frame time frequency seed vector follow emphasize block frame circular shift frequency shift seed ready frame nonzero subsequent norm signal frame prove concern frame coherence frame coherence facilitate mix w eq write shorthand follow algebraic specifically consequence fact nm likewise simplify nm cauchy schwarz inequality since divide expression coherence unit seed seed easy unimodal satisfy coherence long recovery suggests generate unimodal hope discussion design coherence selection frame unimodal seed mathematical research researcher recent year construction specifically unimodal j term autocorrelation decay suited frame bad recently frame seed check consequence
trace potential bold seven gray maximal bar height probability classify random manner perceptron symbol circle correspond choose pattern classify first denote reach potential condition approximate q threshold unless likely agree classification close present generation despite pair neuron spike generation summation spike within temporal window perceptron agree roughly overlap explanation cluster classification large connected error entropy yield calculate volume difficult ir function replica intra overlap domain inter overlap right compatible calculation enable estimate quantity ir character limit effect replica symmetry breaking affect correction behave sensitive temporal conclusion input integrate incoming spike generate response pattern adjust spike spatio fire despite simplicity architecture property superior performance perceptron temporal output spike support fellowship science de paris paris mi paris paris france computational neuron spike linear operation statistical mechanic extreme capacity task perceptron number per finite large size weakly constant concern static intensity neuron activitie spike furthermore stimulus system characterize suggest brain possess extract embed spike power importance decode embed spatio spike integrate spike spike denote temporal u correspond respectively except spike relevant classification fire output spike nan trace rescale fit factor line law circle indicate poisson letter computational standard classifying batch denote spike input neuron duration independently equal correct classification first numerical error algorithm capacity neuron probability correctly secondly important understand capability neuron time dynamic system complex solution arise computational optimization duration neural property easily limit sensible neuron fix capacity independent capacity require qualitative implication capacity bound exceed capacity spike arrive within single carry expect increase fig simulation behavior fast change significantly fast easier distinguish arrive learn algorithm solid large replica continuous mean perceptron capacity two different weight every secondly solution spread solution solution different overlap picture overlap probe walk find walk lead valid auto correlation drop fast
stationary optimum converge need minimal ascent maximization expense involve iterate expect perform maximum multinomial approximate solution cost slow implement per certain useful sparsity decomposition may active option alternative encourage posteriori conjugacy multinomial iterative map induce want responsible log value likelihood may priori knowledge relative component audio might expect map simple conjugate multinomial simple parameter lagrangian zero optimize original choose substitute large approximate optimum x
hard description consistent description description variety english case system bi gram gaussian obtain correct description correctly sentence comparison user average learn social need fuzzy cope ap integrate process perspective interact knowledge publish work movement semantic system language try reduce mean word context actually word learn several different syntactic case different context description user precise combination semi supervise description shape interactive website description human description description user green compound description dark circle green dark semantic learn conference description validation transform description set shape set shape multiple object shape section decide applicable degree label learn shape shape calculate match matching degree form generalize constraint constrain attribute value blue dark green triangle background projection corresponding projection list phase role sentence present description generate word belong project sentence phase new evaluation try sentence lexical phase frequency filter form word class generate composed class front light dark bottom blue green red htb analyze segment extract description compound description compound belong blue red green edge detector color group shape candidate shape pixel match shape feature average blue cb cr difference red bounding position orientation minor minor extension bound box height area pixel differently even size many label htb system relevant associated word share constraint aforementione decide robustness flexibility fuzzy tree classify accord fuzzy decision tree class class class class cr cb select class nevertheless decision frequent decision class else dark see htb calculate degree object obtain example color plot htb htb matching matching projection feature soft constraint fuzzy label calculate matching description ambiguity description degree matching could scene description calculate ambiguity scene description scene thus ambiguity description would description short scene short pattern word build ambiguity description return match else go repeat htb segment find object frequent label high system match frequent find degree ambiguity keep try ambiguity maintain influence propose degree ambiguity feature avoid description description generate light green circle front scene red rectangle generate description page user try guess describe description one come count description htb performing obtain description rank worse little well description obtain description bit rank far user allow system learn semantic class shape sentence correct sentence provide soft web description object allow guess complex work word context step word whose relevance highlight department science innovation program de european social mm mm es david soft word web describe scene guess give accurate user guess describe build analyze class detail model description soft constraint generate system description describe see cover range daily activity social phenomenon evolve complexity language link human capability converse environment lack information reality incremental fashion
fy north south fy north south north south z show determine node nod logical topology datum different consecutive train experiment different period path normalize approach path adaptive randomly rr measurement costly bandwidth measurement term resource potential measurement greedy active path probe iteration probabilistic nature accomplish random specify first entropy measure path path high assign select already follow relation normalize ensure probe call weighted confidence interval basis path path distribution region confidence estimate hand value hard use stop criterion encourage terminate normalize ensure available lie choose distribution probability bandwidth small maximize available hard decision efficacy model link path assign uniform topology probe outcome selection algorithm sequentially path seq estimate capacity lie interval satisfactory term average measurement stand rr seq employ naive outcome spread measurement software code although heavily author describe load prevent accurate load prevent precise train avoid receive train measurement train observe receiver obtain train perform obtain divide receive amount last delay probably consecutive receiver use inter arrival upon valid label follow online different path logical link low display transmission four disjoint path train second encode observe train plus probe clearly measurement estimation occur probe length reduce measurement collect trace use fraction rate measurement maximum posteriori calculate observe perfect chance available exception decay measurement topology path exceed confidence drop conclude determine avoid rate performance vary number network per train suffice significant use display indicate second although methodology overhead use topology sequentially confidence gain measurement second overhead path error estimate despite examine path probe train send highlight correspondence time algorithm bandwidth information probabilistic active informative estimate fundamentally different metric capacity connection briefly review almost focus path exception address path aim available bandwidth path year thorough self train great bandwidth traffic increase way delay receiver available inter approximately equal bandwidth vary converge variability available bandwidth delay employ deterministic inference hoc rule average liu input counterpart technique service service flow et provide end context bandwidth estimation propose elegant min case estimation estimation consume song subset observe load scalable sharing internet link near share behind core define quantity high datum tolerance methodology present address multiple active software software model format propagation marginal path take marginal require passive round perform encouraging many explore need trade take advantage relatively could formulate term resource budget also automatic learn network might need train short train suffice investigate employ scan measurement use informative link distribution currently adopt prior encourage sparse link accelerate software platform validate tool circle black fill thick thick picture know send high transmission streaming introduce define term traffic metric distribute multiple path share process dramatically require application beneficial offer specifically know send traffic induce valuable guide stream choose transmission streaming association content paper send probability almost closely path accurately involve path train rarely overhead acceptable path share load path independently inefficient resource probe information accurately path exploit bandwidth something available share link know share particular overlap estimation estimating capacity estimate large high bandwidth file transfer propose available path network goal multiple measurement approach base concept probabilistic path inference path also quantify measurement sequential measurement evaluate available bandwidth maximize gain path encode model factor well represent available correlation among connect path graphical joint bandwidth path formalism exploit perform inference select collect active learn involve quick measurement indicate whether iteration select path probe uncertainty probabilistic path bandwidth monitoring difference probe fashion significantly reduce traffic comparison sect probabilistic available bandwidth sect assumption sect active belief propagation sect obtain sect contribution relation sect conclude probabilistic multiple probabilistic available metric capacity train two cite estimation scheme employ delay al argue equivalent pass delay employ delay measure similar apply regard bandwidth available probe liu et bandwidth datum confidence stop percent range bind upper terminate satisfy termination stop reach meet secondary sequentially measurement past minimized topology network infer link ip address know incorrect ultimately available bandwidth actual logical topology complexity able limit operate logical topology topology stable minute moderate topology sect throughout different day topology sect link wide estimation employ measurement specify probe indicate whether rate triple binary difference test measurement fuse summarize maintain graph belief distribution easily new would network bandwidth use adopt active choose path probe already approach create probe probe new whenever path receive traffic service cause path ultimately low input employ behaviour trend great display bandwidth path distribution relationship topology link graphical
solely measure try objective unify image segment include contour recently description datum segmentation subject quantization agglomerative segmentation texture merging effective human image preliminary utilize seek image se window around group highly redundant overlap window adjacent entropy image encode pixel use smoothness boundary relationship belong texture entropy justify segmentation segmentation result obtain compression correctly encode texture code encode texture distortion distribution window shape texture image incorporate segmentation encode boundary texture carefully boundary adaptive code principle optimal minimize entropy image code purely yet regression proper quantization achieve optimal segmentation conduct extensive segmentation berkeley method conceptually measure purely objective extremely human compete segmentation texture segment one window around channel stack color inside window segmentation approach filter bank construct texture pixel take neighborhood pixel stack ease reduce dimensionality feature project principal denote principal assign empirically theoretically gaussian mesh model particularly texture synthesis consistent window fill texture nonparametric compression must distortion estimate empirically variance bad compression rate distortion code length describe texture region quantization code distortion vector sum length gaussian vector codebook codebook uniquely empirically exclude window window well model window empirical ht b windows r grid encode furthermore texture represent redundant approximate window ideally rectangular become rectangular code window region belong codebook generic multiple sample asymptotically code scheme inefficient leverage component efficient way group membership image pixel code orientation direction encode three along region image representation chain code code region smooth boundary expect consecutive consecutive code orientation orientation compare original code difference encode eight difference code image human shape web table tend human htb c c code angle change liu coding compression describe hierarchical deal multi texture window simple yet effective regression choose distortion set segmentation image represent boundary region find agglomerative initialize process pixel texture belong maximal texture window adjacent strong interior adjacent purpose find maximally decrease w capture difference boundary merge merge region ht measure denote intuitively small probabilistic well denote ht discrepancy label please refer infer image distortion extract distortion level statistic ki ki sensitivity intensity measure intensity insufficient accurately discrepancy measure optimal estimate around distortion agglomerative specifically either monotonically ground observation discrepancy square attain linear recover training program close learn distortion test ki nevertheless average discrepancy publicly berkeley comprise natural image cover scene landscape database partition testing set provide ground several human subject average five investigate human subject image seek determine color approximate utilize representation check validity segmentation texture perform widely manually berkeley dataset color bit encode texture information compute ground map finally code dataset entire volume thus pixel rescale mean produce range oppose achieve comparison normalize constant across average eigenvalue feature dataset representative examine code length therefore rest convert color metric probabilistic rand index information rand pair label great partition adaptively choose measure sum gain clustering extent nonnegative indicate great harmonic precision metric boundary segmentation boundary precision segmentation measure ground truth pixel adaptively multiple truth ground ground truth ensemble truth image performance segmentation segmentation ground feature vector convert rescale set rescale constitute observe I excellent result quadratic estimate step least segmentation namely compression texture merge good evaluate segmentation human six propose result therein agree ground metric treat ground truth computing qualitatively illustrate htbp human ms noting seem gap index g versus good mainly construct texture region contour edge category literature fail visually inferior thin texture choose texture algorithm fall behind human situation arguably shape texture texture appear break texture geometric pattern thin second enough texture texture region ill condition unstable middle segmentation problem investigate problematic able handle relatively well geometric slightly segmentation category thin poorly mean shift roughly pick point segmentation figure novel use principled texture respectively partition agglomerative hierarchy optimal minimize coding determine segmentation efficient distortion user experiment term region contour aid evaluation website novel segmentation boundary effectively code segmentation image short code boundary agglomerative cluster window texture feature estimate overall true publicly berkeley segmentation dataset achieve
degree freedom fit optimum still nonlinear nonlinear estimate reliably stem value practice show uncertainty may severe intend gaussian parameter value freedom compare trial assess yes normalise accord know estimate normalise residual show fact alternative definition however expectation value indeed approximately root severe comprise task fit well fit single assess true value consequently neither reliably clearly much increase true parameter quantify uncertainty residual mean derivation reliable assessing fit alternative beyond manuscript concern error thing order goodness indeed trivial residual already cf measurement distribution normalise residual need plot normalise residual histogram find residual quantify kolmogorov test ks statistic theory compare normalise fit well fit find may eventually start residual peak happen fitting stop guarantee similarly whose normalised match win truth radial velocity nearby presence claim circular claim additional justified table six every ks model six display normalise residual six distribution imply neither likely explanation discrepancy elliptical assume circular datum discuss normalise subject uncertainty difference blue red comparison gaussian large powerful usually computational give know fit goodness fit sample compute repeat goodness whole multiplying likelihood step require order goodness simply repeat make validation computationally bootstrappe cross validation likelihood bootstrappe bootstrappe discussion draw data replacement sample bootstrappe every th bootstrap contain predict repeat least bootstrapping aim prediction therefore argue completely sect absolutely nontrivial freedom quantification linear model cause become model number degree guess aware reliably degree consequently impossible sect see approximately within model assess convergence consideration popularity certainly apparent justified matter model degree severe use concern explain cross bootstrapping sect normalise model infer model concern want consideration concern correctness convergence thing gaussian acknowledgement david discussion david also couple thank helpful manuscript pm support measure error position one likelihood likelihood manuscript therefore reduce used purpose assessment bad fit whereas consider set ask close one fit stop converge sometimes evolves stop soon reach value one sometimes claim fit certain model error compute already case divide manuscript aforementioned major arise degree freedom explain affect application sect dedicate explain reliable sect degree freedom point I guess however explain degree split discuss linear model discuss freedom number piece concept next give linear parameter degree appear model mean superposition position example cause actually zero highly consequently nonlinear rewrite exist degree freedom introduce concept generalise conclude infeasible practice order concept give fitting arbitrarily short perfect three prior actually modify add influence fit two lose word fit constant
hmm ard ard lag dominant infer indicate turn lag rely lag truth matlab turn order truth variant dynamic instead distinguished base markov switching tracking understand physics section many mode force act specific form place jointly parameter refer choice normal prior detail independent volatility switching process switch underlie conditionally daily represent return stock interpretation non filtering cope log squared daily noise sometimes match mixture gaussian volatility stock stock exchange cite period event match order volatility model target sampler herein impractical training infeasible recursive infeasible leverage filter herein flexible dynamical phenomenon discover experiment dynamic considerable variability inverse wishart portion give hdp ar move wishart sec since time tighter unsupervised examine difference approximate walk raw create dynamic hdp iw sec iw degree freedom mixture measurement noise iw degree expect moment match al hdp compare fig place equal hdp use model table iw initial dynamical cause account observation accounting switching conditionally mode dynamical system utilize unknown persistent mode relevance infer allow vary switching var process develop combine dirichlet utility flexibility sequence stock bayesian method financial target country intervention national appear rarely observe possibility previously unseen motivate develop nonparametric dynamical simple hmms hmm markovian conditionally independent mode switch var process rely mode infer new dynamical paper one agnostic mode return previously prior hmms mode variant hdp crucial switching extend hdp persistent capture wide dependency explore underlie contribute employ ard realization dynamical mode possibly vary dimension switch var autoregressive provide survey recent approach switch dynamical mode ii noiseless switching var process algebraic rely cardinality autoregressive penalize identification simplify deterministic present assume dynamic output find switching subspace mode author also dynamical assume variational continuous sequence evolve dynamical mode state identifiability bayesian adopt incorporate mode cardinality prior complicated describe allow distinguish place simple aic bic manner dynamical system herein previous conjugate induce conclude model synthetic formulation time ss processes covariance process denote var observation var process form though ss model process phenomenon examine behavior model linear dynamical transition index drive gaussian hmm mode conditionally examine relax dynamical mode first analyze simple equivalently concentrated operate space emission parameter take define whose mode dp h space weight sample stick weight weight proportion remain stick prove many see examine draw discrete observation within representation take eq reinforcement extension prove useful define assuming variation global show measure et observation encourage expectation model dynamical mode persistence flexible nature hmm prior hdp add transition specifically expect transition original hdp et learn bias hdp hmm var capture switch unknown dynamic mode model illustrate var generative process table hdp ar hdp observation hdp distribute e hdp place generality fix matrix imply component choice state essence sec hdp ar hmm sec sampling hdp iterate state give mode sample hmm sequences ar hmm exist step involve straightforward hdp hmm construct involve capture underlie state sec prior posterior need condition sampler hdp explicit concept hdp hdp ar utilize definition outline table hdp hmm r ny yy x ar hmm dynamic lag form comprise observation available hdp resampling discuss mode consist k model prior infer single linear enforce mode condition stable matrix straightforwardly derive normal covariance inverse degree update tn k problematic grow identify irrelevant component lag hdp address ard encourage drive presence hdp ard place dynamic zero dynamic amount determine iterative become column whose insufficient imply mode examine look mode imply rank overall realization ard restrict attention mode dynamic must component assumption criterion fix identifiability issue considerably less ard prior may use switch entire hdp dynamic placing prior decompose lag give lag large lag block order examine useful ard q replicate hdp ar replicate recall hdp hdp hmm eq represent recall precision distribute regardless observation remain informative prior upon maximal hdp ar observation place inverse wishart oppose additionally measurement share mode measurement hdp ar hdp hmm sampler state conditioning resample parameter mode step term pseudo specify ar hmm hdp z dynamic ard move hdp additionally sample noise condition sample hdp dramatically improve hdp switch direct correlation variant forward backward backward truncation recursively condition sampling via sampler note encourage dynamical mode hdp dynamical simplify x first backward recursively message recursively tr slow mode analytically marginalization accomplish condition time sampling conditioning mode x backward produce pz forward backward local yx x sequence full derivation information kalman note update parameter sequential still sampler sequence sequential gibbs mode transition hdp compute pt sequentially hdp ar pseudo pt transition transition utilize hyperparameter pseudo mode sequence ard prior hdp specific transition pseudo calculate message initialize sequence compute count sample assignment increment likelihood htbp sequence construct pt associate k k pt htbp pseudo mode set dynamic iterate time construct switch var ard analyze power hdp var hmm hdp var hdp difference hdp hmm fig b display error true estimate mode hdp transition proportion informative hyperparameter generate five switch self mode transition two well dynamical comparable hdp var hdp contain hdp scenario middle ar hdp hmm significantly hdp ar posterior hdp hdp slow continuous scenario bottom neither hdp hmm hdp switching yield significant improvement baseline hmm effective use less rich switch difference ccccc ar hamming hamming ar hamming ar ar hamming ar hamming ar hamming ar hamming hamming hamming observation blue red mode switch ar middle th th hamming quantile trial hmm hdp hmm hdp top middle hdp observation utility ard prior true dynamical model two mode self dynamical equivalently white dynamic directly dynamical white noise contribute equivalently white noise original combine dynamical mode satisfy nevertheless realization still expect set ard recall prior informative size ard superior estimate state component infer fig e sample dynamic identify aa ccccc hamming hamming ard sequence blue mode mode realize solely hamming quantile ard e ard first value non dynamical within order location switch walk roughly straight rapidly body turning involve turn display tx head switch wish six prior hdp parameter sec synthetic datum process pre processing involve center scale dynamic range seq seq seq seq seq seq seq seq seq seq seq color blue head colored label compare detection achieve change number
close time obstacle many spin spin heat establish critical temperature phase regular even dimensional ise spin phase transition finite temperature study spin spin hard illustrate markov chain random dimensional dynamic interact lattice walk lattice integer site nearest periodic understand coupling walk lattice walk evolve identically couple eq coupling thus time also interact periodic particle coupling joint walk integrate yield meet evolve way simple choice walk independently site stay move site randomness mention particle dimensional line coupling particle evolution possible scale whereas behave lattice random configuration function coupling consist sampling walk coupling scheme preserve label meet see particle move entire maximum conclusion walk couple spin regime couple walk previously consider temperature evolve heat dynamic would correspond evolution configuration coupling evolve couple couple monte dynamic diagram carry spin heat choosing random configuration ise dynamic regular temperature preserve guarantee ise interaction although de heat coupling temperature show patch times agree partial coupling show realization coupling time vary couple time entire coupling heat spin phenomenon dynamical phase grow phase dynamical finite although mathematically single property illustrate point verify correlation remain constant heat discuss accept coupling spin update several configuration coupling place opposite configuration stay adapt regular number couple qualitative temperature metropolis couple temperature heat share qualitative confirm dynamic opposite well qualitatively couple spin key physics hard hamiltonian realize molecular dynamic carlo hard couple configuration critical fraction discussion efficiency concentrate couple canonical birth monte particle position life disk one diagram system birth death hard disk disk accept dark reject light configuration yet configuration dark produce diagram survey realization configuration horizontal cut diagram patch sharp regular couple birth death packing monte dynamic random initial life sample exponential dynamical packing density limit patch version birth death label time choose particle random move create coupling birth death closely couple similar choose define move disk disk example show hard consist configuration coupling use follow successful coupling place disk shape region radius algorithm hard pack disk center consist place disk create overlap particle balance generate move metropolis fig succeed coupling couple density conclusion couple importance nature ise spin phase heat coupling
solution literature new combination l evy superior almost apart population parameter subsequently optimisation powerful application problem parameter popular potentially mathematical analyze progress potentially insight optimisation call search cs develop present extensive newly cs problem cs far obtain particle implication mathematical optimisation problem involve design write range nonlinear maximum stress nonlinearity often multimodal landscape subsequently search hill solution modern search genetic particle algorithm attribute million important select good candidate sure search randomization call promise could outperform cs validate function include apply discuss feature study review outline specie though al specie basic direct elsewhere specie world specialized colour pattern specie al reduce species action et nature search effectively walk move state direction implicitly model mathematically example study evy landscape straight l evy style free search feature evy evy al subsequently promising capability simplicity describe search three randomly high solution carry discover either last replace new random maximization fitness simply form fitness rule basic search xx initial get evy fitness say replace fraction location evy good quality solution find current perform step size entry evy step draw evy infinite infinite consecutive walk process obey length tail pointing discover related difference good walk biased supplement reader relatively analytical test list extensive description optima occur landscape final marked figure optimum distribute multimodal optima number optima may become multimodal try population simulation also imply extent sensitive fine adjustment dependent need study literature new validate function de sphere minimum occur function unique multimodal also global hypercube function sharp domain global minima almost deterministic deal test design wave snapshot landscape value landscape multimodal global fact function stochastic extend draw stochastic due stochastic function deterministic hill fail however see recent study genetic ga conventional et al attribute partly lead evaluate evolutionary detail al genetic implement time meaningful less format optima stochastic stochastic generalise stochastic location iteration ccccc ga cs finding optima success rate evaluation instantaneous modern example second stochastic genetic
use tree newton terminal interestingly interpret average thus alg replace boost use sum formulate base class exhaustive strategy derive abc boost develop abc combine boost alg derivative alg differ iteration terminal np r bp bl r bf base number differ abc split procedure alg fit regression boost look hessian freedom fix diagonal determinant zero base base matter list boost concern small study fairly dataset moderate image image boost accuracy learner iteration overfitte mis report mis improvements abc also sensitive upper mis bold upper test mis l boost must exhaustive obviously expensive paper proposal abc boost base illustrate boost far computation reason regression boost iteration base insight step select base iteration introduce gap boost cost fast boost additional overhead boost experiment subsection obvious loss accuracy dataset moderate boost improvement especially large mis times panel mis error abc mis classification robust ratio boost original boost large task boost boost accuracy k accuracy boost achieve boost phenomenon surprising view test present loss accuracy mnist situation somewhat terminate accuracy mnist boost negligible compare boost may produce large small figure experiment present mnist test obviously propose boost abc boost serious boost class prior abc boost require base class base base exhaustive boost gap use sensitive accuracy boost encouraging fast abc boost tool boost exhaustive base boost report however boost prohibitive overcome serious limitation heuristic choice boost well encouraging
replication produce form namely virtue chebyshev inequality estimate upper bound guarantee large grow relative p estimator prescribe level optimality derive efficiency virtue obviously one mind estimator computational come effort drastically split elementary multiplication comparison generation single uniform incorporate estimator say number evaluation square coefficient variation closely relate algorithm back set linear benchmark efficient indeed direct network basically multidimensional walk constrained countable characterize quickly disjoint put simple transition lead subject negative system see transition specify arrival service time arrival take denote job open stationary condition receive job leave eventually th otherwise correspond encode period shall review next section equation gaussian elimination find fraction space state entry fall band show operation g possess efficiency work normalize sense run compare equation efficiency analysis insufficient analysis suggest evaluation solve previous encode two arrival service rate iii stability embed customer epoch system total precisely eq times x q mean deviation theory specify splitting queue motivate early basically constraint constrain random walk motivation deviation split embed discrete chain term type notation transition induce negativity state different region index encode origin empty boundary space empty depend subset eq represent arrival represent q boundary careful I dynamic queue admit walk type give constrain deviation network somewhat walk recognize non create leave aside simply describe large extremely important behind deviation play increment similarity suggest walk surprising increment crucial deviation deviation behavior scale scale queue length evolve eq queue process q negative characterize taylor formally arrive together previous equation characterize behavior game theoretic wang solution weak everywhere coincide calculus variation large deviation see asymptotic appropriate equation sign replace thereby obtain equation guarantee translate said surprisingly construct solution discuss use procedure equation equation use lyapunov inequality give mention rare event growth variation place initial position construct put n appropriately define q total particle reach wish constraint reach n guarantee p suffice ensure relate really particular picking nu suggest weakly efficient estimator precisely conclusion heuristic discussion development property sampler network heuristic method turn precise indicate xy j indicate run simply analyze previous fully branching splitting death zero think branching conceptually particle run sa particle reach update child indicate reach continue particle death particle iteration weight indicator l eq position refined efficiency estimator break particle part deal technique queue quantity reach split direct solving take markov turn queue network use give q x xx path satisfy also part correspond proposition follow consequence also simplify look expect stability one keep mind concerned evaluation involve measure sum particle weight evaluation bottleneck proposition v cp x c tv point measure suitable quantifying need account evaluation generate address expect effort particle remain effort th reach position intensity position particle intuitively level advance next move dominate expect inequality independence facilitate analysis moment add notation analysis exposition self contain generation denote particle disjoint grouping accord parent generation follow moment generation common technique combine term readily arrive take satisfie di order moment dominate turn asymptotic particle begin use particle beyond imply sum weight define stop time k q derive constant proposition expectation dimension increment constant bind finish follow negative since constant leave dominate side obtain finite combine reach lemma allow follow upper term use equality back second dominate equip ready evaluation evaluation sufficient achieve accuracy immediate bind effort per run along system conclusion splitting improve polynomial look network totally substantial evaluation total enjoy store vector encode system importance exist evaluation suffice sort meaningful comparison analysis resort bounding expectation point introduction room refine analysis insight claim conclusion criterion theorem convention splitting rare subset initial position suffice fact suffice relative bottleneck network rigorous splitting directly evaluate algorithm network subject substantial literature last decade specification open reference influential development efficient popular approach efficient rare event sampling splitting involve simulate consideration case network occurrence rare nominal splitting attempt behavior system idea occurrence nest occur occur keep particle reach course particle estimator popular efficiency analysis rare event simulation correspond optimality contribution discuss consider customer fix set reach origin queue length reach level whole intensity weak number replication suffice interest explain elimination system sparsity equation solve many intend exhaustive carlo exponential
evaluation aim diag expect taylor around decrease absolute obtain cf assume numerical algorithm care call diag imply critical order figure west always stay west might reduce slow west take approximately minute four minute perform fairly office pc take double diag recursion cumulative univariate mathematically normal evaluate numerically many lead lack univariate intuitive taylor bivariate cumulative normal univariate cumulative bivariate normal taylor axes overview discussion cf implementation variant although reliable also mainly book double absolute double arithmetic library straightforward indeed double apply competitive trade little often refer survey distribution book bivariate standard normal going use follow recursion convenience define numerically function recursion scheme run apply numerical q formula cf avoid q order favorable q apply correspondingly discuss derive section c quantitative finance field comment reasonable instead inversion avoid algorithm evaluation avoid without cutoff double recursion compute priori increase dropping condition accuracy still implementation always check upper bound sign summation double diag double double double lambda px px double ab px double odd double double odd ab double x double double ab odd odd even odd b odd odd odd odd return max provide comment check evaluate cutoff visual inspection optimize mm double help double double double b double else
difficulty coefficient increase htp avg avg positive group variable component variable hierarchical high choose group avg false positive avg negative avg avg positive avg negative logistic jeffreys diagonal parametrization invariant minimization seem serious example region little effect method regression run repetition respectively give penalization exclude map simulation lead htp avg avg avg false negative avg avg avg false negative jeffreys parametrization order repetition table hyperparameter average false positive negative roughly htp avg correct positive avg false negative htp avg avg positive avg negative reasonably practice bring together correspond ie tend contribution iteratively reweighte methodology resolve issue concavity assess utility estimate posterior mass work secondary contribution generalization penalize practitioner acknowledgment thank schmidt availability optimization code implementation technique popular amongst increase regression involve computationally attention quasi posteriori induce expand date bayesian art provide give hierarchy graphical adaptive recent sparse regression term signal computationally tractable regularization truly posteriori prior denote family involve log penalization penalization penalization maxima function become practice use component penalization induce convex increase difficulty result estimate lead literature reweighte processing hierarchical amount marginally induce hierarchy rise expectation maximization essentially iteratively iteratively reweighte independently suggest user incorporate coefficient flexibility grouping immediately lasso interested setting assume give conditional fy approximate point easier especially solve q equivalently think term construct prior low give ie exponential obtain ie eq prior become induce sparsity propose place inverse integrate call prior compute map prior logarithmic penalization introduction natural resolve difference size model come thin tail concave however one mode latent conjugacy inverse laplace distribution j clear enough value em algorithm mode weight penalization justify partially oracle select asymptotically worth increase remain increase quickly trade generally come exponential distribution laplace conjugacy respect scale normal still additionally concave gives scale adaptive contour plot plot lasso ridge give sparse together high example mapping group procedure prior never algorithm group machine wants solve related issue issue maximize density jeffreys likelihood parametrization methodology multimodal value reduce algorithm converge note still characterization mode open white black black none none e ann node state g node black text black draw none state node node j node edge h edge propose estimation paper interpretation flexible propose obtain minimization however derivation flexibility generalize family positive obtain marginally use prior ie prior statistical literature close whose single family generalization estimate return posterior logarithmic find form motivation smoothly absolute penalization induce prior slowly bias related reweighte penalization basic reweighte except limit case exponential family family penalization reweighte selection hierarchical differ place oppose difficult estimate although computation marginal prior member mcmc improper produce sparse explain prior improper unbounded density figure ep group visualize framework one regression group similarity clear develop iteratively reweighte standardize put improper integrate jeffreys improper
height box become large produce get meanwhile symmetry preserve drawing assign interval node coordinate could replace product although kx help generate graph manner number final box convention color row color accord thus diversity correspondingly generate increase like keep degree subsequent iteration achieve choice use box sized box expression simplify keep increasing increase exponentially characterize calculate measure cluster node row topology sub first sub give generate formalism follow show trivial coefficient near rather simple analytical analytic formula distribution significantly speed measure prescribe degree obtain average symbol plot together obtain show agreement show coefficient neighbor plot panel boundary capable produce diverse prescribed optimize generating requirement number node box actual box boundary self adjust shall follow optimize generate conceptually simple give case implicit way box keep write give note case actually degree like system sort plausible actual annealing also decrease slowly consist change amount energy small change accept procedure generalize principle well measure respect three choose target rather free result optimize respect show agreement adjust typical respect different degree circle mark symbol come law degree correspond symbol experiment bi modal respect different coefficient graph limit network alternatively ever network become would contrast graph analytically see si infinitely relatively simple structure extremely region infinite regime increasingly slow growth appearance si summary demonstrate construct characteristics degree coefficient turn graph hypothesis anneal small observe spirit song thank discussion science office generator biological os institute mathematics os biological physics new conceptual simplicity generate variety degree hypothesis datum suitably define potential allow variety topology infinite network increase present analytic iterated generating distribution parameter measure anneal researcher biology science biology complex enable create becoming include phenomena unit behind network unit connection turn realistic distance distribution modular understanding design control system line increasingly graph representation technique new year recently l dense adjacency et phase variable suppose continuous everywhere limit support develop grow size conceptual rather natural internal organization large school organization playing since complex principle hypothesis measure many successful interpret various limitation degree distribution graph attract remarkable method include systematic approach analyze topology specify degree sized subgraph spin symmetric matrix take value count hierarchy interesting hierarchy os enyi random distribution scale approach around adjacency achieve adjacency link element multiply version element real multiplication element obtain tail eigenvalue small diameter simultaneously kronecker similar point et inspire diagonal give link probability summary plausible generating procedure generate graph stochastic growth prescribe vary configuration stochastically adjacency simple pair connection relate construction fix increase iv assume infinitely limit singular infinitely
respective prox optimize construct multiplication analyze complexity accelerate homotopy introduce homotopy smooth obtain hinge warm next homotopy lp losse ix equivalently replace full saddle function subtract prox prox center project piece wise hinge smoothed hinge error theorem error hinge completely calculate use determine thus smooth arbitrary calculate lipschitz calculate svm define sum equivalently saddle saddle point smooth subtract prox smoothed project explain result smoothed piece wise approximation larger bound completely direction thus use definition nesterov smooth primal rate round construct iteration round represent solution svm auxiliary auxiliary prox whose prox guess gradient auxiliary step iteration round start solution proceed negative gradient round weight round gradient iteration round optimal theorem correspond smooth smooth guess input guess solution termination dual part round iteratively svm expensive computation multiplication step proportion increase addition simplification completely correspond nonzero element nonlinear replace replace require penalize calculate fw ix round reach let accord hence round complete multiplication homotopy lar parameter solve prefer start optimization accelerate start warm start prefer accurate approximation large induce rate accord expensive accelerated term categorization vision task e scene scene ghz processor gb scale solver light criterion classification accuracy perform respective cpu second svm solver test time mean svm solver census categorization scene scene recognition adopt rest wherein feature multiclass one adopt code available solution guess termination criterion census categorization repository census show train training set scalability slow svm solver search svm main irrelevant scalability solver short shorter batch method classification dataset axis pixel sample select group store home public space place color texture datum testing solver value achieve solver optimal descent matrix multiplication irrelevant light sensitive show scene g office include texture intensity scalability svm solvers svm take cpu event image bag split training fig four solver solver light second base expensive cutting svm less recognition event graphic randomly class bag extract accord scalability four svm less solve svms programming square complexity svm gradient combine descent determine lipschitz multiplication require round homotopy improve efficiency norm homotopy adopt warm start time cause solution homotopy competitive efficiency four popular svm svm light insensitive refined efficiency computation smoothed already accelerate future machine svms tool many intelligence sufficiently efficient deal gradient svm compare solve wherein differentiable hinge norm primal iteration round historical vector multiplication require exist solver addition nonlinear homotopy accelerate dynamically categorization scene recognition scene effectiveness available website assessment machines smooth nesterov machines svms prominent machine tool intelligence svm great feature work rapidly dimension addition decomposition sequential cost optimize iteration relevant support vector optimize problem select work set programming impractical complexity slow convergence optimum efficiency svm firstly compute violate constraint training cutting add current working qp structural qp efficiency difficult serious svm scale inefficient qp constraint primal available solution correspond vector plane reformulate classical non stochastic achieve solution convergence second newton method replace hinge differentiable sigmoid function expensive impractical primal second online step update rapidly reduce therefore apply
covariate c demonstrate among nonzero regression true know index fit cox predictor package survival statistic difficulty model correspond effect statistic estimate nonzero figure relatively hard oracle difficulty one datum demonstrate propose classification study solid even hundred united tumor patient nb diagnosis age patient clinical free h microarray include gene site overall survival five array array consideration patient survival information available patient censor survival summarize marginally jointly gene appropriate sis powerful van scad select gene probe cccc coefficient e try significance gene predict survival hazard function correspond log eight gene next eight cox proportional hazard seven log freedom comparison test gene select cox eight plus record repeat average likelihood merely eight gene reduce lot increase develop technique survival dimensionality large focus sure iteratively apply screen filter utility moderate partial select carefully study sure screen section corollary fan partially support grant dms dms grant wu partially nsf dms north state scientific advance huge covariate snp technology clinical understand information clinical survival extend cox study technique demonstrate screen survival biological death failure failure try censor termination study dependence survival covariate covariate goal hazard hazard depend nothing instantaneous rate covariate proportional partially due deal censor assume hazard function note uniquely one identifiability specify identifiability condition hazard introduce proportional reference detailed literature cox hazard need estimate survival also baseline hazard reader recent advance collect huge amount microarray snp information clinical covariate associate clinical outcome quantify contribution data predictor mathematically nonzero model subset reference therein modern survival consider scad concave adaptive event among consideration dimensionality exponentially propose independence sis marginal ranking theory sis predictor vanish extension iterative sis marginally uncorrelated deal sis method non asymptotic theoretical rate although extension cover explore independence screen hazard censor event cox proportional extend sis nonparametric additive carefully organize detail cox proportional variable give cox sis cox demonstrate sis denote time censor associated correspondingly denote censor simplicity conditionally independently identically likelihood fs ft conditional survival conditional hazard respectively failure time consider hazard become consider informative nonparametric observe failure consequently maximizer get censor maximize newly hazard coefficient leave final consequently important variable handle variable procedure accordingly handle cox proportional advanced penalization receive recently penalty function perform optimization capable penalty many scad penalty elastic adaptive penalty function penalize maximize sign front literature scad penaltie study event many sis scad quadratic spline origin recommend argument scad penalty convex scad convex work penalize variable technique great covariate performance penalization variable selection subset mathematically penalize partial lead sparse index sis jointly marginally sis compare sis base make joint sis begin sis penalization base refined true denote utility utility measure contribution include large small covariate top denote respect get update step idea propose improvement idea repeat reach adopt identify note generality two sis estimate screen include probability tend asymptotically due individual number covariate include new variant splitting asymptotic include exchangeability show dimensionality please want remark applicable study bind splitting define new variant original sis screening alternatively choose ensure variant explore compare cox proportional tune different covariate setting generate identically distribute marginally serial marginally multivariate marginally correlation case except serial decay coefficient response marginally dependent therefore expect challenge sis sis dependence condition implementation sis sis var sis select intersection size typically sis scad coefficient whenever necessary bic good censor hazard censor censor correspond censor rate censor correspond censor censor censor coefficient randomly independent though marginally marginally design repetition row median median row label report proportion repetition procedure consideration select application report median
incorrect deterministic purpose prove deterministic satisfy minor difference change fortunately valid compressed sense modify concentration rao achievable constraint valid correspondence organize definition indeed provide form domain rao bind sense discuss show er rao estimator sparsity equal measurement random discussion matrix noisy joint cardinality set denote sub index generalize generate matrix know accordingly lemma additionally assume expression hold unitary extract comprise proof part sake go try random decompose obviously addition show kk k k gaussian generality assume tail take bind say decrease bind inequality bind approach previous decompose unitary lemma gaussian assumption domain nevertheless proof free assumption generalize result area correspondence continue generality first eq addition operator denote element n sake n I kn I km ss eq mean rewrite assume proof use satisfying appendix take respect negative plugging introduce left reader first part far eq important apply upper finally come complete conclude validity valid accordingly theorem rewrite interestingly obtain obtain similar r rao I phase rao bind use two phase estimation process estimator achievable know want er er consider depict form fisher equal er rao er er rao consider one identically eq element gaussian go relation us rao estimator knowledge deterministic concentration seem interestingly randomness fortunately investigate combinatorial proceed joint estimator cardinality estimate unique show constraint mse proof assumption especially variant valid necessary generate additionally exactly minor repeating step apply bound term lemma easily attain upper additionally obvious grows additionally equivalently grow linearly exponent grow polynomially respect compare come asymptotically achieve correspondence er compress sense analysis sort concentration mainly focus build er bind rao bind degree indeed achieve unfortunately impractical er rao open reach find root solution substitute n expression expression obvious hand n prove complete thank ms mathematical department university technology comment author remarkable motivation grateful anonymous associate constructive comment theorem proposition proof height em depth noisy compressed er comprise building measurement randomness generalize drop fact matrix theorem family concentration measure generalized er achievable compressed sense cs vector propose signal usual correspondence signal indeed measurement measurement precisely identically gaussian etc compressive index stand model require present lemma correspondence recovery many theory estimation main compressive estimate noisy measurement effort find solution algorithm relate search existence estimator square estimation much rao bind er rao low mse depend amount knowledge estimator sparsity rao know non contain index location show correspondence structural least estimator second er rao rao kind al differ maximal show uniqueness et support case size er rao equal equal accord estimator limited degree proven state achievable effort estimator knowledge close rao low es factor interestingly use estimator know certain constraint er priori infinity remain asymptotically achieve asymptotically rao achievable bound generally check noisy vector concept compressive estimator solution er typical bound two mention probability event support typical denote second event jointly typical average q rao important mention element randomly distribution consideration
weakly possible make small reduce introduce systematic systematic fig accurate eps eq systematic default function fig eps systematic default straight curve approximately mesh develop code almost code make eps color statistical systematic default red calculation perform broken calculation average calculation average sample cross give use thick break blue curve illustrate average calculation arrange check undesirable datum remove point accuracy value realization average classical default systematic cross give illustrate large systematic fig calculation different realization red spectrum calculate spectra thin line scatter average spectra value slightly energie small eps eps eps optical sample show calculation sample use iterate default function curve iterate default calculation fig eps eps eps eps start default use systematic calculation spectrum default split batch calculation default batch calculation calculation increase average however reduce lead net reduction systematic b realization noise spread substantially vs systematic somewhat vs show different splitting calculation effective averaging reduce net unchanged realistic value actually dependence batch thereby individual calculation calculation increase approximately behave increase classical choose comparable perform calculation discuss curve traditional optimal would next exact opt almost flat curve fig eps eps color calculation est split several batch result opt use find iterate use model iterative cross default error case batch product q average average calculation batch realization difference systematic contribution default calculation calculation systematic two suggest systematic compare calculation calculation latter favorable batch result est opt try improve default iterative recommend calculation default spread large imply systematic reason default statistical calculation calculate nonlinearity result correspondingly case split batch default error drastically reduce iteration substantial factor reduce systematic compare result use close lead iterate allow lead logarithm result analyze behavior expand low deviation define approximation method guarantee spectrum analyze entropy iterate iteration q define contain eigenvector symmetric expansion eigenvector nc rewrite eigenvalue large many close eigenvalue default model show eigenfunction low increase one rapidly corresponding tend weight table expansion default model eigenfunction default crucial calculation close strongly contribution deviation default fail structure correspond since additional show scale make remove fig pattern differ calculation see fig contribution output mainly eigenfunction fig calculation perfect probably nonlinearity due logarithm expand generate function node two also component eigenfunction shift slightly projection operator state eigenvalue iterate eq systematic relative systematic choice linearize pay include operator nonlinear bit different expression expand logarithm eigenfunction couple high eigenfunction node depend component whether different add contribution high component often favorable iterate one favorable study low default expand default expansion unity small eps eps color eigenfunction correspond eigenfunction one rapidly approach define systematic default systematic statistical batch batch total calculation often bad statistical error serious reduction linearize formalism see error deviation default illustrate default potential hard use batch fairly value result sensitive alpha alternatively batch use lose thank want study divide systematic lead statistical splitting batch calculation average systematic iteration often bad splitting batch result systematic max universit energy depend monte carlo quantum possible response ill analytical treat regularize introduce spectrum default control statistical maximum perform analytical approximation decomposition stochastic regularization usually
maximize I application theory large recently quantum mechanic possibility attempt review incomplete useful system perhaps shannon carry state physical might associate explicit rational belief another drive good preferable enhance successfully idea extent wish rational belief information available put explicitly sufficient far capture driving incorporate rational mean belief everything go acceptance arbitrary rational behavior rational exercise piece reliable raise question implication accept belief useful though notion bit introduce acceptable information design handle describe method distribution new constraint value allow problem select increase preference clear preferred ranking assign real preferred possibly equally preferred distribution lead I entropy maximize impose I perform external posterior entropy must produce entropy call entropy redundant drop candidate identify number special prefer shall criterion adopt single entropy elimination approach extent selection applicability reason eliminate quite criterion violate justification entropy inference entropy prior ignore aspect update virtue maximize something keep summarize brief motivation entropy detail refer infinitely class processed refer particular condition update drop multiplicative ranking entropy function sum integral affect criterion entropy multiplicative affect drop function three argument argument locality criterion universe usually value whether coincide leave next turn family choice principle agree maximize entropy constraint multiplier yield intuitively reasonable turn known false straightforward give couple method generalization bayes knowledge relation limit uncertain long know q maximize lead correspond datum derive moment relation prior tell nothing versa treat relation elsewhere one possibility addition know seek maximize lagrange multiplier multipli q multipli corresponding marginal datum take bayes factor reproduce reach likelihood example I issue decide maximize prefer extent update give constraint define space acceptable label manifold write maximize lead preferred question extent distribution believe range assign I represent relevant new extreme product know tell versa retain use original choose assign probability volume choice hand frame coordinate volume fortunately space probability distribution rao metric uniform unnormalize crucial joint select preferred joint normalize convenient lie prefer maximize entropy tell degree result limitation application fluctuation beyond fluctuation bridge deviation adapt lead notion ignore concerned belief rational agent rational belief force agent belief quantitative main relative update I method single inductive inference allow old recent mechanic acknowledge valuable
obtain bayesian parameter estimate mmse sample full sample via inversion outline automate popular co variate numerically approximate target inversion sample stage suffer curse become highly alternative problematic become optimize weight instead sampler variate methodology curse simple adaptively rw rw move deviation tune produce via likelihood information fan variate methodology proceeding algorithmic markov chain impose restriction location proposal j initial prior inversion obtain perform walk element n move proposal j several adaptive distinguish markov sequence transition proposal allow depend many algorithm particularly markov ergodic appropriate stationary propose condition ergodicity algorithm develop metropolis history markov ergodicity bound ergodic prove ergodicity mcmc define convergence derive guarantee develop condition variate proposal kernel know ergodicity detail estimate use markov theoretical choice base mcmc algorithm replace follow variate sampler move proposal di bayes note computationally inefficient involve chain td approach require single factor alternative van van work produce rank compare rank question bayes compare bic comparison bayes factor eq comment numerical implement detail critical handle numerically numerical handle appropriately survey inference involve note average probable probable probability adopt one able reduce potential associate several choice probable turn associate involve popular algorithmic framework model average estimate direct knowledge probability probable mr typically mode careful demonstrate van popular choice proper integral quantity distribution quantity achieve model uncertainty rank bayesian mmse mean methodology develop contain synthetic methodology real data real datum true study p sampler generate conditional sample discard realization mmse deviation sampler point average intercept pre tune local walk perform use mcmc q nh mmse accurate mixture local move global estimate require effort discuss sampler develop actually achieve performance therefore superior gibbs automate gibbs sampler recommend algorithm utilize plot adaptive demonstrate rapid mmse initialize far additionally become sample uniform range identity realization true row first path high dimensional markov chain initialize true rapid sampler true take ghz ram study estimate consider series simulate realization time per run assess paper financial comprise mini mini mini series market daily price sep series datum present mcmc present run sampler initialization split increase datum datum rank model amount converge predict particular estimate suggest distinguish suggest application algorithmic trading repeat perform series comprise mini analysis analysis financial initialization give preference trend analyze evidence co analysis index period period mini mmse step integrate uncertainty bayesian uncertainty accurately study begin segment contain series bayes mmse square series proceeding day present square random posterior mmse perform clearly average uncertainty bayesian selection reflect averaging approach wide selection though assess confirm day demonstrated variate adaptive metropolis alternative local move parameter formulate rank estimation analysis real average unknown develop extend show perspective develop integrate trading trading triple make series p mini perform average select co integration mcmc framework automate two anonymous associate comment exposition university financial theory new york pp uk economic apply trick max l obtain c mmse mmse mmse mmse mmse mmse mmse mmse posterior mmse global ml move global learn simulation start away generate carlo utilize adaptive move represent additionally c index bayes markov length year log chain cm corollary department mathematics south sciences bag email capital mail com capital mail com capital mail com markov auto automate develop mcmc framework model sampler dimensional series block utilize sampler impractical involve full spline ideally suited reflect joint rank practically adaptive sampler also development automate consider make financial trading pair trade able framework demonstrate posterior random trading adjust potentially market condition coherent auto adaptive monte carlo analysis auto several variate improper care implication bayesian structure blind var improper distribution example paper aim admit conjugacy posterior dimensional variate dimension significant correlation random sample involve allow significantly increase typically adopt framework van conjugacy consider variate unknown covariance matrix contain rate mean unknown vector posterior rank dimension large justification direct inference var model set pointed model var model useful property widely use develop base follow utilize along model conclude range posterior triple trading typically note integration
find focus identification fundamental address assumption quantification uniquely associate writing explore issue study basic level close multi discriminate implement discriminant categorical quantification handwritten classification identify accuracy handwritten note quantification study derivative identification system implement laboratory describe document return short system short provide brief overview categorical describe propose leave cross predict unknown classifier predict pseudo simulation classifier discuss evidence long american computationally tool evaluation evidence possibility base need point computer suggest absence identification interested insight set identification comprehensive review statistical discriminant near neighbor appropriate neighbor unknown classified database study computational different together build necessarily document apply subset short different quantification document quantification use quantification select euclidean classifier near neighbor weight weighted segment write sample bin new investigate associate proximity normalize document recent contour white pixel calculate chi square well alone distance provide database correct return improved recent research character potential identify mathematical quantification handwritten convert image individual character manual automate letter character example make letter character skeleton skeleton every identify belong smoothly another shape different letter alphabet appropriate character letter particular letter count represent letter occurrence letter system describe extensive report write sample sufficiently letter alone frequency letter sufficient information individual letter information unknown pool business good mr news week go join col arrive nd letter address dr within five conduct whereby writing collect various year collect ask modify letter l letter paragraph letter letter character character ignore letter modify letter paragraph letter collect write segmentation character manually letter character text paragraph letter character individual segmentation letter association error character association less process another micro feature divided hereafter consist result document miss failure character write reason issue involve micro cause character present usage micro letter character per character per summarize character study facilitate letter th letter th write frequency th l denote place subscript subscript document unknown say unknown count use letter uv count letter let th letter multinomial letter observe parameter number attempt document assume letter dependence classifier accuracy compare principle bayes il suggest posterior multinomial shape procedure estimate conditional document write identify unknown know database plug combine naive rv employ preliminary classification class play letter word bayes rule tend extreme behavior plug literature chi measure difference chi distance chi distance measure suggest determine cluster character part feature single cluster proportion neighbor unknown proportion small document extensively effective range application document relatively chi chi letter combine chi square letter chi square take relative information letter pearson chi measure letter pearson chi square count freedom letter handwritten document degree chi square letter chi square evaluate freedom chi square tend score square document associate letter pearson square way table document database letter chi document number letter chi square chi degree write large reasonable combine pool pearson chi statistic author measurement exclude write author fit example chi chi type use study chi see vector define classification estimate letter il p analogously kl distance th eq write distance neighbor application reasonable document pool evaluate plug naive document classifier document leave predict validation classifier document classifier correctly identify scheme chi square document classifier incorrectly correspond estimate three effectively letter stress would classifier accuracy document explore subsample character write write possible size subsample would limited additionally subsample letter database study implement classifier entail leave entire author classify implement stress sample write sample document unable look simulate document construct letter left document occurrence letter letter letter proportion estimate letter letter left letter generate unobserved document document document leave letter effectively different original
patch texture texture error error use universal instead example texture pixel classified belong class convention even tell belong patch pixel assign majority detection regularizer outperform value incoherence adjust cross map precision show precision positive design model universal coding model approximately subproblem also show prior code patch well laplacian adjust additional flexibility dictionary size weight show practical impact active image bayesian burden several force reason conjugate demonstrate introduction address aspect future design nonzero overcomplete noisy acknowledgment partially wish provide fast toolbox thank incoherent helpful comment integral obtain precisely definition substitute constant mean easily du moment obtain nonlinear technique newton starting significant moment one function weight develop take naturally distribution value iterate function approximation symmetric approximated regularizer ta da da jeffreys exponential evaluate plug function plug desire however close central trivially yield solve part e I moment order one possibility solve u possibility fix conditional jeffreys jeffreys proper assume model plug fisher n last derive result distribution considerable attention lead art regularization critical minimization coding compare presentation classification call learn contribution theory practice learn collection many processing reader sparsity control parameter challenge automatic natural assess describe length framework term interpret bit describe coefficient reconstruct coding develop work denoise wavelet description coefficient compute work previous designing code obtain encode natural universal lead consistently process desirable robustness improved recovery decode signal compressive sense compare practice yield correspond sparse code turn lar regularizer dictionary improvement aforementioned task organize derivation universal present propose denoise conclude j cardinality goal model design dictionary solve problem column usually divide sparse among try commonly close certain condition coincide formulate respective jointly approximate alternate dictionary alternatively sparse code step efficiently update turn code compute good sparse interpretation summary interpretation insight provide norm see regularizer commonly constrain lagrangian form maximum posteriori estimation scale contaminate additive consider term previous regularizer special meaning signal task image compression coding generally response zero patch phenomenon reference subject suppose image reconstruction obtain good compression consist probability stage bit assign known provide approximate shannon modern finding minimize encode distortion reconstruction assume encode choice coincide error encode residual combine obtain l p lead sparse code characterize reconstruction coefficient instance enough interval pa fa pa fa sequel treat density course needs tune interpretation framework offer compare different introduction coding already study early consider shannon solution describe actually sparsity include magnitude uniform coefficient description furthermore actually section size patch well case arise decomposition result careful image coefficient form generalized justification observation heavy tail hoc available possibility close perform family numerical technique derive close coefficient empirical log heavy tail fitting encode patch dc component image histogram variability region correspond scan order contiguous block obtain value atom coefficient explain previous regularizer coefficient different row vary greatly empirical underlie figure occurrence value range associate lead weighted regularizer modeling perspective adjust poor thousand parameter learn model new bad estimation problem unweighted regularizer weighting failure signal atom position estimate impose hyper obtain sample code example compressive approach expensive something choice lack proper theoretical justification derive flexible avoid burden solve sample tool successful theoretic extension compression summarize several possible scenario deal code knowledge secondly identically underlie scenario yet would almost scenario fit original universal code code encode compressed coding assume lead concept class kn find fit simplicity arrange single length sub describe single probability optimum relationship define correspondence scheme assign class parametrize omit write include reconstruct know precisely negligible way universal part optimal value bit model thus quantization p letting require bit write derive complete uniquely equality satisfy l q regret necessarily something often previous assigning coefficient minimum mixture via evaluate exclude result mixture appendix refer evaluate obtain get convention approximation explain code note sufficiently regularizer tail laplacian happen determined fact could bad minima aspect tail bias perform typical use column regularizer induce leave regularizer thresholde shrinkage large coefficient much tune possibility fall model limited dynamic basis exceed true quantization practice advantageous adjust solution exist estimate moment obtain detail deal case jeffreys improper improper jeffreys first sample key observation jeffreys prior integral result appendix conditional jeffreys explain furthermore moment sample give practice converge delta jeffreys approach universal code view jeffreys loose flexibility deal lead prior problematic trade degree flexibility model yield regularizer code regularizer sparse problem convex sparse coding suppose j convex discard term point meet regularizers limit function kronecker delta laplacian therefore mixture point code time sparse code warm begin iteration course j choice choice na code laplacian regularizer discussion around lagrangian code formulation give error since actual term give compute regularization distribution regularizer separable coefficient expression appendix lie total showing efficiency optimization comment estimation exception principle influence need cause dependent laplacian issue compute zero deal properly work patch draw bit channel converted image channel scale unless otherwise overcomplete atom subset seek test sparse dictionary lead show additional encourage atom advantage ability fast incoherence empirically sparse code coefficient prior single matrix compute restrict empirical plot along good conditional jeffreys use moment yield moment model mixture kullback leibler fail improve sense model sensitive hyper estimation hereafter well prior jeffreys well figure vary atom confirm k fit atom globally fit laplacian critical fit atom improvement practical base pursuit ground contaminate regularizer application see measure support fall figure respectively model coefficient image patch dot well fitting laplacian bit clearly properly almost fit tail desire fitting laplacian dark laplacian light dark red fit index atom outperform per atom active define improvement range highly clean project subspace square true recovery
break distribute denote dirichlet process become cluster prior mixture atom provide distribution component partition develop mixture utilize heterogeneity focus couple dirichlet process hierarchical concentration dp hdp aforementioned linkage assumption change dp framework stick stochastic process co infinite proposal suitable problem functional close author introduce dependency dp mixture stick spatially vary variable flexible work focus mostly interpolation infer locally group introduce dependence distribute allow assess cluster extract arise generally index centering multivariate local formalism global dp use dp realization atom group global induced global refer nest yet broad dirichlet induce covariate summary hierarchical specification incorporate dirichlet go fully framework propose distinction hierarchy hdp rich hierarchy take hierarchy bring model place hybrid dp fact serve dp curve curve hybrid satisfactory despite functional require pure pre specify utilize switch random focus prediction instead worth note simple exploit graphical dirichlet spirit quite dirichlet admit independence fundamentally goal brief background dirichlet process hdp define explore graphical dependency sampling characterization also offer global group distribution recent literature brief dirichlet proceed hierarchical measurable dimensional dirichlet distribution concentration around center distribution probability constructive draw concentrated stick break independent beta iid measure positive integer viewpoint dirichlet show dirichlet process perspective accord exchangeable atom likely induce chinese restaurant utilize component study elegant account application dp modeling give brief formalism set group set index covariate let assume exchangeable suggest use mixture identically specifically formalism endow refer index factor prior factor conditionally hdp formalism statistically couple measure conditionally dirichlet base fully distribute base hdp share countable within distribute accord restaurant restaurant statistically couple fact distribution group mix hdp assumption coincide assumption lie product endow correspond borel algebra dimensional whose index modeling specify distribution relate distribution enable cluster associate basis group index nonparametric linkage among govern stochastic spatial measure conditioning vary center amount index graphical index summary collect specification v u draw turn local factor center global across hierarchical involve hierarchy level distinction hierarchy operate nest space hierarchy relate say probability distinct vector explicit place probability obtain hdp provide draw suppose collect choice spatial well many location available graphical also field undirected offer computationally finite collection undirected graphical independence relation connect representation h p h relation dimensionality distribution crucially wide exploit graph express stick break mutually necessarily k g follow satisfy interpret integer stick break atom support aggregated distribution local center local dependency induce dp distribute specific ccc model location independence relation u share q independence relation long among provide interested infer turning factor moreover among integrate partition h e k provide prior quantify finite index ab variation measure define simple derive distinct location due turn local exhibit variation factor extra govern concentration dominate turn stage vb vb across location factor vanish correlation increase either ratio vb dirichlet fully retain factor introduce vector I multivariate kt kk need taking support total factor ik give realization mix increment condition u claim local issue identifiability factor additional u function global nontrivial true identifiability also depth observation specification base inferential behavior essential sufficiently tail domain place factorial job put shall describe nest hierarchical sampling marginal approach dp stick break former characterization conditional leave detail approach dirichlet addition issue dirichlet mixture aspect membership integrate reader notation turn local exclude leave leave denote leave also standard distribution local factor integrate directly reconstruct mcmc construct unbounded finitely represent quantity computation density combine generate previously new value likelihood value use f fy dy tn always proportional become local exchangeable last collection change change mixture membership tf fy main stick break instead likewise measure dirichlet k conditioning equivalently stick break distribution associate location kn explicit g sampling involve variable atom mixture count suffice markov chain end factor disjoint subset conditioning collect item group component dirichlet concentration correspond form let index example tractable long computational conditional density exploit conjugate computation conditional alternatively independence model problem computation readily available tf uk latter still alpha markov inference cc cccc ccc cluster dash solid marker ccccc entry depict cluster average figure set spatially cluster population factor spatially vary mixture normal generate kind encounter tracking problem covariate snapshot particle point move switch identification know move smooth cluster move path illustrate variation number cluster location generate simulate longitudinal draw relatively gp exponential split value previous global normal generate observation clear cluster location give different essentially specification take gp set fig datum fig b specifically support local cluster wider credible band specification factor perform sensitivity weakly across factor despite still estimate somewhat hybrid dp encourage expand concentration extremely reason robustness global second nest equal robustness record logarithm subject cycle day interest day assess time global identify global pattern specification set cluster close addition match group variation day elaborate subject cluster average day interval find indistinguishable sharing range last separate distinct regime cluster sharing drop hybrid process dp perhaps literature global hybrid dp curve subject reveal sensible measure illustrate grouping complex specifie cluster functional curve observe example curve switch two fig probably due overcome proposition consistency worth note hybrid practically directly describe nonparametric global locally nonparametric solution dirichlet virtue global dirichlet center support moreover cluster process local cluster spatially whose dependency canonical dirichlet rich behavior find particularly adapt manner mixture direct local cluster probability atom lie take condition regard hold fact density choose incomplete value gamma probability interval overall away follow west obtain equal variance standard metropolis step prior specification characterization posterior integrating rather index reconstruct think thus markov space principle explicitly play role give data th location conditional previously value takes previously calculate possible f uk fy u dy take tn future always proportional may correspond within treat last index group set tf fy auxiliary draw equal gamma prior
table row average middle row proportion bottom sample package specification w rv scale axiom theorem condition theorem exercise notation summary edu rv tail inverse heavy tail cumulative formula author usefulness introduce implement publicly theory link I error pattern often many white model practice exhibit wind internet traffic datum particularly notable signal heavy develop necessarily tail cauchy stable forecasting long process heavy attractive perspective viewpoint successful assume understand transform rv rv vice versa thus knowledge software still optimally include normality special testing parametric transformation estimate efficiently heavy tail transform choice model convert heavy illustrate researcher make avoid whole base tailed statistical tailed fig modeling nice subsequently technique fig semi parametric suffer drawback sample limit meet transformation restrictive identifiability condition three fold meta tail tail implement statistic author pdf introduce many study tail useful unimodal heavy confirm also benefit remove heavy tail exploratory ray density estimator detect remove finally methodology proof computation simulation publicly wise version jointly still well behaved deal skew cox mle limitation negativity many limitation shift show box cox transformation cauchy fail half desirable underlie process discussion box cox cox lower variance contrast framework tail remove heavy difficulty heavy tail rv rv cdf vary infinity characteristic basis rv heavy tail parameter skew respectively transformation tailed tail tail great inverse express pdf fall specify numerically approximated parameter match empirical quantile recently analytically tractable essential spread long ease result strongly skewness skew adapt skew fig generalize rv rv parameter tail parameter gaussian rv define rv input rv continuous rv rv transformation w rv shape away increasingly tailed value far heavy tailed tail necessarily fig lead property transformation dependent unique remainder state otherwise e implement recently appendix eq transformation available view popularity tail heavy tailed compare deviation see kde pre bandwidth use likely also kde kde kde figure transformation heavy axis remove axis operate degree tail map generalize closely skewed w ease location heavy rv yy f g yy w w z family allow bayesian statistical various different equal solid black tail dash color cdf quantile standard estimate equal family see quantile compute quantile software package useful education tail statistic cauchy stable yet transform via previously method transform tailed world e quantile straightforward tailed cdf equal cdf normal theorem functional student degree freedom often heavy equal student rv scale always match student tail heavy tail w distribution opinion return inference regard moment easily heavy student identically pdf quantile inefficient replace fast usually introduce size use reliable quickly accurate size yy maximize specification eq decompose transform q necessarily decomposition decomposition mle must calculation transform trade transform versus extreme figure show contour transform increase transform datum close input equal likelihood penalty monotonicity red green monotonicity imply black numerically existence uniqueness assume remain form continuous twice etc usual let case mle z principal real satisfy say enough heavy student problem unknown contrary get truth support disadvantage priori specification heavy tailed estimate distributional assumption present base back e analogously skew heavy skewed entirely supplementary back comparable estimate heavy tail rarely numerical confirm fact magnitude property classic standard low sample good mle clearly outperform heavy prove tailed inference limit quantile ml show usefulness demonstrate cauchy sample heavy w excellent daily section heavy tail pattern law know poor location go symmetry cauchy heavy tail two extreme ml fail summary statistic gaussian average excess normality value version fair comparison use well influential affect relevant good already clear location approximately toy work nice use observe heavy tailed transform stand min rr lot financial negative skewness financial model skew student generalize upon implication avoid derive heavy complex far beyond scope direction unconditional figure log return daily table confirm heavy sensitive skewed skewness zero double tail versus computed ratio freedom likelihood double equal double pay lower transform twice give significant fit reject b autocorrelation top right kde normal plot make decision trade negative significantly ignore heavy reject table heavy tail heavy mean respectively essentially scale lead conclusion exist case moment financial literature finite fourth study financial datum actually fit light heavy fit ccccc est pr est ccccc est ccccc est se pr ccccc est pr est se pr study would back indistinguishable gaussian von successful adequate log return trading close truth non even small treat optimistic observed tailed heavy tailed joint heavy tailed optimal researcher improve tailed datum preserve ease usage previous focus remove tail believe underlie gaussian convert assume might interpretability observe unit become helpful exploratory detect tail reveal improve inference image cutoff peak count rate rate datum package approximately day background gamma ray heavy tail make inspection lie drop end drop decrease ray detector sake comparison figure last observation cut amongst visually heavy tail underlie trivial heavy insight optimality cut heavy last heavy tail reveal mean cut equal transform standard fitting component cut tail would gamma ray rate analysis intend gain interpretation statistical ray count adapt skewed output heavy tail contribute unimodal gaussian skewed often assumption research direction perspective distribution view distribution literature discover statistic transform approximate versus tail show direct tail tailed practice I package available facilitate acknowledgment thank attention give detailed suggestion supplementary material list property definition z one relate simplify decrease remove heavy tail gets linearly hold line equal derivative use w gaussian evaluate rest maximize loss multiply yield since square sign conclude likelihood r z sketch stay maximizer occur
aim scope reasoning incorporate programming applicability metric logic long start observation triangular composition area find symmetry hierarchy dissimilarity precision tool mining build generalize ultrametric lead power tool algorithmic relate begin motivate hierarchical general geometry come play metric ultrametric induce datum datum implication conditional analyze conditional spherical ultrametric inductive especially computable shrink sequence generate master program real interval ultrametric systems section ultrametric dissimilarity set define map positive measure dx dx dy dx dx dissimilarity metric map ultrametric need endow metric instead dissimilarity satisfactory hierarchy term dendrogram define embed subset index subset totally require subset ultrametric embed subset constructive induce hierarchical pairwise property I iii q h positive include ultrametric dissimilarity dissimilarity viewpoint triangular metric triplet relationship particular hierarchy innovation capture understand notion anomaly take document close query target unlike query focus ambiguity record appropriate situation query situation illustrate ultrametric triangle small take query sort triplet define define ultrametric study equality side away sort explanation provide query novel treat raise subsequently handle dx dy reading hierarchy inequality ultrametric hold distance close distance ultrametric dissimilarity construct hierarchical newly criterion either agglomerative criterion e connectivity near pair reciprocal guarantee reciprocal prove agglomerative criterion near neighbor article journal les analyse des package information find dendrogram relative rotation alternatively rotation group introduction image case group cyclic shift permutation permutation cyclic group cyclic structure generative level simply subtree define denote product subtree alternatively look shift group amount root us space constant component component vector move hierarchical replace cluster disjoint couple denote invariance discuss haar dendrogram spatial dendrogram successive work build something figure vector inverse determine exactly read terminal detail signal coefficient wavelet dendrogram datum wavelet hierarchy regression entail hierarchy show wavelet input data haar dendrogram give approximation term observation I hierarchical hierarchical way haar characterize read give example give vector supremum sup chain clearly root partially order complete partial alternative closely relate domain endow ultrametric space motivation come monotonicity relate completeness set chain chain ultrametric space rooted ultrametric space consider ultrametric observation pairwise agglomerative hierarchy follow haar dendrogram embed haar dendrogram allow us chain point singleton haar transform call member increasingly approximate correspond two respectively set observation rest note subsection ultrametric ultrametric ultrametric ultrametric partially order generalized ultrametric distance value generalize dendrogram associate ultrametric dominate node could design whereas read dendrogram subset reasoning monotonic rigorous conditional sometimes relation program kind mapping monotonic reasoning describe ultrametric mapping critical usefulness ultrametric ultrametric arise logic force monotonicity monotonicity operator metric study problematic program idea examine I partially order set negative finding monotonic operator logic database introduce imply enhance monotonic long applicable overcome technique analysis argument latter include metric ultrametric ultrametric discuss subsection ultrametric join comprehensive background hierarchical relaxed often object real object dissimilarity attribute characterize object individual etc etc notion join set presence dissimilarity object attribute get distance value prefer treat get three consider
tie inverse draw way define dirichlet random construction draw form increment nonparametric stick break draw put construction considerably connection specie reference readily context bias generalise direct modelling density formulation weight component specific take algorithmic ji j limit joint dirichlet subsection strong statement independently view integrate conjugacy multinomial integer b b atomic joint equivalently draw set extreme preference equation number neutral population factor account allocate lead together familiar formula page partition imply derive consequence sufficient also context integrate take cluster specific methodology derivation dirichlet exploit make notation use label partition lack thorough methodology exploit bayesian highlight procedure mix respect thus dirichlet terminology dirichlet process formulation start implicit locate notable apply rapid power researcher routine computation example visit far exploit bayesian fitting margin substantial dirichlet aspect modern development throughput capable dimensional quantitative genetic biological sample gene expression gene datum replicate typically condition interest specific although variant gene dependence gene condition exchangeable hyperparameter strength efficiency nonparametric counterpart instead describe variation across population gene prescribe model dirichlet consequence atomic share probabilistic cluster expression profile normal conjugate explicit posterior marginal multivariate continue section real essentially univariate unable handle flexible drive consideration mathematical point numerous exploit limit behaviour component rank limit law rather establish limit law see rich volume take poisson pd back stick stick break define gibbs kind effort dirichlet alphabet rich sometimes area perhaps stick break representation dependent covariate book excellent development result surely discrete effective achieved impose probability branch set smoothness approach literature markov mixture without reversible jump demand handle reversible jump obviously appealing integrate chain solely go well reinforcement generalise term p show side far concerned particular computing inference arise define method aim target conjugacy posterior dirichlet begin sampling later hyperparameter together cluster conjugate probability partition partition likelihood expression interpret allocate cluster inverse mixture explicit form limit item require available conditionally possibly independently necessarily identical c sampler likelihood partition multiplicative proportional dp c multinomial er dirichlet ease motivation dirichlet model equally restrict form probability item propose study kind need either fitting demand expect shape methodology level purpose cluster consideration common one foreground imagine cluster belief represent exchangeable would parameter section kind aim exchangeability stress information item draw purely describe class henceforth colour variant cluster colour dirichlet dp independently colour pair mixture different measure colour cluster identify observe cluster stick colour segment stick break I define distribution share content leave atom within colour generate expression simplify significance ignore degenerate ordinary clustering remain analogously draw availability partition gibbs item accordingly colour n py ik expression simplify many gene cluster exchangeable think view k leave label background two regular mass colour p readily adapt regular illustration background vector measurement distinguish wish ss profile j particular probability heterogeneous measure include mass gamma hand density covariate mean variance univariate scale joint eq j km g h tt ks density methodology expression gene record period development system point day stage day obtain totally cluster phase development take represent linear separate phase singleton discard last partition theoretic pairwise negative
increase small mr iii monotonically value hold satisfied right side r meet exist treat r rr give eq take account whereas equality second factor q tend factor tend replace argument tend result numerator converge proposition cm estimation inverse slope interest motivated significance use asymptotic low asymptotic allow number meet prescribe quality irrespective minimax asymptotically estimation sampling asymmetric trial arise branch engineer measure use rather normalize square normalized analyze confidence associate require parameter advance sample sample bernoulli sequential procedure estimator variance procedure efficient rao sign multiply asymptotic different plus observation minimize observation error appeal stop many necessary obtain exactly random observation sufficient stop useful namely function regularity whose arbitrary guarantee exceed asymptotic error associate prescribe irrespective paragraph equal specific costly example nevertheless time second assign accomplish generalize slope weight positive normalize version ratio situation proportional large symmetric name motivate inherently subtract minimum unbounded error symmetric represent scale risk loss symmetric ratio normalize dissimilarity generalization allow multiplicative side natural representative incur production certain device production presence result pixel systematically incorrect expensive discard adopt produce sensor accept correct part process camera camera desirable sensor classify produce sensor depend merely acceptable advanced whereas type production type deterministic sensor number control primarily line sensor require need advance make actual great sensor either resource camera inverse binomial case turn discuss minimax asymptotically proof success random inverse binomial normalize incomplete define follow relationship binomial distribution similarly function parameter pi ip q express estimator risk take identity seen analyze arbitrary follow consider see take follow stem strict hand greatly simplify case apply single namely reduce addition consequence account seen see case close justify worth establish already generalize ratio prove desire exceed irrespective suffice choose illustration depict certain natural consider I risk guarantee criterion minimize estimator address value n include one optimum achieve arbitrary estimator necessarily risk monotone unique determined condition substituting make easily monotone expression computing theorem degradation see far degradation furthermore minimax minimize possibly ba determined tend follow establishes estimator theorem approach asymptotically consequence approximately sense fact commonly mean absolute minimax theorem minimax immediately stem cover comprised curve vertex ensure hold reduce iii strictly increase function increase iv defined express decrease thus j imply prove strictly decrease
parameter mean simulation discard first burn monte burn extensive testing period value summarize accurate parameter parameter mcmc delay rejection observation due fast speed delay produce computational equivalence run monte carlo ratios ratio monte method compute minute hour mcmc therefore expect monte initial benefit univariate quickly monte em simulated call observation method iteration delay value load matrix dataset n dd identity mcmc dimensional delay mcmc contain draw discard monte summarize delay rejection delay high equivalence score delay delay almost monte carlo consistent mcmc vs k replicate plot suggest em converge take hour summarize estimation k low optimization method ratio almost method note jointly exact article find delay stage acceptance table delay use decrease acceptance ratio acceptance high method case benefit sub iteration converge iterate replicate converge delay converge similar delay rejection around iteration iterates generally mix plot iterate initial delay rejection immediately delay rejection benefit block em start method delay fit example likelihood copula sub block agree sub block reduce univariate compare several delay rejection gibbs optimization replication volatility moderate unconditional quite optimization delay mle persistence parameter reason poor could wide suggest may far restrict range allow delay rejection factor score gibbs base delay compare term factor score fast become model apply discussion approximation closely suitable factor several delay loading univariate factor model student delay density tail parsimonious variance series factor economic portfolio asset pricing risk management determine particular volatility expense determine factor expensive propose delay fast optimization determine article simplify approximation estimate apply compare estimation considerably financial market cm cm delay monte carlo report diagnostic replicate acceptance hour minute delay stage ratio loading loading mcmc stage delay stage stage c carlo report replicate across replicate include stage delay stage report replicate monte include calculate replicate replicate replicate replicate replicate marginal mle cm delay rejection mcmc report delay rejection ratio panel ratio replicate univariate exp middle replicate second univariate example right column replicate factor p ratio iterate leave factor cm cm cm pt monte method delay rejection metropolis factor towards stage choose method computational particularly likelihood estimate marginal markov propose estimate stochastic especially useful monte method delay monte use financial economic series capture pricing pricing build existence asset capital asset pricing pricing asset second market asset west asset pricing multivariate volatility excess asset generate portfolio decision way become financial market quickly portfolio estimate literature bayesian considerable limit applicability factor hence article enhance likelihood dimensional difficulty dimension variance covariance time govern consider multivariate stochastic volatility whose distribution student allow asymmetric multivariate govern model model latent coefficient article estimate extend estimation chain univariate transform model hasting proposal parameter block scale mode adjust asset volatility govern autoregressive ar persistence ensure writing unlike error kalman freedom gibbs sampling computationally inefficient normal degree approximation metropolis hasting step write variance article component write introduce latent conditionally univariate equation various summarize initialize jointly density build target maximize density kalman negative inverse hessian accept prior reject current retain next filter step build block carry scheme avoid parameter burden iteration severe iteration consume optimization order proposal h form similar monte carlo univariate treatment right walk proposal likelihood ratio bridge focus feasibility carlo univariate factor model literature latent volatility identify follow factor mutually uncorrelated structure govern first estimating factor mcmc execute mode freedom accept candidate sample space indicator note bf decompose jj jointly sample separate jointly simulation computationally slow delay delay optimization find proposal mode delay rejection build proposal use walk stage value reject load mainly sample discuss dimensional although build two rejection stage efficiency chain monte adaptive stage reduce suggestion first five delay rejection method second split block contain number sub block last update optimization delay delay rejection univariate generalize evaluate represent univariate monte carlo use gibbs burn expected respect f ff ff maximize since conditionally equation substitute practical issue determining choose marginal solution available necessary simulation need given factor marginal choose article use advantage later posterior median mcmc median numerator sequentially integrate auxiliary filter illustrate detail prior decompose marginal alternatively discuss estimate posterior necessary accurate
conclusion couple sound cross comprise coupling detect wrong strong coupling synchronization measure qualitatively result base entropy deterministic multivariate autoregressive var delay distribute lag model dr model well test identify correct depend determine random independent lag form frequency embed show table length htb driving regressor realization add criterion bic shorter demanding absence drive identify direction drive slowly couple detect one two embed vector linearly realization correct embed realization subsequently datum coupling coupling examine significance example positive driving exceed drive lack analytic coupling coupling use simple technique shift surrogate couple time surrogate bivariate series contain chance regardless strength embed patient patient patient b top left panel panel difference channel channel complex differ purpose test system order decide introduce large bias variance distance rather stable far turn simulated termination strict strict need lead quite contribute would detection noisy couple nature information series embed enough embed estimation able detect coupling adequate sufficient scheme purpose invariant cross mixed capable detect measuring demonstrate system multi eeg though channel channel right component measure use measure different coupling significance restrict bivariate apply indirect coupling acknowledgment ed co national general technology european thank pl eeg channel investigate multiple time derive continuous discrete information criterion past purpose causality detect evaluate couple system eeg publication embed setting noise dynamic delay one projection theorem one relate embed context spatially system simultaneous variable location implementation reconstruct later extend series multivariate embed widely strength couple parameter series projection independent thus optimal nearest create vector couple dynamical uniform embed delay neighbor basically extension account characteristic different series univariate embed reconstruction criterion select parameter univariate building embed allow delay simple sound modified purpose multivariate analysis bivariate series scheme sec propose scheme strength result patient summary dynamical forming series pseudo reconstruct delay embed delay topological equivalence original selection goodness fit prediction criterion local delay autocorrelation lag n autoregressive false near embed series dimension reconstruct also choose unique even value minimum uniquely theoretically system thing reconstruction due finite reconstruction marginally observational reconstruction elaborate concern delay range reconstruction time dynamical include series investigation opposite often redundancy address use series prediction reference series optimal consecutive non consecutive component redundant information pair constitute redundancy investigate treat usage lag vary lag inspection combination determination computationally large moderate lag case evident practical need method embed lag collective select reconstruct property dependent dynamic system two time future represent horizon start empty step x I represent maximum j augment embed vector build mutual step inclusion even augment information explain previous univariate series modelling purpose create want mixed preferred quantity mutual vector delay lag univariate mutual appropriate dramatically million embed accurate information mi dimension base entropy near neighbor distance th joint space neighbor dx dd cube metric entropy joint denote immediately mi heavily setup use explain time joint entropy neighbor formula entropy onto substitution distance plus expression estimate note independently mi estimate project embedding rely examine mi eq data estimate bias bias since expect mi systematically reduce truly bias reveal time zero mean series regard fix dimension correlation reference embed accordance time top right denote last next dimensional range setup effect correlate degree comprise series study presence setup series vary identical expect lag second dimension mi fig case htb figure mi reduce fig mi correlation neighbor decide pilot study show though embed form time small mi denote mi theoretically criterion however formula perform well another serious mi criterion reconstruction term stop estimate one mi contribute inclusion component bias negative may theoretically impossible balance bias see bias comparable embed scheme variable xt xt xt contaminate observational white series interested predict variable obtain mi cycle result embed component htb explain mutual iteration solid lag dot represent lag highlight select lag exceed panel embed mutual information cycle panel mutual criterion embed left axis value respectively identical similar embed fourth completely give embed cycle decrease gradually gradually non multivariate embed couple information causality well implement compare idea causality contradict embed accord delay matter couple causality model causality aim embed depend adjustment may threshold increase series different span test quantify coupling strength future eq set near find prediction embed compute increase thus couple contribute varied ahead replace information give measure explain embed mutual opposite direction coupling series strength take uniform embed realization large frequency delay ahead detect effect threshold variation realization necessarily produce select detecting drive also weak coupling realization couple show htb map measure vector monotonic contribution shift small observational repeat select embed occurrence couple map select vector either component strict conjunction presence equivalent regard presence embed information transfer similar c well monotonically system couple identical map strength drive opposite apply coupling frequency occurrence map criterion strength predict expect fact embed conclusion strength turn quantitative measure table detect substantial coupling equal direction take couple respective measure increase weak coupling see couple give x ty use drive account significant delay frequently realization occurrence couple embed vector fail transfer realization frequently embed htb couple accordance form modify prediction monotonically slight start synchronization system coupling comprise component dimension simulation large iteration drive series enter detect coupling vs htb couple r dark gray detect whereas couple large variability change time enter embed form really system even give spurious produce zero system direction regard vector investigate future horizon I r embed either similarly vector weak coupling form make presence small
range code figure scale optical identification peak galaxy distribution intra distortion advantage distinct proxy detect ray candidate explore numerous ray dr survey follow optical confirm likewise arise ray always lie peak scatter ray furthermore contain nsf department contract ac sf theoretical physics ii foundation science foundation max high education web www american natural history university university advanced group university institute nuclear chinese sciences laboratory max institute institute university university university united david david particle laboratory il department university ann mi ann university il physics department university california physics berkeley national laboratory berkeley physics il institute national laboratory stanford stanford ca california department physics il laboratory york galaxy galaxy release identify red sequence galaxy feature galaxy field galaxy correct run dr large ever optical rich range completeness ray cluster range cluster member website decade universe confirm acceleration explain modification gr component pressure adequate perhaps simple possibility challenge explanation retain something dark possibility history growth central physics one growth abundance galaxy peak abundance encode universe galaxy cluster cluster galaxy detect determined ray emission optical galaxy impose galaxy rely physics though often mass ray emission potential high mass consequently relatively free optical require cluster search optical identify optical much dark serious projection optical detection volume survey existence uniformly old galaxy remarkably energy include strong break galaxy shift optical create galaxy galaxy varied sequence prominent galaxy remove field galaxy red galaxy dominate exhibit narrow scatter color sequence galaxy refer reference therein galaxy develop cluster galaxy gaussian identify sequence galaxy spatial galaxy around digital rich galaxy extend completeness test efficient cluster dr single challenge optical detection demonstrate red sequence color outperform step introduce construct dr use match know ray publish cluster test conclude summary future optical survey convention omit detect galaxy cluster precise uncertainty detection therefore optical effectively de calculate plane position galaxy along limited technology year optical galaxy cluster mainly roughly classify de projection cluster algorithm decade de l type projection magnitude smoothing kernel band adaptive band hybrid band cut enhance friend friend band band color band band band galaxy make galaxy method detect massive unfortunately maintain low intermediate contamination cluster create scatter derive technology greatly optical galaxy decade precise survey magnitude galaxy spectra color effective red galaxy narrow scatter color history determine position galaxy basically way de multi color obtain project detect directly color space principle machine color magnitude galaxy train reconstruct reach galaxy sense perform galaxy I compare different well cc cc color compare typical velocity km galaxy small precision possible alone projection insufficient remove cluster population magnitude color dr full right scatter tuned alternative stay color galaxy display color sequence cluster field galaxy red plus provide powerful follow section unique galaxy find galaxy wider include foreground galaxy cloud show galaxy magnitude represent fit purpose measurement proper modelling red traditional therefore correct gmm effectively parametric analyze span range galaxy red curve red blue background galaxy member line deviation gaussian galaxy component intercept sequence panel color color break across informative vary filter red color near band therefore detect cluster span wide color search adopt typical galaxy determine examine require determination candidate cluster broad chance low modal even reduce occur broad window member galaxy search adequate addition member application apparent adopt galaxy cut simplify structure around galaxy addition right band apparent concern define color galaxy color band degree contamination background limit relatively effect detection produce need color mean adjust definition filter galaxy center algorithmic physical motivation focus galaxy potential galaxy potential theory center simplify comparison theory galaxy galaxy center dark potential sense cluster center galaxy choice somewhat uniqueness act determine phenomenon detection search galaxy dominate motivate factor identification play selection galaxy give color list range color wrong choose inaccurate serious problem usually place near filter filter filter apparent adjacent color locate fall band combine red sequence either ambiguity impact estimate detection consider size mass scale keep consistent ideally computationally substitute approach attempt metric radius measure fix everything scale size member circle candidate galaxy error relevant fit color overlap galaxy around gaussian analysis conclude give well hybrid select red candidate candidate foreground galaxy belong sequence red red sequence lie deviation next quantify spatial plane radial kernel important analysis kernel bias shape member galaxy radius project choose regardless essentially peak smoothed field position introduce another weight galaxy galaxy band magnitude band magnitude correspond introduce indicator whether important double check minor negligible quantity calculate straightforward basically step every evaluate galaxy identify red candidate identify cluster finally search list scale procedure summarize essentially peak smoothed height peak find cluster galaxy quantify identify galaxy peak merge criterion contrast previous procedure motivation peak merge peak avoid star importantly peak indicator structure probe internal follow radius candidate fall inside merge set avoid merge galaxy foreground star identification algorithm color optical completeness test match inclusion filter addition vary grid testing select maximize filter statistically motivated feature radial serve filter less biased cluster priori detect advantage algorithm execution neural near polynomial measure color reason easily apply release digital construct optical rich cluster quality cross match ray create dr completeness detail construction digital color imaging dedicated point mapping dr include area unique object identify paper survey area calibrate degree north south input galaxy http server galaxy band magnitude less clean galaxy galaxy neighbor galaxy bad band great principle galaxy candidate subset list color additionally band candidate cut galaxy take color false project cut keep galaxy search effectively color worth note galaxy dr relatively contaminate star galaxy weight reject range color determined galaxy within change background see measure make across whole range relate two mapping however simple color measure narrow range percentile bin bin scale scale figure relation scale affect rank strength unless note cluster strength rescale scale scaling remove much measurement band generate full galaxy dr search reduce effect weight strength much weighted image cut figure contaminate star remove star pass star datum processing mask star star mask add galaxy dr mask fall inside mask full release refer coverage tag public cluster cluster figure cluster b tag name unique galaxy dr gm galaxy inside gm recommend use galaxy public cluster previous section apparent bi situation dominate impose modal potential long dominant modal narrow width color separation vary change overlap overlap project galaxy cut galaxy appropriately figure color deviation sequence cut impose member coincide red equal get select galaxy galaxy contamination weight automatically count cut always count demand leading recommend public tag finding criterion completeness quantify quantify whether however calculate ideally dark matter issue high resolution color interaction dark create simulation prove factor resolution galaxy color term observational realistic cluster put addition check completeness ray cluster uncertainty full accommodate realistic background widely similar step realistic dr rich galaxy input remain keep color property unchanged create whose range match ray visually member galaxy pick number member position color remain unchanged select corresponding galaxy
property bayesian social version segment explain ph distribute say lot region simplex segment simplex iv line set condition statement nice belief outside belief lie outside numerical example iv set twice show iv figure right policy decision maker alarm delay take account decision maker cost pick continue delay operate incur decision social choose decision signal pick event equality scalar actually choice similar constrain optimal bellman hyperplane lie decision assumption decrease continue hold theorem sufficient optimal polytope assume states polytope polytope proof establish omit ph decrease reason even though characterize example consider suppose lie change ph transition satisfy optimal characterize interval ph increase hyperplane preserve structure stochastic obvious dimensional parametrize linear parametrize ph line define appendix lie stop belief iff ii iff increase linear policy leave include sense policy threshold curve threshold nice triangle require threshold example threshold threshold threshold line show hyperplane lie polytope h cc linear policy close resort stochastic aim linearly problem eq initial convenient solve algorithm deal constraint local optima necessary several iteration sophisticated base applicable solve stochastic problem likelihood specify completely protocol multi controller network previous section index act agent act choose stop early false alarm operate mode micro mode obtain observation mode equivalent sensor time state time instant achieve agent micro management detection view macro operate belief micro micro interact decision micro determine decision macro sensor mode avoid solution sec social micro pick sensor convex agent depend mode observation conditional integer unlike view mode belief measurement dependent py probability depend state polytope mode constraint assumption base micro aim detection view subject belief evolve sec agent pick macro bellman theorem detection proof dependent observation observation yield mode dependent consider follow assumption sec recall denote lie c regard decision macro relatively inequality hold j accord appendix present early section present illustrate multiple mention sec ph prove theorem geometric example change change social global choose delay check hold comprise triple compute construct point implement phase change illustrate threshold curve learn ph belief simplex triangle social local cost choose discount model chose geometric since indistinguishable markov transition plot ph transition ph geometric distribution type decision solved imply converge theorem plot hyperplane polytope segment hyperplane lie thereby ph ph therefore hyperplane phase depict example motivated understanding local perform related agent scheduling unlike optimal multiple result detection result threshold behavior simple ph sufficient hyperplane simplex give switch nature straightforwardly underlie current work social order agent market background reference stochastic proof theorem iv optimal social monotonically monotone order restrict simplex order since preserve ordering let stochastically dominate denote suppose increase ii state state order ordering segment connect summarize useful least great great comparable chain e submodular function submodular increase e tp order tp multivariate mass p multivariate say scalar th dominate function associate classical associate kt induction iteration complete induction value pointwise step induction note positively I bellman since concave function preserve concavity therefore concave concave condition filter pa jensen detailed version lem yy c intersect namely p tp b b p increase c b see imply b b definition iii immediately e p e single verify bellman read threshold state update e imply need partition namely introduce interval lem furthermore imply recall straightforwardly verify b b tp imply symmetric claim hold statement straightforwardly statement return proof piecewise four interval vector crucial social since linear concave piecewise result interval follow define proof comprise statement I ph every belief segment belief segment intersect hyperplane straightforwardly establish since statement ii increasing ac decrease appendix statement imply learn belief hyperplane formulate belief polytope statement vertex clearly normalization imply I proof induction algorithm suppose imply pointwise initial step bellman trivially iff two part preliminary public update bayesian filter tp need fix zero non assume say either case equality identical suffice b lr l tp course condition say tp tp prove tp imply ia ia bb iii row first iii ia iy iy appendix tp straightforwardly belief sensor management tp arbitrary tp ph mathematical induction iteration clearly decrease polytope choose increase inductive decrease polytope kt v q k uv complete finally pointwise decrease polytope polytope definition decrease need show decrease hold equivalent iff title title lemma global interact stop decision via agent record private noisy process optimize maker four base detection general decision detection threshold lattice hyperplane curve sensor detection view prior optimal scheduling decision detect optimize alarm delay vast application finance team involve countable agent suppose agent act receive compute posterior subsequent process stop decision maker monotone belief exceed understanding decision consider decision maker achieve change global outline multi recent learn classical act sequentially distribution change decision subsequent agent choose optimize utility local observation agent public decision set decision multi underlie decision stop detection adaptive sense sensor equip local sensor controller multi system act underlie sensor sensor belief decision agent management tracking surveillance case individual sequentially fashion dynamic classical detection agent example generalization decision determine state determine decision continue determine instant decision lead belief decision threshold stop region fig visual optimal policy illustrate threshold policy horizontal posterior continue multi change thus local fig b programming unlike show fundamentally policie global stop continue local decision characterize triple value last decade social study economic financial market social see numerous paper limit excellent exposition learn important decision agent irrespective cascade motivation understand interaction global learn recently several information global paper address utility system design behavior give behavior theoretic involve equilibria ph ph ph geometric ph distribute since ph find ph ph change time social detection change analyze systematic investigation distribution deal characterize policy make perform organization paper present protocol problem formulate social classical kolmogorov classical sec stop term concave theorem measure although make nontrivial need cost entire trajectory classical policy start analyze belief theorem posterior sec probability small explicitly optimal decision policy change zero ingredient proof result fix social update region cascade social characterize policy maker detection ph ph state state systematic markov ph geometric ph distribute bayesian lie multidimensional policy ph kolmogorov question sec sufficient global decision characterize hyperplane multidimensional threshold kolmogorov sufficient characterize switch multidimensional lattice structural result involve stochastic agent belief respect threshold curve monotone ratio partial order policy linear sec agent detection optimal detection agent constitute decision framework sec incur decision maker detection sec sec estimate underlying act view instant act local agent denote local agent define sigma comprise chain point ph ph distribution approximate change time markov state evolve jump jump distribution occur one assume matrix state change appropriately choose ensure state time geometrically agent observation time indistinguishable observation belief agent belief past hide hmm filter denote public local formulate denote non incur agent pick state agent expect cost indistinguishable decision subsequent public belief agent public belief difference belief call likelihood likelihood filter local underlying filter implication belief due explicit contrast explicit aspect pattern protocol private observation instead decision take agent private likelihood probability explicit belief aspect social sequential public belief proof hard argument belief proceed briefly describe public belief interval ph belief simplex simplex simplex course formulate maker eq measurable define policy cost since actually difference social learning filter social expensive classical social access thus compare define policy incur bellman equation recall bayesian social detection belief versus social formulation say belief social incur higher consider time detection classical initial incur small social explanation lose symbol proof appendix observation value concave establish jensen bellman prove optimality policy function detection problem apply incur social comprise list partition convex likelihood specify detection policy rest ii sec consider change geometric double threshold sequential public characterize decision likelihood probability though decision let exclude recall polytope row matrix characterize global recall tp negative tp negative false alarm e submodular sec ph states example classic book distribution etc example iy py iy iy iy trivially numerous example assumption sufficient assumption geometric distribute time view maker need ph straightforwardly obtain matlab local always dominate standard decision yield state e cost submodular important appendix rest partition polytope belief theorem polytope denote decision matrix insight assume matrix define possible decision hyperplane ensure hyperplane intersect simplex ensure intuition hyperplane imply mean hyperplane intersect nice straightforward otherwise hyperplane multi simplex iv appendix hyperplane vertex lie polytope fig social double alarm cost change geometrically proceed behavior theorem corollary plan eq observation optimize satisfy bellman equation interval belief theorem filter characterize state point composite implication interval region dynamic characterize global also cascade salient feature symmetric right represent evolution public sequential hold trivially global q value cascade iii symmetric decision concave implication comprise three interval second public irrespective irrespective public observation subsequent agent notation cost incur matrix denote define interval recall associate optimal transform deal prove existence threshold policy optimal invariant characterize incur global constitutes implication social note cost region region simplify allow tight discount theorem piecewise value iteration element piecewise apply structure also cost optimal double threshold learn policy cost policy almost distinguished policy
precede separate split separate separate independent n nk bind cell constant probability completely radius entire end interested appropriately p I give ball hence radius intersect ball large disjoint cell less know level observe child success second variant covariance definition go guarantee small expectation point cell contain randomization albeit whereas max level dimensional riemannian radius contribute present trace fraction trivially set neighborhood manifold concentrate nice logarithmic improve packing room packing aspect ball radius aspect cell pack lemma moving result demonstrate constant attain mention packing application search however require max primarily right exist alternate space partitioning split partitioning split alternate split forced fraction child thus ensure depth point depth maintain packing space seem theorem leave max structure simultaneously reduction factor level lee point usage version thank research department usa institute technology lem lem lem lem structure adapt notion dimensionality new result level reduce packing structure low manifold local covariance consequence manifold curse direction lead etc almost useful assume turn euclidean freedom available dimensionality several automate capture actor marker although recovery frame dimensional body exploit extensively result example typically assume knowledge intrinsic dimensionality adapt assume structure offer guarantee reduction cell bound two intrinsic variant already numerous recognition resolution structure new family space partitioning bar structure one go cell side box shape cell radius cell ratio length shortest shape ratio etc radius number disjoint intersect prove ratio cell guarantee play approximate neighbor search result max type present bind aspect nature difficult aspect cell instead pack specifically aspect small cell max completely structure datum show dimension result robustness notion brief datum present effective aspect arrive packing cited present intrinsic dimensionality ambient max dimension datum hence take dimension ball cover split lie cell radius direction project assign child z child approximation prove property cell construct subtree root section consider go case start level expect go result great away level radius follow extension radius radius boost level least c radius radius power fact number level solve recurrence typical partitioning guarantee size property build cell dimension construct subtree root every example repeat cover cover ball suffice consider ball center split separate contain ball cell center cell bad apart say child max inside ball child propertie ball inside max split separate split cell split proceed project onto length choose ball get interval getting split choose randomly interval radius interval simplify dx e pair ball contain contain split least completeness pick lie randomly real line let choice random onto projection etc ball project hold apply lie distance show probability projection ball lie far projection ball choice ball separate split let pair describe dimension split work require observation p follows exponentially go center separate radius ball denote cell level contain ball follow k require attribute level cell go split frequent indeed tell contain way inductive split cell depth bad represent notice thus give sd
occur intra differentially locate cluster gene position present differentially arise experiment individual array genome plus publicly yield array probe classify probe take value status test quantile distribution status discovery classify present status around infer site forest minimum decomposable result produce display network complexity infer decomposable model drastically apply around classified figure small contain b three cluster encounter around central cluster examine b figure turn intermediate level water holding stem water hold capacity report global assess array contain probe publicly gene expression analysis way ap absolute pearson correlation logarithm expression probe value significance pearson coefficient accounting criterion classify characterize expression decomposable bic vertex cluster procedure result classify ap large cluster examine uncertainty turn low uncertainty apparent display ap ap calculated model decomposable likelihood yield arise sample genomic probe phenotype group present publicly available expression distribution differentially decomposable present vertex procedure representation cluster present differentially representation display figure total differentially display proportion differentially scenario ht evaluate high variance co procedure describe decomposable model set perfect denote nn evaluate simulate index average improve increase argument indicate essential account intrinsic structure level discuss association expression effect represent infer structure use correlation correlation avoid redundancy propagate spurious gene connect information expression level contain chapter moreover essential construction separation two separate conditionally correlate c separate connect element element essential group separate knowledge expression branch carry expression vice clearly gene special graphical model period adapt minimization aic maximization decomposable graphical act independently b j relevance step towards find genetic microarray profile phenotype trait r evolution establish g mixed aic forest cope probe level york exponential cope probe l visualization paris national university trait correlate pathway candidate water capacity core team foundation statistical g aspect elimination reduce acyclic green decomposable neighbor decomposable scheme analysis molecular diabetes de technique characterize differentially gene co model redundancy information avoid complex relationship allow make thousand typically occur throughput study take internal structure compact identify gene differentially gene less differentially gene locate region notion short relational pattern phenotype simultaneous expression several thousand recently produce gene work technique characterize gene redundancy idea value advantage informative illustrate level determine sample g microarray statistically difference produce naive search association level correlate association spurious might exist change association disease status level gene mechanism disease expression level group gene group association status spurious usually determine express therefore suggest take account level gene experiment I co differentially association association gene gene locate expression whether involve association involve differentially locate less region potential association position gene expression affect interpretation differential informative locate central might aspect perturbation expression discussion summary aspect interpretation differential assess position differentially express take account differentially large gene expression thousand gene possibly form network co expression graphical study connect graphical conditionally gene network similar proposal conditional step graphical expression rich decomposable co expression compact well differentially expression network differentially express informally involve gene expression datum gene graphical way bic criterion restrict decomposable making package aic bic yield optimum condition briefly report sophisticated characterize take internal classic exist enumeration enumeration perfect recursive procedure several sequence perfect respective assume independent realization level mean nuisance concern give graphical decomposable clique greatly simplify calculation instance write clique perfect enumeration clique respective eq cardinality statistical graphical inversion gene individual imply singular decomposable graphical minimum bic consistent estimate ji span forest successively bic take calculate decomposable algorithm stop find stop else find consume technique decomposable cg ic edge set describe representation express co differentially differentially co network connect sub say term exceed pre threshold produce call k adjacent exist adjacent call graph expression graph component clique graph classified algorithm find gene output proportion find ic j j clique dense label red contain label clique clique directly solely clique connect form form cluster drastically idea construct establish differentially express argue
variance estimate carlo correlation cause statistic variance statistic via simulation var variance million replicate statistic behave array use method permutation microarray statistic permutation permutation array permutation column array behavior include study nan commonly need thousand variable create testing procedure control dependency commonly false fdr fdr fdr relax assumption slightly broad negative correlation step arbitrary dependency control fdr dependence dependence step permutation adjust direct estimation account correlation one estimate local fdr fdr average tail study effect row correlation study matrix model four fdr control discovery proportion four base empirical compare package repository structure matrix two row block diagonal simulate three reflect respectively size diagonal block j hypothesis realization conservative estimate proportion number hypothesis reject average ten also report test test test test test decomposition repeat several dependency among fdr besides support correlation simulation c fdr permutation perform similarly conservative control fdr gene reality around dependency conservative hand perform seem confirm dependency among column problematic present methodology inference remainder covariance key model simultaneous row close relationship take covariance covariance share eigenvalue correlation population among make accounting covariance problematic regularize accord model covariance model allow singular row variate place covariance concentration context induce penalty follow decomposition remove ij motivate discuss consideration column datum usually position placing encourage one dimension row penalty partial strong correlation row diagonal secondly gene penalty leave remain establish inconsistent estimate establish multivariate covariance estimation importantly norm eigenvector result reveal covariance de use estimate form accord correlation row remain correlated noise symmetric square let root adding correlate n remain noise row examine example look process datum favorable minus fdr un conservative fdr close fdr estimation datum pre step advantage power gene microarray translate top improvement simulation study nan inconsistent dependency microarray datum thus repetition gene microarray center empirical covariance simulate restrict variate mle mle observe result r test test test repeat time value fdr without study improvement fdr estimation microarray simulation fdr simulation confirm give however still close ten repetition mean accounting appear significant correlation array diagonal simulation among drive pre false discovery variate normal decomposition surrogate datum coefficient estimate nature previously notation account datum covariance whereas additive model respective change ranking unlike capture row c test effect fdr reject method take repetition simulate batch follow zero give ij iid follow ik iid k batch column fdr method standard pre use available package simulation order test statistical power substantially pre specific test reject rejected change true reject problematic standard false positive however fdr close conservative variate processing processing demonstrate use statistical problematic method solve problem technique de approximately row reveal simulation disadvantage fitting covariance penalty may currently propose un gene signal gene detect researcher interested approximation filter component testing outline property consistency remain direct statistic array examine conclusion several correlation major simulation permutation inference conduct despite support many statistic estimate covariance framework fdr focus prove useful variety taylor helpful also partly inspire test test test test test test test repeat ten fdr estimate compare vector scale independent distribute random trivial n w corollary trivial proof since proposition ij k x ij characteristic trace function let consider random nz n independent proof dependence consider form often column interest example array dependent commonly covariance distribution result problem solve covariance simultaneously give statistic scale microarray reveal offer area discovery variate normal often ease pool inference assume test allow assess testing gene simultaneously procedure theoretically measure dependence among able incorrect meaning row significance correlation row account differentially microarray genomic dataset complicated structure pathway correlate gene act negative correlation array many suggest arrange form matrix dependent microarray scale inference testing protein protein array generation sequence testing significance functional significance take subject several begin introduce paper first microarray disease array two microarray filter gene compare nan compare nan could nature thousand another cause could array array study effect separately column generally consider column lead incorrect turn result much problem dependence restrict normal paper devote effect interestingly column dramatically lead fdr seem fdr solve correlation covariance regularize sphere row conjunction standard covariance microarray reveal important test lead conclude discussion study present restrict go microarray gene differential nan correlation section propose study correlation simple matrix variate multivariate distribution independent identically array flexibility either independence correlation structure restrict variate introduce propose variate normal n mean array marginal meaning gene array row decompose eq meaning restrict variate microarray array two notation first array
univariate finally notice also situation interpretation observe element category size appear conditional permutation element sample among reproduce reflect selection speak equally draw number common unit across scalar distribution obtain express practice equivalent population random multinomial variable regard family hyperparameter model express count poisson conditional specific association put characterize product similar assessment variable also sum px b direct acyclic display beta discuss simulate distribution implementation base update step pt pt pt interest draw sec thus quantitie one draw assume population category x computationally difficulty directly simulate easily conditional simulate draw however associate justify mixing cycle order approximately standard update pf f b n k j draw first distribution play multinomial parameter either get two typically former uniform multiply complete obviously simply substantial rest fix eq total value coincide one easily truncate draw beta specify quantity correspond column conditional couple q set configuration eq conditioning draw key method able popular record linkage literature basically consist homogeneous group record inferential strategy record linkage usual record linkage represent create set theory produce match plug first statistical calibrate match ratio false match wavelet currently record linkage procedure pose paragraph record linkage interesting perspective formal posterior expect appear linkage context mean match formal decision theoretic seem necessary characteristic action select optimal decision loss translate measure linkage give define consist zero notice eq loss calculations ab ab ab ab ab ab loss optimal ab b g show easy minimize adopt expected stress theorem consequence additive say decision perspective control type loss account match reasonable linkage actually error miss true practical fact denominator practically data file consist record census population record post enumeration survey example population represent category block represent gender education category total key focus example allow illustrate hyperparameter assume support use supplementary trace distribution measurement probability area record linkage estimate people census outline code hierarchical gender education age category variable distribution match new box plot implement illustrative namely hybrid population plot panel posterior wide interval pattern remarkable fact correction reduce clear conclusion aim estimate quantity rough end panel obtain sum least record survey fact introduce concentrated estimate accounting match large panel evaluate hierarchical study often linkage computer six scenario different population always scenario key category case frequency equal b ib ij j category contingency ib c c variable scenario pt pair hierarchical discard constrain model focus group pair mean close process misspecification element hybrid trend observe fact produce coverage expect dramatically level estimate wider provide partly comparison interesting size three scenario scenario table match comparison method particular criterion well performance follow however consequently criterion prefer constrain conservative perspective favor single measure record linkage criterion produce low linkage pose perspective framework within comparison record methodology statistical also play extra inference framework analysis approach answer strong assumption situation allow assumption inference perspective soon size intensive simulation approach problem popular linkage value error computation reduce computing build actually categorical finite boolean place stress provide survey naturally record linkage model example association important benefit incorporation matching process record sample confident extension capture use advanced incorporate handle multiple exchangeable record addition measurement file may also develop block separately evaluate allow strength extra layer extension future approach handle aspect record linkage address use list provide problem scope believe idea generating might context note linkage entity resolution recent relevant paper statistical model present marginal sum block matrix f f j j f b j bn f f j p f b f f b tn f j pt b b pt f n tn f f acknowledgment comment suggestion substantially version manuscript material file contain file cat file pdf show section di hierarchical statistical linkage size linkage statistical actually available uncertainty plug correct uncertainty size linkage motivate simulation datum source ease lead great popularity merge record linkage refer relevance record linkage highlight significant paper hierarchical record capture size population interest capture current population remarkable mark abundance propose unified uncertainty naturally estimate approach size respectively comprise census census comprise census enumeration datum report gender education perform also number census unit file agree perfectly observe might agree second record true moment summarize distribution produce dramatically different posterior pt n matching beyond relate response analysis perform link linkage entity two file different context happen unique analysis available furthermore may create set survey considerable lack different variable henceforth connect record miss record na common describe paper put kind problem far advance paper among introduce record linkage idea exploit paper record file comparison assumption fundamentally illustrate state paper propose bayesian observe variable always record concerned bioinformatics problem configuration match label situation bivariate sample record get consequence information represent unknown permutation arise material collection describe material come source record linkage attempt suggestion seminal paper structure record linkage monte method need loss methodology evaluate illustrative realistic conduct finally section discussion extension improvement record configuration categorical set data configuration sample whenever avoid subscript notation key variable element set arrange indicate sample population b probabilistic linkage perform assume conditionally py ab ab ab py ab u ab abuse independently comparison vector identically bernoulli assess gold available maximization analytical setup decide whether single posterior pair formalize approach opinion several pair threshold however problematic illustrate record
accelerate carlo carlo sample inverting use monte seem limited place control large demand either accurate importance absolute proposal approximate probability seem overcome frequently encounter proposal tailor nevertheless show correction importance sampling approximation overcome accelerate monte carlo confidence interval correction negligible p test confidence interval approximation guarantee conservative monte carlo still combination turn crucial many application variability control amount importance algorithm nan specify compute carlo analytically distribute support distribution function density approximation extremely importance size dimensionality reader detail importance simple correction importance observation correct approximation value p hypothesis distribution carlo correction reach fail properly ability sample valid generalize discussion become derivative discrete case mass simplifie ratio allow choice statistic permutation mathematically write test argument first argument invariant want sequence way statistic permutation center everything desirable context improve balance precise probability nonnegative version nz nz z z z proof special among nan validity correct practical validity well direct chain generate approximation hold different say distribution generalization marginal distribution random drawback correction randomly increase randomness already generalize theorem proposal require additional marginal place fact valid nan q confidence calculate appeal invert principle weight interval point create clear counterpart essentially practice concern correct behave choice concern target value approximation correct careful express interpolation respective non value shorthand suffer since alternative problem space typical avoid goal correct behave quite bad look closely affect correct interpolation valid validity see generalize abstract strongly conservative interpolation effect versus alternative keep setting often accelerate however play determine hypothesis take advantage ensure rejection favor sensible ensure event heavily weight turn interpolation always correction little power substantial ratio important value classical importance heavily case however functional design whenever proposal range confidence contain matter detail il perhaps dataset interested dependence multiple perspective give tx tv nan permutation test composite nan nan conditioning value become permutation use I single particular ensure probability compute exactly prohibitive monte carlo sampling accelerate sensible value real case label I standard cauchy plus shift tend poorly alternative sensible statistic associate size increase refer likely approximation case final refer correctly c c c incorrect guarantee valid detection drop acceptable furthermore determine large fail error correction useful unless validity correction work properly regression treat nuisance hypothesis remove nuisance sensible q detecting detect combine tail p define direct design sensible normalization validity combine usual way invert test confidence inverting need increase create confidence sampling suffer proposal might practice mixture conservative importance conjunction permit proper practical importance proposal detail hypothesis neuron control family report confirm one qualitative remain scientific conclusion result improper hypothesis practical benefit clear p correction false high variability correction behaved importance advantage interval construct inverting monte always diagnostic poor approximate standard issue p correction approximate sensible approximation correct p estimating permit interpretable avoid indicator appendix additional example demonstrate approximation close converge extremely software generator unable modify software include quite diagnostic purpose expect correction example proposal improve importance budget combine much diagnostic surprisingly maintain significance level perhaps target importance continue gain fit theorems sophisticated monte diagnostic completely ignore rely assume statement trivial assume exist th th begin nonnegative function permutation inside lemma complete two surely need permutation measurable second equality distribution fu wu complete nonnegative variable proof expectation wu wu theorem conditional inference target observe make increase choose binomial leave observed value permutation target assign higher tend label value section inference neuron depend event define set time temporal neuron probability case event case time prefer time lag test matlab execute pseudo generator logistic avoid record nu approximation cdf quantity monte use importance valid valid hypothesis line total plot panel dash cdf experiment involve around p cdf dot plot close since cdf diagonal drop indicate cdf exceed indicate hypothesis correct always valid seem although small computing important time none job even conservative strongly conservative desirable mse describe situation importance extremely slowly sum nan distribution class configuration tend row sum index distinguish nan alternative distribution sum suggest proposal successively sample basic would symmetry value standard although normally particular square mean weight extremely much reasonable direct agree heuristic solid line change seem end take report dash black excellent value plot value importance approximation fix statistic choice thin figure thin line gray thin test case special symmetry sampling choice row remainder equally line despite identical estimate reporting report times sample importance correction computationally demand simulation assertion paragraph illustrate less demanding computation approximate much extremely correct valid correctly preserve interpretation approximate repetition dot line cdf visually indistinguishable second
load negative therefore thousand simple matrix costly furthermore contrast lag involve non factor moderately large identifying contribution reveal loading result one rate loading error exhibit curse dimensionality offset via common factor improve asymptotic estimation strong characterize explicitly index depend relate manner forecasting organize method strength exist section extensive simulation imply technical proof time unobserved factor stationary moment loading matrix white otherwise combination example lag relax correlate past white noise appeal economic know body unchanged loading uniquely uniquely accordingly advantage particular convenient explicitly section measure strength factor instance condition k root smaller introduce small integer practically asymptotic remain constant go full include imply strongly weak factor link explicitly factor slow weak factor facilitate loading column orthonormal follow unless condition obviously negative ready specify loading estimation note unchanged tr definite matrix rhs q lr use load orthonormal eigenvector eigenvalue specify li estimate factor convergence estimator precision original convergence k pn l pn pn l x pn auto loading explicitly k op explicitly strength affect fast strong curse dimensionality offset factor implicit essential introduce presentation sense deal convergence directly two introduce variance vector away assume uncorrelated fix maximum different pool see k grouping look variance model constrain covariance specify convergence hold indicate difference matrix estimator even factor estimator frobenius factor advance numerically however perform sample counterpart provide inverse unbounded factor weak spectral irrespective fulfil error surprising hand study frobenius transform frobenius norm transform frobenius little strength rate estimator investigate presence factor may treat view group jj unless sequel continue estimation much eigenvalue typically practice distinguish presence circumstance strong factor weak propose justification method factor strength ignore obtain ii remove derive assumption ik ik l pp pn pp ik op loading may benefit fast op practical implication residual initial especially frequently method p theorem indicate normalization reflect norm estimate factor derive eq manner similar bad strength differ substantially hold method property via simple illustrate loading element hence list similar htbp cccc indicate theorem sample table column consider level factor distribution adjust strength normalize column th vector component either properly factor replicate calculate standard deviation use estimation right factor display replication show strength good well optimal instead factor fact quite factor strong factor simulation much similar similar pattern show show covariance estimator yield illustrate develop dynamic microsoft imply volatility surface obtain question imply w tu display period imply cross delta htp fact imply stationary amongst unit root reject course perform still favor instead p since perform window day satisfy methodology loading forecast depth well forecasting take advantage lag report htp window ten figure display eigenvalue window side large microsoft right hand second procedure graph apparent eigenvalue number company choose dotted procedure take display cumulative rmse window treat ahead except marginally consistently outperform benchmark microsoft triangular express upper rp lemma factor let factor relation r pp covariance sample hence similarly main q span sep orthonormal basis op definition function since large hence conclude exist hence order specify first ph n pp consider ji bound infinity pi ph op q finally use look rate definition note spectral theorem rate since r pp p need formula j noting note q op eq proof procedure want first find large small note size r op op l proof theorem l hold op also rate proof thus omit form unit eigenvector eigenvalue third note lemma conclude similarly order proof previous idea find use get
directly require sample approximate estimating become slightly context rare random fit chain approximation base asymptotic differential equation equation scheme method system consider change family change minimize time occur sample simulation suggest expectation expression system expression importantly fact basis formulate entropy execute execute change measure retain transition suppose kullback change substitute cross occur program apply lagrange multipli transition notice e direct fact denote start ease times return follow transition follow b iii follow return iv follow reach q hence transition term expression repeat update refer ce ce ce ce opt constant approximation entropy opt ce condition importance asymptotically opt ce opt opt ce upper use third equality illustrate service associate continuous process rare return probability eq transition probability reasoning proposition calculate numerically entropy iterate uniform transition size increase update event relative ce estimator estimate ce ce ce ce opt ce strong efficiency bound opt ce ratio construct simulation key consider rare event problem chain engine importance coincide variance sufficient estimator strong acknowledgement would visit algorithm assumption correspondence approximation simulate entropy asymptotically deal finite fail form set internal internal probability markov good start formally notation denote markov jump state
start dual problem condition entry flip sign modify scalar set vertex arc arc negative arc capacity constraint arc flow arc capacity incoming flow vertex flow optimization uniquely characterize one group ii arcs capacity cost iii arc flow arc iv arc cost canonical variable problem sum cost lead equal arc match show solve include canonical simplified edge edge capacity simplification illustrate quantity present formal reduce speed present draw fill minimum thick blue fill place thick draw black mm bend angle auto xshift g var xshift distance node right si edge xshift right bend g mm edge group xshift mm var xshift edge node edge pre leave var var u pre node pre u bend group h node cm yshift mm var node h node pre u right node pre right si node xshift u pre node bend edge node yshift yshift h var pre cm right si edge node xshift pre grey square group group capacity zero infinite arcs graph zero capacity group hierarchy flow problem flow study simple single solve ball machine research tree overlap group difficult solve cost lead take specificity dedicate share nonetheless perform dedicated parameter flow solution projection uv j v tv u informally return flow proceeding solve replace single sum low bind flow try flow match reach bind reach cut define potentially flow arc arcs arc arc point arc remove yield decomposed solve recursively call formal correctness guarantee bad however problem empirical despite fact guarantee adapt see efficiency arc cost problem look call sequentially use heuristic implementation high graph concept initialize enable warm graph step balance modify step linear dual norm induce proximal duality proper f duality argument equal f duality compute find flow formulation associate arcs problem arc prove graph explain dual gr j uv e v e experiment weight set appendix namely sg regression sg implement run core ghz overcomplete dct organize grid family contiguous generate percent select sg interior method cast qp cp formulation size note qp sg obtain solution whereas sg hour reach gap background try segment foreground object combination plus error w p recognition fact neighbor likely foreground foreground put square image effect regularization maximize pixel channel figure improve remove lack structural neither consistency ccccc cm cm cm method foreground pattern mask image another foreground leave percentage ground bottom use hierarchical learn dictionary signal combination dictionary express vector structure refer admit subtree element idea tree prune irrelevant word follow regularization operate w norm correspond overall penalty combination overlap alternate fix variable denoise patch study whether hierarchy dictionary denoise compare I learn predefined tree patch impose extremely problem sufficiently many rigorously mostly I select validation set hierarchical problem heuristic perform improvement present structured sum norm overlap literature network flow cast min flow propose accelerate wide problem formally equivalence summarize lemma equivalence weight share vertex equivalent follow arc arcs arc path cost conversely vertex vertex arc arc cost graph depend arc cost vector arc direct graph introduce easy arc build flow equivalent arc amount flow carry flow also easy build satisfied flow conversely flow decomposition exist path flow sum unique arc flow arc amount flow arc build show compute proximal operator dual algorithm find proximal introduce essentially condition termination optimality dual variable feasible g g min flow solve min equivalent flow capacity constraint correctness cut find part construction set node conversely arc arc follow arc value would infinite arc flow go property arc arc go thick draw fill blue fill mm fill draw black fill minimum bend source group xshift pre g xshift edge right part var g xshift dot right distance leave distance cm si pre xshift thick leave mm thick gr gr gr arcs bold arc dot scalar negativity constraint converge split process require onto ball max procedure sufficient procedure compute suppose classical flow min cut theorem equality term definition flow min theorem contradiction existence proof hold graph solve correctness induction node v prove initialize simple simple correct compute ball case optimality yield compute scalar situation q rewrite previous positive max flow solved prove next remove see vector respective canonical structure gr gr graph combine optimality arc cut possible condition show relatively arcs arc bite j v gr gr w u j gr gr gr gr u arcs go flow nan flow imply node flow provide gr condition imply u contradiction gr j v detail gr prove see give flow step graph another flow dual definition introduce additional inequality dual g take derivative variable simplify lagrangian correctness compute norm polynomial similar proof require finite max converge correct canonical proceed induction next canonical flow
pac bayesian analysis trade graph practical benefit regularization suggest model quality trade large reasonably practical formulate weight comparison pac co optimize state provide theoretical foundation suggest provide good life problem derive reasonably cluster tool include bioinformatics approach example sm term objective functional minimize whether cut theoretic approach formulate weighted analyze weight rational behind weight specific validation bind address finite order prevent resolve trade co suggest review adapt pac bayesian bind analyze widely row matrix good illustrative co movie rating predict miss triplet movie rate discriminative movie pair expect whether co collaborative space rating star absolute existence px x sample consider predictor along conditional assigning assign label cell c information eq x divergence discriminative clustering prove snp parameterize trade suggest alternate fix minimize substitute back alternatively collaborative adapt node edge generate unknown know node edge weight generate goal build formulation specific enable work immediately belong share edge prove minor cluster edge weight quantization divergence easily numerically co tune substitute validation suggest qx qx measure consider demonstrate superior mutual theoretic inspire distortion ct notational problem without low sampling matlab derivative maximum minimize delta px let variable index prediction reconstruction correspond delta norm equation way appear power repeat projection empirically even remarkably try initialization obtain comparable within follow value fold increment iterate random increase reach note analysis edge interval end round edge projection quantization increase quantization w w w continuous weight kl divergence quantization name edge weight similarity similarity second pairwise protein web edge protein deviation indicate curve coincide subset validation remain train observe parallel minimize substitute result bind perfectly meaningful almost dataset cluster cluster cluster dataset due appear twice node another analysis copy value slightly considerably beneficial edge global cluster first benefit cluster prove measure independently train preserve effective clustering illustrate indicate note
directly constructive strategy bound appear develop round appeal keep expression develop computationally basis development reader state von fan theorem banach space nonempty weakly let nonempty weakly minimax complicated include expectation bilinear equip variation see banach weak continuity semi result compact set measure compact accord example borel banach tight precisely norm example scenario make need minimax see application minimax brevity inf f proof every couple understand move respect infimum equal fashion scale depth cover center calculate invoke substitute arrive upper conclude long follow cube separate note norm apply obtain step also upper finish base bound vector give give q conclude use sequence observe give strategy attain give expert weight w randomize upper bounding constant theorem dominate leave expected indicator loss indicator crucially leaf reach infimum split minimizing let remove priori countable expert randomized manner proposition cl tw lipschitz lipschitz function trivial bound lipschitz universal next low whenever imply ambient next small class number tree analogously cover put piece together rademacher bound game consider exponentially expert countable proof closely include completeness need proposition countable expert thought produce history countable expert initial weight expert play receive update ix forecaster enjoy sequential prediction minimax associate several complexity precise learn particular online supervised learning framework concern probabilistic regard generate well sequential prediction procedure apparent sequence underlie viewpoint analyze value set useful somewhat abstract set well repeat game model choose adversary pick suffer loss cumulative loss fix pair say learnable algorithm time adversary employ origin compound decision algorithm perturb sum loss idea influential follow seminal work lead development foundation code compression information literature science connection economic partial reader prediction researcher include science economic excellent synthesis presentation different refer technique name perturb online gradient may unless tool learn precisely begin underlie abstract develop theory develop rademacher complexity relate note control online turn lead constructive proceed repeat learner adversary need technical separable weakly compact every henceforth notation stand distribution adaptive adversary base history game exchange dual easier analyze choice adversary adapt player measure mixed strategy reduce online learnable require iterate infinite formulate contrast regret possible obtain develop also extend allowed compute especially term prediction information f fx setting study scenario limit paper aim shall keep sequential complexity show characterize uniform law number handle briefly notion mention key sequential unlike I possess dependence dependence classical notion basic look tree capture rooted node identify label tree indicate length rademacher depth value sup stem martingale deviation control sequential complexity matching hold supremum martingale statement hope notion sequential show generalization classical notion necessary picture completely theory key definition class notion check cover analogue eq extend beyond cover eq tight covering analogue q sup p previously description combinatorial depth value depth function exist depth dd sequential generalization binary recover notion definition restrict hence dependence crucially growth covering definition combinatorial class analogue satisfies establish online class nevertheless useful property rademacher prove essentially therefore skip proof next rademacher class complexity individual familiar contraction property immediate corollary g binary value note classical contraction hold logarithmic factor logarithmic remove worth point ahead lipschitz otherwise relate supremum empirical I view supremum rademacher upper sublinear irrespective adversary establish purely base subsection early paper scenario pick adversary pair loss easy classic observe improper equivalence easy verify pre specify alternatively simply observe particular move minimax need apply hold original simply rademacher theorem remove include pass minimax contraction sequential rademacher lipschitz theorem scale supervise complexity notion logarithmic learnable follow learnable complexity function learnable sequential rademacher integrated additionally theorem martingale number property paper binary write investigate control absolute classification universal derive notably bound non cover argument ask able distribution example two framework g learnable online indeed interestingly closely slope supervise online online learnable statement consideration example demonstrate explore decade unify view mirror associate abstract say online online also recognize lose suffer possibly unnecessary nevertheless recent scenario move banach space adversary lipschitz online instead without algorithm trivially extend randomized reader try play norm example rademacher crucial duality smooth let subset banach norm sup z p conclude recover descent usefulness tool develop rademacher bound fairly play big theory classifier space study decision make seminal minimax static expert term rademacher rademacher classic multi layer learnable neural bound transfer function sigmoid hold q eq constructive guarantee neural online efficient statistical margin bound theory margin see recently margin show easily base sequential idea randomize suppose exist sequential logarithmic factor crucially consider tree depth class follow root value decision value decision read leaf tree membership conjunction decision path leaf choose leaf run leave decision denote tree learner reach leaf correctly classify tree minimize
approximately frequency switch p switch probability random follow direct rule dense average contain number one kind bayesian describe happen alarm call whether alarm similarly straightforward estimate full bayesian represent derive chemical reaction addition abstract operational semantic deterministic refined semantics semantics intuitive execution computable sense strategy long case probabilistic inside probabilistic execution em pt pt pt em either singleton fail specific transition execution strategy em em p class set execution firstly execution intuitively say two build equivalent execution get result state equivalent observation strategy pt em unique follow query execution strategy start chance chance execution different c program observation execution refine semantic program strategy correspond program I em em em em fp semantics strategy w every equivalence class final derivation specification execution program define unique distribution depend ambiguity refined semantic program bx cx execute first result accord two outcome ambiguity program every derivation program coincide example consist vice versa consider regular program derivation query end b program execution rule execution consider get therefore execution implement system currently prototype mu naive pure current http people advance logic language build switch exp possible outcome experiment exp value assign pn program part rule fact predicate rule clause allow body clause serve interpretation assign hmm simple hmm next state end state hmm output symbol either b clause indicate fact say value clause string hmm probabilistic event hmm generate string generate bayesian alarm follow yes alarm alarm learn illustrate rule translate graph translate nb true probabilistic simplification suffice remove store soon put rule expect fine computation way implement explanation search explanation create try cause involve fire chance take translate rule code adopt head translate branch actually version variant every probability program implement head tail conceptual point view semantic operational derivation however semantic confusion weight coin example head chance otherwise interpret rule answer head chance weight normalization runtime consider word heavily program localize mean head program tail propagation applicable weight abstract semantic allow execution efficient semantic semantic applicable rule fundamentally refined semantic applicable rule active occurrence semantic differ derivation derivation derivation semantic chance localize depend execution support execution I numerous logic family cp logic cp etc encode compact modular logic language inspire bayesian network give allow detailed description offer language namely thing instance evolution fact might represent logic time elegant apply range field largely extremely suitable example computational valuable many application domain extension exist approach uncertain example refer scheduling reasoning semantic web verification another automatic analysis music past analyse music set strict combine application music specific computation viterbi computation music exploratory formalism combination operational implementation relate language opinion advantage include wise completeness multi rule plain ambiguity relation ambiguity although implementation sufficiently explanation limitation current support would interesting transfer automatic program e obtain support far ground argument add support non query certainly winner probabilistic termination explore program always terminate go since remove program keep never end probability still compute make sense terminate handle infinite loop search stack ambiguity relation ambiguity implementation explanation search consider semantic chance rule applicability rule abstract operational semantic refine operational semantic semantic practical many support probability example feature essentially free challenge yet exploratory corollary definition institute technology mi ac cs institute technology mi predicate build support maximization handle programming rewrite paper formalism level rapid complex language operational semantic notion program distribution probabilistic logic language identify potential handling language originally language implement year purpose language language call statistical probabilistic sampling formalism chance statistical combine like expressive probabilistic implement translate clause early mostly subset bb operational semantic rule rewrite mostly cx like chance conjunction keep conjunction head satisfied body omit call empty call recursively conjunction constraint goal intuitively mean chance store head substitution furthermore head ignore actually rule store execute store one regular drop apply expression ground runtime occur evaluate indicate name execution theory build predicate program transition binary annotated rule start root walk acyclic reach program state probability probability path em em ik derivation semantic semantic partial
approximation select great remain great marker level remain therefore true relatively frequency general positive add toward end sec criterion select cover reconstruct principle selection prefer particularly preference dl entirely express trace node order list compress reporting marker node marker identify parent child identify nd rd require parent since possibility specify subsequent report sum description marker trace describe practice marker define network marker parent every first specify parent marker specify report potential parent specify report third characteristic code explicit define high expensive parent report create describe marker therefore short marker trace require minor covering turn directly network cover edge along explanatory cover add knowledge calculate include along baseline bind positive false positive first consistent covering result search maximally see exceed naive naive never positive away everything marker trace sec would figure verification match true obtain circle fig description point majority true positive limit inclusion naive coincide zero circle indicate cover line reconstruction various clearly positive remain increase contrast presence fig add proportion false edge fix exact network marker demonstrate reconstruction causal underlie branching spread entire consideration way able np efficient greedy two version consistency datum likely control setting propagation version common noisy marker observe reliable tend early good excellent plan direct cost stop another order life datum media propagation acknowledgement thank analysis university benefit centre sciences award partially grant ep fp ac reconstruction node visit branching algorithm crucially consistency local infer neighbourhood optimisation solve reduce cover np hard approximated extend noisy demonstrate sir interest recent reconstruct complex network produce diverse field measurement spike datum field generally nature inherently chemical share challenge causal dynamical stream address challenge reconstruct branching process occur direct discrete infection lie field infection begin stochastically infection initial report site pick reconstruction generally fundamental propagate communication notably medium optimisation set concept description reconstruct contain lose fully nature optimisation present sec introduce address concept outline sec orient node reconstruct infer denote branching occur transfer marker marker network propagate stochastically adjacent along terminology refer node become marker infection point infect infection marker refer marker marker trace marker trace index marker order carry marker infect notion marker marker trace w w marker trace define reporting marker trace clarity future path marker trace approximate make generating assumption generation marker node previously infect notion globally g besides ensure ensures guaranteed reconstruct reconstruct optimisation make sense require consistency involve impractical equivalence neighbourhood consistency reporting marker incoming marker early lc r w e lc demonstrating define construct early demonstrate path node hence trivially incoming trivially path every node incoming edge node claim induction lc formulate optimisation concept consistency I crucially consistency immediate neighbourhood turn optimisation subproblem total subproblem establish minimal incoming explain marker particular node unless specify describe discover node use local consideration consider incoming edge report marker time explain marker incoming explain marker report relate optimisation universe subset minimal cover define universe universe marker income incoming potential explain marker therefore every family define incoming early marker require incoming incoming reconstruct b set follow cover subject v b e repeat allow reconstruct greedy covering cover cover great ie reconstruct cover subset marker trace strategy ensures reconstruct refer positive edge positive fp positive exist incorrectly reconstruction simply find true reconstruct tp give achieve complete coverage marker marker must include reconstruction count number edge determine edge assess performance useful fraction positive successfully recover lower exclude false quantify false positive fp false order make set cover cardinality edge cover finally ratio obtained specify positive cover time useful size subset cover logarithm alternative member cover order cover relate ground cover report way
pattern group vary much sequentially maximally correlate small reference component match wise correlation acknowledge general satisfactory practice correlation match pair ica multi subject functional experiment keep record precede subject inform image whole body study volume slice resolution acquire acquisition experimental paradigm ten subject horizontal visual sentence processing occur stimulus ten occurrence trial give inform functional acquire slice resolution comprise four discard mr paradigm acquisition report acquire group subject acquire successively dataset department cognitive http www ac uk interpolation motion volume template isotropic voxel mask voxel procedure well movement flow cca em map stability ica em ica subspace match cc cc subject cca cca cca cca stability ica map average level level select study ica dataset sequence smoothed extract identify brain cca identify component brain movement bold feature functional dataset functional network fig equation without whiten pattern cca identify material table thresholded subspace fact conversely suitable study procedure stable subspace change tr one thresholded thresholded maps hand back reconstruct quite unstable population split one thresholded map perform component extract number volume result thresholde cc c thresholded rs matching maximal another assessment map give percentile dataset thresholded thresholded correspondence scheme compare supplementary material compare finally subject dataset table improve group cca marker fmri number identification previously activation pattern correspond cognitive study metric subspace run mode linearity give measure dataset metric cca use procedure consist preprocesse sub selection procedure cca implement principal whiten level cca whitening analysis group level component well score canonical perform table size material publish description implement package group subject perform successive average filtering group svd select select level apply impose criterion dataset simple concatenation material subspace canonical correlation less critical retain opposite cca group subject thresholded identify salient thresholding heuristic thresholding heuristic heuristic validation score different seem whereas supplementary material amplitude thresholding depend strongly histogram pattern component interpret functional group yield identifiable extract identify detect effect group two significantly group network understand activate whiten cca sub network encounter ica related visual structure statistical power resolve structure split separate region posterior dataset region show consistently separately default mode eeg measurement form differentiable identify associated display considerable finally state dataset language network map rich network comprise may expect occurrence cognitive vary appear component separate right task paradigm correspond low figure across visual area stand dataset different part visual consider observation also apply appear network level area process sometimes comparison activation topic consistent long produce interpretable much contexts activation correspondence spatial alignment perform preprocesse step subject difference variability smoothed metric correspondence induce score limitation method difference non correspond map validation matching technique purpose marker non nuisance ica ica criteria subject threshold identify one limitation subject lead subspace outline select component consider cca canonical informed ica non less tractable variability extract infer individual component purpose dual extract representative group model rely solely algebra ica loop optimize memory cost dominate group inference cca scale intel important fmri extraction ica step step costly principle component retain subject pattern present multi apply fmri non calibrate automate way meaningful fmri cross associate metric ica unstable extraction mix thresholded one exploratory use pattern marker whiten subject principal remain basis volume acquire wish estimate canonical select resample generate drawing noise correlation thus draw realization give access observe nan cca ica map matching map ica match fig de france team le france france france component increasingly image fmri set region functional marker disease open road paradigm subject group model ica inter comparison propose multi fmri introduce ica probabilistic reduction correlation ica method ica level control state brain gain imaging fmri derive activation fmri correlation distant region distribute activity context fundamental study paradigm bold dependent correlation study identify distant establish connectivity signal fmri correlation signal shape structure bold brain cognitive activity across assume brain function brain connectivity specific useful mechanism disease serve aid clinical diagnosis brain adequate disease study correlation brain activity comprise various voxel yield million correlation seed study potentially network brain correlate seed limited signal spatial identify meaningful pattern prior ica usually easy interpret cognitive often context seed base cognitive salient consider drive constrain datum remain unclear finding dataset sample context statistically seed bold pattern experiment extract exploratory ica comparison however overlap ica coherent report rejection goodness establish ica ica exploratory subject contrary level specialized adopt extraction pattern form challenge correspondence map assess merge statistical along apply ica group extension pattern share experiment novel group fmri volume procedure extract map model component subject canonical analysis cca resample automatic cross compare pattern sub population compare state art method concatenation rely subject thus formulate spatially group cross method ica data base signal base independence blind separation source eq pattern ica notation shape possibly observation depend extract ica guarantee rarely check component pattern solely movement far suit blind fmri right brain functional system pattern acquire fmri volumes ica often acquire ica principal component determine extract context fmri analysis pca basis subspace voxel fluctuation brain group analysis effect ica strategy far group fmri pca group procedure additional individual component inter consider relate level component generate level shape loading acquisition frame pattern load write eq q word matrix factor specific term specific level different ica residual identify mixed effect formulation glm source drive rely procedure extract generative model note pattern successive fmri subject correct slice acquisition motion extract mask brain center index pattern successively hierarchical separate subject pattern principal component pca principal pattern spectrum constitute loading component observation extract describe signal setting retain article select dataset theoretic initially pca identify source fmri many component study conduct influence ica method resample method shape necessary presence assess stability principal principle reader share dataset group interested sub subject purpose generalize compare multivariate successive univariate canonical correlation canonical cca subject variance ic account retain svd row canonical correlation retain canonical form pattern level keep canonical correlation level instead subject interest consist material level amount stability svd step span activation ica pattern lie result keep tail intensity nan thus unit estimate
study undirecte pair suppose denote typical problem interested multivariate particular undirecte value induce lie partition element graph value estimate two use splitting penalize hold minimization theoretical strong assumption optimization attractive cart result value effective covariate sample may estimate glasso one covariance correspond definite develop solve study prove mention condition glasso pt parametric glasso yx glasso smoothing g bandwidth apply glasso appeal nonparametric smoothing require global computationally partition finitely many region glasso take find recursively know leaf node cart optimize cart cart devote detail nx n px value sparse sparse graph covariance cart glasso graph dyadic split dyadic partition dyadic partitioning construct orthogonal dyadic split associate give denote tn kk side denote indicator piecewise mean define definition induced let matrix specifically penalize may always dyadic responsible suitable tuning element way discuss formulate practically split hold log hold cart evaluate require dyadic partition tree dynamic programming computation propose yet greedy hold empirical greedy generic easily penalize minimization form dyadic precision precisely correspondence run glasso large yield small select hold model reduce glasso enforce yield graph greedy start compute decrease hold large hold precisely split hold risk form split increase indicate partition split cart apply element record dyadic implementation pt hold integer estimate determine good splitting l r partitioning classical decade assumption theoretical go cart work oracle note might risk inequality assumption arbitrary induce bl l risk go cart excess r j rt input leaf reasonable inverse matrix unit two enyi maximum four small middle figure c inverse cccc ptc associate distribution size hold base dyadic present node leaf node carlo run partition plane split irrelevant dimension range moreover graph obtain highly dense immediate node simulation tree list term precision score true easy region contrast inside correspond hold significant pt furth ground appendix c deviation value analyze contain span location equally spaced grid month include temperature temperature cloud cover day normal uv detail location glasso connect contradict domain knowledge factor include panel reason edge pool correlation correlation dyadic greedy partition dyadic tree california adjacent suggest moreover factor direct accordance report south central location validation expert graph cart undirecte high dimensional dyadic using finite oracle excess risk consistency relevant partitioning computationally attractive classical advance technique go cart indicate cart several direction denote analyze least assumption bernstein hoeffding probability get analysis give subtracting least q enough subtract side minimal partition proceed different easy sequel need follow achieve plug inequality far conditional glasso lie one space os enyi vertice maximum degree output sure node graph guarantee hold greedy cart dyadic tree structure corresponding display cc compare graph glasso entire term precision score score glasso well glasso graph entire datum false positive cart
straightforward gaussian e adopting convert em maximum estimate finding adopt subspace sensitive effort robustness characterize nevertheless problem bottleneck factorization reveal robustness modify formulation extra nevertheless modification usually heuristic style algorithm get performance corrupt lrr regard generalization lrr solve generalize present subspace describe polynomial polynomial success segmentation certain restriction subspace method due polynomial cause algebraic segmentation resolve robustness difficulty polynomial subspace datum subspace firstly final spectral exist ssc spectral curvature fit lrr possess type method clean sparse sr could sdp within sr sdp ensure segmentation recently zhang multiple exactly even contaminate outlier propose lrr space provably determine lrr matrix capital matrix horizontal resp concatenation resp block vector norm norm use nuclear sum trace support symbol support complement I column obtain resp denote denote belong subspace svd svd appendix create subspace ambient small sum dimension subspace strictly nevertheless simple affinity spectral svd data clean membership independent form entry term shape widely segmentation simply subspace segmentation presence inaccurate lrr contaminate block indeed nonempty intersection pairwise independent still interest segmentation roughly want address store union observation sim analyze success sufficient goal difficulty assumption clean corrupt clean corrupt contaminate characterized fig unlike subspace highlight recover lrr recover cluster deferred order indicate e norm adopt characterize original formulation formulation structure draw actually treat sample much well inaccurate well suggest linearly minimizer obtain z fall lrr use basis appropriate dictionary rank reveal lrr ease exploration case clean rank minimization problem easy practice problem result segmentation minimizer general lrr strongly fortunately minimizer uniquely form summarize feasible problem feasible minimizer also problem low reveal segmentation namely sample sample subspace without subspace sampling basis span minimizer diagonal coefficient x low sr worth property sample group membership generality indice true membership lrr sample replace relaxation norm characterize sample choose show appropriate choice obtain tb inexact alm fix fix update multiplier e z update augment lagrange multiplier alm convert alm lagrange tx unconstraine minimize update lagrange multiplier alm call need solve please base step close thresholding via optimal I smooth alm inexact alm alm present still difficult convergence inexact alm three easy convergence fortunately actually ensure theoretical necessary converge one iteration monotonically decrease k resp ideal lagrange simultaneously easy convert condition satisfied subsection monotonically lagrange function guarantee validity moreover inexact alm reality alternate guarantee adopt nevertheless please boundedness problem size major algorithm svd consume fortunately lrr easily follow conclude subspace span advance column transform replace b recover number row solve lrr quite provide rank dictionary assume converge versa efficiency optimality always iteration utilize lrr address present identify solve nuclear prove lin minimizer clean reveal connection counterpart pca refer pca nevertheless outlier contrast prove lrr recover contaminate outlier subsection imagine datum column support characterize seem contain two choice problem subspace draw lrr support index away one part draw subspace member support denote total fraction lrr succeed minimizer produce space importance cluster notice lrr property refer verify conclusion outli fraction threshold ensure performance magnitude consider affinity matrix corrupt subspace affinity corrupt corrupt treat generate way randomly corrupt experiment carefully determine support identify away member corrupt happen call specific still valid lrr recover empirically conclusion support index corrupt handle deal case similar outlier sample heavily treat lrr illustrate treat non something else experiment create pairwise subspace choose corrupt large error finally total include specific recover corrupt support sparse still relaxed handle support contaminate unlikely exactly recover near inequality prove demonstrate lrr quite notice somewhat loose fig exist alm invoke thus explore affinity perform segment svd affinity assign multiply clean ensure affinity cluster segmentation lrr laplacian nj although generally subspace e resolve due produce strictly affinity firstly normalize singular block reality suggest subspace value laplacian near integer soft thresholding summarize whole estimating lrr detect possibly fraction clean learn approximately support datum affinity possible outlier discard affinity threshold type principle strategy characterize outlier affinity degree advantage easily extend prior lrr art segmentation segmentation experiment paper shall lrr subspace outli dimension std extensive extract total sample manually remove sequence notable level summarize large take resp test lrr effectiveness create database database image extreme condition namely view direction small light degree low rank face non dataset close lrr baseline segmentation subspace segmentation svd utilize difference estimation sim detect outlier improvement pca outli work introduce characterize adopt detect refer subspace case solve parameter lrr appearance sr implement sr compute affinity minimize sr enforce avoid trivial minimizer lrr subspace consensus agglomerative compression ssc curvature fit segmentation receiver characteristic roc evaluating outli detail evaluation metric please appendix segmentation always good part error datum slight evaluation result sequence range segmentation error almost unchanged phenomenon mainly two reason easy choose arbitrarily lrr imply minimizer always satisfie lrr partially stable analysis importance largely segmentation actually sequence overall impractical parameter especially lrr sr lrr sr lrr subspace segmentation give list sr lrr methodology recover segmentation pca design recover design reduction notice use segmentation error illustrate lrr fig sensitive example achieve increase lrr pca theoretically computational regard lrr lrr cost iteration predict rate absolute database also provide good estimate subspace underlie datum lrr illustrate resolve lrr ssc sc c result discard comparable lrr improve use datum dense reality learn space choose dictionary consider unobserve hide z lx extraction achieve subspace long difficult g estimate trivial contain outlier goal face segment rest segmentation outli acc auc investigate segmentation evaluation outlier obtain sr auc lrr see lrr well pca strong sr behind check affinity produce sr even absence unnecessary notably reconstruction sr handle datum contaminate outlier image lrr lrr fig leave original middle visualize lrr size lrr worth note error salient decompose matrix low rank rank lrr extract discriminative salient region low lrr structure correct lrr generalization recently establish shape interaction define different sim row corrupt experimental effectiveness lrr determine lrr recover illustrate choice ensure whether technique matrix issue lrr select parameter segmentation lrr detection use block diagonal transform block whenever two k iy denote k collection subspace subspace ambient dimension assumption pairwise disjoint j decomposition orthogonal singular value keep rv uniquely column resp span column orthonormal basis resp orthogonal since resp determine sometimes refer resp affinity affinity th proof follow lemma dimension column nuclear matrix compatible dimension proof compatible orthogonal feasible constraint orthogonal orthogonal matrix unitary nuclear minimizer second minimizer another solution orthogonal u accord equality allow horizontal concatenation denote partition definition q calculate u problem
whether aggregation mechanism independent perturb mechanism prove theorem order issue majority inconsistent result mild whenever opinion opinion aggregation mechanism consistency result absolute aggregation mechanism preference exist aggregation generalize result characterize independent consistent aggregation mechanism preference mechanism think phrase question aggregation global property section present testing binary classic prove specific aggregation family conjunction exactly constraint technique proof conjunction measure aggregate aggregate consistency independence relax constraint describe describe aggregation mechanism present functional conjunction state motivation deal aggregation preference describe aggregation property theorem similarly individual opinion denote issue opinion profile member vote vote issue vote individual consistent aggregation opinion profile profile simplicity opinion opinion aggregation independence mechanism profile aggregate independence notice generalization aggregation social iff wise comparison aggregation aggregate aggregated aggregation mechanism mechanism measure usually satisfy iff aggregation index inconsistent q aggregation mechanism j section dependency context define index natural multiplication satisfie iff exist aggregation satisfy mechanism opinion profile distribution pick consistent proving prove format bind preference successive relax prove hope aggregation take distribution profile opinion independence sometimes life scenario attack remove reason intra issue dependencie essence accord criterion complex issue without regard aggregation show contradict accept think quantify claim case criterion case change collective secondly aggregation opinion profile vote aggregation mechanism independence strategy player justify independence return easy represent justify vote public justification mechanism decision different place vote voting aggregate vote vote issue definition return deal aggregate notion framework ease presentation logical operator bit return member function pareto sometimes refer follow pareto return definition influence vote eq game define getting normalize measure distance df notation binary formulate two class truth divide type conclusion opinion attain might choose analyze opinion seem family later truth functional conjunction conjunction issue decide issue contract valid making contract valid issue decide consistency answer odd think represent functional study equivalence describe preference aggregation framework individual strict order interested function order framework issue aggregation consistent regard set aggregation mechanism tell reasonable aggregate preference extend aggregation result constraint root science long suggest aggregate mechanism try stay reasonably independence general tailor suggestion mechanism truth aggregation procedure aggregation well vote vote systematic contribute pointing solution leave independence aggregation candidate state aggregation exist aggregation mechanism candidate independence aggregation consistency field deal follow object determine g possibly randomly select whole small failure allow think view aggregation problem test highlight special term testing field case global try test mechanism current separately independence consistency pick uniformly check aggregated opinion issue profile opinion change opinion issue change mechanism distance satisfy main follow consistent box similar ask property property see study question define test one property similarly sub mechanism introduction systematic property deal family aggregation deal truth issue either conjunction b aggregation mechanism exist mechanism direct corollary define either conjunction several far aggregation functional conclusion get well g dependence ff aggregation mechanism affine represent truth functional conclusion lemma issue mechanism mechanism consistency section technique proof theorems approximation aggregation conjunction aggregation mechanism independent case induction framework change get j let independence insight issue aggregate influence aggregation nj nf I read vote member vote member outside return frequent assume issue mechanism aggregation conjunction characterization include h characterization aggregation mechanism proof theorem tight case consistent aggregation technique issue aggregate function aggregation xy df linear permutation exist satisfy aggregation mechanism independence mechanism follow mechanism independent aggregation mechanism mechanism deduce close issue aggregation study aggregation mechanism non conjunction characterization calculated dependency class probably include dependency work preference question inherent conjunction relation depend think assume distribute immediate extension extend complex functional preference open one constraint consistency class mechanism trivial true em proposition conjecture criterion proposition theorem theorem conclusion fact sketch mail il aggregation vote proposition aggregation relax constraint input relaxation notion fit main result case aggregation term truth functional constraint involve boolean influence protocol term test generalization linearity truth economics university author thank participant comment truth dependency deal scenario aggregate independent opinion suggest protocol stand criterion security attack network opinion independently cast vote protocol criterion think violated think enough third think scalable hence although separately protocol pass discrepancy majority vote security scalability conclusion later several logical
therein however concrete require level error quality model height appear denote jump jump simplify exposition error term identically distribute classic normally distribute finite cauchy tail double exponential possess chart applicable change control sharp construction control chart chart control shift standard base exist large simulation expect range underlie specify chart shift let review property chart roughly introduce since binomial variable sequel shift occur long large small positive negative detect testing alternative trial equal binomial arrive chart repeat obtain version chart control limit chart modification item sample either mind chart substantially large delay detection purpose chart count individual move control observation form buffer contain past exclude buffer length replace replace instant buffer whether buffer eq buffer chart form sequel buffer observation lie limit large state chart main chart production count buffer length moving buffer buffer present explanation superiority chart jump slightly version chart signal result extend general outline model jump simplify stop transformation integer small sequel current equal statistic center scale version control section exceed correspond choose value sample decrease call buffer length ensures asymptotically buffer length buffer available series buffer satisfies chart superior buffer strategy example put obviously chart mt consider buffer length suppose buffer run ensure fair let consider choice chart historical beginning classic small well jump q specify control alternative model shall purpose briefly relate underlie give thus sequel limit detail refer work general require martingale ni ni conditional array space martingale array x martingale deal series treat work dependent image consist analyze top origin low corner I array noise neighborhood l ij ij defines follow error equation variable I ij difference pixel neighboring pixel analyze ni ki ni p array respect concern limit theorem modify chart martingale difference hold change weakly time converge eq weakly notice identically bernoulli success say chart stochastic dimensional behave smooth theoretic result benefit modify chart cf case approximately brownian drift yield strictly large classic chart chart jumps detect random drift drift summarize indicate modified chart detect jump right point notice yield distribution jump dominate beneficial unlike chart carefully average false small level experience chart justify theoretical result control practice often chart buffer reasonable buffer fact figure plot buffer length minimize practical select chart desire buffer simulate control reasonable determine small demand reader summarize ensure integer prevent control chart select give jump height buffer length optimal account limit buffer exhaustive expect perform extensive firstly identify buffer secondly investigate normally third chart normally performance study chart table run buffer feed pre control fix chart buffer control provide chart considerably equal rl qualitatively pattern fix large mention handle alarm chart jump double behave quite difficult table c jump rl c jump rl jump c rl jump rl c jump jump jump laplace rl c cauchy jump rl acknowledgment anonymous constructive remark presentation visit technical support grant range science education point array bernoulli ni martingale bt b denote brownian motion satisfy assumption clearly conditional z ni eq set yield third term converge pointwise continuous handle martingale central theorem bt term equal define linearity triangle obtain tend mt imply ensure case obvious put thing height width em pt minus pt lemma proposition remark chart institute engineering technology institute chart exceed financial study modification chart use recent observation chart chart superior shift central limit model explain often arise whether true martingale array applicability firstly time series secondly image primary p produce tool large propose comprehensive review article property chart
contrast approach generation principled combine make reliable walk limited prediction require recommendation detection rank part nsf fa yu foundation microsoft yahoo fundamental china predict occurrence fundamental prediction snapshot would study effectively combine network attribute develop level attribute achieve use attribute guide walk formulate learn edge likely visit node learn facebook extraction descriptor management application term world exhibit interesting property model predict reproduce network research seek develop predict global many highly dynamic quickly edge node study identify mechanism social evolve edge understood motivation snapshot seek accurately add future time predict new problem view link recommendation link scientific facebook friend responsible significant fraction add facebook useful facebook member two point extremely sparse node case million unfortunately predict new near million prediction subtle link social use intrinsic network similarly user gender home edge consider example reason connect party facebook party age probably live link people meet party close people friend circle despite friend party social interact interest eventually principled profile interaction information simply extract common friend short path node combine profile address recommendation supervise principled combine characteristic network unify develop supervise walk bias walk nod often strength walk weighted visit prediction new create node random weighted link score create technical way directly strength function strength formulation view network datum show approach outperform extraction additional extraction combine network evolve addition offer insight network formation relevant social like predict future interaction direct business consequence broadly large directly benefit interaction member link organization research security recognize prediction suggest link form future link suggest work directly beyond predict link protein interaction give suggestion relevant page link network easily next previously un predict include previously come problem cast link moreover walk predict link unsupervised link prediction recently detection link relational challenge primarily precisely link come degree feature edge add together feature heuristic information link appear rich attribute combine use walk edge probability link create likely node walk network recommendation like recommend link link create create link clear create setting appear give positive learn distinguish link recommendation click link anomalous generalized recommendation general classification edge node go create approach imbalance create fraction total hard high imbalance second extract task like age gender edge hard however less clear consideration graph attribute two example might count adjust proximity degree neighbor go give two path thing centrality degree possibility extract trial approach become hard annotation know edge combine get pair link rank idea score walk random walk jump walk walk rank take impact age gender combine random walk powerful tool node edge visit create source aim visit often assign walk edge visit set make overfitte learn assign edge address principled walk set candidate create call edge link label candidate example generalize instance describe node gender interaction attribute edge strength parameterize take compute edge transition exactly learn strength edge walk run stationary walk assign top rank predict probability strength thus edge likely visit set assign visit optimal strength vector strength idea want node node prefer hard constraint practice unlikely satisfy constraint soft regularization violate assign violate establish derivative combine attribute parameter output walk transition jump back seed conditional give currently node parameter walk minimize respect derive gradient recursive introduce eq commonly loss hinge take recursively still compute rule compute arrive iteratively partial score k p kt free intuition weak parameter absence gets find add drop ignore strength continue green regardless vector descent converge validate link social ph mat ph facebook avg four complete facebook predict seed reason person connect facebook triangle walks give facebook thousand incorporate user million co name co area ph matter mat energy th physics ph every compute time create co create span closed triangle attempt make try create give source paper similarity paper co paper number common friend facebook facebook point friend people country friend user randomly node show individual mutual friend become friend facebook neighborhood speed computation mutual unlikely friend practically demonstrate figure create link friend annotate facebook network seven pt cutoff create request communication period common friend half train algorithm test consider curve auc top node actually receive appropriate suggestion describe dataset four co facebook evaluate several aspect strength choice walk weight type choice play optimize correspond metric top optimize pt square q huber margin window q auc function show function sorted iterate sum huber quickly evaluate around evaluate relatively primary perform indeed limit reflect auc order significant translate improved loss fact huber loss obtain unweighted use remainder uv certainly function scalar desire logistic pt uv uv w choice significant impact slight evidence perform version double seem comparable avoid parameter think extreme graph undirecte approach score proportional simply random walk eigenvector score become score add notion strength walk back value short walk away play unweighted ignore strength assign strength see significant unweighted ignore strength weight find overfitte relatively set walk capture idea strong type might friend edge capture idea weight slow divide significant benefit seed node label type type link node six increase move examine somewhat seed could connection path connect capital link end could set make jump correction work friend common apply graph node great small practice introducing help facebook graph facebook formation co long tie connection write people people edge may opposite social capital trust show social norm try fit friend tie evaluate walk examine estimation walk unweighted bfgs gradient notice auc iteration auc random friend degree dt feature dt path dt lr lr node lr lr feature edge common friend dt dt dt dt lr lr lr lr type multiple edge r r co mat co ph co facebook next baseline along machine create test show
context determine row experiment direct seminal sense discrimination vector center target apply algorithm word generate sense word matrix word opinion category word context generation tool lexical intensive whether automate automatically automatic cluster classification specific kind see section several researcher word word typical representation vector token word correspond annotate tag apply frequency unsupervise sensitive kind error semantic labeling semantic role sentence role sentence connection sentence role show word reliably predict lexical refer good level lexical important semantic role narrow lexical expansion query google yahoo document semantic query pay click google yahoo pay display ad query make ad context occurrence extraction field extraction ie entity recognition name entity place relation extraction extraction al frequency facilitate supervise context name pair suit measure pair see similarity angle correspond pattern cosine angle word angle matrix approach examine multiple analogy question college level college high pattern measure similarity construct text pattern infer phrase language retrieval question answer clustered word represent pattern cluster pattern representative pair automatically analogy analogy relational pair strong assess review classify noun class classify cause home classify home locate weather report weather search satisfy search engine cause cancer one relation cancer task search engine relational conventional engine candidate answer relational incorrect answer manually simulate task task attract system automatic automatic arguably relevant generation relation individually use pair distinguish word merely mapping mean analogy proportional matrix proportional select relational however atom contain mass atom complex application approach application alternative section measure semantic query alternative probabilistic model probabilistic retrieval measure similarity probabilistic language information retrieval approach idea view matrix share measure similarity idea represent semantic human expect good performance lexical pattern measure word relational similarity sim aa sim ab measure good approximation hybrid approach combine beneficial stem fact word ignore commonly represent word phrase house house english house house house serve house whereas house purpose house storing estimate english word composition simple become contribution mean sentence hilbert inspire mechanic explore pair pattern raise semantic semantic conjunction yet statement calculus however limitation survey event distributional family distributional usage arguably major progress suggest help person request unfortunately computer force artificial language computer understand human us potential computer enable semantic language semantic researcher semantic conclusion make intuition soon deal organize accord determine processing play important survey show familiar emphasis new believe present matrix semantic human capture kind suggestion journal artificial intelligence research publish ai access national pt pt computer little mean human limit computer computer action computer analyse begin address limit survey semantic currently matrix yield survey broad range three category take detailed project category goal survey semantic provide new perspective less familiar field make full use computer currently understand mean technology language yet impact impact deep semantic space semantic general sense mean phrase language concerned approach semantic survey distributional represent aspect language retrieval system many concept represent space apart distant document document sort order success inspire extend semantic english human school age adequate relation attain multiple analogy question college average work accord context pattern fundamental linguistic three believe possibility introduce type corpus much semantic code basis system measure national resource similarity generally often phrase document lead measure semantic due distributional hypothesis distributional similar meaning often tensor connection space must derive graph represent matrix imply adjacency matrix derive event frequency emphasis frequency bring explicitly connect distributional hypothesis vector cognitive frequency text semantic machine learning classify item vector classify collaborative recommender typical system people correspond item product rating poor fair excellent person many mathematical well term frequency cognitive prototype often prototype membership categorization formalize however usually numerical human frequency extensive measurement usual typically subject item subject item technique analogue relate entirely appear cognitive argue aspect ai plausibility promising area research survey semantic currently comprehensive date survey approach semantic growth ai researcher semantic serve unified encourage area pointing area survey make framework term pattern see importance kind draw matrix address matrix application potential actual exist summary omit matrix nlp cl system cognitive arguably semantic far write survey art semantic introduce perspective familiar area reader basic algebra text book concept perspective semantic good information retrieval recommend reader familiar survey understanding beyond familiar natural reading recommend article accord get corpus text framework involve generating discuss linguistic review process linguistic processing mathematical semantic model plain raw linguistic sense parse section look linguistic semantic document raw frequency operation compare describe optimization concept semantic detail look present retrieval library explore build representative review module build system open application semantic serve short historical view semantic give row section generalize idea phrase book collection discuss context consider occur alternative semantic question limitation discuss state hypothesis usage people mean work define specific hypothesis bag distributional hypothesis distributional notational matrix bold capital letter denote scalar number document convenient discuss document page mathematics set allow bag matter bag bag element element bag row member bag document document bag represent bag word frequency tend relevance word apply capture document document matrix suppose document row unique column let row th frequency th zero since document use whole vocabulary document likely pattern signature tell document seem tell frequently lose phrase document surprisingly seem aspect semantic arguably extract author word section document topic measure document treat engine pseudo document relevance document row matrix document may context sequence character distributional word context hypothesis justification apply measure word may derive occurrence window dependency rich context dependency preference position see various word word reveal thing language primarily interested physical usage context physical building main derive frequency include semantic argue sense co occurrence word say company keep pair row vector pattern cut work purpose pattern similarity pattern find propose distributional co occur tend solve co suggest pair similarity vector relation material material second member tend pattern relation co occur pattern tend semantic word row vector tend relation extend distributional column pair tend pair suit measure similarity similarity suit measure distinction similarity depend correspondence property correspondence degree relation relatively similarity whereas cat relatively relational relational word relation material present semantic cognitive science relate share bank trust company car frequently think kind white kind colour sound prefer relational whereas relational computational share share semantic term relational involve mean semantic occur frequently calculus corpus usually often possibility document triple pattern matrix measure similarity word triple whereas pair pattern triple build grow increasingly rare contain phrase together triple pattern grow increase break tuple triple correspond triple matrix go beyond tensor scalar order tensor high tensor term word tensor preference tensor example correspond word english correspond join row represent slice tensor element english similarity slice question token instance symbol type token consider ever try ever fail matter fail token ever token token type fail line token matrix type token token nine column ever try ever fail try fail try fail token token document ever try fail try try fail matter matrix token binary token document otherwise row integer vector token row ever token represent token vector typical sense deal word token specific context rather mention relationship define characteristic frequency mention five repeat interpret vector work something implicitly assume something like statistical human usage figure people text piece piece frequency intend include word context pair tend tend indicate relevance query pseudo vector term distributional context tend tend similar semantic co similar tend relation similar raw context linguistic raw tf document search bias document weighting tend yet normalize co occur document form method text tf work word variation value replace perform wide measure semantic pair th row th correspond number time row raw frequency value word product expect random semantic relation distributional hypothesis negative give high relation indicate uninformative know problem bias consider I hence increase decrease another event smooth raw replace laplace laplace depend frequency small laplace towards simple way improve information represent occur content however carry little weighting maintain semantic discrimination computation compute intensive share coordinate e vector share frequent precisely little discrimination weighting describe highly occur context keep word conservative show precision matrix compute reverse compare match zero elegant operation term document operation svd thin svd mention truncate question english language indexing apply way svd present way look reduction occurrence three singular form produce corresponding minimize error frobenius discover mean word context svd create capture force context force correspondence improve describe noise think span space span specify reduce rank amount think compose variation high describe truncate svd discover occurrence occurrence appear indirect occurrence similar define recursively low occurrence truncate discover co occurrence sparsity truncate svd k dense sparsity insufficient svd lack svd correct likely another incremental another incremental truncate svd miss treat analogous parallel factor canonical equivalent discover survey empirical tucker decomposition order tensor ram projection subsequent research alternative index iterative scaling allocation discrete four equally well smooth truncate word frequency truncate implicitly frequency explain semantic measure raw cosine angle word two word frequent rare word short irrelevant thing cosine opposite degree cosine raw frequency vector cosine smoothing weight yield converted inversion q ir lexical circle ir measure normalize measure hellinger kullback measure involve word similarity cosine coefficient find similarity focus overlap lee shannon linguistic similarity measure measure group high measure cosine jensen shannon recall frequency sensitive mi lin frequency score similarity tend frequency sensitive prefer word determine measure believe determine appropriate similarity inherently frequency compare smoothing apply problem bad case parallelization multiplication observation scalar vector coordinate cosine overlap decompose nonzero q pairwise row efficient leveraging determine solely share nonzero reduce cost computation determine vector share building indexing change retrieve share nonzero nonzero nonzero coordinate nonzero efficient experiment semantic pair large web average leverage describe coordinate tf coordinate power coordinate interested solely run processing fast open software package implement thousand allow sophisticated parallel execution programming start index streaming part input read dedicated trading increase parallelism increase building index reduce column approximate efficiency project svd perform computationally intensive randomly impact computational little score especially top vector indexing distribute row represent accuracy mostly zero number cosine approximate cosine two compute vector index task locality lsh another approximate projection control tradeoff irreducible polynomial create collection document task remove web lsh general technique map row signature lsh preserve similarity preserve cosine similarity top cosine projection provide indexing lsh similarity task corpus lsh indexing however large corpus outperform lsh efficiency measure cosine near cosine external similarity cosine implicitly internal measuring value linguistic mathematical unsupervised vector general nothing machine aside task literature specific type discuss system source interested reader project text engine library foundation arguably wikipedia offer indexing rank primarily content document image video decompose field store implement content correspond correspond store use field allow string document also text point classified spam retrieve match column document field instance content consist instance index tf store schema identify define function build phrase query date restrict sort update occur search index programming language foundation open searching index present full http web create document index software offer web seed parse web document page seed page index book explain
synchronization process transition generator sense note remove observer directly hide state rather observer internal observing length observer state observer machine length observer know observe simply every finite observer machine word machine abc exactly asymptotically observer exactly time machine since contain almost every infinite word turn asymptotically ref disjoint finally synchronization observer block observer expect observer state machine observer function previously symbol eq observer symbol close close rate synchronization result synchronization consequence lm lm lm w w observer internal state constant know hence q since implicitly conclusion state observer machine rate constant machine alphabet state probability say restrict consist state irrelevant state observer observer currently observe initially possible observer generate similarly observer currently new observer govern machine recurrent strongly connect always original def follow assume order state order case block transition machine row state joint observer symbol joint recurrent observer observer observer observer word k jj w know pair convergent follow k j I n repeat observer state algorithmic theorem topological pair mm box box distinct mark end box already mark mark repeat path initialization replace j pair mark convergent marked box fact prove minimal distinguishing proof omit ref time mm distinct find convergent check either convergent distinct pair exact machine show exact machine observer observer phenomenon map efficient test machine ref similarly synchronization turn qualitatively similar result hold method plan generalize countable fellowship research project physical intelligence project view finding author either express department result alphabet fig name e arbitrary unless follow fact real let radius course radius triangle result linear algebra fact finite version ref linear operator banach define restriction state machine refer know short finally equation take normalize eigenvector maximal never divide eq therefore follow b pt algorithm corollary observer internal use treat exact synchronization observer number treat sequel observer average fast observer well additionally synchronization rate exact machine synchronization state state interest include synchronization machine mean completely symbol observer ever machine internal result consequence enable observer output include stationary particular identify qualitatively distinct synchronization exact infinite case treat sequel alphabet let variable generate future begin observer prediction measure review block uncertainty observer symbol symbol decrease limit observer prediction observer prediction asymptotic interested prediction source convergence restrict stationary simplicity state state label consist correspond node label initially machine pick symbol labeling symbol fashion denote machine visit output symbol generate kernel stochastic sequence state observer internal machine illustrate definition example alphabet transition block
base measurable discrete distribution impulse value independently also probability sample independently impulse stick break r r pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt kind specie sampling attribute view initially call scheme sampling scheme poisson partially admit atomic counter distinction atomic versus important posterior mixed base impulse mixture actual index index convert irrelevant group actual index countable mutually exclusive number assignment primitive tree partition top rgb f rgb rgb rgb rgb rgb bottom display bottom process cm cm cm rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb cm cm cm cm rgb cm cm rgb cm rgb cm rgb list subset order store order need roughly generate generate entry impractical know bin sampling size element order partition occurrence instance order slightly representation set order order bias sequence probability partition occurrence member list follow take probability impulse partition yield form index occurrence sequence index partition represent impulse need probability poisson dirichlet order decrease stick break rgb rgb rgb rgb rgb rgb rgb rgb cycle cycle rgb rgb cycle rgb stick broken stick length break proportion broken stick part remain proportion break stick part suppose random variable poisson sort value call discount literature concentration community inverse parameter use sort act give form break inside mixture indistinguishable ever convenient show log pt plot show hundred effectively say effect discount great show discount roughly like whereas discount geometric roughly common poisson dirichlet extend poisson dirichlet distribution yield finite countable dirichlet formula dirichlet sort definition summation stochastic base index sample distribution long atomic chinese restaurant customer item item enjoy chinese restaurant crp biased index chinese sequence integer index accord crp bias order crp alternative vector get domain property subsequently formula remain index atomic distribution irrelevant bias crp sample table restaurant statistic sample multiple count distinct call size index accord bias denote ordered occurrence size sequence crp sampling gets sort suppose distinct time atomic safe associate sequence biased exist index dirichlet subsequent give dp process random concentration partition parameter dp finite definition categorical say hierarchical broad sample distribution sample length sample independently accord give weakly partition basically true expect slow learn use surely surely use space continuous distribution another distribution atomic posterior sample weakly mass discrete discrete one give model diagnostic version see crp consider finite definition px n hx denote increment lemma term present size partition occurrence list call go chinese restaurant use crp chinese restaurant biased length pi lr occurrence follow definition chinese restaurant distribution note partition represent ccccc partition key definition rgb rgb rgb rgb c rgb f rgb rgb cm k rgb rgb rgb cm cm rgb rgb rgb cm rgb cm rgb rgb cm rgb node second row partition partition complementary bottom rgb cm rgb rgb cm rgb h cm rgb rgb cm cm rgb rgb f rgb rgb rgb rgb cm cm rgb rgb rgb rgb rgb bottom middle top form bin indirect bottom b l order detailed partition represent see complementary change one partition template complementary chinese restaurant functional definition partition let convert conversely convert correspond sequence occur node correspond tree depth partition see partition root node expect size average subsequent parameter discount pt figure instance discount next third label colour child whereas fourth plot label maximally node leave discount schedule split lower also infinite counterpart understand summation start simple draw circumstance follow measurable q inverse explanation start cluster circumstance follow simplifying nest simplify distribution theory improper prior posterior represent improper prior may posterior improper infinite define difficult infinite vector must prior full hilbert extension prior improper good improper improper sub pp measure instance informally believe improper normalise posterior sense property proper plausible standard proof integral exist check improper measure project improper variation define say improper sequence closeness measure variation improper prior sample proper improper detail improper definition use parameter atomic base distribution notation definition eq definition stick ii v n biased order improper prior posterior formulation attribute chinese restaurant process stick breaking stick proper improper sampling crp sort whereas improper correspond sorting distribution discrete identical calculation whenever hierarchical around impulse partition instance cat bias order formula size bias unable version introduce next occur restaurant definition order sequence match latent word column give sequence list cat size instance row word appear map index cat cat cat cat need return corollary prove hierarchical consider finite base sample constraint b one need table indicator contribute compute value start choice discrete indicator corollary indicator indicator appear explicitly forget ratio easier discrete especially moment properly moment base probability vector moment prior lr conclude aa moment apply finite require discount rapidly store log recursion slow requirement discount memory place cache value store save considerable memory repeatedly ratio ratio store recursion table resort storage number ratio present bit calculation proxy linear less accurate ratio recursion ratio mostly proof give augment distribution significantly recommend fitting induce two chinese restaurant scheme generate chinese restaurant dirichlet simplification dirichlet simplification infinite form improper dirichlet finite vector soon proper normalise alternatively probability order dirichlet stick covered dirichlet discount concentration discrete dirichlet discrete base dirichlet present indicator multinomial dirichlet dirichlet remarkably conjugacy apply conjugacy expense derivative unstable accurately compute communication centre mm height dirichlet mm com national
point estimation slight tendency variational place iteration convergence likelihood mcmc method estimate approximate density simulation compare plus minus mu mu mu minus mu tw mu mu mu minus mu minus mu mu minus plus mu minus mu minus mu newton mode update k mu plus minus theorem proposition principle involve focus statistical deal candidate consider computationally approach variational log greedy space step screen large inclusion potentially extend algorithm involve diabetes selection matching pursuit variational mu plus minus mu plus minus mu plus mu mu plus minus mu mu mu vector vector unknown independent mu plus mu minus mu plus mu minus mu mu mu deviation predictor form hierarchical prior refer wide range idea convert bayesian inference fast especially problem approximation independence variational leibler posterior result key contribution proposal role section method selection posterior model probability methodology demand need application greedy pursuit processing lar see excellent family algorithm greedy assumption application serious inefficient parameter chapter number model covariate use involve flexible extension approach demand setting framework develop use aic bic together forward contribution greedy linear currently attractive work efficient section diabetes datum regression consist construct coefficient stop predictor path diabetes stop iteration predictor enter closed maximize novel greedy two benchmark research technical approximation method detailed exposition orient introduction variational refer different idea convert parametric write difficulty application posterior involve integral variational proceed formally attempt kullback leibler mu mu minus mu minus mu plus mu mu mu plus minus mu identity minimize leibl divergence often energy due non negativity leibl low maximize leibl divergence useful bayesian selection bind express mu minus plus plus minus l mu minus mu mu minus mu plus py mu mu plus mu mu expectation put mu mu mu mu mu mu minus minus need iterative fix write design mu minus mu mu minus dy mu mu maximization minus plus mu minus mu plus variational plus mu minus mu mu mu plus minus mu mu mu plus minus mu expectation easy minus mu mu mu nz nz mu mu plus minus apart normalization gamma normal log computation require fitting generalize linear appendix write th diagonal obtain mu mu mu mu minus mu plus minus mu mu plus mu mu optimization retain bind approximate close possibly high matrix avoid follow td mode response x w lower retain specify initialization first perform ordinary fit initial ol use selection ol mention far far helpful drive low approximation apart mu plus minus mu minus py plus mu prior likelihood plus mu mu minus mu mu mu plus mu mu plus plus minus mu mu minus mu mu plus mu mu mu step add previous variance efficient perhaps likelihood condition marginal distribution consideration rule replacement strategy thorough review see bind approximate leibl divergence structure variational distribution product normal minimize kl small good experimental suggest maximize tight present strategy ranking prior give currently active mean variance write predictor independent priori minus plus minus mu plus plus mu mu mu mu set mean model mu mu minus minus mu minus plus mu mu predictor slight abuse plus minus mu mu mu minus mu minus encourage parsimonious small complex one option minus mu mu mu mu plus mu mu minus mu agree extend bic advantage require still encourage add rank possible update part stress factorization purpose inclusion outline correspond row add mu minus minus minus mu q mu mu mu minus fitting extend optimize value maximize respect variational posterior mu plus minus mu plus minus mu mu mu plus minus substitute end plus minus mu minus mu plus minus low predictor write optimize writing minus mu mu plus mu plus mu mu mu minus mu mu minus mu plus minus mu plus plus mu mu plus minus mu minus mu mu mu plus mu mu agree rank residual point detail later value far normal approximation n q l good value variance assume mu minus py mu mu plus mu minus plus minus minus mu plus mu minus mu mu normal mode obtain mu mu mu mu mu plus mu minus mu mu mu vi vi plus mu mu mu series mu plus mu plus mu minus mu mu plus minus available find good newton stop mu minus mu mu minus minus eq q plus mu minus plus minus mu back calculation write mean optimize stop store pc c lc lc j lc describe regard add current hereafter variational short like forward greedy widely scientific field encounter add backward elimination mistake consider burden factorization step bind current model sum base minus mu mu mu minus mu minus plus mu mu minus mu minus replace relevant need current quantity correspondingly plausible current plus minus minus plus minus mu plus mu minus mu minus plus plus mu eq mu mu plausible current plus mu minus minus mu eq mu mu plus mu mu initialize backward elimination stop lc vc lc lc j lc else hereafter might meaningful restrict search inclusion include restrict search candidate algorithm dc v removal elimination j pc j pc j j pc c v j later compare implement model shape allow response flexible bic fisher inclusion rather plausible var burden order var mu plus mu minus mu plus mu ny qx mu mu mu normal plus plus minus mu plus minus v mu mu mu mu plus mu minus minus mu respectively reduce greatly newton get estimate experience rank standardized optimizer rank q I rank absolute standardized residual agree frequentist predictor residual frequentist extra parameter penalization greedy tuning final plus log unlike greedy encourage parsimonious extra tuning penalize bias desire shrinkage well zero confirm stop make valuable high problem var practical benefit processing aim near four water validation serve variable measure spaced nm think range nm second final consist predictor response treat single popular ol fit current first predictor fit assume variable water set obvious see response clear absolute value residual fit response predictor select reflect ols plot nan model analyze assume nan th response predictor predictor visually mean employ result make prediction examine usefulness mean plus mu mu mu yx mu mu mu mu mu minus mu mu mu minus mu minus mu py mu mu mu understand mse sample var summarize estimate var probably reason var water also analyze bayesian transformation transform response wavelet bayesian likely value report comparable wavelet predictor prediction select anonymous separately model rather multivariate compositional justify transformation response mu minus mu minus plus mu mu plus mu mu mu denominator value mu plus minus mu I mu mu mu minus plus
stationary z classifier infinity every depend refer infinite double sided viterbi double side double hmm exist infinite viterbi alignment moreover viterbi satisfied positively nonnegative stationary code operator x rd rd rd rd stationarity fact infinite alignment finite viterbi choose rd rd double alignment process denote restriction integer note side viterbi process variable imply double sided limit smooth j py almost show inequality x r risk let consistently viterbi would hide viterbi follow call viterbi depend function mistake viterbi misclassification viterbi convergence state use surely theorem alignment ergodic viterbi alignment well risk bound go negativity boundedness risk sided infinite alignment stationary ix vx ii element theorem prove rd first term converge surely term surely auxiliary exist proposition moreover hold py almost surely analogously thus cauchy sufficiently q imply stationarity recall unfortunately hold provide surely stationary ensure hence sequence let big cn r remain py v rd constant lemma ready term converge prove let borel shall q imply moment approach section indeed counterpart inequality immediately probabilitie difficulty however viterbi convergence hold n nx nx state introduce sure viterbi borel adjust viterbi conditional integrable prove rhs viterbi alignment integrable integrable theorem state together give likelihood p n v ij v shannon remark entropy implies interpret conditional entropy difference alignment theorem eq every state also notation e let finally let cluster suppose pi py imply state arbitrary exist observe arbitrary loss bellman principle every every q since px sf inequality proof analogous proposition bellman principle cm theorem axiom section conclusion condition conjecture exercise section lemma mail se keyword hide alignment asymptotic segmentation risk goodness double stationary irreducible depend process call non observable emission shall assume emission measurable borel algebra generality hmm literature usually stand regime observation hmms widely recognition bioinformatic processing refer integer drop consist formally look state finding every let problem framework specify shall denote thus viterbi viterbi stand viterbi minimize often preferable pointwise loss classifying count misclassifie sequence pointwise posteriori alignment viterbi shall risk risk minimize clearly necessarily minimization importance apparent minimizer hard enough viterbi obviously integer estimate tuple show viterbi quantity classifier deal risk viterbi value grow viterbi obvious viterbi piecewise shall fairly hmm limit main limit risk constant depend viterbi alignment limit viterbi alignment segmentation sense risk big viterbi alignment classification alignment alignment model know asymptotic risk theorem concern ergodicity sided sometimes call regular ergodic follow state recurrent process couple stationary ergodic stationary version support copy e lattice span since chapter fulfil claim value satisfie theorem hold ergodic since begin limit immediately separately shift trivial consider process together def extended function almost realization statement exist first first element function alignment independently infinite alignment add alignment existence property alignment alignment infinite viterbi restrictive infinite viterbi prove assumption recall cluster intersection support emission cluster consist single state existence
establish show oracle dimension simulate accuracy useful tool polynomial retain advantage suffer curse pattern first incorporate selection paper compare nonparametric moderate include generalise et subject bandwidth fan although lasso recent relate linear include version lin zhang adopt et al aim thereby paper explicitly dependencie pair identically relate predictor smoothness discuss assign increase rectangular form assume bandwidth kernel vary function asymmetric respectively give bandwidth matrix possibly different size unbalanced lead undesirable dimension redundant possibility infinite dimension everywhere fairly exposition generalise find bandwidth become point large value decrease cost variance linear accordingly first traditional cross validation assess need eq lx affect increase variable influence influence remove area reduction dimensional uniformly response significance redundancy eliminate dimension variable significance perfectly dense illustrate adjust dot dash line modify bottom bandwidth entire imply include thus somewhat improve second top consider significant possible selection cross validation select tuning parameter different domain consider hx hx thresholding square redundant extend although unstable approach percentage variable exclude fit explicitly redundant zero lasso problem asymptotically naturally numerical polynomial whether initial bandwidth thus bandwidth bandwidth term function cardinality variance adjustment need expression shrinkage let step bandwidth case define expression measure shrinkage satisfy deal region domain behave somewhat arbitrary result force exclude still final advantageous situation despite ensure estimation increase suggest algorithm intensive work simplify expand equally meet bad grid stop simplification redundant classify everywhere grow easy fix great ability exclude circumstance initial instance proportional use employ offer practical advantage theoretical work addition initial allow variable long constant mention recent attempt assign attractive shrink greedy approach shrink cause change step dimensionality fit local penalty follow reduced penalty suggest comment difference presentation paper performance point ensure active redundant everywhere already strong nonparametric property algorithm technical find restrict attention situation could screening partly also dominate useful depend univariate introduce page uniform consistency estimator h first fairly occur follow purely convenience asymptotic almost everywhere apply cutoff circumstance entire minimum region estimation distinction tail problematic would uniformly good globally global case integrate establish theorem simplify version include hold degree estimate ensures consistently estimate scaling apply proof relevant notice illustrative open intuitively irrelevant thus region exploit small set latter correct form redundant everywhere correctly tend property local adjustment correct separate correct redundant everywhere ensure approach concern trivially correct tend everywhere probability tend result selection result cover decrease remove redundant actually whole fit elsewhere adjust locally follow variable algorithm perform oracle far locally actually square generalise boost approach r package respectively tune choose local include standard obviously fail htbp name description b linear fit relevant local generalise additive spline multivariate adaptive htb example methodology error simulation parameter average poorly redundant incorporate actually well redundant design specific estimate deviation std ols inspection good dependency particular improve performance reasonably nature strict effectively selection represent grid variable irrelevant wrong pattern broadly near boundary slightly large cover remove completely example distribute keep relate cutoff hard redundant dimension cutoff remove result exclude variable threshold need cause compare severe cutoff clear broadly correct improvement htbp use al air plus cube root response york fairly aim predict concentration variable temperature wind smoothed figure dependence part temperature wind flat imply high expand could useful comparative leave one make use observation build suggest model method offer traditional fit variable plot panel non perform curvature ol show use variable dependence fairly complex significant notice label redundant highly wind relatively useful approach std ol engine ratio engine air observation fairly equivalence clear htb std particularly suit produce improve model inferior linear section incorporate bandwidth variable increase understanding dataset potential effect removal partial removal variable redundancy apply property satisfied approach demonstrate classical regression concern predict response predictor dimensional often predictor
compare asymmetric tailed tailed skewness correlation alternative alternative normal beta cauchy laplace c uniform beta laplace beta beta laplace logistic result package package recommend distribution powerful alternative test student nevertheless perhaps cause dominate force close normality sign similarly might alternative recommend test test independence mention independence note complement theorem test powerful version skewness easier common normality widely field overview normality base characterization underlie population independent normal propose coefficient normality asymmetric discuss propose lin estimator skewness sample test simulation indicate notation central replication cube equal coefficient fisher normality sensitive normality skewness tail normality consider moment use manner alternative expression correlation enable consider correlation unbiased formulae somewhat easier standardized excess identically skewness nx standardize calculate moment involve know moment eq expectation expression little standardized tell value allow satisfy statement readily verify equality equality hold ii conjecture never similarly conversely moment obtain replace moment counterpart estimator nothing relate skewness scale test normality furthermore er lemma whenever necessary moment exist show slow prefer carlo underlie close nan hypothesis normality
occur france likely source south france occurrence majority case year triangle generate triangle year method triangle represent dash line period period consecutive week week second correspond link consumption period line stage week week week case age surveillance human surveillance seem valuable tool time cluster could public surveillance develop computer detection still remain necessity ensure public intervention way proceed order quantile promise future also plan investigate extension spatially acknowledgement thank head center provide implementation language none le universit france fr business school paris paris email fr france email system early health surveillance return count particular infection surveillance method surveillance france et theory return surveillance surveillance number event expect enough department death health publish classify three broad control review two step event week day statistical comparison statistical alarm observe significantly expect base count count current count occur g year incorporate regression easily trend model try reduce influence past avoid associate weight take trend past reporting delay incomplete inaccurate report surveillance system delay particularly problematic surveillance reporting concern surveillance encounter report delay surveillance report early surveillance system surveillance multiple statistical analysis daily automate method early cluster essential public health surveillance order generate development surveillance extreme early detection cluster report surveillance system cause human france cause laboratory confirm death occur year surveillance present description observation event develop last contribute surveillance perform surveillance hundred paper mainly illustrative purpose frequent variability event france mention impact comparable year see count within period series illustration give week last occur period five previous year alarm band realization distribute non probability distribution recall represent rewrite return quantile inverse associate able time standard exceed explicitly bound return level return period event purpose extensively assume follow generalized pareto peak threshold context use bound return level develop large small adapt associate upper tu namely confidence delta concern asymptotic confidence v ex impossible whole family function non function choose sub power real number last final follow low necessarily cover enough provide satisfying define alarm consist step explicitly period plot return period return plot read observation return bind theoretical return associate justify lead return period observation well variable interval since new period backward time hence see alarm second notice set one level finish plot associate observation existence least observation pt I illustrate return return period represent upper bind observation correspond respectively deduce return period equal
side sided sided student parameter obvious laplace plot region value small slope student illustration test cc student side solid sided freedom panel panel satisfie recall side satisfy meet student laplace rate another likelihood semi cover student prove proposition entail estimator bandwidth converge summarize introduce mh determine non follow behavior sided value semi differentiable derivative differentiable f side location student side sided student f derivative particular student corollary g laplace rate obtain begin latter require imply table side meet tend tend estimator location connection symmetry property likelihood ratio model side complementary contrary side always consistency sided location model estimator satisfied estimate sided laplace imply consistently estimate estimator derivative order study bandwidth converge rate summarize row satisfy lemma appendix hold bandwidth table gaussian regularity h differentiable convergence table side side sided sided sided laplace highlight test location side make possible conversely side slow property procedure procedure result procedure asymptotically tight fdr critical asymptotic power improvement rate parametric connected test require derivative conclude class light rate plug side laplace slow arbitrarily centrality distribution plug study plug asymptotically powerful procedure procedure far specific application exceed require extend interesting research realistic typical aim two value alternative proposition h h derivative negative x prove concavity x ratio student freedom centrality converge absolutely absolutely dominate j follow lemma entail regularity ratio freedom centrality proposition series entail derivative product series dominate tend concavity student student model proposition likelihood f decrease j conclude proof g therefore go central theorem fulfil yield k h conclude equivalent proportional choice choice prove converge proposition converge obtained hold unconditional asymptotic fluctuation start state procedure consequence converge surely surely go integer enough mm entail number procedure less great write empirical observe consequence fact alternative hypothesis unconditional theorem mt mp respectively consider establish translate unconditional setting replace therefore procedure unconditional adapt theorems unconditional threshold simply unconditional model threshold critical recover proportion plug among lemma converge almost far mh proof almost sure sketch inspire converge surely nan converge item concave fluctuation thus fluctuation negligible therefore taylor converge g dominate finally conclude proof theorem consequence lemma let b recall fact delta g proof x conclude lemma side make assumption differentiable tend simple f onto satisfie differentiable prove extra entail write bt b statistic test statistic false plug procedure control estimator simultaneous hypothesis test parametric wavelet method functional image dna microarray high throughput sequence set goal infer true definition discovery one widely risk multiple testing problem expect false positive among procedure test independent test independent procedure fdr typical go infinity plug construction powerful fdr target study correspond fdr influence plug kernel density statistic testing statistic identity denote respectively symmetric typically fulfil laplace exponential satisfie side function sided well non concavity hold consider sequence test integer side sided test characterize I terminology deterministic setting originally introduce set specifically random independently identically conditional setting satisfy literature independently identically distribution vanish restriction natural unconditional paper situation identity entirely characterize characterized concave generally regularity regularity assumption assumption express density alternative take return index hypothesis reject call testing reject correspond negative ii measure procedure I error measure discovery widely denote among test discovery proportion define risk type error rejection high type may error power testing positive hypothesis quantity depend r v whenever thresholde fdr prescribe order hypothesis rejection fdr previously use another context fdr regardless specific simulate less mt information interpretation mt set henceforth short entail decrease remark procedure sense maximum nan large threshold still maintain natural apply level estimator stage geometric interpretation rejection converge powerful adapt estimator may write value true nan hypothese one converge stochastically dominate uniform include set plug add numerator control estimator view order density integrable identically mx call asymmetric rectangular kernel characterize introduce formally connect depend test test specific testing testing prove multiple almost intrinsic sense regardless multiple obviously limitation g statistic satisfy ratio near critical result asymptotic let proposition asymptotically nan power generalize critical critical multiple critical illustrate sure result extend soon prove theorem cover procedure procedure value rate soon connect fdr control light connection interpret oracle level powerful thresholding fdr value procedure problem thresholde fdr contain procedure include organization extend include modification tend parametric rate section fdr plug achieve plug section test location moreover regular q bias positive estimator decrease plug consistent meet meet shape asymptotic bias variance characterize variance positive h differentiable h bias regularity near regardless trade natural resolve trade optimal choice asymptotically correspond prove convergence parametric estimator density derive g require match regularity proposition kernel estimator optimal bandwidth mse proposition convergence regularity proposition assumption kernel importantly convergence well asymptotic convergence recover envelope assume estimator converge strong regularity minimum away converge additionally proposition lipschitz derivative unnecessary increase estimation regularity rather aim rate plug limit associate false proportion fdr controlling procedure plug depend set unconditional relatively technique adapt completeness procedure bandwidth consistent define procedure asymptotic converge rate dominate result plug procedure proposition differentiable case bandwidth critical procedure asymptotic procedure unlike nan former summarize meet threshold asymptotic procedure particular whereas asymptotic value imply target fdr positive fdr asymptotically plug controlling decrease threshold remark name oracle g g power
kernel parameter select mean law free consequence simulate trajectory limit reach simulate rate result allow monitoring procedure behavior result nan follow walk state discuss behavior series behave change condition shall discuss e omit notation satisfie satisfy ensure sequel notation satisfy let process sequentially update ts define assume satisfied hold sequentially update l process asymptotic kernel ratio limit c brownian central corollary stopping monitoring procedure residual q innovation choose sequence walk correlate change walk stop concern design monitoring yield linear slope detection residual limit thank careful computing simulation final appendix proof result eq functional pz p p ax ax nx ax linear matrix recall ts yield since ts ts ts ts ts j j may eq lemma tr ts uniformly functional easy application formulate change modification exposition map show assume number bind stochastic product topology consequence process weakly thus argument weakly continuity virtue previous position result process associate stop sequentially thus modification assumption theorem may weak combine functional vi linearity remainder induce observe proof pt remark update residual stationary error institute statistic abstract behave stationary important particularly arise financial statistic engineering study detect walk theory monitor control chart test residual central corresponding limit chart finite keyword autoregressive chart sequential classification address institute mail de walk production degradation fail reach threshold financial walk appear price asset random hypothesis finance address increment random walk important economic product sequentially compatible control series answer lead completely wrong conclusion elementary change limit implication rich nonparametric control classic chart discuss detail random brownian detect know mind particular monitoring surveillance detect walk hypothesis soon article investigate monitor relate root detail unit hypothesis stationarity use statistic dispersion sum dispersion statistic et unit lee schmidt stationary study refer al test powerful robust rate walk stationarity versa sequential monitoring stop establish promise preliminary study involve error walk allow nonlinear observe sequentially time observation satisfy motivate polynomial assume detect control favor cover motion drift brownian motion substantially polynomial trend stationary chart stop evidence compatible hypothesis behave walk polynomial establish stationary provide may surveillance particularly signal classic confirm study square residual apply residual often classic computing introduce horizon monitoring stop application monitoring conduct however modification infinite straightforward discuss error sequentially residual constructive representation asymptotic brownian motion asymptotic change fraction random finite property proof result remain form unknown integrate set clear stationary change state walk integrate order specify satisfie stationary ar equation note coefficient lag exactly sec hypothesis assume mild making weak strictly eq brownian equivalent brownian combine assumption yield nonparametric time popular class series recall satisfie put classic time define version tt calculate appropriately statistic formula call nonnegative decreasing increase intuitive recent work condition kernel define instance back increase chart convention horizon stop asymptotic moderate limit monitoring stop may early stop favor alternative control rate favor stationarity indicate control characteristic exposition shall monitor start eq inference begin evidence get walk compatible simply appropriate scale replace place mt vi loss rarely brief exposition weakly yield pf p subtle value define right open singleton closure function appropriate define exist f continuous separable weak study term regression behave chart inf associate turn negligible central theorem process theorem notation intercept base x process former sequel stand ts polynomial fix limit play crucial role main right ts increment scale statistic degenerate distributional process summarize correlation inverse need
preference rate construct subject interest recommender extensively adopt mechanism history transaction algorithm advantage traditional filter music filter hard represent content three primary hybrid show deduce primarily algorithm introduce describe filter base item continue algorithm use recommender amazon com security recommender result filtering email bad associate annotation message annotation rating share annotation message although good drawback query recommendation recommend article find user algorithm similarity similarity category understand algorithms understand similarity closeness attribute tuple help compute tuple attribute categorical vs base pearson correlation item dimension attribute rate know calculus formula another rating show interest good give pearson user rate show pearson cosine et al similarity recommender user item base belong user item user user step first one user user associate weight importance third recommend high create user performance algorithm find user similar algorithm user datum one user pearson pearson correlation cosine similarity approach require item rate user item simply directly captured repeatedly come look define pearson item capture look rating give category database rate similarity cluster find recommend suffer drawback user item website user rate portion rate user item lot insufficient form greedy generation near require computation grow number change database since recommendation conclusion item user recommender recommender recommender make past analyze idea future historical look user make item implicitly rating category item algorithm step step scan past rating give similarity item item algorithm compute item large sure recommend collaborative computed implicit specifically opinion item user rate item formula explicit item average rating implicit implicitly compute item correlation boolean item equal user base major disadvantage build item construct similarity pair rapidly scale accurate recommendation scalability amazon com website book million customer million amazon com recommendation collaborative work rate item item element item similar item record similarity improve performance amazon com build recommender offline create expensive costly item offline item recommendation online rate recommender system reasonable recommend course user properly recommender domain recommender system metric evaluate finally recommender recommender differ poorly word recommender thousand thousand opposite movie recommender vary accord value similarity way extract get system thousand million item time system cope growth suggestion close recommendation user preference recommender system understanding suggest look gain trust evaluate recommender compare recommendation metric mae user item receiver operate characteristic roc metric prediction interesting regard recommender example visit user recommendation useful visit mean recommendation know recommendation metric sometimes recommendation order recommender system reliable accurate reasonable request per evaluate efficiency set compute item big grow quick look problem item trade quality efficiently give rating item knowledge customer memory computation recommender system calculation two perform time run hour speed calculation technique hierarchical search recommendation thousand request efficiency vast item trade prediction scalability consider maintain hashing obtain reasonable recommender rate lot rate call percentage item recommender agent prediction face employ recommender challenge issue rate establish recommender item redundancy user rating propose implicitly behavior approach rely filter call dynamic automatically recommender give recommendation preference accurate become moreover since recommender recommender trust need name birth code email http preference express recommender system user express identify company netflix website every recommender exposition user recommender system also security recommend recommender often target book recommendation book recommend amazon recommendation item may type attack item either attack attack rating rating item affect recommender call recommendation another item recommender way attack know stop study collaborative find propose hybrid combine try efficient recommendation categorization item appear extract cluster algorithm base preference similarity list merge depend item recommendation top recommender system cope item make suggestion approach massive growth ensure term employ user result extract suggest accurate propose deal whole portion due implement apply amount memory much less calculation recommendation term approach item issue apply system able user item profile program name algorithm comprehensive algorithms evaluation cluster association rule feature www os index open http wikipedia http en wikipedia loading file format part file gender student child movie category table table movie movie child provide software sample criterion program modify manually make dataset logical website movie title view rate user generate dataset entry order user category belong pre step file select class appropriate recommender objective recommend title category predict specific would also child movie hand child action movie recommend profile base accuracy experiment classify recommend movie title belong select
relative single process cubic computational preferable gps data gps input collaborative filtering notably winner netflix include drift rating svd predictive use factorization entry pmf possible closed markov mcmc posterior distribution various variable construct carlo typically straightforwardly predictive quantity interest integrate evaluated hyperparameter typically capture relevance covariance control likelihood marginalization process necessary function carlo posterior invariant chain state evolve close e hasting hamiltonian base require treatment detailed subsection slice mix impose several collection typical recently method transition gp number function covariance correlation process play critical effect affect hyperparameter mix due strong constraint despite therefore application marginally distribution hyperparameter specify result conditional mix chain cholesky hyperparameter hyperparameter covariance new unlikely noisy update work well hyperparameter change advanced develop method utility score national appeal approximately censor evaluation player side team play home relevant outcome reason new model game game team team pmf type see score perfect match score negative team tend ten dyadic e different feature contribute contribute enable side use provide relative pmf censor data eight four week ask prediction game past prediction month could entire model metric mean rao winner rmse winner accuracies rmse assign number spread spread yield point prevent tie single unit score take single cause point view evaluation spread expert identity report market force refine line spread imply mention previously conditional entry typical coefficient censor rao store state compute covariance component mixture weight good incorporate temporal fluctuation aspect vary appropriate end handle include slice sampling infer gap mass true different standard bayesian pmf method temporal home number ten chain censor interval year state chain warm start year run start iteration score keep chain component prevent pmf heavily influence previous prevent advantage relatively decomposition improve extensive sampling sampling remainder iterate approximately minute modern hyperparameter span game prediction mild believe effect covariance notable baseline pmf model improve home away alone consistent evaluation interval effect complex clear game home away prediction illustrate model information home paper induce pmf mcmc carry make prediction prediction notable process specify allow variation slow variation star reflect something g address issue nonparametric insight kind latent true indicate winner home temporal home away home away probability winner leave right expert bottom r r r pt r cm cm acknowledgement wish thank valuable placing institute research name probabilistic pairwise relationship collaborative analysis among area actor difficult pmf model side coupling pmf gp function vary smoothly successfully use date model interaction interaction item link salient feature task observation netflix user movie associate rating predict unseen pair sort relational pathway document data treat structure word occurrence pmf model powerful dyadic current pmf pmf often available identity collaborative example rating incorporate low feature however simple generalization probabilistic replace feature place prior introduce pmf relate probabilistic incorporate feature pmf typical probabilistic collaborative set explore later rating observation game use interaction whether user filter movie idea information apply pmf uninformative reduce pmf pmf marginal pmf share one correspond player allow hierarchical hyperparameter strong prediction capture length scale
notation real l undirecte give membership assignment node cluster simplify take connect connect assignment assignment node generally social network definition external node outside relative find characterize commonly definition set property graph give community consider clique community search define property every value sometimes entire pointed often definition community decade see lot topic community good concentrate internal community perspective community belong may describe allow community evaluate expand community maximize whether accepted spectrum clique influential essentially clique recent release synthetic benchmark graph become possible study benchmark generally number world social community overlap like overlap benchmark benchmark place overlap exactly community base explain community avoiding develop scalable highly community structure graph realization model unweighted loop represent connect network review interest work notation describe assume community assignment describe assign graph independent node bernoulli probability community assignment end inter community model draw assignment refer quantity proportion refer difficult em algorithm among thing conditional heuristic maximization search graph cluster use value greedy strategy assume community integrate likelihood integrate create mass mean selection bic technique overlap similar treat parameter integrate allow expand new name overlap blockmodel full column assume draw bernoulli community replace connection sigmoid natural extension estimate use allow restrict consider restrict form clique sbm share integrate cluster allow another heuristic community assignment place allow integrate connect connect independent connection tendency tendency node connect simplification become imagine every treat integrate possible permutation community partition equivalence matrix differ belong use community contain zero q community assign allocate community choosing summing write speak consider bayesian however refer attempt maximum computationally cluster estimating maximize complete likelihood show result clustering gaussian remainder emphasize primary integrate analytically give eq consider nuisance totally yet appear choose triple sample modularity small community calculate million edge maximization add community increase community decompose update remove community order consider allow convenience change eq pair depend whether change simplify fix unknown constant ignore decrease little moreover whether update unique column find introduce multinomial would expect find community output attempt node think community community graph initially community consist node directly add maximize expansion high far small contribution negative clique small community dominate whereby continue decrease unless consecutive expansion improvement subject expand community count community expansion claim expansion informally densely connect removal positive occur edge expand assignment community take place end inspire remove depend current estimate change seed impact fewer share old pair node one extra one share associated calculate node g g make change old ignore search expand factor maximize maintain ratio community update ratio search diagram skip fix investigate logarithm running time fast capable fast get result scalable trivially objective variant modularity fast maximize partitioning find hope progress restrict algorithm facebook five distribution often assume empirical graph strict law average five assume node community facebook user facebook divide six community nod
notational simplicity indicate belong assume throughout design deterministic allow effect large complete deduce q q due restrict likelihood regression add regression fix vector component minimization section likelihood mix section posteriori principle density correspond variable I marginal vector I regularization bayesian bic motivate degree freedom selection example cross criteria bic selection base simulation due lasso penalize may penalize loss study deal knowledge high smooth build upon prove response assume k estimator lie respect since use negative log risk coincide kullback sequel drop require condition eigenvalue bound triangular allow study notation consider support optimality form adaptive also different b automatically order set support appendix could involve adaptive penalize give oracle penalize estimator estimation imply error absolute bind distinguish argument corollary estimator meet restrictive similar result adaptive model version restrict eigenvalue current full recall eigenvalue hand parameter take motivated q penalize assume corollary set appendix main keep calculate employ inexact focus mean consequently omit ordinary cycle involve kind first notation prove achieve defer support information p section p jj value appendix support implementation website every cluster give appendix convexity achieve optimum performance kind mixed effect remark provide support hereafter classical linear mixed package furthermore standard choose overview conclusion effect aspect appear covariate incorporate effect covariate coefficient penalization thereby get large relate covariate aforementione cover parameter simulation remarkable incorporate effect estimator large effect focus effect use diagonal covariate elaborate paper suggest subsequent restrict group penalize intercept assign slightly grouping increase square use suggest approach production covariate measure expression issue covariate specify matrix deal determine apply validation effect coefficient bic lasso include seem reasonable fit wherein additional cardinality fix order fix effect cccc table considerably small indicate might dominate whereas slightly penalize maximum likelihood difficulty thereby deal substantially challenge gradient descent algorithm provable numerical study real datum remarkably incorporate support proof mail consist ensure secondly completion refer parametrization exist inequality norm assumption present result increment low increment constant term constant discussion assume throughout see eq omit deduce restrict notational degree n nj st survival variable eigenvalue centrality claim set eq technique nk formula define proof theorem comprise main present fulfil verification check drop slightly parametrization one proof vector eq make proof appendix matrix strictly positive definite exist correspond excess get eq first term w constant q right q apply restrict restrict give arrive corollary consider appendix solution cross grouping structure ordinary lasso calculation gauss element positive fisher information constant min r p suggestion take fact respect stepsize eq truncate simplification reduce especially setup set package employ cholesky triangular penalize reach choose eq reduce use active specifically cycle reduce stationary check assumption fulfil precisely proper convex continuously simulation study elimination example intercept deviation run tp positive emphasize coefficient indicate h ccccc tp coefficient subject penalization c ccccc penalization active slightly cardinality set select model coefficient active variability penalize bias effect problem z penalize estimation grouping structure provable coordinate descent achieve decade statistical allow large idea pose may speak inference lead asymptotic behave lot dimensional powerful name method reason couple feasibility optimization exhaustive selection square estimation statistical lasso work numerous respect establish equal optimal would would emphasize obtain design get small something aspect involved show lasso consistent selection non value neighborhood stability condition neighborhood rather weak restrictive sufficiently say huge reduction stage method restrictive assumption l non requirement eigenvalue requirement sufficiently eigenvalue stability lasso everything generalize penalization finally prove selector lasso positive besides include subsampling assigning regard homotopy lar coordinate gradient descent efficient datum incorporate grouping group effect besides effect extension concern longitudinal
business parent remain limited conventional analysis would overall behavior opinion conduct medium utilize customer business count sale figure average type role tail apply copy book instance place great importance public transmission sort viewpoint book analysis contrary sort conventional normal sampling internet also transformation past distribution generally inferential benefit technology grow momentum distribution video product video almost service activity sphere call population make age article analytical central distribution sample asymptotically normal useful sample distribution directly precisely calculate convolution considerable technology statistical analysis use distribution benefit individual content service content determine meanwhile software also benefit service enable perspective view relatively unknown become contribute service value introduce statistical software analysis discuss software base research internet population sum population principle calculate elementary size practically impossible size prominent limit kind extremely characterize much concentrated asymptotic probably also great unified research investigate treat distribution population calculate ad hoc recently size tail valuable present specific method tool tool microsoft calculate mathematical explain formation population worth look differently relevant release public drive valuable assess site analyze develop google statistics site department statistical service concept share private share public set amazon service public become next several representative service article organize relationship user service service separately single service amazon com site stock case site often able comment sometimes future service collect general public internet site product service user content video sharing service share site numerous allow comment total user become value piece feature service tag various tag content summarize manually content search separate tag system service social service various improve similar user knowledge resolve computer system sometimes collective intelligence collective intelligence environment normally since certain service measure tag user experience google search perform search service tool audio file allow speech mistake game google element contribution ignore thought drive behind example site comparable user user extension characteristic collective knowledge modern internet service form model analyze growth tool analyze example follow evaluate use previously way video video site team evaluate tag basis importance site proposal tag text site universal various service rather compete section analysis software package essentially convolution arrange file format value store hash hash convolution read convolution arithmetic operation long theorem useful time spaced example convolution memory consumption thought calculation feasibility accurate value calculation include convolution complete sufficiently equip convolution width determination skewness degree central meet derivation derivation probability transformation however explanatory information probability win person change place significance impossible apply hoc convolution calculation hand win first adjacent chapter discuss software practical social service site store involve various tag recommend similar tag service boost site important thing common wide internet service service share site evaluated comment share comment tag site indicator content site website tag comment determined sum model sale business management product customer provide sale strategy sufficiently theorem law meet sale internet rare average customer attempt significant continue stress set comprise user include user analysis aim identify tag would include illustrate tag activity basically expect law number customer decay tag increase tag allow maximum tag tag develop among move illustrate main frequency site distribution treat clearly user large central relative tag histogram long histogram choose tag observe obtain easy rare term value tag assign tag tag parametric test multinomial distribution another readily calculate modification trying determine tag appropriate problematic allow compare assign assign skewness skewness relationship customer total tag number index site tag high relative customer correlation read basically zero variation indicate search site customer site quantitative extracting define site rare figure illustrate rare customer positive increment axis axis recall rare account site rare value account customer determine site rare probability volume customer customer often rank upper percentile lower probably difficult determine threshold line fail student pass fail similar event ordinal tag probability via customer next calculate meet eliminate operation plot common axis sample central limit dominant grow account total majority small substantial axis limit avoid effective analysis item consideration generalizing examine application follow one content receive induce description tag make easy speech recognition correction
action relative game inequality nash payoff maximizer subject instance dynamically maximizer cycle maximizer period period bind crucial maximizer action action paper game player obvious sum sufficient maximizer cycle submatrix probably paper call contain game subject violate directly relative payoff maximizer never suppose maximizer choose action period improve relative payoff period symmetric payoff action non path cycle along cycle relative player action finite game subject positive payoff improvement cycle cycle contradiction g assume maximizer maximizer could payoff already must rule prove cycle converse game payoff generalize cycle cycle play cycle column exist generalize generalize payoff strictly payoff contain maximizer select row relative next period play maximizer action e period another maximizer cycle argue necessary saddle saddle generalize saddle subject paper submatrix profile rise submatrix subject game game symmetric sum game follow finite sum game example game matrix row diagonal payoff essentially game maximizer period maximizer period period maximizer must contradiction maximizer payoff matching counter consider payoff game additive restrictive say function separable let separable payoff symmetric induction suppose step maximizer maximizer merging apply yield maximizer directly maximizer ever relative payoff separable indeed exist corollary proposition application compact function continuous function e pure equilibria convergence totally resp xx difference consequence sum game difference separable payoff totally decrease theorem potential game symmetric potential py py symmetric payoff game essentially proposition demonstrate payoff write apply essentially cost price write ax ac cx benefit assume function study decrease essentially pool resource common resource game two outside activity common pool resource denote maximizer pool resource likewise return resource effort effort game van among individual relationship opponent symmetric payoff corollary essentially arm arm totally concave function player search player search trade another payoff game separable separability payoff counter game game player relative separable yet maximizer zero maximal useful obtaining equilibrium explore implication notion potential besides notion potential potential game symmetric game game potential px px py py ordinal potential game py py ordinal generalized px py x py ordinal potential ordinal potential paragraph ordinal possess nash finite ordinal potential possesse sequence profile exactly player sequential path strict player action strictly payoff strict contain focus relationship payoff game possess ordinal potential cycle cycle cycle cycle construct action action sequentially turn maximizer strict note e maximizer maximizer cycle I converse counter follow possess construct strict game ordinal game subject cycle finite game improvement cycle imply converse generalize payoff game converse true paper strict possess ordinal payoff potential exact weight potential ordinal symmetric game weighted essentially game definition quasi payoff space symmetric exist peak exist space abuse relative payoff game symmetry payoff x x xx x game subject reach finitely step order always large strictly negative likewise upper analogously maximizer maximizer never maximizer maximizer claim exclude exclude maximizer choose choose claim hence simply maximizer element far step period repeat reach reach stationary leave relative euclidean f fx finite dimensional imply model nash game nash demand game nash player demand amount within receive relative player symmetric payoff finite payoff subject payoff remark improvement ordinal game neither paper many game economic possess natural aggregate player game game resource view aggregate study os absolute payoff second totally monotone argument ax ax ax yx sometimes shannon aggregate show action inverse cost require forward game converse single inequality q since relative payoff improve reach improvement possibility maximizer claim action lemma maximizer payoff else new would maximizer choose new step new start repeat stop complete contrary eq contradict prove present extend payoff symmetric demand cost player win proportional bid player bid bid symmetric payoff game al os game game corner finite strictly contrary q contradiction ax ax contradiction ax xx strictly show step maximizer payoff nontrivial maximizer repeat relative nontrivial maximizer take corner show either contrary strict contradiction payoff player payoff throughout nontrivial maximizer corner finitely sequence reach finitely corner reach summarize symmetric game function symmetric complement importantly relevant game complement fair seem game prop payoff heterogeneous public pool resource effort relationship arm game potential payoff prop ordinal nash demand games modular game prop modular generalize thm game need aware limitation analysis primarily game scope paper show game inverse demand write quantity profile cycle induce yield clearly require among thus recall truly sophisticated whether sophisticated experiment payoff go wrong consider suppose start equal maximizer choose price maximizer could marginal cycle example show crucially heuristic class leave conjecture chen interesting discussion national international conference helpful comment symmetric player game decision necessary variety example game competition pool resource game effort arm opponent payoff game ess games games game behavioral stress rule human make capability heuristic decision plausible adopt bad situation long heuristic heuristic look rational currently mutation pool rational exploitation opponent evolutionary follow heuristic even rational look maximizer class relevant play computer human subject easily exception action
say scale tree say law corollary part straightforward convenience define posteriori decode optimality span edge apply coarse exactly cardinality forest formula tight give converge complete natural law different manner regard attribute describe abstract forest oracle knowledge need fraction conclusion
pa array dim array dim pa pa pa nb pa col col pa density conditional e deduce code exp cx col py density normal variance yx yy sx sx cx true col true x independent distribution large conditional distribution large obviously truncate poisson conditional distribution gamma distribution sample gibbs sampler code grey main col rao checking execution program ten correct initialize initialize st col grey st st per se integrable infinity constrain conditional give well simulate joint initialize I exp st exp col add type invert cdf represent gx yy yy predefined sigma mu inf inf sigma precision col col x col col add alpha alpha alpha sigma alpha sigma simulated histogram integrate histogram satisfie detail simulate markov alpha else nice fit concentration stationarity pass fail st fraction nd window var give moving modify alpha x else stationarity var pass fail window fraction window posterior sigma r beta sd sigma mod sigma sd mod sigma sigma beta else beta prop sigma prop sigma prop sigma jacobian probability additional running diag est mix chain completion factor seq seq ks nan le ks converge converge pointwise value connect exercise get example sequence beta check via program ess ess subsampling justify mcmc figure compare evaluation chapter none strongly manual book monte publish solution paris student master whenever appropriate code name grateful student solution manual incorporate manual manual demand strategy find book come ignore book universit paris www books complete reproduce manual stress r come solution solution behind argument effort put manual study introduce algebra conditional notion cover reader lose obviously suggestion manual independently correct almost code page prefer code code student fast code along page automatically substitute code request two odd one access version since become book possess manual please explanatory self explanatory execute seq variation array dim replace interval quantile histogram quantile get region array dim array dim replace confidence interval datum book type environment base information sd function false sd na else na else sd na na else sd look description database modify format recommend assign rather assignment create sep mostly explanatory recommend function save write need transpose internal fairly see try allocation fast inf inf numerical lead fail high enough lattice library produce representation tree index tree estimate std value pr intercept tree plain non false array consider self explanatory instance loop whether integer exist entry box entry remain plain grid sum pool wrong break stop outcome solve cdf u fy fy easy calculation histogram tail pi array dim tail histogram numerical box compare exact range acceptance accept acceptance bind therefore attain easy contour maximum value truly take exactly try code cp array histogram array cs use create alpha cs alpha alpha see histogram use see pareto uniform variate function spread sum sum lambda lambda user lambda user lambda lambda exponential good appropriate return reject obtain maximum since require modification derivation e lead optimal mean ratio maximal lead exercise density square hold derive also chi rotation vector lead variable central chi include centrality nc chi square scenarios ii df df fast since likelihood density improper prior mean mean proposal inefficient credible repeat experiment lead col ty l legend numerator cauchy approximate directly x type col col gold evaluation ratio estimator separately variance get digit normal replace comparison clear normal simulation compare identity see expectation distribution thus approximation pi hx gold col estimate probability compare inclusion exercise book exercise I iid q fy c la la est type px sep col convergence plot clear like appear picking seem produce tail tail straight integral exist sampling eq integral integrable go acceptable quantity x according accept reject empirical mean deduce pi c code exp gold col gold inf pi gamma inf exp col gold col jumps jump differ evaluation absolute pt low e variable accuracy use q approximation simple accounting lead exp exp evaluate w effective preferable choice w efficiency simulation estimate unknown self simulation hence bias harmonic let tell n marginal try marginalization rate ok check look plot seq estimator produce decompose term q result likely correspond likelihood q quite similarly consequence jensen slight read establish since exercise read complete square exponent opposite monotone another exercise exercise exercise convergent estimator denominator exercise indicate exercise programs rao col grey b col grey col sensible b sd type col grey col grey add b col miss distribution beta x grey n col grey gold clear notation replace accept acceptance time associate numerator among probability reject subset uniformly accept choose w j hx hx hx hx hx hx hx hx us density optimal accept reject second moment know approximated instrumental density simple mu add da mu surface sample impact surface binomial constraint find obviously hence therefore subsample constraint simulate simulate circle pi col performance simulation domain via acceptance rate mixture minus mu package reproduce program like log factor calibrate convergence optimization exhibit mode circular program path mode mode mode matrix prox mode c prox schedule sa sa illustrate experiment four outcome mode indicate principle involve power logarithmic involve thus impact remove start point stop base binomial x col grey col gold range true program obviously likelihood apply surface five mixture question capital namely datum relie problem c em produce component notation question density miss normal upon know therefore normal exercise exercise simulate chain sd add hasting accept property transform decomposition derivation detailed exercise integrate x metropolis simulation alpha alpha col lines alpha drop alpha b alpha function c b alpha acceptance change respectively reject candidate max col reject col add simulation give hasting gamma implement col ga efficiency implement g metropolis hastings ga col add quite use early miss correct txt header final add save simulation graph assess col main col strong across rate unstable converge interval question must exponential available run glm binomial pr intercept dispersion freedom freedom aic beta result check col red failure exp add col gold temperature metropolis x sd beta proposal else b b else acceptance acceptable explore c col col l col failure seq exp col grey add seq
absolute distinguish coefficient context test invariance proof hardness inspire game geometric result list concept class learnable noisy consistently explain common address agnostic agnostic class hypothesis require nearly say hypothesis perceptron boost study widely system label linearly determine agnostic agnostic agnostic class constant correctly label exist agree note hardness agreement always example draw essentially subset decision imply hardness proper decision list describe detail hardness learning exception hardness learn rich broad hardness proof invariance hardness invariance principle np hardness cover strong condition unique learn decision list study agnostic equivalent ability come function refer class equivalent long know complete later improve david et result hardness approximate agreement every coordinate whereas work hypercube maximum li hardness approximate agreement david et subsequently finally tight hard distinguish consistent prove hardness allow clause decision hardness hardness problem exception number hardness complementary minimize disagreement evidence hardness agnostic even proper agnostic major al distribution decision list learnable presence random perceptron separate margin robust mild noise hold adversarial et give non agnostic learning give agnostic hypercube analogous algorithm analytical purpose illustration test suffice unique game encode boolean conjunction refer theorem integer connect hypergraph vertex vertex label vertex say strongly vertex sake clarity sketch special games conjecture constant integer positive strongly label weakly fraction strongly agree agree clearly statement convenience hypergraph e coordinate coordinate vertex coordinate every labeling thus label correspond idea fraction weakly fix identity permutation every complicated ex coordinate write procedure output negligible conceptual clarity label f r rf v intend problem linear function satisfy completeness close nature refer hardness approximation notion close influential coordinate coordinate little resolve use critical recursively influential none critical denote say close index influential coordinate influential appropriately counter employ recent independence outside agreement negligible operation query pr distribution carefully infer structural independently bit sample construction pass probability write expand I note large fraction substitute two distribution invariance principle unable c condition value conditional column still invariance principle coordinate j enforce hardness hardness game reduce label cover complicated consistency check encounter hardness commonly map label vertex several label mostly identical correspond edge natural extension unique game hardness game fact execute reduction precisely cover vertex principle certain fourth moment smoothness term reduce np great uniqueness property mention convert game hardness unconditional hardness inspire avoid geometric tight factor relate fundamental space among semidefinite unique game hardness np tool notion coordinate w k w critical contain geometrically subsequence weighted fall interval proof regular define pr distribution fall probability bit bit fall set coordinate regular define pass statement pass least list intersect lemma pass intersect immediately less approximated critical h matching introduce whether suffice show moment match degree influential fix rewrite c possible randomly ensemble c ensemble coordinate one lemma moment moment also unbiased ensemble spread principle average get two influential unlike traditional unclear whether number influential define generality geometrically imply know eq h ok discuss case overall index show conjecture although hardness assume describe purpose proof e r define produce label conjecture integer labeling fraction consider v v r agree agree probability ok labeling vertex randomly reduction good rt additional smoothness problem agnostic give hardness property exist constant integer hard distinguish e un strongly every weakly weakly least fraction addition fix vertex pick contain map find start un e l produce example refer projection follow v output example completeness smooth weakly parameter agree probability combine main remain correctness claim completeness I ei agree label strongly obtain agree complicated section almost edge group group copy almost lemma formal nice appear appear generalization section let agree v vertex nice respect I projection vertex nice vertex nice vertex follow property vertex average argument ok property therefore overall denote generate agree denote projection brevity shall v k v c j notation fix ensemble r invariance ensemble claim degree ensemble spread invariance principle imply eq claim hold average setting ensemble moment degree moment moment agree condition condition c n therefore v I substitute give I tt ok e matching show identical condition without w define I geometrically notation similarly h claim pr g correspond lemma every chebyshev inequality eq recall ok case h q lemma replacement change overall combining agree first statement weakly define suppose agree fraction agree nice good nice exist intersect labeling overall strategy since discussion hoeffding let real random unbiased chebyshev real generate copy generate write claim pr I vector pr claim claim verify apply inequality z independent suffice pr fall invariance principle r ensemble random satisfy ex random ensemble ensemble k c
move proposal random metropolis simplify refine random metropolis proposal add third purpose tail third explore make easy leave mode walk sampler normal component vector identity walk value show random walk walk local mode proposal adaptive hasting density stage estimate target adapt heavy tailed stage density construct iterate adaptive component ten time normal normal obtain stage begin iterate maximum schedule describe tailed density include strategy mode effectively computationally article normal third identify estimate normal harmonic stochastic normal set tune approach report tuning see sensitive particularly naturally distribution flexible density transform proposal multivariate low degree freedom whose help explore copula dt cumulative sequence copula marginal normal freedom copula estimate marginal distribution degree freedom profile copula value small iterate mixture weight component draw location finding copula schedule proposal often chain symmetric generalize allow implement sample multivariate generate respectively accept reject accept third would fix throughout stage component unnecessary third performance without discuss rwm walk rwm metropolis normal cluster mn cl describe independent metropolis hasting distribution copula normal cl sampler proposal term compare take sampler accept metropolis proposal sample divide generate factor autocorrelation lag otherwise lag low sample iterate rate account define interpret sampler attain attain draw sampler sampler take extent language affect sample logistic regression different prior coefficient second intercept double laplace lasso double exponential spike tail normal distribution third prior component normal assume suggest variance example work unconstrained covariate list discuss l year year hour work marginal market experience hour run three target double normal obtain fitting normal prior ht exponential intercept acceptance rate size algorithm min max median max rwm rwm mn cl cl exponential rwm rwm cl cl cl normal rwm rwm mn cl cl rate adaptive walk result compare adaptive metropolis hasting proposal version well home home relate otherwise list ratio american history slow pay pay slow account insufficient history payment record otherwise self otherwise otherwise year otherwise history equal history start initial distribution adaptive generalize matlab normal summarize marginal normal cc normal exponential mean intercept table equivalent sample computing walk high metropolis especially normal prior copula high rate normal multimodal ht min max min median min max prior rwm rwm c cl cl cl rwm rwm mn cl cl normal rwm rwm cl cl cl proposal freedom give copula coefficient reference therein prior parameter take mixture normal distribution worker survey logarithm list carry bayesian independent run double exponential case table marginal posterior estimate age worker year square age education high school post secondary complete college omit cover collective omit otherwise trade service otherwise omit mid west south west east never omit category less year otherwise visible otherwise cc normal normal mean intercept acceptance equivalent size computing scheme distribution independent adaptive proposal acceptance rate low l min max min median rwm rwm mn cl rwm rwm mn cl cl rwm rwm c mn cl adaptive integrate effect response probit cumulative parameter ij covariate q proposal include adaptation iterate mean similarly accept choice design may whether choose cancer presented ask would binary covariate l know otherwise patient due patient cost effect double mixture normal scheme unconstrained hasting algorithm random algorithm standard error identically random proposal importance initially table summarize cc cc parameter normal double normal mean table show acceptance computing time high exponential acceptance least min min rwm rwm mn cl cl rwm rwm mn cl cl normal prior rwm rwm cl cl propose copula scheme generalization design proposal hasting normal scheme reliably study much acceptance rate walk copula copula normal complicate multimodal arc grant dp thank computation intel ghz ram platform compute function inverse cumulative mixture normal file matlab speed normality reject fit density
interest approach tend g composite block exponent agreement reliability bound constant curve achievable rate block signal also compare rate poor curve demonstrate channel complementary function superposition axis give stay block close curve codebook whereas considerably small say square decoder unknown whether decode near capacity decoder convex discuss decoder superposition break achieve correspond residual specify residual difference contribution column attain exponentially exponent slightly optimal moreover least square achieve exponent superposition code level rate capacity least successive aspect power rate capacity need achieve allocation decode reliable decoding rate capacity express square make challenging convex constraint nevertheless constraint decode convex move decode unclear reliability power communication language dictionary linearly combine issue field number reliably zero partition complement recent recovery case channel highlight coefficient value control coefficient minimum conclusion reciprocal well allow converse constant capacity projection constant achieve square analysis least fix capacity capacity capacity input white band specify spectrum decomposition code suffice near need concern communication near channel decode empirically moderately decode rate mathematically prove case channel code base low check code aspect art reliable decode restriction alphabet distribution exponentially small fix contrast code decoder decoder moreover beyond uniform distribution investigate extent regime one alphabet require pack power implication marginally jointly empirical notice superposition idea superposition adapt quantization applicability quantization pack development benefit code pair shannon binary code target available challenge concern code exponentially large step toward practical rate inner drop capacity inner must order least consequence outer noise difficulty superposition comparable outer fraction achieve remain comparison false discovery significance development number arise incorrectly provide additional possible subset selection within section address discovery superposition gaussian user channel send sum put shannon purpose feasibility identification achievable another channel channel power arrange successive decoding relate successive decode superposition code apply individual however part reliability high rate design attractive single amenable channel access brief provide core superposition code reliability discuss composition outer code correction mistake matter henceforth logarithm suitable base calculus log conclusion state base use derivation moment generate function deviation exponent construct normal variance correlation coefficient otherwise understand minus infinity bivariate joint mean let increase maximize near optimize expression simplify composition evaluate ratio expression match occurs include say decode superposition least reliability superposition code allow term first subset correspond zero specify choice subset solution achieve superposition code number section incorrectly channel capacity shannon capacity statistic approximate interval lemma refine mistake send value achieve proceed bound probability appropriately design union give interest subset intersection difference mean give conditional normal govern send respect test term whereas depend express adjustment normalize equivalent density construct hypothesis give helpful rule provide ratio reverse event contribute term side outer among prescribed bind side bring outer iterate involve simplification jensen expectation inside yield recall independent accordance denominator denominator numerator entail choice normal correlation claim exponent mistake occur behavior similar order theory difficulty correspondingly combinatorial analysis denominator pay term interpretation logarithm likewise event interval test split event event event dependence average difference much decompose form standardized normal multiplying maximize consequently reduce quantify function find generate ts give rise appear expression complete proof dotted explain mistake consideration correspond proceed exponent exponent superposition code incorrect expression replace exponential partition superposition bit member bit equivalently note control polynomial size want size order inverse gap capacity instead decompose r coefficient combinatorial plus exponentially v ask exponent difference shape comparable c ce v two likewise v enjoy bound consequently find sufficiently nonnegative small choice end accordingly exclude nevertheless ratio end value insensitive maximum whole limit strict positivity zero within set interior determined ratio end right replace bound accordance ratio derivative respectively accordingly derivative development determine suffice determine whether ratio less certainly ratio case recall associate derivative second third eq evaluate evaluate first give right magnitude indeed take root claim v claim contrast small magnitude produce claim form expression agree argument extend continuous moderately equal near thin error exponentially bottom minimal bind specify high fraction minimal moderately observe reduction require extra via exponentially characteristic gr gr lr ar bound derivative rate equal quantity critical interval sensible establish exponentially reveal exponent stay gap exponent positivity produce match regime prefer conclusion least mistake recall string coefficient previously vector zero freedom magnitude draw send superposition term receive string receiver section implication dictionary least mistake section rt rate capacity bound integer lemma follow constant fraction mistake probability exponentially exponent precede choice fraction rt make requirement c r ref consequently lie tangent derivative see otherwise latter include situation bind line since non decrease development tend tend consequently multiple inferior exponent follow bind control case half less fraction arrange accordingly quantity two exponent ii bind way first choose previous match addition say exponent exceed bound minimum likewise minimum exceed accordingly sufficiently small become least remark tail distribution geometric sum arrange yet mistake small introduction inequality proceed proceed lemma substantial mistake aside exponent proportional mistake small device considerably indeed mistake mistake suitable code thereby small probability compute random sequence individual well ensemble implication review code discuss role rs code outer code mistake rs code element message rs symbol q add find convenient symbol give code outer superposition block first equal view represent symbol outer composition bit symbol outer superposition receive square decoder think symbol rs mistake property code correct since code albeit rs code minimum distance symbol string identical code symbol composition obtain partition superposition positive partition superposition concatenation code obtain composite less equal implication implication section dictionary receive sequence send exchangeability column average write appropriate average conditional expectation likely behave similarly indeed markov p verification simulation repeat geometric implie small codebook leave line communication average maximal say armed empirically dictionary satisfie requirement proof facilitate provide make joint permit simple power condition accordance dictionary match expectation enjoy markov inequality except exponentially control power case formulate dictionary examine role decode power square terminology arise setting transmission wireless equal sign code first sign superposition code section nonzero sequence distribution index choice likewise power across section simplify average deviation chi accordingly chance exceed via chernoff dd normal approximation average outside deviation instance capacity high hold precisely carry mistake long average code likewise subset superposition without sign distribution input independent zero likewise overall need square expectation plus norm input random variance l subset independence draw chi chi equal lb yield average matter power among normal coordinate chi probability nr e nd rr near rely decode sign provide power sign interference produce section lead equal choice sign property conditional subset contribution section mean uncorrelated sign conditioning mean shall deviation equal presence sign lead variance concern square uniformly exponentially union column except conditional allow size small bad show capture typical conditional inner value normal moment equal moment difference independent half x e gr j nd union dictionary conditional equal give concentrate
ray coordinate regard space joint live f response separate overlap split signal especially different log covariance know extract signal inference optimal usually derive principle except rigorous gibbs technique joint space easily spectra reconstruction datum gaussian noise principle help uncertainty minimal approach e power basis disjoint support band band cover space band inverse assume independent inverse exponential cutoff determine jeffreys end calculation effective hamiltonian accord band internal gaussian average expand introduce two free energy ignore correction mean wiener filter spectral assume invertible filter provide unable unobserved space widely field estimator perform irrelevant energy respect template respect calculate integral energy systematic taylor fr expand around energy second alternatively average minimize ks calculate surrogate energy scheme might argue build could markov carlo right synthesis build direct mcmc analytical analytical combine taylor expansion gaussians mixture moment express leave verification synthesis future minimal principle theory maximal relatively straightforward suitably parametrize degree internal entropy hamiltonian surrogate pdf variance free principle tackle normal poisson spread early calculation understand surrogate gaussian correct previously well propose complicated combined verification future easily maximal allow construction reconstruction inference spatially help concept permit tackle perturbation discussion manuscript lin tackle technique within field minimal free understand gaussian full cross optimize three normal background count ray galaxy iii unknown construct free measurement signal encode information retrieve strategy construct minimize prior ground state would function literature entropy result reverse origin certainly sense often numerous possibility much entropy since lack irrespective force inference ideal provide combine posteriori principle energy regard functional pdf minimization entropy latter logarithm signal pdf usage concept freedom approximate gaussian connect theory degree former complicated sect entropy motivate principle derive sect maximal application principle optimize approximation concrete provide sect log sect reconstruct sect posterior obtain sect conclude sect possibility plausibility sure impossible uncertainty obviously theory generalize logic different possibility introduce contain aspect reality aspect retrieve possibility vs datum sd denominator fraction highlight statistical mechanic hamiltonian usual mechanic signal lead ad hoc temperature permit narrow phase space pdf sect field correctness sampling guess suitable energy might hamiltonian minimize hamiltonian signal principle denoting mean thus solution field augment information hessian uncertainty introduce often ie image I ie image equality field ie usually argue ad hoc assumption entropy package plane knowledge lack reveal prior method ideally likelihood ensure maximized constraint temperature parameter weight closely map want enter formalism prior identify assume irrelevant additive constant pixel etc generic either peak several prefer reconstruct commonly signal physical measure spread pdf boltzmann posterior signal introduce internal energy free fully dependent free provide field value entropy entropy functional restrict space obtain entropy state complete lack uniform signal possibility return alone expect analogy use quantity minimize free calculate pdf necessary go sure understand construct free energy suggest partition function temperature narrow posterior importance pdf delta peak locate entropy permit mean temperature low center pdf tail guarantee reconstruction slightly since reveal aspect pdf central differently accurate sect partition read far motivate temperature free calculate connect directly take moment function g restrict free energy explicitly energy internal energy entropy internal posterior average mean dispersion fs fs pdf modify posterior q md solving yield energy gibbs internal therefore gibbs variation eq energy respect optimal map hamiltonian gaussian location set evaluate gibbs hamiltonian q equation seem temperature denominator opposite hamiltonian gibbs free another measure leibl kullback divergence characterize role equivalence cross inference step since last leibler divergence kullback maximal cross however minimize use posterior relation regard covariance minimize principle also hold gaussian later sect relation differ calculate need specify taylor fr expand coordinate integrate energy point calculate analytically field index since result internal odd product ns symmetric internal entropy construct accord optimal internal solve derive depend order hamiltonian match free hamiltonian indeed coefficient case interact minimal use inference noise ray ray count galaxy proportional spatial position permit linearity description support theoretically start count likelihood separable internal calculate analytically put calculation response exactly ray exhibit spread single detector come indistinguishable direction galaxy distance generalize response treatment galaxy case see galaxy problem via map principle spread model
htbp structural extend eliminate column constraint margin corner entry require ultimately stem equation submatrix kl namely nj odd expression hybrid within gibbs know central metropolis eq carry sampling four assume take posterior figure htbp posterior report list credible table credible simple configuration configuration level posterior propagate cost set restriction sample gibbs sampler realization estimate cost credible cover great pattern previously would analyse system cost th estimate posterior aggregated proportion credible proportion evident due section principled narrow gap htbp see belief study relate scale estimation proportion uncertain due natural viewpoint explicitly proportion yield update proportion conditional nevertheless integrate nuisance derivation numerator resort method complexity main uncertainty specify behavior study region variability incorporate pattern instance available region obtain extra defining alternative traditionally refer albeit fully reflect hierarchical candidate multinomial dirichlet mass q informative attain distinction preliminary approach commonly estimate remain count furthermore bayes posterior offer principled incorporate seed distribution count follow multinomial flat adopt distribution adopt gibbs new level iteratively previous section become list sample conditional know walk central step normalize z ij kp kt simplify realization chain acceptance update initial configuration ij ij candidate otherwise reject set random initially take low solely credible arise use observe square htbp informative wider credible length bar square posterior dot see proportion parameter make variability wide credible htbp suppose preliminary keep flat posterior count affect cost summarize show credible marginal compare proportion solution probability solution c conditional display randomness attribute inferential posterior bar square posterior list influential two list c static estimation traditionally regard optimization contingency formal pattern make classical function artificial configuration incorporate principle entropy maximize principle able classical map history behind traditional yet benefit solution solution number alternative preliminary purpose insight region however dramatically traditionally nonetheless fix acknowledge uncertainty make random hyper build contrast informative seed configuration accurately carry besides also able hypothesis explore flexibility bayesian really explore configuration pay need closely generate need assess propose try comprise future direction include efficient scheme improve proposal fast implementation version serve traffic step jointly assignment modeling propagate step aspect refined camera variation dynamic acknowledgement motivation nsf grant static viewpoint novel cast study factor maximum identify solution usually next propose devise obtain several approach highlight source incorporate consider region divide include usually restriction row margin eq constrain immediate static provide broad treatment study many decade contribution generalize decrease cost dc ij incorporate observed regard possible previous study balance factor know define iteratively balance convergence heuristic formulation maximization micro state associate configuration equivalently entropy maximize q would coincide make regard form mathematical program certain constraint second many configuration implicit formulation propose show even instead optimization classical model solution maximum estimate besides consequence generally quantify propagate framework regard usual fully approach consistency satisfy randomness come initially belief observe margin arise next incorporate small study multinomial proportion nonnegative hyper improper informative micro important role area behavioral perspective correspond random multinomial logit model set covariate pair cost incorporate another inference update observe prior self maximum note balance entropy formulation proportion recover define entropy maximize principle maximize uniquely measure amount entropy justify partial consistent familiar subtle difference formulation constraint effectively configuration proportion feasible proportion guide soft argue formulation knowledge seem posterior
operator mm condition asymptotically semidefinite via recover follow one hold psd find psd unique minimizer conversely none psd psd similar threshold psd set hermitian analysis find empty suppose otherwise maximize increasingly tight simulation curve via optimization theory fit almost perfectly suggests quickly limit minimum study need recovery constant suggest trace actually uniqueness suggest psd sure positive interesting need mesh way important recovery threshold weak weak nuclear minimization recently gain problem simulation far sensing threshold suggest grow linearly size need three recovery weak threshold analysis special semidefinite matrix discuss simulation address measurement project matrix dimensional gain attention practical measurement recover fact low program turn replace nuclear heuristic close relaxation nuclear refer program sdp study recover isometry success rip result minimal sampling recovery nan gaussian success threshold establish explicit oppose wise new recovered minimum novel space minimization find threshold basically give separate vector compress positive semidefinite analyze xu basically strength match exact compressed sensing simulation indicate threshold tight shall nan space compress sense call column I e unitary unitary basis matrix decomposition unitary positive increasingly denote call nuclear frobenius act iff ensemble variance circle histogram square converge nothing normalize value similarly normalize note limit support word mi mn degree freedom normalize order set particular hermitian several lemma make later proof large ix particular find obvious norm norm mesh subset unit sphere uniformly haar function introduce analyze strong modification rectangular probability follow fact iw iw rx unique nan regime determine compressed space haar view establish matrix sure nan intersection careful actually equality get optimization clearly hand sort increasingly directly addition one choose nc z c else reasonable lead reason worst basically contribution region expectation separately follow eq gaussian vector singular z fx p value constant iid ph h ns na use exact schwarz large combine upper thereby measurement substitute substitute use otherwise sufficient operator numerical strong plot support recover measurement iff q note suitable write hold minimizer tight find start gaussian analyze nan subspace haar establish necessary intersection bind discuss unitary uniformly I unitary transformation unitary show successful depend assume sec w increasingly similarly let denote hand w w need problem lemma increasingly result let program q combine mesh technique sec sec h circle iid analyze let sec nr f q combine h show sec use combine asymptotically upper use otherwise gaussian n q numerical calculation threshold section weak threshold recover weak simulation strong threshold recovery sense prevent repetition derivation repetition threshold unitary tw matrix left analyze upper diagonal unitary transformation basis transformation assume block firstly sort absolute similarly notation note essentially basically correspond repeat base arbitrarily iid need use let therefore exponentially combine prior yield threshold least mesh freedom nuclear simulation measurement region region include notation symmetric denote semidefinite psd stand semidefinite hermitian denote unitary hermitian entry diagonal variance order create one let unitary order singular define triple limit cumulative definition x ix follow counterpart semidefinite root since psd give program psd matrix matrix x psd matrix want positive semidefinite solution analyze call psd nn via unique unitary hermitian psd eigenvalue submatrix submatrix immediately psd transform long minimizer psd recovery without tw however last small perturbation change hence last psd matrix psd whenever tw tw v tw tw analyze separately nan argue nan restrict iid also imply unitary identical
perhaps I finish stanford would ask know old von day history von generation monte von page publish stanford von expand brief exponential distribution yield intend report joint u discover stanford stanford report mathematics publish issue last publish exception lee publish publish follow von day still publish von illustrate normal von differential von obvious sample evaluation number generator normal von generate enough algorithm von minor interval convenience correct convenient generate sign historical take trial trick correctness experiment well store power reduce reduce call uniform generator von uniform normal von expense describe interval example eq principle necessarily development people give sampling book improve box fair say none well tradeoff depend machine generator pool maintain transformation number function spherical symmetry normally suffer paper corollary power series von elegant compute polynomial refinement
monotonically exist distinguish go section primary weak synchronization synchronization exponentially fast machine ref essentially say observer observe exponentially variable belief states generate state combine synchronization theorem strategy exponential theorem subsequence argument markov chain finite irreducible chain equilibrium fr class irreducible chain state irreducible chain extend periodic block initial linearity lemma strictly joint eq machine constant claim claim edge exist b exist constant large taylor l exist hence q consequence thm exponential exact existence everywhere synchronization establish machine pointwise sense def exist constant thm constant prop lemma machine rhs eqs give word define observer prediction fast observer borel thm asymptotic synchronization treatment involve result exact ref observer uncertainty vanish observer prediction exponentially class countable machine hmms acknowledgment support advanced research project physical intelligence via finding interpret view imply department synchronization synchronization observer source vanish observer average predict ref synchronization machine observer internal state number measurement observer analysis differ qualitatively sense observer machine constant entropy follow provide essential definition result picture synchronization use state section average section synchronization use theorem machine entropy vanish exponentially approximation exponentially entropy sec background result thorough reader ref presentation mm xx jx state depict direct state edge symbol require strongly generate start choose stationary machine pick symbol edge consequently fashion machine visit symbol chain normally observer even extensively derive even consecutive transition even transition probability generate mm markov symbol l machine machine originally probability history equivalent finite immediately apparent synchronization establish ref extension machine edge follow indicator state iw markov whose probability visit move generate symbol strongly well edge relatively output transition distinct relatively fig bottom h length pair distinguishing must finite distinguish observer internal state observer internal study procedure observer know observer able machine time state observer observer define observer generate word observer know infinite set weakly sequence observer observer uncertainty sequence exact machine almost machine necessarily exact observer question thus exact always finite exact word finite quantity observer average uncertainty length output observer randomness turn determine average observer symbol observe stationary monotonically observer asymptotically observer optimal observer symbol induce observer know current machine close machine relate average rate primary study synchronization intuitive picture formula basic observer machine five symbol merge symbol symbol state path observer w p kp p observer exactly merge impossible path observer asymptotically relative path however synchronization concern absolute normalize synchronization quantity probability machine probability path eventually merge state asymptotic synchronization average time likely path synchronization formula analogous previously
overcomplete frame theoretically experimentally overcomplete dictionary ensure optimal orthonormal basis endow property frame propose generate dictionary operator optimization adopt keep terminology duality devote dictionary although overcomplete dictionary pca derivative reconstruct seminal field learn overcomplete probabilistic lead cost perform alternate difference advance compress lead pair decode transformation efficient natural dense introduce encoding module build define penalize augment code product input term time code new vector require minimization rely proximal year empirical evidence proximal underlie algorithm considerable amount devoted topic reference several use sparsity particular introduce tree structure attempt cast experimentally recover dictionary code dual classification intensive hand implementation training goal encoding operator optimality filter learn dual induce sparsity respect reconstruction since know gauss sparse dual dictionary successful towards respect force rely consist convex constraint presence make minimization proceed contribution differentiable part forward backward split convex solution optimization literature dictionary extraction smooth proximity projection descent inner coefficient adaptive discuss section minimizer minimize gradient proximity correspond plug obtain quadratic equivalent denote constraint formalize indicator set resp belong proximity update choice achieve minimize evaluate explicitly choice towards rate similar eigenvalue convergence fista modify fista fista sum two iterate define step replace modification allow quadratic convergence achievable theoretically experiment confirm replace sec break complete outline optimize carry adapt equation popular warm atom reconstruction atom mean element replace achieve iterative stop upon reach iteration find hundred reach case require optimize code confident accelerate easily experiment context discriminative image order dictionary vector obtain learn aim dual control cc converge dot decade synthetic bottom recover necessarily atom first corrupted gaussian algorithm coefficient stability principal converge transpose pca assess angle span decrease image reconstruction superposition element corrupt frame minimum frame experiment frame recover dictionary dictionary apply set image benchmark berkeley quantitative assessment berkeley experiment patch berkeley segmentation intensity center relative tolerance stop set level sparsity low dictionary interesting report code coefficient visual pattern look like tend poorly specific atom seem encode dataset sample atom column least dictionary pair mnist mnist binary handwritten train dictionary atom likely overcomplete pre process literature comprise representative digit figure bottom digit atom extremely middle first report dictionary respect relevance aspect empirical dictionary digits change reach substantial iteration correspond identical iteration iteration group
since orthogonal lead note u theorem establish nuclear norm know use solver become prohibitive hundred variable optimization propose outli pursuit special rate interior paradigm outlier diagonal otherwise column zero quite minimize promising property outli pursuit randomly fix generate entry generate copy random outli htb ccc outlier b identical outli noisy gray denote outlier pursuit succeed adversarial succeed fail pursuit outli sample noiseless matrix result separate subgradient variational result note rx lemma last column lead orthogonal pursuit succeed pair correct column must output pursuit assumption strict analysis solution outli subgradient evaluate part condition strict unique w fu eq strict next imply strict objective proof equation complete exist show two assumption equality hold satisfie satisfie hold strict hand satisfie together establish equal otherwise u orthonormal hence follow remark component successful computable nevertheless well outlier recent consider arbitrarily corrupt pca collaborative bioinformatic agent corrupt contaminated yield completely corrupt outli pursuit mild assumption g exact identify corrupted identification corrupt application beyond nuclear line correct seek rely optimality fail present treatment give zero column aside broad restriction arbitrary know non recover identity zero column exactly efficiently motivated component analysis arguably reduction seek point decomposition find approximate form form standard pca arbitrarily quality persistent corruption failure source mean hence column ask exactly exactly identity establish natural convex column identity outlier non outli note paper alternative entire apply rotation change performance noisy additionally work long find project recover exist identify outli identification outside pca application finance exist robust pca suffer degradation dimension outli iterative inverse regime non point non even combinatorial intractable scale problem size particular seminal recent overall paper spirit difference thing fail handle corrupt technique believe broadly interest significant extension precisely principal investigate exact intend ahead need success use oracle seek convex recover correct noise identity outlier analysis broadly corruption corruption hope capture proof work thus result technique contribution outli consider ambient like b identity outlier column corresponding outlier identically singular orthonormal basis arbitrary recover clearly always impose section interested subspace incoherence column recover matrix corrupt always meaningful impose definition column say axis perfectly incoherent high incoherence column support natural side big essentially recover corrupt meaningful subspace column lie incoherence require incoherence svd incoherence condition require extra recover capital letter represent vector letter etc column column onto column denote row column finally projection span v complementary usual matrix norm nuclear depend context unit identity svd outlier recover matrix goal attain corruption constant fraction point corrupted show mild exactly low lie identity natural exactly via pose significant challenge give pursuit generate row find optimum output noiseless extension adapt noisy outli pursuit surrogate combinatorial stand weak pursuit matrix identity column outli statement noiseless observe incoherence output pursuit recover column space identify index outlier lie corrupt pursuit provide note bind outlier success outli pursuit space convert replace corrupted noisy pursuit essentially tight follow universal constant structure impose state noiseless outli outlier arbitrary rank identifiable separate corrupt guarantee possible prove paper exact successful technique condition subgradient desire solution assumption optimum column column outli pursuit recover outlier intuitively nothing leave column dual pair column correct know follow standard main introduction proof oracle constraint enforce column property dual must satisfy optimality solution obtain condition technical detail sequel norm column v tw ab definition discuss outli pursuit possible construct goal pair pair impose precisely support recall projection onto column result truth optimum nothing outlier arises impose problem solution pair construct appropriate subgradient key optimality svd column thing small must cl eq far correct let strict satisfies convexity argument pursuit imply lemma therefore establish strict w equation complete thus oracle determine condition must satisfy seek paper svd let let u complete construct oracle satisfy consider corrupted indeed set straightforward satisfied order recover I outlier immediately hard require include general condition long orthogonality modify eq definite cone away hold follow establishe reveal correction satisfie require strictly strictly satisfie imply simultaneously five show specify
plus pt pseudo random generator describe g unlikely method processor paper computer exponential normal generating normally random involve use rejection often preferable section efficient vector method serial unlikely competitive processor generator available wish normal mean normally number follow old minor uniform generator exactly exactly never return exactly similar comment satisfactory tr use correct correctness logarithm distribute distribute root implementation discuss variation interval compute go reject else go return circle distribute replace independent normally distribute random number execute root expense implement rejection implementation box processor return less number detect random permit subroutine number library approximation use identity four multiplication computation cycle normally distribute describe generalise generator use length uniformly distribute cycle add cycle peak give main component actually actually time computation generation computation reduce computation way reduce fast method b difficulty function taylor behave near suitably make change degree away experiment easily obtain chebyshev appear requires really want allow computation avoid expense uniform arithmetic generate independent use avoid computation normal gain overhead cause method straightforward implementation result table cycle avoid evaluation increase cycle cycle thus avoid box would work fact similar function replace uniformly evaluate exactly method uniform step reject else section deviation triple advance many number handle level version user routine call overhead cycle p method fast box cost rejection generate million rejection random p simple ratio generate else normally uniform number correctness logarithm proof lie outside occur much way section step go else else go lie inside region almost area execute lie region expense scatter thus logarithm partly r logarithm dominate library routine random scatter although ratio method half many number produce expect fast serial fact fast von sample density generate number first else go hard
observe time trajectory randomly possibly depend depend study randomize alternative code subject collect first treatment treatment code treatment outcome summary code cumulative discrete collect month course eight month study protocol large state art low summary clinical exploratory analysis convenience important development value decision maker summary contain indicator exposure indicator treatment month stage decision h restriction broadly classify indirect direct indirect approximate dynamic programming nonparametric learning indirect search method cumulative outcome pre class maximization outcome indirect generalized series check goodness particularly opinion form rather directly contrast utilize misspecification estimator variance indirect recognize effort variance inference e confidence confidence inferential challenge construct learn attractive practitioner extension assign treatment patient stage present baseline th nonsmooth functional nonsmooth learning plug dynamic h nonlinear expansion open problem principle highlight crucial usual goal note linear make treatment vary focus discover distribution addition b asymptotically nonsmooth implication asymptotically mean signal compose variable effect near poorly fix bias confidence move section asymptotic bias asymptotic bias incorrect level coverage confidence great estimator shrink bad converge asymptotic throughout outcome require design matrix covariance plug let asymptotic second reduce predict soft estimator penalize illustrative bias eq nonnegative positive soft nonsmooth predict first appendix assume c approximate estimator plug fold reduction asymptotic learning result large value prefer indeed soft estimator space fix converge bias drive square infinity chen consider uniform situation look fact viewpoint see actually bias especially large viewpoint provide toy highlight asymptotic play important study nonsmooth asymptotic study nonsmooth manner across use generative close problematic nonsmooth asymptotic alternative converge measurable ny ny n q prove lead large toy bias randomize study denote code high value example problematic see zero variable equality approach reduce thresholding nonsmooth covariate indicator precede small asymptotic lead behavior use simulate perfectly monte carlo replication set plot treatment confidence width roughly problematic large asymptotic cause away zero stay interval estimate treatment difference middle display axis red section decrease plot rescale axis reflect range correspond plot rescale insight asymptotic notion closeness sample plot rescale clinical research essential measure construct confidence difficult impossible ii propose additional extension provide inferential occur probability similar confidence idea regular zero covariance method construct set taylor valid say union appeal simplicity especially fail less group conservative regular locally confidence combination coefficient interval standard method n construct convergent estimator limit nonsmooth subject stage treatment effect subject subject distinguish subject make insensitive bound construct c nonparametric percentile denote percentile become potentially tune demonstrate effective drive describe limit behavior limit distribution sequence infinity validity h np equal limit asymptotically mean probability one treatment almost nan subset upper stochastically experience hold choose equal choose bootstrap follow percentile percentile bootstrap thresholding st driven test wide interval cover nominal frequently cover time wide thresholding penalize approach soft thresholding experiment evaluation model stage follow x feature h h h correctly match avoid misspecification comparison generative influence produce occur occur control patient measure al near example strongly treatment moderate however equally optimal moderate near give extension three stage example regular b confidence interval treatment truth show coverage monte replication bootstrap nine st examples st setting stage st nominal coverage conservative average expect whose double among level dataset estimate mark nr two st estimate confidence treatment nominal level construct draw correspond coverage significantly nominal mark st nr st nominal construct dataset mark nr width interval intercept nominal construct two nominal mark nr r perhaps perspective however view intercept poor attain nominal nine example st nominal fall contrast nominal coverage example average cover adaptive behavioral child trial al subset never randomize treatment subject massive subject provide school year precede odd diagnosis diagnosis baseline diagnosis exposure indicator receive code st stage code correspond behavioral modification patient treatment basis two scale al list behavior month stage outcome initial take prescribe month month school year response randomization randomize month nd code initially correspond increase initial treatment teacher teacher rating score code clinical randomize month month first step month prior study vector interaction action main effect stage least percentile residual show sign misspecification short seem reasonably upper intercept baseline odd diagnosis exposure non st txt stage txt txt dependent assignment treatment available diagnosis subject exposure h contain term action main covariate stage form plot residual obvious sign misspecification provide reasonable estimate intercept baseline odd diagnosis exposure txt exposure estimate function reveal treatment first stage decision exposure behavioral modification subject prior exposure recommend regardless recommend optimal insufficient recommend treatment patient availability individual clinical recommend unique optimal confidence interval response case binary fix patient history would th insufficient nominal interact treatment categorical consequently history recommend treatment history strong leave solely conversely confidence interval recommend evidence clinical making conclusion insufficient insufficient insufficient evidence insufficient evidence discuss asymptotic bias shrinkage stage use form b bc mobile device collect patient thereby research match technical point grow function stationary markov mdp setup algorithmic largely characterize rate limit theory management across location treatment location affect outcome neighbor across spatial often resource location rather need sequence function date spatial treatment recommendation spatial learn suppose treatment option feature treatment interaction feature contain even possibility computationally moderate expect outcome code relaxation replace nonsmooth concave surrogate rule h h h h constant version algorithm h h n mild nonsmooth learn occur outcome standard density proof contribute second display q contribute bias variable expectation give part equal take schwarz triangle intercept first identity behave eq supremum must terminal cover assume reward observe stage seek maximize trajectory restriction line spectral denote member value center uniformly real equip np bootstrap empirical measure respectively characterize derive limit convenience dim eq estimate notation without pf pf f pf bootstrap theorem together ad must guarantee continuity finally continuity next step characterize limit lemma accomplish theorem claim follow weak b last old precede display weak law large number vector na tight covariance f pf pf pf n l vc envelope square integrable follow van close subset limit element f boundedness note f f f van uniformly immediately section theorem expansion coefficient useful expansion individually equip norm b b k alternative expression upper similarly analog replace fashion use proof technique lemma present imply conclusion assume verify h k hold loss h h l n identity hand surely let result argument k class see sequence almost almost sequence surely theorem proof convergent weak law bootstrap w establish strong measure spectral matrix first continuity continuity omit three possible model pa pa pa pa use h c indexing example analogous effect furthermore happen standardized treatment versus treatment treatment versus treatment equal example table interval diameter across nine two stage stage stable quite conservative allow grow
pca perhaps dimensionality reduction pca back early become technique extraction datum theory pattern recognition process construct square eigenvector pc matrix broad attribute primarily classical regime recover subspace existence nevertheless solve well precisely suffer non corruption sensor failure grow slowly quadratic consider principal existence map produce low way point corrupt non emphasize rather g distribution considerably dimensionality signal snr go norm scale dimensional generally recover without main surprisingly contrary nature regime fraction arbitrarily accomplish pca propose tractable provably asymptotically optimal corrupted scale tend algorithm kind moreover pca removal subspace removal probability solution lead good make argument rigorous organization reason classical robust setup pca devote guarantee experiment technical derivation performance capital denote permutation robust dimensional regime pca focus robustness corrupt make fraction outlier regime good addition corruption significantly robust algorithms class maximize variance obtain project serious covariance treat volume cover pose knowledge dimensionality seem crucial subsample spirit approach correct corrupt nothing principal pca maximization become extremely dimensionality local maxima grow fast effectively random exist output dependence dimensionality regime category outli bound inverse interest consider standard setup sample fact magnitude dominate counter intuitive hold principal magnitude principal mahalanobis distance corrupted fraction consider mahalanobis small align corrupted follow roughly nearly large extensively exist robust outli iterative various alternative use mahalanobis outlier depth algorithm propose dimensional neither mahalanobis bad pca sample algorithm mahalanobis namely minimum projection pursuit find fraction determinant pursuit maximize robust univariate combinatorial scale difficult yet volume cover low post algorithm project fine produce estimator pursuit non intractable author avoid examine direction direction nearly direction adapt low rank low entry arbitrarily close spirit motivation completion herein every element hence pca setup pca detail signal realization unknown absolutely respect borel constant easily relaxed expense significant notational outlier denote corrupt point contaminate equivalently top seek collection orthogonal achieve vector another angle express represent portion expect express fraction generate long tail become distinguish quantify effect tail weight margin contribute variance theorem scale paper focus scaling scale happen asymptotic together follow etc slowly zero stands perform h pca contaminate md aa infinity immediately sequence bind explain due removal corrupt may remove account outlier magnitude remove principal component go proving imply corrupted scale corrupt side continuous I contrast existence corrupt optimum enough side strictly maximal never outlier asymptotic gaussian cc gaussian uniform perform dimensionality space know form origin generality mapping pca involve pc note former apply pca pc pca since pca contaminate sample concentration must satisfied decay thank lemma exist least dimensional empirical truncated mean convergence since abuse sample heavy obtaining rely class aa f e vc inequalities one next en notation non variable relate step leave exist constant apply note exist q know ib h q prove second similarly let exist prove show either removal good corrupt precise set remain ss project mainly point denote next true removal corrupt point least trivially focus en true remove corrupted high let event q exponentially high would hence good martingale argument random q detail defer appendix proof due event occur increase factor exponentially sample theorems theorems main term small guarantee estimator intermediate lemma value give leave detail bound h theorem state theorem immediately asymptotic performance pca numerical pca algorithms multi variate project pp dimensional regime robust pca magnitude uniform magnitude report result ill condition report test pca pca htbp pp strongly corrupt recovery satisfactory sometimes bad performance pca increase essentially consider versus dimensionality performance sharp become htbp property make dimensional perform generate choose line trend observe although gap decrease since cc investigate perhaps principal
around ball completely contain ball result boundary valid region valid r n k probability least minimal maximal graph differ additionally mutual degree radius kx kx concentration union minimal n additionally need contain affect value thus continue direct vertex exactly maximum minimum ball ball radius maximal graph proof fix connect large become around near probability inside among inside among neighbor graph high connect additionally connect ball drive example remark conjecture max institute popular walk mild distance namely sense graph plant undirecte expect walk electrical define background read science proximity collaborative supervise image computer science various bioinformatic nice practical function compute close short path shortest connect satisfie highly cluster whereas consequently encode distance behave time simple formula accuracy denote distance vertex representation well expression vertex time ij j relate distance electrical represent edge weight vertex know coincide distance see focus deterministic geometric point vertex neighboring point type geometric two whenever euclidean nearest versa mutual graph connect vice base similarity weight application similarity parameter definition random underlie vertex vertex graph construct order able minimal space connect subset bottleneck regular sense constant b lebesgue essentially spike consider valid couple region arbitrarily narrow constant hx cube vanishing generating volume ball depend geometry constant explicitly comprise two technique restrictive apply kind treat well come price assumption region bottleneck unweighted fix distance depend n c n satisfie case px bottleneck unweighted fix eq statement ij ij bind bind denominator converge various main bound independent gap spectral gap suppose general exist probability surprising underlie enter bound term constant bottleneck intuitively diameter contain walk plug unweighted c di ij px unweighted build knn continuous ij analogous hold unweighted additionally field common approximation fully upper low uniformly result apply treat graph adapt bandwidth principle fully beyond remove bound truncate technique treat general adapt compact build similarity hx pn h c ij px px scale density limit rescale enyi graph vertex certain allow enyi partition degree random random put consider laplacian matrix ed exist enyi graph allow self enyi rescale times enyi uv application plant enyi plant plant split equal put edge vertex cluster simplicity loop plant partition plant arbitrarily slow ij result structure graph time grow slow distance converge result grow might degeneracy visible degree place vertex cf read distance carry everywhere easy prove inverse degree lower weight remove w st simplify monotonicity principle increase graph build adjacent set infinity vertex merge parallel parallel exploit law electrical edge add detailed example lead show computed flow corollary flow weight st evaluate formula tight widely geometric space construct let regular width neighbor least edge grid valid grid point neighboring cell u fit bottleneck give graph geometric bottleneck straight stay inside denote let grid minimal point apart grey cell bound st st n general underlie devoted overview straight edge flow edge bring flow see flow hypercube side center hypercube locate path figure hypercube flow neighborhood reverse flow discuss computation contribution step consider current step unit flow adjacent flow contribute overall neighbor neighbor carry flow hamming illustration compute exploit neighbor let flow neighboring size receive flow sum flow edge might step cube flow hypercube dimension line connect hypercube figure layer cube cell layer final geometric graph geometric bind well graph appendix state degree order simplicity lead final statement near note mutual proposition formula approximate th calculation eq times cm proposition first convenience go eigenvalue pseudo inverse satisfy matrix k bipartite compute thus p p n prove little j j time j u j formula eigenvector ingredient distance gap low geometric geometric general special graph unweighte undirected follow strategy spectral general graph connect vertex make bottleneck maximum average undirected unweighted connect respective e maximum path number spectral bound eq gd b selection see tight proposition sure graph uniform argument absence boundary effect set graph unit cube grid cube grid cell geometric cell contain exact later construction path assume b cb cb deterministic cell path simply hamming path cell g cb ca reach adjacent path cell neighbor interior cell path connect force add point path assume connect path path load cube geometric minimal number construct exist cell cell maximal load upper bound cell grid cell center corner pass c size center per center satisfied possibility part neighbor lie neighboring cell interior cell two cell cell edge cell cell pass one cell start vertex intermediate select point treat pass load cube hx path average graph assume small grid denote ball center point path case let ball clearly boundary clear neighboring cell x l definition part use appendix load cube mapping exist n load construction distance always ensure grid always l mass ball eq cell ball proposition deduce analogously p obtain eq c plug ready apply maximal cf proposition path per cube probability c quantity lead replace deterministic radius radius symmetric mutual graph load load mutual bound converge take minimal maximal length maximal collect corollary gap plug vertex similarly apply minimal weighted spectral stochastic q unweighted replace second eq discussion tight eigenvalue laplacian ij walk example proposition directly follow px px nr ij I right treat bound w weight edge ij treat define truncate gauss ij w gauss converge note exploit obtain expectation implicitly weight minimal also unweighted graph note coincide eigenvalue proposition corollary h
mutual information fitness forest span might may take forest tree liu base minus forest obtain forest paper liu random clearly liu capture section deal generalization consider present summarize state work say undirected respectively respectively connect exist particular undirected forest loop connect pair finite say direct information probability nk although express nx nx wish minimize approximate depend equivalent forest maximize along mutual liu else terminate component find relative yx ix x minimize dp minimizing replace ni np np jx structure could require liu variant easy realistic case px x qx qx n find maximize minimize quantity description notice inequality connect description length become ei ni constant rather maximize loop else terminate connect terminate candidate make loop notice training look simplicity fitness stop rest either connect order structure result forest complete n criterion xx probability section random set set obtain countable operation event field generate borel mapping say satisfy say continuous measure kullback available say absolutely hereafter ni n leibler define minimize ii generalizing dx x dx proof express ji jj ii liu sequence I rx specify add terminate becomes suppose respectively gauss unknown eq connect need number may j j ni nx x ni ni j ni liu algorithm general avoid consider avoid
misclassification correctly blockmodel choose equally meet parameter class class identifiability thus meet misclassifie decay illustrate network datum employ publicly network facebook california technology ac indicate student friend yield gender community output partition covariate identify conclude student algorithmic grouping obtain formation reflect show order student covariate account facebook fit residual beyond choice blockmodel include logit blockmodel incorporation covariate parameterize odd edge occurrence vector covariate categorical indicate plus covariate indicate range analogous blockmodel blockmodel intractable employ approach hold constant holding test initialization consistently fit fitting estimating subset top adjacency estimate leibler bound row student logit blockmodel ranging describe paragraph bayesian sample five fold respectively plot suggest low divergence theorem example leibler divergence ab ab square error magnitude estimate reveal exhibit correspondingly evaluation adjacency assignment show row along correspond covariate student divide naturally interact frequently subject quantify increase concentrated connection evenly distribute membership group fit employ student group constant bottom identify category visible grouping right hand student support national institute health office office additional proof bernoulli chernoff respectively ab ab kullback leibler fix diagonal entry distinct see quantity ab ab ab ab q obtain union claim ab ab ab ex term ij x ex ij jx ij apply independent observe event ab ab side yield pz claim uniformly imply blockmodel condition I realization imply blockmodel formally subset upper triangular blockmodel membership generally p ij coincide blockmodel assignment step refinement induce blockmodel assignment blockmodel membership induce assignment respective assignment let admissible blockmodel assignment realization blockmodel refinement observe definition several membership assignment continue nonempty remain terminate index repeat denote number pair form follow hold index whenever distinct triple manner ready follow assignment refinement node one nonempty half iii condition ii theorem thus eventually triple cardinality cardinality nonnegative divergence refinement indexing let kullback leibler divergence equality refinement stochastic network show misclassifie allow grow size finite blockmodel estimate comprise bernoulli hold assignment fitting logit blockmodel network comprise facebook profile residual social blockmodel biological sometimes propagate yet understand appropriate remain statistical blockmodel salient partition distinct interact network concept connection adapt give rise bernoulli blockmodel misclassifie zero even class advantageous expect size relatively graph misclassifie stochastic blockmodel restrictive requirement closely suggest alternative exchangeability generative blockmodel analogue exchangeable sequence infinite exchangeable whose observed conceptual population fit blockmodel describe blockmodel adjacency edge blockmodel probability far symmetric membership vector blockmodel assignment number maximize asymptotically behave grows fast fit fast apply bernoulli assume amongst success assume class blockmodel fitting blockmodel hold blockmodel blockmodel incorrect assignment furthermore identifiability hold blockmodel class pair conclusion suitable misclassifie yield convergence grow class bound constant condition two eventually single
family include logistic regression p regularization term lead order ingredient establish line al covariate mean tail glm type rsc long scale sparsity allow bound glm address decomposable regularizer decomposable regularizer regularizers impose inactive simultaneously predict vector reader paper structure collection recall section vector simplicity use denote natural consider estimator user define estimator cover commonly often refer ordinary regression provide restrict sparse extension sparse set regularizer discuss block block positive constant consider sparse maxima say mp group choice ask matrix answer design matrix constant depend condition hold great supplement rsc sparse sparse rsc condition rip g close apart rsc impose let matrix associate generalization group assume without rescale noise constant group novel satisfy normalization solve q apply choice ambient refer exactly group derive et zhang arise particular general weak block analog ball optimize eq result generalization corollary reduce corollary consist term ball provide rsc derive class regularize explicit high sample size structural parameter play role isolate notion mean difference nominal parameter fact notion convexity high scale interaction decomposable bind explicit convergence estimator include derive model establish novel approximation describe bound completion framework rate combination nuclear apply leverage convexity state variety work simplicity exposition optimal instance fraction nonzero regularization parameter similarly last example type hierarchical regularizer regularizer combination decomposable regularizer fuse author partially nsf dms dms yu nsf n acknowledge nsf grant support nsf grant thank van de zhang regularizer lemma usa number comparable size consistent unless recent work sparse general regularize optimization combine datum encourage structure unify establish consistency regularize dimensional scaling consistency also identify refer strong convexity many case statistic concern ambient dimension large root old back theory g decade research rapid major force allow measure large throughout science project large survey www sample financial dimensional hundred thousand financial trading advance allow measurement thousand protein lead statistical reference image among image imaging lead estimator constraint impose within impose type study example include matrix component matrix decomposition many base form weight application pursuit square solve quadratic apply program constraint type include regularization regularizer value within statistic obtain high control ambient sparsity degree matrix typically bounds decrease metric place type perhaps consideration recovery noiseless g g parameter selection theoretic method model include consistency sparsity also among random field inverse frobenius selection application among group base noiseless arise nuclear finally although primary emphasis nonparametric almost regularize assumption change unified insight answer highlight key function regularizers estimator specialized different corollary author also optimal noisy noisy en establish corollary convexity group organize define notion main discussion consequence devote corollary model group regularizer supplementary file denote distribute cost quantity user setup derive convex unknown analogous frobenius apply certain type provide measure regularizer classical ambient contrast analysis vector contrast statement obtain finite ingredient regularizer capture might vector particular complement space definition large treat nuclear follow give norm regularizer ideal away subspace deviation always triangle tight deviation away subspace property regularizer decomposable respect complement main pair subspace formalize intuition let projection shorthand interest action desirable small theorem fast concrete begin regularization norm usual class cardinality reflect support complement inner give correspond perturbation subspace capture norm decomposable pair subspace form partition put show claim worth strictly sparsity structure norm group likely simultaneously model partition g n correspond regularization past interestingly lead superior performance subspace define usual precede exploit modification order overlap regularizer correspond identify inner product include collaborative expensive enforce manner study precisely nuclear decomposable rank subspace give notation finally nuclear decomposable problem sum low regularizer form weight nuclear norm regularizer work illustrative consequence specifically implication pm procedure often sample pi need curvature certain definition restrict convexity ultimately interest guarantee vector function formalize rsc particular statistical rsc term arise function high series low constant function ambient instance case covariate control tail follow type provide square norm choice group formalize quantity relate error regularizer compatibility respect reflect regularizer restrict dimensional constant establishing restrict convexity concrete membership definition therefore consequently restrict strong convexity hold paper ready although may abstract result concrete consequence specific immediate corollary design result rate matrix report elsewhere establish model rank sparse nonparametric structure curvature tolerance definition subspace compatibility paper strictly constant consider statement program although convex need strictly convex optimum attain state optima regularizer rsc condition challenge since unknown right note actually subspace decomposable ignore term correspond respectively decrease since estimation usual choosing choice sequel component degree work unknown recall subspace contain strong rsc tolerance reason corollary stated theorem belong rsc hold program satisfie rsc curvature grow compatibility compatibility increase third scale must strictly except compatibility dependence arise regularizer subspace obtain concrete rate use require condition three quantity illustrate follow model py nx pi link via vector focus setup describe give certain sense rectangular consequently identifiable impose section generally might weakly sparse devoted discussion solve quadratic choice loss order expansion case precisely establish appropriately previously discuss decomposable subspace natural equal figure strong convexity satisfy eq pursuit could relate condition less van discussion compatibility bernoulli special imply restrict isometry turn substantial dependency al design form refer strictly depend great
therefore family subset mass set result mean correlation obviously combinatorial greedy poisson combination family necessarily parsimonious determine fitting fast greedy inclusion feasible reproduce certain correlation family limit certain purpose adaptive useful discuss potential copula well bivariate discrete early binary vector copula frank g binary generating copula applicable interesting exercise without much practical application false true true corollary correlated context adaptive generate close target sampling proxy easily quickly exhaustive enumeration parametric carlo applications construction reproduce concept fail suitable approach impractical thorough scalar type bold main matrix determinant denote write closure indicator write ii notation triangular binary produce necessarily weight moment first frame parametric family suitable adaptive monte reason give q probability weight respectively analogously normal binary reproduce throughout rest generic since iff full write call notation moment da immediate dirac delta denote monotonicity respect correspond hoeffding apply factorization permit component conditioning already r I trivial certainly marginal logit family check requirement list parsimonious easily reproduce observe simple impractical monte product target distribution rest deal vector generalize permit vector multivariate auxiliary copula underlie build family work explicit necessary generalized review mapping obtain interpretation representation write eq convenience provide discover allow high try parsimonious interaction term however negative normalizing solution np unconstraine little practical virtue explicit moment explicit mean basic rather give formula yield include normalize write dominate definite value explicit recursive marginal expression sample fit note moment sample symmetric matrix fit fast requirement parsimonious via moment usually rule via factorization family matrix ever definite use carlo proposition derive probability assume contingency refer log contingency well study modeling approximation previous section derive marginal fit univariate probability target function probabilitie logit triangular logistic conditional eq special conditional exponential note possible family component practice parametrization seem impact carlo multiplicative close procedure construct sparse work parsimonious l I variable simple accord unit logistic draw conditional predictor firstly interaction secondly cause even conditional remain component parsimonious identify association intercept conditional family log weight family solve condition reweighte newton w I converge monotone thus maximizer complete separation way algorithm ignore lack proceeding problem jeffreys variance grow beyond threshold logistic family combine approach elaborate cause extra decomposable parsimonious likelihood computationally intensive chain factorization exactly although marginal section turn family copula copula family draw auxiliary distribution dependence rather gaussian natural option gaussian repeatedly cumulative speed parsimonious already sparse identify unit interval firstly correlation really matter adjust bivariate correlation extreme chance definite parsimonious copula identify component accelerate sample easily set task standard denote bivariate newton evaluate
sign negative objective loop correlation zero coefficient sign one eq analyse performance compare six representative dimensionality algorithm principal component discriminant discriminative locality zhang locality preserve database et pca reduction project linearization laplacian linear embed combine pca produce face discriminative dimensionality reduction face variation gender pose appearance consist vary size gender pose illumination expression age randomly select image contain individual gender expression configuration database normalize gray equivalent recognition firstly separate projection testing project matrix finally neighbor test subspace supervise retain number project sample subspace remain image remain test five recognition calculate dimensionality setting fig seven unsupervised pca design less retain neighborhood lda ignore margin level elastic fig outperform consistently keep rate select robustness computational cost proportional select produce computational outperform smooth coefficient path mark circle active mark feature feature head contour feature corner face sequentially feature feature select significant small one powerful obtain matrix term net incorporate dimensionality obtain impose elastic net patch transform penalize square thus angle patch alignment local patch coordinate secondly minimize directly weight group elastic net add rewrite lasso lar solve lar loop sequentially element active direction keep active lar conduct basis advantage aspect margin discriminative improves interpret result perform well algorithm component discriminant locality locality projection supervised preserving still error different retain compressed tool concern valuable norm far improve accurate penalty penalty could lasso smoothly scad fan li reweighte elastic zhang advantage adopt variable still long support grant project university wu science usa sparse reduction penalize directly lar popular learn therefore indirect manifold elastic net transformation lar adopt sample well datum representation margin maximization projection elastic reduce fitting projection various dataset reduction primary focus representation al li et li al al space subspace meanwhile particular original preserve remove decade reduction fisher discriminant regularize gaussians project label find scatter scatter draw gaussian harmonic assume draw gaussians merge basically manifold dimensionality locally et al laplacian le li al hessian generative et tangent alignment zhang linear measurement seek dimensional global geodesic distance pair measurement preserve proximity undirecte weighted pairwise exploit tangent tangent information align provide coordinate obtain eigen analysis build estimate hessian locality embed al neighbourhood preserve systematic zhang al zhang difference reveal ii embed framework g locality alignment liu manifold et wang smoothly scad fan screening fan selector selector dimensionality basis variable reduction develop achieve use coefficient problematic could linear combination reasonable feature selection coefficient property lead response lasso randomly therefore elastic net achieve grouping become dimensional subsequent regression especially important factor original decrease variance fit increment bias generalize interpretation relationship variable important practical minimize subject sequentially utilize acceptable lar propose subsequent principal nonnegative sparse discriminant projection al et definite programming use programming label discriminative class utilize particular manifold reduction g locality sparse absolutely sparse impose penalty loss lar objective direct therefore currently indirect manifold net obtain subsequent impose penalty combination criterion base dimensionality equivalent objective lar apply patch encode alignment alignment unify coordinate patch low directly minimization put new elastic adopt construct lar write correlation lar add set change distance control lar basis basis basis thorough face face dataset dimensionality manifold elastic lar effectiveness face conclude discriminative let objective dx different error minimize base aim low dimensional representation euclidean preserve actually popular criterion maximization combine interpret advantage improve computation reduce model interpret algorithm intrinsic reduction elastic framework obtain manifold discriminative sparse discriminative directly use least angle lar reduction find manifold discriminative reduction via alignment encodes geometry align utilize minimization criterion incorporate geometry reconstruct sample patch framework geometry patch k pf ix df I maximize local geometry k relate group patch representation expect sample possible relate sample class coefficient dimensional consistent alignment coordinate select global r ns rewrite sum alignment alignment worth space implicit adopted minimize therefore function q error model enhance represent error adopt classification however challenging apply classification formulation discriminant multi mild indicator l lda reduce fine draw bayes dimension sample usually flexible l indicator flexible representation near neighbor nn nn reduction point dimensional meanwhile consequence center belong center jx jj pi therefore dimensional representation indicator trade parameter expect projection matrix projection characterize fan projection lasso relaxation penalty although lasso regularize angle lar lar towards zero accuracy lasso follow penalize select care select impose projection elastic overcome retain favorable detail norm increase data response combination grouping effect property projection grouping demonstrate lar efficient lasso penalize penalize penalize least lar lar design particular I simplify instead eliminate objective respect e eliminate objective lar obtain optimal net rewrite q constant accord lar algorithm apply solve penalize linear lar lar entry coefficient increase another direction lar variable nonzero entry loop optimization size lar determine lasso eq simply ignore generality large sequentially coefficient important determine inactive different inactive coefficient sparse conduct follow inactive add current direction coefficient correlation equally preferred direction vector correlation variable loop direction assign sign simplex project correlation active decrease prefer origin space et distance normalize correlation identical correlation variable complement lar mathematically accord I new add direction coefficient accord loop elastic double shrinkage correct lar time huge dimension loop van inverse gram particularly loop accord rule calculate accelerate computation lar structure calculate cost well discriminative six subsection h training tw dimension
divergence gaussian map solve arise year independently show optimal control bellman become admit see relation leave case consideration us briefly recall formulation policy conclude trajectory specifically model dynamic implicitly q make discrete fully solution drawback importantly state latter cf claim show dimensional let q make corresponding stochastic optimal particular solution trajectory hold small though question principle note eq however case divergence zero problem write see condition suggestion back matching maximization consider close relation energy minimize free energy matter correspond partial furthermore choice generalize although interpret policy guarantee log likelihood despite control task equality likelihood jensen inequality observe necessarily tight coincide fundamental finding explicitly present apply give optimal provide interpretation stochastic optimal solution base propose control kl theoretical intended present propose optimal control divergence kullback leibler divergence distribution introduce discuss duality result demonstrate relaxation arise iteration infinite horizon finally relation previous field term multiple slice distinguish control wish infer aim inference horizon take form relate classical choose restriction previously posterior use note case additional strong present subset etc variable extra therefore could avoid introduce formalism helpful develop may state relate discuss arbitrary uniform stochastic one require merely little practical consequence control remain formulation corollary bellman minimize restrict delta argue present policy certainly answer specific suggest update subsequently extend horizon discount derivation horizon recursive pc give boltzmann similarly question control tx preliminary aim near assumption asynchronous converge pointwise constant informally aim finite horizon sufficient policy time easy analog yield therefore analogy old schedule asynchronous schedule give step policy fall immediately guarantee furthermore schedule
measurement applicable many popularity reference therein stage initially subset stage subject idea extend stage approach examine broadly experimental design computer machine type impact microarray contribution gain attain behind ds portion eliminate fraction appear promise consideration component retain previous propose however good quantification initial work manuscript extend previous strong entirely characterization follow review fundamental limit localization localization detection dramatically give measurement auxiliary review detection threshold summary support proof facilitate next baseline noise budget localization false support false discovery proportion component elaborate estimator estimator threshold implicitly establishe limit localization use measurement procedure recent metric utilize noisy sharp asymptotic procedure fundamental limit detection amplitude amplitude zero detection observation hypothesis alternative signal non zero recall exist sum tend conversely test sum false alarm tend possible consider describe process retain zero eliminate step precision allocate precede measurement make budget budget step measure component therefore factor ds localization word amplitude depend level ds capable signal purpose investigation assume zero component refinement rp x w j j amplitude measurement sense kp precision ds algorithm ds I follow ds empty successfully identify slowly reliable require amplitude exceed novel improve initial arbitrary amplitude adaptive gain section main characterize sense ds lemma quantify finite ds I gaussian tail tail mp component begin quantify j z lemma proceed refinement z z assumption refinement ensure event occur union bound obtain z jj require examine situation allocate allocate prove throughout fp fp gp k explicitly ease prove ii detect absence identify location zero follow slight analysis develop analyze index retain ds lemma characterize k ds consist p abuse ds k tail together immediately ds follow apply construct event occur tend event verify quite useful recall consider recall large enough j clearly negligible sufficiently show k know one output ds tend ds z event conclude k bind zero immediately ds conclude recall simultaneously tend conditionally definition converge conclude present ds demonstrate predict quite implement ds procedure simulation theorem lemma allocate precision allocate step provide snr result detection square root first last step equal first beneficial crucial step control r choose adaptive ratio location ds operate output trial generate square ds measurement highly successful happen ds ds successful finally ds average detect demonstrate extension provide ds ds gap arise snr much less arbitrarily course pattern observe upper figure visually arise rational discover sense indicate different accurate occur ds outperform sense examine non sense fdr different signal solid dot number signal allocation case snr sampling threshold result snr exhibit ds threshold fdr ds adaptive snr less dependence interest problem develop theory practice design improvement achieve analyze sense ds ds detecting weak localization ratio snr roughly low adaptive problem interested practical employ future investigation characterize sense devise experimental notable improvement claim sense ds even follow small time indeed choice ensure proof ds alternate measurement combination direct adaptive compressed literature sequentially combination regression show theorem proceed consider separately phase scope begin minimax false discovery coordinate formal optimality hypothesis case retain signal distribute standard fashion ready zero p p q p clearly converge proportion reasoning take p converge show thresholding discovery proportion easier upper necessary albeit insufficient easy control discovery spirit bind false p trivially mean
contamination extreme instrumental density poor rejection sampling use rejection procedure make target distribution optimal yield plausible need acceptance adequate acceptance greater extreme return iteration sampler solve call monte compute trace penalize carlo expensive problem proceed approximate density however approximate form individual multivariate member multivariate distribution procedure choose leibl writing approximation fully q quantity conclusion value pt deviation give deviation mean variance consideration simulation step alternative simulate dispersion triangular element probability define dominant thus partial factor fix minimum eigenvalue work run parameter roc curve run entire computation draw distribution simulate extreme break gene microarray contaminate extreme outlier simulate contaminate equal diagonal contaminate finally compare develop simple many covariance usually robust marginal procedure definite minimum matrix refer algorithm run norm warm start initial iteration lead guarantee unimodal place observe drastically start place performance relative lose simulation good job recover true perform surprisingly moderate discovery perform datum freedom provide rate interest extreme therefore estimate graph dot exclude solid observation extract part observation increase infer level pathway expression measure normality deviation stand exploratory plot gene inverse column include potential outlier obtain find test pair gene draw conditional approach biological interpretation choose run edge low conditional available estimate one remain estimate coordinate manner variational step also leave lead conclusion give take find run procedure use keep believe procedure compare biological find group despite correlation three pathway figure identify achieve plausible pathway original figure perform far relationship behave identify algorithm robust graphical model expensive great gain generally prefer alternative sophisticated monte approximation degree run suggest prior freedom line put increase quickly classical alternative model small line employ degree freedom conditional purpose lose degree freedom run last tune information validation often tend desirable goal rule compare throughout alternative remove penalty small decrease finally remark particularly rather misspecification involve degree maintain gaussian graphical model conditionally give conditionally uncorrelated distribution nsf dms support graphical useful explore structure multivariate interest result include penalization penalize provide use approximation carlo gaussian attract lot interest undirecte conditionally conditional infer nonzero element classical solution ba search optimize penalize likelihood concentration regression subsequent optimization arise maximize penalize algorithmic development deviation contamination drastically graph apply experiment screen become remove entire outlier discard variable fairly concern constraint case outlier contaminate transformation provide selection highly datum build upon use em cope contaminate maximization em penalize likelihood multivariate done analyze expression compete finding independent normal likelihood perform irrelevant lead function cone absolute entry large validation tune operate partial maximization row hold maximization descent briefly review show hold update cycle row diagonal freedom expectation notational convenience vector distribute chapter sampling distribution lead observation arise namely constraint impose correspond factor variance even illustration inequality pt take expectation gamma recall despite still prove pair correspond follow conditionally indicate latent interpret conditional correlation prediction neighbor appeal lack property distribution way issue equip gamma treat use line liu hide complete abuse indicate constant omit complete sufficient statistic follow simple pt update estimate constraint search undirecte however would section penalize avoid put maximize penalize account miss datum e calculate take step step eq yield maximize quantity exactly iterate describe call force convergence chapter concave usual able guarantee obtain well contaminate observation reduce contaminate loss away good contamination deviation normality part long bottom handle situation alternative
applicable scheduling result armed bandit problem rich context medium access simple player play maximize cumulative performance gap know reward reward policy characterize respect player average inherently tradeoff armed bandit sample ensure arm policy well often reward formulate generalization multi armed allow markovian user resource associate evolve irreducible transition allocate resource receive allocate resource classic arm potentially reward resource user resource arm challenge traditional armed bandit resource apply diverse switching input match output schedule wireless need allocate assignment problem learn maximize usage enough apply cognitive come markovian design policy markovian matching resource super polynomial policy static restriction logarithmic polynomial resource organize section present per polynomial user certain describe state arm show still example idea treat across state evolve markovian reward markovian arm assume arm value provide logarithmic extend simultaneous base asymptotically logarithmic arm reward finite achieve logarithmic utilize hoeffding regret recent policy provide asymptotically cognitive bandit markovian reward paper slot reward generate chain transition identical space also bind user independent arm recent liu result single state markovian prove upper element proof however markovian user arm arm cluster provide reward regret arm model linear static number time arm close build bandit formulation storage matching yield polynomial resource time reward reward case work state distribution allow support consider bipartite predefine decision slot resource policy resource evolve irreducible chain unknown assign resource receive depend state allocate user resource evolve resource allocate resource mutually allocate resource denote steady resource interference user zero interference cover scenario setting policy observation select resource user history resource assign throughput policy resource permutation formulate combinatorial multi armed bandit matching resource kn armed bandit share component express weight weight resource design armed bandit reward respect like averaged tend combinatorial armed bandit apply ucb across arm arm store slot decision alone several ucb ucb storage reward arm play analytical motivated policy correlation well use store resource slot slot slot play time clarity necessary step reward play permutation accordingly solve maximum bipartite j nn j accordingly summarize notation htbp index use resource indicate refer resource slot slot slot resource resource steady access resource z k kn k armed traditionally bound expect play take time arm although notice consequently quite loose linear arm novel uniformly theorem irreducible algebra increase reward markov chain denote satisfy lemma matrix j z j ng f resource pair mutually note evolve assign resource play resource index irreducible visit main regret state maximum policy expect policy denote c ic explanation counter slot arm play play case pick arbitrarily say increment pick arm summation equal define false pick arm arm could kt kt kt kt mean pr bind l kt upper upper could extend
moreover address synchronization ref roughly set synchronization ref topic address nonlinear continuous dynamical system stochastic effort close connection work cite symbolic dynamical perhaps intuitive one ref serve synchronization question come state internal organization get randomness issue synchronization review ref highlight synchronization review synchronization ref sign measure diagram ref use joint information x random measurement outcome value throughout limit sequence l outcomes length unlike field course mind symbolic discrete dynamical invariant notion apply spin deterministic probabilistic length eq set sequence bit kind reduce process infinite ref property process solely one property irreducible output one extract alphabet raw compress precisely shannon bit shannon theorem operational meaning channel transmission note dynamical spin mention future generate something one optimize shannon think channel actually channel determine closely random tell share much noisy block specifically sublinear effectively length speak various reference entropy excess merely hierarchy derivative integral discrete length converge excess control speed step see discrete integral determine clear information quickly block intercept ref diagram appear compactly introduce operate slight additionally improve go operator process capture reflected give one attempt calculate hierarchy analytically addition hierarchy distinguish hierarchy trivial example turn conversely process excess entropy reference introduce line prediction communication wish probabilistic minimum good capture information future building good behavior channel mechanic equivalence purpose forecasting process state state fig ref denote matrix consist denote causal state future past moreover optimally predictive know causal word share past structural start symbol sequence state importance reflect representation important component know measurement uncertainty vanish capture minimal store order entropy shannon causal complexity bound excess view memory store basis capture asymptotic causal state normalize transition complexity also directly directly rate produce historical show excess entropy excellent capture approximate job compare description arguably however mind constraint preferred benefit elaborate description model state transition describe address alphabet variable denote machine presentation organization space relax make distinguish several kind presentation notion reverse third predict discuss two discuss recall result theory stochastic complementary entropy typically type output markov use stationary il need entropy convergence stationary il deterministic ref result note serve motivation later generalization derivative integral block entropy derivative ensure consider integral fact block entropy sum state thus sufficient necessary sum state integral sufficient theorem limit curve constant entropy fig take block entropy contrast entropy drop formulation measurement formal entropy entropy offset excess fact derivative recall presentation observer hide internal introduce ref causal observer uncertainty series vanish observer observer answer observer answer though presentation state observer formal synchronization synchronization presentation word word markov admit weakly observer turn weakly state vanish fast l implementation particular state impose sequence due duality synchronization return briefly interpretation give excess internal information sequence synchronization recurrent know transition accord consider useful consider state current symbol observe next converge exponentially add q importantly synchronization starting identify conclude rr lk state entropy define ref converge surprisingly give give far distant ref finite symbol know odd precede making conclude contribute piece comprise distinct contribution piece record estimation length short correlation view amount infinite role contribution synchronization clearly range spin claim spin chain rather introduce ref spin slope l first monotonically monotonically compute entropy estimate entropy analogous technique hold ref focused uniqueness define property process represent entropy excess remain solely lead several capture perhaps quantify characteristic meaning must behavior future roughly think irreducible arise information past future causal reduce presentation curve presentation sublinear block state entropy start work hide future stationarity entropy eq use stationarity ref entropy information share past quantity nonzero complexity intercept sublinear part block proof proceed identically namely take prof prediction must extract perform away representation state justify bi draw degree system presentation state past future past future presentation state uncertainty determine rather presentation intuitively excess later diagram tool point visually sum excess work simple verification approximation limit desire recurrent causal process synchronization happen window tend infinity either always generalize presentation irreducible symbol kind past alone synchronization define ref reference property describe understand block reach concept generalize understand h might particular turn one presentation term term presentation briefly exposition elsewhere observer state presentation regular observation presentation state observer absolutely uncertainty state reach vanish visualize difference asymptotic eq soon observer markov order exactly equal unlike indicate opposite language physics call freedom mention sec observer I look motivation order think length state uncertainty synchronization presentation markov presentation synchronization maximum entropy observer extract bit leave irreducible observer learn observer synchronization presentation helpful presentation synchronization order synchronization slightly let permutation periodic cyclic true equally informative observer observer contrast observer observe instead synchronization cyclic q periodic process sum sum minimal extend recurrent state synchronization give long arbitrarily word despite synchronization repeatedly observe symbol observer consider always order states motivate synchronization intuitive leave detailed discussion properly sequel preferred presentation process interested understanding particular would analogous classification presentation result presentation possibility quantity zero redundant state look exactly ref finally purpose presentation property fact diagram happen define property presentation past state move must addition gray generalized excess dark diagram potentially depict finite random theoretic diagram however component sigma algebra vanish simplify atom dramatically make atom vanish past excess similar calculation correspondingly state induce infinite state presentation induce past define difference excess diagram particularly simplicity derive causal play map state state requirement entirely diagram presentation relationship complex converge linear attention exactly lastly overhead require generally associate presentation uniqueness irreducible process must track bit information correlate future also simplify causal newly integral simple alternate explanation intuition relax constraint leave weakly state infinite minimal causal partition effect fig demand determine diagram indicate consequence nontrivial refinement examine growth reflect curve additionally happens reach since expand way combine variable two expand refinement entropy positive state presentation entropy care complicated follow three presentation fig original causal history straightforward unchanged presentation weakly asymptotically presentation interesting depend integral presentation domain weakly asymptotically long operate recurrent correspond past covering causal every induce weakly induce presentation induce history also finite recurrent product cover capture causal prediction unnecessary unlike entirely call new represent become make synchronization follow weakly l l large synchronization infinite necessary synchronization denote sum piece interesting impose maintain synchronization one mean feature describe straightforward presentation asymptotically history degeneracy break due derive induce rely contribution since finally remove requirement present change whole presentation break much large examine diagram one future beyond think feature maximally utilize additional remark complexity well small sometimes cite one sequel diagram plain make make oracle like without without presentation mixed ref evolve distribution state presentation history transmission complexity discussing indicate one long dynamically lose allow net repeatedly lose converge linear large growth acknowledge entropy block state diagram leave less presentation bit value information copy history cover twice visit coin flip analog reverse symmetry mean development started discuss synchronization block entropy quickly compete reflect give presentation summarize entropy immediately addition track comment development block entropy note analyze directly entropy convergence another important establish final information speak principal subject presentation diagram sigma atom turn measure process c c concrete operating synchronization information contribution information reflect occur length reflect information connection synchronization previously point turn appropriate process beyond describe presentation generic amount finally component implicitly presentation justified concept familiar physics immediate hierarchy go predictor show outside trust presentation make role uniqueness corresponding alternative second wide may calculate information production example convert either ref circumstance resource ready randomness observer prefer recall synchronization control note essentially achieve reflect analogue block entropy counterpart hierarchy ref title usually depend symbol asymptotic property logarithm entropy modify drop boundary give two relate relationship redundancy thm entropy difference l definition
posterior setting strict result additionally cl j code take sample ghz ram burn shape average obtain post mcmc abc cumulative factor specify abc bayesian mmse cl present marginal I turn smoothed great abc high reflect notice aim demonstrate reach reduce distribution change material justify effort require ultimately ensure chain mcmc chain alternatively smc sampler finding paper var predict easily use claim addition predict sum claim year uncertainty conclusion demonstrate frequentist relative demonstrate frequentist bound frequentist lowest close unconditional frequentist residual frequentist free claim novel advanced abc intractable posterior chain chain assess metric abc predict claim via algorithm estimate cumulative claim accurate empirical approximation entire claim valuable reason centrality measure tail var risk complete thank mathematics university award partially dms north finding recommendation express view national cm assumption notation remark bayesian methodology demonstrate classical numerical abc distribution abc parametric simulate directly sample without requirement crucial claim capital value error chain computation chain monte mathematics department email science bag mathematic bootstrapping justification deterministic chain cl model uncertainty cl reproduce cl bayesian abc standard markov carlo mcmc set parametric effectively methodology allow abc methodology evaluation likelihood parametric assumption make overcome present numerical novel abc embed procedure able model us methodology model demonstrate obtain distribution claim point obtain bayesian benefit uncertainty analysis analyse cl unconditional frequentist estimator mmse frequentist estimate addition set obtain predictive classical bootstrap procedure rao integrate parameter analyse achievable bayesian cl summarize contribution cl et distribution intractable result intractable methodology inference potentially poor outline begin follow parameter model construct bayesian present directly illustrate develop synthetic datum via outline development structure triangle give c year claim simplify year period claim time j I underlie rather recursive claim give good involve follow unobserved quantification associate predict analyse uncertainty reproduce cl free bayesian chain bootstrap additive cl j independence density density claim different cumulative q conditionally residual satisfy q p residual slightly involved claim assumption conditionally posterior implication develop make prior distribution prior give gamma cl priors cl likelihood analytically model distributional cumulative claim distributional prior free cumulative claim standard analytic perform long free another use w set abc allow make variance highlight distributional mutually exclusive idea select maintain particular important prior enforce strict develop fail prior satisfie bayesian factor cl cl tail enter discussion simply choose parameter inference context distribution denote posterior allow bayesian cl mmse additionally find approximation line equality exactly cl justify predictor via mcmc bootstrap obtain claim full claim empirically take obtain claim mmse alternatively predictive numerically integrate uncertainty approach result risk security calculate calculate predictor cl around analytical chain carlo mcmc abc free bootstrap previously long moment estimator frequentist approach bootstrap table numerically predict claim associate uncertainty bayesian approach distribute monte carlo require markov sampling distributional regard involve evident skewness instead truly abc facilitate intractable additional encountered working methodology typically methodology see concept embed abc novel back transformation abc intractable abc novel mcmc present numerical procedure numerical abc carlo simulation al description methodology specifically work consider wise intractable abc replace set summary statistic I augment tolerance statistic abc intractable marginal free joint integration intractable statistic converge see reference therein discussion accordingly low possible base sampling algorithms abc informative justification theoretically achieve rejection methodology model paper mcmc abc observation want summary datum assume procedure hard g equal step trivial analytically mcmc method accept denominator posterior computation evaluate extend class analyse measure city block efficient standard especially scale additionally assess serial markov chain sampler autocorrelation chain diagnostic conclude intractable reason normalizing target appear numerator result detail mcmc choice metric study metric comparative distribution unchanged denote generate conditional appropriate choice analysis impact mix joint chain tune additionally round produce acceptance typically design constant datum run version markov show markov chain random analyze behavior serial chain tail due scale euclidean covariance diagonal recommend euclidean trade simplicity diagnostic variable use
velocity via adopt triangle red circle symbol correspond green symbol simulation solution line weak problematic configuration consist wave tx contour lb simulation fourth third version successfully recover observe computation instability application instability engine role natural phenomenon formation domain practical recognize offer physics instability challenge illustrate fig interface perturbation separate wave interface immediately height divide mesh initial interface initial amplitude indicate domain boundary periodic condition bottom boundary boundary velocity boundary reverse one pass interface right reflect wave transmission wave perturbation amplitude interface peak heavy light gradually go light heavy wave reach solid interface transmission wave wave light second wave reach interface continue growth interface instability interface continue grow satisfactory numerical wave air reverse interface observe mechanism mention see medium wave show letter propose flow heat appropriate application heat thereby validate benchmark always show three appear conceptually future xu zhang acknowledge science national foundation china li acknowledge national research program china cb transformation possibility give b way transformation compose iy iy iy ix iy iy ix iy ix iy iy ix iy iy iy ix iy iy iy I equilibrium j j xu li china university mining technology laboratory institute mathematics box china decade lattice boltzmann prominent flow application range pre flow dynamic boltzmann equilibria lb model flow importance wave dynamic lb speed lb lb version flow follow constraint low heat broadly lb flow local calculated expansion velocity line roughly lb lb refer chen al euler well leading refer xu also specific heat degree energy original lb lb lb moment space projection subsequently stream discrete model degree rate moment adjust obvious cause instability effort past speed flow lb enhance stability nearly lb recently construct group theory equilibria expansion equilibrium function regardless heat description formulate flow specific heat express relaxation various moment physical quantity density tensor letter seven g associate equilibrium reference incorporation permit seven equation seven v choice simple detail eight correspond momentum mode stress energy transformation upon gram lead energy consistent divide moment energy two mention group one moment functional one equilibrium basic principle momentum energy correspond equilibrium distribution accord seven relation lb energy
gaussian type handle hyperparameter tractable exception assign gamma augmentation update conclusion product flexibility decomposable encourage induce product graphical practitioner belief problem edge practitioner cluster emphasis place derive model prior examine demonstrate graphical field properly interaction yield lastly american voting amongst concerned problem sometimes learn decomposable popularity mainly tractable graph set use find model accommodate posteriori rely decomposable update give posteriori current accommodate form effort encourage interpretable together exhibit block moment prior handle class product flexibility specification particularly spatial valuable complexity straightforward interpretation examine decision management addition sample important overview exposition let subset say subgraph clique order clique undirecte empty might undirecte decomposable associate decomposable imply decomposable graphical clique graphical model traditionally graphical structure represent conditionally independent give graphical factorize complete cardinality perspective posterior dedicate specify proper decomposable space crucial graph obtain estimate limited distribution bring decomposable often assume prior e prior mass intermediate size binomial number edge maximal suggest motivated result peak beta give marginal default imply interestingly medium sized number decomposable considerably decomposable list decomposable straightforward mcmc scheme prohibitive dimension edge often undesirable string show sample node uniform make interpretation long string tree reality class amongst separation clique move beyond prior model call clique alternatively clique factorial term clique trait interpretability result graph even completely graph preferred independence respectively clique clique value bottom relation clearly demonstrate relative binomial highly separate euler absolute first kind limit exchangeable consider four allow control clique reduce limit partition multinomial use able number clique insight determine policy variability spatially enable management handling thus require accurate effort statistically understand connection yield valuable reason firstly plant management benefit early additionally extreme event rate lastly prefer yield practice portfolio look undirected graph production thousand california department considerable wishart yield give covariance inverse wishart ratio wishart likelihood induce inverse wishart focus specification look expect yield characteristic instance namely prior control contrast put penalization separation clique encourage clique pursuit mcmc length million binomial graphical prior save move clique determine figure high graph mass evenly relative product evident figure form specifically string together plot reach conclusion prior string prior suggest correlation majority plant fall early grow contrast graphical plant decision graph bayesian decision gain understand graphical prediction performance year year simulate evaluated test result evaluate graphical prior sparsity responsible avg turn american
vc smoothness depend one smooth function non opposite interval define obviously former easy describe understand admit part half smooth function inherently complex describe right specify evolution give reconstruct exactly vc dimension smoothness therefore specific visualization different cluster write useful peak description peak position case consumption frame power hardware time switch implement body literature context take dependent piecewise text convenient affine approach inspire technique use instance indexing idea build piecewise level series translate symbol general assume approximate quadrature scheme setting provide straightforward therefore give value approximation segment parametric derive piecewise linear reasonable median provide robust take segmentation resolution computationally intensive bellman showed program different know need segmentation find accord criterion replace accordingly form summation operator aggregation operator crucial minimizing idea quality additive partition correspond partition winner fs fs j kl fs fs fs final backtrack phase index partition output provide backtracking loop figure leave dataset near spectrum feed spectrum segment segment figure segment htbp far naive approach evaluate partition efficient link naive implementation dominate fortunately recursive cost st segment general flexible numerous peak huber etc error piecewise representation former latter indeed easy together piecewise value independence need prevent request drawing approximation belong bellman boundarie interval connect segment piecewise suffer jump could specify segment simple description continuous variation list derive continuity let define make interpolation strictly interpolation belong segment optimize former approximate summary function issue rule address build piecewise segmentation noisy framework relate sample build summary homogeneous tackle case functional error use segmentation set diameter quadratic request quadratic set function homogeneous seem simple measure functional aggregate individual first measure quadrature consist build readily induce mean curve linear bit several curve curve htbp contiguous model approximation partition use measure straightforwardly difficulty efficient curve rule problem simply consider maximal distance curve evaluation fast might compute simplify problem one value observation simplification choose apply optimal segmentation accord nm summary natural cluster previous section application suboptimal optimize derive previous section q scheme independent segment amount global segmentation way tackle optimize sum term programming summary optimize partition segmentation feasibility depend aggregate operator function summary therefore cluster summary distance give partition stable availability programming computing quantity median mean segmentation rule solution tie algorithm cluster distance tie break might never break tie begin imply optimal segmentation unique alternative give obtain piecewise per cluster full spectra frequently piecewise via constant function define rewrite optimize allow section assess distance formulation functional way variation prototype modify mean candidate version version cluster segment regardless cluster fortunately dynamic remove resource cluster segment respect resource partition minimize minimize assignment firstly mention segment segment calculation variant efficient request secondly measure optimize programming indeed segmentation cost calculation segment minimal give computational assignment simple description seem resource assignment segment cost summary uniform reduce introduce arbitrary e time optimal assignment regardless optimisation case piecewise already mention summary compute cost cost scale criterion induce computational motivation introduction avoid rely suboptimal phase alternative representation homogeneous prototype segmentation optimal determination alternate optimum mean partition distinct therefore predict addition algorithm initial configuration three possibility experiment dataset solution start partition piecewise summary initial configuration uniform resource optimal allocation resource assignment picture phase give case result reach resource assignment improve resource identical k give remain favor nevertheless winner help practice favorable case purpose experiment time slow quickly generally assignment dataset study start follow mean k latter iteration acceptable compare alone initial configuration analyst neural base self give exploratory analysis world user build batch help induce acquisition display curve dataset htbp htbp conduct cluster criterion rough dendrogram figure small visual som arrange som analyst decide rectangular superiority som prototype arrange som contain som organize grid horizontal axis prototype however infer grey som summary average segment section iteration display som exception immediate starting summary slow phase cluster follow original map consist consumption record home give consumption minute displays htbp analyse similar clustering som rectangular axis encode global consumption horizontal shape shape interpret noisy nature htbp htbp htbp obtain result som algorithm stable summary segment cluster piecewise constant summary adapt load electrical power consumption summarize noisy counterpart even map power consumption start follow consumption consumption pm day home empty cluster consumption day week end resource emphasize difference resource expect piecewise constant summary figure grey prototype configuration represent load grey curve curve assign rough strong constraint consumption peak outline accurately approximated series segment expert easy addition nature increase segment improve marginally error notation compare error total internal variability dataset relative error quite acceptable strong summary type provide analyst selection guide exploratory functional piecewise linear homogeneous compute optimally program optimally number summary
coordinate coordinate remain list di jk u u jk jk derivative I say dimensional frame define function lie correspondence subset correspondence hausdorff take take prove part curvature identify second lie value frame tell angle tangent v contradict prove claim correspondence let small distinct correspondence bound derivative curvature condition jk jk disjoint counting jk jj follow function claim cover cover restrict domain note distant distant hausdorff vice versa hausdorff minimal ball specifically center curvature cube globally way hausdorff around manifold cover manifold choose hausdorff prove almost entropy need depend uniformly vs vs hellinger hausdorff c b dm dy yu bc hellinger hellinger rate c relate hellinger hausdorff cb bx normal claim x imply two distinct whose contradict claim tangent radius bs b either segment intersect happen c contradict conclusion hausdorff pilot dd manifold cover pilot overlap exist b dx let mx b b nm ij prove hellinger conditional mle manifold n half least observation j relate hausdorff hellinger claim exist v write shift orthogonal equal integral volume radius shift c ab hausdorff estimator choose nn c constant h practical generalization estimator large follow partly r partly partly distance boundary dy dy b dy dy dy dy partly ball outside complement radius tangent virtue partly boundary almost surely lemma imply c dy dy dy establish hausdorff conclude open question assume noise derive minimax rate substantially involve rate near boundary support report important achieve modification estimator hope homology infer reason difficult regression fast highlight distinction topological versus hausdorff exist adaptive acknowledgment comment paper thank comment eq manifolds volume support write define disk q follow u u manifold second coincide b portion radius center sphere summarize manifold see b u upper satisfie v prove ii also since inequality either imply inequality consequence pt minimax hausdorff manifold noisy depend dimension manifold unobserved variable riemannian precise give manifold open center radius risk infimum measurable function value manifold low expect hausdorff bind though exist estimator rate logarithmic theoretical establish tight construct estimator require smoothness vast manifold dimension therein interested estimate literature field computational geometry noise draw manifold specific notion different exception estimator show properly closeness hausdorff distance precisely dimensional ball center euclidean measure lebesgue sometimes volume integral integral measure density hellinger q affinity measure section generic may expression shall concern compact without embed informally mean look like contain denote tangent regard hyperplane regard size large unique onto embedded constant quantity condition reach large intersect sphere space disjoint thus let angle tangent define eq product geodesic suppose geodesic connect unit parameterization curvature mp mp mp v dp n manifold assume draw manifold distributional critical lead simple derive proof report let number lebesgue uniform density recall infimum recall lebesgue mb dy vb dy exist projection want upper line upper minimize take upper simple geometry dy dc locally parameterize surface dy dy ml mb du b du u dy mb upper follow change projection onto latter u du consequence lemma every support projection tangent version le space metric let draw correspond appropriate topic pair manifold le roughly speak subject hausdorff dimensional hyperplane sphere origin create new let constant infimum theorem section lebesgue le set upper achieve appropriate intend establish simple contain maximum distribution lebesgue hellinger distance hellinger logarithm h sequence call actually construct triangle u p result simplicity ready converge optimal maximum half manifold pilot sub hellinger lemma distance
term train competitive well svm contribution order super object overcome yield lie rigorous extend early significance tie application cluster review background widely statistical science comprehensive permutation multinomial computationally feasible avoid rank perhaps exponentially depend case tie incomplete rank largely theory e assume basis probability object preference worth logistic style comparison extend proceed select object stagewise verify sum interpretable incorporate tie rank finally completeness treat symmetric technique learn active topic retrieval g see basic ranking science system often object query pair describe parameter machine divide ordinal assign ordinal label simplify pair drawback order model wherein readily boost simultaneous permutation goal permutation phase output probable permutation appear rank problem object rank object ranking x might document return search response document document ideally contain order return indicate relevance create document rating view assign indicate sort essential contribution perform know object rate consider split partition wherein th rate model among union permutation partition empty subset object rating disjoint singleton idea hard partition super growth behaviour unknown challenge efficient generic tackle order partitioning order among give way partition generic proceed generative subset element remainder select large process continue model wherein reduce singleton advantage partition need specify advance truly brevity clear th remain subset furthermore subset contain possible non partition summation distribution interpret standard significantly small order exponential general computing term alone th eq monotonically increase represent denominator efficiently object towards admit decomposition value essential maximum parameter reader refer substitute w use potential affect local function discard write specific log likelihood carry base take reduce dropping dependency subscript auxiliary array ng k compute imply compute linearly log linearly log mention eq summation rhs summation involve possibly cost often query return object exactly query object relevance query highly issue begin occur rate object enable unseen scoring specify exponential local subsection function case interpret probability possible centre ps resort mcmc start chain subset run centre sample typically choose local distribution idea proportional potential tie worth unique fact choice backward manner shorthand interestingly reverse eliminate bad reasoning limited free kp backward admit place graphical model g probabilistic receive value concept object role membership state space rank rank state subset assignment mutually exclusive probabilistic network markov direct scope handling pairwise basic assign instance preference rao tie tie handle tie create object ordering emphasize advance group super yahoo challenge currently large query relevance irrelevant perfectly relevant yahoo unique contain query two normalise cumulative gain ndcg reciprocal ndcg metric position constant sure gain I put emphasis rank implement several ranking pairwise differ hinge essentially implement variant tie handle implement first implementation result see gibbs hasting sampling handling ignore tie simply document relevance tie order accord sort except bfgs bfgs stop less gibbs mh stop representation normalise roughly mean product order yahoo experience correlate thought correlation pearson whose find threshold ccc order ng ng conclusion first second order feature performance yield either outperform baseline method scope dataset yahoo pairwise train fast slow mh address rank tie generative probabilistic suitable inference yahoo demonstrate rewrite eq empty subset last use non collection object since configuration object appear account weighted contribution towards subset kn kn n kn pairwise ar overall respect w x derivative pairwise describe detail tie rao follow x tie parameter want w e x p w derivative give x w w recall probability px px w w w simplicity optimisation w p j w w theorem corollary address probabilistic super combinatorial unknown order stagewise space
continuously differentiable appendix carlo ideally summary close affect approximate average justification informally globally case sufficiently small argument make try statistic impossible statistic take different requirement solely like abc consider calibrate estimate posterior inferential monte carlo formally calibrate modification call noisy abc calibrate produce recommendation accuracy summary statistic unknown result give previous end firstly calibrate show calibration accuracy abc posterior guide statistic consider abc ignore randomness statement state repeat event probability abc posterior appropriately credible calibration posterior abc posterior idea calibrate noisy involve change algorithm account extra replace abc produce abc calibrate derive variable model link consider use individual inference special calibration inference behave get center furthermore noisy make h regularity abc part show abc uniform suggest difficult calculate result detail accuracy monte abc monte carlo variance monte quadratic want decay trade calibrate give single number summary small recommend combine inference analysis light find approach work abc average noise add guarantee result loss appear approach abc independently estimate theory statistic calculate pilot run abc non use aim appropriate parameter informative avoid value ii uninformative prior implement assume hypercube range parameter range observe pilot run truncate region choice time various take appropriate simple work use neither wish explanatory consider introduce value possibly simplest different beneficial f power produce well response simulate explanatory square I abc use th summary within region adapt run iv weakly pilot composite method linear assume appropriate difference approach approach use abc advantages abc uncertainty get follow abc tends result statistic indirect abc summary refer fair automatic acceptance simulation analysis calculate individual loss shape attractive simulation straightforward inversion calculate likelihood numerically eq skewness typically assume throughout leave restriction study implement automatic possible explanatory explanatory automatic compare compare regression indirect abc also fourth appropriate informally link skewness linear evenly space power choose model range range use parameter appropriate linear final abc first keep subset advantage efficiently uniform exponential perform pilot statistic overall dominate simulation stage automatic procedure choose statistic square implementation automatic abc table abc accuracy correction use abc run stable regression correction great despite poorly pilot abc correction accuracy parameter extent semi comparison predictor except average third investigate pilot automatic implement pilot improve semi regression correction datum set numerically quadratic semi automatic datum set remain unstable true indirect show asymptotically estimator indirect see semi automatic auxiliary initial depend true illustrate produce value simulate set indirect inference many indirect inference loss detail estimate interest parameter identify abc indirect substantially variable ht pilot indirect automatic abc pilot indirect semi pilot indirect ht state number evolve stochastically reference model density intractable simulation reasonable abc base rather give many improve detail set time bandwidth prior state value liu west sample show may inconsistent estimate ignore simple implement algorithm add observed value algorithm abc uniform sequential algorithm implemented choose show bias observation vary three sequential abc noisy abc overall abc appear make observe picture abc accurate abc difference model independent interest initial observation metropolis likelihood summary statistic value affect method distribution mention improper prior place assume negativity automatic simulate dataset lag coefficient square zero semi automatic abc pilot run pilot acceptance automatic dataset comprise zero dataset zero discard training automatically subsequent summary statistic nest explanatory small additionally variance large add observation instead suggest thus summary base synthetic abc regression adjustment produce raw credible synthetic synthetic coverage regression semi automatic model easy often suggest analyse simulation g queue service uniformly arrival queue initially time simulated dataset true draw uniformly service time magnitude avoid situation evenly spaced quantile automatic pilot analysis replicate quantile explanatory construct power minor split form queue calculation indirect result automatic abc comparison latter semi automatic directly abc accurate accurate advantage indirect simulation estimate accurately accommodate summary expensive ht indirect analyse genetic marker thus htp cluster type mutation event appropriate birth death mutation mutation simple sample exist parameter likelihood depend reflect restriction avoid need unlikely restriction positivity posterior reduce distinct number observe retain semi automatic pilot comparison parameter pilot rotation line abc interest explanatory comprise cluster size large cluster semi automatic use abc posterior indicate place less high marginal automatic htp simulate abc abc summary statistics abc average automatic abc reduce automatic summary statistic give bad argue justified parameter approximation abc accurate result abc combine set abc empirical evidence sequential lead biased could genetic combined genomic implement attempt add abc implement rao manner semi automatic implement main summary mean optimal quadratic evaluated method implement comparison semi abc example two give similarly estimate semi automatic abc less accurate alternative motivate approximate incorrect statistic idea liu accuracy sampling normalise mean square estimator size case immediate equality potential gain importance occur vary monte consider transition primarily
little addition many individual circle never customer circle otherwise compare geodesic distance many customer geodesic distance interaction geodesic average expect eight customer include friend friend reach customer customer expect reach interact identify customer span customer c seven customer substantially find customer customer well customer space present refine space computationally feasible extract use rich period might lead really underlying generating depend similarity small cell phone interaction occur among customer heterogeneous practical occur customer among customer treat straightforward update even refer evolve may censoring similarity censor concern like phone call record universe interaction customer reveal people ever may email face face contact connection alternatively phone call cell phone structure another observe useful diffusion innovation occur fashion people one interact regard heart model importance review collect present interest structure geodesic measure latent relation likely area practice identify influential customer opinion frequently diffusion research influence e innovation suggest understand influence among customer focus seem service selective opinion would network latent yield interaction propose bayesian latent dirichlet process well network analysis obstacle involve estimate concept yet fully researcher possess customer maintain heterogeneity interpretability classical machine readily admit latent believe interaction one try recommendation rather observe geodesic distance assume correlated preference behavior test practitioner allow would think might output space configuration latent physical distance sense spatial among people profile characteristic model observe propose correlation function interaction one geodesic distance order conduct infer explain business conduct mobile communication device become available latent behavior select choose specification incorporate nonparametric area abstract nature correct choose offer improve lead fitting examine likelihood use full test characteristic approximation adjust prior normalize rr rr likelihood estimate prefer preferred find difference posterior derive specification open exponentially bernoulli closed remain specification denote duration observation period likelihood way could sufficiently observation period wise unobserved heterogeneity gamma specific shape parameter integrate distribution formulation section definition radius word herein subscript probability let simulate multivariate ability trade draw origin draw clustered origin begin laplace center fold constrained become uniform infer tradeoff placing bound half add add special power balance appear combine transform center covariance matrix multivariate gamma effect sampling general current variable bernoulli trial r simplify combine depend care marginal integrating multiply assumption independence prior normal normalizing simulate walk place likelihood univariate sir might choose base slice draw adaptation direct reader summary terminology distribution take z point number point include person differ singleton person locate people contribution else person singleton call z union vector already set vector set normalizing need thus exist depend yield people mass likely prior singleton probability people one likely interact interaction customer make similarity interaction model scalability infeasible nonparametric moderate scalability find insight latent customer activity interested customer affect relate market among customer diffusion leverage efficacy incorporating model show improve forecast new customer likely interact efficient leverage connect link interaction many accommodate customer simply customer interact share information customer literature thing record observable correlate determine individual interact illustrative parsimonious theory similarity develop output understand customer interact behave model fundamental behind individual space working determine incidence difference clean observational among customer phone call record transaction observe individual two people generate latent interaction similarity identify similarity service paragraph activitie link know customer direct relevance practitioner heterogeneous population segment characteristic attention might contain company segmentation company phone reason model challenge key modeling offer valuable obstacle observational binomial use want break ignore unobserved heterogeneity restrictive interaction intractable challenge similarity characteristic interpretable bayesian nonparametric process essentially scalar purpose salient characteristic realize cluster location mass individual smaller fewer distinct likelihood segment bayesian nonparametric algorithm infer interaction probabilistic allow interact circle customer possibility customer may interact power interaction datum dataset specification incidence interaction distance validate showing improve respect metric literature interaction calibration predict interact unobserve heterogeneous among simply represent scalability nonparametric insight get segment offer finding literature similarity mix effort able distinguish customer computational dp inference wide variety include indicator transaction among interaction treat process govern network differ across call call call call occur phone call might ultimately conditional rate incidence upon similarity likely unobserved individual individual measure space people distance distance way pattern purpose article treat coordinate persistent though interaction phenomenon abstraction reality dimension relative coordinate vector datum observe heterogeneous across distribute mixing would imply sense heterogeneity similarity heterogeneity heterogeneity shift focus individual mostly unobserved trait characteristic unobserved coordinate lie space function distance induce dependency contact positively mean determine evaluating go people function among issue mean iteration value formulation moderately become number discrete latent mass distinct large lead value avoid integrate analytically certain want know dirichlet nonparametric prior distribution although dirichlet back new analyzing across estimate home probability oppose interest context finite paragraph dp think course model parsimonious require mass dp around realization lot dp lot less dp generate discrete important determining cluster purely depend either know though draw directly trick integrate analytically treat process mdp see distinct likely mass illustrate work univariate cdf color realization fewer high cdf histogram draw draw mdp deal dimension rich specification basic idea prior detail introduce restriction concern simultaneously uniquely latent space determining constrain prior distance origin introduce much incorrect prior origin translation define mode origin prior generate specification empirical censor second statistic exploit exponential computational fundamental much dataset never duration choose exponential pair gamma common degenerate hierarchy homogeneous heterogeneity later integrate three latent worth add interaction common maintain heterogeneity appear datum contact homogeneous latent full latent nonnegative latent contact contact select geodesic along short another candidate weight dimension distance space differentially also distance parametric specification equation distance subject empirical context assume customer claim show level parameter ran dimensionality log marginal decide contribution forecast empty allow assess goodness distance histogram call network article week axis log proportion dark dot log actual dataset represent cluster vertical replicate rather value infer close whether empty lot baseline actual dot outside many statistic existence interaction whether contact assess fit perform calibration period nonempty empty interaction duration want ability model lift lift likely contact period completely model percentage top empty great chance lift metric always lift metric variant result empty non remain geodesic distance break tie way lift metric exactly would expect assume empty likely contact sort distance long empty improve dramatically full contact nevertheless assume independence across ability latent behavioral interpretability link formation accounting calibration condition geodesic geodesic contribution model scalability amount dp group every generate outcome dense continuous non empty china mobile distinct likelihood much aggregation homogeneous likelihood available form dp group likelihood pattern dirichlet keep zero happen likelihood zero plus observed latent dp social least large much likelihood must individually latent likelihood likelihood compute requirement extent dataset change ultimately front influence grow small test work practice successively original value weakly compute tp size mass grow incremental change decrease though continue low rapidly though number number nonempty fast increase dp feasible effort likely come aggregate
proof concentration realize latent dimensional condition observe realize volatility take variation process brownian w w theorem law iterate dt repeat process x conditional time frequency ambiguity base interested compute moment term use realize satisfy obtained ignore fast auxiliary moderate presentation fourth two decompose generate simplify without ingredient calculate easily use fact even presentation generate bound I x x q iterate derivation normal second old work term follow condition eq valid condition theorem observation go choice moment function observation final satisfied observation diffusion process bound let brownian motion see theorem write note utilize result conclusion involve q normal presentation grow moderately bound moment suitably bound real replace need iterate expectation normality similarly bound matter need moment appropriately inequality proof omit satisfied proposition fan nj department business management technology mail yu department financial nj portfolio exposure effective increase stability vast require high matrix volatility volatility high portfolio propose pairwise propose vast compare portfolio concentration inequality estimate desirable volatility asset exposure constraint study carefully compare daily trend portfolio time period advantage frequency empirical consist stock portfolio portfolio assessment frequency mean portfolio finance yet practical portfolio selection number know depend volatility lead short portfolio portfolio market introduction exposure non exposure parameter portfolio theoretically optimal portfolio empirically portfolio exposure little accumulation effect empirical portfolio practice challenge portfolio statistic challenge medium period week say realize still portfolio selection portfolio allocation exposure provide reduce volatility capture dynamic volatility adapt volatility expense size wide availability estimation volatility year frequency volatility volatility presence market asynchronous price brownian satisfy process grid become stochastic calculus realized realize quadratic co see example high covariance tend bias toward zero signature affect way bias estimation integrate estimate extend improve zhang separate jump presence wavelet issue address realize reduce achieve integrate non absence first zhang study integrate integrated factor extend study perspective financial engineering topic datum asset handle trade former definite whereas far estimation error control paragraph base pairwise accuracy result pairwise though former implement outperform method adapt comparative advantage rapid demonstrate portfolio govern maximum thank vast overview portfolio datum perspective asset well simulation section technical process denote log price dimensional brownian instantaneous stochastic independent portfolio hold return history short simplify problem literature consider risk practice return constraint avoid negligible exposure exposure problem put equivalently proportion expect unless path current rely approximation even ideal observe continuously window historical reasonably rely continuity vary volatility reasonable rely stationary small hold short recent datum problem usually volatility nonparametric usually prefer million natural question whether result stable exposure constraint problem exposure estimate eq risk portfolio portfolio exposure respectively maximum accumulation exposure drop confusion show risk indeed oracle want portfolio quantity usually grow slowly number reveal exposure approximation semi estimation allow maximum advantageous method high trading former matrix trading several synchronization scheme propose zhang efficiently available datum idea say price asset price obtain previous yield vector available trading clearly stock synchronization integrate synchronization advantage method scheme trading call point estimate definite thank exposure portfolio selection far rich volatility help efficiency portfolio univariate volatility two volatility realize volatility wavelet realize price process pairwise integrate particular element estimate scale realize pairwise price assume actual observe observe transaction price logarithmic stochastic assume assumption mainly replace time process really study asynchronous discuss optimal realize covariance asymptotic normality zhang reveals simultaneously remove due due adequate vast depend replace subsample condition differently integrate volatility case facilitate reading condition integrate integrated process interval condition process candidate parameter condition inequality readily bind log observe market frequency clearly large accurately observation pairwise theorem proof observation time hence clearly see somewhat hold semi optimization yet even symmetric semi definite decomposition p minimum eigenvector remain positive semi transformation diagonal satisfie projection result keep asset study turn decide keep pairwise call covariance covariance pairwise call distortion effect serve matrix risk portfolio compute range profile numerical method number time distribution summarize clear pairwise scheme yet minimum pairwise insight risk approximation p w daily risk compute compute latent range characteristic frequency trading day stock various characteristic portfolio risk portfolio absolute difference pairwise portfolio result pairwise tendency tendency absolute pairwise turn absolute covariance method expectation portfolio exposure allocation risk compute risk exposure exposure agree exposure tight obvious method former latter bind exposure secondly flat due intra day trading condition increase increasingly unstable portfolio become basically actual risk become flat comparative especially portfolio strategy include asset price conduct simulate price trading day day record trading time trading asset mean asset trading day asset capital pool low strategy day use day daily matrix portfolio allocation exposure frequency trading trade use make portfolio allocation frequency integrate projection transform definite optimization portfolio trading day adjust portfolio portfolio hold trading day portfolio portfolio characteristic trading day calculate risk portfolio whole exposure stand portfolio strategy usually relevant exposure strategy present omit standard actual risk optimize profile significantly accurately short horizon provide graphical detail simulate trial intra trading portfolio trading day daily max median number position exceed whose exceed std min frequency covariance estimator frequency covariance short frequency frequency short c simulate intra portfolio trading day daily weight min whose absolute exceed std max short estimator short covariance estimator short c length low range exposure theoretical shorter consistently low portfolio figure slightly surprising low outperform instability realize curve attain figure fall specification couple study exposure outperform secondly risk approach mild exposure explanation accumulation dominate give day stationary portfolio low frequency assign weight around one asset portfolio exposure risk minimization asset stock stock call stock average stock index create co company publicly united market make individual change market condition asset frequency stock highly intensity trading trade median stock day cover birth financial hold period sep stock accord day daily low trading day estimator trading bandwidth since pairwise trading day estimate use exposure risk price arrival news characteristic portfolio characteristic portfolio stock daily risk risk minute return graphical characteristic std max min short short frequency covariance estimator frequency c index std frequency short high short high covariance short frequency estimator weight reveal term portfolio pairwise base exposure low frequency approach necessary short vary local one support strategy outperform
list concentrate many display membership recover matrix actual focus scenario diagonal present group volume bic group membership membership close bic simulation accord probable grouping node show network message along accord membership determine row matrix row order figure indicate inference capable membership prediction lie line inference algorithm email focus send message distinct message send month cc field language transaction result website post link comment chain focus hundred assign comment information include close community interest website resource social comment link comment link comment comment interpretation email represent group link top select discard comment network node per remove transaction transaction send one category comment user activity normalize measure bic suggest mixed membership quite focused element thus deal assign probable ht grouping probable identify appear send htbp matrix figure characteristic cell dark plot band correspond send member group member appear frequency observe frequency suggest remain behavior r probable summary nine calculate probability belong would vary considerably range suggest member group weight likely response member version transaction measure indicate identify identify message message pick predict overall performance message comparison link immediately transaction version send combination fig low htp green hierarchical htp red cluster algorithm compare mention availability observe measure hierarchical transaction use convert transaction count hierarchical apply receive frequency label mixed method hard classification membership ability co hierarchical use frequency multiple membership include communication pattern development variational accurately indicate interesting competitive membership transaction novel performance compare cluster issue study scale transaction cluster network transaction explore relate multiplicative respect varied minute computation cluster fit linear way bernoulli permit transaction impossible email transaction extension exclude nan might transaction information incorporate vary version activity could membership change vary fit partition interval interval extend structure behavior govern distribution membership node version transaction label transaction label label transaction additional message label group label allow topic membership message group category unclear practical support natural science engineering mathematic technology list communication email social network convert node capable model rich node transaction general flexible notion enable accomplished algorithm indicate email extract website clustering superior discover predict email text become consist actor relation relation canonical people relation friend relation always communication individual occur time call e call email involve one additional transaction message content g header network transaction email potentially email obvious develop shall assume transaction transaction thus observable transaction transaction discover structure future combine transaction group role receiver nod social node play role interact assume two role social easily research office propose hierarchical inspire membership detailed network structure develop review section novel performance soft cluster simulation present conclude summary scalability future seek toy adopt convention represent column fourth message transaction transaction transaction message receiver pair convert threshold figure small moderate toy summary count co message thresholded thing directional relation frequency seek edge decrease extension transaction section seek network inherently transaction occur transaction probabilistic efficient uncertainty develop develop send within list represent message node unobserve membership collect node model incorporate additional node belong membership allow transaction membership potential list conditional group membership node membership membership mix draw email select multinomial value select equivalent draw employ elaborate membership nonzero element email random membership potential membership email member tb draw membership draw pick e among node dirichlet interaction interpret differently depend email receive send member member restriction collection allow capture possibility large group high intensity intensity element pattern member notion section illustrate distribution specify variable assignment every row transaction row encode correspond email estimating infer posterior integral pick distribution posterior kullback use factorize dirichlet multinomial variational variational empirical multinomial omit fix simplicity initialize initialize normalize inference order number develop bic compose term term membership node receive email node trial decide receive exclude membership average membership compute probability write transaction likelihood bic eq number propose inspire relation receiver pair seek pair relation observe conditionally outcome membership behaviour incorporate allow relation similar represent edge relation direct require simplification
side et classify spam work lexical work lexical high lexical accuracy build base online use lexical al author outperform focus lexical lexical show algorithm batch work lexical dataset context non draw sum lexical classify al et al chen safe microsoft mention introduction aim web rely base verify manually member site verify month may carry collect appearance characteristic spread yahoo yahoo yahoo generator generate generator collect collect maintain collect mid collect dataset recent different query server responsible level domain implement engine adopt module feature answer could play site site team team server network country complementary former external collection collect collect depend load team extract lexical external feature l com http www com http cn com uk http www de http www http r www form auto www com form file domain ip max file name max yahoo yahoo lexical external feature lexical extract complement break token constitute feature token token domain name file extension token appear argument token part constitute binary bag et address iii name iv follow lexical detect feature classify five dot word similar type name ip number domain name name number use iii token dot token address name case category instance file page file dot file address ii name put name serve write server often include part include value lexical real team date site piece prior learn yahoo yahoo yahoo yahoo bad bad combine yahoo dataset dataset get classification online op weight machine knowledge turn become introduce online indicate otherwise receive train datum give predict op addition sign vs online algorithm train batch train meanwhile label use batch batch achieve perform classify svm construct large hyperplane involve determine lie svms investigate svms discuss operate online receive predict use receive update op update continuously update predict ty update suffer drawback account make adapt enough make binary confident notion address confident ip indicator update sure even domain mistake update avoid lot change formally maintain represent capture new weight pick vector prediction continuously label make close old distribution kl correct datum big require memory category examine considered presence feed slot likely drastically weight domain formally regularizer ty tries preserve valuable old much running update feature memory reader well pt p cm compare lexical svm lexical yahoo lexical op lexical evaluate lexical lexical lexical yahoo noise lexical ii lexical full evaluate effectiveness lexical working summarize pt cumulative svm svm daily compare lexical kernel svms matlab box svm validation yahoo similar examine svm svm svm batch svm svm instead set size svm svm initialize batch cumulative misclassification classified observation update essential svm performance fundamental svm observe third outperform retain high experiment illustrate model continuously lexical feature light weight light memory algorithm cumulative op error rate cumulative lexical gain lexical gain lexical yahoo yahoo bad pt conduct evaluate lexical feature full examine op dataset matlab yahoo good yahoo initialization pair cumulative yahoo due report cumulative rate pair involve note bad pair regardless suggest classifier agree discussion purpose op regardless use lexical full feature lexical slightly yahoo slightly lexical lexical alone achieve summary lexical feature lexical feature alone c c cumulative mis classify fp mis gain gain yahoo yahoo effectiveness lexical report feature boost pair range yahoo look mis classify negative number mis mainly come mis reduce range increase range false arguably spam summary lexical set examine yahoo give number label vice cumulative yahoo various achieve dataset maintain accuracy accuracy summary achieve pose advanced online outperform op pose seek affect performance examine importance long memory dataset notation introduce similarity denote lexical threshold name token file index file interpretability subsequently number similar dataset tail complementary cumulative significant distance size limited accuracy need produce explain batch result classification accuracy batch algorithm batch memory online limitation see effectively explain batch pt head function cdf cdf observe significant mean batch base mean classify essential maximum distance similarity unless update prohibitive similar classify memory behind rapidly op op reflect update section svm op dataset two retain history rapidly implement lexical classify implement requirement loading avoid accuracy thorough comparison classification feature section option divide component core component classify newly model component maintain stand one maintain run service maintain model label yahoo detection add classify split another option core server maintain run internet detection advantage server several drawback amount traffic yahoo scale mobile device life keep service mobile device persistent connection date practical bandwidth mobile device need various feature publicly add implement scheme internet attack detect lexical accurate avoid noisy uci california sophisticated carefully select lexical lexical vs lexical purpose
suboptimal filter inaccurate parameter posterior negligible hasting restrictive tune firstly negligible correlation parameter state secondly surface growth irrespective error modal strong flat mode average novel development use particle mcmc joint process non metropolis objective examine growth complex surface examine via markov capable sampling non careful surface multimodal sampler failure mix concern typical sampler additionally use structure fit precisely address fit first observation modelling abundance source weather detect failure device error bias argue detect model simplification difficult linear allow work presence introduce estimation date reason first non compose posterior design process structure four include strong population section estimation surface important discussion part synthetic newly methodology conclude recommendation development gaussian realization bold bold scalar discrete methodology filter sequential denote th chain particle involve iteration model consider discussion modelling regard density strong effect generic typically population rather mechanism reflect variability assumption transform importantly flexible literature five stochastic dynamic describe individual growth population reflect source variability growth behave exponential growth log rate per birth death discrete familiar logistic growth model density growth strength population exhibit dependence denote growth rate positive equilibrium latent exponential population unknown determine must effect transformation birth rate limit thereby size describe effect mechanism limitation population effect unstable equilibrium occur stable equilibria effect equilibrium equilibrium discrete study dynamic discussion note nest model set density bayesian unobserved time interested opposed estimate state jointly posterior give prior observation process nonnegative give prior c parameter prior observation generic eq latent specify inference model minimum error mmse mean evaluate evidence context estimate follow quantity quantity present considerable challenge dim variance require present recent space sample innovation combine mcmc advantage update entire particle chain thereby dimensional even strong model consider seven space additional hundred state particle proceed hasting static component construct via adaptive metropolis scheme mcmc chain marginal static model improve enable much rapid mixing number second component construct estimate allow sir note sir filter posterior proposal approximate p smc sequence recent review smc methodology acceptance produce generic j mcmc proposal appendix probability design proposal low dimensional monte marginal acceptance empirical law converge filtering particle path adaptive parameter distinguish kernel combination sequence allow markov markov appropriate several recent propose satisfied ergodicity ergodicity scheme two know two ergodicity space metropolis within involve specify proposal involve comprise adaptively line explore proposal empirical chain motivation present sir mutation process tn four subsection subsection accuracy synthetic comparing set recursively rao subsection bayes evidence final sub use real proposal propose additionally markov stage involve posterior increase construction particle stage involve burn non parameter sir particle mcmc proposal use sir mixture metropolis proposal subsequently use one combination chain careful bias overcome maintain would perform one inclusion static adaptation consider involve challenging datum set methodology estimate equation begin analysis number average entire state dimension sampler increase low estimate equation ultimately improve acceptance reasonable produce mix autocorrelation panel diagnostic diagnostic chain diagnostic derive non guide still produce constant occur chain sufficiently trace path sample demonstrate handle initialization parameter sample improvement include vertical dotted involve stage dotted dash top present true observation panel burn sample mmse posterior mmse estimate mmse grey predictive scatter plot pairwise parameter low density parameter triangular ht plot smooth static follow estimated coefficient static sample first follow six static accuracy estimate underlie give rao demonstrate set model recursively er static ip property condition derive modify integrate uncertainty marginalization utilize exist model mmse provide bind correspond matrix marginally static expression resort approximation filter new proposal obtain modify decomposition model assumption gaussian state observation expectation common filter filtering avoid marginal degeneracy hence time current chain particle exponential recursion classic filter particle kalman evaluation bayesian mmse parameter realization mse report average estimate provide methodology burn report per explore vary block calculation effect reflect observation level table increase generate parameter average realization block methodology mmse methodology estimate rmse estimate noise decrease estimator produce large rmse average estimate square block ideal choose integration respect parameterize ratio prior bayes test likely two datum bayes representative consider b b noise variance detailed population trajectory capacity challenge ambiguity actual true consider potentially capture form growth behaviour presence result switch realization bayes ambiguity flexible reason effect highlight challenge realistic observation count importance sampler highlight standard observation decrease exercise factor explore observation ratio focus factor capture vary degree factor process decrease order magnitude datum bayes level demonstrate several relate setting see majority strong significantly reduce reduction incorrect mark decrease perhaps distinguish confirm clear appropriate mix diagnostic parameter ensure study population establish world population abundance census select whereas observation add model strong figure plausible accord l post burn anneal component suboptimal model subsequently adjustment anneal mmse plot solid mmse demonstrate evidence presence comment mmse stable equilibrium go estimate ci capacity suggest fitting contrast aic suggest bic difference aic allow effect analyse time series south well study density dependence show surface process noise estimate surface comparison control equivalent parameter global likelihood local find bayes setting l burn suggest fitting model logistic five sampler diagnostic post require accurately weight proportion sampler mix mode enable assume plot estimate marginal posterior static estimate static factor bayes development sophisticated methodology particle enable robust jointly observation static parameter state model version several base dynamic cite place statistical selection realistic set accounting estimating latent consume design easily sophisticated algorithm population typically correlation
et typically efficient allow nonparametric result estimator major reproduce transaction adequate round li popular additive follow round end model discretize volatility filter series stochastic round round bid ask level book round market maker way past round previous book estimate although condition surely concrete specification g continue surely choose reason also easy less particle need cover pt round book market maker opinion frequency bid bid spread return transaction price produce particle assume support bid maker display axis transaction support log price real gray filter efficient price black maker see state unobserved continuous assumption develop consistency would asymptotic carry filter volatility efficient price specification example simple round ii py j jx x book long bid ask spread transaction observation assume transaction provide limit book bid ask level level really level k immediately transaction unknown past set price small price close book course guarantee realistic bid ask correspond ii lead small round transaction explanation book therefore stock heavily investigate situation contains implicitly trade large execute book large ask small market thick line support bid spread line width line width pt width width pt width width gray gray gray gray circle circle pt pt pt cycle cycle cycle cycle black black black maker available book bid ask satisfy either round view seem adequate need detail behavior market maker market automatically execute price make jump furthermore market maker efficient efficient choice reasonable parsimonious also demonstrate specify image book datum market maker example bid ask spread belong difficult mapping conditionally replace bad ask practical specification prefer finally partial compare round transaction round scheme show price price generate volatility deterministic automatically introduce log similar round bad superiority round section paragraph transaction bid bid evidence round stochastic round maker transaction volatility round include stochastic round end volatility hold formulate section security price aware fact trading present trading process nonlinear form distribute constitute slightly generalize give cause difficulty estimation nonlinear noise know lead log price transaction efficient particle localize estimator mention filter filter round noise localize volatility method volatility carlo et particle j approximation dirac particle filter use sampling technique know importance necessary j j conditional mention early particle filter distribution subsequently sample minimize particle normalize importance weight np suffer degeneracy particle particle I resample carry effective al resampling discuss particle specify show truncate weight model eq sample give difficult constant maximize sufficient modify towards kernel side recursive bt filter end lead I approximation result write new use standard algorithm variant every approximate describe step apply algorithm constant global old particle weight situation carefully investigate similar line eq furthermore turn prefer choice relate decay volatility locally smoothness either time dependent procedure run estimate estimation transaction volatility j line light gray line show red dark see plot new market transaction sequentially price price aspect back particle propagate em particle particle discussion end section merely tt realization process transaction dependence variance unit transaction per continuous volatility deep volatility use relation mention identify time fulfil fill change cf process accumulate obvious intensity estimation inverse eq recursion q estimate define fact bc var often require reason size classical almost change need line estimate j ci ci green gray line green gray log trading denote heuristic stepsize since ct j sign different curve become adaptive first volatility gray gray log trading volatility volatility I gray curve reveal typical volatility trading transaction visible trading intensity transaction decrease volatility trading worth green gray coincide use small period lag comment return lag reason noise small average filter calculation volatility investigate price high frequency hand still still return accomplish think happen consequence lag necessary new need carry correctly particular correctly get lag lag may quality lag however improve estimate presence pattern decrease begin create look estimator stay critical poor day block yield adjusted datum local back modification volatility two curve decomposition curve example analyze shape trading intensity trade volatility special care need affect mean volatility diag observation rescale unobserved negligible whole exactly weight proposition replace volatility finally rescale simple intensity pattern estimator recursion eq correspond bias decreasing propose empirically justify dependent practice line experience set approximately minimize approximate use adaptively select chen set volatility resulting imply computational minimal recursion recursion j critical univariate detailed description chen implement filter step normalize n nc particle resample easy line computationally complexity rarely proposal resample iteration particle quantity particle suffice proposal importance nontrivial truncate discuss extensively relevant reference problem approach sampling multivariate rectangular efficient propose quickly reasonable use g volatility magnitude close start simulate uniformly exclude effect start particle available run volatility consider volatility log volatility initial transaction obtain round volatility apply particle benchmark algorithm benchmark j start later term use price employ price plot plot also sufficient suitable addition large variance estimator volatility compare volatility efficient price generate dash upper lower realistic volatility transaction see transaction price obtain price nearest transaction day stock particle size analogous estimator eq noise function volatility outcome volatility plot low suboptimal describe calculate range plot estimator omit try use give surprisingly j respectively plot plot outperform benchmark general estimate bit nonparametric volatility mention filter covariance jump interval evidence stock price rare compound jump volatility index volatility pure jump acknowledge jump local volatility fine volatility transaction trading intensity volatility large would ds solely explain trading jump also occur although seems jump defer future red dark gray investigate jump take time return show volatility quickly recover jump see recover due adaptive stepsize modification stock transaction maker symbol c extract analysis line transaction trading period transaction transaction condition guide occur transaction constitute single transaction normally use trading inspection multiple transaction reveal transaction preserve decide datum transaction trading transaction correspond recursion recursion number trade fully time almost filter maker work maker match adjustment time transaction trade efficient price transaction price support filtering shown see skew consecutive zero uninformative filter transaction time volatility estimate maker round dark benchmark estimator gray gray middle plot transaction volatility show begin day vary almost constant advantageous practically almost experience feature transaction stock feature u end blue gray show round large realize volatility volatility come stochastic round obvious volatility price first indicator round preferable round volatility time display estimator calculate volatility transaction space transaction price c green plot trading duration stepsize minimize jt despite addition figure compare transaction estimate period occurrence volatility volatility new transaction efficient implementation filter contribution nonlinear bid ask bid price real treat filter third sequential type line estimation vary volatility make distinction volatility transaction model time total transaction volatility transaction turn transaction decrease begin day merely result trading intensity increase shape trading transaction technique model g drift noise likewise decomposition time volatility transaction volatility trading intensity course mathematical think hard achieve mathematically volatility optimal filter determine distribution price recursive em wang asymptotic recursive type present result properly rescale rao property attain estimator side datum even derive volatility grateful anonymous paper considerably express author reference em zhang presence review financial study volatility forecasting market noise manuscript particle partially journal normality asset journal finance induce security journal finance journal financial economic realize study design post j variance jump ti line latent datum journal statistical series chen estimation robust manuscript presence market finance stochastic central limit normalize increment presence round maximum incomplete journal e comparison resample particle international image analysis sequential method ed carlo practice journal machine research fan interface asymptotic power financial fluctuation nature journal graphical rectangular bivariate compute j student
choose replace member operation together produce member member enter fitness evolutionary maintain interact optimization prescribe number complete run provides expect represent formation rule query algorithm account voting relevance make programming follow implement htp initialize randomly determine query fitness initialize individual generation counter prescribed increment generation counter counter population otherwise increment population counter counter individual randomly winner member mutation increment counter count combine population pool member pool copy go count output number population query document relevant irrelevant wish query decision fitness represent either query metric take overfitte avoid benefit use es structured begin es system outline section es compete present benefit term corpus news remainder topic specific r relevance statement available adaptive none available topic description close histogram evaluation document balanced frequency bar set irrelevant variation topic topic large also relevant hundred irrelevant topic extreme r document role proportion well relevant topic provide evaluate efficacy htp implement software gate platform tool preprocesse software implementation es wikipedia wikipedia built publish al suitably integrate framework entity recognition carry field co algorithm toolbox al svm benchmark implement genetic population mutation h name population sub mutation depth maximum depth parameter terminal building concept could single query limit result wikipedia algorithm word comparison evaluate benefit es performance measure contribution integration wikipedia ask retrieval quantify effect wikipedia bag eliminate effect summarize token bag topic es improvement token interestingly compare result recall score well es precision measure query well relate ignore word token expression likely score es idea f es figure indicate topic semantic beneficial score precision splitting evaluation stem topic represent expression es rule tend complicate particular gate topic look death occur due mining first exactly difficult edge token topic favor improvement across htp wikipedia substantially percentage topic r r suggest achieve f previously essentially topic examine token gp precision topic es improve wikipedia concept appear performance es improvement stem ability high turn es strength include constraint narrow topic definition document characterize considerably overall token representation refer svm svm algorithms token es consistently benchmark term cause es rule appear quite relatively es token token svm algorithm es token token es token difference sake completeness token gp quick wikipedia document effect wikipedia es token comparison token vs svm token vs concept irrelevant substantially classical document information document purpose help user relevant concept token difficult retrieval suggest inherently feature beyond concept wikipedia fast technique human retrieval wikipedia database frequently require wikipedia construct query exist information exist framework improve genetic early es perform counterpart promise generic emphasis towards concept knowledge definition equip common world improve efficiency search direction query attempt wikipedia semantics present shift conventional token enhance information retrieval handle query evolutionary es learn co evolve evolutionary procedure base query evolutionary significant extensive study retrieval system system justify token query wikipedia indexing expert systems retrieval ir need significant knowledge knowledge basis expect semantic text close human extensive people concept unit dependency process extract semantic text collection semantic quite currently way document consume engineering trend develop resource semantic consider wikipedia world knowledge purpose user need term wikipedia concept instead query paradigm chen et incorporate semantics wikipedia principle user relevant document learn relevant imply wikipedia structure evolutionary boolean query integrate level information retrieval ir ir enable detect relevance concept identify concept relate contribute towards wikipedia concept base learn co evolve genetic gp call wikipedia semantics automate represent document bag paradigm focus evolutionary c ia development classical boolean ir broadly considerable demand learn run boolean platform restrict level leverage human note concept user want user search newly trade ask human reader economic due relationship appear benefit instead bag es corpus realistic news stream well fit conjunction evolutionary replace token wikipedia concepts precision considerable machine svm well robust furthermore compare produce es gp find query main contribution automate source wikipedia semantics es co evolutionary summarize key summarize document topic retrieve generate token decide perform token document analyze recognize interested matching token document concept token towards rather concept contain human level knowledge provide wikipedia semantics behind author utilize semantic efficacy concept document query search often available document mark entirely avoid relevance document query learn task contribute co evolve evolutionary training document document base single query multiple voting fit happen produce fitness objective space towards relevant irrelevant widely query produce fit token framework evaluate present concept framework evaluation result outperform idea wikipedia utilize briefly leverage wikipedia link development automate query particular genetic paradigm chen knowledge information provide way share despite limited engineering cost manually build resource keep resource know extensive limited need expensive recognize motivated research towards resource speak wikipedia thanks wikipedia rapidly maintain arrive daily wikipedia research use comprehensive review acknowledge et automate close reveal many similarity wikipedia article overall include hyper category wikipedia large al wikipedia solid rare mix recent wikipedia feature wikipedia considerably internal link concept bit closely instance wikipedia wikipedia article broader belong category unite broad category title article addition link represent concept refer concept article connect article bank page semantic wikipedia treat wikipedia concept model query notation wikipedia wikipedia representative certain concept follow recognition several resolve present trivial commonly automatic es question concern concept pair idea wikipedia provide need semantic evaluating measure corpus around quite wikipedia background take technique exist well wikipedia soon measure text wikipedia recent proposal propose internal link wikipedia approach correlation adopted essentially google wikipedia link define uniquely identify wikipedia concept gram concept principle article lot link likely highly link percent financial bank article course article thereby essential reasonably reliable way measure concept far semantic conventional setup concept perhaps relevant ask concept purpose link wikipedia wikipedia term collection wikipedia detect far rather intend document query allow mask concept illustrate document receive large general economic car model still paragraph trade car prototype prevent strongly link evaluate automate drive query need give learn help definition recall grow become solution formulation roughly weight query relevance feedback system feedback remove adjust system document represent current surface examine step learn become range genetic es query seek pick one process step see figure generation topic concern boolean irrelevant I algorithm es account appear one strongly gp es retrieval give evaluate incoming framework task match es pass name entity express term name entity matching rule rule module whether match currently es concept document rule return filter find match es return retrieve document terminate datum use match step ht rough overview closely retrieval wikipedia document continue rule evaluate detail document identify build al et recognize term act concept extend category wikipedia name entity concept name entity modification consider name entity general discuss bank name explicitly mention say sufficient concept exact concept name clearly account specify es document pair collection name entity wikipedia document space document document set name entity concept find wikipedia document document name entity recognition usual first wikipedia separation name concept sequence detect link correspond step recognition fields crf classifier model information construction wikipedia name examine pick recognize rule composition es provide picture es voting query vote weight represent relevance presentation es queries es rule evaluate individual voting system generate es rule summarize es discuss system begin boolean distinguish concept query hereafter ordinary boolean query consist part query utilize wikipedia match document expression wikipedia concept syntactic second document replace query dr k concept purpose threshold threshold function name entity sensitivity purpose name entity threshold name entity definition query request document concern receive lift repeat mr document car car document query point evaluation entity problematic sensitivity consider whether examine recognize strongly relate therefore concept equal acceptance provide recognize name entity reasonably strict level name entity concept high link mixing would serious able entity wikipedia tool due high deduce irrelevant ready fitness measure quality individual voting output fitness query admissible fitness evaluation precision set respectively denote define query benefit resolve document fitness evaluate quality deal contribute voting let finite collection query vote fitness respect vote base relevance leave research voting relevance several alternative query consider relevant irrelevant weighted take account help overfitte document set discussion es es denote admissible boolean query form wikipedia voting evaluate document relevance rule es query es provide query query learn search possible query optimization admissible es maximize collection document let denote
eigenvector eigenvalue corresponding eigenvalue come low neighborhood propose manifold algorithm explicit name preserve nonlinear precede find linear reconstruction reconstruct near weight following represent reconstruct neighbor form non entry th one entry preserve achieve problem show come simply section get matrix eigen consume new sample generating take operation exponentially extremely consume simplify remove hadamard reduce simplified summarize output summarize kernel method vary computational inner product obvious complexity explain linear may taylor nonlinear polynomial assumption approximation mapping test surface embed come space near underlying manifold image person intrinsic rotation sample nearest neighbor right sample randomly recover underlie satisfactory cost support computational image handwritten digit resolution intrinsic freedom sample neighbor train left column respectively randomly successfully underlie shown fig explicit manifold representation apply generic furthermore dimensionality reduction technique explicit preserve locality geometry provide sample simply test real effectiveness algorithm stand bar generate result stand datum versus test f f stand versus testing sample plot dot circle learn learning dot red circle versus number theorem remark zhang wang zhang state mathematics chinese china mail zhang ac cn field science drawback application practical linearity may restrictive explicit dimensional representation far locally derive name experimental result effective nonlinear geometry sample previous work manifold nonlinear draw interest manifold basic assumption sample smooth embed ambient around object high equal pixel aim intrinsic freedom input high sample draw linear laplacian le dm tangent alignment riemannian great success low meanwhile manifold face compressed expression hyper spectral shape appearance main manifold representation output embed procedure contain repeatedly extremely consume sequentially limit manifold linear projection assume datum low dimensional representation locality projection neighborhood preserve projection graph although linearity restrictive mapping manifold learning utilize space mapping implicit depend extremely polynomial datum mapping learning method implicit sample embed mapping specific kernel find projection high dimensional representation space clearly mapping handle sample lie meanwhile experiment combine nonlinear manifold paper concentrate manifold polynomial world illustrate validity review detail demonstrate conclusion exist algorithm linear nonlinear manifold notation use vector normal capital letter represent letter datum euclidean sample dimensional matrix norm vector preserve manifold cast two global preserve locally preserve pairwise distance unfold local maximize alignment preserve dimensional le preserve adjacency mapping preserve geodesic representation extend preserve representation assume representation denote project linear coordinate basis locality projection provide mapping laplacian le training le train dimensional representation preserve optimization algebraic diagonal whose entry eigenvalue solution small projection provide sample high find representation independently projection compute linear training weight solve problem ij neighbor eigenvalue projection high easily map locality preserve neighborhood preserve respectively projection unlike lead easy eigenvector reader besides manifold extension representation unseen manifold base common method manifold technique employ come et unify extend le eigenfunction dependent dependent implicitly le
write definition univariate cdf define many marginal application learn arbitrarily random variable marginal varie deviation unobserve deviation process monotonic parametrize latent shorthand standard univariate multivariate equivalently express model model learn subtle assume time period period condition know normally common assumption relaxed central ultimately wish hyperparameter transform kronecker delta maximize function intractable circumstance laplace integrate make find marginal relate g j kt approximation separately eq deviation goal treatment logarithm unnormalize laplace use taylor maximize entry problematic expectation iterate order cholesky laplace near numerical instability small finding laplace likelihood numerically stable maximum definite since decrease newton size always furthermore condition eigenvalue let newton update expression approximate likelihood evaluate numerically conjugate initialize vary make find approximation integrating give take function iteration newton method converge mcmc later make gaussian elliptical sampling extremely effective posterior correlate update element axis univariate prediction gaussian eq weight exact draw gp take non real deviation monotonic positive infinitely towards practice small extremely certain input wish literature assessment volatility variance follow observation proxy truth g rank compete derive use simulate call make ahead forecast observe make forecast volatility predict time safe comparing use panel true volatility laplace exp laplace historical volatility table result forecast panel accurately dependencie manually decrease replicate behaviour quickly exponential tend peak volatility extremely reconstruct forecast dramatically exponential region low also peak suffer mostly change daily dm great become assess refer return calculate dm day window return day volatility trading day historical mse historical unlike historical assessed see ahead forecast forecast operate suited datum vary exchange learn therefore convenient implement copula develop volatility outperform separate dependency distribution arguably far gaussian marginally cdf marginal probability density learn pdf show parameter stationary attractive advantage rich brownian mat ern periodic periodic learn function volatility copula rapidly become popular copula copula arbitrarily encourage copula bring machine historical exp la mcmc prediction historical dash la ahead volatility forecast dash blue learn marginal thank ar grant definition cm ex copula describe random distribution volatility copula process predict deviation inference laplace approximation monte carlo alternative comparable outperform financial unlike miss incorporate covariate rich structure measure measurement minute minute learn separate distribution separate marginal copula copula recently financial copula role play statistic open copula correlation compare describe volatility bayesian distribution dependency sequence leave get away important index economic economic time volatility arguably volatility index outperform discuss dependency distribution intuitively n cumulative cdf uniform random transform formally cdf nu intuition let exist uniquely determined f copula
unit state conditional conditional expectation dimensional denote simplex segment triangle take policy belong stationary policy stop decision terminate alarm choose variance markov variance penalty stop uncertainty alarm represent change alarm let alarm stop pick state alarm alarm state constant informally multiplier seek cost ii allow choice cost take change state time denote constant depict delay delay e sensor maker need thereby denote speak affect penalty cost incur eq due therefore reduce stop cost belief measurable endowed product algebra expectation ce adapt sequence eq economic discount non guarantee time delay easily delay kolmogorov namely consider solution bellman programming equation amenable c compatible compatibility decrease respect course stop coordinate albeit q bellman program q empty value methodology comment iteration confusion iteration bellman sup banach generate sup metric bellman equation belief practical indeed nonlinearity formulation although value iteration exploit structure switching devise algorithm solve cost stop problem penalty cost possibly observation ph sec namely sec discuss implication main sec approximation stochastic compute formulate ph distribution indistinguishable probability satisfy eq q choice generality translation notation order lattice denote penalty threshold theorem show stop stop choose equally alarm become delay cost assumption discuss sec observation appendix non maker assumption ph detection cost ph ex line threshold switch curve individually connect policy union nonempty threshold line ii distribute threshold trivially appendix intuition behind sec give delay cost stop continue decision maker false alarm f p e programming solver delay stop cost hold ph penalty say e stopping size ph matrix observation describe sec even characterized detection ph stop geometrically coincide total geometric say classical continue nonlinear alarm false alarm kolmogorov additional equivalent geometric threshold kolmogorov criterion trivially convexity threshold optimal case sufficient tp pp maker interior boundary determine theorem determine portion curve lie interior simplex portion lie simplex comprise conceptually eliminate ensure always following sufficient transition element likelihood satisfy subsequent state follow straightforwardly belief compute curve empty discuss sec outline ex maker stop cost assumption decrease decrease condition refer lattice programming policy discussion decrease submodular establish condition ex since decrease ex ph change variance construction ex delay monotone submodular ex preserve update increase iff detection book exponential poisson etc preserve theorem show iff increase sufficient tp order kernel detail satisfied class transition necessary comprise meta involve show ii decrease spirit convexity require monotone update belief stochastic see submodular require line chain restrictive entire line policy belief prove simplex cover line threshold curve illustrate theorem line connect simplex say curve intersect intersect say convex recall state triangle insight visualize say monotone almost almost everywhere decision trivial include belief decision though region mention give region policy increase curve intersect ph curve intersect theorem lie say lot penalty fig satisfy monotone go optimal condition happen hold numerical ex construct region assumption subsection assume ex hold estimating threshold need essential parametrize optimal apply optimal approximation threshold simplex attractive condition necessity linear linear policy parameter hyperplane parametrize linear threshold approximation curve ex hold linear set mean belief states line iff ii iff theorem solution constrained remark constraint necessary increase line define increase threshold yield clearly threshold conclusion slope become intercept lie imply fig slope lie fig show decrease segment start region requirement increase x estimate resort denote aim linearly constrain evaluated I optimization convert parametrization trivially constraint equivalent unconstraine simultaneous perturbation threshold ex social switching initial iteration coefficient pick evaluate number optima try initial condition dimension local one straightforward markovian give score process reinforcement applicable thereby yield threshold policy likelihood completely specify hold reinforcement tracking stochastic observation vary second deal threshold curve transition start state finally jump course since delay delay delay give visit process reach convenience penalty choose expect stop cost cost decision stop tp e sec delay cost conclusion observation degenerate ph time jump ph ph distribute period jump continue model distribute denote state start jump ph jump sec ph start delay decrease cost conclusion maker design satisfy maker solver assume state indistinguishable obviously unable transition matrix long tp follow penalty region deal exponential geometric change consider exponential penalty ph time formulation involve control motivation show risk stochastic state risk ph distribute show switching interpret false alarm delay cost control parameter let geometric cx e cx cx identical delay delay get delay cost additional rest time detection ph denote threshold function threshold nonconvex stop would think continue belief belief policy public continue stop global stop belief local main reason behavior cause update concave necessarily function concave explicitly action irrespective irrespective therefore social action nothing learn take belief comment interval modification composition filter localize interval behind theorem optimum formulation stop threshold optimal action constrain optimum formulation motivate optimum due sequential aid learn act optimize sec ignore cascade rule become denote action adapt sigma detect state second term involve pick action accord rule specifie agent chooses mode immediate cost pick reveal thereby social learning stop state equivalently terminology observation choose belief pa py remains incur equivalent final term define sequential seek policy achieve tradeoff incur act analogy characterize threshold require ce ce ce xy iy ce iy ce ce ex imply cost decrease costly decrease intuitively make accurate private ex satisfie structural switching union social threshold pick pricing implementation view protocol need individual agent store example finally convex stop model markov jump sec jump target evolve observation jump decision stop evolution belief control mention sec bound achievable main theorem ph agent act sequential index belief state choose mode depend mode obtain distribution mode observation obtain mode dependent scheduling accurate cost mode tradeoff observation mode model view confusion communication channel choose mode incur choose since mode optimal overall cost incur mode agent affect subsequent agent problem determine intractable however programming let belief pick low policy appendix usefulness stem rigorous low coincide region incur achievable trivial imply coincide apply dynamical concavity particular proof appendix comprise first concavity trivial cost apply detection scheduling distinct matrix markov question transition matrix phase large optimal type pose problem belief explicit transition matrix I explicit large incur cost optimal see distribution prove dynamic making verify tp ii iid uncertainty say total example change transition small ph model matrix ph change tp show transition order namely theorem transition model conjecture iid tp kolmogorov reason replace order paper assumption appendix illustrate structural theorem detection ph operational form dimensional unit grid length fig satisfie individually connect theorem say optimal stopping assumption hold recall condition fig satisfy condition appendix stop long monotone policy paper stop empty simplex c satisfy stop feasible choice matlab choose assumption illustrate ph probability transition geometric region mark structural distribute variance prove existence switch appendix give threshold several consider social scheduling penalty sensitive stochastic switching order simplex structural belong hold robustness since still useful various detector paper prove detection monotonically clear compare belief ratio order specialized order restrict simplex preserve definition subsequently order belief vector greater replace stochastically dominate I dimensional vector iff coincide state space partially always state simplex state lie segment connect comprise form order great respect chain e use great element comparable l variate lattice operator tp ordering variate univariate say tp p scalar statement row submodular chain e submodular argument decision f assumption major stochastic tp generic iff hold ex sec ex sec ex sec decrease assumption ex ex sec sec sec sec part prove condition decrease family first dominate belief parametrize light follow straightforwardly negative yield condition nothing decrease suffice choose yield ex ii suffice ex mathematical induction v decrease form ex belief e show omit condition respectively monotone increase respectively ii equivalent sec inequality e f iii theorem sufficient decrease imply submodular show induction decrease submodular imply pointwise limit establish submodular policy increase key characterization switching segment connect lemma segment moving segment always threshold note simplex cover consider region segment region intersect region action requirement case convex assume nothing prove line leave intersect action optimal consider iii follow first follow lemma comprise concave uniformly concave belief construct piecewise easily piecewise function compose concave function via concavity iteration fix arbitrary vi converge choice see piecewise see positively I k concave composition concave since linear concave concave concave follow preserve complete step concavity imply concave intuitively segment converge lemma argument repeat demonstrate convexity inequality iff e iff iff difference compare update include continue necessarily concave concave verify dynamic value decrease straightforwardly since concave concavity straightforwardly piecewise interval abuse recall straightforwardly sufficient theorem statement straightforwardly statement return yield since linear lemma interval I prove directly theorem ac c argument union region connect establish condition therefore concave jensen eq decrease introduce proof return rest q since follow take first recall monotone theorem optimal type threshold switch curve distribution order stochastic gradient optimal linear curve consider change threshold switch curve arise detection sensitive penalty stop time agent scheduling change make low optimal varie change impose monotone ratio exponential delay lattice programming making monitor finance deterministic goal delay subject alarm see formulation variable random distribution detection involve need tradeoff alarm frequency delay change model geometric change realize chain policy function belief px u u exist generalization framework dependent markov main generalization type goal lattice programming threshold policy time phase change ph ph use discrete change form find ph uniformly describe ph ph ph states multidimensional generalization comprise alarm variance penalty stop formulate quadratic existence quasi prove exist optimal threshold distribute ordered detection characterize policy stochastic lattice policy govern belief ph design policy threshold
security much want player classical fp good fp strategy utility function eq response give max nash point good mapping static subsection fp fp version fp mix fp calculate player basically exponential time player weight security game assumption payoff decrease attack payoff attack attack decrease attack payoff htp update opponent completely play accord strategy fp employ algorithm evolution fp equations evolution empirical static game subsection player action game admit nash ht response monotonicity fix unique scalar independent similarly pair completely specify mapping suffice write mapping mapping detail similarly transformation mapping constitute static game respect independent strictly increase distinct nash equilibria coincide generality assume strictly two curve equilibrium static proposition fp weight action estimate fp fp calculate writing estimate fp dynamic equation see equation discrete invariant system fix examine stability linearize see jacobian similarly local stability eigenvalue jacobian estimate frequency frequency converge nash equilibrium good response nash empirical frequency run examine hereafter fp invariant frequency size decrease algorithm step either keep fix base previous window initial minimum window frequency opponent strategy randomly play action window decrease compare nash static threshold rhs frequency result dynamic fp figure frequency limitation ne converge thus worth frequency ne fp fluctuation process graph limitation ne fp entropy choose empirical frequency fp fp fp however possible incorporate originally fp process converge adaptive adaptive fp higher less opponent play nash equilibrium dynamic update adaptively player converge fast fp exist yet research extension conjecture department electrical laboratory university st il edu edu security view nonzero sum game play evolution game play opponent vary alternative scheme update examine dynamic play stability equilibrium player theory recently tool security payoff player equilibrium play gain minimize player payoff game play player learn opponent fp current either fast equilibrium strategy motivated examine tool applicable game extensively paper survey
easy obtain bin unfolding calculate especially nuclear consume often calculate element source instability solve difficulty unfold formulate paper place propose unfold unfold training priori contain calculate configuration grid approximation transformation linear demonstrate possibility unfold whole transformation distribution create error component calculate bias bias unfold unfold validate wide nuclear model monte unfold true experimental simultaneous identification priori previous carlo training transformation unfold minimal bias validate unfold deconvolution robustness boost kf experimentally measure distribution differ true particular unfold unfold approach solve priori unfold unfold idea develop identifying solve unfold measure create sample calculate approximation minimize bias restriction shape linearization multidimensional unfold paper organize follow solve unfold formal method unfold unfold system basic unfold example elsewhere unfold run unfold method training investigation reveal bias unfold confirm statistical case identification transform true histogram histogram residual linear majority physics approximate distribution identification determine transform physical experimentally simulation use identification impulse impulse bin impulse input row contain histogram reconstruct output matrix q statistical solution type instability priori identification use impulse use experimental present write content reconstruct output variance statistical reconstruct generate formally square q calculation column reconstruct parameterize use row row copy example sample coincide impulse combine fs include new eq transformation element exclude satisfy maximum c calculation row row rather reconstruct bin row bin criterion relate minimize unfold minimum ellipsoid possibility improve introduce criterion distribution training goodness goodness reconstruct experimentally achieve training satisfy test statistic reconstruct distribution threshold reconstruct threshold level error run unfold boost training sample involve independent realization histogram description illustration elsewhere take parameter interval cm experimentally measure distribution detector acceptance resolution function histogram distribution bin distribution histogram choose bin previous identification comprise parameter simulate represent used identification histogram calculation great element transformation h cm xx basically reflect
implicit define discover behind attack vector briefly comprise sequence current adjust match output profile structure detail subsection form attack token attack finite hmm attack enough describe attack entire view production assign attack represent non terminal path possible production product transition emission capturing obtain implicit generating attack express posteriori maximized dirichlet token sequence calculate emission token bayes balance represent attack structure generalization relation build token come model structure structure merge merge maximum posteriori posteriori merge current merged model repeat token process build attack profile attack profile describe parameter token emission record token attack follow generate attack simulate attack evaluate possible phase start token subsequently token seek viterbi find token hmm purpose generation viterbi attack attack generator able attack attack embed attack attack technique code experiment attack concept subsection metric discover study finally measure attack identify web neither space attack state attack successful effective attack question effectiveness attack configuration environment configuration generate attack attack third attack web allow attack attack application manually check attack report attack really dataset collect collection totally attack several obvious characteristic symbol favor totally token unique token sequence list elementary high school school source attack list generate attack fp worth knowledge attack effective aspect attack interesting attack table attack attack http com prefer double attribute specification attribute separate attack format recognize internet web site attribute attack width height tag attack serious attack exploit body show attack tag second show page microsoft internet google think could social attack attack digital characteristic end tag indicator style attack body text reader third attack tag indicator attack string well third attack form original form block enter separate denote window attack attack hand ever test public rule attack attack detect character attack false still attack expert phase technique include box hybrid strength analysis variable code without however result false positive box input generally tool equip lot test user effective test serious utilize response server attack ignore hand attack box attack leverage usage modify attack back attack generate attack hybrid static verify identify test automate attack work maintain attack public source expert specification attack generate source attack identify web manner attack take attack hide structure thus generation mechanism attack experiment technique effectively attack help provide attack consideration web acknowledgement security center national acknowledge comment suggestion web department engineering national university institute science mail edu tw suffer site attack attack study attack art potential web application attack aim generate attack automatic generation attack hide attack implicit bayes determine model generalizing automatically attack practical attack ability testing identify helpful collect public attack web application security project already web application page attack embed specification tag event situation attack attack body code invoke technique beyond attack device like efficient attack attack page attribute double attack double critical character introduce attack attack vector embed page previous study focus create attack generate path equip attack exploit element attack structural learning mechanism attack vector generating attack present generate attack public attack attack structural module build generalize generate viterbi capability tool attack element structure attack internal string function attack many possible way invoke specification attack attack generation different web aim implicit present attack thereby public handle structure vector attack contribution follow automatically analysis attack ability helpful approach generating attack section conclusion structure generate attack mechanism attack attack attack generator phase phase attack generating phase attack attempt extract attack attack profile relation attack generation mutation attack token attack generator attack attack attack automatic attack attack overview structural depicted profile relative attack preprocesse module decode identification
introduction suitable latent variable present augmentation posterior standard extension model allow home tie em sample popular generalization section apply competition statistically comparison associate datum seek admit enjoy pair distribution rate low time allow pair arrival instead simple shape inverse information augmentation algorithm proceed similarly summarize introduce variable complete assign prior eq log proceed independent first map coincide simple augmentation z conditional augmentation proceed comparison home home disadvantage times home home times play home home total use maximize algorithm flat mm gibbs iteration allow comparison log introduce latent log tie improper maximize update perform win team recently likelihood parameter otherwise update q variable proceed iteration involve individual individual introduce follow ranking position event individual receive augmentation augmentation gibbs sampler model q write rescale dirichlet improve mix add normalize current drastically value log follow update rely parameter suffer augmentation slow arise hyperparameter assigning interesting variable implement conditional give propose accept relate random suppose capture formalize sequence sufficient graph log give require sake brevity augmentation whose density argument feature vector element importance object gibbs derive consider categorical multinomial logit em posterior rescaling influence ensure henceforth algorithm datum study compare property gibbs sampler slightly latter propose update iteration ranking dataset gibb run lag four confidence reasonably sample consider poorly sample car auto united involve find place need predict one base map etc test figure initialize sampler flat improper sample run burn detailed identifiable rating four place effectively ml mmse minimum square together deviation ten h small rating strong restrict international adopt model historical consideration system game either section predict outcome number walk bayesian sampler outcome different hyperparameter em sampler sampler iteration report benefit full report autocorrelation associate large display mix generalization arise numerous special straightforwardly algorithm elegant mix experimentally outperform grateful comment axiom conjecture remark em france mm pour du de dans les mod de inf est l carlo es des de metropolis es des maximization des de pour inf de
associated dpp corollary establishes induce rl convergence rl initial policy rule dpp rl unlike incremental rl involve decay know incremental rl learn step bad choice lead dpp rl seem suffer dpp rule rule dpp dpp rl fast rate dpp basis preference linear column vector preference nan nan case dpp operator project nan minimize expectation equation infeasible estimate q draw also nan nan rx nx x fitting least present dpp operator h n na n kx aa kx x kx aa nan n rx nx nan nan empirically examine property dpp rl algorithm several discrete action iteration vi limit per replacement source algorithm mdps benchmark interior proportional text problem consist cx arrange end chain state nx px na px k l k transition interior select state correspond formally state cx nx probability transition end interior associate reach soon consider stochastic state cx k arrange possible action state state otherwise automatically state intermediate reward upon take wrong proportional difficult goal transition transition mdp leave cx nx center assign remain reward move state long avoid right corner avoid state rl compare like dpp rl action pair iteration vi vi batch learn whole loss iteration consistent discount optimal value iteration polynomial linear otherwise asymptotically dpp rl fix fit make numerically variant subsequently infinite horizon discount measure accumulate product variable current depending keep optimal accumulate replace unique map space radial basis span fix definition next happen replace step compare select unlike action induce action soft policy algorithm implement matlab execute hardware specification average begin run interval area deviation start uniformly figure error observe reach near optimal error estimation solution comparison variance suggest complexity remarkably use increase conclude reduce rely incremental ac actor guide policy ac extend ac actor unbiased estimate incremental rl variant optimistic value concern presence incremental incremental rl problem relative restrict bellman instead approach bellman likewise formalism return fashion low kl another rely relative policy difference actor dpp inverse dpp optimal solution convergence dpp convergence policy programming dpp compute policy mdps theoretically prove performance loss bound dpp express supremum accumulate oppose free rl dpp rl estimate policy prove dpp rl numerically mdps show dpp rl unlike rl therefore suffer decay stochastic policy exploration pac mdp policy different kind dpp rely ac action approximate rely error natural search kind performance bind rely approximate dpp value dpp research pac dpp previous theoretical fit iteration analysis action rule computing idea monte soft lagrangian lagrangian function respect x hilbert science hilbert department definition nc novel iteration policy dpp infinite horizon process performance loss presence bound accumulate error oppose norm policy dpp cause monte compare variant dpp dpp rl reinforcement markov carlo dp action bellman systems tractable value monte programming iteration real world derive guarantee discount w r action policy approximation however supremum variance accumulate consider asymptotically error mathematically justify dynamic dpp prove dpp asymptotic dpp accumulate suggest dependency naturally incremental dpp update error article notation dpp investigate property dpp carlo convergent rl rl estimate optimal present replacement review mdps reinforcement rl standard notation euclidean norm norm ny ny discount mdp denote discount transition next upon state real value reward joint space mdp assume value immediate reward cx determine action call action independent last stationary policy policy action denote discount reward associate therefore discount define notice nan bellman equation eq likewise cx nx likewise contraction supremum norm factor value define linear define na qx cx bellman bellman operator eq derive bellman add entropy induce double loop reduce double dpp purpose motivate characterization theoretically investigate iteration asymptotic dpp new nx penalty term policy expect x bellman cx modify maximize reward policy baseline policy form cx x x cx respectively satisfy function compute maximize interested quantify mdp idea improve policy towards base newly compute regard repeat lead outer loop inner loop consist follow follow derive dpp update preference cx deduce na dpp analytically tractable dpp equation soft max na entropy obtain converge analysis somewhat simpler replace deduce dpp preference soft policy iteration gradually move policy greedy policy finally dpp result solve dpp regardless always incremental whereas double policy policy difference double loop c x kx ax x x prove dpp dpp uniformly cx nx policy dpp soft policy soft greedy policy immediate consequence q induce converge mild mdp unique ca na cx dpp appendix dpp problem generalize dpp problem technique dpp
mean refer method normally involve rejection well old may preferable processor implement fast number uniform generator normal require cycle good generalise require cycle four performance wide new pseudo generators generators stream pseudo evaluation elementary log normally number another normally thus pool normally another normally distribute multiplication inner loop implementation processor processor normally proportional scalar speed comparable fast uniform generator processor aspect processor discuss conclusion idea generator keep pool pseudo form uniformity counting occur bin time seed etc random number sample fourth repeat observed fourth moment sometimes significantly confidence explanation value pool consider pool q correlation successive return reduce eliminate pseudo generator idea processor generalise generator cycle however satisfactory property provide distinct describe appropriately orthogonal transformation satisfactory pool subject inner loop streams pseudo generate processor require thank I also discussion regard generalise research implementation generator propose new class pseudo generator generator require stream loop essentially matrix multiplication
average split conclusion draw figure go optimal well simple space linear wrong feature recursive look kernel specify boost automatic kernel capability linear feature enough produce relax linearity go automatic parameter vertical axis performance ap auc top task c essentially argue put emphasis feature space wrong automatic formula important empirically high high level alone reject hard development abuse notation usual classification training classify long regardless still implicit feature space kernel graph remain describe deep machine conduct commonly often enough apply trick implicit repeat deep recursive correction capability report deep architecture literature science engineering research remark example inspire grow interest datum study classification focus linear classifier kernel produce consider classifier space call architecture machine deep diffusion notation subset unknown respectively study predict protein interaction protein know associate g predict activity product assess idea apply call machine focus end really reason study conventional kernel suit left limitation alone else general instead independent standard problem wide variety simple neighbor near centroid sophisticated machine simplicity predict neighbor distance due machine learn label machine sparse solve cause observation shall simple kernel machine use analyze graph theoretic need adjacent notion often diffusion go roughly similarity kernel short paths adjacency kernel kernel matrix eigenvector compute j mi net respectively average class subscript summation general machine shall mostly kernel easily procedure determine coefficient kernel coefficient emphasize framework svms key machine regard space centroid imply base diffusion intuitively appeal way neighbor nearest centroid machine feature svms possible nothing prevent flexible use kernel kernel density serve distance write radial therefore estimate subscript notation emphasize estimate predict decision kernel whereas summarize say feature use kernel linear sufficient relax linearity choose nonlinear distance metric rise kernel use kernel update put discuss however space machine linearity construct via imply yet except j machine generate recursive deep machine level one kernel present general focused base one certainly base svms appear return return else accord return certainly go reasonable heuristic pair heuristic usa serious accounting attention energy email make publicly unique email account send email account status create email account never email another label status unbalance balanced task l svm collaborative social interaction law ask regard law issue status flexible third work e work rather adjacency friend status two management class highly unbalanced one
model smooth part correspond probability effectively combine generally suffer two drawback computational especially two kernel double tree accelerate drawback invariant potentially since ideally throughout mixture b b table synthetic dataset dataset mixture variable time old perform increase dataset obtain conditional train log double fold kernel subset fold compare double perform mixture cover note use wishart part secondly require wide possible almost though disadvantage bad widely dominate cover small dataset mainly robot less evenly match initially bad kernel high perform mostly case asymptotic tree partition approach markov model fundamental close incremental conditional density double structure result well kernel ability incremental using could obtain invariant would problem conditioning variable cover paper possible tree extremely structure application lattice structure remain tractable cover create distribution cover maintain spirit p thank point gr extremely thank go anonymous provide circle draw mm enable incremental parametric rely upon create sequence cover maintain within remain tractable specify cover fundamental approach incremental conditional cover approach dominate fast tractable incremental bayesian random walk simple variable construction tractable flexibility example estimation obtain conditional informally evidence wish denote borel idea tackle cover sequence cover contain sequence set combine obtain variable appropriately allow state measurable respect algebra contain sequence sequence cover subset partition tree cover refinement alphabet subset partition variable unit generalise construction shall focus conditional estimation structure measure index context context context cover countable sequence cover compose local index specifie conditional though local efficient whereby stage firstly sample derive sequence construct two bernoulli draw rely structure observation random cover stage stop observation context contexts context bound proof geometric markov estimator require walk appropriately markov start refinement let fact k specifically stop parameter contain prior conditional cover tree easily applicable start correspond part tree probability multiple context classical estimator secondly straightforwardly extend density via walk perform thing model context tractable similar nevertheless trivially apart separately ratio hand incorporate estimation context context estimation probability proceed method tree structure tree estimation maintain tree suggest finite set method double
multiple infinitely matter thresholding converge j regression estimator estimate unless specify satisfie mention huber poorly outli detection never outlier leverage reject outlier little shrinkage nonconvex cutoff yield outlier design robust penalty loss tune illustrate threshold follow soft thresholding px x px q threshold j j cost regression qr cost maintain dense give ii outli wavelet problem wavelet regularize step simplified thresholding shrinkage gram therefore recovery reveal outli easy involve operation inversion observation outlier difficult desirable robust I mild ignore clean contamination tune via bic relate suggest estimate square define I estimate known multiplicative help bad iteratively update method indicate characterize observation observation case correspond leverage change outlier leverage combination decrease stands approximately half encounter take criterion second narrow range counter smoothing choose neighborhood determine maxima spline way counter carry way highly outlier affine regression without intercept include outli tune compound procedure behave poorly report library refer provide default point experiment resample py procedure outperform mm term robustness theoretically r robust r version library apart cutoff outlier fully situation time report identification measure fraction label outli simulation outli detection serious cause latter lose ideally simulation table good similar co ht co co mm mm library nevertheless mm estimator identification outlier outlier situation experiment behave outli work unfortunately four low bad large dominate significance mc pair replicate numerator outlier minus large versus acceptable tradeoff cause scale identifie also apply simultaneous outli surprisingly soft thresholding fail challenge sparsity elastic net net severe adopt thresholding ridge refer qx thresholding portion remove influence remove partially extent control helpful shrinkage play bring challenge coefficient choose component j candidate run predictor getting fit later marginal fdr sis run choose selection outli detection concern solution concentrate predictor derivative nm nm nm outside range analyze fitting robust model also screen via proportional ridge ridge ordinary parameter minimize combination rich reasonable experience improve prediction c r satisfie joint kkt equation estimate nc q exactly huber joint estimation two yield huber moderate improvement replace claim argument choice characterization actually norm penalty huber completely algorithm design give thresholde p ft u monotone u ft lemma corollary address university email department stanford university stanford stanford partially nsf grant dms dms grateful anonymous helpful remark literature plus one apply criterion fail correspond soft thresholding identify hard connection bic outlier various iteratively reweighte least square cost avoid extend keyword sparsity analysis refer one routine often go serious selection ordinary least ol outlier regression robust p commonly look well residual detect leverage well way outli estimate base leave value suggest observation outli threshold reasonable n I remove one look residual method fail phenomenon th remaining outli outlier may mask go outlier one serious take observation parameter trivial meaningful thresholding denote exception note regression run good current method preliminary regression point surveys robust regression convex section soft criterion outlier section iteratively reweighte investigate discuss problem detection extend technique dimensional conclusion identification plus goal outlier suppose sparse great paper outlier penalize convex suggest alternate ol estimate soft match detect outlier fit regression soft soft work minimax optimal predictor residual residual probability conservative outlier prefer tune initialize must use discussion initial pseudo outline soft defer converge even section lar shall generalize nonconvex penalize outlier moderate fail remove illustration identify true matter plot soft relevant choose sure
system solution give ill apply symmetric definite subject convex program compatible although submatrix unique kkt length ill solve always condition alternatively monotonically eliminate also regularize equivalent define block limit construction dependent general method incomplete cholesky method return method access might improve condition solve dominant magnitude dominant choosing scale diagonal side column unit see converge condition incomplete cholesky preserve ic ic ic simple cholesky infinity expect small exclude enhance accuracy eigenvector condition small although reasonably accurate system lemma computation machine ghz intel gb ram solver system decrease even desirable rule cause termination accurate similar ls approximate illustrate compatible linear ls v effect round plot residual show terminate directly compute differ monotonically near iteration smaller excellent reach ill condition end mean reach satisfactory claim throughout therefore graph compute throughout iteration condition last residual differ eventually even bad reach soon iteration figure system involve author al retain iterate estimate exceed readily hermitian matrix precision recurrence formulate approximation method numerically future research mention source available acknowledgement van thank support grateful anonymous suggestion manuscript v e bb converge l matlab example c cg solve system singular least square cg solution understanding motivate design compute qr rotation condition system rule residual residual matrix f f iterative l directly complex hermitian ab pose aa ill condition solve system return solver type reliably compatible return solution ill condition illustrate system discretize value problem involve error toeplitz l letter denote cosine angle th unit letter vector letter denote quantity symbol shorthand l review section estimate process arithmetic stop unlikely define ab following keep change scalar shift becomes shift inverse rank might exactly strict singular see subproblem expand qr form later solution substitution obtain nothing else place choose solve compatible singular singular handle describe qr factorization full effect later k tend compatible singular decrease cf r bp ap ap x x unique ar ar v ls solution solution require useful recurrence consider framework relation norm note version take negligible v k ar orthogonal range phase ratio criterion solver large form van round plane good norm keep decrease compute become increase want compatible eventually even long focus involve triangular solve triangular solve final store shall propose consecutive qr column nonnegative element demonstrate extreme important main purpose ensure develop extra approach decomposition obtain process different round see retain perhaps convergence involve product idea understand right additional help show sort red circle plot circle subproblem eq figure remainder solve last update recurrence apply qr stop solve bl ax bx p u v z ax ax ax condition behave ill estimate k transfer w still short recurrence exceed estimate kp factorization e b l estimate ex ex ex k derive several summarize formulate stop condition r l regularization attempt ar recurrence relation norm residual otherwise zero recurrence relations ar ar essentially two follow directly lemma monotonically improve could extend diagonal inspire chen vector figure matrix vary scheme accurate scheme try add little iterative accurate need readily tb recurrence compute last therefore maintain update sometimes simple recurrence ax ax ask construct
receive unit user aim throughput channel slot designing policy could policy know channel reward obtain ts denote channel depend positively correlate channel positively whenever switch channel visit channel correlated channel switch soon channel visit visit case positively correlate channel channel horizon special general base concept policy specifically algorithm treat arm armed bandit one question operate present next desirable slowly duration step arbitrarily slowly channel reward play policy reward ji ji ji jk ji k discrete step q non sequence grow algorithm close dynamic channel regret constant relate chernoff hoeffding difference reveal expect either steady constant matrix omit expectation state prove optimal strong claim logarithmic know optimal chain case correlate open infinite offer bandit arm reward arm chain know seek reward obtain play hard consider hard parameter problem depend value solution prescribe optimal policy suitable treat different non bayesian armed arm optimal demonstrate spectrum channel expect reward aware average bandit access multi armed mab problem tool make dynamic uncertain environment multi armed arm stochastic seek arm reward classify know mab regret know policy desirable maximum particularly variant armed bandit arm evolve chain solve approach regime always tractable index parameter efficient new channel cognitive dynamic prove gap time grow decrease much sublinear average reward arbitrarily option markovian optimal hardness indeed option structure exact true high expect reward contain arm armed bandit operate minimize regret done handle adopt policy length play unknown
plausible finding boundary separate topology show bootstrappe way cluster microarray array effect effect simulate incorporate realistic generate simulated compare author gene percent bootstrap match way provide differently estimate presence absence rich develop tree posterior assess distribution proceed implementation tree combine pick distance multivariate generating geodesic metric arise path path cat unique geodesic tree trees geodesic origin connect path consist segment origin geodesic easily path call split transition transition introduce new meet geodesic tree let root label label formally root though trick include computation label convention consider two edge form uniquely identifies represent compatible add third geodesic present valid valid path geodesic partition property polynomial check uniquely form tree logical edge root compatibility intersection preprocesse classify edge serves share two tree drop think tree disjoint aid division subproblem division share treat part large shared edge classify tree edge work reach take share edge tree store reasonably second bin contain disjoint bin pairwise tree implementation theoretical computation practice dataset algorithm reduce check follow code check form computing form add weighting problem flow equal every vertex use runtime subtree property search flow drop edge subset happen minimum vertex cover point need bin initialize drop subtree path iteratively run follow max max flow represent geodesic give programming bipartite use set computationally efficient since minimum geodesic path final path implementation replicate minute second ghz processor curvature excellent length positively htbp illustrate triangle type geodesic vertex geodesic thin side geodesic thin see colored property close green triangle long actually triangle arbitrarily question try address space whether tree choice spatial representation one clearly pose year context dissimilarity measure quantitative computing section show long multidimensional scaling dissimilarity statistical method zero q offer summary give approximate center extract function come retain scaling see later distance nearly point apart sometimes modification distance tree simply resample overall variability conjunction dna h rr count rr tree rr type count branch simple shannon diversity get tree quite use index type project element project star length analogous interval difference tree region probability coverage package delta uniformly data set balanced bethe run simulation distance max sd bethe bethe raw split tree leave uniform split tree leave split leave ratio calibration configuration would approximate rough diameter scale implemented recently geometry follow could subject comprehensive review evolutionary align pose tree recommend sequence coherent process first proposal mixture tree various bayesian posterior tree bootstrappe mcmc generating make linkage picture combine tree single linkage available tree evaluate pick random pick discard approximated show see explain tree character branch tree run tuple topology colored representation focus far apart point however negative apart exponential dissimilarity sensitive noise around represent neighborhood reason think give ask question meaningful similarity ask combine information similar parametric notion account rescaling case dna far natural ask well branch tree close number sure differential neighborhood sense small induce sense closeness us stability perturbation bootstrappe look tree perturbation change bootstrap tree topology infer compete neighbor original estimate particularly interested ie tree boundary compete tree data star original exponentially contiguous tree close bootstrap topology respective share compete tree sequence interested sequence lead topology give estimate weight boundary position position maintain pair dna dna still proposal reject smaller great accept great accept minima introduce temperature position final tree hierarchical display root evaluate tree bayesian provide variability estimate substitute spread question remain notion issue way quantify variability diversity leave shannon available statistic evaluate approximated tree tree briefly correct statistic choice argue high price diameter geodesic triangle max approximation know distributional statistic dependence valid inferential become evaluate local curvature finite show close tree tree acknowledgement careful reading early pde lrr f pde cx cr r dna est resolve dna est tree tree tree combine tree tree combine bin tree tree bin dna est trees tree tree tree bin bin tree bins tree tree combine bin combine tree bin boundary tree combine boundary bin error bins boundary bin combine tree bins boundary boundary delta generate tree tree tree tree tree tree theorem theorem axiom title inferential summary tree set biology bioinformatic mining mathematical object stability summary distance develop equally show cat compare approximation multidimensional distance tree distance cluster evaluate whether array like cart lead analysis display natural computation advance report describe begin variability microarray plot patient microarray gene patient cross validate gene validate validate tree triangular validate cluster plot tree popular graphical evolutionary biology common standard root tree leave call root build organize leaf occur branch frequency compete tree categorical procedure tree tree tree variation coarse tree take value call refinement come account place question evaluate tree estimation overview current analysis come example include multidimensional scale tree use multidimensional compare validate influential hierarchical cluster detect mixture quantitative tell appropriate tree find branching order path anneal na I become case must together explain complexity evolutionary satisfactory simple evolutionary may characterize represent character familiar matrix tree h root represent tree process process one character generate character root random
optimal know cd procedure would goal exchangeability permutation lose exchangeability explanatory assume take take apply permutation invariant explanatory lost procedure account explanatory manner group exchangeable group explanatory observation cd relate realization aim approximate decision give emphasize cd often familiar terminology accounting covariate spirit currently section suggest naturally explanatory section suggest census involve certain explanatory statistical historical idea setup estimating proportion applying bring normal setup denote find risk motivation ideally sample unknown belong large parametric family np moderate curse belong curse may good estimator respect obtain suitable initially collection affine transformation distribute ty ty define trivial implicitly goal usual follow minor rt opt many harmonic start transform follow outline transform representation transform collection wavelet basis find see chen et hard special parametric ty bb p particular explanatory conditional sequel assume joint covariate follow useful represent confusion later establish follow estimate subsection appropriate affine depend explanatory basis arrive easy handle rough procedure prefer hence indeed account explanatory bring accounting explanatory suitable nearly procedure transform bayes asymptotic induce stein estimator consideration wide correspond consistent estimating stein method examine adapt n every case trivial minimize span square development see bayes follow triangular stage sequence q bandwidth given appropriate three adapt replace estimate x plausible least residual far demonstrate follow minimize equivalent favorable task find favorable reasonable vc regularization may require beyond scope residual inefficient might risk could cause transform structure transform separate original conclude advantage trivial square converge correspond trivial converge point treatment purpose parametric computationally intensive good sequence estimator obtain coupling dominate couple permutation give mean rt sub triangular array assumption equal condition symmetric replace permutation invariant though equivalence remark interested denote mind may explanatory interested may depend conditional ix heavy still try approximate among upon rate slow heavy might however city divide call belong sub include additional proportion area among live area adjust estimate recent census mean shape simulation number area area simulate simulate covariate area temporal analyze binomial setup binomial explain sub entry table sequel parametric may et simulate think suppose use record covariate number people area although well binomial relation response explanatory covariate year scenario specifically change year change roughly shape section transformation square transformation although couple compare risk risk naive estimator apply stage estimate correspond covariate worst moderately naive additional helpful see spatial statistical sub area neighborhood statistical area area belong sub base census proportion people area treat people neighborhood well temporal transformation simulation procedure explanation temporal strong induce isolate extremely smoothing behave roughly naive non parametric e transform option bad simulated risk section spatial introduce transformation also try transformation covariate temporal cause unnecessary among seven estimation kernel kernel denote straightforward apply estimating point close consequently truncation paper bandwidth validation et together improvement sensitive choice em j simple ann parametric compound ann decision chen atomic pursuit issue compound
explicitly whereas generative observable hmm observable embed large translate condition hmm hmm mixture gaussian number mixture invert diagonal case case simultaneously case wherein value use gaussian base k also linear chain hmm px evaluate discrete gaussian factorize computation n hmms hadamard index align rely upon distance computation address sequence computation draw stochastic generation use operator measure dynamic eq estimator previously implicitly learn advantage kde rbf observation evaluation gaussian isotropic gaussian admit integral kde dependency empirical include include rich bayesian network map maximal clique maximal clique observable clique distribution former clique compute empirical clique due variable latent empirical respect conditional expectation embed space clique eq variable clique distribution latent end compute many svm even intractable trick product without yield admits truncate taylor expansion empirically low rkhs feature even distributional space embedding give accomplish alternate kernel learn induce variable identify early maximal clique yield former latter clique backward notation clique cx cx likewise use encode treat identically gaussian latent symbol approach hmms plain kernel hmms fisher identity special kernel special mean exponential gx analog operator derivation require computable clique appear appear clique another one fix isotropic q map explore generalize scalar positive however learn complexity quantify term amenable generalization empirical expectation focus kernel rather high type covering exist cover formally mild abuse b e take correspond operator banach number convolutional convolutional decay exponentially cover act radius ball eq depend indeed decay exponentially briefly summarize decay kernel convolutional transform make fx p fx gx gx ft operator exponentially independent decay rapidly even hence gaussian decay fourier spectrum transform regularization class distribution show svm make cover define risk theorem simplify denote compact measure probability pf restriction cover bind human dna five continuous dataset series repository explore svms discrete observation hmms latent class initialization transition probability initialization emission initialization via viterbi execution rbf map spaced setting linearly spaced length space maintain fisher markov feature string consist manually state symbol generator binary hmms show classifier whereas fisher perform bad loss possibly high run vary symbol formula hmm learn low fisher state restrict low perform slightly outperform poorly sequence subsample l synthetic vs principal triangle circle competitive point series interestingly future plan explore apply visualize relationship sample consist grid grid median temperature specie species kde integrate square form separate certain specie visualization kernel separation toward toward generative distribution theoretic embedding generative operate spectra compare hope distribution discrete outperform bayes technique expressive factorial markov field highly expensive testing rkh letter measure generative kernel measure hilbert incorporate beneficial generalization present model generalization kernel offer elegant clustering particularly iid incorporate statistical dependence information rich machine svms principal component learn particular iid classification observation svms use nonlinear variety sequential sequence use manifold english reproduce kernel I generalize special structural kernel describe distribution provide naturally extend hilbert space embedding empirical mean lack accept mean map provide generative observation generative accomplish measure observation distribution map map widely use extension connection section conclude result species datum concept phrase al observation iid rkhs induce rbf
maximum quadrature confidence root evaluating simulation require estimate core ghz intel mb precision yield van remark number use compute infinitely simple natural goodness statistic draw although present focus straightforwardly parameterization scalar handle usual advantage square computer value simulation suitable thank fan observation support nsf nsf research fellowship fellowship support nsf fellowship extension article level root square goodness fit involve associate black box algorithm level classic test circumstance level monte simulation identically draw member specify draw value accordance terminology bin category work single fully focus parameterized extend parameterization scalar equivalently parameterization natural come use square construct draw empirical page section arise specify root square confident quantify confident let square construct draw arise complement different square various obtain statistic availability computer simple feasible convenient number via monte practical advantage structure review develop utilize level associate rapid section summarize cumulative independent suppose unit cumulative square root branch square root therefore evaluate attain domain integration whole low order double precision root mean square draw fact distribution determine confidence draw root square construct come distribution value distribution draw please simple maximum canonical formulae goodness respective bin particular choose obviously maximum estimate actual determine monotonically confidence level convenience focus classic replace statistic likelihood differentiable occur domain almost surely large draw remark limit joint limit proportional dirac delta concentrated hyperplane hyperplane restriction generalize intersection hyperplane restrict multivariate principal axis algorithm sum square level draw multivariate axis whose vector consist normal hyperplane maximum example take estimate orthonormal basis column construct orthonormal construction van maximum parameter multiply right onto hyperplane obtain eigenvalue adjoint eigenvalue desire axis low computing require appropriate condition sketch proof multinomial distribution maximize define combine desire draw proceed give necessarily number cumulative distribution function obtain algorithm center random variance remark numerically via first detail interpretation model third poisson table poisson test algorithm conduct simulation draw paper concern
generation well policy map context action define convenience payoff multi armed problem devote develop total formally may strategy trial special case contextual bandit since refer type news recommendation view article pool th visit trial arm serve article precisely maximize click turn payoff bandit balance past experience world suboptimal bandit process exploration suboptimal arm payoff refine payoff clearly neither explore purely need roughly class attempt regret grow extensively bandit algorithms bayes index bayesian competitive prohibitive without describe noted method learn goal base algorithm want trial action context interactive nature run likely infeasible serious risk rather offline available payoff arm choose likely evaluate evaluation reinforcement bandit efficient bandit process simulator introduce simulator hard justify reliability contrast sound choose uniformly although considerably allow decrease datum unknown tuple draw I form consist context payoff event crucially partially label bandit mapping history current therefore people evaluate allow life note focus contextual exposition arm old arm may event investigate set simplicity long stream event datum stream one history retain different take entirely ignore input stream stream policy choose retain independent everything else retain distribution evaluate world event evaluate policy stream intuition context payoff event stream say history identical estimate per trial return therefore estimate respective quantity repeat multiple average trial payoff trial payoff respective guarantee event statement mathematical induction event stream retain expect retain event probability expect accurate increase unfortunately situation policy contain per detailed stream similar valid event number reason unbiased trial payoff next show final long enough accurate policy choose independent emphasize fix contexts payoff return close value order decrease case application notation let event return choose side since chernoff bind side statement equation inequality might bandit concentration impossible general contextual coin flip operate choose choose trial payoff always choose style deviation bind dependent history though provable furthermore always repeat evaluation estimate empirically return highly try position offline method world validate methodology specifically evidence guarantee iii effectiveness methodology effectiveness evaluation provide relate offline production yahoo page contextual evaluation unbiased valid event offline algorithms relationship offline three bandit module yahoo front page visit page internet snapshot highlight news article pool maintain human illustrate four article picture title four article highlight picture title click highlight article read click article highlight attractive article naturally model contextual bandit click article age article infer bandit article pool click otherwise payoff click user turn maximize payoff letter randomly generate specify example user letter fall unless offline evaluation million random random article article pool user offline article pool focus position visit click business ratio offline methodology verify compare metric another spatio article high winner serve extract article offline ensure set available note conclusion present differ typical event old article leave enter business rule course visit site still win repeatedly article visit due article click effect assumption fortunately consider consecutive visit winner article whole offline winner online article winner article view treat ground metric offline overall policy article impossible simulator base evaluation evaluation provide accurate offline online decrease evaluation article respectively upper suggest practice rate predict stable algorithm f appendix illustrate offline technique deterministic variant deterministic contextual estimate arm payoff may event run return table summarize deviation find specific deviation demonstrate give natural quite std min section give accuracy method static section view policy particular bandit available article user via serve user article segment age gender note policy fine grain user base age maintain predict user article click estimation model three collect state period exploration run period trial payoff note business online module article fortunately report business roughly multiplicative impact across day cause estimate online business time remain present scatter view correlation slope regression almost policy bandit contextual estimate use maintain article scatter day plot respectively offline business rule retrieval delay daily offline historical presence business rule paper bandit algorithm datum simulator method arm ideally show evaluation method unbiased estimate total complexity yahoo front challenge article verify encouraging suggest usefulness relate online refinement rank ad evaluation however fraction use expensive application uniformly practical constraint evaluation collect reduce exploit specific avoid evaluation progress could policy iteration much baseline policy suppose collect datum act implicit collect new trace choose average trace agree policy iteration score superior superiority superiority although may capture construct appendix simple payoff payoff clearly limit decay regret pick exploration arm suboptimal explore balance exploration trial payoff confidence achieve upper bind ucb parameter uncertainty refine estimate vanishe behave appropriately total bandit extensively understand contextual largely open use achieve payoff contextual shrink assume algorithm empirical weak special strong regret various modeling contextual payoff ucb
hazard hazard leave space take hazard transition impossible prove intensity hazard function underlie ignore product normality hazard rate bound interval support interval show truncation would hazard readily eq cox terminology cox difference situation truncation sec extension convergence terminology translate underlie unbiased may see al convergence cox variation observation scheme censor count inefficient product delay right censor et cox censor hoc rich relationship stationary line assume observe independently nonparametric soon chapter area assumption simple model involve sometimes calculate less alternative possibility four censor version recurrence censor complete right censor observation product situation cox account pe al start define give comprehensive asymptotic limit estimator alternative build particular martingale al available al study robustness gap non discussion start claim pe recurrence ignore al cf fact goodness estimator stationary intensity think axis poisson call individual suppose birth corresponding intersection segment axis observational particular age exactly within death occur time nonparametric maximum appropriate nonparametric inverse easy root estimation possible problem inefficient computable limit type likelihood basically process inefficient easy combine formal censor survival count observation twice censor discard doubly doubly censor theory delta fairly formula covariance easily resample partially estimate extension surface observe observational plane likelihood maximize seem root inefficient easy type van study line segment merely stationary poisson basis estimating equation efficient quasi van fine come full idea early process line quasi case intensity life whether inefficient ad hoc difficult option soon compute inefficient research institute natural inference reference analysis b censor unconditional f york cox new york processes nonparametric censor process survival analysis classical assess huber cox claim life r centre mathematics survival monotone measure length appendix cox pe na weak convergence york soon nonparametric van efficiency segment van van line bias censor deconvolution decrease density estimation problem analysis sample inefficient analysis recurrent event stationary backward recurrence recurrence time cox cox gap propose
prior produce rule horizon specify play capture remain budget horizon feedback value execution trajectory realization goal approximately maximize specify possible dag initial observation obtain rule play state additional observation comparable arm programming computing budget help ignore policy term regard arm lose arm policy feedback horizon I previous change iv feedback time encode single play show single policy reduce statement seem policy martingale proceed via stop truncation trace horizon make ii characterized policy play affect outcome execute induce stop refer edge reward pass rp pr claim induce path policy stop truncate statement preserved accounting integrate result truncation prove lp arm ensemble lp round variant appear arm play point play algebraic sequence curve line part convenience dominate areas jk rhs follow horizon bayesian mab instantaneous feedback reward number play playing play less truncation consequence independence r jt factor arm arm arm distribution lp play optimum observe ever reward time horizon increase optimum prefer play arm optimum keep play optimum ap pa slightly separate delay delay feedback lp collection policy polynomial delay policy enable happen policy step applied policy horizon policy lp scheduling approximation phase size policy order first structured policy horizon behind proof delay every convert policy eliminate know step ensure know outcome next execute play adaptive know outcome observe horizon observe prove structured play make horizon conservative time feedback policy reward factor initially make play subsequent eliminate instead outcome play crucially martingale property arm define structured policy play stop intuitively delay policy play significantly instantaneous reality feedback play policy make delay block pi play continuously contiguous play make outcome simulate play stop decision play pi p pi step compact policy far retain constant block well block play continuously delay free structured free pi block play delay execution couple path decision path play lie couple consider increase root play block eliminate extra outcome plays fix stochastically store follow execution stochastically eliminate block root procedure terminate play eliminate decision number since procedure plays path constant factor arm identical stop step denote path delay block play delay free policy stop contiguous therefore play reduced truncation term terminate truncated lemma approximation policy formulate lp relaxation explicitly truncate meaning begin block step delay block feedback couple two state easy omit consecutive end play make beginning formulate lp randomize truncated block arm state lp relaxation simply structured arm play lp describe htbp begin block play obtain feedback arm collection policy arm policy globally solution remain active aspect novel base play passive complete feedback passive final arm initially active among arm active current play accord allocation become passive follow delay feedback expect contribution observe impact play make make complete since high without interference least linearity conclude pt fact problem definition problem immediately naturally study extensively albeit absence delay set delay provable bandit delay feedback force significant ability significantly prior structural carry feedback available show improve generalize iterate allocation resource uncertainty problem decision outcome seminal develop reference bandit agent compete uncertain reward play arm update focus scenario instantaneous negligible horizon early delay decision make delay obtain upon delay increase bayesian armed delay bandit arm reward specify arm feedback far accord reward reward arm feedback play decision policy concrete framework online determination web present visit page historical page display resolve however characterization goodness visit delay delay system issue budget page display present heuristic base bandit problem stochastic immediate difficulty uncertain controller computable analyze policy output policy rest paper focus mab generalize generalization fact use truncation presence constraint partially execute arm policy original argument space martingale property policy yield approximation horizon absence presence improve lp relaxation
extreme value attribute differently generative permutation object recursive position selection object prefer object distinct propose assign logit refer logit reason axiom show preference utility maximization start early statistical develop necessarily study utilize area primarily example application operation management address generalization notable generalization avoid increase see interested reader refer article generalize sort attractive generally appropriate risk mis model valuable scenario generality applicability maximum impose parameter effective underlie cf scaling dimension maximum impose imply flip learning model summarize either restriction structure limitation choice certainly permutation potentially parametric impose criterion hand approach consider paper simplicity permutation select marginal fraction issue principle distinct permutation need choice information think linear space doubly sparse consistent observation solution computational identify signature show signature noiseless exactly computational scale linearly indeed excellent condition datum available randomly choice signature condition summary signature randomly reality free importantly arise main problem realistic potentially corrupted specifically problem equivalently datum cast linear restrict primarily defer efficient manner describe answer question understand elaborate doubly explain section consistent convex doubly radius von doubly stochastic polytope permutation doubly extreme point observation doubly near choice somewhat establish concerned approximated choice sparsity value doubly next efficiently mention signature play natural ask recover consistent answer question identify original choice approximation sparse signature establish dense exponential family approximate signature designing give noisy leverage structural signature adaptation multiplicative exist signature family approximate well existence approximation search would computation well approximated signature bound polynomial ambient equal model framework compressed programming polynomially effectiveness sparse association rank interestingly basic look projection vote datum capture structural work streaming cf compressive literature mean measurement linear coding correspond receive streaming algorithm correspond maintain algorithmic similarity compressive sense establish generic condition design put view provide non sense efficiently permutation organize precise condition introduce exponential state establish benchmark learn discuss relevance provide search choice permutation component sum choice primarily restrict point marginal doubly item observation generality doubly stochastic else approximate von early exist program guarantee relaxation near put doubly significantly small geometrically translates span extreme point significantly small search run class signature establish signature appropriately dense thereby restrict signature family recall choice say family pair equivalently rank permutation rank support signature question later version model value describe family parametrize doubly family precisely interested reader refer correspondence exponential family moment answer question answer sparse doubly allow explain observation choice emphasize tight dependence require sparsity doubly stochastic zero convex relaxation doubly tolerance choice model precisely doubly pick consistent certainly ignore efficiently restrict improve exponent precisely establish signature family choice remark proof constructive sense propose sparse result signature factor exponent time find sparsity reduction introduce factor worth theorem choice scale scale ignore polynomial ambient scaling sense recover solve polynomial ambient existence signature fit restrictive specifically doubly possibility signature fit precision signature secondly end signature sparsity like scenario happen establish signature family long observation order marginal parameter tuple integer far replace appropriately clarity somewhat weak establish rich exponential approximated marginal thing care marginal sparse signature ignore distribution existence establish restrict signature signature next present theorem prove tuple ease exposition dimension order iff column represent signature family suppose probabilitie signature family permutation signature marginal summation entry column identify signature correspond tuple permutation consistent signature q enforce summary signature index signature choice signature first remainder devote signature tuple signature pick tuple among satisfy essentially approach optimize describe equation justify suffice hull satisfy von establish signature computation cost use algorithm utilize pack possible order sparse signature near signature signature permutation check j tuple signature component towards find signature choice put way interested set satisfy collection gap feasible find provide program signature consistent signature precise notation think satisfy interested essentially try lagrangian relaxation manner lagrangian iteratively initially inequalitie lp problem one infeasible choice relaxed lagrange multipli program negative weight signature feasible solution penalty proportion else impose slack constraint note doubly summation subset sum multiplicative weight solution n permutation signature imply signature component polynomial cost overall priori exist maximum increase value first succeed would well inherently effectively rich describe empirical support question american association choice collect preference candidate candidate vote ranking vote vote win determine common cognitive candidate candidate fact cast complete candidate member simplicity information discuss marginal retain underlie like candidate lot first position vote position vote vote vote vote preferred ranking course manner question however flexibility aggregation retain aggregation choice rank sparse therefore understanding let denote underlying distribution ranking candidate first marginal run algorithm roughly try manner model signature approximate however unable guarantee keep exposition simply refer heuristic description digit candidate position list rank position rank position candidate position full size approximation order average define measure successfully find huge approximate try order marginal draw rich cdf sparse along order nearby permutation sense pairwise need visually well model determine winner determine rank certain functional rich datum winner determination preferred ranking winner vote substantially rich vote permutation approximately additional vote candidate apply win candidate aggregation winner also yield aggregate represent aggregate fair apply turn remarkable conclusion high vote particular order datum fraction position account distinct candidate fourth candidate middle go conclusion line fall behind candidate group wherein least something candidate remarkably somewhat relate model structure candidate vote effectively exhibit preference primarily distinct preference explanation perfect split model important underlie true base make provide nonparametric towards main show first order choice significant expect efficient choice signature oppose force complexity free work robustness signature appropriately recently literature efficient sparse via program effectively linear programming signature another choice efficient result first information theorem order marginal hence reasonably believe family choice extend belief result rely von extend type possibly feasibility respect high direction overcome computational develop approximation marginal inspire exact signature utilize early strongly believe heuristic likely computationally signature family significant alternative speedup relative marginal e doubly accuracy signature sparsity approximate order model reasonable cost bad factor signature describe alternate heuristic outside order contrast signature many employ guarantee research provide algorithm section specifically try permutation marginal unknown permutation represent linear relaxation imply signature however signature option search search efficiently
advantageous mutation fitness increase fitness structure dynamic sampling organization evolutionary equilibria equilibrium correspond rapid innovation organization connection evolutionary innovation organization extend mutation theoretical use sequentially generator capture aim organization crucial dynamic drift innovation architecture subspace question remain jump drift well structural drift organization drift fortunately correspondence precede process within evidence reason optimistic structural drift predict lead acknowledgment physical project via view represent imply department corollary sequential inference learner estimate pass extend neutral structured diffusion organization drift control fidelity lead sequential memory phrase advance illustrative transmission happen knowledge sequentially chain fidelity information lose occur causal teacher pass student quickly motivate sequential game phrase player player phrase win game match original typically interest derive evolve surprising impossible less education human communication error phrase merely product make phrase phrase accumulate set analytical progress sequential select language operate extension evolutionary population neutral drift process population substantial population evolutionary biology require notion diffusion several drift demonstrate drift organization fidelity evolutionary without application effort briefly drift seem available mathematical introduce examine complexity diffusion subspace decomposition structural thereby close structural consequence evolutionary familiar neutral theory skip drift change frequency phenomenon evolution evolve neither phenotype play modern molecular evolutionary neutral evolution mention neutral track population individual space random reduce round biased coin sample memory drift simply bias summarize state drift measurable member state respectively variation vanish state purpose drift homogeneous sampling fact stochastic develop early advance substantially handle realistic effect mutation allele genetic explores relax restriction walk force step identically stochastic iid time probabilistic sampling large learning exhibit like behavior phenomenon organization drift balance structure precise description drift theory begin several alternate gene early color either white parent population mutation allele pass may change individual varie generation generation evolutionary neither mutation assume moreover successive population overlap fisher theory reduce drift version familiar walk receive allele parent previous generation copy generation generation specify drift state drift discrete coin population coin coin generation coin reflect realize walk capture allele individual allele none allele copy population random fluctuation bias represent copy copy expect copy number eventually reach allele states allele drift rate allele selective case mean individual candidate initially plot allele sampling solid line structural sd coin versus mc estimate realization one consequence variation vanish homogeneous population population word establish stop showing theoretically predict predict reference average realization realization binomial random allele count initial allele allele simply generation mc method simulation allele fisher process generalize drift drift consider symbol individual allele drift generation allele replacement contrast structural produce sample string give spatial individual within appear difference evolutionary biology collection individual identically structure mind interpretation life return sampling stochastic drift one lose brief reader value process markov transition give causal allele state leave eigenvector maintain population think generator length string I compact representation string alternate ap ij take allele allele transition one transition generate simple generate symbol enforce alternate period positive transition ba ready describe begin follow first individual infer allele repeat allele drift generation reach population vary net stochastically terminate stop population produce new model infer population generator previous highlight generator finite create sequential closely genetic except analogous biased coin newly bias coin eventually bias towards drift coin replace allow memory transition number presence absence capture essential aspect structural dynamic produce string infer sufficiently mean generation consider allele neither allele current nonetheless alternate prevent despite equilibrium notion say structural correspond become prevent drift however structural drift represent via internal population alternate bit single state suitable biased coin drift single biased coin process generation bias coin become coin becomes alternate variance allele condition vanish population eq allele process call bit per allele entropy bit close asymptotic population diversity form sampling sampling periodic lose randomness generate branching criterion notion occur formally statement vanish structural drift memory population sampling branching recurrent I variation infer lack variation drift process stop entropy vanish allow structural drift stochastic process say analytically drift present structural nontrivial genetic drift coin process process population binomial process coin free sequential inference property explore drift bias h occur final completely occur right drift rapid drop population transition produce later discussion run biased biased coin fig carlo drift modify terminate illustrate allow genetic drift surprisingly interpret genetic case follow low proximity close initial similarly fair coin bias coin represent iid reach consecutive initialize drift library ref alternate transition generation fig remove mean reach transform coin right transition towards reach transform alternate alternate coin pathway biased coin subspace color structural behavior population block consecutive compare coin observe biased coin reach reach different note even process biased coin time setting one substantially drift understand affect ce diagram produce several realization ce diagram vary complexity shannon internal small memory correspondence ce coordinate capture intrinsic population two ce diagram reach transform fix coin alternating reach begin upper left transition merge force bias coin coin stay reach track internal coin process require diagram broad view population structure roughly current transition stochastically since cause correspond subspace fig ref curve subspace process quasi structural drift force change remove state transition quasi invariance break topological shift reflect ce movement often topology innovation return mean biased coin highly mutation drift elsewhere right turn behavior coin weight coin fc ap gm coin bc pass realization reach drift sum visit pathway pathway subspace reached spend structure population break independent time transition subspace jump analytically future drift weight connected pathway spend show bias line along confidence middle sum coin fc alternate pathway transition unlikely pathway fc pathway high pathway frequently grow begin influence fc pathway likely total dominate ap high ap pathway subspace pathway bc spend spend diffusion pathway gm bc maximally far typically jump result small time bc subspace fc calculate know ap fc gm bc jump population realization infer transition pseudo drift drift control trade structural innovation instead infer infer generation generation state create noise transition represent likelihood maximum map retain edge generator way allow innovation zero merge state simplification consider select topology aic correct penalize penalize however may well expense several version drift qualitatively introduce loss panel structural innovation wide diversity sampling create quite produce high variance period period innovation leave drift none phenomenon occur population size jump subspace drift kind behavior select emphasize toward neutral evolution work return example game temporal structure pass language ref ref effort structural drift capture dynamically change semantic drive fluctuation organize
density trick concerned although affect accelerate temperature keep simulation choose schedule minimize distance computational nonlinearity potential curse make convergent include purely method metropolis importance solely purpose therein comparison molecular langevin external carlo introduce walk refer molecular purely langevin dynamic add energy classical molecular dynamic dynamic convergence since langevin multiscale langevin sim anneal introduce anneal accelerate dynamic temperature system occur intercept approach calculate free auxiliary collective stay global annealing perspective distinct accelerate method replica md sampler review temperature general temperature mcmc langevin tune base geometric standard tuning convergent tune accelerate multiscale algorithm optimal acceleration achieved propose handle case negative curvature equal annealing describe bound convergence situation final one sample seek minimize b temperature distance transition convergence rate near schedule refer number langevin detail gibbs need temperature barrier simplicity intuitively interpret landscape tune accelerated accelerated variation however worth st dimensional nonlinear molecular consist fix atom degree system landscape local barrier two right atom start initial momentum long marginal indicator markov converge tune tune anneal individually effect propose fast convergence addition choice investigate grant grateful fast follow system assume system ergodic admit solution fundamental define ode dt cb calculate matrix recall follow discussion achieve minimize hence converge fast opposed immediately difficulty method square could work instance molecular system schedule g ergodicity step repeat statistical total variation triangle equation h energy harmonic minimize nonlinearity temperature g step employ schedule cite truncate inverse fix schedule serve free eq ease optimization optimize purpose error typical temperature linearly ensure number unless inverse error optimal well schedule trivial schedule temperature accelerate type logarithmic popular schedule c inverse shift limit temperature concentrated result show approximated empirical time exp shift inverse setting total schedule increment free investigate schedule rank total agree total accumulation addition discussion dependent error total simulation performance consistent rigorously towards
left plot rsc vertical estimator experiment tuning rsc correct value htp adapt general optimization alternatively bregman et specifically simulation present theorem value tuning calculate calibrate estimator outline selection performance rsc rsc optimally rsc optimally tune validation understand true adaptive rsc version rsc construct matrix generating row normal matrix generate denote th noise experiment r px qx setup require simulated variable strength experiment varied combination result summarize varied correlation mse mean recovery mse median rank percentage p rsc rsc mse mse c mse mse mse mse mse mse mse mse mse mm mean square recovery p rsc rsc mse mse mse mse mse find rsc adaptive excellent behave rsc use large rsc signal moderate rsc excellent mean rsc covariate accurate rsc regularization tune correct experiment improvement support moderate tuning much rsc accurately time regularization threshold mild snr moderate continue true noise build parsimonious via rsc recommend tuning coincide select computationally rsc snr extension rsc penalty induce offset conjecture investigate carefully research suppose much large requirement verify ideal interval give solution construct pair follow outline subsection path recover plot pair conclude tight htp display f iii b xy achieve generalize singular limited generalize page diagonal singular regular singular usual let large notation consist last vector n generalized claim restrict section metric subgaussian inequality subgaussian large subgaussian entry directly subgaussian subgaussian q note element kolmogorov discretization give write subgaussian subgaussian subgaussian subgaussian bind fix obtain claim claim tail bind string self dx x x dt degree freedom et page would like paris thank constructive dms criterion rank matrix multivariate rsc estimator frobenius reduce rsc consistent number value target appropriately classic asymptotic regime stay bound grow either much square rsc computational linear candidate nuclear penalize square inherently high complexity rsc multivariate rsc albeit parsimonious rsc consistent finding extensive class type observation response assume relate unknown equivalent predictor separately multivariate correlate discussion phenomenon restrict equal history back reduce model follow include rao regression excellent comprehensive date asymptotic asymptotic two work develop class estimator adaptively predictor historical latter py k projection onto column singular reveal prominent final describe analysis section sparsity general need count singular turn tuning prove expect bound propose bound ideal penalty penalize proportional estimator al al involve map investigate noiseless study include general spirit comparable albeit condition propose important nuclear less parsimonious offer yield result suggest strongly situation preferable always penalize least denote parameter computationally calculate suggest inverse let column eigenvector correspond arrange small final row obtain first characterize square exceed exceed lemma fit let eq appendix x denote singular reduce py see py consist conclude g k moreover property instance consequence predictor much guarantee describe theorem control singular value control probability consequence pe pe x b pe pe b pe obtain pe appendix state achieve minimum equal assume j j mean inequality hold multiplicative first claim follow pe claim error decrease time expectation bind constant pe free interpret corollary ii large estimator estimator square bias trade restrict f pe pe pe pe rx f inequality pe obtain x appendix evaluate matrix equal proof remark square bias trade rank coincide effective ideal last easier interpret index reduce clearly accuracy small follow eq c recall b pe k immediately I remainder fast ii assume rsc penalty restrict rip plan equivalent assume small large error notice rest subgaussian enough invoke iv error state fact multivariate write
graphical structure significantly difficult necessary response input input snps output phenotype regression relevant covariate correlate output leverage response dependency assume preprocesse step study response group gene identify parsimonious input dense subgraph lasso penalty employ fusion penalty regression across connectivity response guide property success statistical optimization side optimization adopt manuscript response fuse lasso however dimension cone programming qp interior moderate solver exploit introduce proximal qp fusion line rest adopt discuss fuse lasso preliminary fuse dataset conclude discussion instance input intercept denote solve frobenius control recently norm joint across set encourage relevant share across find zero follow q denote combine input incorporate account dependency structure represent computing correlation node edge correlation threshold easily focus denote represent connected edge influence relevant constraint standard follow absolute pair correlate opposite sign subset densely tend fusion effect pair propagation level assume share covariate output penalty qp approach qp newton moderate propose proximal utilize low precisely fusion max bind optimize original adopt accelerate method proximal quite simple recently solve optimization include completion smooth penalty loss handle non smooth find complicated penalty rewrite fusion edge graph fusion penalty wise rewrite penalty eq entry inner matrix auxiliary provide penalty smooth introduce approximation positive strongly original view easily jk control accuracy rate key lipschitz continuous smoothness derivation subdifferential detail continuously adjoint gradient constant bl operator zero project solution kk proximal gradient substitute smooth obtain large descent nesterov minimize th accuracy gradient obtain stepsize combination combination intuitively superior current information current optimize prove present iteration b key decompose part iii gap ii accuracy minimize balance present fast subgradient depend good problem ratio accord assume calculate complexity per edge large comparison complexity iteration accord cost thus cubic edge linear iteration require memory efficiently problem fusion fuse fuse emphasize fusion graph solve univariate norm vector coefficient graph input chain straightforward c relatively applicable analogous fuse estimator function c ml provide demonstrate real superiority gradient exist accuracy scale cross code terminate simulation compare regularize multi scenario mapping simulate input parent panel individual individual regression correlate coefficient relevant select relevant group relevant input induce output assume another correlation subgraph phenotype coefficient simulate entire final select row column input cccc cccc figure apparent f positive reveal clear block structure information across correlate relevant systematically compute specificity average affect element coefficient matrix curve use generate outperform ratio examine sensitivity include purpose spurious include exhibit great threshold correlation close effectively performance approach lasso roc curve entirely overall method output relevant input offer become empty compare prox qp formulation package choose well cpu select qp vary second intel ghz ram vary generate unable qp error due prox substantially qp remove introduce addition increase computation significantly affect cccc c clinical program compare one phenotype agglomerative highly phenotype cluster block row column phenotype phenotype accord give agglomerative cluster figure align phenotype see bar phenotype structure much tend span phenotype regularize e structured graph guide set algorithm problem involve fusion arbitrary simulate demonstrate rich improve performance discover relevant input magnitude scalable qp appendix convex let want differentiable everywhere subdifferential singleton inequality equality conjugate chapter equivalent ad eq function subdifferential everywhere rewrite utilize continuously take reader definition therefore bind find upper formulation eq summation change order bound arbitrary convex smooth minimize lemma present optimize eq b utilize term accord equal b plug side achieve constant minimize notice dimensional holding follow assumption pt output complex graph structured exploit encourage share set input gradient problem smooth fusion structure fusion fast scalable widely cone programming programming provide demonstrate superiority efficiency scalability multi learn greatly advantageous task learn effectively small input covariate relevant furthermore assume related manner closely relate tend relevant goal recover structured coefficient task relate input norm relevant
observation long b inference location even isolate relative yx yx hold requirement isolate strongly derivative linearly intuitive connection strong selective shrinkage previous setting improve isolated observation transform measurement transform measurement selective intuitive nonparametric keep kx lastly carry motivate tail however derivation lose prior jointly let k follow construction view transformation parameter relaxation liu interpretation define could across nc keep constant n label ultimately c hope selective occur pf provide heavy tail lie predictive use laplace gaussian approximation pz combine pz give pz non log pz write nc stack diagonal w resort predictive arrange straightforwardly approximate respect pf x convert average match b laplace log take need account mode evaluate angle discrete angle class covariate pair angular proposition taylor optimize either similarly regularize algorithm regularization objective train ten fold ten fold ten dense laplace marginal perform evaluate predefine sparse contain cross fold dense sparse outperform laplace marginal laplace marginal sparse dense lie region growth eventually marginally report correspond dark copula allow modelling multivariate early song gaussian copula popularity copula literature li pair marginals gaussian modify et al approximate transformation define ng extended work marginally bind copula focus copula give also liu away matrix consistency scenario literature heavy gps elegant region base importantly tail base favorable computational predictive inference utilize tailed idea tail easily build gps heavy tailed selective gp currently covariate discussion give access mm mm science division computer division california berkeley berkeley cs berkeley cs berkeley edu enhance often show tail gaussian via copula shrink dense heavy tailed sufficiently improvement produce competitive result dense region provide predictive region protein particular protein free function protein explore region poor guide view covariate shift differ investigate address overfitte shrink selective strategy end readily could reflect regression augment partitioning advanced kernel problem modify observe uniformity interpret cause address present selective shrinkage replace process tail student tail view outlier I shrinkage notion view outlier I selective far theoretical heavy precisely kind shrinkage structure present construction heavy tailed computation process present biological heavy tailed process substantially dense overview research final mean usually semidefinite location kx kx label observation could towards shrink close mass paper intuition replace underlie section stochastic transform treat kx heavy tailed q center heavy tailed origin note distribution al monotonic transformation structure kx x fx fx take observe kx tail recover process tail easy transform satisfies tail shrinkage selective vary specifically interested shrink isolate dense section marginal heavy induce selective construction induce selective shrinkage investigate additional b gp special shrinkage check build top gp analysis selective obtain focus
indeed covariance large problematic poorly condition instead sample factored exist likelihood elliptical prior weak important regime elliptical moderately obvious much sampling classification hamiltonian monte carlo require alternative elliptical elliptical sampling apply evaluation typically sampler crucially carefully parameter generic slice update although suffer slow gibbs slice assume linear seem role elliptical potentially way achieve search multivariate update sampling update work expect base sampler perform thin straight line point method covariance begin resample control resample move accept reject use processing cost one update roughly elliptical update use likelihood although minor cost might reason give subset proposal conditional prior operator gaussian prior variable elliptical slice alternative standard sampling probabilistic modeling task brief reproduce material model covariance overall covariance apply unchanged likelihood ten dataset input explore application noise shift intensity inference perform intensity bin contribute explain bin distribution mean offset log poisson process cox process bin day day range offset match move trace elliptical slice different make valid make difficult control bar row rescale group cpu elliptical bar height quantitative quality estimate figure take choose maximize provide implementation bit problematic specific detail number primarily low space control comparable elliptical slice sampling less variable fail dimension run unable elliptical increase elliptical slice effective synthetic huge method would elliptical slice along straight involve perform slice sampler many evaluation partly reduce likelihood completely possible elliptical ease human time advantage exchange fix potentially dramatically optimize elliptical simple performance relate scheme wide application acknowledgement suggestion institute advance strong dependency gaussian carlo properties applicable free work well variety property need derive update complex commonly specify priori belief probabilistic gaussian markov marginal gaussian express generally review inference simple carlo apply generally circumstance probabilistic one among poorly contain simple also remove preliminary sampler use metropolis method gp proportional multivariate likelihood tie observe indicate latent also shift p mcmc likelihood point hasting introduce conservative next report fast implement wide drawback need appropriately mix preliminary run usually parameter update desirable step maintain half equivalent rich update slice provide step proposal also select value transition construct state slice adaptively intuitively augment probabilistic slice adjust probabilistic replace still however first operator invariant unchanged effective variable generic implement slice routine cm pt sample update slice make link slice easy validity approach range precise connection technical present include detail slice within update lie plane purpose analysis angle identify distribution quantity vertical stop algorithm likelihood angle jacobian leave probability invariant verify substituting value useful mcmc make initial final rotation return pair rotation reproduce visit initial equilibrium probability draw reverse illustrate ensure angle probable reversible angle acceptable angle draw select use intermediate include initial reverse involve shrink decision reverse transition imply markov chain stationary deterministic unit jacobian probability density obtain variable generate generate remain quantity use probability angle distribution non non enough chain unique stationary distribution repeat elliptical slice start choice sampler
divergence cd ml update hide visible hide respective variable gibbs variable plus cd learn boltzmann machine rbm energy direct interaction unit q visible connection unit bipartite figure different visible unit therefore long show rbms state rbm kl rbm conditionally hide versa logistic column parallel fast rbms integrate variable unnormalized employ visible keep somewhat definition allow individual machine proceed approximation determine isotropic constraint unit connection unconstraine importantly analytic expression available furthermore factorial efficiently independence important visible due generative compose rbms generalization layer multi perceptron become attractive two density rbms visible hide state interestingly result machine undirecte connection evident gibbs top definition easily extend layer another additional prior replace case layer rbms visible unit possibility allow interaction restrictive model exponential rbm unit define replace train log layer log second approximate cd evaluation involve integrate factorial optimize variational likelihood additional approximate conditionally turn discuss layer visible estimate see later density rbms two difficulty arise analytically unnormalized integration hide previous difficulty unbiased contribution estimator applicability partition boltzmann estimate expectation whenever get estimate partition function obtain draw average result point partition minimize distribution well close try proposal approximate act proposal distribution far leave invariant e right integration introduce third hence order simple intermediate precede rbms intermediate rbm whose weighted find estimate factorial entropy still draw carefully chain construct unbiased inverse partition tend short run markov estimate slow bind share conceptually apply consistent equation obtain choice visible wish assign adjust depend layer know unbiased estimate expectation necessary already multiplicative generally rbm unbiased unnormalize second consequence jensen estimate consistency asymptotically unbiased partition tend limit still heavily skewed question whether term reliable formulation become evident unnormalized normalization mention bind readily marginal density rbms state layer state layer generate feed multiply rbms p p unnormalized importance weight use estimate proposal importance easy importance estimate hidden manner however introduce weight take value optimal unnormalized marginal unfortunately importance reliable consider scenario idea likelihood achieve layer usefulness become already rbm generalization boltzmann serve thereby distribution greedy reach optimize log kl kl rbms principle approximate potentially rbm height cm limit none xlabel ylabel legend style anchor east cell west exp exp axis cs lower replace state give conditionally unit bind loss suboptimal log likelihood however rather depict order patch choose thorough analysis effect parameter experiment apply include log center whiten dc patch layer suggest modeling patch model third layer propose possess statistical intensity pair orient edge far analyse likelihood evaluation training initialize layer visible marginal second consist train initialized likelihood likelihood consequence likelihood partition marginal performance measure component evaluation appendix true avg est avg loss investigate likelihood still hide unit third layer train estimate close observation unable visible height cm width xlabel ylabel log bit legend cell south east plot marker width xlabel ylabel bit true legend style cell anchor west blue coordinate evaluate hide blue sample procedure intermediate annealing use sensitive show change loss even unnormalized layer third make observation almost observe estimate log however reason observation likely size third contribute nothing explanation sample require many need satisfactory marginal lead ylabel bit height bar style color ica anchor east plot experiment performance large ica layer cd perhaps interpretation add affect scale gaussian suggest patch unit hand outperform estimate performance optimistic also might improve true improve rbms decrease train parameter connection log indeed ylabel log xlabel number layer minor legend anchor south west anchor west list solid solid mark line dash coordinate approximation ml cd lead improve cd add add lead even worse train cd learn process converge bit cd ask improvement add log likelihood represents achieve greedy additional layer log loss involve integral nevertheless optimistic infer something capability integral thereby encourage optimistic log likelihood log although reconstruction might valid become loss train cd width xlabel ylabel legend anchor east cell mark plot bar cd plot bar cd cd layer log loss minimize perfect learning lead might cd plot could confirm lead well log show reliable estimator important tool evaluation model task effect optimize likelihood toolbox search well effect intractable unnormalized layer still marginal require large one article rbms readily applicable provide evidence suit add improve especially log layer learn low layer would unlikely suggest possible might potential achieve learn greedy might replace different future research improvement natural several multi model substantial improvement beyond layer architecture prove apparent overcome create patch whether due task natural image xlabel ylabel legend west marker color marker plot dash marker width color line width coordinate relevant well experiment layer epoch decrease largely converge epoch visible unit treat momentum unit layer rough encourage layer share parallel used marginal number intermediate distribution anneal schedule anneal determine intermediate equally spaced schedule theoretical effect take layer second estimate optimistic xlabel ylabel bits legend west plot marker line coordinate plot color marker line width width coordinate plot marker color marker width coordinate take unit reveal continue decrease increase code evaluate rgb provide field machine learn way compare model statistical gain increase popularity apply variety complex deep belief limit qualitative analysis due
family concave sense exist functional log function automatically arbitrary complete convex close three nonempty interior satisfie moment inequality satisfy uniqueness density satisfy inequality verify arbitrary moreover theorem affine distribution elementary consideration convexity profile arbitrary side infimum function denote small closed suppose follow concave characterize term function consequence rescale version student concave laplace distribution respectively suffice equal expression everywhere remark apply otherwise log concave interval differentiable concave convex dot mixture f blue interval application understand first mapping respect turn however strong eq infimum also wasserstein due couple quantile present main mention useful fact support distribution moreover imply respect topology supremum depth ready converge entail continuity continuity let density nx fy nx fx dx mode recent result latter contain sublinear describe function estimate maximizer unchanged dx mx define write aim state jensen inequality elementary existence suppose dirac would plausible consist affine design subspace maximizer unless consist function close cone maximum general regularity condition equal equality triangular fix unknown nn zero distribution basic subset write maximizer empirical residual expectation furthermore write thm distance distribution family consistency satisfy far nf n asymptotic linear assumption size point satisfy let let additional refer likelihood turn maxima strength different stochastic search profile mean simulation provide moderately large skewed distribution consider simple linear regression design shape square versus also curve important instance height apply survey united show scatter enhance add seem appropriate neither cubic degree cubic spline say seem quite exact monte fit function kolmogorov residual reveal estimate quantile additive regression curve plus leave hand side curve similar quantile proof elementary lebesgue ingredient pointwise nonempty subsequence second hyperplane suppose nonempty interior sec accord entail small equal supremum next maximizer establishe combine deduce qx b nk boundary lebesgue follow lem nonnegative functions uniqueness maximizer essentially strict convexity concave everywhere theorem nonempty infimum real constant pointwise dt ft dt dt eq whenever dt define analogously suppose log q continue dt ds f ds one equal half dt half line deduce display concavity theorem consequence nn satisfy nx fy fy elementary unbounded converge statement hold equality maxima satisfie similar closed convex lem simplex ccc nu nc nu eq virtue deduce exist pt replace subsequence constant function condition meet moreover dx convergence b h h x equal maximizer approximation smaller dominate satisfy nx fy everywhere lebesgue measure establish essentially cover continuity long vector via side arbitrarily proof theorem equip none dirac maximizer explain estimation attention iw x infinity lem eq combine continuity existence jensen suppose rx entail compact set entail imply n specify view define sufficiently note nx know mx mx maximizer exist large consistency eps nx eq satisfy sufficiently verify subsequence deduce theorem remain argument reveal r side coincide utilize bound distance
exponentially requirement sequence simple indicator general detailed visit expert guarantee policy convex well policy tell upper case trajectory let trajectory follow least policy refined strong convexity j property strongly pick policy reduction treat batch trajectory online concept policy incur loss observe next unknown adversarial fashion go many algorithm find distribution policy analysis need bind variation continue call pick trivial large let ns rl policy surrogate guarantee find limit choose negligible strong classification require strongly surrogate albeit observe induce current observe trajectory true trajectory observe trajectory n ny difference loss random bound martingale order mn l mn mn generalization negligible leverage convexity lead tight demonstrate efficacy scalability problem recognition similar popular current human correct analog linear learner hz b b track star track number per point method total observe baseline training similar help learner recover mistake obtain fall track twice able obtain never train though never fall track significantly baseline smooth look qualitatively video qualitative behavior super platform game character avoid gap run use simulator ai competition stage difficulty gap goal train scenario near simulate consequence action leave hz vector iw performance average complete generate difficulty complete around vary roughly difficulty type gap gap stage hard compare sup choose choice indicator slow iteration report se interval per observe supervise approach controller reason supervise get controller obstacle expert distance next obstacle iterative eventually outperform consider significantly slow improve slightly well indicator potentially due datum generate indicator location collect wide show qualitative behavior obtain efficacy handwritten sequence image character adopt degenerate pass early prediction future compare roughly total character fold experiment fold fold repeat fold term character fold word predict character order predict character predict greedy method well approach simply predict character character always correctly label iteration fold character use svm character character character performance pure influence current character unstable reinforcement small significantly greedy decode benefit respect pass decode state include provide strong guarantee prediction sophisticated strategy decode structured prediction base classifier inverse aid technique present leverage estimate provide understand success online setting acknowledgement support reduced information online incur observe iteration vary adversarial fashion policy regret well regret sequence policy sequence rl regret also regret provide convex use find state policy policy result task enough pick note simply guarantee want show state execute thus rl last inequality fact assume policy n form choose require required online observe infinite I trajectory induce current trajectory would finite trajectory go assume proceed trajectory trajectory guarantee policy induce average iteration random hoeffding mn n rl mn finite policy result n require convexity loss inequality fast corollary list sequential future depend prediction lead poor practice number paper train deterministic set combine policy labeling prediction commonly practice sequence reveal fail must resort prediction learn controller art variety regressor predict expert encounter observation perform expert learner affect crucial assumption statistical ignore make mistake encounter mistake expectation classifier induce soon learner mistake expert guarantee mistake error several input expert learn stationary unfortunately impractical ill train stationary finite policy add practical policy mixture controller mistake cost grows linearly take exist except sub routine nearly expert additionally learner guarantee establish method policy forward induce improve one useful let step initial guarantee follow execute expert j j upper increase bad supervise sub improve
value improper source firstly lasso may covariate step select turn column column produce spurious fit covariate noiseless case exactly expand fit require fail influence produce full fit fit secondly number unlike size high fit simultaneously noise fit vanish higher spurious fit vanish pattern work true pattern recover sparsity attempt predictive illustrate calculate cv fold fold suggest examine thick perfect result however unlike lar individual cv note use calculation nearby tuning value solution extension procedure may may circumstance bag aggregate bootstrap method usually succeed smoothing especially smoothed bag reliably great study aggregate fit pattern select individual bag sparsity treat individually optimisation fit fit covariate great deal term sense natural implicitly effect incorporate though greatly increase shall conduct ordinary initial fit f conduct covariate analogy fit shrinkage covariate fit entirely reweighte encourage optimisation large step covariate mean true fit irrelevant concavity optimisation enhance reject straight repeat iteration recommend cross pose variant scad calculate scad scad adaptive hence procedure conventional covariate know covariate contribute mostly monotonic way know covariate covariate perhaps amongst focus univariate possible check monotonic covariate effect contribution relaxation selective monotonicity case relaxed square variation fit univariate mean follow monotonicity q one monotonic increase fit let knot knot similarly non monotonic theorem k decomposition thresholding style slow however covariate compare check former alternative sign conduct fit treat negative component fit separately decompose fit apply decomposition fit sign p adaptive shrinkage small fit negative positive increase decrease design handle study fit implementation grid dataset estimator plus house price censor version library though discard covariate presentation suggest effect suggest since sign know sign monotonic cross q monotonic monotonically ht variable explain require cross choose involve though greatly characteristic create often contrast fit spam mind sense difficult original covariate include spam covariate finding similar covariate select fairly effect though fit clearly however spam mostly aside monotonicity seem interpretation covariate characteristic residual reasonable monotonicity constraint variable vary generate r zero variance mean take subset replacement correctly amongst quantify look replication covariate reduce additional attempt give lasso recovery slowly thus imply possible fit arise line correct shrinkage shrinkage exhibit end fit see univariate case component fit perform black improve keep correct sparsity recovery add compare contexts repetition plus strong covariate rescale algorithm random forest aggregation tree multiple regression backward shape model spam lasso thresholding penalty sparsity algorithm separate validation separate plug average smoothness tuning run mse response scale approximately scad spam rf rf continue study spurious seem excess shrinkage clearly correction provide implementation thing extent set stick hence handle amongst perhaps adaptive spam though basic perhaps enforce amongst original covariate fairly explanation penalty moderate amount greatly algorithm complex covariate small shift generate l rf scad equally outperform spam rf power covariate variation range spam unable sharp without variability end fit previously method fit sort improve relative scenario save spurious covariate l adaptive scad spam rf case preserve superiority high dimensionality except adaptive increase pick relevant predictor present extend linear behaviour empirically term memory precise success require style oracle addition adaptive improve monotonicity monotonicity know knowledge type penalty effective calculation boundary unconstraine problem univariate first induction suppose far prove proceed full note fit convex function constraint combination solution optimisation contradiction ga would satisfy restricted residual sum applying contradict optimality versus make satisfying imply restrict contradict solution proof corollary strictly optimisation hence solution optimisation domain convex strictly away neighbourhood exist square bind least square proof optimum satisfie thresholded seek note piecewise monotone solution otherwise solution property go independent increase exist component iteration pick cycle continuously differentiable compact subdifferential plane zero plane subdifferential converge observation sort order observation either mean function monotonically monotonically decrease mean zero constraint break q gx gx hx hx therefore solution lie within step function decomposition purpose decompose map decompose vice versa hence one covariate lemma remark department university additive attempt determine multi additive sum monotonically high dimensional versus carry monotonicity keyword describe many parametric may restrictive well know available variable vector covariate identically distribute fit smooth constraint normal justified modelling produce array suggest additive specify restriction knowledge survey turn pool first rapid problem restriction alone retain covariate strictly monotone transformation apart monotonicity smoothing generalize consider cyclic style procedure build around covariate repeatedly fail square convexity allow component fit combine training replicate component impossible different achieve sparse conduct pattern prohibitive context optimisation resolve good amongst identify empirical calculate additive modelling work describe group lasso additive modelling smoothness penalty apply modification restrict solution negative particular calculate correlation mean requirement must
allow incomplete simulation aim matrix low figure completion performance opt fast high quality reconstruction subsequently compare fast achieve excellent reconstruction twice c c pass e simplicity theoretical estimate possible set yield bound main algorithm start subspace global yet investigate automatically performance technique automatically residual grant fa college complete department computer engineering department subspace highly incomplete basic linear subspace gradient slight modification incremental miss entry subspace online system traffic origin flow monitor water activity subspace aforementione first dimensional using identify dimensional even infeasible measurement scale subsample full sense redundant provide intuition algorithm identify subspace highly incomplete traffic spike could efficiency gain consumption large tracking build high sample incremental procedure dimension streaming entry remarkably enable update entry time maintain preference collaborative filter track evolve denote select coordinate error subspace natural euclidean matrix column span submatrix index full rank minimum time subspace average cost steady behavior point evolve horizon equivalence eq write precisely analysis author minimize cost present online algorithm art subspace dimension compact form derive incremental descent short geodesic program partial derivative respect denote use gradient partial derivative equality verify definition step geodesic singular trivial value one respectively orthonormal gradient onto rule remarkable modification current basis along direction taylor approximately serve keep surprisingly additive maintain change vanish converge static tracking rate steady filter explore understand qr svd batch subspace rely eigenvalue counterpart comprehensive find increment estimate orthogonal would span incremental modification exactly subspace work compute excellent proxy energy lie subspace q rotation algorithm compute subspace mixture determine stepsize identifying follow subspace unless series vector accord q whose spam subspace iid variable denote gaussian snr implement stepsize steady subspace vary yield small flat range converge result yet identical stepsize successful show confirm prove residual norm htb cc c except stepsize htb ccc norm track stepsize signature anomaly ability adapt scenario three break select random depend second evolve ordinary sample skew symmetric normally result effectiveness tracking instant red c sensor track density
threshold maintain fit another determine completely fashion lasso improve ols fitness thresholded ols article ols commonly design fit lasso ols thresholded ols lasso ols post corollary ols fitness thresholded lasso near lasso ols ols lasso rate strict ol post fit outperform term sparsity ensure hold lasso oracle c thresholded thresholding component threshold arguably ol suppose c c mn specify f provide characterize lasso characterize level result relative model bad lasso ols post thresholded goodness article sa ols lasso ols post thresholded obeys nonparametric bound suggest sound guarantees thresholded achieve implement practice large inferior parametric lasso ol post thresholded strictly thresholded hold ol post thresholded achieve appendix derivation supplement least result prove definition c take bound step cn x x jk argument eq r ct x r x r x thm x definition nc nc nm condition turn result proof theorem definition follow combine relation n c n b proof lemma bound schwarz master sparse proof modify note proof divide supremum van respect obeys nf cumulative nf substitution nf nm nf nm step step n bind proof nb q apply theorem participant mit comment thank lee numerous suggestion improve thank point usefulness eigenvalue gram acknowledge national science foundation regard oracle sparse eigenvalue monte access theorem remark apply ol estimator lasso regression nearly ol post least remarkably occur lasso fail component true oracle strictly convergence base selection achieve extreme perfectly ols oracle important ingredient sparsity nonparametric act selector apply estimator thresholded rate cover new practical induce maintain certain goodness latter scheme study high dimensional covariate many area analysis author investigate focus act b demonstrate achievable true similar excellent rapid regressor addition obtain interesting article ordinary least ol penalize nearly rate least lasso nice occur sense miss well choose oracle ol post sense rate convergence perfectly select true estimator oracle importantly step selector modification generic step bind generic thresholded perform thresholde lasso lasso thresholding scheme drive estimator finally confirm finding provide summary showing estimator covariate equal similarly intermediate ol post fit outperform post lasso ols post ols post produce occur low large ols lasso improve lasso ols lasso ol post lot kp simulation covariate establish aforementioned ol lasso fitness thresholded problem build establish also build result cite estimator important ingredient sparsity guarantee dimension build reason median large rely maximal inequality primitive sharp sparsity organize benchmark drive nonparametric derive generic post estimator thresholde auxiliary experiment make statement allow parameter index size omit confusion norm denote vector symbol denote standard x bc indexed event say occur increase fix regressor error index namely square square oracle value integer oracle equal achieve risk post selection square accounting mistake select detailed idea oracle previous use problem bound special ol lasso remark later mention balanced say able balance norm square large dominate estimation oracle simplify regressor covariate need obtain task later condition lead post estimator set empirical thresholding fitness near performance post selection prefer lasso estimator fit near well search ol lasso eigenvalue gram give restrict eigenvalue introduce quite post estimator use restrict gram matrix mt isometry nonzero eigenvalue matrix realization primitive well restrict away def high probability primitive condition bound dictionary polynomial similar regressor well plausibility gram equal namely eigenvalue gram q probability estimator model property estimator crucially property lasso dependent extend nonparametric generalizing drive moreover hold drive extend drive derive subsequent function tucker lasso similar imply performance asymptotic performance q remain relation drive drive second generalize obtain general rate compare lasso post first perfectly common post like model possible oracle nonparametric separate parametric thresholded lasso perfect separation near orthogonality perfect model phenomenon possible perfect selection oracle levels supplement parametric sharp begin preliminary achieve oracle sparsity yield sharp sharp sparse two give sparsity bind penalty cn nc main implication cs valid design consequently design sparsity consider bind comparable depend unknown selection post act selector estimator incorrect regressor three implication prediction state ol step rate estimator regressor selector note performance determine
acceptance concern scale mcmc form field literature scaling limit adopt framework author target hilbert absolutely detail form functional normalize precise framework capture arise practice mathematical make amenable dimension product study highlight mathematical generalize hilbert separable full densely adjoint pt trace eigenvalue arrange decrease function paper move particular respect expansion realization independent variable coordinate study straightforward absolutely sure property consequence law behave typical large coordinate preserve change offer hope well however prevent prove consequence product inherent measure sde expect dimensional stochastic thus fact attractive view limit paper hilbert value brownian covariance candidate rwm covariance moreover maximize enable idea limit originally arise nonparametric statistic bayesian see numerically finite target factor coincide respect lebesgue walk enable us sample facilitate clean dimensional walk coordinate pt function change rwm proposal affect identify exercise localize propose move explore rapidly rapidly small inverse poor negligible acceptance critical value rwm distance acceptance rwm proposal acceptance discussion rwm proposal target stationarity show stationarity scale metropolis take explore view home rwm approximation target measure form behave optimally adjust proposal covariance measure stationarity extend high target rwm hasting rwm isotropic scaling proposal conjecture change rely measure pde isotropic rwm start attempt scale question hope future paper practitioner function aware generalization walk standard walk analyze trade explore cost higher several method literature prove invariance rwm utilize dirichlet form appeal another convergence generator process use correct scaling step nearly euler wiener trajectory noise allow often use show increment converge value wiener central limit weak preserve rwm diffusion rwm necessarily weak convergence mala fix investigate mathematical rwm present heuristic detailed outline state theorem high proof invariance principle process proof drift unless letter context explain technique introduce state concern later readily separable space class operator respectively complete assume eigenvalue arrange decrease expansion product norm rr us alternate hilbert space identity operator eq l orthonormal trace hence gaussian cx write form condition may center hilbert measure property frequently subset exponent instance act continuous space decay exist entire distinguished assume eigenvalue order measure q orthonormal x namely hellinger density invertible later impose explain evolve coordinate acceptance chain one conditional random independently see way introduce bernoulli project coordinate random metropolis recall target goal start stationarity convergence dimension outline emphasis prove skeleton outline argument evolve hilbert note time scale result precisely let rwm let rwm explain stationarity explore invariant scale demonstrate measure stationarity maximize acceptance probability implication markov require explore implication three remarkably also study sigma algebra generate expectation one drift notational proposition drift rwm satisfy time markov rwm drift subsequent martingale term converge hilbert state appropriate invariance time mesh step large enough resemble euler simulate finite dimensional drift function covariance underlie chain look like weak euler approximation note important analyze clearly euler scheme identically metropolis preserve stationarity invariance martingale central rescale motion operator appropriate introduce value martingale weakly tend brownian operator converge weakly brownian condition invariance outline sc piecewise randomness stationarity reveal coincide closeness desire drift continuity show weak independent article state precisely argue rwm euler modify brownian go rwm rwm appropriate perform process heart metropolis rest derive drift recall setup start algebra random still define operator drift key accept point lead remainder stay since rx returning suggests concerned one step rwm understand careful critical source drift drift invariance z z ab calculation calculation drift need prove long key give argument orthonormal span denote notational set letting easy involve term hence ix ix see imply lemma identify observe deduce coordinate drift metropolis achieve rigorous control error quantify quantify thereby prove heuristic expect diffusion use simple calculation evaluate expect drift replace fact drift dominate large word drift section make term quantify approximate assumption main decay deviation prior measure derivative measurable ensuring application addition twice limit linearly derivative trace realization assumption ensure weak second derivative assumption globally result inequality thus c three operator imply assume throughout paper handle may show impose give equation constant straightforward imply see candidate furthermore constant functional assumption measurable dx q given repeatedly sequel let rwm rwm weakly diffusion throughout assumption without state proposition prove fix euler let returning produce define sum nu nu du term complete proof r nu nu nu ci nu nu nu nu nu nu tu u r u show rr cp nx prove claim conclude straightforward application pt n generate stationarity show converge motion pt converge weakly deduce weakly since independent precisely law consider lipschitz application contraction unique uniqueness solve subtract globally w z w w space explicit follow concern triangular martingale increment array triangular array independent increment pt array self adjoint limit weakly brownian convergence distribution hypothesis equation imply orthonormal fix give proposition proposition proposition sigma since error require condition proposition lipschitz also n tc fact condition remark stationarity use n b b b b follow x establish verify nt convergence theorem since thus verify hypothesis weakly limit enough converge independent statement adapt furthermore x calculation nearly minor notational e taylor expansion proceeding recursively obtain ns stationarity follow fact e proof lemma pair operator verify notice u I r deduce stationary lem verify corollary weakly brownian operator corollary complete drift proposition lemma preliminary respectively quantity take independent estimate error replacement recall start imply qx rx n lemma variable q lemma qx notice preserve derivative independently first recall deduce since gaussian write hence conclude almost surely strong number surely sure limit preserve cauchy schwarz c n c degree give follow applying follow q calculation lead fix realization converge limit go argument derive obtain wasserstein variable give wasserstein surely respect variable classical estimate triangle claim lipschitz conclude later q could publish fix interpolation interpolation since z difference optimize yield equation drift proof lemma derive control well approximate ix n x
iteration ff practical consequence es biased choice start distribution iteration bias graph node appropriately run rw start even rw follow reduce burn configuration case technique cover component consequently pool independently actual equivalent uniform replacement rw finally expected degree exploration node estimator original sample therefore proportional allow similarly rw trivially degree c rw rw sample sample degree sampling generate heavy tail real estimate technique technique behave average degree cover c b degree apply change degree iteratively left time consequently evaluate turn iteratively fraction cover repeat find traversal coverage k implement simulate technique rw confirm analytical importantly simulation topological capture formulae analytical expectation plain line plot line lie top match theory traversal technique curve initially coincide weight exhibit exactly apply depend type connect similar social detail summarize table user uniformly allocate version rejection uniform regardless actual mainly truth various technique run consists small es randomly allow thought randomly rw straightforward ccccc sample avg degree correct expect correct present facebook average sample contrast rw high degree three rw large drop confirm finding long significantly bias row average distribution expect work rw reality facebook cluster coefficient incorporate model appropriately row apply rw rw work unfortunately reason paragraph hold show facebook rw log take fall toward possible bias precisely practice graph reasonably size rw fig extreme small correction rw actual recommend rw rw method rw really graph diameter short produce view densely calculate carefully affect diameter usually drop grow node conclusion paper traversal technique static model traversal technique walk practice future plan theoretical topological technique question social empirically toward walk characterize date quantify calculate node expect covered random traversal correct broad perspective compare exploration random study easy property bias demonstrate facebook degree bias focus various level web www social network network entire researcher collect www exploration operation two category replacement random replacement traversal first classic web walk either bias rw correct paper baseline c l average fraction sample rw traversal technique c forest full degree distribution rw reference average rw graph traversal technique curve depend decrease second traversal technique completion method example depth forest ff large www well technique understand incomplete particular sometimes g lattice lattice unfortunately intuition introduce towards confirm facebook average popularity surprising return replacement introduce dependency explanation bias date understanding fraction graph central relate traversal technique depth forest ff traversal demonstrate exploration fraction monotonically decrease complete derive addition graph network scope hold discuss graph graph degree various technique correct facebook widely follow al social interaction facebook facebook facebook technique rw empirically incomplete confirm facebook inspire paper analyze far os enyi degrees match random tractable propose assign et operation node union path form include edge traversal discover node origin complete process visit complete trivial body replacement promise path knowledge applicable problem speak term later require propose heuristic degree moderately cover random walks walks web general walk correct walk paper graph unknown begin recursively visit neighbor simply follow classic example classic start uniformly among rw introduce bias node degree uniformly probability depend loop node transition bias rw exploit sampling terminology select randomly typically node visit thus argue compare confirm degree rw tested visit classic traversal seed explore new explore visit select within seed explore visit ff every flip coin probability ff case ff forest inspire name classic sampling visit visit l node fraction fraction node equal depend degree range regular concentrated os enyi well balance heavily skewed decade www internet system assume completely use classic generate call match ignore rectangular interval leave analyze bias degree bias technique run degree node begin walk rw serve reference excellent node stationary degree calculation fix connect eq average square degree show rw make analyze chain dependency handle elegant bias however differs begin define graph traversal interested node integral queue keep follow node discover discover add nonempty discover add remove scheduling implement schedule last forest edge never opposite c initially assign value start queue system already match
observe state easy energy reach observable determine state equal ignore generalize nonlinear advance energy balanced need equation method impulse response balance truncate principal control effectively projection despite balancing assume nonlinear unbalanced coordinate transformation approximation could computational gray mention study define operator rkh provide advantageous though langevin eigenfunction combinatorial coordinate reduced principal component gaussian reduction study outside review balance linear define matrix view instance past energy minimal reach span subspace coincide subspace key also lyapunov q several method solve equation representation align state formally find transformation transform coordinate balanced look singular occur responsible system make contribution idea cholesky note energy solve lyapunov nonlinear hypothesis linearization origin asymptotically equilibrium neighborhood origin solution smooth solution origin equilibrium system neighborhood origin singular sort singular reduction proceed least balance system numerically compute systematic tool solve problem taylor expansions white noise balance differential approach aspect approach analytic balance suitable form q rx xt zero observable linearization also origin asymptotically stable estimating reproduce kernel empirical need particular estimate original idea drive sample trajectory extend generalize lift vector reproduce endow bound use follow important kx reproduce rkh x kx rkhs always rotation suggest xt respective response within finite assume empirical fixing measure response initial great clarity follow generalize high dimensional space taking consider feature working denote center however eigenvalue normalize unit space normalization convention magnitude eigenvalue order nc qx show space kernel kx essence collect simultaneous rkhs rkhs using become simultaneously strictly speak pca favorable continue refer turn collection scale index consist pairwise rank number equal dimension matrix restrictive assumption equal take svd c discard project system well approximate set apply rule map rkh empty difficulty rkhs reduce approximate high degree square taylor need second order reproduce well fact jacobian map capture value q ignore avoid reduce example j pair map training seek function square take form ik z rkhs coordinate function general choose learn validation notational convenience ix j broad separable one function turn approximate appear compute taylor q desire good return kernel derivative improve perhaps use family two kernel respectively case recall dx yx invertible approximation reasonable problem consistent formal rt reduce counterpart original virtue way correspond recognize kernel estimate rkh notion write dynamical eq expression jacobian closed system immediately true situation system essential precise taylor map familiar ix j ik jk rkhs note give compare dynamic controller control input appear pg impulse interval kernel counterpart impose balance transformation cholesky reduction map space next variable follow section choose wave solve polynomial dependent trajectory dimension offset fast leave validation computation remark dynamic explicitly rd degree rkhs appear improve however special version learn wave control match dynamic hyperparameter select system wave panel output bottom leave panel closely system main wave reduce
utilize computation statistic comment show particularly strong geometric commonly occur exploit geometry surface ridge advance sampling constant equally create identifiability namely decrease ridge ridge strong hmc demonstrate ability follow length hmc differ sensitivity step describe hmc suffer presence jump hamiltonian follow step generalize momentum parameter theoretical derivative choose converge size standard vary identifiability step nan subsequent admit normally algorithm affect behavior bring insight adapting provide adaptively devise direction move introduction technical report contribution present journal second comment riemannian manifold present ability adapt proposal proposal case control utilize covariance metropolis mala surveys remain unchanged shape widely shape comment utilize strength adaptive step combine riemannian involve center equivalent additional mc mcmc highly advantageous another curvature updating dependence structure compare
module order shorthand level illustrate module compactly represent diagram infinitely major diagram point sample plane define diagram range persistence bottleneck persistence map diagram figure diagram persistence module strongly q cloud persistence diagram curve show cloud leave persistence diagram wish compute persistence associate module convenient substitute term computable persistence two persistence module suppose module map module family map module diagram pair number rise new persistence module persistence module module homology module arise topological sum way give module family produce topological ambient formally x ambient grow rise persistence module persistence diagram module diagram embed plane along persistence diagram diagram bottleneck indeed diagram upper family space solid line ball circle restrict pair pair persistence module homology group persistence dimension appear dot diagram module fx u r p acknowledgment thank useful discussion thank david discussion sm acknowledge support grant acknowledge nsf dms xt lemma proposition theorem topology great basic cloud ambient space limitation statement manifold consider decompose vary fit together cloud sample clear belong large space unclear everything noisy small scale notion algebraic topology particular equivalence homology attempt persistent homology manifold homology statement cloud infer homology converge homology nature large homology space approach develop state probability belong compute correctness review need background topological geometric statement correctness contain main body discussion necessary persistent homology mostly adapt present material simplify first module mainly
encoding let correspond encoding endow discrete topology encoding easily natural element induce hence follow diagram remove contain cycle cycle definition consistent chinese restaurant crp crp marginal totally analogy encoding regard interpret element ad form group metric model commonly literature consider intersection eq model hence j uniformly independence preserve uniform model coincide distance metric neutral permutation decomposable metric define adjacent transform neutral permutation position discount choose differ discount cycle adequate construction cycle verify construct chinese element place cycle respective create prior projective family projective family application permutation generate guarantee suppose concentrate borel embed canonical inclusion less permutation borel lemma cf proof let projective bayesian conditional parametrize hyperparameter sequence satisfy almost surely countable theorem projective hyperparameter j update describe wise anti permutation deviation decrease closely term sufficient relation may useful parameter instead require eventually occur model represent counterpart statistic update projective respective function analogous give statistic measure exponential prior mean projective limit many construction projective construction field mathematic marginal suitable projective existence expect representation formalize sure conjugacy involve process establish intuition since ab consistently arguably measure restriction inclusion map measurable assign intersection conversely definition function compatible mapping preserve integral proof let projective construct projective h kernels ny q projective limit guarantee theorem projective index projective mapping conjugacy marginal conjugacy projective mean continuous open closed exist measurable draw concept selector function assignment map axiom regularity underlie existence theorem admit selector weakly empty correspondence aa make correspondence measurable measurable measurable analogous index projective form projective posterior p h since model sampling satisfy form mapping mapping proof lemma topology space separable countable countable statement whenever singleton one da measurable mapping simply measurable hand since verify generate compact ball point subset ni I dc nx lemma projective family lemma permutation thus group equivalently image establish lemma projective conditional prior partition establish projective conditional lemma hence borel embed regard first subset sequence tail algebra borel image canonical inclusion composition virtual finite entry cycle cyclic set borel sum converge hence random q either jt jj j example note projective limit conjugate nonparametric conjugate class study include dirichlet infinite projective limit projective respective counterpart application model construction permutation nonparametric effectively dirichlet beta process conjugate nonparametric fundamental conjugacy characterize conjugate posterior infinite priori process conjugacy dirichlet existence marginals exponential dirichlet connection conjugacy conjugacy dirichlet prior show intuitively conjugate posterior family marginal construct conjugacy marginal guarantee analysis projective limit finite use evy process stick break construction transform completely come price technical choice representation probability measure characteristic projective translation invariant infinite process characteristic ill suited question live mathematical object projective limit kolmogorov extension generalization construction infinite dimensional represent finite projective important path projective mapping account path term projective measure mathematical structure limit associate obtain nonparametric bayesian property parametric marginal posterior finite counterpart establish large dirichlet regard nonparametric analogue precise model wide domain regard projective limit mathematical projective use construction topological algebraic discuss construction random construction limit limit projective process obtain applicable unified projective intuitively projective result projective projective limit hold probability fact projective projective infinite theorem generalize conditional conditional projective model bayesian projective limit sec structure sec operation computation posterior diagram word directly marginal analogous projective projective limit applicable projective sufficient marginal resp algebra projective limit sec marginal minimal minimal projective utility bayesian carry marginal projective conjugate projective limit update mapping marginal model obtain analogue model example sec infinite statistical parametric average beyond exchangeable observation projective observation nonetheless projective field projective analogue projective parametric nonparametric bayes applicable conjugacy dirichlet amenable gibbs process law conjugate conjugacy analyze property notable exception family discuss posterior invoke relate process process purpose projective application representation nonparametric posterior projective fact variable share frequently etc mapping field space index denote measure indicate outer assume exchangeable topological space I separable unless refer borel space probability survey stochastic process definition terminology projective limit sense projective limit appendix let topological borel generalize projective mapping induce family projection mapping projective limit family kolmogorov projective family probability dp context represent continuous real represent stochastic collection random distribute index principle projective suitably construction projective address two technical countable word unless countable projective projective space projective projective negativity monotonicity function projective measure function function imply projective construction consist finitely additive contain projective problem address elegant manner space possibly measurable construct stochastic process specifically embed map let mapping call analogously constitute image suitable countable direct projective asymptotically identifiable countable freedom path countable think dimension degree many degree suitably projective projective construct euclidean space subset separability separable modification subset dense see path uniquely establish borel suppose uniquely representation countable mention representation measure algebra borel suppose choose projective countable algebra measure restriction admit parametrization dirichlet fix form projective satisfie projective construct inclusion projective p gaussian borel since restriction algebra induce coincide algebra hence requirement existence marginal parameter kolmogorov obtain satisfy white probability mean theorem projective limit manner projective theorem projective borel space p immediately generalize measure act projective family argument assume order accordance projective projective conditional pa projective conditionals projective converse hold additional condition projective family family effectively parametrize projective limit countable let p projective projective measurable exist denote p projective measurable analogy conditional limit continuity measurable measurable projective projective family projective limit w df regular p regard family measurable mapping projective projective however mapping mapping induce measurable mapping projective index np measurable projective first due projective limit observe limit countable x almost everywhere projective limit generalize regular regular borel hausdorff random variable therefore involve usually image variable parameter kolmogorov continuity function subset sec since algebra wise regular construction construct x measurable namely embed measurable case well projective limit theorem kolmogorov projective system space hausdorff p projective probability measure p compact set eq projective marginal derive result certain preserve projective limit construct applicable construct kernel effectively projective projective space satisfy kolmogorov extension projective probability remarkable requirement impose upon space countable need general regularity support induce marginal random projective random variable relation eq p x imply unconditional measure p sufficient converse generate x correspond projective similar projective conditional family address projective projective projective dp product conditional projective eq justify consist give inclusion p establish implication projective ax integrate section projective bayesian completely define combination projective limit borel embedding allow represent bayesian projective family parametric parametrize regardless briefly parameter exchangeable induce parametric image call call p regard family guarantee conditional probability validity number empirical well image parameter abstract event completely determine contain condition suitable condition asymptotically recover additionally represent parametrize whole purpose sufficient dominate applicable notably conjugacy sec suppose projective sample space parametrize object abstract equip projective mapping conditional index projective limit index diagram constitute model carry projective projective borel p x uniqueness equivalence projective conditional diagram n n infinite projective parametrize largely analogous projective limit restriction abstract assume measurable assume nan probability describe small induce represent canonical restriction respective restrict parametrize justify integrable random family outer turn induce explained infinite dimensional finite g gaussian dimensional quantity posterior censor general formalize bayesian explain measurement draw generate sample j formalism stochastically independent censor projection asymptotically either censor depend application projective parametrize admit projective statistic projective sufficient statistic regular probability kernel ii space mapping sufficient projective limit projective I system measurable x equivalent x draw firstly secondly eq assumption conditional projective x accord I obtain generate projective simply application projective since marginal latter make construction process parametrize borel show fine algebra satisfy conclude question small algebra closely statistic transformation algebra algebra capture inaccurate case point instead demand indistinguishable resolution minimal rather contrary sufficient theorem construct projective marginal admit algebra x algebra imply ac bayesian always exist define deduce posterior prior bayesian appeal nonparametric type g specify family measurable mapping posterior n refer system index loss contain also conjugacy conjugate marginal conjugacy projective projective posterior index projective n index projective model conjugate projective limit model conversely projective open consequently drop either assumption projective mapping lift restriction projective limit family always obtain etc projective mapping closure sampling mapping concentrate countable projective limit preserve conjugacy conjugate conditional restriction preserve nonetheless constitutes define regular ensure abstract immediate definition index parametric example widely nonparametric measurement concrete illustration brevity sec detailed construction record corrupt since space adequate function orthonormal separable hilbert space projective embed projective outer cf process realization terminology projective satisfy denote positive hermitian operator gaussian projective define f projective prior define latter operator marginal choose model eq formally nonparametric construct candidate straightforward verify ii family map coincide projective theorem conjugate assign outer statistic marginal trivially projective proposition projective conjugacy property interpret conjugate characterized canonical prior respect suitable define density sufficient topological contain parameter equip function normalization parametrize model hyperparameter update
subsequence history concatenation string description necessary describe say however lead high bayes try regard length describe resemble kolmogorov combinatorial notion kolmogorov combinatorial provide reduction description decide state still thus bit choose random sequence overlap redundancy sequence representation circular position still point mainly add redundancy somewhat improve datum leave base single bit first section extract sequence set character whose fall string zero character prediction length single character encode sequence character could symbol perspective convert transition entry repeating time symbol last half last encode character clearly redundancy yet obviously source good choice error value select score define instead represent sequence attempt code state transition symbol last transition candidate symbol start either concatenation candidate symbol example symbol agree symbol lead score extra need describe convert process string string encode start bit end candidate symbol sequence symbol form sequence remove right example several source encourage trial close enough even history second run time value length indistinguishable lead predict score compression minimal file never go matter next idea consider extend repeat repeat example qr v g v version string experiment lead improvement history size treat case source predictor source history time predict bit symbol represent predictor represent state bit representation two repeat predict qr next realistic allow order trial change history measure figure average run decrease prediction error start next character contain example figure decrease increase predictor prediction bad steady rich order reach happen predictor produce already problem low already order next inaccurate still slightly slightly pure history reach figure decrease albeit slow error depend increase steady enough combination bit long figure change incorporate dictionary predictive show initially dictionary exponential predictor sure compression encoding investigation predictor compression prediction error plot compression black measure amount score obtain maximize prediction agree say future prediction give compression next symbol text history current symbol skew shorter next opposite suppose predict symbol examine error introduction area information give picture movie data compression shorter search permit remove repetition exist phase first remove redundancy describe phase description encode produce next picture encode predict arithmetic compression compute arithmetic encode pose prediction black change predict symbol motivation practically instance model statistical predict symbol sequence compression principle approach compression question make source sequence incremental start bit add finish predict time predict bit error goodness estimate error predictor compression encode arithmetic permit set
bit distribution start step chain start chance leave walk start exponentially choose polynomially let walk hypercube weight xy xy xy n low hypercube enough complete testing chain computation much mix recall standard machine iff computable function polynomial polynomially every lemma transition chain approximate proof bit query least otherwise yes accept increase see chapter compute enough check vertex possible string run fraction stop amount due step exponential use additive bound use since second run dt ty take conclusion chain reversible reduction computable string string instance input define state space markov generality generality let accept reject either reach markov chain reversible walk weight type connect pair state role play machine transition depend machine particular graph chain polynomial reduction circuit specify polynomially bit polynomially bit secondly tm read weight connect edge complement furthermore weight edge transition loop connect bind take therefore q state acknowledgment thank helpful rgb rgb pt lemma assumption dms work department mathematic support program china grant cb grant dms grant ga problem algorithm iteration stationarity chain practitioner like carry assess lead whether far setting markov rapidly hard zero knowledge start stationarity far stationarity intersect give rapidly hard stationarity stationarity complete distinguish markov stationarity far markov chain monte distribution physics biology image important impractical determining far converge mixing time see carlo integration partition physics effective bound may state time rapid impractical require practitioner development variety try far stationarity survey majority practitioner multiple chain converge public domain diagnostic package behind copy compute various functional guarantee convergence formalize stationarity study main contribution show chain know distinguish much computational problem far stationarity assumption chain diagnostic computer science role study define time measure recall give tv follow mix mixing start formulate think rule circuit rule formalize imagine like diagnostic determine say within stationarity require total variation exactly practitioner diagnostic mixed variation stationarity mix least distance away weak diagnostic within distance stationarity approximate even easier provide correct actual denote realistic formalize diagnostic much strong requirement continue hardness discussion definition diagnostic circuit move chain unary theorem diagnostic unlikely version state input circuit specify markov ergodic yes informally mc start time mix away diagnostic give room term problem hard informally solve time hardness class second statistical see evidence sd completeness restriction put completeness conclusion cover parameter psd flip tail accept reject interactive completeness matter completeness notice inequality tight protocol want statistical case say forget distance via later two strategy long compute quantization compute representative estimate sure exact never run variant protocol proportional minus strategy detect give start identity side equal left hand equal right precisely protocol circuit fraction unary hash accept reject completeness yes protocol chebyshev getting fix number least chebyshev getting take expect chebyshev force second either generality assume number chebyshev conclude repeat sufficiently many consequence protocol protocol protocol circuit reduce constant constant result give claim protocol accept otherwise reject I n completeness rely q similar sum completeness claim none protocol case establish completeness protocol bind use restrict diagnostic start respective prove sd choose suppose eq case sd reduce show membership show sd condition small sd instance circuit output construct state state accord otherwise stationary yx variation
increase alm hour minute hour hour minute alm network breast cancer reference description set alm ht alm n cpu gap alm alm solution sparsity problem primal invert solution fair truncate version complementary th element row number produce method inverse datum alm alm vs vs alm alm alm vs alm table three produce solution exactly note roc positive parameter provide report support grant dms dms de er show follow definition generalization v subgradient subdifferential point subgradient set iteration complement let get let hold summing since instead similarly let get increase yield fy desire solution proposition theorem corollary department great structure maximum likelihood alternate linearization subproblem solution association show practical outperform multivariate analysis graphical gaussian markov meaningful vector gaussian network edge denote independence covariance thus pattern iy hard nature numerically cardinality envelope convex min problem primal problem strictly objective duality semidefinite per propose block descent method method update row programming base subproblem another descent et solving formulate subproblem prox method cd cd inferior project consider art iteration result variant apply apply nesterov primal smoothing nonsmooth obtain complexity optimal nesterov however since performance alternate augment wang impractical alternate linearization alm solve penalty sparsity inverse covariance closely primal plus approximation show method justify intuitive interpretation perspective alm algorithm outperform method organization briefly review linearization complexity alm intuition result synthetic section alm alternate linearization method alm solve convex function effective way introduce new rewrite alternate direction augment subproblem lagrange via jointly alternate version augment lagrangian e g update subproblem symmetric advantage subproblem substitute relation refer technique adopt fast improved complexity almost however apply define property challenge nevertheless apply directly implicitly fortunately proposition n formulate matrix operation need constraint instead impose constraint tight sufficiently iterate remain determine iteration claim hold neighborhood pick solve k x x k condition step ignore constraint change solve effort know z decomposition dominate result definite iterate eventually intuition find fit try propose address objective sparse length recall seek optimize good fit regularization impose prior current guess inverse update take seek solution impose prior whose mean inverse definite pick theory tell present result synthetic efficiency alm alm write intel cpu gb feasible long iteration gap primal objective function terminate alm since decomposition see
recommend segregation high power triangle replicate association nn computed segregation point respectively association solid association alternative replicate triangle estimate almost nan alternative large association imply small notice skew approximate ht leave solid line ht carlo power proportional triangle alternative row middle row right nn mc carlo case get segregation high association level triangle monte power relative density triangle value column association triangle parameter compute density base power estimate get triangle empirical power consider recommend association triangle degenerate association replicate observe imply small power depict density association dash present nan density power imply central similarity value row leave column middle monte carlo mild association small attain mild moderate severe association value recommend triangle middle function bottom multiple empirical present observe monte power tend decrease recommend correspond segregation mild severe power triangle segregation central association proportional consideration index versus efficacy investigation limit nan parameter condition pc equivalent function pc replace satisfy pc pc critical n satisfy quantity efficiency base terminology discussion segregation derivative minimum derivative respect segregation alternative r standard pc pc detail respect q parameter segregation association expansion one relative edge central dash left expansion segregation pe sr pe sr based suggest choose large segregation skewness density suggest test sequence notice explicit segregation respect get note r derivative pc continuity next per substitute numerator denominator present association pe ar pe pe ar pe asymptotic small moderate skewness moderate h calculate segregation case derivative follow continuity numerator denominator easily pc association choose testing moderate normal appropriate due skewness suggest association alternative pe pe relative go rate local hold provide convex segregation asymptotic asymptotic unlike asymptotic variance investigate function investigation tree south rectangular plot near contingency table live breast height record pattern tree specie four comprise stem middle rectangular plot interaction tree take hence tree segregation plot circle black datum tree inside outside hull convex hull proportion correction decrease magnitude raw alternative consider alternative sided alternative level segregation similarity hull correct test circle triangle connect dash line notice horizontal differently carlo randomization calculate value observe replacement trees value monte randomization value determine rank divide divide left side correction determine outside determine convex hull correct monte randomize none significance normality randomization randomization proportional correct solid triangle connect line horizontal horizontal differently segregation live tie nearest nn cell marginal segregation overall deviation tree test interaction vice versa situation test abundance specie cell size cc cc cnn sum right tree distance second modify bivariate rectangular estimating length rectangle since rectangular symmetric theory correction slightly asymmetric bivariate plot suggest significantly significantly bivariate tree line monte simulation independence article consider proximity edge similarity segregation association spatial randomness knowledge theoretic testing spatial expansion central similarity demonstrate express statistic arc thereby normality extensive carlo simulations proportional edge segregation segregation association central segregation association performance segregation relative relative family asymptotic relative proportional edge asymptotically segregation central similarity parameter respect empirical ordering let size point convex hull pattern independence circular point family test geometry invariance property triangle point imply imbalance relative class method appropriate imbalance test construction proportion inside inference ii correction recommend normal approximation simulation might parametrization distribution alternative geometry triangle alternative e might approximate parametrization segregation far away cluster around segregation parametrize expect regardless parametrization simulation execute laboratory conjecture mail tr parametrize namely proportional position density use segregation complete randomness scale properly statistic family theory statistic test monte simulation test alternative assess expansion segregation empirical power central similarity one segregation proportional edge well illustrate keyword complete consistency efficiency proximity relative segregation spatial receive attention proximity direct vertex point distinguish arc vertex vertex class give exact class observation minimum relatively dominating prototype minimum dominating generalize correspond define region triangle increase distance point increase parameterize class point triangle gain popularity analysis move suit connectivity introduce landscape utility explicitly reference introduce spatial integrate tie patch location spatial dimension however usually graph construct datum instance concept depend adjacency object express information allow vertex new reflect intra patch patch quantify among graph theoretic spatial class possibly correlation relative density segregation association one tend segregation observation segregation specie proximity region arc segregation cover many arc thus relative arc arc unfortunately precise dimension geometry neighborhood segregation purpose associate region proximity vertex edge triangle lie article parameterize family bivariate spatial pattern compare finite relative asymptotic alternative triangle propose term hull point family central similarity multiple alternative segregation normality version section performance assess asymptotic propose outside test derivation expression defer appendix proximity measurable consider x nx I identically region extensive treatment proximity cardinality ratio arcs arc complete order brevity asymmetric arcs variance however simplify central theorem stand normal variance asymptotic determine normal iid support illustrative composition scale triangle uniformity transform manner parameter proximity map segment vx xx fall vertex region arbitrarily let orientation vertex opposite xx r define central similarity proximity opposite segment center fall fall edge region assign arbitrarily similarity triangle triangle orientation similarity iii parametrize similarity proximity map notice degenerate density support vertex occur case boundary vx vx vx illustration construction proximity region iid triangle randomness poisson nan analysis relative geometry relative central provide geometry trivially likewise geometry invariance trivially degenerate base uniform nan vertice henceforth invariance proximity mass orthogonal projection edge vertex triangle orthogonal edge projection triangle hence depend need combination geometric mean asymptotic central normality nan proportional nan px n probability arc similarly h theorems degenerate derivation asymptotic arc leave expansion piecewise definition scale function depict monotonically sn pe sn pe explain become continuous monotonically x cs sn cs plot asymptotic function depict also r plot asymptotic nan expansion vertical piecewise function differently h arc expansion asymptotic asymptotic edge solid similarity vertical interval axis differently scale stand convergence equivalently skewness analytically much rr leave right h central line vertical interval definition observe successively condition hence calculation value hold density proportional indicate figure demonstrate severe skewness depicted distribution right histogram replicate density vertical axis histogram replicate indicate skewness extreme indicate approximation central skewness however skewness depict histogram monte approximate differently depict replicate middle indicate severe skewness extreme value present triangle suppose mm result triangle triangle segregation see present identically segregation right r proportional multiple triangle define similarly let theorem conditional rr equation mr p mat mat j furthermore pe pe pe nr gr p x pe rp pe mr pe x mp r mp pe nr ac pe nr p pe pe mr p pe gr q likewise r central jensen I rr phenomenon segregation occur member tendency type vice versa imply unlikely locate alternatively occur tendency member specie alternative parametrize without fp pattern segregation family basic alternative segregation association respectively let b segregation possibility association require occur around class alternative u u j segregation alternative segregation particular expansion association alternative expansion region triangle result alternative family segregation area line parallel segregation triangle away vertex segment opposite segregation similar available triangle segregation triangle case triangle transform thus segregation small segregation nan realization depict asymptotic density hypothesis segregation nan let expectation segregation r association l association sketch ii ii explicit piecewise explicit piecewise q alternative normality segregation alternative normality furthermore normality central segregation normality hold alternative triangle relative proportional replace segregation segregation since segregation association standardize segregation association test triangle multiple triangle one consistent hold multiple triangle multiple nan pattern replicate approximation estimate side segregation association expansion parameter r standardized replicate segregation alternative mc mc mc mc empirical smaller large conservative asymptotic proportion determine significance proportion test significance less conservative great limit plot get level approximation get segregation size nominal seem increase leave I relative association level much size normal base replicate segregation association bottom column column pattern horizontal line locate threshold nominal threshold nan iid conservative size together triangle one triangle nominal alternative segregation side nominal although empirical seem far level side conservative sided alternative ht triangle empirical proportional triangle replicate sided alternative relative segregation side relative bottom column pattern horizontal line threshold nominal low case generation replicate j standardized replicate segregation mc cs mc mc triangle similarity upper limit increase get well right sided nominal test seem conservative testing segregation side nominal size desire sample conservative recommend appropriate get wide large similarity replicate left segregation side association column triangle figure observe triangle level triangle furthermore sided test size recommend segregation yield seem appropriate recommend empirical relative central triangle carlo replicate segregation right e one triangle case estimate similarity close segregation nominal alternative performance statistic uniformly support segregation association alternatives carlo replicate segregation x nn carlo replicate compute segregation segregation degenerate almost rr rr proportional solid segregation dash
output odd pattern local stability pattern change analogy stability pattern stable flip pattern misclassifie flip double learn process strategy search strategy follow ref configuration generate perceptron correctly otherwise adjusted elementary weight flip double flip j correctly solution space reach fig allow construct flip misclassifie choose obvious configuration pattern protocol difference allow contain order integer configuration pattern stability order pair integer learn order th total satisfying pattern one elementary local change process stop classify visit isolate last exceed maximal overcome landscape perceptron remarkable pattern weight one misclassifie threshold process configuration weight achievable storage capacity process strategy ignore ignore pattern hope present work implement ising perceptron unable remove learn stage succeed remove sequential succeed proceed attempt fail stage high large length strategy input pattern strategy pattern stop number correctly classify pattern report appear storage law bad strategy storage less storage systems elementary pattern probably strategie regard walk storage indeed internal storage pattern run strategy strategy configuration measure overlap solution predict replica calculation suggest solution distance theoretical storage reduce hamming solution pattern pattern run histogram almost width ref ising perceptron demonstrate constraint increase shift suggest level solution typical hamming compatible observe double multiple peak histogram simulation consistent random member problem ise perceptron co suggest ising perceptron dominate vanish cluster compatible problem first pattern grow linearly community need walk community maximal process storage capacity make search space point process propose stochastic ise fig input pattern strategy add linearly input pattern work suggest learn random strategy exploit system neurons elementary separate typical capacity study reach strategy solution particularly field isolate space completely fortunately even allow configuration isolate capable separate isolated solution extent strategy replica symmetric give search structural solution ise influence solution ise perceptron input output long uncorrelated perceptron student learn teacher amount student perceptron student match teacher poor foundation china grant china cb laboratory theoretical physics physics sciences china institute china institute chinese sciences china laboratory theoretical institute physics chinese sciences china institute physics china theoretical physics chinese sciences china variants search perceptron study pattern ise configuration modify double weight reach storage length performance give solution construct typical hamming decrease constraint hamming decrease feed neuron perceptron build one learn memory perceptron neuron unit connect value correctly classified perceptron assignment compare ise perceptron two take state perceptron np need grow exponentially weight complete enumeration weight feasible small system pattern unable correctly classify matter weight transition phenomenon perceptron sample maximal storage capacity statistical perceptron perceptron correctly pattern predict step modification exploited expect sophisticated biological perceptron usually read sequential learn new pattern learn biological mechanism help prevent old ref motivated biological consideration sequential mechanism namely space sequential random double weight newly add pattern previously misclassifie later stage simple good neuron perceptron sec walk present
practice principal distinction particularly construct genetic variable repeatedly criterion aic stop path practice go reader main termination force solution deterministic sub optimize aic step arrive ensemble genetic algorithm stochastic optimizer description optimizer despite crucial section aic aic course mid simulation feasible subset show merely exhaustive aic selection article ensemble produce regression cox describe take three variable select st rr rr signal st st consider construction type variable signal ratio relatively low relatively true average type st fit table message experiment st term noise st slightly importantly st significantly weak signal st optimizer stochastic simply st optimizer desirable motivating frequency type notice stability idea start latter graphical shall limit ensemble present limitation dealing correlate present finite shall later control expense signal apply procedure level consist run repeatedly bootstrap aggregate regard randomized randomly select cv cv scale false proceed structure approach st st well ensemble genetic st variable variation lar scad elastic net section comment ensemble tackle describe structured variable selection selection add retain backward already include improve discard continue add add randomly decide traditional assess one often group select assess good contain detailed simply repeat forward step suppose want say group randomly candidate improve aic want number candidate assess size randomly without replacement assess group good objective aic bag random subspace ensemble classifier word increase unfortunately reduce principle ensemble importance correspond trial variance must carefully liu although strength diversity tradeoff ensemble tradeoff explain recall stochastic st st language st algorithm exercise balance diversity tradeoff specify tuning parameter give size say balance diversity variation measure variation objective path optimize strength percent improvement contain base fix quantity plot middle logarithmic also plot st simulation depict fairly typical tend greedy many candidate give evaluate good one high strength search become greedy reduce strength chance find subset explain greedy start parameter control easy reduce diversity middle reach start drop choose look peak course emphasize simulated example merely validation reality must alone choice select decrease desire increase far eventually drop panel st drop clear tradeoff move right corner along panel decrease along course emphasize panel simulate true mainly verification aforementioned ridge rely principle pair along aforementione would subset current marked ridge find strategy move examine size motivating create difficult allow see three type behave reasonably respect noise consistency setting motivate type variable strong noise st sample size smoothing spline fit well visualization motivate c example stability run stability detail control false widely I true signal noise almost major publish ever nonnegative lasso elastic net relaxed lasso group oracle signal scad cross tuning tuning c scad lasso hard subset good oracle stability together e competitive exclude signal together st st excluding signal say relatively behave true signal ability lasso variation elastic net much algorithm exclude performance relax widely time different noise median max lasso elastic net lasso lasso stability elastic relaxed lasso net lasso specifically highly coefficient correlate group correlation highly minimal maximal simulation st min lasso lasso elastic net relax lasso st stability elastic net relaxed lasso together clearly st lasso top much tendency selection quite discovery stability true signal relax st clearly demonstrate highly correlate significantly st robust diabetes example standardize package lar body index response result path decrease enter list diabetes angle st table st agree top least variable intermediate importance age rank middle large early certainly attribute highly correlate st bottom statement tc diabetes ranking map age tc age end discuss nice approach use thresholding ensemble opinion theoretic sparse quite objective imagine search useful medical think rank list provide prefer thresholding must active decision multiple rule produce else concern measure use potential variable repeatedly answer st lasso report paper bootstrapping alone within satisfactory explain randomization mechanism good
high randomness obvious theoretical distribution hamming weight recurrence satisfy thus occurrence hamming sequence lag statistical detect sufficiently serious generator generator since bit hamming recommend recurrence prefer generator different addition mod odd odd return early period period mod vector algebraic structure generator undesirable vanish use generator package generalise extremely long period greater easy align operation output ham simple combine generalise free software period generator generalise obtain long primitive choose minimal polynomial generator implementation shift shift exclusive pseudo number typically say present case shift base eight odd omit fast base state eventually desirable long length generator certain perhaps generator satisfy generator block consecutive bit generator detect relationship segment cycle processor seed processor want segment negligible important generator generator memory cache small bit discuss mainly extension suggest present choose problem sure generator maximal restrict power know q number generator relevant generators example generator mention possible improvement usually distinguish interpret exclusive computer regard shall identify bit treat x eq c recurrence circular array array mod unless recurrence regard block recurrence polynomial primitive primitive power mod verify case irreducible satisfie linear recurrence good number important recurrence full choice choice least transformation mix bit necessary associate could great assume depend want want choose generator parameter choose set characteristic polynomial table search criterion decrease criterion possibilitie consuming criterion repeat
uniform table tree intuitive indexing origin curse let interested indexing indexing partially property base indexing choose recursively value discard answer namely property irrelevant ht terminate discard condition come near two go infinity cf figure illustrate characteristic gaussian one normalize observation common effect moderate point outside region mark bar discard histogram randomly draw vertical mark fewer discard indexing repeatedly pp mention need lipschitz build indexing scheme domain highly concentrate variety geometric object jointly phenomenon link regard regard dataset low combinatorial average sense curse fact concentration explain projection hamming approach get curse entire indexing scheme still discuss briefly conclude article remark dimensionality black lee well base graph informally phenomenon state high every lipschitz different dispersion precise definition probability measure concentration upper complement neighbourhood subset cf drop regular hamming cube chernoff obtain combine chernoff see bind real real one variance function phenomenon admits illustration dimensional cube projection onto subspace cube choose concentrate centre high dimension red outline dimensional projection cube ht shape cube get ever disk difference unit cube diameter interesting essentially dimensional observer precise statement reason choice ball matter much book treat concentration reader comprehensive contain equip agree model characteristic fashion appear realistic metric space measure intrinsic develop slow assumption asymptotic indexing growth note draw dataset point amount randomly draw domain equip index infinite regard product parametrize dimension dimension single domain dataset near function every except cube space etc asymptotically result proof important subset vc dimension denote classical interval family hamming ball intersection member estimate vc hyperplane generally conjecture open banach finite result euclidean borel countable intersection ball borel sigma algebra call subset attention empirical regard normalize finite sample go matter formulation convergence measure whenever theorem uniform interval enter selection book treat mention write metric function lipschitz guarantee view projective viewpoint detail overhead know efficient cf every equip kf ip reduction sequential scan image analyze indexing nn similarity process stand calculation latter computation need separate false optimize correct hamming indexing measure cube addition ball consequently table half distance measure ht intersection spherical nearly high table deduce intersection vc whose near great least empty range query belong notice contain without exponential influential theorem parameter verify sphere parameter claim always fact euclidean domain becomes lead confusion h begin leave inner symbol partially metric type indexing structure consist rooted assignment pruning subset bin cover identify sub subset binary string lf f l xt label range child visit branch amount branching considerable curse metric type indexing scheme content cell read pass computed child follow leaf reach query branch requirement cell cell currently structure early namely say free reason dimensionality use indexing long lipschitz exhibit concentration price pay relevant distance get use exact nn hamming cube binary function space support normalize multiply course dataset point vc exceed accord coordinate hamming cube restriction mapping hamming cube normalize preserve cf coordinate appropriate ann possible range generalization develop project hamming cube transformation assume key suitable large away cube storing nearest answer discovery employ series projection ann indexing pair let draw value ht vast pair geometric explanation intuitive simple half sphere interval contain region projection projection direction subset identify z consist dimensional projection combinatorial vc expectation factor confidence randomly pairwise concentration instead dimension projection pairwise distance remain mean empirical mean lemma large mapping moreover high euclidean even normalize quite good distortion considerably map nn ok histogram concentrate explain search indexing scheme indexing nn article appear time cell probe asymptotic indexing artificial datum belief intrinsic image range area outside great measure concentration spirit paper impossible readily statistical interesting ball vc concentration behaviour really indexing dimension spirit box study lee instance preprocesse indexing similarity search call formally domain classical remarkable nn minimum every cover diameter diameter supremum parameter lee algorithm take query contrary exhibit curse contain unit convert black model set domain metric space uniquely ht finite satisfy remarkable obtain almost surely copy separable reason universal black box finite space box simple produce every query uniquely near initially point whose deterministic execute call distance admit nearest namely search step start define require follow underlie become subtle indexing still adjacent intersect suppose geodesic previously include near else adjacent strictly close short triangle turn indexing exact nearest q move return study metric algorithm efficient vertex observe general specifically prove element exist subspace connect graph translate universal two situation sphere hamming indexing curse concentration consideration would three look like highlight difficulty obtain uniform way bound possible indexing formalize even indexing successfully mine fact investigate two geodesic sense every triangle thin neighbourhood
dynamical together sample prior characterize encode exponentially thus jacobian accumulation fast construct module framework wherein module allow module capable interact module interaction module logic suggest identify resource module brain computational filter per capacity sequence capacity communication network measure per ability unit second quantify mathematically reference illustrative presentation idea statement algorithmic history network sense module module arrange learn ever version overview carlo bayesian planning abstract aim work rational make artificial physics mathematical rare sufficient reinforcement style resource effect resource limitation architecture keyword recurrent reinforcement planning sequence pareto computable monte plan universal model resource replace distribution gold research main difficulty universal obtain algorithmic computable proof hypothesis characterize main environmental must implicit parameterized frequency alternatively approximate bind representation defer equivalence machine network additional available parameterized straightforward sample statistically monte carlo meaningful operation present modify long performance continuously flip fed control straightforward perform way motivation prevent error signal gate directly computation first cell act accord input present output treat input bit distribution network concatenation input subset agent output output next predict predict generate plan specify aggregate reward overview universal prior class operate environmental recent estimate neural accord assign input current horizon generate consist output environmental choose obtain recursively update light prior face sample exponentially rare event technique seek class distribution predictor likelihood length kolmogorov state minimum bit certain precision multiple input go weight influence simplification omit rnn architecture weight prior formula recursively previously however previous paragraph machine bound transition initialize acceptance parameterize proportion hasting construct density repeat boundary level repeat quantile estimate normalization though quantile hasting repeatedly draw multivariate bound variable multivariate laplace covariance architecture previously upon literature especially dynamic block maintain common kalman filter neural latter use relax input bit weight current sample take bernoulli respect observe tw threshold replacement assign form b estimator sample assigning unit suggest kernel density towards mean kernel parameterize translate particle role state plan input joint sequence sequence appropriate stationary sufficiently execute time input sequence state step modify horizon sequence share next
triangle value define totally ultrametric induce various ultrametric non order metric ultrametric tree dx dy cf reading addition ultrametric ultrametric everything ultrametric triangle either shape ultrametric symmetry pattern property circle ultrametric ultrametric topology term iii ultrametric clear ultrametric intuitive notion keep mind ultrametric everything ultrametric permutation row column element decrease circumstance diagonal column ultrametric format dendrogram produce ultrametric read dendrogram visualization figure ccccc width normalization agglomerative figure ultrametric visible appropriate row priori ultrametric matrix one way mode oppose observation attribute column yield order near generalization visualization firstly column index attribute acting set generalize facilitate visualization optimize survey presence element consideration datum comprise cluster reciprocal neighbor dissimilarity map positive symmetric dx dx dy triangular dissimilarity endow metric map onto ultrametric practice need endow satisfactory binary term dendrogram embed object set strong subset index height ultrametric often article refer constructive neighbor chain end reciprocal far ultrametric join review relaxed similarity pair distance unless achieve dissimilarity attribute characterize individual index etc distance join denote characterize absence top dissimilarity matching could euclidean prefer component contribution latter lattice show middle f lattice lattice correspond da da db dc c define pairwise linkage linkage lattice base abstract partially cluster dissimilarity initially subsection usual ultrametric ultrametric generalize ultrametric order generalized ultrametric monotonic rigorous conditional sometimes logic e require monotonic certain important operator monotonic consequence applicable syntactic operator argument order include finally ultrametric ultrametric precisely complete due cluster produce reach requirement reciprocal neighbor requirement access parallel share memory architecture look hierarchical cluster difficulty application mining huge allow consideration agglomerative hierarchical find essential operation million boolean cluster I distance value traditional hierarchical also early proceed benefit metric long metric ultrametric ultrametric chemical chemical precision integer precision assume convenience arrange precision cardinality set generality hence digit left build word equal distance bound ultrametric set determine number st digit partition level nd digit reach fine find successive level pair distance identical number level level projection chemical onto design implicit digit seek impose precision separate equality sufficient w j well cf table immediate sum whether chemical million note identical project read random projection dimensional point equal work literature se guarantee well effectively hash md near hash thereby provide map identically value identically want avoid follow refer stable distribution limit sum distribution example tail virtue high dimensionality lie regular aspect perhaps find work normalize datum attribute small weight projection respectively near map work confirm hypothesis behave identical nearly always discussion remark search sort digit deep induce tree set yield partition member embed inclusion partition digits digit take operation read operation evaluation dendrogram widely hierarchical agglomerative induce article diverse dendrogram possibility map dendrogram important ultrametric space multiplication subsection exist terminal traversal dendrogram root specify terminal embed specify dendrogram specify comprise object traversal short assume repeat traversal terminal terminal choose avoid ambiguity among encode rank binary tree rank binary branch encoding code order ii unique express terminal terminal x branch right branch terminal transpose characteristic branch branching index dendrogram order increasingly node read form dendrogram branching code power fix equation might rank oppose partial consider deal must work expression p somewhat perfectly scale dimension feasible representation terminal arrange metric topology encode dendrogram identical identical look equal code example look look find metric topology string ultrametric infimum long infinite provide hierarchical hierarchical simultaneously ultrametric metric real number integer exponent non decomposition rank first non zero I infinity ultrametric similarity series symmetry operator provide tree consider object uniqueness code concern generality multiplication lose lowest lose tree remain operation level product bottom dendrogram mean cluster possibly singleton remain merged therefore relationship apply application singleton terminal end encoding equal nan element preserve implication take open term inductive inductive reasoning signal way dendrogram invariance wavelet invariant rotation alternatively right application seek shift equivalently permutation permutation cyclic node cyclic term full algorithm subtree subtree product give subtree level alternatively term look amount convenient node say object consider component scheme moving term hierarchical view term term new cluster initially motivate couple algorithm discuss take function wavelet support illustrate haar dendrogram discrete wavelet transform spatial respect work something show figure namely smooth forward vector dimensionality determine exactly read root reconstruct observation signal wavelet level smooth haar dendrogram base figure wavelet detail apply wavelet ultrametric wavelet find treat wavelet recent wavelet general ambient ultrametric say embed become immediate direct dimensionality increase quantify inherent structure experimentally latter gaussian cloud hull sized face convex would expect structure dimension appear firstly structure proximity implication high structure show cluster exploit considerable symmetry dominant picture topological ambient increase dimension ambient proportion triangle sample triangle proportion triangle triangle ultrametric financial exchange stream comprise bid price size case symbol bid report figure embedding window successive start step successive window length window point overlap window regard involve distance find concentration cf great embed dimension dimensional successive term distance figure follow criterion number effectively peak histogram appropriate number histogram gaussians bic approximate bayes outcome well mean respective peak relate histogram want correspond original cluster financial peak distance intra peak histogram peak inter peak histogram approximately co financial signal consistent cluster use multidimensional distance axis account chapter discuss mutually nonetheless relate fundamentally phenomenon case polynomial consist multiple explanation shape see pp cluster capable map curve show ultrametric totally order hierarchy datum know could agglomerative hierarchical see complete criterion sure come explain relate basis check embed read membership point segment dimensional embedding differ inversion dendrogram summarize peak histogram dimensionality expect provide coordinate either constrain clustering original complete imply determine signal volatility explanation average cover article permutation associate analyze approach ultrametric time signal symmetry hierarchy large collection thesis system way theory note symmetry many representation look member mutually similar
subdifferential appendix detail object characterize non involve characterization differentiable usefulness expression functional remarkably learn typically structure expression separable differential different often problem separability correspondence exist compute method cluster describe iterate accord regard equation choose second involve separately iterate characterization suitable coordinate optimality cluster loss element wise component iterate also iteration properly choose procedure give accuracy analyze iteration q emphasize separation role step involve computationally demand input relate fast training prediction speed desirable independently matrix vector multiplication need operation compute observe product operation easily order product significant overall feature significant memory computing might vector prediction input differentiable whose implement choose report supervise svm denote classic classical solve suitably accord q cluster problem strong positive differentiable converge fix matrix definite everywhere unique solution fix problem involve last affect equivalent kernel compute way generally normalize rule observe semidefinite spectral basis norm enforce separable rewrite general new observe memory analyze coordinate compute soon I c ii descent sub last year via coordinate become machine enjoy favorable property supervise functional allow efficiency competitive limitation solver scale supervise coordinate descent technique update depend component two section soon coefficient indeed view kernel coefficient let coefficient optimal th also point involve regard solving equation assume initialized correspondence remarkably implement row recursively accord iteration essentially cyclic group update index pick least report satisfy essentially cyclic macro pick macro notice smooth kernel indeed become positive soon positivity separable form differentiable derive let function continuous theorem compute search obtain hinge function modify subtract coordinate convex risk algorithm solve equation characterize advantage coordinate descent exploit information potential minimizer regularization functional stationary differentiable regularization functional regularization risk review algebra proof endow standard value maps associate point see singleton multi exist modulus lipschitz eq identity finite value function subdifferential value hold convex differentiable singleton whose gradient value function envelope min q remarkable convex modulus fp x x inverse generate modulus converge result page close b prove proof functional continuous bound show solution minimization kernel uniquely decompose belong matrix solution range eq exist introduce inverse equation multiply subtract introduce start subdifferential equivalently multiply side subtract thesis solve solution exist satisfy belong prove observe orthogonal far positive semidefinite hold nan least observe eq subsequence proof rewrite theorem contraction theorem eigenvalue semidefinite condition hold denote differentiable lipschitz easy differentiable lipschitz modulus monotone last consider differentiable contraction mapping theorem iteration converge shall apply descent macro map macro alphabet character observe finite union map union obtain close search direction equivalently subdifferential eq inequality change position macro sequence descent non increase converge size uniformly fix subsequence index th pick essentially observe recall algebra rewrite subdifferential decompose eigenvalue triangular term view imply subsequence consist macro iteration well macro regularization property multiply side finally thesis corollary algorithm kernel quadratic analyze coordinate exploit separable compute solution search close already characterization regularize problem sub differential calculus operators paper gauss development software heavily influence
singular instance entry corrupt function certain allow decomposition perturbation application principal analysis result analysis technique completion significantly purely structural price pay allow relative gap interesting follow study consider support largely complementary analysis review technical tool rank desire decomposition analyse incoherence property identifiability optimality formulation guarantee goal decompose convex optimization trace regularize entry consider norm add constraint image interest hence relaxation core rely constraint robustness recover application refinement characterize target support product eq inner b matrix orthonormal certain norm structural entry balance accommodate balance remark optimization involve spread vector align axis vector consequence take decomposition least conversely decomposition argue matrix non sufficiently spread low singular basis study roughly symmetric explicitly claim possible rectangular norm since must entry fraction zero allow condition vanishing guarantee recover decomposition mild condition recovery guarantee property eq approximately reference dimensional come maximum zero column far proceed satisfied satisfy note analysis formulate require chosen satisfy outlier contrast apply strong analysis fix either process clear recovery perturbation accuracy wise amount serve entry trace simplify choose error possible simply although may introduce slack finally second constraint post recovery guarantee regularize ne let section obtain exact recovery perturbation accuracy entry norm choose choose arbitrary uniformly family one previously enough word nearly entry bad guarantee gap generic probabilistic gap find narrow lie independent entry analysis model infinity z standard probability finally lemma simplify condition mn take mn mn mn mn outlier operator operator vector frobenius linear nn operators composition unique range two define th position also induce norm norm nuclear hybrid parametrize n associated lemma lemma lemma eq concern operator operator invertible invertible taylor expansion claim definition subspace follow proposition fix matrix claim rely x sm orthonormal induce use leave orthonormal non characterize subdifferential non smooth norm subdifferential subdifferential consequence x x x duality note duality give throughout singular singular balance quantity quantity coherence singular coordinate basis constant identifiable prove lemma imply operator complementary imply definition composition contraction norm principle simultaneously small conclusion incoherence construct central optimization system existence arbitrary recall recovery take satisfy constitute subgradient bound eq q throughout fix appropriately accurately recover target decomposition decompose let dual lemma condition eq separately side eq inequality either augment constraint first bound j throughout target constraint condition choose target exist g l lagrangian imply subgradient b j j hold prove dual equation eq subtracting inequality fourth inequality rearrange
system simulate roughly refer simulation monte update phase state parameter call negligible neighboring parameter exchange accord show implement markov chain whose enable system local minima may update desirable replica drift terminal space first merely allow suffice together precise show expand log k distribution determine initial feedback procedure merely overlap though neighboring exist high replica replica repeatedly bottleneck pass merely chain ensure replica round place briefly summarize insight replica visit replica terminal visit replica visit either contribute maximize replica flow fraction decrease old measure make run point tends move small slope towards drop algorithm appear define feedback pt optimize step define within interpolation stop meet result bottleneck present encounter simulate ise unfortunately initial unstable one measure drop number would wide chain replica visit break recursion bottleneck fact require accurately appear chain increase dramatically occurrence merely nonetheless make modification problem several necessary feedback set emphasize fundamentally original achieve line choose geometric circumstance practical experience substantial efficacy number chain space short exchange minimum swap example spin consider typically try would probably place however chain swap compute swap attempt equilibrium swap estimate short need add swap need chain need order routine proceed follow minimum initial swap uniform parameter chain meet practically list initial end sort parameter terminal swap unnecessary discuss begin tackle feedback instability problem severe concentrate replica next iteration implement weight w advantageous estimate become smoothing eliminate though routine swap using accept predict accept result low threshold thresholded result interval add continue grow find around minimize feedback swap parameter swap rate post process k old somewhat counter intuitive normalize smoothed rapid quickly instead implicit analogous properly incorrect reliable smoothing f use feedback implementation aid notice considerably ground state consider correct discuss quantum spin quantum partition classical ise couple slice draw equilibrium system quantum slice well approximation demand hamiltonian purely quantum hamiltonian respectively unnecessary ise appearing crucially spin I suffer issue pt distribution st turn serious transition method essential simulation interested tends concentrate precisely interested paper hundred spin correspond system result represent quantum spin slice perform illustration spin quantum ise slice instance extremely simulate quantum achieve swap bottleneck numerous fair routine rate show demonstrate call target swap chain estimator swap produce interval impossible achieve around achieve target rate htbp show fairly swap rate detail read essential recursion past run realistic length space take gap demonstrate efficacy
estimate likelihood htbp rate mh acceptance explain perfect acceptance diabetes record diabete associate variable logistic size consider transformation diabetes criterion tolerance dl pressure mm mm body mass index diabetes age year specific empirical use eq adjustment maximize likelihood weight calculate aic probability aic include include evidence prior qualitatively reduce inclusion except bic f cf six average fit lead fit mcmc result plot look note interval figure fractional polynomial systematic transformation ordinary polynomial take logarithm power choose collect collect fp multiplication logarithm model uniquely implement bayesian inference fp therein comprise automatic depend hyperparameter manual prior depend transformation equally implement jeffreys infeasible composition nan move successive modification accept mh acceptance ensure f million idea ghz implement package include efficient core author approach covariate comparison inclusion covariate interesting important transformation consider much cf examine marginal inclusion look transformation since map produce uncertainty variable inclusion varied transformation covariate panel strong diabetes odd linearly rare diabetes increase diabetes age diabetes middle participant qualitatively htbp marginal differ average probability visit get decrease also visible range analogously control rise prior could desirable also investigate hyper available difficult glm family important area thorough prior literature summarize performance perhaps explain bayesian fact motivate huge prior marginal likelihood yet suit replace slow careful automatic supervised replace property linear work generalise regression along line hyper possibility drop brevity rewrite score q hence laplace laplace efficiently compute cholesky hold modelling inference model choice acknowledgment sl em chapter priors institute develop classical model handle estimation free metropolis methodology automatic diabetes datum integrate fractional response come glm incorporate ti canonical family derivative coefficient identity iw iy assign distribution indicator covariate covariate vector also transformation indicate prior manual infeasible prior regression normal zero eq locally jeffreys n jeffreys conditional locally scale implement idea accounting observation covariate model nice invariance imply predictor rescale translation covariate automatic situation near act sensitive preference complex phenomenon jeffreys perfectly versus go infinity develop automatic specification correspond fully unfortunately closed hyper incomplete retain closed expression efficient inference handle generalize prior connection inference practical automatic fast estimation tuning monte carlo sampler variable selection fractional modelling discuss research design dispersion improper regression recognize consider appendix standard asymptotic expect g theorem especially size see bernoulli family identity logit mode length fisher information response estimate correlation distribution comparison prior original specific maximize likelihood fully avoid informative regression probit logit link factor improper regard degenerate shape improper factor generalize hyper prior conditional give unity improper prior obvious exception refer order explore accurate procedure laplace approximation plug integrate numerical integrated laplace generally ease gaussian iterative weighted preserve prior high taylor canonical convenient section detail use gauss quadrature approximate unnormalized density numerically routine derivative log gauss quadrature function seven deviation
close boundary support compare bias boundary section detect moment boundary define concentration coverage high ball around concentration choice partition size nn ball pool nn neighbor nn count arbitrary equivalently use fy fx clear imply apply threshold boundary show construct nn boundary point nn volume estimate precise consider let case interior rewrite probability consequently c n ok ok h ok bias nn boundary reduce bias boundary probability variance density correct central correct continue rate interior contribution cross moment boundary correct negligible interior result cross include f kx kx estimator derive concentrate expectation since turn imply therefore x ok bias boundary estimate oracle know boundary boundary know correction exist fx immediately handle f pz pz event conclude z z conjunction schwarz choice conjunction z fy show conclude dx logarithmic follow z g om ok om ok mean ok om logarithmic growth condition assumption observe g bc nn estimate identical set exact method ok lemma z conclude random stress variable process appeal define f fx x theorem borel component independent f f mf z z subsequently observe logarithmic kx derivation strictly parameterize covariance also show kx ig step observe sum conclude proof logarithmic since n bc follow absolute third bind appropriately normalize type main uniform plug appendix correct density density functional list bias plug give fy depend suppose condition plug c fy functional list university school department university lemma corollary neighbor bipartite non shannon r enyi assume estimator functional use piece respectively use integral statistical explicit estimator base optimal decrease square mse converge fast central allow confidence multivariate density application signal processing statistic include application match texture develop component analysis signal internet anomaly uniformity normality empirically densitie gap near neighbor estimator approximation plug nn bipartite near bipartite see sec split construct estimate nn functional integral bipartite approximating exploit geometry automatically knowledge support occur near boundary support set boundary correction attain support since convergence practical include estimator result select derive mse attain certain specific fast achieve nn shannon r enyi positive recently propose estimate unknown realization density wang author consistent author enyi variance distance lead propose normal estimator use paper bipartite estimator truncation near estimator mse consistent mse guarantee minimize specific confidence extended plug mutual estimation distinction estimator shannon enyi latter propose require shannon enyi fast mse fix hold regime remainder paper organize estimator asymptotic consequence proof appendix correction shannon enyi shannon briefly discuss numerically validate discuss section random face functional density realization ff estimator n n nm boundary realization subsequently basic bipartite estimator estimate bipartite graph part density bound lie strictly support however observation boundary intersect describe suitably volume nn distance near neighbor amongst realization radius center kx estimator ball center point intersect consequence ball tend high boundary significant nn estimator kx kk kx finally interior appendix motivate method done identify near show point correct density interior emphasize assume support density interior close realization draw nx u c l relate k x bias general functional specify establish mse z ok z generally appendix idea taylor expansions functional lead bounding remainder taylor series ignore term variance henceforth refer concentration event integral ic density functional estimate might principle estimate ok ok gradient addition variance appropriately asymptotic shorthand limit assumption asymptotic random variable key exchangeable random et exchangeable get et get idea rigorously treat nn r enyi experimental establish functional include enyi interval estimate functional imply unbiased likewise order polynomially plug size functional derivative iii equivalent minimize close define constant possibly opposite sign observe opt analysis require evaluate c k significant mse experimental optimal grow sample decrease explain observe entropy correspondingly functional increase near boundary grow bias decay decay asymptotic sum distance realization support estimator estimator estimator consistent suitably normalize contrast high condition continuity functional nn entropy require bias specify choice minimum restrict decay decay exponent oppose furthermore optimal rate overall optimal decay estimator fast mse propose enyi entropy al shannon correction analyze deduce establish shannon r enyi functional h shannon enyi entropy respectively contrast functional kk si correction incorporate addition list bias estimator iv specify iv specify variance mse convergence mse experimental mse apply functional bias correction incorporate next shannon entropies shannon enyi section reduce assumption note mp u mp k iii derivative great mse density give shannon mi estimate mi xy shannon mi entropy entropy nn estimated remain nn density neighbor estimation I obtain use q neighbor plugging define shannon mi iii require marginal manner estimator experimentally shannon sample draw bias agree well observation finite sample regime predict bc k shannon propose draw fix empirical agreement bias monotonically experimentally shannon density theoretically agree theory section cube uniform univariate beta set shannon bias iii determined theoretically minimize curve even though theory useful finite specify bandwidth significantly mse choosing bias iv bias agreement bias iv increase empirically determined variance express vary choice fix theoretically predict agree u vertical axis population axis shannon sample beta linearity central show fig iii limit length coverage predict confidence interval compare length interval plot bias sample size n bc experimentally shannon draw uniform density empirical agreement bias iv tb enyi entropy sample beta entropy validate iv iv use interval predict determined simulation range accurately enyi entropy define propose correction fast agreement estimator discover structural variable multivariate task recognition machine density parsimonious inherent dimensional configuration discover sample realization priori c six factor true false true entropy play surrogate enough sufficient estimating see list estimate utilize practice form adopt constant vs vs high vs exp vs vs vs constant always test towards entropy irrespective dimensional true elaborate false bias test statistic model false bias nearly surrogate statistic compare result correspondingly compare graph model constant positive bias towards introduce ball knn near relate r entropy unlike dimension e knn estimator functional guarantee among class translate performance realization intrinsic disjoint x follow partition near neighbor illustrated nn approximate density sample give volume nn ball implies rewrite linear respect estimating function different estimate relation relate recognize estimate plug report error property optimize value r give partition constant estimate highly correlate difference k k expectation remain bias modification term original distinct prediction serve upper excellent product beta project hyperplane transformation column orthonormal apply compute partition show experimental e bandwidth fig partition accord show consequence theoretically choice minimize e theoretically serve strict improve expression predict theory modify significantly outperform compare estimate al depend dimension unknown plug et optimal bandwidth selection estimation establish optimal suboptimal theoretically choice choice use bandwidth partition applicable suboptimal knn size choice indicate experiment original marginally inferior compare estimator improve performance superior correlate different anomaly anomaly monitor send pf traffic imply strong behaviour reflect al boundary plug estimation functional bound support bipartite boundary outperform nn term convergence expression derive dimension central develop validate theory finite sample estimator establish density plug density estimator rate density boundary thereby plug achieve optimize discovery intrinsic problem shannon furthermore intrinsic dimension reader use list description correction factor density support unit dimension x n x nz ni tn c constant appear theorems decay condition event kx kx throughout section rx support denote seek realization sphere dimension uniform region density estimator define illustrate density ht neighborhood small equivalent coverage taylor cx tr fx chernoff estimate binomial taylor expansion imply random density estimate since variable uniform integer distinct expect positive close far ht disjoint set complement ball intersect disjoint disjoint ball x r ball x therefore concern disjoint let positive mass chernoff l power density power condition bound f cauchy schwarz derive stand ball ball moment nn euclidean denote euclidean near neighbor amongst volume region volume coverage define beta follow identity chernoff binomial variable exp p satisfie similarly xx x aa arbitrary variable event neighborhood lie continuous neighborhood close boundary neighborhood intersect boundary term volume ball expand taylor write taylor around notation volume monotonic divide side get substitute rhs tuple positive cardinality finite cx fx h rx op x k event nn whenever density fix turn also explicitly finally nn let use turn give mp implie next identical let error define event note cx tx rx mx beta k turn q trivially express tx tx tx le bound concentrate form conclude prove seek nn ball define
magnitude require especially problematic since involve approximation original sampling operate column matrix study theoretical computer community matrix om particularly range crucial involve low extract subset explain negative instance extreme dimensional vector although approximated subset unless theoretical property often loose nonetheless deal kernel sampling recently characterize extract show theoretical evidence tie nystr om singular span capture field research compress sense completion use coherence motivated show class randomly generate matrix low coherence uniform arbitrary help basis determine attractive bound hand singular prohibitive cost calculate precisely motivation behind numerous theoretical related address number remainder paper introduce definition brief estimate formally section coherence experimental synthetic analysis provide algorithm wide excellent basis th th entry vector thin svd x orthogonal right singular correspond define u orthogonal onto projection orthogonal x semidefinite start accuracy use frobenius e briefly common form nystr om rectangular first x take contrast sampling nystr om deal without sample nystr om k complexity nystr om extent previously mention analyze sense robust pca om use variety notion column notion coherence follow coherence contain coherence matrix coherence u q coherence nystr om singular completion rectangular two provide moreover use deal noise completion pca coherence low deal x dependent coherence occur u statement corollary relate coherence rank theorem column method x unchanged second event lemma proof lemma orthogonal onto span relate projection assume column span view subspace span span orthogonal orthogonal onto statement l yield statement difference always inequality apply statement former x coherence event coherence dependent coherence matrix select imagine lead high coherence force completely illustrate synthetic generate matlab ht worst extensive empirical perform variety vary coherence suggest address match rarely encounter remainder f presence compare coherence level low spectra rank matrix decay value decay singular vary manually use vector induce singular singular value minimal coherence set use baseline mid vary column coherence deviation trial although coherence estimate coherence matrix recover note influence singular induce examine scenario matrix medium decay experiment create rank qr orthogonal left singular singular repeat small noisy deviation presence clearly estimate coherence fairly accurate function quite varied value need capture show coherence dataset converge slowly trial next nystr om reconstruction illustrate connection coherence exhibit significantly performance remain l c essential protein bag word nm robot coherence type error equal coherence estimate low tied choose algorithm efficiently
regret policy quantify degradation distribute policy provable accomplish note lower derive good result good distribute centralized access consider distribute bad centralized policy learn whereas sense use centralized regret different future user regret growth fix varied sum centralize decrease increase bound monotonically prove regret access increase hence suffice bad channel second consist increase extent simulation reveal actual regret centralized scheme channel since make bad exist far increase anomalous fail account among user compete bad channel increase regret central allocation pre allocation centralize eps bind l eps allocation low c central scheme allocation scheme low central allocation distribute centralized eps simulation scheme early vary availability characterize bernoulli evenly centralized scheme channel bind centralized insight incorporate cognitive open problem user sense relaxed unknown incorporate dynamically leave dynamic traffic secondary node formulation author anonymous valuable comment point liu extensive version manuscript sharing mit union bind regret break term n bind user slot eqn channel use fact mn channel sensing could modification realize slot new user original allocation coincide scheme orthogonality orthogonal slot reciprocal nothing way ball position allocation user configuration slot probability orthogonality increase absence reach orthogonality eq union chernoff imply h q eq bad eq slot good number good channel event run top channel hence hold bad good channel line define event event slot bad channel good correct rank hence decay choose edu channel access cognitive multiple availability channel initially secondary sense explicit exchange agreement secondary user policy cognitive throughput successful secondary learning policy total number distribute learn regret low regret unknown estimate asymptotic regret grow cognitive armed bandit distribute extensive cognitive decade resolve challenge encounter communication main challenge heterogeneous typical cognitive user primary user primary user secondary cognitive spectrum resource hardware give cognitive secondary channel transmission slot sense beneficial user channel high mean channel less channel availability priori secondary since secondary require sense estimate channel access transmission throughput designing goal paper require converge correct availability available sense strong throughput due perfectly desirable fine throughput sub regret respect throughput additionally framework exchange agreement secondary introduce throughput secondary competition channel channel policy hence unknown distribute primary user l secondary cognitive eps contribution two learn policy guarantee one achieve transmission policy requirement incorporate design distribute access secondary self secondary prove logarithmic transmission distribute policy regret also logarithmic secondary achieve secondary regret characterize verify good exploitation competition medium sufficiently discussion engineering insight towards practical indeed markovian good primary traffic towards derive scheme extension arm bandit markovian result markovian channel complex exploitation bandit multi armed bandit medium access extensive cognitive medium armed bandit employ selection establish availability compete consider channel partially feedback information spectrum investigate difference availability probability secondary consider work centralized access access access exchange user channel theoretic cognitive medium learning game weakly equilibria nash equilibrium assume equivalently random observe recently consider combinatorial bandit channel assume secondary channel respect requirement decentralize propose still user contrast remove requirement albeit decentralized access secondary scenario secondary pre allocate rank analyze detail logarithmic regret addition secondary bound scenario liu rate uniformly decentralized work another schedule user channel achieve discriminate user policy extend incorporate sense organization read deal system deal secondary centralized solve classical result multi armed bandit distribute access provably section consider section low simulation scheme conclude since deal multi jump main high vector rest bad channel alternatively ease abuse kullback channel slot width slot slot availability vector mean channel initially secondary learn decision exchange primary user user sense variable slot sensing obtain value indicate user record channel channel none receive whether general policy user feedback result design successful subject interference primary user policy secondary channel high entry access sn regret distinct incorporate expect throughput policy time nk centralized access appeal classical multi round base index entry channel sense free secondary user policy minimum process armed bandit regret henceforth arm arm high index slot summarize transmission mean policy n jt I select channel jt channel tradeoff channel predict availability throughput sense channel exploitation exploration term channel exploration statistic channel exploitation regret optimal mean jt policie statistic logarithmic regret good channel satisfy spend channel statistic secondary centralized central policy avoid centralized sense centralize good spend availability hence achieve algorithm availability high sense channel channel arm learn allocation policy first bound satisfy channel channel availability section access policy provide guarantee uniformly lose transmission due selection bad performance due good armed bandit distribution variable analyze technique learn access observation target multiple user communication user access avoid channel avoid contribute avoid channel rank slot slot converge free adaptive randomization feedback otherwise generate retain channel rank statistic input round high entry loop select channel c ensures allocate good channel transmission go infinity regret policy every implement number spend spend channel result expect ideal user availability attempt reach free stochastic process markov markov number channel chain self uniform channel exactly consist channel transition chain assume appendix knowledge allocate rank lack communication channel occur reach configuration define rank logarithmic appendix incorporate number spend channel perfect expect good u appendix channel spend bad channel logarithmic access logarithmic u access secondary user regret explicit communication imply lose successful logarithmic design scheme distinguish sense equal good channel logarithmic simulation demonstrate phenomenon secondary implementation policy entail truly exchange user unknown duration channel learn availability designing channel feedback policy bad scenario regret linearly bad channel slot execution algorithm update sample high entry slot current transmission loop stop jk ji mn
sequencing framework short inaccurate unique identification present generation sequence large read constraint present enable reconstruction specie approach simple comparative discussion read support order experimentally preprocesse convert measure represent see preprocesse measure sequencing reaction resolution approximately nucleotide sequence reaction higher later sequence reaction amplitude position division bp position normalization size bin sum bin column result input preprocesse describe preprocesse nucleotide frequency amplitude constant sized square transformation sequence mixture division total amplitude sized bin iv obtain preprocesse align predict effect position sequence mixture nucleotide database peak peak nucleotide th peak trace peak model center peak position height nucleotide peak peak peak width sequence evaluate space thus trace basis trace transform square preprocessing sequence sequence display basis highly bin position reconstruction initial bin offset reconstruction distance root predict verify know composition offset circle peak peak peak position red correction peak red correction local employ ht root measure position align predict use specie position database cm cm conjecture assertion equal contribution unseen million comprise enable small number tool composition single sequencing reaction compressive deal many comprise specie simulation sequence base may reconstruction mixture thousand realistic measurement promise reconstruction toy mixture may availability everywhere population environment create complex system body cell order cell typically specie sample specie characterize human high species cavity community composition physical disease broad understand interaction composition technical limitation scale survey conventional profile relatively inaccurate incorrect identification rely availability laboratory identification much attention standard direct sequencing extract gene use sequence identification sequencing therefore require hundred sequence depth mixture sequence result reconstruct human dna microarray identification review microarray platform aim sensitive sequencing still microarray scale microarray array specie detect recent universal account single sequencing obtain read region read sequence composition reconstruct method dna enable throughput community detect length basis limit obtaining long read sequence sequencing reaction mixture sequence constitute sense sequence equation database frequency frequency experimental compressed wide variety fact natural certain compressed information small thought need cost simplify various single camera computational biology design sequence drug cs combination specific mix microarray specie probe cs solving determined great uniquely accord cs condition notably rip uniquely logarithmic instead furthermore exist size thousand intuitively rip similar close orthogonal detail requirement matrix representation cs application pool sequence community note numerous characterize level enable relatively certain compressed use database represent dna coming step reconstruction reconstruction simulate mixture specie thousand biological addition applicability sequence species community reconstruction composition hand gene assume specie gene purpose specie mixture frequency vast sequence gene sequence region well serve universal sequencing gene region ability distinguish dna abundance mixture constraint composition try characterize frequency specie specie vector hundred specie sequence matrix nucleotide sequence length specific position four mixed sequence composition certain frequency th position nucleotide similarly nucleotide sequence therefore eq hence sequence reaction typical sequencing reaction considerably free hundred thousand hence system solution crucial cope reflect formulate cs seek sparse solution small set species cs sense concatenation matrix concatenation notation general np hence optimal remarkable theory replace lead require measurement frequency linearity mixture fortunately cs paradigm cope enable reconstruction account measurement utilize importantly problem vs sparsity lead fit possibly require vanish enable error tolerance experimental achieve rather sparse specie frequency fine accuracy measure frequency threshold reconstruct design desirable order enable unique reconstruction isometry rip uncertainty principle briefly rip perfectly property invertible sparsity furthermore achievable general system thousand suggest reconstruction mixing represent database specie database sense far advance version check reverse align match difference unique sequence input enable pc sequence unique sequence database mixture universal r dna gene mix dna obtain sequence reaction describe position specie sequence peak correspond nucleotide identify peak complicated sequence previously perform sequence multiple dna peak sequence nearly impossible locate preprocessing identify nucleotide sized bin see apply sequence position score base database predict matrix step measure performance propose specie select frequency random normalize result frequency later input figure mixed figure reconstruction specie well original large positive measure rmse precision rmse root rmse I measure account frequency rmse score present vector reconstruct sensitivity define fraction present reconstruct predict continuous recall rely minimal present reconstruct mixture precision mixture specie uniform nucleotide mixture mixture use circle mixture successful reconstruction need incoherent accordance rip determine though thus aid bring example even base reduce previously feasible information mutual coherence coherence column coherence pair specie correlation center exist correlated pair show high mutual coherence place sequence highly distinguish difficult sequence relate specie distinguish acceptable still small measurement translate read sequence sequence dot product expect exhibit coherence around even typical algorithm length line plot rmse specie enable run subset sequence number sequence large lead rmse cumulative position typical basis basis sequence species reconstruction reconstruct distribute mixture specie database black randomly generate sequence bar derive sequence vary species mixture blue specie show incorrect identify black reconstruct frequency present effect sequence coherence mix mixture sequence compose value coherence sequence rmse sequencing species present species reconstruction rmse positive specie frequency sequence successful reconstruction mixture percent frequency specie distribute specie test reconstruction mixture figure rmse law b specie large tail frequency sort specie frequency reconstruction mixture specie simulate distribute minimal reconstruct present measurement practice turn effect minimization determine solution nucleotide figure slowly real black green approximate sequencing performance specie noise reconstruct rmse present predict show sensitivity precision attain realistic degradation performance one reconstruction realistic level rmse specie frequency uniform sequence length green line represent vs curve sequence minimal inclusion experimentally mixture introduce mainly sequence exhibit variability stem activity standard sequencing qualitative require maxima may combination variability peak overcome utilize peak local accurately predict sequence database sized bins single sequence peak peak predict effect height sequence perform position peak local correction deviation height prediction feasibility reconstruct sequence dna universal result gene proportion preprocesse figure peak measure dependency bin root error stem peak reduce accuracy reconstruction successfully three remain identify basis sequence sequence manually add addition identify presence another identify space highly sequence differ basis test sequence mixture solid line five proportion correct nucleotide position range reconstruction result runtime frequent specie frequency quantify specie sequence reaction study amount database ability information
rule edge everything sense helpful distance distinct graph different define thought graph remark geometric detail chapter develop toward algorithmic application geometry everything need careful four nest check familiar edge dx city q city inside diameter segment city graph figure illustrate http com edge construction become intersection open boundary iii open invariant axis ax ax transformation translation rotation scaling take template associate relative neighborhood origin instance proximity subset extra origin radius edge e vertex corner proximity monotone family template one analogous geometric graph neighborhood graph intersection two radius pass open center think one fix configuration vertex say cost one subgraph graph probability spirit study use design vertex exactly north east se conceptual previous edge create imagine infinitely long thin able log position space instant define position process page successive jump remain place axis axis part follow west replace single west straight procedure successive collection path city random edge finally realization poisson process need external randomization initial deduce external randomization near boundary square directions ne se immediate adjacent trajectory figure intuition confirm say road network network familiar impose deterministic efficient general phenomenon infinite sense consist pattern area network study random loose remainder city unit exact limit convention per normalization natural regard write q straight number possibility context however statistic descriptor world inefficient subtle drawback length collection line connect exclude discuss characteristic shape slowly characteristic hold calculate quite insensitive discretization characteristic shape common imply alternate prevent grow typical city straight network difficulty city road thus minor insight city normalize enforce constraint visible world quite arise structure city one lattice nearby pair paper exact analytic formula calculation resort carlo obtain explain value minimum span version interval suitable configuration deterministic eq carefully I prove background strong attain network infinite low boundary believe skeleton attempt well ad construction close skeleton look something value tractable say know exist clearly rigorously unable rigorously continuous toy road intuitively place road city nearby city city distant proportional context consider optimal network benefit functional rigorous viewpoint assertion nontrivial sufficient neighborhood prove fact quantify normalized return around correspond square decrease increase decrease increase economic prediction normalization close substantial literature relative graph deviation give extend rather focus different instance focus critical certain maximum little seminal various statistic closely detail proximity seem object attractive alternative connect remarkable result find road network line mathematical question spread contact process random interesting usual lattice critical q contact asymptotic make loose particular particular deterministic equality view realistic real world road mathematically progress discrete imagine finite consistent mathematically natural structural process ii translation rotation invariance assume constant analog kind arise scale invariance acknowledgment nsf grant dms thank anonymous helpful comment remark lemma sep motivate one proximity come possible question attract attention year example www edge geometry concrete visualize city road dimension would graph purpose reader review connectivity attractive review geometry potentially interesting purpose imagine specifically point road primarily interested short length turn subtle innovation section theory quantify network efficiency amenable calculation tractable carlo simulation particular
expect provide leibler kullback q easily turn majority vote binary decision term mixture discrete continuous consider factor relate threshold consider attribute priori choose last pose learner identify select stochastically learner classifier choose interval offer another kl divergence posterior limit kl continuous kl small whenever kl divergence suggest small guarantee risk minimize group direction refine reflect hence close sg ad expression except replace risk gibbs majority decision consequently difficulty main previous framework obtain various optimization detail ideally conjunction bound use cover classify example misclassifie penalty misclassifie positive remove repeat reach heuristic compression cover machine sc however approach utility aspect strategy subset let cover choose maximize u p use maximize decision decision reach greedy learning determine cross attribute small keeping utilize greedy cover gibbs low gibbs measure piece indicator use utility instead need utility partly fall partly versa follow far first jx assign jx new vector attribute ix I jx cover decrease decrease decision suggest remain example decision cover cover cover contribution decision divide example cover amount example soft add gibbs add reach totally cover greedy totally cover utility number increase analyze attribute number cover take attribute attribute microarray remove cover training kind guarantee cover example algorithm run news prefer threshold oppose fix alternate threshold choose attribute however previous c test real microarray expression tumor gene identify high intensity across level gene patient set microarray sample gene set contain failure gene expression value breast tumor various level example ex sc pac framework attribute first name obtain point contain svm elimination present svm state mining validation cv five set gene training perform nest fold selection criterion fold nested deviation side confidence fold classifier interval adaboost cv gs adaboost frequently respective choosing boost inconsistent boost round attribute experiment frequently gene far exceed dataset stop follow boost run algorithm use fix table pac respective microarray bind provide uniform fold refer classifier testing fold respective illustrate current c c note quite relevance observe limit factor microarray large give tight currently dataset percent current percent hence limitation come availability bound tight sc find classifier able acceptable classification two try use separate hence classifier compression minimum encouraging domain alternate explanation suboptimal extremely offset pac perform combination approach pac competitive add importantly bayes bayes reflect utility try report observation adaboost notably marker cancer factor commonly give insight worth investigate experimentally identify pac include prominent marker disease prominent marker gene identify disease discover gene cancer breast b gene biological regard finding study follow protein complement nuclear protein classifier breast er interact breast cell also discover element second gene gene identify microarray md express er cancer confirm rank eight gain ig sm statistic tt sum one cancer identify rank four criterion ig sm similarly rank sm ig tr tt provide strong cancer dataset marker division perturbation one finally discover gene regard relevance system dna aim attribute characterize three formulation different principle sparsity generalization small size trading addition seem extent allow yield competitive classification utilize significantly traditional feature selection approach approach need basically furthermore generalization practical potentially guide dna gene propose find validate various justification approach throughput provide approach yield marker utilize gene microarray validate rt costly impractical full gene finally mention wide relevance significant implication design justify approach combine generalization guarantee result classifier property assume significance limited microarray limit approach domain array within acknowledgment science engineering research research grant research medical objective performance approach successfully goal analysis size limit give bound far expression learn conjunction sample bayes identify reliable identification dna much give tight guarantee unlike propose approach design application microarray feature accurate depend attribute far associate huge obtain guarantee acquire biological microarray investigation genetic parallel front result two type normal base gene dna focus give insight microarray quite variety reason easier oppose gene microarray deduce interaction number facilitate make technology gene indicator disease subsequent lead well disease attempt yield instance involve subsequently combination gene another centroid e appear traditional filter attribute perform conjunction acceptable empirical approach theoretical justification work come formulation algorithm combine feature selection consequently selection tight combine algorithm classify microarray correspond measurement part motivated variety strategy extend immediate classify framework lead class guarantee exist error absence sample three optimal code string compression attempt classifier separate subsequently empirical microarray task string predefine bit one respective minimum attribute definition attribute take way need fall bit identify value risk depend receiver priori possible give priori string length preference threshold string decrease bind rapidly dominant contribution definition finally attribute
close hyper k rp exponential less used investigation reveal occurrence substantially small reduction limit singular recommend approach acknowledgment thank anonymous useful suggestion support natural engineering research r krige j genetic convergence w circuit buffer factorial hypercube f compare capable frank rigorous surrogate convergent finding solution simultaneous equation grid van md bayesian process numerical investigation phenomena krige fidelity evaluation distance optimization expensive function blind assessment evaluation highly computer simulation simulator comparative traffic architecture bayesian ill system regularization computer code deterministic approach contour estimation complex mit j w constrain computer stein simulation use hypercube stein l ny engineering pattern search solution incorrectly formulate van ed electrical theorem example mathematics ns department expensive deterministic computer desire statistical spatial commonly simulator determinant computationally due close overcome introduce along inclusion cause unnecessary smoothing interpolation gp inverse use model physical engineering process expensive consume simulator say replicate decade deterministic widely computer al design deterministic deterministic circuit simulator analyze reference prefer demonstrate preference stochastic counterpart paper simulator deterministic confident deterministic simulator realization gp simulator desire deterministic computer simulator numerical solver differential accept simulator representation paper gp simulator gp maximum technique approach inverse several positive definite also ill study conditioning quality fit experimental design ill conditioning krige use singular near use sum gps overcome near stage krige model lee predictor second develop new approach reach tolerance effect iteration require desirable popular overcome unnecessary section iterative several illustrate remark recommendation practitioner portion usa high portion difference water low period energy extract hereafter notion propose electrical energy consider infeasible variety economic recently rapid wind place diameter ideally find potential extract greatly portion power optimally place numerical examine power simulate water grid see triangle differ model set triangular center triangular david institute chen power location simulator average location give objective turn gp fit simulator detail undesirable interested simulator maximizer function put location prototype roughly smoothed power helpful lead computer simulator denote response respectively simulation hold simulator gp distribution al like square sense popularity radial krige discussion stein exponential discuss vary slightly structure gaussian gaussian develop structure replace et al detail fit numerous often use determinant ill condition large norm correlation unstable ill precise computation likelihood ill often pair close close k neighboring design size setup design et al pre contour quantile popular ill conditioning condition condition white vary z z gp model produce gp viewpoint interpolation desire tolerance achievable section along major fails overcome ill conditioning fix close introduce unnecessary unnecessary smoothing ill singular condition threshold objective threshold behave van diagonal shift small function point close expression hence arbitrary unknown like often pre process design prefer goal follow near expression infeasible obtain eigenvalue course compute numerically matlab build compute low threshold getting behave near sufficient follow stein choose consequently simulate compute proportion near figure matlab find simulator suggest ill condition system recursively iteration regularization final iteration one cholesky decomposition follow forward propose interpolation accuracy lemma iterative generalization von series version taylor series th predictor popular first von log correlation behave define converge also result near singular behave even near iterative practice propose choice estimate change portion surrogate increase secondly different value numerical change unstable recommend optimize profile regularization outline depend key implementation computation lower design replace optimize profile compute use compute iteration depend interpolation one build stop specify use predictor measure predictor von lemma tend converge appropriate achieve matrix original whereas application illustrate even good popular example simulator eq illustration hypercube ill condition turn condition contour simulator outline section implementation close fitted significantly reality gp fit mle fit iterative show summarize term von accuracy summarize fit hypercube design matlab build design interpolation surrogate fit behave small fitting interpolation forced turn computation propose lead improvement interpolation summarize result build matlab design iterative improvement approach denote denote applied simulator give variable scale fit use propose section generate evaluate median tail fit gp set hypercube design matlab specify estimating approach c popular value process row table indicate behave size correlation matrix behave interpolation increase design propose require fix ill much small get singular become less likely dimensionality accuracy point hypercube example h popular denote realistic simulator flow surface head head variable compare median obtain gp model surrogate fit hypercube simulator candidate keep c expect popular approach optimization last near need improve table summarize value choose accurate approach certainly popular apply c simulator coordinate present hour processor computational cluster cluster sided increase
rs selection match snp rs report though criterion turn strongly snps report namely interestingly gene snps lie many snps share clear must gene genetic variability testing correlate snps represent snps perhaps remarkable detect snp approach use modification bic statistical argument preferable comprehensive confirm individual project important trust marker influence correlation causal snps positive power might widely discuss phenomenon discussion miss snps effect indicate really still study include snps rest mainly sample complex would study individual difference testing deal huge potential rather strategy simulation manuscript strategy present mapping might useful discuss certainly remark theorem conjecture proof association publish marker elementary statistical consideration clearly trait bayesian criterion deal comprehensive simulation snp substantial proper tend false link causal complex aggregated snp snps believe explain power advantage datum publicly project year wide study genetic review involve g major detect trait quantitative categorical genetic marker snps snp million snps within marker lead paper report single marker test snps review recommend significance family wise error significance understand likely correlation include permutation like control recently interest trait age snp individually account fairly design marker marker generalize model due notice classical criterion criterion even bic marker bic suited situation marker marker signal informative prefer relate multiple version correction series base recently property section ease presentation consideration demonstrate single marker stress marker test incorporate causal section particular selection criterion dimensional multiple deal closely range great apply huge search particularly huge marker strongly associate trait perform strategy simulation study rather surprising insight procedure snps detect causal marker procedure publicly quantitative project detect snps lie region snps snps report able several case motivate discussion context regression principal conclusion might study quantitative trait snps l k index snps person snp q sake covariate complexity intercept deal snp additive model index order marker large marker model elementary tell square x usual residual square ie f nan none causal snps statistic calculation rather straight figure model model snps sum residual chi test ratio much considerably incorporate effect power test effect fix orthogonality power situation snps marker consider kk testing expect causal orthogonal complex trait influence concern marker variance marker orthogonality thing slightly correlation might joint causal snps problem snps false loss marker complex justify favorable marker assume model denote statistical like aic function form aic bic linear coincide minimize parsimonious aic go infinity consistent explain assign model dimension much likely choose bic formulate criterion model interpret causal snps snps chance incorporate distribution minus logarithm causal snps orthogonal closely correction rule control family recently new extended criterion prior dimension coincide bic model prove assumption maximal dimension fix trait sparse undesirable encourage pick large article consider equivalent regressor context work thorough relate similar suggest criterion extra select asymptotic snps question selection interesting search multiple advanced strategy simulation whose step modification fact snps snps initial single marker test snps take snps consist search bic marker p proceed snps marker decide bic enhance snp consider snps snps practical selection lot causal principal bic square section subset snps population manuscript p gap file txt subsample study comprise homogeneous population snp snps snp neighborhood snps snp predict two trend large yield huge association use relatively explain procedure order snps respect entirely order snp substantially detect influence snps square detect causal snps give numerous positive snps quantitative trait simulation threshold first snp runs causal explain snp really positive detect snp positive c snp snp snp snp snp snp snp snp snp snp snp snp snp snp snp snp snp snp snp snp snp snp snp snp snp snp snp snp snp snp snp snp snp snp snp snp snp snp snp snp snp snp snp snp snp first observe snps simulation clearly distinguish snps believe instead report snp snps report value fairly small decide consider suitable exception detect classified relatively snps detect twice time might classify causal snp expect nature frequency table frequent snps several practically snps detect positive correlate prominent snp time false correlate snp explanation effect centrality rewrite square q proportional individual centrality become number signal pairwise snps chance fp first show snps centrality regular behavior causal power snp crucial question report detect multiple snps p might effect false snps trait fluctuation sample correlation different false positive figure give snps occur positive correlate snp turn positive snps relatively sigmoid plot detect causal snps vanish association influence missing gene chance detect false snps functional testing procedure trait et analyze expression population european background china major objective find association snps snps mb probe permutation obtain additive exclude individual consider population datum four population base permutation originally col number tag association col snps match account structure col value col rr rr snps cat e hmm hmm hmm association snps pooling gene detailed supplementary result comparable additive believe would interesting population snps link list snps decide snps snps snps project accordance snps snps snps bring information marker marker replace regressor modification snps snps pairwise choose accordance snps selection search procedure local step strategy perform selection combine snps detect snps perform backward naturally snps strongly correlate comparable apply snp accordance
constant low fix integer conditionally q case indeed realization pair belong I pair act proof modification assume q take conditionally see guarantee kullback leibl expression analogous property completion close establish frobenius upper uniformly let inequality addition note get thus prove denote rr x cf p assumption satisfy least large entry constant side extend replication determine reformulate gaussian eq give denote restrict canonical basis q ab oracle necessarily dictionary value inequality replace modification improve inequality brevity follow immediate bernstein matrix define least norm let dimension constant easy hermitian self cf give statistical surely eq apply distribution follow recall n x xx separately lemma c deduce proposition absolute eq condition exponential union variable corollary scale dms part linear corrupt general sharp isometry apply estimator admit simple form satisfie fast logarithmic factor low coincide constant recovery rank also statistical find approximate restrict estimator sharp inequality pair observation though obtain motivation matrix convenient model independent mean scalar product bilinear denote uniformly canonical form orthonormal matrix particular interest clearly isometry however isometry usual cf since space operator general give orthonormal subject type reference design replication nonzero zero column identity r generally learn interested consider identically model reformulate longitudinal design matrix I replication subgaussian root compress sense either I rademacher noisy setting study analyze treat dimensional dirac design become usual accordingly become us deduce consequence general inequality improve sharp successfully emphasis paper example suboptimal matrix active rapidly penalize nuclear von penalization also worth point application exploit procedure involve regularization norm mainly coincide usual emphasis paper set thresholding factor simple matrix absolute intuitive bind error suboptimal obtain suboptimal slow hermitian nuclear motivated density optimality derive prediction error frobenius finally discuss frobenius class prior main section derive sharp matrix rank low bound rate briefly implication devote appear fact rectangular decomposition orthonormal orthonormal quasi trace duality property subdifferential exist q discuss introduction satisfied equality role q follow brevity matrix belong normal cf easy represent belong normal cone arbitrary representation arbitrary monotonicity duality identity fact deduce mp p mp p mp q l l l frobenius matrix satisfy condition recall support denote cone linear equip frobenius intermediate linear cone dominant low selector vector eigenvalue restrict design design let yield l also characterize minimize penalize risk minimization coincide random become lasso I sum bound bernstein union implication imply matrix completion explicitly singular singular v form thresholde subdifferential minimizer characterization matrix consider representation understand property always preferable computational standard become view oracle inequality probability almost surely inequality next theorem denote absolute constant possibly uniformly distribute pair satisfying almost follow lemma theorem improve concentration choice maxima indeed maxima respectively remark form q equal useful minimax depend view suffice normalize frobenius error large
physics stanford university keyword lie code transformation scene lie fit eigen basis reduce allow inference encourage discovery sparse minimal distance affine operator video sequence show video description frame difference standard decade research learn image sound representation efficiency wavelet representation observe pattern video video code content change approach largely translation encode motion result occur quite motion change onto plane translation large prediction bit seem change transformation dynamic lie transformation may smoothly large visual transformation affine transformation intensity spatially localize version transformation capture affine temporal despite simplicity lie part evaluate gradient previous full video change frame full transformation coefficient technique perform operator lie describe inference local extensively transformation translation show overcomplete arbitrary initialize inference coefficient initial pyramid solution resolution seed piecewise coarse minima searching proceeding part constitute indirect smooth robust estimating transformation directly operator demonstrate infer learn reduce tractable smoothing transformation inference test contain operator movie implementation transformation frame generator occur seek ensemble video infer minimize rule naive operation computationally reduce approximation eigen perform term matrix consist complex hold eigenvalue matrix facilitate periodic transformation rotation orthonormal benefit eq therefore replace multiplication non many white transformation generator translation perform problematic overcome map transformation replace minimize coefficient along transformation direction u effectively along transformation direction u along translation operator rotation single inference match coarse way dot show value local minima translate transformation change visual transformation index transformation order maintain due constitute lie group transformation generators structure affine transformation capture though encourage learn operator learn patch direct move sum distance act code encourage occur long transformation pool patch video transform range pixel degree scale horizontal inference transformation fraction recover recover reconstruct db transform evident fraction recover coefficient image code simultaneously recovery patch reconstruct majority patch reconstruct ability transformation translation rotation skew patch transform used affine operator motion horizontal rotation compute learn operator exhibit property correspond pixel location array influence instantaneous intensity pixel basis show image patch block pixel location therefore location contribute instantaneous change intensity pixel note block spatial motion translation patch b c intensity interpret pixel image consecutive frames video move patch apply central patch wide buffer transformation pixel wide buffer act pixel horizontal vertical translation expect focus transformation contain full field translation translation provide useful basis exist motion base translation variety transformation learn intensity contrast spatially transformation learn capture video patch reconstruction transformation frame pixel pixel pixel bilinear central compare pixel translation smoothing horizontal allow vertical horizontal translation smoothing translation transformation operator unsupervise fashion learn fashion hard code translation figure increase become operator frame also translation substantial pixel suggest useful employ standard describe image movie build previous operator transformation make key eigen operator allow tractable operator reduce minima infer video translation motion attempt image translation
proof convergence euclidean nonsmooth apply explicit regularization simultaneously regularization statistical comparative empirical error vector similarly define dd excess comparative theorem exist lemma require moment nonempty x therefore desire assumption q conclusion write right hand bound unlike set expect mean due thus still term depend decomposition control let sample denote entry replace verify role desire denote easily rademacher variable complexity es desire q constant q direct application rough follow bind I directly rr apply sharp desire assumption conjunction conclusion lemma p convergence analysis analysis manifold behind gradient controlling term regularization excess exist directly decompose excess define structure true set lemma proposition rr sharp estimation omit argument prove derive problem overall splitting algorithm positive n rkh completion span lie dimensional span convert finite functional span function function k transfer infinite optimization coefficient c j pi nc nj nf nc column non optimization descent newton method etc develop backward convert change semidefinite k identifying follow set sparse gradient simply similarly note forward commonly process split backward splitting estimate proximity operator q frobenius need term relatively therefore q problem subdifferential calculus equation q subdifferential subdifferential element set see optimal iteratively convergence minimize operator correspondingly derivative small threshold variable keep unchanged norm norm furthermore convergence theory split converge regularization sparsity sparsity large solution correspondingly select practice gradient solution none obviously minimizer large iteration minimizer eq desire initial initial focus vector derivative wise thresholding entry wise involve weight could small introduce transformation main note rank high use singular decomposition unitary notation involve number calculate nr use greatly involve inefficient sample computation kn n kk decomposition v nr k ki pd perform I singular decomposition desire sparse briefly introduce py output value also binary define fitting ny I odd posterior mild regression taylor pf jj j consider expansion extra term formulate minimizer nj set minimization reformulate p split backward splitting become unitary dimension omit artificial expression method nonlinear setting equally detailed mention simulation study minute gradient lasso point lasso assume view linearity equip simulate five uniform form contribute ten uncorrelated lasso symmetric respect select norm limitation bandwidth median pairwise point regression select regularization vary able select five summarize repeat choose return five variable select lasso select fail treat contrast much great frequency advantage lasso c lasso h k partial derivative regularization respect dataset sample lie dimension half point spherical dimension noisy draw sphere space implement distance htp level vary correctly capture emphasize generate two report return dimension reduction call capture original dataset compare derive preserve explain plot norm derivative method sparsity small empirical ef test surprisingly fail omit variable dimension expression gene expression thousand sample become important biological widely accord tumor type difficult datum build datum code response represent variable unit normalize variance apply extract reduction compare loo datum implement choose distance point leave lasso leave one loo table implement although method perform build addition lasso lasso illustrate pathway distinguish type pathway selection many selection simultaneous automatic reduction optimization algorithm generalization regularize regularize integrate advantage previous refine one point lie rather implement calculate occur semi supervise several direction often efficiency use implementation term j mainly motivate paper introduce I regression know laplacian approximate support marginal intuitively smoothness penalty supervise view generalization remark proposition approach traditionally treat integrated call selection gradient impose gradient select derive covariance error gradient one develop forward backward practically scalable medium extraction method loading efficient biological science biology common million snps modification million site increasingly deal response many variable selection fall metric combination strong selection aim drawback evaluate group problem function term accounting prediction term control elastic net widely implementation perform try overcome base smoothing spline make impossible high dimension commonly approach deal belief real concentrated manifold accuracy visualize therefore likely suboptimal finding onto capture go sir explore subspace sir impose distribution regression easy limitation sir yield classification include variance save contour limitation direction quite direction include method function methodology derive term worker estimate gradient use reduction offer tool dimensional available variable notable exception principle analysis produce principle sparse loading mainly use linear dimension reduction variable nonlinear motivate combined microarray expression normal tumor microarray likely responsible expression cell dimension feature subset gene improve remove noisy focus extract biological extend worker selection supervise optimization gradient prediction irrelevant partial expect variation p hilbert associate optimization use norm sparsity component make norm widely impose possible value gradient fidelity term term propose optimization refer q key ridge appear significant many component potentially derive comprise primary depend impose help eliminate improve infer relevant et case learn view space directly since impose regularization remove limitation gradient special lasso invariant regression learn make linearity thus extension nonlinear framework
randomization sequence increase square finite write general form unbiased standard unbiased provide integrable n e eq alone positive lagrangian determine constraint finite entirely believe least tail sequence deterministic convergent complex shift geometric minimize wish constraint occur variance stop mse approximately plot interpret bad scenario indicate rate little relative increase mse eps optimisation computation generate integral since minimization budget minimum occur otherwise somewhat fast achieve large budget produce choice unbiased toy wish require initial iterate eq course step estimator use estimate start close adaptive choice permit large sequence quadrature rule integral particular variance therefore variance estimator compare easy l interval efficiency monte integral advantage replacement exist create obvious neutral price asset volatility brownian drive asset run volatility black price option volatility rate option yield price option neutral conditionally usual bss even exact latter unbiased raise choose functional page euler order sufficiently continuously drift case eq suggests shift reasonable shift price standard error evidence simulation agreement option price condition positivity fail volatility may question value minute run matlab intel quadrature equation bias bias simple problem finance provide extension allow theorem axiom conclusion exercise theorem process sequence deterministic numerical integration obtain carlo involve approximation unbiased value include integral root find pricing volatility finance
like anonymous suggestion nsf fellowship nsf theorem goodness fit division bin goodness division bins powerful availability computer efficient box variant problematic many circumstance feasible confidence level chi goodness fit test norm identically distribution model consider value accordance terminology class common bin use root construct empirical page draw fact root statistic confident draw square draw root square draw come confidence specify significance namely unfortunately mean statistic distribution avoid average root probability associate various availability computer direct box calculate monte carlo statistic would standard statistic easy circumstance extensively root mean asymptotically statistic example present article detail level goodness computation discuss involve part confidence sum independent briefly illustrate power square direction research detail section come specified confidence draw mean draw unknown specify root statistic draw begin notation statistic analyze associate bin obtain bin perform number obviously expect statistic statistic focus goodness fit replace contrast use multivariate central joint limit dirac concentrated origin restriction origin multivariate define sum variance axis variance square particular draw restrict along principal project onto complement consist clearly adjoint construction eigenvalue axis diagonal computing require usually efficient unless hard accommodate homogeneous accounting analogous impose extra describe cdf center tool theorem cdf evaluation gaussian variable number addition random variable cumulative root take characteristic principal branch function define cdf number principal almost cdf identically calculate yield cdf value branch cut though contour obtain eq contour thus decay fast cdf gaussian obtain goodness level draw arise cdf draw figure versus six formulae summarize attain display tail describe cell probability draw specify quadrature require evaluation figure total second display figure bin constant positive real great involve run example ghz intel mb l cache less digit take precision yield digit exploit precision digit suffice computer high fast example section fig vertical axis horizontal axis e k statistic classic statistic comparison complete much comprehensive treatment constitute root observe bin bin whereas draw take actual root statistic square associate classic pearson statistic label ratio statistic convention plot hellinger line limit number draw example however differ level simulation rely say draw draw model mean least simulation simulation simulation draw I generate successfully arise significance level simulation confidence great significance define show root statistic fail plot draw specifie bin draw increasingly increase example success example statistical plot remark distinguish bin root opposite fig draw plot define remark specify example specify q via draw consider estimate likelihood specify
correspond krige analytical formula give local indicator distribution new sensitivity maximize condition case moreover particularly deal computer select experimental informative reach test also sample independently well hypercube variable xx mean mean obtain scheme reach variance form poorly enhance fill dimensional one powerful distance etc criterion lead design conceptual popularity application uniformity initial volume interval interval exist kind form functional measure norm easy compute measure remarkable center discrepancy good term amongst exchange genetic find simulated anneal temperature slight result criterion point ht space design robustness dimension indeed ensure proportion contrary create dimensional project onto design reveal non step sample solely include retain input compare initial reference consider lead criterion behave two keep criterion value different size three conclusion computer ht numerical determination percentage xx involve dimensional provide result time initial output toy evaluate mean efficiency robustness design discrepancy optimize optimize increase show conclusion design property much variability important lead width width htb dimensional analytical several comparison design concern repeat toy evaluate coefficient design discrepancy optimize summary furthermore conclusion distinction quantitative sensitivity study gp even dimension guarantee point projection type result systematically less design fit course design one step instance simulation fit estimating issue safe sensitivity precise capability input fit computer compare prediction residual analyze measure coefficient call validation require calculation computer face point cpu sufficient tool validation answer test approach monte localize example near point leave large domain fine strategy could avoid proximity proximity optimistic bias validate cross propose divide sample learning sample residual obtain sample residual use global leave validation become geometric cause new adequate might quality many prediction optimistic therefore solve main drawback minimize test discrepancy step away point perform process choose prediction capture additional idea computational efficiency method mean error bias vs xx sequence pattern low discrepancy discrepancy center discrepancy additional sequentially design point discrepancy difference initial new low discrepancy require especially design advantage size add point compare design validation analytical toy represent ht gp sample range wide variety initial increase sequentially point keep optimality maximize distance design choose design contradiction objective validation reference coefficient leave point leave process curve paragraph minimal repetition greatly satisfactory case maximal value interval obtain increase contract fit coefficient design add close compare give confidence rather large choose carlo evolution basis range dot blue sequentially carlo illustrate poor monte contrary sequential validation size precise monte blue dotted blue line nuclear water break peak scenario part model operation light water propose economic operation development implement illustrate give q exercise peak temperature scalar cpu minute iv pc lie consider law essentially material either normal variable fit dimensionality make thank devote include center measure range design give estimation begin result case estimation clearly validate confidence extremely large variation dimensionality sample lead optimize red relevant expensive I resolve uncertainty sensitivity test concentrate year widely use objective discrepancy measure adaptive design gp design adapt availability predictor secondly leave use design put required test real application analytical minimal necessary validation effective useful sequential acknowledgment support p france fr computer instance often expensive directly use sensitivity robustness
prevent incorrectly outlier correctly outlier carry variant real situation scheme primary period failure gradually teacher curve student topic machine understand algorithm make flexible function favor outcome although ignore simply maximum outcome prediction discuss boost source computation naive classification major limitation fail feature take take represent similarly replace independent gate pair form overall product interestingly come criterion replace bin arbitrarily sufficient represent number choice bin increase feature become code bin construct memory bin fall bin link datum bin bin always train read express bin intersection count bin unity distribution function theorem initially uniform fail increment incorrectly bin incorrectly show iteration increment fraction increment factor learn paper name network central rule compute equally outcome problem range hereafter research high uncertainty observe default optimal entire appear straightforward outlier issue represent illustrate distribution prediction call clean example green distribution sample classify without high confidence estimate prior cause cause figure region red red incorrectly observation alone red dominate red example overcome estimation difference become classification class may use training remove difference classification might straightforward effect become nontrivial optimisation argue school method go round describe section balanced subtle difference feature classification pattern equally likely prediction generally flat pick one training close take training test comparable fraction sample criterion fail wrong fail test data failure compete incorrectly round failure round cause fail sample add round bound fluctuation likelihood converge unseen test sample remove difficult boundary outlier test add give datum start train classifier training file test add train classifier classifier save pick add datum fail look process collect test incorrect label incorrectly cause subsequent fail similar good example propose teacher student correct come example important note optimally select rest datum unseen training estimate difference identify would good produce boundary fuzzy classify example deep project unlike laboratory infer object basis namely pseudo curve learn curve test example pseudo level gradually fail boundary example label practical detail increase rapid prediction green accuracy sample time boundary class account spike curve prediction accuracy fall mostly cause incorrectly label object fail maximum confidence turn incorrectly pick boundary get however long network accuracy represent pseudo curve continue remain low incorrect get rather assign dataset training continue note spike degradation examine time mostly update continue eventually optimal training less realistic accuracy entire uci repository consist neighborhood pixel predict code respectively grey grey type grey dataset available uci training test uniformity evaluation well report name training original select optimally datum sample optimal selection maximum optimal selection ie dataset provide uci repository optimally datum result accuracy comparable unseen training obtain slight test mix training uci datum maximum entire uci repository result show attain uci repository good accuracy good heart cancer classification accuracy interesting fail subject tool get level give confidence level use take like reality center galaxy although band many believe massive emission integration large device survey one extensive survey discover high simple filter call colour five colour colour input correctly predict spectra appear star object also spectral universe introduce drift efficiency region shift identical add sample region overall classification reduce assign confidence belong overlap patch identification observer overcome improve observational figure histogram release methodology know concentration blue correctly red incorrectly star since galaxy remove remove fail example fail confidence cut important point note object resolve colour outlier totally object identify object object exist colour another predict verification light surface address light evaluation trivial verify ray many precise measurement either observer thousand star different magnitude across star broadly class name basis nontrivial measurement star galaxy assume ignore case spectra impractical method straightforward state predict type spectra belong class sample correctly network test table table accept illustrated spectra good entire
graph topology propose average minimize local efficient sharp network walk update include stochastic communication protocol protocol communication provide update dual body agent maintain gradient update q jt proposition arbitrary xt combine distribute composite dual finally lipschitz composite projection ex ccc department electrical sciences berkeley berkeley berkeley berkeley decentralized network form nonsmooth convex localization sensor analyze average sharp topology allow clear separation effect arise structure gap network prediction confirm low well optimization development problem network structure arise variety domain science agent tracking sensor naturally distribute common necessity decentralize locally light network periodic failure example quickly processor challenge canonical arise minimize average e vector desirable small subset datum different processor minimize loss entire dual paradigm know significant seminal analyze process important network fast convergence general processor must agree recover sharp consensus study topology path argument underlie e reference allow stochastic gradient average minimize equality rate shift focus processor potentially whereas recent al project subgradient distribute minimization non contribution algorithm distribute maintain average essentially facilitate issue mean topology careful demonstrate close spectral analysis term optimization technique nesterov splitting issue constrain optimization stochastic elegant averaging scale chain protocol deviation comparison previous characterization scale term often give paper network tight polynomially topology require paper guarantee optimally obtain optimization essentially topology connect graph network tree protocol attain dependent structure current maintain optimum online expect comparison give network scaling scale spectral graph factor obtain solution cycle iteration two bound simulation excellent cover begin study communication protocol cluster fix dependent randomized communication protocol well failure communication tradeoff transition theorems structure regularize objective often remainder section formal statement whereas main consequence main section depth basic derive depend network treat communication present collect throughout one vector means write formal statement distribute averaging optimization specifically undirected vertex edge subject belong convex sub need assume generality translate maintain impose directly immediate neighbor ni nature arise domain motivation follow sensor equip device environmental application sensor temperature computation minimize scalar function variance quantile estimator motivate learner empirical processor suitably processor cluster processor directly small case computational dual averaging design minimization nonsmooth discuss extension set dual strongly satisfie make generality proximal quadratic convex il cost satisfy lipschitz lipschitz denote generate step step receive eq type iterate first approximation proximal stepsize seem originally nesterov relate appropriate novel dual distribute maintain pair associate node subdifferential receive node neighborhood base weighting doubly stochastic notation update projection compute compute projection stepsize sequel convergence local optimum locally manner definition sequence weight see average equation version investigation corollary follow dual averaging provide define basic sequence generate access locally sum term common subgradient third estimate deviation extra significant concrete convergence roughly deviation tt v asymptotically approach statement optimization subgradient incremental subgradient distribute algorithm advantage per rather highlight subgradient machine statistic minimize randomize incremental access every lead disk seek algorithm avoid gradient albeit turn investigation communication occur doubly singular summarize projection pp connection method quite since know rate walk tie walk graph probability explicit interesting four topology first placing connect analysis connect near neighbor panel grid topology network panel random geometric place random connect node separate pattern device wireless node extensively generalization position distribution finally show belong sparse good property attractive option distribute computation spectral gap graph typical probability satisfie see construction design construction interest specify matrix choice adjacency ni invertible doubly stochastic graph need order doubly illustration protocol dimensional connectivity let symmetric construction moreover doubly technical relate convergence summarize network topology cycle grid graph high probability bound ratio factor optimization remain term topology instead state size topology corollary define actually sharp mean centralized optimization iteration address issue spectral follow establishe network function graph give hard require conjunction implies predict quadratic match varying matrix still obey impose communication want decrease network refine communication incur hardware failure provide doubly stochastic theorem upper bind consist last shrink stepsize tradeoff compete stepsize minimize yield boost hold modify penalty communication establish convergence bound whenever consequence contrast stochastic communication fix dependence scale fast finally none gradient correct straightforwardly result case corrupt wireless observe field information satisfy special subgradient add gradient give discussion note average theorem stochastic update agent receive addition l uncorrelated communication cover essentially replace result result introduction researcher design maintain update minimize stepsize give clear slow quite proximal address euclidean geometry example dimensional simplex e become technical difficulty update precisely work essential fast ease extend stochastic incremental maintain token determine token neighbor let update q optimal ip pi bind never weaker tight dimensional grid well define quantity use via sequence show average evolve q matrix consequently evolve almost like subgradient subgradient subgradient average avoid linearity challenge early work proceed regard average give dual averaging non restrict easier analyze centralize convexity break piece remain control lipschitz continuity projection divide side convexity give concrete dual averaging walk doubly crucial cluster location sensor sensor corollary adopt notational simplex frequent eq brief relevant frobenius refer control namely sequel via write evolve notational clutter break cutoff term second consist step summation concavity sum basic see appealing allow note thus ft statement proof bound spectral matrix upper spectral devoted bounding note control walk imply random walk equip address graph cover recall place neighbor imply th q perform taylor expansion node neighbor right kn thus regular connect substitute case grid specifically horizontal vertical analyze grid eigenvalue product small precede cycle boundary nk grid corollary b immediately see von small particular say obtain geometric graph factor gap remove corollary give dependence tight objective convergence slow eigenvector matrix eigenvector eigenvalue equal walk generality eigenvector loss generality otherwise flip sign indexing need define lipschitz see evolution q one guarantee eq little appropriate matrix theorem failure sum tt show sum communication agent occur underlie network relax vary doubly update usual namely analysis make use however still evolution unchanged eq communication control uniform claim modify simplex arbitrary recursion symmetric chebyshev replacing similar random index high probability break separate throughout structure ease q make remain sum doubly stochastic ns q master outline sampling giving procedure consensus quickly edge drastically communication network yet fast topology asynchronous protocol computation proceed round communication clear adjacency diagonal matrix identical send round edge graph cluster centralize select round vertex still achieve factor graph relax function agent stochastic simply combine type suit completely decentralize environment communication protocol edge edge edge failure protocol psd psd hermitian upper average communication protocol node pick neighbor pick double matrix random adjacency pick dt adjacency underlie stochastic define definition dt slow factor proportional maximum degree amount factor algorithm edge fail independently use communication dt underlie apply factor rate generalizes receive subgradient receive easy virtue simplicity dual network prior care analysis pass nonlinear thereby obtain derivation nothing assume h old imply recall proceed put around norm couple argument q complete statement show statement x
independent orthogonal another distribute course claim origin qx independent generate transformation arise jx overcome choice sign transformation expensive number multiplication per generate take special use plane rotation vary compute angle convenience compute matrix multiplication one division detail loop implementation similar inner generalise generator observe desirable value pool pass transformation experience aspect implementation generator several appear plausible produce acceptable fail regard value additional hoc odd mod permutation odd appear satisfactory seem ad hoc little theory sum number would chi return user sample scaling ensure permutation phenomenon become less increase entirely pool rare variate occur I pool thus event adjacent devise correlation significant reduce easy discard generator every pool generator generator conventional generator although generator entropy random number uniform orthogonal etc uniform per normally pool rejection choose generator period least normal orthogonal random generator great uniform random generator contrary spirit long period certainly avoid generator could guarantee period period extremely unlikely care need normal generator end discuss generator conventional generator ignore speed acknowledgement comment david version unfortunately pass version computer journal outline contribution recent generating normally without rely number generate random number idea uniform generator discuss avoid many graphic anneal carlo substantial pseudo number dedicated contribution several aspect hardware hardware generator mention software generator normal uniform idea appeal transform consume give number pass construct product make public exchange hardware device stream device connect map appear content technology idea depend availability volume stream obtain generator close word require stream bit require popular normally random number rejection von recent reference elegant generator rejection distribute pseudo average normally generator slow involve slow uniform pseudo method five machine generation discovery depend generator idea pseudo fx distribution statement calculus bayesian entropy annotated idea speedup percent slow generalise random generator report probably well restrict comment one arithmetic point
fouri insight recent thin tree proxy optimally thin bethe entropy rather sample derive develop thin tree bootstrap method bootstrapping analyze tree experiment validate implement theoretic test machine ghz processor analyze begin experiment quality issue understand approximately vary relative able approximation despite comparison run split recover perhaps surprisingly split unbiased factor return effectively ranking far sample fourier capable handling order run fourier running set structure simulate draw jointly thin vary able possibly require recover full underlie correct trial indeed also note recovery correct model log likelihood draw distribution figure structure thin know generate chain know eventually enough also note jump jump success rate leaf long correctly run simulation datum generate structure mean recursively partition sized figure also interestingly discover discover nearly balance less chain analyze dataset full ranking ten compare twice many item ccc sake roll roll study unlike set divide directly sample use split evenly group distribution biased plot help significantly lower bias useful behavior show approximate first ranking bias marginal people people provide distribution sample somewhat say interpret certainly related type cluster typically choice away remain item understand small bootstrap size partition recover leaf correctly leaf leaf recover sake leaf apply large house use candidate five candidate fine detail well candidate party fine english fine p party show party candidate party candidate notably party low rank portion may necessarily candidate independent minor party candidate order independent visually marginal marginal significant matrix belong candidate approximate plot principled see visually exhaustive running second include mutual fine leaf leaf small stable smaller learn recover leaf major agree original tree insufficient candidate belong group party voting hierarchy thin chain likelihood achieve hold training improve one overfitte practice learn correlation suggest rank correlation crucial run similar support north west show consistently leaf dataset consistently group dataset potentially exploit independence complexity idea throughout show bayesian problematic indicate main immediately pt pt retain advantage real item rank generalized assumption sample answer explore recursively real currently method depend ranking however compose easy ranking rating extend parameter algorithm handle valuable mutual already rank interesting estimating ranking learn understand careful small place structure considerably able simply identify independence independence analyze rank potential give new insight believe crucial develop procedure form acknowledgement support n provide dataset idea upon item since number absolute necessary abuse count datum equivalence generated rewrite composite evaluating likelihood first item item absolute rank item count along input optimize optimize thus datum objective equivalence rank independent rank natural objective certainly necessary simply generate ranking ranking recurrence write recurrence obeys recurrence delta permutation take support size recurrence know recurrence binomial array structure p computing binomial indicate collection fourier transform write recursion uniform particular give fourier fourier coefficient branching see detail linearity convolution fact domain arrive fourier recurrence irreducible detail irreducible implement recurrence careful dynamic sure thing triangle binomial graphical adapt variable estimate triplet probability amount triplet use triplet h union bind mutual triplet within define complement let subset strongly n kn triplet internal internal define triplets eq goal connectivity strong triplet inside lie internal e b k eq bind formal argument lemma plug connectivity triplet edge ba establish desire pn root complement strongly ab uniformly k almost require simplifie behave I bind hold probability conclude theorem empty theorem fact permutation recent storage argue full strong structure expressive family distribution reduce complexity draw permutation game form rank ranking formal fourier theoretic framework automate discover ranking dataset decomposable class machine ranking reasoning preference survey retrieval certain ranking many problem ranking way ranking yet preferable tractable overfitte achieve exploit independence structure naive rank base constrain two item rank rank novel relaxed rank item rank contain ranking political vote voting typically constraint histogram recursively ranking unlike estimate structure outline contribution section ahead study summarize relation distribution introduce contribution intuitive novel permutation base ranking appropriate notion independence rank evidence relation approximately rank independence relation complexity novel independence transform biased perform scalable use theoretic factor factor empirical evidence interpretable item iteratively large recursive stagewise section partition item partition objective subset partitioning exponentially space structure dataset method assumption effective voting paper ranking association item rank assign convention think low rank rank item place mean prefer also mean rank notation certain concept express notation rank rank rank rank order notation mapping difference ranking notation permutation ranking permutation order set ranking hand permutation item item rank item item order composition permutation collection permutation object simultaneously rank run throughout analyze known rank american association candidate name five candidate year figure proportion vote ranking vote occur vote also visualize represent matrix level overall high far winner notice candidate vote place throughout via vote distribution percentage vote ranking marginal reflect rank ranking pose challenge ranking store array second marginal probability item example computation issue nontrivial impractical ranking realize dataset aware problem lead exponential algebraic fouri probabilistic independence briefly probabilistic ranking rich day expand upon body detail think analogy paper ranking allow expressive conceptual bridge popular rely family independence old ad hoc maintain good hypothesis update sense discuss permutation either distribution successfully less object detail among object mass vote figure rank vote recent research around maintain pair store number example store fig rank last follow vote divide one store order tuple perhaps encode joint rank require storage fourier transform permutation primarily marginal correspond sense frequency fouri low fourier matrix frequency fouri fourier theoretic reasonable view principled frequency contrast sparse scalable making exactly reconstruct order marginal moderately scale demonstrate exploit dramatically track confusion association define say store keep size general typically recursively without independence constrain rearrange row impose thick gray factored permutation order despite argue permutation restrictive independence mutual allow rank subset permutation complement ranking rank preferred refer restrictive block structure marginal permutation identity position field reasonably potential identity confusion track red team track team rank condition force third place seem quite approximate vote factorize solid factored true capture candidate poor assigning permutation receive vote support infinite next section condition flexible notion cut successively drop inspire novel independent subset object ranking form final object intuitively complex relationship within allow correlation generate decide prefer within decide fig fig fig prefer stage ranking preference item result offset left denote offset way independence first random walk rank drawing mapping formula h mh commonly probability distribution convolution answer question property cutting preserve relative rank say preferred assign permutation preserve rank turn permutation preserve description distribution assign nonzero permutation preserve four notation everything else formally independence fully independently fully independent drawing stack prefer position might cut independently q convolution define relative ranking denote rank fig rank notation write possible distinct map follow algebraic uniquely decompose compose stack ranking mean number show think coordinate uniquely item perhaps relative ranking every factor definition assume definition direction q ranking use element q satisfied convolution view also analyze concept remarkably ranking require convolution fact definition reader also theoretic description fully factor along ranking would capable place ranking symmetric extension draw independent sense item special independence distribution ranking delta assign delta interesting delta never delta respect thus independent fact complement later reflect even preference set relative item amongst delta ordinary independence strict figure marginal independence regime rank delta think ranking useful incomplete ranking candidate candidate first approximate distribution like independence commonly would rarely independence small sample size indicate almost always ranking ever independence dot accurate kl factor distribution visually interpretation result approximate candidate winner independent inferior partitioning however candidate examine order marginal approximation show factored candidate cc case storage full require storage family distribution simple assign relation within simple correlation across amongst amongst completely subset recursive drawing pick drop pick pick setting first marginal figure region recover preferred generalization drop write pointwise fourier theoretic call join simply sample memory bias one low employ recurrence linearity fourier transform proposition detail appendix transform would apart show relative perform deconvolution algorithm show independently split ranking set q ranking respectively rank consistent ranking eq summation exactly marginalization compute function fouri step establish first fourier transform apply convolution dual establishe notice compute mle factor know normalize split fortunately normalizing divide complete intractable computation instead receive parameter distribution quality result fourier fourier reconstruct marginal fourier reconstruct marginal likewise fourier marginal return marginal know theorem state give join reconstruct marginal joint distribution since joint operation pointwise fourier domain enough reconstruct marginal reconstruct fourier coefficient frequency run join split bad cubic fouri directly appendix must compute transform uniform biased computed plot experimental running understand binary partitioning explore natural simplification subset run example one imagine consist consist fig draw secondly set rank express hierarchical decomposition visualize tree item leave hierarchy leaf example decompose tree impose assumption suit possible structure encode common consistent call level partition partition ranking together consequently hierarchical way establish write independent leaf decomposition decomposition line since know desirable way require decomposition hierarchical thin chain thin refer hierarchical express thin thin chain analogous thin clique never scale polynomially ranking thin sequentially marginal thin describe clinical opposite political somewhat within verify conjecture use independence remove perform search nearly kl divergence identify candidate belong group hierarchical datum kl divergence group set biased mixture biased parameter indicate since lie impose independence available rank obvious particularly really next structure rank relative rank subject constraint graphical optimal distribution base address alternatively want partition tree base ranking subset complement automatically determine independently sense formally solve kullback reasonable training infinite truly independent actually single evaluating globally relative already hierarchical propose locally cluster show tractable compute prove fig six absolute whether prefer formally subset absolute assign denote interpretation equation ranking ranking ranking ranking guarantee detect independence necessary absolute ranking argument establish reverse equation evaluate converse rank determine thus equation optimize intractable however mutual triplet item mutual mutual value triplet internal triplets triplet set aa mutual triplet show graphical finding term invariant index triangle bar instead mutual view involve computation triplet time instead tuple reflect rank tell know prefer fig absolute make commonly whether objective sum problem triplet low weight internal triplet sum triplet symmetric result poorly dataset candidate easy visualization set form since highlight correspond mutual mutual conclusion show candidate mutual information surprising candidate align know rank preferred like cut graph objective tendency prefer partition balanced partition unbalanced cross avoid optimize optimize useful thin alternatively objective encourage balanced variation intuitively denominator equation exist encourage experiment proposition accounting dependency involve pairwise independence look marginal factor upon examine practical variation objective quantity triplet q measure prefer tell preferred sum arise detect insufficient ranking yet moreover would consist measure measure nonetheless subset use almost partially rank argue function access triplet approximate remain reasonable denote use regularize adapt triplet mutual estimate accuracy
several two network thing size roughly go achieve gradually indicate get gradually bad community large substantial beyond traditionally size scale become phenomenon large network network consist moderately large community notion maximal removing associated rest average importantly though remove core indicate interest identification common largely way confident domain specific conclusion responsible low partition optimization network participant recent machine draw area combinatorial scientific implicit study often lead good boost fit additive objective computation parameter somewhat fundamental diagnostic crucial development perform regularization well prediction emphasize neither algorithmic consideration computational black closely couple statistical understanding seem problem design constraint implicit statistical consequence decision discussion dna discussion co numerous participant form recent idea scientific interact increasingly development scientific dna nucleotide idea area serve exploit complementary order solve apply recent advance internet broadly term chapter I trend increasingly area trend appropriate way body diverse distinction former science adopt issue algorithmic roughly application strong specific concern include phenomenon reliable datum effort union depend increasingly etc business perspective perspective national security perspective find force union thus look versus substantial shift away issue broadly term explicit involve participant vice versa substantial shift generally great detail computational extent statistic relatively learn ph mechanic molecular water protein water science lot later scale thing I transition conceptual remarkable class computer software engineering remarkable lack come understand datum confident conclusion output fast chapter like thought work take complementary versus statistical understand make claim justify choose problem represent dna microarray dna nucleotide historical trend identify wide range micro market query internet apply previously specific internet idea internet issue would algorithmic hope two statistical algorithmic consideration apply black coupling seem scale design understand statistical would like detail modern although perspective statistical account everything customer transaction course month consist site perspective goal patterns associations association also quality fraction database frequent hard much algorithmic devote exact solution unobserved pattern achieve variability around proceed noise well former necessary event yet assign give give particular event ask question common whether undirected unweighted edge interaction model value encoding describe two novel section column exactly scientific linear motivated large computation deal computer science sort often describe genome sequence base code protein microarray device measure genome protein individual environmental variation numerous amenable nucleotide human genome nucleotide negligible snps occur genomic marker track gene population use infer population human either encode people snp encode individual computation attention application matrix svd dna microarray dna snps gene snp perform eigen capture eigenvector heuristic eigenvector certain lead heuristic may justify happen tend ellipsoid along justification domain mathematic reason easily process linear snps isolate one interested one reason task input find good dna snps pathway reconstruct etc common algorithmic include look large variance maximally uncorrelated intractable consider call subset choose exactly eq spectral frobenius span deal focus several include algorithm currently spectrum decision hope sophisticated rule practice call qr emphasis question backward issue whether run multiply result essentially involve enumeration top singular onto problem general focus typically make algorithm column fail desire typically constant big notation order exact also demonstrate column importance column fast small additive could disadvantage immediately applicable scale even uninformative heavy graph adjacency network quantity decay heavy tailed power manner frobenius bound matrix compute top proportional span accord detailed prove bottleneck subspace compute span rapidly motivate probability recall look relative directional singular hidden reason since euclidean encode encode suggest choose sampling structure send subsection sampling algorithm scientific analysis describe previous subsection describe intuitively reconstruct quantify well look l arise moreover interpretation natural interpretation viewpoint arise measure spectral amount sense algorithmic perspective relevant depending number cholesky qr full question right thing answer outcome predictor adequate rely assumption perfectly sure violate response column nice sensitive influential determine good interpretation coherence gain insight consider visually ten intuitively stick particularly leverage magnitude leverage suggest diagnostic regression investigate course turn might point statistical score square fit leverage ten point mark leverage visual inspection call color code leverage red leverage document construct mail leverage metric popular gene axis star gene red dash unsupervised leverage supervise algorithmic consider leverage singular vector span leverage equal euclidean norm column uniform hadamard extremely column identity row element use solver black subproblem vector solve opt ax opt opt highlight essential bad case l see statistical diagnostic regression analyst tend bias toward ls input suffice nontrivial many subsequent subsection singular acceptable generalization case hadamard transform hadamard tend score reason localize orthogonal projection lead development l problem run implementation deterministic thousand hundred describe leverage come perform note combine look collect heart hybrid algorithm deterministic qr perform hybrid phase let span top importance accord rescale qr return qr matrix non uniformity speed algorithm complicated sampling bound p ok ok interestingly make importance probability concept describe subsection worst critical qr matrix bad case respect bottleneck matrix subspace qr apply million column implementation traditional fail quality bound select comparable constant good previously goal dna microarray snp snp diverse non trivial goal select column generally diagnostic unsupervised selection problem algorithm behavior score typical call edge laplacian notion related tend edge community stick leverage span present code gain uniformity term document publicly mail choose plot score high leverage nearly order large size high score suggest surprising email corpus though successfully plausible generative associate reason expect nice phenomena concentration occur microarray snp present plot normalize leverage describe different cancer remarkable dna snp present evidence two phenomenon responsible term domain believe property like occur history stick express axis snps back etc snps elsewhere mutual metric particularly prominent datum maximum axis datum course strong bias model quantify condition dimensional unsupervised like supervise issue observation inner usefulness qr algorithm decision version qr qr phase one algorithm behave low tend surprising practitioner observation implementation scale preprocesse randomized phase perform room qr tend make sophisticated less randomized direction select qr keep column qr preprocesse get run qr norm directly column tend make thing thereby deterministic setting rule randomization result practice choose behavior deterministic qr choose somewhat column improvement column randomize statistical statistical leverage concept bad traditional answer seem datum score informative worst reliable bad effort informative application answer seem intuitively scale application assume base consideration surprising stick relative model suggest score statistical reason diagnostic application perspective extensively scientific interested balancing vision computer interested primitive divide network interested meaningful specific search contextual ad top engine page search typically text important construct bipartite graph discretization discretization keyword bid bid phrase perform mining graph quantity click problem example identify worth analyst well coherent thought useful testing new query market place match phrase also occur specify bid ignore game theoretic issue imagine original middle homogeneous large similar phrase nearby topology advantageous engine phrase phrase nearby original bid perform expansion application applying analyst constitute community intractable exactly algorithm heuristic problem set thereby look otherwise modify iterate might singular nice hierarchical intuition nice noise adversarial less intuitive problematic reasonable readily wrong whether improper correct intuitive insufficient resource noise etc visualization apply large lead largely illustrate visualization problematic situation network principled algorithmic statistical understand explore ccc toy nice realistic network illustration network low responsible typical community method flow find flow ask look piece equally sized opposed moderately like leave way formalize refer family objective approximation involve cut partition quality weight cut balance two lot chapter cut balance pay give possibly denote end complement cardinality alternatively substantial normalize great case eq could replace min denominator replace product formulation slightly big big preference equivalent function achieve cut formulation importantly partition intractable learning cut version partition general approach include laplacian cut cut degree expansion meaningful application robustness divide graph far deal include focus cut return node graph time analysis achieve observe basically imply task eqn piece objective note improve application million arise locality nearby seed piece kind nearby cut say initially two method low cut whole view spectral complementary strength find multiplicative well relate amenable implementation community detection begin observation experience network precisely community connection amongst rest
every every super set rank set learn submodular difficulty submodular discuss proof sketch discuss intuition prove jj prove interesting maximal basic could set case disjoint constraint third necessary similar must additional small nearly disjoint theorem prove eq think non special construction family generalize partition unfortunately achieve reason collection polynomially super polynomially case relevant goal super polynomially reason weight well code super unfortunately small plan combine observation mean intersection expansion turn perfect strong ratio nearly nearly idea family suffice theorem insufficient prove modify precede introduce sort truncation truncation ordinary ordinary rank away keep large get integer call function define large family family family cover know truncation size broad appendix partition correspond section general suffice ti property commonly note property sufficient unable construction empty satisfie axiom maximal disjoint replace I every use empty show define apply priori hand side function modular submodular applie issue constraint tight submodular cf ic ig minimizer intersection minimizer j second satisfy consider two lemma I gd construction let index denote construct instead take common parameter I start bb achieve g set bipartite graph additionally construct family u v satisfy last allow concrete probabilistic construction follow al match requirement state match proof require slight edge independently nod achieve parallel edge edge decrease family define additionally claim b whenever since definition immediate b da trivially remain prof desire submodular submodular distribution compare chernoff bind arbitrary roughly additionally view submodular prove result prove paper survey naturally median expect submodular let product distribution application corollary function construct column corollary imply submatrix concentrated expectation row technical provide theorem submodular theorem supervise paradigm draw I unknown may perform goal real small stand probably function error pac boolean special submodular note pac submodular natural problem distribution building learn class begin useful submodular lipschitz distribution provide monotone function value sufficiently approximation return example product label training begin expect output corollary constant carefully zero monotone submodular union close boolean zero implication approximate factor assume l together approximate fraction mention recall monotonicity furthermore imply inequality goal definition statement zero event hold probability least idea proof set suppose example time formally argue measure amongst none hand n n contain rank remark case minimum simply modify output modify show new show monotone formally draw draw approximate low point ratio learn sized even make formal probabilistic guarantee note function possible hypothesis approximate within underlie distribution query word augment let draw underlying query probability approximate within least theoretic hardness slight modification way exist learn submodular polynomial algorithm support family hard though prove algorithm open question construction submodular submodular normalize non negative w submodular symmetric q let fair coin flip label constraint output sn monotone submodular approximation use multiplicative learning instance I passive supervise slight passive u xu md repeatedly fair coin label coin label label linearly say claim every variable independently condition belong hypothesis fs sn first program feasible discussion fact draw label approximate correctly fs sn proof fact similar prove n reproduce probability least inconsistent incorrectly produce approximate within simple et bad robust handle extension clear algorithm general assume submodular assumption factor negative learn use describe fs label extend agnostic case agree target fraction arbitrarily inefficient submodular q learn approximate multiplicative proceed error mistake agnostic learn see appendix np hard mistake result procedure inefficient statement surprising rank proof let concave surprisingly partial converse follow sufficiently small idea value value uniform concentrate concavity say probably sufficiently get henceforth variable set cardinality finally identically distribute appeal great recent work fy variable obtain calculus non concave fix scalar concave implication follow pick independently remove element call easily monotone approximation q median define hx k argue complete immediately imply since whenever l complete original section hardness illustrate minimize formally eq submodular running compute construct minimizer lattice survey far minimizer encodes minimizer lattice lattice minimizer combinatorial much hard impose submodular cardinality variant submodular submodular analog tractable main minimizer perform query output query factor size minimizer large return negative monotone function approximate factor jensen minimizer strong construction sufficiently slowly set u u define query attempt cardinality bit correctly represent error determine whether whereas suppose query I apply choose neighbor remain vertex random neighbor randomly distinguish vertex call cut min cut modify graph vertex monotone perform structure represent minimizer moreover query almost proof require disjoint vertex neighbor pick independently analyze repeat neighbor v n du follow remainder theorem let path minimal cut word let apply neighbor perform cut whose correctly prove perform query determine whether second every vertex cover return f et submodular restriction seem unnecessary et cover ratio many query modify construction theorem bipartite monotone submodular minimizer within factor compute minimum form matching cover contain independently probability suppose query set first perform query multiplicative theorem consequence show satisfy property substitute open economic briefly economic function economic combinatorial intuitively increase price certain demand change formally price gs price old contain preferred price structural implication extensively many efficient condition necessary economic
independence diagonal however correlation thousand reasonable generality fisher discriminant depend increase sign turn rule naive depend put visual inspection equal discriminant discriminant perform away rate tend analytically accuracy impossible select apply classification depict well bayes restrict discriminant outperform restrict fisher powerful enough variable rise feature represent closely discriminant whole implement impossible empirically among reason focus ideally possibilitie counterpart consider naive method penalty way scad mcp primary procedure constraint classifier solution fisher portfolio reflect exposure preference drive accommodate application coordinate implement road use spirit diagonal regularize eq road fair study independence road road fisher marginal also incorporate road track performance variant road road road along theoretical property correlate feature discriminant alone two feature employ though standardized mean difference word mind large absolute objective standardized mean difference st rd hand covariance simple calculation lead st rd lagrangian problem propose constrain descent tailor minimization constraint optimization solve piecewise common affine replace pool enforce affine serve normalize fisher discriminant regardless confirm solve descent popular descent direction denote element vector search cycle meet coordinate particularly attractive explicit enter analogous use warm solve suppose need optimize become calculation form soft thresholding operator convergence coordinate strictly decompose therefore coordinate wise strict convexity differentiable theorem descent converge derivative computational cost operation cycle warm start use denote cycle enjoy road similarly replace version require third road sample classification misclassification question address assume pa pa nd misclassification rate say misclassification road view objective fix discriminant version eq prominent technical challenge multipli method reduce utility much complicated projection oracle discriminant constraint fisher discriminant pre feature road advantage two rule selection specific screening sure screening demonstrate road discriminant road road drive motivated fdr choosing screening permutation nan carry calculate statistic feature intuitively index let quantile make whose alternatively know road track sub fisher space include road achieve road loss screening block next verify theorem kn simply accurate scope pp word piecewise path reduce include differ affine purely spirit particular stress property trivial believe dimensional constraint replace involve composition lipschitz compare road road road road screen road version centroid discriminant fair independence naive generalize covariance oracle study testing repeat stability generality mean set dd size sensitivity due subsection path road realization clear index cutoff road dramatically road road road road road fair nb result range would mention subsequent around guess screening version road road select road road pre set median oracle decrease phenomenon dimensional go contribute classification road reasonably oracle method fair nb fail discrepancy employ fail road road rate road emphasize screen mainly lie pre road perform even lot table road almost road fair independence road substantial road recommend quite microarray well substantial road similarly thresholde standardized difference fair select marginal road penalize summarize feature road fisher coordinate number select road road road road road setting subsection road median broad range choose almost primary simplicity subsequent l classification road road road road road road road setup except block correlated matrix block examine varie percentage select road road road road road road road road perform road look contribution road show table express advantage highly correlate road pick take road recommend road road setup size equal correlate pairwise performance estimator vary percentage road road road road nb nonzero evaluate stability road structure give integer matrix definite unity signal nonzero element summary road compete fair road road road road error nonzero road broad expression cancer first come set contain vector cancer datum vector microarray phase project gene expression profile gene trial nb analyze event year year diagnosis year information positive negative select subject positive negative one total reader find set standardized road road road fair nb road desirable contrast road competitive select close performance number vary three robust tool dimensional road road gene road road fair nb error select gene road road road fair nb simple dimensional employ un regularize difference vector curse accumulation choose evident simulation discover pattern resolve part introduce variant road control worth explore selection easily extend nonlinear order polynomial spline kernel learn community transform road challenge root stochastic role completely preliminary proposal extend road section outline road suppose mean fisher approach classification dimensional centroid project centroid onto observation centroid close onto necessary project population centroid spread multi class scenario select coordinate th maximizer coordinate determine b binary regularize coordinate regularize discriminant coordinate additional discriminant find coordinate base centroid span road topic associate comment greatly scope financial support grant dms gm greatly theorem w w w c w set consequently lipschitz pa nc
replace modify likely tail idea behind heavy could possibly help speed work develop use approach approximation unnecessary base way statistic way would decade k perfect simulation coupling perfect simulation process dominate context develop purely purely process multiscale area capable modelling point pattern vary topic regression suitable coefficient coefficient perfect within promise compare method independently couple monte birth chain issue markov reach problem past perfect simulation therefore need rigorously burn appropriate error carlo identically distribute reduce simple price long difficult code many give present dominate locally point potentially literature go context modelling point pattern wavelet particular area relate zero coefficient wavelet simulation begin coupling dominate spatial justify perfect standard introduce describe infer turn attention problem review index perfect appropriately modify approach feasibility address investigate example section offer brief intuitive introduction principle behind formal description see suppose irreducible intuitively go infinite chain run chain couple chain I minus infinity chain thus state enough coupling two coupling stochastically order whenever stochastically ingredient attempt notably discuss truly method process couple allow simulation interact soon type general locally subset define locally stable exist point obtain poisson evolve markov fix death mark process recursively low process process evolve suppose death happen accept birth happen event remove define identical underlie process mark keep calculation calculation costly version calculate algorithm general dominate partial ordering induce preserve markov whose wish locally value function monotonic respect partial induce intensity interaction step write computationally deal clearly simulate dominate expensive least linearly practice burden follow alternative evolve way process death occur happen accept happen remove event obey coupling suppose wish simulate calculation dominate coupling past involve calculation write identical attractive multiscale theorem construct refine clearly necessary several class cox stationary area interaction process produce cluster moderately order clustering fill gap produce neighbourhood area configuration regular poisson configuration constant compact neighbourhood point addition process configuration case reduce flexible yield regular spatial randomness cluster unfortunately small distance display sort behaviour distribution tree patch physical physical law behaviour behaviour class q equation ball scale measurable integrable extension standard area perfect multiscale process already uniformly substitute extend multiscale might indicate scale existence process perfect datum multiscale appropriate circumstance fit systematic adjustment pseudo parameter sample spaced interval assume independent normally problem transform large behave distribute identically combine wavelet discard small since wavelet spread evenly discard noise cross discovery bayesian prior population wavelet wavelet thresholding capture may dependency area interaction section idea appropriate approach coefficient subject noise value wavelet noiseless wavelet level coefficient level transform independent equation extension consider discrete index wavelet wavelet nonzero specify binomial lattice negative integer assume point concentrate pure variance extend consider reasonable produce make natural structure clearly discrete wavelet represent tree configuration mind intensity expect reasonable exactly theorem tell couple discrete complete specification area prior need interpretation neighbourhood location index possibility decide adjacent coefficient child coefficient adjacent make nine illustrate capture time periodic show discard part unless neighbourhood function include child immediate coefficient variation wavelet boundary modification practical close posterior coupling past advantage normal lattice equation convolution density simulate ignore mark amenable abuse third refer monotone subset condition process process constant intensity dominate lattice maxima minima maxima minima global maxima low consequence feasible dominate maxima modification essential chapter give dominate location intensity dominate lattice start point section probability upper remainder carry way birth dominate although reason rule give ease assume jk jk jk jk jk algorithm feasible value birth necessary issue live point location whose large purpose nearby assume strictly location large problem q monotonically monotonically dominate location good gain taking dominate
inferential ratio popular program user pair drawback comparison tends favor rich model aic criterion commonly aic compare nest nest favor criterion select ratio penalize increase accounting uncertainty aspect factor bayes incorporate uncertainty intuitive comparative evidence probable probability compare popularity publicly available likelihood complexity computational burden high compute marginal straightforward integrate subspace branch substitution eventually sum extensively condition nest bic firstly sometimes recent appeal laplace neither option implementation require tuning suffer extra point sampling harmonic integration review context large advantage applicability protein need highlight integration reversible accurate interesting alternative mean estimator estimator share simple make easily substitution tool complex introduce formula estimator nothing normalize unnormalized negative integrable define unnormalized instrumental integrable contain q sample introduce consider hence case harmonic posterior sampling probably explain original importance arithmetic q generally reduce integration harmonic harmonic estimator tool exception argue theoretical explain yield chance region large standard end yield approximation fact raise sometimes whether reliable generalization improve alternative review likelihood estimator harmonic approach basically share density formulation particular choice instrumental originally instrumental perturbation total target eq arbitrarily perturbation multidimensional density perturb length origin visualize act perturb instrumental density density respect instance yield condition full moreover perturbation extra computational mass perturb depend asymptotic via delta method square ultimately estimate key role infinity kk act address minimize estimation consume one obtain perturb original purpose computation easily http site google com site home shown recommend simulated covariance density origin frequency transform entire real call jacobian density successful simulate output simulation software specifically upon request author publicly software benchmark harmonic indeed estimate marginal though infinite unstable sample posterior must aware quantity surrogate rigorous bayes factor example simulate comparative evidence alternative benchmark synthetic panel implement couple simulated evidence whole order improve rate equal marginal know true scale relative mean interval autocorrelation correction since correct perturbation logarithmic scale harmonic arithmetic relative similarly estimate sample quantity monte carlo estimate although condition error sufficient guarantee accurate q denote bootstrap replicate formulae account autocorrelation correct relative error small arithmetic mean simulate point really marginal nonetheless distant corresponding value strength compare method estimate mc serious reliability magnitude robustness range specie show topology model couple evidence approach repeat time table somewhat relative carlo relative correspond factor logarithmic consistently model bayes possible extend deal select compete advance compete evidence bayes evidence favor topology continuous evolutionary nuisance belief comparative summarize compete substitution subsection density fix indeed hadamard topology aim compare topology marginal likelihood also exhibit performance evidence possibility simple evaluate evidence compete bayesian eventually compete probably date model context applicability computationally light burden appeal explain option routine simplicity match estimator biological provide effectiveness marginal class harmonic estimator share simplicity feasibility unlike enjoy theoretical moreover simulation appeal evaluation like stream burden reduce also search optimize verify substitution comparative respect posterior estimator circumstance outperform precision candidate evidence estimator even fine appropriately cm cm cm em molecular bayes marginal marginal solution term simplicity burden researcher far simplicity arithmetic result estimate bias inaccurate reliability conclusion generalize harmonic simulation share infinite alternative fully satisfactory estimator currently publicly sample marginal model state related connect specie infer specie study genomic chapter substitution address fully propose organized briefly review concept focus substitution tool popular tool harmonic estimator harmonic section name model numerical comparative discussion dna specie consist specie site relate replacement nucleotide specie describe replacement tree among call leave external represent day species internal genomic branching branch period substitution dna probabilistic modeling change evolve site
dirichlet regret kullback propose optimistic aim observe kl reduce probability vector maximize resp easily lie p neighborhood neighborhood case ball vector dimensional represent represent direction simplex maximize maximizer vary continuously p consequence optimistic assign compatible optimistic mdp transition really exist small never equal therefore optimistic ball assign positive even true accumulate optimistic favor transition experimental indeed assign strictly positive transition eventually unobserve transition potentially work except difference neighborhood together illustrate represent evolution example take bottom cm section explain solve arise maximize lagrangian maximizer follow np imply sum iv denominator satisfied way determined v define function play maximize decrease mapping jensen easily function denote z solve initialization series iv iterations paris fr consider reinforcement decision optimistic mdps favor kullback purpose studying kl term solve optimistic extend kl benchmark significantly improve mdp reduce observation element keyword reinforcement decision approach kullback leibler long model decision mdp assume agent need face fundamental trade experimental consequence action act order reward consider state maintain running estimate expect approach exploration exploitation armed context extend several instead act optimally accord surrogate collect policy armed prove optimistic logarithmic mdps optimistic use optimistic extended transition highest remove less transition elementary interpretable extended lead undesirable may optimistic importantly optimistic assigning probability connectivity persistent transition accumulate impossible optimistic kl kullback leibler kl pseudo indeed smoothness metric first issue thank relationship simplex kl optimistic trade promise accumulate search kl consequence build logarithmic kl traditionally number practice environment property kl paper brief description discuss kl mdp rx action agent receive action reward consider mdps mdps states reach mdps policy respectively reward fix optimality policy focus reinforcement problem know e probability consider jx optimistic receive reward count proceed episode start episode length th episode episode episode soon action policy follow episode optimistic take optimistic solve equation extend maximization action maximization rx c dot product probability radius confidence maximization appendix algorithm lagrangian depend current value probability ir f newton newton argument accumulate reward play adapt kl neighborhood start constant appear theorem eq dependent logarithmic upper quantifie depends inspire every proof rely concentration sum pair probability regret write episode matrix optimistic episode otherwise summation denote resp resp transition matrix resp third written th x addition combine complete compare kl randomly environment model transition six
optimality noting give word side constraint substitute optimality already solve explicitly calculate solve cubic follow maximization solve part keep maximization cubic equation repeatedly alternate alternate specify care implement overall thing nd step lagrangian multiplier evaluate nd admissible overall satisfied part check objective zero invertible candidate acceptable acceptable guarantee satisfy suppose candidate part number side admissible side write basic cause instance update initialize newton give pt dirichlet find optimize objective find root newton quasi newton initialize step repeatedly instead iterate solve question problem connection bss differential version bss mean without mean second
action team behaviour simulate general confirm recognize set represent team b front forward b front forward forward front forward leave front forward b front front leave forward b b forward front forward front front forward forward forward front b forward front c forward b front forward b forward front b front b forward front b front forward front b forward front forward forward front b front c front b c forward front forward forward front front c forward front front b front forward b front front forward b front team behind forward
r p functional line propose monitoring perform use simulation q distribute term parameter root innovation uncorrelated monitoring unit setting numerical corresponding monitoring rule limit take define nuisance parameter weight past nuisance west point carefully yield investigate rate hypothesis delay signal analogous conditional delay quick method table average delay result monitor curve simulate law overall seem explore figure trajectory immediate detection negligible yield hard detect
locally first state optimally strategy directly train latter concept consist simplify estimation extension classical normality problem know normality quantum learning problem proof depth sufficiently likelihood side rao inequality estimator example coin q statistics concentrate rao unbiased find mechanic situation essentially dimensional version er rao quadratic overcome adopt modern perspective provide instead problem idea statistical sense know restrict adaptive rough estimate step sample rest estimation parameter normality smoothly converge model observe single mathematical two statistical quantifie dominate distribution densitie analogue quantum define infimum le
agglomerative algorithm deduce agglomerative hierarchical linkage subset application cluster result search function quality function diameter number call problem strategy agglomerative agglomerative go see drive force classify discuss agglomerative agglomerative bottom cluster cluster cluster create hierarchy two possibly useful property cluster bioinformatic cluster priori agglomerative specify distance object equivalent use distance object cluster agglomerative complete linkage hierarchical polynomial length input diameter approximation time although agglomerative widely consider factor analysis guarantee linkage agglomerative diameter closely problem search center minimize distance center center come cost optimal solution know hierarchy center provably center know metric diameter problem euclidean also metric space approximation discrete center diameter clustering
hypothesis probability transition eq contain one ask control whether control law pa tt operation pa obvious rest essentially stream htbp probability want converge rewrite denominator obtain reference pm illustrate simultaneous divergence process controller measure unit htbp characterize depend apply hence growth depend
mathematical arguably theorem integral real value distortion wang call distortion distortion parameter fairly suggest distortion start transform new assume eq therefore call parametrization necessary loading assume net index therefore become natural notation infinite depend cumulative intuitive value always later shall hold load basic loading satisfied whenever non vx vx
illustrate calculate mm x x imply apply third second reciprocal x surrogate g x x x x x x mm obvious check mm iterate accord second iterate whenever converge infimum identity imply hope much surrogate reduce separate mm x limit function complicate unbounded component coincide
posterior involve know distribution wise model easy implement except gibbs draw variable color need component combine produce feature simply relate follow dimension equal dimension
project proof contraction continuous almost proof rademacher lie point draw dotted geometrically onto crucial examine figure clear point still project seem necessarily lie corner right corner believe almost map affine exist hausdorff lebesgue zero simply sense invariant different unique intuition degree freedom matrix dimension boundary boundary sign neighborhood trace zero stein conclude pt sort detail statement involve involve argument indeed continuous see lasso boundary quantity invariant everywhere apply stein formula result x divergence x space interpretation fuse also freedom correspond fused graph degree connect component extension connect component slightly modify degree lasso fuse fuse problem freedom generalize correspond two connect interpret dual correspondence decompose contain coordinate last define coordinate yield keep correspond coordinate yield furthermore coordinate give conjecture degree fuse equal fuse cover much general result nan one corollary omit sake brevity list along fuse sake completeness degree
component compute simplify logarithmic term construction distance kernel rely dimension size nd error come place feature probability invoke chernoff hoeffding x q kp time sample k approximate subset np integer determine subset reduction transform size q ct I qx x q valid distribution deep measure embed possibly infinite paper algorithmic fast computing point near provide technique reduce convenient set set
subset refine try belong neighbourhood solution neighbourhood build neighbourhood obtain modification start generate solution improve otherwise end optimum important biology since protein fold logical secondary domain predict five fold alpha beta protein b bind protein protein beta alpha beta class class round treat separate classification example fold value experimental fold validate conduct impose applying indeed validity
cauchy finite use induction almost singular bound vanish almost consider proceed check theorem condition hypothesis empirical bound variable converge finite almost surely indeed q hence induction hypothesis variable z show combination pseudo fy side pseudo function jointly surely transform due symmetry difference cr ts td limit hypothesis form definition q form h almost write proceed easy check variance induction variable note show variance combination pseudo c g tb ty e side equal z surely z z surely statement equivalent jj distinct check immediately I let z theorem trick avoid repetition algebra generate coefficient form column converge lipschitz limit exist
replace surrogate instead optimize optimize mm denote last discuss numerically optimization simplicity objective mm read science besides specify explain plain way discuss monotonically function prove decrease instead step argument compute global minimum last equality expand omit assume appendix subsequence converge I meanwhile convergent subsequence k x k proof besides obvious contradict get base two lemma model sequence generate x upper surrogate tangent taylor accordingly evaluate know point x model could impossible claim converge find always satisfactory result numerically solve optimization constant convex objective mn ij tr equivalent rewrite
variance ever recommend matter provide remarkable mcmc importance oppose importance density must support like hull simulation completely probit ml obtain replication compare harmonic significance covariate simulation compare approximation harmonic diabetes simulation source explore rhs select approximation harmonic solution preliminary dimension model attractive natural rao estimate simulated estimate unbiased approximation conditional probit obtain replication approximation usually reliable dominate normal imply harmonic estimate compare bayes harmonic importance diabetes rank order worth marginal good rarely generic solution whenever available approximation generic model compare model jump reversible constructing method try limitation simulate markov existence stationary immediately produce chain include besides target enough
wind drop summary exponential smoothing write truncate normal easily ahead forecast function expression one iterate ahead forecast apply approach wind simple benchmark benchmark persistence random forecast forecast truncate persistence ahead location hour forecast location set variance consider persistence constant forecast describe benchmark forecast third benchmark forecast unconditional forecast exponential smoothing move empirical computational estimation density day density appropriate conditional forecast origin ahead forecast parameter forecast keep show decrease j maximize conditional forecast summary benchmark approach generate forecast compare minute hour ahead forecast tn forecast tn forecast lt lt tn tn lt stand tn stand truncate generate forecast forecast forecast density particular forecast form
call cluster denote contain output k asymptotic target consistent pf sample work first find empirical distributional maximize already cluster assign point cluster assign notice iteration initialization initialize I jt jj k two complexity distributional tuple tree efficient furthermore therefore double weight zero way joint ergodic
fire present three strength unity strength two interpretation graph connection direct thus specify specify second regardless inconsistent evidence specify neural remain value typical solution neural mix source represent traditional network sum
distinct pair sample distinct propose recursive way express except therefore allow straightforward marginal marginal observation point closed likelihood compute irrelevant computation bayes factor practice derivation recursively I component update update j update nu jj nu nu
e sparsity pattern interested computationally characterize network structure degree chemical encode various chemical involve hundred stochastic task consider proximity fluctuation specie count network would chemical area tackle state stress graphical sample infinitely continuous course subsample space raise depend sufficiently approximately stationary latter large intuitively limited information confirm efficient support discuss relation exist literature
number exploitation epoch player play spend go step epoch arm sample increase epoch go show achieve reversible eigenvalue sr I arm reward policy condition end epoch appendix policy nontrivial bind knowledge choose logarithmic state basic th exploitation epoch length play length arm irreducible sr arm reward condition detail order similar replace regret choose nontrivial available achieve regret close formally irreducible reward b decentralized player play markovian play notation adopt transition passive arm former evolve arbitrary even play arm due know player play adopt receive former give play reward player otherwise ideal perfect player arm effect set strict achievable ease global exploration exploitation player offset share eliminate
vi direct proposition prop argument exist far prop p continuous identical sequence equivalently consequence relate somewhat proposition imply first p sup norm subsequence p prove see ex measurable real nf element satisfy l lf continuous value sup norm compact non value every additionally satisfied use convention supremum empty claim hypothesis claim additionally extend logarithm interval establish b prove analogously b continuity continuity extend logarithm element converge logarithm already establish claim follow argument already establish b every part theorem law large number separable banach fx establish conjunction sup clearly variable every evaluation borel see e van compact borel mapping hypothesis prove analogously similar separable banach continuous already conjunction take function x away show compact establishe give part essentially sup claim intend dx inspection analogous fx f ff f uniform proposition appendix application hence probability theorem space unique maximizer order sup choose proposition prop constant cover fix r mb f j note
agent unless around estimator change step converge assume span definition unbiased observe estimator estimator recalling agent happen step long converge stop unbiased iteration least unbiased limit simple private measurement social rational raise complete structure future relax common network pick agent proceed calculate expectation
module error time solver minimize influence factor affect intel processor gb graph node enyi clique edge triangle create complex create random represent accuracy efficiency test clear create part part pick entry compute harmonic compute square solution harmonic another rely residual pick entry form stack matrix square harmonic definite magnitude semidefinite magnitude nonzero number conjugate substituting satisfie require gradient use laplacian spectrum include graph free involve property type connectivity edge remove single conjugate graph formula easy upper degree connectivity path graph acyclic vertice vertex degree regular star degree connect center comparison actual graph star enyi graph fewer exist graph cell complex embed boundary place vertex dual connect edge boundary surface sufficiently complex possible complete euler characteristic complex boundary lead solver gradient cg residual solve rectangular form solver mathematically equivalent mention early semidefinite nice nontrivial effective quadrature differential solver formulation number triangle complete rectangular solver list indicate tucker formulation exact identify homology lead harmonic component report harmonic part require create hierarchy linear decrease hierarchy call coarse
universe function submodular lipschitz building bound self bounding lipschitz submodular expectation expectation structural theorem together concentration inequality structure say decompose submodular one domain prove begin simple monotone block bound submodular even follow submodular submodular lemma submodular ordering b vs fs respect define submodular submodular satisfie vb bs submodular well release strong vb gs submodular always terminate size vb bs shift g mapping complicated want able choose give former define deterministic carefully procedure gradually child procedure differ construct single influence ps fs intermediate state fact property completeness ps b sx fs sg item reject element reject
variable large dimensionality toy series free widely different field economic finance effect crucial activity vector autoregressive
envelope affect let lebesgue subset vc corollary cover collection loss generality represent define essential supremum analogously routine b cover thus argument unbounde integrable envelope major truncation g precede f covering let ergodic every average converge
actor method treat super accounting interaction nature community membership determine number method agglomerative cluster analyze citation network examine recover agree know network simulation ensure feasibility structure identification great hierarchical formation start enable inference role actor link possible link prediction process actor position infinite feature goal al annotate discovery nonparametric issue annotate infinite blockmodel recover branching among grid offer dependent membership link network mechanism latent actor interact crucial call membership coefficient actor actor generalize
eq lipschitz solve programming however moderate size cost newton linear application attract interest relatively implement challenge huge achieve convergence descent successively nuclear must solve solution descent formulate include fused projection fortunately usually compact nesterov construct objective descent fuse numerical utilize instead work people gradient sometimes one typical
sum cardinality namely configuration ks thus obtain instance result development degenerate finish case structure come write l prove rely state permutation notation permutation equality constant permutation expectation back may namely q j l l j non argue negligible negligible theorem write central limit triplet chapter differentiable classical almost sure parameter compact criterion jensen hand side attention lead inequality far far usually taylor expansion derivative quantity almost asymptotic function omit almost fisher invertible obtain fact convergence proportion know theorem describe first consistency normalize composite value preliminary necessary uniformly consequence almost sure consequence assume f eq leibler entail positivity establish permutation label continuity theorem
theory small enough follow induction distribution obtain randomly generate application interpret basis extraction phase generate large result observe increase close expect framework extraction universal testing cast rank easily effective constrained optimization
analogue te rank scalar te ty conditional two sum use derive vector consider continue component ty account last ignore case entropy probability te position vector rank joint possible possible represent pmf pmf white find double distinct fig probability white bit bit assume time follow future sample possible possible instead neither joint example form single rank ahead white augment vector
time service job job enter job time vector similarly general useful treat complex consider later independently arrival observe arrival equivalent service regime framework type transformation job job service job job queue minus interpret job regime come employ service processor queue single within queue request remove queue manner queue simultaneously request queue request arrive call processor density arrival service service processor auxiliary job indicate job server clear time solve system equation jacobian arrival jacobian triangular depend arrival queue processor job job process queue service arrival queue queue service time equation set notice job immediately queue service latter number service joint q reasoning processor sharing design computer simultaneously share way understand queue imagine system slice consume job queue service slice remain service time service time drop ps queue limit job precisely queue independently arrival compute equation solved iteratively hold hold procedure converge
discussion hoeffding deviation independence bivariate hoeffding hoeffding page hoeffding interpretation f px x f f x axis analogously four argument two drawback firstly five secondly certain continuous formulation characteristic replace square straightforward version correlation give preference due simplicity preference ordinal literature note case family formulate appropriately empirical coefficient ordinal utilize ordinal nature
spread thing equal limit experimental covariance characteristic length hypercube pseudo input adopt standard prior parametric gamma space similarly random remain directly walk proposal parameter measurement model identical say scheme stage gaussian simple one analytically function keep explicit advantageous factor need since parent pseudo function issue jointly everything define vector sample univariate conditional v dd dd k dm plain metropolis entry parent walk proposal parent proposal justify follow row submatrix standard
energy would fail near material atom may geometry another examine atom sometimes atom eq near neighbors cubic lattice atom edge material neighbor contribute equally neighbor atom atom state atom behave solve hamiltonian
aspect measure symmetry large sign relate convention horizontal handle describe band aspect angle htb b band side arbitrary sided co show five sided figure htb side aspect angle band incorporate need model ideally train started scenario versus evenly space verify increase aspect object set start evenly distribute point type final random angle rotation figure object aspect angle versus final datum upper moving sphere
note could two additional notice setting g supplementary great satisfactory weighting cm induce dedicated incorporate predictor result simulation investigate follow enhance induce capability extend concept norm factorization la project european project characterize value minimization use order order simply twice conjugate definition property consider compute mt j fa sa j fa indicator fa fa maximum fa fw fw sa minimize hypercube convex envelope conjugate
question similar exponential lasso jeffreys normal gamma prior observe light natural I hope well conditionally local greatly simplify write write preferred error scale prior prior marginal double obvious px draw version version enter level expand local
estimator root consistent pilot estimation n bn constant tx vx ex vx tx vx vx tx e tx xx vx nx recall nh ni place result vx tx accordingly whole square estimate although exist quasi importantly square directly less nonparametric involve quasi limit vx pp dimension limit quasi fisher information fisher help estimator small theorem adopt tv proof indicate estimator plug suggest estimation classical quasi also bad position element correspond uncorrelated classical quasi likelihood asymptotically equally
mle lead limit scale corresponding average however subsampling show correspond consistent effort identify optimal issue write diffusion dimensional estimation limit equation multiscale theorem drift I system appropriate asymmetric hausdorff hausdorff distance drift estimation depend maximizer discretized mle slow averaging describe diffusion average minimize clearly system detail
small genomic recently system analyse variety factor size disease variant complex disease marker identify replicate phenomenon attribute underlie expression biological drive phenotype review describe additionally inherent rather genomic analysis biological interaction genomic cause disease crucial mechanism single amongst attain association identical may see additionally necessary activity interaction share subset note take advantage multi snp literature article snp technique statistic sum use frequently amongst snp marker statistic systematic yet subtle purely snps reach issue detect association search space multi effect remove snps significance novel identify association status degree reveal would type scheme drawback snp put limit snps examine arbitrary marginal notion control motivate interesting detecting dimensionality reduction snps combination examine cell minor find interaction expert limit must limit combinatorial recent powerful taking processor make feasible genome pairwise still remain computationally snp rather restrict pairwise interaction combinatorial exhaustive search narrow combinatorial discover
pt rectangle abundance rectangle rgb rgb rgb rgb circle rgb rgb cycle rgb cycle rgb rgb rgb circle cycle cycle rgb rgb cycle rgb cycle rgb rgb cycle circle rgb cycle rgb cycle rgb cycle rgb rgb rgb rgb cycle rgb rgb
behave probability lemma cause part nb nd suffice n argue f finish splitting n slight yu van behavior satisfy assumption q leave unchanged estimator b low come eq bind multiple virtue c inequality deduce q nz n n iw I n great equality chebyshev inequality concave maximize derivative fix real w pt notice tm g
norm otherwise weight mkl mkl norm mkl mkl net mkl allow interpretation posteriori eq regularization hyper regularization correspond marginal kernel bayesian mkl rewrite weight regularization model mean term quadratic analytically integrate marginal use bounding rewrite mx mx base regularization regularizer mkl unfortunately block regularizer however hyperparameter maximum update minimization respect weight section respect
auxiliary asymptotically auxiliary result combination variable central limit normal cm lin optimal require spectra priori used inference within framework field concrete application reconstruct unknown five separate spectrum measurement reconstruction maximum posteriori power spectrum maximize joint iv spectrum wiener flow filter operation exhibit jeffreys spectrum spectra exhibit statistical filter superior wiener correction smoothness noise like vision enhance stream artificial since signal permit filter focus mode sufficient available permit signal straightforward available exclude purpose reconstruction simultaneously due trivial general original scale well homogeneity knowledge spectrum permit construct filter linear galaxy distribution signature matter non scenario put capable would property avoid fidelity tuning structure key matter spectrum field verification reconstruct thus spectrum map make immediately optimization soon critical signal filter wiener linear wiener filter knowledge response differ covariance spectrum wiener spectra filter effective wiener effective differ wiener exhibit operation account critical implement behind frequently kl jeffreys threshold
eq bipartite graph satisfy eq constant observe factor see follow design set expansion tuning construct unbalanced denote adjacency degree observation satisfie satisfie restrict invoke finish observation compress sense derive result polynomial exist unbalanced approximation
bandwidth assume discuss rather assumption encounter setup I asymptotic put determine approximation observation kernel smoother study allow monitoring process asymptotically process ts satisfy functional limit limit special turn limit asymptotic depend question bandwidth respectively sequential calculate past criterion sequential achieve square prediction integrate distance avoid omit good fit sequential estimate eq statistic regard sequential leave define put asymptotically put distance want response q array tn
solve optimization submodular show monotonicity property monotonicity fa fa fa sum equivalent unique solution proximal submodular unique proximal u jj property prop fu fw optimum proposition solution single solution proximal unique minimizer minimal minimizer proposition set know large may problem restrict separable base many minimization prop describe detail certain say denote give base say set tight tight everywhere desire tight exchangeable tangent prop exchangeable dependence function non tight pair prop tight tight exchangeable conjunction property pair exchangeable pair let exchangeable tight proposition check support maximizer tight prop tight decrease true tight tb ta fa w prop prop deduce base optimality cone exchangeable give proof prop vector tangent cone cone contain tangent cone weight
rest paper integration system deal kb motivation investigate develop preserve crucial e integrate relational database dl cl paper implement solution semantic degree integration say kb dl combining cl indeed allow describe two case integrate dl hybrid former safe hybrid body rule allow resolution calculus deal run constrain sound answering query atom language devote dl I answer answer logic extend rule concept occur answer rule answer iv combination require dl log answer constrain resolution modify version calculus logic logic programming concept logic inferential mechanism induction prominent concerned automatically induce inductive clause form induction presence classify accord orthogonal discrimination characterization ground clause
nm usa many vertex topology attribute choose query learn much vertex stochastic block choose query two attribute heuristic consistently network represent vertex represent hide take way topology simple assignment likelihood q number tu tv direct loop self loop connect may reverse kind community specie interested
component distribution similar component decision make unconstrained project decision reach eight demand decision observe state decision weight equally available underlie hour ahead wind decide energy amount wind energy energy expensive price make excess lost day wind history two hour contract price c l current price hour current wind variable energy wind receive variable know hour wind data north american assimilation system location tx wind pattern contract price contract price price generate varie day year analyze separately year train wind wind weight base rule package function hierarchical day von exponential family unit sphere
mle perturbation turn fmri study movement intensity easily likewise foreground surveillance across represent another sparse high intensity magnitude
artificial investigate mixture problem include protein start cut iii depend task achieve theoretical analysis mkl novel base perform sparse scenario appear additionally offer complete derivation additional experimental extend previous publication novel illustrative improve publication mkl subsequently researcher project optimization mkl view unweighted svm six bioinformatic generalization bound analytical optimization also mkl composite kernel learn small medium scale structure sparse mkl discuss introduce conclude view cast multiple unified framework present norm mixing show comprise mkl include dual make regularizer loss latter cover easily structural prior incorporate regularizer begin sample unseen h form hilbert constrain regularization allow mapping consider minimize optimal avoid overfitte regularize note approach regularizer non arise inherent parameter adjust favorable convention regularizer arrive problem primary investigation regularization contribution show exist optimal multiplicative switching refer proposition proof prop prop satisfy regularizer contrary yield infimum regardless contradict yield show mkl implication ray class addition couple trade versa optimize implicitly search problem preferable model begin expand slack norm lagrange incorporate introduce lagrangian multiplier incorporate lagrangian saddle denote derivative reveal yield attain arrive ct conjugate maximize subject let ignore moment
band scale band transformation identify log band locate structure ht illustrate loss band first band simplicity select band first widely see frequency business line type year horizon select extreme heavy interest infinite suitable claim finance paper particular close analytic expression poisson compound property form es derive analytic es ii gaussian stable location restrict stable process include infinite skewness tail loss year stable denote univariate determine degree location term characteristic large imply heavy tail respectively member special admit close typically proceed characteristic discussion intractable setting efficient estimation datum random characteristic analytic mixture bank model positive stable expression es result linear proposition lose truncation implication general
modification demonstrate divide four modification nonconvex convenient two nonconvex lie partially outside justify overlap boundary even nonconvex correspond mention sec body body overlap intersection nonzero volume outline uniformly intersection bin change histogram mark index consideration tackle normalization probability could justify normalization unit convex eq disjoint integral normalize include four density component quasi common principle decomposition straight determine appropriately refined index additional produce combine symmetry distinguish histogram symmetric directly generalize
observable discuss misspecification estimation stand number another chen examine suppose observed contamination another independent measurement univariate provide measurement expectation measurement error panel
program check correct practitioner ad ad module modelling mostly due difficulty multidimensional difficult treat model noise package measurement extended kalman approximations order task multidimensional conditional work rigorous time require specification effect restriction theory point view variability precise dynamic follow diffusion triplet th expression log expansion symbolic calculation value dy dy du k du iy iy iy I j I j I I I p j j ik expansion j iy j iy
etc decomposition case vector energy rotation linear find neighbor question happen want concentrate conclusion empirical motivation code
phase perform letter kolmogorov exploit different reduce need low method sort cdf edu novel method
number predictor pairwise predictor indice fit spaced rule figure number percent explain total tend fairly numerical looking take fine grid stay solve notational convenience rule discard predictor discard predictor package safe rule elastic net discard actual extremely excess assume logistic letting likelihood discard inner kind investigate analogue regression max use sequential broken curve classification binary indicate particular http stanford global
approximation imply novel smoothing straightforwardly method determine distribution smooth filter smooth proceed filter high sufficient smoothing result derivation smooth gibbs sec provide concept numerical evaluation conclude dynamic measurement tb otherwise accord x purpose filtering smoothing approximation bc observe assume p soon measurement extend compute state given infer posterior map
input group note reduce close applicable however conventional sparse sparse consider treat case wide regularization parameter stochastic process mutually make remark derive general specialized valid input result weak deterministic great section demonstrate evaluate performance evaluation five
scheme value alternate complete linkage suffer inherent concept category construction concept section conclude remark section let denote metric point distance equal follow definition relative correctness encode multiscale persistent pair rr tr tx associate persistent underlie partition consist agglomerative hierarchical distance relation block easily notion discuss linkage usual definition agglomerative definition merge useful mathematical construct encode nature object together formalism extremely class share structure consist g vector x corresponding transformation etc composition assume ff f category side indistinguishable except category consider go refer category must identity two object three identity exactly exactly three identity must set equipped element composition
define include unit arguably implicit modification see external specify however partition novel coherent realization weight beta lead well random call generalize regard distinct sequence natural specification characterize application dirichlet prior beta outline basic compare beta hmm suggest beta exchangeability heterogeneity virtue beta hmm unknown structure section analyze genomic array investigate among al hmm partition tumor dna copy et extend latent markov dependence heterogeneous markov single adjacent du et dp state impose persistence dp mixture respect proposal assume persistence beta weight specie mechanism adapt intensity link breast conclude proof theorem modification rule characterize sampling beta characterize distribution define randomly
could seem global adjacent chain swap chain suitable confirm good ess temperature overlap prior size acceptance replicate delay rejection exchange range operator range surprising automatic acceptance increase prior ess discuss detail ess report simulate marginal ease drop subscript index precise marginal model replicate order ess order record large burn vector compute lead large large replicate ability deviation search strategy least replicate stability report time across ess matlab gb ram marginal inclusion average replicate triangle vertical blue triangle perfectly covariate contribute covariate blue surprising less situation hand small small effect evident second respectively posterior replicate small blue solid line replicate ess small posterior replicate triangle vertical two evidence ess able ex account detect ex surprisingly assigning replicate mark ess ess model addition receive marginal red triangle replicate vertical red dash line ess capture coefficient contaminate effect ess present variability red line solid vertical receive contrast posterior agreement respect see surprisingly main difference ess reach well stability expect towards drawback far apart compete ex ess
laplacian denote vertex inequality loose well cut expense runtime bregman much guarantee associate non eigenvector laplacian laplacian subdifferential value moreover subdifferential denominator eq constant eigenvector laplacian q sum anti symmetry positive component zero median odd even median eigenvalue laplacian constant constant direction subsequent result constant median sequence terminate find zero f contradiction terminate iterate second direct compatible eigenvector eigenvector nonlinear modification
merely behave define rate admit entry rearrange obtain hadamard e entry wise nonnegative definite eigenvalue interval equivalence theorem component eq formula definite lemma wise hand nonnegative sum frobenius chapter
algorithm datum arrive systematic trading trading intervention trading trade relate portfolio optimization idea make algebraic classic variance ordinary least construct mind certain regard trade financial attribute adapt stationary dynamically incorporate portfolio financial counter estimation mechanism technique consideration idea allocation devise trading technique dataset asset allocation multi period portfolio exhaustive investigate trading strategy computer machine standard finance article technique portfolio finance financial improve appear finance exposure typically et correspond instability encounter mean asset mention adaptive adapt environment finance two algorithm mean variance idea process
unique maximizer expect log contradict inspection fix maximum section e mention exchange exchange log near conventional em concentrate record system program request start uniform mle em galaxy fit model mean grid equally spaced deviation mle em dramatically reduce computing add conventional em appear increase per decrease appeal implement strategy censor review problem proportion numerical collect unit accord censor inspection failure subject govern
finish schwarz claim start adversary tf tf argument vanish regret guarantee vanish rate adversary one solve lipschitz optimization optimally function subset lipschitz conclude game set set game game play eq corresponding minimax player game minimax playing theorem corollary second conclude note get conclude cover appropriately extra application add break inside nest three arrive mention theorem loose supremum draw assume simplicity supremum equation follow bound pass bind repeat arrive bind next quantity conclude proof cover member close sub use triangle quantity last bound supremum value tree depth corollary element sense thus proceed assume arbitrary enumeration crucial non indeed member eq remainder shorthand hence union hence since n appropriately statement sequential rademacher proof payoff class moreover class cover eq define fix depth assume enumeration pair zero path cover
region train layer variable code learn mrf show state art denoise generally clear variable regard circular symmetry coefficient conjecture model cosine phase structure complex pyramid phase always corpu image however concentrated variable dependency difference I phase distribution identity express cosine bivariate term pairwise statistic angular
computable computable differ computable computable computable computable computable measure set computable let computable metric space let computable computable computable computable measure measure computable abuse computable restriction continuous topology almost continuous density measure almost computable almost computable measure version describe pair computable computable conditional description conditional even almost rule space enumeration variable variable random assign equal fall variable computable coin enumeration version calculation eq side consider function suppose continuous exist interval contain image fundamental obstacle admit able pair computable version continuous even distribution computable existence denote machine standard enumeration map index computable computable computable coin computable mutually greatest uniformly distribute expansion computable variable x lebesgue probability integer admit
net interpretation graphical section following grow xt xt case affect point indicate direction child conditional parent separation configuration child xt net clear reasoning separate conditional irrelevant generally irrelevant model generally speak case inference remaining obtain update low gamble guarantee inference coherent positivity hold also right smallest large calculate sign recursively root calculate follow equal eq counterpart first possibility message gamble tuple number ms belong eqs course reasoning form replace lead course take strong turn recursion tuple possible single equal node local message tree transform node local simplification sign greatest close continue eqs reach eventually ab moreover eqs find cx ce conclude separation draw anchor west node unobserved root x x root use extension message pass calculate towards root
analogous alternate section sparsity number informative would enforce entire well dna strength signal sparsity desirable microarray gene goal require exclude different gene active factor finish enforce pca explore x u v x discuss datum similar first signal simulation
acceptable recommend method slightly high nominal partly dependence partly get true believe power uniform event apply ks repeat show figure ks generally test somewhat
exploration standardize let mode eq treat marginal integration demand accurate select approximated th omit simplified expectation identity follow quantity suppose exist denominator let approximate q eq combine sum weight include take account convolution component several model medical proximity study belong implement goodness
sparsity case strong sometimes moderately suggest base powerful optimality variety balanced way modern design arise effect similar effect normally class testing relationship response fisher trial tool statistical clinical microarray important consider simple th th treatment assumed goal formally test analysis software package test nan identification nonzero reduce independent effect testing variance reduce chi square power maximize alternative invariance versus apply implement minimum compare adjust variance nan exceed max might reader margin small max substantial minimum alternative class kind likelihood close alternative distinguish entry absolute single particular biology
w use obtain term
order coefficient snr snr make pattern sort condition significantly boost section practically motivated generative arbitrary varied tend degree transform study capability attain detection threshold introduce generative model hierarchical multi ise denote internal characterize dependency vector characterize parent probability parent tree govern contact infection disease latent agglomerative hierarchical covariance leaf proportional denote root subtree contain cluster contain thus covariance easy verify covariance agglomerative perfectly recover graph covariance unbalanced haar lead binary pattern draw
join empty intersection join maximal next elementary full vc collection subset full join dim establish c disjoint element measure let finite follow cover
equation equivalence market strictly convex regular scoring proper exist convex function immediately imply regular strictly convex see imply n define equation equation suppose equation score particular meaning equivalence exactly realize market price occur know optimal I function kkt require market scoring market probability cost base finite score maximize regular achieve market price prediction market represent n recall l maximize expression n equation trade regularizer market regularize cost yield stable bind bind playing role previous imply market scoring regularizer conversely strictly regularizer view strictly rule perhaps scoring rule score scoring rule rule market market distribution treat market correspond understand stable
carlo events pdf belong carry histogram carlo reconstruct experiment simulation value pdf initial value event accord event th bin value estimator setup adjust variable physics consume simulation relate medium complex calculate histogram pdf shall
homogeneous state simplify value space case since henceforth chain random identically similarly matrix depend current affect state c pt traditionally define since observation suffice adjust triple finite observation definition model state observation make since permit time wish determine use past sense define sequence discard potential leave represent actually make random write hand hide process actual process situation make hide underlie actual mean practical physical underlie need take deterministic need way underlie observation precise recall probability letter letter underlie definition q naturally eq distribution probability banach measure variable maximal observation independent symbol symbol take place underlie observation independent union definition conditional represent information make hidden prescribed policy fix mean induction see function write aim
possibility dense analytic turn reduce chebyshev determined study stable deal take generate pz result say normal iff rational satisfying condition deal clearly indeed generate normal strictly stable even power zero within let generating represent sort
function conjugate smooth smooth derive proof therefore ball large summation v technique obtain project accord construction value bound therefore bind tight lemma arbitrary lipschitz lipschitz constant th iteration bind eq bound f simple require acknowledgment like thank verification breast cancer pe related order associate constructive comment improve quality remark grant nsf gm fa high structured sparsity induce structural either widely adopt kind example graph fuse efficient general structured spectrum induce combine proximal method subgradient scalable scalability simulation genetic feature arise science engineering typical response lie interested truly fitness nonsmooth lasso coefficient limit capture structural advantage encourage closely related idea explore structure response mapping output structure relate relevant development remain address develop structure impose order e group
derive multiplier bregman iteration mild irrelevant choice split bregman explain method popularity get equation inversion check explicit constraint square symmetric eigenvector root eigenvector replace correspond approach slow fast separable separable easy solution update negligible third equation straightforward run mostly update update definite symmetric possible demand decomposition newton
example gene investigate sparsity latent formulation small loading latent loading hierarchical incorporate sparsity prior independently result marginally priori loading develop context type dimensionality factor selection choose significant manual reversible jump mcmc many dimension approximate latent could mcmc define appropriate ibp allow specify number dimension nonparametric development equation th dimension include model inverse factor delta distribution infinite infinitely nonzero entry
trade relatively potentially formally kullback leibler third illustrate population desirable information discrimination framework nuisance density framework comparison parametric density value comparison realization component density outcome g g comparison common assigning usually correspond biological feature conversely thereby normalize since integration main arbitrary leave order issue arise sample bayes popular take arithmetic average minimal iid would likewise
high throughput flexible experimental protocol demonstrate feasibility towards automate throughput functional biology early wang extraction automate software preprocesse highly dependent report successful novel amenable root parallelization seminal et variation direct set typically show vary preprocessing depend acquisition vary growth kind hardware standardize remove issue ensure optimal standardized outline gray basic follow image outcome desirable
perform infer signal variance plus minus observation gaussian random centering effective much ideal input uniformly hypercube signal ten method coincide six process hyperparameter two day square mining run mean show group bar rescale always bar height bar cm problem work baseline involve derivation expect method realize
mapping extend create generalize document concentrate discrete relate document spirit influential differ fully local parametric attempt make visualize track word oppose collection important external resource wikipedia google wave framework tool present representation control space simplex tf vector applicability predict predict operation highlight significant structural change benefit collaborative edge discover
discussion united ep conjecture example case study solution abstract generalization variational theory physics problem relative entropy leibler reason measure family absolute show property characterize optimal dual absolute separate transition play decision deterministic transition strictly information resource dual illustration deterministic infinite unbounded infinite measure solution information theory physics mutually discuss simplest index measure normalize element correspond exponential course set banach compact later family multiplication exponential compact compact hermitian operator hilbert characterize absolutely absolutely imply mutual continuous property important measure direct normalize uniquely define transition absolutely non within deterministic transition another perhaps sense mathematical variance attain physics distribution maximize entropy maximize capacity geometry banach construction quantum optimality family measure one seem time criterion deterministic transition strictly sub importance communication algorithmic complexity transition input mutual
maximum md ir n cg inspire framework restrict learn cg form cg machine technique often fast solver solution stress gradient approach learn via approach mention abstract partial least square euclidean norm partial competitive benchmark approach usefulness examine particular rule optimal rate main next
pl proof denote problem compute success argue proof directly transform compute formulae disjoint therefore various tackle pd engine inclusion principle propose symbolic report diagram thousand differ avoid disjoint require proof depend bind proof drop proof become expensive ultimately infeasible path algorithm slight propose use formulae obtain probability work context method remark describe set mean proof never formula formulae subset query success probability shorter always probability information decrease carry include branch motivate rely threshold stop formulae tight level proof current formula derivation stop due reach goal threshold probability shrink algorithm proceed iterative
sdp draw define concentrated sdp generally way start procedure effectively gaussian sdp provide sdp natural function distribution diffusion walk sdp want parameter relative convex infinitely invariant semidefinite think convex von concentrate vector possibility main implicit via computation matrix sdp play role relate structural previously
weak symbol follow real symbol exist symbol self adjoint lead give word depend fouri sense h unitary boundedness mean hausdorff uniform globally operator boundedness localize function construction modification prove set open mean increase flow neighbourhood neighbourhood necessarily unbounde modify follow modify compactly way estimate specifically state construct notation scale angle depend enough contain angle access complement formula decompose block provide square elimination invertible triangular formula effective operator auxiliary operator matrix operator invertible say pose successful reduce nonlinear spectral problem zero valid left verify denote integral center effective review formalism later banach check property immediate recall extend flow consider near operator replace scale confusion likely occur denote quantization letter focus subscript aim theorem canonical neighbourhood kt mutually allow energy layer use arrival notice different equivalently end dynamic energy map fully
project g state classifier sample feature extraction step integral report several publish biased prediction scheme randomization design sophisticated subsampling etc gene vast amount gene datum last decade analysis apply biology g nucleotide snp array analysis generalize sciences health financial etc like relevance gene microarray experimental result appear dimension gene expression level show dimension reduction binary extended multiclass prediction g regression straightforward currently extend latent g etc author support application e increasingly handle
choose correspond root rest sufficient ensure perhaps obvious solution cover lemma incomplete intercept rational last design polynomial system parameter estimate guarantee write consequently polynomial constraint simplify design satisfie ensure argument design polynomial brevity omit scaling choose corollary also upper numerator numerator optimal count hand root nonnegative zero distinct zero interested specific parameter high degree generality information pseudo compatible develop compatible equivalent note optimum complement characterization semidefinite zhang eliminate finally simplify essentially linear criterion semidefinite come whose coefficient main appearance show rational rational design union find
iii tail fy fy short medium tail long generality short tail tailed satisfie von satisfy von monotone satisfy short short tailed satisfie failure failure hypothesis medium hypothesis test es medium tailed alternative hypothesis es es significance favor reject test long tail tailed sub classify asymptotic hazard super moderately short slowly vary theorem provide consistency test weakly short mild need theorem short tailed assumption theorem statistic short tailed condition mild distribution function denote log long tail short tailed condition satisfy vary long e seems demonstrate many unable long tailed eventually test test vary result
pseudo derivative jx jx j jx yy jx jx risk argument prove nf right nx f equality w display ny p n definition integrate p well duality argument expectation
clearly advantage list heuristic fit fit neighborhood neighborhood bind cluster choose cluster identify cluster outlier case report sensitive removal naturally respectively htbp misclassifie three traffic ms ssc r c pt median median ms ms standard deviation misclassifie database r pt mean median median ssc l c ms ms ms ssc data dimension fix follow mixed fixing generate hybrid linear code http edu subspace uniform ball multivariate generate outlier cube maximal former origin ssc mixed support specify dimension subspace mix unnecessary instance misclassification record algorithm e various artificial hybrid modeling affine obvious outlier g ms case unlike affine particular non ssc fit heuristic effectively estimate noise running time algorithms ssc algorithms function kf
performance unseen signal dictionary suit representation bind depend number problem minimization erm prove convergence admissible bounding sample rely consist naturally represent similarity inner product diverse text represent extend usefulness set representation practice diverse field etc dictionary motivation coefficient typically compression commonly store representation condition unique dictionary chen atomic sense sparse representation retrieve need implication dictionary simultaneously choose prior motivate dictionary use compression denoise dictionary extensively
rescale forward backward normalization approach recall recursive ji sx k rescale backward jx manner sx limit label align mt mt decoder base base transformation break first claim concern write upon require eq original recursion handle straightforward consequence continuity respect claim fourth claim maximize arrive characterization viterbi practice thus decoder side decoder transform beyond decode let v break rule exercise calculus positivity make note decoder unnormalized limit decoder normalize transform let cm align leave I I decoder transformation decoder use break reduced factor numerator denominator expression induction variable proposition jx jx recall numerator observe kf sx sx give I example last hmm emission alphabet initial distribution emission f suppose show original bottom transform decoder power hybrid indeed satisfy also realistic hmm indeed convergence hybrid decoder viterbi naturally generally summarize purely algorithmic align fail viterbi viterbi since understand interpolation choice seem practice except trivially underlie normalize well aspect mainly rescale indeed work analyze clear general member regardless explicit make interpret complex cycle g viterbi cf apply decade aware idea inference plausible reason see mostly lack illustrate viterbi task predict structure purpose base entirely first hmm particular allow significantly elaborate include attain
scope current argue statistical challenge ask classical approach dual type investigate encode character alternatively genetic fix geometric multivariate character specie variation closely challenge selection address statistically discussion suitable bayesian
anneal long history trace maximum twice free strong aic number minimum description principle mutual closely source approximation quite space cluster dissimilarity weight vanish dissimilarity clustering rely ultimately fail need identify theoretic perspective uncertainty measurement resolution hypothesis program set
tx belong class henceforth integer stand convex continuous non negative ball relax subgradient projection mapping employ scale path carry view study project minimizer subgradient subgradient class quasi mapping large strongly mapping give save reader metric onto weighted ball j component follow sequel regard definition assumption subgradient projection mapping c n lx k jx j j stand convex hull hold true hold relaxed mapping save calculation standard easily choose w l establish subsequence example section function sequence already online learn class strongly construct wide applicability even large usage algorithm sequence nonempty close user index sequence slide
decrease gold proxy air likely take proxy identifiability include proxy somewhat average coverage fig discrepancy diagnostic reasonable diagnostic expect scale scenario discrepancy large scale scenario scale discrepancy diagnostic estimate cause identifiability suggest diagnostic rather discrepancy diagnostic diagnostic distance drop diagnostic scenario fit separate month mid proxy four km grid spatially overlap km assign cell fall take cell purpose hoc retain essential character computation miss due cloud smooth additive cover month location cloud operational model nearby road smooth spline road three road four km spline point source five per year source km consider calibrate height scale comparison co relate proxy month fig find relationship pm ignore proxy pm explain discrepancy fig consistent exclude prediction correspondingly assessment proxy predictive worse exclude likely assessment quantify qualitatively hold suggest area informative strength gold able exploit proxy pm bottom month proxy mid panel u panel prediction proxy domain analysis p inclusion mid without proxy isolated proxy proxy proxy proxy separate output proxy km area east give
adaptation account positive corollary adapting double laplace scale gibbs sampling augment p relevant posterior section treat marginal mle describe facilitate bayesian cdf pdf representation follow likelihood parameter posterior conditional truncate normal complicated infinite prior I combination highly inaccurate even evaluation derive conditional rejection prefer integration helpful mh obtain alternate cdf normal proposal accept mh good proposal improvement acceptance rate high posterior therefore scheme preferable acceptance increase mh rule thin draw fast fold sampler speed evaluation arithmetic square root
indicator regular design coincide design design regular design non fractional factorial design fractional factorial design design regular proper design q indicator coincide indicator indicator indicator complete constant small give fractional factorial design full factorial desirable identifiability add real factor relation purpose basis factor formal include let fractional factorial suppose additional among new q conversely
dependency correspond evolutionary breaking markov slice sampling perform method flexible expressive family unobserve end task address arise latent nonparametric constructing provide posterior explore variety density nest mixture depth tree branch also useful multi task vision motion discover develop unbounded depth live node se unlike exchangeable
specific simulator contour gp sufficient run simulator must complete ei yield good improvement multimodal ei estimation minimum branch outperform sequential step dramatically computer experiment estimation employ simulate realistic consuming evaluate desirable whereby initial surrogate spatial surrogate combination along suitable subsequent set trial great improvement interest popular trial expect criterion varie point exhibit local optimality greedy trial interest high search away previously design small simulator point change ei require another ei find focus branch contour simultaneous estimation maximum minimum bound branch
construct equation step could construction strategy construction third say game scheme finite half payoff base might game resp choose generate player receive mapping extend si iy si distinguish player stress realization law realization observation nature choose outcome pick choose right label payoff payoff law receive probability cc cc payoff signal action response generate generate action player response say behavioral resp player nj external framework player average payoff uniformly player might pure mixed pure bad
ks chains simulation split mcmc batch quasi limit ks experiment statistic result beta truncate modify autoregressive order besides precision influence parameter order c precision rmse c rmse c square rmse acc effective ess average ks diagnostic batch precision precision parameter behave differently autoregressive truncate rmse value get third column list parameter acc acceptance rate improvement acc autoregressive ess draw markov order ks last acceptance hypothesis convergence display chain get outcome prior well truncate autoregressive rmse calculate modify prior acc slightly ks
simple method diag lastly diag replace every give may tune validation cv suggest graphical advantageous simulation decide select cv pay attention choose cv bic un replace alternative bic select sparsity k binary standard coefficient consistent odd preferable section well previous likelihood small conclusion confirm sparsity namely positive correctly identify false positive incorrectly association well identical positive harmonic oracle regard conduct penalty neighborhood coincide alternative would approach worse select bic recommend line package adapt allow un standard logistic un method value small
size especially cccc linear setup lin whether code totally response generate plus minus mu mu minus mu mu minus mu mu mu minus mu minus glasso adaptive wang frequency correct replication wang glasso tune aic degree freedom lin equally spaced result glasso model contrast produce work cccc glasso composite ii lin create previous code true totally group composite sometimes wrong interaction correspond fitting always order theoretically yu cccc glasso novel aspect use shrinkage give consistent ensemble produce averaging inference unify group constrain penalty yu implement mu plus adaptive lasso
empirical image image address separately frequency factorize q explicit law law robust paper gaussian consequently observe q naturally embed classical choice practical algorithmic update law law conjugate law parameter pdf law improper informative jeffreys insufficient reason nevertheless choice limited law possible easy point law build multiplying law explicitly rule law normalization describe however preserve frequency addition prior eliminate integrate accord appendix law parameter remove penalization law denominator remove dirac equation integration respect denominator
similarly detail classification sim via author sim classification use gp sim integrate improvement combine statistic derive illustrate constrained arise health policy combination include sim function classification context design support sim package sim contain use hierarchical difference similarly orthogonality identifiable orthogonal matrix care rw proposal reversible mcmc finally gp sim gaussian acknowledgment grateful foundation education whose constructive comment lead three sign sample gp sim solely identify label aspect direct true gp discuss employ possibly unit involve collect reference find sample obtain collection sign wrong index plot average index easy much one rely predictive perhaps reliable automatic involve look directly
allow step hence possible evolve fix learnable element essentially see set choosing reduce function rather process always high unless distribution efficiently learnable every access set polynomial circuit circuit access construct output access f imply learnable concept efficiently learnable exist inverse evolution monotonically tn pn sn circuit representation theorem require point algorithm explain uniformly nee sure high present mutation denote representation high candidate generation pn nr mutation sn rule tn fr fr fr empirical mutation empty fr tn sn establish slightly representation increase candidate influence empty pn sn improve decrease substantial transformation transformation
satisfied optimum optimum identify solely henceforth though convex feasibility incur computationally problem establish rs problem linear equation np instance integer rs remain rs rs establish search optimum computationally adopt square alternatively consider error handle every per reliably identify sensor envelope large g use hard efficiently implicit norm albeit unconstraine norm residual circle sum euclidean norm optimal location constraint either interior relaxed rs reflect link interpretation section versus ls rewritten mention sparse establish connection assume non reconstruct determined equality minimizer pseudo block sparse degenerate single reduce regression cf cs pursuit thorough instead substitute close namely convex yield tight norm apart logarithm determinant propose box kt sufficiently strictly tend
confident drop precision baseline core cascade exponential power law generate cascade edge cascade choose cascade point three much break particularly especially careful cascade cascade cascade edge cascade performance regardless work reliably regardless refer similarly network power law dramatically drop heavy exponential problem much hard break drop performance break axis transmission reliably recover achieve forest bit important note network free core remarkably seem depend try type cascade sharp drop greedy start low marginal gain edge probability make mistake cascade spread easier never identify transmission edge identify cascade examine cascade cascade break cascade transmission event cascade transmission event cascade amount cascade transmission recover edge big produce cascade interestingly cascade cascade large infer parent parent break find opt right therefore opt computation per algorithm localize hierarchical exponential localize improvement order solution practice use infer node matter hour gaussian experiment far true draw break add cascade even function add time reach increment interestingly break real cascade able cascade jump technique miss
generalize suitable demonstrate finitely ordinary differential regard change transmission behavior process gray rgb analysis action potential cell detector ensemble possible number follow delay enable process generic response well phase jump transfer periodic class process applicable framework process event basic enter device typically discriminate arbitrarily detector nature upon arrival action might release component many network
inexact line search require one variant search iteration dimensional acceleration dynamic depict difference three since em implementation small iteration obviously cost run mainly relative next set compare time line search optimize detailed line unconstrained appendix nine algorithm set simulation effect performance accelerate em need need implementation fast fast term cpu vs flat quickly flat clearly let covariance matrix
may rescale actually without generality keep rescale therefore observe property fast asymptotic behavior function present result fix assume differentiable taylor take g odd z eventually yield achieve prescribe theorem depend vanish super polynomially observation distinguish ridge function b ga neighborhood f r u fx ga u ga point evaluation respectively hand notice condition impose give function class generalize approach ridge obviously ridge appearing hold motivate algorithm access direction respectively collect directional similarly
quite complicated quantity asymptotic exist give equation solution mean table stock market usefulness share package bank portfolio package monitoring share quality realistic statistically desirable market share package portfolio choice package characterize bank bank invariant share portfolio share order package priori qualitative characteristic stock simplify real assume possess financial capital portfolio share resource package bank analysis modify package quite portfolio market investigate theorem exercise compete portfolio statistically strategie markov
median self exposition omit literature logic discuss idea search database content content discuss database structure learn computational dedicated structural updating phenomenon language formation exposition evolve already list human thought memory last way read numerous replace space field exist grateful advance comment acknowledge due abstract set partially endow implie say borel natural sub easy hold say say subset say element resp mean relate possibility observer sub object graph state embed structure memory long implication question structure unchanged property favor contain big interpretation concrete return arc formal tb vice versa relation choose consequence sure n w would change relation hold complementary mean relation hold see hold graph paragraph originally terminology fit intend application coherence coherent coherent call selection mean relation iff symmetric back endow structure w visualize simple cube answer every turn incoherent adjacent cube symbol generate proper coherent skeleton cube exercise exercise finite nest element cut exercise converse know exercise disjoint immediately kind vertex vertex pn union set shortest connect
insight ps bounded rhs fy fy exhibit ps mcmc score ps show continuity sum finitely point rate mix gold apparent extent practically big prior particular remain normally start finite moment posterior likely moment make function integrable suggest near converge may slow posterior degree rapidly contribution large provide insight mcmc converge mixing support figure maximize remain make transformation parameter go high maximize nest parsimonious inherent showing prior stand issue maximize likelihood nearly concentration around mode namely range space range lot fluctuation mcmc mcmc output tell proportional see remark lot explain score decay spread bf reflect figure ps factor detail check point accuracy well test remark two nest modification ps new work quite differ log discussion would effect theoretical path ps notational
involve cancer reference ratio intensitie sample plot position segment various particularly level possibly encode assign segmentation copy several provide apply popular deal resolution assign discretized unit clearly array platform interest copy protein nature dna piece piece relevant unit association clinical simplify terminology array although approach enable consider resolution region region neighbor discretized profile contiguous direction clustering consider spatial pattern perform simultaneous cluster wise idea multiple testing
much stability properly select learn big rate demonstrate usefulness five microarray gene expression commonly scientific classification gene varies implement effective reduce classifier case issue associate factorization issue scope issue specify cluster origin knowledge one stability varied guide whether cluster consider recently al interpretability latter correlation original variable microarray currently go record
selection band jointly kernel spectrum decay spectrum mkl model nine toy problem goal spectra goal subset kernel well kernel
term unlikely fraction step additional step bring fit threshold variant amenable accordance role likely fraction adjust likely correctly allocation acceptable allow correctly section mistake role formula mistake extent positivity formula favorable reliability target code moreover evaluate mistake arise threshold standardized product
cardinality gap introduce slightly form allow replace gap suggest david establish elementary relating gap weak name sensitive result x let family borel stationary ergodic asymptotic dividing minor modification positive equivalent stationary process measurable importantly use proof key nevertheless family condition nice measurable countable sub immediately ergodic extension drop exclude example process illustrate rotation measure composition xt stationary ergodic easy see f give sufficient case show ball df process give
variable follow next martingale w k martingale q variable sign follow hoeffding conclude case modify old inequality word argument finish q real choose bfgs bfgs hessian gradient goal function mkl model apart investigate mixture coefficient weight single highest sparse cover entire setup mkl elastic
two nature consume interested surface shape site common protein efficient search scan surface two candidate bind site mcmc method small confirm match difficulty use efficiently htbp htbp htbp htbp sample last sample variance calculate hold constant conjecture school sciences uk recently shape markov monte improvement exist convergence jump configuration match bind connection gibbs configuration point challenge application bioinformatics vision matching set matching may relevant object
integrable partial thus optimal smoothness randomization digital nets rmse integrable partial derivative result necessary net describe digital net underlie randomize stem function vector ds sd f monte sample aim convergence write bad q c star choose z integer b
determine min I unlike particle perturbation ergodic readily parameter achieve rate present employ langevin mala ir qr density perform step retain average acceptance ratio ergodicity mala perform equation computational expense exhibit mix physics ratio successive density obtain update importance identity estimator multiply normalization require section basic propose inner subsection compute loop cardinality expansion gain exceed tolerance desire temperature break equation p describe calculate temperature often interested landscape temperature goal fact firstly landscape nearby
friend event neighbor low friend event group high overlap group friend determine diagnostic diagnostic across different diagnostic friend group neighbor diagnostic iteration hand type reach group event group five friend event indicate customer observe k correspondingly converge show isolated reflect convergence approximate equilibrium reliably equilibrium bias e connectivity discard number run collect verify remain discard reach stationary contain user keep remain formally qualitatively burn determination also inspection run fig reveal walk note fm method stem status network rich base connect isolated individual user tie call walk base particular reach graph might typical belong friend friend neighbor friend friend friend
intra level shift particularly affect trading consider tailed inter day case shift ignore occur real abc observe run sampler perform model innovation analysis synthetic study setting table estimation mmse mmse mmse mmse mmse variance simulation markov away parameter ignore shift significant exact non abc mmse intra shift propose consider identical stable tolerance abc table versus basic mmse mmse mmse mmse acceptance mmse set start away model incorporate summarize mmse inter day section accuracy important algorithmic trading price series deviation trading basic mixture abc cd sample min interval market perform contract worth produce length raw circle represent open inter boundary h transform deviation perform batch day batch average sampler table c c var var mmse var batch demonstrate account
large plug code word modify plug sequence redundancy code unless since must case must member code almost lebesgue I rough sketch detailed theorem special say pp arbitrary constructing source rewrite redundancy convenient family g take divergence trick
way prediction treat possible art machine use prediction sized pair short imbalance predict link anomaly dendrogram connection prediction numerical al co occurrence evolve compute centrality evolve aggregate adjacency similar netflix rating predict focus rating change preference time et filter tensor web link email communication exploratory several link author conference service context application internet traffic temporal combine slice single matrix summation also bipartite graph analyze matrix fully temporal signature cp temporal cp gain computing use matrix see datum auc tractable however behind new link tensor tensor illustrate predict link underlie process tensor incur
significant amount expect bregman iterative usage wide require differentiable treat explicitly see proof detail theorem subproblem involve optimality solution order p v formulate compare denote subtract side subtract fourth side q substituting sum subgradient involve therefore lead eq prove whenever contradiction subsequence height bias hinge complicated observation l ns subtract k k ty k inner product second third fourth sum subtracting
strategy select include intrinsic another default informative change comparison comparison performance oracle assume tuning selector selector relaxation normal two group limitation net ridge design six dataset uncorrelated component iid simulated ii response multivariate simulate example multivariate gaussian gaussian correlation simulate assess tuning selector elastic minimize one simulated measure performance error mse
relationship bi graph special direct marginally bi like dag conditioning vertex graph page domain vertex acyclic remain variable dag bi suffer example graph parameterization introduce bi vertex contain probability parameterization operation parameterization form complete sense however price exponentially conditional table bayesian cumulative convenient direct additional term factor graph purpose bi direct cdf parametrize sufficient cdf direct graph
modify memory angular momentum evolution alternatively correlation term take physical contain problem autocorrelation beyond program author upon support scientific fellowship cluster maintain national job grant grateful ann discussion anonymous paper institute university nature system measure angular body reliably relaxation rate angular find angular momentum walk slow
bound follow strictly continuously continuous weak topology weakly continuous attention continuous functional weakly definition topology directly weakly continuously banach convergent bind dominate g whether weak concern hilbert weak continuous set weak bound weakly compact moreover subsequence weakly necessarily vanish uniqueness convergent note reproduce kernel space pointwise thm thm thm example thm predictor present banach intensity parametrize reproduce result representation log latter additive specification estimation functional linear enter specification inspire multivariate
variance extent residual another correlation numerically row total give mean generate connectivity zero equilibrium correspond exclude deviation divide generally stability divide figure across outlier confirm simulated generalize genetic could unstable ga ga computed fitness select removal exist follow sufficient fitness e still highest obtain across property unstable respect examine sum row heterogeneous system stability computation rather together confirm stability ratio system numerically term review causality simplicity extend describe satisfactory decomposition stationary lag eq transform x lag characteristic lie outside circle condition residual split eq spectral decompose part causal address transformation leave causality uncorrelate whereby split causality frequency ref establish fundamental motivate relationship g
measurement affect traditionally affinity show define measurement prove unitary replace compressed dimensionality problem dimensionality datum costly dimensionality tractable apply transform feasible support domain transform detail compress random pose sparsity recovery result compressed prove exact recovery ambient compressed isometry q bernoulli fourier random sensing provide measurement uniform let x eq take root give q divide subtract satisfy c ci measurement rip close gaussian find bernoulli row fourier
dp dp paper define prior choice prior encourage natural dp normal induce estimation interpret regularize covariance regularization arise introduce arise introduction encoding cluster independence unstable similar although specific generating translate pair conditionally conditional independence derive regular arise analytically tractable graph resort mcmc extensively year divide gibbs sampler exchangeability explicitly mix jump sampler paper marginal sampler option indeed integrate precision sampler dramatically burden proceed joint indicator conditional j l th exclude cluster predictive cluster
user interference whereby secondary channel channel optimal channel match user sum throughput maximize scenario two user e link assume show throughput secondary user optimal secondary allocate channel user allocate channel aware sequential bandit matching channel throughput channel allocation random convention variable reward permutation channel slot slot apply storage update slot polynomial matching problem bipartite channel tight short source transmission delay minimum sum I maximize maximization problem clarity
example k deduce give range reduce estimate large almost entirely close hand lemma unconditional distance order since mean begin mean eq poisson formula albeit rather reach contribution area great st research uk limited national nr scale section evolution molecular different sampling
hamming set eq proposition attain exponential particular exponential sufficient q appendix ham ball relation coding theory convex hull code mm let strictly convexity mm bit elaborate part boundary dc c cc c induce homotopy sphere interior mp mixture must number p family without row hausdorff
review validate formal presentation count traditional argument reproduce sketch completeness qr factorization rearrange arrive imply lead deal inequality partition adaptation pp sort singular resp resp follow unless require amount lemma result show proceed qr transform complementary dual rank continue towards constraint always would inequality increase unless uniqueness conclude factorization affect uniqueness within explicit distortion version respectively general provable change much solution assume theorem towards confirm leave norm specific practice however assumption uniformly chance
short memory fractional artificial sample ii estimates ar spike lag pattern fractional lag inspection indicate period seem may decay period strong structure I apart estimation different table parameter calculate frequency display empirical square ols small ht c mse short accordance may reduce bandwidth however task practical behavior value may indicate short part model fractional affect whereas fractional affect estimate small regression
uniquely copula function dependence translate function variable transformation preserve novel generate multivariate specify distribution nonetheless marginal correlation frequently strong generating marginal equal organized idea simulation accommodate plausible range detailed example bivariate family uniform beta minimal correlation correlation call al never summary
rarely practice algorithm concern rest illustrate et fisher choice algorithm start uniform respectively advantage count potential algorithm display
claim two induction yy theorem diagonal sufficient yy xx dependent standard vc origin necessarily lr yy yy yy right side sphere require lemma norm variable psd yx u ds v eq matrix fourth moment psd exist depend lemma extend n column far psd matrix ij center ij ik
regression aic bic summarize bic outperform second explain cc ccccc ccccc size bic aic behave candidate dataset nj keep dataset bic table summarize comparison uncorrelated ccccc ccccc size aic bic aic bic single e set keep working certainly true nonlinearity aic model summarize conclusion aic cc ccccc ccccc ccccc c aic aic bic new criterion add variance model assume keep rest assume model model aic bic summarize result frequency aic criterion indicate usefulness capture
let need convenience continue moment trivial counter conceptually length entry vector random skew stable I analysis stream matrix entry demand stream prove estimator unbiased see theoretical property moment skew generate maximally skew random variable avoid q simplification explain show accurately represent since straightforward stable define cc sample harmonic moment simplify unbiased asymptotic adequate unbiased mean
neuron former fire fire post stimulus almost spike even stimulus per presentation entropy spike paper determine train priori take optimally markov spike train length confidence check demonstrate quantify algorithmic content quantify spike quantify spike simulated train train record layer demonstrate increase structure practical technique quantify structure intuitively neither highly order strictly periodic spike train think since possess generate order spike organization neural activity would imply calculate stand assign high complexity train reproduce high algorithmic rate sort entropy desire state define condition event entropy differently use symbol string train string appear spike symbol string markov bernoulli exhibit possess extremely future process note train hmms state macro macro graphical hmms different linear kalman encoding decode recursively constitute minimal drive formalism generating influence implement spike train neuron locate reflect neuron internal nonetheless neuron neuron state appear stimulus detect stimulus
suboptimal study use digital communication orthogonal multiple channel estimator digital quantization scheme analog transmission unknown symbol symbol mle exploit channel complexity rapidly mle performance low communication snr result suboptimal digital communication outperform transmission channel quantization decentralize multiple quantization superior strict constraint quantization quantization relative high com suboptimal decentralize wireless sensor access digital transmission training mle implicitly reduce propose estimator signal datum level redundant traditional fusion diversity communication observation perfect message analog digital transmission simulation digital communication estimator analog transmission channel energy quantization superior quantization medium observation noise ratio level wireless sensor physical processing decentralize decentralize fusion sensor process observation fc inter sensor
mean assume ergodic generate cluster consistent sample variant require regime length formulation literature model hide model example bayesian cluster formulation generalize one homogeneity two test generate distribution cluster process I markov show ergodic binary ergodic demonstrate asymptotically joint stationary mix consistent distributional distributional
function cardinality variable independently frequentist freedom factorial lead submodular submodular function marginal likelihood associate highlight factorial exactly function extensive factorial provide trace statistic w large lead norm however analysis apply algorithmic sparsity norm norm unit ball potentially number vertex face cm get descent compare differentiable particularly consider accelerate variant apply efficiently p equivalent minimization namely fast submodular show solution moreover minimize directly unique submodular also
jacobian density differential privacy remainder excess minimize convexity objective regularize follow excess minimize classifier objective risk minimize perturb optimality q use cauchy schwarz combine probability least prove tight trade interested classifier perform new source private perturb function risk classifier expect value
pearson assess result produce network entropy reject pearson slightly mutual deal contingency table shrinkage score criterion grow hill combine test figure hill algorithm date large display variability min hill confirm make instability h statistics monte carlo along fundamental
note condition write event trivially assumption f consequently condition turn trivially union coherence modern processing arguably become nevertheless still think million entry design seem reconstruction restrictive variant thresholde regard carry thereby fail optimally worth point past variant perform illustration regard key model asymptotic deterministic vector nonzero result use multiplication solve regularization trial amount average take selection order second apart thresholding selection analyze agnostic understand true selection point conservative nature small scale still room reduce threshold mainly loose conference reduce experiment regard specifically kf f respectively set carry prove somewhat conservative nature constant contribution low complexity matching pursuit succeed generic deterministic average used establish carry recovery irrespective phase nonzero collect concentration
overlap consequence strength weight learn imply learn variance govern induce furthermore pattern duration obey time basis capacity draw replica cluster probability threshold pattern compact shape typical accord classification yield spike representation ir implement ir entire solution
encourage induce penalty cost option unnormalize low unfortunately
meaning integrate expression relation aim human like ap user share different language different color position play describe game object user try select figure contrary image user guess describe root language constraint mean computing learn semantic description step need collect description shape description feature generate new description learn web user sort semantic share constraint
induce rate drop significantly specify bandwidth direct connection link represent start store forward switch delay know remain duration traffic path traffic link notation denote interested determine large traffic flow almost seek define complete transmission specified period denote rate available available relie develop bandwidth fact moment link relationship rate p r establish satisfy determine satisfy pair tight imply l link formalize employ train measurement form network available bandwidth model available train evaluate outcome whether train instant path outcome specify probe link sequence form credible interval tight
code length locality image image constrain agglomerative merge region adjacent texture effectively deal majority texture window intersect boundary region degenerate window neighborhood region reliably estimate window size degenerate marginally deal propose size start window ever degenerate reduce please hierarchical htb window nevertheless new much accurate image much segmentation result summarize texture encoding distortion max window stacking construct maximal maximal r r w mr l j require determine segmentation measure segmentation image scale optimal solution distortion parameter segmentation assume user ground truth
parameter analytically natural equation employ lead quantitative us quantity case np definition rewrite analytic account prediction translate prediction matrix linearly column freedom remove degree fitted remain necessarily linear independence linearly linear compose arbitrary discussion conclusion give practice freedom cm fit motivated argument fit prior must negative figure demonstrate two prior flexibility
examine daily return gaussian dp concentration use rely iw sec hyperparameter process noise return fall start agreement give package fall bank joint economic fed cut rate exchange bank posterior display variant towards self rapidly redundant align bar bar roc bar bar roc hdp towards hdp hdp hdp table raw daily overall point hdp hdp hdp ar non roc maximum window probability use advantage hdp hmm sequence hdp return observation process datum center around roughly take outlined hdp regime case dynamical physics motion complicate dynamical description force appropriate herein target tracking application dynamical know target define transition dynamic equivalently acceleration explore along experiment multiple demonstrate flexibility al variant et structure switch dynamical describe phenomena utility hdp ar hmm one able volatility stock exchange ard sparsity induce flexible variable
markov spin dynamical nearby apart dynamical follow term dynamic nearby initial indistinguishable algorithm closely term coupling lose initial configuration coupling chain sampling coupling applicable carlo coupling element entire coupling avoid couple large large space avoid ise heat spin patch inspire rigorously generate
look algorithm carefully exploitation exploration evy well commonly good cast exploitation solution use walk obeys adjustment search evy potentially efficient step large far fortunately solution neighbourhood effectively diverse evy evy good lead simulation insensitive dependent tune specific
efficiency boost class boost boost additive tree multi logistic I adopt flexible weak learner tree parameter log otherwise typically adopt build gradient newton first information loss construct derivative respective split
interested beyond enhance develop proposition see paper steady queue length g scale version use proposition mean close p shall stationary say steady queue length see give subset guide construction efficient splitting develop behind splitting shall divide position target put j nc nc induce discuss intuitive becomes place splitting proceed follow weight initial particle say reach integer child parent child particle start particle particle either level apply weakly splitting algorithm unlikely particle motivation construct balanced convenient formal mechanism indicate sa
draw distribution draw uniform implementation function evaluate double library double diagram figure quantile double double apart due taylor expansion zero peak due peak error
solve optimization q relate together ie covariate model version graphical correspond prior incorporate place laplace ij b zero definite central obtain iterative construct lead procedure modification benefit generalization group hyperparameter negative belief focus keep tendency observation optimization particular distribution map estimate motivate primarily deal bayes estimator loss issue address perhaps
average degree degree degree predefine link turn characterize frequency give sub increase heart possible etc assign related sub proportional sub graph parameter study maximal model approach study frequency object simple order sort scenario stop network higher computationally hierarchy self describe nature turn theory construction adjacency randomize locally matrix
situation homotopy smoothed parameter solve solution next prefer homotopy large smooth warm start often allow homotopy constant update weight vector typical svm svm programming l unified form lp svm calculate lipschitz loss therefore smoothed hinge solve algorithm regularizer hinge loss hinge calculated smooth parameter eq initial homotopy l svm regularizer eq sum quadratic hinge loss lipschitz ls solve demonstrate efficiency
usefulness deal dimensionality rank absolute marginal rank covariate result rank covariate name screen sis marginally iterative sis begin response covariate sis sis meet sis sure independence sis linear robust several important sis cox proportional index true cox sure procedure satisfy property select model tend partial single covariate x large marginal utility correspond survival rank covariate covariate index especially formally show noise covariate expand sure property lot
valid asymptotically valid later er rao tend introduce concentration inequality depict one find independently arbitrary vector tends exponentially tend straightforward show eq want concentration want simplify construct index exponentially approximation asymptotically approximation orthonormal index probability tend exponentially orthonormal eigenvector eigenvector approximate substitute q say enough exponentially concentration similar state substitute
relative error probable make unnecessary set batch instead possibility iterative split batch statistical reduce entropy lead linear positive spectral eps default focus optical typical easily transform transform axis modification work start recover exactly deviation different approach fig optical default model optical peak transition exact default contain default output calculation similar result broad spectra organize introduce formalism
available domain coincide coincide consequence normalization concern without assume universe important include incorporate interest crucial independent identical whether term factor affect rank summarize rule make inference information piece know relation likelihood update process emphasize relate contain completely insight smoothly incorporate
alternative sampler vector time efficiency markov utilize sampler large become computationally impractical involve variate full sampler difficulty utilize reflect variate variable ensure update acceptance mcmc methodology line mcmc several mcmc basically mcmc online way though markov simulation preserve result variate gibbs typically bayesian automate alternative adaptive sampler additionally bayesian perspective tackle bf analysis rank e lag lag work lebesgue denote unit vector markov chain transition realize adaptive sequence index variate model formulate discuss integration rank include discuss justification select respect explicit
high document plug naive classifier accuracy would common identification address compete application letter unknown also three review across letter approximation numerator factor quantification procedure application writing collect measure document kullback kl chi square verification difficulty create write accurately observe score potential resample subsampling writing contribution equally stage development writing acknowledgment manuscript give support part contract investigation laboratory name inclusion necessarily support ic fellowship study document sample document categorical classifier large writing
depend concentrate universal code derive mixture index specify weight thus scope mix sequentially well state state long weight positive unimodal great impact sample derive model purpose side counterpart absolute back sign computed close conjugate prior side respectively plug full abuse prior two instead know code theory model laplacian distribution tune row unknown apply model see central coding induce center differentiable regularizer become robust alternative regularizer regression consistent identify context compressive well recover signal compressive sense regularizer arise approximate pseudo element report partially confirm regularizer show regularizer use non regularizer recovery coefficient patch well jeffreys determinant jeffreys hyper parametrization lack reason jeffreys regret code jeffreys grow jeffreys attain
increment draw u draw increment increment draw kk jt illustrate group course box group customer come customer join either upon group contribute popularity box box whole factor indice pool global moreover induce index exist take choose increment mechanism group generate member sharing note global relationship category main university various popularity share manner frequency popularity although describe bring distinction viewpoint viewpoint inferential identifiability finite view limit finite model weakly bound hold counterpart shall distribution due randomness place density call weakly finite group normal mix q place rectangle place arbitrarily sufficiently spread dirichlet omit thank relevant viewpoint interested recover local identifiability mixture union mixture mixture component marginal density group component distribution give consequence
transform characteristic know note though test particular optimistic however require purpose explore property compare base replication show fix unique either solution right column error rr c variance joint remarkable extremely heavy tailed typically lie particular standard consider realistic consider tail interesting gaussian fourth heavy tailed estimate row bottom rr mle w c r rr c na r gaussian mle ratio r rr median w na ratio tolerance replication proportion middle deviation around truth mle mle implicitly see fair include median impose critical affect solely directly result deviation fourth moment exist grow contrast heavy tail median outperform median finite location inferior estimator unbiased mle continue unbiased extremely heavy moreover last show na
constant interval since seek hierarchical successively approximate example follow subsection framework symmetry operator symmetry also role transform encode terminal root encoding contribution coefficient set imply code x x non concern generality lose lowest lose operation object effect one level product bottom dendrogram either cluster result apply operator singleton give terminal end element preserve implication let take term inductive inductive define subset lead
profile infer plot biological attribute relevant grateful literature break random distribution begin result dirichlet close identity exchangeable form model computational readily adapt partially gene profile gene profile partition h four close probabilistic might encourage statistical worth connection methodology important volume dirichlet satisfying mixture broadly characteristic mixture cluster identity shall call identity character unit call item unknown parameter subset conditionally exchangeable covariate simplicity write
separately let addition meet suffice monotonically tend monotonically small stem second character establish iv side expression reduce value yield see factorial q solution lie condition let monotone monotone decrease interval monotonicity follow analogous function definition kk uniform proposition use determined hand particular establish
mr auxiliary report simulate example replicate p respectively likelihood factor due believe base delay copula approximation similar burden increase give h base later reduce evaluate complete monte carlo auxiliary language ghz job assign worker believe code factor contain stock major uk france continuously return nominal obtain capital index delay rejection one estimate sub gaussian copula sub marginal suggest provide evidence list due consideration level discuss powerful hasting either single block multiple slow delay rejection discuss delay generalized block factor gaussian factor survey inference
bivariate decision significance measure couple ordering significance give sided value surrogate level correct statistic graph rejection coupling vs though driving detect great series rejection confidence achieve agreement apply embed procedure patient generalize eeg transfer anti brain channel channel improve resolution potential reduce activity neighboring channel correspond p record length patient patient take channel one segment second duration scheme mixed vanish delay mi lag right channel channel number vector pair direction patient patient htb conclusion patient seem exchange indicate almost always channel channel channel patient indicate observation exchange embed panel record agreement brain patient diversity regard channel dimension c substantially component indicate area seem embed channel explain top interval minute dimension respective profile agreement profile
threshold ray detect subsection able ray dr ray identify cluster consist cluster ray cover limited band compose band confirm optical cluster area span dr combine ray algorithm effectively completeness cluster separation project ray match cluster ray cluster ray separation ray match reliably ray separation ever optical go wide range identify galaxy cluster detect galaxy feature galaxy line contamination universal preserve exclude feature particular form even blue use band exhibit tight galaxy cluster spectra regular break color show large use use red color galaxy getting detect match precision algorithm within serious disadvantage use computation produce dr hour core long detect plus red note good weighted optical mass count member galaxy direct galaxy member determine formation cluster work room deep work dr reliably guarantee implementation color red serious preferable color span color additional improvement deeply co income dark tm acknowledge nsf grant er
change decision social terminate false alarm penalty incur alarm event happen alarm penalty alarm course false alarm incur expect false alarm alarm penalty obviously ii delay learn protocol continue incur event change delay remark public belief depend determine decision terminate link view modification decision define include incur measurable correspond sigma algebra denote determine economic discount guarantee kolmogorov criterion geometrically alarm vector discount objective assume unlike classical belief give public belief cost maker minimal policy summary distribute time specify eq transition observation cost alarm discount factor decision social extension standard incur detection large classical policy programming belief characterization analyzing stop bellman eq maker belief define convenient rewrite bellman define bellman programming equation transformation algorithm captures involve unchanged coordinate maker public change stop iteration use previously time result confusion iteration bellman proceed bound real generate continuous bellman yield computing since bellman value computational view section devise stochastic illustrate detection trivial belief belief update mathematical induction bellman positively denominator equality clear concave necessarily concave concave map distinct explicit function stop read denote incur optimal detection function preserve concavity easily piecewise linear value
point condition curvature surface quantify curve notion curvature second fundamental informally restrict region manifold less hyperplane implicit riemannian seem fraction tangent plane apply manifold turn manifold let q x close property consequence imply x nr nr x r rx combination hull point x q rs means fx consider random provide improved number however
throughput reasonable resource section theory graphical describe procedure inference place expression model propose present conclude remark gene graphical expression gene point gene characterize v vertex edge vertex adjacent connect conditional vertex contain edge connecting carry carry unweighted introduce range iv iv connect term directly indicate specific vertex cycle graph compose tree vertex adjacent tree cycle call span tree maximal forest span span span tree clique maximally separate path vertex vertex contain ks ks ks k clique intersection clique
affect statistic first statistic change numerator statistic give denominator statistic datum certain n statistic define ij follow covariance column correlation degree originally independent statistic row scale proposition however penalize major observe diagonal calculate value statistic central portion outline denote quantile indicator central central portion test central matching since follow variance estimate conservative directly test common evaluate performance many simulate example pre process center method normal observe dependency microarray robustness compare dependency select five portion note refer column center first row test test fdr without subset
file separately assume posterior posterior constrained matching value panel summary match display panel probable record quadratic give quite produce match compare whose frequency marginally distribute simulated update matrix metropolis hasting step report summary probability match display panel case record obtain loss constrain provide latter seem uncertainty c solid line latter guarantee estimate figure step consider ratio simple plug also display probability great high potential match almost certainly due assumption absence match information retrieve provide datum multiple match maximize produce figure likelihood use constrain illustrate exercise enumeration census enumeration include person census survey record people census parallel enumeration survey block individual note block survey census enumeration
design particular correct respectively similarly upper via nan invert fix p behave numerically describe detail set repeatedly dataset interval confidence maintain coverage probability become value low probability agree c importance currently confidence inference dataset size exact unable often fail hour unconditional interval poorly electrical brain neuron electrical activity various spatio process challenge pair neuron much complicated demonstrate accelerate hereafter value mark mark value k fire near ms time increase recorded period minute perform ms conditional firing window use test power alternative minor speaking fast test type temporal motivation nan carlo unfortunately test challenging size need successfully conjunction
arrive pp n omit nr pp n similar pi op op see complete theorem omit involve p axiom case conclusion condition theorem exercise remark dimensional loading carry loading loading result original series consistent independent curse latter case procedure prefer asymptotic propose together far report title factor model classification primary h h phrase curse dimensionality precision modern age practical series finance economics environmental understand dynamics key portfolio pricing risk management economic phenomenon environmental
calculate relation rare parameterize previous measure asymptotically entropy probability iterative rare first markov property retain ii associate rare program iteratively path chain simulate bad state
duality require evaluate sum moreover brief map vector strong solution kl k k w f parametric max show flow reduce efficient parametric max version basis cosine transform dct compose pixel software implementation available conduct core ghz execution c correspond regime ex example proposition theorem conjecture axiom million several video demonstrate applicability scalability become popular linear address combinatorial selection problem rely develop reference primarily encourage spatial hierarchical effort design induce capable allow successful bioinformatics topic vision sum involve nonsmooth prove penalize overlap induce within make show thereby scalable solve efficiently relevant various structure dictionary image patch differentiable
parallel distortion compression parallel worker operate rather minor degradation formal trade equation finite aspect experiment focus derive minimization cluster trade experiment alternate projection alternate derive minimization distortion ct writing provide box derivation x tc tc
choice come appeal distribution subgradient lipschitz subgradient objective simply write point reasoning distribution understand supremum range convex minimax minimax proceeding conclude past bind convexity supremum convexity technique basically theorem theorem repeatedly claim supremum briefly attain supremum attain loss lipschitz value rademacher combine inequality eqs observe choose depth rademacher conditional distribution irrelevant dimension prove proposition infinite say online learnable supervise finite bind online learnable online learnable immediately adversary jensen simply regret correspond randomized contain deterministic one minimax player dimensional rely fundamental order expansion convex also shall prove old inequality upper equal achieve carry mirror reverse two
change manually use learn name evaluate runtime yes fail syntactic ignore example b syntactic yes omit shorthand contain probabilistic pn otherwise determine probabilistic special case experiment name implicit kind choose choice later result em em pt dd kp p assign h r gp em abstract operational semantic resemble operational semantic execution state analogously additionally fail execution want fail symbol abstract operational semantic identify serve sequence tuple solve match note conjunction goal tuple number prevent trivial termination counter use state
calculate reconstruct network new subsequently give length generalise marker sir marker separately drop fashion fall section marker binary adjacency indicate direct enyi total average incoming per relatively sparse likely infected recovered begin I determine stochastically show table parameter marker reasonable frequency marker report lose parameter marker infect node infection recover infection infect naive covering explanation infect precede good guess therefore imply interpretation marker trace capable produce data marker explanation true since away marker ensure positive
dependent ica toolbox software mode thresholding individually I amount posteriori variance threshold pattern see equation actual implementation generative differ reduction step successive ica observe image common mixing pattern impose outer subject share set pattern map vary ica multi external course cognitive subject tensor ica low hand subject loading independent describe specific level group concatenation observe subject ica structure mix mix em group limitation ica compare provide statistical framework compare glm contribution comparison fit mixture ica approximate ratio compare discriminate model essentially variance systematically assume across voxel tractable computation loading different population important cognitive strong comparison complete perform difficulty stem ica global exist decomposition ic feature separate ic achieve subject variability bold independent component common span generative pattern via give pattern represent subject pattern
least inequality hold minimization go construct hold criterion definition excess r j rt r contrast depend strong correct conditionally dyadic tree dyadic recover true clearly rt two estimation note fine establish assumption assumption specifie dyadic either impossible boundary true element parent I small moreover cart hold minimization form result fine assumption control result synthetic dyadic integer
nevertheless segmentation class label try label match truth inefficient namely hence suggest give ground label index contribute maximum number local efficient produce outli receiver characteristic roc widely binary roc positive rate false value roc evaluate auc range r section em plus minus member member yu member engineering university china visual group microsoft china laboratory university china electrical laboratory usa electrical usa subspace union multiple subspace subspaces propose novel name lrr low candidate combination lrr solve clean structure contaminate certain corrupted lrr approximately row space theoretical membership provably determine lrr correction outli process often enable need parametric characterize know mainly effective type visual face texture subspace reproduce model much recent year principal pca establish completion recovery hypothesis draw lying namely consider show strictly draw subspace draw underlying subspace group subspace cluster numerous vision image clean draw subspace solve
commonly accept aggregate general scenario accept conclusion preference happen accept grow work economic political science survey field abstract field aggregate formalize need outside scope individual opinion answer issue j issue vector ease presentation like acceptable accept logic consistent conjunction example literature issue logical set consistent opinion assignment achievable logical search semantic proposition state prefer uniquely opinion aggregated issue issue majority present state member consistent nf return opinion conjunction issue majority consistency accept agent require decision aggregate member act unit independence guarantee wise aggregation justify
chart fast shift detect moderate jump call nonparametric covering far combine classic chart run chart shift inferior chart propose chart small shift normality comprehensive chart extensive carlo follow although power considerably increase extremely investigation surveillance detect result quality take analyze real detect enough engineering image grey mean dependent complex take account important behave important namely application statistic asset uncorrelated square correlate conditional see difference mind propose chart classic chart chart buffer store recent modify
u entry straightforward strength must differentiable jk easily derivative like quasi method minimize first method resolve several different add rule derivative mostly omit mark notation rest parameter predict make different extend source graph pair example slightly modify gradient independently alg optimize individual less likely final descent impact eigenvector speedup achieve position initialization walk bfgs derivative solver improve converge optima quasi synthetic graph edge triple try model free start create add exist create strength uv exp uv uv random way first already vector variable interested thing model classification accuracy whether edge deterministic create add perfect recover close weight area auc report mean auc mean perfect figure show model blue perfect news noise drop reach actually bad algorithm perfectly
process three raw decide text convert string character car normalize third raw text mark identical string annotate vb noun nn good linguistic linguistic surface syntactic syntactic attribute extraction english simple approximately work sufficiently well sophisticated accurate english know g versus recognize wish ignore stop word word include separate break text character match often correct accurate task language normalization string character want word reasonable normalize variation common word reduce case english problematic language difficult word distinguished mean problem sometimes information whereas common kind internal word compose form past compose ed kind stem english heuristic relatively concept combine word analysis language sentence english performance system truly say query truly normalization variation recognize similarity thing increase sometimes variation significance variation cause normalization also token corpus selective normalize text normalization information retrieval annotation string identical string depend annotation part sense accord intend parse sentence word sentence role annotation precision program noun able search act computer noun precision noun say even never may gain ir performance result syntactic syntactic annotation query segmentation speech semantic association annotation word discover row vector word contextual parse annotate first generate adjust element frequency matrix way similarity give mathematical transform raw smooth term word situation event practice complicate corpus step scan sequentially event hash engine idea surprising event event surprising measure semantic discriminative context like high expect document tf weight tf document tf weighting evaluated demonstrate tf weighting
symbol row correspond equality block radius establish thm observer machine state average symbol decay thm eq w entropy symbol although slightly formula publication hide machine suffice constant eq establish constant also eqs finally know constructive arbitrarily say distinct pair path every distinct mm convergent machine
non unseen generalise task generalised attribute biased ordering index give size else expression recursion n n n aa kind closed present unstable partial derivative thus lot remain figure tx pt distribution skew upper shape location horizontal spread pt expect detail size posteriori sample b bb symbol posteriori b bb n st almost sublinear roughly posteriori deviation approximately posteriori approximately b nb distribution follow size sample identically notation series series quite experimental evaluation close approach dirichlet behave like dirichlet exponent case behave factor parameter multinomial rather partition give assign n use symbol bring class sometimes subscript symbol use create far use create second technique
minus mu minus mu mu minus mu predictor generate interval standard variance note variance var correctly rate coefficient prediction score replication independent prediction manner var result various var job large model simulation summarize table note current consuming compare var work package design variable large var var mse var var var mse c var mse come mu plus mu mu mu plus minus mu minus mu mu simulate rest rank forward default estimate surprising probably discuss var tune select cross carry package maximization greedy widely pursuit suggest methodology efficient especially approximation mcmc problem framework direction simultaneous currently research extend group appendix mu mu plus minus mu mu minus nz z mu minus mu minus mu minus mu q mu mu evaluate variational posterior generating bind simplifie plus minus mu mu plus
viterbi reference result subsection section surely viterbi alignment analogue viterbi process alignment prove method run convergence different subsection imply alignment need general reasonable programming minimizer apply optimal coupling develop main use couple stationary ergodic limit viterbi main preliminary risk classical g notation shall let version z delay recall measurable hold chapter sub algebra
redundant variable everywhere gradient adequate performance presentation also approach similar demonstrate save impose htbp ex oracle redundant dimension increase ex greedy applicable dimension estimation calculate significance point however make prediction likely efficient grid calculation require useful establish power include variable selection oracle redundant linear interpret oracle two select converge property predictor may attention weak oracle regression converge error would advance property nonparametric converge order asymptotic variable achieve correct parametric scope
critical result table little limited study asymmetric alternative high asymmetric alternative test whenever high sided test often powerful
less week come involve consumption three implement version call package surveillance apply believe method list surveillance system indeed necessity modern surveillance system deal analyse handle count method treat past since count past particularly count period past method language allow automate intervention although et incorporate several
differ assume hypothesis alternative interact level differential need essentially relies identically distribute seem could marginal distribution still extend setting convergence establish proof rely formalism extend composite provide distribution function assumption spirit result recently suppose test infinity fdr drive shape whose centrality parameter result plug hypothesis infinity therefore rate near slowly enough may strategy estimate fast rate powerful incorporate plug estimator characteristic estimator location semi regularity specify recently lebesgue measure estimator extreme arise case occur one sided laplace x reach vanish nan little room non proposition reach distribution test laplace proof omit sided location statistic side one e side side g probability e concavity two
firstly give secondly method stop limit later run correspond control effect positive set common yield chart type regard testing viewpoint partly horizon horizon run substantially yield associate rejection length power ratio residual design simulating limit case stationary additional rejection chart robust parameter determine increment behavior consistent finding monitor early quite change detect quite may quite reliable regression rejection rate right change slope point change thank associate excellent review
huge set explore filter item identify user identify user compute recommendation introduce recommender way recommender system paper item recommendation moreover simple conducted system propose enhance quality recommender storage become increasingly cause people pour internet contain thousand article valuable resource item successful recommender technology predict particular item n user recommender system create collaborative successful create item preference recommend target similarity compute base common call recommend approach recommender many randomly item available randomly great selection great failure algorithm recommender example customer frequently pattern minimum require recommendation express rating recommendation preference merge item list user
unobserve view task must prediction draw parameterized write say row independent feature member row rating may natural ordinal prediction may pmf necessary explicitly include pmf rating nuisance dependence pmf parameterized representation capture idea side nonparametric latent ordinal bayesian often specification definite machine restrict gp prior deal reasonable scalar gp cholesky decomposition inter function relevant hyperparameter across member intend variation tend member learn
social network scalable evaluation synthetic superior especially overlap demonstrate social link student seed find algorithm complex facebook user red blue many diagram community visualize algorithm find community node assign graph accept community clique partition preserve recent overlap community repeat capable capable detect scale empirical graph subgraph compare edge method spirit exist objective community proceed use simple heuristic finding allow objective work overlap
column effect coefficient random coefficient normal matrix fix effect non state set table coefficient intercept previous example fit mean deviation run therein tp positive cc tp indicate fix cc ccccc tp indicate cccc model low appendix set whereas restrict biased covariance approach parameter gauss cycle fix without show whereas estimate variability around model table effect half also observe towards notably concern know drop variance use methodology group performance procedure validate validate lasso covariate measure generate table differ
standard consider rating add web content population may produce discrete application fourth characteristic site tag probable future high user visit visible conventional arrange sale ranking ranking customer ranking effect rare article sale rare interesting future idea investigate service structure possibility often long tail central limit theorem convolution statistical effectiveness consideration plan explore item besides sum expand release providing present analysis package internet small population limit theorem large effectiveness mine modern internet uniform information material manner report sale amazon com book conventional
etc notion repeat game opponent opponent period maximal period notion consider concept say opponent achieve infinitely game arbitrary rule play possible opponent dynamic maximizer payoff maximizer infinitely look importantly dynamic payoff know opponent strategy loop assumption certainly maximizer include absolute present subject repeatedly game paper exploit maximizer question paper submatrix number ordinal potential also provide sufficient essentially separability game gain q start play look opponent straight stay straight stay opponent stay suppose dynamic payoff maximizer payoff differential opponent sum payoff maximizer play case receive opponent heuristic theoretically experimentally mostly mistake allow al generalized games os payoff payoff
inspire estimator result converse
col integer polynomial expand integrate produce end density gibbs package sigma sigma clearly average rao grey n col define distribution conditional normal mean modification program l grey example conditional grey sigma bar sigma col example additional grey iteration col code type grey alpha col plain red original need markov see assumption miss shift cauchy schwarz ergodic almost surely small estimator necessary condition exercise regular enough modify program bootstrap j ts ts grey col figure variance iid iid exercise posterior closed form read show marginalization marginal albeit marginal gibbs simulate q use loop mu alpha mu alpha alpha sd alpha alpha alpha mu stationarity start var pass var pass nd var reproduce kolmogorov ks alpha alpha alpha alpha alpha seq ks visible pattern indicate uniformity compare sigma alpha exp
weight geometrically subsequence probability geometrically w I differ existence decrease w r pr hold condition bit bit fix equal pr invariance principle setting principle application contain derivative exist say bound ensemble variable match element invariance variable ensemble moment bound ensemble ensemble roughly second ingredient hardness distribution exactly bit distribution follow parameter specify bit independently bit know express linear feasible calculation bb ex accord eq q calculate conditioning coordinate match case replace moment conditioning reduce case integer random bit example positive test completeness pass h pass probability k x
normal update bayesian regression double prior quantile burn iteration prior burn probit first stage burn density sampling random metropolis algorithm burn correspond normal double normal data se university concern iterate variable mixture normal walk refinement perform multimodal compare scheme realistic prior metropolis walk copula good keyword hasting normal chain monte carlo extensively generate proposal example tune draw deal adaptation successive proposal converge theoretical adaptation make metropolis mixture normal proposal body theory adaptive construction adaptive sampler receive
computation time introduce new moderate square square approximate square computational feasibility similar address closeness capacity good superposition reliability least achieved mention dimensional setting discuss code discuss development reliability subsequent section framework code linear list book vector coordinate vector accordance keep terminology call arrange value choice term send small somewhat superposition message sign coefficient algebraic book section size likewise split indicate additional freedom partition code desirable arrange convenient size power bit length give index zero index say code per channel close superposition code number alternatively allow size match allow additional simplicity simplicity analysis partitioned code sign code string split section specify zero bit specify control computationally advantageous code decode number number codebook direct impractical extreme coding sign generator code combination subset concern converse channel capacity close independent result draw yield want obtain associate sum coefficient coefficient henceforth coordinate non magnitude decoding consist
adopt jeffreys include correction correction correction identical neither still however process band way channel permit jeffreys margin less obvious seem however imply fact set regard reconstruct spectrum result uncertainty dispersion correction way know sect construct accurate map peak asymptotic perfect never combine appropriate way improve correct permit moment end combine gaussians formal existence scope note accurately gaussians often approximate peak describe gaussian use use practice assume well pdf tail want distance kullback measure practically surrogate introduce un probability degree freedom enforce read
central metropolis available perform realization gibbs integral posterior mode conditional marginal mean produce extend solve balance factor seed system inference case survey come certain might discriminate specify bin range know range count fall depend cost aggregate ij conjugacy another approach inform censor dirichlet prior lack conjugacy regardless substitute dirichlet next update explicitly proportion clarity also joint posterior set informative dirichlet flat sampler two step alternate
order large similarly increasingly large eigenvalue w note result exponentially reason previous yield need show concentrate minor become get n psd measurement modification uniqueness program let recover negative eigenvalue n nn psd recover measurement program psd curve weak uniqueness give psd strong solve measurement although resolution consistent white success various give exist suboptimal lemma carefully paper generalize estimation tight tight significant compressed singular nan allow suggest
average number point need expensive special suppose stop word expect series rejection generate accept probability odd exactly correctly provide
l l expansion line sufficiently imply know machine several respective fix take relation mm x gx see eq combine eqs exactly know exist number sum expectation compute omit machine machine rv rv nb order convert need entropy decrease primary synchronization first use denote prop equivalently monotonically follow
take feature sign available regardless sparse easy use vector proximal overcomplete discriminative representation multiplication believe valid building robust expressive dictionary di universit scale di pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt algorithm universit di v http www devote learn
identical outli outlier outlier identify observation motivate let eq promise research understand case observe entry success rate vs observation report experimental outli pursuit anomaly dataset digit correctly note stage digit exactly decomposition large outli thresholding sample identify htb complete partial average consider decomposition outli pursuit pca setting outli pursuit exactly result innovation whenever concept believe description quite goal collaborative filtering partially obtain tight identification orthogonal pursuit condition orthogonal pursuit succeed pursuit succeed hold succeed choose first outli pursuit succeed pursuit
uniformly library routine number evaluate describe standard idea contiguous array call scatter scatter overhead fast straightforward implementation scatter give normal generator sx implementation method sx effective peak time two sx hardware root encourage table method
decision dynamic treatment rule clinical intervention map patient recommend treatment review approach area index dynamic nonsmooth functional unbiased arise treatment illustrate asymptotic sensitivity asymptotic perturbation treatment use child illustrative discuss regime policy create health relate compose sequence treatment decision regime decision via rule dynamically evolve patient recommend dynamic expectation cumulative population technical challenge problem clinical guide regime estimation nonsmooth functional generative consequently estimator asymptotically bias standard primary inferential distribute form denote subject collect receive outcome measure trajectory collect traditionally observational inference dominate method call illustration uniformly either drug intensity behavioral modification remainder month assess occur teacher concern behavior occur child current treatment behavioral modification long child meet figure trial rand cm txt edge south west north west node cm txt north west txt north ad ad txt rand ad node rand cm ad txt ad yes south txt cm txt aa txt
corollary upper third hold least defined let case true definition sx next due sequence martingale hold claim substitute prove result stage substitute eq corollary simplify exist universal low lemma lemma eq universal depend denote iteratively hold c cn cn cn cn cn tc cd therefore remark find contaminate observation arbitrarily contaminated result achieve unlike achieve limit proportion corrupt statistical dimension analysis robustness outli dimensionality comparable
small bottleneck flow start layer detail phase ai cell bi uniformly phase ai layer adjacent least cell consequently ai ai z z bi upper phase flow neighboring simplicity assume move argument ai contribution step bi ai bi last partial harmonic series parallel cell go carry contribute n completely analogous completely sum overall st proposition dimension give construction high step avoid everything hypercube always figure place really grid cell harmonic replace consist dimension contribution couple remark presentation simplify couple strictly flow reason direction leave flow magnitude final result stick rough consider turn loop contribution consider straight piecewise path constant add corner construction work bottleneck small diameter couple apart take care principle cf corollary decrease graph give
dy dy similarly abstract liu version generate forest length balance fitness simplicity constructing express acyclic require exponential attribute eventually choose even avoid address efficiently attribute liu liu
class whenever observe comprise proportion class stochastic analogously understand finally blockmodel force accordance maximize kullback fit stochastic trial average subset respective la z ij z former distance kind confidence finite blockmodel fit comprise trial blockmodel log least ab ab argument term assignment bernstein minimal restriction couple union hold growth restriction whereby blockmodel whenever
well working previously discuss dual inner product v u upper hold equality maximum v norm picture show cone still shape dual example term duality group lasso lead norm dual nuclear dual respect trace product correspond matrix reduce norm role specify differentiable decomposable follow decomposable belong geometry relation constraint cone star shape require requirement parameter nz fairly mild guarantee equivalently rather closeness depend curvature high curvature around excess desirable pt function illustrate loss error flat via notion strong since first taylor series expansion enforce require twice amount hold uniformly loss function mild population strongly hessian correspond statistic size argument show regularity neighborhood whenever way drastically
multinomial enumeration avoid iterative proportional fitting interaction fitting procedure require storage fit infeasible moderate dimension interaction family marginal sum chain sample degree marginal parameter degree taylor q parts come class family exponential quadratic goodness control repeatedly degree propagate use precisely define fit parameter adaptive carlo transitions fitting requirement parsimonious minimum family control
even train low attribute supervise irrelevant bring unnecessary effective projection bring well interpretation low list dimension face arrive good rate dimensional feature recognition dimensional representation reliable lda lda seven dimensionality dataset box plot method median box extreme robust property classification inter unstable lda label projection seven three face vector basis basis pca call basis et grouping grouping effect adopt basis interpretation face retain g contour part face contour face face fact explain face less lar loop list column zero set active increase norm track lar track coefficient path change another active keep make correlation equally along greedy proceed direction objective loop
constitute one control relate view delta rather cause intractable relaxation restriction show minimization result immediate policy ever decrease cost rather iteration let policy generate formulation schedule cover assumption obviously leave
sample ds ds reliable detection localization support possible problem amplitude exceed localization amplitude arbitrarily grow dimension nan decide methodology problem fundamental limitation set clear adaptive measurement differ flexible address flexibility gain arbitrary localization zero decide whether common entail give exceed exceed context thorough review process sequentially collect step measurement quantify adopt convention crucial sequentially adaptive observation precision measurement example collect observation sample collect exposure comparison non
computational little interestingly outperform alternative distribution recover without substantial noise explain observation extreme throughout offer improvement much signal valuable manual screen difficult offer clear simulation misspecification surprising similarity input weight covariance one think set weight conditional degree freedom go get assumed current weighted matter less gene normality fact gene assess adjust somewhat focus exclude graph exclude extreme recover edge
since play possibly grow happen match another probability c calculate still arm case optimal arm play several lot explore sure optimal arm present resource storage yield grow resource evolve resource discrete static consider hard explore scheme information remark edu combinatorial armed bandit bipartite user resource resource pair evolve irreducible markov occur user allocate resource receive depend
spin sure material show diagram ease presentation quantity calculate entropy joint entropy atom diagram everything want pass want ask know atom diagram involve enough atom delta information diagram atom diagram fast calculate entropy corollary theory observer states stochastic determine internal organization observer analyze convergence entropy compare block along introduce hierarchy integral hierarchy introduce entropy draw synchronization process keyword store rate complexity excess information dynamical generate device utility device chemical amount capability intrinsic remain intrinsic another practically service storage address processing characteristic system key aspect control concern internal stationary exactly know observer ref give design process series reliably desire internal design say synchronization synchronization observer incomplete typically start complete extract indirect set duality observer control key intrinsic computation dynamical leverage intrinsic useful computation circuit attempt circuit income essential digital must even amount digital change reliably reach elaborate device fail device digital operation concern memory line properly risk happen simultaneously component device quite
risk numerically popular taylor aggregated bootstrap cl deviation constant generate fluctuation text resample calculate residual estimator observe bootstrap conditional resample cl unconditional version generate cl calculate cl j repeat step obtain empirical bootstrap uncertainty frequentist mean choice parameter study fluctuation point uncertainty come formula appropriately scale correct fact introduce novel embed parameter capital present abc frequentist concentrate conditional term frequentist estimate estimator involve bootstrap estimation error variance observation tc setup choose unknown term since full distributional
equilibria ii nevertheless lb lb read number discrete subscript nf particle distribution velocity space obviously map velocity space transformation bold n lattice set discrete velocity cyclic permutation level besides freedom degree freedom correspond molecular original publication simulation evolution discrete
result clique clique ability induce clique sparsity prior clique increase increase mass put clique plot limit give small put mass cluster therefore select gain literature partition instance may select intuition regard verify carlo cross select potential computational form admit several connection prior put prior partition
interesting analyze curve basis link interesting recently variable group explanatory power reduce extremely provide analyst tool curve able select explanatory summary curve mask detail variable thank anonymous valuable improve paper functional function prototype g piecewise constant user relevance cluster programming problem finite know world spectrum online hardware good associate physical rate another consumption curve load describe load million france period year every summarize load typical daily period daily understand consumption weather new price monitor exploratory analysis curve analyst set classical cluster prototype map come symbol time transform contiguous interval interval actual guarantee error
generality without generality axis plane lie horizontal axis lie horizontal angle great circle radius lie projection fall z xx z contradiction let du intersect face contradict two geodesic geodesic within ball orthogonal point geodesic angle part must like g jx cut claim jx g jx g x eigenvalue qr exist angle manifold intersect bottom surfaces dx step create hausdorff ny xy lemma net
array k form gradient mention clearly expectation intractable evaluate make sampling binary stage field metropolis hasting meet remain stage randomly object currently select include criterion choose number distinct k wherein th step technique produce indicate estimate importantly full presentation probabilistic modelling order partition rank need return relate return unseen slightly standard object identity object find rank find unseen query q rank decrease establish return sort carry combination rank
simulation indirect monte application partial easy simulate calculate intensive simulate set least replace simulation consistent however negligible abc combine likelihood abc popular population possible increase importance see current amongst flexibility easily simulation evolutionary really require available recently implement within dimensional specific parameter value abc map iii posterior abc closeness closeness reason consider ability future exposition impose density bandwidth importance abc simulate assign possible introduce extra carlo thing valid value abc remove occur abc deterministic variable uniform ht mm h bandwidth define ng output implement abc input bandwidth integer ng abc error posterior
remain constrain inside place customer expression spherical component radius origin location surface show factor combine distribution sampler derivation step mdps sampler mdps allow relax many distributional assumption detail process reduce burden substantially specific model provide china mobile large phone operator china phone self report ideal base consist contact record phone call message member gold company customer contact date contact take purpose ignore customer customer divide period month month period statistic summarize nonempty dataset appear nonempty nonempty calibration customer empty coefficient mean geodesic distance call share friend non speak small large number coefficient assess extent small geodesic distance number friend person approximation geodesic expect size cluster coefficient mobile observe quite clustering see graph would possible reason distance cluster network type distribution notation contact end length period follow logic exponential core never first like close follow link way contact rate observe observation
keep rich expect positive accuracy positive typically entail semi always realize building easily impact pairwise approximation upper control number time approximately distortion projection leave much pairwise market price dynamic know profile tool quantify sample behavior demonstrate frequency profile throughout portfolio avoid ambiguity call risk optimal version stochastic latent log price brownian obeys integrate give ds ds analytic formula conditional decide brevity slightly keep make well model price scheme gaussian price gain asset price year us trading intensity mean trading time asset spread arithmetic benchmark portfolio portfolio grid second latent asset second simulation asset price trading day base price strategy assessment base frequency portfolio one price minute study observe minute use empirical position therefore price jump section latent price estimate covariance short regard method noise employ method
comment table activity row unlikely receive except column group member group highly group column message exception probability message send group belong last block evident primarily amongst although membership individual belong several tendency group tendency receive group inspection profile also ii membership rr estimate examine predictive link considerably due email communication communication receive message member message receive traffic group constitute member class group send member receive member calculation carry entire multiply column member display active expect size
learning maintain classified user click classify oppose server long load experience page loading web classifying type classification lexical name external acquire lexical external feature full feature lexical name introduce due life bandwidth nonetheless rely comprehensive lexical accuracy seek answer well one lexical lexical properly achieve achieve lexical lexical vs feature art specifically follow op weight adaptive lexical feature decrease full however lexical trade moreover lexical boost classification online impose less overhead outperform environment comprehensive working include google ii attack mis label insight gain classification operate use lexical implement desire
instead implication static prior follow wherein capacity observe induce strong linearity apply hand formulation much give unlikely population unlikely impossible term general useful way consideration constraint show restrictive presence error observation impose finally note value concern opinion mix efficiently complex surface least clear improve population dynamic reflect couple measuring device reflect particularly fitting beyond complex surface potentially slowly particularly metropolis hasting currently generalise smc incorporate realistic reason believe comment suggestion research visit contribution capability research proposal th weight evaluate j adaptive component proposal chain capable sampling state weak test examine observation error may effort novel involve combine sir hasting slowly consider er expression rao population growth notably multi modal
volatility transaction precisely change volatility transaction trading influence trading volatility nonlinear time transaction intensity financial frequency topic practical relevance participant execute high frequency market risk trade large market make volatility management often immediate strategy automate fact participant pricing option second frequency much complicated cause filter volatility paper describe efficient security treat relation price transaction computationally develop price transaction base filter volatility sequential algorithm work transaction contrary transaction price extensively zhang et al suggest localize version integrate wang schmidt noise essentially regression estimate side transform article change transaction conditional evolution unobserve log walk transaction time low frequency transaction treat lead decomposition volatility volatility trading intensity influence opinion advantage continuous aforementioned volatility volatility transaction constant volatility et relation unobserve observe transaction market noise round eq past addition maker complementary paper try round order book maker fairly possibly restrict deterministic cover complex situation close bid form observation transaction speak volatility view realize particle situation complicate particle article
reflect probabilistic vector modification et query take context fuzzy boolean determine boolean operator join text learn word type functional query free al es mainly view query set admissible admissible boolean set well follow syntactic atomic single composition composition admissible way semantic remain specify query boolean straightforward concept evident query find es difference base document query base inductive chen relevance feedback explicit relevance user guide sample document boolean obtain modify exist iteratively run learn execute boolean retrieval consider characteristic system relevance description architecture present system separate outside view space usually rely genetic gain robust choice framework examine mostly appear inherently application al popularity evolutionary largely parallelism argue less perform fail retrieve document probabilistic permit query evolutionary justify query rarely query try automated learning lot past year majority improve examine representation wikipedia enhance learn es evolutionary semantics automate query paradigm intelligence program induction evolutionary computer program individual get machine problem explicitly learn relevant document keep irrelevant document drive evolutionary pressure individual among query candidate es architecture
degree stable make accurate already unfold surface embed manifold randomly neighbor figure generate datum mapping fail variance entry lower similar new come first evenly manifold sample time low fail embed also select fig far validate performance procedure time versus test residual sample dimensional give b three method cost linearly increase sample experiment
suitable observation infer flexible try related gibbs conditioning alternate find perfect unobserved exponential exp covariance except amplitude hyperparameter hyperparameter covariance great train maximize laplace likelihood laplace discard pass variance variance sample gp laplace extensive review competitive sophisticated matlab toolbox implementation make forecast volatility historical volatility volatility cause fluctuation stock proxy truth predict
risk belief description sensitive show e define bellman condition element decrease p example verify elementary calculus geometric always provide stop ex algorithm optimal appendix poor convexity imply degenerate check theorem exponential recall conclusion delay straightforward social learning formulate threshold motivation subsequent subsection brief act order index also discrete act social previous section private agent current private observation let private define comprise state nature sec except instead agent record private observation iii private public time private belief action take minimize k ce incur pick agent choose social finally subsequent agent public action apart update eq pa ib result term cascade lead information cascade public agent pick make decision protocol system global system agent detection threshold social probability model q dominate sigma include history stop sigma agent belief social continue stop pick stop minimize delay decision choosing continue non incur stop public eq global decision learn policy bellman parametrize scalar partition interval belief readily verify observation tp main consider problem agent tp optimal b stop point interval iii interval region cascade cascade filter example illustrate stop construct double behavior perform learn due addition cost consider total social constitute total social public belief provide decrease set zero policy value double
observe opponent stage play generate mixed fp process player play pure mixed furthermore fp player nash player time repeat nonzero attack node payoff adjust fp player estimation see update opponent security couple exact weight
unfold divided initialization equation validation initialization experimental selecting bin pick step correlation bin must experimentally solve system choose generate expect row relate use square calculate accord matrix initial calculate variance calculate give adjacent combine reconstruct effect resolution leave procedure distribution procedure yield distribution error least transformation
attack attack attack accord token viterbi algorithm attack raw corpus attack contain attack test oracle target program previous attack string positive fp attack invoke false positive successful attack summary attack fp fail attack attack equation attack fp attack attack identify application capability identify achieve attack manual ignore database convert retrieve database convert page title adopt strict positive occur page would page two attack cm attack long plain text token limit
extension simple efficiency variety repeatedly rate numerous application mention publish paired comparison entry european build output classifier extension handle home advantage multiple popular name define prior permutation rank multiple possible rating simple refer algorithms converge excellent generalize guarantee towards bayesian approximated propagation develop suitable
replace dpp rl result policy algorithm mdp action benchmark time second lead solution approach file intel core gb specific library superior standard cpu performance run action preference interval cx correspond amount dpp achieve dpp rl outperform benchmark attain mdp grid respectively rl significant order magnitude mdp grid dpp well concern vi cause obtain table dpp rl vi deviation substantially small vi deviation lc lr lr mdp sec dpp rl show suggest dpp simulation cause rapidly dpp rl vi three benchmark illustrate limited sampling modification
combination number vector processor pool outline generalise generator uniformly transform way pool observe exponential standard uniform batch pair
quite mention architecture lead initially reason back neural deep deep deep deep network necessary realistic also architecture necessarily decision journal science author net use different architecture little decision ask article person answer tries pick entirely satisfactory think limitation deep advance learn problem scientific entirely satisfactory answer deep except
light idea context least contain context denote walk stop stage next observation tv precisely tractable recursion see follow give measure observation walk mainly straightforwardly recursion finally prove mention depend cover computational complexity case scenario context walk relate contain cover
regression provide coefficient estimate even presence multiple base graphic outli alternate estimate identify model clean initialize ol well something occur outlier know main challenge describe broad direct indirect procedure include backward selection indirect residual robust regression identify include median gm necessarily efficient fast publish example outli high exponentially py robustness although property worth closely quite identical problem coefficient still overlap coefficient two hand perfectly identify outlier task obtain support section predict
present illustrate key feature method hermitian e distribute value test hermitian part poor unnecessary error sure solution well first laplacian order eigenvalue interval negative component c denote associated converge cf otherwise relaxed condition total column list final reproduce classify linear solution twice summarize solver solution solver reproduce run eq truncation strategy justify orthogonal last element treat treat ar solver backward prevent bring back residual continue
computable identify hard unknown applicable bayesian partition single treat policy different multi bandit single selection policy well approach dynamic sensing secondary channel sense maximize primary model
conclude really outer convex project star belong idea topology close modification want give distance recognize tree bethe lattice mechanic length give reconstruct root topology character bethe lattice plane scale compare mutation trend phenomenon open plot could mutation data bootstrap mutation useful leverage detect jump context tree transfer subtree resolve tree event dna sequence since without gene tree exclude supplementary material distance tree basically group tree group rich effect overall give picture gene ht cx tree complete cat triangle thin compare satisfactory posterior tree give distance four four point large four three sum criterion among l ij kl jk er er
proportion compound empirical explanatory idea review paper zhang compound observation desired estimate major concentrate procedure bayes assume large note estimating resemble stein bayes member recent zhang advantage elementary grow high problem need stein problem estimation technique e rao seminal modern explanatory symmetric elaborate motivate certain addition one close area example economic size age exchangeable follow covariate statistically aspect sample rough permutation
kernel rbf derivation come embedding arrive rely upon model dependency discussion affinity fisher score parameterization computable without heat parameterization rarely hmms kernel another rational alphabet kernel treat distribution disjoint kernel produce leibl incorporate measure smooth property theoretical rkh induce kernel analysis kernel univariate probability distribution restriction ensure translation respect family parameterize parameter view space location bound hypothesis induce rkh corresponding family unit
great fully extremely small associate proceed finitely bin estimate number draw retain associated p proceed finitely say fraction experimental draw retain bin confident arise distribution compute ff third observation e table ccc table f k efficient computing confidence
score newly generate news especially little service individual distinct feature application partial nature observe article display exploration exploitation choose business article collect user feedback strategy contextual problem important application ad etc conduct serve recommendation expensive require substantial negative experience furthermore metric time contextual bandit valuable online recommendation benchmark supervise uci repository valuable collect benchmark reliable offline article recommendation yahoo front visit store click offline feedback article recommendation display candidate raise difficulty bandit algorithm algorithm create simulator run system unfortunately drawback simulator consume artificial reflect approximation contribution describe enjoy valuable guarantee result record yahoo
censor death occur occur unknown moment death occur censor occur unknown moment death occur censor poisson equal poisson process birth nonparametric likelihood parameter maximize proportional van van calculation ff nonparametric come observation window van datum behave
induce delay allocation compete collection number pay allocate type bid per engine click favorable maximize event occur contribution classical allocation ii click relaxation assumption I greedy refined click constant delay present study mab feedback instantaneous delay algorithm discount presence delay state feedback change play exponential finally delay balance along exploitation schedule delay across standard relaxation arm execution delay policy treat independently arm arm exponential delay decomposable efficiently question delay exist near optimum policy decomposable
doubly model von result draw sample end let denote sample permutation put index permutation draw identically marginal denote randomness therefore equality doubly marginal within underlie per shall assume satisfie exponential regularity approximate approximate well permutation manner inequality follow satisfie signature satisfy signature manner wish pe establish two condition belong family satisfied bound hypothesis pf pf item item map map map hypothesis fact signature existence signature theorem choice arbitrary shall think hand need implication start equal see permutation assign equal bounding involve choose permutation onto cardinality map turn pf pf j pf j item need item map mapping know conditioning assignment space permutation wish use therefore use argument complete theorem suppose signature signature signature solve program signature
process mechanic drift population exhibit population evolutionary structured population represent individual give reproduce study behavior graph find structure sometimes effect advantageous investigate direct self demonstrate range complementary evolve node neutral unlike sampling process combine approach examine affect population example desirable shift result nonlinear dynamic formation genetic drift coin stochastic allele develop drift memory examine substantial evolutionary population rather structural drift replacement genetic drift latter population propose tie language perspective population hide call infer generate population otherwise make latter much exploit architecture structural combine process generally structural occur form measure drift well structure outside subspace lead exhibit structural greatly simulation process dependence complicated theory nonetheless show decompose sum subspace spend entry predict component global diagram drift structured substantially short coin jump coin without matter spatial memory increase restrict sequential structural drift
empirically large therefore rate calculate parameter preserve take time figure value confirm optimal fix example h choice indicate negative present throughout trajectory total time constant apply rarely optimization determine nevertheless purpose outperform demonstrate really temperature annealing
generic singular index proposition py k py py pe pe q invoke indicate consistently estimate enough effective relative appropriate hope mean regard indicator strength large subgaussian interestingly matrix independent entry simple independent space equal pe consequence instance supremum noise list main rsc recover rank hold stay lemma remark theoretical paper remark need conclusion subgaussian give proposition ij corollary inequality remain independent subgaussian f pe eq inner operator
influence phenotype output gene pathway share genetic task activity brain word activity brain correlate necessary brain region region share stock prediction stock price correlate contribution fold fuse problem introduce fusion encourage sparsity optimization proximal smooth general fusion penalty employ fusion regression connectivity guide penalty parsimonious relevant subgraph graph structure output provide consistency snps phenotype adopt fuse fuse lasso learning fusion penalty develop challenge cone programming quadratic qp moderate chain structure input penalty addition point guarantee exact manuscript flow applicable addition make gradient
optimum rather derivative equation c extend angle plot measurement shape color dark region producing indicate region produce straightforward albeit optimize note underlie well change detailed derivative pair angle pair angle outline figure dependence angle plot angle indicate color dependence estimate give angle one energy model minima protein predict region serious problem state
dimension length typically validity intend slice select new state never current unless adapt cholesky subsequent compute drawing variate case elliptical slice hasting algorithm minor improvement previous need routine draw choose cm cm cm receive variate define slice draw whole edge reject make shrink slice return configuration discuss final move elliptical slice setting algorithm index visit angle
dataset likelihood leibl divergence negative motivation kl stem theory entropy cost would correspondingly kl divergence additional code cost code assume success setting unfortunately exactly image van learn train patch attract lot attention recent belief deep belief generative together greedy approach long network successfully task character recognition particularly patch image face image network natural belief figure relevant belief estimator estimator applicability evaluate image patch thorough belief consideration particularly respect even train offer explanation observation analyze commonly deep network chapter review remainder constitute construct deep relevant likelihood reader want skip notation statistical assume denote h bend bend left cm cm h v bend boltzmann restrict boltzmann form boltzmann contrast
certain concave interesting comprising parametric indeed lot reference estimator consistency behave aim deep approximation somewhat broad open class probability density density view fact integral rewrite minimizer leibler well unless estimator regularity bound convergence applicable identify unique want class goal mind denote small show converge weakly entail well show wasserstein sequence converge large entail strong maximum surely respect
reduction online scalable outperform challenge car play image human treating prediction degenerate competitive introducing consider horizon execute policy step furthermore average policy necessarily know observe base well expert surrogate learner action expert goal surrogate loss induce input policy convex previous train policy guarantee task problem grow sup hence traditional guarantee due instead prefer guarantee growth near surrogate upper forward train non stationary policy iteratively iteration policy train
similar principle spline propose modify additive fit conduct fuse covariate haar wavelet basis univariate allow additional constraint producing discuss explore leave let assume increase define term subtract intercept univariate monotonically covariate decrease function additive model involve half close space function k objective term allow neighbourhood origin objective solution value strictly unique likelihood covariate total variation take account covariate attain observe covariate flat beyond value observe covariate monotonically mean optimisation space fit simplicity represent knot shall equal identifiability term component without previously focu univariate optimisation ordinary linear pn amongst
column iterate relate familiar linear algebra information gradient prescribe show least computation perform linear algebra package equation bad vector compute zero complement norm subspace computation time overall subspace orthogonal entry predict compute case converge satisfy analysis ode guarantee convergence
often lasso lasso additional regressor define thresholded corresponding ol selection gain lasso section thresholding coefficient separate oracle separate nonparametric small nonzero may goodness fit motivate goodness maintain depend select ols gauss apply select thresholded third fitness thresholded property post model circumstance post oracle selection occur interest application key quantity value noise choose small level possible dominate namely propose drive datum note literature impose obeys possibly dependent mild upper possibly datum drive example satisfy bc sa last ols finite sample traditional theoretically motivated selection
pt pt wasserstein lipschitz ks distance wasserstein ks estimate see follow give complete side equation converge pt e bound weakly deduce n pt thus lemma complete rigorous estimate coefficient proof quantify make cauchy schwarz qx qx therefore hence make index n used mention go therefore prove claim j acknowledgment thank anonymous reading clarity presentation nsf grant dms diffusion stationarity date diffusion naturally occur measure hilbert absolutely gaussian metropolis infinite hilbert value sde complex inherent structure quantify naturally study behavior dimensional space focus metropolis hasting metropolis rwm target rwm move propose walk although pt somewhat naive within hasting rwm application cost acceptance langevin mala needs evaluate gradient analysis complexity mcmc metropolis one author consider proposal space
node biological e significantly connection degree pair value purely graph simulate strongly affect traversal technique indeed densely purely discover inherently interestingly explanation node wave surprisingly opposite degree tends degree discovery former contrast walks expect distribution graph simulate cluster extension graph however turn preliminary result decide work planning incorporate ccccc rw absolute length collect life example facebook idea life system facebook social million facebook social topology facebook make implement collect facebook rw technique
capture output demonstrate technique model comparison show nonlinear strongly nonlinear dynamical direction definition sketch remark recent balanced truncation carry reduction belong reproduce kernel hilbert characteristic benefit simulate may simple lead circuit light process reduction dynamical system success date detail linear reduce essential input behavior nonlinear involve scheme nonlinear control exist idea space balanced reduction work convenient analytical theoretical understanding
algorithm matrix slide state could strong requirement adaptation study recent fit proposal covariance observe
subspace generalize principal reduction mixture space focus pure mathematic sample address homology specifically group relate sample topological belong local persistent homology condition sample topological characterization topological intuition reach finite bound
pose technical challenge heuristic investigation future sake conjugate conjugate marginal sufficient image finite interpret suppose statistic censor observation affect repeat describe individual interesting regularity admit finitely similar evy bayesian finitely remain precise projective limit limit measure use neither though appendix brief survey relevant projective limit limit projective measure projective theory object subspace construct marginal specify connect object generalize space large object proper infinity index direct set relation let di mapping projective set limit canonical mapping projective product canonical restriction projection projective structure canonical preserve induce projective space relevant measurable carry topology algebra system projective projective make mapping analogously projective measurable measurable borel topology define projective generate canonical manner projective limit structure family mapping projective respective projective limit g projective index mapping unique I word diagram diagram preserve projective limit projective system continuity projective system preserve projective algebraic structure notable even manner projective domain range kolmogorov projective countable j exist uniquely projective generalize kolmogorov originally space automatically satisfied projective limit countable index construct subset point negativity continuity address set aa occur
pose prediction compression would pattern recognition statistical prediction interpret principle follow let symbol represent datum symbol require short additional description short basis description obvious compression let concatenation symbol symbol denote sequence box symbol candidate represent predict next symbol xy interpretation paradigm paper check interpretation reality capable binary source b transition respectively distribution
slightly restriction show sd gap completeness protocol point fast start hard start strong circuit markov ergodic yes measured point let let q hard second part unlikely polynomial hardness hardness unlikely contain hard restriction rule relevant research physics sampler spin convergence follow circuit state yes difference thus informally case additionally restriction diagnostic decide decide hard least hierarchy circuit assign distribution associate pair circuit constant sd circuit satisfy satisfy constant theorem
linearization alm k fx way iteration function current approximation alm add prox term step fx subdifferential replace qx qx kx identical fail lipschitz constant thus converge iteration function algorithm iteration improve nesterov obtain acceleration combination iterate extend et
function asymptotic normality asymptotic parameter relative triangle expansion triangle triangle segregation alternative density q edge segregation relative score segregation alternative expansion multiple proportional solid central dash vertical axis leave conditional give unlike one triangle pe sm j pe sm pe sm triangle figure triangle case cs sm cs sm sm segregation alternative relative skewness recommend central large asymptotic skewness moderate around recommend edge similarity segregation alternative relative central cs sm pe since cs sm pe sm compare segregation present realization figure pe r triangle density tend asymptotic relative testing skewness compare similarity association central pe central proportional finite carlo carlo finite require design nan hypothesis possible nan case simulation mild severe deviation extremely result comparison agree segregation association furthermore compare agree carlo similarity segregation segregation proportional parameter agree conclusion power parameter central mild segregation respectively alternative central segregation agreement optimal extension edge proximity region higher restrictive might information denote suggest effect restriction proportional similar correction outside repeat outside carlo adjust proportion p adjustment affect estimate rectangular hand segregation correction favor right sided alternative segregation alternative expect hull favor side e illustrate
replica broken spin theory confirm large predict achievable strategy increase expect perceptron huge different two successive single configuration classify act local hard reach technique landscape simulate annealing also study landscape ising perceptron annealing indicate delta limit make advantage weight perceptron value uncertain weight enumeration adopt pass ise perceptron able propagation study hide discrete add system
remarkable pre entirely fundamentally change easy sis ensemble enhance point design characterize diversity give phenomenon false false reduce mechanism list care false false necessarily negative versa balance objective valuable practical algorithm mechanism objective mechanism find leave future main give description ensemble point ensemble tradeoff strength ensemble well st demonstrate many theorem article mechanism ensemble pick care construct compare numerous boost bagging solving shall selection ensemble ensemble variable candidate independent entry typically variable vote type considerably
easily integer pseudo sign vector follow take transpose fix choice recurrence give recurrence define left shift similarly matrix idea product take see recurrence bit length shift operation
numerous every indexing structure sensible parametrize compute computation take type condition essentially compute particular bounding bins string length asymptotically negligible assume generality indexing every bin family vc bins skewed concentration neighbourhood neighbourhood bin centre verify empty lead contradiction course applicable indexing scheme wants validate curse hamming cube exact space structure understand cell probe abstract indexing consist mm cell index alphabet view rooted cell selector computable define think string value hold bit leaf near weak problem whereby range distance cell yes preprocesse consist store bit string indexing initialize leaf level begin
serve communication module filter environmental control responsible computational module serve serve consideration capacity individual module module location relevant environmental input sort brain planning resource separately architecture agent build environment permit agent observe decide act conceptual analyze observation capacity unit computational measure bit
endow determine transformation leave gain endowed entity way element configuration certain describe ultrametric topology provide comprehensive number agglomerative ultrametric chemical section rise deal embed target dendrogram embed haar dendrogram wavelet filter transform deal relate especially set example financial exchange trading introduction hierarchical cluster family automatic discovery fundamental unsupervised classification generalizing decision making statistical generalizing unsupervise classify event phenomenon huge cluster algorithm visualization ii partitioning include article last mention agglomerative hierarchy consideration pairwise common approach comprehensive useful number deal euclidean geometry start fact number natural take consideration follow view carry quantitative survey approximate topology convenience fine take alternative norm besides infinity label p theorem via endow locally compact multiplicative haar p zero set p integer
iteration consist iteration alternate macro iteration index macro cyclic rule randomly macro computed component variation introduce coordinate x pz iv scale linearly happen speed technique implement svm one loss observe theory coordinate objective functional point differentiable optimality coordinate descent prove reference therein notice lemma state next theorem entry matrix coordinate descent recursively rule solution functional also denote provide machine
lemma note bind trace acknowledgement thank proposition matrix recover individual observe application numerical latent analysis via trace minimization programming strong assume spatial pattern outlier stand contrast analysis decomposition outlier decomposition modeling setting matrix low corrupted matrix observe latent visible precision visible condition general dense dependency visible hidden conditioning hide small exactly matrix impossible sparse
problem instability post prevent show difficult spin ise test system satisfactory acknowledgement thank valuable discussion com describe property choose poor especially ise spin phase transition start careful iterative system appear enhance pt know ise spin worker molecular interested optimize serious eliminate replica suppose experiment difficulty make ease presentation
intercept average flat intercept implement center treat generally proper bayes nan hyper use prior standard close constant computation derive however simultaneous hyperparameter gamma precision matrix normal scalar variance factor produce fix hyperparameter moderately inference link observational fisher precision eq except hyperparameter nuisance nan center covariate zero ml prior approach author maximize marginal eliminate
estimate error true vanish dominate yield even e g nonparametric elegant solution optimal tree nonparametric nonparametric estimate finite consistency qualitative property clearly quantitative characterization structure size graph optimize free interval estimate choose direct optimize early compare class graph bipartite use represent consider gx factor seek set kullback leibl distribution induce geometry configuration kl cross specific instance discover markov cross pairwise mutual empirically mutual take information estimate effect finite factor denote graph cross follow eq shannon entropy give disjoint cross entropy test estimate entropy term nn plug illustrate entropy test replace cross entropy elaborate v entropy partitioning variable theorem statistic variance entropy estimate independence individual weak surrogate term dependent factor factor furthermore mse grow graph entropy test mse surrogate test factor graph want bias dimension dominate case rise notion dimension graph index integer count factor equivalence graph model zero factor equivalence model high graph discovery knowledge translate maintain use expression fix choice mse weak law theoretically entropy implication denote beta let x x dd ensure dependence f
row norm maximum notion relate focus context column original variety exactly low presence present algorithm coherence arbitrary ultimately interested contain singular mention impact algorithm focus closely subsequently leave apply low matrix perturb submatrix ht store u rank u onto denote projection orthogonal complement statement monotonically furthermore
allocation loose tight upper allocation regret centralize however distribute account among optimality scaling explore impact secondary policy channel increase centralized predict centralized number bad decrease result increase channel increase bad channel evaluate performance different channel varied fixing channel channel along channel channel situation quality increase agreement number bad channel keep fix logarithmic nature number channel availability statistic know channel agreement theoretical number gap know increase cumulative statistic scheme differ simulation optimal performance enable provide central allocation allocation scheme centralize low eps favor user chance channel evaluate characteristic assume channel depict allocation approximately allocation fair learning channel availability statistic channel secondary cognitive secondary provable guarantee combine low optimal analysis
finally predict constant sized final purpose scheme produce predict sequence later basis stage predict predict real peak height preliminary contain preliminary step compute adjust statistic use represent peak position local context database generation table peak process perform require match process specificity depend incorporation therefore dna peak precede local context result statistic sequence run obtain sequence present diverse long peak position height choose top four trace represent basis peak peak model context precede nucleotide different represent nucleotide nucleotide occurrence sequence approximately instance possible accurate prediction formally peak statistic measure first peak peak measure peak peak eq peak height table position height context height range position independent linearity peak consecutive peak move nucleotide represent peak peak position next peak peak distance distribution around show local
connectivity careful strong build road nearby indirect disadvantage apart say nothing length discussion length long straight sensible provide short technical level advantage disadvantage measure efficiency meaningful different calculate realistic criterion depend issue population incorporate optimal create short weight tailor sense exactly distance city form average interval width recall take area calculate proxy shorter assign regular study problem explain
approach support machine couple feature adaboost preliminary appear basic notion learn characterize hypothesis upper alternate encode bayes explicit sparsity propose learn evaluate select case publish finding dimensional gene set classification example input draw otherwise measure risk attribute attribute input conjunction direction decision finally label example towards conjunction code bit ultimately guide zero binomial classifier observing least observe point tight also bind decision suitable moreover valid discrete value continuous bit let conjunction prior string let denote possible dimension pd motivate come final specify choose decrease increase reason decision small eq q
build cholesky check near singular number right panel figure contour average h behave proportion contour leave panel rapidly somewhat intuitive region dimension design need conditioning correlation high design design follow number behave choose criterion lead behave correlation behave accurate compare section force model undesirable smoothing singular increment methodology simulator enough interpolation may ill simulator noisy bias mis require attention smoothing undesirable computer simulator confident desire interpolation depend develop behave choose iteration attain desire interpolation recommend inaccurate ill conditioning new rewrite profile replace ill behave get parameter behaved substitute spirit evaluate accurately
frequent value individual basic procedure treat final set snp imputation snps snps causal range snps correspond correlation snps indicate snps equally distribute individual range aware set phenomenon large snps play procedure multiple testing wise adjusted approximately procedure perform fdr model see calculation control approximately correction correlate regressor adopt detect snp causal snp absolute threshold false decide threshold absolute snps procedure frequently snps correlate causal snp threshold detection individual causal snp fdr tp figure fdr replicate threshold procedure fdr direct consequence fact selection procedure much large power discovery rate though small effect fdr procedure reduce positive section tb detect level respectively snp evident also power
deal reformulate distribute know underlie describe far small distribution realization get p n section frobenius completion imply logarithmic completion norm set corollary immediate let q constant completion matrix introduce distinct contradict small numerical already mention remark obtain inequality achieve lasso trace regression q ia diag without gram rescaling become usual little abuse large
distance transformation express patch assume adaptive single closed approximated linearization around put component include speed employ expectation maximization step video patch frame hold parameter latent use note degeneracy due eigenvector rescale remain unchanged scaling row row length ill condition detail rescale degeneracy scaling prevent multiply correctness know transformation construct apply specify construct
respect impose gradient natural interpretation variable irrelevant norm derivative respect norm importance selection motivate encourage add variable automatically derivative sparse learn data major innovation construct organized section certain proof delay describe implementation infinite backward splitting effectiveness gradient extraction world value draw response variable direction dimensional span zero depend function respect quantity et al gradient span development method direction regularization datum fit x f x jj taylor observations ensure locality example control bandwidth implement tune fitting might meaningful dimensional datum explain lie relative specifically embed well give taylor expansion expect j iv v yield sparsity motivation
converge discuss example appear stop order retain sufficient therefore simple powerful rule determine shift establish simple evaluation denote rule fourth shift quite estimator shift distribution parameter
remark evaluate section attain roughly digit good unlike alternative number point depend bin easy quadrature range rather quadrature double roughly digit subtract contour draw proceed form define cumulative positive sum center positive remark describe evaluate numerically illustrate via numerical example statistic limit statistic draw use cumulative
code experiment remain challenge model scalar depend high sensitivity propagation avoid huge calculation computer code mathematical polynomial neural gaussian gp extend krige principle two response numerous powerful code response particularly difficult design fit range hypercube discrepancy moment theoretical lead fit numerical answer fundamental important address set simulation accurate degree two approach paper gp analytical example evaluate performance fourth look validation problem minimize test sequential summarize
spectra classification mean similar model spectra incorrectly fail actual failed incorrectly spectra subtract spectra spectra fail highlight wrong miss update object real spectra obtain spectra model detail select school pattern show valid application evaluation discuss wish comment suggestion thank due centre provide computing support organization grant algorithm base primary school methodology continuously repeatedly uci produce ability rare neural network algorithm reflect generalization unseen depend explicitly quality training good hereafter implementation play crucial role popular large close logic machine
analysis gradient several interesting question explore kind combine different acknowledgment aa microsoft fellowship fellowship careful reading provide helpful basic distribute averaging recall without indicator function let conjugate uniquely strongly convex uniquely two important first bind integration upper q product note thus use eq q rearrange early thereby obtain upper bound eq exploit fact q second bind continuity triangle lipschitz tt lipschitz continuity projection state completeness continuity give short proof completeness denote obtain imply review singular gap connection tp tp remove calculation tx simplex chain node volume graph laplacian e path normalize k invoke conclude k node minimum easy assume loss generality calculate assumption low note direction pick direction node path adjacent adjacent since let leave right end share k kk result section generalize dual incorporate objective derivation brevity work conceptually equally write negative define composite operator conjugate lipschitz dual
sum square pool return routine free much q cubic buffer provide sufficiently exact suppose return pool differ pool previous pool undesirable pool square transformation could subtle simplify uniformly write unit occur although undesirable pool
vice versa drop instead refer various form recurrence fourier consider version bias ranking vice versa bias control one reflect preference subset object vice versa recover straightforward generalization bias rich drop prefer discuss certain operating single rather entire consequence conditioning operation certain map posteriori decompose straightforward inference decomposition independent pointwise respect bayes compose subset come form pairwise prefer argue pairwise depend whether prefer object object say factor rule less observation prefer affect intuitive conditioning rank like subset respect respect except post prior sketch factor uniform relative factor distribution conditioning item require associated rank condition condition modify show comparison involve update cover intuitive ranking estimating item would independent raw statistic second probability discrete consist form count thus mle simply give formula eq like compute interested estimating definition may interested know marginal compute item rank ranking quickly intractable main remainder compute explicitly compute perspective raw probability reader jump directly fouri theoretic exposition unlike analog real fourier transform take fouri order respect discuss analog coefficient order rough frequency reconstruct marginal example first marginal reconstruct familiar continue paper fourier ii fourier scalar paper consider approximate truncate set fourier inference operate fourier marginalization conditioning ever fourier generalization tackle generality discard later section follow fourier domain join conversely compute return h refer theoretic b I domain fourier join tell fouri convolution
network different degree homogeneity semantic concept inter versus intra behave differently relevant algorithmic statistical I discuss chapter role determine network extremely enyi roughly concentrate sufficiently fully close node simple mechanism reproduce qualitative deep cut well balanced deep cut law due variability analogous qualitative new randomly mechanism reproduce qualitative implication ensemble return spectral versus manner know operate piece set interpret scale nod piece interpretation flow permit piece bag combine flow require piece flow become spectral find piece long path cut obtain coherent substantially small figure local spectral consist piece conclude section verify understanding course cluster smooth recall solution pose theory view bias substantially variability computed quantity typically natural regularization optimize avoid fit lead hard think regularize least regression discuss interestingly large intuitive notion tend optimize community score instance obtain intuitive rather notion implement introduce systematic compare much intractable bias case spectral regularize community incorporate cut internal implicitly incorporate statistical benefit formalize note algorithmic ask prominent include statistical probabilistic central recent case otherwise
improve submodular result even bound distributional et game theory economic build al efficient new section structural et submodular show possible obtain computationally inefficient within within coverage e prove expressive submodular problem preserve analysis learn answer submodular technical piece build allow lipschitz monotone access appropriately normalize stability obtain algorithm statistical query target error appropriately consider submodular represent pseudo membership ok simplify graph also chi van grant grant fa discovery microsoft fellowship ex plus minus ex ex minus plus ex minus plus minus ex plus ex ex ex plus align ne ne also interesting property illustrate via example collection lie lie
subsequence imply imply infinitely sign infinitely conclusion singleton neighborhood versa indeed know continuous notice sign monotone l remain jump c parallelism subspace translation versa left exist c cp j c cc hc hc ce ce p uniqueness origin move segment continuous paragraph continuity theorem financial engineering york ny well perform lead due spectra accumulation researcher sparse accumulation biological responsible extent misclassification discriminant independence finite regularize optimal discriminant road road select narrow pool road coordinate study theoretical advantage variety structure result piecewise road interpolation impact scientific throughput thousand sample challenge overview challenge dimension discriminant rule perform
suitable configuration simulate towards accept eq accept construction dark either light configuration attractive whereas configuration add large plot follow moderate scale scale among plot wish multiscale give result use use value moderate scale seem slightly large scale possible interaction envelope solid transform remain point datum remarkably small scale provide appear define analogue process transform fourth subtract yield value process plot thing firstly choose perhaps scale ignore gain behaviour nature main advantage perfect exponential feasible
automatically publicly assess apply differently shape normalizing produce fully result simulate several variate allow variation case asymmetric multimodal obtain apply normalize different shape real implementation marginal involve substitution branch define building integrate hence method need two tp evaluated ingredient monte simulation code prune straightforward prescribe density must kind obstacle lie half eq become obviously call jacobian like substitution follow simplex set constraint entry rate constrain space call call additive transformation nucleotide
base act transition conditionally tx probability consider set choose probability respectively estimate transition rely kullback divergence seminal prior mdp need kullback leibler divergence convention sequel dramatically behavior significantly performance section advantage use kl divergence instead norm illustrate mdps optimistic detail tn jx tn
resort possible approximate requirement simply reduce noise gaussian mean co eigen contain q let submatrix correspond eigenvalue al ml mix ml arbitrary estimate source source et et suppose entropy worth constrain correlation seminal et approximation entropy work expectation solve together al differential constraint give normalize version differential case odd value average et entropy density nh entropy empirical minimize constraint date sometimes correlation modify maximal introduce transform maximal impose therefore development square source exactly exactly uncorrelated word vector ignore correlate nd term drop correspond correlate case estimate I subject element orthogonality important orthogonality enforce orthogonality
sequence sequence sequence indicate progress frequent team thank collaborative overcome towards goal area configuration action recognize sequence take predicate meaningful pattern team use per configuration regard configuration make meaningful team fisher use power datum average occurrence pattern class discriminative measure gain fisher discriminative two discriminative power limited power suitable mining represent frequency calculate among different frequent team reason enough adequate discover frequent team fisher team b c b b c action collaborative behaviour sequence team purpose ball relational action behaviour team predicate description agent act consist predicate action represent considerable state situation abstract difference behaviour robot front front pass typical significant forward front forward characteristic team front
convergence ts ts imply q obtain eq yield assertion use c lag west estimator satisfie weight limit along proof since continuous chart e df df control chart particularly weakly ks dr dr correction alternative interest situation unity depend horizon limit asymptotic unity alternative affect nuisance account array observation ar satisfy brevity drive brownian motion assume unity process process crucial argument weak numerator sketch tr ts converge uniformly exponential
unknown except plug measurement probably become dimensional bayes pure symmetry one base explanation neighbourhood prove beyond scope discussion support ep theorem quantum school mathematical sciences ng rd united science mail ac pattern theory numerous recognition image diagnosis follow feature unknown feature quantum copy outcome classification future find asymptotically typically estimation excess excess theory broad research statistic devise pattern practical recognition computer diagnosis paradigm information quantum carry potentially fast engineering develop accurately quantum engineering statistical procedure pass purely status practically orient interface put inspired
diameter metric embed hence proposition yield satisfy metric exist input proposition agglomerative problem note every cluster equal diameter corollary norm input time compute norm follow time embed without agglomerative instance state upper bound factor simultaneously precisely sublinear raise doubly factor center diameter limitation extend general distance improve lower bind proposition observation appear publication available department bl foundation grant bl subset subset cluster compute approximate agglomerative complete linkage decade practitioner agglomerative dimension norm analyze closely center consider give maintain input show agglomerative agglomerative agglomerative notation doubly exponential approximation level compute optimal clustering necessarily hierarchy approximation independent technique prove center center logarithmic
policy respectively core however unclear whether ever hypothesis undesirable could right htbp hypothesis operation mode policy select region dynamic know region analysis policy agree operation iff theorem law mt tw mt pm almost surely choose probability q proceed member similar inequality apply multiplicative
e negativity become throughout loading decrease also borel paper stating imply load lead new loading loading monotonicity specify outline research consideration answer subsequently number interesting mathematical house hence depend form might equality exist whether answer naturally rely turn variable sort goal distortion pose justify section definition whenever facilitate submodular
penalty beyond avoid instability function nuisance demonstrate penalization illustration constrain programming constraint symmetric quadratic minimizing tend calculation define convenient appear quadratic handle diagonal sign check reduce h ij mi avoid ignore numerical x minimum quadratic program x list iteration acceleration constant fortunately acceleration conditioning problem convergence inner loop
need specification role convenient conditionally change u gibbs consist alternate draw update update directly possibility often attractive alternative categorical probability full conditional full conditional mean full
connect remove remove edge nan connect span vector component connect projection group coordinate fuse boundary graph change dual part coordinate value boundary group correspondence note primal dual fuse coordinate leave dual primal fuse apply boundary remove corresponding edge boundary remove connectivity happen remove coordinate path leave dash general last focus covariate problem much solution suppose py u u u definition fourth rewrite transform rewrite new problem dual study column constraint treat px px nd xu path dual mean also piecewise logic fit conclude working would need expand newly hold fuse still boundary coordinate leave graph x deal fix p apply put aside concern may preferable instead may example fuse prediction analogous perhaps look rough sketch computation rewrite constraint svd basis kkt modify incorporate become subject instead appropriate block satisfy leave consideration discuss solution primal rely path expression
rotation g gs p q p gr rt q translation note p order pair within order matter argue one pair grid ir jj ir composition rotation affect translation error subsequent show recursion affect rotation point since runtime order give term recursion grid power size rotation multiply approximately preserve problem surface range space continuous kp weighting capture surface compute metric computer image
product assumption write hence discriminant decide function relevance determine boolean present classify optimisation formulate construct pattern find subset examine impractical obtain explore subset find adopt discriminant initial optimisation characteristic pattern make give seek method quality solution combinatorial greedy empty candidate add good local improve search well local optimum greedy adaptive iteration local construction neighbourhood algorithm procedure perform use search
surely almost surely cf correct precise apply fact law number triangular array identically distribute variable form state immediately triangular nx algebraic conjunction sequence number next distribute ni dx stein jointly zero converge kn lemma function independent absolutely conditional random denote immediate follow prove strictly variable convenient span sequence limit element wise characterize algebra p recall formula vector I I onto random dd mn rotation see dd b lemma lemma appear note argument easy non ta tx equivalent simplify drop conditional denote multiplier lagrangian usual impose yield multiply multiply obtain leave f ff aa correct vanish induction let property inductive hold hold
refer correctly foreground represent distinguish video omit almost performance foreground video regard comparable low score rank lrr application stock clustering make try solve follow derivation z rank structure cluster effectiveness application apply platform video summarize cluster video correspond task lrr consensus affinity subspace ssc presents assume linear ssc lrr liu model comparison norm lrr provide thorough default suggest lrr segmentation post low seek good lrr get processing enhance low sc contribution subspace exclude post effectiveness result method video three report error two know slight improvement lrr lrr already promise highlight problem consider htbp
markovian markovian reader point book single perspective computational challenge generalise standard selection goal impose count response generalise explanatory partly conditional transpose sample explanatory generalized specify belong family expectation parameter possibly dispersion link relate bayesian proceed outside model prior distribution extension discuss chapter motivation factor scale avoid exploit generalise binary corresponding thus easily handle chapter illustrative diabetes probit presence diabetes predictor bp pressure diabetes residual min coefficient std pr bp degree freedom degree fisher column star characteristic exist covariate strength bayesian impact special associate discuss competition say bayes associate hypothesis approximation link competition exclude covariate matrix bp individuals bp notation
nonlinear gaussian particular wind wind contain chain zero jump nevertheless outperform forecast forecast review consider forecast hour forecast mention wind time highly nonlinear wind power appropriately aggregated generation relevant power wind forecast aggregated generation study consider aggregated wind series argue utilize correlation wind forecast aggregated wind power multiple time unless generate aggregate wind univariate series ahead density forecast wind wind wind demonstrate normalize aggregated wind describe density truncate forecast simultaneously step forecast normal although perform power forecast computationally forecast obvious normalize organized wind explain approach concern logistic benchmark performance various proper paper benefit research direction aggregate wind generate every total wind wind facilitate comparison wind
remark ergodic consistency exist parametric notion homogeneity achieve assumption neither assumption case example quadratic goal formalize concerned generating often cluster similarity assume similarity good even one reason cumulative seem goal np financial observation case ergodicity consistency achieve within distribution datum ergodic assumption stationarity intuitively
discuss mutually might code significance utilize child represent probabilistic take x utilize node nx joint probabilistic density neural x minimize make property rule nonzero contribute update come child child activity
occurrence em last sum allocate update update nu jj nu nu nu nu jj ji jj jj nu ji ji ii computation replicate last overall show cccc sufficient correspond correspond large statistic computational requirement normal
contain state distribute accord mild restriction converge portion ask independently transpose relate complete thus get integral xt rt residual term shrink structure refer stand maximum small non row diffusion lyapunov equation regularize state assumption submatrix incoherence let defined observe enable reconstruct network state support trajectory square recover support logarithmic polynomial roughly
exploitation epoch lead th exploitation epoch reward exploitation epoch upper cause denote arrive arm regret arm exploration cause mistake epoch play exploration epoch cause play time spend arm dt cause mistake exploitation part total f incur incorrectly identify player explore occur occurrence event slot mistake player logarithmic cardinality exploration thus expect logarithmic occurrence logarithmic prove also logarithmic consider contiguous consist player correctly identify good two arm randomize player good arm order singular period logarithmic theorem transition edu liu player play play evolve accord unknown passive reward player know arm good construct exploitation structure achieve nontrivial certain available regret player exchange show decentralized preserve centralize adaptive
assign nx irrelevant asymptotic consideration k arbitrarily minimum distance md mapping minimize whenever arbitrarily otherwise course closed serve auxiliary device asymptotic furthermore whenever satisfied view well turn md estimator fact tend follow strong suppose minimize n furthermore converge minimize assumption event value compact kx early p b remark tend proposition b remarks iii observe tend coincide therefore b subsequent immediately n nx event tend event md md coincide md underlying satisfied attain particular mod inclusion satisfy minimizer outer md remark outer detail see assumption always minimizer uniqueness consistency result md estimator lie show md asymptotically normally matrix md show carry estimator standard non partially differentiable condition hold always possess proposition theorem furthermore hessian minimizer mi additionally sense proposition ii open space probability tend mean value evaluate mean invertible outer proposition proposition continuity satisfie central lemma n p appendix interior view consequently converge proposition eq
iteration diameter begin agent distribute deviation agent observe neighbor action normally distribute graph know linear coefficient value add memory already span space span belief memory keep unbiased calculation involve invert linearly independent take alternative improper measure private different q finite normally since turn give otherwise
simplex rest symbol well value space functional become analogy functional space note elementary elementary elementary calculus view often differential discretized define order situation abstract absence geometric vertex need geometric differential capture star operator first take either purpose edge edge elementary triangle correspond triangle entry match appear triangle approximation manifold e partial differential play important boundary arrange zero operator operator decomposition describe chain arrange reverse deal value operator require chain write relation write elementary clearly analogous topology determine recall definition tool distinguish space homology notion topology value boundary come diagram call equivalence cycle cycle say chain coefficient homology respectively homology homology real homology one piece information integer homology turn space consequence coefficient simply side dimension algebra fact four subspace chain combine laplacian combine diagram duality one box leave geometry operator denote act numerical analysis increasingly diagram geometry consist form vertical identifying via duality abuse three clique definition operator study clique homology number algebra combine vector space desire recall clique decomposition analog analog negative divergence two two analogous
learn intuitively every agnostic learner sample technical consider penalty weight method close update step predict expectation qx qx qx distribution universe px qx step reach return recall multiplicative weight qx b qx remainder far else indistinguishable qx tx qx qx finish long proof equation every union stage occur probability assume equation satisfied put reach point subroutine concept q assumption whenever conclude concept clarity probability release grow call union agnostic reverse let least label use simulate condition return approximation qx np qx x simulate run obtain answer except prove monotone polynomially monotone learn
future pt symmetric rectangular occur problem economic statistically meaningful especially factor involve possible systematically find
cardinality arbitrarily set denote intersect complement depend dependence family finitely x family ii extend restrict case simplicity work conjunction vc sequence measure establish conclusion corollary mutually exclusive family vc depend borel measurable clearly infinite vc finitely
level explanation actor path know relax place prior allow membership mm actor identify level interact actor stick prior actor number stick construction stick length let remainder stick length remainder stick fraction general draw influence mean influence hierarchy learn stick interaction fine level expressive expressive actor indicator level mm vector pair specify hierarchy denote contribute actor propose full conditionally goal finite depth community entity connect node link level actor involve identity level interaction community identity specify
lin school business chen computer science school business optimization compose smooth smooth expectation type interesting propose incorporate method first proof problem decade program lasso lasso optimization especially bad exact formulate objective problem reason mention solve unbiased fx gx subscript present constant convergence rate show convexity function utilize inspire
journal another lead symmetric work display structure left side figure homogeneous intra weighted find homogeneous intra inter distinction interpretation find exhibit small intra link play reference indeed top left find american review journal journal economic journal political economic review economic review economic statistic health health economics journal economics natural journal economics economics environmental economics management economic history exploration economic journal economic economic history dedicate topic economic less reading decompose let moreover number sure given establish identically event q establishe come prove limit central apply appear side establish limit decompose sum product denote denote map obtain singleton give whereas indeed value negligible term converge infinity suffice first term rate indeed go
define exist test universal hoeffding test family support sequence family next bound exponentially answer ask approximately design suppose function dimension divergence approximate alternate keep universal tradeoff propose class
seem te suitable maintain te give worse order give low strength te couple system embed time small time te unstable within radius te small discrimination coupling embed show three coupling ahead noisy seem flow increase couple te three reach discrimination still reach te whereas small pattern small strong free te te decrease lower obtain give good b r driving strength vary
examine arrival job service time invert detailed request receive request day reason build library source performance system system possibility whenever probability choose call record call arrival queue job whenever task easy time job task observe sophisticated observation certainly example appear order detail information random task percentile approach work key difficulty interest job divide service represent delay service service queue capture response job cause missing goal mcmc time invert equation approach alternate likelihood approach design sampler complex subsection difficulty rejection metropolis design even simple conditional varie arrival see two processor horizontal vertical arrive enter service finish service time service exponential rate arrival condition wish sampler queue dominant service excellent inexact change poor heavily conditional proposal decay exponentially precisely consider unobserved arrival modify job force modification service later arrival keep terminology graphical model markov processor queue panel depict value
alternative variable shorter continuous understand essence may wish shorter real continuous x g use take g iff f x x vanish denote g h g x g g g g g g g g negative iff identical prove another p x x x pt g g I x easy x b theorem pt x
done evaluate statistic compare variability within chain latent ht cccc two sparse robust variant measurement sparse instead parent box latent embed dependent initialization issue inference practical sparse model statistic structural parent gaussian term interaction parent perhaps use rbf determination pseudo study predictive respective set height indicator variable variable indicator original neighborhood correspond within fourth air due data file non give use neighborhood refine neighborhood neighborhood skew indicator among direct accord
zero contribute kind new step new arrive moment make neighbor examine examine material perhaps effective one illustrative atom atom atom atom atom atom atom empty slot non atom atom
fold cross kernel page threshold affect support classification purpose experiment set testing full full first fourth band band band band moment see text variety way vary object object result decompose object decompose experiment r text r r result mean deviation text reflect object identify object depend visible observer point testing raw repeat object decompose object result result well raw visible datum distinguish
single add outlier average call function node see right presence submodular depend fw w behavior decrease explore term cluster together show allow allow strongly plot scalar replace cm lead variable piecewise great middle figure leave outli present optimization regularize lead problem tackle proximal subgradient inefficient piecewise affine consequence present subgradient one proximal cm cm proximal cardinality set piece level total case still affine agglomerative
prior plot expand history expand gibbs sampler prior expand figure three px sampler sampler pseudo default start seed code reader replicate unfortunately expand offer explore small improvement show shrinkage prior appear model bayesian quantify sampler run combination
course facilitate use estimation condition vector strong point augmentation problem piece point nuisance regard result nuisance parameter useful discussion shall detail motivation present present methodology extension main suppose sample point weighted write dispersion almost form estimator unknown need estimation simplicity throughout quasi likelihood linear normality taylor presentation quasi estimation residual regularity condition simplify next subsection remove section identifiability stage describe one assume parameter dimension matrix great nothing quasi title subsection
n line w en p n n n dt dt put everything pz k tc p tc outperform define small problem scale go estimator normalization substitute become behavior behave assume outperform separation estimate corollary proposition conjecture multiscale multiscale system review result
snp association case status significant genetic play disease genomic together association jointly locate snp feature selection identify feature interest weighting near construct one whether neighbor neighbor value randomly neighbor weight strongly distance attribute drawback significance far find interact instead exhibit snp average one propose idea subtle systematic difference individual population identify pool breast close pool pool control suggest control building idea technique pathway set control pathway distinction analysis snp relationships pathway expert biological closeness pathway snps show genetic disease pathway pathway seek identify differential heterogeneity control return quantify case relative pathway snps control pathway quantify manuscript detail first permit pathway issue snp abundance significant marker well pathway whose leave return odd conjecture pathway disease exhibit distinction design snps close control close systematically snp snps compute remain snp relative distance statistic question amongst adjust hypothesis find snp
mean truncate specify redundant parameterization length burn phase length value random effect poisson largely major uncertainty due evident pooling uncertainty shrinkage evident estimate uncertainty shrinkage evident year either credible theoretically practical describe describe section material evaluate look diagnostic quadrature
finite approximation mle depend tradeoff nonparametric detail process admit eq kn principal infinite sequence finite model ny kk rest vector purpose notational b b rewrite mle cutoff accord variance maximize likelihood cutoff note much exist universal characterize slope assume course essence stage lemma serve build establish main proof section unknown write lemma notation straightforward parameter lemma state
nonconvex overall objective mkl early set relevance correspondence process structure start strategy kernel weight furthermore framework numerically mkl categorization task contribution classifier extend framework jointly literature concern formulation theorem formulation mkl belong space classification kernel specifically rkh combine rkhs rkhs represent rkhs correspond function zero sec lemma intuition supervise combination write hinge might introduce
sec drop distribute accord translate function localize mark three without sense treatment incorporate induce incorporate virtue pure wiener case add uncertainty add operation spectra filter finite amount pure filter correction describe wiener filter pure spatially statistical inference discover independently theory quantum mechanic dependence theoretical extract partly smoothness controlling derive smoothness gaussian signal covariance know wiener filter spectrum characteristic wiener filtering gap spectrum extension exist problem assumption implicitly beneficial answer several optimally incorporate filter really assume deal relevant answer question accurately example wiener amplitude signal pixel free distribution infer spectrum prominent rigorously spectra scheme trivial analytically interesting work uncertainty problem spectrum filter principle go beyond pure fourth generic provide specific application pure spectral fidelity sec smoothness present finally conclude sec briefly extend uncertainty well terminology find spatially signal measure treat spatial abstract vector observational dealing precise field freedom therefore call usually bayes marginalization field function
compatibility eigenvalue depending restrict follow adjacency unbalanced degree quantity c eigenvalue parallel give selector construct graph precisely say design requirement hence design satisfy distortion universal show unbalanced satisfie adjacency unbalanced degree satisfy however subsection
e follow weak epoch condition put weak large number hold restrictive class indeed take cross conduct illustrate approach real module quantity assessment simulate power module regard acceptable denote standard represent hypothesis validate sequential smooth depict approximation apply detector conduct monitoring rule obtain monte ensure run length simulation mean real chart jump l ccccc delay dr uv financial foundation exchange service remark anonymous presentation ts ts k
flow dimension obtain decrease submodular include cover include source notation b net get max flow cut theorem decrease type decrease belong flow flow get method submodular induce submodular function one fa g g compose subset connect contain capacity figure submodular fa hx extend consider differential lead matrix relate positive function concavity lead submodular p submodular thus decrease submodular find rectangular consider covariate unit consider non increase k certain subset cycle component subgraph column support grant de european author like thank discussion pt pt pt plus minus pt plus minus pt plus pc appear area computer science apply vision operation submodular space principle good knowledge book article material mostly order present material relate research paper
kp ir ks lrr variable dl atom integration notion every variable least atom force variable atom reason generality adapt dl kb student mix dl say student say notice weakly safe safe semantic semantic nm semantic distinguish head atom body h r semantic stable dl predicate still predicate particular ground query analogously answer perform problem rule atom problem linear nm semantic treat verified ground rule always applicable act sure rule inconsistent conclusion dl predicate moreover semantic nm aforementioned boolean dl dl adaptation model semantics gr obtain dl dl appear rule dl replace similarly partial except rule occur atom denote
fig active information average agreement offer new deal computationally expensive however expect truly question mi aa use scalable selecting centrality performance type deal network heterogeneou low mi aa correct block acknowledgment helpful web j support cs edu com com edu find node integrate independent
know impossible monte integration condition weighting q allowed condition partition combine unconditional conditional weight nearly therefore perform integration q show gibbs markov monte obtain partition constructing equal implement gibbs sampler collapse include iteration sequentially remove chain follow base measure partition observation cluster partition gibbs markov chain limit change hold let fix normalize base conjugate criterion rule number burn discard burn thorough sampler n base initialize discuss weight convergence concern dirichlet produce distribution neighborhood true like examine numerous conjugate normal continuous condition provide simplex base weakly true weakly convergent almost surely weakly convergent
affect scenario challenge perturb scale brain signal intensity need paper know insensitive perturbation cp small problem
dual conjugate translate employ substitution translate hand translate combine lead unweighted sum regularizer call sparsity induce regularizer considerably entry hence mkl favor solution note recall side optimization translate maximum subsequently expand slack result mkl mkl kernel employ norm mix identity norm dual special hinge optimization study primal sketch equivalently express group dual small formulation handle solver however quadratic constraint non demanding propose unconstrained norm verify notice differentiable memory quasi descent anomaly detection svms description cast rise align c often expert instance interaction form compute frobenius dot product scenario pilot kernel inferior one scenario handle consider isotropic regularizer verify desire isotropic straight fashion instance standard approach mkl optimize remain recent mkl solver set svm commonly albeit appear kernel cache pre compute carry repeatedly induce heavy memory always optimize end many require certainly suboptimal suffice master mkl kernels set commonly application bioinformatics database security present mkl subproblem decomposition embed extend sparse problem analytical update typical version defer algorithm first strategy op group hand derive operate coordinate descent gauss thereby carry basic
bank identify mesh ht ht accumulate extract information bank var break plotted policy quantity provide bank capital equality ht observe attribute percentile subsequently risk extreme significant value begin coincide dependence divergence sufficiently significant risk provide discrepancy figure attribute call understand provide set derivation subsequently great occur risk equality policy reduce tail tail justification frequency distinct difference fact extreme decrease comparative capability extreme loss extreme decrease decrease probability dominate become event consequently two bi variate figure low aggregate impact coverage exposure limit rare low figure figure display exposure dot line bank exposure dash risk bank demonstrate bank dot line coincide light
straight line necessary source point calculation ref rewrite intersection six fig rewrite may express rewrite nonconvex expression nonconvex body explain expectation us b due coefficient nonconvex distribution measure straight constant absence analogue nonconvex body bl l bl bl body average multi early sec prove possible preferable voxel small surface smooth surface sphere carlo straight boundary tangent distance along straight intersection consideration orientation distinguish often
lead denote q denote thus solve advantageous generalized reason restrict absolutely ordinary convolution via fourier transform restrict transform choice regression subject
interval increase verify pooling unit deviation estimate value around second second intel cpu expansion determined parameter simulation average ci skewness skewness skewness ci skewness ci skewness mean ci skewness estimation population incorporating propose simulation rarely study trivial estimate difficulty derive density numerically transition approximation inference estimator several decade e review suggest multidimensional result study move quadrature integral effect latter grow need count statistical quadrature option quickly grow reference review mixed framework devote integral g decide symbolic calculus long symbolic calculus software ad already
converge perform decomposition find couple original formulation inherently nmf formulation write one way update value fig
snr range limited complexity use show cause contrast purpose cm phase perform perform classifier sensitive function kolmogorov achieve
powerful suppose computed wish discard strong sequential giving motivation demonstrate example safe strong value panel population negative four panel zero value plot stage safe exclude begin remarkably rule scenario dense bottom panel zero plot along plot screen slope plot penalty panel predictor panel outperform still winner except standardize give motivation strong rule sequential kkt subgradient let derivative ignore conclude eq
distinguished computing stay distribution p bayes time approximate x approximation compute analytically symbol predict predictor measurement next measurement prop predictive tt transition predictive approximated exact exploit compute filter closed eq integrate p measurement exact mean marginal measurement exact marginal measurement distribution give respectively
mutually therefore induction start calculate conditional q iii edu family square properly provably square deviation expression choose advantage convergence assumption regularization least identification interference rate ease implementation robustness scenario example impulse performance several early active
behave map define lack degree exhibit interesting seek could category cf manner simple example sep collection space eq empty construction follow generalize seem make rich addition particularly quite constant finite equivalence define check yx x xx xx yx x xx yx name come construction metric later construction context group restrict ambiguity desirable call existence pick choice context refer partition subsequent application ax xx ix family invariant recall x xx xx yx xx ix ix yx explicit apply space depict metric non permit turn collection isometry metric space clearly specify manner statement state obvious
keyword nonparametric prior increasingly popular last wide interest bayesian approach inferential nonparametric group see specie ss allocation non atomic collecting distinct non weight exchangeable ss characterize predictive characterize dp exchangeable appear literature g covariate derive predictor dependent generalize mechanism relax exchangeability predictor implicitly special maintain exchangeable sequence notably hence marginally stationary representation introduce rule reduce known case choice instance coincide characterize beta random call choice specification specie simplicity later section allocation also scheme individual characterize mark individual determine cluster assign tag tag subject tag cluster observe tag discuss induced specification weight define block
apart replace exchange move computationally calculate exchange acceptance pair one l maintain balance condition must chain whose select pair denote temperature chain I accord relatively instance finally adopt monitor delay choose batch add delay rejection liu end period precise first total batch burn laplace approximation specification case line easy drop subscript response distinguish posterior guarantee recall posterior transformation avoid boundary square calculated laplace find throughout mode root de jacobian calculus log ty r real solution moreover c c real solution one summation pair product coefficient especially become proved guarantee large middle tend zero fulfil solution equation c define c great two root order admit positive equation unless mle bound positive axis propose logarithm current every acceptance distribution batch arbitrary batch period burn able set impose batch burn restrict inside indicate batch figure report ess analyse ess independent enable implement independent fair perform fix ess ess secondly ess define implement former latter proper exponential ess challenging
objective ratio quite restrictive modeling perspective homogeneous quite interesting standard eigenvalue relation unsupervise application spectral cluster prior work nonlinear graph cut recent improve considerably cut oppose eigenvector second principal motivation pca pca difficult interpret nonzero kind study recent reference pca natural characterize min principle several modeling restriction functional positively pg gx sf homogeneous eigenvector easy functional standard functional gain differentiable operator
give explanation rate partly al ensure numerical shall establish strict monotonicity yu show rate eigenvalue combine convergence yu monotonicity iteration strict monotonicity monotonicity emphasis strict
fast theory introduce discuss possible transpose symbol one write identity write give vector say compose old trace financial observed construct optimally manner completeness theory matrix return objective problem portfolio weight straight multiplier covariance data portfolio literature write et minimum portfolio construct subsequently arrive early often computational delay trade stream allocation information allocation assign constant portfolio portfolio allocation technique study al outperform allocation study multipli fix window slide ensure length selection conversely slide q ridge formulation detail fall give recursive estimation able devise asset allocation
feature censor collect mixture writing seek current conditional step ei iteration maintain monotone log likelihood mle ar potentially exist heavy auxiliary eq eq appear twice formulation indicator component component proportion become step routine algebra reveal mixture proportion equivalently derivation indicator collapse version density observation proceed calculate formula expectation iteration correspond fraction datum augmentation van improving call obtain subtract component slight show much
third third argument upper finite payoff hold subsection average throughout additive norm side lemma sequential martingale difference sign close invoke prove general lemma generalize two integral bound wish scope upper bound sharp start obtain bound particular time q smooth norm identical lipschitz smooth suppose proving upper lemma martingale banach space tree concentration smooth smooth tt calibration existence arbitrarily outcome norm numerical exception control quantity consider game norm maximal packing belong minimax upper upper bound mass element packing cover second theorem supremum let theory note indicator radius parametrize length arithmetic vc case parametrize value membership vc effective let place action finite supremum value pack everything give rewrite deviation q almost result sure guarantee confidence show uniformly fairly calibration play player adversary lift instead closely value trick almost sure guarantee discussion grant dms
impose tensor unit mixture unit unit optimal choice beyond focus tractable select subspace model code amplitude model model radial edge produce dependency spatial least implicitly angle dependency phase angle edge c v v phase phase couple denote matrix hide contribute dependency cosine wise cosine identity difference phase dependency may
formalize random want especially make everywhere everywhere unique support argue hope exist proceed classic bayes possible computable density exist literature hoc technique circumstance constructive definition rao sensitive issue recall measure theoretic approach notion computable sense conditioning computable probability computable clearly computable question average string david theory extend general ai set detail set fail call return sequence everywhere elementary concentrate argue restrict conditional admit continuous everywhere study computable kolmogorov complexity conditional rather character point l respect conditioning setting computable present paper work study continuity derivative integrable computable show equivalence case moreover ideal provide detailed situation section negative conditional construct computable probability everywhere even elementary dirichlet computable continuous fundamental barrier construction possibility natural question whether conditioning
infer applicable notable case arise expert ideal net resource precisely expert make fail independence situation attractive observe belief make assessment belief reduce set long probability belief suited independence notion imply inference interpretation justify net give fully answer consideration direct finitely towards tree chain address uncertainty tree play build chain base define tree general model joint way conservative define number conservative model marginal independence alone show natural extension crucial tree section turn recursive leave show one global one consistency criterion bayesian net separately see important requirement separation criterion go develop justify make inference compute belief conditional call treat system remarkable work node compute expectation net formalism condition lead inference inconsistent start inference comment separation paper amount rather
several interpretation provide encourage sparsity base approach instead use suggest problem formulation begin regression satisfy ii depend characterization add q element present recall take algorithmic randomly formulate seek subset x b view randomize approximately subset column describe optimum multivariate regression approximate
near neighbor near approximate binomial distribution distribution much negligible estimate nan join event repeat say nan idea also great choice question
follow binomial link correspond ff j ji ki f integrate latent approximation fast mcmc exceed comparison assess fundamental health service management map category clinical difference variable modification happen road access structure age range patient identify place may allow subsequent determine match patient reliability group address patient determine detailed
obeys constraints balanced balanced upper exactly magnitude complement simulation design I fix high max compare available predict asymptotically sufficiently risk trial plot scales simulation sparsity correspond average case trial empirical sum figure n n indicate trial red square plot scale identity improve quickly dominate improve small accordance opposite interestingly competitive across sparsity correlate error section multiply brief issue generalization design grow infinity variable weakly correlate predictor design matrix researcher research toward acknowledgment discussion early manuscript van help anonymous support part testing significance assumption modern actually contribute response instance dna level
read probability pdf sources
probability q p cp follow l conditioning value level relative chernoff e recursion cd p cp upper bind canonical sparsity q transform orthonormal see false alarm p e c alternate empirical transform bound x x p x fact also arbitrary low employ coefficient coefficient correspond taking hold covariance recall covariance auto covariance small pairwise leaf cluster covariance covariance less covariance covariance first integer moment odd moment
thank van point proof corollary work nsf grant theorem set family vc uniformly approximate immediate corollary fact vc number law average result law
maker instantaneous function convert learn set expert algorithm instantaneous security share share security learn valid market must property loss expert bound loss market maker learn bind trade maker price show standard bind converse result bind learn restriction market market price quickly suffer naive capture stability define price differentiable j allow slowly price price point quantity price maker collect security market instead share price ready derive assume behind partition period exponentially increase period lead extra factor stable price let maker equation sequence expert loss simulate outcome share step simulated market completely control simulate market set share
pdf parametrization reconstruct monte event pdf event formula monte event choose bin bin histogram investigate quantity calculate parameter statistical positive bind contain realization bind test fraction program calculation asymmetric reduce event
consider evolution function f ps ps maps restriction markov countable state almost sense convergence positive policy recurrent construct nonempty probability state must state reach call composition correspond nonzero partition q recurrence transition nonempty nonzero chain eq positivity constant lemma direct finitely many finitely depend thus entry independent copy homogeneous property dropping condition distribution convenience banach sign define finite sum ps ps ps fc map inequality pick positive condition convergence invariant ps chain irreducible recurrent hence measure pick let restriction mass divide since variation norm pick triangle norm q information variation ergodic transition still take increment lie take step increment small example information converge make sense consider sum work finite quite restrict remainder state observe exclude contain correspond
tail central probably wide relate generalization instead deterministic include call present strictly financial mathematic strictly literature depend stable hold index normal property definition strictly infinitely exist powerful tool investigate distribution generate
cancer pathway investigate optimization problem estimate pattern structure induce penalty structure penalty fuse lasso difficulty show optimization penalty proximal problem show enjoy desirable scalability direction reduce convergence rate hard gradient easily careful investigation method far boost accelerate jacobian framework jacobian strategy idea nonsmooth function conjugate proper function strictly lasso separability nonsmooth composite gradient method leverage directly separable coordinate leverage order algorithmic structure impose example order tree group fuse idea group thereby introduce enable incorporation prior structure standard fuse extend fused graph structure tailor fuse readily apply optimization solver interior could always solve either second cone qp prohibitive great devise numerous propose induce survey short reach unified induce structure fuse lasso penalty motivate although common form optimize directly use
solve cpu roughly quadratic appear htp total relative equal next learn gene gene gene profile microarray diverse chemical across lead final summarize spend consistently derive graphical objective algorithm converge solver htp c sample much free select explain idea matrix maximum problem nontrivial likelihood propose bregman
include interpretable since small factor observe allow ibp generative many assume isotropic option noise complete graphical b derive define infinite simple dimension tell whether contribute source independently contribute binomial conjugate beta take ibp model conjugacy binomial integrate find limit nonzero take limit feature nonzero harmonic binary correspond infinite source arrange line customer observe sample customer right customer sample reach previously
reliable question acknowledgment thank manuscript management innovation innovation thm thm david institute biology department road normalize justify discrimination hypothesis namely result di bayes evidence vanish asymptotically weak regularity condition hypothesis unlike di require minimax averaging pseudo extend unweighted leverage side hand nan involve involve di robust weight suggest sample indirect selection maximum reduce weighted quantify area science
acceleration growth growth acceleration observer feature trait development depth framework comprehensive reader refer website use quantitative movie assign step towards quantitative trait classify mention appropriate signature mutation learn come call machine flexibility svm tailor rbf refer radial rbf movie describe calculate show class movie precision employ growth acceleration htb understand
sampler treat arbitrary make proposal multiply jacobian give propose metropolis acceptance confirm acceptance acceptance probability similar apply hasting propose new pseudo posterior rather implement surrogate current require poorly fix true gaussian point various gp hyperparameter four one cox full code supplementary material summarize follow section run different seed iteration
max location finite either accuracy measure regression well baseline edge base distribution b segmentation assume paragraph ignore consider minimum gap information measure close boundary direction aggregate distinct segment base proximity location cluster base geometry weight balance contribution display boundary segmentation apply largely final future smoothed present predict concentrate
intuition numerous deterministic outperform valuable early measure motivation work later exponential family leibler divergence logarithmic precisely reason composite imply conditional probability exclusive family geometrically interior positive mention variational information employ geometric introduce optimization recall fact abstract represent functional distance establish prove relate mutual continuity strict convexity resource convexity separate measure requirement information generalizing axiom continuity apply set quantum several fact channel separate transition deterministic broadly constrain utility unbounde deterministic utility end work represent transition space output utility conclude theory measurable measure expect argument kullback linear follow program figure measure package conjunction option either package graphic terminal graphic macro ltb lt ltb lt lt lt lt lt lt lt bp r show dash belong maximized physic call represent lower irreducible
modify stop satisfie present rule inner surely consider discrepancy stop threshold follow bernstein discrepancy stop outer except replace assumption bound sc discrepancy satisfy material surely criteria intrinsic parameter bind bad rate regularization knowledge dimensionality inner factor observe obtain outer lie reproduce hilbert
stochastic program markov directly define proof query investigate promise top typical fact program use module fact unlabele ground interested answer interested query ask likely would approach exact approximation explicitly reason probabilitie individual carlo program execute probabilistic execute call require fact stop current substitution proceed next tool implementation execute goal logical fashion similar logic programming logical perspective pose fact efficiently call program approximation transform discuss mechanism label fact example logarithm create specific ground retrieve update path queue benefit source scalability fact engine allow access indexing contain order call non probabilistic fact ground maintain within query would queue update
problem graph application tend quite matrix definition powerful extract sort norm constraint optimize computationally original clearly problematic one large heuristic sometimes exactly nontrivial three viewed implicitly compute interestingly regularization form semidefinite powerful extract useful smooth interest meaningful solution pose exist datum generalize
put admit proposition solve summarize follow q inverse eq enough operator series write eq follow appropriate hilbert q maps height follow fact last simple compactly support right scale prop lemma prop open open st hence study hamiltonian c hamiltonian range energy example surface dimensional show first explain operator eq classical flow generate vector associate classical shift focus easy quantum energy analyze reference lead evolution recent refer green operator absolutely interval continue disk sense integral view flow ensemble abstract topological mean intersection totally tangent surface split neutral stable decomposition property find flow call fig formulation stability also neighbourhood eq neighbourhood characterize quantum map family open fourier integral rank q precise version involve space respectively give full control cutoff operator family reason often hamiltonian simple existence
class course dimensionality arise microarray evaluate knowledge extensive address microarray reduction widely use base handle gene multivariate large therefore unsupervised supervised use score class prediction performance average test consider reach although benefit preliminary gene aim value performance confirm discretized response level indicate discretization minimal expression work demonstrate effectiveness microarray may improve prediction follow discretization gene three low low medium medium etc ii mechanism qualitative statement discretization introduce high microarray potential
behind semidefinite global optima function manner nice define instance polynomial nonnegative interval optimality coefficient polynomial nonnegative semidefinite reasonably well statistic completeness appendix assertion even characterization necessary criterion handle level exist optimizer semidefinite begin relationship main semidefinite respect side inequality positive definite admissible non empty representation matrix optimality function admissible every information criterion see semidefinite quasi representation admissible empty optimality hermitian matrix rp semidefinite let plug k yield representation form optimality fit framework limitation literature respect geometric eigenvalue mean rational semidefinite invoke symmetric also admissible set fisher optimality criterion criterion approximately design recent introduction
fu write c pf x x pn n x pn pa n surely right go fy cn pn pf f pz fashion consider pz pz pf f pf n n pf pn x f pn pn argument go life writing consider ft lx lx immediately since converse since expression fr cx tc recall prove fourth identity tail medium tail immediately tail result tail vary vary represent rapidly vary rapidly vary prove case follow slowly technical challenge smoothly smoothly consequence neighborhood represent expression one expansion u h h
true noise constant version estimator follow uniform probability aggregate upper risk page give good magnitude temperature improve oracle sharp sense front require procedure selector impose impose design p eigenvalue coordinate complement
well initialization figure pick realization initialization lead misclassification simple select neighborhood determine prove local neighborhood state accuracy speed hybrid manifold handle manifold expect together group fit theoretical guarantee hope quantitative alternative dimensional affine outlier affine fraction neighborhood good look locally though globally curvature pure optimal represent affine rh fact immediately obtain whenever point observation conclude verify generality rhs satisfie orthonormal pass span one eigenvector prove notice combine direct applying inequality use integration w r r rhs simplify proof q follow last indeed rather sufficiently answer question regard ssc code provide version code public edu mathematical sciences affine begin form affine subspace size automatically geometric
choose indicate obvious sample pca similar learning sum square close dictionary location dictionary multiple represent sample address relevant coefficient representation np hard understand despite decade aware generalization learn address discuss section relate identifiability dictionary give dictionary recent somewhat kind identifiability result except otherwise contribution bound complementary used length uniform order q case polynomially minimization compatible main significance achieve rate regime question leave due learn class bound result
function family path parameterize viterbi parameterization fractional discuss wide principled compare decoder efficiently newly define usual memory viterbi recent advance theory main risk review practice discuss conclude hmm decode sound statistical notably broadly prominent also property viterbi discover several claim suggestion explain give leave within framework within forward backward framework viterbi argue analytic family possibility analysis cm align leave clear show accuracy path still path maximize correctly recognize block size propose design rich family establish key general functional markov newly family explain idea viterbi fail viterbi establishe regard algorithm viterbi optimal incorrect forward scale operational decode hybrid propose replace transform operational algorithm viterbi optimal accuracy viterbi decoder give risk specify incur decision predict actual popular shall risk minimize viterbi sequence stand correspond suitable break viterbi advantageous logarithmic lead generalization section pointwise additive form element every commonly stand set related distance stand bayes relative misclassification maximizing maximize risk subsection explicit definition validity give refer path prior aware publication state path positivity prior understand positivity positivity guarantee positivity probability often decoder constrain priori happen entire emission alphabet enforce properly admissible minimization prior assume classical forward recursion viterbi constrain path constraint long equivalent term viterbi decode
value record surface view function value trait extend trait along role consideration indexing consideration indicate branch covariance unique natural evolution see two statistically trait topology evolutionary root call process generalise
f cause give condition define expectation formula index realization uniformly average random object index receiver jointly receiver logarithm entropy clustering side logarithm receiver clustering logarithm joint entropy size intersection error asymptotically free communication error suggest control derive expressive clustering rate resolve grain fashion select
mapping could linearity working become classical amount system possibly particularly machine euclidean hilbert offer discussion hilbert unknown way disagreement quantify order popularity optimality robustness wide variety employ design nature stress well apart training employ system approach close attack minimization sequence call initial stand metric projection denote previous recursion time classical deal smooth besides online indeed let substitute identity previous recursion well knowledge robust priori close convex usually empty set analytical intersection avoid popular solution strongly demonstrate potential wide learn task classical adaptive classification priori motivation couple observation nonempty
highlight resolve critical proxy minimize proxy keyword fusion assimilation numerical spatio spline substantial proxy information sense particularly air term fusion combine goal air management exposure sense spatially surface spatially correlate contamination cause spatially correlate term proxy relate proxy effort reason model relate gold proxy latent process I support define multiplicative identify scalar take coordinate additive bias moderate scale structure sense play proxy spline discrepancy fitting function fit function computational recent moderately specification quantity number basis dimension flexibility spatial properly short spatially discrepancy proxy spurious implicitly fusion proxy reflect discrepancy gold help assess potential discrepancy scale relative discrepancy small scale proxy effort lie flexible discrepancy discrepancy efficient mrf specification sufficiently flexible scale hold sec thin precision mrfs autoregressive proxy critical mrf specification variation scale stand contrast basis omit efficiency car variability possible analysis realistic spatially correlated discrepancy unfortunately latent discrepancy scale flexible may poor attempt proxy signal open question proxy sufficiently inherent constraint improve improve prediction improvement prediction krige find
full write distribution likelihood may less response term vector length section say correspond original scalar conditional unchanged diabetes machine include outcome diabete value reasonably missing remain treatment treat follow estimator posterior fixing apply obtain mle glm estimate apparent mle panel converge map increase accurate particularly intercept term considerable near highest surprising map ht illustrate panel posterior rapid confident way consider infer ig typical default convergence spread half binomial cdf binomial logistic sample rr
basis indicator treat fractional factorial distinction design designs scope fractional design application gr basis theory design topic relatively algebraic algebraic recent computation become feasible statistical fundamental field fractional factorial part factorial design properly factorial design orthogonality design algebra factorial design regular fractional factorial design algebraic statistic simply distinguish fact algebraic treatment resolution
topic model also hierarchy combine live hazard practical base demonstrate datum increasingly evolutionary diffusion author thank provide cifar valuable inference helpful institute advanced name propose flexible prior nested stick break allow live infinitely provide pseudo require understand infinite interpret part evolutionary structure example area
design begin quickly ga turn get maximum maximum function contour ga static design add average hypercube display contour ei ga algorithm accurately decrease contour rapidly branch competitive ga estimation contour well simultaneous minimum ga par function ei explain feature interest branch contour compare two approach demonstrate careful ei save significant amount simulator version bound rectangle investigate use branch bind computer still time consume explore surface simulator computer sampling strategy use study implement contour main contribution generalize expect estimation develop branch contour simultaneous
respect actual payoff actual payoff convex rely payoff linear indeed optimistic get evaluate consistent move player conclude monitoring improvement calibration aim link notion extend calibration appropriate derive full monitoring thank great acknowledge appear proposition perform auxiliary construct auxiliary game converse calibrate tool framework game monitoring player action define internal monitoring calibration introduction notion link calibration repeat prediction outcome probability calibrate empirical outcome predictor specific forecast close payoff great payoff call set characterization convex set
extend first framework work special beta case suitable purpose due necessary procedure follow beta likelihood eq approximated forget initial know initial autoregressive process constant belong set specification consider multivariate indicator chart identity easy uniform positivity truncation simplex return parameter constraint carlo procedure prevent take boundary parameter boundary simplex distribution fig bivariate create boundary simulation real contribute considerably density hessian need carlo fact behave term allow algorithm constrain modify beta naturally independent multivariate stochastic way easy representation beta satisfied chart
lastly pt lastly standardize covariate motivate mention option adopt instance binary interestingly yield similar via slightly matrix great variability performance r acc score cn cm pre acc score cn cm cn l save space via bic diag show lastly rely approximate ise likelihood save case pt consistent r c pre acc cm c n l cn measure select intersection two denote edge compare agreement disagreement agree lot graph extreme ratio expect respective agreement resp disagreement model high model quite different question model
mu minus mu plus minus minus mu mu mu mu mu mu plus mu minus mu minus mu minus mu plus mu formulation bayesian formulation normal gamma mu minus mu minus mu mu mu minus minus hierarchical mu hierarchy mu plus mu mu mu mu minus mu j mu plus mu mu mu mu mu j r mu mu plus minus mu derive conditional mu minus mu mu mu minus mu minus plus mu mu mu mu assess usefulness framework three brevity bernoulli compare performance coefficient weight shrinkage validation present result sample coefficient linear signal adopt
deconvolution factor case error slightly non coherent concern close fall evaluate accurately hyperparameter compute estimation exhaustive choose wiener solution sec report smooth variation optimum value good wiener improvement negligible report tool work true knowledge true image characteristic histogram histogram fig marginal report image parameter uncertainty histogram concentrate around non explain system system law reliably estimate second small consequence non value knowledge histogram quite hyperparameter fig interval histogram parameter manner informative parameter finally visible interpret ambiguity primitive deconvolution manner shape
six gp sim well canonical scope experiment gp sim sim modular section generalize exception extension already r package gp sim sim fitting suggestion package add sim add poor motivated idea bayesian tree infer partition region divide package far come convenient gp model enjoy extension give rise sim possibility upon run sep sim benefit align counterpart partition lead order unfortunately interpretation aspect sim translate interpretation nonparametric try sim lead improvement partition reduced sim partitioning partitioning categorical pair flexible nonparametric categorical predictor sim detail application gps response consume thus minimize subsequently extract experiment common design heuristic fit update simulation repeat maximize relationship
apply set value namely define define analogously easy dc follow concept observe contain contain function setting distribution basic agnostic recent agnostic recover advantage approximate h specific agnostic equivalence weak agnostic concept weakly learnable agnostic boost easily translate characterize strong dim polynomial pn pn dc weak briefly completeness agnostic contain hypothesis polynomial tolerance follow tolerance approximate gx finally convert agnostic agnostic dim characterization pn b dc dc concept let monotone learnable index conjunction variable n agnostic uniform learnable analogous agnostic noise theoretical hardness reduction hardness agnostic simple survey agnostic characterization present brief detailed behind mechanism resource performance mechanism measure evaluate mechanism ideal ideal
cardinality partition binomial base notational simplicity corresponding denote ii scale invariant study exist facilitate specifically lipschitz expect function cf n lemma focus ignore yield q substitute nontrivial value explicitly expect imply uniqueness scenario acquire corrupt unknown sensor stand zero ambient precision analog digital communication measure link group originally propose recover setup present solver noise useful uniqueness identifiability issue practically high sensing application suitable additional sensor noisy counterpart rs state aforementioned minimum incur combinatorial since problem solve outer relate l small cf solution satisfy readily show solution problem subsection establish robust estimation building remark subsection sensor rise outlier corrupt decade huber huber cast cutoff
tree exponential tree cascade present cascade edge si graph dag order construction triangular determinant upper product instead super exponential time require build determinant cascade super graph goal find search graph propose ad hoc heuristics hill combinatorial nature maxima leave alternative optimize cascade formation cascade tree moreover devise provably find optimal section informally approximation concept edge get infected influence network mass media cascade similarly influence tv phenomenon think create external small connect external influence every probability get medium additional influence infer hard capture external influence introduce concept edge add make clique union create clique edge role thus connect disjoint edge via analog transmission diffusion first solid line role edge edge fail label bold dash bold iii edge fail solid fail cascade product eq edge real cascade magnitude benefit introduction edge consider edge still diffusion later become monotonic treat edge possible cascade cascade aim simplify consider likely cascade instead consider possible cascade compete concentrate might case concentrate extensive structure exhaustive indistinguishable approximation equivalently maximize log occur
explain low harmonic population conversely profile count detector periodic input distribute neuron drive produce action fluctuation hyper potential delay pool active event existence ensemble either active depicted fig component fraction nature event arrival potential neuron exceed release assume happen stochastically generation indicate fig detector actual model describe fix duration poisson duration randomly
em factor correspond eigenvector bivariate em display increase corner represent bottom nine dimensional information effect display increase display smoothed curve thm thm effective acceleration propose allow acceleration choose dynamically lead widely applicable theoretical neighbourhood numerical two factor cpu popularity publication expand scope area time run widely stability recent various method accelerate tu liu wu extension r pl compute observe estimate replace step
appropriate formulation stochastic analyze approximation follow dedicated motivation introduction formulation proceed analyze vector finally address dedicated discussion extension follow deal real matrix letter write singular orthonormal value semidefinite vector singular eigenvalue instead nonzero frobenius also convenient introduce norm ball radius lebesgue first approach may form together restrictive one polytope return introduce invariance row long apart avoid polytope ball consider far twice domain sphere get require satisfy row actually unnecessary rewrite particular direct g q c row orthonormal
valuable share package markov integer phase thus construct valuable criterion matrix optimal share package break moment choice bank choose calculate function share package transition analyze subsection let rewrite fact respective trace distribute conditional analogously expression sequence employ threshold lk l accordance kolmogorov obtain markov desirable share first convenience function obtain quantity inequality algebraic result procedure impose fine agree volume markov sequence sum find strategy respect share package transaction portfolio
task task update observer response retrieval observer model whether consistent discussion realized maintain consider observer practical observing thought maintain database purpose reveal question fire observer provide reality represent observer neuron maintain label label labeling vertex nothing edge label join element non detector feature mechanism observe algorithmic implication observer model observation situation reaction observer date change coherent incoherent close containing begin switch coherent coherent false prove coherent iff contradiction raise turn situation actual meaning decide necessarily right current case easy satisfy intuitively think candidate take cell realization graph equal physical signal propagate imply chain hope course propagation play processor unbounded computation reader notice propagation signal think upon therefore exponentially possible element satisfy remain wave contradiction detector coherent incoherent throughout reason rarely coherent ultimately remain observer also update increase wave though instead approximation update retrieval database management believe possible answer way observer I partial neuron every immediate fire neuron connection every wave propagate tool probabilistic algebra appear range definition observer algebra
ps ps perform bayes factor bayes ps sc ps whether example try impact unable change respectively model n ta ta still improve see bf ps bf bf explore ratio example link concern could conditional laplace satisfactory considerably method worth possibly modification propagation ps mcmc path fail smoothly marginal due poor estimation also fail ps gm intensive mcmc standard implement moderately choose correct bayes ps sc sc factor comparison c ps sc efficient real life real life long processing explore whether ps reduce ps sc behavior high real factor notice set early discussion reliable bf set study also model ps ps sc differ lot rigorously laplace analytical difficulty point approximation method search narrow field model use ps sc ps see ps ps wrong partly grid path
genomic dependency independent region unlikely region combine testing testing allow significance cluster control family wise rate prove permutation permutation cancer array comparative genomic array design resolution copy number image voxel snr ratio gain fmri spatial feasible voxel array possible splitting independence offer fdr value smooth realistic meaningful rate apply suitable hierarchical procedure control testing array cancer datum set dimensionality common throughput genomic large traditional note generating allow suggest
section second perform factorization minute speed ghz ram gb special code sequel application guide namely concern classifier notation xx specify th element zero henceforth xx brevity expression leave estimate otherwise bx data x reduce bx bx bx bx point optimistic bx side bx replace bx estimate
differ obtain whereas result dense advantage approach function image classification combination factor sift namely scale sift scale sift px sift van de al
rate consideration splitting decode successive decoding send procedure case let statistic associate positive threshold step likely quantify fit step inner form quantity easy somewhat indeed direction maximize wrong complete decode previously pick need step briefly describe increase reliability vector receive associate
relationship show family suitable finite process conversely connection notion find reference show cover bound weak paper cite satisfy variety characterize note uniformity class respect provide condition uniformity early element function subset vc result family show f theorem direct construction technique uniform core contained state theorem outline key diagram provide let ergodic process f replace generality elementary lemma omit ergodic exist ergodic process marginal suffice case equip real analysis measure x precisely
recently vary formulation tailor inherent characteristic approach ii fact solve mkl primal insight difference purpose various compare empirically formulate regularizer regularization incorporate formulation modular dual criterion separate practitioner plug adjust flexible mkl mkl elastic match cast kernel unify supervise label sample n iy unseen return minimizer f
rotation parameter simulate reliably likely variance method effective match false essence posterior similar pairwise configuration extend al green way mcmc exclude possibility match et match run convergence match match adopt tool way find optimal correspondence surface surface shape region correspond bind et among comparison consist site protein protein protein beta site protein site protein protein suggest protein protein probability often match represent run
apply gs lebesgue along net generalize include order large integer large dy exist proof immediately estimator hence need b iy iy iy sf subsection digital construct net coefficient digital net b q proof n subsection depend
specify reaction connect task furthermore sampling along advanced quite force capable dynamically utilize trajectory current facilitate set discover reaction dependent priori reach although work langevin free landscape kullback appropriately framework provide energy obtain collective analyst free energy landscape efficient scheme rely carlo enable potentially multi modal clarity generalize later reaction molecular boltzmann role temperature free additive estimate parametrize adopt statistical approach functional successful reproduce kernel hilbert definite adopt order fix z literature type thin spline function
goal apply walk size community member node run walk length bottom decrease walk alternatively way example close clique argue er good capturing real member quite member participant technique clique combine random clique metric qualitatively benefit approach graph consider cluster confirm fm orient user community interest fm build internet service user fm fm popular social site relation fm mainly music use social make reach likewise music music empirical despite challenge show sampling representative via relation well mutually something group connect last fm connect fm match activity direct consider adjacent collect user random friend event
identically covariance identify observation address conjugacy day innovation transform covariance product covariance eigen decomposition eigen orthonormal eigen vector trivially vector give recover independence property identify transformation identity lemma give identity mean factorization solution mean block factorization decomposition theorem tensor obtain decompose admit handle impose situation factor th dimension th element element choice ensure conjugate transformation variate matrix matrix comprise give whose remain element fact specifically select diagonal identity show explicit identify uniquely transform eigen invertible unique utilize conjugacy fit price shift asset conjugacy wishart define q define conjugacy direct consequence develop conjugate choice equation later demonstrate conjugacy utilize stable novel algorithm intractable variate admit model series markov mcmc set mcmc sampler literature
think predictive though distribution posterior density interpret universal code base estimator code plug code estimator code establish redundancy closely code plug conjecture code essentially conjecture family redundancy become large conjecture fact location look like hence code look plug code bayes
blue link figure present roc procedure mention cp high cp auc score last return accurate period investigate cp experiment vary amount value experiment instance test score show auc link training randomly increase advantage cp period depict predict correctly link decrease figure varied cp correct top detect level perform extremely large percentage signal crucial computer lose context service auc tp tp average result size forecast auc predict seven tensor tensor generate median box middle outli red conclusion prediction cp accurate link link numerous task include temporal suggest summary graph time incorporate seminal co network except link comparable variant note truncate approximate recommend
attribute detector detector difference total amount profile patient preprocesse htp error htp fuse fuse lasso site predictor vector solve fuse spend ten fold approximately ten case able solution tumor cell large dna segment phenomena technique dna differently intensity dna calculate gain copy probe fuse signal detect array datum array htp parameter htp brain tumor show cpu spend regularization fast improvement art copy detect
minimize prediction competitive free prior valuable nonetheless acceptable selection tool validity model grateful associate valuable comment suggestion greatly improve quality comparative es es es author de grateful participant bioinformatic comment pass influence field continue universit paris france mod de universit universit paris france benchmark bayesian frequentist simulate real comparison build free proposal numerical highlight
maximal subgraph keep construction firstly grow clique cdf variable unbounded graph probability restrict family advantage extra extend acyclic direct describe factorization cumulative member definition kind former call parent parent subgraph set respective remove relationship graph connect connect entirely bi direct trivial associate consist node I pa gx gx x x notice external singleton c subgraphs density mass parent already pa gx
star angular momentum evolution clear physical interpretation appropriate detailed extensive currently apart initial plan angular momentum analysis lead understand fractional generator similar repository compare autoregressive inferior describe long determine arise angular momentum toy essential momentum statistical simulation dynamic centre dynamical mass section characteristic existence position
sg sg c twice differentiable g show bounded lipschitz lipschitz bound set theorem induction iteratively prescribe induction bound ray proceed h h h f side non iteratively entire wolfe together cauchy schwarz continuity angle always combine angle wolfe h h induction finish proof gs low bound banach modeling occurrence genome neuron claim financial activity occurrence mention mark seq model genomic organization seq process linear model possible computation terminology furthermore spike train consider train
whereas equivalence iv transformation linear univariate admit satisfactory consistent relationship residual natural decomposable preferred individually significance take emphasize comprehensive extension multivariate consistency prefer individual equivalence transfer robust wider univariate variable contaminate multivariate differently scale constraint additional robustness practical example eeg detect neural differentially satisfactory establish numerically frequency result obtained form preferable high existence univariate counter unstable simulation confirm derive novel potential substantial new light complex density system dynamic integration extension dynamical causal measure dynamic capture aspect non visual capture world effectively prominent compete measure multivariate causal significant deal application third helpful consider one approach respect micro parsimonious macro function relate micro g idea macro micro independence characterize macro
place person cluster perturb perturbation eigenvector present section justification affinity section completion compress measurement small coordinate span perturbation section dedicated synthetic reveal object linearly pattern make spectral eigenvector form reveal structure choice kernel eq walk graph normalize symmetry lose find eigenvector spectral prove eigenvector extend clustering use multiple early use second perturbation follow perturbation provide eigenvalue equal separate form connect cluster separate second affinity ideal underlie point eq perturbation
previously context graph show final graph sensible trading european join link single clique connect notably large connect integrate european increase variability impose evidence effectiveness graph method author low variability exchange day datum run burn five completion assess verify figure belong comprise roughly regime comprise deal difference display model chance inclusion regime structure broadly tight grouping quite clique connection clique amongst economic area similar european track long european exchange mechanism erm
intuitively would regret time average reward armed bandit linear maximize slot reward arm across store independently information alone show policy require storage grow grow exponentially number intuitively dependencie motivate efficiently exploit refer reward np nn nm maximization q accordingly htbp l map index slot slot observe description store random rather whole operate exploitation gain arm two vector slot value slot arm one time slot
eq distribute much appear calculate reason unknown sufficient appropriate approximation define take give along line take remain set conclude least key nz b k nc nc bc nc use p bc bc bc range uniformly finally formula form model provide asymptotic joint
dimensional family indicate distribution blue measure appendix necessarily family greater equal believe many model ham minimum distance binary cube hamming individual simplex every cardinality distance vector odd one x binary every contain distribution cardinality furthermore n convex sufficient statistic ham remain statement assume
lrr psd scheme instead sdp solver poorly derivation psd scheme lrr lrr future proposal validate theoretic optimization real affinity thresholding deal setting formally segmentation dense k vector respective regard vast range basic mean method elegant sophisticated spectral sc towards strong believe exploiting enhance basic framework sc extensively review employ image segmentation vision remarkable fail simple method sc partially explain connection method easy freedom method sc form affinity sc affinity kernel affinity affinity eigen analysis laplacian normal sc intuitive perhaps fitting aspect classic expectation
popular empirical asymptotic well establish bandwidth et basically distinguished choice follow fractional period multiple assumption sequence satisfy eq ks otherwise assumption q j simple expression reach mx k ol theorem ray one ar parameter display replication carry short dynamic case ar bandwidth bandwidth use short component investigation approach ft possess memory harmonic frequency frequency exclude frequency spectral ft see fractional ft estimate parameter summarize first
well provide constructive correlation advance marginal mixture hierarchical rely straightforward initially example costly inversion rely order limit additive poisson hybrid produce source otherwise function ask whether attain answer back fr introduce set minimum coefficient call upper hoeffding characterize follow f h h inverse
multiplicative optimal design implement criterion yu reference therein important optimality accommodate uncertainty naturally bayesian simple context et monotonicity
learn base learn gap bind ng actually easily establish instance coordinate bernoulli gaussian spherical x learn calculate sample estimate center near gap generative summarize margin rich characterize adapt algorithm semi feature construction characterize sample believe obtain answer thank therefore k theorem complete sequence every column mt necessity theorem set column exist matrix rank
derive obtain derivative respectively asymptotic expansion fact lagrange hereafter drop drop quasi tend kl regularity n expectation well argument expansion share generalization incorporate effect misspecification maximum indeed term n predictor glm reflect aic extension work cross weak cross log suppose set compete popular put nonzero bound principle choose I correctly specify exponential n seminal include make argument assumption response lebesgue divergence lead regardless additive choose divergence
I dynamic eight web randomly make rare word consist l l business name th frequency moment propose geometric mean moment find stable geometric implement function gamma matlab fortunately sufficiently estimator together estimator projection dash overlap numerically new numerically present result normalized decrease panel roughly flat capture trivial geometric new middle decrease latter moment shannon entropy entropy
force parametric however learn something extent external neuron subscript fire firing describe mis indicate apparent actually external driving quantify bit drive force stimulus stimulus drive relative leibler theoretically neuron run actually distinct predict future stimulus quantify predict neuron technique neuron spike train input entropy appropriate stimulus drive entropy predict fire neuron spike record neuron iii train without external begin neuron whose rate hz spike follow hard period spike twice baseline decay see figure history dependence intuitively period reconstruct resolution bit internal entropy residual randomness bit bic exception spike transition spike happen transition however transition state fire fire internal entropy spike conditional subsequent emission compare fire solid line square model generate spike dash calculate spike except spike fire discrepancy arise firing plot since spike emission panel entropy complexity high emission baseline state probable entropy third panel spike impose
mle communication interesting mle symbol symbol good suboptimal mle use training symbol estimator symbol inferior low snr mle know sensor see enough compare analysis unknown estimator approach mse lead evaluate record carlo show second step mle suboptimal computation suboptimal snr mle implementation exponential simulation r db mle communication fc final base receive communication error traditional assume perfect communication reference therein exploit redundant sensor dramatically ignore wireless aim achieve capacity improve reliability example bit communication signal appear motivate communication orient diversity combination fusion orient fc strict decentralized sensor bit quantization discuss channel noiseless noiseless channel propose universal isotropic quantization adaptive method mle gaussian introduce suboptimal decentralized snr
tuple value triple formal useful concern ergodic although summation approximation consistent suitable empirical estimate cluster point maximize minimal point assign contain close assign simply put mix rate finite probability incorrect notion assumption try optimize parameter may interesting preserve hypothesis concern without
iii submodular contraction soon equivalent stability equivalence straightforward n jj jj jj jj jj jj jj j jj x otherwise complement thus non affine representation support show path surprising finite point submodular associated maximizer minimizer j jj moreover non algebra exercise desire j j c jj c j jj jj jj jj thus w jj jj x property eigenvalue j z
framework extend support separable slack parameter extension change characteristic algorithm semidefinite program equally simplicity discussion gaussian hard margin modify margin satisfy perturbation objective lead classifier preserve differential privacy size perturbation frobenius norm method norm direction random propose minimize function
display variability associate value variability hypothesis relate matrix assume form value correction account variance asymptotic hypothesis assume g correction associate instead significance hypothesis become example significance value associate multivariate specify covariance completely marginal estimate monte significance test statistic distortion
recovery frame frame guarantee entry contrast recovery literature around section detail proceed develop facilitate regard recall concatenation generality permutation equivalent state measurement express collect general notion relate orthogonality permutation orthogonality respect derive name trivially norm relate case average heavily specifically x I follow theory next apply tuple thing linearity condition define easily order obtain every obtain similarly j regardless establish martingale assumption see inequality difference routine k identically rely begin note j ii construct martingale thing suitably km complex combine fact follow identically uniform ready provide writing proxy let note need verify notice trivially event follow use threshold establish claim regard therefore together obtain combine previously finally eq q conditioning fact loss
decay clustered overlap auto correlation walk inside volume surprising counter intuitive since close neuron similarity understand property ensure zero unit potential pattern equal probability potential induce evaluate obeys distribution whose factor threshold tc ct supplementary
objective optimize argument maximize iterate reach point
relevant use isolate use use web language game game object among scene another guess object need accurate user guess far provide description allow semantic perform sentence correct sentence average provide sentence rest paper structure soft description main inspire present description scene compose overlap square nevertheless turn realistic aspect description user familiar include shape triangle circle allow
variable path add outcome connect belief propagation update newly acquire terminate marginal intuitively similarly well close step probe likelihood value co jointly constant single good fit minimize sigmoid rapidly decay available prefer less outcome procedure sensitive specifie decay sigmoid moreover topology day observe available interval observation execute latter procedure slow rate swap swap fx fx fx fx fy xshift mm fy mm yshift fy xshift fy xshift yshift fy fy fy fy fy fy fy south fy north south fy north south fy north south fy north south fy north fy north south
nf author interpret either imply research laboratory reproduce purpose cm il mail edu rao usa mail university california berkeley mail berkeley edu berkeley computing microsoft china partitioning natural texture commonly image widely accept crucial understanding content segmentation application largely visual e object understand dominant qualitative quantitative comparison segmentation explore principle good segmentation texture image multiscale cut cut ms seek color distribution contour edge shape method combine color contour segmentation local homogeneity salient image feature scale area image segmentation practitioner mainly two
verify claim nonlinear act counter claim always method estimate degree freedom well behave taylor expand number freedom nonlinear taylor freedom view issue really help sect sect degree reason thereby upon consequently uncertainty propagate draw degree freedom argument far see absolutely nontrivial degree usually number degree however
mode comparable et supervise inference hold truth parameterize ps supervision learn nonetheless performance hdp manual supervision hdp var performance seq seq seq ar seq seq ar hamming sequence g represent red hdp var hmm viterbi variation sequence truth inaccurate specifically typical turning note tracking pattern dramatically affect indeed create specific mode attribute mode within achieve reasonably discrepancy performance approach al hdp var mode five six jointly infer sequence hold see partially improve head head pre unsupervise specific modify switch vary body noise place detail hyperparameter prior c var hmm unsupervise var partially dd dd ard ccc ard seq ard seq ard seq affect ard assume switch var likelihood compare hdp conjugate hdp var ard prior g ard avoid ard switch dynamical consider
scheme transition sphere algorithm birth death algorithm inspire canonical label regime pack canonical birth death critical spin hard configuration feasible application perfect method much patch entire nature coupling chain start configuration probability couple construct may lower naive correspondence paris france couple
far promising optimisation designing sometimes interest use widely standard diameter length weight constraint stress limit detailed please study compactly limit use solution design optimisation four length area cost constraint stress stress load problem limit bound exactly et al cs find
moderate test accuracy gap promise abc boost classification criterion order find se expression weight response mp end split shrinkage normally terminal alg
relative benchmark solving would interesting study method procedure suitable even behind procedure paper goal splitting probability analysis contribution properly splitting evaluation relative complexity small theoretical justification believe beyond recently generality class addition dependent connection user event algorithms split total propose base depend state weakly turn construct recent base lyapunov evaluation achieve guarantee run time choose importance bottleneck sharp believe understand connection organize efficiency deviation asymptotic require splitting theory concept efficiency event discussion context event p without exponentially design rare construction number
else x b diag diag return double double double return else max diag return diag help call code choice purpose else besides exp two everything else elementary
invariant adaptive problem setting standard htp c avg positive avg negative htp avg error positive negative capable give setting however average evident mainly penalization run table result superior incorporate penalization improve performance drastically likely small htp correct avg positive false negative avg avg negative ie n hierarchical iteratively solve let run repetition hierarchical adaptive clearly
simultaneously complexity increase advantage topology degree degree simply prescribe distribution define stage approach define generate next generating couple link draw link generate measure axis splitting must normalize next multiply take associated product generate convention stand generate link measure point interval generation fig show small p leave interval respectively height box multiply middle
frequently encounter computer vision present analyze fast typical svm model I classical svm ls prove complexity refer nesterov acknowledge fact nesterov compressive covariance differentiable hinge primal solve iteration round auxiliary weighted combination current historical two multiplication easily census categorization scene classification scene event four solver light experimental indicate short svm solver write svms minimize hinge nonlinear framework saddle subtract
repetition correlate covariate sis well worth sis reduce serious correlation covariate implement cox adapt code recall cox convex subroutine system laboratory university convex quadratic take much long especially dimensionality confirm strictly sure screen median large absolute select coefficient size nonzero yet case performance c survival ensure marginally sis confirm select rarely marginal screening challenge nevertheless demonstrate van take hour van finish minute huge lasso case demonstrate performance
element element unnecessary compressed look stable method measurement assume vector know measurement additionally call take generate difference ordinary compress large modify perhaps prominent identically distribute force gaussian entry realization role communication compressive set I compressive generalize er rao bind asymptotically achievable matrix condition depict sense literature correspondence rao noisy compress accurately lemma compare tail building equivalence
formalism framework error function input true spectrum choice calculation discuss simplified equation formalism discrete optical factor choose eq quantity use give probable weighting calculate term sufficiently logarithm low due solve equation define noise deviation default exact write equality term contribute average data systematic eqs apply iterate iterative figure illustrate behave ill behave statistical huge
discussion theorem axiom conjecture exercise notation ny essential argument belief rational agent logarithmic unique relative two entropy bayesian scheme tackle question nature information discuss concept directly rational agent argue uniqueness entropy I relation entropy bayes hand design form whether compatible situation beyond individually explore I
co integrate van include intercept time trend intercept include methodology co integrate series denote integrate relationship vector gaussian covariance error correction give lag express produce detail parameterization trend multiplier multiplier important quantity model root integration integration relationship univariate adjustment adjustment equilibria likelihood long multiplier indistinguishable standard unique restriction van still enter chain variate occur product direct carlo var informative var illustrate van present conditional use conjugate h variate prior degree freedom definite variate likelihood trivial variate trivial marginal variate
nature generation count across letter make classifier document summarize accuracy result suggest naive high accuracy plug character character leibler distance chi seem leibler new dirichlet kullback leibler high degree smoothing see pool naive dirichlet large chi classifiers accuracy random plug classifier classifier rate character suggest write accuracy accuracy match exceed rate currently publish identification summarize near accuracy leave evaluate determination document actual grow interest determination system tend known verification determination document refer know list search database
improve however next make life well task analogue previous approximate active denoise overlap patch image simple consistently improve depend method patch compare encode via repeat variant regularizer two variant variant summarize figure although quantitative improvement coincide one report noisy provide case extra denoise obtained produce reconstruction appear white l cc code c code average cc cc show coding combination clean recover result patch value average cccc cubic tool right summary cubic result reflect improvement summary noise previous image tool simulate miss technique wiener filter real averaging fill miss pixel summarize improve apply propose class use regularizer repeat baseline universal improve accuracy begin classic texture patch belong database actually
proposition theorem corollary technical report ann mi locally university analyze heterogeneity multiple infer arise covariate domain formalism dirichlet relate notion global cluster provide efficient inference utility include analysis local process model common group cluster observation group often interest change covariate primary extract sort cluster aggregated tracking covariate snapshot group cluster really movement individual evolve primary path global infer locally observe functional information mean functional reality conceptually individual behavior number day cycle local group cluster typical behavior evolve throughout aggregate day cycle might typical medical cluster privacy subject neither collection level example index different consider individual gold substantial body sequel unless specify covariate heterogeneity group assume cluster covariate handle variant process measure call concentration base center dp stick
hold candidate minimum negative increase derivative approach zero never equal transform heavy squared condition mle large therefore showing must line proving decompose monotonically decrease input monotonically attain maximum unique mode obtain theoretical example time tail back remove heavy tail flat estimator remove tail equal match property assume result back transform formally skew important lose transformation w monotonically also identity slope decrease transform unbounded rarely also back transformation affect triple iteratively package divide deviation k step normalize output pass obtain update new back transform well k start pass tail w double replace remain rv slight excess estimate median rr gaussian w mle na c c base
find lattice cardinality lattice vertex lattice da db c db cluster linkage pairwise linkage text formal lattice discussion base hierarchical abstract lead cluster dissimilarity measure note see computational logic ultrametric embedding topology ultrametric hierarchy tree linkage logic ultrametric generalize ultrametric logic review ultrametric ultrametric review allow geometry topology information start develop metric ultrametric generalized ultrametric logic particular chain ultrametric ultrametric
view htbp htbp show infer functional general additionally dominant class show take random treat whether principle kind formulation strength unit parameter control capture idea may nevertheless something option follow specific rather hyperparameter turn variation determine flexible several approach might nonparametric together related nature modelling approach nonparametric begin early intend nonparametric yet capable tractable posterior arbitrary eq simplex density proportional give discrete draw exhibit
strict taking numerator attain equal thus I nonnegative nonnegative consequently strictly change part take account establishe theorem case substitute stem value except accord account give leave zero equivalently inequality hold function monotone bound regard attain stem establishe follow equivalent small meet condition satisfy sufficiently
obtain reasonably good split small block put splitting block equation reduce integer equal block reduce computational time necessary carry reduce evaluate deal kernel density copula normal cdf cdf equation represent use simulation example compare sampler define autocorrelation iterate tend factor autocorrelation analysis decay lag simulation mathematically factor ratio accuracy preferred may account sampling fast need less second iteration ratio parameter note although factor sampler univariate delay replicate sample step iteration true nest prior specification range volatility positively time shape scale
coupling delay select difference always driving also drive htb system use time show legend b driving driving also behave similarly subtract vector though couple component couple drawback neighboring frequency scheme observational variable vector couple system system embed mostly component another component criterion choose couple wrong criterion bit measure give take small predict scheme transfer measure generalize synchronization equation drive drive complexity individual mean great try regard investigation first limited realization three f n lag give strength conclusion proper choice especially great lot strong drive driving give reasonable system third choice respective value increase multidimensional lag computationally demand alternative ultimately restrict mutual length component form see coupling drive reduce drive
base actual realistic background galaxy run search completeness firstly sort match match match cluster exclusive match detect match completeness fall bin cluster match number cluster match total completeness plot plot cluster sample input give look gm recover though systematically low primarily due low cluster essentially rich retain low consist cluster dr survey cover small perform cluster uncertainty difference scatter similarity galaxy careful cut make center ambiguity time minimize match due projection yield accommodate scatter appear counterpart match ultimately determine quantify agreement uncertainty difference appropriate selection window radial separation separation appropriate generally speak matching match low cluster therefore match respect cluster cluster word completeness execute match yield match place match leave panel separation create result figure compare match color large among new lambda perform original estimate lambda hereafter dr
algorithm discretize grid previous multi behavior social social threshold ph model yield threshold kolmogorov criterion comprise result condition characterize hyperplane ii characterize ph simplex use preserve involve social definition order consequence appendix describe sufficient condition policy iv sec optimization ph ph constrain belief similarly decision social learning section make following polytope say treat outside stop agent treat identically leave hyperplane obviously ph change would degenerate result maker hyperplane straightforwardly vertex introduce follow assumption p relevance apparent q ph sufficient ph begin sec social ph geometric entry note linear function non unlike stop numerical illustrate region denote bellman say region ph polytope states lie line correspond region lie segment region depict polytope assumption theorem numerical show fig concave policy social detection example though single detection single classical problem observation ph subsection curve stochastic decision maker vertice social base detection ph ii convex polytope almost everywhere iii geometric distribute identical kolmogorov polytope boundary coincide classical optimal incur always threshold polytope consequence obvious together ph polytope state mapping trivially cost deal concave ii stop argument straightforward decision state thus detection kolmogorov uniformly distribute proof iv
give recurrence pack max formal ball max disjoint radius great intersect structure give bind behave r dr rd show probability radius intersect ball invoke theorem aspect ratio radius show split one argue see cover radius clearly ball suffice without center ball lie get get split cell radius split separate show useful whereas tell split cell get get bad get split time
network differentially gene essentially differentially associate locate extreme none express differentially express reduce suggest classify disjoint way gene construct connect clique decomposable since adopt convention vertex intersection two gene opposite arbitrary classify reciprocal convention would maximize clique classify differentially gene clique express differentially express number gene network resemble entropy measure uncertainty uncertainty standardize divide find way uncertainty differentially represent illustrate might differentially gene decomposable model however allow identify gene visualize pattern hold capacity differentially express proportion differentially present situation
signal write make model analyze standardized iteratively scale row mean covariance column capture keep separate process two test decomposition model instead sample population gene iid whether shift throughout variance common n go know statistic n l ij n vector center state array correlate statistic column correlation within assume microarray alternative centrality correlation distribution square pool long random denominator correlation small four structure scenario block within block j j within ij column distribution see correlation dispersion statistic distribution explanation dispersion see microarray statistic statistic appear affect confirm table variance
assume file procedure might record record consequently extra necessary linear preliminary estimation alternative incorporate assume become unknown record record linkage knowledge linkage literature matter dependent suppose necessarily dependency among redundant comparison function independence key fail disagreement key introduce correlation absence independence mean sophisticated dependence specify analysis diagnostic gold propose introduce subject categorical categorical account measurement structure linkage conclude stage explicitly introduce measurement misclassifie record parameter reasonably simple call level variable conditionally record model second one see assume replacement unobserved natural count due determine frequency correspond hold write role matrix variable unit linkage problem match note exactly denote match true need facilitate section illustration
original use importance correct case rare importance gray line show varies use uniformly follow though sample statistical illustrate testing uniformity specie order sum sum relevant select occurrence sequence sum statistic denote transpose statistic competition specie generate approximate approximate importance sample new reporting even error guarantee guarantee national foundation dms author statistic university nsf dms lee comment example importance sampling common include correction value create create nan correction use original observation give valuable problem evaluate accuracy approximation invert create nominal significance large exact conditional logistic nuisance technique include carlo approximation besides sampling
op k l hence eq eigenvalue argument similar q proof theorem ti p fm f establish model arrive use series dimension many vector practically proper make reduction observation retain dynamical frequently way dimension multiple early include result large time effort mainly focus economic financial phenomenon identify common factor common series factor series white identification asymptotic series infinity identifiable adopt consider depend substantially different aforementioned series part dynamic conceptually bring identifiable furthermore
return mind failure highly reliable markovian rare small make need reduce simulation importance system largely measure rare importance implement change markov chain transition bad
two possible discuss stop know scalar necessarily max max flow reach optimal constraint feasible arc j new solve satisfies arc replace compute cut describe fista duality stop without look datum duality obtain satisfied pair variable consequently nonsmooth one often choose encode knowledge prediction interpretability learn penalty index index scalar use literature norm piecewise individually overlap still set set consider general criterion context group proximal sum continuously differentiable lipschitz smooth therein linearize current w w keep solution hold constant call unique convexity iterate proximal obtained develop extend overlap wide spectrum norm propose compute
bind side five split training constitute slightly consider iteration test perfectly tight mainly meaningful cluster result could improved chosen loss minor cluster grow considerably due lower much tight bind scale experiment tell help problem enable graph
n number online represent cover scale pointwise easy verify claim depth simple ensure sufficient covering bound fix learnable learnable consider value vc lemma ensure small argument call sim model describe sim du lipschitz iteratively make mistake possible round elegant computationally perceptron variant even ask basic sim learnable deal question interest particular necessarily decrease evident composition square lipschitz decrease proposition covering increase class learnable improper regret computationally method exist theory rademacher classical integral prediction round information static put binary side write loss independent loss rademacher appear hand side tree verify scenario appear need pay tool minimization work
observation answer query q iff refer q derivation form allow right shorthand execute choice query q result learn base occur shorthand repeat observation number first toy program coin head head chance query head head tail head tail result query head outcome head match program paper fix player distribution however thing player third cf interaction dr ex dl dl play player play make situation partly observable game derive move information less straightforward estimation presence plausible learn build indeed
marker report element relate marker trace bind initially less opposed logarithmic marker different continue marker marker trace rapidly tend marker propagation network amount heuristic alternative false positive determine marker hence reconstruction match set use distance identical value match bad upper number true positive limit marker trace constitute case reconstruct perfect free reconstruct reality adaptation datum assume consistent perfect perfect report marker assume infection infected require presence marker marker incorrectly suggest presence edge really increase quantity reconstruction
maximally direction nan select exceed specificity distribution develop instance mixture modeling voxel salient pattern method currently ica model determination generate select device separate group variability thresholde absolute voxel mostly discuss expression describe variability ica motivated fmri display validation criterion ica unclear feature subject across subject split subject ica map overlap thresholded map map overlap quantify study identical quantify reliability dataset extract full subset compute validation coefficient average measure compare concatenation implementation toolbox tensor ica software effect separate cca group equation component level independent component run perform non thresholded ica implementation thresholde separate thresholding thresholded thresholded thresholded little interest voxel specificity voxel statistic map define stability subspace span map overlap subspace group frobenius different normalize dimension quantify subspace span map span dimension dimension instability
dimensional design generate previous space two os enyi maximum construct precisely either edge select care maximum underlie graph generate learn dyadic glasso entire partitioning achieve apply note middle line connect great variability method b proposition graphical encode dependency vector estimating value paper build space cart optimize cart cart cart dyadic partitioning establish oracle risk consistency cart tool analyze complex let way
u tv form simplify z tu u tx tx ta lemma four compatible cc f bind proof optimizer problem ij vector subspace uniqueness minimizer uniqueness minimizer ij lrr solution theorem z e v z way classification problem challenge possibly exist follow cluster sample contain error draw correct segment respective noise perturb perturb whose sample fraction phenomena fraction subspace generally subspace exhibit outlier three subspace shall concern b belong way recover term lrr basis lrr representation computational procedure regularize problem convex solved dictionary lrr solve clean lrr exactly recover lrr original well data error approximately subspace membership lrr term lrr image segmentation face recognition subspace subspace handle subspace clean theoretical robust share considerably exist four main category mix model mixture distribution degenerate
connection scenario characterize aggregation consistency mild opinion profile aggregated opinion aggregation issue iff consistent characterization find aggregation mechanism point think even result quantify number profile conjunction prove five condition close necessarily consistent aggregation return aggregate opinion return profile far return return opinion profile aggregate opinion profile aggregate opinion conjunction truth functional divide conclusion conclusion opinion attain conjunction mark restrict mechanism almost almost define consistency aggregation aggregation similarly quantify define mechanism version ham certainly trivially consistent whether aggregation aggregation aggregation mechanism question equivalent
uniformly shift still normality assumption computer system control chart reduce slow shift classic chart shift argue question arise hold classic chart drop interpretation dependent namely triangular array array respect chart also effective monitoring dependent organize introduce control chart chart appropriate change viewpoint main appendix chart extensive present providing recently chart detect shift allow chart tune reduce control lead quick jump limit alarm refer chart powerful change order move average chart satisfactory shift cite
curve ph unweighted curve high predict area care facebook user top error bottom compare walk plain common decision tree pt friend feature seed table method facebook network random auc surprisingly recommend nearly supervise significant unweighted walk gain logistic unsupervise logistic term auc walk near facebook near walk improvement supervise extraction many relate edge random determine network make supervised walk compare logistic regression art outperform hoc feature examine assign sense recently friend co amongst walk graph feature rough run ghz processor facebook put category learn take iteration quasi converge minimize compute graph partial parameter iteration converge derivative iteration derivative minute increase facebook new link recommendation utilize edge walk co random walk improvement random walk machine require
integrate gate semantic semantic implement measure similarity create term document reduction use example projection view operation vector package emphasize efficiency argue indexing updating encourage research development create source convenient modify software incorporate stanford project projection vector modular design kind instead support building addition build operation calculate search operation search search document operation include quantum provide package public http google com module source implement library semantic lexical pattern together corpus word pair follow sentence effect channel vs sentence pattern frequency smoothed occurrence expand seed alternative expand south south store detailed survey aim reader reference scope group involve pattern matrix semantic document cosine document retrieval develop devote core idea document cosine angle vector vector variation retrieval retrieve document technical discovery although system smoothing expense document collection indexing help collaborative document similarity across cluster flat may hierarchical group group soft differ similarity minimum average similarity document task label sentiment positive versus spam infer thus classification notion document involve document matrix similarity automatically student assign proportional student highly get focus series cosine topic shift drop view document document block question answer answer find question corpus typical many big retrieval extraction retrieval present automatically base question direct automatically question measure semantic similarity similarity two word cosine row document choice question achieve human word average english us college document seem long researcher shorter prefer word center word remove word context important distant
reach machine control hmm alternate alternate think alternatively bias depict hmm break isolated leave fig depict hmm inspection redundant machine analyze synchronization edge hide machine symbol symbol pair x markov machine give even abc np example machine note machine ref machine whose infinite definition definition present state apparent
university increasingly model image rich distribution chinese article review dirichlet build partition infinite partially use chinese crp article theory notion partition distribution crp conjugacy well relate property consistency interpretation improper infinite dirichlet posterior conceptually dirichlet present refer non distribution remarkable conjugacy property basic chinese restaurant process distribution bayesian dirichlet know name chinese restaurant crp elegant analogy incremental model prove way tool modelling want flexible useful phenomenon compression well perspective generally task associate model partition tree hierarchical structure concept along statistical develop general form partition crp crp tree e nest bayesian improper infinite improper dirichlet readily obtain sampling science partition express posterior second play role occur important reasoning mathematical community kind consider require
mu mu mse validation var properly compare seem transformation account potential compositional positive impact overall result agree literature diabetes ten body pressure six response disease necessary regression model ten accuracy predictor variable predictor input five age evidence solution forward implement inclusion quite interestingly var show evidence literature diabetes select form remainder predictor var include assumption predictor var respectively average respectively number deviation poorly selection backward move generalize aic package parameter information present approximation approximate var selection default impact encourage variational describing consider aspect consider fit device conduct gram four variable initial pressure pressure mu mu minus mu minus mu mu plus minus mu minus mu minus mu x mu mu minus range fitting term respect indicator correspond center multivariate zero estimate posterior mean estimate posterior construct variational variance use hasting draw burn
statement iii hmms viterbi cluster primitive state meet satisfied since irreducible primitive condition general mixing imply versa infinite alignment possibility hmms assumption relax finite viterbi alignment side hmm viterbi alignment viterbi satisfied alignment positively recurrent process respect time infinite alignment piecewise recurrence construction r theorem obviously consistent viterbi predefine break property chapter vi argument identically necessarily
fitting regression relationship response make accurate technique attention particular response also investigation notably smooth spline exist nature exclude naturally estimate local influence may elsewhere level significance definition local close one would predictor influential treat treat incorporate influence accommodate significance certain discount treat locally know ignore predictor lie final consideration particular particularly informed often cause nonparametric interaction vanish neighbourhood consideration regression adjustment locally algorithm motivate example property
positive skewness large negative alternative similarly normality reject alternative tail clear perform compare tailed test comparison power nevertheless logistic mixture alternative
week six year order alarm could case adopt alarm week observe apply disease surveillance center al count author compare generate week national reference equal past seem quite nevertheless comparison replace systematically compare week gold represent count triangle alarm triangle line represent method c line see
achieve procedure increase asymptotic slow parametric oracle prove plug whether fdr critical procedure depend apply model one side study laplace double laplace may appendix correspond situation fulfil specifically decrease side additionally appendix arise test gaussian unknown variance central degree freedom location write translation ratio convergent formula dominate well integral central distribution dx student test imply test side test proposition model parametrize centrality ii tail student freedom behavior crucially characterize primary interested note location student likelihood test centrality tail yield xx laplace location parameter ht value satisfied panel bottom panel slope cumulative panel fdr fraction procedure laplace panel ratio f laplace location g pg laplace statistic appear laplace h
condition invertible ax vector sequence theorem explicit process appear case process residual observation sequentially extend tr ts r q notice limit sequentially appear influence summarize us r dr r base feasible use simulate appear statistic brevity exposition sequentially residual modification agree put residual calculate sum definition reasonable plausible eq formulate fix e brownian residual chart appear consequence chart corollary mind worth discuss issue nuisance
movie movie would rated recommend recommend movie experiment recommender try reproduce provide age gender student yes child category science h movie name movie yes movie follow use build recommender user scalable recommender system mainly way scalability combine rough slope smoothing build user system recommender privacy issue deal recommender system system add business keep dr paper produce template recommender technique available item vast growth internet system recommendation handle efficiently vast growth recommender system high even list express opinion algorithm attempt recommend item past treat recommendation problem used select
covariance determination ard scale ard prior popular situation periodic unit allow variation perform restrict via restrict sign improve model less expressive clarity generative come denote prior uninformative function transform appropriate cross transformation apply inner product compute model ignore model view pca pmf view inner observe movie collaborative filter link latent dependent another factorization stochastic provide participant dependencie model
four summary facebook dataset runtime bp community network belong community exist low overlap facebook seven community demonstrate node community demonstrate community community thank table subsection b section clique social need algorithm capable overlap recent assigning performance contain highly overlap highly structure introduce facebook five benchmark benchmark great consider community undirecte loop letter realization denote letter g realization letter
effect vary induce important cope surprisingly establish understand likelihood present estimation size penalization smooth convex log hand could discovery method design coordinate numerically focus covariate model fairly intercept pre specification covariate penalize effect develop perspective setting address truly scenario scope present datum model effect illustrate empirically effect comparison penalize mixed describe detail simulation procedure proof defer observation group let grouping observation group response effect fix unconstrained positive possible may multiple remark fit nonetheless sake
difficult detect continue forward payment add tag inferential subject able tag tag text thing social service help tag development allow visualization tag importance frequently tag see customer pay see previous correct correction view user speech recognition correction case correction contribution service extraction multiple user refinement independence well kind want currently service overlap characteristic tag create convolution unique carry basic book store addition case service aggregate material service framework medium business diverse avoid idea trend web tail public historical method social
round hard prove hold hard class symmetric game highly economic game public games pool game nash organize give section provides two player endow payoff player relative payoff player payoff relative every relative game since introduce player opponent action strictly formally period action player period action dynamic maximize payoff give maximizer strategy accordingly prefer maximizer payoff payoff yet maximizer even number use maximizer prefer payoff begin payoff feature rational opponent would advantage assumption regard maximizer maximizer infinitely patient look mistake importantly assume opponent maximizer decision absolute maximization action run period essentially maximal one concept evolutionary stable prominent role
implementation consideration cf moreover
le iterations b col slow mean exp graphical metropolis hasting involve produce alpha alpha last alpha exp alpha exp alpha rate exp alpha b c est col gold give simulation le last high acceptance b sd sd type normal high seq le j eps eps b last last type main laplace walk increase however random acceptance performance decompose depend moreover truly chain normal conditional implement sampler I I col main col duration time impose truncate component eq chain constraint program alternative solution keep plain normal conditional constraint j prop p prop p prop p prop j prop marginal long normal coherence exercise constitute e proportional density z n acknowledge dim I sd sd col grey false sd col grey false add posterior information table individual multinomial whose clearly break complete extend sampler
prove let I suffice eq rewrite inner fact degree fact therefore establish lemma absolute constant approximate bound xx q state bipartite smoothness I e labeling say satisfied maximum absolute distinguish pick neighbor projection pick labeling generate know ji opt define label assign every vertex fraction pick vertex least vertex neighbor recommend label nice recommended recommended fraction nice side fraction contradiction pick pick pick projection define analytical corollary remark university wu work university hardness hard constant agnostic hard allow big hardness result previous include immediate corollary result weak agnostic decision list previous hardness positive first invariance regular
mixture normal mn cl mn sa multivariate normal third term accept draw stage give detail burn update update successive acceptance logistic datum normal burn update mixture less fill gap proposal also acceptance refine hasting heavy tailed mode easily well example refine sampler initialize hasting proposal normal proposal challenge realistic distribution long alternative version difficult way generate generate take condition iterate converge regularity iterate draw describe walk dimensional initial constant take identity
choice sign power enough near david helpful edu channel constraint decode least square tailor assume communication show reliable probability small shannon capacity channel shannon superposition code superposition code achieve small communication capacity provide development merge linear information theory familiar problem require bit string string real number constrain across add receive string event bit event analogous reliability requirement sufficiently error small average communication communication channel traditional white interest mathematic version sphere pack code codebook moderate prior probability sum square problem value partition superposition range give solution probability identify produce bit heart differ unlikely provide polynomially sufficient determined partition superposition code rate capacity probability mistake fraction require section polynomially reciprocal reciprocal undesirable less complete task sufficient outer code arrange tailor partitioned code outer code fraction mistake end rs rate correct associate superposition total rate mistake distribution fraction mistake superposition regard composite superposition least separation decode
describe location rate separate signal whereas recover former diagonal hamiltonian diagonal internal calculate close due expand logarithm separable first term reduce vanish general taylor expand logarithm expansion stay small percent uncertainty locate position instrumental low expect mostly informative solution substantial location reconstruction expect simplify hamiltonian dropping reconstruction suffer region affect poorly already ignore term find minimize correction keep ignore change point spread sign term always suffer knowledge ray ray point neutral fortunately signature knowledge formalism take background count physical space isotropic
eq mean average respective oppose simply pick linear combination feasible normalize z assignment address offer motivation margin intra practice however simple posterior adopt walk actual position leave metropolis summarize obtain execute otherwise accept reject follow desire q obtain carlo use mode unimodal interestingly probable fraction configuration come grow single role proportion guide configuration however principled relation proportion origin margin distribution along margin feasible
desire hermitian replace intersect nan rewrite theorem draw equivalent product use previously hermitian firstly entry denote increasingly order h w w previous program entry program maximize whenever word result consequently program except repeat step previous show follow lemma decay use psd figure analyze h ct concentrate around get sufficient psd threshold gaussian operator nn q formulation function semidefinite threshold gaussian
idea give generate number development mention background people mathematic project gene uk take away finish stanford day communication
contexts ref completeness somewhat ref separately calculate directly step rule step entropy say observer predict intuitively suggest observer vanish mm kp h lm lm concavity state statement establish proof exist let unit word respect distinguish length know step follow eqs contradiction distinguish rv rv eq contradiction
impact dictionary build assign specific class practice compute c result obtain dictionary encourage computational code fix image second overcomplete learn require code order drawback propose aim leverage jointly code sparsity representation penalty real image capable recover dictionary art intensive superposition every orthonormal would seem signal
u c u step lemma first represent whose coordinate fact continue u strict q need lemma incoherence incoherence incoherence recall hold cv space orthonormal I cv orthonormal orthonormal I thus last prove condition indeed certainly since hand depend establish main outlier strictly succeed note imply namely prove observe copy show modification specifically observe approximately follow replace q essentially noiseless outlier succeed succeed column counterpart succeed slightly strong requirement noiseless constructing outli succeed noiseless solution pair define x g n cn hold
implement processor require implementation generator restriction split rejection would implementation even slow box normal slightly box consider normally ratio show processor dr comment implementation generator computer agreement pt plus plus minus plus
estimate nominal level effect intercept estimate stage example nr regular estimate denote intercept estimate construct draw stage nr r algorithmic use bootstrap draw turn bootstrap compute interval estimate cover rgb draw thin fill blue draw thin green em minimum em draw thin draw corner sep fill blue rectangle corner sep thick corner sep inner draw thick height em sep inner draw none none proposition thm thm grant science university support sciences engineering university york city ny ann support mh da treatment regime grow clinical science clinical aa south west txt aa north cm ab ab ab north west ad txt south west north west node cm b txt ba txt ba south west txt ba north cm r bb txt bb south west txt bb north west problem asymptotic alternative bad method estimation propose index clinical trial school simple estimation stage salient challenge furthermore subject
analysis tractable optimal pc pca subsequently variance theoretical favorable performance effort robust remain algorithm design dimensional observation apply consistency lack sufficient motivate propose new robust pca take inherent difficulty high proof support inequality let q upper due equivalent last prof set eq side inequality hold decrease hold substitute decrease property I increase attention decade spectra observation range thousand practical high microarray financial trick linearly infinite transform effort extend tool design regime fact cope
c h last px factor term quantity h plug assumption graph minimal order volume constant proposition ij constant eigenvalue converge plug spectral np corollary probability j graph approximate soon disjoint path minimal degeneracy time random operator underlie even though limit evaluation limit distance degree speed fast raw moderate might remainder explore grid lead tight distance grid explicit grid variance good distance collect implicitly tailor definition tool geometric graph know appear let p pn n n compute maximum graph ball serve template later count collection px ib resp b graph often expect distance quantity proposition rule degree valid region maximal n converge px px part part come argument
various approximate leibler utilize maximize terminate result liu mutual large connect connect make connect large terminate process candidate loop mm task rather use maximum
meet follow go present investigate blockmodel moderate instead explore visit theorem empirical sufficiently os enyi blockmodel achieving likelihood identifiable achieve identical empirically remainder investigate os comprise independent ab ab ab record respective respective loose error whether necessary sufficient os network growth edge number fig prescribe density closely require size shown dot solid test generate correspond
guarantee property low matrix satisfy analysis extend substantial column simplify entail generality since noise vector mean instance corollary consider sparse normalization although error bound form past g norm complement entry compatibility final w specify choice x n valid claim corollary regression well one formalize ball radius weakly form case cone rather center compare panel ensure tolerance term state allow rsc ball conclusion normalization sub belong parameter strict generalization naturally toward al consequence lasso generally rate function state rsc thresholded follow supplement sufficient convexity subspace rsc hold corollary approximation q r ensure claim outline family model glm suppose parameter
might repeatedly boundary preferable gaussian switch di elliptical attain idea evenly correlation feasible implement end product minimize matrix propose alternate algorithm problem far projection expensive experience parsimonious might effort importance sampling carlo evaluation q combinatorial binary vector rely usually combination copulas family sum family auxiliary eq cross
loop induce sparse step reconstruction patch definition manifold calculate whole coordinate calculate scale new indicator column calculation fast calculation update projection matrix patch calculate patch part optimization accord unified alignment compute accord pca class center define eigenvalue conduct lar loop matrix define consider lar conduct several column sparse ready sample lar subsection nonzero lar lar computation sparse via unified popular learning patch alignment local retain penalty add sparse combine term elastic net elastic superior reduction algorithm powerful consideration well square apply lar find regularize lar lar optimize general regularize special lar solve problem asymmetric constant lasso least penalty special define lar optimize penalty loop change consecutive lar
weak policy relate present great development control history part discover reward history interpretation probability proportional utility discuss recall formulate describe aim pass expectation propagation original close message claim optimality control specifically make
also converge z tail p r last infinity z z chebyshev inequality expression rp probability necessary convergence finitely many capable control discovery discovery proportion conclude proof let sequence fp gp last conclude fp established fp fp fp fp rely account establish fix kp kp gp kp gp j kp gp kp formula q theorem theorem lemma initialize edu noise measurement precision budget hypothesis satisfy adaptive
illustrated variate coordinate alternative symbol coordinate covariance imply correlation constant surprising appear marginal correlation imply interpretation allow latent simulation confirm consider correlation inference alternative model present difficulty likelihood however em conditional trace entry cycle index draw iteration full q normalizing use detail simple formula integer freedom form divide write cumulative standard moderate lead instrumental drawing accept repeat suitable multiplier important ingredient rejection tail focus multipli minimize rejection draw
grow decrease grow achieve logarithmic present decrease expect specify stand sequence lt show depend system consist resource evolve irreducible resource table c stationary resource state reward pair arm optimal arm great reward figure logarithm bind proof empirically however allocate resource learn reward maximize minimize define reward expect reward per step achieve logarithmic user resource broadly
l h difference entropy claim sum multivariate measure vanish fact h limit actually generate conditioning current alone determine future atom vanish vanish past vanish generate acknowledgment intelligence helpful manuscript include currently bit tool dynamical system analyze classify dynamical kolmogorov shannon lyapunov characteristic invariant logic result short fast bridge theory characterize short synchronization connection synchronization nature give duality synchronization control term one notion synchronization though interpretation different converge differently either process representation give insight elaborate apparent employ related review shannon asymptotic aspect hierarchy call systematic convergence short possible description note description model analyze entropy entropy entropy introduce block latter entropy synchronization summarize synchronization relate back introduce derive notion order spread control synchronization explore increasingly emphasis conversely class widely statistic elsewhere start optimally relax redundant synchronization class relax stage convergence property entropy turn induce contribute largely apparent tool advantage various sign display diagram dynamical history see control dynamical broadly go symbolic dynamical synchronization
increase step chain window recommendation diagnostic mean divide converge chain early st window lie within material chain city diagnostic run parallel simulation post rate statistic demonstrate distance superior secondly test aspect markov abc euclidean markov discard sample cl model associate map mmse j posteriori posterior knowledge mmse quantity unconditional joint f acceptance markov mmse factor estimate marginally obtain standard estimate bootstrap classical cl corresponding chain example datum loss turn claim divide analysis previous justify simulation coefficient variation
xy xu yu xu yu xu yu heat equilibrium tune belong reason stability ns lb sound instability compare pressure pressure ratio propagate sound pressure jump use visually subscript indicate numerical heat ratio relaxation boundary reach end equilibrium
vote occur vote party visualize voting question role determine operate explore posterior binomial graphical prior binomial string graphical place grouping sc al ga tx arise close ar tn ny ks consistently node variable consider indicator vote bayesian approach place sparse close examine therefore exhibit neither
finally embed prototype refine summary alternate flexible quality aggregation strategy show meaningful summary help greatly analyst get quickly good curve list variant illustrate numerous distant note look unique support summary piecewise contiguous interval series average turn label step segmentation basis solution merge two build mean prototype interval prototype prototype assume obtain prototype optimally respect specify total number piecewise represent section segmentation overview segmentation introduce segment mostly independent standard technique build connect building view goal section material relation segmentation point approximate simplicity concept article capacity learn rather visual analyst smoothness e base
entropy relate hausdorff distance hellinger hausdorff contain restrict pilot control hellinger hausdorff vary greatly hence pilot within nice normal cut stay parallel second define distribution lemma entropy quickly hellinger hellinger distance hausdorff imply get thin tight hellinger hausdorff true thin hellinger within rate hausdorff simplicity data pilot let estimator step relate allow entropy finite cover manifold net cover small covering call hausdorff hausdorff cover dc hypercube cube within manifold translate natural coordinate domain
learn hold yahoo competitive know human express preference consequently statistical comprehensive survey intersection retrieval area learn review learn retrieval broadly speak rank order natural phenomena movie book web page wherein focus model tie previous pair see ignore simultaneous interaction strong return query alternative object partition amongst amongst permutation partition wherein partition object generative partition object partition specify partition substitute potential choice potential potential normalise potential mcmc demonstrate application hold yahoo besides second
either ignore stationarity come average algorithm acceptance rejection posterior transition p come make change condition imply taylor give k k modulus right suffice expectation choice case denominator numerator k integral argument combine give ex rgb rgb abstract many modern easy calculate abc method replace calculation involve simulate simulated summary semi manner aim parameter mean parameter mean within empirical analysis ad choice summary literature demonstrate advantage affect implement abc implementation turn understand alternative posterior variable summary density give conditional carlo function numerator calculation importance mcmc rejection summary discrete summary statistic marginal summary finite ii
aggregate latent course scalability bayesian hierarchical parameter line observe connection among customer panel several adopt new product around customer customer likelihood model choose person position individual share realization locate coordinate aid visualization add label include interpretation considerable distinct super space cluster customer quite customer figure represent configuration small space provide become examine segmentation central strategy behind e price copy study two interaction practice interaction customer service help potential customer result observe basis graphical highlight among customer quite identify line individual place circle common link line potential customer useful reach link depend customer homogeneous segment represent similarity proximity circle customer observe occur certainly interaction among well example college people college share trait age education gender close relationship nothing else common college share common friend place student close close friend aware represent base observe alone unlikely behave way incur
low instability realize exposure portfolio increase drastically high reason fact produce low attain local portfolio risk moderately exposure financial comparative need exposure constraint high frequency profile improve e position short approach maximum portfolio exposure constraint reach comparative portfolio concentration daily stock carry return period avoid hence affect price jump financial portfolio numerous jump jump news move hold exposure sensitive accumulation expect hold approximated accumulation pairwise precisely element pairwise semi definite projection need pairwise strategy financial volatility estimation window theoretical observation simulation outperform one exposure constraint portfolio allocation theoretically far need integrate integrate simply time process process v xy condition impose little estimate integrated volatility need random variable finite generation realize volatility proof generate consequently hence variable specified continuously bound note part term far c paragraph eq calculation
raw relation node message correspond simplification simplification message model link strength facebook comparative feature binary relation transaction work block frequency relation take account group assess measure situation truth assess soft developed clustering measure apply later mixed membership membership normalize propose evaluation measure overlap output soft propose overlap cluster metric precision data data cluster belong true extension assign count number vector estimate membership membership precision measure present network corpus another website simulate recover interested
online overcome base propose name compare approach online online enable continuously evolve become increasingly tool sensitive internet report anti record target payment service cause internet community put mechanism currently service site safe service microsoft provide already appear else spam email provide full low impose overhead classify new I clicks rely maintain discuss describe dataset extraction describe algorithms dataset feature discuss explain base attempt include account somewhat adopt party trust party find distinct technique manually select lexical feature google page work build identify feature common string design automatically google classifier lexical contain google content describe maintain server side
understand posterior denote metropolis gibbs vector latent highly inefficient correlate moderate sized mix poorly metropolis irrespective proposal simple would gibbs markov mix slowly problematic require estimator complicate sequential positive evident markov path sampler mode adapt mode demonstrate methodology effect rare accounting moderate dependent suggest negative easier parsimonious include chance decrease observation absence include noise effect find evidence datum strong agree observation numerical inspire multiple mode suggest relationship growth sampling methodology mode static instance analyse hand scale algorithm surface dependence highlight ability reproduce search evidence metropolis recursive low model space time avoid jointly estimate latent jointly exclude special realistic sub kalman filter linear
rd ed american statistical association realize market journal economic statistic quadratic create stock variance serial covariance quantitative security bid ask journal financial infer quantitative trading finance li approach process normal journal american quadratic variation journal liu imputation miss journal american gibbs generation truncate conference realize volatility j quadratic price increment manuscript li bernoulli simulation limited journal schmidt frequency corrupt g kalman ed fluctuation market trading quantitative finance volatility simultaneous bernoulli normal high volatility volatility manuscript g efficient constrain regression technical volatility round convergence line jump business economic incomplete journal integrate estimation noise journal financial wang almost convergence recursive asset price filter finance zhang time volatility noisy journal american association frequency data exchange journal business economic volatility estimation volatility frequency technical thm proposition mm pc pc cm volatility transaction volatility occurrence transaction computationally price volatility neither transaction also volatility noise particle sequential volatility transaction volatility implementation result application could replace notation realization decide stick notation exist log additive zhang
es es fitness see define learn problem es stem reason correspond document thereby explore direction es voting solution make decision es gp therefore individual population rely solve evolutionary es detail aim propose fit inspire behind population pressure cause rise fitness fitness improve mutation ensure diversity evolve es rule form collect population vote remainder structure begin query query population fitness production new query evolutionary present formation es express act concept figure act four concept boolean operator document contain concept tree depth representative depth branch reach terminal terminal depth meaning boolean operator drive principle population randomly set depth individual start root place single create repeat grow depth terminal terminate growth tree place terminal complete individual equal size assign fitness individual mention create relevant concept give compose member boolean query document predict irrelevant fitness search maximum possibility therefore fitness produce query mutation parent choose create subtree root parent subtree root parent parent figure produce operation produce mutation operation mutation consider primitive primitive mutation operation mutation primitive
nystr om come investigate principal reduce extend approximated use theorem extension theorem reformulate address explicit nonlinear mapping low mapping follow manner integer stand indexing vector coefficient assume polynomial mapping unknown unknown lie manifold kronecker hadamard define prove framework find embed high reduce use polynomial explicit substitute substitute eq equivalent simplify th entry solution
define bivariate cdf cdf copula default substitute gaussian marginal long capture dependency reason contain copula imagine choose drawing encode specify univariate cdf gaussian process value beta obtain draw beta gaussian inverse variable dependency encode generate many distribution dependency call describe copula index distribution let joint base inverse previously gaussian copula copula mapping
structure underlying parameter still chain ph introduce substantial parametrize scalar completely distribution order ideally suit preserve expectation trivial characterize partially order within totally order subset threshold show delay alarm existence compute hyperplane several iv reader description various starting jump detection existence switch exponential sec generalize poor time consider novel stop threshold mild delay stochastic geometric time ph change belief control use finance seek optimize accumulate seek term risk control delay generalize lattice vi stop problem learn amongst economic market agent optimize local private observation social agent make stop automated system sensor decision reveal decision subsequent agent time belief belief strong threshold policy result decision local decision involve deal constrain social social rational eventually pick irrespective private stop enhance relate reveal observation stop stop intuitively stop state accurate private constrained propose switch public belief stop implement system scheduling change bound transition probability question markov stochastic problem formulate consider agent scheduling optimal policy rigorous intractable markovian large formulation allow stop problem subsequent comprise chain distribution phase ph ph hence construct markov chain denote assume evolve markov state correspond view composite one determine transition equivalent power appropriately occur special geometrically q integration counting measure integer decision choose
application network relevant play comprehensive exposition organized follow provide overview fp process fp invariant frequency update time fp conclude player game concept fp fp subsection security game pure simplex take first static select mixed instant player boost mix
fluctuation error component investigate unfold procedure measure calculate ix bin number ii cm present fig comparison demonstrate superiority previous comparison literature unfold h cm algorithm eight table perform robustness bias eight problem widely measure frequency pass resolution event acceptance unfold replace original use slide unfold problem low smoothed filter unfold
attack automatic collect public web information security attack token handling first attack seek attack attack output token attack attack attack attack follow subsection responsible attack familiar attack detection code double attack directly also attack decode character decode attack attack accord splitting recognize six token responsible transform token token attribute plain text comment token receive attack attack original raw corpus information token attack attack increasingly attack attack profile attack vector automatic sequential attack regard
iterative estimation bayesian monte tailor hasting show associate set property understand poorly scenario demonstrate fold introduce suitable home potentially like differ interpretation efficient idea generalization model include comparison base em sampler perform inference knowledge ever context great allowing perform well organize follow basic
explicit require possible simulate optimal loss norm dpp standard approximate dpp dpp preferences round preference approximately dpp monte cx nan begin dpp presence establish dpp bind integer accumulated dpp performance bind dpp define asymptotic error unlike critical dpp dpp near large consider law number dpp converge result dpp simulation carlo well presence sampling variant dpp dpp simulation surely need problem instead simulate dpp sampling dpp dpp dpp rl replace bellman nan nan ax kp cx pseudo code dpp rl algorithm h cx nx cx nan rx nan equation result dpp
processor processor outline idea variation time box uniformly distribute parallel computer require normally since translation give
belong another management status list management create relatively balanced unbalanced use average ap ap community unbalanced balance receiver characteristic roc auc reason auc ap affected table summarize c task adjacency ap auc machine hyperplane fit svm specify slack often cost parameter package fit wide benchmark understand benchmark conclusion reality svm cross
fall write conditional estimation mention pseudo normal wishart conjugate density set initially inference form construct structure online partitioned large easy depth branching depth maximal leaf reach consequently examine depth consequently ht figure demonstrate firstly gaussians bivariate angle consequently highly normality
able extensive technique inherently detection promise adopt nonconvex paper operator iterative substitute appropriate thresholding multiple procedure use somewhat surprisingly soft thresholding hard thresholding hard challenging pick start eq coefficient hard directly ol clean question arise try arbitrary converge condition converge answer question discussion rule threshold include soft thresholding define odd unbounded rule version vector assume huber soft thresholding correspond generally multiple threshold hard thresholding find small throughout pilot j cm penalty q
previous singular preserve symmetry length system avoid must positive symmetric compatible system possible equivalent denote vector root multiply equation get consecutive term recurrence addition recurrence multiply original new write also output associate sign equivalent solve z produce c second last element k k k x w x compatible length compatible return solution necessarily ax compatible dy ax bx minimum
duration novel meta policy spectrum access version problem provably near logarithmic achieve reward optimal system user try access availability evolve chain see channel user select channel sense channel
fair character evolve tree pass edge probability long branch mutation molecular thus throughout length mutation character character leave little mutation h factor tree two generate balanced tree sequence larger easier precise estimation method group non parametric method presentation remarkable year problem implementation posteriori estimation tree designed take extra computational viewpoint strictly rate markovian criterion individual favor distance markovian simple hamming distance compute building procedure follow one heuristic agglomerative tree procedure linkage update pair find build distance simple dissimilarity leave drive evolutionary dissimilarity either gene word hierarchical clustering inform case tree clustering create score distributional clustering tree thank bootstrap aligned tree resample character
decision bayes spatial stein bayes paper ann place stein small area proportion design value high dimension journal theory decision compound rd rao berkeley berkeley statistical ann zhang h zhang compound method ann remark assumption compound spatio incorporate covariate normally involve empirical view generalization method prove compare spatio certain relation identity exchangeable procedure permutation permutation q abstract space exchangeable realization belong function loss
kernel work two briefly describe kernel distribution rkhs iid fix induce clique empirical map maximal clique clique universal kernel induce clique q specify exponential limitation map operate handle dependency note zhang optimize explicitly document necessary well modify mean modification induce q pd completeness positive definite pd feature map product element pd benefit distribution currently open algebra identity mean iid extension dependency assumption whereas generative mean
convergence straight slope provide numerical create take specify confidence sort level sort vertical horizontal number draw straight slope dot slope correctly mean square compute equal distribution continuous cumulative list describe conduct parameter generate class
generalize example china yahoo com contextual bandit popular systems yahoo news online user practice simulator environment hand algorithm simulator create simulator paper contextual different simulator easy adapt provably unbiased empirical article recommendation yahoo front page offline contextual algorithm show accuracy effectiveness computing web recommendation service yahoo yahoo user activity clicks identify front page suggest solution create real unbiased thorough investigation improve theoretical study exploration medical web different classic arm particularly interesting contextual information make contextual call reinforcement bandit side bandit arm contextual trial trial know context arbitrary context unknown ta emphasize feedback payoff expectation define
delay left offer scheme observe leave complicate survey possibility restrict use consider process equilibrium absolutely minimal hazard hazard apparent later fix say recurrence bias proportional triple result let correspond distribution marginal marginal length consider truncate follow similar truncation model
subsequent decomposable technical policy make initially load play reward develop outcome index without delay final show decomposable complete characterization somewhat index highlight accounting martingale prior bayes traditionally result play yield reward current encode play accounting policy clarity approximation extensive mab know compare scenario typically comparable motivate instead moreover play true delay delay exploit outcome however constant regret arm underlie draw distribution prior specify input reward play make arm play available step observation posterior serve
within increase dominate encounter effectively computational lemma property claim capture preference key outcome effectively view permutation datum turn permutation choice distribution permutation near consistent approach seek choice permutation marginal seek establish choice admit exist choice whose relative agree signature relative force empirical american set useful choice signature road internet arise human choice particular behavior assume individual rational maximizer collective behavioral entire population guide g build country product put interest refer simply discussion choice crucial input make effective reveal preference via assume behavioral mis choice ideally behavioral structural absence need criterion select preference natural criterion simplicity model sense seek sort consistent simple choice model marginal literature devote partial parametric diverse name follow provide history reference simple set permutation parameter distribution probit realize logit
symbol ref identify role isolate symbol write english notably structure responsible dynamic symbol amount way shift architecture drift correspond change diffusion within subspace jump subspace correspond loss drift behavior explore periodic formal short human chain communication though distant begin periodic capture semantic sequential language drift mutation future prevent preference extend iterate evolution evolve production acquisition agent stimulus effectively force bottleneck agent generalize pressure bias traditional view human language brain propagation valuable evolution language approach neural bayesian drift serve agent linguistic latter framework quantify trade learner transmission bottleneck evolution ce decomposition linguistic present serial communication channel structural give drift population light kind behavior extend scope phenomenon exhibit derive population environmental political lie population area proper drift question model evolution progress rapid fisher drift process view pose fisher fitness connect fitness
superiority linear top pointing although anneal significantly priori constant need investigation conjunction determine concerned langevin sampling canonical ensemble choose local hamiltonian also temperature function propose accelerate langevin boltzmann gibbs follow potential positive wiener boltzmann gibbs canonical geometrically natural thing enable
denote procedure statement theorem remain unchanged projection section construct adaptive term employ notice challenge value p complement p nm appendix remark value decrease rsc employ rsc minimize work provide corollary constant error subgaussian replace suitably generalization lemma next obtain oracle resemble rsc stress fact entry furthermore inequality hold constant x random plan model thorough assumption nuclear rsc comparable point restriction estimator rank recovery show rsc achieve trade
real world closely likely relevant account effective genetic goal discover genetic proximal exploit fusion accelerate desire scalable solve involve fusion penalty univariate fusion penalty structure briefly review proximal optimization preliminary result task regression space flexible module qp descent grid work real world datum mining mapping example genetic problem quantitative trait attempt discover association snps million candidate clinical phenotype g problem formulate problem regression regression identify parsimonious multivariate treat independently adopt structure among response
analytical work location sparse x kx let kx kx interpret space densely kx real posterior distribution derive choice ensure agree satisfy b inspire dot solid increase become gradient carry induce lead p derivation intuitively marginal x figure g linearly yx yx analyze relevant interested cluster location
propose angle satisfy reversible edge place opposite randomly could lie acceptable recommend elliptical slice accept effectively place check location slice cover representative posterior proposing point one another width tuning method proposal away current towards slice move previous weak limit prior algorithm initialize draw angle initial first coefficient exponentially common offset entire still element sort tend draw
motivated circumstance article estimator computationally practice deep natural image patch investigate patch contrary claim qualitative present investigation particularly good image statistical occur constitute tool many learn lee prediction statistical image understand include approach assumption concerned goal important assess compare instance tell everything decide also visible unit boltzmann mass give zero mark boltzmann operate reach conditional unit eq logistic boltzmann see network unit otherwise increasingly magnitude deterministic boltzmann interest build boltzmann partially state state variable ml conceptually term right hand gradient draw distribution follow energy state energy connect hide interpret anti anti evaluate computationally intractable carlo typically slow measure machine feasible replace simple former lead rbms
kolmogorov real theorem close respect kolmogorov r ni nh half ni ni nm utilize know vc arithmetic c imply acknowledgment constructive theorem institute mathematical foundation study space mean minimize kullback type support hyperplane approximation estimator concave density fairly condition us model show respect convergence result independent regression identically distribute concave zero likelihood illustrate real defer proof detail hereafter
expert additionally mistake step back execute drawback impractical different sequentially adopt policy train iteration initially start policy expert interpret remove iteration action stop return expert problem aggregation iterative train deterministic simple proceed use trajectory collect add proceed collect iteration train aggregate collect intuition input likely execution previous experience interpret pick good trajectory well expert learn policy query expert collect mistake visit state irrelevant typically expert behavior could expert decay
involve due restrict turn calculate estimate univariate strictly optimisation also attribute fast pool hence solution monotone decrease function solution permutation index x iy arise naturally apply shift response deal separately observation tie covariate work affect tie covariate general solve dimension covariate extend iteratively word component store marginal fit value pm residual kx k na adjust result necessarily converge estimate converge monotonically locally quadratic converge define th step objective direction great evaluate solution derivative follow involve covariate step exactly check simple note towards drop become facilitate fit consideration corollary great thresholded hence small
svd simulated level monitoring water variable chemical available ambient collect minute compare data stepsize stepsize small stepsize large stepsize sampling decrease sampling near normalize yet comparable reconstruction sensor reconstruct instant section subspace column rank identify incomplete onto examine excellent
fail step relative induce select constant uniformly uniformly inequality constant post estimator step allow derive estimator model post post lasso despite post cn constant ol characterize ols post lasso perform strictly comparison support comparison repeat definition discuss post pt grow lasso post coincide occur fail subset however separate coefficient dominate ol post part perfect case sharp ol strict provide select ol fit condition f provide performance ol post fit lasso sparsity lasso ability ols ol post fitness
nc qx qx lemma fact independent identity observe globally approximation ix r prove follow proof normal claim since cauchy schwarz estimate last follow ix control claim obtain weak replacement control ix pt mn following eq notice wasserstein use distribution bound ix mr n gr ix tr ix z ix dt mr ix ix e ix pt proposal lead scale sde recover identity old denote cdf banach continuous define endowed supremum invariant sde density lebesgue interpretation start rwm take explore proposal mala langevin constant maximize limit sde explore invariant quantifie proposal rwm information develop mathematical metropolis hasting fashion criterion practitioner optimize instance rwm mala yet perspective measure class target measure retain inherent simplicity metropolis valuable key acceptance limit rwm maximize
index match plain interval already sequence e loop discover fix configuration fix fix unfortunately combinatorial adopt therefore alternative iterative every execute uniformly principle two dependency iteration pick index interpret continuous determine queue trick dependence make tractable fig execution ready index expect fraction vertex degree sample observe unfortunately interpret proportional match discover node grow e except special case rewrite q traversal ff subject index apply might significantly discover select sampling traversal select show node k concentrate degree
simulate system frequency collect success driving case balanced truncation suggest dynamical ready present case simulate emphasis empirical broadly applicable apply begin high infinite simultaneously identify direction observable assumption linearity exist become nonlinear closely linear feature reduction state design simple half dynamical behavior problem rkhs balancing adapt balancing propose determine reduction approach system control finite control balancing
curvature local version mcmc may extend riemannian reversible jump extension geometric extend efficiently community geometric
homology group induce topological topological homology background orient module arbitrary unity restrict immediately subset together explicitly unless real say persistence module map otherwise persistence module critical module critical value
hyperparameter k theorems representation nonparametric marginals p family sufficient statistic form projective projective index analogously model back posterior dirichlet process base next instance parametrization detail illustrate symmetric construction projective finite likelihood projective iii concentrate desire interest measure infinite mean conjugacy read marginal conjugate observation draw multinomial distribution note way dirichlet distribution generate set adequate choice countable algebra extension finite partition partial index space dirichlet hence singleton jj j sum probabilitie j hyperparameter space projective space collection random c vx I additive probability measurable ga projective dimensional projective marginal define multinomial affect henceforth omit multinomial family projective g measure eq additive hyperparameter sampling define provide suitable borel embed set v dirichlet instead marginal vi da ig proposition statistic bayesian hyperparameter index statistic dirichlet base parameter sequence group permutation finite element infinite potential model approach motivate permutation datum model censor symmetric task realistic movie movie cycle induce induce partition prominent restaurant permutation projective limit projective limit virtual permutation group condition limit symmetric sequentially inclusion projective system projection mapping mapping intuitively raise consistently natural vector
context default use yet predictor another length bit predictor see refer subsequence history value binary string investigation choose control first attempt predictor predict next single bit subsequence cover time instant two phase initialize prediction candidate score score subsequence decide count cover unchanged repeat decide candidate parameter repeat value increment follow score
tv verify set yes complete yes complete use choose set yes ct see yes next theorem arbitrary give polynomial follow relate let markov graph denote let weighted edge chain stationary vertex protocol let protocol sd protocol completeness protocol let c co hard vertex hypercube edge weight chain let constant edge add add markov walk loop polynomially couple
expensive decomposition thus check iteration alm adopt alm update project subgradient method schmidt comparison meaningful randomly similar al li create sparse hence iid gaussian matlab alm stopping solve dual additional cpu alm c gap gap e e e e e e e alm increase
line leave replicate density segregation alternative kernel symmetric much nan large monte investigation kernel imply nan imply high density skewed segregation dash ht triangle monte power proportional case asymptotic value segregation top bottom middle empirical base use function expansion mc n mc due magnitude severe segregation mild segregation power moderate power however power within right sided small moderate segregation alternative normal yield triangle value segregation alternative column column bottom uniformly segregation triangle corollary approximation monte get empirical maximize mild segregation severe segregation recommend segregation seem recommend segregation triangle monte replicate consider repeat segregation alternative degenerate relative nan segregation line replicate segregation alternative separation kernel alternative imply density segregation leave right separation nan imply skewed solid segregation dash ht triangle carlo segregation middle row column right mc mc cs present carlo central carlo power estimate get severe segregation high estimate power power power recommend mild segregation severe segregation alternative multiple triangle monte carlo estimate triangle value segregation alternative column middle bottom compute formula corollary normal tend increase get segregation power around moderate severe segregation power empirical seem appropriate
sec sec performance summary discussion sec random investigate add random mechanism easy nontrivial instance complex learning I regard perceptron patterns open configuration circle represent configuration flip ise perceptron depict unit perceptron association classification pattern actual perceptron modify configuration pattern space ise perceptron configuration ise perceptron paper binary pattern uniformly long solution quite configuration appear message transform transform
generate certainly moment algorithm quite develop st kind systematic mechanism genetic st bootstrap subset selection simply pre screen st sure screening sis variable subset impose upper st path include design multivariate normal mean simulation zero minimal median simulation signal min median max elastic net sis competitive make signal large reader st sis quite generating lie contrast selection measure average independent measure beneficial attractive many approach decision rank considerably option variable average thresholding rule call look
satisfactory check criterion polynomial candidate primitive apart thus polynomial parameter optimal generator web recommend fail test c easy implement since operation shift unlike primitive characteristic
thin geodesic triangle long organize complex manner loss normalize distance result lipschitz without concern particular prove fashion statement domain exponentially grow confidence every technical observation contradict converse radius concentration around fix constant depending consist statement exist banach regard integer see page respectively function determine lipschitz property preserve exist accord rademacher reference p everywhere regard lebesgue differential denote property though number observation banach banach combine classical constant banach choose distance banach corollary remark scheme exact link histogram lipschitz function get concentrated observation concentration structure dimension indexing scheme curse lipschitz concentration
discount group perform highest discount future algorithmic parameter principled reduction representation carlo select output present input unnecessary analogous input error planning effort prototype module convenience equal output mode reverse backpropagation effect analogous apply model modification need determination encoding mixture permit recurrent map dynamic
ultrametric highlight capable address face provide good tool application face old focused way regularity massive data mining determine express measure reality hierarchical cluster mining finding determination understand express hierarchy intrinsic interest review many ultrametric topology ultrametric discrete focus analyze massive illustrate finance publish study keyword multivariate recognition storage retrieval ultrametric topology economics human science frequently hierarchy scheme mining often instead pointed fundamental complex reality work mathematical complex present make symmetry architecture principle range notation division number field form integer binary define field see practical extensive consider dna encode u offer uniqueness encoding digit expression default digit start section common discuss explore lattice ultrametric distance hierarchy influential paper survey al ultrametric topology ultrametric hausdorff motivation complex give study field
presence constraint high optimization whereas technique coordinate considerable devote make feasible important modern obtain typically differentiable functional chapter analyze functional differential calculus operator study minimization empirical plus squared kernel hilbert reproduce namely unique combination characterize coefficient family equation introduce two general large regularization regard non linear gauss kernel solution symmetric result semi separable separable call dimensional whose follow parametric method characterize problem subdifferential
lemma combine fact fourth combine decomposition guarantee identifiability condition identifiable row column vector consider also recover certain program program constraint entry natural convex surrogate sparsity rank optimize formulation suitable enjoy recovery guarantee work closely sparsity application characterize incoherence sufficient identifiability analysis characterization yield strong favorable
interested simulate evolution spin represent spin obtain discuss spin monte improvement initialization strategy introduction mechanism carlo physics orient concept briefly review quantum simulate skip simply imagine intractable binary value idea contribution improve discuss quantum spin present excellent suppose distribution member dependent implement naive mcmc metropolis heat mixing
gauss z n model mh proposal proposal proposal construct j unity many density suggest sampler generate inverse straightforward piecewise quadratic coefficient z precision compute contribution reverse value current besides mh compute estimate section review new choose high detailed ensure estimate draw acceptance error variance ordinary square laplace sum form truncate yield incomplete gamma whether include nine covariate linear
r x ball disjoint intersect follow decay exponentially state nn ball region pr ok note ht sample disjoint note correspond furthermore kk p x joint power x identical derivation lemma define cx e cx r ok cauchy schwarz pr ok cx tx cx tx cx I tx u mu cx tx ne ok cx r follow pr ok eq note cx rx cx tx le rx also use clear I tx le ok disjoint conjunction pr ok observe schwarz subsection imply ok k imply integer c uniform identical result coverage ball density depend kernel x volume intersection let coverage estimator ball estimator begin establish density give pr q expression coverage neighbor field point definition p iii use cauchy schwarz cx cx cx cx estimate estimate cauchy schwarz obtain term ii iv ex cauchy locate term previously establish ball lemma obtain arbitrary continuous function x ok moment density x nx kx invoke point invoke moment nn density estimate density truncate volume ball ball volume
coherence approximation approximation applicability sample provide far fit corollary definition university california berkeley berkeley approximation algorithm coherence ability extract matrix completion question paper coherence formally new analysis propose whenever coherence across wide coherence estimate excellent become variety method spectral cluster manifold technique ridge algorithm order
update user count serve learn channel channel channel user denote total far transmission slot channel statistic function horizon collect discard policy regret maximum u regret transmission bind policy part asymptotically combine policy regret upon rank channel channel perfect would increment hypothesis test current alarm need decay asymptotically select count first user u n secondary implement bad u none estimate ensure go threshold count trivially occur rank user wrong rank line maximum start user configuration generalize user transition probability define note surely otherwise time spend wrong stochastic go threshold none give main section threshold function condition policy decentralize setting regret user user
information present long specie present sequence reaction hundred amount suffice million enable redundancy extraction mixture sequence two contribute nucleotide enable level comparable measure indicate enough sequence mix reconstruction decay prominent challenge sequence amplitude position peak correction raw context peak incorporate effect complicated peak cumulative calculation overcome sized peak dependent nucleotide position current preprocessing peak position position ordinary tool need problem force remove develop improve cs solver cs greedy pursuit approach might without removal reconstruction consider remove frequency explicitly utilize accurate rip reduce coherence novel dictionary sparse redundant easily sequence reaction sequence increase accuracy case noise sequence easily sequence additionally several sequencing enable enable single universal sequencing region align sequence identical identification via sequencing specie clinical remain species population specie
show trade normalize minimize abstract must deterministic giving limit length follow look practice focus city position certain network large state give bound tractable bad perhaps design network provide main city standardize apart distance apart elaborate outline elsewhere conceptually define network concrete language regard city line segment define applie rule configuration configuration euclidean randomness city euclidean plane though configuration city
message classifier indeed risk conjunction large attribute message string conjunction small string basic compression datum reconstruct small reconstruction return training subset contain need reconstruct compression define set index denote present arbitrary compression reconstruction consist message classifier return message message consist define attribute threshold compression set consist per threshold compression reconstruction map train information equation specify message conjunction specify conjunction attribute threshold value attribute attribute threshold choose subset specifie threshold assign eq complete compression aim obtain sparsity minimal encoding string examine large separate yield pac formulate dependent partly decision gibbs select assign risk
computational expense factor make individual simulator costly generation examine simulator deterministic simulator grid obtain display goal build computer fraction budget simulator coverage constitute contour surface true simulator approach outline take figure less interpolation gp fit well approximate predict power surface point popular maximum predict power obtain maximizer whereas assume underlying computer simulator correlation fit determinant inverse section conduct specifically hypercube design propose new accuracy lemma zero reach important remark worth note methodology also several realization use square theoretical may lead mse sometimes fit recall near design say
threshold table tag snps associations furthermore association detect different also snp snp population population associate pick arrange accord category column collect variable gene category gene category snps majority category quite category snps detect use surprising marker hand gene hmm hmm hmm locate strongly correlation combination report snps rs three snps locate indicate region strong four gene summarize snp furthermore four include snps proximity ambiguity snp correspond gene agree snp include snp snps evidence level future multivariate trait rr rr rr hmm hmm rs rs rs c rs rs rs rs discuss three close actually report characterize snps position kb take report pos fairly think region snps variability within snp
quantify sample size successful note side oracle remain multiply constant al error satisfy probability knowledge furthermore rectangular I consistency detailed rate matrix ball bound spirit general weak usual isometry isometry fix coincide scale restrict scaling remark valid replace positive involve restrict isometry observation brevity assume element difference element exist subset containing imply conditionally leibl q satisfy small
code substantial improvement frame intensity contrast spatially localize translation improve inter video potential compression tradeoff explore distortion beyond natural response infer transformation operator neural observer could change transformation learn could provide extension variable response movie might following decompose generator another diagonal following represent degeneracy degeneracy choose minimize practically every institute california berkeley department
regression sparsity gradient nonsmooth apply investigate suitable choice require short euclidean joint exist mean old condition specify boundary common piecewise smooth terminology treat component necessarily constraint framework describe later rate greatly lie manifold learning exponent euclidean let short denote vector exist satisfie estimate confidence except replace euclidean intrinsic dimension restrictive avoid introduce notation somewhat complicated proof base gradient zero natural specifically variable follow set f empirical covariance large variable directional derivative view datum along represent maximize maximize minimax repeat important effective direction eigenvector outer product provide approximate covariance estimate appear zero entry coordinate identify direction refer
seed interested limit scheme period argument advanced use mc numerical estimator consistent produce statistical advanced offer error argument generally consistent unbiased merely introduce completely eliminate price pay
distinguish fig subsection method consider specify distinguish fig statistic sensitive distribution bin bin recommend asymptotically see goodness limit briefly handle multinomial handle infinitely bin observation furthermore handle weighted root consider separate statistic advantage variation sensible availability computer level monte algorithms efficient easy acknowledgement would like thank discussion
expensive model cpu two essential first initial provide adequate compare type design optimal fit around particularly well suited second evaluating initial optimize validation comparison keyword computer compute investigation code realization code output treat realization center computer code degree polynomial sufficient sometimes capture process characterize spatial correlation stationary focused write one function correlation analytical correlation q wide shape predictor formula denote predictor variance
spectral unlikely due difference within like observational noted label object datum show network class classify case validate test argue matter chance contrary possibility inaccurate enough issue problem machine use logic feature use opposite correctly fail close look look failure occur spectra actual likely acceptable unlikely address recognition identify find datum active good free nn incorrectly label sample correctly remove basis citation alternate instead remove fail pass classification retain opposite scheme clean outlier
martingale previous analogous assumption lipschitz conclude decomposition divide probability appeal f thereby complete proof concentration uncorrelated martingale inequality eq extend n martingale equality uncorrelated q quadratic equality dividing complete last report result dual illustrate excellent agreement behavior versus graph grid expect scaling scale minimization hinge pair dy classifier machine choose minimize hinge associate shorthand hinge associate linear classifier sum setting minimization consider form generate qualitatively similar effect graph topology namely cycle grid range iteration vertical horizontal dimensional size spectral gap dot line corollary error versus grid demonstrate define number obtain function shift goal gain understanding discuss cycle grid follow quantity average panel three panel graph type grid blue dotted exhibit exhibit panel show network paper experiment present incremental currently figure optimal stepsize analysis distribute stepsize therein fit plot connectivity averaging gives reach vertical axis horizontal axis incremental show
test hardware traditional processor compare processor implementation old fast appear clear landscape change publish implementation intend efficient implement recent implementation three method think processor generator uniform apply number normally apply linear avoid consume generator
show unique partition triplet independent strongly let use necessary respect certain distribution even belong mutual case complement multiple equally design complexity example objective recursively partition meet leaf simply stop split search subset time partition item evaluate independent use thin chain structure thin discover thin chain partitioning item thin item n optimize exhaustive observation could advance nonzero second sort know unknown threshold infer item advance fix repeat candidate setting method search exhaustive third strongly remark method somewhat conceptually connect anchor exhaustive path nonetheless effective necessary information triplet sample triplet exhaustive minimizes subset evaluate triangle cross triangle total partition reality large element partition paragraph evaluate optimization big time require cache hierarchy discuss practically confident amount estimate adequate learn would bootstrappe offer approach replacement estimate resample typical bootstrapping structure lie discrete summarize compactly summarize hierarchical fraction set hierarchy confident small tree summarize large size final time run plot solid agree force set partition item structure dataset even hierarchy agree hierarchy tree rarely case entire structure agree make ask agree count correctly leaf meaningful identify hierarchy uniform consistent concentrate even case simple structure much fouri theoretic machine community graphical lie exploit inference operation bias drop formalize recurrence transform dynamic programming branching use recurrence compute
topic work typically far perfect probably combinatorial bi mean objective benefit although exactly combinatorial eqn intractable exist wide range heuristic strength understand approximately optimize method complementary bad flow method spectral bad method piece cut difference flow road might scientific easily well cut bias social property analogous piece regularize suggest novel partition apart insight perspective define perform instead couple noise particularly problem instead much reveal look ensemble cluster property heuristic interested analogy intractable partition less non determine dna look etc structure put x ray one protein physics input hard visualize protein large numerous large piece procedure employ reconstruct visualize adopt interpretable community principle let small scale social function minimum cardinality intuitively surface area aspect capture community community size illustration hard order compute different flow follow processing provide strong heuristic good piece find tight community like surprisingly compare behave graph road datum non manifold identification flat connect moderately graph perhaps surprisingly common model reproduce community flat thus view perspective meaningful piece moderately like graph size fact advantage analysis information network
agent exist question describe researcher ask exist amount sketch represent substitute representation substitute admit perspective structural submodular bound submodular distributional implication theory economics combinatorial present approximation achievable example simple achieve product behave fairly submodular behave technical question economics immediate use complicated approach technical precise achievable submodular upper improve function proof description central submodular central issue distributional bring usual model structural provide understand natural technical gap improve improve likely work real would natural class learn well perhaps distributional assumption learn trivially extend lipschitz concrete value satisfie integer normalize monotone submodular converse also rank property normalize monotone submodular edge monotone rank formal value lattice subset whereas large close na set follow property
pa proof w use w w hand complete definite inactive c small affect onto dimensional clear mapping minimal c j denote topology natural subspace topology fix definition c c c sn j sp sc p ps cp j j j cd ccc cc without spectra show rule setting demonstrate use noise accumulation centroid fair select feature report microarray ignore gene centroid employ microarray essential datum correlation information case whether covariance significant accumulation setup consideration precise suppose normal p setup performance rate pseudo classifier fisher discriminant difficulty discriminant whose first arise noise accumulation centroid challenge
unnecessary sometimes advantage simulation established investigate kind signal arise test spaced test signal scale root ratio replication posterior rule deviation parameter practice estimating significant trial dataset several establish wavelet reconstruct signal ordinary discovery wavelet block signal haar signal analysis package perform wavelet mean interaction ss bt false fdr signal replicate error rr rr ss cv fdr ex ss ss bt replication table present extremely estimator thresholde naturally cluster transform perform moderate reasonably
state markov nucleotide index represent instantaneous substitution nucleotide nucleotide transition change state substitution homogeneous substitution site reversible proportion amount flow flow opposite follow notation nucleotide nucleotide specification substitution distinction substitution rate evolutionary reversible substitution substitution simplify illustrative look wide range substitution cardinality specie nucleotide substitution propose maximize adopt endow prior update follow mx suppose competing substitution eq favor really year monte carry ad hoc currently depending
state current lead current agent receive reach offer mdp require procedure valuable environment seven among leave agent reward otherwise receive monte replication constant ensure hold reward deterministic slightly algorithm benchmark environment agent environment four environment generator environment environment environment average five state
organize begin stochastic problem give description main make place distributed complement proof collect norm norm f fx fy gx recall free delay sequel analyze describe two closely first average nesterov far collect giving dual vector primal mirror dual approximation mirror proximal essentially often recall lipschitz particular imply lipschitz continuous respect convex assumption show optimization function lipschitz smoothly differentiable loss g bound standard assumption sharp choice factor extend mirror update receive stochastic gradient point simple delay uniform delay analysis admit delay long mirror descent simple replace averaging mirror descent follow asynchronous asynchronous vector rule method involve potentially different delay delay smooth arise drastically slightly significant overcome delay smoothness delay perturb since variability result delay essentially second penalty asymptotically negligible set tt delay gradient stepsize mirror sec theorem corollary average convexity addition hold satisfy home corollary theorem asymptotically negligible favorable implication stochastic scenario distribute delay gradient delay abstract away procedure simplex though leave value mirror combine sec tt theorem consequence powerful turn develop application include assign sec scenario simple dataset sample among worker sized subset streaming worker receive stream make simplify worker receive pick replacement base master topology delay protocol compute master time lag distance latter section convergence rate node describe architecture master scheme worker compute master gradient parallel gradient time parallelization alternate delay worker master maintain parameter round update begin computation worker master worker parameter pass update worker worker delay early apply delay cc master worker master node time master gradient compute master communication toward node information store level protocol combine delay delay average worker master work span rooted master node phase leave master parameter neighbor simultaneously parameter depth child fig receive communication iteration leave leaf gradient parent parent gradient leaf average tree take gradient vector round average current gradient pass span description formally delay date vector child master child child child compute random node distribution hierarchy leaf iteration receive gradient parent master root receive delay entire give rise update architecture corollary section achieve asymptotically procedure use synchronization update characteristic network also assume explicitly centralized cyclic protocol update update fx allocate centralized cyclic delay delay compute gradient cyclic assume communication compute sample theorems beneficial master receive locally delay master centralize cyclic local algorithm unit take centralized number average architecture cyclic delay plug count corollary provide architecture unit cyclic figs cyclic compare cyclic locally average algorithm locally always guarantee well quantify improvement path cyclic tree distribute ignore logarithmic cyclic possible modify communication computation compute gradient machine learn natural language reasonable cyclic delay strong convergence guarantee protocol rate delay though focus theoretical present understand aspect real cyclic delay specifically solve regression problem news article economic feature article otherwise optimization delay cyclic method master term worker cyclic delay several worker figure stochastic convert assume define take master master worker compute master receive centralized gradient perform optimality centralize discuss delay enjoy centralized number negligible similar demonstrate linear small investigate network asymptotic communication delay benefit parallelization nevertheless allow delay asynchronous significant improvement section collect technical key choice assumption follow identity hand sum bregman equality straightforward equality lipschitz recall definition negativity replace bind recall strong xt convexity sum proof decrease stepsize delay convexity lipschitz continuity q xt xt non probabilistic xt imply consequence remain second conjugate addition sigma combine xt xt combine early inequality bind essential eq completely prove mirror descent average control involve differently expand condition following leave gradient cardinality chebyshev inequality theorem term proof eq proof theorem term eq non write bound second give expectation happen first condition term bregman delay preserve benefit stochastic relax synchronization specifically resource failure asynchronous penalty delay omit brevity microsoft fellowship support science fellowship program grateful communication like read manuscript collect continuity property proximal operator average dual xt result essentially many context property mirror frequently difference mirror descent minimize mirror q thus q convexity complete update recall lipschitz continuous easy thus schwarz last slightly tight lemma triangle inequality delay indexing know expectation increase h old substituting complete proof essential equality unconstraine upper bind simple side hand side evaluate hand inequality rely application boundedness boundedness iterate gradient without iterate iteration delay berkeley edu department electrical engineering california berkeley delay development master perform parameter worker compute local parallel delay take problem huge internet contribution delay asymptotically achieve overcome communication synchronization requirement architecture asynchronous scale iteration delay additionally statistical stochastic convex computing xt xt analyze receive gradient central asymptotically stochastic delay delay gradient distribute master
sp sp suffice assume moreover r r rp rp rp proof condition exploitation phase attain exploration price price kn fraction offline q plug pricing quantity use pricing limited achieve near bandit style algorithm key designing algorithm price estimate exist respective base see contribution reduction base pricing pricing style setting well well particular use crowdsource conjecture conjecture problematic particular bandit algorithm hard code amount time exploration rise immediate general demand desirable extend possibly respect offline second theorem extend demand demand distribution price one fix follow randomized benchmark fact explore resource pricing iid immediately extend make direction demand time alternatively promise apply algorithm final price small neither likewise price price sale increase price surprising main near regret help question grateful slightly purpose regular distribution least offline benchmark recall agent fix price maximize offline player sp price regular symmetry offer function jensen bind agent never multiplicative strategy compare demand environment sampling yet player correlate exceed include therefore agent draw I excee item happen environment moreover definition correlation q always therefore combine regular price kp pa kp n kp omit kp kp us pass immediate demand approximate offline benchmark several sale denote begin characterization sale moreover log concavity sensitivity support fact plug complete version detail online appear version microsoft designing maximize mechanism offer leave possibly agent maximize mechanism know scenario compare mechanism offline free mechanism whose less offline assumption relax match demand multi armed pricing mab intuition mab set level limit treat price armed bandit another small agent leave price account iid fix distribution bayesian demand tailor one assumption avoid know costly likewise significant demand easily like demand know mechanism call sense specification mechanism spirit demand integral mechanism face mechanism benchmark specific demand paper mechanism offline mechanism depend demand price demand satisfy strong monotone hazard one satisfied price commonly appeal reason first agent needs offer human agent former much easier reveal entire private reveal private third dominant side price mechanism particularly useful demand advance consider item sequentially observe manner influence make item independently demand assume support normalize f whenever agent item item compatible never value observe select henceforth call design pricing strategy compare benchmark assumption demand maximal allow distribution henceforth demand distribution constrain demand fit mab round set arm payoff maximize set correspond mab round exploit mab payoff specifically price behind applicable goal mab converge price wrong maximize treat exploration instead explore arm price schedule payoff much arm suboptimal even index assign history round arm depend history index estimate expect single elegant prior apply section bandit algorithm respective near base pricing limited agent pricing strategy offline pricing trivially follow detail free pricing demand offline minus emphasize pricing know mechanism compatible price focus power well fix offline henceforth fix strategy technical pricing strategy achieve expect benchmark surprisingly demand price minus demand moreover mab sufficiently small improve free demand offline minus constant hazard price item recall price moreover price pricing strategy whose expect offline minus constant condition meet demand hazard property satisfy distribution within offline demand monotone hazard depend achieve pricing vary parameter improve theorem nice trivial arbitrary parameter distribution large depend directly comparable dependence distinction constant fact essentially match latter benchmark expect pricing pricing regret demand demand informally hope theorem uninformative provide another online price meaningful big free pricing demand expect offline minus maximal offline pricing management literature see overview prior price without set study detailed early iid theorem assume depend imply special provide super case continuous specialized equivalent prove upper bound benchmark inferior distinction pricing strategy exploitation demand regret demand parameterize parameter parametrization rely know parametrization current upper upper apply demand improvement demand distribution bind pricing demand dependent pricing round mechanism two benchmark online mechanism unlike logarithmic multiplicative work paper consider opposed agent price design mechanism multiplicative elaborate price mechanism regret al online arrive period designing result offline online multiplicative mab duration reward mab rich literature operation computer science economic proper discussion beyond background relevant work prior free mab payoffs mab lipschitz stochastic payoff e call round arm index pick index arm index bandit arm index essentially available confidence accordingly new price ucb strategy exploitation sample exploitation explore price accord schedule payoff define adopt obvious elegant since expect payoff price word ucb specify standard price sale analysis suboptimal way elegant unfortunately nature adopt trick appropriate proving event rely standard framework round alternative observe maximize payoff mab independent price exploit special mab payoff price arm rely regret neither behind limited informally mab converge highest wrong quickly main conceptual set appeal separate exploitation explore arm continuously observe payoff suboptimal arm choose rarely assign score confidence payoff index depend payoff arm provide reflect limited interpret price ucb payoff strategy obvious elegant rather respectively number define ucb technical analysis sum via elegant limited pricing pricing numerical pick price breaking tie recall fix price approximate round ucb ucb sale price round current sale rate division equal define hold namely least suitable radius want subject quantity observable standard literature use elaborate bad sale see perform well price sale imply proof sale specification nk price set optimal fix price strategy choice parameter regret item think experiment consider pricing continue version realize realize item latter round sale realize realize give execution sale high event describe rest argue loss low probability negligible round via concentration guarantee focus sale respective importantly rather event hold sp second indeed generality th select pricing happen play price union long hold happen pricing price round eventually regret total price round consequence follow immediately sp radius round price price price line bind third sale rearrange bind key instead realize effectively constraint brevity denote sp kp k later respectively set select price plug claim summarize finding far active price active fact active price let price respect tie break else rest note simplify remain take regret improve demand informally formal sale demand easily claim price regular moreover constant demand achieve distribution map sale regularity pick hazard maximizer bind imply regret particularly arm regularity third claim sp np p therefore improve final desirable theorem use pricing strategy pricing set trivial bound arbitrary demand pricing
practice carry approach acknowledgement early stage manuscript grateful anonymous reference part european community reflect universit taylor information technology bayesian analysis sequence tool hoeffding concentration integration hoeffding inequality introduce feedback combine tool although regret bind yet regret potential bayesian tool encounter pac decade st contribution supervise approach lie flexibility bayesian pac optimize resolution pac bound linear classification pac learning pac bayesian long treat suitable almost hoeffde canonical pac handling combine bayesian sequence certain sequentially dependent expectation prominent field reinforcement advantage pac learn recently include suggest state regularization mutual incorporate bellman bayesian justify guarantee batch reinforcement able knowledge informative confirm irrelevant algorithm case pac par situation exploitation batch minimal root number state applicable want bad action reason difficulty pac exploitation observe rest density within evaluate usually situation reason action minimal action pair size bad round resolve weighted strategy commonly bandit usage introduce difficulty influence play influence subsequent variable pac bayesian approach future work sampling grow enable take variance explain gap result bound combine bernstein work survey main present pac bayesian derive bandit conclude variable function lemma precede pac hoeffding tight certain derivation dependent belong distribute special reference depend concentration bernoulli variable type theory ct proof find approximation factorial bernoulli variable empirical average divergence function ct convex convex interval sequentially inequality lemma ready hoeffding verify minimize ia simplify make equal almost equal worse contribution however low lemma tighter relax preferable pac basis take root back physics relate posterior arm distribute distribute irrelevant lemma fix probability great lemma expectation substitution bind obtain simultaneously well tt key ingredient proof ta alternative pac hoeffding inequality theorem martingale ia ta ram hence go bind ta e ta derive obtain great base way furthermore special reward action expect reward eq provide section adapt provide section assertion last substitute back choose get integration conclude technical lemma proof regret tight n unable
select sequence experiment end integer site specific bias denote bias reads observe probability accord view replaced length length error concave infer translate relative per case bias reduce equation slight difference name describe read end appear formulate equivalent directional rna seq confusion pair seq strategy first follow length appropriately site close preferred protocol alternatively site double unclear read analogous modeling error read beyond address issue allele remark possible consider entire compatibility rna seq principle impractical compatibility practical allele infer paper formalism describe utilize probability unknown infer map previously equivalent formulation describe close nice know numerical unique seq address appear biology literature relate property mathematically mean different relative testing equivalent compatibility full certain assumption read typically em read gene three pair initially abundance read assign assign read abundance consider red green read compatibility see subject notational z yx must maximize conditional maximize em initialize expectation pair property every illustration figure em theory derivation multi important case e square equivalent count variance possibly variance multinomial well count suitable computational inference formulation furthermore square approach constrain non heuristic approach except publish bayesian rather infer dirichlet mle use em information theory abundance estimate important seq rna seq differentially term frequently abundance desirable model equivalent virtue condition sum multinomial find analysis observe even biological behave phenomenon refer dispersion alternative multinomial instead binomial poisson thorough differ beyond scope key single uniquely drawback differential gene quantification quantification differential poisson model quantification virtue binomial equivalent quantification different relative abundance generalize read done read quantification although view model optimize seq rna seq advanced year review discuss publish rna seq technology reduce bias sequence technology result throughput development lead rna seq relative abundance introduction eventually single read long rna solve sequence reveal future seq modeling utilize rna sequence require assign short question light remark model relevant practice sequencing technology believe read length read essential accurate quantification differential gene family map read issue read bp long read multi issue effective correction denominator affect abundance protocol expense read quantification fewer note seq uniformity across yet see therefore paper well seq protocol bias model correction datum possible bias affect rna seq protocol sequence establish quantify progress rna seq relative abundance review benchmark cast seq systematic benchmark complexity aspect seq fully connection relative abundance estimation abundance estimate analysis perform reference almost rna seq novel even extensively annotate crucial time abundance accurate ability relative length currently length local estimate abundance available distant reason communication efficiently remain open helpful discussion understand rna seq lead interpretation formulation seq lead seq comment equivalence multinomial rna seq appendix provide preliminary version valuable comment insight equivalence poisson multinomial rna seq model pair end read model extra integer composition note part make convention induce composition composition range element composition equivalence position end align induce weak composition crucial equal factor expression weak eq q number seq mean distribute generalize read replace derivation easy notation name structure experiment primary three l together position alignment length compatibility abundance rate poisson bias weight effective length give thm corollary definition example remark thm conjecture berkeley edu rna seq rapidly become technology application seq quantification accurate measurement relative read review describe approach also explain formulation publish rna seq explain relative abundance crucial model quantification seq sequencing rna seq array include genome annotation comprehensive gene genome seq resolution probe rna seq bioinformatics draw primary quantification topic identification significant compare difficult impossible scope constitute rna seq single quantification hope rna seq ultimately success seq accurate abundance focus problem abundance quantification review begin rna seq quantification rna seq rna seq rna expression refer generate although case consist rna abundance perform translate rna seq amount rna seq allow measurement absolute infer relative able currently common explain outline section model special approximation model individual relative likelihood computation describe rna seq next discuss rna seq examine development yet resort reader rna relative abundance count region length total proxy abundance count understand correspond relative organized rna seq rna seq model another word label publish author model model contain end uniquely read pair end read read uniquely simple modeling bias read read equivalently multi read read formulation pair pair read correspond end pair subsequently uniformity library first model pair empirically random strategy bias bias model content addition modeling effect error read mention ad hoc way review connect dash line read different abundance multiple gene read replace comprise genomic basis genomic number prove projective demonstrate present normalization read uniquely interest abundance identifying replace read count suitably feature restriction read major valuable omit many consist uniquely adjustment equation relevant abundance estimating abundance gene infer frequency pool sequence sequence species rna seq abundance however read show equivalence apply paper publication abundance rna express tag term rna seq assumption est issue comment likelihood describe read read otherwise compatibility eq denominator directly abundance make suggest substantially due addition length derive denominator become apparent inclusion probably denominator length scalar likelihood correct since abundance evaluate qualitatively presence absence denominator result seq denominator read read reason read
inequality inequality immediate production production production sequence consist member I positive integer list latter inequality well eq many certainly ordinal theoretic ordinal formalize part type ordinal smoothly obtain consist color edge color black consist size graph former sequence bad subsequence bad contradiction graph induce subsequence one conjecture follow corollary actually present comparable great element closure quasi category system computer logic type adjoint see induce finitely branch whenever exist finitely branch quasi relation branching rx mx linearization linearization put v fy fy fy order monotone category category category quasi order branching simulation order branching simulation composition z rx category quasi identity rs category resp category category space introduce function originally attempt difficult category section seq object object object former prove part production sequence exactly write assertion monotone define rr lin lin lin rx r rr branching sr x aa transformation transformation resp call lin natural identity lin composite inclusion assertion whenever author thank dr anonymous aid scientific c education science remark teacher type symbol quasi equal mind iff short theory index recursive language learnable monotone inclusion preserve various closure closure monotone direct image type embed category finitely branch simulation subspace space linearity closure branching game target language set symbol short mind topology reverse mathematic finite learnable language learnable system pattern language combinatorial preserve motivate example bring teacher language concatenation observe l follow preserve type game another red x q system finitely branch relation closure preserve image function monotone inclusion regard topological space copy finite discrete topological monotone plus characterization function mx rx coherence stable rx v nothing direct monotone question parallelism trace continuous finitely branch stable closure continuous unbounded finitely order finitely branch set system quasi send branching send coherence question establish monotone continuous organize next part closure language computational section quasi system quasi language computational indexing ordinal section monotone answer category categorical category record sequential category computation duality operator class set relation sequence possibly short quasi bad upper order lattice sequence mean segment say infinite node ordinal supremum immediate extension root number accord tree great tree say system let class module finitely generate sub extended pattern language bound language elementary learnable alphabet alphabet empty definition n u u v closure study algebraic language b I language closure closure language put fact alphabet close positive integer complement nice lemma finitely branch prove preserve alphabet class class subset viewpoint algebraic regard learn system teacher learner teacher hypothesis notion complete lattice element lattice equal lattice whenever finite subset lattice quasi lattice every lattice assertion equivalence obviously assertion lattice ic ic ic iterate process construct elastic closure question preserve closure property sequence advance theoretic topological operation set preserve employ theoretic type argument nonempty set yx arbitrary union intersection generators generator build mean finite sequence boolean formula assignment element generator generator intersection generator hausdorff distinct n ij gx gb say contain set equivalent monotone boolean oracle finitely branching relation define every positivity boolean equivalent ml rx sequence element ii ny ny n n n useful lemma proposition conversely r topology mx finitely branch therefore corollary mind language hold topology positive topology respect topology monotone boolean formula respect positive recall must l monotone simply write monotonicity obvious continuous open preserve range subset inverse image contradiction system fix alphabet belong closure guess closure closure monotone useful derive proof case empty word machine otherwise try oracle partition prove every production sequence empty word production infinite production contradiction done prove counterpart similar
unknown finding solve advanced field however dependence conservative belief evidence element evidence interval avoid taylor algebra initial interval uncertainty ref uncertainty box taylor also review interval polynomial bernstein ref propagate ds close interval support intuition provide extract solve expansion bernstein range bernstein polynomial mathematically separate characterize dynamical uncertain parameter deterministic solved numerically obtain polynomial initial general type order use transform find stochastic condition ie ji expand wiener polynomial variable basis polynomial coefficient polynomial response sec orthogonality property polynomial obtain differential numerically expansion evaluate quadrature nonlinearity basis numerical integration sec integrate however bound ref bernstein compact thank polynomial ds structure ds probability approximate pdf density function two ds white autocorrelation body sec normally distribute find induced structure moment time evolution propagate ode solve central moment structure ds box envelope c use estimate use confidence estimate singleton utility theory function confidence trust cdf cdf obtain variable vary laplace insufficient reasoning pdfs probability million cdf present quantity present approach randomly transformation pdf propagate pdf fig evidence measure indicate much trust lack threshold depend consequence drop two f construct point wise offer use theory action function decision make purpose paper propagate achieve propagate approximation evolution box incorporate author propagate uncertainty polynomial arithmetic moment structure estimate interest probability addition make scalar conjunction cm mm cm author edu lack model uncertainty structure close interval propagation propagate propose uncertainty order uncertainty parametrization approximation probability evolution equation knowledge combine different singleton transformation decision evolution arise deal require propagate segregation create propagate uncertainty evolution evolution govern kolmogorov analytical pdfs literature number technique mixture closure method pdf precisely perfectly practice value amount information available systematic uncertainty great decision system choose representative approximation model evolution equation polynomial propagate uncertainty combine propagation finite box structure uncertainty quantity response classic whenever decision close make propagation problem conclusion future consider uncertain initial vector time variant moment characterize distribution condition characterize structure quantify uncertainty cumulative distribution uncertain interested follow three main looking determine cumulative utility make secondary include order ds structure close represent ds structure x xx cumulative box close induce unique box uniquely body cumulative plausibility function thus cumulative function compute expectation belief finite distribution indicator expectation needed si give construct singleton pdf utility theory applicable way quantify amount take relatively define summarize deal uncertainty variable mean thus confident low
associate unique htbp xshift rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle black yshift xshift rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle node xshift rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle yshift xshift rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle yshift xshift rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle partition easy analyze z z share minimize set partition partition htbp yshift xshift rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle node yshift xshift rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle node edge thick sample retain sub submatrix figure algorithm graph graph retain graph sub un adjacency matrix adjacency see yshift xshift rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle yshift xshift rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle yshift xshift rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle yshift xshift cm rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle thick yshift cm xshift rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle bl correctness adjacency matrix independently appendix expect complexity establish ideally attribute balanced otherwise ny ib iy converge replace eq go size legend south east width height marker xlabel ylabel color mark cd marker value blue dash bind partition restrict define however depend ny nc probability pos east marker node ylabel title blue bar cd explicit red thick color marker thick table scale legend south east height xlabel nod title mark thick bar mark red dash black marker table pos east height pt xlabel ylabel title thick error mark marker thick dash marker thick dash legend pos south east height xlabel color blue bar marker table theory color black marker size value approximation tight assume chernoff sake remark report usually overall advantageous high behavior observation attribute configuration configuration representation represent phenomenon visually pos east width xlabel configuration rank ylabel occurrence marker table color txt color table x color attribute configuration concentrate log select whose configuration occur attribute occur partition attribute occur set attribute say uniform node edge graph graph probability remain require empirically evaluate question graph behave implement available experiment machine ghz processor run three use matrix guarantee valid empirically various xlabel node ylabel black white mark bar prop title xlabel cycle name white color blue mark bar cd mark prop report edge graph grow constant graph confirm observation furthermore strong indeed scale xlabel ylabel node cycle blue error cd prop title xlabel nod cycle list blue thick bar cd prop scalability size various compare vs trial edge sampling graph hour graph million node graph million well least time term number node title pos south xlabel ylabel mark bars cd error cluster mu mark thick explicit mark naive south east blue thick explicit dash mark table x naive naive scalability plot run across range generate therefore grow empirically title pos north east xlabel ylabel mark mark x cluster red dash title pos north node blue thick bar cd explicit mark table mark thick dash cd naive theoretical guarantee algorithm time vary towards run run legend cell anchor east legend pos north east title xlabel ylabel relative name white color blue mark thick mark thick green mark triangle thick color thick table color black mark legend style anchor east pos outer north east title xlabel attribute cycle list thick color green mark table color triangle thick thick table color black mark triangle well configuration diversity hence tendency run since interested plotted factor increase growth reasonably million feasible value title xlabel node ylabel pos mark thick title node legend pos color thick function vary particular fix vary effect section increase exponentially title xlabel ylabel run mark bar mark ds cs xlabel color blue mark thick bar mark cs cs highlight sampling run million node hour currently work rigorously prove guarantee currently investigate high search technique locality lsh apply contain attribute configuration therefore set least partitioning produce first partition without poisson chernoff plug q pos north ylabel coordinate naive legend pos north east xlabel node ylabel coordinate e rectangle cs title legend pos north xlabel coordinate east legend pos north east xlabel nod run coordinate scale legend north east xlabel ylabel title attribute ylabel cycle black mark color blue thick scale title xlabel ylabel cycle list scale title xlabel ylabel run name coordinate scale title xlabel attribute cycle name coordinate scale title xlabel title xlabel ylabel relative run white node attribute node th digit representation problem sample desire prove figure yshift xshift rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle yshift xshift rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle
refinement step accurate approximation probabilistic bring variance sampling parallel monte carlo simulation two possible implement step decide final augment failure krige involve second prove applicability basic reliability example involve mean standard successively influence result carlo reference result agreement monte failure correction also correction increase dimension krige misclassification surrogate krige refinement input instrumental cost induce gradient design search approximate number call number call carlo inspired concern dimension figure longitudinal support fix zero nonlinear elastic surface consider critical load prescribed service hybrid post krige smooth deterministic failure account krige quasi optimal importance algebra meta factor original krige accurate estimate refinement stop krige surrogate build parallel turn application reasonably multiple include reliability author engineering convention national project denote variance sake clarity variance variance also denote coefficient eventually read coefficient range technique variate pdf read note translation fundamental simulation whose key curve easy show eq equivalent slice procedure conditional distribution identify interesting algorithmic improvement pseudo pdfs slice normalize start seed pdf generate uniformly distribute mm mm reliability system prescribe modern engineering resort require run surface expansion krige original substitute cope impossible make substitution krige optimal density function ensure bias estimation meta importance krige active assessment probabilistic dimensional pdf make mathematical estimate simulation failure failure indicator function convenient derive monte nn limit asymptotically estimate decide dramatically failure rare carlo technique intractable world problem expensive black box code frequent event property different alternative force might surrogate amongst surface recent often impossible substitution second reliability taylor expansion mostly hold difficult investigate infer set carlo random attractive fit tail failure probability make monte carlo efficient monte carlo surface probable failure preliminary simulation compute failure estimate monte although much demanding case current practice evaluate probability demand rely hybrid importance krige build adaptively provide krige surrogate build optimal importance failure surrogate compute function krige present quasi introduce adaptive refinement sampling parsimonious describe detail phenomenon interest consideration coherence observation experiment order retrieve large amount relationship input sequel attempt build failure refer define response quantity interest whose spread variance spread source uncertainty model structural reliability family svm krige present make kriging could krige process essence gaussian cast gp correspond basis stationary read define widely class autocorrelation blue point follow gaussian variate moment g read easily surrogate use apply group evaluate krige function author surrogate reliability complete see similar measure krige krige still equal switching shall interpret predictor prescribe negative concept two performance read figure original represent black represent build sign krige prediction state dash fully accurate misclassifie safe accord krige failure domain surely krige I triangle accord center krige probabilistic cumulative center zero whose spread characterize state krige classification propose probabilistic original indicator failure uncertainty sequel matter fact prediction reduce impossible quantify uncertainty motivates approach probabilistic conjunction importance reduction indicator function failure density dominate instrumental density rewrite follow compute instrumental definition nn draw instrumental central quality estimation instrumental instrumental pdf practice probability denominator art build instrumental density quasi instrumental suited problem instance use center standardized inaccurate krige instrumental quantile variate instrumental read illustration instrumental use choose quasi instrumental pdf expression mean failure product augment failure correction factor indicator function krige fully correction identical augment estimator general krige fully probability accounting probability monte carlo sample pdf instrumental pdf central limit theorem normally distribute finally failure read rely reasonably variation read instrumental sample observe constant indeed exact simulation usual multidimensional pdf consist whose asymptotic use technique distribute significance optimality instrumental refine function quasi instrumental counterpart relatively krige approximation g call maximum minimum region sign predict criterion together discuss comparison analytical propose state surface improvement global criterion usually argue reliability region misclassifie propose trade achieve pick well according thus optimization introduction instrumental performance suppose step fill criterion pdfs normalize proceed population criterion technique slice reduce cluster center cluster experimental design krige accuracy refine batch optimal point thus solve
augment sdp extra favorable optimality guarantee greedy branch explore generalize convex approximately adjust guarantee relaxation condition write plus size condition polynomially spherical low scan constant space identify lie candidate search always retrieve component identity identity x optimization since nonnegative matrix decompose present solve along nontrivial rank maximize nonzero loading index loading denote lead term large attain subsequent development spherical lie surface obtain initially follow low rank continue internal maximization rank absolutely behind concept every function absolutely retain expect formation index absolutely sort sort two change point point intersect intersection determine cell construct among candidate check cell retrieve surprisingly exactly intersection count possible combination element cell interval sort point intersection create great value sort region interesting yet check however exploit possible determine index two identify algorithmic intersection pair element intersection sort intersection point value solve problem however ambiguity coordinate resolve visit intersection identical depend point resolve way intersection point combination examine element absolute large ambiguity examine cost main polynomially rank rest constructive proposition identity positive semidefinite maximize constraint begin constructive spherical define set similarly cauchy inequality maximize equivalently rank give element element notice magnitude area retain magnitude expect formation hypercube sort remain sort around change formation region expand sort change determine cell even well set small exhaustive candidate give resemble motivated map set optimal belong number pair comparison magnitude order switching occur v convenience use pair generation di omit v j cells index lead vertex actual sort interior sort may area cell addition show computation lead cell finally associate vertex cell ignore unless determined hypercube hence already examine l guarantee intersection bad correspond cost determine array consequently build recall conclude overall complexity upper principal
straightforward vector produce hyperplane encode inherently quality share carry first reduce symmetric geometrically embed need appear constant clause involve guarantee section formula formula formula moreover clause clause pair formula clause formula assignment satisfie clause clause variable clause variable clause hardness clause appear clause constant clause reduction add global assignment add clause result formula many clause break per impose achieve clause remain connectivity property maximally satisfy assignment would achieve variable clause satisfied assignment clause reduction instance clause construct formula example set clause denote existence construction clause conjunction clause computable clause variable clause indeed theorem completeness fraction clause assignment clause completeness straight assignment clause maximally clause maintain clause reduction change clause clause boundary least assignment contribute least clause satisfied satisfied clause clause bad clause fraction clause assignment theorem question efficient margin try algebraic lead author believe acknowledgment thank optimal solution induce label hyperplane large whose later reason denote give completeness universal unit immediate get guarantee proof lemma spherical origin area estimate correspondingly sphere compute surface area sphere odd low equation fact success occur hyperplane random unit view sphere x vector direction probability sign w occur require semidefinite relaxation simply coordinate sdp achieve sdp gap regardless always yield solution value equally integral question add result answer draw sphere formally coordinate independently deviation separation side linearity cauchy th fact whose entry w rearrange hyperplane entry fact h sufficiently bind close hyperplane change arbitrarily spread sphere normal choose hyperplane trivial bad section claim theorem remark dms paper hyperplane problem machine hyperplane passing maximize separation hyperplane input achieve provide low bound hard margin tight sub separation margin admit present reduction extremely study theory refer thorough reference setup hyperplane maximize subject intuition concept intuition extensive beyond scope ellipsoid time solution hard go beyond scope consider knowledge dealing purely proof possibly constraint xu et coin name maximal indeed behave cluster group assignment produce solve mixed integer use solver encourage xu show well suggest local improve optimality author proof unsupervise side hyperplane appear hard optimize offset claim distant hyperplane yield iterate pair observation reader keep algorithmic claim also begin describe exponential first force look unit sphere vector right hyperplane analysis assume solution unfortunately hardness show allow discard hyperplane separate see discussion ok efficient hyperplane distance optimal singular version prove within preserve get exact optimal sub notation set lie endow inner denote state throughout solution margin refer hyperplane slight abuse usually invariance convenient otherwise base assignment labeling labeling margin margin feasible use pass origin ellipsoid say labeling obtain undirected expansion large absolute adjacency construction infinitely many say enumeration algorithm output hyperplane state feasible classifier vc enumeration label iff one point label degree edge neighbor check separability perform thus labeling exist hyperplane rotation conclude enumeration separable net set hyperplane margin induce enough obtain optimal obtain hyperplane correct labeling optimal labeling I enough hyperplane normal belong deterministic construction net know argument hyperplane dimension attempt margin project dimension preserve apply reduce turn simple pick sphere maximize margin correctness labeling somewhat deferred lemma seem surprising spherical fall spherical however merely induce lemma hyperplane straightforward randomly compute output label hyperplane admit margin solve corollary hardness unless exponential produce optimal approximate one hyperplane fraction point time margin hyperplane whose maximize whose row separate
find moreover possible le recommend separately intercept gives minimize square remark david w robust criterion influential outlier towards perform criterion parsimonious avoid minimize meet challenge small algorithm simulate supplementary robust elastic net high become routine range finance latter array activity help refine disease genetic disease immediate statistical address likelihood fitting inspire relate simultaneously fitting variable generalize elastic nonetheless recover parsimonious attention extend handle outlier datum majority wang discuss penalization model regression response paper hinge loss primarily piecewise hinge illustration apply highlight outlier despite fast computing path investigation surprising outlier outlier influence identify circumstance motivate robust logistic elastic simulated datum regression consist estimate equation extreme alternative approach contribution consider fitting von outlier robust penalize coordinate review logistic regression potentially effect variable penalize loss algorithm fit summary direction adopt follow assume design matrix notation element wise seek predict explain pn expression microarray single nucleotide snp py outlier addition show robust estimator produce panel implication depict penalize logistic regression show path coefficient function penalization add quickly noise variable coefficient relevant covariate combination covariate bottom panel path robust estimator insensitive covariate chance select robust next increase irrelevant add mle second le mass pmf py pmf solution close log minimize impossible minimize unbiased estimate expand summation expectation random draw summation summation mind loss familiar density estimator parametric estimation trading efficiency estimation fact previously divergence mle member divergence parameter explicitly efficient member le represent efficiency le two benefit aforementione robustness median member possess fast solution section seek minimize measure logistic extend sample sensible minimize namely additive constant compactly write divide remarkably loss produce regression close equation intuition le observe value namely predict value far extreme discrepancy free sample predict zero predict tend extreme sigmoid contribute little robustness logistic sigmoid le covariate change accordingly transformation response theory like reader move use line nonparametric solution quadratic minimize minimization easy mm adapt behind mm function instead choose goal mind decrease easy state value real value k always take increase step respect simpler minimize l convex quadratic ii mm material quadratic sharp curvature le supplementary material implication control size solver consequently low speed express kn descent direction compute theorem immediately jj intercept parameter regularization penalty intercept update regularity solve le guarantee follow convergence mm differentiable hand guarantee sufficient convex add ridge meet le net ridge lasso elastic penalty include tend correlate exclude covariate advance lasso could useful perform generate iterate discuss practically regardless solve follow start converge material extension locally experience le issue practice ridge elastic net imply penalize logistic le also group correlate predictor wu coordinate round optimize coordinate hold th round descent th residual threshold nest find section compare follow property first pi f outlier run centroid scenario add ti scenario describe regression le summarize result le mle position outli suffer come spread mle add summary fit supplementary material scenario population pi pi regression le issue address choose amount material tucker optimization scheme choose parameter perform net penalize package robust classifier hybrid machine detail piece linearity path parameterization net supplementary material h table positive scenario heavy contamination le demonstrate sensitivity specificity mle interesting tend drastically close compare three cross validation curve find supplementary first consider regime repository describe consist patient classify belong group normal patient disk patient patient six patient incidence pi slope ss continuous pr pi ss gs disk patient table show correlation attribute disk normal deal expect scenario disk role however mle le show result path mle le similar method show point le examine genome employ study snps current ever ever discovery snps snps rs rs significantly cancer snp mechanic tend correlate rs rs linkage logistic current univariate adjustment take analyst dependency subset restrict miss snp li miss keep retain summarize le rs line dark thick great correlation control path rs behave penalty predictor exclude outlier commonly maximum know attention bias material standard influential cause thresholding mechanism selection predictor response perform method minimize estimate parametric reduce whose coordinate warm logistic regression immediately extend related generalization multinomial straightforward observation otherwise element vector curvature also subroutine perform tensor decomposition common decomposition block alternate lee block minimization regression clear le computationally run le mle occur insight interesting lead predictor algorithm algorithm guarantee result complement characterize speed great importance material supplementary material include detail g criterion estimation variable proof derivation supplement result show along relevant make file datum reason file thank cancer make grant er department david dms national supplementary material curvature strategy quadratic derivative ii surrogate follow observation equality continuity follow derivative argue attain thus ensure existence global minimum global argument eq less sufficient suppose locally notion singleton consist
sample spatial infection report estimate credible period week estimate credible infection less l city statistically different period implication introduction movement observe change incidence movement table amount infection move later none difference statistically plot movement plot incoming figure city movement gp income small community infection reach small introduction infection link distance income mostly distance shorter relative big factor exclude possibility introduction neighbor useful represent phenomenon phenomenon spread disease aspect complicated process necessarily typically inferential traditional approach develop paper address provide flexible inferential fit disease dynamic addition flexibility tractable section employ approach summary construct exist bayesian literature work idea system biology main paper summary normally unknown function utilize model easy summary normally kind consider normal use flexible variance fact utilize transformation statistic process process model function capture complicated relationship parameter multivariate summary uncertainty approach uncertainty fact discrepancy biological account obtain point contribution inferential summary statistical simulation study context pre interesting appear change school enough period separately movement incidence movement identify transmission seem spread infection regardless movement infection might inferential like worth infeasible great around widely expensive realistic likelihood approach relevant characteristic tractable inferential acknowledgement grant foundation edu edu edu probabilistic useful spread infection function expensive may computationally intractable traditional characteristic inspire recent complex motivating focus scientific process model summary via computationally inference result period insight produce poor computation interest disease provide system investigate question biology understand management disease disease epidemic infection account inherent form becoming increasingly allow challenge make furthermore traditional inference may lead poor estimate capture simultaneously address computational challenge inferential dynamic inspire computer computer develop approach disease obtain simulation demonstrate reliable fit motivate infected dynamic receive lot availability rich well issue transmission infection cause persistence population may enter neighbor spatial coupling management infection study disease investigate transmission model demonstrate likelihood expensive minimize likelihood develop infer characterize uncertainty carefully issue arise infer partial allow bayesian uncertainty show challenge problematic reveal example setting reproduce address issue aspect underlie develop simulation choose capture important characteristic dynamic mcmc feature apply investigate scientific change school period structured allow construct amount infect period movement infection seem estimate display infection help transmission generally situation unable scientific forward simulation make approximate abc efficient organized model act motivate inferential challenge describe new alternative traditional method application summarize scientific general disease understand affected model extension infect recover explicit transmission disease division human infect recovered infected individual disease generation city time assumption reflect movement infection movement city take generation length serial transmission positive time epidemic parameter local dynamic henceforth indexing reflect take piece accommodate variability transmission repeat year see move come city computational poisson distribution exploratory fit binomial affect change infection period time age infection balance infect infected unity exploratory fit infection lie full epidemic previously transmission equal assume primarily depend major continuous week attack unity attack infection one could prior dynamic infer infer significantly increase crucially already assume dynamic know call focus description suggest expert inferential tend high pre study capture biological city infection incidence city associate dynamic difficulty integrate high dimensional translate datum pre web algorithm discretization describe simplify assumption detail web note approach require simplify imputation simulate datum location time section datum resemble unconditional likelihood use apparent fix true value ridge value demonstrate ridge move change plot two respectively computational challenge summary statistic simulate parameter gp uncertainty discrepancy approach inference challenge liu begin summary interest proportion let obtain simulation space design word grid element return normal predictive obtain substitute appendix statistic datum connect follow discrepancy discrepancy point discrepancy match model infer consider another parameter note discrepancy reliable fact fit term thought adjustment fact always original use discrepancy informative prior help reduce identifiability inferential give distribution mcmc inferential simulated example pose traditional approach carry describe section provide bayes rely sampling slice continuous grid update analog walk mcmc chain sample core processor process update slice generate length adequate monte summary interest intuition summary informative regard bi make proportion zero course epidemic infection enter neighbor big movement infection case dynamic infection become infection may statistic demand base exploratory analysis conclusion statistic trivial disease link address limited summary important capture statistic bayesian construct non increase summary thereby regard however inferential expensive select ordering summary statistic whether inclusion substantially criteria informative summary technique transform parameter discrepancy term always uniform grid cube equal use permit computationally simulate grid simulation calculated realization highly insensitive model little much resource setting repeat realization cube follow traditional simulate bayes another answer figure confidence region solid gp region contain base gp traditional method traditional bayes set base bayes outline solid shifted truth use gp dash correct contain reproduce estimate gp plot proportion new improve model fit substantially bayes fail capture feature htbp term approach also try wider contain incorrect importance approximate parameter simulate
quadratic decomposition non whiten know row covariance discuss offer proper context whiten naive whitening interpret assume model component important operator discuss finally connection small yield interpretation operator appropriate imply common employ structured field spatial operator structure adjacency mesh connecting represent smoothing embed operator operator smooth variable think weight together structural potential certainly incomplete provide flexibility notice closely spaced series laplacian except boundary second use functional datum move additionally gaussian mat ern class derive function graphical operator possibility diagonal connection employ operator belong class method pca way two us column give difference example show two half smooth svd take another smooth svd outline smooth solution like smoothing half svd half contrast smoothed converse true inverse smoothed operation essentially smooth operation operation share similarity reduction spectral scaling relationship small sum f ensure variable operator relate small concept multi dimensional scaling place proximity provide additional quadratic indicate quadratic operator take unsupervised summary various encode employ interpret orthogonal smooth find reconstruction structural simulation performance operator encode hyperspectral image environmental sensor quadratic operator wide methodology relevant pca pca regularize massive well spatio temporal fmri many spatial location brain inactive extremely automatic smoothing context pca contribution place framework demonstrate general penalty place factor problem approach frame place computing regularize constant penalty bi concave mean versa monotonic converge method penalty scale factor parameter return factor restrict factor avoid employ material prefer former fuse unclear penalty use implicitly factor penalty wish employ penalty smooth adopt matrix speak lagrangian keep mind interpret product induce tractable many alternate maximize step algorithm homogeneous let state one give penalize regression note norm necessarily order function lasso fuse generalize elastic penalty commonly square root similarly minimization algorithm penalty piecewise penalty satisfy avoid problem regularize permit use within algorithm greedy orthogonal sparse pca sparse schmidt update penalty penalty method symmetric norm factor principal interpretability principal factor voxel truly contribute signal hence commonly encourage penalize apply soft simple positive diagonal minimize lasso minimize iterative coordinate coordinate jj row claim obtain find solution compute single warm start iterate lasso fuse norm concave penalty scad mention penalty solve lasso wide range factor sparse framework literature data penalty encourage functional analysis penalty smooth factor factor quadratic fourth penalty minimize penalize group penalize generalized gradient descent note elaborate first solver work solver onto decomposition denote difference row degenerate linear transformation set norm penalize write solution correspondence consider norm op step employ pca quadratic operator necessarily upon structure return smoother yield excellent investigate feature operator respectively receiver roc achieve vary present regularization via accounting spatio bic spatio demonstrate fix r false sparse smooth autoregressive arise matrix arise former size five diagonal element vertex connect vertex variable row covariance column respectively method operator assume unknown structure laplacian replicate dimension perform rank diagonal covariance svd pca functional svd diagonal covariance svd pca functional functional row equally compare compete notice however well un penalize quadratic operator act yield fairly positive positive positive factor row factor sparse decomposition column factor simulate percent false positive method dependency know structure identify substantial demonstrate effectiveness functional classic structure understand spatial autoregressive fix operator pair operator large variance consider neighbor voxel power laplacian five two example study publicly fmri read one yield process voxel dimension reduction unweighted laplacian window ten present three pc pca spatial pc illustrate eight slice pc dot red denote noise correspond experimental period characteristic bold pc incorporate sparsity spatial consistent subject image sentence inferior inferior characteristic stream pathway object identification pc pca advantage reduction variance classical pc initial reduction recognition fmri directly accounting structure variance present incorporate fast massive type factor example regularize powerful present error use measurement grid spaced coordinate environmental spaced structure encode operator close unsupervised covariance behavior measure structured surface particular estimate quadratic via factor separation much future done determine well greatly drive learn optimal quadratic operator class determine fix option explain error always structural actual spatio temporal additionally issue carefully application would develop property structured beyond initial author explore issue well direction closely incorporate way technique frobenius additionally relate pca theorem penalty convergence local need initialization comprise convergence consistency work need broad wide massive common area hyperspectral image sense environmental finance structure recovery exploratory pca regularize pca massive area author reference attention helpful manuscript grateful associate helpful comment suggestion partially support support award svd minimize objective easily verify constraint complete corollary result property equivalent power svd last proposition simplicity problem maximize svd recall amount kk th notice proportion variance result hold analogous kkt necessary follow p c condition put together kkt condition discussion norm occur nan exclude easy satisfy kkt prove solve optimize respect jj j put proposition show algorithm method minimum generalize f respectively op operator linear separable block employ put converge penalize predictor impose orthogonality subsequent factor compute cumulative variance explain k k proportion state result k k k k manner sample assume penalize arrive replace proposition college electrical engineering brain stanford university department statistics stanford massive arise imaging series study technique ignore relationship often poor decomposition svd pca massive dependency generalize relationship decomposition two way penalty sparsity smoothness computational massive set brain reduction keyword sparse singular svd upon form multivariate know poorly enforce sparsity factor consistently recover column relevant dependency two exploratory massive structured set image environmental series pixel voxel hereafter activation location voxel multivariate thus spatial row column apply fmri find spatially contiguous group activation region rarely fmri dependency brain activation interested furthermore component dependency activation remain unclear dependency understand poor examine independently frobenius sum error frobenius norm svd perform poorly strong matrix product loss frobenius permit weight error ii ij good approximation develop decomposition account structural weight element via generalize attention multivariate community gain popularity statistic text publish english review relation mathematical statistical need dimensional flexible mathematical pca account manner computationally analysis data contribution matrix allow dimensional previous review weight applicability imaging engineering section framework generalized section weight quadratic operator discuss interpretation exist methodology quadratic appropriate datum reader intuition aspect future pca pca assumption implication extension singular svd structural massive previously begin write good svd value frobenius generalization weight definite norm nan leave inner take norm svd call notice frobenius motivate respect likelihood frobenius proportional spherical proportional variate norm assume row column covariance check know operator precision account know discuss structured solution state svd decompose let brief regard decomposition similarly eigenvalue decomposition l theorem variate covariance yield identity word value variate row svd pre whiten svd
result noiseless capacity compute shannon size capacity constrain channel local plane shannon capacity plane zero column shannon capacity although remarkably constrain channel yield approximation low capacity order mm extension fig reduce output therefore draw lp lp admissible configuration channel rewrite run fig convergence region belief approximation sequentially basic draw accord proportional involved region belief sample order graph finally kernel vs horizontal dotted show estimate noiseless read size db noisy horizontal dotted noiseless capacity channel average realization information symbol db white symbol snr estimate mutual constraint dimension noiseless estimate finite capacity capacity case constraint additive gaussian information different acknowledgement acknowledge first author thank comment also suggestion presentation physic know free energy introduce basic parameter choice basic plausible since validity array determine slide row array constraint region graph free estimate give operate region beliefs region region energy region region constraint f cf f factor constrained line show result apply capacity shannon capacity bound improve nine digit identical shannon capacity plot versus fig estimate noiseless
rapidly decrease show integral px separately use inequality estimate complete note integration formulate estimate introduce generic context expression also define suppose constant q c h h u r line rely know pseudo x px induction show k since calculation finish distinguish elementary obtain replace bound principal since partial integration twice similarly integration r ix iy complete definition adjoint asymptotic summation pseudo differential operator cf suppose small integer r e application loss generality fix write h follow iv elementary otherwise proof relate note u u satisfy exist constant treat together conclusion dominate expansion proof part r imply r h suppose substitute e assumption uniformly r r ds upper norm r integration integration set h lemma let recall cauchy schwarz see together remark show extend function supremum side motion value brownian two step ff f fp fp f fu fu ff let standard brownian satisfying first weight derivative r equal conclude strictly increase w x fulfil whenever function ng last multiscale rich index suppose ds center standardized decay maximum behave criterion far easy q similarly illustrate multiscale discuss order level value wavelet k remark universit universit unknown important example framework monotonicity simultaneously moderately ill pose fourier density multiscale testing calibration modulus continuity brownian motion theoretical major work x goal distribution estimation deconvolution decade fan selective qualitative study fact rate induce ill band attractive adaptation scale difficulty aim derive simultaneous confidence statement qualitative pseudo differential density assume moment important check convexity concavity monotonicity property moderately ill pose meaning decay polynomial know fan cf pseudo differential combine assumption important error gamma e deconvolution interested increase deconvolution smooth partial ds multiscale follow analytic pair number q negativity decrease tend infinity able simultaneously asymptotic implie allow region prescribe solid kernel low display statement thick line thin figure simulation panel display inspection apparent difficult find monotone display increase decrease plot pick horizontal monotone monotone overlap contradiction statement hold empty plot besides thin uniformity sense confidence statement simultaneously illustrate panel display three reconstruction kernel unimodal kernel surprisingly reconstruction mode completely know want mode plot vary ask mode display density decrease horizontal line monotone increase interval reflect behavior pointwise mode confidence interval merely conclude local local maximum simulation four segment decrease find systematically numerical simulation size statement tool analyze conclusion differential ix l depend class differential fractional identify pf projection subsection implicitly operator monotonicity confidence region shape scale simultaneously generalize need confidence eq h multiscale standard key result distribution free critical compute identify pf multiscale method statement strong statement testing deconvolution cover class ball al paper focus deconvolution deconvolution derive significance view treat sequence tend limit couple multiscale deconvolution multiscale method make attractive choice multiscale interpretation statistic number property mode factor small multiscale differently unlikely detect desirable neither number penalty necessary independent infection interval increase emission detect laplace paper show free approximation shape deconvolution statistical consequence statement performance multiscale numerical expression function stand variation put clear integer norm h shall fairly convergence use structure model observation I uniform q constraint value localize around statement refer distribution free approximation multiscale assumption model process bound generalizing utilize convergence give lemma whenever lebesgue verify sample cf contrary q general cardinality instance make set address carlo start differential operator pseudo fractional integration pseudo differential eq continuous operator paper integration eq define operator identity q differential fractional us mention proof essential integrable formulate symbol however restrict throughout paper axis obtain uniform kind number location root identification specify conclude attain want attain conclude overall sequel empty problem interval analogously whenever interval way solve interval root section collection interval qualitative might interested density random instance transform suppose concavity smooth strictly monotone denote differential therefore confidence problem ball estimate statistically speak observation density iff measurement observe constraint way say operator furthermore arbitrary denote adjoint product formally equality symbol mu formulate symbol number characteristic bound moreover fouri density moderately cf fan real smooth test proceeding consider multiscale notation operator suppose exist side approximate approximate statistic furthermore choice simple ill kf ss r pseudo differential pseudo operator composition compose operator composition heuristic argument theoretical investigate cf section relate restrict confidence result iii confidence link behave coincide density speaking return reject next give h event argue pair large pf h return compare integer old continuous relate confidence statement root location know number trade clear control number root trivial need root separate eventually property simultaneous say zero simplicity root rich neighborhood behave define calculation corollary construction eq length confidence confidence zero therefore become disjoint show number root pick probability observe localization mode log coincide al special density deconvolution comment mode restrictive smoothness contrast rate square strong confidence band comment multiscale theorems calibration multiscale l evy modulus supremum attain uniformly particular attractive construction adaptive multiscale exclude statistic free exclude weak important detect zero outline discussion study deconvolution notice deconvolution inversion operator therefore take deconvolution problem natural variable far thus multiscale statistic particular display repetition concentrate imply boundedness multiscale solid subset complement investigate confidence equal estimate simulation monotone investigate numerically signal average decrease way decrease localization confidence statement instead prominent one good something mean line subset exceed line hold confidence find derivative much achieve statement value come multiscale corresponding belong therefore view superposition statement different qualitative quantitative statement construct band mode whereas qualitative account scale multiscale analyze operator work refine multiscale operator strategy apparent harmonic operator spirit theorem deconvolution model deconvolution viewpoint harmonic multiscale know qualitative construction root require continuous highly impossible statement interval strong band multiscale distribution difficult ill pose include deconvolution become pseudo differential good knowledge formulate space treat subsequent restrict associate limitation curvature handle within allow linearity qualitative integral pseudo fourier handle research acknowledgment support joint research national science partly acknowledge grant comment well associate lead replace eq fairly classical never describe exception brownian bridge gaussian bf f bf coincide brownian uniform define particular empirical brownian universal theorem state supremum behave supremum symmetric hull bridge sequence depend brownian bridge constant refer event cf indicator absolute theorem constant bridge bf x nx let brownian covariance define bf ii standard show surely establish result obviously xt xt h xt q bind application surely almost surely prove c
therefore rule hope good rule decode use regular sentence token length string token vector mt calculate marginal sum possible token generate word string concern computational complexity take sound probability clear next mathematically segmentation show corollary probability structure segmentation unknown give estimation ensemble method method would provide thus serve decode good hypothesis input follow interesting turn distribution certain pr pr hypothesis constant extent method generate segmentation I separate token mean yes lemma source token string translation string number token pair appear prove neither time cover total word let easy picture rest suppose assumption token assumption upper bind assumption proof give also proof appendix token use obtain corollary assumption proof accord small assumption hold pair token similar reasonable conditional ensemble proof monotonic translation length source target side translation try string assign tree node probability input appearance way structure free skip procedure approximate theoretically fact need explanation assumption connect inequality simplification token base give assign oriented parsing size build block tree framework pair represent thus tree probability quite translation justification formalize widely induction nlp cope uncertainty well future field parse field nlp acknowledgment inspire discussion author however belong I appendix lemma proof variable binomial assumption nlp exhaustive advantage translation give exponential work justification nlp view heuristic pattern future language nlp solve induction formalism context dependency substitution etc translation mt learn transfer string tree structure language state aspect show oriented parse bias inconsistent mt almost mt rely method example like parse way heuristic overlap structure train exhaustive cope uncertainty statistical far mt phrase word obvious outperform building need datum valid algorithm investigate empirical large article mathematically sound building show diversity explain many rest article organize formalize show corollary conclude core idea monotonic translation introduce justification monotonic translation monotonic string length monotonic simplified word ignore impact alignment effort incorporation monotonic enough nlp task pair string align word let length string show segmentation token token least one source side
slot policy solve one problem regime trivial tractable index hard chain main bayesian develop treat arm armed bandit problem observation performance problem dynamic spectrum sense secondary select channel maximize reward transmission primary model identical unknown develop channel sense dynamic policy gap obtain model slowly non decrease numerical know reward mab toward unknown treating update observation bellman generalize mab mdp independent fully observable arm activate time arm change offer dependent reward mdp naturally lead programming solution induction incur open present forward arm depend current state optimal reduce exponential arm several researcher come index variant basic classical mab include bandit bandit bandit switch particularly variant classic change optimal play condition find offer check exist analytical optimality regime also approach test constant approximation bandit contribute basic bandit optimal positively channel policy bayesian make identify mab dynamic basic sequential arm associate process performance measure define could obtain know observation know player reward essence thus identify explore sub growth reward first policy characterize single arm parameter markovian logarithmic ucb finite know focus parallel much definition regret regret well sublinear definition imply deviation arbitrarily obtained play know yield pick arm reward arm markov keep activate armed bayesian process priori first describe option markovian say exist finite policy despite finite although player treat multi armed bandit policy sequentially positively stay channel switch visit correspond switch soon channel among visit step channel circular circular circular channel circular circular slot tt odd treat classic armed goal give question arm present next show increase integer ucb policy positive play denote reward constant record non grow dynamic spectrum channel unknown constant relate fact lemma hoeffde trivial chernoff hoeffding allow expectation variable constant eq c na stand generate inequality stand either belief play steady steady throughput belief average policy circular order channel slot order channel j order entry policy channel q theorem eigenvalue generality steady policy different different bind regret c two policy switch constant exposition index possible play ft I equal define play contradict least time similar policy time select time play policy must apply lemma eq difference vice follow readily translate simplified policy however something claim logarithmic policy positively channel whether infinite horizon answer conjecture conjecture offer initial belief initial finite system channel transition different correlation first channel positively transition sequence show simulation quite quickly practically happen great infinity exceed slowly long grow fast though converge quickly also trade bind generally linear ht problem learn treat arm develop spectrum access channel policy strong result bayesian meta state identical arm identify fill finite option structure u edu arm reward chain seek arm
n argument lipschitz derivative cost control cost stability function impose nominal model nominal reason relax oracle robustness learn arbitrarily bad impose nominal subtle maintain oracle polytope polytope strictly large whenever intuitively bad learn nominal reduced oracle update advance scheme denote point input endow feasibility constraint close loop sect control constraint reasoning begin n I n ii n n n mp constraint property n invariance property gx x prove property sect loop system provide b robust trivially varying allow stability system modeling follow prescribe discuss robustness function proceed constraint get maximum continuous continuity generally cf convex reason practice benefit suboptimal hence convex feasibility nonlinear bad behavior formalize stable type controller approximate converge apply approximate bound asymptotically prove key e identically oracle provably bound assume sect feasible exist continuous lyapunov law e satisfy let minimizer unique assume strictly similarly approximate construction proposition imply exist check argument strict convexity satisfied sect certain condition check lyapunov lyapunov check quadratic bound definite lyapunov require linear pa lyapunov converse lyapunov indicate meaning exist second n show begin note positive positive minimize side inequality linear furthermore value equilibrium condition immediately lyapunov situation decrease input answer proof prove theory proceed certain study consequence name reference computer generic continuity boundedness consider take tool identify system natural question example construct oracle secondly law know begin class oracle conclude address law minimizer discuss sufficient ensure minimizer minimizer limiting function often argument trajectory difficult compute generally problem call hill equation type pi set reason situation simplify require give function mx whose intuition correction nominal high nonparametric input without assumption mathematical technique non traditional relate sufficient set lie denote n minimizer constraint converge constraint time relevant result purpose interested theorem let composition ax gx u converge compose dynamic theorem convergence control input trajectory mode activate control ensure key reinforcement nominal sufficiently explore controller ensure se difficult system open se se hold law se regularity correctly follow find se control know u se ergodicity hard verify sample ball center radius inter less consider generic pointwise form decrease make control probability know present prove control nonparametric regression stress meet controller decrease know summarize main law control know n experimentally simulation feature build berkeley air name berkeley energy room use generate energy warm day room able load form adjust name seven office laboratory hybrid control achieve eight energy american home temperature load load adjust action achieve system four steady property kalman correction coefficient make perform successive nonlinear provide robustness amongst display controller overcome phenomenon ground make close ground display mis improve generalization integrate ball nonlinear illustrative purpose engine exhibit type region air engine air engine operate engine possible active model ode describe predict instability flow pressure rise control transfer function n choose take approximate discretization linearization equilibrium r u u linearization unstable pick ensure closed loop choose still linearization small error performance nonlinear lyapunov l use incorporation improve reduce significance setup satisfy feasibility maintain error close loop simulate demonstrate condition check interesting system operating point linear vs perform nonlinear require compute code solver multi toolbox bound deterministic guarantee robustness many type identification tool use require use satisfy robustness simulation improvement translate real amongst future speaking nonparametric work property local globally regularize would acknowledge ram discussion material foundation laboratory agreement nf air office scientific research agreement fa control controller design face reliability practitioner focus however interest growing describe control scheme robustness identification identify rich order optimize design variety tool insight performance reasonable condition framework minimize ensure check input subject furthermore compute face trade stability approximate accurate drive adaptive controller control refine controller design handle input b performance respect c identification tool uncertainty challenge combine purpose deterministic show difficult convergence introduce adaptive predictive base predictive control insight framework tool particular update ensure robust check online difference robustness nonlinear form robustness use learn nominal model focus level define deterministic robustness prove incorporated nonparametric provide convergence know discuss engine estimation filtering note compact transpose distinguish system system state type strictly v lyapunov x mx v difference important transformation tf nr nf pr nf fr qx na nonlinear dynamic modeling lie polytope restrictive determine quantification fitting uncertainty simultaneously perform dynamic measure
design truncation equation fourier specify power spectra differential real net domain order desirable requirement probably frequencie one approach inverting spectrum nonzero coefficient matrix cost learn memory possibility closely approximate spectrum nonzero relate spectra must superposition coherent space central also suggest order take inverse transform differential difference net inspire introduce scale function scale delta function gaussians derivative case forward behaviour generalise elastic net point order cholesky primary train consider orientation origin centroid cell arrange grid stimulus densely arrange map online take long adjust value respective net sequence anneal high value centre axis geometry stop surface note kronecker product critical net axis set order simplify stimulus position order nd net avoid sharp corner hull always intermediate fitness term effort frequency width order periodic net centroid therefore link remain inside hull centroid lie convex hull fitness increase isotropic encourage centroid maxima outside hull net lie hull tendency end net exceed hull end produce corner heavy model would centroids convex hull stimulus preference outside range modulus anneal enough stop training centroid hull exceed keep net inside hull difficult task net investigate map dimension elastic self apply space decompose region diagram diagram obtain forward map figure slice space leave phase diagram contain map reflect contain net tend contain characteristic decrease contour towards orthogonality gradually contour align increase map experimentally match region narrow critical phenomenon map model nearly really phenomenon call present loop smaller attract fast anneal slow phase diagram anneal region contain practically elimination wide another sometimes part cutoff argument forward difference twice square modulus assume net frequency equal map family line slope use logarithmic otherwise evident note slope single correspond thin close curve horizontal geometric relation still differ rigorous explanation phase anneal learn interpret principal along direction freedom annealing become freedom interpretable strength depend small visual dependence connection thing incorporate modify define manually parsimonious define parameter periodic net initial net anneal net bl bl bl c r bl l bl l bl bl bl bl bl bl cholesky map rectangle loop loop elimination net converge training grid net centroid net appear initially continue loop eliminate separate c lb lb lt lb lb lt r phase diagram enough map anneal slow anneal region necessarily contain boundary approximately dotted line region represent curve along start arise curve line mark map curve ex ex ex periodic depend elsewhere ex ex orientation ex ex contour net width orthogonality explanation characteristic periodic although analysis rigorous net effect fitness term arise angle net independent formulation analysis net gradient descent optimisation dynamical analysis type order net periodic self map elastic equilibrium infer consideration eigenfunction operator wave frequency difficult close analog limit space forward difference e transform continuous formation term loop elimination linear relation map require net research elastic define generalise depend define neuron either map centroid related suggest al explicit correspondence model indirect generalise assume binary elastic centroid centroid compare see elastic partly motivated appearance net define limited connection connection proposition far formal represent preference centroid square term add convolution preference play suggest model rather abstract neuron equivalent reproduce orthogonality centre column periodic fitness function type time near zero euler descent fitness net come fitness term elastic trade formation chemical light act however correspondence term operator represent scheme use partial boundary problem regard whether stability whether size tend sparsity coefficient quadratic preferable coefficient desirable memory storage high depend b step solve forward quadratic stability semidefinite whether remain show penalty admit suitable smoothness ill pose nf mathematically concern define second difference constant centroid place centroid annealing system consider latter locality implement spatial work dominate weight wish unbounded cholesky slow operation necessary construct several difficult besides gradient advantage toeplitz toeplitz toeplitz product fast fourier also toeplitz toeplitz extension near structured dimensionality variable density mapping centroid noise dimensionality mode dimensionality reconstruction centroid interpolation centroid traditionally elastic use reduction unseen goal replicate net space perfectly could computer vision graphic density closely som mapping brief som define principled algorithm convergence som reconstruction mapping pass centroid also define prior way derivative net version curvature parametric practically penalty rule momentum seems alternatively define som penalty likely computationally three model suffer curse whose centroid grow manifold locally embed manifold pair term close consider local similar truly construction optimisation method anneal euclidean short elastic try sum distance distance different elastic net lead heuristic opt replace arc locally iterate optimum usually correspondingly generalise likewise complete opt applicable multiple contain city city company home office city city visit length quality splitting start define advance submatrix complex centroid central centroid index get elastic centroid two net disjoint qualitatively justify area dimensionality short approximately proportional depend ff share fix city add centroid extend centroid centroid centroid centroid clearly separate net net perspective globally optimal net matter anneal elastic net e net loop anneal slowly similarity neighbourhood cost space centroid city adjacent equivalent elastic choice accommodate term product pairwise accommodate seems differential general problem difference family elastic family choice spectrum efficiency nonparametric suggest rotation though discrete also particular natural solution work nd equations physics also nd th order case provide constraint operator penalty surprisingly example suggests rigorously consider test sr delta maximally rough increase must make sure square derivative term dominate volume concrete net partly represent partly size complicated difference modulus derivative high discrete triangular height delta net net delta somewhat resolution one centroid small modelling motivation development visual generalise elastic give fitness centroid one sparse cholesky result convolution net approximate differential periodic boundary fourier equivalently however space family repeat order derivative high pass filter filter tend infinity see compete net discrete case qualitatively rich present case rewrite way form alternative subject high context previous elastic net demonstrate equivalence square stimulus look sparse nearest implement connection strength decay distance bring elastic dense connection cholesky annealing explore term phase diagram map interaction fitness term cutoff predict periodic structure strength spectrum map explanation particular geometric thank helpful grant gaussian proper improper semidefinite normalise prior determinant covariance rest take along row improper along convention integral frequency must low impulse delta filter discrete power frequency delta net different piece different besides turn approach integral grid dimensional length w voxel net space delta function real origin coordinate irrelevant wave superposition l concentrate single period compute note cosine term get eq n l expression period valid boundary voxel w fourier difference use forward net physical dimension g piece assume amount modulus forward modulus central decrease know property fourier continuous text proposition section elsewhere integral gr substitution change toeplitz plane wave try wave inner associate mn mn row dft power spectrum matrix note shift correspond invariance row theorem note nn proof frequency differential must identically wave dft proposition obviously matrice tt proposition delta hold delta hold construct divide incorrect gr statement trivially take simplify proposition induction q nm analogously vanish odd function around integral become gr theorem lemma proposition prop conjecture conjecture thm center road dc elastic optimisation fitness term implicitly net operator give generalise elastic net difference derivative map relate prove elastic net generalise elastic net discrete differential area graphic optimisation solution problem subsequently essentially preference square relative success elastic net ease probable attempt beyond specifically complex perhaps approximate cost distance context model whose unknown connection neuron examine relate connectivity suggest consider detail motivation primarily extend model area computer graphic type generalise elastic class generalise elastic third map result net mean original elastic square distance elastic collection centroid dd mm centroid semidefinite prior improper purpose appendix elastic recover possible scale role temperature centroid centroid could shape curvature implicitly centroid centroid semantic typically base scheme statistical training dd posteriori iid ignore independent however anneal alone minimum fitness solution combinatorial problem system latent net iteration index evolution unseen consider hyperparameter validation bayesian g investigate accord criterion differ objective sign multiplication fitness positive latter influence term decrease hereafter however simulate train three cholesky gradient invertible ng training centroid three algorithm cholesky considerably sparse iterate small need elastic net centroid centroid towards centroid confirm simulation short step would obtain related modelling assumption definite since semidefinite depend point fix update guarantee familiar order explicitly compute question ill operation computationally solve g iteration follow compute implement explicitly relaxation trial extra nonzero element quite matrix elastic net iterate equation cholesky find good matrix triangular strictly necessary usually although band structure fill minimum ordering descent certain net net centroid centroid large problem particularly fast anneal term final iteration gauss matrix cholesky generally numerically stage cholesky try cholesky direct gauss iterative try cholesky require cholesky twice high bottleneck net weight take solve cholesky mean cholesky gauss go deeply annealing particularly net centre contrast descent fast anneal considerable cholesky train drastically practically split robustness efficiency cholesky choice allow investigate behaviour range simulation describe practically convenient separate extra copy large case remain define multiply old reduce make take place period fitness elastic mixture associate centroid however non rectangular shape specifically nonzero equivalent implicitly condition type appropriately modify complicated force fitness likewise remove appropriate gradient equation rhs option since operation numerical instability iterative cholesky method involve singular overcome proportion energy become remain centroid modelling approximate primary may patch inactive separate net force net central elastic example centroid periodic elastic net equation nm nm nm nm elastic anneal descent gauss mm definition search incorporate model elastic net concentrate operator ignore dm dm map term fitness represent tensor penalty still restrict secondly note enough symmetric form use lower without generality frobenius specifie centroid net curvature specify centroid one show change centroid line plane etc typically change net via net topology neighbourhood region net represent multiplication periodic b matrix represent generality concentrate particular identical assume invariant consider appropriate arbitrary invariant one q invariance rotation require operator way follow matrix zero choice origin desirable naturally work case positive matrix square matrix sort rotation obtain rotation sign permutation row kk derive transformation since backward difference central elastic sum I element specify boundary simple use modular linear combination periodic since successively toeplitz leave idea elastic net dimension complicate toeplitz analysis simplicity periodic notational remark assume convention central column etc obtain successively ambiguity e assume centroid net dimension coefficient often rather circumstance indexing convention easy later separate tm convolution discrete net ki jk reverse turn basic finite differential difference derivative small curvature consequently element must otherwise would consequence eigenvalue eigenvector centroid centroid rotation fitness invariant newton quadrature operator origin constrain elastic obvious course operator operator contrast readily continuity smoothness geometric accept uniformity term opposite g operator nonsmooth net simple placing parameter mapping net ridge force magnitude represent well net parameterized mapping centroid location coordinate second topology prior spaced essence operator reconstruct net strategy appear repeat along toeplitz slightly delta net modulus net equals construct pass along dimension penalty times modulus eigenvector cosine discrete frequency case bound play describe condition condition net centroid condition differential net zero pattern black sum zero white square analogously term example different net centroid wave wave satisfy condition sample nearly point must wave enough frequency power frequency incur negligible wave wave penalty monotonically net look different truncation error character obvious power spectra happen typically monotonically unimodal view secondary importance net order net different alternatively modulus centroid relevant net plot family question error zero require nonzero order combination order question iterate analyse backward forward difference convolution reflect respective matrix high writing convolution call p p also successively composition derivative associate eigenvalue cosine eigenvector mean convolution one cm p n n repeat irrelevant omit elsewhere backward draw family exactly appear obtain appear except convolution diagram eigenvalue derivative eigenvalue correspond string periodic mode proposition form triangle modulus along otherwise forward curve since nan consistently family practically qualitative explanation behaviour though actual preferred simulation net net idea fitness frequency frequency largest less call cutoff frequency power exceed cutoff predict net expect frequency dominate centre set single decrease gradually decrease fitness energy series minima net way imagine cutoff power horizontal dash line cutoff net centroid centre mass cutoff small cutoff thus behaviour note cutoff proceed energy uniformly decrease cutoff frequency appear develop frequency local one start cutoff high net little confirm annealing understand without curve fitness dominate centroid draw training set fig cutoff cluster differ confirm qualitatively different net cutoff argument effect increase cutoff result cutoff map net cutoff explain plane prefer direction contour approximately circular power cutoff equally result preferred direction fig contour line near become thus occur isotropic result preferred c relatively stress curve cutoff see define central proposition family obtain order divide nm p p frequency low frequency practically fitness generally elastic net result family contain low net part wave value one centroid odd centroid alternate nonzero however structure difference horizontal quadratic pattern map also understand combination consecutive either coefficient overlap term avoid map net show individually fig b visit give discuss differential operator desirable centroid eigenvalue dft delta penalize integration opposite though integral smoothing c c etc net domain look curve look forward central
incorrect however occur neighbor number denote frequency figure frequency neighbor list note neighborhood histogram neighborhood useful determine likelihood give frequent element likely neighbor big neighbor additional ordering form list let symmetry neighboring neighbor neighbor vote receive suppose however identify indeed think average bad fold cross validation procedure see net value rather compute pick empirically vector pick entry node interpret rank elastic neighborhood estimate order weighted neighbor additional correlation neighborhood paper paper ise penalize penalty recovery recall model addition rate confirm maximum edge wise neighborhood produce way node structure lemma example david attract considerable sense biology language protein estimate gaussian ising extend work include penalty numerical introduce leverage introduction sparsity neighborhood graphical simplify let vertex simple concentration underlie neighborhood random neighborhood set importantly variable give nr nr employ pseudo measure compare penalty base enjoy property domain treat variable generalize model reconstruct full edge graph allow al structure gaussian field model neighborhood regime max improve examine build upon work expand scope graphical multinomial discrete lasso drawback moreover highly correlate variable incorporate elastic net retain introduce union estimate idea neighborhood pair node necessarily neighborhood pair design usual neighborhood undirected encode clique basic form comprise interaction node max write normalization function mrf sx inverse covariance correlation node zero neighborhood reveal neighborhood expectation represent mrf distribution describe full distribution form term eq hessian fisher much correlation entry extend expansion describe parameterized indicator represent particular variety relationship despite factorization simplify parameterization denote agreement ise x binary note transformation discrete indicator classical multinomial regression ns l equation selection approach take condition penalty relative term preserve sparsity elastic net original package multinomial library al evaluate mrf distribution enough force pe positive experimentally start increment value sampling computationally expensive long temperature overcome wang augment iff formulation node vice versa value otherwise generate sample update substantially vertex chain explore outcome rapidly wang regression net setup observe rate number neighborhood large bad matter small neighborhood counterpart plot vertex degree even sample scale test star star star connect center add neighbor star maximum number impact fix recover significantly hard density maximum star density dependence degree additionally equally sized plot error rate top edge see plot bit bring error rate bottom time sample clique star edge maximum degree recover exceed figure density increase sample simulation achieve consider group vertice community connect belong community complex group rest application community individual page modular however available achieve rate increase perform regularization produce ht figure elastic mrf ising degree run range trial ise clarity note represent neighborhood neighborhood neighborhood multinomial penalty benefit neighborhood sample value penalty chance penalty allow select exhibit effectively average parameter provide essence figure dimensional curve consistently provide precision recover ise model introduce drop
unique main color colored green solution hyperplane strictly separate red rewrite solution system substitution variable substitution lp common hyperplane separate hyperplane separate random separate remain hyperplane draw leave distribution let subset unchanged permutation equally vary uniform come first vary step experimental x success z axis curve correspond three label success vary vary space rate z axis label set success transition suggest conjecture entry absolutely distribution lp recover also tb conjecture system give solution integer entry sufficient recover lp relaxation minimize exchangeable well geometry knowledge compressive empirically distribution lp recovery succeeds always fail conjecture binary entropy interested grid wish retrieve connectivity three power series hour send customer single phase principle measurement phase determine customer empirical collect time customer imply underlie system reduce problem binary checking difficulty et one uniqueness lp norm lp recover compute generate instance almost study condition entry minimize recover sufficient satisfied recovery albeit lp relaxation definition constraint relaxation lie lp lie polytope face uniqueness tight entry lie polytope
eq follow uniquely identifiable string identifiable already distribution string long uniquely determine extra work employ date binary value stationarity stochastic row resp column translate column summarize whose function rise matrix translate zero probability string equation string collect insight length sum string avoid string infinitely crucial write ideal comment point independent outline string computation confirm example q ideal characterization variety apply markov model due necessary remain whether computation also base early follow determine submatrix setting refer string length necessity definition express plug close let alphabet determine parametrization state generic parametrization hide return routine compute parametrization subsequent note invertible apply work invertible eigenvalue lemma choose b bm x recall decide incorrectly form incorrectly algorithm proposition relation membership thereby p n exclude iteration infer parametrization respective process finally eq e unique column thank algebraic great stay berkeley special present algebraic berkeley author would thank participant thank david private stay berkeley thm notation observation thm thm remark thm thm generic decide infer subset answer process markov process root algebraic statistic algebraic set algebra allow algorithmic elementary statistic concern finite alphabet refer process hide infer representative list contribution exhaustive list estimation practical practical argue explain contribution form extra center around exception cone stationary solution algorithmic date string exercise hide introduce turn attention identification string due hide course value root algebraic draw particular concept variable uniquely determine rise hide answer yes yes stem process density infinite state read algebraic unique course ideal characterization process arbitrary base argument hide relationship note review summarize result turn note correspond note algebraic treat convenient require stationarity characterization binary novel basic definition algebraic serve treat algebraic setting formal definition markov definition statistical algebraic model hence binary markov formulation alphabet string letter concatenation letter stochastic process generate string technical convenience confusion write throughout none exceed string length alphabet polynomial complex affine zero iff imply correspondence irreducible counterpart complex value boolean stress algebraic list auxiliary virtual string short elimination necessary fact algebraic family algebraic statistical call process take value variety string length generate stationary rise unless process stationary often remain unclear difficult determine computation stationarity mention stationarity stationarity geometric implication among stationarity technical among remark practical process stationary evident particular establish gold example speech classification hmms gene certainly early identification linearly model finite markov runtime say iff sum whose parametrization immediate parametrization process process parametrization admits provide condition notation rank integer choice parametrization state tuple emission matrix hide state write refer write hide relaxed follow consider proceed initially observe probability symbol subsequently addition use correspondingly eq technical computation reveal reflect forward backward observable state make obvious hide admit dimensional parametrization definition process process act state rank db b invertible write process parametrization rank admit would algebraic family algebraic model write zero reflect every parametrization parametrization obtain algebraic complex require row sum make eq row imply process analogy hide hide string write variety associate n h reflect state translate invariant hide work one thereby obvious immediately equivalent elementary argument variety insight rise process statement q imply statement equivalent rise straightforward generalization previous let q quasi projective standard applying choose example process exist invertible transform dimension invertible lemma existence transform summary invertible form variety cardinality generic reflect permutation state yield observe stationarity value argument prove identifiability also value identifiability
regularizer asymmetric estimate maximize joint similarity thus strictly psd wherein general zero structure show theoretically ground truth uniform many real application estimator report demonstrate variance report gaussian fix draw th integer run draw th distribution sample ti single average stein also compare randomized fold validate stein validate choose minimax stein low leave minimax stein validate mean q cv comparable span propose minimax datum set axis random draw percent mean risk stein reduce black blue green line draw risk vs single mean estimator average draw percent estimator risk stein overlap identical blue green line h percent vs mean half x axis mean apart average stein task estimator provide gain dominate apart outperform stein estimator performance indicate benefit level early bad opposite minimax optimize cross validation albeit cross green blue dotted summary variance wide margin stein achievable oracle variance calculation pairwise separate separate issue use always minimax eq exclude cross validation performance pairwise well indicate estimate improve th real application use application final student student application project constitute single experiment student treat task never student cross pool student pool class risk average across student dataset percent vs cv denote standard deviation cv low percent stein percent validate version bad counterpart minimax estimate similarity rare impossible student result never bad task gain stein performance surprisingly student score good task estimate wrong draw choose task sale impact business amount customer spend time period customer order customer amount customer amount compare task ground compare fold choice statistically two sided test observation stein statistically outperform estimate counterpart cross non estimator customer task stein minimax minimax cv cv mt kde averages recall task kde assume estimate density query un average average estimate mt kde kde laboratory adversarial open primary create rich region attack security relate risk event goal km market event attribute group seven treat kernel kernel bandwidth minimax also expert school assess seven period similarity force unknown separately kde grid grid leave validation assess kde mt kde kde mt sort seven reciprocal ideally worse kde datum big inferior preference perhaps multiple fashion improve test stein cluster kernel denoise acknowledgment research event dataset helpful united award united states office admit rewrite matrix laplacian notation asymmetric derivative optimal exist prove lt tb tt clear every disk plane real eigenvalue invertible negative clear entry true part eigenvalue already prove wise negative sum third sum zero entry wise derivation similarity minimize limitation q parameter g version replace plug simplify eq q minimax derivation estimator minimize estimator prior minimize risk prior prior find need follow minimax solution minimizer favorable constraint least favorable constraint bad put mass constrain make write clearly weight minimax guess bayes indeed expression risk page tb u risk minimize minimize proof proposition formula write inspection clear strictly general al cs task jointly task result task minimax estimate outperform estimator stein motivate leverage evidence stein stein square mean surprising perhaps often provably nice effective practice particular amount stein true estimate sale error report need apply average section task th task th mean pairwise solution variance problem task iid task key pair generality minimize loss jointly together multi regularizer separate minimization produce normalization estimate empirical dominate formulation regularization set experiment restrict error formulation many generalize trivially rather scalar notational similarity often convert task appropriate choice case minimize estimator background stein learn manifold body stein strategy simultaneously draw stein th dominate stein replace unbiased stein towards regularization shrinkage shrinkage implicit assume stein estimator become q work r decrease degree see number stein multiple vector task large stein give admissible theoretically estimate eigenvalue trace covariance matrix replace effective result stein separate effective maximum stein estimation use effective address explicitly mean special task fit main regression task task regularizer minimize empirical matrix norm row alternate feature closely relate constraint estimation estimate notion task provide task matrix joint joint alternating degenerate simplicity enable similarity general form et regularize least square supervise ny leave laplacian negativity strict positivity investigate list collaborative recommendation task et diffusion test graph svm application task also inspection estimate combination task average combination analyze sample respectively suppose variance iid finite variance without loss fix simplify mse single low separation variance sample approach infinity mean helpful differentiable specify minimize obtain non specify task ideally task minimize derivative effect varied error similarity use use analogous bayesian prior precede design minimize method minimax estimator linear result recall generalize result try risk minimize approach fold tractable freedom considerably try minimize g analytic task estimate solve
hierarchical likelihood identically goal new showing mixture great relevance posterior computation user far specify prior prior fundamental augmentation gamma associate turn upon evy fisher extensively likelihood mixture normal result parsimonious compare involve example random utility logit logistic table odd use logistic model closely extent new ratio logistic analysis chapter issue poisson table paper contingency include focus factor multi cccc center total total present concern logit way begin representation computation illustrate propose jeffreys prior remark generalization defer trial design control behind trial observe denote success control product advance write odd easily analytically traditionally use approximation hasting intractable mixture bivariate logit likelihood lead describe conditionally gamma name closely relate distribution fact moment special mixture inverse gaussians interesting straightforwardly gamma main normal latent v n arise treatment center patient let correspond single success correspond log odd ratio favor independent assume bivariate apply v estimating employ em mode begin pre iteration I estimate converge log odd ratio odd marginally augmentation normal give justify maximize conditionally maximize trivially use essentially weight least interesting comparison method run plug step compute apply clear whereby current obvious hybrid either I converge estimate log odd ratio sample relevant equation far draw gibbs density still incorporate normal wishart improper applying theory wishart figure part sum gamma work make important sampler examine robustness inference generation gamma merely simulate single modern computer far less consuming appear environment rapidly become norm task core available pc initial v draw eq sampling odd assess avoid odd ratio center vertical central credible dot posterior ratio treatment center observe draw posterior indicate wishart describe choose match identity hyperparameter marginal correlation pre specify result figure compare likelihood pool treatment effect center produce support efficacy quite uncertainty incorporate many pose tail tail logistic widely inference west particularly case logit sensitive default informative log lie informative particularly model use right tail approximate agree reasoning lead modification framework augmentation directly match tail extra compare student distributional odd odd definition distributional many logit prominent example boltzmann logistic currently comment affect em algorithm parsimonious updating quite situation arise inference precisely exponentially lead moreover strictly generate ever appeal bayes evy relate brownian motion offer advantage simulate step situation applicable insight gamma random generate inverse gaussian detail relate independent moment namely arise observation rearrange kk exponential density result euler write moment generate moment generate gamma proof result straightforwardly use expression recall write normal conditionally specific turning form arrive result straightforwardly distributional evaluate reduce explore property odd ratio odd odd ratio lead distributional author odd ratio ratio augmentation likelihood recall representation mean gaussians augmentation odd mixture theorem inverse moment generate e gamma also gamma direct avoid calculation simple representation parameter augmentation scheme integral via identity easily exponentially version distribution fall euler use generate require derivative integral consider inference logit suitable constant conditional note multi consider size odd transformation transform gamma gamma distribution normal gaussian usual bayesian calculate define sampler therefore gibbs sampler augmentation conjunction identity map odd consist step extend way default mcmc regression possibility design also penaltie conjugate likelihood interpret moment gamma gamma log z I lead exact sampler repeat issue use fisher define jeffreys jeffreys logit tail prior west make penalty matrix z pseudo avoid influential row motivation place exchangeable logit help lead prior likelihood match exactly pseudo logit code design b pre stack iterate step update agree drop covariate imply posterior use multiple odd ratio exchangeable logit parameter meta number odd ratio family functional form likelihood penalty vector odd ratio give ii parameter hyper active total set variable moment normal stack covariance combine conditionally posterior style algorithm iterate estimate em inverse wishart hyperparameter estimate repeat point emphasis weight though weight appear second new consider place categorical multi usual lead shrinkage sometimes require propose alternative log odd norm version hierarchical van scheme conditional augmentation model study normal wishart tr likelihood b family natural b observe wishart add
derivative upon loss term view market lot trading strategy loss even view coin flip chance stay coin section present help course rather practice compute mathematically result product skip large direct exposure expect side replace add explain give example bring market agree everything apart variance market kind plausible mathematically reader immediately unbounde bend justify leverage see give propagation leverage belief leverage avoid substitute realize payoff swap swap theoretically well familiar scenario succeed turn express extreme fall finite cover look exposure outside range practical traditional calibrate sophisticated express consider designing course question finally motivation process vice bank thank former bank present author herein author necessarily view email com bayesian probably logical optimizing turn separate presentation summary happen remove case capital eqs depend whether thing normalization end return probability one payoff careful numerical asymptotic remove capital risk grow e mathematically odd ref risk free paradigm denote normalization coincide capital risk free ordinary accept research market really exist market available belief payoff drop leave proceed sense market mm mm around traditional modeling extend propose quantitative framework creating meet relatively simple last world financial development quantitative method remarkably little innovation derivative business compare product recognize payoff financial challenge solution quantitative innovation serious quantitative innovation purpose introduce elementary argument rational sake clarity generality principle back trading derivative demand solution reason natural fair serious reason belief much economic treat fact belief convert payoff imply belief payoff structure check behind position agree obvious rational view physical ability product result depth one raw consider represent market stock everything know summarize depend whether knowledge powerful probability modern knowledge mechanic certainly describe finance belief material impact reality heart introduce distinct imply realize role outline market represent familiar derive distribution price entirely realize reality market predict realize realize variance auxiliary definition surprising language intuition market might trade indeed imagine market variable put opinion separate express view probability question redundant growth objective support subsequent section general present imagine expert market confident high return formal near return people would risk analysis simple limit imagine absolutely certain lie distribution belief fall within pay variation payment view digital call spread believe future outcome summarize probability illustrate nontrivial question maximize utility formulate allocate mathematical connect think win well understand understand return proportion choose subject constraint maximize expect return realize simplicity start odd market imply th spread rewrite derivation allocate oppose logical make mathematic appendix remove logic reader speak use price market probabilistic market return probabilistic interpretation odd people odd true even bid offer imply imply pricing exist belief probability payoff bayes tell update research regardless actual q spread end consider underlie market payoff run imply rearrange eq fundamental payoff rational market market logical market perform assume bayes theorem explicit posterior leave recognize case logic equally growth interesting sake brevity focus couple practical paper shape growth eqs payoff function payoff match importantly payoff even actual growth section payoff expect rational imagine justify view must adequate nothing pure justify relatively growth imagine variable market imply still agree illustrate growth mention choose payoff close follow risk optimal payoff fall analogue introduce happen ignore beyond answer boundary realize position never intend matter many scenario potential side relevant tuned boundary product really example ensure market naturally market classic payoff current pressure simplicity imply european digital option belief must far consensus justify leave reader become
qx real risk ii convenience remark map objective depend mcp risk thus concept simplify selector mathematical finance way extend therein definition application subtle scope similar implicitly depend merely risk maps markovian since mcp assumption homogeneity see policy construct replace expectation determine know introduce correspondence preference convexity concavity example economic represent uncertainty outcome first risk preference minimize preferred intuitively convex categorization objective maximize categorization convexity suggest see confirm reasonable preference situation safe state policy conservative high uncertain even apply within mixed preference risk map complementary obviously risk since real value measurable map everywhere power mixed preference map exist map property exist several map economic mathematical finance literature merely space bound restrict definition mostly mathematical finance verify coherent neutral name also field optimal sensitive control preference everywhere convex everywhere everywhere equivalent besides tradeoff expansion e minimize avoid contrary variance prefer intuitively coincide categorization concavity transition exactly probability contain kernel cost adapt apparent verify intuition scenario consider dynamic state also notable regularity unbounded deviation whole chain preference map mean variance tradeoff idea generalize measure analogously additive additive integral homogeneous everywhere behavioral economic interpret behavior risk integral model analogous classical eq discount follow control sensitive markov rest convenience objective notation solve cf discount rest shall operator simply accordingly tv define selector action deterministic fx cx assumption selector economic discount reflect valuable outcome effect mathematical property multiply widely economic finance discount map homogeneous optimal optimize well define discount multiply discount immediate discount see homogeneous equivalent discount homogeneous merely discount indeed discount objective merely risk map convex merely contrary later objective rather define show limit exist nonnegative constant w x obtain proposition f iterate proposition assumption v selector thus proposition u f banach discount show selector hence due let arbitrary f ii exist extend mcp framework prove existence measurable ii x aa aa become condition lyapunov widely reference therein optimization unbounded deterministic comparison map study risk mcp literature map countable therein discuss sup merely risk case follow connect note value change hence set obtain weak condition state existence sufficient sup condition kb assumption condition cover assumption average fu w eq xu xu yu repeat switch xu yu optimal selector selector ii jx jx lemma iterating apply mcp yet white value measurable transition kernel mcp dy ba weight indeed next k hand mcp hold proposition hold acknowledgment careful constructive suggestion des tu author b author support remark measure risk context know risk measure finance behavioral economic weight infinite horizon sensitive discount solve optimization discount propose conventional lyapunov generalize chain existence solution bellman risk stability lyapunov stability control total core mcp framework description mechanism switching action immediate action however policy mcp usually incorporate replace immediate utility whereas require sophisticated commonly cause environment employ probability finance decade therein temporal horizon infinite sensitive apply coherent base suppose rational limit make risk economic measure one make overcome limitation mention behavioral economic maintain existence infinite horizon albeit map stage discount depict discount minimize objective select two objective notice discount objective objective risk neutral expectation similar obtain treatment context horizon risk consider finance behavioral economic space prove type objective discount optimize discount discount conventional coherent measure generalize lyapunov literature markov ensure existence solution organize follow context borel generalize finance behavioral economics markovian temporal call accordingly risk control parameter subsection construct simple one discussion example subsection infinite horizon sensitive objective discount satisfied introduce framework control follow concept state borel space xy markov ax x component borel space feasible action x k ac r random capital variable letter e g denote choose markov rx x say ii resp coherent resp concave slight abuse terminology maintain monotonicity translation invariance risk map investigate risk module fx f gx l map let real value iii sublinear r v consequence remain prove iii r measure counterpart construction real map I sublinear ii coherent rv l xu v lx xu xu xu value map module axiom risk upper apply control growth iteration specify consider sign u norm rv w w due homogeneity key role investigate originally study ergodicity lemma reader v reverse give vx vx q risk important role study property mainly trick real value measure whenever r pp remark probability ergodicity state generalize map subgradient
must find solution estimate counterpart kolmogorov nonparametric record arise size obtain statistic order record perform poorly suggest basic statistic distance region evidence record arise simplify w ny df df n df df distribution record come ml distribution function equation depend desire result calculate record nan hypothesis exceed present simulate critical provide record order level early reduce alternative hypothesis leave open testing obtain level estimation restriction eq degree freedom sample go degree minute company al follow al record complete mle assume model calculate table let approach accept testing statistic support et give basis example air conditioning time ccccc approximate therefore support assumption simulate record arise ccccc mle assume kolmogorov goodness fit test record goodness suggest record first accept procedure arise exponential accept exponentially reject reject parametric result arise sequence sample obtain sequentially record sample induce statistic power conduct proposition ac ir department statistics school sciences box reduce cost running check inference chapter p record testing case summarie substantial determine within consequently check kolmogorov von goodness fit record data mm von exponential goodness reliability primarily fail course record continue service fail case long long long situation terminate failure item test observation item fail obtain call censor variety censor example censor encounter data shape exponential appear frequent life short portion parent limit parametric mind exponential record obtain estimator point inference area goodness fit base record publish direction check inherent bound variation consequently check motivated checking record record
include trial track dynamic therein mab receive provide tradeoff exploitation holding potential future arm address bandit achievable sublinear play arm propose fine ideal play reward know achieve reward sublinear regret reward arbitrarily great logarithmic achieve regret growth reward distribution simple policy know distribution bound assume support develop two calculate past play arm tradeoff exploitation reflect arm intuitive logarithmic need index confidence sufficient ensure develop mab deterministic sequencing exploitation classic separate objective exploration exploration sequence exploitation sequence player arm fashion play properly mean exploitation reflect see sequence exploration spend bad nevertheless need ensure ensure exploitation caused incorrectly identify exploration reward tail logarithmic use sequence cardinality heavy tailed reward achieve reward th moment tail classic policy ml approach knowledge reward classic support range apply know knowledge difference arm demand distribution achieve regret order sublinear heavy tail high classic ml adjust hardness term observation variation decentralize mab player observation mab often arise minimum spanning dominate weight decentralize player observation player affect player arm necessarily refer player whether involve receive reflect immediate communication multiple unknown successfully channel search collect target location agent way player agent deterministic however exploitation carry reliable ideal centralized scheduling grow mab arbitrary rank necessarily solely markov model combinatorial arm involve study extend classic policy mab setting ucb policy order heavy reward basic ucb estimator chernoff hoeffding result however high observation differently achieve heavy tailed mab tail heavy tailed also offer variation discuss option sublinear heavy sublinear knowledge reward decentralize player regardless observe reward ability system logarithmic classic mab division fair construct decentralize mab address set reward liu general interference light tailed reward arm address I propose decentralized regret reward focus light bandit play arm draw player observation arm permutation policy sublinear average reward performance tail heavy tail divided exploitation player play exploitation play choose past sample sublinear htbp notation denote include sample tradeoff exploration exploitation balanced exploration cardinality exploration set minimum reward loss exploitation order cardinality reward light tailed tail gaussian light tailed sub gaussian extended chernoff hoeffding mean subsection truncate regret tail carefully cardinality sequence variation regret memory significantly ucb store sample truncate time instant idea satisfy truncate lemma result include exploitation play truncate arm policy argument substitute consider r logarithmic regret certain address certain model cardinality exploration sequence calculate mean extend variation include objective decentralize incomplete mab dynamic arm arm objective arise multiple player constraint cost arm ml example ucb arm large tend due combine bind arm complete arm rank ucb arm however directly extend rank sequence consequence general simply choose arm specifically assume define regret cost choose theorem sample mean exploitation theorems key play arm exploration sample non small ensure properly alternative incur whenever play similarly theorem satisfy player arm large sample truncate exploitation sequence relaxed need due distinguish arm specify select rank vary objective extension class mab incomplete arm arm arm time player involve focus good cause take unknown dependency solely exchange player involve equivalently receive reflect arm local arm policy history concatenation total reward compare centralized player mean play policy similar sublinear sum reward reliable obtains solely utilize decentralize construct sequence player play fashion eliminate player calculate either fair sharing across player randomness reward limit carefully learn fair sharing player need address fair sharing round fair share sharing achieve average reward player exploration exploitation decentralize policy regret decentralize policy determine good player happen least incorrectly arm consequence analyze exploitation consider proof classical mab formulation arm one arm span dominate weight path dependency arm ignore exist naive often grow exponentially combinatorial mab polynomially exponentially maintain properly detailed rather mab formulation assume model dynamic practical scheduling continue evolve player continue evolve grow target continue mab arm continue evolve change markovian transition arbitrary unknown extend mab centralize equivalently decentralized distribute player logarithmic call weak detailed derivation address fundamental tradeoff exploitation mab separate objective clearly handle furthermore exploitation variation decentralize player incomplete
interaction participant define framework occurrence direct count change arise occurrence event kind occur specified generation implement specify generate actor termination condition semantic state responsible change termination achieve closure event fulfil consequence new state event therefore constitute trace representation process formal translate equivalent knowledge language follow convention calculus action language component atom atom atom say support create ab ic atom intuitively atom know body call head constraint indicate satisfy body program theory conjunction semantic set assignment atom program minimal consistent solution mapping part model component framework independent component deal generation predicate responsible single observe atom identify describe time establish state event event event occurrence fact translate expression translation formal augment specify length trace interested occurrence event specify incomplete trace complete trace set matching trace detail answer linear model course complete trace detail inductive programming concern prior observation contain fact trivial accurately language head write schema indicate output example bias compatible mode iff schema head replace replace atom implicit body rule hypothesis head body well form rule compatible tuple set property mode inductive approach incremental system support synthesis rule case theory monotonic inductive logic programming exist minimal tr system bias similar define transform program differently different transformation task call program program theory theory task distance ii let main availability system system intend behaviour incorrect require trace specify event denote expect output case relate instance avoid effect trace time expect body body head rectangle draw minimum height cm cm height body minimum width height cm body minimum cm body cm right cm translation must split form trace case framework tuple default static relation mode schema crucial increase computation conversely choice problematic iterative representation describe language perform event indicate answer acceptable conceptually e use satisfied set step ultimately choose rich enough demonstrate description agent find digital object block large entity file agent copy copy agent restriction request share generate agent terminate sequence event generate trace share initial case propose addition leave rule yet intend additional request set feedback refine specifie occur answer improve set include rule body combine change original specification change since include leave intel gb ram b correct add variable intend learn case detail describe integrated program reasoning execute derive available constraint construct provide perform transformation describe conceptual step reader phase example meta hypothesis rule exception case body add head body learn mode newly predicate pre try I mode execute describe phase generate informally exception rule condition body condition plus exception empty body exception condition pre phase syntactic transformation preserve solver use monotonic transformation recently particularly suited reason default observational causal dependency monotonic hypothesis essential none exist mention learn within semantic permit provide solver contain ultimately section transform logic programming notation obtain replace type replace omit top head definition predicate true whenever ib ib atom label argument argument use third limitation completeness true translate inductive match requirement case specification requirement converge course case practice system accurate solution case input instant block b b try try exception rule level lr level lr l b sec motivation behind upon correct incorrect behaviour practical experience able verification seem broadly framework al system check case logical capture completeness modality capture new assimilation time agent framework form indicate insufficient participant unable frequently course agent prefer evolution framework either put forward management whether norm purpose develop additionally explicit likewise actually choose purpose presentation line much discussion address effectively mechanism something yet software engineering undesirable behaviour background specification perspective logic program support tailor particular design requirement author propose complete differently employ stem framework high nevertheless human identify specification framework process formal program achieve mean inductive working representation inform instance behaviour actual coincide propose modification correct trace process foundation properly connect system aim criterion suggestion provide currently investigate select likely minimal e
ij equation unit equally iteratively proceed maximum ml mle hessian straightforward depend easy hill involve estimate intensity quantile estimate term mcmc approximation value provide htp estimate cc dyadic intensity obtain glm cycles jk ki ji grey bar place feature demonstrate efficacy real united directional flows united number people state datum allow consider require transformation away continuous parameter flexibility function thick tail network independent specification glm previous suggest population significant dyadic change cauchy glm restriction provide degree freedom significant statistically effect cluster effect flow offset find leave warm substantial increase population year evidence away state cauchy far depict level glm theoretically would exhibit anti anti spike place likely anti reciprocal modeling framework greatly scope technology expect tool span science across network production knowledge relational phenomenon ability structural network graph limit introduce exponential capable network greatly expand analyze examine network vary binary unbounded important graph capture generative yet limitation network binary tie exclude gene various transaction network inference thus subset specify broad gibbs sampler strength limitation apparent specification adjacency direct parameter network permutation give much contain statistic capture covariate challenge model apparent specification come specify evaluate represent proper constant distribution support infinite every ij specification capture sum product subgraph particularly unit reciprocal process likely value vertex star edge transform support multivariate transformation xy j derivative diagonal way inverse parameterized covariate specify elegant feature distribute case gaussian covariate reduce allow ratio upon interpret dependence modeling via discrete edge interpret
partly analytical etc yield valuable insight significance statement along associated great non may limit limit purpose allow derive detection would n possible bin may varied constitute trial statistic would hypothesis include composite hypothesis framework consider maximization likelihood give analytically amplitude detection series maximize also maximum statistic hypothesis independently variable cumulative cdf independent essentially signal independently distribute random cdf eq cdf parameter amplitude pd phase likelihood show particular posterior consider amplitude amplitude dominate bin main kind integration simpler comparable illustrative case instead amplitude pa observable since amplitude prior simply frequentist limit particular since also upper event happen quantile background probability large amplitude already upper imply fact intend frequentist bound suppose fall maximization integration parameter primary origin maximization behaviour affect little crucial amplitude contain limit behave ratio snr apparent amplitude amplitude frequentist confidence amplitude value simply quantile true amplitude chance frequentist limit happen zero trial elsewhere snr differently constraint amplitude actually conservative realistic wave much something improper frequentist limit usually via numerical bootstrapping frequentist desire would probably monte carlo integral get marginal amplitude quantile frequentist snr translate constraint snr amplitude present amplitude consideration parameter generally frequentist suggest bayesian complicated frequency restrict fourier sec construction may case yield detect frequentist require specification alarm connect also commonly upper limit
account x z substituting yield condition corrupt group maximum optimum procedure outline overlap need lasso derive overlap group group draw display function child scale wavelet length haar wavelet decompose image bind recover compressed sense pay group overlap tight subgaussian penalty result toy agree acknowledgement comment correctness proposition definition plus em minus width compressive signal measurement extremely exhibit pattern pattern predefine group I measurement reduce measurement need universal group group hold overlap configuration many field recover high relatively possible signal indicate one measurement exactly recover signal knowledge sparsity arrange pathway highly correlate coefficient model belong tree child property spectrum display e pathway tree knowledge help recover theoretic variety ensemble author need lie union version recover derive iid need signal sparsity analyze emphasize assume gaussian subgaussian constant gaussian case highlight main keep nature short generic group variable overlap partition ambient distinct theoretical asymptotic group similarly condition lasso author derive lasso author derive group provide group vector complement distinction make overlap complement union union set complexity compressive framework measurement gaussian focus contain restrict singular operator measurement first grow coefficient close per consideration g etc disjoint remarkably bound regard somewhat group challenge support use group sparsity loose upper group overlap specific case make tight derive overlap group derive take group require recovery correspond rest signal sparse multidimensional coefficient set I lie assume subscript convex hull wish solve paper recovery signal atomic govern formalize notion atomic group signal block define eq atomic simple representation hence look minimize atomic subject equation q standard norm atomic aware correspond atomic lasso norm group atomic group substitute give atomic overlapping case norm respect tangent cone guarantee recovery main recovery unit sphere width width certain specialize mean gaussian program optimum provide empty cone gaussian euclidean jensen obtain tangent cone cone versa thus cone atomic sphere cone quantity disjoint overlap recover group iid lemma variable freedom ii monotonicity merely optimize application particular magnitude duality suffice g follow random construct bind distance support normal cone
insight connection multiple mcmc monte carlo integral mcmc markov stationary metropolis hasting mh mh mode try chain independent identically I enable sampler make jump portion molecular chapter chapter candidate pdf accord directly target fix scheme pdfs produce identically candidate extension technique analytic weight acceptance probability find two correlated formulate rule detailed balance condition sampler exploration improve reduce proposal pdfs apply introduce procedure candidate generate early construct improve follow recall metropolis novel rigorous novel detailed balance numerical advantage relationship classical mh draw movement approach correlate constant correlate different distribution eq brevity notation vector choose pdf sequentially since z rewrite consist pdf calculate normalize accord step kernel balance need weight j draw weight normalize accord remain set repeat step weight weight criterion improve computational target possibility choice follow pdf remark recently clearly great flexibility construction weight statistical cost study need unclear classical propose eqs different acceptance probability reason eqs eqs step coincide reference different fix variable recall q multiply member notation write repeat development detailed balance pdf show generic pdf draw technique pdfs step weight variable pdf kind weight point pdfs weight compare run multi point candidate calculate acceptance movement compare use use statistically long moreover observe candidate become step j analytic weight coincide depict solid normalize figure average weight use depict dash dotted triangle rate result always correlation htb solid normalize histogram b acceptance try dot line triangle solid line circle estimate scheme correlate function analytic arbitrarily balance draw approach novel flexible correlate different instance procedure pdfs different pdfs tune proposal candidate successive proposal pdfs improve technique weight unlike point metropolis weight improve computational avoid check existence selection broader observe depend proposal important independently pdf use namely proposal separately theoretical study good
dc u r copula popular page gamma rule respectively score associate dc z estimator dimension parameter function system version idea inversion solve obtain previous inversion estimator parameter iterate inversion parameter system four omit estimator consider straight asymptotic normality fulfil dc r u u u dc u tend solution converge proof correspond assumption verify copula satisfy q u dc u dc u monotonicity dc u dc u q assumption natural parametric copula second check copula family k u dl dl u dc assumption iterate readily obviously integrable denote matrix view q family give us context evaluate estimator simulation study iterate evaluation bias th outline repeat different size repeat copula kk select parameter copula choice assign moderate dependence select copula specify reach choose true moderate dependence couple numerical corresponding summarize proceed follow copulas estimator software simulation reasonable result notably strong dependence family size reasonable family method look subsection family respectively size inversion md inversion md criterion clear table well far concerned hand case inversion md rmse rmse become reasonable observed md estimate hour notably execution term minute md system equation study inversion contaminate copula mixture refer proceed contamination model copula inversion md estimator parameter computing bias summarize contamination rmse inversion well bias rmse sensitive indeed contamination equal contamination md sensitive among md formula bivariate copula introduce moreover simulation minimum md focusing bias estimation perform reasonably rmse worth quite md one author grateful improve mm statement mm bivariate bivariate asymptotic normality carry compare rank perform well bias concern root sensitive outlier cite copula secondary g copula dependence multivariate copulas author copula construct copulas area simplicity restrict two df copula copula df j u g generalize quantile arise topic copula include maximum margin present method front produce result univariate interpretation possess feature moment extension heavy tail mention method relative sensitivity propose copula compare approximate apply multivariate copula univariate bivariate copula example performance give first begin classical moment yy statistic moment analogy moment skewness respectively functional representation quantile p shift polynomial sequel transformation orthogonality term summary fit brief moment water science especially moment point relationship definition moment show wide moment redundant moment drop moment suffice statistical definition moment medical application see pearson exceed g type iterate less equal iterate maximal family minimal iterate discuss give copula iterate iterate equal bivariate tool copula moment iterate copula three bivariate iterate copula frank table twice differentiable generator example generator one parameter frank instance copula continuous concave copula generator generator c generator page bivariate copula moment give formula bivariate moment obtain corresponding system bivariate copula moment counterpart see semi parametric
grow linearly increase pattern also look cell relationship strategy naive develop maximize usage locality demonstrate benefit obtain naive partition strategy partition schema schema compare less intermediate reduce intermediate usage locality surprising state capacity software experiment graph speedup rate low speedup law machine reason practical upper fashion conduct experiment check pointing balance speedup low job large whole rest resource correctness implementation parallel implementation setting text range finally wikipedia corpus page wikipedia english website comparison test default optimization separable implementation accuracy besides linear promise enjoy soft experiment train subset instance start kernel multiplicative table major suffer svm attractive also grow example acceptable acceptable adopt scale interesting column dropping cause terminate guess believe heart binary cm p training kernel full capability scale factorization symmetric performance two consider multiplication interest also sparsity pattern find list dominate reason large partition cost reason dramatically drop accordingly general fit computation although computational current may local machine environment cm first dataset adjacent university website web edu corpus adopt calculation sort descent plot notice since small edu reduce logarithm verify see fit side remain suggest converge evaluate execute wikipedia within hour provide plotted figure certain notice large shift regard interpretation leave chart list sort although wikipedia release popularity rank measure life experience list implementation also entry drop list top sorted mark c wikipedia entry united united english negative successfully paradigm scale problem extract discuss concern parallel setting configuration schema sparsity various scale main distance multiplicative update individually experiment successfully multiplication encourage speedup reduce schema discover crucial speedup performance show speedup computation relate svm achieve comparable matrix wikipedia entry hour encourage work suggest sophisticated involve instant communication online version development collection algorithms time fortunately prototype design version experimental widely reveal potential may far improve schema discuss prototype http advanced centre http www uk rgb song liu multiplicative suitable paradigm implement version large scale multiplication characteristic focus fundamental algorithm promise speedup multiplication design considerably efficient maintain core produce encouraging programming keyword multiplicative deal algorithm require learning paradigm remarkable representation question whether paradigm fashion wide major similarity dominant position attempt variety algorithm available paradigm solution several neighbor early stage good report individual implementation graphic unit locality hash lsh google news broad effort establish generic solve et algorithm introduce inspire realize implementation greatly model interest et propose methodology web non multiplicative lee two multiplicative machine include negative factorization support machine svm multiplicative paradigm organized follow setting three adopt extract common parallelization conclusion summarize implement machine engine google service page repository news google suggestion direct evidence google google number idea effort novel way factorization series multiplication multiplication strategy stage balance maximize parallelization huge permutation resource handle extremely acceptable plan algorithm et investigate processing pattern take lead iterative advantage limitation immediately research fashion illustrate community query property theory summation form fit sufficient query aggregate core responsible examine study generally notable exception illustrate graphic processor fail pc minimal may medium sized iterative overhead cluster implementation fail handle programming use report typical reconstruct view nmf divide observation nmf nmf lee soft k I uses please refer iy recursive quickly get single represent eigenvector chain effective direct markov chain principal eigenvector eigenvalue stationary method employ multiplication generic extract dot involve dot product vector multiplication require multiplication calculate measure distance handle euclidean distance sparsity communication stage e piece intermediate group computational call locality maximize multiply summation partitioning determine piece go locate input naive p however cause severe hash hash parameter uniformity unfortunately locality violate three parameter secondary stage factor affect stage aggregated note schema matrix result summation dense partition large profile matrix partition intermediate parallel utilize aggregate partial I implementation straightforward machine implementation computation communication random piece long row cache regard small among allow load two general essence nmf decompose multiplication nmf similarity multiplication consider unique profile utilize four multiplication generate dense sensible schema across short plan largely partition similarly schema list matrix multiply generate splitting across prefer column unity feasible conduct row entire row go calculate file format retrieve column computing read single sequentially store final output actually computation kernel matrix represent kernel calculate schema dimension account challenge storage result intermediate grow partition also prefer conduct see
tail covariate interested year affect univariate location tail trend mix estimation parametric parametric know parametric introduce fitting use author focus estimator result extend estimate propose regular twice continuously similarly author obtain base move use value weight log select advantage regularity covariate multidimensional require define normality highlight discuss present hill address minimum practical arise illustration associate metric conditional eq fix q precisely want focus design non random us ball radius tend go response belong design concentrate go belong z n tt order estimator situation covariate latter increase negative function tail discuss universal asymptotic normality weight asymptotic normality sequel fix first slowly vary function say vary detailed cumulative constant satisfying uniform condition regularity distribution slowly normality tail normality hill close call index estimator necessarily introduce approximation continuous satisfying extreme value analysis simplify rescale rewrite main establishes exponential rescaled type pareto variable covariate approximation hill sum exponential random refer near use study main normality define multiplicative weighting similarly proportional multiplicative force index impose negligible standard estimator evaluation theorem particular slowly hold arbitrarily slowly condition lemma problem usefulness two extend classical estimator covariate second choice give overcome restrictive first introduce tw tw w ts tt corollary estimator knowledge estimation order well estimator plug subsection arbitrary choose estimator replace second direct w sign misspecification figure extreme second consequence misspecification efficiency unconditional case view easily remainder paragraph devote partition define concern variance plane reason represent propose cubic centre united available http www uk daily build year distance hill estimator eq heuristic functional idea pair estimate approximately select smoothing contains rescale validate distance goodness locate interval tail day month extreme flow rest sake sequel sufficient constant verify well slowly account exist x entail conclusion sufficient integrable origin since absolutely monotone verify origin covariate rectangular ball lattice suppose approximated jt jt jt assume uv I z tt uv enough thus tt uv see theorem q I tf h introduce theorem sufficient integrable focus finite conclude theorem n k w w k conclusion theorem assume thus hold straightforwardly choose arbitrarily verify conclusion normality
ratio fall balanced use require sequence balanced set mix normalize nest sampling set self need ahead accord disadvantage property deterministic integration upon nested fall ahead present combine include nest posterior ise present discussion ad ask operate set away center take move poisson allow approximation show introduce describe balanced simultaneously member family set discuss area exploration four bb bb aa describe way describe py slowly fall attention condition vast mcmc critical sample reader reference therein turning ability execute line well demand beyond modify simulating draw key let identically exponential mean function call let reasoning py dominate observe natural uniform mean say run homogeneous describe number point run furthermore point process run use ise physics inverse call way either hx add auxiliary lebesgue center mind work yx hx go z graph add change continuously find normalize constant known estimate nb z connection finish job suppose time call mean give approximately interval build similarly easy find give prior lastly output acceptance follow bind tail special start difficulty refine estimate final estimate within ii phase create estimate success distance failure right hand chance phase expect need complete running collecting run poisson call ei simply sum iid suppose simultaneously doubly intractable arise bayesian process form independent generalize necessary far chernoff markov probability upper bounded poisson taylor series expansion simplify bind yield yield tail yield must term let prove taylor consider draw condition ise wherein adjacent similar eq integral hand run straightforward perspective come approximation valid technique answer present method lattice return suffice right node node mean circumstance instance fall give constant make solely
uk college example corollary proposition brain human claim make static multiscale modular letter relationship substantially random variation correlation decrease topology increasingly edge letter synthetic phenomena module potentially multiscale unweighted modularity weight sign adjacency c topological lattice module keep throughout relationship edge module regular lattice graph module modular different color simulation unweighted edge show tend topological unweighted characterize regular time type module graph decrease add entirely nest modular brain cast temporal organization reduce window discussion letter
objective location polytope doubly stochastic gradient backpropagation range gradient demonstrate information retrieval straightforward document supervise define goodness rank build algorithm capable query label lead aspect vary ordering function rank rich order feasible query therefore produce permutation term label piecewise gradient training learn without become infeasible supervised rank doubly relaxation permutation ranking preserve doubly interpretation propagate doubly lead choice pre normalization approach integrate powerful suited development network rank set size item label query rank permutation document aim learn permutation rank document study group ndcg indicate ndcg persistence maximize aggregate subject difficulty change order piecewise expectation objective ranking gain objective sum e kronecker delta call objective previously one remarkable aspect objective capture entry expectation doubly matrix nonnegative special doubly every doubly stochastic permutation think incorporate uncertainty appropriate gain row properly position probability expect leverage interpretation distribution doubly interpret entry ndcg permutation describe counterpart permutation matrix degenerate differentiable objective remain produce single ordering doubly natural maximize maximization bipartite problem solve time cubic bottleneck short bipartite document pass imply rank sort order permutation procedure suit rank ndcg heavily influence rank focus computation sensible second way act row training objective efficiently backward discriminative training backpropagation proceed provide computing layer input normalization gradient column index amount differentiable optimize unconstrained square piece family example probability computed define edge define construction cumulative row mass appear great preference rank reasonable probit logit cdf fast compute matrix document alternative sorting recover imply one discuss everywhere differentiable derivative tie practical primary optimize value vector possibility network sophisticate seven benchmark five fold distinct split number large query smooth anneal initialize early stop ndcg prediction mle select reduce small query turn derive document document perform normalization five short cut seven baseline figure ndcg graph art substantial build rapidly expand early employ surrogate gain aforementioned evaluation measure include gain gain ranking rank entail sort rank use ranking score view optimize ranking concentrate mass peak select variety also method approximate scale recently normalization minimization permutation directly incorporate normalization balance time post rank unlike optimize objective half balance optimize conceptual expectation certain rank objective objective doubly stochastic polytope appropriate possible projection remarkably call
estimator graphical lasso graphical inference treatment reason difficulty sample fully bayes decision theoretic involve autoregressive section propose prior imply computation shrinkage estimation covariance matrix symmetric unimodal write scale strategy prior traditional mention scale distribution early usage include sensitive regression error seek shrinkage prior flexible wide shrinkage potential highlight salient feature augmentation obtaining draw shrinkage none permutation invariant estimation carry solely metropolis hasting involve move metropolis high experiment point prior autoregressive model framework encourage integration assessment consequently paper organize outline shrinkage covariance construct augmentation conduct simulation multivariate discuss vector multivariate normal wish unimodal mode symmetric control motivation often real individual incorporate shrink clearly flexible posterior precision main unimodal density unimodal form q shrinkage variable express v may scale gaussian distribution scale uniform example popular prior applicable popular construct exponent gamma extensively scale power shrinkage ht px pareto prior shrinkage distribution shrinkage property infinite spike heavy tail precisely shrinkage normal standard half distribution close denote scale representation provide simple way distribution fy fy augment former involve sampling truncate often break gibbs shrinkage uniform latent parameter cumulative advantage cumulative distribution density provide use cumulative whenever inverse shrinkage introduce outline density inverse cdf student double pareto logarithmic cdf extend shrinkage parameter direct approach propose direct cholesky metropolis gibbs step sample first sampler systematically matrix note ij jj truncate truncate normal detail strategy generate method step complete class feature sampling constrain undirected set useful information involve multivariate modification ig start scenario involve normalize choose substitution integral hand scheme single shrinkage element towards idea rate entry hyper conditional sampler cumulative section model prior towards normalize constant ij fix represent prior hierarchy deal constant therein approach utility compare three frequentist wishart zero b model yield risk standard loss stein loss tr pl lasso validation shrinkage specification reversible jump fit wishart default total scale mixture shrinkage power double iteration rapid good autocorrelation lag mixture wishart one dataset core ghz run mixture respectively lasso compare loss thus indicate relative relative indicate method report shrinkage outperform less sparse mass prior model encourage sparsity power double pareto prior illustrate indeed ep ep wishart generalize double logarithmic constitute powerful proximity cx ip ci popular autoregressive distribution row constrain maximum ensure location set satisfying determine prior estimation uncertainty conjugate wishart flexibility model assume wishart provide author recommend model proximity prior high mass close encourage equal theory autoregressive place prior leave extension far flexibility incorporate knowledge example similarity easily diagonal functional form parameter gibbs choose plausible range association specify range recommend light recommendation prefer increasingly favor close association smooth across region concern study cancer analyze count type record state plus year collect national health count miss cancer surveillance number tumor follow spatial intercept tumor mean associate hyper place cauchy expect robust shrinkage simulation let ji neighbor heavy tail assess sample experiment divide count bin bin follow mean sample discard burn cancer report measure square mean wishart wishart maintain variance overall credible sensitivity plot validation display surprise seem influence appear tight influence implementation hour validation take day runtime core ghz flexible htbp var non shrinkage double logarithmic property construct mixture recently shrinkage receive proceed via prior model regression coefficient author independently propose base kernel bridge bayesian posterior suggest introduce sampler implement simulate normal simulating exponential posterior exponential analog bridge challenge posterior mixture normal logarithmic prior advantage many specifically independently identically standard normal parameter conjugate mahalanobis report error logarithmic iii outperform performance exponential logarithmic prior median report bootstrap error cb iii ep ep three recommend generalize ep exponential mixture unified framework shrinkage matrix wide uniform insight advance
financial time series arise proper inverse development indicate time besides evy pareto paper analyze brownian inverse difference property analyze mainly moreover asymptotic gamma large exhibit motion motion inverse increase evy transform evy exponent drift evy moreover characterize dirac delta differential define memory kernel laplace transform classical anomalous brownian length period distribute describe period besides stable case namely gamma review function stability skewness distribution skew assume simplicity property sum stable tail stable distribution c moment finite attractive physics anomalous phenomenon physic define exponential gamma infinitely tail gamma mean tb stable stable l evy laplace q form moment brownian stable stable evy increment evy laplace operator proportional derivative fractional see process case calculate function generalize memory formula simple consequence calculate stable derivation increment evy laplace also namely evy transform consequence moreover inverse gamma calculate eq consider summarize c stable stable ts c tt te du process plot panel observe period tb stable ga stable equal stable focus one popular characteristic record trajectory particle analyze motion scale matter stable stable gamma scale square ensemble obtain curve gamma whole obtain law mean trajectory process plot stable ga stable law plot gray line tb ts ga motion property analyze process point asymptotic squared consider show case small appendix gamma define gamma
condition probability theorem obtain decrease curve shift right problem hard certain sense error versus rescale sample curve align different panel b show plot rescale axis predict stack establishe appealing choose large condition although theorem guarantee minimizer nonconvex program minima impossible local minima global optimum nonetheless program reasonable geometrically project gradient constrain update stepsize assume statistical consistency optimum constrain universal gradient descent bound let denote objective optimum gradient update contraction claim deterministic probabilistic condition enter corollary involve surrogate noisy extension descent high state impose apply nonconvex note nonconvex program addition minimizer considerable understand upper iterate polynomial compute focus involve verify right essentially term optimum similar guarantee composite descent apply lagrangian project objective decrease geometrically experimentally prediction simulation show apply additive panel panel random apply projected method project gradient iterate statistical trace trace statistical geometric rate straight regardless start iterate exhibit converge statistical geometrically point theorem require meet statement corollary universal positive constant corollary triplet scaling mean instance draw scale pm theorem signal covariance ratio measure note covariate agree know corollary compare derive tolerance sub particular notation arise must noise use sophisticated chapter replicate form n corollary note sample example sub quantity conditioning fraction establish uniformity bind matrix curve align suitably rescale perturbation figure rescale curve miss rescale autoregressive additive perturbation drive point excellent agreement scaling run appear corollary additive value run project gradient q plot rescale roughly b rescale error rescale miss curve roughly constant showing point trial noise run plot rescale finally type arrange linear except two set rescale rescaled come randomly first entry probability condition number rescale generate sample appropriate corrupt miss error run chain additive plot missing panel rescale rescaled prediction qualitatively similar star technical corollaries supplementary pt minimize indeed regularize imply equivalent remainder derive condition combine low left side side triangle piece conclude hand sparsity piece conclude inequality inequality combine stronger regularize returning obtain q piece rearrange claim side final use combine giving prove claim project use state require proof show argument hinge restrict smoothness apart exploit update appear paper tolerance condition hold final assumption similarly combine turn lagrangian version correspond within contraction take previous remain verify put piece compound formulate sparse corrupt may project guarantee statistical draw estimation exist type dependency covariate perform corrupted replicate point interesting nonconvex broadly discussion grateful associate anonymous improvement section notation remark part foundation fellowship grant dms air force office scientific although standard involve draw involve noisy miss high novel datum inherently nonconvex difficult establish guarantee practical approach nonconvex program optimum project minimizer provide miss dependent condition project converge geometric global minimizer standard prediction partially observe voting behavior vote behavior survey suffer miss question tend noisy partially variety deal well involve book reference become noisy consequently converge minimum guarantee optimum study sparse develop despite still error optimum show scale perfectly fully observe miss corrupt g well reference therein mean require relevant allow miss converge close covariate noise selector multiplicative estimator past dimensional model remainder description estimator descent section devote main include optimization corollary case noisy miss section confirm theoretically spectral hadamard background description motivate class discuss descent link vector via observe covariate include independent say vector entry observe hadamard multiplicative noise however interested probability random consider fix accord vector autoregressive predictor grow exceed impossible unless endowed structure instance sparsity also infinity class examine simple constrain program long radius unique program know matrix certainly try estimate nonetheless provide suggest principle quantity consider regularize user note equivalent lagrangian duality objective set shorthand row calculation concentration inequality sequel well program involve contrast matrix loss appear unbounded make eq generally impossible presence minima remarkably issue project descent program close optimum illustrate serve unbiased noise pair noiseless semidefinite regime cause equal consider miss random extension entry use diagonal cause consequence semidefinite convex multiplicative generalization application row I noise arise reader discussion q calculation past observe contain various closeness focus closely understand condition type matrix satisfie eigenvalue condition support bound magnitude conversely since convenient analyze fully row n choice early also probability previously discuss generally negative finally analogous upper restrict eigenvalue formalize upper restrict tolerance high project et loss rsc smoothness random scaling choice minima possibly program polynomial procedure optima simple either program quadratic function application method generate recursion stepsize rp procedure update variant project regularization also onto detail semidefinite converge global objective generic iterate family program iterate actually close statement
utilize stick break study break beyond dp identically distribute observation convergence infinitely compactly continuous derivative optimal require intervention derive old well old stop short deriving derivation require class development assign useful recall two dp distribution partition stick break interpretation first approach characterize measure measurable characterization say independent base measure determine admit lebesgue density strictly positive constant variate lebesgue equip borel qx p identically iid establishes calibration induce infinitely density rate compactly twice continuously differentiable result section probability compactly continuously n tool condition slightly two number kp px px qx qx kp ball cover hellinger positive hellinger call sequence existence quantitative lie stick break dp mass precise fix ml dp b b let net let simplex h ds h h small equal h assertion multiplication hellinger root break z hz h describe paragraph md md formula assertion subset adapt nearly super finitely differentiable function arbitrary super n n n l hellinger sm prove second fourth bound bound possibly n c old n hellinger correspond old derivative care establish class ba cn n immediately proposition include first tackle density big challenge smooth proof minor need handle leave gap intend ordinary compactly ba cn n density z du kk du n u mm dp fu fu db u diameter u gamma super along simple development n n n straightforward extension display distribution point propagate extension discrete n p support p supremum continuously differentiable let closely need dy x integral form remainder dy
correlation perform strongly suggest fast range presence hardness dimensional frequency parameter frequency model quantify change response bm natural characterization essential bm generally short response correlation frequency also experiment letter infer keep recursively must overfitte enough important conventional expansion pseudo intend implementation function boltzmann ising model eq bm ip expand later spin loss impose come increase allow calculate large maximal size hardness obtain individual frequency expansion ise grid noise spin allow entropy entropy length short entropy common interaction sign depend absolute separately entropy smaller discard low neighbor cluster select reach summation partial entropy vs perfect sampling label dot configuration b deviation entropy tail interaction differ affect entropy absolute value correlation first extensive overfitting make secondly entropy universal coincide system behave perfect accurately cluster entropy path systematic enumeration pass pl fail interaction critical size error decrease rao fig system cross core processor ghz performance seem interaction recognize pl vs grid obtain grid infer grid dependence boundary condition vs b spin spin understand calculated bm fluctuation configuration equilibrium expression poor correspond criterion justify fluctuation comparable bootstrap find fig merely bottom leave vs unity underlying size analyze second activity bm cluster reach fig c error infer correlation fig interaction find select cell small cpu computer expand alone entropy generally achieve fluctuation fig coincide good expansion system rather dense severe version include base ij even bm problem exhibit correlation due accurately ise range decay interaction consider even direct widely closure change dr r correlation redundant redundancy origin locality property cluster entropy useful biology advanced nj paris france l paris procedure correlation build successfully recover
intrinsic notion improvement tail replace slightly weak tail negligible tail inequality inequality sum inequality moment tail use shorthand concave due result give inductive first inequality jensen inductive combine inequality n inequality corollary estimate proof technique first subgaussian subgaussian q require bernstein bind surely almost eq q eq q tail dependence dimension moment trace roughly form apply full tail improvement applicability inequality subsection disadvantage quantity replace theorem bind early tail case point elaborate far rademacher equal x remove factor side perhaps importantly deviation whereas former latter provide tail roughly put optimality share inequality exponential moment relate nearly moment technique count contribution beyond entry worst denote small eigenvalue subgaussian constant tail state previous work tail infinite subgaussian finite example give error moment matrix I copy relevant high x rather spectral generally spectral error appropriate spectrum show dimension inequality sometimes strong vector ix dd potentially dimension n tail tail cover appendix comparison scenario compare rapidly decay time square factor factor roughly example latter much weak subsection independent concentration expectation put main factor give approximate fix column computation prohibitive alternative say q allow failure give sample outer product identity inequalitie b union b bind grateful useful pointing subtle early comment reference tail small large subgaussian result explicit originally vector surely application lemma simply completeness subgaussian subgaussian certain combination relate tail tail particular almost chernoff bounding lemma therefore
novel name normalization assessment embedding assess similarity correspond low dimensional embedding caused eliminate aspect embed still geometric structure manifold preserve global assessment evaluate skeleton manifold preserve local assessment evaluate preserved tool selection embedding optimal specific include validate effectiveness literature describe independent embed depict conclude assessment review presentation notation table throughout paper euclidean lie embedding lie datum local index near neighbor low near neighbor identity frobenius norm motivation embed quality assessment global work review pm enable quantitative translation match finally assessment result show pm provide quality point pm neighborhood although modify pm scale neighborhood address coordinate low pm follow neighbor work overlap representative lc meta serve diagnostic tool assessment sum consist index neighbor accord euclidean quantify embed embed exploit develop agreement ar shares ar assessment france rand index reduction lee propose name co ranking criterion aforementione rank neighborhood cast unified framework visualize extend dependency high assessment pairwise preserve pairwise long keep degree inspection residual variance diagnostic evaluate standard approximated geodesic distance value embed nevertheless normalize embed geodesic distance preserve method manifold ground truth within euclidean geodesic ground truth set assessment case hold manifold short construct ranking branch length overall assessment normalization aspect assessment assessment measure scale coordinate subsection demonstrate normalization motivation description subsection state dot black origin blue neighbor origin embedding randomly normalize embed mark dot origin mark circle dot nearest origin mark red square meanwhile near marked circle see origin lot overlap meanwhile clearly observe distortion overlap cause normalization topological preserve neighborhood structure manifold neighborhood view subspace ambient normalization rational choice measure assess coordinate formally index motion rotation translation assume stand goal find optimal match solve follow constrain matrix square transformation formally q normalization item introduce eliminate scale optimization admit alternatively orthogonal manifold riemannian embed multiplication convenience presentation ia square form yield nn rewrite form let objective proposition trace expand derivative strict convex substitute solution fp db jj mb jj since make follow substitute fp ib jj jj ia jj jj let rewrite tm j tm tm column stand hadamard product transform descent problem function gradient follow calculus dimensional stacking column derive w elsewhere tp iw w mh mh mi tp matrix calculus formula length method vanish iteration update q qr triangular high decomposition greatly possible project densely optimally manifold method step step otherwise decomposition assess embedding need evaluation manifold embed preserve geometric neighbor embed assessment address normalization local subsection low characterize neighborhood preserve assessment index geometric preserve topology embed word treat representative pairwise geodesic preserve geometric motivated assessment motion scale computation global step respectively neighbor treat length euclidean graph short distance count short path finally denote define geodesic obtain optimally preserve position equal one correspond define global corresponding need experience state replace yet complicated method subsection implement suppose namely well locality vice versa say well topology vice versa compute specific combination assessment score test benchmark use select method embedding compare commonly assessment unified assessment indicate embed method description embed surface embed surface notation description measure residual measure assessment eq assessment truth manifold le number near embedding give shape le recover geometric structure training c embedding method state bar plot learn embedding bar correspond bar visualize fig bar embedding embedding learn affect normalize still report reasonable design give overall reasonable method local neighborhood le good completely match inspection demonstrate effectively bar matching match intrinsic degree learn name bar learn character share intrinsic convex rectangular geodesic connect training sample neighbor learn bar plot approach distortion global shape bar fig report quality assessment match illustrate embedding evaluation geodesic short approximate distance intrinsic degree freedom various method bar low bar evaluation training embedding bar observe except manifold global isotropic report evaluation still fail assess correctly successfully match isotropic name learn embedding experiment code intrinsic freedom graph construct learn neighbor bar see give orientation successfully extract intrinsic despite inspection bar fig indicate experiment lie geodesic short geometric manifold value intrinsic freedom manifold subsection application important near first section select manifold
time sum obtain choose action sp kullback horizon arms reward max q recommend arm confidence newton strictly tie maximizer choose random idea suggest section main paper asymptotic state problem optimal regret maximal ucb choose arm bernoulli ucb asymptotically kl ucb generalize tend kl asymptotically chernoff obtain ucb ucb maximize choose choice sub logarithmic proposition concern kl ucb kullback omit ucb study difference compute arm soon exploration instead case quite large simulation suggest ucb reward ucb large principle rely bernoulli makes present bernoulli ucb choose appropriate arm belong density respect reference partition twice differentiable poisson gamma easily check upper replace line kl choose reward state apply remain object generalization possibly different arm technical topological discussion conclude observe require upper policy price slight loss theorem canonical exponential precisely observe cx b concave point poorly concentrated remain correctly almost begin arm estimate play good mistake important bandit design event decay fast skewed slowly vanish arm reward suboptimal time logarithmic compare five tune suboptimal represent plot six panel clearly highlight effect mention simulation reliable typically ucb instant note show ucb ucb ucb proposition arm scenario hence kl quite actual ucb criterion whether play criterion ucb ucb estimate compare arm correspond upper consequence arm play ucb share common effect ucb generally preferable elimination instead outperform conjecture ucb term ucb line remain preferable final comment tune perform though slightly bad kl ucb right panel fact control ucb ucb differ ucb simply asymptotic bernstein inequality draw grow exploration significantly difficult inspire reward nine arm different reward regret scale tune reach par kl significantly significant cp kl binomial confidence easily verify quantity satisfy property pearson ucb differ ucb arm compute interval leibler easily adapt proof show bound cp ucb kl often actually decision term ucb achieve marginally guarantee cp arbitrary truncate reward ucb ucb clearly ucb account reward value distribution ucb horizon yet ucb tune ucb variant kl ucb kl ucb prescribe easily slightly performance excellent consider positive arm generality denote bt kl ucb confidence expectation last lemma prove appendix positive provide two lemma ta dr dr self present improve ucb make policy simple even binary f common field hold proceed make trick slice geometrically slice trivial nn nn kt x state deviation moment ts france analysis kl ucb algorithm online horizon bandit distinct bound reward kl ucb uniformly bind reward reach furthermore simple optimal unbounded exponential study compare ucb main ucb tune kl ucb remarkably stable ucb rely arm simple agent slot arm try maximize choice draw opponent subject bandit choose receive conditionally arm choice reward rule observation measure reward horizon reward would accumulate arm exploration exploitation rate play sufficiently since work several variant solution see reference therein distinguish family distribution belong family bind policy determine optimal policy multi draw optimal maintain work refine recently ucb term parametric bandit online call kl ucb horizon tune recently together learn ucb denote kullback consequence upper arm fact divergence specific case reward rescale
hardware fuzzy circuit consist three part create work procedure part since currently work fuzzy consequently layer layer specify gray dash purpose rule describe work fuzzy consider become strength input big degree reason circuit confidence treat circuit value confidence degree concept neuron create fuzzy number system represent concept pixel interpret aforementioned concept intensity degree hand concept first become specify activation concept fuzzy universe one big terminal represent concept big complement terminal represent small confidence validity application detection pixel represent minus input dark fuzzy concept purpose simple circuit fig vertical act universe universe weight membership universe fact specify membership function circuit weight fuzzy big small universe shape circuit depict side specification shape membership support similar evident function implement work procedure simple column locate maximum input row create membership define universe variable low generate universe fuzzy function x x fuzzy circuit correspond number fuzzy column circuit second layer specify call create role circuit compare column perform dot fuzzy layer weight form column middle therefore fuzzy pre create output correspond weight negative say column implement input implement output variable maximum equivalently similar implement rule output fuzzy column implement variable big complement configuration two part rule distinct fuzzy strength output value compare way recognize concept simultaneously input apply rule input want determine strength activation fuzzy actually product function hide vector finally artificial neuron fuzzy create consider difference neuron kind activation another operation fuzzy part conjunction simple part happen evaluation rule evaluation result rule tie reveal similarity rule basis last circuit layer consequence evaluation per visible system method summation triangular fuzzy primary fuzzy fuzzy intend system simple herein note sum operation kind degree determine degree concept assign neuron neuron represent output neuron concept big assign increase big output aggregation neuron big neuron output unit variable simply consider final unlike fuzzy system circuit fig single output properly get fuzzy number circuit want gate briefly image locate intensity word intensity intensity edge otherwise consequently pixel application logical neighbor intensity indicator existence pixel one detect kind gate work input pixel intensity output function intensity output increase output proportional build consecutive strength higher weak system generate gate fuzzy horizontal vertical fuzzy loss generality circuit fuzzy merge pair consecutive fuzzy fuzzy manner construct extract vertical use circuit vertical extract degree repeat horizontal extract circuit fig analog extract consecutive pixel applicability hardware fuzzy system fuzzy structure pre membership numerical specification output require present easy modify specification shape force probable smoothed smoothing extract present fig merge single final simply fig circuit extract intensity degree directly concern view detection e smoothing fig edge counterpart area edge visible analog therefore operate fuzzy surface different versus behavior fuzzy function clearly area similar intensity zero difference begin I activation hide circuit fuzzy inference method fuzzy surface describe fuzzy function next propose extract image extract structure apply illustrate stability white gradient algorithm noisy result information neighbor hardware fuzzy express fuzzy fuzzy implement newly I fuzzy extract edge simulation show edge detector noisy concentrated detection obvious structure implementation fuzzy rule inference fuzzy system lack hardware tried overcome fuzzy inference fuzzy achieve introduce fuzzy realize detector analog edge detector advantage edge image integrate circuit successfully however limit extend addition small device powerful device computer beyond limit scale equivalent circuit currently work digital gate array construct digital logic perform make former separation inefficient later lead consume computing establish flexibility also limited precision arithmetic operation carry limited publication scientific progress compute system front digital fast counterpart almost logic recently capabilitie use fuzzy logic digital addition fuzzy logic concept analog completely analog fuzzy advantage way implement fuzzy introduce fuzzy analog fuzzy edge fuzzy fuzzy construct fuzzy consecutive extract simulation meaningful edge benefit fuzzy detector hardware implementation analog advantage note system use even hardware artificial construct binary network demonstrate devote fuzzy fuzzy inference system fuzzy edge eventually section passive field artificial intelligence become clear passive many analog neural building analog circuit hardware soft computing implement digital circuit mathematical passive neuron connection build output neuron connection create logical say neural fig implement traditional circuit capability weight fuzzy relation activation layer fuzzy probable similarity logic usually interpret work another figure connection layer weight usage denote locate matrix connection locate layer neuron represent linguistic concept neuron neuron active logic concept mean output time neuron consequently consider work fig network membership connection connection matter neural network fig
rate finitely rate always converge finite subset candidate recommend support determined reflect mixture relatively modify approximation finite location mixture focus admissible substantially decrease combinatorial solve reasoning calculate consideration pick good score maximize profile essentially marginal integrate reasonable sort along mix support fix discrete choice henceforth dirichlet measure linearity without explicitly evaluate via sequential imputation function sense define reasonable properly account preference two support contain justified assume restriction closely support grid advantage nonparametric become parametric fact possible estimator drawback however maximize combinatorial satisfactory computational second bayesian fraction value nonparametric know density borel present show process computation mix absolutely continuous measure almost surely depend shall counting application continuous large two density kullback divergence respectively mix weak closure km show conservative obtain near see base likelihood density marginal recall f dirichlet hierarchical mixture prior single act marginal bayes work maximize restrict moderately set support cardinality despite give anneal combinatorial problem anneal move current new tend simulated decrease temperature take follow give acceptable discussion subset associate determine suffice anneal set generate define characterize else visit herein choice success annealing move move likely flip prevent get temperature make shall exactly component encourage number assign great mass q equivalently sample uniform next I likelihood reduce dependence marginal likelihood predictive averaging computationally herein stable keep specifically corresponding example distance consecutive run choice option drive force behind simulate subroutine marginal likelihood suppose subset recursion convergence rate present focus case focus galaxy set consist velocity galaxy consideration reasonable second version liu imputation likelihood study mixture seven consideration respective list table support method take point complicated take aic scad model estimate single tend example bic ccccc bic bic aic bic aic bic aic bic bic proportion estimate size take procedure able lattice higher become costly kernel outline construct space collection quite advantageous introduce simple mixture could reduce size space important location interval respectively location fine grid mixture unimodal cover mixture student fix degree vertical accommodate restriction simulated location adjust indicator characterize enter pair admissible subset optimization restrict subset reduction structure section change chance respectively maintain encourage accomplish encourage entry select likely algorithm mixture anneal flip coin move accept move bad candidate help simulate annealing get location scale galaxy five overall good elsewhere five need six computation panel mix simulation component make relatively detect distinct component vary size location anneal optimize collection admissible subset subset second comparison include estimate seem increase large quite competitive method good across rw rw rw true take present novel hybrid stochastic annealing algorithm finite introduce unknown bounding approximation simulated annealing maximize likelihood candidate benefit rate approximate work example competitive exist experience method fast spread support bound bound bound need since explicit mix simulated annealing proposal implicit anonymous incorporate approximate candidate consider grid fine nearby concern want unknown algorithm replace profile version experience expensive computationally acknowledgment grateful lead go liu suggestion work complete author university recursion anneal efficient grid anneal method employ maximize likelihood estimate novel annealing compare predictive recursion complicate describe mixture difficult specify mixing assume observation density eq
function respectively I qx te put give bind partial centering iteration tolerance p ij x I v I p ij q qx ij c il conjunction center tolerance k ij q p q qx ij p variational low example conjunction initialization initialization observe initialization work cluster four replicate array analyze al ng version approximately ng four go adjust index partition four category illustrate mixing unlike go weight length cluster high among five axis axis profile covariate gene attempt ng et category addition gene observe category consistently separate remain gene category probability fit axis probability substitute pz represent belong mixture belong go impact center five component marginal attain take average second average efficiency clear hierarchical center hill tool journal k genomic analysis systematically perturb ng ng effect
x qr growth theorem notation restriction observation available method section order order reduce dedicated variable order perform procedure variable way order sort value variable technique improve perform make asymptotically appearance variable number select high give penalty order frequency decrease false discovery rate require indeed see relevant place order successively eq stop accept set relevant generalization since multiple sake understanding deal procedure order testing test successively nan accept section recall k ts v q introduce space first upper multiple I ks kt kk u simulate choose accordance follow ensure procedure next fix estimate differ part indeed lie order essential chance occur restrictive appear family stochastically order define derive let set eq reject explicit model orthonormal cardinality let accord procedure k p already define tn k tv unknown nan hypothesis introduce relie define kk tv kt let accordance reject second procedure ensure next give condition notation follow theorem tm tm tm nx consider state b I powerful step important procedure combine perform right bound constant depend rewrite theorem remark context procedure show call almost estimate consider begin section section applicable modification concern account denote dimension successively collection alternative x k apply comment result table selection procedure present implement project six method procedure order denote order table section fdr method comparison several framework orthonormal latter fdr variable compare adjust fdr fdr calculate variable onto introduction fdr biology order selection calculation avoid schmidt orthonormal le kx l l j lot redundant calculation gram schmidt case order unknown family chi degree freedom define simulation family unknown concern simulation coordinate orthonormal orthonormal principal example mention consider reflect well condition mean column percentage actually label variable replication record select average mse square calculate cross threshold pe also objective distinguish remain assume reach frequency frequency stay decrease assumption wrong procedure display order variable accordance far powerful order unknown lastly context give excellent dimensional table result surprising choice powerful almost concern test orthonormal orthonormal compare fdr weak fdr indeed nearly procedure outperform combination induce decrease become satisfactory l correct replication record record correct average replication zero l l fdr mse truth c mse replication record replication average l fdr c correct truth truth mse replication column truth time record select replication mse non sparse sample hypothesis procedure prove powerful signal show outperform especially h k k h obtain eq conclude construction k condition u bind quantile u n condition q x log upper u inequality hold positive number n lead j k orthonormal j k upper z explicit let gaussian order v u expression upper part k denote proof condition verify quantile n give event n f n ta n g k g u tm mb ba n nx combining k k k n tc c mc two lie theorem remark estimate linear new order variable procedure inspire propose new procedure powerful condition give discovery fdr recent provide biology throughput still major challenge fit also purpose discover variable lead high linear probably penalization set lot consistency lasso penalization induce pls pls kind aic information penalize portion high develop powerful high selector recent show selector exhibit behavior criterion model selection procedure
work ensure step triple correctly orient v describe application orientation path l four vertex two successive conditionally dependent hx ix jx jx kx x hx j applied lx search length consecutive dependent structure conditional relationship satisfy orient one relationship triple create depend path input orientation rule open possibility path yield skeleton skeleton order infer skeleton need determine trade slow obtain modify instead consider subset output identical output output equivalence however detail suppose independence ccc dag skeleton skeleton skeleton denote initial skeleton final skeleton skeleton absence skeleton x skeleton find pt step possible result mechanism triple orient consider check triple check would contradict information output correspond interpret dag latent output output page read dag b namely reason independence skeleton sep appear output encode consider adapt add pointing modification conditionally belong conditional read class interpret specify graphical underlie identical theorem output infer edge output induce extend jx jx induce connection induce see page x shorthand skeleton vertex three induce induce induce moreover induce induce path give relative pt ix must adjacent occur member j ii dag condition c c role formulation illustrate adjacency matrix entry replace variable nonzero entry strength mutually random variable assess half parent child restrict particularly scenario oracle version output identical oracle sample version version processor ghz gb ram r first version simulation average observe integer assess case number additional compare give difference additional almost difference next performance version compare simulation value dag run parameter replicate replicate perform slightly large replicate mark replicate replicate mark replicate see text dag average dag set size run computationally graph eight hour termination occur times nine latter nine lead algorithm compute number extra mark b remain first sep time sep vertice consider combination run sample version simulate possible replicate new use max drastically consider subset cc still next modification setting run second replicate text show running replicate scale version run correspondence time fast graph fast second test adjacency vertex large see really fast independence exist small interpret output weak mean identical algorithm similar output cause structure find version rare consistency considerably modification show use assess impact use building causal observational start reason open investigate completeness investigate edge mark maximally section remark e nsf grant ai direct dag arbitrarily selection explicitly infer independence causal setting computationally infeasible fast output informative independence causal setting package consider acyclic system background outline system satisfy causal causal acyclic effect cause possibly indirect cause direct path dag read dag dag equivalence markov give shorthand equivalence dag alone markov equivalence dag uniquely complete partially independence relationship exactly imply suppose causal independence consist dag idea prominent example pc sound complete I maximally informative causal implement r asymptotically want dag case dag calculus intervention calculus represent dag conceptually causal idea form basis observational datum generate causal dag stand intervention calculus algorithm validate expression practice variable record speak whether measure sample speak see several problem inference pc incorrect relationship observe dag pc algorithm would lead cause neither direct box latent circle represent observe latent dags marginalization conditioning follow dag conditioning dag variable example x imply entail exactly solve introduce observed graph dag latent transform observed page describe infinitely dags latent selection relationship mark selection variable possible underlie observe separation separation see definition describe relationship partial selection relationship cause selection variable dag true imply mark reflect independence arise dag cause arise cause markov information among alone pc algorithm originally orient induce path sound presence et introduce extra orientation output interpret despite intensive modify less cut typically tail sound size situation latent causal conditional independence variable much order hand compare define prove consistency setting sparsity need strong exist supplementary document section simulation error similar modification package implementation graphical compose edge six bi mark graph direct direct bi presence mark graph one present say pair graph vertex vertex call sp ne vertex say path path together vertex definition vertex x form cycle three adjacent adjacent path three ii vertex adjacent adjacent mark star path pair vertex graph maximally subgraph call graph almost undirecte parent dag entail relationship via call se separation say middle structure mx x joint density conditional qx p qx implication conditional infer separation separate dag separate ix selection selection variable e disjoint correspond conditional relationship page path definition moreover ix encode independence variable selection separate separate importantly preserve relationship dag remainder event distribution conditional relationship want causal relationship underlie dag represent purpose new first make distinction partition c mm mm g iii absence two exist presence vertex pt edge j jx vertex gp iv condition x strong several consequence dag skeleton skeleton correspond equivalence class dag example worth may skeleton graph also discusse propose modification speed remain sound discuss several algorithm give skeleton final skeleton structure mark conditionally ix jx j I mean j x conditional call pt k jx x h pt sep vertex since sep skeleton pc starting test adjacency vertex responsible final skeleton triple mm kx kx x information sep thus algorithm every element x j arrange edge remove independence complete orientation necessarily iv orient algorithm replace circle tail orientation computational effort two set testing subset set infeasible contain sep set play important defining must sep first modification sep decrease edge type mm j x jx hx hx kx ix ix ix line replace modified modification possibility use conservative structure fewer especially algorithm v bi edge set orientation work follow result subset separate separate separate separate
paper pareto front locate deep weighting dissimilarity efficiently linearly theorem pareto validate theoretical advantage real acknowledgment labeling thank suggest computing nf present proof theorem measurable conditioning py complete assumption x hence fy dy dy substituting integral special simplify recall note large complete event q interior hull condition independent let pe b pe q eq calculation change proof theorems two correspond theorem deal asymptotic suggest hold even though experimental rigorous two denote absolute coordinate result experiment suggest grow figure show curve regression give half power growth dot align experimental criterion induce domain test current pareto front denote near connect near dissimilarity detect anomaly specifically anomalous nominal sample small sample near anomalous anomalous may much like hand choose sample nominal mean nominal near neighbor connect near neighbor e isolate could anomalous undesirable keeping try problem nominal deep retain connectivity require connect implicitly consist proximity work return situation design construct dissimilarity connect choose independent probably choose jointly could perform choose dissimilarity poorly example problematic previously paper area heuristic include trajectory low trajectory nominal detect criterion speed trajectory anomalous anomalous anomalous anomalous perform quite auc advance example identify pattern exhibit anomalous often anomaly detection anomaly single euclidean dissimilarity capture anomalous case test anomaly advance algorithm need execute choice combination parametric anomaly pareto anomaly criterion propose criterion provably criteria anomaly variety diverse detection detection method parametric typically dissimilaritie high multiple dissimilarity measure anomaly detect trajectory multiple criterion use anomaly order anomaly combine dissimilarity advance difficult dissimilarity choose furthermore change execute anomaly detection concept pareto detect anomaly choose typical define multiple compare item item say pareto well equal criterion pareto necessarily hence anomaly form create corresponding dissimilarity dissimilarity pareto pareto front depth dominate pareto front depth remove front front continue remain front depth see figure nominal anomalous locate pareto front front discriminate nominal grid exponentially number criterion assumption multi criterion realization pareto front precise use combination theoretical validate several art anomaly synthetic set tp illustrative training black dot pareto green pareto induce concentrate around triangle concentrate follow front relate pareto anomaly lead detection algorithm experiment several utilize optimality previously propose overview typically formulate problem pareto quite difficult pareto optimality introduce rank create datum pareto correspond dissimilarity sample pareto another feature view case view improve view disagreement investigate criterion could severe disagreement performance another differ multi area learn anomaly detection survey distance th neighbor score distance neighbor nearest adaptive anomaly span bipartite anomaly nn anomaly scheme define aforementioned setting execute unlike anomaly section utilize optimality area economic science social sciences pareto pareto front criterion item function criterion non negative weight combination minimizer linear involve find item another item criterion write pareto dominate combination include item combination pareto front pareto front pareto dominate item member front convenience pareto front seminal much pareto front relevant I pareto belong cardinality general pareto framework feasible vector common construct criterion combination easy identify pareto hull pareto good many pareto pareto selection weight linear compare paper pareto point identify anomaly anomalous reject much anomaly pareto linear front two domain induce open density vanish outside large density strictly asymptotic z pareto point normal convexity identify pareto optimal geometry panel theorem enough pareto neighborhood attention randomness even convex front method pareto let tp pareto front induce geometry randomness pareto black experimentally close likely pareto pareto nominal sample anomaly anomaly significantly criterion associate measure dissimilaritie dissimilarity compute ki ix convenience strictly dominate another pareto front dominate second pareto strictly section front correspond locate pareto dissimilarity combination criterion nominal basic criterion pareto require front connection anomaly test near neighbor choice jx jk nearest say front near distance anomaly mention multiple criterion property know pareto pareto linear pareto weight independent dependent support appendix pareto observation near anomalous anomaly mean depth easily use phase calculate pairwise dissimilarity create pareto front testing dissimilarity near create training sample anomaly show provide heuristic practice require anomaly detection mention
hide vector set information perform activity mutually exclusive across note hidden vector translate require across vector connection variant illustration dot connect layer constraint valid activation single j conditional sampling efficiently show softmax vector pair train use cd generative gibbs initialize vector probability divergence compute copy useful feature set relevant desirable robust situation maintain connection target accomplish copy energy dependency constraint activity least call illustration constraint example activation j exception condition tractable generative perform layer hide cd update implicitly pool soft input definition softmax pooling disadvantage begin essentially activation unit actually hard operation definition apply gradient class approximate likelihood classify input learn classification problem drug categorization year propose parallel diverse expectation extension citation knn support mi decision loss expressed accordingly present rather maximum generalize classifier without function product radial base family function gaussian convenient kernel kernel instance kernel max mi tend quadratic compute kernel set use probabilistic convolutional rbm propose see pooling rbms mail rbms set baseline describe size feature idea baseline indicate gray bold student classification accuracy good variant never statistically worse importantly outperform maxout confirm usefulness level oppose input purely discriminative scale pooling however categorization piece mail page page handwritten letter form white mail thousand day process couple day document management page process handwritten produce feature correspond presence word page vocabulary recognize word resolution gray scale regular box handwritten compute statistic predefine page detector bank kind paper output recognition high noisy extent mail expect page label label assignment page natural enough imply label one page individually develop vs experiment collect dataset mail mail page piece mail validation different important reject piece mail uncertain piece mail process human whole threshold goodness multiclass micro b cc observe emphasize large fast boltzmann could investigate achieve also success mail categorization experiment confirm usefulness pooling oppose deep neural architecture acknowledgement derivation simplify derivation assume size necessary sum use softmax go e softmax softmax softmax f recover unit free energy softmax h provide free assume size use mutual use inequality hence c softmax c softmax c softmax general function cm ia sa paris france ia com department computer science edu classification occur problem mail page web site propose generalization restrict machine rbm explore relationship rbm experiment dataset rbms variant incoming mail classification vast machine develop vector process one setting consist find incoming piece mail document represent example simple approach would away mail useful pre computer another sequence relevant dynamic take e informative speech content popular classify multiple belong originally motivated drug disease drug shape label bind implicit recognize appropriate partial observation enough restrict boltzmann multiclass set information present learning extension deal competitive result dataset mail rbm binary joint input target correspond label take belong position
assume discrepancy median log within fit highly lead ignore discrepancy verification uncertainty perturbation control explain discrepancy acceptable inherent uncertainty input conventional sampler abc rejection mechanism adjustment inference may view high combine adjustment marginal distribution posterior dimensional principle abc replace univariate precision particularly summary parameter dependence initial sample poor however accurate approximation extreme increase marginally approximation roughly constitute obviously poorly joint point anonymous linear summary marginal allow method moderate beyond abc likely abc acknowledgement support thank discussion adjustment contribution water reservoir reproduce mixture normal file contain entropy txt contain reproduce file library analysis fan mm linear computation abc complex connect demonstrate regression moment approximate adjust bayes adjustment role exploratory problem low abc first rough abc separately dimensional provide joint illustrate material article regression bayes linear tool analysis think fundamental primitive quantity first article method model write discuss computationally intractable generate smoothing evaluation regression framework transform independent error coefficient weight give estimate estimate interest writing adjust approximately model hold globally extension generalise model feed forward neural perspective regression adjustment adopt adopt regression quality occur reliably obtain complex obvious strategy summary possible quality largely dependent curse argument overhead increase imply date moderate article primary contribution first abc summary adjustment motivated contribution propose regression adjustment approach method obtain abc univariate estimation improve estimate explore literature extend applicability beyond abc practice structure introduce adjustment describe adjustment abc regression adjustment simulation study real present conclude suppose summary statistic construct vector covariance adjust expectation posterior coincide obtain specification full first key interpretation bayes abc full outer side distribution relatively bayes hold globally locally unweighted hold appropriate summary ordinary ss ns abc population fix sequence infinity ss es justify adjust demonstrate adjusted method interpretation regardless globally bayes connection surprising monte prior square present purpose interpretation helpful exploratory adjustment high anonymous shrinkage implement bayes adjustment continue kernel weight first second adjustment replace matrix diagonal possible take square element adjustment outside bayes framework however nonlinear appropriate expansion give certain regression adjustment suggest transformation bayes broadly applicable adjustment fitting choose set statistic construct truncate bayes aspect paper summary sampler smc rejection importance strategy increase dimension force summary cause sampler abc suffer hand often linear difficult validate may preferable low adjustment essence rough approximate abc marginal easier full reduce statistic parameter estimate base margin joint marginal rough estimate order statistic adjustment maintain multivariate original reasonable expect fit sample without exclude marginally informative abc distribution sample space replace quantile precisely write margin sample generate save suggest sample summary statistic something leave work point marginal compatible ordinary approach joint statistic quantile spirit procedure theoretical improvement parametric marginal joint beneficial normal copula mixture normal motivation come determined projection arise joint distribution characteristic adjust adjustment relationship locate negative slope summary statistic line marginal posterior summary estimate use rejection rapidly curse restriction fixed accept within increase simulation return sample curse beyond density estimate poor adjustment sample rejection clearly adjustment improve estimate quality approximation increase albeit sample follow adjustment contour improvement regression component appear density marginal dimensional margin adjust figure improvement location place adjustment transform message estimate figure abc true kullback pd draw compare computational conclusion rejection sampler adjustment rapidly summary regard gain though marginal adjustment marginally lower marginally beyond sampling adjust posterior largely represent uniform able degree improvement correlation lose beyond benefit adjustment use lower accurate abc discrimination incidence consist zero one incidence represent random observe lattice field rs symmetric function elliptical thresholding geometric set element fairly fraction probability expect one west lag respectively approximately probability describe lie sensitive reason prior variance field package summary similarly number pair matter estimate n v n summary adjustment perform r nonlinear method incidence solid denote individually marginal margin component line margin adjustment solid estimate individually estimate accurately verify rejection predictive wrong interesting insight gain figure scatter versus informative statistic order appear change function adjustment help overcome versus incidence method computer aim uncertainty high discrepancy approximate treatment assessment view regard model input modelling subset type input g sampling ignore involved involve assessment due exercise see different aspect computer external function uncertainty abc simulation accord precision handling round abc water water design management water three area store input produce add split fractional area flow store water stream total fix beneficial vary dimensional great input denote run
reduce disk record string generator seed simulation code original string whether accept proposal acceptance calculation involve ratio proposal probability degree freedom care number must stream particular evaluating acceptance discard stream platform corruption computer processor architecture source eliminate code point platform valid original run nontrivial mh spherical particle particle spherical two surface sum potential particle center site intra particle particle center coordinate site file source file material run runtime version execute highlight straightforward degree particle obviously inefficient store update value change degree format transition simultaneously particle format potential generally method code would enable reconstruction without compare storage scheme information sample sequence sample change energy string sample proposal calculation iterate take enable analysis histogram angle intra particle pairwise accumulate string evaluation efficiently change original computation particle bit string record energy evaluation iteration degree htbp operation second record bit string string enable long spend proposal compare ratio fortunately satisfied sample determine degree freedom component implementation minimum achievable consumption execution record sequence reject analysis expensive instance physical heat capacity would additional bit much alone far reduce acceptance string corruption single incorrectly subsequent solution encode error storage consumption low check persistent device would constitute redundant non freedom sample equality hash apply bit string retain accept reject expand technique replica exchange metropolis simulate efficient analysis require system theoretic perspective broad algorithms foundation national medical corresponding author edu prototype carlo transitions state accept reject random gets record practical mh implementation discard disk writing contain acceptance rejection desire also nontrivial impose restriction simulation compatible mh carlo representation kf monte enable importance sample complicated metropolis hasting prototype proposal applicability ratio easily broad usage since scale resource capacity accumulation discard advance mh algorithms experience illustrative simulation equilibrium interact atom
distribution coupling binomial negative binomial motivate beta mark regard process evy mass measure mark beta produce kb binomial set count atom count bernoulli draw like distribution replacement beta bernoulli process atom since count generalization version rather simply select already evy measure express define include show ibp beta bernoulli beta ibp customer walk try previous popularity previous customer additionally customer try place ibp take unlike ibp posterior occur conjugate binomial compute atom produce poisson q finite hence go simplification consider restrictive desirable construct evy non infinite limit evy measure q conclude approach bp k r also stick representation infinite count hide encoding encode sample analysis useful assign latent derive inference exploited compare characteristic eq cf characterize observed section count factor distribution equivalently gamma count gamma naturally beta gamma hierarchical structure formulate hierarchical strength group exploit binomial beta gamma poisson eq x x relationship r k c pr k k newton metropolis hasting mh stepsize rejection gr strictly p r strictly gr strictly unique hierarchical way connect result gamma support require restrictive mention investigate binomial process examine subset table include factorization allocation lda topic generally follow dirichlet distribution characteristic infer infer nmf lda let pa substitute em algorithm non nmf kl kl gamma poisson gap summarize special impose case change ik x ip ik lda variational equation brevity impose prior essentially distinction beta prior framework conditioning ki topic ibp compound prior actually give construction however arise sense ss uci research toolbox restrict vocabulary occur document corpus include journal unique comparison keep remove corpus detail corpus holding hold pi five testing partition discard similar nmf gap inference lda closely compare inference prior prior gap thus prior toolbox setting iteration corpus ghz pc show infer variance factor count draw active binomial close close factor case corpora tb infer factor assign result mcmc dominant topic example dominant characterize effect stop dominant topic keep diverse prominent prominent example capture adjust negative binomial infer factor stop produce readily interpretable remove show dominant easily interpretable infer iteration top corpus increase sign upper automatically fig last test count topic result setting factor automatically infer influence size accuracy generally support specialized factor loading binomial beta poisson modeling count augment evy convenient efficient poisson binomial reveal factorization demonstrate factor whose variance via capture common corpus acknowledgement thm thm proposition multi process gamma gamma poisson approximation beta evy convenient computation augmentation marginalization encourage document count setting medical care specie model corpora poisson binomial univariate repeat measure count extension incorporate variable principal factor analysis discover low incorporate variable tend prohibitive restrictive count
point algorithm scale impractical large moment scale old thank admm estimation make auxiliary develop exact choose soft upon may discard decomposition value parameter plug know slightly efficient run linearly default good consider great arbitrary application unnecessary uncertainty slowly simple fix scale beyond variable ideal concern sparse expect poorly see issue note convergence characteristic date find open use appear efficacy regularize discriminant overall plot lambda covariate diagonal diagonal also group nonzero upper use vary simulation unbiased misclassification l l lda signal snr adaptively third entry misclassification adaptively sparsity account outperform roc unbiased run realization enough perform though lda label choose train one classification peak middle end cv match nearly indicate solution converge concern reach compare lda choose simple maximize st regularize drop minimize bias nonetheless appear motivated estimating sparse efficacy plan admm rewrite affect motivate move finally arrive ease notation unfortunately necessarily property could calculate converge original unfortunately calculate minimize fix admm employ calculate first true initialize iterate q q instead step step update begin must take q c ignoring solve soft eq eigenvalue require reason hundred linear common distribute within pool covariance separate rarely insufficient covariance regularize model adaptively adaptively pool paper propose fit moderate problem efficacy simulate penalize discriminant interaction class observation covariate classify far independently normally mean come classify propose simple discriminant classify observation find high class classify class discriminant analysis mle constrain rarely covariance matrix decrease increase discriminant note pool find tradeoff convex combination generally partial wise equal covariate interact identically group give natural constrain ie eq likelihood unfortunately require relaxation sparsity among sparsity graphical recently joint partial many reader convex later pool discriminant wherein assignment forward bayes simplify term logistic identically nonzero nonzero forward model method sparse interaction recent interaction basic model continuous response entry often simple q show nonzero term estimate differ estimate boundary discriminant lda partition estimate back decision quadratic rather expected decision hybrid number pooling fill idea
simpler stop establish satisfy regularity mu strategy state preserve polynomial assess although mat ern old brownian motion old view characterize sequential resort make fine sound corollary lemma thm axiom thm thm thm thm thm remark thm problem thm principle pt ex france mail fr fr let formally set mu mu consist deterministic arbitrary sequence amount denote class sequentially class bound behave global justify proposition combine bad may whether correspond question view path theoretic random knowledge make quantity interest evaluate widely explore optimization computer assess article brief concern article part assumption deal let assume krige observation projection onto evaluation n mu n solve equation estimation q mat ern assumption equivalent eq domain empty assess quality mu mean mmse mean q screening optimality parametric design mention case adaptive even process view know mmse induction select mmse prove claim case linear criterion bad error ball denote ball adaptive mu n equal nx krige isometry concern mmse strategy gaussian satisfying lipschitz interior condition rate decay criterion prove non
probability direct crucial capture small irrespective probability drug worker utilize relationship member target facilitate chain expect participant disease typically conduct stage survey member attempt biased sampling study typically multiplier estimate population elaborate difference since briefly strict capture service almost decade past little justification interest survey complete survey hide mean attempt exactly present ht paragraph design restrictive adjusted analyze estimator show adjust desirable converge stage various empirical focus mostly survey also additional set regard design consideration contrast discuss way could apply easily circumstance unknown twice individual appear unfortunately positively correlate stage mark sample consequence study stage comprise record associate likely infect take hoc analogous assumption capture individual capture individual rearrange covariate uncorrelated particular know trivial therefore correlate use sample proportional degree good homogeneous heterogeneous stage perform walk result walk person capture thus capture exclude possibility consideration plug jensen equality upper bind simplify give q conclude homogeneous e naive low opposite true relatively yielding figure strategy prove numerator denominator sum like concentrated close expectation find concentrate close prove capture matter capture individual observable concerned remark capture twice thus stage demonstrate heterogeneity capture sampling stage heterogeneity individual capture sample calculated value true decrease naive decrease relatively across range similar irrespective simulate seed follow construct repetition focus repetition population mean bar plot vs combine yield population adjust similar test select population note accord encourage weight naive repetition general bar deviation plot black bar notice survey capture capture take record age discuss highly privacy record record age stage uniformly sample stage repetition group plot age group variability probably e qualitative survey reveal head average age yield relatively behavior attribute describe one might size g variance group head naive project simulate evaluate time complement make university source construct network census project assess adjust evaluate time obtain agreement finding ref bias large adjusted estimator true record conduct group record check white appear rest tag look explanation convenience seed group close size record slight consideration record ref population c c black common application sampling individual proportional concern direct regard implicit application incorrectly report possibly second note population assess deviation generally problematic sample nice capture preference like population weighting yield marked individual correction give age datum idea patient study population reference hand estimating exclude incidence list suggest sec even first stage non random individual privacy record thus suggest adapt use diverse utilize sec chain individual sample manner detail x exp nc stage generality mutual denote sum k easily apart sum cb centre modelling department population school white ac ny cb es study new drive capture size
ed dd ib ib b lb iv ix j z learner tree ensemble full pure full tree ensemble tree training subset attribute forest decrease forest total attribute drawback ensemble member classify agree case sample member sufficient prediction high via evaluation evaluating need voting stop probability evaluation voting classify ensemble initially step randomly remove vote run vote accumulate member criterion safe vote safe member draw safe ensemble member agnostic base vote vote process vote irrelevant disagreement I many way ensemble voting ensemble member chebyshev vote observe binary categorization random vote member algorithm invoke binomial infer around observe unobserved vote agree ensemble fall outside side terminate q critical usual finite correction account model stop simulation describe require upper power computing side far treat treat vote stop show error al stop early inference vote frequency model distribution dirichlet vote proportion update compute yet member current exceed specify class approximation exist class ensemble fit memory memory ensemble fit memory rule classify send read sub evaluate map test receive one test reach decision file copy read base read job vote vote output second every input explore efficacy vote behavior stop threshold method number vote need ensemble threshold practical ensemble threshold compute care avoid numerical avoid table mm factor loop produce complexity threshold take difference ensemble ms ms vote generate vote otherwise vote rule prediction evaluate label accuracy repeat million million report stop increase error compare five ensemble evaluation five approach gaussian tail g population require vote importantly benefit vary value evaluate show ensemble perform web web page compress language categorization predict web page english feature proportion character character use extract feature randomly divide full hour create file gb contain nearly randomly extract block gb us reference count broken species location count record describe environment much effort observer spend scale hundred american base environmental american united complex vary region make task testing count convert create intend appendix detail testing block example gb storage include file implementation nod one intel ghz block speed without accuracy serial train range ensemble train across ensemble member serial range min serial total ensemble version evaluate ensemble specify always vote require uncertain figure train block varied varied likewise gain parameter except ensemble block show bag node construction randomize e set evaluation ensemble datum use full single figure decrease average number figure size provide speed entire speed top row relative less ensemble need ensemble drop less bottom evaluating finally evaluation scale stop vast majority require require machine unchanged parallel way disjoint partition find voting tree partition enough produce boost tree accurate partition learn partition simple vote job selection choose voting aggregation partition et al work empirically create merge comparable classification serial fast distribute bag size benefit easy datum memory failure incorporate ensemble large ensemble particularly important wu train tree ensemble evaluation evaluate boost train disjoint combine fast experiment boost ensemble distribute overhead creates independently create massive tree accurate ensemble ensemble overfitte uniform evaluation train sub appear bag unlike unclear cost incur yield accuracy expect building build benefit node split local controller choose split communication exception construct tree pass construct involve fall collect sufficient controller final building include overhead task job job et al build framework support easy ensemble specialized specify test avoid file system scale gb remove unnecessary ensemble pruning study dynamically time decide evaluation stop differ reliably fix order voting ensemble evaluation pruning base decide meta train reliable different accuracy lead fast pass build block appropriate scale fit compare subsample serial ensemble combine ensemble efficiently evaluation cost option easy requirement prune could remove could network ultimately small accuracy domain future contrast tree pass source version consume direct comparison imagine trade acknowledgment thank david particularly grateful david suggest learn task national laboratory national nuclear security contract perform preprocesse step record cover rare group count type respectively attribute attribute thousand spurious mistake miss attribute token outside handle record categorical multiple tree handle long short build forest ensemble distribute block bag gb time subsample serial propose dynamically member reduce evaluation cost x tree ensemble evaluation integration technology daily life massive volume practically computer memory massive website transaction throughput biological sensor massive process streaming sized subset c datum computer analyse parallel often stream benefit make process large consume processor construct massive volume datum evenly partition construct ensemble form ensemble vote classification complexity resource scheduling use require approach minimize disk overhead short prove novel local employ importance instead usual bagging train ensemble bag forest feature example local ensemble ensemble thousand large point classify use ensemble confident easy implement approach follow divide massive single decision use approach bagging implement fast compute bayesian approach bag ensemble build partition phase job point evaluate large build ensemble variant bag difficult however vote equal weight trivially merge ensemble single easy vast majority
realization transition expression countable j j due assignment arm independently great choose state arm break favor arm center optimal thing probability matrix subtract construction break tie arm number center belief finite repeat none belief exist hold hold hold symmetric illustrate next consider arm center since optimal would arm center belief p reach never arm play infinitely long arm play center never reach finite reality never belief integer hold action optimal exist transition conjecture reason player arm affect arm evolve arbitrarily reward arm claim hold probability reward distribution word arm transition subset facilitate iid problem reward hence bandit assume arm state slot arm say reward k reward arm reward arm k kx correspond constant belief kk kx equivalent assumption iid set arm symmetric symmetric appear subset assumption sufficiently dependent regret player know choose assume start know player set proceed separately admissible compactly represent rhs degree denote arm transition transition extension clear admissible policy individually exploitation c count dependence drop bound play time arm least time take place exploration l chernoff hoeffding estimate transition true constant transition estimate sufficiently close great real q continuous action e transition solution true exist depend player appendix k appendix appendix bind hold l least least away true minimum threshold let constant arm strong hold hold lemma depend know exploration ensure exploitation logarithmic know theorem true transition probability type regret instance free open regret transition probability reward expression probability prove continuity equation system exploration player achieve know step solve finite mdp form estimate state finite transition original mdp except play computed fp sublinear regret fa show fa fp inefficient practically state space comment place pls transition c arm choose term reward similar keep use whenever arm explore player choose high one separately choose parameter unless constant fa policy fa transition know use algorithm average reward show fa perform equally well information state assumption state never fa assumption violate runtime fa comment elaborate please although regret bound perform policy total identical suboptimal decrease make equally fa see prove c almost fa reward arm policy high reward know two arm sufficient optimality horizon discount guarantee scope fa potentially c c c table reward fa propose sublinear learn poor fa fa arm step fa total fa exploration exploitation continue final sublinear use multiply fixed see computation visit fa step guarantee fa online matlab policy exploitation computationally practically policy policy avg run fa constant hold fa choose impact exploitation reward decision step poorly much great exploration step gain exploitation step em c bandit regret horizon player solve often costly practice belief discretization solution player structure greedy policy effort performance hold continuity property w substitute time approximation action begin measurable transition kernel q indicate variation countable correspond chain ergodicity ergodicity uniformly chain perturb kernel c c chernoff hoeffding bind frequently reward range let relate belief player true upper difference product size unit pairwise number set consider induction clearly observe player select arm observe next arm happen estimate u v l induction eq hence radius hold b compact guarantee implie exist give liu paper problem optimal policy bandit play select player maximize horizon arm transition time dynamic policy contrast commonly study respect single action policy achieve mdps partially variation performance online exploitation discrete rule user reward observe current state arm control selection system however arm player exploitation state design optimal lie balance optimal horizon discount stationary policy contraction stationary optimal transition probability knowledge arm arise many sequential channel channel wireless initially track movement design fast minimum horizon reward give statistic model causal system commonly policy static policy arm determine need regret show arm player know logarithmic regret finite near would iid bernoulli prove iid logarithmic knowledge attempt extend decision process mdps process sub parallel aggregation prove regret problem reward arm horizon reward regret logarithmic regret specific see remainder paper countable finite bind strong algorithm increase variant aim assumption respectively extensive prior armed start seminal arm iid policy asymptotically parameterize logarithmic order refer order optimal logarithmic consider index easier optimal et policy parametrize arm parameterize iid bandits states exist achieve regret prove bandit problem upper strong algorithm numerically index iid logarithmic regret process uniformly asymptotically extend iid bandits markovian markovian whereby arm activate whereby arm change markovian version bandit know discount bandit different arm arm drive optimal policy policy always play arm weak greatly arm reward arm observation arm evolve static aforementioned difficulty optimal bandit know criterion design bandit typically compare static always play arm expect reward logarithmic arm extend decentralize merge continuous path liu al sequencing explore exploit block length geometrically version iid problem work design bandit identical arm whereby finite player consider feedback bandit study computationally dynamic outside mdps possible strong policy logarithmic regret estimate reward kullback kl divergence show optimal term instead divergence regret find probability another logarithmic reduce computation mdp reward literature apply regret contrast mdp learn logarithmic regret bandit regret take estimate transition learn play contrast action transition sublinear take pair infinitely index arm prove mutually markovian arm index whose evolve unknown transition arm simplicity without generality true either perfectly play reward perfectly observe receive uniquely space space arm probability arm transition ergodic exist notation let represent unit vector th element denote natural number integer product vector element whose transpose denote arm get select arm reward reward horizon player select state need arm probability reduce exploit acquire high carefully player player optimally unknown vector variable represent state choose step select simplicity assume label player observe action evolution random represent action observation matrix policy policie program player probability player know state system initially take take sublinear regret sublinear convergence rate mention early probability matrix learn approach section problem often belief simplex observe select arm belief arm choose eq reward optimality expect assumption existence markov chain probability space follow true consider function finite bound condition existence solution prove prove integer state therefore hold boundedness belief state adopt belief form arm select write state state observe state note countable subset player exploit continuity player method thus distinction state space statistic calculate subsequently belief subsequently analyze upon completion information possible state player select arm step select matrix knowledge state denote state set probability state information ic iw ft w w arbitrarily u arm ti j average exploitation phase phase player play phase accord transition player exploitation state arm player clearly q non negative player explore player exploit time player keep play player depend state probability exploitation player issue q advantage action gain index player action action
epidemic manner capture must result benefit extensively potential impact evaluation inform decision comprehensive transmission traditionally formulate deterministic equation partial increase computer performance prominent particularly low possibility infect ability epidemic ordinary ode prefer develop type impact discuss incorporate incidence specific study methodology ordinary equation epidemic consequence serious inclusion currently availability incidence pattern behaviour furthermore phenomena transmission clear area efficacy clinical transmission duration duration currently test duration nature acquire relationship contact population demonstrate uncertainty calibration ode epidemic statistically outcome contribution individual uncertainty population monte carlo sampling gain essential perform review review extend statistically challenging relevance interpretation calibration response incorporate tune typically result follow rate stationarity chain example widely hasting markov proposal tuning performance mcmc typically chain stationarity regard theoretically derive act complicated discussion ode epidemic ode ode parameter space provide proposal fact incorporate stage ode therefore construction avoid computationally expensive tuning process hence incorporation mechanism carlo demonstrate improvement automate avoid mechanism approach classified mechanism distinguish generate name combination state sequence depend history markov chain kernel ergodic stationary distribution recent propose ergodicity prove design adaptation satisfy law posterior ensure law large number reader refer detail successfully metropolis formulate transmission treat incorporate whether agree available calibrate model key modelling framework paradigm couple ode automate methodology base incorporate response predictive adaptive assess collect source specific dynamic view distribute stage disease present intend describe move recover move epidemic assumption discuss assumption model people five age group absence people range year survey trial model may grow furthermore activity cause infection serve contribute age number activity age e transition vice versa model population people belong least group group never leave activity level old activity drop active risk group individual age gender choose opposite gender age mix treatment become aware status lose upon state correspond use response conversely general long infection infection support whereby infection lead complicated formulation ordinary capital people activity group denote derivative kronecker delta gender I take since comprise year member move age maintain people term obtain divide individual year reach g evenly add distribution mechanism enter leave propagate age period leave discussion transmission infected subtracting leave go either g role infection linearity system specific behaviour usually subsection construction discuss construction development member infection transmission feature developed age group community certain formation process refer mix find medical literature parameter throughout calibration ode epidemic behaviour survey serve reliable source information number participant survey survey survey likely irrespective experimental design population survey understand result take assume specify model formulation individual gender group opposite gender group age gender age opposite gender challenge take calibration reasonably practitioner worker social worker alternative involve simplify uncertainty specification suitable calibration mixing importantly present transmission use approach degree specify infection equation transmission gender become infect infected person gender easily incorporate several extent reflect formation pattern individual thereby make extent old prefer tendency old gender opposite gender last become whenever gender meet demand individual gender degree activity group uncertain specify matrix unknown parameter low complexity enable scenario epidemic ode addition epidemic specification ode purely conditional trajectory ode conditionally specify differential literature purely epidemic form ode detail forward obtain ode evolution ode gender age activity time component conditional ode forward ode population note fine study subset system observation several solver notice choice solver efficiency depend parametrization parametrization one system repeat generic computationally inefficient write solver intel differential equation library solver distinction explicit implicit degree system solver employ solver explicitly modification stage implicitly fourth jacobian interface remainder model two matlab solvers solver matlab solver ode system point speed considerably minimal obtain discuss option jacobian specify model epidemic extension mix matrix make choice work choice basic describe assign report specification completely set period use twice uninformative specification estimation trial limitation outside report incidence covariance accord study unobserved analytic solution trajectory transform age gender follow detail transformation specify situation population serious use mcmc mcmc learn recursively reject mechanism adapting thereby discover enabling mixing improve adaptation explore dimension epidemic model present detail iteration proposal construct markov simulation ode propose parameter vector ode correspond observation acceptance evaluate methodology internal base variant global proposal involve metropolis covariance learn explore distribution iteration proposal empirical parameter chain provide optimality present recursively detailed forward simulation study omit decide assume life long population simplification would procedure initial simulated system step year calculate generate age specify statistical ode take run forward obtain observation figure present posterior estimate ode epidemic parameter construct quantify statistical incidence prior specification derive mcmc sampling dimensional mmse posterior generate demonstrate work accurately synthetic study estimate trajectory disease state figure step disease observe trajectory within aggregated trajectory true aggregate furthermore versus fit incidence simulate agreement observation gender h ode real maintain duration individual choose well measure performance posterior associate gender incidence consider calibration except appropriate group think life incidence age acquisition rate age high capture incidence though incidence especially similar result develop model calibration calibration posterior take marginal forward result carlo forward calibration forward project ode solver sample state analysis distribution incidence post scheme receive year infection individual compare individual belong present predict response calibration epidemic level incidence develop ode transmission statistical modelling datum model ode model base projection ode static parameter together estimate require perform rejection demonstrate application estimation capability calibration incidence novel perspective studying estimate around equilibrium epidemic describe disease epidemic ode group poorly understand implication view relate epidemic infection attempt interest available aware publish calibrate incidence account confidence real epidemic consideration allow beyond currently available literature combination reason insufficient poor uncertainty interpretation notably well somewhat status disease short year ode appear use slow solution quality calibration concern separate group necessity deal nine prior want question ability accurately amount important address relate assumption illustrate property addition extension methodology trivial incorporate estimation decide maintain parsimonious like methodology mcmc automate additional suffer curse dimensionality metropolis hasting sampler mechanism solution affect change perform burden perform serious practical design facilitate framework approach epidemic ode grid increase epidemic represent place additional space suffer curse dimensionality volume add would ode solver sparse grid computational must explored contrast adaptive procedure computational effort ode epidemic mention straightforwardly bayesian paradigm selection select scheme perhaps goodness utility flexibility framework adaptive methodology appropriate effort statistically consistent weighted plausible calibration space sensitivity ode produce note statistically directly interpret advantage approach dr david discussion motivate conduct mr work conduct early project project ik research linkage project grant national medical program institute department south matrix widely disease define activity rate typically notation gender gender opposite individual belong activity group age brevity equation describe infection term person gender infect gender individual form acquisition call mix construct matrix construction study health survey despite limitation currently representative one acquisition age acquisition risk overall whole individual pool gender rate acquisition rate follow calculate gender individual gender gender calculate gender activity group opposite gender age group mix activity group individual gender activity active member gender become activity product side age suggest group able adjust sort depend need specify q help extent age activity fully age fully age fashion age popularity old reduce age age reduce age group age increase probability age belong age increase old group new population gender acquisition age multiply group balance balance demand want acquisition individual state acquire multipli thing imbalance establish demand gender gender gender e meet demand form h getting infect individual recover recover risk infection
ensemble expert weight forecaster forecast weight prediction weight bad expert expert individual combine draw ensemble expert track well place via introduce implement quite suitable mind however start low none much rather expert statistical actually share forecaster limit depend past grow ensemble add expert step stationarity expert fit keep update course prediction ensemble grows fit successively fit thus expert whole stream expert nearly rule whole nearly expert would fix share provide regret introduce exponentially forecaster grow ensemble modification fix share forecaster result two previous discuss significance sequence deterministic forecaster expert prediction forecaster expert forecaster prediction loss aim forecaster predict expert track cumulative good forecasting strategy expert ideally forecaster convenient sub observation likewise regret expert forecaster randomize forecaster little regret regret reinforcement negative evolutionary time fitness anonymous mention member ensemble actually expert forecaster keep even number forecaster describe next one manner update forecast initially eq control except weight share expert ever fall fraction weight exponential expert provide choose expert expert divide epoch length add expert c expert epoch epoch expert expert train hope cope course stationarity obtain tracking ensemble construction inductive write ti te I move weight te eqs expert within epoch track grow forecaster expert key prove e eq q action assume bound substitute latter familiar length achieve run modify fix forecaster make remark share probably tune well ensemble since term illustrate time great practical united record bank somewhat arbitrarily make year allow evolution growth stationary ensemble weight quite mis economic forecast smooth however autoregressive move fit residual give accumulate grow memory take advantage flexibility parametric non spline calculate good ensemble allow produce profile comparative loss also family weight new whose fixed minimum expert maintain share pre drastically cut dominate ensemble expert attain contiguous bound assume expert time ordinary loss able high model turn literature detect walk model specify seem ensemble capture process parsimonious concept drift machine deal input output unlike assume well nature expert incremental window grow expert handle mild specify able strong ensemble modification share still expert account grow arbitrary technical counting expert loss vary get introduce epoch evolutionary generative low deal character long even come fix system specify trend behavior well adapt automatically non il amazon com goal low expectation powerful guarantee often stationary ability expert expert combine plausible deal historical evolutionary system modify track cope grow fit new become available grow stationarity record gene prominent example problem anomaly sometimes process conditionally stationary historical though historical turning subtract trend analyze trend systematically
compact game heart matter minimax weak characterization either section opponent close study suffice deterministic high dimension ss complete closure manuscript term game term tuple player simply opponent similarly bound determine payoff payoff prevent trivial restrictive discusse satisfy state explicitly define follow exception manuscript iff iff analogously reflect suppose force force force repeat access selection tx deterministic suffice make rich tuple satisfy family opponent note player opponent assume play effectively strategy consider quantification opponent construct force analogously opponent quantification move move say move clarity definition effectively move correspondingly round criterion strong property respectively replace matching game grant every scalar final convention run strategy combine lastly hull notice straight force iff force iff force force property force force satisfy existence minimax suffice force construct b example demonstrate restriction fit becoming value say force subsection difficulty arise close nonempty q equivalent statement via duality z scalar hold convex provide player type existence nash equilibria suitable grant player whereas nonconvex structure allow side introduce major note repeat player projection equivalently hyperplane definition whereby either pass refer tuple force mean force force strategy opponent may force opponent greedy distinction truly early start allow opponent measure impact minimax equivalently force extend global property characterize compact continuous tangent grant set guarantee relaxed quantification strategy parameterize player eq opponent strategy reveal require convenience inductive term vanish otherwise grant inductive grant contain run unclear suppose union another nearly small piece miss way detect quantify benefit may force grant sequence derivation satisfy boundary away correspondingly ball boundedness pass interior player force regardless future produce allow various tolerance define remove strategy tool limit albeit construct opponent nonempty approach fail must infinitely choose opponent strategy follow compact hausdorff exactly nonempty opponent small provide next empty grant iff follow compact nonempty nonempty hausdorff nonempty os j contradiction grant way depicted eventually entirely complexity define player depict determine consist nonconvex theorem primitive opponent build opponent arguably constructive opponent importantly structure every encounter fact grant middle verify condition nearby actually location recall union center make thus take triangle strict treat cause fit contradiction since suppose extra guarantee nonconvex manuscript iff without generality compact contain contain gain minimax implie game statement grant strong minimax b piece throughout convenience map analog deterministic choose additionally strategy family augment member exact history clarity tuple eq result family effectively randomness reveal simply family adapt stochastic concentration independence suffice adapt strategy never care value rather sequence surely apply hoeffding inequality vary define even may across iteration independent apply direction establish opponent exist converse note bilinear compact grant present nonconvex small demonstrate relationship duration game perspective player minimal q minimal set proper consider every grant game grant particular eq play strategy determine inner remove arbitrarily stay piece begin outside opponent provide necessary piece require careful merely come z z consider boundary intersection satisfy second discard scenario take suffice third suffice satisfy b potentially perhaps adjusted length similarity triangle along away satisfy meanwhile mean grant property divide set intersection triangle rearrange desire mean similar triangle q b z every hull union h partial boundary actually pass meaning follow ball radius k exist force grant whereby strong force apply generality segment form adjustment placing force force appear presence optima lie plane lie must exist force force suffice symmetry lie appear plane force every satisfying set shift via shift grant guarantee satisfy nonempty close fall within force grant end inside appear incomplete albeit
group dynamical lead hamming quantile bp ar dynamical see share ibp behavior variation seq model trial e estimate mode sequence trial ar bp ar three top bottom panel display correspond colored place prior prior use contiguous predictive generating series comprise datum take mcmc iteration hold series likelihood conditionally independent bp hmm summarize hmm consistently identify ar mass ar shift positively roughly ar see hdp ar skewed towards low log whereas hmm key ar hdp hdp ar bp library infinitely behavior hdp assume select behavior transition ar subset behavior transition probability examine generate transition behavior ar exactly dynamical mode hmm aspect bp ar dynamical dynamic hdp regardless hdp predictive illustrate benefit switch dynamical successfully visual dynamical gaussian identify behavior amongst task ar benefit specify behavior illustrative examine six exercise combination arm circle raise two reach skeleton plot display trajectory contiguous second reduce box segment behavior color behavior analysis split behavior skeleton modification toolbox position angle select measurement informative behavior capture body angle angle angle angle frame preprocesse step window observation equal use data scale prior observation maintain hmms dynamical system idea herein relate limited easily specifie emphasize condition infinite reduce switch flexibility computationally future split merge proposal improve initialization time help splitting behavior problem upon switch consider splitting behavior occur split issue mix behavior dynamic behavior var problematic grouping address model behavior idea appear rate observation model grouping component portion allow flexibility increasingly important acknowledgment grant grant fa preliminary version detail present conference graph hmm mode bp ar hmm paper base feature ar hmms describe constrain along define running recursion desire likelihood compute sum message ar message hmm specify time birth move propose feature series namely previous contain parameter ratio note definition probability simplify ratio acceptance death move hyperparameter let recall definition write reduce compactly metropolis sampling simplify bp hmm series pt aa ff matrix create count plus appear appropriately index share consider pt plus create transition variant ik ff ik ff ik f calculate unique pt birth set death likelihood propose discard mm remove appropriately variable transition fix aa ff pt distribution update pt compute q working time start count plus increment imply implementation eq ks yy criterion nonparametric approach jointly model related time dynamical beta size carlo base process relying truncate sum hasting acceptance explore dynamical behavior death proposal base synthetic dataset promise beta markov markov switching series bayes generally focus inference daily stock index wish infer regime volatility grow field time financial eeg contiguous epoch complex e autoregressive exhibit among dynamic segment stock return might model volatility eeg pattern dependent type one like behavior among essence would combinatorial form behavior library behavior motivate paper multivariate velocity place go exercise routine series exercise g goal discover type behavior occurrence stream multiple multiple exercise take overlap discover modeling yet behavior assume series transition latent behavior individually model via dynamical system include switch autoregressive switching dynamical system prove useful diverse target track motion capture switch var underlie encode behavior var condition evolve discover collection behavior individually exhibit subset behavior relate series discover amongst series indicate vocabulary prior vector aim allow flexibility behavior encourage similar behavior motivate possible behavior model process indicate implicitly property beta induce encouraging share bernoulli specifically coin behavior series though clearly identical e ibp computationally var process partially review markov switch beta multiple markov switching process carlo truncation finite dynamical system feature reversible birth death proposal sampling rely ibp interpretation beta beta ibp outline section relate benefit propose dataset challenge motion process markovian markov conditionally independent emission modeling conditionally insufficient capture dependency conditionally hmm capture dynamical switching hmms many domain simplify property hmm autoregressive dynamical arise special case series represent realization dynamical phenomenon time switch dynamical mode var behavior behavior convenient shorthand behavior intend series dynamic behavior exhibit imagine represent exercise people expect behavior solely switch behavior time list set exhibit time define discover via interpret time one structure infer behavior share within maintain unbounded behavior feature nonparametric informally think coin bernoulli realization coin beta coin implicitly encourage sharing process realization inherent conjugacy analytic observe e beta process know completely measure sigma correspond real completely poisson completely assume yield collection completely arbitrary sigma improper distribution refer measure draw necessarily completely normalization constraint atom interval guarantee realization fig atom atom cc corresponding cumulative draw realization dot coin associate customer indicate feature provide realization atomic atom example realization show continuous independently sample realization series realization allocate bernoulli realization often summarize infinite ik feature variability employ I f I possible var behavior prior scenario beta behavior bernoulli bernoulli implicitly feature infinitely encourage flexible illustrate collection feature draw draw collection switching var define define dynamic behavior govern particular fact normalize gamma series delta define transition hadamard product define integer assign index amongst indicate precede generative distribution contain hyperparameter place extra component self analogously hyperparameter hdp imply j j jk really abuse infinitely reality examine distribution notation reader value useful likelihood hmms markov henceforth motivating series latent mode order ar hmm ik r var dynamical mode behavior pick feature behavior conjugate wishart cf place share comprised wishart conditionally mean define covariance separately latter library dynamic posterior pooling amongst consider bp result hmm set transition ar spatio comprise dynamic specification summarize eq graphical bp ar eq constrained transition j switching var develop binary observation hasting simplify auxiliary programming efficiently computationally rely predictive feature ibp ibp outline key birth death improve sampler ibp know ibp ibp arrive line case customer customer associate show fig q distribution prior process beta simplicity least separately location atom currently simply imply currently already portion generate follow argument simply choose ibp exclude series simplicity behavior ibp separately k ibp exploit exchangeability ibp binary metropolis hasting proposal mix fast give hasting proposal compute likelihood combine exponentially mode sequence variant ar hmms unique associate choose I unique integral intractable early ibp limiting process infinitely bernoulli trial metropolis proposal however move birth death reversible jump propose either single new feature exist eq proposal encodes create lead transition death reject maintain birth death eliminate associated gamma transition hasting compactly either birth move recall birth death modify jacobian term arise reversible share simplified treat assignment sequence message recursively backward pass detailed transition dirichlet prior conjugate derive transition mode generate transition equivalently solely update wishart comprise wishart normal prior I observation inference presence conjugacy equal prior hyperparameter ff conjugate follow hyperparameter hyperparameter assign prior generative conjugate hasting mean hyperparameter let recall definition eq hasting simplify bp hmm appendix bayesian building regime space recent series na I employ model couple share hdp assume space e demonstrate strict sharing ability discover behavior infer hdp hmm describe coarse
sample scalar impose require impose constant define ft trivially obey state need extreme opposite perfectly result generate trivially obeys important example instrumental follow density integral similarly absolutely dependence randomization permit specify absolutely nature admit either orthonormal j step eigenvector set preserve lebesgue dominate measure kolmogorov countable hold assumption final b j orthonormal r old side multiply iterate expectation continuously f w value lie intermediate orthogonal subset jk j b b j ta ta ta ta ta ta cauchy q enough triangle ta ta ta ta c ta g follow satisfied conclusion g bm differ row conclusion conclusion conclusion g g conclusion proceed condition check continuously zero theorem extension theory operator leave cone banach space borel equip consider point operator adjoint uniqueness example outline hilbert track derive outline uniqueness note remove complex need five cone nonnegative shall existence closure hull consider modulus eigenvalue least closure hull zero like associate simple shall compact solution constant sign vanish uniqueness mean contrary also theorem eigenfunction vanish r eigenvalue positive zero vanish everywhere imply eigen triple obeys multiply side integrate positivity everywhere appendix maintain cone ill pose inverse et relation among give fourth conclusion cone identify identification conclusion cone conclusion identification side find similarly conclusion conclusion jacobian rank local identification role restrict necessary sufficient relate subtract section classical identification moment condition differentiable true parameter model dimensional condition importantly show avoid linear give restriction value identification primitive identification condition model include quantile instrumental asset pricing asset pt important conditional eq functional work motivated restriction identification chen fan paper identification condition nonlinear include quantile restriction economic far identification identification condition local fisher identification pointwise sufficient identification restriction give curvature also primitive addition identification euclidean usefulness illustrate example primitive primitive regressor another identification identify condition consumption capital asset pricing literature condition conditional restriction restriction give identification apply moment instrumental general identification sufficient parametric model primitive condition apply quantile model section index asset pricing example conclude contain help explain brief identification parametric let vector context derivative exist condition state condition parametric local rank equal extend value depend structure banach banach I restriction local identification concept note identification ball strictly open bound calculate sometimes require fr functional empirical strong space analogous condition linear follow familiar identification completeness continuously mild model think regressor instrumental variable weak condition deviation linear moment chen discussion completeness bound completeness fr local open explain identification fr continuous fr neighborhood continuous unfortunately assumption strong nonparametric moment inverse ill pose survey obtain identification locally ill problem infinite consequently nonlinearity identify zero restrict restriction deviation relate nonlinearity q condition include number old continuity derivative continuous derivative condition represent nonlinearity restriction specifie happen include linear fr constant impose give link curvature scalar twice continuously differentiable bound derivative value r l r restrictive identification neighborhood richer restrict necessary real number let twice bound bound h moment norm moment onto even example locally kk k bound satisfied give f locally identify provide necessary condition focus associated locally easy result nonlinear complicated absence assumption involve nonzero result model assumption satisfied restriction impose possible give interpretable hilbert compact mild operator mild compact condition compact eq eigenvalue eigenfunction denote adjoint assumption part operator restriction boundedness smoothness assumption chen chen sense analog right follow select countable orthonormal equal accord marginal lebesgue measurable properly obeys example highlight include impose boundedness positivity restriction use randomization employ economic previously notion argue rich induce iv inspire absolutely need lebesgue marginal absolutely case perfect require density one choice inferential j characterize coefficient necessary fouri old j j ellipsoid nonlinearity allow well environment impose fouri coefficient smoothness derivative chapter sense identification impose impose van generalize hilbert scale chen reference parameter illustrate scalar space integrable conditional pdf pdf x link assumption complete add primitive local primitive identification discrete lee condition rate nonparametric estimator identification possibly decompose component assume banach eq vector residual inner g g also banach inverse theorem semi parametric moment ai chen identification consider lead identification parameter even derivative identification imply neighborhood effect identification g local neighborhood model often suffer curse practice thus dimension q instrumental apply show identification instrumental variable nonparametric model part separately projection vx w jk simultaneous equation still hence identification require completeness conditional instrumental give primitive condition variable hold completeness one necessary one dimensional strong restriction dimensional consumption capital asset pricing provide example lee chen apply identification utility consumption easily specification consumption consumption growth consumption give substitution asset q one external special case chen formation risk asset price consumption singleton economic market substitution solve asset return return impose helpful restriction moment restriction multiplying moment uniqueness differ nonparametric iv restriction equation integral nan operator dimensional kind example need completeness integral second kind arise model inside integral regularity give eq compact direction close specify identification give precise impose regularity condition apply map b primitive suffice let c c ac bc c hc hilbert compact impose completeness version previously note find separable weak identification away necessary small consumption growth amount infinity
research control dms dms award finding recommendation express view identity projection th matrix position zero elsewhere index position elsewhere decision counter subtract side expectation side obtain useful throughout proceed tuple update note depend let expectation final moreover convex turn line intersect schwarz multiply independent first cauchy jensen inequality final jensen obtain simplified recursion involve structure nonnegative sequence complete linearization put steady note sufficiently concave yield satisfy long processor fast set final step hence side suffice target value choice automatically suffice inequality follow proof section conjecture subsection computer sciences department st popular achieve state performance variety several researcher recently propose memory synchronization use without scheme processor share memory modify part scheme magnitude keyword gradient small rapid stochastic descent intensive sgd scalability inherently sequential nature processor web parallelization many large process parallel much recent parallel sgd focus tool develop extract web tolerance simplify ideally suited online numerically intensive ensure indeed google size machine preprocesse necessary statistical consist problem system performance advantage high throughput share processor share physical gb typical read less frequent achievable parallel synchronization amongst scalable overhead overhead call allow memory scheme might processor sgd step modify memory speedup processor occur problem formalize speedup canonical machine collaborative product convergence demonstrate implement one sgd practice sgd recently method significantly slow respective free variety induce hypergraph induce row edge hypergraph simply graph whose throughout natural function learn interest act induce node induce edge example suppose support sparse rewrite arise distance popular resp resp problem involve frequently arise comprehensive survey nonnegative matrix index goal graph similarity arc correspond nonzero list associate cut entity resolution string index author e propose precede formalize define edge determine fraction intersect intersect sparsity regularity feature appear clustered hypergraph one assume provide divide linear relatively discuss parallel setup memory processor processor processor read store share addition atomic perform processor hardware operation single atomic exchange purpose processor require auxiliary processor standard range equal component euclidean matrix onto coordinate I equal zero elsewhere multiply coordinate component know component index processor sample gradient processor leave alone stepsize processor processor decision processor bite careful definition break cycle compute subgradient yield individual asynchronous incremental hypergraph isotropic number counterpart since nearly speedup turn protocol analysis tractable update follow replacement processor uniformly subgradient stepsize completely notation replacement scheme gradient compute one scheme component carry decrease perform modification procedure begin processor respective mechanism analyze replacement tractable analysis replacement replacement step replacement sampling state result parallel scheme follow simplify lipschitz lipschitz modulus minimizer even ordinary main summarize lag integer reduce achieve sgd protocol achieve processor form bind plug bound minimize minimize equal set error protocol still logarithmic initial stepsize robust protocol curvature arbitrarily dimensional convergence factor curvature rate occur round implement agree processor eliminate band message processor reduce processor allow synchronization idea present seminal instance gradient computer master worker setting convergence extend sgd convergence variety delay computation stepsize protocol procedure theory theoretical demonstrate protocol provide propose setting machine average show serial scheme involve via distribute protocol propose implement machine communication require core average scheme et order decision writing speedup lag severe magnitude type c set netflix e epoch core implement fair code rr identical schedule update notable software release rr mechanism need round order magnitude increase discuss ground protocol identical line fine grain induce undesirable slow code run x core gb ram software tb never store disk file disk share read factor run pass epoch large look pair percent describe text training test demonstrate suggest nevertheless figure factor speedup rr worse indeed gradient implementation train average pass parallelization train gradient computation serial run serial ten parallel averaging run large netflix datum row reveal column reveal set note memory reading disk attain speedup take speedup rr completion experiment quickly rr serial serial netflix slow way segmentation cut popular two cut scan voxel associate connect see cut rr twice slow serial entity recognition entity list website entity datum entity string compute individual quite sparse core slow rr speedup computation slow outperform ccc three massive speedup parallelism epoch
et rank element moreover derive order gain great understanding mechanism formulae rank page every area threshold cut double ranking introduce triple estimator unique follow percentile rank side subset maintain rank non uniqueness classifier sd plug conjugate gamma level average replicate loss optimal column scale express indicate p p pt pt g ig ig ig n n ig ig ig ig ig close entry percentage percentage small percentage page report percentage regret particular plug proportion posterior simulation scenario plug plug performance ensemble compound plug compound four cb gr ensemble estimate scenario notable cb exhibit triple goal classifier compound gaussian superiority cb estimator gr trend cb plug outperform triple mle bad although plug rapidly increase poor plug describe observe increase result compound compound plug cb classifier improvement compound plug systematic parameter ensemble classifier plug model equation level replicate first column scale pt pt scenario n n ig g ig g n n ig ig ig ig ig close digit entry percentage close entry percentage point document different overall within percentage point optimal posterior cb classifier fact identical performance simulation page due cb plug preserve account estimator gr plug good cb surprising rank gr triple rank conditional upon optimisation therefore gr expect plug well experimental result substantial plug compound compound model regret severe compound bad increase explain term relationship factor large sampling prior plug note plug substantially systematic plug contrast different simulation ensemble spatial highlight however choice evaluate plug function spatially ensemble complete highlight aspect classification spatially sc compose isolated area isolate sc surface varied heterogeneity modify generating produce true recover risk laplace l fully finally level count modification analysis spatial loss criterion posterior moreover performance plug cb triple ensemble point plug interest weight iii function evaluate consequence posterior loss dependent therefore classifier respect optimal yielded rate laplace sd five plug estimator expect level scenario isolate isolate isolated spatial sc posterior indicate pt sc sc sc sc sc sc sc high sc sc sc sc truncate digit entry close integer percentage spatial simulation plug quasi conditional varie depend scenario cb plug sc classifier classifier behave poorly simulation plug outperform variability heterogeneity produce plug apparent examine posterior level heterogeneity plug appear ensemble percentage area sc easier classify plug consider extent cb benefit dispersion mle extent gr appear benefit plug prescribed observe systematic improvement performance ensemble note car car model spatial category scenario consequence quantile procedure assessment classification sc form evaluate classification chapter entire heavily dependency gr visible gr classifier increase five plug spatial scenario isolate cluster isolate heterogeneity sc spatial hide percentage pt sc sc l sc l sc sc sc sc sc sc sc sc sc sc percentage close decision adopt increase particular weighted introduce conservative decision giving false large area risk medical loss quantile different plug interest result car laplace plug percentage finding report plug consistent scenario overall sc cb medium observe two marginally cb estimator sc cb gr behaviour outperform counterpart heterogeneity fourth section constitute plug estimator counterpart positive addition false page plug term performance point cb classification necessarily find gr plug outperform mle spatial mention find small posterior ensemble risk cluster estimate reformulate theoretic posterior whose already focus chapter plug theoretic indicate constitute candidate routine may aid report public health estimate note interest normally posterior mean converge note weighted penalty give great negative plug experimental shrinkage constitute naturally conservative choice produce positive false negative gr cb produce posterior percentage classifier decision paradigm posterior substantial describe page optimisation adequate order reasonably precise estimation value make thereby candidate plug loss positive converse issue false rate area threshold area equivalent rule positive natural identification justify sensitive health media suboptimal classifier risk classify public health conservative strategy report surveillance finding function page pc est case expand risk arbitrary relationship determine follow directly consideration chapter review main finding thesis extension consider thesis generalise sophisticated serve address thesis thesis adopt formal inferential issue arise firstly heterogeneity secondly pressure report research finding various use ensemble inferential analysis estimator parameter qr contrast quasi plug inherent difficulty associate ultimately objective thesis set nonetheless investigate formulation decision introduce framework inferential objective allow preference one goal another manner thesis introduce non spatial estimator perform par mle hierarchical improve spread parameter estimation lack another specifically difficulty specification loss see symmetry weighting ensemble classification identification exploration chapter propose penalty negative potential classification loss ranking originally author suggest generalised penalty point framework basis classify element ensemble basic principle thesis extend fp equation page family loss produce estimate zero regardless correctness side amount amount proportional proportional permit vary penalty negative moreover adjust directly weight relative although theoretical family would implement practice family respect rank generalise necessarily generalise therefore need behaviour generalise affect processing ensemble interest good lead substantial heterogeneity ensemble already several instance dependence quality ensemble concern lead empirical prior assumption heterogeneity classify thesis three prior conjugate conjugate proper prior non improper prior interest one tend accuracy ensemble often qr inferential encounter ensemble element preferable classification addition natural modelling problem chapter cut simply interest threshold procedure interested se sophisticated adopt category instance use reversible decision ensemble chapter calibrate threshold differently different median may nonetheless specific mixture parametric context ensemble level could aid ideal exercise conduct spatial simulate datum simulate present normal level different average synthetic pt ig ig ig ig ig ig ig statistic equation gamma average set appendix simulated synthetic simulation sf controlling report correspond cancer adjust occur extract cancer pt pt scaling risk variability medium control scenario sc sc simulated model sc sc sc pt sc sc sc sc sc choice scale sf variability medium four spatial scenario sc sc p low sc sc sc medium sc sc sc sc sc sc low sc sc sc sc sc sc sc sc sc sc sc frame line car normal car v car weight alpha scale line car car convolution n u rr car car u alpha v v line label I alpha rr alpha v lemma conjecture remark proposition proposition ensemble application supervision david thesis college constitute modern hierarchical theoretic framework ensemble report important particularly ensemble unit interest estimation may vary inferential since satisfy range investigate set meet inferential thesis produce quantile ratio classification purpose review decision theoretic framework optimisation ensemble cb square triple gr ensemble maximum estimate ensemble posterior latter firstly set plug qr synthetic spatial spatial regret loss choose plug estimator plug quantile qr area weight unweighted false positive negative converse unweighted unweighted framework quasi scenario five also literature estimate constitute plug loss approximately cb gr functions surveillance report uk demonstrate classifier optimal study plug classify risk conclude chapter thesis specification tailor serve inferential goal pt dag direct iid identically york model car autoregressive cb ig gr integrate laplace york maximum odd ratio normal odd ratio pm quantile square qr ratio loss square smoothing square square parent acknowledgement write thesis always present long chance distinguish I great I understand stage thesis dr de grateful particular would like van suggest support uk research acknowledge college like thank addition department university lastly great thank throughout concern report mapping wish cancer cancer area public health compare different may interest indicator task summary statistic constitute complicated variety goal goal estimate optimally area alternatively select histogram true ensemble related point heterogeneity public health covariate naturally exist criterion report ensemble point estimate yield solely estimate certain ensemble within former inferential goal permit interest optimisation function point loss resource become area increase collect public health expansion especially use prior strength permit variability estimate bayesian optimal mean interest readily summarie choice author particular researcher demonstrate mean estimate true unobserved limitation motivate suggest function produce match empirical ensemble ensemble certain part specify element successful inferential objective achieve triple constitute good empirical ensemble goal however appear qr element amount qr good candidate quantify related absence dispersion ensemble frequentist little goal wish thesis interest education lead routine surveillance drawback method make surveillance particularly significance need uk public acquire development reflect journal advance disease surveillance publish international surveillance focus monitoring count classify ensemble classification could assign quantile qr optimally meet inferential goal central thesis inferential behaviour plug evaluate compute posterior specific plug estimator spatial non simulation thesis chapter describe principle specific commonly spatial respective estimator thesis plug chapter dedicate optimisation empirical qr dispersion ensemble classifying surveillance consider possible extension technique thesis emphasis serve inferential goal chapter briefly decision theoretic attention commonly absolute definition provide family type throughout thesis model emphasis parameter ensemble issue relate review brief discussion adopt theoretic ensemble estimator cb triple gr plug consist ensemble thesis particular optimal within inferential dispersion parameter ensemble ii throughout thesis criterion evaluation theory special difference frequentist point issue arise thesis throughout chapter rest thesis restrict introduce thesis assume real decision sense constitute opt among decision utility strength theory fully theory firstly denote state refer act reason generally term decision link define loss state consequence know theory replace order provide several property demonstrate game probably accept also decision thesis especially simply decision identical also line assume moreover consequence one albeit discussion description utility bound precede theory point naturally game attempts modern game select optimal decision follow rhs differ handle frequentist traditionally describe perspective sample basis denote estimator estimator take perspective rule loss population call traditional specify state reflect world specification bayes prior entirely distribution rule equivalent via equivalent bayes risk term action notational decision implicit thesis adopt inference operator respect posterior generally denote except define precede assume improper well case well context refer generalised commonly improper generalise bayes decision bayesian model world frequentist perspective true set constitute discussion decision put nature may specification consider everything else equal decision however facilitate problem function accept throughout science three constitute key development square ii iii review turn especially therefore respect statistical euclidean candidate strictly mean compare make quantify loss bayesian posterior median gray fill gray corner font rectangle draw circle draw circle draw node draw circle circle circle minimum denote parameter ensemble control dependency random variable traditional effect probability directly different hence joint specification hyperparameter specify distribution distribute acyclic dag refer iid nuisance albeit typically sampling simple assume compose function iid modelling relaxed thesis priors identity simple normal compound compound gamma eq gamma proper prior model logarithm risk second level sometimes conjugacy example conjugacy study link conjugacy specific sampling evaluate improper linear count dependence popular prior inter dependence intrinsic car car marginal ensemble therefore generalise detail thesis study population refer affected case divide briefly present conventional assumption model known binomial represent disease population thesis especially interested modelling rare disease rare relative reference count likelihood ensemble refer tend flexibility type specify thesis hierarchical formulate implement software improper flat vector capture define ni area normally area proportional tail specification smoothing area car laplace laplace car laplace identical car refer thesis refer parameter structure ensemble several parameter interest wish ensemble loss form eq use notation adopt ensemble optimum bayes denote sometimes compound compound use entire loss multivariate posterior constitute ensemble property empirical ensemble generally define indicator analogy context iid random effect iid may ambiguity posterior base discuss sometimes ensemble quantify distribution ensemble quadratic interest instance empirical ensemble bayes turn key study although theorem case generalise relaxed distributional retain whereas conjugate compose normal relationship density ensemble control two describe equality hold present lot total variance e total sense former conditioning side equation dispersion posterior mean ensemble commonly encounter shrinkage bayesian towards desirable property little justify modelling limitation especially affect count theoretic produce empirical ensemble concept sequel introduce definition element ensemble define equation ensemble notation function sequel percentile eq quite reference quantile percentile rank practitioner percentile quantity percentile rank quantity quantile inverse cdf monotonic among rare quantile cdf argument discrete variable right convention last infimum variable quantile variable ensemble define parameter abuse spatial structure iid satisfied understand reference mode distribution wish vector respect notation sometimes useful allow accept argument monotonic decrease particular transformation monotonicity integrate function sequel quantile percentile rank introduce different technique derive quantile particular permit address highlight three decision theoretic specifically construction plug space order address hierarchical point est subject mean true eq q function generally refer cb albeit empirical ensemble lagrange multiplier played explicitly transform equation resemble estimator element mean unity index compound cb estimate interpretation ensemble differ specification conventional posterior I constrain similarity formulae mean every equality controls weight take oppose produce point ensemble cb cb estimator gaussian cb suffer limitation influence functional particular cb first two cb ensemble skewed approach attempt limitation ensemble triple constitute cb solely moment successive goal turn result successive gr rank note gr goal good consecutive step equation order obtain ensemble posterior follow secondly rank rank ensemble est rank true rank wish vector compose integer used estimate rank arise successive produce triple estimator triple heavily rely parametric permit case joint conduct use simulated rapid inclusion quadratic loss denote element strictly constrain satisfy square take weight reduce weight rank entire notational convenience ii bayes whose element formulae law every equivalent addition moreover trivially generalization inferential goal adjust shape extreme quantile adjust emphasis parameter symmetric consider highly describe specification middle rank specification thesis ensemble estimator function variety straightforward several inferential thus performance particular use way compare sub difference associate incur formally loss hold concept although expectation specifically point concept setting finance consider plug value plug classical loss denote review conduct parameter take constrain triple goal moreover ensemble useful benchmark compare clarity experimental vary useful achieved generally report distinction sometimes absolute ensemble firstly determination either demand unit high risk surveillance cut interest follow choice heterogeneity chapter treat element ensemble threshold point summary heterogeneity present instance indicate quantification heterogeneity link ensemble chapter heterogeneity qr quantity function loss estimator ensemble obtain cb also measure evaluate plug non spatial simulated gr plug scenario performance noted car tend optimal quantity basis recommendation qr frequentist main effect attention turn ensemble quantify heterogeneity present whenever quantitative statistic constitute useful group nested model n heterogeneity general measure generalise generalised interest response parameter effect dispersion ensemble interest generalise effect include crucial risk modern non function effect unit area exist control unit specific variance several constitute heuristic solution quantify interest model commonly heterogeneity attempt heterogeneity effect conduct study potential dispersion individual model effect constitute dimensional factor control develop disease exact relationship variance level difficult infer consideration address assess amount variability computing ratio choose absolute iid normal variance k moreover choose random suggest aforementioned specification truncate gaussian logarithmic dispersion ensemble meet singular literature adopt disease monitor health setting available available rarely distribution exhaustive evaluation require quantity within function mean compute median absolute randomly fitting become rapidly grow intensive quantify dispersion ensemble ratio parameter measure heterogeneity show statistic datum modelling especially generalise linear regard dispersion qr amenable standard optimisation decision approach quantify interest quantile ensemble computation describe goal thesis optimal ensemble conduct comparison posterior regret chapter assess simulation experimental particular construct use spatial simulation sequel reproduce structured surface evaluate impact plug estimator estimation dispersion ensemble analyse describe important hypothesis status recently family history may area year consistently rate city relationship report suggest set task hand shrinkage use ensemble provide dispersion estimator construction plug first theoretic estimation quantile empirical qr simulation plug conclusion introduce two loss ensemble quantile qr straightforward easily addition discuss evaluate quantity empirical especially relevant dispersion optimal quantity quantify loss function take note posterior expect depend joint quantile q ensemble dimensional expectation posterior plug quantile ensemble solely quantile plug quantile optimal bayes ensemble cb gr quantification dispersion ratio qr quantity first denote ratio likelihood likelihood combine intensity qr relate section qr obtain qr formulate quadratic qr argument clarity refer particular quadratic qr define ensemble qr posterior interest plug compound numerator denominator estimate alternative qr quantity q obtained estimate qr posterior numerator distinction ensemble qr dispersion constrain normally element however context estimation conduct loss mean qr quantity estimate qr latter difference posterior dealing ensemble replace quantity posterior loss optimal qr compare assess lead comparison optimisation qr describe follow take empirical empirical wish qr equation become estimator equation sequel optimal qr qr regret function introduce qr cb gr qr respectively quantile qr evaluate using assess quantile firstly well addition level heterogeneity iii concerned assess quantile qr estimation distributional generate estimate generative identical description test effect qr compound compound skewed conjugacy two case regard conduct prior every bayes give shrinkage every simulation control specification however specification study compound gamma ig specification conjugacy obtain posterior simulation choice detail synthetic model report table simulate observation variance different effect distribution choose interest size simulation choose take experimental factor variability aspect make select ratio sequel formally compound compound gamma level similarly model order shrinkage variance compound gamma respectively choice approximately variance keep comparable compound empirical gamma adequate choice prior result replicate factor aforementione identical model compound burn conjugate compute quantity various plug distribution non simulation page estimator compound compound gamma different replicate set entry scale percentage p pt p scenario ig n g ig n ig ig truncate percentage truncate small point page compound compound posterior percentage posterior percentage correspond denominator formulae posterior remain regret gr plug quantile gr estimate simulation column plug gr cb mle ensemble increase outperform point page impact b increase behaviour explain ensemble recall set choose tend since control hierarchical yield phenomenon visible panel tail distribution explain rapidly increase plug perhaps gr compound gamma model percentage indicate compound exception trend gr plug systematic trend identify ig optimal ensemble study page point remains center thereby tendency distribution ig skewness make affect cb plug plug increase systematically effect term add variability associate directly cb effect term hierarchical shrinkage subject large low sampling variance change overall shape cb thereby lead trend gr gr difference definite conclusion plug increase regret ig finally size ensemble result systematic increase percentage plug early relative estimator empirical ensemble plug quantify ensemble table remain cb gr plug absolute ensemble ensemble scheme qr plug model replicate estimator first scale express p pt pt ig g ig ig n ig g ig ig g ig percentage entry percentage qr table page compound loss replace order plug percentage qr order function exhibit percentage across scenario except n plug worse notice qr size plug explain modification increase ignore estimator bad mle base qr cb plug triple goal match simulate scenario systematic percentage regret identify gr estimator q level heterogeneity mle cb estimator systematically decrease increase cf behave direction whereby yield gr appear level gr qr regret ensemble gr loss model strong mle cb five nine table ensemble estimate compound gamma yield qr systematic identify estimator dependent level triple plug ig plug contrast low percentage compound gamma ensemble bad estimator cb plug goal systematic estimator gr plug performance qr scenario simulation evaluate plug realistic modelling study firstly spatial parameter secondly consider modelling parameter overall cancer west occur expect vary across simulation assess influence plug plug interest spatial main simulation smooth car scenario construct single isolated area risk simulation scenario situation contain relatively spatial structure sc area sc spatial occur overall count keep scenario west uk generate protocol one isolated cluster sc isolate isolated pattern function pattern covariate produce medium sf locate west uk protocol medium sf west uk protocol variability four spatial construct situation isolate five isolated isolated area pattern spatially three level across scenario specific scenario replicate consider isolate area randomly select cluster area sum count entire remain cluster denote whereas area e level stand risk choose thereby create rr datum create situation heterogeneity simulation risk risk firstly area index contiguous area describe first scenario ensure expect count variability vary every define rule except risk sc randomly area adjacent overlap buffer around region exclude ensure adjacent third spatially risk surface specify symmetric denote region convention total particular moderate low twice value specify spatial structure third set overall variance spatial vary surface heterogeneity region create q social intercept throughout produce set generation rr variability simulate multinomial trial count denote scenario variability produce replicate thereby set spatial set posterior parameter ensemble evaluate scaling factor sf combination factor additionally varied q generation consequence scale datum sf example produce simulate true ratio medium sc sc sc sc table page table statistic model use combine effect car car fit car prior stationarity specify flat car normal car hyperparameter specification constitute common choice code report appendix simulate qr estimator quantify estimation estimate respect choose choose evaluate theoretic interest express parameter dependent square qr respect qr qr square qr discrepancy laplace scenario q posterior level variability spatial isolate sc isolate area sc heterogeneity sc surface covariate sc average set express low sc sc sc sc sc sc sc sc sc sc sc sc sc close first digit entry truncate close simulation car denote car posterior plug note symmetric model overall gr plug estimator outperform plug derive ensemble cb well empirical poor estimate outperform cb scenario sc simulation heterogeneity sc lines sc sc different one sc spatially risk scenario page typical discrete nature sc ensemble behaved shrinkage function moderate whereas cb gr result ensemble different shrinkage maintain typical property change dispersion disadvantage quantile ensemble sc sc property true superiority level variability substantial plug percentage scenario spatial cb estimator heterogeneity ensemble mle contrast appear performance although trend mainly restrict sc effect decrease percentage plug systematic trend mle plug positively increase rr ensemble car laplace mainly affect cb estimator performance car sc simulation trend verify cb estimator produce bad regret gr plug marginally laplace column page indicate inferior normal qr qr plug column variability four scenario isolate isolate isolated sc highly structure heterogeneity sc surface generate covariate replicate scale posterior p p pt posterior sc sc sc sc sc sc sc sc sc sc l sc entry percentage close percentage qr plug level experimental table page precede dispersion sc variability property advantageous variability converse cb three plug percentage sc increase triple find outperform ensemble yield car result loss consider qr percentage cb plug summary behave similarly qr percentage simulation htbp q five posterior report spatial isolate sc isolate isolated sc highly spatial heterogeneity surface average replicate factor posterior percentage loss optimal pt p pt scenario posterior pt sc sc sc sc l sc sc l sc sc sc sc l l sc sc sc sc sc sc sc sc l sc sc percentage close integer htbp qr six loss first spatial isolated sc isolate cluster isolate heterogeneity sc surface covariate sc replicate scale factor posterior optimal pt pt scenario sc sc sc sc sc sc sc sc sc sc sc sc sc sc sc sc sc l sc sc sc sc sc sc l sc sc sc sc sc sc close percentage expect page qr percentage table qr small result optimal cb plug comparison scale percentage gr plug qr function sf large counter comparative loss estimator although gr plug correspond posterior loss qr describe low induce high base episode design investigate differential episode ever episode conduct uk date analyse solely centre individual area contact service probable broad country incorporate wide variety address pass scan history schedule schedule record international disease broadly detail provide count deviation area count car laplace specification five datum discard burn computation plug estimator previously simulation plug estimator spirit direction thus separately qr qr five ensemble point qr similar fashion qr empirical qr equation estimator pt pt pt pt qr laplace qr optimal qr column p pt p pt car car laplace qr car normal car laplace truncate close digit percentage page family histogram ensemble car empirical distribution ensemble distribution estimator sensitive hierarchical cb gr balance ensemble cb gr behave distribution gr modify behaviour estimate observe panel car car also page car summary page empirical estimate modelling plug quantile whereas mle plug quantile opposite relative hierarchical quantile posterior counterpart plug quantile cb gr function difficult cb appear empirical qr plug likely qr overall ordering plug estimator qr follow order report non spatial simulation exception parallel cb empirical qr qr modelling albeit posterior qr slightly car importantly interval quantile qr car laplace thereby suggest quantity car page report posterior qr quantile order plug estimator study plug estimator yield large posterior triple empirical quantile quasi cb plug outperform triple qr goal cb term however large ensemble set region affect plug remain qr appear agreement conclusion particular counterpart analysis heterogeneity qr yield approximately upper risk exhibit heterogeneity area variability finding good quantile qr triple gr plug behave throughout spatial
permit derive production rule structural show syntactic arc syntactic branching rate language present syntactic syntactic generalization model give syntactic filtering band numerical study syntactic experimental syntactic trajectory pattern use conventional processing sentence structure dependency make hidden stochastic naturally language major tool biology dna sequencing protein hmm insufficient capturing spatial effort enhance incorporate state attribute target span engine utilize target trajectory velocity cross attribute track track feature also find complex recursive example plan infer generate computationally intensive surveillance track sequence directly level inference drop person person lot target turn infer measurement track coupling track loose temporal inference single perform surveillance sense identification via ground move array processing filter conventional wiener form single hand wiener filter form component receive move dimensional although use conjunction base motivate syntactic model syntactic tracking algorithm present overview tracking language behaviour syntactic syntactic pattern express decompose descriptor trajectory primitive pattern line primitive target syntactic support syntactic syntactic aim ground surveillance move classify trajectory pattern motivate syntactic tracking syntactic approach security gate turn around circle around build move track recognize spatial identify pattern syntactic tracking example high description motion surveillance characterize certain formation level inference geometric interest pattern filtering enable characterization identification trajectory stream approach interact recursively compute q exponentially merge instance hypothesis mode merge merge syntactic filtering keep apply prune specifically syntactic filtering term probability keep bayesian probability likely trajectory track syntactic pattern computation probability discuss sec perform syntactic system framework syntactic five processor move target optimizer assign measurement track keep track target compute target sensor pattern knowledge base production pattern geometric pattern enhance characterize transition refer characterize geometric remark already exist association probabilistic evaluate track association tracking assignment solve optimization field syntactic modular well suit data move stop target complete detail syntactic trajectory provide discusse estimate syntactic geometric pattern prove specific parse classify sec motivation outline spatial description syntactic formal theory finite set production leave paper letter letter string markov termination rule specified terminal generate denote production form specify denote exclude case length string include indicate terminal side production rule independent contrast free embed embed represent chain show compactly generation segment production use symbol symbol string derivation production rule eq b illustrate possible acceleration depict mode model ground velocity transpose uncertainty direction indicate orthogonal noise switch remark ground target compare acceleration acceleration semi jump ground exhibit model equal direction mode assume different mode ground target move particular observation matrix element range rate angle measurement platform motion sensor time mode sophisticated pattern clarity focus line line arc generate regular target move arc associate syntactic processing save arc align axis trivial trajectory extend etc chinese character language include string string derivation hide arc compactly matching mode stre number forward mode basic arc parse inclusion cause parse case production rule arc regular self need eventually move language rectangle string rectangle side comprise turn side trajectory coincide reason comprise second view recognize target robust rectangle end coincide begin may language operation arc opposite continuous syntactic arc comparable identify location together within moreover event syntactic tracking trajectory trajectory identify open detect move toward ready formulate syntactic filtering mode filter model previous starting explain generation language encode rule line respectively generate arc point fig counter turn length production arc capture note illustrative exhaustive application development production rule practice production assignment keep next provide language use language generate sized sized arc course regular arc amongst trajectory algorithm incorporation syntactic semantic summarize process current scan geometric input sequence keep likelihood mode evolution production rule syntactic filtering track sequence iteratively build geometric estimator compute implement particle nonlinearity need describe mode feed continuous mode denote transpose approximated associated mode resample avoid degeneracy step involve generate base matrix involve transition calculate importance mode yet normalize equation use becomes normalize generate resample replacement necessary effective compute eq resample degeneracy occur certain one kalman process q noise convert convert measurement sec terminal parse store mark enforce consistency two interact mix match probability combination ready syntactic processing need integrate tracking describe system framework illustrated assume datum association module syntactic extend keep parse robust data module track syntactic unlikely track computational largely specific syntactic measurement extension later parse introduce production track scan scan let spatial correlation many spatial correlation experimentally production move production generation add modify include map module parse give intuition parse short string bb production list production parsing extract state entry table applicable string state place build detail produce previous operation state contain operation search marker marker symbol operation predict production rule please state look index marker terminal terminal input string index position index marker state please entry terminal lastly index marker state advanced position please completion completed completed discuss associate line applicable explain string empty add production predict closure leave corner compute significance reader state discard forward threshold value track form detection capture start instant operator string state generate filter update update parse operator advance marker state string marker end rule state predict generate state derive correspond match terminal explain marker accordingly associate accord closure unit production compute cyclic complete threshold trade track completeness parse incorporate knowledge human operator logic add probable add whose production yield symbol compatible string word purely parse incorporate parse parse collect experiment setup pre summarize numerical finally sec level syntactic substantial possible collect band mode wide imaging collection mode surveillance feed e carry enable collect surveillance unit near centre embed system centre undesirable deviation ideal hz angular increment yield velocity gps internal kalman result position velocity external kalman give stability phase correction trajectory result coherent collect scene move knot position discuss ground move trajectory geometric ground point radial velocity could continuously move point angle angle length frequency coherent duration acquisition second fairly snr velocity move consider feed consecutive tracking input target need similarly require several eliminate eliminate although tracking range range respectively use component sensor platform provide give range rate angle local coordinate plane coordinate order tracking plane centre syntactic illustrate deal arc numerical filter kalman similar extend kalman show track illustrate fig trial real track dot line extend perform quite turn constraint impose mode noise state kalman syntactic parsing mode bottom panel four mode easy display syntactic parsing mode soft soft hard hard parse soft parsing focus parse arc pattern parse trajectory arc two arc detection track parse sec arcs arc identify arc irrespective orientation illustrate rectangle panel maintain high support segment terminal string drop drop certain terminal syntactic syntactic could completely could greatly parse signal processing geometric syntactic fed syntactic arc useful algorithm syntactic sec fed mode weight mode output instant offer mix equally mode geometric move dotted line correspond time sharp turn turn covariance syntactic tracking goal syntactic filtering making behaviour syntactic signal geometric trajectory word syntactic geometric pattern form trajectory provide target parse trajectory implement parse parse bayesian tracking track extract track anomalous spatial trajectory model move move stochastic context free adaptive parse tracking context
solely university ny depth arbitrary without skeleton configuration correctly pose degree freedom depth dimensional pose advance image convenient challenge provide training estimating pose arbitrary skeleton pose estimation evolutionary algorithm rather belief train solely explain vast skeleton survey two method particularly successful human tree approximate pose core hour million randomized pose domain explicit additional comparison motion pose complex model underlie structure representation object tracking estimation optimization skeleton model observe angle branch link parent robot camera depth sensor drastically subject capture robot link second link degree arrange robot distinct pose collect range angle pose set eight subject pre background run evaluation approximately core intel processor minute evolutionary successful pose place qualitatively score survey hill baseline superior sharp evolutionary hill optima drastically
parent option depict grouping coefficient explore htp new represent pattern commonly adapt advantage pattern fast compressed improvement encourage coefficient sparsity parent persistence regularization child group denote group group application disjoint couple lasso offer deal deal introduce coefficient group certain overlap propose analyze subsequently column replicate shrinkage overlap lasso treat coefficient parent group persistence transform motivate cause scale modify master copy copy variable force agree encourage strong persistence scale previously overlap lasso quadratic penalty apply henceforth lasso lasso fig organization right invariant preserve encourage child persistence organize penalty child lose randomization ratio less group ratio small indicate strongly reconstruction htp evaluate real equation haar wavelet sparsity induce transform explain htp illustrate deconvolution fig compressed variance corrupt htp improvement penalty noise scheme variance varied error toy fig average trial toy trial employ applicable reconstruct piecewise length jump child measurement signal implicit oppose conventional see propose coefficient natural coordinate wise penalty traditional group overlap address new penalty demonstrate penalty compare partially support university statistical deconvolution sense reconstruction work issue greedy suboptimal reconstruction propose modeling group solve perform deconvolution coefficient deconvolution sense commonly tree graphical superior situation inverse deconvolution sense linearly mix dependency exact algorithm past issue greedy iterative pursuit modeling group penalty exactly compressed efficient standard approach lasso modify dependencie nature structure widely use g structured admit pruning pass operator strategy general unlike motivate main pattern sparsity wavelet capture model persistence large small
rule set existence uniqueness bregman entropy projection geodesic affine arise third taylor relative accept unique make appeal derivation cox might derivation accept relate mathematical update operational description specify choice decision consistency person principle choose arbitrary inductive inference opinion inference oppose construction accept member give range decision information crucial observation see always frame specification outcome allow experiment variability fact determine consideration inferential experimental operational relationship entity theoretic scientific fact relative decision individually frame decision fact instance completely consideration outcome configuration range allow outcome theoretical provide inductive inference outcome historical development separate operational define operational inference justification rule give operational language justification choice constrain assumption express operational language operational description latter operational language provide operational experimental inductive theory mean shared decision construct experiment verification prediction particular give model setup type inductive inference simplify operational definition fix operational use theoretical operational experiment correspond theoretical plausible outcome interpretation passive static define relational relationship construction use experimental setup logical neither experimental jeffreys capable solution neither objective justify evidence procedure behaviour relationship operational construction experimental well theoretical operational relative user agree agreement meta theoretical character describe rule inductive experimental arbitrary scientific objective remove criterion coherence experimental establish link inductive propose inferential insight kolmogorov approach replace measure measure statistical directly define specify deviation affine prior eq algebra quantitative integral represent triple specify give triple lack existence say initial prior general consideration distance operational conceptual meaning require accord information description provide operational construction experimental give description correspondence impossible inductive operational verify instance mutually abstraction twice stream abstract structure criterion category experiment type give category abstract subject consideration active configuration input outcome association outcome course instance experiment experimental map treatment act configuration outcome decompose part part provide allow deal category experimental observation projection layer equip triple hypothesis assign equip geometry preserve hypothesis respective inspired interpret passive active passive layer call respect context rule iff replace absolute evaluation correspondence analogy object entity geometry consist integral understand quantitative describe setup type description determine inference relationship inference mm section probability aim mathematical conceptual logical reasoning sentence provide mathematical difference inductive form logical procedure specify base inference inductive depend reason inductive quantity choice statistical element everything single leave arithmetic around frequentist probability outcome limit influential year sound separation formalism inductive theory statistic frequentist interpretation consideration subject change change moreover inference frequentist approach principle justify convention possess mathematically justification frequentist beyond successful theory theory quantitative used need choice particular drawing require justification requirement conceptually relate experimental justified amount justify nature reality theoretical construct syntactic amount calculus formal sound calculus make syntactic irrelevant early jeffreys cox e inference attempt provide observe le replace description term integral function boolean normalise integral expectation characteristic hand finitely additive equip dynamical bayes continuous domain concrete provide inference notice bayes laplace precisely validity necessary frequentist model lagrange multiplier careful look note cox type derivation bayes equivalently
break degeneracy posterior observable central prediction inclusion dot uncertainty helpful break degeneracy conclusion analyse situation rapidly curse dimensionality markov carlo represent respectively universal choose wrong model parameter wrong take sign strategy refer permit detector plus subsequent counting one construct prior except hard adapt improve analysis accumulate realistic assume reference compute however long realistic generate multi rest prior mathematically propertie difficult begin simple construction together function proper physics study serve subsequent density index build reference yield credible frequentist robustness inference weight multi exponent permit reference flat physics model hyper surface three illustrative question mcmc seem little tune markov chain suit severe accumulate remain bayesian discussion grant de interpretation term parameter bayesian prior construction bayesian reference likelihood physics reference induce physics model posterior density continue indistinguishable model start difficult practical physics depend multiple landscape possibility another construction model hundred thousand million shall difficult task accomplish another physics become applicable multi three availability increasingly computer become routine technique monte development show qualitatively use base benchmark study frequentist method frequentist construct confidence project parameter flat albeit construct parameter profile likelihood fit profile fact conceptually detail extract credible region datum accord uncertainty irrespective systematic good guess etc conceptually coherent unified construct investigation difficult circumstance parameter place physics model choice prior therefore purpose paper prior difficulty conclusion draw new conclusion extract difficulty obvious prior lead successfully recent top production cdf namely multi intuition bayesian approach view extremely powerful put mathematically place method call use invariance excellent latter bayesian credible moreover perturb control way check robustness conclusion significance analysis prior publication analysis parameter propose rapidly prohibitive parameter solution begin proceed describe detail step compute interest prior marginal posterior study clearly step apply single count yield simple possible calculation prior parameter detail count multi prior sec three parameter universal generalization conclude describe count pass expect give purely additive event physic expect value letter encode function density pdf discrete expect signal lie interval result count remain cut however physic devise event rather background hypothesis readily generalize count count experiment event make q mean factorize two way ref likelihood marginalization permit avoid technical ref evidence expect expect signal derivation arrive next section nothing expect signal flat act ill equip parameterization flat new experiment single interest prior maximize intuition construction great separation quantify separation kl invariant zero identical gain count wish influence prior depend enter density twice satisfactory integration practice frequentist approach un thing however inference sense perform unknown completeness key algorithm appendix simplify considerably coincide jeffreys prior model adapt ref coefficient reference eqs reference integrate detail assess physics traditionally propose reference composite signal hypothesis favor alternative background thing normalization pn ds incur background outcome experiment significant decision reject thereby accept alternative physics choice kullback leibler divergence hypothesis specify ratio counting characterize search take bayesian analog calibration independent count count signal significance generalize fact product task map predict predictor determine plausible specific class would every indistinguishable look constant hyper arguably however information surface simple something challenge functional practice simulate cut f term sect application bayesian model important experimentally describe suppose wish rank candidate observation model distribution aspect need specify chance however reach possible rank model necessary use prior integrate improper latter prior proper construction reference publication prior physics dimensionality provide simplicity illustrate consider free fix lm observe grid lm
integral fy dy continuous function equip order corresponding eigenfunction function consequence representative furthermore verify yy consequently product linear rkh operator pointwise comment scope involve pointwise assume rkh co norm f precede definition example recall orthonormal underlie hilbert operator act generalize k hilbert domain scale give variation obtain I x case uniformly spaced approximate development various previously square matrix involve principal equivalently operator case strictly sharp minimax meaning hold take infimum bound infimum hilbert iv chapter meet illustration truncation bound theorem tight show fouri truncation geometric interpretation tb l f generalize operator general although sharp semidefinite eigenvalue trace q relatively partition sharp semidefinite c leave consequence different hilbert illustrate subsection reproduce rkh operator assume eigenfunction k condition decay decay exponential decay r corollary concrete meet sufficiently constant sufficiently correspond express uniform operator let e lebesgue explicitly eigen integral q write easily verify n example share involve periodic eigenfunction away introduce periodic row column periodic vanish row k q constant class find particular imply c eq display type periodic absolutely convergent uniform define simple explicit practically function integer elsewhere identity obtain principal index row column periodic suppose example decay polynomially apply combination kernel decay hence obtain correspondence diagonal eq define bilinear whose argument space decompose similarly term partition partitioning argument yield assumption control inequality duality put piece together note notation show note rhs problem detail particular change proof geometry tb convex hausdorff toeplitz range projection bind piecewise display bound norm ball restriction act norm particular estimate norm certain infinite eq eigenvalue depend sampling fourier hilbert fourier case sample eigen decay n technique sharp acknowledgement aa support grant devote reproduce base auxiliary begin state eq claim j x symmetric adjoint bind consistent look whole principal submatrix k let principal previous notation name block similarly denote event event k boundedness k apply c summarize show extend lemma pc sampling pi satisfy th basis note boundedness n eq bounds union q impose exponent furthermore eq hold sum furthermore bound recalling use converge partition index submatrix submatrix combination perturbation write fix argument note double sufficiently double get part quantity simplify look slightly version r convert low bound properly invertible hold contradict b constant noting right translate subproblem discuss explicit formula generality solve picture arise depend whether fig case plot equation claim generalize fourier truncation u ts select v appendix derive inequality use consider semidefinite partition q q matrix observe condition differently find another sufficient corollary example remark norm acting surrogate study subspace include usual empirical encounter rkh implication pack quadratic space function l metric regular borel continuously within design
distinguish time target time property track joint filter particle filter give position candidate instant decision neighbor sophisticated approach important disadvantage determine drawback motivated conceptually heuristic usually limit scenario difficulty presence trajectory significantly decision available exclude cause batch avoid track deal come even furthermore require evenly process gps apply gps component role gp responsible I force gp approach inspire association particular multiple perhaps object construct gp associate target component ambiguity target reflect simple model scope localization mixture gps propose similarity tackle trajectory demanding oppose trajectory standard variational standard tight organize brief review gps introduce discuss hyperparameter year gps attract attention nice art performance brief summary input predictive gp observation model noiseless plus independent proceeding power k yy property function unnormalize one fast correlation output decay input grow unknown function nn conditioning predictive computable time inversion typically evidence analytical available carry take pc set correlation expect covariance overlap mixture latent trajectory association latent determine trajectory entry model trajectory trajectory handle trajectory matrix place prior q e multinomial gp different trajectory multinomial general form impose hold clarity omit conditioning assume moment analytical intractable resort approximate hyperparameter jensen approximation search distribution maximize analogously analytically latent nm increasingly therefore compute term divergence affect trajectory computation gps mn inversion assume however practice select change slow solution maximize tight improved bind propose low divergence kl correct arise jensen constitute correct analytically depend possible decay amount posterior process equation implement instead numerically though orient towards use datum grow new arrive start effort use update latent fully markovian property hold update instance slide rank option behavior association regression show implementation ghz gb yielding time experiment perform association toy circular available source observe source trajectory circle center source make difficult successfully identify illustrate specifically decrease whenever source come air tracking scenario observe interval unity detail refer pose tracking estimating scenario pass one source instant sir filter sir joint particle filter combine technique instantaneous operate sir initial vector regard ht trajectory sir initially perform difficulty source come one trajectory mistake mainly prove insufficient multiple solution perform entire trajectory rmse trajectory assign wrong trajectory observation version sir furthermore sir initial state knowledge l rmse rmse sir batch interestingly find multi communication third experiment wireless interference ia wireless network idea ia spatial interference align receiver interference filter possible smooth frequency implement datum association setup illustrate fig datum distinguish smoothly noisy matter fact implement ia set project application multimodal come source htb observation three gp fail multimodal previously mixture restrict gps depict b interpret along one dimension horizontal snapshot noise measurement snapshot gp capture gps ard covariance noisy identify mean finally correspond mechanism produce observation gp track previous mixture rely
inference conditional investigate surface location variation much remain main difference occur location height height height height height height location height height height height height height height height height height height height result simulation setup paper exception fix knot figure compare residual surface knot knot surface knot model knot describe generate mean dataset knot knot use knot loss knot model knot improvement knot covariate fix knot equation theorem axiom conjecture remark division computer science link se mail explore sensitivity inference variation aspect prior shrinkage influential prior flexible display density variance row large knot partly three place prior mean
strictly point lemma invoke return rx e suffice turn n j rp e form neighborhood around fact complexity noiseless uniformly dimensional possibility calculation assume similarly total variation contain contain radius upper bind similar noise region give volume calculate respective notice identical reproduce particular suffice eq claim derive denominator numerator nb depend increase small outside keep keep apply conclude choose select condition choose return nn particular large procedure correct homology least deconvolution bernstein hoeffde crucial clean remove point region notation analyze indicator mean case case eq remove bernstein case bernstein inequality region remove bernstein put piece procedure least x measure deconvolution provide completeness let z px qx qx qx px qx remark lie understand geometry homology group manifold topological provide algebraic contain connect estimate homology manifold noisy bound risk bound ball appropriate radius establish le dimensional manifold homology group homology manifold connect cluster order homology topological extract beyond topological homology group application homology rapidly homology application medical imaging shape analyse microarray book study range homology homology appropriately bound homology show condition homology manifold perturbation wasserstein study homology geometric prove theorem homology homology induce literature focus aspect manifold homology reference therein upper bound mainly generalize henceforth establish sample dense thin region size homology high variety clean essentially remove sample thin sample fall deconvolution concentrate around sample procedure deconvolution reference use deconvolution cluster tree support knowledge homology exist previous result outline describe brief homology ccccc noiseless paper unknown dimensional manifold whose homology seek depend describe manifold clarity positive constant different specify riemannian volume manifold ambient paper regularity impose condition open bundle self collection manifold volume clarity treat constant low match noise noise clutter observe distribution clutter noiseless case noise q qp dimensional whose add restriction manifold comprise manifold use along consider vanish get let difficulty sample small permit call use phrase homology ball usage discuss briefly see detailed computation homology group topological might imagine union geometric underlie infinite computation tractable computation homology particular see discrete classic homology identical homology complex describe homology e triple triangle sum integer ease form denote simplex boundary simplex boundary chain p group z z ht boundary collection chain cycle cycle chain th homology homology cycle homology correspond next homology cycle loop homology equivalence homology collection homology cycle cycle boundary chain triangle fix radius ball since field compute homology polynomial size reduction variation supremum show respect make le le iid n q infimum le member close two different manifold homology manifold establish task underlying manifold lemma describe apart smoothly end red manifold pair apart radius inner radius smoothly end manifold manifold correspond identifiable recover homology impossible establish bind case thin region ball carefully around sample far away manifold threshold region include edge concentrate manifold deconvolution densely construct ball homology high review statistical fan modify account deconvolution kernel noiseless take dirac generally specify construct union ball around appropriate homology probability homology one matrix homology compute homology approximately homology quantity minimax similarly write constant analysis typically upper factor correspondingly factor despite two manifold le result manifold get carefully upper specific estimator homology rate provide upper bound carefully choose manifold specify manifold describe make low calculation show low sketch noiseless densely concentrated manifold union risk straightforward result reproduce inference choose c appendix clutter sketch lower via construction calculation variation le sketch preliminary clean far away close analysis show probability ball appropriate radius give homology outline invoke clean careful region show retain thin b rx thin dense cover homology formally dense upper minimax resolution appendix noise get width around highlight phenomenon dimension concentrate around care reconstruct homology rate identifiable depend tight address dimension identifiable consider manifold place manifold center manifold clear distinguished old however important manifold differ appendix still recover identical bind noiseless sketch suffice involve two sketch ball compare ball inside mass concentrate around manifold large disadvantage homology around right homology balance consideration homology cover n homology bind minimax resolution argument case derive additive somewhat separate mixture collection distance clutter clean ball consider manifold deconvolution draw clean around show homology satisfy density j transform deconvolution require divide transform bound behave noise say broad deconvolution satisfy take fix bind aspect manifold accurately reconstruct homology use thus draw measure appendix ball radius remove ball radius
edge mesh onto solution eq q equality property whenever complement inner property noise integral assumption functional basis compute piecewise function basis computation appear structure work hard relate problematic good number mesh require sum piecewise mesh row basis consist basis zero everywhere tp I solid kernel dash approximation principle function job careful depend check sensible ern respect space get effective range knot obtain furthermore computation integral figure non stationary sphere still explicit element compute extra ability stationary continuously markovian assimilation trick equation integral integral sum functional quadrature infer primarily replace progress overlap mat ern markovian mat ern model smooth integer markov essentially powerful non mat ern material surface field demonstrate property greatly boost modelling capability ensure combination modelling markovian random continue produce useful markov field frequently unfortunately traditionally traditional property markovian practical markovian gaussian offer parsimonious interpretable practical viewpoint statistic computationally speak spatial advantage linearly rather success series markov spatial field connect jointly equivalent iff problem usually concern spatially domain fundamentally reason field paper space markov use markovian barrier conceptual non possible construct sphere show difference infer cox observation infer non convex multiply sphere flexibility construct field paper review deterministic approximation insight theory markovian field review take detailed gaussian begin lead differential markovian gaussian insight discuss gaussian field possess spatial facilitate quite explore markovian investigation one quantity instead significant quick computing special amount dimension key cholesky efficiently book cholesky cholesky sample dense occur instance however conditional usually significantly dense conditioning technique whereby unconditional conditioning krige cholesky krige inefficient sparse conditioning datum subsection construct note krige update solve auxiliary definite reveal krige augment system tucker constrain unconditional unconditional measure method cholesky mind iterative solve system work need property conditioning discussion tie especially play set bayesian jointly joint implicitly equation eq sparse structure availability inference come section likelihood field group mcmc sampling posterior easy exercise marginal give take compute gaussian perform integration moderate inference show construct scheme integrate laplace successfully spatial art user interface aim statistic infer field gaussian gaussian I non dense like transfer computational outline gaussian random barrier classical tie discrete discuss broken barrier spatially markov simplify directional nature distinction future allow less exactly generalised obvious informally field markov set separate independent spatial obvious spatial property ignore markovian gaussian gaussian show markovian stationary markovian elegant tool correspondence differential operator variate k lf infinity covariance call operator q l follow previous isotropic stationary markovian operator symmetric dt univariate critical markovian sense value neighbourhood depend everywhere covariance locality field previous form markovian random field set greatly noise field proposition stationary integer markovian random field somewhat surprisingly back part show mat ern mat isotropic mat markovian ht consider sensible approximate necessarily class approximate reasonably demonstrate deterministic linear mesh mesh mesh functions grey pyramid figure piecewise jointly gaussian approximate markovian mat ern ern field stationary let right hand side integral green formula derivative vanish boundary sensible
asymptotic construct plug preliminary allow projection regularity condition ensure convergence estimator subsection shall normality efficiency notation integrable cube decompose dx dx triplet belong resp denote convergence choose subset index sequence partial projection three assumption subset find constant ellipsoid necessary condition control square queue grow decay density basis assume great regularity property function function derivative exist assertion regular conditional two allow infinity asymptotic respect control normality entail theorem er er rao bound consider density estimator family neighborhood rf efficiency define summarize estimator asymptotic p extend result whole half column matrix formally x generalize estimator taylor lowest precise handle also cause er rao er rao taylor help expand normality aim estimator context similar result behave reasonably despite article project would semi regularization constitute simplify explore like take derive equation get argument decomposition eq first part constant term shall classic variance establish q converge remain h use usual derivative equation imply I dy I dx ij result quadratic locally normal b eq uk er recognize follow random central cf n tf j dx dx mu nh dy hoeffding nk nk term give constant equal equation prove end proof da du paris universit france regression set methodology joint du paris france universit france noise face call develop overcome reduction density projection refer effective dimension approximate covariance denote transpose therefore refer parametric form vector lastly method element asymptotic high quadratic integral semi estimate estimator alternative plug organization section notation coordinate estimator state expansion technical lemma proof section square joint density index triplet density finally dx dx variable center give aim similar whole extend write integrable functional specifically parametric identically sample split subsample
tolerance tolerance final tolerance properly algorithm algorithm tolerance smc couple replicate rejection start particle tolerance support bin lead grid cell sum difference histogram study yield unimodal model fig fig logical number child straightforward partly census peak approximate toy intermediate value lead recommend require reach gain simulation algorithm versus simulation replicate bar replicate circle triangle plain triangle smc grey plain square grey star plain rejection depict plain circle smc target equal tolerance square time depict average replicate choose appropriate interpret stop sequential performance large modal distribution benefit improvement automatic selection summary step regression processing contribution directly comparable would straightforward different publication collaborative project european union contract description cm simulate sx ti nx acc simulate u sx sx acc ti x k cm keep particle I acc acc acc mini k j fx acc xu proof exist associate tolerance define identically define particle aim level easily interpretable toy example complex social fast abc currently computation particularly relevant easy lead proximity approximate lead run demand exponentially tend limit paper minimize run reach abc subject scientific version regression improve informative chain improve sequentially approximate use sample set focus part avoid systematically apply abc rejection method however bias approximation population hereafter perspective sequential deriving difficulty remain tolerance bad modification monte abc algorithm population monte hereafter tolerance stop furthermore approach avoid particle mcmc algorithm intend stop value apply toy significantly less simulation population monte carlo hereafter hereafter smc hereafter smc implement package approximate present abc currently available limitation population hereafter carlo sequential abc algorithms appendix predefine tolerance parameter get target particle methodology step use transition randomly proportional inverse eq reach procedure draw prior particle introduce newly density probability reach e attribute newly particle major decrease tolerance optimal sharp importance sampling chance possible indeed arbitrary level smc algorithms level sample define smc concern smc algorithms move particle mcmc jump accept probability metropolis mean limitation smc mcmc kernel drawback particle jump particle accept keep new particle occur particle appear strongly solve jump evolve course weight attribute draw algorithm generate scale step throughout ensure weight produce distribution stop criterion proportion particle particle stop rule additional marginally change ensure proof toy toy study structure comparison indicator perform density histogram particle tuple sized bin choose compute indicator choose tolerance level equal sequence tolerance algorithm explore perform average deviation simulation twice effect particle distinct course show number cause decrease particle evolve towards ensure relatively distinct particle maintain reasonably run see versus average replicate represent replicate blue circle plain triangle tolerance smc grey plain square plain depict black plain circle distance times abc smc target tolerance distance algorithm cell depict ccc per accept new make couple couple distance net distribution impact studied improve quality total slowly numerous stop toy explore good acc
I f calculation across subject describe classification column equality appearance fit parameter age cognitive test score school intensive make outline essential acknowledgement thank associate suggestion correction respective length constant eq q minus derivative derivative minus example fitting marginal marginal lagrange produce identically covariate infeasible size base categorical penalty constraint year marginal generalize parameter multivariate flexibility enable conditional especially graphical na I algorithm estimation several challenge closed equation raw log consuming newton scoring try general constrain context show equivalent second less deal level unless size small variation marginal model penalty section marginal basic algorithm effect individual method categorical joint determine probability entry strictly positive frequency entry arrange likelihood p vector whose design determine matrix score expect take eq enable simultaneous modelling specification suitable complete restriction family ordinary group subset interaction margin possible margin call hierarchical later fitting describe study likelihood show exist outline maximize constraint derivative multiplier proceed suppose current reasonably replace second minus quantity exploit solve explicitly get eq sort length multinomial fit constraint constrain jacobian invertible rank thus crucial completeness necessary condition smoothness choose remove remove exploit computing note design equivalence matrix follow parameter respectively score alternate combined column definite u v rewrite second give update proposition step neighbourhood function respect neighbourhood expression add subtract least solution design implement constraint dimension parametrization computation making usually ordinary take detailed asymptotic existence maximum might observe zero converge jacobian matrix ill make unstable concern note modify newton modification elsewhere update converge stationary addition consequence tucker ensure look know local efficient observe matrix give model concave value order one probability need define parameterization example matrix row homogeneous simplification mention everywhere ordinary extension first equation substitute similar covariate available
boost receive ix ix ix x mm output q deterministic cutting deterministic proximity fx fx become extra jensen strongly strong convexity side inequality condition optimal deterministic quite programming case stochastic q assume clearly ix iw prevent asymptotically cut plane apply option low q accordingly guarantee constant lose cut plane validate clearly already deterministic unbiased bounding bit either condition independent bf algorithm summarize convex stage yield upper constant call first optimize least time option previous instead decompose side oracle updating prove asymptotic behavior apply algorithm rate hand recursively induce worst four gd cut plane method fast sgd result corollary yield markov loose demonstrate probability strongly convex respectively expand right w immediately sum consecutive property accordingly eventually together convexity fx f even easily decrease exponentially reduce average parallel distribute order four strongly norm continuous higher discover loose possibly fail likely fy n isolate parameter q unfortunately alternative g smooth reveal completely principle good ever still time raise
dynamical visual repository illustrate ability develop extract storing object speed architecture connection feature repository formation recognition capable development experience stimulus extraction som system ability extract feature frequently collect aspect extension neuron inferior modular like topological representation domain capable operate multimodal integration moreover modular imply flexibility architecture plausible configuration area capability stream superior etc serve platform network like mirror system intelligence school valuable comment comprehensive discussion focus development object recognition system general module module correspond connection level match stimulus short stimulus distribute spike reciprocal spike initial wave spike test information consecutive extraction accumulation rarely activate update repository update illustrate dynamical topological connection area broadly discuss visual object subject artificial intelligence ai create exhibit achieve parameter architecture replicate brain responsible visual recognition include area inferior temporal structure visual area visual size field neuron neuron decreases tune bar orientation tune various form degree translation invariance visual recognize consecutive area along stream object system ai architecture resemble visual area either task vary ms human process feedback speed implement feed furthermore rapid neuron produce feed top circuit reciprocal brain efficiently concept reciprocal bottom top connection spike reciprocal numerous present area broadly several address matter development orientation primary visual modular structure inter development key mechanism formation implement self extraction area object mean self grow feature reflect inclusion development regard growing establish intra development neuron hierarchical extraction level process visual recognition consist module som radial neural network level inside area group neuron mutually head line send som connection head line schema translate rbf grow som unit receive input grow layer well rbf exist signal dash unit mark head double solid som serve visual development som head dash processing intra layer ten fig black white describe v orientation wave equation filter relationship input different therefore four neuron cell densely implement result orientation map orientation contain neuron selective orientation location employ v area connection indicate orientation prototype specify empty absence orientation v neuron several neuron support neuron integrate orientation process complex different neuron type neurons rbf current rbf prototype preferred stimulus neuron term variance origin feature visual vertical integer rf vector c map wave wave serve feature repository update reach indicate signal spike inter connection inferior neuron consider rbf area address various area visual experience change apply neuron neuron neuron neuron store neuron perform stimulus stimulus center tuning estimate neuron feature dimension absolute depend activation actual stimulus normalize threshold detection store recognition processing however code store regard store neuron cause neuron rbf class densely cover produce response grid densely spike response grid scientific paradigm stimulus additional stimulus confirm reject process formation refinement orientation hand object hand spike step curve corner plane component show axis axis horizontal vertical dimension original image correspond magnitude information wave spike object study step follow grid store neuron calculate identify object neuron value present back purpose coherent identical neuron map actual localize map detect localize activation belong wave spike exclude call identical element signal stimulus internal stimulus spike major accumulate wave spike form occur refer hand object spike produce stimulus distribute spike shift spike initial hypothesis store stimulus effect stimulus rbf object feature stimulus significant effect take stimulus rbf center many common acceleration neuron spike send generation retain change procedure match identical element repeat back iterative continue spike wave wave dynamical intra inter connection change extraction mean exist rarely hand processing stimulus distinguish frequently counter good counter less chance remain feature extraction extract
definite approach efficiently surrogate concerned matrix might rank exponentially trace argue norm sign situation quite surrogate surrogate control e entry explicit section norm norm norm specify entry bound trace entry magnitude fix maximal minimize reconstruction guarantee absolute reconstruction measure impose scale word think measure relative constant obviously multiplied constant notion magnitude magnitude entry simplicity replacement whether choose whether possibly apply entry magnitude summation repetition uniformly magnitude ss either theorem hold probability without replacement nm obtain sdp potentially requirement error exponential subgaussian yield guarantee complexity similar ij previous yield max upper actually identify predict entry reconstruct observation observe entry n equation high uniformly assumption replacement positive always yield remainder organize prove sample bound square norm compare establish replacement set replacement follow regard determine either norm norm particular hypothesis error lx ij absolute rademacher trace ball rademacher trace norm index pair bound give prove loss immediately ij detail reconstruct max trace norm choose trace norm replacement square loss least infimum predictor upper size class ab rademacher last assume increase set theorem remark dominate instead square might hope similarly square use unfortunately fact demonstrate square rely specifically arbitrary reconstruct choose regardless control expect reconstruction regime relevant generalization obtain nm oppose favorable theorem dependence dependence technique rademacher even give norm factor come elsewhere literature familiar max norm square max factor theorem replacement without observe intuitively replacement sampling replacement state briefly notation denote lx sx os without replacement class define rademacher derive sx take must imply well replacement similarly rademacher lx hold subsequent remark replacement replacement error j draw z set aim observation intuitively replacement although tradeoff independently prove matrix square error entry lm ss depend noise future zero reasoning q turn without replacement notation sample add time without show nm long nm might require different sampling incoherence see g relative maximal magnitude nm compare approximate difference require comparable essentially guarantee rely compare guarantee discuss approximate theorem near exact state recovery exact otherwise identifiable matrix reconstruction aware immediately excess bound reconstruction restrictive guarantee replacement replacement entry e entry identical I independently difference quality sample recovery literature error minimization penalty method cite prove approach exception mostly trace norm use approximation entry elsewhere remove three provide nm generality subgaussian sample replacement sample time give independent recovery meaningful specifically size bind assumption bind error actually meaningful subgaussian replacement weak subgaussian similar fairly different method rectangular matrix see scale improvement much hand include compare theorem pt recovery assume identically subgaussian recover possible estimation significantly approximation know believe technique quadratic result match norm relax scale absolute strength replacement include entry noise replacement observed entry draw mean exact near recovery exact corrupted subgaussian near recovery result recovery work fundamentally bind definition approximate recovery type guarantee link generate nonetheless comparison magnitude recovery near approximate rest follow literature complexity section section recovery complexity result incoherence sufficient getting norm section incoherence condition clearly approximate recovery norm recovery svd also denote improve prove recovery low set subgaussian incoherent improve noisy give additional meaningful therefore regard subgaussian reconstruction algorithm find observe subgaussian simplicity relaxed require ignore sample ignore dependence case exact incoherence complexity establish exact less recovery bad recovery approximate recovery km light complexity result rescale contrast recovery incoherence number g may attain incoherence enable approximate relative average incoherence entry perhaps extremely subgaussian mild dependence complexity complexity several max excess method result obtain loose align writing number rank slightly singular perturb matrix complexity trace norm matrix carefully paper string rank show rademacher combine argument guarantee rely point superior commonly able reconstruction study replacement establish generic relate setting bad approximation although dependence rademacher consequence rely mean assume max rely rademacher max careful rs complete proof matrix time bound lemma draw replacement element appear exactly time obey convenient might write regard list order format let follow equality arise recall next etc time appear rescale treat therefore eq observe rs conclude except entry follow statement sample vector sx unlike lemma therefore apply index
obtain vc conjecture theorem department query optimize execution operation major contribution interest vc outcome dimension operation join operation individual query collection collection tuple database query build database high database evaluate query need major change compute small store present analysis advantage technique complex microsoft server advance technology collection storage vast need work database crucial query execution plan optimization database task efficiently obtain range table database yet powerful commonly inherent involve table query datum cost query collect disk large storage medium therefore expensive provable running sized database concept vc dimension novel technique query hypothesis sect theoretical vc dimension class indicator product dimension number join adapt vc query database execution query provide query database hold collection evaluate query major change query measure vc experimental accurate estimate use surprising concrete memory significant execution apply vc dimension uniform practice efficient store table construct present experimental validate technique microsoft server give accurate prediction create multidimensional histogram join estimate sample possible solution rest paper organize relevant work sect sect introduce develop analytical contribution bind sect sect query plan explore range offline histogram datum mainly tuple arrival discard sampling database disk operation cardinality significantly sequential keeping improve apply cumulative inversion still expensive offline et use systematic require tuple tuple et use obtain belong expectation query inference bind conservative join sampling point number table table priori together common join sect explain pre compute statistic predict histogram used construction histogram examine rigorously size building estimate query extensively although offer query efficient storage need histogram inherently limited query drawback histogram intra uniformity assume distribution inter independence correlation suggest use multidimensional memory updating literature seminal computational encounter success related database context database aggregate vc need unbounded privacy purpose column tuple appear select join operation define consideration impact column operation tuple tuple combination clause exclude discussion category table column return tuple join basically join condition contain involve mean tuple join one correspondence correspond join set table return subset tuple set selection return join direct whose elementary join operation decompose node output table operation definition combination select join nevertheless define query plan leave internal join throughout tuple tuple join join table execution sample operation execute function equivalently subset bind vc bind learning outline adaptation specific work vc define space range finite infinite point range set query input tuple query combine join involved class vc range range vc cardinality let range vc range query database table tuple query range set family interval set point interval vc observation vc dimension half dimension theory relation size sample let ax rr b construct vc subset q relative least interesting sect probabilistic challenge vc dimension range fundamental result size define integer growth lemma range space vc finite sect combination range range space intersection member combination intersection q hold vc dimension start complex attribute query join query join operation table column tuple tuple equivalence previous paragraph range vc dimension tuple extend query query selection combination selection predicate note query either rewrite clause either vc number predicate clause space align box output include disjoint align intersection overlap collection technique see combination query intersection operation imply let table select tuple product order pair form join simplify query predicate r r j join query v jj car ap j ar r identify b c c p r eq hold plan select internal join tree range select query table join table define note extension join query set car ap ir r vector see I ia tuple actually choose choose operator join q join operation involve boolean operation suggest result vc dimension concrete optimization compute class class involve query execute query execute def eq within table plan query rooted internal query run obtain define correspond thm application prove maintain easy independently write tuple product table assume table tuple form tuple tuple ix query join count tuple ji kt op tuple tuple op alg query table table add index tuple concatenation tuple element note major point general impossible join join require size result join vc size identify plan may standard plan generation execute multiple common candidate overhead execution approach likely technique well overhead store intermediate significant extra present microsoft goal usefulness theoretical assess run query database size use random join operation previous estimate thesis large query sample commonly histogram briefly sect histogram fine grain histogram soon column long give modern database system query optimizer histogram statistic help determine efficient use histogram information item frequency default store bin list histogram histogram list table tuple histogram tuple column develop sample require join attribute therefore complex database million tuple distinction run selection query join run different category independently treat contain normal I value correlate join query consider join common table treat tuple replacement table join tuple table independently tuple base tuple form sect table original sample fix thm million tuple vc size fix
grant office two grant powerful effectively extensively area trick inner feature without know trick extend spectrum space eigenvector sense eigenvector explicitly lead eigenvector base subspace spectrum sense spectrum simulation correspond lead eigenvector matrix segment kernel pca generalize ratio sense cognitive availability band secondary user without interference primary user spectrum match covariance spectrum nothing receive hypothesis user involve present sense alarm rate alarm extensively apply especially support machine svm counterpart kernel diversity algorithm inner implicitly infeasible inner point become product mapping generalize space nonlinear well hardware pca white wide signal prove stable paper spectrum lead propose employ map inner lead know eigenvector say eigenvector pca generalized algorithms spectrum propose hyperspectral detection background subspace span target background projection assume combination column kernel match employ consideration background hand operator assume generalize space determine simply speak lead pca product value projection space spectrum organization spectrum sense lead review eigenvector propose eigenvector kernel modify base match simulate receive primary user signal transpose lead eigenvector large assume eigenvector eigen decomposition diag corresponding eigenvector template pca likewise number receive simplicity denote lead eigenvector sample I pca desire alarm eigenvector call pca kernel classical employ kernel implicitly feature need increase computational receive kernel obtain x last linear imply combination side ij eigenvector matrix positive eigen normalization eigenvalue point know however compute template detection x coefficient lead explicitly lead eigenvector sample know likewise eigenvector lead product orthonormal vector white obeys receive hypothesis q conditional gaussian eq approach explore cast eq substituting take operator onto subspace detection threshold value operator feature eigenvector nonzero projection operator assume project pca eigenvector nonzero eigenvalue covariance x x eigenvector nonzero hypothesis gaussian claim still gaussian kernel employ approach center spectrum gaussian kernel consideration center kernel j normalize receive eq compute determine threshold alarm detect presence chart spectrum kernel detection threshold determine ec ec obey ec optimal hypothesis ec totally blind without design maximal minimal pca matrix primary signal assume amplitude receive length receive ec implement implementation kernel kernel pca varied pca ec fig pca method kernel ec know knowledge notice affect varied snr ec still ec ec gaussian actual perfectly subspace model factor affect calculate kernel kernel divide maximal method test choose kernel operation center similarity kernel linear test verified correctness capture sense first sample user matrix test framework divide segment segment eigenvector first segment hand varied snr compared ec fig pca db db correspond ec ec fig kernel
topic relative frequency query access plot probabilistic ir oracle produce vector evidence section design end refer ir report term occurrence exclusive e presence ir pearson occurrence partition disjoint include frequency rejection document absence region subspace span entire symbol partition include accept rejection observe operation subspace span subset explain vector illustrate law vector space span plane dimensional note provide since vector occurrence due must measurement finding via device occurrence physical device read text frequency sufficient optimal vector much difficult physical measure vector report comparable effort oracle text limit actually informative content observe document question automatic indexing retrieval van book introduce formalism ir within uniform space theory book quantum phenomena ir interested investigate ir probabilistic powerful book provide theoretical foundation vector particular optical communication optical signal improvement domain quantum theory relevance parallel detection weak relevance play role quantum statistical ir paper obtain characterization measurement sense distance relevance justification angle unitary justification excellent quantum particular explanation illustrate difficulty however mention e interference superposition retrieval find quantum phenomenon aim leverage ir quantum formalism ir abstract formalism exploit formalism illustrate improve author conjunction angle quantum interference interference suggestion probability induce different induce traditionally concentrated extract combine evidence accurately ultimately scalar structure retrieval implementation thus implicit achieve possible ir incremental achieve answer end suggest ask proposition accord document high optimize retrieval accurately principle subset theory name vector classical subset mathematically verify suggest retrieval effectiveness condition measure define probabilistic ir probability event distribution use set measure axiom state theory stem prefer event set classical probability latter interest whereas ir scope replacement ranking ir report result date ir result rank irrelevant often unit document list besides weight result research ir view new perspective describe probability accordance effective rank accordance classical evidence effectiveness correct result prove mathematically experimentally quantum show superiority retrieval investigation phenomenon argue quantum ir ir give intuitive contribution quantum outperform classical central introduce vector exploit rank probability occurrence optimal document address theory outperform model feasibility tell vector occur document conclude appendix proof depict intuitive relevance occur presence absence symbol sake clarity side ir b act detector b relevance probability appropriately receive symbol ir implement relevance transformation symbol figure whereas implement transformation symbol straightforwardly receive symbol equip ir implement classical theoretically hold ir describe ir compute symbol index relevant document relevant detector symbol occur non call document retrieve rank document relevance operator hermitian unitary operator rule equivalently namely document subspace optimal document end define represent relevance uncertainty condition take upon illustrate appendix probability acceptance leverage rule high optimal span relevance determine geometry decision geometry correct decision probability relevance relevance geometry vector clarity generalize angle vector angle orthogonal angle relevance rotation hold vector locate relevance note ir like detector compute symbol occur relevant document call mutually exclusive yield latter refer refer relevance non suppose relevance p vector eigenvector main show latter exception probability latter term find detector sake clarity e bm slightly theory bm illustrate find underlie normalization estimation depict ir equipped bm differ ir evaluation already ir detector would oracle
denote also xx jj proceed construct outline extra ensure recovery key recall row define magnitude excess residual element either least element role role block matrix construct pair follow significance define ss subspace matrix orthogonal complement ss projection motivate bb subspace support projection define p bb index achieve sign p technique consist construct dual candidate theorem step differ condition impose follow primal candidate design candidate restrict problem subgradient specifically optimality oracle problem substitute step would require recovery result since candidate pattern make follow specifie problem problem p assumption c ball respectively fact eq saddle point strictly dual equivalent uniqueness matrix solution show primal condition differ complexity theorem result assumption positive guarantee property suffice hold hence unique inequality result hoeffding part p equivalent hoeffding happen k condition primal projection eq moreover eq thus nx state regularizer nx happen probability pr state condition distribute tx hoeffding bind result least rsc solution unique b nc min provide proof part separately result eigenvalue hold share portion large portion assume equality dual recover strict variance turn fix j row sub norm j conclude assumption condition hold constant projection notice concentration theorem union conditioning mean value gaussian get eq j fraction large task gaussian q go substitute ds rv union discussion solution f require investigate characterization convex derive sub differential differential cc rt belong sub denote jj condition uniqueness jj provide proof case j j j j prove establish contrary j index k contrary exist element except j entry j j trivial suppose exist row j j conclude ratio regularizer since optimality integer unique row equal except row contradict optimality similar uniqueness fact j kk j solution contradict characterize introduce take tuple initial guess warm stopping update subroutine update block matrix pd x j break cyclic descent keep unchanged k mi ib return cyclic keep unchanged block sparse cycle unchanged vector fix subproblem vector j kx correctness directly correctness descent argument correctness unknown measurement like multiple sparse several partially recover whether decrease line recent actually separately ignore sharing depend level parameter pay differently show theoretically except case perform well multi learn dimensional increasingly problem high hope statistically consistent lasso rank high dimensional context task task estimate leverage aspect feature sparse problem simultaneous arise variety context signal range sparse represent column row block sparse zero mostly lot focused encourage sparse example norm regularization encourage sparsity assume share suffer arguably realistic depend feature addition concern row nearly identical far original feature method regime support vary widely parameter thus ask leverage overlap parameter identical instance general fall block might bias first focus simultaneously superposition search overlap sparse theoretically extent support remarkably follow sec basic setup present sec denote row sum absolute regression k task independently target notational convenience interested estimating leverage extent simultaneous sparsity certain entry share row row would feature adapt level guarantee successfully recover sign sufficient merely recover row support particular support column error interested superposition feature task two encourage simple would sparsity regime interestingly block sparse clean statement recover support scaling regime provide sign row scale require sign recovery third explicitly quantify lasso block find early theorem correspondingly follow case task design two fraction interesting phase transition scale successful sign particular rescaling sp rescale sign converge scale converge regularize rescaled sp show probability success near rescaled sharing perform share perform show rescale sharing sharing everywhere else detail respectively outperform require sign support consider specify analyze normalization incoherence denote incoherence minimum consequence finite regularizer require regularization hold unique false false b b remark guarantee relevant addition require recover exactly need enough matrix ensemble graphical row condition curvature impose matrix row regularizer require regularization pr least guarantee estimate finer precise quantitative gain experimentally regularization except share match result sample also generate submatrix regularization scale choice c reflect balanced share synthetic datum consequence theorem theoretical digit task digit show practically via cross discuss generate use search regularizer program find three sign sign decide program successful sign explain simulation three generate entry support multiply cross k kk train generate recover descent appendix tuple guess regularizer set good guess stop algorithm inside update objective algorithm completely optimality thus partition logarithmic filter coordinate minimum run recovered otherwise element different sign predict stack observation sign regularizer lasso less observation performance grow well lasso still change theorem versus
variant detail follow mod sec reconstruction result key sec iv cardinality also value mean clear sub element great strictly zero element less equal pattern transpose matrix denote also reconstruction problem reconstruct length denote denote contain signal estimate along restrict c sc restrict orthogonality roc small real mc ss frequently roc matrix rank mod put explicit order small similar constrain suggest mod achieve reconstruction exact mod put encourage close nonzero instead distance add least exact recovery mod cs mod bp recovery mod bp exploiting result discuss implication sec lemma lead proof sec iii b outline inactive find set maximize constraint bad size unique u notice large reason fashion notice ensure first ensure practical active nonzero deal signal take signal sec iv take expect hold occur recursive video original video project base foreground person move camera large correlate noise similar look slowly video rate common small extra requirement bad case mod cs result hold recursive mod recursive recursive compression extraction correlate moreover active mod anonymous exact knowledge recovery small reconstruction error mod mod cs recovery subset observe probability reconstruction increase size check remove element hold check complexity roc condition result loose bp mod cs set sufficient unique next measurement fashion theorem later suggest tucker kkt set necessary condition keep us condition let find recall mod cs measurement roc lemma help hold ensure third apply iteratively notice ct notion satisfy mt b appropriately condition bound size ii ks ks ks c positive similar give disjoint condition iteration disjoint theorem satisfy probability good nonzero mod compare mod cs higher alternatively significantly reduce probability compute various exact recovery also second simulation signal step equally scalar variable range notation mod cs weight mod bp mod mod bp bp mod cs mod rmse normalize unit norm size uniformly element support n cx tx ki thus set mod bp mod weighted choice package primal equal choice record table record record good root mean squared rmse record next thing half increase increase decrease increase mod cs realistic scenario bit set x zero clearly draw finally n since expect mod well bp mod mod work sufficient exact recovery mod active subset active satisfie give mod achieve weak happen estimate weak mod cs either bp simulation mod bp minimizer fact follow scalar equality eliminate equation tw w hold equality equality support xx simple fact let minimum I semi definite matrix mt mt mt ba mt definite b b numerator denominator mt mt mt na mt prove try find construct lie nan satisfy infinitely closed mt mt mt mt mt definition theorem satisfy lemma similarly equivalent rest follow beginning follow ii mt iv mt u mt mt x otherwise take e mt thus satisfy material finally theorem zero combine simplify summation absolutely convergent equation w rt give condition satisfie theorem definite ta symmetric let b since form I thus
finite due result possibly separable hilbert space except adjoint definite identity outer linear u uv operator frobenius mm denote large vector form orthonormal since let eq copy square uniquely eq specify condition dimension covariate moment apparent size accuracy technical merely remark naturally ridge analysis kernel easy product equality expectation mapping q leverage finite surely sure bind relaxed analysis sake simplicity subgaussian requirement subgaussian lead ordinary surely satisfying typical approximation error eq subgaussian subgaussian normally distribute mean minimizer approximation surely q sure relax sake appear low ordinary requirement exist surely surely inequality triangle third schwarz argument satisfy eq ordinary study design covariate response independent analysis review section control bias ridge essential compare ordinary estimator random design square covariate bayes relative error plus predictor approach beyond analysis often additional dependency spectrum moment comprehensive survey excess empirical satisfie certain boundedness bound objective approximation result square computation work provide ridge regression bound essential assumption fail give review analysis assume surely assume dominant explicit assume almost surely explicit surely lead factor specialized estimator depend moreover ridge therefore vanishe bias analysis expect sharp ordinary mild assumption bayesian ordinary probability least rely assumption tail arguably reasonable quantitative establish comparable although understand ordinary mention number work considerably weak section present result square ordinary square review design review set deterministic response variable x eigenvector v design modeling decomposition proposition ridge effect fix error effective notion level ordinary square fix analysis ordinary exist specify bound inequality simplify constant spline ridge also specifically periodic smoothing spline th map condition satisfied remark datum behave lack approximation reveal crucially control fix overcomplete approximately solve exactly square prohibitive approximate satisfactory randomization rotation replacement n square problem rotation randomly matrix apply create square quantity solve sample leverage accurate leverage direct generalized orthogonal rotation matrix arise randomly row na arbitrary rotation operation distribution see considerable retained ridge operation suffice approach treat regression produce pair row choose rotation matrix problem b ordinary guarantee probability consider constant proof fix random orthogonal q th coordinate nu vector union simple condition distribute random follow jensen eq last hadamard vector rademacher component especially present operation significantly fast run na I multiplication therefore last hoeffding bound lemma due slightly argument reduce difference recall basic used bind design ridge employ matrix matrix contribution probability well effect proposition inverse normal w j follow w expand square equality shrinkage error consequence second norm define condition pick probability consequence tail adjoint ab product condition pick definition q claim observe condition triangle assume observe ni jk define I nonzero eigenvalue observe spectral version definition cycle property denote von lemma inequality cauchy schwarz acknowledgement david inequality choose order availability subgaussian generalize gaussian vector form subgaussian value matrix let variable q expectation moment generating degree fact tail application bernstein vector bernstein vectors eq spectral accuracy moment bind surely independent copy ordinary least regression estimator set mild sharp reveal error neither decomposition concentration inequality matrix set linear regression sample population vector random typically
recovery universal nearly optimal universal bound necessarily rip qualitatively rank component reconstruct nuclear norm frobenius frobenius f good state reconstruct frobenius lastly rip insight design measurement setting repetition experiment essentially element small operator rip actually strong result appear elsewhere rip gaussian strong rsc somewhat weak show completion follow cover argument concentration failure union seem randomness phenomenon weak union result account favorable correlation behavior different matrix behavior positively correlate good transform bind use compressed rip fouri key nuclear norm ball use entropy duality also sometimes set I interest call state gain complete element nothing element turn rare body work regularize program see organize discuss appear section vector p denote act product element measurement incoherent speaking must I proceed sketch quantum dimension semidefinite trace convenient experimentally feasible matrix tensor kronecker matrix expectation matrix scale incoherent general uniformly unknown minimize nuclear constraint note convex relaxation thus efficiently approach selector alternatively regularize lasso discuss estimator isometry rip say rip denote norm notion rip rip large measurement orthonormal incoherent depend let rip x remainder section rank previous obtain selector measurement near measurement incoherent operator rip result sketch couple quantum real arise situation state low instance pure decaying typically obtain good ignore secondly inherently noisy observe entry copy state copy average approximated gaussian noise noise one reduce repetition experiment possibility choose elsewhere reconstruct frobenius let estimate allow mechanic distinguish error implies contribute quantum sketch apply value residual part ideally estimate selector frobenius say strength seem norm particularly strength nearly fall tight nearly theorem couple besides f even nontrivial result incomplete consider tail know nuclear small bound let choose sampling operator mm rd selector bind interpret bias second term norm tight well decay term adaptation bound covering summarize x adjoint norm banach respect suffice first hilbert schmidt inner adjoint prove rademacher small fix k universal algebra gets find root dr desire result concentration bound rademacher norm comparison iid get index upper bound supremum cover metric need cover actually elementary calculation I cover notation unit norm observe u give number restrict next use universal covering number cover banach modulus let eq duality reduce dual basic denote ball need cover empirical dual covering inequality detail lemma banach norm want work infinite measurement restrict isometry property rip low technical nuclear ball et al manifold space one strong rsc manifold embedding try generalize rip metric embedding rank likelihood variation notion g fully david anonymous suggestion work california berkeley grant institute technology measurement supplementary overview claim defer later section involve proof cover use restrict isometry claim rip suffice imply adjoint suppose rip rip norm element term precisely standard basis vector norm write ab ab cd ab true must ab ie cd write q argument get straightforward finally banach complete respect return concentrated around claim rademacher view hilbert schmidt adjoint element equation bound fix lemma uniformly norm mc lemma operator operator side rearrange root bound simplify write c c c concentrate concentration banach jx x copy inequality use constant write eq plug inequality failure decrease rademacher quantity iid lemma equation index supremum fact pp cover radius metric need simplify norm actually semi b upper metric simplify I gx x x cover number plug change covering introduce ball observe simple eq
agent b lastly check reveal always satisfied imagine pool play repeatedly desirable failure avoid agent resource sufficient equally utilize common would solution issue protocol communication decide channel slot channel occur need access protocol protocol probabilistic work consist power transmission power able otherwise transmission straightforward independent constant play measure utility agent agent update update agent keep track perturb payoff history agent select provide satisfied independent identically refer ni topology borel field measure endow topology topology value chain denote let banach q verify admit stochastically state cardinality identify dirac characterize follow collection probability eq finite chain asymptotic therefore ergodic process proof require proposition refer govern element sequence notation induce initialize operator x decrease satisfie eventually hold initialize path let c x prove define tt define recursion length rough induction eq side transition eq positive q evolve necessarily yield property prove let obtain decompose perturb also define similarly transition simultaneously lf invariant invariant side multiply side invariant iii limit right convergence dominate ss unique eq constant transition support also note continuity theorem q definition show restrict irreducible proposition unique finite incorrectly strong characterize asymptotic behavior note game characterize stochastically end denote clearly correspond profile important strict game decrease dominant facilitate start govern introduce payoff profile dominant profile receive within let game hypothesis u c induction set recall simplify follow obtain secondly possibility profile decrease value play gets define thus strong hold pick level perturb level satisfie lemma let profile respectively j j j obtain sn collection pure satisfie player kk kk game mutually coincide accord profile terminate let j contradict assumption partition strict game either h partition family transition decomposition equation positive get last proceed complete combined theorem provide game behavior theorem formation strict apply htbp h nod neighbor dominant network single network network plot typical setup level well denote convention graph plot run inverse I dominant profile play show pool access resource remainder establish provide invariant irreducible chain extensively show stochastic argument particular ratio etc satisfie equivalently point define hold state every finite distribution chain theorem game profile two profile k kk equivalence profile write u kk correspondence profile strategy pure identify path provide enumeration path moreover also path since game state profile sequence profile lead therefore hence graph use game pool set successful state pure strategy action I mention respect relation since pool pool game lemma I recognize disjoint invariant put weight either establish moreover put state outside state theorem behavior game accord percentage perturbation failure setup action demonstrate learn either agent succeed increase joint neither succeed zero htbp pl ac algorithm analyze game multiple contribution relation allow apply payoff become arbitrarily correct learning game two player demonstrate simulation network formation game network interest condition outcome player show multiple utilize limit expect resource divide among increase remark ari outcome game base player continue exceed memory become level proportional chain characterize include pool game game profile outcome desirable game common pool game fair sequence highly simulation fair outcome game include pool game game learn interest engineering distribute formation medium access control wireless utilize resource desirable formation immediate possible link schedule I resource avoid achieve adaptive distribute couple adaptation motivate agent endow utility depend agent previous experience receive challenge optimization impractical inherent number action lack form reward theoretic obstacle utility include adapt agent consequently effective issue consider agent scheme simple stay shift successful drop desirable return level update prior experience agent learn play history scheme explain decision simple decision stay reference action select similar contrary incorporate update effort game payoff profile payoff action bad profile player game keep track satisfactory action satisfactory payoff benchmark stay benchmark pareto efficient payoff play b make stay shift level payoff game payoff converge neighborhood pareto payoff surely paper also focus achieve pareto efficient action agent characterize game condition select characterization asymptotic induce chain extend game player also call particular unique put weight action level sufficiently formation formation primarily response dynamic dominant action profile desirable game interest profile might action profile similar interest weak direction change also existence equilibria furthermore definition write profile b payoff profile namely desirable action profile namely nash profile outside row bad satisfy definition might multiple choice selection desirable table alternative profile case property word game desirable action profile exist terminate profile u repeat generate sequence necessarily terminate consequence claim acyclic games formation interest wireless communication model development also network formation model introduce formation motivated plane agent contain combination link link establish point direct direct link start end e integer joint action nash equilibrium show unique path neighborhood action profile set network payoff dominant exist connected link dominant
semi method first demonstrate supervise methodology label allocate label functional procedure value figure ratio assign unlabele seem expansion supervise select criterion simulation microarray show modeling strategy relatively rate supervise functional work support education aid cm cm center q smoothing consider procedure assume independently normally distribute function dispersion dispersion follow spaced knot regression maximize log semi smoothing criterion matrix diag methodology circle discrete solid solid line et give label predictor previous indicator first logistic label functional group al introduce expand logistic describe depend multinomial introduce label unlabele functional l k belong unlabele n label unlabele ill pose problem functional employ regularization regularize positive numerical identity log explicit via label n help fisher method probability lx replace likelihood parameter scoring satisfied statistical include selection regard introduce construct semi supervise logistic theoretic propose aic evaluate likelihood field various estimation supervise functional selection criterion diag diag diag block diag l z hadamard product bayesian schwarz bic viewpoint likelihood bic cover et criterion estimate regularization generalize model al statistical model semi model tuning criterion derivation selection investigate simulation modeling demonstrate model discrete two g u g w I case implement supervised randomly divide unlabeled label semi functional logistic model et rbf svm knn functional machine et al consistency yu inductive transform smoothing semi supervise et al five fold cross knn select leave validation comparison average parameter observe evaluate superior method almost seem error case method outperform svm knn situation
vanish involve outlier outlier base new analogy q indicator cell definition base freedom goodness outli base anti diagonal ideal base table ideal ideal increase instance without key algebraic lie mean linear model way cell avoid indicator cell model cell parameter behaviour immediate connection cell definition pattern outlier parsimonious set procedure step residual definition flexible show analyze form classified gender social network small count ccc base write log base markov basis form move quick inspection residual suggest run test monte carlo case notice helpful test show behaviour pattern namely separately exhibit strong evidence homogeneous additional cell interpretation model common cause scope description geometric technique table exercise logit multinomial contingency table refer contact influence management ccc ccc medium medium medium medium medium base degree fit poorly form computation ti analyze residual table base residual absolute cell count low equal bold two approximated carlo value outlier look table cell special inspection relevant useful address outlier contingency efficacy direction recognize view algebraic outlier theory research question remain mention need problem level multiple widely article cite see statistic yet explore connection mixture characterization outlier yield useful structure correspond outlier basis already currently acknowledgment acknowledge help several suggestion clear algebraic anonymous associate valuable quality collect presentation result algebraic definition ideal ideal ideal generator exist write parameter act constant notice sufficient define ideal ideal generate associate know ideal finite pure generator computer implement ti ideal major side markov contingency sign eq markov move log associate identify eq actually polynomial generator relation ideal following hold statistical form corollary pattern table goodness test outlier essential clear applicable several example word important contingency recent author estimator cell count high expect count present account application find difficulty contingency table contingency present appropriate later notion outlier detect standardize note notice respect univariate gaussian sigma criterion outli contingency less variable categorical cell contingency count model one use quantile use outlier presence use adjust residual chi square detect decade algebraic statistic research major table natural contingency structural actual algebraic statistic contingency cell quasi symmetry reasoning algebraic contingency table outlier term fit table algebraic tool test easily describe structure outlier useful example analysis candidate appropriate goodness later material recall study single monte carlo notion pattern two conclude remark work help experience polynomial decide main avoid possible group technical fact normalize probability classical sample contingency cell contingency extensively section wide statistical contingency table scheme cell count random log lie give linear design obtain parameter representation minimal sufficient simplex table issue e implicit eliminate paper follow connection implicit contingency outlier present contingency table residual column suitable approximation adjust residual detect outlier useful ml poisson tail appropriate analyze residual test high residual critical evidence cell level region adopt outlier contingency define goodness nest algebraic statistic understand outli give contingency cell table model representation name outli base well negative base eq indicator th goodness fit compare avoid sufficient goodness test point impose candidate easy algebraic statistical variety state negative row virtue proposition show system generator definition inclusion theorem outlier goodness fit log statistic maximum must chi alternatively observed contingency contingency eq contingency
gradient prefer iteration obstacle reason describe piece block correspond realistic reason describe arrive datum network equally necessary problem characterize use cd remainder mix claim relevant outline contribution basic algorithmic strategy name linear linear gauss subspace domain impossible reason iteration focus single block conceptual cd literature reference therein survey block cd semidefinite never focus cd recently area machine machine learn regression closure partly change describe efficiency cd depend balance time spend choose update one possibility descent prohibitive due compute seem compute away surprisingly cyclic cd yet know author proceed sequence theorem iterate gradient cyclic describe realistic well effort perhaps block intuitively randomize strategy preferable randomized case possibility block goal speed adaptively availability block choose current partial derivative usual sometimes computation seem promising candidate describe line implement entire reasonable value algorithm use problem smooth nonsmooth proper precisely cover decompose block machine elastic solve consistent linear enjoy global author claim conjugate motivate study give randomize cd quadratic function coordinate proportional define interpret derivative onto dimensional consider lin iteration several descent smooth apply coordinate nesterov descent smooth improve way understand result rare good cd composite analyze case unconstraine sum c rewrite box extend simplify nesterov treat sum convex nonsmooth separable focus accelerate accelerated scale problem website author solve iterate composite block coordinate descent case ht ccc objective complexity composite convex c symbol precisely far encode iterate minimizer strong convexity nonsmooth briefly outline expand find composite cover minimize make detailed use argument remove nesterov norm approach norm experiment focus regularize support machine powerful adaptively change speedup shrink organize basic state describe generic uniform variant composite study cd regularize support follow identity vector write u I euclidean identity matrix vector wise lipschitz separable decompose readily eq ready describe generic iterate pick block update upper directly closed example ix k iterate random clearly scalar define subsequent denote element eq iterate technical enabling improve ii suffice expectation case notice well precisely since hence albeit result run process technique shoot two argue choose use case technique ii example name ix compare therefore define minimizer use repeatedly generate fx ix ix tx ix k estimate fix ready need logarithm attain counter choose f q lemma result follow c theorem assume optimality lemma useful prove convergence strongly convex define lemma analogue logarithm respect convexity random theorem first add thus regularize approximate regularize quantity advance fix regularize convexity rest subsection show recover minimizer analogue apply compose suffice section much simplified section smooth norm let usual constitute euclidean believe smooth euclidean nevertheless subsequent claim give depend rewrite ix fx fx h fx purpose analysis come objective euclidean result target kp us decrease fx u ix get fx rearrange result first notice states lipschitz set assume strongly respect definition convexity estimate objective accuracy decrease ix fx f rearrange fx compare paper endow table look smooth nesterov contrast brevity strongly convex nesterov n ix fx nesterov lx euclidean strongly convex comment table detail hence lead logarithmic line coincide lead term note term table cover set eq ignoring bind note additive decrease result probability uniform result cover fine logarithm general term fashion hence result summarize characteristic complexity coordinate case cover general grows increase randomized give method constant block constant schwarz coordinate variation choose lipschitz c choice yes greedy expensive cyclic separable yes impact estimate approach individual constant coordinate fu fu l c iteration counter section sparse group size gb ram problem induce result moreover lipschitz constant explicitly thresholde give modification direct throughout refer generator control instance generator propose generator detail refer reader paper versus complete iteration iteration table roughly linearly depend per iteration proportional computation htp typical performance start instance big huge first size ccc c c time r column correspond iteration coordinate counter coordinate correspond respectively residual iterate residual decrease additional factor first look problem table half support solution residual decrease factor surprising linearly iteration decrease decrease factor convex iteration guarantee able hour performance strongly less minute take second convergence htp b nesterov constants vector power instance produce generator correspond htp tendency small choose switch effect influence effect coordinate vs lipschitz work multiply constant interval exhibit find tolerance fast fact lipschitz choose well order phenomenon lipschitz constant whereas able quickly get update method solve eventually iteration htp position resp lipschitz remain resp summary solution accuracy recommend switch shrink speedup shrink increase encourage increase setup certain predictor
economics unite objective conventional evaluate efficacy numerous fail cycle period formulation fit absence prefer aim distribution observable series term feature dynamic cycle long short want name challenge misspecification paper attempt develop systematic address aim match rigorous propose conventional fluctuation direction development dynamical widely describe change salient feature observation capture demonstrate sir newton newton explain concern motion series make interest long future salient approach often prefer available mention discussion evolution population delay take death specification suggest reciprocal unknown nonparametric estimate consider similar discrete biology see nonlinear time cycle cycle cycle suggest although possibility consider dynamic underlying cycle sir sir investigate many disease member variation individual epidemic fall infect enter recovered disease break back death birth death sir dt dt contact recovery investigate successfully time version birth rate basic transmission disease understand birth force rate transmission disease also guide policy maker spread pt concern objective observable salient cycle secondary mean interest truth model fact improve capability innovation recognize generally matter note setup different error happen coincide ahead condition minimize mathematically box wrong know boundary wrong unlikely remain restrict estimate preferable box spirit diagnostic goodness device recommend box device development development however spirit stage rather worth recall classic autoregressive ar capture business term letter respectively observable stochastic constitute realization observable throughout start model say match series perfectly conditional almost surely call approach version subsection name quite possible definition broadly step intend answer step involve model economic unlikely economic calibration matching difference methodology methodology bayesian usually wrong seem end methodology private requirement vector practice value prediction norm ahead idea depend speak complete ahead model composite q expect arrive suitably multiple ahead accurately sense measure weight innovation difference instead development worth exploration suggest speak shape tend emphasize pass filter slowly vary si tend filter balance pass filter pass filter series square ahead cox xu chen clearly former ahead different extension build shift effectively panel square error stress primary good sometimes medium long member panel recover formally setup weight unity leave zero observable stationary weak form difference distance explanatory difference two difference example distortion discussion difference observable fy ar ar setup classic describe way implement criterion innovation observable estimate second reason estimator lead series innovation e shall illustrate minimal observe give analysis case involve measurement relate linear skeleton special misspecification measure version measurement measurement old least early assume recursive namely equation write give independent mean variance long denote series yy determine estimator equation p matching incorporate lag beyond may minimize denote minimizer regularity condition far smaller substantially decay slowly decay decay cause shall return exactly equation difference show amongst mse method lag extent lag method find autoregressive plus additive white noise approach essentially treat attract attention li autoregressive match wrong consistently possible lag contact call next stop ahead estimate typically reasonable autocorrelation observe property discuss dynamic measurement apply system example example skeleton observable employ ease explanation start dynamical lyapunov exponent predict lyapunov exponent system x admit cycle derivative suppose lyapunov neighborhood x summation take noise origin fy fy x fy fy used mle indicate ahead noisy second part suggest reduce bias cause also past offer remove statistically interesting skeleton value nonlinear white ahead consume numerically ahead implement suggest ahead skeleton sometimes helpful especially turn statistical parameter parameter sense tend absence specifically achievable observable precise infinite parameter define unity ease exposition infinity see panel figure useful time panel observation nonlinear eq show figure skeleton replication match period column four infection massive considerable prediction year trend http sl txt panel error average ahead error take mark select mle ahead density function method fan estimation matching capability investigate span ahead step period display bottom mle superior short reverse b pt pt surface cycle notice cycle cycle fall regular helpful weather help historical recorded world number period http www autoregressive lag higher suggest well notice dynamic self threshold autoregressive regime delay equal recommend try delay horizontal solid respectively post skeleton period model frequency fit fit generate skeleton draw short one ahead panel long fit step beyond ahead appear line series avoid unstable observe time line datum consist population control laboratory datum employ obtain daily biological major series population bi population bx bi day differently usually discretize ahead mle base skeleton cycle extent day bi day cycle capture period grow bi day bi day last panel clear cycle period light population twice period panel mark axis vertical color code blue higher thus indicate period infect sir ordinary qualitatively behavior however make bridge early epidemic general form liu di sir force bi base infection year explain massive statistical improve directly observable distribution reporting propose reporting rate line equivalent reduction adjust multiply un show panel unknown panel adjust birth panel line skeleton list calculation simplify solid panel figure show two panel bi important birth middle birth massive relate cycle birth rate relationship source theory york year year cycle birth piece cycle five equivalent birth transmission massive quickly change birth fail capture worth aid show satisfactory matching investigate cycle birth peak code peak correspond birth birth become birth cycle three year cycle even cycle fitting model perhaps capable birth rate thereby support sir box either parametric free instead way match notion absence early attempt estimation ahead method one ahead prediction application absence prediction secondary evidence stand medium aim horizon say rest nothing take observable give estimate minimize term sample design k rare condition estimation alone extra information exactly indeed exploit approach concrete approach issue specification suggestion experience would connection bayesian statistic need experience especially area relevant reference include al al form criterion model comparison multiple series among fit might appendix theoretical justification however se strongly mix sequence exponentially moment w marginal fy moreover linear fy x marginal g h old cx w ease otherwise large x e note w cx distribution hold triangular array due lag introduce denote quantity pn
appropriate value freedom play response vector modeling produce freedom fit define element estimator combination response matrix effective select tuning parameter ridge lasso penalty express form unbiased degree regression take proof suggest e unbiased expect model select minimize estimator freedom criteria consistency bias correct cross cn generalize introduce furthermore modify ease burden algorithm generalize path closely variety condition condition member absolute value example penalty include elastic minimax elastic net many convex include denote tuning path start solution value continuous coefficient monotone jt jt remain unchanged e jt jt condition first jt I update direction apply produce iterative algorithm loss kt orthogonal ng jt orthogonality times absolute monotone square follow degree freedom iteratively calculate trace initial mt kt freedom freedom easily derive orthogonal orthogonality define small close get degree increase degree freedom degree equation little square th coefficient become kt ng kt coefficient appropriately freedom algorithm freedom jt ps jt kt jt compute equation suggest step matrix inefficient simple cost want generalize degree freedom store qr implement write orthogonal triangular write th equation freedom extension bayesian criterion generalize validation via degree package comprehensive investigate effectiveness generate datum set correlation simulation degree adjusted tuning compare degree freedom penalty selection criterion give elastic elastic net detail present generalize degree et freedom freedom selection result deviation sd se th zero say coefficient et procedure tend incorporate et deviation sd percentage non propose mse sd selection criterion correct aic bic generalize cross information yield paper bayesian criterion need square complex leave validation expensive validation compare penalty elastic net elastic net elastic net elastic eq produce ridge elastic approximate penalty difference elastic similar lasso al directly elastic net deviation correctly say criterion elastic follow yield elastic sparse correct good mean perform g dense performance bayesian bic cv elastic family excellent validation estimate error unfortunately generalize elastic solution penalty case aic sd nn sd mse sd ex sd mse sd nn ex nn sd nn ex cv sd sd nn ex sd mse update na I various change penalty carry os produce solution speed base na I slow degree large na I average I na I modify I na I modify diabetes ten baseline predictor tc response baseline penalty elastic elastic degree freedom decrease penalty increase degree path nd helpful degree rapidly update algorithm large make substantial change degree estimate standardized diabetes elastic net elastic degree generalize elastic net yield elastic intercept age map tc procedure via method wide penalty carlo simulation procedure tuning yield especially penalty elastic parameter include generalize linear multivariate tuning parameter future research large mathematical appendix us te new expectation te equation new algorithm first consider write small function sign jt programming mm principle international statistic l heuristic instability reweighte p apparent error rule penalty cross least angle fan li tool h ann classification report university regularization descent j software fu regression regression improve freedom multivariate department stanford team language foundation schwarz ann assessment stein ann n correction shrinkage wang li tune smoothly absolute deviation li shrinkage tuning measure effect model lin group zhang nearly penalty ann g yu ann regularization net adaptive lasso oracle r degrees ann guide journal author manuscript option head source file please examine carefully follow formulae reference etc closely please acceptable requirement large amount manuscript delay publication attempt correct production team paper please department country mail address address publication put california sp publish side probability drop beyond paper elsewhere document web essential place side subsection use divide appear subsection end paragraph mark extra character line line correct production english use mark direct attribute mcmc ml dna ordinary upper efficient method method well special symbol text mathematic rather symbol comprise letter english autoregressive capital letter aic macro start publication delay failure journal write idea clearly reader time sentence nr cg alternative force energy parse focus underlie mathematical expression please carefully display display expression brownian motion derive proper name square error mean square minus sign uses join double person user range join name people minus mathematic remark place phrase cut equation refer number place formulae save line text display likewise expression leave expression avoid g matrix unless vector avoid preferable capital care please iterate sign power complicated quantity multiple avoid symbol align var trace expectation please point use list give range integer denote whereas range number appearance expression display place display mathematical reference production elementary comment find standard reference axis label page use avoid axis graph environment roughly usually page ensure label letter font production panel type symbol appear please figure colour essential care journal figure refer reduce error sentence example pc refer check effective page avoid reason character wide include minus sign character arrange without table improve multiply power ten respectively save effectively code example table digit justify gain clarity error measure common phrase standard census percent percent r black white black american u united comprising stop description line symbol appear verify agree line correctly line symbol body brief
broad insight range achievable suitable review relate binary parametric family accommodate motivate type quadratic adjust family specify compare correlate index sub vector denote arbitrary I moment coincide first moment inequality know several article map I I ii dm ii exactly fr symmetric definite derive moment maximize valid specify da multiplier complete coefficient valid elaborate compatibility seem easier express binary coefficient feasible correlation coefficient difficult statement notion independent dimensional uncorrelated uncorrelate p p p mutually uncorrelated independent since correlation binary form ff u use term coefficient x I regard metropolis hasting mention result proposal auto metropolis hasting let metropolis hasting auto definition expect value obtain triangle inequality yield ix structured dependency average ij article correlation family sampling correlation high require therefore conditional value triangular conditional monotonic straightforward wise one auxiliary allow come back idea ht q p discuss conditional truncate link structure matrix fails accommodate complicated conditional valid structure conditional structure differentiable cross value triangular link probit dy proof moment b moment r monotonic jacobian proceed induction scalar conditional construct conditional since suffice show column contain inductive family general suggest motivate logistic arise triangular h design parametric marginal distribution family q identity non q quadratic procedure logistic conditional original exponential behave function absolute thus exponential hard fit correlation dd cross moment propose fitting dependency space limit low dimension sequel structure enumeration secondly approach adjust cross add new moment triangular exact expectation expensive replace remark ij become limited solution fail converge conditional moment rather usually virtue lemma version feasible cross assess di adjust parameter newton suggest bivariate distribution proxy might start solution copula attain small alternatively near frobenius integral therefore easily incorporate introduction conditional logistic copula moment fit parametric record sample meet alternate sd dm di dd dd replacement consideration inverse dim di dim ii ii ic dim dd I permutation approximately uniform draw matrix sampling moment might scope parametric introduce difficulty shrink uniform cross matrix entry moment parametric adjust might numerical experiment find frobenius qualitatively picture conditional replace exact carlo concern conditional family explicitly sample cross feasible loop dimension order random median show group family scale axis represent em suggest scope conditional far practical logistic accuracy compete conditional family fast grow flexible conditional besides allow evaluation conditional far finding comparison carry particular part author ph thesis supervision family specify mean correlate pt pt definition procedure class family
coupling succeed recall event coupling fail imply combine prove proportional coupling pt comment second coupling prove pre cutoff careful actually prove cutoff continuous total stationarity choose since coordinate time improve bit vector start chebyshev observe entry prove mix worth point argument go change large distribution include step cdf show total essentially argument analogue indistinguishable know analogous markov box hold ball every putting remainder give greedy give wide couple create practical way obtain point simplex obtain simplex much hard past monotonicity anti monotonicity begin detail choose start markov chain space couple single couple markov coupling eventually bad inefficient property property chain easy coupling track start maximal minimal markov infinite course keep start monotonicity obvious tracking little copy start couple chain chain jt jt jt tm jt jt jt jt tm jt n j x j give q apply q tell extremely second identical run choice representation attempt chain merely substantial coupling use variable perform determine couple otherwise n n j analogous coupling subset probability measure metric coupling chain couple subset fail perfect coupling succeeds start epoch record information fail epoch rigorous rigorously estimate fail run distribute binomial stationarity author thank david author thank lemma support determine gibbs sampler conjecture couple contraction markovian coupling perfect gibbs sampler hard analyze measure obtain monte mcmc period analyze difficult receive exposition sampler general geometric coupling technique build technical understand simple detailed analysis nice analysis walk walk application analyze namely addition use powerful non markovian coupling past initially toward sampler complicate contingency author thesis technique paper improve analysis still substantial room improvement paper popular variation measurable start simplex whose stationary take move chain choose uniformly first show confirm demonstrate cutoff moderate simplex satisfy ccc step transition measurable hand prove bind closely relate chain idea wider marginally work coupling coupling assume accord tx throughout marginally sampler joint evolution time method markovian final move likely distance large amount probability coupling describe coupling coupling always update entry coordinate describe couple update complicated chain couple always specific perform proportional coupling otherwise assume generality condition list read start coordinate chain burn copy markov proportional generate measurable pt expand collect induction obvious n decide simplex go infinity first pair vertex jt jt jt repeat define construct partition let nest either define mark mark marked time whether graph connect follow place enough graph fix connect ng ignore vertex p couple coordinate update update coupling mark coupling otherwise proportional coupling mark coupling proceed accord walk time coupling begin show probability coupling satisfy bf assume succeed fy n f bound remain close closeness
bi local recent approach involve type sometimes bi degree work multiplicative merely way two multiplicative relationship frame video also class relation multiplicative bi circuit multiplicative interaction static thought relation product intensity example interaction transformation modern term mapping briefly feature discuss model feature graphical bi connect observable connect pixel unobserve one separately parameterized variety boltzmann machine learn hidden extract image capacity simple become application constrain capacity activity capacity mind common generation synthesis vector minimize reconstruction combine encourage optimization end alternate optimize inference minimize test fix avoid eliminate define implicitly common linearity encoder minimize reconstruction code one penalty term encourage alternatively train auto de corrupt achieve corrupt turn feed technique encoder rbms amount mapping linearity tractable gibbs order training ica one complete encourage response becoming enforce inefficient practice eigen generation mapping typically refer appropriate train advantage stacking eqs affine shall bias avoid clutter alternatively think extra size around pixel reason computationally demand important image deal bag interest sift corner add representation vector classify classifier variation histogram similarity collapse descriptor vector classify classifier scheme local yield competitive performance recognition approach extend encode oppose naive two coding receive would transformation suppose detect net receive show equally logical see example content every constitute act detect equal c less receive spurious activity depend pool amount compute value datum pdf shift secondary evidence hide compute model suggest transformation typically variable figure alternative sub slice connect pixel matrix stack gray intensity commonly define slice tensor leave image turn map shall correlation pdf coding consist could bias connect shall drop clutter simple bias bias dimension coding model sparse code almost code group variable example infer recall code turn function form show inference amount define one simple linear bi inference differ standard meaning versa image analogous expression quadratic function commonly represent function representation factorial factorial contrast mixture make cc xy assign quantify useful calibrate achieve use training dependency discuss detail section standard difference code outer way utilize view conditionally input case contribute possible similar cf iterative coding like rbms machine change normalize consistent define joint training difficult involve auto define learning encoder acyclic use train see figure probabilistic make quantify image model change way rbm simplify long gibbs train model relational auto q force like auto gradient optimization reason allow image intensity optimize joint contrast hybrid train unit higher learn show toy example boltzmann image dot copy shift center plot figure infer vector infer infer strongly figure transformation top image illustrate decompose factorial end parameter thereby parameter cubic assume pixel easily highly suggest reduce inner q factor like choose illustration factorization interesting note factorization law factorization group onto cc reduce project claim find frequently restriction optimally allow multiplicative interaction example factored encoder factor multiplicative interaction restrictive connectivity equivalently advantageous show factored filter optimally class component rotation figure affine top bottom natural training field obtain light filter take page correspond detection plausible motion project image onto shift function add square sum square model temporal sum square component energy thus cosine pair quadrature shift set different translation pick get pdf suggest like layer filter may along function image view layer use suggest adopt ica avoid degenerate ica cf approach shall hide layer analogy factor factor boltzmann art motion closely connect term eq activity mean unit think unit variety simplify interaction practically field filter slowly maintain way maintain normalize every connect top level full connectivity help unit virtual grid space neighbor popular mainly learn common patch center contrast usually also detect rotation train transformation code represent shall restrict attention transformation identity also know practically translation shift transformation important eigen complex eigenvalue multiply amount complex plane pdf orthogonal thus equivalent project real eigenvector rotation invariant iii projecting decompose rotation contain secondary elsewhere fouri rotation subspace amount fourier norm fouri signal translation part matrix fact come commonly refer quadrature pair detect eigenvector rotation refer eigen transformation central way transformation angle rotation within share well set eigen transformation less spatial rotation set eigenvector transformation eigen differ angle result may angle rotation eigen projection onto denote projection axis plane projection cosine angle normalize projection cosine rotation need normalize amount divide unfortunately case invariant rotation close axis rotation ignore provide recover impossible angle q rotation allow preferred angle denote maximal angle rotation idea well subspace rotation meet projection represent subspace via ability suboptimal representation detector take compatible detector image computing sum subspace encode subspace rotation pool stack eigenvector respectively subspace appropriate meet row row filter rotation inference cf pooling identify thought multiple learning involve orthogonal subset learn dimensional subspace form filter albeit canonical correlation cca canonical tie pca tie weight model simultaneous transformation imply length impose non compute mapping interpret provide content representation help noise rotation detector equivalent square subspace thus projection contribute transformation make response response invariant projection detector prefer rotation detector concatenation model transformation modify term adjacent markov alternatively energy compute square implement train video move dot auto encoder concatenation constrain frame dot move speed speed vary movie auto two effectively implement absence structure implement figure model learn spatio temporal fourier frequency orientation cccc concatenation random dot main utility feed network may play task cross requirement currently correspondence task extraction descriptor removal system replace single train help visual variety homogeneous module correspondence invariant input orthogonal encoding amount graphical manifold distribution feed effectively transform canonical pose represent canonical pose help explain filter recognition selective similar rotation feature show feed recognition somewhat recognize static biological move pose compute product automatically associate object train frame analogy argue analogy making phenomenon question make module building capability sparse energy standard cf
n xshift q xshift cm n n xshift cm r edge double n double xshift edge double n edge n xshift edge present obtain p select path fully length first line enforce choice completeness make heuristic identify make enforce among left pt maintain throughout return minimal input equivalent lemma ps take unary view substitution replace decompose atomic brevity rule child edge p atomic identify long would discover incorporate query convert line contradiction argue lemma p qp learnable example set challenging unary unary select indicate path query example boolean query construct problem devise satisfied recall query class head boolean boolean query boolean would path xx xx xx xx xx ps ss trees l p r r return I path result element arbitrary late presence singleton ambiguity xshift offer n item n n offer edge clarity presentation path item item query whose string choice negative item sp boolean query tree sound completeness algorithm sp completeness sp prove class learnable resp return minimal boolean necessarily query minimal query boolean conjunction fix q ci boolean path em example q consist another path query reconstruct query speak formally path query note become embed ignore htb xshift double edge xshift edge double xshift double edge free element path algorithm present extend query xx xx xx xx xx ss l return satisfied never sound exactly time query illustrate consider article book xshift cm article title article author title title return yield give title author title consistent title em query minimal produce minimal perfect tree child parent path completeness construction minimal element algorithm query fusion moving never leave finally exist leave subgraph path leave claim sp xshift cm xshift yshift xshift xshift yshift cm xshift cm x edge yshift yshift xshift cm yshift yshift xshift edge yshift cm cm yshift xshift edge x x block v x separate one construct consist formula tree tree encode leaf ensure separates form encode input remove example part technical separating hold edge free consistency pattern np adapt np remark expressive interpret unless class learnable polynomial learn language inspire tree area language learnable g language language learnable fact class finite learn restriction g reversible language occurrence expression important query come expressive path query relatively responsible learn path query unary query inspire survey extra character instance match nonempty possibly empty match unary need use match instance pattern query behind inference unary node infer schema information pruning handle annotate advantage expressive properly query drawback scenario heavy formalism exist easy visualization infer include class path language technique infer convert unlikely successful translation task consider beneficial avoid line restriction would easy translation approach would require modification would constitute language learn begin construct generalization pilot generalization similarly word inference query incorporate example program inference use direct membership query refine infer framework query contextual query contextual language language condition exactly relative child tree language order finally study relational database language offer structure database instance query construct condition tuple table study infer query investigate several boolean two one example sound complete path path query hand inclusion sample believe informative investigate advantage allow query example learn produce direction add union quite decide clear technique interested operator impact restrict path define semantic query match connect path query semantic notion match produce moreover finally enable take schema cf schema explicitly result algorithm testing general hardness modify limited schema tailor section institute investigate learn query query take annotation must general user annotation one always return every practical unary learnable add picture application basically leave store easy document need require exist core query allow file language framework user formulate knowledge specialize address gap remark however set exchange query define mapping specify pattern new source discover another query annotate query select accordingly annotation produce annotation may current identify type annotation node must select annotate every term computational setting one instance document annotate htb library book edge book edge r title author title edge capital edge n n edge edge n receive annotation properly annotation fit work consistent consistent annotation mean influence computational inference learnable take I query sample trivial discussion learn query sufficiently informative precisely role teacher rather add satisfactory commonly sample I return extend restriction impose ensure unary query investigate unary document boolean certain case purpose boolean positive simple feed offer web site annotate htb xshift cm edge edge xshift cm offer item edge edge xshift cm list type n edge pc boolean annotation select unary query example presence path query learnable algorithm inclusion minimal algorithm try construct respect algorithm learn positive reversible regular language occurrence return consistent sample query consistent minimal learning algorithm query see approximation path query identify enable path give annotation trivial example query select consistent result consider query restriction admit positive define establish theoretical addressing additionally investigate construct minimal check consistency learnable unary path base nontrivial way remain nontrivial organize introduce notion define learnable query present learning discuss negative outline direction restriction proof acknowledgment would anonymous helpful comment thank insight improve support education de region project education research n throughout document order extend define canonical iff document tuple tn child acyclic require non relation tree cardinality node tree leaf path node view path word term n n n n xshift represent answer distinguish distinguish tree select node indicate rarely make ambiguity structure query query may additionally distinguish child correspond axis unary distinguished selecting n n n q q acyclic require exactly boolean query boolean distinguish edge line line box restrict unary query one unary capture exactly positive sequel use unary node unary query virtual way mapping query say relation symmetric match embed match simply embedding tree htb n xshift edge bend densely dot bend bend leave densely dotted bend leave densely xshift bend densely bend densely dotted bend densely edge bend densely require embed node query map path query note self typically semantic unary query nod unary define query naturally notion closely belong notion extend condition replace n p converse general b complete whereas identify minimal tree term inclusion ss standard adapt query comprise concept learn instance concept case tree possibly semantic concept tuple example unary tuple define setting auxiliary notion query take learn two return special query extend often learn fitting requirement insufficient eliminate instance return universal sound require complete analogously sample later exponential size section learnable essential enable implication search refinement logical capture property polynomially query serve characteristic property learnable merely direct learnable open question example set form learnable query match characteristic construction match generic symbol construct match contain child every path label contain unary ab edge edge xshift edge xshift characteristic construction seem artificial match simple sample path query path query node main reason work embedding restriction node boolean query impose technical lemma impose restriction edge unary query b restriction impose node boolean ab unary query properly query limit query basically discriminate depth alone restriction boolean query equivalent note however boolean query query technique unary boolean lemma claim structure path b boolean unary path query occurrence claim unary let construct map root map b node belong map preserve label follow take therefore node
utility ir observable property observable usual observable value intersection hypothesis use state formalism quantum non state ir relevance call alarm call false alarm false alarm acceptance induce acceptance maximum power note pearson ml define powerful acceptance decision provide nan alarm detection q space vector field set linearity set scalar vector vector transpose product dirac notation reader refer brief illustration dirac observable observable vector correspondence correspond observable vector state define distribution hypothesis play role ir suppose variable relevance alarm document principle ml test cut false recall detection equivalently document region rank relevance recall observable observable vector probability relevance however scalar clarity binary relevance relevance belong either term basis establish relationship observable orthogonality superposition physics superposition observable much relevance illustration distribution binomial approximate observable document mean term document poisson give large equal observable term collection indeed probability frequency encode parametric term pt false alarm acceptance relevance difference subscript swap correct decision coin discrimination alternative minimum minimize maximize yield power minimize curve region relevance region rejection document possible I term alternative complement pearson subspace orthogonal subset yield operation orthogonal suppose yield new subspace reformulate observable something three subspace ray subspace plane dimensional span provide l l thus hence subspace key optimal way powerful region subset yield ml powerful state high eigenvector optimal region observable vector spectral vector geometry decision state suppose observable vector define span relevance correct angle observable vector vector figure illustrate geometry observable reader generalize angle observable relevance relate angle angle vector hold illustrate replacement observable angle correct q cosine angle poisson shape correspond shaped curve plot relevance relevance less high relevance prove side lie angle relevance poisson apply tell whether observable even estimate tell false alarm correctly acceptance induce observable poisson term thing equal estimation outcome observable term either term relevance function induce observable corollary compute optimal computed acceptance region observable observable distribution document false coordinate orthonormal basis observable coordinate express angle provide angle error error minimum observable observable vector detection maximum observable thus accept p p observable span optimal actually sort every test illustrate aim realistic experimental computed state measure rather aim probability every superposition curve like label probability shaped curve part different word never shape curve curve label error label figure topic type exhibit pattern issue optimal observable finding absence device frequency straightforward occurrence physical measure device sufficient much physical correspond frequency document outcome correspond observable answer automatic indexing retrieval observable retrieval indexing interpretation observable provide superposition interact place observer upon whenever trace induce valid observable subspace span however logic combine subspace combine subset say relevant relevance measure say possesse measure observable set describe document term index orthogonality vector implement mutually observable hence say union intersection complement ir capable observable occurrence document specify thus achieve accurately possible ir capable observe observable ir observable document decide observable shall pay attention would ir van book book theoretical foundation optimal book quantum quantum retrieval find formalism ir aspect paper research
isotropic therefore u estimate logarithmic moment line let eq inequality fix subgaussian design response satisfy eq subgaussian zero mean invertible square eq jensen inequality q define triangle inequality follow bernstein martingale follow variable follow give tail freedom q eigen decomposition orthonormal eigenvector thus ax freedom random variable proposition remark quadratic form subgaussian setting quadratic spectral following provide multivariate appendix invariance tail random due slightly weak sharp gaussian tail sum sum martingale I square much large fall show completeness appendix difference proposition give constant subgaussian result subgaussian bind deviation simple proof explicit
tt tt tt u tt straightforward covariance function pointwise quantile identically representation sampling lin validity enable confidence band one band substitute investigate sample procedure design normal conditioning failure censor censor study pointwise moreover simultaneous band outside present survival censor smoothing necessary usually square plug lead infeasible say minimize integrate behind censor validity censor use censor table average coverage pointwise confidence computed variance tend coverage deviation decrease numerical reason compare subject availability expand hence bias indistinguishable deviation become satisfactory empirical interval good performance select automatic point probability censor reveal pointwise nominal simultaneous coverage coverage probability simultaneous confidence band stay entry diagnosis death depend receive patient prior rest receive study patient association decrease marker death meaningful analysis simplify marker dependent n far variation provide dependent simultaneous band week summary confidence band classify patient time within non conclusion detect week week respectively figure decrease patient close figure accuracy measure time along line appendix derive converge pointwise simultaneous reveal figure negligible word might lower discrimination ability cd classify subject week explanation homogeneous survival time cd count estimate survival cd irrelevant receive variability significant enable traditionally derivation load expensive develop technique rigorous justification estimator well variance application detect propose stable estimate sample censoring marker censor propose simple easily derive bivariate estimation estimator always meet advantage totally censor empirical censor survival form occur pair usually focus survival period marker censor make practice flexible assumption conditioning estimate article quite study appendix ix tt tt cf direct calculation side express u h theorem asymptotic establish hadamard result delta van weak van eq taylor expansion mapping theorem imply q reference distribution bivariate censor wang expansion statistic consistency neighbor berkeley auc diagnostic marker measurement parametric roc curve censor diagnostic marker lin functions journal theorems mathematical van asymptotic method diagnostic york sd se cp se time se sd cp sd se cp se sd se cp sd cp sd se cp sd mean sd pt se cp sd sd se cp se cp time sd se cp sd se national assess diagnostic result operate characteristic roc curve widely summary measure generality life extension event usefulness disease detection compute expression process provide essential evaluate cd cell patient survival clinical inference dependent patient receive signal detection diagnostic clinical beneficial sake improvement diagnostic compare roc false purpose diagnostic test advantage curve specify moreover invariance characteristic roc scale suitable base move application auc area roc might entirely capture roc curve auc totally performance furthermore drawback range practically sample property estimator comparison future address estimate develop procedure loss derive censor represent censor status ty perfect interpretation rescale explain value marker censor
return high feature lead place lead high whereas retrieve observation pair thus design translate provide occur event partially satisfactory cause reach optimality without cost superposition coordinate unit quite provide expression reach optimality basis therefore feature new value region superiority pure rank induced region follow pure case differ answer mixed region acceptance whereas pure counter rank discriminant associate associate pure equation rank detection false alarm state find density define coordinate weight estimation occurrence hypothesis ir whose state bm weight drawback weighting bm tuning database thus rather problematic illustrate follow accept true false alarm threshold formulation rule indexing currently bayesian estimation quantum mechanic book quantum example interference processing start area ir inspire report within paper however management context classical quantum principled retrieval ranking retrieval request decrease request effectiveness system user principle classical probability keep subspace acceptance rejection probability unit yet address paper although perhaps rank probability interference estimate dependency relevance rank rank quantum effective one ranking contrast address interference estimate classical probability effectiveness probability stem region acceptance quantum interference contrary ranking optimality upon subspace relate framework view paper formalism feasible ranking formalism comparable bm classical database query survey query plan execution classical concentrate answer tuple assign probability database paper incorporate management information analogous management future unit may insight quantum investigate crucial understand paper theorem management database index retrieve information tuple document pattern profile meet relevance usefulness unit unit thus law admit total law perfect optimal question ask improvement obtain theory set unit differ know false quality ranking give false alarm detector classical within management implement effectiveness recall estimate probability management retrieval ir ie store retrieve tuple document system management uncertainty rank crucial deal uncertainty aim assign relevance usefulness unit place measure relevance theory measure end necessary represent prediction management retrieval database measurement predefine hypothesis calculation probability false alarm management thank rank perfect unit theory event kolmogorov axiom theory describe hermitian quantum entail classical set region rejection indicator base detector detector quantum imply phenomenon investigate ask whether quantum theory outcome principle available known theory compare quantum detection classical ranking quantum rank section interpretation region provide work also quantum view definition exclusive event event sake clarity introduce simple occurrence binary relational usually represent mutually exclusive scalar scalar product dimensional vector event must inner event binary vector symbol representation must event difference start point paper algebraic hermitian self adjoint operator use quantum mechanic prefer matrix sake clarity prefer operator hermitian transpose hermitian important hermitian quantum eigenvalue hermitian correspondence vector conjugate transpose event mutually trace axiom function hermitian particular space collection event space unity orthogonal latter term resolution unity dirac notation appendix represent mutually exclusive hermitian event system particle system system color ball internal device e color management physic management query click tuple attribute state review allow algebraic approach rule introduce consider event occurrence alternative list arrange matrix matrix zero probable otherwise mixed pure correspondence outer transpose concentrated certain help event provide hermitian finite rank mutually spectrum theorem pure one say hermitian decompose pure matrix hermitian probable mixed pure whereas estimation another management predefine categorization whether display engine page put engine page database decision computation associate illustration necessarily brief simple aspect bring elementary retrieval quantum information unit number review make number example calculate engine function clarity frequency interest customer store mathematical implement decision unit hypothesis generate hypothesis system hypothesis density label management customer irrelevant engine item customer irrelevant user depend density decision pearson instead without whether hypothesis threshold reject otherwise accept nothing accept high alarm threshold pair give size curve receiver operate characteristic pearson partition distinct region accept acceptance region observable region pearson instead utilize acceptance subspace follow region spectrum identify theorem function positive observe place usual deal one however mixed eq eq q absence acceptance correspond respectively never accept accept feature accept topic categorization furthermore build describe either include unique feature occur yield pure replace distance density cosine justification fact angle space subspace unitary pure mixed detection alarm pure curve counterpart yet evidence alarm interpretation interpretation tie pure follow rule decision rule estimate alarm let probability order pass lagrange interpolation pass lagrange power plot produce decision suppose ten management binary h computation follow interpretation latter
various optimization plot result size pass method poorly even pass hybrid solution visually nearly indistinguishable still possible obvious deterministic stochastic inversion van image place surface emphasis application analysis objective function prescribe depend accuracy implementation rather intermediate approximate particle particle thus provide acknowledgement thank suggest le valuable grateful anonymous comment lead detailed mistake theorem l l grant schmidt author grant research involve offer subset progress approach solution contrast steady expense hybrid exhibit benefit incremental gradient rate quasi experiment illustrate potential benefit often function measurement choose square q gaussian logistic rise equation often amount uniformity measurement mean evaluation unnecessary progress motivate method evaluate iteration gradient time incremental achieve rapid initial progress accuracy higher full dominate incremental exhibit benefit two incremental preserve incremental version iterate update q residual gradient deal typically converge suitable gradient estimate residual specify iteration characterize convergence deterministic noise context refer influence scale account curvature g quasi newton minimizer overall strongly continuously ratio number imply value characterize every iteration asymptotic example characterize emphasize note imply converse imply convergence analyze iterate sufficient strong convexity imply eq rate uniformly positive definite constant euclidean replace scale divide imply sublinear convergence describe ensure rate achieved arbitrarily require convergence deterministic gradient sublinear show maintain sublinear size control achieve effect choose size allow control error implicitly idea line implementation fitting compare incremental base spectrum full expensive fast incremental sublinear iteration dependency give stochastic oracle contrast convergence satisfy iteration incremental gradient converge incremental achieve close spirit convergence treat pass gradient difficulty et memory grow strategy practitioner explicitly mention grow aware present propose quasi newton previously analogous weak extend work require strict decrease allow realistic lead asymptotic stochastic source file reproduce consider basic iteration iteration inequality q obtain eq low left hand side give aside require convergent sublinear reflect noiseless convergence absolute suppose iteration recursively choose trivially arbitrary implication linearly zero q yield scenario gradient appealing achieve nesterov strongly possibility bind imply limit error sense counterpart expect expectation proceed construction come require convexity gradient lipschitz trivial current optimality positive control strong bound imply case computed expectation low linear follow expected rate bound fast extreme mean long bind seem allow iterate maintain strong iterate solution calculation fast trivial bind valid although practice well norm gradient deduce proxy magnitude gradient residual decrease heuristic decrease increase likely remove deterministic sublinear assumption sublinear preserve average sublinear rate bound lipschitz continuity convexity deduce hand simplifying obtain result side nonnegative show auxiliary enough ic recursively recursively side get eq imply thus get imply sequence convergence achieve deterministic constitute typically single element choose cyclic sample sublinear implicitly improve sublinear complement term portion sample thus leverage result next optimality linearly increase induce sample towards q obtain furthermore nk difference consider control size ensure bound iteration fairly use irrespective manner g sample obtain tight however suitably modify related gradient use eq q q side bind initially zero illustrate sample size requirement e b size schedule cc far make modification scale search attempt curvature quasi newton approximation hessian system approximate maintain use limited maintain recursively formula recursive bfgs update ensure bound condition scale hessian gradient analysis requirement part ensure sufficient decrease design attempt balance rigorous none procedure quasi newton method initial trial represent observation yield decrease increase eventually reduce conventional true guarantee eventually couple choice newton hessian local convergence summarize incremental gradient grow sample series fitting application summarize structure random general first outcome choose likelihood maximize regularize satisfy function upper available particular application least last datum central relevance digit structure crf noun crf image nonlinear compare conventional limited memory bfgs hessian study type memory quasi newton logistic incremental constant method prove advanced crf compute satisfy interpolation number grow linearly include deterministic plot progress pass evaluate model application feature outcome goal parameter inner give q typically add equivalently take value compactly form lie nonnegative hessian rank combine binary logistic regression describe token email whether email spam datum plot figure across pass power give sensitivity advantage deterministic hybrid hybrid make rapid progress unlike method make steady progress deterministic agree multinomial
cm xshift cm elliptical vertex box box specify box specifie uncertain calculate calculate see right leave plot next find fortunately see minimum conclude proposition node leave x node interestingly low actually cumulative expert say along although name water water level study length parameter uncertain expert scale calculation introduce uniformly transformation coefficient mode uncertainty water measurement triangular distribution actual equivalence box box calculate depict immediately apparent node log form need safe side traditional instead box independence expect much wide lead lead weak maker verify method box assume analytical uncertain course box operational modelling inference box totally represent coherent natural consider box totally space thereby box infinite box space number paper calculate box prove natural extension box additive component respect topology equivalence corollary completely calculate natural suffice follow proposition box whose mapping box particularly attractive correspond region shape combine box box rule combination thereby dependency consider extreme theorem formula derive arithmetic obtain approach demonstrate inference harmonic assessment calculation generally open box even marginal choice lead dependency arrive bound p box possibility measure acknowledgement grant grateful extremely suggestion various finally constructive comment presentation substantially example remark cdf curve I I curve I type I I I type I I I I curve point type I lower cumulative function box theory expert quantile formal already new construct inference theory heavily extension independence particular arbitrary totally thereby p bound set dependence fr extend arithmetic practical feasibility uncertainty information allow include upper possibility necessity probability coherent unlike classical probability complex functional description possibly expense ease upper box cumulative distribution essential aspect box include arithmetic efficient inference box connect technique applicable box arbitrary yield problem signal coherent work generalize many equivalent set finitely study box lower enable p box problem concern come tool reflect consequence extension many extension box new tool inference arbitrary bound quantity box low mention box totally box box box continuous perhaps even importantly handle consider admit p box space box build dependency express construct unlike box restrict rather box space exact inference extension arithmetic real provide multivariate study model point natural extension event topology equivalence natural extension gamble via box real multivariate construction box coherent dependency model special unknown probabilistic approach demonstrate infer harmonic parameter box expect box end introduction model shall also brief summary call gamble denote context value elsewhere confusion subset interpret transaction assess collection gamble argue subject belief usually conjugate upper subject infimum price low gamble coherent see ff f pf linear event finitely additive lower let dominate coherent agree envelope see consequence envelope lower coherent pointwise coherent agree pointwise natural extension conservative coherent thereby minimal consequence extension natural extension set gamble gamble include natural gamble usually coherent interest lattice gamble wise wise formalism totally finite space instrumental natural extension restrict box write exactly similarly simplicity always moreover mass happen make sense topology yet denote induce element cumulative distribution definition effectively impose impose decrease thereby box interpret interpret box straightforward box arbitrary gamble natural extension box circle circle draw smooth node node node node node yshift node cdf smooth yshift xshift node node function envelope natural extension gamble conversely belong belong coincide cumulative belong shall proof sensitivity regard knowledge cumulative trivial yet useful box natural obviously extension instance extension dominate least conservative devoted natural simply expression clearly event field extension introduce element however apparent model stick coherence nothing gamble shall finally gamble precise box finitely additive finitely include proposition box eq cumulative secondly satisfy kx k satisfy deduce fx k expression calculate aa use conjugacy determined box coherent field unless shown given define aa deduce eq coincide even define total eqs denote cumulative box arise event calculate cc supremum event arbitrary calculate interior closure respect closure also component operation commonly addition complete component interval immediately yx z constructive calculate case connect case usual case precisely continuous immediate take immediate eqs conclude form component complete full particular square naive way decrease constant element think write box triangle whose diagonal box need define marginal box partition topological interior eq illustrate inference probability ordering poorly choose approximation event interest cycle xshift cm node xshift cm cycle scale cm yshift right cycle instance ordering square course entirely obvious discretize natural strategy reference modal correspond interest p box totally natural example give gamble express simplify low gamble topology show totally space finitely completely complete monotonicity envelope monotone necessarily completely envelope monotone box result box space coherent clearly monotone denote restriction completely satisfied monotonicity gamble gamble immediate event coherent monotone gamble coherent low monotone turn every fortunately way topological eq natural interior gamble easily interior closure ff eqs use calculate must simply component cut upper calculate integral monotonic way mapping whenever element think cumulative function think box type eqs topological closure call multi see sometimes inverse full c corollary eq result union equivalence component clearly large large must contradiction gamble immediately q section construct box coherent hoeffding assume factorization arithmetic special natural coherent low construct box induce perfectly mapping induce marginal box induce roughly speak relative effectively mean affect inference choice nothing prevent theory induce p box quite complicated see feasible box represent coherent low consider rule coherent coherent gamble couple eq conservative theorem box whose natural dominate follow envelope distribution linear compatible completely structure rigorous discussion eq conservative box natural immediate box box outer close domain consider extension event monotone box monotone contrast independence refer rigorous box dominate joint box outer approximation joint domain extension writing dominate dominate
dirichlet monotonic strictly monotonically define ff f mx mutual depend concave eq eq lemma let q upper increase trajectory minimize eq little irrespective process namely expect information thus subsection bounding subsection lemma np horizon namely abuse time horizon follow inequality decompose capture difference second capture side decrease simplify let operator look k maximize deterministic history trajectory probability ta clearly put constant thus central follow eq compare let proof difference decrease however alone converge environment consist connect component environment markovian transition kernel ensure go half imply always state positive propagate state iv choose decrease increasing imply finitely simplify fix visit happen finitely imply visit probability get visit finitely ia must reach discount visit choose number turn last union nan event nan one hold surely value bellman let initial follow bellman contradict law union combine produce iv agent act illustrate environment consist densely clique agent action clique generate goal clique consist state compare select information gain reward iii action iv programming exploration follow mdp reward gain four clique greedy dp clique exploration clique reason gain decrease attempt plot early accelerate history lot attention artificial progress divergence back optimize future expect differ explore environmental design focus query affect environment principle highlight necessity balancing gain progress conceptually motivated agent environment external permit present exploration environment center sound foundation design dp optimal mdp gray need initially answer environment send environment agent carry build select greatly measure shannon agent least optimally choose cumulative maximize special mdps experiment exploration approximate programming follow review basic establish terminology exploration focus exploration mdp briefly review suppose agent environment cycle action receive string refer facilitate environment initial capable light summarize bayes dp environment worth require certainly sign gain history grow kl respect write gain additional action view distribution represent encode element sample optimal coding cause update recursively information proceed action mean action environment life extend agent action action agent gain action formally recursively note definition action reinforcement particular h expect gain notation equation immediate situation end cumulative give history optimal namely monotonically length increase always close never reward accumulate p recover mass dirichlet kl compute accord gain gain increase exploration constructive readily implement finite maximization compute respectively immediate large look ahead infeasible heuristic practice restrict maximum span define infinite limit life without coin cumulative increase simplify discount reinforcement force environment without discount ahead exponentially policy eq optimal discount add discount value horizon case however scenario discount fail
ahead horizon cone specialized set horizon cone large hull affine denote small contain hull cone transformation convex cone get cone corollary transformation close b bn b k n bb k b k b v k corollary subspace theorem characterization effective domain represent many relative iy iy side converge stay clear union finitely many unbounded subset elementary must integrable tu cone k kk yu b b cone lagrangian feasible cone point since cone active column rewrite explicitly slack slack nonzero equation kkt entail operation reduce triangular always ip iteration follow diagonal positive diagonal invertible proof eq immediate diagonal diag diag diag I ia kalman smoothing context diagonal density moreover invertible finish reduction upper triangular definite tt block invert form time remark conjecture penalty optimization huber interior reconstruct dynamical classical penalty robust fast system dynamic e jump addition penaltie huber loss paper model computational kalman smoothing develop establish interpret penalty present kalman maintain extend computational efficiency broader contain mutually interval unconditional estimator column become compute kalman provide circumstance linear penalization difficulty reconstruct jump great impact machine selective shrinkage compressed circumstance smoothing replace insensitive support regression interpretation machine year lead review representation relate scale fundamental condition allow view ensuring penaltie huber penalty penalty derive new smooth penalty measurement deviation point perfectly degeneracy solution theoretical foundation interpret generalize formulation theorem tucker kkt penaltie necessary result support theorem remark recall definition hull affine cone cone cone horizon quadratic penalty nonempty familiar penalty huber hypothesis proof quadratic density always contain characterized th piecewise piecewise function dim true density affine adapt order consider case sequence notation definition also q block kalman problem mean def decomposable prove derive recall extend work tucker interior ip relax specifically decrease ip relaxed theorem main show computational ip time kalman come e penalty appendix show solve preserve usually practice motivate huber verify ml huber respective kalman nonsmooth penalty
apart past explicit quantify decay move definition mix strength dependence mix regularity speak precise lag separate independence numerous learning rely dependent focused derive via consistency approximately prove iid remain pac bound rank finally stability input generalize exist iid mixing mix coefficient knowledge researcher specific coefficient mix risk empirical generalization compete prediction machine know satisfy explicitly mix accord mix begin derive addition convergence markov application result desirable series mix rate certain see overview rate give mix regime present mixing time define coefficient consistency estimator intermediate convergence histogram input conclude proof fix sigma distribution denote many definition however intuitive variation say absolutely speak say total joint separate supremum unnecessary time negative integer subsequence nearly iid block entire odd even block sequence see main two stage recognize finite lead version infinity observe dimensional histogram joint dimensional construct histogram detail grow bound induce approximate establish consistency provide bound error measure theoretic mixing mix coefficient process know case mix integer n histogram leave proof proper choice prove process consistency prove datum histogram let sequence respect essentially apply mixing block widely spaced block particular expectation mix density consider center let union bin absolutely absolutely partial derivative ik taylor fp integral bin proof histogram estimator nh indicator field block sequence allow proof distance density respect dominate let joint process create separate distance target theorem specify rate infinity histogram histogram lemma eq optimal determine n n nd nh nn k gives plug necessary grow supremum unnecessary sigma rewrite notation consist close dimensional marginal eq restriction nest sigma field monotone sequence additional rr supremum negative loss generality theorem exist must triangle eq remain consistent learn know ability despite obvious exploration well drawback convergence provide well understanding immediately quantity define rate tail variation notably mix estimator trick well use kernel joint density
connection identify redundant appear reflect structure interaction value connection demonstrate total correlation measure illustration use information analyze al available activity detail regard production analyze minute minute information create examine train identical label spike record throughout spike joint calculate measure double group assignment group order fire rate development information statistical create randomized randomization accomplish splitting spike remain spike almost preserve fig day essentially total develop information decrease slowly increase maxima day interestingly g steady fig mutual measure among mutual significantly interaction positive exception value group positive investigate relationship redundancy fig goal however redundancy portion redundancy provide distinction redundancy consider appropriate perform unlike develop show correlation throughout redundancy logic find like interaction unable redundancy able detect unable subset variable interaction partial decomposition provide useful unlike measure gate find redundancy interaction indicate perhaps system decomposition entirely highlight define redundancy redundant quantity decomposition similar show redundancy development able several attempt explore property produce similar gate examine difference gain able information illustrative interaction information produce hope system goal would thank comment rewrite add subtracting entropy substitution substitute total applying probability text show ex quantify rise although shannon commonly upon measure develop differ subtle review examine apply usefulness analyze neural early stage aid seek specific research goal user apply area include data compression code name broad applicability part rely value variable although measure approach one relationship measure physical biological information differ subtle way inconsistent throughout array measure clearly system difference information information goal facilitate information measure matlab software available calculate herein previously crucial related information redundancy regard though researcher begin understand provide know something portion know say receive know instead redundancy portion alone alone provide redundancy receive measure quantity researcher create distinct different result overall clearly measure limitation use attempt difference multivariate discuss multivariate theoretic name literature consistent name name name involve call entropy entropy quantify probability near uniform examine quantify give kullback leibler wherein product marginal measure among grouping treat value mutual calculate way consider information variable activity neuron consider relationship amount information neuron neuron consider group condition yield conditional quantify information variable attempt quantify among three attempt gain information knowledge third give eq widely refer redundancy positive imply among interaction imply redundant interaction thus information correctly multivariate redundancy take exclusive entropy entropy interaction denote interaction refer ci clearly interaction even equal odd become co information number variable present co generalized conceptual mutual information information gain mutual arrive correlation tc introduce vector entropy spatial integration correlation write series correlation dual introduce entropy beyond entropy condition refer excess introduce coding relate another signal stimulus compare associate act beyond conclude relevant correlation assume act state q bayes independent give leibl yes know correlation would question guess one nothing everything name change redundancy create redundancy index maximal provide negative index equal redundancy refer redundancy al original refer refer possible information positive say provide decomposition al partial overlap provide partial produce unlike second possibility redundant unlike brevity partial general original four variable describe redundancy work motivation specific quantifie specific provide individually minimum take consider separately redundancy calculate partial note information expansion information decomposition interaction imply contribution great contribution partial interaction exclusive accord redundant remainder article partial accordance term interpret interpret redundancy redundancy refer new expansion total decomposition contain variable contain equation interaction information summation multivariate measure simple system similarity system contrast measure produce offer note due boolean logic total equal boolean redundancy directly information measure ccc c result gate exception total interaction redundancy index partial indicate bit might expect know gate gate decomposition equal bit entirely determine confirm table decomposition difference gate produce bit variable amount decomposition find mutual individually redundant uniquely subsequently information amount interaction redundancy return gate produce beyond mutual individually take together beyond individually also gate gate examine value clear use present conclude chance chance specific must equal subtle multivariate measure interaction value whereas discussion correlation gate example demonstrate crucial difference dual treat case gate bit dual great redundancy portion redundancy surprising redundancy redundancy incorporate different quantity section discussion crucial intuitive variable immediately well examine express rhs though explore fundamentally complete alternative measure correlation datum emphasize provide useful information fundamentally ccc c interesting information decomposition indicate gate variable bit information redundant despite fact distinction variable amount contribution redundant mutual individually partial conclude unique bit ccc ex c c example interaction consider relationship know uncertainty additional gain lose know individually simultaneously example demonstrate consider produce eq total correlation relationship unable dual information pass individual together value variable measure ex c entropy either relationship focus interaction involve logic example comparison redundancy decomposition gate produce gate information interaction able difference
algorithm depict visually allocate produce rejection express rejection free present minimize rejection next construct kernel describe maximum arbitrary configuration weight one rejection reversible break depicted maximum subsequent reversible net heat spin transition probability concrete always minimize average rejection rate method construct reversible first method construct show net stochastic matter efficiency however flow introduction extend importance powerful tool degree bioinformatics economic method satisfy principle central concern achieve autocorrelation key effective choice extended method replica exchange method successfully protein problem selection wang loop overcome critical configuration physical determination focus probability total balance impose although attention transition principle minimize rate stay implementation metropolis hasting transition balance balance simple elementary easy find reduce autocorrelation concentrate within far however construct graphical instead surprisingly satisfy balance impose balance sufficient distribution find beyond balance balance picture explain concrete reversible spin configuration locally run whole discrete elementary spin ise candidate next weight configuration equilibrium balance express weight quantity correspond raw flow law total rejection flat w detail I symmetry task minimize visually move amount weight box picture ise fig allocation heat average rejection balance fail rejection mention besides easily geometric
figure show dimensionality reflect class structure clear rand load mat n noise tp tp nan manual eps eps manual eps eps converge noise noise sx end sx sx sx sx sx alpha alpha alpha sx ny ny w diagnostic ny lx w lx w ny w noise sx w w diagnostic converge limit converge false alpha cca end figure alpha color legend xlabel ylabel legend location xlabel ylabel eps figure hold alpha color alpha alpha ylabel retain eps seek representation spherical noise partially explain g covariate leave residual variance decompose call cca algorithm illustrate pose probabilistic analysis covariance datum assume impose center maximize variance principal principal independently place element likelihood form recover isotropic inner rather express model gaussian involve log likelihood low spherical dual paper form motivation partly study residual idea representation form information wish include motivate covariate could expression patient environmental well remain unchanged see place invertible log entire eq respect proof substitute eq see rely regular form substitute singular value dual follow solve symmetric definite invertible define via generalised algebraic likelihood easily primal maximum canonical covariate solve rewrite cca matrix individual covariance generalise generalise eigenvector direction pair space respectively feature rectangular projection equivalence generalise inspection generalise diagonal show cca maximum graphical correlation rotation likelihood subtle incorporate noise project covariance see cca block diagonal component explain two covariance choice residual case expression predict partially explain variation space graphical standard via turn share illustrate treatment series cell alone come group point gene group complexity uniform medical gene profile could represent imply covariance matrix covariance compute residual force find residual component function simplicity provide bandwidth roughly add noise project profile generalise norm projection retain decide cf increase eigenvalue become retain similar definite always ground list bind tp compare area roc curve tb microarray detail explain
begin specification section element domain database output count description qp ad class algorithm finally release later access private release private definition class predicate set thresholded sum set nf tn thresholds output output work distribution class distribution draw example definition full formal reduction differentially release learn description predicate result database depend mild polynomial release threshold simplify universe description distribution release provide free threshold release general statement overview note obtain query distribution accurate answer clarity simplify assume work access label enable get distribution addition access note literature particular harmonic release provide restrict slightly rich allow new release work reduction add grow private theory work connection technique correspondence database want target explicitly e view release set correspond release satisfy requirement distribution release obtain conjunction count threshold throughout fix query consideration monotone conjunction iff general monotone unchange original datum monotone locality conjunction unchange monotone arbitrary original free release differentially private accurate release way monotone runtime size release algorithm query boost release accurate big formally release free release database release differentially accurate release monotone conjunction database boost improvement work consider differentially release marginal contingency table corollary mechanism et release way table run however note guarantee distribution specific general database distribution specific bad differentially yield algorithm expensive database roughly privacy show mild release hold release release database k consistent range formulation give data release algorithm answering query polynomial fix datum universe description semantic description learn target use fouri view query count query fix many database use release w harmonic release query differentially query database size runtime q quite class namely query circuit data release database database polynomial release counting query boolean circuit differentially private release query database runtime q result majority circuit release structure non box note class predicate quite rich example include approximate count polynomial uniform query set query description amenable self answer w leave large query predicate may amenable computation hardness al data release count synthetic universe database universe order consist sometimes think tuple item distance map abstract range differential privacy pd pd class query specify fix database fix item query item satisfy database indicate satisfy close concrete equivalently query u section release set stage definition handle formal definition non interactive giving reduction give allow give item specifie receive back type query however potential access access release universe query description distribution u sampling database database database database release access release unchanged get sometimes release focus release release differentially private must randomize release differ free release boost datum release whose boost result release distribution single distribution uniform query access simulate low need domain unlabele predicate tn access take accuracy parameter integer output g oracle establish private release threshold next capture notion access definition sampling evaluation black oracle oracle access let restrict evaluation access access example answer remark release evaluation weak appropriate simplicity perfectly evaluation access hope threshold need learn pose stronger define next smooth extension reduction combine result concrete new release result private via threshold universe description threshold tn bn smooth running release next point two modification theorem use label use sampling query precise hold requirement e g query answer task predicate database predicate want allow bound query scale laplace approximation bound query differentially tackle learning predicate ii noisy answer sum ignore concern straightforward predicate give oracle thresholded predicate produce approximate thresholde sum privacy consideration thresholded predicate use oracle sum noisy answer threshold sum close restrict margin say threshold w still high reason every threshold threshold h specifically call roughly speak threshold oracle formally query suitably scale ensure return return large output oracle return condition remove conditional distribution useful oracle noisy answer allow privacy fact subsampling threshold privacy problematic complexity predicate predicate try predicate fact sum predicate close subsampling argument hope may namely rather notice threshold oracle goal well roughly query differential privacy avoid margin oracle n continue threshold denote algorithm exposition detail distribution release access denote oracle case sample margin approximation accuracy threshold contain extension theorem suffice receive oracle access overview section formal proof formalize analyze begin describe oracle purpose ensure differential noise access give threshold enable argue agree set terminate ask otherwise previously create fix two query answer differential mechanism answer give answer database database existence subsampling argument main state element far suppose event datum sample oracle magnitude tail laplacian happen two statement either thus else convert private threshold privacy preserve function call distribution learning threshold require denote ik il terminate pose hypothesis equal parameter description parameter reduction keep fix satisfy assumption make query call make query query ever prove produce formalize overall satisfie distribution chernoff terminate put terminate satisfied terminate put mass terminate choice third terminate assume terminate remainder generality sequence place particular learning reject step union bind induce enough lemma choice randomness randomness think answer variable remainder example iteration note since condition z p answer query precision chernoff fact choose assume argue condition internal randomness guarantee probability example sample correctly query ask learn sufficiently invoke claim query case first note put query desire claim direct consequence conclude probability randomness complete proof assume oracle occur three event claim proceed three event occur claim follow property must differ one classify spaced claim finish conclude proof privacy factor label universe description provide learn threshold g u bn algorithm indeed sense privacy correctness identical theorem release theorem release conjunction base threshold throughout query monotone I result differentially private monotone probability accurate runtime class boolean provide boost accurate establish thresholded value arbitrary section boolean multilinear denote hamming main subsection degree lemma follow claim give quantum give claim slightly univariate chebyshev polynomial desire degree denote corollary every integer desire polynomial integer consider previous subsection imply represent polynomial threshold entire boolean hypercube degree closely difference need use notion place width suppose proof follow construction subsection place throughout variable contain sketch directly width conjunction fix predicate replace size width represent proof identical theorem change width leaf get threshold pac see class sum terminology universe query description monotone conjunction distribution threshold bn approximate threshold conjunction learn q u bn ok label recall together reduction release begin data query reduction base algorithm free section count subsection query query differentially uniform accurate database theorem denote datum universe description threshold u u learn algorithm harmonic explain harmonic terminology smooth extension black sampling access sense constant output powerful allow recall precise difference black oracle access priori point query allow harmonic never query fortunately harmonic work box actually extension section equal harmonic return formulation subsection private release broad query namely circuit computed circuit release let differentially release database observe
many type see law rather number power law plot effectively nonetheless compare simulated right simulated point connect value distinguish display across unlike predict lie curve since process necessarily small small blue represent green probability beta actual examine probability blue represent classic black upper due fact process aspect simulation randomness bernoulli power law fast two law predict q merely create horizontal law beyond rank decay fast poisson formulation easy section model empirically handwritten digits repeat process handwritten sample digit mnist handwritten handwritten digit project component apply beta mcmc original two parameter represent three hyperparameter iteration draw left parameter draw discount autocorrelation parameter discount burn model show see round exactly index eq length round indicator rewrite v usual indexing expression th point express feature quantity step feature indicator step integrate break proportion round remain product factor write stick integral sum trick beta second factor dependence k r feature agree calculation value normalize feasible value fall pre latent integrate conditional likelihood integrate n nx nx final second factor pz k pz numerator integral beta discount three parameter indicator assume scale equal generation occur g round indicator variable calculate monte approximation discretization monte sampler tail fall pre complete discount lie behavior p prior describe sampler feature conditional variance proposition involve feature draw beta collection probability draw stick beta representation dirichlet break directly motivate law process beta stick set arise population involve covariate capture within assumption assume exclusive model parameter former principle model dirichlet typically whereas typically combinatorial encode binary capture bernoulli extended draw draw distinct value treat value random subject inference deal connection obtain stick collection distinct obtain connection place drive development wide algorithms model prior suggest generalization slice possible bernoulli bayesian family measure beta survival model hazard focus beta contain measure view infinite repeatedly entity random view integrate analogy chinese restaurant combinatorial previously process binary collection break particularly important algorithmic exploring generalization yield recursive case explicit recursive size evy yield construction stick break break ready power generalization stick recursive sized biased evy rather recursive representation stick break making break beta conceptual derivation elementary permit unify perspective generalization beta behavior particular law illustrate concrete previously factor loading dimension via bernoulli th binary encoding infinite modeling remainder organize beta conjugate stick behavior review break power law law come section stick break expand beta theoretic poisson process process aspect beta exhibit illustrate power simulate experimental result mcmc inference appendix bernoulli random measure measure poisson let denote product realization countable measure denote discrete measurable set completely random construction completely random measure deterministic completely characterize surface illustrate uniform line segment plot poisson process line segment realization denote continuous function refer nonnegative density illustrate draw beta draw index exchangeable indicate horizontal axis make white indicate infinite coin correspond draw infinite treat potentially atomic component define bernoulli discrete necessarily atomic corresponding atomic non link bernoulli draw random hyperparameter draw discrete draw draw bernoulli equal know construction overall beta draw depend exchangeable class exchangeable binary underlie beta process analogous chinese restaurant dirichlet process stick break broken middle broken remain stick recursively stick v form partition break simple dirichlet update stick extension yield power turn consideration break law beta stick break recursively break interval countable fraction remain break break stick break fraction remain stick stick broken stick break figure sample process heavy exposition wish obtain law break representation law notation draw draw dirichlet associate obtain cluster exactly total cluster behavior law constant limit hand value power behavior law law recall variety world problem specie construct internet document paper publish prefer reflect distributional attribute neither power law though dependence stick representation cf note bernoulli provide kind study power law model consider exactly elsewhere place non surely elsewhere sum case type law number say positive last type type procedure include inverse evy procedure stick process analogou procedure however worth note tuple subscript across additive one generalizing stick break breaking arise process poisson proof rely argument shorter demonstrate ease incorporate analogous thereby motivation stick toward break begin three generalization side notation say discount show mass turn represent process identically distribute poisson countable union poisson poisson poisson measure finite distribution enough end line follow monotone convergence term outer equality I substitute back copy bias take measurable size stick stick biased rearrange du du du stick break conclude assignment law big three parameter beta distribution obtain asymptotic behavior constant case type power law surely appear subsequent derivation obtain convenient proof apply beta eqs eqs represent term atom law part asymptotic derivation extensively feature mean law proposition obtain surely behavior proposition law define process construction start top illustrate poisson line rate bottom illustrate infinite put minimal necessarily sum eventually case atom one generate see poisson positive henceforth define exactly eq definition principal integer jump observation addition find convenient assume poisson approximation fundamentally tie work able mean count case draw equality power property behavior vary lt lt key laplace give asymptotic integrate integral integration laplace part laplace assume integral thereby part transform link result show count relate law know probability rate law translate relate mean process relate almost behavior count concern feature count almost unbounded statement monotone convergence consequence generate poisson process establish inequality consequence finally consider wish law sure power law poisson wish borel q sufficiently lemma expression type I law imply representation beta process eq x
study runtime example decrease runtime sample ignore extreme bottleneck definition empirical denote paradigm learner minimize learn call return hypothesis throughout improper derive runtime formula boolean mapping follow approximately perform former hardness learn feasibility improper formula rewrite v conjunction problem polynomially cast programming simple however theoretic variable conclude important example know theoretic yield sample reflect reality curve know scale amplitude segment thick node l erm erm exhibit distinguish applicable since assumption work result one let string map polynomial possibly permutation exist treat length permutation formal agnostic learnable size exist polynomial learnable improper whose implie get runtime illustrate node leave define treat pair bit one way example simply hard distribution h contrary efficient free learner hypothesis r trivial soon learner algorithm least pick bit get attempt see work pick event predict inefficient value return see error likely frequently example appear hold pr hypothesis small good one pick least example efficient follow early inefficient rather work implicitly compute product even slightly replace non continuous transfer transfer lipschitz label expect rademacher complexity analysis theoretic erm convex due show erm reveal perform search example identify solution runtime exponentially erm idea hypothesis b l contain respect solve erm list dependence like online learning demonstrate runtime price need paper formalize discuss phenomena dependence discuss restrictive perfect exist involve hypothesis agnostic improper learning half construction phenomenon natural seem introduction base available computationally size open believe outline existence example hold additional happen leverage large efficient surely asset machine application eq result draw bernoulli variable whenever subspace span assumption problem price correspond achieve error round receive arm environment pick learner tell feedback correct correct label frobenius restrict multiclass implement feedback problem multiclass classifier perceptron exploration tradeoff running round require small c round inefficient follow I structure support diagonal thresholding take return index prove perfectly complexity diagonal sort runtime sophisticated sdp scheme clear tradeoff note gap polynomial c definition microsoft research grow understand reduce runtime study example result exponentially growth range every day task genomic recommendation meanwhile world algorithm construct algorithm speak datum reduce situation hardness original computationally problem reduce flip statistical class original might overfitte though keep overfitte check example polynomially example however extra learn hypothesis goal model study construction demonstrate similar phenomenon hypothesis agnostic learn still synthetic continue arise availability upper raise open first study tradeoff runtime sample theoretic complexity latter need learn present efficiently learnable learnable polynomially learn efficiently construction extended decision list large gap focus mostly focus domain crucially provide term technique rely assumption permutation similar sense even however return correct agnostic compute improper predictor hypothesis example consist natural employ carefully decision list
background introduce functional paper part quick hilbert refer reader therein reproduce short hilbert positive throughout hilbert eigenfunction eigenvalue expansion term eigenfunction involve parameter operator illustrate discuss material generate rkh generalization book paper operation bound adjoint mapping order adjoint mapping fa play important positive orthonormal random span eigenfunction eigenfunction smoothness eigenfunction pt reproduce embed lie problem involve time might collect eq fix collection noise vector observation projection inner scenario unified introduce analog analytically convenient study dimensional classical truncation way choice representation time model reproduce property mf rescale yield map vector basis f equivalent truncation remainder shorthand throughout analyze subspace span sample let projection distance orthogonal subspace bound schmidt important form compute matrix special operator thus time hence reproduce truncation k semidefinite throughout definite eigenvalue prove small estimator functional theoretic receive function lie within assess norm approximate measure bad semi norm uniformly radius operator particular second equivalence eq q see background place description representation span goal dimensional subspace span version let orthogonal via z hilbert schmidt notation ny straightforward computation might give approximation objective however account eigenfunction calculation precede lead recall compute trace arise column slack also feasible supplementary claim subspace hilbert interpolation suggest define minimal sense map linearly linearly preserve space aspect qualitative accordingly solve reformulate sdp know equivalent solve moreover lagrangian duality multiplier regularize value path algorithm dyadic one possibly evaluate optimal around standard ordinary pca keep amount variation apply order uniformly space four true signal row observe poor roughly minimum observe particular one considerably optimal oracle value estimate regularization smooth reveal reveal accurate version estimate noise smoothness signal embed noise vary zero smooth increase turn high estimate case begin theorem theorem derive corollary rate z state rate regularity assumption measure vector straightforward upper equation accord signal strength deviation go convenience eigen part involve constant low prevent degeneracy eigenfunction space would define whenever hand go infinity grow subject small satisfying inequality c r sampling sample early property rkh I order kernel integral expect operator decay usual kernel achievable addition condition suppose decay c pair depend small hence one rate minimax reasonable choice although guarantee error critical statement upper nr piece trivial satisfied time model turn consequence achievable truncation operator hilbert decay condition basis truncation base rate consistency closely closely approximation capture define equation bind bind case scale condition namely pt supplementary claim theoretic applicable state abstract form obtain concrete illustrate return case condition need appendix material namely guarantee choose moreover bound polynomial decay verify supplementary truncation sample assume condition interesting become long model truncation satisfied moreover imply follow decay comparison trade basis functional reduce whereas reduce fast regardless minimax achievable standard unit ball range low polynomial truncation corollary similarly truncation optimal become bound eq identity approximate inner product eigenfunction universal kullback leibl relative different condition reproduce kernel eigenvalue q truncation low q calculation time rkhs frequency truncation trivially corollary minimax norm term imply regime begin prove corollary begin heart matrix addition r z define matrix rw algebra eq b establish existence early manifold matrix describe convenient version consequence cs decomposition exist orthogonal matrix theorem work instead version choice properly align align decomposition bind hilbert recall central proof probability shorthand optimal p rearrange sr rs cf supplementary provide concentration well put piece together inequality problem study establish inequality hold constant nontrivial task depend derive law subspace properly align inequality suitably linearity trace separately quantitie equation supplementary material show consequently fix follow lemma supplementary material proof delta recall linearity trace obtain since high bound upper rt supplementary material claim theoretic viewpoint hilbert rkhs sampling operator observation plus consider generalized frequency possible subspace mapping trace condition influence rate convergence subspace work detail cc truncation mn sample rate exhibit sample rate qualitatively interact fast interaction frequency optimal characteristic minimax lemma support grant dms pca generally model map functional lie reproduce subspace span lead model attractive rate minimax functional establish statistic great see book time treat term functional various applicable dimension component one mathematical abstraction reality treat functional continuous arguably practice resource force true sort truncation instance understand justify corrupt noise study affect estimation presence share domain
uncertainty direction encode proxy marginal gaussian determination variance currently wide thousand million arise explicitly store full infeasible gaussian random loop sophisticated develop purpose class variance study variational unless propagation sensitive demonstrate variance computational routine context build f factor follow sample suffer systematic sample suffice capture reliably need application easy variational several fully requirement conventional point gaussian sample draw processor show energy hide response capture induce potential form aspect blind deconvolution unknown kernel evidence partition sensing amount lead application point provide uncertainty estimation overcome capturing consider imply evidence variational mostly focus concave induce potential e variational induce super form super potential quite rich include member super replace potential variational bound evidence derivation demand point map capture difficulty lie exploit concavity vector naturally globally convergent log concave variance upper newly w leave concave minimize inner obtain variational completion loop variational parameter subsequent outer loop bound variational mean field adjust minimize kl shown mean bound sec expectation parameter true match various iterative sequential message pass optimize focus convergent loop bottleneck highlight sec repeatedly variance routine z impossible store explicitly far estimation build monotonically reveal rough structure scale run relatively yield variational prove estimate qualitative follow reliably sample translate uncertainty model enforce constraint use estimator illustrate variance variational fig sample last double variational marginal variance linear iteration runtime variance variance scale space dimensionality htbp monte free energy purpose course note sample reliably extra eq case good reference crucial solve million image deconvolution build glm ie matlab toolbox bound propagation extend implementation future glm ie assume spatially degradation motion blind variant know blind blind deconvolution parameter maximum penalize likelihood determine integrate maximize carry maximization inference complete quadratic computing efficiently suffice exactly estimate poor add extra sparse minima various fundamental domain heuristic blind deconvolution thus software may kernel system perturb typically poorly condition convergence plain gradient design would stationary stationarity fourier employ dft fourier domain beyond conjunction dramatically conjugate gradient fig course conjugate typical bounding attain reach conjugate substantial improvement overhead relative problem aware exploit benefit real camera motion shoot camera still order laplacian induce ks scale employ double describe bound vb propagation ep vb ep fig blind ep reliably vb ep db vb db db ep posterior pointwise blind criterion blind bound leave ground initialization kernel estimate image central area account variance variational allow variational cost stochastic sub random key mcmc systematically bayesian acknowledgment work n education science research foundation suggest sec feedback ie toolbox token department california department cognitive engineering university heavy prior probabilistic viewpoint sparse probable sense shift towards capture latent quantify parameter variance turn constitute main bottleneck variational scale leverage recently map contribution estimate variance much standard fully scalable allow extremely memory conventional prove tv modeling thresholding compress linear random reduce scalable million variable viewpoint compress violate filter response exhibit heavy measurement operator impossible
factorization q mapping mean fouri certain fourier accord formally component arrange order say radius bound mind function need ball condition element equivalent large coefficient qualitatively strongly influence covariate influential subset wish soft empirically suitably g basis attain rate rich density bias rule begin q fourier coefficient form thresholded real value indicator observation square incur thresholded eq impractical criterion unbiased lead estimator q stand improve chen thresholding shrinkage disadvantage thresholde impractical thresholding allow whole coefficient rule approach ds observation usefulness observation purpose hence depth coefficient square magnitude exceed square coefficient leaf exceed internal yet visit tree without explicitly I leave develop shall rely follow proof suggests estimate biased property rather former require scale good balance variance attain minimax rate convergence strategy estimate sample specifically set case case always thresholde sum see expect result describe routine desire update factorization n kx u j I tune estimator risk bound positivity issue orthogonal necessarily happen handle expensive suitably sparse carry approximately sum many logarithmic bind threshold small conservative hold attain logarithmic factor moderate regime minimax risk minimax particular outperform histogram orthogonal attain statement problem estimate statement constant final result time fix constant km mse controlling rate decay case severe reject lot lead complexity computational viewpoint preferable rapidly decay threshold complexity decay practice low tend especially achieve balance mse moderately decay threshold become propose operation explanation possible incoherent compressed testing canonical achieve al group strategy boolean collect tool exist need hoeffding inequality q rademacher u index additional needed need contain countable exist converge pointwise separability ensure sequel class element separate fix element positive consist dimensional statement hold set contain kullback divergence relative entropy exist decompose squared error start necessarily define apply concentration bound list first cauchy schwarz q handle conclude bind eq absolute q argument write inequality inequality eq put eqs second threshold use theoretic low test I estimator pack write shannon observation next upper bounding suppose convexity substitute satisfy lemma conjunction suppose moreover assumption satisfied recursive recall call recursive correct call incorrect recursive q express supremum suitable hence accord see recursive call give recursive call contribute exactly bound call compute scope present low truth comparison strategy enhance regime synthetic mixture hypercube product choose truth shown profile bottom estimate probability sample cc scheme run second retain result five run deviation study thresholde constant twice whose value reject coefficient reliably distinguish zero spurious retain sample theory constant show time second recursive platform complexity keep mind call overhead also call scheme detail lead computation increase polynomially logarithmic considerably recursive become cause expect making later decrease finally number retain number coefficient perform alternative exhaustive threshold see far alternative cc exhaustive search mse time exhaustive exhaustive ten standard therefore optimize related challenge set indirect near bottom remain coefficient subtree sensible number level reduce binary branch trade possibility branch optimize recursive many computer limit recursion restriction recursive typically stack deep queue open branch stack number node free sort latter criterion simulation pruning frequency give hamming measure impose ignore branch resource significant gain branch high sequence level control trade impact technique provide appropriate yield computational mse computation multiple aforementioned bernoulli use ten dimensional component bernoulli average ten string eight expand open queue decrease achieve decrease force datum population effect formalized observation attain rate run polynomial complexity behave real another promising direction investigate form neighborhood coefficient fourier coefficient amount something field govern see sparsity serve correlation whether introduce binary analog something like account localization fourier recursive hypercube factorization property factorization property q leave uv prove case fourier coefficient leibler divergence chi q item combinatorial theory correspondence binary hypercube map eq let fm care purpose state follow lemma follow property associate fourier taking determine sign consider suppose latter consequently q prove item supremum key theorem end copy cauchy schwarz prove unit ball argument bind theorem nf binary density vector bernoulli expansion covariate estimate conventional prohibitive certain underlie moderate flexible trade consider density identically density binary hypercube wish observation arise variety profile yes presence marker recent methodology occurrence medical quantitative science survey panel answer correspond co event intelligence could search engine database store website like component
index draw nash generality see uniform raise nash equilibrium nash polynomially family every two theorem base construction pay ensure nash equilibria close nash equilibria subset let cardinality imagine game whereas complement get player always payoff payoff choose element approximate game nash equilibrium row uniform equilibria formally nash equilibrium row player uniform distribution distribution assertion straightforward player since activity component game averaging imply player payoff equilibrium player strategy payoff element player assign mass distribution argue payoff distance measure nash mix induce mixed assigning mass super disjoint counting argument appendix imply player game run recall anonymous section anonymous game also give anonymous exponent exponent specialize anonymous game define anonymous game index tuple player strategy profile player playing carry flow expect success player among approximation anonymous type sketch appendix tuple probability normalize anonymous game player nash player depend game enforce indistinguishable us player regardless choose play ensure equilibrium every player get payoff get enforce equilibrium challenge include player type player mix player player prescribe nash equilibria construction outline anonymous games family nash equilibrium strategy ensemble anonymous collection strategy equilibrium quantification appendix approximation omit section provide anonymous running course way collection tuple represent moment strategy collection strategy collection recall exist player anonymous payoff nash player turn max argument heart sum indicator sum indicator mean anonymous strategy per player variable replace one change payoff experience indicator nash equilibrium strategy search becomes prune search nash provide rather quantification total variation first collection collection condition sum expectation polynomial moment indicator theorem provide setting moment theory indicator expectation variation indicator quite ess sufficient heart result complement indicator expectation binomial derivative value derivative correspond sign e two sum indicator condition first sum derivative polynomial upon nash equilibrium treat easily polynomial exhaustive search flow anonymous equilibrium integer determine reason nash equilibrium must nash equilibrium player guess pure mix probability handle separately theorem indicator take corollary guess represent strategy player mix strategy mix remark actually find step answer follow could player play equilibrium exploit achieve multiple choice player aggregate player regret multiple aggregate multiple reject mixed profile additive come go equilibrium virtue set regret test nash equilibrium proof assignment strategy player player player player solve trivial dynamic assignment constitute nash player anonymous bit require payoff observation nash discretized value find nash test choice accuracy nash another lose full showing sparse computable sum independent variation cover cover sum indicator ik sparse value kn kn kn q k size indeed collection form collection heavy binomial specify indeed total collection binomial choice choice index precise since sparse permutation need etc index remove aforementioned collection collection tv I I collection give cover produce perform operation moment arise collection operation additive total distance argue total indicator choice claim vector l upper bind possible moment collection indicator k finish proof remain obtain cover produce follow claim time invoke r I hence algorithm produce cover possible moment moment collection indicator binomial add cover nash equilibria simple take anonymous algorithm elaborate moment bernoulli barrier familiar anonymous game many remain nash equilibria game program truly practical interact anonymous nash equilibrium graphical game game alternatively graphical case mit mit berkeley berkeley edu lemma corollary remark games equilibrium either zero polynomial time somewhat surprisingly game equilibrium payoff complete kind sample strategy nash deterministic bring question equilibrium game quasi recent anonymous sense appropriate game fix collection mix exponential prove must exponent player type contrast devise game number strategy set collection player work probabilistic two sum indicator random decrease moment two indicator establish existence cover sum cover nash equilibria nash equilibria equilibrium year progress small corner anonymous strategy different play identity strategy player divide type choose report progress new nontrivial computing equilibrium game entry row interesting randomized nash nonzero quite surprising special easy suggest logarithmic roughly first outline anonymous fix pair mixed strategy look game whether strategy constitute nash definite develop expectation game recently discover anonymous game player game utility player play identity sophisticated case player partition type type playing appropriate anonymous game player identity nash nash equilibrium briefly strategy th mixed nash player every nu nu games player type play type class class row chen find nash hard equilibrium game strategy player contrast class nash show equilibrium mix hence strategy long point simple always nash difference give player payoff payoff nash optimal optimize player payoff class guarantee
statistic respectively typically mathematically commonly represent vertex wherein distinct undirected prominent fashion wide vertex web page direct represent link point page protein network biology vertex undirected edge affinity network vertex represent social great literature focus variety range largely mathematical interest cite short review paper model factor relational hence complex vertex quite geometry modeling phenomena problem quite hundred hundred vertex edge vertex particularly network pair alternatively think setting institute north usa fact three response I network answer despite model knowledge formally pose analogous question ask notably series latter maximum estimate dependency obtain combination might asymptotic characteristic practice interpret scale variance insight arguably real network scale vertex I regime asymptotic corresponding response popular notably derivation utilize tool accordingly serve straightforward illustrative effective trivial answer subtle basic model assumption organize definition main edge give illustrate implication study asymptotic sharing examine extent world find support variant background chapter history practitioner network specify follow statistic despite appeal exponential handle computational concern random graph wherein assume identically arguably small still variant introduce situation practical quickly insight effective network use statistic model vertex edge tendency wherein model network hold draw configuration realization effect realization tie vertex preserve preserve importantly bernoulli bernoulli question parameterization introduce shift adjust realization produce vary configuration typical realization realization like baseline asymptotic configuration would produce shift mean recognize large distinction fundamental implication effective main parameter vertex degree infinity stay tend degree perspective traditional offset asymptotically equivalent enyi alternatively social odd value network able maintain shrink maintain beyond observation effective fisher respectively calculation implication immediately apparent likelihood estimate maximum model finite consistent largely asymptotic constitute consistency estimator argue theorem consistency estimate proof normality usual technique due exponential normality statistic case array require normality vertex triangular array rather array need derivation provide appendix size assume perspective rescaling induce notably subtle fisher analogous similarly asymptotic property analogous previously satisfactory call poisson hand behave tend limit vanish fisher network infer reliable suggest actor contact small suggest exclude varied consistently conservative level tendency interval proportion nominal level prominent rather strong confidence select level simulate study percentage point percent section establish question tie establish sparse detailed beyond initial explore question face point require similar collect ask list precede neighborhood find significant share find construct overlap consist neighborhood baseline correspond realistic expect slope lastly realistic slope around color indicate coefficient clearly magnitude close although possible difference nevertheless enforce tie explanation magnitude substantially argument network close stable tie likely share little two comprise tie neighborhood close assumption violate respective tie lose small small per figure decrease slope tie less adjust decrease increase reduce slope unlikely unobserved contiguous color tie discussion conventional effective sample associate depend case affect meaningful simple already sophisticated say suggest effect arguably type next index likely require complex treatment yet effective sample insight literature unlikely directly method interval network begin gain context begin exponential well present consistency relate contribute lack understand property commonly network particularly package package compute general formulation report estimate unfortunately practitioner seem aware construct theory confidence test formal perspective appear necessary foundation justify practical interval discussion material derivation key result expression main paper begin establish preliminary expression section section offer proof recall model family fisher offset expression order magnitude twice appropriately parameterize parameter model find capture simplify behave consistency normality interesting follow conventional various might technique requirement however behave behave large likelihood amenable demonstrate study behavior pointwise condition lipschitz true easily continuous function compact mean allow standard writing numerator recall proportional average link variable worth note allow towards infinity increase define employ asymptotic suitably normalize standardized variable easy fix bound theorem sum normal note behave numerator mean calculation calculus tend v tend zero establish consistency argument partial show consistency reasoning proof gradient set compact version
purely assumption greatly network influence classic topic economic agent agent major positive seek decision anti game make positive several preference benefit effort depend preference critical sequential vote sequential voting candidate project probability success payoff project succeed receive project benefit voting agent decision subsequent agent make aspect outcome decision combination hand become anti agent seek differ reward nevertheless agent belief uncertain social reward game rational player maximize reward play many different research cognitive storage service selection cloud compute secondary user access share user available storage service reliability availability use service service cloud storage platform receive customer negative product agent make possibility utility access service decision player learn focus design experience device design system learn non approach applicable rational choose action benefit follow design chinese restaurant unbounded chinese restaurant customer restaurant customer restaurant share customer table generally customer customer join process systematic unknown chinese restaurant formulate social restaurant customer sequentially customer request table prefer big big table since table request customer experience key chinese game customer table enhance involve experience moreover table customer receive signal game theoretic chinese restaurant game wireless ii rest provide detailed description chinese restaurant game simultaneous game customer behave perfect sequential game general chinese behavior customer uncertain method response customer game limited knowledge state decision agent agent influence negative characteristic game game draw financial network cognitive simultaneously work effort attempt investigate binary study game decision private unknown game propose attack enough status payoff payoff study stage agent regime change dynamic binary state heterogeneity simplify position game game simplify moreover mostly propose chinese restaurant game game framework study network chinese restaurant table request customer request table table restaurant table restaurant I restaurant exist restaurant decision let state selection game action may mean customer choose table customer utility decrease regarded degradation utility customer number group restaurant restaurant exact may exact may review restaurant relate restaurant information therefore know l l pr subsection describe customer sequentially customer decision later customer customer exclude prior customer belief represent probability receive probability belief bayesian naive updating rule rational bayesian implicitly customer predefine capability information rational customer restriction fix customer rational follow update belief updating learned generally affect like discuss simultaneous customer decision e restaurant time scenario learn request table investigate game understand customer behave simple customer table state indicate table state signal reveal indicate pool customer immediately customer know opponent perfect size customer decision opponent rational customer table opponent depend severe network customer learn table depend severe customer signal customer since customer rational action customer influence describe player chinese restaurant game customer customer recalling stand customer customer rational customer action describe response represent action customer well consider utility player action customer equilibrium popular concept predict game rational customer informally nash equilibrium customer profile none formal equilibrium give nash equilibrium nash necessary simultaneous chinese give customer table current nash equilibrium chinese restaurant perfect equilibrium action player grouping generality customer utility j ix x x nash equilibrium condition nash table nash nash equilibrium customer utility would another table another customer eventually table size customer restaurant table size nash customer table customer equilibrium exchange action customer nash equilibrium nash necessary grouping may state nash equilibrium game perfect inequality hold grouping signal current customer give current grouping table player j sn impossible reach nash equilibrium nash j eq due last customer utility table response customer propose nash satisfy group result grouping grouping customer choose observe customer customer customer candidate customer choose contradict customer customer follow lemma perfect equilibrium customer nash equilibrium grouping propose customer customer equilibrium group customer customer choose another utility become never bad customer choose instead reach contradict n grouping never customer final customer equilibrium thus choose never equilibrium grouping grouping customer grouping achieve moreover show grouping nash equilibrium perfect nash perfect equilibrium sequential game bring advantage decision make decision early large utility equilibrium customer choose table reach equilibrium choose subsequent customer customer show sequential perfect customer advantage get table thus signal uncertainty arrive right table arrive later eventually table collect right decision customer choose later therefore trade choice accurate playing later trade discuss signal model unknown customer table express denote probability customer customer private f customer condition customer uncorrelated restaurant customer make sequentially number mind subsequent decision customer rational learn make decision collect well estimation reveal update updating base update customer reveal expect group customer choose customer customer reveal signal customer see utility key final grouping system belief denote information system state reveal customer belief utility customer close generally impossible impractical recursive approach customer accord ss give ji predict eq customer choose customer include recursive recursive x eq eq x calculate ir response derive customer customer easily verify recursive simulate chinese two customer customer receive begin conditioning system receive give close likely reflect state customer decision sequentially customer game last customer make decision customer number customer table end optimality customer customer customer choose response customer confirm customer increase customer customer mistake choose table customer benefit mistake customer quality previous size ratio table utility choose table odd customer utility simulate customer fig make significantly affect show signal size customer region utility customer large small case choose advantage since signal table even third form belief true less customer decision customer customer choose negative customer likely belief collect customer advantage get chinese nevertheless restaurant game non show customer increase scheme utility quality size customer low customer utility customer large provide equilibrium customer table customer customer correct third customer choose table customer determine size generally state table effect high argument argument customer fig customer utility peak consistent table ratio equilibrium customer decrease customer large utility table size equilibrium customer customer customer size large expect utility ratio next resource pool user act sequentially maximize mean utility scenario scheme customer response resource pool customer average high scheme quality customer quality view customer customer receive signal quality follow table customer choose customer empty table customer dominate good customer opposite receive customer customer equilibrium accord customer utility large equilibrium customer expect share another customer c show advantage customer play less customer scheme large utility customer utility phenomenon customer try identify customer nevertheless observe late customer likely customer illustrative response quality resource pool still play significant
elegant random improve exponent dimensionality use qualitatively insight active thorough empirical batch mode demonstrate impact importance empirically scalability mnist query fast good reflect underlie problem bias process weight differently proceed round probability entire pool sample round round label hypothesis gets weight weight unbiased denote denote guarantee pool round come point uncertain probability mass h hx iy calculate depend current learner possible loss function show conditional use convex loss labeling p tx n j tx tx say h tx hence retain property allow interesting look behaviour square hypothesis return invertible elementary establish integral solve proceed odd substitute ss assume matrix exponentially expert commonly learn interpret prune soft exponential get weight ask answer analyze squared function yield analyze extend logistic exponential assumption excess risk learner invertible finite exist shall boundedness dimensional cube suffice though kernel infinite mapping find learner active excess proceed assume invertible upper bound excess risk motivation square term reduce maximum bind positive semidefinite nothing eigenvalue reduce give upper eigenvalue rank lemma spirit analyze eigenvalue sum bind tool quadratic moment quantity calculate bound establish use bind suppose represent active learner active learner round next quantity excess shall inequality get inequality excess risk assume invertible matrix cauchy use probability bind probability get since get upper probability last provide similarly bind proposition prove follow let theorem tp tm nr get since equation place equation inequality use use inequality apply theorem q substitute lemma tm first square last lemma invertible sample get z enough provide eigenvalue positive definite rearrange q equation fact round square equation inequality equation next column aa subgaussian arbitrary left exponential definition form deviation martingale sum second lemma boundedness upper condition eq datum show lead last exploit hoeffding hence prove add get desire pool employ strategy none unbiased simple strategy point uncertain text classification calculate mostly probabilistic uncertainty close agree label maintain member query boost bag discriminative framework cause reduce one thorough al provable label agnostic unable trivial hinge loss least loss entire member unlike choice implement passive pl variant call learn loss logistic motivated logistic code possible similar uniformly currently pool however implement potential unbiased check helps batch al superior empirical pool primary pool like proceed select pool pool point suggest monotonic optimize point potential dot calculation new query far round subsample value algorithm numerical implement regularize show current importance pool fold perform reduce dimension uci namely axis implementation c error average gap pl see case performance pl always success match reflect performance pl well never pl time huge gap pl query performance random perform via especially round step algorithm problem also solve problem fix budget mnist tend scalability experiment set find budget speedup run dual core active algorithm show scale constraint theoretically prove true view establish tight excess excess high interesting calculate round calculate bin ball randomize pool bin ball equivalent round unlike ball bin round expect bin form subgaussian almost surely q prove et al bernstein symmetric exist q surely dimension version al dimension useful work dimension space
institute mit currently electrical laboratory system mit research interest include engineer electrical engineering receive mit thesis transaction journal ph electrical receive california receive award work software apply technology ph fellowship application communication electrical receive ph degree mit degree electrical stanford move dr work hold wireless california berkeley laboratory university work laboratory research coding signal process security application computer architecture device dr award mit tucker award intel fellowship stanford award department fellowship edu systems institute technology usa email mit edu establish information estimate take sharp reliability decoder appeal low probability union scenario amenable decode average number require random literature fact motivate minimum distance code succeed sparse code ensemble one reliability code attempt area community year minimization code endow hamming herein provide investigate code completion noisy require variety collaborative filter netflix obtain singular generalization completion problem entry arbitrary inner product nuclear matrix also measurement exceed product yield sense observation reconstruction field exactly albeit rank code endow metric concerned check minimization problem besides measurement field code want th string furthermore know bar show linear work work consider herein graph property code linear channel induce assume array rank receive matrix succeed true tend infinity code characterize dense deterministic code code motivated array arise technique unfortunately highly success subspace decode technique pattern work able distance property random code correct derivation similar spirit start rather combine exponent result seminal uniform give multiplicative derive exponent decoder summarize use converse technique require obtain product sense contain procedure min match theoretic lower require hence sufficient condition concern computational recover rank compare exponent min decoder de bind estimate reliability error example upper show fact code comparison achieve theoretic thus much reconstruction main possibility sparse check matrix draw code code derive analog binary block code analysis obtain geometric decoding perform matrix guide strong along et al able seminal work unknown many application information al matrix mirror sufficient additionally analyze inspire derive theoretic derive condition completion argument measurement logarithmic measurement converse match extended analysis conference case measurement compare decoder another author channel symmetric study generator sparse capacity achieve code sparsity make decode amenable technique code decoding pass perfect ellipsoid min code rank analog binary code equip hamming metric minimization finite field various reference table elaborate comparison notational choice describe measurement use reconstruct unknown rank uniformly select random sufficient reliable recovery reliability de extend noisy reliable recovery understand theoretic perspective search section defer rank dense section notational describe notation font font deterministic quantity face respectively deterministic scalar respectively font set event denote element identify integer matrix entry let hamming use mn stack one use throughout definition reader paper definition rank scaling section square ease exposition rank limit assumption converse linear trace sense mass pmf measurement estimate depend e multiplication interested represent precise measurement setup fill capture choose sense general although minimization analyze et whereas unknown general measurement ill pose measurement particular pmf deterministic recovery recover recovery would familiar distinction compress suffice w sense operator matrix rank less rank matrix example random follow tend allow state low one strong recent sequel matrix number matrix q fact present necessary reliably estimate converse necessary object unknown deterministic k nr event q emphasize random follow converse fix rank less equal assume independent rank less recovery reliable I tend degree noiseless section involve elementary inequality k bind convergence statistically matrix validity assumption restrictive practice sense linear reliability recovery problem select decoder denote singleton optimizer analyze singleton equal I error optimization intractable unless sense ar decoder exponent constraints section pmf measurement exploit idea weak fix measurement recover theoretic low rank sharp converse proposition rank decoder converse side min simple albeit rank q equal fact precisely independence uniformity uniform continuity integer depend satisfie loss assume simplification precision argument decoder asymptotically require match proposition rate decay decoder code sense matrix random connection problem theory detail reliability function exponent exist unlike reliability normalization min decoder reliability function upper de inequality convenience event need compute argue latter independence neither logarithm simplify er proof reliability complete appeal min follow may unity lower express bind note yield reliability event essential uniformity independence obtain analogy show capacity inequality exploitation independence error error exponent ensemble linear code large exponent pmf conjecture introduction theoretic perspective compare code literature distance third construction reliability normalize simply quantity instead code random decoding give code singleton eq rank error uniformity randomness code code scheme reliability whereas give rank reliability regime furthermore deal encode matrix amenable low decode exponent general require search propose reduce precede generalize assume deterministic situation minimum generalization min decoder let noisy measurement model choose monotonically measurement increase increase intuition decoder part uncertainty location remark proposition reduce different model technique measurement match converse decoder matrix whereas derivation converse reconstruct unclear also amenable noisy also converse vector assume represent every independently ask also proposition converse converse inspire theorem decoder notion recover proposition code non endow matrix statistically I pmf code theory element belong field adopt approach direct rank codebook satisfy na cn connect directly literature basis j nj discover metric code matrix constraint metric code term correspondence code guarantee self check usual metric element extension field establish correspondence code matrix quantity take deterministic code remark analyze analog index rr turn amenable random generator instead moment mean satisfie namely exponentially compare converse chebyshev immediate eq generator derivation define minimum rank minimum distance rank distance cf pmf let k surely proposition consequence bl code code prescribe decay light code denote minimum minimum go serious implie may hence small matrix code probability say sequence equal denote concentration sequence rank n rank code apply hence n proposition exponentially code contain interval analog substitute definition typical remark code metric derivation require assumption check generator linearly knowledge previous study distance rank property derive inequality decoder succeed typical exceed rank correction derive conclude compare proposition pack sphere pack code n ball ball rh total deduce sphere perfect analog proposition check matrix pmf analyze matrix ham hamming code justify way hence omit allow tight analogy metric code code distance code minimum condition relative linear code remarkably property dense statistically independent fact bound distance ensemble coincide explain decoder match theoretic rank linear recover ask measurement reliably strong recovery measurement uniform min rank probability weak min decoder recover thus say many needed limit increase factor recovery context compress sense finite field derivation precede subsection relative small recovery decode word distance exceed see illustration minimum translate distance code minimum rearrange limit proposition number prescribe recovery decoder recover equal one use follow line disjoint devise procedure complexity decode exhaustive inspire technique mention belong matrix minimum clarity exposition vector rank integer exponentially elementary operation multiplication affect reduce check exactly equal generality read justify nn system partition part equation ax solve decode know write refer coefficient expand solve I employ basis basis linear coefficient terminate increment go second number less since successful equation attempt increment complexity I distinct linear elimination idea dramatically na I solve loss number basis reduce indeed decompose r contain precede enumeration na I different equivalence find equivalence class lagrange non singular inequality consequence equivalence class matrix note corresponding belong class hence previously consideration complexity note symmetry connection introduction table conclude contribution suggest future solve min rank intractable characterize code admit receive polynomial singleton decode however particular line guarantee suggest decode work matrix draw sparse factor strategy hamming loss adopt subspace message pass strategy cause assumption additive value channel field also achieve however uniformly fig fundamental limit code metric construction analogy channel code low code probability recovery conclude analysis complete remain table minimization field combinatorial demonstrate subgraph number clique ellipsoid semidefinite fact minimization characterize information limit recover field give noiseless noisy measurement even sense decode dense code theoretic succeed herein interest analyze require motivate sense matrix fix property sense isometry joint use programming necessary sufficient guarantee et sense random linear lp decode lp compress sense reach sense new analogously whether derive thereby provide reverse grow literature design tractable interesting decode code interest analyze possible tradeoff reliable low analog finite solve optimization acknowledgement suggestion improve acknowledge discussion thank liu fig joint precisely definition inner subset equality condition
practically useful n respectively main procedure complicate give long family replace respectively multinomial approximated degree jj around accord light approximate eq update analogy update accord type q compute size fortunately tune definite answer tune concern number nonlinear shape initial fitting case filter informative likelihood simulation logistic nan matrix regression nan imply covariance make support simplicity well theory stochastic observation increase importance sampling budget particle plain smc specifically adaptation update adapt proposal fast enough less twice bad initial iteration run typically iteration concern exact optimum slowly let framework stochastic em constant step mention smc successfully filter k unobserve space transition sometimes initial conditionally particular density respect development straightforwardly optimal record recursion filtering perfectly g q omit brevity time associate approximate follow design adaptively describe gaussian toy optimal available vanishe expert value hide state filter separate initial algorithm observation belong family expert expect adapt accordingly importance particle produce zero spread confirm look proportion particle sort decrease proportion mass number adaptation plot row propose visually normalise adapt shift row show adaptation support thus span importance balancing drop near nan proportion unchanged look evolution fit result couple serve lack close htbp focus away single adaptive particle filter bootstrap simulate record adaptive filter proposal entail filter symmetric rotation shannon two filter shannon improvement surprising light consistent particle infinity relative effective sample chi exact briefly kernel stop partially sequence mutually gaussian state observation leave censor nan display row kernel mass third un normalise kernel reference depend relatively line particle adjustment multipli weight adapt lead adaptation un normalise middle bottom leave normalised match save sample un normalise widely match non finally drop family large illustrate gaussian student achieve allow kernel kernel half purpose thank concern case highly relevant adjustment logical smc design open rely algorithm proposal instrumental fit version apply relatively approximation every evolve among nonlinearity adaptation imply limited markovian possibility adjustment multiplier thank comment value jj jacobian mapping consequently thus conversely complete figure filter approximate integrated logistic enough strongly skewed mixture leibl divergence target instrumental single optimisation illustrate successfully nonlinear leibler shannon entropy decade smc simulation rare liu flexibility method efficiently step composite particle markov dynamic datum assimilation material normalise dimension equation smc density recursively weight f measurable function state main former particle specifically mutation particle transition particle update correspond measure ratio eliminate importance particle accord factor introduce adjustment multiplier weight particle likely contribute state target locate choice adjustment affect let particle dynamic closely choice adjustment multipli yield perfectly weight call adjustment express close one refer approximation deal proposal approximate approximated nonlinear always e g multimodal cause severe center student go log unimodal nevertheless approach optimisation particle thus recent state art meet success implicitly sequential sampling minimize choice definition maximal completely single carry minimal coefficient variation smc instrumental shannon instrumental specifically use cloud update resample view standard proportional tend tend asymptotic addition shannon variation give sound theoretical shannon measuring suggest shannon entropy purpose adaptation matter adaptive design smc adaptation adjustment kernel henceforth implement auxiliary enough hand way simple integrated let weight choose proposal allow partitioning region kernel whose student proposal closely appear learn community large em procedure smc decrease minimum gain already minimal computational overhead organize notation throughout adaptation mixture treat optimisation paper result several unless th trace c transpose quantity define equation r smc sample notational argument wish move particle adjust obtain weighted yield introduce normalise q partition ideally update drawing particle independently firstly close secondly form normalization cope aim take importance consist drawing position importance resp adjustment introduction importance absolutely proposal multipli transition resample schedule give methodology simple framework describe foundation smc development recall space method replace auxiliary particle induce term discrepancy induce index term quantity family instrumental distribution member mixture expert take approximate already closely resemble weight partition smooth assign region eventually dominate region weight sometimes resort mixture let without partition transition cost broad class flexibility ease state space resp statistic method proposal illustrate multidimensional multidimensional mind fulfil discuss optimisation algorithm paper recursive parameter key smc em gain simulation describe scalar sum unity treat auxiliary specifically convenient state integrate satisfied student eq sample addition statistic mapping note additive subproblem assume problem fundamental case integrate regression hierarchical plain possibly deep expert overhead possibly overhead reduce flexibility somewhat expert ji definite
vector suitably collection well list order gene diagonal via monotone transformation diagonal list list obtain position value well suited list desirable weight depend exchangeability carry exchangeability gene give list sided exchangeability score e diagonal list gene actually contain consequently affect diagonal possibility contain present exchangeable list position exchangeability gene position change general support similarity kind expert biological e g concern list positive gene biological global weight influence may probability list relevant list microarray gene symbol probe two datum gene patient good cancer outcome contain gene patient subject main gene list list use gene experiment far rank usefulness exchangeability apply set bootstrappe account modify accord snr compare patient gene positively outcome place deviation outcome gene extend list list way list vector describe accord contribution list position snr eq comment subsampling time keep snr vector side mean exchangeability gene define list magnitude matrix exchangeable position list two extreme end final gene list snr median subsample top create aggregated list product subsample ranking product list extend list describe step positive data exchangeability gene list set five step among result set depict variation underlying list among stability ranking overlap five appendix top figure plot list obtain clear exchangeability list list variation exchangeability list indicate correlation bottom row figure list figure stability note gene share feature discrimination patient poor outcome correlation expression gene estimate exchangeability next distance rank adjusted contain exchangeability exchangeability score position construct bootstrappe compute extended list distance extended list variation list vector distance compute extension much extend vector estimate incorporate comparison diabetes extend list imply dissimilarity set dissimilarity also diabetes appendix f histogram stability ranking desirable interest ranking assign gene position rank extremely therefore ranking obtain examine rank gene list discriminate two patient fold validation performance classifier compute five describe gene level center standardized value train standardized level gene classifier classification receiver operating split classification bottom gene gene ranking rank always ability top gene list considerably observe expense decrease biological extend median aggregated univariate multivariate genetic ranking criterion gain biological knowledge ranking highly unstable list much framework variable list order pair visualization g multidimensional collection list stable list list several list exchangeability concept quantify redundancy gene incorporate list way gene list choice dissimilarity new used gene response htb panel right label randomly aggregated ranking coincide indicate ranking aggregation original ranking variation list interestingly part list stable thorough position gene subsample modify set gene ranking rank collect position ranking place position provide used exchangeability distance exchangeability j bs j overlap exchangeable obtain eq consider sensitive sample exchangeability two estimate integrate part negative difference integral note differentially express sided exchangeability distance position length sided exchangeability v exchangeability score datum therefore repeatedly uniformly form exchangeability distribute take exchangeability score main article motivation treat positively poor snr somewhat adapt exchangeability fast high exchangeable gene top list decrease trade effect choose rank construct case similar general present measurement information example reference list contain resemble document field retrieval weighting could position list gene often position influence vector position part give ranking part expression gene exchangeability plot article ranking give overlap gene set aspect since row overlap gene top list overlap indicate exchangeability relevant ranking absence two gene list list top rank htb study stability distance figure ranking independently list extend ranking find important list exchangeability dot list gene position list extend position contribution cr measure exchangeability coefficient relationship expect exchangeability highlight relationship simulate rank use last exchangeable contrast also exchangeability relate response subsample keep group pair exchangeability correlation average realization exchangeability gene group consist group moreover highly exchangeable last detect mm average simulation simulate synthetic matrix sample sample relate group mutually situation compare two sample exchangeability score vector time group part average exchangeability take variable exchangeability score mm mm represent exchangeability positive exchangeability variable pair value provide brief propose list compare list order list compare order review dissimilarity measure list essential part short checked simple commonly compute overlap gene differentially gene check significance simplicity method make software package gene spirit base diagram score recently relationship gene experiment check gene bottom modify kolmogorov statistic combine statistic average gene ranking compare list bottom list adjust list high stability middle permutation propose permutation metric somewhat possibility module rank within module matter method compare list rank account formulate list choose suitably note association universal explicitly compare list similarity score cosine similarity coefficient geometric describe quantify overlap gene gene list positively correlate list exchangeability account set reverse role draw let give list compare experiment list gene gene correlation metric exponent control choose position list gene wish ranking finally list matrix create list q list maximum deviation repeatedly label calculation estimate two rank q vector also feature module gene argue module penalize difference within module practice put module order list value list rank q compare ranking similarly preliminary overlap list gene contribute overlap list accordingly replace desirable one pt set frequently list interest biological often complicated instability sampling variation may partly redundancy imply exchangeable exchangeability functional account variation exchangeability representation list support list ranking incorporate data microarray cancer patient robust sampling biological microarray throughput pattern copy output throughput consist quantitative trait order response gene association exceed study gene subset interpret understand process hypothesis inherent interpretability list change redundancy gene thereby depend list substantial gene overlap apparent instability small ranking extract experimental exchangeability redundancy general incorporate exchangeability list gene microarray list quantify list gene gene mean conventional contrast method tailor specific
present characterize correctly identify discuss associate group approach breast recovery block machine signal process attractive domain interpretation data moreover inference dimension amount traditionally approach involve penalty prove theoretically practically penalization estimator model correlate convergence refer covariate jointly lasso set define restriction model group author encode recovery subsequently notion natural generalization recover display first group overlap support formalize propose construction support stable diverse pattern tree greedy penalty formulation group norm group support encoding idea support combined regression call notion learn problem regularize study notion strong recovery recovery typically always exactly group sufficient consistent group weight group disjoint question potentially application problem cancer identify molecular signature gene pose group biological connect reliably empirically explore application simulate contribution extend preliminary publish group predefine support asymptotic regularize estimation length play overlap address error pathway gene structure latent group work several formulation correspond section notion section toy illustrate consistency discuss present variant covariate finally experiment artificial gain well real breast gene article c g usually say article usually pm pf p convention whose ball convex equivalently lie sparse usually adjust minimize loss level group group neither decomposition remove group latent partition covariate belong group covariate group select precise effect differentiable covariate leave full nonzero lasso select overlap however entirely illustrate overlap third neither one formally extensively study risk general assumption surely q support complement equivalently induce group covariate introduce feasible assume least g illustrate solution component satisfy precisely group immediately therefore sparse solution likely interestingly reformulate cost penalize reference formulation group overlap group equal standard overlap figure unit shape circular consider coincide appear propose group nonzero covariate nonzero reduce penalty enforce impose covariate belong penalty intuitive classical empirical risk remain article investigate detail group theoretically empirically idea component separately appear multi decompose regularize norm share latent group coefficient associate robust decompose trace norm sparse regularize decomposition relate idea interesting length induce length set penalty latent group norm note support consider consider case function relaxation submodular naturally note theoretical analysis study paper propose view complementary useful prove section valid formula subdifferential penalty context optimization satisfy proper convex classical base penalty norm homogeneity decomposition g consider form dual norm equality due convolution convex p g plug effort variational rewrite solution optimization optimize lead I g formulation positive case respective express reach deduce problem g solution incorporate rewrite I h I see admit convex hull suggest visually ball hull horizontal disk define g g eq show hull terminology union convex subdifferential sum conclude g deduce g regularizer penalize lasso covariate belong group implication regularizer stacking restriction note occur group stack successive loss operation consider particular consider restriction eliminate reformulate group lasso overlap expand overlap trick hand extend consider endowed definite kernel section implementation implementation section show section consequence lasso restriction overlap present mkl disjoint concept present consider gram function p kernel multiple find minimize kernel reproduce kernel reproducing show precisely form let sense objective solution equivalence overlap turning introduce problem show mechanism original introduce one extension overlap combine function input mkl use orthogonal typically correspond paper mkl formulation binary mkl reformulate view structured mkl interaction obviously derivation priori lose rbf relationship covariate expect non within non algorithmic structure applies present explicitly computer memory access require large alternatively efficient algorithm proximal proximal operator via dual associated norm regularizer natural ask answer latter suggest express formalize informally notion characterize induced decomposition formally strong decomposition g term dual variable support follow immediately support definition support variational constraint group weakly constraint notion resp support resp group determine j g another variational formulation j definition consider illustrate situation strong entire abuse notation write explicit decomposition correspondence relatively include note support group note fall back overlap decomposition unique unit group necessarily cover support group convexity length illustrate w w dual strong represent line separate triangular colored support weak strong group coincide color group adjacent result never motivate uniqueness obtain vector variable true particular sparsity induce regularizer coincide interested result group lasso disjoint support complicate several exactly express might notion concept situation hard consequence notion characterize study generate finite hard imply consistency however require study notion continuity end particular correspondence correspondence element space notion continuity space correspondence say open resp correspondence singleton value correspondence notion main hypothesis denote g clarity eq h tend subproblem standard argument restrict tend rest construct condition probability whose computation gradient enough variance get correspondence contain show contradiction condition b hold condition consistency outside support converge high imply support show nonetheless support solution union ensure situation necessary sufficient usual disjoint favorable j mutual incoherence overlap motivation g difficult outside set result generalize convergence bound consider version sense show dimensional neighborhood optimum focus group recovery relax give ball bounding image paper complementary recovery compress sense understanding associate motivation different solution test context definite kernel context context overlap significantly arguably consider group indeed notion support norm accord guide ask support show useful attempt characterize prefer small scenario discuss relevant false informally fact contain small enter cover large enter change dual group unless g g natural weight exactly general consequence lemma choose latter group group cb previous c w b prove behind geometric imply intersection redundant associate unit norm redundant exist unit ball g gd h impose redundant pi g g like sufficient redundancy might restricted family soon group group without size become special necessary redundant redundant find g quasi weight dominate dominate include suggest without large group active correspond possibility get ball impossible select support would union detail bottom set possible g intersect characterize discuss require single group group result weight dominate entirely assume dominate constant satisfy eq duality inequality dual corresponding follow group could trivially gap small select give critical dominate equivalently possible trade surprising interpretation penalization expense support support put simply support support variance reduce note consequence pattern analysis propose rip take generally subgaussian thresholding mapping support sufficiently assume absolutely retrieve support provide redundant never g g g select show pose consistency write support never weight select positive negative outside select condition see possible ensure group hand usual easy probability arbitrarily uniformly accord large small possible fail guarantee theorem summary control select uniformly satisfied furthermore c k want incorrectly select incorrectly base similarly choose noticed analysis ignore overlap incorrectly select address sufficiently large nonetheless allow positive ask contain element one could negative previously element pattern kkt none select k kp cc c chernoff group interpret group number put thing differently negative correspond certainly process active group another discount group select contain enough depend long tail two aim specific group solely motivated need criterion guide fine analysis scale dedicated collection require definite recommendation note view weight adopt relate vertex connect wish connected graph clique subgraph alternatively experiment subgraphs length formal choose control connect typically connect overlap group priori subsequently assess synthetic match support datum cover group union group offset set express group latent support b ht mm variable axis represent gray penalty band report frequently support regularization pattern small path hand time third root method path replicate test average fourth replicate help recover decrease connect simple successive index graph correspond protocol experiment group sub report choice cc axis level gray axis blue band support improve result consecutive group exact boundary influence penalize extreme group extreme weight root size small covering effect weight scheme form create nest covariate level setting compare weight scheme replication extreme regime way select small possible entire path regime ideally would theoretical ol obtain ol hold lead simply return case across correspond corresponding size recovery recovery path pattern covariate quantity reach weight right bottom blue band belong last table illustrate effect weighting result correspond recover term selection fourth uniform encourage lastly suggest performance mse point create selection spurious regime figure illustrate behavior expect large extreme active intermediate group active allow except yield adequate choice lead large path hard regime fourth behavior lead pattern reason fine pattern precise term mse reach optimum grid fraction covariate dimension show suggest helpful use large reasoning size mse ds ds min ds motivation possibility microarray use prior gene modify various mechanism biological gene involved amount interaction empirical biological database involve small set hope gene likely involve estimator practical implication expression small gene select number noisy addition select functional lead increase interpretability signature goal predefine gene set various database canonical pathway gene restrict gene indeed keep pathway poor break section combination group biological breast dataset cancer gene least pathway unbalanced use loss weighting example validation specificity regression pathway keep training practice microarray result gene keep select cross validation experiment noisy notice change lot split often sure cause choice choice accuracy observe improvement use weighted lead consistent improvement outperform
dataset ard hull term gp ard contrast ignore ignore additive irrelevant order example additive relevant r r ard add learn ard irrelevant variance variation subset repeat kernel however necessarily axis align combine rotation axis align process sum gaussian covariance process converse hold positive along exp put mass complicated hypothesis unable difference marginal try taylor remark relative degree matter huge subset cardinality specify sum kernel future likelihood cc department engineering university gaussian process low gps generalize generalize additive gp hyperparameter see expressive efficient increase regression example logistic generalize easy interpret extension smoothing spline add fit increase end allow simultaneously se gp much flexibility input introduce process allow interaction interaction kernel parameterization kernel constitute powerful interaction significantly efficacy major advantage interpretability implement recently call learning explore interaction suffer fact validation use necessary train dataset outperform se solve classification kind capture determine difficulty specify gaussian represent medium sized impact efficacy ccccc kernel st order kernel draw draw gp st order nd figure compare gp model high order gp order gp range depend house approximated price part small house capture unseen precise dimension define additive dimension assign interaction covariance base kernel full necessary specify additive hyperparameter kernel variance order control interaction heart nh vary widely come st come se gp specify degree model decomposable discover suitably flexible advantage allow order additive na sum quickly become intractable term th let recursive polynomial formulae computing trick additive base merely remove polynomial trick evaluating se evaluate gram additive interaction allow invert gram matrix typically order interaction computationally simply limit dimension recommend approximate order hyperparameter experiment support spline allow active closely relate smoothing spline spline along plus spline triplet weight number exponentially order practice ss via penalize fix sparsity hyperparameter procedure method separate individually allow automatic determination gaussian mat ern scale et kernel emphasize locality local argue care method solely require st interaction interaction rd exp additive additive structure distant space example strong similar dimension provide geometric comparison kernel additive non problem gp demonstrate gps data shape area peak training gp green function come first gps discover suited deal collection additive refer additive base gp first order gp gp ard additive base set hyperparameter fit mean inference propagation table split specify concrete nh gp square exp additive l concrete nh gp gp breast exp heart exp bold along bad sometimes model large experiment se perform order well
precision procedure qualitative procedure systematically htb change binary cancer micro jointly frequent dna profile dna run treat cancer group individual stage tumor smooth whole segmentation constant detect segment theorem propose detect multidimensional statistic generalize procedure knowledge observation free efficient programming simulation particularly keyword statistic segmentation estimation series contiguous stationary segment appropriate analyze time driving jump known segmentation arise range eeg processing square adequate instance presence criterion optimize recursion without require prior knowledge suitable first distance relevant rank analysis approach limit point way multiple group propose statistic amenable dynamic optimize term dynamic test change point criterion estimating point instance penalty add location adopt reference therein original approach norm propose rely slope approach preferable bic calibrate paper homogeneity section procedure derive describe observe transpose testing group convention rank j rewrite cumulative f denote continuous course direct maximization exponential maximization dynamic algorithm eq optimization start use procedure number rarely focus paper number principle change two trend rapidly increase hence least square regression fit residual minimal treat significant non alternative dash line performance assess arise report dimensional piecewise predefine four feature well simultaneously moderately db ratio jump amplitude assess replication correctly estimate point use tolerance whether correctly conclusion qualitatively ht programming multiple detection compute intra scatter add indeed approach slightly
ml knn ml knn knn knn knn k knn multi label several correlation improve correlation mapping paper novel method label sample subspace structure decompose wherein row consist matrix residual space correspond estimate via group coefficient concentrate sample belong evaluate real aim wherein belong label relevance early lp transform label several task label task lp treat unique label share label although bp lp variant multi multiple problem exclusive thus necessary discriminative correlation lead imbalance label decompose multi problem demonstrate exploit label group multi structure imply randomly final prediction thresholding relevance parent binary several subset build hierarchy classifier subset cc predict feature predict group method separate I correction result subspace formulate extension knn dimensionality feature preprocesse multi linear multi always decompose formulate problem correlation study label learn structured decomposition sparsity assign label via training estimate subspace stage matrix correspond nonzero space subspace explain decomposition low group cause specific linearly subspace coefficient concentrate label able method build mapping label decompose sparse multi label increase imbalance compare different several sample decompose residual explain reveal linear I wherein coefficient thus multi subspace representation correspond label belong wherein multi rank wherein stage group multi thresholding label preserve mapping information encode explore label matrix correspond matrix bound indicate cause monotonically iteratively value complete represent characterize label residual main time require impractical multiplication decomposition accelerate letter projection w I wherein matrix include multiplication require initially distribution firstly calculate adaptive algorithm summarize stage acceleration htb kn kl py ny l ir I introduce group representation stage decompose component sparse space component label lie subspace group wherein define corresponding coefficient nonzero concentrate belong accord solve problem integer final prediction vector thresholding group sparse although via select coefficient threshold guarantee summarize htb evaluate dataset annotation scene music categorization web compare knn five evaluation evaluate effectiveness cpu experiment server core ghz intel processors gb ram prediction five metric ham score prediction give label wherein one rate operation exclusive accuracy scale medical music select yahoo website table sample vector yahoo yahoo education yahoo yahoo show cpu knn table matlab train classifier parameter knn knn yahoo hamming score cpu second ml knn education ml knn ml knn regularization uncorrelated subspace
c mrfs cross scale coefficient across texture moreover global illumination classification model learn powerful mechanism classification affine space pca handwritten digit recognition oppose convolution learn algorithm lie group rotation intra variability operator clutter responsible intra variability define linear network wavelet modulus locally signal affine compute art handwritten texture descriptor classification appropriate include complex form invariant transformation rotation elastic manifold kernel operator address space high property operator lipschitz relatively log invariance linearization operator invariance interference class domain penalize rotation concentrate translation invariance carry difficulty application review cascade wavelet transform modulus operator affine training describe art digit database software build transform begin wavelet mapping coefficient modulus operator operator operator lipschitz continuous wavelet orientation angle large frequency function operator q complex wavelet orient complex wavelet numerical gaussian finite q derivative amplitude frequency fine scale translation improve frequency frequency modulus modulus eq concentrated may frequency modulus wavelet concentrate frequency xx interference combination depend exact location wavelet modulus modulus wavelet transform thank towards coefficient wavelet modulus locally q convolution carry wavelet j insensitive reduce phase modulus provide co occurrence scale corner edges texture one frequency fine wavelet average procedure eq co occurrence family angle satisfy verify invariant key x eq term signal locally linearize lipschitz class computation source material illumination variability build affine path large affine first covariance affine model empirical empirical mild covariance belong dimensionality learn affine computational estimate affine dominate calculation eigenvector thin operation signal let select affine family embed affine l ik classification class centroid affine space highly discriminative dimensional far adjust discrimination dimension approximation adjust dimension penalization penalization factor parameter optimize intra class variability penalization thus handwritten recognition texture discrimination illumination wavelet length mnist digit provide good classification compare currently table compare optimize coefficient find optimal compatible classifier yield deep apply pca validation linearization approximation affine digit intra class function compare space belong intra fast decay affine
discrepancy name context call approach root statistic exploratory datum choose simplicity include regular histogram string estimator discrepancy principle approach also penalize deconvolution discrepancy exist weak rather old integrable density decay slowly inconsistent extend exact version quite asymptotic even sample size asymptotic mostly previously suggest wide size last conclude existence distance kolmogorov define continuous depend usual easy note probability sign dirac sn show surely condition analogous statement prove find pp surely n n df show solution previously sample explicitly underlie surely already function monotone bandwidth principle unique suggest subsequently seem sample show realization mentioned numerically possibility discrepancy usually criterion know formula kolmogorov function function law iterate logarithm see discrepancy principle need obtain behavior estimator similar theorem hence part h tight page continuity interval identically around continuous consistency bandwidth bandwidth go rather let denote old exponent follow q lebesgue borel set integrable old continuous exponent similar old exponent next consistency estimator discrepancy old sufficiently exponent go slowly hence density fulfil yield chapter imply fx dx surely almost consistency threshold consistency rather inconsistent rough vanishe quickly iid draw compactly n f ab compact almost surely surely example density discrepancy lead consistent inconsistent estimate uniformly function iff fulfil w trivially case elementary since compactly directly consider threshold version discrepancy propose order need generalization nonnegative part second df df give small surely discrepancy intuitive case slow principle law iterate logarithm introduce correct nonnegative propose go chapter version high almost surely triangle theorem order form choose risk depend accord interpret threshold suitable density discrepancy principle guarantee asymptotically optimal sketch asymptotically yield depend invariant rescale kernel translate rescale standard small choice lie extremely severe sample size lead version principle work mainly conduct mention large estimator include discrepancy included find suggest extensive describe good whether discrepancy principle perform discrepancy principle specific applicable method quantile kolmogorov ks propose fulfil method l nr formula extensive simulation variant standard brevity largely size density package density figure opt variant typical sample bandwidth fairly bandwidth k obviously large small fourth nr asymptotically arithmetic density size table density ks ks nr cv lr ks nr ks ks nr lr risk although risk quantile kolmogorov ks ks kolmogorov statistic beneficial risk mainly nr although asymptotically choose bandwidth poor seem capture multimodal density bandwidth e surely density density e bandwidth estimate large discrepancy nr distribution density select lead perform worse discrepancy except nr small fulfil discrepancy principle consider inconsistent state bandwidth cv lead inconsistent applicable discrepancy relatively principle choice perform lr consistency guarantee density generally simulation consider much largely base iterated discrepancy easy branch mathematic rarely estimate density infinite peak work surprisingly discrepancy principle size much choose normal nr poorly take recommend ph thesis tu university collaborative nonlinear author support discussion lemma corollary foundation department science university de investigate discrepancy
variable consider base width variable evolve environment influence trait influence fitness evolve convergent regressor regression suggest hand suggest fit display evolutionary along available www c regression trait dynamic trait accord sde generalize study generalize yield research focus evolve markov highly correlate system adaptation treat trait meaningful importance maintain algorithmic predictor herein optimum sde imply predictor appropriate sde computation ease presentation sde diffusion random process initial solution e x let conditional sake pair specie sde x jt j jt j trait time e successive increment take isometry equal j acknowledgement anonymous associate anonymous comment allow manuscript substantially partially national science award roc second author work foundation award trust fellowship national mathematical synthesis science department security nsf national foundation ef pt studying trait relationship develop herein adaptive estimate novel square algorithm implement process comparative comparative group relate specie comparative evolution specie evolutionary history view evolutionary specie often incorporate list study reference describe evolutionary relationship specie evolve statistical dependency tree trait trait brain paper consider response trait toward optimum precisely trait stochastic differential sde eq measure adaptation toward mean change evolutionary dynamic responsible trait toward indirect force optimum correlate evolution trait brownian motion bm optimum bm ability central tendency dependent evolutionary predictor response find curve relationship trait value develop identifie rely free univariate make computation knowledge methodology set evolution adopt adapt base overall pattern paper describe provide implementation summarize suggest direction topic foundation let trait sde change predictor predictor variable appropriate evolutionary trait evolve evolutionary trait optimum evolutionary curve trait regression tb specie whose pair trait correspond branch evolutionary root specie evolve herein see evolutionary sake unless otherwise result trait deduce common eqs estimate thus need residual lemma trait covariance eq adopt variance regressor vector method observe value specie represent trait value regressor entry equal coefficient optimum mean trait incorporate process rely predictor enjoy close branch length accord depend optimum therefore estimate adaptation trait
cost apply map near neighbor centroid centroid high preserve separation shift generate centroid minimal deviation optimally denoise one level hard low noise method cut optimum temperature flat maximum like report mutual gibbs distribution expectation easily transformation calculation integration mutual around except modification analog mutual term transformation depend auxiliary truncate value decomposition close select svd principled cut spectrum convert code optimal simulation theoretic determine state selection decomposition exploratory thereby unitary essentially induce rotation quite rather interested justified underlie cutoff define truncate svd method approximation coding approximation capacity enable compute capacity channel originally discrete problem select rank truncate thereby investigate challenge space method indicate svd remainder derive svd problem principle page description optimization di satisfying temperature minimize play central give interpretation hypothesis approximately define code receiver via transformation asymptotically vanish channel achievable encode effect lead high patch overlap decode possibly approximation capacity select maximal going list quantity truncate frobenius optimizer convenience frobenius svd instance relation basis linear basis rank cost minimize encode mutual eq originally space binary string sum first volume hypothesis infinitely receiver distinguish union reason capacity analytical calculation transformation r weight u sum np condition temperature distance solution temperature suggest high constraint temperature infinitely htb discussion infinitely infinitely negative infinitely transformation volume practical criterion svd integral finite space hypercube isotropic identity way grid experimentally investigate study influence integration capacity mixture isotropic lead capacity sphere standard finding fig n none could possibly codebook indistinguishable capacity contrary capacity analytical solution infinitely serve htb transformation mutual grid increase increment preserve transformation density impose large dataset
spatial model investigate generate instead randomly add lattice generate add link proportional q connect degree link fig obviously adjust spatial network link e super link connect therefore tail long heterogeneity large heterogeneity confirm indeed heterogeneity suggest heterogeneity real modify spatial perfectly preserve see fig length identical reflect general conduct affect optimize process report always become shift small dimensional imply exponent previous homogeneous distribution heterogeneity shift value however optimal enhanced introduce heterogeneity heterogeneity favorable degree average short however shortest overcome effect degree heterogeneity keep average independent introduce heterogeneity spatial law shift far decrease heterogeneity explain law distribution exponent system heterogeneity explanation wide length empirical heterogeneity short explain heterogeneity range sense understand design thank zhang chi helpful suggestion work support science spatial range long constraint short spatial network propose spatial constraint result heterogeneity exponent shift short length synchronization reproduce play internet phone network grid past decade system model locate plane distance generally reproduce spatial associate link introduce network link link site dimensional choose distance length reach exponent control number formation link length short author optimize aspect availability customer company carry traffic constraint observe degree network total system many work reveal network propose degree heterogeneity heterogeneity affect exponent link shift short heterogeneity degree heterogeneity place phenomenon exponent briefly describe locate lattice near neighbor pair site receive site e separate regular long control average length long correspond vice versa path network minimize certain optimal accordingly exponent b usa average receive proportional therefore space equivalently nod
model use slide window window far union separate clarity mining relational weighting temporal view active current step relational treat additionally relational instance temporal predict attribute weight recent highly decay rapidly define ed w historical long lie exponential historical inverse w could e traditional appropriate temporal learn temporal since diverse simple use validation learning extend naive bayes heterogeneous relational subgraph model topic topics nn ga nn rl nn ga design mirror conventional assume attribute independence label calculation link calculation incorporate temporal attribute instance heterogeneous algorithm search feature average mode dynamically relational create split use standard zero conditional computed sum link attribute standard attribute weight select appropriately augment traditionally prediction vote propose ensemble method ensemble information relational vary link temporal ensemble propose ensemble five temporal ensemble sample discrete strategy perform construct temporal temporal weighting additionally bias toward past strategy time link method ensemble transform temporal space temporal weighting learn parameter temporal ensemble noise temporal dimension node observe past link temporal learn distant temporal randomly relational likely due diversity correctly predict instance additionally temporal assign dataset attribute I communication email communication extract www list period email user report contain comment user size text message dynamically individual discover centrality cluster prediction task pt conv tool tool window module ex count email email topic eigenvector count topic email count topic citation database citation information research paper extract automatically web seven machine paper ai reference addition work author temporal relational evaluate relational area describe model weight union window otherwise information dt relational ignore temporal window besides vary classification evaluate different relational vs also different centrality team demonstrate temporal representation discover attribute temporal outperform ignore dynamic significant improvement traditional relational ensemble vary apply mining discover link temporal topic feature correspond evolutionary classification effectiveness discover vary incorporate pattern task brevity compete non dt auc relational model show case appropriately assess prune primitive temporal relational relational information use model various explore relational relational compare primitive relational classifier dt dynamics relational interestingly improve weight learner suboptimal dt see improve indicate focus attribute explore small representative set framework representation choose correlate class influence link attribute learn explore appropriate show attempt apart superiority outperform necessity temporal optimize select strict representation link model worse significantly scalable include search exponential link relational representation depend g citation temporal construct robust reduce relational increase temporal bias temporal relational representation aid mining evolutionary improve explore selective e motivation link decay rate learn temporal ability selective temporal attribute find selective temporal simple task temporal relational classification vary relational relevant relational repeatedly almost temporal relational ensemble traditional ensemble outperform temporal information link sophisticated temporal ensemble instance investigate ensemble use temporal wide ensemble ensemble traditional technique ensemble clear lot ensemble aim reduce significantly provide information utility information increase significantly information compare ensemble auc randomization primitive apart accurately attribute team centrality topic primitive temporal representation help minimum temporal relational relational complex find team localize change frequently unlikely temporal relatively percent temporal ensemble ensemble pattern indicate temporal notice change project frequently could responsible increase accuracy importantly significant model performance temporal example randomization attribute additionally randomization identify two significant randomization attribute thereby value association step due attribute change ensemble standard traditional ensemble figure traditional ensemble step even rely attribute relational past present use future behavior attribute ensemble topic useful indicate topic way understand among vary technique relational discover pattern relational weight attribute vary temporal insight temporal discover temporal consider link distant successively window link attribute node attribute successively past successively include recent attribute conversely successively auc increase increase drop temporal also quickly since paper publish distant generally show justify auc ability predict instance future might always noise unlikely past relate determine past behavior anomaly decrease back stability modify temporal classifier model network whereas stability lead structure unstable tree gradually evolve ai relatively stable addition also find perform amount hypothesis complementary advantage especially temporal justification method select representation ai ml use yet insight measure year ai ml task figure ai cite majority year year paper paper indicate transition paper become researcher factor paper influential omit brevity ai lag interestingly probability begin indicate past behavior use communication discover communication interested classifier version latent estimate gibbs representation justify complexity latent topic network investigate topic communication evolution use discover pattern explore relational c cc topic c topic code file os type mail fix release build day amp pm module home file support module check main thing good exception van ms list topic find positive related sentiment exception social appear van various representation classification necessity relational representation link attribute see relational perform indicate meaningful appropriately exploit remove brevity temporal relational outperform clearly annotate effective communication effective moreover time correlate ensemble mining insight network effectiveness scalability flexibility ensemble mine temporal acknowledgment research nsf contract number research make support air office national science fellowship reproduce purpose notation view
nash correspond equilibrium game see wireless sharing decentralize equilibrium game distribution far none player know strategy actual nash article information whether actor competition present principle ne finally access compose amount bandwidth token accounting protocol exchange service token spectrum datum transmission type spc spc spc resource sharing request actually spc communication kind allocate bandwidth spc thus allocate introduce bandwidth need bandwidth allocate free allocate however allocate token accounting protocol exchange service token represent depend whether green connection token bandwidth token indeed green due fail ht describe spc customer spc share connection green green determine indicate sensitivity spc tolerance degree towards risk spc sensitive green spc big say spc spc customer characterize parameter want spc resource tolerance indicate tolerance sensitive degradation price able pay connection sensitive amount spc spc request utility depend request request spc decision allocate spc bandwidth split competition receive bandwidth spc spc utility request spc competition spc rise focus competition spc competition access could model spc could motivated fact environment regular connection need spc access could game thank run software good strategy article game denote spc green part would played stable player mention spc spc share regard request request bandwidth amount elastic delay file connection price file transfer transfer file denote file spc amount bandwidth require file spc bandwidth minimal request amount green user signal strength spc request bandwidth limited since request spc connection could besides bandwidth amount request avoid amount aim interval request actually discretization focus request accord profile depend sensitivity degradation fix accord ht sensitive great propose cost pay maximum element permit green connection couple amount parameter highlight minimum bandwidth green request spc spc bandwidth allocate request accord request spc triple amount spc resp mean connection spc receive connection spc spc eq answer respect request spc equal formally spc bandwidth availability green bandwidth spc outcome consider spc connection maximized compete formulation describe player player strategy define pure corresponding couple integer quantify game since several allocation decision spc utility maximize spc gain spc allocation represent triple represent connection spc express cost connection utility nash equilibrium nash potential admit least pure nash equilibrium response pure nash good response correspond profile let profile pure nash equilibrium stop player change response new profile strategy mean spc end spc bandwidth game restrict admit one equilibrium want whether know converge pure nash equilibrium maximize step request use spc spc compute repeat profile converge prove consider game pure equilibrium sufficiently value nash decentralize learn nash equilibrium incomplete principle follow player discrete update strategy local represent gain spc gain technique else u correspond respectively player margin percentage step mention restrict pure nash equilibrium article try exist nash nash characterize spc file category specifically simulation present take consideration extremely gain sensitive spc want file communication article simulation strategy simulation strategy notation analytically compute present fill highlight nash payoff rise verify pure detect strategy converge expect expect expect remark strategy match pure vary checked pure nash equilibrium keep vary h c iteration fact consider ht ht nash eq gain even strategy per still pure nash equilibrium nash equilibrium gain pure reach stable game make represent rapidly fix respectively connection maximize number propose duration connection connection mean duration software update slot q focus per pure nash analytically permit game restrict pure nash equilibrium profile learn pure equilibrium strategy strategy necessary user give minimum amount file competition model game spc
tradeoff term monotonically decrease argument standard point newly assign never increase cluster pay penalty still decrease finite clustering perhaps objective past criterion aic penalize mean motivation constructive monotonic connection dp finally datum log choice sampler cluster formation cluster text extension dp arise dp top define prior mean mean share appropriately hierarchical reader detail hdp inference dp straightforwardly hdp outline datum mean share local cluster dp I hdp index define base mixture dirichlet base yield follow analogous global process prior extend employ hdp summarize derivation determine hdp require threshold work global association set assign create cluster close start global penalty distance plus sum local whose specification hard hdp initialize step else g dp mean cluster cluster intuitive minimize k whenever create appropriately cluster across monotonically minimize theorem monotonically local convergence two objective hard relaxed large eigenvector relaxed indicator suitably process clustering mean relaxed dp matrix stacking whose correspond matrix optimization note relaxation orthonormal standard argument cluster eigenvector relax cluster cluster common thus relaxation mean dp great possible develop dp mean normalize briefly review straightforward map j note express space expand c c unnecessary explicit assign cluster near implicit hdp point weight objective k function vertex edge adjacency disjoint collection cluster cut normalize seek minimize cut find cluster minimize minimize v c equivalently prove normalize mean normalize ratio let extend w k c objective optimize construction utilize distance cluster perform penalize normalize set dp k car balance scale breast brief goal enjoy many unlike scalability k ground output application match cluster ground truth parameter potential way clarity utilize desired iteratively find distance distance among repeat distance utilize hdp replace sum gibbs covariance fix inverse prior yield though bayesian consider two place via validation key terminate within run contrast figure clustering first learn use validation tune cluster poor validation sample converge contrast dp reach additionally dp benefit among gibbs comparable accuracy uci labels truth cluster datum set gibb yield run gibbs run ground truth table dp achieve datum scalability vision datum patch second versus dp fully obtain gibbs infeasible demonstrate highlight baseline ground choose uniformly covariance set generate gaussian per individually share result score across set baseline mean datum dp mean whole obtain score yield hdp form cluster mean dp mean individually share via hdp versus dp scalable retain benefit bayesian model several future basic mean relaxation iii nonparametric process process generalization comparison helpful proposition corollary offer cluster hierarchical utilize multiple flexibility method viewpoint inspire mean gaussian hard result monotonically elegant like analysis multiple argument dirichlet extension relaxation thresholded cut statistic impact field learn instance dirichlet document dirichlet result gain simple preferred bag collection motivation algorithm implement work variety attempt design scalable viewpoint connection mean mixture gaussians means view maximization matrix mixture show dirichlet nonparametric argument simple hard except form point cluster centroid monotonically include take hierarchical model hierarchical dirichlet set take hdp novel cluster result cluster across cluster turn mean local cluster global cluster additional extension mean arise relaxation compute compute eigenvector highlight connection bayesian graph early algorithm monotonic unlike formulation cluster conclude result underlie chain indicator proceed point perform indicator cluster z ic normalizing start new cluster gibbs point currently assign often g g infinite particularly dp define covariance mean prior compute closed calculation z assign z happen obtain assignment
use measure rather minor since always valid valid valid old existence univariate find kernel consider assumption density function exponent level density kernel extend assumption give correspond observe theorem give depend lemma case term dominate assume star shape generally star shape corollary equation risk note hard estimating level cut logarithm also assume contour old low smoothness plug although construction apply cut interest illustrate prediction practice bandwidth drive density intuitively estimator near kernel quite estimation guarantee contour bandwidth candidate idea applicable validity small preserve validity correction input level construct satisfie use q use conservative coverage much big level sized subsample finite validity loss negligible although excess minimize symmetric small loss difference detailed require illustration well present bandwidth sample select table coverage lebesgue repetition coverage regardless decrease cut tune excess loss support corollary agree figure realization dot show sample splitting blue curve inner set respectively grey clear three capture close panel require parameter practice estimate prediction picture hull construction ideal region least fast depth excess axis stable bandwidth especially moderate contour dense allow moderate phenomenon interest construct combine idea statistically level method compute near without regularity validity algorithm popular device comparison conceptually stable future aspect bandwidth selector well study procedure smoothness possible develop deal nonparametric n yy yy level estimate view process nest empirical vc kernel suitable formally define easy nest empirical theory eq eq triangle result exist constant conclude part eq subsection neighborhood exponent enough q allow switch q applying empirical definition inequality suppose event exponent condition event enough consequence proof observe therefore suppose event condition right side depend example nsf support dms air force fa university e investigate region establish region regularity condition approximation simplify bandwidth selection theoretical demonstrate prediction want prediction product pc ny name prediction region beyond anomaly item fall prediction previous indicate sample investigation mass concentrate choose give rate lebesgue contour formulate region thorough introduction book criterion efficiency region prediction region formulate notion validity however evaluate probability work prediction hand free efficiency closeness provide contour unknown benchmark set prediction estimation difference loss eq region minimal propose validity constant cost check linear rate formula smoothness behavior near contour special combine idea prediction estimator sample validity nature region analytical form carefully tune value efficiency approximated density level always region validity finite validity computational cost refine rate convergence tuning bandwidth density drive demonstrate construct region method general construct prediction exchangeability prediction equivalent block value coverage prove order completely nonparametric rank n pp finite coverage usual focus aspect discuss subsection augment region algorithm example investigate histogram mixture plot middle middle three estimator bandwidth lead valid lebesgue bottom plot bandwidth whose corresponding minimal prediction region give closely characterization level lemma also region introduce density denote cdf version sample respectively eq outer set proof accord much validity next investigate efficiency region subsection efficiency term convergence level value mass exactly continuous equation equivalent contour outer plug ny nh regularity condition consistent refine modified condition
swap swap example like uniformly graph similarity enyi swap metropolis toward empirical qualitatively show value eqn expect world dirichlet scalar vector compute sort repeat replicate average normalize nonzero eigenvalue replicate procedure average result behavior qualitatively quite prior shape shape separate become concentrated around swap top eigenvalue quickly merge eventually go infinity become regular relative frobenius show swap regularize sdp regularize construct laplacian sample wishart eqn laplacian edge implicitly sdp observe criterion relative appendix present criterion spectral error intermediate value plot replicate replicate implicit regularization improves regularize sdp correspond stronger improve strong intermediate level figure improve wide level illustrate obtain level figure similar explicitly case implicit regularization parameter illustrate depend illustrate optimal proportion converge high value like agreement interpretation quick laplacian exactly regularize present instead result make implicit extension suggest extend characterize implicitly eigenvector statistical application though illustrate become increasingly analysis department mathematics stanford university stanford quick nontrivial eigenvector certain regularize program provide manner regularize often ridge lasso respectively interpret gaussian laplace regularize sdp estimate conversely imply run base nontrivial eigenvector eigenvector correspond exact heuristic time output sense recently formalize context graph top nontrivial optimize equivalently optimization program objective program semi nontrivial interest usefulness graph image segmentation semi etc ask diffusion run heat quick nontrivial eigenvector algorithmic tradeoff box solver optimize interesting regularization sdp rather diffusion sdp constraint analogous regularize regression regression prior laplace respectively detail whereby interpret population laplacian drive random define inverse population posteriori estimate prior assumption regularize sdp heuristic say sdp interpret regularize regularize computed sdp solver laplacian heuristic nontrivial eigenvector laplacian background section describe section describe implication importantly light certain decision brief appendix define set symmetric case combinatorial laplacian semidefinite one often eigenvector eigenvalue define normalize version laplacian laplacian graph laplacian laplacian compute black also walk state walk diffusion base negative evolve dynamic three dynamic following evolve equation evolve seed evolve term add penalty interpret incorporate prior observation constant depend encode computed maximize equivalently value minimize regression respectively ridge impose interpret impose problem normalize view laplacian induce observe formalism follow observation specific another laplacian node population degree degree laplacian matrix analogous wishart trait accurately one however laplacian appendix provide justification precise denote support eqn analogous eqn play object function eqn eqn eqn linear program note sdp particular program except prior estimate sdp appendix prior heat prior invariance subset semidefinite set eqn support orthogonality sense empirical eigenvalue laplacian equivalently normalize scale factor simplex exchangeable permutation neutral degenerate possibility dirichlet must eq one implication prior nearly
interactive mechanism modification multiplicative interactive object section private interactive online mistake analogy analogously multiplicative weight normally maintain distribution element element universe update query element ever need update key pick universe let element initially hash mapping update ever implement run neither result depend interactive polynomially compute inner product project project vector normally represent however limit representation also become standard solution vast differential exhaustive et answer arbitrary sensitivity interactive mechanism make however answer query database interactive answer sized family counting improve al improve differential result arbitrary query comparable multiplicative generalize reduce release interactive unfortunately discuss private bound suggest private release query substantially run guarantee bad query give release mechanism give size discretized set far efficient sized simple relax notion long bad case efficient algorithm sphere respect point algorithm attribute run et bound expectation give average private boost although require I algorithm error complementary error guarantee require database interactive algorithm mistake implement multiplicative analogously set thought universe involve multiplicative small universe give universe difference select universe run adaptively whereas subset linear randomly database contrast bad utility query projection family independence previously use stream limited streaming element universe mapping universe define database think sometimes evaluate linear normalize qx universe polynomial quickly query list universe class even many accurately linear interactive one time output capable answer interaction mechanism interactive mechanism query answer next interactive mechanism let query interactive abstract qr q interactive mechanism mechanism adaptively stream query query polynomial size require define notion neighboring database database individual I differential randomize act database range differentially neighboring database interactive output choose adversary interact adversary differential privacy laplace center fundamental privacy sensitivity laplace preserve neighboring database procedure preserve privacy understand individual privacy prove composition universe ss tx tx class sparse small weight analysis use must database sequence define hash table argue never run argue apply infinite ever attempt consider private potential state number tx x x tx si expression argue every drop least begin know drop x x tx tx tx tx tx update q never report observe variable ed therefore last follow recall choose proof update theorem differentially mechanism interactive set run query sparse bind prove recall small application algorithm lead universe conjunction release multiplicative weight follow differentially interactive release set time roughly moreover per query super polynomially class interactive release algorithm privacy polynomially sized class constant interactive release query together database vector obtain multiply require projection use hash function family use independence prove limited wise independent also preserve inner product integer random matrix vector ax ax ax ax ax identical ax compose output kt wise independent hash tu hash wise purpose integer finite select represent function hash last bit wise independent hash mapping write u therefore composition preserve magnitude matrix consequence chernoff limited independence random query independent entry projection recall query I qx dominate rademacher equivalently write mb mb b markov wise independent dt dt follow chernoff plug recall union query query prove sparse wise entry denote q also use laplace prove I scalar prove logarithmic projection implicitly random using introduce laplace analyze source together first probability conditioning event occur complete briefly mention super polynomially conjunction say conjunction sparse size differentially private interactive release polynomially run accurate super polynomially sized interactive super polynomially sized also achieve interactive interactive class query release run polynomial unfortunately fast release random projection seem powerful tool query comparable pt fact pt accurately answer differential privacy class relax guarantee zero polynomially universe interactive interactive technique sparse universe runtime mechanism universe universe universe specify database record think infinite task statistical query database provably preserve individual record use become accurately answer privacy date come type accurately arbitrary family query database unfortunately size universe database impractical technique query rich class sensitivity answer axis align query vc conjunction offer choose query open privacy release know exponential polynomial restrict universe typically database polynomially universe sparse query still entire universe vc dimension rarely ask analyst medical study might analyst disease sparse analyst knowledge overlap participant analyst database beyond merely universe information privacy preserve matter way analyst algorithm accurate preserve differential act adaptively choose stream arrive interactive shot output datum structure encode interactive query answer sequence take
compare four consider range compare reflect bottom panel full observation full mle substantial fold standard case low usually perform well cell use cell useful right real practice variance ratio substantial whether substantial improvement leave fraction interestingly standard hashing b bit hashing poorly hash bit hash scenario improvement b bit involve contingency prohibitive sum achieve improvement consider permutation provide formula convenience assume large verify q q q ratio without std standard binomial really multinomial define probability event solely however combine ensure accurate ratio even hash mainly focus estimation estimation application use text column characterize tool user understand context contain column describe compression coordinate reduce message pass small estimate recent bit hash storing bit prove bit storage bit target encouraging result improvement application page adequate substantially increase permutation similar work svm hash good estimator hash apply three q convenience presentation intersection unbiased provide variance delta straightforward skip lemma less obtain resort likelihood mle equation unbiased result theory variance estimator panel reduce variance improvement fold magnitude bottom confirm well know may right panel well pair occurrence real web page contain present verify pair mle two magnitude unbiased mle simulation bias mle vanish hash low bit b available form form low practice convention estimate hashing tool multinomial estimation cell e summary conduct permutation achieve improvement review basic probability
distribution count different context indeed invariant reason favor assume permutation note associate order table new tn n tn metropolis probability sufficient positivity instrumental employ stay feasible candidate proposal l un see positivity condition propose candidate option candidate cell count recall free permutation cell equal count cell count th mn j mn c modify table generate order table return free return belong mn mn cell using eliminate cell attempt describe local move union various order translate mn mn mn mn candidate table instrumental distribution mn n mn hasting need n n effort result chain irreducible positivity discrete performance sample jump move small less move around specify tailor arbitrary table produce proposal whose uniquely determined particular bound constraint contain table sn tn sn necessary sn computation substitute sn reduce upper perform binary search index u jj b r table never n algorithm return sequentially guess calculate si j si ss ss cells line tn return equal purpose sufficient repeatedly iteration proportional keeping free result successful different generate move explore make adapted table generate call dynamic basis example zero eight table effectiveness describe section compete unable generate basis algebraic importance sis chen calculate cb henceforth implement package table structural zero produce burn chain cb sis generate sample run replicate define hasting proposal distribution monte carlo mean batch p estimate sample batch carlo error value batch iteration throughout batch process mac intel report compute c sis chen al sample sis implement solve optimizer code replicate material individual group business self teacher employ survey horizontal structural ten education level similar manner freedom interaction subtract zero correspond zero al must increase marginal table level freedom interaction lead exact section sample agreement second million iteration table fix e markov chain dynamic two show p increment calculate sample table dot represent mean incremental estimate solid represent way datum table classification eight relate economic page yes child yes education school yes education yes cell mean observation table contain observation applicability relate limit due marginal two interaction observe interaction degree value markov markov cb cb exact cb monte markov cb error p recent chen report version sis use sis chen chain replicate run second markov million second h count vary fast varying total value show exact number iteration increment sample dot incremental represent number markov basis sample exact instrumental key tm candidate would c instrumental increase obtain effective determine lead chain dynamic good running finding properly adapt table procedure integral part dynamic markov basis need complete practical determination one step question study basis algebraic spirit list relate basis material computer material contain file run base markov basis importance sampling description file txt acknowledgment grant sciences university thank author comment generate markov basis contingency cell marginal low zero framework find markov table move structural zero eight code article keyword contingency chain structural sampling contingency perform arise nuisance minimal sufficient validity nan approximations raise cell large structural e g occurrence one way importance chen chen chen chain thompson al literature table concept behind understand pairwise n tn tn general computationally solve simple markov move equal connect primitive extend log basis basis determination table cell arise limitation inference lower induce theoretical basis include unfortunately quite carry principled assessment algebraic dedicate software far paper dedicate basis preliminary complete start perform basis contain three repository useful move simple decomposable avoid basis advance instead connect current applicability markov basis example handle approach present set basis table algorithm chen et applicability zero conclude variable contingency observe ki ci km cell order array ni ni ci cn cross c kn un bounds specify cell structural zero express take addition constraint satisfy minimal consistent odd observe induce bound li programming constraint exact count cell cell determination need comprise integer array order equality system e assume determination complete analysis constant constant perform volume arise conditioning multinomial term straightforward computationally infeasible induce markov front expensive walk impractical reference table basis contain number move difficult handle algebra necessarily need entire chain basis generate ahead consist connect neighbor union neighbor two algorithm always valid reciprocal cell include h uv u l strictly neighbor e move one correspondence table expensive value give increase array equality correspond stack along determine reduce row gauss number system efficient programming small translate determination bound represent vector version linear cell cell n determination use bound system line algebra gain determination cell calculate application certain combination integer linear constraint associate sample cell make sample positive key sis calculate estimate quantity exact various desire table distribution sis describe chen chen many completely numerical involve way table table contribution chen previously cell value version sis eight table argument example situation sis distribution importance sampling procedure section c method table marginal limit applicability index cell free I imply
outside preliminary notation map identity associate composition restriction matrix operation complement form column rest unchanged na ni ai n q way projective system act remove row edge obtain vertex label edge n projective projective restriction map restriction map projective system restriction permutation map I permutation monotone monotone uniquely determined subset monotone set say subset subset symbol subset distinction become restriction ga km usual composition collection class edge whereby without correspond clarity restriction respectively projective relationship collection elementary category category c must satisfy correspond composition must represent object need make object implicitly restriction category object give element object define monotone cover n preserve aa show infinitely infinitely exchangeable project subset look subset either restriction intend notion subsample subsampling depend way make chinese restaurant crp permutation modeling represent among population observe social reasonable intuitive sampling subsequently edge restriction accurately reflect removal remove without node subsequently u radius less depend choice removal nx nx necessary infinitely action ar r collection exchangeable mean every b exchangeable suppose finitely cardinality cardinality permutation restriction ne ne ne ne consistency restriction reduce eq restriction element infinite endowed implication start exchangeability negative b infinite exchangeability direct corollary exchangeability doubly sequence non negative exist space graph denote vertex induce hausdorff hausdorff know throughout triple vertex adjacent argument os account comprise edge graph information implication miss link absence know monotone set ccc five lack involve affect cluster involve description overlap subset onto also helpful nature naturally certain statistic individual network undirecte let collection equivalence relation every network implication parse link social among individual clique due unit sample assume point imagine infer observe image monotone cover collection monotone exchangeability g element population presence within reasonable inference cluster context cluster sample restriction newly classify already exchangeability gauss allow restriction realization poisson observe complete nothing
decomposition therefore nx n jx conclude proof hold q precisely proposition linearity true interesting reference reference product compactly department university also useful theorem remark remark width nsf grant dms statistic al united adjust langevin mala move incorporate mala start stationarity suitably chain mala imply dimensional compare walk mala previous target applicability correlation rescale mala weakly hilbert scalar show decomposition suitably resemble eul discretization martingale continuous numerous arise affect quantify cost part develop mcmc arise application product gain class hasting propose accept proposal brownian langevin proposal langevin quantify increment proposal rwm adjust langevin mala aim analyze markov determine optimal increment precise suggest existence accept may large proposal large move large move still order satisfy dimension question concern choice suitable required stationarity proportional diffusion research along paper concern rwm mala direction converge determine parameter rwm proposal mala require target rwm mala quantify efficiency gain mala rwm employ inform feature optimal maximize rwm mala analyse huge concrete tune proposal rwm mala algorithms acceptance probability practitioner criterion given ask extensive see distribution subsequent beyond slightly variance diffusion limit contain diffusion employ considerably complex form limit diffusion perspective motivated nonparametric condition separable space inverse additive noise absolutely first measure subspace reference normalize interested mcmc finite approximation project onto eigenfunction brownian operator rwm apply dimensional weakly furthermore target scaling exponent speed limit I remarkably case fundamental method dimensional measure develop limit numerical derive limit wide hasting local consider mala context field model behave measure contribution output mala suitably apply dimensional approximation limit diffusion acceptance regard remarkable langevin form add demonstrate develop numerical obtain technical estimate conclude state develop fashion theorem measure induce structure finite enable mala scaling probability follow notation sequence expectation sequence constant notation real satisfy notation let separable hilbert trace family hilbert refer function represent via often move expansion see allow variable product norm rr rx rx alternate space identity r k b rd rd orthonormal say trace j include every belong furthermore induce alternate measure condition formula valid eigenvalue existence impose direction comprise functional however identification identity avoid first derivative operator exist constant ensure trace full measure full derivative x x h h refer interested approximation end span eigenfunction notice subspace approximations reference introduce law random normalization support nx equal schmidt u strictly repeatedly give langevin leave similarly define informally sequel lemma show quantify various repeatedly sequel consequence proof satisfy functional define satisfy sm satisfy order formula sm estimate constant lemma lipschitz similarly globally mala langevin diffusion motion operator mala proposal euler proposal n time euler discretization derive reject ensure average grow detail choose diffusion stationarity lie markov chain evolves implement acceptance coordinate form langevin proposal acceptance define eq expect acceptance stand also bernoulli success may bernoulli key behind quantity behave like summary markov describe project mala lebesgue start stationarity solution informally scale mala proposals exponent exposition simple explanatory dominant inclusion carry n proof lemma evolve stationarity n n show acceptance acceptance diffusion summary choose probability infinity exponent move jump expect adopt efficiency exponent analyze describe mala nx acceptance mala rigorously prove corollary accept rigorous theorem show rescale hasting mala mala converge weakly follow result accelerate stationarity explore explore stationarity maximize choose acceptance one mala algorithm acceptance first implication mala explore invariant implication law speed invariant fast practitioner tune acceptance express mean probability speed function maximize precisely product acceptance outline drift martingale statement paper pointing proceed full subsequent simple scalar scalar markov weakly acceleration scalar variance success sequence converge acceleration diffusion original idea arise instead work hilbert value coordinate present precede account variable autocorrelation bernoulli variable markov mala start stationarity invariance principle mean acceptance depend randomness quantity approximate approximation lem lemma bernoulli limit mala drift martingale constant define ensure array drift decomposition read exploit behavior version approximation resemble euler prove martingale difference q process brownian strong brownian outline continuous additive define closeness see eq continuity continuous x infinity prove approximation chain evolve space drift eq go infinity rescaled diffusion follow general diffusion chain separable chain suppose stationarity martingale converge converge globally lipschitz rescale converge weakly dt brownian motion independent define map equation accomplish nz n ax nz nu nu nu nu nu last nu z nu n first check satisfied sequence mala key lemma weakly state satisfy nx need satisfied establish asymptotic chain quantitative every infinity verify let n proof independently consequently j lemma jump globally follow normalize uniformly functional measure converge weak contain prove need function normalization uniformly bound convergence surely converge surely prove sure result state exponent repeatedly sense principal come function converge law z rigorously generality quantity expand quantity n ny pt n n purely change involve simplify constitute q expand remainder taylor x x lem ny put together state globally nz ny simplify demonstrate definition tx x nx nx give equation b n quantity lead term quantity eq
proximity fact composite valuable regularizer problem product induce every shorthand p proximity operator convex proximity operator exist recall subdifferential subdifferential nonempty compact subdifferential gradient establish relationship proximity subdifferential proceed positive define fact immediately namely norm search simple soft thresholding otherwise last large example recall fact theory refer exist mapping call define well fact fail state tool fix mapping I appear fall class every penalty prescribe penalty e form overlap apply formula separately proximity operator compute function z n di fuse consider immediately fall intuition fuse vector case choose incidence consider composition norm specifically absolute permutation xx rr form singular order correspond proposition proximity base von call fan uv right vector von sometimes von trace fan state system diag suppose contrary obtain th component positive contradiction form task nuclear consider gx r assumption convex occur build proximal use compute proximity special case proximity operator develop apply minimizer unique minimizer subdifferential inclusion formula combine inclusion term proximity formulate convex minimizer conclude add subtract fix conversely hold proposition b spectral eigenvalue iterate without need use provide proximity mapping bb part proximal corollary motivate proximal approach lipschitz idea behind therein approximation point simple iteration accelerate nesterov specific property objective optimal achieve certain interior iterate reweighte square apply successfully nonsmooth hessian cost system accelerate involve desirable accelerated see restricted computation proximity method exhibit far empirically accelerate influenced nesterov update achieve recursive fista computing operator sequence combine compute efficiency nonsmooth problem demonstrate art case exactly computable overlap proximity know hierarchy acyclic graph order improve moreover report incidence penalty aware composite apply variation lasso build nesterov software advantage proximity regardless linear scalability methodology simulation case square run first consider synthetic set simple embed generate uniform randomly nonzero group group assign discussion choose coefficient group normalize like zero noise run datum correct zero due objective reach efficiency iteration different indistinguishable conclusion cpu show depend almost grow higher comparable proximity insensitive require cpu grow remove show nesterov match tree structure lasso berkeley segmentation extract dictionary place branching practically indistinguishable synthetic draw explicitly connect cluster dimensionality accelerate decay overlap monotone decay graph make progress close long future accelerate optimum verify fast time matrix quick addition incidence total without datum nesterov identical favor solve class nonsmooth give nonsmooth term composition example cover regression term nonsmooth feature deal rich regularizer efficient exist demonstrate specific fused group handle composite accelerate yet theoretically suggest simulation room acceleration case method include range learn linearly composite problem wish useful discussion support air force grant fa grant h nsf international european cm
apply determinant newly covariance datum aggregate entropy within gp integral covariance eq whereas scalar maximum datum define solution next candidate adaptively domain overlap one high dimension large sampling computationally feasible monotonicity determinant follow counterpart collection approximation computationally costly computationally variance lead rough widely heuristic minimum discuss point collection proposition nonlinear dynamical evolve action control state nonempty evolve scalar often observable mapping nonlinear system relationship possibly simplification observation noise model variant respective control nonlinear discrete partially observable input nonlinear maker maker maker limited goal describe dimensional reference model dual control formulate system output output variance function x good quantifying observe adaptive control maker start zero little put emphasis quantify section dynamic shannon information maximize base discrete dynamic relationship u goal control system reference let system control objective follow side unlike static close approximate reason reasonably dynamic costly purpose utilize methodology approximate objective adopt candidate solution sum utilize visual iterative account newly receive start exploration action reference estimate gp associate approximate strategy solve couple make regard present firstly turn control unknown system step depict figure note mapping htp htp htp unknown map affect controller relationship obviously hard control initially increased gradually summarize greedy accurately process long twice control less keep mind identify control htp htp htp cart classic system case position cart nonlinear state period position cart cart angle angular standard available step look strategy control cart result provide benchmark htp htp cart control use ahead controller act external dynamic choose obtain show satisfactory within htp htp book provide valuable insight relationship discuss however focus book problem play important introduction machine literature gps get increasingly popular characteristic book comprehensive treatment gps additional area active neither discuss quantification build shannon theory use gp discuss collection discuss objective measurement subsequent active gp heuristic measure old attract interest research community dual focus adopt dynamic reinforcement application identification dual present address focus limit quantify serve quantification allow joint two illustrative cart develop static way influence control action identify select static constraint control present mainly future research investigation exploitation trade elaborate weight person theory constitute control decision priori posteriori identification exploitation select action provide point identification control quantify serve art regression quantification identification control objective illustrate map position control cart make limited obtain accurate often infeasible time fast change controller collect underlie dual obtain control controller controller must controller perturbation differ amount information controller controller aim stationarity prohibitive perturbation action observation provide single identify discrete continuous perspective process underlying equation time training adopt allow quantification goal explicitly combine objective pose multi control control iteratively objective resource approach uncertainty additive likewise variance additive observe provide incorporate simplify provide supervise gp regression rely gp describe iteration observation determine optimization take objective information measurement objective issue infinitely collection incorporate many concept implicitly build upon information learn specifically field acquire control adopt bayesian use process iterative capturing meta control despite heavily utilize different aspect image lack expectation inherently information play datum develop strategy obtain scalability worth note problem seem social economic information variety fundamentally decentralize resource allocation decision change quickly characteristic decision nod another security management costly another biological operate system summarize present utilize control regression gaussian modeling observe learn specific gp provide follow spirit trade one hand try minimize try estimate unobserved end fit follow predictive unobserved balancing prior capture basis observe gaussian regression preference
site merging course run mean current estimate run strong suitably unconditional site suitable site lattice magnitude large cluster site cluster size site probability idea essentially inside another simulation list function grid run header change empirical cluster different estimator produce subsequent run run run estimator generate neighborhood generate unconditional manually simulation n run pp generate use binomial input success p run run remark update six neighborhood provide tt tt run scheme permutation belong run maximal edge tt edge edge cluster neighborhood order connect l l calculate either construct neighborhood cluster cluster edge tt tt lattice corner cn neighbors n ci b neighbor ci cn interior ci generate unconditional cluster integer generate determine single modify furthermore simulation manually site probability inside estimate simulation site giving position site give site xy subset corner c c c single est sim p estimate est run run modified remark update step neighborhood site size outside edge site site belong list list maximal tt label order magnitude remark moment triangular lattice random connect neighborhood edge else l merge remain tt tt size tt status store tt status active store add updating step special neighborhood complement vs complement list list single subset tt neighbor edge label order edge remark always label connect neighborhood edge edge calculate maximum cluster merge edge cluster large tt size tt size l exceed site inside size output n list inner site outer site binomial probability bin site determine entry les indicator k bin would like di discussion mm remark phone phone author develop algorithm shape presence boundary object condition interior paper method write digital image noisy image paper efficient quick detection object noisy use indeed look first question ask question diverse automate moreover picture procedure picture surprisingly majority image statistic skip stage reconstruction object permit nonsmooth mild interior detect automate analysis cancer material regular precisely shape advantageous object testing approach density assume continuous tail connection object seem spatial consider triangular periodic linear pixel top exponentially object detection build datum stop view machine learn valuable field machine indeed unsupervise even rate convergence describe algorithmic nonparametric processing also research language programming implement substantially free language advance already build type describe store processing image implementation algorithm explain image method behind introduce suitable modification allow fine image subsection implementation modify image processing package convert sequel store image grey entry vector intensity pixel convert display vector mat mat mat mat mat mat mat mat follow code generate lattice n neighbor n ci ci cn neighbor neighbor n analysis find thresholded remarkable algorithm linear find linear respect pixel depth construction span explore order encode neighborhood neighboring site belong site store site explore current site unlabele get store site site new proceed site unlabeled neighbor seed point label site span tree whose display root please slow much generate span span tree span r pos rank pos l pos I else pos pos pos current something nothing backtrack pos current else return cluster image nan find object test extremely finite site minor change make cover well let probability site
many detection exist within number change mean change situation change parametric simplifying perform efficient contrast book describe many often detect change rank statistic method compare empirical paper source presence zhang produce offer detect require side contrast figure well point real problem extensively review poor rise essence page consider offline match length likelihood approach language word length word change point count homogeneity text review variation uniformity appearance use calculation length adopt stre finite simplicity write subsequence string define prove length consistently true length markov surely though complete suggest class however non fast convergence perform plug estimator insight shannon stationary ergodic state ergodic finite alphabet string length typical mean expect matter formal dependency involve involve return take possible directly e px normality condition corollary extend condition simpler analyse partition match avoid parse block see cover understand result iid even source partition fix block long return source iid similar prove typically ergodic concatenation parameterization independent sequence length change concern contrast common algorithms quantity help set match position length position among detect tend tend direct look suggests simply normalize martingale address expect find hope easy rate relative stationary ergodic relative rate role analogy argument probability md string set typical string graph generate n theorem prove consistent theorem build understand behaviour establish martingale tool martingale consider develop control inequality mention process write nx kolmogorov kolmogorov brownian fact supremum principle sense dimensional appendix prove consistency martingale illustrate behave length model see look affect analysis language figure form english switch author natural english two language language write english distinguish suggest english author typically consider text already distinguish show text marked vertical change estimator idea information work variety source related toy direction hope establish likely return dependency exist time distinct regard towards behave case expect perhaps suggest decaying suggest toy decaying case fast toy scenario multiple point change believe real come direct theory work issue point stream spirit burn believe act proxy analogous figure n control envelope fluctuation see toy fluctuation fluctuation fluctuation overall first technical regard operation enyi binomial autoregressive discrete ar recursively take u j nd j martingale definition prove induction bin bin bin u e write argument since birth death standard martingale jensen inequality since bin n j n equation deduce martingale z c c k n j bin n j lr deduce variance lr c martingale characterization essentially version stay minimum minimum figure htbp change function stay function martingale bin explain see equation mean leave change point part similarly equation mean concave right two follow exact curve lr lr rl accord small limit proof assume curve either interval know mean standard conditioning decompose term mean nz mean term function mean coefficient decrease similarly equation since add contribution example lr rl together deduce take tend zero tend adapt slightly loss generality equation interval empty suggest empty deduce empty otherwise divide use bn divide union interval q make mistake imply p bin work grant mathematic thank pt pt conjecture condition pt alphabet detect change property motivate idea length information novel parametric enumeration form example show chain distribution require avoid consistency relate toy establish martingale argument string symbol alphabet source piecewise source section offer perspective substantial consider match length length elsewhere string class process length describe position choose place create direct link formal model change believe
freedom desirable node jointly schedule wireless computationally intensive multiple become extremely challenging study available report wireless author power effort schedule early paper direction technique apply energy generation wireless solution node spread wireless wireless nature ideally every coverage allow receive limit effectively form graph note interference continue irrespective transmission towards network call wireless provide late transmission via intermediate connect loop flow source flow depend availability transmission consumption choose slot restrict node instant point transmission form transmission network one though final throughput gain capacity involve activate simultaneously define activate power link power scheduling scheduling become gain centralized setup channel notation channel link channel gaussian calculate area spend retain power capacity calculation whenever statement setup general modify exploit objective maximum statement contain link average power form statement refer versus sign value refer represent effective rate choose flow mode mode link power refers call spend mode correspond power spend slot represent availability rest pure system statement applicable mode interference vector capacity capacity interference channel mode mode spend one problem user sum noise interference assume matrix network clearly know optimization link iterative scheme link simultaneously continue link sum monotonically link capacity link spend active perform namely capacity optimum capacity interference instant namely link channel link initialize li constraint loop one choice see mode turn million note ignore calculation require get need solve network generation claim node thing variation network even network large basic solution start choose consider choose new improve master mode scheduling master attain improve iterate converge suboptimal solution suboptimal problem optimum search evaluate infeasible avoid heuristic sub optimality example heuristic ready implement denote among link mode solve solution optimal obtain next depend initially discuss early well convex chapter multiple optimal solve problem similar claim performance node ht number channel number gaussian random source rate flow avg availability unit approximately optimally get solution problem unit achieved give fairly amount heuristic solution unit optimum ht direct source flow nod heuristic multiple initial point see follow solve achievable user unit unit network
imputation offer assess imputation use validate code categorical surprising imputation validate imputation cross validate grey significance pair investigate first already categorical yield rare present protein array status gene find prediction mass experiment novel search child product systematic heart heart adjustment health life assess observation comparison perform imputation categorical generally amount imputation ill nearly dependent e binary category statistical successful category variable number nearly introduction lead removed variable make implementation statement bar bar grey black three order magnitude level pair encode significance child result run imputation imputation simulation lot behave scenario estimate imputation minor role compare tb three simulation assess compare imputation previous show far fast however run considerably require code categorical variable knn tree comparative tree high tree forest tree strongly increase note imputation number equal node tree run perform always miss number tree simulation imputation basically multivariate consist require distributional aspect come biological near multivariate imputation imputation quality need subsequent represent potential include relation dimensional greatly excellent uci repository molecular thank medical centre thank child center child rich provide thank anonymous constructive comment science program biology disease disease motivation modern acquisition throughput technology depend imputation offer however majority imputation variable type handle separately ignore relation cope several imputation evaluate imputation tree constitute multiple imputation build forest able imputation without need biological introduce range handle comparative method relation imputation error additionally exhibit attractive cope availability article journal bioinformatics rd version miss fully biological enhanced measurement technique field provide multivariate greatly exceed type arise technical diagnostic opinion additionally often contain structure parameter parametric exclude imputation restrict appearance field imputation base combine categorical idea book assume detail imputation unlike restriction logical domain record motivation imputation structural aspect forest mixed type allow interactive nonlinear effect address miss imputation train rf miss proceeding iteratively completion thresholde condition dimension complex structure rf suit apply furthermore rf allow bag compare imputation lasso perform proportion competitive use source error computationally attractive error imputation average deviation hence need pn absolute imputation imputation method ten categorical type imputation weight euclidean choice large effect imputation implement imputation mean imputation knn mis good original paper standardize constitute gene apply vary scale center imputation matrix imputation cross standardize require different regression comparative categorical algorithm imputation contrast specification default setup mainly simple miss imputation imputation assess pool choose procedure imputation control transform error imputation categorical variable comparison code categorical categorical deviation apply validate variable back categorical completely rate versus follow publicly include patient pd health remove shape energy patient follow level time give perform even well data dimension
supervised fitting cross validation type monte report integral associated estimate integration number sample additional computational bias true sample sample distribution fitting optimize condition severe degradation slightly trade nominal wide range rarely never optimize operate interested summation generate expensive evaluate standard quadrature mc powerful mc technique drawback expense prohibitive question technique exist lower combine mc statistical technique traditional method brief monte stack introduce technique call stack monte carlo stacking error problem problem show reduce return source truth set many bias estimate bias euler equation run deterministic return run lead square must error mc refer extremely mc second mc converge correct answer decrease carlo kind concern fluctuation monte estimate fluctuation carlo generation help sample alternate eq sampling measurable use mc several million accurately importance unlikely occur reduce sample location monte leave location method sample generation algorithm unit hypercube technique seek supervise fitting make integral polynomial fourier expansion compute quadrature fit piecewise polynomial induce correct answer increase exhibit fitting impossible make difficult know additionally datum exhibit overfitte datum inaccurate location result fit evaluate address issue validation several fit fitting stacking stack sophisticated winner strategy amongst stack view version nonparametric technique therefore argue generalization use stack single reader stack apply stack gain netflix competition combine prediction stacking partition instead correct stack close comparison stack stack monte incorporate avoid introduce poor carlo function necessarily perfect fit equation instead term fit properly mc optimal deviation intuitively correspondingly low perspective alone add thus incorporate emphasize fit poor one fit nearly ignore heuristic ignore fit inferior extension however set introduce fitting change depend hold accord special approximately calculation fitting modification option fit integrable expand calculate replace significantly computationally attempt concern five fold point training minimize inverse leave calculate monte give fit alone either first example analytic examine set obtain sample test fold cross unless plot versus green sample blue polynomial form fig ten number point high outperform carlo range perform additionally see validation bias generate fig fit whereas outperform example fitting fit upon carlo fit accurately contour generate hypercube fourier expansion low individually ref program synthesis conceptual future certain frame eight probabilistic effect technology represent burn apply order easily follow eq generate take standard find twice apply respectively x exact trend two sample magnitude calculate calculate cost sample expect error uncertainty quantification test response surface noise signature output recently propagation tool robustness pressure signature minimal specifically fidelity field roll angle uncertainty pressure like true space fit great fit increase despite fit still strength additional cost number introduce stack monte reduce carlo unbiased thus monte able effectively use extremely generic generate mc also monte furthermore discrete result regime examine explore application fidelity incorporate form fit curse cost entire evaluation assumption
ambient bold symbol capital product denote strong convexity recall private tt step problem utility algorithm generic differentially private guarantee framework convert private generic privacy sub private differ one privacy formally define sensitivity th mention early requirement degree adapt private formally sensitivity decay linearly convexity typically satisfy output magnitude learn private algorithm along concentration bind privacy guarantee framework convert private guarantee bound tb algorithm assumption change sequence noise output lemma c projection I function differential see measurable addition see apply eq good definition differential privacy composition argument differential privacy order needs add intuitively large incoming lead arbitrarily bad exploit iterate algorithm union probability differentially private technical lemma stage preserve privacy entry observation privacy sensitivity differentially private proof noise let z tb hoeffding get depend private private algorithm see output diameter lipschitz time add chebyshev inequality function sequence private diameter two private strongly lipschitz show restrict practically exposition privacy obtain update eq elementary algebra theorem algorithm see also differentially private variant private regret key observation behind differentially differential obtain add structure preserve propose sum add see lead well next iterate function tx trees structure differentially l b guarantee privacy state sequence rl regret lipschitz schwarz bind private fact eigenvalue bound ta observation use problem differential provide goal output time privacy treat concatenation sum partial change na noise privacy get generalization provide description label root label node add noise node thick node b depict change partial denote leaf string follow level label right concatenation string label two summation vector root j rooted leave tree root tree correspond leave root output add perturb node summation node level illustration code ts ds root ss ss minimum form string string q privacy utility partial sum entry consider ratio inequality equation combine formal utility guarantee proof inspire technique tw dt output differentially tw vector tree select sum differentially private sequence differ noise independently hence thus lemma ratio use differentially deterministic hence differential reasonably broad drop albeit perturbation modification analysis sub fit private spread practice inefficient completeness technique analyze differential privacy framework differentially online algorithm good show good bound offline well differentially large class offline offline later practical describe offline one observe minimize formally consider specify data minimization provide via execute incur comparison produce private noise output detailed presentation framework underlie privacy bind differentially private prove privacy need perturbation sensitivity output produce execute dataset entry triangle maximum continuity lemma guarantee utility rewrite incur derive convexity utility bind number dimensionality inequality lipschitz equality follow vector least lipschitz continuity bind rhs plug exist offline method differentially framework wide namely however significant advantage pt learning theorem utility loss popular svm require lipschitz minimize fix constraint bind bind dimensionality believe primarily usage add practical iterative guarantee differential privacy extend force propose differentially framework offline compare wise optimum algorithm approximation privacy two practical practically regression online offline online regret provably preserve guarantee specifically fashion minimized problem variety finance apply differentially private guarantee logarithmic dataset year dataset fix generate dimensionality contain offline use ridge non private normalize number incur synthetic note log close requirement weak pt ccc iteration incur level privacy plot scale privacy meaningful provide privacy especially low logistic regression square logistic private logistic point purpose private show average algorithm privacy reasonable private point reduce underlying sub maintaining show bounding sensitivity consider algorithm private differentially private differential special cost logarithmic regret guarantee show private differentially private algorithm offline learn offline obtain error regret open optimal similarly privacy logarithmic research direction differentially private various course lemma claim conjecture microsoft com university edu edu preserve learn continuously change preserve privacy challenge subsequent etc learn customer behavior customer etc practical use privacy measure critical order sensitivity I arrive regret goodness utility two convert privacy good popular variant regret linear differentially private offline offline art modern privacy customer generic scenario engine ad serve ad relevance past search two key query goodness ad engine try guess history available ad get online engine several reasonably pose severe query ad click guess past query thus engine correct guess relevance privacy etc concern learn one generic privacy preserve formal notion programming notion interesting lot progress however without privacy contrast g click relevant ad produce roughly analyze output produce privacy hard differentially private handle typical programming several theoretical algorithm convex algorithm incur exist private offline notable notion private online specifically two differential privacy whose privacy record privacy current set count differential typical arise practice contrast consider practical namely handle offline result particular expert differentially answer count offline datum entry total answer start subsequently answer
consider due analytically latent consider simplicity vector length grid note however dimensional large infeasible extremely slow mix recent gaussian hyperparameter slice general mix gaussian element burden impose typically use drive heuristic recalling devise accordingly evenly knot local bin spline compute cholesky spline autocorrelation choose truncation level computational consideration gibbs inversion operation sampler step computation dimensional draw distribution draw non model apply methodology handle choice diagonal towards shrinkage towards shrinkage redundant straightforward gibb present infer adaptive covariance diagonal become distant nearly band limit upon invert band limit efficiently versus naive issue maintaining analyze covariance regression compete alternative covariance nonparametric replicate discrete length additional additional precision draw sim covariance ccc sim residual sim residual covariance sigma set prior parameter section evenly space determined round experimentally sampler insensitive initialization lower dimensional one analyze mix initialization form cholesky initially take spline location sample iterate couple iv drive newly sample indistinguishable present burn run discard true component display noise covariance density mu sigma jj sigma mu jj sigma sigma ij covariance high interval show row represent mean represent error bad magnitude see able capture component worst contain density interval interval small band representative capture observation standard build instead benefit exactly mean regression model wishart analyze dataset element decide whether remove chose predictor region result remove compare miss likewise sample mcmc regression base actual model result gibbs discard clearly indicate regression predictive improve naive depict constraint fact lead improved kullback leibler element base true predictive predictive kl regression regression replicate dimensional choose evenly spaced knot sx diagonal maximum value covariance show ccc est beta analogous nonparametric wishart mean replicate large near replicate exactly truncation replicate gibbs thin examine display aggregated replicate average covariance range norm indicate good commonly volatility financial wishart discrete account slowly ty specify discount observation maintain wishart construction integral run backward frobenius depict wishart approximately twice furthermore towards error accumulated high degree force cc x nonparametric covariance wishart replicate analogous apply nonparametric capturing spatio temporal united states surveillance grow although increase public surveillance include rapid distant spread drive medium coverage surveillance control health www site us surveillance key report population week plot show report six state available http www media fs severe dr david united human predict million n united ccc vs trial possible nonparametric google trend dataset thick trend united york covariance line scale united state show show plot indicate period event shoot severe aid rapid researcher search query predictive methodology search logit transform query examine rank variation perform query explanatory region base result rate closely track actual advantage datum http www google publish ip address fine aggregate reporting finally google trend methodology search search activity raw search query track quickly various medium week week vector element consist google health human service block observation omit reporting end year number respectively exploratory figure plot period state move window aggregate f event due dimension simply observation dimensionality ability correlation cc california allow temporal correlation important predict function define rate week capture x ix model google line particle jointly univariate joint lose individually predict rate select simply analyze pattern substantial miss solely google trend ability introduce methodology cc trial trial california california trial map trial map map map trial map trial plot capture compare figure clearly demonstrate temporal google entire dataset length hyperparameter examine truncation level run chain discard burn chain examine every initialize parameter assess perform diagnostic index york california south event spatially correspond state factor period imply correlation shrinkage component trend follow google estimate national york plot notice display key mild distant area distant york california highly expect south fairly state infer south finding display specific note dimensionality solely level unable crucial redundancy report form covariance miss dataset also ability method cite observation observation readily handle limited matlab take machine four intel core ghz processors gb bayesian nonparametric potentially formulation collection herein induce dependent methodology tractable dimensional simulate google trend interesting direct propose fall address limitation collection observation predictor collection cope framework hierarchical propose building complicate additionally continuous response employ probit lead probit current include augmentation impose covariance defer another consider possible approach avoid specify discuss draw process large become infeasible scale large predictive process directly use integrated approximation dramatically affect hyperparameter address one thus allow formulation datum image voxel spatio dependency represent covariance imagine replace element spline hierarchical allow variability subject interesting future domain acknowledgement author helpful discussion covariance theorem every cover repeat second continuity sure almost sure continuity shrinkage prior surely enough wishart x surely let pairwise surely almost xx inequality sup ij diag prior imply recall e probability symmetric semidefinite wishart conclude combine positivity proof ns sure exist almost convergent nk last equality e lemma recall matrix distribute imply iterate moment location eq wishart conditionally n fourth moment slice really covariance derive accord zero drop derivation notational summing index cross arise share process moment derive eq dictionary conditioning rewrite dictionary conditional take definition although literature regression vary relatively little focus develop allow induce covariance dependent loading loading unknown induce highly flexible tractable conjugate miss theoretical nonparametric capacity grow within collection portfolio likewise temperature etc record spatio variation imagine arbitrary potentially multivariate setting although literature univariate multivariate I capture correlation typical inference address decomposition precision propose predictor row problematic exist alternatively submatrix coincide submatrix additionally involve assume definite factor still flexibility approach rank dramatically increase parameterization volatility discrete volatility volatility linear return suffer curse typically limited dataset dimension volatility assume autoregressive process approach recently latent conditionally wishart precision limitation wishart process challenge theory description intra relationship review dynamic linear conjugacy cite volatility ability dependency often lead additionally value within typically herein volatility modeling covariance wishart marginally inverse spatial dependency arise computation scale accommodate spatio temporal upon factor symmetric positive without xx establish decomposition explore vary predictor factor relate model related conditionally work
belong skew term geometry r entropy divergence family multinomial beta study unified family divergence family generic furthermore entropy calculate exponential distribution among shannon entropy entropy divergence exponential family communication field know entropy measure amount accord close expression shannon entropy theory seek code underlie language since practice true unknown observer rather entropy unknown leibl divergence orient notational convention rewrite enyi entropy modify one axiom characterize average enyi single prove l fx classical illustrate rule later discrete entropy q since enyi tend shannon enyi entropy keep shannon decrease close entropy report multivariate technical motivated multi generalization shannon derive entropy entropy non tend shannon shannon enyi entropies entropy monotonic function admit density denote natural convex since f p fx ff characterize member later say observation sense show factorization regularity condition class statistic family lebesgue distribution order family multinomial exponential family canonical decomposition prove exponential family follow exponential r enyi entropies particular standard center closed entropy shannon entropy enyi entropy shannon fx illustrate generic let start family exponential successive canonical decomposition family r enyi entropy use rule converge shannon univariate exponential natural kx enyi shannon eq tend calculus complexity base may mean calculus result enyi tend shannon entropy h tx conclude probability define divergence geometry also rewrite ar case coefficient hellinger family measure enyi divergence follow skew jensen gap let member exponential enyi exponential skew enyi divergence bregman divergence exponential direct appendix natural log member enyi
round possible trivial rich fact minimax allow contain corruption regret low proposition appear minimax bound key corruption learn task follow regret map restrict mirror imply vanish corruption dependent class parametrize matrix may simple parametrization corruption define affect determined possibly parameterization q drop subscript simplify introduce throughout parametrization problem start corruption cast within base power weight feature absolute sign correspond round base loss want enough magnitude parametrization allow strategy correlation elaborate induce classification weight thus pick monotonically discriminative sign hypothesis restrict op pp z scenario impose soft corruption pattern nature corruption process observe define weight array example sensor wireless array sensor fail situation sensor measurement surrogate neighboring feature neighbor say miss ki put neighboring sensor specify graph predict feature vice versa entry would add constrain diagonal mirror frobenius norm function could simplify presentation shorthand incur define exhibit regret theoretical recall setup imputation mapping eq q retain observe feature predict encode framework expectation product like quick inspection result optimize hypothesis play framework restrict relaxation relaxation natural restrict simple existence follow denote consider rr overall interested q imputation optimize hypothesis bound imputation optimum easily find jointly take saddle concave n k drop convex relate replace relaxation piece define eq strictly semi definite convex defer lack arise relax involve ensure relaxation relax correspond point solution saddle problem equivalent proposition behind replace rademacher justify predict like class purpose give hypothesis rademacher possible triple lie optimization problem note hypothesis correspond relaxed deal reasonably dimension thereby still provable rademacher inner outer sample bound brief sketch defer hypothesis prediction use lemma straight bind come us control gap risk follow sample term covariance vector online show rademacher relaxed ridge use baseline miss zero gradient ridge evaluate several uci dataset dataset artificial pixel central remainder corruption corruption base deviation pattern tune corruption present use type force additional choice regularization task entry correlation coefficient entry pixel allow corruption imputation perform relatively poorly corruption imputation expect provide imputation rich corruption use perform least imputation offer significant vs method omit top corruption fraction indicate corruption across sdp solver instead find well regression see improvement least well imputation corruption continue perform corruption figure display improvement cccc corruption table corrupt significantly imputation able outperform imputation naturally miss perform well cccc regression corruption bottom naturally occur introduce I motivate construction theoretical empirically result imputation strategy accumulate task index contain distinct associate classification different show total regret von duality distribution clear optimize set alternatively vector individually minimax regret accumulate measure complete order regret convexity last regularizer finally old yield step non negativity substitute simplify yield theorem formulate problem rewrite ridge problem maximization find regression component hadamard contain dot product instance fractional problem eq due quadratic substitute result show precisely long necessarily semi fractional add additional add make replace complement positive semi complete cauchy schwarz follow rademacher term cauchy schwarz separate remainder second term expand apply expectation schwarz cauchy schwarz supremum use follow complete analyze wish bind separate depend cauchy add summation inequality divide prove corruption corruption corruption corruption choose induce corruption instance choose tune induce show corruption sum available corruption average trial fold test applicable trial random corruption pattern across update optimize fix writing give equivalent optimization respect respectively equality couple optimize subject norm work case deal solve minimax q eq positivity constraint matrix optimization concave von minimax max observe minimize maximize direction normalize appropriately I n leave constrain equation imputation view natural know even corrupt hard alone miss expectation start simple imputation strategy regret setup weight equation derivative loss first control natural
pairwise yield kronecker tensor aim formally product edge generate gaussian kernel generate approximate type armed arrive kernel assume universal like product pairwise well product theorem say suitable rkhs contain close relation likely concern outperform kronecker domain symmetry relational formally indicator return element recently alternative pairwise kernel limitation generalize pair graph impose lead direction connect repeat measurement reciprocal symmetric briefly summarize learn reciprocal let start definition binary hold relation every multi unobserved opposite transform reciprocal reciprocal relation represent preference win direction interpret interpret edge ordinal ordinary interestingly framework reciprocal give proof immediate domain least product obtain represent reciprocal relation relation every relation specific relation constitute application symmetric relation relation relation call preserve arise domain special setting turn easily incorporate mapping mathematical kronecker look obtain kernel reciprocal kernel protein bioinformatics latter enforce every edge twice edge method kernel represent impose remark well flexible represent space symmetric every every reciprocal version reveal kronecker product kernel reciprocal symmetric require kronecker relation lot play competition name show symmetric relation despite application characterize relation relatively preference modeling argue reciprocal preference certain rational human decision make well play etc extensively theory reciprocal relation traditionally increase reciprocal relation call v special case moderate relation notion put minimum form numerically fuzzy term learn reciprocal relation reciprocal call reciprocal rank reciprocal acyclic graph rank reciprocal relation figure yield interestingly rank reciprocal relation simplify joint simplify r respectively reciprocal follow decade reciprocal notion powerful reciprocal relation reciprocal property relation rather applicability need root euclidean space euclidean ranking hold transpose euclidean basically see multidimensional restrictive one embed realize rank object reciprocal nevertheless type sometimes situation euclidean occur existence ranking relation metric square symmetric reciprocal synthetic standard reciprocal kernel kronecker product pairwise kronecker pairwise reciprocal kronecker product kernel illustrate kernel learn family base measure consider measure family include coefficient coefficient member first coefficient coefficient originally correspond conversely member analyse give member generate statistically bernoulli mention feature feature last vice illustrate score experiment set validation hyperparameter sample replacement relation predict missing relation experiment set set unseen present result test use pair compare multiple testing rbf node mse training explicit grid conduct regularization table outperform relation method demonstrate easy relation generalize node enforce outperform symmetric kronecker clearly exception experiment successful probably enforce apart difference pairwise case experiment conclude knowledge really help predictive pc hardware window learn correspond number occurrence document experiment second connect node use size start node except gradient mse fail decrease rely early experiment kronecker pairwise kernel exist aim achieve around show succeed kernel lead prior symmetry relation error large small instance enforce symmetry training performance ordinary reciprocal kronecker metric explanation decade imagine resource dominate limit define fitness advantage draw disease etc specie specie dominate relation reciprocal simulation specie simulate specie thus specie represent limit specie dominate function specie training validation determine validation testing factor try probability specie ordinary reciprocal pairwise kernel regularization grid obtain repeat sign significance conservative difference statistically rise worse prediction relation ordinary pairwise reciprocal kronecker extend approach handle relation kronecker feature propose feature pair object constitute addition relation learn incorporate property experimental synthetic world really help improve recent development fuzzy look grant metric closed universal rkhs dense every input accordingly hence universal approximate metric separate vanish compact space write accord write dense accord confirm separate vanishe point consequently arbitrary rkh belong rkhs obtain replace observe connection immediately arbitrary write single proposition drive bioinformatics retrieval relation recently investigate relation standard preference framework introduce extend consider exist relation model reciprocal relation establish important link development fuzzy theory usefulness demonstrate relational modeling forecasting winner computer protein interact protein document mining friend site example application machine mining develop relation specific learning scenario summarize object variable usually consist infer unknown relation predictive unseen predictive advanced new assessment like relational preference learn learn relation edge relation aim preference relation also development fuzzy theory article elaborate connection contribution typical real ordinal relation relation simplicity learn relation possible binary example infer protein interaction bioinformatic world relational furthermore recent logic literature algorithms mathematical relation impose domain knowledge learn relation even already domain example social notion person reciprocal neither like person person b approach one model distinguished mean loss object setting restriction arrive classifier relation restriction mathematical framework fuzzy theory use notion relation account real scaling relation somewhat middle actual matter provide learn different domain satisfy distinguish symmetric domain similarity require triangle similarity learn undirected reciprocal bioinformatics preference relation win probability gene formal rescaling relation convert reciprocal symmetric property preference preference learn reciprocal relation interpret relation application symmetry relation see constraint framework object study edge weight edge node indicate
approach early ard share four penalization solution ard dictionary approximately ard ard except middle bottom decomposition ard divergence al show generative statistical relevant audio explain equivalent power investigate decomposition short second long condition sequence play pair combination subsequent temporal analysis window ms overlap two frequency temporal log signal sound string sound truth run ard initialization return low nmf multiplicative ard nmf initialization nmf solution fit use matlab implementation good run factorization wiener reconstruct produce component inversion decomposition nmf ard produce ard nmf al order fig display ten first component produce ard nmf axis two comparable histogram standard nmf ard al histogram fig ard nmf confirm relative relevance upon weight drop component individual component sound string sound contrast tendency piece split component leave component visual reconstruct split distance contrast flexibility ard nmf choose desire ard fig et component resemble closely attack merge note weight remark ard ard nmf retrieve choice decomposition point nmf desire experience correct kl kl leave ard plot yield inferior ard nmf cost apply stock section price comprise american g stock price rd th trading day stock price representative stock price company display leave plot capability set entry indicate miss integer perform nmf incomplete estimate multiplying infer basis activation normalize stock datum estimate n miss stock well ard nmf termination literature column normalize unity compute relevance weight initialization order right observe trend increase infer addition ard components ard ard penalization method dense prediction rather fig large observe ard perform uniformly standard nmf miss stock price kl nmf ard modeling know choose ard kl nmf though retain ard component retain ard nmf nmf stock fit assumption well ard well comparison return ten run clearly inferior demonstrate predict analogue analogue across ard calculate choose flexibility ard nmf tie prior exploit ambiguity induce term accounting present monotonic result multiplicative complexity update preserve initialization efficiency approach validate propose method offer flexibility exist deal prior moment rule recommend work fully seek variational carlo handle principled concern tensor acknowledgement like acknowledge work discussion share also thank improve theorem support project factorization paris france mail fr matrix nmf divergence family include leibl important fidelity automatic determination dictionary tie parameter minimization posteriori drive course prune spurious component efficacy robustness perform synthetic stock prediction factorization model order selection entry nonnegative factorization find factorization dimension dimension early reference nmf work seminal contribution lee become technique diverse process financial usually minimization q nonnegative separable scalar parametrize square kullback kl divergence divergence short detailed crucial conversely overfitte seek find solution fidelity information bic set scale linearly bic assume relevance determination ard pca computationally monotonicity auxiliary directly update leverage technique decomposition correct order produce fairly limit reference monte mcmc strategy nmf evidence high reversible jump computationally intensive another reference close principle irrelevant component zero qualitative significant author conference publication firstly paper parameterize show flexibility quality image secondly herein decrease local whereas mm bayesian detail ard nmf extensive numerical efficacy ard denote matrix matrix denote thus consider nmf review originally introduce later definition limit divergence another square euclidean parameter control observation either learn cross map parametrize respect greater separable function kn kn kn kn kn h kn kn kn kn kn kn kn kn kn kn kn kn standard nmf recover mm building everywhere function whenever replace iterate mm minimum attain key thus auxiliary approximate objective decompose sum construction consist jensen denote result generally previous iterate minimization thus follow exponent common drive drive prior constraint observation prior integrate element set activation integrate analytically prior activation prior x x assign note tie together column vice relevant norm fidelity common row finally impose relevance relevance every related note familiar family mean mass pmf form pmf vary generally gaussian coincide coincide shape b aa base log ratio divergence scalar cost whenever negative form posteriori statistical half observe monotonically third force tend serve pruning kn describe determination weight update previous k attain strictly choose ard ard division dispersion tradeoff fidelity term knowledge validation case optimize literature estimation factorization fix kn audio power multiplicative model upon choose principled focus selection sample write shape parameter v real law large number element exponential inverse gamma expression empirical ard calculation defer supplementary material sometimes fall moment observe ard ard informative section confirm small conclusion robust draw automatic pruning map develop independently extension array column coefficient exponential prior square euclidean kl divergence major optimize solve author change multiplicative along treat update occur row may nonnegative pose parametrization model tie converge reach automatic projective al projective seek nonnegative span fit ard originally describe et al additive half relevance describe adapt relevance nmf multiplicative noise address employ nonparametric set large hyperparameter place assign sparsity enforce factor contrast herein unify nmf decrease section music decomposition stock price demonstrate decomposition model nonnegative generate sample weight inverse gamma sample half depend ard reason define noisy matrix kl noise db noisy poisson element datum db gaussian observation db number choose set iterate converge limit ard dispersion fair initialization
solely exchange exchange play source network bias network contribute arrive generalize class adaptation far ahead original strategy allow source noisy step diffusion two adaptation signal network reveal network mean analysis lead far choose improve steady combination metropolis context consensus suffer degradation since ignore noise profile node combination outline far ahead mobile move neighborhood evolve even critical adaptive track variation profile order cope dynamic environment issue vi letter denote letter plain deterministic radius argument form argument stack column noise consist scalar successive use dependence scalar measurement form denote phenomenon study hybrid adaptive minimization problem several diffusion type node problem adaptive manner learn time review combine second small size whenever node broad diffusion diffusion selection example recover identity node evaluate estimate rely measurement left vector mean receive neighbor describe weight introduce quantity step involve share information node generally subject quantization likewise share subject perturbation objective perturbation diffusion weight enhance happen link additive fig observe information comparison reference noise without li compare ki k four source appear appear measurement independent zero covariance lk I lk lk uv lk white spatially independent covariance noise perturb diffusion continue symbol avoid eq denote node matrix worth respectively scalar zero notation ki easy lk substituting get recursion compare compare block q arrive follow presence exchange consist nm w exchange adaptation exchange weight recursion ready strategy exchange assumption verify assumption matrix th submatrix driving term driving term convergence whenever see bind size steady namely argument jensen induce hermitian ii iii combination determine solely regression note perfectly compare dynamic range mean filter analysis tractable diffusion exchange derive excess square analyzing error flow global error positive hermitian choose notation stand quadratic denote link ignore approximation side rhs third term error independent rhs express fourth rhs relation depend w relate linear ahead verify guarantee select conclude diffusion stable mean small size convergence exchange variance steady iw notation proceed evaluate first rhs q use identity rhs give likewise rhs approximate eq ignore steady rhs matrix vector representation mc v ki r lk lk di lk I lk semi hermitian arrive steady state positive hermitian steady say select evaluate q link bias section examine simplify sharing sharing within neighborhood expression recall expression optimize impact arrive expression eq source performance b I obtain expectation steady give term summation source contribute combination denote q minimize stochastic rely well also fan hermitian positive semi scalar norm numerator motivate separate node noise covariance rule particular scalar minimize expression manner note eq minimize bind network algorithm variance combination product neighbor r l lk propose rule estimate close allow estimate realization lk lk ki lk factor arrive adaptive diffusion benefit step vary analyze able adopt walk stationarity weight vector change accord zero initial recursion verify continue steady stationarity steady lead network define early hold environment consider network topology adopt size fig randomly generate fig white link top link collect collect k mml examine simplify namely share rule metropolis degree node iii rule rule fig also result see relative rule diffusion achieve steady uniform relative diffusion algorithm exchange assume change along circular fig dynamic express generate trace randomly db noise variance simulate fig algorithm denote average estimate fig horizontal axis along red trajectory blue trajectory along trajectory exhibit ability high investigate source exchange environment link bias mean noisy exchange derive motivate choice help also scenario change simulation illustrate finding
use even obtain strictly speak equation I really alone believe error implication therefore converge limit importantly order apply boundary normalization order laplacian happen point explore boundary laplacian spaced panel scale interior laplacian scale factor panel log would boundary boundary gradient normal normal direction half negative slope compute pick near plot expectation rigorous square compare decrease edge going see reflect intuitive nn go pde boundary boundary laplace hope regular boundary laplacian partial x connect element point boundary axis fy fy fy interior g along direction inside pde fx h fact point condition pde add point normal h h fx become fx boundary use implement method pde laplacian grids laplacian laplacian suggest eigenfunction satisfy condition density outside normal need boundary positive integer eigenfunction unnormalize laplacian see eigenfunction x boundary condition laplacian density eigenfunction panel eigenfunction eigenfunction boundary eigenfunction fact second eigenfunction correspond sign symmetric normalize eigenfunction correspondence eigenfunction eigenvector ix therefore xx boundary along normal direction asymmetric towards test graph nn graph study cluster panel shift less fix notice nn graph eigenfunction near boundary reflect near neighbor asymmetric graph decrease regularizer limit study point compare laplacian regularizer boundary follow thin quadratic manifold control distance near replace constant independent integral integral become use conclude term fact test several density value laplacian ratio figure ex ex test suggest maximum report table result theoretical suggest pattern decrease numerical precision h degenerate unbounded boundary behavior happen bring e dominate satisfy boundary safe product essentially short importance regularizer reproduce hilbert rkhs reproduce kernel unit uniform expansion green laplacian reproduce orthogonal nan expression hand without finite green reproduce expansion kernel approximate obtain scale quite due boundary eigenfunction engineering algorithm graph data attention certain continuous operator research topic exist interest evaluate boundary considerable paper analysis discuss convergence implication volume manifold scaling elsewhere manifold manifold implication manifold laplacian well amount analyze theoretical aspect manifold different bandwidth operator appropriate understanding object light property guide selection practical versus unnormalized suggest preferable practical algorithm converge fix suggest iterate superior laplacian limit cluster infinity kernel bandwidth study operator arguably significant coming since explicitly interest simple pixel gray scale image zero boundary image motion configuration human body sensor limit range boundary process boundary graph boundary operator direction normal boundary choose bound laplacian near boundary interior fix function appropriately scale derivative likely correspond influence laplacian ignore boundary laplacian confirm provide way laplacian regularizer application bound norm would minimizer boundary confirm related investigation different unnormalized graph px boundary seem view illustration boundary theory boundary effect reproduce dimensional manifold take boundary laplacian regular grid pde proceed compact riemannian satisfy necessary implication discuss review away define instead geodesic hand discuss discrete normalize weight function depend three discrete degree degree unnormalize two define accordingly l nd include unseen l fx notion apply sample become close close result interior need boundary converge weighted limit find dimensional manifold small neighborhood equivalent boundary neighborhood key fact near argument notation rest use without subscript subscript sufficiently unnormalized laplacian random laplacian normalize eq expectation limit help result sufficiently distance thin different study manifold point small comparable tangent whole tangent half ball radius center coordinate fix local coordinate
riemannian survey function robust tracking step compute gradient gradient short geodesic illustrate idea geodesic seem would robust lagrangian previous corrupted order geodesic formula equation equation respect correspond respect gradient equation easy rank compute geodesic singular orthonormal left set singular step approach fact lagrangian exploit leverage success estimate subspace help dual gradient augment lagrangian step estimate admm recover sufficient admm fortunately iteration subspace update achieve practical experiment show produce corrupted outlier take depend tradeoff step track subspace predefine stochastic literature proven however identify subspace change obviously change rule shrink dynamic adapt change need track steady adaptive rule produce empirically achieve fast adaptation gradient e take slightly step otherwise mean subspace slightly along step besides sign step inner proper equation update inner consecutive gradient min max f control control estimate identify subspace precisely change dramatically take adapt subspace undesirable conservative update increase limited large subspace change subspace change drastically accelerate step though demonstrate much level change prescribe always equation variable far calculate level select let exponentially linearly new follow discuss hand level new quickly change really dramatically idea sec e sec application prominent separation foreground background surveillance imagine stack column several frame fact subspace separate foreground background track suited application task challenge video static background move foreground video third simulate camera examine dynamic video background use foreground background static foreground select train frame randomly video real may video confident select pixel set experiment separate foreground simply subsample frame resolution cycle frames stationary subspace subspace second foreground separation stream fashion deal frame perform separate second time separation high video effectively video background foreground separation separate separation quality experiment admm track resolution separate virtual virtual c alg eqn alg alg alg virtual alg full virtual video change dataset frame adjust track unlike run algorithm pixel pixel separation numerical result matlab virtual virtual virtual camera choose camera width resolution virtual adapt frame virtual camera track frame separation pixel second camera camera frame pixel track pixel separation computation second frame robust subspace low incomplete datum question constrain global great minimization alternate triple estimate without useful situation fast know business explore one promising separate background surveillance video may wind result kind movement movement foreground acknowledgment pure mathematic internet together suggestion college university com usa edu chinese university edu algorithm subspace robust stationary corrupted completion foreground perform high quality separation move popular achieve per keyword alternate track surveillance long application communication localization medical imaging leverage reject reliably collect modern standard setup difference massive ever people example estimate surveillance city netflix rating million thousand movie day amazon com second survey whole equally example indirect really sensor response inconsistent corrupted fast tracking challenge outlier use manifold fix operate amenable streaming contribution work propose online subspace tracking algorithm adaptive tracking combine augment solve via discuss dual optimization triple use admm inherently successfully highly successfully recover outlier significant state art completion component analysis algorithm nature separate video surveillance frame matlab motivate robust subspace tracking give subspace tracking familiar robust introduce subspace robust detail discuss critical implementation limitation compare several world video surveillance conclude future direction build problematic corruption noise detect computer like would need anomaly order anomaly subspace datum often anomaly rely tuning heuristic principled seek sense combination residual outlier without outlier sensor network collaborative video surveillance monitoring experience failure signal move foreground surveillance camera include recent seek whose datum majority computation perform pca slow consequently identification develop robust emphasize effectively low dynamic literature tracking previous consider vector object exactly incremental descent algorithm minimize vector fit subspace variable operation outlier early survey track subspace come literature signal process vast literature qr context fast incremental svd modification thin work tracking aim large useful direction arrival estimation introduce target receive array follow music music tradeoff array time track incremental thus make suitable follow improvement analysis algorithm conduct ambient operate along algorithm require conjugate descent tracking oppose careful give nice compare none subsection address issue robustness miss address problem fraction extend handle outlier relate differ may outlier great subject investigation netflix matrix entry rank complete prove norm minimization recover incomplete low algorithmic singular thresholding column fast completion order evolve subspace orthonormal span equivalent outli entry may white
receive simply order statistic n x order fall histogram suppose x hold probability z thus technique enjoy wise bound regime upper linear scheme besides histogram differentially sensitivity analog technique release sensitivity enjoy dp allow sensitivity demonstrate dp q logistic g hx x give diameter note draw first require give except consideration valid give empirical cdf independent close quantile cdf inequality small give apply statement probability rearrange take achieve quantile eq namely hence concentrate probability stem triangle inequality demonstrate mean polynomial examine choice statistic eq cdf evaluation analog behavior quantile differential privacy exploration boundary differential privacy conceptually reasonable differential privacy whether adversary really access histogram cell reveal adversary application two dp synthetic histogram example histogram bin well privacy disjoint ht example histogram empty technique relax differential differential result arbitrary suggest privacy replace relaxation privacy gain deep guarantee release extend bin allow besides relaxation report reference thm proposition thm privacy privacy database effect release add analog property procedure histogram show histogram accurate histogram differential privacy show analog global release privacy dp computer science privacy give strong mathematically rigorous disadvantage guarantee come statistical propose notion differential privacy privacy differential could concern ordinary differential time privacy loss great explore version propose privacy tradeoff introduce ordinary differentially technique context histogram introduce concept identify differentially differentially histogram low relaxation privacy enjoy nonzero thus histogram relaxation analog composition database example input consist database column individual differ coordinate neighboring satisfie privacy measurable intuition dp space privacy differential strong guarantee essentially effect differential key therein much simple differential noise call interactive goal database user query one way release private database release arbitrarily histogram histogram focus mechanism e value rather mild partition space lattice take simplex histogram observation bin essence element laplace rate histogram norm valid histogram satisfie correspond subset differential privacy private histogram although set restrict space histogram demonstrate least hypercube demonstrate present risk point result differential upper move give result relaxation I tend fast differential privacy subscript whenever improve risk uniformly rate question problematic large remain open technique property would wise differential privacy admit release mechanism keep histogram namely cell contain privacy view random draw certainly denote strongly restrict invariant permutation strongly affect replace instead affect draw random privacy randomize differentially private analog differential algorithm differentially decrease inverse dp relaxation quite namely privacy meet take randomized algorithm strict relaxation
detect avoid detection probability require calculate marginal multidimensional whole estimate integral computationally easy one complicate multidimensional integral may difficult problem typical need assess reliably estimate integral statistically density assess metropolis hasting determine integral newton despite well known simultaneously density hasting receive integral test posterior mixture calculate accurately criterion size posterior whenever make assumption regard derive exist harmonic property metropolis require simultaneous give detailed summary also performance case undesirable integral demonstrate collection relative number probability calculate integral likelihood truncated estimate integral select select parameter posterior importance conversely asymptotically estimate convergence therefore integral eq impact slow also key ensure rapid practice certain unimodal likelihood reveal posterior need unimodal symmetric reliable reflect skewness asymptotically mixture various scenario prior parameter increase measurement reliable estimate estimate assess converge estimate accordance definition scale simplicity converge convergence practical except aic converge use integral integral radial velocity limit amplitude reference amplitude short hyperparameter density multiplicative proportional correspond model integral insufficient detection term note choose lead choice simple corresponding parameter becomes set prior decrease prevent call improper way unit invertible system way transform prior constant superposition radial variance become variance chance posterior always use convenient prevent undesirable define use retain statistical allow result radial velocity choose density artificial build still hold free statistical reason dimension space effectively amount turn decrease integral integral simple reliably receive posterior series different velocity noise suitable assess nearby report period day cat epoch enable reliably bayes close close provide slightly great support rather accurate estimate use receive reliable estimate integral assess lc epoch reliable integral estimate aic odd well lc estimate aic demonstrate four artificial radial determine number parameter conclusion use improper conclusion artificial epoch epoch later hour amplitude zero describe standard every simulate measurement set table contain factor signal model signal direct bf approximately bayes detection periodic weak detection bf bf bf see could detect chain converge clear maxima none probable rest converge clear periodic broad factor estimate integral unit prior factor signal clearly undesirable side effect prior yet bayes factor turn term enable detection weak coefficient density consequently prior integral generally challenge integral limit statistical method integral truncate calculated draw posterior density deriving field restrict problem certain dimension reveal know choose star know enable angular namely anomaly aic reasonably accurate well rapidly somewhat biased complicated possibly test current use parameter suitable value essence yield converge rapidly test probability small practically respect odd great though bayes select possible cause large cost low receive correspondingly effect deal unity make support european author acknowledge significant improvement assess model eq essential calculate posterior model sample dropping notation marginal value marginal estimate integral choice posterior denote eq practice undesirable property instance require likelihood information broader correspond dominate except simple harmonic extremely usage limit large impact sum extremely slow reason approximate marginal integral representative
stop iteratively square due instability penalize nontrivial simulated multivariate normal zero draw ridge along double non differentiable induce tail conditionally make choose full keeping penalize likelihood initially compute thereby zero grid use show iteratively fail small algorithm numerically penalize distinct logistic interact poorly way coordinate wise gibbs situation multimodal favorable signal moreover tractable univariate thresholding analytically narrow handle double penalize square differ subtle checking maximized confirm iteratively least yield optimum understand circumstance recommendation use logistic beyond scope penalize quantile fit corresponding gaussian scale quantile th value sufficient normal distribution percentile whose three model traditional package along regression double pareto penalty error loss new th percentile straight significant difference double pareto double systematically sparse pareto error goal conditionally representation mixture implementation maximization together acceleration many variant improvement collapse choice explore option fact relevant posterior argue predict future well predictive generalizing version symmetric prior quantity p original along discussion formula typically chain scheme identification logistic active anonymous associate comment generalize special view pseudo generalize yield th quantile maximum machine represent convolution lead distribute marginal improper necessary mixture still apply multinomial logistic modification indicator ik follow write conditional independent indicator coefficient product regularize multinomial update eq sign divide function equivalently obtain identity derive expression classification collect z recall row mixture since rest corollary example variance mixture regularization generalize prior regression expectation maximization wider demonstrate include quantile acceleration augmentation normal regularize objective multinomial predictor unknown log phrase minimize unnormalized negative consequence penalty discrete generalized miss problem unify many include negative outcome support machine robust key derivative sufficient expectation conditional variable estimator generalize disadvantage maximization slowly fraction information large basic exception substantial gain quasi acceleration combine good maximization newton robustness super convergence research correspond example include bridge estimator machine jeffreys pareto augmentation decomposition p working code estimate although may specify user notably study specific include support logit thing estimator refer issue machine inverse approach mean representation likelihood normal density combination mixture likelihood choice fact avoid deal distribution mode hyperparameter mix exponential integral identity density use involve represent expression lead identity improper limit density loss first pseudo support third regression canonical improper multinomial category exploit guarantee penalty check presence mode return compute mode gaussian mixture conditional row exploit variance maximization quasi newton remainder conditional remainder numerically newton like omit formula iteration increase density explanation acceleration offer mixture I wish regression assume outcome code represent update estimate stationary I row resemble iteratively square due subtle difference iteratively least matrix entry rapidly numerical failure nearly initialize solution illustrate run pure goal standard different draw factor loading standard nearly orthogonal acceleration conjugate different hypercube latter case
indicate proportion outlier ph bold noise x outlier ph bold r input level noise x bold indicate r r proportion c outlier ph value r proportion proportion train cpu cpu proportion outlier train cpu ph train train test cpu ph outlier cm r r set dim size ph ph remark corollary edu school filter building replace half start examine fitting contaminate criterion prevent unnecessary removal training challenge non optimization derivative free outlier correctly nonlinear global robust approximation task science processing application rarely provide reflect nature quite efficiently sophisticated huber basis wrong phenomenon typically occurrence routine range need eliminate examine notable cell least infinite phenomenon popular discard half discard small asymptotic residual something maximum likelihood estimator outlier residual stand much devoted regression paper deal see estimator affect chen et square robust backpropagation anneal fitting regression minima ann training square criterion high minima applicability traditional fast backpropagation article fitting fit combine ann removal outlier fine clean backpropagation second improved prevent unnecessary removal undesirable prevent unnecessary impose removal introduce definition several optimization section ann exist present comparative study briefly introduce origin global discussion intercept explanatory identically goal determine term absolute deviation mention sensitive outlier affect estimator small proportion sample size overcome ols estimator introduce least deviation contaminate datum detect account high follow discard build residual evaluate partition square residual small residual advantage heuristic method evolutionary semidefinite method mention achieve contaminate outlier easily detect either eliminate closely case interest mention minima instance estimator inner minimizer permutation distinct determine efficiency efficiency ols estimator fully ol efficiency method efficient high efficiency preserve initial provide robust cutoff beyond absolute residual exceeds remove adaptive weight ols step normally initial combine efficiency final ls nonlinear nonlinear vary model data output hide respectively input bias term represent ol residual backpropagation weighting method combine multiple minima criterion outlier demonstrate unlike regression shift towards attempt use huber criterion estimator residual growth large effect quasi term quasi huber either priori estimator scale latter estimation estimation mean evaluate robust criterion ann ann discard treat build wrong several introduce allow criterion mention criterion need discarded mention numerically sufficiently ann decide design removal subsequent backpropagation approach fully efficient ol ann mention objective clean remove large backpropagation objective ann clean consuming multiple iii ann execute use ann study negligible cpu implementation language library translation complexity achieve competitive cpu calculation graphic gpu c valuable traditional cluster core gb ram execute thousand limitation identical parallel parallel execute primitive calculation gpu sorting gpu summation parallel gpu artificial datum set paper artificial set example consider take segment indicate take subset outlier outlier divide subset center random variable coordinate combination euclidean norm explanatory segment robustness ann investigate deal model explanatory early explanatory variable numerous article robustness ann investigate follow explanatory investigate deal explanatory early one function surface datum standardize network publicly consumption regarded value use water consumption set water consumption real world take difference value optimisation fix cpu start package ann default datum training subset contain noiseless domain function ann average cpu procedure depend table rmse backpropagation bold apparent ann view row train figure reflect large rmse consistent method practically filter remove contaminate clean outlier repeat across datum set furthermore mean cpu backpropagation long backpropagation increase explain rough gpu core quality cpu appears contain rmse cpu time world consider backpropagation good comparable backpropagation absence reach artificial look figure correct robust note origin lose undesirable criterion bottom give
shift instead notation general assumption operator robust nr nr proof state uniform function measurable b define separable q universal uniform totally say ba refer next list need bound instead lipschitz define b follow statement em closely theorem instead n enough ns qualitatively robust continuous case get imply due weak even topology kernel space qualitatively neighborhood measure ns ns assertion step step continuity svm functional bootstrap svm qualitatively compact totally bound continuous shifted assumption compact metric lipschitz map triangle definition shift definition continuity combination functional continuous respect topology obviously separable therefore qualitatively every bootstrap qualitatively b complete proof axiom mm van complete robustness shift nonparametric unknown known borel separable qualitative robustness special show vector stable functional svm generalization proof mention borel algebra borel metric borel algebra borel complete borel bn value n ns nz nz nz ns nz ns nz nz bootstrap bootstrap value value call bootstrap approximation sequence transformation q valid continuous neighborhood nz qualitatively neighborhood ns n ns
distance recover subspace base subspace clearly linear occur image identify highlight conduct show scene hyperspectral paper estimate towards bayesian draw operate conventional mmse manifold mmse approach minimize distance manifold formulate entail along rank unconstraine coordinate q derive give make k since ty previous equation stand wise vector q k ks scheme suitably like university comment point also lead kn db kn prior prior snr snr snr department france university subspace operate manifold usual adequate metric manifold alternative propose carry minimize distance estimate consider metric eigenvector carry illustrative include von obtained analytically implement monte provide hyperspectral processing signal span evolve consist mode estimation play central recover resort kn kp estimate arrival multiple snr swap occur subspace circumstance bayesian enable estimation investigate approach herein assign minimize yield illustrate new performance assess conventional pure material hyperspectral conventional minimum wish range stand operate subspace mmse systematic mmse minimize euclidean I distance natural natural distance two subspace orthonormal svd tp unitary seem adequate mmse span intuitively appeal face minimize square angle argue subspace mention estimation author hence parameterization since subspace span eigenvector word notational convenience case derive approximate monte carlo constitute schmidt low difficult approach mixed need jointly circumstance coincide differ mmse note meaningful unitary relevant investigate occur final comment unitary range although address grow set manifold excellent reference interest geometry attempt herein illustrate conventional condition first step derivation consist distribution accept von trace matrix arbitrary function argument respectively observe thus view manifold manifold span orthonormal matrix possible namely proportional proportional cosine angle sum cosine angle close purpose display fraction figure identical increase additionally close distribution angle exhibit show figure density practice condition assume vector assume knowledge condition condition recognize p see therefore form projection maximum posteriori estimator contrast use refer von fisher np pp although knowledge exist burn unitary successively unit nk therefore draw interestingly arithmetic set differ truly minimize require compute prior posterior distribution mode close matrix gibbs distribution condition zero eq orthonormal noise follow eq express eigenvalue representative information th signal second facilitate derivation condition priori distribute interval choose low resp posterior thus problematic gibbs draw p p conditional recognize distribution generate accept reject straightforward show condition still need initial illustrate monte matrix signal ratio burn sampler estimator mmse priori figure evaluate conclusion draw estimator well prior snr enable subspace prior mmse poorly since average remark mmse well restrictive conclusion precisely uniform db make well propose subspace decade receive great interest purpose monitor mapping concern analyze decompose signature retrieve signature band obvious physical consideration kind proportion satisfy positivity constraint hyperspectral induce abundance pixel subspace simplex vertex strategy hyperspectral literature generally identify well linearity context occur consequence hyperspectral image capital image notably assume pixel hadamard reduce fan generate signature library assume interaction interaction display fig resp white represent low linearity image increase left corner note contain pixel linearly belong conversely local pixel live linearly subspace determine detail span refined estimation develop matrix suppose subspace contain computed form expression e evaluate projection state
coordinate coordinate update gradient coordinate calculation operation cycle dense per qp number outer cycle ie across column update associate converge characterize precise estimate warm start accurate moderate experimental glasso inversion note operate explicitly compute keep retain glasso optimization upon consequence panel precision produce update square plot right precision produce glasso minimal value life problem might viewpoint approximate sparse glasso purpose glasso maintain update entry principal return figure typical glasso operate optimize regularize require dp glasso working maintain glasso estimate track simple rank update describe unlike glasso positive precision column iteration advance compute warm guess solution dual warm necessarily address states row glasso maintain warm glasso column maintain work glasso partition th remain condition furthermore w box qp since box e qp combine violate row update glasso maintain matrix encounter counter warm tuple option converge easy numerical choosing example encounter briefly describe iid sample glasso take warm glasso fail warm note sufficient update glasso eigen surprising w light need guarantee iterate remain pd glasso converge establish pd warm p glasso glasso ensure warm glasso glasso every glasso dp consider update update diagonal entry satisfie update pd glasso coordinate simple lemma arise regularize box valid program respective quadratic pd glasso glasso converge definite warm start due primal glasso require initialization construct dp glasso require half update example glasso series play problem start dp glasso wish early glasso starting start dp glasso diagonal trivial algorithm glasso glasso warm start describe example micro example model concentration zero follow generate iid entry set one spaced article glasso algorithm coordinate glasso initialization wise version glasso warm suggest converge due fine grid glasso diagonal path wise glasso warm start glasso comparison glasso dp glasso glasso block dual include dp comparison glasso glasso update glasso box dp glasso report qp glasso constrain computation cpu ghz glasso operate glasso operate dual example base successive q across row tolerance primal glasso inversion comparison compute expensive experience base precision glasso matrix glasso present grid show eight evident glasso warm algorithm large typically sparsity become slow end viewpoint warm glasso probably far warm glasso design glasso warm section base plot path plot vertical dot estimate precision population minimize green blue dp glasso seem prominent latter table present type time level glasso return moderate glasso across entire path high precision glasso glasso see warm winner among decrease winner glasso warm starts glasso warm start see dp warm starts glasso warm report comparison example primal warm winner observe primal warm fast warm type difference matrix htp warm start dp glasso warm start grid panel covariance vertical line correspond htp solution dual warm primal primal warm dual warm glasso warm second combination glasso warm start across example htp solution average warm primal warm path glasso table dp glasso warm start consistently experiment process filter gene gene thresholde microarray threshold gene large form gene pool subset warm warm warm accuracy warm perform htp warm warm explore apparent glasso explain leverage glasso dual problem dual equivalent course maintain definite inverse tight essential inverse glasso solution sequence warm glasso former maintain operation dp glasso glasso solve update qp property end maintain start glasso package implement dp glasso make thank group stanford discussion comment presentation experiment warm glasso take seed seed gaussian matrix denote absolute solve glasso warm start eigen produce glasso clearly undesirable glasso seed generator iid gaussian matrix solve warm glasso fail converge example row covariance eigen compare glasso glasso follow set eigen matrix definite manner value scale perform primal primal warm warm result show result overall glasso warm explanation though dp dominate dp glasso warm start dp glasso warm scale index acknowledgement discussion statistic stanford department stanford university graphical undirected zeros glasso popular allow one efficiently glasso converge warm start explain outperform glasso study glasso solve likelihood p glasso dp operate coordinate sub conclude realization definite covariance task unknown especially ordinary mle poorly behave regularization framework estimate precision correspond algorithm graphical regularize negative q sample amount shrinkage glasso analyze propose suitable implicitly insight use absolute frobenius denote normal sub write equation solution component wise sign glasso use partitioning way partition form glasso hold get plug glasso operate read fix stationary see glasso solve rest except current coordinate sparsity easy update move onto next converge glasso outline successive glasso keep initialize cycle repeatedly implicitly solve warm round save row convert produce glasso iteration value zero successive glasso produce monotone dual value curve produce glasso confirm plot block behavior lead problem correspond equation qp constrain qp kkt optimality stationarity kkt box qp substitution relate result theory dual qp denote primal relationship duality qp qp glasso column unlike require glasso ascent constrain dual box glasso optimize lagrange dd
generator hold configuration plan fixed run heuristic planning plan heuristic planning learning thus serve slow policy improvement approximately improve learn step reasonable curve never policy started show high episode improve rate episode present find trajectory time trajectory algorithm correspond path behavior planning planning regard curve present find start step drastically episode reach optimum per episode find path show several trajectory heuristic planning episode algorithm successive episode strategy bad trajectory use trajectory extremely learn curve evident comment short instance help close focus visit reason exhaustive work long although planning contrary policy distribution branch however propose algorithm planning planning planning episode around episode suboptimal significantly reach episode case identical policy significant planning find policy deal save resource achieve good reach length quite interesting plan novel reinforcement planning well planning module incorporate ability make take heuristic strategy apply three conclude heuristic planning therefore role since scenario inform sampling problem work planning environment software open support c heuristic play play trajectory system key batch resource option nevertheless rely complete search scenario application decision paper plan finding select branch likely outcome branch proposal excellent suggest analogy role human behavior business activity make may rapidly change system importance business design adapt rapidly change work decision make rational rational recent heuristic year issue sensitive learn variety application total benefit effective call engineering decision seek achieve environment action influence thereby choice take indirect delay consequence planning decision I entity simulator role play game game quickly although establish paper game select target point entity interact avoid trajectory automatic carry different search literature nevertheless computationally especially environment decision rely exhaustive planning strategy heavily precise complete environment game scenario apply suited incremental mechanism action mechanism application strategy literature exhaustive cite author use sensor able use application reinforcement learning activity even education application policy student fit survey search branch branch branch selective previous novel planning module bad express action particular example distance goal architecture grid environment treat markov sequential decision work inform sequential problem go experimental devoted conclusion far go briefly combine heuristic solve agent homogeneous planning method plan current heuristic search encounter tree alternative node root tree max stop state action rest conventional heuristic save value design result algorithm agent reward reinforcement learn view asynchronous dynamic dp reinforcement act optimally markovian domain consequence without receive domain similarly td try consequence immediate one discount straightforward greedy choice parameter exploit value agent assume planning take produce interact approach planning path cause another function terminal execute r model ss r terminal plan new gain intensive available computational resource easily reinforcement planning experience model directly sometimes indirect involved plan next action reinforcement planning learning plan act rl planning rl step world record next reward sample pair conceptually plan rl agent propose planning strategy incorporate advantage shortest like environment g heuristic search branch branch successful selective incorporate heuristic make search reinforcement contrary consist environment bad priori bad reward lead priori trajectory simulate interesting behavior hypothesis human brain orient learn experience trajectory use strategy jump law thing analogous heuristic system grow system behave activity connection agent base aspect behavior relate motivation intrinsic publication
kernel see latent output strength entry old connection connection form network structure bayesian dynamic write conditioning understand output correlation couple condition covariance influence distance point point gps view amplitude mixture kernel entirely square motion kernel switch addition signal correlation volatility model gps influence account small factorization instance case dataset marginal dataset introduce signal node magnitude result fitting bayesian weight constant switch switch scale fix value predict value different approach mcmc w kernel compose function q diagonal since way order matrix weight use bayes incorporate gp volatility volatility volatility likelihood p p volatility minimal procedure component significantly change bayes use bayes mix poorly correlation node mix impose elliptical recent mcmc correlate joint costly numerically unstable highly w joint construct construct infinite generally heavy assess long take fit posterior kullback leibler variational computational nj nj f use deterministic pass vb form q f nj nj vb available ard however require take obtain message descent lower see straightforward contribution mainly take block covariance operation regression function evaluation volatility normally invert numerically unstable allow scale complexity vb dominate iteration function give covariance weight calculate weight mean per iteration vb reach overall account compare multi complexity volatility also new model compare variational vb chain inference keep comparison date task set account account change correlation volatility gp co krige output gene account dependent specifically volatility prediction make generalised wishart estimate suited exponential function function measure gene expression level hour hour replica gene activate protein focus least influence level factor output training replica replica average testing dataset alternative scale sparse accounting volatility volatility standardize small used replica replica training testing create similarly vb comparable outperform likely high indeed may mcmc mcmc level toolbox typical second km region access location wish predict standard correlated enhance location enhance function weight vb output correlation heavy figure generally around structure result learn spatially vary beneficial able predict dataset show three dimension learn marker use absolute mae predict experiment mae krige unclear log transform dimension unit skew include mark give mae gp cost accounting correlation moreover scale method run method intractable volatility dependent making incorporate generalise wishart correlation output volatility prediction ahead forecast historical understand financial follow exactly predict five process especially become benchmark assess make ahead forecast forecast predict compare generalise wishart original wishart table vb competitive even historical learnt value less meaningful encourage ahead forecast likelihood historical prediction covariance fully volatility outperform vb volatility perhaps suggest r vb mcmc vb mcmc mcmc mae vb vb co gp historical mse forecast mcmc empirical mcmc empirical gaussian process interpretable structure extension gaussian output heavy predictive scale scalable procedure empirical several benchmark structure easy gradually periodic hope thank valuable gene briefly process regression introduction define etc exponential display long trend length interpret learn useful determining past forecast covariance input covariance introduce velocity motion correspond time ar process markovian independent mat ern eq time model special recover many periodic period gibbs allow length scale datum noisy note gaussian integrating input regression predictive marginal amplitude noise section online book website length reduce mode posterior weight vb explicitly constrain representation find extension improve multivariate volatility weight propagation vb need straightforward analytically fit q
prevent class order versus technique successfully multi extend classification build expensive perform test member observation test classify entry extend binary label sign real return one positive value take sign vector entry indicate event I label prediction sensible since confident label observation affinity bag ensemble repository ensemble bag forest grow five label predict label use rule compare bag bag classification predict ensemble bag majority figure vary level predict easy easier distinguish ensemble suit versus consistent strong ensemble neither boost bag ensemble figure apparent rule ensemble apparent ensemble solver easier large percent correctly difficult classify l rule bag vote breast w htb approximate coefficient solve constrain advance magnitude fraction absolute direction change expense importance avoid prevent coefficient keep basic rule figure rule yield complex deeply subtle within depend close ensemble use tree complex far ensemble variable large tree tree tree variance rate general distribute change tree ensemble variety tree bottom also rule rate allow result method substantially use less ensemble chance phase subsample experiment take decrease subsample terminal lack difference enough use tree term build term less linear regression ensemble nearly would complex nonlinear significant predictive variable control many thresholded descent figure include range capability threshold decrease use experiment package return elastic net within regression elastic wise method nan cycle partial soft update modification also allow decrease exponentially meet increment warm prevent cause train inherent interaction subset clear solution prevent overfitte capture enough characteristic decrease generate experiment use coefficient method change find false twice negative probably negative rate experiment use iteration iteration minimize operation experience scale figure accuracy solution increase high coefficient experiment weighted sum leave control weighting minimizing importance become thus problem choose multiply extract central exploit property reach shrinkage spee many zero choose solution fix initialize expand contract user variant generate show figure roughly accurate solution generate solver thresholde dramatically behavior weight square reasonably build selection difficulty risk build different plot simple come tree use build capability coefficient figure successful correctly classify build attribute importance correctly important save expense indicate coefficient magnitude rule sift look consider rule order large table repetition attribute repetition vote influential repetition vote high rank form attribute available rule repetition indicate magnitude rule large contribution attribute look different order receive example rule order rule try set rule top rule decompose rule repetition influential attribute attribute repetition available rule repetition test repeat figure compare rate attribute number attribute model successfully rank attribute essence extra attribute act attribute accurate capability well fewer allow extra exclude datum collect save store vote influential try influential least small attribute train figure htb namely boost bag variety class ensemble extension method highlight ensemble well tree decision tree ensemble return ensemble relatively number like class build would separation make training rule control correctly capable produce finding difference coefficient method consider ideally solver would way important method indicate magnitude share similarity trivial reflect return return return rule method identify attribute contain method benefit return weighted rule rule rule important rule technique attribute error building model set traditional ensemble decision bagging provide weight relationship certain tree complex deep tree limit affect fall correlation simplify post processing capture rule need regression correlation exist computational ensemble matlab write reduce training portion coefficient grow ensemble substantial unnecessary quickly method machine computational ensemble solver programming implement language acknowledgement suggestion appendix method detail fast negative take coefficient negative constant minimize trivial second term square want derivative substitute gradient function rearrange switching update iteration coefficient take everything act predict scalar use gradient simple move proportional large negative gradient absolute length fully update affect update coefficient function update follow stepsize exceed step th substitute add turn subtract indicator turn negative david berkeley national laboratory road popular accurately predict class learner offer capability offer little insight ensemble accurately advantage indicate predict rule give multi bag tool classify high automate grow powerful successfully disadvantage interpret ideally like generate quick give insight understand attribute response dependency ensemble method fast base learner capture capture bag test refined dataset neither bag boost strong modify extend bagging boost attractive combine importance regression quick collective make learner build form grow node combine ensemble utility modify coefficient test wide ensemble class classification repository dataset ensemble look dimension large th predict class future unlabele belong binary two classify risk map predict label function develop problem consider risk train expect set combination learner coefficient risk minimize take solution many solution interpretable like possible prevent influential use impact control receive deal enable coefficient brief gradient tree th pseudo entry subsample memory iteration build note intermediate rule training pseudo unable multidimensional regression termination behavior capture control dependency build learner past build model need approximate construct distinguished approximate
parent east child anchor west east distance child circle child em em child parent child parent child node child parent parent child rectangle em parent child node edge parent parent anchor east anchor grow east em child distance node edge minimum child anchor east em child child right anchor east grow scale shape node write gamble final chance us branch decision strategy subtree compare gamble move branch cd consider cost cost insufficient decide cost option create unclear follow policy take predict involve normal decision easy check induction apply six gamble tree work know agree invoke relate terminology method provide theorem state coincide method yield strategy entirely nature take arc reduce form decision unlikely specify eliminate subset tree form reward entirely normal form set decision tree gamble early non empty event let follow equivalent satisfied fairly straightforward notation prove convenient normal gamble chance gamble gamble formalize method start find gamble gamble solution normal form formally definition strategy say calculate check whether optimal gamble xt tu tu tu tu operator popular reason wish even trivial decision tree least path child lot gamble impractical particularly avoid implementation backward property adapt traditional informally difference focus represent generalization gamble backward require clarity prefer rigorous induction operator move tree subtree set form remove retain gamble decision normal move next node root reach normal node call gamble empty gamble gamble unless optimality gamble affect non element attribute gamble call irrelevant state non gamble non gamble contain together check family empty gamble path appear frequently investigation axiom satisfy backward gamble let equivalent label satisfie section coherent proof strong let non gamble bx b satisfy empty gamble bx ax z immediate proposition consistent immediate proposition corollary decision fail property gamble contain envelope cc gamble clearly dominate dominate interval backward induction easily see consistent tt backward induction theorem property induction fail serious inferior backward induction gamble example subject decide provide site yield pay site optimistic minimum size west grow east level child circle node distance parent parent near edge parent edge parent child rectangle parent circle level child edge parent child node node child edge parent edge draw draw circle em parent child parent parent parent child parent child child node edge parent near node parent low table et incoherent correct ccc form decision uniquely identify gamble gamble notation baseline anchor grow transform node circle em child parent node parent node parent em child node edge child em node parent child distance right parent east child anchor shape circle pos child pos child edge parent parent east anchor east shape child node x x child em east anchor west east transform shape x depict backward chance node report follow outcome good marginal great x xt x eliminate dominate dominate calculation maximal maximal corollary differ al detail trivial give probability gamble backward induction gamble except computational benefit form knowledge assign coherent low set function plausible maximize solve apply list backward induction help solving differ cast result coherent fail interval surprisingly low property fail fail precise property conditioning independence want satisfy argue solution specified reach one acceptable subject mind advance carry function cause undesirable example subtree moreover always computation burden secondly gamble stage eventually even perhaps surprisingly normal exactly tree intermediate since coherent investigate uncertainty present gamble order path empty gamble empty I family empty gamble property property definition empty gamble therefore gamble gamble event empty gamble empty gamble finally satisfy property gamble property empty empty gamble lemma solution typical choice large tree find backward induction choice yield work backward induction instance classical decision maximize probability limited maker may handle include amount probability problem expect utility suggest systematically systematically decision criterion admit solution decision problem induction main contribution backward induction form option uncertain consequence option base subject seek represent maximize advance choose maximize utility specification fortunately induction utility previously reach backward coincide probability form criterion strategy generalize backward induction summarize replace move strategy go de idea differ many expand low find reason first mention feasible induction eliminate strategy hence far efficient secondly might paper present formally define characterize discuss large conclude low section informally event reward root chance node leave grow happen last either also cost weather leave outcome predict figure size em parent east anchor east draw circle child em child em child child level child node edge parent parent parent child rectangle em em child node node child parent draw parent node parent edge node child node level edge parent node node edge child em child parent distance em child parent child parent edge child draw circle em child child parent node parent edge level child em node edge parent edge node child draw distance child node child edge parent edge parent assess utility next solution demonstrate backward allow identify instance state element subset call capital arcs chance function uncertain reward state yield reward whose sum finite express close assumption whole expectation empty mean probability obtaining follow
decision incorporate validation answer systematically find address prediction reproduce model validation inform calibration rely understand limitation maker quantify failure satisfie maker requirement framework quantity assess become necessary activity decision computer state validation accurately capture behavior uncertainty effect quantify correctly approach partition next calibration calibrate predict small discrepancy improve principle incorporate goal detail specifically examine situation value available scenario impractical still want incomplete directly example situation computational aim characteristic wind nuclear assess environmental impact failure predictive systematic call validation suggest hierarchy validate phase update pyramid ability use high maintain author split experimental validation pyramid experiment calibration rigorous propose partition datum calibration average split reference instead way disjoint calibration choose analyze split choose term split optimal find validity replicate assess reproduce quantity interest information must produce satisfactory second fundamental issue validation set challenge model confident prediction concept mind satisfy inform calibration reproduce data validation quantity able answer work receive intensity intensity decision reliable explicitly step algorithm section reduction iv short h demonstrate quantity interest updating prediction make update newly assess predictive capability drive require metric calibration decision may perform suitable tolerance quantify ideally metric provide easy metric percent nominal measure determine tolerance procedure flexible choice threshold choice application inference probabilistic input function briefly capable reproduce work mutually suitable demonstrate inform inverse metric analyst system develop partition split admissible inverse problem impossible area improvement approach subject future treat calibration obtain depend could evaluate calibration detail visualize metric model measure consider split threshold ability reproduce must change perhaps next optimal highlighted figure assess ability predict partition threshold fail threshold use make prediction tolerance decision maker conclude valid demonstrate may perform metric maker partition stress procedure extremely advantage allow range specific reduction assess prediction solve markov chain monte sample equip hybrid briefly next present result camera gate camera count gate gate linear regime predict raw receive gate range micro second quantity interest processing gate require data nonlinearity gate introduce gate width incorrectly rate camera respectively calibrate problem algorithm mention unit horizontal cutoff tail cdf datum include correspond percent characterize value gate include calibration validation use datum tolerance gate leave set fact point provide gate keep single unit inverse prior update compute result requirement find plot show predict thus look represent first split form receive micro second might optimal split contain low gate close calibration
ny sf sf tf sf consider I random ny ny z apply keep notation use without b kt k h k n kt h b h kt k h h h r k h rely finish proof ny computing functions unconditional characteristic nn z independence bootstrap consider value account mapping theorem argument recognize show case q follow similar pick account eq hand know eq q argument similar least last identity argument arbitrarily similar manner brevity omit proceed write similarly q strong prove vi proceed prove consider square envelope maximal inequality consideration subsequence display happen surely compact k inequality hold subsequence mean different constructing interval smooth asymptotic argue bootstrap confidence bootstrap remarkable study procedure show consistent also condition procedure external influence encounter every science relation finite point simplicity compound crucially among nuisance page build bootstrap nuisance generally scheme build employ illustrate issue arise smooth nonparametric regression therein bootstrap asymptotically valid inverting bootstrap bootstrap use form estimate slightly suggest complete validity go cite evidence suggest nonparametric bootstrappe residual bootstrap build certain process suggest absence weak limit addition smooth approximation residual finite superiority smoothed procedure triangular validate achieve first argument version develop conclusion broad function specify work nonparametric procedure ci neighborhood change jump stage rely approximate ci immediately organize scheme notion section prove generalize constitute section residual bootstrap show consequence general point proof sequence parameter convention suppose concave leave third piecewise w interval maximizer small stand small maximizer nm property square estimator page n two process way distribution brief review function usually root break simple generate nh nr say weakly consistent weakly strongly choice mostly essential smoothness expect perform phenomenon root x follow property bootstrap draw bootstrap scheme bootstrapping widely residual n predictor response z smoothed density successful scheme choose nonparametric n j n j bootstrap usual bootstrap regular number z n available satisfactory solution vary framework convergence prove scheme procedure give theoretical evidence scheme widely page typical see failure situation investigate estimation situation give rise asymptotic compound poisson asymptotic distribution estimator independent dominate procedure usual problem crucially depend generalize page triangular array array k throughout operator constitute bivariate nm estimator k nm convergence consistency result estimator assumption iv regularity property like fact hold independent easily converge uniformly calculus aid theorem page v proposition sufficient measure triangular achieve would highlight satisfy suitable assumption z n vi q converge two independent sided compound continuous limit element compact terminology leave limit topology topology q endow metric become subspace page value refer reader difference prove easy iv find tight tight compact rectangle sequence process weakly mind definition z continuous poisson give mapping theorem lemma weak weak jump piecewise continuous integer jump compact see apply first show define h square notation eq argue scheme residual definition note I state sequel strongly refer second let interval z still n n n side probability translate proposition k prove estimator conditionally surely bootstrap vc hold also lemma recall process compact rectangle iv bootstrap scheme follow lemma evident situation know hence therefore think convergence measure conditional distribution say mean measure characteristic fail limit hold converge probability n two ny measurable random moment ny aid able main compact neither weak probability weak enough pick observe e h converge weakly probability weakly eq converge probability note imply hence e limit probability latter happen iv nh characteristic probability weak weak make unlikely rigorous depend note conduct depend argument illustrate bootstrap approach unconditional centering process kk n k small inconsistent random sequence I random addition mutually jump I e nh bootstrap analytic compute approximate limit two distribution immediately random resampling arise bootstrappe fix inconsistent speaking resampling fail basic residual compute bootstrap conditionally surely strong argument bootstrap scheme introduce give start center empirical variation almost smoothed bootstrap result prove converge compact r nm define paragraph minimizer seem complete statement limit prove bootstrap scheme bootstrapping achieve proposition regularity bootstrap smoothed want fulfil procedure estimator construct choice kernel see bootstrap random next scheme turn hold probability scheme framework number condition remark subsequence p pair size take approximate bootstrap provide coverage length scheme make estimator kernel density kernel reference bootstrap refer fix bootstrapping residual scheme c coverage avg length coverage avg smoothed avg coverage avg c coverage avg length coverage avg length coverage length smoothed c scheme coverage length avg smooth fdr outperform coverage increase length fdr bootstrap procedure inconsistent obtain scheme single smoothed right panel top histogram indistinguishable guarantee efficiency choose tuning smoothed also require smooth insensitive bandwidth certainly asymptotic right bottom bottom broad change two unknown twice w least problem least bootstrap bootstrap produce build interval contain empirical replicate w iw iy j j j smoothed procedure must difficult method adapt case additive form smooth like helpful comment state characterize kk third let w next maximizer small topology converge topology open ball radius maximizer belong contain rectangle function observe jump every pure jump functional statement jump suppose jump h k express jump jump topology wise continuity page integer see large thus finitely jump happen finitely jump correspond jump jk supremum concavity unique supremum maximum write statement topology lie c proof imply second inequality sequence consequently imply n account lemma aid proof vc subgraph envelope vc index satisfy envelope sample probability notation lemma page z vc index class iv z also replace z z z z z z ny iii ny inequalities ny n ny z z n ny b similar iv lemma prove rearrange term nz z n decompose n n unique maximizer page theorem prove nm take iv make inequality consider constant expression st rd note add display get note expansion admit n lemma enough precede consecutive expansion see take define bind maximal imply see inequality z n expression n completely tight j ne imply since function z vector tight process agree number choose least jump jump form vi happen exact z j tight fourth tight uniformly show distribution end consider linear grouping rewrite imply note ne n z imply follow argument ne vi write characteristic put arbitrarily device prove
track node individually internal environment external include noticed believe indicate also world consider point choose extent share political b change political role choose em measure influential reflect opposite initialize assign validation degree learn influence sort influential area research criterion include since comparison ranking impossible study influential influential list compatibility matrix em diagonal intuition share learn one direct seem political year comparison year voting record year turn highly regard historical trend political increase strong every unbalanced entire majority comparison step agree show increase window measure quantitative highlight correctly eight term present co evolve mixed membership blockmodel model able reproduce hide able detect record different world explicitly pair result complexity network approach introduce greatly enhance efficiency foundation grant grant nf grant fa dynamical structure model co evolve network specifically membership blockmodel probability observe two membership vector membership evolve network change variational real social important scalable modeling world network inherently dynamical system attribute network change often mechanism node illustrative social influence mean interact evolution neighbor properly characterize selection subject extensive suggest continuous time agent base network evolution agent characterize depend well local evolve markovian choose select maximize serious missing attribute observable network grain suitable dynamical model blockmodel extensively blockmodel block node assignment influence relationship account role co evolve blockmodel mixed membership modeling suggest impose assume membership drive parametrize membership aggregate mean trajectory separately describe track whole shift membership experience change political experimental membership node form connect process induce sequence normal lead tractable equation compare dirichlet prior update step q qp influence vector p account sample role indicator multinomial tp bernoulli kb rs node role interaction account node role influence benefit node specific describe easily influence conversely node co evolve datum p write simplify pair describe resort parameter eq q factorize variational distribution multinomial compute problematic upper variational em iterate calculate expectation variational updating locally maximize tb guess initialize variational minimize take variational normalize eqs couple covariance similarly generally depend q equation simple iteration process converge compute expect compatibility block compatibility compare evolution dependence use
hybrid monte carlo hmc algorithmic detail broad exhaustive relevance application body derive relevance ard iterative optimization nice ard variational sparse cca relevance real method exponential yield product paper relevance ica blind deconvolution develop also basis pursuit engineering pursuit useful uncertainty spike sparsity establish statistic describe gaussian type model hierarchical uncertainty necessarily spike network unsupervise unseen appear handle miss observe miss create test select case predictive nlp create data metric aside benchmark type add image probability common optimisation optimisation need demand execute many combination approach separate make sensible quick overall learn setting variable demonstrate tradeoff run spike iteration well reconstruction human budget spike use figure well reconstruction various poor sparsity show spike compare induce shrinkage relevance problematic certain reconstruction restrict contribute difficulty set regime ep issue hybrid ideal leaving demonstrate consider accurate reconstruction paradigm broad develop generalise provide sampling importantly comparison unsupervise norm demonstrate diverse application model sparse prominent wide modelling advance cifar nsf fellowship unsupervise diverse area signal collaborative work method performance infer unsupervised latent variant induce bayesian factor accounting principled unnecessary shrinkage practice outperform budget need assess care unseen last decade parsimonious significant significance tie scene sense importance efficacy sparsity among provable relate optimality property increasingly diverse application norm behaviour compete intractable norm closely match prior spike spike spike similar impose penalty prominent vast modelling completion amongst toolbox real underlie observe gaussian unsupervise desirable underlying introducing framework develop comparative contribution generalise strong sparsity provide sparse sect mcmc efficient naive sampler sect present comparison optimisation spike bring benchmark sect interestingly strong outperform concerned search factor weight datum often isotropic covariance model explain explain datum increasingly deal heterogeneous consider probability shorthand exponential family distribution natural parameter subspace take distribution belong prior notation conjugate item notation distribution form row generally family latent recover family correspond consider latent variable indicate highly heavy tail gamma laplace mixture encourage notion mm parameter none element zero exactly spike place suit achieve learn average rather search good use continuous mass enforce place penalty thus binary latent dimension contribute bernoulli spike form component specification denote applicable sequentially variable step slice decide contribute integrated pz nk nk exclude slice evaluate computing q integral family approximated integration laplace bias approximation laplace latent behave show require definite information matrix boundary parameter latent alternate integrating describe result fast reversible specification indicator e slice thought proceed sampling slice randomly draw interval relationship gibbs full easily derive encode connection density appeal optimisation broad provable exact recovery rip optimisation linear semi definite sparse laplace norm generalise modelling n negative use framework control latent variable loss unsupervise
plus minus sampling estimator n certainly since expect numerical approximation bias true occur bias move actually fall another suggest idea gain employ estimate dramatically impact respect bias shift design bias overall chemical spatially homogeneous dimensional frequently leave couple ordinary specie specie specie molecular heat capacity pressure production rate define elementary reaction rate side reaction reverse th reaction activation k change standard pressure reaction mass express compactly subscript ratio fraction specie perfect pressure experiment consume expensive conduct design base employ statistical foundation inference indirect mechanism incorporate source construct measure expect monte gain stochastic approximation feasible intensive setting algorithm demonstrate model parameter inference detailed quantification bayesian shannon chemical role physical example knowledge discriminate compete observation laboratory datum expensive even consume perform experiment dramatically model design question experimental question receive community application depend criterion experimental design write functional include optimality minimize maximum variance instance utility shannon optimality may derive optimality counterpart nonlinear exact evaluation optimal design tractable criterion obtain impose additional model approximation lead involve locally require select unknown model maximize fisher though broad significantly normality assumption prefer rigorous theoretic criterion throughout seminal gain information provide justified maximum shannon range criterion introduce gain material kernel equivalent chemical criterion design design limitation expense design space infeasible gain require discriminate design application simple design objective chemical criterion objective strategy suitable broad theoretic formulation propose curve wherein intuition utility surface space markov chain combine integration idea extend simulated annealing expect utility metric direct information enumeration consider design space information via coupling criterion open issue realistic advance state art yield design formalism assumption parameter inference shannon gain criterion naturally incorporate parameter probabilistic relationship experimental need generality computationally tractable capture dependence nonlinearity dimension quadrature exploit parameter dimension link approximation result objective experiment design rigorously figure component embed design cycle upper box focus experimental formulate construction surrogate intensive section come experiment tool chemical experimental reflect relevant consideration specify objective objective reflect infer appropriate would reflect discriminate consideration motivate notion course expect utility develop goal add cost broadly resource interest cost constraint formulate offer inference noisy indirect incorporate constraint heterogeneous information assessment natural decision exploit discussion approach bayesian paradigm treat probability space measure I measure endowed appropriate hence uncertain observe realization datum change give bayes evidence reasonable vary simplification experimental design form utility support function reflect usefulness value precise choice utility root put kl generic support difference carry unit inference note utility involve internal represent simplify equation yield prior decrease amount informative maximize parameter design apply optimality maximize allow multiple simultaneously well repeat gain experiment equal experiment incorporate likelihood give carry condition expect information gain simultaneously objective often experimental would always single carry experimental e help next approach design experiment approach necessarily design programming sequential experimental design least design due gain must numerically rewrite inside special shannon drop maximize sampling retain equation accommodate example likelihood error carlo e evaluate analytical yet carlo sum bias estimator proportional control control sample see monte outer inner monte produce zero small contribute bias effect numerical bias value grid clearly exponentially monte carlo objective available na I optimization expensive effectively monte evaluate thus suit function simultaneous perturbation stochastic nonlinear simplex stochastic receive estimate two perturbation estimate regardless problem dimension recommend value inverse gradient justification gradient average proof randomness difference allow position feasible perturbation nearest omit discussion magnitude gradient minor modification improve noisy function simply project point near base take objective require step finite however level slow constraint false approximately optimization pose sense expect utility stochastic optimization complex embed equation evaluate repeatedly value task expensive enter function discrepancy leave drawing evaluate calculation replace support entire design option reduction polynomial expansion latter exploit output uncertain extensive define space field generate measure random variable measurable infinite multi expansion coefficient orthogonal distribution expansion convergent purpose infinite dimension truncate order truncation influence degree nonlinearity relationship freedom availability pc output depend impractical depend suggestion proceed increase dimension put component simplicity affine transformation uniquely associate vector inside hyper uniform convergent expansion pc coefficient alternative difficulty step strongly character equation prohibitive arbitrary expansion project quantity interest function deterministic black box need flexibility functional may depend smoothly orthogonality pc coefficient simply pc index analytical expression quadrature model term projection non forward essential numerator equation evaluation regularity dependence quadrature sparse quadrature quadrature rule make formulation quadrature quadrature take advantage quadrature rule overall use quadrature cc especially nest weight require little difference formula difference quadrature rule using plot span space maxima appear pattern understand examine noise batch adopt respectively single parameter symmetry contour identical second experiment locally design intuitively slope output instead slope great examine restrict prior experiment figure design surprising winner explain combination design yield note optima original could experimental essential role refinement chemical indirect relevant develop model regime new great demonstrate design framework behind reflect sharp rise pressure suitable delay delay carry process experiment describe spatially pressure chemical ordinary individual chemical detail subset reaction associate reaction mechanism chemical nonlinear variable consist temperature specie mass equivalence production depend factor energy reaction table reaction equation help open chemical software formula parameter reaction branching lead net species reaction reaction relevant help target range nominal positivity easily appropriate temperature expect equation maximize choice state state affect delay due front state either dependence desirable smoothly goal easy release intermediate chemical specie peak heat release characteristic actual implementation several order magnitude function design ode deviation error value constant resolution technology note depend implicitly influential expect approach different ode system practice construction evaluation worth total evaluation optimization expect exceed surrogate tradeoff gain input support expansion polynomial expansion projection pc numerator expansion truncate stop degree expansion examine section accuracy rigorously observable ode pc affine pc expansion expensive minimize evaluate use level isotropic sparse quadrature contain show contour dominate exponential observe smooth ideally contour plot accuracy gain difficult contour use ode estimator pc surrogate contour pc less variability due importantly illustrate since design reflect large ode datum perform contour ode surrogate posterior full pc utility design informative three ode precisely model ode exactly generate noise use two second value surrogate remark characteristic informative peak greatly influence even observable full observable force mode observation selection make argument lead designing experiment infer unchanged efficiency pc dimensional grid entirely introduce objective might objective optimum noise existence non negligible balancing understand impact scheme fix maintain loop set optimization evaluation I noisy evaluation perform plot run cross red pair design choose design result less determined influential group final figure indicate case quickly affect evaluation take evaluation shape utility observation far assessment history utility final really end histogram utility final point negligible employ histogram indicate persistence create support bad good design study term quickly reach factorial experimental list design lie experiment use design much well experiment factorial factorial pick corner good experiment factorial would exponentially become impractical quickly experimental capable produce quadrature high quality gain must quality similar pc position history final fact appear even tight surrogate practical reach require ode speedup order full experimental sufficiently experiment posterior contour involve average broadly explore range sensitive optimal experimental issue polynomial experimental employ thousand million combine computational surrogate construct polynomial perform optimal reach construct pc inference expansion capture thus smaller fewer expensive pc easily paper systematic tool design incorporate intensive rigorous theoretic criterion simplify give show cycle experimental criterion prior posterior couple stochastic optimization would otherwise prohibitive accelerate flexible demonstrate experiment nonlinear delay magnitude illustrate design information investigate utility scheme overall experiment informative
go reason decompose batch observe q nc properly scale onto clean part involve utilize analysis validation crucial handle vanish fraction observation clean entry condition simultaneously equality b construction inequality observe step uv te tw four tw uv te tw uv te f uv te uv te write eq ii first use hold second argument thus utilize sign similar self hoeffding te event k high probability k polynomially different prove inequality use ii use separately term I use assumption fail dependence use though random subset un corrupt independence operator term define k ie type bound lemma term turn sign bind need distinguish case term definition expand product term k first type term q q np use theorem collect five give w apply part twice p apply later sphere property operator xy xy I hoeffde inequality xy nd make exponentially union summing sum together e therefore inequality complete line write sufficient optimality convex impose condition indeed optimum part way devise less frobenius repeat time decrease jump phase go increase probability adversarial success change experiment predict deterministic rank corruption adversarial lie hard recover adversarial spread bernoulli possible go deterministic predict grow linearly bernstein useful sequel later let l n away follow te ie proof one simple lemma since incoherence f f triangular fact index np lemma sum zero bound small moment random z ie te ie ij ij te ie ie te ie te ie ie te ie ie ie f ie apply part variable index index satisfy p ie z p set fix j te p te ie np te ie z te te np next lemma te eq column repeat desire condition te three incoherence entry np choose cn similarly side finally iw assumption finally ce independent sign decompose ce b separately te b te express e e te e bound argument te nc te term lemma inequality te b te te te f np side c complete low contain entry observe location unified minimize rank succeed allow component one hand corollary single way work completion hand rank analysis dimensionality reduction area pca collaborative either correct simultaneous presence fraction corrupt recognize fail fail even corrupt light studied fully set guarantee support wise strong present theorem presence scale corollary rank logarithm provide exist provide guarantee support case vanish grow application fraction error assumption recover work allow vanish observation deterministic work locate besides improve proof high neither location constant allow fraction program notation value intuitively act norm surrogate sparsity specify early appear incoherence optimum I impossible identify add prevent scenario approach incoherence proper subsection guarantee adversarial error recover constant quantity set additionally observe entry generate entry column unify universal solution nn conclusion treat treat miss apply weak result study refined manuscript handle case randomly locate provide strong allow small publication conference also random observation good exist completion adversarial prominent adversarial recovery improve sign sign constant equal nn fast agree intuition easy error location sign arbitrarily provide strong require fraction usually quantity dependent deal error e unobserve correspondingly tradeoff improvement provide recovery satisfy satisfied result guarantee matrix moreover another improvement square tight manuscript close completion paper prove unified line low matrix simultaneous deterministic dual proceeding additional notation clean clean element write entry span share projection complement orthogonal sequel might simplicity case square proof five elaborate next five section suffice assume sign arbitrary sign error obey
introduce rw among neighbor node loop move move essentially transition high degree bias rw recently exploit network contrast graph visit terminate collect traversal seed explore visit seed similar except iteration explore seed ff flip coin probability decide ff already forest grow name accord classic name neighbor visit visit hide population drug social survey comment l node fraction cover depend range regular graph concentrated around graph balance skewed several decade www internet ip level give graph graph arbitrary capture approximate topology well classic give match fig ignore rectangular left random run high respectively relevant particular rw serve reference point analysis walk excellent converge equilibrium degree calculation true connect average degree show transition show consequently rw system analyze chain crucial dependency handle adopt elegant connect bias comment work c real interval next node index match plain uniformly two match degree sequence loop begin traversal technique node compatible model queue keep discover yet initial yet discover nonempty discover remove depend scheduling implement schedule last forest line never never edge discover uniformly grow exponentially construction execute uniformly defer theoretically equation bias number sum depend iteration process determine search choose trick dependence tractable ready degree choose independently vertex sample time expect degree normalize eq unfortunately interpret proportional neither match edge discover goal express calculating expect visit although numerically rewrite distribution fraction node expect degree describe bias equation insight nature graph traversal technique ff etc discover select degree therefore sampling simplify process traversal equivalent rw th select node procedure follow concentrated imply say x ff short biased number configuration process stop cover possible efficiently graph previous expect degree biased node potentially arbitrary try estimate node straightforward node rw proportional unbiased proportion simplify bias leave node densely purely easy inherently bias affected explanation likely wave surprisingly connect walk rw distribution fix graph regardless topological recall analysis approximation network property technique graph evaluate approach range life fully internet topology graph average q real node degree well difference visible topology fully capture rw simple alternative coincide rw systematically degree node description matter email large european facebook facebook wikipedia network com web google c correct rw circle node line correct plain line rw average thick analytical line calculate bottom correct eq precisely degree distribution c k facebook fraction average fig facebook seed seed full degree facebook analytical guess explain life fully facebook implement facebook follow rw es node time collection insight collect describe facebook sampling observe k consist relatively indeed start finally short compare facebook reason drop yield almost identical methodology collect describe entire fig facebook underlie connect nevertheless k approximate degree collect report report agreement subject bias relatively facebook rw bias correction apply particular life internet estimator fortunately arbitrary far parameter interestingly unfortunately variance section incomplete traversal start follow uniquely show appropriately requirement account calculate mean whose value order ball size around sampling technique obtain various topology sample extend trivial sample stage half feasible radius try approach k appropriate calculate half improve arbitrary topology feasibility sample translate difference summarize arbitrary estimator huge life topology concentrate precision trade terminology arbitrary whereas inaccurate correction rmse topology email topology facebook topology web google arbitrary topology average seed recommend rw arbitrary topology assume practically rw contrast rw diameter short case attractive plausible carefully affect example drop grow good restrict community component collect full facebook completely www domain recommend introduce metric particular bias random walk monotonically analysis base procedure bias technique broad ready implementation calculate diameter acknowledgment discussion unbiased facebook proposition corollary widely large internet topology advantage random walk plausible observe incomplete biased observe degree traversal technique forest bias procedure capture life base internet topology evaluate demonstrate unbiased internet topology community focus internet topology ip connectivity web online network restriction entire graph impossible topology facebook likely impractical collect representative sample particularly interested naturally www operation category walk category include classic walk rw walk web graph walk bias rw collect rather topology c sample walk rw traversal technique calculate average rw bias rw increase complete sampling traversal technique lead shape degree calculate category node visit completion graph vary visit include search ff sampling popular widely internet topology www use popularity plausible path coefficient walk course seem lattice small lattice unfortunately fail high node confirm facebook degree topological despite popularity bias relatively formal challenging sample node mathematically characteristic main contribution function node describe second propose bias input collect cover arbitrary common graph technique perform broad facebook implementation finding propose correction arbitrary although characterize scope life topology restrict attention self unweighted social link dynamically interaction graph scope bias correction procedure design analytical diameter future outline follow work traversal study briefly paper evaluate introduce correction practical sampling recommendation section explore et social many et interaction facebook constrain facebook facebook rw bias observe facebook motivated field social sampling
view imply laboratory major source information wikipedia serve role may subtle behavioral amount page cc score account people try potential idea easily change behavior become behavior status stable suggest work well community become nearly subject aggregate create strong biased wikipedia formulate form persistent attention power page user wikipedia ill attention wikipedia systematic quantitative investigation nevertheless suggest serious publication medium camera accuracy middle east report ask keep become anti member goal significant wikipedia page prominent retrieve far camera wikipedia may influential source break wikipedia wikipedia even retrieve addition difficulty belief even subtle outcome enforce wikipedia neutral viewpoint viewpoint leave viewpoint vice versa nothing page worth sensitive small wikipedia page page mathematic generate page deal quantitative evidence behavior paper present examine behavior page work assign page identify article fraction minor page attention focus page score particularly assess behavior page especially page broadly across article party past clustered cc score account similarity page user cc particularly focus user act interest behavioral interaction heavy equally validate analyze simple score also tail public topic large cc become status status cc request find focus become population fail well sense score pre low indicate fail actually significantly wikipedia introduce behavioral whether wikipedia historical determine use investigation behavioral rely beyond wikipedia discover focus behavioral status wikipedia behave change upon suggest wikipedia scientific far identify difficult incorporate deal user focus extremely user unlikely automate detection wikipedia candidate choose stand go request user stand late user finally article begin discuss describe cluster page wikipedia article page article unless proportion article fraction minor page page anonymous quantity mean article score transform produce page one manual user page page expect page remove page measure hypothesis concentrate article go able focus entirely broadly number party rest across interesting far interesting concentration could parameter change correct topic concentration article user cluster page page page page page category similar incoming link user page base divide cardinality intersection equal category consider page entire like concentrated average sigmoid condition prevent final reason measure measuring relate ranking extent concentrate tool raw cluster score page page page wise page say fraction broadly wikipedia cc quantity compute wide email message processing similarity evaluate validity provide measure wikipedia goal gain wikipedia score use amongst camera people wish point wikipedia try look cc contribute behavioral metric quantify use perhaps modification behavior deep second wikipedia evolve along
compose choose cause aspect classical cubic order smooth curve able take region monitoring activity make difficult stationarity trend first smoothed curve covariate degree spatially evaluate variability structure input theoretical fit case functional demonstrate characteristic h paris p des j spatially functional technical www co file pdf r krige spatial publication h random ed rand evaluation di spatio york spatially functional environmental di universit functional em pt environmental spatially correlate analyze spatial find group similar center optimize good representative propose evaluate study simulated feature environmental environmental landscape science daily environmental phenomenon area explanatory essentially analysis perform functional contribution tool datum focus spatially knowledge hierarchical group dissimilarity functional two alternative univariate functional datum characteristic approach kind define alternatively suitable al method spatially dependent achieve krige spatio prototype cluster minimize spatial variability krige curve get curve minima local krige propose krige base definition site representative several attempt discover linear regression two spatially organization relation residual instead dispersion unknown decompose semi large scale spatial variability although handle densely domain rigorous present difficulty belong environmental area spatially prototype optimize group function variability issue procedure environmental application stationary model accurately paper organize spatial illustrate real spatially measurement observation dimensional et distinguished marked point focus functional site spatially dr compact tt site nh nh small empirical involve simplify expand coefficient function functional obtain consider tt tt dt tt gram identity spline basis integration distance due empirical ordinary ol weight validity spherical domain change within sensor sensor potential error describe spatial variability component center lag estimate nh nh average center estimation center express manner functional clustering find characteristic find representative name fitting formally find partition cluster set cluster follow cluster representation measure prototype represent heterogeneity dissimilarity goodness perform allow element concern tt ts spatially locate partition spatial optimize center center dependence curve site lag evaluate membership mention random reach curve ordinary function compute allocate variability consider location pair separate spatial distance functional variability partition allocation rule tendency spatially correlate especially consistency allocation criterion criterion verify base euclidean cluster prototype optimize study first spatio functional random tt ts scope test performance procedure detect spatio structure largely separable purely purely temporal apart schema control temporal scale parameter six make locate field belong three cluster include spatially
cluster exceed proof proposition modify fashion word cluster go exponentially consistency see surely decision incoming signal precision determine value level determination error nan hypothesis accord plain simulate critical region triangular lattice consider level table critical c region boundary region simulate site display h cc low fact ii p p denote configuration associate detect cluster determine nan level conservative implementation picture depth thresholded signal find cluster detect reject statement terminate linear standard distribute give signal correct site given help algorithm threshold follow way perform repeat reject advance make uncertainty finite simulated cluster correspond optimal separate visible noise however signal may infer type sense threshold author like thank discussion appendix explain asymptotic nx let tend infinity definition phone method noisy detect shape presence boundary shape impose weak object interior algorithm linear formalism explore important theory addition behavior finite noisy digital pixel reconstruction quick mathematical look ask diverse automate detection picture run intensive image reconstruction picture majority skip start impose permit detection nonsmooth object especially one highly usually field automate material regular shape advantageous image require reconstruction analysis performance drawback subject wavelet paper nonparametric hypothesis completely necessarily smooth continuous nonparametric term exponentially pixel algorithm stop stop paper noisy focus image serious limitation affect object object detect picture colour boundary differ colour moreover applicable simple rescaling colour organize detail statistical power small subsection critical size present devoted proof auxiliary theory consist connect site site neighbor site bernoulli mild graph qualitative behaviour phase connect finite qualitative size connect site site precise size order infer intuitively regime quite sharp locate critical deal mostly discussion triangular sketch emphasize choice lattice discretize decide six denote think pixel discretize indicate denote indicator subset give variable noise noise signal signal signal paper detection statement hypothesis alternative statement thresholded terminology site thresholded site configuration complement choose critical lattice paper hope cause statement intuitive picture reasonably large equal example triangular subsequent mild condition arrange point approach state make qualitatively formation accord description immediate arise threshold discretized picture say site question super threshold determine detection however choose lattice properly lattice universal satisfying concern start condition entire remainder degenerate critical degeneracy simple degeneracy always probabilitie signal time site white obvious reason pixel site give address favorable black one support inside c free crucial observation let symmetric white first degeneracy consideration assume except degeneracy symmetry round infinitely triangular lattice complex additive root unity eq thus please hold triangular rely assumption logarithmic change quite critical triangular particularly locate site triangular lattice cause packing unit discretization choose ask obtain triangular lattice describe bound type ii mild shape completeness refer attempt triangular lattice lattice monotonicity lattice resolution right statement result please mean make necessarily cluster setup lattice site contain detect subset equation collection inconsistent want suitably depend connect weak contain site ng type contain square type consistent estimator estimator large issue detect correctly noise high fact precise explain size large begin infinite lattice depend contain triangular lattice consequence mean know define introduce ordering say analogously inequality read follow inequality decrease estimate cause lattice precise among configuration mark black depend distribution
flow diagram steady reach extraction pca respective euclidean histogram histogram obtain respect dynamical structure estimate observe orientation eigenvalue provide reference eigenvector adopt subsequent e rich though focus regular much attention drive integrate unfold network model characterize relationship respective dynamic remarkable application methodology dimensional pca integrate unfold average integrate realization uniformly steady axis observe frequency colored degree result along example respective signal interestingly time term pca signal maximum frequency fire instant move cycle along frequency cycle harmonic dynamic weak coupling depict projection also color illustrate significance value simulation reference integrate dynamic histogram log integrate gray fit log distribution significance level confidence specific illustrate methodology strongly version complementary figure histogram histogram figure nan couple simulation strongly couple strongly understand versus dynamic sis map topological separate shape histogram measurement two group signal eigenvalue centrality three measurement tend interior shape network da grateful financial support support grateful author thank cluster work author box reading focus characterize overall average account distinct properly represent identify aspect influence work integrate sis identification highly dynamic sense present network way compose interact effectively map dynamical extract typically quantify topology geometry g angle focus try structural provide real world system grow devoted dynamic global investigation focus aim address comprehensive systematic step principal pca important dynamic unfold probe structural organized box henceforth consecutive order characterize completely uncorrelated optimally align along direction variation principal axis focus attention dynamical relevant aspect signal influence aspect measurement index neighbor markovian assume therefore depend three function topology signal network simulation dynamical feature estimate network node node introduce define node significantly dynamical understand structural dynamical predefine density resort system namely sis box steady enyi er investigate neuron integrate fire node record pca project onto projection color remarkably fig average entropy pca projection supplementary main densely dense complex observe peak low suggest dynamic order sis use investigate node figure f pca sis map color respective plane result simple projection see dynamic yield signal tend supplementary illustrate significance nan condition integrate density small influence f k greatly affect integrate sis remarkably vary result sis dynamic mostly affect strong coupling dynamic result network identify methodology stronger verify coupling obtain entropy explain
generalization ds offer powerful aggregate uncertainty interval structure vice versa discretization propagate propagation build expansion introduce intuition response extract homogeneous represent generalize scheme characterize beta compact polynomial mathematically attractive separate polynomial orthogonal polynomial particularly characterize uncertainty dynamical represent ordinary equation uncertain solve polynomial transform bernstein range bernstein response optimality interval demonstrate interval nonlinear function algebraic challenge investigate ds interval conclusion future discuss primitive evidence represent give set understand evidence evidence member define interval structure world quantify amount amount complement measure argument frame combination aggregation mass due derive belief aggregated formula mix reliability source xx represent cumulative box induce box uniquely structure exist body n cumulative plausibility similarly complementary plausibility thus upper cumulative plot h evidence b assignment find propagation solve develop conservative evidence automatically evidence problem propose bernstein polynomial ref arithmetic inaccurate present index transformation power bernstein th bernstein bi polynomial ref box transform intermediate operation power give bernstein bernstein bernstein expansion range form estimate experimentally ref bernstein small univariate tight choose minimum coefficient sub efficient range computation vertex get structure interval interval element body way response prove concept select algebraic challenge literature use ref concern correspond ht obtained aggregate source use ds basic assignment polynomial expansion total great numerically quadrature bernstein provide genetic ga box bold indicate small large upper genetic provide ds table wide box represent narrow box bound return line interval ht fig interval properly include sum properly interval intersect region box ref give along region reason due large low interval quantification polynomial random support way find basis exploit map transform expansion bernstein form dependency accuracy propagation proposed investigate et propagate obtain basis build bound function polynomial expansion intuition expansion discard information provide randomness range bernstein form property bernstein polynomial propagate nonlinear algebraic
c rectangle pi north south north document description north pi edge pi w direct model dirichlet allocation identical construct generative vector dirichlet value mean potentially evaluate deviation proportion word topic everything part correspond lda part challenging pass connection share everything include estimate topic use analogous cite accumulate gaussian likelihood connection main introduction derive broadly speak inference collapse form belief benefit opt lda factorize entail loose slow integrated since form topic integrate per adaptation brevity construct given assign dirichlet q divide dirichlet dirichlet distribution approximately independent approximate graph gaussian latent product independent py h mean marginal posterior gaussian q writing message inference contain particular without change generalise hyperparameter noise likelihood evidence amount pf f separate evaluation less evidence numerically stable optimisation straightforward algebra identity ht leave softmax basis approximation correspond laplace mode softmax simplex basis link part inference connect belief task amount uncertain form probabilistic belief show softmax jacobian proportional real number parametric arbitrary ensure restrict basis unimodal mode fall aspect good laplace numerical convenience q hessian kronecker stem dirichlet derivation identify mean gain analytically introduce fit capture dirichlet ht wider unsupervised learn topic gaussian process dimensional generative learn mapping document define topic generate usually assume kernel distribution discrete sample dirichlet topic perform conceptually cite model topic consist state union address joint annotate identity combine feature discrete one identity fall power drift topic represent radial function evenly period topic rational hilbert identity time additional author rational square kernel assign nonzero scale exponential finite function expressive section optimisation difficult consequence expressive interesting american interior development red topic fast development american external linear radial author either predict date office figure see optimisation every visible negligible effect intuition hyper optimisation two dataset require numerical invert external external external iteration kernel document cause optimisation directly topic point subsequently dataset show considerable optimisation relatively elegant topic experiment document take wikipedia list document set element short link document link assign arguably present nonparametric kind allocation conditionally efficiently laplace cubic corpus offer sized corpus analytic elegant side effect laplace marginally paper replace estimate bayesian uncertainty infer maximum evidence acknowledgement like thank david david circle minimum text width pt sep fill fill text draw black fill black text white white fill size sep size latent dirichlet allocation discrete belief document mixture weight document hierarchical social document challenge efficient cast around laplace approximation transform type variable topic feature work latent allocation lda comprise collection collection discrete distribution document constitute popular corpus treat bag ignore collection document word specific real exist product communication corpus amount author augment additional popularity vary west poor old lda extension sequential document description function space link softmax assume stay change round link
process demand importantly target basis em analytically multiple anneal idea converge step counterpart chance maxima copy draw monte therefore apply multivariate probit option draw hand may lie ensure identification meet relation probit expand simulate suggest multivariate probit acknowledgement thank associate greatly manuscript thank implement numerical six probit give normal constrain domain inverting integral respect independently normalise simplify domain lead write smc sampler smc random walk metropolis p weighted mh smc loop incremental normalise weight degeneracy ess covariance mh update current particle available multivariate student sign n algorithm cycle draw effectively mean ensure proportion preserve truncation sure particle initialization proceed update metropolis run particular automatically implementation resample adaptive ess smc key smc probit possibly weight eq core repeat noise loop step cycle conditional p implement current smc move likelihood sum probability smc noisy zero keep expand sum towards normal sum fourth error quickly variance correct pt probit model appeal dependence multidimensional binary modelling matrix latent multivariate probit regression carlo avoid intensive multivariate normal gibbs sampler normal proceed stage draw truncate far towards em nature exploit em update iteration computational conditional step iterative em probit embed algorithm equivalent method validate widely high simulate approach correct treat probit monte carlo multivariate probit originally bivariate particularly useful dependence typically high dimension likelihood analysis previously augmentation nature expectation iterative classical optimisation iteration ideally implement analytically version truncate chain monte gibbs particular art different option sequential weighted particle mean weight though originally introduce dynamical kalman smc static art multivariate normal gradually truncation normal tail student particle require truncate normal gibbs main difficulty standard optimisation large simple extension guarantee identifiability constrain normally difficult identifiability issue intrinsic regard expansion approach em fact probit make idea smc version parameter burden validate previous high example monte sampler produce sequence obtain refer particle interest static scenario purpose element control degeneracy resample perform effective fall move next normalised formula incremental weight backward kernel respect suggest convenient backward walk metropolis hasting give iteration moving proposal scale practice mh sampler arise wrong art extensively investigate mcmc literature original adaptively monitor acceptance target metropolis adjust gold realistic situation purpose diversity sensible avoid move degeneracy lead smc metropolis type quantity stepsize logarithm computation maximum likelihood posteriori problem rely let datum us estimate eventually correspond term mm certainly decrease comprise respect e replace sample augment suggest iteration analytically difficulty multi turn ideally wish time probit formulation vector component associate observation probit parameter density random identical setting care identifiability probit variable discretization multivariate gaussian variable think obtain unobserved vector z I function probit component elaborate inference matrix lead positive ensure positivity eigenvalue probit trace ignore correspond completeness derivation provide expression provide trace convenient intractable gaussian density expectation weight j normalised sample provide smc truncate multivariate particle obtain sequential manner split obtain zero condition value approximation current trace write condition satisfie already update step remove intensive approximation generalised lead impose step probit model phenomenon link invariant change decompose perform space large conversely ensure identifiability could high art wu principle local zero decompose practice likelihood though space subtle proxy likelihood unconstraine would conditioning change arise exactly create unconstrained preserve inequality increase lead though necessarily conjecture agreement example seek incorporate expand examine neighbourhood maxima expand converge standard suggesting give significant demanding appeal probit rely smc sampler evolve take iterative discuss identifiability consideration probit almost computational walk metropolis towards tail exponentially method involve normal inefficient low rate acceptance tail distribution respectively replace denominator correspondingly actually acceptance allow degree grow variate unconstraine student distribution truncate intermediate target smc nc probability c ultimately probit reach student freedom enough truncate distinguished perform could truncation since reason student aid region overview truncate gradually oppose increasingly truncate distribution student relative conceptual adaptation framework notice section artificial address functional form section move intermediate target priori determine dynamically local difficulty adaptation evolve parameter quantity solve q efficiency inspire page stochastic dynamical design keep threshold update observe introduce scale maximum correction term target end ideally smc sampler fraction slightly define resample resample threshold number reach reduce small also volatility binary smc algorithm advantage sampling truncate multivariate normal proportion probability reach behave like use target dimension perform one single smc target cutoff interval fig smc form give diagonal element reduce include rescaled element p inside correspondingly log essentially tie apparent though independent inside constrained keeping eq expand impossible likelihood preferred remove ambiguity replace replacement effectively invariance rescale whose element root appear result higher see unconstraine preferable define nature invariance obtain identical identity obtain depend fix diagonal vanish find solve repeat identically allow involve invert multiply relative optimisation grow fast diagonal dimension package parameter identity start method fast time improvement vanish along translate likewise solution turn depend constrain unconstraine associate drop scale matrix element invariance mean transformation shorthand form ex ik p c ik identifiability invariance identifiability give crucial constraint implicitly attempt formulation probit model clarity table probit model section vector coefficient diagonal response model may keep special case share single allow slight row give reduce yield invariance write compact alternatively covariate may share dataset covariate response represent compact row correspond response coefficient fix probit model fix account impose modelling reduce invariant seem six impose unnecessary identifiability vector sub transformation positive constraint avoid sub vector multiply factor unchanged rescale direction fix consideration slight change trace need constrain impose ensure model formulae smc sampling truncate multivariate normal gibbs chain draw truncate variable efficiently reject efficiency near gibbs correct region slow converge correlation extreme smc move truncation sample become preferable truncation truncation occur four variable correspond dimension moment gibbs sampler statistic distance rejection sampler burn sampler plot inter fig rapidly smc sampler moderate correlation smc notably well large smc gibbs pass obtain sampler provide discard roughly correspondingly growth still smc rather value course computational time efficiently sampler sample error scale treat widely longitudinal study air probit bayesian geometrically multivariate oppose probability likelihood readily produce sample far expectation six model status age consider analysis refer child child value indicate present covariate namely age child term probit j cumulative distribution standard normal example space sufficient covariance correlation constrain em invariance stochastic stepsize gradually importance innovation learn weighted reduction long within neighbourhood monotonicity convexity practical cause issue estimate value multiply correct likelihood obtain increase particle tune acceptance root likelihood run comparable slightly correction discuss run close likelihood region evaluate feasible accuracy smc find notice seem close close exact use sample optimisation sampling optimisation involve evaluation require form numerical routine page variable efficiency indeed exploit significant complete conditional single complete iteration benefit smc method particle update necessarily column smc estimate much reduce computational updating particle approximation fast draw latter factor keep number particle strictly six correlation reason dimensional impose invariance fix
expense burden quantity carlo repeatedly entire appear bootstrap encounter massive quite demanding costly modern trend seem ideally suit straightforwardly leverage processor bootstrap parallel massive cost independent processor compute node operate even resource largely devote reduce bootstrap complexity eliminate need comparable subsample book closely introduce bootstrap appear repeat consideration drawback sensitive subsample subsample variability rescale rescale knowledge drive optimal great eliminate gain reduce cost conjunction however series expansion estimator automatically execution bootstrap motivate assess scalable result bootstrappe original bootstrap subsample little subset individual weight form sampling subset replacement computational rescale overall favorable profile quantity much small dataset suit modern generic favorable property correctness show bootstrap formalize statistical discuss subsequently share bootstrap study compare bootstrap subsampling discuss distribute present superior massive hyperparameter finally bootstrap real series sample corresponding indicate compute random underlie estimator computation estimator assessment notation place standard practice base knowledge estimator directly addition operate normalize example compute distribution statistic obtain direct dependence nonetheless involve allow standard deviation development generalize straightforwardly exposition notation drive compute generally amenable straightforward repeatedly subsample subsampling book subsample subsample form approximate correction prior knowledge increase bootstrappe give uniformly disjoint q analytically general compute manner bootstrap repeatedly compute estimate b substantial benefit nominal particular generate suffice vector point count estimator respect storage space hold use estimator result resample tb disjoint subset predefine b n avoid problematic repeat computation contrast contain contain mb dataset tb gb subsample gb substantially total multiple fairly suffice small size amenable distribution allow distribute parallel enable additional simultaneous compute partition large cluster thus quite costly computational computation quite indeed straightforwardly permit multiple process naturally leverage intra parallelism subsample thus bootstrap assess natural single possibility prohibitive full assess point compute returning inferior average discuss empirically applicability favorable bootstrap addition bootstrap subsampling choice subset size statistical correctness identical bootstrap bootstrap consistent degenerate limit study asymptotic procedure center additionally account carlo approximations standard estimate generally take hadamard subspace measurable pf continuous space metric practically hadamard regularity condition satisfy compute also generalization assumption work g beyond consistency characterize high great devoted showing order many case converge driven limiting follow degree correctness choose importantly value bootstrap via asymptotic expansion power expansion compute expansion term expansion expansion fisher expansion full expansion well function mean version define obtain moment p nb enjoy correctness bootstrap natural power estimate represent moment like bootstrap sufficient additionally k p probability decrease applie highlight substantially slowly disjoint datum q enjoy correctness consideration divide standard hold generally empirically characteristic statistical exist experiment datum correctness property bootstrap subsampling setting parameter vector model procedure compute marginal parameter compute boundary confidence wise define averaging assessment task ground generating realization dataset compute collection fidelity realization assessment parallelization record produce bootstrap subsample deviation component confidence estimate confidence width repeat process realization trajectory quality relevant variance interval control quality assessment procedure probability large execute maintain notation subsampling regression I I draw j skewness vary quadratic vector encourage numerical dimension estimate parameter approximately bootstrap middle row ss trajectory linear generate right linear succeed converge middle fail bottom perform bootstrap subsampling relative large suffice implicit axis range large quadratic beyond order estimator quadratic regression via method penalty encourage numerical confidence parameter linear approximately row bootstrap trajectory logistic regression middle logistic show result classification varied appear set comparable bootstrap converge higher still converge fast robust fail even superior qualitatively bootstrap experiment implicit axis require range relative subsampling performance bad legend leave plot logistic generating examine converge relative explore relative vary value procedure report achieve many output increase consider accordance indeed grow achieve relative substantially lower quickly precede intend investigate statistical insight see figure processor require time less bootstrap ability via platform exceed storage capability individual use parallel architecture assessment tie ability effectively resource exposition due bootstrap partition node least potentially feasible bootstrap incur associate overhead repeatedly compute system website disk memory physical size constraint exceed memory incur extreme cost read disk disk magnitude read disk read acceptable slowly compute prohibitive bootstrap require permit simultaneously implementation enable gain small process subsequently intra parallelism compute subsample full read disk store disk potentially store implementation benefit advantage available parallelism parallelism compute multiple cost super subsampling worth subset relatively disjoint access distribute large platform modification accommodate partition allow truth substantially realization dataset numerically single base resource require storage prevent degeneracy large logistic bfgs large implement process simultaneously processor compute node simply computation upon partition utilize resample avoid though little qualitative outcome amazon implement either disk available gb leave show disk full worker store disk disk quite worker gb memory compute core cache repeat access bootstrap improve disk exist bootstrap process result computation work bootstrap largely address computational issue generally process reduce subsample amount dependence influence provide achieve seek value study value confidence leave plot insight influence set datum particular small low select ci min hyperparameter linear vs parallelization adaptive trajectory mark trajectory hyperparameter interval ci component wise case expect hard large avoid hyperparameter validate inner whereby subsample continue note form series well behave many case unknown suffice increase use processing increase value adaptively independently batch parallelism output tb z adaptive hyperparameter representative set early parallelization selection illustrate hyperparameter compute converge low unnecessary computation degradation though priori interpretable yield generation apply conjunction different easier help hyperparameter selection value efficient desirable could bootstrap subject future fairly investigation seem choice real absence correctness assessment output various procedure knowledge dimension yield figure bootstrap uci connect uci study use bootstrap change procedure qualitatively six slice range see selection bootstrap legend move stationary liu bootstrap setting variant bootstrap identical context briefly procedure stationary suitable assess series extend subsample mechanism generate particular suffice select length generate bootstrap subsample length hyperparameter bootstrap select subsample follow series length probability next beginning reach subsample subsample execute describe bootstrap introduce task rescale approximately stationary bootstrap stationary improve performance characteristic ccc method standard stationary series rescale aggregated trial rescale approximately provide powerful alternative automatic quality suit modern compute architecture share property consistency generic applicability bootstrap computational demonstrate computing platform subsampling analytical correction enhance adaptively datum open extension remain though hyperparameter subsampling resample validate beneficial adaptively select may reduce note
data constructive unsupervised regression dimensional manifold reverse formulation space optimally reconstruct fit manifold problem point problem hard unknown review knn regression topology learn linear kernel project hilbert locally principal unsupervised regression start algorithmic parametric e denote unsupervised kernel leave loo cv n argue unsupervised regression memory accelerate introduction neighbor output consist pair knn regression compute mean knn locality neighborhood label label close pattern sample knn g section iterative regression datum matrix space seek optimally minimize define optimally manifold unsupervised manifold contain manifold norm word consist reconstruction knn example extension knn move effect still near neighbor extension penalize avoid limitation knn fix far knn absolute position position perspective neighborhood solve high dimension combinatorial iterative strategy iterative assign propose variant test intermediate look embed distance choose remove repeat element low comparison position test near datum neighbor overall constant practice step computable compare neighbor similar color b embed point part topological sorting although observe local optima variant variant show beginning b embed begin color correspond embed see assignment experimentally test problem sake pixel handwritten h digits digits embed digit assign digits digit latent achieve digits show three setting lowest highlight exception achieve exception role fit dimensionality reduction iterative strategy high latent speedup restrict solution reduction heuristic turn experimental analysis achieve slightly fast constant work analysis optima strategy extend global extend latent topology dimensionality simple stochastic strategy employ randomly p ii department von universit scientific high
appropriate prior estimate model commonly prefer solution time family criteria bic widely datum give measure function constant investigate consideration principled thesis publication finding experiment investigate external internal validation alone sensitivity initialization parameterization variation internal bootstrap validation approach internal validation compare real external investigate independent exploratory throughput thousand generate hypothesis intervention laboratory highlight assess uncertainty currently widely genome constitute thesis overview technology section preprocesse measurement crucial chapter thesis utilize genomic database microarray gene chapter organize overview various throughput microarray study external sequence database model evidence across microarray quantitative probe throughput summarize novel throughput come level biological technical add variance natural biological individual nucleotide add technical source measurement rna experiment platform laboratory significant array come activity give probe differential gene expression highly contaminate add variation number affect probe publication elsewhere portion microarray moreover although design uniquely intend nucleotide affinity content likelihood cross target alternative cause position effect contamination probe source poorly understand importance challenge technical process analytical ultimately publication tool probe relative probe level contamination probe probe level detect probabilistic differential target indicate line show estimate source genomic alignment probe rich associate publication datum help reduce improve preprocesse statistical raw throughput preprocessing array thesis microarray array step microarray quality correction microarray quality array remarkable remove degradation array microarray array vary smoothly array arise technical background detect array probe background correction array help differ significantly array average preprocesse background correction global intensity observe gaussian b correct bs background comparable array quantile force array quantile normalization assume concentration population human systematic array final individual probe level single estimate preprocesse difference characteristic systematic probe probe significantly affect widely preprocesse utilize probe probe probe preprocesse probe level effect account improve quality microarray probe expression probe method probe algorithm probe specific characterize probe bind systematic difference capture probe probe help level summary probe outline I probe level probe bind affinity noise probe probe j model need absolute gene drawback order identifiable probe average yield well show accurate extension probe utilize data microarray probe probe preprocessing array design expression gene expression investigate expression level difference experimental expression level method probe perform probe affinity probe potentially suboptimal publication demonstrate calculate already probe improve need formally publication probe publication summarize probe gene specific outperform widely probe preprocesse publication probe specific effect probe preprocessing utilize external microarray design date sequence available rapidly evolve genomic sequence reveal probe annotation body knowledge grow recent include publication detect uniquely match remarkable microarray match database publication exact vary utilize thesis microarray analysis background database microarray collection probe verification increasingly preprocesse microarray probe genomic database construct interpretation array genomic annotation accuracy cross microarray publication elsewhere publication combine probe novel compare combine microarray huge microarray available come microarray microarray prove source probe sequence probe matching array technical verification sequence database confirm confirm publication publication utilize genomic probe remove genomic external database contamination publication probe base datum collection reveal characterize source probe contamination gene average theoretically justify prior analysis verify match gene remarkable portion sequence microarray pearson correlation array biological show probe match array comparison good probe array advantage probe verification array probe collection compare I comparison technical replicate match set measure pearson array publish publication investigate probe genomic publication physical reveal alternatively sequence collection microarray public valuable study publication extract probe level microarray collection preprocesse assumption probe datum collection independently external level preprocessing expression publication explicit formulation quantify incorporation concern probe reliability estimate probe reliability differential expression expression external incorporating microarray publication summary suffer probe affinity contrast probe expression remove advantageous gene level preprocesse probe investigate probe publication assign probe probe guide probe consider probabilistic introduce publication ultimately expression vary collection ground assess signal assume characterize target normally probe quantify probe reliability probe array probe probe distribute probe reliability probe parameter probe affinity probe level differential expression avoid probe key probe probe preprocesse affinity user array array array probe j probe array write noisy observation true probe probe affinity publication partly explain level improve analysis probabilistic preprocessing method aim quantify effect interpretation probe reliability probe reliability differential expression expression probe specific variance simultaneously prior select control array latent obtain search relate gene connect correlate across potential global interact gene gene sensitive role gene predefine decompose predefine pathway module rely predefine classification gene drive gene directly prior genetic need refine suboptimal enable discovery demonstrate measure differentially express measure activity characterize change publication address de tool increase cost false finding gene discovery function prediction visualization widely cluster approach hierarchical patient novel activation association poorly gene self visualize coarse collection gene problematic whole genome study reveal sample small coherent subspace particular interesting detect gene module database subset coherent define relate tool condition compact summary aid interpretation collection enable discovery expression signature central global signature describe expression establish signature however establish signature typically classification particular signature association underlie process publication enhance signature ignore use feature technique connection gene typically handle hundred insufficient throughput use guide interaction gene coherent cluster model condition include orient additional include orient utilize information discovery module instance publication introduce complementary concern relation gene genomic applicability limit network gene construct gene genomic collection collection enable discovery genetic probabilistic dirichlet computation tool characterize expression set molecular interaction network thousand protein small molecular cell reflect gene response general drive proxy network distinct functional activation information gene global context pattern publication introduce general provide model interaction response subset cell biological focus analysis introduce publication first publication base gene coherent response coherent publication state detect find subgraph network gene identify step suboptimal condition response publication introduce purpose algorithm criterion publication search coherent interact gene interaction guide classification involve amount protein gene context state reflect unique expression signature expression level state gene precise give state state observation r include specific measurement state index association treat gene model leaving justified considerable gain response primary gene cascade expect help distinguish partially cost triple fluctuation bayes interpret soft membership hard deterministic distinct agglomerative use interact merged gene singleton merge neighbor joint gene value gene likelihood comparison give merging size possibility constant dimensionality effect criterion increase optimal g cg pair direct pair iteration continue merge modeling yield genome part support power agglomerative find step lead high merged mixture note globally response often publication activation pathway base database pathway provide package gene expression publication outperform unsupervise use search method figure publication response coherent suggest potential functional confirm detect p highlight exhibit term response alternative highlight connectivity publication response co gene group individual global activation use context particular condition reveal help formulate novel context specialize cell biological induce co gene produce specific datum condition specific proxy identify distinct network potentially functional role publication tool genome scale predefine classification bring miss drive readily applicable pathway protein available therefore scope rely implementation implication require result highlight response provide activation human disease j chapter present thesis genomic measurement organization understand organization genome ultimately various level genome system biology key property integration evidence across discover mechanism interaction statistical integration heterogeneous genomic variety technical challenge approach accord investigate measurement limit genomic suitable become increasingly house public observation highlight novel modifications micro rna bind nature biological uncertainty prior biological subject issue need principle minimal knowledge model overview standard throughput integration connection develop category evidence accurate ii order guide primary source characterize dependency order discover new contribution chapter publication cluster category introduce exploratory tool dependency tool guide bayesian publication cluster dependency source applicable integration apply dependency activity evolutionary human study occur observation genome micro valuable resource investigate functional property layer genomic tool relate develop biology community sequence genome structural organization accumulation research challenge create picture genome volume genomic intervention element historical public challenge intensive associate complex system availability observation solve challenge combine power public exploratory material genomic concern underlying phenomenon robust facilitate provide theoretical framework thesis exploratory integrate evidence genomic mechanism investigate small contribution improve throughput ii activation iii integrate measurement genomic contribute challenge genomic develop understand biological genetic evolutionary variation thesis carry publicly access guarantee tool original thesis extension develop integration functional promise line future readily genome wide disease occur datum concern various genome become micro gene modification study layer organization contribution health fundamental challenge biology wide exploratory rich idea mm science sciences public university school technology university technology computer school technology sciences science p box series http mm thesis http www accordingly free display purpose assume figure separate http thesis school science integration exploratory advance throughput sharing genome encodes program functional biology organization thesis computational strategy cell computational extract genomic observation probabilistic multiple genomic thesis key preprocesse genomic database collection throughput exploratory cell activation pattern functional across information genomic interaction database derive help gene approach occur source mechanism evolution analysis layer genomic functional measurement source implementation facilitate far rt ta te py sis lt n min sis sis ep te ne mi ty si ty ep si sa I k te I ne ne mi ty I ne I en la work carry network centre adaptive centre laboratory science computer school science technology work computer science university year I also part institute information technology school engineering project research school bioinformatic support thesis work give truly field scientific work necessary essential learning like express thesis expert feedback research biology excellent traditionally distinct science biology particularly grateful daily laboratory institute year belong co extend former member mi I department provide valuable sharing publication datum impact thesis express contribution thank scientific life I demonstrate rational go science friend science want thank discussion grateful share life child evident remain understanding nature life I year share aspect grateful parent accept I I create tie thesis overview probe level array probe reliability array transaction biology bioinformatic network bioinformatics detection constraint international machine page machine conference machine cluster dependency transaction bioinformatics bioinformatic summary author contribution author author thesis effort key contribution author thesis summarize publication genome study genome author study external manuscript probe gene combine multiple microarray experiment gene analysis probe improve microarray author derive formulation probe level manuscript open implementation review computational biology introduce activity genome network provide tool collection genome design perform author publication introduce dependency cancer major role task design jointly author author carry open publication introduce principle integration dependency author design manuscript publication extensive consideration work sensitivity analysis comprehensive bioinformatic design comparative experiment technical list thesis symbol denote vector capital symbol symbol vector scalar trace mutual parameter vector parameter vector ac comparative cca complementary dna dp dirichlet algorithm ib bottleneck kullback posteriori ml maximum rna rna lead introduction life process new technique small object molecular dna lead pass sequence human dna publish researcher large volume concern accumulation genomic database accelerate research structural organization poorly understand characterize regard system paradigm biology transform genomic collection new bring challenge throughput mind phenomenon challenge relevant statistically uncertain volume observational make life study massive set thesis principled exploratory central functional rna genome time protein synthesis cell level gene order associate genomic available public combine heterogeneous background public relate statistical uncertainty well form picture genome starting point research poorly observation toward proceed particular question support refer visualize identify facilitate new otherwise poorly characterize adapt extract automate oriented require process may require wide thesis increase high throughput combine statistical database across microarray ii specific interaction normal body information genetic guide iii integrate measurement occur strategy recognize challenge obtain first strategy side genomic microarray collection probe microarray preprocesse genomic sequence use remove extended publication introduce principled framework probe probe combine probe microarray number contaminate probe level well poorly contamination unknown could control probe level principled incorporate prior introduce outperform alternative activity genome interaction publication contribution thesis search genomic interaction database detect finding activity biological third thesis human genomic novel datum detect dependency publication use association knowledge use constrain latent improve introduce tool open contribution thesis wide access tool science proportion embed traditional valuable public thesis organize chapter genomic exploratory paradigm chapter contribution thesis chapter reliability throughput microarray present wide dependency introduced investigate functional activity conclusion thesis summarize law program double feature life purely world year evolves maintain adapt environment external hierarchical organization replicate reproduce life mechanism molecular suggest common evolutionary genome encode program technology science view organization genome molecular biology investigate organization genetic thesis new investigation functional overview concept resource background molecular genome life carry genetic program cell carry copy genetic code genome genome cell constitute dna rna ordering carry property synthesis paradigm molecular biology organization life dna set gene activity consequence code code carry protein synthesis despite identification protein gene genome detailed protein key entity cell protein synthesis refer cell biological protein product synthesis double proximity gene gene dna sequence convert code code segment respectively form rna convert sequence universal genetic triplet form sequence constitute protein stage protein synthesis fold post protein ultimately key synthesis key synthesis translation pre modify produce carry translate universal code nucleotide triplet correspond organization material nucleotide pair double carry material locate http file rarely attribute ultimately activation change cell biological environment activity level synthesis major functional genome protein gene refer chemical structural modification dna instance bind modification pack dna around cell combinatorial modification access source protein synthesis bind element fashion post modification variant variety protein contribute diversity cell several mechanism degradation micro nucleotide sequence specific complementary base degradation modification degradation mechanism cycle protein protein biological interaction complex life functional organization genome rapidly genome overview human level resolution nucleotide attribute dna traditionally point call nucleotide snps protein dna increasingly recognize structural variation genome contribution variation structural variation organization nucleotide large variant genomic modification directly influence health publish genome pair gene code less human genome million year majority half genome highly sequence genome sequence element mobile dna genome recent dimensional dna remarkable human highlight evolutionary specie human protein synthesis gene contain gene law law copy small express thousand copy source microarray study discuss dimensionality size form challenge throughput microarray greatly exceed sample study sample leave considerable uncertainty analysis concern phenomenon different part overfitte multiple form considerable challenge automate thousand hypothesis concern finding characterize prediction dimensionality functional challenge challenge control remarkable technology development procedure thesis combine increase rigorous genomic individual remarkable extent contribution characterization population genomic mechanism disease discovery medical throughput disease mechanism genome disease take sensitivity signature reveal lead diagnostic disease cause change chemical genomic new response genomic genome disease genomic disease understand genomic change continuously micro rna locate genomic relate non dissimilarity p give mutual kl theoretic employ thesis inherently couple different euclidean potentially lead equally naturally invariant variable demand refer scale transform suitable far select stand technique problem inherently equally feature interpretable particularly important genome sample phenomenon advance dependency consider interact model centroid publication selection genomic decompose reveal publication maximally set central dependency biological goal exploratory analysis technique summarize facilitate hypothesis research poorly characterize knowledge analysis toward hypothesis test data procedure exploratory traditional hypothesis light exploratory particularly standard biology discovery differ target mutually exclusive generalize partition unseen approach without cluster discover mutually cluster popular computational biology comprehensive application beyond thesis expression profile cluster suggest formulate concern discover novel cancer visualization another exploratory compact summary projection principal optimize self cluster probabilistic popular set sample uncertainty account rigorous particularly sample infer employ learn mixture process alternative infinite relevant extent infer variable component tractable characterization generative model distributional relationship probabilistic interpretation pca observation model normally distribute latent variable parameter noise define subset typically task instance describe process irrelevant nuisance integration marginalization marginalization latent give marginalization find account yield speed marginalization marginalization goal thesis utilize publication discrete publication latent dependency occur observation specification distributional practical task parametric principled structure gaussian publication proportion example relation describe theoretically principled exploratory flexibility potentially overfitte less moreover interpret probabilistic stochastic process prior structure process prior space dirichlet process dp dirichlet partition ga ga pa chinese restaurant crp provide intuitive description dirichlet crp define cluster crp customer arrive restaurant infinite table choose subsequent customer table state restaurant table customer proportional customer table pi probability proportional table customer analogous cluster intuitive cluster potentially infinite ultimately base method number component datum potentially infinite let replace intuitively call stick break stochastically break process I stick truncate stick break stick assign prior proportion learn help increase observation use increase model certain cluster selection include comparison theoretic seek insufficient approach uncertainty uncertainty lead refer ignore element posterior characterize encode match particularly train considerable parameter informative sample converge provide robust uncertainty
partial specific include model somewhat contextual multi bandit provide round determine pick still action policy choose adversarial select choose adversary choice round regret simplicity focus horizon advance expectation sequence choose multi armed bandit get reward receive construct unbiased action jt jt jt action essentially information action member depend assume advance intuitively think viewpoint remainder split clique meta expert action present build exist deal round clique choose reveal unbiased reward clique exponentially action expert standard exp meta action combine expert provide appear appendix round forecaster regret stick clique clique correspond clique action reward action correspond attain empty reward action standard attain graph regime follows fact clique computational hope well bad specific class compute clique relatively easy disadvantage structure stand exponentially arm exploitation uniform action carefully exploration solve program name bind regret concern action information theorem j kt lk jt jt j jt jt jt well choice undirected graph discuss regret undirecte via compare thm partition attain change case computational involve require np create large course advance solve treat copy run armed construction meta action incur factor deal undirected choose versa set condition choose equal thm rely tight conjecture case note weak superior graph clique determined quantity compute trick graph provide term number link randomize every least proof appendix intuition independent armed bandit play low armed bandit result follow undirected match regret upper logarithmic direct difference rather huge briefly discuss concrete notice performance partition choose choose seem reveal everything current improve feedback essentially undirecte gap third learning probability fourth case devise inter center market split exponentially forecaster action forecaster pick receive observe reward update unbiased estimate reward actual reward completeness forecaster define convenience hold q taking sum series fix rearrange get pick forecaster meta action exp meta action incur meta single lemma first definition combinatorial unable occurrence third lemma examine particular denote independence number size large adjacent node hold prove allow convention contrary exist follow either repeat guarantee eventually arrive show kp original fix vary adjacent everywhere namely node increase discuss state lemma graph define adjacent include exist otherwise choose value adjacent large since value either lemma rather armed bandit e get definition take fact sum rearrange get expectation side slight use simplify get thm difference subsection identical upper invoke long counter opt weak small upper bind independence let set two expect use resort know lower bind armed bandit game action side pick action assign action show learn cumulative adversary armed bandit describe independent choose one armed bandit feed reward back neighboring feed back well node round neighborhood exploration fix choose stochastic matter feed obtain reward identical implement round achieve goal provide bind choose action outside whenever choose receive highest big increase round time possibly pure round exploration round regret exploration round expect expect regret overall q rearrange standard multi armed bandit select obtain plug maximal node exploration rather replace allow oppose corollary microsoft adversarial maker observe action get side encode graph link naturally decision maker armed bandit maker reward practical provable regret theoretic information basic setting framework assume action performance term reward iterate round total good action accumulate framework generating might full reward many world canonical ad display whether ad constraint lead choose set adversarial price provable factor factor bandit expert rather bandit receive theoretical reward study rich various reward formalize bandit set intuitively assume obtain reward reward action way round feedback direct action action sufficiently good reveal action well complete expert set scenario motivate web standard armed bandit ad two ad package ad display contrast ad run ad unlikely clique sort motivate sensor certain sensor cover may cover centralized controller receive model cover area sensor reward choose obtain sampling come select transmission noise channel adjacent frequency band result picture provide fundamentally superior guarantee theoretically
evaluate show situation aspect nontrivial exploration induce exploitation balance location uncertainty accumulate important insight usefulness reinforcement secondly system trajectory necessarily follow full assigning low good trajectory amount reach slice control control inner sep node white l td kalman filter achieve measure cart study arm compare average controller access kalman gaussian controller approximation td radial free assumption frequency influence nontrivial rarely reinforcement accumulate differ discount learner controller control balance necessarily curvature value well potentially compare exhaustive quantity exploration vary bayesian gaussian nontrivial reinforcement equation easy differential field raise reinforcement solution approximation functional note argument dynamic process inherently determine may loss incorrect replace even concern well uncertain probabilistic allow similarity work extent inference widely utility acknowledgment author express crucially improve pt black minimum black exploitation trade central reinforcement bayesian intractable statement space subject infinite dimensional differential equation give result helpful learning thing spend irrelevant incur unnecessary classic consider trade policy classic reinforcement ad hoc method thompson thesis know exploitation require trajectory along situation depth tree propose approximate ad hoc optimal parallel idea reinforcement area structural recent et al exploitation trade kalman reinforcement loss function restricted invariant system core reinforcement dimensional new approach paper result prior section description system yield amount control control exploitation control rely concept reader reinforcement control action transition reward optimal control attempt reinforcement description optimally control uncertain dd qx qx analytic concern dynamic seek simplify uncertain sample function note g uncertain acquire stochastically sample belief process k word draw infinite vector demonstrate inner point describe uncertainty process scale unit change location utility gain task horizon trajectory equation control law discount definition uncertain rather generation somewhat ad hoc simple assigning acquire dynamic continuous dynamic still system cart realistic exploitation representation reinforcement usually construct bellman sec analytically solve optimal phase belief function nontrivial swap integration controller actual reward access true controller system disadvantage average optima actual ignore integrate drop controller follow adopt greedy effect amount step look ahead imagine extension future analytic bellman explicit upon central affine optimal reinforcement order interpret sign comprise immediate drift first effect phase subspace effect augment former latter dominate govern physical cause control physical knowledge statement reinforcement objective exploration even equation remain nontrivial two although product look integral read remain solve least choice integral solve exponential nontrivial equation numerical answer arguably constitute value future trajectory differential equation raise hope analysis claim hard transition belief infinitely condition joint control form assumption form clear way infer without address specific analogous use break section k approximate nonlinear radial restrictive widely reinforcement kernel convenient integral serve
methodology situation anomaly psd integrate drop allow euclidean string describe class contaminated future investigate asymptotic trade impact different begin first third distinct linearly positive definite subtract multiply multiply equation turn g compact continuous continuous space calculate differential case come lemma solve iw first fact decrease convex imply condition inequality strict inequality strictly happen happen distinct strict monotone define surrogate function define argument q next monotonically lie limit come inequality g sequence indice contradiction give fs generalize however thing care deal summation suppose give probability q exchange integral valid h g strongly well therefore come fact continuity increment still apply need find exchange valid dominate convergence lipschitz g h g x combine df f df simplify substitute x x linearly compare n diag eq found therefore imply sg triangle inequality open ball center radius f unique j sg ann mi usa email nonparametric robustness contamination kernel kde classical interpret kde radial associate reproduce sensitive robust iteratively least sufficient give global minimizer density trick kde multivariate relatively little improve kde situation method contamination training international conference machine learn follow contamination eq nominal contamination assumption nominal distribution many nominal density spatially relative although state condition precisely capture intuition behind motivate application anomaly imagine multi collect example volume traffic along instant measurement collect nominal detector unfortunately often anomaly vs intensive estimate nominal distribution desirable propose robustness density estimator kde radial definite psd hilbert rkhs yield robust density weight necessary first weighted kde influence exact formula sensitive contamination outlier third conduct several dataset anomaly motivation kernel know without kde tends low motivated dependent bandwidth adapt dense aim aforementioned outlier estimation psd refined treatment adopt hilbert embedding approach attempt match design contaminate mind estimator square problem rkhs estimation develop song et optimize squared space whereas integrate square design contaminate primarily robust surrogate loss apply feature around spatial median depth contour influence function contain proof matlab implement available estimate ensure kernel condition e borel q example satisfy property kernel laplacian dimension psd exclude psd kernel hilbert completion reproduce thorough psd purpose critical property reproduce state evaluate inner radial kernel kde kde sensitive presence find q unlike huber huber loss quadratic huber loss detail huber many tb htb argue valid density contour f x iteration sequence weight experience initialization view optimization transfer perspective characterize every grow quantify recall influence scalar influence represent assign probability contaminate mass measure bound consider concept scalar express estimate function standard kde empirical influence kde influence fs huber find matrix ii addition strictly extend matrix solution near contamination kde agreement robust view less sensitive mention boundedness influence robustness htb show univariate tail value kde additional kde kde decrease training confirm setup present conduct diabetes f originally www uci set randomly partition f contamination one choose nominal digit contamination mnist nominal anomaly amount nominal compare kde density loss use kernel neighbor well omit c terminate three study outlier third task detail th set rank absolute let rank rank statistic test level study first measure influence exclude comparison x change density impact experiment learn nominal mean performance sign state affect c kde kullback leibler kl eq divergence whenever nominal kde separate estimate infinite
duration population birth rate disease furthermore age differentiable easy eq ode table population furthermore relative know ht hand type ode I p ode analytically life disease rate age equation stationary describe introduction derivation age incidence age usefulness report publish health year whole population member percent take account percent sample gender stay diagnostic code associate gm report age frequent report confidence decide whether group significant due ode general office year order incidence perform derive calculate incidence spline function uniquely bound perform software version table input follow describe mm age person incidence disease ode relate age specific incidence rate ode incidence cubic spline choice one incidence extract sophisticated restrict optimization intensive perform analysis group generate incidence treat six hour pc ghz cover reporting period author try estimate rate member get diagnosis third potentially positive early see case account diagnosis incidence ode ht age person per person compare incidence visit newly incidence rely free hand systematically increase year diagnosis instead value hence death incidence proposition diabetes incidence rate relate age incidence allow study applicability age agreement publish disease cubic spline equation basic characteristic disease population characteristic fundamentally incidence observational assess population classical incidence somewhat group patient examine point whether meanwhile investigation mostly complex expensive study fact get lose question incidence know health service resource patient example may population number number see equation later complicated analytical transformation ode reduced equation take incidence solve incidence paper organize newly discover age distribution incidence health incidence age group
interesting density approximate consider recover level close nn estimation technical requirement recover requirement level provably cluster retain nn index discover neighborhood retain sample single linkage nn differ retain result namely level recover cluster procedure correspond note trivially guarantee return salient mode empirical treat false cluster consider cluster provide pruning heuristic typically consist define strong tree size moreover assumption spurious easily appear sample rely prune instead return value empirical notion unfortunately finite guarantee removal spurious smoothness require upper density distribution start operation unless radius center contain nn graph say every continuous call tree denote notice form infinite mutual either versa simplify set pruning control show sample level prune cc contain tree keep look lower tied tree give tuning connect component subgraph tree I assumption continuous connect two pruning essentially large remain subset remain envelope suppose nn subset belong cc recall parameter separate da term apply nn tree mild illustrate practical pruning wide range consistency compact n da separate path satisfy ga disjoint thus high hold establishes remain part becoming create spurious cluster removal spurious letting discover removal spurious cluster one guarantee set actual underlie spurious cluster remove upper light surrogate consistency maintain expect discover salient mode salient salient separate separate every salient theorem follow disjoint na salient leave n sample maximum sparse increase average max setting mode tree guarantee lemma describe happen subset remain prune interestingly although interest sample statement literature supplement intuition combine find probability nx ix provide l say low region remain mutual nn subset r point level path show salient salient contain set point vc establish condition subset denote mode satisfy nn nn mode empirical salient mode set ball whenever bx r kx salient mode contribute map iteratively start establish therefore leaf root leaf subset connect empirical prune near connect path depict path depict connect must sufficiently consecutive whenever scale apart various near neighbor create way get choose possible near ball weak keep track scale path nn belong cc lemma show possibly x x ip bx nx ball continue adjacent must argue must ball center show establish edge relate first imply two imply g ny must bx dr nx iy iy terminate procedure remove spurious hence spurious pruning let disjoint na na n nx low n thank often chernoff ball nk eq choice fix nb bx nb word chernoff lemma combine bound choice long infer event inspire relate center ft combine prove combine bb c fx r consider eq integrate probability kx finally
curvature noise perturbation tangent leave mostly tangent tangent classic cause interpretation suited noise appear u triangle geometric control seek regime careful employ concentration present appendix matrix unbounde care ensure hold ambient dimension avoid dependence ambient hold analysis concentration hold maintain main mean curvature rescale curvature model natural strictly positive formulate follow number sample point observe linear position sampling result result q random realization follow definition ease presentation finite correction numerator finite correction denominator term simplify choose demonstrate accurately angle true computed decrease noise bind minimized tangent recovery optimal scale may neighborhood ensure condition note denominator analyze bound recover angle subspace unable require conditioning impose spectra curvature perturbation quantify scale curvature eigenvector tangent condition requirement sufficient meet numerically track violate bind tight recovery error demonstrate main quantify need geometric principle impose tangent solve satisfied require real derive full allow geometric uncertainty natural perturbation curvature curvature compare curvature ball require note principle manifold less intuition correction interestingly tangent careful perturbation would compute manifold clean perturb remove compute homology noise topological main track scale require optimal recovery practical main demonstrate intrinsic parameter level local uniformly tangent curvature recovery plane norm utilize practically trial indicate deviation hold empirical relevant chernoff far provide geometric encode principle tight unchanged trend regardless accurate condition tangent plane radius flat vertical see discussion show linear subspace increase scale axis monotonically green track blue nearly red b show free three curvature case due monotonically slight numerical predict reach curvature infinity true indicate orthogonal tangent violate long spectra predict compute contain orthogonal true tangent version panel large curvature scale bound track error principal observe behavior large scale encounter free saddle embed demonstrate principle mixed sign red track true tight case except order understand height dependent plot height red curve result nonzero experiment track behavior show parallel logarithmic multiplicative constant far indicate triangle bound norm tight decomposition tangent dash indicate minima dash dash green location occur indicate yield tangent particular location occur flat panel error compute error angle tangent stable examine coarse scale coarse indicate miss exact optimum dimension important experimentally sensitivity tangent error follow experiment sensitivity track one parameter vary hold radius neighborhood radius manifold embed detail hold scale intrinsic shape direction principal panel panel varied panel neighborhood right main show blue track angle hold curvature incorrect value range right way curvature hold thereby green main result compute inaccurate radius entire indicate properly intrinsic sensitivity estimate mild shape principal blue track recovery indicate optimal radius hold dimension range green vary within bind indicating noise relatively stable small curvature set mild curvature manifold normal insight avoid stable b blue track subspace radius hold large incorrect curve right remain stable bind variation curvature high principal principal expect experimental indicate perturbation provide algorithmic tool directly tool translation distance tangent plane estimate clean origin equip two bind space free practice presentation assume recovery origin decompose recognize compute side realization sample size observable ambient approach determine proceed calculate ball give ball radius effective insight gray scalar curvature manifold volume small ball plane volume ball vb vb distance estimate eq approximately capture effect relationship previous derivation volume noisy concentrate surface radius convolution manifold gaussian flat ball center account along beyond follow volume conclude radius plane radius large precise computation remove confirm noise less lead prove along author confirm thorough next rough accurately track figure first embed normal geometry consist noisy manifold embed principal great geometry tangent plane blue ambient show reference geometry geometry indistinguishable relevant geometry compute tangent true tangent reliably therefore observable tangent plane blue equation indistinguishable scale ambient repeat subspace recovery ambient tangent radius ambient radius tangent plane bind track error present ambient ambient manifold c like like vertical line indicate logarithmic axis specification give exclude linear geometry implement choose give principal set gaussian experiment ambient use tangent plane error true perturbation utilize represent good hope mean error mean bar deviation origin show behavior counterpart compute decay axis track true blue nearly indistinguishable red manifold three normal direction ignore instability scale match panel observe due curvature matching geometry principal track dash vertical location curve blue dash fall quite flat figure scale collect ball ambient radius tangent plane show exhibit direction impact geometry green curve normal situation ball ambient necessarily point large small radius plane amount curvature geometry see ambient radius grow tangent grow ball ambient tangent lack geometry indicate orthogonality scale indicate provide sort discover ambient green order project tangent plane red order identical geometry curve principal equal geometry exhibit great curvature note possible future nonetheless present indicate main accord track tangent ambient user result explained previously propose system analysis seem perturbation source perspective alternate use reader apply noisy compute simple require decomposition worth expect universal multiscale present intuition role close clean plane since exist coordinate direction coordinate align goal move normal framework rotation axis merely convenience discuss close right close define blue normal red neither perturbation require trajectory radius grow coordinate recover scale ball n ib r mn b square scale explicitly precisely tail scale reach coordinate uncertainty densely sample manifold apply rotation conventional coordinate unitary coordinate slight modification observe coordinate intercept origin order leave coordinate coordinate proceed without generality show small coordinate derivative least square enforce model estimate anchor examine detail confirm scale smoothed stable must observable radius radius presence informally measure point let denote offset calculation similar show yield scale hold introduce presence yield hold high take indicate necessary geometric trajectory slightly consider scale point let note rescaled curvature encode scale eq probability trajectory decay replace rigorous analysis mean g understand scale produce result replace discard discard behavior scale trajectory quadratic explicitly interval compare choose error could carefully could compute one expect satisfy decrease procedure scale interval small trajectory center dense algorithm demonstrate parameter implement sample dimensional embed manner origin support reference specify seven experiment origin reference error trial report interval use c c e curvature saddle saddle offset table accurately origin setting curvature initial offset large occur high setting produce curvature relatively noise quality beyond careful driven dimension noise coordinate origin error decrease error pca manifold receive assumption growth work focus recover therefore confirm closely pca spectrum estimate free develop multiscale detect sample analysis level much result set angle regime although curvature treat drop main result form recover curvature way noise perturbation datum density neighborhood analysis analysis yield density lift proceed limit requirement associate separate regime correction recover require expectation imply agreement size choice ensure small denominator decay numerator limit infinite absence density study multiscale pca spectrum detect manifold generally cloud random empirical localize noisy close population covariance high author estimate population noisy noise effort center multiscale noisy appropriate present population control center localize lead moving allow curvature affect point experimentally main experimentally cause theoretically center rescale radius localize close localize origin recover center true origin neighborhood offer algorithmic practical curvature analysis recover suggest intrinsic dimension introduce approach recent pointwise fashion perform svd determine examine growth value interesting remain coarse exploration approach dynamical perspective exist present set experimentally several neighborhood multiscale value suggest denoise method estimate curvature g vision explore tracking center rate pca spectrum tractable plane choice conduct experiment plane radius manifold tangent plane rigorous need tangent tolerance may chart ensure requirement meet similarly able neighborhood may tangent neighborhood local global name optimally cover national foundation dms department de acknowledgement grateful anonymous comment suggestion manuscript inf mit qualitative chen little advance multiscale wavelet analysis ed pp j adapt wavelet j eigenvalue number shrinkage curvature measure transaction american determining flow find intrinsic dimensionality manifold r deconvolution hausdorff mathematical introduction several variable geodesic riemannian journal computation eigenvalue principal ann dimension sample ann component analysis manifold process tangent plane th international mathematics tangent plane noisy avoid curvature plane statistical pp mathematic surface pp functional ann lin riemannian manifold pattern multiscale ph thesis university multiscale multiscale curvature mit unify conference surface noisy cloud perturbation ann composite locally science orient filter journal intelligence global locally neural inf pp mit vector isotropic conference pp wu vector diffusion map connection tail matrix smooth embedding matrix compress ed pp wang j finding dimensionality fast active contour alignment scale patch dimensionality linear modeling pp zhang manifold appendix calculation particular probability norm norm tight bound ensure independent ambient dimension proceed bound result start proof comment notice sometimes simple interpret merely upper interpret notation often eigenvalue square standard event seek nonzero nonzero equivalently ignore utilize find self adjoint expectation eq yield set eigenvalue notation great hold large great soon necessary eigenvalue remainder spectral use eigenvalue eq frobenius delay section tight however hold proceed denote small singular respectively result give tight control let whose standard normal variable divide coordinate soon reasonable sampling similarly soon direction curvature quantify note bound matrix nc center expectation distribute original involve frobenius merely euclidean unique modifying slightly arrive complete compute tc tu set expectation take matrix u nc q finally eq great random seek form realization dimensional wishart since block indeed let partition equal frobenius differ rotation explain next first entry extract norm control norm matrix get norm frobenius loose equality probability random proceed conditioning coordinate gaussian matrix prefer row euclidean conditioning know diagonal copy eq bind derive replace full small finally happen moderate mild condition realization last consider depend tell likely e imply depend finally bind l tu fm previous proceed precise envelope te tp correlation coordinate term clearly upper top eigenvalue variance measure along expect rigorous value singular orthonormal column size define project vector technical difficulty check rotation proceeding realization eq concentration te v fashion overlap block row orthonormal orthogonal matrix zero circular shift right construct stack construction nonzero every column allow action everything together conclude orthogonal previous norm proof remove condition p conclude n indeed great joint selection random realization q nu te tu give brief outline get q nu tc tu component great realization department mathematics usa set lying consist parameterization tangent plane return optimal basis sample nonlinear manifold small scale stability pca space adaptively reveal plane stable purpose provide theoretical real tangent principal model curvature noise lie manifold parameterization datum represent parameterization may inherent discover geometry sample remain research study parameterization give svd datum near pca linear case subspace higher proceed nonlinear embed consider fashion linear latter subject linear parameterization manifold tangent tangent plane corrupt curvature local pca tangent characterize local tangent perturbation theory estimate randomly within neighborhood adaptively neighborhood give analysis proper recovery intrinsic curvature however pca locality partition g adaptively size perturbation vary locality hand approximately avoid large effect simple criterion tangent plane set color angle form plane define adaptively accord neighborhood neighborhood exhibit tangent recovery curvature vary adaptively orientation due curvature stable quantify high angle recover subspace
rsc glasso rsc rate rsc regardless new rsc joint rank row proportional reduce set nonzero row generic optimally square coincide construction knowledge bound model obtain immediate restriction single estimator satisfy q inequality constant select countable rank pattern essential matrix satisfie n computational responsible selection existence force address propose norm two estimator define minimization restriction glasso synthesis strategy refer lasso mild say satisfie index iff row design literature depth diagonal pt eigen tuning minimizer large enough nonzero pt stay adaptive predictor prior contrast regular glasso differ glasso need refine three suggest consistently rank rsc singular exceed span glasso two mild restriction strength detection problematic prove rank technical rate let satisfie restriction small pay efficiency initial analyze validation consistent square estimator true discuss present computationally two construct b conclusion provide satisfie q indicate comparable perhaps stage select glasso rsc adaptive select consistently prove follow consider row rank estimator spirit involve becomes couple c j minimum j matrix equal directly nonconvex constrain may way lasso orthogonal abuse notation optimization r n v refer iterate statement monotonically stationary suppose fs k grid rsc estimate single tuning criterion select em block convergence stationary require crucial need optimization denote kronecker I subproblem instead perform low thresholding v soft operator jt k resource need flexibility uniquely strong arbitrary accumulation monotonically simulation I normal matrix entry row feature resemble thus rank constrain variable report similar consideration even correlation rsc glasso minimize influence observation tune similar lar restrict result correct path suitable pt set table summarize mse histogram turn asymmetric computed goodness comparison l r experiment pt automatically useful surprisingly yield new predictor associate fold cv find complement analysis score rsc variable order associated eigenvalue instance important accounting roughly simply quantify cognitive set infection cognitive cognitive performance typically measure via cognitive five domain work processing explanatory contain clinical imaging measure several matter initial rsc massive especially derive run new score select aside mean rsc newly propose method demonstrate existence cognitive establish suggest perhaps fractional derive appendix generic coincide column furthermore space favor hold penalty writing c imply b inequality k b inequality conclude entry consequently write q find reasoning side row notation imply second e e b f p e inequality complementary case inequality hence eq ep conclude proof fix globally respectively tucker j j take combine ep find decompose remain projection column chi degree use schwarz right invoke complete theorem yield term penalty b r take proof global lemma map contain outside convergent subsequence solution pt refer characterize let k fs composite set v analyze minimum globally start pt fs glasso attain von inequality svd globally uniformly pt fs fs f describe point prove set r fy x sx j sx fy fy sx lf fy f fy sx similarly close map theorem far converge begin context view unconstraine manifold explicitly fs page continuous since uniquely attain directly lemma denote limit converge fs j fs point suppose fs contradict strict apply algorithm system recall minimum stationary without scale kt fs fs pt know describe similar ms j jt j ss argument pt accumulation fs lemma apply verify equivalent minimizer give global remark remark support nsf grant dms support nsf propose reduction regression exceed size sometimes complementary predictor matrix article gain prediction obtained consider motivate new rank least square penalty impose restriction prove extensive analysis response q measurement response subject unobserved set need denote index nonzero counting observe free furthermore span column either parsimonious model propose popular rank tailor approximation value second reflect belief method class effective parameter higher especially study model suggest analyze strength reduction square tailor article impose sparsity restriction coefficient aic univariate penalty rank result estimator coincide rate cf
fitting potentially region unique region error paradigm turn effective behind region output combine overall partition input combine expert general well derive variable derive necessary gradient reformulate address expectation maximization consistent responsible paragraph standard learn activation activation classification output layer node hide unity competition different competition nan probability combine superposition usually together mlp observe try take strength account discuss implementation show network stress sake simplicity require partition space color follow regime observe dr vs express source compact lie vast fact break optical arise bi particular source color fact two inside window change sub domain paragraph regime region heavily mostly determination employ fuzzy fuzzy hereafter sharp oppose fuzzy counterpart give metric find centroid minimize cluster maximize reach belong different sharp work sharp counterpart find centroid belong distant particular identify centroid membership determine coefficient equal partition membership large threshold assign soft introduce redundancy translate pattern allow belong optimal although address mlp introduce negligible identical train prediction choice randomly couple normalize unity plot network variation precede optical galaxy optical optical use determine number network determination case threshold reach reach optical optical optical digital survey dr confirm optical retrieve survey galaxy source classify galaxy clean estimate survey band compose retrieve dr table kb optical source subset sample database source kb confirm optical source counterpart optical band kb subset kb query database method encode magnitude correlate color represent measure thus discuss change color encode partially distribution dr plane error color evaluate individual correlation color color case window correspond error color corner error vary finally spread plot high color symbol source exploit contain color uncertainty uncertainty experiment produce less color color yield experiment involve determination optical galaxy magnitude color error magnitude galaxy obtain matching source fit two image extend source band profile well fitting magnitude magnitude correct accord provide third magnitude use optical color uncertainty far magnitude attribute source kb distinct class perform vary parameter one yield term statistical discuss output well use galaxy variable hereafter give word hereafter evaluation accuracy experiment table physical motivation characterization mean paragraph choice total respectively experiment involve hereafter galaxy diagnostic equation diagnostic select criterion separately respectively class optical optical clustering mark optimal high optical galaxy optical optical uv expert expert expert expert neuron gate epochs gate learn gate gate expert treat explore expert experiment evaluation galaxy retrieve color single magnitude training describe single distinct c mean cluster perform kb uncertainty color uncertainty magnitude determine fuzzy dimensional space add color uncertainty fuzzy member account membership show member kb galaxy histogram optical confirm four color associate uncertainty determination cluster kb source color expert generate optical color uncertainty fuzzy fix determination carry feature distribution kb histogram optical uncertainty color optical filter filter magnitude therefore whole available color determination final reconstruction optical experiment show figure galaxy database obtain galaxy way kb galaxy clean band observational retrieve identification sake completeness format give uncertainty service name description unique object right degree color double double color double double error belong determination galaxy extract show figure appendix galaxy kb certain galaxy kb galaxy galaxy significantly galaxy hereafter call band employ deep galaxy case evaluate contamination high galaxy census galaxy galaxy release include span survey overlap galaxy cross galaxy apparent magnitude filter galaxy fraction assign method consistently uncertainty quantity evaluate statistic bar magnitude cause low statistic dr galaxies apparent band symbol optical candidate extract compose source type classify extend clean filter band source query retrieve galaxy change candidate extraction additional quantity derive method candidate common available retrieve matching original optical candidate dr service long unique double degree degree long double r I color color double kde estimate relative double kde estimate p double kde kde f opt uv optical extraction include candidate rely characterization confirm employ cluster achieve separation combination algorithm includes follow perform salient confirm extract kb source extraction source close candidate associate dominate candidate candidate confirm probability associate distribution star kde refine reduce completeness extract yield source first third subsample optical available optical optical detail contain source query retrieve reliable counterpart appendix describe cone c unique error error long double double color r color color double double kde double kde p p double opt uv characterize observational use thorough description bias comprehensive z average number overall variation reconstruction measure spread source z z galaxy hereafter use estimate alternative accuracy paper wide ground survey optical survey optical optical optical galaxy table galaxy optical knn achieve method bias method reconstruction optical knn much aim determination empirical color optical optical consistently method exception normalize uv optical column describe determination optical optical galaxy optical optical statistical discuss diagnostic exp exp belong kb kb possible large coverage reconstruction value show sigma function source show axis percentage source kb source set optical galaxy optical set randomly draw subsample minimize effect source behavior experiment increase set create kb exp exp exp exp kb always evaluation accuracy advantage ability statistical useful scientific error represent difference evaluate color evaluate absolute train carry approach slightly different use expert except employ table reconstruct source belong distinct plot variable low panel panel color code evaluate source vertical dash represent emission line characterize spectra shift plot show color code dash emission characterize spectra line filter resolve l c min opt cluster max epoch expert neuron gate gate learn gate gate error involve error estimate lie inside degeneracy vs plot upper occur characterize shift system shape globally error compatible c optical right average profile black plot emission similarly vs plot optical optical sample contain region optical plus error estimate real uncertainty extent reconstruction heavily yield completely phase generate plot absence belong hereafter provide object optical unlike error source sample evaluation inside kb spaced bin error spaced presence lie prominent source reliable involve plot associate optical optical effectiveness reconstruction depend value explore determination efficiency completeness source value completeness third maximize efficiency e product efficiency completeness couple large efficiency second experiment optical n respectively optical reliable estimate interval determination retain reliable inspection kb optical color symbol quality histogram subset difference reliable vs experiment concern estimation optical c exp exp diagnostic reconstruction source determination contamination exp show plot right source expert determination working galaxy able distribution kb template besides give description work also determination optical galaxy optical accuracy optical galaxy express dispersion variable optical reach thorough apply provide perform result experiment optical galaxy produce optical optical associated detail method relatively optical achieve priori convergence newly acquire process acquire new kb requirement become mining cope stream optical survey produce amount total collect part drive framework new old particular traditional provide hypothesis accurate galaxy survey put nature call group group appear well fact drop employ technique drop non template hypothesis drive fitting link employ complex consistently use together expert base gate exploit hybrid architecture learn traditional physical neural interesting architecture different integrate strength belong domain dm consistently predictor dm technique training gate predictor hybrid approach empirical classical method method template review template acknowledgment take author acknowledge mostly library foundation http www project retrieval publication tool protocol international publication publish cone search service service center ia version anonymous comment improve retrieve whose use describe server system g galaxy example query retrieve dataset query dr server p bad retrieve counterpart p join join p distance center department sciences university associate california institute technology usa availability survey become crucial expert datum technique technique exploitation base compose optical galaxy optical square accuracy galaxy extract optical uv paradigm virtual field observation galaxy mining survey ever digital survey carry extended human approach mine advanced technology etc stands recognize scientific theory shall task galaxy etc observation still demand still abundance observation branch stem long rich technique example colour colour classical consider number member uncertainty possibility extensive effort year candidate instance determination visible scale source trade derive quantity e arise reliable select advantage obvious latter select digital release dr extract effective source magnitude range find theoretically lie limit spectrum galaxy determination galaxy type structure investigate reality physical characteristic galaxy worth less template differ among aspect knowledge kb kb accuracy hereafter interpolation modern wide mixed survey combine band thus number significant subsample digital survey remarkable mixed survey benchmark optical select characteristic effectiveness evident encounter critical tune galaxy template fit part degeneracy bias final mining degeneracy minimize derive belong kb shall relevant exist color see emission mechanism approximation example present dimensional training
q message miss factor message marginal generalization factor cycle bc bc bc indicator box later consider version channel function one go fp generating kp ideally gibbs produce sample remark introduction base address turn capacity channel constrain method turn resort describe layer importance q verify importance useful feasible feasible obvious choice factorization helpful base gibbs acceptance require find may address several auxiliary part inside outside importance also importance j particular efficiently gibbs follow index fix factor cycle cycle partition fig configuration x backward draw chain irreducible gibbs fast naive analogously tree simply drop e estimate graph fact gibbs sample importantly direct cf easy noiseless capacity horizontal graph fig likewise follow easy pass factor graph noiseless bit vs plot path fig capacity vs several symbol fig issue mix improve vertical variable fig product still equivalent time mix result reduction computation fig respectively bit figure single loop loop carry task draw unconstrained I however draw p loop compute q follow fig principle estimate handle slow channel additive noise horizontal dot noiseless channel noisy constrain channel additive white signal snr define specify db symbol vs db different path information rate symbol vs sample noisy constraint plot different grid compute db capacity noiseless bit symbol fig db different path snr snr decrease isolated mode good importance value require show db number limited plot limited constraint db function point temperature partition snr low snr layer temperature note carlo computing rate model method channel open case input constraint without belief information guarantee gibbs de product method knowledge channel
actually least probability ce corollary several constant line phrase depend context lot sub row contain matrix index row contain row index contain column actually column adjoint denote element support define fix set cardinality introduce deep absolute provide give scheme introduce david later constant value first two replace provide hold sufficiently small assume throughout prove appropriate requirement inexact optimization explicitly inexact duality exist vector satisfy x f plug moreover construct construct f block divide c mention deterministic let absolute replace high provide sufficiently construct imply independent moreover calculate know follow etc orthogonal satisfy recall incoherence ie ie j suppose pp slight proof inequality hold os subgradient nuclear q h construct also prove notice l high enough construction q large enough uv ie ie iid random high n z uv z l z z provide provide therefore z j sufficiently notice similar scheme method factor least square part dual actually apply acknowledgement grateful ph help manuscript pt corollary example mathematics stanford university stanford exist compressed sample introduce recover signal tractable fraction corrupt provide fraction corrupt nonzero entry tractable fraction keyword robust isometry cs approximately acquire numerous field cs represent establish solution guarantee original iid occur provide dft discuss cs compressed sense totally absolutely entry want recover signal accurately mathematical nonzero coefficient recover discuss frequently acquisition device typically nonlinear occur component portion sensor measure outcome typically measurement report totally measurement collect error investigate recovery cs noiseless cs rank suppose square year nuclear sum original low meaning iid guarantee probability positive numerical dependent concern cs broad applicability wu et typical true model defer section random gaussian variable choose numerical everything else nonzero need imply adversarial moreover lasso pursuit model iid obey model introduce matrix support cardinality restriction concern sign symmetric solution provide constant support recent randomization first fix sign de randomization row dft numerical proportion sign know would hold general emphasize little strong free important remove sensitivity introduce good however little decide noiseless rank write originally support fix either specifically k model numerical sufficiently available approximate follow recovery recover actually slight modification argument large suboptimal completion tight noise noiseless take validation compare exist common choose notice motivation satisfy bit li set sub sensing x recovery obey restrict exact recovery sparsity requirement standard however go piece later appear form matrix incoherence independent uniformly randomization sign nearly optimal word extra condition optimal model discuss tight continuous paper plan include common assume similar require condition least impose bad differ prefer may proof scheme exploit another matrix always assume interested strong condition al
construction central limit variance observe evolve odd consequently evolve geometric parameter assess criterion order reversible kernel consider apply irreducible acceptance set result reversible stationary metropolis yield maximal inferior variance assume distribution transition allow euler discretization advance diffusion set hasting bernoulli execute however hasting alternative hold claim metropolis view acceptance order example sampling algorithm normal example transition geometrically reversible trajectory setting cn normal normal normal sample q sample normal sample cn normal markov report variance corollary stationary substantial normal become inefficient cn order cn big operator operator asymptotic walk bound increment metropolis metropolis metropolis thank helpful eps stroke department university uk department transition integer transition chain derive spectral formula limit assumption e depend theorem fundamental chain reversible ergodic chain associate refer spectral markov probability nonnegative integer sample context set irreducible proposal move accept metropolis ratio big roughly speak motivate recent advance equal reversible stationary nonnegative integer kernel geometrically analytic spectral kx explicitly corollary reversible ergodic transition kernel markov dominate stochastically dominate pe moreover conclusion another direction study markov chain bound pt chain bound bounding hold geometrically meet bound
paper whereby hypercube connect close error bound provide rigorous particular robustness notice determine rotation translation inter matrix position shift throughout one position original position center note sense invariant transformation generality yx yy xy yx f cn hold high distribute connectivity necessary localization problem connect h dimensional vice regime far surely nonzero semidefinite localization follow sdp minimize compute ne th th abuse notation sdp step gram node key obeys constraint solution eigenvalue minimization coincide remove degeneracy eigen center reconstruct point origin high p converge hypercube connectivity sdp coordinate conversely constant general stress see next use stress matrix random graph see prove background localization attract significant past name propose localization guarantee prove noise group distance reconstruct matrix distance short pair localization scheme sdp crucial existence sdp efficiently check whether uniquely find realization maximum unfolding sdp problem local metric interpretation lie manifold ambient low representation attempt apart ambient maximize total distance give large method broad paper model relatively broad class remainder organize brief notion property implication application theorem contain proof use prove reader convenience symbol table give distance finite node coordinate transformation brief overview definition theory interested thorough discussion undirecte correspond stress mention establish geometric preserve satisfy call equation anti derivative span account freedom transformation correspond degree correspond dim say dim framework scalar stress stress set edge length configuration recall coefficient connection follow result prove stress globally stress stress geometric graph lower singular value stress prove basic number hypercube volume node defer bind ball symmetric vertex eigenvalue stress computer present geometric let laplacian degree node eigenvalue constant restriction notation subspace span orthogonal denote throughout constant frobenius singular set edge convention adopt represent vector convention notation center position match note connectivity threshold logarithmic numerical idea show namely immediate bad measurement perform dx c r reconstruct position question motion set hand vanish hypercube sdp recover graph namely poisson globally w already establish phenomenon geometric globally globally phase geometric exist globally high area determine position sensor hardware rule system sensor acquire interest global environment semidefinite network localization develop advance inaccurate shall place ambient dimension either connectivity various interference nearby accuracy measure technique measure wireless device signal arrival measure receive ratio proportional receiver use dominant receive measure receive denote received measure magnitude per nr remarkably accuracy average compatible connectivity time signal difference distance receiver velocity measure lead proportional theorem obtain dx nr suggest design system stress bound prove adversarial similar ambient lie implication sdp equivalent assume little generality unit hypercube estimate pair try find position reproduce geodesic nearby euclidean reduce mathematically localization whereby distance measurement estimate depend curvature manifold vary unit radius minimum equal show lemma ij support claim estimation paper focus sdp noise direction take zero variable instance another due consider improve way instance manifold maximize pair add constraint short reconstruct geometry cloud incomplete inaccurate local angle arrival gram base lemma numerical w q constant p proof lemma frobenius assumption triangle take eqs eq q prove claim term psd stress construct psd psd stress combine eqs next construct psd node clique clique define establish property clique true ready stress zero next immediate psd almost last fact h corollary low bound p finally remark node thesis turn author show smoothly vector remark claim claim markov proof next claim clique defer appendix ij w q near I I n nm x establish graph appendix theorem decompose I x u j obtain matter explicitly constant clique degree claim defer probability di prove suffice claim obtain complete da xy yx ty tu u tu yx ty tu di xy gx compare key ingredient subsection proof appendix prove define e gs step whose tuple entrie tuple use set whose tuple set adjacent graph subgraph see illustration chain consider chain choose choose number line illustration vertex construction partition bin h bin vertex choose obtain perturbation width setup lemma chain appendix let exist defer position theorem width force flow number chain edge term accord describe contain bound p therefore equivalently xy chain node side desire number set vertex number x lk lk thesis notice value convenient generalize system variable write anti eq force impose force state net u interpretation mind find constant simplicity proceed defer appendix assume direction easy solution show force suffice observe force value write lk dd one form lipschitz function satisfy node node contain hence force respectively low map hypercube curvature radius instance slightly hypercube map let distance measurement measurement also ij crucial point step dimension input return gram eigenvector eigenvalue algorithm project estimate position separately taylor provide let unitary refer small defer appendix p tm p small eigenvalue addition small entry eigenvector small applying remark small z iid surely eqs consider worst uniformly introduce map claim algorithm run sdp bar configuration define accord let summarize fig respect stanford award nsf dms grant thank comment width dx plot width plot delta dx plot reference otherwise application right side become bin non overlap length total bin hypercube mn n lb apply union get line bin imply thesis orthonormal k w k l ti ki therefore vector j configuration generic imply adjacent graph dropping prove nr dc markov sequence consecutive possible choose follow bin side discuss node path node bin bin bin uniformly
observe th moment fourth observe var jt proof continuously differentiable real e first q triangular first pt first triangular inequality substitute item substituting give I differentiable work change amount I sum difference order expansion martingale q drop convenience ease ten p tt proceeding e n hence dominant note ta second term however order remove ny dy dy dy eq moment dy dy satisfy lemma prove divide respectively give n cn hence give time continuously x nf diag var entry nf use eq approximated li give uniformly q proposition lemma give proof variance mean variable assumption imply first equality second start give remain bound equality proposition obtain u tu j r jk n identity term twice v establish five cauchy schwarz establish bound upon partition define eq property fundamental pair say break pair fundamental index pair contribution symmetry representative assumption case contribution discussion concern fundamental impossible fundamental fundamental fundamental eq q turn break pair simultaneously break requirement break change contribution break assume pair break must contain remain assumption pair supplementary law iterate scale corollary thm robust es anonymous associate first la la sciences research author school economic finance material cm universit e du mail tp weak selection box statistic consistent critical lee tailor autocorrelation coefficient provide many coefficient simulation experiment classification primary secondary keyword white inference automatic test optimality white context autocorrelation regression confidence interval correlation residual residual improper autocorrelation diagnostic tool finance test noise hypothesis fan wu derive white extended goodness fit test kolmogorov er von test hypothesis refine residual setup sum lee white wu contribute propose nan nan test implement lee extend lee include origin kernel improve result appeal feature er von detect directional converge nan parametric detection type slow local spectral conclusion er von powerful box type universal exist er von point magnitude detect still describe type alternative multipli correspond scenario term policy effect term alternative expectation past give process close difference alternative finance alternative test intuitively explain wu normal critical suggest suggest box provide nr box provide enough order square box correlation smaller show er von important limitation critical ad hoc detecting instance unlikely give test power detecting need properly development drive various test detect alternative converge fast class unknown chen nonparametric hypothesis relevance le equivalence spectral continuous ready white exception fan maximum standardize box et drive choose order selection procedure west simulation selection justify west although optimal testing differ therein choice base aic adaptive white drive white noise alternative mention suitable optimality compare er drive wu report propose calibration automatic test drive west order conclude material observe covariate stationary mean nj residual density test kernel division evidence grow normal shall subscript n multiplicative propose select maximize ep k n penalization penalization differ aic penalty term term follow base maximize equal differ side critical drive covariance base formal statement variance e pn n bias trade value statistic lee run lee residual twice continuously motion rejection use sample variance tu j tn rejection region turn mean variable z alternative na n n dependence state nan follow wu stationary satisfying moment contraction wu measurable univariate vector copy change assume fast mixing use et al satisfy nonlinear volatility bilinear wu reference main away differentiable critical twice continuously differentiable support regularity c c nu maximal process brownian full ii admit u n n assumption interval impose coefficient condition come high order finite base cauchy schwarz implement propose test principle alternative role proof restrictive suggest focus since various process comment ns tu consequence wu verify ols version b lee show condition restriction value exclude next issue restrictive describe requirement white alternative important stay test statistic hand level nominal substantially address nan asymptotically sequence asymptotically asymptotically therefore impact impose compare element since impose either hold technical contribution without power allow bic drive view potential negative see side behave standardize exact b supplementary achieve alternative go alternative similarly sequence alternative exist consistent lag construction statistic latter sequence admit penalty sequence condition impact impact use fact alternative situation detect correlation well np favorable sparse coefficient allow coefficient converge cover independent variance setup possible detection achieve wu n wu test lee region preliminary replication quantile respectively penalty behavior white noise standard report level replication simulation draw small ensure observe size close slightly less test use benchmark drive lee west lee pilot bandwidth remain differ valid use standard er von distribution three chi chi lack chi square reveal sensitivity skewness size slightly strong chi except chi experiment consider white martingale examine I normal coefficient expect behave uncorrelated report bilinear label examine process uncorrelated dependent finally residual test adapt thank test even small suggesting distortion critical lee behavior either pass behave follow adjust desire normality alternative expect alternative especially simulation impact decrease decrease lag similar characteristic power test drive surprisingly outperform test outperform illustrate nine example undesirable correction lee consideration compute preliminary label figure nine except process suggest test nonlinear white noise power noise well bilinear second randomized simulation moving shape coefficient p infinity covariance scenario report experiment test show drive test lower really affect propose automatic noise observe residual new test statistic lee estimation alternative coefficient autocorrelation moderate lag enough alternative average autocorrelation test wu scenario impact may cause finance rule deviation martingale alternative alternatives simulation new cope white empirical finance also confirm regard alternative nonlinear process goodness test goodness criterion autoregressive integrated move average series university empirical test check goodness test process detect martingale automatic serial test significance diagnostic uncorrelated white drive specification regression minimax specification consistent serial unknown conditional form parametric without estimator bandwidth autocorrelation presence covariance matrix spectral origin bootstrap noise goodness fit adaptive testing wavelet asymptotic strong principle dependent red red dot lee coefficient range sample replication cm robust pt pt supplementary material proof lead line integer part copy condition ensure supplementary material proposition impact deal residual thank proposition proposition k cp l kl kl suppose j q let p nr ensure enough give ensure retain value level critical p np chebyshev inequality proposition eq stay away first nan show imply lemma eq observe residual give alternative enough q eq set spectral density center coefficient define old constant spectral r ik b g j j ij md eq enough kk f give correlate alternative p pp lemma log brownian process depend family sequence behavior lemma satisfy level test since hold old equivalent brownian motion bayes base choice schwarz op nj n n therefore
sense aspect bound recently area extensive contamination svm synthetic briefly heuristic amount contamination mis collect contamination understand co situation context mapping yield many contamination datum contamination model bind statistical rule throughout function decision decision replace rule denote substitution fix among call bayes risk distribution stand surely simplify quantity distribution bayes learn minimization classifier differ sample totally question classification train testing generate wish access contamination vanish classifier state different state theorem sharp achievable contamination eq contamination uniformly lemma rf x rf contaminate xx gx continuous e completely membership hold imply lemma conclude contamination induce apply generality decision center write let take universal consistency imply risk arrive sharp contamination indicate contamination make small suffer contamination explain svm fraction label rely fortunately currently popular early stop statistical contamination research statistic back survey extensive estimation primarily problem relevant sensing however al investigate image mis work purely underlie mis form forest form patch type additionally al impact mis xu error location richer broadly divide stage stage roughly deal contamination learn bag work include related deal broad contamination explicitly difference source long small whereas contamination numerous publish domain adaptation beyond give reference particular david follow error indicate source nature replace vc rademacher theorem nature learn vc often loose still loose asymptotically vc bind small underlie vanish way component contaminate easy contamination establish contamination david et particular david bind quickly approach whereas apply consistent classifier dimension polynomial neighbor classifier without consequently svm kernel boost bag nested decision dataset test average generate aa set test respectively classification contamination figure ignore second adjustment eps scale eps class originally boundary contamination repeatedly multi bind contamination simulation eps eps total dataset take repository detail htp c set training include split test size training rest aside train rest contamination result contamination average image dataset use example reference cite description linearly slow optimization include contamination svm plot htp eps eps eps eps eps image image interest interval day year optical relate acquisition sensing image scene offset cause bilinear index mis mis ii mis roughly contamination sample mis sample take fold validation fold c I mis relative area correspond pixel homogeneity homogeneity number mis since field effect mis half may cause far svm adaboost applicable collect table adopt result htp convert original contamination contaminate contamination leave future mis heuristic pixel mis thus pixel serve underlie boundary roughly mis say proportion pixel fall boundary inspection result train contaminate boundary follow pixel patch center two patch pixel patch mis capture formulate contamination nice feature bind distribution free datum extensive contamination tight adaptation literature classifier infinite already contamination mis beyond use contamination useful empirically show boost training carry success co ht eps small amount example originally high build noise f w clean denote clean co clear case rate label typically small potential benefit training co train feasible example training help limitation contamination image I mis image way desirable account work derive contamination desire impact datum classification leave interested reader acknowledgment thank square department environmental science management california google view problem mis determine result pattern contamination phenomenon mis model applicable many measurement etc contamination classification study statistical contamination apply classifier extensive simulation replacement random derive motivating example mis occur mis phenomenon map wrong usually cause acquisition device underlie mapping map datum collect scale take angle mis image eps eps primary sense monitoring acquire characterize liu classification xu etc never perfectly make mis acceptable al achievable thus important assess mis round weather additionally type part occur amount contamination broadly contamination cause datum
branch scientific mining analysis decade author theoretical practical base amount interest numerous investigate art technique hierarchical cut method certainly methodology regularization theory author use canonical define differential apply potential task two joint spectrum treat layer respective believe concept spectrum share base processing way classical certainly focus future partly author thank figure grateful bfgs de mail de mail observational multimodal nature represent address layer propose combine eigenvector result joint multiple cluster social dataset superior art common baseline summation layer extensively year usually object unweighted edge equal weighted value object subset one wide numerous reader survey mobile social bring challenge modalitie interaction modality represent whose set fig mobile phone mit reality mining mobile phone different proximity physical movement iii phone communication contribute meaningful combination possibly layer mobile leave cell assign edge phone seek propose novel spectrum spectrum layer view eventually generalize decomposition apply laplacian framework eigenvector share graph vertex cluster spectra enforce regularization characteristic cluster graph alone theoretic generalize evaluate world compare art technique baseline base summation graph result show term cluster outperform baseline competitive art contribution limited cluster spectrum multi layer lead rest motivate iii review building block section consider layer weight undirected associated aspect combination index explicitly point embed reveal intrinsic mean easily layer build mit reality mining vertex graph participant mit mining represent relationship mobile phone user term different proximity proximity phone call relationship three layer indicator blue nature insufficient achieve cluster mobile fact membership isolate vertex two entry rich information layer well unify joint combine provide addition effective group spectral novel method joint inspire popular spectral cluster main novel reader familiar skip section spectral become promise describe undirecte represent laplacian degree vertex along unnormalized combinatorial version laplacian define keep version detailed discussion choice adopt cluster algorithm spectrum dimensional form transformation eventually cluster illustrated toy show weighted graph number compute walk compute correspond small eigenvalue contain th row algorithm assignment first eigenvector new representation cluster trivial theoretical process theory multi find joint spectrum perform embed eigen eigenvalue eigenvector eigen contain diagonal eigenvalue graph layer eigenvector provide decomposition laplacian layer multi minimize write represent play role capture characteristic identity dimension function fidelity error add third constraint enforce inverse additional finally regularization balance trade convex alternate find optimize fix consequence give initialization suggest informative eigenvector initialize inverse loop solve variable quasi newton memory bfgs namely multiple layer spectral eigenvector eventually layer work eigen base average information tends treat equally layer propose target degree l w get eigenvector column th cluster assignment section novel method cluster treat base consequence help preserve layer examine eigenvector laplacian matrix weight eigenvector constant mapping mapping line vertex stay mapping satisfie scalar minimize orthogonality introduce view view condition rewrite equivalent usually illustrative mapping construct cloud keep connected vertex importantly view value objective graph eigenvalue illustrate b see stay quite mapping form embed property imply special smooth function graph process combine multiple equally highlight propose smoothness laplacian vertex enforce layer jointly smooth share layer spectral process graph follow eigenvector seek scalar function smoothness smoothness regularization fidelity term term objective eigenvector jointly algorithm worth role eigen regularization layer generalize propose clearly discrete membership information graph end jointly layer weighted target compute walk laplacian w kl n ki spectral replace cluster mean prove propagation one whose vertex propagate neighboring vertex make inference exactly regularization clearly solve iterative initial represent value parameter update convex value neighboring notice initial value relaxed graph problem membership interpret cluster way take make base task disagreement source source example algorithm disagreement multiple enforce layer disagreement reflect optimization fidelity term explicitly disagreement solution come regularization term structure indeed small two end function therefore total disagreement multiple formation disagreement model different individual respective performance three social compare mobile phone one dataset explain graph mit reality mining mobile phone three location phone specifically physical location service aggregate month weight addition phone call assign weight take ground cluster self medium school student group intend mobile phone currently research include mobile work source mit difference pair mobile layer graph third object mobile user still reflect human activity experiment come natural mining manually label one category truth cluster represent title abstract take corresponding citation reflect citation paper paper create mit much truth cluster observational intend cluster dataset reflect physical proximity phone mobile user make difficult moreover imagine dataset even mit email nevertheless choose ground well rich mobile phone challenging explain briefly propose eigen iv two solution experiment enforce inverse choose select part mutual layer mit dataset cell mutual combine act cell third phone incorporate final relative importance information share second mit dataset representative spectral apply adjacency weight layer use summation normalize summation vertex represent eigenvector graph laplacian regularization regularization propose art multiple graph
linear acoustic array communication relate objective use predictive location regression observation classification briefly theoretic relate seek expect location test monotonic entropy point include yield experimentally expensive posterior inductive theoretic require integration probabilistic close active active svms volume vs proxy improper volume vs fact posterior become intractable observe vs propose approximate approximate laplace ep vote vote disagreement posterior disagreement vote confidence exhibit mc ep ccc width title xlabel dim ylabel dim axis axis leave mark option coordinate black mark square option title xlabel dim ylabel dim line color mark mark option coordinate height title xlabel dim ylabel dim color mark coordinate color mark consistently amongst match good performance condition fig approach test line range expect knowledge noise dataset g reduce rapidly bad noise free indicate strong algorithm often small often poorly noisy dataset fig criterion maintain notion inherent uncertainty exhibit poorly greatly yield choose cluster influence point point reveal perform access objective empirical weight picking assign weight fail poorly hyperparameter fix fair select location maintain hyperparameter done mcmc far pick point randomly demonstrate theoretic gp date apply learn directly naturally learn hyperparameter active notable include ep online thereby trade theoretic computational appendix supplementary taylor expansion even term preference compute laboratory theoretic active widely probabilistic policy tractable however task classification hard achieve gain entropy apply make information experimental performance compare active low computational well decision theoretic secondly develop preference extend preference inference design internet expansion vast quantity costly call former minimize decision collect minimize risk know test advance want extreme exploratory hard motivate agnostic hand inductive seek reduce possible either heuristic margin study quantity entropy although decade criterion complicated infinite entropy perform present yield kernel interesting theoretic theoretic approach apply extended approach compare care contrast approach machine address datum directly goal active key learner choose query observe response within dependence output parameter infer active reduce maximally datum problem np made optimally seek data decrease expectation unseen g parameter often compute entropy eqn become poorly avoid exponentially dimensionality entropy introduce bias approximation calculate computational difficulty potentially update eqn insight eqn unknown insight entropy space eqn entropy usually entropy require intuition seek uncertain confident outcome learn disagreement apply build entropy around posterior approximation minimal additional knowledge algorithm represent fast discriminative gps tool challenge information quantity entropy infinite function consider probit give cdf inference intractable assume sparse though care indicate exploit query eqn expectation posterior mean variance compute quantity eqn handle probit second f pf eq reflect integral intractable perform material approximated curve convolution finally close eqn depict incur tend zero yield author previous context approximation monte consist apply predictive mean point select query practically function gradient maximally square parameter set nuisance I setting want maximally integrate nuisance gp spatial maintain posterior hyperparameter parameter certain determination hyperparameter variable primary interest preference label prefer denote task predict relation special building pair latent preference predict preference denote become define inference entirely probit perform operation gp supplementary k
suitable kernel statistically likelihood perturb define q analogous abc mle smooth nz ask analogous hold smoothed smooth noisy abc mle careful read analogous continuously reference measure condition smoothing note comment remark smoothed noisy mle smc conditional law drop law hide particle smc respect likelihood smc approximation use mle approximate perturb hmms smc I random extend perturb hmm since conditional give state likelihood standard smc evaluating likelihood computation likelihood k l lx k n density lebesgue abc likelihood manner particle see reference detailed analysis smc resample sequence price hmm economic factor directly price become distribution return asset price likelihood stable difficulty financial noisy toy transition conditional stable growth bad ie log asset stable distribute drift asymptotic bias abc mle true intuitively due perturb hmms hmms position lastly small size seem ie order theorem investigate abc mle suggest fisher abc mle decay large value indicate fisher abc mle decay fourth power large accurate use achieve use particle abc small value even bad volume around quickly abc truly practical neighborhood rapidly overview investigate abc hmms mathematically collection arbitrarily noisy mle noisy mle increase finally mild abc result help extend exist investigation firstly paper weak assumption find mathematical relax secondly suggest mle technique sample current support lemma sequence continuously cauchy uniformly second concerned identifiability additive give zero lebesgue measure zero denote characteristic follow concerned state measurable suppose probability connect see section support probability add observation general complete common dominating support equality let variable observe variable appropriate dominate identity density variable apply jensen inequality immediately jensen measurable hold v measurable measurable p since curve contain sequence open ball less b k measurable correspond dominate continuously p proof observe dominate measure computation follow dominate convergence c b establish property component hmm perturb hmm k let hmm hmm identity assumption follow b certain well property hmm extend sequence law consequence hmm extend dominate exposition give conditional obtain py l r thus bind express hand corresponding hmm leave hand conditional conclusion corollary replace property infinite concern collection hmms extend hmms l p n surely define corollary eq far first part boundedness density bound collection finite positive mass still since positive finite measure equip doubly kn equation follow letting apply function clearly continuously derivative derivative uniformly corollary sufficient inequality appear analogous manner side bound individually use fact straightforward history likelihood converge infinite likelihood continuity continuity r mapping rest fix dominate mapping g thus dominate immediately map dominate theorem also term shall prove proof completely identical difference value likelihood identity eq imply converge definition limit bound lebesgue theorem almost follow lebesgue dominate continuously differentiable eq uniformly uniformly bound eq satisfy condition conclusion identity continuity follow l p sufficient perturb respectively process stationarity observe perturb eq stationarity last dominate eq function p likelihood converge hence apply obtain finally law one prove sequence integer assertion follow order assertion since connect law probability finally densitie analogous definite sequence show every ti stationarity follow straightforward consequence apply law integral w gradient g since dominate assumption remark physical sciences college abc popular approximate likelihood often likelihood analytically intractable abc many investigation result estimator abc base estimation markov normality discuss provide implement markov bioinformatic e also recent often hmms observation particular maximize q unless simple analytically variety sequential wide conditional despite sample process parameter g lead approximation carlo another technique problem indirect hmms filter density inaccurate extend kalman filter likely expensive deal computation abc reference methodology summary statistic assume metric fold reflect estimate intuitive volume radius around factor ignore likelihood order result behave purpose issue investigate ease exposition paper result continue statistic see end section conditional state likelihood sequence radius markovian simple monte smc implementation abc experience compete see reference deep approximate parameter firstly could commonly take frequentist maximize approximate henceforth bayesian computation mle abc become estimation either frequentist particular mle distribution concentrate true abc seem mle converge place mathematical purpose bridge theoretical develop justification mle standard normality likelihood perturb hmms abc behaviour observation asymptotically bias arbitrarily small mle rigorous justification complete picture mle noisy always raise abc result suffer efficiency perturb additive independent finally study herein help provide field notation approximate mle abc overview smc implement give qualitative estimator support shall letter letter observation variable various infinite brevity shorthand notation integer variable sequence give integer denote denote shall use sequence notation combine doubly infinite random j j l essence property operate hmms ensure consistency space hmms compact furthermore denote kernel dominate state space conditional density dominate chain recurrent write law expectation law dominating mutually absolutely respect long term kullback present arise try understand behaviour extend unfortunately require perturb likelihood w dominate measure essence despite associate perturb sufficiently need order take limit constitute conditional perturb process give past expectation stationary hmm far let asymptotically let accumulation perturb alg conditional law law bound conditional log part argument mle bias show arbitrarily sufficiently provide justification mle standard mle small theorem whose mapping q continuous exist sequence follow result additional decrease positive constant theorem whose give invertible continuously differentiable hence case markovian abc summary ns markovian moreover suppose mapping identifiability system hold hmms preserve reasonable choice summary statistic show perform abc choose maximizer likelihood likelihood inherent sequence noisy nz law correspond perturb estimator standard hmms result mle asymptotically light remark observation noisy noisy method investigate abc section mle consistency normality mle provide mle
implicit improve multivariate assumption form prior restrictive appear function objective reader adopt improvement candidate acquisition rate convergence prove thompson option portfolio strategy combine appear optimize acquisition function version project find acquisition objective acquisition evaluate observation parameter draw policy policy gp call boltzmann sample candidate transition one sample generate optimizer find maxima unnormalize boltzmann exploration slight sophisticated greedy draw optimizer function follow contextual bandit dimension popular block gibbs system trivially vice interaction positive tend opposite sign landscape interaction consider behavior gibbs wang regular model boundary dimension periodic boundary square interaction bias phenomenon arise experimental protocol trial compete sampler store energy visit energy comparison analyze energy trial rapidly markov parameter discard sampler connection unit visible vice versa visible illustrated learn one variable figure well wang perform slowly carry compare popular like would connection tree address present considerably gibbs bit lattice computer depict wang minute ise model minute minute second minute minute second computational spend flip length sparse possible speedup store l updating flip entail replace result variable flip contrast densely connect om om simple despite rbm densely densely easily proposal parallel unnormalized flip unnormalize flip flip divide evenly among processor speed updating unnormalized near take sampler sampler simply rough could happen find small move sampler affect degree connect compete wang demonstrate extent indicate sampler range model already nonparametric optimization parametric carry point ise develop acknowledgment thank modify slow research support joint wave proposition wang specialized value avoid walk state bit mcmc self dynamic integer remarkably broad task boltzmann machine arise computer ise also boltzmann machine deep apply rao integrate example effective discrete great statistical efficient monte inference ise vast domain challenge sampler make notable hamiltonian monte effort deal space domain wang well densely graphical latter acceptance trial simulation possible rejection accept move favorable get present specialized equilibrium monte move self avoid walk mcmc unconstrained system tune trade exploration bias proposal distinction seminal author concern inherently namely system another priori idea impose self generate rapidly state class know yet propose sequential coupling importance boltzmann machine sampler mcmc proposal enhance apply consider meta wang system consist state boltzmann system statistical learning particular energy mcmc gibbs heat sampler suffer mix minima dramatically rely mcmc idea present relate mcmc local previously focus advantage particular state ham clearly agree I I extension act sequence uniformly via acceptance low single flip move tend move incorporation requirement asymptotic procedure force exploration equilibrium take type biased length sequence visit procedure bit select equivalently f bit flip call show element sample follow likely sample principle choice behind energy bias propose configuration final state imagine uniformly high extremely unlikely accept begin bit sample yield reach point self avoid process imagine space lie flip avoid walk state induce acyclic subgraph lattice move twice stage process obviously avoid note construction impose transition occur neither seem ask less molecular try yield traversal landscape avoiding visit substantial portion reach I multiply straightforwardly move delta result term ideally mh would evaluate propose f accept mh unfortunately length marginalization massive k special case obtains follow accept state care detailed balance still mh reverse straightforwardly map example see somewhat involved support coincide turn balance balance side enforce balance k summation ready chain balance propose accept computational evaluating accept ratio order proposal numerator completely involve proposal look transition kernel evaluate path right side accept call accept strong f reader detailed balance long nonzero example visit fortunately set visit trivial collection flip set consecutive integer separate occur two length enforce correctness section discuss choice discuss matter ise type interaction propose move idea continue familiar note term mind present equilibrium note restriction unnecessary readily consider principle concatenation straightforward iteration move flip proposal intermediate avoid evaluate flip generate distant favorable unfortunately priori guarantee intermediate state visit often sequence especially potentially pass automatically proposal segment iterate procedure operate value encourage new two bias unfortunately likely rejection rate numerator ratio enforce particular low temperature deal three type p frequency choose sample encourage local towards desirable minima last seem somewhat since reject purpose acceptance move carefully reject due help accept ideal candidate explore effectiveness adaptive tune free parameter length respectively previous group free symbol define probability tune parameter fortunately challenge candidate increasingly stochastic obvious function minimize auto specific lag previously
furthermore specificity ratio low alarm respective alarm observe network attribute limitation structure induce effect evident specificity decrease grow confidence addition appear grow sample specificity sensitivity show fig also alarm hoc usually exclude dynamical level size ad hoc certainly conservative choice impact sec incorrectly significant ad hoc fig plot systematically ad hoc low threshold comparable specificity negative across hoc attribute separate entity concern identify biological set abstraction underlie mechanism pathway context pathway full commonly ad hoc identify significant hoc correctly aspect strong dependence hoc effectiveness propose expression datum al package determine significance significant sample empirical cdf early across identify fig respect edge false positive spurious application et contrast et study identify influence cell arrive posterior learn present flow vertical dash investigate functional record molecular intervention perturb perturb study sec fig estimate threshold non significant present sample size big algorithm exclude use et conjunction direction edge direction direction edge datum graphical considerable biological medical community especially interaction entity identify significant ad hoc edge learn noisy negative propose statistically identify graphical cdf cdf effectiveness synthetic expression define setting learn uk technology sciences national library lm also thank useful suggestion reference association molecular pathway mechanism graphical especially ad hoc often conjunction significant propose statistically association identify significant association graphical edge counterpart effectiveness demonstrate datum publicly molecular protein sensitivity specificity result also demonstrate across vary specificity linearly logarithm systematically maintain level specificity reconstruct paper study use structure ad hoc threshold significant association ad hoc significance association motivate hoc spurious conclusion association model average molecular statistical relationship compose refer express relationship refer graph express relationship semantic principle graphical imply variable find acyclic nature focus univariate former parameter interest usually estimation structure determine encode present coincide structure thank optimisation difference terminology group class goodness score al et al assess arise algorithm measure datum unknown systematic assessment particular identify bootstrap resample average original datum edge present eq mu mu degree another structure accept unknown serious limitation lead use ad hoc define assessment study setting first minor accounting edge distributional require addition make latter either hybrid computing use offset propose statistically cdf observe confidence cdf subsequently al al q intuitively clear element ideal either non significant arise happen learn element equal significant edge provide separate importantly datum provide motivated one amount cdf way depend common norm divergence leibler grey change straightforward programming common q norm compare robust variety configuration identification thought follow follow even individually example model undirected q grey edge great satisfy test synthetic proportion edge correctly proportion miss correctly proportion edge set vary commonly use benchmark alarm network provide alarm message intensive care possible severe structure edge network design evaluate car risk compose incremental constraint child network grow algorithm independence test mutual information unlike mutual consider separately hill hc score equivalent arise uniform approach consider min hill combine max parent hc test illustrate performance measure network alarm possible bootstrap similar edge ignore edge network specificity compare threshold approach
separation bss bss exist prominent worker achieve sense hyperspectral image ica bss term matrix describe boltzmann shannon temperature internal principled free minimization free energy much suitably generalize scope extended compression communication accurately model commonly biological paper formulate variational source maximum dual entropy normal average via unsupervise formulate aid independent superiority separation g single information duality implication duality permit say infer logarithm derive previously ref possess property exponential form important concern expectation expectation define linear employ satisfactory molecular approximation successfully utilize iv describe employ separation observe pixel scale prominent utilize characterize exponential exp entropy introduce note equivalent g entropy govern calculus apart provide analogy expression calculus statistic derivative logarithm wise clearly lagrangian relation sum condition q j eq canonical n ij entropy substitute result potential relate substitute aid aid cast henceforth information update matrix ascent result taylor yield characterized exponential one face perturbation un ref additionally derivative replace expansion turn procedure double pseudo force dual break logistic vi follow pseudo value provide constitute experimentally fact former initial merely input far simple unity signature number iteration separation model b readily statistic exhibit correlate separation employ paradigm
interior surely weight previous close convergence make close surely slightly sequence allow satisfy make root rate eigenvalue unclear eigenvalue helpful case sure available consequence taylor almost surely compact become fast considerably fast albeit support weight message vanish convergence previous support know gap pr handle consistency example illustration simulation elsewhere finite accurate generic produce estimate mix whose simplex u leibler role support argue appropriate minimize adjustment start finite huge element optimization minimizer model cluster galaxy velocity galaxy datum finite location mixture mixture galaxy methodology component consideration component interval grid candidate pr method identify six galaxy cluster closely count substantial pr present show aic scad hellinger distance count five include besides find zero include grid let decide pr fit c estimate aic three take ccccc exp exp exp author suggestion department mathematical university portion complete recursively design pr simplex boundedness trivially assumption theorem exist lyapunov ode differentiable eigenvalue part corollary chen recursion pr distribution know pr mix mixture consistent condition misspecification finite theory converge well kullback nearly pr know modify pr estimate compare rate kullback function approximation challenge algorithm fundamentally different way hill learn sequentially like pr dominating unlike surely pr depend dominate pr density goal little shall approximation author knowledge fully therefore shall analysis pr possibly support case pr surely parametric close leibl also one choose algorithm naturally suit finite mixture unknown show pr yield estimate unknown support establish unknown example illustrate detail reader refer nonparametric borel present estimation sequence compute return pr connection distribution take connection pr unknown pr marginal section support pr recently become let set mixture density build show respective topology show close leibl divergence identifiable almost topology bind pr suitable rate leave suggest upper bind correspond boundary nearly root conjecture hold finite fy py xu two assumption mixture model identifiable one define leibl henceforth shall infimum closure lemma ensure particularly rather pr asymptotically pr generic mapping pr design root nothing conditional expectation density equal consequence martingale investigate ode ode limit purpose definition plug follow vanish equilibrium ode converge regardless useful lyapunov ode differentiable neighborhood f lyapunov lyapunov equilibrium show slight kullback lyapunov context map lyapunov ode calculus reveal fu
simulation analog digital cs synthetic digital algorithm approximate achieve algorithm depend approach solve interior ls investigation produce accurate digital time possible significant issue implementation scope cs parameterize signal depend variance priori empirically observe task variance improve multiplicative desire ensure display inspire index simulation depict relative calculate evaluate solution essentially digital algorithm term optimization digital compare calculate rmse meaning variability comparable digital solution difference row objective bottom plot solver normal digital solver well hard recovery synthetic value mse simulate parameter previous medium produce good converging second whereas converge order individual average case basic though circuit many include power consumption constant analog solver converge approximately simulated speed order magnitude fast solver real recent implementation especially accounting interface analog circuit right convergence hard fidelity difficulty recover fidelity converge support investigate medium cs problem display plot show simulated reliable interestingly digital g multiplication significantly analog like multiply circuit size increase time implementation demonstrate synthetic achieve digital solver improvement several order magnitude digital analog circuit potential image cs recovery section simulate speed compressive short scan improve patient throughput scan furthermore compressive mr medical perform high without cs acquisition dynamic subsample image transform wavelet transform fourier solution transform transform coherent subsampling poor result synthetic two section signal large image wavelet digital image reconstruction mse relative quality simulation take second translate concentrated explore many fitting form process basic variety analytically determine program approximate norm modify norm well group zero coefficient coefficient explore function note program theoretical wide impose condition analytically translate cost function require activation result case behavior imply huber scad nonlinear activation regularize program consider square form widely function include e count regularization benefit norm cs activation special analytically activation large huber piecewise cost calculate activation separately region interval activation put piece thresholde show function converge early barrier norm scale mean poor amplitude cost one draw intuition various norm jeffrey determine q show activation activation cost early increase signal community coefficient also block purpose encourage sparsity block penalty cost pointwise input output effect calculate cost yield directly simplify back relationship simplify eq range yield activation thought type group input group recent sparsity solve tractable program minimization update q coefficient drive small increase weight update q establish interpretation drawback require immediately iteration save significant time digital solver advantage gain modify analog involve weight modify steady represent analog scope system reach digital method plot relative recovery measurement sparsity variance signal iteration modify system achieve plot dynamic weight dynamically clearly iteration traditional digital algorithm modify optimization iterative weighting scheme digital comparison weight modify discrete solution evolution simulate term dynamically converge approximately play processing toward computational tool toolbox difficult application power analog analog circuit demonstrate real solver solution scale beyond implement wide result analog dynamical system certainly investigation cs expensive computational bottleneck cs wide variety increase incorporate sparsity structure improve would dynamical already establish analog circuit traditionally benefit illustrate achieve actual development analog issue analog circuit platform preliminary simulate issue future inherent load implementation work include solution explore design exhibit system hybrid analog digital achieve benefit note prototype achieve less rewrite equation formulation essentially depend write constrain program barrier convert approximately program strategy algorithm barrier solve increase norm increase activation derive ideal soft thresholding operator relaxation fit used solve find invertible straightforward soft relaxation function soft figure plot extend give occur less activation function back l macro indexing vector coefficient coefficient dimension tradeoff sparsity symbol derivative derivative vector coefficient index subset index inverse inverse energy log edu pg berkeley mail com present grateful valuable child improve significant effort algorithm specifically problem advantage system sparse analog recover synthetic acquire compressive sensing system scale support fast digital furthermore analytically problem basic norm classic approximation analog rely apply incoming circuit significantly improve processing strategy noise common formulate thought search signal e improve signal image rely linear linear eq represent signal transform fourier represent corruption fidelity solve find signal frequently assume result separate cost p decade focus number zero draw significant interest appropriately actually problem simply count coefficient recent heuristic approximate performance guarantee relax date guarantee involve function optimization name include pursuit surprisingly many recover tractable compressive cs performance inverse highly zero show certain generally take random recovered signal use despite long optimization field application utilize highlight optimization solver real mention real power imaging channel wireless communication application problem art include computer vision well research focus dramatically reduce solve challenge smooth function recent fast specialized solver unable solve moderately fast storage time requirement time steady design efficiently induce circuit thresholding etc analog significantly power example quickly recover thereby optimization process cs main highlight benefit wide applicability analog efficiently analog system digital context recovery demonstrate analog architecture support order magnitude digital arise community norm norm classic mention provably optimization program smooth system locally competitive algorithm comprise drive approximated steady way generality
segment delay nonlinearity sequence reservoir construct record reservoir state often helpful mask symmetry mask present also mask nonlinearity exploit intrinsic chose reservoir delay loop allow explain detail mask periodic piecewise e value mask reservoir dynamic approximate regime reservoir correspond dynamical similar become couple rich instantaneous nonlinearity transformation reservoir linearity provide feedback delay turn representation architecture depict nonlinearity implement place light generate adjusting intensity change bring system instability dynamic variable obtain rescale lie interval version eqs derive material stage process reservoir experiment round usually reconstruct reservoir recognition concatenation wave wave top reservoir line bottom essentially well system obtain discretize check discretized version choose architecture experimental code take component agreement measure dynamic efficiently explore validate supplementary reservoir task use reservoir art digital implementation train move drive noise precisely reservoir produce output detail method measure normalise similar obtain digital size report reservoir consider receiver reservoir digital reservoir obtain apply reservoir isolate digits reservoir community performance computer reservoir present report yield reservoir detail material reservoir computer art digital implementation relevance speech channel flexibility computer new operating reservoir change possibly reservoir experimental reservoir computer successively dynamical system speech nonlinear introduce relate report input respect reservoir use internal introduce low pass filter reservoir enough processing convert analog process multiplication mask output multiplication digital constitute towards build optical reservoir computer computer implementation help understand analog processing acknowledgment van suggestion discussion begin project researcher work reservoir project acknowledge de task nonlinear auto move average simulate recurrence uniform aim predict know task reservoir community channel go channel yield yield range task misclassifie ref metric isolated digit recognition task ti ten record time digit reservoir take mask take mask ten associate target otherwise winner digit word train reservoir four average word fraction misclassifie correspond digits red green part optical optical setup operate nm determine operate feedback input signal arbitrary generator computer generate input signal task record optimize read feedback adjust average inside loop optical output adjust operating point operation fully automate diagram depict plot reservoir label bottom go mask reservoir get get new multiply add desire ph panel normalize ph blue take account red square three agree within statistical bar bar point might material practically obtain reservoir progress accomplished rise science digital analog apart remarkable find science incomplete cell process elementary organize system issue conceptual huge machine progress approximately order consumption computer presentation analog biological system rise artificial neural abstract develop machine attack reservoir propose paper date exceed reservoir analog particular biological rise computation present exist review rather point build reservoir computer reservoir provide detailed road building system architecture understand computing reservoir road experimental computer reservoir consist internal evolve evolution variable external evolution reservoir independently instance gaussian distribution variance call reservoir fix reservoir large proper reservoir past traditional mean aim reservoir computation internal state reservoir denote given header digital another would predict input electrical load reservoir computer state reservoir time reservoir perform input winner take optimize logistic discriminant analysis function support vector end optimisation prefer accomplish e ridge reservoir ready input reservoir input correspond datum check reservoir new reservoir target draw normal feedback gain gain necessity understand coefficient state behavior adjust appropriate contribution term input reservoir eq find choice matrix happen try coefficient optima linear highly advantageous list biological neuron interact sparse resource add reservoir modification behave dynamical time could come evolution system trick outperform predict time noise simply reservoir remain beneficial add ridge use simulation well linearity digital simulation give realization may experience extent sub continuous simulation far evolve system natural reservoir operate continuous write progress line reservoir analog system new insight operate reservoir computer base present look digital high consumption programming method first nm standard c band optical time pass light minimum light adjust rewrite light optical enable adjustment loop variation value mode optical detect integrated rf produce intensity experiment hence round act rf act filter unit transform cutoff intensity intensity lie becomes experimentally change experimentally tune range typical optical optical optical digital increase intensity diagram observe diagram evolution diagram reach affect diagram simulation way setup measurement setup verify branch diagram addition term input evolve discrete mask continuous reservoir mask may changes rf place rf intensity combination intensity experimentally vary amplitude generator take place rf affect filter time negligible sn depend exact intensity integer physical upon effect discretize equivalently dynamic coupling line appear reservoir experimental discretize component nonlinearity preserve operating system moreover discretized error traditional allow validate nonlinearity topology computer continuous matlab experiment signal discretize generator transfer operate gain response exact measure configuration dark add agreement simulate diagram agreement simulate performance continuous easily sensitivity shape nonlinearity always reach post light intensity loop convert record discrete duration associate affect efficient synchronization system estimator computer send reservoir reservoir reservoir task detail performance system random sequence discretize mask interval reservoir size wave ideal experimentally obtain perfect agreement simulation practically study error misclassifie nonlinear channel reconstruct noisy nonlinear wireless output channel gaussian yield signal ratio range reservoir computing could mask uniformly reservoir task mse obtain estimator discretize close estimate symbol fraction calculate form set set study noise figure text bar note number bar close suffer measurement db study give reservoir snr system give reproduce nonlinear input distribution target task mask reservoir train step system input reservoir average pair sequence performance discretize regime digit recognition isolate record time five digit reservoir time input mask probability mask reservoir ten train one digit average time winner take digit divide train reservoir repeat system digit give recognize digits reservoir reservoir reservoir size winner take report experimental reservoir measure diagram vertical optical measurement de optical multiplied gray optical intensity feedback feedback gain unity light intensity possible feedback gain unity intensity gain nature transfer lead behavior determine inside branch diagram experimental reservoir reservoir reservoir depict measure normalize lie whole length red discretized reservoir output show reservoir reservoir leave panel panel input feedback reservoir instantaneous input influence previous recurrent neural technical report technology term memory technical national center nonlinearity predict system wireless york processing chapter make dynamical mit stable perturbation neural international conference cat page david reservoir journal international reservoir advance recurrent european page www competition com david reservoir speech international joint conference page advance process optimization network neuron network international advance computer science edge advance neural systems mit david van toward optical reservoir van dynamical complex complexity network transaction apply delay large dimensional delay letter delay optical system physical review letter k scale physical recurrent neural network automate page bilinear david isolated word state european artificial page source recurrent transaction adaptive processing page mit develop word isolate corpus ti speech international conference speech signal electrical k cat bag sc new york usa david capacity dynamical david versus linearity international conference page david intrinsic temporal neuron visual advance memory neural international gray rgb service universit information universit cp author work author ac reservoir computing processing datum basic reservoir computing recurrent couple architecture non node delay importance nonlinear remarkable capability attractive decade exploit computation concern optical optical digital suggest flexible approach call digital reservoir inspire intelligence reservoir computer architecture element nonlinearity integrate intensity store internal reservoir report implementation task practical channel speech reservoir highly provide notably financial speech task reservoir computer nonlinear drive reservoir computing reservoir depend experience case computation randomly choose
pearson paradigm nonempty note enough check state small integer solution eq go hold right hand zero go message motivate standard learn independent couple eq let I subsection previously symmetry lead q condition satisfy moreover implement pearson binary classification connection constrain reference constrain optimization problem vector robust latter problem essentially simplicity take value function chapter convex surely chance derive sufficient call copy consist big author address unlikely term control attempt follow instrumental convex problem conservative replace f nonnegative take choice loss name idea employ cast simplex however directly scenario eq theorem entail fix moreover cosine complete proceed property eq q two display complete build upon r proof h proposition together imply since increase convex hold previous probability decompose treat proposition apply indeed yield combine bound find hold statement theorem definition complete n q eq complete lemma binary two binomial parameter hold bernoulli hoeffde yield binomial easily particular resort derive introduce scope fix binomial variable take derivative eq interval eq inequalitie variation hard admit hence inequality increase follow imply q fact scale theorem definition edu problem anomaly detection implement error combine specify ii minimum possible solve technique consequence chance program binary paradigm anomaly constraint empirical chance optimization pearson classical latter focus minimize slight abuse language type ii essential kind expense illustration medical diagnosis severe consequence spam machine true error enforce hope dependent goal design bound pre high show good performance excess type review main classification emphasis framework main proposition theorem proof secondary technical illustrate couple dy therefore define indicator expectation joint clearly reference rewrite replace decrease commonly convex surrogate hinge logit loss hereafter relaxation treatment problem overfitte mapping one small empirical risk resort way first classifier specific sometimes candidate collection setup result remain asymptotically meaningful naive tree satisfactory classifying individually decade boost exploit suitable consequently restrict classifier combination flat em rule sign rule majority restrict search natural measure bound formally inequality hold scope candidate classifier class result know bound vc dimension however argument bounding rely proposition convexity classical em type error paradigm constrain significance user np closely concept work theory provide thorough treatment statistical two statistical hypothesis determine randomness come randomization bias coin arise occur occur pearson amount significance minimize ii solution sufficient respectively lx kp lx merely proposition nevertheless paradigm recall classification goal solve directly unknown I x n x n mutually assume deterministic subsequent investigate subsection extensively proposition paradigm remain well theoretical np paradigm find vc solution pearson high bounded consider family accordingly setup sensible np beyond kind detection np work matter paper ensure pr follow principle pearson constraint concern difficulty seminal justification program seem unlikely estimation proposition confirm opinion define unknown base regardless sample excess pseudo fact come technical view go resort surrogate particular empirical bound involve summarize type construct type high excess result process optimum optimization consequence np chance programming explain section solve classification subsection simply counterpart classifier empirical type
plot sparse able recover nonzero prove small nonzero procee turn compute schwarz inequality nc distribution subspace constant small almost sparse solution nonnegative nc section euclidean c appear recovery n subsection connection robust correction compressive sense compressive sparse sense multiply q compressive sparsity perfectly noiseless compare almost euclidean noisy geometry tool stability transition compressive sense mainly focus error perturbation measure remarkable sensitivity phase provide compressive average gaussian paper derive arbitrary limited compare restricted isometry bind correction happen good isometry small proven sparsity illustrate almost show sparsity outcome system th measurement nonlinear nonlinear true nonlinear iterative nonlinear subsection performance recover correctly recover q part h let differ pick cardinality certainly optimization h lead exist certainly computationally costly nonlinear may non set h variable ideally minimization function convex apply h optimization jacobian state estimation update reach specify additive additive recover true condition true secondly correction additive condition jacobian nonlinear jacobian need assume iterative start generally additive precisely algorithm guarantee v state local jacobian aa k iteration nonlinearity capture term estimate follow plug denote gap generate jacobian suppose norm function reasoning k respectively h local know algorithm true additive jacobian satisfie local point aa verify correction nonlinear local jacobian simplify focus true state though first jacobian subspace condition hold fact difficult convert check explicitly condition error correction verify condition local jacobian trajectory input check gets guarantee converge neighborhood system neighborhood every kk bad requirement solve subset objective take solve large k upper objective enable relax semidefinite second nonlinear recovery jacobian state row local picking value fix jacobian property meanwhile h h inequality norm h nh long get summary parameter neighborhood true h decode algorithm true get size specific fit linear jacobian measurement noise also apply recover nonlinear bad power gaussian measurement independently denote choose estimate I dimension dimension randomly denote averaged run choose reduce recover even percent contain percentage fig power monitoring operation base measurement state magnitude angle operation power device measure limited current real flow contain error modern technology exist attack incorrect system result consequence massive scale measurement power angle magnitude system one reference angle reference magnitude rest function minimization test fig vector magnitude minimum nine angle take flow characterize level randomly contain independently noise apply iterative minimization recover nine take final error relative contain example system prove condition exact next indicate indeed case first know weight iterative test detect measurement pass test potential repeat pass statistical run verification method jacobian state row figure cardinality always bad datum around true iterative conservative broad predict recovery nsf theorem claim xu university wang recovery bad detection mix locally measurement property error bad noise sharp linear subspace mesh solution though linearize convex nonlinear electrical use programming inspire power additive nonlinear static power vector magnitude angle indirect send central state power common error sensor communication adversarial introduce call bad datum usual estimation need detect eliminate measurement power us problem suppose make function additive measurement dimensional zero element assume nonzero entry real error reflect nature bad generally party may sensor moment adversary party able error rare know ls effect estimation try minimize l bad magnitude bad denoise decode sharp almost mesh estimation error propose programming perform measurement guarantee iterative result network convex programming linear characterize decode optimizer restrict
group bound early cite logarithmic assumption dictionary design oracle aggregation weight achieve prior discrete development start general oracle sometimes overview oracle derive opposed turn mirror finally quite sparsity suggestion langevin carlo inequalities noise deterministic design use drive truncation characteristic one add group compute term resort exploit fact mh g describe mh note example hypercube define instrumental neighbor mh instrumental simplicity speed another potentially accelerate mh generate generate chain define pattern estimator mh solution estimation prediction integer block value elsewhere readily recall define triangle fuse version fused report measure figure screening poorly illustrate yield miss fact minimum zero sake risk minimizer risk yield fuse fuse fuse prediction performance turn eq theorem entail nonempty realization weak isometry constant nonzero let brevity enough risk random variable infimum test take random back selector denote acknowledgment author part dms corollary fuse gaussian potentially exponential take derive sparsity popular framework include ordinary sparsity fuse aspect principle successfully review several weighting emphasis construct scenario exponential weighting deal estimation cf presentation framework consider pair gaussian norm notation dictionary function preliminary estimator detail possibly give measure average square wish satisfy property remainder term characterize aggregate characterize aggregate often consequence measured coefficient inequality indicate dictionary reflects also lead systematic since wish small possible important obtain lead oracle find literature however infinity additional assumption oracle call truly salient reasoning goal appropriate whether viewpoint parameter indeed inequality offer coincide dictionary sense risk way attain minimax sharp oracle mixture exponential weight select minimize major theoretical organize weight penalize risk combine combine function estimator oracle adapt sparsity derive inequality popular sparsity implementation pattern procedure empirical minimizer broken exhibit intrinsic minimization minimization inequality selector selector exist follow function sense exhibit term turn well cf choice convex namely oracle inequality minimization receive lot choice consider obtain oracle front oracle sharp overcome limitation combination dictionary combination belong potentially good penalize look since proxy inequality replace combination carefully choose ideally remainder term difficult simple eq q follow minimizer kullback penalty lead flexibility result associated let probability measure divergence sequel adopt convention weight explicitly kkt multiplier kullback solution selector achieve desire average select dictionary aggregate aggregate balanced side contain gaussian necessarily exponentially aggregate infimum kullback leibler divergence take discrete consequence restrict vertex precisely denoting lead remainder term prior uniform section suitable methodology worth terminology average set aggregate preliminary hold initial randomization analysis aggregation enough work aggregation deterministic limitation randomization carry many estimator aggregation aggregate observation aggregation estimator affine direct independent identically distribute first randomization still lead expectation randomization big remainder inequality square kernel ridge estimator filter long list equal row rather deterministic exponential modify risk deterministic risk weight long unbiased linear aggregate naturally nf consistent family aggregate bind immediately mainly estimator particular general spirit apply projection mixture suitably however element square terminology come fact interpret indicator feature pattern give square moreover variable gaussian least guarantee square pattern small least square aggregate clearly function projection na risk selection exponential weight thus select resort probability distribution performance last projection pattern fact remainder term balanced inequality choice depend information candidate exist piecewise prior knowledge fit frequentist setup prior candidate nonparametric candidate difficulty meaningful basic favor prior main difference exponentially polynomially exponential least plugging valid penalize another sparsity yield screen aggregate sparsity screen oracle often classical one method fuse idea mix sparsity appear reformulate difference fuse invertible possible difference combination
diag extend logistic regression belong penalize likelihood logistic mixed ig genetic effect statistic matrix make approximate variance statistic function degree linear kernel model treat differently corresponding statistics university california division public sciences research logistic machine derivation response affect include intercept snp snp goal score statistic
refine intuitive trust agent sophisticated machine learn capable similar trust trust base differently focus transaction agent overall viewpoint group share useful feature perform machine efficacy generic system machine transaction set transaction work common discriminant analysis many mechanism rely specific agent historical predict use make approach quite party local sharing e traditional feedback aggregation system mechanism possibility avoid privacy save communication overhead surprisingly positively discrimination feature interaction interaction trust especially party reliable quite rely local approach detect future work investigate tool discriminant etc improve accuracy direction concrete site etc grateful dr technology interact carry ability assess risk provide safe transaction involve traditional trust trust transaction knowledge transaction capable distinguish transaction algorithm transaction previous transaction experiment accuracy especially specific agent rare inaccurate system trust ingredient reliable participant diverse system multi agent dynamic interact past trust mechanism traditional approach rely knowledge instance past trust agent agent insufficient trust trust absence agent learn aim complex decision exist may transaction argue investigate distinguish transaction one apply sophisticated transaction successful transaction transaction use give beyond validation explain trust behavior want say restaurant guess possible restaurant past friend give restaurant recommendation restaurant friend among friend friend restaurant drawback friend feedback restaurant mechanism experience make restaurant price location know city country bad want essentially proposal rely success assessment hypothesis experience learn new transaction experimental scenario proposal predict proposal trust machine integrate work effective machine discriminant lda presentation proposal belonging find example measurement physical characteristic prediction provide tree popularity visually human language lda generic framework discriminant may trade paper proposal agent local past describe feature generality successful interaction set feature quantitative historical interaction group successful perform lda two group allow likely classify root move produce investigate recommendation please design decentralized agent might involved framework local repository multiple trust assessment knowledge propose construct information generic trust design suit generally rely e g dt argue transaction efficiently good moreover confidence recommendation reliable information compare base trust local thus inaccurate party much need knowledge difficult suitably also information simulation evaluation trust party quite demonstrate efficacy proposal detect particularly evaluate transaction conduct trust know trust management organize trust framework proposal dt study subsection discuss sharing proposal real regard future unlike mechanism behavior decide interact base local avoid risk inaccurate third party agent request service agent customer service transaction transaction happen quality service transaction binary transaction customer generality unique transaction transaction happen new virtual characteristic transaction record agent trust transaction transaction take online camera transaction collect country physical distance transaction phone profile average comment number friend etc item category comment different place bid age already place bid regard collection note aforementioned maintain exchange platform ii historical successful trust extract transaction suitably historical trust directly historical trust feature traditional request entity much effort difficult transaction proposal attack trust fig depict proposal component storage sc trust engine knowledge responsible transaction party transaction record transaction relate engine item already dataset past transaction item trust application correspondingly machine trust calculation engine responsible predict knowledge conduct trust calculation otherwise dt suitable restriction suitable across confidence measurement section suitable reliably transaction however insufficient learning address propose traditional mechanism share exchange result sec advantage easy privacy identification difficult bad specific avoid local knowledge detail trust assessment transaction agent transaction transaction repository extraction engine machine discriminant recommendation obtain transaction transaction offer transaction represent historical transaction reliability potential transaction historical transaction disjoint group accord transaction successful transaction transaction please perform linear transaction transaction group service transaction mathematical one group average across transaction centroid averaging transaction lda class external separability extent successful distinguished transaction internal centroid scatter matrix internal fraction transaction regard external actually two represent mean scatter variance sum lda aim projection inter minimize maximized eigenvector associate group similarly transaction transform classified measure transaction group centroid transaction collect transaction filter distinguish transaction one final classify sort root leaf classification object object put classification classify child node node process continue transaction past agent tr rule potential transaction decision tree np heuristic generality hereafter select classify tree gain binary categorization successful past transaction denote transaction transaction q characterize collection transaction belong class successful past transaction evenly measure entropy feature enough corresponding gain high appropriately choose transaction transaction otherwise classify different datum e past gain respect transaction classification correspond decision transaction transaction inaccurate accuracy quite recommendation less impact also trust learning algorithm bootstrappe join interact propose dedicated exchange share help agent transaction transaction request knowledge share p fig depict give share weighted agent trust provide knowledge trust local trust fall trust evaluate usefulness update trust please since trust although discuss trust reliable sharing transaction transaction trust service vice versa provide act transaction transaction agent transaction historical information available rely aggregation web etc scope paper collect request agent trust provide useful maintain list agent provide knowledge initially past request explore request familiar e friend agent request super hybrid key special agent e agent detail use lda issue share among agent etc detect potentially transaction two knowledge knowledge help likely successful projection direction centroid successful group eq weight knowledge determine knowledge projection combine centroid potential transaction euclidean transform centroid behavior e provide perspective share update trust score adapt trust knowledge transaction inefficient unnecessary produce prediction transaction adapt trade computation overhead transaction evaluate local distance transaction centroid transaction predict successful mean trust score trust rating trust score statistically beta function posteriori trust priori trust denote trust score eq prediction correct incorrect transaction party avoid unnecessary transaction successful party good adjust third party site transaction million transaction successful feedback fraction performance dt study internet transaction conduct unknown I historical available transaction perform transaction environment transaction label explicitly environment trust synthetic consist service behave service target transaction target transaction successful transaction outcome feature construct node maintain list record generate transaction describe let transaction factor tune transaction transaction completely mean successful separate transaction base feature approach proposal three approach reject network aggregate node trust false form experience derive trust rating group belong please refer related section exist model lack metric performance trust transaction predict transaction would rate experiment bar add deviation transaction past one transaction start evaluate transaction successful vary investigate effect volume fig demonstrate dt evolve expect increase transaction grow prediction quantitative threshold individual past note logarithmic quickly minimal transaction transaction much two transaction experiment empirically transaction transaction relatively reliably overall rate base dt simulation thus capable quickly transaction online knowledge insufficient initialize accord item item third party knowledge helps predict transaction introduce inaccurate information instance transaction transaction inaccurate demonstrate lda number increment trend false false increase transaction collect inaccurate hence accuracy able achieve corresponding trust local three transaction rely integrate low well service keep negative heavily environment thing false large reason amount increase transaction false negative accept reject benchmark evolution trust management outperform aggregation especially predict request thus relatively accurate lda feature quantitative low outperform
e stay bound see impose measurement variance also verify arise mix derivation error expense complicate asymptotic estimator band obtain asymptotic survey issue quantity community theorem check rigorously far os enyi prove thompson asymptotically size extend chen quantity hold equip supremum g van say converge bound functional existence limit follow state normality zero provide variety condition size soon regular flexibility proposition controlling bias establish normality local linear smoother control uniformly easily handle property estimator prove entail number inequality van finite convergence smoothed process establish argument converge consistency distinct fulfil expectation remark fulfil section band process taylor chapter set difficult stationarity fulfil band coverage compute mean showing stem weak practical importance via notation conditionally one simulate one obtain conditional gaussian process performance width band curve consumption simulate realization basis orthonormal main consider random without replacement population number consider quantile consumption unit half determine allocation suggest get thompson trajectory small discretized trajectory control situation independent random variable variance ar illustrative discretized curve corrupted smooth trajectory corrupt noise deviation noise l trajectory case study strategy smoothing crucial performance drive compare noisy consider denote build easily check weighted least weight new justification kt minimum validation estimator far criterion value bandwidth furthermore suggest criterion define denote bandwidth keep partitioned size take value curve error bandwidth sampling l lin lin different median choice select mean median lin lin third improve interpolation smooth oracle concerned cross adapt select interpolation hand validation effective bandwidth dominate moderate choice lin lin focus due summary summary impact individual trajectory roughly load really interpolation criterion sample account choice median lin lin level bandwidth median lin pt band bandwidth unit l median median lin pt lin band area high turn band quick implement satisfactory bandwidth band application raise straightforward error build band functional quantile automatic however take make lead substantial improvement reduce principal shaped sampling relate score al normality approach confidence band appendix letter vary place decompose stochastic measurement bias e h old turn see pointwise square uniformity write local weight kt kt fact weight smooth lemma deduce prove process pseudo metric hold sub van cover ball radius packing number inequality smoothed stem normality random boundedness increment cover number satisfy get detail eq guarantee convergence nd logarithm look establish finite dimensional start limit function sample function derive fact dimensional light show check boundedness trajectory preserve well asymptotic normality inequality corollary van notation kt increment constant observe maximal weak factor integral employ cover number satisfie integral x c adjust deduce terminology van covariance pointwise pointwise uniform e lie compact equip sup norm close maximal establish mean inequality strictly index vice versa mostly thank slightly simpler reduce triple sum l ki kt iv calculation already main ks kt k moment finally like ks call inequalities van first norm increment justify hereafter mesh index van ns nx arbitrarily first inequality k theorem namely grow dimension affect consequence obtain term start ns ks ks reduce van pseudo ks kt diameter classical n cauchy schwarz moment mc deduce diameter covering c tend probability term result simultaneous dimensional trivial similarly theorem metric arbitrary moment constant vary across dominate thank sum form denote root vector identity vector study remain variate wishart freedom hold one apply away deterministic tend calculation one nx apply van theorem increment usual integral covering tend hold real sequence anonymous review lead improvement r collection survey quantity discrete noisy smooth trajectory mild population observation time regularity trajectory limit establish consistency former band band attain nominal simulation covariance account highlight sample development automate sensor collection exhaustive transmission storage analysis assess appeal trade accuracy cost particular competitive compression explanation although survey long establish motivation recently regard examine theoretical principal component sampling relate population read minute week assume interpolation discretized signal consider thompson greatly utilize see however property rely contribution present interpolation smoothing interpolation improve accuracy prove simultaneous inference estimation vast literature van therein ball exploit goodness fit functional regularize estimator build conservative confidence derive equip supremum confidence band ball approach adopt bootstrap band vary obtain bootstrap band establish rough supremum unfortunately generate correlation cause empirical coverage differ innovation easy attain procedure bootstrap lie contribution provide nominal coverage covariance mean desirable bandwidth survey necessity account usual cross survey total aim set base devise cross validation weight proportional simple replacement study paper organize
take either length scale length general principle dimension couple typical technique go high dimension infeasible high elliptical slice overall scale generality prediction generalise wishart process make since popular arguably predict volatility return mean dimensional white noise condition model specify parameter part length parameter hard positive discusse definite point effort lead general notably conditional variance covariance correlation entry full implement toolbox choose general variant use order low triangular multivariate see predict follow rigorous except make ahead forecast historical forecast prediction account historical covariance different financial distribution parameter parametrize place length scale posterior align though procedure wishart classic like generalised wishart process assess mse always never true proxy component intuitive thorough univariate analogue proxy use assess log likelihood forecast although proxy use historical also sort simulate common financial daily repeat critical easily reconstruct historical datum forecast table identify accurately poor change mse historical forecast forecast financial proxy square forecast datum point forecast thin marginal variance unfortunately assess natural mse proxy historical assess consideration vary daily five period multivariate figure return behave like index return normally distribute compare make advantage another critical benefit true forecast historical square forecast suit generalised wishart capture variance compare advantage procedure complex volatility whereas stochastic process generalise wishart alternative easily depend research apply extremely dimensional hope effort volatility general interpret high dimension acknowledgement thank cm cm ccc engineering university university ac uk wishart dynamic matrix capture diverse dependent readily parameter interpret introduce outperform multivariate process accounting correlation imagine price composite index dramatically rise discover etc rise anomalous index make prediction concern model price security volatility risk management financial key risk come leave fluctuation capital indeed economic economic time volatility think change zero generalization arguably return multivariate volatility understand correlation index univariate portfolio allocation say return asset may wish return volatility receive understand transmission financial another science otherwise useful know correlation importance volatility generality estimation difficult impossible constraint multivariate volatility volatility unlike generally effort lead simple make correlation leave vary machine introduce wishart marginal semi recently multivariate wishart limit dependent scalar restrict velocity particle motion autoregressive learning procedure generalise address arbitrary covariate like one miss easily smooth make aspect require reason scale yet provide general multivariate specification section wishart review wishart present procedure prediction experiment include composite set subsequent present dimensional also gp account correlation predict correlation review process construct detail joint arbitrary kernel mean smoothness kernel square popular function trend value time also generalise wishart conjugacy could inference learn transform formulation outline volatility kernel control variable covariate interest introduce first procedure prediction wishart new base regression generalise wishart process dynamic posterior datum posterior distribution relevant low cholesky decomposition scale model show dependence generalise wishart cholesky scale sample posterior monte cycle successively discussion assume step dimension finally also change describe posterior value fix freedom increment dimension gram function gaussian covariance form kernel depend dimension degree short posterior elliptical update especially posterior prior place axis align slice dimensional metropolis hasting reversible simply wish take divide cholesky decomposition parameter distribution must joint covariance pair otherwise row change freedom dimension dimension represent conditioning construct avoid explicit parametrize
information audio metric similarity metric music song query song traditionally set information ir query query song receive back relevance isolate example may focus new aim collection intuitive advantage importantly mind advantage specific song ir operating quantify detect item coherent isolated answer query contain group advantage guarantee alone usually increase song system advantage rational item correctly detect retrieve community identify song group music employ reader detection cover song community inside set matrix dissimilarity couple assign cover song g originally rank content involve network never apply retrieval task previous work considerable improvement overview outline article build analyze song apply similarity house characteristic expect reasoning cover song detect cover art contribution assessment computation improve obtain group stage sec increase particularly confirm intuitive exploit answer internal organization song perform author cover study first show tendency community conclusion sec house music refer consist non label stage per expect cover song piece content therefore compute couple present measure allow track piece promise far far processing brief outline descriptor employ thesis feature amount short window component ambient independent specific independent piece volume fluctuation characteristic song strategy song delay representation recurrence crp recurrence finally extract sensitive song characteristic previously adapt allow track potentially trace crp restrict application song fig compare song around perform interested comprehensive overview cover measure pair quantify previously black trace maximum symmetric similarity fill proceed dissimilarity duration song song threshold pair cluster cover inside evolve threshold correspond density component node display tb look evolution threshold link network cluster top right represent formation cover song begin form maintain bottom network note isolate remain expect threshold middle group nearly cover community evaluate unsupervised approach implementation asymmetric dissimilarity dissimilarity explain classical computation solely operate distance advantage noisy drawback algorithm implementation package incorporate use try provide implementation agglomerative hierarchical cluster method test single linkage sl linkage cl group linkage default check inconsistent maximal clustering experimentally next algorithm collaborative network outperform maintain pm link unweighted dissimilarity certain low node experimentally cover I show dissimilarity computational approach improve triangular checking component pm try triplet coherence triangular suppose g result triangular one cover either create triangle measure objective triangle incomplete triangle vertex connect penalization incomplete triangle experimentally tb implementation sequentially situation create maximize keep adjacency update necessary process set pm substantially whose seem pm dissimilarity extremely song identification detect accordingly edge weight closeness margin experimental aspect evaluate cover identification collection include cover add cover wrong accuracy estimation analyze cover set cover cardinality setup fix cardinality classical weighting go bad average measure song typical measure entropy representative song positive belong community positive e cover negative actual detect optimize grid search try definition cluster community parameter algorithm turn obtain accuracy broad range near km pm report accuracie high majority nearly reach effectively detect possibility coherence answer enhance retrieval sec accuracies pm comparable pm tb km sl cl pm pm detail denote world music collection qualitatively evaluate spend setup see completely process collection spend hierarchical raise usefulness algorithm huge music cluster well km take matrix music million handle aforementioned exception pm pm pm furthermore subset link notice sparse e link million link investigate community dissimilarity refine ensure community refine rank song initial accuracy common measure average depend precision song retrieve answer sort list relevance song otherwise increase assess optimize instead threshold different one imply method high particular cluster community community achieve increment role positive optimal might illustrate reasoning regard positive rank answer query real song group belong cluster suppose algorithm recall precision answer refined answer evaluate relative increment eqs high table overall reach pm poorly km sl cl pm detail denote within audio cover song identification evaluation international task assessment song identification participant black accuracy collection publish participant tune algorithm collection version obtain accuracy achieve respectively algorithm comprise plus pm dissimilarity achieve former increment task might detection improve particular final accuracy high music basic device represent human exist evidence brain category around category prototype piece would prototype feature prototype form membership cover song gradient category song firstly cover prototype lead song song good manually setup discard song song mark state actually original avoid song popularity regard cover follow cover set employ direct graph support evidence song community fig strong see define usually community song cover make blue dashed line tend dissimilarity tb ability automatically version community extent consider
mdp play crucial minimization empirical loss proportion satisfy previous compose transfer error transfer third dimensionality decrease linear account target task bias towards wrong approximation target average target still effective bellman two operator g dy solve task bound logarithmic use much big estimation suffer additional transfer whenever advantage great bias sample come introduce estimation small select source report thorough arbitrary cp gram dx let whose correspond greedy bind display difference single bind account decrease well bellman mdps enough carefully design feature eigenvalue whenever well tf bellman reward transition might introduce tradeoff approximate close error display similarity introduce run iteration play crucial role proportion bellman operator define define minimize transfer case learn quantity training transfer fig simplex iteration pair return return bound achieve transfer large auxiliary fact task whenever task relate task particular small call generative fair dimensionality dependency optimize simplex may transfer depend space well necessarily transfer well previous show notice depend action slow state bellman operator error approximate monte auxiliary possible advance access source verify target source begin auxiliary learning include auxiliary compute mdps introduce source sample share notion try identify target transfer consider obtain notion try always image bellman operator notice minimization convex quadratic simplex task run iteration prove well approximate bellman operator nonetheless implicit solve tradeoff reduce proportion source properly solution tradeoff task remove favor pool task tradeoff maximum sample vector n optimize tradeoff error first account transfer induce error performance bind main difficulty previous consider task assumption def constrain proportion sample accord multinomial whether experimental effectiveness tradeoff parameter reward parameter ccccc task reward preliminary transfer introduce previous finding challenge variant walk action move make affect intend define interval region elsewhere transfer algorithm eight function report function radial spread episode start uniformly bar single iteration transfer problem use source leave plot compare performance transfer task sample refer auxiliary error thus task add run notice target auxiliary good policy existence linear produce approximation reward four task combination reward able transition task different drop among significantly task happen leave plot transfer beneficial bounded quickly target sample auxiliary training together source select proportion significant behavior possible preferable actually match much performance highlight tradeoff task provide address sample reduce large transfer enough interesting notice improve keep unchanged source avoid comparison sample task formalize first performance rl show access source task consider target set principled algorithm confirm finding work future problem increase make transfer effective adaptive transfer direction future balanced target action see definition generate task access source design specific acknowledgment project education de european receive european fp agreement besides introduce introduce norm pair empirical x orthogonal unique n tf already detail explicit since iteration explicit straightforward regression quadratic call fix design set n ny nr x qx single fit satisfie fig noise observation project function variation inequality put together analyze performance generalization action nx nr value measure space bind rewrite minimizer big variation together q finally proof sample l q qx p v f final due thus affected term refer inherent bellman error bellman relevant error recall bellman previous interval bellman image projection finally possible relate transfer statement let q return probability chernoff true bellman inequality previous series inequality statement available available transfer represent iteration consider source scenario target obviously task percentage sample task constant task decrease reach iteration behavior attempt early still suitable force drop lot source small introduce task optimally poor q reduce proportion weight try lack task approximation proportion change favor finally use task decrease start iteration effect tradeoff realize available tradeoff realize tuned analyze performance value
collect eqn precede display cd dd cd g cd exponent g g divergence infimum measurable conjugate dual convex variational characterization convex mix inequality via desire claim w imply almost identifiability condition contradiction immediate consequence sufficiently small boundedness boundedness thereby extend claim admissible way outline step suitable modification inclusion hellinger condition measure proof existence turn contraction wasserstein apply le discriminate give infimum hellinger p rw g complete turn ii maximal packing fact c r p g p p first fact p packing equation test due monotonicity proceed least w condition satisfied equation x term precede display bound thank display large probability vanish third vanish np gx g np gx bind denominator theorem combine display thereby conclude proceed suppose form cover denote dim simplex shall couple display construct combine number probability time volume k e tp kp argument dim g wasserstein metric combine cover simplex single kp q kp mi I kk conclude px p sr acknowledgment author thank associate anonymous valuable comment suggestion point version lemma grant behavior infinite wasserstein metric relationship wasserstein distance space hellinger kullback leibler distances mixture provide natural rate distribution notable mixing help relatively rich year extend mix construct model fit datum several book distribution include mixture measure atom vector probability combine dominate kp distinct behavior heterogeneous datum probability proportion behavior compare assess mix direction cumulative measure chen result set mixture unbounded multidimensional even distribution see past decade model bayesian set b primarily convergence hand concerned latent quite arise deconvolution mainly zhang fan note recent consistent parameter certain primary wasserstein mix model establish rate number well wasserstein although divergence leibl variation hellinger distances wasserstein utilize g atom equipped wasserstein q infimum wasserstein atomic assessing model worth note probability atom must converge permutation cluster illustrate mixing interpret convergence relevance distance chen special wasserstein mix know divergence functional wasserstein mix strong identifiability density entail mix wasserstein theorem divergence mix measure atomic generalize unbounded convolution latent mixing nonparametric endowed generate accord study namely truth contraction theorem prove notable rely likelihood typically formulate term density wasserstein typically weak hellinger rate possibly number include mixture multivariate distribution dirichlet mixture convergence logarithmic minimax dirichlet ordinary smooth achieve ease place place distance include dx hellinger kullback divergence divergence ball cover entire mutually distance equip distance inequality measure equip borel sigma algebra abuse write atom likewise probability atom measure denote discrete measure infinite support ij q denote discrete measure g take wasserstein employ may metric role mix dominate ease use combine function relationship wasserstein divergence divergence play role hellinger leibler instance broad divergence know divergence let f divergence distance hellinger composite distance kullback finite highlight aforementioned mix measure evident inequality enable test establish bound leibler mixture density probability metric latter former identity j similarly kullback leibler kf kp kf kp g g lemma choice yield topology induce space may imply convergence distance property specify identifiable strong version discrete support finitely identifiable almost imply rate notion herein adapt identifiable q finite identifiability satisfy chen identify broad gaussian p p suppose compact wasserstein family finitely identifiable suppose likelihood identifiable thm ordinary thm pt cd ingredient prove measurable indicator function I subset wasserstein ball write highlight hellinger metric fix totally bound bounded exist condition need loss lack capture packing ball pack bound exploit hellinger information wasserstein test interesting control pack wasserstein ball quantity arise addition pack wasserstein ball whose hellinger lemma latter packing appear analysis lemma provide core establish general contraction theorem concern quantify estimate entropy metric several kullback likelihood leibler hellinger characterization finitely identifiable constant w use function useful handle nonconvex follow family identifiable tend sequence g g probability statement eq theorem augment deduce obtain move class cover bound n kp km bound finite mixture process let euclidean metric kp need kp g bound distance away hold family gaussian density contraction wasserstein kp kp kp assumption atom must g g I let condition trivially provide w g w w assumption bound n identifiability nc bound j large assumption specify bound arbitrarily constant ensuring metric logarithmic univariate g kp kk probability base pack metric dirichlet g rd dp denote ball pack take gamma center dirichlet gm ig support contain measure couple I inequality desire place density constant kf kp p specify g specifically ordinary likelihood likelihood take main first prove condition eqn kp handle ds packing hold cn trivially turn lem lem e hellinger previous verify note respect
initialize eq run budget update reward number totally decrease correct simulate always amount theoretically budget outcome transform constrain price call equation price also x total budget independent outcome eq appendix determine price observe cost everything one monotonically x kf condition price function satisfie inside function least budget unique budget define constant logistic budget could find computing employ price potentially double f nf c converge experimentally usually step number converge use equilibrium happen instance I budget simplify market use method non guarantee exist degree flexibility choosing function way choose logistic regressor market constant market constant equilibrium equilibrium aggregation eq artificial aggregation forest aggregate aggregation however adaboost forest aggregation aggregate forest prediction market constant participant also analytic form budget classifier online aggregation variant model market cx c b x x resemble logistic factor sure constraint observe multiply market thus participant example u x x contract price classifier price give verify reasoning carry market train rbf kernel online boundary rbf kernels participant market likelihood obtain online budget prove artificial market perform maximum version likelihood total batch present market sequentially constant market ascent incremental update maximize ascent ignore dependence gradient market approximate find constrain differ step divergence carlo case much desire maximize weight exactly approximately eq overfitte important market market optimal market participant conclusion artificial general market market representative prediction market capable available market participant train usually reason aggregate could perform instance performance subset classifier train behavior prediction market aggregate classifier participant contribute specialized opinion specialize triangular triangle negative market participant three participant negative outside three lie outside triangle high budget fall triangle budget classifier agreement evaluate positive negative market obtain many construct specialized depend language specialized classifier propose specialized leave decision leaf domain rule obtain linear verify aggregation participant currently different aggregation classifier idea economic bring classifier economic economic market aim market estimator class entire budget participant win outcome market information fusion artificial mechanism odd take strong updating odd instead odd allow price allow relate market market analytical price formulation allow maximum finally evaluate predict class evidence online try adapt observation instead solve regard market solve implicit criterion asymptotically observe market misclassification implicit type reject instead reject aggregated decide roc reject could market define overall reject outside detect rejection instance overall desire classifier market participant leaf simplify take decide relatively would classifier market participant overlap define generic instance type leave however specialized bid discuss artificial implicit online adaboost two mechanism participant market sequentially call maker predefine scoring machine deal participant price utility form price mean participant belief experimental market adaboost implicit online artificial market section market leave first market participant forest specialized leave initialize market market binary market price equal validation bootstrap train market aggregation capability market market simplify market address edu investigate market term repository dataset market incremental update divide thus behavior incremental unless otherwise ht average run see incremental similarly incremental preferred less handle market test incremental market bad experiment probability frequency distribution direction desire bayes error training analytically bayes mm market compare practice approximate market setup error relative forest market obtain forest estimator significantly value misclassification four predict correct difference figure forest see market well error behave conduct uci repository number meta depend take validation market market cb lb ab forest c cb lb ab breast cancer letter recognition voting record balance car connect band hill vs hill tree split name rf rf random use specialized participant market cb lb column respectively market dataset respectively significant pair market rf implementation bad market significantly rf ab significantly outperform outperform implicit minimize update bregman divergence convex function minimize learn rate bregman unconstrained minimizer bregman use market price truth euclidean distance bregman valid update implicit cb cb rf offline breast recognition diabetes voting car connect hill vs vs gene table permutation fold cb epoch cb offline offline epochs cross validate different pair bad implementation implicit cb cb implicit perform offline cb cb offline aggregation capability detection position adaboost construct classifier split participant observation fall participant feature initialize namely budget market initialize adaboost constant bin index much small negative sum weight weight negative adaboost node dataset ct region node inside manual solid segmentation segmentation solid show training test false positive acceptable false training epoch epoch gradually roc adaboost market epoch detection positive per volume improve constant market difference pair present theory artificial market purpose supervise learn conditional novel learn easily certain artificial market inspire life specialized subset comparison real market usually forest implicit online learn artificial market simple update binary classifier train tree meaningful instance manifold market involve face classifier etc involve orientation hypothesis evaluate participant decide combine approach observation currently extend artificial market extension market include future specialized participant adaboost tree cluster could instance specialized acknowledgment thank innovation market grant total hold divide budget exist proof strictly mean assumption thus kn kn k kn kn kn kn price become
section main inference sample depend partition ensure act preferred direction configuration field enough configuration align along direction pattern field model implicitly note positive analyze configuration distribution define scope pattern pattern encode priori infer actually leave coupling example attractive pattern invariance entail infer pattern statistically model pattern statistically distinct matrix define distinct pattern invariance generator one set break convenient throughout paper dot attractive vanish use inference impose amplitude three interaction spin correlation call normalize unity introduce reverse small guarantee respectively small unity index comprise integer range maximize hard introduce scheme derivation expression attractive pattern pattern calculate field aspect coincide presence secondly dependent unity effect easy pattern vanish must low pattern closely inverse mf interaction interaction contribution value depend generalize overfitte avoid consider locally likely expression matrix substitution fluctuation field bar calculate pattern tell instance bar statistically compatible zero formula expect bar rescale eigenvector orthogonal likely arc radius drop geometric criterion reader calculation attractive rescale positive orthogonal rescale virtue likely value indeed zero surely vanish hold variable vanish square represent difference expect norm non zero straightforwardly formula angle fig pattern pattern number uniform break constrain amplitude introduce gaussian presence effective strength term particularly severe transform unity regularization eigenvalue criterion attractive q correction identical first correction find note correction pattern interaction mode interact overlap priori vanish correction projection extract finite amplitude ratio exceed critical refer discover learning compare order order low formula place pattern configuration order pattern vanish order component low formulae first correction require minimal formulae field synthetic various ise attractive pattern specify later component field ensure weak intensity average simple ise vanish connectivity spin correlation estimate equilibrium monte carlo consequence perfect depend spin configuration block growing allow systematic formulae depend amplitude pattern dense hard attractive inference ambiguity infer energy continuous pattern show low eqn sample align along uncorrelated size direction conversely likely align pattern small suffice systematically systematic study inference bottom configuration deviation true infer bar calculate unity pattern pattern deviation amplitude configuration generate pattern due invariance fig size amplitude decrease block ten value error error twice large confirm limit correction j low formulae priori ten block circle square respectively eigenvalue low coupling find contribute coupling dash configuration eigenvector spin penalize line small model compare infer configuration standard low pattern pattern generalize bar true pattern infer bar infer pattern compatible zero discrepancy quantity calculate error bar fig generate extension sampling close correctly confirm interaction infer calculated pattern moderately fig middle pattern add excellent sampling allow inference fig surprisingly systematically correct account correction enough vs dash unity right vs small unity attractive pattern retained differ mf correction test perfect error infer fig section expansion next correction test field zero instance orthogonal pattern simply accord error average four large formulae formulae pattern circle correction test corrupted noise compare component low formulae strong pattern strong relative infer formula order decrease decrease infer correction sample improvement low weak carry expect configuration correction effective full low circle formulae black formula bar small perfect fig correction c vanish second correction correction qualitatively low infer close correction slightly apply biological boltzmann learning procedure top number come f consist wave precede pattern analyze activity binary calculate correlation attractive hereafter one expansion base use select agreement close rather mf clear interaction correction sufficient change sensible next analyze commonly encounter domain bind protein extract interaction sampling site frequency multi briefly speak pca site poorly representation material amount alignment otherwise keep track contain alignment infer pattern recover agreement good attractive number also calculate discard precisely distribution keep red site result interaction denote compare attractive interesting whether affected remain site differ account chain interaction go site nevertheless site see fig strongly retain weight middle panel correspond optimal intend derivation maximize respect minimize appear energy configuration configuration reasonable mechanic systematic expansion entropy power calculation explain formulae reason clear make hereafter pseudo value comprise hereafter infer energy obviously identity fulfil energy square spin dominant maximizing following expand cosine power expansion cosine exponential nx expand cosine low partition low approximation cross accord cosine entropy order entropy magnitude system typical infer order correction straightforwardly generalize pattern pattern model dynamic adequate interaction fix system configuration align along field vector determine along field pattern root saddle solution one saddle point vanish configuration mainly determine behaviour locally stable eigenvalue matrix correspond zero correspond calculate field one affected phase system happen equation equal equal contribution illustration field opposite latter give contribution infer minimization correction coincide value identity condition fulfil limit pattern diagonal vanish cross subspace large square vector large attractive pattern obtain notice pattern unity keep scale accord couple pattern assume add quantity surprisingly add fit determine center attractive denote leading coupling give much small probability inverse mode particularly correlation covariance fluctuation report cross vanish find assume determined come bayesian bic decrease pattern add cost decrease value eventually balance depend correlation need represent bic mathematically justify compare always datum hereafter consideration fluctuation non vanish quantity marginal sum run respectively integral integral dominate contribution root repeat define infer onto eigenvector directly maximize fluctuation eigenvector square maximize vanish mean coefficient fluctuation angle reliable picture draw expression square naturally look correction low order expression first cross low cross entropy inverse hessian calculation correction eqn shift vanish dominant determine draw independently equilibrium pattern limit expect self averaging depend want fast decay infer specific great analytical hard treat field vanish pattern infer measure unique physics zero phase e pattern number reliable opposite reconstruct pattern make arise pattern restrict spin map pattern play role dual spin configuration spin configuration dual dual decay exponentially theoretical entropy infer pattern spin pattern back find overlap configuration overlap random state opposite limit keep interact applie vanish remarkably pattern infer pattern scale denote self e replica result detail solution entropy analytical simulation small evaluate function enumeration entropy one compatible analytical existence effects b reach critical range concentrate pattern existence meaningful phenomenon discover unsupervise replica break negative fig nevertheless may entropy decay ij suggest identical dual semidefinite calculation replica break extended pattern regime remain qualitatively unchanged critical high result sufficient context connection section admit solution spin decrease sufficient zero simply coincide know consist single pattern component readily calculate q encountered send break avoid state suffice come ideal perfect result inverting pattern affect correlation contribution boost large eigenvalue absence pattern eigenvector component ml perfectly recover presence sampling finite corrupt contribution physics mathematic phase check coincide transition regime large eigenvector uncorrelated identical correlation whose eigenvalue mp continuous component mp relate noise large eigenvalue finite converge square analytical formula analytical transform square rescaled identity fluctuation rescale pattern onto illustrate pattern fig formula eigenvalue correlation eigenvalue spectrum ratio spectrum eigenvalue mp agreement formulae pattern overlap vanish repeat find entropy strong pattern accord agree discussion pattern state fluctuation discrepancy correlation entry dominate contribution informally correlation accord pseudo infer field value component equal pattern tangent clearly discrepancy nice illustration correction recall large presence small ratio unity correction quality infer study value generalize interaction encode set technique systematic calculate estimator field variety regime low component large correction validate expression synthetic pattern ise criterion compare study depend sample configuration elementary infer pattern pick sampling scale several consequence exceed secondly large confirm present intrinsic calculation cross inference local suffer study particularly e inference know modify first correction linear number possible
field similarly index operator index correspond axis fields result recursively give formula causal would could utilize equation ultimately algorithm determine much field section give additional field model exact four technique mixture leibl distributional behavior extend field expand fairly specialized adapt vector treatment use length regressor grid lx thing compactly matrix various density field dft field call frequency also dft center frequency proportional dft satisfie unbiased estimate emphasize require assess field treatment differ fourier frequency shrink asymptotic expand domain define term explicitly regression parameter result toeplitz section careful toeplitz approximation assess kullback discrepancy series kl depth treatment field define convenient effect parameter order utilize spectrum assess proximity guarantee minima true cf write kl call minimize assume nontrivial spatial time essentially corner separate corner volume ratio zero corner increasingly dominate lag effect f prefer utilize dft positive frequency although type kind flat discussion bias spectral lag window modify bias fast utilize dft approximate obtain integral kl mesh dft minimized practice accord convenience convenient fourier handle formula prove supplementary propose minimize depend formula although exact log maximize square standard argument q calculate replacement field seem utilize objective effort inversion together estimate proportional recall denote use iterative computed update either exact update iterate approximate biased consistency provide rigorous treatment formal auxiliary datum recall equal limit theorem result expand unique pseudo exact asymptotically mean v h assume fourth expression involve fourth spectral estimate theoretical refine hypothesis correctly exposition misspecification asymptotic likelihood specify parameter fisher particularly elegant case equal twice identity mirror symmetry except equal except lack efficiency building procedure low order jointly coefficient might zero negligible large likelihood ratio utilize ultimately asymptotic parameter residual covariance behave examine residual equivalent correlation hypothesis absence autocorrelation comprehensive regard limitation li supplementary evaluate goodness fit statistic spatial model rigorous process fair theorem gaussian assumption list asymptotic lattice asymptotic paper like together utilize broad process formulate treat marginal moreover provide condition iii highlight frequentist contribution concentration assume likelihood work toeplitz spectral block integrate first entry z w stationary produce structure admissible field consider sum h h f f z spectra minor derivation spatial write stationarity upon difference index sample spectrum spatial field require absolute fourth spectrum via dot bivariate impose product condition regressor specify write correctly adjust mean assume regressor require follow spectral twice continuously differentiable process pseudo exist interior must field model skew index argument extension argument see argument decay boundedness partial see move condition problematic move distinct independently distinct entry interior convexity kl bayesian identifiable imply ensure parameter subset euclidean lie interior accomplish transformation easily accomplish coefficient important extend spatial toeplitz limit average extension quantity g hold continuous ii estimator bias give correction assertion require extend series generalization dimension seem mechanic considerably theorem estimate exact hold regressor n normal l v f denote transpose independent zero process simple application series yield normality condition focus skewed gaussian specify identifiable worth automatic make automatic furthermore together model automatically entail asymptotic normality frequentist result concentration seek estimate exist away infinity restrict range effectiveness estimation outline accord calibrate size constitute grid example denote mean asymptotic therefore asymptotic place c c ii column experiment response column equally space follow effect analysis supplement simulate multivariate simulation circular though account log simulated square model high agree provide difference standard parameter increase go mean finally average great simulation parameter increase extend direction recursive formula setting extremely notable establish consistency normality rigorous platform independent expand simulation supplement readily able acknowledgment associate anonymous provide comment substantially author I statistic interested encourage statistical census supplementary material section nsf nsf grant nsf census central role spatially correlate significant broad paper easy computation comprehensive likelihood field parameter theoretical generally facilitate building greatly inference refine model result yield spatial environmental science area spatial broadly mainly field process spatial among comprehensive stein therein grow spatial stationary impose hold inverse covariance hold generate upon impose structure demand generate define ensure way specify field variance appeal contrast move average identifiability present information hypothesis briefly estimate comprehensive field formula field move field likelihood guarantee move field identifiable without impose quasi field parameter expand particular frequentist propose inversion primary express axis furthermore field spectrum nonnegative value
least determinant class generate st list usual computational switching take start column switch dual remark dual dual matrix walk eight class size find size list mark least exclude class total assume additional sample estimate imply view two cause inaccurate estimate interesting know small st reader bipartite acknowledgement author program hadamard visualization mathematical sciences square case hadamard maximal maximal case maximal gram hadamard class solution give switch optimal hadamard maximal optimal concerned determinant change hadamard hadamard hadamard order hadamard say sign say self say hadamard equivalence class self equivalently class know conjecture interest hadamard determinant order certain candidate generate generally hadamard switch induce switch new order odd order eq due show sharp sharp bind complicated sharp order sharp odd odd may gram via determinant gram gram gram want row row gram involve find solution satisfy omit merely reduce possible search use gram gram matrix determinant generality assume search regard searching level correspond search typically search child leave deterministic searching solution exist decompose may better small search per experimentally unlikely find node child recursively child gram due give determinant determinant maximal fail hour explore reach depth size way strategy matrix determinant sample lack uniformity introduce uniformity advantage uniformly operation determinant hadamard equivalence hadamard equivalence one idea introduce close possibility switch switch close row sign interpretation bipartite preserve product preserve preserve preserve switch four class equivalence switch case equivalence denote equivalence exhaustive know two sum preserve switching normalise ht st consist split yes yes new st dual former case say st split st self st know label paper compose associate st order order occur list least determinant lie least switch website give st generators implement find class generator generator determinant upper plausible maximal improve despite optimisation technique prove technique order search determinant gram matrix st st class gram follow ht st website st class class switch generator generator table class split two c c split ht h small class class order order c candidate gram product thus plausible conjecture difficult reason h determinant even prove incorrect technique class determinant start matrix find duality number start iterate row find program explore switch walk walks switch row column switch connect walk intersect size expect walk intersect probably geometry store check vertex
short set system since quasi order every jt operation prove countable ordinal rr q smooth u tf tt b barrier contradict immediate useful develop language alphabet concatenation sequence sequence concatenation infinitely iterate concatenation closure language closure type require linearization quasi order linearization linearization linearly characterization say say provide infinite conjunction preserve operation converse assume infinite contradict n among set nonempty px I infinite immediate inclusion subset finite principal inclusion quasi order assertion order satisfy condition hence fact assertion since assertion partially aid scientific education technology thank anonymous thank go definition learn teacher abstract independent order tree system theory nash theory quasi introduce system correspond monotone inductive unbounded pattern language system member viewpoint characterize member system inclusion continuous similar positive linearization unbounde target theory quasi order neither employ algebra sometimes employ learnable study lemma consider nonempty member study learnable employ unbounded restrict motivated systematic learnable set teacher datum learner abstract type family learnable positive short consist construction inverse quasi order maximal define order ordinal nonempty discrete represent say continuous identify function many binary unique topological class series title paper enjoy useful continuous language operator closure application induce speak transform correspondence quasi set order image useful investigate language supremum conjecture proposition investigate order type type study e decision net logic characterization definition order upper ordering viewpoint system number analogue review order short closure topological relation infinite closure please hereafter ordinal identify integer denote resp identify exist sequence type eq barrier ordinal quasi order say barrier countable ordinal barrier define countable explain barrier exist small element less ordinal barrier ordinal countable ordinal order barrier countable ordinal barrier barrier quasi study net verification countable
speedup matrix aim successful particular infer internal black box know measurement uncertainty exploit underlie statistical acquire accuracy cost especially roughly since relative map appear thank anonymous comment manuscript imply sufficiently eq average lin tt diagonal routine necessity especially image use improve cost generalize wiener field significantly estimate precise functional exploit autocorrelation develop show situation imagine might necessary engineering give computer act operation probe might infer due internal reconstruction like multiplication rotation solver etc output mathematically linear perform space paper interested input identify g correspondence output vector box operator box scheme probe sequentially nonzero certain pixel repeat read expensive reconstruction deal sequentially always prohibitive accurate feed box output input component find efficiency obtain hadamard improve calculation likelihood extensive corrupted investigate scheme come price result uncertainty completely measurement achieve long computationally expensive spatial pixel etc beyond average generalization wiener determine suitable amount cpu respectively accuracy examine signal reconstruction enable computation marginally prohibitive remainder review sec sec propose derive verify example real sec covariance equal hold zero field define space transpose understood elsewhere refer grid voxel entry square uncertainty effective reconstruction tool filter review review far emphasize also signal eq linear maps ref thing forward signal scenario measurement wiener assume stand respectively gaussians solely simplify mean often forward datum r pn r actual estimating covariance encode result filter straightforward read field signal covariance need covariance spectra project band spherical degree spectrum might whereas often isotropic wiener ref assume alternatively derive logarithm degree freedom spectral band characterize option former derive classical contrast exceed line mean equally probable component sophisticated ref number trace estimation ii regardless choice estimate eq want period residual latter law applicable variety want bayesian exploit knowledge interested confusion quantity appear marked form signal matrix element originally point ref treat separately requirement estimator matrix square underlie generic formula base sec state diagonal adequate suitable since entry considerably already estimator trace distribute diagonal chosen eqs covariance require sec show covariance present ref classical filter equation iteratively compute accord ignore guess spectrum step extreme limit give nontrivial whereas generally exhibit nevertheless present contribute marginally accuracy therefore calculation numerical pose map maximal computation perform open system package estimate estimate dotted starting dash nine case explicitly ensure matrix e sphere norm respectively efficiently expensive computational complex sensible number argue however converge couple pure time amount fig evident estimator reach progress datum finitely diagonal realistic characterize two represent smoothly h smoothing spectrum qualitatively overall gain decrease diagonal rough estimate calculation ten estimator random vector
value function evaluation reproduce throughout semi product duality mapping banach bound operator operator denote great linear operator language value belong exist together semi inner respect functional operator requirement uniqueness satisfy coincide usual kernel reproducing explore reproduce kernel investigation banach bilinear prof cauchy inner duality mapping eq recall mapping bound adjoint use inequality second immediately q property kx bf reproduce enjoy reproduce rkh difference semi firstly additive reproduce pairwise distinct q although still reproduce sampling exceed give vector map converge converge suppose converge get pointwise bound introduce notion adjoint banach banach identify indeed position characterization reproducing value reproduce banach define reproduce reproducing fulfil shall compose eq impose also banach function moreover eq evaluation remain reproduce compatible call banach theorem pair map value banach space endow compatible value give always scalar input value linear complex inner product reproduce via choice q satisfie value show reproduce satisfy sake finite banach compatible inner completeness reproduce first space respectively adjoint corollary form equation span end vector linear get reduce search matrix reproduce value sampling exceed appropriately purpose conclusion value section sensing consists consider find banach banach simple example always imply wise th convex norm uniformly reproduce compatible duality equation translation namely scalar reproducing value banach determine subsection class invariant banach equip space endowed bilinear banach nf l r old notice continuous eq adjoint hence compatible inner value translation reproduce dual product bilinear form equation remark endow gaussian rkhs confirm validity application vector unknown point observation could shall follow regularization methodology form topic form regularization output one choice positive tolerance remain converge regularizer admissible continuous regularizer least minimizer value case topology equivalent continuity topology equivalent norm dimensional admissible deal loss form respect admissible set weakly inner uniformly minimizer uniqueness minimizer subsection follow proposition value reproduce case subject cite therein establish closely relate minimal interpolation start examine fix sampling function interpolation notation theorem subset functional vanish u nonempty good origin known banach characterization approximation banach complete main without effort regularizer satisfy increase let follow satisfy lemma thus strictly consider solve regularize try model parameter substitute convert original many part usually generally additive characterization together constitute system stand function continuously differentiable convention next dimension continuously minimizer product imply suppose point linearly minimizer equivalent linearly independent similarly characterization equation minimization due respect finite basis reformulate leave minimization nonlinear cm example accomplish author zhang banach space propose notion reproduce kernel banach basic property construction concrete especially minimizer scheme value reproduce banach feature map regularize equation establish kernel banach demonstrate rkh machine target vector reference learn task mathematical foundation find framework function output hilbert constitute special banach space space reach banach variety structure norm intrinsic make embed hilbert base hilbert space finally banach banach banach mainly lie banach functional representation discover extend type argument banach substitute product banach illustrative theory basis banach inner product lee study margin hyperplane banach value reproduce banach space scheme theory attempt build mathematical multi banach shall map representation concrete respectively investigate concerned shall space deal banach inner product banach linearity positivity conjugate homogeneity variable inner say compatible banach inner cauchy linear dual functional value semi might role banach space instance q endow
unit slope simulation versus likelihood generation draws consider fast draw increase significance via number draw take model estimate parameter goodness statistic via use square converge exact estimate goodness fit statistic via likelihood generation root converge far fast draw via versus limit goodness via use generation draw converge ratio possible accelerate via asymptotic acceleration four discrete parameterized bin meaning draw suppose underlying experiment significance level repeat calculate realization parameter level less limit repetition experiment reason approximation remark underlie repeating calculate hypothesis distribution somewhat scheme many favorable property goodness form pearson statistic division often lead serious even present illustrate via numerous fortunately availability computer avoid easy problematic division take goodness statistic interpretable black program rapidly calculate identically may consist either parameterize take value finite countable accordance cell common whether I draw statistic statistic bins root difference distribution draw arise root confident quantify confident give square come significance level define significance concern term significance alternative term distribution pearson replace weighted average bin classic however weighted division divide power error divide nearly arise every bin main thesis article use classic statistic long computer root mean square conjunction ratio goodness generally preferable ratio division somewhat fact kolmogorov variant example powerful root certain circumstance discrete kolmogorov complementary square largely compute confidence root number draw trivial via maximum square please report present exact level recommend complementary multiply carry sure subtle bin much problem yet black result goodness fit careful make many advantage substantial draw least every bin subsection please treat terminology old concept long computer technology table fit significance compute really reject threshold significance instead significance probably correct whether large fluctuation outside science typically exactly test validity purpose due contrary goodness even suppose exactly remarkably original article statistical arise set testing association independence contingency cross two review goodness hellinger good member read divergence family bin associate respective bin denote root statistic root pearson standard likelihood common refer hellinger distance define draw substantially analyse monte simulation rely draw discuss testing parameterize family draw actual alternative draw nuisance unfortunately devise distribution none standard likelihood hellinger significance number especially useful draw realization experiment measure repetition repetition course testing seem feasible probability bin typically fit test concern define please mind section nuisance always repetition amount statistic significance level significance assume seem experimental yield combination possibility parameter contain permutation maximum entail sort frequency need freedom monte draw consideration run draw obtain estimate statistic mean square step take simulation level fraction calculate conduct detail procedure work draw measure vector root distance statistic entropy divergence draw respective bin draw necessarily equal occur regard leave probability random observe confidence error monte conduct produce estimate produce statistic significance level produce statistic great corresponding binomial standard great standard fraction monte estimate calculate investigate goodness root perform discuss briefly significance remark accuracy fit toy example q q draw bin unlikely arise like exactly goodness fit discrepancy whether computed monte evaluate take root square extremely find evidence level statistic realization classical classical arise large reject classical statistic display unlikely even suggest sure arbitrarily significance irrelevant bin report statistic contrast square reliably classic behave agree bin bin unlikely arise like various statistic discrepancy plot whether arise monte draw model square classical little evidence model agree draw bin draw bin law analyze datum reference page present subsection goodness english repetition word choose corpus bin plot frequency sort statistic significance hypothesis integer sort first bin contain great bin choose great draw bin contain great remain bin sort proper proper bin plot significance must appear word display significance goodness fit goodness bin word clear priori dictionary appropriate significance root mean several digit classic contrast classical depend know proper produce result report english word repetition word obtain bin frequency sort root unlike statistic discrepancy nan hypothesis seem fortunately essentially irrelevant indeed mean sensitive bin potentially inaccurate inaccurate root whose refer logarithm unlike mention facilitate good relative world monte root square report model match american report various rate physical generating score health significant simulation symmetry square note reveal significant issue symmetry could significance level simulation testing root square calculate table powerful table sensitive model sensitivity bin recommend ccccc good poor excellent good fair poor cc ccccc fair poor excellent fair cc ccccc excellent good fair excellent specie exclude readily identify specie collect via number specie sort appropriate must bin sort subsection sort less likelihood sort please carefully truncate common level monte carlo simulation root square discrepancy datum substantial draw solely log ratio unable detect discrepancy determine discrepancy full include incorporate analyze report detailed report unlikely sample belong order figure bin sort permutation furthermore test alone tail parameter bin contribute goodness fit aside summarize parameter sort real specifying geometric model seem sort frequency great number divide fit remain maximum sense permutation us sort order allow ignore great fit bin fit plot number sort well fit distribution associate great infinitely bin provide aggregating draw calculation goodness fit without draw computing simulation parameter infinitely carlo simulations root figure empirical model draw fluctuation detect section draw detect parameterize quantify success detect draw actual draw distribution compute accord significance level simulation draw say mean confidence subsection principle maximum fit generating plot would level calculated three remark much approximate level parameter subsection remark exactly equivalent confidence time approximation please remark statistic statistic associate square distribution associate equivalent variation eq plot actual arise meaning yield confidence level I root successful fail simulation actual distribution mean number statistic draw require increasingly bin draw figure plot draw actual specifie require increasingly exhibit opposite let specify q draw distinguish distribution define specify root mean uniformly us distribution draw distribution specifie mean distinguish classical square specify figure plot draw required remark distinguish begin present specify truncate poisson draw plot distribution define draw require distinguish distribution specify next subsection time estimation parameter see specify distribution truncate consider several maximum likelihood specifie distinguish specify maximum consider actual parameter specify distinguish specify truncate poisson parameter consider several clearly distinguish likelihood remark distinguish consider q q require distinguish maximum remark
coordinate cut dag construct schedule could obtain graph cut normalization b theorem edu posteriori submodular structured function find cut propagation mp energy mp suboptimal fix scheduling mp cut apparent showing scheduling point mp schedule proper graphical vision biology algorithm solve empirical choose however clearly outperform compete algorithm cut max belief propagation mp precise mp graph cut mp map analogous scheduling mp cut pass message bottleneck capacity let connect decode strong statement mp submodular energy mp show suboptimal submodular wrong solution depth implicit previous scheduling converge bad point give characterization always exist fix energy point mp fix alternatively consequence product tight binary submodular energy point proof believe fix novel due come run ordinary algorithmic degree scheduling construction suboptimal bad avoid inference distinct improve scheduling insight cut maximize assignment energy sake exposition choose present energy energy submodular graph variable potential whose value ij represent potential form energy express thorough discussion notation energy polynomial flow cut computer appropriately construct know combinatorial cut machine computer reference therein convert weighted direct source add edge map direction initial create terminal terminal terminal terminal energy function every terminal graph cut optimization back repeatedly flow equivalently hereafter maintain capacity amount edge opposite edge residual capacity residual minimum path phase path e connect determine source respectively tb unary representation capacity effectively flow energy unary potential pairwise potential unary potential optimum set potential flow identity ensure flow make optimal coefficient potential path phase component path directly positive residual determine directly terminal capacity possibly connect label either within label practice terminal typically strict belief strict algorithm map structure employ algorithm min sum update simple structured energy mp update beliefs b ix ib x variable quantity strict mp asynchronous use mp algorithm formally family belief mp message passing arbitrarily possibly order possibly manner long point strict mp broad exclude fundamentally program reweighte max scheduling include string work scheduling lead improve detail mp cluster definition mp variant scheduling dynamically node flow set potential entry potential useful sequel terminal potential edge potential two factor graph edge potential graph map jk jk connection graph begin energy function balance energy path path energy decompose structure important balanced energy subsequent section convergence suitably energy close linear time rely equivalence cut section energy one path function along message fig balanced compose energy section scheduling cut phase return pass hold message message pass standard apply leave unary message propagate across particularly way leave pairwise factor message terminate strict mp convention term unary chain increment desirable accounting accomplish first unary path constrain increment backward direction increment forward receive increment pass receive similarly direction nx must n else impossible backward message calculate message unary constraint choose capacity unary factor bottleneck ensure message increment propagate receive increment propagate increment path ft ft want unary factor increment message capacity unary potential unary message yield never denominator non iteration approach leave unchanged see message increment propagate chain analogous dynamic message increment backward later schedule combination edge contain residual path minimize potential specifically end iteration residual terminal ij ji cut formulation terminal construct potential unary iteration schedule return path bottleneck implementation residual cut execution since cut finds run finding routine message potential find bottleneck reconstruct note direct cut execution perform residual formulation ij f ij f ji ij ij fact affect contribution c ji ij clear relationship value residual quantity bottleneck capacity find capacity pass pass message strict mp continue reach mp essential reasonable scheduling message phase assume variable receive incoming message neighbor ix variable message increment normalize next condition factor propagate preserve factor message incoming factor ij plug message material jx jx lemma allow message pass execution belief structure unary belief normalization unary unary without normalization propagate factor propagate message exist message value preserve maintain message change product version message use compute parameterization message parameterization cut tree equivalently view mp cluster energy change potential current potential potential product message energy component far remainder message everything else unary potential assign potential component use unary message leave ix remainder potential remainder unary potential ix ix message energy analyze belief parameterization unary pairwise belief consider message message consider potential potential adjacent belief change change could depend variable belief forward potential parametrization structure pairwise begin iteration iteration apply supplementary material detail unary group message leave add end phase unary leaving mean unary change x x unary potential involve total change parameterization pass chain correspond equal flow unary ix b put change exactly phase phase tb potential message pairwise strong draw message give max cut construct product decompose term u message graph modify discuss modify potential change else execution change tt modify potential subroutine schedule path bottleneck capacity pass modify potential maintain see material amount entry vice mp forward backward terminate modify potential every less quantity maintaining cut unary pairwise begin schedule increment increment analogue message reduce negative change unary potential maintain equivalent potential potential vice u ij ij modify potential ft rt ft interpretation begin potential reduce pass backward next message equivalently return send equal case previous run mp holding begin unary potential message pass along pass run mp equivalent easy analyze cut tb arbitrary message normalize belief view edge equality potential potential ij ij involve leave variable message equal incoming message explain fig message edge mp message hold calibrate calibrate calibrate potential pass along receive factor plus receive left factor increment change propagate variable sum incoming message neighboring potential message propagate leave illustrate fig inductive message actual send exact propagate message unary clearly message plus increment increment reduce backward along chain message right neighboring factor essentially lemma message send send backward calibrate reduce section pass forward calibrate potential edge potential base message produce lemma unary belief computing belief increment match message increment belief end mp path edge calibrate follow edge unchanged mp normalization message potential edge iteration additional interpretation working potential specify schedule incorporate potential careful ensure pass lead graph new message free one potential message perform equivalent product tree representation view tree potential max product even message completeness supplementary citation conjunction q message initially edge clear message compute min could initially identity alternative parametrization add one message discard back define edge however mp message thus express parametrization potential recall path increment parametrization group message treat potential justify division suppose message mp converge tu full message step initialize message message initialization pass reach iteration u purpose know guarantee parametrization relative potential leave difference end proceed perform message divide potential potential change potential neighboring could possibly affect since belief standard normalization change message potential case parametrization message restrict also final belief change parametrization potential parametrization pass message backward parametrization message iteration unary b ij know unary belief potential message lemma potential minus reduce computation variable update parametrization initial ij complete proof change parametrization parametrization pass forward parametrization cut difference pass cut simply change cut analysis throughout apply equivalent end phase equivalently work potential presentation simple path unary potential unary capacity cut first use run strict optimal fix existence mp binary constructive rely define term connect belief message amongst vice inside unary illustration homogeneous consider cross max minimum cut path cut supplementary material lemma mp independently homogeneous zero boundary reach internal convergence iterate homogeneous lead belief unary entirely unary potential connectivity eventually variable mp loop message cycle loop unary add pass message stop acyclic structure strict mp obviously monotonically potential I strength unary loop converge pairwise potential prove point submodular homogeneous belief initial seed cut decode arbitrarily previously suboptimal give assignment mp fix submodular function return submodular energy potential define suboptimal reach scheduling fig however message fig decode belief map pass version cut thus update map algorithm view block section normalize potential constant zero show cut see either cut ascent chain structure assume potential standard form chain canonical unary entry know problem primal optimum bottleneck cut path unary value dual block cut follow execute normalize potential dual absolute unary cut local unary potential behavior
depend mdp nature states reflect much expect action optimal approximated parameterized dot action optimal compose sequence sentence n number sentence make content mdp problem triplet document currently read correspond sentence already currently previously assign read iff read category assign read read transition act process reward stop bring case assign classical accuracy measure otherwise must purpose compare text tf global local read already assign category global feature concatenation read sentence trick project high dimensional inside dependent easy classify carlo amongst states size average performance approach able properly training tune generate initial rl svm robust hyper figure comparable outperform dataset visible metric set small reading behaviour explore big corpus hard properly right histogram group notice document read read clearly capable read less different baseline read deal multi remain sentence particular due large multi learn classify sequentially read document collect enough show tc learn document whole document able behaviour fast system performance baseline set new imagine mdp classify read national research sequential reading sentence sequentially learn stop soon reinforcement learning label corpus propose set read tc act label document label unlabele document tc extensively tc generative discriminant mainly lose compute score look document svms classification binary entire decide category inside little word correlate suit category concentrate additionally case associate entire document inform drawback attempt process et base network use linear svms string development text less approach sequentially read sequential goal relevant sentence label reading document last learn correct category read paper fold sequentially read sentence assign topic additionally reinforcement focus stop read classified useful expensive document popular corpora tc baseline portion ability read sentence classification easy training set task small organized follow formalize detail label tc corpora denote document document training document compose iy assume tc category parameterization reduce provide overview formally present manner propose classification sentence sentence decide read document belong possible category choose stop read document correctly compose begin read contain reliably classify document sentence read classifier classify classifier reading document additional four entirely action pick classify denote action denote see choose good action action high score minimize loss document action document illustrate stop action action set remain action repeat training document model
alternative cluster factor neighborhood one assign member cluster neighborhood rv value map cut influence neighbor away whose neighborhood learn par structure discover par together par involve appropriate par score structure application par algorithm graphical graphical par identical tie form graph par factor tie graphical next overview basic tie detailed reader case fully mle aim observe training maximize interested helpful undirected separately bayesian network consist probability value parent value observe undirected mle calculate close use parameter per expect current estimate equation extend tie done tie share count node share parent analogously equation modify share arise compute statistic database sufficient statistic computation cast discussion one dramatically optimize pseudo pseudo compute multiply reduce train fashion h possibly refinement set last skeleton set build correspond inductive logic view procedure parameterize potentially bias derive one start candidate trivial par proceed exist candidate score candidate carry refinement specific kind incremental par graphical literature several clause across relation learn path appropriately relational relational path crucial aspect relational hypergraph form cluster agglomerative intuitively entity cluster kind structure clause commonly occur pattern densely domain identify perform relational discover start relational perform walk entity thresholded include include clustered time cluster agglomerative via hypergraph analogous cluster long via depth adapt structure area approach structure modify methodology place fail focus discover methodology also follow clause observe way clause transfer goal representation source module source predicate constrain extremely structure target alternative approach general clique template logic clause quantification predicate take clique template possible mechanism template structure direct link extend analog pc causality graphical stage first skeleton second place skeleton consider orientation rule set orientation domain link graphical review par exist relational representation formalism algorithm include inference answer review come decade mix entity noisy hope synthesis idea provide point researcher field would early version ci fellowship nsf computing grant recommendation author reflect view entity type relation entity individual relate via biology frequently model chemical interact social medium user interact page reason word reasoning entity attribute unobserved entity web page improve categorization develop relational friend molecular biology domain researcher interact application learn field growth detailed overview define graphical omit discussion program impose probabilistic interpretation scope able focused discussion exist representation pass illustrate structured define recently available define generic template discuss particular aspect way relational study efficient reasoning uncertain machine setting entity entity iid relational entity type multi attribute entity entity among iid domain uncertain relation entity representation support essential need language dependency entity b reason noisy environment notation terminology rest survey logic describe word rv logical avoid example rv bold letter assign rv letter letter x logical entity x relational focus rest logic flexible expressive kind predicate describe entity entity act quantification g represent g relation category category predicate predicate operate entity convention name predicate capital letter argument call apply atom atom positive conjunction formula quantify follow typical specify understand express positive called contain whereas remain constitute clause variable replace possible type specify constraint present connect ground atom entity assertion friend reason logical parameterized rv parameterized rv alternative relation orient specific entity entity entity attribute commonly orient language refer category whereas author refer set category paper write author note author orient language mean write attribute chain par relational store self relation relational schema attribute entity type entity natural represent statement follow name name selection constraint heavily therefore introduction graphical reader describe store configuration assignment conditionally many redundancy explicitly general tuple factor typically draw consist rv necessary e define conditional assignment computing sum assignment necessary factor generalize graphical represent whose vertex distribution parent via network convert represent product automatically normalization sum network compute maximal argument function compute x feature capture characteristic evaluate clique variety learnable factor clique define advantage discuss bayesian network true hand beneficial separately undirected representation graphical representation structure language graphical second impose interpretation logical allow group convenient representation describe start par short parameterized define terminology analogous generalize treatment regardless undirected par factor consist par parameterized operate evaluate positive par par par specify par graph par together assignment set equation par par popular discuss view par group graphical undirected hybrid list highlight subsection model relational network orient section par par establish par take tuple select constitute thus appear return tuple markov par log tuple arbitrary however share par property language language generalization illustrate document present goal clique assignment document category document document statement clique function incorporate example page page category encourage page et favor clique tractable closely markov specify par logic par rule establish boolean value implicit truth clique potential feature weight human similar people capture order logic rule par par establish clique markov g entity clique par factor constraint implicit variable entity name want constrain rule predicate time suppose know inference infer people entity friend correspond trivially regardless assignment ignore extend predicate hybrid contain conjunction logic similarity logic object orient language par weight whose par par rv interval rule continuous value manner unlike support similarity entity set formula mixed term consider infer document similarity wikipedia attribute state text eq refer entity par rule rule define par assignment par done generalize logic due combined boolean interval rule distance joint assignment arbitrary pick par use functional type sub relation type variable novel link list par template determine par factor template rv par template class template template compatibility entity assign mention template mention def mention entity mention yield mention entity string canonical describe representation consist child par rv parent par equation specify care ensure would cycle restrict par atom parent pa pa implication probabilistic distinction ordinary logical clause clause logical restrict par factor probability pa example copy x dependency example whereas par complete provide combination aspect suppose predict quantity rule represent logical par rv atom par formula correspondence pa implicit evaluate pa formula indicator tuple logical variable combination boolean operation combination slight two formula evaluate sub mean learnable range undirecte discussed specify logical entity name background model take relational perspective orient language relational entity one citation attribute paper reference par attribute go chain specify par pa pa par topic topic direct parent par paper vary aggregate correspond example reasoning perform relation reason presence uncertainty extension deal link uncertainty well link known existence logic specify represent atom dependence par rv parent specify child model entity uniformly capture citation author title cite entity domain advance instead statement type drawn entity task advance standard poisson discuss define either direct undirected advantage graphical direct need express causal hand suit dependency care ensure direct fast typical dependency word parent parameter adjust parent undirecte adjust entire structure undirecte straightforward mechanism par share par rv simply multiply whereas direct require combine aggregation independent cause par causal exploit factor represent conditional normalize efficient direct undirected dependency variable parent immediate neighbor however dependency cycle necessarily coherent marginal dependency similar set lift relational represent child rv parent dependency represent effort undirected model provide advantage direct combine undirected draw analogous graphical type variable refer marginal map posteriori likely unknown variable state overview inference graphical inference strong emphasis place discuss graph model directly efficiently knowledge extent necessary adapt undirected property graph answer need conditionally independent variable briefly refer simple algorithm factor elimination would like particular rv par rv rv call proceed order establish factor contain factor contain multiply effectively remove algorithm heuristic normalize sum proceed contain value gave marginal propagation product bp marginal contain cycle frequently know bp operation send node message reader message send send node message send expression neighbor use message receive message product leaf neighbor message together variable send leaf end incoming message neighbor bp graph sequence iteration correct frequently converge happen typically variable elimination summation product viterbi inference popular randomly carefully pick one iteration iteration condition break ergodicity assignment visit ergodicity violate closely deterministic dependency slice auxiliary slice slice jump region slice sampling derive algorithm slice sampling mc identify set assignment appropriately clause factor select clause mc slice orthogonal concern efficiency care take keep independent one par linear program construct graph introduce program constraint enforce factor value share consistent find f general factor case graph integer rv whose formula simplify replace par whose par rv correspondence formula establish logical effectively rewrite formula convert form carry cast cone rv par rv corresponding correspond include constraint represent description page hard rv rather boolean variable similarity program integer np procedure inference formula convert reality procedure satisfy par rv par rv whereby yet rv relate cut cut plane optimize constraint proceed iteration solution bad case consider meta relational originally memory efficiency maintain formula maintain thus decrease requirement initially set assignment value activate active activate active carry inference technique describe par graph algorithm regardless exclude consideration need case low employ identify ignore take advantage particularly large problem prohibitive memory inference exploit obtain repeat multiple exploit gain technique would repeat perform approach variable elimination significantly extend ordinary obtain entire constrain rv par way result sum rv brief discussion several elimination summarize apply detailed treatment reader excellent introduction example operation auxiliary par par par apply par splitting operation contain par eliminate use essentially together rv eliminate facilitate par rv try inversion elimination completely inversion elimination
c vanish inequality since zero rhs hence general greedy suppose run nd degree false equivalent neighborhood characterize via likelihood generate least loss loss adapt yield threshold initialize break property small symmetric entail result satisfied third derivative dp nd parameter satisfy probability local mle lasso neighborhood analyze regularize mle probability sample greedy regularize impose minimum zero allow broad minimum neighborhood algorithm smoothness condition impose star node fig center node parameterize entry edge induce setup impose regularize entail equivalently regularization parameterized parameter correlation node check induce desire mle require case greedy depend impose fig let check induce graph mle equivalently allow since equivalently unlike previous impose dp figure greedy dp neighborhood suggest figure show neighborhood recover outline simulate achieve optimization relatively operation selection fast calculation entail close simplify single formal zero estimation learn global chain size greedy size structure performance support zero inverse use stop parameter well gaussian discuss implementation al lasso generalized optimally cross validation successfully control graph star illustrate greedy art definition theorem task pattern mean iid recover underlie sparse novel zero matrix series combine intermediate neighborhood principal rigorous consistency pattern surprisingly greedy graphical lasso require smoothness greedy weak extensive compare greedy well high increasingly field considerable dimensional specific vector inverse multivariate set associate set impose concentration require art line gaussian regularize entry result determinant program strong stagewise forward add backward removing greedy appear various community boost function pursuit context greedy show lasso forward general guarantee approach backward greedy iid structure require sufficient condition impose gaussian vector write problem recover determine drop abuse notation element correspond graph graphical include entry degree self vertex symmetric correspond underlie greedy begin gradually add remove basic forward find candidate add meet algorithm terminate
structure estimation term neighbor search separate markov compute empirical exceed threshold number low establish two common set mild assumption dimension graph successful graph complexity positive graph employ theoretic augment graph result various graph length potential consistent estimation establish relationship degree consistently estimate also infinite degree observation liu also guarantee family enyi enyi scale degree average recall sample tree learn parameter regime impose presence sparse parameter require potential certain establish effect path decay graph feasible conditioning ise criterion condition polynomial knowledge ise establish avoid tree ml efficiently liu ml maximum span complexity liu scales exponent liu acyclic maximum algorithm hence enyi graph selection approach approach incorporate term encourage logistic show algorithm incoherence incoherence np hard verify convex search rely propose spirit factor author greedy graph degree restrictive maximum albeit control rate consistent wang graph distribute degree state degree degree define relevant rest paper basic notion alphabet kullback leibler eq similar mutual countable know empirical define use empirical refer conditional testing family accordance undirected say vector mass hold say set ab ga markov positivity state positivity clique clique serve normalize important pairwise model ising take value know potential convention correspond say ise absolute uniformly bound potential ising define distance conditional replace distance ise belong ensemble graph regime grow growth emphasize probability denote distribution give graph draw asymptotically randomness assume every ensemble guarantee notion intuitively vanish ignore graph amenable estimation end characterize figure illustration denote subgraph span addition remove edge respect addition local separate length characterize def ensemble satisfy separation belong scale family satisfy enyi random world graph criterion separation novel graphical express separation width width asymptotic required exploit local separation potential supplementary explicit ensemble graph satisfie involve ensemble ensemble q set lebesgue also characterize specific remove edge supplementary edge potential relate marginally fail potential limit attractive model path material success generic potential potential lebesgue similar assumption direct underlie edge threshold conditional variation distance find conditioning minimum need size conditional inverse dependence increase establish distance decay moreover sample certain sample base variation asymptotic recovery graph eq supplementary material consistently recover graphical tend extend result guarantee family require free case scale alternatively prove hellinger kullback leibler mutual assumption term discrete consider variation analogous also applicable pairwise nature ise recovery uniqueness impose conditional hard complex low statistic graphical relevant high potential outline statistic base statistic similarly variation threshold version concentration bound provide insight ise threshold define guarantee threshold least number scale proof supplementary characterize distinguish variation detailed description family depend degree choose world subgraph corollary edge potential suitably pac majority ise logarithmic class deterministic local property neighborhood separate separation threshold potential also paper relax sequence maximum grow end structural bound path describe distinct disjoint local separation henceforth local word overlap cycle ensemble local path property length short satisfie separation property ensemble lead bipartite graph efficient propose short neighborhood cycle term locally graph node pair pair locally grow albeit tractable precise later supplementary enyi equivalently overlap scale free law relationship supplementary along ensemble regular denote ensemble cycle require graph supplementary establish simplifie obtain degree threshold fairly threshold incorporate ensemble represent natural node path threshold consistent degree learn path tree accordance successful recall liu enyi two occur property establish degree ensemble potential threshold requirement simplify practice separation former short cycle constant latter cycle degree short cycle property augment chapter graph union short cycle graph small graph cycle enyi local hybrid section supplementary grid ensemble threshold world p consistency small potential entail minimum sample provide threshold section improve world bound sample sparse graph regime complexity input assume ensemble without characterization bounded tailor exploit property underlie complexity incoherence algorithm guarantee improve sample enyi world random degree large strong moreover method selection logistic practice equip success np condition relate imply vice versa appear weak incoherence enyi require incoherence enyi similar weak wide parameter r enyi world algorithms ise model enyi graph previously bound loose ensemble enyi average degree sample enyi necessary estimation find supplementary material along line theorem remove converse depend sample expand albeit involve converse instead converse provide supplementary material thus higher intuitively grow learn converse tends merely converse maximum degree exist result mass enyi graph dependent field outline na I meaningful since realize another identify graph cardinality large lie condition probability typical tend typical set almost recovery belong various ensemble bind model markov family give necessary family follow ensemble g k ensemble local degree k material family family necessary ensemble slightly substitute graph characterize necessary synthetic implement variation different regularize logistic compare matlab regularize regression package http www di fr software www ac groups synthetic implement ise software order ise cycle enyi degree size penalize er cycle er cycle er er
categorization object vision web multiclass example later accurately classify input object feature many rely value predictor show linearity require recent year sparse sparse hardness norm compressed rely greedy g boost pursuit processing maintain class since class overall number possible vector row single equivalent non prove zero grows linearly demonstrate method selection approach column generalize rely object classify example image predefine feature patch equal inner learn object predictor parametrize follow map score maximal element maximizer break zero column pair norm respect surrogate base follow verify loss multi hinge support whereas result perform soft equality nice convex convex order order upper use know indeed close function approximately problem q among column parameter section unseen constraint hardness formal sparsity attractive multiclass engine efficiency centrality approach straightforward approach multiclass construction predictor eqn extensively great construction share pair mapping construction number exponentially e predictor mild tackle multiclass prediction like plausible go construct specific feature mapping adequate class form give train frobenius g point regularizers many non alternative mix like solve convex eqn advantageous optimization approximate goal mix yield optimum find solution error bound paper establish guarantee mix norm may seem strong strong particular rip contrast would generic correlate desire sparsity run easier whole eqn work large long extremely still contrast mix regularization hardness solve predictor multiclass sparsity group greedy analysis technique obtain greedy recently indeed similarity wide application objective cost norm rule toward square far seem predictor show learn constrain disadvantage pure subset rely heuristic boost allow real allow sharing class rich expressive identifie class rely also lastly finally share merely across propose share sharing solve eqn greedy column derivative eq rewrite maximized optimize far result example feature runtime step nesterov accelerate smooth accuracy runtime runtime minimize mixed smooth nesterov accelerate runtime chooses show sufficient decrease feature may lead decrease except runtime almost iteration incorporate prevent norm form verify adapt possible overcome define norm main rewrite replace version overall version e tradeoff quality smoothness predictor correspond non concrete mapping decision widely boost piece appropriate mix yield mix norm produce sparse let block two type uniformly zero second matrix denote equal note w expectation jensen mix norm prefer possible show block substantial mix tie way aforementioned well demonstrate scenario share demonstrate mild feature class follow experimental outperform predictor regularization dense show accurate experiment handwritten digit art runtime motivation derive multiclass slowly image digits letter vary class digit capital letter require class space describe rest binary namely select fig overall achieve set easily mild vs compare norm eqn surrogate section fair follow experimental code path regularization run original therefore expand product respectively much regularization preferable rather surprising controlling prevent overfitte one incorporate regularization experiment follow setup run implementation heuristic display decrease much rather surprising add runtime short discussion neural et error goal digit consist digit see example mnist extensively study multiclass handwritten digits rate advanced algorithm mistake challenge mnist nearest nn approach represent entry match x majority label naive produce advance naive run cost shape similarity introduce bipartite matching descriptor compute shape challenge therefore run top mnist feed translate million multiply well geometrically generate expand training million mac test summarize stand original example well svm expand training accumulate ten classifier describe corresponding produce center spatial patch pair pool template plus vector entry template fc response template template template converted maximum going template example template match digit top template produce encode horizontal expect digit fig misclassifie round round template dot product mac round mnist pool type descriptor achieve competitive mnist effort feature cc since fairly cc lead select feature correspond convolution mask might remove piece section mnist come gaussian anchor vector piece anchor anchor point together compare kernel framework subset corresponding point piece wise predictor share principle
intermediate fall break receive node expect link calculate vertical ratio segment contribute height illustration b degree illustrate process multiply since link actual become link generation hand randomly node weight take wise spike fig draw achieve take convolution also correspond isolate original choose ratio isolate rotation fix calculate scenario solid distribution red original logarithmic correspond delta accord deal line dash red logarithmic original setting obtain whereas frame rotation angle degree respectively plot distribution salient explain sect consist spike increase significant shift large fig wider well tendency much stay ratio isolate rapid tendency tendency display angle overall around degree point previously seem similar unique attribute area overall degree isolate change angle argue sect examine sect curve original affected rotation angle fig obtain slope curve curve slope low measuring obtain rotation angle comparison curve setting scenario rotation seem determination within summary investigate new dimensional determining projection question particular vast majority projection slight variation involve rotation rotation determine effect lack reasoning generalise related degree numerically fraction isolate original without science office research sciences h write furthermore line equation top line diagonal express eq parallel shift green write location calculate length area intersection box box triangular square intersect box fig either fig line three intersection four boundary intersect box remain position share fall opposite share intersection opposite intersection fig fall adjacent side fig change coordinate give skewed scenario accord slight rotation preserve degree almost completely rotation skewed nature slowly decrease tv os tool create embed isolated node relatively realistic dominant limit infinitely relation avoid rotation natural phenomenon become connection decade turn highly trivial distance combine average anomalous distribution modular structure play structure extremely understand principle test fine network past aspect newly due concept attract great interest hide systematic study entropy generate wide type coefficient distribution heart mapping measure square self graph become infinite parallel increase slight iteration grow infinity isolated si detail usually construct graph improvement light convergent sequence network size propose rotation sect continue propose avoid sect sect sect generator inspire early worker infinite dense graph adjacency unit square similar introduce used phase transition variety suppose continuous predict replace self define generating measure unit square identically rectangle normalise normalize probability measure integral advantage generate measure time analogy generate factor original convention stand generate factor term stand division analogous write simplify point uniformly interval link pair process correspondingly iteration standard limit e degree node appropriate follow area sized expression increase increase exponentially p iterate multiplying box centre obtain assign draw green construction make standard symmetric unit square although result kx preserve come setting increase degree obtain preserve probability degree natural way however multiplicative generating would column statistically degree topology compose individual row width row give ratio node accord sect fraction isolate depend measure equivalently projection projection belong shall small origin set equal sized box generate equal sized rest shape construction stand step generate evolution unit box length order generalised ref box inside box behave denominator rule isolate node crucial value curve point give occurrence link concentrate one majority isolate self govern expression agreement picture produce multiplication equal general value monotonically thus mean generating node modify construction way projection determine meanwhile kind angle long coincide generation construction set main cut square measure shown coincide measure point along side link pair examining adjust rotation link long arrange straight unique indexing inside triangle various line give angle node origin expression analogous
eps game walk landscape red line start middle black profile reflect white show representative landscape free landscape research intelligence dynamic system thus landscape www generality computer human blue http www game describe random player devise move contrast solve play optimally suggest applicability stochastic game game initially see reaction evaluation quantitative various factor material example material factor subtle form reaction htbp eps dash red sub optimal reaction game describe equilibrium solid red framework rescale coefficient unity reaction optimization position completely number transition direction game exponent dynamic scale sub dynamic project anti reaction alone reaction coordinate indeed poor reaction white subtle phase game exchange piece position computer perform extent maximize gain force optimize iteratively result high coordinate high fig indicate markovian reaction coordinate specify e reaction protein useful generic reasonably good reaction similar description sufficiently time markovian similar sub exponent time original scale distance dynamic equilibrium trajectory represent underlie fig emphasize robustness complementary markov formalism describe dynamic transition profile network agreement way game realization diffusion start continue either end reach profile initially switch many describe constructive game try opposite unlikely dynamic soon advantage barrier much piece roughly half barrier barrier win probability white w w kt agreement compute htbp probability white position play suboptimal reaction game compare win position profile reach protein calculate dynamic network coordinate notably neither naive win collect every position impractical configuration markovian coarse onto construct reaction coordinate simple showing win probability initially black analyze sophisticated make entire try optimal reaction move conclusion move game impossible game move reaction diffusion conclude reaction suggest blue indicator improve win good playing improve heuristic sophisticated treat phase game reaction tailor perhaps reaction game dimensional landscape game assume stochastic coordinate description free reaction coordinate alone specification allow analyze phenomena characteristic impractical play central role construction cancer landscape description construct indicate markovian sub essential missing fellowship appendix give reaction coordinate reaction coordinate trajectory profile energy transition assume constant obtain correct correction entire trajectory move significant net zero detail profile introduce equation steady obtain expand reaction coordinate coordinate diffusion unity numerically integrate free energy sub absolute scale cx different reaction cut value generalize transition minimum position reaction reaction different exhibit coordinate close coordinate maximize flexible reaction attain minimal reaction part another hx dx kt assign profile give langevin attain reaction reaction sum component weight evaluation position iteratively randomly construction reaction reproduce order absolute goodness describe construct partitioning reaction bin transition bin bin reaction convert equilibrium population connect component terminal discard protein reach diffusion energy directly trajectory bin end black ensemble trajectory game play phase game exchange piece accurately evaluation initially piece game evaluation evaluation position white fourth black coordinate avoid phase piece change coordinate threshold discard
library goodness formula ref pt pt pt pt program unweighted unnormalized chi test title computer pc
meta fraction cover clustering iii redundant clustering overlap maximize minimize fig clustering clustering order tell nothing bring novel content clustering way node clustering clustering clustering clustering clustering pairwise avoid triple overlap practice title etc physics article citation apply process paper edge ht article connect bag article connect author article article detail graph composite cluster compare clustered clustering block diagonal clustering clustering set considerably exhibit modularity imply cluster statistically quantum algebra scalar ad world dirac string statistically dirac challenging article attribute country mail country statistically clustering show separation various domain expert clustering vi large far quantitative inspection cluster graph represent influence many many bias may instance clustering clustering bring graph multiple set clustering sphere ht illustration multiplicative clustering raise question improve multiplicative oppose illustrated account cluster language produce modularity graph edge modularity significant modularity clustering modularity score edge pair display c single name modularity vi modularity vi section maximize cluster graph look type distinguish high quality distant clustering distinct distinguish almost clustering sufficiently distant clustering membership achieve choose representative drastically another always group invariant cluster together l address multiple type aggregation scheme finding represent find working prevent rich cluster exist cluster compactly explain show significantly significance improve look intersection increase due multiple algorithmic tractable draw research community challenge room growth nonlinear see section importantly work give chance well method cc edu type main motivation similarity paper author publish accurate describe similarity object metric information datum recover clustering become addition deal clustering graph multi remain good clustering efficiently clustering cluster distant clustering give ground couple community fundamental effort effort construct object represent two graph type graph object scientific base author keyword publish relationship people nature business communication phone etc file group name create graph relationship subject multivariate construct convenient lose aggregation crucial believe working edge accurate analysis importance community sample svd generalization identify factor ground recover aggregation efficiently cluster significantly clustering community challenge problem propose technique rely value detection edge metric quantify clustering result file system group project rest review paper variation clustering aggregate truth multiple type aggregation ground method address latent clustering introduce concept meta cluster discuss represent structure efficiently seek cluster edge type weight represent tuple iw kk multiple intuitively cluster break well mathematical formula algorithm pose significant quality continue lot challenge pose concept graph without get formulation study scope software et al top recursively agglomerative optimize modularity et core discussion similarity call distance clustering metric compare variation measure expectation learn mutual define variation intuition locate rewrite lose know instead second opposite clustering independent cluster multiple type aggregation scheme ground crucial finding community fundamental scheme truth available apply I formally cluster addition difficulty define good matching truth discuss problem find weight quality cluster problem scientific finite observation prediction since try compute random guess cluster clustering quality guess within close ground truth disadvantage reason consume graph aggregation maximize ground cluster quality individual instance quality move ground truth misclassifie truth vertex compute maximize vertex vertex belong large remain strongly proper hold power maximize hold change ex limit step encode hold positive routine benefit gradient gradient nod hold power lose cause hold near extent parameter function resemble position justify individual vertex overall maximize potentially use purpose measure cumulative edge internal edge set cluster cut edge objective rewrite trivial assign exclude yield within feasible modularity random setting modularity modularity score denote vertex graph correspond hypothesis cumulative edge optimization solve hybrid package national problem recover weight experiment aggregation justify et propose benchmark graph size edge perturb independently identically average edge weight three histogram hold power blue example perturb poor green optimal perturb preserve vertex bar correspond hold bar correspond hold presentation portion hold power mean red bar show hold power ground hold power result similarity none vertex hold c ground system classify file belong take similarity modification file file show sensitive correspond hence number positive hold power lot node far account take energy consider similarity citation link share author edge article mail uk proxy truth cluster author edge come distant intuitive article common link topic nearby text encode factor plan nonlinear aggregation state goal aggregation function position pareto objective axis cluster modularity percentage node hold modularity modularity number since aim modularity fig show trade objective axis range modularity modularity range power importantly look hold power preserve reason connection within cluster cluster complex network matter hold power file system aggregation clustering vi truth measure vi clustering objective second clustering experiment produce promise scalability propose aggregation graph operation synthetic graph present correspond runtime depend evaluation since function different run informative grow expect cause scalability linearly metric result omit space edge share redundant combine example connect text similarity cross etc edge document influence similarity redundant article share author tend field nearby say location attribute explicitly much relatively clustering document meaningful class illustration perfectly embed vertex weight euclidean visually nine cluster arrange along axis would along two physics mathematic west east example article datum set directly journal aspect provide partial view underlie structure geometric feature partial depicted fig green red projection provide diagonal partial column partial metric provide however ensemble picture goal view paper conceptual weight combination graph depend particular graph characterize space clustering homogeneous identifiable boundary clustering etc exhibit community present method
minimize remain instead perform minimization w internal external n n concavity exploit positive involve new result version difference present henceforth term counterpart lead threshold cluster associate non early convergent sequence guarantee characterize point beyond scope yield computationally algorithm deal identify cluster operation per cluster relatively data presence imaging application wish dna microarray rarely occur sample expression thousand avoid store process operation hence preferable offer dimensional identify separable next subsection soft algorithm modification entry pairwise update product lie alternatively involve induction every centroid q scale scaling depends latter readily compute tn thresholding prove include initialization column lie exist update n update summarize f exploit algebra stop actually ignore store need purpose input acquire cluster initialize update limitation shape cluster assume linearly separable share limitation mean mapping map term space mean separable partition feature partition able operate two mean trivially th entry term replace must even even reproduce hilbert typical datum non well string graph tailor input regime straightforward compute mn carry nonlinear regime high nonlinear recover input I facilitate dimensional input enable simplify presentation spherical gmm remain dimension implication updating entail become come second aware degenerate overcome limitation fix induce also setup mixture remain valid input property know product update n randomly via c alg whereas store reweighted reweighte introduce readily numerical dataset assess latter adjust rand ari partitioning search remark thank warm start comparable bivariate distribution belong lie outer outlier multiscale spherical outlier outli curve fig correspond initialization zero outlier assume exhibit identify value transition outlier value outlier outlier show slope indicating identify table square center average initialization contamination point approximately apart novel k em include robust mean em low weighted weighted rmse hard soft soft n robust plot point identify outlier proportional kernel circle radius proportional large test united states handwritten digit recognition corpus image intensity digit although label inconsistent digit classify digit digit combine sample image initialization common choose assess ari exclude vector ari table ari assignment already interestingly digit monte c kernel k polynomial cluster obtain yield outli image become nearly dataset cluster fig centroid fig identify identify outlier small trait similar homogeneous kernel ari robust score important observation outlier improve outlier fig use partition identify outlier social link adjacency identify group connection kernel mean spectral connection conventional spectral k specific partition poor local depend cluster initialize symmetric laplacian depict identify identify marginally outli structure yet outli interestingly degree gender dominate unobserved show share cluster college college division games node team link divide team game conference often spectral tune outlier ari coefficient yield outlier sort central three namely mid american conference many game conference game play mid american conference conference east mid american mid american conference american conference ari coefficient partition outlier outlier identify principled accounting outlier cluster exploit appear connection aware processing lead development efficient provably convergent version well suit among developed validate numerical contact author popularity conventional sensitive outlier meaningful structure outcome robust rely translate outli domain robust lie identify sparsity outli choose iterative algorithm robust algorithm develop numerical applicability block maximization lasso mixture robustness subset unlabele minimal challenging yet dna microarray bioinformatics mining label costly k gmm conventional relie thereby minimize cluster scatter fuzzy tailor identify belong gmm consider observe arise ml estimation gmm ml method separable cluster popularity gmm cluster inconsistent appear reading belong rarely phenomena cluster parameter motivate approach outlier robust cluster investigate cluster intend centroid assume respect outli sensitive output cluster approach robust statistic minimum ellipsoid huber contaminate extract cluster root robust linearly contribution outli vector fact rare outli vector compressive methodology deterministic second contribution work comprise cluster develop hard mean gmm coordinate descent close optimization variable particular group solution close operate counterpart application bioinformatic social call aware dimensional involve cluster accommodate probabilistic algorithm update section develop test handwritten digit recognition result letter column letter set expectation pp cluster contaminate develop probabilistic section vector exhaustive mutually exclusive vector euclidean assign history centroid introduce instead compare point centroid distance consider mean introduce membership valid membership coefficient apart satisfy empty pose centroid assignment even suboptimal drop constraint check set one iterate stationary motivate assume c easy c offer merely blind least square ls assignment constraint widely mixture pdf n controlling suggest minimize positive definite solving n membership conditional guarantee pose avoid spurious even matrix mixture grow unbounded g let possibility common common make conventional mle gmm unbounde derive outlier jointly nonconvex aforementione variable us devise minimize iteratively hold ls index however express show uniquely minimize strictly simply c n equation positively scale side yield compactly cost minimizer number exceed cluster point th minimize similar soft regard hard solving distance note randomly f computational resource next suppose cluster e g large mean per store scalar per operation store structure convergent increase sequence iterate follow boolean otherwise problem unconstraine continuous empty set per point
optimization correspond penalize recovery tractable approximation aside condition seem check synthesis validity may become condition purpose sense kn kn nh satisfy toward sufficient validity upper function satisfie provide follow entry equality satisfy recover way matrix augment satisfie condition satisfy efficiently tractable satisfy explicit system computable matrix simple efficient norm issue r tm simplicity associate satisfy explicit computable convex efficiently function context see tolerance difficult since compute relation slack component system tractable block derivation article symmetric matrix sufficient satisfy condition satisfied contrast sense essentially cf q sense range standard gaussian rip imply essentially range performance severe proposition mention validity cover role partition question provide quantity mutual incoherence compressed see sparse provide sense block incoherence contrast conclude pair satisfie suppose satisfy relation pt incoherence exceed pt satisfy assume provide block incoherence proposition imply gaussian norm enough condition fix k nh norm recovery norm block exactly ensure validity signal value consider correspond addition mutual incoherence block structure respectively candidate choose powerful recovery follow find large quantity sparsity matrix result contrast problem block sufficient recovery pt ccccc pt randomly submatrix hadamard h k entry entry take structure arise matrix four scale select norm display level penalize contrast goodness recover ax pt example block sparse recover latter experiment make two conclusion recovery level candidate experiment upper goodness h h r noiseless yield reflect run series simulation four sense block recovery test contain randomly recovery instance nonzero corrupt gaussian block tune actual signal instance apply recovery measure error divide error error routine routine good current rate rating close current simulation routine process experiment rating routine surprisingly second good routine describe guarantee recovery reflect table favorable guarantee purely model observation proportional plot figure recovery favorable performance lasso theorem business office nsf dms handle overlap block noise promise recovery verify condition emphasis recovery condition computable recovery respect oracle link recovery estimation utilize develop block close present linear unknown symmetric w random w k sparse sense estimate signal advance block refer advance representation transform appropriately implement block relate goal compress indeed sense nontrivial therein number band signal measurement share sparsity pattern plain recently relate lasso recovery euclidean block plain lasso lasso selector extend structure cite recovery concentration true nonzero sum magnitude magnitude typically block isometry introduce es block block recovery minor mapping precede handle identity minor nontrivial ax py mapping hand impulse correspond introduce add matter cost nothing concern however emphasis transform provide question note verification condition rip hard study theoretical exception incoherence time restrict restrict eigenvalue property efficient meaningful believe recovery bound error optimize method recovery consider case bound error third specifically utilize kind show tune attain addition condition designing possess usual main organize recovery sense magnitude sparse restrict relation validate isometry property version introduce norm block norm observation guess error guarantee transform everything contaminate notable norm condition among efficiently besides tractable mean recovery satisfy sufficient recovery signal validity build quasi also relate incoherence incoherence conservative latter limit performance alternative either require provide guarantee regular present nx short block magnitude tie vector block denote w sl interest q know advance approximated block observation compact origin know introduce instrumental subsequent construction sense n evident let satisfie know sparse whenever condition closely sparsity satisfy euclidean norm fix play crucial role selector bound eigenvalue assumption l recall sense restrict isometry recovery b define slightly replace weak condition let recovery pair let see thus small small regular good guarantee extract good component whenever say recovery build quantity independent assumption automatically quantity numerical support favor penalize disadvantage routine tune recovery guess tune loose rough grow infinity block block observation error well validity efficiently sense difficult give candidate pair satisfie fortunately become fully computationally necessary optimal contrast induced argument
ks n apply estimator recall converge proposition apply p I k go estimator select perform obtain argument n n inside integral convergence treat n nr rewrite integral trivial conclude every number globally result small observe hold consistency immediately ls suppose clearly p entail closeness ls prove second claim l l I I ls ls n ls ls kp kp b large k l np choose proposition appear far bound sn na hold complete similarly tight already subsequence argument case n arbitrary observe exist eventually par nm n n I view select par remain observation distribute n boundedness necessarily l hold select par observe n n h n identifying immediately calculation eq result follow elementary making distribute front indicator inspection cdf converge part move par unknown proof show atomic theorem move par continuous n convention real I respectively absolutely part see mx converge involve n iw select n continue converge nz nx furthermore n dominate n h n ls n I distribute note establish rewrite display p display n establish complete subsequence n normally use move reference l event tend p stochastically collection note par minimization norm bayesian procedure g shrinkage li penalize fan penalize nd ed york frank unimodal fu asymptotic selection estimator versus fact estimator return estimator penalize adaptive lasso zhang concave converge sense distribute variable freedom method converge variable mx since cdf associate unimodal true distribution lemma argument axiom theorem theorem criterion exercise notation em linear depend addition investigate estimate degree slowly tune perform tune conservative furthermore discuss subject classification e keyword phrase lasso variance high thresholde soft soft regression number orthogonal design soft frank adaptive see wavelet contribution concern thresholding fu tune act study estimator selection fan li fan asymptotic call smoothly absolute act selection fan li fan papers partial except fu paper parameter highly penalize maximum estimator sample penalize estimator adopt move post estimator paper variance contrast regressor depend case variance create trivial consider differ asymptotic variance degree result vanish limit tuning imply distribution thresholding slowly idea rough exposition variance hard parameter denote denote use instead consider variance thresholde conservative threshold estimator infeasible estimator case soft thresholding estimator move tune limit h convex absolutely infinity limit functional finite next tuning possible limit combination location depend arise infinity fast sufficiently slowly weights picture adaptive thresholding well infinity constant absolutely point consistent uniform soft light theoretical orthogonal sample latter non paper thresholding combination absolutely continuous normal component non parameter line expect limit behavior coincide infinity case conservative uniform convergence simplify simplified assumption numerical study thresholde qualitatively adaptive long follow treat consistency rate study l adaptive q vector allow support shall notation regression model situation element square least estimator shall infeasible infeasible counterpart adaptive soft thresholding l cc ls n l ls I ls n l infeasible specific factor interpretation value always infeasible version note hold estimator unique minimizer function exist diagonal reduce coincide thresholde lasso selector independent solution specific iii obviously hold infeasible version exist least event specific depend case hence case lasso soft thresholding therefore adaptive immediately correspond obviously thresholde nonetheless estimator p easily result spirit result thresholde principle zhang minimax mcp penalize apart tune mcp depend shape turn thresholde mcp cover mcp analyzed form mcp thresholding namely least estimator brevity exclude seem mean regressor get least subsequence wise impossible use endow usual topology shall cumulative probability degree centrality write convention selection estimator I consider drop probability infeasible version suffice probability study selection fix n c gives correctly detect nonzero correctly detect coefficient part part except little interest case slow cn n approach lemma kn kn bound least converge cn n c bound incorrectly unbounded converge p n product apply asymptotic depend sample shall concentrate pointwise shall case respectively light extended suppose satisfy I I ir imply limit properly capture move analysis limit uniformity arbitrarily neighborhood holding entail tuning go discussion essence approach speed cn cn convergence case slow remark variance finite allow replace chi random degree make perfectly enough necessary condition k e k conservative tuning know converge eventually pointwise unknown reason variable move eventually generality consider pass converge satisfy n km p n n r I n ir conservative furthermore variable selection case know freedom infinity eventually turn known variance case case expect rate fast slowly b average degree look average respectively variance later par apply see quantity sequence select along quantity proposition convergence theorem par one essentially free accumulation point discuss conservative get behavior tune case state consistent consequence hold little move par probability along purpose follow immediately sufficient every uniformly sense display neither display counterpart follow consistent I sharp conservative one precede tune thresholding matter pointwise equivalent immediately tight proof infeasible hold conclusion continue supremum replace display statement turn arbitrary h I equivalently h x I n multiple absolutely generally case independent hence entire particular cf remark scale apply cdf equivalently cdf n h variance indicator function lead function absolutely density soft thresholding indicator involve shape qualitatively finite entire conditioning provide lasso lasso cf remark estimator framework asymptotic lead conclusion regard cf version give satisfy true n converge reduce n n ir h weakly weakly measure b k n h weakly measure e e tx dx tx e I distribution b n n factor proposition conservative consistent uniform essentially distribution demonstrate parameter framework satisfactory correspond property parameter hard regardless soft observe discrepancy limit proposition uniformity convergence limit locate hard thresholding essentially mean tune component variability randomness two seem connect limit hard sequence recall issue soft thresholding problem high thresholde infeasible parameter variation supremum degree freedom infinity enough apart instrumental subsequent factor satisfy q eq conservative weak general remark none constitute show limit conservative k n cdf se se se se reduce hold weakly thresholding scaling n n n converge atomic normal soft enough factor true n n km weakly cdf measure se I n always reduce proposition conservative tuning limit discussion apply obtain limit limit limit limit essentially set property limit always soft thresholding finite especially ie consequence uniformity distribution degree freedom limit distribution move framework tuning thresholding I I I converge weakly ir n ir r n r consistent satisfy n n convention soft unknown hold tune enough factor km cdf cdf I jump height ms particular weakly converge weakly distribution sense show soft thresholding violate however hard violate limit different seem thresholding continuously fix limit case freedom limit absolutely component seem estimate thresholding soft parameter limit limit know thresholding soft result center scale correspond next asymptotic arise degenerate condition parameter soft fact thresholding estimator property mind long stochastically move cf theorem oracle call statistical extensive tune additionally inspection use I n z n I asymptotically chi distribute reduce I stochastically unbounde adaptive thresholding parameter conclusion apply infeasible simplification I illustrate moving limit n analogously proof subsequent analogous p suppose I I n ix x I ix I ir adaptive thresholding large enough I n ix f n w oracle proposition characterize mass bias proposition whenever save proposition fail different theorem I I fact consequently also parameter lead center identical proposition result n n immediately extend whenever extend fail proposition theorem free reason explain remark
bayes factor often differ bayes discuss sufficient behave abc without ideal achieve tolerance odd feature besides inconsistent choice case assess divergence measure beyond application summary rather reason storage dataset several hypothesis handle solely k relevant factor instance outside domain sample infinity similarly compare point statistic connection appear current paper condition either converge answer precise consistency statistic wrong except nest statistic summary rarely candidate formally state quite convergent abc computational checked summary statistic also summary connection whole bayes factor sufficient solely furthermore collection random conclusion necessarily true outside imply performance illustrate impact double mean equal sample median deviation statistic median absolute deviation statistic explain fourth expectation always artificial computation laplace bayes step show distribution normal oppose data outcome predictive base statistics abc deviation quite hence summary abc median satisfactory statistic fig outcome statistic abc solely fourth moment slowly fourth case occur use distance euclidean lead experimental fig term observation centre either summary abc abc table tolerance quantile ht illustrate fundamental convergence e bring validation abc realistic situation contain behaviour behaviour likelihood section result criterion evaluate summary discussion dd ng n dominating state result hold brief letter occurrence write denote number resp resp symbol converge distribution constant asymptotic mean satisfy define set compatible mean assumption claim mild easy check illustrate versus laplace realistic likelihood lemma marginal compatible equivalence g detail result corollary inference compatible provide yield lemma go tail follow help assumption compatible I n spirit criterion regular dimension non regular abc illustrate present implication relevance behaviour bayes drive value model study behaviour testing probability bayes generality actually enough compatible generality belong compatible irrespective true solely embed go bayes convergent behaviour neither compatible satisfie lead consistency compatible behaviour irrespective therefore select effective dimension compatible essential merely drive set bayes behaviour neither light summary rescale asymptotically leibl distribution variance pdf evaluate purely realistic formally statistic differ example statistic explain discriminate fig expectation differ hence occur must hold fix iid identical truly toy constitute exception case q bayes statistic merely bayes else belong phenomenon consistent illustrate section practical relevance sense assumption summary condition concentrate low g type like statistic chi square also even though control moderate deviation mean model mean typically weak set assumption behaviour often find condition usually vanish marginal become invertible near continuity condition deduce imply assumption term first secondly density close become check case point whole informative informed tail depend recall trivially consequence central verify address mean equal ball case mean address fourth expansion model equal belong bayes due special illustrate lack consistency quantile quantile quantile scale skewness well easy function abc statistic consider model perspective set sub abc tolerance quantile empirical quantile agreement bayes g irrespective factor statistic make prove hence bayes consistent quantile obviously fig one empirical quantile right quantile empirical level row quantile third abc algorithms proposal tolerance quantile distance examine carlo relate abc namely population population divergence effective population define behaviour loss genetic variation act proxy assume mutation occur repetition probability configuration term common single parameter mutation choose parameter abc number allele let copy past copy branch explain mutation adopt model mutation eq apply mutation genetic realistic relevant statistic satisfy statistic moreover couple poisson order theorem least g verify compatible satisfy compatible use uniformly argument bayes converge indeed expectation table second either distance factor expectation fig analysis medium indicate summary model abc software ht summary statistic algorithm tolerance operate consistent summary compatible propose run practical check relevance hypothesis compatible statistic equal relevance mean eq summary compact continuous nan relevance proximity indicate discriminant quantify produce sample posterior choose go infinity asymptotically even case convergent covariance adequate toy run three summary statistic lead respective model quantile distance approximation produce derive mean evaluate experiment base quite satisfactory approximately fall within nan hypothesis include
transition probability measure take respectively environment control probability event policy blind policy expect utility action define value process sequence denote policy handle mdps mdp lie outside thesis reinforcement use framework whereby distribution mdps represent mdps consider hybrid rare derive robust bayesian mdps mdps policy without mdps share state abuse optimal find bayes generally note policy necessarily optimal policy mdp draw computation intractable polynomial belief utility optimal policy tight optimal expect mdp bind refined branch technique employ policy belief variational propagation two moment none return stationary mention interesting belief mdps utility complete utility report condition belief tight mdp mdps belief significantly principled approach mdp optimistic policy expect problem tackle compute belief finite mdps induction arbitrary monte bayesian approach optimal restrict draw analysis probability mdps reason policy respect far belief condition history since policy measure small follow induction together policy utility aa ts ts r uncertain certain belief step utility future past easy alg correct expect bound gap bayes function belief small posterior let dominate final result geometric policy simplicity hoeffding imply policy combine achieve expect alg policy lem lem require belief reinforcement simple approach follow alg calculate return accord computationally via new mdp use rather policy stationary incur small loss htb discount mdp act accord policy mdp heuristic step mdp reward belief act bad htb b alg alg reward expect relative sampling curve alg curve act mdp reward mdp algorithm commonly reinforcement traditionally discount expect reward discount estimate generator transition dirichlet equal parameter reward total hand side difference reward optimal total reward monotonically detail behaviour distribution get sub environment explore less run remains run c percentile interval alg algorithm cite paper interval base report present alg reward enable alg surprisingly actually outperformed alg would bad alg long induction near policy mdps generalise expectation polynomially arbitrary reinforcement sample mdps regular principled calculate near interval experimentally perform make tight result perform surprisingly attribute learn multi account uncertainty dynamic application reinforcement loop employ planning tree tight hope consider wide
adaptively normalize available surveillance camera evolution frame static exploit code encode size bilinear encode scan encoding procedure common coding depict produce regard shannon encoding column laplacian estimate laplacian constant purpose precision short also expect redundancy dimension column also index predictive sequence ij v encode part together length column index non part code bernoulli code model encode row pixel frame location hence parameter use code decomposition decomposition sequence simple augment alm algorithm alm computing base repeat consist surveillance point pass frame stack background frame matrix appropriate long obvious summarize top figure two curve show bottom leave plot scale right recovered background change illumination combination eigen low combine low decomposition theory able promise description graph turn right figure rest capture second eigenvector matrix increase vision recommender theoretical establish portion measure hypothesis nature device regularity estimate address theoretic datum demonstrate complex extraction sequence robust success processing application dominate pca develop alternative review true value recover exactly power demonstrate modeling tool achieve goal want approximation balance estimate generalize adapt overfitte selection formulate balance work assess ability regularity regard practical implementation state short ability model result selection capable capture video surveillance illustrative bring theoretical perspective problem another information naturally incorporate assess matrix matrix iid entry pca vary introduce type I framework mean consider family use use denote description bit description shannon scheme value fractional shannon code naturally lie define assignment possible describe pair description term reduce decomposition rank discard description n safe magnitude encode arbitrary positive integer sum stop non add requirement uniquely diagonal map round virtue assumption unit encoding manner encode
consideration belong low I message jj bound involve potential subsequent iteration lower indeed belong guarantee upper quantity presentation primarily bound I associate compatibility matrix equation entry frobenius guarantee maximal eigenvector scale entry stochastic eigenvector guarantee value proposition simple often denoise potential edge tune smoothness type smoothness bp update quickly motivation consider illustration singular upper sufficient condition intuition node potential degree guarantee contraction inequality refine previously proof main result note ordinary write randomly mass state somewhat e immediate uniqueness bp iterate bp iterate hence uniqueness bp direct consequence accordingly unique iteration claim almost sure consistency claim global stacking rewrite representation side triangle via recursion martingale equation representation martingale martingale difference martingale equation conclude zero vector surely extend argument consequently deterministic iterate piece almost thereby complete part claim martingale concentration know hoeffding integrate turn inequality upper dx manner expectation substitute bp fix immediate realization obtain mapping index notation write compact suitable application conditional know past step apply need verify hypothesis constant lastly define averaged verify euclidean product message lipschitz strict conclude upper corresponding increment recall q side recall update recall obtain combine piece since hold recursion product equal upper product substitute dx fast rate algebra yield claim begin subtract equation denote recall substitute recursion yield note martingale namely cross product term vanish show move martingale expectation finally put piece inequality fact consequently chebyshev deviation specific thereby conclude simplify last obtain mass function depend generate rewrite bp block place q frobenius know corresponding equality representation upper somewhat complete variety confirm theoretical provide simulate denoise simulation edge potential node potential variable interval h cc path couple panel node potential topology collection bp run average run cc b curve examine chain structure particular instance algorithm step trace square versus number plot contain confirm strong give particular observe typical concentrated amount sample path next increase dimension simulation square edge potential trial chain guarantee iteration straight slope exhibit exactly vertical slope grid panel convergence predict theory convergence bp iteration bp c total graph top iteration bp total discuss previously update time algorithm dimension indicate scale linearly bp fair also measure either tolerance comparison fact bp iteration approximate nonetheless table chain graph significant running become large type ccc b gray study instance computer vision two node graph pixel base noisy scale enforce model potential respectively c illustrate square denoise despite jump reduction substantially low marginally job substantially shorter gray deviation experiment vision popular base original et enforce observation potential paper dissimilaritie maximum dissimilarity vision per however comparable bp since standard dissimilarity develop analyze low complexity dimension oppose bp oppose belief also require number usual distribute algorithm main contribution graph ordinary update contraction provide expectation suggest exploit generalization also operate product semi include reweighte bp bethe free bethe free minimization natural idea analyze stochastic posteriori graphical markov undirecte discrete reduce form decode useful variant directly general likely requirement could work approximation phase associate careful reading suggestion improve recall relation substituting bound algebra yield degree inequality constant claim construction convex conclude message particular jacobian derivative continuously continuous set sub direct denote integral value triangle upper eq diameter graph let indicator straightforward verify positive pair direct edge induction claim entry non direct path complete tree structure graph note bind logarithm claim state differentiable apply integral form value show suffice quantity vector bind inequality final defining hence final argument repeat valid take algebra piece definition find bind conclude lemma ccc department statistics electrical computer california berkeley propagation widely message graphical core bp update apply transmission dimensional vector involve dimension complexity bp belief propagation pass neighbor complexity quadratic linear without assume per establish theoretical guarantee performance converge graph condition provide asymptotic decay slow yield communication various graphical belief provide among collection field bioinformatic formalism mean computing distribution subset marginalization computationally intractable efficient graph without marginalization solve product belief propagation bp computation neighbor form message cycle long nonetheless extremely effective marginalization reader e nod discretize estimation sensor network store message scale researcher bp therein pass multiplication graphical exploit check code complexity perform use fourier reduce linear arise computer involve fast transform computation accelerate property absence researcher quantization bp update non propagation certain method consistency particle particle negligible proof researcher technique decode encode message bernoulli lead decode hardware bp novel refer adaptively practically order potential communication require transmission oppose bp even converge stochastic bp quantitative rate provable computing point tolerance precise structured meaning converge asymptotic upper maximum cycle bp update contraction condition rate around simulation showing reduction begin background bp turn statement theoretical consequence devote proof aspect proof defer correspondence prediction practical background belief graphical define joint vertex collection edge among particular clique edge graph clique graph index eq clique dimensional show clique vertex edge take factorization form pairwise discrete variable convert suitably product message easily translate vice ccc unobserve inference basis observe conditional probability write control code value additive white denoise vector corrupt bp incurs require real number pick column message probability costly ready precise potential ji pre make sequence quantity vector message complexity calculate mass weighted operation operation find multiplication regular degree reduce significant message communication bp gain require dimensional edge summarize feature appeal practical edge dimensional bp remainder paper understand message intuition behavior index expectation effect recall perform algebra equivalent lie show despite update bp point stepsize precisely stochastic variant bp update randomness raise ordinary per significantly incur answer certain provable gain tree choice message update bp uniqueness develop working decay high iteration concentrate around potential graph structure provably correct substantial gain obtained begin case markov figure structured integer say degree refer background definition graph diameter state let notation ready
part understand difference area survival problem decomposition class life bias regularization deal survival event indicator event occur censor survival analysis understand accept modeling index measure interpretation censor observation failure order proportion order pair tie rare proportion index allow survival classification commonly ec bias x py py set class bias closeness class variable estimate decision set sensitive shall function especially case high depend size function many algorithm associate sir david cox concentrate hazard conditional survival individual cox strong hazard assumption imply individual proportional modeling actually dependent hazard advanced survival make survey regularization penalize cox attractive produce interpretable path regularization select performance regularization variance variation intend bias step overall describe evaluate increase life directly sum evaluated sample aside size evaluate randomly test average training evaluate experiment conduct conduct characterize dataset hierarchical associate identify cluster single gene gene feature group patient expression signature aggregated figure method figure ph dataset opposite offset additional due dataset regularize cox ph fix may recommendation preferable please file size letter example option file year reader available machine recommendation pdf file acceptable please http www pdf files figure otherwise please check contain type shape implement solid shape complex sometimes problematic include files eps eps clean figure black user file microsoft save office www microsoft com en b ed save office file os drop box save user file computer file http www ps click advanced font option font outline select click ok file ps create file ps file
daily leave change apparent forecast model pick risk bind square give select strategy use risk pick high small dramatically ar truncate daily ever none include mis specification see reference therein control cover autoregressive family believe elaborate family average conditionally restricted stationarity strong variant particular specification theorem theorem proposition edu department statistics university edu mark pa edu univariate autoregressive impose stationarity enough risk demonstrate variable wish another time predict measurable measurable go infinity building learn past forecast ideally function minimize estimate minimization train optimistic risk grow tend prediction strategy restrict risk pac confidence confidence bound time prediction fairly development extend important problem set series stationary decompose predictor finite part memory index oracle finite complexity amount incur due loss approximate infinite finite provide pac non regularize learn sort weak serial square close strictly generalize apply stable sub iid generalization iid generalization learn literature svms complexity risk autoregressive conditional model keep use stationarity autoregressive ar allow application bound without system series theory explicit applicability forecasting interest direction develop need explain idea effective related dependence mix technique risk must source effective investigate reader notion strong stationarity finite vector one imply distribution definition event variable restriction restriction norm one make mix increasingly approach probability typically supremum unnecessary use complexity e g think measure seem white gaussian necessarily everything supremum nz take risk intuitively seem fit nz f ergodic complexity slowly unless flexible nothing predict gaussian base bound present datum predictor space loss loss stationary coefficient bound straightforward control first describe complicated increase small calculate expect tight third term point train reduced process accomplish spaced block asymptotic independence quantify treat independent frequently economic lie interpretability combination value evolve finite ar amount estimate square generalization characterize complexity mix truncate ar bound jj slight ar least norm ol solution despite autoregressive stationary check complex root
term since discrepancy affect hold suffice failure repeat runtime construction p topic fix integral usually shift kernel bandwidth bandwidth write arbitrarily allow www nx n nx n hold bandwidth affect discrepancy imply completeness shift via set point locate locate range small point th would great ball leverage bound instance exist point dp translate size outline absolute reveal hence reveal assumption instead point matter sample subset kernel convolution e input input represent range simplicity focus smooth notion space turn benefit greatly kernel slope bound accomplish study space improvement plane ball set kernel formally start pp define kx kx broad example kernel gaussian normalize kernel pt geometry kernel binary seem heavily result adapt discrepancy imagine space ball range pd kx xx drop apparent book pp range binary range recently show kernel super level approximate proxy space constant vc achieve approach outline book base idea discrepancy repeat point leave colored much size tie classic range resp although intended range discrepancy cost minimize sum point match color discrepancy p algorithm construct know factor appendix combinatorial result pair color long result constructive simple still cost difficulty prove single kernel least section slope space range space boundary slope lot boundary range slope chernoff bind discrepancy analyze kernel j ball extend section extend generalize distance th relate reasonable admit sample sorting sort range align variant fix necessarily vc super increase provide distinction important small quite rarely many simplification family range near net improve low bound question regard improvement logarithmic mainly approximation choice focus determine original highlight book error book generally kernel typically first chen kernel random require require binary completely completely last year although similar literature focus sampling show ball sample space open go range answer size space correspond super raise space simplify analysis range possible quite bound focus write rotation invariant would presentation assumption generalize slope kx simplicity since instance ball volume radius dd dr let pair match main volume ball contain length associate edge volume shape edge edge proportional volume ball minimum sum length shape disk apply ball fit inside length back assume sum handle appropriately next long angle differ edge point towards geometric fact low swap pair type contribute construct cost matching color discrepancy insight apply say bind straight forward point apply attain bind eq start kx slope replace since jensen dd kernel kernel application kx kx p b dy I would replicate bound volume width th edge situation count towards volume could point close close edge either entirely entirely contribute respectively
initialization explain lack accuracy situation miss algorithm remark point outside discriminant group c gmm com diag pca accuracies deviation uci c mean accuracy percentage uci experimental experimental em apply detection mass technique useful disease assess tumor evaluating drug treatment particular detection cancer cause cancer year estimate control spectra spectra spectra patient hereafter spectra control cover read supervise experimental protocol da spectra fisher cluster em pca mixture apply ask remark cluster among deal cancer l cancer control table em fisher em confusion compute fisher principal axis axis em appear em significant negative classification symmetric conversely em point medical negative acceptable false positive em provide understand indeed load high highlight fisher correlation arbitrary peak discriminative axis plot spectra cancer red triangle variable surprising cancer big discriminative surprisingly cancer spectra value extract unsupervised framework power original original subspace spectra indicate triangle discriminative subspace intrinsic dimension parsimonious group estimation estimate discriminative determination procedure discriminative unsupervise furthermore discriminative em em cluster dimensional mass give prove least work visualize cluster estimate discriminative version gram kernel finally could ease discriminative axis author suggestion greatly article index fisher indicate matrix first complete expectation well group th mixture previous cost reformulate us notation definition rewrite operator projection reformulate associate complementary py iteration orientation write ts reformulate tw ty scalar quantity point k column provide maximization follow equivalent lagrange lagrange multiplier consequently th group maximize log partial ty partial formula logarithm determinant td leave covariance conditionally imply du j rewrite q derivative estimation equation follow partial derivative tc already expectation complete partial respect du tc equation partial du equation proposition universit paris paris france universit france high recurrent many fit datum orthonormal intrinsic original parsimonious algorithm propose mixture discriminative subspace simulate dataset provide method visualization parsimonious scientific popular show behavior suffer know mass intrinsic dimension traditionally extraction popular could method fisher discriminant framework powerful span discriminative subspace past year method practice specific provide cluster helpful result difficult visualization cluster addition scientific economic actual accord overcome curse model classify discriminative combine cluster goal discriminative introduce name objective firstly subspace secondly discriminative cluster review mixture link discuss base estimating also algorithm exist simulate present fisher real world conclude give statistical aim observation homogeneous group review scientific field require task conversely refer live efficiently classify firstly review deal widely model group time consider divide homogeneous realization vector density often parametrize unfortunately parameter function particular dimension numerical ill furthermore without restrictive dataset small overcome several strategy subspace dimension tool reduction certainly project axis maximize project projection refer alternative similar spirit find map grid relevant variable selection cluster approach past new approach group two category search subspace bottom discriminate reference algorithm iterative start original variable remove heuristic conversely probabilistic group live data generative space linear relationship recently well mixture family parsimonious partially technique turn consider enough account visualization fisher pose discrimination goal fisher linear separate assume fisher look project observation class small discriminative subspace different criterion constraint traditionally ts kn km vector column maximization eigenvalue optimization problem solve determine discriminant analysis classify solution sensitive noisy focus discriminative subspace compute subspace visualization introduce call latent aim find parsimonious fit clustering visualization idea firstly actual live dimension observe discriminate want group observe realization independent realization discriminative dimension strictly describe dimension latent link transformation orthogonal group center noise follow assume density latent space eq therefore gaussian space complement finally sequel summarize parametrize proportion orientation basis subspace notation model model generate constraint matrix instance similarly diagonal k discriminant view noise variance acquisition process introduce refer variance within latent outside parameter rise independent mixture model nb com diag gmm gmm text parameter gaussian model com diag give number specific decompose number term classical parametrize conversely diag gmm gmm parsimonious model require com gmm intermediate diag gmm prefer turn whereas comparable model establish exist literature close flexible refer parsimonious isotropic principal parsimonious model year well parsimonious loading isotropic remark family parsimonious parsimonious loading loading variance despite difference parsimonious model common orientation close loading subspace orientation good visualization particular axis great parallel discriminative illustrate introduce model probability group orientation probability parameter hereafter fisher sir r discrimination simple fisher combination classification version conditionally current come let observation belong eq space u interpretation mainly mainly depend discriminant complementary simulate initialization frequent datum number parametrization parametrization one penalize likelihood criterion bic criterion certainly popular selecting criterion favor model section criterion context stop computational em discuss condition orientation maximize em view demonstrate supervise fisher criterion covariance expectation assumption maximize u u step em algorithm condition guarantee em procedure rarely fail converge initialize decide algorithm converge detect advance em sometimes slow practice necessary estimate converge user criterion maximum provide stop check whether maximum experiment fisher check procedure somewhat big ordinary require schmidt suppose small notice observation propose consume usual actually pc scale problem fisher appear slow average second necessary fisher em fisher estimation procedure practical interest among visualize axes deal problem discriminative theoretically equal strictly dataset possible extract besides axis may certainly visualization cluster indeed operator visualize actual look equal visualize discriminative visualization obviously axis discriminative one axe visualization relate indeed visualization fisher may bad mini strategy discriminative interest visualization point interpret axis notation look matrix column loading axis discriminative original select high loading highlight relevant loading set remark interest application economic frequent sample problem overview larger available frequently modern mass generative impossible indeed ill well bad overcome step require determination last deal partition column orthogonality fu tu tu theory theorem maximization space reduce maximize orthogonality gram reduce procedure axis dataset sir apply fisher illustration discriminant collect correspond specie discriminate consist specie length fisher community datum latent em datum unsupervise width axis supervise fisher apply course method panel projection show perfectly three reach model secondly show monotonicity stationary present matrix partition provide result confirm within loading stand discriminative axis estimate case case first axis em discriminative subspace axis sufficient discriminate besides turn accordance variable discriminant via visualization simulate right gmm diag gmm fisher ccc ccc gmm em com diag gmm gmm experiment compare efficiency fisher fisher standard model diag gmm gmm gmm simulate simulate unbalanced group group model space complete noise experience whereas axis observe
time approach invariance provide lyapunov framework element work realize relation consecutive allocation reward among player well aforementione mainly uncertain element add robustness capture design allocation base observation allocation specific core game priori direction design rule partial extra cone convergence prove via lyapunov connection lyapunov theory contribution delay flow turn robust control aspect lie fact lyapunov stability idea turn tu theoretic novel represent far main organize conclude denote transpose denote th scalar boundary nonempty utility tu characteristic nonempty tu core denote integral formulate elaborate role among characteristic ergodic set value coincide average dynamic tu player tu games whereby tu nonempty game game dynamic instantaneous game tu vary core nonempty assumption average nonempty introduce steady average game subject fluctuation instantaneous game instantaneous tu player instantaneous assume budget allocation within eq priori starts central know bound know function know observe give line paper goal difference integral reward excess give answer question allocation yes core game nominal rule converge priori say dimensional motivate think stability sense provide statement henceforth symbol f r ft cv look require full know sign relaxed network flow material resource different production serve one demand value realize demand hold therefore also particular play individually cost single serve dash cycle demand ht play agree equal total among serve reference comment apply cycle v topology figure describe clear look subgraph vertex except subgraph connect e player share part conversely subgraph serve represent capture player nonempty reflect nonempty q derivation player also st bound tu model generic dynamic description game discuss encourage play core run occur edge hypergraph player game edge per player generic player associate incidence describe arise naturally allocation demand correspond vertex translate satisfy specifically last satisfied sign condition admit limit driving reach note tb dynamic xt state capture characterize control induce previous full rule observe detail subsection core study allocation satisfy probability solution structure generic pseudo inverse allocation also depend control obtain converge core average subsection elaborate structure highlight feedback review sign since xt xt threshold generality reference component tell choose every accumulate return yes actual ellipsoid optimization cut feasibility us comment condition certainly excess imply content generic development full case know intuitive last positively speed nominal nominal reflect aspect demand stay fulfil theorem next principle share similarity section let close euclidean discrete time analog excess dynamic consider result controller strength result light convergence one let version continuous translate principle constitute guarantee player repeat game payoff root production application player cumulative payoff sup vector payoff player share notion refer describe set proper control independently notion follow sketch formal proof aforementione notion reader continuous differentiable condition strictly formulate prove control set condition payoff instantaneous payoff player h cumulative payoff condition vx towards interior use lyapunov stability start lyapunov establish condition martingale surely tt tt b tt consequence assumption condition imply therefore far integrate last yield u u claim lyapunov true tt xt b tt x tt also conclude cv k cv controller invoke condition evident kx tx conclude invoke condition attain evident thus equivalent tu game value interval convex interval know long run e balanced simulation instantaneous game behavior random r r tp else go go construction interior generate game probability pair simulation run pair take size nominal ensure min u instantaneous allocation allow allow game threshold robust illustrate far law ensure illustrate crucial ensure translate instantaneous allow great next probability behavior corollary law converge core instantaneous lie neighborhood nominal uncertainty nominal study instant unknown robust use lyapunov lyapunov law scheme gd know gd allocation design average capture expectation question like thank explore laboratory university usa mail consider dynamic central continuously instantaneous subject game know bound mean ergodic long run hand generate allocation reward convergence average drive priori cone allocation vary nature highlight extra core converge priori allocation observation
prior imply eq g denote denominator prior write infinite bernstein density mixture normal evaluate side mixing construct alternate augmentation difficulty symmetric build classic represent make proper mix crucially mix monotone moreover completely bernstein return bridge positive focus refer usage west robust densitie connection kernel symmetric integrable let line kernel p constant mixture normalize constant mixture drop originally work stable quite bridge analogy slice form likelihood prior normalization example slice auxiliary interest marginal able already monotone invert slice uniquely course calculation simply case invert wider well know variable latent else choose kernel mix wide bridge especially look like additionally match fact component beyond pose become representation turn case may simulate extended case strategy important bernstein monotonic sx measure implicitly evaluate mixture limit inversion formula expression give discussion lead main could mix bayesian bridge recommend default package support recommendation describe study briefly conclusion exhibit well interaction substantial design matrix orthogonal representation lead roughly principal covariate expand arise discuss finally power conditional orthogonal alpha place appeal omit long normal lead simple introduce far implicitly apply slice latent bridge region define sampler start iterate step generate update truncate normal least simulate e p come invert define back fact center usual naturally account case fact component represent mode run triangle identify try show recent draw marginal draw far plot recent two repeat identification mode parameter directly favorable evident give take hyperparameter draw gamma distribution transformation consider useful proposal approximate posterior introduction show ability local scale crucial ahead reflect shape beta walk level diabetes patient available lar predictor amount lasso concavity diabetes random sampler autocorrelation posterior draw fit bridge lr qr outcome encode desire within introduce term likelihood logit mixture gaussian detail diabetes solid penalize validation dash line predictor standardize bridge diabetes bridge default prior bridge generalize em predictor response center step mcmc relatively difference bayesian classical fit density notable joint bayesian bridge distinct mode coefficient extent predictor case satisfactory summarize classical force coincide attribute ignore classical fundamentally posterior difference mode posterior tc bayesian coefficient say predictor sign objective sense tc copy mark mode mode conclusion posterior different clearly involve median house census plus interaction quantitative concentration package predict environmental predictor use create train split split estimated square bridge bayesian bridge method case standardize bridge regularization bridge jeffreys bridge set implement experiment file compute test case bridge estimator choice nearly square error different bridge bayesian estimate involve construct coefficient simulate correlated residual loading typical simulated bridge bridge assess square error convergence algorithm assess identical machine carlo bridge estimator jeffreys gamma square estimating square classical bayes classical factor tool bridge bridge posterior model summary importance predictor substantial estimate squared loss existence generalization mention scheme bridge virtue directly capable orthogonal mix strongly limit normal method work stable perform global model study file concavity incorporate naturally implement r available explore bridge across wide commonly situation clearly density decrease continuity q fx gs beta give conclude mix regression key bridge normal stable component turn complementary efficiency representation orthogonal problem avoid need exponentially notably wide explicit slice normal explore classical variety datum simulate fitting exhibit excellent mix parameter favorable analogous algorithm sparse bayesian package extensive provide file analogue unknown vector bridge minimizer shrinkage lasso find recent focus asymptotic strategy compute work adopt bayesian perspective bridge arise product power proceed markov stationary bridge group exponential yet penalty crucial concave prior meet bridge dominate although oracle property relevance correspond distribution important yield desirable situation avoid coefficient oracle bridge explore multimodal surface view argument full thing summarize multimodal surface matter serious computational target convex approach seem explore good augmentation strategy mcmc behave anneal never addition present cross validation bridge orthogonal supplement iteration start difference rate control sparsity equally difference bridge form account vertical axis model object comparison rich well uncertainty quantification tail mcmc compare heavy prior advantage free bayesian model
chinese avoid completely world illustrate e facebook information typically available know technique visit challenge depict connect walk rw proportional simplicity illustration rw sample population face resource sampling edge graph setting might poorly mix shown determine fast walk practical heuristic balance relevant part example visit facebook irrelevant resolution category contribution weight relevant measurement facebook time simple walk outline rest paper follow graph sampling combine present unified take account various trade simulation college facebook present conclude scope node list neighbor weight volume weight q volume collect sample contain copy briefly weight build uniformly random node independently frame publicly allocate nevertheless comparison walk possible connect metropolis hasting transition probability desired randomly move rw outperform comparing rw rw choose proportional weight stationary basic design next family sec rw weighting walk sample estimate whenever china take china bad naturally survey partition overlap next select budget prop q another opt minimize every carry may application variance category mean average allocation sometimes give allocation n easy variance translate length opt prop prop long opt order gain gain evaluation section efficiency equally translate interested alternative maximize precision comparison lagrange multiplier gain assume know size application allocation gain many want interested node white similarly facebook interested student account cover prop optimally become gain calculate compose category gain allocation let look work many typically simplification simplify compare node degree category rather entire section allocation independence typically impossible graph available primitive ng minimize although analytically specific topology address limited paper arise graph every undirected edge edge fine collect sample e individual simplification category ideally enforce strictly sample exactly category node discard natural mass equilibrium weight draw try many count category edge weight advantage weight evenly across central factor probability modify arise graph c ic sec resolution fortunately easy category volume information enough run pilot rw derive category weight plug toy interested find red dark middle distinguish type eq white similar show appendix fig due irrelevant benefit contribute estimation need fast goal lt optical system fundamental exception toy clique node weight sec relative category behave black category suggest sample category small fig length goal replace formulation pilot rw sec modifications weight inter edge weight end edge node weight intra category edge inter opposite assign enter stay short intuition induce draw assignment take geometric mixing alone avoid effect hybrid edge assignment pilot rw category fortunately additional especially page sec college neighbor friend pilot rw consider node neighbor facebook sec robustness sec pilot rw facebook set black natural interpret exploit pilot appear poorly node relatively high extreme however induce node connect recommend use facebook maximal category relatively sec extreme grow graph skewness degree rw rw almost equally cluster rw advantage node nevertheless later sec affect significantly fig e course grow weight likely visit rw drop monotonically u shape confirm indeed plug optimal w ec ec minimize already increase high demonstrate black discuss presence tight course vanish relevant control obtain consequently gain achieve exploration reduce factor much length cluster rw perform nevertheless show significantly sometimes significantly outperform walk improve far focus set gain equivalently fortunately line drastically nevertheless drop address category volume estimation sec treat volume affect weight sec unlikely even bring benefit avoid ii size still category form tight community effect choose translate community rw hybrid arithmetic hybrid c college f concrete facebook motivate purpose facebook member college membership publicly available interesting question college college college college evenly default facebook name friend together membership facebook interface inform pilot rw sec pilot visit volume among neighbor greatly outperform volume cover several resolution college student pilot rw fraction irrelevant small phase collect edge resolution hybrid rw baseline table user time bandwidth cost perform rw hybrid geometric arithmetic total college rw come vast majority effort collect geometric arithmetic sample value low target suggest hybrid reach agreement sec finally discover unique rw seem rw facebook look facebook friend user rw advantage rw avoid node irrelevant category one feature important differ facebook college exhibit heterogeneity fair discover rw actually rw span order heavily confirm small per college around rw college rw college well size facebook medium college come walk three rw fail mit college majority middle sized small number rw walk contain rw agreement fig produce reliable estimate finally aggregate gain error ground collect type rw length expect gap rw version rw l pilot rw naive relative size sample base plain dash collect effect recall describe resolution choice try shorter run ten sample small college rw facebook member greatly range member aggregate sampling rw perform rw collect college average sample typically per college length per college outperform rw overhead pilot rw attribute college compare rw factor important robust way resolve implementation minor bring benefit www variant however introduce towards degree impossible later walk rw rw remove bias walk alternatively rw rw rw web therefore rw walk outside walk efficiency rw walks walk large equal orthogonal side fast mixing target applicable knowledge estimate sec perfect assign loop likely importantly face poor sec statistic relate flow random walk equivalently reversible markov state knowledge design explicitly close discuss online employ edge link sample approximately uniform population paper extract something unknown world wide web technique follow page specify interest avoid instead queue queue suffer regular strongly able sample
balanced subtree require inner product comparable product trajectory cost negligible define deterministic element u preserve include exclude u r u long exclude generate equation build stop either final exclude since start satisfied full build stop leave meet state subtree j u u j r j build r building resample initial first explicitly store momentum slice momentum visit indicate meet subtree subtree repeatedly call stop new return satisfied position union return clearly leave leave r u resample momentum simulate hamiltonian begin low encounter slice choosing position height subtree introduce boost jump require operation stop practice dominate algorithm however store momentum memory transition large jump stop easily soon stop correctness save use transition another leave uniform kernel require momentum position disjoint hasting satisfie I leave empty set iteration return final old replace sampling uniform invariant propose half invariant build jump consider reverse try jump fail try jump rejection avoid associate able uniformly return contain maintain exploit subtree explore leaf subtree multiply choose observation build subtree layer build sample subtree two give pair represent subtree subtree continue complete subtree return weight encoding store momentum usually algorithm implement improvement matlab implement part package step attention parameter propose adaptation specifically adaptation base implicitly vanish stochastic statistic aspect behavior metropolis average acceptance boundedness iterate meet converge schedule satisfie satisfy long impact adaptation unchanged say enough stationary therefore burn phase burn quite ideally quickly shift sampler initial regime size iteration issue algorithm nonsmooth find subgradient straightforward adapt mcmc adaptation want towards schedule define clearly go converge update slightly elaborate introduce issue mcmc adaptation conventional optimization setting improve stability iteration prevent computation simulation length iterate forget produce burn apparent setting originally nesterov computation allow affect eventually rewrite feature need try hmc specify produce reasonable well consistently hmc test entirely setting may work neither small computation high rejection rate tune probability strong simulation produce metropolis acceptance probability position momentum th proposal equilibrium accept reject acceptance iteration reach state chain momentum chain understand acceptance hmc momentum explore mr r r r j r base r recursion implicitly build leave r j n r r averaging scheme target convergence recommend heuristic english value langevin result enough amount recommend dual algorithm preference less value save algorithm hmc specify incorporate averaging derive initialization scheme acceptance require implementation package examine effectiveness outline average sampler hmc hmc distribution allow adapt averaging update first iteration hmc space test successive large try evenly space evenly hmc simulation length target iteration different seed hmc term sample ess gradient ess distribution precision ess worth purpose give large ess computation evaluation proxy overhead hmc gradient estimate experiment discard burn ess ess inherently test multivariate ess sampler dimension ess central moment take long reporting size well length hmc anti yield low variance ess evaluate hmc precision matrix wishart target correlation uci repository customer customer receive regression normalize give variance predictor datum customer parameter exponential expand vector make challenging dimension remain customer weak relatively stochastic volatility day return generate refer integrate speed mixing versus target statistic post burn job somewhat attribute regime stochastic algorithm appropriate value since rejection cause lead slow convergence iterate volatility burn converge histogram trajectory length power turn equation complete intermediate desirable half trajectory balance state visit rate turn back compare efficiency length choose length automatically tune evaluation seem occur suggest occur seem seem reasonable problem produce hmc volatility hmc well ess expected test good simulation hmc varied factor optimal hmc usually preliminary figure efficiently preliminary tune hmc section hmc demonstrate advantage rwm gibbs run rwm section run first burn rwm normal whose produce theoretically cost per rwm effectively identical cost algorithm run gibbs long rwm multivariate cost evaluation nonetheless rwm visually independent mcmc algorithms rwm leave independent visualize relative rwm moderately correlation rwm rotation appropriate rotation expensive rwm gibbs highly optimistic parameter multiplication rwm result require time per expensive transformation rwm efficiently tune present sampler hamiltonian monte hmc hmc ability effectively parameter hmc make run place mcmc stochastic paper extension several review consider hmc introduce covariance bad hmc window simply trajectory step size expense window tune lack accept responsible gain introduce manifold hmc hamiltonian space effectively mass although bad inversion expensive dimension function stand ability adapt matrix explore hybrid seem hmc unconstraine value target everywhere handle trajectory make hmc present since tie hard eliminate change probability hmc stop momentum vector region short point visit make progress dual make hand desirable hmc valuable experience particular without intervention suit largely much currently develop core value able effectively sample posterior order magnitude fast summary inference minimal intervention allow researcher datum target markov chain mcmc produce set number would lag estimate ess compute eq estimate precision separate chain try autocorrelation serious fair necessarily lag bad yield ess give interval autoregressive come expense costly quality cutoff find precise cutoff hamiltonian monte many inform feature converge dimensional metropolis sensitive step small undesirable walk behavior turn hmc step recursive algorithm start perform efficiently sometimes tune hmc method require intervention costly adapt dual averaging automatic inference efficient sampling chain monte carlo hamiltonian carlo monte model machine rarely researcher practitioner resort inference review rather series sometimes deterministic counterpart method walk metropolis gibbs require long converge tendency via inefficient walk continuous discrete hamiltonian monte mean scheme transform target simulate hamiltonian cost hmc roughly cost hmc come hmc posterior impossible hmc step simulate hamiltonian system literature set costly tuning hmc challenge hmc general inference http net contribution sampler resemble hmc choose problematic tuning hmc make tuning version sometimes cost find tuning hmc bring generic unable mcmc hmc momentum usual variable standard unnormalized augment system position dimensional denote momentum particle energy simulate update depend coordinate preserve volume region remain map hamiltonian monte describe identity denote covariance momentum generating proposal position necessary simplify addition analogous slice seem present trace process build leaf correspond figure subtree overall start double e start make u stop simulation care trajectory implement derivation follow motivate build intuition use
heart gmm base decoder gmm describe mixture gaussian simplify mean gmm gaussians function decoder posteriori map decoder posteriori decoder heart therein understand concentrate gmm involve gaussian orientation e distribution sort follow next us selection oracle situation signal generality decay dimension understand kl second compare monotonic analyzing help understand lead cyclic maximize direction increase gaussian basis common second give ratio increase check maximize orthogonal writing observation angle principal component gaussian go eigenvalue indicate large lead check anti diagonal zero elsewhere give bottom decrease take eigenvalue correct carlo simulation figure principal gaussian go illustrate behavior similar divergence increase increase roughly rapidly quickly towards cc gaussians go correct oracle hold orthogonal plot give increase eigenvalue decay fast gaussians rapidly show gaussian high htbp two decay gaussian one signal follow via monte expectation draw investigate result number sense correct selection reconstruction going assume gaussian eigenvalue decay random go random dimension even remain high rapidly increase converge stand certain note close far fix indicate accurate low energy concentrate principal interested error energy go obtain reach value cc measurement b normalize plot correct mse normalize ideal energy eigenvalue decay mse respectively converge go htbp cc different eigenvalue signal gmm influence geometry sense selection gaussians higher concentrated assume covariance j real posteriori maximization em iteratively signal gmm map apply sense conventional cs iterative two assume j following assume signal signal code estimate index use assign gaussian well estimate complexity map cholesky map describe iterate map observe coordinate natural image geometry motivate gmm conventional practice decompose patch patch consider follow patch decoder em initialize geometry capture algorithm parameter standard berkeley database contain image dictionary show decoder calculate image house sense slide regard extract subsampling realization sampling outperform sc gain db gmm measure use ideal improve db high subsampling low rate sense gain learn dictionary compare slide regard patch house patch sampling sc gain db rate substantial illustrate truth show first outperform cs contour rd htbp reconstruct image individual aggregate whole illustrate patch remove considerably image sense patch dramatically increase rate nevertheless reconstruct computable sense operator matrix diagonal operator figure typical far remove improves reconstruct generate subsampling reconstruction rate former db rate cost rate compress oppose aim sense reconstruct collection cs depth sense measurement small conventional cs optimal implement pursuit bound constant failure rip upper term efficiently piecewise estimator selection heart decoder term sense compressed present gmm compare considerably compress formal compressed study compress sensing generate nsf author thank conjecture definition compressed aim introduce depth follow sense require conventional cs implement via filter fast pursuit orient yet result calculate multiple gaussian unknown heart decoder analyze sense expectation maximization iteratively signal application show conventional considerably sensing aim rate small interest consist encoder reconstruction reconstruct ill pose require signal frequency classic shannon theory conventional whose column approximation amplitude random minimization greedy sparsity reconstruction obtain approximation upper achieve dictionary basis investigate sparse well novel framework cs oppose cs aim efficiently collection signal reconstruction instead restrict work general bayesian pdf encoder decoder error bound relative conventional signal implement moreover mixture describe decoder motivation first control average real effective example signal overlap short signal signal signal signal real signal art image miss mathematical adopt conventional cs reduce optimal linear significantly cs error cs sensing time bind extend gmm introduce accuracy heart analyze property sense measurement general selection compress presents posteriori gmm calculate map algorithm apply improve cs cost reconstruction discuss rest eigenvalue error orthonormal pca pca vector generality pca bernoulli universal analyze canonical cs perfectly kk belong condition perfect must exist possible construct size requirement mm compress sense reconstruction measurement signal reconstruct position non measurement thus suffice concentrate signal gaussian full eigenvalue decay analogy conventional mention simplify without generality gaussian one always center signal prior e e mse pursuit calculate signal fast via filtering store since decoder seed change equivalent nan index condition decoder optimality hold hold q nan constant nan decoder follow choice decoder necessity let otherwise split vector deduce proceed mse comparing require good requirement thank linearity good prove decoder nan consequence theorems gaussian mse constant optimal instance follow mae mae decoder consider instance optimality isometry rip measure preserve conventional rip order requirement rip support positive integer let index rip rip block rip sparsity consecutive relate distribution let satisfie decoder assume e rip something conventional sense signal index inequality rip fact realization verify rip address concentration eq rip outside great rip require next multiply rip fail subspace let draw accord rip great prescribe matrix rip fail whenever exponent exponential enough ensure prove nan rip linear rip coefficient pre consequence measurement gaussian bernoulli rip rip improvement signal gaussian sense example
also maximum likelihood two average svd mle via distribution near page near object http bin fit fisher analysis near focus direction direction direction close direction object pair lt unit lt lt meaningful identify treat plane sign preserve svd q analyze either statistic preliminary size hypothesis uniformity nan uniformity statistic chi let column similarly uniformity almost fisher two clarity fisher expansion add mle normalize truncate maximize implementation bfgs start gradient find reject reject normalizing series next gradient direction add descent respectively agree mle mle improve mle expansion aic criterion minimize statistic degree ccc aic show numerator check nk generator generator look coincide module theoretic nk derive let long odd number respect lebesgue value eq odd mm mm langevin rotation group gradient descent expansion normalize constant estimate compare manifold illustrate keyword algebraic gradient method series evaluate normalize constant rotation manifold exponential family manifold family nice normalize I generate calculation however define expansion integration monte iterative recently introduce likelihood normalize integral involve module equation constant simple normalizing von respect assume simplicity maximum method update accord dc gradient modify know give point obtain numerically key give compute numerical repeat converge determine need direct normalize partial contain let numerical paper apply gradient descent rotation orthonormal manifold study number orthogonal illustrate develop integration distribution normalize normalize organization rest section fact group manifold section equation satisfied normalizing section series normalize constant datum notation summarize preliminary fact interested orthonormal denote matrix transpose orthogonal group volume probability call denote two prove similarly qx qx ie rr value decomposition svd let svd also preserve singular fisher distribution fisher fact chapter distribution r matrix argument group correspondence projective show one dimensional integral normalizing constant argument section derive differential column question distribution fisher fisher distribution redundant however fisher state fisher dimensional smooth measure open neighborhood function hull sufficient see zero p ie ie ie e ie ie ie l e x v r let svd solution preserve explicitly remark normalize sample preserve ia ij jk il ij cd polynomial denote ideal diagonal diag operator rational proposition dimensionality gr find ode apply maximum ideal ideal dimensional prove utilize gr generator function representation generator ideal span field rational function proposition large gr basis program website conjecture dimensional operator respectively denote generate matrix rr differential rational equal span proved computation program conjecture consequently differential well differential equation differential differential polynomial differential operator equation show extra differential put normalize satisfie equation utilize expansion normalize next subsection computation explain auxiliary operator ideal normalize find follow q e find ij obtain coefficient polynomial solve candidate also ij n analogously eq normally order system collect ij put analogously method q straightforward calculation theorem explain evaluate derivative truncate series expansion extend equation ode constant restrict differential satisfied apply broad guess search point mle guess system become current heuristic grid make exhaustive
applying alternatively verify choice satisfie radius appendix fix high extend program recall expanding imply triangle ii schwarz h old conclude follow apply cm lemma assumption ccc department berkeley berkeley version technical report berkeley solve matrix noisy transformation sum matrix complementary statistical share give bound pair norm impose incoherence result study plus plus frobenius stochastic noise matrix approximately low approximately identity operator establish show result theoretical confirm decomposition problem suppose recover pair course ill pose necessary denote allow form sparsity matrix motivated classical reduction matrix problem robust covariance describe plus arise decompose decomposition arise collection task common weighting preserve model discussion motivate study use linear operator simply map task linear operator deterministic stochastic exactly assume instance observation version involve noise matrix form share across across observation decomposition analyze past noiseless case entry give sufficient adversarial zero analyze uniformly xu analyze aware detail yield oracle structural impose decomposable decomposable regularizer theorem solve class convex program composite regularizer frobenius sparse asymptotic sparse corollary corollary broad class model deterministic rank observation addition general identity operator error estimator see feature impose incoherence condition norm sparsity dual bind proven set identifiability exact noiseless provide degree identifiability arise devoted corollary decomposition bind devoted proof technical aspect defer discussion convenience ordinary family form act reference therein applicable regularizer satisfy property particular work motivate factor random generate I load project onto guarantee need span nonetheless ty l relatively constraint via see give problem j write multi application share across type share model impose extreme enforce however learn complicated subset task subset substantially task amazon book include user numerical mean category kind namely differ significantly baseline instead row zero row discuss appropriate enforce column sparsity general notation obtain linear corruption straightforward perform index subset corrupt corruption z sample covariance center wishart remain write column write column entry sparse example observation consider estimator solving regularize negative regularizer guarantee property estimator although reasonable yield attractive property practice additional noiseless setting indeed recover unless incoherent position decomposition low sufficient exact impose condition introduce past common statistical realistic observation contribution mean substantially regularizer quantity moreover norm estimator intuition sparsity sparse motivating include factor non sparse formulation robust pca gauss sparsity regularizer choice general take form involve serve type control prove relaxation appropriate signal maximally position lie decomposition impose low rank component singular vector quantity coherence singular canonical remarkable make exact goal noisy concern recover singular impose moreover bind incoherence poorly behave small number small include covariance xu et norm regularizer analogous verify constraint serve component manner natural noise ratio stay fix extreme maximally zero column lie discuss consequence apply belong decomposable regularizer restrict norm decomposable regularizer strong appropriate result devoted consequence complement convex show special operator show fail notion term subspace need special example subspace think choose complement guarantee deviation penalize decomposable subspace pair relevance example decomposable appropriately subspace kk decomposable define compatibility compatibility frobenius regularizer subspace yield convexity establish quadratic loss function convexity lower strong convexity restrict convexity weak condition norm regularizer weight correspond minimum well convex see strong respect norm operator choice non choice rsc establish error error estimating identifiability rsc state observation later subsection result result use across matrix notation model low frobenius consist three rsc rsc curvature exist regularization universal remark clear theorem deterministic statement apply convex program whole bound choice optimize obtain upper condition multi rsc hold correspond complexity sub associate correspond represent similar interpretation correspond index adaptively simple rsc condition low lie within decomposable subspace vanish theorem guarantee square frobenius error regularizer decomposable respect note decomposable regularizer decomposable program constant matrix index note theorem decomposable verify jk claim index subset integer far low natural approximation vanish guarantee noiseless approximate namely eq guarantee incoherence singular singular factor identifiability model operator consider fact lower precise improved impose restriction incoherence impose exclude attain bound however estimate soft entry component interestingly understand first step co convex step component minimize sparse constant observation regularization solve optimality descent method operator rely inequality hold trivially first introduce example show return consider one give small singular let consequently good mean exploit fact reasonable tail concentrate expectation large consequence method decomposable keep presentation relatively brief regularization arbitrary cardinality directly previously example norm discuss decomposable subspace rank column vanish bind eq slight modification exploit much small difficult compare xu exact concrete matrix suppose satisfie program great corollary upper bind reduce corollary interpret freedom somewhat subtle estimating column embed within frobenius selection parameter sub involve multiply usual discuss consequence column wise consequence theorem consider covariance sample satisfy column solve sdp great comment motivation concrete parameter involve operator factor case achievable namely achieve frobenius error turn complementary fundamental algorithmic independent limit question analyze infimum observation model give jk interest family corollary minimax risk family observation q agreement guarantee corollary respectively discuss corollary logarithmic careful relaxation worth involve noiseless analogous column program adapt excellent agreement square vector form sparse choosing position zero uniformly nuclear motivated matrix study study scaling predict universal fix square complementary scaling sparsity vary since plot agreement theoretical ccc frobenius corrupt entry growth square theory error e theory neighborhood around ccc plot frobenius error plot error predict curve dimension reach sparse matrix dependence interest different phenomenon report vary dimension zero plot dimension choice decrease scale error produce linear plot panel b moreover main proof defer assumption regularizer decomposable throughout convenient shorthand deal weight decomposition q upper result restrict algebra show slack frobenius convexity taylor complete algebra h triangle allow upper condition obtain inequality side yield combine lemma rearrange special corollary regularizer requirement pair little z w j requirement column gaussian variance bind union state turn rsc multivariate show rsc careful bounding reader wishart standard hence tail begin recall define error proof lemma component proposition optimal writing inequality side obtain choice claim addition low bound somewhat appendix bind order condition early upper noting concentration measure conclude probability replace refined detail noise singular wishart high quantity bound claim proof reduction packing collection pack q packing consequence component bind choice different bound technique begin prove matrix decomposition involve radius non identifiability namely scale construct packing matrix low verify matrix distribute pack pair pack equal piece imply eq pack subtle one component modify moreover matrix modification set interest packing integer conclude exist cardinality apply thereby minimax suffice divergence suffice exclude degenerate suitably sufficient thereby follow argument packing set bad denote construction pack verify distinct frobenius packing ensure control guarantee set characterize packing set sparse pack integer k n cardinality satisfy suitably adapt rate lemma apply state packing claim theorem analyze relaxation general class decomposition goal
distribution convergent motivation come shape attempt use shape distance explicit shape space next space linear functional word denote simple discrete row fix particularly form simple let suppose verify satisfie result dual us norm case large merely onto return norm yield metric return maximize project actual mind build take eq value linear linear interesting pick particular subset know hilbert distance rewrite simplification familiar us kernel section compare specifically surface viewpoint treat shape easy surface discretization yield invoke shape surface information location point hausdorff curve require continue surface geometric measure tool geometric surface encode orientation surface generalize form whose cross product vector imply simple combination vector generalize tangent capture one expression differential understand linear form use denote appropriate tangent vector surface action understand integrating point capture action definition continuous generalize orient manifold include relation orient manifold relationship schwarz distribution space back variation total variation variation manner example induce irrespective curve distance disjoint ball retrieve kernel instance vector expand represent transpose precisely lead alternate current introduction distance review provide tailor reader background computer exposure analysis geometric aspect shape g surface space rkhs structure elegant mathematical recent decrease simple kernel use root symmetry transformation transformation general induced multiplicative need construction theoretically symmetric difference set expression cardinality play kernel kp kp p act square view distant similarity similarity point another kernel distance naive would sharp similarity uncertainty feature kernel beyond set surface merely generalization intuition deep mathematic weight merely sum cross integration distribution usual let consider convenience continuous entirely rigorous think merely space vector whose coordinate entry interpret generalize look eigenfunction generalization definite positive kernel induce hilbert useful operator operate kernel take virtue integral analog decomposition positive positive definite eigenfunction coordinate eigenvector describe span euclidean instead euclidean capture definite euclidean map
frequency classification build shape fourier coefficient reveal sub set neuron unique spike unique word know neuron vice versa micro record neuron gmm detect neuron every spike necessary window tt detect spike entire micro long far spike since signal local window spike thus know tell look characterize spike visually quantitative automated rule spike spike detect outlier bias spike favor spike neuron exceed spike fire neuron characterize spike spike rule move spike identically iid signal slide window reveal spike histogram fig simulate play reason assume brain background noise neighborhood micro iid histogram slide background iid signal due lower indeed though know slow know clearly bottom move spike spike concern separate boundary extract spike spike fire firing interval lower good call neuron spike unit extract possible spike step take avoid slight spike sort adjust position show apply slide exclude visually spike seem noise low spike signal dynamic structure induce yield density series define sense process decomposition hence indicate important variance peak contribute integral spectrum fourier frequency frequent generate variance representative system correspond nonparametric raw series log signal stationary large sub system thus plot log minimum wave minimum figure separation match close overlap raw fit show axis represent fouri frequency variation series frequency cycle cycle year generation short business cycle two show growth center right code red green intensity equal n connectivity cluster confirm separate three major highly persistent slow business highly persistent affect business peak flat frequency short cycle slightly business red matter global affected rely heavily production technology affect global happen map effective policy boost effective public affect global try fit spike summarize greatly accelerate computation fit mixture gmm logarithm spike histogram peak correspond differently shape spike assign spike posteriori model accord high bic choose run gmm spike show conservative shape represent gmm neuron domain advantage domain signal fit monotonically separate spike relevant spike behavior simulation prominent occur shift easy reveal five spike domain spike introduce novel detect classify dynamic signal dynamic shape stationary signal auto stationary spectra research area process sort pattern economic demonstrate usefulness present also introduce neuron spike differently shape literature robust threshold chemical band hz dataset average entire rate remove overall us baseline rv finite alphabet kullback leibler kl particular sample dirac delta rv estimator mle data divergence intuitive empirical plugging kl log maximization conversely em spectra allow compute model em algorithm claim theorem exercise theorem summary department maximization em group time class dynamic magnitude fourier dft spectrum pmf similar pdfs pdfs parametric robust model shape stationary auto correlation spike sort stationary pattern economic usefulness stationary k example brain speech stationary economic researcher study send understand fast brain neuron necessary send neuron economic public market characterize country recover region formally entity economic divide homogeneous dynamic different neuron economic market need many dimension reduction correlation show carefully distinguish series extremely impractical observation fit average study test distinguish two necessarily independent datum see detailed although small suffer model wrong sense hard metric clustering distance similarity spectra treat signal pmf circle algorithm avoid cluster series macro economic decade law economic country us state economic help provide economic support overcome certain state show year business heavily analyze auto order adjust growth select base unlikely different dynamic particular model business dynamic spectra growth adjust store environment must visual back brain neuron front aside neuron measure top signal brain spike two spike visible macro measurement help
ccccc space decomposition mixed piecewise linearity comes induce link strategy consider internal bi piecewise linearity write belong additional assumption thus mapping partial mapping polytope inverse piecewise subset chapter vertex state lemma fix belong simplex belong possible decomposition associate convex get linearity conclude proof piecewise lipschitz norm contribution rely bi piecewise mean decomposition thus exist piecewise pp lemma bi provide show satisfied one p p argument monitor strategy compute mapping compute set ap estimate distribution exploration usual monitor ingredient unbiased extract I take conditional expectation equal first take argument precisely sake simplicity provide fix next simply proceed regime time round consider depend satisfied bi fed eq fed efficient turn respective indicate finitely round possibly successively notation th statement immediate linearity definition assertion direct element remain assertion q forecaster th block hoeffding sum hilbert space martingale get assertion q since lipschitz norm union block application assertion get put thank triangle length exploration need average result action pure setting rely parameter play form compute projection set ensure q slight modification one square third boundedness state inequality follow compare sum integral therefore conclude extension polynomially obtain indicate polynomially average consider variant block polynomially weight average call hence version ensure depend proof closely resort hilbert martingale translate begin lemma regime union regime substitute suffice original convergence robust guarantee get put thing obtain end constant ensure result minimization game result vector game monitor bi linearity satisfied consider sometimes scalar payoff player section q action second round regime external player round define maximum regret indeed first bad player play action induce law action actually play theorem explain however achieve question could strategy game hand proof partial monitoring strategy external latter case payoff consider j actually mapping fix coordinate true convex hull linear piecewise function exhibit consider therein end lipschitz denote soon belong latter norm define claim see follow indicate indicate statement component entail argument piecewise equation refer rp evolve way piecewise piecewise equation onto hence internal monitoring follow internal stage action play action player stage partial evaluate payoff pure element stage pure play convenient first mixed thin g measure round stage internal first player measure minimization swap regret definition swap regret easily construct direct swap internal two present strategy strategy grid complexity exponential idea reference provide explicit strategy player corollary idea role player proceed show assumption swap fourth final possess bind cone g q q suffice piecewise piecewise accord separately project equality bind swap follow p rp g r rp g rp prove norm bound j convergence conclude claim statement prove end l triangle inequality fact probability conclude simply resort write triangle schwarz final conclude exist game monitor follow bi mixed action play play marginal show set game dirac denote concavity strict side product also bi piecewise linearity show exist per linear consider conclude thank make argument notation therein payoff mapping mapping refer substitute satisfy interpretations hold note component construction definition show work positivity component payoff eq extend without piecewise linearity strategy rate complexity bi piecewise game devote scalar payoff piecewise linear equation replace extension polytope intersection original necessarily bi piecewise game payoff define eq equivalent sense translate hold lemma least cardinality vertex increase quality playing regime close strategy round fine around grid bb nan q form payoff element locate position close resort characterization pure necessary q rewrite correspond action form positive square equal schwarz last signal probability put signal indicate martingale whose generate since show sum variance sum get union onto convex proof paper appear conference page paris paris paris france paris en condition become adversarial setup game ambiguity belong rather tackle game partial monitoring algorithm regret partial monitoring strategy theory tool learning algorithm game numerous theory geometric situation easily quantity regret minimize game whose duration fix analyze perspective monitoring monitoring maker get maker setup include reward setup tell decision nothing obtain repeat situation gain situation one signal reward statistic maker regret minimization paper learn bad maker thank extra vanish procedure case base regret minimization extension partial value importantly concrete problem work constant seminal theoretical study concrete matter recall fact theory decision maker opponent propose setup term value decision maker obtain represent ambiguity concern reward linear maker robust value monitoring type property bi exploit game rich enough useful section constructive constructive highly inefficient sort probability mix require refined number external partial simple similar simple mention explain special efficiently convex although recall value payoff consider two player maker player refer reward payoff mapping whether pure action denote strategy action obtain accord denote monitor take opponent round monitoring indicate reveal action player strategy player ensure value payoff almost uniformly form often strategie player approach soon sometimes player strategy soon sequentially action monitor bandit monitoring reveal exist deterministic close simple consequence two pure mixed action observe require monitor play depend take projection norm solve equation inner case minimax determine easily zero associate enjoy follow theorem mix take pure strategy least imply monitor modify lemma constructive probability since equivalent equivalence indeed state mp e one arise former pure action mix include p repeat lemma indicate translate rate rate instance robust projection favorable g minimization external monitoring play mixed action play pure obvious bandit monitoring play ta previous value pay loss efficiency function uniformly x yx norm differently continuity q lemma indicate boundedness argument entail uniformly continuous argument close extend consider hoc contradiction assume satisfied necessity apply suffice follow concavity robust mixed action approximately player introduce calibrate first consider constraint denote ball choose forecast l choose denote nonnegative player score consider consist ensure probability existence calibrate base calibration illustrate proceeding regime dyadic round carefully term length modulus continuity argument calibrate associate element
process replace absolutely continuous lebesgue change specifically index whereas induce dynamic adapt satisfied integrable h log ratio version sufficiently understand locally additionally log admit suffice due end shift clearly remain martingale theorem pre follow vanish condition integral sense definition stochastic delay measure term pre optimality continuous generalizing sufficient brownian satisfied fractional brownian fractional drift brownian motion exponent way drift brownian motion section corollary proposition rgb sequentially test version special optimize criterion fractional index exponent quick problem application area track surveillance security rule raise soon occur detection small false formulation change detection trade goal develop change consider formulation exhaustive sequential book poor cumulative page quantify delay change criterion subject upper identically criterion attain prove optimization optimality continuous brownian drift case signal noise product obtain optimizes modify version diffusion satisfy criterion diffusion path process admit thus clearly exist optimality particular optimal detecting change fractional brownian fractional detect process polynomial term exponent rare recover point view presence application phenomenon characterize long memory result build traffic finance diverse great fractional calculus study drift coefficient process rao study version mle rao optimality process vanish coordinate every tt mutually restrict algebra denote change deterministic mutually stop follow consider constrain quadratic problem coincide explain inclusion redundant proportional eq formulation detection delay period actual accumulate variation latter appealing leibl detection dynamic detection define solve define satisfy correspond false alarm introduction detection diffusion full condition additionally diffusion ratio optimal prove reveal vanishing quadratic pg u stop event notion leibl consequence close expression function let introduce statistic consequently clear give definition u consequently moreover condition side monotone increase condition tt iterate p otherwise become infinite due contradiction go back simultaneously monotone convergence right side interesting terminate surely sure alarm explain constraint impose false alarm occur continuity increase word bad scenario alarm threshold generality time false constraint stop stop time second complete side suffice way standard motion random brownian motion two equivalent ds reduce constant proportional
reach event q convergence proportion imply define eq update eq without markov irreducible sigma rational q counting come detailed rely require satisfying proving seem finite irreducible proportion converge vector since mean visit frequency number x k proposal accept make leave use exact contradict go infinity toy toy consequence walk arbitrarily split desire frequency result wang use figure proportion bin use dotted line indicate check observe converge toward figure similar plot side equation theoretical limit example occur update convergence frequency ptc ptc vertical left proportion bin update update dotted represent toward consequently meet theorem allow irreducible measure require equivalent walk know precisely transition differ hasting generate different bin position metropolis hasting weak recurrence property fix flat reach frequency criterion note eq measure irreducible lebesgue time strictly lebesgue process visit time mh formally denote nonempty go statement chain choose proof go come let go define path reason wish construct put together prove point reach author thank lee p author theorem research science aim sampling region state use wang reach flat never reach variation bound context consider sampling suppose density constant proposal mh mh let denote indicator obtain ergodic algorithm wang eq choose multimodal distribution region attempt choice monte algorithms proportion choose might refer desire set typical bin guess frequency wang introduce penalty time act define distribution return index bin x jx mh wang generate past update widely physics practitioner flat convergence property criterion finite variation miss section wang penalty argue convenience section certain flat criterion might relax several algorithm schedule schedule temperature q typical wang pseudo code kernel I penalty visit chain decrease otherwise obvious converge use since update practice shall schedule along kernel fall mcmc hold literature wang schedule shall flat decrease schedule predefine threshold iteration intuitively criterion meet observe desire histogram proportion always finer flat decrease step meet know meet counter iteration meet counter something else case update give indeed need wang widely physics show meet one meet hold know meet occurrence schedule share invert plug limit proportion hence update proof go remainder define eq eq consequence run necessarily update corollary consequence validity proportion desire flat make simplification make bin empty compact mh acceptance side eq assumption verify proposal compact wang four propose proof propose increment bin increment depend whereas update prove go strong leave interval probability start decrease keep process event k dy qx qx dy qx dy exist mh ratio qx dy qx dy I define take strictly probability start increase u b increment increment return imply leave visualize process horizontal go time stay integer let take z first probability indeed markov state hence u
powerful gmm ability growth framework attempt er binomial process generate coin form random binomial growth need fix fact classic fix number rule er growth base unit interval node pseudo er gmm classic gmm growth rule grey bar density randomly er gmm pattern experiment classic er random build software network evaluate gmm simulation follow result describe purely create edge degree distribution binomial minus loop subsequent er model binomial distribution fit er top grey bar binomial classic goodness regression calculate wherein densitie approach count chi square case due abundance distribution assess error rmse well std rmse aic gmm er gmm gmm report measure quality fit aic quality level simulation basis er gmm successfully recover er er cumulative comparison parameterization encourage gmm easily er say er describe network people gmm classic attempt mention short average path node localize short length localize cluster clique densely lead hence produce lattice base additional incorporated lattice lattice iterate achieve desire localize graph measure comparable paper large theoretical attempt enter argue describe network regardless remain degree connect component pseudo report graph compare representative world lot specification benchmark compare gmm classic er variation experiment represent simulation line classic gmm simulation dot dot characteristic length low world gmm simulation gradually increase wherein flat short gmm may represent ba gmm vary base upper panel highlight strength framework panel gmm ba gmm gmm panel produce varied scaling seem gmm affect interesting variation produce well result parameter graphical rule produce scale average produce network graphical however undesirable show classic ba piece produce systematically precise maximum fitting law immediate considerable reduction simulation moreover mle ba estimate produce scaling figure finally size affect panel gmm encourage concept exercise gmm able successfully recover structural classic reveal gmm introduce alternative evolution assumption gmm assume evolve actor enter network growth current rely gmm count subgraph structure computational simulate basic gmm package programming rely scientific package specification near gmm test gmm classic small successfully outperform encourage gmm many many political science state much model often model lack flexible human interaction subtle complexity gmm able capture affect social outcome note technique limitation draw gmm sensitive conduct nature growth termination treat fix construction interpretation initial rule rule furthermore potential poorly suit human many growth contradiction biological poorly protein network gmm primary establish evolution human social limitation limitation order belief network structure problem therefore scale poorly complexity relatively result speed beyond research technology improve scale could remain static mean subgraph need count compute version improvement account growth base simulation run utilize consideration quality large ability compare fitness comparison constitute large portion future appendix termination si binomial section subsequent technique social one largely political discover primary relevant science represent actor interaction macro include micro study comparative party american finance contribution political science application variable measure centrality whereby relative actor type political science international relation inter key actor central actor vary etc similarity among international pattern american method political behavior outcome centrality member analysis causality static aforementione co difficult influence co behavior affect influence outside political insight process alternative viewpoint preference actor actor attribute structure simple one member tie technique coefficient group appropriate treat innovation highly unbiased network application political go suggest correct international relation causality wherein determine systematic science remain paper research trend continue advantage separate coefficient centrality co difficult affect relationship analyze longitudinal research much focus mle panel assumption determine relationship mle specify extremely envelope incorporate dominate give availability present exist network dynamic mle introduce tool method heterogeneous remain valuable social science especially political interested organization historical position actor simulate actor role attempt possibly something organization actor join function attribute always inherent security concern reasonable occur actor assign role network localize network leverage way political collective relationship two inherent collective action gain may make structure context consideration network collective much experimental game actor match color choose treat type reach equilibrium study variation vote collective act color structure vote evolution precisely base propose modeling tool remainder proceed graph modeling growth formal specification software package recover classic binomial world simulation characterize phenomenon social seminal decade social phenomenon technology improve ability complex mean vast majority heavy late structure model mention preferred network networks monte estimate model address closely gap concept social alternative level structural occur random graph provide insight structural remain specifically mean model design dynamic adequate modeling understand dynamic countable constitute practice generate achieve degeneracy implication limit interest complete connect empty model vast majority social actor model enter structure except simple actor actor structure immediate growth network rich complex social people meet change social create social friend worker simply create exist increase bridge relationship visually difference concept considerable ambiguity figure dyadic binary dependency metric include diameter likewise level rigorous yet assumption ignore inherent complexity self evident reality natural interaction provide modeling leverage graph overcome limitation random graph social network graph gmm new two key actor network actor bring build network process enter observable structure force strict requirement share random derive belief model one random base constitute despite degenerate base structure similarity note exhibit increase first mention preferable network relevant unweighted describe graph set describing may gmm however applicability unclear hypergraph wherein limit graph great account singleton entire gmm node ordering become critical describe let tuple increase node correspondence among subgraph put another subgraph np certain approximations belief structure order generate belief aspect growth specify evolve subgraph calculate count generate evolution simply define discrete count give mass element tuple element element dependent pmf purpose specify pmf exclusive example pmf satisfy complete exclude set pmf enter subgraph wherein zero mass subgraph base specify probability pmf gmm explicit latter pmf subgraph provide discrete structural proportion subgraph provide belief new structure subgraph base never enter possible pmf limitation structure exist maintain bipartite structure assign utilize element structural specifically alternative mean natural motivation occurrence increasingly event likewise enter graph around base assumption reflect generate however specification specification pmf pmf direct gmm require subgraph limiting define pmf distribution assumption probability gmm structure graph take form draw add growth rule construct assume decision applicable fundamental subgraph design growth rule form belief belief mass likely grow dynamic calculation continue termination gmm growth beyond restriction termination strict model gmm framework flexibility technique detail one gmm specify proceed graph describe attempt current propose wherein rich describe evolve assumption base upon form belief model base graph represent useful proxy model order tuple define subgraph pmf growth repeat termination gmm
vote give strong weak strong perfect next subsection first great ensure dictionary weak weak opposite opposite rule counterpart opposite decision ensure great classifier index formulation imagine space condition specify q pair final group opposite classification condition set cover output interpretation illustrate entire sort value weak classifier fraction vector th minus classifier fraction incorrectly classify illustrate minimal consider one suffice weak satisfy possible situation like shall minimal overlap condition arrive term illustrate pair show region incorrect line top pair provide classification incorrectly vote vote classification accurate vote bottom weak vote neutral vote majority vote entire vote term combine vote vote correctly make third overlap overlap amount vote substitute condition yet satisfy satisfy modify weak classifier condition interpretation condition classifier subscript add overlap separately completely accurate majority vote vote first element fourth pair fourth omit extra vote remain correct strong weak classifier similarly correctness overlap element pair accurate sum correctness everywhere eq select evaluate vote give classifier comprise give classifier case output vector strong classifier weak belong classifier comprise weak satisfying suffice correctness namely equality ns j set event great randomly correctness sum least tell correctness sum classifier sum equal correctness comprise solely correctness condition namely q proceed great imply sum three guarantee must violate way could replace minimum overlap weak q output overlap prove classifier everywhere namely eq select imply theorem replace weaker choose uniformly correctly comprise classifier satisfy perfect suffice correctness overlap condition definition correctness correctness sum exclude pair sum classifier partially exclude virtue tell always cover majority vote calculation alternatively could condition way weak correctness pair correctness include weak correctness thus conclude great probability correctness turn probability perfect classifier tradeoff turn weak subsection correctness classifier majority vote replace weak classifier weight know shall ideal determine weak classifier new write second term double sum solution weak classifier define correctness classify incorrectly member classify penalty classify incorrectly accomplish assigning pair alternate translate ise hamiltonian section variable yielding omit direct translation hamiltonian represent rather assign combination classifier pair set strong find shall henceforth input decide quantum speedup testing exponentially input output regard formulate space indeed perform parallel return state state recall concerned vector behave whether ideally behave combination eq vector use replacement purpose strong classifier specification construct training label produce output implement result task identify implement stress relax positive negative eq unfortunately relaxation trick classifier vote function combination gain receive exist condition part ideal program likely specification positive low lie input portion care specification enough program ever occur fall outside implementation try case classifier opposite definition receive examine formulate detection minimization need amount behave incorrectly program implementation insufficient relaxation energy hamiltonian hilbert span state candidate procedure never negative would positive input either find output boltzmann boltzmann contribution thus expect lie state return nearby implemented detect undesirable low probability small provide member identify classifier set member beyond scope write opt ic opt yet weak classifier p function logic intermediate I z z z z I I I I x x x I z z z z I I x z x I form subscript column e relationship output evaluate bit input measure classifier motivate correspond interaction weak boolean boolean concern input bit wish quantum dictionary implementation three perturbation bit local include devise product logical weak relationship specify form I x x follow body intermediate bit tie intermediate introduce penalty modify translate value modify amenable note act intermediate bit directly dictionary three bit function number need implement processor correlation relevant classification intermediate function sum add penalty equal ax intend behave f interaction quantum computer hamiltonian term implementation classified classifier whole weak classifier bit case hamiltonian inclusion tie product associate penalty appear hamiltonian intend outside instance nearly two simultaneously verification region limited classifier fit intermediate involve four bit bit weak classifier keep instead choice boolean become third index possible tie ground optimal derive detailed subsection consequence interpolation state superposition h implement starts grind boltzmann distribution state design find contain logical assume quantum processor preliminary achievable classifier implement toy design critical system redundant could particularly important different system indicate consistent redundant thus suppose implement majority vote simple explain implementation monitor routine variable frame implement shall logical tracking frame fail frame variable facilitate quantum program essential quantum three redundant frame per snapshot whether reflect logical bit whether redundant routine nine bit fail fail fail variable variable incorrect previous frame challenge recognize specification implementation classifier find objective hybrid quantum quantum technique classical resource example classifier computational study routine quantum set classifier appropriate set evaluate weak typically discard feed ideally quantum optimization hamiltonian place classifier reality quantum come classifier accurate dictionary address ground discard space weak classifier yet consider classifier include optimization weight weak classifier dictionary clearly strategy combine subset weak classifier could genetic eq precede eqs classifiers eqs create final spectrum portion preprocesse order small bit output cost ground simulation effort well parallel classifier simplification implementation frame result fig subsection produce comprise weak member figure perform amount time computer algorithmic perform error achieve quantity total incorrect classification eqs classifier analyze implementation classifier plot fig fig classifier consistently specification thing outcome limit note dependency simulation address describe increase far perform number simulation vertical horizontal axis arise incorrectly respectively classification determine observed iteration intensive computing plotted assign uniformity simulation weak classifier even misclassification classifier place wrong neither rather application datum quantify error iterate encourage generation challenge improve break intermediate light ask satisfy sort expect yield relate light pixel output horizontal red divide represent give weak black incorrect classification problematic aspect bar white accurate detailed completely fall classify incorrectly impossible bar classification span height weak incorrectly weak use correctly classify overlap correctness fraction weak arrive classifier overlap eq classified correctly minimum weak otherwise weak correctness force classification see fig correct vote pair weak show correctness overlap weak minimum weak misclassifie weak minimum classifier apparent short ideal weak classifier overlap horizontal correctness maximum order come estimate accuracy achievable classifier correctness overlap minimum correctness ideal weak overlap completely classifier cover space overlap weak extra vote fraction three minimum overlap relationship significant large classifier fraction input incorrectly currently produce classifier quantum apply classifier quantum parallel input select classifier first quantum quantum hamiltonian first inspire classifier weak selection optimal condition second quantum strong entire output quantum intermediate track execution tune weak classifier overcome limitation restrict state path initial hamiltonian world development characterization strong involve proposition machine learn anomaly via quantum training classifier form classifier superposition certain phase execute quantum evolution software verification validation computational range complex stock market concern address speed concept quantum computer show quantum require much classical handling classical binary assigning example whereby combine alone separation distinguish two specie pick letter letter alphabet base effort quantum advantage boost formulate use detection trading change verification classical software anomaly anomaly detection intractable exhaustive piece set input give software variable although exhaustive lost testing resource widely effort focus considerable new testing attempt available combination cause considerable software parameter formal verification phase software absolute software implement exhaustive consuming checking validity state solve np require correction provable check verification validation processing quantum use test quantum boost apply consist sort sort classical testing step turn classify likely error translate hamiltonian allow potential examine return candidate test use quantum encodes ground computation perform simple slowly system evolution hamiltonian universal overhead protocol degree begin problem define eliminate resource section establish implementation learn develop ideal alternate quantum section detailed result future formalize vector occurrence consider ideal mean perfect instead real refer wish verify operation eq generality think finite without generality limit length string within move specify specify input output binary string space output vector program output space implement program ideal element fix input trivially ideal map correct q every course implement ideally reality may mind simple software norm program purpose existence error program order incorrectly pair count number incorrect classification strong classifier classify prevent manner balance weak comprise classifier formal tune unfortunately discrete evaluation amenable error correct pair already interested incorrectly correctly classify formal replace classification make sense equally sr good training set equivalently find optimal weight ix think symmetric offset drop wish follow one define
apart compare actual value via carlo identify epochs iii initial put ignore fa bound involve involve give use b replacing involve far combine th sharp exact however constant constant problem importance identify drift vary specific example drift drift typically present tool computable space might apply successfully example state therein hierarchical paper cite failure hour assume conditionally I hyperparameter gibbs follow cite satisfie condition establish drift sm x vx sm vx obtain calculation iii prove jensen shall define assumption respect consequently n p proof repeatedly iii vx corollary rearrange write vx x jj directly term condition corollary nn n vx vx desire I drift induction k vx f vx x assume vx n ii ii n term v vx nn term put ii fx v x v acknowledgement comment acknowledge help comment associate paper partially education grant partially bound square mcmc estimator valid ergodic encountered computation unbounded variance bound geometrically polynomially ergodic chain condition corollary confidence distribution borel objective quantity high bound arise often solve monte average explicit reliable prescribed level begin ergodic chain admit sharp sense lead asymptotic central proof rely technique sequential bound geometrically polynomially appropriate quantitative transition mse geometrically polynomially ergodic establish drift utilize ergodicity use mse uniformly consider ergodicity polynomially ergodic reformulate bounding immediately confidence chebyshev trick inequality mse upper geometrically polynomially example hierarchical bayesian poisson normal toy mse derive geometrically polynomially chain section applicability proof defer markov various brief derive result ergodic explicit trick exponential conclude give turn concentration dependent large see example reference use result motivate metric expression ix iy set suited functional ergodic ergodic chain detail explicit bound apply ball condition detail chain ergodic verify rather dimension seem tractable tail ergodic however constant explicit may chain chain establish cl approach directly geometrically ergodic computable paper generic approach chain polynomially ergodic secondly computable bounding variance surprisingly estimate mcmc curvature geometric ergodicity wasserstein bivariate approach appear applicable different employ coarse assume curvature complementary rate convergence geometrically investigate quantitative quantitative computable together need importance translate allow burn irreducible construction exist bivariate way rule give qx chain moment epoch x specify sum bind always markov prove lead assumption ergodicity section explicitly computable quantity set lead estimator refer detail recall eq excess entail middle significant portion proof j version q I pair identity identity q attention mean sum invoke follow elegant excess obtain account proof section geometrically quantitative defer standard establish definition specifically drift exist model practical geometric geometric computing bound appear hold v pt theorem note th order theorem involve follow simple bound v f mcmc always case iii inequality might improve add burn computation discard initial part justification equilibrium technical reason effort inequality upper f polynomially ergodic chain drift deferred section follow drift counterpart assumption use establish polynomial ergodicity markov constant drift mcmc sampler walk langevin sampler example section expression pt pt pt pt pt pt establish analogous iv necessary involve depend difficult simple complementary inference small present simplify hierarchical example design actual quality literature poisson demonstrate realistic drift nevertheless obtain model van multivariate student bayesian effect model algorithm student regression body relate devote drift set kind paper l substantial establish quantitative cf geometric validate argument qualitative suited derive quantitative conclusion simulation experiment design compare prove actual normal reciprocal eq unknown thing improper q abuse symbol draw conditional start convenient letting rhs
form analysis miss entry know study I missing modify objective square cp second stay compute partial term respect conjugate update search toolbox opt accuracy randomly tensor whether underlying factor extract denote extract different scenario couple couple third mode one entry normal matrix example first factor r tensor r j adjust factor r entry experiment opt extract quantify match follow matrix denote mode vector norm similarly rewrite indicate weight original extract extract value change opt number iteration set two divide gradient tolerance generally stop due change function rarely opt stop reach maximum e run opt furthermore opt stop ccc c c alg success p opt opt opt opt e e e opt present result experiment factor normalize generating factor ratio experiment factor parameter run compute pair statistically scenario opt significantly outperform opt quite accurate recover underlie factor noise affect accuracy hard component near accuracy true datum first accuracy slightly change greater average around extract factor c c c c c c c c e opt c opt opt shift focus explore structure source formulate problem couple address couple tensor square opt factor extend couple extended incomplete well fitting algorithm opt alternating fit note current formulation couple factorization drawback r r formulate couple f scaling take consideration scalar address part modeling area future bayesian promise future loss incorporate interpretability laboratory direct development laboratory national security contract ac briefly couple construct rr assign plus entry member versa member form form factor matrix r normal construct g source improve restaurant recommendation system rating customer rate customer social facebook restaurant category well recommendation consist high rating tensor interested capture formulate tensor factorization tensor manner traditional approach optimization opt joint analysis tensor handle couple incomplete demonstrate access amount data advance internet medium device genomic medical diagnostic restaurant site access customer medical patient eeg monitor fmri functional imaging laboratory test restaurant category chinese couple high tensor discusse analyze heterogeneous simultaneously entity represent movie review tensor show rating similarly tensor collaborative filter example suppose dimension mode common model cp list cp factorization tensor show least bregman metric illustrate motivate couple tensor source fine grain capture set suppose customer customer store customer period customer live customer group group people live interest item imagine discriminate svd customer plot svd separate conversely imagine tensor discriminate cp albeit illustrate middle factor use group separate bottom plot recovery different miss limited amount single long enough datum recovery miss still use fail couple mode compute r r random km miss recover entry extract recover extract common mode letter subscript entry denote entry ni ij k kronecker result size k kronecker product property kronecker rao product tensor I hadamard define I tensor frobenius give nr matrix fusion collective multi field decade briefly datum couple source attract considerable community netflix competition movie rating order rating rating exploit tag movie movie collective simultaneously couple matrix rr factor general extend loss early study collective matrix scheme collective factorization back common variation later follow set factorization matrix work moreover simultaneous develop blind separation speech microarray data analysis tensor analyze multiple tensor third analyze nevertheless couple heterogeneous address different heterogeneous formulate first show al tensor couple mode
spline space approximate kernel solver classifier incorrect evaluation maintain multiplication look unlike dense change simple form initialize repeat follow large consist dimensional inner loop algorithm refer easily solver exploit practical embedding somewhat image histogram intersection kernel classifier pyramid histogram orient gradient website instance zero entry set dense image svm often take machine sample b spaced basis degree quadratic cubic spline offer smooth fit figure show degree accord spline fit uniformly spaced bin various additive expensive training additive linear svm expense additional training train vs label high response find cross validation model significantly outperform classifier closely match c svm spline spline spline fourier compute train spline additive outperform svm hour b spline spline b spline spline batch additive outperform additive embedding enable memory overhead see suitable store outperform significantly spline work svm high spline fast spline library publication train also new class classifier paper current kernel svms linear spline suit ease compute overhead connection spline intersection kernel spline intersection kernel svm enable solver slow order fast train parametric classification become attractive svms desirable view offer significant shift semi shift approximately train overhead naturally setting e kernel computer vision additive histogram intersection map lead compact feature estimation line explore feature learn additive propose construct spline estimate statistical ever additive introduce practical formulation extremely spline embedding efficiently embedding underlie learn linear embed embedding spline function spaced basis adjacent embedding svms develop train additive embedding kernel particular arise spline basis advantage control choice desirable moreover fit go prominent spline spline modeling space b basis difference adjacent spline formulation key whole discriminative training involve use hinge implicit reproduce hilbert rkhs additive shift invariant kernel feature base propose efficiently kernel propose model typical dimension feature enable dimensional derive one overall concatenation embedding additive dimension space spline difference adjacent spline let spline consist construct th difference repeat dimensional everywhere top result spline svms additional embedding approximate choice matrix invertible therefore k equivalent inverse triangular excellent spline spline ccc cubic various refer spline basis truncate polynomial provide interval x regularization overall additive embed practice small form classic derivative orthonormal orthogonal sequence three family make well purpose
integrate curve continuously differentiable tangent agree field curve interior approach fully infinite computationally intensive empirical component consist detailed field represent nonnegative window tensor field play role imaging calculate representative eigenvector correspond eigenvector use monte spatial point advantage quantify uncertainty effectively structure field strong alignment arise many different study densely one dense long rather connect relatively point window new consider stanford galaxy appear cluster along interest surface extend surface along open ridge locally parallel close arise align along reconstruction dominant orientation overcome construct dominant orientation directional ambiguity stanford principal generalization principal em choice smoothness number difficulty reconstruct region signal datum galaxy empirical density galaxy thin compare technique see ascent path path analyze useful insight method brownian motion enhance contour edge digital along structure around model segment integrate encourage area prior model bayes death mcmc formulation use exploratory focus dominant spatial modeling point ordering process homogeneous unknown cluster henceforth one object segment high dimensional space contain discuss appropriate identify spatial window overview propose appropriate outline sampling compare issue implementation describe segment orientation curve segment say orientation segment agree mixed poisson cox drive independently point generate cox point character tendency regularity along distribute distance often spline fit integral agree field orientation specify see characterize parametrization arc length poisson along construction introduce bias consider adjustment correction make negligible part seek field property finally background independent homogeneous onto generate directed acyclic dag dependency show acyclic henceforth denote reference arc list write arc arc th orientation possess may signal cluster anchor point isotropic bivariate normal spaced arc distance adjacent point dirichlet encourage either evenly spread along independently point allocate th proportional approximately add store variable resp allocate allocate across window total poisson parameter noise point sake simplicity assumption assume total poisson distribute estimate proportion point assumption proportional suited noise detection along number example prior common include suggest alternative instance indicate omit distribution make represent estimate regular grid point estimate orientation evaluate small distance orientation adjacent segment discretization course calculate advantageous field integrate datum high field bayes mean aspect datum alternative treat random identify derive theory see appropriate smoothness field field computationally calculation relate lead complexity huge difficulty ensure properly explore use easily suitable orientation produce orientation arise estimator likely field section vector mapping pattern apply euclidean field definite eigenvector eigenvalue indicate strength orientation assign eigenvector eigenvector unique say equal dimensional field commonly diffusion imaging use analyze scan water use water tensor orientation eigenvalue proportion water let transformation represent method set create principal eigenvector field interpolation see extended account calculation tensor matrix zero eigenvalue indeed intend one dominate calculate due point far round become zero remain error log tensor least eigenvalue uninformative suggest lack take potentially field smoothing kernel metric hence field principal eigenvector give integral drawback create rapidly orientation field analyze magnitude bias find smoothing b smoothing bias occur field unbiased allocate trivial j proposal death desirable add extra move utilize move amount reference keeping occur predefine hasting improve section propose noise propose beta move find deviation update field degenerate configuration allocate exclude state inspection show reach death process irreducible visit infinitely motivate low inspection estimating occur happen point short exponentially distribute region within burn assess spectral density diagnostic reject mean sample sufficiently last assess whether death death rate event birth death etc algorithmic stationarity estimate birth sum additional move implement pt mind complexity show record constant rate sampling reciprocal carlo remain mix decrease approximately proportional prior inspection datum continuous determine detailed balance per fix relationship actual hardware hardware describe birth rate adjust propose join unit time evaluate four neighbor window cluster server intel processor gb fully ram http www hour datum yet explore perform show brevity first simulate pattern facilitate stanford signal family allocated allocate higher allocate indicate allocate sample point enhance clarity symbol indicate indicate noise pair nonzero round c c c c percentile distance point birth death sample sample hyperparameter half well albeit curve fit probability property point closely correlate chance chance percentile point percentile origin bivariate normal variable percentile simulate partly explain slight beneficial less desirable evidence dense dirichlet density lie drawback multimodal proposal birth anchor good clustered background datum set usa find http www edu cat birth death unit sample state hyperparameter dispersion half table numerical distribution http edu cat window dispersion probability nonzero round cc pt percentile b indicate allocate indicate allocate find smoothing area density allocate sample enhance clarity symbol indicate different cluster often associate across one limitation assume number particular width influence signal unit figure central dispersion apparent width shorter wider effectively th percentile distance table overcome could cluster feature sample indicate agglomerative arise interestingly positively suggest additional arise part preserve lie depth set allocate indicate allocate estimate reduce enhance clarity estimate cluster consider birth death unit discarded take time near neighbor picture small scale phenomenon partly inter contrast thank smoothing step field succeed fitting indicate area location near edge show ideal fairly interval identify nearby could gap reconstruct data property posterior distribution window noise nonzero round noise point percentile new process monte instrumental arise nature investigate around line galaxy visible universe galaxy know align turn connect web overview different location place straight line aid currently prominent curve nature curve sufficiently smooth must angle smoothness desirable identify curve desirable computational integrate use algorithm em
construct coincide function elsewhere show eq construction continuous lipschitz constant version operator define htp technique adaptive loss function canonical radius combination ty threshold dual algorithm regularize variant proof differentiable derive proposition prediction let tune quantity unknown forecaster figure individual eq unknown forecaster two obtain without cf since otherwise adaptive regime losse high curvature corollary loss regret unknown forecaster adaptive original loss dependence obtain grow grow hence improvement property stem another benefit optimization regret gradient improvement correspond forecaster ask ball forecaster ask ball case regret good simplicity assume framework r tr sub prediction work access sub simultaneously perform regression forecaster aggregate simultaneously regret concavity two sub regret known every assumption average forecaster expert lemma fact nonnegative xy ty inequality xy cr u finally get conclude claim second follow bound scale knowledge automatic adapt quantity use update past adapt adaptation replace sequence r grid ensure carry trick however avoid use automatic term negligible author would thank comment suggestion national ec yu partly fellowship le la et les perform stochastic trick employ follow notation space orthonormal family dx space integrable ts independent precise xx eq remark entail comment since argument elementary inequality get cauchy schwarz follow random get choice technical page rely elementary fast hold constant eq conclude entail inequality fact xt get conclude bind regret keep eq supremum moreover regret regret latter remark conclude proof theorem past identically f expand via notation c denote infimum multiply inequality conclude follow loss negativity b construction adaptive therefore corollary particular loss gradient case function resp continuous draw consequence inequality gradient substitute combine result inequality rewrite solve suffice first bind elementary rearrange conclude recall round account q follow elementary cumulative forecaster satisfie next prove elementary exponentially forecaster base infimum supremum treat separately regret bind fy tu denote linear output forecaster satisfie step exist forecaster upper forecaster orient slightly ridge forecaster forecast forecaster access base forecast predict temperature forecaster know weighted forecaster satisfie q proof straightforwardly latter exp assumption argument proof integer canonical j rest dedicate bound last sum equal zero equality k combine remark inequality conclude variant reproduce convenience combinatorial cardinality k k k elementary tuple lattice path straightforwardly corollary sup paris individual forecaster sequential prediction one linear present dependency exhibit transition furthermore algorithm I knowledge adaptive arbitrary competitive good predictor extend task aggregation include analysis dna sequence filter country growth diameter predictor dimension provide algorithm ball linear follow environment choose forecaster instant environment reveal forecaster incur dimension relative step case sequel inner product du forecaster shall hold opponent bound may unknown forecaster linear linear ball refine express absolute transition bind partially type argument deterministic yet stochastic refine setting ty stochastic batch make statement order extend achieve inefficient unknown overcome term forecaster tuning prove confident cumulative efficient logarithmic regime contribution bind nearly optimal prior issue modification automatic tuning algorithm nearly section generic transform lipschitz technique generic prediction yield main improve derive curvature result grow naive way uniformly knowledge neither upper bound efficient adaptive known aggregate achieve uniformly discuss automatic tool minimax term intrinsic quantity bind definition ambiguity call f upper base prediction observation infimum supremum secondly partial question ask gap prove see term intrinsic quantity relate intrinsic quantity theorem term intrinsic bad regret choose continuity main relie type another apply sequence bound achieve ridge forecaster forecaster therefore third sequel argument refine bind e look discretization study grow instead naive first fact competitive dy yx remain aggregation forecaster predict b mp u forecaster tune elementary put together corollary quantity improve already dy c absolute infimum sequence lower bind small proof batch employ lower bind take inefficient dimension achieve minimax modify forecaster tune nearly achieve modification regression forecaster computational nearly regret technical loss inefficient version call require oppose original version automatic tuning suited loss arbitrary forecaster environment choose loss incur online radius dt function proposition general function gradient majority expert let adaptive sequence differentiable bound straightforwardly linearization argument regret
plus sec hence occur intuition like various ensure know inequality fulfil e explain propose practical via carlo following set run randomly draw independently bernoulli center perturbation unique trajectory continuous know behind angle regression homotopy condition ty tx ty result inequality tx ty immediate gaussian distribution recovered estimator top strategy increase solve equation globally condition initial refine simple compute solution first correction recover monte lasso false positive result lasso use well lasso available estimator deviation slightly figure top htb level derivative maximal see vs fidelity find decreasing summarize compute lasso l n find recover monte properly false component lasso penalty tradeoff value newton discard line trust region correct price number component vs fidelity tradeoff constraint see respect false positive figure propose simulation encourage evidence dependency practice ahead result confirm snr know large configuration lasso exactly true setting perform poorly component carlo experiment strategy snr much surprisingly word appear practical suitable snr snr choice could base occur varie question let except decide refer interested globally convergent paper finally remain question proceed support aic criterion recall result provide high trivial modification condition perform product lemma ec instead simple inequality e true component instance w obtain whenever actual x row inequality write hold compatible I previous th simply second take obtain z section prove sufficient optimality admit unique l hold decrease proposition solution property derivative moreover show immediate consequence concavity ty thus interval form continuity assume contradiction continuous sequence respectively l l uniqueness continuity imply sign due least interval uniqueness imply multiply ty tx tx tx tx obtain exist exist calculus contradiction bound bound fix also large subspace n imply imply tx increase nonempty denote infimum take tx continuity var admit trajectory generic generic moreover interval x tx combined tx ty tx sx decrease study deduce read tx tx since tx tx differentiable use author grateful thorough comment presentation argument france mail fr address issue generic jointly minimizer square functional parameter tune parameter choose enforce fidelity similar one plan ann support simulation show enjoy lasso signal outperform false dimensional unknown gaussian read denote unknown error study regression impossible study extensively context compressed paradigm design matrix number positive estimate selector refer simultaneous b control sparse introduce sample resp common incoherence coherence coherence appear basis significant recent context require whereas bind require coherence pattern uniformly op index signal condition possibly suboptimal address author aic use practitioner procedure avoid enumeration subset covariate intractable covariate motivate provide joint variance compatibility problem support provide explicit strategy present paper mainly aim understanding extend right magnitude estimator consist replace main difference summarize satisfy condition numerical viewpoint iteratively overcome estimate fidelity precisely enforce fidelity complex estimation regression natural constraint observation support probability explicit coherence assumption readily currently concentration property singular yet concentration value criterion difficult impossible oppose property make nonzero regression coefficient magnitude great require bind result suggest snr end paper strategy snr present give proof technical intermediate usual notation x transpose symmetric real matrix maximum resp order symmetric stand notation resp distribution line degree bernoulli distribution submatrix index orthogonal onto column norm j selector h diag fashion support main strategy discuss study particular tune numerically find support resp coefficient satisfy small slightly require may argue strategy experiment empirical signal experiment standard lasso ensure enjoy special continuity say ix tx lasso problem support non singular prove generic hold surely implicit var estimator implicitly exist sequel mention uniqueness show give increase prove precise interest existence scheme discuss satisfy bind component gaussian set order design column column haar unit sphere scalar x phenomenon sphere u concern require beta deduce sharp ahead require matrix imply sharp advantage sample sharp increase sliding imply usual beta see want specify may risk interest practice plain variance many lar instead order model aic bic etc vary support recover incorrectly detect component range quite vs noise compare constant various set r choice impose less allow end limiting allow make fair divide consequence condition prove oracle full e tx intersection discuss next study enjoy know ahead follow oracle b r might interesting tx formula th recall assume seek var ty p var orthogonality var var tx var p henceforth x tx particular subsection show tx high choice var p property tune great virtue deduce p yield r writing p satisfy r conclusion recovery pattern need rr hold proof hold high slight improvement iv probability appendix conclude strategy deduce proxy feature pattern sign
base delay unbiased jk delay note discovery indeed delay order moment delay provide delay discovery delay variance suffice variance variance additive henceforth variance distance estimate distance abstraction imply block discover node definition l eps pairwise la lb lb literature since la lb v detect graph practice relax equality tuple path path hide connect length middle h length linear end sum along respective algorithm merge employ additionally introduce modification addition discover random outline short idea behind reconstruction cycle test summarize intuitively avoid testing fail reconstruct limit avoid obtain tree reconstruction short edge length thus maximum bind diameter choose carefully cycle least point relax distance node attempt possibility consideration already nothing merge create cycle cycle create need firstly cycle guarantee merge list bad secondly cycle need create short whose attempt merge without create attempt merge create new cycle check distance original list processing attempt merge bad account bad towards spirit merge handle presence cycle distance length short uv uv uv merge topology short path distance second short length uv uv uv uv uv heuristic discovery shortest short I path uv b j merge create short cycle short join consistently add inconsistent uv attempt create new long new path else contract least hide uv uv node compare length assign miss assign connect already assign b w b path create exist split length uv create current node short short short tolerance compare length kl kl shortest available agree exist create cycle path split verification fig else add uv create cycle outline short distance addition short propose summarize use short distance shortest carry shortest retain distance specify second short distance short middle edge retain rule minor check multiple different length simplicity assume distance node algorithm l l dot line length path short middle cycle detect bad eps eps eps fail succeed wrong show test recall satisfied internal path outcome incorrect outside bad outcome bad use internal merge procedure detect wrong internal outcome result error give procedure edge length additionally equality constraint fig equality constraint instead cycle event occur length far analyze exact delay input instead propose require achieve distance delay variance delay delay imply discover topology distance topology node require low distance realize construction end prove low realization graph distance graph adjacency random belong decay graph reconstruction less otherwise output achieve distance node graph participant short enough distance lemma every great threshold impossible uniformly strong converse say short short path algorithm performance sub participant lower cover argument cover high l eps l reveal shortest greatly improve discovery graph accomplish end counter fig fraction distance yet identify reveal fundamental topology subset key tree species tree series occur specie low g correlation decay range estimate delay long delay delay sophisticated however reconstruction short distance discovery topology sub uniformly participant property obtain end path participant scenario measurement available topology sub participant participant efficient explore algorithm edge length explore locally well model employ explore change join interest guarantee minimal plan centrality analyze plan develop develop thank anonymous uci award fa cycle cycle length two cycle overlap cycle length cycle argument expect q edge overlap cycle obtain deal representative degree first condition enyi event result least modification minimum degree minimum proof line cycle length overlap cycle denote number enyi event three give node overlap cycle prove characterize good event addition distance reconstruct minimal graph concept bad length criterion middle generalize cycle accurate merge consideration edge short length short successfully part induction initially empty correct path yet add merged node add merged join point path create correct join point join part cycle join point imply middle cycle accurate merge correctness middle middle bad reconstruct contribute wrong amount reconstruct three correct number middle node middle edge event v b v lemma occur generalize cycle bad chernoff b v g r expect distance proof succeed accurately candidate join short path edge middle middle short cycle expect result delay sample take obtaining reconstruct short q know c require asymptotic obtain nk obtain journal use node participant rest information topology consider discovery participant exchange message exchange second short uniformly participant strong sub linear uniformly imply participant bind reconstruct original graph demonstrate tractable graph participant graph discover participant availability end path participant end characteristic network topology network many rely failure infer social knowledge useful infer characteristic information flow possibility traditionally topology however tool require node protocol request privacy security concern topology request scalable discover protocol switching path increasingly discovery network topology e without flexibility increase popularity detail topology end may discover place g population drug survey discover network fraction participant many topology need computationally provide fraction inference topology desirable low sample definition measurement achieve accuracy scale size achieve objective discovery topology participant issue phenomenon participant identifiability desirable topology guarantee performance r perhaps provide reasonable explanation network social address graph fraction provable kind participant useful discovery participant low distance discovery achievable address insight sparse provable end second information participant discover extremely participant reconstruction graph redundant consist delay short path participant refer test tree topology previously topology roughly participant end short node maintain demonstrate achieve participant thus discovery sample logarithmic end end sample needs state discover participant exploit locally random graph enable guarantee know topology control use test time exploit cycle obtain tree topology leave accurate topology use participant error participant specifically roughly reconstruction algorithm also general identifiability discovery participant reconstruct path participant contrast discovery end property applicable discover tree like social extensively propose instance mapping network prediction link consider consider infer latent propose survey discovery development topology discovery various topology discovery availability kind previously path internet network formulate discovery random query law node network protocol relate consider topology short several edge whether edge select query edge subgraph query return two vertex query know unlabele unweighted discovery extensive end end delay refer previously traffic thorough local test inspire previously topology accurate reconstruction tree recent temporal exist similarly notation probability measure say hold almost surely equivalently property generalize length edge count definition union short denote subgraphs subgraph topology enyi graph arguably denote edge occur regime exhibit contain component super critical regime regime discovery regime real world extremely participant discover topology discovery graph topology let node exchange message amongst fraction desirable reconstruct node decide random regime mean node discover message provide presence need goal topology experience delay delay sample metric depend consideration scenario delay pair end delay message message experience delay identically direction assume delay delay delay along participant delay topology message exploit efficient discovery end delay assume message participant short path path delay also scenario participant short alternative short fail
solution interior formal cutting solve initialize find interior respectively duality dual new interior optimize first statement violate dual constraint relaxed associated weight know np iteratively add maximize summation return constraint must discount terminate duality gap program add primal add theoretically interior primal computational finally iterative bregman guarantee solution relaxation note practice never need explicitly within product basis efficiently provide application effectiveness community purely art clique arguably simple evaluation sum inside clique clique clique element volume htp cc detect team team repeatedly clique detect interaction ideal scenario since indicate player concentrate equal matrix b les social social character les b clique identify clique overlap detect follow character co character relationship social regard figure social spectral result red cut three cut community network ht box plot clique volume clique approach volume clique clique meaningful clique see table verify ground novel correctly separate clique treat character small clique important clique clique necessarily satisfy bad clique volume slightly clique character ex n n social community cluster note think large clique score clique frequently clique seven contain six node medium broad field part clique clique volume clique identify pursuit comparable clique volume identify clique hundred ht science show identify behave persistent exactly cluster combine persistence clique clique meaningful bipartite clique cluster identify parent identify clique c network clique method clique volume clique parent identify paper clique heavy persistent c clique program clique track rank contain give rating rating top score ground truth top count whole path capable htp top curve select fourth persistent connect compressive adopt algebraic tool homogeneous formulate detection compress construct characterize clique heuristic level community model compressive result area create usefulness new framework clique network compressed sense algebraic basis group compressive interactive information management rank bi variate pair look compressive clique turn recovery time approach world acknowledge grant nf nf nsf google support basic program china program cb microsoft research university thank support package university compressive liu stanford modern acquisition produce amount though analyze largely statistical processing present framework connect different area compress sense perspective network consider clique tool homogeneous recovery conceptual solving keyword analysis compressive sense pursuit isometry property clique detection past decade research include scientific study link citation network connect citation relationship frequently modern application domain lead hoc exploring help scale drug control network principled analytical crucially researcher predict fall static static snapshot dynamic dataset index os enyi blockmodel blockmodel dynamic though analysis learn unlike usual measurement collect relational prevent directly art analyze bridge framework assume sparse compressive compressive system reconstructing due contribution adjacency node problem clique connect new space regard study sparse clique basis address exact algorithm recovery apply setting restrict rip paper new exact sparse choice root pursuit compress sense also practical usefulness content present polynomial clique example conclude paper general compressive nonparametric notation denote euclidean edge represent associate generality upper triangle triangle model vector evaluating infinite nonparametric without dictionary sequel element column index construct sparse thing simplicity pre section clique problem identify community arise management social application typically network pairwise clique frequency govern community clique formulate clique answer rigorously motivate situation sensor identity belong grey pass interaction observation due player mostly difficult involve team member infer information belong rank item set item partial ranking understand organization direct appearance activity typically network character exploit clique network basic interest group clique often govern community formulate relationship community social social majority study community base partition nod modularity base frequently practice overlap structure clique address clique model clique node compressive clique turn community structure clique explore network light multiple aspect frequency social bivariate function pairwise order intuitively interaction belong map low order subset fit compressive previous demonstrate basis purpose programming formulate name pursuit construct clique perhaps clique weight adjacency clique clique matrix transpose suitably space exploit transpose matrix transform dictionary observe discussion scope paper entry many strong realistic instead study universal seek signal pattern sequel clique clique could clique extract less rank need number column linearly expect signal observe complement extract follow characterize self include assume invertible cx come kkt equivalent lagrangian lagrange multiplier kkt give program two sufficient minimizer minimizer condition j j w inequality less since independent necessary free setting necessity span condition respect invertible sup lie condition sure condition must relevant basis irrelevant depend span mild necessary selector like subsection verify important clique let clique let example thing verify fact really clique clique may large recovery behind enough check define intuition try technical entry set condition exactly give fact contradict condition disjoint every set remain equality generality belong contradict member member remain say contradict proper condition case enough happen large choose large disjoint set say I ta show row correspond row construct long often clique reasonable network exist join together make community belong partition belong hold inequality sup norm note belong overlap size entry clique bound establish second inequality observe fix time belong intersection fix must fix belong otherwise far satisfie see implementation basis clique basis actually submatrix subset clique submatrix clique intersection respect suffice firstly diagonal dominant recovery impossible
calculate complete operation complete qr qr perform complexity enough size realistic away scalable brief discussion iteration qr factorization nonzero element number overhead iteration total parallel feasible pt localize range three peak apply basis resolution wavelet wavelet four vector sparsity stick loss test thresholde sparse pca specify theoretical stop compete recommend c c two margin large margin algorithm spike loading seem select coordinate well next take spike figure spike spike loss average spike threshold recommend first spike relatively separate compete separated method estimator imply estimate pick right subspace summary spike principal successive focus purpose low dimensional projection devote proof divide major step pt well approximate interest oracle sequence estimating sequence sequence various need actual sequence oracle forces oracle road extra quantity oracle construct rest organize oracle matrix replace pt formal guarantee full th factorization k principal satisfie identical step satisfy actual three section subspace approximate break part oracle lemma principal regard feature nj supplementary material key proof claim require well estimate since analogous eigenvalue nj supplementary material claim inequality oracle high satisfie end characterization sequence evolution role point denote show oracle j nk material claim require condition much high claim j evolution large lemma nj supplementary material ingredient subspace characterize foundation proposition error maintain decrease continue slow slow fashion interval inside proposition previous element high study sequence k consequently soon nm nm directly aspect sigma hold proposition take error sigma uniformly oracle give material complete rely actual sufficiently probability least supplementary theorem actual special theorem proof theorem lemma sigma c jensen event sigma l author like thank discussion claim section remark principal subspace eigenvector large propose model recover lead eigenvector consistently optimally many row usually thousand hundred issue analysis large space principal project onto span population variation capture dimensionality addition dimensional visualization covariance traditional plus recover vector generative factor loading vector factor logarithm spectra spectral component component number spectra collection reasonable asset stock people usually simultaneously scale hundred addition big signal processing generality high suppose rank covariance eigenvector therefore th spike literature spike make dimensional spike practical difficulty eigenvector challenge side sample eigenvector estimator sometimes even nearly different generality examine result recent start pca span dimensional explain approach start penalty become normality spike prove pca feature large variance eigenvalue lead eigenvector exactly loading nonzero location method al consistency fixed propose augment pca eigenvector show attain range lead separate span opposed find vector individually reason eigenvector identifiable eigenvalue dimension great new iterative estimate addition orthogonal basis model sparsity characterize lead subspace adaptively wide sparse appropriate moreover lead eigenvalue rest eigenvector attain optimal derive sense result large visualization avoid construct implement last principal examine normality demonstrate present produce table figure author website index denote dot submatrix norm say orthonormal might differ different occurrence number constant use observation eigenvector equal subspace object always primary circumstance interested visualize want consistently principal part convenience normality note onto projection function possible discrepancy onto range geometrically square lead orthogonal iteration matrix upper triangular schmidt dimension orthogonal matrix qr denote orthonormal eigenvector terminate iteration classical pca could problematic orthogonal dimensionality interpretation accumulate impossible span one sensible focus zero heuristic incorporate orthogonal effective feature screen orthogonal multiplication estimation summarize theoretical conduct normality datum threshold level orthonormal matrix multiplication k nj basic add user satisfie thresholde result column specify unchanged across indicate size apply range qr amount basis qr although estimator initialization involve without radius spike last establish wider simultaneously spike allow order spike constrain radius outline extension special allow recall impose growth large spike spike satisfie ratio j j part spike large magnitude interesting spike flexible spike grow infinity mild end coordinate ball radius rapid sparsity entry extend follow decay unified notion sparsity uniformity depend though require grow radius appear example bound away arbitrarily constant exist verify condition discretized eigenfunction eigenfunction isolate wavelet belong ball wavelet moreover determine eigenfunction grid away functional datum type define lead large magnitude compare coordinate coordinate actual equal coordinate let complement stand dependence notational convenience convergence cardinality show bound large cardinality constant parametric term interpretation discussion theorem turn ad spike allow depend ad eigenvalue spectrum principal say th satisfied satisfied large spike convergence establish convergence subspace relaxed generalize probability least state appropriately probability uniformly vanish constant vanish uniformly consistent quantity could elaborate little focus coordinate appear far accumulate later focus thus variance tradeoff nonparametric term vanish condition procedure rate adaptively sparse yield bound away iteration precisely stop could note nm follow direct setup property switch selection motivation iterative trade variance signal specifically
enough object recover representative analyze analysis rank write shorthand correspond case denote identify average main prove easily unless note average query choose collect comparison pairwise assumption query request correspond rank previously collect comparison fashion pick inefficient also version error persistent probably audio prove idea interpretation problem derive seminal learn crucial position dependency analysis dimension easy dimension make sort bad object initialize request object dot new label rank significant gap permutation assume passive pairwise ranking adaptive full quite inefficient ranking full recently note theory adaptively select need learn cause concern since consume pairwise example researcher collect pairwise comparison preference object understand due expense require collect underlie informative pairwise query perspective component primarily passive query admit space comparison especially subject highly rank familiar need million possible well worse inaccurate accurate intuitively intrinsic use binary response object embed distance learn comparison correctly object embed study main contribution specific standard unfolding familiar relationship distance space structural constraint ranking use generate arguably ranking assumption rise interpretation view rank ranking request whether consider connect hyperplane define close close thus equivalence query pair hyperplane possible intersection recall ranking cell indicate refer cell cardinality set ranking assumption assume cell hyperplane I recursion partition cell recursion addition fashion prove positive q suffice ranking low algorithm require comparison bit query provide full bit ranking example dimensional object order binary achieving impossible higher induce hyperplane bad require query conclude bad situation instead reveal randomly inefficient answer ranking cell consist cell query pairwise random replacement integer query rank hyperplane bind probability unless query need ask infer probably correct rank label call cell pick random query informative fact hyperplane point lie check fall p free recall reference equivalently response represent hyperplane observation see equivalent primal interpretation dual interpretation point label linearly scenario assume comparison exist hyperplane still separate point pass point primal separate hyperplane correspond define associate pairwise sequential initial object nontrivial partial ranking probable denote object equally probable partition cell partition cell cell partition equal cell comparison choose j conditionally event separate pass hyperplane representation cell see hyperplane object bound constant see correspond probable object divide total cell immediately query query eq algorithm situation response query probably correct label incorrect response equal fashion algorithm exception query encounter several equivalent vote accurate respect ranking adopt popular comparison convenience time report incorrect pairwise closeness response random group people realization decide vote identify request deduce exactly recover condition one need request comparison possibly incorrect persistent human ranking distinguished guarantee good hope recover rank object merely probably correct recovery object approximate henceforth persistent persistent ingredient design voting encounter suppose query set identify object rank contain define rank rank contradiction furthermore rank uniformly initialize order rank uniform ranking explanation sequential encountered call closely sufficiently large accurately determine threshold draw replacement call decide voting otherwise list determine next enjoy favorable approximately rank initialize k jk l l output object consider algorithm end object rank query pass guarantee mention object pass one three chance mind constructed rank query average intermediate stage full initial randomization voting pass uniform assumption object prove sequentially random large one ranking least e pm robust pass first object first pass ordering randomness voting algorithm figure recover least ranking initialization repetition correctness correct ranking rank quantify estimate ranking consistent probably correct additional uniformly correctly know comparison combine lemma figure n initialization sufficient object passed arbitrarily recommend believe early greatly affect present free represent hypercube experiment repeat new simulation reference response identification ranking plot exceed twice agree deviation query noisy response solution dimension std error algorithm symmetric similarity represent similarity audio row possibility repeat persistent suggest relationship approximate non multidimensional embed pairwise label sequential fraction similarity rough guide report average suggest hope agree many make perform binary implement query lemma n dd simulation permutation object embed cell rank hyperplane point denote cell keep one unbounded obviously similar construct upper induce hyperplane object hyperplane hyperplane partition intersection partition intersect general way cell hyperplane partition hyperplane ease pairwise comparison st object event conditionally follow binomial query relevant sufficiently none sort implement query th ranking random randomness assign th rank probable ranking placing straightforward follow argument equal generality may cell hyperplane relate observe word uniformity constant trial vote chernoff query see I j random py calculation place may aid see converse possible ranking probable choose partly theorem statement
express regularize incomplete computer algebra trial process know success system sec voting interval yield agree derivation principle straightforward mathematically least rigorously justify bootstrappe implicit distribution error regime bootstrappe say seem mention indeed well replacement confidence coin beta bias result estimate problem ref percentage percentage book se compute confidence method sample near red rest white sample take replacement red yield confidence wrong fraction probably certainly instead nonzero describe call sample medical trial case might consider interval never several obviously wrong result exact describe detailed want final eqs exact value order discrete case stand whenever limit coin tail specific tail head specific contain way motivate main trial later infinity coin coin trial eq trial coin coin example n mean observe head coin coin head confidence computer large continuous probability head
absolute relative resp da da da f da extend frequent mining q absolute association definition rule resp w w w c relative close solution mining rule property def point bind dimension sample approximately outline detail dimension sec science infinite member range range cardinality space cardinality subset large arbitrary range interval define subset vc range vc dimension sense subset relative range subset eq construct see resp cardinality resp assume draw replacement constant depend show constant currently know thm interesting range space size sufficient approximate solution market transaction subset transaction transaction empty da range empty vc transaction follow corollary let dataset support set exactly definition di rp ds vc transaction subset exist di transaction di di contradiction transaction correspond space maximum transaction transaction build easy transaction transaction compute bind range efficiently upper maximum integer transaction length one dataset item index anti anti large anti transaction c chain determine reason anti transaction subset transaction contain also contain contain easy transaction dataset case vc dimension dataset def transaction clearly contain transaction transaction appear would transaction short transaction appear long must anti transaction transaction member contain labeling get transaction contradiction therefore false strict vc formalize vc different transaction length transaction transaction transaction range transaction since dy transaction argue vc dimension two exactly anti build maximum anti compute maximum polynomial slow take solve match hence scan memory htb tie break arbitrarily easy bound contain constraint transaction anti fashion transaction keep integer transaction memory transaction avoid already argue transaction transaction deal correctness compute maximum contain transaction length less maintain tie break transaction integer different length complete thesis size transaction read invariant transaction begin loop iteration transaction examine loop invariant condition fail begin transaction equal would neither invariant transaction contain th iteration transaction transaction great transaction hence end transaction least transaction indeed line end th different transaction express invariant instead th transaction transaction contain transaction mean see begin transaction strictly transaction p follow sx dx algorithms absolute execute mining estimate depend integer thm frequent least great property def def property size absolute need compute need approximation integer dc time easily element dx sx k I def property theoretical concern lemma build cover deal constant correspond thm know association definition anti w association rule pf otherwise anti monotonicity property therefore def property def thesis size close absolute extensive experimental approximation analytical size theoretical motivate market previous work artificial come repository moderately effect size frequency dataset build transaction possible thm create follow generator ten transaction use fp growth association rule compute resp resp reasonable characteristic ar currently value find work practice select range frequency threshold top association rule size measure rule sample deal extract frequency real real figure plot take association rule collection single resp quantity run rule always collection indeed theoretical guarantee collection collection interest collection return least relatively false collection output remain extract pos ar acceptable positive distribution real frequency dataset within frequency highly probable pattern real high positive low threshold depend algorithm real exact false positive scan tp goal error obtain look close correspond pos index picture bound analysis error size derive seem suggest room improvement report compute artificial dataset transaction million transaction draw absolute absolute close problem tp evaluate extract note possibly association frequency two appear transaction frequency similar conclusion motivate intuition market mining entire mining transaction frequency running make useful frequent experiment artificial sufficient speed grow perform create scalable dataset become mine grow dataset generate associate report bound transaction show actual relative e vc associate dataset exactly transaction length behave equally fig fix dependent evident suggest present always vertical logarithmic encounter size pos derive extract approximations frequent association rule linearly vc transaction tight family theoretical statistical develop solution computer well guarantee moreover ar random aggregate frequent quality exploit adapt resource achieve parallelism quasi correctly account guarantee technique small dataset believe apply mining traditionally use important discovery vc dimension dataset rank suggest conjecture computer science university edu discovery association fundamental computational primitive application market database heavy find great identify frequent association define confidence solution multiple dataset expensive explore application sampling sample tight relation size satisfactory solution analyze frequent discovery item frequent exponential distinct item begin represent chernoff difficulty simple application therefore sophisticated work loose bind novel application dimension tool roughly collection complexity sect formal major vc dimension indicator obstacle apply vc problem range theory presence transaction contribution characterization vc range tight vc quantity transaction large vc time require greedy fashion analyze frequent rule tight extract mining top ar bound absolute relative sec quality mining table compare technique see vc consistently dependent minimum dataset extensive experimental evaluation advantage work first extraction random sample believe exploit data k sect formally define goal analysis main derivation bind vc association rule sect mining sect extensive conclusion find sect mining extraction dataset start problem generate rule imply frequent association rule improve ar reader survey time heavily depend use present empirical validate random build contain frequent candidate frequent candidate frequent pass entire suggest ensure union drawback sample linearly static chernoff possible analysis suggest work try sample technique theory hybrid chernoff derive loose useful corpus extract select add transaction self fast evaluate suitable satisfied major guarantee another approach give frequency use sample mining guarantee recent first analytical size transaction also present approximate present well apply problem top extract
desirable machine storage rather bring additional complexity natural reason distribute decentralize typical user click storage process avoid bottleneck powerful server relatively distribute platform amazon ec oppose server constrain hardware improve size past decade several reason question cluster broad pass optimization delay distribute consider batch version distribute article technique parallel framework close centralized knowledge implement scale report implementation fair compare exception leave something large measure run interface speed fast achieve throughput gb interface throughput achieve cluster example time course per throughput difficulty parallel efficient parallel occur platform transfer support generality force deal cluster unlike programming effort essence call one key across node computation robustness rapid synchronization overhead good environment core evaluating provide intuition work discussion open become abstraction ill machine researcher iterative algorithm abstraction operation start number name impose proceed phase phase average entry typical main architecture simple architecture section spend time attempt optimize parallelization bfgs gradient accumulate locally average exist moreover implement little address compatible span server cluster job process connect connect span create ip address ready service desirable reliability fail fail trick reliable enough practice computation hour idea computation gain abstraction similar primitive hybrid batch approach pure desirable drawback overcome attractive optimize objective rough precision pass sequential algorithm however make drawback attempt batch newton quasi newton bfgs optimal reach rather require aggregation every attempt drawback start one pass adaptive modify good rather locally accumulate square maintain square confident e assign node feature indeed call routine diagonal bfgs global summing gradient locally initial rapid guarantee quasi point communication small hybrid strategy repeat learn similarly pass different online section get moderately test fast reach strategy share carry iteration break phase vector size size application order communication much second naturally nature share open invariance see jj jj j across weighted start bfgs typically avoid execute job job finish framework handle span topology create span net trick hundred length highlight benefit gain experiment online page key matching page visit click ad get improvement user result click ad example result click negative click good visit represent user ad visit end element illustrate conjunction imagine place bit hash string conjunction unbalanced subsample negative set day dataset find human public dataset effective strategy day induce learn never result sequence regularization test hash feature zero explicitly impose implementation introduction small site recognition curve drop subsample substantial thus optimal roc curve precision log likelihood report iteration node record spend spend spend finish communication minimum interest speed median outlier time slow execution successfully cccc max without execution execution study measure need one run repeat turn speed slow oppose aspect pass median time function experiment bottleneck main degree slow also time version display datum example pass minute describe speed throughput result distribute boost run per report core factor processing report throughput range appear slow investigate speed hybrid interested fast objective fast learn online pass optimize plot gap define I pass strategy bfgs hybrid consist bfgs l bfgs appear site bfgs yield explicit representation create overhead algorithms job job scheduling transfer datum parse implementation table confirm speed extremely remain plus compare gradient tune sgd site online stochastic accumulate mini batch minibatch gradient carry yield mini gradient site mini batch pass update overhead update hour less finish pass much superior run communication instance conclude update possible mini reach similar much batch batch size communication overhead expect time reach pass small theory site inferior prohibitive tune evaluate characteristic communication computation aim hope design choice scalability consideration strategie clean simplifying restrict weight extend scheme additional detail convergent bfgs bfgs provide gain substantial certain regime comprise example uniformly node example objective couple continuous neighborhood around bfgs understand pass hybrid analyze pass perform pass node approximately jensen yield eq complexity arbitrary minimizer function combine inequality fourth remainder discussion denote switch contraction bfgs pass data bfgs hybrid amount overall pass ensure computation cost quite observe bfgs batch similar quasi alternative ever relatively hard local pass online averaging notation approach discuss beyond special one hybrid virtue phase overall approach offer first level level argument clean pass hybrid oppose pure pass regime extreme noiseless desire impact certainly strictly bfgs site typically moderate accuracy evident rate overall competitive computational identical weight per pattern cost modern cost variable total nonzero pass way relevant format pay across
general hereafter limit kernel interval corollary towards first introduce complementary additional whenever stand ii family function satisfy usual hereafter display consider distance assume function differentiable exist condition satisfied function class van norm whenever radius cover set condition easy display set lipschitz index take take diameter condition satisfy smooth let nonempty possess integer derivative set van page inner evaluate quantity u dy dy taylor expansion around everywhere smooth consider clear interior moreover make subsequently clear maximize function easily eq give function f increase notice xt eq follow eq similarly whenever proof straightforwardly principle continuous consequently speed first contraction principle eq low suffice nh l nh lx lx establish lx constant away lx lc nh nh lx lc proceeding proof lx lc decomposition statement l l lc lc lc lc lc lc lc lc lc obvious nh nh lc lc lc r lc nh r nh lc nh nh lx lx observe first real x consider statement obtain lx lx lx lx lx lx lx lx lx lx lx lx account shape lx lx lx lx lx lx lx r lx r lx family observe x whenever similarly therefore whenever e l whenever therefore value take follow consideration definition consider function whenever tt whenever therefore establishe achieve processes york york normality nonparametric de la dans des semi es paris functional series international conference trend characteristic process theory new york inference practical aspect nonparametric nonparametric ergodic deviation principle university york cm remark universit universit paris france abstract devote functional index vc chernoff state deviation index functional deviation vc class g great interest motivate great time nonparametric continuous explanatory finite play important role account reference therein due availability phenomenon modeling year view worth increase number recognition studying model continuously present work al al recent therein introduce lebesgue measure banach distance function kernel go go define notice estimate stand whenever deviation derive asymptotic case xu xt xt f xu speed good integrate simple whenever uniform rate display notation assume whenever differentiable uniform explicit function whenever display whenever differentiable end set complementary large
message update conjugacy message along inference recently apply channel interference decode bp speak exactly bp pass graph bp check code lot even tree reweighte recently develop reweighted geometry decode use projection find mf mf convergent message rule compatible cycle compatible constraint code constraint especially probabilistic bp mf combination drawback unify combine approach equation bp mf minimize kullback region derivation pass bp mf main technical theorem message pass equation bp mf correspond stationary state couple bp mf correspond pmf separation bp mf pass point represent whole factorization pmf interference problematic research community posteriori probability feed receiver coincide propose channel thorough justification message factor factorization pmf stationary constrain correspondence belief arbitrary point marginal always bethe observation message namely factorization pmf certain organize fix devote energy approximation recall bp em approximation use briefly extend avoid calculus message bp lemma pmf bp real convergent implementation message equation graph pmf special combination bp mf joint decode advanced architecture numerical find application communication direction capital cardinality write convention set e capital discrete realization pmf convention representative realization run realization realization pmf define realization convention denote capital letter stand th row x iy I stand proper pmf variable ia aa set ia ap pmf generality factor positive positivity factor connect depict subset region associate number number approximate normalize variational normalization energy kullback leibler base rr belief give two valid set count show region count q bethe free energy bp bethe bethe free energy impose marginalization normalization lagrangian marginalization belief bp point positive belief versa solve schedule message constant drop belief indicate normalization ratio message update accord ignore belief change rescale message irrelevant rescale belief obtain solve lagrangian elementary publish rescaled solution suppose n ax ia rescale solution see tree message run forward backward interpretation approximation free factorization constraint good approximation error plug expression region correspond belief normalize normalization mf exist convergent belief simply use iterative belief converge note order I ax transform pass mf case lebesgue message pass interpretation special mf belief rewrite minimize message except message eq q pmf ia ia factorization bp mf furthermore define I aa counting region define compared count number bethe guarantee valid counting approximation aa normalization marginalization need belief marginalization ax I ia bp part backward belief ax available mf ax ia satisfy normalization constraint equation free proceed describe free guarantee converge show compute simple message enough class complex architecture together simulation bp gain convenient work hard mf approximation message mf apply bp intractable cf pilot symbol respectively represent bit respectively represent code bit n lx ni remove cyclic receiver symbol vector pilot symbol multiplicative represent nz time channel set condition imply factor htbp mf split mf utilize advantage mf refer figure note fulfil bp factor graph convolutional initialize eq ix bp part bp bp compute update message eq get parameter consideration message bp mf figure pass mf pass bp part whereas message pass mf part ambiguity ambiguity reflect family replace decrease compare single complexity mf approximation discussion ambiguity make evident message simplify exploit conjugate mf gaussian factor equivalent apply bp factor I admit closed form term consequence message term message bp combine approximation comparable message complex alternative figure notice perfect consecutive approximately bandwidth channel twice bp inference like g comparison mf find mf yield complexity tradeoff instability bp mf combine describe receiver pilot number evenly pilot scheme symbol convolutional channel channel coherence bandwidth realization replace exist message additional gamma conjugate mf estimation message pass equation mf approximation correspondence solution system point equation split mf bp node result pass point belief stationary update certain show mf bp demonstrate efficiency computationally intractable propose bp computationally demand message extension bp contain generalize lagrange marginalization objective fr differentiable fr definition book continuous bp mf self localization another region message pass reweighted correction mf reaction author thank ia set equation message positive belief combine fulfil pdf integral region verify mf apply discuss q minimize formally significant simplification make marginalization normalization marginalization q shall compute stationary imply ix ix ib point stationary rewrite normalization fulfil ia ax ia ax j jx ia stationary point belief reversible rewrite fulfil analysis remain exclude exclude vice versa prove contribute see ia finish run backward algorithm base income mf bethe free minimizing accord allow plug
classic classification matrix video dictionary mp formalize section especially learn tool recent review device size critical effectiveness publish model determine bayesian g impractical aforementioned lie answer question know selection information aic minimum description principle cost search minimize sense regard practical state description phenomenon simpler usually good length principle give data sample describe bit define describe practice define ml choosing model tool mm familiar explore wavelet denoise corrupt additive white datum exploit iid variance coding encode define art aforementione work follow way dictionary learning atom adaptation low critical deal successful sparse pose particular item systematic fit naturally result category robust thus information learn check meaning treat rigorously natural naturally add dependency feature art bring fundamental understanding bring world detail introduce different part encode describe actual datum mn error active pz I refer matrix say achieve notation extend quantization solution either use greedy pursuit mp convex commonly body certain mp case property denoise wavelet domain parameter choose optimal respect orthonormal avoid cost guarantee alternate optimization dictionary patch patch decompose overlap patch image noisy patch variant user algorithm keep patch patch belong plus small despite method later stage patch partial problem assign however still increase burden introduction practical well angle selection subject follow call class nest p want objective criterion selection mean possible class encode traditional broken guide possible obvious learn patch parameter embed fashion well thus free advantage practical slow one critical clarity presentation encode principle call refined author date extensive reference subject compete coding good capture description avoid description select short description translate problem ideal shannon common lx assume constant quantization encode include cost depend come summarize fundamental establishe model encode depend geometry initial encode separately use expression develop bic significantly main difference early block shannon fully know classic theory encoding produce need produce example describe instance compare pick parametric likelihood ml well encode code base require something laplacian dictionary consequence art maintain encode separately use universal part next encoding scheme describe model approximation noise model regard description main incorporate prior information thereby include certain transformation markovian encoding model quantization component respectively order realistic precision drop simplify example write quantization several mind encode separately rest end discuss single signal form three independent need cost extend sparse discretized laplacian wavelet encoding depict use sparse coefficient encode model number encode ideal smooth parameter result encoding scheme code describe bit universal scheme actual sign coefficient encode magnitude continuous quantization determine cross validation handle stationarity variability universal universal mixture standard mix density informative shape expression zero density ideal shannon quantization coefficient reduce coefficient practice quantization natural indicate advantage attempt optimize keep mention solely actually clean follow nn reliably sufficient component due call show figure purpose verify derivative influence type estimator see theoretic experimental distribution unknown encoding call universal employ come component gamma eq guarantee informative also reliably say parameter free numerical formula estimate coefficient atom th atom west markov atom pixel markov random assume unbiased f obtain case two zero previously universal model statistic typical see learn dictionary observe row encode respective kt plug encoding scheme sequence encode family coefficient th recalling give markovian adjacent dependent example use estimation depend value image process markovian neighboring west one encode occur depict take encoding encoding nest describe call well however code approximation constraint alternative pursuit another relaxation function brevity describe relaxation sparse coding publish elsewhere minimize mp selection empty value give dictionary current coefficient coefficient high add contribution enough produce part implement candidate increment variant turn slow compression gain per sample stop iterate previous assess validity variant stop mp residual coefficient evolution iterate marked black circle note describe pursuit p pl make datum base make easy class dictionary speed forward describe start empty approximate depth subsection algorithm learn frequently atom initial dictionary dct lp l p give set traditional alternate cost need regularize estimation current iterate accumulate iterate process along accord mp correspondingly fitting term none differentiable triangular efficiently use backtrack variant fista focus experiment code actually datum bit gray selection bit gray decompose patch overcomplete dct obtain ability compare apply dictionary database measure backward variant add backward dictionary compression cost convergent forward yield require result implementation ii ghz three reach similar backward cost significant partial fast yield task clean observe corrupt known contain overlap patch denoise patch backward selection global start clean atom second stage admit distortion distortion case describe prescribe c consistent derive dictionary obtain j develop j portion clean think clean problem j markovian dependency occurrence previously encode neighbor denoise summarize detail figure case improvement relevance limitation rd comparable carefully tune scale image aggregation patch rd variant well tend rd visually rd case include cccc rd rd denoising rd clean noisy learn dictionary recover estimation residual portion add back final sample task assign texture patch patch encode dictionary center pixel short patch rate consistent picture figure inconsistent formulation adapt patch respective texture expect decision cumulative patch patch basis compare dictionary patch success comparable see explicitly markovian improve dictionary automatically different patch wise correspond texture success rate classification map average rate extension mn equivalent certain able recover noisy arbitrarily corrupt framework camera surveillance
prototype cluster give convergence address perform baseline hundred scheme adaptive wasserstein start class datum second show fashion e distance artificial show outperform experiment quality paper deal distance histogram histogram complex description phenomenon spatial environmental wasserstein compare wasserstein histogram skewness histogram histogram propose base wasserstein cluster contribution histogram cluster kind variability descriptor whole account descriptor obtain partition measure index function wasserstein distance real collect represent characteristic image usually field description phenomenon flow bank account well statistic aim collect respect deal cluster number histogram introduce context symbolic contiguous histogram symbolic domain discovery relate pattern intelligence cluster factorial describe cell weight proposal dc suitable dc proximity call prototype element cluster prototype dc general cluster look good dc dissimilarity phase dc criterion dissimilarity schema around line factorial comparison distance dissimilarity vision color texture worth histogram pixel intensity wasserstein probability wasserstein permit internal distance spherical possibility identify orientation term alignment cluster step include tune weight associate distance histogram schema easily extend wasserstein present adaptive wasserstein allow compare euclidean account support wasserstein histogram two variability histogram decomposition adaptive approach decompose globally cluster cluster tool ratio base cluster sum inter cluster square histogram compute wasserstein organize wasserstein histogram dynamic non criterion base adaptive wasserstein distance histogram application usefulness propose cluster histogram end conclusion histogram discover phenomena similarity choice relate capability wasserstein histogram representation assume represent histogram contiguous bin different partition contiguous interval frequency let histogram ccccc var I n histogram function dissimilarity introduce histogram distance distance distance wasserstein permit result distribution wasserstein derive wasserstein distance natural euclidean metric quantile respective mean correspond histogram word square wasserstein wasserstein two latter difference wasserstein decompose location shape quantile represent problem calculation need computation quantile histogram avoid computational drawback identification wasserstein define measure histogram quantile means respective quantile histogram within decomposition use property define empirical standard object specific formulation wasserstein equip wasserstein joint histogram wasserstein multivariate wasserstein give weight variable distance equip general induce several associated partition set decomposable weight weight squared wasserstein wasserstein first propose well fitting individual algorithm cluster proximity function individual homogeneity minimize wasserstein distance histogram dynamic proposal wasserstein distance weight define describe histogram histogram description prototype histogram dynamic cluster partition fitting cluster locally minimize update accord cluster follow weighting allow component assign class proximity convergence criterion reach criterion minimize wasserstein histogram prototype criterion assign wasserstein distance prototype eqn general assign cluster dependent wasserstein prototype center eqn base stationary eqs minimize mean represent histogram belong resp result intra inter heterogeneity partition algorithm index intra tool non euclidean distance interval use evaluation histogram square global prototype cluster criterion pp k accord bss distance wasserstein prove additive term within square bss eq descriptor describe histogram descriptor adapt wasserstein data eqn general prototype histogram quantile average variable prototype accord prototype histogram description prototype description sum minimize criterion algorithm recall adaptive square q evaluate index define hold wasserstein index follow equal decompose inter wasserstein distance consider clustering unsupervised clustering evaluation quality specific quality limit synthetic crucial lack public describe histogram histogram compose repository test data generator monte experiment allow quality index agreement partition one china index performance describe performance dynamic square wasserstein distance control variability datum dispersion component histogram dispersion well dispersion histogram wasserstein distance decomposition end set monte carlo histogram choose skewness set assign set belong obtain four parameter generate random number parametric histogram criterion datum measure distance validity adopt generator experiment compare agreement whereas negative agreement chance correct classified end statistic cr accuracy variability generate histogram variability variable globally variability normal describe st cccc st visualization show baseline parameter compute cr deviation cr std clustering adaptive outperform classic performance cr base histogram cluster globally fix baseline sampling distribution skewness cccc parameter mean st skewness order visualization generation baseline cr deviation htbp cr std adaptive outperform dynamic see cr structures dispersion list begin dynamic cluster distance pressure wind people china recorded purpose month month contain histogram propose describe statistic relate global square variability histogram mean percent percent pressure mb pressure mb wind total mm show report description correspond standardized moment spatial coordinate relevant except cluster active c st st mean c st skew skew st st skew skew skew st fashion choice method cluster cluster index within case execute initialization three index fig reach suggest cluster
study univariate logistic model sis seem table contingency estimating way contingency range exact goodness contingency satisfy give problem basis mcmc state via contingency state already easy program intensive drawback bottleneck markov way contingency table show difficulty basis allow entry trade run markov independently take assumption lastly clear general converge importance implement sampling contingency table proceed contingency final terminate cell sample identically iid proposal thus sis expensive prohibitive basis table sis guarantee long sis present new problem compute lp sis reject sis contingency due interval table satisfy sum moreover way contingency sample sis distribution rely sis contingency give excellent interpretation sis fact lead target subsection table satisfy marginal via sis procedure notion short sis rejection limit I rejection conduct logistic model study procedure rejection even table cover sis detail matrix show contingency input table give sis rejection sis rate logistic contingency contingency integer integer generality view contingency n set eq write simply mathematical sufficient usefulness enumeration theorems rao extend element th rewrite simply count let set condition trial draw sis sample count compute integer notice always constraint sequentially apply sis programming ip solve integer q already sis equation lp however importance sample sis sis procedure compute solve integer sample integer interval return distribution mean big probability number equation inequality x exist nonnegative cone call feasible encode rational sis rejection polynomial fix namely generate q rational showed size rational function herein index size rational generating bit rational short rational generating suppose assume term input table sis state fix term short z fix rational generating input size short rational generating form lemma generate ij z j apply unbounded polynomial short generating respectively compute form sis rejection sis pick generating via generate find return pick side input solution thus nonnegative thus proof polynomial size step since count lattice sample implement seem function sis arbitrary rejection integer programming integral assess rate univariate bivariate choose sis seem rejection even generator poisson generator item integer output table cell pick else assign else assign randomly pick pick integer table random
learnable first statement classical argument estimator lemma prove restriction distribution game hard game relationship batch online focus depth study supervise game complexity bad surprisingly adversarial vc learn indeed irrespective regret within rademacher logarithmic factor algorithm blind game blind learn supervise adversary supremum satisfies pass associate game whenever lipschitz rademacher scenario distribution translate restriction mixed strategy interestingly upper bad hybrid function ty tp armed composition say rademacher hybrid scenario lipschitz rademacher complexity analogue contraction class lipschitz eq rademacher choose necessary corollary follow absolute irrespective bad rademacher matching bind attain choose variable irrespective matching rademacher bound define absolute adversary allow coin purpose prove enough adversary pick whenever restriction history tx restriction bound case second bind hold restriction allow rademacher restriction supervise learn protocol sometimes adversary yet observe subsequently reveal protocol allow yet equivalent modify improper allow choose side cumulative cost inside decade arguably optimistic bad smoothed see realistic measure smoothed explain despite exponential section analogously concern former batch sequence adaptively argue neither reasonably world smooth bad sequence well I simple learnable supervise scenario yet online case supervise dimension dimension threshold infinite mistake choose adversarial world opponent adaptively bad case smoothed scope greatly smoothed infinite tractable conceptually smoothed smoothed smoothed sequence technique smoothed yet hypercube example choose random smoothing pac consider corrupted error formally perturb measurable example noise correspond generally consider move smoothed adversary generic online trivial upper guarantee enjoy smoothed adversary adversary player observe move perturbation proceed nothing restriction study adversarial parameter smooth somewhat surprising learnable exponentially noise adversary learnable show bad notion restrictive supervise function binary learnable online trivial particular complexity appear formally eq uniformly adversarial follow setting whose move corrupt bin well smoothed fall bin take discretized threshold threshold difference generalize threshold correspond argue adversary bin supremum supremum cost bin sign range result exhibit martingale difference fix hoeffding observe discretization coincide element interval ensure fact threshold sake element fall bin mind choose throughout game bin q use need infinite indicate half learnable perturbation half classification smoothed note upper value game inefficient recover formulation problem smooth half space discretization job directly bin ball loss non discretized acknowledgement nsf grant dms proof proof couple understand move inside arrive repeat step expression q way prove fix tp sequence functional expectation q b fix infimum strategy past eq q bayes response supremum infimum strategy depend minimax strategy move upper martingale employ technique introduce sequence construct copy independent condition true hold plug hand obtain informally indicate conclude think path determine outline worth mapping draw independently sense define tangent supremum last equality verify understanding correspond supremum successive distribution irrespective contribution successive distribution condition value path irrespective clearly contribution argument alternatively expand claim verify simultaneously term generally perform center expand q pass upper tt p increase equality pass coin define constraint pass supremum tree path tree apply path rest tf tf linearity last lemma slight pair equation arbitrary tf tf linearity pass step however equation range rademacher associate quantity select pass arrive eliminate outside increase without follow appropriately proof proceed sequentially property start towards mapping supremum fact depth note rademacher bind need justification reverse argue supremum supremum supremum provide removed supremum long appear conclude notice play copy distribution rademacher draw whether via selector produce sequence restriction protocol define bound proposition expectation expand stochastic allow conclude proof adversary rademacher unlike sequence define conditional tree lower concern path similar lemma classical statistical scenario adversarial pick learner neither possibly due game adversary move capture stochastic building approach range deduce adversary supervise hybrid way consider smoothed half space learnable smoothed adversary decision turn infinite learnable continue line array assumption notion theory complicate nested tool unify bad learn scenario two present make progress towards instead measure external nature restrict play notion place nature assumption adversary yield associate name player name far traditionally refer decided keep whenever problem sequential adapt game language think sum adversary learner minimax heart statistical half restriction put adversary allow aware similar sequential minimax central estimator minimax function rough capture vast array consider associate minimax supervise loss infimum goal predict function combinatorial vc uniform risk minimization unsupervise learn erm quantity zero formulation long make generate opposite manner ahead game reveal frequently minimax write randomize randomized adversary pick bad sequence course protocol adversary choose move proceed next move player instead write quantity way sequence adversary round stochastic capture assumption online scenario move various smooth present minimax many question adversary organize minimax rademacher see case main careful devoted rademacher behavior adversary batch online complexity irrespective pick learning dimension learnable small introduce extensively use close metric distribution learner capture adversary restriction scope adversary restriction adversary restriction adversary player write define restriction name restriction strategy deterministic point support x tx constraint allow budget pick bad case corrupt equivalently restriction adversary choose distribution restriction shift gx pick force restriction eq adversary restriction restriction might point equivalently write q crucially strategie mapping q value main object payoff mapping tree extension bound q statement particular center relative shift adversarial move restriction denote banach play allow increment condition increment depend walk understand rademacher well rademacher complexity equal sequential sequential rademacher upper apparent reader minimax formulation mathematical necessarily yet difficulty become familiar next classical tp writing path equality fix path rademacher make expand inequality supremum interesting case hybrid adversarial refer proof dependent rademacher distribution sequential sequential tree depth result state maximal upper function mapping control integral function lipschitz constant z cover composition play useful scenario yet satisfy pick include game move previous move variance generally adversary specifically game adversary sequence round constraint play adversary view adversary conditional way game constraint compact necessary corollary satisfy necessary hold range set tree minimax q tf armed extend result say adversary allow away decision player exploit learner paper function consider constrain
reduce criterion netflix burden perform ignore remain rank approximate enforce constraint metric step within manifold rank lead art general address minimization riemannian update riemannian riemannian straightforward gradient tangent update geodesic direction intensity hand equip usual product straight note procedure intrinsic independent easy calculus symbol need much easy map tangent yield update give idea sphere endowed consist follow operation avoid calculate geodesic riemannian critical subsection remain converge size manifold manifold rotation real subsection continuously differentiable use instead exponential subsection slightly modify riemannian take account effect tend mild convergence prove manifold e disk half plane trajectory compact riemannian uniformly step exist compact build usual remain parameter compact measurable hz hz hence imply converge like fact use sum trajectory variation denote non bound variation e n n e sum quasi converge absolute converge converge taylor expansion ep f k right martingale convergence value map riemannian bounded continuously differentiable size satisfy condition compact gradient rely twice small exp riemannian martingale converge variation exp right unchanged second previous convergence long remain converge hadamard manifold hadamard manifold connect manifold curvature effect size yield relax even case hadamard map invertible usual case opposite towards curvature denote function manifold converge mild assumption imply upper appearing finding may require suppose converge build go trajectory easily expansion half riemannian geodesic bound hessian square eigenvalue tw fw w generate fw us taylor triangle fw fw v v v prove necessarily h quasi martingale mean thus fw continuous ball radius stay inside fix compact trajectory belong center compact moreover weak imply theorem derive used proof present illustrate theorem interpretation third throughout precede track eigenvector several application want principal e principal reason suppose vector element gr subspace ambient identify manifold column cost minimal basis dominant rotation state manifold gr study event consider stochastic riemannian application states geodesic vector input sequence gradient manifold zero compact prove point prove invariant subspace prove dominant subspace average invariant covariance application particular decomposition argument follow ambient particular already general present result directly stems illustrate viewpoint geodesic step computational especially know disk tangent conventional product diagonal angle riemannian distance move disk reach illustrated arc equip fr riemannian unique hadamard manifold grow interest filtering manifold propose closely optimization parameter randomly pick squared tensor riemannian redundancy equal riemannian require operation redundancy randomize reduction computational burden besides used filter measurement track slowly see low pass filter mean current new figure verify radius open geodesic contain hz positive quantity definite asymptotic similarly span subspace stochastic kalman theory gain become recommend gain gain make estimator initial sufficiently fast avoid concerned insensitive final could generally use compare behavior average gradient descent hz latter estimation decrease second slowly converge iteration experiment discrepancy rapid error require initial comparable norm gaussian input identity via semi rank matrix asymptotically top plot versus iteration plot solid estimation hz line coincide choose line gain method full rank technique parameter difficulty maintain definite thus attack algorithm project semi definite cone proved perform equal illustrate however algorithm little stochastic advantage convexity average full propose apparent operation become moreover rank lead thus low address emphasis refer variant extensively art variant test netflix riemannian multi distribute replace exchange various limitation gain popularity procedure neighbor average node reach intermediate value little distribute play consensus well know consensus problem consensus space wants agree motion phase adapt g consensus receive procedure address decentralized measurement distribute computation assume possess estimate allow exchange node accord pick randomly represent availability node reach value implement interesting cone riemannian view symmetric information tangent distance agree kullback leibler symmetric denote kullback leibler geodesic amount information separate understand statistics rao bind accuracy state draw look sample unit increment whereas identify variance discrepancy admit strong invariance mean attract ever image year follow procedure tackle step pick randomly neighboring node towards geodesic half geodesic select cost manifold explicit expression advantage viewpoint nice property merely geodesic proposition mean new accordingly unchanged yield orientation axis versus merely theorem assumption endow complete find geodesic ball geodesic ball step towards lie current remain convexity belong compact moreover radius away appendix cone definite riemannian behavior always illustrate figure run show fast initial outperform usual versus diameter superiority simulation robust together property average riemannian graphic superiority node heterogeneous bottom graphic ht versus run riemannian converge top node heterogeneous manifold reasonable convergence numerous control manifold enforce constraint intrinsic concern lead substantial source cast successive propose riemannian parametric natural asymptotically intrinsic intrinsic rao future explore reach rao bind detail future aforementioned completion prove critical lead mathematically identifiability possibly consensus stochastic hope extend algorithm consensus fast author thank subject l discussion geometry realization parametric joint log parametric law conventional space definite chart assumption w w term w expectation law basic rao prove statistical efficiency prove space endow gradient blind bss performance gain recently intrinsic rao replace distance fisher usual update could replace intrinsic paper achieve fisher reach beyond geometry let g riemannian carry metric length unique path minimal geodesic geodesic position cut roughly stop circle cut geodesic ball radius riemannian tangent kk along geodesic paper develop extend stochastic gradient point numerous potential novel test numerically provide great practical minima optimization demonstrate idea toy briefly mention traditional trajectory gradient correction correction angle tend reasonably hope
training test base index smooth marginal may benefit weight let smoothed sx appendix criterion convert inductive hypothesis establish guarantee smoothed inductive furthermore might nr analog smoothing need cubic root guarantee investigate advantage perform unweighted smoothed empirically weight choose work uniform sampling emphasize benefit even might weight attempt reconstruct noisy random signal ensure error perform prof throughout trivial entry recovery cite define since combine hold case step true since inequality fix write ab eq continue denote smoothed empirical generality matrix apply theorem bind probability empirical marginal cn define expectation split obtain si yield result weak requirement bound product expectation prove combine proof theorem define define argument two department brain cognitive sciences technology title trace norm might fail distribution index present empirical know empirical collaborative filter small reveal match well theoretically understand case nearly row require completion motivated issue variant entry show superior guarantee clear trace theoretical norm recent report e index product distribution seem realistic case netflix user rigorously trace correction netflix indicate helpful rigorously empirical weight advantageous arbitrary rely instead present consider arbitrary target matrix reveal replacement like prediction expect respect trace norm marginal distribution trace particular pi although certainly estimator sx sx empirical observe although inductive draw future state fix theorem set use trace rademacher complexity pair value modify trace sampling index empirical rademacher sign possible random use bind yield provide unbounded product marginal restriction fix sx elsewhere guarantee expect complexity duality spectral I mean matrix combine r calculate row assume marginal potentially unbounded truncate class spectral sl pz extremely sl p sx px rl px sx result marginal lower precisely satisfie result row marginal marginal unweighted trace none perhaps trace give arbitrary excess give trivial weak guarantee discuss lipschitz loss require amount require entry boundedness lipschitz fix rademacher product uniform excess error cube give cube setting whether improve theorems factor construct degenerate use product non uniform special cube good hold arbitrary error unbounded construction arbitrary unbounded regime error demonstrating let rl py sn sx rp ax summarize guarantee tight c unbounded loss uniform degenerate example generalize need enforce sort uniformity computation lie large small row improve guarantee smoothing provide guarantee suggest smooth large advantage situation check smoothing weight beneficial denote smoothed hold loss fix sx x p apply essentially identical definition si apply result eq think significant order square trace low rank column magnitude much place require dataset netflix netflix rating user netflix rating sampling scheme user deal test create rating aside set rating movie aside rating rating zero via reason truncate trace optimize regularization choose cross mean k smoothing smoothing even differently sample
family probability inferential lemma say universal universal family inferential equation per conservative hold pa connect concept density universal pp universal g equation pa aa equation say odd terminology change odd use original surrogate g recover hypothesis g apply measure aa may either section type minimax bad among since average solution density thereby finite jeffreys originally invariance whereas parameter describe reality default serve inference scenario imply support observation review asymptotic population density universal optimality subsection minimize minimax normalizing act convergence consistency since asymptotically indirect evidence incorporate define q notational parallelism bad hold I w accordance population observation consider logarithm commonly finite cancer transform abundance conditional absolute protein simultaneously estimate maximum obtain appear fig sample minimax approximation could achieve comparison simultaneous I amount support nuisance comparison argue support one bayes support quantify odd odd hypothesis pure previously replace maximized parameter constitute composite hypothesis thus strength evidence union fx fx perform nuisance parameter extent intuitive principle viewpoint l primary generalize pure predictive make applicable hypothesis measure difference empirical bayes proportion compatibility even example odd calibration compatibility information uniquely measure protein standard odd measurement protein come simultaneous inference nature discrepancy support increasingly increase dimensional application overfitte support penalty factor acknowledgment research partially innovation innovation university david biology department road thm david r department frequentist hypothesis accurately ideal support observation second minimax normalize compatibility normal sample expression datum closely odd available protein random parameter reliably indirect evidence description likelihood science observe p belief ideal salient easily term value hypothesis quantify odd odd available frequency although baye assign alternative unless hypothesis factor improper conventional divide test generate proper expense specification sample concept measure applicability routine scientific present herein another odd rely relate probability hypothesis biology genome protein determine nan hypothesis hypothesis operate distribution nonetheless retain model across include necessarily datum paper reliable often parameter measurement available often reliable estimation argue due microarray less maintain reliably record statistic meet explain framework one odd thus report support enable reader roughly determine either use value minimax address support abundance protein analogous multiple protein finally conclude density parameter hypothesis call member nan density reflect randomness specify value employ eliminate nuisance parameter interest identify thus nuisance eliminate distribution strength statistical theory effect real uncertainty nan respectively pp indicate hypothesis sufficiently
fair al publicly crowd use amazon color color ask crowd indicate ground color comparison distance color distance check validity collect response comparison task response similarity worker answer subsample would number response per crowd amazon quality indicate since phase around algorithm perform majority scenario assignment assign priori collect limit task worker latent draw provide minimax result minimax factor term budget core concern answer cost query budget achieve target good task assignment inference crowd parametrize minimax range range assignment ask query total probability assign query achievable oracle prove minimax case low rate worker low rl mistake task flip fair coin half convexity jensen necessary crowd assume query actual assign choice hold bad assignment minimax regardless worker term number query achieve necessary adaptive query algorithm average worst worst analyze however result prove worker quality budget establishe minimax pay factor necessary simple voting approach voting provide achieve voting least I achieve voting need ask per voting costly term budget well operation require prefer practice process parallel switching assignment hope adaptation help careful requirement much gain allocation identity worker reliable worker manner crowdsource amazon identify particular worker persistent identifiable open exploit worker adaptively batch task next worker base thus far hope adaptively collect less perhaps surprisingly gain adaptive worker error good task allocation true total exist assignment scheme query budget error worker scenario achieve average less case compare corollary significant achieve factor scheme limitation adaptation strongly fact worker exist design corollary worker improve constant fact family worker algorithm succeed worker exist achieve probability show adaptive algorithm vanish increase simple adaptive show go algorithm worker task group repeat reach allow mistake run allow query finish prove allocation scheme worker ask worker among response run query worker order terminate condition concentration result prove vanishing grow scheme prove non instance adaptation significantly worker fail crowd apply produce error half grow section explain vector reveal answer fill zero prove constant numerical cf due result scale know effective perturb mt conditional expectation mt reliability exactly perturbation mt spectral signal correctly extract use crowdsourcing introduce prove scale result analyze spectral gap spectral conditional denote prove lead randomly initialize normalize converge sign v power iteration identical pass task worker iteration message upper scale analysis technique singular crowdsource graphical weight bipartite connect edge posterior ia ia abuse denote probable graphical bp maximize approximate probable closely standard line provide support bp operate set message message worker message message bp task correspond worker message worker standard framework iteratively message follow randomly initialize ip fp p aa k aa x ib ib rule indicate proportional algorithm produce end predefine I ip bp ib ib I either tell always give wrong answer rl bp update iterative I ap update cf belief propagation despite surprising perform optimally task prior discuss implication research always suggest stop transition discuss provide iteration iterative voting algorithm notice meaningful half inequality half main emphasize behave try error iteration increase algorithm generally section scale always cf suggest iterate help cf gain maximize exist worker bad intuition variety algorithm identify ask whose answer also gold standard unit assess quality gold unit seed inform gain gold utilize lot gain embed gold estimator still include strategy utilize seed seed gold pilot gold pilot question pay question benefit pilot gold collective crowd worker pilot describe gain pilot still pilot crowdsource crowd might worker crowd collective quality payment use mix say worker subject per immediately maximal implication pricing scheme crowdsource crowdsource platform collective crowdsource platform quality ratio least good crowd possible paper factor difficulty worker response worker question task reliability bias towards answer represent reliability formula worker binary model sign large answer likely regardless correct answer response latent study crowdsource early ij another ij ib ib model paper task share worker worker distribution proof symmetry result estimate average task probability randomly task assume density task worker edge worker distance root inference worker pass task run worker message grow subgraph evolution error make actual next vanish regular bipartite bind error condition variable whenever variable negative distribution decision know density code theory distributional probabilistic equality density operate whose randomness randomness graph realization variable variable condition regular locally regular tree distribution message initialize worker variance initial worker iteration ij symmetry density evolution equation definition also variable rl p represent worker distribute worker conditionally independent initialize message incoming message response worker quality incoming message response variable chosen numerically compute computationally message take heuristic novel sub recursion upper prove sharp decay exponentially message behave evolution accord p p p simplify p substitute evolution variable r assumption bc v k q second k k var k x chebyshev hand prove sub weak message fundamentally regime good regime converge grow q get tight gaussian following hold e k r km precisely definition distributional independence k follow satisfie k k get km r write formula evolution mean substituting follow proof inequality regime recursion k k bc c worker node configuration half edge match worker let denote worker nr similarly way half half probability r total assumption bound p apply fact majority voting majority agree formula neighborhood random without rescale random variable node assumption p monotonically increase assume value odd use closely approximate error low expand substitute bind node distribution task e follow q prove even know reliability worker th worker collect reliability decision next assignment stop criterion meet label maximum make worker assign task follow p rl worker assign stop worker assign oracle know computed worker reliability budget achieve allocation necessary since total definition pt hold focus task reliable worker estimate worker worker let reliable worker answer therefore jensen worker reliable jensen maximize result notice change ensure variance step variable binomial gaussian calculation limitation general crowdsource model mistake accord worker make worker reliability task equally challenge inference crowdsource model bias heterogeneous answer approximation generalize algorithmic interesting solution might modification belief propagation probability small worker approach scenario worker counter achieve term budget around well majority formally crowdsource study theory mechanic rgb rgb crowdsource numerous piece worker solve character recognition nearly devise confidence answer typically assign answer majority vote general crowdsource total achieve target reliability worker infer worker inspire outperform majority voting comparison know dynamically assign task worker might hope surprisingly minimum manner scenario scenario rely worker exploit build worker design crowdsource system human annotation optical character recommendation crowdsource amazon market batch worker benefit crowdsource highly task even among worker collect strategy reliability answer worker pool reliable exploit fashion crowdsource amazon large anonymous trust particular correct correct answer may never truly make hard resort worker irrespective answer aggregate majority reliability answer crowdsource worker neither persistent identifiable may identify particularly reliable worker nonetheless worker answer draw weight however worker aspect unlike batch plausible ask pilot question worker decide ask question answer decide ask understand variation define probabilistic capture question collect simultaneously may adaptively worker answer provide task error achieve error establish task optimal base characterize collective reliability crowd achieve target reliability necessary replicate interest assignment ask achieve rate somewhat manner adaptively help setup crowdsource integer categorization example state correspond label child solution worker fashion crowdsource two task allocation query sequentially accord choose subset constraint choice batch worker batch worker reliability parametrize rl character clear next adaptively answer collect thus far assignment typically query meet inference estimation answer say task answer collect answer adaptive batch batch process parallel however switch adaptive investigate aware worker consistent real crowdsource worker go persistent solve new particularly game sequence trial pool identify multiplicative crowdsource hope particular capture worker difficulty hence performance across task discuss generalization worker distribute example answer mean different literature model random label crowdsource parameter proper normalization crowd role capture collective crowd worker indicate crowd case achievable population distinguish sample quite meet batch requirement prior execute reliability determine time task many iteration reliability discuss simple way overcome perfect crowd correct justify truth task crowd agree consensus without crowd definition ground worker quality proportional accuracy want crowdsource minimal deviation make crowdsource identifiable worker worker batch crowdsource efficiently worker identity worker imagine crowdsource platform identifiable worker choose explore exploit one significantly simultaneously estimate worker select worker good worker phase al voting identify good worker exploration exploitation however test crowdsource use pre track worker market popular crowdsource amazon world crowdsource impossible track popular crowdsource market worker crowd reason assume worker provide algorithmic worker show worker technique furthermore worker track develop worker start account attack closely formally address crowdsource payment infer accuracy worker worker pay worker task amazon optimally crowdsource contribution technique inference introduce operate message priori overcome novel establish recursively prove class message certain answer query regular answer achieve order task precisely approach low rate adaptive minimax factor necessary rate bad worker propagation allocation scheme worker amounts design bipartite node indicate include batch crowdsource platform batch complete task work batch bipartite many worker resource g task worker typically task automatically edge achieve low bind graph degree thing significant worker budget collect decrease grow immediately affect performance rate however worker degree generate regular want graph slightly propose simple random task half edge half might edge node graph number double hold result analyze performance sharp intuition behind graph spectral follow vector weight adjacency excellent spectral gap enable low allocation connect assign index worker node node edge correspond worker worker introduce novel inspire approximation connection operate value message worker message reliable worker message initialize sensitive positive initialize message need extra degenerate accord task neighborhood worker task update worker give sign believe message represent hence answer worker iterative I jx max belief approximate bp detail crowdsource density evolution bp crowdsource numerically evolution cf density develop density iterative analyze broad message upper achieve iterative configuration decay dependence effective gaussian tail inference r follow bind run iteration crowd collective
implementation ease state along generalize range restrict e tensor square problem method tb input b bb output histogram histogram recovery briefly discuss extension multiple histogram convert sparse dimensional orthogonal transform version wavelet apply transform e omit refer interested reader wavelet coefficient non zero coefficient turn practice cluster significantly histogram similar matching pursuit inverse transform represent histogram correspond coefficient unseen cardinality section present histogram batch load level comprehensive engineering believe conceptual choice address question extension algorithm modify query maintain appropriately histogram step incur consideration similar solve small non wavelet propose modify say accumulate modification remain unchanged histogram maintain online characteristic however old information adaptation might assign recent discuss learn formulation involve empirical assign alternate assign old modification algorithm detail algorithm present current histogram histogram training dimensionality histogram attribute evaluation involve one histogram percentage average relative number use experiment detail present histogram report relate dynamic census repository dataset essentially mixture pt census various attribute synthetic varie clearly range linear dimensional synthetic projection census c synthetic gaussians random synthetic ii five gaussian around attribute census age census database synthetic census spherical dataset mixture gaussians random census age work attribute choose attribute status education describe range query generation dependent query center total uniform model data range hyper generate around volume query train test various detail dimensional implement experiment well use implement matlab average solve solver solve scale pt query incur synthetic converge around error synthetic type incur incur error test age attribute incur error incur histogram vary learn increase train first compare incur attribute naturally incur increase query however able around query incur primary round fit query dependent fig query converge boundary discover align peak true frequency mis peak contrast accurately boundary incur type ii query converge accurate specifically incur incur incur inaccurate bin boundary much wide boundary interestingly query histogram histogram method align boundary true fig census uniform incur less able learn ht cccc c dataset model incur error query query dependent incur census query incur less vary synthetic query respectively figure incur incur error incur error incur however query query model significantly well perform next query vary compare three three method incur incur incur error census query draw vary incur study vary compare synthetic well theorem heavily query trend query summarize converge incur incur synthetic consistently vary demonstrate advantage two small incur multi census ht synthetic vary query incur incur error incur incur census vary query incur error incur incur incur test histogram incur incur plot incur skew able reasonable histogram incur follow figure report finish census training census incur significantly cccc pt frequency number work attribute census estimate histogram census training query work attribute capture accurately dual core ghz gb ram mostly finish finish stream incur version dataset database update query train dimensional compare vary dataset generate datum dependent histogram outperform reduce rapidly query query incur incur next three vary outperform small incur around experiment census dataset recall census project census attribute database work concentrated try empty consequently incur incur error still approximate underlie incur estimate density peak contribute incur report incur census vary incur relative scalability increase conduct experiment synthetic gaussian show incur vary incur incur incur implementation terminate day entropy problem round need iteratively number run method minute figure training error incur number significantly incur repeat census consider education attribute trend accurate inaccurate incur incur performance presence datum wavelet keep unchanged effective converge robust synthetic generate batch batch around similar th converge update database summarize enjoy number number remove end fit high error contrast well run solver form large even though require well high scale fairly high inconsistent extend update simplicity critical query histogram use work histogram histogram reader tune first accurate split bin density provable restrict grid high case grid however grain impose feedback small estimation call learn entropy state art tuning recent involve distinct dimensional histogram effort seek execution wavelet tool haar popular wavelet histogram piecewise signal haar wavelet extensively histogram estimation wavelet probabilistic method decide wavelet provide guarantee tuning contrast introduce appropriate introduce theoretic tuning cast learn well histogram approach limitation still good histogram self histogram width many empty region haar cast adapt pursuit omp transform easily extend dimensional update effectiveness method scenario provide variety result especially critical world database able reasonably census future difference histogram expressive individually tb standard fu follow f formally assign furthermore I intersection last cauchy schwarz record q attribute select use inequality selecting hence incur solution histogram recall regularize function hence fu along axis use follow q last follow property theorem corollary theorem axiom tune histogram theoretic feedback histogram small memory minimize study histogram highlight width histogram histogram histogram haar reduce histogram sparse multiple scalable multi scalability histogram feedback natural incorporate handle advantage art histogram modern database query typically histogram approach limitation first uniformly histogram might space construct scan sample histogram nontrivial update inaccurate histogram build limitation idea collect execution refine query together query expression feedback collect minimal execution plan characteristic histogram build histogram feedback arise year select histogram boundary fix exist theoretical dimension current self use feedback feedback step valuable information quality scalability another limitation update update feedback theoretic histogram informally goal histogram standard principle advantage leverage translate efficient inherently inconsistent strategy recent old next begin study dimensional width reasonably approach well practice analysis incur know histogram fail contribution recovery problem informally wavelet histogram cast vector provide adapt width generalization histogram extend haar wavelet transformation show query maintain presence extension incorporate database histogram include propose dataset prior term accuracy cardinality database outline histogram denote column algorithm generalize categorical domain record histogram consist estimate partition estimate use range notation represent query histogram bold matrix letter term b u kk represent range histogram histogram arbitrary histogram next width histogram optimal relax histogram search search record search induce correspond search convex observation avoid function lead relaxed specify mb ir mb distribution section instead empirical solution optimally standard optimal multiplicative problem l I width constant note show property fact generally nr hence histogram consider confirm intuition histogram query however factor lipschitz lipschitz function
value become constant evidence eqs total eqs live live arise return possibility evidence accumulate estimate live versus red series spike live dominate live volume live likelihood decay volume continue decrease represent nest want continue nest ratio illustrate live mean take extra effort uncertainty therefore interesting effort comparison live stop fractional sample present uncertainty histogram agree well bayesian volume sum yet quantitative whether agreement equivalence two yet apparent depend likelihood nest sampling moment currently foundation theoretic continue result problem establish estimator compute statistical maintain core nest draw inside current surface determining mean effort acknowledgment thank valuable discussion suggest error us national uncertainty nest valuable determining volume I evidence estimator new make evidence constraint valuable challenge explore compute evidence dimension complicate possibly multi modal shape carlo popular mcmc sample great range yield evidence intensive recently nest bayesian speak away volume layer evidence volume statistically focus evidence yield challenge nest surface nest pick new discuss surface likelihood include multi modal importance moderately hamiltonian evolve new keep differ pick call nest always step proceed computational nice statistical purpose evidence nest numerical choose applicable nest nest establish detail likelihood space normalise prior volume span allow range incorporate flat bayesian high monotonic function rewrite increase word examine step heart method idea straightforward difficult associated volume treat statistically slightly statistically small volume uniformly word distribution large prior define volume relevant region prior distribution begin live live nest live point estimate associate draw replace extract live point prior iterate sampling estimate technique nest mainly iii associated posterior much difference gain leibl see integral straightforward approximately live dominant statistical fairly uncertainty distribute evidence rigorously variance volume carry estimator uncertainty like small volume eq correlate independent density q volume sampling volume q eqs term since statistically volume obviously evidence convenient begin component include include one index average write product collect rewrite care distinguish appear twice product notice back moment partial evidence combine usual new estimator uncertainty current notation yield two expression interesting contribution enough expense make expression nest nest remain evidence live uniformly live point live live average live eq moment uncertainty account involve term evaluate put together live total give step live negligible end formalism assess since sufficient also flat prior coordinate align principal axis q prior cube origin box test examine gain last box dimension conclusion live estimator fractional analytic volume mean simulation predict average yield agree indicate uncertainty small analytic accurately volume yield
h idea recall selection complexity vs answer heterogeneity ex conjecture axiom comparison search target manner ask select similar object list present selection repeat point terminate strongly world network focus database popular case demand heterogeneous heterogeneous demand novel heterogeneous demand comparison mechanism bound intuitively demand constant capture topology interesting connection comparison classic search problem search user ask object list new object list typical unable query exploratory life cite human subject example pose automate method feature fashion human person select mind unable query mind able express identify among close web priori post present let search comparison amount determine object user possible model choice output goal object object query recently attention content embed edge minimize cost variety database unlikely frequency follow world demand early homogeneous novel problem context comparison bind mechanism also low content adaptive content meet heterogeneous demand intuitively appeal relate content important distribution target capture capture organize formally state search comparison adaptive devoted proof extensively early embed intrinsic net deterministic support metric embed satisfy certain sphere pack restrict connection constant mechanism restrict demand advantage object metric space rather require similarity object object rank performance cost call capture triangle play role demand work search heterogeneity restrict analysis additional distinction existence explicit question oracle datum phase incur contrast scheme adaptive drawback search lie incorporate history arguably first membership ask belong interactive handwritten access human moreover cost designing dataset prohibitive contrast behavior ability limit mechanism seminal condition embed explore heterogeneous demand investigate propose structure expand definition notation assume embed represent mapping feature object object abuse keep mind object partition equivalence respect empty close close formally imply basically aim capture human target picture similarity associate pair equally human stress place target analogy user mind present relationship character hold oracle order order order pair call demand vary different target eq demand play quantity search distribution introduce notion define define connection content access object accord find need oracle average occur frequently require identify entropy target bound search also topology capture radius probability distribution minimum nc n note contrary determine cube arrange uniformly cube arrange dimensional entropy l order demand entropy local give expect formally object embed although constrain directly distance access comparison section serve start greedy content propose object object process return something happen greedy content search process terminate analogy actually never reveal oracle stress priori present oracle content try response oracle response include object content one history object current content search allow policy case reveal locate history reach select number particular target policy randomize expectation comparison define demand select minimize note small consider assume direct property pair distinct object exist object lead object content goal start oracle oracle q greedy message repeat greedy terminate property eventually reach currently moreover close comparison object oracle neighbor close typically call determined proximity arrange rectangular grid gap distance every rectangular indeed locality satisfy edge property goal possible select greedy message edge random subset expect greedy heterogeneous target select accord demand embed edge demand greedy essentially obtain technical appearing problem restriction select second exactly edge go object connect joint edge eq minimize cost cm content sample draw suppose problem search start policy generate source see one vary across neighbor local content move object target message move effectively movement reader proof rigorous relationship optimize greedy np found section reduce version interestingly instance distance remain np hard even consider suggest problem solve motivated content restrict particular cost term entropy content whose object suppose define property
analyze set contribution learner sensitive supervise control sensitivity ensure learner bad assumption fail invoke feature assumption manifold density metric diffusion analyze degree unlabele control strength strong use sensitive kernel provide number adapt degree choice learner fail preliminary confirm alpha risk hold fail reference therein knowledge paper allow choose strength assumption outline section error used adapt xy p yx px py state sensitivity follow distance modification curve speed make correspond smooth respect smooth distance recall derivative exist degree condition real projection boundary boundary small condition inference additionally access access let bandwidth unlabele support boundary mx follow use sensitive characterize performance sensitive proof negligible unlabele enough sensitive mse joint condition supervise depend inf supervise estimator couple term integrate intuition lemma need take advantage keep familiar smoothness cube add series two number implying design specifically low easily essence boundary establish square sensitive estimator know sensitive parameter attain well outperform index parameter control strength semi hence take advantage unlabele semi extra preliminary alpha supervise fail report possible density sensitive relax besides elsewhere state repeat appendix indicator unlabele mx write measure curvature volume follow h give point combine component part grant fa dms sensitive distance q outside must proposition characterize plug behave assume px x x triangle mx x xy must similarly proposition mx sx mx mx x write apply follow condition dimensional euclidean plane argument proof q constant compact compact connected quantity let nd q lemma euclidean even result base vector affinity supremum measurable semi mi mi dirac also measure define product note measure denote logical arbitrary define clearly e easy satisfied easy see complete ef rf
sde dt x motion ergodic tx chain reject probability pointwise family hasting hilbert provide metropolis hasting markov k k main value sde equation ball probability maximize radius ball minimizer generate concentrate minimizer possess almost fine scale reflect operator brownian motion brownian quadratic draw invariant measure almost sure finite sure property reflect metropolis limit sure asymptotically paper limit ode globally quantitative rate algorithm motivate create noisy flow dimension random variable evaluation come complexity suffice walk metropolis rwm increment condition condition require analyze restriction tend independently indeed diffusion paper demonstrate langevin mala condition step prove closely introduce nevertheless analysis chain evolve theorem temperature increment gradient minima standard heuristic neighbourhood global minima stress asymptotic effect schedule hilbert simulate annealing present challenge distribution mutually singular brownian contain statement theorem proof quadratic result often term limit conclude section notational concern real function canonical inner trace eigenfunction assume orthonormal like rx x rr rd operator say orthonormal outer z operator denote norm l l satisfy satisfying notation sequence constant notation randomness present notation fact hilbert brownian hilbert may self adjoint class gaussian cx expansion x notice variable denote diagonal rx r equivalently r frequently functional arise exponent fix distinguished exponent change formula notation form orthonormal c continuous gaussian stationary increment real brownian define brownian motion equivalently brownian equation express functional connection bound act must full underlie decay eigenvalue domain define appropriately formalize element dual comprising may identify derivative weak localize assumption operator domain exponent satisfy derivative satisfy imply x functional behave sense make dx dy second remainder expansion clean exposition highlight central derivative localize assumption cost considerable technical argument present version formulation precise show weakly lemma reversible measure temperature hasting acceptance use mean acceptance position via formula reversible chain variable metropolis hasting x repeatedly proposal p give conclusion horizon piecewise linear markov article refer weak markov sequence value differential motion conceptual clarity consequence diffusion sx martingale drift chain read x k notice martingale array increment rescale drift limit sequence rescale brownian sequence piecewise converge weakly diffusion globally lipschitz x process weakly motion priori priori state rescale process tend generator next exploit arise limit particular convergence together explicit idea appear literature article context markov drift martingale decomposition define converge weakly differential dt equation x du brownian equal unique value continuous process du quantity approximate rescale piecewise k continuous argument weakly cauchy employ weakly du weakly converge weakly verify process go condition exist p condition priori last conclude proof prove sde read weakly mapping end proof order lemma suffice verify satisfied quantitative drift x linearly principle prove principle let rescaled define equation weak covariance priori metropolis approximate property see introduce satisfies quadratic wiener variation like almost sure piecewise solve limit ode whose globally stable quantity quantify equilibrium expansion every definition quantity n possess possesse possess quadratic speak condition terminology behave quadratic number vector possesse surely possess vx let prove suffice surely hold borel every readily temperature discretization propose dynamic show vx finite time temperature mcmc variation let hold converge vx accepted position x happen indeed use borel exist define gain insight behaviour ode metropolis continuous piecewise interpolation process probability trajectory non equation assumption let start converge already indicate showing go prove lemma proceed accept move bernoulli variable acceptance converge compare show number accept use trajectory quadratic behave accept ingredient low acceptance piecewise vx represent accept function differential consequently theorem suffice accept note u k u vx vx k go supremum content conclude minimization functional note h competitive jx ds ds second functional unique euler lagrange minimizer sign different exist display blue minimizer minimizer whose ability evaluate gaussian emphasize lagrange develop lagrange solution dot maximum minimize recall finding precisely demonstrate densely define regard viewpoint although course measure together identify see basic give bridge law brownian bridge x brownian bridge differentiable algorithm implement define support embed locally set develop paper relevant version temperature time discretization move move accept space start global minimizer functional dimensional set distribution lebesgue proportional show log scale finite htb rescale full behave dotted concern concern detail dominate study require ability parameter behave control systematically construct dimensional metropolis proposal take infinite rwm rwm multiplicative component identical rwm infinite matter apply approximate hilbert choice diffusion sde decrease target contrast produce diffusion limit function mathematically last may limit dimensional hilbert diffusion sde limit variety two define hx zero prove give control quantity x corollary jensen prove suppose indeed x hx x p x integer lead martingale x let cauchy schwarz independent source randomness consequently x follow suffice x x give describe x sx fx consequently fx fx x cauchy schwarz fx sd fx proceed give give ball radius xt sake completeness equation differential xt consequently xt xu du xt proof algorithm divide find finite markov lipschitz lower bind finite stay xt stochastic prove finite universal thus imply prove mcmc proposal x least ball rr bind give bind accepted b acceptance k bind k hold every sx since trajectory mcmc stay bound bernoulli I k behaviour exploit x k prove instead control lemma author thank constructive grateful david frank helpful discussion concern behaviour discussion theorem grant dms grateful financial department statistics university thank department conditions mathematics institute uk department university cv uk hilbert gaussian
semantic pixel sift benchmark well maximum markov exact appearance pixel paper experimental similarly move beyond demonstrate incorporate information scene take scene patch tag model strong scene label image global capability model among image share pixel label bank detector demonstrate consistent crf part object capture label elegant hierarchical classic benchmark principle scene inter relationship scene ps undirecte model part connect specifie specify human pose specify plane part configuration follow specify unary potential part specify form potential express interpretation part vision firstly star object video context pose appearance enhance robustness localization appearance part mechanism np tree graph interpretation normalize function energy permit principled estimation structure significant limitation general may miss precise require rather strictly present g five part pixel define basic individual patch specify semantic car label maximum labeling element nonparametric comprise jointly also semantic pixel level description appearance part shape plausible significant passing long relation among support structure scene plausible upper typical g five dataset specific unary binary individual plausible explicit form plausible spirit sparse appear similarly relaxed full need indeed mixture section underlie part potential specify definition unary term capture appearance shape term mm model appearance space map filter bank filter follow optimize filter appearance particular specify foreground background histogram model use foreground histogram specify appearance measure fitness foreground vice denominator measure fitness class histogram numerator denominator matlab right shape mm capability shape pixel member centroid coordinate height part function frame otherwise store call part induce membership finally shape shape road class show object modeling labeling object location gaussian centroid centroid location covariance mahalanobis term ht angle pairwise connect part coefficient map pixel support centroid sophisticated plausible capture structure part evaluate parameterized distance ij angle von unimodal angle relate von direction concentration parameter variance jointly relate configuration essentially cast unary potential part seek define configuration respective connect general finally part potential mle contain current intuitive explanation accord role sample foreground background vice versa although clarity inside outside converge initialization assign ratio appearance mh annealing process posteriori sample add pl restrictive schedule sample challenge result unless resolve principled schedule manual basic desire acceptance shorthand consider move schedule principled manner manually tune begin estimate assume ratio needs manually acceptance acceptance impossible experimental sift flow sift gold manual percentage label gold standard split note split sift flow drop th different global n ij color intend legend mm mm quantify gain part quantitative comparison mrf case assume method basic mrf basic element make good structure reader assumption know quantitative among display three allow analysis aspect basic overall clearly outperform independent mrf per labeling mle dataset improvement building strong global increase exhibit modeling bring visual character sift intra dominant sift global bring car person rich category sift mrf description insufficient accommodate nearby phenomenon case quantitative graph shift crf crf nearly class label crf extension hierarchy field great labeling directly interaction rather object paper label labeling accuracy numerous careful literature assume important comparative furthermore unary potential simple color yield power respective perform crf immediately evident compare class mention one type road cat body tree water road show pixel object error group tend stable appearance shape uninformative object informative claim key allow emphasize global shape look comparative accuracy explain model follow infer location appearance class allow rich potential seminal firstly segmentation incorporate significant secondly plausible partition seem seek mode whereas know part plausible carefully mixture configuration mechanism return spirit highly team player set pixel sparse make scene labeling label rich remain tie pixel label common experimental restrictive benchmark sift flow solve assume need extension method jump plausible full logic two potential global pixel dependency principled big respective class incorporate grateful provide grant mind nf finding author plus minus plus minus plus ex plus ex minus plus ex ex minus ex plus plus ex em plus minus ex scene community visual importance scene vision vision limit global problem semantic labeling propose approach pixel structure overcome limitation part directly pixel support permit toward pixel modeling report first
characterize employ fine covariate fashion difficulty cell small hard big dominate partitioning allow prove logarithmic bound static mind improve remove encounter bandit previous sense reveal somewhat surprising price pay partial information pay obtain setup good round one arm observe arm information fall family oppose minimum detailed difference similarity two type side online side discretization employ worth cumulative paper define weak compare literature reward propose certain policy bound depend type receive suggest arm notable stream work covariate convenient notational elimination setup oppose ucb policy adaptive attain regret policy well reward arm improvement ucb policy static armed bandit see pt assume optimal analysis traditionally denote indicating observation strictly measured drive random bound high operate round decision time subset arm decide eliminate hypothesis significantly remain arm keep repeat make obviously quantitie policy arm prescribe horizon horizon perfect horizon horizon choose always lead horizon iy note exploration arm exactly time reward denote average arm convention essentially high deviation arm bandit pseudo presentation also complete time regret policy armed bandit horizon random random reward se exhibit otherwise contribute introduce quantity go accumulate suboptimal arm quantity accumulate accumulate arm induce eliminate happen choice define eliminate round ki arm contribute decompose union k k complement event side decompose follow side c hoeffding every yield right eliminate round arm I k bound I xx put display yield regret equation follow slight variation allow run respectively eliminate arm see page unclear elimination matter arm side dependent show unknown spirit small aforementioned ucb logarithmic previously gap gap nevertheless recover near bound much strong gap since hold reward necessarily independently none superiority arm expectation case bound dedicate nonparametric bandit denote yield throughout lebesgue respect reward expectation take place sequentially machine arm observation complete oracle arm large break pointwise oracle measure arbitrarily subsection describe natural regularity condition possible achieve sublinear policy nonparametric impose smoothness condition pointwise function every consequence smoothness old control weak closely employ terminology setup naive policy bandit machine describe partition initialize arm partition collection define decompose independent denote bin policy accord pseudo bt nm nb tn expect policy difficult parameter kk study suboptimal bound ucb policy running policy thus theorem indexing constant measurable integer appear margin proof collection nm j associate yield large smoothness bin bin indice eq well behave otherwise ill bin h old inclusion hold fact contribute w number weakly bin indeed yield behave part step bin condition sum define ix j f ji x ix ji ix c f f j j together j r nm ji j j generality gap order way ki ji ji j low generality indexing ji f j satisfy smoothness parameter together fact c j inequality yield j hold prescribe point specify bin g reader potentially factor appear upper generally expression light significant limitation surface arm bin regret least bin bound away arm could reduce overall bin logarithmic definition bin definition operate refined root tree child node bins form partition subtree construct partition leaves sense construct time set bin initialize initial covariate bin live replace child bb round intuition policy run birth arm average end round high smoothness eliminate satisfy suboptimal bin keep arm uniformly tn policy consider expect time q minimax sense low bound imply improve except round multiplicative say dominate imply exist bound arm operate static discard covariate theorem recover regret imply keep track constant might differ section newly create new initialize reward obtain successive remain round integer define parent q recursively bin incur define leave active policy arm remain leave bin adapt convention go treat incur live differently quantity decompose regret accumulate live accumulate live rely event b ix point great result hold hand also remain probability define occur play bin control note b interval treat arm event arm probability put together nb display together margin right side dominate leave bin proceed pt proof dominate lemma difference sequence moreover denote average yield integer martingale q argument p grant nsf grant dms dms armed bandit noisy reward realization depend observable armed bandit setting dynamically reward describe reward capture maximize introduce adaptively elimination suitably localize static construct partition use bandit optimal regret seminal extensively field computer science economics multi armed population arm point receive reward property devise reward population high regret population sample homogeneous identically policy regret small regret see
apparent two edge point point ji em em em em identical point pointing pt recursively remove point line obtain em em em em point line since remove point line matter remove pointing form em notice refer path notice since point notion minor inconsistent look path mix hold path introduce use base definition path would remove obtain point removal must turn arc inductive either adjacent line adjacent remove conclusion else adjacent connect path subset path separate induce replace criterion extension separation introduce separation graph line connect graph compositional compositional axiom connect connect contradiction connect short connect contradiction connect give contradiction close connect contradiction connect shortest contradict addition inner contradict connect suppose connect short symmetry enough non node contradict short contradiction connect type call path suppose contradict focus pair nod composition separation previously chain four markov graph discuss classify ii iii dual amp property iv property consist entirely arc multivariate induce separation typically interpretation graph loop however modification mc dag conditioning graph independence graph generate dag deal graph definition follow arc case em em I preserve say subgraphs illustrate straight simple cyclic establish graph identical sequence remove point full previous starting unless line another preserve cycle point arc still therefore reverse induction identical clearly illustrate markov equivalent connect path connect path give preserve edge connect path conversely suppose connect non instead thing remain preserve path point member preserve argument direction preserve edge instead connect ensure edge become absence essential induce induce independence model cox paper without bi edge direction preserve path internal let node take connect intersect connect follow let direction simplify graph acyclic mixed check primitive induce maximal identical pair adjacent reader establish maximal graph contradiction primitive induce node short primitive induce trivially node unless direction preserve turn identical contradict primitive induce markov model far also compositional result undirected graph independence model property separation lemma theorem pairwise maximal establish lemma compositional disjoint subset compositional six compositional fact six property marginal independence notation give sufficient condition connect graph graph suppose connect connect mutually exclusive b pointing pointing edge obvious connect deal two pointing connect connect connect node pointing path preserve kl connect pointing edge set compositional r compositional global markov pairwise compositional axiom composition far observe establish moreover property follow result induction trivial first I subgraph independence model c compositional pairwise adjacent property g l l mi cg connect node ij property inductive consider connect path contradict point symmetry pointing preserve path imply connect contradiction node lemma contradiction conclude symmetry compositional composition union obtain get base part inductive ci first cl l process lemma contradiction connect cp lp lp cyclic adjacent shortest connect contradiction path case symmetry c independence get follow independence composition axiom pairwise markov r markov six compositional axiom mention six axiom show composition compositional axiom intersection axiom imply axiom imply axiom imply compositional axiom imply axiom pairwise markov imply intersection violate undirected five axiom equivalence global statement associate compositional semi axiom markov I proof intersection conclude state independence model define compositional property w global property grateful comment version corollary theory graph mix separation compositional graph case acyclic graph graph pairwise independence compositional widely recent relation node random range simple mixed similarity motivate attempt mixed graph criterion cover graphical independence independence form exception mixed ensure reasoning indeed behave motivation define mc summary graph relation conditioning acyclic dag study graph graph obtain mc dag summary case pairwise latter edge compositional model ensure independence represent miss support intuition concept pairwise context independently several abstract satisfying axiom graph use separation condition equivalent prove compositional mention conjecture theory compositional mix mixed associate separation mix compositional section introduce class graph concept concept demand independence markov property compositional graph section notation triple consist relation edge distinct write say refer order every hence em edge neither loop edge induced edge edge walk repeat graph uniquely throughout node sequence describe path edge apparent belong say path j jk j general path path path may intersect path
informative dependence parameter orientation consider et al subject guarantee two inner fix use affinity closeness subspace maximal root perfectly factor ssc recover point random ambient fully detection ambient differently ssc achieve previously understand depend obeys assume point subspace exponentially result hold property bind course explain general shall discuss restrictive well difficult dimension regime concern many reflect comprehensive effect slightly version detection probability fraction hand probability point ratio early dimension grow linearly ambient note statement factor relatively assume unit apply sample detection ssc decompose solve expansion threshold make belong subspace dimension short expect make rigorous show outli iff remain outlier value together n ne nc numerical subspace uniformly de ne n de subspace outli detection scheme reliably number root ambient orientation point similar hold shall succeed notation subspace exclude hull introduce concept q point euclidean show dual direction dual arrange correspond denote radius euclidean incoherence subspace point property local solution column subspace incoherence subspace affinity incoherence imply cluster become spread distribution point direction difficult skewed toward blue special point huge subspace situation would successful sparse recognize set lasso subspace orient spread ssc geometric perspective concrete subspace model definition affinity subspace angle k orthogonality alternatively column cosine angle large affinity point subspace subject hence ignore square subspace assume simplicity subspace perfect occur ne notion affinity match sure close define affinity cluster less allow subspace intersect ssc still provably cluster discuss subspace small number subspace inherently reflect intuitively small increase probability modify hold probability ne similar manner presence claim make suppose outli threshold correctly constant semi detect q point outli threshold multiply ratio dimension increase get hold probability small need able inherently small prove believe conjecture support conjecture couple theoretical advance definition dim direct disjoint geodesic subspace dimension property long independent formally q detection singular rank sub appearance principal angle hand particularly paper introduce deterministic restrictive obeys r checking np slightly tractable assume subspace since column unit side strictly look entirely clear achievable exclude ease side subspace intersection dimension subspace intersect long fraction angle explain ssc disjoint simplicity subspace subspace see impose fully cn would put restriction comparison subspace ambient improvement come insight apparent ssc succeed inner product another zhang study effectiveness recover sort minimization simultaneous subspace minimize convex semi zhang subspace dimensional hand typical analogous outlier number combine therefore say hold general result zhang perfect recovery even noisy manuscript address outlier suggest outlier exceed possible gap correct subspace follow similar ssc il feature detection belong far choose neighbor term equal value cluster build ssc clustering take correctly otherwise eigenvalue ii check detection property hold detection vanish wish intersect end generate dimension uniformly among subspace denote inside another uniformly set feature detection uniformly three instance property ssc vanishing intersection show success ssc subspace upon point subspace hard affinity per decrease trade great experiment every point express point subspace basis angle linearly affinity affinity evaluate ssc criterion value affinity sufficiently large figure display proxy interesting cluster ssc generality achieve perfect subspace increase subspace random subspace different subspace normalize laplacian evident property subspace case gap always effect spectral ambient dimension point per sphere noise normalize noisy I value level correspond laplacian evident figure regime noiseless correspond evident decrease present subspace correctly advantage ssc ability broad circumstance subspace eigen gap discuss quickly review subspace start rank affinity interestingly also affinity perform perfect affinity approximately block diagonal empirically presence assumption violate affinity put subspace none violate ssc approach eigen broad circumstance demonstrate sample dimension ambient affinity ssc gap plot ssc subspace robust possibly add case methodology noiseless setup noiseless dd l turn subspace choose random outlier equal point plot correspond see gap appear solution correspond outli argue detection small work detect outli correctly consider able outli detection prove threshold perfect detection worth concentrate prove would ratio different value heavily concept exposition sphere norm eq cauchy schwarz deal appear page concentration follow lemma modification geometric face polytope face positive banach banach eq body concern volume compact I n n primal dual hold primal primal equal first variant support c identity solution identity optimal primal dual feasible subspace set check satisfied definition small ball consequence symmetric notice condition thereby conclude incoherence state choose assume numerical parameter modification subspace union establish independently furthermore dual distribute unit uniquely uniformly random justify orthogonal dual point problem know random sphere transformation prove apply union sphere unit positive eq assume begin map upper plug low step furthermore direction sample sphere know area spherical union pair theorem relate norm body variant x conclude relationship bb polytope theorem step use equality inverse one another eq apply identity establish uniformly random derive unit sphere column well sphere radius family norm shall plug volume sphere proof mean lemma eq pt lemma bind expect combine union part b part proof es thank discussion paper plan grateful comment stanford fellowship award fa grant collection assume near union many dimension develop name provably ssc ambient prove ssc data intersect develop ssc succeed corrupted outlier insight numerical demonstrate effectiveness fundamental step analysis reduction approximate low pca collection plane furthermore know approximate list unsupervise build representation input make predict input unsupervised union manifold furthermore well approximated handwritten handwritten character recognition simple transformation rotation shift character reasonable model insensitive change characterize invariance transformation digit approximate et al subspace problem visual corner million web visual development appearance motion segmentation multiple move approximately need move object hence subspace computer vision include image face propose researcher work comprehensive review reader reference therein disease kind specific factor test test construct row factor level contain cluster group suffer cause disease subspace together factor subspace cluster hope clear lie dimensional subspace class may surprising study science begin framework develop
nonnegative factorization inner polynomial separability meet segmentation additionally separability condition nmf unique separability hence work sense separability fairly practical context learn various retrieval david nonnegative r rest organize exact sf section prove sf nonnegative throughout factorization inner factorization must full first impose context scale column factorization interpret column introduction column topic nonnegative explanation document express column lemma hull matrix disjoint instead combination entirely disjoint nmf task retrieval even set extract useful would whose rank solve maximum rational semi value variable al rational approximation use fact column let matrix factorization iff basis factorization conversely factorization let basis row correspond matrix row let check condition become make polynomial full factorization factor matrix case algebraic reduction complicate goal cast nonnegative polynomially obstacle entry minimal span function transformation span transformation satisfy choice possible partition algorithm question constraint inequality constraint subsection minimal structural informally maximal corresponding column element order iff nonnegative matrix dimension minimal minimal respect respect nonnegative dimension proper choose basis use nonnegative repeat condition satisfie transformation recover column recover subset eq equality identical replace respectively remain technical polynomially many choice efficiently even two transformation column order definition minimal choice polynomially demonstrate restrict partitioning establish regard partition virtue arise chain partition set collection hyperplane chain row w iv I contain I j tw imply say important partition reduce partition hyperplane specify specify partition generate domain hyperplane polynomial time constant vc dimension hyperplane imply distinct partition fact result give upper bound efficiently structure checking result intend partition partition column disjoint separable partition hyperplane lie subspace give improvement remove condition algorithm encode hyperplane choice slight hyperplane separation define separation hyperplane mapping hyperplane hyperplane hyperplane independent extension hyperplane already contain perturb non eventually hyperplane contain remain agree define hyperplane nested apply obtain hyperplane correspond implicitly column contain large recursive add column contain choice initialize active run hyperplane partition hyperplane correctness algorithm follow partition find semi theorem nonnegative factorization cast nonnegative existence question algebraic nonnegative compute rational l nm r rational nm lemma partition partition decomposition row transformation span row span respectively long nonnegative transformation similarly recover nonnegative lastly factorization write constraint choice formulate constraint define matrix inner factorization call return al elimination bss bss bit find rational approximation quantify determine give polynomial exist set inequality particular quantify give polynomial also give number bit solution special time binary search find approximation note result sf structural special positive nonnegative rank give factorization solve time hypothesis hardness reduction unable reduction range goal e recently prove exponential fact exactly state definition distinct bit solve reduce intermediate intermediate point simplex point simplex reduction intermediate simplex versa preserve value immediate result e universe intermediate figure intermediate simplex reduction represent recall input inside dot except simplex contain triangle figure instance angle determine edge length denote coordinate place intermediate simplex get triangle thin triangle intersection triangle intersection triangle vertice triangle segment respect rotation symmetry situation illustrate triangle later specify line boundary two line extension thin difference intersection segment segment high line also inside since always distance large inside intersection edge intermediate simplex triangle possible hull angle sum symmetry shall case interested contain direction angle move intersection view vertex coincide case show figure observe take generate generality every triangle particular triangle outside get know intersection hyperplane intersection check vertex figure triangle contain also problem define construct number use dimension sum ensure box conv x origin simplex plane choose close constraint enforce large recall ab sure value constraint e gradually relax extra proof observe almost effect dimensional box simplex specify intermediate let possible still point reduction sum establish completeness reduction completeness straight triangle choose include choose equal point line contain line hull clearly contain next point convex hull triangle correspond three coordinate choose recover coordinate lastly hull point vertex combination partition triple set let contain contain hull two dimensional cone like plane plane act intersection origin affine know convex triangle know number abuse use section make number ab ce strict sake contradiction since place weight equal want use three dimension contribution convex weight imply enough simplex sum lemma intermediate must contain solution choose know know lemma one k value give evidence nonnegative consider allow give provably non trivial condition widely cite identify condition factorization database give assume one separability right usually separability factorization separable entry intuitive context column specify occur nonnegative coordinate thousand topic appear happen simplicity still normalize preserve factorization writing factorization norm eq unique nonzero separability nonnegative hull nonnegative separable inner dimension row convex row modify non coordinate coordinate nonnegative column let construction separable say column zero everywhere else operation end condition condition end output apply lemma loss separability appear row plain say row copy hull remain row iff equal vector index row convex hull conversely hull row separability hence convex hull determine exactly thus row nonnegative separable equal inner separability assume input add separable alternatively notice separability satisfy column column row entry satisfy condition unknown instead small hull remain unit condition separability separability generative model column distribution pick identifie seem suitably generic pick column vector tend satisfy robust property separable robust polynomial time nonnegative factorization row separability index whose nonzero entry column claim since convex row find robust upon ignore row convex hull linear claim robust robust claim show hull leave row row least close hull conversely least robust robust row cluster row w rather approximate unlike factorization assumption nonnegative rank well truncated decomposition e solve approximation note frequently outer product normalization main equality intuition behind decompose responsible little ensure nonnegative remove onto less singular vector singular triangle norm v ta w enumeration programming enumeration vector lie w find appear svd choice unit suitable multiplicative solution convex separate column denote close enough v r substituting claim argument still w solution value claim contribute second bind proof complete program candidate right find square w w lemma term bound choose nx rx xy randomize may ensure svd let inverse near q row column multiplication entry enumeration entry cause carefully deal keep row approximate candidate namely entry orthonormal schmidt orthonormal solve
apply edge setup path level efficient integration focus undirecte contain loop edge unweighted triple value generalize direct unweighted belong measure make notation side path naturally direct set eq restrict graph conclusion range topological computation short matrix choice topological measure quantity derive consideration length path vertex unweighted neighbor g function topological interest graph unweighted graph topological express function network density unweighted realization low take element follow equation write follow topological computation path integration equivalent computationally expensive update adjacency strategy need invoke algorithm level rank entry weight path successively add topological short return topological three stage firstly compute rank interest true tie randomization extract rank run rank order eq suffice pair recursively collect short finally integrated metric difficulty algorithm proceed addition edge exist short search around vertex conduct check neighbor secondly check short south black south anchor south south background rectangle every execute cell execute cell execute cell row style style row column style node anchor south matrix south west west anchor south anchor south black anchor south south anchor south anchor south text node style font execute execute empty style row column column style row column row style row row row row style anchor south anchor anchor west anchor south anchor south circle anchor circle anchor south black circle south edge time edge graph component conduct respect accordingly degree neighbor represent red panel respect blue first degree corresponding modification path inclusion short storing efficiency efficiency case code well efficiency search efficiency would coded short path thus combination interest code recursive shortest graph integrate topological metric respect generalize direct order pair vertex check one topological availability describe paper become fellowship uk centre health foundation trust college foundation valuable pt remark example recursive path application integration institute centre sciences research centre health south college institute send centre centre college se uk email send uk theory generally constitute weight topological density proportion graph topological short interest integration usually density short recursive short edge updating replace procedure first search code adjacency iterative among seminal systematic calculation various topological characteristic biology several
look give detailed online protocol datum base input forecaster time round reveal input forecaster choose linear environment forecaster dy protocol forecaster hold choose adversarial environment later regression random bold letter u u q endowed borel denote kullback integer equal thresholded natural regret forecaster advance trace gram analysis game forecaster bind study variant variant exponential weighting introduce stochastic tailor individual heavy heavy tailed point early heavy tailed sampling approximately one quite temperature scale associate tp satisfy jx case dt follow pac loss constant consequence theorem loss aforementione recall exp comment remark get inequality restrict infimum note therefore truncation prediction square previous define natural pac perspective crucially particular shape thus outline change inequality technical tool convenience reader last take translate namely symmetry term rewrite inequality corollary yield inequality approximate combination sparse appendix regret actually term small former empirical bound proposition assume forecaster gram matrix requirement prove sparsity bind initialization get purpose differ threshold temperature allow time forecaster idea forecast past regression set unbounded square ingredient perform truncation automatic previous several corollary derive access quantity forecaster batch analyse tuning tuning scale proof corollary forecaster previous proposition pac satisfy argument change spirit little comment replace prediction zero hand convex duality kullback cf recall sum replace let note concave next dedicated upper bound particular definition fact loss exp jensen take divide bound happen b concavity version precisely set property ty jensen concave elementary calculation summing put together pac inequality yield complement distinguish distinguish case x b ty b convexity line except instead restrict infimum cf get adaptation possible long gram forecaster trick repeatedly algorithm regret multiplicative round regime instance call round date begin regime particular past tune require begin game fully automatic possible viewpoint automatic extra viewpoint without repeatedly like vary might question price involve pac appear since change proof visit tuned instance regime summing result see upper uniformly forecaster proof stochastic design design prove risk logarithmic variance question sequel online dictionary regression forecaster copy whose estimate regression set pair I I surely measurable satisfies oracle risk begin game treat use estimator sequentially tune therefore depend knowledge tune bound extended technical base forecast ty enable base main deterministic jensen corollary jx satisfy expectation apply jensen proof amplitude later question next oracle lemma l almost subgaussian comment stress term avoid least achieve quantity respectively correspond classical design type aggregation rate automatic online bound unknown adapt sequentially bind risk bound batch ty tail recover still question heavy tailed output tune automatic purpose third variation note several assumption fx type assumption subgaussian subgaussian conditionally subgaussian factor avoid together q key particular subsection noise subgaussian upper slightly tight corollary twice oracle eq x get conditioning conclude bind oracle inequality notation straightforward jensen seem since case choose reasonable make free another simple analyse tuning assumption r q ball whole advance ask drive answer positive still quantity risk correspond bound infimum risk however contrary thank drive open deal prior inverse practice author ask answer batch forecaster almost small mean fx like consequence sequence tx tx ty tf tx deterministic theorem main setting jensen appendix design almost subgaussian variance assumption hold hold assumption remove idea appendix algorithm require factor la european provide state corollary need comment begin regime past past ambiguity regime convenience bind empirical gram fact apply period infimum get xy yx sum substitute inequality note conclude view check extended remark elementary holds non corollary pair definition figure remain expectation therefore concavity infimum side note right conclude definition constant individual risk appear front equal would previous vanish assumption appear avoid multiplicative factor prediction interval change see sf corollary loose fulfil assumption replace without elementary bind assumption thus sketch main argument sequel expand square tf might comment design case divide exactly state distinct convexity square definition jensen take expectation conclude exactly instead several inequality kullback leibler notation measurable set expectation bayesian reproduce case translate probability linearity integral sum side bound indeed expand equality prove u concave q conclude next corollary theorem respectively center real constant eq random constant subgaussian q entail least remain precise get line rearrange conclude proof definition associate show convex xx base type maximal xx last jensen inequality convex exponential moment conclude jensen moment sup sup paris france ambient round sparsity regret weight drive truncation version algorithm solve question open bound adaptive gaussian sparsity individual sparsity extensively stochastic decade dna netflix amazon hyperspectral high dimensional cross country message impossible statistically feasible situation focus many practical work decade bound express oracle fact guarantee consequence statistical possibility scenario deterministic namely regression sequence newly deterministic prove online sophisticated automatic preliminary parameter apply deterministic nature thank imply unknown noise logarithmic open introduce main motivate online viewpoint main respect statistical literature machine arbitrary deterministic forecaster output assess goal almost forecaster sequence form small sublinear sake omit set version ridge forecaster
limit statistic node kde computation begin reference compute far moment reference use node moment node compute dm leaf r rr r show include rp r determine query rr dr core dual routine computing kde g l r tree recursion structure kde query current tolerance reference approximation continue fine terminate consider avoid exhaustive leaf achieve accuracy computational cost query change reference regard locally give query entire subtree reflect change immediate child query fine g g rv computation query pair call exactly refine hence query pass node reference value sum note add onto query sum refine bind g finally post level pass child scalar point post routine g r low upper maintain properly call maintain properly main call point q incorporate passed contribution un visit call query consideration change two correctness prove call leaf subproblem hypothesis call maintain low child r argument query incorporate pass change correctly among leaf call maintain low upper hypothesis contain bound query account either exhaustive exhaustive incorporate node belong recursion global simplicity limit available exhaustive denote centroid leaf node whose contribution centroid set sum r b r ir snapshot low sum subsequently triangle four available method p naive naive c x x naive naive e e position color dimensional dataset angle dataset last scalability kde guarantee criterion table symbol tolerance common validation conceptually visualize store conceptually dimension iterate scalar give scalar bottom storing ensures meet time global include take recent automatic tuning tree method automatically parameter fast base fast combine kde gauss gauss gauss expansion computational implement implement inherently query moment similarly node moment allocate array imply mapping digit number position array base p position linear index dd index position basically convert representation multi expansion basically form single reference x p pm p basically compute multi see implement r c p implement far operator doubly accumulate implement r compute h univariate function ad e k df evaluate expansion implement structure outer loop iterate far query dot far computed figure evaluate far reference term p pf w r c q implement translation doubly loop doubly translate accumulate moment term translate local implement j pf h c p implement accumulation basic doubly nested loop accumulate inner position implement equation r l p implement translation reader local accumulation doubly moment apply implement j p evaluate outer loop moment among reference local accumulate query cp p pz q theorem thm kde achieve optimality computation kernel summation algorithm transform derive additional algorithm utilize hierarchical demonstrate truly gauss second within dual compute user kernel density bind wide procedure density kde point choice spherical density th th reference assumption true mild estimate kde need find initially dataset validation bandwidth query kullback subscript point maximize bandwidth sense cross square yield score gaussian kernel base force make reference practitioner apply commonly evaluate sum efficiently kernel expensive sum many cross kernel algorithm develop expense precision consider criterion measure error absolute value I error bind bound hard initially quantity many focused bound relative criterion achieve specify tolerance build upon relative barrier evaluate grid flat briefly strength gauss method rigorous bind bandwidth bind widely contexts sum grow dimensionality cause bottleneck center another exponential box empty gauss similar utilize flat advance point group proximity whose propose expansion translation expansion algorithm multiple offer user fully absolute still inefficient section b neighboring sum use discusse kde extend handle structure specify raw compute coordinate dimension discuss type rule recommend neighbor implementation recommend I h reduce fouri dataset zero grid count matrix two key ingredient fourier transform convolution count matrix corner grid point inside c c l k l calculation point boundary interpolation grid provide moreover quantify incur kde discrete algorithmic summation algorithm tree consider query point recursive structure reference variant centroid fast kernel summation cross optimal bandwidth optimally evaluate demonstrate dual bandwidth large improvement dual first summation gauss transform derive extend demonstrate truly hierarchical gauss tree gauss transform integrate expansion approximation compute algorithm world dataset bandwidth selection guarantee relative offer fast range cross validation build tree fast transform add thorough comparison general computing sum extension dual provide kernel query computation point construct index component enable tree structure computation single flat clustering collection collection tree hierarchical location recursive r b b x x procedure hyper n u center split two coordinate continue point threshold compute bounding involve kde computing summation formalize query tree traversal demonstrate reference tree compare reference box partition record bound highlight recursion upper distance reference tree r compute summarize contribution node query query query traversal thereby exhaustive computation univariate crucial mechanism gauss integer time continuously field expansion query four term r c involve array query array expansion kernel separate reference centroid far field region part sum approximate moment node term dp sum query centroid reference query point location expansion sum representative query denote disjoint function remain reference point location moment accumulate store within query coefficient generate third q involve take product array coefficient dimension far field window function whose reference main summation pre compute evaluate point dot vector require far operation compute show local accumulation operation moment see centroid translation call local translation operator far lemma truncate far centroid term fr p taylor centroid c proof must moment accumulate c p show operation bound choose order section wrong correction evaluate truncate evaluate truncate truncate error reference evaluate far reference reference r r n c expand expansion function x intuitively evaluate field reference form bandwidth centroid node inside hypercube truncate local centroid query accounting contribution reference l taylor r respectively c achieve multivariate bind lastly evaluate truncate form truncate require node local expansion convert already centroid eq expansion centroid fr l p r n q p geometric propose use node constraint large possibly derive incorrect derivation determine need far answer bind nevertheless lemma question maximum term achieve within tree partition show field expansion error allocate far field reference algorithm ratio length twice absolute error side determine expansion form reference query node twice determine error evaluate determine form reference p necessary
obtain proof bind set note write derive simply impose prove c ignore fail begin ir follow l low c identically choose variable conditional lyapunov cumulative big explain calculation calculation point variable obvious subtree uniformly neighbor local dimensional tt child eigenvalue expression variable inside expectation whenever recurrence hard child expression denote measure restrict try since addition expand power decay interesting condition hence consequently ss hard tell ss nontrivial notice solution expression expand expand connection remove write regular expand define write law second introduce mind recall expand expression inside take inside simplify finally first notice eigenvalue immediate c family already discuss explicitly incoherence ss proof class succeed reconstruct show fail reconstruct dimensional everything use expansion power one expansion priori first notice depend fix independent zero eigenvalue reconstruct enough note differ require behave numerical function expression along understand behavior concentration incoherence turn tell incoherence violate enough question success minima due symmetry pl path constant high axis graph solution yield reconstruction exhibit make curve path eps plot separate include show tend value high finite plot positive region curve reconstruction also unless fail vanish identically scale allow converge require make fail scenario graph surprisingly graph correlation big neighboring notice fact consequently simple use prove large neighboring great non neighboring node algorithm operation range prove g node great expression calculation concavity easy furthermore condition assume case get inside plus pt lemma definition conjecture fact structure ise field concrete low often fail field range correlation phenomenon appear apart mechanic ise seminal find numerous computer distribution dependent interest introduce stress follow study structural qualitative encode identifiable unbounded resource class operation sample general emphasize definition alternative reconstruct correct consider definition result qualitatively spectrum general understand trade devote tradeoff structural ise fact independent instead conditionally dependent particularly large connected discussing get toy illustration learning study figure family vertex interact directly interact numerous conditionally since fix choose ise distinguish unbounded covariance statistic infer assume graph distinguish covariance match x ix remain involve subset number spirit achievable even marginal confirm need ise distribution formulae word although let match eqs correction ambiguity arise weak indirect graph phenomenon correlation connection reason edge correlation graph bound degree system distinguish graph former vertex trivial configuration counter away regularize regularize regression rest paper bounding pseudo method defer general pattern fail toy demonstrate trade via maximum interaction strong answer gibb predict behavior uniqueness gibbs far roughly independent vice versa family far apart dependent polynomial exist arguably hard uniqueness phase dependent compare analyze mention provably fail raise question structural provable dependent question overview finally paper hard learn logistic fail regular specific family indeed mostly analytically tractable try likelihood ascent gradient log likelihood require ise carlo reason first output approximation zero overcome suitably regularize never value use mcmc fundamental limitation yield polynomial sampling apply dependent case develop computational provably maximum field degree present limitation author exploit view consist regression ise model vertex appropriate problem variable logistic structure extend early notice short present neural system two explore challenge forward average structural decay generalize analysis adaptive family mathematical strength simple thresholding thresholding correlation compute choice numerical bounded degree fail result confirm idea complexity correlation apart decay exponentially learn happen family range characterize advanced behavior algorithm strength challenging degree exploit conditional independence encode consider fix whose want reconstruct rx neighborhood assume produce condition remain value produce conditional subset vertex motivate select denote minima maxima contribute follow consequence analysis omit degree exist constant first implie reconstruct impractical restrict decrease rapidly graph distance mention assume small enough constant exist take consist likelihood fluctuation select often regularize likelihood v r x p j regularize logistic calculate numerical degree exponentially hold success write converse make say probability neighbor isolate sequence successful particularly encodes let vanish order asymptotic converse whose deferred section degree regression fail regularize logistic regression indeed fact describe fail enough fail reasonable dimensional fail high reasonable call necessary successfully similar lasso necessary model ise model intuition behind quite solution quadratic center add regularization analogous lasso plus error control dominate contribution lead incoherence expectation ising check graph contribution consist gap mechanic ise grid weak ising explore relevance result carry use logistic among algorithm balance ise gibbs change temperature take day processor tree weakly couple regime case p report success subgraph obtain independently probability average scale satisfactory clearly illustrate phenomenon page despite threshold irrespective indistinguishable plot versus figure random probability sufficient reconstruct analytically compare proceed convenient notation make form whose lie submatrix index whose try reconstruct hereafter shorthand subgradient e variable distribute accord convenient hessian limit denote matrix argument clear quantity evaluate clear subscript introduce copy recover exactly throughout expectation connect least hoeffding ji ei possibility impose right hoeffde expectation derive bind avoid walk self avoid walk imply theorem prove degree vertex add extra node converge exist fails obviously choose use leaf node show complete failure first establish small ss ss min min c reasonable sense unbounded choose sequence star node edge central increase star condition ss violate rely convergence prove regular degree exist positive exist proof lemma enough c q v generality eq exposition even differ exposition incoherence bound eq show stationarity r n ss notice q recall absolute value index relation know hoeffding yield conclude ss min min n c ss theorem expression ss I subgradient take ss iw straightforward computation bound bound detail result mention lemma regular root associate ise leave unique prove expectation ball whose neighborhood rt pp node expectation measure expectation g rt consist
real self configuration still difficult get big solution prove rich enough especially stream environment science cm cm configuration http www gm de contribution self machine train data experience come exist achieve tuning detection pattern self tuning advance construction boolean forest increase current barrier discuss systematic partly solution potentially new human adapt rapidly situation environment generalization environment work case game mining game ideal self system world complex like mainly formulate immediate goal playing well learn rather program expert opponent discover reaction rl solve feature remarkable td show alternative evolution follow many contribution start case rl games reinforcement toy wrong one interesting tuple recently game remarkable g uci special robust forest larger careful detect aim numerous especially area stream mine recent development parallel feature bring optimal use feature systematic model development rely test confidence finding high input look configuration self w teacher look n tuple sec configuration mining dm discuss desirable flexible dm feature dm finding research question remarkable example game learn behaviour play master td reinforcement complete solution dramatically simple piece piece winner pick right detect leave opponent formulation trivial successful td learn raw human index white black hypothesis tuple similar trick relate dimensional indexing input situation indexing perceptron system architecture original perceptron unit form random input strength problem perceptron tuple perceptron signal concept vast amount capability near future machine considerable progress decade forest robust support number build implicitly knowledge complicated usually build c tune model preprocesse able want system mining mining streaming underlie configuration calibration daily routine adjustment configuration although well quickly get become curse contain boolean integer modeling often consume adjusting run budget red line training among competition clearly high tune column high winner rf build help krige optimization run es bfgs dm result novel ultimately pattern find numerous infinitely way input fourier transform transform gp genetic pca assume linearity assume linearity dim form random one belong general principle good could successfully htbp gauss alone rf evidence work require module
ask person world nuclear anomaly change detection frequency alarm anomaly method point detection change detection anomaly frequency detect job job keyword early thus algorithm four collect twitter wide spread job quick propagation news conference response promise detection approach conduct framework planning scale handle stream content boost microsoft core topic interest rapid text video dynamically new anomaly sequentially aggregate anomaly relationship network demonstrate twitter mention approach detect ill anomaly discount facebook daily life since text video datum text contain may rarely may friend time receive every mention sense mention like number interested user topic like discuss friend conventional approach concern ambiguity cause may preprocesse apply information form unique little available social first nothing many friend friend twitter friend topic something new show link user user occur anomaly use reflect behaviour anomaly obtain apply technique detect topic effectiveness four set twitter good show propose link anomaly topic frequency mention detection tracking extensively task one topic none topic text mining factorial research concern seminal firing inspire change momentum topic social content content utilize citation citation analyze lie focus social content document flow service user mention assign anomaly post feed analysis stream contains mention formally mention training compute predictive assume beta integrate beta integral numerator beta predictive far follow total number derive predictive likelihood set handle appear anomalous instead chinese crp crp addition keep probability accordingly compute anomaly post past trend anomaly obtain discretization time include aggregated anomaly point statistical time series monitor new datum employ layer length autoregressive ar scoring length know optimal code employ point detection procedure outline follow anomaly aggregate sequentially learn smoothed sequentially appendix change density alarm score exceed threshold histogram probability exceed positive histogram n th update histogram threshold alarm collect service collaborative service belong detect recognize people set job organize list list participant change occurrence frequency topic manually topic describe sparsity seem bad method experiment detection anomaly anomaly fire rate alarm drawback relate must advance always think detect datum participant job person student anomaly detection frequency
cpu program know consequence requirement gpu device computational capacity theoretically certain usually apply perform realistic subset compute statistic independently distribute sufficiently computable typically compute exceed value value think extremely pass test say test fail whole test perform small large present thorough test quality produce early variant suggest particular design carlo mind generator algorithm distinct generator request bit bit popularity element sequence express parallelism exploit algorithm hx hx last produce yet sequence produce careful specific generator sequence recurrence direct quality distribution library framework library default generator library generators generators generators generators still pseudo integer right shift exclusive shift architecture typically operation division generator retain paper generator period period software package generator advantage period memory requirement instead subject criterion design give good linearity generator comment proper generator suitable parallel family comment combination step perform linearity generator shift class generator simple odd recommend odd integer generator generator shift ensures omit periodic period addition mod linear operation operation allow pass fail increase factor increase extend nontrivial consideration parallelism represent recurrence consequently count generator circular operator circular buffer reveal structure circular buffer denote access circular buffer element replace element calculate parallel examine flow within buffer produce sa sa sa sa ib observe maximum inherent provide versus define generator parallelism class generators generator consider approach create sequence number independently block architecture technique logical subsequence mp parallelism thus space generator period advantage make dynamically allocate generator whose exist gpu implementation gpu device present table generator state period generator small requirement generator great period short next throughput device generator generator generator fast old architecture order generator fast generator design current initially event speed platform dependent compare generator library benchmark fail benchmark fail test generator design fail interestingly fail two test generator none none none none none test consideration generator fail test combine generator generator pass generator test approximately probable relate consecutive seed value block within avoid unclear take parameter set value explore overhead increase memory generator consequently reduce generator conclusion show perform comparable speed require space device frank com generators graphic generator rapidly pseudo gpu statistical demonstrating generation graphic processing monte interest carlo mcmc mcmc numerous physical demand large random quality acceleration carlo gpu component bottleneck gpu accelerate gpu statistical
potential unlike generate return unnormalized computation likelihood many conduct inference stream nevertheless hard parallelization attractive part exploit advantage parallel hierarchical markov chain monte variant importance datum parallel issue demonstrate non mcmc direct argue need like chain autocorrelation generate independent absence parallel chain acceptance ds rejection common demonstrate improvement resource applicability failure separate prior another moderately sized et high conjugacy method speak share ds direct focus alone approximation ds proposal draw proposal ideally improvement improvement need concerned estimation advantage maintain conjugacy common generate target entirely parallel placing core permit likelihood start mcmc conclude discuss issue implement limitation like ds unnormalized let mode proposal obviously inequality hold negligible little uniformly yu marginal simulate marginal du write equation repeatedly et proposal construct continuous bernstein polynomial poor extremely large tackle strategy sample variable denote cdf draw proposal proposal draw density proportional segment partition probability fall multinomial proportional draw standard first random sign expression sign find unnormalized log compute proposal draw draw repeat draw place proposal draw draw multinomial variate draw single draw choose naturally efficient principle spirit dominating implement metropolis hasting multiply nothing trial concept tuning happen mode know importantly analysis collect investigate take advantage et notable al evaluation resource execute linear markovian dependent draw technology generate require hand parallel dimensionality motivate latent prior mean robustness outlier tail one joint distribution give contour around uncorrelated tail extremely poorly indeed note diagnostic chain chain attract modal uncorrelated chain tail chain posterior mode proposal draw area high pick correct shape acceptance collect draw collect mcmc mala constant hessian iteration start draw predict jump ever gap move tail consider covariate intercept coefficient gaussian prior weakly simulated simulated intercept value case population parameter summarize replication sample multivariate normal mode hessian multiply roughly average proposal collect accept reject phase take posterior mode start transform line take acc take execute accept reject collect sample total require algorithm conduct mac cpu core ghz ram core allocate h proposal proposal acc assess examine estimation draw generate core responsible collect parallel however one use chain independently arbitrary rejection burn reach also worth implement reject begin comparison adjust langevin drift choose commonly hard mcmc general exploit use posterior start million iteration several diagnostic density decide burn minute acceptance rate collect adaptation proposal also sparse hessian block grow turn ability computation researcher popular newton acceptance method sample pseudo propose dominate substantial effort likelihood dataset mcmc marginal accept express expect draw rearrange treat proposal draw shift geometric acceptance count draw remain integral proposal draw also convention put estimate follow accuracy eq q multivariate compare conduct covariate intercept linearly spaced collect density number draw factor precision density case exclude proposal proposal draw present along harmonic percentage take draw mode summarize table likelihood remarkably robust factor appear approximation negligible improvement note comparable harmonic compare method one fall density importance run mcmc collect illustration advantage scale sd mean sd sd time sd sd discuss implement entire mcmc continue insight gain experience provide suggestion mention investigation search inference example standard package find mode estimate difficult tool however appropriate immediately mode barrier adopt ill problem density unimodal one newton trust region instance algorithms search method subject nearly flat method stable trust short find optimizer posterior neither trust gradient find approximation newton ad ad write compute ad library small take otherwise estimate mode covariance software hessian work expensive heterogeneous unit hessian diagonal dense right margin substantial storing compressed format zero estimate computer computational exploit sparse offer implementation library exploit matlab store consideration multimodal posterior mode discuss unimodal instead normal global well mode unchanged long global proposal guarantee optimization token statistical statistical language literature facilitate efficient multiple mcmc guarantee distribution high happen general algorithm offer distribution selection proposal multivariate center mode hessian think proposal scale gradually experience rate still collect draw remain time try
low preserve geometry project far utilize due standard next top implementation develop speed scale consist uniformly replacement row independently sample without form compute spectral reconstruction submatrix return indeed require factor mf assume projection step generate compute parallel multiplication parallel execute generalize perform repeat computation ensemble improve approximation straightforwardly leverage parallelism modern error parallel time project onto rather project onto onto choose random submatrix base mf submatrix generalize parallel average highlight empirical offer great benefit unite accurate expensive completion robust factorization admit poor repeat costly sec attractive scalability thorough estimation section give rise mf formulation guarantee present intuition proof entry outlier advance informally information entry correlate limit extent singular correlated coherence miss notion let standard define coherence contain coherence coherence coherence eq definition call sec coherence well property capture entry entry mc mc solve bind mc guarantee thm noiseless incoherence coherent prototype notably preserve degradation require e whenever maintain coherent entry suffice nuclear step base mf moderately coherent moderately allow base coherence present lem introduce draw upon randomized spread responsible accurate fidelity reconstruction projection guarantee projection rise master coherence thm error error step master estimation guarantee subproblem new establish mc high present present well consequence mc sec present coherence mc mc coherent eq replacement satisfy probability positive assumption reformulate parameter error uniformly distribute noisy incoherence problem f f provide rate parameter coherence mc make precise next algorithm proof build master return either row cl coherent proportion quantity always simulation follow result utilize incoherent fix large subproblem coherence submatrix demonstrate noisy consequence thm solver operate much mc algorithm solve follow coherence master incoherence coherent uniformly fix parameter base choose achieve f f fraction reveal column whenever understand conclusion base algorithm thm satisfy scale scale key control success noisy noiseless setting missing corollary guarantee solver base convex master theorem thm incoherence moreover q fix rate rescale suffice tc place mild grow comparable quickly exact noiseless base uniform mc exponential q entry accurate imply precise zero thm base sec defer sec master guarantee I small high base subproblem submatrix use control subproblem demonstrate thm estimation noisy thm probability suppose mc solve master theorem base mc constant observe algorithm solve suffice decrease entry sample whenever understand base apply f high exhibit scale exhibit plus non matrix incoherent matrix sec variety simulate world accelerate algorithm mc algorithm base report ghz core gb memory default suggest error square comparison execute subproblem since subproblem execute plus compare two base carry focus decomposition similar scheme create matrix entry uniformly support generate take average ten noisy eps eps figure eps vary percentage outlier gap perform figure figure slightly match performance observe optimal frobenius match omit explore speed outlier row mc nearly identical result present mc curve overlap fact highlight minimal cost mc eps eps recommender collaborative interpret rating infer rating publicly collaborative filter netflix available ensure remain split notably outperform time present well slightly outperform problem mf inherently easy estimate netflix rmse rmse cm cm background model practical activity surveillance background object background change illumination evaluate video foreground variation include illumination frame video pixel value reduce video rmse run time eps eps f eps eps frame sample sample scalability factorization leverage compute architecture divide trivially suit environment low provably maintain super suggest first complexity guarantee master alternative theoretical practical benefit open ground incoherence replacement thm require randomized proof thm rely heavily accord distribution slack probability relate incorrectly statement matrix multiply transpose case matrix tolerance failure satisfy trial th column whenever th lem probabilitie binary diagonal rescaling replacement lemma lem given value lem lem immediately yield sample thm lem lem typical prop involve sample slack term allow guarantee projection sufficiently incoherent probability desire incoherence eq immediately lemma claim assume consist first row write fourth full define old norm norm coherence claim old norm definition coherence claim column projection incoherence b since prop yield minimize minimize must incoherence notice thm q noiseless admit note recovery appeal factorize decomposition approximation low randomly rank submatrix iff treatment hence lead vector lem rhs rhs bounds statement independence prop prove statement proof gaussian eq fr coherence suffice bind hold lem probability union projection result result block submatrix lem begin event event coherent master thm whenever holds desired reveal q uniformly cardinality variable eq q hence ip master coherence master thm aa moreover eq identical yield begin prove let eq coherent coherence master theorem probability least hold hence definition number satisfy bernstein n n assumption fact n ip identical master thm coherence cl q spirit even noiseless reasoning replacement appeal svd complement f sufficient condition suppose write f hence orthonormal old assumption high four lemma establish entry provide state uniformly multiple eq least infinity q assumption entry replacement admit identical replacement note bernstein inequality replacement lem construct consider replacement set consist location sample replacement replacement show location event location sample replacement batch replacement batch existence sample replacement conditionally replacement satisfy second condition imply first second projection final exploit coherence conclude lem lem fail lem fail replacement hoeffding sampling desire projection row argument conclusion column row replacement proof two key relate multiplication thm column second probability lem failure matrix lem master whenever suffice lem event probability establish fp mc f l nc n master guarantee event event entry variable distribution hoeffde yield bind follow identical manner master engineering award stanford edu stanford statistics stanford cs berkeley electrical engineering computer berkeley berkeley california department electrical engineering science berkeley author
penalty incorporate singular specify bic joint pattern association explore association gene gene correlation figure show gene implementation gene association additional score gene nonzero entry closely panel sign distinguish suggest sample driving appear gene square target load absolute loading load colored red loading color proportional loading display component component neural panel indicate second capture association apparent correlation alone association capture joint biological gene display gene interaction construct large loading gene link predict expression prediction module database current target inexact gene predict interaction far contribute joint gene joint expression frequently report tumor cell facilitate cell link survival loading study number set consist multiple type relatively general integrated provide powerful activity type account individual versa tumor provide tumor interaction point associate estimate outlier exploratory suffer see interesting computing account structure property bootstrappe regard however factor must carefully focus general integrated finance improve within market study acknowledgment wish thank constructive especially associate concern existence uniqueness decomposition rank algorithm discussion property simulate set grant nsf datum several genomic tumor paper joint integrate rank approximation capture variation structure type noise quantify variation type dimensionality exploration individual popular canonical gene expression tumor association provide tumor type software analyze measure set increasingly include diverse set distinct mode computational protein abundance spectra atomic sciences temperature traffic link page motivation particular application biological study number diverse amount available expand collection publicly database molecular cell biology genome collect scale project cancer genome type establish separately measure individual analysis critical association relationship type unique statistical association source inference research collaborative national cancer institute national genome institute characterize molecular multidimensional genomic integration information genomic comprehensive understanding focus tumor sample brain tumor systematic tumor classify neural expression copy addition clinical difference across copy gene recognize important aspect biology biology analysis gene biological biological suggest function primarily target gene list relation research partly responsible known tumor gene investigate individually important relation interaction gene expression joint variation gene expression lead tumor publicly use gene expression share shared vice versa may biological many individual signal joint exploratory capture joint type identify potentially type account allow illustrate size algebraic identify sample gene structure complex share subset identify gene structure variability joint relevant biology blue correspond similar pattern green object row matrix combine often baseline helpful subtracting differ variability total frobenius contribute rank structure matrix represent joint unify matrix independent model impose furthermore row joint structure responsible responsible individual constrain orthogonality structure orthogonality joint uniquely supplementary material remarkable orthogonality constraint rank rank important accurately quantify amount supplementary describe selection minimize error account joint structure minimize little present red yield contain loading structure load loading score individual factorize summarize explain loading score row summarize pattern type loading sample score structure reveal expression principal structure individual two gene present effect visually apparent color initial consider appearance surprising apparent joint plot interesting apparent suggest biological remarkable variation joint standardize joint structure permutation conclude four distinguished structure gene represent type play great role biology think unsupervise integrated analysis investigate component survival direct identify distinguished association association alone identify global mode association type perform popular examine set first canonical loading weight vector maximize geometrically interpret pair canonical loading subsequent find enforce direction set cca overfitte pls similarly cca covariance pls data examine variation vice drastically pls call pls seek variation linearly vice versa pls find explore common standardized loading extension pls develop mf pca group functional spirit mf decomposition group variability type group mf among main level variation return introduce job find joint weak association principal component response pca variation figure analysis pls pls score joint response panel association indicate pls individual structure structure cca score panel f individual tendency color within
situation become human poor temporal clearly indicate largely build ahead become considerable improvement moderate time incorporate open include become costly branching ahead become consume ahead ahead cognitive aspect sense mind sometimes understand lack branching become reward circumstance advantageous event formalism next subproblem rl subproblem solve factored support robust dynamic description state module choice module first module evolution optimize complexity serve factor markovian large rl problem polynomially optimize macro ii markovian detect measure state macro apply macro use follow ms temporal reward table ahead look ahead estimation immediate reward discount macro behavioral look ahead since look ahead change memory work need solid promising ahead factored scale property extraction le motivate hierarchical spatio appeal individual offer selective globally optimize method give rise I behavioral evolutionary pre learn td optimize rule td macro state decision surface factor look ahead predict behavior control learn inverse dynamic improve control solve agent model lack knowledge agent enter especially factor machine description mdps laplace approximation different tractable problem reinforcement extraction close concept transition control mdps spatio temporal modular state description markovian achieve markovian factored look ahead argue factor introduce ahead factored look ahead markovian description scale cognitive working memory evolutionary try separate aspect give property individual ahead markovian macro state td description collaborative planning considerably thank help european financial project grant agreement www home page reinforcement inefficient markovian environment policy investigate propose architecture utilize combinatorial test behavioral run deterministic factored finite learn illustrate property deterministic ms view decision insufficient old yet still markovian markov process mdps solid whether description whether state description relevant rl obtain raise whether compression world unless mechanic inform laplace activity necessity markovian state optimization mdp formalism know within turn time description markovian world markovian change agent restrict consideration approximate finite state factor practical factored mdp tractable relatively description question certain checking property increasing occur consider expect discount temporal markovian direct policy markovian state ahead might intelligence demonstrate good state quickly overcome poor value estimation state improve represent learn agent save evolution use may lead markovian behavior actual world start behavioral information td td develop sensor surface improve agent overcome estimation markovian utilize ahead method deterministic ahead poor estimation follow review ms pac illustrate concept fig ms pac proper read available enable characterization prove favorable factored rl aside problem elaborate conclude decade reinforcement large include review subject g reference therein agent action rely mdp formalism markov action state assign policy value value discount discount denote large value ps practical utilize rl know comprise improvement may short temporal td td matrix regard rise state error r ts td ts learn rl g description non markovian large size markovian property extent rely representation maintain episode search description optimization action combination act simultaneously enable representation discrete fitness fitness particular goal correspond fitness maintain family parametric solution subsequent mark n adaptively high specifie item determine actual sample fitness sort way update kullback leibler divergence decrease selective resemble genetic online treat factor tie supervise control rl create convergent reference therein use correspond individual case factor process factor probability require exponentially exponential often simple sampling polynomially exponentially large space stay convergent tie factor concept action module I complex state whereas activate module state two enable module macro combinatorial flexibility factor description behavioral upon optimization dynamical rl planning estimation example new dynamic sum upon exploration rl factor rl planning pac game later restrict transition easily describe ms pac walk dot since machine approximate risk factor ms cost sake ahead planning look game ms therein ahead rl learn face method task feature available logic algorithmic component reinforcement possess action extraction property use pac illustration player pac pac particular dot worth dot try pac life initially extra life reach corner pac power dot turn blue short period try pac pac worth player would four dot worth study achievable original ii accomplish entropy combination rise frequent building powerful ms life dot follow consume group simultaneously behavioral pac list value behavioral e td code description nd digit digit code apply ms action behavioral effective average optimization action macro behavioral upon average description behavioral pre selective evolution pre rule improve performance thought macro formation macro sequence mean abstraction basically behavioral pac agent turn power dot select rule low away rule achieve performance give close look ms dot chance may simply selection policy rise proper macro behavioral pattern module optimization episode optimize far macro macro compute td histogram predictor decision surface td new alternatively ahead procedure state information introduce within option state via rise behavior increase collect ms pac satisfy greedy replacement new however markov show improve change optimize poor average slow overall drop even optimization estimate column td fig exclude frequent td behavioral fig reward pac dots peak broad dot third fourth peak pac four would ideal ms pac avoid td collect one build purpose collect forecast define rl starting collect extraction solve rl execute categorical therein factor come back search factored look ahead argue decision make decision surface list encode surface lose world human recognition utilize category categorical example base change decision surface version useful easy component environment furthermore component surface even artificial pac slight actual type big sensitivity change component high dimensional easy categorical categorical appear mostly low example color example decision surface ms long learn ms game dot coordinate ms dot behavior ms pac randomness deterministic observation pac behavior sometimes deterministic factored branching potential world different speed easy observe branch ahead la commonly e ahead collect approximate closely relatively close end exhibit ahead model branching reduce heuristic branch ahead ms pac ms iii ms pac stop turn graph simple tree backtrack possible branching factor
optimize scenario real statistically background discovery physics wrong level guarantee nature obey something think yet even way free explore parameter value consume major systematic error uncertainty background separate signal new like b train wrong would regard datum contain tb search try beyond look deviation physics search whether idea forward cdf use music scan deviation algorithm search physics supervise anomaly oppose detect deviation standard produce use analyze physics method inherently multivariate free dimensional histogram among measurement statistic solve semi anomaly design deviation label experiment semi detection solve independent classify observation anomaly fall region background tail region anomaly solve exploit physic occur anomalous deviation estimate collective deviation background emphasize datum maximize parameter recognition anomaly classify anomaly use probability discriminant anomaly proportion anomaly anomaly could far section anomalous anomaly statistical nan statistic enable discriminate statistical background anomalous statistic bootstrapping replacement background fit new corresponding compute collective anomaly simple illustration model show generate gaussian density set simple anomalous model contaminate anomaly find anomaly mix proportion model gray illustration histogram excess histogram estimate black line anomaly gray number covariance model background maximize computational carry proceed step q estimate th equation step improve anomaly anomaly model anomalous log background em free simply skip fix complete equation base idea implement overcome issue choose detection detector realistic benefit event event produce association decay bottom look slightly show new physics consideration single simulated cdf detector neural facilitate datum principal detect signal background unlabele contain reality weak limited artificial toy detect contribute percent magnitude background statistic cross validation evaluation log maximize contour background fix model algorithm converge anomalous dimensional solid contour gaussian low would enough close attention figure receiver roc regardless high mass lie background low anomaly train mlp compare roc anomaly show signal mass anomaly similar start likely anomaly successfully identify experiment efficiently train severe consequence anomaly limitation tb semi identify priori knowledge train network supervise scan focus think standard significant anomaly background reconstruct observable reconstruct physical anomaly physics interpretation likely anomaly detector determine introduce region anomaly repeat anomaly stage anomaly explain adjust study physics computational semi clearly could tool limitation curse dimensionality seem perform relatively possibility parsimonious gaussian handle
descriptor descriptor descriptor rapid molecular property atomic associate descriptor store library molecular descriptor summation atomic technology provide rapid descriptor traditional descriptor topological autocorrelation descriptor surface integral atomic path topological path topological autocorrelation descriptor without dimensional minimized structure density energy correspond molecular van nuclear potential feature detail elsewhere version protein score score score represent characteristic construct difference similarity linear notion dot product covariance dot achieve dot hilbert possibly nonlinear entry kernel size require computational effort work call kernel produce kernel square regression square response cite work column equal maximize assess round center zero absolute round execute code adapt maximize retain cv pls calculate validate cv permit pls hyperparameter combination calibration loo obtain maximize force matlab routine regression supplement descriptor exploit gaussian loo cv place finish supplement descriptor result prediction place moreover second place third round descriptor derive descriptor row descriptor one model chosen loo cv across finish post look dataset tight high descriptor despite prediction assume parameter loo cv calibration loo calibration prediction improvement pls autocorrelation address generate pls consistent feature generate descriptor set present descriptor find close place performance response retain upon calibration performance call descriptor descriptor loo translate noticed prediction coefficient ignore moment descriptor outperform high descriptor round exponential perform input descriptor performance across calibration prediction winner back feature possibility pls find advantage use question descriptor bind contribute descriptor descriptor contribution work p fellowship reference square pls journal statistical reproduce journal affinity global journal computer molecular reliable class bind affinity explanation complex protein column energy use van descriptor atom j thompson modeling system computer method quantum r semi molecular discovery virtual throughput institute york sciences sciences computer ny paper obtain blind bind part comparative prediction least pls outperform pls incorporation capability keyword least square comparative prediction algorithm cv square pls rkhs atom
framework easily stationary identifiable stimulus assessment assume bayesian assessment specification assess broad alternative without make direction advance highlight attractive predictive bayesian assess fit testing normality since multivariate normality assumption build two assess tree alternative normal dirichlet offer compute easy univariate base computing density may inefficient estimation alternative sub detection normality simulation claim dirichlet normal smooth dirichlet normal know variation possess adaptive rate normal line normal beta collection process normal normal correlate alternative scalar parameter parameter underlie parameter determine potential component carefully map parameter avoid identifiability despite slightly different mixture normal amenable computation sequential imputation posterior bayes computation factor computation adapt liu sequential imputation parameter propose type frequentist test univariate moderate size test classical test address refer bayes go go technical section sample size simulation consistency dirichlet mixture model independent draw variate cholesky p pp triangular testing improper along natural parametric provide develop stress maintain balance nan nonparametric specify alternative nan alternative normal wishart modification scalar beta normalize constant write distribution dimensional proof ensure alternative model discuss degenerate mass breaking write independently consequently give independently nx v stochastically volume see fine evenly shape curve center likely parameter alternative nan control base shift volume argue otherwise may weakly degenerate chapter alternative vanish concentrate remain alternative away make irrespective vanish nature limit choose bring useful flexibility value stick representation make normal close component shape allow curve possess sharp scenario satisfy limit remains carry monotone reasonable use specification reported determine little overall shape possess sharp broad shape notice intermediate dashed alternative spread away nan intermediate issue study nan magnitude alternative always way normal right haar light choose weight alternative insufficient justify need obtain proper either predictive property notation fold I surely absolutely respective lebesgue collection space df df family call invariant random law absolutely respect measure absolutely haar p p consist distinct observation calculation associate gx gx f h carlo draw unbiased root depend conditional density choice approximation later draw give ss choice justify conditional density partial sequential calculation therefore simplify gx nh n x due write suitably adapt implementation order randomness tractable approximate make approximation embed density expect tail way likelihood importance simple section sample oppose scalar inverse density wishart component tuning efficiency code website likelihood choose recommend conditional recommend gibbs rao alternatively smooth technique suggestion min st rd max summaries replication bayes normal refer monte sampler fairly west posterior base smoothing package bandwidth computationally expensive choice result improvement refer run posterior gibbs smoothing run importance substantially latter help extent competitive illustrate pressure minimum showing normality minimum three synthetic respectively bivariate bivariate student freedom bayes factor drop little normality outlier detect heavy tail dataset elliptical scatter sharp decrease minimum normality leave right consist take epoch multivariate variance residual mahalanobis deviation fail reject figure minimum bayes indicate strong large factor wide respect nan unique alone univariate produce mean spike extreme hence non pick approach bayes fit normality classical subject calculation simulate nan normal specification run compare test test derive similarly minimum factor partition minimum calculation normal student simulate result power non normal mixture produce outperform surprising tree alternative process frequentist property bayes nan whenever normal kullback prior substantial literature indicate mixture normal distribution broad support mild continuity expect nonparametric formal challenging nonparametric densely around prove simplified mixture simulate standard evaluate bayes sequence display path factor appear converge appear alone scenario substantial du nu u accord eigenvalue complete form change find characterize
consequence stop lemma extend technique obtain bind greedy rsc hold sparsity terminate lemma noting hold greedy substitute backward stop characterize terminate stop stop support backward step stop support stop q rsc stop notice satisfied value useful graph depend stop stop nd parameter far satisfy false condition corollary rsc note conditional logistic covariate bound analyze rsc logistic order remainder exist exist constant argument function rt rt w verify stop parameter well neighborhood recover union completes impose require node contrast regularize counterpart greedy observation result pairwise model variable take multiclass greedy backward greedy would add group greedy rsc far lack experimental several structure outline experiment b near star assume binary type generate learn structure match range sample batch suggest experiment via validation n coupling logistic suggest greedy less star mixed sign success versus parameter demonstrate terminate fail entail consequence upper stop support q optimize rhs stop sub range carefully arrive contradiction eq hence algorithm backward fail go entail stop stop parameter eq contradiction reach forward backward kk along similar hold terminate iterate variable completeness reach rsc consequently entail suffice forward optimizer j latter contradiction provide k claim conclude begin forward lemma optimizing conclude reach begin carefully k rt rsc eq fact proof structure graphical study property special case discrete model neighborhood estimation general sufficient size recover edge high greedy strong convexity numerical end undirected variety domain statistical among concerned markov mrf identically encode independence subset thus broad mrfs estimation greedy estimate successful exhaustive quickly estimate neighborhood set conditional need solve expensive large another approach score scoring candidate typically indeed discrete mrfs search large intractable thus heuristic local procedure come focus consistent possible even dimensional relevance sparse zero show practical strong cf paragraph induce nature program log likelihood recent model focus stagewise greedy add possibly parameter yet strong guarantee estimate finite greedy variant appear counterpart observation graph node pairwise specify e largely ise denote sample vector form infer sample estimator neighbor equivalent characterize define goal condition give neighborhood greedy stagewise condition combine neighborhood add occur respective node could use selection succeed recover neighborhood rule exact describe general next apply general next section outside graphical suppose estimating assign cost ease shorthand factor
c pure warm table number iteration value utility intermediate iterate algorithm contrast global different completion namely soft iterate simulation use code due matlab numerically expensive operation burden singular decomposition likewise implementation soft operation al accelerate involve solve lead burden carry approximate et proximal stronger accelerate three heuristic rank project estimate accelerate algorithm soft iteratively replace element involve post like performance vary computation iteration simulation dominant descent ability solve fix soft without like truncation assumption sample remove uniformly tr soft stop either iteration stop relative duality ht plot behavior greatly affect soft suffer moderate outperform iteration behavior optimization training kept entry descent tr soft initialize fix initialization include suggest entry stop relative variation stop duality fall convergence later exceed similarly stage tr rank trust region ht scale vary value generate random initialization iteration take stop number exceed stop absolute variation relative error plot converge take ht two regularization comment completion summarize complexity suit moderate required iterate scalability well descent tr benchmark suit low need compute move strictly go warm regularization coefficient minimize response multivariate optimization low rank motivate follow although smooth directional euclidean complexity term like cost full low note duality regression define x w trace numerical dominate numerical generate unit deviation white noise add cost regularization validate error rmse ratio snr fix minimum apart fit claimed gap similarly trust algorithm stop fall ht c snr main present minimization rank analyze convergence criterion duality problem numerically thank simultaneously ensure decrease cost function term p space riemannian technique geometric devise order proper trust region convergence contribution predictor step geodesic descent approach predict performance superior warm idea low completion encourage result n r u b x sub trace norm stationary uv v w global optimum virtue strict derivative constant update result function obtain maximize introduce x trace norm q sup lagrangian dual prove cm address low equip particular riemannian trust guarantee solution maintain parameter naive warm manifold completion regression know nuclear attract year relaxation intractable decrease unbounded consequence whole minimizer proxy minimizer rank dimensional propose second control svd orthonormal matrix span contrast allow unique rank long riemannian devise trust generate converge local rank minimum reach select decrease convergence minimum rank also thereby transform global minima procedure path numerical surprisingly literature primarily idea semidefinite framework riemannian develop improvement discriminate thank choose combine one appear particular context completion spirit set global tight trace norm priori use operation demand potentially singular one potential single efficient warm sake illustration comparison application matrix iterative numerical favorable represent manifold definite stress redundancy implication key success illustration optimization euclidean manifold manifold proper point whereas scale positive invariance necessary optimality solution optimality condition differential trace throughout either write lagrangian space section local optimum op operator I singular fact local identify global threshold criterion check global closeness optimization duality gap duality solution dual dual singular dual trace sup x f expression duality gap operator op operator conjugate easy propose problem first optimization early trust idea behind region quadratic solve iterate depend decrease objective function reject detail rewrite u notational convenience important second minima isolated rotation v pp symmetry total count mp equal rank remove symmetry identify belong equivalence class manifold conceptually optimization minima isolated computation manifold tangent tangent representation equivalence product eq z z subspace vertical skew size horizontal space pick positive horizontal tangent characterization b projection horizontal solution solve lyapunov manifold special convenient riemannian horizontal lift due equivalence class horizontal expression element directional lead representation likewise horizontal lift well riemannian positive cone derive riemannian connection eq euclidean directional derivative horizontal lift trust manifold guarantee quadratic rate propose trust region eq trust region radius solve subproblem lead direction quadratic model iterate mapping mapping refer map case mapping metric known virtue mp op omp mp mp op mp show linear operation algorithm start alternate fix manifold table ensures decrease belong stationary rank update ensure dominant leave right f fact project onto subspace stop point step obtain rank value orthogonal projection write orthonormal constant justification value give good stop rank guarantee fact fix unconstrained descent iterate rank theoretical however always contrast propose tight iterate problem separately increment make trust addition priori solution interpretability iterate global different motivate path n initialize warm regularization argument paragraph regression extend idea u I factorization perform warm scheme show minimal computation sufficient add multiple square x optimality condition look geometry rank obtain dual consequently smoothness smoothness decrease solution first order path compute exact solution step ht I geodesic extend need u v I refer logarithmic numerically expression mapping numerically approximate follow horizontal projection accomplish operator v p u eq approximation approximate eq predict step backtrack motivation observation line numerical denote descent tr low completion regression penalization recovery full ghz intel core machine ram illustrate case optimization completion behavior search p p mp diag diagonal suffer impose generic complete entry addition problem denote frobenius matrix multiplication convex high assumption
seminal paper approximated quantity appear optimum parameter inference exact implication oppose approximate implication justify optimum goodness reduce family generate optimum serve compactly lemma log likelihood number reasoning assign prior schwarz proposition bayes optimum large poor criterion aic aic bic evidence measure compete tell economic give goodness selection evidence maximize true nuisance substitute use theory estimation economic finance reference therein histogram fit theoretic popular economic particular physics theoretic analysis student model ac ny kolmogorov solve regularity condition subdifferential calculus duality infinite duality mathematical mathematics york ed bic york free test fit fitting nj second ed york behavior mechanics mobile university york des paris york university communication criterion correction equilibrium reasoning transaction university non likelihood ratio edu recently optimum kullback leibler subject approximate obtain model unconditional oppose method axiom nonparametric optimum information already measure kullback information theoretic reference therein gain popularity theoretical foundation provide product obtain closely bic bic pick compete measure tell poor I take compact believe unbounded compactly ball radius etc denote generally lebesgue moment define represent inference choose uniform hereafter carry set drop subscript minimize theoretic point justification special like finite ii regularity discussion regularity integral compact always substitute duality algebra understand development later unique dimensional affine tx strictly maximization solution concave solution newton optimum verify substitute equality optimum family guess conjecture true maximum tx maximum show imply coincide maximum maximum valid truth put wrong one exploit estimator family bic unconditional conditional solve population counterpart kullback asymptotically freedom optimum estimator unnecessary exponential arbitrarily
set convenient inner consideration abstract form establish ensure perturbation set perturbation work depend nonempty relatively complete bound perturbation give must complete hold form eq denote scenario random vector constraint discrepancy value branching convergence building work let decision extract scenario pair represent stage branching structure formulate optimality number decision jointly recall decision beneficial represent expand statistical attractive nonparametric process relatively particular space update mean stage function coordinate notation kernel semidefinite similarity also dataset noisy optimal stage optimal infinity cover decision select radial bandwidth tend coordinate policy replicate decision optimal act numerical regularizer confidence make prior learn literature deterministic problem nominal extract decision parameter affect regularity update exist policy exhibit usually possible procedure output know process density predictive distribution mean decision maximize feasibility program feasibility figure induce option correct feasibility heuristic maker near policy denote correspond ultimately ability explain scenario output lead problem true simulation scenario scenario finite estimator policy moment normally inverse guarantee true basis could eliminate rapidly investigate methodology three factor variation scenario approximate relatively solve accurately play role predict decision experiment evaluate criterion spirit application product distribution resource represent observable contribute reveal decision quantity decision allocate production composition demand numerical conduct simulate pure programming decision solve horizon decision fix benchmark processor run compute set absolute optimal two adapt production available demand reveal feasibility procedure describe variant report obtain must one radial bandwidth variant cumulative radial complete tune behavior feasibility heuristic adapt coordinate create reach quantity consume proportion namely heuristic generate order permutation shrink horizon procedure implement shrink horizon compare branch quantization assigning minimize probability integrate cell exponential growth scenario branching accuracy branching grow exponentially net percent recall structural property guarantee branching cm branching second test set scenario new horizon scenario tree branching stage choose optimize online scenario tree branching horizon use simulation processor result feasibility gp test collect tree branching well make treat overhead fast possible explanation tree thus much solve heuristic speed table benchmark program gp attractive form policy decision simulation eliminate completely call program achieve feasibility mostly solution problem scenario tree concrete branching structure turn study present consider study risk parameter budget generate could stage speak otherwise solve branch already decision turn branch branching scenario deterministic induced scenario branching scenario negligible know advance resource branch day realistic value motivate branching purely randomly sense optimistic bias turn scenario tree fast enough several tree thus essence branching base well branching structure lead produce tree assume node create randomly large number create eq iterate recursion small recursive mostly level tree expectation effect ex scenario create number child branch branching increment step otherwise branch tree approximation compact feasibility summarize table overall simulate scenario matlab processor randomized benchmark policy neutral cccc tree cpu somewhat surprising multiplying scenario compare set randomized decision discuss tree ensure good good obtain good one probability low program decision maker improve risk besides expectation possible empirical scenario paper approach infer policy rule stochastic present lead nonparametric investigate scenario tree generation develop use useful scenario algorithm exist current acknowledgment present network dynamical office scientific associate financial carry department engineering university li grateful suggestion presentation parameter test pt stochastic research university usa department engineering li combine run sample independent solve program technique choose scheme scenario branch parallel could ultimately retain test excellent stochastic scenario powerful rapid growth scenario grow limit size make possible optimize first look ahead compute problem propose assess tree sample pose span planning horizon relative sampling updating span time remain previously process find sequence decision value accord procedure produce unbiased unfortunately simulation demand stage restrict technique relatively update combine tree return statistical tree decision supervise repeat exercise produce policy function tested determine primarily approximate reinforcement community use actor computer low dimensional constraint approximation constrain feasible minimize deviation policy scenario optimize scenario could view near regression however context recognize direction makes quickly perform approximation create use scenario safe guarantee build fundamental implement particularly reach pass node scenario tree formulate optimization associate scenario associate optimization path root identity among fx k subject optimal first stage regret implement machine mapping generalize value feasibility set per simplicity scalar parametric value linear build stage index advance sample new distribution
extreme identically value max generalize extreme fr member generalize margin estimate generality assume margin fr transform step multivariate identically replicate maxima degenerate exist sequence extreme necessarily method max stable infinite multivariate theory hold possible max process fr practice generalize parameter location transform often spectral often negative replicate max process fr margin represent max process function intensity stationary max unit choose family correlation mat ern cauchy modify third order force reasonable environmental paper use parameter serve mat ern form let denote point poisson intensity max stable fr margin variance increment concentrate widely year simulation realistic feature stable marginal bivariate write function statement motivate max field margins joint describe max eq measure location explicitly joint location correspond independence think consideration available coefficient form directly manuscript estimate pairwise unit fr move triplet location distribution close follow pairwise triplet serve bayesian free aim posterior computationally prohibitive approximate area facilitate auxiliary dataset integrate simulated interest exactly point else recover likely occur probability practice summary sufficient limit familiar occur density simulate form uniform favor straightforward accuracy challenge enough computable produce identically draw role draw approximated approximation closely resemble practice highly informative remainder implementation motivation stable margin show relationship extreme parameter coefficient gamma margin relationship transform margin usual fr margin unbiased minimize squared square equal square mathematically ol procedure utilize approximate algorithm ols obtain remain eq integral difference two curve take final suitably collection particle correlation analog mean pointwise pointwise performance study curve pairwise coefficient parameter sum residual ordinary square proceed use summary pairwise composite approach improvement computing method move consider order tuple triplet explore max stable triplet equation basis utilize triplet estimation triplet rapidly increase triplet pose computing summary decrease uncertainty estimate triplet coefficient large natural homogeneous ideally homogeneous heterogeneous triplet group measure triplet euclidean measure two triplet permutation triplet identical length rotation translation two set theoretical isotropic triangle respective length increase entirely actual estimate triangular size distance choose cluster balance maintain homogeneity ensure triplet average method assign cluster cluster choose minimize increase sum center merge location requirement practical dissimilarity computable consuming average begin draw except draw stable fr triplet coefficient absolute collection value filter final distribute collection compute error simply identically draw empirical construct dependence mat ern conduct specify range dependence preference suffice error year location uniform grid dataset composite example approximate computing draw substantial computing need simulate computing around iteration pairwise constraint replication simulation run filter percentile accept posterior distribution threshold specify percentile relative smooth opposite produce abc behavior comparison correlation pointwise h focus comparison region higher great way stress pointwise abc use use handle assess manuscript computing report compute composite abc composite reduction vs composite two follow approach mse low abc essentially tie composite likelihood within five standard remain large conclusion view whole favor outperform composite give composite find great abc pairwise utilize essentially finding section carry computationally approximate bayesian closely performance approach simulation increase adaptive abc produce approximation approximation second abc allow implement algorithm specifically simulation particle filter percentile repeat times filter percentile ensure particle accept call weight produce weight modify estimate parallel initial cost performance expect result three f approach benefit range aim loss united management loss month daily temperature site center united historical http daily sites north daily region jointly four responsible year daily temperature month risk transform unit univariate extreme maximum extreme scale heavily relationship spatial coefficient triplet describe group consider ern correlation criterion find mat ern range selection abc range spatial process scale draw run pointwise accept approximate pointwise credible interval pointwise centroid centroid census http www census www specific extreme value standard spatial krige whereas location fit matching replacement stable mat ern correlation centroid centroid temperature threshold incorporate uncertainty intermediate exposure million percent fall threshold degree provide evidence respect additional support financial product loss show beyond calculate interested calculate expect policy model stable
natural generalization side q minimax sense attain eq I address pre belong occur post grateful useful discussion office nf air office scientific research grant fa national grants california mathematics section definition mathematics california california g department mathematics university california usa mail edu mathematics california usa mail nan composite either embed exponential stop minimize rule approximation rule verify asymptotic test open test test identically whose sampling soon possible favor absolutely solution testing test eq level eq stop call test terminate surely whereas terminate almost furthermore stopping time alternative hypothesis side exhibit optimality measure associate alternative p exponential versus observation likelihood side delay latter approach also ratio statistic give lebesgue bound q bind attain mixture continuous information trajectory favor minimize maximal kullback leibler favorable motion mixture family particular choice density lead attain theorem emphasis hypothesis df ix df ix assume mass required belong exponential parametric setup approximation likelihood ratio discretization main motivation setup sample channel population possibility correspond attain non obtain approximation see even additional may include problem actual control channel typically advance thus case misspecification naturally approximate continuous discrete evaluate point allow mix design attain kullback identical latter attain asymptotically order asymptotically e ratio converge I rule attain natural criterion e kullback leibler distance express minimax mixture highlight show constitute use random time family integer theorem impact develop limit stop walk family stop probability play role one reason connection tool rule theory decompose statistic long change basis order additional rigorous say satisfie growth uniform nonlinear understand perturb walk consequently variety sequential asymptotic normality whereas purpose useful quantify focus rule optimality simulation sequential conclude call stop sided assume refer expectation set e kullback leibler versus walk whose denote walk brevity easily show e designing side sided attain infimum eq eq consequence predefine asymptotic term stop obviously active close assume exclude index decomposition fact slowly able rule break presentation without insight need exist jt ideally like minimize task attain distinguish notion follow use theorem asymptotically positive weight fully main subsection lemma sequence iy ip condition q definition I slowly obtain suppose q asymptotically process due constant j increment mean eq iy I arithmetic therefore measure yield prove assertion furthermore eq decomposition slowly weakly recall complete arithmetic either walk arithmetic follow nonlinear long p n aa integrable satisfied converge consider first lemma inequality first q long chi square variable eq everything obtain negligible either condition eq usually nevertheless case corollary bound order attain content theorem arithmetic finite auxiliary bayesian optimality nan therefore cost observation leibl divergence prior h integrated stop moreover indeed small minimization establish optimality test infimum important particular scenario stop arithmetic asymptotic define mixing way mix consequently remain construction whenever mix test third asymptotically follow weight every q mixture mix q every perhaps choice implementation reduce uniform work criterion reason optimize condition thus mixture rule attain mixture symmetric minimax rule inefficient order unless suppose control attain asymptotically sided meaningful minimax appropriate bound definition ratio write define recall case mix respect lebesgue positive moreover see complete substitute mixture however continuous asymptotic remain minimax maximal leibler asymptotic mixing attain bind must continuous density exact exponential rate imply mix completely normalize close therefore optimal mix mixture computable minimax show continuous use minimax discrete optimal mixture interval take eq corollary asymptotically asymptotically third minimax corollary theorem write kullback divergence q p arithmetic see condition
trick kernel tractable student novel machine arrive carry performance method demonstrate capable distribution reconstruction nevertheless unsupervised task image cubic fast another extend define g graph develop novel kernel methodology investigate limitation decision certainly test frequentist characteristic trick marginal likelihood enable issue potentially include numerical would song helpful discussion le song part grant ep ar college popular nonparametric rely present trick derive regression despite process kernel aim demonstrate wide machine obtain popularity past decade rely chain monte approximation ever grow form approximation would desire many perhaps process predictive exactly method adopt frequentist remarkable algorithmic clarity widely trick limitation apply representation observation product product increase increase computational notably allow popularity bayesian kernel trick rarely tool rare new machine kernel computation rigorous framework reproduce rkh offer hyperparameter mean incorporate model methodology find mm simple observation bayesian inference dot crucial finding fortunately orthonormal generative underlie principal pca show distribution preserve orthonormal transformation sec review discuss apply trick predictive consider sec experiment high bayesian dimensional euclidean task nonlinear ambient lie sample consider estimate distribution importantly require result still amenable kernel trick discuss guide distribution trick term scalar invariant orthonormal space orthonormal want express rotation permutation kernel conjugate gaussian meet invariance wishart gaussian normal wishart restrict two way firstly secondly wishart set spherical sensible restriction ensure define observation q dependence scalar invoke inversion q reality however assign actually make trick product say previously interested manifold geodesic density input space implicitly invert mapping define multiplicative jacobian correction relate input change variable density appendix pt pt illustration estimation embed observation real line multiply give scale exactly apply trick simple unimodal multimodal possess draw complicated model linear fig map dimensional fit inference map student predictive manifold observation multiply jacobian fig final fig remarkably jacobian include mapping feature essentially student predictive restrict indeed simple although improper observation decide ignore jacobian pt student se vary panel generative want require embed hilbert potentially infinitely measure model space address misspecification stem project manifold limitation closely underlie generative bayesian inference effective popular perspective induce sensible recover toy degenerate feature along roughly axis fig intersect manifold twice delta degeneracy recover component clearly degeneracy integrate degenerate topic kernel element observation space exploit way feature interpret therefore characteristic help task gaussian incorporate unsupervised construction trick method rely chain carlo call doubly intractable use intractable unsupervised model simulate convenience improper space point invariant subsequent k condition different setting process chinese restaurant like eigenfunction induce performance label table constant past include deep belief net hard directly unsupervised reconstruction absolute assessment introduce task able model recognition test unsupervise task use detect handwritten digit training label gray handwritten model ten student length median point digit dimension label point threshold consider baseline dirichlet mixture gaussians operate three choose search auc separate ten test compete significantly complex investigate carry sequence average auc size amount becomes consider
unnecessary regressor leave advantageous power insufficient consideration lack flexibility original regressor explain regressor alone reveal concern ratio regressor generate regressor relevance response drop little regressor probably variable higher less regularize build flexible generate performance stable rbf possesse describe may optimality cost variance experimentally rbf additionally indicate first present synthetic noiseless use handwritten character define handwritten character canonical difference target resemble desire like comprise font character difference handwritten original font generate font character handwritten character would font font canonical system use font vector field comprise vector evenly font express handwritten explanatory font response explanatory comprise respective output unit draw drive respective probability feedback connection marginally activation high connection become unstable drive internal discard internal collect row observe mse modeling stage mse generate regressor response variable variance regressor select moderate steady variance regressor select regressor give ratio regressor half contribute vector examine cause weight show half contribute little explain response purely reveal analysis relevance regressor original locally linear regressor train mse stability robustness thus test model half little explain response high locally rbf try performance rbf evident locally figure show regressor significant dimensionality improve robustness dimensionality tested attribute strongly variable add sequence transform training sequence reservoir sigmoid teacher force step sequence use formula trial variance code able compare result period leave difference net formula teacher internal repeat test interesting free locally worth training plain fluctuation smooth minor smoothing beneficial investigate stage always sequence table outperform plain linear fit similar sequence local work well regularize outperform linear ten trial may benefit regularization analysis provide deep present fig flexible explain probably room improvement extracting regressor g extract delay sum suggestion research present locally improve penalization regressor linear transformation rbf regressor rbf helps understand state consideration construction mechanism matrix activation give regressor enable concern mechanism mechanism internal concerned adapt topology rather adapt world free yield interesting activation state space equation besides sigmoid radial rbf activation test stability insight construction mechanism delay detail delay regressor analyze delay importance analysis temporal underlie whether rbf enhance room improvement effectively combine leave automatically regressor automatically desirable recently propose coordinate find significant improve rbf well evaluation present generalization alternative traditional rbf improved importance regressor via analyze therefore straightforward improvement improvement likely design model parsimonious well aid non expansion rise regressor relevance teacher often certain regressor effectively explain teacher concern generate contribution relevance locally depth underlying regularize build may improve relate suitable hand limitation sometimes insufficient flexible parsimonious excellent alternative feed rbf net state local forward variable radial rbf novel network easy appeal attract output dimensional vector internal element sigmoid state temporal teacher expansion carry diverse input teacher name state diversity appropriately teacher traditionally usually trial diverse successively simple construct generalize expand parameter mostly expansion adjusting regressor linear variable desire importance quality regressor generate expansion state generate regressor various relevance regressor variable less regressor hundred regressor contribute instability regression large response unstable usually easily fit overfitte regressor point couple therefore model traditional regressor improvement address issue regression combine pruning test pruning decrease result prune exhaustive square mse regularization delay sum bayesian approach effective delay improve memory deep model parameter smooth regressor relevance smoothed teacher alone fully indicate usefulness regressor experimentally regressor orthogonal individual jointly variety appropriate robustness effectiveness later text term model flexibility insufficient analysis able determine whether usually require linear approximate forward neural rbf nets rbf effect input least iterative base considerably flexibility require training meaningful contrary appealing estimation support rbf considerable popularity decade may rbf easy train possesse recently however suffer instability deal datum perhaps regularize constructed rbf flexible excellent possess paper regressor locally analysis provide locally locally present rbf model noiseless research discussion modeling conclusion forward matrix regressor variable th model express notation orthogonal triangular minimize least triangular obtain computing coefficient concerned orthogonality useful transformation carry orthogonal regression orthogonality regressor equal regressor regressor orthogonal regressor regressor alternatively algorithm sub select regressor reduction ratio ratio alternatively regressor gradually variance selection certain significant introducing regressor cause marginally regressor contribute ill conditioning model build overfitte selection purely appropriately regressor term regularize follow error diag normalize regressor criterion tolerance produce sparse regressor parameter unknown set carry full result sub use use iteratively unchanged two iteration regressor often dramatically suffice parsimonious enhance cost determinant improve combine selection regularize eq stop iteration regressor regressor parameter quick system output follow form eq step time delay lag white identify datum possible approximate rbf regressor rbf centre norm rbf nonlinearity thin spline function choice write I set parsimonious generalize generate iteratively parsimonious centre effectively make centre combine optimality regressor select manner iteration state update equation desire output discard rest store design eq common zero traditionally square l regressor various desire shape govern pseudo weight nonlinearity mostly experience approximate response desire accuracy obviously pseudo rise regressor high regressor contribute instability variance fit noise issue desirable regressor address regularization regressor response noise alone generalization regressor regressor
depict hmm simple break isolated indicate right history p induce observer remain nontrivial induce possibility direct calculation output thus past least positive comprise machine technical arise conceptually similar machine generator history hide markov characterize finite alphabet history generator correspondence finite generator machine generator machine history machine generate machine history follow gm hide markov key prove come study synchronization generator machine terminology generator symbol random x ts x observer current observe machine word machine relation low maximize observe symbol describe symbol formally define cross realization primary exponential constant synchronization differ time shift length refer observer instead observer index work symbol exponentially unlikely observer small generator n exist lemma tt w x x iw ni appendix equivalence measurable measurable ie ne class history generate preserve ix j directly point take I x w w ix generator assumption machine process generator history history generator machine priori clear indeed hmms hmms generator uniqueness long hold machine reverse finitely finite history hide also generator machine note appendix know characterize ergodic hmm generator hmm construction claim throughout n equivalence entirely connect component strongly connect existence x hmm transition define essentially fact separately two case equivalence l sm give ii word nan convention take always analysis well w claim ie set sum I mention take equivalence word x distinct lemma hmm strongly define via contradict distinct class two component process generate generator distinct argument lemma w stationary machine distinct one satisfy demonstrate history generator machine however recently also proof improve intuition equivalence new come recent bound state large machine number result countable generator unfortunately countable decay long unclear whether countable either hold countable machine entropy acknowledgment award nf fellowship establish trivial alphabet sequence nontrivial measurable set martingale hence length measurable equivalence class equivalence alphabet past recall past implicitly follow claim several various word tw n tw x w tw regular symbol conditional eq regular regular equivalence fix x claim eq hold equivalence e equivalence measurable set throughout ergodic alphabet past proceed measurable countable intersection p w p class definition w w measurable establish probability equivalence claim proof proof theorem assume ergodic alphabet define set length e x length equivalence nan class prove bind guarantee need claim application expectation version claim w x e e e claim finitely finitely alphabet probability past class class symbol finally equivalence stationarity fact transition hmm pt machine stochastic process originally machine hide analyze though alternative definition hide state key difference machine generator machine let whose infinite sequence correspond structural dynamical system subsequently context thorough definition original history discuss machine oppose history derive generator process hide obtain initial underlie establish history generator formally generator implicit technique substantially somewhat section argument implicitly concrete directly result fairly elementary state thus useful providing intuition terminology generator show consequence helpful order parallel generator machine study machine machine alphabet generator stationarity history definition stationary process introduction machine come physics substantial symbolic dynamic review development reader refer overview symbolic model help machine understand relationship reader skip type first shift present direct label alphabet consist walk present vertex edge restrict essential vertex along thus graph occur present state accept clearly essential accept thus unique say word shift irreducible irreducible strongly present say symbol irreducible shift unique present graph symbolic dynamic presentation refer irreducible shift minimal deterministic irreducible label allow future x vertex consist equivalence vertex vertex past necessarily minimal deterministic arbitrarily start state irreducible graph cover recurrent minimal three closely present irreducible irreducible necessarily recurrent irreducible component cover cover extension purely history analog analogously cover infinite sequence allow history probability cover recurrent history equivalent abc process example exist ergodic process support irreducible infinite even example shift right leave cover associate generate shift extensively rich theory many characterize stationary shift induce past finite length positive probability future natural machine exist infinite latter alternate process unfortunately quite well despite alphabet sequence sometimes name corresponding normally assume theory strong old particular relevance g restrict show requirement surprising perhaps construction irreducible associated infinite give may past sequence equivalence history machine g induce probability symbol infinite future converse necessarily bad nice history machine derive finite complete stationary hmm output word state overall next symbol q respectively associate hmm markov irreducible transition word choose process ergodic irreducible ergodic ergodic x algebra bi infinite stationary measure stationarity uniquely finite consistent measure two preserve edge label edge irreducible hmms converse hmms also concerned generator machine irreducible hmms important property distinct hmm strongly connect symbol symbol iw check hmm distinct condition separate distinct check distinct states generator associate generator machine ergodic process probability define generate transition generator symbol hmm path nonzero long start symbol equivalence history model whose class past consider induce probability take specify clarity verification defer reading overview separately entirely self none note ergodic parallel generator although requirement stationarity actually need ergodic alphabet past sequence w symbol tt trivial word however probability well equation past normally define conditional like uniquely intuitively keep mind understand conditional probability intuition justify construction history set regular equivalent prediction simply precise drop subscript infinite see equivalence equivalence appendix equivalence transition relation well x regular also equivalence symbol independent equivalence class formally distinguish function history machine appendix say equivalence finitely characterize slight
note yield represent term price deviation term second return long return minus convenience parsimonious perspective term long price factor unobserve variable term equilibrium growth price follow additional assumption speaking point convenience greater discount price cost additionally introduction level review demonstrate issue remove potential positive additionally admit time vary yield analytic contract unobserve develop convenience proportional square root instantaneous yield cox yield proportional instantaneous convenience yield level analytic price multi several extension question relate give neutral time return convenience yield mean coefficient term rate volatility modify function exponentially affine family derivation price consider exponentially affine neutral allow additive account volatility imply volatility remainder xt dt dt dt dt v correlation structure addition risk neutral ensure discussion real neutral develop without variation respect factor short dynamic price contract time contract tt price neutral detail derive space formulation bayesian novel parameter specify panel denote addition parameter use process neutral discretized formulation achieve scheme process order finite taylor otherwise scheme euler two firstly discretization secondly process euler increment scheme mix variable innovation bivariate wiener discretize space truncation specification take develop sense generate transform sir filter methodology zero normal noise long linearization state approximate extend kalman filter formulation develop novel introduce formulation convert neutral pricing proceed posterior distribution real neutral g parameter specify prior world neutral specification random choice hyper uninformative specification section though consider specification specify give deviation space equation form addition admit affect observation ft b detail equation simulation recent sampler state jointly latent methodology particle filter parameter embed fashion construct adaptive scheme use static sequential rao via kalman introduction markov distribution adaptively learn static significantly factor develop involve kalman importance resample sir variance optimally latent sir version filter particle mcmc utilize approximate one interested self well set static latent hundred thousand depend frequency ideally suit simple univariate component sampler achievable especially problematic chain achieve typically lead markov large optimal distribution carlo static proceed metropolis probability require acceptance metropolis hasting recognize particle mcmc mcmc proceed obtain candidate previous chain acceptance key advantage design problem design carlo monte detail distinguish kernel kernel proposal depend past particularly important paper propose must ergodicity adapt markov ergodicity adaptive mcmc scheme use static know ergodicity present metropolis mcmc factor specify static give comprise component present methodology associate particle covariance I aspect stage bayesian adaptive sampling rao consider study demonstrate calibration represent panel detailed context aim systematically assess estimate perform prior static prior datum parameter utilize process utilize noise path day present panel present day generate accord curve contract day time follow contract contract latent curve ft additionally state run sampler present discard burn interval daily discretization hyper setting additionally assume contract observation around variance jointly different note framework risk neutral parameter estimation filtering latent day simulation rao particle mcmc estimation mmse interval neutral estimate true day contract panel consider day contract day contract improve panel contract panel separate trade neutral trace plot uninformative ht figure kalman filter filter long dynamic trading volatility well include within ht log price versus mmse contract panel contain trading day rao adaptive truncation particle simulation study generate equivalent varied ratio observation section trading day day contract day contract mmse posterior four sir proposal calibrate deviation estimate trajectory estimate four day panel contract day panel ts mmse band next result top panel right price latent state result gray correspond interval price predict contract row contract close contract middle series daily price market sample mmse mmse price model secondly estimate forecast mmse estimation extend short firstly allow short factor secondly develop structural model develop allow discretized volatility derive closed price novel jointly filter long volatility regard rao chain methodology calibration real price neutral process need free trend account price contract time contract denote expectation neutral process must first denote v obtained write backward equation drop condition price multiply backward refer allow express pde enter pde fashion substitution ode substitution grouping divide price still keep option ode constraint get solution ode q equation nd kind obtain solution ode turn ode integrate explicitly expansion dimensional remainder adopt book component j useful integral equal easy integral p gaussian variable study recommendation provide positive bivariate e specification discretization I standard mixed expression series perform strong taylor aspect proposal mechanism sir htb simulation bivariate scheme discretization evaluate approximation obtain sample j give adaptive metropolis rao sir conditional metropolis acceptance increment j perform di v I I py x run sir filter cm lemma mathematics sciences bag north statistic work multi price long volatility secondly additive discretized volatility develop equation develop advance sequential carlo monte calibration jointly regard methodology adaptive rao deal accurately synthetic price stochastic carlo filter rao history economics instrumental introduce attempt inter price price account cost basically inter temporal price storage
identify bi directional arc connectivity construct datum edge group dyadic neighborhood vertex week dyadic arc strength count call make number call arc strength call neighbor neighborhood feature include number neighbor common indicate neighbor call make neighbor consider exist look feature dyadic capture indicate call mark window small indicate old temporal indicate higher active long well consider predictor see predictor short tie could indicate edge persistence decay partly relatively short memory build model predict period time conversely operational decide divide period define decay occurrence period evenly divide build final persistent criterion communication method mine class persistent note task build take derive feature building validate divide subset line build test effectiveness period randomly period set remain training ht model features predict simple decay easy tool ease interpretable classifier social discussion discuss strength network well well mining output disjoint rule assign edge persistence reader familiar approach present result reader familiar discuss decision tree divide subspace axis show circle tree series split dimension attribute good gain formally first ht recursively divide subspace criterion leaf meet generate split classify appropriate branch leaf reach assign classified else circle primary task course reasonable examine classification leave additionally importance define feature description network range call number call call call go proportion call go level neighbor neighbor direct call nd call nd order temporal call time choose correlation network predictive membership persistent adjacent window build predict window predictor question examine among question theoretic predict address edge decay week call median week time substantially lower note early omit great eliminate range edge edge asymmetric call therefore view indicator indicate call proportion total make middle call neighbor turn vertex median share neighbor appear edge direct edge call finally temporal week occur one week median call call call neighbor number neighbor number neighbor number call show time blue correlation figure indicate vertex among dyadic raw direct raw call neighborhood feature indicate simple common neighbor good look note two independent old time look category feature dyadic temporal one exception normalize edge neighbor go neighbor must range trade exception pattern correlation correlation second order agent going connect essentially reduce indicator vertex edge value old edge weak old edge strong indicate make appear really relatively independent dyadic multiple indicator exception temporal remainder wish determine persistent quantify usefulness several approach importance theoretic feature quantify fine grain structural feature link problem gain track decrease conditioning randomness quantity persistent perfectly entropy priori informative conditioning dataset proportion among take similarly negative achieve return feature instance instance log ix appeal intuitive class feature feature edge attribute reveal gain calculate direct extent communication concentrate send receive along drop produce feature dyadic time edge predictive follow predictive ability neighborhood number frequency among vertex feature vertex minimal unable strength tie critical factor raise important merely surrogate strength important concrete c four metric correctly three classifier persistent tie proportion tie belong persistent extent theoretically recall classify tie persistent could perfect classify confident define harmonic precision recall question classifier expect majority decision tree classifier predict tie tie classifier decay precise tie predict decay tie predict tree job persistence term table use regression decay persistence task decision logistic tie model correctly tie big job slightly job predict however precision decay class decision logistic indicate persistence decay pattern model yield fairly prediction relatively present logistic decision predict persistence network ht odds call make call call proportion proportion neighbor neighbor call neighbor call parameter odd odd ratio standard statistically parameter begin gain direct strength persistence additional make odd net call call odd effect member sign vertex chance decay edge vertex straightforward actor persistent effect actor vertex effect adjust raw appear process edge persistence decay turn neighborhood effect measure note early general odd edge slow rate common increase direct call j odd persistence indicate tie value tie positive persistence activate likely inactive mention insight recall subtree implementation attribute subtree mean attribute great choose branch call direct another membership early table stand tree direct weight dyadic helps predict level neighborhood level factor neighbor hand side edge cutoff persistent actor week strong odd classify follow level activate recently persistence drop direct relatively non reciprocal incoming direct weak strength chance cutoff period persistent improve substantially explore decay phone determine determine time adjacent million cell phone decay prediction structural predictor observational range take account total dyadic temporal incorporate relative associated phone emphasis weight find metric total predictive decay temporal indicator edge weight reasonable edge build predict short decay phone contact perform know persistence tie likely activity actor actor something stability network instability evolution degree actor result email tie likely decay recently likely show tree classifier dyadic determine network high decay classifier give combination information gain strength edge range persistence logistic say persistent phone characterize couple activation edge high finally persistence edge gain flow persistence evolution boundary direct old relationship early stage balance relationship interaction guarantee persistent case relationship long survival stage balance dynamic consideration effort framework quick component reference anonymous helpful comment suggestion edge likely address people extent embed age decay large scale phone call week determine importance decay relative power assess active continue period direct type persistence prediction strength agree network fundamental obvious centrality essentially classic behavioral exchange theory edge classic balance analysis suggest observe shorter likely party similar behind influential weak edge argument rare embed fully likely receive process dynamically selective relationship embed transform accounting edge empirically edge consideration make persistent level whole network community actor level volatility relationship major relationship identify circumstance weighted think false positive behavioral email phone rely exactly unclear able predict decay circumstance edge increase availability longitudinal social beginning receive increase recent development actor analysis longitudinal couple evolution behavioral review centrality link phenomenon pattern decay interaction report connect need constraint design limitation validity strategy limit relative site limitation flow use think formation relationship limitation importance contain thousand million actor human communication examine analytic primary second actor produce survey selective reporting significance comparative draw conclusion effort characterize either level analyze formation link little actor leave individual edge decay social prominent prominent within organization ask year year organization frequent substantial business contact year
large dimensionality million recovery classic orthogonal pursuit choose support easy method indeed run omp rip show omp recover restrictive minimization compressive pursuit sp inversion pursuit family hard name suggest stage htp decide one support size notable sp back desire solve algorithm typically exception add remove restrictive rip locality replacement however algorithm analyze unified family hard family positive algorithm htp novel replacement think omp instead replace element prove sparse rip provably use locality lsh provably sub require rip local optima family sp able guarantee sp hope unified light kind iterative modify add element set replace set surprisingly method minimization recovery orthogonal locality sensitive hash lsh element support advantage unlike rip default step optima though direct omp relate extreme member partial operator htp end light thresholding include enjoy omp locality hashing noiseless setting provide least condition guarantee sp omit body paper find omp classic algorithm recovery inner residual square current thus also briefly eq thresholding decrease absolute formally next denote denote matrix denote denote index support vector orthogonal matching pursuit replacement omp instead size control relative two coordinate replace correspond combination iterate iterate note satisfie rip large well solve reliably iterative solver tb input size z j input sparsity level b omp recover appear restrictive condition pursuit sparse provide compare heuristic compare rip take rip say restrictive rip condition recover noisy case iteration converge fx ax ec theorem convergence algorithm turn attention family mild ensemble soon large constant particular integer member replace current correspond connect cardinality hard thresholding operator clear operator reasoning hard search partial fact precisely give operation elementary conceptual current ta ta ax ty solving nice guarantee rip note restrictive experiment degradation recovery increase recover step satisfie e fx ax k dd sketch appendix ta residual proceed element name lose use f ax choose expression monotonically optimum rip satisfied bind rip normalize need need sufficient lemma reduce function hence obtain within least element special immediately time iterative newton thresholding pursuit htp call view newton objective satisfie satisfy newton namely stage element support solve drop form iterate least square support technique proof algorithm hard size recover measurement two provide rip condition measurement provide subspace pursuit pursuit require supplementary corollary discuss hash intuition behind find column view sensitive lsh well near neighbor retrieval lsh near guarantee neighbor able neighbor lsh lsh hash randomize hash create hash hash processing independently stage index hash key function retrieve index state lsh lsh lsh directly guarantee hashing require find setting lsh weak rip hash sub however lsh transition diagram omp experiment robustness exact third lsh implementation require omp pursuit inefficient independently normalize unit select optimize omp newton matlab hash routine file ccc omp phase diagram commonly compress sense literature size problem generate apply recover vector diagram omp newton show code different success blue success gaussian rip universal fraction diagram successful recovery phenomenon respective diagram omp plot next empirically compare diagram recovery newton fairly sample recovery measurement binary gaussian figure level outperform perhaps guarantee fixed varying finally show difference newton level error around ccc pt error various incur provably incur increase ccc kb b incur lsh hash measurement construct offline hash goal signal accurately report require method run increase measurement vector minimize hash bit hash incur hash newton input newton hash perform able newton number recovery newton newton monotonically least minima contrast hash increase table figure compare incur hash converge hash slightly time proceed element name lemma square third line assumption schwarz equation least element case let furthermore every md fa fa add get along get imply element add use since element case x f lx furthermore large multiplicative e neighbor neighbor neighbor nearest simplify hence approximate neighbor follow simplification md md md still constant decrease probability iteration lsh ensure lsh neighbor lemma text noisy q assumption c next value term ax ax dd low q last imply eq use cauchy schwarz one new find furthermore cd analyse exhaustive lemma get define imply definition top hence eq provide lx furthermore choose base entry plug use multiplicative reduce thresholding
trade steady behavior serve template briefly q eq order group regularize square approach mix norm wise vector norm eq index norm encourage sparsity group group coefficient penalize group recursive update approach work use homotopy problem solution eq fix accomplish regularization homotopy recursive group solution ease propagate homotopy apply provide group complement active two part index finally differentiable reach sub satisfy eq notational convenience drop leave implicit eq subspace definition sign comprise comprise equivalent therefore invertible eq directly class set write constant share eq correspond sub theorem provide remain path contiguous piecewise segment propagate segment segment seek maximum ensuring remain within increase condition come condition exclusive move q update relate form determine provide great point homotopy calculate calculate cost point table bound solution action per critical number critical solution critical group lasso zero call recursive lasso sparse vector coefficient location shift indicate gaussian matrix create recursive group boundary average square implement standard sparse regularization also achieve low steady outperform steady sparse superior mse occur across group system cluster shift group lasso recursive critical accounting trajectory comparison method homotopy base trace critical yield complexity respectively develop homotopy solution acquire use previous warm homotopy streaming measurement predictor steady square non partition future overlap derive remain q calculated formula simplification define define definition proof ease range monotonically ensure calculate one one q eq q therefore critical value concatenation comprise part q condition indicate critical value accord critical eq solve develop fast require c positive critical general q exist denote slope segment piecewise generally denote base linearity show sort x n ix nh x seek edu group penalize predictor compute exact penalize recursive minimize develop homotopy simulation outperform regularize group identification direct homotopy system filtering prediction processing field acoustic wireless interference
compute ep end minute result cpu assess condition generate sample compare response stimulus separately position compare successfully answer function sample glm link predictive confidence fit reaction versus reaction grey dot line offset right dotted capture data reaction look mean distribution reaction indicate abc posterior cause concern behaviour abc explore suggest problematic shape posterior bad problematic begin toy multimodal posterior iid hand variance line distinguish thick thin versus ep abc ep abc course vs pass vs st correction ep abc ep abc iid obtain thick pass stop execution completion site negative definite slow update line figure slow show type assess correspond site perform pass site seem slowly try run pass invertible behaviour thesis run ep abc fail cause failure definite problematic principled thing could correction ep relatively distribution use build correct toy modal example correct serve third gold standard available manner goodness distribution answer much separate ep produce estimate manner ep might lot mode exist might previous unimodal behave theoretical convergence ep behaviour adapt reaction would scale probably reaction model imagine pick reaction category high accumulation speed create speed vs threshold generate plot still mcmc inference accumulation rate two category uncertainty uncertainty run strategy ep abc probably condition matrix arise eliminate choose million acceptance pass minute stability abc reaction mark kernel density check summary statistic reaction time space composite also context replace posterior marginal observation negative exist straightforwardly abc treat make describe make concrete class likelihood intractable hence marginally may take illustration alpha tu stable fix skewness scale parameter gaussian suggest transformation fourth create successive block composite approximation composite take possible subject time alpha average abc abc posterior constraints metropolis abc filtering running time three six hour iteration particle alpha stable especially reasonable source consider gold approximation different set constraint ep abc cl apply size prove model could likelihood version ep composite likelihood spatial distribution tractable ep abc deal marginals ep approximations block dot histogram posterior limitation likelihood posterior poor multimodal one scenario mathematical ep assessment understand start recently address error certainly important remark first ep empirically free result base abc designing site summary must ep away completely take seem quite application summary limitation abc future ep helpful author author ep generic transform p exponential family distribution describe simply compute p create hybrid natural site moment generic exponential family stable update list family essential derive hybrid h product case hybrid hybrid calculation yield hybrid dimensionality correction ep approximation henceforth correction correction derive expand correction truncation might everywhere although arise application un normalise correction particularly ne il integration easily obtain product quantity ep converge happen site matrix correction expectation improve correction nearly abc expensive discuss social science efficiently bayesian still thank approximate abc abc routine often rate introduce choice make quite user ep know fast magnitude standard produce ep abc replace statistic iy iy point possible summary entirely ep likelihood free novel abc word bayesian free quasi science example include biological choice economic evolutionary biology environmental intractable like usual statistical traditional tool directly introduce explain often quantitative analysis reproduce free context version abc conditional keep otherwise reject vector summary statistic quantile sufficient vanish p suffer two error density play small induce increase lead somewhat although increase acceptance e g matter analysis may take tune real problem automatic recently problem still one sometimes pilot introduce ep abc adaptation advantage ep may hour day ep requires decompose possibly way sequentially ep abc convention way operate ep replace possibly abc simple dimensional summary whole iy I iy small course amenable least detail paper discuss genetic collect recommend construct theoretic figure supremum equivalent impose simultaneously abc simulate dataset abc resp minute minute number transition obtain ep density obtain ep run apply use pseudo actual sum perspective equivalent process indicator replace abc resort mcmc sampler design walk marginal approximate marginal sampler big situation approach markov report day cpu simulate transition correspond practically ep abc black latter difference abc corresponding bit close reaction behaviour subject subject stimulus move move alternative observe measure reaction random reaction time information decision reach accumulate variant evidence illustrate reach boundary determine response wiener independent wiener process reaction corrupt represent time subject execute answer r r credible capture reaction basic experimental describe allow vary trial trial mechanism reaction assume boundary stop high determine need account subject
g institute brain regularization model combination approximation effectively even numerically lasso recently pmf show pmf yield reconstruct inconsistent input correlation globally fast pmf problems input scale cubic introduce multiplier formulation cubic close linear dimensional inverse variance least square ol vector covariance problem approach typically uniquely addition ol well get replace rank optimize validation improve interpretability lasso linear ols interpretability regularization prior shrinkage ridge result optimization problem long approach variable narrow spike center distribution high thus mcmc bayesian approximation posteriori tend slow propose integrate binary selector remain analyse form absence prior motivate organize variational variational term solution define control prior insight solution compute close resort approximation variational variate sample sparsity phase plot solution behavior close argue variate map lasso regression pair pmf recently pmf outperform example correlation level irrelevant globally fast pmf tends regression p input x factorize eq q since affect identical spike contain representation detail compute due variational pd likelihood spike selector selector bit n spike substitution w use identical parametrization parametrization require prior equivalently involve quadratic bp solution base bp variational approximation energy find bind width simplest factorize specify expect term first term line prior give set equal fix eqs algorithm variational procedure non solution minimize n non eqs expect replace variational equation would expect ols normal ol I variational mechanism depend dynamically adjust thus rank rank rank remain note fix choose validation prior help step obtain increase implement anneal sequentially empirically minima far low energy eqs well alternative dimensional eqs require repeat summarize minimal cross validation pn I pass cross validation input uncorrelated map without resort sort otherwise optimal eqs interpret variational eq explain total reduce explain interpretability factorize although coefficient eliminate correlation number sparsity illustrate appendix transition solution unique line exact right bottom low corner corner unique dot solution solution minima stable well low indicate order variational solution co separate variational result effect compute decrease solution variational easy modal small negative inaccurate multiple minima region dash dash interesting variate ridge lasso ignore deviation depend ridge prior negative solution depend difference ridge shift identical behavior interpret soft variable identical plot extend variate orthogonal breaking except term increase except depend way plot qualitatively variate compare regression pair pmf input variate gaussian specify structure optimize optimize ridge quadratic lasso pmf regression pmf optimize dataset input pmf ensure pmf take input teacher minimize correct range solution fig plot minimal variational run solution low error fig component correct plot versus minimize fig feature bad remain large sparse row bottom row regression middle b solution ridge versus r r train lasso significantly outperform ridge prediction parameter non difference lasso pmf lasso solutions pmf input multi gaussian compare pmf instances r ridge pmf pmf outperform lasso ridge prediction pmf lasso pmf randomly correlation weakly input fail pmf limit pmf keep constant correlate input get minima instance yield instance generate randomly component two strength correlate correlate pmf initializations lasso inconsistent consistent consistency different respectively fig inconsistent versus range bottom leave bottom right r b optimize trial inconsistent yield large quality lasso bad example suffer find remarkable might sub minima pmf pmf approximation consist predictor include value pmf pmf denote hyperparameter see pmf perform fix initialization random pair sampler pmf pmf pmf approximate truth error pmf together confidence initialization show obtain small observe pmf two solution soft initialization showing initialization agreement contrary pmf always evidence pmf analyze input practical relevance genetic scenario active active run plot area roc definition method except pmf train pmf low theoretically generalization target figure operate roc calculate weight inactive roc positive fraction threshold area classify perfect function number pmf regime poor regime column shift value pmf regime pmf slightly well area roc pmf ridge lasso pmf correlation average run plot roc mse pmf validation pmf inputs nucleotide nearby highly correlate distant snps strong prevent size pmf preferable comparable pmf interestingly difference pmf minima pmf feature conclude analyze method scale cpu figure show pmf describe appendix fig pmf constant
estimator see proposition give influence q remark influence function ht unbounded limit reliably form calculate generate normal distribution sorted iteratively repeat value value mle parameter robustness influence clearly greatly example involve nice unimodal excellent exhibit histogram maximum ht conduct explore generate robustness median well divergence hellinger estimator include huber location show analysis model estimator case good mle term theoretically note asymptotic size perform mle comparison various table high contamination small outperform mle decrease dramatically aim investigate divergence univariate exhibit behavior evaluate show solid normal keeping demonstrate amount contamination provide good maximum paper divergence theoretically robustness leave study open estimator divergence location member offer attractive maximum robustness key phrase divergence estimator robustness estimator divergence widely model framework context divergence appeal bandwidth way continuous smoothing benchmark condition full divergence robust test approximation divergence censor introduce estimation copula refer divergence estimate function location concern article suitable capture robustness insight attain huber without efficiency background concern devoted deal remark divergence absolutely divergence kullback give divergence divergence chapter class overview origin interested divergence recent simple I interest underlie dominating function exist eq finite mp define property mle coincide mle leave divergence keep parameter divergence contamination crucial outlier illustrate empirical contamination shift contamination parameter maximum remain close divergence associate divergence mle affect outlier robust divergence equation improvement estimate robust estimate ht contamination outli outli term stable evaluation value situation assess robustness estimator
way counter multiply bootstrap get conservative large come quantity weight create bootstrap weight prefer yield thing equal prefer bootstrap variance well weight uniformly nz nz naive simplicity observation implement replicate quantity weight index replication represent possible stem delta usual iid holding fix nz nz delta method bs double stability proposal give example get share pt reweighte mean delta z refer take product derive exact introduce k z u un fraction match exactly triple match matching effect model coefficient ideally would bootstrap biased typically conservative sometimes interesting instance trivial negative bias column resample bootstrappe name page method way paper noting model call weight exact gain interpretable netflix worth point arise form sample level factor greatly dominate internet factor country aid interpretability construction match netflix datum dependent people name phone many even phone people phone fraction simplify may expression retain explicit constant gain find bayesian ordinary bootstrap variance setting reflect additive nonzero coefficient reweighte bootstrap variance hand ed bootstrap gain coefficient nonempty mod term netflix variance extent differ variance bias inference unbalanced entity systematically realistic produce error tend un next development effect mod give exact gain interpretable bound gain term q gain reweighte q bootstrap conservative dominate every analysis use small sum exist fold random share index index replicate nest within index nest outer ordinary word include outer nest another subject outer contain repeat measurement ratio outer length facebook user highlight substantial presence additional effect three reject would quite interval portion conservative worked conditionally hold clear informative thereby introduce mean correction netflix company rating unobserve rating people netflix try rating make pair predict bias facebook difference comment comment accounting involve infer comment make comment conditionally describe stability comment length adjustment kind mechanism necessary value sampling bias adjustment build bootstrap adjust alternatively statistic bootstrap seem effect bootstrappe identify suitable bootstrap plain practically important example click rate usage bootstrappe biased effect set product weighting replace acceptable properly calibrate variance multivariate replace variance covariance consider bootstrap extend smooth expansion use confidence interval estimate central properly calibrate correct bootstrap percentile confidence conservative percentile way via estimate define mx mx practice histogram set loading depend let describe l loading make could contribute generalize svd extremely contribution handle bias bootstrap phenomenon bootstrap inference accurately appendix contain easy theorem statement preserve proof q numerator z z model u naive bootstrap naive resample theorem stability naive variance effect x z n effect expand reduce z w z sum vanish establish variance hold nz nz variance bb bb w n n theorems approximation coefficient theorem show factorial line eq un un establish interpretable variance reweighte z w ii w distinguish next turn proof k effect ed gain nonempty exist contribution similarly u theorem begin analogous expand assume prove n z establish proof theorem last z model coefficient product reweighte next r reweighte dominate j eq j r j comment support nsf dms bootstrap estimate take set independently observation independently conservative biased variance unbalanced apply computation parallel illustrate comment facebook effect commonly arise internet service automate draw netflix rating neither iid interaction internet easily two factor account string document place web page measure spend read load page version rely balanced necessary might mechanism specifically independent mean random broad jx ij pt approach bootstrapping independently bootstrap bootstrapping put level factor w r j j j bi j get bootstrappe bootstrappe row usually relatively balanced missing remain conservative sample unbalanced random effect every column interaction resampling analyst variance product I ji scale computation bagging reason fashion multinomial bring substantial synchronization cost expression dependence distribution easier develop effect main variance easily computable give explicit formula naive bootstrapping estimate instance netflix resample column close n comparable acceptable contribution naive strategy fact resample generalize become reasonable component roughly simply describe dominate random every nonempty random bootstrap effect distinct variance bound away outline introduce observation define seek naive bootstrap unless high case bootstrap factorial closely match resample interpretable consider combination variance effect dominant bootstrap closely match variance repeat nested one reweighte facebook uk long comment mobile device reverse interface significant factor structure appear product effect discuss among reason variance random effect contain interest integer notation write mind level internet ad priori compose value observation practice order estimate apart brief subset sum name hold random depend u u x v writing general expression denominator goal resample different observation index convention quantity important dependence pattern j j match say highly perfectly customer phone specific motivating define pair match
purpose must hash orthogonal hashing projection rely pseudo practice hashing replace hashing modification time trivially conduct line pure research notice truly loading dominate memory testing svm benefit hash gb preprocesse cost loading hashing reduce bit fraction loading cost still load subsequent report experiment certainly would experience mining community hash especially similarity mainly work view either two set use one apply hash bit storage prohibitive truly application efficiently express inner product immediately hash inner extremely concatenation total indexing use development bit hash simple solution store low bit convenience datum highly example small highly large replace bound extensively independent permutation encode often e provide logistic popular package regularize solve optimization regularize important penalty purpose demonstrate bit simply conduct approach store bit value way store merely bit run suppose originally binary digit expand feature fed solver experimental follow hashing randomly select testing sample gb dataset expand dataset product combination product choose tool effectiveness experiment conduct cpu ghz gb ram windows l gb testing expand datum also test expand training wide fine mainly accuracy hope entire gb resource accuracy accuracy plot result one figure verify svm merely accuracie h dimensional sketch algorithm variance rp review rp convenience vector product general multiply e g generic ij er q satisfy small elementary know point distribution probability refer count sketch algorithm vector introduce write biased inner correction multiply entry situation multiplication hash er er er vanish essentially option pre multiply course variance become number dense probably bit need time hash square norm inner size gb logistic solid representative bit hashing achieve substantially bit dash panel plot bit advantage bit hash hashing training hashing suggest bit hashing far apply additional bit indexing extremely large number conduct notice indeed reduce hashing widely hashing require practice well understand use good hashing simulate permutation preprocessing require time gram trivially hash substantially consumption store example hash disk task near etc example learn logistic truly large datum loading dominate loading gpu loading without gpu preprocessing time magnitude train gb expect loading time preprocesse gpu preprocessing cost loading gpu gb conceptually hash require mapping store mapping application however store permutation would infeasible permutation require store popular hashing uniformly store store hash e avoid modular gram gram location walk parameter deterministic simple permutation store permutation practice permutation hash solid svm right curve simple lot bit applicable hope draw interest research search microsoft microsoft microsoft com work use bit gb may compare access paper study expand report merely per
contrary seem execute effort routine mathematical device teacher truly system fully supervise supervised reinforcement ai ability attribute new teacher letter suggest architecture way prototype dynamical shape external process biological knowledge scalar time represent certain origin visual sound pattern algorithmic adjust strength weight pattern profile form phase space nn fix new occur nn evolve towards technical occur formation spurious unsupervise considerable propose construction dynamical shape external without supervision stimulus ergodic dimensional probable new like place profile move possibly affect space base loose analogy slowly shape pressure assume initially flat profile fig drop distance elastic factor capacity forget deeply fast try forget online memory external stimulus moment drop position stochastic nature arbitrary derive profile consider interval shape describe g function q however show shape condition leave behave decay fluctuation trend sort change evolution profile b circle apply delta combine go infinity tend dirac delta kind peak shape line consecutive correlate actual circle eventually uncorrelate fast dynamical algorithmic restriction evolve recognition phrase child song six song involve three choose illustrate principle wave file agreement speech slide sec roughly peak extracted note frequency slightly frequency respective note introduce see individual automatically hz save stationary ergodic consist song finite line automatically probable frequency show hz hz hz hz hz hz phrase phrase signal phase ft ft ft ft purpose finite multivariate visualize way fig take coincide axis connect feasible pattern point whose probable five scale also wave file create machine digital computer neural device close propose unsupervised traditionally however supervision I regime ai device pattern combination major curse complicated ai device grow become connectivity curse work duration calculation device curse acknowledgement critical comment play approach construct
method tree interaction associate associate freedom though forest offer construct tree quasi random property underlie causal causal predictor adopt subset ridge limit penalty arbitrary alternative preference reflect pick category interaction contribute relationship accurately describe decision model identify logic spline restriction product fall category place restriction convenient formulate consider relationship f g kx disjoint predictor allow restriction partitioning expand copy accordingly require belong useful order group f determine predictor nan group interaction term vast hope relationship determine fortunately detect contribute usual formula partition nuisance pt base construct g association multiple copy keeps assign describe multivariate response trait response logit link marginal parameter require exhaustive partition infeasible sized use chain monte carlo statistic propose component propose processing spend posterior almost speed methodology provide section material carry design various comparison exist detail simulate remain material typically predictor causal response ask three examine predictor scenario concentrated underlie relationship ii iii particular frequency predictor multiplicative third involve present relate underlie average causal different predictor figure interpretation reporting successfully detect causal well clear increase essentially maintain drop second iii come perform simulation prove robust relationship violate generality performance pt previously study individual examine set http university www ac uk supplementary individual nucleotide expression level dr di interaction snps million present snp display bottom report circle run triangle dash vertical mark snps interact provide threshold top association test detail text promise rs rs association search highly correlate often fine genetic snp rs former snp latter posterior interaction rs indicate project pilot trait focus gene decide eight original phenotype correct population choose miss approximately unobserved plot bottom plot main text pt method strong association suggest single appropriate mark vertical suggest strong association allow contain explain variance variance examine deal encounter project expression know snp association within region mark achieve genome association approximately suggest spurious gene association rest top truncate project release across five snps subset g contain reflect event snp allow miss infer reliably et decide adjust true plot figure result vertical find association give great recognition possible causal allow identify concern know case hope verification count heterogeneous stock snps length using design value unobserve provide gender code genome decide cd link g decide copy demonstrate top due linkage rs indicate top snps repeat show run offer plot snp pairwise gender association significant test interaction snp gender snp act gender specific agree allow multiple copy option material fitting predictor group interaction unable distinguish say way interaction copy predictor recommend tune superior power iii try believe simple model method compete fairly strength demand many search interaction benefit consider existence strength derive generality perhaps suffer complicated disadvantage underlie certainly mcmc estimate scoring visit error seek estimate use bayesian mechanic generality posterior correct space partition unfortunately small force visit force instead able try variability introduce compare expect strong association repeat random highlight obvious trace plot additionally permit repeat association scale linearly require base true suitable predictor pick high certainly true association standard pick perhaps top biased predictor reflect simulation five supplementary almost heavily demonstrate drawback treat predictor natural overcome application pursuit develop additionally influence inclusion neither fit chance fit offset inclusion imply unlikely predictor appear consider would moves resort exhaustive predictor suitable transformation reduce continuous measurement neutral increase decrease hope whole detail let nan belong unique order irrelevant ordering single set model predictor expand copy predictor alternative keep predictor relax alternative discuss interested observe fully predictor calculation nuisance pt partition construct probability predictor equivalence contain partition predictor ensure marginal assign equal calculate predictor way th contain calculate recursive fashion bs bs q weight member place even straightforward could favor interaction must largely due couple reasonable allocate calculation ensure extreme size practice vast amount unnecessary assume summation equivalence achievable condition hold involve prior ks ks g p ks sp g ps ks sp n cumulative value find copy predictor allow association copy set copy associate multiple copy predictor element partition partition copy effect prior relationship increase necessary specify minimal recommend therefore underlie write regression f k functions assign penalty categorical belief control extent smoothness residual integral incorporate reflect preference small improper binary logit
algorithm contrary value jump opponent opponent jump drop slightly finally next jump approximation last opponent tracking iteration two behaviour opponent approximated number iteration factor play become identical fail adapt equal play opponent jump play drop account lead opponent article performance geometric play player hill hill game risk dominant nash c ccc episode payoff end replication parameter track pre ia simulation smooth allow factor whereas play play nash replication concern geometric play payoff factor play nash equilibrium difference depict propose play geometric target assignment game agent coordinate common total target specific action target target many choose action target value utility target product express utility produce life receive target greedy simulation inverse target target sample fix end information update strategy target action depict instance geometric play instance step score instance score good equal need reach reward management simultaneous occur area people collect people capacity capacity formulate player one allocate variant utility objective save express utility add people player one player help follow available scenario reach generally action time need reach one people trial scenario easily unnecessary approximate response solution algorithm branch base ratio could variation play gain mean variation bad percentage overall percentage people c c c scenario geometric play coordinate percentage group play regard percentage trial performance adaptive play decrease observe play scenario include structure one time penalty great utility variation play initially search utility need local optima influence instance stop play geometric c opt stop play c opt time geometric factor play percentage collect step even people geometric area iteration result play perform adaptive factor play perform rnn percentage overall people percentage rnn collect percent play percentage factor play percent evaluate penalty penalty inefficient allocation exclude ratio evaluation especially play classic form incorrect give recently stream datum examine impact play choose carefully induce poor combine low weight estimation drive adaptive factor hill assignment management play play planning use empirical play intensity play david department mathematics uk game learn game play play implicit variation allow realistic opponent streaming literature observe geometric play variation theoretical optimisation optimisation sensor traffic scheduling domain communication complexity optimisation optimisation term nash versa maintain belief belief choose expect reward strategy action equilibrium game practice assume particle predict filter difficult paper adapt action empirically step play need hence communication overhead optimisation demand propose classic start description game section present result hill game target management scenario finish games optimisation optimisation element game joint player choose accord select action deterministic choose act strategy action strategy mixed element player gain choose resp resp decision player choose action expect belief strategy strategy response show least equilibrium equilibrium equilibrium imply possible player utility change strategy select equilibrium action pure pure nash equilibrium particularly category game agent game utility follow stand player player game equilibrium life design individual utility potential act utility obtain select obtained arbitrarily choose eq optimisation game prove equilibria player increase play game play choose response belief opponent player belief choose action player update belief good belief beginning maintain arbitrary weight function formula opponent follow belief play nash classic play strict play game steady play payoff game game nash equilibrium become period use use response action choose action strategy case player rare player mix variation originally poor serious change filtering mean depend descent respect residual error online stream generalise factor adaptively change ascent log new sequence stream parameter write arrive factor express fitting update instead classic play maintain belief place time action discount weight belief play similarly geometric play close action classic rule consider opponent form identically independently opponent opponent strategy respectively update recently observe eq order derivative j adaptive ensure leave player action belief response play player carry weight j accord response opponent factor lead poor ascent lead algorithm area result big jump solution evaluate play combination opponent stream opponent strategy repeat time time measure square range combination depict dark less value great approach wider observe great square suggest value behave play game easily opponent independently poor estimation opponent
exactly none fix true marginal algorithm bp perfect marginal exploiting assume correct turn gaussian match one component performance ensemble accurate still available hull reach equilibrium bp consistent increase bind observe equilibrium propagation learning proceed bethe algorithm cause converge learn marginal project cycle circle onto space color inconsistent black belief precisely marginal red even amongst fix marginal bp red concentrate comprise h statistic first ix positivity appear ij interaction bethe hessian figure ise couple bethe hessian width pdf marginal bias marginal measure bethe select bar median bp performance encounter bethe iv last gaussian distribute statistics iv generate ise target select bp ensemble bp method parameter run message time step message change belief bethe learn performance ensemble iteration bethe learning belief poor target loop get marginal guarantee selection target belief give excellent use give order limit gaussian ensemble make bethe belief ise unstable bp hessian throughout bethe convex bp fix yet might hope belief propagation systematic bp hope provide work fix point average preserve locality scalability bp raise possibility inference generalize novel novel input belief use fail example happen bp failure thus address marginal available marginal exist variable hide phase engine brain inference perhaps belief undesirable circuit big blind reasonable inference draw precisely occur bp blind eliminate mechanism perhaps fluctuation average thereby conclusion mechanism acknowledgment university york ny york definition belief loop one might hope adjust purpose explore previously claim marginal belief probability marginal propagation whenever hessian bethe energy definite produce inaccurate belief belief parameter perturb learn mean calculate generally require sum exponentially np hard one belief bp product pass algorithm efficient merely approximation model loop appropriately achieve belief case ensemble marginal p write index interact variable sufficient statistic irreducible write sum linearly depend collect match marginal algorithm energy use converge belief must graph equal graph belief guarantee ib globally joint belief marginal belief simply belief circumstance belief bethe energy kullback leibler energy optimize energy appropriately depend entropy minimize recover energy entropy entropy neighboring factor structure graphical bethe entropy bethe energy bethe marginal nonetheless bethe energy often enough gibbs free minima approximate marginals minima bethe free successful bethe free write satisfy positivity consistency propagation matching marginal simply condition bp reach bethe span belief bethe hessian width pdf slice bethe free energy line axis space parameter energy add minima second free energy identical parameter slice bethe colored bethe hessian bethe learn belief blue along point region jump region exist binary energy symmetry target marginals bethe eigenvector thus bethe marginal occur minima bethe marginal exist actually quite multinomial loop definite small correlation definite bethe pseudo moment matching fail descent procedure descent leibler free bethe divergence energie bethe energy optimize propagation bethe entropy energie belief propagation obtain belief current change opposite direction generally bethe free belief decrease allow draw
issue action policy round subroutine application sensitive oracle work context modify argument use probabilistic net cost thm sensitive require tractable would common reward feedback delay round straightforward modification proportional delay policy definition domain let arbitrary round choose reward action round choose learner choose reward short learner set enumeration policy impractical general illustrative second option oracle correspond learner policy reason sensitive interpret negative cost policy denote maximize learner produce nature history take policy follow relative learner internal randomness regret policy denote context shall give context shall action etc correspond randomize x b tx choose ta step policy induce value theorem find specified method algorithm step project onto smoothing finally determine analyze first game player produce find allow randomized policy correspond space randomize policy hull feasibility randomize inner expectation convexity expectation max without value hand satisfy non existence see constraint policy elimination quantify show policy yield eliminate yield second fix inequality denote conditional obtain analyze monotonicity yield least round history approximately value satisfied slack choose reward minimax keep explicit good policy version eliminate chance recover perfect address short present ucb keep action choose explicit tracking avoid implement suboptimal frequency use previously effect follow round appendix variance policy quantity incur round induced quantity draw high probability bound value round direct round mostly policy bad bad policy allow estimate analyze show relate reward defer section solve describe polynomial optimization ellipsoid method solving equip separation separation oracle produce side ellipsoid standard center decide empty give two empty processing use ellipsoid oracle fix contexts kx px convex program eq equivalent optimization optimal follow require interpret distribution equivalent solve search testing feasibility detail complicate need requirement lemma leave oracle separation oracle first consist separate hyperplane constraint convex set separate separate point oracle check expectation least section show argue class close result positive write average delta function approximate choose drawing denote least let q v follow gm fix eq eq q distribution z x mean z mp chernoff third fourth final display observe maximum second display fact inequality follow display lemma respect add existence proof determine fix guarantee eq factor bound reward quantity check index round effect allow slack constraint constraint slack e kb state slack somewhat arbitrary slack appropriately eq bound pick follow deviation pick least let easy since union bind pair next lemma imply since lemma imply satisfy slack combine k inductive deviation suppose contradiction fail fact contradict must immediate show policy large round round assume pick third inequality lead existence feasible non small eq compact condition otherwise prove round trivially sum condition hold lemma union approximately ellipsoid fix drop subscript become relax ensure region constraint set point cauchy schwarz fact etc b form find point call relaxed program ellipsoid recall requirements radius radius ball region contain feasible feasible region feasible consider point assume eq thus hence give ellipsoid perceptron one candidate point feasibility check constraint constraint separate violate hard recall follow follow iteration hyperplane separate policy satisfy hyperplane put algorithm candidate need shift origin perceptron update separate hyperplane assume perceptron separate hyperplane hyperplane fact run algorithm convex hull perceptron step policy euclidean projection quadratic finally give else rewrite define xx check constraint ellipsoid run ellipsoid program hyperplane separate separate ellipsoid fy ellipsoid conclude infeasible feasible notational convenience define separate ellipsoid start ball solution constraint check construct separate hyperplane else check separate else point perceptron policy lemma ellipsoid lemma every lead algorithm lemma thm corollary claim definition problem observe receive optimal oracle setting enable delay previous bandit follow loop context information action action world present reward bandit reveal choose several feedback prefer essence set difficulty avoid reinforcement bandit half half motivate contextual interesting news article ad internet naturally model bandit medical generally imagine future essentially bandit consist distribution choose algorithm set success compare policy regret computation policy policy space computational hope hand reveal classification sensitive regression classification efficiently bandit similarly large bandit sensitive oracle
number write exist similarly write let let correspond shall assume self many time denote embed throughout fix positive distributional fall depend manifold noiseless allow manifold nontrivial fix compact set hausdorff I measure gm attempt idea model model noise estimate homology group minimax case noiseless fix satisfy tangent plane uniform measure item g state manifold proof use sphere top result manifold sphere around bottom construction make rigorous q tight case omit point seem seem actually occur estimation circle indeed rate smooth boundary remove effect recall clutter model give denote direction tangent positive mod tie maximizer vc hyper least let high see radius empty contain ball hence lemma sm ns mx sm constant support hausdorff compact dimensional gaussian gm since manifold hausdorff could unbounde truncate q enough tm u u picture density bound identity g g I dc vector c du du u du kt dirac delta generalize corresponding integer kt kt hence le ball center origin cauchy schwarz q u e conclude conclude eq special function estimate agreement however technique quite proof technical deconvolution zero around origin simplify considerably thought estimator let define calculation j define define let dt dt dt dt dy kb bx gx gb fix h u ga pt define complex dm nn follow combine h dy lm lm lm lm n prove deconvolution particular slow consistent wasserstein case form regression deconvolution rate conjecture estimator might rate slow manifold deconvolution error purpose section recall model singular manifold recover usual deconvolution support dimension regard deconvolution favorable bound ordinary versus show typical favorable bind ordinary deconvolution top top plot distance density bottom plot nearly indistinguishable favorable pair need density favorable manifold density rather plot perturb circle hausdorff density nearly indistinguishable fact variation support relate manifold regression measurement observe usual convergence deconvolution indeed nonparametric measurement error let however moreover manifold lower favorable follow convolution refer convolution generate favorable pair use case favorable fan subtle way eq choose q drawing reduce error special like usual upper construction manifold except try methodology realistic additive deconvolution know least seem realistic proxy work remark nsf dms national support nsf dms air fa risk hausdorff several connection class area machine yet basic manifold minimax risk hausdorff estimate assumption riemannian include compact nonempty noiseless noise far usually call study singular put mass
separate treat calculate I kde represent statistically source heterogeneous population kde diversity support diversity still support variation heterogeneity power dependence help determine composition variation likely versus contribution diversity important concern f exclude pdfs differently relative bin study evolution vary important like population include also convenient keep point represent pdfs graph gauss quadratic gauss interpret diverse variety circumstance map understand unit gaus non domain set include quadratic series set gauss set baseline pdfs within versus totally effect bias calculate quadratic support diverse amongst pdfs identical thus index save computation analyze l l l c pt source quadratic gauss map value gauss decay expect support variation detecting support diverse support representation disjoint set six table notably maxima homogeneity support totally furthermore average aggregated different conclusion sum aggregate base metric diversity uniform purely location information compare intra nearly inter agent intra highly disjoint intra show case nevertheless detail pdf aid first consider variation pdfs sensitive sensitive considerably pdf metric slightly interpretation component population scale versus population dominate overlap instead disjoint much close compare identical system support normalize useful proportion diversity diversity var boundary range analysis variation support contrast support variation especially quadratic gauss aggregated set homogeneity heuristic detect pdfs meaning detect pdfs moreover like diagnostic expect none mix effect support effect compare map set point metric diversity homogeneity quadratic end metric diversity show homogeneity perturbation especially one however primary set upon contrast diversity among pdfs entirely surprising great pdf estimate size highlight display motivation base moreover converge analog rather decrease mean sense add always relative likely datum considerably base therefore aid point analog analog target datum induce add infinitely string existence introduce divergence estimator true thus exist sample say string bias bias aggregated datum homogeneous structure represent infer whether population ref presence correlation simultaneously population composition difficulty previous datum particular patient repository patient measurement range measurement patient collection patient least visualize normalize pdfs population pdf plot wide pdfs fig motivation choose moreover value likely homogeneous compare snapshot difficult situation existence extremely broad patient likely substantially say second representative small effect kde considerable patient measure see time consider conjunction say differently relatively member fact reflect variance order versus range value pdf diverse less sensitive pdf variation also population nonlinear evolve measured population must care constant know representative temporal five notable peak interesting population outside window informative hour peak presence human appear less scale long drop work third comparing observe difference homogeneity amongst combine qualitative somewhat homogeneous population enough resolve signal considerably fourth aggregate peak usefulness aggregate measure evident compare histogram kde estimate fig identical aggregated calculation display strong error evolution valuable otherwise interpretable context hour day fig kde extremely periodic evident fig bias kde imply presence daily clearly evident kde average time theoretic reach population hypothese seven heterogeneous homogeneous seven patient regard population code act proxy composition assign patient two frequent member six occur pay patient drop patient frequently code drop code drop homogeneity broadly speak code conclusion draw theoretic static reveal heterogeneity algorithmic computer diverse non measure length diverse likely interpretable population diverse picture particular calculation imply information homogeneous filter frequent measurement yield homogeneous correlation patient likely population require nevertheless analysis result paper address within framework I multiple iii distinction stationarity case handle section ii difficult detect state ask amongst fail multiple issue need multiple piece standard paper series differently single source appear know single state amongst individual support relative paper mean whereas likely heterogeneity say pdfs point per one upon variation source perform calculation future generate individual eventually integrate scheme mathematically population nevertheless detail remain list include regard technical claim etc relationship imply propose practically complex series leave interesting question thank carefully read useful comment acknowledge financial grant lm begin recall recall pdfs define abstract pdf pdf next convenience ip ip write term collect form begin recall average abstract relative pdf convenience define follow never value summation collect collecting support diverse well represent support diverse disjoint support homogeneous determine heterogeneity well population etc contribution quantity source set delay composition origin primary delay mutual delay aggregated tool detail delay understand heterogeneity population nearly time university health record repository picture composition aggregate set pdf understand series normalization deviation demonstrate study human series practically theoretic insight composition time dependent degree homogeneity human measurement aggregate measurable system measurement way treat lie average ergodic require gain much aggregated population element advance make physical statistical mechanic aggregation context fact often average quantify problem homogeneous produce aggregate whether aggregated population mind delay mutual understand density quantitie iii diverse manner apply possibly diverse need aggregate ref short focus receive care center clinical measurement human population analysis limit broadly split primarily background focus characterize second characterize propose metric demonstrate read theory devise interpret delayed series motivation come understand health complex biology record repository university datum collect future human contain regard million contain laboratory difficult particular correlate state population type disease phenotype population understand disease disease evolve define completeness medical record benefit wide spread carry practical gain wide biology represents term physics motivating complexity laboratory science nearly datum control many easily dependent context g begin assume discussion pdf note approximate estimate density specify support pdf lie always exist pdf may seem critical pdf auto delay average aggregate member interpretation employ kde calculations develop ref kernel bandwidth estimator relatively insensitive qualitatively moreover use point amount permutation order pdfs detail address sum individual represent essence integration perform define note random want hold interact statistically necessary interact copy independent verify merely apply conceptually correct note product dimensional orthogonal thus theorem conclusion individual average understand pdfs value individual element aggregate pdfs length treating series calculation add mathematics source collect individual specifically pdfs individual include point individual specify specify delay calculate intra column denote dedicated quantify implication aggregate compare aggregate sample versus aggregate methodology numerically iii bias aggregation estimating density proportional poorly measure compare understand datum point broadly average aggregate computationally calculation calculation suffice pdf style bias follow note datum carefully quantify estimator versus bias pdf aggregate population differ evenly time average consider scale linearly contribution eq obeys law difference bias estimate say satisfied element poorly population population importantly converge cardinality aside overall effect presence kde name estimate close kde contrast histogram estimator probability empty yield finite simply kde histogram working understand context poorly measure aggregate population estimate carry calculation minimization bias attempt fundamentally estimate random permutation mix data replacement random population thus preserve inter individual finally exist bias exist aggregated context randomly regard individual replacement vector population correlation relatively pdfs think estimate support pdfs population individual bias estimate discuss dependent point general paper dependence apply use pdfs collect exclude use pdfs calculation whose frequency filter consider example population differently individual population hour second subset sample month represent patient month calculate month represent graph two population month complicated patient particularly health care patient patient possibility filter calculate estimating pdfs change pdfs population estimate notation represent pair individual total pair iii individual population monotonically note normalize square quantify composition member uniformly one composition entire population close composition subset possibly individual quantification percentage individual contribute homogeneity sometimes always correct really give specifically bin representative population resemble median among among abstract definition mean relate get focus q q explicit go relative support population represent write briefly auto information calculation series define thus tend toward understand aggregate ideal stationary source circumstance obey represent intuitively say create aggregate pdf pdf closely resemble helpful concept relative actual population separate naturally conceptually abstract support aggregate one spirit pdfs roughly imagine achieve goal relative severe aggregated define abstract quantify relative abstract pdfs difference thus difference define arrive drop explicit appendix go zero width band pdfs decrease overlap individual homogeneous population band pdfs disjoint population say aggregated population follow second difficulty interpretation calculation yield explicitly completeness define aggregate analog q contrast tend information individual pdf mean information section speak broad practically ii least element leave conjecture accurately represent individual population identical statement population briefly prove conjecture stationary rely eqn homogeneous essentially one population make understand understand source tie population important differ contrast population split broad support iii difference estimate support pdf aggregate collection understand graph pdfs difference permutation population roughly estimator regardless support small next individual permutation plus estimator contribution bias approximate wise zero order visually become gaussian amount primarily drive imply support circumstance estimate average versus aggregate interpret estimate overlap support similarly element instance support lead homogeneity poorly population define less diversity close great diversity worth compare difference reveal principle behind depend estimate time diversity particular variation collection due intuition g offer justified inspection remain happen note act extremely unlikely concavity depend nature whether general less clear diversity amongst situation understand pure individual graphical pdf estimate size well estimate static section dependent contribution due entirely limit aggregate component intuitively intuitive diversity support contribute induced fig begin relative average pdfs p clear average pdfs relative relatively trivial support support primarily moreover individual vary event variation support affect individual pair point effect likely effect correlation time population reflect diverse population establish interpret three homogeneity homogeneity address population ii support address support iii homogeneity address variation pdfs graph pdfs quantify homogeneity moreover homogeneity exhaustive simple intuitive method right moreover least quantity supplement
dependent assign every alphabet string family sequence computable outcome ergodic markov symbol source bernoulli induce variable outcome consist sequence sequence divide probability every note large outcome essence computable repetition contain computable go definition occurrence character string character unless occur order reconstruct induce bernoulli th computable entropy entropy stop string copy pattern reach sequence copy outcome concern bit family alphabet state reach process state state state require program bit bernoulli process frequency binary typical sequence bernoulli computable outcome string know bit generate successive bit variable string bit compute program logarithmic consider case ignore english experimentally compression technique modeling set symbol stream next symbol stream shannon consider alphabet character alphabet since hx high knowledge kolmogorov entropy entropy knowledge kolmogorov complexity observation computable shortest compute add program short program compute sequel extend notion alphabet string computable probability outcome finite respect joint mass kolmogorov complexity theoretical direction reference metric normalize proper manner free measure great impact parameter distance variant test database major turn heterogeneous detection range forecasting music paper string machine like kolmogorov upper metric distance satisfy normalization binary minor ignore string family variable computable computable probability mass function moreover substitute kolmogorov complexities entropy family computable mass family computable probability mass hence ignore computable computable variable singleton mass markov computable clearly function entropy family bernoulli gaussians process consequence bernoulli string identify relative source stre static bernoulli total leibler mean comprise header header ignore joint sequence shannon probabilistic version e hx since mutual variable entropy cite connect two empirical entropy definition string computable appropriate family computable computable variable outside framework approximation entropy operational formal definition intuitive notion domain integer value extend argument rational argument call upper computable countable recursively countable computable example computable kolmogorov recursive example recursive informally string string reconstruct machine constitute program program deal kolmogorov matter plain call formally kolmogorov short input output unconditional kolmogorov machine additive absolute character thesis ability machine simulate execute kolmogorov finite object kolmogorov absolute quantification shannon hand deal average produce former theory much surprising complexity somewhat weak variable string view indeed input another similarity mutual finite string remarkable information plain kolmogorov complexity conjecture national center mathematics science computer support address email compression version empirical alphabet previously consider possibly description induce involve relate entropy kolmogorov basic shannon alphabet interested message receiver receiver consider message message entropy source notion author intuitive traditionally notion finitely outcome string alphabet message entropy receiver reconstruct therefore draw encoding
continuously ny constant nf appearing contribution product w recall multiplication r deduce satisfie simply lebesgue continuous couple weak twice vanish expansion yield deduce mse variable risk regularity estimate unbiased replace reformulate chi put everything together specific transform representation notably wavelet block discrete cosine l via denoise strategy long prove reduce various type possibly redundant pass treat procedure generic j arbitrary perfect submatrix channel complementary channel w coefficient free l actual l rule sophisticated processing pointwise continuously differentiable piecewise differentiable partial derivative introduce hadamard element pointwise proof straightforwardly develop transform denoise prove band need shrinkage minimum pointwise freedom choice replace mse recent degree optimize generalization square estimate directly close yield equivalent potential realization display transform estimator previous square orthogonal explicit remarkably particular haar derivation mse possible unnormalized haar wavelet channel resp channel implement unnormalized haar implement filter z em give unnormalized haar scaling wavelet haar wavelet chi chi square random degree freedom empirical coefficient chi jk since filter wavelet process j jj unnormalized haar scale variable unbiased ease ii involve ii successively equality omit natural wavelet representation introduce justify contrast case removal wish poisson unbiased requirement continuously differentiable approximation show good value pointwise cc right thresholding sophisticated denoise dependency wavelet call show quality poisson unnormalized haar wavelet unnormalized haar wavelet delay sign magnitude coefficient scale advantage persistence let account similarity neighbor thus intra dependency naturally arise haar wavelet parameter optimize via advance process k sense linear finally jk magnitude image image magnitude measurement magnitude n dimensional datum freedom magnitude mr necessary technique magnitude mr rescale eq three quality metric signal visual metric introduce see variance reliable signal background sophisticated various level set mr cccc cc intra inter unnormalized haar unnormalized haar art mr evaluate pointwise apply transform domain intra inter unnormalized haar wavelet baseline provide unnormalized haar intra scale shift invariant sophisticated variant unnormalize haar wavelet cycle technique fig show improvement average cs observe cycle invariant transform cycle sensitivity optimize appear address common choice mmse choice benchmark retain mr denoise code spatially mmse filter mean mr respective mmse optimize mmse support variant cycle intra haar wavelet thresholding pointwise overcomplete haar block discrete cosine note span overcomplete basis previously invariant table observe basis overcomplete consistently denoise relative state condition dependent obtain consider overcomplete dictionary e em cs cm haar haar haar let cm output average realization c haar cs haar let em haar let average various denoise denoise fine confirm higher obtain denoise algorithm execute matlab run os equip ghz intel processor quite image within transform dedicated reconstruction haar cs haar cs file denoise magnitude mr raw image acquire mr weighted signal n treat fig denoise various noise efficiently bias cc em via haar haar let derive magnitude mr denoise comprise continuously differentiable focused transform domain pointwise coefficient consider specific unnormalized haar wavelet multiscale allow adaptive joint inter intra simple soft algorithm test image qualitatively finally mr efficacy mr image chi estimation datum implementation online rgb theorem ex wolfe chi unbiased magnitude mr denoise article derive unbiased expression expect square associated chi chi degree broad class shrinkage transform unnormalized haar art simulate secondary fundamental medical imaging provide contrast ratio acquire mr physical structural strength average encode development focus primarily inherently high image reduce system pursuit objective acquire post denoise essential visualization meaningful acquisition fourier sample random primarily fluctuation degradation pattern straightforwardly inverse discrete space resolution reconstruct image determined frequency work accelerate mr acquisition trajectory nonlinear consider axis work image consider apply wavelet mr real article chi unbiased estimation denoise mr image two
cause thus algorithm construct dataset construct coordinate synthetic random tend irrelevant use try lipschitz repeat normalize observe rather across difference normalize ht lin sim concrete value report squared error mean glm lin sim community normalize perform uci choose dataset glm standard simple heuristic sim index procedure problem fix via perform fold fold table square across ten fold normalize variance fold many slightly illustrative transfer function plot fit lipschitz result piecewise intuitive smoother find reader notation show error current x x w u literature rademacher rademacher entropy follow class standard utilize square rate union particular element sense union easy right pick verify f hilbert integrable inner yx verify prove minimize therefore convexity require establish inequality statement inequality convexity square w dimension begin lemma note u well least simultaneously invoke get cover analogue somewhat bind dependence let x tell empirical risk minimizer follow lemma lemma argument combine invoke usa microsoft usa microsoft generalization regression target dimensional problem often provably efficient glm provable provide efficient practice linear function value assume flexible extension existence link expectation immediately include regression aware rate provable square challenge practically question non achievable simultaneous estimation former continuity possible effectiveness appropriately attempt good note problem significant body heuristic improper learning type sim agnostic setting flip complexity polynomially provably recently provably efficient common assuming model computational perceptron regression unfortunately datum request empirically assumption glm hardness theoretical provable seek address theoretically algorithm monotonic lipschitz practical parameter free provably computationally moreover feasible original ran case lipschitz generally assumption glm efficient despite non assumption learning sample support feature decrease role lipschitz equivalent restriction measure equivalently counterpart write construct decrease algorithmic condition presentation result constant problem properly simple perceptron algorithm close arbitrary decrease appear algorithm input tx show iteration nearly independently decrease iteration appropriate pick line somewhat rough idea iteration square within reasonable hypothesis iteration accurate unknown correspond parametric single index model main difference compare transfer must track iteration fit monotonic maintain get training decrease lipschitz literature method large reader one randomly subsampling argument beyond input tw I I turn formal formal subsection difficulty simultaneously plausible sharp theoretic reason dimensional low setting independently support unknown lipschitz follow iteration dimension sufficiently minimize base formally lipschitz entire linearly sort function suppose decrease value constant optimality easily formalize kkt constraint I must bit additional training input define clearly call ti relate actually value conditional somewhat require result need space probability heart
monotone incur error ignore negligible care take restriction naive approach generic monotone learner inefficient require closeness sample complexity monotone complexity fortunately monotone distribution learner agnostic handle monotonicity naive give monotone distribution monotone overall show suffice handle additive besides term hypothesis overall bit learn total modal run hypothesis make modal completeness far decrease yes analogous testing whether modal correctness learn access modal fix sample distribution partition j j r j ki ia j I ia repeat start interval atomic continue interval bit ready explicitly exposition neither non suppose extreme fact either first fact failure atomic atomic step satisfie atomic kb case interval thus reason every step invoke contiguous interval enough atomic otherwise atomic k sample standard argue atomic contain ball bin desire ball completeness execution step run contiguous atomic interval contain extreme return failure leave extreme point modal failure repetition moreover claim increase orientation follow increase contiguous output yes far every union interval partition either bit negligible point expect event discussion precede lemma partition negligible total variation ps ps tv bound rhs negligible contribution contribution close monotone hence observe monotone interval hence desire expectation ps ps ps suffice term claim binomial claim expectation j q monotonicity mapping claim apply e ps claim rhs old j ps ps old use j consequence concavity require clear claim easy polynomial analyze testing access modal use sample property yes vote quantity capture iff average average distribution pass consecutive sum test sound modal ingredient procedure convert input access modal distribution output make long option triple point either neighbor theorem establish correctness use property start completeness say every collection interval good henceforth b write consider similarly I term sum term e completeness imply I say yes probability least crucially let modal non I first show let modal far exist I latter assume rhs decrease domain point I I c achieve sample fall empty iv combination desire thus prove let modal distribution n c high stage stage reduce mode one modal appropriately show triangle axis domain informally identical left mode maximum work height equal right flat reduce proceed convention mass cut quantity mass interval point unimodal decrease mode right interval indeed define mass exceed mass unimodal interval mode outside decrease latter interval recall decrease argue b construct recall lie interval continue iteratively remove may conclude establish complete step trivial order even naive would programming run want decide use dynamic jj running completes run low think accuracy sample weight region accuracy need least level strategy develop namely decompose modal simple monotone context efficient motivate kind log concave monotone hazard sample would construction proper modal priori anonymous agnostic learner also agnostic claim explain follow special theorem sample require arbitrary let conceptually three partition carefully interval weight assign entire give reduce correspond obtain hypothesis follow inequality output hypothesis I interval combination third tv entirely explain show fact semi agnostic claim clutter description non algorithm succeed average satisfie close show assume desire distance imply distance deduce complete test routine choose winner candidate hypothesis hypothesis high routine select winner candidate close run candidate chapter build difference approach competition hypothese symmetric compete hypothesis support competition competition carry draw return mp draw return check competition competition competition winner intuitively winner winner intuitively moderately winner variation let indicator iff mutually chernoff p h winner competition ii competition winner winner theorem require I take account probability learner binomial hypothesis place resp increase increase provide boost algorithm hypothesis distribution candidate generate would boost success appropriate distribution formally let suppose collection use output chapter distribution cover notion competition choose cover algorithm choose achieve cover instead run choose output never exist output argue never competition consider draw union never competition argue competition failure suppose perform construct hypothesis generate true variation conditioning assumption time run additional run remark run produce run description distribution pair description former case description output consist monotone interval construct candidate pt claim corollary conjecture claim mit mit edu university ed uk edu modal discrete peak generalization unimodal study perform modal probability give run factor prior case crucially subroutine easier consider modal section precise g apply naturally simple unimodal distribution study discussion aim modal variation give access theoretic goal bind computationally run size input contribution nearly rich restriction researcher unimodal among probability mostly deal theoretic monotone unimodal central learning cite mention monotone unimodal complexity asymptotically discuss detail ingredient relatively develop algorithmic aspect rather contrast interesting issue arise learn modal modal output hypothesis give distribution argument adapt discrete case show generic distance monotone sample copy length monotone nearly precisely bit running time factor aware modal know simultaneously fix exponent depend section give efficient distribution concatenation monotone increase learn modal indeed guess run inefficient moderately naive roughly separately since interval monotone used interval contribution failure efficient totally give polynomial turn polynomially roughly moderately naive unlike instead successfully distinguish property versus variation property testing output yes crucially use able identify ii handle thus roughly run algorithm algorithm understanding note testing preliminary diagnostic whether expensive differently modal monotone monotone aware successfully property decompose learn may find elsewhere element vi correspond increase decrease nonempty point min extreme extreme interval resp denote resp modal distance ps kp kp tv capture subset variation identify recall interval vc deduce several sample vc value successful partition domain mass next step close distribution monotone monotone interval mass contribute though interval affect focus monotone overcome monotone algorithm testing interval confidence distribution satisfy probability boost confidence overhead give bit description distribution hypothesis decrease detail algorithm simple bit modal input access modal domain atomic j j light interval conditional easy claim draw sample run simple step failure disjoint rest simple good interval except potentially interval construction step light point heavy interval heavy sum proceed contribution modal
arrange triplet visualization pursuit long really macro entire inside micro interesting work normality prefer experience index micro attract micro pick cube know analyst come macro cube structure subject confusion unfortunately subject matter knowledge criterion important really exploration process may decide guide projection towards meaningful similar automated parallel pursuit structure like ever half measure discrimination mirror normality capable really general question place answer incorporation modification index possess benchmark even yet asymptotic constitute coherent applicable organize section recall traditional pursuit algorithm design overcome limitation three real pursuit literature clear precede flexible exploratory begin try exactly usual exploratory projection break user index projection project substantial optima index specification constitute interesting implicit projection normality specialize like apart scope intervention direct incorporation pursuit dimension benchmark index benchmark possible essential feature notion user without necessarily thus input incorporate require specification though number differ search similarity dissimilarity search support structure way occur use previous could arise old drop expand could start explore analyse case subject available two arrive factor level easily interpretable benchmark exploratory discriminant always possibly thus less powerful look discrimination neither benchmark traditionally find projection pursuit fairly simple contain naturally still comparison synthetic reveal original close meaningful synthetic dataset ad principle benchmark wide utility split distribution hope projection remark obtain cluster highly structured difficult benchmark otherwise reason motivate utilize course investigation second important projection extent similarity projection multivariate dataset exploratory analysis kind distributional principle adapt index requirement power scheme yet projection appearance cloud dramatically cell scheme choice generalization multivariate kolmogorov multidimensional multivariate von statistic rather serve index index nothing absolutely like cumulative characterize however advantage continuous statistic interest propose estimation plug evaluate numerically region integration convenient estimate location serve circle centre spatial fit parameter take parameter experience preferable least fast requirement possess easily apparent none possess easy hundred take time whole search search require kind hardware available computer naturally useful statistic preferable adaptation preferable essentially exploit relaxation integral accuracy evaluation way accuracy also kind quasi stability search finally functional point evaluation ks type multidimensional thus evaluation also involve kind sort easily computer three spatial next index projection relate trust package carlo package optimize throughout algebra adapt package encounter wish illustrate matter come number generator package benchmark rarely deal benchmark suggest number sophisticated generator however always attention dataset generator uniqueness different seed generator view different seed simulate optimize initialize allow start radius multipli solution though projection reveal subtle generator part primary interest though clear either exploratory investigate could use relate relate intensity dataset quite high would build argue two kind result vast majority solely gene individually separation need benchmark comparison class demonstrate interested projection figure separation appear solution projection exploratory analytical article come vast identify ratio present gene retain largely exploratory projection pursuit entry gene order row gene entry magnitude could importance gene less retain change bit separation albeit important identify separate exploratory pursuit separately exploit linearity pursuit quick follow solution drop euclidean criterion reproduce original well preserve figure obtain separate plane draw however misclassification something great concern neither projection encourage pick subtle fundamental pattern relate produce contain production region identify broad south east west four south north south section none available arrive benchmark recommend identify obtain pursuit plot finally isolate comprehensive structure graphic static understand h pursuit algorithm qualitatively projection display cluster degree separation symbol break south west readily symbol break cluster bottom remove region reduce none show shift three achieve well projection show north precisely method box modify interact exploratory even currently level interaction incorporated hand always dynamic graphic drive exploratory pursuit balance highly usually however black
q every without generality p together require reward state reward draw adversary however subsequent round adversary force reward need rapidly amount adversary market maker call bandit maker vanish previously maker know distribution market maker asymptotically cost know advance fix function market maker achieve reward obtain performance maker uniform belief initial first period switch draw price towards half potential adjustment direction lead market meet price attractive bad maker asset trading preserve guarantee market literature optimality relative class market understand work far ever grow work explore suited market environment substantial evaluation market life market interest modular combine bandit algorithm obtain free space well understand practical behaviour market market claim free case automate market free analysis outcome cost function price world pricing payoff decompose two proportion occur price market maker first handle cost providing provide overview mechanic explore vs price reduce demand conversely reduce contract bundle demand maker demand select sufficiently exploration tradeoff well literature maker use cost mapping size bandit reward bandit cost modular grow bring development market motivate price extreme situation maker situation period asset occur quantity exposure market market maker since incur loss perfect knowledge outcome tradeoff price draw know centre maker still price extract belief centre observe still trade trade optimal maker extensive automate market market expert made market mathematical base market follow learn stochastically draw belief mirror descent see market maker outcome reason onto belief market maker motivate extensive separate prediction market make maker balance extract automate market sum market extract market maker case rather narrow equilibrium criterion function market sequentially proper scoring compatible player belief action proper reveal interact maker act enough compatible mechanism paper arrange interact maker proper compatible bandit price back economic consider case modern method specific online price adversarial overview component subsection exclusive period period maker cost pricing behaviour sequentially market maker sequence period shift vector describe introduction exploit trade maker round view bandit formalize market pricing behaviour maker assign vector represent market maker component case event market position want portfolio share price pay security vector detail construct important bad bandit vanishing regret fix market maker attempt receive face market obtain difference market maker outcome abuse maker cost outcome use q complementary maker condition also function offer price exist crucial potentially never offer price market maker c imply consider lipschitz continuous continuously price family cost fair price discuss logarithmic scoring construct single price market price cost straightforward verify condition satisfy monotonicity boundedness hold treat game closeness metric supremum cost indeed pseudo triangle carry difference show market make outline salient motivate allow cast adversarial armed market play round player arm predefine choice continue large total reward crucially round player force balance action repeatedly play game central action receive reward play typical analysis derive sublinear bad adversary function adversary choose play result reason latter set available compact subset mean choice make difficulty relatively work action space tend place strong restriction require require technical broadly vary say restriction follow definition state lipschitz adaptive interpret reward make locally view impose difficulty inherent make realistic scenario trade two choice round mean sequence necessarily history function thus interpretation market must assume adversarial non market maker cost function meet condition rich
robust pearson correlation nonlinear relationship view especially suitable situation relationship speak extent tend require relationship coefficient consistency covariance establish matrix via hard predictor eliminate explanatory irrelevant actually significantly independent feature marginal utility frequently correlation propose marginal select screen additive generalize model implicitly depend explanatory change correspondingly relevant regressor fact diagonal equation conditionally average also reflect permutation cluster explanatory variable utilize score decrease order kx k k kx generality assume form perform variable construct overlap construct selecting continue index step equation grouping weakly correlate new predictor actually dimension reduction pearson screen second explanatory second feature thresholding try discover multi could utilize discover benefit order economic financial construct index science break edge keep correlation show np hard sample covariance correlation order matrix permutation simpler regularize limited nonzero r simultaneously explanatory combine especially suitable modeling method financial thresholding regularization remove irrelevant remain one avoid np hard clustering base sign estimation procedure implement table rank correlation unlikely economic disjoint give ideally correspond w motivate refinement consider x correspond lagrange contradict requirement role increment extract simple sign require minimize choose otherwise value iterative achieve step present minimization exist also model dimensional group third approach fit great deal fit face fast macro covering include production capacity price start stock price rate exchange time except rate nd difference raw denote variable index mat component u item less care pr pr u services pr service correlation threshold interest price indicator percentage effect price magnitude value besides economic adjust economic model display relevant detailed digits index provide forward backward construct overlap overlap come th originally correlation th variable th item medical implicit consumption identification mainly concentrate link function space price implicit consumption respectively interpret service actually interpretability economic view keep parametric coefficient table rd economics correlation parametric price unlikely sign corresponding parametric coefficient column table explain variation positively mean eliminate except lar estimate least angle include economic indicator economic activity estimate variation lar improvement lar already perform paper spatial quantify consistency dependence level resample cross estimate utilize link graphical spatial backward structure method spatial panel economic financial series superiority function directly correlation define except use introduce address pearson zero correlation zero brownian independence correlation functional mutual correlation detect dependency want point step screen explanatory descent residual square regression construct thus could f jx sure correlation screening define error help rewrite study focus cumulative investigate motivated grateful li xu discussion particular thank stay california berkeley subsection definition assumption inequality see dependent yield tm go complete bs tm jj since v b lead eq q integrate rhs minimizer imply jj condition lemma well spatial panel economic financial first spatial hard latter j quantify consistency discuss validation consistently utilize screening implement forward relevant modeling apply large panel economic financial price superiority technology create economic financial genetic panel economic begin variable view assume estimation play role important numerous finance limited management interval forecast grouping via reduction recent tool low frequency variate good eigenvalue support thus attract lot attention recently one construct impose structure model among order weakly panel array invariant thresholding covariance high extend iid series setup important go recent understand especially serial correlation temporal dimensionality begin complex parametric adopt nonparametric curse disadvantage maintain curse however prior work carry discuss one specifically explanatory high single perform selection technique eliminate gx unknown assumption additive actual structure sample size specific class approach character economic financial near expect unstable sensitive minor perturbation expect selection economic principal component regularization method weak matrix compose requirement denote set index eigenvalue gram cardinality submatrix gram satisfy integer derive upper view fact find penalty ideally penalty coefficient shape nonparametric high among end develop speak try could cardinality number univariate parameter parametric tx sx tx possibly moderate balance flexibility general additive linear view rhs know link disjoint element say identifiable every identifiable group immediate extract second question paper answer mainly various resample scheme prove cross validation approach cluster estimate high spatial estimate matrix utilize link graphical standardize explanatory label explanatory spatial reduction sign dimensional face help dimensionality group information knowledge available spatially dimensional propose criterion grouping try intensive practical grouping drive without sign sign coefficient economic law organize present main notation conclude start concept covariance notation define matrix average eigenvalue thresholding thresholded notice preserve permutation preserve positive uniformity mainly future suppose without introduce view dependency subset fractional cover j w spirit note unless process consider mix related sub coefficient v kolmogorov appear mix process article coefficient notational affect tc state consistency rate get slow level maximize reach offset behind correspondingly slow accord jt retain question ask extent dependence
manner case median svm median smoothed version result consistent predefined I estimation step l call difference svms carry svms package implementation svms loss also result loss think early consistency svms jx reproduce hilbert corresponding assume write f would gx estimate conditional median residual loss function smoothed version smoothed loss estimation estimate one assume unknown smoothed continuous l parametrize logistic purpose introduction newton measurable function say consistent assume predefine median uniquely g quantify r quantify absolute predefine iii plausible nuisance median svms calculation proof integer c measurable complete space borel obviously unique converge sense surely sense ne f existence uniqueness combination risk immediately sub argument combination q q svms svms exist heavy tailed assumption trick shift sense l see function minimize svm e immediately uniqueness consistency difference difference svms immediately uniqueness choose median median conditional quantile quantity conditional well regression svms occur quantile wrong occur prevent kernel view appropriate estimating median absolute residual conditional median conditional need svms type happen approximate g rbf kernel jump happen easy estimate complicated advantage estimation flexibility quantile flexibility type estimate quantile lemma nx h probability follow e apply lemma close separable dense notation define f l nx gx nd g fulfil calculation f negative function reduce definition apply triangular l g r triangular separately four converge superior large part term note define nx boundedness well see easy follow f nx theorem x inequality converge imply therefore guarantee pt ex pt sketch pt ex pt pt ex definition remark definition mathematics department main goal statistical single svms establish estimate like median svms median volatility median absolute value problem conclusion generally location consider two statistic
polynomial one polynomial contribute provide model straightforwardly expert convergence quasi maximum estimator obtain leibler bound generalization pm mn nj total number weight function identifiability permutation unique maximizer remove attain approximation estimation expectation I expert consistency expert specification target belong parameter kullback leibler divergence model number multinomial logistic consider smooth continuous variable satisfy rate assume least finding yet maximum mixture study rate paper long I I case throughout notation h inverse link mixture model glm kx mh kx class density derive consistency restrictive multinomial logistic poisson among function satisfy estimation maximize likelihood necessarily identifiable permutation restriction expert kullback kl divergence minimizer index kullback work consider I straightforward next generating sequence random vector ensure compact np xy almost surely demonstrate identifiability maximizer estimator consistently index I maximum converge also ergodic ergodicity function identifiability distinct every find sufficient identifiability binary adapt specification set adopt identifiability unique maximize taylor around maximize hessian identifiable technical allow flat pg likelihood often maximize expert complete likelihood maximization put q expectation maximize maximize divergence side approximate estimate good approximation follow use kl divergence present shall introduce partition partition fine partition partition abuse notation justify sequence definition bind growth see sub idea approximation control inside fine partition inside cardinality sub geometric employ notation expert expert structure expert actual loss actual expert approximation constant direction derivative polynomial expert result enable important mix many general specification density expert dispersion modify agree assumption therefore sharp deduce model expansion term approximation combine unique maximizer able summarize rate divergence true convergence particular proportional result derive obtain assume identifiable maximizer price pay generality localization process neighborhood favor smooth enough dimension favor simple possibly small big ii iii show quite deviation optimal optimal size situation ii achieve convergence drawback rough polynomial conduct study impulse polynomial increase exponentially preferable remark table compare approximation distinct hold expert know total fixing want table small error h cccc quick exercise conclusion small large compare improvement section mixture expert polynomial link sharp mean rate estimator pm far uniquely specification effect approximation estimation conclude balance inclusion error proof assume include glm ii class instead iii calculate parametric convergence estimator expert sample increase vi polynomial development light acknowledgement author thank expert comment previous version appendix justify drawback work kullback leibler divergence hellinger hellinger distance hellinger density divergence bound hellinger basic inequality relate hellinger leibler stand respect leibler bound square hellinger hellinger rate leibler boundedness support overcome choose hand side large bound estimation convergence I hold inside measure cover moreover universal show task g kx f gx lipschitz cx first inequality take prove likelihood bound rate decay small kx kx ki convexity logarithm application enough combine previous use hellinger maximum likelihood completeness pg estimator let choose sum continuous proceed existence identifiability satisfied assumption boundedness continuity integration limit expansion bound satisfy theorem compact condition remain integrable bound dx kx kx finite assumption kullback ensure dp dp p term choose sc first bind tail e inside eq hellinger inside arrive result rate substituting parallel precisely inside remove allow proof consider two increase minimize decrease minimum iii straightforward kx far first differentiable kx reasoning definite tell let logarithm consider consider unchanged next satisfy metric metric corollary proposition expert give
tend density get hard function favorable easily approximated evaluate real world test use outlier positive house speech contain record extract assign contain word take class dataset write digit consist pixel take level regard assign auc useful systematic auc choice summarize overall small tendency agree bold dataset sort l r diabetes heart speech news news news vs vs vs vs vs vs outlier label py n py test discrepancy input ultimately generalization error py practice covariate setup plain minimization empirical shift q shift tend large importance propose call correspond plain minimization importance empirical give intermediate trade model regular model one drawback hard estimate poorly adaptation cope definition exponential indeed relative importance play weight plain minimization minimization trade consistency importance agree weight unlike exponentially reliably propose adaptation effect behave covariate shift employ ls exponentially change plot show realization ls l give true well figure error l comparable accord next change realization test sample learn l ls test ls tend outperform l effect transfer world transfer human ask perform task hz axis plot stream slide manner slide position reason decide take acceleration step invariant situation want available unlabele obtain train j unlabeled test significance specify bold walk run walk vs activity plain without plain importance different depict three accuracy unlabele sample plain without instability weight useful transfer covariate propose use divergence give direct approximation theoretically parametric show preferable pearson furthermore asymptotic divergence complexity estimator experimentally practical homogeneity outli transfer covariate addition homogeneity test detection transfer machine include multi independence dimensionality cluster probabilistic thus relative homogeneity detection transfer simplicity pearson non pearson pe possibility bound norm rkhs satisfy eq property kernel dense another canonical square derivative term impose rkh spline rkh speak density derive rate book l quantity function large rkhs note rkhs condition pg denote auxiliary sg g pg side entropy show f relation entropy jensen concave similarly show last combine eqs simple b moving complete pf eqs eq also therefore substitute observe complete variance model use follow correctly eq note eq probability likewise sup standard sup asymptotic denote xx regularity linear likewise asymptotic confirm negative increase non negative see negative upper therefore complete yield q expansion complete theorem estimator ratio go denominator apply machine task detection homogeneity always smooth ordinary ratio favorable complexity estimator experiment usefulness pearson outli homogeneity square importance fit fundamental would divergence kl divergence plug know approach unless good approximate without ratio parametric mini max linearity cope divergence pe pe kl thus pe divergence vanish pe estimate give pe computed usefulness various homogeneity dimensionality first non pe govern sup denominator density speed ratio paradigm overcome fundamental comparison quantity call pe q direct q notable unbounded relative density ratio approximation ordinary ratio relative pe divergence pe complex extensive homogeneity pe estimator compare follow pe estimator pe experimentally paper contribution identically distribute dimensional paper let reduce ordinary pe thus regard pe divergence pe relative transpose kernel third expectation obtain easy reduce direct call relative reason refer criterion matlab implementation public acceptance estimator pe divergence pe divergence express pe far similarly particularly simple numerically toy denote denominator contain c c ratio ratio graph relative smooth ratio tend analyze pe divergence estimator specifically section insight mathematical rate appendix analysis density describe space statistical model parametric convergence pe practically dimensional rkh norm represent precise pe tend eq shrink speed technical detail follow pe fp coefficient lead term asymptotic become large preferable term tend infinity pe would advantageous plain pe divergence lead depend accurate large furthermore vanish fast depend another divergence regard tend zero hypothesis approximate value kl pe divergence adopt homogeneity propose experimentally pe two homogeneity contribute hypothesis different correct hypothesis keep hypothesis plain reciprocal adaptive behave two sample homogeneity scenario dataset test plain reciprocal show dataset nan hypothesis plain correctly error control hand acceptance rate toy still reasonably small p critical different want reduce incorrect setup reciprocal setup dataset density reliable smoothed distinguished completely ratio reciprocal plain accurately setup large perform dataset plain reciprocal trade correspond tendency middle overall plain reciprocal work unstable reciprocal tend instability plain reciprocal provide wide assign wide setting work support systematically issue p two correct acceptance specify l r mmd c diabetes heart ratio denominator density positive negative nan acceptance rate comparable face l mmd diabetes heart homogeneity binary dataset pe
straightforward middle always mean unstable reasoning point locally depend one flow dynamic critical exploration finding game action reward h decide pd payoff dominant beneficial nash players ne pd nature guarantee rate contrast observe start regime lack behavior attribute next mp zero player action pure ne ne ne globally position rest term player action decide pure ne ne rate dynamic correspond game furthermore equilibria unstable rate diagram find risk dominant ne game show fig strategy dominant payoff opponent describe game strictly risk show meet critical ne call anti beneficial anti game one behavior game ne mixed ne ne anti game condition anti symmetric ne c xt ct ct unstable consider ne outline parameter value see small single ne game similarly rest critical exploration rate three anti consider additional ne elaborate appearance rest appendix diagram plot diagram contrast payoff different player payoff matrix dominant player payoff mostly make picture preserve light grey hand player role range vertical critical rest behavior grey region parameter region strictly dynamic boltzmann mechanism exploration two action exploration dynamic guarantee reach demonstrate depend game exploration rest structure stable rest positive analytically examine exploration asymptotic behavior ne structure globally observation numerically ref equilibrium fact boltzmann dynamics game dynamic qualitatively game ne exploration rate allow rest ne games single ne rest dynamic rest exploration tend dynamic exploration various mechanism indeed study far limited dynamic insensitive exploration anti fine grain sensitivity rich employ agent manuscript aware reporting use local demonstrate finally analytical diagram possibility rest game ne complementary present thank discussion national foundation grant fa derive rest sake intercept pass intercept exploration happen yield multiple symmetry whenever rest analytical repeat reasoning depict ne learn sake region graphical representation rest fig becomes find require calculation yield xt numerically figure critical solution intercept game converge game nash comprehensive characterization structure game exploration result games ne demonstrate single ne range player tend reinforcement behave trial error single rl extend interact difficulty stationarity learn agent vary agent convergence except well issue perspective dynamical note boltzmann equation additional account similar greedy exploration mechanism schema select high number develop understand behavior categorization learn dynamic game insight body recent boltzmann softmax mechanism decision play competitive indicate decision lee observational human seem reinforcement softmax spectrum behavior important conceptually make prediction outcome analytical technique complete rest necessarily seem previous slow cycle systematically examine exploration asymptotic behavior hand noise inherent aspect human softmax mechanism perturbation show depend one rest point structure vary critical remain rest globally rest describe connection boltzmann non conservative nature asymptotic behavior different game type example conclude remark provide brief connection rl optimally repeat agent choose action receive reinforcement reward agent act cumulative adaptation mechanism call agent parameterize interaction environment agent finite available denote action action need agent greedy selection globally solution one explore less selection temperature tradeoff agent maximum exploitation agent interested continuous scheme game dynamic correspond ne furthermore unstable dynamic generally ne limit insensitive reward mix uniformly behavior exploration rate volume make dimensional dynamical implication specifically dynamical crucial cycle interior rest globally stable nature variable modify read rate phase eqs action games action dynamic eqs introduce point trajectory interior simplex rhs eqs examine interior equilibria interior plot side monotonically increase assume guarantee inspection show type solid curve rhs temperature non increase rest temperature exactly rhs eq alternatively whenever sufficiently branch well separate increase critical branch meet saddle type examine useful interior rest equation graphical easy thus accord consider decrease function state consequently sign side monotonically decrease thus
time select rigorously heuristic currently e subset patch consider involve flat dictionary large heuristic increase perform deviation improvement proximal sparse problem involve sum overlap light connection network proximal norm cast quadratic min algorithm resort certain modularity dictionary come repeatedly loop norm operator cost flow demonstrate apply wide class address formally summarize graph canonical sharing vertex say condition arcs source cost arc cost arc cost every arc conversely arc notice flow arcs flow feasible value arc path node lemma arcs arc flow flow flow equivalent amount flow amount sum flow construction build capacity constraint conversely give flow path decomposition decomposition path correspond arc flow path decomposition arc amount arc flow proximal operator find proximal essentially termination optimality condition primal respectively solution primal problem dual w g g w intuitive flow amount u h old inequality prove convergence correctness also min capacity find cut divide part construction path path arc whereas arc figure arc go cut would arcs inside construction flow go circle draw blue circle thick blue fill place fill red black bend angle auto xshift leave g part xshift u xshift pre dot pre var leave u edge var node right pre leave thick xshift control gr u gr gr v arcs arcs flow cf scalar negative add polynomial operation split process projection flow sufficient call compute min max min j existence jj u equivalent prove correctness correctness use canonical number induction e g since induction canonical algebra suppose ball instance use max scalar u sum optimality condition lead consider case j empty remove notice graph group gr gr optimality derive lemma optimality notice arcs cut equation j cut arcs arcs residual decrease side bit show g disjoint part gr v gr j gr gr gr gr u u want necessarily reason go observe flow nan receive v j gr j contradiction gr u similar summarize g gr u equivalent algorithm give graph perform dual dual g j q introduce g maximization primal imply strong duality respect primal lagrangian sign correctness algorithm show call proposition non empty call canonical induction simple algebra show correct j constraint necessarily inequality follow cut exist set feasible value necessarily algorithm vertex part remark arc go arcs solution induction hypothesis solve replace constraint arc go result equivalent argument exploit show cut section fista duality stop criterion generality look matrix typically compose observation primal place argument duality gap w optimality gap dual algorithm proximal convexity precision duality f k kt k procedure f choose max min compare max experimental namely dataset build basis transform dct respectively compose parametric max flow conduct single ghz follow execution algorithm run statistic correspond sec experiment consistently outperform berkeley california berkeley usa fr de sup paris france consider induce whereas put develop hierarchy proximal polynomial allow proximal splitting address alternative learn tree dictionary video sequence wavelet learn optimization code factorization flow method various supervise combination regularization powerful tool address regardless relationship e spatial temporal hierarchical effort devote designing encode non bioinformatics vision sum norm appropriate variable solution usually difficult part involve nonsmooth strategy proximal ability nesterov handle differentiable paper regularization splitting build upon overlap group hierarchical sum norm solve thereby establish connection literature context fuse lasso maximum flow convex combine method norm gap present proximal regularize demonstrate practical algorithmic introduce sparse build link draw decomposition regularization algorithm different video image natural patch short version publish advance process denoise present method discussion letter pseudo nonzero however sake keep notation I q x r p present work devote proximal gradient algorithm present experiment application paper coefficient zero encode overlap tree structure basically deal computationally concentrate programming method impose priori nonconvex w design induce norm next interested induce generally function fitting structured variable represent coefficient case basis partition e individually know organize explicitly priori regularization improve interpretability penalty set set consider challenge induce penalty see encourage small though structured sparsity overlap purpose union predefine factorization pattern group use rely subgradient scheme approach mention smoothing norm parameter control accelerate depend solve problem parameter norm add smoothed encode g reweighte scheme mention sparse assumption example logistic reference community therein ability nonsmooth nesterov procedure extension gradient objective function minimize p f close constant equivalently attain fast rate proximal reach precision nonsmooth generally define proximal unique generalize projection proximal appeal decomposition operator soft operator u norm soft u onto effect u g norm either include closed form g b q lasso lasso general proximal reformulate propose dual formulation dual proximal problem jj w solution without generality optimality sign sign entry solve flip dual negative magnitude sign know scalar dual variable number primal remove overlap interpret vertex arc capacity arcs arcs flow arc arc incoming flow flow except value flow let group canonical vertex index size simplicity index refer vertex arc arc capacity zero arc capacity flow arc infinite jj canonical figure flow identify non arc cost lead cost minimize graph arcs flow arc arc g equivalent problem canonical simplify edge edge remove capacity simplification quantity compute appendix reduce draw fill minimum fill red mm rectangle thick draw fill bend angle source edge pre xshift mm pre xshift edge pre var u distance node si u leave edge pre node bend auto cut balanced factor practical complexity serve zero optimum duality converse description along computation give z problem derive standard duality efficiently equivalent arcs arc solve amount small flow arc appendix computation dual initial gr uv either sum admit form v ff know svms structured one scope slightly offer regard since operator discussion schmidt choose advantage proximal parameter ability proximal consider variable associate amount lagrangian equivalent r g admm find saddle iterate fix ascent respect summarize respect w influence efficiently cope exploit ni x pi I equivalent problem methodology know usage possibility additional add l compute issue adopt replace quadratic v augment lagrangian proximity w symmetric positive w still demonstrate applicability compare proximal present admm lin admm two sg sg lin run cpu overcomplete dictionaries cosine dct dimensional grid family spatial contiguous sequence case square generate vector w nonzero level sg admm optimize provide low respective sg note step size form choice lead lin interior cast qp program formulation lin except set objective sg second lin lin scale lin admm similar qp lin sparse whereas illustrate single large sum task orthonormal wavelet wavelet orthonormal orthonormal candidate norm sparsity wavelet adapt wavelet wavelet version gaussian good c wavelet run different realization fast dimension proximal approximately take second ghz tool fact subset generalize inverse particularly interpretability expression dataset insight patient decomposition work decomposition I solve lasso problem induce penalty parameter solution submatrix readily identify component bring problem sampling gene expression sequel center run set row involve randomness deviation five initialization conclusion match already partial scheme worth way imagine add low thank latter report deviation initialization dataset background task frame fix camera try segment foreground new pixel w p make account pixel background foreground
without replacement individual sm sequentially new individual combine sample reproduce please always complexitie htb section subset matrix maximum scale matrix correspond fitness scale level several controlling criterion difference typical structure sm probable connect dependency remove multivariate curse dimensionality strongly restrict problem increase dimensional report sm partition group small subset prove sm try dimensional complicated computational sm dimensional sm try perform estimation fortunately control implication sm additional consuming reduce imagine global successfully learn sm outperform controlling section sm flexibility model discuss complexity select representative univariate likelihood study experimental sense include representative gaussian matrix easy implement fair comparison computation implementation comparison within template share flow utility computation building module list select benchmark function shift optima omit find landscape also unimodal separable optima multimodal htb make domain shift shift way also transformation sphere shift nz nf nf nf x z n nf z z bias x nf shifted bias nf b x nf f expand plus see page application fitness evaluation varied besides vice versa people tradeoff balance vary knowledge achieve instead choice every give dimensionality final moreover initialize initialization one individual generation newly sample individual constitute new widely study illustrate population problem algorithm average execute ghz mb ram point calculate pair correlate context experience observe small value result I make may dependency release set constant aim benefit issue value sm preference normal large large vice versa give estimation approximate combination time require user physical implication determine pre preference easy apply later difference good fitness optimum negative small multiple population report fast correspond cpu cpu fig htb marker l e e e e e e e e e e tail tail tail compare benchmark adopt mark bold r separable unimodal facilitate univariate case estimation degradation well good overall significance global shift search cpu size although different cpu good performance cpu grow keep high clearly population performance unimodal problem except bad performance shift performance bad optima shift worse absolute always robust cpu although bad size next sometimes grows explain since apply maximum population keep absolute problem far among global optimum require good approximated subspace dimensional approximate estimation combination well combination experiment experimental fig combination model poor global find significantly scale much time result run show bold nonparametric htb f cost always big superior performance evolutionary curve answer evolutionary curve proceed imply fact converge perform apply therefore large little size perform bad failure population cpu htb f result result include comparison nonparametric gm gm c gm word group fail dimensional quite large optima lead landscape make hard solve estimate although study local optima perform adopt apply shift surprisingly still outperform become intuitively result multimodal perform pose enough group sm common optima present experiment concern characteristic prevent perform characteristic necessarily failure success probably use dependency degree high long failure attribute failure degree setting exactly dependencie consider see essentially include influence indicate algorithm probabilistic matrix replace base four implementation model among summarize table htb nonparametric significance marker marker l dim e e e e e e tail result tail candidate switching dependency help performance promise matter three resource maximal utilize effective long minimal search huge optima increase serious nevertheless although always complicate landscape estimation perform well scale discussion size guess budget fast fast complexity requirement outperform increase also curse optima computationally expensive univariate resource discover efficiency optima separable well three propose begin gaussian usually base cost fail summarize low landscape optima fail case univariate effective success curse dimensionality less control necessary population size rely high traditional high really solve still design investigate several optimization involve whose dependency shape bivariate conditional acceptable code provide dr al variable problem name evolution base framework partition strategy activate although group subset instance code author es several hundred es study g sep es well estimation major es path current usually require es experiment lead sep es deviation derive set result population please refer population keep whether keep pressure test high confidence summarize g es test regard g deviation result result sep es nonparametric marker report leave tailed demonstrate available equivalent significance post hoc sign rank test outperform significance significance es l c e e e e e e tailed result simple separable sep es group separable es covariance separable outperform sep sep es es little well average shift hold much bad shift landscape g group function analyze effectively one behavior huge optima exception explain optimum also partly likelihood estimation high observe perform well curse dimensionality simple univariate handle problem difficulty speak robust optima shift optimum disadvantage compare sep notably consider far successful multivariate significant superiority tune potential even newly test population adopt test htb htb l e unstable current implementation subset dependency eliminate definition become since generally separable good combine empty separable solve non cpu analyze dependency eliminate partition separability scale small separable besides recommend bring speak good impact may cpu cost ask sophisticated partitioning cluster intuitively enough e g available curse dimensionality sm greedy result htb ni remove loop specify subset maximize x xx variable since sm fig sm partition subspace pick outside add variable reach strongly cluster rest outer keep generate subspace manner partition still univariate dependent htb run significance marker marker significant l tail compare algorithm summarize difference test sample dimensional limited population partitioning subspace effective illustrative experiment exclude outperform cluster relatively high contrast partition scale property learn model finding ability characterize advantage recent discrete interaction structure graphic sm also record procedure generation analyze result record strong element evolution plot variable also plot matrix variable row ranging indicate examine even graphic add read omit small purpose evaluation horizontal page length report result effect sm role mutual effect sm separable low grow strong become high interpret sparsity space fix experiment separability grey fitness function value consistent sphere strong correspond plot element show correctly shift dependency chain determine second determine see important structural htbp specific experiment shift condition fig help dark checking table fix among mostly ji ji ji partly speak effect onto hard analyze variable rough figure second negative coefficient third th compare last size etc rough experimental function correctly dark htbp especially test show shift result help explain examine find similar analyze recognize optima look like problem learn htbp shift average plot variable remarkable characterize show although find characterization problem underlie structural remarkable regard valuable aspect however number optima limitation notice implementation try possible univariate gaussian problem property try effort explain problem capability thing address solve run problem sufficient allow aggregate multiple trial role sm interaction sm implement sm compare version role save comparison respective sm experiment population size well sm find magnitude sm weakly dependent simple slightly sm bad cpu sm solution comparable quality cpu time acceptable much speak sm show much robust cost perform slightly sm sm bit partitioning sm fail marker e tail sm htb sm contribute nothing sm sm result strong function global affect eventually partition also modeling become cpu sm characterize find solution sm help model help effectively help properly strategie weakly characterization robust performance sm difficulty due curse population traditional model separable improve reduce adopt identification significantly traditional dimensional optima requirement also besides exhibit characterization solution user variable far valuable black extract design also space available offer may effective appear separable huge local gaussian discussion implementation still restrict test adaptive still issue leave suppose build denote variable dimensional select individual generality g model select select individual estimate complexity new need repeat create cost joint select individual complexity need generate primary multiplication repeat create cost complexity ignore multivariate sampling select individual calculate building identify weakly building building build strongly build building sample sampling time thus grateful yu ray wang comment code support china visit centre computer university uk mail cs ac improve continuous high dimensionality computational scaling continuous necessary explicitly discover useful great benefit box propose novel complexity subspace sm performance population characterization successful model effectively class newly design specifically strength carry kind benchmark compare traditional evolutionary genetic ga neither mutation model promise search space present global statistical well actually model usually evolutionary classified instance originally optimization research domain progress focus decade branch continuous study major branch validate low dimensional rarely ignore continuous difficulty high space rely multi increase computational make impractical name continuous adopt weakly identification sm curse dimensionality reduce overall comparison effectiveness advantage traditional separable optima traditional appropriate motivation advantage discover learn explicitly possible learn simple base ignore deep dependency adopt explicitly control attempt dimensional penalization remainder organize follow traditional present sm model adopt traditional penalization discuss experimental sm investigate sm compare sm advantage partition sm final along typical loop refer evolution p initialize population meet select individual primary adopting form normal define mean matrix base simplicity level solve strong multivariate likelihood gaussian represent normal gaussian graphical factorization introduce computation maximum computational reduce conventional must base performance superiority another however involve normal idea make later poor ability criterion find besides base adopt multimodal hybrid interestingly although accurate true distribution promise nevertheless offer multimodal problem satisfactory explanation phenomenon literature multimodal recently analytical base univariate histogram base flexible consider bin exponentially make multivariate histogram hard together control weakly dependent sm call control covariance covariance variable respectively accord coefficient see covariance evolution multivariate coefficient nearly mean dependency behavior much example coefficient case univariate gaussian significantly hold fact firstly identify
recommend quantity reflect appropriate acknowledgment support award es national institute environmental health sciences view national institute environmental health reference proposition david nc statistical nc years rich variety prior address massive prior normal form cauchy mixture generalize many prior special compete connection develop class variational massive major area rich variety lead interpretation posterior different induce prior induce parameter problem perspective normal choice family penalty appeal strong tail avoid double tail many double exponential near phenomenon new prior widely bayesian framework rely bayesian selection prior mixture mass equal exclude distribution non paradigm appeal accounting learn correspond subset one average subset inclusion predictor include predictor predictor recently may conventional prior attractive bayesian shrinkage mixture gaussian update substantial improvement carlo rapid desire inference many interesting reveal connection bayes approximation truly massive conjugate allow yet advantageous form straightforward sampling intuitively adjust shrinkage control non sparse structure inherent selection brief background primarily possess appeal lasso tail drastically shrink signal heavily interesting representation later form conjugate potentially computational complexity prior multiple hierarchical denote half parameter appropriate represent special name magnitude desirable result tail unbounded create elaborate back cauchy origin explicitly behavior prior trivial exponential ng form yet lack tail user implicit behave see distinct formulation prior able prior formulation furthermore gibb update make flexible class normal appeal name function denote distribution wishart function second q show th gauss propose compound propose denote beta becomes becomes consider behavior equivalence prior observation cauchy mixture half mix parameter place half complete hierarchy help formulate complete hierarchy bring analytical procedure much relatively hyper prior give proposition proposition ab draw dashed line pn normally hierarchical shrinkage common infer hierarchy identical hierarchy huge reasonable exist appropriate reflect underlie coefficient adjustment discuss also error formulate conjugate likelihood sampler conditional b laplace approximation low likelihood may variational posterior posterior distribution iterate reach deterministic make attractive quick mcmc conjugate take straightforward prior regression sophisticated scheme towards flexible normal scale focus many reader sparse hence follow brief posteriori expectation maximization accomplish obtain hierarchy modal term heavy careful hand estimation admissible sparse apparent essential prior density marginal illustrate tail origin drastically jeffreys np ij pi ratio approximately regression variational bayes calculate median model value bootstrappe regularization tune cauchy attain clearly particularly rule choice well due set set randomly component index place due reflect limit sample note adjust global priori whether sampler hour
day train idea rnn fix value keep mind direct layer model seven mixture expert feedforward expert expert expert expert architecture expert mechanism difference expert expert adaptively l py vector bit likelihood bit give denote code logistic online first hold constant kalman dynamic consist observation linearization investigation wu use computational cost seven weight initialize bit jacobian yx table via modify tradeoff compression performance eight achieve bad compression memory achieve compression l change weight layer change initialization refer neural change significant computational cost use manual tuning file difference tune tuned file book book news paper compression result document string string application system compose acoustic component could directly replace recognition prediction string people people slow input device mobile source predict character new create character generate character create file train observational text continuously online character capture syntactic semantic spirit decide soon little whole existence b china name kn name fortunately control example exponential ref show past gauss mu lambda alpha r conclusion prior theory h mean figure long live rnn compare rnns rnns bad rnns difference method continuously beneficial opponent move predict move opponent next ai play reason human human round practical text categorization spam image one sometimes way map several researcher compression usually preprocesse concatenation piece compression compression rate set use near neighbor use achieve competitive rate categorization shape compression classification procedure length approximate minimum neighbor store file class file concatenation file run compression file compress assign compression compression file ff file size minimize fa fa file bt assign class method datum comparison consider time bit get note difference many dataset test file large order magnitude extremely compare compression program access source modify source code essentially copy allow file continue adaptively modify file still parameter default classification experiment performance measure file file fundamentally two file store disk non arithmetic file disk cross entropy compression performance subject neither access source code file categorization assign evaluate news partition evenly category j corpus message retain count os ms windows pc hardware mac window correct classification randomize seem preprocesse methodology percent naive output code train language train split multiclass svm split multinomial naive publish dataset comparative note several dataset evaluation protocol publish result directly example zhang dataset seem dataset shape recognition contour within dataset available dataset binary image five category example orientation resolution ht count seem image text categorization slow object task compression option effective classification al shape contour representation decide percent classification c actual methodology percent nn leave leave leave nn approximated cyclic nn time feature leave nonlinear split convert leave demonstrate convert one first centroid shape project ray centroid point ray around contour measure angle ray convert single rounding run measurement cross protocol parameter surprisingly adjust resulted accuracy adjust classification exhaustive perform exhaustive confusion parameter classification publish show result decrease al image invariant procedure euclidean metric rotation angle euclidean representation try run alternatively invariant small classification achieve compression specifically design image compression perform keep representation perform compression create set learn patch patch scan patch store filter example compress figure compression still create file image visual compress advanced compression exploit human variation high frequency degree sensitivity exploit limitation leave future filter outperform method visual compression hope exposition method area weight try technique include kalman implementation promise technique filtering implement rao filter future combine prediction expert ensemble forest technique gap rnn seem relevance architecture reduce rnns conjunction front remarkable tackle broad metric anomaly detection speech equally remarkable challenge beyond require store well architecture apply univariate seem trivial rnns acknowledgement would understand provide compression would thank discussion available ai run program source compression compression benchmark report detailed show understand improve intuitive module remain hope description increase understanding facilitate transfer recurrent modeling adaptive text playing compression gain great domain unsupervise audio researcher understand intelligence close perhaps machine machine open closely prediction partial compression compression benchmark compression memory record breaking ratio expense memory mb wikipedia specialize web website compression well act concept intelligence number pf book news average file text format reference book crowd principle computer point file topic knowledge format arithmetic code compression format source compression character field compression corpus compression benchmark corpus file appears table test pf implementation compression huge success rarely compare machine core difficulty lack scientific inner good incomplete description source close language extract examine provide explanation learn community lead inspire research develop propagation character character alphabet day make neural contribution demonstrate enable machine contribution novel text classification outperform show achieve develop compression feature problem arithmetic description improvement section conclusion access create consider character alphabet character character character compression two stage first prediction give character encode file scheme arithmetic code arithmetic code preferable arithmetic produce near predict solve compression arithmetic arithmetic two encoding encoder character compress character compress output original character decoder decoder reproduce compression create predict compression generate character case compression character file try character arithmetic need seed encode character arithmetic encoder essentially store process work suppose alphabet character arithmetic character visualize number see assign character arithmetic encoder region expand assign arithmetic measure directly code character character bit code character another common text prediction entropy character suit character redundant contain less individual pixel maximum variant partial compare length ratio text character pixel scan bottom another mapping maximize locality al metric anomaly cluster variety file music li distance string kx kolmogorov string length length program kolmogorov good kolmogorov compression approximate cx cx compress compressed compression close kolmogorov et cx justification use compression without modify fortunately open modification purpose define ex symmetric always e x report al metric perform anomaly well theoretically although use model context match unlike noise long specialize make prediction level sequence make bit detail depend version detect overview architecture general recognize preprocesse modeling ht model predictor pass secondary
discover remove fortunately correct original convex penalty strongly broad mc strongly differentiable hence smooth k bfgs name differentiable lagrange smooth hand tends become respectively recovery impractical fortunately empirically enough recovery theoretical lead mc algorithms fast converge rank positively literature back linearize bregman approximately solve pursuit compressed phenomenon discover converge whose pursuit parameter condition condition equivalent mc rough limitation generic mc exceed matrix mc respectively explicit improve firstly strongly programming rank recovery broad range exist optimization literature secondly prefer objective algorithm give word suitable organized follow section summary exist strongly programming rank recovery explicit conclusion notation define transpose euclidean nonzero equals denote summation singular denote verify important recovered mc problem ambiguity singular positive right vector orthogonal completion let respectively q save notation subgradient nuclear function form subgradient whose shall act letter operator mean symmetric operator subsection exist via give mc impossible unless author decide rank incoherence characterize principle sparsity incoherence obey incoherent eq canonical assumption namely incoherent guarantee exact programming strong assumption easy exist assumption obey incoherent inequality statement mc assume uniformly point convenient zero probability rely bernoulli sampling simplicity ready result completion principle analysis convex let fixed incoherence unique distribute among cardinality write numerical least unique programming enough exactly recover mc problem computable bind result discuss strongly program completion like rank partial show unique dominant exceed explicit sampling unique standard theory assume matrix programming eq multipli strong convexity optimality lagrangian unique sufficient necessary exist shall condition list least obvious satisfie valid dual via assume solution strongly programming theorem high probability verify abc event construction verify practice quantity hard computable bind summarize sampling ratio discuss recover rank fraction fraction programming provide mc present state sufficient unique strongly idea follow feasible perturbation unless arbitrary arbitrary hold eq construction always eq triangle schwarz inequality imply q convexity bind assumption theorem uv sign strongly verify go theorem numerical checking suffice parameter final expression inequality notice inequality choose get low side monotonically easy maximum attain eq ab lemma proof see give upper use q cauchy schwarz matrix convex matrix analog convex numerical recovery principle explicit lower involve complementary result counterpart guarantee observe corrupt least therefore direction completion robust component zhang chinese visit university china multi advance process z completion j pp linearize bregman iteration compress z linearize bregman norm pp linearize j sciences relaxation transaction theory pp plan proceeding li analysis journal uncertainty signal incoherence decomposition journal chen atomic pursuit pp compressed sensing theory pp ph thesis stanford principal physics biology pp recover low ji xu robust video imaging sciences pp ji liu xu denoise pattern liu interior journal pp lin j wu chen fast convex recovery technical lin chen l wu augment multipli exact report min keyword pursuit proceeding conference information collaborative completion cognitive chen bregman iterative mathematical positive program pp linearize compressive sparse pp via minimization w minimization conference decision pp xu minimization mathematic pp simple r nj accelerate gradient least optimization stream international journal vision pp rao robust exact matrix convex optimization journal available w
ii optimal optimizer table take computed design dimension second consider turn point quadratic scalar popular modeling disease cancer efficacy assume setting locally e depend optimal take equivalently bayesian apply locally design et optimal design next description cover model optimality reader design design transform via transformation place sake although red quickly design transform back give interior thank work national foundation dms security state design give formulae optimal prescribe li follow hold respect c lemma far cx subtracting since positive give q last second expression get independence multiplier easily minimize prove claim claim p design li paragraph assume linearly strict li previous apply ix I q complete small eq conversely constant ix vb conjecture section design formulae unknown design single illustrative design give elegant sign sign chernoff chernoff design state dimension involve search sign notation assumption method explicit formulae thus computational design greatly give current sign fast along line without search sign formulae sign practice polynomial regression turning point wu generalize reference recent show design standardize optimize estimation design characterize certain generalize use design clinical carlo efficient scalar normally transpose parameter take independent li li kk satisfy small give nonzero achieve e class admit singular follow elegant origin ray origin design put take
erm unfortunately enough notion online choose sample slightly stability batch however set furthermore know notion review threshold batch online binary randomized uniform show learnable notion loo begin notation definition notion might necessary binary stability notion notion uniform loo learnable problem future open question batch set unknown find minimize define empirical asymptotically regularizer rate whenever mention monotonically I say notion common datum loss risk learnable erm recently much set hold learnable learnable erm turn notion loo stability measure latter general notion shall see commonly notion loo stability measure leave look loo stability remove dataset algorithm distribution index universal average loo loo stable loo loo average stable opposite counter implication direction exception mean matter loo stable loo stable stability notion name slight strong loo simply similar loo need smaller hold hold uniform loo show erm general learnable loo stable learnable erm loo erm nice consequence set attention loo stability loo natural analyze online algorithm batch stability notion notion measure another look mention necessary batch weak another stability except loo stability learnable exist stable addition allow learnable exist stable always potentially learnable convex exist deterministic strongly set observe mention online thought thought supervise classification datum regret definition online say lipschitz convex algorithm regularize erm regularizer functional measure special mirror descent see majority type online stability relate uniform loo weak online difference stability loo change datum point algorithm stability loo weak uniform asymmetric obvious loo online learnable achieve loo case strongly loo loo loo stable well regret loo currently interesting non supervise online mini pick want loo stability sufficient performance unfortunately always exist online loo rate stable online learnable particular I believe technique always long stable able notation regret equation allow online stability rl lr fs I I fs z rl fs fs fs fs j fs fs stable fs I fs fs fs fs fs extra double summation erm easily see fs fs erm regret stable general summation fs fs j fs j I fs z r prove always hard double negligible algorithm eq rl fs combine symmetric loo stable stable loo uniform stable rl fs fs fs fs I I fs fs I fs fs loo stability fs corollary either imply loo erm loo lipschitz rl erm obtain z z hence I l lipschitz loo rate lipschitz regularizers z follow minimize j j rl j z conclude prove alternate regularizer necessarily lipschitz strongly proof ax c cx aa I prove sequence online method method mirror descent mirror low point previously choose rather regularizer previous stability refer surrogate loss minimizer point surrogate loss fs r r broad bound h prove instead stable z exist stable learnable rate apply corollary previous observe dataset randomize weighted generalization distribution online know online learnable learnable randomized adversary problem fall category variant interpret randomized loo stable analysis useful existence randomize formally I functional complexity satisfy might mean tuple gaussian case finite h z ensure algorithm incur hypothesis sample advance definition extend fs average distribution go instantaneous sample probability go g change z randomize introduce play instantaneous apply z loo expert regularizer instantaneous randomized uniform loo finite expert iteration pick satisfie kl regularizer regularization make kl di randomize long point randomize lagrangian choose ji consider choose lead c j tt must loo regularizer expect lipschitz p h regularizer lead lm l c integral statement theorem establish instantaneous learnable uniform loo demonstrate finitely respect online learnable randomized loo restrict symmetric loo stability sufficient give illustrate loo learnable set erm loo deterministic deterministic loo randomize algorithm learnable stable loo loo stable regret hypothesis space loo stable dataset occur show loo stable adversary pick algorithm incur allow allow linearity make hypothesis distribution convex denote learnable uniform uniform loo algorithm stable still uniform loo make universal loo hypothesis change algorithm achieve erm loss odd equal erm pick would thus follow randomize subgradient pick uniform loo fs fs z I loo stable previous plus minus removal achieve whenever pick pick pick pick seek track boundary increase incur loss incur average loss average way learnable learnable learnable show loo stable stability achieve regret stability loo necessary loo strong loo loo stability batch show condition exist learnable learnable set deterministic hypothesis define binary vc finite batch learnable batch learnable erm loo online pick binary learner iteration learner iteration number hypothesis achieve entire sequence even effectively learner iteration learner entire regret regret randomized limit go hence conclude learnable potentially loo argument characterize set learnable know loo learnable potentially loo stable expert play round randomize loo construct expert
km virtue convexity equation concavity factor decomposition gradient matrix eq consider whenever virtue sub histogram computation algorithm computation follow ij simplex cost warm start simple minimize project subgradient linearization subgradient nest loop loop parameterize subdifferential metric subgradient concave part taylor expansion inner loop gradient function increment terminate outer loop realize computed objective far return dc minimize minima link point crucial I pm q qp p z minima good global guess far point interpret r popular initialize minimizer way initial replace discuss q histogram need trick satisfactory seed near phase trick yield explain experimental minimize sum either depend value feasible set unit sum coefficient intractable norm ball pseudo straightforward three condition consider metric algorithm point applicable negative entry solve replace approach dot product handle cone constraint define leave technique section arbitrary table dot candidate table center instance easy table trivially independence approximation tend histogram definite positive call table table briefly explain result write small typical histogram polytope directly tractable large unique minimizer program hessian diagonal compute metric coefficient negative uniformly histogram contrary cone unit ball must order computation arbitrary center independence exhaustive dominant machine bin independently jensen hellinger divergence usually well histogram straightforward euclidean illustrate instance experimental bin enough histogram either co occurrence form prior color bin bin bin distinct support regardless support describe form arguably matrix mahalanobis euclidean root directly information recent review literature follow learn mahalanobis semi criterion candidate model neighbor inspire consider possible learn ground metric little conceptually mahalanobis operator learn operate worth although design vector histogram statistical motivate propose bin parameterized propose naturally operation perturb histogram accordingly describe perturbation semi histogram side key distinction algorithm follow present ground normalize histogram optimization require flow warm gain approximately problem toolbox projection loop objective unconstrained solution positivity optimize slow constrain newton image histogram obtain implementation feature mid computed dimension sum result binary task class form form test amount point hellinger coordinate root public mahalanobis default representation histogram originally euclidean histograms hellinger hellinger mahalanobis build upon euclidean learn mahalanobis significant observe simple confirm classification class class normalize neighborhood directly stepsize experiment fact normalization comparable step inner loop loop progress step tried grid claim select metric table consider machine estimate use train svm bandwidth distance set range fold fold cross select distance general amount regularization minus eigenvalue matrix consider distance recall neighbor quantity average classification point consider select average close neighbor gap compete retrieve task compete svm agree usually near support vector machine hellinger geometry usually intuition far validate mahalanobis perform well histogram show vary use typical table appear impact additional curve point despite overhead table seem table average please figure except hellinger figure simplex explain metric directly progress agree intuition metric neighborhood parameter comparative parameter neighbor classifiers experimental average overlap set classification despite typical independence tune adaptively unique dataset type provide good ground project difference compare compete superior feature histogram ground lot recently argue computation matrix distance metric thresholded attractive compute suggest learn improve accuracy believe distance structural looking minimizer criterion algorithm lie call compute pair warm carry fast implementation could provide computational improvement accept would considerably optimal neighbor candidate learn histogram available u decade machine metric parameterize distance distance choose date practitioner knowledge limit considerably scope lift metric ground follow use descriptor image metric histogram arise frequently language computer vision bioinformatics involve object simplify occurrence frequency feature represent histogram histogram color sift bag word gram follow principle distance propose histogram theoretically popular influential work distance think vision histogram histogram ground machine motivate prior problematic problem knowledge argue universal problem machine ground adaptively organize distance similarity obtain minima subgradient mahalanobis metric inspire much propose unnormalized histogram program sum resp resp element remove sure constraint rank
expansion wavelet expansion truncate whereas thresholded wavelet practical toy realistic inspire study show good moderate wide alternative hypothesis choice publish procedure yield method possibility variety interesting analyse contain measurement galaxy particular event ray sphere event sphere object depend production density universe field interest origin arrive particle interact atom huge cascade secondary secondary detector ground permit measurement direction arrival ray particle energy observe rapidly low numerous composition responsible acceleration least phenomena high energy event decade addition accelerate particle recent extremely recent alternate hypothesis concern origin star extremely yet many origin energy energy extent propagate effect energy cutoff ray physical identify account chemical high energy present knowledge energy order range completely alternate origin alternate production decay big year old publish primary impose limit kind attempt origin statistical signature production likely two rather source second correlation direction arrival nearby attempt plausible production hypothesis probability direction like nearly hypothesis distribution correlate local universe could superposition distinguish understand production year several arrival sometimes difficulty lie source locate ray permit acceleration depend path small small atomic number typical small neither ray factor error direction arrival degree degree heavy long come source distribution analysis sure direction total number event completely constrain meaningful analysis arrival experiment correlation nearby active locate distance km obvious investigate new insight statistically well publish particular work focus question although seem active goodness directional wish discover paper organize ray propagation carlo parameter present supplement devoted investigation tool analyse carlo physical ray emission propagation beyond decide qualitatively simulation although representative toy ray emission one source arrival energy alternate assume uniformly volume radius origin indistinguishable direction arrival identify easily direction address uniformly identical energy draw random energy ray square distance produce study high energy correlation arrival assume coverage exposure compute straightforwardly simple consideration detail exposure display cutoff simply sphere ray subsection model use exposure incoming event direction important medium impact optical range via foreground star align line major star reveal general near align emission infer direct field nearby dense three field direction amplitude measure rotation amplitude amplitude reach local outer galaxy align plane optical scale field disk inside impact energy model physical component arrival induce model center superposition justify central limit theorem sphere irrelevant truncation typical atomic regular atomic alpha propagation length velocity income regular field axis typical field regular part field typical ray come galaxy income ray maximum typical disk distant location universe qualitatively strength less know correlation length large energy although shape region index exact shape spectrum impact various energy field implement ray first ray exact induce refinement easily change practical application high help light particle factor account energie outcome extreme many source source right event different angular capability orientation respect equivalent direction display exposure exposure exposure map assume accept angle incoming period exposure right effect exposure perturbation would measure currently arrive whole subsequent handle choice take acceptance exposure incoming event direction density nan observe direction position physical phenomenon observational need reformulate sphere function directional algorithm small may collect incomplete optimal way pose nonparametric alternative derivation view shall twice old exponent space speak shape neighbourhood nan exclude constant critical alternative essential estimator nonparametric like pixel constant estimate count event sphere could procedure precise contrast nice various context detection uniformity goodness fit test homogeneity uniformity account directional model noise uniform exposure rotation adaptive ideally moment along arrival direction handle formalism sphere clear good especially usual respect sphere consider remarkable concentrate enable traditional basis usual event false second alternative separation optimality eq infimum test collection mean nonlinear analogy homogeneity procedure index respect make implicit know optimally argument concentrate would probably mathematical regularity achieve hypothesis individually define adaptive loss prescribe test interesting procedure plug density property view prove argument suggest minimax exponent estimator approach favorable measure minimax critical radius detection see asymptotically hypothesis respect nonparametric infinite popular minimax technique thresholding post processing constant difference situation especially consider drive density interpretation thresholding quantity one compactly line number actually sphere concentrated replace type finally prescribe deal benchmark er von function high dimension sphere sake comparison run simulation test near point sphere draw take value admit exposure case quantile big exposure uniform quantile point test angular geodesic counterpart evaluate detection typical maximize priori consequently care quantile compare section procedure tune prescribe exposure detector figure carlo replication choose probability scale c pt pt c section power alternative test table online supplement power norm online supplement plotted supplement observation use exposure investigate small choice publication speak plausibility alternative alternative unimodal would uniformly repeat physics inspire rise rich frequency give alternative density put density unimodal observation display family alternative repeat emission source explain source uniformly distribute realistic type matter generalization straightforward source error incidence angle namely exposure detector kind least one ray ray describe source randomly spherical radius assume simulation value namely play incidence assume resp sample resp statistical much energy go detect become simulation scatter source source specifically multiscale alternative understand respect source draw table table alternative percent prescribe level practically band vary j round bold namely case refer supplement curve operate roc procedure along wide procedure curve concave envelope accordingly analyse table represent choice method roc complementary relevant first sensitivity use regular expect unimodal roc supplement procedure figure appear behaviour radius vary case ns illustrate sample produce shape display curve logarithmic highlight comparative performance bad sensitivity case mainly group standard whole near nan distance near sensitive discriminate vary alternative consistently slightly one illustration size low detection supplement less table highlighted multiscale method appropriate future datum set behaviour respect sample soon remain display density alternative power remain roughly separation analogy euclidean space numerical consistent claim base bound density consideration become adaptive pre remain angular sake comparison tune clear give precise close parameter plot alternative figure structure scale observe respect truly efficient parameter level alternative alternative incomplete test exposure column reach point quite taking efficiency test public arrival direction directional event distribution length along perform reference notice early reach minimal interpretable computed value significance isotropic data table datum c c quite sensitive except take theoretical expression statistically monte carlo suggest consider significantly soon turn rule multiple methodology appear evidence realistic alternative
multiclass multiclass website correspond randomly remain problem two version unconstraine constrain classical averaged figure lar classify multi class lar svm learn sequential set fixed note experiment state use stability different value sparsity model c ccc sparsity l show two representative curve summarize sparsity due lar equivalent svm note give linear interpolation compare believe provide ccc corpus un un show sparsity outperform svm true similar result corpus configuration classify comparison figure dataset right good svm less fact solve select feature entire approach propose advantage decision take consideration rank function associate purely redundant inter feature individually induce group tree also although similar filter embed fold describe brief good drawback search perform sequential embed practice classifier remain entity sort black box additionally entire combinatorial resemble similar goal use sensitive similar model optimize quantity decision mechanism inference tie heuristic article classification classify take consider combinatorial inspire reinforcement show problem additionally work easily extend overfitte experimental maintain black universit france technique representation approach whole induce standard classification model classify wise extend inference traditional svm lar class equal performance machine direction modern selection encourage sparsity goal improve prevent overfitte sparsity representation entire choice vary classify infer look difficult underlie balance optimize balance result high preferred second extraction able space dataset onto subspace wise automatically classify use subspace classification iteratively choose classify wise introduce inspire reinforcement contribution propose new sequential obtain term classification wise sparsity e classify classification problem series corpora lar qualitative define explain interest approach approach detail also give qualitative problem supervise want associate category parameter commonly empirical risk solution moreover feature classify risk penalization encourage obtain feature possible classical feature perform feature classify input use classify easy classify one predict label vector convention feature classify category wise advantage feature encourage qualitative explanation extension classifier classify number classify general select classify input differently cc right circle one terminal bold bold red illustrate reward receive reward define equation original decision mdp classify attribute feature classify classify category deterministic currently classify select possible action state set currently correspond stop sequential decision terminal action define q scoring action score quality action follow obtain started follow parameterization reward reward episode obtain see reward initial correspond empty pick classification choose reflect take action state reward feature avoid classify incorrectly choose feature explain goal find parameterization describe let show maximize equivalent wise empirical risk mdp equivalence mdp classifier detail due computed possible state take state action state represent note may representation propose projection restrict select attribute intermediate representation manner action easily distinguish classifier project higher dependent offset amount find optimal parameterization reward wise regularize explore state action policy consist well bellman compose main iteratively choose state policy classical regression obtain generalize estimated action visit iteration parameterize use classify core suffer select regularization classifying training general allow overfitte allow classify input constrain constrain action vector handle ignore type action p cm feature unconstraine un select choose force choose classified
school temperature degree capital city life high temperature cp school list exp aic bic life cp life exp l l correct estimator restrict scenario state aic estimator uncertainty scenario estimator unbounde correctly restrict fold ten correct validation exactly diversity row seven row covariate represent interest specie represents find specie area area km km km original miss full sub aic bic specie area adjacent shrinkage along restrict shrinkage ps error error show thousand represent bic model bic likely notice bic compete uncertainty consequence cause sensitivity restrict estimator outperform estimator predictor capture variability monte conduct positive shrinkage x si si si pi nn initially repeat calculate estimator norm various measure compare amount rmse unity superiority list shrinkage cccc clearly restriction unbounded decay horizontal positive move event small among estimator neighbourhood unbounde fast rate rmse neither suggest maintain superiority estimator wide range shrinkage panel maintain superiority wide range panel suggest shrinkage preferred remain specify statistical correctly one go wrong assume wrong case estimate expression distributional distributional setup much alternative meaningful asymptotic generally simple squared expectation loss write covariance matrix evaluate compare risk prefer say dominate however hold every risk local cumulative eq dispersion dominate asymptotically context propose distributional regularity variate expression form quadratic let dispersion asymptotic distributional present local statistical property estimator restrict gain substantial near review shrinkage estimation multiple bias expression estimator covariate full hand base suitable criterion subset covariate become nuisance step full estimate repeat cross rate restrict superior shrinkage consist respective close misspecification conduct monte characteristic shrinkage size nuisance carlo numerically compute restrict shrinkage study fact outperform estimator unbounde unbounded restrict however approach rmse unity shrinkage perform wide nuisance subset panel either select restrict shrinkage towards subspace space penalize member pls perform estimation simultaneously estimation shrink towards sub space towards change sign practitioner although estimator take care part coefficient eliminate shrink introduction half decade shrinkage around performance shrinkage lasso partially comparative shrinkage absolute find review literature front communication proposition remark positive shrinkage study sm subset covariate contribute restricted covariate may subset combine estimator outperform validate preliminary validity restriction thus outcome preliminary illustrate shrinkage estimation positive shrinkage degree uncertainty carlo keyword phrase estimation rmse branch estimation considerable attention statistical model linear unknown whether parameter obtain insight many situation information contain positively nevertheless advantageous rather parameter datum reliable useful whether accept confirm result mind depend usefulness uncertain may incorporate uncertain prior depend restriction restrict preliminary later estimation stein type take alternative estimator utilize sample prove useful shrinkage attention analytically demonstrate shrinkage preliminary usual maximum study give description large error analytically numerically preliminary dominate stein estimation uncertain various location uncertain available shrinkage certain apply real life purpose affect interpretability sign type stein estimator context preliminary indicator test practical shrinkage incorporate shrinkage life form shrinkage estimator outline estimation utilize model combine sub subspace response prior nuisance subset usual nuisance covariate bic restrict shrinkage prediction shrinkage sub fold cross validation divide roughly size aside term subset use raw correct prediction validation estimate bias random varie
determinant confirm know set maximal determinant extension resolve small resolve order search matrix decompose spectrum maximal determinant hadamard large determinant change sign generality open since hadamard experimental code mapping vice versa confusion thank correspondence convenient determinant split determinant unless subject investigation small three class four order sharp order q factor arise sharp work high gram block see tree order mod open odd order resolve upper use early author infeasible thus computer describe gram order low odd order order determine upper order integer largely take definition design determinant design odd normalize normalize iff iff show odd convert sign regard equivalent row column determinant suggest sign hadamard iff sign permutation matrix gram iff sign order definite furthermore call candidate gram matrix summarize gram gram increment detail admissible gram good step rely follow originally use maximal gram set member diagonal take furthermore minor equal potentially unfortunately multidimensional set expensive restrict diagonal element last entirely latter search done result approximately list candidate complete gram describe involve combinatorial regard row know find row row correspond principle hadamard rely member family gram family gram build column general gram clarity algorithm gram tree application constraint recursive search search implicit subtree root search increment solution simultaneous solution recursively relevant subtree justify sign permutation partial order call level create nonempty contiguous collection frame frame consist width refinement frame level frame consider number entry may weight column output otherwise variable subject follow w zero entry call search recursively subtree elimination always interval variable uniquely non discard preferable basic proportion elimination illustrate candidate elsewhere yet associate vector comment translate together entry entry impose give linear child lead solution outline gram equivalent column relation efficiently know matrix polynomial correspond constraint gram principle new pruning matrix appear block know row matrix depend especially large arithmetic need number case fail backtracking succeed existence decomposition rational far design three corresponding design gram computational maximal determinant describe determinant take hour find nine equivalence class nine twice matrix website g seven candidate matrix decompose run nevertheless attempt replicate involve tree nine matrix characteristic determinant decompose give three hadamard order differ three switch decomposition search decomposition equivalence run program give second similarly second equivalent three design determinant much case give bind previously corresponding equivalence maximal indicate figure correspond maximal factor hadamard least backtrack take hour find website characteristic characteristic polynomial second gram matrix decompose equivalent stop reach explore search decompose find take hour get hadamard order solution expect website upper order order bind attain equal thus last gram cc bind pt pt maximal determinant summary pattern continue case see gram must since attempt construct hill construction base hadamard order fail plausible may resource candidate gram improvement gram take processor candidate hour show decompose construction hadamard appear elsewhere determinant set spectrum include metropolis stein gap gap later result spectra spectra find shorthand computational use find finding run second gram appendix determinant apply minor non diagonal minor block block principal minor submatrix principal determinant q assume consist depend take first thing prove explain introduce replace I expand submatrix operation arise result q leave question empirical search empty induction theorem hold want form subtract expand find suitable element write n r r let satisfy condition true element last column claim determinant claim positive symmetric subtracting
study framework would bias proposal toy aid splitting bivariate shown figure three hasting firstly structure might secondly low separate proposal display log reaction effect improve ability plot create expensive integration separate area low exploration first target rate energy expand parallel recover chain spread target chain accurately mode reach outer mode initial mode hasting exploration successful bottom evaluation scatter normalization sampling scatter adaptive metropolis algorithm half illustrate figure plot bin split bin stay constant show histogram evaluation point show bin bin run uniformity within hence reaction movement paris exploratory stage preliminary stage much attention propose automatic combine upon component adaptive method wang heart interact feature decrease improve explore density parallel adaptive wang interact illustrate modeling lastly overcome encounter spatial contain full pseudo modal remark technology introduce measure device ever grow accordingly linear several decade complex integration growth largely interest distribution arise allow practitioner increase algorithm density invariant current proposal parametrize state accept chain reject state distant chain mode sample outside ensure hasting study sample practice even run convergence accurately approximate reasonable currently available power illustrate highly define high correlation tune manually possible issue monte carlo refer exploratory discuss trait multimodal high connect trait process wang improvement adaptive spatial image distinct goal firstly proposal past sample largely improve already mode might different explore adaptation prevent adequate exploration alternate solution adapt secondly whose goal encourage include energy wang latter distant potentially unknown mode often consume practitioner code merely exploratory believe room exploratory learn particularly would ideally continuous space associate interest user consume model without purpose carlo various idea work aim automatically algorithmic method able much fundamental mode highly multi instead proportional peak idea behind parallel employ chain subsequently dynamically move sequential approach whereby move strategy phase transform considerably across temperature relate partitioning partitioning auxiliary iteratively fundamental wang energy sampler convergence reach temperature metropolis move temperature chain interest distribution use mode move wang partition along reaction energy chain invariant iteration instead chain various wang heart discuss combine energy sequential carlo counterpart central sequence successively reaction coordinate manner largely select monte choose target smc mostly hasting gibbs move increase popularity exploratory adaptively paper dedicate solely namely review result call literature create mcmc principal past sample encourage sampler mcmc consume practitioner save automate exploratory nature complete proposal distribution careful employ encourage exploration exploiting previously explore mode proposal visit prevent reach direction axis additionally combine wang desirable grow recall constitute core improvement task interact encourage end term answer previously generate time admit biased iteration distribution converge infinity restriction coincide restriction multiplicative mechanism improve would analytically invariant validate b straightforward bias univariate biased partition state along left plot partitioning case integral area coincide constant however integral estimate wang generate estimate infinity replace constant wang reaction coordinate choose decrease typically invariant behind increase towards visit tradeoff exploration hasting conjugacy wang use criterion meet criterion meet close last criterion meet control criterion already meet describe generalize wang partition reaction sample sample transition flat ix normalize I explore region decrease frequency useful follow notational simplicity already answer reaction coordinate regardless model reaction coordinate one reaction far wang increase flexibility efficiency sampler know bin partition reaction coordinate depend issue one sample evaluation first wider allow wide initial must decide bin due difficulty select bin bin bin normalize important maintain uniformity within movement realize reaction bin artificial within strongly skewed artificial leave bin bin difficult line within sophisticated outline contrary new create right bin bin start equal specify bin half bin exhibit chain bin bin close split bin bin split check flat meet meet mean easily bin hence keep automatic distribution bin certainly would never reach demonstrate strategy bin allow bin allow bin say reaction wish bin induce tail find new mode become bin include low state take define inner time bin split bin reaction coordinate algorithm idea transition detail alternatively exploratory seed mcmc code wang adaptive mechanism proceed certain choice reaction coordinate inherent example model application could employ quantity interest demonstrate include fourth application walk propose explicitly algorithm preliminary mcmc reaction parameter state interact increase encourage within wherein level variable average measure across calculate consume study convergence towards include describe likelihood employ follow represent select induce explore select noting option would ensure different size emphasize hasting flip high explore ability poorly preliminary value space biased importance use seed traditional mcmc calculate exactly examine number parallel wang aspect examine chain mention target target hasting explore specifically explore much latter run indicate partly seminal paper describe take benchmark smc wang move gaussian mixture weight call follow take invariance likelihood permutation component lead switch label mode replicate emphasize study induce computationally increase parametrization along reaction easier explore refer article choice reaction coordinate default component weight explore multimodal unnormalized weight handle straightforwardly simplex smc parallel detail naive improvement instance plausible reaction propose mcmc chain iteration point draw parameter compute quantile conservative spirit equally divide go twice initially explore instead robust iteration terminal number situation point confirm parameter overall cost acceptance meanwhile adaptive proposal rely evaluation initial number ess consecutive metropolis particle choose induce evaluation depend computational cost mcmc plane restrict plane replicate symbol check visit project dimensional clearly chain modify wang chains final importance bias importance put particle point proportional quantitative admit mode mean run mean highlight context smc confirm explain distribution mode expectation might though high ht method hasting monte smc initial quantity consider hyperparameter simulate high unchanged chain close another likewise particle illustrate degeneracy smc though important exploratory knowledge region parallel mcmc give initial estimation mean parallel hasting smc concentrate average independent algorithmic change number evaluation iteration chain fail mode result huge chain precision suggest unit available l number chain quantity average posterior plane finish identify region identify ice employ likelihood employ posterior similar original neighbourhood interior block mcmc propose flip fail however demonstrate power demonstrate metropolis hasting run preliminary hasting explore divide evenly split time split stop reaction bin conclusion run due flip mixture calculate worst taken example whereas require case wang add slight additional wang encourage alternate explore chain induce mode top leave region bridge central flip metropolis hasting explore absence ice encourage presence pixel tailor overcome explore ise purpose automatic exploration core community literature obstacle implement wang bin speed modern interact algorithm demonstrate density wide discrete mcmc unified practitioner field
bt h conditionally order conditionally remain small conditionally value k event consequently paragraph depend normalise vary hence v view enyi conclusion proposition consider v n calculation imply entail v survival mention eq collect n conclude let sake proposition application conclude let remark quantile eq condition asymptotic obtain heavy tailed explicit simplify lead separately since asymptotic impose whereas impose entail eq collecting imply finally mm st team france fr address estimation heavy tail functional covariate quantile range near boundary depend extreme introduce investigate quantile extreme value dedicate extreme quantile quantile tend propose heavy tailed adapt recorded period covariate consider fully model observation similarly spline estimator fit penalize likelihood covariate establish besides covariate curve come illustration quantile covariate hand smooth functional deal move window hand extreme comprehensive methodology framework make tailed parametric amount survival decrease polynomial three situation zero enough quantile quantile locate boundary outside define finite infinite denote cumulative distribution function cumulative quantile speak quantile rate vary characterize tailed account regular distribution satisfy given precisely want focus end point tend go infinity estimator use move window response ball role design concentrate go covariate belong quantile mean estimator adapt highlight unconditional examine de consider summary result slow wise conditional quantile interpolation slice c thus rely extreme extreme quantile tail estimator paragraph note adaptation achieve magnitude estimate tail index heavy give notation condition establish quantile control respect variable ii asymptotic distribution satisfy eventually situation arise result tackle satisfy situation eventually small appear sense application next paragraph family tail assumption simplify paragraph heavy tailed family introduce base log large integrate extreme quantile function let situation arise lead follow hold weight hill weight detail simple example heavy decrease case vanish quantile example proportional fr extreme value theory furthermore distribution extreme quantile european express around physical water answer dataset transfer co proportion transfer see spectra function discretized dataset co support machine regression see overview proportion perturb perturbation eq q identically fr furthermore value estimate conditional quantile end distance derivative denote base b chapter discretize depend index select thank estimator extreme two function associate denote
hierarchical wishart use link analytic technique variance make agglomerative particularly balanced agglomerative criterion produce cluster criterion dissimilarity thereby obtain partition partition p c variance criterion l vc c dissimilarity class note singleton early efficient hierarchical respectively cite early make mid present briefly exactly computationally way construction neighbor mutual chain rnn consist nn follow necessarily pair reciprocal rnn dissimilaritie tb nn irrespective rnns soon arrive use traditional store dissimilarity store algorithm criterion previous ambient particular something normally tb q dissimilarity linkage centroid either account des select grow pair rnn cluster update rnns step rnns return day find sum square latter term monotonic variation criterion sequence unweighte general dissimilarity also method optimally pack address include construction whenever reciprocal near meet distribute implementation parallel dissimilarity method et chemical database reciprocal near hierarchical assess al scientific social science cite linkage pass make create subset linkage analysis comprehensive agglomerative property agglomerative method induce hierarchy identical observation cluster important software package quite handle limitation impose adjacent near neighbor feasibility take self k et application hierarchical self review include hierarchical map term follow grow discuss local hierarchical wang model analysis cluster et identifiability next hierarchical gaussian cluster analogous cluster identifiability criterion criterion refer agglomerative feasible cut concerned chapter closely eigenvector sense interest innovation development eigenvector reduction algebraic give min frequently somewhat direction cluster focus xu classified select base density split split information dense region dense step create structure partition cell sort density center traversal neighbor cell category wang spatial rectangular cell represent hierarchical child level space difficult cell child grid cluster database noise partition recursion multidimensional grid cut minimal dense half use grid turn clustered topological search algorithm main step calculation sort e traversal block cluster define uniform dimensional datum represent datum set grey scale treat create grid cell image great deal transform segmentation grid lee et base algorithm dense low therefore arbitrarily distribute important advantage noise important discover arbitrarily shape since find et lee et integrating apply cluster cluster large spatial database et distribute cluster arbitrarily deal divide consider cube near neighbor neighbor inner capable find spherical computational detailed presentation wang see study integer expansion advantage system system case string common us metric ultrametric example bound order index place split stre leaf cell assign successfully retrieval root domain hierarchical long agglomerative development cell algorithmic computational domain area back decade decade early mr york analyse la paris rd ed ms cluster behavioral application york massive collection thesis hierarchical distance liu xu th international conference database dc pp agglomerative hierarchical journal classification des des structure hierarchical self hierarchical map journal signal processing xu discover database nd international conference discovery pp wu algorithms journal ad classification review statistical international conference parallel electrical engineering dc computer p history span tree history la agglomerative automatic document journal da base discover cluster noise proceed international conference discovery mining york towards curse dimensional international conference basis surveys mf relational cluster scientific de par en les de analyse des self rd hierarchical journal mathematical imaging ic analyse paris le b correspondence von survey map connection journal art f multidimensional cluster self bayesian segmentation image vision correspondence haar dendrogram journal institute massive ultrametric embed journal scientific nh lee clustering stream record semantic multidimensional behavioral sciences york cluster span computer journal xu density hierarchical th international conference zhang database journal optimally single computer numerical hierarchical constructing manifold principled transaction pattern intelligence van nd ed wang wang proceed conference mining rd international conference basis pp principal subspace visualization transaction visualization ed science technology hierarchical cluster visualization j de ia na red de ia agglomerative processor journal mode analysis reduce effect ed xu survey algorithm transaction xu computer xu international conference dc computer pp xu j fast large database mine spatial international agglomerative algorithm discuss software environment look mixture focus hierarchical finally develop grid agglomerative dominant embed scheme aim reader attention practical method point view effective view helpful target attribute numerical vector space formulate form rectangular inconsistent relation cover cluster interactive user storage retrieval recognition survey coverage include xu review include role retrieval make van et various mathematical view look normalization historical remark motivation agglomerative formulation wide theoretic reciprocal near neighbor near hierarchical overview self survey development grid particularly us consideration consideration relate comprise group decide group similarity pair symmetry iff triangular triangular space distance family positive integer chebyshev distances special cosine retrieval match vector query query close cosine dot product mahalanobis review metric distance distance map datum determine widely optimization proximity many salient answer dendrogram agglomerative hierarchical algorithm hierarchical group linkage method unweighted stage clustering hierarchical member
entry cone equivalent positive cone cone configuration jj ii ij configuration true dd property dd condition cone relax monotonic cone relax positive condition huge cone unknown positive cone configuration simple prove lemma rank n dd size matrix express lemma read dd k k recursively dd dd dd dd dd dd remark keyword true le university universit universit des des pr universit sciences ef le pt cm university sciences et france mail fr pt ref pt ptc david j er yu wang compress paper square absolute hyperparameter lagrange multiplier great profile tradeoff penalize condition optimizer increase also generalize norm take propose level signal condition homotopy lar norm dominant compressed solution property cone decade compress signal nu common compressed sensing theory elementary residual formulate operator concern apply straightforward evolution typical colored entry find theoretical view profile tradeoff help l example correspond plane pareto curvature tradeoff discovery algorithm homotopy angle lar develop advantage piece reconstruct whole hyperparameter interpolation colored evolution homotopy lar solution obvious homotopy lar usually start decrease necessary homotopy maintain update entry zero contrary previous work strictly yield iteration homotopy reduce word homotopy homotopy yield condition sparsity paper sufficient signal organize sec exist total variation denoise sec extend total variation call row row meet notation n k size transpose k n dd principal minor nm subdifferential system piecewise piece constant permutation nonzero entry omit sake brevity multiply since n increase piece continuous straightforward absolute decrease zero remain exist sufficient obvious hadamard monte satisfy whose find utilize high low column random enough recovered optimization intrinsic bernoulli dd within distribution step give kk correct homotopy property instance obey mutual homotopy run stop solution
bind improved previous follow theorem proof small restrict bound real see arbitrarily large entry triangular big consist row diagonal observe tm achieve margin dataset nevertheless margin row guarantee bottom derive imply doubly technique previous admit quantity technical corollary together bad price pay next end completeness norm decomposition lemma quick every lie om light optimal loss although bad dependence optimal need key decomposition margin finite conclude invoke similar dependence dependence modify second technique om omit long conjecture om rate converge state conjecture feature therefore c w bt induction iteration rewrite term product second since initial round need suppose non negative constant induction base hold thus side u equation separately quantity matrix ai ji proof first suffice without column k kk kn orthogonal act subspace recall linear algebra without loss independent submatrix form finish suffice k follow k vector pick coordinate set suppose point coordinate contradiction reduce solve om bf addition slack standard everywhere else know norm slack yield strict inequality entry segment therefore arbitrarily next matrix decomposition multiple possible nf positive banach f equality item mi mi bound adaboost combine weak slightly strong converge exponential previous minimizer exponential adaboost norm round bound depend round achieve within descent perform slightly adaboost understand focus simplify precede easy functional gradient iteratively example descent move adaboost direction adaboost choose add updating value exist asymptotically converge loss minimizer address state polynomial hypothesis q round rather weak typically low without additional assumption minimization aware bind convergence adaboost namely within adaboost achieve situation constant may within proof call margin loss far loss class prove determine proof consistency impact adaboost converge practitioner wish exponential well violate appear finite depend measure minimizer case variant also hold generally point compact space improvement exponential exponential loss round adaboost converge descent section conjecture associate section provide hypothesis compute hypothesis choose hypothesis specify adaboost pt weak hypothesis tx td hypothesis refer combination hypothesis adaboost define contains write compactly coordinate maximally decrease coordinate coordinate elsewhere recursion see term also adaboost hypothesis round divide side exponential drop depend ti ti weak rate criterion weak hypothesis absolute line long drop rate hypothesis however choose enjoy say small continue hold leave explicit simplify adaboost attain parameter norm serve reference bind thereby pose show section follow rate adaboost loss high behind large round indicate large guarantee adaboost grow proportional either way progress idea concrete lemma notation measure logarithm exponential combination achieve interested r notice loss round trivially also attention boost edge polynomially compare adaboost edge round monotonicity non negative great l otherwise next put loss negativity adaboost create summation rewrite target last combine complete assume lemma prove grow fall growth large prove firstly rearrange complete prove b proof show suffice chain tr tr complete provide desire believe tight slack decrease improve directly never decrease monotonicity holds obtain absolute constant fast rate convergence except round scale back largely hypothesis j ts great decrease exponential ts instance distribution e modification adaboost loss round continue exploit adaboost improved back effect weight becoming show write summation add true gs denote minima throughout minima finish round together never occur adaboost identical rescale imply adaboost would hope show achieve argument tailor adaboost coordinate decrease connect solution require round show wide achieve dataset example get wrong correct round row boundedness hypothesis predict magnitude lie contradiction loss round thus assign mt mt statement directly weak entry feature property achieve fix every row example row triangular diagonal row complement loss lemma round reach least picture construct loss loss might achieve add therefore least margin combination satisfy imply attain exactly therefore hypothesis confidence rate arbitrary arbitrarily construction requirement integral norm achievable achieve norm two therefore margin least fourth triangle round boost cause cause rate increase drastically fact arbitrarily absolutely rate weak arbitrary confidence purely suffer highlight rate solution important convergence dependence round parameter solution achieve target dataset realize want round decompose part solution loss however introduce section depend additionally show necessary converge previous although respect polynomial upper adaboost reach depend technique upon adaboost mainly particular hold achieve adaboost combination optimal round situation fail yet optimal technical hold solution make progress progress always sufficient next result quantify nature use empty c z exist example finite deferred immediately denote c achieve zero finite margin thereby name l proceed finite illustrate wrong solution serve round adaboost proof appropriate subscript exp example sl combination end round edge mean loss unnormalize affect immediate decomposition recall put weight get x round matching bound henceforth loss margin example use lemma together show good progress use progress make solution early appear depend generality subspace nan nan necessary matrix along rescale bound derivative suffer I let schwarz may put quantity inside everything adaboost round solution lemma pick entire c edge last inequality step k mc complete boost loss dataset k claim apply sequence achieve round suffice throughout otherwise state boost simple rigorously define loss admissible combination weak zero subset loss complement margin build final subsequence empty pick example ever away subsequence attain begin subsequence repeat subsequence sequence converge call loss finite margin show extract combination item suppose combination
estimation observation equivalent alignment fact claim derive work space criterion consistent definition proper eqn f restrict result fisher rao metric f fisher rao fisher compare probability rigorously focus family metric nonparametric find important curve attribute preserve simplify calculation modify introduce interval go qx cc study velocity include parameterization absolutely integrable vertical obtain invertible understand reader metric smoothly tangent fisher rao riemannian dealing function cumulative classical fisher form rao sign metric fundamental riemannian metric invariant play important role information geometry geodesic geodesic geodesic connect simplification motivate represent elastic rao compute simply negativity fisher rao distance find compute involve calculation represent elastic step elastic represent set metric use induce elastic q domain belong distance proper satisfies negativity pseudo distance quick relationship rao metric item fisher rao dt w dt f geodesic numerical straight sf use framework align improve match peak across cross function align template derive template alignment element define individual function elastic geodesic minimizer function name become manifold rao understand geometry represent derivative transformation dt root geometry sphere simply length great connect rao define mean fr algorithm fr function initialize compute mean square rather dynamic programming align although prove global th iteration optimal q decrease iteratively converge template align function condition identity cross prove existence depict condition definition minimize fr fr fr apply setup utilize algorithm finding template align set note simulate transformation simulate give f second panel align function panel panel deviation ccccc std std tight alignment peak effect remove remain mean band alignment simulate function variability show leave peak peak mixture panel function well next estimate panel original practically cccc simulate gaussian phase variability shift tight alignment function leave remain align compact ccccc std std family multimodal phase show mean show huge apparent amplitude align phase variability perfect alignment ccccc std panel plot occur different discover bottom panel top row mean deviation alignment function mean fact average similar perform curve fig consist original growth deviation tight peak suggest several std std std std signature datum effectively elastic handwritten signature acceleration signature time study variability alignment show signature analysis panel align suggest align function much peak alignment due cccc cccc original std std spike sequence sequence train language focus convert spike train domain st dirac train spike smoothed spike train ft kt panel spike train perform path movement deviation neuron peak mean standard growth signature increase decrease deviation observe std align std example temporal microarray time cycle measure period minute fully cluster related phase fig gene expression many goal focus subproblem deviation alignment panel right panel align strong functional publish conceptual utilize discuss section criterion obvious criterion alignment three provide comprehensive denote function cross validate total variance align original correlation pairwise pearson well alignment least time least criterion variance align original alternative synchronization well achieve compare rao curve principal expectation package self method present match simulated result alignment datum even easy perform alignment visual alignment sometimes three fail evaluate alignment performance shape signal shape wave job evaluation number involve choose challenge scenario original signature method table representative complexity c r matlab matlab r matlab sec sec sec sec sec automate basic fisher define elastic distance template select identity align distance separation variability achieve framework template include development amplitude class functional corresponding idea relatively limit apply mention albeit context proof ft qx directional directional derivative tangent w dt rhs eqn proof dt corollary lemma dt q important last q strictly lemma minimizer minimize everywhere ft ds ds qr dr fr fr mathematics university north geometric separate frequently study framework rao metric convenient square transform rao simplifying distance align estimator scale translation idea real data berkeley growth handwritten curve spike train signal superior several publish alignment application nearly range processing easily align problem principle hilbert space compute distance cross function serious challenge arise inherent variability variability possibility tool increase elastic keep mind allow value function differ peak location term variability extract correspond function right function observed variability parsimonious consider height berkeley growth growth subject highlight growth growth rate discover broad pattern would technique show gene train relatively proper align across g require manual natural comprehensive alignment unsupervise fashion principled align elastic important unified fundamental idea treat process different scale observation vertical translation seek use mention introduce brief summary
solver author address fit approach search web contiguous dataset dimension common english may require dictionary available approximately mean rarely thereby unlikely occur even empirical study achieve binary fit memory extremely however often load training svm much severe arise memory situation publicly available dataset gb disk input format exceed document small dataset bit sparse text hashing use issue matrix solid foundation nonlinear effectively sketch hash definite linear svm bottleneck partition block repeatedly update however bottleneck memory loading iteration number disk approach work hash either widely apply permutation repeat time hash prohibitive bit problem store low bit convenience th bit low bit approximate formula numerical comparison probability permutation apply hash theoretical property hash foundation bit hashing behind construction learning call pd follow pd whose th entry ij nm ij te b hashing pd dim pd pd bit pd expand expansion dimension exactly svm popular representative package include sgd solve regularize logistic important demonstrate effectiveness bit hashing simply achievable approach permutation feature store store merely bit expand originally binary digit expand vector fed solver inspire bit hash pd theorem note total datum exactly work closely conduct public randomly sample demonstrate effectiveness gb ram system format fit bit hashing substantially advantageous testing merely mb try train result hash second accuracy match test therefore benefit provide hashing demonstrate able tuning parameter logistic conduct extensive experiment std deviation test achieve randomize experiment report mean illustrate produce averaged repetition hash solid accuracy use dash red computed repetition become extremely e second training take minute load take course study collection store multiple near neighbor hashing testing see merely include loading efficiency processing hashing may often color available accuracy bit hashing figure standard deviation train second second summary effective reduce testing svm training interestingly bit curve represent repetition red color original method projection although dimensional related min dim estimate inner multiply sample er v eq include projection provide small satisfie equal I paper refer correct version sketch hash vector element task estimate product suggest generate remove bias variance difficult mention major heavy pre multiply original vector equal correspond apply multiplication hashing e er follow see interestingly choice vector element wise paper test bit substantially size time example bit small projection storage hash fold bit bit substantial make I size zero preserve mean vector number exceed easy zero zero focus reduction introduction relatively often zero develop useful excellent tool indexing due property extremely indexing reduction reduce original data technique achieve huge expand point binary like vector use substantially especially expand actually generate binary hash size may provide insight straightforward formula b kt datum vector generate bit hashing counting exactly denote estimate unbiased complete reduce trade figure verify intuition bit bit quite accuracy additional bring accuracy bit bit practice understand practice use simulate permutation require stage
solve numerical linear solver essentially result incorporate improve several provide rank notably underlie amazon random moreover hide hadamard projection projection run approximate leverage dense brief review relevant main algorithm contain main conclude approximation main environment column column ax square square alternatively finally identity let thin general singular finally orthonormal subspace range orthogonal onto frobenius norm provide call l set transform projection probability matrix draw let accuracy strong transform vector project orthogonality quickly specifically find let randomize hadamard use efficient construction recall unnormalize hadamard transform hadamard equal simplicity generality numerical property apply energy need access transform vector distribution formalism denote imply property generate summarize combination use arbitrary fix orthogonal probability additive approximation leverage start high recall goal orthogonal span fold second multiplication application projection bottleneck computing eqn compute approximate matrix preserve idea uniformly sample fail return meaningful matrix identical row find row preserve rank row see formally recall let example approximate approximated take efficient bottleneck need euclidean norm thus reduce specifically vector sketch essentially dot different randomize approximate leverage matrix output idea improve sketch let svd let note implementation approximate approximation specify basis choose leave approximately compute norm next state matrix lemma say eqn compute qr rather multiply triangle inequality obtain lemma dot product score approximation within first follow pairwise product achieve compute use improve leverage result theorem algorithm subroutine heavy provide sketch proof theorem proof subsection point lemma event estimate improve run since orthogonal prove preserve alone need inner relate actual cross efficiently product preserve condition event additive approximation rescale I e full rotation unitary invariance eqn vector preserve let vector j expand square algebra multiply throughout u u j homogeneity together return return squared row norm implementation r nd nr q simplify product among denote pair schwarz call norm heavy first pair notation heavy cauchy schwarz sort initialize initialize stop otherwise increment first heavy none pair heavy occur add loop pair next decrease otherwise occur whenever pair norm heavy norm heavy number operation pair heavy whether heavy return heavy pair apply n tu nu tu eqn j f conclude return satisfie extension computation leverage score case specifie arise compute capture interested low majority capture parameter score normalize k ill degenerate leverage well moreover leverage leverage trivial degeneracy help singular singular example play role solve distinguish singular th get leverage score leverage score leverage might spectral useful feature develop algorithms care score notion approximation give approximation rank instead eqn ill pose hand instead normalize leverage good seek number leverage remove leverage popular spectral frobenius approximate leverage score input namely approximation draw I statistical input close write essentially prove computational detail appear see conference remainder technical report purpose section detail sketch gaussian matrix k consider negative x algorithm approximately leverage score assumption span hold clearly normalize leverage leverage rank summarize normalize rank parameter approximate rank output namely rank rank draw compute orthonormal compute vector leave q return worth ta ta lemma close member constant unlike spectral provide closed formula return approximation normalize gaussian I eqn standard linear prove rearrange take square side row matrix orthogonal follow eqn leverage normalize matrix take computation discussion give compute leverage conclude broad related constrain variant streaming environment statistical leverage proceed tw argue leverage obtain nontrivial provable compute truncation negative perform order final return algorithm truncation leverage score maintain positivity matrix notable approximate separate compute dot positivity estimator manner thresholding provable albeit weak guarantee bad direction considerable evaluate empirically leverage score score quite column well time given argue compute use order make statistical leverage proportional leverage main result failure solution least square eqn probability satisfy constant right vector q diagonal strictly positive full apply singular value hence ready theorem opt u tv derivation drop term change fact simplify imply opt compute computing sum satisfy eqn rt p opt remark first constant clear leverage approximate leverage hadamard multiply hadamard analyze follow line interesting evaluate experimentally score relate statistic way stream stream pass space depend integer size stream compute qr decomposition compute outline effectively row euclidean norm equal find row idea bit demonstrate concern sampling vector treat matrix along copy row set linear matrix ta norm multiplication serve magnitude streaming set additive reduce entropy bit pass compute effectively bit natural obtain leverage importance sampling sampling score compute finally procedure proportional bit pass identity row definition proposition corollary david square top singular popular problem completion nystr om low statistical develop precise I relative several practically important cross extension idea streaming vector correlate basis usefulness recently randomize relate popular nystr om base approximation leverage compute singular leverage statistical leverage outli score bad amenable implementation useful domain detail I well dominant value qr projection span randomize compute qualitatively arbitrary algorithm assumption precise opposed na I corollary coherence addition practically underlying definition denote singular th
differentiable w scale norm bregman divergence compact ergodic mirror iterative maintain gradient specifically stochastic receive select dependent project descent since continuity bit fx denote measurable element field sample norm consequence expect though still function definition essential presentation distance hellinger factor different supremum square hellinger fact hellinger metric hellinger denote mixing probabilistic exist time weak version assumption mix field markov uniformly ergodic chain space indeed assumption mixing randomness process stochastically mixing allow wide process assumption begin expectation section sharp numerical factor sample algorithm assumption arbitrary assumption hold eq assumption bound immediate jensen hold satisfie method similar least theorem obtain additional arise care couple nonetheless corollary clear uniformly hold mix assumption make broad turn slight bound intuition attain process stationary process uniform geometric variation mix hellinger condition assume assumption update stepsize integral simplify stepsize multipli parameter corollary generally stepsize argument choose shorthand choose descent mirror generalization attain sharp conclusion hold high replace step al ergodic multipli penalty scale bad classical approximation choose nonetheless average yield see class ergodic remark homogeneity reasonably work similar assume conclude process geometric simply present slowly process concern optimality result informally oracle query return oracle th receive distribution xt xt xt random set set mix return stochastic assumption definition minimax norm minimax oracle satisfy minimax complexity implication match discussion al dependence bound optimal brief definition proximal theorem constant attain collect consequence statement begin concrete abstract principle complete mix ergodic previously example previously incremental descent al ram et come optimization scheme processor computer function goal minimize procedure work pass processor token indicate hold iteration update token move processor draw distribution algorithm update token evolve doubly stationary case true th doubly stochastic q denote spectral addition recall significant consequently follow corollary evolve evolve doubly consequence theorems corollary mix walk hellinger distance rate somewhat original incremental gradient update euclidean geometry essential obtaining method mixing return sharp finally ram rate et algorithm convergence rate never weak strong random walk bind uniform web receive click impose result ranking order lead total order impose certainly challenging sample rapidly develop permutation partial transition order show chain follow mix consistent objective evolve update multipli turn example broad guarantee generalize markov gradient random communication consider autoregressive require expect total distance assumption appendix procedure token processor transition token doubly token et al notably k take apply define example applicability agent communication link algorithm failure suitable obtain doubly logarithmic roughly increase suffer bound suffer example statistical standard chain apply space simulation phenomena autoregressive em assumption markov ergodicity difficult require essentially allow difficulty focus moving model identify linear area paper particular condition assumption obtain sharp universal corollary contrast ram apply property ram et exist strongly connect example motivate corollary fast rate brief discussion process establish ergodic assumption markov exhibit rate hasting sampler stationary assume simplicity hasting sampler markov denote condition density transition accept hasting mcmc generates associate away metropolis hasting uniform mix main update distribution give fix statement statement multipli apply along enter minimize derivative inspection slowly mix incorrect set incorrect substantially slow mis suitably choose demonstrate provably yield convergence slowly mix require mix incorrectly note mis ergodic suitably quickly nonetheless corollary stepsize hellinger total regardless notational infimum term decrease yield probability borel argue occur suffice ft present experiment algorithm guarantee essentially interesting natural understand benefit mirror descent problem natural identification surface pair bi variance system q minimax ar study alternative generate sample case call replication specify repeat one sample analysis limit infinite step system generate accord autoregressive process replication sgd sgd begin use sgd proximal yield analogue descent n fx taking computing yield figure simulation obtain sample figure replication approach theory still convergence stepsize enough multiple performance gradient plot figure convergence plot sequentially rather draw efficient cc task robustness modification stepsize second numerical experiment study motivation instantaneous distribute incremental mirror draw sample sampling vector flip sign perfectly uniform analogue offline use objective effectiveness non euclidean robustness stepsize nearly denote analysis al yield I norm whose corollary simulation distribute left plot connect cycle cycle mirror circle denote dot plot th optimization set predict theory mis take spaced plot right figure show optimality mis certainly degradation capture behavior analyze necessary subsection proof expect prove give collecting consequence make proof represent subgradient expectation fx fx make substantially easy proof essentially early care establish relevant optimization setup understand impact equality may take expectation essential idea allow nearly parameter lemma stochastic unbiased formalize whose proof hold stability show value apart let non eq hold nee apply precede lemma use proof turn lemma sum require control final either lipschitz obtain complete statement holds begin point expansion show small stationary mind appropriately behave approximately martingale hold provide appendix combination proof start obtain remain probabilistic hold guarantee lemma guarantee accuracy probability inequality guarantee consequence state hold description intuition returning employ online identical minimization packing hypercube classical fact cardinality sequential pair otherwise construct uniform sequence step tu yx denote inspection al enough sampling set different block mutual entropy sub within block apply al lemma construction see coin oracle lemma subgradient descent extend elegant desire difficulty reasonably fast ergodic stochastic generate show strength analysis believe version carlo sampler clean derive convergence full g provide low numerical constant special nice property thank question thank anonymous suggestion begin control make result subgradient take
smoothed matrix follow static select number static evolutionary heuristic many cluster number cluster accomplish review modularity heuristic cluster employ choose affect face challenge cluster matching cluster cluster weight common object general multiple merge splitting multiple beyond scope reader address problem affect framework tracking measure criterion seek rand agreement ground rand rand spectral propose evolutionary experiment track gaussian distribution dimensional first step size new membership run cluster cluster incorporate mse plot estimate oracle calculate cluster membership application fig sep alpha oracle increase nearly walk oracle appear oracle alpha rather steady estimate continue objective close fig draw identity mixture proportion draw initially initially proportion cluster unchanged sample tp time use mse experiment vary membership choice mse significantly exclude oracle cluster membership move perform stationary experiment plot cluster compare simulate two filter short memory rd order memory fig good rand poorly begin overlap overlap slow filter index rand place historical membership change rand affect tp plot iteration notice visible outperform move membership change finally cluster perform appear converge low oracle effect natural phenomenon phenomenon propose object govern try centroid try keep move experiment initially regular interval time path note behavior simulate change change membership one tp tp rand static iteration run experiment linkage previous true computed membership various display memory experiment rand various list see cluster simply move toward position rather switch cluster affect modularity equip run membership maximize rand tp estimate iteration tp tp rand method two rand separate summary list outperform good drop notice iteration rand perform unlike interesting observation affect cluster contribute rand leave step experiment mit reality phone activity student mit phone record medium mac address nearby device device proximity student proximity divide week partial truth namely dominant could proximity business school likely school cccc school iteration cross experiment real simulate instead fold believe substitute rand entire school year list surprisingly static cross validate leave step estimate similarity membership contrary object leave six important mit day class notice estimate drop physical change student physical similarity time begin break estimate matrix notice physical particularly change similarity fall estimate factor appropriate evolve set namely price daily stock exchange operational construct stock coordinate price day vector subtracting divide deviation day period cluster stock ground label stock list stock evolutionary rand affect standard error five leave cluster stock list rand c ccc capital stock non service stock finance health stock stock rand index static iteration iteration estimate drop mit mining happen market occur suggest tp evaluate scalability affect varying object cluster stock compare affect evolutionary algorithm ten intel processor fig computation affect run affect consist iterate static cluster notice affect stock cluster iterative nature fast increase decrease fast great iteration affect propose track cluster accurately track recursive control order square allow evolutionary selecting unlike exist framework synthetic good outperform evolutionary future factor converge improve finally plan proximity optimize perhaps certain dot sample mixture calculate oracle factor section drop simplify arbitrary independence eq calculation involve expression variance confirm indeed possess assume structure acknowledgement thank anonymous suggestion improve national science office grant nf xu partially award science engineering application evolve evolutionary typically static propose often smoothness tracking follow present evolutionary adaptively parameter shrinkage improve I additional include evolutionary synthetic indicate evolutionary algorithm scenario obtain tracking find time vary stock market mining machine processing object change short term variation na I datum extremely sensitive produce cluster unstable inconsistent long short variation several evolutionary cost static penalty prevent cluster evolutionary commonly agglomerative penalty manner paper evolutionary treat track static dissimilarity mean view involve past perform static state estimator improve raw estimate past call formula optimal evolutionary factor adaptively adjust dynamic propose evolutionary tracking extend dissimilarity handle cluster accommodate object leave time demonstrate affect three namely cluster spectral synthetic set outperform recently evolutionary cluster algorithm adaptive evolutionary cluster framework several advantage evolutionary static enable extend proximity input evolutionary algorithm outperform static cluster exist evolutionary increase static iteration extension evolutionary propose static insight effectiveness experiment commonly use algorithm affect term notation assign cluster datum store dimensional th feature vector create proximity object dissimilarity represent adjacency denote edge vertex edge agglomerative cluster dendrogram dendrogram certain obtain flat variant agglomerative hierarchical general vary dissimilarity common dissimilarity object cluster dissimilarity complete linkage dissimilarity low dissimilarity dendrogram attempt minimize centroid object cluster close simply square euclidean object close rewrite dot product algorithm dot product integer calculate centroid compute centroid cluster similarity positive definite similarity similarity laplacian spectral cluster association aa np solve relaxed relaxed variant optimal consist eigenvalue similarly solution consist optimal relaxed typically normalize tp li unit algorithm normalize ratio cut eigenvector instead ignore row average association large ignore step contribution area constrain evolutionary incremental type cluster type stream focus stream object target evolutionary cluster incremental incremental could apply type consider focus incremental low computational expense quality incremental often bad static already introduction evolutionary concern capable constrain optimize fit objective use hand use preference evolutionary evolutionary historical result evolve object step unlikely cluster divide segment differ significantly one remove unobserved matrix reflect mutually filter correspond filter impractical model secondly covariance enough evolutionary simple recursive define expect static rather proximity objective static estimate also lead cluster result I static use disadvantage unstable inconsistent adjacent incorporate potentially step allow rate allow amount smoothing smooth unbiased combination bias representative term statistical estimating similar variance convex suitably notice shrinkage smoothed proximity proximity shrinkage intensity manner shrinkage estimation frobenius proximity smoothed optimize try estimate know shall henceforth refer mutually independent risk eq note conditional derivative set rearrange minimize risk lead minimize risk knowledge proximity trying suggest replace w tw tn belong cluster membership propose proximity use object two object two distinct belong structure proximity fig short inside possesse assume ti gmm gmm parameterize mean manner sample determine object draw gaussian change component membership stay show dot similarity observation structure although rather information shape block model order framework beyond scope area work model assumption block obtain variance know cluster membership around membership membership logical choice membership static substitute estimate substitute static clustering result result refine improve quality clustering find rarely
birth death general previous focused birth death often nonlinear birth death rate particle provide mean sophisticated realistic biological example population sometimes growth capacity genetic allele depend allele rate researcher assess death progress birth death researcher longitudinal observation observation likelihood partially since write close challenge researcher progress none development likelihood general birth death major likelihood expression expectation miss relevant expectation em researcher derive unfortunately expectation pose challenge joint distribution generate able expectation certain rate depend particle development promise technique sophisticated realistic review little applied researcher parameter apparent provide derive estimate laplace transform fraction continuously em describe expectation expectation transition laplace transform expectation technique costly integration previous provide example em fast compete validate open question evolutionary particle time happen happen instantaneous rate homogeneous efficient birth death rate complete likelihood transition ordinary parameterization possible transition term orthogonal polynomial special find even rarely outline additional specific insight specification advantageous essential maintain probability compute expectation necessary laplace transform equation laplace transform rearrange recurrence relation exact laplace write compactly numerator denominator rational function fraction combine although transform transition close fraction fraction representation fast series transition probability fraction control generate stable probability unobserved conditional expectation discrete wish find maximum parameter birth death state realization spend let step realization denote total particle count amount demonstrate concept figure likelihood maximize em complete maximize step surrogate clarity omit parameter assume uniqueness expectation death unobserve state path adopt combine author derive resort approximate relevant expectation simulate path recognize conditional expectation statistic conditional expectation formula appear markov chain whose integration product prohibitive however integrable laplace transform easy laplace property inverse laplace formula offer substantial integral inversion involve inverse amenable acceleration method transform method death representative example complete analytic maximization simple unknown solving update estimator continuously probabilitie expectation algorithm sometimes population enter arise model point dna sequence suppose new arise process rate regardless many exist represent become derivative concave maximize take preserved maximize respect expression weight proportion state illustrate specification evident state examine typical population often incorporate limitation environment model growth stochastic previously roughly deterministic growth support growth beyond balance restrict growth suppose parent decay death interpret capacity roughly appear q preserve place epidemic infect become infect currently infect sis disease specify birth death number rate new infected population allow assessment novel way associate process birth death rate linear possibility covariate ease arrive surrogate maximize step surrogate differential process note become ill condition difficult newton option perform carry avoid ascent check newton present application evolution outline subsequent algorithm usually simple evaluate fortunately replace summation eq expectation greater slow near appropriate exploit acceleration introduce implementation give figure likelihood accelerate newton guarantee achieve ascent advantageous efficiently formulae analytic expression analytic generally termination em also outline purely technique hessian formulae achieve exist expectation various time table employ rejection reject method I convolution use compute domain outline implementation effort share code sis list table fast method term time stand achieve logistic finding construct markov method finite process result aware affect simulation hand enough path upper however condition fail sis model issue quite largely computational rejection condition quantity convolution logistic kt sis several draw start integer trajectory record glm simple parameterization table simulated observation regardless find factor generate path start finite size partially stochastic point se linear sis glm short character dna molecular responsible repeat researcher analyze evolution however many depend nucleotide addition rate fit composition generalize examine evolution evolution common repeat size nucleotide find human control extended inference evolution draw introduce novel modeling deduce implement insight must acknowledge evolutionary human human evolutionary separate human mild reversible assume stationarity stand justify evolutionary regard vice evolutionary human equilibrium line kolmogorov therefore observe human evolutionary separate scale unity evolutionary evident small number necessary mutation interpret justification avoid g past researcher grow suggest equilibrium distribution length assume addition depend present birth equilibrium close form nonzero addition always evolutionary likelihood equivalent distribution also incorporate achieve impose barrier iteratively barrier composition heterogeneity covariate define control model together surrogate become eq report infer size evolutionary great least unique largely consistent justification nonzero distribution formal statistical repeat mutation cccc covariate se birth correspond nucleotide composition control difference birth birth death death compare mutation statistic flexible realistic become observe establish require birth death rate exact ingredient hessian numerically previous completion typically numerical simulation rejection condition comparison convolution conditioning markov chain sis parameter laplace offer chain nucleotide substitution availability yield parameter update model mle close majority return update available one numerical section likelihood slow surrogate lie far newton require inversion surrogate condition speedup laplace convolution acceleration slowly toward na
label thm lemma de universit de universit b f france criterion recursive adapt deal allow arrive converge surely particular attention average automatic descent step average compare term speed estimation partitioning means finally cluster profile every hour dimensional recursive approximation fast domain biology computer vast literature recent argue computation procedure large slow focus deal fix non sequential sequential require find local element belong cluster drawback base consequently may represent fraction get develop consider norm spatial proved certainly partition around local consequence reduce subsampling computation distance accuracy scenario outlier distance cluster center simulation study sort bad execution idea estimator dimensional propose recursive advantage one recursive nature automatic store descent value empirical step sequential order run present third proof rely technique sequential technique small number initialization point competitive moderate small major search element consequently adapt deal temporal indicate appendix copy take mean aim finding follow non take present new notation recursive equal set point recursive start arbitrary group allocate allocate recursive fast rely exhibit denote check partial stochastic suppose measurable denote gx gx classical descent choice estimation set point realization cluster minimizing follow risk classical descent step adopt asymptotic could suitable practice attain parametric algorithm experimental define sensitive carefully simulation particular show inaccurate always even choose prove geometric corresponding classical averaging around prefer slow provide center start one value converge surely absolutely lebesgue measure n absolutely almost h fulfil stationary towards converge surely converge randomly absolutely moreover get decompose nan probability scope establish convergent provide continuous version estimator relate property fulfil satisfied datum deal fulfil infinity simulation recursive function default make average recursive parameter far consist approximate recursive justification automatic tuning work well important beyond scope however intuitive certainly group would consider value present highlight strength drawback recursive realization bivariate random vector cc cc contamination sample dimension also perform vary multivariate covariance structure autocorrelation play draw dimension directly cluster evaluate comparison term small contamination equals estimate even contamination level secondly nearly even perform replication equal present figure performance well look solution among sample explore enough look provide error around attain consider multiply factor mean presence stable term empirical recursive value drive average chosen drive summarize center mean evaluate formula run average classification measure partition indicator place define equal perfectly design detect automatically replication size contamination term drive fraction outlier figure clearly affect performance even well performance strongly large contamination median unchanged meaning fraction small one term effective recover code call average fast get mean allocate computation drive recursive sample size large ccc critical recursive mean take second second average
auxiliary lemma admit z let star initial admit unique z moreover z since yield second yield h z ph sup des paris france france sup dr award general dr interest communication theory wireless gray rgb pt corollary conjecture remark remark interference wireless digital line multi division access decode channel multiple parallel channel orthogonal successive interference cumulative signal ratio response reinforcement differential tucker ordinary paris may analyze non game channel available channel possess unique distribute evolutionary theory stochastically game unique equilibrium employ stochastic still theoretical novel nonlinear dynamic nash potential view decentralize wireless distribute assume protocol goal resource etc question whether policie deviation equilibrium reach require accordingly attract wireless communication concern allocation viewpoint subject optimal reach boundary rate assume knowledge central hand allocation capacity achieve power unstable decentralize consist operate channel focus answer point ac analyze despite apparent relevant channel allocation throughput control analysis focus scheme separately treat incoming signal instead low decode overhead latter consequence decode suffer property rate theory channel iterative converge special conditional fail static thus capacity region understand attempt carry allocation admit nash equilibria potential equilibrium uniqueness fail uniqueness surprisingly turn though static nash uniqueness characterize system behavior get regard consider pricing interference similarly show local channel overall subject mild interference water equilibrium game result enhance drop modify water water present base often player require solve water study extensively game property understand nonetheless unique open fast adjust learn payoff achievable ergodic game theoretic game admit star convex game equilibrium fast convergence establish notational span canonical measure finally indice player one simply setup consider wireless receiver orthogonal typically may subset denote channel subject represent user channel power analogously pre k k denote clearly achievable rate allocation gain much duration power extreme coherence short correspond interesting intermediate block instantaneous game channel efficiency user bandwidth gain draw transmission spectral efficiency user high lead player player scale simplex allocation vector space strategy player payoff game nash player finite even payoff multilinear hand possesse examine channel remain transmission know gain la evolve evolve channel regime evolve transmission relevance power ergodic counterpart payoff admit whose depend calculation crucial numerical begin notion profile resp resp admit potential existence equilibrium already practical view equilibria crucial evaluation etc static potential profile strictly might singleton admit detail fact quantity call scenario receiver access almost always strict convexity promise determine admit construct channel coefficient admit equilibrium sketch proof whose vertex kp kp kp impose acyclic assertion version potential game accordingly realization admit equilibrium potential give nash equilibrium theorem clear whether decentralized environment distribute cognitive present attract point view player assume policy optimally unfortunately monitor large large decentralize limitation static water channel overall interference point interference except number rely player know possibly hard algorithm discrete adapt action space one start aspect dynamic suffer drawback algorithm game q lead term ensure utility behave nash equilibria stationary instead seek player alone water user kp update scheme clearly unlike water close develop power instantaneous iteratively little overlap coincide popular total unit calculate base appear shall set equilibrium power spectral configuration coincide black rest converse vertex necessarily nash nonetheless game dynamic interior solution dynamic kullback leibler divergence nash equilibrium rate see dynamic review kullback divergence channel thus power quickly fitness specie phenotype fitness equilibrium see extend dynamic really water water contraction contraction fail equilibrium state least sake equilibria equilibria power equilibrium q k k grow follow deterministic evolve stochastically dynamic p kn kn knn concentrate channel uncorrelated case expectation gradient mean asymptotic follow represent notion block channel satisfy uncorrelated nash equilibrium interpret either discount interpretation purpose limit device evolve convergence note ergodic fast instantaneous information update power little static user learn obtain equilibrium ergodic exponential equilibrium mean static fig equilibrium channel fig channel theoretical begin rate achievable aggregate capacity game equilibrium similarly equilibrium static channel equilibrium unity attribute equilibrium alone fair simulation much channel rate equilibrium reach begin channel realization time channel user within ergodic block plot system user system ergodic equilibrium slow follow version dynamical deterministic space consideration possess long stationary simulated well user km fig run plot power evolve instantaneous channel quantify evolve remarkably dynamic evolve equilibrium within ms ms km ms number channel previous tracking channel time present
open loop turn another formula request ucb policy good item ucb step go sequence uniformly report good good ucb discover request adapt high missing mass reference therein heuristic principle reinforcement prefer ucb mass time step make request expert high start brief good strategy rely new interesting item make expert issue address effort subsection version discrete element element mass item occur instance error eq denote modify applying get moreover conclude confidence bandit good arm tx tx tx tx tx order ucb tune cn tp ucb design general iii explicitly outcome request request bound validate ucb behave meet focus show number know every information make choice expert item note expert index n k k n nk find expert every step horizon close strategy loop strategy request high interesting step alternatively request expert step expert successively shall remain expert besides require collect time go infinity together converge go deterministic compute strictly decrease mapping define q mapping appendix go ii infinity sequence infinity go nf nf lemma proof omit expression oracle policy allow consequence sufficient upper bound request proportion item yet satisfy I line least good optimistic make request request time collect good ucb round close loop policy theorem proceed step expert obviously missing denote expert distinguish either belong easily request expert draw twice otherwise nu il u il absolute th request expert place I u il u u kn il q il n accord surely sufficient show converge loop policy respective number request appear limit recall n f proportion interesting find request go almost proportion allocation define technique eq lagrangian denote get first conversely item open loop policy q ki performance ucb balanced side ucb algorithm proportion arithmetic unbalance simulation behaviour ucb illustrate satisfying assumption proportion display item ucb solid balanced simply alternate dotted representative run averaging remove variability obvious moderate oracle take rather course improve ucb probabilistic geometrically ucb sampling remain ucb oracle former sampling entire discovery process choose tight estimator well suggest proposition value discovery expert simulation extend first analyze behaviour less restrictive optimality good ucb remove fairly oracle policy clear remove though assume ucb good ucb complicated explicit mass whether convergence whether sense another interesting item infinity possible good estimator contribute experiment decrease comparison show eq hence converge write nf nf il nf reciprocal sequel suffice expert almost infinity elementary geometric distribution eq fourth develop markov permit converge infinity accord eq proof proposition assumption engineering nj usa universit ac paris france fr analysis probabilistic expert address base optimistic paradigm estimator strategy uniformly attain probabilistic provide suggest behavior still weak optimistic algorithm request expert request latter draw discover request expert arise real security power often amount credible may security perhaps e g country power hour
noise level weighted group fuse identifie point asymptotically without position scale unweighted group fuse first point tend outside fuse lasso find first tend e robustness method fluctuation position point classical multidimensional video likely rule profile come genomic tumor gene although level change point lie change function result focus detection however effort conjecture fuse analyze check point single point kkt correct change strictly point single case weighting estimation change independently may although multiple change confirm indeed correctly multidimensional select segment multidimensional find would expect square criterion successive value error approximation appear intensive impossible subset programming strategy ease notation physical reality implement slope derivative real processor gb lasso behavior profile length dimension run fused lar record run descent slope exponent fuse linearity curve initially sub extremely practical limit technology critical second fuse perhaps hard surprisingly increase eventually deterministic fuse lar suggest say version later slow run lar relatively converge final lasso increase complexity cubic already slow lar fix axis test empirically fuse multidimensional profile jump position add theorem trial fuse identify unweighted case occur across theorem variety resp unweighte fused weighted vary change location weight fuse remain fluctuation extend share length consider profile draw center profile run one implementation weight record define percentage detect resp group fuse outperform lar unweighted although lar reliable lasso experiment exact may worth burden accurate demonstrate change point number profile nine center either length possible application frequent gain among copy comparative genomic technology purpose adequate joint check segmentation profile remove fused lar subset post processing piecewise share calculate green c segment red large segment believe gain analogous loss obviously statistical carry frequent segment joint loss hmm outperform several benchmark jointly help gain constant interval point segment figure segment red positive believe loss segment profile profile exhibit region performance roc curve auc follow run default auc second weight fuse lar take auc method time cancer cell originally cell weight fuse lar fuse lar number change programming total second point hmm give author hmm look top panel stochastic code case common gain suppose advantage hmm predict gain fairly genomic weighted lar hmm consider publicly tumor profile give relative quantity dna patient fuse lar take second take minute profile version result smoothed version tumor profile fuse selection transform score correspond hmm vertical boundary comprehensive common genomic table hmm frequently arm q frequently lose region loss h figure select difficult verify frequently arm useful precise figure clearly gain loss hmm find perhaps account h weight fuse propose extend multidimensional approximately theoretically empirically group fuse change likely several theoretically empirically profile share change encourage accumulation genomic profile share try union present profile reduction detect point profile post fuse modify g tv fuse constrain signal frequently finally view solve proximal constraint multidimensional tv biology indeed theoretically help commonly conjecture hold technical successive beneficial oppose implementation follow lasso fast lasso would nice exist criterion change strategy result carry implementation claim define group lasso center zero mean compute cumulative formula get namely compute I u complexity significant compute step satisfy u j j ni n st conclude multiplication memory q easily check admit convention define row expressed similarly recover translate point locate reach row euclidean compute hand dirac jointly covariance particular similar computation equivalently tends union select union select converge lemma tend asymptotically check deduce whether select tend always result index th u happen denote eq convenience notation rewrite let event clear event q probability derive union bind lemma shift independence tend let support correspond possible weight unweighted move towards symmetry also otherwise lack generality treat suffice never e france paris france paris present detection share occur signal constraint multidimensional piecewise approximately consistency evidence simulate array comparative genomic place dimensional change way question several field common want find multidimensional processing detect computer economic time analysis another situation dimensional believe genomic profile patient increasingly biology variation genome microarray biological share individual cancer genome measure segmentation multidimensional signal dimension profile signal fix technology individual increase patient statistical view develop identify signal increase number exist vast point segment multidimensional approximate piecewise constant quadratic know segment dynamic prohibitive technology alternative global point segment change detect point signal piecewise signal global minimum benefit detect base multidimensional multidimensional signal approximation multidimensional increment reformulate yet problem lar design point length correspond signal genome design benefit though within signal towards consistently able correctly identify change notation section lasso empirical comparison copy number preliminary version publish two integer frobenius case vector indice b ji pp profile length store profile th column profile signal noise location tend share profile detect benefit possibly profile change profile function quadratic solve eq dirac otherwise jump solve although impractical current computer reach million genomic profile combinatorial relax replace jump variation tv recent solve fast like find change add pi tucker kkt expect strategy brief maintain remove kkt condition result optimization line check outside active fulfil group active kkt therefore need convergence group line must several time compute takes bind correctly would times kkt active provide coordinate strategy center inactive c computationally intensive interest implement lar regularization lar approximate solution path affine intend solve straight add line justification original lar require storage design benefit lar descent step need equation overall change line memory change center initialize point wu next theoretically extent estimator recover
proof monotonicity actually independent prox increase language operator objective associate whose infimum attain differentiable pointwise infimum index derivative objective develop obviously strict negative progress ensure convergence eq derivative whereby subtract follow cauchy flip sign apply conclude bound gx fx k x sufficiently large help crucial eq eq note scalar plug side since inequality obviously true scalar limit work allow reduce case applicable claim say norm write prevent enter region fall behavior pay strong assumption vanish variant apply scale composite objective function sequel decomposable advantageous differentiable analyze backpropagation incremental aware generalize fail composite function disadvantage even incremental proximal splitting induction combine shorthand inequality complete constant whereby requirement incremental allow invoke aim illustrative application nonconvex nmf nonnegative add allow nonsmooth regularizer study covered stochastic like method analysis vanish deterministic rewrite form amenable eliminate attain subroutine ii nmf rank factorization google column remove dataset objective nmf art toolbox dense optimize fair unlike equally matrix fig expect time line speedup numerical compare stochastic start well stepsize tuning good stepsize tuning substantially return sparsity htbp cc cc bar plot high well factor plot bad objective nonconvex composite permit error practically specialize incremental large scale factorization indicate state art numerous algorithm example incremental theoretically open analysis remark nonsmooth nonconvex objective includes extensively study convex nonconvex introduce powerful framework avoid assumption error within new incremental splitting even application scale nonsmooth factorization form continuously differentiable possibly nonconvex semi nonsmooth compact make written reach machine learning formulation extremely familiar blind deconvolution neural primary contribution algorithmic specifically new split smooth nonsmooth proximal splitting notable allow capability critical scalable incremental knowledge incremental nonconvex splitting distinction lie notably require realistic inherent error make simplify perhaps remarkable nonconvex choice general allow nonsmooth idea proximal prove projection nonsmooth proximity operator attractive important implementation associate split similarly nonconvex subgradient method fails exploit structure iterate batch recently briefly discuss nonconvex monotonic introduce generic nonconvex nonsmooth gradient algorithm exploit composite despite benefit splitting raise especially allow scalable descent base simplify presentation replace formulation primary via approach define component proximity function nonlinear introduce orthogonal notably algorithmic minimize alternate forward proximal formally suitable tie tackle nonconvex
regardless datum dot box represents learn permutation close learn clearly relative associated parametric outperform test permutation conditioning sample provide adequate probability structure alarm network size number capture arc likely relationship would data conclusion test goodness network new model preferable shrinkage permutation cover network shrinkage fit bic positive network learn size contingency therefore value soon test large shrinkage coefficient mutual classic increasingly shrinkage sample shrinkage systematic behaviour produce shrinkage many lead parametric affect bayesian discrete shrinkage test goodness network parametric permutation close dependence sciences ac uk network structure always heuristic algorithm dependency base algorithm cover overview develop shrinkage article whose publish communication taylor communications available online www graph node refer direct arc connect stochastic connect either marginally independent conditionally specify principle choice function aim random univariate random variable usually task network artificial intelligence consist ideally coincide correct terminology classical application probability optimisation despite terminology fit combine approach deal global distribution structure extensive study heuristic technique influence strategy independence associate I threshold equivalent sample pose serious conclusion decision heuristic therefore conditional present behaviour tune appropriately investigate behaviour permutation test discrete datum class test base pearson information mutual consider global multinomial latter independence frequency xy level condition classic contingency information proportional ratio differ relate pearson contingency table respective require permutation statistic norm considerable know curse cause exponential issue maximum multivariate discover investigate increase overall maximum likelihood n usually call closed mean derive multinomial improve definition shrinkage likelihood compute classic reason classic behaviour shrinkage test observe gain additional shrinkage freedom independence shrinkage test indicator goodness fit score prior bic learn well score dependence indicator estimate true distribution alarm network alarm arcs network hybrid use conditional investigation combine hill since brevity asymptotic test pearson mutual bic score
follow readily conclusion acknowledgement grateful improve early greatly careful reading discussion subject thank de france universit et paris france construct asymptotic divergence bootstrap include particular choice several propose subject phrase consistency weighted bootstrap parametric estimator empirical bootstrap set modeling flexible provide powerful variety therein good source reference research limit divergence crucially propose general sophisticated confidence student useful tool inaccurate attractive one procedure datum population draw inference confidence receive study topic main summarize follow estimator objective identically bootstrap draw mention alternatively exchangeable weighting resample propose extensively name vector order consider empirical subsequently index example nonparametric regression statistical procedure purpose paper survey formulation present drawback namely observation difficulty formulation weight bootstrap computationally many distribution smooth statistic stein bootstrap usefulness survey bootstrap refer organized procedure base divergence property section censor propose flow presentation mathematical development recently introduce sign absolutely continuous divergence kullback leibler respectively le respectively divergence sign correspond real notice whole strictly example chapter mention real kl well survey consider unknown basis shall satisfies whenever class divergence divergence assumption duality divergence represent optimization procedure accord function fix put eq display reach naturally divergence instrumental divergence estimator suitable robust maximum mle robustness influence treat numerous model keep definition bootstrappe measurable weight rewrite bootstrap exchangeable sequel transpose vector shall condition exchangeable ni ni nonparametric satisfied nonparametric condition list condition restrictive nonparametric double two source quantity e rigorously divergence space order relevant canonical randomness throughout independent order outer reader cite follow open contain assume hold q definition refer function brownian bridge brownian covariance separate optimum well separated define class possess randomization multiplier randomize admissible page normality q class eq limit may state precisely consequently take capture variance weight refer follow quantile bootstrap stand dirac take show nh shift contamination position global divergence associate maximum mle affect subject dual refer mind reference example divergence divergence associate detail kullback leibler x x divergence estimate example define parametric closed exponential gamma pareto unfortunately also ml practical use algorithm powerful since objective concave estimate nh remark consider power divergence calculus lead maximum estimate kl n simple calculus exponential eq equality consider divergence gamma gamma imply power divergence routine divergence pareto eq simple calculus give last equality n lead maximum upon choice divergence criterion divergence value indeed leibl hellinger divergence infinite negligible corresponding finite also classical therefore automatically contamination effect misspecification parametric reasonably property progress automatic mention minimum estimator involve solve multiple root consistent estimate necessary survival survival f ng censor observe pair whether censor q time death risk generally weight cf integrable sequel follow write censor situation replace estimate recall formula censor power divergence x give define infer lead independently censor open conduct examine provide coverage implement indicate divergence divergence hellinger consider simulation take estimate limit indeed coincide mle detail subject refer mention weight table mse various normal expect produce close look simulation hellinger mle term produce ht table estimator notice inferential value nominal ht subsection simulation censor censor censor take censor robustness ht
countable trivial one probability person information know extra agent information extra agent first new get inequality agent x prove finite countable consider set disjoint exist finitely one conjecture minimal assume define whenever disjoint j n h h ij px k concept neighbor efficiently site minimal efficiently neighbor neighbor coincide neighbor positivity every row equally positivity neighbor cm cm mm hypothesis section categorical university road bc neighbor field field site site belief site condition give neighbor positivity keyword field categorical spatial process definition field show joint various site neighbor positivity hold uninformative set field site belief agent site change conditional information site sense site question less event suppose change belief might well conjecture wrong probability take shorthand sure denote equality extra hence well general positivity imply positivity mean categorical field
viewpoint involve class discrimination framework drawback map discrimination prove function popular boost pose weight act study involve framework address allow discrimination label method pose tradeoff estimate fashion avoid act classification dictionary incorporate avoid involve technique modular guarantee module incorporate organize follow map section methodology validate framework digit art benchmark give probabilistic representation optimization formulate setup seek maximize couple prior parameter dimensional combination atom type need nature e explicit identity dictionary belong sharing simplify sample encode class membership linear classifier classifier vs setup classifier completely scalar quantify assign label functional section seek linear model denote formalize consider note determine keep represent code dictionary matrix combine classification unify allow consist dominant peak determine pp simplify representation prior behind encourage training representation encourage respectively classify sum prior respect encode representation enforce representation obtaining lead regularizer regularizer select effort estimate context code however representation dictionary develop specialized parameter tune require dependent descent alternate learn fix procedure estimate r weight term form exponential optimize adaboost optimize incorporation additive e pursuit omp refer require represent sample one vs vector element update facilitate define use however easy apply project newton derivative around p pz z p z pz z well outlier overcome cost positively correlate moderate cost correlated cost outperform htb l logistic z c logistic hinge fix update avoid inversion require efficient svd iteratively generate atom case iteration reader ji sample overall scheme experiment randomly scheme produce similar dictionary scheme compute use scheme summarize stop initialize classifier test infer seek maximize belong respect dominant maximize p inner exactly label testing account dense g identity compute independent speedup computationally expensive algorithm omp exploit parallelism suitable dramatically speedup training number initialize previous moreover framework sample modification update representation handwritten digit face digit standard benchmark mnist acquire write technique error handwritten rotation need shift version face vision sparse traditional treat estimate winner outperform method achieve significantly outperform baseline training dataset comprise pixel baseline four type plot indicate yield minimum reporting square low clearly comparable adapt label yield suit increase overall performance reliable square performance classifier variation predict value mm cc mm figure learn form observe change notice close indicate reliable reconstruction take contain significant corresponding linear visualization interestingly atom particular proportional variation require representation comprise improvement comparable comprise image implication discriminative much significant loss cccc c digit recognition exp log hinge exp hinge c address discriminative linear sequence study art thm address linear probabilistic framework single discriminative dictionary learn jointly oppose capable classification cost avoid technique update code validate apply digit face benchmark signal popular processing learn sparse column dictionary significant progress predefine recent result face recognition solve np hard empirically underlie improve upon art denoise refer refer
b entry dimension time summarize dimension three sparsity element fix artificial generate unit however suffer point generate much ccc cc structure solve regularize nontrivial log propose three time solver method expression consist thousand measure experimental interesting model consistently consider correlation consistent module prominent assertion proposition definition remark matrix tractable concentration rank matrix represent marginalization bregman mild fast artificial gene correlation observe explain latent class covariance dimensional arise draw poorly behave often essential robustness gain popularity recently covariance matrix covariance covariance concentration precision encode variable conditionally sparsity equivalent independent estimation covariance estimation formulate tr denote positive explicit definite method concentration efficiently however real provide instance prominent movie recommender influence latent social densely correlate marginalization therefore sparsity alone may gaussian simply submatrix concentration component specify latent marginalization hide variable impose underlie maximum latent formulate decompose denote hide whose parameter et show grow observe strictly high due appear penalty positive problem use purpose log program solver order involve produce believe efficient fast large derive adapt close subproblem bregman method expression find latent explain well correlation gene explain latent aspect analyze follow split bregman also consist utility expression datum split bregman include method admm multiplier increasingly become method recovery proceed bregman reformulate emphasize although bregman solve firstly bregman fitting adopt secondly appear provide formula divide symmetric update subproblem term couple make amenable bregman detailed although bregman demonstrate multiplier admm presentation split augment lagrangian define lagrangian augment term lagrangian dual apply gradient ascent multiplier ascent efficiency iterative algorithm largely solve note augment lagrangian solve minimization run time equation split bregman alternate iteration rough eq convergence convergence bregman quite mild irrelevant involved check square root symmetric root definite whose square first eigenvalue correspond adopt routine divide eigenvalue decomposition routine fa sl use symmetric eigenvector thus together diag easy routine divide full eigenvalue together latent graphical initialize kl u k bregman trial implement bit art semidefinite need problem demonstrate generalization ability datum guarantee theorem would affect involve artificial
interesting multivariate vector show adapt theorem need hold inclusion ii finally sign mean divide part show complement union give fact bind line last focus c spectral great inside line stationary jensen conclusion follow far tx line projection us line inequality line tell lasso choose zero ol know write tell adaptive biased equivalent converge optimal converge already cr cr device central limit z treat increase theorem remark asymptotic adaptive lasso regression weakly asymptotic distribution ol present study method access become main area traditionally test theoretic first fit later adapt general sequentially unnecessary number regressor method spurious huge assigning want distinct choose quickly greedy another face candidate model feasible identifiable shrinkage successfully matter leave shrinkage method several eq parameter matrix entire path handle relevant modification property modification lead adaptive estimate candidate large weakly enter coefficient enter enter effort understand lasso advance little series economic autoregressive choose variable weakly dependent framework integration paper suffer candidate small size large technique framework lasso factor forecast goal order predictor forecast possibly explanatory extend regression converge distinct rate parameter consistent estimator impose already integration number number sample size candidate possibly condition impose error candidate finite panel understand price portfolio know object include interesting evolution time series predictor lag autoregressive lag present show estimate final remark delay index meaning process weakly weakly stationary process hold mix rate mix jt jt z assumption common model particular derive multiple integrate invariance decide directly relate candidate coefficient nonzero stack matter case letter parameter assume without know result interested space problem concave minimization satisfy tucker condition paper include lead ic condition presence examine violate comprehensive ic section converge oracle assumption satisfied perturb standard perturb adaptive minimizing minimize estimate issue lasso iterate consist weight precisely initial ridge regularization stable provide penalize prove adapt consistency conclusion adapt implement cross projection adaptive dynamically fix small affect report study want accuracy four specification performance mean large effect except variable consider moderate effect moderate among highly error tail select correct select coefficient resample estimate mean correct coefficient zero coefficient c nz nz nz table adaptive frequently select correct set zero coefficient candidate variable effect moderate affected structure proportion select particularly model correlate error correlate large correctly zero accuracy parameter accuracy square estimate standard use resample table mse affect large oracle square quickly oracle mse expect mse moderate observation decrease indicate really negligible observation ols oracle ols ol ols ol c ols oracle ols ols ol oracle ols oracle c ols oracle oracle ols table estimate calculate resampling assume knowledge reasonable interested formula data
h take toward tend specify thereby enable prior toward toward little information inference correspond confidence credible presence physical parameter inference inference intermediate extent ambiguity utility discrimination level extent difference pointed process science quite bayesian suited try progress optimistic modeling frequentist frequentist bayesian try frequentist aim acceptable conclusion stand minimax attribute attribute idea motivate place facilitate make inference definition formalize face toward mathematically economic identify distinct type agent toward ambiguity vice toward ambiguity differ ambiguity balance statistical distinguished belief latter act bad consistent motivate frequentist technical literature conservative interval coverage rate term assign operational definition ambiguity ball color ambiguity action accord display gain accord utility display toward ambiguity action ii would concept require extreme minimax red utility nature red action iv ambiguity generalization conditional utility posterior posterior posterior insufficient spread physical variability ambiguity study decision approach minimax frequentist sense strategy minimize maximize member traditionally special utility take action maximize minimize review extend posterior physical cg reduce action complete similar spirit call cg strategy drawback prevent cg impose parameter cg cg necessarily frequentist cg procedure reduce procedure complete notation theoretic identify limitation cg demonstrate generality note call exchange brief discussion frequentist preliminary vector model realization p inference simple possibility nan posterior understand every triple probability call knowledge yield inference plausible posterior corresponding prior prevent confusion system would purely plausible posterior posterior admit inference inference p denote surely eq random posterior scalar borel encode nan device confidence probability probability give observed loss confidence posterior coherent associate denote set represent posterior necessarily rather derive confidence posterior law probability recent simple distribution benchmark respect moderate inference absolutely continuous many literature kullback information statistical gain replace plausible physical distribution specifically gain call plausible oppose bayesian posterior high follow make inference define bad inferential minimize maximum bad gain moderate work solve let write define decision take loss consideration q extreme decision work posterior minimization lead framework fact q finding moderate imply thus thereby reduce immediate convex moderate eq one posterior work bayesian nonempty confidence close posterior p nonempty unless nonempty whenever posterior lack example involve use bayes depend confidence posterior verify p xx quantify prediction cg dp imply entail substitute unique drop parametric plausible prior quadratic contrast moderate involve typical hypothesis selection parameter hypothesis equivalently term work correspond work let side versus sided example ix posterior nan equal widely applicable bind nan hypothesis restriction eq bind hypothesis great two similarly stand apply ip I I omit brevity posterior recover interestingly case simplify evident unique say equation uniqueness moderate close work p ix work plausible balance bayesian encounter various bayesian posterior improve lead frequentist illustrate unclear would lead work benchmark attractive whenever corollary word inference irrespective posterior posterior plausible posterior specify member many involve member derive impose procedure average member explain posterior simplicity version posterior moderate three applicable version summarize generalize beyond version inference upon
ik note resemble specification stick stick break product fix reason specification construction different second concentration marginal aspect follow section sake clarity discuss specification independent thank result marginally precision measure hold true precision component base process h assume two process marginal use along precision process cluster observation cc h h row purpose analyze let correlation component q fig stick break right graph white gray area parametrization interval stick assumption eq q parameter bivariate beta breaking measure worth mixture globally exchangeable distinction limit h independent obtain consider block exchangeable case assume independent marginally take I ig give proof appendix recall process parameter take section dependent marginal random proposition extension slice shall sampling beta subsection propose extend product poisson stick distribution start introduce variable finite latent indicate finite crucially introduce allocation take value slice already conditionally vector ij hyperparameter law assume hyperparameter v exploration monte carlo gibbs sampler iteratively multidimensional trivial generation allocation multivariate stick break efficient conditional let q shall k subsection conditional u conditionally conditional strategy full depend kernel joint metropolis walk acceptance k employ independent conditionally important remark slice full almost conditional precisely small new dependent dirichlet normal apply simulate production united states european h synthetic base normal bivariate parametric section shall describe sake omit indicate full hyperparameter order sample n conditional inverse respectively previous parameter break obtain gamma respectively v alternative independent simulate accept turn inefficient rate th previous simulate walk proposal conditional proposal parameter acceptance rate simulate independent component alternatively model component normal mix cycle even feature regime cluster growth could expect regime cycle dependent multivariate dirichlet process literature usually consider day adjusted index regime py forecast shift shift intercept volatility dirichlet prior product inverse gamma var consider improper normal gibbs cc frequency period logarithmic solid line dash empirical note non oppose capture non linearity business cycle distribution row histogram allow conclude cycle result coherent result switch consider growth finding economic inspection posterior mean see economic business phase substantially pi coherent suggest regime nevertheless finding matter research four consequence solid fig tail dash ty predictive density conditionally evaluate effect correspondence realize similarly expansion phase atom fig approximated period expansion posterior distribution expansion fig second column one volatility concentrate expansion order identify dp mixture posterior associated country originally successfully allocation mcmc one cc st us marginal l il minimize sum square q specifically show cycle column vertical dark gray band correspond growth true gray column fig sample every mcmc iteration two make allocation comparable dependent estimate probability one cycle another cycle belong white light gray share atom allow find cycle fig identification expansion group atom interpret strong phase cycle phase coherent business exception present growth dependent apply dependent context non mcmc computation particularly cluster joint analysis business cycle parametric able highlight issue gibbs describe principle sample infinite proceed number check summarize pt algorithm step initialize suppose involve come sample metropolis gibbs old eq direct calculation obtain hence measure h sake simplicity place rs addition set eq give conditional definition proposition remark context may across motivated non modeling dirichlet multivariate process stick break weight effect provide efficient illustrate simulation united european index keyword paper concern statistic rely probability measure measure process stick break stochastically weight break construction beta dp paper generalization dp dirichlet take generalization stick break construction dp univariate bayesian non univariate dp incorporate prior provide extremely specifically model kernel availability simple use field aim process naturally generalization model divide different shall observation exchangeable exchangeable recently dependent dirichlet specification incorporate dependence atom process covariate exist dependence structure dependent random upon hierarchical structure stick g probability normalize vector measure employ bivariate dp stick weight tractable inference procedure multivariate poisson show appeal dp new dp posterior model data augmentation extend slice conditional series non time increase employ independent dirichlet process stochastic flexible stick breaking capture extend bayesian parametric model allow accounting clustering parametric usually introduce breaking process property introduce dirichlet process modelling propose chain mcmc dp simulated united european conclude value divide may specify assess information introduce ic iy dependent stick break stick break propose vector atom hypothesis stochastically probability determine via break stochastically independent dependent measure vector dependence measure affect underlying density
simple picture reduce dynamical half exploit helpful system organize provide background model reduction extend balance rkh set determine nonlinear control control input approach observe lot require small energy determine coordinate state reach significantly coordinate generalize approach nonlinear control advance suitably balanced need lyapunov equation balance impulse truncate balancing approach assume study circuit unbalanced approximation operator rkh provide laplacian build trajectory reduce system principal carry control balance control observable two generate input column span coincide many reduction lyapunov several idea behind balance align find f balanced look gap singular responsible output relationship assume negligible unstable integral however exist lyapunov balance find transformation positive balance unstable balancing method exist computing cholesky problem energy lyapunov development around observable asymptotically stable stable equilibrium origin eq origin asymptotically equilibrium hand normal output system coordinate theorem neighborhood transformation value case sort sort remove important balance systematic expansion statistical white gaussian balance transformation differential topology combine drive analytic approach balance xt linearization origin origin treat estimating operator reproduce rkh primary contain original pca extend generalize development lift reproduce brief overview reproduce theory heavily let hilbert functions rkhs kx kx call property reproducing reproduce unique kx kx kernel kx definite reproduce satisfy every sequence strongly pointwise convergence positive kernel hilbert feature feature fx kx use iv feature rkhs positive covariance matrix essential go existence reproduce rkh important define neither computable eq play minimize empirical eq solve eq hence minimize possibly infinite uniformly sample input rotation however signal xt see column observation response describe fix measure response q coordinate separate convention lead clarity understanding briefly review relevant background pca generalize pca dimensional work product center accord subspace show zero leading sort magnitude q similarly sort eigenvector feature principal space kx kx essence collect perform simultaneous component estimate rkh space worth transformation favorable practice center consider additional note view scale similarly collection q equation kernel simplicity choose terminology kernel root matrix system behave linearly assertion prove immediately proceed center order consider distinct convention center separately let similarly build feature representation c mn feature quantity c center immediately remainder drop quantity balance dimensionality state truncation calculation original behave order reduce discard project balanced reduction balance accomplish empirical feature map center space let mx space counterpart similarly since kernel balance rkhs last transformation onto eq map recall form top root system linearity yx yx c zero eigenvalue nonlinear close original accuracy nonlinear construct correspond reduce space refer appropriate inverse difficulty approximation get difficulty approximated element rkhs state assume series last might direct account reproduce case ability capture nn taylor approximation avoid state ix u th characterize typical behavior even trajectory seek ik z nk true regularize square notational convenience far ix j broad characterize separable estimate component solution order taylor approximate update dynamic desire expansion length eq ones calculation certain choice exploit fact derivative perhaps polynomial polynomial degree dx particular polynomial recall dx yx invertible would potentially empty reasonable convex pre image formal noting trajectory reduce counterpart virtue way formulation reduce recognize representation final theorem derivative kernel approximation need close dynamical jacobian notation jacobian see give close nonlinear control solely reduce p reduce exist expect input leave approximation appear future dynamic possibility approximate familiar ix ik hx rkhs different coordinate note map compare define define close simple linearization preserve first paper empirical datum rkh system brief reduce linearization system linearization observable reduce approach linearization follow control let example x rearrange write u truncate affect output response space time degree kernel retain retain sake eigenvector system follow control choose hz square wave peak sample map problem kernel output variable bias account offset use fast remark rd rkhs fact reduce instead simple square wave peak cycle zero demonstrate square wave input summarie taylor
algorithmic implication problem constant might solution entry minimizer one support minimizer follow minimizer local minimizer contradiction concern minimizer theorem minimizer trivial local minimizer illustrate theorem minimization datum part minimizer minimizer low minimizer easy vector satisfy eq order minimizer vector satisfy claim give desirable regularization problem tractable efficient unfortunately strong however minimizer basic solution exactly stationary difference follow hardness facilitate np np hard hard prove useful technical minimize optimal unique optimize maximum first partition optimal must partition present polynomial reduction strongly partition subset sum consider minimization remain strongly np smoothed problem form suffice hardness change eq ij n equality come hold z q one know argue reveal hard original complexity global minimizer hope problem enough desire nonzero theorem choose bridge estimator sample tend nonzero bridge design matrix typically standardize small covariate constant find sparsity minimizer estimator become meet efficiency bridge model eq q hence minimizer likely could choice consistency estimator unconstraine theorem example wang unconstraine minimization minimizer extensively least square minimizer concavity minimization np carefully certain desire sparsity regularize keyword nonsmooth nonconvex variable reconstruction bridge minimization least dimensional datum original entry subset regularization regularize deal derive continuity boundedness nice property oracle theoretical distinguish entry approximation bridge advantage minimization nonconvex analyze approach develop globally convergent guarantee computational open problem attempt draw hardness hard np polynomially bound admit time np objective admit polynomial unless penalty precisely find hard smoothed hardness side sufficient desire minimizer global local estimator encourage global organize follow meet requirement minimizer minimization hard smoothed version even though lipschitz gain advantage term finally literature choose regularization
intervention control passive joint signal processing community concept probabilistic causality several causality causal kernel representation causal relationship distribution w acyclic dag correspond causal system naturally think diagram overview paper motivate system framework motivate intervention sequential recursive whereby natural induce factorization accord correspondence kernel correspondence direct generalization quantify strength compare distribution theoretic interpretation back conditional analog statistical diagram communication without figure message map deterministic intuitively clear message cause message joint eq q factor statistically dependent indeed inherent channel decoder message causal message opposite break symmetry end assume suitable sequential happen make assignment look input assume mapping message assignment hard assignment one replace decode map map effect assignment q clearly random action conclusion decode absence causal dynamical multiple feedback loop influence system couple equation eq specification sound sense equation dynamical system loop use seminal system generic dynamical feedback loop arbitrary dependency causality limit system order system causality markovian sort factorization direct vertex direct dag graphical heavily sequel associate dag let direct eq go impossible markovian simple study causal effect variable examine assignment main start call intervention let distribution add value assign claim effect upon would intuitively intervention intervention original equation original intervention define intervention reasoning follow intervention consider intervention intervention eq intervention intervention intervention distribution induce follow write outside following suppose observe system evolve accord belief graphical markovian offer visual way correspond dag let couple consider joint intervention word upon diagram eq depict channel sequence feedback intervention represent turn construction distribution capacity tuple order course subsequent depend equivalent conversely way q step view mapping tuple define channel specification intervention system originally direct complete encoder decoder controller contrast causal various causality oppose markovian system let point sx coincide strength intervention divergence expectation w marginal induce obtain distribution recognize information turn context communication channel information causality conditional reliably ordinary must along direct path dag pointing toward definition arbitrary markovian markov factorization product dag marginal accord edge dag correspond numerator denominator exploit independence relation encode dag point status causality involve disjoint define conditioning eq sensible structure let direct quantify flow markovian dynamical contribution ordinary happen direct information give disjoint q brevity equal similarly eq fundamental discover influence observational datum reduce canonical causal structure chain direct relation structure direct cause moreover direct go since direction reverse active study causality concern causal effect markovian dynamical whenever causal feasible become index express disjoint dag distribution
however neighborhood portion propose allow understanding first generation markov monte draw correspond present sample procedure equivalent drawing try draw measure initialize graph g lr von fisher distribution edge size converge graph topology think employ assign yet approximated employ process uniform unobserved small graph spherical basic lk example generate lr g adopt study fit statistic simulate network fit simulate either degenerate secondly edge graph neighbor divide compute census proportion set lastly minimum median network node outline graph random network show capture statistic generate resemble observed geodesic indicate cccc indicate line effect high starting depict model generate sign misspecification cccc summary indicate summarize benchmark whether degeneracy phenomenon network summaries degeneracy severe mle specification correspond original generate cause degeneracy feature vector relative convex polytope issue spherical belong boundary interior outline social degeneracy sampling spherical g von fisher individual geometric mode spherical realistic author thank v c nsf award generating degeneracy locally graph contain relationship bioinformatic interaction researcher area develop network network long history generalize social nice suffer issue degeneracy instability place distribution empty result purpose degeneracy discover cause graph hull become mode second insight embed surface sphere convex vector belong hull could serve avoid degeneracy issue approach embed surface sphere distribution spherical possible thus degeneracy additional start present generate realistic exponential synthetic realistic conclude future similar self loop undirecte adjacency zeros diagonal label fairly random network feature explicitly define used triangle star subgraph pattern feature show edge triangle star model use vector partition much feature view mass refer weight q exponential family polytope statistic graph p interior hull possible essence preserve observe extremely fact approximate e pseudo likelihood instead often suffer degeneracy type degeneracy first mle exist mle type happen reliably significant graph degeneracy consider viewpoint undesirable justification place mass vector place little undirected graph feature vector graph display diameter topology cell count accomplish diameter give mapping edge table sized space graph pair count cell feature distribution hull subset black estimate different pmf triangle estimate indicate indicate mention degeneracy unless otherwise degeneracy mode focus achieve attribute give structural edge triangle graph gender however degeneracy know minimize degeneracy task knowledge accurately reasonable make geometrically geometrically dyadic share specification avoid dependent specification degeneracy entirely surprising mode investigate degeneracy suppose set boundary hull hull observe log observe include boundary maximize attention illustration node specify solid black number boundary correspond mode unique tt vector mode outside cone maximizing explain many place large graph degeneracy change geometrically weight geometrically distribution arise fact match concentrate mass grid uniformly spaced model evident exhibit degeneracy issue describe cc right plot indicate indicate provide ii degeneracy sensitive geometry geometry place little mass place e onto belong define section spherical sampling technique graph ng von hyper feature spherical new model generate approach propose locally embedding spherical short execute outline algorithm graph possible otherwise walk step graph away result embed mapping spherical approach dissimilarity distance euclidean outer beneficial preserve map locally small graph spherical rest neighborhood preserve estimate spherical position unknown radius sphere x u n kk von first member unlike distribution mode determine concentrate see detail wish single also treat control concentrated obtain
possibility index exploitation epoch player cause mistake exploitation epoch part lower decentralize armed bandit unknown multiple player markovian play evolve accord passive player arm suffer loss reward decentralize arm decentralize construct achieve arbitrary nontrivial logarithmic find financial engineering mab independent arm arm play choose select large reward select selection case grow certain lead constant regret growth good result accommodate multiple simultaneous reward evolve play classic mab refer ucb achieve time ucb extend liu formulate study decentralize classic I arm reward player choose information player player arm one receive desire decentralized mab player fair share logarithmic centralized player share eliminate centralized perfect scheduling bernoulli decentralize mab player policy player decentralize markovian unknown markovian play accord address ucb achieve reward know play well epoch epoch length balance optimal markovian reward model latter long know strict decentralized model former arm action player state evolve accord arbitrary play extend logarithmic stronger know couple single player liu adopt policy achieve weak system available use structure propose outside epoch learn strict mab specifically arm govern stochastically identical policy arbitrarily logarithmic positive l decentralize mab arm player choose activate offer amount reward arm arm time arm arm markovian irreducible arm arm occur play obtain reward reward arm permutation specify history th play notice random share obtain arm play mention sec performance evaluate arm regret follow random induce minimize growth growth rate slot epoch slot epoch arm decentralize r l exploration exploitation index index epoch exploration exploitation epoch epoch decentralize base divide disjoint epoch epoch illustration epoch player index arm play obtain exploration epoch exploitation epoch sufficiently accurate fig beginning select high index arm exploitation epoch player play epoch detail obtain exploration player epoch play epoch decentralize exploitation epoch epoch information phase observation exploitation epoch dynamic epoch e point epoch depend epoch htb exploration exploitation epoch begin exploitation exploration play exploitation epoch play epoch every spend exploration epoch go exploitation epoch index current sample accord time high index arm divide player exploitation epoch step divide epoch arm play agreement different subsection agreement eliminate logarithmic regret player join global synchronization epoch agreement arm consider player play arm elimination achieve join system schedule regret order markovian irreducible reversible markov chain state I satisfy condition epoch get decentralize epoch epoch agreement decentralize also appendix dl achieve arbitrarily formally arm finite irreducible reversible markov chain increase conclusion hold model appendix decentralize multi armed bandit player aim term
part recover cm cc section structure prior e document goal probabilistic representation semantic collection latent dirichlet document represent predefine number vocabulary usually vocabulary e representation corpus fundamentally argue lda multinomial cast problem formulation present call representation vocabulary e component fix constrain entry nonnegative decomposition interpret ensure describe structure regularization light decomposition document also share force conversely deep dictionary typically non extra inclusion acyclic dag select induce norm present section eq variable direct acyclic introduce norm associate application find estimation consider resp active predictor aspect add step efficiently collection able appropriate possible term potentially doubly blue dag review structure convex norm norm readily norm introduce dimensional unsupervised interpretability increase predictive european project grant nsf award anonymous whose comment greatly method aim parsimonious combinatorial admit relaxation norm structured deal structure dictionary concept scientific take form selection two interpretable admit look sparse predict variable tool benefit reference therein develop theory selection individually regardless exist relationship structure hierarchical merely improve form functional voxel discriminate state localize area face show plain norm fail encode background order come bioinformatics diagnosis profile genomic feed profile characterize profile specific genome discriminative feature contiguous improvement accuracy find present nucleotide snp gene target share gene hierarchy information design induce scheme capable encoding sparsity mention available pattern nonzero support induce induce norm structure way e convex optimization theoretical analysis easily traditional norm popular select overlap group different type introduce application namely section variable vector letter bold one j nj w row extensively norm review induce norm learn predict typically observation usually regularize suited indeed pair follow sparsity typically nonsmooth non reader description regression norm pursuit lasso take pursuit write correspond limiting span counterpart correspondence notation notation describe variable represent correspond notation express usually representation consider regularize solution depend ny belong normal cone sufficiently ensure solution ball favor correspond shape pattern double pyramid exhibit point align subspace tangent sparse ball norm section norm cm axis align origin norm induce norm display priori reflect ht cm sparsity arguably priori disjoint block ignore form positive zero square regularization interpretability model block share task learn group size depend desire discussion seem reality collection encode quite structured key present construction overlap complementary section overview encode relevant setting group show sparsity remain entire reflect extreme point unit pattern obvious overlap interesting tie belong group solution obtain form mean take intersection group moreover mild possible hoc solution belong norm adapt figure select diagnosis profile since genome constrain support grid set relatively half may pattern sparsity inducing build upon group performance background wavelet denoise fourth hierarchy variable assign forest select hierarchical exactly instance wavelet hierarchical model model order bioinformatic exploit network task scale mining fmri cognitive contour tree represent group set zero gray remove root subtree obey ii select possible set aforementione example complicate topology consider discretized discretized slice application direct encode developed overlap theoretically addition group interval weight globally structure model support scalar group generalization norm group solve norm equivalent interpret expand careful zero keep zero component lead select convex relaxation encourage knowledge take graph connect prior co gene favor neighbor select variable formulation point lie align circle correspond group hull ball ball leave without rely fa therein sparsity induce convex envelope convex ball structure far submodular thus lead different type level mainly convex define viewpoint context focus without theoretic non convex community statistic name fuse lasso w signal recover piecewise function sparsity penalty select exclusive inside mention grouping priori penalty term encourage thereby form review solve regularize technique adapt constitute take advantage simplicity present proximal least gradient logistic assume process numerous reference therein therein chen provide strong convexity assumption function take q stay close solve iterative constitute name proximal technique split iterative furthermore rate function scheme extend accelerate case proximal converge optimum rate rate order non overview core define rewrite regularizers proximal close make turn norm compute operator efficiently group operator close g variable computation proximal composition g overlap proximal expensive technique norm proximal implement open software generalize structured proximal done traditionally analyze load denote solution traditional consider model e x possible criterion consistency assumption generate loading especially characterize hardness refer oracle regime consistency regularization validation large try collection along usual model ol hybrid without validation phenomenon characterize compatible additional encode assumption sometimes relaxed interpretation global formulation orthogonality dictionary signal also impose terminology observation coefficient product coefficient correspond factorization formulate constrain pseudo norm say typically sparsity matrix assume convex versa convex
motivate em provide mean start list lead experiment cluster yield list method continue guess c c c c mse runtime sec ir cr na na km na na motivate follow mse th singular fall formulation utilize use next parameter start form individual think include set mn tt estimate list minimize mse separately fit least square regression index column corresponding response predictor z consist least find similar th let individual estimate span whole estimate individual km perform joint regression perform explicit various along ce discuss runtime dataset compress sense projection replace dimensional sketch consist split dataset relevance ce dataset calculate ce fraction vice versa method base respective matlab implementation intel core ghz machine number km choose parameter mse mse ce summarize km requirement cr perform individual regression run km method km report mse cr improvement ir far see great km runtime performance ce associate algorithm appear arbitrary never know eigenvalue unity suggest cluster propose show substantial ir method cr query take cluster sensitivity match remark shall snr regime figure cluster size method linearly super growth near mse ce runtime km sensitive compare estimation compare method runtime km scale bad complexity section order ir th operation multiply transpose operation multiply matrix cr operation complete em e iteration operation vector compute operation km need well explain cluster experiment group close regression km multiply thresholding operation thus operation individual regression compute need similarity operation individual operation ir per see cubic growth consistent seen figure growth km growth unlike real km algorithm much compare em ir cr growth shown mse load also understand introduce end vector trial ir cr noise noise level low find cr less ir hard neighbor pick cr phenomenon clear close level total filter pooling low balance well level since em iteration converge optimum trial something close similar optimal snr use ml regression point following denote fisher last choose I normality estimator tell normal mse empirical even db predictor introduce various experiment yahoo challenge near optimal performance relatively insensitive associated mathematical understand prediction surface hope present deep future manuscript general non appear th experiment notation begin kk p represent eq find I maximize em pick pr mn z ny mn mn thus denote r eq maximize derivative complete em column mse constraint relevance whose th block g dm obtain stack one problem usa experiment cluster experiment relationship experiment cluster relationship predictor exhibit regression focus several applicability yahoo associate study diverse adaptation maximization inspire mean value thresholding adaptation collaborative filter dataset simulate datum good near good reasonable method light choice regression relationship response variable predictor rich employ field often collect reason share relationship example yahoo challenge query multiple relevance score relevance score many query cluster consider problem cluster already unknown study arise outline main level find mse factor collective give notation matrix bold vector transpose
datum histogram point heat leave white fall common string figure look find equal rd digit digit us plot figure leave take value share digit digit thus highly correlate share digit share digit digit digit plot digit figure share plot share value share part match third digit show find level precision word example value least percentage share five retrieve fall build figure depict frequency gaussian surface three dimensional top panel first concentrated precision uniformly gradually go precision peak digit distribution peak discriminate peak top digit peak traditional ultrametric ultrametric useful case digit relevance share digits ultrametric common digits distance show store advantageous digit instance distance basis chemical comparable due semantic tw email house safe tw ex uk induce ultrametric quadratic agglomerative apply distance digital survey costly neighbor keyword ultrametric agglomerative axiom triplet point dx dx dy dy consider ultrametric triangular ultrametric dx max dx dy addition positivity triple metric map onto ultrametric dissimilarity often metric ultrametric agglomerative hierarchical pairwise merge place cardinality closeness distance closeness singleton singleton inter cluster dissimilarity define wish step agglomerative hierarchical pairwise dissimilarity show reciprocal near neighbor quadratic time present carry linear theory bad node term dendrogram set embed totally strong subset hierarchy ultrametric show embed constructive vector involve series pairwise propertie iii h I ultrametric euclidean use minimize change dissimilarity agglomerative constructive construct ultrametric search ultrametric inherent agglomerative furthermore help greatly find inherently ultrametric view coding seek string could determine compression partially express infinite common close distance value digit exponent notation range digit integer order order place take example position distance ultrametric dealing number hierarchy dendrogram dendrogram general non integer ultrametric present rank geometry shift focus impose encoding summarize aim rather agglomerative quadratic individual seek read number observation p read scan dataset interested encoding define digit bin digit bin digit level neither deep rise suffice bin require operation value constant read cardinality digital produce position million galaxy al
candidate gene time disease respectively promise ability global disease curve begin ht top top disease select set unlabele complete pathway table disease reasonable disease cancer diabetes breast successively disease size p gene list top list precision unknown list training namely see cancer limit large list test c c disease cancer diabetes gene disease know disease across still retrieve disease validate gene list database report ten cancer cancer diabetes list cancer tp diabetes col c col disease order cancer diabetes cancer breast cancer publication find relevant disease gene extract pathway tool discussion introduce gene unify able information rank causal disease disease gene share real database outperform art method disease dirac prevent disease generic disease relevance enhance dirac influence final believe much room improvement phenotype could integrate induce question recall top choose ready addition think disease known situation gene already disease intuitively need information disease dependency global bring local number check light illustrate plot standard disease least known surprising difference disease criterion suggest practice interested rank list top soon go gene novel learn score fitting augment disease identification unlabele bioinformatic good pathway cancer patient high protein protein protein us disease gene aim gene typically human genome disease include expression annotation assume source product gene think accord representation kernel gene collection phenotype know phenotype gene disease complement far disease available short candidate gene disease disease retrieve gene disease gene disease near list single disease source list must candidate gene exist quantify disease learn motivation behind scoring initially gene represent retrieve scoring usually try gene could dash letter green negative area represent solid consequently positive density practice assign binary discriminate label assign point logistic training implement positive unlabeled proxy building set binary lead biased bag equal practice bias weight fact represent confidence negative namely add bag contaminated speed scalability take specify successive aggregating positive element decrease detail observe dramatically affect default iteration multiple datum source gene expression formally encode result source first kernel gene similarity widely often heterogeneous integrated score method building non optimize weighting differently mkl formulation discard give gene straightforward mkl instead svm train disease disease treat disease know characteristic similar disease often gene suggest instead treat disease beneficial disease across disease disease attempt disease causal property disease obviously jointly disease problem adapt gene candidate disease scoring order instead positive gene disease disease kernel disease share precisely consider variant various dirac two disease disease treat disease strategy treat disease disease define allow basic sharing disease causal dirac disease extent disease gene disease share low information disease exploit knowledge capture disease share disease many work prior phenotype cause gene across principled disease practice disease plug disease mining measure measure gene kernel slight variant phenotype add dirac eq motivation disease phenotype incomplete characterize disease disease similar addition allow distinguish disease give importance associate phenotype result four gene score variant summarize c l name kernel share disease across disease share share share identity differ share disease third column disease variant apart disease variant follow gene easily translate gene mean kernel variant restrict simple average leave dataset association extract disease train score positive pair training list implicitly know pair remove candidate rank sample receiver operate curve formula therefore gene method cdf association rank list function gene first mkl outperforms method mkl gene kernel mkl website compare design disease function query gene disease similar gene similarity disease label score smoothly description v multiple status leave gene expression est functional database activity text consist data product define create ensure phenotype measure measure automatic text mesh medical subject similarity disease mesh disease finally collect association disease gene association involve gene acknowledgment grateful yu make grant centre f france paris france paris france france basis goal molecular linkage analysis modern throughput long list hundred candidate time consume candidate propose disease implement strategy allow integrate various share disease search disease datum show outperform human decade considerable genetic rare disease diagnosis towards biological disease discover disease identify contain linkage population identify hundred gene gene consume sequence whose activity disease sample genome scale identify list candidate gene among agent important promising gene likely information biological pattern expression promise candidate availability genome set variety heterogeneous store various obtain candidate expert perform task via approach previous work attempt identify promise without gene annotation phenotype investigation successful disease gene share know phenotype phenotype perform use art integrate heterogeneous information candidate decrease use label protein protein interaction able know disease gene disease web disease association already new bring main first paradigm score like gene exploit know disease gene candidate disease gene machine paradigm positive disease play disease phenotype disease disease allow rely disease relate disease machine implement share disease disease heterogeneous include scientific literature datum integration differ approach limited scoring database disease disease gene gene share across disease assess ability retrieve disease disease share across disease extract association nine source expression profile functional annotation interaction activity encode pair accord way source mkl compare differ unlabeled mkl comparable pair sign integration though begin picture behave come refer train panel global curve panel begin visualize scatter directly rank small rank rank list many mean dirac phenotype kernel phenotype concentrated side indicate favor kernel generic although plot different rank share disease beneficial across disease beneficial restrict associated share disease one training procedure leave disease association show cdf share information retrieval behavior depend look globally mkl rank systematic difference pair sign summarize picture indicate significantly confirm variant mkl cdf likely serious investigation cdf curve mkl increase curve mean yield truly additionally proportion identify top list show
infinite neither convex function set requirement banach sign bound continuous value function vector call pre banach everywhere satisfied pre let q condition unique another consequence hence banach functional possess regularizer q point minimizer obtain borel subset exist index countable norm put transpose former result minimal interpolation minimizer form hold ready satisfie hold lemma x bf recall x inequality lemma interpolation minimizer q exist constant bridge reproducing introduce bilinear say regularize square literature compact measure definite reproduce open ball radius cover q consist eigenfunction eigenvalue cx minimizer problem hypothesis regularization choose enable discard immediately ef z ef proof regularization error get since turn decompose z bound law surely suppose lemma reach rate cx eq discuss lemma maximum achieve substitute higher impose symmetry might strategy thm thm thm thm zhang regularize bind error reproduce linear bring discard automatically reproducing banach reproduce banach space least square reproduce banach linear theorem recently purpose rate regularize square sequence row reproduce network extensively rate respectively programming attract much attention mainly sense able yield sparse result reproduce estimate regularize least explain machine learn fundamental instance measure minimize eq conditional unknown competitive precise intermediate banach nonnegative decompose quantity hypothesis carefully reproduce space definite regularizer regularization need
algebraic bad dimension believe obtain number k artificial present extraction employ artificial method approximate randomize oppose expensive fact comparable space suffice create via run dataset formally would obtain dimensional run state notation obtain running mean low space eqn projection drop without frobenius equation eqn provide also k optimality svd first eqn fact appear argument proof eqn rescale accordingly preliminary extraction present work prominent feature mac core processor ram empirical finding exhaustive extraction satisfactory rather far believe constant analysis cluster select construct execute feature compare feature method summarize follow modification matrix calculate naive multiplication first contain singular vector fix particular matlab execute evaluate allow use practice employ experiment namely every mention mean initialization repetition ht whereas right ht whereas right column generate center uniformly dimensional center point center center include center provide optimal use handwritten uci repository point normalize take object image image grey pixel ten take homogeneous position tolerance object database person pose illumination dataset pose expression dimension identical datum cardinality correspond normalize mis base label example correct finally report important report run dimensionality propose five ht dataset whereas present relative dataset mean demonstrate approach respect mean effective apply separate method world dimension result decrease increase dimensionality compare laplacian dataset laplacian superior notice naive poorly notice necessarily increase take evaluation run nearly separation gaussian thorough evaluation would far finding quite encouraging topic consist approach explain dimensionality approach dimensionality reduction wants explain connection text modern reduction mean test practice encourage fast dimensional provably novel extraction interesting research design provably efficient relative mean follow proof rr k drop without change spectral ii side singular value inequality assumption proof multiplication involve sampling subsampling row notice observe inequality norm bind apply together bind failure rest assume algebraic variable matrix rescale four wise independence construction chebyshev side conclude definition transform row eq failure notation rescale matrix set appropriate applying index j multiplication claim say row contain entry random independence bind drop limited repeat rescale consider respectively crucial bounding failure last k apply k happen follow event let event k obtain bind add spectral sub individually bound lemma k last eqn k markov theorem axiom theorem false green yahoo new york ny computational department two select apply construct feature construct despite cluster heuristic address provably provably cluster random decomposition towards understand cluster provably extraction method upon feature extraction fast svd factor objective cluster engineering web perhaps know attempt address euclidean point positive integer distance point center minimize cluster application modern massive considerable challenge cluster dimensional datum computationally inefficient existence feature allow practitioner address selection subset construct rigorous selection mean mathematical study point belong collection jj ks jk approximate cluster feature construct actual dimensionality extraction construct artificial original formally optimum partition reduction construct compute project plug optimal notice mean study clustering technique helpful observe favorable rp svd rp svd despite address provably accurate feature extraction method know describe projection imply row immediate mainly due prove pairwise distance multiplicative factor pairwise correspond within reduce artificial clustering briefly section corollary suffice provably accurate selection algorithm present k achieve theorem compute present feature working approximate approximation approach compute synthesis paradigm paradigm extraction probability result show small obtain approximate outline rely paradigm paradigm describe extraction employ decomposition construct dimensionality almost show svd switch algebraic mean notation eq represent belong non zero belong cluster slightly indicator identity hold identity I jj pick scale cluster formulation minimize quality evaluate cluster framework new reduction along give denote number indicator denote formalize approximation correspond run cluster run run dimension considerably fast algorithm though although admit well employ algorithm section dimensionality compare similar run arise one dimension key subset column algorithms I new combination extraction algebraic perspective main analysis approximate rank reduction cluster procedure orthogonal short proof completeness completeness value rescale procedure close orthonormal r prove leverage rd conclude rescale column randomize close intuitively subsample affect frobenius define tt j r linearity definition worth note set equation rescale rank orthonormal sampling correspond compute rank two another rank factorization theorem proof switch bound inequality almost state point project multiplicative precisely show orthonormal research transformation gaussian random recently fast transform construction prove rescale computed utilize summarize interest defer proof matrix norm square scale comparable analog fix effect sign singular rescale construct simultaneously w analog analog lemma rescale matrix random worth note random require dimension close ok r c c construct approximate right singular technique section one replace algorithm exact present corresponding proof replicate replace working considerably algorithm kk r ok suffice roughly point theorem formally introduce obtain partition fact run k first drop equation short svd q eqn eqn failure triangle inequality drop eqn drop
forecast pi successfully several forecasting kind input yield concern combination sequentially sequentially remove motivated flexibility deep exploration initialize previous concern winner test winner combine alternative average proportional error present eight forecasting show respective forecasting variant forecasting loo variant forecasting autocorrelation criterion forecast avg strategy value weight testing assessment compare strategy forecast measure general comparative two stage compare post perform high significance rank average tie hypothesis nan degree show improve statistic nan accord degree hypothesis difference least hoc among eq asymptotically normally compare comparison possibly pairwise incorrectly reject procedure adjust value well adopt correction strategy statistically sort post competition selection procedure obtain phase equal quite good competition forecast series illustrate forecast htp deduce pre mostly competition phase overall weight comparable improve big perhaps version strategy consistently consistent reason put burden model forecast pattern input beneficial mixed whether perform construct concern combination winner winner finding forecast combination persistent finding base competition true competition phase true finding concern selection persistence validation select good strategy good pre datum intelligence competition competition hard forecast horizon uncertainty comparative review ahead apply forecasting competition comparison promising finding output impact persistent selecting testing method could make rgb rgb ahead forecast deal literature extensive missing aim fill gap strategy ahead forecast theoretical term attain strategy large experimental forecasting forecast combination ahead forecasting follow finding consistently support multiple strategy perform forecast long forecasting forecasting learning forecasting playing role field science engineering economic finance step forecast ahead accumulation increase forecasting influence however increasingly adapt many useful model bilinear autoregressive autoregressive bootstrapping assumption distribution nonlinear forecasting development last decade machine attention serious box drive historical stochastic find artificial neural conduct conclude appear neighbor create area mining forecasting forecast attention forecasting forecast critical topic however ahead forecast point good knowledge ahead forecast ahead forecast fed input contrast strategy return value combination strategy behind direct previous introduce preserve value characterize time series multi multi scalar single strategy call preserve direct strategy flexibility present sometimes thorough unified review well comparative ahead fact collective outcome regard forecast little use favor strategy fact strategy strategy show strategy give direct recursive strategy theoretically author theoretical experimental evidence favor concern strategy et al provide bad forecasting recursive forecasting strategy dependency forecast previous forecast finding find truth relative paper experimental different ahead forecasting competition pose usually multi forecasting existence series dynamic experimental comparison configuration regard comparison recommendation make forecasting conduct rather show forecasting influence forecast adopted successfully forecast task also forecast competition goal assess compare intelligence competition potential present strategy describe forecasting methodology apply discusse finally give summary conclude ahead also series task next historical observation forecasting horizon give presentation comparative strategy common notation dependency refer dimension series predict intuitive forecasting perform step forecast apply subsequently use one ahead continue forecast give depend time forecasting suffer ahead especially true forecasting observation reason potential recursive strategy accumulation horizon intermediate forecast propagate forward forecast subsequent limitation recursive strategy forecast many serie like neural consist horizon series learn model compute independently induce independence affect forecast complex dependency forecast break forecast large model use multi ahead forecast near neighbor combine principle strategy like input forecast like learn forecast outperform direct series set research need far single see introduction strategy existence dependency affect learn model return model preserve characterize time series conditional independence accumulation successfully real ahead forecasting preserve structure constraint flexibility forecasting appeal direct strategy forecast horizon block fashion thus ahead decompose size task direct value number equal intermediate value allow dimensionality dependency maximal provide preserve large dependency predictor strategy strategy clarity series forecast learn successfully forecast nn five ahead forecast task strategy forecast relationship forecasting strategy link combination direct recursive combination present multi ahead forecasting ahead forecast task series horizon forecast n forecasting obtain forecast give task number computational recursive allow strategy computation hand need equal follow conclude five forecasting cm computational series accumulation accumulation trade recursive reduce flexibility trade forecast forecasting forecasting estimate scalar value represent dependency paper multi ahead strategy forecast require setup instance local since series forecasting present learn section type forecast future value past series value parametric describe value analytical whole nonlinear regression assumption divide combine divide two different module cover space base inference radial basis tree modular intermediate scale identification basis global excellent complete rather predict also call point local locally weight combine proportional distance predict nearby weight distance well know intelligence give rank competition rank use work algorithm learn query contrary local parameter extensively model technique defer forecast number neighbor next local discard repeat query reduce consideration motivate multi step ahead context definition namely ref adequate configuration paper number selection essentially control bias strategy length second future consecutive section series pair output different select associate leave loo loo disadvantage repeat effort fortunately powerful sum calculate set aside loo training sum perform index replace already calculate method loo error neighbor loo error prediction return query sort increasingly neighbor look multiple response quantity criterion compare model different multiple output strategy sort respect index th closest j e look loo loo lk extension loo cross capability model loo aggregation error note output happen single e direct horizon suppose large time forecast quality forecast measure autocorrelation consider autocorrelation query symbol concatenation represent pearson discrepancy two part use note partial part discrepancy autocorrelation concatenation training series sequence autocorrelation evaluate minimize discrepancy training series number preserve property sequence return q sort increasingly vector k equation consider prediction testing consideration prediction query exist mainly
subject control failure hypothesis tight calculated parametrize define ball provide later short length shortest embed relate replace lagrange multipli ml constraint controlling cost restrict hypothesis f cover calculate derive later cost handle accord regularize beta formulae norm decrease decrease generalization broadly fact derive constraint specifically intersection operational equal bb bb x b regularize incomplete beta influence operational able specify something operational guarantee able operational result use volume spherical space cover uniform improvement lead arbitrary totally result cover number cover volume theorem cover convergence lf e relate cover number covering follow l first lipschitz cover fx l supremum empirical involve number cover metric follow norm relate loss view function function mn substituting use relation spherical b complete several decision operational framework power grid abstract necessarily explore application hypothesis operational quadratic constraint arise current type lead intersection currently shoot decision decision epoch mdp improvement present call framework advantage quantifying size generalization ml operational potentially school management technology usa operational exploratory theory knowledge implication reliability cost failure estimate also present framework essential influence explore range reasonable operational smoothed different distance belief cost minimize operational combine step sequential process operational term operational simultaneous process thus simultaneous simultaneous process organization simultaneous require subproblem solve solve vary full simultaneous know vary range cost operational belief operating cost unlabele instance sense close connection reality regularizer company give answer policy rely accurate abstract deal calculate deal us department grid document engineering inefficient future need without operational substantial capital decade utility implement inspection whereas new york city electrical go half time separate include inspection electrical service program extensive replace possibility serious company scheduling inspection project characteristic history past failure serious serious event repeat failure failure rare probability fail framework make fix assume terminology indicate time encode electrical previous whether within specific feature unlabele substantially carry physical permutation failure estimate form b logistic lf e logistic send change long operational cost repeat failure visit come make first make failure process discretize approximated bernoulli cost application application fail convenience begin return differently mean first visit visit definition equivalently start visit start start proportional failure visit visit occur within unit failure determine random thus distribution I failure visit visit trading visit failure l give become visit relax assume vanish failure node would expect failure visit node return term indexing cause failure visit cost reflect failure govern occur probability ip proportional similarly influence visit early incur reach schedule visit later node high chance visit total failure present failure probability cost derive cost quantity visit visit failure failure summation cost mean exhaustive define operational minimum problem cost proportional node operational proceed operational property operational true operational unless implicitly major objective program md md k different goal minimize traversal case need visit goal visit np total view item customer stop customer remove goal customer start program later flow flow drop yet visit variable flow edge integer restrict self loop form ensure one going come start constraint quantifie relate define estimate change sum failure form integer choice model version intuitive minimum particular individual serve purpose depend operational necessarily map nonetheless subproblem completely subproblem alone discuss solver simultaneous generic solver solution bind relate exact np hard cycle hence unweighted mixed integer formulation c show property motivate large accuracy triangle point end figure color black also plot color simultaneous failure cost model label possible lot choose enough training process cost unit illustration predictive operational cost exist feature triangle unlabele right indicate number edge indicate highlight company set give problem show simultaneous second experiment find predictive operational third issue experiment predict failure course year failure arbitrary interval failure hour fine scale develop secondary electrical design predict represent encode type electrical past source encode source prediction period test operational equip relatively distance obtain google driving note distance coordinate drive especially city limited inspection probability spend outlier simultaneous generally prevent many logistic misclassification size prediction roc auc increase tend cost fix range know could see substantial reduction varied away sequential process auc solver dramatically failure cost probability increase unit eight operational yield failure probability estimate due uncertainty cost nm marginally horizontal auc obtain simultaneous increase increase horizontal represent regularization failure horizontal horizontal regression different first provide I probability sequential failure depict I view I solution sequential solve electrical grid simultaneous substantially na I little understand truly believe belief combine justify varied simultaneous expect small operational cost simultaneous showing operational helpful regularization term much effect sequential similarly observe experiment seven problem hold decision pick whose size solve simultaneous involve sequential pick coefficient subproblem simultaneous achieve value good encode right per per type figure auc hold varied different training decision infer test small size simultaneous knowledge operational cost simultaneous sequential perform nonparametric versus auc median auc
present boost objective converge even issue demonstrate experimentally model standard good result widely detailed select direction error previously study kind primarily overcome projection final rest operation within hilbert discuss quantify term exist discuss descent recently natural match representation vector convenient present hilbert boost restricted descent input class lebesgue measure fx hilbert natural quantity expect perform need compute gradient functional say space iff functional functional wise pointwise differentiable respect outline boost replace descent necessary optimize space computationally generalize restriction directly primarily first descent second quantify explanation direction minimize generalization al special multiplication directly projection operation way direction restrict projection operation schedule subgradient onto hypothesis nearest f restricted quantify relative restriction refer boost weak convergence generalized notion equivalent norm search direction later restriction project gradient either h projection though hypothesis interpretation specific function traditionally weak classification project equivalent classification problem norm gradient output notion quantifies performance advantage hypothesis equivalent notion refer improvement result multiclass weak learner modify statement edge satisfie extend version analyze descent descent direction boost meet approach rf analyze extend space smooth functional result number work rely smoothness learner performance tailor pac weak learner learner train two unconstraine optimization theorem strongly restriction iteration size smoothness requirement upper guarantee progress select gain require iteration adaboost tight reasonably convergence result applicable wide derive benefit broad weak learner perform problem equally albeit efficiently require much approach quickly case objective objective two always give final space find f second optimizing take keep residual leave force past possibility functional residual functional restriction ff restrict like extra come serve mechanism error projection analysis norm residual derive observe increase decrease projection requirement residual present complete repeat weak algorithm weak learner evaluate preliminary task classification problem plan policy learn feature policy smooth loss minimize cost demonstrate adapt boost naive algorithm know three planning domain weak learner tb second experimental microsoft learn specifically web use learner learner final algorithms uci repository letter dataset experiment multiclass hinge multiclass learner interest naive fail letter rate strong case repeatedly learner optimization acknowledgement helpful feedback conduct laboratory technology adaboost weak classifier statement equivalent non negative achieve error inner formulation examine break incorrect weight side inner finally adaboost converse give edge requirement proof show imply increase relation adaboost style notion boost framework strong training multiclass multiclass classifier output baseline requirement extension classify simply weak output label convert weak learner output otherwise adaboost style cost reward requirement fine modification edge performance multiclass framework multiclass learner sum sum max multiply show existence implication broad strong goal formulation norm use n n sum adaboost requirement inner strongly definition strong examine objective subtract apply f rearrange conclude norm strongly functional restriction tt potential descent sum q q give final repeat convex restriction g ff f start convexity bind final theorem potential augment step multiply get restriction let ff f similar theorem boost powerful learner boost framework analyze descent boost introduce performance generalize
maximal matlab duality glasso ghz intel processor create p entry entry kp table set component take large give precision computation block machine improvement report operate block loop across sec speedup graph glasso glasso glasso glasso glasso glasso screening reflect list component thresholded give fail table improvement propose strategy glasso glasso write demonstrate help improve baseline screening glasso large parameter problem observe glasso probably associate strategy make quite attractive time thresholded connect microarray around obtain problem range micro observe vary thresholded split non connect size machine budget graphical micro heavy regularization solution analyze array process filter expression stanford patient body gene expression gene third gene signature breast cancer expression thresholded covariance arrive entry grid thresholded range value b color bar gradually decompose size become isolated differ great size appear appear cb ct cb cb encourage speed screen glasso appear glasso run independently choose emphasize screen c speedup partition glasso glasso range value leave column maximal average decrease thresholded negligible example size beyond scope glasso solution average glasso grid sort absolute diagonal thresholded glasso partition novel characterize solution lasso surprising pattern concentration thresholded seem insight solve large around thousand notational kkt order notational convenience already thresholde graph q represent block eq v kkt note thresholded satisfy kkt entry construction kkt condition block problem solve problem ij ij partition fine partition component proof direct establish estimate precision graph thresholded thresholded nest within connect component vertex thresholded graph inside component thresholded sample covariance conclude component definition department stanford stanford covariance regularization form decompose component component thresholded induce connect solve use graphical lasso proposal infeasible scalability synthetic life microarray comprise realization positive ie inverse know achieve sample variable use parametric criterion penalize develop efficient active field convex machine appear special property ignore paper surprising equivalence induce thresholded focus aforementioned observation screening insight behavior simple screening arguably challenge convex use sparsity separation connect introduce notation terminology entry denote also theory undirected order tuple set undirected edge symmetric matrix convention adjacency zero maximal connect v g relation graph k partition note label throughout refer component admit say vertex connect graph along relate algorithmic numerical conclude pattern edge skeleton eq v namely edge suppose imply one perform covariance admit construction operation perform arguably message connect obtain path solution understand screen within edge concentration graph numerical non monotonicity edge remark detect screen operate isolate covariance author show isolated solve paper concentration admit connect isolated treat connect unit learn also discover screen glasso improvement exist glasso glasso screen immediate block update glasso exploit glasso operate fashion zero iff pointed screen notable node screening glasso glasso check go implementation go optimize regularize fairly challenging proposal depend b subproblem form handle compute thresholded graph size graphical simulation observe
portion project call would piece ad hoc manner help especially adaptation specification use technique near neighbor computation retrieval contrast retrieval expansion expand module maximize eq permit mapping satisfying goal compute ic ic quick instance discovery retrieve currently compute possible graph combine match hypothesis structural consistency still event become computation score approach retrieval implementation cognitive science cite influential analogy computing provide mapping domain previously information give meaningful structural pick analogy sum score possible discovery concept label probable relation domain tag relation operate fall make mapping retrieved use match expand possibility stage technique retrieve would domain modify update retain newly retrieve case base instance cite even apparent argue limit analogous resource scheduling fact category cluster international resolution structural similarity international incorporate international desirable social international field effort international project maintain international research agreement third party together management aim classify approach lack knowledge basis contain detailed information enable fall arguably limited category technique future large fitness structural make suitable modification graph account ability support base feature highly advantageous law reference important practical life diverse domain among law resolution international management form situation component meaningful effort ai development base generation mention improve concentrate ai perspective lastly framework domain fellowship grant de next thank constructive reasoning une school mathematic international conference reasoning method reasoning reasoning allow satisfy requirement expect utilize domain transform issue utilize reasoning employ flexibility circumstance contrast specialized incorporate range international reach agreement fail field law e resort include type explore solution experience draw similar handle instead within field intelligence study develop together recent al theoretical computational mention tackle variety approach stem incorporate matching component recall past different extent innovation retrieval stage score structural new knowledge mapping retrieve component stage reason present idea illustrate discussion end conclusion solve ai utilize past experience view type make generalization derive domain model decision tree lack generalization particular inherent base keep decision valuable explanation support structure base describe form cycle module detail cognitive onto newly encounter similarity play role aspect cognitive capacity memory collection inherent nevertheless almost restrict indexing matching develop implementation capable retrieval domain especially adapt past target research international b dash edge reasoning correspondence concept pair domain base exist domain link dash correspond successfully address reasoning module expansion concept relation concept analogy possibly semantic relation base target want embed division goal allocation reasoning ability maintain regardless semantic fundamental capable create view retrieval successful adapt current utilize middle retrieve goal description implementation together flow agent newly acquire base solution holds successfully fully describe denote respectively associate goal solution even subgraph concept relation embed list indicate concept relation correspond indexing purpose possibility
entry clarity satisfie consist apply forecaster matrix round achieve round adversary choose different round imply trace rademacher thm round apply require trivial explicit bad rademacher variable recall rademacher shift replace convert outcome set shift one respect outcome apply rademacher author support ec fp online di universit di microsoft research usa online receive recent attractive lack informally speak prediction choose learner goal attain excess predictor precise standard batch convert method effective batch distribution instance needs reveal fix already compute online analogue instance need probably technique extend use prediction static expert forecaster minimax outcome expensive easily randomization case binary convex lipschitz random randomize round unseen way depend result trading regret polynomial online interestingly focus link concept forecaster use method subroutine forecaster determine comparison class measure statistical setting explore recently idea imply extend design learn trace constrain matrix straightforward application learning mirror match know setting analyze inferior rate pose whether forecaster rate wide loss efficient online thesis emphasize forecaster minimizer prohibitive whenever erm efficiently claim practical nevertheless seem tool online often work technique appear technique employ derive online context formally prediction play forecaster adversary outcome forecaster reveal forecaster predict almost expert irrespective outcome sequence horizon horizon advance simplify forecaster derive problem quantity one explicit tf ty erm scope outline optimal prediction work calculate round etc remarkably crucial rademacher probability vanish round explicit complicated recursive little class variable simply cumulative adversary forecaster predict receive suffer guarantee summarize static expert forecaster forecaster erm round variable ty randomized forecasting least statement statement statement expectation algorithmic easy infimum sequence oppose infimum different outcome minimax forecaster outcome limit applicability outcome trivial deal convex loss function outcome propose essentially forecaster subroutine carefully lie interval round randomized round forecaster describe version subgradient prediction forecaster present upper tp ti ty z z tp tp ty tr convex lipschitz condition drawing observation convexity use z tt martingale difference step relate z might whose influence method q hoeffde probability finally equal forecaster prediction round outcome thm rademacher straightforward rewrite simplify theorem empirical approximation reflect precision improve runtime trade note algorithm single forecaster constant know readily generic initially new actual equal enjoy regret moderate multiplicative factor union know game end horizon mild function forecaster suppose loss assume supremum easily find infimum contradiction suppose adversary adversary adversary aforementione compute round adversary regret round least hand end play prediction forecast prediction expert far permutation unchanged irrespective adversary consider sequence example round must provide reveal learner incur learner tp fx main class vc dimension exist empirical whose minimization imply learn hard boolean major pose rate learning batch thesis also non prediction vc dimension online forecaster achieve respect rademacher computationally risk minimization forecaster unlabele example reduce remark mapping function expert remark care forecaster thm use thm online dimension dimension vc turn present online collaborative regularization matrix well consider weighted stochastic observe pick free fail practice stationarity online distributional assumption require
md md mse top panel md sampler ratio md sampler md demonstrate effectiveness dag md sampler mse time md huge probability domain estimate domain representation bn moderately complicated local mode fast md investigate keep mode term comparable domain confirm indeed major burn period cell properly basis response flow construction towards various treatment disease use network pathway natural activation cause b network imply change reaction refer chemical physical modification exist dag protein flow flow throughput technique cell cell cell huge al protein network naive cd cell perturbation pathway intervention infer perturbation measurement discretize level low naive continuously extensive annotate among causal edge report flow md burn predict probability edge predict tp md predict annotate unnormalized dag md md algorithm table term mode md reflect high standard deviation md sampler jump show network true threshold notice mean threshold identical fact close size edge predict md sampler demonstrate md underlie compare advanced md md sd tp fp fp tp fp tp fp dr run result probability evaluate likelihood p g dr annotate predictive expected mean low domain literature superposition please give boltzmann superposition approximate function often physical respect sum domain employ md carlo sampling estimation mode superposition md sampler carlo deterministic method li carlo liu metropolis use find minimum improper space point mode promise md utilize recorded proposal md far enhance future construct sample md domain collection candidate direction current appendix appendix establish convergence md conduct doubly sampler adjust equip generate nonempty denote write measure j q notation target distribution accept scenario involve I e doubly adaptive doubly fix routine place norm ad working theory apply md remark update translation md establish main mathematical always mh jump either multivariate multinomial c ia jk c aa c imply liu routine decrease almost become deterministic drift projection give p irreducible condition imply verify component straightforward calculation verify global lyapunov u algebra I ensure interior c assumption immediate outline drift exist c assumption al essentially kronecker infinite desired convert proposition corollary routine distribution unconditional mean decompose space mode domain probability expectation informative unconditional expectation construct domain representation multimodal network throughput model landscape accuracy multimodal protein wang unknown observe posterior belong distribution chain monte carlo bayesian include metropolis hasting mh metropolis sampler review recent monte computation chen expectation approximate sample however unconditional expectation offer summary multimodal mean locate conventional multimodal major mode calculate statistic neighborhood achieve partition domain unimodal parsimonious minimize domain use rigorously later illustration partition trajectory domain various expectation domain summarie unconditional expectation dr practice sufficient efficient multimodal construct representation arbitrary multimodal sample utilize wang far generalize however accurate reason partition accord partitioning method complete nontrivial job detect mostly rely simple move coherent move dynamic partition sample domain devise utilize iteration enable fast transition feature multiple domain md md particularly problem field mainly concern structural causal network monte carlo zhang acyclic dag md construct landscape insight study protein cell highly pathway pathway constitute network critical disease recent advance allow measure collection rich statistical inference md build network discover pathway reveal distribution approach domain representation md ergodicity test example study human article conclude discussion relate future differential mild basic descent maximize index eq local start k set zero default article integrable write domain representation array dr provide expectation decompose contribution summary multimodal distribution every domain reason mode index define partition md k geometric posterior nf leibler regard may manifold potential index drive form domain representation necessary identify differentiable application gd local space structure naive quite carlo belong partition domain identify domain simple drawback efficiency design monte carlo version generate major mode inaccurate addition information overcome drawback md multimodal representation computational wish majority density neighboring often exponentially mh sampler across multiple allow region domain consideration partition md sampler mode include mode give density partition domain empty empty nonempty indicator dd immediate representation commonly strategy multimodal stay g substantial jump domain representation local move multimodal step size overall estimate respective domain design jump one one proposal md burn initialize mp lk regard visit decrease visit choice follow design employ sequence et al originally wang adaptive modify wang md initialize give c sampler visit since update default set normalize sum see covariance use mode mean accurately gd proposal proposal domain give work mixture multimodal efficient global well approximated identify mode proposal covariance single cause big domain covariance incorporate utilize guide summarize partition wang sampling gd complexity step move jump mode covariance domain move verification regularity furthermore iteration roughly converge acceptable iteration jump stay may divergence small md local mode update dynamically burn algorithm scheme construct representation real application evenly identify iteration mode routine new k tm tw tw routine record record mode replace old density adjust interval update routine help explore part kl kl tw j input note record update routine routine burn search density powerful finding mode md scale cover around low dimensional default propose mixed effect mode later representation liu however post burn carry convergence md document unnormalized please supplementary discussion section effectiveness md statistical estimation especially step example routine burn md algorithm update modify w iteration consequently work effectively q gd domain estimate highlight effect md sampler md basis evaluate mode positive combination five mode respectively mode permutation sign md independently iteration million burn numerical integration error mse estimate conditional mean layer expectation save cccc cccc mse md md md md md naive md show md especially domain sampler estimate partition contrary md domain improvement domain jump md md md default fold convergence md sampler without mixed jump slow reflect five fold iteration mix jump move md sampler joint construct code bn connect variable via joint area equation randomization model experimental intervention parent conversely fix intervention affect perturbation take state take state parent bn give parameterized infer type fix intervention infer causality intervention extensively context calculate mix observational intervention collection parent structure specify
give expectation forecast loss straightforward use law complicate rademacher iid rademacher still way iid introduction sample expectation expectation rademacher lead dependent quite rademacher differently online depth dependent follow introduce recursively start discuss q rademacher introduce z sequence path adversarial thorough treatment z z structure structure choose leave child order probability take selector become scenario past bound example aid three aid independence case simply note datum difference control hoeffding trying predict sure bind thus entirely result side independent imply essentially datum define everything iid hoeffding give decrease probability bad iid control predict future order handle rademacher complexity expectation need adversary unnecessary clean form datum rely application certainly case choose surely loss undesirable requirement rely infinite depend amount theorem department pa edu department mark edu generalization wherein outcome result iid concentration concentration behave version much machine prediction iid ergodicity may predict precisely control error sort behavior situation predictor obvious past observe ease remainder possible empirical risk result datum optimistic erm sample restrict model rather function class observe characterize include dependent derive give standard affect conclude idea future section stationarity flexibility throughout measurable measurable limit go field input notion stationarity variable stationarity measure complexity predictive model rademacher seem fit draw joint eq q independent everything expectation term inside supremum large empirical minimization fit outcome sample ergodic overall absolutely nothing predict past iid developing derive random behavior priori knowledge
requirement lack dimensional scenario still setting nlp coordinate motivate descent advantage give quite useful optimizer simplex pair density weak corollary strong take advantage euclidean geometry nonetheless term dominate rate high gaussian problem clear geometry compact smoothing remark corollary algorithm return fixed essentially corollary benefit satisfy condition corollary set achieve instantaneous dominate grow exploit assume oracle expectation output supremum similar hold set query bound corollary corollary dominant term issue query similarly strategy logarithmic application experimental illustrate result parallel computation imagine assume query imply consequently optimization error briefly simple master worker master maintain access uncorrelated smoothing master sample aggregation speedup stochastic exploit improvement convergence serial implementation centralize adaptation improvement second interest calculate proximal expensive update may beneficial several proximal concrete statistical semidefinite cone form include covariance completion therein problem metric belong desirable discover guarantee illustration involve cost reduce appropriate project simplex benefit randomize technique gradient cost stochastic sample update unit randomize stochastic experiment section describe experimental explore benefit study solve describe essential whether smoothing device ccc b number achieve use result b scale average algorithm use uniform dominant dominant invert small theory predict assess accuracy prediction robust identification denote experiment index dimension setting similar behavior decrease number allow roughly increase batch linearly decrease qualitative complexity c achieve measure learn sec mean induce pseudo consequently solve stochastic give choose pair uniformly give experimental optimality gap performance even require clear give predict improvement objective suggest indistinguishable achieve simple descent reasonable smoothing necessary point simple dual averaging answer though experiment suggest smoothing experiment iteration optimal query oracle averaging accelerate randomize average plot stepsize favorable indistinguishable complexity grow corollary proof corollary full lemma follow recognize smoothed fx close non apply rate approximately appropriately complete corollary relatively tight smoothing convolution lemma fy moreover lipschitz couple proof identical appendix lipschitz fx fx eq universal f vector sub lemma claim recall strongly remain assumption obtain specifically q proof auxiliary result show provide sigma assumption prove appendix constant ensure involve corollary smoothness add challenge essentially idea fx xt modification partially auxiliary sub collect ts gaussian consequently sub parameter recall jensen replacing complete linearize indicator adopt shorthand triple regularizer bind substitute simplify yield write tx z side combine inequality bind eq divide equality since consequence successively xx recall note f f x eq use z sub define step return establish later bind book yield claim involve invert bind prove zero recall see thus invert return intermediate claim replace x eq property smooth sufficiently x density already somewhat smoother differentiable finer f continuous notation proceed almost everywhere differentiable abuse inside integral expectation everywhere differentiable measure abuse probability end uniform lipschitz continuously tight iv tight factor satisfied provide smoothing respect b u lipschitz moreover continuous iii let f uniform ball b lipschitz lipschitz exist simultaneously lastly lipschitz respect use normal smoothing de consider function norm purpose carefully quantify norm continuously lipschitz continuous iv illustrate bound show distribution smooth f continuity begin lemma replace lead term factor also corollary build accord norm density symmetric attain shift huber loss turn inequality fx fx since use jensen estimate f du directly imply f symmetric calculation complete auxiliary lemma end let u volume eq see claim proposition theorem triangle follow dimensional trivially fix show minimum one index zero logarithm clear minimize derivative decrease increase increase decrease though distribute uniformly jensen fx jensen inequality early part consequence follow consider function early throughout give f verify f continuous apply lemma control order technique make inside invariant variable exploit combine early inequality constant consider f c symmetry imply dependent remain normal modify lemma convexity difficult constant z z q take arbitrary complete see follow equality symmetric inspection purpose book moment n know satisfy let exponential rely complete density completeness note side complete hand side hand integrate similarly multiply side grows follow gaussian result union chernoff van behavior hoeffding adapt lemma martingale establish sum realization quantity control martingale martingale value defines bound martingale since conditionally sub slightly define exponential zero variable equivalent characterization expansion variable whose square jensen recall take combine bind em j berkeley edu berkeley department electrical computer sciences department statistics california berkeley berkeley rates procedure gradient method smooth provide decentralized yield distribute procedure problem value close potentially throughout paper domain satisfy convex almost include regularizer procedure throughout mostly function mainly reason many evaluate usually throughout work access consequently focus stochastic address difficulty several researcher deterministic regularization method proximal method convex objective smoothing reference difficulty require impractical except algorithm solve subgradient et gradient stochastic stochastic relevance instead return unbiased exploit work rate oracle optimizer box update oracle subgradient optimizer issue iteration technique non smooth stochastic start note particular perturbation smooth lebesgue smoothed whenever measure analyze procedure smooth density main contribution issue theorem tail consequence iteration achieve domain constant implication statistical area discuss remainder organize achieve randomize several appendix outline smoothing approach main specify fx denote subdifferential shorthand notation ff fx bregman divergence value denote large modulus transpose draw motivate state convergence behavior descent though omit reader assume sequence mild guarantee instance strongly moreover generate update refer reader paper aspect subgradient obtain thereby result stochastically accelerate base improvement accelerate stochastically objective stochastically perturb iteration slowly decrease perturbation algorithm quantity rather oracle point query draw step stochastic compute throughout distribution procedure ensure scheme accelerate strong objective version series point query receive sampling evolve average scalar accelerate scheme vary smoothing scheme number iteration extra
site loo avoid know site tf site bind representative loo calculate base tf loo cv representative repeat loo sample repeat loo compare rank loo procedure obtain loo loo cv mark loo cv experiment mark expect rank second cv comparable first perform statistical rank evaluate collect loo cv compare include probability scoring centroid compare include nucleotide pair centroid non content loo framework rank consistent marked horizontal bar bad centroid fig tf perform tf centroid well significance observation test centroid outperform outperform outperform value cut fig relation median content method utilize low tf loo however similarity tf nucleotide incorporate bind rank tf rank tf among tf method except tf centroid combination measure optimal tf reasonable similarity tf achieve cv content relationship rank median datum centroid aside intercept median relationship scatter straight represent relationship median simple content view measure site tf bind site predict reveal perform fig centroid centroid content pair tf divide term three factor median ic length bind site comparison tf I tf median ic tf significantly great ic tf perform interpretation centroid perform centroid tf ic short bind site comparison centroid bar suggest tf centroid statistically column display centroid centroid report four improvement centroid always significant centroid identification centroid outperform centroid confirm hypothesis tf median ic employ site rr rr rr centroid centroid centroid centroid centroid centroid ic well whether use nucleotide nucleotide nucleotide tf support test value relationship denote centroid second fig content position display information content improve scatter site bind site respective scatter bind bind site respective centroid bind site bind separable non result fold improvement scatter plot bind site respective centroid identify scatter plot bind bind site centroid solid nucleotide abuse moment become nucleotide factor since therefore nucleotide pair score look first nucleotide pair transform variable pair find correspondence also implication vector obtain tf signature bind site centroid display method imply difference gave give average lie weight letter position similarly investigate example utilize centroid compare state art rely loo bind representative set loo single good search method measure tf method tf cv experiment nevertheless pair centroid tf tf low median identification centroid effective bind site couple conduct factor tf genome claim centroid site search database show matrix vice properly space bind aim extend method handle bind length seek sequence alignment consume seek similarity investigate support national foundation edu lee factor bind site identification decade utilize discovery bind site search understand role bind limited centroid propose benefit negative centroid scoring method perform method argue tool translation protein determine certain gene cell crucial tf considerable past decade bind double dna bind genome bind find gene locate bind site factor bind site identification discovery search may contain algorithm predict pattern discover discovery algorithm latter assume learn supervise know devoted evaluation search evaluation article therein search dna bind site tf bind tf bind site score bind site tf encode score letter count position observe position extend nucleotide pair information loo cross validation conduct totally bind improvement use representation mutual matrix scoring call background method bind site loo tf bind site superior example vast amount bind specificity site may site bind previous study role improve review negative benchmark positive benchmark rely recently wang fold vector require human artificial tf tf investigate novel extension centroid sequence employ incorporate centroid rely upon centroid performance assess cv experiment factor discuss method accurately bind bind advantage couple organize study novel leave cross establish conclude conduct datum first bind site tf set one contain bind tf genome release tf bind length crp ns crp introduce centroid similarity two sequence letter indicator set content position probability observe letter share account similarity indicator set content letter respectively apart embed dot vector convert gs illustrate l transform measure preserve sequence embed embed bold e bind tf centroid score l centroid bind site non tf centroid centroid bind centroid employ centroid bind site bind illustrate interpret bind measure similarity bind site length equals angle form illustration computation equivalent computation equivalent onto centroid centroid rise describe bind bind tf exceed constraint non stay flexibility threshold introduce distinguish non difference two equivalently minimize scoring background length pair distinguish take background nucleotide nucleotide observe separate sequence chain observe position transition background consider background markov show experiment section conduct loo cv section comparison step briefly loo cv tf region construct negative sequence bind comprise loo scoring negative test sequence forward reverse produce
evaluate however reproduce could forget model eq four exhibit dependency product model mass true carry large analyse importance dependency particle already naturally particle strongly concentrated want keep diversity system spread entire figure set explain sequential product logistic acceptance product rapidly half contrast logistic particle diversity product kernel decrease stage regression carlo algorithm acceptance rate odd rate step jump ahead however see marginal posterior easy reproduce distribution move decide fail ever logistic simple completely approach unfortunately incorporate monte alternative strategy fail provide suitable section parametric monte function interaction additive fact allow formulae ever author like much cite unfortunately impractical coefficient remove term eq exponential simple recursive factorization propose approximately compute approximation logistic fit logistic fit conditional binary family parameter option repeatedly literature first function unit moment adjust feasible replace drawback latent variable wise probability model carlo context potential random attempt binary vector construct precisely frank family sampling binary unfortunately multivariate non therefore yield sequential mention sequential monte carlo recent approach evolutionary difficulty modal thorough advanced need consideration high particularly modal build polynomial carlo present tend fail multi yield something nothing include logistic supplementary posterior distribution monotonic run smc particle sequential monte need evaluation problem markov distribution observe significant change update monte recommendation adapt generate package run ghz scientific source numerical evidence along publication source work provide project format result run window obtain marginal linear visualize variation plot monte varied run figure white box contain carlo box add bar smallest obtain otherwise indicate monte precisely run chain day monte extremely run data set algorithm major chain markov monte carlo standard chain adaptive chain outlier box start probability completely wrong outlier black chain confirm intuition make robust sampling constrain difficult key performance respective less running markov term computational chain set average complementary figure standard acceptance rate smc I I key complementary mc mcmc computational acceptance move concrete smc concrete strength set key indicator complementary mc mcmc min acceptance chain I concrete smc I concrete I concrete compressive strength restriction indicator complementary sequential adaptive min evaluation protein smc protein average complementary mc min min length move I smc mcmc effect complementary mc min acknowledgement n support thank two valuable acknowledge uci machine repository provide corollary definition example sequential sequential variable say past define present high binary practical motivation variable bayesian version raw version sequential monte easily vector satisfactory key provide carlo life instance sequential technique carlo algorithm adaptive sampling besides sequential carlo adaptive carlo aspect adaptive need parametric distribution property family performance quickly reasonable past space clearly practical analogue family binary space allow exhaustive enumeration whole order construction suitable discrete continuous counterpart major regression model encode linear explicitly constant approximate like expect value rich find monte approach variable view problem firstly grow global track initially spread robust chain carlo latter largely illustrate gain year multi central graphical computer regression motivate section briefly principal monte carlo commonly integrate ingredient algorithm family target construct rich core discuss completeness construct example variable problem fail reliable successfully write vector index sub model explain include drop linear choose conjugate prior bayesian q follow flat interval variance quickly eq normalization minimize criterion schwarz criterion basically coincide structure integration approach least absolute shrinkage optimization computation large compete essential bayesian model feasible alternative practical reason markov monte rapidly go carlo approach advance parallel combine elaborate move evolutionary ess thorough comparison scope metropolis hasting stationary distribution estimate expect markov chain discard give chain towards stationary stationary trajectory theorem apply chain work move metropolis hasting copy shall kind transition trajectory stationary modal ik iy parameter case chain iy refer acceptance likely interested chain explore mutation single drawing uniformly permutation kernel sampler proceed wise involve sequel gibb sequentially marginal correspond acceptance mutation copy state approximate predictor matrix ensure vector row trajectory update mutation gibbs largely computationally evaluation adaptive already produce adaptive proxy evaluate comparison propose current state component mutation probability modify move average gibbs component simultaneously suggest word value propose far mix property problem numerical benefit update sequential proposal dy markov chain kernel refer kernel current practically coincide obviously work choose close high acceptance average reliable shall sequential carlo general transition particle tailor work reader familiar sequential rather look correspond implementation ingredient sequential carlo smooth purely instrumental support smoothly theoretically initial pilot recommend simplicity reliability sequel associate convenient natural geometric alternatively sequence procedure sequence produce kk k nu weighting step reach instrumental idea control degeneracy move track particle degeneracy sample eq effective particle bridge size merely degeneracy length step effective value solve bi see procedure stable newton involve solution advance fix tuning parameter always speed sequential n draw nc n resample resample unweighte multiply size demand improve rao scope repeat weight resample rapidly particle reservoir reduce different particle totally inaccurate key decay move provide unweighted contain multiple copy particle idea monte carlo draw kernel measure particle change target almost move kernel particle within therefore locally review practically impossible hasting kernel proposal sufficiently acceptance particle convenient sequential come determine move system health particle distinct particle diversity quality analogue recommend keep system diversity uniform particle target reach beyond particle lot would identical change firstly aggregation multiply sample state aggregated size presence particle concentrate seem distribution find secondly keep sort way repeat move break particle copy move separately split aggregating might copy summarize complete method sequence index counter procedure n recommend store unnecessary latter systematically particle copy particle multivariate parametric need construct metropolis use want expensive binary moment want wise entry already generate rapidly evaluate metropolis hasting ratio metropolis satisfactory rate explain stem multi contingency table interaction theory completeness brief binary model reason sequential provide family meet requirement difficult important part paragraph mention unfortunately closed fit conditional suggest regression conditional valid regression
learn continuous space say acceptable minimizer important property pairwise element every acceptable lying span lie equivalent ensure minimal notice function nc xx nx direction q linear theorem suffice interpolation minimizer show k form particular cardinality finitely minimal interpolation f x g f ingredient preserve topology compact metric universal input admissible introduction requirement admissible construction requirement easy present requirement demand kernel arise brownian bridge bridge admissible space direct xx tx nk n requirement requirement function suppose hence clearly function side equal take side equation distributional sense infinitely continuously lebesgue measure nonzero sufficiently rate appropriately constant linear relaxed loss regularization satisfy characterization rise admissible x may x k allow equal vector singleton conversely satisfied shall discuss paragraph x n x completes regularize xt constant kernel exactly demand lebesgue bound regularize infinity specifically prove lebesgue constant therein invariant kernel show lebesgue constant commonly satisfying include radial mat ern compactly support kernel shall future numerical experiment sparse learn exponential rkh restrict number loss shall compare model eq numerous solve employ close minimizer equally q compare measure number c sparsity max three type gaussian random noise run mm section thm thm thm thm il usa mail edu part national grant dms banach space property possess norm banach integrable measure continuous linear bilinear example construction reproduce banach pursuit brownian bridge know sparsity extract dimensional datum live apply compressive sensing refer pursuit purpose establish research rkhs many reason account success rkh hilbert evaluation functional usually model therefore space hilbert rise functional rkhs kernel theorem rkhs determine many one evaluate theorem find point desirable force regularization rkhs unknown result banach go case aim technique functional represent kernel possess continuous point semi inner bilinear product natural substitute banach reproduce via semi space fr banach space guarantee inner represent functional shall approach point functional bilinear briefly paper let prescribe start hermitian construct admissible construction set banach respect note every sequence namely constant kx prove section admissible banach evaluation functional bilinear scheme nonnegative continuous q coefficient conversely organization construction banach reproduce kernel describe study brownian kernel final reproduce start banach necessarily definite norm functional explicitly construction uniformly fr differentiable banach fr impose existence reproduce kernel functional accommodate alternative point evaluation functional vanish everywhere banach evaluation plan check abstract completion consist bound functional yield let sequence functional denote cauchy give let pre need invoke space function every cauchy nx x make pre necessity norm validity consistency define sequence f follow f banach banach dense moreover immediately banach limit side kernel end bilinear well define reproduce bilinear whole kernel functional functional begin suppose functional k fx functional take evaluation functional necessity suppose let functional observation satisfy consistency discussion endow extend bilinear tell evaluation functional satisfy property hand side need direction arbitrary g q note sufficient reproduce functional form banach bilinear kernel banach reproduce norm bilinear thus reproduce banach proposition convergent consequence combine dual clear bilinear form necessary property mapping proper nontrivial necessity proper closed subspace kernel proper closed subspace enable find ff f present x property hold complete evaluation functional consistency complete reproduce via bilinear embed subspace nontrivial follow bound imply kx finitely many clear functional check find imply assumption satisfie necessity cauchy sequence hand suppose cauchy sequence nx n finitely definition suppose direct f nx complete satisfying later hold remark banach programming observe say remain check norm consistency shall consistency requirement calculate function supremum hold kx j cx xx xx g x satisfie cauchy nx go infinity eq imply n proposition conclude bound satisfie inner integrable lebesgue sufficient almost everywhere lebesgue point reformulate q lebesgue continuous transform discrete measure imply nontrivial regard wiener therefore fourier transform nonzero everywhere except lebesgue measure argument kernel standard norm kernel almost everywhere page b kernel spline give radial basis wu function compactly support satisfy corollary fourier compactly support indicate compactly lose nontrivial infinitely continuously vanish expand proposition construction dc proposition q yield procedure compactly borel measure satisfy radial function dirac delta assumption regularize construct requirement useful consideration give rise regularize scheme say unique possible acceptable regularize lie dimensional x kx convex fr reproduce kernel semi inner rkhs rkhs purpose interpolation prove follow minimal namely bf obtain
precision error small parameter tend formula consistently whether accurately say numerically absolute scale logarithm scaling stay decrease picture error compute although absolute tend consistently rate logarithmic application good tail tend rapidly uniformly good bound increase look support approximation formula median quantile zero increase varies rewrite fast approximation median introduce relative shape increase keyword median readily formula closed relationship exact median incomplete inverse review ha median value show scale median beta form satisfy symmetry relative median variate ratio unit look median median denote
ns ok issue seed occur submatrix seed seed without represent assume rank complete observation column assume detect candidate secondly still complete successfully draw seed seed redundant subspace completion column subspace candidate lemma determine assumption single subspace span span subspace otherwise neighborhood set column seed near contain certain span union small candidate true contain observation subspace describe sort small large subspace candidate contain sequential strategy far determine span proceed order list identify subspace subspace column span subspace ease span matrix determine subspace determine correct subspace subspace incomplete close bind matrix completion surprising require recover internet connectivity importance load poorly internet recent send passive internet internet resource measurement internet passive internet ip address observe link locate count incomplete fill infer characteristic analyze subspace structure rank internet exist behind probe send particular count address offset relate distance ip rank completion topology measurement count passive locate randomly subspace observe completion procedure experiment improvement miss completion approximately element completion h count result synthetic network cumulative show observe element internet delay ip address imputation significant performance matrix completion imputation ip address university university cs university complete miss entry multiple subspace rank completion union miss version lie union assume mild assumption perfectly incomplete usual incoherence subspace numerical experiment internet topology identification consider assume subspace interested situation total propose novel matrix completion mind arbitrarily focus quantify perfectly complete column complete translate extremely suppose entry mild perfectly probability computationally procedure depend incomplete entry location exceed also rank determine consider matrix subspace yield theory constant completion bind per column significantly conventional matrix rather must subspace subspace challenge provably subspace algorithm column highly dimensional application internet matrix identification device record computer monitor internet determine network computer point traffic portion internet use burden place network disadvantage monitor normal control observed pose completion incomplete distance potentially rank computer cluster access internet limit submatrix computer distance lie distance idea completion one ingredient completion particularly require element reconstruct build assume subspace differ concentration incomplete near paper involve intuitive step outline detail section column call seed neighbor reliably even portion subspace span seed completion agree seed discard seed neighborhood enough accomplish show complete observation find correct use detection complete section neighborhood subspace illustration sample subspace depict small dot seed observe seed neighbor seed near seed overlap incoherence play coherence main make subspace coherence coherence pair intersection distance away belong ball draw column subspace key seed formation algorithm suppose column perfectly methodology apart total sufficiently need fix definition end perfect factor individual low final completion perhaps choose per column detail methodology four outline entry uniformly replacement without replacement assume relation note result unique let size random pe pe pe pm pe pm pe column select random nearby neighborhood design probability assume seed great one seed chernoff select belong contain column yield note seed accomplish seed belong column must column bit challenge determine seed show seed index reliably column common denote probability sum replacement bind assumption replacement note variable change equivalently yield observe let proceed seed column observation seed precise moment uniformly choose ensure seed need uniformly seed return later requirement seed column ball seed pt seeds probability ball expect chernoff bind r seed requirement procedure neighborhood observe entry seed per input discard seed find seed form column seed produce least seed seed lemma column uniformly seed probability seed within seed entry observation seed certainly observation index seed least within show must determine low common theorem assume since seed entry imply seed chernoff bind suffice guarantee enough take index random index subset n following size
fourier sparsity w cp variable selection journal american selection average statistic average manual research american li penalize institute nc mail university nc mail edu department national mail com proposition pc pc pc pt pc double shrinkage minus inference jeffreys prior spike tail behavior straightforward role result prior word phrase dimensional relevance robust prior plus great work selection rich analyze fu fu fan lin yu li article popular coefficient normal ii develop relevance sparsity come drive student fact drive jeffreys hierarchy jeffreys prior prior variance propose maximization maximum laplace jeffreys jeffreys lead substantially improve shrink coefficient shrink improper shrink light tail laplace property yet analytic design zero heavy tail jeffreys carefully mixture spike shrinkage prior analytic alternative shrinkage facilitate straightforward result joint good job quantify uncertainty propose double pareto mention analytic possesse appeal student characterization mixture normal straightforward sampler inference prior penalty regularization rule continuity estimation motivated application genome wide association study lee prior include double pareto limited contribution beyond I formal introduction thresholded ii pareto central work jeffreys limit hyper along incorporation sampler vi detailed analysis double pareto thresholding rule analytic maximum posteriori posteriori case step connection generalize generalize pareto contrast parametrize generalize pareto reflect assume symmetric pareto cauchy tail compare cauchy standard pareto zero double pareto density suggest property posteriori cauchy like tail illustrate different represent lead shorthand proposition reveal analytic form bridge two laplace jeffreys setting imply place jeffreys yield normal jeffreys item dirac delta note tail size understand observed level proposition tail grow variance hence cause though strong increase around increase converge density shrink typical specification like tail desirable robustness motivate default shrinkage factor et shrinkage towards generalize pareto upon proposition double pareto complementary prior behave behavior strength small tends unbounded suggest signal standard prior form prior adjust value prior function jeffreys observe conjunction unbounded effect well clarity increase shrinkage place posteriori hyper parameter trade robustness jeffreys variance normal augmentation sampler gaussian mix absence prior datum pareto hyper prior location scale shape median choice exist transformation suggest pareto posterior embed form interval calculate normalize set obtain fixing treat unknown determined establish tie frequentist approach analytic generalized pareto analysis hierarchical straight posteriori estimation double pareto induce imply follow fan li denote minimization fan li unbiased thresholding coefficient instability penalty first term estimation appeal desirable desirable rapidly imply shrinkage coefficient get fact convergence jeffreys prior control pareto tail bias regardless fan li thresholding minimum imply thresholde p derivative negative two root absolute elaborate root j fan estimator unstable change result signal minor hard thresholding unstable stable fan li minimum ensuring create tail robustness region wide yet nearly hard thresholding double pareto posteriori likelihood formulate expectation maximization expectation respect kk let refer estimator exploit laplace integration prior mix laplace require expectation ease c respect step li via representation mode li resemble algorithm use refer path form bridge reveal use representation I laplace scale representation illustrate iteration mixture representation mix integrate li estimator possess oracle normality selection generally hold n deferred estimator prior average fan li denote pareto prior hyper parameter provide table prior obtain hyper j j calculation fan package default package ridge scale default place parameter shape package default package employ et tuning lin li cross validation report median model calculate median calculate standard model three show great flexibility adapt signal computationally use sampler procedure user let may dense constitute similar add thresholding ability although take value adjust method laplace inference hyper take inference hyper restrictive posterior posterior error higher move particularly prior quite room small accommodate dense sparse I I median retain estimate hence model yield well somewhat worse design obtain improve attribute advance good treat hyper appeal flexibility sufficiently error account uncertainty appeal contain model predictive box acceptable practitioner interpretable mixture normal prior pareto thresholded prior inference feasible provide uncertainty estimation prior allow frequentist posteriori argue model mix hyper
hundred thousand knot automatically canonical knot knot sample another fix knot low adaptation value one fix knot make infeasible run markov sampler loop finite cholesky triangular satisfy I give construct recursion remain time first triangular negative cholesky approximation knot result process ii therefore incomplete factorization available knot burden identify training variance need case stay restriction employ process stop acceptable check tolerance tolerance increment repeat produce update change increment subsequent clearly reduction expect proceed new th row give approximation finding lead package cholesky produce permutation triangular permutation cholesky result cholesky covariance process knot comes replace computing fraction compute need element incomplete cholesky factorization user tolerance absolute tolerance tolerance kk mr lk lk kk nr lm illustrate forest contain axis vertical axis reference surface forest covariance function xt orthogonal relate landscape projection location encode range decay knot knot need meet nine vary hold top leave middle project right column project axis indicate pick knot spatial small give canonical metric surface directional topology knot along horizontal horizontal knot axis project show nine vary take shape rate landscape fix record slope value location top row row large make close narrow south west variation slope feature pick knot figure choice tolerance meet tight coverage entire maintain throughout knot nature try pick bias greedy offer highly attractive knot indicate meet tolerance grow yet available relationship knot extremely bayesian markov computation covariance sparse exploration space fit show efficiently adapt change cause change regard adaptive clear advantage handle knot datum independently eq exposition assign normal independence across knot clearly infeasible place knot axis count place knot learn drastically reduce likely poor ff gibbs knot suppose knot find knot density knot would look figure strongly knot b line interval dash scatter along th gets adapt knot value restrict search knot covariate tolerance likelihood py py pd predictive adapt approximate likelihood compute knot formula posterior py explore sampler figure summary metropolis sampler exploration level united relate education autoregressive relation respectively population high education number log response three relate spatially regression x ii vector formulation zero tx jt ji normal priori fit use count exploration py pp distribution unimodal knot two country unlike substantial meet bind summary fit bottom predict combine figure scatter fitting dot remain one red dot predictor find relate home spatially vary influence positive relate education moderate effect relate weak absolute predictor interpret ordinary include equal methodology substantially conceptual contribution emphasis approximation infinite priori provide approximate posterior distribution theoretical knot fundamentally deterministic knot drive stay limit close process approach model knot stochastic well task discuss process smoothness employ autoregressive lattice approximate case need approximation yield smooth pose additional irrespective ill computation solve algorithm ill norm bound z packing easy knot define calculate integral assumption cb bind side sd st knots predictive cubic complexity process crucially knot retain approximated knot domain present calculation coverage process place knot change change control covariance present algorithm toward cholesky concept already lie implementation predictive result offer substantial fitting keyword gaussian knot cholesky sampling nonparametric gaussian ability incorporate smoothness spatio computer quantile etc thorough overview theoretical bayesian therein view value finitely many characterize involve possibly conceptually straightforward
continuously divergence take quantile simple limit relevance leave open keep good provide leave problem divergence open investigation argument argument remain establish arbitrarily small latter satisfy second show q exist term q hence p tend pt theorem bayesian estimation th universit paris paris com technique estimate establish divergence markov particularly attractive increasingly attractive handle difficult divergence use bayesian concerned measure contaminate outlier difficulty encounter dual posteriori major require estimation smoothing produce estimate contrast empirical purpose estimator reason commonly mcmc outline property divergence discuss estimate section law remark dual divergence general recall respect well divergence divergence chapter divergence problem underlie attention strictly satisfy class define satisfy represent elegant prove divergence connection convex condition represent supremum replace measure q class indexed specify instrumental consider sample ratio robust robust divergence dual censor introduce test copula performance estimator turn function estimator integral type equivalent generate condition access asymptotic use criterion risk incur form density integrate rest bayes expect form loss choose loss square dual estimator integral exist familiar mode divergence define divergence dual divergence divergence modify keep get upon section posterior evaluate use mild satisfied circumstance derivative integrable derivative integrable nonnegative convexity open neighborhood taylor expansion tend usual integration regularity require remainder
surface meta often impossible essence technique aim robust regard though demanding implement reader book develop base hereafter apply involve model world speak oppose physical consideration coherence order retrieve space attempt indicator function problem meta able interest spread lack observation two paper krige extended svm krige interest base essence krige path would model stationary read class autocorrelation unbiased variate stage set observation group solve analytically assume thus solve likelihood solve turn give surrogate reliability consist meta meta second taylor surface surface may accurate quantify substitution failure probabilistic krige instead since express close read follow note quantity deterministic propose probabilistic concept basic limit represent krige meta build represent meta triangle misclassifie fail also red blue fail safe deterministic decision classification smooth subsection propose refine indicator mean confidence surface area blue line margin safe point margin uncertainty due nature prediction read maximize bring statement many author literature decide use global algorithm name equivalent propose add point regard normalize constant either choose pdf difficult usually structural original reader refer mapping pdf indicator chain monte simulation generate sample maxima predict uncertain property center span function propose real indicator failure rewrite definition argue latter equal sum random hereafter g approach probabilistic classification function conjunction technique efficient technique indicator bias failure interest instrumental dominate failure rewrite subscript recall mean read central limit estimation read variance estimator instrumental pdf instrumental pdf many allow reduce variance strategy propose instrumental pdf specific use normal order failure probability may krige build instrumental suit extreme variate indicator optimal instrumental propose quasi pdf read quasi instrumental pdf instrumental follow failure augment ratio real function krige fully correction unity augment propose krige failure account monte first quasi optimal instrumental limit normally might propose monte normalize instrumental interest present use slice simply read calculation coefficient final thick north east cycle thick north cycle reliability inspire concern reliability mm diameter equal mm end stress suppose elastic behavior due boundary poisson though modulus model variation assume square autocorrelation field strategy propose independent group simulate order retrieve von stress meta base importance reliability krige predictor build generate section new refinement procedure stop estimation may provide stop refinement procedure function respect krige estimator failure table compare form estimator implementation matlab
entirely consequence place functional shape highly undesirable limited available crucially select parameterization lead shape underlie uninformative viewpoint approach interpretable distribution context mixture transfer distribution precision assign hierarchical variance necessary space shape however entirely methodology construct metric space regression simulation would compact p ideally define reasonable closeness seem adequate distance parameter euclidean plain impose notion metric construction describe packing application jeffrey prior adapt situation notion need distribution distribution limit discrete spaced notion equally underlie constructive apart lattice calculate unclear whether notion packing packing metric lattice packing pseudo minimum distance aa packing space constructive explicitly practically finite local approximated root imposing hold early distribution obtain consideration riemannian manifold case riemannian consider riemannian manifold parametrization impose parametrization special case taylor appendix calculating transform twice transformation obtain technical metric space limit sequence grow integrable develop general like uniform continuously function back obtain subset transform td apply end desire jeffreys approximate fisher situation residual lead density interpretation jeffreys rule rare among jeffreys rule principle viewpoint jeffreys universal prior weight density statistical model depend covariate undesirable application directly numerically space easily triangular simple yet jeffreys density parametrize hellinger px obtains observe uniform hellinger reference uniform kolmogorov result nonlinear shape reality usually unclear prior suit metric compact vector derivative result functional equal fx improper extend exponential example calculate square normalizing base figure mass desire advantage uniform parameterization particularly choice shape cover small shape extend uniform choice variety choice could fact jeffreys case situation jeffreys surface weight measure obstacle functional fact analytically integration however need observe modelling situation propose nonlinear trial trial variability large often prior test assess formally regression take range compound treatment four adjust score total complete balanced allocation assume response relationship maximum effect give percent clinical exist illustration improper uniform prior necessary improper bound boundary cover account one extend density perform apply introduction induce shape large almost shape shape induce shape shape iterate implement see observe distribution bias linear happen particularly shape fit might expect functional acceptable investigate detail prior study conduct report use hence subject per use scenario power behaviour misspecification scenario compare jeffreys nonlinear uniform prior prior jeffreys proportional analysis tune suit simulation mcmc case mae cp jeffreys power power posterior median mae mae maximum display repetition interval interval simulation length simulation run jeffreys functional improve uniform jeffreys slight jeffreys interval jeffreys keep level linear uniform achieve power interestingly large estimation estimation message parameter jeffreys roughly equally jeffreys simulation conceptual covariate design sequential situation cccc cccc mae mae mae cp model design lead design optimize prior restrict space optimization design functional design whereas essentially look efficiency opt design shape plot shape functional scale observe shape uniform shape efficiency shape motivation practical jeffreys prior want calculate find trial distribution uniform underlie function achieve early simulation reason uninformative metric distribution prior uninformative particular aspect reflect nonlinear argue functional shape depend consider assumption regression situation adequate occur extremely often adequate reasonable rather build uniform potential shape might priori theory
response around neither series causal effect enable direct strength remove resample depend autoregressive resampling effect equation significance g however cutoff remove effect absolute quantile instantaneous acyclic figure graph direct edge orientation dag edge member undirecte direction influence figure price seem connect two price instantaneous difference direction edge go price information flow keep reservoir wind already causal small remove effect lag small cyclic dag equilibrium find contrary generally important market price expect market generation grid price local pay price partly price volatility price naturally peak demand influence affect market partly price play role european price game view advantage previous identify might properly instantaneous forecast decomposition impulse instantaneous dag combine implicit error residual common create see scope deal could include investigate price fine price work statistic innovation innovation particular grateful helpful comment help methodology dynamic major generation series advance answer reservoir price price adjust price vector correction market causal direct acyclic non data market market use advance causal model relationship market govern line different market similar market integrate market dynamic flow major peak price price among european market major full european market market pool dominate pool price market hand dominate production complement assume price causal price dynamic water reservoir level production treat logarithmic direct acyclic dag instantaneous dag joint dag indistinguishable influence edge dag class direction paper acyclic recently allow identify dag able causal relationship dag identify approach direct influence var describe instantaneous causal effect random depend basic previous coefficient integration stationary case var possibility correction derive difference correction long relationship vector relationship series make sense equal inconsistent compare general causal center cause early corresponding estimate dag observational receive infer dag observational greedy implement software fit equivalence reach dag distinguish alone equivalence assume term triangular imply unobserved sense causal let rewrite ica find statistically limit pose use entropy vector density h entropy non efficiently see estimate constant scaling find show estimate available difference var instantaneous effect correspond strict contain autoregressive correspond cyclic var representation coefficient residual find time flow causality causality reduce prediction combine significantly non cause thorough market physical partly explain wind production water level ideally far back wind secondly six time quite rapidly integrate european market market available could include transform common induce exchange price fluctuation influence include exchange series use week try accumulate likewise accumulate consider overview give table price represent united market market expect important formation close market treat wind reservoir influence wind term wind water water ideally price correct log reservoir logit transform price subtract square ht l description pool average price price price exchange price national balancing price uk price exchange average water production wind production htb htb use schwarz lag var even stationarity series perform lag ht rr rr difference also series series reject significance reject level test trace schwarz determining indicate significance schwarz criterion indicate inside schwarz constant within outside lag var repeat lag var outside conclude critical r schwarz combination possibly component combination series stationary since test suggest several reject indicate consist value fit test test normality univariate
markov maker directly access relate state minimize problem optimal find mapping space various controller stochastic quadratic programming complexity know computational target achieve controller controller period class action np guess controller incur linear np even easier observable version extend deterministic allow controller article show hard long controller choose action recent observation np probability action considerably effective deterministic generalization blind article np controller controller unobserve mdp deterministic blind controller action regardless history straightforward evaluate deterministic blind controller action one chain deterministic blind trivially check action good controller action article allow construct np discount factor horizon mdp states cost probability action mdp pair bellman map consider constrain allow stochastic blind distribution simplex mdp blind state relate controller encode cubic graph cubic three cubic column construct mdp matrix action indexing state transition quadratic graph cubic maximum satisfie contain system tight attempt list counting recognize hard np resolve computer argue similar reduce let construct action self start state input state action transition bellman read rewrite jensen inequality note cost reach minimum tight achieve large application jensen establish blind clearly whether reduction case controller trivially correspond transition doubly cost depend proportional scaling arbitrary bilinear real eigenvalue definite see also complement definite simplex corner controller deterministic take operation policy researcher turn tractable show membership np resolve np assumption recently address case publish stochastic relate address grateful point author
aspect present cf algorithm prevent feature perturbation derive form mis rate mixture cluster membership stand variable matrix specifically decision let loss define take measure define distance cluster seek population call strong feature exclude interpretation include cf noise proof calculate expression equivalence recall similarity spectral compute laplacian diagonal degree notion similarity way decomposition cluster form accord eigenvector e eigenvalue partition criterion meet analyze mis cluster rate cluster affinity random cluster generality mis affinity produce block diagonal four mostly perturbation matrix insight cf mis e proportion assign wrong cluster characterize perturbation model main analytic expression perturbation let operator express contour exclude interest mis formula simplify mis derivative hand side unimodal minimized clustering size cluster unbalanced finding play mis analysis cutoff base second shift correspond mis nature perturbation affinity mis cluster two synthetic design several several metric subsection subsection simulation feature capability assume generate observation generate I test include clustering criterion growth exclude clustering rank accord contain five scale eps include plot eps plot first noisy feature identity next generate finally noise useful coordinate occurrence note plot produce competition total accuracy conduct segmentation heart breast cancer robot execution failure lp robot provide feature eight partially satisfactory ccc heart instance cf metric respectively indicator set label idea performance assess cluster different one metric favor observe rp rp calculate cluster bc rp accumulation implementation adopt little try cluster agglomerative rp linkage match throughout run package unless stand base scaling difference feature time grow vector possible empirically particularly rp target conduct proceeding suggest sometimes result replace linkage suggest bc robot table report take produce five illustrate figure competition feature competition chance cluster instance boost table table feature competition h cccc dataset rp angle eps eps histogram strength feature plot eight dataset inspection present choose good thus could feature could measure discuss leave possibility work robot row heart robot row explore vary gain case cf base cluster pre grouping neighbor however statement extensive future empirically compare rp cf base compare angle eps angle heart eps angle eps eps angle eps scale eps explore compare cf cluster eight robust run cluster project automatic search different cluster metric table cf almost outperform eight cccc mean heart robot propose new ensemble include bc accumulation rp cf base cluster boost level provide support formula mis spectral particular spectral devoted deal population cluster e explain competitive certain completeness rule sense figure mixture shift invariance invariance rotation transformation preserve geometry optimal determine eps eps overlap component gaussian mixture star population mean cluster underlie calculate ss ss assume feature exclude two spherical affect addition feature simplify lemma eigenvalue eigenvector eigenvalue curve eigenvalue eigenvector mis dt ii ti ti ii ti dt pn pn let column one verify pn pn conclusion pn part direct rely law omit matrix proof omit forest rf context forest randomly cloud clustering via dataset local clustering cluster cf resemble world cf measure possible desirable close mis aspect cluster dimension include dimension reduction dimensional beyond full subset separate choice attribute projection reveal probe space detect good aggregate separate membership combine many different component direction involve recover projection potentially useful tend huge feasible conduct whole exhaustive randomly probe projection explore compressive however forest rf classification rf improve root start collection variable root become strong strong split cluster perform control achieve eventually produce pursuit therein methodology achieve make recent year treating define notably spectral vector form competition heuristic plot grow cluster corresponding clustering regularize matrix regularization thresholding level less nonlinear grow base vector partition construct
eq follow contradict strong optimal action frequently contiguous construct construction play sequence use identical eq playing subsequent time step let environment arbitrary discount environment r asymptotically later weakly take action function previous maximum sequence algebra contradict exist explore consecutive infinitely often choose step time definition algebra asymptotically infinitely function weak optimal policy exist ucb bandit know explore detect choose environment probability accord explore history consistent policy countable class deterministic environment pz ti kk agent optimal definition weakly deterministic computable environment remark necessity easily generalise arbitrary technical difficulty construct policy l deterministic environment extend introduction see go computable reason rely second operation environment asymptotically computable access number exploration construct environment use require difference environment history different exist satisfy make inconsistent history difference n definition ii pe infinitely ie tp finitely probability triple play policy follow reward recall environment approximation ready environment tt explore first environment inconsistent inconsistent claim bound monotone history optimal asymptotically suppose defined probability start phase explore h I k h lemma history accord inconsistent contradiction indeed policy least recall independently h k therefore close policy environment lemma equation theorem difficulty discount satisfy exploration ensure policy asymptotically discount insight eventually environment change discover weak policy limit turn part asymptotically optimal computable computable policy weakly unlikely even weak asymptotically discount theorem policy discount trivial surprising discount often contiguous dependent discount theorem problematic intelligence construct asymptotically problematic seem reasonable counter able asymptotically optimal assume would counter environment world environment result complexity arm exist weak asymptotically example stochastically environment recursively computable already satisfy markov process mathematically behaviour behaviour usually result formal intelligence satisfying accept even optimality strong something weak order complexity quite similar policy whereas show eventually environment believe asymptotically strong rather actually optimality self possible version n existence self weakly weak self either discount function suggest restrict environment satisfy theorem lose present challenge discount ensure countable section countable reasonable analysis computable environment thesis computable stochastic environment computable discount discount kolmogorov nice question discount prove stochastically discount environment modify policy class stochastically believe possible complex extend part discount admit policy environment weakly augment intelligence valuable feedback early research cm lemma research school national ex artificial intelligence aim agent capable interesting two optimality agent exist asymptotically intelligence reinforcement artificial intelligence knowledge eventually learn mean play car consider existence precise optimality environment unknown necessarily explore two agent play optimally weakly optimally difference asymptotically optimal eventually explore optimal agent decrease asymptotically deterministic computable environment deterministic sake already sufficiently thesis hypothesis compute physics universe stochastically computable large environment universal agent weakly computable environment prove agent weakly class behind proof eventually stop somewhat surprising assume principled way passive extend bayesian computable weak optimal agent class computable environment computable reasonable discount agent exploration component learn ucb bandit explore sufficiently adequate environment exploit balance depend agent discount enough environment surprising exploration understand case decision process various satisfactory ergodicity else satisfactory similar environment return finite alphabet string alphabet set denote string length define condition agent asymptotic interesting discount sequence reward start I limit discount computable computing discount function horizon computable note represent effective horizon agent stand optimal policy tv x guarantee appropriately geometrically strong q weak strong asymptotically obtain environment policy asymptotically bad mistake must fraction optimality imply weak optimality appear somewhat vanish serious infinite believe environment would serious error would notion optimality still discount version
aim agent instead learner observe whether within multiplicative nearly tight gap bound furthermore reveal target number learnable finally tree leave learnable establish bind much construction small class provide approximate class classical class learnable factor start information theoretic showing learn approximation polynomial set pairwise intersection size via construct probability bound union follow large ia ia j nf nf polynomial sized mass great learn claim factor ss know equivalent fs recover optimize direction involves tailor formally therefore quantity lp indicate dual exceed coefficient equal optimal proof fact pf ft correctness structural sketch briefly appendix idea claim linearly try learn technique linear must ensure achieve relate first flip fair coin I coin tail mention find example come correctly function key fs sf non empty chance nan sample see dimension imply reason f value outcome fair nz fs polynomial number complexity easy learnable principle interestingly show within learnable example simplicity structural root leaf ji maximum value power value ls follow tree hand side immediately subset item formally set multinomial expansion tree add parameter prove correctness theorem fed lf least set ms f desire approximate appeal function submodular tree function learn extent interesting namely tree factor location stand small good function tree learnable small non demand hypothesis construct set prove approximate function inequality definition imply desire finish ff demand clearly target result sequence example section start arise item example present plane room requirement leave example company component large million company type company leave tree tree guarantee learnable large take learnable example learnable n proof represent leave leave class learnable use argument unit demand training see unit demand tuple constant pac learnable show representation approximate linear within ground define satisfie demand fix item map ks last reasoning consider fundamental price set item price customer price demand item price price collection prefer gs e gs property prefer old price old price early apply quite economic point query even tool gs characterization marginal fs sf gs maximizer among f f sf sa sf f sa sc sc f previously via concept concavity query relevant set able polynomially target value polynomially output fs fs et submodular restrict lower interesting consider factor gs learnable query learnable query leave tree tree submodular query family uniformly leave value analogous fourth nf concern cost apply submodular low al learnable family simple introduce paradigm many application learner random query framework learner learner price set confidence set value price agent demand price price model variant offer learner respective clearly bound still price reduction learning approximate theorem learnable factor increase convenience family power approximate ss reduction function long set price price always assign price way many induced hypothesis high target specifically l lf l f l notice decision q lx l always q nz lf learn linear hypothesis output size specifically bundle handle separately draw price mistake pn occur thus predict equivalently pn yield fraction pricing algorithm probability recover everywhere bound sequentially price item item depend learnable price tree leave tree leave commonly economic quantitative encoding concern approximate upper factor bind algorithm representation size time highlight importance new submodular distributional leverage structural provide new al query introduce realistic economic decision price observe directly upper continue despite receive less david discussion support part grant grant fa microsoft research fellowship correctness provide allow separately analyze zero subset convenience idea reduce standard binary passive passive unknown u sample flip use linear u moreover think draw belong construct classification linear program find fs correctness feasible early use fact remain show approximate point easy fs fs f finish fact k chernoff give argument inconsistent probability hypothesis approximate prove pac I pac learnable example show solve I reasoning demand sample eq fs training mean generate output one tuple function tree train example uniquely set formally q closely relate item demand item meta pac pac meta guarantee example rank learn everywhere leave warm simple ease fs fs tree leaf every element reader different let completeness tree get reader element leave imply leave fs fs r ts rt fs rt conjecture sketch wang core item rather often complex relate consider hierarchy submodular al realistic setting learn focus submodular approximate level distributional pac style nearly important show complexity representation polynomial time establish hardness class prove distributional setting novel everywhere et finally realistic economic bundle provide bound distributional understand customer assign price bundle product carry company understand preference economic model function know company distribution give customer estimate customer pay package become survey customer price internet particular class bundle henceforth terminology class submodular class hierarchy analyze natural distributional learn polynomially goal q use pac predict high view approximation represent e g g alternative essentially depth max sum leave expressive represent submodular function item include include class present hardness provide finally application receive monotone
non show p f column decomposition use lar software proceed prevent entry becoming stay normalization project update function function well function p project become orthogonal single fact extension admit indeed ii method adapt extend project accelerate work nesterov attract machine processing prove first ability nonsmooth whereas gradient descent solve smooth convex difference maintain iterative past theoretically rate often practice simplicity search scheme indeed fast gradient descent increase exploit nature iterate illustrate good initialization initialization rescale ratio invariance dictionary task structure dictionary come sparsity induce penalization change introduce small overlap easy form rest stay unchanged simplicity single shape dictionary indeed patch define large slightly shift invariant dictionary regime image denoise optimization question already mention flat initialize dictionary overcomplete dct admit choice otherwise initialization collection common initialize pass filter extract use denoise qualitative first move visualize image contain arbitrarily rescale work figure initialization may initialization framework image size seem aspect figure experiment learn patch detail large induce structure show area case regime use white denote pixel arbitrary dictionary patches approximate patch match omp clean patch address clean pseudo norm pixel clean estimate estimate quantitative scale six image five level noise denoise peak pair regularization mean db single scale house I second bold report three elaborate dictionary bm competitive compare seem explain flexibility state art exploit sophisticated new extend shift regime possibly invariance key achieve good invariant mapping partly european grant appendix equivalence define moreover projection vector matrix binary take patch operator create pixel contain correspond indeed entry number overcomplete representation orthogonal paris france call adapt prove task traditionally dictionary flat dictionary choose shift invariance propose learn illustrate image introduce generative call summarize patch image image reconstruction task domain object removal face sparse framework call denoise image combination sparse highly redundant dictionary overlap represented prove useful texture synthesis invariant shift pattern appear time signature idea main contribution framework establish dictionary dictionary shift invariance image signature collection manner q encode overlap patch integer context traditional flat generalize wide structure admit use far relatively family exhaustive naturally new characterized image
speed many order factor implement requirement approach output technique advantage depend solve overhead art per evaluation number zero apparent identity approximate exceed rate additional notable bandwidth radial find convenient feature summarize hyperparameter tune additional hyperparameter let implementation technique k g current consider stop stop operation implement efficient expensive precisely assess identity simulation simulation matlab numerical cpu ghz ram memory window proposition normally consider intractable conclusion application tuning role achieve capability nonconvex demand must usually gaussian hyperparameter dataset identity spectral employ reduce quantity involve computable represent several state art problem solve hyperparameter resort identity exploit optimize verify computational new appendix follow simultaneously mean q hold diagonal additive efficiently derivative consider orthogonal entry q recall square read q element q matrix whose entry derivative n mit em em depth em national scientific setting strongly datum would capabilitie candidate distribution additional need achieve performance operation efficiently presence optima resort optimization respect hyperparameter implementation every novel identity jacobian prove identity computation hyperparameter notably advantage kernel study validate hyperparameter tuning process regression apply wide spectrum notable easily find economic advance science generally intensive application reliable representative probabilistic characterization unobserve former latter valuable qualitative aforementioned probabilistic yield probabilistic unobserved paradigm trade see model bayes rely process critical achieve distribution maximize respect achieve score demand nontrivial maxima employ overcome challenge require apparent combination remarkably decade reduce demand equation develop jacobian hessian exploit notably make propose solution amenable aforementioned follow hyperparameter regression derive art aid conclusion proof result present probability let variable entry suitable definite parametrization rkhs kernel beyond interested observation likelihood derive notably describe well ridge yield combine marginalization respect value constraint transform reason minimization notably derive resort obtain minimizer notable example include particle usually rely minima approximate exploit jacobian score converge evaluation span space iteration characterize q inverse jacobian also bottleneck due store storage local optimization pose severe dataset employ hyperparameter art commonly rely likelihood maximization complexity application represent constraint dataset section identity quantitie specifically perform step describe different first order define let order eigenvalue cost identity summarize main support th eigenvalue proposition exploit pair matrix proposition jacobian hessian valid equation include appendix remarkably employ identity jacobian initial overhead convenient propose whose simplicity
approximation differentiable enable adaptation auto lag paper involve trivial main practitioner bayesian argue optimization freedom strategy design trade exploration exploitation optimize use set adaptation find randomized policy optimization approximate minimize hybrid mix adaptation adapt constrained space sampler state discrete sense sampler hamiltonian hybrid space sampler typically tune even expert require often often want ise boltzmann optimal significantly topology sufficient ensure adaptive vanishing grow large prohibitive reason mechanism adapt run mixture mcmc lower choose building mcmc draw propose move parameterize q influence example discrete connectivity random different setting approach adjust connectivity propose discuss refer certain sake simplicity adapt proposal distribution successful propose restrict adaptation proposal motivated covariance restrictive sampler approximation need later replace mh algorithm ig observe field mcmc base stochastic typically surely adaptation also clear proposal use auto certain lag seem task used assess difficult objective section adaptive adaptation phase accord randomized phase function however value mcmc set approximate entire history noisy gaussian noisy obtain value show review htbp chain noisy evaluation objective I gp sufficient optimize acquisition accord adopt ard intractable invariant mcmc work hypercube hyper however quadrature integrate objective run index indicate implicit q distribution observation restrictive smooth sophisticated reader high candidate acquisition asymptotic alternative thompson upper portfolio strategy combine appear recently several newton method acquisition evaluate location gaussian construct space step several way sampler transition kernel parameterize generate tend ergodic optimizer unnormalize boltzmann metropolis sophisticated version greedy policy account optimizer strategy bandit auto lag respectively characterize value state energy experimental slide minimum l ai move I recently sampler tune boltzmann machines boltzmann boltzmann z e x e e normalizing rest regularize I propose state avoid walk determine control experiment sampling differ pick state flip flip I manually draw naive draw uniformly optimize I parameter sampler parameter baseline ensure bayesian optimize I well naive contiguous come parameter bit three study note ham none bit machine application might percentage regularization selection strategy grid ising arrange rectangular boundary boundary grid exactly interaction weight cube ise arrange two periodic boundary bias visible exactly correspond filter run additional step adaptation consist overhead involve additional far I sampler tune manually ise trial burn correspond burn phase include function begin phase sampler fair mean suffer many long string consecutive evident length parameter choose drawn figure imply landscape cube make manual trivial tune intervention length essentially perform performance confirm ising find rapid inclusion show however
q biased construction overall round conditional index proportional metropolis hasting index equivalently notational pmf distribution eq positive sample conjugacy hasting proportional proportional random may use approximation component index sample variable plus minus plus pt minus bayesian individual combinatorial infinite appropriate posterior beta asymptotic exhibit present domain segmentation traditional goal induce latent class use mixture associate capture mutual observe pattern population variety domain model build model treat exchangeability within domain intra marker marker probability population document individual underlie topic encode image individual characteristic location extract car distinct probabilistic different particular draw draw proportion hierarchical share treat effect focus context literature bayesian model component focus method inferential modeling generally concern although repeat draw situation perspective trait characterize trait possess mutual focus individual exchangeability natural mixture count mixture wish count wish linkage count base survival hazard function latent draw collection coin obtain description conjugacy inference countable coin binary total modeling variable conjugacy conjugacy involve binomial beta conjugacy analogue negative binomial current investigate negative binomial rigorous conjugacy nonparametric beta hdp model allow individual justify modeling choice characterize binomial empirically generate infinite vector count nonparametric gamma connection stochastic process far clear process connection beta process remainder framework measure discuss bernoulli process conjugacy devote study beta binomial summary attack domain automatic segmentation measure tie construction variable present construction use dirichlet present additional independent component atom complete randomness poisson randomness infinite sigma light criterion extension independence describe specify ordinary beta purely atomic purely separate mass notational convenient description deterministic infinite finite parameter atomic poisson sigma number specification allow though ease exposition nonetheless draw process union ordinary atom beta name beta atomic poisson intensity ordinary improper beta infinite parameter countable process interpret though beta appropriate context atom index feature occur form beyond classic law say discount parameter atom infinite finite atomic ordinary intensity focus homogeneous straightforward beta beta process determine part poisson beta weight atom problematic beta associated atom constant restriction position reason prefer component relation dirichlet concern let depend parametric perspective specification classical atom conjugacy parameterization strictly atom purely atom simple specification process introduce exception parameterization atom intensity discuss favor intensity exposition extension depend location beta parameter value quantity rather purely atomic atom atom weight parameterization gamma purely measure ordinary process gamma express l classic dirichlet atom frequency must normalize gamma particular poisson process finitely many atom mass gamma process almost infinitely atom divide mass thus finitely fix choose fact almost atomic gamma far dirichlet provide countable set view countable cluster introduce prior value couple real process weight natural binomial frequency nonnegative rely survival model context construction construction bernoulli analogous conjugacy use couple single process matter bernoulli need even poisson may process bernoulli countable atom differ feature equal weight individual derive ordinary component completely number feature total mass bernoulli parameter atom ordinary finite number important finitely constrain beta conjugate process component conditionally j note posterior beta hold integrated recover condition independent l apparent parameterization atom conjugacy binomial conjugacy preserve conjugacy couple classical parametric conjugacy negative binomial two measure say negative binomial atom construction geometric interpret assigning particular assume binomial process datum word binomial beta process three beta ordinary component infinite number atom atom binomial process fact negative specification conjugate binomial likelihood scalar member scalar measure negative draw atom r j j j post n l piece model recall basic g belong binary one component class zero integer view vector count word draw finite mix view component draw component integer component data parameter individual point draw individual alternative couple poisson convert base follow individual atom location individual might complex perspective focus consider individual becomes achieve remainder bayesian nonparametric individual proportion mix couple achieve via range note global drawing measure therefore weight share atom individual specific proportion specification draw index proportion might th stochastic generate couple count across construct beta base measure independence count multiple individual flexibility mix proportion proportion proportion flexibility hierarchy level atomic coincide atom specific structure decompose decompose draw beta beta explore bernoulli propose prior continuous associate process atom failure beta contrast offer assign individual beta weight leverage hierarchical beta suited come analysis integer factorization motivating lead challenge algorithm develop inexact monte approximation include conjugacy behavior belief interest datum diversity atom process component size cluster asymptotically dirichlet extra parameter size generate grow indeed popularity prior attribute power highlight subtle priori number datum potentially effect dirichlet use process mass determine however treat case number number number case case far establish full statement table cluster find cluster expansion term asymptotic expansion upper table cluster growth basic power growth expand model theoretical simulation law beta simulation perform binomial evenly space generate beta atom binomial sample count number note finally simulation behavior law figure scatter tuple triple tuple leave case agree upper agreement contrast exhibit law cluster negative simulation see right cluster plot asymptotic behavior plot red red law exhibit logarithmic growth cluster deterministic grows generate see yield yield asymptotic growth group claim al al al party experiment model posterior code remain default mcmc confusion obtain model identification notably document group group stem distinguish name c group actual c actual group great difficulty usage frequency across plot density document heat usage frequency pattern nearly group align majority make result similarity ten probable intuitive salient vocabulary organization organization name l organization claim armed claim claim force claim group claim claim group claim party claim front united divide meaningful semantic core content tracking jointly image comprise patch generate image patch label infer object tackle inferential performance patch modality complementary modality texture fourth opponent angle descriptor grant invariance variation geometry dense sift histogram orient single patch cover patch index cover opponent descriptor discrete visual raw descriptor sift opponent descriptor mean four component descriptor modality conditionally object patch define generate microsoft wise image wise ground labeling pixel label truth label lie boundary learn ground task remain nine semantic pixel space pixel image visual vocabulary assign represent patch perform divide label evaluate infer image object hyperparameter dr dr sample patch label high lda standard variational semantic generic figure pixel patch center accuracy report generate training prediction every save object showing provide classical c actual label tree actual car test misspecification hyperparameter hold summarize hyperparameter maintain predictive performance vary order model hyperparameter specification segmentation recognition hyperparameter vary remain hold report test patch average class randomly patch predict posterior sample hyperparameter accuracy latent count characterize relationship asymptotic mcmc document latent vision aim many object genomic seek underlying event responsible repetition segment acknowledgment support national science foundation national science fellowship beta process arise model section deep new stochastic construction lead novel comparison nonparametric clustering conditionally random poisson process intensity negative dd e motivated strong construction cluster random stage consist gamma mostly highlight classic relation study change prior derive process distribution start beta distribute gamma distribution though rest result process gamma process new proof nonnegative define measure specify location atom typically sigma ordinary define l name reflect improper beta beta atom distributional result beta derive transformation atom weight process process suppose ratio construct variable gamma suppose analogue construct beta variable gamma gamma component derive connection stochastic emphasize perform process alternative mix ordinary tell collection tuple come draw intensity variable atom since completely ordinary component measure assume poisson associate tuple mark distribution collection intensity match ordinary intensity component manner inverse change component gamma process tuple process tell marked process uv intensity classic distributional eq start briefly establish cite move concentration satisfy cluster size discount growth range discount beta satisfie beta number beta mass concentration parameter number next asymptotic growth discount parameter growth cluster end establish growth number growth combine asymptotic diversity expect point asymptotic growth growth cluster cluster proportion dirichlet draw measure describe appear asymptotic ratio result statement apply line iff equivalently iff poisson intensity intensity result q follow conclude proceed poisson find atom binomial integer integral desire draw consider let monotonicity atom beta associate binomial count equal previous r process intensity intensity completely atom component atom atom variable binomial input correspond process measure three repeat atom atom old atom normalize q new atom normalize deterministic measure beta bernoulli conjugacy completely measure sigma algebra atom associate sigma induce let marginal counting measure introduce process special finally note ordinary atom total atom atom component atom location together count measure location distribute independent identically case ordinary weight ordinary atom atom restrict location restriction proceed marginal may recall distribute location distribute yield describe next particular atom occur ordinary locate atom locate note atom break eq atom count notable special write single atom likelihood evaluate measure analogously calculate quantity
randomly generate environment generate employ softmax policy addition error bar display b function show underlie mdp percentile bar policy demonstrate mh metropolis alg mh hybrid alg sec first examine derive reward domain mh near mh reward estimation reward inferior nevertheless track suboptimal baseline mdp set attribute poor demonstrate see policy increase plot state space model mh consistently mh significantly rl perhaps attribute mdp optimality policy walk rl suboptimal performance performance theoretic slightly much never possible preference reinforcement procedure estimation flexible policy agent preference although require adjust closely sampler outperform inverse demonstrate simple discount promise hard difficulty maintain see give belief mdps hard preference prior firstly environment different preference tie reinforcement learn easily achievable reinforcement interesting promise already useful modelling agent experimental preference preference addition many automate opinion complex planning experimental useful acknowledgement thank anonymous partially bernstein project I fp project result principled inverse preference relation statistical method learn experimental respect obtain demonstrate preference whether event among event preference determine event prefer utility hypothesis numerical event preferred maker gamble utility relevance cognitive behaviour reach direct user model customer apparent task expert set preference act obtain reward environment functional optimally respect preference framework allow inverse structured reward significantly fairly main contribution formulation reinforcement preference prior determine policy obtain policy inverse reinforcement compare relation preference estimate give behaviour theoretic reinforcement preference setting relate abstract preference discuss preference conclude preference dynamic specifically environment control process environment action convention observe act action wish learn infinite horizon try utility discount choice correspondence reinforcement markov utility decision denote distribution define chain subscript shall notational paper similarly denote mdps mdp abuse different speak reward discount agent belief shall task additional structural policy obtain well assume reward policy easily relax additional define reward policy policy reward reward reward policy datum right reward reward reward reward policy depict basic model introduce model reward obtain leave allow draw policy give arbitrary pure reward agent one valuable quantity disadvantage k ts preference attract lot uncertain preference problem design query generalise additive relation application multiclass generalise multiple user decomposable multi utility discuss introduction act model static agent finally experiment program order good action elegant suffer hard demonstrate visit state frequently mdps gap example could equal every structured implicitly policy exponential correspond entropy approach rearrange computable metropolis employ mh walk exploratory examine determine crucial consistent initial trajectory obtain n guarantee consequently guarantee bounding side high neither suggest require statistic feature entropy policy efficiently particularly notable aware probability demonstrate low bind may link try expert b namely random mdp discrete four action arrival uniformly state arrival pair define
symmetric justify every somewhat complicated good empirical rademacher linear functional vector n sample represent rademacher mutually uniformly distribute uniform error result loss let slightly main finite generalization regularizer mx I nx logarithm course finite conclusion priori pass recover exist considerably rise novel datum retain condition replace corollary eq key result independence dependence turn rest organize present result mention conclusion comment give great simplification range member norm simplify occur simplify remainder generic iid point iid member operator factor dimensional I agree bind dominant whenever corollary need remark lasso replace supremum elastic net penalty priori appropriate operator th coordinate q far almost r eq let j projection span orthogonal regularizer vector whose know desire previous lasso moment moment x previous always mutually give appearance range orthogonal complicated almost linear functional complete lasso satisfying disjoint writing rp quite loose structure nonempty convex show supremum attain extreme closure rademacher class simplification hilbert arise group lasso definite background read finite agreement infinite eq auxiliary hilbert collection recall schmidt letter tuple object ny inequality bound back state write random let iid simple lemma read positive homogeneous homogeneity w finite linear combination subspace therefore also consequence iii infimum together refer bound norm v member orthogonal range member onto take reverse bounded concentration linearly random vector variable iid transformation orthonormal satisfy j q f triangle k difference jensen ii calculus rt r part e integration inequality essential compare hand give insight fine appear infinite variable eq notation part dt inequality variable use dt follow lemma version substitution small since mx calculus imply theorem obvious completeness let norm fact omit version paper analogous recover multiplicative sample somewhat set normal q q jensen give dt
measurable average filter usually would actual actual suitable measure generate weak convergence multiscale signal drift fast sde diffusion slow represent slow change brownian motion suitable multidimensional convergence periodic use probabilistic chapter nonlinear representation determine limit behavior give precise sde theorem multiscale diffusion give marginal unnormalized filter filter sde possess fix filter nonlinear variation component apply variation obtain convergence technique terminal govern dynamic stay time tv nice x behave notation work brownian therefore write instead facilitate read key article expand rigorously terminal converge term formal expansion term omit facilitate read solve terminal exactly uniqueness solution equation apply superposition showing achieve backward doubly differential estimate us component vice existence stationary distribution semi matrix unit coefficient diffusion g finally introduce norm usual supremum metric generate prove section conclude converge backward solve explicit precise allow convergence idea precise probabilistic representation form brownian motion general equation denote equation characteristic advantage permit would characteristics independent solution measurable forward doubly differential equation bx ds dy x ds gs ds cover random side remainder give precise reading able existence result classical theory square smooth degenerate see uniqueness continuity define introduce adapt jointly measurable outside inequality every coefficient time continuously partial derivative continuously partial uniformly time differentiable polynomially classical nan integral indistinguishable derivative bound theorem corollary claim give space deduce thing need verify polynomial growth coefficient pt dx combine analogously neither measurable z ft ty write multidimensional one ultimately show f ty ft ft z associate differential strong sde bx ds associate assumption unique reader suffice q solve gradient x z sx drop sx sd w ds p ij sx ds sx iv z sx p ij sx ds depend p tx x du sx sx sx x run hence pt sx z sx last inequality therefore next sx ds integration sx ds kp x du pt u du kp u b iv sx sx kp ij sx pt du since center ordinary cf sx sx sx ds x sx ds kp du kp u tx du du sx sx follow result high derivative tx existence derivative sx z ds hx x sx hx drop measurable b note independent jensen inequality sx z hx sx sx schwarz combine ds c z ds z ds hold theorem track condition solution state polynomial growth satisfy polynomial growth proposition combine homogeneity proposition obtain eq right hand calculation facilitate read transfer result cauchy schwarz inequality long brownian isometry b moment describe lem x dx dx filter follow exactly chapter sake completeness bound p completely analogue jensen tb bound test eq third pt lemma since replace q countable strongly separate every ix iy sequence topology measure follow metric determine generate topology complete development particle filter state estimation multiscale systems approximation grain coarse grain use particle filtering accounting dynamic incorporation datum coarse grain slow apply multiscale though deal incorporate realistic separation process govern condition vary flow cause fluctuation cause fluctuation whereas drift fluctuation slow limit distribution away solution develop exclude example convergence filter existence solution bound smooth work entirely rely argument might restrictive however explicit terminal avoid building stability filter wish express anonymous reading manuscript comment presentation p n engineering grateful conclusion recommendation express necessarily view support grant ph nsf grant nsf school universit universit e multiscale dimension converge filter equation correction practical weather prediction consist enough class minimize call filter distribution effectively simplify observation filter realistic evolution measure impractical dimensional finite approximation extend kalman filter approximation observation extensively quite strong science provide comprehensive issue see use particle problem dimension completely difficultie particle signal observation filtering capability result order magnitude scale science engineering field example evolution govern slow dynamic address effect multiscale equation canonical way study present filter attractive equation filter useful hence numerical partial rigorous support numerical stochastically qualitatively develop low demonstrate filter kalman differential let
minimizer opposite point regret point act recognize discard recognize great either discard large similarly center since require cut ensure simple center incur start device loss total let l l rl rl lr x rl r proceed feasible region aim portion work suboptimal point round point know stop query suffice ensure regret epoch subset l guarantee epoch point portion work contain point discard suitably exploit query equally separate identify either right contain region discard algorithm continue depict look confidence around identify contains discard possible ci early algorithm sufficiently work feasible region hence epoch query interval three example ci case rely function contain avoid fx remainder condition maintain point query run lipschitz subgaussian least algorithm adaptive noisy apart mid work feasible order section key idea show epoch bound per flat never show point end round epoch terminate l x former latter imply convexity q mean analogous replace fact lemma incur single epoch first incur epoch trick epoch incur epoch continue incur round detailed incur fact incur round epoch suffice continue round round iff contain analogous epoch incur entire regret epoch lemma incur round know query recall query incur round final epoch incur q feasible epoch epoch perform proof epoch would confidence lipschitz epoch end definition give claim rearrange follow combine epoch hold incur epoch end round eq overall epoch recall thus condition event construct round make hoeffding epoch uniformly query upper algorithm dimension would construct cover unit along cover know along scale exponential encounter optimization direction polynomially query define pyramid point form pyramid base see graphical build capture direction early approach fail idea case regret combine center feasible noisy box allow r construct simplex vertex let continue k pyramid epoch angle define cone ellipsoid contain pyramid region optimization domain begin begin epoch epoch end discard set way retain optimum epoch apply affine volume ellipsoid define constant within epoch round successively let round center query ci pick depict diagram pyramid epoch angle diagram successively successively construct identify region discard book angle pyramid orthogonal pyramid always sphere diameter sphere angle pyramid depict pyramid center pyramid ci pyramid include large denote ci illustrate know next pyramid continue pyramid cut event depict figure center pyramid sample current pyramid terminate step pyramid illustration pyramid ci angle pyramid let pyramid show pyramid step correctness perturbation conclude epoch b pyramid base angle define center pyramid cone thing isotropic position ellipsoid ellipsoid brief discussion regard clear step cone cut isotropic analogous ellipsoid know furthermore update ellipsoid correctness epoch fx sample remainder condition correctness proceed cut indeed couple construct center least cone discard epoch round ci pyramid epoch assume construction pyramid graphical see pyramid pyramid ci pyramid convexity simplifying know dr therefore complete guarantee discard mistake discard final check correctness lemma new pyramid form round epoch cone center angle vertex center dx ff since enter know fx line use suppose incurred address early theorem scale noiseless random walk idea question future incur together round epoch play pyramid encounter pyramid operation base angle construct center reach pyramid ci net incur evaluate consequence inside pyramid pyramid vertex upper reach value brevity shorthand center convexity rearrange bootstrap vertex center center incur sampling center pyramid ci complete total face b dr ball center net incur round substitute value complete critical us pyramid pyramid exception lipschitz visit round case bound lemma modification fact evaluation little round pyramid construct simple geometric exploit pyramid equal pyramid case enter regular simplex radius contain enter simplex round simplex guarantee unless simplex pyramid construction enter one b terminate unless ci insufficient resolve thing construct sufficiently lemma focus control construct convenient incur round every net round round ci terminate net incur round construct instantaneous pyramid note pyramid cause know pyramid construct round function bound instantaneous incur sample pyramid simplex sample vertex point query point ci geometrically total put together proof state incur regret round terminate ci net control round description terminate exception round ci proof simplex pyramid round instantaneous regret incur pyramid end case bound lemma ci general ci pyramid query net pyramid construct complete put piece show incur ci come far geometrically regret incur ci get regret epoch epoch end ci bind show cone cutting understand volume discard reduction prove method suffice intersection ellipsoid discard cone exactly origin cone height inequality suffice distance origin ellipsoid intersection sphere hyperplane volume cut ellipsoid statement epoch show end contain regret least next ci least terminate condition case epoch proceed cone cutting show equation terminate pyramid dx lemma point complete lemma number epoch play total ball radius around instantaneous volume epoch instantaneous regret epoch maintain radius algorithm algorithm claim together observe guarantee analysis far condition round design complete proof theorem present possible build low point algorithm crucially demonstrate optimal rather dimension dependence dimension noiseless walk successful improve noiseless investigate scenario acknowledgment work aa support google support grant nsf nsf grant dms pyramid dd pyramid center base r center vertex base figure inductive note vertex distance claim node node black north black node south circle east east pyramid mass prove claim q lemma ccccc ccc microsoft california berkeley university new berkeley pa cm address lipschitz bandit function interest value query demonstrate generalization incur classical armed formulate arguably sequential decision arm maker observe accord associate performance algorithm sequentially cost expect much bandit action rely cost reward arm make tractable paper space
frobenius translate tail proof lemma function dependence use loose upper weak explicit directional deviation additive hide kx g v u observe minimum size inner mm black hide minimum size sep mm draw name v uv inner draw circle minimum sep name w condition eq matrix singular large value case relative observed style fill circle inner sep name name h orthonormal km cm circle minimum size sep mm z name h minimum cm fill style z name name name hide r name z uv schwarz also strict pair return must hold I z z imply ji combine return prove useful allow suppose induce iv return allow imply return next characterize root algorithm subroutine return main maintain loop terminate achieve show loop claim properly parent subroutine whether child claim definition sample condition threshold test return event need hold union henceforth next ignore direction subtree ignore direction root subtree imply child relationship child etc exception root rise start subtree subtree say subtree fact super tree maintain super collection disjoint root leaf super next relate opposite bottleneck correlation bottleneck induce z scale observe style circle cm style observe name observe name z notion accord edge direction effectively depend relative exploit z cover size cm sep black style circle inner z name z hide name name z orthonormal inner fill sep name z name name h g choose column u sep mm fill minimum sep black name z name name style circle minimum draw black mm black name h h choose orthonormal u two lemma case subroutine collection disjoint root leaf far relative leaf pair u u x return note neighbor relative path undirected node u u follow observe style sep style minimum name observe name hide name g style circle sep black fill circle draw name name without undirected node node pass root undirected leaf x v topology observe cm inner sep mm fill size name circle minimum draw black style size sep draw black name observe name x v x upon return disjoint rooted root leaf exist share neighbor least neither v u u u hold neither common leaf neither uv uv inner hide cm mm black u dash least moreover xx xu yu induce mm draw draw name name hide observed observe name u give follow style black style circle sep observe name hold inequality analogous x undirected circle inner fill minimum cm sep mm observe name name name hide u x claim second undirected observe sep draw black fill circle sep draw name name since sep black fill hidden style mm name name name x u prove hold respectively root must v u uv observed style inner draw style mm hide name u dash leaf u yu u xu u yu topology sep fill black hide black name name name observe name observe observe u p u observe style size cm sep hide inner name v observe name name observe observe p x suppose iii generality leaf root uv inner sep fill hidden mm name u name hide name name dash argument prove finally loop consequently object loop u appear subtree moreover loop prove initially cardinality lemma final subtree root loop initial inductive start terminate failure loop begin iteration u pair neighbor leaf adjacent component neighbor existence two neighbor component leaf child exist u pair v claim terminate subroutine satisfie v return claim relationship common u iv share subroutine first leaf claim child leaf relative u u u q adjacent leave leaf loss generality argue subroutine symmetry subroutine call construction one leaf component leaf collection pick leaf child loop subtree imply u except change add degree one leaf consider continuous markov markov evolutionary tree certain goal tree recover sample recovery reveal furthermore sample explicit algorithm applicable dimensional heart determine relative topology graphical central tool learn methodology success ai language processing vision bioinformatics statistical challenge graphical model rich parameter understand involve certain expectation em recently understand learn np greedy focus model tree hide binary evolutionary tree set multivariate observe graphical relevant vision scene co aid structural reveal tree configuration four additive metric induce unfortunately ultimately robustness estimation quantify various neighbor serve basis method work area mathematical focus evolutionary basic effort deal allow oppose polynomial recent learn extend evolutionary domain observe style inner hide inner name name z name hide z scale circle inner style sep mm draw observed name hide circle sep black black circle sep mm observe name observe style mm fill mm draw name z observe z extend model scalar address multivariate may scalar handle wide need characteristic discrete multivariate latent core multivariate classical test spectral canonical spectral success useful test recursive tree confirm hypothesis relative topology properly probability efficient manner also considerably restrictive address provable spectral direct every term internal moment either though configuration possibility return degenerate fail motivated singular km test true tree condition strict deduce correct topology q confidence sense reliability z z z j j z spectral singular lie length event hold remark valid variable vector iid copy return induce topology important redundancy correct among gap z z return quantify observe tree ii condition iii return use structure iid globally grouping procedure build fashion root discover
immediately carry notation proof find proof markov substitute geometrically mix show essential mixing basically identical corollary come form boost make clear nontrivial space strongly individual function scalar old inequality example regression latter strongly logarithmic loss guarantee per discussion logarithmic regret requirement modification et combine forecaster sample prediction specifically iteration receive suffer al logarithmic appear drastically encounter update stochastic recall fact q difference stable require outer matrix mahalanobis stability linear generate derive follow assumption online apply linear prediction specifically regret remain q assumption argument minor martingale previous measurable difference adapt modification involve guarantee noting complete proof conclusion ignore size space probability geometrically follow geometrically analogously build largely identical bound excess extend martingale hope proof without require expect believe open question work raise guarantee underlie necessary couple strong regret bound acknowledgment thank careful greatly aa fellowship google national engineering fellowship bit simple form intuitive turn bind nesterov begin take supremum continuous mahalanobis g z z definition update plug bind complete norm note result give begin thus convexity apply h old organization accord recall outer construction proof q update inequality consist achieve begin eq inequality divide proof ex em berkeley edu sample dependent online computable probability error assume loss sharp bound problem logistic least svm application martingale convergence need rely attractive point fix predictor hold assume statistical regularity sequence ask something probabilistic function example algorithm good distribute et al output predictor loss play probability ask ergodic online regret fix predictor natural lead researcher online change distribution computing fix process fix meaningful reasonable practically include problem learn difficulty encounter researcher setting scenario generally mix imply yu law direct approach convergence stationary natural localization self bound exploit machine statistic sequence extend bernstein inequality due generalization dependent particular predictor favorable regime geometric loss lipschitz loss hypothesis prediction fast regression boost expect predictor zero show demonstrate generalization guarantee online answer give unseen say question regard dual averaging broadly establish suitably sequence history area book dependent probability guarantee markov bound mirror optimization non build martingale random bernstein geometrically mix process version prove far relatively convergence argument datum stationary suitably convergent receive sample play algorithm suffer generalization goal low algorithm respect hypothesis require variation distribution density respect suitably converge grow absolutely weak mixing respectively assume mix entire slice result either mix strong mix regime mix mix markov chain recurrent example process example arise metropolis hasting sampler lower state instantaneous boundedness common literature boundedness first final center sequel strong presence bound strong strongly lastly generalization iterate assumption stability least analysis quantify algorithm possibly sequence produce sequence dependent measure sample conditional loss sample I excess risk course set slightly definition assumption place suitably substitute completes remain consist probability throughout algorithm result satisfie depend add sum give divide f jensen recover theorem generalization leave development reader stability make indeed ask analysis insufficient guarantee p rule trivial consequently expectation cumulative result expect guarantee guarantee online expectation would like strong setting martingale argument use concentration remain predictor guarantee bind arise sequence around mirror algorithm idea martingale ergodic block random variable dependent previous directly moment generate sum different proof index random associated define martingale adapt subsampling define first boundedness difference hoeffding term representation well concrete corollary begin corollary mix assumption assume constant mixing process generalization set somewhat slow define et al concrete error mirror extend convex function average satisfie stability immediately universal least obtain corollary assumption weak nonetheless quickly predictor apply proof treatment piece require care observe via see though thing seem make stochastic geometrically corollary geometric mixing corollary unless desire argument polynomially mix somewhat well regret learn due reader recall gradient mirror average stability f term fluctuation martingale scale sample property construct martingale self bound martingale
even marginalization search correctly even propagation could energy mc propagation physics long rapidly topological distance bp pair conditional cavity instance probability remove assume neighbor message fix normalize iterate reach point take time free bp z ij c aa stationary likely fix maximization group several free decrease phase diagram phase transition non arise benchmark agree except effect plant actual assignment configuration energy module soon close message fix typical first phase fact locally stable location easily small random perturbation point dynamically attractive hard ground marginal exactly hand impossible hard know retrieve hard unlikely realistic world test common benchmark fix correspond energy large node initial good splitting actual identical marginal term finite tree principle asymptotically strict phase free energy landscape infer community problem number exact size generative performance variety acknowledgment grateful mark foundation universit paris paris france computer science department asymptotically detect community module noisy plant cavity transition assignment split easy translate practical module underlie hc network genetic community module social connection community propose connect letter generative modular provide popular module cavity develop module sparse network assignment nod intuitive topology retain membership algorithm unable previously approximately size addition unlike minimize entire boltzmann distribution believe exist propagation size bp however ability module marginal marginal aspect approach applicable discuss briefly restrict elaborate block consider node specify member label group choose model leave rescaled affinity block assignment literature plant fundamental cavity block however monte carlo crucial contribution compute cavity bp roughly network assignment prior available maximize assignment network generative
send receiver reverse symbol probability receive ease inferential active symbol symbol digital frequency variance vb conjugate remain unit circle serve conditional location directional signal processing literature phase loop von explicitly von hence various normal symbol exact basis symbol wherein apply htbp symbol marginal symbol give characterize incoming get simple phase vanish symbol ambiguity symbol map decode distribution multiple period sequentially decode output model reasonable inter consequence period employ kullback intuition functional posterior force independence learn artificial intelligence recently result posterior latter partition node kullback vb consider situation transfer period infer symbol phase symbol expand unknown phase posterior integrate summation resort discuss include author assume make sequentially infer marginal symbol symbol independence symbol assume combinatorial step von symbol expand unknown eq section iii apply transmission characterize vb figure describe proportion various develop batch symbol regime db db algorithms well decode use exact ignore fast offline though general ambiguity pilot get main paper primarily able synchronization generalization resolve ambiguity show independent phase thereby apparent concentrated presence unimodal phase invariant symbol relatively long overhead angle scheme pilot symbol invariance simplicity observation deal essentially delta significantly low snr snr vb wherein virtue modelling problem ambiguity lose multi would grant minus ex ex ex receiver noisy message receiver receiver rotation receive plane impossible process synchronization important aspect art phase synchronization vice decision soft decision consider aim fully flexible varied uncertainty procedure variational
decomposition ccc linearly sparsity example keep compare toolbox point interior point method centre tree indicate tolerance converge obtain tolerance grid bring remarkably increase overhead pass constraint grid reach well vary iteration contiguous value contiguous whose noise tree use recover coefficient reconstruct image method h background right new algorithm else sparsity form contiguous proximal penalty derive computing result highlight advantage greedy indicate study convergence whether sparse institute college college university college structure extend group incorporate straightforward prescribe contiguous overlap optimization limited build point algorithm class norm rely proximity operator subproblem regression optimization statistical question definition regression parameter result interested word structure sparsity certain configuration prefer several denoise discussion structure recently nonsmooth vector auxiliary constructive sparsity specifically formulation involve variational incorporate magnitude sparsity contain vector optimum extend formulation sparsity terminology sparsity contiguous embed scene learn limitation technique describe easily associate regularization proposes coordinate feasible limited contribution general combine inspire fista recently significant research subject grow reference therein mainly penalty former focused extend possibility hierarchical structure penalty upon improved form whose computational per iteration comparable descent simple combine acceleration gain certain importance admits form proximal appeal proximity operator computable structure sparsity organize section structured numerical method induce otherwise structure vector prescribe positive attain infimum attain auxiliary vector term sign insight upper namely tight fact arithmetic particular encourage function norm encourage desired well understand contain sparsity pattern indeed since pattern reflect moreover lead end case convex whereas possible convex cone shall constraint specific understand pattern choice encourage hierarchical regularizer explicitly overcome difficulty may unit ball encourage subgraph correspond penalty recall conjugate function subdifferential subdifferential nonempty compact set differentiable subdifferential generic computable deal end recall proximity value proximity operator r proximity operator define minimum various way regularization reproduce hilbert regularizer write reproduce indeed eq view function belong prescribe encourage convex interested cone norm sparsity early ff cb solution hold duality condition supremum attain solution f combine norm duality yield interest quadratic norm yet equation solve denote constraint p problem variable norm impose cone mention penalty dual automatically employ end see example solution q condition self dual solution p x accelerate first scale respect computation proximity special computing operator apply wide variety constraint proximal r solution attain equation simplify define ts solution proximity composite solve seem difficult proximity map task notation eq iterate despite fix modification point fix point iterate proximity give require vector state domain attain set make root motivate numerical problem nonsmooth nonsmooth respectively replace linear specific lead computation case w simple
et underlie simplify merely furthermore kalman complexity grow time behind improve start whose linearization non wise analytic path adequate analytic grow multiplication output substantial multiplication dimension well dimension dimension whose denote deterministic ordinary differential omit behind firstly wiener approximation differential secondly replica couple henceforth index space replica input input calculate conditioning solution replica assume couple denote strong replica couple stochastic differential path read determine green chapter path order exponential function translate adjoint green read function two replica two find symmetry consider dimensional subspace component replica replica shall matrix suppose integrate degree eq square matrix whose block equation determine replica simulation therefore split replica online note replace remainder chapter recursive ordering eqs imply variance former relation latter recursion relation calculate pre h relation order use recursion multiplication dimension interested recursion order analytically case piece analytically q derive eq last section call vector water store water fig equation bf dp fraction model equation read intersect respectively eq input dependent euclidean metric eq reasonable noise diagonal entry obviously calculate conditioning statistical solution series dynamic path analytically resort wise present integrate step piece additional quality exact present eqs numerical multiplication smoothing need multiplication disadvantage one huge I many r code presentation apply sciences simulation intensive calibration ordinary develop
shape part extra part assign shape define value gibbs follow sum omit unary potential unique additive transformation consist potential normalise important notice call homogeneous marginal coincide present reveal verify converse chain homogeneous admit homogeneous appearance assume colour space shape popularity ask identical respect occur imagine complex structured latent show variant expressive want able model unsupervise informally task necessary segmentation task task assign penalty number misclassifie pixel lead max calculate probability currently infeasible belief propagation overview guarantee exact sampling potential task comprise unknown latter applicable situation distinguished format element format supervise consist learn cope well e event potential train logarithm substituting give derivative potential co occurrence random exact calculation expectation ascent posteriori replace expectation calculate potential completeness mention simple view posteriori perform interaction far neighbourhood unfortunately know option interaction despite complexity lead discrimination neighbourhood give possible variant likelihood exhaustive search would prohibitive successively include structure successively remove start variant greedy texture start unary potential edge include previous potential assume gibbs neighbourhood likelihood orthogonal proposal vector kullback divergence euclidean estimation proceed neighbourhood successively remove impact impossible gradient variant situation supervise potential gibbs potential denote nothing family expectation among remove neighbourhood see write arbitrary latter nothing potential sub subgradient uniqueness potential potential unique estimate removal neighbourhood neighbourhood gibbs potential small relation segment relation segment input segmentation middle bottom priori code colour image full neighbourhood whereas ability capture neighbourhood structure show training scene appearance segment assume multivariate gaussians neighbourhood neighbourhood baseline semi fix rectangular priori potential appearance model learn observe show wrong neither simple simple investigate shape generate neighbourhood standard neighbourhood vector gibbs short potential long edge modular potential short anti modular heuristic structure discuss generate approach iterative run show grey code histogram may essentially correct shrink neighbourhood estimate conclude neighbourhood least model composite shape row shape capability capture simultaneously shape spatial manually corrupt accordingly model appearance model gaussians experiment growth variant estimation describe fig upper row object part part shape capture time segment correctly number edge relation grow bottom necessary relation neighbourhood neighbourhood reflect adjacency learn potential occurrence vector responsible potential red express characteristic potential potential mainly anti correlation position part composite shape encode experiment demonstrate possibility segmentation conclude appearance learn fully unsupervised shape combine model statistic eq bit detail shape fig obviously potential generate easy shape detail us shape background label joint set shape vector fig informally correspond joint shape statistic extend component latter gibbs statistic joint correspond shape shape type appearance part final object composite share shape often understand object property nature follow modelling shape expressive segment shape aspect simultaneously shape understand shape recognize whether simple part desire spatial part explicitly important issue useful particularly valuable regard structure education project support grant author education new distribution two corresponding difference potential q function fairly general modular unary function second step claim non edge respective coincide vertex equation eq hold tuple sum unary consequently omit type value vertex coincide eq vertex eq subset span whole space analyse shape modelling expressive already shape simple part motivation goal characteristic major visual colour pathway cognitive way human simpler coherent spatial part principle formulation assumption concept convexity stage visual feedback layer lead whether composite shape learn aim question mathematically principle vision divide two shape
violate close approximation insight simulation plot dot smooth introduce concavity covariance implement concave classification estimation correct incorrect breast cancer aid diagnosis future potential breast instance defer theoretical concave study concave overview enforce study hyperplane assume eq piecewise affine tuple distinct think plane attractive derive cite introduction sample base moderate appear datum typically good idea log come corollary concave likelihood estimator moment however modify variate smoothing automatically concave log concave analytic agree nx dx nx dx aim smoothed concave preliminary compute convolution transformation region integration set furth u distinct use argument small similar onto unit simplex integrate simplex integral propose apply note one variable see consider integrate simplex concave estimator straightforwardly quadrature variation package method close zero way slow case inversion complex fine grid transform convolution multivariate draw observation draw return probability sufficient existence unique semi continuous density fact certainly kullback concave density play behaviour smoothed suppose dp x nx impose theorem ensure slightly version close concave concave consistent exponentially variation sublinear despite turn quantify bind moreover new insight enhance understand behaviour estimator result show concave preserve component concave writing smooth concavity definite statement concavity develop density large upper log instance concave log projection density mixture integrate smoothed concave likelihood replication log kernel concave offer substantial analogue particularly outperform considerable estimator violate smoothed concave however fact smoothed log concave remain situation concavity violate cause misspecification variability concavity another simulate none multivariate motivated procedure pt density distribution follow compute test justify concave draw instead run small first simulated fx setup proportion reject report critical omit critical trace critical replicate setting bivariate unimodal log correspond present table ht c confirm trace error appear concavity compare bandwidth note critical bandwidth due bias estimator quite outlier also region somewhat issue well density support change slightly previous section independent identically pair conditional random measurable assign q classifier risk define small analogue coincide classifier concave density classifier give class concave smoothed analogue classifier reveal bayes encounter parametric classifier quadratic discriminant parametric modelling concavity x log somewhat practical analogue outside hull datum positive avoid consider study recommend curse dimensionality circumstance dimension remain applicable part breast cancer fine breast aim aid capture variability figure curse dimensionality facilitate plot ht density class smoothed treat serious may seek incorporate class notion generalise incur assigning continue observe modification require smoothed loss package demonstrate boundary vary purpose illustration plot course considerably set interested simplicity exposition return situation accord log concave usually apply sampling outline describe denote norm q probability whose approximation strong continuity functional measurable associate two anonymous grateful support fellowship early theorem empty follow notation dominate independent let total probability let convolution surely yield log concave generality semi generality seek semi concave hand uniquely semi concave quantity x uniquely semi continuous density side write independence structure give dx ty follow denote generality may suffice f px xu moment density must integrate part calculus continuous upper continuous log concave concave semi density metric
occur sentence title etc structural svms predict structure structured input map submodular scoring structural example input document summary typically document annotate multiple manual summary denote summary optimize eq call determine I structural formulate ensure score objective learn trading qp sentence polynomial via cut plane step document worst solve quadratic step require care appropriately loss intermediate slightly target normalize train structural maximize manual summary easily different loss tie structural prediction test empirically conduct contain document four manual set article use determine good report use single recommend performance organize group construct stop sentence contain word far refine threshold weight form variant pairwise cosine similarity word word create coverage coverage consider sentence e instance instance look cover word sentence combine feature word tune baseline use weight cosine follow result evaluate score work apply function evaluation predict summary supervise approach hand summarize pairwise result increase manually pairwise result performance test conjecture dataset note coverage hand report model pairwise feature strength greedy largely reflect model pairwise model perform coverage coverage limited flexibility move document adaptation manual evaluate effectively make give amount already example increase improve rough estimate scenario manual summary inter subject disagreement hold summary reasonable restrict summary use select drop drop may summary look report drop finally come drop overfitte fidelity promise l human affect strength final score remove basic remove result role I basic one improve rich essential feature sentence make intuitive usually group none location send stop feature actually score later reliable clear whether use manual manual arbitrarily four summary label otherwise experimental subsection document slightly show figure similarly drop conjecture manual effort summary expensive component help multiple benefit svms scoring pairwise coverage base solve plane empirical evaluation svm conventional ability provide building fidelity member nlp feedback nsf computer usa ny edu score structured optimize measure learn apply demonstrate effectiveness coverage art enable fidelity beyond automatic short text describe summary news article paper summary hard sentence art lin inter sentence hand score reward summary coverage hand summary sentence optimize however address select inter sentence similarity leave select trade redundancy manual overcome problem propose similarity lin apply model due function designing develop learn parameterized scoring document summary maximum unlike approach learn optimize method dependency learn submodular tune particular tune hundred fidelity tune counterpart show investigation source identify approach start known approach selection consider redundancy extend support incorporate pack centrality graph system identification use sentence sentence extraction scoring pair graph include inter document sentence diversity diversity summarize set document integer encodes compression removed allow part preserve structure explore general model document document sentence sentence maximize give score subject character restrict submodular iff say submodular section return reflect redundancy submodular q similarity include summary sentence sentence scalar amount summary exclude repetition simple cosine similarity show document retrieval apply document notion coverage
optimisation decompose separate mostly provide collection put restriction follow lead consider optimal distribution mh mh mm mh q kl variational optimisation need directly family conjugate em restriction optimal achieve optimisation weight proportional result approximation posterior conditional estimate sampling equal posterior weight vb although come classification hide chain emission normal emission whereas collection mixture datum see al semi density approximate hmm latent variable hmm hide binary aim estimator wang mixture model wang algorithm hmm emission article author variational rate obtain likelihood belong family state markov involve despite modify framework decompose stationary parameter belong update hyper parameter collection distribution vb pe average mostly estimation eq variational estimator focus average classical hmm throughout paper describe probit within configuration homogeneous simulation mean alternative consistent exponential conjugate proportion performance consider reference provide vb pe variation quantify pe population well separate simulation pe estimation approach mix pe among tend average similarity vb method clearly appear vb estimation pe become vb simulation estimation get close estimator become easier low hmm vb include misclassification misclassification calculate bold correspond smallest pe similar rate rate averaging average seem table show vb result equal base optimal low misclassifie rate vb provide good hmm averaging bring estimation weight entropy vb pe seem select take account oracle term hmm highlight similar vb provide mse pe misclassification closeness misclassification condition furthermore average vb compare dramatically focus real dataset collect health surveillance al fdr false compose show group group moreover describe hmm aim population want interest influence within varie c cc ccc ccc cccc ccccc ccccc every transition article due keep four model low weight influence classification component propose al differ correspond approach vb display estimation comment approach estimation great comment averaging refine probability area consider difficult estimating probability mainly binary within variational approach selection theoretically prove average mse modification log require datum show estimator mse highlight approach classical moreover weight close problem high significant carry refine epidemic remark corollary proof sketch paris france france cr cr cp cr cp france want retrieve precisely know distribution develop fit sense estimate less relevant information estimation name aggregated collection provide approximation dependent study public health surveillance system average bayes unsupervise classification want retrieve wide want equivalently pure noise situation observation precisely retrieve unknown label associate label label far know perspective model consider finite likely information bayesian averaging develop provide improve et demonstrate provide gain term fit determination weight ingredient reasoning stand calculation schwarz laplace integral size monte provide joint
jump reconstruct slight fig cut short mask spike mask cutting estimate interact side cutting trajectory inverse branch two branch associate respective distribution interact variance large pointwise global jumps value reconstruction reconstruction reconstruct add jump therefore short even centroid mode mode strictly case range traditional method map close miss successful missing possible miss application density conditional present rise dynamic search exceed miss finally seem nonsmooth mapping correct mapping nonsmooth define g forward end robot region end constant length transformation joint angle analytically general trajectory formation sophisticated derivative forward map continuity configuration robot angle position angle toy uniformly mapping normal spherical see train mlp unit space result isotropic mlp toy mean regression continuous sample point obtain trajectory apply obtain sequence toy perform often close bind high occur hard pt pt trajectory trajectory dot space end choose branch trajectory jump report method alone allow incorporate length path planning operate stage candidate whole p sec p continuity constraint normalise random term np continuity period attack directly reconstruction logarithm variable n n standard fitness weight elastic sophisticated prior miss take mode restrict operate right reconstruction section think unimodal ambiguity reconstruction maxima many maxima mode n certainly difficult perhaps optimisation anneal helpful another candidate reconstruction mode weight mode towards pointwise reconstruction expense tradeoff reconstruct datum separate construct pointwise reconstruction find short joint offline reconstruct many programming complexity always mode finding confirm crucial amount miss affect mode accelerate mode discard speedup increase mixture potential mode variation shape large mode spurious smoothly spurious happen rejection spurious search wrong mapping mapping spurious mode spurious give prevent spurious mode locally component width globally smoothing cost mixture centroid loss density log low mode remove relate gaussian represent method likely effort reduce nonsmooth note mixture consider avoid mode mode make gaussian mixture framework long outlier attempt density aspect posteriori application trajectory respect approach differ markovian kalman filter speed trajectory play make experimental trajectory reconstruction depend trajectory sensitive slow speech space time series n useful constraint would lead besides reconstruct appropriate sense several branch around branch branch multimodal value take reconstruct concentrate possible select would decompose mixture kp model gaussians attain mode probably average missing operate suboptimal miss want compute acoustic viterbi feature miss example thing pc miss variable unknown give reconstruct contain delta superiority context recognition reconstruct classify speech frame benefit wide reconstruct whole segment classify reconstruct n continuous variable noise latent latent expect correspond sequence first reduce pick representative map onto multimodal besides dimensionality observe equality conditionally convenient continuity observe countable happen variable geometrically joint value span approach nonlinear system variability inside manifold embed observe take care need conditioning mixture provide working mode thus act manifold appear exponentially mode variable miss save possibility computational centroid principle lie area space fine centroid problem reconstruct long receive reconstruct speech reconstruct pass operation unbounded horizon greedy unbounded problem recommend programming datum stream request reconstructed risk get stream though long reconstruction effectively subgraph whenever dynamic computer point detect experimental recovery field generalise multidimensional experimental elastic net high curse multidimensional algorithm exponentially feasible dimension efficient multidimensional use continuity sequence experimental intrinsic pose challenge modelling difficulty dynamic search cause reconstruction robust local past aspect approach learn use mode reconstruction definition geometric aware attempt reconstruction full perhaps set error test model inference population come inference small interested inference e variable miss imputation imputation multiple instead single repeat imputation carlo require ignore dependence take constrain result candidate reconstruction pattern multiple imputation method reconstruction section difference imputation whole imply avoid exponential dynamic multiple imputation average branch mapping method branch identification mean conditional joint use random sample centroid high well mode miss predictor mapping use invert train net implement must descent provide minima gets find return mapping square reconstruction drawback need mapping combination separate mapping grow result branch review extension approximate forward mapping describe miss solve approximation mapping member attain inversion task partition subset mapping try set restricted branch separate branch restrict global costly run inversion mapping disadvantage getting cluster difficult difficult neighbourhood branch without priori branch detect geometric manifold curvature sensitive wrong region computationally costly density represent branch implicitly determine topology branch mode mixture combine net multinomial logit model expert process expert expert network expert large mapping forward inverse robot result map conditional solution nonconvex convert inverse satisfie first forward result unchanged result portion network description whose train branch multivariate method map kind constraint particular never trajectory contain branch incorrectly reconstruct inverse mapping contain jump branch similar feedforward net mlp depend recurrent net feedback loop unit unit delay n attractive modelling net high feedforward net learn point reconstruction drawback feedforward net reliably unit apply autoregressive datum greedy expect suboptimal universal mixture spherical variance etc e centroid high mode rest conditional ignore value pointwise propose method joint curse dimensionality function treat asymmetric miss reconstruct mapping construct codebook codebook candidate onto return codebook vector codebook reconstruction codebook programming continuity forward programming option configuration produce acoustic many huge high codebook time construct codebook difficult among reason codebook manifold space codebook search really several reconstruct finite though codebook limit require neighbourhood crucially preserve reconstruct missing exploit temporal smoothly propose miss secondly reconstruct obtain candidate give plausible reconstruction actually plausible one constraint analogy miss pattern treat unlike treat predictive distribution deal mapping branch always branch average branch consider inversion high simply input general extremely constant miss reasonably contain spurious suboptimal reconstruct inversion mode forward distribution definition insensitive arc length geometric invariant also reconstruct mode offline pointwise distribution pointwise reconstruction ex c greedy net apply unknown situation relationship mapping inversion problem arm inverse estimation pose movement video inversion decode activity miss reconstruct speech acoustic perhaps grouping principle scene multimodal speech want read vice versa type feature pointwise mapping variation datum applicable case construct make mode every independent exist I original fine joint though use replacement mapping efficient grateful helpful discussion cm college road email reconstruct multidimensional contain independent involve take give vary redundancy redundancy live one continuity constrain reconstruction mode present obtain programming toy reconstruction mapping inversion programming reconstruct component speech corrupt sound instant band corrupt consider problem band whole give net usually mapping mathematical determine give one invert forward map robot arm uniquely determine angle vice versa mapping occur traditional miss compatible flexible generating mapping subset via however impossible break ambiguity prior choose result kind redundancy component pointwise redundancy vector rest paper explain explain mapping reconstruction section experimental version appear conference publication part reconstruct examine definition give rise consecutive point sample point observe necessarily collection redundancy mobile wind field wind colour several problem vector live want reconstruct verify walk trajectory region allow take trajectory live dimensionality look relationship since use sequential unless note generalise write sequential case mean variable scalar measure otherwise say whose reason multiple measure lose depend rather uncertainty recognition speech stick miss classified though associate set act mask complete problem approximation mask varie miss reconstruction mask missing describe pattern missing datum completely pattern miss mechanism take account missing may miss reconstruction provide reconstruction information terminology seek reconstruction set single pointwise reconstruction measure reconstruction vector component index pt convenient missing miss set reconstruction base functional relationship idea define functional relationship conditional pick representative multimodal mode reconstruction pdf contain fill perhaps mode several peak functional relationship mass concentrated construct mapping particularly general probability low spread say principle bl joint area dot density manifold mapping multimodal obtain define probabilistic extension manifold dimensional curve surface concentrate near purpose distribution discuss issue measure conditional informative uninformative require unimodal single depend domain differ median mapping usually via see section mean multimodal lie area bad may support since even unimodal pointed mean common representative respect point ease variable locally probability general calculate know computing analytically find bar locally though absence information keep mode likely representative pick sample manifold effectively attractive mode much sense unless conditional return corrupted may fall area mass body reconstruction serious set certainly miss correct wrong affect global via constraint consistently bad generic missing model observe relation variable appropriate p estimate long estimator density offline training even pointwise criterion greedy tendency point jump reconstruction offline gaussian mixture density time sequence n miss gaussian reconstruction candidate reconstruction easy traditional method map one gx xt gx see forward mapping sometimes reconstruct square reconstruct five mask missing miss miss regression apply mask complete miss sample curve additive train b perceptron mlp layer square reconstruction descent mapping baseline basis separation function interval see implement reconstruction pointwise mode mode likely pointwise euclidean course lower achievable tell reconstruction conditional pointwise reconstruction pointwise mode continuity base unweighted conditional unimodal intend skewed unimodal pointwise reconstruction follow slightly branch branch chance contribute without appearance compare coincide unimodal density training trajectory density manifold lb lb lb mask mask mode bc pt bc pt bc bc pt bc cm cm r view colour panel panel follow text detail dot manifold dash contour joint fa mode circle fa fall reconstruction noiseless original several mask trajectory trajectory mask e reconstruction fail centre corner occur point trajectory gaussian nonsmooth box panel several red line circle widely reconstruction noiseless nonsmooth panel mask confirm reconstruct far full gaussian mixture mlp report aspect draw conclusion bad mask forward practically true mask mask mapping inverse branch branch symmetry branch happen branch symmetry inverse mask predict theory
curve simulation outperform pair positive definite entry quantile aa mn equality remainder sample implement assume comment section optimality choice valid dimension case provide intuition competition separate large difficult discriminate desirable reducing tend bring close hard fact measure term kullback divergence consequently seek dimension preserve pass transform distribution dimension project classical project statistic k draw haar manifold k invertible probability consideration property projection k eliminate variability projection refine average quantity remainder consider p p k diagonal indeed confirm theoretical quantify relationship integer p data random nominal asymptotically order p kp possess nevertheless concern k g qr distribute algebraic section generate number depend well precision eliminate average fluctuation illustrate simulation case roc statistic average characterize statistic theorem comparison relative past work section asymptotic sequence index size mean implicitly vary although derivation restrict condition hypothesis assume ratio tend divergence alternative shift satisfy asymptotic power result let denote order pair define exact twice kullback project determine satisfie correspond blind advantage blind kl divergence discrepancy du rp function form entry denote alternative similar tend arbitrary compare function van term inside define ratio term explicitly asymptotic compete test advantage theorem covariance formula interpret gain insight rp natural interpretable shift power likewise natural extreme adversarial orientation direction haar sphere du equal compare encode idea shift direction direction easily follow factor emphasize power relative usually context asymptotic lead clear proposition theorem comparison state scale power appendix maximum minimum quickly eigenvalue shift spherical limit hold kp ok include variety sparsity entry support pattern capture shift shift scale pt demonstrate dimension proposition indicate scale fluctuation formally lead maximized provide condition optimal class along limit optimality limit straightforward numerical roc several refer case statistic compute projection construct slowly decay spectrum eigenvector haar induce shift vector accordance proposition curve fluctuation sphere similar proposition five setting correspond matrix draw uniform sphere choice equip chen theorem end fix inequality p serve reference asymptotic asymptotically interpret pt structure long grow orient test pt measuring indicate rapid decay project map low subspace chance variation great instance contain mass interpretation eigenvalue contain total mass calculation satisfied grow effective another decay occur connection fourier p dx pt computation integral easily hold decay rate associate lead competition hold must theorem direct proposition recall k k k event tend event tend replace turn sd recall shift pt limit k pt pt condition unlike play role sd test frobenius large datum large small uncorrelated let prevent small effect think statistic take correlated datum enhance relative suppose block structure block diagonal entry dr imply q consequently conclude copy dominant assumption theorem example illustrate involve powerful test bs correlation generate eigenvector haar orthogonal impose sufficient condition non dimensional e pt low grow fast confirm gaussian matrix eq involve norm oppose frobenius norm condition interest pt hold write r z consequently combine conclude inequality replacing compare broad compete simulate theorem roc normal roc curve datum dimension roc curve reflect choice choice c decay block shift shift theorem draw draw describe haar orthogonal group see whereas select spaced slow control rescale fixing amount variance plot spectra figure last use block along diagonal interpret negligible greater slow rp implement bs sake completeness procedure mmd kernel label statistic curve rp test procedure rp independent power gain slow diagonal covariance fast qualitative compete compare panel panel roc unchanged correlate similarly advantage diagonal panel agreement remark slow spectral versus panel compete insensitive spectrum take remark theorem quantitative assessment curve determine c c decay decay pt pt pt r involve obtain average theorem sd pt average figure set average increase remark second n projection novel implement addition derive interpretable correlation compete specifically well interesting regime furthermore realistic case condition comparison work dimensional discussion suggest case fellowship de er cancer partially support grant dms proposition proceed state main lemma concern p kp lipschitz gaussian pt eq substitute pt obtain second use tail namely eq eigenvalue k pt p variational cyclic letting eigenvalue jensen spectral follow ki diagonal column square equal kp pt guarantee ensure bp pt pt pt gives let precede equal almost eigenvalue zero semi matrix p ok imply sufficient write lemma limit prove limit converge expand notice realization kp kp kp kp nan gaussian strategy conditionally pt pt hold turn later next pt precisely eq follow theorem deal pt need kp k simple definition orthonormal invertible upper positive thin qr factorization white wishart claim substitute verify limit cyclic trace jensen inequality extract exchange since may note paper check uniformly variance mean bound integrable formula jensen twice yield norm reasoning work irrelevant piece tend uniformly ensure k uniformly integrable pt imply eq q adjusting event k c k ce piece tend upper piece replace frobenius schwarz inequality k pt
impose analogous computer image discretize image call picture mean array number observe alone detection object colour colour colour pixel within value assume object image pixel call pixel always noisy transmission medical example image scan always body observe noisy image optical begin pixel noise special noise normally normal mean know noise symmetric image rectangular behind present various analogous possible related role mostly analysis need plane black pixel interior point like interior pixel colour boundary type noise explicit method complicated situation pixel colour different formulate formally denote therefore accordance stress dependence assumption even general variance adjustment quantity proceed quantitative estimate black pixel correspond omit notation identically colour original standardized black colour ready describe main thresholding colour colour pixel grey grey procedure colour observe grey add reasonable pixel colour white pixel black end call analyse real follow small hold obvious principle formally first transform picture pixel call colour white vertex edge vertex side connect connect point black picture collection lattice call graph efficiently picture graph lose consider presence get gray make sense overcome definition goal probability site get picture black formation cluster estimate size shape split vertex correspond pixel original background subgraph denote observe black detail form cluster little black difference method derive efficient randomized happen relation issue observe black white cluster outside moderate value make look make colour several way suitable maximizer use white picture decision possibility test h nh object maximal probability picture fact terminology working algorithm picture image object detection problem example even make completely side pixel eq obvious relax replace shaped figure two asymptotic character valid nevertheless remark asymptotic consequence assume resolution infinity mm pixel therefore could detect fine formulate false run thresholded find report detect step find pixel satisfying connect rigorous us work assumption linear exceed nc implicitly mean comment think work place detection devoted crucial convenience predefine operation see analysis chapter operation save pixel cluster position complete analysis show false q white pixel theorem constant event black lie mark black site theorem book explanation terminology measurable large obviously subset lattice side increase measurable decrease ci ci p denote coefficient nn ci k assume part true site lattice rectangle vertex leave side number right rectangle slight lattice pp picture satisfie thresholde picture observe site less take certain happen
customer make history customer response write customer restaurant need collect prior pr l belief ni beside customer decision effect customer customer use l recursively ir n customer response customer induction two make decision restaurant part paper restaurant game research area four response strategy utility dynamic spectrum cloud deal application formulate chinese restaurant response system simulation well random customer table probability signal customer purely signal regardless customer reveal table give strategy extension learn system reveal customer strategy belief customer maximize customer decision signal reveal previous grouping except bayesian subsequent customer evaluated treat rational behavior traditional dynamic access identify available spectrum sense potential enhance efficiency available sense share member within collect access individually spectrum increase detect user primary user access secondary user access network decision secondary estimate primary secondary cognitive secondary user access access primary active user slot primary transmission secondary activity slot secondary individually sequentially channel going slot loss make decision secondary access user user policy slot transmission interference transmission cognitive model chinese restaurant hypothesis primary user detect activity activity detect addition sense probability updating belief sense point access channel slot slot primary secondary choice choose secondary user channel secondary choose high follow response recursive simulate cognitive channel primary access user slot secondary primary three beginning slot primary channel false alarm channel channel channel within fig response secondary factor user secondary channel loading channel equilibrium loading secondary make secondary secondary secondary secondary difference channel successfully identify channel offer utility secondary expect number secondary secondary six make decision utility secondary early secondary secondary user decision subsequent secondary eventually benefit make early user case mistake make average utility secondary secondary scheme average secondary agent action user large scheme take decision secondary make decision later likely primary likely choose channel look secondary hand good response high secondary effect secondary access channel channel hand scheme since tend available channel spectrum phenomenon consensus primary make finally interference fig involve learn scheme primary signal efficiently channel primary application service cloud reliability availability major concern cloud storage two enhanced platform software hardware reliability availability affect transmission capacity reliability platform may decrease software hardware storage platform service potential platform growth platform let cloud storage say maximum pricing long term platform since due service availability binary respectively platform platform reliable probability party platform service however prior platform platform form platform receive public discussion know assume decision short compare service ignore availability decision service end platform receive platform otherwise platform platform respectively platform vice belief rule function platform probability service platform service reliability let choose platform response recursive simulate storage type reliable low reliable offer failure per one first decision simulation show utility decrease decision take negative effect utility need predict decision platform reach number reliability low platform effect dominate reliability platform advantage choose dominate platform platform platform case advantage platform reliable collect utility decision however difference scheme load balance platform reliable provide high serve reliable platform exactly different scheme fig see high balancing reliability platform platform reliability platform platform high signal scheme reliability reliability well load balance reliability platform without loss platform network social show business offer especially platform significant discount deal deal limited purpose deal network facebook twitter observe successfully likely receive response service product restaurant restaurant serve huge customer restaurant exist customer possibility service quality deal mean customer deal customer contrary restaurant customer visit restaurant review internet binary model high uncorrelated customer deal customer along public customer customer accord review customer positive restaurant represent quality customer collect belief restaurant review customer restaurant customer customer discount degradation customer function significantly simplify due linearity number customer derive belief customer customer customer customer response customer recursively customer response customer compute j deal customer deal offer restaurant restaurant crowd factor restaurant equal quality restaurant conditioning restaurant customer receive positive restaurant restaurant restaurant receive restaurant customer receive choose deal reveal customer sequentially response four result customer random scheme customer customer quality high major determine advantage collect review deal utility contrary restaurant determine utility case early customer choose restaurant customer customer expect subsequent customer customer customer become indistinguishable scheme customer also reflect phenomenon social deal eventually reduce decision consider network high learn customer make different since separate crowd prevent severe quality degradation however well network social finally study restaurant price deal restaurant game new restaurant try enter put deal discount customer give review restaurant become customer control signal control restaurant depend signal restaurant game restaurant deal website deal restaurant restaurant restaurant already customer restaurant unknown probability restaurant conditioning restaurant customer receive restaurant restaurant customer receive review restaurant new restaurant deal restaurant simulation restaurant simulation expect see customer always deal price regardless signal restaurant increase customer restaurant high customer decrease
significantly train row flat represent nn train flat flat node perhaps important running analysis processor complete time require nn nn call give result average time appear one train likelihood likelihood evaluate equivalent decrease ccccc ccccc equivalent nn speed prior fast analysis know knowledge nn prediction value variance hessian weight prediction calculate validation initial predict uncertainty calculate error great nn continues nn nn confident make confident enough set justified way bar train use old quickly multiple average bar nn product future demonstrate rapid blind multimodal inference combine artificial network reduce significantly run computationally toy nn surface produce accurately log non flat model case nn sufficient nested predict able speed comparison mcmc f ph ph constrain hamiltonian carlo gr ph w ph ph r p j ph ph r p gr I quantify user manual entropy white h neural ph ph ph theory machine von nest york laboratory jj cb uk institute road cb algorithm combine nest artificial network blind multimodal bayesian nest learn function expensive rapid approximation order begin ability complicated surface ability probe observation valuable increase addition obtain train function combination problem expensive particle task traditionally exploration tune accurate additionally efficiency affect multimodal selection need accurately chain multiply expense evidence peak fail multimodal degenerate situation nest method design provide allow implementation nest multimodal particle physics evaluation million able call ideally task approximation precisely nn give widely technique function variant conjugate gradient optimum hessian towards blind accelerate multimodal inference combine use train function predict likelihood original make accurate place original future sample second user obtain network train evaluation near nested find good weight show statistical estimating hypothesis write posterior dimensionality ignore estimation inference un normalize achieve monte hasting hamiltonian publish operation begin live likelihood live remove replace new high removal replacement live continue difficult find discard point algorithm go volume small inefficient problem new decrease live surface contour contour constrain distribution able multimodal separation calculation evidence substantial physics equation run output node activation smooth purpose linearity training mapping original additionally percentile training calculate mostly original require network accurate multimodal degenerate surface toy peak must able likelihood problem narrow region lastly long degeneracy require reproduce nn problem require simple analytic expense dimension must able region peak structure run live recover method value analytically agree analytically compare return two lowest remove identical reduce log nn gain would thin circular section magnitude prior dimension live obtain manner return nearly identical nn end explore use degeneracy dimension present function log dimension figure prior perform sample live node live hide return analytically value analytical compare posterior dimensional distribution identical posterior nn decrease computationally expensive cc solid outer contour toy example usefulness surface critical parameter standard code paper performance require evaluation temperature value code second compare computationally limit factor likelihood function full benefit particularly physic peak around location set parameter range table parameter range flat analysis alone galaxy df survey ia datum resolution website live training likelihood
kt signal case preserve process sensor exploit exist straightforward refer result svd svd operate block finally svd cast source power spectrum source power modification estimate expectation modification base method location index peak k detail inversion update h nonzero correspond location source entry correspond location set zero truncate hereafter kn zhang direction arrival development reconstruction advantage conventional one practical situation estimation study modeling develop base joint exploit assume prior snapshot snapshot simulation maintain high grid sensor array research decade focus far wave front assume angle music conventional estimation prove realization large uncorrelated source advanced year development reconstruction compress cs candidate assume unknown grid formulate constitute recover vector snapshot guarantee recovery prove recover isometry property rip column highly incoherent measurement sparse share support exploit average incoherent learning method cs formulate perspective exploit laplace posteriori signal possible recover g inference heuristic extent offer accuracy compare array include compressive combination outperform music cs though exist show still practical grid necessary gap near point estimate constrain dense coherent signal standard coarse notation conjugate transpose vector trace entry denote element take real part estimate organize grid introduce conclude far delay represent shift observation q tt ty sensor reader refer discussion unknown mapping relationship kn approximation measurement write validate approximation error noise measurement gaussian closely observation approximation small modeling error adopt grid grid higher dominant grid adopt considerably comparable find formulate bayesian perspective develop complex value simply jointly white denote precision pdf circular symmetric assume conjugate prior broad among share two row
expansion subgraph first fuse fuse eigenvector graph eigenvector subgraph indicate value eigenvector amplitude slow subgraph half show plot fuse become eigenvector graph impact spectral time vertex slow belong subgraph relatively fast term subgraph vertex subgraph value expansion decay slowly large belong subgraph slow come heart section close slow fast realization fast mean vertex fast valid nevertheless ratio similar within fused see fast graph separately slow fast subgraph fuse subgraph axis subgraph fuse result confirm embed concentrate vertex much experimentally average patch graph study graph clearly transition fuse slow slow slow slow subgraph slow graph fast order magnitude transition display slow separately component fuse b scale lastly transition slow dynamic slow confirm embed fused vertex subgraph effectively divide subgraph away subgraph preserve demonstrate identify subgraph fuse implication anomalous rapid away baseline concentration anomalous patch happen patch fast embed segmentation become patch local extract requirement exhibit geometry fuse section synthetic prescribe autocorrelation low frequency yield signal smoothness argue type change class quantify time fast portion show four four compare numerical regularity autocorrelation autocorrelation control local partition autocorrelation keep create regularity homogeneous poisson intensity adjust autocorrelation frequency increase figure signal appendix give decrease regularity figure covariance parameter time describe brownian motion specific maximally patch time regularity compute embed eigenvector frequency principal show signal display extract high segment slow blue extract section regularity signal patch smooth visual confirm color frequency embed code match plot signal patch patch patch leave quantify noticed patch ratio end therefore study lipschitz patch patch slow mutual short eigenvector design gradient measure indeed eigenvector therefore minimize quantifie restrict see whether fast however belong fast patch compute square either slow study ten realization slow regularity shape patch remain compute choose show smoothness exhibit rapid local associate fast concentrate parametrization analysis true patch realistic signal realistic probabilistic argument walks provide explanation analyze patch base reveal presence contain rapid change another leave slowly change along low dimensional interested local scale become live curse informative large patch continuous patch nontrivial graph patch weight distance requirement intuitively reasonable patch represent discretization nonlinear manifold close another distance conversely poor approximation geodesic information available us patch trust observe contain walk fast patch jump avoid walk distance large choice random large avoid connection choose mutual project sphere allow local force irrespective distance consider vary adaptively patch self weight matrix consist patch patch local behavior involve indicate optical also exploit construct dictionary representation g embed slow fuse dataset exhibit concept clique correspond subgraphs texture synthesis super reference patch patch near analyze patch use provide evidence provide justification experimental embed organization natural patch extract patch contain intensity work provide explanation success patch denoise author filter interpret evolution diffusion patch duration rely properly toward backward eigenfunction one perturb patch assume image piece experiment anomaly frequency content change furthermore et derive time use energy existence high patch consequently low finally patch argument add slow interpreted function support manifold potential narrow agree reach present investigation present work use reference therein area physical neuron fire instead process slow fused rely aware decrease reason rely loose upper provide geodesic tight could number path length able spectral choose ultimately vertex q last inequality approach bind slow compute pair vertex bind standard bound nash inequality nash electrical adapt introduce concept disjoint separate path connect edge weight loop walk two disjoint slow attention edge show leave removal edge prevent connect diagonal green entry walk move connect connected edge generic edge entry figure go height therefore l sum sum put everything nash summarize slow observe graph divide side simplify green self show rely relationship electrical random walk graph begin assign connection consider potential difference result electrical flow equivalent effective connect necessary maintain unit current express side choose obtain weight express eq connect distinct rewrite binomial effective geodesic distance scale expression os graph utilize geodesic distance yield euler simplification give electrical network node parallel add edge decrease autocorrelation nonnegative fourier equality apply binomial frequency fast frequency identically unit check autocorrelation linearity mean together problem understand success base analyze art technique denoise work explanation metric model geometry patch graph parametrization graph time mutual patch correspond patch correspond expand concentrate rapid local change would otherwise patch numerical diffusion patch address success patch metric patch signal organize connect patch similar reasonably patch measure distance patch edge short geodesic associate walk analyze art denoise consequently geometry geodesic two vertex geometry analyze study walk metric derive efficiently patch manner reveal notice base concentrate rapid rapid contain furthermore contain spread metric main contribution explanation model geometry patch patch smoothly patch exhibit anomaly rapid change parametrization graph vertex relative change effect result explain parametrization patch eigenfunction concentrate patch would patch phenomenon exploit classification large analysis indicate develop intuition patch study several example patch allow parametrization embed experiment finish discussion without sequence patch need extra notion patch contiguous extract collect patch patch patch organization organization presence patch delay specifically theorem allow replace dynamical system equivalent form observable organization throughout think several originally series think also keep mind understand patch set connect patch patch patch graph vertex connect along vertex sensitive change measure detect change smoothness scaling define edge weight rapidly topology appropriate two location explain analyze patch h h become patch region patch edge patch along weighted weight diagonal reader intuition geometry associate help motivate provide sketch plan signal image visualize patch patch onto large variance patch analysis display size around image quantify local patch color magnitude local red blue encodes temporal proximity arrival wave proximity baseline arrival illustrate detect patch maximally third horizontal collect patch image image local variance patch patch distance patch correspond intensity vary varie little concentrated surface visual mutual patch explain remain overlap close coordinate vary slowly curve argument rapid change neighboring argument allow understand characterize accord patch appropriate normalize transform rotation patch region blue contrary content change middle try organization patch represent activity expect rich content slowly baseline mutual distance compose part expect sufficiently long generally patch extract distance another associate display indicate close gain organization patch patch note diagonal close dark near left slowly vary end diagonal variation column exhibit rapid local center tend lack prominent diagonal entry concentrate column correspond center intensity smaller extract apart indicate relate preserve indicate patch concentrate slow patch blue green align along surface display embed patch map patch low part concentrate region figure define patch time measure goal explain patch embed figure characteristic observe compose patch subgraph fast patch extract confirm analysis applicable model graph without generality patch compose patch affect conclusion interpret proximity proximity patch corner weight edge patch diagonal slow fully connect self connection require distinct regular last regular since signal periodic comprise patch demonstrate size throughout away weighted graph model os self connection weight possibly less one edge green w n slow appear low right fast right patch exhibit region fuse size subgraph probability weight ensure connect validity parametrization edge allow patch combine create subgraphs subgraph connect patch construct adjust edge connection subgraph vertex fuse connect distinct binomial vertex pn n vertex equal vertex realization graph vertex vertex understand fuse subgraph slow tractable fused complement fast slow provide subgraph fuse confirm study see understand within subgraph study subgraph would appear straightforward compute reason low time sufficient rapidly even rely connection electrical electrical circuit vertex connection circuit circuit key state moment rough slow fast quick version slow self self time compute path add decrease nevertheless walk distance lead conjecture average average regard graph analyze clique vertex
prove differentially query release packing extend wide mechanism date query error machine perform database draw particular result view draw predicate sufficient proportion half main order preserve construct approximate respect yet distributional interactive handle theorem computational release domain vc dimension algorithm query usefulness release privacy create answer alternative definition relationship database tuple database string length hypercube tuple endow copy mechanism arbitrary synthetic concept notion definition differ symmetric access differentially neighboring database section alternate strong privacy simplicity privacy proof notion difference neighboring database release evaluate fraction counting query neighbor analogously predicate restrict count paper linear query vc predicate vc predicates query predicate collection one counting predicate vc exist cardinality abuse write collection predicate et al give mechanism sensitivity differential privacy laplace query return variable draw mechanism mechanism answer query level query answer composition useful tell differentially private et differentially mechanism string mechanism access shoot interactive public arbitrarily seek preserve privacy imply query seek release database usefulness output satisfie derive net class net cardinality net general release mechanism map quality score state fix user prefer mechanism select proportional mechanism privacy large implement super preserve exponential net mechanism privacy differentially private net usefulness necessarily net mechanism useful quality exist definition union output n gs c prove condition therefore differentially query net query question mechanism mechanism count upper net count prove crucially finite query eq q sample database mx standard chernoff database satisfy complete exist prove class count straightforwardly analogue counting strictly strong utility mechanism useful sufficient set concept preserve privacy view draw extra usefulness guarantee result discretize efficient remain section give mechanism respect count tight namely count differentially private fix counting query predicate denote universe vc database answer simple class line many follow preserve database contain value would able percentile point usefulness answer impossible privacy answering class generalization answer percentile fall percentile real privacy database contain element contain value answer must point must mechanism query class generalize high axis align differential privacy database preserve differential privacy construct answer impossible differential binary since around definition domain section another relax definition interactive mechanism query mechanism mechanism non mechanism second margin private database note without utility query denote margin introduce fact projection norm dimensional purpose preserve projection pairwise integer matrix entry apply theorem ax ax ax ax ax ax ax ax x identical ax complete proof synthetic collection structure random mapping query write evaluation u h ji project projection collection net choose approximate answer guarantee shift answer projection bind need discretize uniformly vector j privacy composition mechanism differentially call database except dd polynomial let margin quantity structure corollary iy iy u apply take plug choose recall distribution assigning draw except therefore condition union plug conditioning privacy universe say draw database example treat patient disease guarantee privacy informative actually motivation definition particularly perspective sample draw reveal inherent whereas whether strong differential provide single guarantee two database hand distributional privacy make element exponentially probability probability database draw nevertheless possess distributional privacy strong database neighboring satisfie privacy mechanism satisfy distributional meaningful database satisfy differential privacy draw mechanism preserve distributional privacy preserve privacy differential impossible preserve privacy note imply differential distributional draw distributional query behave draw preserve distributional preserve differential privacy neighboring database theoretic existence accurate differentially private mechanism query query database linear dimension class discrete interval main left algorithm utility comparable net database question query thank david theorem definition database query size net represent particular counting query guarantee grow query query simple generalization preserve time slight relaxation guarantee release capable represent answer query dimension notion collection privacy increasingly might useful sensitive correlation cancer collection medical financial reason sensitive release dataset quantify privacy mechanism interactive interactive differential privacy removal element amount intend information private reveal cancer almost nothing information motivated preserve mechanism conjunction count say class answer change building et show discretized admit net release release count computationally efficiently interval axis dimension guarantee range discretized definition impossible differentially private even quite simple natural usefulness release information answer approximately correctly approximately correct relaxation notion margin theory hyperplane fraction concept mechanism reveal database database change probability distributional privacy satisfie accurately satisfy database element distributional privacy formalize separate privacy issue outside database learn database notion privacy database connection compatibility formalize substantially linear call adversary original private mechanism fraction remain privacy counting answer query preserve differential scale linearly mechanism et consider learnable model learnable polynomially sized interactive differential privacy query perturb laplace sublinear number query answer counting grow class allow analyst query linearly learn private database ignore pac learnable pac learnable build
easy solve difficult derivative follow partial derivative eq get equation solve arise yes element zero matrix definite definite force product univariate normal possible center corresponding since model simply acceptance jump tune address obtain model effort require mixing figure show jump frequently compare implementation within almost identical minor attribute jump posterior model probability jump sound mapping reversible apply selection problem particularly plot ratio year year exposure ratio number loss unit exposure collective collective exposure value normal denote time adopt process precision inverse evidence introduction loss confirm behaviour sort ratio negative could theory way restrict ratio within adopt serious failure ability detect provide useful check equation use gibbs posterior parameter distribution implement full distribution gamma ne q shape scale variance q full use dependent turn conditional difficulty general metropolis rejection stationarity flat interval trace small mass marginal reveal union disjoint contain interval identically conditional density sample whether reduce effective integrate nuisance difference implementation first largely integrate parameter result reduce posterior make posterior show conditional density likewise depend conditional form integrate density conditional gamma q conditional readily scheme variate current implement updating proposal interval fine proposal difference improvement table mean interval variance important diagnostic tool mcmc autocorrelation figure autocorrelation autocorrelation great except lag big lag significant autocorrelation significant lag plot trace implementation integrate could indicate indicate proposal small current chance b integrate b plausible alternative paper examine detail discriminate adequate algorithm notably simulate iteration compete vector jointly simulation construct include reversible jump diffusion birth remainder jump specify reversible countable index great probability estimate modelling include reversible reversible state space problem eq move move achieve deterministic matching ensure match discussion function acceptance probability q respective jacobian acceptance chain reversible balance sufficient existence distribution reverse move transformation use uv acceptance probability move regard run reach move augment model mixing improve increase address reject attempt proposal generate new previously proposal mix reversible propose extend extend give motivating reason term reverse move govern result easy density unknown use uv centering center expand uv proposal new identity acceptance probability become convergence assessment chain parallel diagnostic include chi square kolmogorov hasting model within context particular reversible jump algorithm simulation order propose square kolmogorov kolmogorov compute critical value kolmogorov significance reversible jump selection additional distribution exactly simple eq distribution remain posterior conditional necessary describe correspond simplify respectively simplify posterior distribution equal probability propose ease addition reversible parameter keep remove density ensure dimension matching density simulate reverse move move jacobian u density parameter theoretically choose scale poor acceptance move try result posterior marginal mean posterior jump posterior estimate jump candidate reverse simulate q change move description similar introduce density marginal reverse acceptance move w acceptance move transformation notice probability well discrimination posterior like place posterior instance place increase algorithm probability since offer table great
computer propose decision take account learn duration decrease fusion scheme compare propose decision fusion suitable problem drift track change non describe decision scientific technical grant european grant fp network project multi network risk extreme weather department university email university education city image interpretation foreground background segregation functional adaptive framework vision compound sub yield decision real around zero decision combine fusion projection set describe sub algorithm human feedback fusion video detection oracle security present test active call propose compound yield final decision take real number level sub perform projection projection onto describe sub projection computer decision classifier combine useful difficult especially procedure projection convex collective recognition middle last decade lead large classifier achieve improve employ individually classifier recognition characterize simultaneous individually effective base face pixel texture significantly integrate texture window technique evaluate give texture window article automatic video five video object colored wavelet iv range camera separately together adaptive fusion forest video projection hyperplane monitor case security whenever false whenever occur security decision incorrect update decision security security support tool help attention typical security minute monitor feedback interval run without feedback organize entropy decision section previous part propose entropy update sub section security five detection experimental fusion universal weighted algorithms method uci repository nn training decision compound compose sub algorithm input decision center detect otherwise event type sample region vision pixel incoming vector value algorithm step simplicity rest define input step advantage propose feedback produce incorrect correct iteratively specific subsection scheme ideally decision sub eq q dimensional hyperplane convex next determined project weight hyperplane orthogonal hyperplane close hyperplane formulate minimization solution multiplier next weight hyperplane define plug obtain hyperplane new decision project hyperplane iterate intersection hyperplane rate introduce guarantee set hyperplane track oracle support establish point successful sequentially h yx iw ix x reconstruction compressive sensing everywhere approximate minimization bregman algorithm widely bregman provide globally problem framework cost represent convex bregman start successive projection perform hyperplane step iterative generalize projection onto convex cost problem hyperplane tx yx projection equivalent point e orthogonal norm projection hyperplane functional hyperplane lagrange hyperplane equation hyperplane globally convergent update hyperplane notice orthogonal equation reconstruction positive code fusion error yx ix yx snapshot typical capture camera km fig like segment gray pixel indexing curve instantaneous histogram decision deviation energy currently available decision propose camera different spatio nearby far nearby describe accordance weak intelligence ai ai ai address engineering area move ii colored wavelet region smoothness pixel incoming select combine determined produce false number around camera output sub confident first four describe add review sub deal color discriminative calculate pixel location pixel horizontal use filter horizontal vertical region vector covariance region formula total component covariance half first region predict actual label train actual confusion matrix image software library use posterior library sigmoid large descriptor contain compare result point final pixel threshold issue alarm alarm oracle give decision alarm variation area temporal region weight evolve manner operate mode scan constant adjust classification reduce scheme intel cpu ghz processor test surveillance capture forest weather stable throughout entire happen successfully detect forest also independently california false snapshot present forest ht projection linear decision linearly update assume decision govern minimize cumulative loss normalization indicate n scheme approach compare table hour surveillance video video range km capture good knowledge stage european project mention method forest detection comparable hand fusion reduce alarm feedback describe fig alarm move cause background appear algorithm ht depict bound false alarm video frame km frame rate universal base weight v c video error duration universal weights v v data video contain cloud cause especially frame except low pixel classification scheme compare video weight stage gradually frame number software forest propose method
satisfied design finding solution system compress lasso various monotonicity position denote use equivalent stand notational simplicity review sufficient x differential ty optimality preliminary remark simultaneously c conversely deduce belong p hand index argument imply index sign complete monotonicity important function use condition begin lasso introduce vector definition x subtract together desire submatrix ty establish always uniqueness general position resp resp equivalently remainder proof rely linear programming ensure extreme solution completely extreme basis unique way couple determine x immediate deduce therefore assume uniqueness part give equation uniqueness either assumption equivalent exists decrease since infimum concave singleton thus moreover last assumption surely non divide increase boundedness notice ty norm continuity boundedness two sequence converge problem b uniqueness prove goal fidelity compare penalty vice intuitive tend tend decrease tx immediate consequence interval lemma divide small let sequence converge recall l write union subspace dimension deduce small thus x imply tx moreover hence tx tx interval continuity result partition sign pattern nonempty interval multiply ty tx calculus step increase nothing prove result continuity deduce tx tx continuous moreover tx tx non decrease france penalization crucially fidelity provide general proof well know basic generic denote assume briefly scalar denote notation submatrix whose large discover sufficiently estimator due absolute stem least estimator remain nonzero predictor interested overview relationship concern extension strategy instance uncorrelated several oracle may oracle often emphasize support ahead
cm cm home page http ar intelligence information national increasingly important network widely formalism handle technology sound whenever representative work survey state discuss limitation produce quality efficiency realistic domain store world support refer literature popular reasoning uncertainty statistical calculate model represent assign add abstract domain tuple limitation requirement variable domain mathematical nonetheless restrict variable second usually conditional variable representation semantic pattern proposition represent distribution people influence may figure table row assume mutually n mutually table parameter address exponential storage requirement several simply formalism compactly joint compose numerical structure encode present numerical defines quantify detail graphical encoding distribution network distribution dependency influential last decade graphical wide field present image give address complete exposition field domain area datum random field present spatial random wide biology economic diagnosis heart disease propose discover gene application graphical describe individual fitness recently propose retrieval dependency markov contextual dependency propagation list summarize example reader solution certain cm computer vision example cm object cm cm method biology spatial economic environmental disease biology discover cm search network term dependency modeling dependency support capability highlight graph provide conditional efficient computationally tractable compact graphical exploiting understand semantic give graphical highly task marginal usually marginalization compute conditional posterior prediction sub routine learning task elementary sub exact working structure magnitude fast joint extensive topic popular work open approximate recently return graphical model really knowledge author tool use drive provide automatically large claim approach purely review many core challenge technology independence infer independence representative target art discuss current limitation potential current document discuss relative advantage conclude open domain overview network consist qualitative quantitative denote qualitative know domain distribution quantitative quantify relationship distribution two knowing tell I nothing already know independence dependence undirected independence distinct node random domain encode represent first regular spatial typically spin model mathematical mechanic read conditionally variable conditionally conditionally give correctly represent graph short disjoint conversely map connect map empty map characterization relation separation graph concept basically graph graph distribution say graph necessary condition axiom axiom network omit l p eq disjoint stand axiom valid strictly list axiom represent relationship hold summary represent encode well inference graph undirecte represent acyclic direct case encode introduce predict application concept denote domain influence identical graph strictly markov variable state every axiom strictly axiom section quantify relationship encode address learn quantitative network distribution subgraph whose call maximal clique among edge clique clique clique clique graph assign negative potential measure compatibility configuration assign clique show include clique gibbs normalize product combination system eq compute use general second calculating discuss difficulty historical condition analyze format machine file per assignment solve know computationally challenging technique markov parameter perfect encoding every learn represent desirable densely exact inference intractable method limitation reason less main goal learn underlie evaluate distribution use perform never improve answer fraction segmentation image feature discovery structure motivation correlation discovery assess prediction extent identify medical diagnosis discover certain disease markov usually tune propose require estimation computation combination although possible guarantee optimum result estimation simple ascent sophisticated unfortunately partition across network propose alternative variant another robustness belief propagation avoid scoring need good running stage validation broad learning network approach generally suit inference complete structure mind independence suited goal discovery independence feature since algorithms domain purely tool section structure work complete one base perform accurately high space markov learn search start atomic potential create candidate currently model potential model second potential model compose candidate assume remain parameter candidate evaluate much log end perform search candidate structure automatically induce log however report variation highly optima couple induction force select potential reason involve clique long potential evaluate potential approach markov match nearest drop generalized improve score incorporate loop end slow space intractable evaluate necessary require numerical explain constraint discover semantic constraint input underlie goal encode query variable assertion dependence assertion example independence practice information pearson bayesian test test compute statistical triplet decide instance actually assume true rejected reject statistically significant elegant several independence call strategy generalization previous outline theoretically sound step ix piece together construct learn al bayesian local learn global variable member exist publish algorithm pc independence appear work network series appear ks incremental association parent mb mb parent algorithm child mb important conclusion review cm sound specify mb advance gs sound first phase grow variant cm sound variant gs implement poor variant well mb cm sound trial enhance efficiency sound add candidate speed poor mb sound make topology much compare slow cm sound theory efficient topology information distinguish distinguish mb algorithm distinguish distinguish parent child algorithms learn markov algorithm publish markov grow inference independence appear network particle algorithm framework improve independence algorithm detail independence contrary sometimes complexity sound statistical correct correctly represent correct outcome reliable third algorithm learn sufficiently sampling underlie incorrect good contingency table example must cell count less another disadvantage learned approximation publish literature independence review independence appear review grow independence structure network algorithm adaptation learn show algorithm gs outline h gs maintain initialize empty line contain markov variable line gs association unconditional proceed two dependent conditioning contain false positive positive conditioning prove theoretically correct independence statistical reliable efficient structure disadvantage use cascade incorrect outcome generate incorrect test markov bayesian incremental association prove gs modification sort positive grow reduce phase condition conditioning exploit union axiom disadvantage prove quality gs literature propose network evaluate grow work gs inference theorem reduce learn structure expensive introduce triangle axiom sound infer far eqs every triangle rule triangle rule test independence triangle check independence assertion infer infer store determine visit inference run obtain comparable particle filter structure independence review improve work iteratively select major bayesian select test correctness test distribution converge domain independence case domain respect comparable algorithm triangle unnecessary test outline show reduce dynamically select state knowledge select inference help test require evaluate markov subroutine comparable section independence focus structure statistical reliable deal reliability independence error incorrect test axiom direct axiom depend target power axiom knowledge incorrect propose resolve reason robust call markov network evaluation simple statistical accuracy discovery gs disadvantage formalism first proposition
store cycle constitute active allow another cycle weight update begin iii discuss sampling depend pathway direction region extend sampling initialize phase less flow indicate obtain unfold reconstruct unfold pathway interestingly reaction pathway pathway occur break around occur around pathway pathway pressure path trial path suggest probable close two pathway locate place first encounter pathway even unfold pathway region unfold pathway equal pathway total unfold ensemble histogram unfold ensemble peak peak state histogram ensemble peak histogram pathway unfold ensemble path respectively unfold ensemble e flow clear pathway contact entry unfold pathway construct contact difference subtract contact entry list ensemble contact entry unfold contact map figure characteristic structure figure divide group b along associated agree order unfold rate counter path increase unfold flow flow gradient surface loop apart unfold decrease flow trend comparable average time go amount time also make trajectory list compare flow ns relatively flow increase rate field illustrate enhanced unfold trajectory start unfolding reach usual increase probability observe straightforward trajectory ghz intel processor simulation processor parallelization processor employ trajectory unfold unfold event grain far unfold rate mechanism significant flow effect flow suggest contribute moderately state lack rna prevent unfold transition extremely occur slowly dynamic unfold pathway one secondary break break p loop one parameter sampling pathway precise competition pathway pathway flow currently parallelization increasingly space grow acknowledgment thank award sciences engineering acknowledge resource fusion compute laboratory center laboratory support office contract ac unfold reveal freedom system molecular phase separate solve enhance divide phase integrate region method facilitate parallelization trajectory aside initialization termination enhance suitable drive coarse grain nucleotide rotation dynamic unfold long dynamic unfold range rate pathway biological force induce unfold complement follow evolution distance force optical experimental probe degree molecular dynamic simulation position assumption prove tool elementary condition representative computationally costly employ extreme alternatively computational effort priori prevent enhance sampling without rely relatively uniform different region space degree freedom acceleration convergence trajectory portion space relatively develop paper version improve significant association weight copy motivated segment integrate independently make strategy processor simulate unfold grain nucleotide rna flow single study loop rna enable rapid lead question dynamic compete pathway depend flow compare reversible unfold simulation straightforward coarse grain enhanced need study end overall describe phase simulation parallelization ideally parameter degree relax relatively quantify unfold backward degree freedom separate unfold pathway enable simulation order parameter region uniform region evolve natural average copy region configuration list neighbor choose choose list partition active copy active copy continuous incorporation motivated partition branch ensure set lattice situation study know configuration apply flow configuration principle simulation configuration unnecessary correct sampling start run record configuration serve entry visit configuration unconstraine region employ copy describe visit activate region activate neighbor trajectory activate result directly study account differ require thus region parameter token boundary forward distinction importance dynamic lead rapidly principle similar permit control include detail branch prune weight slow weight large region convergence issue fact space number region procedure accelerate initialization phase interface interface scale weight scheme region total element value nontrivial solution simulation copy require limited communication parallelization implementation high computer parallel array implement global address access side sided communication address storage set entry node region store enable atomic update modification instance prevent global array communication initialize collective rate break cycle end cycle region segment choose trajectory step region per trajectory region list mean computational end rna simulation tractable coarse grain atomic interaction nucleotide rna treat potential potential potential connect pairwise potential namely nm ensure prevent integrate velocity fs construct full coordinate isolate full carry coordinate mass structure coordinate add relax without force add form loop contact model rotation dynamic method particle group cubic cell comprise stream particle update velocity particle random likelihood combination water calculate equation move flow show allow include particle make ml rna particle use periodic boundary cell cubic lattice employ generalized cell extra locate prevent move along cell boundary condition periodic sphere flow direction acceleration cause extension rna direction flow particle box rotation acceleration parameter range fs figure flow ratio
information mathematically validate model insufficient general conclusion stage opt handle exploratory bayes evaluate carlo evaluation selection measure comparison stress repeatedly validity testing setting operate follows produce distribution abc call summary distance tolerance indicator justification tolerance necessarily insufficient sufficient dimension loss information pay computable quantity provide identifiable handle abc arbitrary specific posterior probability tool marginal eq valid naturally choice proceed create parameter model denote straightforward challenge available abc represent choice mutation justification behind abc acceptance proportional probability practice posterior straightforward derive widely e illustration model analyse north initially subsequent area abc genetic belief decade process categorical comparing involve logistic approach widely another illustration popularity availability abc abc smc abc biology include propose smc implementation mc population molecular mc g incorporate principle correspond summary statistic concatenation elimination pseudo model follow motivate base software fundamental posterior say representation q simulated simplification without generality uniform f converge namely versus current abc contain use abc odds bayes method convergent regularity condition bayesian inference insufficient impact indeed contain sufficient namely statistic gibbs field detail sufficient immediate mc approximation discrepancy limit inference therefore abc wrong express answer general arbitrarily grow base example si abc may differ derive bayes possibility theoretical model currently costly must standard numerical gibbs detailed case differ inference bayes derive approximation ratio setting necessary bayes factor choice absence stress validity model choice abc instead abc frequencie inference select suitable complexity likelihood abc mc comment apply software whole validation markov extend rely observation contradiction computation gibbs competition abc converge field allow vector exact field statistic property exponential set sufficient statistic concatenation statistic sufficient compare optimistic complex beyond family happen realistic normal si insufficient loss happen wise bring light huge abc factor formal understanding discrepancy discrepancy impossible expect illustration wise statistic come geometric counter gibbs across give poisson geometric negative binomial geometric bayes show produce nothing produce factor log versus binomial negative binomial replication discrepancy fact increase selection iid sufficient create sufficient mathematically provide choice outside formal structured devise mechanism across detect rather specific abc introduce apart since increase biological study include choice insight study specie g module population importance sampling order discrepancy abc base scenario two content problematic enough order evaluate divergence abc experiment population third scenario first second pseudo individual assume evolve mutation occur increase mutation effective population assume choose particle scenario provide posterior computation reference million simulate provide simulated close allow summary subset reduce collection yes average population allele allele var yes allele population population average sample sample population allele population allele population yes allele dm yes distance dm yes population two two third population simply pseudo experiment individual mutation contrast analysis include high scenario abc analysis experiment evaluate abc decision boundary discrepancy demonstrate opposite conclusion experiment fig first genetic experiment summary experiment validity sampling obviously display across different proper per see si distinction abc provide model conclude evaluate carlo genetic experiment evaluate independent carlo evaluation simulated dataset population two scenario sampling abc extensively involve realistic exception respect sufficient conclusion analyse bayes statistic sufficient situation insufficient rapidly increase worth estimation need produce approximation stage
fill outcome six six player must manner cause player real world performance advantage implement transition ease twice home run basis entry probability home run block every player possible base event represent c represent event play represent th th transition zero calculate calculate game current single calculation represent already therefore whose run column maintain cause run distribution calculate eq subscript nine nine row run end occur apparent heavily datum outcome stop player goal compute team establish need player instead use scoring run use nine production nine index similar ensure distinguish scoring markov becoming score triple scoring somewhat offset matrix getting walk double triple home run markov refer score score way ability optimal write difference omit computer permutation produce team near criterion team high high scoring fourth scoring position second position position low scoring position score scoring four low scoring four seven ten possible show comparison team whether yield run consequently method require near regular determine yield test datum index nine team team article affect significance abundance promise evaluate difficult output consider different team want maximize affect situation play surface play away home eight common situation player average opposite ground ball ball home versus play versus count count ball refer count score first half star player whether play home away home performance home th contingency denote home player home bernoulli success accord binomial respectively transform q contingency express player representation logit represent attempt player situation flat nuisance nuisance effect priori use distribution mean degree reflect lack effect assign construct home variable representative upon home away simulate primarily datum spread give situation shift ten day reveal behind count ahead high opposite point eight home result pattern player individual nine extreme year play helpful exist mention availability visit measure evaluate depend question take attempt exploit whereas bayesian evaluation phenomenon heavily sample statistical limit easy spatial movement decision ball fix reach base shot infer player shoot complete six game shoot chart shoot center shot shot player shot purpose covariate certain apparent reader article lastly success shot location convert angle shoot al shot attempt game explore affect could shot attempt shot attempt column effect select variance gamma proportional ig analyze confidence median time early develop relationship angle shot location shot location divide shot shot often drive net directly shoot location take four shot cell share effect shoot multinomial shot assume multinomial ap jj regression logit coefficient car normally neighbor towards neighbor share realistic establish cell adjacent cell angle smoothing quantifie type normally car mean angle respectively angle distance angle computation stable lie distinguished distance angle sparsity beneficial issue amount location successful shot attempt cell represent behavior covariate cell change team strategy team point production due player player covariate percentage regression smoothed shot draw distribution shot article methodology spatial player shoot et effect shoot selection drawback entirely thereby method affect smoothing grid insufficient cell angle smooth analyze accurately aspect separate poor jensen player ability current metric rating evaluate play player state difficult accurately player ability area insight illustrate roughly player front behind evaluate metric jensen et success fit idea et al playing relationship distance angle home name spatial fit two model ball model direction velocity incorporate plane depend velocity direction measure angle dimensional arc position reason formulation intuitive position almost base base model fit respective remain position rf lf aforementioned fit author fit year separately result total article previous instead location shoot ball responsible fails similarly shoot previous mathematical formulation model play play success outcome success model model distance indicator direction move forward move cumulative equation control directly backward recognize probit regression permit direction angle bernoulli change location velocity move leave right still quantify previously share player position issue draw specific share player position contain player component posterior lastly player wish find contain player position separately unknown sampling sharing across cell share focus shot shot sharing cell et representative rely share extensive safe safe safe seek evaluate number play observation player respective yield surprising standardized constant add player maintain consistency noise safe position second base safe average position safe correlation surprising safe elegant intuitive another pose safe model adjust safe lf ss lie ss positions b adjustment account ss eliminate ability case article statistical equip article cover perform paragraph safe year computationally infeasible another beneficial artificial intelligence artificial intelligence numerous algorithm analysis review explore study intelligence see modelling use programming medium infeasible reason computationally due discretization article seek represent also sampler single infinitely team maximize team team goal number reach either team terminate new fail play team begin achieve receive team achieve number achieve team random depend diverse play maker option attempt field attempt reward minimize net mention state mind goal abuse triple vice lose team state reward chain minus score team start position yielded ball arbitrarily obtain varied attempt make wide recommend pass attempt large enough fourth diverse team goal play goal goal attempt recommend produce agreement employ programming show optimal plan play opponent advantageous policy policy heuristic good reward point even policy result simplify version combine technique intelligence hold policy realistic play different goal player situation model promise representative require validity aside previous article computer possess ability greatly statistical possess elegant elegant property amount computation analysis situation effect base situation precise quantify chain th transition p correspond upon mathematically k x dimension refer period state give chain important paper review difficult sampler generate act generate sample enough say converge random enough say deal general reward maximize state immediate state start long discount state looking often refer bellman achieve bellman determine system value start guess define successive ki intuitively speak function approach way start evaluate policy converge long evaluation article discount provide quick frequently count four base appearance end team score team team consist least appearance pt algorithm people come fan serve fan detail eliminate various idea chain player environmental could dynamic programming control paradigm theory idea theory winner team opponent opponent contribution non scoring team team category player refined reflect retrieval player ball team shoot team retrieve ball player playing seek player make heavily team assess result difficult assess team situation investigate impact team outcome loss team could
gap vanish obeys generated obeys imply entry r ii also condition make give quantity w ready thank geometric hold result bound explore performance match disagreement independently observe run optimization successfully equal underlie cluster vary repeat experiment time trial rescale curve align show indicate one predict control curve fashion run follow figure note parameter choose scale align figure quadratic tradeoff I p compare popular adjacency zero first adjacency generate fashion probability imputation scheme particular scheme consider zero b symmetric reduction graph show guarantee find optimal disagreement clustering succeed miss exist effectiveness method validate application expensive consider deal connection lemma non inequality give auxiliary operator entry operator p tc high indicator ij bernstein provide symmetric appear define symmetric n large constant random symmetric relate entry provide fix random satisfie obey bernstein obtain yield norm bound equal otherwise row u u sx inequality assumption right hand u j complete argument size equal size size approximation hold partial unobserved denote p piece chen xu chen xu observe know edge remain edge want node relatively dense observed yet focus cluster edge cluster use optimization disagreement minimization recover sparse partially performance plant block cluster characterize tradeoff density equal observation undirecte unweighted disjoint connection mean pair know field across engineering search query phrase yahoo represent similar background find well document database focus application facebook accept user pair thus obtain e regime interested mathematical potentially performance relatively provable clustering observe graph additional input require section different pair total type note specify formulation problem bad compare section result aim combinatorial disagreement splitting section represent matrix corresponding cluster sparse correspond return cluster cluster natural strong matrix presence field course vast survey therein scope correlation cluster plant stochastic block partial observation mathematically graph lack define correlation np work relaxation relaxation rounding emphasize focus yield without round enforce triangle inequality relaxation tight constraint program however argue tighter practically since deal graph practically small high complexity cluster plant block assume inter fully recover often directly comparable area provide table cluster density cluster difference needs exact plant soft notation recover except handle partial directly miss probability apply side require condition wise thus restrictive observation show recover nice consider adjacency entry similarity show similarity hierarchical seem structure split cluster tree active control observation know convention pair find define analyze observation study derive algorithm either b never suboptimal present describe already ideally cluster edge clique second augment sdp graph interested produce correspond ideal cluster validity output rank optimal clique column order validity check elementary identical block insight yield disagreement solution valid cluster adjacency graph initial empirical fraction pair develop tailor splitting take observation solution contribution provide find minimize among entry characterize plant describe plant partition observation partition let rank adjacency otherwise fraction across k cluster support recover show condition universal cluster q disagreement sufficient gap observation parameter side condition impose decrease require mention cf demand weak vanishing tradeoff density side mean four time gap consequently treat weak agree handle miss entry easier whose unknown would point capability handle isolated node connect classify identity low row correspond appear theorem consider establish fundamental density gap observation correctly cluster plant partial clustering correctly complexity characterize logarithmic significantly improve use complicated picture know rigorous recovery impossible requirement gap probably rigorous tradeoff unclear regime theorem appendix produce e node minimize feasible valid semidefinite obey disagreement zero hence subject edge relax feasible minimization clustering
induce usually apply co use available proper subset way initially largely machine literature split flexible lead co size hence divide sized also call slice unless state refer random split special construct self classifier co randomly pick partition alternative split co training meanwhile classification section optimal partition heuristic test future accord mit essential ingredient classifier label closely strength coupling classifier determine preserve connection list supplement gaussian mixture indicate ratio separation original quantity preserve q carry substantial fraction contain u indicate quantity feature separation closely relate rate know result loss column write subset see suppose wish affected general classification power vanish I assumption I technical later proof let cholesky triangular matrix possess locally dependent may avoid assume assume eigenvalue permutation assumption induce indicate supplement mainly work closely state gaussian separation mixture center original difference carry nontrivial transformation separation euclidean original separation mahalanobis comparable empirical tree randomly half construct formal justification half provide support aspect comparable classifier meanwhile set tree good result co training conventional reduction redundancy feature become nuisance get clearly redundancy redundancy coupling co carry splitting feature type redundancy assess stanford microarray see http stanford edu er protein cancer er marker express assign scoring represent definite dark positive cancer cell unbalanced assess proportion split first various option image set section relationship train co experiment additional experiment default reflect image er marker occur accord moreover feature derive co derive experiment range far expect substantial distinguish subtle gray ensemble node value suggest package fed rf set report rf find apply score blind set image two evaluate self necessarily grind truth serve test set argue example represent neighbor rate refer theoretically good training rf around twice bayes around sample variation bayes original close pt image patch image conduct patch achieve indicate report score salient detect salient highlight highlight verify highlight pixel oppose identifying highlight score two copy along score reference accuracy evaluate proportion score see assess proportion repeat score consensus issue obtain image avoid range course one desirable automate rf svm rf alternative network set boost boost svm naive adopt idea maximize posterior ix seek detail rf make conduct co correspond combine split combined number example fix design make class carry reduce without interesting training rf consistent sample partition lc rf marker algorithm marker surface additional stanford image marker cd large miss exclude experiment automate refine recent er cell present manner include count statistic spatial regularity heterogeneous image salient score incorporation patch cell type achievable set pixel scoring patch insight co slice whole classification power coupling co train natural lie population seek evaluate potential marker may score poorly manual score hundred hour score marker manual provide type appear subsequent statistical determine clinical potential marker regard score two pilot reveal observer accuracy define reference image inter observer attribute include lack lack surely upon highlight inherent provide clarity confidence marker localization er marker characterize cell ease marker software request associate make project acknowledgment thank anonymous constructive suggestion proof theorem separation skip skip grant gm microarray technology medium throughput diagnostic study grow manual limitation throughput greatly variability expense co occurrence quantifying phenotype base regularity summarize inter pixel relationship train tune salient pixel contribute score via training marker training train high redundancy flexible scoring evaluate clarity study er marker performance describe throughput technology assessment group image display indicate score begin distance rectangular core contain core block micro array capture display grid fashion separate probe sample rapid dna rna protein expression large number clinical remain common method localization expression quantification scoring validation assessment target clinical panel image construction automate large limit throughput intensive intensity density provide quantification problem inconsistent highlight thus grow rigorous actually without consistent automate scoring sophisticated focus commonly system rapid device newly scoring rely various several reaction detect typically marker nuclear dedicated person propose co pattern texture pattern rely color filter segmentation base expert patch categorization interpretability note design diagnosis tool clinical thousand require score prohibitive automate essential purpose concern necessarily cost time adopt substantially explore natural split readily available fairly support theory slice carry condition organization remainder section follow insight address lack easily quantify cf template retain idea image require common scoring selection expert biological feature involve substantial gain interpretability machine knowledge select benefit manual difficulty setting beyond b svm sigmoid report lc rf svm boost use underlie classifier classifier dimensional rf performance argue rf tree instability create bag node randomly feature index measure bag proportion class rf pruning example fall tree vote class vote rank index alternative permutation base valuable property salient see pixel rf variable since entry count associate pixel pixel associate entry position pixel treat pixel salient important g rf pixel rf salient quick work manner salient pixel
approximate family switch fractional help level sensible evaluation likelihood integration require gauss quadrature computational moment function evaluation mode cavity modal case mode integration limit around hyperparameter monitoring require marginal numerically fraction first cavity remain update care entry become due outlier present suitable option cholesky update laplace parallel site decomposition definite site negative cholesky ep step evaluate uncertainty outlier increase heavily hyperparameter handle regular map initialize hyperparameter small result outlier unable explain induce plausible outlier mode case flexible enough divergence fractional update discuss property increase posterior uncertainty input region priori estimate insight negative laplace I I I regular observation increase far clearly increase uncertainty situation arise hyperparameter many covariance matrix positive posterior ep behave disagreement cavity observation modal moment posterior decrease sequential run smoothly remain site multimodal standard outlier provide narrow small apart significant posterior uncertainty input neighborhood smoothness govern equally much neither label contrary site correspond posterior despite outlier update become problem sufficiently example force unimodal converge case hyperparameter panel increase panel hyperparameter separately calculate analytically draw gaussian unimodal ep estimate cover posterior laplace one nonlinearity observation close strong nonlinearity much strong near approximation localize standard ep mode example modal short py py panel correspond fractional update site discuss illustrate dimensional site update ep start dimensional marginal contour fit around near site get negative expand toward outli site long update cavity precision happen support current posterior reduce ep cavity site first posterior center couple loop negative cavity variance problematic site get small need expand cavity variance keep large uncertainty small site hand induce may algorithm update decrease require sequential convergence fraction include correspond fractional site initialization still modal less distribution consecutive fractional marginal variance site large regular ep small keep divergence measure allow localize fractional ep hyperparameter unimodal panel small uncertainty put objective illustrate hyperparameter marginal ep mark panel site keep parameter value iteration negative cavity become also decrease nearby fluctuation prior amount gradually still compare cavity double loop loop proceed loop ep slow iteration get parameter ep attain much switch drift visible site estimate parameter marginal may slightly inaccurate implicit gradient evaluation ep without apply localize see positive update show approximation ep fractional ep approximate fractional mae region degree contour log contour double loop converge hyperparameter sequential converge marked dot hyperparameter mark ep hyperparameter mark example problematic ep previously ambiguity area problematic large unable strongly region unimodal clear artificial hyperparameter sequential ep fail initialize evaluate lie area visual accurate contour standard uncertainty see fractional property match hyperparameter fractional except fractional ep run dot coordinate la vb four use involve create five irrelevant variable generate house price prediction concrete predict quality distribution fx method fold approximation mcmc predictive deviation laplace ep plotted numerically integrate statistic clear see indicate derive somewhat approximate density la deviation slightly together method hyperparameter depend optimize section mcmc measure mean predictive test variable datum fold validation compressive large bootstrap described ga ga ep vb optimize density em variational augment challenging issue la ep vb vb robust alternative outlier tail ep ep robustness estimation linearly log run optimization hyperparameter approximate integration require speed target variable scale zero mean freedom initialize magnitude optimization initialization direct hmc mixture sample sm combine slice result well give visual calculate autocorrelation burn period exclude begin remain form mcmc central credible observation ga reference student ep scale sm number method stand integration comparison sm credible show sm figure four illustrate figure densitie sm difference student sm sm compressive significantly difference concrete ep actually sm explanation wrong possibility proportion observation pairwise comparison reveal ep ep vb except strength compare perform la vb compressive vb prove challenge sensitive initialization hyperparameter often likely style la vb integration objective whereas la bad la compressive vb significantly well decrease vb compressive pairwise comparison integration significantly compressive strength never significantly give ep marginal line mae student ga concrete give perform hyperparameter optimization ep others compressive sm ep concrete otherwise difference la vb ga ep sm max c c require posterior hyperparameter prediction cpu give time fast compare baseline ga ep slow double iteration achieve difficult clearly ep repeat value slow compare adaptation la vb optimize guess hyperparameter average set exclude mcmc mcmc clearly mcmc randomly show much research ep accurate application problematic utilize difficult mixture modification fractional well loop good ep interesting give interpretation negative site increase posterior site turn problematic update even hyperparameter optimize example ep unless converge discard hyperparameter situation relate cause away unimodal note moderately ep work fine multimodal globally unimodal true underlie think approximation false uncertainty prefer also design approximation ep also cavity loo latent site loo site approximation cavity reason fractional student omit robust mixture gaussians moment analytically adjust contain gaussian tail adjust increase infer estimate turn consistent base modification improve robustness ep concave likelihood ep likelihood package consider regression observation student several ep find problematic contain site student illustrate standard fail algorithm occur type posteriori ep may robust implementation primarily ep update utilize matching loop compare laplace variational bayes chain propagation observation include outlier strongly failure absence robust require study extensively describe observation lead observation however rejection outlier posterior rest outlier becomes give tend state combine normal prior robustness ny negligible student heavily commonly influence matter far outli rejection reject locally nearby already multimodal illustrate adopt student gp heavy tailed distribution degree freedom robust model laplace student model analytically use see show method burden uncertainty assume function variational vb un low independently detail comparison relate describe student freedom divergence kl describe vb special kl extensive comparison performance laplace kl ep choice since kl ep establish student one discuss ep update ep loop primarily update utilize moment loop adaptively size stationary solution implementation implementation laplace approximation carlo three world zero q control notational simplicity variable function produce semi covariance magnitude correlation increase analytical approximate limitation robustness marginal p analytically method approximate review give short description well vb way problem draw represent posterior numerically implement markov student gibbs representation variance slice sampling hybrid sampler may reduce implicit model latent give negative diagonal py I inference hyperparameter laplace marginal hyperparameter gradient log scheme laplace essentially name gaussian integrated field thus later implementation posterior approximation ensure moment convergence share moment bound consequently stationary small free choose first concave maximization concave equivalently substitution constraint respect proper fourth student site moment match cavity keep positive cavity distribution regard loo ep simplify moment evaluation robustness family flexible due cavity f site moment standard recover ep cavity another divergence compare force mass consequence fractional ep tend minimize represent overall reverse kl divergence fractional fractional benefit problem approximate measure fractional update help cavity cavity decrease cavity make numerically cavity become combine
near sequence value part sense interest similar observe order start segment historical price asset pattern scale asset window pattern profile see look euclidean small increase kt construct appropriately ahead asset neighborhood pattern order q adjust represent term value inverse asset process expect worth character sense hyperparameter fit information indicate change provide non maximize log indicate event occur definition asset return occur way horizon contain scenario return price probability price satisfy q therefore evaluate estimate conditional respective value eq expect worse design informative take asset portfolio thus change return base price arithmetic give divide integral price asset quantile normalization dx evaluate get evaluate calculate index recommendation establish currently quantification condition examine daily year value index calculate window e year consider acceptable accord compare financial market difficult agent permit correctly fluctuation thus exchange increase worth would apart difficulty combine access volume computer software advantageous simplification view assume asset predictive center asset without necessity assumption evolution return beneficial ease include benefit evolve similar initial specify evaluate correspond though present correspond whose fall evaluation useful design aspect wish sim de measure financial two asset evolve piecewise process supervision result indicate satisfactory tendency assume series asset density point practice really
meta passive learning cm passive simplifying arrive extra constant vc learnable rate learnable directly claim aside disagreement dependent directly prove kind meta theorem simple possibility bound request meta unlabele disagreement budget formalize lemma note unbounded bf relevance insight arrive follow show generally constant without label appendix exist achieve cm interval improvement disagreement coefficient like improvement passive disagreement subsection least little strictly general subsection active meta analogous disagreement meta dramatically meta meta ms mx mt x k mx v ms mx mt l l variety way long converge fast true result definition meta however disagreement focus conditionally I feed break stage rather share stage chernoff depend complexity wish architecture sometimes complexity corollary replace return termination within value analogous label infer update unlike mechanism motivated contain arbitrarily classifier discussion label step potentially informative sense meta key obtaining improvement label guarantee overall build intuition behavior meta toy uniform ignore fact effectiveness threshold nonzero interval round meta essentially identical meta grow threshold width period I find meta must look round improvement sufficiently process label correctly request reach furthermore circumstance appendix l label interval single nonzero disagreement improve passive round meta provide improvement round reasoning sufficiently label separate consistent argument pf relevant go scenario study analyze number target difference subsection quantity disagreement improvement achievable meta analogous disagreement characterize disagreement core integer define h c far key role concept classifier q disagreement represent generalization disagreement coefficient measure measure f f k additionally target union implication label corollary asymptotic always might define restrict never discuss extension vc mention restriction expense complicated theorem disagreement coefficient describe variety behavior toy example interval see x bf bf consider see f union interval width j purpose h h interval label interval nothing interval nothing interval label element minimal contain bf z z ks j bf bf rr latter point union jk z z ji inspection reveal exact see coefficient particular course quantity able describe family beyond toy fortunately disagreement much euclidean linear c b learnable exponential establish similar argument px px c hx hx bf rx bf pc pz z surface li dt regularize satisfie indicate mass least mass xx bf rx bf r hx yx aforementioned spherical reveal height surface spherical every bf hx little reveal height mass r r bf bf p rr albeit follow reasoning use opposite example advantage indeed disagreement subsection characterize magnitude passive specifically immediate corollary proof vc achieve complexity satisfie passive distribution achieve complexity passive plugging simplify arrive vc c learnable learnable first follow meta actually closure obtain label direct cut return original algorithm obtains bind passive see possible separate classifier return meta let corollary significant improvement represent good gain achievable answer meta corollary strength modification might potentially improve possibility theorem careful analysis instance meta f tight see refined complexity disagreement additionally significant constant quite careful meta possibly exponential learnable countable vc element except instead infinite path one depth path countable slightly replace eq quantity term sometimes possibility improvement ms likewise replace v state meta remain require modification proof gain theoretical arbitrarily room arbitrary summing constant possibly sum furthermore prevent set function large improvement active capable passive access unlabele exception major issue inherent achieve unlabele practical concern unlabeled data efficiency disagreement base method meta introduce trade though trade factor definition indeed desire actual theorem appendix unlabele follow set might ps implicitly ps ms x appropriately modification dramatically example ps costly unlike serious meta achieve corollary consistent classifier returns classifier replace return become run invert clear guarantee case absence meta large obtain evaluation remain corollary previous learn move make interesting perspective significantly version passive complexity nontrivial conjecture regard need focus develop technique noise disagreement active generalize meta meta sake disagreement active subsection develop agnostic relate disagreement relate meta represent disagreement agnostic several result sense less elegant potentially promising direction subsection interesting direction subsection joint hx xy xy xy py xx xy xy context denote xy h xy h ph ph simply c g hx equivalence compact recall totally closed monotone close nonempty learn ix ix xy define request time sequentially request joint specifically definition label em may classifier rate particularly interested passive generalize analogously definition learning label em agnostic set passive meta agnostic active xy cc agnostic vc class agnostic even exist passive classifier nh n xy behave expect result include agnostic threshold expect case leave open family passive agnostic interesting agnostic reasonable excess family passive help value ignore redundant request discussion passive algorithm fact vc active meta conjecture vast grow literature complexity risk minimization empirical effective conjecture remain remainder view label passive subsection interested state active learn explicit initially purely nontrivial depend passive passive subsection commonly use property margin specifically formal p depth passive literature passive imply variety case satisfy p study depth wherein xy xy xy way space realize surface depend surface surface assume xy xy density surface quickly work interpretation study passive risk minimization return classifier mistake label satisfy achieve work nontrivial xy xy xy give infimum complexity learn passive refined active improvement satisfy condition disagreement active analyze generalize specific refined generalize label since entail hope purely nontrivial xy xy xy satisfy value range complexity achievable follow subsection establish disagreement agnostic present originally analyze label agnostic beyond disagreement disagreement analogous meta replace passive also vc namely know find offer general complexity achievable agnostic exhibit dependence coefficient role discussion vc unlike eliminate classifier merely mistake good take mistake underlie discuss present empirical excess learn literature disagreement slightly consider representative disagreement algorithm concern disagreement method label budget mm request generalization passive inspire application learn dependent follow rademacher independent operate label disagreement space mistake since infer establish implicitly round loop update step include somewhat behavior vc suppose condition parameter output least essentially simplify idea also immediately implication concern vc class dependent em result algebra achieve require procedure focus move beyond disagreement active theorem result art label complexity advance understanding capability active next improvement natural meta benefit label improvement passive whether meta agnostic theorem whether f modification analogous see often compare represent active toward algorithm input mx l v I k estimator usual rademacher principle meta proceed repeatedly label size significantly agree infer rate remain equal difference empirical passive literature bound excess eliminate situation subtle principle case space explicitly step eliminate neither classifier meta meta exist version space specifically label active exist active f extension perhaps generalize possibly multiclass structured label particularly problem scenario substantial improvement structure claim passive preprocessing clear instance specific bias semi splitting index interesting characterize number request unlabele active algorithm really adjust example aside many would try unlabeled example reflect label example behavior increase label interesting agnostic exhibit trade concern agnostic vc empirical risk algorithm agnostic positively capability active passive well factor classifier h hx bf repeatedly define l passive achieve meta achieve satisfy p bf n argument end bf chernoff n ng nf imply label choose term identically distribute algorithm pl probability vc p pf nontrivial break essential event f claim note return necessarily decrease define nonempty return minimal completeness ix fx meta know nontrivial since happen small theorem claim definition lemma proof relate meta slightly parameterize state vc behavior meta suffice passive theorem importantly relaxation subsequent namely lemma substantially general version eq convenience convention classifiers iff iff x p ks ks ks ks rate hx ss adopt short wherein everywhere almost variety way definition possibility though access unlabeled partition preprocesse ix statement hold state serve partition three eq indicator every indicator whether define definition eq remain q certainly definition interesting property general trade present instance could probability would possibly drawback alternative priori many expect k h bound might sequence address modification elegant definition follow primary proof set aside request label return end support regard serve core probability zero difference lemma primarily extend basic idea surrogate sub label request lemma subroutine request output least n em completeness total request sum size batch label px large jx x pm en union must h px jx one hoeffding en describe limit see probability establish decrease except zero lemma large except zero p fm hold fix infinite h ix bf rs rs follow ps mf mf rr dominate first event probability note establishe claim event take claim monotonicity law imply event monotonicity extend event give implie note specific implication take bf latter bf eq complete final bf k f bf h h right left equal examine denominator numerator fact side let let random implie equal fx particular result imply monotonicity bf bf h v otherwise inequality claim claim continuity measure generally dependent imply fs fs line bounding chernoff imply since cm em integer define bf k bf bf bf bf bf eq extend constant event q clearly hx bf ks bf hold take h imply lemma imply satisfy eq second kx claim lemma bf bf x event convention proceed chernoff bounding technique markov independent bind total h claim define q bind imply remain first term k convention eq eq monotonicity q establish iv iv iii iv iv g q imply inequality namely inequality combine yield represents set meta specify integer q suppose r begin u w independent chernoff law total union event establish union bind establish run lemma imply n imply f term note l I furthermore always meta infer label eq q f pl know p p establish arbitrary meta immediately lemma arbitrary passive certainly every implication modification replace step kk imply appendix lemma expect request meta unlabeled request mass disagreement although lemma insight disagreement coefficient include throughout fix notational proof hx I ct eq represent index unlabeled request aforementione em prove lemma disagreement unbounded bf bf request make assume likewise implication region bf bf bf bf bf xx mx bf rx bf rx cx equality independence x inequality indicate bf bf inductive unlike lemma mn cx mn last mx mn mn mn x mn inductive note lf lm f infinite ir bf r run upon reach continuity bf bf monotonicity union imply lemma bf bf bf ia bf bf bf bf bf achieve vc particular x h f result replace meta specific take definition appendix lemma previous throughout reference final implicitly continue denote fx repeatedly must consistent meta quantity quantity represent upon reach round end correspond event bf hold lemma monotonicity imply claim also principle hold constraint redundant decide request dependent f f iii f kk kn k last simplify iv k kx km total chernoff imply inequality thus nk k nk probability sum redundancy establish label obtain meta budget data ii em n n ii nk serve base consider value meta induction imply ii choose sufficiently budget exist dependent iii iii f iii e iii event iii truncation f f iii iii eq plug iii thus f ni f fx w n chernoff total ii n fm n union imply sum trivially hold ii ed run passive iii v recall l ff classifier return meta f ii iii ed ed p f l unconditional meta take agnostic show achieve algorithm xy nontrivial xy specifically xy pa characterize xy classifier furthermore p xy exist z q z px h z z z h px hx fact label xy xy xy xy xy pn n p xy achieve label complexity xy establish correspond bind active prove proceed estimate toward useful nonempty sequence first let independent internal proceed bar p combine construct hypothesis test define ib ib p b p ib ip mass imply eq establish nonempty range random independent define ip b side ready result set achieve active learning xy reduce task estimate iid take request already repeat request label give independent return run return context denote z z sequence value k xy xy essentially adapt section fix joint marginal complexity purpose suppose satisfie furthermore continue appendix though proof general threshold replace sequence point sequence run internal denote power additionally establish concern definition define purpose e rademacher excess lemma derivation minor follow borel purpose budget apply independent rademacher I imply claim union existence probability n confidence event satisfy f true serve suppose condition let integer note bf ii induction upon reach claim lemma inductive since reach every ii last combine inductive toward end recall upon h principle budget I fm recall fx fx bf total request constraint event inequality invariant event n fx bf iw q toward end h imply expression chernoff law union exist j ii nj satisfied combine c finally ready break tie remain thus update remove minimizer exist budget confidence fact statement consider passive method union xy xy xy xy xy preprocesse un un shift reference label un u let examine hoeffding least xy satisfie noise xy p xy n xy p xy acknowledgment grateful discussion study classifier vc class learn asymptotically strictly nontrivial characterization magnitude generalization also presence guarantee strict improvement selective sequential design rapid growth datum source rapid extract bottleneck annotation accord instance straightforward collect training significant effort way example refer label expert emphasis design however go beyond allow example label select interactive designing effort toward informative redundancy establish provide significant practical term accuracy understand capability yet sure meet characterize active behave yet identify principle active design design linear decision assumption design passive learning improvement well magnitude improvement toward capability additionally motivate practical concern date decade behave understand hope equivalent amount discover algorithm particular leverage vast passive extent design desirable active algorithm prove variety common even namely improve guarantee passive obvious dominate exist test heuristic active active algorithm passive subroutine passive execution active rigorous look develop reduction study passive general transform significantly fewer result active compare label passive reduction noisy find problem capable explore generality entirely design active complexity quantity algorithm supervise protocol characterize teacher access learn query human expert question nature query simulator protocol type sequential common model set allowed pool request observe label algorithm another unlabeled pool request continue sequentially intend generalize behavior collection label objective return approximate true future previously hope example achieve accuracy label passive many modern learning unlabele abundance annotation pool set labeling examine website survey application discussion multiclass indicate advantage active primarily interested request understand topic largely year progress advance group category know linear decision tree well know pac passive instance strategy request label label example determine phenomenon think line exhibit general strategy elegant refer meta behind unlabeled consistent request inspire disagreement sometimes active never example label attempt seek particularly informative example disagreement far ranking disagreement quite obviously strategy setting know random unlabeled analyze achieve information gain gain exponentially small threshold certain far interesting implication bayesian characterize complexity achievable active learn tight concept analysis match generalization higher homogeneous near distribution notion beyond simple employ disagreement describe pair regardless provide elegant request understand improvement active general particular clearly illustrate quantity vary target issue example come slightly perspective active sufficient natural generalization learn set quantity relate easy calculate though opposite seem next progress toward label disagreement algorithm characterization complexity original strategy state disagreement direct discuss disagreement substantial detail base active sometimes suboptimal disagreement large analysis disagreement coefficient surprisingly disagreement coefficient practical fairly calculate geometric relatively smooth discuss use label learn noisy reason ease work label favor disagreement coefficient label wang significant paper focus extend generalize maintain make disagreement useful particular passive type namely adaptively label request need achieve outperform learn index disagreement certain respective reflect label analysis note algorithm simply label request find good classifier result vanish concept vc passive learn superior target certainly advance active simplify elaborate statement active rather strong remove replace calculation dependent whether dependence claim passive concept theoretically active practically direct access available complexity particularly well passive threshold many leave open characterize improvement achievable even achievable type leave far involve label pick progress question addition advance method misspecification topic root agnostic know perfect set linear objective whose bad well strictly agnostic hope achieve passive agnostic progress publication disagreement base essentially effectiveness show uniform improve complexity learn strategy find extended bound world general achieve express disagreement result hold arbitrary vc dimension case passive disagreement problem soon strategy agnostic set new disagreement base reason establish disagreement coefficient b improve dependence coefficient set computational algorithm develop capable essentially loss large class loss term loss encouraging reflect well constant passive fact label bind depend improvement detailed improvement passive study refined noise well passive special restriction improve passive type threshold class result show agnostic also improvement complexity far bound disagreement apply arbitrary type classifier reflect improvement disagreement coefficient wang later noise identify weak noise improvement exponential certain simplify principle class reference publish learn typically complexitie verification might whether might significantly well complexity label build extend contribution meta active algorithm label nontrivial target exist characterize lead new type algorithm set issue mention disagreement coefficient achievable active learning case small result new improvement passive achievable case include publish literature purpose active base aforementioned complexity achievable agnostic often previously formulate type throughout protocol procedure active superior review precisely characterize scenario disagreement learn desire improvement beyond disagreement set procedure section begin bound disagreement disagreement coefficient somewhat set passive define disagreement procedure us improvement extend allow noise term proceed learn term generalization disagreement coefficient present general concern algorithm conclude investigation suppose borel space usual borel algebra though generalize measurable classifier characterized refer simplify focus counting sequence special denote px informally access access unlabeled example request select request observe integer return algorithm attempt give study sufficient active label distribution originally dependence complexity achievable illustrate achieve accuracy note sufficiently label active come problem need small label label rather number label algorithm request application active label extreme label target equal true mean notion play concept generally design intend relate notion complexity learning include verification interested sequence label passive learning internal passive passive unless state complexity take strictly complexity passive active passive complexity guarantee error also discuss guarantee formulate asymptotic single probability guarantee extract much explicitly employ notation include asymptotic always consider form mean notation follow passive meta passive active call meta passive establish universal exist vc bit explanation interpret strongly little seek learning function toward target trivial passive wish implication strictly nontrivial bind label complexity passive algorithm f serve purpose framework require trivial scenario focus toward nontrivial scenario scenario truly proof give slightly broad nontrivial discusse regard scenario passive algorithm finite intend framework trick aggregate always allow analyst reasonable passive expect c function interpret particular measurable hold make mention notation throughout convenient discussion hx label sequence ph p ph b ph x x h h hx equivalently h problem space space throughout repeatedly canonical although toy important type problem fundamental type example drive understand complexity mind provide passive universal passive sort increase request median repeat update x request least integer request procedure maintain invariant likewise equal active algorithm label example request achieve label simple vc subtle problem b universal request example sequence immediately return constant point nx mi jx u repeat update accordingly request accordingly let label request phase fx otherwise maintain every sequence mean pn n ph bring phenomena analysis learning much strong target fundamental avoid analysis active learn highlight explore verify observe rate base observable essentially issue example handle initial value subtle consider x em sometimes issue effective search perhaps identify point identify perhaps choose request disagreement candidate canonical discuss disagreement subset candidate request request would information substantial attention natural way noise misspecification set disagreement frequently share discuss complexity achievable active let type representative disagreement base achieve passive disagreement property sophisticated complexity passive guarantee vice versa use request label isolate aspect active involve improvement region disagreement formally specify meta passive budget mm request ny mx meta estimator divide stage reduce sample might observed indeed albeit constant property passive conditional choice example second stage trick employ explain implie behave classifier simplify ignore address second phase algorithm request label mass l unconditional complexity return meta f label universal space threshold contrast example one h x h x l label simple passive expect meta passive meta scenario subsection prove meta algorithm tie particular refer reason certain em clear disagreement request disagreement core therefore build perhaps bf h bf z z decision slightly r b disagreement illustrate boundary geometric disagreement core label region derive sufficient condition behavior intuitive request refine interesting reason core correspond boundary intuitively happen surface could probability nearby rest turn case important nearby tie disagreement scenario disagreement insufficient sophisticated overcome along corresponding refinement definition disagreement case meta large expect request around core thus expect request passive request become focused fraction passive factor intuition formalize class meta universal f idea correct also grow classic shrink converge strict improvement passive fed hand unlikely label example represent sketch prove meta achieve strict improvement passive nontrivial f disagreement difficult always hold countable effort class dimension see probability disagreement core particular studied aforementione aside nontrivial disagreement discussion scenario pf thesis wang disagreement core always whether simple could passive vc unlike exist vc handle disagreement request meta become therefore grow speak meta algorithm scenario request specifically meta request prevent improve passive restrict surely large split disagreement fx accurately x perhaps subset long need request agreement label request request shrink asymptotically improvement passive meta describe already improvement address nonzero target address x nearly f repeat argument time infimum agreement surely thus label disagreement repeat many need shrink disagreement obtain improvement passive simply x continue clear shrink determine iteration sufficient shrink probability maintain possibility result kind comparison multiple majority infer generally estimate certain probability detail implementation later algorithm reasoning refer meta state passive budget classifier mm request nh l mt l n jk en jx kx request set meta estimate variety way universal seem appropriate respective state take request batch one third subroutine unlabele chernoff redundant mechanism motivated reasoning check whether new disagreement increase vote request mention property since intersect vote correctly v reasoning dependent
coefficient see conjecture open influence generalization boolean function eq prove conjecture conjecture tight answer consider weak state almost weight concentrated entropy extreme fourier concentrate level fourier fourier entropy bound suggest could decay proof entropy use level discuss section upper bind conjecture influence formulate conjecture conjecture show contain copy complete induce assume conjecture base let concatenation translate q first relate influence second relate fouri coefficient third middle norm describe relation proposition depend since ready equation apply conjecture q random subgraph threshold choose value range graph property show order simplify computation copy nice hold range choose otherwise constant upper necessary union portion fourier concentrate show correspond follow equation consider copy fourier induced measure contribution include contribution equal negligible combining get proof note inverse polynomially conjecture measure boolean generalize universal measure show cube bound implie since preserve equation polynomially statement claim differ conjecture constant factor section discrete study boolean fourier simplify replace character thus simply fouri weight concentrate particular induction assume assertion let fourier degree concentrate low level let discrete th fourier expansion definition thus fourier possibly work strong eq f beyond fouri weight concentrate appropriate use discrete inequality decay assertion important allow observed conjecture q f f prove conjecture power proving deduce sufficient weak enhance decay concentrate around expectation become almost inverse slow majority weak enhance reach conclude mention easily prove deal cube accord first n conclude strong note boolean indeed conjecture g show fourier concentrate one hope fourier resemble theorem conjecture strong conjecture boolean influence fourier contribute grateful recent theorem notation conjecture conjecture seek relate measure fourier boolean claim generalize biased product
first random let suppose follow see pay know situation situation price tp pp past lag additionally dependence gram depend thought complement measure increase minimize reach offset correspondingly increase expect minor interpretation property p zero lag convenience p lag lag significant lag analogously c c c facilitate introduce notation theoretical normality square multiply without confusion lag correspond coefficient lag coefficient consistency appendix regressor consistent associate regressor tend consistency selection assume diagonal assume consistent another challenging cholesky triangular upper entry transform select show affect entry entry set show entry much small correlation weak grateful participant comment would stay california berkeley thank ba proof tp intermediate result show tp random p upon martingale limit p dominate probability local minimizer complete proof minimizer kkt th employ central term equation analogously proof lemma consistent taylor sufficiently normality indicate stand suffice show complete section remark nn economic financial forecasting economic financial lead improvement forecast dynamic lag mixture serial dynamic spatial dependence lag appeal study auto regression estimate treat variable lag different lag lag select lag consequence series consider propose tuning scheme illustrate superiority regularization c e economic broadly speak forecast structural economic theory hence fall year forecast forecasting exploit time series little theory work continue improve univariate series analyze ar moving move autoregressive var many challenging lag omit mistake create omit variable consequence structural forecasting point reaction price forward shrinkage auto improve add additional illustrate example forecast primarily multivariate model e jump switch mostly series bank fed basis adjustment policy heavily variable forecast interaction finance come information variation aggregate default reflect general economic find event affect affect finance finance also example analysis finance grow trend financial series forecast impulse response structural see economic finance economic behavioral finance one imaging fmri brain identify finance dynamic volatility risk pricing field large analysis portfolio management price international finance among low multivariate method might relevant temporal suboptimal forecast rate forecast dynamic deal macro cover production order exchange span summary come correlation temporal moderate lag address simultaneously appeal problem factor datum assumption dynamic represent factor dimensional series well principal auto natural factor procedure dimension high dimensional series variable impulse correspond feasible since var involve lag primarily augment var recent theory regression analysis large dynamic therefore proceed bayesian view grouping study subsection coefficient autoregressive shrinkage relative article regression large regression selection theoretical exist meet practice contradict regularization later propose serial correlation dependence together dimensionality moderate enable consistency scenario reveal selection carry lag lag lag additionally lag repeat procedure lag varied stay drive include vast financial weight seem restrictive base package angle regression package work prior organize next var estimation method motivating outperform contain conclude proof appendix discuss optimize provide lag b p ty tu tu ti j justify variable standardize also carry diagonal discuss see large moderate macro effective much original attain representation parameter avoid fitting follow mild try balance ccc cc loss generality matrix say dynamic lag diagonal term different pattern w associate variable primarily practical choose unit preferable specifically group type penalty impose lag lag respectively generic control lag less lag large assign lag lag diagonal one dynamic drive vice versa reflect different lag small get belief amount distant lag amount shrinkage towards one importance distant lag one decrease however datum drive correspondingly especially multiply side p b fit already lasso type pose realistic economic one hyperparameter control relative lag lag lag lag meet practice correspondingly forecasting average instead suboptimal forecast lag always lag general row implicitly variable group estimate column consider th column use lag lag idea subsection emphasize call estimate b lag lag tune forecasting get disadvantage term drop common panel financial rate time r price index segment segment generality index segment also denote situation lag lag lag segment correspond ideas ii segment b I neighboring view grouping penalization selection hyper grouping scheme real forecast denote tt ahead equation ahead forecast compute spirit forecast forecast relative benchmark walk parameter observation word desire magnitude fit perform grid minimize j group computational first loose grid perform grid use angle regression package provide www stanford edu experience motivate jj w bx b omit generate corresponding solve grouping b motivated one penalty mix iterate penalty standard generate w u j output j minimize step iterate multiple package www stanford edu matlab parsimonious exactly time retain use thus implement ordinary
speak show occur positive suggest generic establish stability markovian expense result expand projection stay surely inter dependency various reader fundamental theorem establish abstract lyapunov satisfying set assumption speak instability infinity point focus noise condition trade solution ergodicity proposition prop final prop smoothly random step comment stability one independent metropolis hasting ce existence subsequent adequate define inspire due page many see setting boundary unbounded contain line twice continuously differentiable subset constant family random whenever w I constant satisfy I introduce lyapunov modifying derivative certain extreme present induce spurious boundary notice field lyapunov projection assume satisfy continuity involve role control ergodicity hereafter relate growth ergodicity establish topic h eq trivially eq introduce infinity enough whenever respectively remains prove eq order achieve deduce x since condition imply w imply conclude taylor pt c since note lyapunov condition quantify drift lyapunov assume hold hold q tail trajectory eventually contain visit infinitely often thank hold shall suppose contrary condition expectation leave hand side necessarily almost surely contradiction unless go na imply claim condition time w I imply conclude provide stability ensure abstract condition expectation poisson require shall verify geometrically case old allow consider old continuity continuity recover c vx call comment relevance expand projection choice must condition efficiency grow slow establish convergence choice constant property easy optimality state simplify decide growth quantity power whereas small weak condition inter proposition admit old continuity appendix constant projection denote different value sum x p last summing term g w turn sufficient notice cv x condition jensen imply yield term hessian condition w consequently x condition c last independence I I condition respectively variable independent term observe term right side assumption side index scenario geometrically ergodic satisfied numerous chain practical regularity see norm kernel norm reader exist evident poisson condition bind clearly establish lemma vx vx vx p vx iteratively ergodicity rate hold finally state condition simultaneous drift condition exponential decay contour irreducible invariant borel set vx x depending follow r proof practically interesting possibly old natural quantitative manner applicable old continuity exist commonly encounter slightly handle start p f f r write p f hold r kf k p provide verify constant hold I p h independence jensen sequence establish proposition imply observe h condition old continuity establish condition involve enforce independent clear exposition power practice period induce manner condition hold condition hold poisson equation I v condition uniformly satisfied I imply imply decay otherwise converge context monte stability stochastic process expand finitely often typically stability convergence strict neighbourhood applicability often lyapunov stability practical lyapunov function yield close possible lyapunov establish formulate convenience set field exist differentiable exist inner h di without sequence unchanged theorem contain finitely projection sake detailed establish noise size sum poisson practice either size must ensure z maximum likelihood hasting order intensity employ measurable give taking assume markov value latent likelihood iteratively open application require one log precisely focus use mcmc particle filter particularly suit state space denote output value output consist variable see hereafter recover function n n k introduce function trajectory latent return complete parameter statistic choose stand computed particle rewrite f whenever integral define note possible within stochastic static diffusion theoretical essentially thing stability apart expand projection random sequence allow consider intensity determine autoregressive determine follow law brevity keep unknown datum lx cc proposal eq convenience artificial initial associated perfectly quantify ergodicity drift shall bind proposal weight determine likelihood jensen recall particle hasting overall generate proposal distribution give ordinary hasting proposal particle eq q number give stand particle bind directly bound establish ratio proposal density ergodicity analyse behaviour unbounded average overall filter one proposal overall density particle finite path jk jk v establish expand projection intensity identify lyapunov statistic purpose constant p x symmetric except enough numerator n convention rearrange write independent u eq overall deduce constant exist leave bound variable calculus q deduce choose sufficiently stability distribute variable independent constant proposition convolution c I hold establish straightforward check generality geometrically ergodic imply drift assume imply soon sufficiently proposition condition establish proposition relax certain fix em dash correspond induced logarithmic control sufficient start final rate run unstable behaviour rely vx consider claim q let vx rx rx c rx vx rt proof sufficient show claim p r p martingale inequality imply employ
commonly find similar control develop systematically vary experimental conclusion term amount relevance difficulty closeness solution assess confident sample synthetic set irrelevant redundant hypothesis scoring account amount relevance suboptimal yield artificial opinion mention properly let cardinality feature call feature refer feature relevance feature jointly relevance carry although way latter keep differentially contrary interested subset likely equally instance optimize account inspire need technical etc call whole set assume imply scenario fix minimum cost meaningful amount among find among restriction subset exist feature policy may solve resp scenario addition shall shall described feature inductive implicit induce logical example place useful particular embed pre mode evaluate subroutine favor mode goodness disadvantage burden briefly genetic exclude review none allow work include comparison equal consider feature count instance equal equal q monotonic evaluate hash particularly set arguably advantage feature filter frame equally set well allow repeat end incremental algorithm describe portion instance inconsistent portion iterate intuitively big add current portion hence similar author suggest proportional experimentally choose may fail noisy datum consider irrelevant htb output portion portion repeat stop else make inconsistent find instance latter instance among separate near separate version original feature htb frame array initialize zero random instance near near end iteratively check nested subset measure simulate type look assessment irrelevant principled discrimination correlate account several similar type sequential generation iteratively try backward counterpart describe evaluate preferable relevant otherwise report contrary practice frame line b output find xx backward generation operate sequential forward possibly essence remove among chart characteristic size backward counterpart backward follow effective popular drawback desire subset generation current sample quick branch hybrid compose branch stop branch branch variant full basic idea point remain search efficiently automatic branch find var list state end begin queue list allow element htb quick branch label monotonic evaluation small element arise design aspect certainly maintain well trade sized solution assess time practice task greatly advance ability respect relevance redundancy family whose influence relevant set provide subproblem redundancy front identify redundancy situation something find report something aware set problem redundant add instance multiply mean depend measure capture obtain match criterion sense similar denote partition relevant irrelevant redundancy let general r x ax flexible divergence redundancy end collect feature enough feature weight redundancy redundant valid feature break correct equivalence define solution every split equivalence form feature redundant define equivalence original redundant give relevant denominator let three redundancy x severe relevant feature redundancy reflect irrelevant miss relevant redundant well one fact choose could precise equation suitable reasonable though depending need twice count difference practically overall section parameter know relevant size function illustrate fig accuracy respectively automatic criterion cut sort great weight discard idea low cut bt total three odd nn odd classic set group follow p boolean condition check divide group explore second explore group different instance relevant indicate twice specifically involve analogously reason representative graphical fig random fig example relevance redundancy vs relevance represent ratio explain vertical represent htb vs relevance vs relevance f c example redundancy vs sample fall dramatically perfect top almost number present good tends poor total feature fig interesting provide increase example study fig score fig b average difficult high algorithm top bottom also algorithm end also useful reading graphic show end fact show surprisingly quite general poor compute independently f picture close view way execution order respective detect redundancy presence size light conjecture fashion well fit option like bring different performance way scale number feature reliably tell also like outcome entirely precise another good deal limited sample dependence feature regard outcome yield outcome resample recommend final specific evaluation know solution expect score performance experiment different use na describe solution well evaluation
focus contain favorable relatively association observation health conditionally limitation independent large health path environmental support etc need health remain essential health predictor provide person limitation support many people survey percent severe rate health hypothesis confirm individual component support health general health variety environmental environmental factor upper favor false negative issue choice team particular individual already appear background restriction macro unfortunately exchangeability violate disadvantage implementation issue subsample choose categorical validity finding mixed seem runtime depend estimation execute core roughly hour child complex heart cognitive behavioral impact daily routine severe heart due heart disease upper expect cognitive conditional validity child common likewise minute latter connect score minute identify insight open genetic factor mostly work child already upper bind plausibility potential imputation utilize forest exclude identify value original exchangeability memory minute random forest satisfactory mixed type exchangeability responsible conservative dag true small poor mixed case feasible modification multinomial response penalization continuous task scale health survey consist cluster macro cluster affect bind connect confirm tendency toward health evident individual many fail identify factor connect risk genetic xx j equivalent thank anonymous valuable proposition among graphical novel estimate conditional tuning graphical obvious constitute help positive health order reproduce health organization international health risk suffer heart perform stability control positive graphical mixed forest selection response predefine predictor association set remain pairwise possibly I appear absence conditional application include largely focus graphical base see binary involve method lost deal mixed ensemble notably performance importance forest allow rank conditional suggest overcome definition response rank derive dependency framework allow specify false random appropriate forest compare performance comprise possibly health health environmental organization international health identify independence conditional realization cumulative dimensional assume proof trivially analogously level express bernoulli moreover conditional let motivate nonlinear regression whether include regression indicate variable rank edge rank mixed type ranking criterion find comparable instead ranking perform local analogous among forest regression decide e ranking select regression tie rank th edge tie remainder tuning outline guide choice allow positive subsampling n thresholded construct impose subset theorem select subset false positive depend precisely give edge vice versa actual minor different fix throughout choose formula specify desired accept false exchangeable note interpret threshold solely assess stability undirecte whose forest date convenient type incorporate observation predictor predictor assess regular forest random fit within relationship response choose package goodness continuous majority fit response local rank response conservative assign bad finally aggregate stability subsample stability graphical henceforth variable many forest favor influential inference unbiased tree overcome implementation forest conditional party package lot drastically reduce become feasible produce positive instability forest ensemble allow fair random forest linear response predictor predictor coefficient also logistic regression median aggregate category discrete consequently suggest estimate lasso remain penalty sequence zero select conservative penalty j estimate regression correspond subsample selection denote stability acyclic statement node represent connect parent entry percent zero encode dag simulate ji present variable least multinomial predictor opposite total effect category multinomial predictor restrict definition link relate previously sample purely bernoulli purely multinomial alternate sequence multinomial ij b ix j ij sa sa scalar choose multinomial association half pairwise dependency ise realization parameter conditional absence see triangular uniformly triangular upper size interaction without averaged repetition give observe false repetition average small achieved bernoulli ising seem figure third many positive rate cover rate return satisfactory poorly cause perform gaussian multinomial set especially procedure nevertheless one indicate potential recover counterpart forest estimation ranking consequently lack various provide selection forest besides enable deduce across hence stability condition mixed setting explain error however return indicate problematic behavior unlikely exchangeability argue appear grow minute hour run core comprise core ghz gb ram package false positive average simulation report average raw counterpart false positive average third report average true third report false rate raw counterpart stability relative stability false average third raw counterpart selection false positive true raw counterpart average repetition similar finding positive arguably small drop surprising ability incorporate whereas explicitly false positive average column rate n average repetition outperform positive positive control especially apparent raw estimate counterpart signal factor positive average third average false raw counterpart remain clear forest par selection apply ml estimation burden positive bound well similar perform somewhat raw stable average column report false positive organization international body structure influence factor gender age environmental include relation support property macro recommend world descriptor conjunction health conduct interaction among environmental health variable conditionally know observational health survey office base private select survey complete percent participant mostly collect far available elsewhere include various limitation respective sometimes ordinal mass health outline consideration plausibility index check module index
fix denote eigenvector furth eigenvector leaf include obtain eigenvector remain two establish taylor minimizes mean error mse conditional distribution q estimation note achieve identically unbiased measurable h x h minimize minimum mse unbiased root finally prove broad establish let vertex vertex root sample correspond top markov topology contain information state high probability distance vector use resp close identity matrix indeed identity variation distance vector total gm couple since statistic structure describe state leave I v vi kt estimate datum eigenvector stationary distribution first eigenvector orthonormal respect eigenvalue description difficulty however note deviation concentration enough ie small use relate eigenvector eigenvector define include eq use proposition length since recursive setup weight child constraint bias condition setup paragraph satisfy far concentrate internal z z establish u equation moment argument apply factor cauchy schwarz e c proposition expand v hold lx x bias accurately estimate take enough bound equation probability calculate x lx observe e lx exponential suitably recursion let eigenvalue e lemma j show ks question conjecture may coefficient technique extend non homogeneous tree weight discretize tree result would control state combinatorial rgb rgb rgb corollary definition corollary proposition theorem conjecture conjecture hide inexact latent tree widely biology efficient procedure latent improve previous requirement regime tree tree transition mathematical processing e reference therein seek evolutionary datum specie assumption molecular sequence sequence evolve independently tree key parameter tree evolutionary branching past matrix mutation alphabet arise biology generality entry encode state branch edge branch roughly mutation think evolutionary involve section naturally context estimation tree topology give fully unobserved statistical theoretical computer science provide insight parameter weight symmetric ise critical parameter section detail contrast tree depth regime know ks connection specifically regime label optimal factor conjecture discrete furthermore polynomial requirement need several reversible threshold may little rigorous dedicated regime prior rate edge discretize require sensitivity root inexact design new ks provably robust discretization precisely ks regime discretization reversible know evolutionary far know previously threshold statement work sample tree neighbor metric leaf follow work dyadic technique high degree conclude root label dyadic integer set balanced leave markov tree leaf variable discrete evolutionary rate reversible balanced rate respect move away root parent special symmetric ed biology generalization include channel asymmetric let eigenvector correspond markov field gaussian signal balanced covariance leave ensure identifiability nonnegative affect convenience leaf edge choice denote extend leave leaf covariance detail close think pick distribution independent globally markov disjoint subset separate independent failure leave go infinity tolerance model model satisfy moreover tolerance establish equivalence threshold tree without sample tree satisfy weight tolerance general critical leave give sketch discuss let balanced generic use framework basic initial sample loop build infer root reconstruct previous step heart step assume correctly correct need issue address topology call ks effect notable ise discretized estimation structure distance estimator eigenvector define define note leave quickly accuracy failure constant replace approximation infer natural mutation estimator flow variance follow strong moment level obtain estimate accuracy lead bias hidden term close exponential moment provide ks bias distance bias adaptively recover estimate eigenvector build eventually eigenvector identically subtract careful procedure prove balanced assume explain building state sample everything else leave build correctly construct convention form satisfy amongst procedure minimize level adjust deviation accumulate branch child big constant estimate seek construct treat weight coefficient recurrence particular weight estimate compute similarly let constraint bias prove satisfied concentration uv markov third moreover gaussian estimate type explicitly note suffice independently assume hold z z z similarly direction hold bias x x x x x line inequality e e e enough recursive estimator q x take enough proposition rely know topology know estimator leave let topology split
connection indicate display sometimes mathematical description take include exploratory algorithm label box connect side point motivate long characteristic procedure within theoretical world point prior treat inference human making never satisfy logical framework purpose go make important look back really high ground inference logical cost thing begin introduction neither avoid connection inference hold inferential frequentist act normal involve conjunction equation speak accord equation bridge build theoretical statistical partly see pure argue succeed data problem complicate aggregate uncertainty scientific investigation context distinction practitioner understand analogous believe figure population concept drop understand option large development terminology avoid confusion offer recognize pearson make contribution introduction behavioral seem think far particular hypothesis upon valuable evidence nonetheless behind lead inference valid mechanism chance small situation quantum physics chance variable carefully head david imagine absence explicit chance strict I however precisely reason vast mechanism fisher jeffreys many cox extent like stanford fundamental theoretical align hand gap seem big offer scientific goodness heart acknowledgment support grant grateful comment frequentist assumption interpretation confidence assumption bayesian frequentist interpret suggest serve foundation statistical argue jeffreys take goal exclusive failure nearly old statistical go way significance aside concept meanwhile part ignore possibility inference event despite occurrence interpretation posterior belief confidence frequency limit use introduce practitioner open view mode aware suggest place logical world frequentist fundamental paradigm take separately understand important student right imbalance describe dominant modern think valuable inferential chance count fair basic immediately mathematic come calibrate fair another say odd mathematically assign fair cover consider run may regard consequence interpretation important use reporting interval statistical inference model illustrate figure scientific distinguish world call world random confidence interval probability live world implicitly datum capture reasonably world real apply statistical construct conclusion theoretical inference implication phenomenon scientific concern scientific implication involve new somewhat model large weak careful statistical note random live thing normally proceed kind shorthand assertion purpose variability variability occur sample linguistic framework distinction aside treat apart inference good way reason inference development elsewhere quantity counterpart mathematical important viewpoint bring center purpose provide interpretation believe practitioner stimulus histogram fire neuron two ease familiar situation issue arise conduct human construct light intensity light vary intensity subject determine intensity subject light observation yes intensity day tool analyze datum maximum light report report al involve fairly large answer bayesian probability interval come spline fire rate study ultimately differential display fit fire rate maximal firing bar follow frequentist posterior base bar similar bar either inferential interpret frequentist paradigm sample mean interval confidence frequentist assumption infinitely random interval cover confidence world nonetheless substitute useful conclusion align world produce remain statement apply variable random prior become eq inferential interval statement remain nonetheless draw conclusion world real produce although world relate object live inference distinction random inference statement introduce happen distribute reasonable describe accurately important student principle notion picture illustration figure extremely population work subsequent drop analogous concept try claim population random analogy independence response suppose student whether reasonably reason
tag tag document share tag proportion tag tag document topic proportion tag tag word replace latent topic tag tag multinomial tag order document largely specific vocabulary tag bag exclude label exclude subsection modeling mrf assign semantic label explain bag vocabulary corpus label topic index word group model often theory mrf assign label posteriori estimation mrf map framework vision specifically mrf attempt topic label configuration word maximize problem latent mrf use labeling reduce labeling configuration encourage smoothness neighboring topic neighboring system exclude denote label associate index exclude lda factor design potential local specifically encourage labeling type hyperparameter complex inference mrf factor connect connect neighboring labeling direct undirected fig parameterized function multinomial smoothed hyperparameter pseudo count multinomial resemble collapse gs treat perform collapse recently reformulate constrain mrf causal indeed reflect lda generative latter mrf neighboring mrf extend visual topic model labeling mrf expressive directly emphasize generative loss factor representation direct world slightly graphical neighborhood function fig neighborhood system connection bp efficient fig calculate turn calculate marginal message normalize efficiently computation message proceed convergence iteration adopt sum infer message pass scheme em infer hidden graphical mixture backward hide probabilistic labeling message base infer almost gmm detail book fig variable message q normalize message turn pass factor message neighbor labeling evaluate dependency topic imply configuration q often cause close avoid arithmetic product approximation mrf acceptable convenience shorthand notation subsequent mrf clique prior knowledge encourage high pass neighboring message document message message word index vocabulary make message comparable across different word word except sum possible message normalize locally message converge multiply count topic message perform figs message derive equation employ multinomial dirichlet conjugacy hierarchical maximize mrf assign topic accord rule p collapse gs variational bayes topic alternative topic model bp gs resemble bp sample token update base currently sample topic label gs word token bp message vocabulary document word bp gs addition bp gs keep complete information vb use objective function maximize maximize variational resemble minimize leibler kl vb differ involve function learn average token document word indice small token vb corpus hypergraph fig bipartite undirected hypergraph equivalent bipartite c denote connect tag factor fig c hypergraph tag connect neighbor solid black tag share tag three relation among document label configuration influence separately relation meanwhile influence neighbor high result tag fig encode order among semantic tag usually account part document image associate document appropriate tag probabilistic likelihood form label base message message factor subsection tag content topic initialize tag per iteratively normalize topic tag notation hadamard wise product notation indicate document tag hadamard similarity tag pair tag encode message tag message pass word tag depict high dependency tag order relation tag similarly base denote total tag obviously triple capture representative order structural dependency contain loop develop bp estimation subsection calculate message subsection involve pairwise high fig constraint factor denote tag current document document except document tag pass message document joint message tag influence pairwise across tag product operation neighbor arithmetic product calculate input flexible message influence play high rewrite balance factor sum term tag message pairwise high relation automatic require work manually tune message fig summarize use relation image engine area paper engine name tag fall broadly category collection contain kind range manually label appear picture colored pattern appearance visual slide window visual word vocabulary index quantization summarize total number vocabulary vocabulary number tag per document constitute constitute c manually relation comparative study model order topic among link experimental benchmark model handle relation document contrast additionally relation induce connect tag lda topic lda tag link tag document test word topic unseen test low correspond topic consistently low ability unseen test benefit rich relational information document pair tag specific pairwise potential capture subtle dependency document specific show compare order role worth prediction performance predictive topic latent interpretability involve basic idea identify topic document define inconsistent word knowledge ten qualitative topic share university third show interpretability top ten moreover link predict share tag link document author name link retrieve document tag proportion link decide link link compare encode document contrast link outperform link reason rich proportion tag differentiable without share unseen word enhance content similarity document alone account additional may well document classification mutually may document proportion feature discriminative ability document end proportion seven randomly select choose water tree people use associate tag image proportion inconsistent word lie treat tag equal link reality tag topic structural document thus encourage tag correspondence tag class limitation encourage proportion tag modeling furthermore high tag constraint pairwise performance generally partly tag individual component equivalent label topic enhance tag recommendation classification suggest tag document world find annotation recommendation image annotation recommendation tag multiclass svm image proportion tag tag tag tag train tag binary svms tag tag connect tag tag encode tag tag predict likelihood tag balance linearly good training annotation protocol top query tag use connected tag refine tag recommendation tag tag suggest include recall rate tag tag image set tag tag recommendation system give tag recommendation recommend tag tag test ability achieve recall tag precision lda recall h lda compare two tag recommendation rate lda show tag recommendation web page advantage annotation problem connect tag information major role rule false positive enhance precision still superior fig topic tag performance effectiveness smoothness pairwise document mrf bp inference parameter four lda mining many vision application unsupervise activity complicate multiple encode discover motion another historical interaction pattern work grant china grant modeling problem higher extract reliable tag topic structural dependency framework representation latent mrf belief propagation approximate hypergraph relation learn encourage label smoothness among image extensive confirm incorporation high state many text
case histogram unnormalize weight homogeneity monte comparison experimental program cm title language library histogram unweighted weighted solution formula ref histogram pt
accordingly preserve partition let index eq q projective xx constitute result construct let mapping take particular outer measure image restriction later borel constitute x element space finitely q conversely x satisfy hence regard theorem measurable measurable well first relate functional evaluation functional generate mapping evaluation functional set weak topology coincide map cx express large projective algebra deduce regard mapping trivially remains borel countable projective topology borel borel projective obtain rather manner ensure surely projective limit proposition gives formulate projective projective limit projective space p expectation element additive surely additive criterion term countable projective construction content additive along imply reduce countable subset derive result countable algebra generate ball sequence countable sequence additive finitely additive mapping projective ii surely mx mx form endow convergence imply mn whenever hence conversely assume surely sequence nan q convergence satisfie conclude finally obtain establish proof first projective proposition define probability conversely assume proposition f complete provide description problem list address reader measure theoretic obvious measurable partition axis hence encode family dirichlet product kolmogorov well assume dimensional event event consist simplex euclidean marginal live form sec projective measure dimensional space limit apply kolmogorov extension embed product properly formalize marginalization require construction illustrate marginalization merging event subspace kolmogorov natural formalize node scale use fill black body minimum width cm stroke fill body black minimum fill white fill occur equivalently x projective construction axis set useful implicitly algebra algebra algebra resolve particular illustrate set il axis shape direction situation illustrate assume shaped plane measurable obvious reason algebra kolmogorov closure countable set algebra axis parallel countable overall countable interest however event joint countable would countable arise measure assign event countable subset behavior measurable infinite suppose countable sequence hold even though along set aggregate substitute countable resolve acknowledgment thank associate valuable pointing corollary grateful helpful comment correction lemma corollary theorem bayesian construction dimensional marginal prominent dimensional dirichlet show difficulty construction probability distribution whose projective limit countable dirichlet projective finite distribution stick break overview construction exception construction key representation account surprising purpose projective limit modify prove put main dirichlet derivation sufficient stick specialize latter approach provide currently tool arbitrary prior make readily comparable prior notably process technical arise problem review detail product space kolmogorov adapt construction dimension label borel resolve event specific construct condition projective limit feasible requirement topological sufficiently accommodate measurable space useful without topological natural whether parametric existence regular probability validity de address generalization kolmogorov countable set substitute projective iii set space remainder apply going summarize sec brief overview construction relevant additive review notation relevant notion marginal topological topology underlie abstract call measure weak topology context correspond borel algebra partition disjoint probability ib jx vi x xx xshift circle fill b x x illustrate mapping image constitute intuition dirac roughly think analogue regard evaluation v keep mind topology topology euclidean space excellent exposition pa following construct suitable marginal law borel moment discrete sec example condition serve finitely sec contain probability turn distribution need p additive almost vx constitute random measure technical restriction mild purpose illustrate concrete separable banach banach metric space space domain topology compact cube distinction may dependent process process background construction three example marginal distribution let let dirichlet additionally problem ii generator provided resolve iii behavior know aware well follow specific dirichlet projective limit invoke tailor real dirichlet throughout mathematics dirichlet derive mix tree stick break remarkable construction list require arbitrary stick projective process trade impose restriction applicable represent phenomenon encounter theory example mean theorem factorial product space choose across subspace topological kolmogorov component process purely stochastically regard analogue factorial measure factorial measure general limit construct mathematical set projective small sense precise totally sequence whenever direct partially inclusion projective limit manner formalize define mapping space measurable regularity assume algebra generate underlie continuity direct di function f kf topological satisfy f f topology algebra projective limit space topology algebra make canonical measurable analogy f projective projective p analogous impose precisely coincide family projective guarantee f space countable direct projective projective stochastic refer theorem index ensure projective arise whenever projective theorem theory regard product
compute trace let sort decrease equivalent determine large uniquely present method typically order within second admm level reasonably large effectively b illustrate compare see differ produce see indeed bias determined ensure negligible implementation result n h h I relax previous subsection hyper write semidefinite matrix widely arguably iterate x iteration involve estimation generate select next paragraph denote produce result without first define residual variance diagonal refer applie may relate see formulation say denote resulting stand also formulation solve though involve quality multiply set x x become constrain unit determinant motivate solution identify explain variance uniform feasible convex optimal solution ascent alternate optimize compare aforementioned synthetic second historical price stock kind data model unit sample take factor variance orthonormal vector mf x n use plot represent availability variable difference outperform scenario largest well performance h synthetic similar effectively control residual tm plot moderate residual outperform tm variance grow elaborate important finance covariance risk guide experiment involve historical daily stock represent period dot daily compute price describe detail trade stock active set normalize log daily day application day day assess day begin day day evaluate daily day test slide try regularization time specifically select day estimate per parameter evaluate take ten ten day average define tm dominant natural historical observation superior allow involve lead difference alternative start simple residual identical let matrix move theorem suggest reason tend loading assign large large variance variation performance eigenvector load one factor load insensitive residual dominate eigenvector eigenvector imply preferable residual differ tm suffer similar tm tm incorporate tm produce desire strictly formally proposition give monotonically bias variance contrary accurately recover favorable incorporate ie estimate deal context assume among conventional provide computational experiment involve synthetic pre direction datum thing estimate guide subsequent interesting quality relate subspace interesting understand suitable finally body research robust variation pca pursuit sparse corrupt would explore connection work v rewrite associate multipli lagrangian denote kkt plug trace permutation g v v diagonal entry multiply side generally rearrange l finally plug complete sort denote note write n impose vanish imply j ik theorem eq fix scalar n therefore numerator limit rewrite law proposition sample inequality chi distribution l rewrite straightforwardly since eigenvector solution optimization f equation plug back yield result monotonically furthermore q therefore c r eigenvector differ symmetry arrive discriminant distinct root root great root great eigenvalue great eq q increase algebra mr mr sides r mr derivative describe experiment cross validation split whose candidate maximize likelihood select fed full tm choose rarely implementation em terminate I suppose algorithm evaluate equivalent uniform trading day stock return adjust close stock day stock day let great interval volatility stock week select tm range value proposition ex learn linear synthetic superiority exist theoretical explain bias feature algorithm requirement enjoy part due notable economic finance combination consider factor observation entail estimate loading residual seek explain approach learn principal variance pca efficiently maximize likelihood datum simplify ideally treat attention uniform consider number factor select portion datum baseline residual propose number approach maximize restrict trace model covariance serve model parsimonious validation similarly synthetic demonstrate estimate recent bias practically relevant assume stock demonstrate accurate residual aside aforementioned analysis efficient program sdp solve exist interior alternate direction multiplier long practically obstacle arise sdp formulation large solve efficiently typically multiplier pca wide extent essentially problem computational acceptable trace important difference however establish perfect asymptotic regime make trace penalty deal residual variance demonstrate treat semidefinite demand provide solve though research relate order graphical lasso detailed although share like fundamentally graphical model induce whereas result address bias algorithmic front develop solution build demand without estimate sample x think generate w residual small sample factor loading result model seek matrix leibler choose maximize maximum simply maximum sample accurately unless
adjacent possess social specie node explore far several simple heuristic node centrality maximize mutual next expect long history intelligence first couple generative discover network label gaussian field harmonic technique likely learn relationship community node label collective community differ information uncertainty case link end sign propagate class node generate correlate label simple could depend event independent classification tv undirecte modify pair wish undirected machine community assume community node node direct edge web classification likely classification order define integrate product particular bayesian edge hyperparameter dominate small beta allow user community dominate remain however entirely assume likelihood since graph take fix averaging overfitte paper determine include learn resource field resource node guess label explore mi node distribution difference q expect information entropy uncertain strongly correlate entropy sampling classification gibbs markov node explore allow collect conditional entropy write probability average label offer markov graph exponential real try marginal say markov condition explore explore stage quantity might explore classification define whose nod correctly could agreement gibbs correct classification know assume draw explore classification gibbs event agree numerator heat sampler except classification start chain class consist community community nod network commonly occur occur connect word appear point precede exclude direct simply noun focus attention uncertain early nothing perfect elliptical last algorithm explore explore stage use average markov chain performance gibb stage unlike mi aa specie point type specie namely value result stage use choose heat markov average type explore correctly remain perform somewhat mi include explore explore classified node early show confident wrong type variable aa correctly word wrong explore leave argue node perform specie large diverse specie type level account base specifically specie likely course regard mistake connect another extent consistent update specie specie iterate six every specie type topology suggest away occur four year world reflect hidden algorithm solely matter sophisticated probabilistic use active explore subgraph node centrality path go subgraph node choose heuristic heuristic variable leave high high heuristic early quickly mi aa heuristic perform process high high classify explore heuristic classify node surprisingly early bad mi explore costly one lexical algorithm good job explore relatively small certainly generalize probabilistic grateful mark web grateful institute foundation california usa ne topology label could choose word novel make even structure assume connect way category collective labeling biological attribute link view tend much correlate word internet business review define definition community group connect might english follow even divide know trying classify political social try infer focus discovery community community make assumption
pca mnist clear include explanation increase dimensionality generative glm properly overcome well regularization c c mnist mnist mnist scale letter glm summary computationally achieve order magnitude speedup mnist minute report combination section baseline learn k discriminative replace conventional rbf metric point specific linearly baseline optimize algorithm display average misclassification experiment result report table detail nonlinear combination metric poorly table metric attain good method understand metric normal heart baseline heart euclidean proper metric unsupervised unlabeled address iterate global metric apply significantly performance exploit learn nonlinear metric result euclidean metric mnist colored identity metric help well exhibit identity approach discriminative build upon semidefinite metric improve well combination compute combine ip space drop subscript clarity c linear term solution space I q eigen combination metric another combination probabilistic point estimate depend metric iterative list combine compute estimator kernel tp bandwidth parameter maximize gmm class assignment trick learn classify datum gmm eight method metric learn margin near method review learn metric review distribution modeling procedure classify metric finding much combine described speak define difference point near neighbor low purely base local weight density estimator compute tune outperform euclidean efficient sound glm glm c c avg norm heart glm scale shown generally reach train dataset mnist mnist letter euclidean scale count scale letter scale c c heart c c mnist scale letter metric represent normalization factor scale inverse bandwidth metric use desire formulation scheme constructing adjust accordingly metric combination simplicity us svm refer optimize combine coefficient solution depend include aspect ie simply local metric improvement metric promise kernel impractical heavy cost c normal heart baseline dataset heart identity method normal euclidean mean euclidean mean cluster metric metric extract mean euclidean truth class generative compute cluster use learn distance iterate long simple cluster measure metric essential compare distance demonstrate usage dataset metric well euclidean describe rand label rand return metric compute regularization overfitte generally validation ratio tuning cluster compute rand score validation tune learn use set rand performance rand average find mean improvement reveal department electrical pa science california ca lee department electrical engineering specify manner generative metric framework specifically learn optimize parametric base minimize training criterion combination achieve magnitude speedup distance datum learn approach discriminative metric serve block minimize combination extensive discriminative significantly competitive baseline train method magnitude metric distance interest machine aim improve training technique try point belong increase mahalanobis metric semidefinite use semi triplet nn interest discriminative technique metric aim try increase distance metric mahalanobis solution programming sdp positive semidefinite optimization learn computationally appeal asymptotic limit underlie metric near upon bias class conditional glm optimize simplicity desirable infinity classifier attain twice optimum asymptotic calculate training nn term metric glm technique knowledge empirically learn attain competitive training moreover glm metric learn solve sdp metric difficult distance distant unclear address issue metric global classifying view metric base kernel benefit generative combination highly kernel outperform compete method magnitude encourage technique metric need every significantly capable exploit generative adapt learn metric summarize average metric single reduce classifying empirically illustrate subsequent compose base base metric secondly extend intuition combine base gaussian kernel identity replace address important provide derive building benefit conduct extensive lead improvement performance furthermore computationally speedup magnitude organize review previous combine train present extensive follow derivation comprehensive approach briefly start metric margin near neighbor examine glm parametric attempt classification conventional mahalanobis metric follow terminology literature square mahalanobis transform arguably neighbor exploit identify structure metric significantly increase secondly glm conditional generative suggest attain competitive rigorously quantify relationship generative unclear generative adapt improve performance nearest classifier learn metric issue kernel classification consider metric define learn locally treat function intuition linearly kernel learn q coefficient kernel global metric combination metric simple combination empirical strategy note view apply transformation datum transform average computed identity near neighbor space p metric convex combination induce ip exploit close combine radial rbf covariance q bandwidth goal learn rbf kernel learn mkl convex metric semidefinite include reproduce hilbert represent close mkl choose rbf matrix difficulty properly base kernel problem formulation refine optimize specifically low use classifier vector machine benefit discriminative optimize combination preliminary indicate extensive reliably simple uniform present form appendix find appeal effective algorithm dominate calculation cost local metric add little overhead contrast learn optimization examine roughly moderate greatly outperform speed later competitive report linear comprehensive include
dependency lda small number non bad logistic method multi date largely machine drawback drop increase exhibit skew distribution world advantage respect number rare discriminative approach document labeling range probabilistic compare skewed text dependency decade publish document discuss limitation classification common dataset label power advantageous multi label corpus assign document corpus task dependency much set relatively instance label corpus occur relatively axis three multi label axis relationship log axis benchmark log scale plot point fall equivalent researcher restrict example popular yahoo yahoo fail however conduct yahoo construct dataset category yahoo leave classification yahoo dataset dataset mesh contrast research real contain thousand skewed frequency illustrate corpora thousand axis plot corpus vast label document example single document corpus stand yahoo yahoo health bottom occur frequency summarize dataset drastically world label rare previously notably multi drop dramatically real dataset skew illustrate discriminative poorly dataset full dataset classifier label positively document state vector machine classifier effectiveness neither flat hierarchical svms need skew yahoo many rare category svms substantial improve difference traditional dataset relate compare label document style large typical dataset label three yahoo health median per document document distinguish particular label little training illustration extreme additional assign document relevant reduce example feature label able leverage word token likely identify within likely away word word purpose remove wise since label relevant learn word construct document learn simultaneously separate training dirichlet allocation lda document mixture topic word primarily etc version learn label give corpus assignment rather learn time treat assign word frequent learn introduce label tend well associate associate word belong investigate version helpful type rare label pose purely svm york article show advantage lda rare text news article take new york human law game classifier learn lda learn contain etc rare discriminative game later rare predict nonetheless well job relevant probability benefit away effect far additional arise total label per accounting dependency label prediction classify document example assign document large like label ambiguity account word also motivation beneficial probabilistic dependency feature elaborate dependency label problematic past literature take correlation growth correlation consequently applicable probabilistic dependency type approach statistical task label document emphasis corpora model lda flat employ various prior novel dependency lda extend account label two variant vs variety specifically document ranking relevance yes ranking yes label contribution novel multi dependency improve performance simple accounting dependency competitive popular approach extensive label benchmark generative svms multi svms statistics svms svms prediction dependency outperform lda method remainder handle label text incorporate frequency inference perform extensive present task five corpora discussion vs statistic task adapt lda supervise lda label lda lda account correspondence restrict document document document lda illustrate certain qualitative advantage ability interpretable certain lda competitive svms differ firstly lda label include account label frequency additionally account lead improvement lda conduct large systematic include number label skew generative particularly method l relatively primarily yahoo sub see idea model compose document view early lda propose recently demonstrate probabilistic field compare test benefit able naturally unlabele semi crf particularly accounting label classify label pruning pruning upon ignore restricted dataset unlikely able account power hybrid generative discriminative label classification learn learn take document parent parent specify body prior elsewhere binary multi classification binary classification solve use suitable classifier employ task handle notably svms knn classifier vs since commonly discriminative multi propose work discriminative another build classifier label combination due flat generative assumption make lda tokens corpus multinomial capture dependency assume token sample topic document topic section describe depicted notation extend provide first describe multi inference description describe document model document multinomial type sampling word three model process label extension lda flat regard generate lda incorporate via wide label prediction lastly dependency lda label label token corpus topics lda dependency also unsupervise extract type corpus represent document flat document corpus document formally represent multinomial vocabulary document label hyper topic generate lda document observe assign token label label depict notation similarity flat present author lda author condition author l condition document learn l description bernoulli assignment e variable l lda reduce additional assign unlabeled document reduce despite generative label generative description lda employ distinguish flat frequency within corpus observe document label account difference law traditional flat generative account difference observe document wide multinomial generate multinomial distribution label think single vote hyperparameter label sample formally define word count document hyper weight label smoothing contribute label zero fully generative label sample hyperparameter multinomial always e zero infinity may observed label binary count bernoulli testing note bernoulli tend negative absence document label bernoulli slowly document turning label turned assign elsewhere flat model multinomial tokens j dd di graphical panel lda account frequency lda extend label label topic dependency prior dependency corpus frequency model induce topic dependency investigate past model relate dependency topic multinomial lda compute accord prior lda distribution topic j tt j labels di label w z graphical figure flat lda dependency estimate observe describe unobserve require multinomial distribution additionally test hyperparameter prior lda lda assignment document see label greatly serve upper low namely conditionally label furthermore estimation three dependency consistency evaluate one source factor difference attribute qualitative difference improvement could regard hyperparameter collapse sampler collapse gibbs sequentially update token collapse lda lda label long token time assign entire label document subscript token word integrate distribution term label distribution present indicate long chain single sample current assignment assignment word average chain several corpus document compute estimate document arm political house north nuclear flat lda collapse gibbs unsupervise lda assign across entire set time denote current token remove count topic integrate dependency lda topic label corpus chain run burn iteration assignment compute train ten chain mean chain set see learn corpus document present distribution document probability first proper inference lda unobserve proper inference document type token word token dependency lda involve inference label additional involve assignment token equation learn training assign document dirichlet document arise sample simplify dependency lda dependency lda description fully provide irrelevant flat label unobserve document exact token token token topic assignment must term word assignment token update document term elsewhere derivation equation label token assignment conditionally assignment assignment label topic rather eq label integrate document variable conditionally independent information propagate back assignment token label token sample token topic token pass efficiency act mean bottleneck increase token substantially compute reduce amount require careful storage still slow long give similar bad lda describe suggest require substantially similar proper explicitly level thus avoid information bottleneck create token also avoid costly step achieve directly pass treat substitute assignment document label conditioning motivate multinomial distribution topic compute label learn use parameter compute explicitly variable term lda approach expression step token label assignment token hyperparameter back token pass equation ignore actually influence assignment vector sufficiently assignment provide dataset even optimize sampling per fast inference computational lda base present note often incorporate label dependency read etc flat uninformative difference current token document reflect relative priori topic assignment addition dependency lda dirichlet reflect give infer mixture assignment relationship document label example four assign ten bold dependency correlation lda dependency improve rare learn training flat assign word probability label flat rank two include rare prior improve flat lda exclude evidence small rare label rank flat rank whereas rare high stay prior united relations dependency lda lda force attribute arm sale united labels force united international middle binary svms frequent rare label learn introduction key binary emphasis contain skewed annotate article annotate label assign manually york times indexing service document commonly benchmark yahoo avoid method employ lda represent word word document one document label prop ex health present statistic multi statistic power explain per document divide divide equivalently label divide assign label combination label occur average document set document combination label cardinality reflect degree truly label corpus cardinality frequently occur median reflect many exist sparsity clearly quite group relate label tell average unique label tell combination type deal dependency handle dependency unique three small document unique combination unique building combination reasonable hand mean nearly effective task metric performance lda base svm result show table svm condition lda advantage discriminative detail binary classifier comparison base scheme document count svms training vector svms parameter value parameter default weight penalty misclassification certain especially label desirable instance ratio instance hold set select close determined entire setting default numerous evaluation metric label broad focus prediction know prediction label consider make yes item irrelevant together choice task provide basis illustrate informative fair comparison lda possible binary prediction document prediction traditionally task increasingly grow etc literature adopt many metric literature base compare publish difficult multi version benchmark consensus evaluation prediction discussion early c one rank document predict rank notation bar item bar viewpoint set label rank predict rank broad document irrelevant document root general four bold curve alarm document document macro document document area area document combine macro document average document percentage document rank pt percentage rank irrelevant high rank relevant document relevant label percentage comparison roc curve statistic simplify publish rank pt binary macro average micro average traditionally label perspective averaged label however basis calibrate ranking etc approach harmonic label eq item summarize micro average macro compute individual matrix take micro confusion sum across confusion matrix confusion micro weight item instance weight item frequency must careful difference frequency increasingly skewed like ny macro poorly label vast power poor macro reasonably micro illustrate binary prediction see direct extension rank test predict rank sort transform set binary either assign top rank issue choose particularly selection comprise research selection emphasis thresholding rank cutoff calibrate cutoff predict break cutoff median cutoff expect instance frequency label train train document assign document learn svm note test document calibrate item pt break optimize item commonly refer break select information comparison theoretical calibrate tell high predict maximize average search close score micro fact optimize label generate false assign positive impact micro calibrate score prediction fix cutoff document prediction four evaluation rank label relevance document ranking prediction positive cutoff evaluation prediction show auc roc result ease publish calibrate absolute difference micro macro score evaluation oppose number label lda dependency perform well flat across outperform yahoo yahoo evenly case flat much dependency lda difference demonstrate achieve simple frequency tune svms non law outperform significant svm approach outperform tuned rating area svms svms prediction metric tune svms largely comprise thus overall svm prediction exclude ranking across benchmark svms outperform svms percentage label score much influence low rank see distinction performance power law dependency outperform svms skewed large cardinality lda outperform svms law mix metric yahoo dataset dependency outperform five ranking outperform measure label error measure evaluation dependency lda outperform svms health bad svms indicate relative without center see variance dataset plot label document train datum performance relative drop eventually flat tuned svms dependency svms lda drastically bad lda prediction term median label per label center per decrease e document improvement dependency flat document increase flat boost attribute inference test time dependency result intuition dependency lda metric document number document increase right center around zero dependency lda increase seven aspect ranking six partition cutoff base evaluation show suited numerous document perfect label include completeness calibrate represent proportional benchmark literature comparison lda base document lda outperform lda flat lda document document whereas gap relatively yahoo document even dataset automate dataset strict assignment tune consistently svms except equivalent notable dataset pose binary svms fix dataset intrinsic dealing answer introduction difficulty relate surprising outperform statistic improvement parameter conjecture difference dataset first svms difference actually secondly likely relative whereas test since include label evaluation split somewhat de performance rare assertion support performance tune svms perform svms evaluation metric overall intrinsic difficulty svms rare observe tuned difference relative lda outperformed lda outperform tuned svms measure macro macro equal regardless power dominate rare macro reflect svms macro measure model outperform support handle law svms generally dependency lda svms inferior health subset worse v method outperform base three amount datum datum despite dataset dominate lda label lda fair well interest dominate macro macro macro score previously discriminative publish label appendix additional reason vs binary sparse lda across score frequency macro score significance pair significant svms dependency svms frequency less five significantly level frequency svm dependency label lda frequent dependency lda svms frequency except prediction seven metric evident lda significantly lda scale depend dataset notably significant binary play however comparison across wish model discriminative vice versa purpose dependency tuned svms focus specific performance tune svms four prediction dataset task achieve fall qualitative yahoo multi amount dataset yahoo law dataset unlike dependency overall svms yahoo two v tune svms outperform lda relative performance illustrate evident else well suit seem suit base dependency lda outperform overall prediction quite yahoo health dominate label explanation fundamentally lda learn model direction label direction binary svms label across type whether label suited look lda experiment lda improve flat accounting label frequency improve flat lda prior accounting gain accounting dependency addition
figure draw claim meet establishe satisfied argue draw except fail return put mass run unimodal conditional output unimodal follow e ignore right least sample learn need right chernoff notice algorithm list number assign linear point represent maintain thus turn overhead run run explanation since atom remain justify form say differently unimodal change ok argument learn fail follow correctness either inside triangle inequality target return two guarantee close translate poisson probability plan cover must mean cover natural estimate translate poisson show highlight fact close variance bound figure output translate poisson value proof cm draw k produce eq treat obtain achieve guarantee allow expense omit routine boost reader time specify separately independent chebyshev variance correction check q excess e ii chebyshev imply proceed translate mean next guarantee output distribution heavy binomial follow inside cover translate poisson conclude use suppose remain distance translate remark priori whether inside simply produce routine competition candidate distribution draw select winner hypothesis quite chapter competition make draw return least return competition carry cm pair I mx p return either hypothesis immediate lemma proof suppose winner intuitively likely winner intuitively moderately unlikely winner mutually chernoff stop competition part competition winner otherwise competition winner conclude claim finite moreover algorithm choose output never tie output probability competition output triangle give show competition argue close competition prove pair cover notion competition tie cover recent improve choose pair distribution evaluate pdf take priori point binomial binomial return return absolute lemma describe part modification distance argument poisson learn replace choose appendix numerically within additive support pi tv p remark wish purely exactly hand hypothesis mistake define make unknown choose need much overhead particular running hypothesis instead proceed sample correctness immediate consequence choice lemma run run compute suffice need compute output learn poisson inspection bit hence cardinality spend produce merely involve subtract explicitly fall inside rest choose run theorem proper part modification proper learn output add step poisson translate proper proper distribution choose return return return constant cm sort list constant return put cover hypothesis sample procedure proper sparse explain beginning claim imply case statement output fail return put observe x sparse bernoulli ax proper construct run describe place event complexity run proper follow correctness happen close order translate xy routine search sparse cover distance fail close succeed lemma learn succeed form close learn succeed see routine indeed would k explicit j ks k sample complexity claim remain run competition carry efficient amount competition competition kb n k k kp w tuple tuple dynamic subsection peak efficiently weight bernoulli alternate asymptotic construct sum bernoulli distinct exponential algorithm give problem ok running time mode weight distinct unimodal sum weight algorithm hypothesis choose uniformly call relevant draw fraction constant error argument probability uniformly equal target easy exactly sufficiently must easy chernoff ever sample target entire learner ever go equally likely element note prove initial conference publication binomial conference concave distribution poisson binomial accuracy sample question subsequently poisson binomial mutually integer give binomial goal obvious come bit learn ideally output theorem except additional statement property involve construction reader contain note subset heavy permutation cover theorem cover cover subset cover distribution large cover small variation instead small also proceed argue cover review collection indicator collection filter stage respectively denote collection filter stage tv possibly heavy binomial form make cover heavy proceed cover define denote satisfie heavy concluding well convenience let define round expectation index additive expectation expectation indicator expectation variance k proceed separately symmetric index obtain proof stage filter expectation depend heavy stage sparse look distribution heavy binomial collection produce eq learn distribution briefly explain moderately familiar variation hypothesis though denote distance unimodal take bound target hypothesis expect pool hypothesis term remain carry bit begin true mode identify stage search function proceed appropriate analogous recall empirical come element point lie compute adjacent point lie explain paragraph paper minimize additive difference exceed find finally output convex cdf cdf simply collection conclude follow mass operation able input number produce poisson distribution assign run separately take follow motivated approximate within enough approximate particular involve satisfying bit complexity able claim indeed k algorithm te logarithmic q approximately combine q monotonic construct grid adaptively perform find known fraction exponent within error time e bit operation search te argue additive distance two grid additive thought give want distinguish output e te apply give hence run view show compute step recall say exponent term suffice error approximate expression try first digit multiply integer use bit approximation approximation bit constant exponent multiple fraction bit logarithm fraction within complete additive division clearly integer description om om rational evaluation rational division operation approximate truncate comprise arise combine claim claim remark conjecture claim remove mit ed ac uk cs unsupervise poisson binomial arbitrary expectation poisson generalization familiar surprisingly work basic far essentially class respect bit thus example quantity absolute draw positive result extension sum somewhat city week city pick copy course week pick many production analysis etc like detail snapshot pdf describe reader week answer yes sum random need thus binomial rich class poisson consider extension refer poisson binomial call discrete indeed arguably distribution nontrivial statistic tail important chernoff use survey control analysis survey understood literature natural essentially draw divergence use framework output surprising result bit output description follow property draw least output define bit string bit complexity indeed observation learn result namely sum sample weight sum sum weighted variable weight different draw use run complement distinct weight simple sum unknown draw broad study introduction via eq subsequently many proof range line extensive relevant approximate binomial distribution see approximate via simple fall approximation moment target attractive moment approximation perspective show unimodal unimodal understand computationally unimodal state continuous unimodal distribution argument easily highly one complexity assign mass could point estimate na sample could see chapter target sample still cover remove refined understanding refined aforementioned binomial outline approach section structure roughly speak say support sparse close translate heavy cover run unimodal make translate construct hypothesis translate whose distribution cover hypothesis output capture learn non translate translate poisson hypothesis distribution handle alternate approach unimodal algorithm show exist size apply describe cover size generic approach density work distribution high account versus stress proper algorithm many technical implement high careful detailed specialized sum sum many distinct direct distribution pdf density cdf denote conditional distribution sometimes mean often distinction support kolmogorov kx xy ranging kolmogorov denote think observation domain every sometimes vice versa poisson poisson represent go collection binomial lemma denote mutually
nonlinear cv method knn regression method project prohibitive large data approximation gram cholesky reduce method kernel reduce memory approximation gram x tr mf g n dimensional propose find project projection matrix onto repeat result expect call second class encode gram limitation problem include slice problem propose partitioning matrix eigenvector tb x rkhs denote xx mx n appendix note assume convergence eigenvector use verify variant estimator true choose kernel knn distance cv regularization b sir iteration work propose also require gradient well sometimes fail cause large dim heart breast uci repository visualization represent parameter choose cv knn intel core ghz sec result separate cross validation unstable mm mm mm evaluating method project heart breast cancer uci repository rate knn project slightly cancer hundred thousand fast computational sec breast cancer sec sec heart disease cm breast two difficult handwritten digit gray pixel although subspace separate reasonably classification knn estimated subspace compare cca baseline table give show error improve dim mm uci repository provide classification knn classifier effectiveness estimate subspace classification dimension save computational I uci repository c comparison knn show competitive dimensional dim cca variant dimension applicability little algebra sec degenerate true work focus unsupervise employ extension nonlinear feature important apply give interesting involve direction discuss xx literature terminology completeness assume c xx ab side xx c xx pn xx xx xx xx c xx eqs proof complete expression yx sufficient follow suffice rate derive assertion operator side eq yx yy yx xx yx yx xx n xx pn gx c xx hc yx yy xx yx fc xx assertion note eq side assertion thm purpose base reproduce space comparison exist applicability without assumption result involve modern handle dimension reduction preprocesse datum expensive expression focus concern purpose reduction effectively dimension effective although I extraction choice nonlinearity method ii nonlinear transform combination give classical cca independence wish span without parametric unlike inverse sir employ lie statistic slice computationally strong slice number slice class another kernel require careful apply dimensionality gradient regressor sec contain one estimate limitation high limitation method use characterize problem without require convex descent gram large novel reduction unlike exist estimate kernel virtue method careful solve call mean assume x uniquely rkh characteristic gaussian rbf measurable rkhs dense direct imply rkh rich advantage kernel schmidt discuss fact express general easily make kernel nonetheless empirical estimator appendix prove rigorously expression method derivative open euclidean space continuously
fast shown compare ht passive known exponent rate reflect price pay directly learn density helpful minimax consist pay class put extra denote joint triple class measure p aa infimum density htb n decision subscript control attain adjust p determine hellinger probability divergence inequality hellinger concentration let bound lebesgue condition exist number correspond lemma complete fx h fx qp h qp h qp h thus verify first since r fx fx h theorem leave reciprocal right information estimator reciprocal fisher unlabele slow small definition detector estimate probability knowledge distribution probability aim quantify label training show lipschitz probability detector estimate probability locality general maximum converge favorable detector base mle converge optimal parameter typical risk flat near insensitive error mle bayes error achievable matter unlabeled detector minimax hypothesis prior unknown disease know proceed pearson pearson one detector base bayes detector provide train binary hypothesis detection theory extend handle criterion view case machine learn knowledge density hypothesis density terminology detector special know approach develop mle signal testing detector minimize detector denote risk incur place produce difference quantifie express joint identically unlabeled estimate stand base great operator performance excess concerned label prior risk property extensively exponent nonparametric estimator attain van around excess margin exist rule converge super rate view special take matter deduce determined locally convergence proportional smooth smoothness actually mild condition limit flat near insensitive error mle rate minimax optimal fig depict case smoothness htb b b label convergence unlabele training discuss detector train density probability minimax take triple class conditional remove mle parametrize risk explicitly thus n general result pmf dirac assume reason assumption accuracy relate lemma describe lipschitz property satisfy tp q lemma constant parametrize margin exist case boundary reduce interpret eq fact derivative first neighbourhood smooth derivative satisfy near make leading fast corresponding domain show determine assumption consider derivative boundary reduce infinity illustrate minimax also set constant r satisfie detector increase increase increase decrease mle constant lemma hoeffding constant accord know fast bad show rate standard passive consider lie go infinity proof theorem rate degree assumption mathematical excess q measure vanish chernoff always guarantee lie finite often arise practice rate relatively speak likely label convergence rate attention meanwhile also
law law rescale projection denote rescale selection sequel compact call old smooth derivative interior small th old large gaussian covariate obtain observational study differ across contexts brief value df measure observation design dependent indicate probability partially assign subsection say convergence distribution set old inside coordinate converge rate kk f kk partially logistic I q rate whenever h old kk kk independent negative spatial intensity assign x dx hellinger say converge fast draw probability density satisfy g g kk r g kk vary intensity along model entire density model conditional let isotropic function b qx db b prior g n ix kk x n kk later refine posterior independent relate two existence receive prior relate exist set nn b condition map prove condition subsection extend ingredient calculation reproduce kernel hilbert associated rkhs closure v rkhs measure rkh hx dimensional ball banach equip supremum ball stand rkh vector old real exist w b w theorem universal b b nf f proof calculation one operate relatively take however support behave compact make relate rkhs rkhs x eq follow application cauchy hx q theorem notice w n clearly w qx qx qx qx qx q last probability show sd complete assertion set inequality replace therefore b b kb b q b nr b v n operate common way rescale law prior distribution parallel explore point view study extension present verify restrict finitely differentiable infinitely differentiable regularity condition rescale model nearly depend lead seem essentially define joint lebesgue measure thing joint view purely prove preserve interested useful give place absolutely hope issue joint fast projection shorter credible band amount joint formulation delta delta ball around invariance haar delta ball integration formula radial radial delta eq imply integration estimate mistake root integral approximated count taylor expansion exponent term taylor term case limit observe determinant replace solution kind integral know particle integral eq determinant result q final proposition know equip rescaling nonparametric posterior class rate dimension function could potentially obtain rate low amount loss general priori explore classification density density rescale process equip variable low exploration novel without convergence nonparametric density process widely analyse function spatio temporal nonparametric thorough likelihood process bayesian remarkable nonparametric common specification equip posterior
exist derive ar source process learn ar temporal correlation slow deal bayesian source space make operate effective low computational mahalanobis matrix datum local cost noiseless experiment superiority algorithm counter intuitive may motivate derive analysis finally discussion notation bold vector dimension evident principal diagonal square block principal diagonal column represent kronecker column trace deal temporal easy modify optimization base find bayesian initially regression rule hyperparameter perform implicitly exploit framework source hyperparameter capture need let ml transform block elaborate I block bayes obtain give block zero clearly associate zero associate block evidence maximization estimation refer solution hyperparameter bayesian framework framework temporal via way learn matrix hyperparameter assign source correlation good source totally importantly minimum property unique solution confirm hyperparameter maximization maximize yield effective hidden maximize simplify result thus challenge model consider temporal elaborate either environment basic nonzero hyperparameter contribution dictionary column identity identifiable nonzero worse arbitrary instead however ambiguity covariance obviously excellent learn slow speed sparsity develop source convenience operation attempt adopt snr show adopt quite broad condition consider term base note use rule rule ambiguity snr number nonzero row explain presence form invertible low medium case suggest add namely ensure definite element zero noisy modify name modify rule observe incorporate source load rule feed suitable rule load learn definite low note zero follow insight since generalize root theoretic essential modification order original analysis nonzero positive definite overfitte change conclusion equal probability irrespective still except set global deriving avoid overfitte explain recover source noiseless add mean instance attract minima discuss local function respect present result composition concave equation degenerate local minima loose meaningful minima result role local minima satisfy role whitening source source minimum role whiten us motivation modify state reweighte extensive computer conduct representative informative trial trial create uniformly draw lk index experiment source ar th source rescale noisy unit noise adjust measure indicate percentage total noiseless fail recognize index fail recognize index source index case mse use measure suggest experiment identity suggest small communication code matrix matlab accord suggestion communication reweighte iterative reweighte extension iterative reweighte algorithm count adjust terminate attain otherwise increment implement toolbox noisy exhaustive search practically pick give small failure could nearly noisy besides choose estimate experiment code benefit discount source vary source easily observe ar process sufficient temporal six different level multiple surprising observation excellent matter source environment behavior algorithm noiseless save case correlation performance b htb source noiseless dictionary source generate correlation level accurately much advantageous source compare highly number vector snr db ar ar large error trade increase algorithm performance large source point ar sufficient experiment maintain superiority experiment snr kinds process e coefficient feasible examine showing outperform performance gap increase replace averaging source random imply algorithm maintain superiority source ar outperform scenario noisy case derive sign temporal beneficial answer dictionary measurement varied experiment e reweighted snr estimate thus reduce nonzero prevent closely snr three exhaustive previously outperform noise imply want emphasize snr propose near performance algorithm use zero poor performance indicate advantageous impossible think correlation always temporal ar coefficient result expect surprising well achieve correlation phenomenon observe noiseless observe helpful appear close recovery plausible explanation interact combine determine correlation help limit dimension infinity location noiseless vector almost one show noiseless first row trial form dictionary measurement vector process common coefficient varied see behave curve interesting phenomenon closely seem provide almost counter intuitively close change condition number form e nonzero condition increase ill source behavior matrix gaussian matrix emphasize importance exploit temporal motivate ill issue correlation vary exploit work algorithm presence start consider temporal issue unclear trade modeling extend one classify source assign capture source one far source group grouping accurately capture area work play role whiten regularization indicate one way estimate ar process common snr add estimate achieve many form advantageous issue channel extend assume different noise parameter work author channel noise instead symmetric variance know clear inside framework framework load pursuit denoising suggest method modify l choose environment see algorithm adjust speed channel channel extend source support vary assume slowly interesting approximated concatenation several support change appeal work algorithm basic effective learning rule superior basic especially sparsity show poor practice solve problem block temporal derive base latter extension mahalanobis distance extensive superior desirable would like thank considerable help help experiment mr writing code mr md provide author helpful especially thank outline equivalent globally noiseless equivalent minimizer state noiseless adopt notation indicate globally minimum e achieve summary equal rank write l lemma optimize concave obviously local minimum minimum chapter extreme indicate convenience element index element il ii il let definite e kl finish proof zhang receive electrical university china ph degree department electrical engineering university california interest compress sense blind cognitive receive electrical institute technology
notation pair wavelet subspace orthonormal q last term vanish general geometric scaling translation lie term projection difference fine equation multi subspace edge connect say analyze approximation dimensional using wavelet result sec dimension embed continuous measure let chart asymptotic decay coefficient decay smoothness manifold quadratic smooth manifold high geometric wavelet seem benefit high asymptotic affected distortion measure depend vary location estimate thresholding would wavelet third constant may price replacing may achieve appendix generalize manifold intersection intersection manifold cloud affect neither eq think matrix discrete e small k rough course geometric sample dyadic wavelet construct dyadic cell center scale j j kk accord multiscale variation weight near weight molecular practice choose iterate st repeat iterate construction performance reader ii constructions iv construct dyadic definition display code code construct dyadic wavelet scale construct dyadic tree cell convenience htbp wavelet illustrate construction present manifold dimension construct achieve average wavelet geometric wavelet select scale approximation coefficient horizontal index arrange tree index coarse top scale block display wavelet index wavelet row wavelet wavelet rate error use compressed wavelet wavelet fig wavelet coefficient good reconstruction e scale threshold compression dyadic accordingly leave reduced coefficient magnitude corresponding manifold essentially say consider mnist image handwritten digits digits digit digits three reconstruction fig dimension leaf magnitude coefficient stop decay certain wavelet future work efficient coefficient sec index element right convention magnitude scale fit last closely part equivalently digit reconstruct scale dictionary image handwritten digit successively point dictionary quickly orientation distinguish feature face database extend illumination note background variation decide solely face discard display image reduce complexity keeping algorithm compress geometric keep leaf function magnitude see indicate lack reconstruct corresponding wavelet basis orthogonal lack orthogonality direction scale efficiently dictionary encoding construction geometric wavelet coarse fine seem situation analogous also direct construct wavelet construct construct scale encoding may perspective roughly vary fine scale projection well set choose local cm dyadic cell local wavelet construct dyadic center cm c kx k c remove wavelet basis complement projection htbp end band limit leave dominant sort coarse scale bottom fine geometric transform space smooth characterize energy dyadic I care fact smoothing frequency issue observe band gaussian comparable norm smoothness g unit space band family roughly block dyadic band expect geometry ellipsoid axis length dyadic dyadic approximate transform discuss technique dictionary encode need extra geometric wavelet goal encoding precision intermediate approximation version correction sec encode encoding include cost dictionary define element multiply coefficient nonzero precision local achieve child optimally encode cost example leaf optimally plane corresponding size way encode datum separately use parent approximate encode direction parent encoding produce iii combination pca direction scale lastly parent encode wavelet encoding encode need child node cost cost wavelet complex parent node top correspond child wavelet intersection store encode cost small correspondingly section prune practical realization define node quantify encoding bottom start encode pca minimal let parent determine encoding minimal remove child separate tree new parent discard tree examine wavelet basis accordingly repeat htbp dyadic mean wavelet node construct dyadic cell tree center tree compute basis encode cost achieve encode parent wavelet good child child form discard parent node accordingly let svd encoding cost various think sort fouri multi real science news comprise document model word dictionary set pruning threshold rate cost encode wavelet curve way svd error pca second svd coefficient correspondingly discard curve split intrinsic create dyadic tree kn j summing point stop precision reach strategy jk cost leaf project correspond geometric scaling term performance vertical axis axis increase computation show ambient noiseless noisy handling time computation curse ambient computation vary scale multi roll red point generate cloud hausdorff dot line hausdorff cloud roll scale map generation svd p p hausdorff similar distance ambient space look hausdorff function imply variability almost along similar experiment point shape noise term unable advantage intrinsic seem much construction take approximately sec technique measure support approximated multi construct report publication cloud distribution training belong fix scale drawing define way point plane database train project principal component encode run compare quality hausdorff randomly measure generate quantify capture variability generate multiple point hausdorff call hausdorff variability tradeoff hand requirement correctly fall greedy region correct bias geometry spirit publication refined application currently develop user interface interact geometric well set technique pruning construction cost slightly near minimal give approximation datum depend locally thereby approximation box one cast probabilistic subspace wavelet prove upper start every j dyadic respect dyadic calculation spirit coordinate write hessian calculation high pass pass hull curvature direction last bound formalize span top high tangent plane pass provide obvious assume interpolation curvature multiply quite mr thm exercise remark interesting exploit project example point dictionary either dictionary dictionary dictionary vice versa contrast nonlinear multiscale dictionary depend geometric resolution analyze space regime setting important various application gene array eeg manifold value datum investigation several old estimate intrinsic manifold construct application multi scale efficient precision particular b space organization geometric set signal research harmonic typically class term dictionary atom dictionary desirable highly concentrated require processing interpretation motivate construction fourier basis wavelet dictionary representation suitably define certain class simple originally prefer towards dictionary frame library cosine fusion transform usually organization dictionary trend motivated signal structure allow function construct optimize signal becomes typically complete give rapid possible construct sample harmonic approximation machine construction seek certain dictionary give current add size column fix minimize alternate refer reference therein construction dictionary svd bayesian involve heuristic intensive dictionary precision analyze mathematically challenging dictionary multi resolution analysis inspire multi dimension fashion analysis guarantee algorithm growth control depend datum wavelet vector space geometry wavelet share similarity wavelet transform etc crucial arbitrary nonlinear manifold albeit consider crucial construction may algorithms use wavelet paper organize geometric wavelet fashion sec introduce orthogonal variation two section efficiently cost sec distribution point future direction borel measure paper restrict theoretical section compact riemannian embed endow natural volume discrete metric small ambient typically unknown practice also additional assumption geometric resolution geometric cell linear efficiently encode
actually complex base base complex network concept partition detect shape present drawback identification provide division consider metric connection every distance accurate identification traditional chebyshev fu distance artificial expectation verify rate complex seem able clustering base potentially traditional complex topological connection power undirected matrix zero account present strength different cluster coefficient measurement allow reveal far purely addition modular cluster module brain function community develop community also partition basically group spectral g agglomerative choice depend complexity describe quality division term metric construct fraction connection modularity calculate value division network modular cluster identify understand region high community inter property object attribute length width group feature belong high object concept object connection quantify pair vertex vector similar object quantify adopt automatically necessary modularity complex provide many account execution database take organization modular measure natural similarity measure expect measure inverse distance value interval interval distance assume interval chebyshev values chebyshev fu fu assume limited divide former modularity repeatedly join pair merge great small decrease modularity division result high method lie walk community consider time execution method discuss result fast optimization account breast visualization project analysis cluster database range necessary attribute transform equal deviation call cluster small error among note limitation result chebyshev distance community obtain case equal provide specify database value modularity partition nevertheless proximity modularity cluster error case chebyshev metric analyze error without verify error case dendrogram chebyshev modularity c error first database result small produce distance community nevertheless produce modularity separation imply methodology cluster breast cancer database l comprehensive complex approach discuss section allow database cluster point separate well case cluster automatically maximum modularity traditional result complex chebyshev provide small b fu chebyshev clusters accurate well traditional algorithm go case chebyshev among mean database symmetric equally small rate circle figure present become tend take base mean complex tend figure mean complex based identify f chebyshev distance chebyshev distance obtain modularity determine density adopt measure chebyshev chebyshev distance cluster proximity measure represent adopt suggest improve new compare small account proximity measure chebyshev inverse chebyshev similarity feature vector community reveal small real world approach artificial application constitute promising possibility da algorithm theory graph distance partition spectral automatically overcome account algorithm complex quantifying find world database datum traditional chebyshev distance similarity identification rate intrinsic activity facilitate huge receive matter categorization human almost material due
discover cycle change cycle reduce schedule give experiment schedule gpu demonstrate key benefit exploratory work produce experiment smc sampler particular decide cycle perform mix relation quickly scenario produce particle allow practice mix desirable dominate perform general want smooth picture successive posterior fraction intermediate control recommend scheme convergence sampler sensible note aim distribution give run benefit smc exploratory plot function informative understand predictive fail full key aim report bayesian exploratory tool posterior describe statistic contain heavy run smc gpu sampler generate plot weight smc move true coefficient color one observe path mode jump away concavity marginal density mode decrease global mode red posterior global jump find iterate median plot cumulative quickly origin precision decrease scale scale marginal evidence q indicate fig concentration posterior tolerance correspond covariate appropriate important together plot highlight influential evidence strength change freedom fig moreover map much dispersion plot highlight plot plot four credible mean weighted function show marker weak association nan sparsity induce tail smc use uncertainty moreover calculate concentration away fig summary fig plot overview importance prior plot smoother investigate detail tail induce great rapid concentrate zero away retained fig plot fig evidence variable predictor comparison observe support coefficient fig summary plot snp turn adjacent snp marker leave ambiguity fig partly point association around marginal fig see red hold marker interval green considerable actual value association explore well present path aid understand genetic association summary coefficient range complex indexing prior provide scale likelihood computation demand would day worth time conventional cpu processing make core gpu producing improvement run benefit within working day acknowledgement acknowledge computational biology acknowledge trust centre figure solid line estimate mean blue red reduction shrinkage htp mode use htp green median red htp htp credible green median red htp htp htp plot credible median black mean red htp double htp htp htp htp green black red help nature association region wide association adopt exhibit attractive double tend pay attention obtain scale shrinkage precision heavily concentrate around prior coefficient distribute around map generate distribution prior computationally challenge amenable carlo smc scale parameter smc processing unit efficient inference generalize obtain genome challenge marker univariate test test marker marker highlight region genome motivation decompose span marker linkage lead marker regression priori marker responsible hence prior induce tool aid understand phenotype interesting signal consistent causal attention phenotype although likelihood via gain popularity seminal interpretation posteriori map prior penalty tend increase zero perspective little justification full bayesian exponential monte mcmc penalization coefficient towards evidence induce penalty increase mention statistic reweighte concerned publish article date analysis context recent map gamma sparsity sparse mixture adopt contain spike component broad classified relate irrelevant setting explore sparsity benefit allow visualize scale regression student absolute attractive analytic form interpretable prior task believe much explore form posterior path towards provide heavily origin distribute maximum stress model probability interesting predictor phenotype exploratory challenge suited carlo index graphic graphical processing first suggest context sparse likelihood double pareto prefer easier interpret logistic generalized context develop gpu run pay make hierarchical em take conjugacy inverse gamma different find q standard exist appropriate perform estimating asymptotically sparsity datum smoothly step method examine expectation asymptotically unbiased satisfie worth bayesian represent belief irrespective may receive recently deal solely exponential prior sparsity mixture list mention table compare prior know understand prior less amenable fast parallel gpu use freedom student researcher exponential ab form development use anonymous centre snps genome originally control identify cancer space snps column plot marker strong make multiple pseudo five coefficient realistic size generate bernoulli individual individual marker marker coefficient marker create explore posterior sequential smc prior dominate large limited impact
coordinate look mutually exclusive nonempty indicate single block index define quantify hold among block find partition multi identify variation theoretic early multi state system dynamic start start dynamically divergence transition specify generalization couple brain cc function partition provide uniform start bi h rest dynamic interaction identify modular information system great interaction every subset select stochastic favor unique due author propose normalize justification hoc work identify module principle penalization modular clear interpretation theory statistical assign maximize choice equivalently distribution kl term reach excess error minimum specify call state start prediction system possess perfectly previous factorize parameter condition impose depend quantify divergence take value previous point learn evaluate new infer mapping start state future risk attention component arise consequence minimal excess optimally interaction capture term train arise maintain value space parameter uncertainty simple approximated number p possible beyond distribution low partition provide predictive risk offer well small generally induce present principled weakly module amount trade emphasis complexity stochastic interaction group variable learn block search factorize parameter select decomposition generate multiscale infinite partition minimum form dynamic variety possibility exist product chain dirichlet priors b start index variable support dot bottom accumulate model graph total modularity cumulative risk system probability assume variable maintain amount variable value couple illustrate fig coupling modularity decrease modularity independent modularity grow without proportional flow partition choice pp uniform subset lowest factorize optimal decomposition start risk decomposition become causal organization lead decomposition compute copy previous show plot column calculated uniform starting choose whose independent state risk distribution show induce decomposition partition optimally start identical system dynamic risk decomposition architecture modular indistinguishable causal connection highlight interaction handle utilize semantic intervention recently theoretic direction distribution impose dynamic interest causal boolean frequently artificial life biology organization state mention start system change formally risk eq need start uniform state kl divergence b whole block reflect perturbation block dynamic partition consideration less statistical eq p partition assume risk causal interaction modularity organization model infer modularity predictive treatment connect theoretic provide identify module causal train dynamic test give rise measure identify causal framework modeling framework produce total quantifie predictive advantage modularity process note cognitive proceed suggest modular limited learn gain power term depend model form product model heavily parameterize think generalization module search decomposition probability depend fuzzy modular variable module overlap generally identify modularity biological modeling detection inference organization feedback identify understand modular study several many term statistical modeling amount simple trade simplicity predictive multiscale decomposition dynamical weakly couple module version dynamical gene network modularity broadly speak modular compose weakly numerous complex suggest effect perturbation great operational module arrange argue modular adapting change combination integrate arise result adaptive process fluctuation network pattern modularity acquire central vary scientific science reference formal approach apply discrete multivariate whether boolean dynamic time recent community static graph modularity organization dynamically interact argue analysis life light notion utilize domain modularity next provide outline modularity system
pyramid consideration solution ode practical example calculation apply age duration due case group year report claim health year tend incidence reason age pyramid statistical office peak decade peak age test institute diabetes center calculate duration disease incidence equation applicability datum ordinary differential deal disease duration disease calculation incidence formula mean age disease develop apply basic incidence prove helpful model mean respect disease consideration respectively transition intensity rate incidence rate intensity duration disease state suffer disease disease duration henceforth depend system equation patient eq play system derive ode relate change age rate close analytical solution special expression solution age incidence ode direct infer incidence rate addition incidence cross study incidence terminology duration disease total spend incidence age exceed first age age correspondingly age age specific ode completely rate remarkable age pyramid inherent consideration dependent function number people age simple rate birth rate depend age population office capture age replace incidence separately general population life office reference patient type incidence knot ht integrate initial via fourth r foundation statistical year age almost early incidence rate age almost twice high lead age year age
retain variance project onto pca average coarse search case equation interact eq step root interval error normalize energy c c c purpose potential rich reproduce extra double precision method line give percentage lose projection
class tree minimum principle space use store case compose associate encode induce represent total graph probability need ignore candidate network dag store parent include parent enumeration cardinality parent encode bit node encode length store description length depend encode training probability code depend length description dl px variable occurrence encoding count represent table parameter minimize rewrite hx entropy theoretic interpretation representation bit encode combine total length score statistic score proportional structure satisfy independence relation encode pg pd constant structure follow structure address difference penalty assignment namely datum parameter distribution pd nx ip decomposition p observe sequence te tt assign pd criterion use logarithm posterior pg pd ratio equation log structure posterior probability drop express assignment parameter amount gp configuration degree taylor approximate plug equation approximate diagonal get simple approximation term increase linearly approximate bic bic use without assess measure predict minimum except length high certain prior decompose x score base structure search whenever score structure part remain simple lead maxima constrain list follow simplest hill hill modifications modification modification accept hill list acyclic add remove graph choose high besides hill search anneal procedure exploit space structure example hill compute score check large candidate hill solve hill return turn parent key parent early mutual determine existence learn propose evaluate dependency mutual information xy px qx information discrepancy estimation qx qx number expensive easily probability utilize node define ix metric introduce j ix metric empty measure iteration incorporate structure candidate hill stop criterion usually terminate long increase update network candidate terminate remain unchanged score monotonically bound score guarantee stop candidate enter need sound identify true parent iteration acceptable approximation parent parent allow risk discover suboptimal unnecessary efficiency imply impose either search max hill explain parent problem impose algorithm capable hundred hill search usually optima list record loop minima solve local optima follow pick arc globally arcs arc search generate operation ordering iterate randomize order change conventional hill relax max parent author domain model optimal procedure modify structure benefit add parent single hill propose decomposable markov network iterate link set mutual hill list section imply property order would parent fundamental ordering scoring network order degree equal ordering order second current already imply order perform potentially disadvantage statistic ahead parent discrete simply count structure parent node thus chain carlo typically network structure reliably hill list define score good start order swap operation thus branch swap search order find prevent swap execute try parent node follow exhaustive programming dynamic feature pf unlike programming acknowledge feasible fitting usually add loss e importance value square algorithm encourage towards zero towards invariant test generally prediction logistic regression regularizer also loss quadratic minimizer close dependent regularizer regularization network coefficient drive play role equivalent equation see figure avoid density decay increase observation contribute fix setting non zero tend exact parsimonious property regularizer lead sparse graphical popularity year mainly choice list objective model n assignment subset give log partition aggregate feature order dot product prior z ise x minimize pn estimate graph nonzero weight neighbor gaussian author choice regularizer weight circumstance optimal choice certain full author direct graphical always multiple px least joint dag pd px distribution I precision precision prior I pp associate integrate sample th px ki p x map ki regression advance incorporate undirected graphical optimization kf f f mapping linear high th training category identification multiple branch system relation infer system use differential algorithm key illustrate steady g substantially approximated differential x tt gene network residual coefficient x w regularizers ij first decompose training constrain interaction explore graphical large covariance precision sx x efficiently nesterov description try minimize author propose find parent apply parent neighbor solve follow regularization log parent parameter hybrid approach since incorporate bayesian network incorporate selector encode bayesian encode algorithm author max hill algorithm bayesian algorithm share hill step learn discovery algorithm parent conditional max min variable minimum association parent step hill within constraint skeleton algorithm impose author parent node score parent neighbor discuss parent identify skeleton structure create mb mb mb replace candidate procedure potential parent application search mb sc exhaustive besides list similarity may euclidean boolean assume boolean gene try profile employ reconstruct network correlate measurement indirect interaction two thing unclear approach elimination justification discussion limited loop edge cycle category matrix interaction factorization max factorization reader refer corollary graphical modeling graphical intuitive structure insight inference sophisticated carry graphical field bioinformatics science analysis cope combinatorial space structure paper notation represent direct edge parent child call direct parent domain may attain probabilistic graphical condition formally localize identifie predict node imply theorem reverse direction node graphical essentially divide two group undirecte ht mrf call factor clique potential clique graph potential domain joint element non clique fully connect maximal clique maximal clique probability calculate normalize clique current clique variable maximal practice w function real training sample representation especially convenient evaluating difficulty partition ise node spin clique parameter represent external particle representing px variable normalize follow gaussian zero direct representation joint distribution network underlie compactly advantage network direct acyclic dag correspond random describe configuration x independence together probability via px px ht way critical specifie dependency parent clear size grow node take tree exploit node parent vertex split vertex condition root place softmax parent summarize contribution parent node common many associated choice density incorporate determine node sigmoid belief px j logistic sigmoid root take parent noisy popularity category well employ test conditional attempt constraint pc try pair possible size difficult reliably independence constraint lack objective try directly structure framework name bayesian determine existence independence pseudo slight modification undirected graphical vertex subset undirecte write adjacent separate adjacent adjacent direct path edge edge adjacent possible condition super graph become infeasible sparse besides reliability determination conditional reliable name
contradiction c determined rectangular coordinate system origin old rectangular coordinate dimensional less otherwise definite convex ac ax contradiction integer put denote p pt fx px py positive cardinality p jx I ir I jx x jx jx vx fx jx jx vx lemma dimension great easily combination naive linearization subspace precisely however linearization seem establish px bound degree aid scientific research technology second aid university establish dimensional ellipsoid set class employ asymptotic estimate dimension exact euclidean ball dimensional da section theory theory probability real integer component belong open great tx converse ta b b ta
variation model upper receive imply indeed difference account assess combine datum great posterior rv rv amplitude outer velocity contour rv period outer contour arise plot also euler turn noise also possibly cause say report rv appear uncertain indicate uncertainty show report ms ms euler ms ms euler ms ms claim six rv challenging combine rv rv combine set combination show five check latter factor order probable another parameter map set limit ms amount ms ms ms ms rv star star rv surveys advance recently fourth signal analyse combined rv model criterion however start check rv published rv publish additional rv bias bias combine set imply rv model bias corresponding period fourth day day see contour contour show amplitude parameter top bottom contour contour blue amplitude differ also contour publish less periodic datum parameter table rv day ms ms ms ms solution significantly solution estimate period day whereas day fact rv table ms rv latter estimate excess include likely roughly contain rv likely assess differ adequate bayesian rv make rv describe use uncertainty cause model magnitude suggest rv uncertainty force uncertainty parameter uncertainty considerably likely base turn table conclude uncertainty rv rv data criterion unfortunately possible turn rest note consistent four rv differ significantly solution period stability estimate period could coincide could stability investigate able system successful variation measurement model commonly whether statistical describe respect measurement whose dependence physical derive principle description numerous effect account system insufficient whether valuable extent assumption exact nature construct prior early use select formulae posterior model great probability describe model case star european david present simple whether describe mode derive criterion set usage radial adequate describe two datum set model reasonably radial update generality assumption need apply probability radial rv detect nearby nearby star rv survey rv update little bias individual determining analyse rv bias receive purpose method naturally commonly use jeffreys efficiently compare model whether great accurate measure bayes goodness set measurement goodness reliably however model assess contain accurate measurement reliably several adequate author determine source aware single discuss single describe importance signal limit sensitivity target survey model superposition trend corrupt analysis rv variation usually despite effort model magnitude dark rv excess explicitly statistical two measurement measurement way practice describe finally combination probable probable arrange system calculate magnitude determine odd determine hasting also several rv existence work play important assess nearby star commonly tool assess probability likelihood density parameter model interpretation assess confidence view probable strong adopt need solid ground probability especially probable possibility explain probability likelihood description independence measurement model model result marginal marginal probability therefore describe measurement bias description practice wants determine fact model bias measurement datum three number estimate e receive adaptive density sensitive robust enable rapid metropolis assume reasonably sample roughly period posterior density nonlinear rv however th member file chain indeed converge five integral digit chain likelihood sake uncertainty semi axis rv draw mass calculate axis density rv reasonably model well motion negligible practice though several rv understand adequate cause whose period noise cause surface refer aspect rv statistical lead analyse
base fractional brownian motion reason come mathematical underlie fundamental describe background stock trading inherent stock fundamental noise model involve fractional brownian motion study paper drift originally observation present differential equation fractional brownian fractional several drift propose fractional parameter differential involve brownian motion compare mixed standard non formulate strong estimate need behavior fractional derivative fractional brownian growth organize fractional establish concern strong sequential result strong consistency drift fractional brownian fu fu bf l notation statement concern introduce eq generalized eq b f l integral respect evident us stochastic wiener brownian parameter lebesgue integral lebesgue integral coefficient assumption hypothesis differentiable exist assumption exist satisfie satisfy specifically old start consider recall fact drift estimation hold together continuity space process mm integrable h martingale two eq observable condition c hc integral process functional convenient ei sd likelihood integrable sd result r tt construct preserve return assumption exist lebesgue observable assume generalize traditional estimate right provide shall investigate strong stopping time form version estimate technical maximal metric space separable element minimal center space exist exist constant follow increase suppose ab random assumption turn separability separability ex statement ready state asymptotic growth series h old take converge case immediately chapter directly efficient sd consistency evident evident non bound ds inequality yield example correspondingly linear version bound function exist non zero consider assume derivative integrable sufficient fractional derivative locally integrable arbitrary locally limit representation true condition omit random hold identically decrease derivative preserve zero estimate estimate strongly assumption addition consistent shall strongly alternatively strongly fractional interval likelihood however hold case present follow form accordance consistent coefficient bound dt strong assumption strongly check obviously dt bound must moment gaussian random calculate attain maximum maximum specifically follow apply formulae wiener integral fractional divide assumption vanish separate therefore boundedness relation behavior let denote several positive sake technical simplicity omit multiplier contain old q admit derivative se bound behavior standard satisfie eq consistency establish consistency case version sde evident construct let estimate se q proof fractional fractional parameter let way rewrite rewrite fu u follow combine ready fractional motion integral du opposite explicitly evaluated similarly eq derive similarly relation handle
survival splitting class early vs failure reduce bipartite survival outcome simplify though class model produce associate test bipartite method strong regularization naive bayes build univariate avoid multidimensional naive bayes curse sophisticated term strong approach smooth method cox well large study type computation real life exclude set size artificial keeping advantage datum build predictor problem build weight vote several implementation observation two survival calculate predictor smooth calculate outcome scoring implement experiment class survival censor censor exclude cosine transform make evaluate spaced default naive density error g smoothed use procedure stand function polynomial two weight formula outcome class tie ci step post improve dataset threshold selection use optimize make update calculate relatively filter tuning marginal class eq frequency ratio posterior conditionally equal prior point association influence otherwise kernel procedure fix ensure smoothness unlike advanced risk modeling method parameter tune survival sir david cox risk hazard cumulative individual failure cox hazard ph hazard dependent hazard rather score r package popular cox ph cox include feature exceed method build zero change aic aic training miss patient free survival diagnosis three characteristic overall among nominal compound fu moderately thing model recurrence death treatment option fu tumor survival patient advance cancer score patient perform usual activity characterize loss dataset record dataset patient feature bc breast record primary description disease al cell contain expression associate microarray cox apply data expert relevant along include patient author aggregate signature signature gene modeling aggregated patient value feature dataset split train split method test split index cox ph regression record miss neighbor imputation cox mean ci bold font features measure small smooth good low advantage prominent high process version original value include signature equally aggregation aggregation penalization use cox ph time almost identical ph model lead selection advantage indicate superiority regularization confirm advantage conduct control aspect instance series experiment artificial gradually dataset experiment conduct list miss near neighbor draw training set size ci number high set small dataset smooth experiment confirm smooth rank quality half year confirm cox ph regression good survival available select artificial logarithm distribute assumption depend variable uniform half assign censor censor censor time indicate observation occur censor formula every reality make less clear record size multiple sample show method train cox ph regression add achieve cox ph tendency method overfitte good accuracy smooth dataset datum serve justify survival study easy factor affect outcome noisy high address analysis bipartite ranking multidimensional avoid curse smooth marginal smooth proved comparison survival cox test method life dataset systematically instance sample artificial gradually increase indeed number make valuable study rank motivated risk bipartite application another study traditional associated sir david cox dependent hazard event survival cox proportional hazard regression popular survival hazard unknown individual result hazard cm bar e mail medical modeling dimensionality simplify algorithm aggregation predictor
atom sampler synthetic bernoulli patch value th stick break prior factor collect iteration variance process roughly fast second present integration two process region analyze poisson process use fall interval poisson alternate calculate replace result break stick break r program david nsf p foundation google breaking construction specifically beta derive truncate beta tight develop process part nonparametric prior collection binary obtain ibp focus dictionary motivate ibp chinese step stick ibp stick breaking finite limit ibp process mean show provide paper derivation addition tight literature section poisson provide immediate beta vary concentration mcmc stick efficient b process review beta process evy review break construction beta evy weight lie discuss generalization atomic measure value contrary measure measure beta parameter bernoulli process show ibp ibp clustering beta follow ibp condition least sampling measure example draw beta poisson generate count poisson pairwise underlie q base goal notion stick break general discrete stick breaking play thank largely seminal stick follow representation previously atomic sequentially receive draw stick break atom stick keep illustrate reduce construction q approximation process show stick distribution derivation derive beta poisson prove beta differ beta process collection section generalization beta basic poisson lemma process lemma concern variable second superposition poisson underlie countable collection independent process superposition I fix let I ic superposition countable lemma dd calculate summation group group use calculate atom weight I df df stick breaking process role truncation mcmc evy measure measure form solution calculate poisson decompose follow eq complete stick underlie process dd beta stick break superposition poisson truncation construction center contour bind plot corollary process arise representation characterize beta process discard underlie corresponding measure closeness truncation bernoulli set parameter could globally share closeness process measure truncate beta slight modification round say less minus bind account atom truncation construct process mean probability distribute truncation additional integral increment give definition constant base poisson limit present partition poisson transition kernel superposition modify construct round atom draw atom draw weight use break break union new process idea stick poisson atom consideration likelihood integration computationally sampler derive quantity specifically collection bernoulli atom care exponential atom multinomial atom atom belong
demonstrate constructing allow hide infinitely typical approach compare alternate compute likelihood generating model ratio likelihood may even hypothesis otherwise choose derive regardless standard algebraic therefore influence therefore exceed interestingly observation bound trivially need rule latent possibly occur number sequence rewrite must put test line weight observation exchangeable k pa simplify domain constructing simplex form hull weight coin requirement normalization exchangeability refer extreme point form extreme boundary region convexity distribution positivity seek example exact hull half polytope suffice description convex instead distribution determining find outside explanation hand conclude example tt outside demonstrate hyperplane dot suffice produce course experimental visualize test problem large develop tool find algebraic finite polynomial inequality conditional parametric inequality equality constraint positivity observable write measurable like represent call hull set construct relaxation converge ultimately alternate representation contain formulation implicitly translation also contain convex hull hull finding comprise amount relaxation set q converse sum square body semi completeness describe polynomial deal simple relaxation degree sum polynomial guarantee negative write semidefinite amount problem semidefinite technique demand positivity bound thing easier set cone default polynomial form sum x g I also cone exclude provide impractical guarantee convergence hyperplane maximize format translate semidefinite matlab convert sdp code large solve difficult construct rigorous example correspond call weight coin hand increase produce constructive specific polynomial statistic direct acyclic reflect rigorous conditional independence equivalently specific distribution term although graph allow intervention rarely inequalitie test quantum specific realization inequality introduction formalism briefly imagine detector pass summarize basically party measurement outcome measurement local variable formula nontrivial reproduce expect party reach hand large evidence favor achievable measure quantum particle extend space observe polytope np whether distribution look tight relaxation individual behavior highly correlate effect influence actor neighbor individual similar suppose friend time begin would certainly influence friend typical attempt covariate either substitute case come relevant take latent actor action various attribute e edge possibly unlike consideration asymmetric give correlation write restrict possible difficulty transition change essentially freedom static demand crucial stationarity transition look stationary coin unlikely independently influence coin intuition precise see mapping observe distribution structure combination e directional variable value parameter take outcome principle consist outcome would like pick mapping q hull visualize statistic correlation vice versa state symmetric fix latent take hull region fig outer dot want e hull polynomial omit brevity hull line construction test explanation correlation alternate reason alternate produce latent second could produce correlation impossible loose nevertheless deduce strength usefulness start real online news com edge semi know model started pick copy slice change apart consist indicator possible outcome distribution ia hoeffding give increase bind suffice quickly increase
analogy mechanic annealing generate stationary state initial increase put mass global crucially prove schedule use schedule guarantee reach surely anneal common geometric schedule schedule anneal sample anneal instance construct sample readily anneal level use accord proceed sequentially modal way straightforward mcmc distribution statistical mechanic learning field many recently transition importance metropolis hastings monte paper mcmc bayesian three describe follow importance employ sampling distribution essential ess measure correspond mcmc sampler name subsequent aim derive efficiency illustrate concluding remark distribution parameter respectively distribute markov generate intermediate sequence subsection adaptively sample obtain anneal motivate rigorously generate size correlate desirable rarely property mcmc stationary importance use independent eq importance reasonably accurate condition analytical distribution markov chain form describe part dirac form metropolis understand example walk metropolis hasting proper transition simple illustration consider panel interest j panel hasting j mention density continuous overcome take j j motivate j n j n n iv define us markov chain generate j ji indeed chain enter accord n n calculate acceptance involve replacement ergodic markov continuous proposal calculate discussion height mm state proposal total generate state generate accept stationary anneal chain aim application e normalize sample usually readily suitable distribution previous sample posterior description aim choose anneal affect aim advance require often available adaptive sampling degeneracy ess ess imply sample ess j prescribed characterize eq level produce ess small reason select aim anneal adaptive annealing give rise height threshold ess anneal level level n I stationary aim j increment anneal total distribution anneal generate chain algorithm description markov generate annealing density choice aim absolutely monte carlo generate sample accurate estimate like highlight difference role hence thereby sample sample become aim increase generate markov chain refer critical safe posterior use assessment numerous diagnostic comparative none one finite mcmc assess run chain requirement understand easy multi exist theoretical unimodal affect speed anneal intermediate inaccurate aim intermediate importance prescribe threshold behavior regardless speed evaluation important relate high speed need follow ergodicity ergodic take place anneal chain notion call integer uniformly satisfied kolmogorov nd equality center ergodic always fulfil hold give enough next reasonable gaussian aim produce practical case recognize acceptance jump neither picture property high acceptance implement aim bound expect associated annealing probability proper transition transition bind equal accord accept candidate step side nothing else acceptance expect candidate basically says never exceed acceptance rejected automatically reject repeat dimension modal dimension model modal mixture denote vector refer display posterior cluster overlap reflect space level posterior I chain anneal b black annealing figure independent find grow aim hasting fair comparison metropolis likelihood I chain aim trajectory markov aim successfully mode interested mean vector component aim variation average display posterior factor run interesting nearly I demonstrate truncated density sample markov sampling posterior five examine challenging detail implementation run outperform capable generate mode five scenario outperform table dc sample level proposal whose spread monotonically high local large local sample neighborhood far look local scaling run observe peak global suggest small anneal indeed natural concentrated attention table anneal monte variation sample aim decrease affect total proposal improvement correlate variation illustrate feed forward neural network model approximation goal function compact base example feed forward activation input unit hide connection connection weight unit properly sum unit cube architecture input unit component introduce prediction output principle produce subject probability prediction initial assume posterior framework obtain integrate nuisance mean aim per threshold ess aim total anneal sample approximate hand intermediate approximation plot visualize approximation plot paper scheme sampling distribution aim well importance simulated annealing behind draw intermediate posterior approximation independent hasting algorithm sampler generate draw name ergodic derive show condition often fulfil parameter recommendation value theoretically demonstrate three modal acknowledgement science foundation award california institute technology finding conclusion recommendation national foundation remark proposition question computing sciences science institute bayesian many posterior monte carlo applicable posterior distribution popular sampling annealing call uniformly markov recommendation demonstrate three example markov express quantity interest respect monte expectation estimate average often encounter multi explicitly strategy importance markov briefly review method play nearly old readily computable sample estimate converge almost surely number condition hold prior stand error especially nearly although make preferred major instead especially know multiplicative constant depend call importance yield poor estimate efficiently importance prior different therefore inefficient complex challenging minimize variance early optimality normalize integral independent posterior distribution physic use
put mass consider conjugate chi restriction ensure wide support beta assign belief highly origin score close fine straightforward recursion analog hierarchical predictive writing let predictive likelihood write predictive run value produce implementation maximization bfgs pr produce recursion methodology factor order score upper pr choice nature induce visit replace number add stability parameter estimation experience permutation negligible reduce permutation maximize discovery represent argue local false rate fundamental pr readily false example choice investigate simulation benchmark result fourier take ensure find nz du tail nz nz asymmetric portion mass concentrate origin nz away zero six form time pr regularize average permutation table summarize simulation accurate omit pr specifically practically namely smooth zero time roughly second rate rate nan figure oracle oracle fdr message test across sparsity similar non test relatively false rate spike large theoretically bayes bayes observation may interesting dataset careful versus microarray consist differentially score fit clearly fit fdr thresholding report express score standard z report hope comparison oracle method pick reasonable bit oracle false plot black black solid gray fdr value gene fdr gene fdr oracle fdr restriction component nan purely verification kind seem could insight entirely formulation accept notion produce z magnitude j selection zero ability still perform case yield model argue abstraction indeed test score magnitude origin give order interesting ordering wider issue conclusion focus work behave nan score indeed identify impose strong theoretically nan suppose base assumed uniform purely verification could insight biological entirely nan justify formulation make parameter identifiable likely encourage flexible situation concrete average identifiable wide abstraction score whose magnitude shift order wide nan breast study exist fail discovery point dimensional set classification gene collection classification statistical shrinkage whether interesting reveal limited resource force interesting employ exhaustive calibration infeasible multiple vector exploration intensive multiple testing software software find website acknowledgment author grateful associate anonymous suggestion discussion portion department mathematical university see take e degree satisfy therefore recall say generality imply rearrange modulus unbounded bound contradiction symmetry equality relation easily complete variation predictive pr pr development consideration mix detail find focus text unconstraine e kernel pz u nz gradient compute return identifiable consideration empirical nan case use computationally recursion nonparametric mixing lead estimate false simulation demonstrate handle tail turn realistic conclusion predictive recursion testing arise abstract score define derive treatment control characterization rejection insufficient perform major development shift treat independent treating exchangeable testing model share separate elegant testing problem assume mixture describe distribution z reason adequate choice various draw throughput particular responsible approach offer substantial advantage technique currently formulation application typically link phenotype exist two take encouraging close conservative detect majority interesting microarray study breast study method produce cover tail score leave nan zero gene make number tail study breast group representation likely magnitude belief score wide range exist fair interesting method study mention find classification similar simulation score range base group efficiency recent stochastic recursion estimation mixing dominate address use mix marginal strong dirichlet calculation group model specification parameter plug present section non nan example artificial microarray oracle likewise breast produce fit tail consistent sophisticated technique nan lebesgue version identifiable general identifiability nan identifiability shift tail correspond nan incorporate similar belief component strong tail histogram report comparison separate tail important drive force
derive pairwise associate weight image optimally align optimally transformation transformation dimensional shape orthogonal linear construct principal map dimensionality heat heat field transformation hilbert particular metric distance application meaningful transformation organization connection manifold way manifold point cloud dimensional specify vector heart mapping construction way embedding laplacian cloud sample dimensional manifold operator function result convergence except geometry achieve basic connection laplacian explain report diffusion nystr role play heat heat relationship diffusion geodesic distance nearby application reference alignment extension topological though datum cloud high dimensional manifold manifold cloud consists view orthogonal cloud local plane pca fix keep convention except confusion arise neighbor satisfactory choose boundary curvature due slightly define data neighbor purpose disadvantage proof pca monotonic diagonal define vector purpose emphasis nearby singular advance neighbor exactly b estimate number zero singular value curvature singular practice dimension value account enough percentage threshold estimate singular singular integer dimension manifold round sum median intrinsic q minimize sum estimating compare step facilitate notation write svd column orthonormal know define vector column establish eigenvector corresponding interval never actually form requirement perform intrinsic use try simple able alignment suppose nearby parameter later manifold boundary tangent space space exactly matrix differ orthogonal equivalently operator copy subspace usually exactly necessarily orthogonal close orthogonal local basis optimally align orthogonal refer optimal orthogonal basis alignment matrix basis align nearby align graph vertex correspond point basis align iff weight construct matrix either multiply zero w regard length vector tangent average non linear single transition sum end use complete eigenvalue diffusion mapping eigenvector usual dot diffusion weight start affinity consider connect sum path transformation length orthogonal multiplying transformation order result analogous operator geometry path point curvature thus add may happen affinity affinity affinity consider matrix affinity square path measure connect also transformation agreement differ eigenvalue eigenvalue decrease tv li nd ti affinity finite dimensional hilbert manifold embed invariant dot diffusion dot product v li li ti inner space mapping consequence well large truncate equivalently ti ti non manifold would real truncate diffusion slightly different diffusion mapping matrix eigenvector another vector schmidt embed datum hilbert proper vertex define diffusion ti ti ti unit angle embed point mapping suggest distance ti ti mapping obtained suppose define symmetric degree nd tv sd tv next diffusion discrete walk diffusion manifold graph converge pointwise operator smooth function laplace operator point graph laplacian eigenfunction pointwise convergence sample manifold laplacian converge high error case converge laplace state potential depend operator additional terminology thus specify way generalize map map heat operator heat laplacian diffusion eigenfunction th vector laplacian formulation far riemannian manifold assume dim riemannian metric canonical j k x accord support connection laplacian lx di lx dd lx I hold slow compact however dim riemannian embed induced way choose dx x I choice appear consequence term discrete connection homogeneous satisfy remark boundary approximate heat dim induce laplacian geodesic square integrable vector diffusion eigenvector detail obtain wide purpose numerically embed bar numerical rotation calculate theory agreement bar diffusion distance distance manifold embed mapping compute vertex correspond truncated diffusion eigenvector case embed embed numerically compute due sampling truncate embed embed similarly numerical h square transformation result usage want eigen connection embed truncate embed embed dim sample equally spaced point interval diffusion embed dimension embed truncate embed figure h sample spaced grid fix truncate calculate truncate diffusion calculate eigenvalue embed smooth xx xx xy arrive real nystr extend local alignment mapping approximate alignment find subset project embed represent orthonormal decompose suppose step orthonormal basis approximate plane local use among point inside ball center use eigen field finish early graph converge generator heat heat connection laplacian adjoint tangent bundle accumulation distance heat smooth analytic vector mapping hilbert pair need vector choice property map compact manifold dim basis field laplacian vector diffusion continuous note compact smooth xx xy xy tc xy xy x mx ny tx tv tv tv embed diffusion theorem diffusion distance diffusion behave geodesic distance smooth dim close similarly expansion coordinate denote possesse close bundle tangent bundle also equation thus eigenfunction operator rewrite q heat kernel laplace operator laplacian trivial bundle also heat put fact besides robust reference alignment object dimensional periodic shape describe problem dimensional arise comprehensive problem reference arise graphic structure image ct determine structure channel protein determination ray ray since far attempt ray ice thin typically disjoint promise structure typically whose pixel image proportional line path image highly impractical projection ct shot maximal ice partial homogeneity set hereafter random rotation structure collection noisy rotation orientation orthogonal transformation three rr tr describe orientation exclude intensity plane integral along path image potential fix laboratory coordinate know ray column orthonormal plane angle clean rotation rotation basis plane rotation plane rotation extremely group noisy raw within single double within result average angular average raw clean projection white noise simulate choose visible denote angle plane average noisy alignment snr clean frank invariant mean procedure identify similar invariant define euclidean optimally align rotation center rotation operator image angle prior distance radial angular rotation result center true pixel refinement procedure challenging assume image worth note proximity average cross distance proximity practice distance although emphasize effect find average share angle perhaps plane ideally ball center neighboring alignment angle happen match plane lead spurious neighbor average image poor image invariant neighbor know direction image angle indicate identification neighbor angle outlier snr outlier direction neighbor whose angle indicate angle way perform much na I near neighbor average distance share reference distance angle utilize mean algorithm snr experimental averaging far improve detection information plane rotation angle base powerful noisy snr near diffusion pair neighbor angle degree histogram angle figure panel neighbor identify invariant distance identify angle number neighbor apart algorithm show robust introduce diffusion map algorithmic consistency transformation along connect affinity inner point hilbert distance point distance orthogonal transformation scalar procedure seek alignment also rotation diffusion correspond orthogonal rotation organization structure graphic shape framework become collection manifold orthonormal tangent space mild manifold prove transformation procedure approximate careful pca prove manifold sample uniformly lie heart framework connection graph prove operator sampling prove asymptotic heat generalization diffusion determine latter double basis tangent space example vector mapping topological cloud extract use modification vector order product high act form topological harmonic form laplacian topological therefore modify approximate laplacian instead laplacian basis use tangent may parameter effect vary problem multiscale resolve recommend incorporation mapping difficulty face deal locate necessary limiting estimation space recent method improve result heart diffusion whose block matrix tool analyze eigenvector eigenvalue may also matrix haar orthogonal presence example matrix mention early mapping framework alignment process transformation focus utilize point probably ask transformation utilize remark diffusion mapping framework group compact less obvious arise extend diffusion compact group partially support award dms nsf award fa award gm foundation wu discussion regard express university institute stanford university air force appendix mathematical background reader familiar operator surface rate directional derivative eq extend field way notation directional denote direction vector field look first generalize embed dimensional surface smooth curve affine span collection tangent denote embed tangent plane tangent please plane affine figure tangent vector differentiable map abuse usually understand embed rigorous field please field derivative difficulty make belong easily change bit curve guarantee theory face make sense obvious differential play essential role fix field parametrize q ordinary along curve denote back address derivative say want direction comparison sense live notion derivative point tangent plane always mean know field vector axis definition generalize field detail operator heart equation property setup roughly speak manifold surface theorem state throughout smooth dim compact embed metric induce canonical denote dx np step expect large hold automatically tensor embed denote ease notation sequel denote connection whenever divide form approximate embed tangent plane prove approximation crucial proving geometry x high orthonormal dim subspace orthonormal minimizer column p well near seem bit radius indicate observe improvement relevance deviation error pca near boundary without choose manifold choose remark expense slow orthonormal alignment procedure connect boundary crucial proving proof lx orthonormal accord third approximation section tangent bundle parallel theory lx lx fourth expand laplacian operator plus term first third vanish vector field sufficiently smooth potential vanish laplacian geometry field x put theorems theorem theorem theorem get I ip ip lx I upon divide vanish specifically dominant surely proof please fix identification p taylor expand exponential map x tv xt eq obtain conclude next embed q evaluating give pd taylor follow calculation small taylor similarly put get please fix sufficiently choose enough neighborhood expansion together fix elsewhere properly translate pca coincide leave singular rewrite denote geodesic radius identically number q evaluate moment substitute apply taylor lemma integral odd power vanish symmetry sphere consideration become move establish estimation purpose establish bind define calculation similarly provide quite say inside identity size due curvature finite statement note rewrite identity symmetric orthonormal eigenvalue order dd matching note generate sampling column dim span point without full integration domain jk odd power vanish expansion case bernstein inequality provide
gaussians mean could explain single address change sample program depend specify program transformation frame replacement program abstraction abstraction remove abstraction replacement abstraction abstraction abstraction abstraction abstraction program abstraction abstraction abstraction body remove assign body instead value pass remove abstraction abstraction variable abstraction program abstraction instance abstraction body abstraction abstraction abstraction make name abstraction abstraction body abstraction abstraction abstraction application list abstraction abstraction abstraction map abstraction body pair body program abstraction abstraction application location deep nan primitive else abstraction program abstraction abstraction abstraction abstraction abstraction list function abstraction change frame remove abstraction abstraction empty abstraction abstraction change variable position define recursive argument application argument remove argument argument variable position abstraction abstraction change transform abstraction program remove application transform compactly represent identify induce structure place noisy two call replace abstraction abstraction variable instance replacement abstraction program shown begin color color uniform choice identify instance replace abstraction change color node size program begin v node datum color choice application big simplification affect program search improvement fit datum principled apply less generate remove v color color color argument replace choice indistinguishable value look program abstraction create argument abstraction advantage potential redundancy merging argument transform frame abstraction instance match abstraction abstraction variable program abstraction possible match find program abstraction match pair pair equal car cdr cdr match replacement abstraction variable match abstraction instance match similar application object number explain first choose abstraction choose abstraction whether abstraction match variable remove argument similar argument indistinguishable unlike program start v color datum apply program match variable true begin gaussian v uniform program size color gaussian compactly represent transform call pass function program terminate induce recursive illustration capture abstraction valid variable recursive call filter abstraction application abstraction recursive call abstraction valid terminate terminate abstraction call nan valid recursive call terminate call valid instance flip recursive call uniform choice recursive call terminate abstraction non define abstraction name hash abstraction name list abstraction hash abstraction abstraction checking status name hash ref abstraction name define set name new status hash abstraction new status abstraction abstraction status terminate status abstraction f status abstraction begin abstraction name program abstraction name abstraction terminate base branching list abstraction abstraction map base base call illustrate transform abstraction transform f remove change definition former outer body remove get flip datum start demonstrate compactly infer process call represent program flip color color color generative explanatory size color use gain noisy use instead construct frame third noisy replacement instead return sample return call replacement abstraction instance map instance gaussian induce noisy use program three third show node three incorporation three structure color color node merging generate node make stochastic noise color node f probabilistic implication solve take run want generate remove within problem step set program noisy occur program dimensional recursion program generative one induce simply call overall merge procedure posterior define depth transformation depth top transformation sort transformation program search depth lp program depth program well take sort program perform recursively program transformation depth filter program frame depth transformation transform program filter depth program apply map depth program reduce preserve program transformation us program semantic preserve transformation program semantic preserve apply transformation semantic preserve transformation change program transformation semantic semantic semantic change program define transformation semantic preserve program program transform transform program program program semantic program program compute frame sort program program program semantic semantic program log program semantics semantics program program posterior prior program program semantic posterior list new illustrate example domain colored example program sample program pattern show pattern recover representative merging result differ color gaussian repeat ten incorporation learn create width color f program ten mostly mostly flip merge abstraction depth begin f uniform choice tree true flip repeat width recursively call stop build begin f f f flip f observation reach trade prior recursive adjust occur merge demonstrate width define color color size begin color flip uniform demonstrate parameterize recursion place program pattern branch run width set trade likelihood introduce program branch would amount program compression define node define flip branch branch define color program program function single color node size pass branch end begin v flip uniform size uniform merge induce model central translate program program identify repeat compression goal include efficient computing furthermore consideration systematic machine algorithm world capture rich limited human engineering pursuit plus minus report approach program pattern choose language program extension abstraction generalization extent merge induction explanatory use abstraction form find key nest list might series tree green branch branch either pattern intelligence human rise form rich language capture program probabilistic program play prominent modern learning free lead pattern able capture feasibility learning class much limited develop tractable investigation take explore might proceed expressive identify abstract program probabilistic language represent program program programming program parameterize recursive searching merging simple colored algebraic extend guide use probability begin transform explore transform program explicit move formalize transformation learn probabilistic idea color e status contain detailed code illustrative example document progress complete inspire high aim illustrative example merge search accurately merge transformation posterior give datum building incorporation set generate hypothesis program transformation model generalization successfully artificial represented gram extend merge rich program represent context parameterize use transformation compression describe implement program give form use program type lambda abstraction operator probabilistic program objective program program program estimate em move move abstraction transformation improve search strategy objective function move algebraic point translate derivation sequence datum nest expression attribute list node tree color color size visual way represent interesting merge tree var flip primitive primitive uniform color apply program merge different list etc merge data incorporation incorporation going create expression evaluate algebraic combine uniformly list tree tree expression rest datum easily convert expression tree question leave incorporation structured feature program true data color color node size gaussian node size node like illustrate generate program pattern improve meaningful branch body branch body branch branch branch body branch branch size flip flip branch branch datum combination flow lambda program branch body connect branch branch connect small node color create identically color reflect capture branch program initial incorporation structured form program represent create symbol function name frame abstraction body make name abstraction body define name abstraction name body abstraction name body abstraction name abstraction define abstraction body fourth program consist list program body define keep track motivate idea computation transformation affect program program log semantic preserve list semantic program program third likelihood fourth program preserve define program incremental forward choice sampling observe evaluate tree smc core core smc argument repeat symbol find symbol member repeat sample mcmc use parameter compute node child score data parameterized attribute parameterized attribute parameterized list attribute original variance attribute attribute parameterized inf gaussian variance third incorporation program pattern prior abstraction abstraction aim syntactic pattern program call newly create remove act computation merging transformation potentially lead well generalization property successful implement lambda anti matching pair program occurrence program program program nan program cp size cp compress compressed program abstraction body program define list unique map anti abstraction abstraction transformation simple size node node b true f node v behavior transformation semantic equal transformation partially match contain common example common create potential abstraction transformation partial pair partial match expression anti tree expression list interior primitive element number tree partial match expression find subtree representation represent anti common match anti proceed recursively primitive primitive return list return frame anti begin define variable var primitive primitive primitive add add else pattern reverse anti list match root match match since match match list size match variable match find list primitive final lambda create partial match attempt abstraction abstraction body replace match replacement abstraction frame
configuration algorithm krige surrogate function nest reliability analysis evaluation function optimizer extensively numerical paper article rectangular load moment respect axis load refer neutral additional material elastic perfectly stress originally whose original paper gaussian small coefficient respect objective formulate follow failure choose optimal limited satisfy constraint minimum probabilistic denote form index cost line row reference coefficient variation failure carlo second approach slightly lead turn failure dependent example self quantification third reliability krige surrogate limit krige constant regression refine per refinement iteration mm reference krige comparative reliability illustrate index behave acceptable occurrence acceptable mode structure due load failure cd exceed yield performance read stress member ab load read euler load force member ab read probabilistic variation along kn gaussian mm gaussian find rectangular cross member reliability requirement respect result ab cd ab ab cd ab ab reliability form reliability check reveal reliability index implement approach plug krige surrogate confirm krige save provide error reliability oppose reliability h aim develop reliability design engineering consume expensive evaluate probability krige strategy counterpart indeed convergence evaluation quantify sequentially onto final numerical efficiency property krige surrogate build call simply surrogate one thus performance also worth refinement make add thus availability platform krige efficiency amount real latter investigation publish author grant present design g nonlinear finite use starting failure quantify estimation failure advance krige surrogate error surface failure sequentially krige surrogate reliability analysis refinement reliability reliability sensitivity analyse adaptive surrogate estimation finally classical order problem krige surrogate reliability thus nest literature three structural mechanic mechanic aim find cost often lie boundary account uncertainty realistic despite attractive field limit mostly simplify assumption intensive attempt efficient bring field sophisticated real word safe within evaluation quantify minimize induced development devote formulation literature review argument surrogate resolution section introduce krige specific emphasis put refinement krige surrogate nest reliability loop model describe read objective bounding call soft constraint consist prevent negative infinite constraint evaluate additional system failure oppose involve evaluate box finite probabilistic minimum acceptable probability failure term integral define word joint later gradient design might consider sufficiently argue full eventually account randomness extensively thank analytical simulation c straightforward consist reliability analysis loop conceptual often lack since require consume range performance weakly reliability nest able formulation often refer nest despite concept reliability transform reliability inner reliability quantile argue much nested resort reliability optimization towards refinement criterion note reduction overall run demonstrate applicable grow increase availability resource pc double loop sequentially reliability optimization perform assumption problem simple efficient closely relate notion partial certainly soon deterministic build design simulation design uncertainty design probability closely reliability explore possibly refined minimize whereas approach consume task surrogate surrogate build refine evaluation various surrogate reliability surface polynomial expansion machine krige classical reliability theory literature surrogate krige krige surrogate address mathematical call space simulator present consume krige spread depend prediction purely lack discussion distinction source reader refer property krige surrogate essence krige start uk process read term part require mean autocorrelation function read accord autocorrelation autocorrelation generally autocorrelation know common zero polynomial ordinary together dramatically output fidelity choice may low fidelity analytical build simplify construction krige subsection aim gaussian correlation mean krige indeed ii mi symmetry relation prove krige autocorrelation common statistical krige methodology field turn methodology well suit purpose methodology maximize respect one optimality numerically tractable within toolbox application consist weight improper pdf volume though explain hereafter note pdf sample mean well highly concentrated property cluster failure extreme failure uncertainty standard unfortunately failure section spread useful surrogate accurate reliability stop refinement reliability define application criterion refinement stop propose criterion additional replace respective accordance simulation reliability refinement g generate candidate improper technique sampling reduce use newly point update krige autocorrelation update reliability g perform onto failure order measure tolerance k new I refine illustrate apply nonlinear limit surface show contour population slice give pdf several mode section black krige black bound line interpretation blue point adaptive strategy previously krige surrogate augment reliability krige indeed build krige reliability would refine krige call augment reliability standardize n suffer argue cause performance present augment reliability keep equal vector indeed augment account design consideration read pdf assume illustration provide augment reliability space axis simple augment make limit surface potentially optimization precisely accurate choice onto space marginal vector able compute univariate assumption contour region tensor confidence interval margin order quantile interval involve margin parameter margin quantile level follow problem function margin domain rectangular derive margin analytically numerical solve mean gradient property quantile respect location improper volume refinement krige double loop mean proceed optimization nest reliability reliability perform reduction krige surrogate simulation sensitivity detailed advantage failure indeed point state failure joint pdf analytically probabilistic distribution copula trick unbiased read follow sample estimation estimation additional run simply estimation concept easily subset simulation detail toolbox matlab sample refinement improper pseudo refine optimize refine optimize refine j j ji refine c region bound
trajectory origin consider quantum markovian quantum study planning describe spin p warm support grant aid science reference reveal recall approach uncertain model situation kind heat heat several concern govern temperature might quantum paper term stochastic quantum markov quantum neural us extensive asymmetric build basic network early pointed energy treat spin researcher matter pick material remarkable theoretical mathematically concept storage capacity phase retrieval spin phase utilize replica introduce prevent build compose namely spin boundary control capacity temperature heat storage capacity energy function monotonic consistent signal analysis enable derive couple self mention theoretical neuron heat artificial might kind conventional model add classical hamiltonian shall quantum especially investigate structure diagram replica method static equilibrium understand already theory evaluate recall powerful point equation take account correlation asynchronous dynamic dynamical replica utilize sub self noise derive quantity theoretical recall quantum systematically candidate dynamical deal shall quantum investigate differential respect overlap paper organize follow origin artificial quantum clearly quantum carlo recalling dynamic quantum deterministic build pattern master monte static apply case namely pattern via asymmetric conventional last summary divide namely model put noise quantum system hamiltonian identical hamiltonian network eigenvector classical mainly consider dependence however even numerically hard reliable become huge realistic brain use quantum computer recall dynamic quantum algebraic calculation due operator hamiltonian namely order classical system slice simulate attempt flow master regard equilibrium neuron strength classical go symmetric hamiltonian transition specify k denote neuron neuron obtain immediately imply classical slice master spin flip operator pick overlap build states sum substitute introduce notation get around complicated expectation k flow average equation local field realization sub k notice appear nothing effective neuron neuron state along axis differential form substitute tm tm side equation integral subsection carry overlap slice call inverse study numerically validity approximation simulation successfully validity recently argue path immediately mind
decompose eq estimate access second measurement q simulation incremental determine cycle cycle average trial size size incremental set incremental cycle algorithm fair gradient exchange metropolis gradient average give respect minimizer biased away factor second derivation eliminate result due replace curve incremental db gain factor steady value steady state mse large long regularization fig deal localization problem diffusion strategy know component cost multiply eliminate node minimize individually instead seek arise communication diffusion manner node access case available base estimate target neighbor simulate stay size incremental average exchange metropolis gradient close mse expression diffusion step become performance diffusion mse furthermore localization local gradient next algorithm scenario along incremental step continuous track decay step tracking diffusion optimize adaptation distribute interaction track state localization incremental cyclic side sum iterate q show guarantee right converge side eq require recall definition respect arrive block block block stack vector top block x nm mx x symmetric note nx diagonal inequality generality maximum define large magnitude q obtain theorem radius matrix stability matrix examine satisfie eq norm stability mn diagonal also block diagonal substitute within require obviously guarantee condition edu global individual allow information real help effect noise continuous error steady apply diffusion study incremental method require cyclic link diffusion network enable biological category sum individual component spatially distribute maximizer biological network interaction resource allocation online latter underlie manner network technique optimization manner notable incremental consensus incremental cyclic cyclic manner determine cover np cyclic along path share cyclic stop consensus vanish consensus optimizer steady state however environment prevent step early motivated introduce adaptation use global problem process process continuously diffusion encounter resource cognitive pattern recognition optimization wide class function approach tend algorithm size condition datum necessary adaptation result exhibit organize sec use taylor expansion optimize alternative strategy sec square statistical perturbation gradient sec application problem localization finally conclude paper notation vector regressor take simplicity letter denote quantity regular font letter denote diagonal entry stack symmetric difference collaborative manner minimize individual function differentiable minimizer function network work attain track chemical physical frequent context biological group location location common problem process individual study challenge nevertheless converge taylor approximate optimize alternative descent within neighborhood spatially motivate approach node represent value assign neighbor leave coefficient new combination nonnegative convex actually guarantee strongly treatment ahead approximate taylor subsequent second result substituting expression ignore approximately expression relate original function second side wish hessian node expression lead explain kk share reasonable initially summation adjust early term function approximation correction step proceed development practice say observe vary index approximation common view eigenvalue replace show stage scalar embed rewrite scalar nevertheless localize constitute point development minimize move namely step size adapt close get optimal mse sufficiently size lead add estimate update correction add one intermediate use arise examine optimizer neighbor would replace help bring influence arise generally estimate therefore replace incremental widely literature describe q note nonnegative coefficient arrive combine diffusion structure originally square satisfying intermediate involve receive neighbor use step intermediate generate perform similar order motivate alternative adapt diffusion originally square error square ahead strategy albeit vanish ensure towards choose sharing optimization node agreement size local word coefficient weight doubly performance every ensure across address strategy treat diffusion three mode interact choice derive version exchange enforce doubly decay satisfy one error introduce q subtracting give relation twice hessian symmetric component apply introduce across symbol kronecker product introduce assumption cost gradient follow exist ensure function strongly minimizer exist denote past definite x similar subgradient method norm uniformly bound restrictive allow unbounded condition allow grow instead noise regression u case easily state note random absolute appear necessary gradient quadratic cost common square linear instantaneous assume illustration purpose arrive strategy originally extend solution square sequel square towards steady state steady performance rate answer largely influence constraint examine diffusion variance weight evolve evolution reasonable profile form expression characterize relation evolve hand denote actually truly recursion nevertheless relation recursion regard however via turn challenge reason alone filter analyze filter adjust argument enable state node ahead expression jensen derive norm eq upon weight symmetric positive definite matrix k establish note yet n meaning lead combination hand inequality sum obtain introduce error write notation denote component entry vector nonnegative combine inequality certain condition mean bound use subsection state small size stability satisfy condition entry appendix examine conclude validity establish strategy stochastic gradient absence tend vanish interesting free impose provide size stability expression since norm entry worst verify enough nonnegative factor define eq sufficiently size denominator substitute step step
individual recall hoeffding use martingale martingale easy define great lemma proof martingale eq combine great inequality entire pick dependent simultaneously despite great compare inequality corollary corollary imply n hence corollary match corollary derive minor general mention imply bn bn martingale corollary least tight hoeffding simultaneously project tight corollary hoeffding application value drift tight q tight hoeffde whenever actually practice manner obtain bernstein bernstein opposite case estimate inequality bernstein pac average kl bayes inequality grid vanish inequality martingale difference variable variable enable reduce study simple study application analog hoeffding hoeffding logarithmic case drift walk region tight hoeffding bernstein inequality importantly present concentration average multiple simultaneously inequality control standard one important tool evolve process useful domain analysis importance line convexity imply first note expectation side order last concentration bernoulli information bernoulli variable factorial restrict tight since function convex corollary inequality variable markov take normalizing proof theorem variational root back theory physics relation generate respect bound use theorem respectively change measurable distribution eq since exceed reason finite jensen nh h r convexity divergence hold great inside nh nh n normalize minimize distribution simultaneously proof value many possibility instead relevant range note bound cover grid ni relevant need value strictly take condition pick exceed substitute factor know obtain relevant range minimize pick weighted complete substitution individually simultaneously intermediate technical condition martingale fact apply reverse conditioning result author valuable improve presentation european community european publication reflect author computer science institute systems science college research reinforcement supervise unsupervised processing bioinformatic ph mathematics universit e thesis solve seven international distinguished mathematic universit work verification system pac computer universit di interest theoretic author games taylor ph mathematics subsequently complete sc advance publish college development inspire theory boost show analysis co introduction support book analysis publish receive mathematic symbolic university technology learn california accept area theory symbolic computation research project european union interest include exploration lemma college uk mail universit mail universit di mail mail ac present set weighted learning set importance reinforcement interactive encounter comparison difference shift expectation independent derive tight analog hoeffding bernstein pac bayesian fundamental modeling study hoeffding present inequality shift function analog hoeffding index martingale represent martingale denote everything clarity possible average simultaneously evolve illustration inequality infinite individual high averaging law depend sample inequality importance weighted widely estimate draw proper reweighte control expectation base unweighted form order observe average law round application technique reinforcement average generate hoeffding bernstein bound martingale sequence shift lemma variable lemma p analogous theory probability combination analog great explicit bn similar obtain hoeffding however certain less tight hoeffding tight bernstein provide control average multiple evolve result
value work posterior plot produce stable estimate replication seven quantile typical left panel plot link right plot fit visually smooth fit value significantly autocorrelation plot autocorrelation appendix nearly cause numerical numerical place evaluate ht plot jj conduct replication table error next exponentially superiority setup except error set illustration error follow estimation value estimation illustration generate fit case index general limitation far satisfactory median increase demonstrate setup tn table save compare show still ht ht finally tc pose economic statistical tc datum occur wind north early quantile analyze influence wind fit quantile extreme three tc intensity tc intensity index qualitatively reliable level especially column histogram figure imply equation performance simulation suggest fitting obtain sim unstable quantile algorithm superiority modern carefully partially collapse simulation quite encouraging exponential possible explicitly incorporate mean function normal thus consideration zero exponential flexible possibility model mainly spurious underlie come likelihood particular may accurately accurately estimate reader incoherent inference quantile intersect proceed separately nothing prevent overlap term computational ordinary pc fit dataset setup minute burden slow speed mcmc algorithm variational desirable scope index pose serious challenge thank associate three anonymous helpful manuscript education detail n e mathematically distribution full inverse nearly singular integrated avoid inverse matrix parameter consider q conditional modify metropolis manually tune manual simplify transform run partially collapse sense specifically obtain partially collapse modify sampler intermediate marginalization improve example corollary division university school business usa distribution mechanism develop approach single quantile nonparametric vector popularity monte carlo careful consideration conditional partially collapse frequentist gaussian chain quantile sim provide estimation curse nonparametric covariate compare offer nice fitting base spline literature sim study velocity acceleration study trying identify incidence stable estimation inspire work area frequentist long bayesian spline process gp gps subsequently sample work limitation supplement description response relationship response tail robust tail article single quantile quantile response implicitly quantile dimension univariate variate treatment quantile adopt laplace alternative support hand constraint identifiable norm mathematically impose keep parameter advantage specify note matter heuristic sign norm prior choice put component sometimes interest recent identically prior attractive far generalize normal g generalization variable generalization straightforward
ia jj array equation norm I matrix order pdf distributional linear moment marginal independence well array variate eigenvector provide optimality estimator merely check variate consider paper variate assumption rule variate covariance state problem put restriction last matrix attain array variate flip square assume n l ia calculate root st array observation q array diagonal array let known estimator consistent hand assume kronecker delta develop size considerably assumption covariance kronecker estimate obtain could ccc ccc ccc ccc plot compare piece general array slice array assumption variate estimate eigenvector eigenvector kronecker small reduction dimension obtain ordinary n assume repeat result summarize matrix center delta covariance regularization expression tumor compare tumor normal tumor definite model theorem hold testing covariance inverse propose rank calculate reject figure array htbp compare true eigenvalue right compare true kronecker delta estimate use scenario reasonable variance covariance identity diagonal kronecker delta finally figure true kronecker htbp htbp htbp observation classify tumor misclassification covariance estimate normal tumor summarize finding misclassification practice addition permutation reduction covariance parsimonious model propose covariance like issue important glasso package implement shrinkage conjunction covariance individually sparse selection sparse glasso glasso estimation glasso shrinkage glasso aid model expression dataset order variance expression tumor color high little among discrimination remark cluster extraction paper introduce call essentially assume kronecker finally implication high collection capability lead dna micro array million easily less characteristic bioinformatics field variable usual sample covariance method classification estimating essentially covariance realization kronecker delta kronecker corresponding matrix
quickly nearly demonstrate increase sequence marginal converge set payoff follow feasible saddle point restrictive carry admm relate method decade field reader refer spirit fundamentally adversarial definition robust department electrical stanford university email stanford optimize structured uncertainty problem game nature subset variable try assign maximize von minimax randomize strategy player several structural property minimax assignment choose satisfy dimensionality introduce pass solve minimax generalize max belief propagation whereby strategy player call strategy optimize opposite game capture situation agent limit pool remarkably applicability economic theory learn theory estimation game try design good statistical nature choose close engineering reduce maximize appropriate course inherently importantly real design nominal try bad player theory control present take complementary explicitly index let pure index parameter robust course hard indeed adversary special case approach follow factorization shall objective sum local whereby node control naturally abuse terminology shall objective robustness graphical model effective relationship think instance subtle disease medical diagnostic system normally model family probability g logit expect distribution justification prediction robust robustness implicitly never carefully investigate apart graphical simplification strategy introduce established game device play try expect utility try consequence linear program von existence saddle point strategy pair form nash equilibrium order nature indeed remarkably bad expect utility pure achieve pure hence key general formalism pass minimax finally work ise ise alphabet unnormalize inaccurate challenge find uncertain strategy depend model variable distribute two finding model tree classical check figure summarize result different run increase perform give consider nominal response experiment compare strategy strategy strategy optimum strategy outperform long adversary nominal value control node neighbor denote analogously control discuss condition distribution small since belong lp graph writing take play role problem generality field mrf marginal pay product thereby loss generality graph distribution degree specify mrf completely marginal minimax definition belong fact simplex necessarily point possess separation tractable relax marginal denote polytope q exact alternate multiplier solve optimization problem transform admm guarantee case admm short presentation admm reader q indicate norm admm try solve optimization closely augment indeed despite
ad full present iteration phase avoid division little matrix algorithm sample cost strongly influence computation need well run rough computational empirical iteration chooses add objective derivative choose coefficient compute e value alignment continuously parametrize base parametrize good early kernel restriction solve encounter program matlab simplex center k kernel real see list method cr continuous multiple kernel search follow list cr da kernel provide follow combination kernel improve achieve single family dimensional performance illustrate h design interval label show experiment dirichlet kernel investigate classifier think good single good frequency misclassification frequency rate kernel frequency figure cr misclassification close frequency show indeed discover frequency sake parameter space dirichlet kernel define insufficient easy discretization dimensional method serious space parameterization multidimensional require inferior wish always obtain frequency design second vector seven measure repeat experiment misclassification alignment running version nd search bandwidth bandwidth average modify parameter kernel bandwidth constitute find material large number since able cope drastically small idea scalability nd large tune increase multidimensional running nd htbp finite base kernel matches generate frequency kernel intend value without come kernel kernel appear parameterization kernel continuous table uci repository rank contrary letter uci letter experiment take du expect fastest cr stage method da cr around dc cr fast rest continuous iterative algorithm solver whereas terminate algorithm require fast alignment novel method learning span kernel ascent forward stagewise correlation kernel method show benefit compare successfully deal dimensional kernel space experiment start might performance think design competitive interesting future whether exist provably although directly natural dictionary search however thorough investigation work secondary alignment large necessarily misclassification completely implication help construct synthetic potential avoid continuously parameterize example multiple significantly alone datum competitive multiple show multi parameter growth kernel notice limit directional derivative rule calculation cyclic uniqueness notice denominator solve root root contradict value work maximizer detail iteration objective function sign local minima mnist uci letter experiment specify iteration function kernel select uci need find cr employ matlab multiple multi ca particular run space expensive equal element kernel weight soft validation tune validation value decide experiment I factor choice bias though similar cr also validation set together result positively run everything training classifier validation nd expensive alignment readily multi method see transform performance alignment kernel alignment breast cancer diabetes heart lemma lemma pt ari department compute science ab base select step demonstrate art variety world explicitly demonstrate importance dictionary kernel base set however amenable demonstrate positive continuously parametrize implement ascent stagewise alignment technique several strength case multiple learning parameter heavily influence method function idea consider kernel learn linear belong parameterize continuous e extremely gaussian set available literature kernel paper reference priori problematic contain moderate say since number moderate discretization accuracy furthermore independent might explore avoid continuously future gradient directly main alternate jointly convex excellent avoid forward stagewise robust implement expect suffer minima belong group stage kernel two stage kernel fact alignment metric objective technical implement compute objective solve nonlinear optimization problem solve overall excellent completely boost greedy cf chapter new exist efficient closed determine add serve purpose explore potential advantage limitation technique reliable despite potential implement successfully simple combine kernel issue weight fact recently finally method cost stage method dataset previous method experiment new competitive less reveal insight method performance kernel give rise overall use metric bad like necessarily well primary overcome use combine seem mainly comes discover fact method search avoid conclusion dictionary important dictionary dictionary certainly already discretization infeasible dictionary competitive art kernel considerably control construct example nontrivial family achieve method gain correlate predictor set give rise center surprising derive favorable method method learn discretization publish force multiple virtue lag illustration difficulty parameterization deal show observe considerable gain algorithm outperform accuracy comparable require conclusion arise study discretization discretization exist finite kernel hilbert kernel interesting idea run method find spirit stagewise alignment align desire output empirically learn technique example combination continuous powerful priori performance world suggest dimensional kernel g exponentially multi kernel learn differ whether simultaneously happen far concerned start restrict search stay algorithm restrict lp qualitatively different statistical restrict rule tune limited option explore finite kernel kernel reproduce kernel learn happen happen single stage drawback work usually choose dramatically case continuously parameterize family derive natural kernel might prefer final however picture surrogate stage parameterized base study family kernel optimize require kernel positive semidefinite combination show semidefinite sdp semidefinite program computationally expensive simplified show sdp solution compute standard meet claim generalized learning kernel parametrize kernel choose number propose iterative optimize add general belong dc technique example function rkh formulate problem learn large group learn base kernel kernel parametrize control practice base value learn predictor base kernel usually base continuous alignment similar case alignment learn alignment outperform shift intercept alignment dataset let positive semidefinite apply alignment alignment use program alignment non negative kernel learn maximize label kernel ridge learn kernel alignment center semidefinite px semidefinite center kernel alignment dataset kernel emphasize linearly separable linear perfectly expect target point long alignment vary class conclude center definition alignment combination
existence uniqueness ensure uniqueness definite definite unique show optimality showing prove lemma sum implement centroid carefully computing distance centroid vs fig illustrate expect eigenvector log close slow centroid matter application application riemannian metric riemannian metric retrieval explore instead minimize seek prefer geometric solve make median differentiable ignore obtain necessary iterate nonlinear correct mention author contraction thompson claim deduce uniqueness contraction fortunately valid choice metric large generalize eigenvalue section projective ia iv iv projective prop analyze calculation prop third prop iv strict contraction start must attain minimum definite solution complete strict contraction generate iterate stay compact construction satisfie whereby empirical behavior fp different collection compare fp gradient fp turn remarkably manifold implement intel riemannian collection wishart study positive definite qualitative similarity hermitian definite notably divergence define divergence derive main present property metric hilbert develop mean median useful hope encourage investigate new acknowledgment grateful department university visit remark conference positive definite matrix attribute strict function draw view cone definite divergence sequence result connect riemannian divergence distance several property riemannian demand nonconvex use theorems matrix bregman divergence stein divergence jensen bregman geometric curvature hermitian definite manifold study manifold curvature possess ii closure form close convex cone view enjoy convex nonlinear algebra manifold view role see manifold drawing motivation view cone definite matrix connect divergence riemannian distance importantly divergence common riemannian demand build view hilbert usually transpose hermitian denote usual x f riemannian riemannian q logarithm counterpart formally name metric definition already divergence alternative simplicity think image exact paper highlight crucial speedup times compute latter toolbox time take matrix compute implement state art average fast alternative popular euclidean compute make cholesky slow much limit reader aware cm aspect pt hard connect riemannian distance connection cm question relate positive scalar cm necessary solve result global optimality proof establish previous jointly present substantially outline difference space support appear corollary similarity riemannian table well concerned theorem jointly prop remarkable contraction corollary corollary lipschitz inequality study q claim fix thompson metric formally viewpoint differentiable equality side inequality behave certain choice chen hermitian let arbitrary nonnegative typically asymmetric view dissimilarity q unnormalize occur two multivariate gaussian divergence theory applicability divergence symmetric attractive perhaps representation formula use advantage reader alternatively jensen bregman multivariate eigenvalue aa equality observe iii iv v prop enjoy convexity concavity equality ii convex divergence hessian prove rich distance result fail multiplicative harmonic prove metric prove chen b yield thm rely available alone nonempty inequality subset n help e imply inequality prove suffice number observe tx lemma another ce nc define converse deeply decrease cone thm claim let follow matrices ii definite theorem positive kernel problem use list quick l ref ref b ba ba c nb ca ta ba b ba ba ba b ba x kk u easily connect gm gm give numerous attractive among variational important surprisingly enjoy characterization even ba bx bx translate whose saddle thus go observe principle realize actually show prove geometric prop includes result special gm concavity e thm jointly continuous suffice eq monotonic determinant multiplicative combine identity establish fail quick suggest attain strictly share contraction term depend elegant metric divergence determinant reverse geodesic positive curve satisfie albeit weak result observe illustration theorem exhibit strong contraction curve straight empirically fairly gm arbitrary geodesic prop theorem help claim power monotonicity let scalar satisfy prove auxiliary prop vector decrease write relation usual follow general monotonicity mean scalar equivalently recall monotonic monotonic equivalently inequality obtain symmetric immediate prop basic analogue important property derive solve nonlinear similar monotonically decrease equivalently eq see operator prove shorthand see semidefinite follow plot show contraction display eigenvalue translate shape curve powerful compression connect thompson key
bring together york explore potential respective great success book advance mine advanced technique vast amount understand universe attempt window vast field like spend contribute year easy well website thank thank go space study york usa nd co york sciences strong research traditionally people science mathematic believe mathematics university york invert process aware scientific also science book fourth article make I narrow great field science discuss book agree field look data set discuss look expensive several progress people contribute hard science forget discuss possible innovation author field history support thesis title book west thank come york york ex institute studies http institute introduction pe universe gauss mining challenge analysis source machine understand universe high exploratory partition science understanding time series scheduling observation inference ari flow dynamical reconstructing
define perturbation closed curve large trace norm operator norm small norm unstable convex surrogate complexity property reweighte correlation net scientific domain statistic interpretable admit look sparsity knowledge many combinatorial inference induce lead framework year body lasso optimally correlation prediction estimation support however datum exhibit block diagonal sparse involve case toeplitz lasso although lot lasso instability elastic net add elastic blind group predictor correlation line focus ideal intervention net modify behave precisely follow propose trace nuclear norm show reweighte around relate include correlation synthetic trace lasso regime row note whose outside support norm singular value maximum value matrix norm equal consider predict assume training n widely predict avoid especially ridge square estimating drawback thus behave np norm estimator pursuit later reference therein unstable select correlate residual hand together elastic estimator adaptive predictor adaptively norm square introduce comparison group sum norm effect group norm introduce regularization depend normalize normalizing reweighte norm norm predictor support lasso adapt setting dimension span equal surrogate lead trace lasso property norm canonical orthogonal hand correlate behave like uncorrelated predictor behave statistical towards lasso convexity highly always minimum respect penalize eq appendix consist show flat loss trace section penalty trace allow newly special norm matrix line choice therefore q homogeneity triangle consequence linearity state norm special introduce overlap group lasso normalize family norm trace lasso unit see unit see bind upper norm elastic norm elastic fact single particular covariate strongly covariate uncorrelated behave like pairwise net important quantity unfortunately norm dual norm dual time estimate extend index optimize subgradient inefficient subgradient subgradient iteratively formulation infimum attain use optimization minimization infimum attain reweighte equal iteration warm lot converge start empty decrease eq start op use prop obtain second rewrite abuse second term interesting fact term pairwise net synthetic classical draw identity block eight experiment toeplitz design observe elastic lasso perform model almost ridge elastic net adaptively variable couple selection regularization trace approach
cx bx b cx function maximum system connect length elastic external noisy force model newton velocity mass sde write calculation write function specific parameter might two evolve observe trajectory learn reconstruct interval structure actual length observation effort devote develop complexity model devote learn use model pseudo paper gaussian sde upper bound prove match autoregressive however develop complexity substantial address question sde line yield quantitative scaling sample generality parameter random unknown distribution subscript omit interested property define value variance identity tx regard denote scenario mutual depend system account tx tx bound theorem follow tx var var fx fx simplify apply sde understood process interaction matrix first provide dense transpose matrix support matrix per row support large enough sde dense shall fundamentally regime sign sign tx together sde task square dense indeed upper sde vector denote jacobian non stationary small tx ax assumption existence solution sde energy prove throughout sde admit useful establish linear sde symmetric interaction stationary sde var ax tr conditional estimator expression form substitute proof show trying support require average unless bind follow uniformly regular generate independently define small maximum eigenvalue guarantee calculation law notice unless entry limit second compute low jensen step law define distribution since claim denominator multiply bind numerator denominator write stationary q arbitrary rescaling factor give probability matrix symmetric pa pa pa evaluating prove non family var x xx f ix ix e individual component therefore partially nsf award nsf dms fa fellowship later proof recall define define symmetric gx g g dividing
cost integrated indicate topology significantly irrespective connectivity aspect regard qualitative reflect expand discuss remark form cost topological differ express topology exhibit integrate mask subtle computationally integral mc require another integration population require comparable need possess vertex fmri however hold recommend selection denominator ng thus cost reflect several relie rank may topology tie rank value cost level create sparse network arise however non occurrence counting tie differ tie rank artificial limitation integrate several omit take topology unweighte arguably cumulative importance edge edge thresholded graph retain remain especially metric monotonic one also address cost control monotonic ignore contribute topological integrate metric combine weight respect use thresholding exhibit undesirable tend cost topology provide well dash cost specific plain cost average subject line suggest effect metric conclusion apply topological main prove general set independent topological usefulness integration difference topology unweighted weighted short main proposition general equivalence topological measure flat space cost development sophisticated effect consider correlation easily higher preferable emphasize build large integrate topological put network whose cost nature structural necessary integration justify usefulness control interest weight network thresholde artificial clear mean graph architecture ensemble thresholded fellowship institute health research research centre health mh south foundation trust college south would also anonymous valuable input integrate first integral method advantage provide interest advanced order mc theory first expectation convenience drop express straightforwardly possible omit order converge almost strong number provide integrable follow special mc permit refer mc density variate especially integrate integrate sample topological nature specify underlie mc topological care substantially however need rely context imply tie rank recommend resolve tie rank assign order tie rank random introduce amount tie high restrict simplified division purpose standardize percentile rank result rank rank order formal indicator similarity function cost verify unweighted verify equation discrete integration notation monotonic modify rank integrated suffice simplify requirement tt however complete contradiction conclusion suffice least weight short satisfy path short connection vertex denote invert entire clearly contradict prove claim p conjecture proposition run head weight separate topology e ed college institute health research centre health south college institute department school correspondence concern send sciences box p college se uk statistically principled population topology inherently unweighted evaluate limitation compare population efficiency comparison network differ key integrate controlling eliminate monotonic weight unweighted cost difference difference constitute valid global simply recommend report weighted cost integrate topological analysis provide cost topological finally limitation integrate subtle topological integration global last decade biological physical adopt approach interest originally seminal world free network idea adopt experimental research classify topological brain network different cognitive behavioral task common brain difference control patient author evaluate property task brain study spatial modality eeg fmri compare subject level spatial resolution question arise rigorous network network arise connectivity strength network topology topological edge draw comparison therefore secondly issue division weight unweighted graph difference topological depend whether unweighted likely unweighted concentrated graph theory biological originally nature interest consider relation element unweighte regular weighted correlation likelihood different population readily identical systematic manner straightforward network correlation interested correlation thresholding matrix approach uniqueness since language discrete mathematic correlation real value one concept originally unweighted consider weight firstly evaluate use unweighted secondly integrate possible network unweighted cost edge analog probably author explore integrate topological respect substantially differ relate cost topological coefficient derive integrate topological successful manner although way therefore literature network formally integration compare topological quantitative manner qualitative adopt network generative regard os enyi model common unknown refer type topological enyi lattice qualitative rely category contrast wish perspective whereby specific topological os enyi different level global therefore qualitative approach topological quantitative topological generative latent topology appear well suited definition weighted become therefore situation conclude weight share integration useful cost section basic two family topological measure network integrate quantity contribution outline describe fmri investigate us monte finding science close conduct weighted report arise exposition comprehensive complex find follow measure refer quantify metric mathematical definition topology term unweighted graph network order pair node connection I refer element edge cardinality denote graph describe topology short undirected denote triple index entry necessarily satisfy matrix draw unweighte one range development paper apply include synchronization likelihood simplicity lie interval standardized weight strength association indicate straightforwardly practice pearson standardize weight lead major since transform positive subset may positively secondly link positive negative take spurious close likely weight explicitly direction standardized correlation matrix sequel refer specify discuss limitation cost integration graph topological propose interest refer I metric former quantify transfer globally capacity information locally characteristic global local although metric wide range analog useful measure retain interpretation cc applicable wide range specifically efficiency metric irrespective graph neighbor family topology efficiency one remarkable local information unweighted summation short path node subgraph neighbor interpret information transfer value transfer similarly efficiency interpret efficiency subgraph imply unweighted distinguished edge set denote quantity distinction notation expectation probability technical general apply topological level interest quantify unweighted generalize term density unweighted quantify relative connection proportion match implicitly definition eq element represent cost generalize unweighted matrix order value association formalize natural choice entry formally standardize weight starting unweighted concept connectivity depend upon choose standardized respect different measure consideration fully two type weight cost equivalent currently quantify topological aspect weighted I integrate metric define family quantifying translate unweighted measure include weighted version rely unweighted short subgraph pair letter stand edge weight set path take notational introduce normalize association value use choose shortest regardless exist path every index topology integrate transformation level identical create vary compose regular topology corresponding association value build layer topology every regular layer integrate graph approximately contrast substantially integrate range hybrid topology globally efficient gray fill corner ex background scale every text node anchor west anchor west node anchor anchor west west anchor west anchor west anchor west west west anchor anchor west anchor west west anchor west anchor west anchor west anchor west anchor anchor west world transform regular hybrid exhibit lattice layer regular entry arrange facilitate integrate cost potentially substantial topological however population provide pool find statistically significant comparison network full yield identical group naturally opposite direction seem subset integrate full cost fixing suffer firstly generally justify practical secondly consider cost level network conclusion cost subset report nonetheless difference topology irrespective mathematically strength may reformulate difference proportional v w standardized interval trivially efficiency cost level cost attain integrate nan cost cost crucially difference cost return case exact two proportional integrated integrate respect point proportional thus identical derive hold invariance cost true monotonic metric unweighted argument formally undirecte monotonic weight version satisfie q proxy solely depend diagonal ranking set independent monotonic transformation argument topological I originally set unweighted integrate hold level function preserve original ordering tie rank adjust advantage topological topology roughly irrespective successful topology connectivity singular advantage measure invariant ensure comprise interval cost integrated topological measure demonstrate limitation potentially mask example addition integrate topological metric network rank cause induce topological structure rank allocate cost level discuss limitation connectivity characteristic directly topological efficiency equation serious limitation potentially type show setting weight efficiency whose proved contradiction situation real data difference zero nonetheless add weighted relationship theoretical therefore highlight various topology recommend report cost topological illustrate publish sup sup sup sup inf inf inf inf back back wavelet hz frequency band fdr correction base location correspond centroid inferior orientation axis analyze task base imaging fmri mc procedure cost integrate brain cognitive solely give description use basis fmri working subject letter second ask whether letter trial previously nan subject automatic template series wavelet wavelet hz main network fmri paradigm condition repeat wavelet concatenation simulate find final analysis subject functional correlation specific coefficient construction network illustrate univariate edge subject test thresholded discovery connectivity subject experience cognitive load description support mc mc propose cost convergence medium sized derive working memory task figure minus report formula dash grey variability twice mc indicate study compare integral computation calculation reasonably good quantity mc error constitute negligible mc derive computation could uncertainty associate interest
unsupervised machine valuable well unsupervised method far rate convergence assume image unknown pixel pixel propose procedure vary intensity test threshold picture choose consistently detect size one establish uncertainty present paper distinction formulate type literature knowledge research wavelet relation stronger hold detection object generality detection give test black object develop object vary intensity nonparametric block uncertainty relation mathematical analysis theory review proof relation detector device start notion infinite sense site connect site neighbor site q say assumption critical statement infinite even finite difference size site see say site precise cluster infer whether site configuration distinction locate near critical infinite lattice white modify cover detection intensity construction basic case graph discretize topology neighboring paper consider signal denote indicator function give identically variance measure detailed detection mean e previous construct test compute explicit mild proof denote finite triangular consist site mean arbitrarily resolution think write eq depend way define pixel side view inconsistent pre let suitably weak condition square site n follow type arbitrary detection actual error remarkable type object suppose constant type error exceed type cluster pixel intensity exists sequel realistic signal precisely give non degenerate respect define q device intensity properly display assume sufficient incoming explain appendix test perform read versus analogous fashion hypothesis strength bound detector device explain intensity detector likewise scale threshold sub detector device consideration attempt eq nan probability black cp ps ii let essentially situation lemma n suitably function theorem may contain length correct scale right indicator completely hold pick strength overcome scale positive intuitively clear ratio horizon prove like error explicit valid noise give q thus proper estimate triangular lattice depend black contain origin please asymptotic statement instance sharp sequel depend triangular lattice appendix upper eq eq intrinsic clear something statistical procedure therefore something signal distinction explicit condition literature gaussian wavelet strong wide threshold close lattice statement lattice large lattice nan reject relate strength ratio assume detect sufficiently large sufficiently matter principle lattice give automatically effective type ii error word derive necessary ii study interval fulfil however justify time lattice exceed exist might continuous sufficiently detect exceed circumstance principle minimal detect provide size bound detector device intensity signal rf signal super account property able potentially threshold begin statistic terminate nan reject reach repeat cluster time overall asymptotically slow original unknown color carefully rejection indeed perform single test occur threshold analog type tend exponentially tend fast pay able get plan paper acknowledgment read manuscript collect going prove basic introduction event increase indicator whenever state add completeness decrease subgraph sigma configuration occur q inequality let q ff disjoint site finite increase q statement denote event sequel write measure ss marginals sa p pa e ps ps p finally q depend closely connect lie size connect expect note write q connect adjacent path connect connect meaning site path site switch change disjoint finally dp pa p nx ny p ny ny ny p ny integrate differential detail see mention proof finally material matching finite probability site mark black mark white black boundary mean polynomial number extend infinite consider per power series size clearly infinite graph self polynomial triangular match probability triangular lattice instance actually special polynomial thus construct match equation similar discussion device device detector device able signal intensity detector device contain cutoff intensity cutoff lose signal happen view notion bound detector device equivalently reformulate still conditioning event course yield important
term cope interaction logistic exploit motivate attractive regression output linear coefficient reasonable value remove may stationarity adaptive entail considerable burden iii raise instability curse modeling recover equation unique readily find infinitely many hardness program basis pursuit motivate l l uniquely ls continuous least invertible grow translate numerically l ls determined may resort regularize readily store invert computing regression th call trick compute also neither estimator lasso weight improved price require ridge isometry albeit regression generalize model regularize logistic probit regression successive lasso problem polynomial expansion generalize polynomial certain improve coefficient level toward aforementioned constitute generic solver tailor estimator method coordinate scheme outline next core iteratively w scalar ii component minimization th entry sample iterate simply define eq update constitute coordinate alg guarantee apart computation alg coordinate unlike counterpart offer memory enable track vary l ls solve factor invariant variation exploit priori suffer instability overcome limitation follow polynomial recursive lasso call instant burden reference therein time spirit batch single form alg present hereafter cyclic convergence regression set alternative setup approximate expand basis formulate solver polynomial cast homogeneous n p cf objective insensitive though kernel concern primal domain section specify whether problem capable identify polynomial expansion study though tool call rip involve restrict isometry isometry initially identifiability noiseless uniquely recover unique analysis extend constrain bn bn bn rip describe hold properly early rip establish identifiability compressive setup definition suggest derive one ensemble rip bernoulli possess constant identification toeplitz latter bind versus scale former attribute dependency rip involved system filtering input independently practically frequently limit gene beyond consider possesse rip exist condition form combination possess rip favorable matrix unity norm unity quantity coherence rip appropriate normalization least enough desirable vanish requirement inherently inner positive desired soon study rip equivalently intercept dc toeplitz parts toeplitz q quadratic submatrix readily verify satisfied transition raise rip insight scenario actually substitute upon expansion hold translate adjust carry answer pose modification solve modify resemble replacement wiener polynomial wiener module white distribute analysis appendix define form c suffice scale bind agree filtering whereas due involved dependency describe sparse input recover comprise noiseless filter setup row statistically independent rip polynomial tight deal expansion e measurable endowed set tt kronecker delta involve entry rip rip universal rip basis trick apply basis upon stack properly replace one thus translate sparse rip rip possess straightforwardly independent input probability observation scale compare scale increase matrix explain structural polynomial paradigm admit product input typical multilinear setup comprise coefficient inequality rough bind sample phenotype additive alphabet latter alphabet analytical entry mean rip exploit appear linearly mass multilinear entry fact characterization readily aware relate difference input alphabet oppose one rip multilinear draw multilinear positive constant whenever n often phenotype scale previous section probabilistic bound identifiability representation evaluate applicability sparsity real indicate attain accurate impulse describe system model corrupt counterpart plot follow estimator ridge scale agnostic outperform condition offer accurate even assess setup mse carlo recursive conventional recursive system draw carry moreover fig aware closely study effect phenotype trait height seed follow gene effect marker determine trait pair trait since population regressor determine trait analysis multilinear paradigm synthetic detailed population evenly marker phenotype linearly express intercept effect leading marker note marker ii effect regularize center intercept fit parameter tune fold pe unseen data v estimate small pe determining estimator mse provide keep estimator software take min final see attain pe pe column ridge yield model yield avoid coefficient entail real north american outline height population consist phenotype marker cm genome marker cm tr allele value main involve leave fail handle large weight pe attain parsimonious spurious peak infer magnitude offer effect ii effect main marker iii marker main exploiting estimate abundance allow parsimonious expansion estimator need meet analytical solve efficiently find computational load enable analytical rip show high order interestingly generalize generalize regression aforementione numerically verify develop adaptive exact outperform simulate extend high utilize considerably genome tool surely essentially around dependent useful sum dependent random partition exhaustive exclusive subset respective minimum intra independent also uniformly minimum may yield correspond always construct way dependent degree vertex degree assign every vertex differ partition intra key guarantee dependency sum necessary realization possesse rip thus rip event rhs exploit bind triangular yield rhs entry exhibit component linear bilinear nonlinear system notation indicate th inner product start upon hoeffding multiply account n yield bilinear diagonal diagonal sum generally two split partial sum example entry express sm collective return entry splitting trick contribution regard immediate hoeffding whereas cb already nm equivalently depend sum bound include trick yield n exploit splitting trick sum k mx lb simplify presentation slightly consider sum three use degree dependency associate product entry together write k ij kx mx k bb entry associate dependency bound diagonal imply arrive translate whenever yield acknowledgment thank update ir ii r ridge ridge lr corollary support international fellowship european nsf grant award mn play identification range value meet method compress offer cs common sparse initially pose build identifiability principle sparse sufficient measurement rip commonly meet generalize result counterpart verified phenotype lasso isometry appear frequently engineering speech name nonlinearity smooth offer memory impulse expansion mapping filter approximate expansion apart extensive character recognition recently valuable phenotype relationship wide association jointly investigate albeit nonlinear input linear via bottleneck grow raise computational record reliable dimensionality issue however admit expansion exploit nonlinearity nonetheless polynomial expansion attribute parsimonious underlie parameterized sparsity expansion formulate motivate polynomial expansion drop contribution adopt sparsity
passive large present achieve excess smoothness involve question relate implement extend result answer logarithmic coincide derive argument every one iterative transform goal work ph numerous discussion grateful anonymous carefully read claim choose enough mean partition dyadic since cube exist dyadic cube q attain point eq concluding simplify brevity additional notation recall excess union event eq q imply section section lemma partially support arc nsf grant dms mathematics ga active investigation provide bound generalization achievable underlie obtain rate selective confidence band couple design denote goal predict practice remain often happen obtain associate label pool almost suggest classifier active devote modify property previously passive learner obtain design difficult collect situation learner outperform constant convenient characterize investigation discover generalization zero fast well passive however available noise decision boundary computationally adaptive field standard polynomial covering derive regularity decision assumption restrictive h old bind excess active label budget upper namely certain h old satisfie show plug learn regression excess second propose attain possesse within organized next introduce specifie make qualitative work statement govern observation sample sample usually measure respect measurable excess indicator omit minimal infimum measurable function restrict attention belong old continuously taylor distributions property bt cause impose two behind deal regular grid unit partition consider nest dyadic partition unit cube regular eq definition nature minimax excess sup sensitive local assumption another useful characteristic regular certain subset fit say belong derivation piecewise use define dyadic cube note view span haar nest desirable selection projection decision mind sign design support similar sigma follow appearance assumption algorithm nonparametric subject mention impossible adaptive confidence band subject follow satisfying allow away decision boundary lipschitz convex satisfy proposition appendix finally satisfied brief since definition appear naturally emphasize known restriction estimate desire base plug piecewise mention passive plug classifier attain iteratively improve construct shrink confidence band piecewise procedure h estimator use active clearly band classification potentially serve modified concentrate domain tight band control require priori noise adaptive proceed main recall risk active framework rate goal output excess converge fast build upon minimax constructing similar present reader infimum estimator kullback go back let cube partition follow belong center always integer infinitely distribution marginal independent mr define support union dyadic spread tend respect smoothness requirement check satisfy noise consider sphere next step well mm stand distance follow theorem nx briefly denote admit choose way set n eq finally lower since noise sign tractable show attain rate factor cover interesting decision proposition order complete address remark regularity assumption budget replace define simple resolution iid follow around bind consistent procedure play appropriately choose ready active detailed budget lb nn km lb k dx lb every sample divide k ks ks towards general iid absolute enough bind hand ready satisfy remark fast rate passive main b depend act claim active set finish satisfies q easily risk inferior show prove final finish projection onto
strategy process candidate determinant follow counterpart approximately strategy process covariance give collection candidate costly greedy term gp hence sample simplification dependency choose reduction consistent entropy collection candidate maximum lipschitz maker limit costly maker point call iterative function quantity simplification observation basically formulate use uncertainty model difference noisy noise entire second solve note good detail evaluate numerically describe replace candidate htp objective respective specific formulation constitute sum state substituting approximate search weighting describe meaningful objective order transform normalization individual dependent estimate function observation utilize value specific respective df provide combine thus provide weighted address objective minimize convert constraint many real life normalization counterpart constrain optimization scalar respectively provide objective practice emphasis estimation learn maximize method disadvantage prohibitive problem bind simply identify make method along scheme part balance vs search implicitly however variety function time decision make framework set noisy past observation change capture noise dynamic capture stationarity change algorithmic summary specific provide bound objective variant variance function error next action maximize objective state next proposition observe multiple issue space visualization purpose hence choose linearly I estimate consist show peak figure point eq htbp htbp htbp quantity well correlation depict result bound objective give sum process price rectangular region grid interval estimate weighted sum utilize space sample minimum price use depict optimum although optimum value still satisfactory consider price million percent depict htbp iteration third setup brain show rectangular peak location show location point htbp six function respective real location htbp datum make limited article theory search recently substantial historical valuable inference measure version subsection focus communication machine book decision make play statistical learning complementary discussion topic foundation gaussian favorable book treatment gps include understand nonconvex method gp focus amount hand model iterative search particle random nature distinguish apply latter quantification connection shannon utilize state theoretic problem likewise box account information criterion discuss subsequent active gp heuristic rejection foundation adopt experimental observe priori reality methodology work indicate already domain actual adopt present method replace specifically kolmogorov complexity priori probable accordance hence describe simple gp incorporate kernel theorem different consider problem obtain costly scheme obtain entropy shannon theory provide metric aspect optimization framework allow vs trade vs obtain amount point illustrative grid dimension e resort carlo case address issue include anneal resolution candidate biological analogy maker biological priori meta gp refined environment domain preserve resource much evolutionary fast member explore meta refine meta evolutionary assume competition framework present address decision account content datum theory relationship optimization framework future exploitation adaptive sampling relationship evolutionary make game theory acknowledgement support thank discussion subject make decision maker posteriori nonconvex except costly observation present information maximization quantify information acquire entropy nonconvex maximize model adopt approach art result maker preference world decision whether obtain information consume characteristic infeasible option leave maker develop strategy collect miss concrete maximize lipschitz except small may start observation may make decision location unknown paper capture pose observation multi optimization aspect structure manner enable aspect explicitly incorporate concept scheme build field machine specifically concept quantify acquire theory modeling system adopt capture aspect objective formulation heavily aspect application domain vision handle datum make play play important develop strategy time advantage scalability gaussian process frequently encounter seem method mine mid due decentralized resource decision complex system wireless local decision security relate decision effort action relate security management costly operate part large system limited concrete definition motivate helpful aspect generality nonempty domain fully adopt provide reasonable unknown possibly simplify assumption lipschitz main distinguishing limitation observation cost information assume objective nonempty strategy q may domain adopt approach beneficial allow usage incoming alternatively cost relax formulate scalar iteratively location conceptually strategy unless rarely complicated strategy collect information derive gradient heuristic explore step direction randomly jump also flexibility entire objective belong interpret maker parameter separate slow capability search fundamentally impose explicit priori learns manner available computation search opposite end utilize extent almost regardless valid wide field security economic management system human close spectrum advantageous search former end predictive unobserved balance usually achieve balance belief function fitting focus process within framework develop multiple reason preference elegant combine secondly gp well immediately model drawback namely heavy really amount available comprehensive treatment gp therefore decision make set observation gaussian formally completely characterize since noise matrix use infinite real accordance gp point outside training x scalar characterize gaussian define distribution objective furthermore uncertainty provide subsection subsection discuss possibility maker address quantify obtain precise answer principled manner example search shannon information readily framework measure discrete variable quantifie quantify final entropy conceptual shannon good obtain information constant action less threshold uniform case simplify derivative clearly roughly ignore quantization boundary effect middle prior provide process repeatedly quantization learn search strategy make limit present address problem
common poisson random let independent poisson multivariate poisson large focus bivariate process bivariate case characterize independence independence innovation distribute notice diagonal still still joint expression derive matrix expression poisson innovation normal distribution infinity numerical optimization maximum previous run simulation study behavior example two parameter two parameter interior size smooth function set size table standard converge go figure table though valid size sample use approach count propose account autocorrelation occur create serial within contiguous statistical exhaustive risk across require reference north american split east west chinese south american decide together west east secondly past entry magnitude database period mention website mid affect cutoff test st th keep period investigate first autocorrelation autocorrelation count measure degree order range hour propose serial autocorrelation independent thus pair show ratio poisson compare show sampling follow combination along quantile proportion significant thus provide useful add really table dependence contiguous serial correlation important independent expect mechanic one count sample influence interval hour occur find first degree autocorrelation autocorrelation finally occur second panel percentage importantly contiguous similar model model combination large part contiguous percentage far apart often sufficient table combination statistically combination fit contiguous different area frequency significant line show unconditional number moment west table part table daily three day poisson generate day observe day west autocorrelation autocorrelation thus average number day quantity west potential management west compare diagonal context compare model situation west e pick significant analysis one directly value influence hour frequency west converse west count causality poisson hour hour hour hour hour hour hour hour hour hour hour hour hour hour mean hour hour hour vs hour hour hour hour hour hour diag vs west vs hour hour hour hour hour hour hour hour diag mean frequency hour hour context predict large medium likely observe five propose independent poisson fit diagonal independent statistically sampling model finally diagonal autocorrelation explain future size medium case weak frequency hour table explain temporal decay rate hour frequency last unconditional medium hour frequency causality significance set illustrate impact hour medium autocorrelation period indeed cross account expectation much section cross finally approximately consistent name hour hour american east chinese south american denote number medium magnitude period name mean north american west chinese south medium management application price total area city important total area question section area propose see significant south american south american locate limit observe west assume south american occur half bivariate bivariate occur table leave focus day especially tail first increase auto count increase week whereas probability long occur k n take lot effect probability model tail conclude scenario confirm effect tail useful short term risk pricing derivative market model c model american day day day day day day west american day day day day day day k west south american model south american day day propose south american day day west south american vs matrix large distance contiguous perhaps day take account period overall medium theorem corollary proposition universit du pr email de universit du pr email finance represent occurrence integer occurrence extension cross autocorrelation fit various bivariate count many contiguous cross autocorrelation seem count multivariate point apply reason series model excellent autocorrelation autocorrelation investigate operator application treat car order independent derive multivariate attempt random variable notable poisson application paper autocorrelation policie cat cat cat offer periodic component various location upon poisson cox process self release review markov hmm toward location location see g purpose autocorrelation location count another site area main investigate management management consideration view example cat influence dynamic lack count prediction occur well causality count vector random component process diagonal chain probability satisfy associate positive assumption irreducible recurrent strictly eigenvalue td variate autoregressive matrix variate finite
estimator possible scope involve quadrature adaptive grid base capability smc overcome limitation smc base ei value dataset gaussian unknown soon smc however quickly algorithm function reference always lag behind new algorithm kind study evaluation ref ei ei ref ei ei proposition real past evaluation difficulty use article decide find maxima use ei deal implementation idea ei view define sake generally mean ei distribution conditional algebra ei expectation easily see follow impact applicability deal integral deal issue dirac use method therein sequential carlo smc care follow small population evaluation region x objective particle nx ng nx reflect past consider exceed initialize sample instrumental pick construct nx I c ic j nx
involve particular partial linearization incorporate fista several synthetic strength compare relative induce norm among fista good fista appear alternative fista variation different discussion chen matlab completeness convexity iteration apply hold let let get q sum instead hence n yield stop criterion dual outer iteration gradient iteration iteration optimality primal feasibility vanish e residual fx line fx k residual satisfy every inner residual derivation satisfied fista optimality condition feasibility vanish clearly z k x residual consider iteration primal feasibility q feasibility primal residual residual lx cx l cx l v cx lagrangian evaluate augment lagrangian outer readily outer c cpu avg e fista fista e fista fista e e fista p c c cpu avg fista fista fista fista e cpu avg dct fista dct fista fista fista p fista dct fista p fista e e fista c set sub dct fista dct e fista dct fista e dct fista fista dct fista p fista dct fista fista cpu avg dct fista fista e dct fista fista dct fista fista fista e fista fista e dct fista fista c c data set cpu avg fista fista fista fista dct fista e e fista dct fista p fista e dct fista dct fista fista avg sub fista fista section problem space regularize induce incorporate pose considerable challenge non sparsity induce norm unify augment lagrangian variant efficiently build block linearization splitting accelerate algorithm alternate fista method norm sparsity alternate direction augment lagrangian dimensional back classical lasso particularly appeal gradient take sparsity show assume define feature allow array application image extension adaptive paper minimize loss regularization eq index element possibly overlap pre without penalty cast overlap close approximation nesterov solve thresholde fista upon practical convergence liu fista proximal operator penalty solve descent gradient et al use unified tackle base linearization alm fista solve equivalent problem contribution augment lagrangian block alm splitting linearization fista serve key fista tune every evaluate rich breast cancer video unify base variable splitting lagrangian method solve problem obtain repeat component group group structure sparse equal time include appear loss term equivalent elastic reformulate merging penalty net loss solve augment lagrangian algorithm augment lagrange multipli update parameter amount place view x cx l guarantee exactly instead minimizer abstract subroutine theorem inexact version mx lf converge condition augment lagrangian subproblem solve cx l update call outer subroutine index penalty regularization without well direction overlap approximately minimize minimize update lagrange multipli procedure serve multiplier accommodate example splitting fx iy solve simultaneously simultaneous multiplier partial parallel proximal algorithm solve subproblem soft td jx side many set highly determined stay store subsequent large mention cholesky system conjugate comment section fista linearization alm apply sparse regularizer alm lagrange multiplier define linearization lagrange multiplier alternating implement fx fx k variant fx eq hold matter violate minimize ensure convergence execute likewise ready lipschitz satisfy optimal regular iteration among first hold easy satisfy entirely equivalent linearization variant lagrange multiplier redundant subproblem hence thresholding subproblem solving give optimality gaussian yield system section cholesky factorization amount comparable straightforward accelerate partial present algorithm apply alm split solve write even gradient compute cx alm need backtrack line fx adopt scheme initially decrease number reach minimum prevent hence leave factorize splitting potentially linearization lipschitz side allow alm line become intuitively job load hardness augment lagrangian define variant set property hence implement subroutine linearization k kx equivalent lipschitz iterate easy equivalent exactly difference outer lagrange multipli stay inner delay lagrange multipli load balance section obvious full linearization gradient minimize take large size due constant present version fista th outer termination stop iteration primal relative inner table expression directly gradient residual th outer outer inner objective gradient residual fista k c k k z k c x z z except fista fista hold solution return become increasingly evaluate quantity lagrangian experiment primal two basically terminate iteration subproblem high inner criterion slightly alternative stopping loop context fista require th iterate update outer heuristic primal residual subproblem subproblem adopt kind keep appropriate work dynamic scheme primal residual yield small hence outer iteration fista propose arrange adjacent overlap three output vary increment b version scalability base direction fista perform slightly show obvious fista base large cholesky fista system solve plot right fista fista due number dct approach dictionary cosine dct contiguou five non increment considerably hard heavily exhibit run c algorithm leave whose already fairly figure tolerance gap keep cpu versus fista outperform regularization fista perform growth fista trend outer show fista large factorization fista cholesky factorization fista subproblems performance fista fista model serve update fista significantly improve scheme scheme work turn case fista suboptimal fix report well algorithm numerical compares stay fista job gap fista obvious usefulness test world breast cancer tumor measurement goal good experiment survival patient pathway naturally group regularization summarize table fast tune scheme speed solution keep constant return instead identically fista fista behave fista number inner error different leave rmse compute low rmse yield well magnitude return group pattern rather background main segment foreground image frame frame camera pixel form ax noise combination video frame foreground lasso regularization patch still group consist element alternative net yield well linearization fista p lagrangian also solve solve mask truth foreground ground truth maximum special
kernel rkh characterization observe reproduce machine onto span w inner linear fourier adopt fouri lebesgue satisfy tell integrable prescribe rkhs lemma paper evaluation shall characterization infimum make convention proved fact get kf h lead another feature moreover nontrivial adjoint argument particular choice countable prove translation reproduce invariant characterization invariant translation exactly borel measure inclusion relation translation invariant kernel q borel topological space absolutely subset function denote function norm later hilbert borel translation kernel happen everywhere clear pay borel continuous lebesgue integrable measure essentially equal everywhere radial kernel variate area area define borel define reproduce borel vanishing equip argument rkhs borel k dd absolutely lebesgue measure inclusion six invariant kernel learn apply mathematic apply norm transform identify page combine fouri function spline spline inverse give kernel denote singleton exponential application kind hold np h b p relation rkhs separate let hold use one define right infinity h unbounded origin neighborhood origin therein thus estimate yield go inclusion rkh form constant relation hold break prove step clear possess almost consequence handle way h b nb p prove eq everywhere method ii combine explicit e g e r equation laplace delta singular measure borel absolutely common vary nj yield tell corollary hold equality achieve h eq claim lebesgue dominate origin corollary h monotone equation h continuous zero ix upper tend convergence everywhere positive hence conclude measure absolutely continuous choice result argument straightforward calculation hilbert schmidt represent recently construct multiscale wavelet introduce hilbert schmidt nonnegative denote hilbert give belong schmidt associate nontrivial space similarly exist constant arbitrary imply denote delta equation suppose cn bb give example inclusion hilbert schmidt schmidt necessity contradiction nonnegative clear schmidt kernel assumption proved sophisticated mathematical radius reproduce fall proposition result explicitly sequence distinct nonnegative example include periodic see eq b n n nz nr b assumption especially finite kernel inclusion let reproducing hold eq contain matrix denote hadamard product still definite kernel notational shall definite k g complete kernel restriction sequence remain kernel kernel respectively j get remark remove last let limit reason proposition ready result analytic nonnegative coefficient origin nj nj nk ne proposition able prove k investigate inclusion relation small positive constant refinement relaxation refinement accommodate reproduce investigation characterization let hold simplicity satisfy f f f k word conversely since lemma imply finite introduce clear move applicable schmidt example borel absolutely introduce kernel characterize inclusion measure respect nonnegative denote exist constant schmidt dense refinement kernel increase chance polynomial kernel kernel positive proposition refinement weak refinement china science program air force office grant reproduce science hilbert maps correspond reproduce full table invariant example operation reproduce briefly inclusion equivalence refinement reproduce
disease early throughput genomic genome wide disease genome study dna copy gene throughput disease gene annotation phenotype example know tool david others annotation gene poorly often association phenotype availability molecular disease high genomic phenotype mine clinical phenotype molecular protein network provide relation tend interact utilize module disease study disease gene gene query similarly set select quantify relevance phenotype phenotype coherence top rank phenotype phenotype target query connectivity connect phenotype discrepancy phenotype rank target phenotype query coherent upper phenotype phenotype phenotype association phenotype gene utilize phenotype gene network gene retrieve phenotype assumption rank relevance gene set coherent association disease phenotype disease phenotype framework phenotype gene laplacian fig laplacian result gene seed walk propagation relevance phenotype similarly laplacian random fig ranking phenotype rank gene phenotype highly quantify coherence real problem phenotype unknown design phenotype combinatorial ridge close form select disease phenotype variant phenotype good match gene gene phenotype rich reliable phenotype share gene purpose gene phenotype phenotype relevance phenotype phenotype walk simple exploit network phenotype gene map phenotype know association utilize propagation heterogeneous combine phenotype explore gene module phenotype phenotype interpret good formulate query phenotype discovery consist network phenotype gene retrieve phenotype association phenotype query denote membership query gene phenotype target phenotype coherence laplacian utilize global gene laplacian normalize gd g laplacian connect receive ensure gene neighboring consistency query global relevance query contribution penalty close eq empirically iterative eq phenotype phenotype form solution score infinite laplacian modular graph use scoring count direct set measuring query information fully coherence coherence whether query phenotype association disease association relevance graph phenotype phenotype whether give connecting phenotype similar propose coherence regression couple close relaxed simple score phenotype reconstruct neighbor cost equation norm small take standard ridge real vector simple post one phenotype significantly score phenotype full solve require matrix inversion thus j j enumeration enumeration disease phenotype phenotype query phenotype score phenotype use pearson disagreement relevance strategy conceptual simplicity incur calculation phenotype find multiple full inside loop line overall complexity want retrieve phenotype explore configuration exponential leave one task predict recently discover disease gene association validate finding copy network phenotype represent disease phenotype edge phenotype text clinical record calculate association connect node version may experiment association disease phenotype gene association connect disease phenotype experiment use copy expression two protein undirected functional dna contain around million weight reduce cutoff weight generate million weighted evaluation label propagation shortest sp association average walk label propagation phenotype choose balancing method cross query disease phenotype phenotype performance receiver operate characteristic curve interested phenotype near area roc evaluation select rank phenotype coherence top gene phenotype rank large cost small penalty quantify connectivity disease phenotype know gene association select fold rank query gene phenotype rank disease phenotype compute phenotype phenotype plot threshold score assume evaluate well disease gene association association study design disease phenotype phenotype since gene experiment phenotype newly disease disease retrieve report sp new phenotype network necessary accurate far compare disease third mark denote multiple trait category disease trait rank cell high diabetes diabetes disease predict disease phenotype disease individual disease influence implication pathway annotation case collect disease phenotype discover disease trait phenotype disease trait phenotype subsequently disease trait example predict phenotype breast gene novel rank disease gene gene also include gene similarly disease phenotype rank breast cancer phenotype top gene phenotype association phenotype set disease gene disease trait disease phenotype disease trait match multiple phenotype report match phenotype well rank disease trait among within top notable cancer breast cancer disease phenotype gene within accurately breast phenotype gene rank rank disease around fold phenotype fold connection phenotype rank disease phenotype phenotype breast cancer gene directly rank disease phenotype gene directly interact rank association phenotype result association suggest gene disease phenotype network disease association statistical significance connectivity fold association obtain interesting also disease know association disease phenotype poses reveal rank cross validation cancer interestingly network compare disease share disease study disease gene reveal disease suggest incorporate topological association successfully discover association case disease phenotype dna copy copy identify target disease phenotype disease copy change region collect dna copy change copy study dna copy measurement snp copy change detect tool default detect number region query gene phenotype rank disease top rank rank target disease state copy gene previously target human reveal disease association find htb include human cancer copy trait rank rank cancer cancer cancer tumor phenotype frequently gene differently gene experiment algorithm predict disease differentially express gene gene expression collect cancer gene profile array differentially express differentially express disease quantify differential propagation predict disease differentially moderately encourage neighbor differentially association disease microarray gene second disease trait cancer algorithm disease association gene retrieve phenotype rank association fig disease gene currently gene c pt identify display pt discussion set genome throughput disease detect association phenotype significance molecular analysis fail find disease experiment phenotype report improve phenotype gene association gene well association global comparison disease gene algorithm utilize hide gene network label explore gene phenotype quantification relevance score association evaluate association bias utilize datum handling association shortest ridge association phenotype enumeration encourage limitation heavily gene association target disease phenotype introduce thus useful disease characterize well understand disease closely phenotype
medium support get corollary remark ex address minimize optimization hard approximately singular value certain calculate completion rank matrix smooth generally equation speak base instead infinite index respectively sparsity minimize minimize application reduction since infinite cast find singular matrix analyze robust approximation side benefit bind completion efficacy movie datum equation various context trace surrogate closely low singular extensively mainly context trace encourage rank produce study several guarantee rely assume incoherence entry trace involve program large problem minimization selection mp omp formal guarantee norm however choose choose rank omp algorithm sp come elegant rely e isometry assume smoothness solve psd turn wolfe well important change though tackle enforce neither fully difference analogous wolfe fully greedy selection discuss involve approximately improve approximation assume v scalar simplify real du finitely many u v input tolerance initialize I v tb tu u mapping eq easy convex fr problem optimization equation arbitrary forward start find maximize magnitude partial assume obtain maximal though element even lead singular expensive maximization implement element see analysis value increase note approximation whose contain set v equation write write whose svd tell unconstrained optimization minimize minimize practice maintain pseudo code runtime runtime function describe later runtime choose section lead improve still increase section find direction possible verify describe simple start find candidate matrix small let value rank increase decrease analysis tell sufficiently restrict replacement rank guarantee increase guarantee rank constraint guarantee regularization vector term equivalent norm add pose trick singular singular step reasonable effort step expensive completion runtime diagonal word change solve variable competitive perform solution competitive theorem smooth eq zero constraint low trace competitive convex strongly implication several problem example movie find agrees square minimize low rank easy otherwise element row element write u take make least runtime completion parameter j rank predict unknown high low assume frobenius da sensitive outlier replace frobenius norm constraint propose minimize norm require work smoothed replace absolute huber otherwise smoothness huber bound therefore entrie huber discrepancy lemma matrix completion dataset rating movie integer first try decrease singular matrix inspire proof decrease scalar like one yield examine balance
detect pick happen say practical issue purely guarantee behaviour white goal look colour maximizer maximizer q noise possibility design question j nh formally maximal work noise statistical terminology randomize crucial within pixel case moderate easy strong size obvious algorithm relax square triangle shape formulate picture find depth thresholded black large report object black cluster search deterministic description rigorous complexity classic book picture probability nc dependence implicitly mean think image detection pixel point algorithm remark simply detect object sort thresholding avoid although work possibly heavy tailed example heavy detection thresholded detection thresholded picture maximal cluster safe much efficient calculate moderate applicable cover contain square size consistently neuron pass detect neuron acknowledgment van discussion proof devote proof also convenience theorem suitable predefine take operation step chapter standard also save position pixel false denote white pixel outside black additional lie marked cluster thresholding prove open origin conclude book terminology increase large large contain obviously symmetry affect increase decrease ci j p p j n ci j p n k immediately detection follow detection first follow site lattice path leave maximal vertex rectangle slight modification pp object picture square picture square site take cluster big proposition phone phone correspond propose method detection image detect shape appropriate prove result algorithmic quick noisy indeed look noisy question ask primary question field cancer automate detection picture intensive procedure particular picture surprisingly vast engineering skip crucial impose nonsmooth object object noisy shape advantageous detection one practitioner require reconstruction detect usually method rigorous performance free drawback though study subject technique present detection nonparametric number asymptotically consistent pixel stop human stop could serious limitation case affect scope object vast majority detect background colour intensity object thick colour background moreover background case paper simple rescale colour section new section main testing computer work interested colour colour colour background noisy black black background white belong meaningful object within pixel noisy pixel observe begin formulate model observe correspond stress level way continuous moreover easily arbitrary noise statement let black omit since identically white colour image q standardized colour noisy image look colour colour account intensity grey pixel pixel black colour obtain grey value white pixel colour previous procedure would call analyse picture paper let exist satisfy satisfied small pick need research formally definition step transform picture pixel picture call vector moment pixel picture furthermore colour graph black white far edge follow pixel
high profile operational loss supervision discussion management find survey conduct supervision capital risk market risk capital working adopt change international adopt operational capital requirement broad seven event business business failure process management risk fall risk treat separately result bank include term consider nature assess model understanding modelling framework capital internal deal quantification financial bank around bank model statistical tailed infinite family stable loss stability combine ii requirement agreement aside capital implement framework requirement occur expect loss rare bank methodology operational area finance broad approach bank capital advanced model bank adopt comprehensive internal risk quantification flexible believe environment quantitative criterion sufficiently account impact lda frequency simply fitting parametric mathematical compound matter situation expression compound class stable develop paper compound affect setting operational company growth forecast year economic factored scaling threshold record choice distribution binomial binomial generalize pareto stable family important distribute heavy tailed scenario tail extreme providing process likely financial introduce give stable family enough incorporate light tailed mean loss incorporate expert opinion scenario approach adopt model select combine compound compound capital compound fit business process business wide correlation typical frequency framework adopt lda ii requirement business year horizon section framework present cell business compound year event count typically year nature rare particular infinite propose classical available modern processing calculate algorithm recursion long history quantile function heavy recent development class admit analytic solution bank risk calculate requirement seven eight business depend drop index risk unless utilize loss business convolution density sum calculate convolution thus q nz convolution analytic integral form consider generalization close stable ensure stable possess useful skewness year cell four parameter parameterization appendix tail decay determining skewness exponent heavy light cauchy tractable family stable admit closed evaluated point typically characteristic discussion intractable simulation efficient function basic require lda binomial poisson doubly poisson bank positive considering analytic es case lda asymptotic comment capital aggregation model contain dominate time increment month distribution model variance less arrival loss increment arrival process poisson clearly year loss loss extend model doubly derive binomial year process extend doubly mn doubly binomial standard beta convolution lemma mix compound consider bayes conjugacy know hence doubly negative cumulative distribution close conditional application mix compound derive conjugacy property know hence recently poisson begin poisson process exact loss poisson cumulative close eq convolution random weight compound conjugacy poisson q consideration study capital binomial doubly develop analytic sum representation determine sum theorem demonstrate paper compound asymptotic truncation finite tail compound median allow term lda compound searching value integer specify n w r p n nr l sum compound express analytically eq determined demonstrate truncation discussion section table derive frequency setting frequency addition matlab intel ghz cpu source code edu file c cn cp binomial beta gamma ga frequency beta binomial beta poisson ga loss compound correct simulation study simulate develop via obtain year obtain expression point cover represent capital model compound doubly beta model derive frequency doubly theorem frequency low poisson high frequency gamma compound doubly gamma derive frequency frequency carlo widely generation compound demonstrate accuracy derive estimate excess estimate example demonstrate truncation binomial doubly binomial binomial doubly poisson versus estimate cdf square cdf mean demonstrate squared index simulation monte equivalent ccc carlo low sec min beta negative min min poisson min min binomial min beta sec min poisson min times versus monte estimate next result provide distributional combine lda distribution present closed form trivially result lda lda compound I jt analytically comprise mix specify q apply density cumulative analytic doubly lda model ability heavy typical aggregated compound model loss first throughout year intensity result compound scenario consider increment increment capture binomial success change derive estimate simulate exact low high frequency scenario term develop truncation
heavily depend let procedure poorly effect come da procedure path matrix scale likelihood recognize poor behave look validation organize section show weak part local degeneracy weak element posterior von asymptotic sequence law chain stationarity guess distribution impractical good mind say always work inefficient good bad degeneracy exclusive degeneracy degeneracy poor behavior sometimes kind convergence mcmc weak call interpret large iteration semi jump embed markov law almost hand identically general take lemma n follow ergodic tends claim measure assume always strictly procedure component perform well modal switching degeneracy weak consistency assume close bad illustrate obvious relation write integrable take scale q consistency local degeneracy localization da ergodic see also convergence process weak order invariant rewrite convergence probability procedure direction asymptotic high small asymptotic particular latter local result da independent hasting compare da procedure mixture elsewhere briefly procedure iterate f hence obtain proposal close kullback leibler posterior n consistency da consider mixture illustrate difference da procedure fairly unlike trajectory da behave weak consistency effect difference quickly size da da suggest check effect different value order da htbp check mcmc le suggest comment difference may article present suggest order verification chain diffusion probably da procedure probit numerator denominator propose grateful suggestion ng gx rt side q argument I negligible follow contiguous vice versa pd n probability prove numerator nf numerator nc bernstein von come distance bernstein assumption dx da diffusion limit become sequence respectively moreover nx x read remain term expansion n p next show p previous n n n n n n h nc n equation come term side n nr n n n write give convergence study markov reference therein apply ix da tend proposition continuity representation may space theorem assumption assumption remark assumption commonly year result convergence convergence
use validation set well benefit output binary node filter use class decision node arrange bottom testing proceed leaf computation logarithmic filter similar study reward context know contextual bandit include health care task context challenge past proportion take either former leverage overcome policy evaluation value good consistent doubly low variance well policy doubly become environment receive feedback action internet whether user ad receive ad health refer patient version assume kind address contextual bandit dm use incorrect data model reward model reward restrictive hand usually possible quite suffer especially differ evaluate technique dr estimation overcome doubly robust statistical estimation incomplete dm correct unbiased drawing inference age family ask survey census statistic condition weight estimate form form predict outcome available doubly estimation accurate predictor doubly bandit analysis model deviation doubly style doubly robust beyond study evaluation doubly estimation therein internet effect feature focus evaluation address doubly asymptotic various assumption make paper idea language reward estimator bad variance offset tree think offset case estimator describe sophisticated contextual new reveal choose concatenation precede reward observe collect estimate maximum lead well classification challenge previous reward datum limitation dm condition good refined instead approximate action proportion old new function argument since understand policy direct variance issue become gets take doubly take advantage previously study learn informally baseline correction estimator accurate perform estimate truth dr deviation multiplicative express expected value clutter fix derive identical magnitude contrast direct variance policy reward suffice significantly however direct leading estimating evidence dr feedback benchmark average internet armed contextual bandit allow compare evaluation dataset letter datum draw iid class cx kl policy label loss select reveal component form summarize adopt repository letter dr rmse dr dm letter page dr ft offset investigate accurate policy context constitute policy test obtain dm dr estimate dm dr estimate loss ridge result bias rmse squared predict dr estimate accurately suffer apparent enjoy effect come policy dm focus dr apply dr time partially decision ridge partially run policy optimization advantage great evaluation dr includes offset specifically optimization implementation version rather weak dr version offset tree tree finally one descent induction different choice number internet website record million associate feature age gender calculate summarize unique random true however situation impossible scheme special formulate arm formally partially label sample define adopt denote univariate deviation th randomly dataset subsample
henceforth simply describe ball ball respect avoid confusion conventional definition uniqueness radius available radius set interior minimum continuous compact nonconvex present minimizer acceptable axiom compress sense rely provide certain definition state asymmetric previously format rip rip isometry large vector symmetric minimization accurately accuracy estimate sp guarantee solve least rip proceed form rip compressive measurement rip estimate coherent kk obeys eq parameter approximate ideal feasible f term error ideally residual merely step small desirable restrictive choose control increase decrease imply measurement increase iterative rip I algorithm reduce function determine error nearly sparse power law choice suppose hold obey x thus apply inequality inequality note q use yield desire case inequality please pl prove define projection onto thereby recursively statement theorem descent sparse square norm onto ball statistical requiring small method surprisingly leave unclear intermediate x x projection ask design develop provide building method may find future sophisticated nesterov method constrain learn f property projection every proof contradiction ix ix I lemma therefore simplicity assume real onto coordinate another onto x j I jx jx jx fact gradient constraint uniquely define entry q partial derivative desire pp pt statement hold regard root root respectively illustrate thereby since case zero peak strictly fact p exactly claim yield fact peak suppose obvious contradiction w otherwise identical merely algebra always sign contradiction rely isometry entire guarantee iterative thresholding suggest group increase accuracy compress restrict isometry project problem signal observation govern equation linear simply collect achieve merely singleton least describe imaging absence independent ideal extended norm convenience throughout scalability functional merely merely count theoretical minimization however alternative framework regression solve square eq tractable include thresholde pursuit compressive pursuit solver solution approximately iterative outline henceforth limit recognize soft originally pursuit also solver consider shrinkage iteration iterate sufficiently contribution comprehensive regime unlike feasible satisfy therefore relie fact long extra condition convergence name therein ignore exhibit sublinear optimization guarantee decomposable norm include linear possess objective
home sale work stock price market unclear market collective web daily volume stock correlate daily volume stock particular query volume many peak trade day carry dataset web engine enable user show dynamic collective finding contribute early financial www introduction activity leave digital transaction mobile reality field science physic several social phenomena direction concern epidemic people usa activity keyword actual care paper address issue obtain early financial market issue financial ultimately phenomena signal complexity anomalous collective great interest policy intervention appropriate start however context market show volume shift correlate price yahoo engine list exchange assess hereafter relate daily trading hereafter mean cross correlation mean causality analyze search activity collective hereafter consist stock market financial stock price query correlate transaction stock lag week week query volume correlate week trading volume yahoo engine volume stock aggregate volume suggest trade stock pointing investigate market daily scale even volume proxy market address forecast market novel analysis traffic activity market frequency volume volume stock trade company top plot trading volume company research motion limit material simple inspection reveal time series tend peak fig cross trading query pearson q trading volume range cross correlation positive broken line volume trade volume day beyond lag day trading vanish cross trading volume clean cross completeness support clean table stock spurious origin volume volume trading volume confirm finding also vision market influence activity contrast appear much short seem resolution result coefficient present query volume trading volume appear large coefficient opposite assess significance set company randomly pair trading time company average permutation range small present explain term general remove event trading time series verify table stock report support value cross stock observe stock consider drop distribution underlie investigate series tail fig support discussion validity correlation responsible correlation stock percentage explain turning towards volume trade volume well trading volume volatility appear volatility exponent range therefore volume trading volume fig effect respect correlation absolute return result branch volatility even trade origin due volatility cross correlation volatility autocorrelation support determine provide trading volume find volume cause want causality argue cause interpretation causality link wrong result direction trading claim volume observe volume tail investigation fig support picture come focus yahoo yahoo expect user stock interest user market display important news corresponding page among link various window interestingly month whole robust along certain restrict high correlation trading volume amazon com netflix month year user check portfolio stock suggest financial expert search uniform way addition trade price drop drop show appear evidence query trade user bring surprising picture volume user store yahoo engine assess stock trade stock focus trading volume price volume compare trade stock computing delay existence web traffic trading stock confirm analysis user query stock month suggest market crowd finding explain market proxy furthermore query reflect composition find assumption portfolio composition know empirical market confirm take bring incorrect disease portfolio balanced could market straightforwardly epidemic financial market fact latter differently ordinary news financial agent market short disease reason early carefully effectively detect sign also believe promise currently kind semantic material overview contribution previously say user yahoo event place market basic market activity stock correspondence activity stock trading volume variation volume relate search query volume volume compute conduct analysis perform take consideration contain stock string company query volume trade permutation causality several analysis assess significance correlation statistical user work user issue query finance refine extract typical look one look stock analyze compare volume volume national association trading market united precisely analyze market amongst index contain financial include outside united list daily financial yahoo finance focus attention volume yahoo engine span year mid action user interaction engine page return decide volume extract aggregate daily text contain stock yahoo text company name incorporate log moment engine aggregate volume different annotate represent query activity count daily distinct company daily query distinct user query volume yahoo finance volume trading volume compare every trading volume stock separate query company company extract source volume series interval range mid mid contain stock financial available trading thus uniformity working day work stock series volume series trading volume company coefficient delay choose maximum lag week five correspond cross coefficient consideration coefficient case lag range consider worth individual observe stock support information worth stock stock stock hold stock also table table lag remove time trade interesting correlation seem event news second company name observe weak average cross correlation query spurious query origin query nonetheless stock represent string natural language life completely study level query company stock support information filter query noisy spurious query restrict report correlation lag volume volume query relate company clean log correlation volume trading volume show one volume correlation user volume trading lag statistical permutation test randomization permutation validate nan permutation outcome true among determine value least extreme reject verify significance company volume cross query volume trade high company company merely consequence market activity market activity pair trading volume cross permutation ensemble series volume company pair series trade company pair randomly compare macro real dataset company pair company get value test find separately understand deep actually one scenario company company trade company company form every company empirical value statistic sort similarly volume company trade company correlation company trade company query volume real dataset include p reject value case stock query volume global finance relate search traffic activity general worth present trading volume volatility trading volume volatility volatility source volume trading autocorrelation back volatility price proxy query trading define price day stock price return build series series positive return rt rt rt rt length similarly involve trading volume stock cross volume company break report correlation basic price return reflect stock exhibit variation web search concern stock show correlation volume volatility break significantly query trade solid branch case volatility observe autocorrelation expect origin correlation query trading return negative volume reason consecutive clean table one get causality whether test cause reject apply causality volume trade volume trading volume company cause trading volume company stock whether causality opposite direction include however query volume noisy extract perform manual result table summarize outcome specify volume volume compare company cause column provide fourth reject report observe opposite strong time cause trading company oppose user row examine reject much opposite direction reject already observe experiment weak consider user volume case trading cause volume autoregressive trading volume volume volume cause trading also interpret follow query volume help predict trade reverse argue principle term regression instead deal query
without receive feedback action play otherwise action probability play result correspond confidence action play figure action exponentially call tb game horizon confidence horizon v v v g tv building worker routine routine discuss round update receive change subtree store node tv v tv recursive play tree v either exp optimize corollary tuning tuning book show figure exp transformation second action outcome v pl v v w h er exist outcome throughout play measurable universal give er become let stay index tt jj j far least bound martingale associate denote play time history recall construction algorithm explicit th unbiased random x hence bernstein inequality achieve desire bound depth either exp satisfy exp induction hypothesis true depth action action furthermore word loss contain event know last last switch step even inequality outcome assume step exploit hypothesis sum root call child eq action euclidean non trivial loss game monitor game exist minimax regret present work adversarial ensure dominate action small outcome close finite dominate action simplex cell set action space polytope denote induce interior small pair cell cell decomposition cell hyperplane I characterize follow action dominate kind cell ic dominate game mean action cell decomposition correspond non dominate clearly cell coincide pc p pc pc f choose lie two fix combination dominate cell compact convexity contain closure well length also p eq choose p v part adjust necessary continue lemma quantify kullback divergence define divergence divergence eq depend proof lemma notice sum coordinate modify generality coordinate write logarithmic second taylor expansion around universal substituting let third term dp always lemma action satisfy avoid indexing action later randomization time choose outcome monitoring outcome outcome appendix imply ki ta imply continue collect averaging get lemma proof carry state action close partial outcome pair outcome n action put ensure game hold remain obviously take corresponding trivial present outcome condition lower bind er constant minimax let important minimax degenerate degenerate action find degenerate generality dominate reduction section far non get generate outcome outcome later replace variable internal outcome depend depend algebra depending prove inequality rearrange substitute statement subtract lead lower bind low independently give choice ensure separation clearly trivial b choose monitor minimax question degenerate degeneracy category game minimax nonetheless conjecture include important open result generalize game outcome finite monitor restrict opponent two outcome result hardness regret condition believe generalized situation play respective generalizing result algorithm exploit two yet outcome bind exploit assumption opponent ab bc lead proper continuous minimum action whose large action action non dominate dominate loss vector vector da action lemma compact imply outcome regret know thus tc depend c c c I ic I sp ic ic regret take outcome randomization clearly omit prove deterministic denote history history section since determine general feedback action game action product sequence last divergence notation none action degenerate play action th x tx al de ec game mathematical framework decision repeatedly suffer receive feedback signal minimize progress game classify game finite regret either computable four toward classify monitor online monitoring constitute framework decision feedback arise make expert multi dynamic pricing pool convex partial monitoring game opponent receive feedback signal neither outcome reveal learner player learner assumption opponent opponent learner keep loss opponent learner suffer loss ask cumulative competitive viewpoint take cumulative compare I round growth sublinear per round approach design focus monitor optimal view measure hard motivation behind determine minimax achieve large game action set action outcome full game determine lie time therein game loss lie bad regret least monitor game loss action loss lie inf know exp know optimal strategy regret bad learner bad recently design possibly action game game step classify minimax monitoring game exclude later minimax fall call four game respectively give efficiently characterization four class admit efficient regret factor design et trivial game choose algorithm partial monitoring game number outcome notation denote outcome outcome denote alphabet real learner opponent game proceed round choose opponent outcome receive nothing else reveal learner particular remain simplify assume opponent deterministic learner equivalently allow learner feedback outcome internal random learner incur learner constant ta eq supremum outcome infimum also identify probability dot product characterization preliminary partial let one action dominate dominate pi outcome neither hold outcome game action bind monitoring game action bandit example armed bandit game feedback instantaneous advance classical bandit loss constrain game condition type action game hard classical bandit game regardless loss vary action action dominate game minimax fact want sequential test suffer loss formalize monitoring game cm q outcome correspond correspond dominate minimax regret result also notice picture translation picture bandit efficient situation sequentially spam email output additionally request label email incorrectly request suffer email request feedback loss formalize game c correspond request third spam request outcome spam chain neighbor game reveal outcome q action dominate minimax game regardless choose round guarantee regret cm non action thus degenerate game cm
embed plane lattice various embedding site triangular lattice occur statement directly proposition site big least constant hold prove acknowledgment like thank van helpful phone phone novel statistical algorithm noise wavelet system noisy digital pixel propose efficient technique quick detection noisy detection one look image question diverse cancer automate picture make intensive procedure skip immediately impose object permit nonsmooth noisy mild object interior detect aforementioned automated cancer material procedure work well precisely image method advantageous object use practitioner medical perform reconstruction image object rigorous analysis error differ substantially subject wavelet technique nonparametric necessarily asymptotically pixel algorithm drive stop human stop appropriate white focus serious limitation essentially vast differ background colour colour object paper applicable rescale colour organize statistical detail crucial theorem testing devote main numerical computer discretize setup colour colour background mathematically assume black pixel belong meaningful image pixel noisy pixel pixel paper always begin analysis assume array value stress nan fully necessary mean restriction quantitative original corresponding free dependency throughout degeneracy symmetric since symmetric ij p p ij complete ready describe thresholding pixel colour colour pixel grey observe grey begin think pixel colour grey value noise pixel colour transform picture mathematical main fix exist condition pick vary transform set pixel picture call pixel colour edge add edge two connect vertex crucial definition lead testing procedure property view square subset view black white picture lattice triangular thresholding theory formation crucial explain split vertex belong original left background vertex subgraph observe cluster go detail theory introduction mention relatively difference theory quantitative behaviour efficient randomize detect phase simultaneously appropriate mild noise triangular lattice statement proposition explain lattice applicable nonsmooth nonparametric observe design question ij nh formally probability noise time crucial kind picture small observe precisely object moderate easy strong contain least pixel furthermore purpose relax triangle shape character finite ready false run step find picture first thresholded picture large find size cluster define define search graph procedure deterministic classic book chapter symmetric e picture exceed n nc mean order think work image pixel detection square use consider size neither sort thresholding indeed noise tail tailed thresholded value false detection fine fig neuron advantageous gaussian independently pixel image picture version picture run simulate neuron detect picture pixel value grey picture thresholded study consistently image nan true picture algorithm thresholded consider moderate size algorithm consistently detect pass neuron run shall site triangular cluster contain origin triangular satisfy eq otherwise need
know assume illustrates data four bayesian mixture mix prior cluster later make mixture choice model mixture generative define assignment assignment assume observation conditionally return rt example cluster rt interested assignment grouping switch care label care preserve ignore rule assignment fix assignment denominator sum group become salient limiting discuss contour finite specify cognitive advance cluster allow previously unseen assign call partition independently discover rational categorization crp derive name imagine table imagine sequence restaurant customer table probability restaurant table customer customer partition show generative crp customer circle crp crp mixture model assignment customer sequentially assign class customer number table intuitively large customer crp invariance order seem counter customer assignment accord chain term come equation group place customer customer assign customer cardinality first customer contribute customer contribute probability customer third contribute probability write rewrite equation per notice identify customer assign exactly group simplify denominator write joint equation reveal exchangeable group group configuration depend order customer crp model table though exhibit finite active customer draw observe concentration control table customer treat return rt crp us place cognitive process set process specify mean transform force notice skewed outlier examine infer assignment cognitive process addition crp away parameter partition group predictive crp predictive distribution crp previously unseen example time force decision experiment color gibbs factor influence way model intelligence application latent usual component small factor much contribute see representation primary mode popular observe property latent recent review specify preference kind g intelligence assume factor intelligence loading influence noise extend factor loading product two binary mask variable indicate continuous sometimes spike spike zero bayesian approach infer factor mask ica fit likelihood classic application intelligence argue intelligence intelligence test participant highly test highly several pattern arise unitary intelligence resolve factor intelligence convenient reality intelligence avoid specify instead allow row correspond intelligence infinite correspond generative ibp customer circle ibp customer select factor note ibp rather chinese process component ibp infinite similarly customer dimension infinite arrange line sample customer sample never customer encode customer ibp ibp crp unbounded crp one latent ibp tend ibp key crp ibp crp ibp illustrate return intelligence intelligence intractable approximate one g give one typically estimate crp ibp analysis collect reasoning task participant histogram factor posterior sample indicate combination around although general intelligence investigation right display first load ibp demonstrate example histogram factor infer generate burn relationship loading infer ibp describe factor ibp generative analyze give latent generate datum thus computational many close way treatment beyond scope use link package implement markov chain equilibrium draw chain eventually sample markov construct thank exchangeability property crp particularly amenable consider use distribution term excellent survey crp gibbs factor inference different figure two drawback assess idea posterior family search member close recover unless typically crp ibp operate crp ibp factor operate available alternative selection parameterize broad array bayesian variational provide direct space model find convenient modeling number component search efficiently explore widely limitation worth past year idea diffusion idea volume concern across group similar virtue group provide cognitive characteristic degree coupling set extension beta share individual document word share across return rt imagine subject infer underlie cognitive suppose cognitive share different proportion precisely kind hdp limitation dependency capture markov class hide sequential employ class infinite hmm hierarchical dirichlet alternative linear dynamical autoregressive moving evolve linear linear dynamical explore nonparametric switching mode hdp another type dependency arise dataset disease location also occur nearby way capture dp variable spatially couple generalization dp segmentation allow nearby pixel pass devise specification dependency covariate people age risk disease recently author ideas ibp factor discover hidden output variable discrete supervise output non linear function place supervised proceed function gaussian inference process analogous mean parametric predictive develop prior input linear mixture relate marginally linearly within output model gaussian process output input build flexible grow review model potential scientific noting address choose general may preferable finite capacity factor treat tool work model argue human categorization adaptively number observation domain ibp human argue human decompose feature strongly object treat part find change include segmentation theory acquisition recent volume introduction review statistic acknowledgement grateful ed comment version manuscript manuscript suggestion suggestion fellowship nsf nsf google crp ibp combinatorial way understand construction review perspective crp ibp factor dirichlet dp parameterize distribution denote appeal measurable measurable partition dp average play high value mass concentrate process infinite atom atom independently property connect restaurant consider draw examine repeat note repeat draw exhibit define partition distribution chinese restaurant process parameter share independent dp assume exchangeability de say exchangeable conditionally collection prior generate reasoning equation theorem dp add step dp mixture crp mixture good break stick beta stick break dp via provide constructive stick break segment proportion remainder stick give draw representation corresponding admit atom note dp weight sum
close quantile value nominal significance report quantile become nominal especially cauchy affect distribute conclusion table detect consider ordinary noting performance especially case competitive c c cauchy mm c cc represent true denote total combination threshold parameter multivariate normal mean matrix experimental try three dimension non gold combination discussion list variable linear combination combination cc cauchy rely normality addition table deviation small diagnostic simulation apply propose tumor diabetes tumor set tumor patient continuous percentage except code scale accordingly diabetes consist subject subject variable body mass ratio progress diabetes addition diabete also investigate apply avoid variation variable table case cc diabetes set tumor ct great association tumor tumor datum set among set diabetes table cc variable little bit large set linear marker tumor diabetes diabetes diabetes c ct tumor data diabetes diabetes diabetes diabetes tumor diabetes diabetes diabetes tumor index ct cc type index age diabetes datum index ratio diabetes diabetes diabetes index paper first evaluate potential diagnostic individual variable gold measure share threshold independent roc curve variable threshold existence index useful practical maximize measure normality valid realize lar lar selection scheme conduct binary standard available variable unknown use density estimation propose variable computational large combination moreover ordinal suitable cutting elsewhere random pair suppose assign control give sample index gold distribution cut end instead explicitly define weight critical define width smoothed lemma distribute converge find page identically random identically gold marker condition triangle due tend generality tend let density complete theorem acknowledgement partially support science roc theorem remark operate roc useful tool diagnostic power gold gold become less useful evaluate diagnostic finding diagnostic auc index good potential diagnostic scheme addition descent maximize new even huge estimate property performance roc area gold roc tool diagnostic study moreover combination diagnostic gold gold standard evaluation classifier roc curve choice threshold informative index continuous organization national health suggest cut finding diagnosis cutting point interest new advance criterion evaluate necessary may changed conclusion evaluation diagnostic standard gold prefer motivate standard affected cut gold outcome gold lot report roc curve still gold maximum roc problem bayesian gold roc gold construct gold however approach issue especially variable biological genetic study auc propose gold study multivariate lar normality violate nice lar selection study performance range cut property next evaluate diagnostic individual variable measure find example summary detail appendix novel auc type gold standard briefly roc relate real variable example disease subject primary difficult look surrogate association develop association gold threshold subject member auc random respectively subject disease non disease threshold hence threshold diagnostic association independent cutting point gold monotonic property roc auc diagnostic integration cut weight support weight cut cut point treat suppose conditional rewrite distribution dimensional continuous linear multivariate normal z iy independent identifiable therefore combination linear coefficient standard identically gold generality centralize datum subtract tp z I regression clear regularity dominate convergence consistent estimate omit variable calculation matrix calculation numerically unstable sample relatively overcome obstacle small shrinkage lar lar regression lar propose combination lar adequate rely normality lar omit lar instead start linear combination variable follow extension sigmoid sufficiently window identically distribute continuous gold marginal conditional detail smooth summarize theorem suppose identically distribute vector constant hold need choice depend part curse method maximize positive anchor vector dimensional component anchor generality coefficient anchor identifiable base pt calculate derivative coefficient dl pl find
already powerful learn rich set expert vc expert pac factor expert reflect divergence flexible allow treatment expert policy expert experience treat shape property contextual reinforcement follow survey paper discuss result apply derive instantaneous per bind bandit problem derive instantaneous concentration result martingale sequence martingale difference variance define pac bernstein reference independent satisfy apply definition reward play round action independently respectively reward action round high expect good action define martingale cumulative martingale follow round q probability simultaneously translate logarithmic theorem improvement due bound apply generalization important state art alternative previous art technique rich infinite contextual bandit pac enable involve mutual bandit plain expert dimension complexity analysis prove co cluster unsupervised appendix theorem lemma bernstein bernstein inequality martingale second statistical physics bayesian ready take tt union fact take technical due new left bind instead negative decrease without positive apply hand modify tight tight identity verify q want get theorem enough requirement ok great substituting choice thank support network european publication reflect coherent exploration improve precede combine pac bernstein combination study multiple simultaneously evolve exploitation reinforcement selection empirical datum fit question good knowledge trade simultaneous consideration illustrate follow imagine pool additional know bandit imagine contextual consider increase simultaneously exploration trade simultaneously extend pac feedback sequentially pac decade make development method pac lie successful model pac pac bayesian prior bayesian derive generalization classifier margin beyond supervise domain surprisingly limited domain potential advantage apply bayesian reinforcement recently researcher suggest use regularizer reinforcement incorporate bellman justify provide guarantee pac batch reinforcement learn reinforcement difficulty pac limited reward action take estimation hypothesis evaluate sample size hypothesis limited action action bad
minimum across vertex hyperplane place piecewise average model smooth robust highlight minima design minima response surface method approximate programming involve unconstraine additive approximation robust regression broad resource portfolio management article novel nonparametric show consistency approximate show frequentist regression sampling need test stochastic semi scale well moderate limit could extension stochastic extremely tool sequential iteration fit search approximate set many resource wide perhaps combine preference product tend likely covariate age education function great economic reinforcement grant es national institute result cover need time cover place measure place space net constant g I numerator prove lemma norm main one jump chain balance satisfied hyperplane hyperplane addition hyperplane hyperplane hyperplane jump hyperplane initialize draw posterior draw sensitive space hyperplane order efficiently region create region describe create create proposal region subscript indice j define component proportional obvious hyperplane high start hyperplane adaptively add follow hyperplane along produce collection along linear combination covariate interval covariate assume choose partition mixture define observation fairly evenly section corollary proposition economic research reinforcement function constraint sequential state decision propose nonparametric characterize unknown hyperplane induce prior consistency norm reversible jump posterior computation function py n nf easily modify allow concave regression negative economic operation research economic production stochastic optimization know advantage estimate first place condition derivative substantially solver multivariate little piecewise support hyperplane solution infeasible observation generate regression weighted kernel subject solution find sequential programming convexity condition dataset propose convex partitioning split fit like hyperplane scale covariate guarantee univariate poor minima hyperplane intersect location sensitive hyperplane hyperplane could averaging produce smooth rely implicit hessian translate increase dimension discretize covariate normalize slope point place prior polynomial restrict spline prior place restriction basis dimension closely regression convexity enforce projection create set project back prior guarantee hyperplane define reversible markov carlo convex dense subspace determine space numerical produce term outperform objective potential suit objective support hyperplane support point semi hessian place restrict meet function automatically meet specifically hyperplane arbitrarily maintain straightforward follow factor hyperplane give prior spline bar place parametric prior number parameter reversible jump markov chain bar introduce knot within dominate detail consistency assign converge one small neighborhood grow rate size contract maintain despite property multivariate frequentist framework show show recent literature relate estimation show consistency mle transform estimator restrict asymptotic attention monotone spline extend prior let true throughout rest paper denote quantity counterpart eq examine actually convergence dimensionality full bound compact generality assume first compact restrictive unlikely occur bind cover plausible design compact range plausible choice atomic p x f ensure sufficiently hypercube let lebesgue suppose exist least point atomic give compact covariate continuous stochastic convergence uniformly bound compact covariate derivative every theorem use result put non requirement positivity leibl neighborhood existence test positivity construct create assumption pointwise determine empirical make model function matrix convenience sufficient algebraic actually subspace situation arise example covariate combination require know simply keep hyperplane b covariate theorem showing give b achieve term leave open bound setting rate achieve general extend accommodate reversible sampler model assume often violate shape locally outlier region globally poor prediction accommodate induce flexible introduce separate variance modify poisson misspecification propose determine metropolis proposal candidate block hyperplane update area unconstraine model candidate distribution proposal covariate partition subset n kk hyperparameter proposal distribution necessarily parameter high hyperplane covariate current removal component hyperplane hyperplane create region proposal distribution mixture along subset observation direction full mcmc sampler extremely fast unlike converge right component reach four construct strict block ensure autocorrelation drop three sampler lead situation restriction sampler noise admissible acceptance drop estimate unconstraine counterpart competitive art behavior objective suit
comment shall proof subject order apply regularity realistic case modification allow subproblem end two subproblem number generalize lagrangian henceforth assume sequence repeat assertion repeat replace precisely follow sequence keep mind q suppose continuous sequentially weakly compact weak k saddle lagrangian far turn subsequence weakly low semi estimate yield solve proceed saddle observe observe u p v prove part assertion see q subdifferential section thm thm lemma definition alg concern automatic adaptive function noise end mean type constraint alternate multiplier orthogonal projection intersection convex prove capability illustrate imaging relation observable unknown shall formalize identically distribute r encode functional inverse inverse apply estimator speak invert regression application primarily mind signal g characteristic structural assumption broad claim regularization recently different branch mathematic consider therein recently image neighborhood relation model framework straightforward reconstruction typically consideration imaging situation high dimensional respect minimize functional subject coefficient residual modification aim regularization variation cf norm hence convex smoothness texture relation account selector increase paragraph efficiently class estimator end incorporate possibly fit quantile encode distribution know parameter sound least constraint estimator rich signal visible suppose significantly differently statistic bound reveal application area estimation mind mainly normalized indicator residual resemble white set word feature illustrate example among residual resemble white noise prevent parsimonious reconstruction mr statistic various context incomplete overview classical mr consider indicator mr statistic call q reduce weakly establish image denoise employ diffusion equation spatially vary jump piecewise regression consistency general hilbert another attract function reduce particular multipli cf trivial choice consist segmentation translate testing nonparametric qualitative monotonicity similarly mr density multiscale sign test curve cover framework generalize selector interpret consist regularization functional lasso selector small consist scale sense constitute extreme practical cardinality set denoise million simple mutually perform solution iteratively poorly reconstruct mr approach allow put differently paradigm redundant mind convex convex cone euclidean realize employ appeal scale cone cone straightforward capable previous approach programming homotopy method author stress projection computationally application mind locally reconstruction use aforementioned graph large constraint bring cone exhibit face compare w fact turn cone projection onto simple linear algorithmic framework numerically many slack primal feasible slack appeal effect employ without consider alternate direction case occur form tackle orthogonal projection intersection merely increase considerably decompose contain never put position method locally visually algorithmic linearly problem lagrangian alternate assumption prove convergence give qualitative iterate theorem occur paragraph illustrate study problem nonparametric question linearly discuss notation subsection paragraph alternate multiplier admm alternate solve unconstraine penalize euclidean reveal modular replace projection paragraph stand grid moreover convex notation rewrite average arbitrary could useful different image transformation residual consideration separable product induce bound mr agree upon bound discuss follow saddle assumption refer instance dense need order requirement fulfil gradient base regularization slack rewrite equivalent characteristic region assumption set technique recall definition name stems augment saddle point saddle lagrangian already saddle versa existence usually hard assumption summarize set assume saddle due continuity minimization follow explicit third variable perform convergence subproblem sake simplicity ht size tolerance approximate iteration uv u v r kk generate weak cluster point statistic modular projection section algorithm penalize least square extensive overview tolerance terminate output inversion know introduce constitute efficient case euler functional fail serious try euler lagrange condition simulation regularize order hand standard smoothness signal end introduce regularization functional bregman satisfy semi example bregman incorporate word bregman differently symmetric bregman total sufficiently tv tv formula complicate mean symmetric divergence give approximate empirical mean trial regression estimator close estimator define bregman oracle practice reference secondly constitutes drive spatially adaptive regression become independently upper solid application mind arise spectra entity signal choose finally mr consist interval quantile mr statistic except special statistic usually hand henceforth argue smooth small exceed interpretation evident peak stay contrast level additionally maxima take indicate maxima imply c c l similarly reference concerned even superior indicate meet smoothness much result visual index increase side side condition interval intersection group simulation minute need step considerable parallelization section apply dimension typical noisy scale black h variation semi define range agree specify henceforth particular simulation estimator test favorable preserve estimator preserve essential detail piecewise reconstruction smoothly vary undesirable oracle visually perform bregman bregman smoothing image area pattern recover u h reference distance inconsistent human take contrast use author lie perfect match simulation list suppose superior mind evident rather suited distance remarkable equally bregman unknown far structural concerned superior natural proper weight function achieve substantial part g etc obvious standard image depict signal pattern yield power test order image mr average sake consideration precise center colored indicate location smoothed region incorporate local residual comment normally distribute distribution set scale certain dominate mr statistic multiscale compute normalize constant alternative approach would turn purpose eq group side intersection correspond hour parallelization deconvolution assume lattice zero circular deviation primal amount solve solution approach application technique model depict marker image area resolution pixel spread optical full half nm note fall range cover modify division understand pointwise involve firstly nonconvex apply secondly project onto intersection project modification iteration consequence modification consist square side length overall normally step iteration take minute need depict choose number appeal locally adaptive concern protein mark concentrate basically gap image emission constitute relatively capable image resolution reference depict comparison reasonable regularization relevant become one mark box visible mention aside transformation normality intensity
cyclic fig worker master choice either satisfy q straightforwardly mm long satisfie delay bound large compute sample say cyclic large delay penalty may dominate parallelization address drawback attention locally average architecture delay height span synchronization cyclic architecture worker communication master worker give section master simplex put theorems assumption gradient uncorrelated simply receive ii condition set jensen asymptotically number calculate delay graph give corollary irrespective adapt slightly give well rate expensive focus cyclic setting though principle average scheme denote master worker delay consider two regime rate attain roughly gradient dependence increase stepsize nd fast cyclic different
mechanism mab adversarial learning number include recently design offline sequential price approximation obtain multiplicative large dynamic pricing sequentially interact agent private value reveal agent price whether mechanism mechanism round output observe draw denote round item demand furthermore maximizer know sale define differentiable hazard price pricing price item stop case expectation pricing demand design price mechanism depend demand offline mechanism mechanism give seminal price let total price additive regret pricing demand price benchmark benchmark provide expect minus obtain multiplicative pricing regret offline regular focus benchmark address price benchmark fix price price price eq demand p price variable chernoff near price provide rest price p nr sp benchmark demand distribution price benchmark trivial bind price price mechanism upper free pricing strategy demand carefully optimize trade exploration exploitation
random eq subsequent bandit machine variable encounter action play sequence take round random random define two random q p hoeffding present regret bayesian play prior bound action stand leave absolute eq ct result present combination pac hoeffding definition
strictly map adapt model bias mean way impractical remark hierarchy box mathematically quantification expand important implement may implementation trivial program drastically careful tailor rna seq sequencing technology quantification consider read end uniquely begin read random length denote read note adjust generative choose read select among read read change reveal read note lemma supplementary condition equation transform unique map obtain read mapped formula measure exception equation derivation think relative abundance factor derivation relative abundance reveal evident abundance estimate absolute rather relative normalizing map read replace read sequence mistake apply improve abundance drastically address implement software package rna multiple ambiguity assignment necessary
explain example coincide introduce computational language positive production sequence l production clearly denote let order ordinal dimension ordinal recursive recursive ordinal conversely ordinal without notation k l l ij obviously preserve ordinal constructive recursively initial segment ordinal recursive range put di j finite inverse system quasi l x preserve order turn closed indeed quasi assertion l il x yx lx conversely actually l infinite anti viewpoint algebraic atom atomic
consider uncertain two cumulative f ref f ds response define ds structure box structure deal instead singleton cdf one need belief finite ds construct cdf transformation body law apply much early quantity interest measure cdf total amount integrate cdf decision denote deal interval drive
grid grid nod configuration require also scalability however well circumstance ex axiom describe first attribute graph exploit suffice sample together restrict technical scalability evaluation graph interest scalable also goodness generate compare statistic original graph hypothesis predict large graph estimate stochastic scalable graph previous graph scalable large model attractive fail capture characteristic order generalize attribute provably model power furthermore able real issue open currently aware scale bad observe edge determine exploit graph number know absence sampling adjacency trial generate real world graph million node paper restrict significant permutation sample graph target kronecker product np x kronecker take care discard later use th kronecker henceforth see rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle
may denote outcome load determine finite mesh inspire independent describe shape material modulus material stress linear euler mode independent objective shape maximum amplitude around surface half random field coefficient expansion field path squared exponential choice analytical detailed field random discretization involve cccc reliability table propose importance lead probability agreement strategy equal reveal configuration identify demand maximal whereas random configuration failure allow third provide prove problem reasonably rectangle rectangle usual reliability instead reduction consume expensive black
span capture orthogonal direction offer euclidean purpose pc decomposition disadvantage eigenvector nonzero sparse favorable interpret easy pc slightly requirement constraint maximization aware sparse pca high cost np literature rotation pc thresholde
clause clause gap reduction np hard hyperplane reader construction facilitate constitute hardness formula clause clause two whether variable follow clause four also importance become constitute correctness unit unit whose margin clause set coordinate integer true claim appear satisfy argument exist clause call absolute good one bad appear clause show many imply clause unit include rearrange w ic ic iy since
regularization increase face multi make le minima kkt minimum equality covariate discard instance goal exclude fold scheme fold use reduce nonzero adopt strategy lar framework adopt henceforth le choose le influential measure prediction minimize table simulation perform le increase std std mse outlier std mse detail replicate validation curve three replicate figure replicate le minimum vanishe regularization path mle consider binary class label employ notation construct linear net mixing derive entire regularization lead parameterization vary author website begin taylor follow minimize return problem chain rule relationship leave hand constant intercept regression coefficient
true interest maximize repeat reveal analogous contain simulated intercept ridge difference create thereby result poor parameter correction discretize surface discretization generate simple capture biological characteristic effect surface save true simulate mcmc use traditional confirm shift simplify appendix due inherent problem examine whether reproduce important characteristic fit data value paper use mode summary plot summary simulate actual term b compare summarize region suggest may fairly rich inform biological parameter shift vary show estimate contain truth vs traditional bayes develop characteristic process review gaussian web new several vector simulated case pose serious
smoothing penalty functional smoothed factor dense commonly store run gradient step projection show use penalty incorporate analog closely relate consideration address number factor extract amount scope several select factor extend imputation straightforward possible select amount two factor long variance explain first k proportion note algorithm component control sparsity smoothness seek way propose bic selection bic bic estimate freedom freedom iterative converge stable simulation achieve highly approach related achieve assess effectiveness simulate real example simulation generate k p snr center bic first spatio two consist region temporal activation pattern snr autoregressive autoregressive process var var demonstrate implement pca variation region activation pattern subsequent mixture never instead know equally spaced respectively possibility spatio temporal graph neighbor smooth near neighbor laplacian mesh space denote analogously explain spatial smooth recovery
width region array experiment shannon choose basic slide window start apply cluster depict count constraint capacity channel horizontal dot
ill detection pf ill deconvolution form function gamma essentially evaluate assumption paragraph statistic principal symbol order approximate theorem considerably impose asymptotic fourier fan et al recall ai exist positive previous variable negative principal symbol error asymptotic coincide usually multiscale r statistic almost ignore formulate differently asymptotically band moreover moment kernel negativity continuity non empty intersection solution argument I comment deconvolution scale simplicity band locally pf bands converge constant band situation confidence band bias error multiscale scale confidence display obviously say existence confidence denote confidence account construct symbol act derivative appear density study statement explicitly width influence pd reformulate easy additionally lagrange calculus polynomial restrict scale I q locally powerful summarize find kernel require define theorem prove weak
token later hold hypothesis interesting tend generate token theorem lemma prevent expect individual solve divide part contain internal separate box keep internal separate two define separate sub set inter separate binomial fix separate font variable token
operate try minimize regret adopt one subtle must handle adopt meta parameter duration policy slowly increase follow novel meta policy problem regret achieve secondary try channel availability channel evolve channel find receive instead channel get reward aim reward throughput slot designing difference reward know matrix express obtain policy ts show channel e positively
adaptation constraint method stability separation robustness neural use form one type feedforward neural neuron coefficient x sigmoid consider unless estimator bound ensure differentiable allow estimator bound design difficulty intuitively think interpolation sample suitably convolution designing boundedness thus serve modification wish emphasize subscript match convention statistic let finite estimator act ensures parametrize convex nx lx characterization kernel value hypothesis result nonzero characterization need robustness numerical calculus deterministic conjecture careful show stochastically true stochastically know vary pointwise minimizer limit
use systematic deriving net dimension obtain taylor expansion symbolic find see expansion derivative centre align r pp p condition conversely approximate truncation obviously analysis seek example derivative forward difference central mathematical software raise expect derivative behave surprisingly answer proceeding know derivative pose pose net h yu mu refer figure mh laplacian evaluate grid separation u value omit convolution operator net analyse character compare net noting degree freedom introduce oppose subsequent analysis fourier spectrum periodic behaviour behaviour width map also rewrite order elastic net part net carry dimensional case joint fitness character minima energy explicit continuous formulation du du norm square integrable operator order p du since term keep variation likely result belong incur incur polynomial fitness take generally fitness unique spline fitness mixture variational penalty transform continuous domain fourier act pass cutoff frequency increase fig likely fitness inverse write convolution make explicit discrete however continuous operator introduce extra consider finite approximate derivative mention delta cr cr cr cr r r associated area high low define toeplitz hold large computed directly involve modulus transpose hermitian transpose symbol implicitly produce align centre dft appendix toeplitz toeplitz represent rest successive rotation follow property matrix eigenvector plane easy basis also matrix mm dft since identity odd pair distinct conjugate eigenvector allow spectrum give call mn eigenvector hold eigenvector latter invariant permutation rotation origin eigenvalue odd span cosine n span span eigenvector generic eigenvalue nan incur compute superposition plane uniquely net eigenvector contribute eigenvector apply operator eq discussion fix eigenvalue quasi toeplitz polynomial later useful proposition show modulus tp modulus
noisy edge boost good correct edge describe entry think rank neighbor build row early neighbor degree way select frequent regular degree variance around top order look idea illustrated frequency decrease node drop experiment well type ii balanced error come error false pick threshold trade l type ii rely histogram neighborhood neighborhood degree one
satisfy another subset equally thus imply random conduct carlo simulation generate nk match start random record lp succeed repeat experiment
assumption write submatrix recall relationship p p expression string decompose image combine theorem eq apply would imply contradiction show determine therefore imply hence p p p v key p p therefore th analogous suffice generator induction yield iw iw argument denote closure algebraic variety sec def simple proof theorem first consequence closure closure agree topological closure agree follow equivalent n u j iv j consist p
label adjacency feature rkhs oppose regularization rather global interpret task estimate manifold helpful laplacian graph matrix diagonal differentiable laplacian think measure smoothness give eq small write formulation graph laplacian matrix function useful eigenvalue show reveal structure laplacian literature walk reach various property derivation eq average vector w st infinity inversion continuous I law let l laplacian generally matrix generalize positive machine th interpret th motivate context answer give graph proximity taking oppose interpret vertex connectivity entry jj ji metric path connectivity expansion equality side heavily heavily generalize
fisher jeffreys odd distribution lead e b lead update interpret failure reasonable default choice jeffreys possibility table prior match bivariate normal perhaps entail minimal odd lasso well contrast choice bridge extensive specification center multi hierarchy regime indicate subject assign treatment full exchangeable matrix prior treatment matrix respectively imply lead dependence notational ease gamma framework e l examine posterior ij p multivariate bivariate appeal may express combine kernel covariance hyperparameter conditional loop center conditional show inference datum exploit exact modern day machine
go indeed basic gamma payoff recently single stock insufficient explore payoff realize still trend development designing product acknowledge limit available information via payoff concept design simple asset financial product natural create exist argument mathematic research want last future unless appear unless belief market good probability payoff derivative similar third volatility product side payoff connect likelihood transform research payoff structure tool utility technical concept independent
action bold exist deterministic step say objective stage discount discount give expect select notice objective decompose vx successive state incorporate risk replace obtain objective objective analogously remark compare dynamic history allowed apply type risk due motivation markovian optimize risk history include behavioral economic measure originally value mapping monotonicity whenever translation moreover within economic cost reflect high axiom translation future viewed consider sure time axiom l homogeneou risk measure define finance neither especially mixed preference state win neither coherence without mcp first map without control chain investigate time
measurement make sequentially record record record value record break inverse scheme sequentially convenience scheme examine successive record q exponential study variable cdf characteristic percentile determine hour cycle pure determine
chernoff hoeffding light tail prove extend chernoff light tailed sequence sequence policy generality incur sequence easy exploitation incorrectly identify event easy rank correctly arrive exploration give horizon horizon technique partitioning epoch geometrically epoch logarithmic regret good need bind arm hoeffde logarithmic cardinality necessary identify reward available cardinality achieve logarithmic exploration see combine approach logarithmic large tailed heavy tailed chernoff true lemma chebyshev consider induction jensen performance heavy tailed exploration give consider see inspire
trace possible scenario current case framework specification undesirable behaviour framework logic social rule describe framework virtual automate reasoning consequence acceptable monitoring participant generate follow framework specification make crucial behaviour analogy engineering requirement describe see automate mechanism framework couple logic gap normally behaviour intend capture case case existence answer failure solve specification inductive suggestion improve specification secondly novel integrated logic semantic support purpose nature essential create domain demonstrate show iterative present introduce use
derive flexibility edge generalized operate joint among vector specify dependence attribute value network specify joint capture produce illustrate edge formulate represent feature distribution vector sum subgraph product
probability limit zero distribution statistic mapping illustrate behaviour limit amplitude limit behave one rule amplitude way zero quantile straight intercept true behaviour amplitude frequentist tend especially limit
take measurement sufficient lasso signal see show compute regardless scenario group configuration overlap yield length scenario apart last almost overlap apart element element common one list identical odd disjoint group overlap remain case argue become overlap ii overlap iv transform group account parent
case z pz z coincide multi recall x xy yy weight coincide acceptance multi obtain satisfie detailed guarantee target algorithm detailed I move simplicity generic satisfie detailed balance symmetric want generic inside integral describe section
two margin sequence respectively analogy moment central couple v df margin dependence copula use respect eq coefficient th moment copula moment bivariate bivariate page need reason likely use copula copula deal estimation bivariate parameter coefficient counterpart estimator three copula copula accordingly present estimation popular family copula copula copula correlation
dot summation cause storage avoid implementation locality many problem aim costly gradient address qp solve descent gradient obviously computational dominate suggest greatly benefit power demonstrate calculation iteration multiplication extract handle large often vary multiply parallelism adopt consideration demonstrate type multiplication prove solution multiplication adopt multiplication multiplication partitioned column multiplication partition partition summarize definition illustrate input partition group stage perform multiplication sum part summation result procedure often job responsible partitioning grouping intermediate detail ki step group individual sub partition schema often partition intermediate multiplication consider
hill covariate formally expression normality consequence propose adapt slope belong extend fulfil distribution address variance combine precisely follow result straightforward two define theorem entail permit asymptotic combine hill asymptotically estimator result direct convergence hold w tool minimize conditional hill provide asymptotically unbiased estimator continuous minimizing tw
balanced automatically create balanced fast function carlo integration arise carlo create variance unfortunately must good present create random mean random obtain output carefully analyze number b unbiased estimator multiply final error
national health research research centre health mh south foundation trust
rank behavior relax doubly long learn doubly doubly construct wise softmax projection know nonnegative row formally column hadamard division iterative normalization matrix convergence unique summary guarantee non negative doubly suffice reach matrix facilitate normalization incomplete term square whose constraint nonnegative normalize interestingly
fitting acknowledgement nsf dms give q l jj jj expression truncate two method sampling cumulative ex minus plus ex remark model wang south sc department flexible prior enjoy normal prior characterize sampler lead new approach basic autoregressive spatial structure appeal new improve predictive augmentation autoregressive estimation statistic large efficient advantage parsimonious often inherent high approach reference deviation flexible hierarchical prior conjugate prior typically metropolis hasting matrix identification zero zero inverse covariance undirecte attractive gain attention imply
simplify integrate part recursive obtain grateful suggestion work european european period series analyze anomalous model type stable similarity difference brownian model describe diffusion finance diffusion time period
corollary simulation wish empirical number entry per denote result section turning case stationary vector var process zero mean driving corollary miss var drive suppose vector row drive parameter let theorem corollary corruption deviation extension identical omit problem relate via lasso recently bound fully I row var devise corrupt remove exist vector I estimate additive observe additive noise pair j covariate covariate response pair obtain j j jj matrix cc project descent nonconvex case versus rescale missing covariate autoregressive represent trial column finite uniformly scale bound corollary technique hold
adaptation stick breaking process asymptotic parametric receive lot attention recent year recognize bayesian collection range seminal rescaled parametric discuss estimation mixture prior number normal remain univariate case surprising study dp high
example size existence path strong connect entropy spin birth size common underlie principle path connect entropy discard create contribution entropy interaction stop build recursion sum respectively ise grid bm huge mean message
dimension infinite principal approximate multiplication sum hence tail carry matrix augment thompson demonstrate technique adapt bernstein yield version tail prove construct simplify probabilistic sum prevent infinite arise
name normalization quality assessment assessment effectively preserve embedding second global overall assessment therefore natural task learning validate effectiveness manifold advance store set issue science understanding cause curse dimensionality complexity pca multidimensional lie linear embed ambient meaningful embedding manifold family method diffusion map dm tangent alignment variance unfold riemannian manifold great interest nature intuition successful detection recognition hyperspectral crucial natural learn embedding label method fully unsupervise freedom underlie dimensional unknown quality learn inspection intuitive qualitative quantitative dimension address assessment cast sample pairwise preserve neighborhood neighborhood
arm must idea availability reward bound primarily nonetheless ucb match binary among bandit family interest consider policy draw arm ucb arm ucb call optimistic act instant reward high possible value compatible past ucb call ucb latter ucb online horizon prove constant ucb variant needs tune tight regret bind arbitrary expense front proposition correctly ucb numerical ucb ucb tune
direction current stopping change nothing else analog variable properly direction duration device analog store basically consist set usually thin change device circuit typical circuit depict explicitly suitable locate decrease first vertical connect negative dropping across cause consequently passive element decrease similar increase state application time adjust suitable analog store goal propose fuzzy binary gate primary artificial purpose logical gate depict binary receive datum come via strength correspond efficiency neuron neuron thresholding write total set logic become logic two input e create neuron weight logic layer neuron create logic
marginal likelihood small divergence sense might indeed finite maximizer eventually theorem remark order originally motivate match rate however improve difference come ultimately subproblem version third regard complexity considerably space correctly specify converge identification would arguably close prior hierarchy propose prior incorporate effectively reflect unknown likely reasonable recommend consist binomial conditionally prior denote prior include able component say information first problem poisson quick comparison context example gaussian
functional category gene near original contain miss capability profile experiment replicate mixture component five five
value variable variable threshold cite quite small set major drawback deal index hypothesis free tuning model length whose set let set index distinguish framework order variable asymptotic hypothesis method order procedure procedure prove powerful procedure let subspace denote organize simulation suppose suppose focus hypothesis test subspace nan alternative collection subspace collection level level reject procedure collection subspace successively soon kx k
graph output dag condition l occurrence path extra illustrate equality establish present dag jx circle dag p dag variable function size restriction throughout correlation version section contain consistency weak correlation partial correlation nan ai denote version simply adapt decision conditionally role impose maximum skeleton satisfy correlation allow grow polynomial sample represent set pose maximum size oracle upper correlation tend avoid identifiability bind assumption outside range assumption condition similarity evident difference dag version unknown datum maximum denote step strong skeleton algorithm skeleton contain sep definition sequences sep grow linearly low study consider require additional tuning directly tune adaptively modification apply simulation estimation version significantly moderate feasible graph procedure dag size generate
involve criterion sample yield sample figure versus versus give simply increment increment run sample show detector present training create dissimilarity require number pareto dominate sort non dominate sort sort construct pareto comparison involve create need calculate involve find dominate test front binary front training front comparison bad anomaly threshold determine phase sort criterion experience requirement obstacle sort order pareto searching list dominate front criterion pareto front hence average occur pareto front attention canonical anti dominate sorting criterion pareto sort nonempty add front
predict target apply logit maxout perceptron maxout applicable binary baseline problem result task report give adaptive hyper parameter tune loss kernel kernel kernel kernel average kernel maximal experiment perform fold fold model optimize report fold svms train descent use rbms hybrid c average hybrid maxout mlp maxout logit maxout svm public drug annotation task time repeat fold
adjust expect rough estimate distribution beneficial adjustment computation adjust variance conservative credible rough importance g quickly determine reader comparative construct toy gold comparison understand property method mixture component dimensional evaluate b marginal two mixture combination component generation proceed independently probability specify support prior computation simulation close rejection follow adjustment marginal adjustment inference package abc mixture relationship observe summary panel b c rejection follow adjustment indicate dash quality kullback grey rejection adjustment panel illustrate bivariate contour indicate true rejection rejection follow regression adjustment panel addition
limit requirement execution important aspect implementation ideally format interpretation hasting rest discard enable post degree retain freedom considerable storage capacity capacity disk throughput limit sample write dominate execution second pattern average moment histogram accumulate simulation record storage
atomic hazard binary process underlie de ibp extend introduce negative ibp line choose select ibp customer control negative binomial general use may control count allow representation particularly simply contribute variable poisson enhance flexibility place gamma construct count make connection time appear contribution mark process count topic diverse corpora poisson commonly model place produce binomial gamma large thus
show alternate admm note new subject semi definite vast norm subgradient minimize substitute back condition lasso pool unchanged give pool average find optimality satisfied solution n path recall full solution far assume assignment henceforth backward approach
context know concern wiener set error procedure derive limit adaptive suggest average criterion restrictive available old mu achieve mat
mostly combination stage random population age capture use record thus implement method general census record head part methodology survey carry current detail briefly day seed seed datum collect report extensive degree sequence degree experiment obtain group panel panel population tree combination addition participant say preserve clarity combination panel panel eight panel nine evaluate circle
processor gb tb disk execute map make run serial worker ensure contain block mb vs default mb block size accuracy allow expense worker variant use bag instead distribute illustrate block block different line size bag instead block train run different measure accuracy large accuracy started yield result ensemble member core size size ccc size straightforward replacement number ensemble member evaluate evaluation may ensemble evaluation ensemble start thus per first subsampling random block datum accuracy use full code train size member training train block ensemble total time local member size ensemble vary point comparison good serial model total
large choose first section term finite except due transition initially assume know transition lie cl function regret know incorrect calculation become independent player increase regret transition except slowly linear horizon term slowly version prove regret horizon fp fp take threshold mix exploitation fp player pick pick arm arm end play optimally else arm play optimally play large function due suboptimal bound theorem give fp fp threshold tc horizon fp upper exploitation step select possibility know optimal imply either enough vanish gap belief fp belief correspond get bind clearly diameter regret number partition proportional tradeoff theorem balancing set set suboptimal balance tradeoff sublinear proportional player future research section inefficient approximately quantization relative iteration modify exploitation solve observation transition increase percent example solve transition exceed policy
education etc model equilibrium describe observation calibration epidemic importantly collect population equilibrium result follow likelihood present transform trajectory practice point spaced model age proportion percentage individual give number population person model year incidence collect new south validate apply date collection result health cross case gender age normalise roughly capture health person health incidence rate obtain gp observation spaced assume full activity available though parameter observation vector vector ode system st gender assume observe assume beta furthermore gender bold corresponding incidence category state vector incidence incidence gender code code decompose incidence apart would system model ensure point state least finer solver group gender model specify beta parameter matching obtain comprised model age explored perform bayesian estimator correspond mean mmse posterior mode explore resort draw paper procedure non ode adaptive monte history consider present present monte carlo combine posterior give particular introduce background essence construct trajectory ode epidemic
limit expert expert switch expert expert time course expert length define sequence summation abuse ti virtue direct forecaster expert track record loss issue return conclusion set dependence condition expert restriction sequence keep share turn eqs initially static always tw f tw share assign exponentially forecaster apply base expert sequence ti eqs comment
payoff value player opponent aim game interesting existence consistent calibrate name abstract basic structure central manuscript payoff subsequent establish geometrically characterize player usual payoff convex compact equilibrium scalar determine dependence simply organization summarize intuitively standard theorem fx scalar interpreted state player affect carry cf later consider thus minimax cf player may minimax remove game natural target project normal always appear choose orthogonal attempt force manuscript role play turn manuscript organize present background
run examine segmentation minimize chain method collection sequence mcmc compute empirical mean test sequence expect hamming empirical rely across compute distance sequence optimal sequence high aim provide address issue display skeleton depict learn contiguous segment box group behavior label behavior two category show common ar identify group dark various behavior appear movie four split green motion category raise attribute raise one exercise slightly counter motion subject three run motion category split also correspond body run subject perform subject splitting phenomenon gmm gmm initialize component pca primarily bp ar hmm hmms frame six movie hamming frame hmm ten mcmc demonstrate ar hmm difference display map gmm bp hmm component hmm state behavior namely hmm assume see strong band imply bp well specific portion bp note bp merge behavior hmm discover dynamical formulation prior subset binary time novel ibp utility bp focus switching approach equally
vx vx v x vx g vx vx thus first follow completeness conclusion final conditional projection completeness vx second hypothesis measurable positive definite g e mapping obviously cauchy inequality hc te hc w h also note cauchy inequality give h b g h w expectation g g let tf dc tf c tf tf cf hilbert schmidt mean square square g subtract hand multiply subtract additive separability c eq eq q part condition follow implie know I completeness part hypothese r hc iterate expectation hc g follow divide zero eq distribute imply chen global identification assume completeness rely
recursion demonstrate depend curvature determine bind follow constant stepsize sgd serial case fairly pay stepsize proportional counter result slow demonstrating eliminate complicated protocol decrease number run stepsize follow exponential standard precise real function constant iteration appendix discuss gradient less sufficient tell iteration give eliminate stepsize phase exponential converge shrink stepsize iterate ball suppose choice stepsize suffer much stepsize algorithm sufficient assume stepsize factor epoch epoch require previous epoch final q select run combine guarantee rearrange step protocol stepsize serial incremental eq put recursion
yield order plug find outperform counterpart first scenario isolate cluster isolate isolate heterogeneity sc covariate percentage pt p pt pt sc sc sc sc sc sc sc sc sc sc sc sc sc close percentage truncate spatial simulation page cb counterpart spatially structured finding rank cb triple exhibit performance closely cb plug mle bad classifier sc sc outperform mle classifier sc spatial albeit variability estimator sc discrepancy mle may sc sc plug find mle chapter sc weight scenario well htbp sd five plug present different spatial isolated isolated area highly spatial heterogeneity sc spatial structure sc well estimator p pt pt sc sc sc sc l sc sc sc sc sc sc sc sc sc sc sc sc sc sc sc sc sc sc sc sc sc close percentage truncate close percentage htbp five estimator different level variability spatial isolate sc isolate isolate area sc structure spatial heterogeneity sc spatial structure generate hidden sf expect count percentage loss p sc sc sc sc sc sc sc sc sc sc sc sc sc sc sc sc sc sc sc truncate second entry percentage close percentage heterogeneity systematically note variability ensemble estimation plug cb gr plug heterogeneity substantially increase dispersion effect increase heterogeneity simulation car car scenario indicate table page different plug expect benefit level function show level plug systematic trend nonetheless notable whereby high similarly cb plug although triple goal term percentage expect consider column latter scenario heterogeneity play substantial whether sf observe sf plug affected level expect count appear percentage estimator sf systematic mle notably count albeit heterogeneity chapter also next ptc pt p pt pt threshold number percentage truncate close small uk year surveillance publicly available classification health service medical practitioner probability contract evaluation use full retrieve uk health set cover four period thesis longitudinal classify accord year department health period day member plan department health rate day seven discard national disease compose statistical illustrate inherent change level mainly concern risk oppose trust year threshold expect optimal estimator scale factor p pt loss threshold c truncate close second digit percentage close small percentage observe case trust per day ny rate uk day trust thousand use assume dispersion cause disease variance gamma normal close datum fit cb triple vector specify discuss chapter plug choice third threshold scale classify identify substantially level choice high national also substantially low positive become solely classification affect posterior percentage sd sd threshold dash indicate red sd sd threshold dash indicate red particular poor sd sd threshold dash red htbp sd panel dash indicate red plug threshold remarkably yield great classified exception trend conservative ensemble indicate conservative classification particular gr find cb estimate show cb gr plug behave however small misclassifie area gr plug yield set plug estimator page aforementione across cb plug exhibit classifier behave plug mle classifier increase order posterior regret page show ensemble estimate consider trust panel figure page classification ensemble compare figure page plot distribution see general classify rr posterior mean visible since produce identical classification class
pose check radius language regular least break language string middle since piece say string may show arc rectangular regular suppose language language contradict rectangular trajectory b language contradict deal language rectangular free formally construction rectangular trajectory length sufficient represent language trajectory rectangular trajectory free string contain conclude free apply deal rectangular trajectory conclude modeling issue state reach state self self string production rule never terminate instability desirable derivation terminate finite criterion define matrix number production number constraint terminate derive mode syntactic classify target arc generate iteratively process parse parse top sec give syntactic discuss mode sec syntactic estimator version operation infer production string call stochastic parse context syntactic track sequence state state semantic tracking algorithm recursively sec string store define marker end partially probability incomplete illustration syntactic trajectory estimate string target syntactic produce trajectory syntactic build hypothesis detail provide specifically syntactic multiple mode
standard image optimization numerous optima evolutionary population iteratively select combine produce increasingly skeleton acyclic
norm formulation norm motivate problem section measure data image operator lead write denote wavelet regularization tradeoff minimizing serve lasso penalty wavelet coefficient reality pattern plausible commonly small wavelet scale
sound conceptual justification neither frequentist problem justification experimental evidence idea probabilistic role play inductive inference variety laplace borel kolmogorov build apart selection mathematical implementation example abstract relational might bayes updating etc borel kolmogorov notion probabilistic model build theoretic equip reason frequentist claim probability frequency update probability update fact first lead abstract boolean allow semi measure ever theorem algebra
satisfy moreover chain explore surface also surface automatically incorporate toy experiment dark light dash influence evident density prior wrong location indicate dash degenerate plot peak near wrong seem lie peak point lie hyper small value hyper surface though hyper surface figure happen multiplication likelihood sharp contrast observe count strength whereas constrain multiple degeneracy expect c integrate observe noted purpose however happen wrong show relatively constraint merely able albeit integrate wrong method
abuse terminology dependence clear goal sharp bound discuss relatively quantity picture special operator functional study q square quantity additional factor involve control map correspond sample sample theory relate consequence example useful general recover noisy observation square unit hilbert convex fairly derive translate upper norm example subset f j l mm useful pack remainder begin background set
integrate context correspond make hyperparameter hyperparameter iterate keep respect keep fix step become course maximize find em well provide expression challenge need take explicit system numerically operation triangular case association cluster hand trajectory task location distribution analytically express gaussian approximate posterior mix probability hyperparameter typically input correspondence equation belong current sometimes
variation hard specify change display location surface prior height height height height
hold consider cover ball clear radius volume easy ball center cover manifold ambient ap ball every point bind invoke lemma project onto tangent action clear manifold onto ball invertible distance point derive point use lower get c consequence bind decrease noiseless proceed dimensional constant rapidly rapidly grow invoke hold manifold desire large invoke homology start result tell kernel positive volume appropriately clear lemma conclude around use clean point denote remain estimator analyze eliminate eliminate since within contradiction keep nb
krige poor choice piecewise approximation true mean bad another choose directly high well variant right hand three stationarity evident advantage fields define expand weight change interestingly replace incorporate dependence general drift concept behind markovian transfer care need take advantage stationarity underlie well ts describe direction random diffusion stationarity define locally flat field manifold still
asymptotic normality semi parametric rao equation form presentation x j simplify explicit q j dx dx f moreover note equation case asymptotic efficiency new type estimator functional choose taylor expansion dy behavior
particle least parameter propose particle ratio jump number go adaptive approximate abc line tolerance tolerance generate compute weight sample decrease reach feature see generate particle twice particle combine constitute ns algorithm stop result retain particle step heavily equal quantile simplicity case avoid particle suppose negligible compare algorithm particle x kernel yield
survey interaction depend fit marginal adaptive penalty equality penalization vector penalty form parameter estimate zero parameter zero need separately concave differentiable coordinate wise coordinate ascent cycle take minimize approach sub section local perform computationally
article seek iteratively approximate strongly optimization smooth domain accurately point unbiased independent estimator subgradient independent identically scale measurable scale dimensionality oracle
record sufficiently dim distant enough neuron center prototype total neuron neuron neuron densely rf size neuron densely extraction neuron densely present distant stimulus neuron map collect stimulus visual stimulus finding match vertical horizontal map contain correspond match response neuron contain fig whose visual fast neuron response neuron exclude neuron spike stimulus spike extract eliminate feature neural activation map feature match actual stimulus stimulus fig white color indicate reach spike reach spike
remark spectral sum independent matrix surely equal ni ni mt q plugging get like bind random regard prediction result excess expectation rademacher theorems part rather lipschitz function sample derive derive depend argument definition remark institute consider approximately reconstruct partially approximately receive much attention surrogate reconstruction trace max present reconstruction rademacher unit publish specialize approximately reconstruct unknown rank interest reconstruct partially observe noisy quality observation search error np work focus trace nuclear norm surrogate rank trace sum norm value singular quality reconstruction largely drive compressed sensing condition empirical
multi server column suppose help one histogram evident join histogram experiment prediction within prediction histogram real figure deviation paragraph histogram fine grain big wrong histogram join histogram predict hundred tuple output tuple observe deviation prediction method small join operation often contain tuple tuple prediction accurate I contain tuple query database soon behave percentage prediction correct make join add column thm evident guarantee spirit conclusion al online query properly database demonstrate use significant improve histogram join advantage within predefine range collect pre capture collect histogram join multidimensional multidimensional histogram exponential representation join operation join database exponential interesting highly
without give leading ij kernel f evaluate chart spectrum lead eigenvector kernel simply kernel snr feature zero center commonly kernel polynomial kernel radial basis heavy rbf rbf paper obtain eigenvector take basis match hyperspectral background interference spectrum modify match sense interference linear subspace lie hypothesis
belong depend evidence probability estimate accurately bm illustrate subset effectiveness cut false alarm include retrieve high collection belong subset indistinguishable fact optimally rank way index share value decision subset retrieve disjoint subset subset occur document term orthonormal either represent respectively term clarity weight depict vector relevance relevance non relevance relevance vector relevance occurrence belong common basis space vector random collection vector vector correspondence
show satisfied solution bind eigenvalue condition nn concentration get nc min immediate consequence construct dual pair r pc projection notice ratio triangle distribution probability kn result zero variable concentration bs state notice ds assumption sign j go provide assumption lemma eq q union bind theorem say magnitude large magnitude magnitude theorem hold trivially corollary constant specify equal concentration optimality get random concentration vanish exponentially fast success recall dual variable satisfie satisfied hence rest direct consequence prove lemma exist contradiction variable equality orthogonality projection inequality
condition compressive pursuit bp mod briefly early recovery support denote sec mod try outside among solution enough may recursive recursive compressive cs compression cs instant bp practical reconstruct measurement bp actually linear q rip geometry recovery bp work sparse reconstruction partly knowledge call compressive sensing obtain condition mod show weak mod cs try e solve refer mod mod cs introduce retain also parallel support solve threshold
condition variance q ordinary least error triangle differ design apply ordinary require dimension covariate slightly direct argument simple characterize excess ordinary hold show fixed bind bind lemma appear similarly ny come therefore analysis estimator expectation bias triangle paper fix pick condition q various aspect simplify ignore treat overall relatively mild essentially effect isolate simplify view nn second noise unique covariate suppose
intuitively infinite see note prove recovery adapt crucial ingredient nuclear mb easy suppose choose necessarily satisfy hold easy obey well mr selector lasso choice q slight modification bind thm proposition reconstruct unknown application quantum analogue sense recover sparse fouri almost isometry rip recover norm nearly optimal class orthonormal obtain entropy low reconstruct machine filter netflix remarkably turn choice matrix uniquely reconstruct nuclear element case measurement suffice provide incoherence
profile usually describe game pool multiple user utilize resource propose convergence fair outcome common relevant resource allocation control wireless communication player specialized game involve profile simpler applicable almost yet specify closely present different require game specify profile organize define presents game namely present multiple outcome also demonstrate formation game establishe establish finally conclude remark finite action profile product player preference induce utility complementary pure nash equilibria nash nash always refer pure nash equilibrium first action profile value game satisfied payoff dominate j n form interest large accord one game satisfy payoff difference player profile difference establish
extract perform contrast panel superior second method unlabele classify al unlabeled randomly unlabele example supervise set average repetition test rate prediction result suggest improve prediction functional supervise functional basis
algebraic statistic design collect appendix mention consequence presentation ideal polynomial finite symbolic without medium sized table strictly vanish binomial vanish log odd versa odd algebra output computation emphasize usefulness definition eq therefore obtain normalization define integer call markov goodness geometry basis goodness independence represent q quasi independence encode except represent force diagonal probability typically use notice issue outlier model zero view detect mention
speedup htp figure effect start segment red line line generate uniformly thin plot leave right htp origin classifier training vector class cast q l detail reader reference survey advance htp svm logistic setup continuous remark l svm htp svm jx tx jx able gets update ii derivative perform dataset instance selection list testing ta htp ta dependence ta iteration find datum half second let remark scale htp use train approximately second l loss lemma remark paper method minimize nonsmooth prove linearly nesterov efficiency smooth composite remove importantly contrast author achieve
regularity positive detail autocorrelation slowly alternatively assign experience function degree choose mle feature suggest regard e short xt carry interpretation notion loss obtain different example let p newton method many goodness context even theoretical clearly distance difference shift sample efficacy parameter obviously number assume ahead model separately pt misspecification integrate shift stationary memory example aic figure observation close strong close well matching interesting well comparison matching size dot solid attract considerable admissible distribute stationary identically aic still lead line solid dash kalman pt kalman utilize likelihood method em dot figure kalman filter unstable sample aic kalman produce outside range model truncate lie replicate simulation first estimation true realization panel correspond result panel panel observe pt pt demonstrate substantial figure pa time perfectly specify parameter
regularization tuning view use parameter selection degree freedom allow criterion evaluate regularization variety penalty carlo simulation situation datum degree freedom fundamentally important dimensional traditional procedure information criterion unstable inherent drawback impose simultaneous model considerable lasso penalization regression elastic net penalty minimax elastic net solution close present efficient entire solution modeling identify assign include criterion cross validation freedom g directly sparse analytical stein specific show degree freedom degree fuse base differential parametrization enable freedom minimax procedure via bootstrap approach unstable estimate present iteratively degree extend wide net furthermore obtain degree algorithm perform remainder briefly describe regression iteratively compute degree generalize path section artificial dataset conclude
play role life study ise weighted link information log graphical field distribution lagrange multiplier ratio interpret log odd ratio conditional correlation despite similarity exponential relate moment difficult compute marginal family repeat marginalization conditional let triangular matrix conditional third family regression family marginalization assess logistic original may da dd
say subset coupling fail happen fail coupling note variation uniform uniformly variation fx always couple w imply give rough description strategy describe coupling evolve couple markovian couple sup record specify coupling show keep occur consist difficulty partition showing remain worth point dependence coupling get correct analogous markovian walk couple essentially modification gibbs
inductive favor multiplicative research question transformation ability suggest robustness pre energy would transform dimensional fundamental many vision task include role multiplicative play infer detect rotation share multiplicative code method light review operation perform cell implement multiplicative interaction suggest utility arguably primitive tracking view scene invariant description action recognition relationship single carry contour consider correspondence namely area pixel go depict almost understand movie understanding decoding frame correspondence task keep mind build vision system lot progress building task reason help eliminate recognition consistent feature plausible feature overcome engineering domain degree place construct module raise whether amenable may open road perhaps beyond vision paper recent task introduce mapping mapping illustration unit interact
boolean learnable positive unary free positive example learn represent query xx xx xx xx let l l r subtree root return filter construct serve filter subtree root node select part select invariant one choice library n library book attempt bottom two common root return path title move author title expression presence reasonable schema algorithm schema line least desire completeness completeness choose free unary learnable positive e set symbol positive learning setting exist consistent sound need query follow decision positive consistency query consistency np outline hardness consistency use minimal example indicate symbol yshift cm xshift yshift xshift c yshift cm xshift yshift cm edge edge yshift xshift cm edge yshift xshift cm yshift cm xshift f edge yshift cm yshift xshift yshift xshift xshift edge xshift yshift cm xshift yshift xshift c yshift xshift edge yshift xshift f cm xshift edge edge x xshift edge edge yshift edge xshift f xshift cm c yshift yshift
e word ultimately scalar structure retrieval implementation answer implicit good ask another question lemma example value relevance optimize estimate key subset recall subset recall result prove retrieval ir system decide relevance relevance subject measure end define research ir describe axiom perhaps crucial ir task ir reach good thank classical unit top probability operator classical hypothesis express quantum investigation phenomenon
I bernstein inequality state sequence let martingale difference sequence I entirely together
finding treatment portion apply practically area roc curve assess intermediate auc roc good easier determine range several among thorough roc find extended status disease death represent diagnostic marker study disease classify subject generalize time wang nonparametric estimators develop inference procedure process correspond facilitate dependent summary organize follow section
diagonal correspond hermitian trace pure course example occurrence eigenvalue assign certain event latter event space algebraic represent matrix event rule trace q compute allocate may odd diagonal sum assign state prove every space dimension great space probability vector basically probability calculate trace call use superposition suppose event occurrence two probability multiply occur multiplied occur law interference range usefulness mutually exclusive describe interference violate kolmogorov axiom admit space general store acquire eliminate precise exhaustive nevertheless former amount set hypothesis decide topic must relate
acoustic source problem minimize correspond observation create measurement receiver location solve equation differential regularization typically achieve nonconvex figure measure especially dimension accomplish strongly increase pass hybrid outperform deterministic good nearly hybrid perform although show pass practice involve prohibitive make quick progress solve good focus inexact gradient method optimization objective inexact nesterov case convexity assumption optimal inexact bound author notably error encourage achievable might analogous proximal convex non composite noise subsequently convergence rate bound certain individual concentrate us concentration hold high select element bind individual far refine
non show event topological respect topology infinite equivalence topology topology except hausdorff unless anti separate every also vice anti topology reduce discrete close open open topology class every every topological union class closure give union intersect aa desire establish field subset monotonicity aa aa constructive natural obviously every observe additional completeness order every supremum full full case prove make completion element satisfie show similar result follow natural additive component recall possibly family number outcome subset super additive bb bc write component component taking hold arrive conjugacy note box alone additive full box check extension low envelope mass moreover already event explain proposition
h h maximize assume v value namely form namely eq life span maximize gain remain step time agent fix life span gain maximize recursive clear reinforcement horizon type worth point behave differently reinforcement signal assume signal together meaningful plain reward sum theoretic meaningful namely gain reward planning ahead immediate expectation grow
symmetric possibly capture penalty transformation requirement shift transform use characterize denote set positive piecewise example huber notation definition inside inside maximize huber obtain take huber representation take need ensure call turn ensure penalty def subject denote integrable measure
similarly constant less exponential go histogram convergence early density estimator estimator attain argue metric study far aware target analogous rate asymptotic target order vector value stationary prove sequence asymptotic histogram yu doubly asymptotic histogram iid
plot slightly redundancy total information decrease increase relate decomposition note redundancy give position fig near find group redundant simultaneously interaction obviously set vary bin use illustration discuss herein address system change simple interaction focus interaction boolean gate beyond variable individually relationship decomposition show magnitude indicate presence node indicate redundancy develop interaction contrast interaction show sum word correlation interaction variable feature apparent total varied approximately linearly regard whereas measure variable neural mutual information development correlation total varied approximately connection dual highlight gate correlation variable gate evaluate interaction know variable compare act correlation
flow far boost efficiency present lattice continuous transition calculate autocorrelation several autocorrelation variance estimator consider autocorrelation bin size fig comparison conventional autocorrelation become metropolis heat
explain yield latent pose ground truth frame yield square iterative standard general small aspect iterative self regularity impose proportion satisfactory describe residual datum give covariance novel difference dimensional variant cca include characterize rank powerful
preserve release highlight technique thresholded predicates polynomial threshold learn base utilize view privacy analysis sensitive extract individual individual record offline interactive setting query mapping answer release extract approximate answer query medical analyze release grow explore show permit surprisingly least dimensionality central private develop hardness release efficient tool datum task predicate derive release answer challenge access access get oracle query answer theory give access query release face apparent access impose release release ensure privacy many efficient run another often sub size oracle query description work explore learn private release release relate task obtain differentially release algorithm briefly way differentially decade aim back work similarity work insight future view connection technique release set explicit modular reduction release conceptually arise release include dependence database release aim conjunction contingency query study differential literature take universe attribute count item database attribute answer past tailor
concentration discount time illustrate parameter long vs iteration exhibit autocorrelation figure three parameter computational reason accordance methodology concern nonetheless seem explore discount behavior power expect expect concentration instead number model roughly achieve seem concentration three top nine feature way three look learn collect last iteration express iteration nine express distinguish digits version competitive reach burn iteration run per iteration average iteration stick break stick break essentially complete stick break chinese restaurant motivate three stick break type law discover process power law useful discussion helpful suggestion experimental national fellowship knowledge science foundation award stochastic carlo many conditional form require notable step discount integration discount
instead return efficiently build require subspace improper checking elimination probability improper previous chernoff bind overall collect conjecture inverse time example bound demonstrate runtime pair preferable hypothesis write therefore perfectly erm time agnostic minimize mistake obtain learn preference idea hypothesis class addition complexity solve erm overall erm erm predictor predictor map
take decay adjoint map embed hilbert rkhs eigen decay relate interact give rise functional number versa costly function various author lie ball span sample together theoretical pair sharp rate track asymptotic accounting introduce smoothness use among principal recent devote noisy consider via approximation derive rate estimation certain paper trade lda approach inspire special observe oppose aware pca working framework per optimal eigenfunction analyzed sample al derive rank couple scale invariance combine extra weighting also estimator eigenfunction lie rkh different interesting question whether work emphasis sample
bind albeit word despite error end inference suppose expectation moment matching error break carlo leverage efficient although already suggest effectiveness bayesian reduce solve system q perturbation diagonal conjugate matrix product avoid costly cholesky factorization simulation length vector store g matlab also note conjugate gradient employ estimation use marginally chi freedom imply unlike drop reach estimate sufficiently accurate even expectation propagation work
question group panel participant participant group panel consideration call effectively cope thin sample estimation adapt possible function estimator attain infimum choose effect certain risk hellinger kullback risk naturally hilbert turn derivation refer work risk density dominate divergence dominate risk translate square total variation kullback leibler application binary naturally slow prohibitive covariate yet decay slow orthogonal series achieve near decay estimation computationally instance fast hadamard estimate basis operation appendix therein orthogonal recursive particular entail estimate basis coefficient compute stage decide remain likely significant allocate high significant spirit originally
strategy equilibrium anonymous sized mixed strategy nash anonymous strategy player base insight anonymous game sum bernoulli first second vanish exponentially player iterate involve dynamic implement nash equilibria sum ess weak slow analysis approximate equilibria existence efficiently computable cover variation player play player player player set pure player payoff player payoff player literature equilibrium normalize mixed pure strategy choose player iff among vector nash x strong notion support nash simply nash anonymous sense dual player play strategy play utility player player way play player play
likely regardless network direct tie offset odd capture intuition limit degree poisson tend limit additional offset asymptotically analogue n appendix theorem effective interval argument develop primarily establish insight question effective focus asymptotically distribution derive little foundation formal illustration interval interval estimator v examine desire study e v word mean expect number actor level substantial natural model desire value simulate evaluating individually coverage sample interior convex realization mle mle mle frequentist report
propose greedy construct nash grouping since action nash equilibrium another nash customer action customer customer choose action choose group customer greedy table customer customer since customer customer choose accord next prove contradiction inequality equilibrium grouping suppose equilibrium equilibrium grouping contradict equilibrium grouping affect customer request find customer nash chinese pre customer sequential chinese restaurant queue outside restaurant generality rest customer know customer customer know decision customer current customer table restaurant notice grouping customer customer eventually end number customer predict decision equilibria chinese restaurant game equilibrium chinese game begin formal chinese begin customer group customer nash equilibrium perfect nash nash equilibrium refine nash equilibria perfect equilibrium chinese restaurant game construct equilibrium grouping function customer equilibrium grouping notice equal equilibrium grouping generate remove expected customer grouping procedure implement j choice customer
prove semidefinite idea ex mm theorem lemma definition base active unbiased estimator hypothesis give weighted forecaster exploit recent spectra random hypothesis active learner active empirical access sampling oracle provide predict unseen cost obtain quite task speech speech need fairly extraction label datum also good unseen label oracle unlabele active algorithms oracle learn provably broadly active membership query algorithm stream pool query query label might necessarily
converse consider sense give increase probability converse fig scaling possible recall random model fix measurement tuple satisfie noise sufficient condition reduce match converse bound technique admit critical minimum numerically see fig critical small converse converse achievable increase probability converse scaling respectively increase number phenomenon noise per hence cf degradation depict fig setting converse analog however hamming choose say program decoder need decoder free constant choose decoder decode converse disadvantage converse corollary previous section draw sense analogous product sparse set restrict channel model noiseless sense precisely pmf unity derive reliable challenge long informative rest dependence explicit ease exposition decoder reliable weak fix measurement appendix utilize split albeit control allow success seem plausible min decoder see surprisingly require sense draw match theoretic analyze set work study distance proposition firstly decoder event tight case speed theory large deviation large deviation speed n bind speed open secondly small devoted code theoretic interpretation understand geometry optimization correspondence minimization rank decode form code element whose rank exist whose identical vector assignment interpret rank
improve particle mass curve proportion close optimisation bring histogram final weight concentrate around unity mean large size proposal adjustment particle practically adjustment make improvement compare evolution several magnitude step smoothly impact look particle weight prior kernel challenge sequence value ensure prior set state equation provide concern invariant rotation shape kernel figure likelihood em em adapt initial iteration algorithm constant visually particle updating filter nk nn draw particle distribution spread particle negligible particle weight proportion leave carry mass adaptation highly adjustment weight particle specific adaptation use display whose choose constant result
decade address directly application obtain ranking overview well aggregation ranking gene statistic position position ranking aggregation extract list chain assessment stability ranking compare g order list gene brief list suitably triple multivariate x call permutation permutation mean false exchangeability definition exchangeability much exchangeability weakly exchangeable permutation e absolutely continuous exchangeable weakly implication exchangeability give exchangeability variation iff turn space algebra consist weakly weak define hausdorff q variable give clearly ed ed max x x mainly exchangeability exchangeability random visually plot plot weakly exchangeable overlap exchangeability exchangeable pair panel provide terminology exchangeability define exchangeability universal set gene consist patient rank test ranking position gene exchangeable weakly without subsample ranking
compact infimum attain k u I I I correspondence c since show I correspondence apply continuity pz singleton unique uniquely unique neighborhood correspondence notation er h uniqueness continuity u denote column row g g result j j g g g show inclusion unique case statement unique imply neighborhood h lemma product l h cf show hypothesis suggest support recovery consider strictly w w w kkt contain decomposition support sequence converge contradiction assume extract subsequence subsequence assume n contradiction support however overlap dual take form q l point assume reduce simplify w substitution eq role form w yield unit ball face unique write q one reduce situation active analysis obtain group admit optimal use duality complementary equation share yield add side expression derivation uniqueness incidence q invertible span positive unique pseudo latter necessarily remove consider situation decomposition necessary sufficient support lemma incidence support row index whose index element uniqueness uniqueness notice contribute objective convex problem admit kernel consequence solution kkt condition sufficient ensure uniqueness objective strictly row conclude sup paris france berkeley university california berkeley usa centre
accuracy additive experiment standard toolbox author website construct gp th interpretability flexibility full experiment intermediate contribute relate particular degree quantify et previously show learn call learn set use sum number x pre weighting search also force hull neither difficulty hard contrast optimize
extend multivariate classical univariate indeed replace u cumulative reduce multivariate without normalize fix limiting assume vector function infinity chi degree segment start simple maximize boundary
label sample label sample belong separate row constraint minimization global subproblem obtain hard svd thresholde global solution first singular zero solve subproblem subproblem variable prove keep minimization subproblem decomposition subproblem round round optimality global yield therefore decomposition error keep decrease round
successive path index modulus modulus path eq q cascade modulus convolution pixel yield coefficient convolution modulus intermediate computation computational complex wavelet energy decay numerically depth obtain iterate wavelet satisfy signal translate modulus become translation
implementation discrepancy fit simple kernel estimation choose extreme quantile discrepancy principle quantile combine kolmogorov distance mode suggest distribution lie since almost surely prove bind fast behave although become estimation sufficiently kolmogorov detect kolmogorov leading principle goodness form result large threshold law iterate logarithm analogue slight essentially pp detail sn lower go fast criterion choose
sde appendix predictor loss adaptation essential cost section key lemma base trait evolve dynamic describe initial trait value root equal adaptation optimum pair trait transpose eqs bivariate employ argument q trait reader treatment consider involve eqs standard regression theorem present establishe
truncate select rank maximize capacity thereby reliably subset like geometry difficult challenge comparative order establish sparse acknowledgment partially center fp project htb several mutual simple create quantization error
distribution spatial result average respectively minimal study introduce heterogeneity keep interestingly network size spatial network conjecture ref network process exponent far due heterogeneity aspect network enhance reveal master analysis quantify laplacian strong link
domain link important conjecture discover attribute relational static largely ignore utility incorporate relational relational attribute value might g research author secondly might activate throughout person email additionally change time predict anomaly prediction attribute set relational change link depend predict decay link window temporal representation temporal classifier temporal dynamic link present datum social biological work select relational increase temporal relational temporal ensemble transform leverage constraint finally explore discover temporal pattern scalability temporal representation ensemble mine temporal static temporal assume strict temporal ignore relational propose flexible capture link moreover evaluate static work discover pattern attribute centrality network temporal link weight past temporal window link node form
player address spc game static behavior spc software access strategy gain know article present incomplete software game spc connections necessary equilibrium keyword nash issue mobile wireless solution power business motivate localize extend network area macro network weak cell access datum part member member share start member share member share share member member make associate lead exchange technology moment share mobile sharing mobile mobile access spc sharing problem
hope line hard method begin discussion mixture gaussian dp mixture arise coefficient covariance bayesian set observation initialize alternate step value follow ic j ic assign k function datum datum attempt minimize popular simply cluster minimum index centroid updating quite precise connection equal straightforward cluster equivalence explore differently ultimately obtain apply nonparametric briefly equivalently choose generate observation cluster cluster discrete arise place dirichlet covariance dp vector mixture utilize state
require shape density level indicator clear provable require compute principle proper prediction agree well first sample prediction distribution shape crucially pre rich intensive organize construction combine kernel density discuss practical choosing bandwidth present possible technical give construct method measure agreement multivariate depth see test construct sample distribution argument random exchangeable let yy exchangeability closely relate level prove augment could sample compute inverting
formally evolve walk graph input seed either stay node evolve hold seed random dynamic thus interested graph seed random set brief discussion global spectral arise generalized determinant matrix norm conversely solution sdp run procedure notably sdp nontrivial heat walk see regularize eigenvector intuitively act penalty ridge regression intuition precise mind context predictor pair ridge minimize
differentially output abstraction show automatically efficient release mechanism interactive mechanism maintain give sense datum sequence sense distinguish may set formally capture database database database tuple property construction iterative database every tt imply database construction database exist sequence differentially private release query exist query release interactive time adaptively efficient private interactive first multiplicative structure sparse multiplicative compose write sx sx si hx hx counter add element add element cause structure present sparse work universe run multiplicative update variable delay universe necessary simply multiplicative cardinality universe depend carry
detail h analysis remarkably expect involve probability need double summation form routine numerous database hash estimating similarity equivalently practice store small bit hash care g involve pairwise care
permutation define obtain accord cell cn n sn ct sn common increase sn si n sn si sn si si jj minimum induce table integer demand integer counterpart real bound calculate equal procedure approximation give bound seem perform table generate count count precede fix define sample define bound induce define determine employ iteration importance sis determination sequential adjustment cell appear early table et currently ss li otherwise n u n sl n u I sn l sn sf return combination integer properly recognize chen call interval gap good knowledge tool tool integer discrete
removal along simply projective characterize implication exchangeable action permutation exchangeability consistency example infinitely exchangeable exchangeable uniquely characterize dimensional projection follow cd induce cd simplify infinite exchangeability measure poisson variable ignore realization random x cover x straightforward
integer nx h h sd nx self operator sd nx h nx h h h h arbitrarily section start denote weak brownian operator motion converge weakly consequence show converge limit natural sake write condition satisfied prove algorithm preserve sd nx sd triangle prove weakly motion show complete mala algorithm hilbert suitably scale markov infinite dimension follow show mala approximation mala probability reach target target motivated differ study diffusion rwm work hybrid hmc explore invariant case extend challenging start stationarity ode combine hierarchical herein challenge direct applicability measure exist rwm mala acceptance degenerate explicitly rwm study ask metropolis
build first order numerical method match mm corollary section supervise optimization fit penalty term regularizer encourage r matrix vector machine processing range composition prescribe regularizer induce mixed identity matrix choice give rise kind prove composition e fuse well norm multi many cost per iteration descent acceleration number reach value essential method certain
constitute aim concern whether get weight system partially observable thus effort achieve htbp meta initial set reference control action function update newly map q parameterize dimensional discrete step iteration illustrative property visualization purpose limit cycle htp controller know simplify problem significantly relationship controller action take function constitute reference signal start action state time point trajectory logistic map depict figure act limit explore map standard value htp htp htp analysis behave
I mat since lattice site intensity neighborhood structure assign site index neighbor rectangular grid give neighborhood rectangular mat generate rectangular grid consist column properly mat neighbor list corner cn cn neighbor ci ci cn interior neighbor j ci neighborhood corner c neighbor cn cn n ci neighbor ci I neighbor cn neighbor cn ci neighborhood triangular mat list corner neighbor cn cn
close hope htbp diagram bound red green curve envelope source form iid vertical red line would hope fit model close except end interval smooth concave another concave function pattern might mark line symbol iid symbol algorithm repeat graph particular represent algorithm partition concatenation transition symbol fail essentially uniformity skewness graph htbp mark source generate draw markov chain iid show stationary estimator detect figure
ten well heuristic performance increase node almost achieve number mm mm joint scheduling power problem wireless multiple single present scheme interference achieve wireless essential computation currently wireless world hoc wireless wireless employ wireless area wireless pose recent convert careful communication scheduling
datum use miss build routine frequency initially complete part separate four mis x likewise guess miss imputation sort amount variable random forest response criterion meet algorithm representation vector sort index store previously mis mis stopping meet newly datum present categorical assess normalise root use notation proportion around observe part variable get meet imputation
restriction prediction value analytically integrate volume see value goodness could cause lead inaccurate use technique fold cross validation exclusive slightly set contain sample create set hold sample hold time function cross choose adopt stacking hold eq use hold variance value finally one greatly calculate prediction hold alone low quality evaluate goodness represent monte give fold
cost cost reveal regret total add loss incur offline cost portfolio management regret incur cost incur solution use differentially private want provide sub convert online private sub regret two popular namely ascent fairly strongly framework differentiable regret differentially private online privacy preserving algorithm private bound offline differentially large learn world critical issue several ad hoc notion stand break netflix challenge publicly two instrumental discard hoc relatively sophisticated notion diversity attack pursuit theoretically sound notion inspire definition notion accept standard privacy notion privacy year community several interesting problem concern among interest offline program interestingly handle handle well error mention
evolve order column field formulation evolve discrete dynamic temporal process walk paper nonparametric multivariate covariance covariance readily specification dependent loading sparse gaussian function quadratic dictionary element methodology advantage employ collection cope rely another fact prior model handle presence limit observation cope computation update finally prior covariance addition theoretical google present value discrete tractable categorical letting goal good setting accurate prior collect subject analysis section space represent aim building nonparametric covariance dependent limited loading build loading index factor predictor despite latent loading challenge large far element coefficient comprise dimensional th column characterize predictor location combination tractable formulation factor induce decomposition function increase latent predictor factor ensure constrain loading triangular however task decomposition unique interest characterize decompose dimensional finite
jensen divergence hellinger distance also yield show enyi entropies shannon cross entropy formula whenever enyi entropy multivariate calculus neighbor although applicable kind intensive enyi close exponential available close
collect sensor subject design learn adversarial fashion provably optimize objective exhibit generalization publicly corruption encouraging indicate efficacy superior baseline method miss imputation fill well imputation work address issue local feature batch setting theoretical online useful corruption problem corruption well component wise product give give corrupted subsection batch type hypothesis arbitrarily reveal learner ask simply distinction fix weight making assume predict hinge see address change corruption give corrupted corruption predictor make provide implicit depend consist hypothesis function
bound incorporate fr e rewrite correspond ij ji symmetric symmetric establish ranking strongly differ establish relation exhibit relation set adopt joint feature mapping explicitly outperform main structured object string pairwise kernel paper pairwise compatibility bioinformatics various kronecker kernel discuss article utilize within naturally set run regularize square kernel apart mention solution predict dyadic setting movie argue specific well exploit relationship space object simply train dataset multivariate structure pairwise kernel basis dyadic pattern match relation intuitive near learn protein like r contrary consider learn nonetheless course prominent place special kernel relation enforce end least
irrelevant lead instead row fact optimize relate variant function discuss connection material algorithm objective symmetric two auxiliary upper introduce ard nmf half prior regularizer behind regularization term fact easily auxiliary objective proceed auxiliary j close first monotonically lemma rule ard regularizer take regularizer iff auxiliary recalling update term denominator tolerance vector fix value calculate kn case regularization auxiliary mean q replace table upper new function summarize algorithm update implement regularizer loose convergence result show bound original function g choose solve cubic equation rational derivation ard counterpart regularizer reader sum ard ard ard w input data hyperparameter nonnegative relevance nonnegative tolerance calculate hyperparameter
noisy weight play role network adaptive quantization error reveal distinct square adapt diffusion adaptation exchange collaborative strategy enable real incremental strategy cyclic cover node generally enforce hamiltonian cycle cyclic robust failure incremental network adaptive diffusion originally extend encounter also online consensus agent main emphasis role adaptation network original quantization study degradation particular perturbation incremental strategy extend square analysis stand alone counterpart explain useful along link exchange estimate work account algorithmic
build assign tn j ii unnormalized normalize instance commonly asymmetric walk normalize laplacian govern normalize unnormalized walk normalize correspond diagonal graph normalization closely connect limit graph primarily limit typically fine local proper without build provide theoretical harmonic semi label point aspect study weighted laplacian paper exception manifold boundary space specifically series involve normal reformulate implication laplacian believe popularity machine boundary necessary main laplacian explicit also interior manifold example
novel size discussion corrupt algorithm subspace solver augment lagrangian gradient subspace state main adaptive tracking solver least derivation admm extract residual dual rotation track vary behavior subspace change true new random figure successfully identify change track h ratio corrupt inexact multipli write want comparison rank previous entry matrix select value maximum oppose bernoulli challenge outlier outlier outlier noise perturbation perturbation admm little reasonable computational outlier fraction second second subsampling second sec e e sec e sec sec e sec sec sec sec
hold refer perturbation method histogram concern subset make hence concrete motivate relaxed privacy differentially private term statistic risk eq private randomize risk function achievable hardness describe differentially private mechanism use derive minimax hypercube neighbor hypercube namely differ eq q neighbor upper e obey kullback bind affinity take expression differentially
enable posterior various far datum analyse radial make survey simple model numerical integration epoch integral integral function uncertainty deviation consider reliable see aic biased marginal however biased converge estimate extremely logarithmic grey converge density aic uncertainty estimate aic however estimate reliable make posterior much investigate accuracy combine epoch one clearly close omit fig receive combined estimate grey blue
maximization expect value log maximize aspect highly user lead monotonically increase disadvantage disadvantage avoid quasi acceleration yield second constant expectation complete therefore hierarchical conditional f characterize require likelihood function moment numerically care naturally sparsity find handle start remove get prior mean work practice disadvantage enter perturbation assess whether enter optimization
discard large residual unnecessary keep residual lipschitz continuous discard residual time median correspond criterion ol objective aim indicate median absolute parameter control removal residual linear interpolation coincide criterion everywhere huber affect discuss consequence compare equally indicate landscape ols criterion complex objective minima backpropagation minima ols backpropagation work study optimisation start free bundle start free base dynamical quadratic objective piecewise proper optimisation minima speed become study good objective try free simplex competitive
assertion fix robustness theorem special borel follow robustness convenience value yield complete algebra metric b totally see see respect weak combination
require enable knowledge information snr contain hyperspectral draw convenience respect denote relate remain towards general theorem due matrix p nm complete indirect reference parameter matrix show unitary explain successively generate unit explain present situation need bring modification handle statistically rewrite p kk give k set core see vector von dimensional sphere sampling strategy rank derivation
incorrectly entire work glasso update monotone glasso minimize glasso coordinate w equivalent minimizer simply need achieve covariance matrix update identity know simple update operation primal lasso glasso cycle around column repeatedly perform implicitly compute solve warm start column update work glasso update every actually procee establish glasso solve coordinate reach conclusion elementary argument align intend ascent dual objective plot though coordinate update solve instead present framework consider primal stationarity condition sdp rewrite q wise eq symmetric denote one operator observe box transformation solution notice optimization suggest glasso hold optimal exclude diagonal hold fix equation express optimization p glasso solve dual
widely game environment grid node combine heuristic first find later necessary keep track list heuristic queue node effectively couple path heuristic admissible never remain heuristic among node turn flexible go back stage artificial intelligence game playing reinforcement observe state generate word transition bring reward short later contrary given receive immediate make enter reinforcement environment agent receive current
infinity way subsequent motivate correlation response adaptive unit ability lose extend multiple correlation output make well imagine process able measurement task method make enhance enhance prediction change similarly extend variance dependent multiple output extension include distribution rejection process combine flexibility process adaptive signal correlation input length heavy tail distribution expensive numerically unstable computation bayes carefully multiple gp model gene model financial review notation
introduction use justification coefficient build present ensemble classification learner bag forest boost choice bagging grow combine boost sophisticated try build manner equal weight ensemble rule rule base learner eq q indicator evaluate attribute constraint assign attribute vector meet constraint single attribute constraint indicator rule emphasize parameter rule set terminal root sort decision tree build cart regression algorithm contain terminal node make expensive control diversity observation subset decrease potentially less extracted tree clearly tree terminal node size prevent capture subtle distribution size determine grow branch termination meet terminal node select cutoff attribute choose split child split diversity time huge avoid capability boost diverse
depict circle branch decision place tree decision appear hand branch decision small decision tree subtree subtree immediate problem root decision combine node combine chance subtree partition write instance fig utility recursive definition past configuration arcs event therefore associate tree event represent arc possible decision tree easily extend tree convenience tree event arc precede event consistent tree really restriction inconsistent brief overview standard probability calculate expected chance utility well subtree expect procedure take arc expect either would em parent west east em child circle child height child circle em child node parent child node draw circle child node parent edge parent solid edge child draw distance node
performance aspect model notation prediction denote selection fold leave validation indeed however say impact final care particular concern provide enough parameter inform issue may analysis calibration size validation measurement partition outcome partition unclear relate admissible partition provide insight influential behavior know priori frequency capture behavior consider
compound process eq f respectively poisson process rate result fourth second asymptotically respectively match proceed prove convergence use device omit proposition stochastically bound unconditional h brevity omit technical bootstrap unconditional weak consequence unconditional asymptotic sum distribution follow zero surely immediately ng total characteristic one x side immediate third number sure side conditionally bound upper x conditional nz nz b index therefore measurable hence n integrable page equation replace nr far previous vanish show page proof proof statement compact point whenever imply characteristic z n iv dominate almost zero strong n z automatically w condition vi event imply condition vi condition imply vi result follow proposition validity regular vi observation follow equation true compact interval achieved able vi probability increase sequence whose subsequence subsequence borel vi almost therefore
party get verification american rate vote score conservative vice compare rating give former range relationship score interpretation correspond qualitatively infer agree remarkable base vote co course interested compare infer trajectory score score compare measurement rate year compare show trend
latent factor contribute nlp benchmark method develop binary predictor nlp bit optimisation base approach spike small bar real attribute count patch generalise bayesian spike overcomplete basis low case model overcomplete reconstruct hold specie property result dimensions nlp rmse data nlp nlp popular consist spike apart spike deal provide effective reconstruction good predictive behaviour model comparative spike much well
consecutive level subtract quadrature entry dimension quadrature rule adaptation user correspond sparse quadrature product quadrature original divide old set new candidate candidate backward validity expansion formula difference index intuitively contribute integral error candidate multi iterate active member tolerance propose find drawback parallelization parallelization also evaluation nest quadrature proceed advantage quadrature yield integral simplify notation inefficient quadrature surely overlap integrate total local indicator indicator lastly summation manner original experimental exploring characterize posterior ideally narrow range pointwise constant grid number increase arbitrary monte instead must resort monte sampling pointwise construct reasonably approximate posterior perhaps burn mmse simple powerful metropolis mh metropolis generalize improvement mh delayed combine mcmc exist active involve derivative posterior balance even proposal million estimate acceptable accuracy mcmc thus computational surrogate describe first illustrate use algebraic nonlinear use estimate designing experiment leave surrogate consider nonlinear scalar parameter denote additive measurement infer uncertain utility equation appropriate figure
exact paper completeness condition condition solution dual distinguished ambiguity rank guarantee clear exist matrix projection mn n contradiction innovation way construct minimum equality lemma let establish well establish infinite subsection also exist e sum converge n w consider candidate satisfied condition show subsection equality projection orthogonal complement next q eq last sparse incoherence conclude two inequality assumption power simulation agree low success minimum generate adversarial matrix method relative
k rw trivially inclusion node include proportional thompson analysis rw require fraction equivalently feed result obtain correspond repeat close degree implementation estimator publicly propose characterize far technique cover node degree average real c rw sample rw function graph tail correct behave analogous average node square b degree average node configuration pairwise edge bring well theoretical finding life graph formulae analytical expectation thick plain background average result line lie perfect estimating degree traversal follow coincide rw monotonically decrease exhibit bias estimator bias course life substantially different depend node connect similar indeed social tend degree degree quantify coefficient computed graph value indicate purely affect traversal
evidence literature aspect wikipedia well wikipedia article high article grow collective wikipedia influence opinion analogy google huge portion page site dominate political subsequent google major stream relate wikipedia automatic detection characterize wikipedia characterize page page article measure page level c measure user serve evidence usefulness previously automatic topic engineering perspective
optimize criterion test evaluate model spherical fitting evaluated measure table test trace spherical input convergence discover split include east variability ht correspond third variability exploratory functional cluster term prototype representative cluster spatial cluster process association among analyze spatial exploring similarity curve datum alternative spatial
square signal low signal paragraph illustrative concentrate application vision see compute positive whole mark simulated maximum type c c omit c error prefer contain square result signal give square however directly table good power already follow probability tend zero exponentially improve rapidly simulate ii simulation center lattice simulate principle shape concern triangular lattice site cc threshold signal maximal certainly effective pixel please
system find integrate neuron fire tend integer period find dynamic er structure favor uniformity reflect remarkable include marked cluster signal uniformity er explain along us spike element instant th spike element express size window respective eigenvector pca integrate sis dynamic observe result curve different parametric dynamic
bivariate scale define argument specificity application compact input six multidimensional polynomial interested characterize variable orthogonality one ref joint integral quadrature bernstein maximum thank bernstein polynomial output transform basis expand polynomial eq identify multi index
rectangular kb ab strictly diagonal invertible lemma analytically invertible hessian inverse positive dirac hard good important previously investigate differ considerably cite evidence neural amount generate x try infer gain dirichlet belief opposite belief result approximate compare monte carlo elliptical slice return sampler laplace estimate deviation approximate gaussian wishart bridge two apparent bad
though offset dimension slope stability algorithm recently investigate depth sampler normal behaviour rare event early set truncated type via rejection truncation recent guess available truncate smc hand sequential step monte truncate distribution initial sequential monte easily define estimate perform provide newly previous particle truncate move approximate truncation covariance particle lead shift imply region prevent version backward art rescale multiply long scale positive transformation scale mean truncation smc covariance multiple depend two single step start observation initial distribution possibly require smc therefore provide depend simplification derive argument trace provide though component order start set rewrite function depend replace next matrix way iteration yet however next complete iterate numerically n even decrease approximation conditional complete iterate equivalently multivariate normal turn single mention demanding case algorithm remain space probit comprise rescale check likelihood unchanged root
polynomial raise argument eq c x bx assume proof lemma write raise similarly term outer hand g additionally summation distinct correspondingly term algebraic plug sum expansion subtract remainder eq ne apply three find remark moment p hence lemma permutation size apply conjunction lemma k theorem eq expression panel plug ht mark large right marked square plot hyperparameter convergence mark plot show use large square bootstrap use adaptive output mark plot proposition conjecture remark electrical engineering science california berkeley electrical computer science california berkeley department department electrical computer california berkeley simple assessing involve increasingly demand computationally subsample bootstrap robust specification often rate bag incorporate subsample computationally assess modern parallel architecture furthermore applicability favorable statistical bootstrap bootstrap subsampling demonstrate superiority massive empirical extension series inference exploit basic capability
lead topology restrict subset illustration topology variant arrange line element intermediate latent choose embed illustrate sample exist circle example element position neighbor complexity take take I intermediate embed latent
likelihood eq independent flat conjugate prior gamma form incorporate probe set probe variance probe signal allow probe maximize limit sample ordinary parameter limit microarray robust overfitte optima compare moreover take explicit manner principle prior probe microarray probe hyperparameter external microarray collection previously side improve gene publication provide treatment publication validate compare preprocesse preprocesse probe probe level contamination comparison publication preprocesse probe effect probe contrast probe probe standard preprocessing provide array algorithm local background correction utilize non array level array summarize probe expression robust statistic probe preprocessing method experiment target spike receiver characteristic roc publication publication spike preprocesse estimating outperformed provide explicit genome array probe know probe error error genome alignment probe presence snps sequence good assess probe reliability detect result relative different source probe contamination detect source seem fraction highly contaminate detect remove probe level external information genomic fail portion perform publication rigorous algorithmic tool investigate understand probe probe contribute reduce probe noise array present chapter strategy differential introduce utilize probe analysis microarray database probe lead verification alternative microarray preprocesse analysis interpretation microarray tool preprocesse method publication probe strategy array experimental probe preprocessing array convenient access algorithmic microarray preprocessing probe view wide availability investigate genome control normal disease source valuable unique expression need traditional large scale collection chapter thesis exploratory response genome scale across wide collection summarize observation activity reflect exploratory research hypothesis material detailed widely cluster dimensionality visualization collection open possibility investigate discover previously connection gene expression traditionally relatively set particular disease cell detect gene differentially express predict disease outcome potentially increase collection cover thousand drive analysis formulation question traditional insufficient approach propose study gene group predict gene network base increase integrate genomic detect process disease mechanism implication perturb cancer cell line enhance detection drug functional genome wide lead breast remainder section particularly closely contribution thesis detail biological activation pattern diverse database contain concern interaction instance functional molecular classification human category micro chemical perturbation joint related increase statistical build activity typically ignore interaction gene pathway database molecular cell relational gene guide interaction improve activity predefine utilize paradigm genomic pathway
see super helpful factor endow sphere action sphere non overlap number rapidly large sphere packing action place element metric packing contrast sphere cover sphere third os enyi form every action independence clique translate algorithm performed simple os enyi observe neighboring graph randomly except high armed bandit observation run number axis run variance algorithm side utilize observation improve standard multi armed bandit gap clique tend almost action study problem broad regime bandit provide upper dependence
derivation drop form entirely analogous dimension stem n k ks ik move chance poisson expect eq define vector scalar consist uncertain change truth leave wiener measure function change rate consist deterministic deterministic mean sample process gps beliefs obeys operator go bellman bellman expansion eq integration remove loss correlate
plot depict contour kde mixture c point region much affected htb decrease lipschitz hold huber express explain translate robust necessary let hilbert necessary characterize principle satisfied differential condition name represent combination kernel suppose furthermore q give interpretation I robust sense additional also satisfied convex since assume strictly satisfy I strictly n imply convex convex fortunately iteratively square classical weight least generate iterate intuitively
suffer disease constant patient incidence overall call inverse duration eq correspond often statement odd incidence disease duration example disease read equal product incidence duration incidence article overview new ode express transition model
corollary nearest nn widely mining application aim understand reveal spurious variability contribution statistical certain subgraph graph point finite tradeoff conservative pruning able guarantee removal cluster level salient nearest nn link mining interestingly particular like understand graph reveal underlie spurious variability spurious structure contribution graph sample level satisfie boundary unclear might mild recovered subgraph
sample account fact center around clean manifold explore devise analysis assume confirm origin practice tool cloud et condition noise arbitrary model contrast explore accord corrupt crucial result careful analysis nonetheless also analysis present population eigenvector despite difference technique bind recover free multiscale pca study parallel develop address examine recent wu tangent basis neighborhood absence hybrid zhang assume collection recover flat see idea denoise smooth surface al graphic problem well tangent work approximation top value neighbor work often refer size answer consider perturbation tangent trade seek small avoid curvature tangent plane eigenvector tangent space curvature trade subject decade dynamical main true tangent define orthogonal distance angle compute perturbation theory ease presentation order measure tangent ex aid quantity denote due subspace general useful subspace encode subspace curvature recovery curvature trade readily apparent small value neighborhood small ill intuition dominate approximate neighborhood radius denominator bind control numerator intuition serve enough curvature contribution conditioning denominator force curvature enforce bind principle curvature approximate must curvature become large ensure probability curvature uncertainty principle intuitive notion combine curvature perturbation accurate tangent uncertainty principle computation homology manifold explain detail section principle geometric perturbation review accuracy estimation present modification account noisy introduce practice apply theoretical algorithmic consideration technical riemannian locally reference tangent manifold origin rotation align axis direction principal coordinate axis plane manifold
pt report predictor median preferred incorrectly variable ideally discard glasso setup rsc achieve recovery definition rsc inferior three result confirm glasso rsc number unlike large explain slight method however tune infeasible practice serious finding selection inferior experiment conclusion find winner term speed appeal reduction provide evidence cognitive carry world publication paper control paper variable take predictor yield intercept make response result neither penalty impose website structure dimension reduction rsc describe find assess performance leave square newly
year overcome several pattern aim galaxy neighbor neural support map universe galaxy add near optical kb deep survey like systematic specific region emission often method use statistic evolutionary observational shift filter specific dramatically value previously mention probabilistic increase computational overall hereafter design accurate relatively scalable produce survey method type tune automated ease exploitation survey outlier limitation mention region happen section worth never account possible structured feature principle discuss implementation use provide description selection find determination final galaxy together thorough performance method reconstruction determination error outlier describe find dm aim case local reconstruction otherwise measure order dm machine hereafter ml method problem extraction point ml derive case source knowledge kb useful unknown treat analytically dataset ml three extract base kb kb degree run implement avoid like extraction knowledge without target say extraction practice approach drive validate subsequent aim object subset consist belong say unsupervised determine cluster represent dataset tend clustering figure efficient clustering define supervised mapping regressor thus target explicit provide training regressor map response variable
horizontal dotted capacity width compute channel channel interference section wish compute case channel factorization follow many analytically available case analytically available see bc bc extension x q accord cycle
desire requirement tp l f n tp p condition g p sufficiently l la k independent hoeffding absolute inequality probability nm f mainly scheme bind term necessary incur sign capital letter symbol font matrix zero respectively large singular nuclear value phrase numerical use proof fix w ij ss obviously whole setting
without reversible spectral self adjoint finite one argument obtain claim prove integral observe gx gx k write integral spectrum spectral geometrically ergodic
ga denote point clique scalar equation translation without loss x l k easy q eqs firstly j kt secondly r sequence surely obtain surely therefore part equality without chain vertex chain bin bin chain equally likely node bin number average chain average chain procedure demonstrate special discuss ij j h let hypercube direction almost align small direction xu exist bound follow eq constant task node vertex hypercube define unit constant b eigenvalue notice due thesis number node coordinate vector node sdp k r ng nr xy dy tx x ip coefficient arbitrary conjecture claim cloud euclidean distance various area localization closely dimensionality
moderately alternative satisfy two main alternative satisfy restrictive critical value statistic critical value available assumption nh consider nan alternative sufficiently moreover lack drive specific alternative choice propose statistic correct estimation elegant optimality necessary suggest hence possible restrictive allow absence theorem establish optimality alternative p n sequence theorem detect alternative detect popular process variance constant alternative polynomial short term statistically negligible multipli order covariance alternative j coefficient process tool may contexts alternative case test q statistic study wu er von statistic introduce observe asymptotically critical test alternative nan white gaussian hold consistently asymptotic ii imply standardize establish involve due statistic equal simulation aim propose valid penalty test preliminary practically white process investigate latter order give box label second since kernel k
clearly restriction different test availability widely software flip flip replace observation cauchy datum replace randomly select label keep scale two class circle htp eps eps scale eps scale eps eps plot contamination class represent circle simulate contamination mis capture aspect mis nature tail additionally attempt simulate extremely error scale center cauchy multiply typically datum contamination occur uniformly I calculate cauchy gamma interval empirically effect contamination nest boundary
graph smoothness work addition branch devote analyze spectrum good introduction particularly emphasize promise constrain suggest way cluster task work graph view intuitively efficiently graph analyse effort idea essentially spectral walk mixed random walk graph author permit efficient clustering result post work close sense use unified find representation multiple graph directly develop co regularization conceptually similar adopt related paper first despite nature exist effort combine layer mostly way art graph equally layer role addition respective layer theoretic beneficial processing multiple graph network believe effort mobile phone dataset field represent technique
mark mark black nine method vote member empirical random member theoretic complete plot dataset perform approximate gaussian exponential enable estimate gain extensive gold evaluate loose monte carlo objective preference select continuous eight version decision expect three challenging decision boundary second uninformative uninformative capability multiple disjoint use depict uci multiclass dataset distinguish letter vs process preference aggregation fig significant random preference domain
point indicator additionally square shall frobenius tv distance real mle initial hide process order keep instance various mle existence invertible normal interior yx yx remain open ball center mapping space state tail theorem good qualitative guide mle even hmms sense asymptotic mle quantitative abc eq consider q analyse abc define mle measure corresponding take one immediately law variable crucially thus statistically identical q denote perturb estimator reveal mathematical tractable similar spirit author view hmms try section likelihood perturb abc mis raise still general answer let directly law asymptotically consistent though value maximize estimator long asymptotically consistent follow abc mle accumulation belong subset accumulation lie neighbourhood neighbourhood go finally one mle spirit
experiment adapt critical temperature lattice wang design efficiently surprising must temperature many peak large peak comparable wang might seem sampler achieve mechanism learn choose length sampler sampler thus acceptance peak learn length problematic adaptation length manually wang figure principled addition temperature change investigate bias ise auto trace considerably wang effective clustering show exploration ise cube ij sample low speak space visit simulation densely wang substantially reasonably cube spin hard worst would hope arise run train likelihood rbms bipartite undirected variable side visible unit unit
high frequency network combine handle direction determine structure recent study c gene investigate profile bn relationship program reflect expression rna isolate month old real rt implicitly inherent uncertainty justify author parametric resample level bn structure edge several reveal relationship normalise algorithm et selection edge line represent threshold limitation computational fact permutation computational complexity second large extremely value positive gene study choice
duality employ average principle rule formulate aid calculus role learn principle free reduction also principle formalize principle brain optimize representation identical optimize complexity free
simplicity assume mixture support contain handle get state next many exist modulus vanish pointwise summarize surely range finite root I abuse notation I vanish nearly theorem middle difficult mapping drive rate must finite mixture optimal rate allow rate restrict subset ultimately illustrate
introduce amenable determine minimize et match limit case norm intermediate specific minimize interval soft restrict show figure value approximation relationship quadratic activation solution activation valid people undesirable bias many signal processing statistic attempt statistical norm smoothly different capture desirable many norm basic architecture modify continuity near demonstrate norm constant avoid bias one compete goal smoothly deviation scad penalty region cost function define scad individually region give activation activation argument thresholde restriction condition scad attempt something close coefficient calculate cost activation invert cubic root calculate generate thresholde positive correspond outside range analytic function obviously fitting circuit aim modify robustness function quadratic smooth transition
obtain reservoir international competition future evolution reservoir extensively speech recognition outperform isolated recognition reservoir complex recognition comparable good method couple road building basic reservoir suggest new computation system reservoir loose reservoir dimensional turn turn idea reservoir computer fact important good reservoir rather good eq change one neuron experimentally dynamical reservoir design remark computer build dynamic couple dynamical external much come dynamic experimentally adjust instability adjust external read construct weight reservoir subset dynamical satisfied experimental realization reservoir standardized test evaluate linear capacity often benchmark reservoir computer memory input input memory reservoir unable task task task reservoir auto move average system reservoir drive two widely see instance predict trajectory benchmark publish recognition digit recognition exhaustive benchmark road master isolate digit moderately hard imagine recognition nonlinear memory give independent reservoir road cat couple semi optical recently delay feedback loop demonstrate analog reservoir task isolate recognition question road map leave open reservoir noise exhaustive remark distinguish reservoir noise reservoir computer continue
bernstein approximation approach knowledge chance constrain knowledge moment study constraint high hybrid surrogate let take convex surrogate copy objective chance constrain optimization infinity obtain essentially theorem structural assumption inherent address indicator investigation beyond scope classifiers eq eq yield yield find sufficient low begin extensively use sequel argument classifier simplex theory marginal rl lipschitz em convex successively contraction
large state eliminate bad datum bad view share mathematical compressive nonlinear goal sparse recovery bad network consider analyze measurement mix norm bad observation linear mesh improve almost mathematical early error generally generalize iterative convex programming noise measurement iterative measurement program iterative compare propose minimization additive consider estimation data attack formulate attack rely signal compressive sensing extend restrict isometry sense overcomplete error correction rely condition rest bad detection
integer result one remarkable feature mh ratio example mh greedy trade decide add subsequent employ speed observe safe exhibit aim subsection estimation replace employ result screen nevertheless qualitative coordinate show pt know sparse theoretically entry depend choose numerical norm value run replication readily lasso regularization fold fold ten calculate cross validate package prediction b case right sparsity pattern recover report deviation particular illustrate well right illustrate evolution hypercube correspond sparsity
reformulate absence consider statistic
gain every past determine choose decision feature value variable denote transaction clearly gain become classification transaction consider f transaction dominate past transaction branch past transaction gain tree construct decision construct transaction describe transaction simply tree transaction gain branch add node past transaction successful terminate test perfectly transaction classify transaction compare root tree discuss decision avoid overfitte etc transaction trust able return recommendation transaction confidence recommendation input e transaction perform algorithm knowledge quite recommendation confident recommendation equally random recommendation completely confident denote successful transaction transaction input recommendation discriminant analysis refer transaction input transaction space recommendation greatly power separate successful clearly minimize transaction otherwise transaction may overlap potential transaction confident word determine lda centroid transform transaction measure distance transaction centroid potential transaction successful transaction group lda perform discriminant analysis distance clear eq classified transaction safe trust recommendation construct accord information gain accord gain
level bandwidth choice observation sampling pt median lin lin band build describe sample covariance draw realization variance explain equation band highlight smoothing validation validation lead empirical get always confidence band estimator smoothing coverage smoothed interpolation high pt area lin bandwidth band band interpolation coverage strategy weight empirical expect one population transmission cost network noisy thompson population sample curve consistently although present estimator band asymptotically correct band conditional difficult survey good rigorous theoretical reveal far superior performance interpolation even low smoothing beneficial building confidence band
easy depend autoregressive gaussian definite definite wishart multivariate gamma chi square integer square univariate chi square distribute likewise outer product wishart wishart distribution definite chi freedom wishart mean wishart wishart assume dependent though effort value everything write replace suppose kt nk ij kt u wishart constraint function univariate wishart generalised wishart time variable major axis eigenvalue like draw collection index time collection definite process matrix index draw
extent decision detect come alone insufficient essential aspect historical go probably accuracy importantly one place epoch indeed digital concept article build song audio content community algorithm achieve comparable even provide acquire community valuable improve raw query cover song identification outcome consider cover song community song within show measure centrality discriminate original community light stand promise cover especially acknowledgment author thank review project reference complex cluster music retrieval cover song explore cover song automatic underlie piece pose song match similarity song community answer start art system embed similarity content recognize cover
include source transfer different enough sophisticated transfer introduce source encourage measure supervise effectiveness measure performance interesting transfer benefit target existence limit source whenever transfer whether adapt transfer perspective work summarize transfer also method whose solve tradeoff identify good source formalize set finite highlight task advantage identify combination task task confirm adaptive transfer tradeoff organize notation transfer problem report challenging report conclude appendix consider discount process close bellman v action span basis orthogonal projection transfer source build
w g notion key generalize chen identifiability compact strongly identifiable fix fix assertion exactly distinct point e proof subsequence distinct strong identifiability specify practically condition eqn next unbounded support convolution function thus fx p smoothness tail behavior I ordinary density density normal full symmetric dd md g measure discrete subset remove density p g laplace density unknown framework endow measure measurable eq study neighborhood van analyze behavior divergence hellinger density mixture view notable concentration term oppose divergence mixture separate family simple hellinger metric ease consequence
q writing c I ib ic q gradient decompose update conclude proof life predict outcome paper mathematical artificial market supervise probability feature fusion contract price outcome market inspire real prediction market reasonable efficient linear aggregation well logistic aggregation specialized instance experimental market outperform synthetic extensive evaluation detection ct improve adaboost positive volume online market information market yield interest department health care make informed price market probability outcome market capable introduce mathematical simulating purpose mathematical solve contract show market participant forest turn linear aggregation market regardless market contract furthermore section market regression classifier respectively certain accuracy predict outcome region classifier obtain whole become sort ad hoc trend
range genomic finance distinguish produce interaction party component infer exploit concept statistical causality maximum principle major theoretical difficulty reliable analysis interaction comparable present establish quantitative correspondence boltzmann machines physics graphical analysis bm mathematically computationally challenging infer correlation scope letter symbol correlation configuration bm field ise equilibrium many design tackle advanced message pass expansion graphical lasso running procedure quality popular principle one rescale fluctuation separately projection uncorrelated contribute say notably large pca determination retain mp correlation sampling configuration generally spectra coincide retain explain achieve time pca may illustration pca look principal vanish conceptual bm appropriate detail necessary mechanic model originally rely informally speak attractive direction tendency align norm characterize attractive pattern auto bm direction orthogonal generalize attractive bm interaction equal know priori generic bm allow infer parameter component small limit show ml attractive correspondence pattern correspond normally discard remove address many inference pattern due analyze pattern send possible supervise rigorously study see review validate ise
present regression effect provide rigorous building expand lattice random impractical little advantageous remain one fourier log spectrum arrive example date example page make mention context separability thresholding model obtain field formula random proceed describe recursive estimate maximum result asymptotic estimator section illustrate study conclude extension imputation supplementary begin basic concept field focus field hold index lattice field may regression correct upon weakly stationary effect lag center variable second order define depend let via spectrum inversion notation integral frequency expression compactly
corresponding equivalence write transform st transform st equivalence equivalence matrix ht st equivalent define class finding small however end vertex connect column switch st graph contain connect switch st gram symmetric sign without form block three compose ht solution hill local optimisation follow decomposition find conjecture
member sequence linearization characterize arbitrary viewpoint quasi write false sequence mx false conversely false condition infinite sequence condition condition assertion theorem hence condition theorem desire conclusion enjoy construction conjecture
filter band insufficient critical filter threshold iterative correlation signal recover knowledge structure analogous cf signal encode underlie structure matrix importance pixel order critical generic diagonal interpret accurate scheme filter interest unfortunately complex operation inversion often matrix box routine read calculate seem e r obvious multiplication vector invoke example instead canonical
strictly sense verify explicit wise row transpose section th singular belong class invariant unitary look reproducing endow respectively satisfy proceed q reproducing derive explicit complicated regularize dual norm choose respect homogeneous pair straightforward eq equivalent satisfy compact subset recall strictly positive constant exist exist satisfy norm constant conclusion positive shall contradict proceed convexity contradict banach
rank frequency plot level monte distinct must also include bin zero list please display significance reference goodness fit testing hold remark know root powerful perfectly bin great number draw law corpus randomly bin indeed consider model integer nonnegative number sort fit bin bin aside draw bin fall three frequently occur law level via simulation great summarize classic whether first bin mean total little bin demonstrate experimental bin statistic significance monte simulation bin set four goodness fit distribution statistic mean relatively insensitive truncation particle observe bin dot square display count plot along significance root significance monte poisson simulate report reasonably ccc square square dot poisson line suitably proportion nine population proportion genome formulation suitably entail draw model goodness confident suitably figure c calculate
cut mp combinatorial structure need cut term cut solve graph cut pass elsewhere cut moderately special case analysis technique novel rather rely map combinatorial case mp execution map prove strong statement fix energy construct magnitude belief correspond apply broadly could region strict mp mp connection cut cut idea path strategy scheduling capacity scale graph cut broad cut high high separate dual regard delay message safe believe concept explore non throughout project thank anonymous valuable lead paper message preserve ij simply plug update ij ij b final lemma tt message x ta except position message I b subtracting belief final prove message homogeneous binary factor tm income form true message opposite plug plug message jx matter homogeneity homogeneous pass mp belief pass belief incoming message pairwise factor minimum sum pass belief previously still case analogous lemma homogeneous equivalent max flow min find phase belief variable message incoming message would opposite thus nonzero edge iff merge encounter cut assume eq cut bottleneck capacity easy position capacity
learn discriminate action due lack detail model four dataset dataset process manner stop character baseline weight representation dataset corpora corpus corpus compose corpus large category corpus page ng compose l corpus document nb category nb sentence document compare measure
direct employ operator add hill scoring annealing begin take step decrease undirected learning structure possible strategy candidate keep step hand short try clause move candidate clause algorithm pseudo pseudo atom prevent predicate dominate graph selection markov network regularizer norm impose force extended technique first learner clause perform one learning contain thus focus address challenge group exploit sophisticated whereas search performing design pre address potential sophisticated avoid maxima discriminative structure learn alternate type move locally current order optima alternative increase structure al domain attribute employ constrain search table attribute single entity table describe relationship proceed stage dependency local second dependency join require dependency stage dependency join attribute space constrain orthogonal characteristic use bayesian network convert graph structure structure modify score pre roughly find promising space approach constrain potential parent par rv proceed stage form potential parent par chain potential similar candidate add beyond capture currently parent parent par rv parent relation lead high network group learn structure network identically process induce template clique markov network template dependencie ordinary clique par par rv consistent template far search relational promise community search
likelihood greedy neighborhood conditional log square loss neighborhood recover elementary entire recovered contribution rigorous algorithm condition also local greedy require significant improvement global weak marginally greedy mle impose explicitly impose condition condition entry theoretically
conditional variation tractable family representation high variety vision financial social network graphical contextual occurrence opinion formation technology social involve estimation draw np high observation typically design structure seminal structured span tree various broadly classify two category combinatorial typically solve penalize ise algorithm exist estimation high tend graph consistently complexity demonstrate fundamental local wide enyi r enyi graph almost indeed whenever random law cycle small applicable elaborate relevance aforementione extensively topology model topology various phenomenon social formation technology concrete use ise voting reveal political decision scenario e network popularity new regard distribute model finding imply
predictor construction recently binary predictor may take trick note feature sort calculate motivate consider wise piece wise function see piece piece represent piece domain multiclass shall predictor apply variant piece linear piece practice example train multiclass predictor pair piece advantageous generalization provide defer number error iteration analyze mention solve eqn np follow tell accurate illustrative find fail example exist entry gap equals denote eqn solve q pc matter regularization regularize rather produce use show
isolate angle big advantage generate graph examine affected rotation beyond goal article example power sect promise rotation drastically ratio isolate especially rotation angle check effect sect deduce isolate limit isolated node become dominant proceeding note deal size case iteration lower bind bind size cover measure describe sect width column adjacent slope united segment fig application measuring carry low segment yield much small interestingly frame curve consist two subsequent segment setting
simplify include physics index asset price economic citation serve completely current dynamic principal generalization contain temporal configuration ignore future variable markovian conversely describe construct reaction hereafter preserve upon protein reaction energy dynamic optimal reaction system give reaction cut energy half transition reaction time complementary superior histogram invariant reaction coordinate rescale insensitive detect reaction
statistic
function graph maximally clustering naturally multi criterion information independence need clustering impose style graph want cluster something know criterion information independence second far independence clustering variation metric section aggregate ex world country model force service etc structured format language unweighte people trait american weight skew correlate trade relationship clustering substantially optimize linear combination edge allow free package balance modularity et clustering group independence modularity vi distance china cluster france clustering modularity statistically thus aspect like less aspect share cluster trace highlight spread contain root root group explain alone combination potential drawback aggregate clustering necessarily aggregation explain adjacency weight axis eigenvector length matrix significant flat without cluster hand close
sensitivity residual explain account outli nonzero estimate minimizer nc ml nc n denote membership satisfy cost rewrite nonzero yield outli reduction np hard mean hardness latter line iteration guarantee due practically feasible remain eq robust mean convex jointly solve resemble lasso recover interesting robust cluster couple mahalanobis covariance euclidean mahalanobis distance c outlier identify outlier replace entry whole iterative solver develop present limitation constraint membership assignment assign cluster vector fractional membership soft differ hard binary alphabet box th replace leverage sparsity set present section still soft assign cluster soft interpret unobserved centroid subsequently outlier probabilistic draw membership
limit state proposition recovery alternative algorithm minimization implementation sparse section satisfying pursuit kk ax dd assume situation integer nj entry block representation loop approximate convergence analysis upon situation true approximation quantitie bx bx bx bx applicable exponentially along since follow bx cover block theoretical sensing b block sparse absence observation small possible pair answer aside condition validity optimize allow yield recovery problem may think norm recovery validate provide sparse course noiseless discussion besides assume mainly sake notational
allow formulae finite give integral finite substitute bs ds ab sample elementary yield finite density calculation adaptive analogous analogous reference analogous proof whereas analogous analogous recall part density obtain n n ns dx u sn ns ns ns n ns use trivial eq every b nc nn since arbitrary show k already converge zero closeness argument q b n dx x soft du use substitution elementary verify nc nn complete z c observe lipschitz n n I clearly assume bind consequently nc atomic hard unknown density weakly atomic par I atomic converge absolutely next absolutely total mass absolutely limit mass density se se eventually get dominate ie absolutely preserve complete closeness soft converge weakly view move fact I atomic mass atomic turn part total total absolutely absolutely part sn sn converge convergence absolutely soft since total preserve complete immediately closeness convergence cdf converge large cdf prove nx x x prove immediately proposition n l normally
technique issue worth investigate constitute seed author thank warm author paris author partially grant grateful whole stage helpful recall assume infinity go constant define upper bind write let impose finite disjoint n ij n I kk inequality follow go go use exactly combine arbitrarily choose enough equation together go n n j equation corollary statistics abc nonetheless algorithm statistic bayes asymptotically true quite appropriate author bayesian pick necessary
optimal bound policy optimal mdp tight mdps much mdp reinforcement learn policy finite alg approximation alg look perform time step operation step utility loop operation inner loop perform time mdp p ni expect calculate stochastically simply sufficient mdps accord loss mdps subscript
occur alternative merge cost lagrange multipli true rank set goal underlie dimensionality perturbation appeal clear typical cross final error rate term way stage chain well mean intrinsic desire provide actual theoretic
draw object factorize new compatibility fix functional thus distribution involve apply general marginal field paper mean marginal summation computation graph propagation belief propagation product algorithm pass exact tree cycle bp update pass apply general approximate source order message update require notation local function detail clear constant choose ensure provide flow cc message iterative equation local update global update pass meaning structure unique mild potential least long type contraction uniqueness compute marginal potential normalization see structured graph define graph description multiplication original iteration inspection incur per prohibitive although certain exploit special structure purpose bp communication cost update number along limitation sensor cost stochastic propagation adaptively randomized form usual bp update motivate observation namely pass along formulate incoming
line citation acceptable long consistent reduce font size point year use cite pp mit book realistic new recall journal error compare area introduce survival experiment cox hazard cox series
large insensitive say large instead predict occur mix data assumption homogeneous markov mix transition markovian ar error coefficient transition address parametric elsewhere assumption constrain ols common assumption enough moreover predictive bind analysis traditionally model minimization residual see aic really prediction
radius region factor present straight bound appendix brief binary sample unbounded discrepancy believe discrepancy sure super polynomial purely jensen inequality th well hard constant kernel describe page kernel principal shift invariant include kernel nice sometimes negative kernel super set unbounded infinitely min run hence attain run discussion lead I length david
subspace distance reformulate figure seem obviously proportion e computational view important provide stability algorithm paragraph discriminative aim good discriminative axis discriminate group criterion situation interest unsupervise furthermore model discriminative must approach axis keep orthonormal conditionally soft partition discriminant extend sequentially select subject first matrix soft conditionally q ik dataset preferable ny maximization problem remain fisher aim optimization procedure construct first complementary subspace discriminative axis solve problem iterative eigenvector associate assume orthonormal discriminative span discriminative axis lie gram schmidt allow basis linearly independent discriminative large soft matrix subspace stop axis model likelihood conditional provide conditional expression vector counterpart discriminative axis step remain result vector ic em fisher em efficient selection work popular keep high partition also generate covariance multivariate normal parametrize empirical directly fisher adapt strategy determine
toward preliminary average express cone describe see rewrite w balanced conclude w worth use make distinction main work relate allocation core game result core highlight solution principle principle solve infer possible select say derive augmentation fluctuation value formalize fluctuation fluctuation convenient fix u generic inverse q square augment independent run dynamic satisfy equation let formally q obtain mind ready controller drive zero allocation allocation converge direction previous controller
reasonable considerable average uncertainty variance bridge logic concavity even information typically predict square optimal solution realistic example bayesian outperform counterpart estimation conclusion reach example arise merely bridge marginal case risk conclusion penalize particularly regard bayes classic diabetes lar conclusion wrong specific merely practitioner broad dimensional case mixture many share favorable induce tail include relevance machine normal exponential gamma inverse pareto mix convenient lead poor miss paper discussion include supplement extensively poor behavior sampler difficultie something bridge excellent mix favorable mixing
small cause discuss sec rw recommend set category case set c equivalent volume proportional sec set one size sec resolution discuss increase decrease small category weight rw unbiased note collect weight attain weight increase empirical facebook suboptimal sample cluster show two base c j latter rw show rw optimal plain gain compare rw come irrelevant demonstrate facebook among category observable facebook well evaluate allocation gain control demonstrate insight nod densely h partition category nod dark light extreme scenario partition scenario purely cluster arguably remain ec ec resolution category respectively normalize square obtain short pilot estimator show plain curve advantage category neighbor naive dash moreover advantage
accept reject reversible preserve simulate hamiltonian dynamic preserve volume theoretically algorithm name carlo updating via hamiltonian updating depend energy hamiltonian always energy discrete step l discretization hmc proposal though distant previous hmc simulation inaccurate acceptance take successive undesirable slow mix hmc generate trajectory loop back doubly close side markov ergodic result chain slow hmc powerful usefulness limited tune parameter preliminary problematic simple short rely autocorrelation preliminary run turn extension averaging devise take us step long initial use convenient dot initial position system quantity proportional progress algorithm run quantity proposal move unfortunately guarantee converge issue slice black solid solid exclude possible sample position correspond exclude satisfy balance subtree momentum vector final trajectory process stop since blue x ignore begin
sort rewrite pca basis fundamentally different proceed approximation signal follow observe q keep monte simulation decay plot ideal energy linear approximation patch size example decrease e decay approximation small comparable ccc gaussian approximation comparable approximation numerically evaluate mse minimal proceed independent realization apply mse ideal good approximation decay fast linearly plot figure eigenvalue typical almost htbp ccc indicate number show good term parameter typical ratio mse approximation mathematical analysis decode impractical assuming describe single analysis conventional failure order cs decoder error term certain respect optimality say mse equivalent instance optimality index constant otherwise side one draw similarly
fisher maximizer maximizer right hand write fix evaluate however invariant respect th th column transform satisfie maximizer correct value determinant limit covariance column determinant indeed opposite formula deduce evaluation expansion initial describe refer algebraic ode ordinary satisfied gr rank ode call ode reciprocal reciprocal numerical procedure system toward along direction differential first study group manifold type differential operator delta analog
unit applie ratio bound dimension hold rescale rank satisfy convex program regularization deviation slightly mean computational achieve much worth observe interpretation rank consequently numerator noise second arise identifiability avoid restriction turn previously involve plus sample equation tail norm specify choice regularization regularization eq great pca see bind rank dimension roughly degree expect see order state regularization covariance concentration establish population version differ focus decomposition noiseless focus identity exact bound nuclear program enforce difference et three see incoherence consequently provide noiseless condition guarantee interest observation distinct leverage notion restrict sparsity rate plus minimax et sub involve bound limit decomposition nearly fraction zero analysis allow high simultaneously inspection corollary go set regression know goal share perturbation share sparse matrix maximum rescaled design uniformly mean note design program regularization q great leverage deviation bound base perform might relaxation provide norm
match point vector curve point construction surface difference tangent yield let set three origin fix thus kernel inner theory explanation simple get euclidean definite decompose bilinear define original explore form consist eigenvalue generally I theory carry
avoid stationarity figure non stationary harmonic signal signal model classify fit parameter bad true parameter maximum wrong improve separability harmonic series space black dot show avoid bias simply model describe white mle solution exploratory ar example bias good would identify correct tune however example large overlap remain red triangle green panel fig parametric subsequent comparable axis series right color shape represent section applicability growth neuron spike less stationary transform
consider instead compute norm minimax modify enjoy follow exercise pure ensure player resort modify derive result payoff round simultaneously accord mixed monitoring observe subset payoff ambiguity uncertainty follow convex dx first behavior also rewrite compact linearity extension case mean tend surely strategy round player respective mix full take end get account yx ba empirical joint action round interested rewrite linearity strategy player player proceed provide reformulate indicate repeatedly probability entail claim cauchy schwarz value continuous yx yx mm lemma entail let prove continuity read stand value norm pure action robust satisfy necessity hold exist set define distance give individual compact indicate corollary attain minimum whose action denote show pure action concentration argument hoeffding
whose absolutely lebesgue definition side complete condition fractional condition affine fractional analogue reduce take eq consequently interesting corollary optimal clear clear proportional previous one implication drift fractional brownian establish pg brownian motion case corollary
accelerate stochastic increment change reach implementation penalty normalize form follow collection behind ratio update average fact control control generally way prove frequency unfortunately transition find drift joint general able irreducible section prove directly case assumption restrictive imply implication frequency bin suppose schedule go meet notation use prove go imply follow proportion reach precision satisfy except proportion
world assume initial gmm effectively cluster localize turn decrease network gmm flexibility specify tailor evolutionary intuitive third final experiment gmm ba ba heavy act massive vast majority connection many feature skewed exhibit web email country country tend fast rich get rich node inherent attribute make likely web massive primarily web site connect google dynamic piece ba graph always begins iteratively enter tie already form tie ba produce tail degree si ba graph simulation leave panel red parameter cumulative gmm specification curve gmm ba si ba gmm counter increase figure scale likelihood simulation ba parameterization gmm recover generate classic ba gmm connect base structure base ba gmm hold constant classic ba gmm conduct ba base increase gmm function base pseudo code implementation include ability like fit method fit slope law ba piece ba gmm scale estimate figure
set input program input output implement program correct correct correct pair complete intended specification output concerned conversely correct output possibility belong care correct satisfying definition space find fig vector imply implement conversely fact capture definition role quantum setup studying introduce map error formally replace space consider refined comment capture anomaly detection perspective anomaly detection batch anomalous pair definition weak efficiently train quantum speedup map detector aggregate average turn quantum formally weak classification output weakly classify classified advantage classifier minimize heuristic input n use construct representation weak cluster computational set input training training easy idea perform weak clear sign hypercube label indicate assume assumption advance weak classifier weak training set process create accurate classifier know classifier open purpose include identification overfitte space quantum approach boost term accuracy speed select shall generalize value introduce weight dependent classifier difference strong express definition correctness pair average opinion know whether compare strong pair versa correct formally high set challenge construct beyond training solve find weight assign penalty
posterior root suppose start simulation well via vertical shift put recover calculation also proposition finally constant paper improve early geometrically chain applicable illustrate improvement analyze normal toy put drift normal alpha elementary compare chebyshev minimal minimal ep base magnitude remain less reality proof auxiliary note assumption special assumption relate calculation drift x v x geometric drift lem well bind entail qx let fix apply write sm
recover behaves miss cp less however sharp error factor recovery recovery increase tensor datum reconstruct tensor factorization literature alternate entity square fitting commonly speed suffer fail accurately component correctly ii missing imputation suffer poor therefore alternate least simultaneously contribution develop opt couple opt incomplete miss opt numerical notation discuss analysis opt algorithm miss entry compare factor couple euler letter denote capital matrix denote
fourier fit along various degree training bin split dc lead solver time spline high degree regularization cubic bin unnecessary embedding compute generalize fourier normalize train relatively expensive store cc cc cc fourier degree accuracy accuracy accuracy encode fouri embedding split dc training
estimate idea paper notice correction take weight construction tensor point becomes weight likely field effect I tend seek pattern include locate direct hierarchical draw distribution markov carlo interest estimate rough estimate point knowledge within density orientation orientation field evaluate orientation choose proportion mix property monte indicate knowledge alternatively little known nature extend start continuous birth death control enable range birth death brevity key detail fix birth death rate maintain detailed birth death indicator indicator signal point parameterization describe event occurrence birth reference length distribution respectively integrate field length field integrate assign choose birth density include update calculate death rate ensure hold death variable allocate
make clear bind orthonormal family tf take mass fx theorem times recall main difference argument show probability low application argument minimax batch infimum forecaster unless omit tf apply constant online forecaster dependencie take achieve instead standard gaussian jensen get proof present page f tf define lemma prove definition last meaningful elementary true equivalent c imply c dc tc c c choose eq remark fact multiply side conclude concentration argument stage sequence lie outside conclude dedicated term leave output note j
variance strategy analog var thus basically estimator implicitly existence strategy b estimator strategy type restrict study little suppose ask prove strong impossible plan overcome occur support version chernoff involve set incoherent analysis prove know ahead condition estimator probability prove automatically support highly quasi isometry property obtain sign jointly already deal unit norm sparse uniform among subset sign last concern nonzero define introduce x range range strategy recover pattern sparsity e hold support great reader relevant us coefficient strategy x strategy exactly sign normalize standard gaussian coherence proof
scenario middle side wrong edge length bad number discuss section bad affect merge serve cycle lead guarantee short path affect performance correctly recognize either short combination shortest detect middle cycle long shortest short use locally graph establish every diameter strong distance guarantee analyze count hide least recover event occur diameter decrease increase event cycle increase state choose c fraction node imply nr function enyi recover representation v accord homogeneous edge length approximately length distance length select effectively dominant presence subsequent shortest occur meaning enough second event occur middle bad analyze cycle short strong compare recovery minimal possible node component parameter equivalently consistent topology recovery length participant recover participant suffice recover availability short distance make discovery proof theorem test middle short
characterize indicate clique node program ax basis consider differ selector reason lie fact enable computation network try represent identity people frequency interest clique every view clique receive measurement order interaction represent reconstruct provide figure observe clique htp illustrative example scenario signal correspond assume denote interaction frequency row entry whether set column equal normalize summarize canonical transform omit mean matrix transform call transform row exploit transpose matrix transform construct observation basis way partially rank usually interpretable information approach sequel compressive compressive correspond rip I every subset column constant column index sparse unfortunately basis small failure let inequality every depend problematic constant clique rip try arbitrary
orthonormal pt denote bb coordinate q bs output suggest available could incorporate subspace specification tend theorem terminate give one stop iterate iterate sufficiently typically zero intuitively simulation stop always interesting question algorithm eigenvector analyze property assumption establish yet principal rate suitable normality assume though assumption could orthogonal factor dimension spike initial apply algorithm constant study iteration theorem probability state spike eigenvector hx x least satisfie decompose component coordinate average coordinate term factor arise separate first eigenvector rest regardless establishe understand upper compare eigenvector cp note bound
show ranking demonstrate select slightly adaptively comparison instead comparison propose address rank pairwise ranking mean rank uniquely pairwise pair primary objective need correctly structural constraint suppose may embed space wish discover knowledge practical restrict pairwise comparison view rank ranking specify ranking bit many sort human pairwise obtain pairwise comparison impractical situation thereby induce object object rank audio difficult section object specify rank select comparison select need pick select impact form response denote indicator tie allow quantify query reference also reference point object satisfy relate embed limit rank learn
development coin infinite formally bernoulli process know problem replacement pool chance unknown coin distribute equally likely look
argue end great transaction transaction contain end hence th replace transaction long transaction case transaction invariant th iteration stay contain begin algorithm transaction begin argue transaction begin enough transaction transaction begin iteration argue th complete invariant thesis perform online greedy grow assume add transaction build conjunction class constant vc always build transaction item interested use vc much dataset equivalent vc also tight bind empirical vc monotone vc construct algorithm guarantee collection construct obtain algorithm exact appropriately formalize transaction build ground index thm know mean particular satisfie f sx guarantee def def stress absolute interesting know advance minimum build use sample dataset transaction depend transaction less different relative constant pf sx x let sx
factor cost mini optimization minibatch minibatch pass data minibatch pass suggest imply communication small minibatch large cost mini cost per yield overall still empirically note minibatch another algorithm use algorithm single family delay minibatch minibatch backpropagation differ substantially communication cost format run complexity since compatible yield combination training programming rapid online high batch percent performance reveal drive benefit gain bfgs effectiveness figure discuss loading scheduling affect approach oppose nonetheless scheduling overall paper choice linear system upon training predictor ik ik ik present learn predictor dataset million hour careful synthesis conduct describe show choice see grow
involve area book function involve histogram sequence topology notice main together rate estimator assumption hereafter differentiable nonnegative tend argument uniformly f xu vx number cm iii iv usual infinite space dimension ball al ib hereafter illustrate condition induce principle abstract metric element together
show finite lagrangian eq derivative respect belief lagrange relate lagrangian solution beliefs lagrangian combine approach belief versa constant ensure appendix continuous derivation present lebesgue whenever correspond random continuous pass value suggestion factor pmf restrict real find code however base always always sequel solution approximation want minimize constraint setting suppose ix ib aa constraint fulfil stationary exclude lagrangian belief exclude vice versa hold include special appendix contradiction marginalization indeed x marginalization constraint together imply cycle message pass iterate belief bp bp cycle point initialize
ideal differentiable everywhere desirable purpose estimator vanish encode digital error simply uniformly digital practice atom example piecewise image atom atom efficiently encode atom mb residual mm triangular causality aspect consequence depict along atom patch causal bi atom linear prediction residual assume describe dictionary update compute iterate later know image patch dictionary patch arrange bi predictor template outside patch atom k example quantization dictionary learn alternate estimator mapping optimal ml parameter part significant fix late atom prediction see use atom term share decomposition assumption encode sequentially encode distribution encode suppose encoding coding encoding pre encode th laplacian form universal counterpart original convex support able minimize zero example already encode encode
resp resp mean description additive criterion kt ji histogram histogram worth standard wasserstein distance accord accordingly scheme criterion weight criterion satisfy relate dispersion high respective weight criterion restriction cluster restriction close respective weight perform accord use lagrange multiplier method sake without prototype accord rule md property convergence stationary convergence partition criterion decrease converge inequality inequality hold finally hold finally decrease bound converge stationary equality know minimize equality operation computation wasserstein distance operation weight allocation operation order algorithm histogram htbp eqn accord prototype eqs win cluster k
generate sampling ip mean bivariate logistic column fourth one show fix row sample marginal sis rejection polynomial time term arbitrary implement method interesting table condition thank section definition question sequential bind notice importance sis constraint equal distribution sis table uniform contingency
paper restriction adversary end necessary separable tight state topology topology allow require restriction player minimax restriction necessary q distribution tx x tx joint distribution restriction adversary round come apply nevertheless restriction strategy satisfy bound answer question even though supremum immediate question regret play minimax adversary latter case yet holding achieve importantly infimum move minimax move learn restriction online external obtain restriction sequential arise affect roughly speak introduce sequence tangent selector tx sign arise eq mapping conditioning make sign difficulty sequence bad notational seem employ version depth select sequence notation write tt understand restriction satisfie restriction pair recursive eq gain understand tree root next say child via p reveal greatly simplify nevertheless describe allow unified notion rademacher
away hz negative triangle inequality object general find know however manifold obvious point always calculation disk choice benefit least utilize online identify apply tx come semi matrix identify mining mahalanobi tw yet need remain tractable spread consist work approximation factor greatly typical operation complexity development mahalanobis distance geometric rotation identify r r induce tangent identify horizontal tangent tangent vector horizontal satisfies riemannian element class elementary computation hz sequence normally invariant dominant complete fact subspace stable equilibrium average invariant indicate unstable equilibria simulation loss coincide horizontal apply course hadamard way nevertheless violate moment begin preliminary matrix ij I tend parameter average origin ii ii ii assumption curvature positive
code adapt perform per empirical weighting noise standard sx repetition range weight experiment guarantee analyze rademacher complexity marginal degenerate netflix dataset empirically fill zero rip rip rip ts recall maximal q proof case row turn possibly marginal uniform modify proceeding j fp mp f mx fp r assume lipschitz bound sample term expectation ij sn identical argument proof
conjecture measure conjecture normal next approximate even proximity simultaneous describe extent measurement abundance protein true difference difference unknown protein protein compare bayes abundance measure laboratory institute breast er pr breast cancer transform positivity abundance abundance protein iid health condition cancer condition cancer respective distribution likewise standardize
research increase cost crowdsource lead worker worker less isolated effect keep still need extension view worker another example worker leave choose want stop batch middle systematically worker come repeat task amazon difficult task start practice way condition payment complete batch assign take batch complete task want worker task assign might worker finish restrict model crowdsource time tail law finish optimize pricing continuously stay available fast complete worker response time strategy crowdsource assignment crowdsource systematic crowdsource paper well crowdsource together importantly establish optimality voting majority voting majority worker crowd error since regardless voting provably upon reliable worker response learn worker worker idea expectation maximization consider label response give algorithmic base commonly crowdsource classification step probability maximize answer estimate probability recently number algorithms variety crowdsource em apply classification find get considerable advantage despite popularity give guarantee initialization make difficult task understand matter question task allocation inference together devise algorithmic total cost want strong performance fundamental contribution design reliable crowdsource allocation allocation optimal low propagation bad case distribution oracle estimator target crowd collective less task inference runtime factor conversely worker bad constant budget main show non bad use strategy attribute fact amazon worker exploit worker message worker truly benefit
unary form discuss histogram define ease exposition domain overlap frequency rectangle kind pl lipschitz lipschitz f convex r r application n minimize provably achieve goal set r incur minimize erm solve pt prove bound relate function fu l fu solution least serve proxy original decreased haar wavelet wavelet serve tool regular signal purpose haar wavelet piece wise vector parsimonious representation haar orthogonal transformation piecewise dimension wavelet transform several exist life mostly nearly learn theoretic tuning architecture tune histogram operator maintain feedback record abuse expression although available period assume independent initial histogram theoretic tuning histogram width histogram present learn setting handle reduction recovery static histogram multidimensional histogram formulation query fix goal incur unseen distribution query n ns rf cardinality query l problem solution
live negligible run could high evidence properly familiar uncertainty live sampling procedure converge measured uncertainty increase live explain independent live merge live simply merge sort step reach uncertainty necessary validate analytic verify indeed analytically scale cube box I fractional analytic examine evidence volume volume analytic uncertainty respectively agree gaussian nearly examine
determine content local object propose close roughly current compute move say phase greedy keeps find object close write comparison phase note phase terminate set probability happen sum use terminate word selection object define geometric fact number h x law learn call take place return zero occur source follow relationship proof show search cost upper us part stochastically comparison assume strong object content lift object embed target leave include upper np characterize consider approximation algorithm space exploit heterogeneity
choose might sensitivity select supervised learner sensitivity marginal risk ef rf x f notational f using let hx fx n constant h denote define everything adaptation h logarithmic powerful certain boundary support number supervise
section follow horizon integer consequence conclude use deferred expansion probability quantity analyse rigorously quantify next quantity appendix probability integer explicit result quantification mean drift next approximate u z z w thus conclude use computation give proof estimate express equation suffice bound corollary p p right hand equal zero consequently schwarz schwarz end covariance sd hold martingale rescale finite time generality integer satisfy k convenience suffice establish k iterate computable constant increment bind integer hold prove suppose state drift bound k schwarz inequality suppose binomial equation conclude noise converge weakly motion weakly prove sd sc priori lemma term proof conclusion condition cauchy readily consequently conclusion
potential estimate weight significant however pixel nature part finally combination potential main determine configuration former future contribution herein could determination configuration alone significant problem option handle configuration pose minimization general hard approximate decade graph differ substantially class complex pixel take move part effect appearance shape drastically one consider label label say scope reach addition centroid ideal relative nonparametric convenient ideal location tie directly pixel adopt hasting mh mh eventually detailed clique gibbs exist walk probability configuration propose markov act move say proposal dirac normalize part
say margin marginal margin information marginal function conversely value close essentially reflect subsequent intuitively margin condition opposite indeed ensure whereas quantify extent condition latter nontrivial policy straightforward nontrivial show condition bandit solve covariate point wide range local surely piecewise subset covariate partition theoretic sense collection hereafter call bin analyze convenient bin th time bin arm expectation successive arm sequence therefore observation static lead policy precisely integer fix static bandit arm successive yield mean parameter policy initialize initial arm outline
motivated arcs line indicate conditioning figure illustrate associated diagram chain implie unless state independence model g graph independence change independence pair directly figure path ci em trivial extension primitive induce note primitive induce primitive induce lemma property maximal graph side pointing towards imply primitive hold ij contradiction connect path must ji j l identical edge contradict short li primitive induce connect necessary analogous contain primitive induce adjacent node primitive induce connect give internal internal may pick
scheme result even section one present introduce crucial polytope eq unit take since always program introduce express convex body definition give hull word perspective minimization face pass ray point face reveal index nonzero figure concept figure see hold close face geometric consider construct hull face figure red plane origin grey space intuitively subspace outside face far lie outside face lie see close lie approximate restrict finally sufficient subspace direction depict dot line blue point coherence direction onto polytope help understand blue draw polytope blue almost volume red total volume propose understanding limitation ssc analysis restrict understand spectral gap ssc ssc reader indicate ssc corrupt trajectory research subspace ambient beyond spectral metric subspace detection hold intersect ssc affinity subspace
significant find separable plausible impose hope study nonnegative great machine currently heuristic approach popular nmf separability hope concept thank david discussion lemma proposition fact restriction constraint remark remark theorem support nsf grant nsf grant support part grant dms nsf innovation fellowship nonnegative factorization problem nonnegative make sense minimize norm rich mechanic etc past decade popular variety heuristic without restriction optimally study problem give constant nmf small complement substantial give identify algorithm provably believe nonnegative entry matrix refer factorization factorization nonnegative rank goal nonnegative rank note make ask norm inner dimension minimize nonnegative problem exactly decomposition fundamental context heuristic find extract relationship segmentation history name self resolution biology g research natural factorization nonnegative e chemical law mass optimization call slack boolean polynomially deterministic complexity equivalent real rank matrix application statistic quantum mechanic run factorization small nmf problem large
modify full edge code adopt storage graphical edge update integration available platform package frame edge code v v around set end break gs gray fill gray corner font node execute execute execute cell anchor south node anchor west west south anchor south anchor
regret sense assumption extensively sparse advance dimensional j oracle roughly thus sublinear sublinear thus seem ideal support paper regret achievable logarithmic indeed derive call bound belong learn discuss see deterministic online counterpart set latter express number forecaster observe pair fix capital paragraph integrable conditionally forecaster regression depend risk fx p tx notation typical probability thus trade coordinate df scenario oracle derive via selection argument fix design body dedicate computationally nearly prove reference work account regularization focus weight sharp design reach guarantee regularization efficiency algorithm sparsity oracle inequality almost approximated numerically reasonable online inspire langevin properties technical focus sparsity bound comment rewrite form
differentiable close differentiable lagrange term throughout index tuple negative integer pp univariate taylor theorem generalize eq polynomial define multiply side q like shift dimensional express product similarly follow eq multivariate representation property recurrence one taylor description algorithm notation point query tree r point query give notation relevant query g rr obey additive disjoint specify type notation represent change change use four different I moment centroid field form representative point inside accumulation inside like expansion way light choose format inform science perspective structure discrete aspect one expansion kernel sum express taylor infinite term tolerance series representation kernel region expansion geometric u expansion reference query expand thus impose along expansion center far expand small truncation order incur discuss evaluate
grid fail reconstruct graph toy root ss belong short path connect distance write equation row equal ss semidefinite hard trivial periodic connect four close neighbor ss p expansion taylor expansion power label node neighboring denote neighbor node denote periodic symmetry know write expansion write calculation expansion fashion term appear thus irrelevant computation configuration zero contribute two basic configuration must negative front picture configuration count track type produce counting express c ex ex basic counting configuration figure ex thus negative spin neighborhood second picture take count state type record ht negative boundary number ex thus write basic state one build positive negative table ex expansion e e write follow expansion put everything together obtain expansion put everything expansion put everything involve
bar question tuple system game consider chain piece chain look table possess tuple entry entry tuple simply form tuple output network sum approximate function I probability train td tuple generate playing since game van need td game seem organization behaviour present
could topic appear release propose two change whereas frequency area probably stage people individually another alarm dynamically value increase false meanwhile early topic value job advance practice far concerned text image video etc topic stream focus reflect behaviour model per detection
assessment may attempt truly use package test speed repeat work processor small critical conventional challenge produce stream release library generator library work graphic work generator small size short order exceed limited individual statistical quality adequate case statistical application default package feature perhaps adapt period optimisation quality gpu brief
draw represent sample suggest advanced algorithm region guarantee definite regardless collect quickly model online complicated anonymous website course user visit website ever website result objective ad site sale version week ad effect ad build ad decay week week consist five ad stock ad stock depend ad along nonlinear representation give structure high attempt estimation prove work conditionally data likelihood algorithmic software posterior mode hessian proposal mode covariance time inverse assume log generate efficiently dramatically collect draw
solve effort problem algorithm distribute exist multiple parallel build directly scale guarantee algorithmic challenge computationally guarantee subtle spread may retain divide divide expect preserved face divide problem whether design relate statistical property freedom overall machine algorithms factorization wide practical filtering recommender reference therein link web video surveillance graphical modeling focus rank noisy factorization recover matrix corruption outlier arbitrary significant propose factorization problem completion vast majority tackle formulation solution formulation rely nuclear admit variety matrix completion inherently repeat truncate decomposition limit scalability previous attempt burden approximation justification factorization propose divide randomly factorization subproblem subproblem parallel regularize formulation combine subproblem provide statistical underlying variant set new novel characterization illustrate experimental collaborative video finally sec matrix th
accomplish structure minimize svd individual first remove individual iterative supplementary reducing via red value monotone decrease thus improve joint individual structure algorithm pattern depict object common independent add represent size row joint visually phenomenon contribute structure correspond measure present example effect microarray show estimate true correspond negative variability scale permutation material rank joint permutation identify variation explain figure amount share variation type show responsible variation gene considerable amount variation influence residual order reveal share pattern joint
version deterministic require movement optimal reinforcement learn agent stochastic path thing description bit dot position ms pac position whether blue second kind feature necessary rl action north south west however typical consist hundred still need mid knowledge module hand combine module combination pac attractive illustration environment life characteristic task environment task separable mutual rl utilize technique extraction architecture shall consider kind advantageous break tie e decide like go dot direction near go towards go opposite example construction publication dot distance blue extend rule rule easily decision high condition fulfil execute make action module intuitively activate important rule module execute rule create assign action rule
potential analysis signal current search important propose anomaly show physics search feasible manner grateful access financial institute physics box classification use high energy fall machine event systematically inaccurate mc dependent search semi additional gaussian recognition anomalous excess misspecification supervise fail correctly anomaly suffer
chemical biological medical goal software derive reconstruction topological descriptor topological descriptor show improve pls input guide selection approach much sample correlate least pls nonlinear limit linear combination largely outperform pls remainder follow introduce pls discuss
simply condition satisfied argue p random measure also absolutely respect lebesgue rotation result hold absolutely apply desire matching scalar precision discuss amount recommend side free evidence nan various evidence obtain plot particularly summary bayes course reject achieve search favorable prior know alternative explain bayes factor harmonic overall bayes specification efficient imputation inner approximate different adaptation liu outer show stick breaking eq label track tie three set n
machine approximation basis pursuit forward forward greedy algorithm perform forward weak eigenvalue due circular requirement require control taylor behave extension occur address first part analyze backward statistical general reader case backward log come strong guarantee objective outside model well counterpart theoretically experimentally condition impose strong graph recovery node
scalability large comparison fix case large accelerate complete ratio os imply reveal use rigorous deal question broad trust minimization scheme factorization gauss superior rank viewpoint trust implementation guess stepsize plot trust region scheme trust scheme stop x trust competitive although superior per iteration suffer scale iteration scalability issue bottleneck rank iterate low issue accelerate additional heuristic lead thresholding project fix technique stop either absolute trust region addition stop max op create recursive relax criterion trust stop increment duality gap suggest competitive rank considerable rank rank increase perform surprising rank effort full moderate however heuristic truncation truncation iterate move compute
goodness compete conditional evidence maximize exponential family optimum posterior theoretic goodness fit frequentist since bootstrappe estimation need thereby evaluate related model nest apply give goodness bic nothing theory incorrect book
work branching represent uncertainty planning horizon limit planning especially exploit water reservoir option price resource also complexity problem schedule paper branching generating scenario need ask scenario tree tree draw scenario branch detection change scenario generation scheme tree inside procedure paper structure procedure good specification amenable fast quickly generate decision large conduct various guarantee potential decision stage impose successful remainder follow explain procedure implementation numerically describe propose report conclude explain say scalar constant distribution problem assume scenario represent determine transition arcs determine path particular obtain multiply arc
way implementation question high efficient sampler improve approximation serve abc computational abc compete computing day despite drawback abc sufficient method principle beyond triplet tuple fall uncertainty helpful feedback calculate bayesian likelihood rely estimate high application risk unlikely free max modeling motivate predict environmental heat impact event natural spatial spatial statistic model design suit focus increase probability high event modeling take maxima temporal block year maxima block environmental maxima location develop several location theory rapidly spatial infinite maxima estimate promising limiting field
I occur soon occur avoid every link rule side repeatedly side detection sense attain quantify delay change apply repeatedly choice mix open plan nan hypothesis composite rule arise multi decide population hypothesis advantage counterpart minimize order suboptimal usually continuous rule main find discrete sequential maximal kullback leibler negligible sequential hypothesis nearly detection proof lemma h well definition time therefore without restrict ct tc c suffice acknowledgement us paper
shall call course toy polynomial feature complicated g mode would relevant statistical property expressive power bayesian integration overfitte conjunction rich perhaps allow parametric restriction difficulty lebesgue derivation respect measure rigorous may issue feature jacobian correction fully determinant product exponential nonparametric shift ignore n important exponential laplacian fall issue multiplicative derivation implicitly dimensionality involve improper however argue space improper
regressor rbf result outline suggest task optimize nonlinearity matrix non significant regressor analysis improvement entire discuss differential equation time series mild behaviour follow stepsize generate training depict last sequence used unit modeling comprise sigmoid matrix randomly assign respectively unit feed constant respective linear feedback uniform include state equation correlate regressor space weight first discard design train ordinary find show compound regressor variance training training fed reservoir cause generate regressor low mse design analyze easy variance drop regressor autoregressive generate training observation probably generate rbf flexible likely autoregressive may performance ht regressor free examine regressor table locally regularize smoothed compare sequence order locally rbf
hide several perhaps common however convenient construction space consist bi sequence course employ time define history convenient adopt finite set word symbol exclude word space bi x primarily interested shift sure convergence probability w follow fact concern word probability throughout mention uv uv primary type state simple focus restrict hidden markov output although generalization set alphabet symbol matrix represent state state normally I xt overall overall symbol label edge depict length reach two symbol label transition matrix state alphabet single parameter operation think random walk associate direct current determine probability hmm output symbol labeling generate matrix interested symbol generate observer hmm direct access symbol state symbol
adaptively learn explore posterior proposal distribution give covariance theoretical motivation choice scale dimension present recursively online recursion mechanism path metropolis hasting framework posterior design algorithm smc develop allow achieve first rao via kalman factor present idea discuss secondly know sir latent latent perform weight adaptive rao particle appendix detail rao sir construct discussion reduce variance importance sampling conditional gaussian kalman filter increase advantage formulation importance likelihood beneficial perform give convolution integral algorithm rao present path acceptance involve development kalman scheme conjunction sir particle decomposition filter filtering seek exploit utilize sir particle resample kalman filter easily sir particle context sir
interact collect consideration precisely ignore fact communication network differently depend amount artificial ability predict remain behavioral phone record link decay advantage fact link dyadic communication suggest bias self foundation fact research social phone share strong connection without necessarily phone mention tie behavioral observational really produce observational behavioral interaction produce dynamical pattern formation decay social structural property already validity cell phone interaction accurately predict measure traditional close one cell phone communication study rely yet usage email cell phone display basic small community connectivity paper insight bring mostly due fact binary weighted supervise science social discover scope million million communication traditional regression technique tackle dyadic structural link without incorporate assumption form homogeneity size remainder organize briefly social connect identify factor decay largely literature computer tool hand describe basic distributional feature choose task decay section analyze
iteration least solution ti ta let furthermore last eq next subset equation rip furthermore q fact j md optima although fast optima pursuit pursuit corollary paper compress goal recover propose hard thresholding lead iterative extreme like htp novel pursuit like add exactly residual unlike omp remove simple prove term restrict isometry property omp rip locality hashing provably proof technique novel algorithm pursuit provide experimental provide vector demonstrate learn greedy call pursuit add large gradient method guarantee omp isometry popular locality provably connection thresholding recently popular extend
residual prediction filter steady application impulse system divide sparse whose impulse sparse system impulse response include acoustic wireless channel compressive sensing attract sparse recognize signal communication propose estimation work group homotopy lasso ii recursive manner
ever quantile rescale accepted category latter acceptance abc color contour ep posterior across diameter run ep abc exhibit extremely poor triangle posterior computed stress later scale end value several million auto case show fig contour ep ep abc hope fairly since acceptance enough adequate obtain varie range truncate dimension locate variance third ep arguably rather poorly without help produce gaussian already may propagate wishart matrix consider accommodate treat constraint ep abc simulate ep simulated model maximum alpha distribute innovation incorporate certain easy discusse illustrate composite technique intractable review sake
denote b solution depend solution must small local discard two eq condition solution unique see decrease point variational eqs fix require efficient new lagrange multiplier iw iw I remain denote similarity recently pair pmf variational pmf variational consider govern relate present input layer factorize specify relate make thus model paper pmf distribution place contrary factorize approach restriction posterior delta function optimization find search solution forward hyperparameter validation initialize warm dm b I decrease sign linear ignore eps
exist neighborhood divergence follow unique upon define divergence maximum belong obtain mind get formula instrumental divergence type element dual divergence divergence normal location p n x
friend friend comment link produce united site american english produce united site english analysis uk additionally user web interface facebook mobile character comment q character country mode index user share share comment character uk american english include examine uk post comment country regard estimate combination mode write comment user long comment mobile interface user difference search cause datum analyst likely want hypothesis comment length mode software base compute four mean level seed generator compute comment could factor different four comparison analysis difference factor inspection confirm attribute chance account random factor confidence quantile variance
conduct line combine nevertheless preprocesse issue preprocesse incur cost process many cross validation set truly large dataset cost preprocesse magnitude loading gpu preprocessing easily load hashing replace need mapping hash practice bit hashing inherently discuss training unique interestingly difficult researcher obtain truly dataset experiment hash gb may overcome gb format still
stimulus stimulus numerically onto grey stimulus density stimulus front consecutive stimulus strict sense process probability stationarity ergodicity realization characteristic ergodic show forget learn multiply integrate consider average prove ergodic ergodic replace brevity integral convolution dirac minus
variability base software thank provide thank stein anonymous helpful early article acknowledge cancer uk supplement explanation study apply fellowship grant grant identify individually affect involve binary property suit differ place implement search define group predictor jointly influence testing obeys violate offer introduction study drive genome ever detail aim detect genomic variant link detect understanding variant effect pick fairly readily subtle association obvious view regression predictor either association predictor absence mutation variant neutral association abundance examine currently available characterized influence fit possibility affect specify examine
player strategy advantage good exploit whereas large play play player select response action rule strategy use belief strategy treat environment game stationary implicitly observation poor play treat weight play opponent probability play formula section automatically adapt opponent player belief stream available class fitting stream fix multinomial assume frequency datum describe real stream adapt player strategy simultaneously widely impact old window jump segment window form approach window complicated window method another introduce geometric discount old recent distribution evolve advance result drift
use directly bethe usually analytically contrast use move location bethe energy pseudo moment overcomplete representation convenient express affine rectangular gradient bethe eq zero linear free place marginal realize converse whose exist poor sometimes surprising claim contrary matching reach fix point target belief set belief parameter belief propagation marginal sufficient bethe local hessian bethe hessian subspace consistency hessian
violate hyperplane separate via check constraint automatically separate check algorithm origin run positive aim hyperplane put policy candidate hyperplane wrong side hyperplane find optimization negative direction hyperplane policy perceptron number collect policy perceptron rewrite xx u constraint ellipsoid program separation oracle separate hyperplane get complete carefully iterative call infeasible policy reward delay choose present reward action delay modify incorporate delay tx w ta tx tr delay unchanged thus replace acknowledgement several reference value variable
challenge give rise class precisely interested risk infimum hausdorff equation find hausdorff use metric assess beyond sometimes differ phenomenon deal hausdorff distance sup eliminate additive manifold relate deconvolution problem support noise deal purpose field geometry paper draw way quite exception estimator define ball choose share certain topological closeness
amongst actually include follow verify order l l c separation able thus considerably mean great meaning present individual aggregate population determine nature heterogeneity base point uniformity range entire conclude interpretation population heterogeneity support diverse temporal due aggregation population bar time hour great mean aggregated metric uniformity reasonably represent confirm concluding homogeneous present population population l l quantitie source datum eight heterogeneity see magnitude argue interpretation homogeneity study well pdfs respective could careful population constant fig kde confirm lack var observe quantifie support diversity variation justify patient bin measure homogeneity support correct diverse less pdfs point homogeneous totally figure confirm visually minima integrated essentially see pdfs see relatively small homogeneity independent detail diverse information make overall finally homogeneous small table really interpret uniformity support range give imply population diverse move include appear great apparent contradiction imply somewhat homogeneous calculated diversity diversity section confirm patient patient ten bin conclude population support temporal aggregation exist individual population diversity reflect represent represent finally patient patient estimate scale represent population filter relatively essentially essentially individual proportional contribution aggregate heterogeneity homogeneity calculation difference particular substantially information include patient heterogeneity population estimate member overall sample mass pdfs bin filter
select message length notion empirical bernoulli plus message mass computable bernoulli variable description kolmogorov normalize finite distance two new notion explain former individual kolmogorov object former finite receiver review kolmogorov string finite string alphabet string denote string letter string identify emphasis convenience alphabet encode theory neutral alphabet cardinality encode bit length
complex signal corrupt independent complex mr imaging visualization sample magnitude remain contrary noise mean variance magnitude consequently design magnitude mr gain attention past strategy distinguished treat square mr chi freedom proportional noise magnitude appeal aspect noise rather enable follow maximum likelihood locally field make standard effective follow filter adaptation filter develop effective approach distance magnitude mr relate transform significant reduce sharp mr denoise account subsequently unbiased coefficient square coefficient exhibit signal easily thresholde transform
large I decrease tw get bind notice w w h reduce applicable use simultaneously actually probability fix small refined deal integer assume cover w dimensional w pass point whether grid upper combining lipschitz scale get fourth unlike w lemma combining lipschitz bound simultaneously tell cover class cover
satisfy unknown modal hypothesis algorithm confidence tool suffice learn fundamental fact kolmogorov distance asymptotically support kp vc independent drawn follow mi mp vc also follow subset distance say close increase resp ready state agnostic learner hypothesis aforementione domain interval hypothesis obtain tv stress monotone perfectly agnostic easily result agnostic non final distribution one routine place know whether hypothesis another hypothesis testing accuracy hypothesis generate hypothesis routine sort draw hypothesis hypothesis sake algorithm modal distribution result property subroutine ignore mass use learn matter identify bad mass interval ignore use take kind deduce bin get probably ok give sample essentially logarithmic factor output theorem confidence partitioning consecutive disjoint empirical roughly care need support mass may error roughly global error across
principal try analyst quantify promise central definition quantification scalar seem fairly quantification first exploratory correctly central answer either justify interaction several issue subsequently appear literature arguably dominant thought projection pursuit school certain structure interesting design preferred direction maximize away central school argue search expect view literature development methodology making point unimodal modal fact development pursuit non index note index similar normality development serve demonstrate realize nothing normality exploratory pursuit success really multivariate normality simply
transform reward game uniformly locally lipschitz choose carry reward map continuous bound bind take bound map local reward smoothness total maker strategy much bad choice function measure implement exp behaviour maker performance sequentially interact budget belief period range include range market maker market market start asset behaviour belief note knowledge price interact market share oppose infinite case market bandit maker exp
disjoint also generate overlap index among use vertex represent block connect block dependent vertex correspond instance additive various linearity consequently include graph special graphical correlation establish graphical matrix construct specific link large cluster explanatory form structure corresponding focus assume grouping time invariant study belong unknown function unknown spatial another modern robust noise structure meet sparse eigenvalue modern however hard different grouping proper grouping medical care service item item less medical price mat material implicit price consumption service fourth group grouping close actual put service service grouping drive lie situation operator sparsity quantify rate novel context
kernel propose svms rkhs parameter quantity describe paper interesting area volatility finance consider concern range chemical logarithm ratio receive source single input reflect detail show leave svms look black middle value
achieve optimal estimator allow increase generate consideration hold impose remove optimal kl true estimate go denote likelihood allow assumption increase glm whose true derive I combine many grow input target smoother generally encourage expert ii study increase derivative polynomial pay bad optimal evidence upper derive polynomial fix rough statement true minimize ii choice vary rate constant proposition achieve parameter smoothness smooth target
matter happen term favorable accuracy investigate artificial numerically estimator tend perform sample compare theory variance pe parametric parametric relative exist let come standard book express upper let denote asymptotic normality book depend ratio affect therefore occur pe note superior pe divergence imply overfitte occur relative ratio pe eq holds bound monotonically approach pe advantageous plain pe divergence mm define express ratio thus include density minimized deviation model significantly overfitte long include produce experimentally evaluate homogeneity task propose estimator two homogeneity test sample p homogeneity different homogeneity two test
toward divide interval average policy g see latter action play rarely update rarely might convergence possible normalize obtain equation describe express strategy toward rescale set term increase tendency profile exist show emphasize system energy temperature maximize maximize entropy relative recently free principle suggest review learn receive joint generalization idea two reward select second action agent yield term bi equation examine proceeding elaborate rest theoretic equilibrium ne ne increase equilibrium ne
xshift xshift right pre edge pre xshift bend angle auto edge cm edge yshift pre edge edge leave si leave pre xshift node distance cm bend angle auto node pre node right pre node yshift var node var u cm leave right u distance si pre xshift pre grey red link maximum capacity link infinite capacity j group structured hierarchy flow literature solve projection ball community sum sum problem depth overlapping show efficient exist generic exploit cost significantly well practice similarity network simplify equivalence priori proximal operator regularization explain section compute flow cm step jj uv tv e j u solve weak optimal cost flow capacity constraint arc flow exist algorithm return cost reach flow bind achievable cut part exist detail optimal cost arc remove optimization call correctness far appendix min max flow experimentally essentially max despite case guarantee weak adapt graph complexity
practice although effort improve model gaussian study wang li regularizer large microarray study parallel sphere et attempt limitation evy describe utilize optimizer train constrain effectively broad parallel performance term speed execution open optimization rely model sample must know flexible yield reliable enough curse amount spatial unless simple obviously satisfied practice grow fast performance try population large population selection pressure maintain course size level univariate solve estimate good necessarily grow validate dimension complex considerable since gaussian curse dimensionality usually model freedom histogram freedom however notice previous unimodal multimodal limitation curse specifically suppose population move search guarantee variance although develop likelihood improve effectiveness high search still validation curse restrict application dimensional exclude fitness building cost however multivariate rapidly increase step whose fitness evaluation consume multivariate runtime concentrate analytical data complexity last generation individual complexity please l model share differ parameter share computational via matrix factorization factorization likelihood estimation step computational complexity cubic whereas graphical factorization relevant algorithm state idea variance cost complexity say idea different manner analytical calculation difficult offer analytical computational consider fact representative sufficient solving grow univariate overall grow multivariate fast normal idea practice overall multivariate thus experimental comparison cpu nature search computational population univariate find complexity speak gaussian effectively criterion advantage
shrinkage allow conjugate lead gain onto model flexible prior shrinkage obtain scale variable representation tail shift strong shrinkage density neighborhood tail equivalence three hierarchical mention early ab equivalence work mixture lead flexibility shrinkage make worth prior take mix identical ng first fixing mix global act hierarchy choose mix
efficiently range encode arithmetic decoder character arithmetic tell character character arithmetic know character point arithmetic decoder need typically character know reach technique store additional along file length although arithmetic achieve compression factor prevent file disk overhead storing limitation prevent overhead arithmetic string ccccc ccccc cccc cp ab ac da algorithm consistently compression create character alphabet character character string represent character character character likely occur character assign predict character entire one history input occur character character immediately match history character character short one match less occur chance create entirely context match various context partially responsible implementation technique model count context character low model character bayesian nonparametric model share similarity implementation compression good way compression performance file however metric use
zhang project matrix apply mathematics pp zhang l z low analysis zhang z nuclear problem mc rank great science especially publication al nuclear low suitable strongly convex rank recovery sufficient lead recovery rapidly recovery rank approximately occur science range machine vision attention publication recovery nuclear matrix low fundamental impact engineering recently network keyword low capture recovery strongly convex exact guide choose namely completion
curve correspond transformation support visualization possibility extend optimal design another design constraint design design convex origin ray candidate difficulty approach difficulty constrain problem acknowledgement author
handle instance hz rational cover regret rate mirror method majority analyze purely stability underlie batch pick introduce notion online stability relate derive setting number open question sufficient rate guarantee show counter otherwise example learnable exist loo verify online potentially loo exponential learnable randomized loo stable open covering existence loo stable online section institute university stability quantify output small characterize convergence necessary sufficient necessary stability online hypothesis good stability stability uniform set online popular learner category mirror majority guarantee property example suggest might set sufficient class learnable learn sub exponential online stability exist observe sequence pick hypothesis incur loss next functional mh goal
compete conclude bold letter number upper letter stand write stack successively triangular express must coherent upper vector frobenius dot b tb fact histogram practical application particular case scalar histogram represent column vector canonical row column research contingency two margin histogram define real matrix program positive homogeneous matrix point convex highlight parameterize notation mmd mr two argument distance whenever distance name variation wasserstein recently extend un histogram histogram carry whole onto extension later
take decision statistic say say separation blind decision kind exist indeed critical test sensitive grow critical quantitative regular uniform rotation theory chapter kind invariance therein invariant rotation detect deviation process preferable completely ignore mask must localize wavelet alternative spherical spirit wavelet angular spectrum noise although purpose ask kind regularity procedure regularity sphere coefficient supplement precise construct approximation meaning compare function q modification characterization wavelet xu second distance sake see ray ray nearest rely norm require smoothness framework type efficiently handle test adapt usual wavelet never recently tight frame orthonormal produce enjoy fouri obviously space sphere basis basis spherical lead invariant wavelet define analogy sphere extremely diverse inverse especially full context region happen sphere supplement empirical coefficient jk f simulation dyadic lead concentrated wavelet choose profile represent available supplement
appropriate dataset split amongst appropriate input dataset overfitte version suited learning problem seem fact lar learn category example feature classify classify classify exactly un qualitative left histogram decision illustrate ability detect svm lar baseline mainly show adapt behaviour difficulty classify confirm histogram decision input always classify input seem identify good behaviour direction concern acquisition feature main filter embed involve classifier classifier
consider response error denote distributional cumulative two regularity I vector vector contribute significantly predict response situation arise study cancer consider main seminal regard nuisance situation inference shrink coefficient full towards restrict utilize nuisance covariate plausible generality restriction constant follow shrinkage life present positive shrinkage validation describe section square define eq degree stein sign could unity view
prune gram level frame compatible sense invariant permutation correspond column take choose decomposition variant attempt find second variant decomposition hadamard output remove transform attempt tree example complementary properly proof rational quadratic form theorem long history design projective candidate gram matrix block rational rational signature agree criterion rational signature divide signature follow fail gram sometimes backtrack existence pair back tracking explore decomposition fast search
mechanism split propose scalability parallelization diversity use interact year create proposal induce range sampler development parallelization constrain chain generate distribution iteration move chain emphasize chain entirely proportion update proportion therefore indicator mcmc law similar employ approximation chain costly particularly graphic iteration single chain distribute iteration tradeoff chain multiple processing consequently collect update processor graphic wang available additionally multiple processor benefit hence flat enough chain explore mode method reaction chain provide reaction move potentially reaction mechanism space proxy movement acceptance acceptance make move rate move adaptively encourage recommend although find example test stochastic standard rate accept move particle walk chain history covariance structure target proposal help space mode yet chance eventually act standard deviation large chain computation storage history chain recurrence formulae compute covariance justify certain
log respect define let large sample exist eq hellinger study asymptotic smooth I large dedicated position conditional extreme observation slice c conditional slice variance remain stable extreme gaussian behavior statistic slice eq explicitly sake situation distribution drive opposite
object distance correspondence space quantitative rank score read agglomerative hierarchical greedy structure merge view output construct fine trivial object binary child node terminal node commonly refer dendrogram ultrametric topology ultrametric define strong topology compare geometry ultrametric symmetry though note triangular inequality ultrametric triangular ultrametric discussion linkage hierarchical terminology subgraph threshold linkage single linkage arise dissimilarity member hierarchical characterize linkage dissimilarity hold cluster employ contain dendrogram inclusion relationship partition early work field first major context dendrogram sufficiently common interpretation hierarchical type interpretation interest level approach semantic attribute approach significant hierarchy apply dendrogram wavelet ultrametric sense
signal denoise choose eq necessarily derivative nm n q def problem lagrange read lagrange multipli optimality reach two substitute dd get sufficient condition number optimizer penalize square also condition gram matrix connect general positive
optimal next empty target suffice size notice achievable q ensure step new optimal finite length end polynomially norm compete finite different rate optimum loss suppose fail constant adaboost respect loss convergence rate first convergence achieve round show polynomial minimum solution upper rate adaboost near kind derive decomposition intrinsic value feature rate doubly example rather training table c reference bound om norm reference parameter study national foundation grant helpful discussion appendix optimum table adaboost step bound round get loss round add begin edge round generality pick c pattern claim round fact strong claim recurrence prove claim verify round edge correctly incorrectly round update add
justify estimation concentrate observation previously paper aware formal proof lemma positive denote prove step back last equality isometry equality last fact hold actually unique next mean mean rather g identifiable unless additional constraint go denote lebesgue first compute distance minimize lemma asymptotically signal align population I cg c ccccc estimate illustration I generate sequence panel align function fourth finally examine error compute estimate panel converge size grow domain specifically focus alignment dataset berkeley berkeley growth subject case use growth leave top leave
microsoft microsoft com kernel integrate adopt transform efficiently scale massive extremely dimensional especially compare hash cm sketch projection empirical usually bit storage bit hash far especially internet inherently store context translation corpus practice long capacity example potentially unique architecture consequence emphasis learn technique rely solely parallelism expensive requirement often induce additional approach challenge pose leverage area increase make storage prohibitive representation compact
rank modify subtle leave implementation element eq suffice qr norm run comment regard first role randomize algorithmic leverage score column clearly unless diagonal span relative leverage cross qualitatively approximation construct randomize euclidean sketch approximations cross see theorem provide parameter statistical leverage section return least q relative assume treat running time score coherence rank ji ij c ij score satisfy within treat overall running application scale generally either preprocesse develop problem roughly hand score discussion decomposition compute importance computational probability bottleneck leverage depend fast sampling approximation problem numerical randomized algorithm traditional implementation generally quantification case algebra implication
bind statement observe assumption imply measurable apply jensen assumption see conclusion conditional summation entire define noting otherwise expectation martingale sum proof note complete common inequality see ct see equal hand berkeley support aa support microsoft fellowship google fellowship support laboratory office contract nf descent situation optimize long suitably quickly distribution enjoy strong implication optimization dependent stochastic paper analyze solve statement collection close contain statistical convex wide explore literature procedure possible receive instead index relaxation circumstance know receive may sample efficiently combinatorial design converge computational access source randomness study effect assume distribution point incremental apply problem objective access certainly control stochastic statistic study book nonetheless
evolutionary time objective view set attract significant recent extension mean expand propose cluster quality preserve cluster framework static spectral association normalize cut framework smoothness cluster similarity cluster choice temporal membership simply optimization example association q adjacency cluster membership find discuss approach share similarity pair association history work cluster attempt modify cost snapshot none aforementione address determine ad hoc manner accord preference temporal beneficial allow choose adaptively check test satisfy evolutionary preference temporal undesirable exist evolutionary agglomerative impulse sum filter calculate filter output affect adaptively dissimilarity cluster centroid time accommodate cluster deal adapt finally evolutionary addition aforementione involve mixture method semi evolution treat evolutionary problem follow already convert proximity proximity matrix denote realization non stationary discrete assume evolutionary algorithm identity row add describe propose
setup hessian available limited easy imagine comprehensive study nucleotide covariate might additional insight nearly ideal burden interestingly generic start variety I optimizer fail converge similar curvature likelihood em advantage analytic derivative surrogate numerical death flexible derive discrete hope sophisticated realistic death apply researcher estimate parameter structure currently available maintain publication grateful comment grant gm gm nsf dms california usa edu usa edu california usa edu birth death process continuous track particle biology particle death limited particle death times augmentation find algorithm close form remarkably conditional express computable probability arbitrary representation transform probability efficient
cl cl thus much day minute opposite represent sample rate day cumulative minute intermediate consumption distinguish occur cluster represent sample cumulative minute proof light everywhere four adapt gx ng converge surely check condition fulfil absolutely surely consequently gb ga get kn r n deal one get eq h instead anonymous valuable suggestion company allowing illustrate
potential theorem follow strict intersection quadratic ignore l qp decompose k kp qp z kp h tp kt kp kt qp recall decrease first invariant small see game possible assume positive say inequality follow receive major university sc degree mathematic ph economics department currently dr research interest lie system game university universit paris paris france post flexible sup university universit de france sciences fellowship candidate science receive degree ph matter physics physical division wireless technology laboratory currently physics associate system game theory statistical theory communication physics sc
usually one security credible indeed set credible credible resource contingency security frame address rapidly could analyze computational resource output main point significant turn distribution generate draw strongly sometimes engineer possible identify engineering distribution however try draw contingency instant base security correspond describe believe web attention expert random sequential expert pick
capacity tv denoise propose generalize tv multidimensional sum euclidean increment see reduce tv intuitively increment variation case imply increment profile piecewise profile share point natural classical investigate affect penalization differently position uniform unweighted suffer position choice group sharing profile additive profile length though leave weight signal change capture formulation profile lead error amplitude apparent profile dimensional tv denoise reformulate change reformulate convenient variable ii rewrite matrix attain jump center specific design group solve purpose solver want reach make sparse memory number select statistical criterion however none rely affine change section respectively suggest group efficiently place perform operation repeatedly implementation follow group optimize turn group converge report amount n
appropriate operator choice example reduce classic convex detail proximity call unconstraine major penalize constrain analyze convergence generalize differentiable begin match behave main must term bound increment compute helpful ease notation extend last z z r j j last tx
eq degree freedom property limitation deal sparse situation often small setting expression mutual test condition statistic nan compute detailed compute denote time usually observation present conditioning table configuration test permutation advantage permutation large assumption condition parametric
define infimum keep mind assume without generality make use consistency confidence define fact vector recall w quantile theorem along consider ensures far prove type set percentile summarize condition bootstrap routine bootstrap problem entire confidence detail well coverage limit leave open keep good empirical criterion nh independently divergence agreement show divergence regular exponential satisfied without parameter crucial presence criterion generate
many site site site depend note positivity concept neighbor suppose pi study minimal first case site
z z p take concatenation classifier base derivation omp capable recover even sequentially convergence almost termination stop criterion compute p form namely four proceed need replace traditional hinge hinge solution obviously algorithm quadratic impact determine weight influence cost impact previous classifier determine relationship hinge classifier high yield positively classifier imply correct misclassifie make representation classify
exp sparsity ability gaussian use high testing log difference use easy log likelihood validation graphical outperform generalization model tend moderate explain tend predict explain currently generalization latent variable expression encourage know gene interaction high wide observable marginal concentration decompose concentration matrix reveal model
zero assume estimator available ic time need statement kkt adapt refer kind regularization oppose realization exist state ic introduce q invertible inside provide bound eq inequality paper select subset sign consistency oracle asymptotic ols straightforward carry traditional actually grow polynomially sample selection consistency tell relaxed tail hold stationary tend oracle weight
continuous measure posterior consideration use work benchmark represent specify discussion moderate simple cg wide setting cg uniqueness quadratic restrict space restrict space restriction moderate principle derive frequentist statistical inference organization example bayesian confidence posterior represent frequentist represent pre modify procedure encode confidence moderate incorporate frequentist see provide undesirable inference undesirable little plausible posterior framework remark acknowledgment innovation innovation frequentist david institute
two mixture component normal mix leave histogram right histogram datum exercise break specification informative si mix mix correspond setup amount information exchange marginal posterior empirical evidence si exchange si mix mix mix concentrate si present previous raw chain burn period discard solid distribution gibbs set ergodic setting mix mix second mix h production business cycle great advance allow period regime seminal dynamic observation expansion phase business successfully estimation regime cycle verify contraction use high number business cite selection regime joint number regime multiple dirichlet vector autoregressive international economic united european
nonlinear high space carry truncation reduction suggest combine representation capture characteristic pair simulated comparison show propose nonlinear machine learn offer dynamical many estimate reduction unstable preserve stochastically perturb find remark sketch remark l department mathematics department college usa ac uk control draw progress statistical reduction method behave infinite dimensional balanced truncation carry implicitly nonlinear belong reproduce hilbert capture output approach control dynamical long benefit controller simulate cost simple important might light underlie linear system detail model proceed dimension system preserve essential directly balance system considerably
second one global decrease root stationary first generality transformation problem transformation invertible give vice versa therefore consider rational disjoint subset equal number instance x j j equality always equality apply equal
still effect identifiable condition disjoint relation x back graphical check know conceptually ordinary author like corollary model signal acyclic dag one occur probability characterization inference focus formalism particular notion intervention kernel direct identifiability passive back thought causality recently attract attention signal process researcher state imply relationship theoretic conditional
geodesic pair classify social interaction formation node leave component synthetic graph observe discover center generate extend treat set dp parameter set control concentration package figure depict summary box display degeneracy relate empty complete graph sign degeneracy show sign degeneracy vector lie relative hull generation realistic family extend right initial htb cccc observe line box plot include variance network degeneracy htb cccc observe indicate box
arm player player lead upper exploitation spend exploitation epoch fact exploitation epoch combine regret get reward cause mistake lead upper b theorem imply part regret cause epoch cause mistake exploitation epoch lower arm regret cause play arm epoch cause exploitation spend bound dt lt I thus
bayesian paths lda illustrate constrain dictionary learn topic coefficient manually expect conference section induce efficient consider function many sum additional interaction order make express potentially subset main many difficulty vector reproduce hilbert space complexity address impose structure among see reproduce space arguably since learn belong necessarily predictor reproduce hilbert space predictor parameterize consider state function admit f expression eq whose element k long assume space look j j j norm turn solution detail mkl define set pm desire functional limitation complex expansion functional expansion different subset theory function sparse predictor feasible theoretically
setup detail associate two come mind fit experiment experiment share wish briefly outline algorithm iteratively vector km classical iteratively centroid replace centroid computation thresholding formulation rank regression correspond constraint adapt multiple response regression modify speak neighborhood worth collaborative scalable motivated local identify estimate regression report well complexity mean ce method near exploit estimate coincide size method individual collective
digit share digit common ultrametric also ht digit first digit digit ultrametric produce classification see measurement say speak level level node course potential empty observation number precision point big storage give fall advantage viewpoint store fast easy
learn removed turn disease rank remove list disease purpose disease I gene curve rank mkl hide positive confirm level mkl variant illustrate learn formulation variant roughly one association mkl set assigning source try mkl mkl consuming mean decide restrict experiment refer kernel mkl heterogeneous information information sharing disease run experiment assess allow disease test disease weight across disease disease variant additionally control disease description disease differ limit disease association dataset association gene variant share disease propagation cdf compare average disease four fact outperform confirm knowledge disease phenotype weight sharing across disease fact outperform surprising illustrate fully description beginning method low rank generic practice available confirm approach comparison fair improvement run picture
lemma useful regularize satisfy exponential bridge kernel kernel verify lebesgue nonzero almost everywhere satisfy condition compact belong lebesgue argument similar manner brownian satisfie clearly brownian
implication dimensionality give algorithm construct kk opt cluster indicator correspond row opt matrix eq orthogonality opt opt drop eq combine conclude opt f opt k kk norm kk slightly give nm jj yy independent property strong version work introduce turn prove svd value clear matrix k svd certain algorithms frobenius norm svd approximation find target denote compute almost rank k give take input whose denote inverse follow projection random use event hold respective probability hold union event let r sampling critical replacement rt ip tt tt randomize sampling take sampling three lemma
show reproduce competition phase pre competition phase forecasting available different forecasting scheme training validation set partition three exclusive figure day testing figure use tune select neighbor perform performance suggest origin denote ahead forecast origin forecast day day day forecast three point forecasting ahead criterion I learn test competition generate forecast competition advantage design make pre competition model final competition b tune present discuss strategy forecasting criterion section provide average rank nan state equivalent hoc present strategy forecast configuration selection amount computation time hoc error htp post availability make pre competition sake consider move exponential smoothing nn avg exp sm configuration present hoc summarize forecasting configuration round rank htp
cover x class p md sufficient show md ip slope bx bx bx f show adapt collection eq functions bound slope bx bx bx e bx b bx b bx x bx bx produce cost bb mn mn b b cover function b bf mn x b b bb argument volume hyperplane origin complement ball ball hyperplane spherical hyperplane volume volume ball let half parameterized let spherical denote measure complement incomplete reference relationship pack every use lemma cover number b b x bx b argument part volume complement spherical minimal boundary maximal packing b follow h yield
perfect arbitrarily attempt able like obstacle project version switch optimality f show optimality decrease approach restrict true online effectively difficult online scope projection convergence guarantee obtain schedule subgradient project direction issue boost gradient single hypothesis small step learner error convergence functional restriction iteration rely describe restriction ff f gradient add dependent projection weak threshold train learner schedule
cost thresholded order magnitude furthermore graph amenable parallel separate optimization problem sub number may impossible operate distribute nested suffice operate independently path distribute architecture operate graphical small connect large split exploit induce thresholded covariance graph induce thresholded permutation appear decomposition graph labeling appear structure match every v solution component nest increase regularization within component formally address
integration seminal framework reasoning recall different base together ability bring knowledge adaptation address requirement reasoning possess active basis company institute technology mit prominent lexical maintain cognitive science laboratory university relation mit discover middle involve facilitate reasoning thereby illustrate reasoning briefly essence seminal resource point reveal want resource entity dash
entry reveal still vanish regret seem general loss outcome never one obtain batch arbitrary appear use assumption forecaster outline reader derivation forecaster actually refer prediction first forecaster minimize bad eq work start round deriving outcome reveal tp plug get q show repeat minimax recursively recursive exponentially expansion sequence bernoulli probability plug also constitute forecaster note equal definition
report ratio md dr k adjacency mean call local result sampler threshold identical local parent domain meaningful alternative pathway pathway contain table experimental cell pathway pass indirect mechanism exist widely indeed report enhance although unlikely may enhance dr landscape domain insight dominate global mode log md sampler reach mode contrary detect much negligible demonstrate md sampler burn mode mode dominant mode produce cc ten fold partition size nine calculate point subset learn repeat time dataset threshold performance md imply robustness tp md method compare panel average probability md sampler ratio utilize calculate regard use network store dr variability domain method probability calculate probability calculate md application new experimental datum nine validation cell cell one sampler dataset edge result ten experimental uncertainty determine
adversary response branch future precede e two construction tangent equivalence path allow term good rademacher therefore generalization rademacher handle range elsewhere machine know rademacher complexity furthermore rademacher class argument application determination
sequence function appropriately state give give priori number state technical gradient f generate update term tx claim sequence satisfy straightforward recall follow inequality jensen f claim consequently f f apply lemma imply turn simply probabilistic bound key take expectation randomness time lemma thus side eq fact g fix replace see remainder achieve probability rate term consequently notation consequently probability equip recall eq q lemma apply remain control complete non optimization develop provably oracle model knowledge technique smooth think least question remain necessary achieve convergence question smoothing whether corollary answer smoothing tight outline technique give provable method experiment show qualitatively develop stochastic anonymous feedback comment science engineering fellowship program support award dms
short bind site well tf outperform tf high median ic tf less utilize example tend tf high centroid factor bind fig ic ic centroid compare relatively ic property reveal content tf tf contribute median tf high median ic identify tf individual site tf bind site two scoring matrix use score assign search superior searching single cross validate couple centroid identification different similar embed bind clustered centroid centroid couple score denote embed centroid centroid binding assess centroid centroid counterpart without leave validation set tf nucleotide ic indicate weight nucleotide content impact centroid introduce centroid
conditional standard chain secondly markov produce briefly problem compose datum range per yet treat discrete variable interesting problem specifically logarithm correct enhance interaction covariate add explain construction dependency predictor multi modal short name business unit build weight distance radial full value proportion status second treat uci repository concrete strength strength compressive covariate add covariate table augment predictor explanation water aggregate aggregate age day analyze implement explain protein activity enhance interaction covariate detail predictor reason prior name ph buffer protein temperature want include interaction incorporate challenge exploration main prior feasible carlo implementation environment suppose fair sequential monte carlo stop chain carlo speed make comparison sequential carlo scheme markov effort evaluation hand however algorithm move analogue speed context chain detail report
converge therefore requirement fulfil brownian second exponential ern check computation xx xx remark gaussian consequently happen satisfied aim requirement accommodate kernel grateful anonymous approach let let validity examine role linear theorem coefficient observe positive z j form variable probability minimizer general predictor usually measure function respect precisely expect large approximation zero fast increase quantity error respectively satisfie immediately estimate start stick theorem replace relaxed point simplicity fix error accordingly factor keep advantage therefore competitive input similar result represent regularity decay still hope satisfactory
formula mode equality mode fix say median formula mode approximate
give situation incomplete subspace upper coherence vector j tu without observe wish chernoff size draw replacement second similarly p u unique implie use along j x union least denote column complete least residual subspace u allow entire since may prefer level simultaneously prove rank completion completion examine match draw span subspace incoherent generate span procedure seed software procedure implement completion result message provide rank complete term simulation
angle fan wu sis manual fan li journal american transaction pattern machine fu penalize regression bridge versus journal graphical p natural efficient inference thesis california berkeley asymptotic b monotone package manual adaptive penalization technical gamma prior lasso manual fu asymptotic statistics lee generalize biology bayesian variable journal american statistical partially adaptive american inference extreme globally act statistic gibbs sampler journal american e minimax mean r shrinkage machine machine learn r tune absolute mixture lin estimation model journal american association yu model journal
process invert reduce mostly learn successfully method lie stochastic representation obtain knot law approximation location go complex process model rich become understand difficulty approximation knot determine accuracy concept knot coverage intuitive need emphasis metric geometry help knot base knot offer face severe difficulty automatically geometry cause covariance enable geometry call predictive predictive add two knot candidate dynamic determine knot connection predictive approximation cholesky factorization well view
use generic eigenvalue hold enough complete central write prove tend divide integration part integral substituting since integral arbitrarily suffice integrable sufficiently consideration
krige use smooth classification importance derive meta converge impractical importance estimator refinement stop progress require factor acknowledgement grant engineering financial also importance simulation rare pt universit assessment system failure numerical probabilistic failure define event failure define follow introduce equal turn carlo read asymptotically unbiased variance
inner lower derive approximation packing number explicitly give pack local approximation term continuity know exist upper bind uniform h hence lemma consider topic distribution major depend shape back primary article regression idea might beyond parametrization invariant utility nonlinear clinical finding real nonlinear medical biological instance prior distribution remain external quantify example absence jeffreys variant pi
lag time lag influence influence nothing else effect quite lag fewer strong except play advance dynamic flow among price price generation source apply wind water reservoir estimate causal price relationship price well price water reservoir wind power similarity difference peak price peak peak due flexible innovation response causal link price market stand alone exchange rate price stand alone stand european may case price price adjust
involve bilinear general nonconvex bilinear np global particular program tractable show stochastic controller address discount stochastic blind controller incur sn via bellman denote np undirected loop restrict
methodology criterion analog rf incorporate empirically cf compare base cluster support cf manner formula mis perturbation insight relevant remainder paper organize detailed description relate spectral section mixture well spectral benchmark comprise phase one discuss cluster phase aggregate problem instance base feature vary vector govern quality cluster cluster square feature currently use denote iteratively expand consecutive attempt expand cluster initialize competition f bf datum space project get f setting reduce mode growth competition procedure feature competition motivated prevent noisy
xy mx x concatenation common represent string string pt minimum height input input let observation reward illustrate diagram receive new observation another goal agent discount reward consider environment action reward deterministic xy observation pair implicitly computable previous leave dependence action policy history action interact play tuple play kx define infinite reward discount geometric reason consistent discount discount horizon human grow old existence depend discount discount discount discount purpose discount prevent discount
everywhere query al adaptively pick sequence unlike accuracy requirement lower introduce realistic variation price setting agent interested establish almost tight upper low bind within lower information sample bind function learn show approximated polynomial conceptually highlight learn polynomial tree would immediate care novel result per number also low apply derive new low introduce observe agent agent interact interestingly query preserve theoretic function price query paper paper tree paper paper bind learn construction natural construct economic setting sum allocation focus set approximate learning paradigm submodular main class recent paradigm necessarily limit less class read toolbox polynomial specify price prefer bundle price contrast price bundle class distributional
qx jj x x jj introduce formulation usefulness classical formulation try dictionary follow norm control amount prevent arbitrarily belong norm become unconstrained formulation formulation remove replace solution normalize knowledge equivalent empirically behave prevent degenerate formulation constrain dictionary original equivalent unconstrained size
predict goal jacobian average execution time scale computation time estimate remarkably overhead especially evaluation estimation surprising twice score evaluation arise order author internal conduct computer yield result half time computation allow per local global optimization step depend evaluation per step number without new identity
need mechanism efficient physics machine probability carlo algorithms parameter tune often domain expert consuming manual adjust algorithm reader recent excellent review field various adaptive firstly theoretically chain update history markov ergodicity ergodicity sampler ergodic level vanish asymptotically space adaptive approximation practice limitation stochastic approximation approach rely know optimal disadvantage
evenly space simulate plot datum blue red expect present posterior focus exchangeable infer responsible parameter component hereafter early posterior analytical marginal develop draw challenging aspect countable develop cope way control thereby sampler posterior describe alternative finite base measure neither sampling successfully bayesian nonparametric contexts beta serve adaptive level stochastic proceed associate auxiliary positive sequence binomial represent gamma conjugacy component slice u k round represent conditional sample share process beta slice sample detail defer biased finite beta continuous converge may leverage approximate posterior sampler fidelity detail conditional gibbs behavior sampler toy process utility approximation component sampler gibbs slice sampler toy bar ten component beta weight exceed sampler underlie sampler attempt iteration practical unsupervised inferential task consider observation word vocabulary generate topic underlie intensity parameter vector mass binomial heuristic exact slice sampler let document tracking attack transform summary field add account seven vocabulary group document organization report claim list remove organization randomly aside next independent organization specific drawing classify document document confusion ten display slice sampler finite approximation nearly claim organization experiment
see induction take integral appear state entirely optimal utility assume subsequent calculate expect give restrict max policy simplicity policy temperature reward policy choose perform chain monte policy fig metropolis mh performing mh proposal rise describe alg enable two alg sample thus utility conjugate distribution case condition reward simple sufficient statistic continue hybrid
factor show setting moment satisfied kernel worse countable kernel interesting note result also oracle modify another scenario entropy wish thank van de ep international grant theorem conclusion theorem exercise uk regularize regularization lasso group multiple novel countable transformation
I ds fs ds gs gx justified independent polynomially hence also fact let mesh notation act help deeply c l p p bind backward z ds prove collect x assume center condition statement get claim course constraint derivative z brownian position copy lemma coefficient guarantee smoothness coefficient respective remainder derivative function respect suitable probability measure note prove c q solve existence growth solution sde ds generator sx ds
cost action parallel focus difficult adversarial arbitrarily round round leverage non bandit setup next section work connection give algorithm special small box specifically subgaussian uniqueness value point henceforth ci ci width ci separate cope unimodal unconstrained author variant rate non optima derivative vanish derivative optima yu study unimodal bandit setting al armed bandit function algorithm regret convex lipschitz grow exponentially case cost hard adversarial point good regret adversarial setup include function exploit setup bound attempt solve convex instead specify setting find feasible membership oracle convex fact noisy elegant solution random walks separation oracle membership oracle mostly noiseless much drawback often explore observe closely noisy whereby query domain produce minimizer
note generally multivariate terminology random tree structure v class consider hide hybrid combination model tree include take value variable take take vector represent relationship markov markovian addition assume recover th condition identifiability latent condition estimator rank matrix introduce additional limit rank redundancy hide furthermore hide node correlation correlate hide redundancy two node reduce soon direction ensure latent generalization condition consider remove subtree remove triple effective logarithmic achieve though variable though conditionally relationship neighbor effective depth therefore polynomially large second isotropic denote ax x determine topology subtree four possibility induce subtree return subtree output guarantee subtree topology subtree correct subtree integer respect four confidence tune apply empirical
enjoy high generalization respectively insufficient guarantee distinct setting proceed development intuition insight mix almost sample stationary et technique building condition far sequence idea combine theorem start hold first recall second use lipschitz part begin first inequality inequality expectation coefficient relate expect predictor stationary field small risk lipschitz assumption mix key test mix loss remainder complete proposition control respect apply confirm hypothesis random point remainder let denote expand sequence regret summation three summation boundedness guarantee note q use increase
become illustration use plant partition free landscape overlap assignment result marginalization permutation assignment generalizing difference produce asymptotically attractive illustrate varie observe overlap situation arise true co transition fig locate energy phase
parameter achieve wherein good bayesian consideration offer therefore decision advantage von number pilot symbol generally time deal course elegant computational synchronization prior model finite unique substantially expensive
compare show average grid performance run estimate contiguous two centre leave centre four region contiguous rectangular single three figure estimate outperform behavior consistently confirm experiment brevity cccc contiguous different centre contiguous centre centre four contiguous group underlie pixel foreground random plus noise due uniformity hard grid compressive instance image compressive besides sparse haar
scale multiplication state point demonstrate keyword path integral apply hand many intensive control purpose interpolation response slow simulation simplify account due fast produce
neighbourhood baseline relation like etc fully unsupervise upper row model full neighbourhood see experiment semi learn variant complex prior completely unsupervised posteriori complex priori model complex see belong segment generate priori reflect correct force model simple shape input image show baseline right complex well perform leave dna cell segment circular neighbourhood diameter neighbourhood grey two per segment supervise tree well complex complex figure see prior capture lead segmentation model
ac uk likelihood probability smoothing concave maximum estimator concavity estimating use justify use breast diagnosis classification smoothing constrain great deal tuning idea characteristic shape log exponential moreover drop convex smoothness drawback however circumstance attractive visual justify substantially hull rather section concave likelihood smoothing convolution density preserve concavity shape concave fully nature idea upon challenge convolution integral take figure
select representative sentence pairwise include redundancy similarity select summary pairwise summary sentence rely citation sentence give summary lot binary sentence dependency redundancy arise rank decision introduce dependency dependency sentence hmm hmm produce summary sentence expressive crf disadvantage sentence model range method closely use retrieval label available diversity coverage problem approach thus wide range add simplify furthermore closely
estimate approach aim vb population distinguish observation statistical put aside observation probability belong interest high mse evaluate average dataset mse performance achievable aim obtain weight oracle view negativity constant contain allow numerically take constraint account achieve vb oracle display different first vb good provide pe provide comment
redundancy observe variable multimodal principal easy conditional mixture model density form constrain keep reasonably ability density mixture approximation paper hereafter distribution gaussian mixture row find mode model joint course redundancy point constrain mapping absence information answer mapping turn case problem constrain obtain reconstruction redundancy nearby nearby accord distance depends continuously result continuous bounding numerically approximate derivative apply condition end sharp sharp near norm typically euclidean different contribute also form difference another often reconstruction problem forward map small candidate reconstruction effect reality reconstruction difference incorrect reconstruction spurious generally help discard incorrect reconstruction introduce involve whole define reconstruction break reconstruction analogous add constraint sequence eqs forward experimental sample length trajectory pass combination constraint global variable product candidate reconstruction combinatorial optimisation express short figure pointwise reconstruction total exponential order fortunately local bl layer define leave leave path highlight fully link associate reconstruct short case short acyclic programming dynamic optimality graph decision node layer keep figure give forward recursion dynamic programming global due symmetry backward definition set reconstruction minimal length layer il I I unlikely tie return programming link node adjacent layer achieve mm mm il mean concatenation select edge close point improve proceed edge undirected algorithm need path term
use k fact complete local iii record white wishart chi variable degree freedom value definition asymptotic iii show recall obtain integration thereby obtain since work expectation hand front give eq w local thm thm proposition university california berkeley hypothesis testing multivariate size classical working derive achieve great state roc curve simulated test parameter determine substantially situation cumulative occurrence sample become increasingly application molecular fmri analyze fmri fmri measure order hundred thousand investigate sample large several may applicable accordingly grow develop procedure well deal dimension b fundamental manner mean definite problem know define pool singular nearly seminal short study test suffer subsequent year bs sd
computationally intensive procedure reconstruction picture surprisingly vast majority image skip immediately impose nonsmooth object noisy mild assumption suitable highly work well method advantageous modern practitioner medical perform preliminary reconstruction drawback substantially study view object within algorithmic testing pixel top accuracy grow drive stop rule human stop image focus serious
wireless channel take negative network account secondary interference secondary primary user sense user platform cloud compute customer automatically load accord knowledge loading optimize derive chinese restaurant social phenomenon quality deal traditional strategy chinese customer restaurant deal price effort restaurant quality try restaurant come high restaurant quality low customer department electrical college md usa institute communication part two part chinese restaurant learn negative agent chinese restaurant influence agent study illustrate restaurant cloud social formulate chinese restaurant learn agent chinese restaurant well decision furthermore agent different decision order conclusion influence agent avoid make maximize various dynamic spectrum cognitive service
collective speed run likelihood train repeat range prediction two train validation datum prediction second precise set unable right bar trust determine final away properly parameter choose nn accept show quickly accurately likelihood thereby overhead choose network probability posterior sufficiently require hessian product slow large time calculation follow analysis obtains even likelihood large flat spectra calculation long regardless ccccc datum set
hereafter sense constrain grid grid sampling account grid square yield cause show satisfied grid propose bayesian case true bound gaussian idea value svd computational sensitivity noise power propose exceed music
weight location image order equivalent scan would read page repeat dark patch little local content dark like structure near main roughly mirror section fact anomaly g rapid etc patch detection extremely difficult patch surface anomalous usually one new parametrization patch anomalie rough patch anomaly see quantify study rough region suggest study parametrization provided introduce slow patch notice patch anomaly fast matrix entry patch follow extremely graph conversely lead concentrate diagonal walk slow patch slowly previously replace notion distance quantify time diffusion distance see section patch patch term remove evolve seminal equivalent walk characterize therein therefore patch distance notion proximity patch homogeneous markov patch evolve slow patch start g narrow bottleneck visible upper matrix diagonal take slow patch quantify computing start vertex eq I symmetric version walk fully eigenvalue label vertex emphasize transition diffusion vertex distance compute walk initialize decompose volume follow scale regime come patch agree euclidean original distance graph consequence idea manifold foundation indeed accept error first dimension patch embed
dimension guarantee count every tt x last equality follow fact class query database select accuracy bound constant database claim complete uniformly note obtain element random reconstruct occur symmetry recall private also find size usefulness extend axis align space clarity discretization query preserve proportional easily bound array bi repeatedly perform partition approximately fraction mass terminate round database mechanism differentially private query laplace query differentially sensitivity composition differentially mechanism differentially union database none laplace great every step error synthetic interval contribute contribute low axis align class nevertheless exponentially interval query mechanism possibly discretize
move center simulate prior likewise density birth death reversible ratio reason move center likewise acceptance show variance reader mean proposal acceptance choice center choose accepted move accept probability use proposal improve actually general model reversible iteration proposal implementation horizontal axis time number transition reversible respectively increase transition seed total assess reversible jump chain reversible proposal clearly efficient proposal reversible
n ix nx tracking capability active dynamic nature due weather condition change illumination adaptive strong nature keep track adapt accordingly adjust importance decision widely around detect surveillance surveillance area furthermore computer security circumstance leave capable automatic necessary security forest operate forest six feed system whenever alarm reject active fusion classifier function real slow every sub linearly weight adaptively red detecting system capture compare regular camera unless forest long distance
end last assume support lasso notice term let sufficiently deduce second must develop x replacing tx tx hand tx g case support different satisfied
independence quality equivalently implement advantage motivate far sound increasingly open expect technology discuss series markov gs use gs greedy outcome discard inconsistent indicate gs cascade error produce error approach optimize generative compute interestingly quality quality advantage avoid cascade assigning learn independence execution massive test contingency record strategy decompose markov insufficient edge add mistake edge comparison experimental publish experimental quality publish quality score versus fitting independence adaptation recently improve quality gs pc mb fast mb section develop improve accuracy axiom basis detail independently black box independence mutually large test test determine test become dependent useful example show insufficient relate axiom improve data analysis technology independence network low quality learn however motivate independence sound availability grow increasingly expect solution pose open
path unfold overlap break unfold pathway pathway mind unfold trajectory trajectory question analogous reversible p break trajectory causal unfold mechanism contact obtain subtract contact map contact reveal break pathway colored circle division number average list every unfold ensemble iii pathway representative rna description panel contact green circle secondary region b note pathway unfold pathway table respectively ms ms unfold pressure unfold obtain pathway sf cccc unfold ms ns sf ns unfold pathway sf cccc unfold ms sf first unfold event predict path average e step mean time
discrepancy accord express discrepancy size alternative statistic conjugate bayes base distribution respective therefore sufficient computing power close bring approximation realistic informative population experiment per per tree per scenario national core processor experiment approximation assess fig approximation hard assess though furthermore universit de paris universit france abc tool model al rely inter property show wide model choice information induce insufficient summary statistic effort spend necessary representative bayesian
home show player run cover accurately create central define player position start double walk walk generate run effectiveness measurement run per game team possess player base measuring depend reach advance cumulative statistic compute six outcome single triple home number average sequence performance game consist position take digit position denote base ignore markovian recurrence represent outcome state also use probability percentage express take expectation expression differ mean classify triple home home home run home use require player require result home three base score one assumption triple require walk base calculation sense statistic drawback statistic wide scenario average wide average well player average calculation show fluctuation run affect player scoring attempt quantify player production entire allow wide change rule average contain plus walk triple home technique analysis
low cluster block diagonal one course order rank exactly low rank look block one idea depict propose perform convex drop structural low entry unchanged state observe entry know surrogate k norm singular objective surrogate objective semidefinite sdp cluster tradeoff determine exactly relaxation add constraint entry constraint definite even guarantee automatically imply relaxation formulation reason importantly recovery cf section important regime tight relaxation seem benefit
criterion scoring localize characterize pattern insight heterogeneity exhibit strong regularity visually pattern regardless cancer cell type breast marker g etc capture texture image patch construct mask relevant scoring report salient contribute intensity etc description image toy indicate occur three value panel right matrix cell correspond illustrate bar successful image matrix frequency pixel intensity across neighboring pixel particular spatial relationship define direction distance distance interaction pixel involve interaction pixel choice extend involve relationship appear sufficient spatial common summary employ classification typically thousand however capability random forest subset entry order mask scoring realize choose image patch consist
communication moment enable computationally term posteriori variational robust student essentially approximate model determination kl result latent gamma parameter maximize likelihood expectation current hyperparameter drawback determine minimize reverse kl divergence uncertainty previously mcmc laplace introduce gp find form bind marginal student likelihood parameter ep hyperparameter optimize package definition ep general py un normalize iii cavity site refined approximation match marginal finding equivalent matching moment finally change repeat site normalization df refer recently scheme ep site calculate site although cost ep computation cholesky number require roughly scheme easy hyperparameter enable efficient map initialize non likelihood arise discuss improved ep robust loop site I follow controlling amount view small saddle converge find double double loop fail model consideration parameter cavity site approximate distribution respectively min condition consistent ec free energy view ec ep bethe energie double loop fix outer inner affect current
conditional price asset deviation price expect deviation uncertainty management course distribution price look design process process function technique temporal attractive view index
agree classifier eliminate classifier eliminate bad excess classifier round essentially conclusion request label request growth final feasibility set implicitly update perform calculation constrain include algorithm elegant additionally improvement algorithm well modification certainly reasonable another meta agnostic thesis natural advantage aspect yield agnostic meta way complexity achieve algorithm often publish literature extend include finite output cm implication appropriate setting follow algebra represent disagreement coefficient f improvement learning expect often significant improvement unlike author even maintain guarantee sometimes small key exploit label disagreement learn shrink proof passive break furthermore disagreement algorithm improvement passive leading region even first p n thus wish achievable term two distinct refer characterize xy xy way least rate must occur interestingly set query behavior infer agree classifier phenomenon threshold uniform satisfy scenario never focus query improve passive sufficiently point either two implication dominate vote infer agree label agree label quickly converge neighborhood request converge attempt satisfy convergence entirely general providing however early meta behavior set thesis thesis explore unlike label never fact region limit whether lead noise xy diameter bayes fraction learn difficult infer fact particular versa get rate purpose explore note specifically interested label achieve well handle one part detail follow fix achieve learn main theorem extension entirely label xy nontrivial active noise case agnostic sense work raise contribution fundamental capability active establish vc make possess favorable property passive passive label dimensionality space margin similarly whose dimensionality wang preserves dimension passive preserve passive active passive input preserve request phase reduce passive guarantee fed algorithm conditionally dependence among fed clear whether slightly request might allow passive clear passive example improvement theorem algorithm possess vast majority publish style passive subroutine construct execution learn label whose passive heuristic method passive example whose expect type passive passive satisfying early base trade nontrivial scenario ease allow factor scenario make proof careful inspection proof reveal converge remain open nontrivial definition underlie passive trivial algorithm definition implication complexity problem nontrivial label passive various select among trivial may counterpart nontrivial many label rather require nonetheless question label complexity label complexity relax specifically relax nontrivial still maintain existence class least replace say nontrivial relax achieve via reduction coin via label xy restrict single relax nontrivial possibility outcome conjecture issue achieve acceptable simply nontrivial alternative definition interesting concern consistent sequence run largely issue adapting whether efficient bind corollary request universal reduce construct label disagreement batch nontrivial universal space batch meta question whether increase universal request every might disagreement base another search sign interesting specialized execute two instance component usual guess bias via response query might require regularization construct universal space infinite vc usual condition run algorithm round especially whether continue strict improvement passive achieve specific label van improvement passive question
discrete seek relate measure fourier boolean boolean entropy fourier boolean square fourier view distribution entropy intuitively influence replace unchanged influence computer mathematical physics social theory etc survey express fourier fourier normalization coefficient
consistently indicate satisfy tune nonzero component lemma eq submatrix corresponding imply tune result establish avoid illustration dataset macro indicator cover production price start order stock exchange rate apply express special real activity economic price rate lag regressor series nd logarithm raw forecast forecast year table column unlike robust primarily lag know lag begin flexibility ahead outperform behavior universal forecasting performance purpose figure summarize classic estimator scenario able lag lag lag confirm come typical come business cycle economic development spirit motivate normal equation extend consider much difficult consider
vx related condition page page old p condition define page ergodicity ergodicity I acknowledgement helpful comment author advance fellowship capital support al foundation centre dynamic research condition remark box stochastic occur diverse unstable expand focus stability general condition incorporate possibility sequence setting theory sa find measurable measurable numerous behave suggest recursion find integral need approximate numerically focus rely exactly possible instead monte probability distribution sa dynamic obvious dedicated study section occur ergodicity term make vanish hereafter situation fail mapping zero priori projection induce difficulty sequence form cover x stable markovian noise x
redundancy support conclusion inductive set learner generalization benefit number understanding loose learning definition common sense enable fundamental literature compute match give rank amount relevance synthetic discrete feature illustrate prevent available point principled algorithms assessment subset survey main categorization feature comment tool cover experimental end conclusion work selection comparative study demonstrate datum typical lack wide fully control environment possibility redundancy availability another issue assess normally encounter feature take place leave aside aspect namely subset affect
require heart limitation protocol provide picture factor association group heart disease heart prior child homogeneous total describe family information characterize child factor I characterize child post quality ease one adjacent type factor outcome interest score cognitive development assess year group miss birth birth birth family birth family birth birth length family birth birth school education economic status birth number risk max low low head weight year length year post head year year post post score group percentage value percent completely observe observation score child head birth child seem infeasible result loss use cognitive development heart birth
give leave extra leave sufficiently allow log distance q symmetric denote corresponding eigenvalue orthonormal note symmetric real use eigenvalue eigenvector somewhat large enough second eigenvalue provide large enough uniquely classical e show eigenvalue possibly pass eigenvector deviation enough infinity pair small threshold pick estimating could take empirical indeed distribution build everything else leave node state build recursively child root reconstruct construct estimator convention form estimator u turn assume priori eigenvalue decay recursively presence
move frequentist depend review statistical mention approach inferential interpretation fact publication analysis frequentist two sometimes fire intensity time frequentist incorporate bar force inconsistent interpretation statistical general approach model statement advantage allow interpretation split framework seem I inference statement partly eliminate bayesian eliminate jeffreys subsequent hand really matter frequently focus challenge high capture theoretical figure text path break excellent
tag connect document relation relation tag author subset circle mine author intersection subset circle focus combined vision mining possibility totally biology three exclude fig quite union multiple tag distinguish tag separately document content tag prior effort topic rarely specific topic dependency tag topic mrf relations generic allow theoretically mrf topic modeling problem mrf model develop classic belief hypergraph bipartite fig pairwise important modeling structural many estimation object segmentation infer mrf structural labeling property co document tend high configuration smoothness higher equally need topic labeling encode structural design encode representative smoothness make efficient parameter rest follow introduce mrf bp inference estimation propose focus model among extensive several mining broad interest conclusion lda circle probabilistic topic text mining art lda basic label hyperparameter
unweighted weight
g uniquely determined marginal lebesgue simplex z derive direct imply let p inverse uniquely normalize almost applicable construction continuous measure convenient example projective cumulative interval nk ni ng ni ni represent way indexing index unique node pass child binary representation arbitrary associate beta measure suppose particle tree reach recursively beta ny applicability set b np satisfies b hold continuity property tree unique p tree measure measure absolutely difficultie recognize indexing space ingredient draw dirichlet recognize construction measurable excellent survey construction approach
scalar side towards word dimension compare factor estimate admit constant take eigenvalue produce constant offset illustrate explain give eigenvector maximize specifically h mis follow theory relate scalar eigenvalue negligible factor sample performance modify correction satisfy narrow range pointing well see quantify factor model bias relation analogous em equivalent requirement behave explain intuitive understanding phenomenon magnitude residual variance impose additional b straightforward scale scale outperform question tm believe
test three undirected indicate circle respectively around node weak tie community highly density gibbs distribution assign stage initial heat chain average step produce marginal show least threshold stage explore five correctly label node belong aa achieving explore nine classify perfect perhaps algorithm choose sort bar node stage central node argue uncertain learning
classifier point class discriminative learning margin learn label force away great margin slack variable formulation additionally directly metric classification classification nearest neighbor result unchanged show invariant transformation limit rate near infinity scalar laplacian trace hessian choice glm optimize local rate approach solve optimum compose eigenvalue matrix eq constant determinant unity neighbor distribution bias equal metric glm bias rise note close attain minimum minimizer identify semidefinite stand eigenvector stand diagonal negative optimizer scale determinant attempt rate generative discriminative neighbor solve result metric depend specific metric specialized
unique addition control complexity indicate set topic label approximately count example token give topic add topic approximately lda increase consistent document keep hyperparameter term sum two power prior benchmark total benchmark dataset short document length chain seed burn iteration take assignment sample average run single estimate multiple chain unsupervised set store test model lda run lag reduce autocorrelation dependency lda incorporate ten estimate chain across set run compute change sample flat lda average prior add estimate distribution conjugacy multinomial combine weight I choose use word prediction would mostly influence assignment document word assignment derivation document token need compute remain factor assignment label beta numerator abuse denote argument cd beta function function range change equation analogous lda analogous word estimate comparison highly benchmark alternative area put classification publish publish literature evaluation metric version compare able publish value least evaluation metric utilize lda goal put area demonstrate svm similar tune svm elsewhere demonstrate competitive state comparison score rank good publish publish version yahoo publish use set train split yahoo numerous discriminative additionally label least introduce account label demonstrate use competitive present consider lda worse good discriminative yahoo dependency macro performance frequent lda discriminative micro score yahoo bad method yahoo label dataset discriminative dependency lda macro micro generally bad yahoo provide power strength macro score type depend dataset lda competitive even advanced discriminative publish benchmark numerous us wide consider consider style macro micro score prediction prediction thresholded micro optimize macro single macro optimize micro tie macro micro lda dependency lda perform svms however outperform macro additionally micro
trivial save sparse complete detailed moreover sparse use dynamic distribution overhead interval argue final sure guess exactly learn sparse succeed output try sparse close hence overall routine poisson form unknown tp place work correctness routine eliminate value define note ensure proper intuition binomial distribution future reference case satisfie tp distance certainly lemma observe exceed claim claim hold first last assertion cauchy rearrange follow inequality last binomial valid bind distance lemma notice fact learn ready describe similar proper modification translate poisson hypothesis choose hypothesis convert close tune analysis guarantee overall unknown conclude generalization problem unknown throughout know learn sample th simply independent time bernoulli efficient draw bit time sort result independent value aware arbitrary theorem show test subroutine target cover
estimator need numerical applicable demonstrate operator idea base contain derivative gradient local reduce desire one datum difficult iterative accurate reproduce space rkh value definite kernel dense k reproduce hilbert f definite boundedness positive kernel rkhs yx k xy vector reproduce derive operator statistical definite property
verify obviously q expression analyze design assumption p uniform excess lemma relate excess risk measure introduce x c c symmetric measure know proceed apply lemma variable analogously kl kl n p divergence bernoulli verify dp p j p n
rescale offer rate class rescale counterpart ensure class article explore utility exercise independent draw assign rescale old rate without sense coordinate information cast fast coordinate would cast gp perfectly dependence explore extend gp variable low question equip rescale offer
research receive receive award speech student international conference speech conference well song r award communication high estimation dr rao member technical processing communication process signal remark rao edu nsf grant zhang address nonzero correlate consider correlation performance correlation propose block derive bayesian superior recovery especially temporal highly extensive interesting motivate sparse recovery compressed sense correlation recovery compressed processing basic mn unique available noise source measurement wide arrival estimation imaging eeg localization estimation measurement measurement l unknown solution row possible source often application orient model index identical row solution nonzero compare case rao advantage relax mild exponentially rao analyze
wavelet coefficient reduce splitting complement orthonormal basis q parent child dependent necessary fine encode large variation approximation correspond first correction advantage tangent correspond geometric disadvantage really encode precision claim equation wavelet point lie exactly outlier lie tree close point unique enough scale calculate encode multi compute two set coefficient fast associate advantageous construct wavelet generate approximation technique process thresholding multiple haar smooth wavelet tree understand suitable suitably tree try tree metric euclidean operation independently tree balance suited purpose detailed investigation publication cloud store cloud cloud cost resolution manifold sample point fix interested theorem even encode cloud large expensive cost geometric count wavelet geometric take independently datum geometric wavelet wavelet basis add correction term geometric cost encoding wavelet encoding
fast optimization perform artificial database artificial separate point gaussian circle former consist two equal one axis show increase identification artificial database classic semi density unit define produce define three configuration artificial database dataset breast cancer compose fine breast database compose traditional method tendency spherical method work rate
particular display summary part informative exploratory representation exhibit provide brief phenotype section concern sequential algorithm simulate deal display plot predictor vector phenotype adopt coefficient density family distribution arise double exponential central independent across coefficient simplify freedom refer double prefer think generalize early context also understand direct derive density prior hierarchical prior whereby coefficient priori tail generalize light tail tend clear infer posterior exploratory examine map estimate generalize problem associate prior local posterior
alternatively area risk accurate suffice explanatory risk plot reach well overall optimize partition simple treat merge block finally partition far modularity parameter measure dynamical definition n next assign suggest interpretation successive training accumulate error see grow decrease cumulative risk cumulative risk select modularity overall gain value total indicate simple significantly early cumulative modularity network cumulative risk equivalently total cumulative risk modularity
reason high incidence far hand predict characteristic diabetes highly health service level distribution state differ quite function incidence fulfil calculation
able reproduce weight
probabilistic name dependency combination markov probability component dependency bayesian set parent cyclic distribution px dependency network collection separately specific would regression network sampler apply dependency joint dependency causal author network basic module gene gene module function employ module reduce similar module gene level coefficient genetic network description like class entity associate attribute fix relational describe entity consist dag attribute conditional special world characterize edge neighbor connect let denote average reflect block link degree decay compare power neighbor property study structure appear network behave connectivity small network group free major approach method approach attempt set attempt drawback identify structure base explicit try fit probabilistic score score indicate fit space intractable evaluate structure heuristic find popularity year essentially
definition belong population choose b ta transformation
inconsistent respectively factor imply description consistent investigate whether inconsistent rest bayes calculation use probability set rv datum inconsistent rest regardless three analysis conclude bias together bias rv residual analyse amplitude roughly correspond period despite analysis rv old set contain signal remove need purpose calculate say calculate correspond inconsistent give combine reliably rv rv restricted probability
h b place shall assumption satisfy consistent accord variable case wiener integral brownian motion condition integral simplify hold denote n pn immediately residual lemma assumption strongly proof theorem condition satisfie bound shall look hold unbiased separate consistent function estimate separate integrable martingale
time occurred discover end censor occur remove patient lose follow cancer could location understand fact early censor individual partially j survival analysis ci call tie tie thus goal formalize order way build scoring pair put survival analysis class supervise ranking problem quality medical longitudinal study call censor degree unknown factor clinical feature put emphasis survival commonly treat usually event occur
inaccurate believe largely condition spread widely round variation would unique atom would slower algorithm despite learn initialize underlie dominate inference since use process beta consequence sampler stick break offer flexible representation superposition countable poisson may generalization appendix f df lead exponential simplify third equal dd n nk
alternate possibility hypothesis graphical either consider instrumental applie model usefulness correlation social commonly hmms find variable goal one acknowledgment thank discussion national science fa quantum see write obviously sec rigorous measurement choice possibly ax pr ax without generality hide hide share pa r pr pa pa form combination write graphical minimum special
easy ess mathematically prefer criterion essential weighted reduced density usually reasonable choice correspond see anneal efficiency density determine neighborhood metropolis hasting mcmc sensitive depend scale parameter determine satisfie potentially minimize scaling consist adjust scale whether optimization finally coincide intuition intermediate concentrated ergodic aim discussion anneal involve delta verification need first prove describe generate explain reasonably unique limiting annealing aim anneal follow condition generality trivial case second account prove aim probability example chain distribution irreducible positive assign uniform etc aim generate irreducible therefore denote recall simulation course measurable surely
discovery discovery information differentially two microarray dataset mention breast classification summarize small identifie identify figure finding level validate accurately figure breast cancer study gene connection breast cancer gene expression level validate light surprising fail mark blue bar height bar posterior include dot z breast breast cancer gene expression compare identify tail histogram figure identifiable nonparametric empirical bayes testing include oracle particularly differ finding distribute work behave normality identification
x l eigen smooth eigen subspace fix q conclusion boundary smooth conclude sense invertible next norm expansion thus together equality come complete proof eq module haar laplacian operator adjoint uniform operator eigenvalue dense topology g hermitian differential module count copy inside frobenius law branch irreducible form form note calculate module call frobenius k copy irreducible calculate irreducible dual lattice unify fundamental power standard representation irreducible representation high weight power irreducible representation split irreducible high please irreducible irreducible module irreducible module run integer irreducible module run simultaneously irreducible module high weight please base g precise side split k high weight please step relate representation consider endow bi invariant semi metric enjoy relationship highest irreducible high positive root induce inner please note combine irreducible representation particular decompose irreducible module eigenvalue inside character formula calculate highest irreducible I pm irreducible pp n please original odd os nn odd otherwise eigen put eigen irreducible list list happen eigenvalue em eigenvector conclusion see compatible thm example wu vector massive image shape generalization dimensionality heat heat low space field diffusion near dimensional embed laplacian operator dimensionality heat alignment connect point quantify past decade reduction locally
program transform semantic preserve make program semantic preserve program generate prior program generate prior program frame program alpha program search towards program mean size program body frame abstraction abstraction abstraction abstraction abstraction body body abstraction size track particular produce important quantitative large make furthermore give point account exactly describe computation fact likelihood mean case program estimate sequential generate discrete extend selective point tree frame tree evaluate program single correspond generating selective averaging frame program tree replace gaussian program program topology parameter score topology score trees topology tree map compute tree tree inf topology inf apply score score fact function program mean code syntactic implementation replace return list replace abstraction abstraction make name abstraction abstraction return abstraction pattern abstraction abstraction convert replace abstraction program body body convert convert body define list choice uniform rest test list replacement replacement abstraction abstraction name abstraction name abstraction transform abstraction pattern abstraction abstraction let convert map abstraction convert transform body make convert converted body force program detail beyond short
likelihood parameter posterior formulation quantify involve nested purpose might variance empirical uncertainty induce uncertainty propagate sometimes though krige provide interval refinement krige surrogate refinement interest introduce krige technique krige probabilistic krige prediction refinement consist reduce optimum sequentially achieve reliability interest limit subscript clarity krige denote might onto state surface correspond uncertain spread achieve locate margin express finding maximize quantity bring well krige decide criterion optimization paper propose refinement criterion include easy exist platform fall multiply weighting consider weight original pdf confidence region pdf reliability equivalent gaussian reader discussion maximal justified numerically indicator q
explicitly neuron topology connect follow denote embed stand number build mind energy lyapunov neuron determine overlap cosine pattern term appear hand equation contribution whereas concentrate comparison extend classical quantum rewrite local stand
weight add importantly time vanish critical network otherwise stop track constant show agreement noise present attain agreement tend square error mse bind sufficiently agent behave agreement allow individual give node flexibility operate end enhanced proceed detailed diffusion strategy important consensus decay set combination adaptation evaluate incorporate neighbor diffusion actually mean square diffusion achieve rate consensus dynamic algorithm vector gradient gradient component denote symbol expectation randomness node gradient express diffusion q observe quantity highlight algorithm arrive convenient carry assess estimate minimizer square mse sense necessary force agent acceptable flexibility beneficial biological network adapt forced requirement agree small behavior
new paper think think weight h directly free parameter inequality choice therefore average allow relate average law reference sample know closely moment moment proof hoeffde variational basis introduce pac high set expectation set expect define randomize randomized round apply next pac generalization guarantee influential support machine cluster name extend hoeffding et al tighter apply pac domain average inequality
quantile nonparametric regression posterior computation automatically incorporate remainder quantile focus efficient partially collapse integrated drawing explain detail conclude discussion section th quantile asymmetric assume q skewness parameter direct likelihood scale mixture satisfy link distribution zero hyperparameter write constant identifiable exactly
much covariance matrix variate kronecker delta rule array variate structure parameter describe simulation dimensional gene study array element say array array array principle technique multi result correspondingly
optimization potential bound strictly feasible program lagrangian saddle saddle similarly saddle v optimal u e saddle lagrangian nash equilibria pass equilibrium equilibria complete within game literature player control single player two consider player player game uncertainty author already analyze sensitivity message study
dataset experiment handwritten digit recognition task dataset pixel range due dataset set pick validation consist test repeat experiment plot section give rank experiment report describe overall counterpart suggest kernel range width measure characterize versus four mnist htbp suggest mnist experiment choose cr affect change however finite improve outside small base well base kernel set expect c nd cr da du rank mnist letter dataset mnist letter experiment evaluate uci letter recognition include capital letter validation misclassification measure mnist kernel however experiment dimensional available table build examine learn target supplementary material achieve task experiment high alignment translate accuracy aside mnist see cr outperform repository median
following let positive scalar invoke diagonal ii ii important positive eigenvalue sort decrease order likewise result proof hermitian finally definite satisfie triangle symmetry obvious show prop triangle arbitrary preserve prop triangle consider prop z immediately computer show scalar hilbert may ask previously embed different kind admit embed negative definite suffice eq arbitrary unfortunately quick although answer positive characterize necessary essentially treat matrix definite prove map follow product whenever cover invoke
people idea science greatly via people never work year way use process mathematic science lead
ab side equation q outside interior equal center radius singular note integral second say eigenvector eigenvalue outside everything loss strongly penalize minima let convex ss w ii could path use since equal zero term minimum
denominator find definition remark theorem parametrize dimensional long need complexity mutual apply interaction lower dense match qualitative behavior bound achieve stochastic differential sde dimensional brownian px unknown dimension
subset range integrate entire second refer efficiency discrete finite useful order arrange increase index network unweighted discussion describe percentile rank tie order knowledge specify exclude nan cost purpose irrelevant population mass treat level discrete element probability occurrence every weight kk define every computational generalization open one due opposed case handle particular topological cost interest become integration treat integral probabilistic treatment integrate particularly helpful consider estimate monte scheme readily integrate level order answer question briefly consider fix cutoff fix iii four example datum extension idea several simple topology correspond association specific discrete topology naive network proportional association weight v association denote association simply proportional illustrated element standardize cost solely interestingly easy use equation since naive topology say straightforward quantity show additional efficiency cost metric increase thresholding graph fix cutoff weight tend level cutoff point separate cost adopt author condition topological thereby calculate quantity small since topological arise consequence transform integrate fail distinct functional connectivity parametric transform produce regular simply choice
object interest shape vary negative boundary impose shape unknown base applicability via achieve prove object noisy problem image quick noisy mathematical object noisy basic look ask primary diverse automate detection application random picture make intensive procedure surprisingly majority skip shape permit object interior aforementione automate material detection regular method advantageous object problem extend tail give implementation program statistical program language statistical object seem spatial triangular periodic case complexity pixel
expansion estimate advance sampling toolbox estimate well implementation adaptive constitute analyze measurement identifiability sampling call restrict isometry show bernoulli uniform toeplitz appear bind deal rip rip scale extend contribution rip regression involve tool utilize rip bound counterpart sparse regression scale also applicability simulation aware estimator cope curse dimensionality yield parsimonious record work letter denote stand density covariance pp expansion notion multivariate problem nonlinear respect curse inherent involve motivate highlight input mapping truncation practice yield condition capture term module module reference goal expansion memory extensively present vector pn np tn ph arrange accordingly tn linear generalize filter aim approximate nonlinear tn several multilinear prediction problem natural science economic choose carry setup henceforth easily
wise suggestion presentation acknowledge foundation nsf grant dms randomness institute arc proposition continuous q lipschitz twice satisfie cube minimal compact possibility second dyadic follow vanishing require regular gradient denote subgradient representative possible lipschitz remark dyadic cube note subgradient assumption choose independent hyperplane contain integral remain boundary center cube eq q impossible exist dyadic edge
candidate datum equivalent amount adopt previous provide quantify theory issue possibly noisy discuss extension next objective nonempty convex objective function search I multi fx loss estimate represent e additional constraint cost observe almost orthogonal closely related purely unknown observation try maximize amount experiment worth note exploration ensure balance robust objective visually depict htp l exploitation ex ex robustness multiple trade list exploration versus exploitation put exploitation e trying achieve observation computation capture risk exploitation present utilize framework address present discuss shannon information infer machine constitute involve error function alternative follow spirit try try estimation point
market long confirm major impact occur close region plot km take use multivariate multivariate already derive sake completeness additional moment auto causality test process causal effect present specific bivariate monte carlo maximum behave magnitude confirm close contiguous short period hour perhaps subsection medium arrival sum assume consecutive day natural might poisson regression derive autoregressive denote series imagine operator eq values compound binomial autoregressive process q markov chain integer moment interest high dimension provide
mean ern smc approach variance mat ern covariance jeffreys endow prior log thank
occur every fx z iteration fista theorem lipschitz although leave stay follow essentially backward substitution following iterate alm fista give backtrack line start fista inexact linearize quadratic fista version fista iteration surprising fista gradient handle label scale impractical fista serve subroutine involve section example group solve notational subproblem ball norm simplex time et euclidean onto euclidean follow replace problem absolute take projection onto take test fista fista framework interface provide core matlab alm find loop experience show slow even system perform pc intel ghz processor gb framework subroutine termination criterion criterion primal dual residual criterion appendix outer tolerance outer tolerance
model desirable necessity evaluation functional bound dimension differential equation generalization shannon rkhs possess reproducing reproduce able save birth kernel trick rkh underlying application field reproduce kernel base theoretical issue devote inclusion rkh reproduce interested rkhs rkhs understand reproduce relation multi rkh study reproduce kernel critical success avoid overfitte principle overfitte large relation inclusion establish reproduce rkh application map past purpose
measure association gene phenotype observe association phenotype disease phenotype association indicate rank phenotype disease phenotype association roc roc c c pt report disease phenotype phenotype cross phenotype associate phenotype retrieve phenotype gene phenotype disease phenotype experiment disease phenotype near reduce report roc leave validation algorithm outperformed achieve well sp score global rank disease phenotype rank rank clearly achieve ranking query case ranking phenotype phenotype query target disease highly good fig compare query disease phenotype phenotype fold threshold suggest short distinguish phenotype neighborhood utilize interestingly detect association high set include gene disease dense disease gene module neighbor tend dense association phenotype query gene fail
norm employ high constrain trace norm small achieve precisely scale large application multiclass fast acknowledgement discussion schwarz science need generalize let assume denote hold smoothness tell e eq multiplying side support combine rearrange equip ready let vector value thus operation monotonically increase addition lemma imply
case section outline method picture neuron shape advantageous situation picture level pixel algorithm picture typical picture relatively fig algorithm simulate neuron detect detection describe experiment h pixel value black grey picture detection choose see consideration reasonable thresholded symmetry least noise practice consistently strong numerically phase nan picture noise
process contain stable extend model addition four parameter n property parameter lose truncation support implication general lose truncate exception family illustrate result discussion numerical likelihood software implement numerical procedure perspective stable context practical take restriction ensure skew result satisfy closed expression function distribution illustrate loss increase number software stable website american stable c c already practically number heavy even fitting heavy tailed represent tailed demonstrate therefore stable sensible select stable admit sub model present expression consider assess
spectral transition approach eigenfunction calculate exact tight relatively approach difficulty low bound without calculate different comparison positivity application area call augmentation da procedure expand see exact comparison mcmc prevent use traditional theory theoretical comparison among index tend hold consistency
separate conditioning though line sum doubly past past stationary simplify stationarity equality dr neither dominate doubly estimator whereas dm require similarly dr section argue compare dm look fast treat past policy suffice moment variance derivation write sum let policy doubly eq account reward second variance weighting
onto suppose coordinate equal coordinate zero feasible p continue obey coefficient rhs suffice pp desired sense x straightforward extension vector suppose order claim become without none vector vector follow inequality obey furthermore use yield q ready
distinct look month observation year show user month look look day week month conduct focus correlation query volume trading volume amazon com netflix make whole day month whole year fig month volume trading volume observe significant stock trading activity typical user suggest profile people financial expert remarkable trend order confirm analysis yahoo yahoo yahoo yahoo visit user yahoo complementary standard engine include page query search yahoo yahoo statistics limited line yahoo user yahoo gender country seek behave differently yahoo user set least half whole property aforementione respectively report worth observe user small representative yahoo concern gender include slightly country get find come california new aforementioned states united acknowledgment support grant cm cm yahoo research diagonal le south uk institute advanced study mail live leave
none call action dominate dominate action action none theorem two context allow generality action vector keep action remove keep remove action correspond keep equal dominate dominate action outcome game draw vector action lie bottom boundary hull action rise connect non dominate action convention order accord loss outcome introduce chain dominate action consecutive degenerate degenerate soon separation distinguish game merely technical proof exclude minimax game outcome monitor finite monitoring outcome satisfy separation outcome satisfy er dominate action regret satisfy otherwise respectively prove trivial minimax dominate four dominate action trivial appendix prove entry change change achievable
rigorous consideration statement definition complete note q site lattice side vertex maximal vertex rectangle exist constant inequality close closed left denote cluster contain obviously proof help pp everything product apply thresholde operation step standard step operation save position position white exponential false iii prove object picture assumption thresholde
limit finite capacity assignment place assume mask variable correspond create coin bias determine whether show ibp appendix package available package implement cm p crp mixture crp crp matlab ibp matlab david ibp latent factor matlab key choose appropriate appear choose nonparametric method side step determine introduction nonparametric use question model select one metric usually include fit second term simple one provide different rather grow observe traditional clustering analyzing nonparametric previously mainly random chain monte nearly later tool nonparametric central formal mathematical using hide model make posterior hide likely bayesian complexity advance survey bayesian bayesian mention mixture
discrete countable cutting cutting becomes cut iy z ny ii nz subject control group assignment fix exist assume consider cut estimate cumulative however cut end say denote extend observation identically ij auc converge density follow dominate variance use calculate gold require computation one diagnostic variable motivate maximize auc study good multivariate aim usually involve issue since combination give
result still room improvement usage variance distribution exp tc action hold hold however fraction suffice direction pac combination pac bernstein inequality enable finite exploration exploitation selection simultaneously direction result evolve possibly
environmental sciences national fellowship contain show pointwise let define assumption b design metric denote assume satisfied assumption every b put break truncate positivity mesh intersect parameter make mesh subset covariate region hyperplane b constant loss generality take bound theorem respectively random design equivalent case require care rely let condition nf nd nf nf f f f quantity cover supremum norm iv meet work linearly transform suppose b
break guarantee know construction validity let show approximate reveal choose possibly constraint quadratic hilbert densely selector computation subproblem classical choose weight subproblem orthonormal rather saddle particular situation slightly different decomposition characteristic function admm projection step program quadratic algorithm q set close onto intersection accord set create intersection successively onto individual projection ht primal find constitute explicit project decrease estimate note application appeal even set rectangular image detailed geometric interpretation mr projection problem however enter slightly sophisticated present q onto index set property continue necessarily illustrate capability regression yet different denoise apply aforementioned henceforth moreover forward define amount implicit diffusion time solve
maintain objective master computation researcher delay asynchronous synchronization issue certainly optimization decade least seminal minimize prior assume throughout however al stochastic possible aggregate low modern stochastic et network possible asynchronous subgradient incremental take date illustration asynchronous subgradient suffer rate delay procedure elegant play delay delay convergence centralize attain asymptotic objective approach provable asynchronous delay parallelization delay update provable cyclic worker compute gradient pass date master master diagram gradient build processor compute stochastic common convergence processor appropriate hold independently remain show discovery update suffer delay delay show different range range either centralized delay paper
pricing achieve demand regular respectively note regular pricing detail pricing strategy detail regret demand distribution constant choose construction reduction case pricing sequence strictly increase item fix instance sale agent correspond instance agent round remain interaction agent since happen demand sale problem pricing instance achieve realize instance let ok ok kn part roughly bind become pricing monotone hazard price p np sp pricing proceed consider price price pricing essentially close parameter minimize mechanism approximation regret offline benchmark demand satisfy monotone hazard rest devote far appear difficult prove else lemma lemma additive benchmark pick multiplicative compare turn offline demand regular let price kp technique also detail least easy go observe hold examine stop choose pricing encounter sp get satisfy exploration claim
analytically minor lemma adapt assertion proof fact z otherwise rearrange together fact sequentially hoeffding martingale pac martingale sequentially long contribution possibility sequentially encounter apply positively limit feedback exploitation although art yet bandit whole reinforcement bandit contextual reinforcement domain incorporation beneficial
may genomic coordinate secondly gene segment throughout consistent l relation exist allow relation specify closure closure denote denote set element also alignment alignment end align read distribute compatibility parameter read start distinct generative seq poisson count distribution multinomial equivalent likelihood maximized discuss seq clarity detail nz priori arbitrary obtain combinatorial useful summing suitably leave maximize independently right unconstrained maximum equation eq interpretation poisson interpretation formulation adjustment bias overlap genomic coordinate equivalence distinguished overlap genomic whereas map present rna seq includes alignment location relative restrictive simplify description select site within depend abundance site relative determined content position length
return try every every corollary tree finitely corollary subgraph adapt dimension integer integer edge color complete subgraph color call call sake every ordinal positive lm l positive assertion km vertex red otherwise assume u jj u u il j j jj l li n apply n rx v show inequality I condition theorem system distinct obviously fa great verification gb gb n side rs side linearization
propagation equation set ordinary differential ode uncertainty condition polynomial handle quadrature nonlinearity model moment condition give low moment depend thus infinite hierarchy equation moment one closure interested first order moment closure variant transform closure interested find induce knowledge independent ds describe uncertainty use convolution di di
rectangle rectangle rectangle rectangle rectangle rectangle rectangle piece edge become rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle generalize close introduce notational straightforward adjacency independently independent observe ij effort kronecker exploit independent bernoulli approximately one follow recursive initially proportion expect north west north east south west south east via integer proportional repeating time reduce probability time reject code v vx nk start kt start end kt end ks start length digit integer edge value multiplicative attribute bit vector even bernoulli ki additional reveal representation call sequel ease analysis initially assumption relax counterpart sampling attribute
locally indeed fill several propose strategy interpret improper local cluster mode performance computing platform check optimality criterion model design experiment l build learn onto context leave krige build minimal reason refinement krige predictor mean costly state factor contrary evaluate estimate get classification counterpart consequently instrumental towards counterpart propose stop refinement stop problem limited refinement variation happen problematic define x denominator cause argue never exactly krige equal zero precision figure divide three independent step require coefficient estimate number
auxiliary polynomially optimal nonzero element stand constructive principal component polynomially identify principal introduce spherical variable exist index argument polynomially develop degree well study tool pc computationally modify lasso elaborate nonconvex combine tackle nonconvex technique program present
x I guarantee give high produce w I adaptively hyperplane small iteration combine hyperplane gaussian obtain good set na n di iw ji ji main loop terminate iw iw j j elementary calculation ji terminate know end eq last get claim output w least fail markov fail proof hyperplane separate analog well motivation perspective machine hyperplane correctly point separate separate exist provably hard hardness admit np
one n objective regularity great consequently locally lipschitz meet recall q strict exceed curvature r meet sum argument quadratic continuous meet minimizer converge finitely descent minimization finitely iterate connect iterate accomplish iterate guarantee differentiable coordinate elastic net penalize change continuous differentiable define prove apart subdifferential differentiable expect function iterative solver give pair symbol hadamard active speed computation initial change active tucker meet kkt violate repeat sn compare regression rapidly warm start regularization start iterative subsequent regularization apart sufficiently intercept come small regression q grid equally spaced
shift recover related discrepancy explore term try prior work around vector take base set purpose description closely report pre coverage interesting epidemic understand report complete reveal inter infection feature infection frequently locally persistence couple average reporting take analysis use represent period pre standard therein birth period area rough infection part birth effect conclusion biological want answer transmission school period versus non period period movement infect people time period question period infer hence key almost indicate change infection period n kt interpret amount movement represent amount infect leave city th infect people come city
wrong whiten whiten correlation problematic solution whitening case whiten covariance section without take root computationally whiten way pre whiten sparsity undesirable mathematically way naive whiten property solution corollary global nan right value unique addition rank value factorization comment operator semi definite global non obtained immediately result stem relation svd decomposition subspace rank possibly solution still great flexibility conceptually computation massive obtain storing permit massive structured k solution global problem variation calculate high dimensional set approach discuss name vice versa problem follow scale allow high quadratic operator wish decomposition recall finding combination linear combination account structure transform norm arrive loading orthogonal right factor proportion find one massive section outline method relate methodology help quadratic operator class quadratic statistical structured leave quadratic operator connection interpretation well correspondence connection variate covariance interpretation separable
theoretical low bound fig estimate noiseless capacity capacity dimensional constraint eq noiseless child channel width horizontal dotted shannon capacity
strong version require symbol follow directly theorem sufficient theorem condition remark may lemma v total decompose rv ii verify decompose cv property h n yield finally choose eq q conclude bind follow xt path view space iv lemma j ba b k lemma exist theorem rx rx r pa self adjoint adjoint therefore decompose argument n jt jj w n w w brownian motion hand side zero boundedness approximate constant apply replace course brownian motion w p moreover similar finish done cut h pt q replace assumption assumption easily may term right uniformly imply r show thus finally check embed space symbol differential finite proof subtle appear differential universal denote change p establish ap p integral space
share segmentation token build token instead short token apply way mean token token string token mle token token token token example segmentation mle follow unknown cope string token possible generate certain follow count token give time string q
reward play challenge problem work unknown develop original applicable bayesian setting propose bayesian meta treat different develop channel achieve logarithmic average first literature generalization arm bandit mab decision uncertain arm player seek time maximize reward obtain challenge bandit arm evolve bayesian reward model mab update parameter probabilistic objective know regret
case focus state theoretic extension presentation replace polytope present terminal robustness property terminal important prove terminal feasibility maintain variable model mark uncertainty state nature reason science input know gradient yet mathematical recall fact first steady suitable matrix k k fact terminal reach admissible control constraint input property formalize eq q component parametrization arbitrarily maintain invariance scheme even appropriately use feasible nominal nominal trajectory feedback grow initial nominal initial horizon width nominal lie trajectory lie imply control keep optimization define q feedback
c forward central arise third table twice difference central difference arise central arise net laplacian fig grid map original net forward city even centroid dot line good solution multiple periodic nonzero anneal fig city many cyclic permutation number show shift sign produce semidefinite want power general infinitely must agree modulus fix dft eigenvalue correspond complex power several thank dft dft long dft divide inverse state symmetric semidefinite even eigenvalue matrix sign prove matrix decompose associate first matrix cosine diagonal summary always eigenvalue obtain inverse dft verify forward central difference family next imply symmetric p even real pd pd formula centroid confirm q consequently even original number nonzero already k number sign proposition odd even table central element centre high plot order coincide odd strictly though carry analogous sign c spectra section define odd decay finite pattern difference even index odd shall statement another different option combine net generality combination differential accept high order forward central result convolution fourier combine domain spectra since consecutive forward equation power negative row square pass matrix maps forward n mn mn express square mn relevant taylor vanish constraint derive horizontal pass net heavily direction contour around align circular however basically corner would column interesting show behaviour methodology property periodic c detail additional way perhaps use idea particularly domain rectangular adjacent domain investigate simulation b elastic nonzero fall net centroid discard size net dimension toeplitz g eq hard formula eigenvalue plane become contain analysis many extension derivative periodic grid think derivative area even derivative linear centroid net
large benefit curve trend fashion curve uninformative additional yield substantial relevant ht neighborhood reconstruction regular node neighborhood one vertex vertex may adjacent estimation neighborhood neighbor node group term approximate approximate sum remain necessarily adjacent ij ij list neighborhood repetition concatenation paragraph pick frequent appear list exactly frequent error contain pick neighbor appear node appear incorrectly node
zero lp recover binary give sufficient signal link uniqueness still open unique solution lp use unique lp eq denote
string name early convenient treatment due require relate far statement input proceed negative lemma perform third subsequent create necessary link section letter alphabet find irreducible coincide conclusion algebraic process light process imply string process process compatibility model def call rank string alphabet finite string example process let process take characterization process submatrix ask agree rank string generalize equivalent trivially arbitrary argument proof know argument
solution precisely try eq similarity produce refer entry order constant compute stein expect risk less formally minimax maximum bayes minimize risk bayes risk bayes favorable prior minimax estimator bayes specify set estimator reduce minimax follow extension minimax work simulation draw matrix different minimax q since visual summary set green many know estimator stein variant fall prove green show generalize pool proposition familiar regularize choice choice classic section entry diagonal non entry set task follow pool entry choice pattern towards write appendix use similarity nothing initially however asymmetric regularizer
taylor evy x keep return return posterior rejection rejection sampling expansion expansion axiom theorem conjecture exercise theorem theorem paper propose estimation center clinical heavily gamma relevant augmentation first gibb need analytic traditionally come logistic example allow incorporate technique bridge compute regularize odd augmentation naturally suggest default jeffreys method case analysis center fit way contingency table augmentation focus table common meta analysis center practical approach problem nonlinearity augmentation address issue need analytic approximation hasting describe involve rely upon mixed topic table trial efficacy arm treatment suggest treatment
option meaningful display scale visual comparison informative question reasonably rare human intuition poor look call option imply tail look experience tell inspection tail reason formal payoff likelihood provide unbounded paradigm outcome g research belong implie bound infinity neutral use indistinguishable growth view research law probability say product choice derivative huge market product good thing substitution see extreme view expression view instability expectation reflect market place extreme limiting end
state state suppose positive set b xu b definition originally convex concave however counterpart risk contraction hold rv v rv c add change side inequality vx c proposition iii first inequality rx vb monotonicity replace let tv tv tv b tu u tv b rv f tv suppose f tv tu tv f due f v w tt tv tu map b theorem relation endowed induce equivalent fix conversely point exist th ht h due hold risk axiom rv risk measure map control mcp ax
cdf may adequate rest article organize variety application mle unknown test goodness suitable statistic statistic study exponential model obtain example either take logarithm denote natural logarithm show
mab unknown arm arise short path span dominate unknown design generality performance though offers show inferior classic policy target specific whether one finite performance generality edu armed mab problem arm unknown select play sequence exploration exploitation arm policy tail reward achieve total ideal reward heavy exist th knowledge moment tail offer logarithmic mab general parameter cardinality variation mab include mab objective player observation mab dependent arm arise span dominate unknown multi armed decentralized armed bandit combinatorial armed mab sequential arm choose arm obtain different maximize reward mab range
heuristic level extension interaction framework matter scalability error particular mode grow construct give language bias experiment department college gate uk mail ic ac uk ba e mail uk principles division national mail case drive iterative design framework call virtual use open semantic logic programming one desire property comprise event specification sharing discuss system section describe principle action bind member proper acceptable state matter cx terminate term state context outline formal framework adopt mapping answer set event bring state concept capture evolution
standardized intensity characterize dependency latter resemble joint deriving illustrate high htp x represent density trivial great integral denominator variable positive mean proper denominator order denominator use gibb random
solid amplitude sensitivity observe statement specification alarm define uk measure statement limit frequentist notable trial look elsewhere probabilistic statement one property illustration simple signal encounter nuisance
nonempty x gs verify property gaussian entry decompose finally vi cone except loose surprising dependency among group signal group much large signal measurement refine block see theorem group pay extra measure group signal noisy observation noisy solve atomic relaxed constraint take
choice furthermore weight candidate general powerful illustrate among scheme accord proposal performance reduce produce would version manuscript work project ref ref ref project ref corollary es algorithm advance normalize technique weight specify analytic arbitrarily great mcmc also give
bivariate underlie copula belong page inequality identifiability e van page satisfy condition iterate algebra family already check page sample rescale df correspond df q compute solve iterate q estimation estimation whose q absolute hand copula absolutely density base consistent root main copula moment improve score l moment statistic fact interpretable special one
power method engineering discussion phase similarity multiplication experiment explore reveal matrix final objective guide choose design adapt conduct purpose implementation parallelization evaluation give runtime th parallel generator non zero index width height demonstrate relationship generating compute load locality requirement compare experiment list multiplication square dominate particularly actual complexity multiplication square remain figure generally assumption plot show summation beyond storage multiplication line linearly plot theoretical benchmark illustrate theoretical prediction section curve curve prediction gap explain store total running equally stage discover cell plot vs whole picture
first rewrite invert sum sake follow ti q recursive simplify expression weight difference deduce prove consequence proposition minimize constrain almost everywhere tail pareto estimator mild illustrate finite index move tail denote survival zero refer comprehensive extreme various framework numerous dedicate
theoretical notation logarithmic conjunction continue bring acknowledgment nsf finding recommendation material necessarily view mark huber college edu high relative often large make nest count percentage fall bernoulli shrink balancing consideration provably ideally
solely static brain call two main conclusion brain
effectiveness seven retrieval future area ability gradient objective enable wide possibility retrieval document particular recent advance gradient discrimination speech general broadly gradient operator correspondence finally linearity expand prediction representation acknowledgement rank ranking present difficulty fact permutation include precision cumulative expectation characterize marginal expectation
one zero prior combine element another vary useful analyze area finance environmental science investigate platform biology finance environmental sciences inference conjugate wishart place element wishart parameter definite matrix normalize cone matrix constrained indicator wishart successfully limitation wishart prior element computationally challenge recent advance model rely completion intensive decomposable graphical expression unstable graphical determination determination place prior prior non decomposable fit jump metropolis hasting unclear essential absolutely continuous shrinkage subset element represent flexible develop
behavior incomplete gamma modify integral behave eq satisfy relation call dataset field evolution observable area anomalous diffusion model find system include inside period also process might
consequence constrain satisfie certain condition result theorem quantity nonconvex obvious costly concern rigorous iterate run project descent controlling assume previously surrogate surrogate aid follow deviation satisfied version sometimes easier derive tight inequality second corollary verify various form quantity change result global surrogate apply claim probabilistic enter analyze statistical satisfied lasso choice past allow surrogate example arise noisy dependent satisfied probability miss nontrivial inequality principle optima separate optima triangle inequality guarantee similarly program scaling optima ball whose nonconvex additive plot rescale align rescaled plot addition specific simulation additive state
sx sx p p dy sx sx dy inequality page assumption normal mixture explore rate class densitie construction offer study break keyword lie non mixture density construction part derive class main construction desire scalability
benchmark ise correlate homogeneous fundamental physics finance natural correlation direct ise physic constrain capable reproduce average configuration infer ise model mathematical implicit equation interaction average boltzmann various inverse ise bm learn message method likelihood
chernoff twice claim small apply cover q proof assume triangle inequality corollary exponential sum dependence explicit matrix dimension chernoff inequality analysis analogous dimension quantity small bernstein moment appear regard
datum embed embedding set training integer fig illustrate fig embed quality quality embedding improves increase validate visual manifold learn wide first new motion coordinate effectively normalize embedding topology set validate propose improvement future state meanwhile accelerate manifold densely datum lie manifold outlier efficiency solution implement denoise manifold work different acknowledgement partly china china grant cb zhang current natural quantitative measure assess learn greatly limit assessment normalize embedding le euclidean meanwhile scale quality name scale efficiently configuration
bound kl ucb significant kl index low reward ucb kl ucb solution adapt ucb law simply change definition build great elementary independent interest sense concentrate well bound significant advantage ucb ucb even ucb various compare bernoulli organize follow section kl kl ucb case kl bound optimality extensive experiment show superiority kl devote elementary main interest action expectation step action observation possibly randomization get
configuration implement logical function first set neuron equivalently percent logic absolutely occurrence output neuron compute put become matrix connect fuzzy inference input neuron example become occurrence neuron neuron neuron belief fire neuron fire fire neuron neuron high neuron locate neuron put thresholding network two output thresholding neuron eliminate value affect rest neuron thresholding value neural determine process fuzzy logic know necessity thresholding function use aforementioned instead common determined threshold modify create hardware implementation fuzzy way exclusive gate show version note base system base rule base way utilize
correspond weight nonparametric maximum maximize goal fit seek consideration another modification typical whose choice penalty bic penalty discuss sophisticated scad context factor practical posterior distribution mix draw dirichlet dirichlet imply surely well quantitie flexible structure algorithm fit dirichlet unknown approximation approximation efficiency suitably search essentially parametric problem anneal novel annealing call asymptotic property asymptotically subset measure kullback divergence act like distance
cluster expression profile bioinformatic genome article material li david term denote indicator gamma
hypothesis procedure fisher reject level calibrate ensure collection introduce throughout section successively nan level accept note reject q hypothesis reject collection accordance collection whereas final consist successively negative upper theorem notation model family testing accord hold imply derive procedure follow inequality accord matrix appear orthonormal orthonormal projection thus explicit namely coefficient hand reject point family x order define successively accept consider
separate separate triple satisfy may lead possible sep set step oracle version conservative use finally step algorithm triple result pt triple structure triple version conservative step use proof oracle version obtain well version modification give oracle let prove output maximally orient tail orient result oracle orientation orientation really causal modification give graph additional path size graph consistency weak section level independence find initial set list update step step initial skeleton definition triple orient base lem triple rule ix jx jx ik x ik detail step list store triple find condition triple conditionally check step subset conditional kx ix k j removal triple add moreover triple remove testing removal edge find set example show
execute use present problem since four near neighbor anomaly compare hypercube hypercube four nominal one dimension draw nominal nominal anomalous anomalous distribution experiment measure square euclidean curve auc simulation grid six select require select choice search ccc method best nn nn thousand approximated spline seven represent use trajectory dissimilarity instantaneous along histogram manner dissimilarity trajectorie speed histogram trajectory uniformly along dissimilarity euclidean roc k training along experiment consist consist auc compare spaced combination auc criterion worst show perform false weight auc find
assume binary generalization normalizing ensure illustration tractable ph strategy minimize average achieve necessary cd parameter note cd neither performance use objective refer hybrid discriminative perform separate rate wish layer expect contain predictive vary layer activity large issue generalization copy layer approach make present different nature input believe nature
discard discount assume daily cover area prior write describe uncertainty true specify model specify ik lead treat specify term approach nuisance round mm challenge mode input log effect consist point intuitively residual residual optimisation objective provide robustness show dotted sample marginally ccc transform indicate observed value summary joint posterior implement adjustment use kernel scale individually margin select estimate understand adjustment prior informative statistic similar incidence change complicate moment difficulty adjustment marginal wrong place inferential purpose cross validation usual predictive involve simulate posterior high nuisance abc raw output available
model expectation accumulate initial storage subsequent analysis perform simulation interest hour option suboptimal completely avoid describe simulation bit sample operate proposal suffice preserve complete influence reject mh view channel bit operation necessary writing bit also help
discrete nonnegative limited range present flexible beta completely model discussion binomial correspond binomial create evy aa assign infinitely borel set sign generally evy compact evy notation nonnegative l evy additive bp space define product evy concentration parameter infinity satisfy measure atom atomic mixed sum appear location atom unit atom construction attractive distribution diverse modeling binomial
discriminant combination path lda change basis change often extreme good believe difference propose diagonal matrix mean implicitly explicitly look decision suit remove decision suit generally regression interaction main hence relatively accuracy condition convenience subject psd recall one semi definite program
interior hypercube interior cone condition nf l sf h distance rkhs norm mu fill adaptive mmse difficult problem dimension address course suboptimal
well observable covariate discuss property detail appendix various empirical focus population thus estimator use size population heterogeneous biased latent approach circumstance interested heterogeneity explicitly individual thought variation size age put finite population collection method parameter covariate observable therefore classifying avoid thus covariate thompson employ comprehensive comparison specify despite justified problematic capture conceptually parameter nuisance capture
agree member determine ensemble rest grain compute assume distribution accomplish processing phase learn aggregate together final implement could framework distribute file define map pair reduce execute assign file map case correspond pair pass node function receive key associate produce map partitioning scheduling failure file assign show train ensemble local random show thus receive approximately output file step use importance sample build repeatedly tree induction produce comprised base incorrectly repeat bag get prediction ensemble see training outside base training differ receive equal ensemble simplify merging ensemble show contain incorrectly avoid possibility draw ensemble mb
possible state transition time may update subsequent regret time suboptimal infinitely information state bounding play introduce remainder denote know way ensure belong subset indeed exploit continuity start introduce state lag state play define state single infinitely partition set state contain infinitely belief belief stationary distribution belief although depend arm set infinitely belief lie outside belief increase element partition rewrite contain l infinitely information define extension belief belief diameter diameter decrease enough action follow l gap define diameter hull infinitely state exist diameter small extension hull conclude optimal action player matter close respective action suboptimal know policy suboptimal grow challenge lead first almost always optimality control horizon choose false state arm set state n g simplicity analysis true action optimal I draw player play begin probability zero subset draw bandit f hold
becoming infect per recovery patient approximately treatment recovery average duration infection become individual equal average development become transmission transmission probability average period infection duration infection duration dim scale degree age non linear linear prior c period median infection median infection h ccccc cccc c cccc h age ccccc activity university south south grant bb g university methodology forward carlo calibration system ordinary differential epidemic type involve age gender relevant infection type two paper extend population formulation suggest mix matrix allow unknown mix matrix involve epidemic describe member population bayesian allow duration treatment duration gender age group unknown parameter relate trajectory markov markov chain recently human far classify potential risk basis association transmission infection causal factor majority cancer head attribute develop clinical infection subset type high publicly national national costly decision health economic infection small resolve cancer many decade acquisition employ incidence benefit calculate relevance transmission population force infection infection individual infection reduce benefit order course
trend need well trend fluctuation series trend state fluctuation trend identify procedure smooth many evolve low may old become try point model little reason take opposed transition nothing evolve totally fashion period predictive history lead forecast e contrast forecasting collection matter path process prove forecast well hard see face arbitrary adaptively combine
set hausdorff compact nonempty completeness compact nonempty grant subsequence limit nonempty subsequence contradiction note exist minimax state minor applicability space semi continuous semi continuous apply semi iff monotonic grant monotonicity analogous continuity low semi continuity monotonic scenario concavity iff affine since affine repeat author thank discussion manuscript relationship stochastic establish set minimax theory grant spirit opponent reveal opponent repeat select play game game value single play essential need precisely second case
scale temporal behavior common autoregressive dynamical series exhibit extent unique alternatively use generalize transition emission class broadly address attention treat combinatorial parametric variety align interact univariate use approach series independent factorial hmm underlie widely ibp evolve markovian instead focus modeling series series align problem detect anomaly share among time hmms trace state represent dynamic bp ar hmm discover series ar eq fig truncate ibp sequence color mode one autoregressive white dynamical order autoregressive ar cc estimate average mcmc series autoregressive indicate define tail ar dynamical mode positive occur define ibp estimate mode sequence map mode map maintain infer dynamical series discover cause commonly dynamical green making nonparametric incorporation additional dynamical bp ar hmm hierarchical ar hmms tie together share parameter hdp ar mode hdp prior hdp hmm ar dynamical third hmm assume share matrix match
nonparametric condition positivity linear eigenvector equation frobenius matrix functional analog uniqueness discount factor let pdf q matrix analogy clear eigenvector condition conclusion nonparametric iterate inclusion equation identification discount factor identification utility consumption homogeneou positivity marginal utility support risk use positivity correction formation utility analog stationary compact extensively long compact general cite paragraph apply consider generally identification return equation restriction instrumental additional restriction singleton chen provide sufficient condition identification restriction important model usefulness primitive quantile iv consumption base asset pricing appendix square h e h open assumption hold open ball invertible banach theorem map conclusion list suppose operator follow step select basis perform step sequence
average speedup achieved serial delay measure get nearly linear speedup computation raise rr ever slow gradient question run simulate figure c time achieve delay take long able million gradient per hour gradient intensive rr method take advantage sparsity machine enable linear variety implementation instance problem moreover speedup even computationally intensive scheme occur certain particularly add update computation sgd memory processor recent propose bias avoid processor enable acknowledgement support nsf award cr support air laboratory contract nsf award award
scenario explain satisfied triple goal point quantile qr ensemble particular highlight potential benefit triple research note qr respectively estimator consistently plug therefore parameter readily obtain dispersion ensemble model chapter highlight limitation spatial non spatial observed range exponentially parameter aspect calibrate accord moreover note poorly compound symmetric rank unit choose control skew asymmetric deal skew assign low value ultimately applicability automate preliminary conclusion chapter superiority car car quantile spatial car laplace quantile qr normal car yield plug finally analysis quantile concentrate mean therefore car dispersion specification laplace car variate distribution fair car comparative spatially structure qr parameter however empirical tend skewed quantile regret formulate plug point could joint quantile differ rr change produce preference another need triple plug function indicate estimate specific wide rank risk gr heterogeneity triple estimate constitute area specific risk report summary heterogeneity interest inferential objective however gr highlight function cm chapter parameter ensemble different rank total amount answer threshold classification threshold total specify element review research rank adopt concerned case spatial spatial unweighted decision originally mean plug necessarily base decision surveillance datum uk good classification false problem point several statistical classification datum search pre class essential ingredient classification dissimilarity classification dissimilarity theoretic arise accord area posterior particular wish classify mass necessary allocate accord attract france study cancer breast cancer plot table cycle north anchor west dp p consider great general classification choice apart dependence therefore ultimately particular inform risk surface statistically problematic sensitivity specificity encounter frequentist adopt note discrimination undesirable consequence cut optimal threshold decision vary choice rule different mix choose chapter ii discuss calibration consider mix spatial draw extensively rank substantially differ one easily similar oppose lin et al despite inherent similarity classification substantially quasi theoretic advantage classification require optimal specific classification simulation plug estimator spatially section illustrated uk chapter discuss broad light theory point give decision framework estimator estimator link concept sensitivity specificity advance cut terminology express positive negative concept misclassification I false misclassification parameter denote reverse base specific parameter interest threshold classification loss unweighted threshold advantage quantify ensemble normalise pc p correct none element loss precede obvious context section threshold quantile equation distinction quantile quantile chapter function chapter quantile proof estimator fact solely consequence respectively indeed follow proposition weight unweighted note early immediately proposition describe identical trivial former whereas chapter unweighted relationship supplement simulation evaluate plug rest chapter focus unweighted except otherwise unweighted page expect formulae ci classification correct incorrect penalty vary whether classification distinguish false diagram former cut positive threshold threshold observation diagram attain expect great zero proposition justification gray corner anchor cycle anchor north east scale north plot cycle anchor west dp scale anchor north node anchor north north east node north anchor north anchor anchor west dp representing rank interested plug rank concern proportion estimate ten percent condition classification percentile cut denote high wish percentile false positive function percentile percentile percentile refer percentile except otherwise notational distinction fp percentile base unweighted analog unweighte describe estimator report refer reader original proof
target conventional system mode model eq velocity observation random process observation mode summarize mode cause ground move trajectory tracking target mode finite compute typically art interact multiple variable move road type markovian sequence motivate level abstraction call suppose surveillance interested pattern track infer possible loop circle exhibit construct characterize various linguistic main devise polynomial syntactic parse spatial pattern conventional track perform syntactic filtering abstraction track compatible design conventional tracking four form production stochastic regular sec regular regular polynomial practical hmms dependency embed syntactic filter system several formal production human easily pattern building permit high trajectory ability encode important lack data trajectory model difficult handle consider model hide branching regular hide parameter regular multi branching rewrite calculation target indicator platform syntactic filtering etc vast amount motivation develop yield interpretation track main result describe track formulate trajectory arc describe syntactic fit complete
ground result overlap superior determining pose support part da nsf thank support program content
coefficient specifically wavelet coefficient certain location scale small neighboring fine child magnitude accomplish overlap function child depict child group leaf coefficient orientation vertical orientation fine many option grouping
algebraic calculus base evidence function assumption probability inference algebraic additional justification allow consider positive integral inductive updating oppose aim lattice relation evaluation transformation algebra particular form inductive dynamic determine inference logical inductive justification impossible practically convenient axiom constrain relative candidate inductive inference evidence appeal expectation bregman entropy metric expansion ar entropy iv frequentist paris kullback leibler relative updating
decay generate fast detector plus energy event event take ref expect surface simply curve indicate dot induce integrate several region sensitivity low mass prior note curve fact show infinite inference choose carefully sensitive prior manner irrelevant break small expect count property define prior parameter beyond count numerous test argue interpretation test prediction risk reach conclusion example region complete opposite access result consistent systematic address prior
define base sum generalize precise dimension hilbert space continuous space act semi sequel minor statement main illustration consequence space finally cone semidefinite square g vector hilbert consist sequence equip inner begin background hilbert space example hilbert basis positive decrease converge k k k choice inner function namely element equal converge special symmetric function open subset semidefinite
rely framework provide variational hyperparameter consider work offer move section establish adapt gps applicability advance approximation enable thank university interference alignment author science innovation grant collaborative effort provide gps association source propose mixture novel characteristic gps mixture global clustered space variational recover show problem problem arise position association surveillance tracking htb human observer effort
knot locate posteriori consider location fact knot correspond surprising inference location
measure place nn implicitly calculation together give minimax rate minimax bound homology inference various possible ambient implementation homology become thank advance topology homology several sound drive volume lemma tangent ball collection minimal contain manifold dimensional piece add smoothly join inner piece two construction volume curvature evenly manifold guarantee otherwise construct outside manifold differ remain spread evenly density disjoint clear bind calculation main tell except volume homology least notice minimax straightforward case noise identical noiseless factor manifold identical step clean datum eliminate point radius homology probability
field mat ern unfortunately equation arrive function triangular mesh easy computed ij ds sparse therefore ds essentially numerically increase q zero matrix correspond ern field define sense piecewise nature vertex centre triangle advantage piecewise basis lot work leverage show functional derivative functional length edge mesh functional random space square fs ds fs ds q constant large triangle
covariance expand taylor third moreover expansion operator functional equation decomposition split three part estimate main quadratic equation build approximation conversely term procedure give suggest build size choose onto lead f x term negligible asymptotic
acc toy quality performance yield algorithm require illustrate criterion small smc european project simulate scenario job evolution activity virtual individual central france model second intel ghz proximity service initially census economic job job outside drive four parameter simulation census matching criterion census discrepancy observe eight infinity generate prior hypercube select move parameterize twice weight previous parameter study particle
interest specify non restriction let individual vector stacking alternatively contain table subject contingency table response configuration ignore either approach stack span complement fit constraint computation addition n almost infeasible canonical g contribution likelihood level log become
reduce believe long predictor getting want dr manuscript want li reading asymptotically time derive exist readily convergence sgd
object collect ten stimulus throughout object illustrate evolution illustrate ten development universal stimulus illustrate collection object begin network level empty grow intra module object add threshold intra inter take area spike collection feature map update implement previously dash coding cause reciprocal feature aspect change cause intra inter connection schema area inter level system principle organization recognition spike structure feature repository model single neuron grow feature storage
rademacher ball view predictor entry maximal singular zero elsewhere amount analyzing expect meaningful generalization indeed rademacher complexity know regularization fail focus uniform complexity ns magnitude argument analysis reduce logarithmic recent bounding tp combine various rank guarantee regularizer receive regularizer superior art uniformly correspond also assume predictor sample nm compare three avoid dimensionality slightly bad make incoherence guarantee trace trace magnitude uniformly show assumption regardless use focus optimality condition work ball trace mostly year denote frobenius surrogate give max row norm optimization involve constraint trace norm possibly
parameter good vc query column query boolean clause thm combination run join query size tuple combination join dim vc dim different due limitation increment number store list estimate histogram impact intra uniformity satisfied help estimation combination join create query predicate involve clause operator clause wide compute histogram query histogram method predict especially independence default histogram well prediction situation percent predict analyze quantity query deviation percentage decrease technique predict histogram hold fig histogram soon method prediction keep improve sample grow interesting deviation prediction histogram suggest high prediction fig marginal impact estimate sometimes give prediction less explain histogram reason line
kernel apply extensively effectively machine tool linear datum space high dimensional computation feature possible pca leading eigenvector take become match identity perfect pca whether simulated signal signal ec choice depend kernel ec knowledge kernel kernel manually kernel lead eigenvector detection case subspace know similarity work national
detector outperform aim measure difference realistic set compute within mean absence thus probability occurrence relevant improvement relevance advantage availability relevance make depict always every vertical meaning word depict idea relevant relevant query word topic indistinguishable close event explicit lack event although compute latter valid estimation depict depend depend discrimination query word confirm increase decrease large nevertheless either become small topic
five phase transition model regularizer sign exceed interpolation close show phase low transition regularizer sp sn complexity sharing validate handwritten digit illustrate accord handwritten extract collection dataset handwritten handwritten digit totally digit image provide digit provide total class provide record shape transform profile integer regardless learn ten digit make comparable range appropriate number perhaps large feature sure dynamic division care divide table divide coefficient sample digit sample row digit setup block handwritten ideally solve tune result cross digit get error support average theorem notation proof prove theorem rigorously norm
rhs fashion exactly analogous derivation upper rhs since recall set take ks find proposition corollary partly support department electrical engineering ia usa email edu regularize modified pursuit mod bp condition weak compressive discussion available partially sparse obtain condition mod similar another wang iteratively improve obtain support repeat cs cs residual recover signal early cs cs filter cs kf residual size result
relationship true distribute estimation interpret accuracy design contrast response treat need simplify standard design primary concern initial population population already concern unseen detailed random setting quantify essential design reveal approximate regression effect design soon enough approximation vanish ridge moment particular choice covariate enter except immediately square aware characteristic optimize appropriately theoretical modular probabilistic bound outline discuss excess ordinary discuss design computation inner restriction
restrict integral interval sophisticated equation proceed integrate back substitute follow number appear banach modulus convexity let reduce covering follow complex ball ball method covering banach consist norm measurement basis fouri recovery measurement generalization sparse state quantum physical describe density rank measurement carry measurement previously lead statement successful show sampling operator obey rip isometry acting say sampling distortion rip low work gaussian application rip hold open rip result
player dominant network strict nash payoff dominate utility economic context transmission apply wireless practically constitute measure connectivity utility path formation dominant empty agent pick agent become agent action establish agent become less payoff continue way satisfy game restrictive cf payoff dominant distribute payoff dominant network pool game refer interaction need resource rather resource share agent game pool pool view finite analog pool game common game player call action player failure pool game desirable profile example action correspond action payoff dominate game game moreover equilibrium profile pick increase either utility action
propose method knn svm knn describe semi supervise contain medium time cell cycle phase classify al sample synchronization discard consider discretized realization datum use use
sufficient table distribution see derivation sample reference implement hasting chain uniform sign uniform move chain stay markov basis ensure proportion table great analyze contingency table evidence conclude asymptotic value chi monte dramatically see eq differ eq less package example one procedure run cell case computation basis symbolic computation independence symmetry quasi independence basis theoretically numerical consider markov form move connection variety vanishing
sparsity pattern enough zero stay iterate numerical reference zero advantage indeed randomly eventually proportion spend non zero visible much htp blue iterate grow value start reach stay increase reach perhaps look thing via eq indeed suggest policy accelerate let adaptively change uniformly variant fix change particular illustrate effectiveness shrink apply start origin shrink htp notice nonzero element shrink shrink shrink modification iterate plot exception plot choice point initial choice b reasonable intuitively former preferable start shrink true numerically huge machine block coordinate relaxation alternate support necessarily measure prohibitive space gradient execute usual
pn replace jointly variance normally first central en n pn pn k w k k pn pn n pn normally distribute lyapunov neighborhood fy n summation take limit state let k take value origin p x x lyapunov lyapunov exponent x k exist contradiction period j e k j g x e contradict unique complete skeleton w nonlinear pt k k wu taylor expansion exponential fan page w w k mix mix mix sequence exponentially decrease fan page eq hand pt x e x g ty k mx mx x mx k mx martingale lyapunov complete acknowledgment support grant management institute university anonymous constructive grateful mathematical national nonlinear section width pc school
reference text current style preferable possible guide reference author al three correspond book page chapter necessary reference book first page article refer state report available include reference accept publication citation passive citation environment give place proof argument restriction conclude paper need repeat acknowledgement body information contract reader exclude material material technical code example appropriate please file material title acknowledgement material sentence give paper material style reference contain remark division mathematical science engineering university chemical co engineering mail mail com com u ac select parameter remark without response center change transformation predictor te penalize square square solution g penalty tuning
take model conditional ise exponential parametric vector example estimation longitudinal stochastic material let longitudinal explicitly parametric compute h dd enumeration rely vast majority involve transition metropolis hasting accept walk often mix mixing self avoid potential encourage family accommodate combination correlation background prove correlation connect exist
couple time coupling coordinate x always set show proportional time couple along edge loss since I imply analogous kn independent simplex immediate p certainly give check less prove possible fact event equality time hold part partition assume equality mark successful x equality k st coupling occur inequality
analogy interaction research question cross product advantageous computationally plausible course simulate feed sigmoid activation however abundance matching provide dynamically variable likewise variable product contrast common ica machine auto network interaction interact value weighted product closely model think multiplicative interaction energy rotation predict product mapping unit energy model receive cross model type multiplicative model motion relate frame vision relate relate projection encode independently content energy cross entirely practically field square somewhat fourier component analyse transformation case encode phase variation model know invariance polynomial compose unit sometimes pi unit pi stand sigma multiplicative interaction make build symbolic pdf triangle value three sum filter represent early subspace put sum compute share filter parallel bi linear
label boolean case occurrence embed leaf label pt implication right assume claim give follow query sample combine query path query path boolean convenience boolean query see boolean consist path query representation semantic characteristic also infer boolean point keeping contain redundant path query label query boolean query mention previously query conjunction infer path set path root query need reduce path boolean restriction relax unary unary expression path unary path root select unary iff unary root query begin path unary query class belong practice advantage set store leave leave pattern equality text value rarely path query construct essential result show restriction finding setting inspire extend stand begin universal consider select node sample construct stage htb xx xx xx xx ss r decrease replace let l edge find stage attempt every path refine invariant mutually every stage take last occurrence symbol create copy replace unchanged third attempt execution xshift n edge
abstract formalism exploit formalism author modify probability relevance intend interference traditionally concentrated extract evidence accurately belief ir superiority document ir ranking accordance vector classical available measure vector ir relevant threshold minimize definition use introduce aspect relevance subsequent explain relevance poisson observable introduce observable rank optimal observable confirm observable make remark actual refer l ir concept observable color state recall alarm threshold observable frequency color relevance
establishe independent decomposition matrix leave vector value
conditioning suggest estimate tt xt yy yy py xt estimate tn zero st possible lead mention asymptotic irrelevant least irrelevant measurement difficult censor truncation provide might time range limit large complete ty disease vary converge h h replace propose yy tt
e whereas absence quantum term mathematically information unit suggest customer unit index represent informative interpretation implement indexing time internal physical number answer correspond occurrence index index represent occurrence occurrence something geometrically vector superposition interact place vector observer move subset assertion element universe event belong subspace subset belong span basis relationship g plane span information give whole x maximize eq note connection even interpretation tie together criterion assign whether algorithm implement perform follow read occurrence e feature include relevance etc reject precede ranking
separate concave satisfie know digit trend logistic crf multinomial value discrete naturally multinomial scenario assignment tie q configuration write involve straight logistic simultaneously predict correlate chain crf define normalize computed variant forward backward hide multinomial see crf noun share word combination group plot regularization sequential dependency order use edge variable wish pt c original case know runtime lattice tree minimum structure degenerate structure optimize pseudo conditional form logistic compute use define function denoising different illustrate method
envelope desire follow factorization box monotonicity real multiplication simplicity stick three case along probabilistic arithmetic yy box number dependence unknown fr fr explicit formulae arithmetic operation provide inference marginal box box save reasoning cumulative result interval induce usual strictly rely bit induce cycle cycle cycle cycle achieve still strictly without notation conservative extension dominate natural valid coincide similar hold cumulative arithmetic operation probabilistic constitute example concern harmonic toy concern height issue simple harmonic determine mean fast engineering complete determine actual design account expectation uncertainty box seem quantile necessary rescale take supremum simplify reasonable distance calculation modelling fill node cm cm rectangular xshift cm leave
outcome information gain give time bandit horizon always grow give infinity increase intuition growing bound expand h simple action input markovian determine probability agent try learn corresponding observe perform estimate environment array gain observation accord precise infinite recursion discount initial investigate gain expect
process penalty necessary interpretation regard condition characterize play central condition example viewpoint insensitive state ensure pose smooth control complexity second part kalman formulation operation execution finite value degeneracy support low subspace computational computational practitioner powerful believe model development number application
somewhat entirely estimate stationary time suffer major analogue histogram increasingly cause choose increasingly integration numerical become increasingly kernel however question certainly drawback derive principle violate seed literature series asymptotic generating mix mix rate single stationary independence active series
appear information variable add basic gate three fact interaction indicate gate information focus despite fact gate gate interaction variable redundancy return indicate cccc c default state node receive drive note connection network pr correspond topology c ex ex represent node example partial indicate entirely redundant counter node act independently structure partial information redundant return result magnitude redundancy note interaction exception several correlation amount result total correlation dual reflect scale interestingly unique example return redundant interaction intuitive driving driving note less redundancy interaction result redundancy unlike solely partial return redundant provide accurately drive
picture optimal make detailed describe name swap swap swap reversible minimize rejection configuration iii w jj j start use swap operation construction therefore whole
assume covariance latent specifically graphical space cca advantage low model allow optimize consider come produce pose summarize pose representation software choose similar probabilistic noise latent dimensionality vary give pose computed variance cca square
result algorithm majority language universe denote query description circuit restrict bn call run tn note learn circuit say majority depth boost learner exhaustive negligible harmonic membership extension argument one harmonic harmonic majority depth circuit evaluation result privacy language parameter work threshold reduction release improvement release new improvement central open question differentially private release work database run way release way provide rgb definition question release database contain sensitive answer computationally release seek sub exponential primary reduction private release query thresholded predicate general learning threshold release differentially private way conjunction counting way contingency release bound ignore dependence database database bound database ignore run database answer way run run add answer way distribution threshold sum relevant predicate polynomial threshold overview fouri database value new polynomial release counting query depth counting query harmonic circuit elaborate preserve release threshold section loose release simple reduction query release threshold
du x du final law j type law eqs feature success z tell data exhibit exponential q three note finite subsequent beta bernoulli hold law atom choice discount keep total mass concentration carry extension stick break generate feature bernoulli empirical growth right number derive line theoretical theoretical line leave power law number law type quantity number show classic black case black point asymptotic quadrature plot compare asymptotic asymptotic ultimately prove respective show curve straight behavior describe growth expect feature represent line normalization grow expectation many first change connect middle green eqs index assignment great type iii law type law somewhat easy visualize iii law lack difficult visualize
goal gap problem computational distinguishing enough bottleneck life line research joint stochastic descent potentially demonstrate via importantly advantage example supervise learner past target spam email spam target x natural loss classification otherwise algorithm I unknown distribution domain training assumption prediction goal risk tell risk agnostic pac learner expect risk return denote upper example main propose
aim paper sampling fairly functional theoretic treating procedure treat truncation unify function hilbert subspace functional covariance estimation square penalty tensor rkh rate argue logarithmic factor eigenfunction resemble principal line treat various smoothness functional component translate ellipsoid principal significant substantially sampling difference explain lack lead identifiability issue difficult order exploit empirical well concentration exploit localize reproduce hilbert technique norm finite bound remainder organize material reproduce space study functional statement discussion consequence section result technical defer material supplementary devote lower sharp hilbert schmidt inner e ij introduce
latent quantify associate processing bring idea powerful describe image bayesian model probable posterior latent uncertainty practical compressed sense framework allow principled employ prior heavy filter probabilistic signal compute heavy tail sample gaussian close computation adjust parameter tight mostly quantify quality give rise bound ep relations variational sort integration contrast base order around algorithm require variance sparse point routine approximation capture
tag give situation hypercube possible large meaning shape underlie influence reliably identify look class work panel answer adopt membership reflect estimation near sparsity recognize accurate qualitative influence number covariate binary positive negative medical test outcome determine diagnosis hypercube series therein expansion reason various length short central coefficient widely boolean harmonic hypercube sparsity hypercube describe thresholding recursive thresholded analyze prove illustrative technical denote string use concatenation string kx return empty index arrange order real number denote generic whose change absolute throughout function inner reference introduction instead generalize
nash large search support game say clearly surprisingly exploit technique game exist subset trivially subset small game game equilibrium assign strategy least constant broad equilibrium game ny yx show strong theorem provide game random define pair nash e game small equilibrium universe size form take uniform nash equilibrium go uniform argue polynomial nash define player similarly define nash equilibrium justification lower bind contain indeed least belong moreover evenly satisfy appendix uniformly random succeed equilibrium inverse complete strategy enumeration sampling come game mix
result consistency asymptotic normality leave asymptotically side second asymptotic normality leave er drive let variable true bivariate mean algebra behave therefore establish side right hand direct addition know p p theorem effective size pt theorem program award award foundation science david model year researcher various remain pose address yet question illustrative exponential random answer question depend rate estimation magnitude practical sharing normality random dramatically sciences biology bioinformatic economic mathematic physics contribution come spectrum see perspective economic
role using even find likely table since customer utility choose table choose customer e accord belief construct available restaurant combine game learn chinese restaurant game new analyze predict social recursive achieve tradeoff decision decision table customer advantage playing dominate learn contrary table play later well thm department electrical college md usa communication decision either decision agent learn make decision play role social still limit restaurant random structure decision chinese restaurant process chinese restaurant formulate analyze chinese restaurant game derive achieve network setting learn field communication device adaptation cognitive decision achieve may need may external market limitation expand experience help mostly case decision enhance take account belief belief maximize social impact reward impact e one action reward state state becomes learn agent nevertheless reveal preserve partially reveal network subsequent agent former
version martingale probability poorly handwritten labeling image point decide label tend computationally change pool unlabele sample budget constraint allow query informative pool al overcome query discover query necessary pool al margin minimize provably estimator risk unbiased stream algorithm importance speak round put pool pool query learner minimize risk concern e weighted forecaster expert weight loss pool forecaster prune soft manner place determined pool true linear
j case q set event occur recover clearly pair n k union bind argument noiseless r rs respectively bind subset r sa monotonically second bounding ensure binomial monotonically entropy far exponent apply bind stochastic third bound pmf symbol note low main proof proposition typical delta chebyshev bind law zero arrive bind continuous second exponent prove denote say furthermore upper hamming partition hamming distance loose distance optimize free later use fact monotonically decrease k cardinality differ term cardinality hamming weight position upper rank lemma large equality objective sufficient exponent sufficiently small denominator justify nc negative obtain particular sufficiently differ choice also remark restriction serious claim complete direct circular convolution pmf fourier dft dft probability evaluate whose linearity dft power yield decompose q complete code single linearity since complete appeal pairwise lemma sum eq integer define corollary field element draw pmf constant version also integer imply sufficiently dominate satisfied apply inequality since borel lemma receive electrical engineering college ph electrical
lead eq mean common covariance mixture dirac avoid enforce optimisation lead updating importance replace student robust alternative freedom degree freedom choice among cast section thank shape student distribution know gaussian case moreover covariance pooling way prove clear optimal optimisation sufficient nevertheless express em root use approximation specifically decrease size importance estimate set currently fit associated form note size define lack role latter constant target recursion obviously normalise unity effect kernel usual thus sum unity measure execute iteration decide user parameter instrumental work formulate
list ranking list response gene ranking attract considerable collect position gene ranking vector sample use estimation adapt rank list behind want exchangeability appear ranking gene exchangeable always gene situation exchangeability pair new exchangeability random rx pr exchangeability variant maximal exchangeability distance analogous introduce exchangeability introduce attain exchangeability gene exchangeability score random hypothesis uniformly cardinality normalization score analogously side exchangeability uniformly exchangeability gene exchangeability leave irrespective number gene distance exchangeability panel synthetic framework list list position give rank list dissimilarity comparison specifically certain list determine important list order weight rank universal gene microarray order subset characteristic list create list order denote position list characteristic exchangeability global list use similarity angle
unweighted unweighted since pathway preprocesse keep gene standard group typically select group gene involve since enforce reasonable table select learn version mention coefficient allow remove lead support suggest min fold signature fold choice weight weight fold microarray drug target identify disease form densely information interaction biological perform edge group breast datum present pathway reduce almost suggest informative fold select isolated graph connect table large average deviation might increase connectivity merely cause overall select select gene connectivity gene gain connectivity new drug target cc generalization group sparsity group group connect graph give condition recovery decomposition induce parameter highlight weight correctly empirical characterize collection group group efficiently encode collection case analyse consistency comparison lasso formulation important continuity order prove start section notably theorem ingredient lemma several lemma technical lemma correspondence resp correspondence h c correspondence l u check c moreover product state maximum continuous real correspondence consequence indeed definition apply subdifferential view direct consequence maximum correspondence correspondence easy verify value need compact therefore first ia md ik u c classical f paris france norm sparsity predefine overlap call group latent
explain high order se perform due hyperparameter become performance benefit integrate hyperparameter gaussian process generalize class additive structure indicate additive dataset allow gp recover arbitrarily flexible modeling efficacy interpretability procedure hyperparameter additive gps state regression acknowledgment like helpful repository follow wishart construct fx x sum add well variable ignore
preferable contamination demonstrate contrary suffer ht db background display change estimating point present iteratively change point determined stage apply value decide segment stop large result
reduction many use experiment competitive performance explore increase random projection accelerate small gap imbalance multi several rank accomplish projection minimum row mapping label representation via predict label nonzero representation coefficient concentrate find label correlation preserve mapping explore label robust imbalance relative mapping feature knn cpu ml knn
discrimination affine much intra digits service handwritten digit pca coefficient kernel validation c tangent human texture class texture pose illumination surface variation classification illustrate large intra across always well optimize markov field useful texture algorithm digit exist rate optimize polynomial
density distance prove estimator principle crucially old bandwidth furthermore proposal reasonably least principle choose kernel like risk distribution distribution call order monotone kernel density distribution depend however functional aim minimize empirical discrepancy bandwidth choose depend take kolmogorov generalize distance metric suggestion differ multiple discrepancy principle introduce widely choose recognize problem discrepancy chapter chapter detailed account
work incorporate ordinary least mle variation exceed word observation establish regressor iterative evolutionary accord trait comparative regression cm give tx correspond incorporate step evolutionary curve different regressor search conduct exhaustive search space high search notion evolutionary regression curve explanatory
transformation integration properly isotropic contribution transformation fig illustrate analytically compute mutual influence capacity marker correct htb limit svd perform approximate evidence probabilistic pca score exhibit principle cross learn svd compute
arrange lattice periodic boundary connect near keep randomly cost repeat cost reach follow path procedure appear exponent distribute fig system far example chinese truncate degree show usa obeys capture introduce degree heterogeneity
herein author interpret represent policy temporal evolve change node many static propose framework relational evolve vary predict time relational exploit temporal outperform sophisticated ensemble issue temporal relational evaluate temporal classifier outperform compete ignore necessity relational ensemble dataset evolve temporal relational representation temporal relational ensemble temporal temporal relational internet citation social network arbitrary temporal apply construct temporal relational attribute select repeat step relational relational tree provide possible temporal model special node attribute email attribute attribute attribute remain existence add refer tw refer assign c pt c pt pt traditionally classifier conversely classification three attribute
objective belong category sensitive prove algorithm player pure equilibrium consider share access access represent restricted level also wireless optimization main traffic capacity coverage operator capacity area macro able serve demand mobile operator solution reasonable cell answer problematic sharing access mobile operator customer problem model service customer actor indeed amount bandwidth bandwidth pricing determine agent interest actor especially spc equilibria share bandwidth pricing et al game bandwidth require existence primary article profile potential game classical pure
additionally write start simplify n generate probability numerator dominate furthermore dominate receive value k mean point close distance great start assignment must move amount cluster gaussian posterior assign posterior c asymptotic go mass limit penalty put everything behave exception cluster form whenever away exist centroid initialize cluster specify dp algorithm depend cluster dp datum process one future consider adaptive method choose ask objective loop mean simply objective number threshold control
plug density accurate around extensively nonparametric condition various level third turn contour formalize modified exponent efficiency prediction class nonparametric inferences negative taylor expansion degree old differentiable function require essentially require roughly contour fast contour exponent condition modify exponent density condition level constant eq differ interval contour contour level measure constant unless cut indicate contour allow enough contour contour smooth away
eqn explain crucial conditional give prior map estimate sdp give eqn support set eqn relation suppose sdp show solution estimate satisfy eqn regularize laplacian truth observe importantly family qualitative feature varied procedure wishart density eqn misspecification world generate ground truth equivalently graph width height lattice necessary edge
interactive require evaluate point interactive interactive interactive follow differentially private release mechanism interactive run per adaptively non interactive bind release mechanism interactive set eq theorems notable universe sparsity give inefficient mechanism query since typically view polynomially database improvement fact work universe entry string attribute specify length universe element deal infinite universe would length universe element encounter run universe query may still possibly infinite run resource interactive match would perturb require trivial trivial sparse course query super exponentially advantage interactive interactive answering query preserve differential give answer algorithm answer problem
assume parameterize cell proportional mle solve straightforward variance relate hashing matrix burden numerically first probability intensive cell rest include cell bit hashing
covariance marginal method table cell partition cell free near free value contingency determine upper sample algorithm replace distribution variance bound unfortunately guarantee identify successful line bound condition work successfully justification strictly lack table define induced marginal rely calculation algorithm produce slightly modify fix random free cell present substitute rigorous applicability expect seem calculate way table subject linear markov algorithm sample context algorithm discrete proposal chain tie target moreover end importance identify present reference set feasible free value bound translate low upper receive permutation probability could candidate vary considerably theoretical small mix state stationary
infinite consistent thm thm conjecture infinitely subset natural number poisson measure monotone undirected show infinitely projective permutation mapping infinitely system subset restriction show connection machine bayesian inference infinitely exchangeable point set number infinitely exchangeable area analysis previous ng
q n n one n x n n eq end quantify independent current independence lipschitz term satisfy define highly concentrated sense nx z last nn positive also I conclusion drift drift limit drift assumption drift acceptance equation express x nx prove suffice let prove remark lipschitz choose independently prove converge lemma x nx dominate prove follow zero nevertheless quantify bernoulli independent position prove satisfie equation give proof independently definition readily nx q jx nx n sx j j x martingale error introduce approximation hold pair satisfy eq martingale proof equation essentially identical easy globally sd x equation consequently suffice nx suffice show equation
pt university department university college place bt uk department mathematics ny usa xu mathematics ny usa machine obtain lasso total method regularizer proximity advance approach practical admit appeal proximity choice work group fuse regularizer solution certain proximity efficient finite accelerated method order technique class argue simulation efficient state art overlap match fuse structure lasso
firstly elegant nonparametric gp discussion concept drawback computational heavy apply already limit overview value assume zero variance number choose characterize since noise well continuous function accordance outside follow scalar mean variance key mean used provide subsection observation discuss subsection address quantify provide first important provide amount search gp shannon necessary mathematical measuring quantify information quantify avoid conceptual communication ignore conceptual shannon
site neighbor site call cluster thus notion determine configuration I probability give site active site probabilitie graph influence via adjust active alone fix random consider simultaneously permutation site visit consist cluster denote coincide permutation hence active site active site site first permutation estimator active permutation storage consume run permutation check site merging already check site large site belong active belong label update site
distribute tend lie direct look normalize define definition q enumeration subtract model prove consistent related capture stationary uniform prove depend appendix evidence graph model source even output markov chain stationary iid non illustrate text would find hope markov source situation count similarly require prohibitive appear situation stationary source illustrate language length hold definition two case behaviour range example mention bioinformatics finance sensor write style security approach work alphabet suit bioinformatics analysis style review
optimal show gain multiple system shown see observe dimension consider communication multiple recent exploit optimize design wireless network concept irrespective directly depend neighborhood thus power scheduling jointly schedule scalable even centralize receiver receive diversity reduce thus wireless bandwidth feasible
continuous reach imputation validate possible number imputation hour available white four miss order encode error compare significance encode hash miss capability categorical deal categorical see datum emission set pair dna domain patient different record mainly nominal fashion nine level always miss namely decrease data decrease range hand except point primarily tailor
make predict analytically example effort eliminate lead improve exception change need implement weighted fit currently partition fold imagine partitioning carlo term recursively fold could confidence vary spatially despite well mc fitting point transition number average mc know never sample close post processing trying estimate integral david uncertainty quantification engineering calculation mc suffer sample stack post exist
subset small one reconstruct fraction noise need preserve subsequently provide privacy privacy see add bound avoid barrier exploit formalize privacy preserve online privacy privacy programming underlie gradient ascent private online class improve regret exploiting include differentially online offline practically problem online online section convex online player adversary select pay hence tf player minimize cost fix select move incur offline solution provide let select convex regret select random convex regret notion privacy context privacy convex differentially give hold change amount dataset much reveal extra privacy affect td
framework prior induce covariance continuously combine computation inference specify dictionary cope require seek control propose stochastically shrink toward flexibility zero shrinkage explore property nonparametric focus regression support property nonparametric constraint choose aim condition specify satisfie statement row towards enough induce place prior function arise dictionary function weight specify assumption property condition wishart variate wishart specify satisfie provide condition prior choose assumption prior inverse denote induce element conclude limit predictor towards decay go zero infinity diagonal equation length property regression wide sense solely order stationarity follow stationarity recall solely imply truncation propose derivation update conditional conditioning associate hyperparameter gaussian specify define follow examine standard hyperparameter shrinkage update h incorporate induce dependent sampling sampling x ik eq condition
cross distribution belong relative entropy exponential family separate term f bregman generator divergence leave deduce explicitly constant kx fx e kx f kx e fx close expression enyi divergence
corrupt online missing arise comparison round extend hypothesis relation value miss bound several uci baseline algorithm example suffer life corruption every adversarial situation arise medical diagnosis array medical constraint incomplete non response task deal image setup accord distribution minimize h scenario imputation imputation unobserve feature value x make elaborate I rademacher thereby also hypothesis reason imputation analyze general hypothesis definition regret control rich linear predictor
object unknown figure toy symmetry depend try pair present dataset edge b pair side show relation cover framework relation become clear example start notation relation moreover relation mapping define inverse value relation value kernel rkhs rkhs transpose cardinality formally follow select appropriate quadratic rkh accord admit dual representation form dual correspond alternate edge graph establish every consist couple node norm q mention make different case loss hinge opt function function result type regression loss like insensitive case specification offer couple restriction relation specify kronecker individual kronecker
ard display solid run deviation firstly ard recover hyperparameter datum use correct db data top bottom robust recovering secondly assume however statistic observe estimation significantly ard bottom much right initialization deviation zero demonstrate section report introduce invariant value body show fig truth position background form associate four equally arrange adjusted black small large ard hyperparameter initialization return ard number basis iteration ard see perfectly recover adjust log scale log ard show return ard manual correct low indicate presence minima consistently ard inspection learn dictionary learn ard decomposition despite residual noise inspection reveal despite fully relevance evolution spurious component
well nm maximum block give x norm block block maximum unitary invariant block diagonal unitary mn mn stochastic q block complete hermitian size radius submatrix decomposition unitary u u eq hermitian matrix lower complete nm nm mr stable matrix satisfy diagonal lemma express complete replace submatrix assumption evaluate expression evaluate term rhs knowledge proceed invoke principle approximate q arrive theorem tu h adaptive agent enhance estimation reason level help reveal process important get
tangent ready term integral radius big approximation product determine high mean enough ball radius around change ball radius radius replace integral ball interior function take order odd therefore vanish term inside integral order next interior near boundary normal term normal normal near boundary direction show local result generalize walk interior l u limit easily maximum variable bound high region region maximum analysis apply inequality notice come equation
subset entry denote submatrix index whose entry index critical quantify incomplete span give incoherence control paper detail vector influence demonstrate quantify fix classic absolute deviation admm alternate multiplier therein accord introduce constrain manifold neither estimate could triple admm refine approach detail evolve problem relate corrupt let index rank matrix span version rewrite offer efficient differ major column time greater say discuss piece variable adaptively choose estimate observe optimal find lagrangian equation admm alternate quantity invertible guarantee soft discuss admm detail
dp dp procedure converse everywhere unknown whereas require hold everywhere show compose give constant composition allow rapid allow interactive eq union combine composition release datum release together result composition first histogram privacy differentially count private histogram partition sx mechanism satisfie demonstrate take output neighboring meet either complement partition fall
vector markovian member integer exactly truncate sample correspond truncate signal radial assess correspond calculate apply receive radial velocity mixture estimate calculate marginal use reliably suitable selection amount parameter purpose extract several search nearby star method therein radial velocity reference therein therein search rotation reference current determine model assess ability explain also assess set term statistical different number star relative
location calculation processor core avoid pre operation iteratively implement problem logistic r accelerate quasi newton bfgs factor exploit operation r much precise conclusion nearly iteratively least square converge reflect quadratic likelihood poor iteration sometimes severe cause unless trust basic version expectation maximization robust combining quasi equally robust fast pareto penalty simulate least grey line
fact nearly perfect unnecessary removal impose penalty removal figure visible end optimisation method concern notice detect cpu significantly two cpu average cpu start fail correctly cpu rmse detection classical least criterion regression sensitive investigate find various derivative method dynamical optimisation factor variable ten criterion removal ann backpropagation undesirable feature loss impose remove criterion accuracy outlier reliably ph bold indicate r ph value bold r outlier ph noise dimension ph mx mx mx bold r r ph bold
immediate given obtain valid qualitatively generalization case statistic separable infinite vector machine statistical special vector rbf kernel svms space rkhs
section matrix map depict panel notice quantity information estimate associate obtain image subspace svd whole hyperplane locally image comment make svd panel reflect contain conversely svd bottom panel large linearity summarize evaluate estimator linearity quantify property exploit hyperspectral follow hyperspectral acquire field panel scene black area compose green pixel abundance estimation pixel live dataset refine locally subspace bottom compute svd panel local unable non scene bottom panel
block graphical lasso paper original glasso operate dual directly box use interior operate quadratic problem every row dense qp lead attractive kkt derivation entry optimality box qp consequence w update qp equivalent require column last implicitly solve solve return though problem sparse use solve boundary glasso primal glasso glasso glasso operate regularize appear choose use coordinate use warm available work copy way appear generic descent algorithm easily detect stay zero get pass solution zero hand procedure effective coordinate update gradient column experimental glasso glasso zero per glasso dp small qp particularly attractive
along line planning city distance effective heuristic planning module guess heuristic heuristic action follow e heuristic initialize terminal execute ss terminal term search short state typical simple ignore need position immediate plan advance play gray position neighboring depend situation another state depend figure planning figure plan planning planning planning step initial action select tie break episode run randomly constant experiment free
mean analytically ex ac ac uk combine structural bayesian neural flexibility correlation response heavy monte variational multiple volatility model demonstrate multiple multi volatility expression popular expressive interpretable avoid fit predictive thorough machine learn network show bayesian hide value aim correlation particular assume could input imagine represent heavy laplace something would vector latent additive kronecker gaussian network random label square function label vector gps adaptive show generative hyperparameter
discuss lasso solution maintain accuracy initialize intercept minimize eq squared iteratively calculate st calculate iteration evaluate scale choose coefficient single enter iteration avoid effect eq parameter update expensive accelerate fast effective particularly compare used formulation technique encourage little I tolerance binary goal three process individual original statistic attribute dataset object dataset total negative binary use select use time cross assess overall misclassifie total misclassifie observation misclassifie experiment show solver overall ensemble numerical ensemble bag three repository fold validation test dataset equally proportion class testing bag take test name breast ghz processor rule ensemble many contain multiple class identify applicable ensemble easily
event see gamble x ax generalize describe link unconditional must classical theory generally subject example gamble maximal decision comparison gamble gamble conditional think operator gamble subset note social event function option gamble choose relative option necessarily single preferred option choice choose gamble present partial coherent low gamble whenever rise function gamble eq recall envelope maximize gamble admissible maximize utility least admissible maximal strict partial non event gamble write order call empty gamble event select gamble maximize gamble usually worst depict coherent example contamination observation q find gamble apply suitable list gamble reward determine shorthand notation l gamble induction require gamble obviously utility
must gate validation group see split calibration remain problem perform without analyze understanding plan experimental expensive investigate may able experiment challenge newly split increase capability paper demonstrate systematic assess extend reproduce experimental datum use ultimately capable gate fail quantity partition constraint allow determination ii requirement edu
continuous mapping lemma imply every path u expression gets maximize argument show follow iterated logarithm walk satisfy hypothesis show c apply notation w application lemma expand n numbers z class vc subgraph integrable envelope notation four converge since measurable assertion immediately family almost surely see density equation show satisfie lemma n know corollary n om let note n om bootstrap distribution poisson h u denote great bn bn sufficiently dx dx another borel bl l easily turn take prove bc characteristic prove permutation page subsequence converge z sf surely subsequence proposition nh b increase write n n kt k kt kt h ks kkt h k k permutation random I h sure choice subsequence proceed analogously compound distribution
across diagonal node elsewhere describe role across homogeneous value initially membership around around membership center third compatibility matrix give diagonal variational original ground trajectory mix show simplex three verification dynamic membership stochastic though simplicity trial evaluate actual infer membership method membership capture
setting specify loss allow configuration variable find refer optimisation establish extensive regard exist equivalent solve lar algorithm cone solve approximation amongst high excess used model prefer sparsity give weakly specify laplace loss laplace contrast optimisation parameter latent positively constrain exponential distribution nk sparse explore specification place rate shape respectively model accomplish carlo base
heuristic surrogate offer full three order magnitude test moreover accurate reveal correct requirement promise mean adaptively stochastic optimization offer considerable gain incorporate experiment hierarchical structural successful design resolve current surrogate provide thorough bayesian natural extension rigorous incorporate sequential quite effective research energy office office advance research rv r htb different reaction r species body specie explanation delay peak release characteristic peak characteristic peak occur characteristic release peak value peak peak htb bind prior expansion heat release validate htb htb htb outer iteration surrogate utility course terminate expect utility output factorial htb pc htb surrogate f htb htb different size line outer loop red dash bar show deviation gain information gain individually fix perform conditionally design marginalization obtain differential individually use fact equality remain expect experiment never gain equality conditionally equality information two equation bias bias reduce evaluation result source list question nonlinear information estimator one outer loop blue marked marked bar many
recover sign sign arbitrarily fix sign sign moreover recovery hard refer reader proceed easier construct four entry quite standard literature decomposition probability provide notice generalization involve random utilize sub condition follow define imply uv inequality schwarz notice p f hold assumption theorem strict must assumption existence variation idea approximate te tw error uv te consider observe clean also small uv te largely still natural subtract remain one set repeat correct geometrically
analyze analyze large classic r enyi node uniformly tractable assign interval idea graph discover perform graph path form node depth traversal present discover origin study visit correction procedure correction sample small methodology consequently bias correction independently two paper fundamentally final heuristic generation base generation implement situation generation expensive generation capability reduce graph sampling unfortunately none applicable problem speak use either sample iii average previous work change successful correction scale life internet topology procedure property complementary ready use implementation stress base differ without graph except seed exploration seed recursively visit technique walk walk walk technique start choose
wikipedia behave surprising wikipedia cluster cc build find page metric platform validate score presence long tail page among score solve issue highly score identify interesting wikipedia phenomena level medium page score single aggregate shift less due role work alone score highlight group improvement language processing aspect platform collective interested thank nsf award laboratory nf
set belong capability rand rand range measurement partition indicate agree partition compute rand schema partition latter partition report rand repetition l rand reflect appear high ht result since discover highlight figure highlight institute mathematic report approximately locate us period test series consider
site consequence lattice threshold infinite lattice infinity site cs cs translation invariance infinite thus site call leave analogous pixel black exponential factor mean error correctly site leave right connect hence imply addition prove yield cluster infinite lattice possibly cs cs cs infinite lattice hence essentially pf en en en enough therefore eq help convergence remarkable tend exponentially fast object exceed ii
small conditional entropy integrate dynamic centrality affect principal see fig topology value sis directly also find remarkably present peak similar variable sis topology dynamic feature investigate relationship complex report critical concept dynamic level along allow effect approach despite uniformity er topology dynamic uniformity article investigate
challenge interval fundamental find monotone operation interval computation due dependency effect optimization robust introduce numerical computer interested method propagate function contrast define irreducible derive incomplete increase information great importance uncertainty model separately probability assign evidence conjunction combination
describe maximum posteriori yshift rectangle yshift xshift lda document topic term feature topic replace hilbert space linear softmax simplex generalize replace expressive semantic rich dynamic belief tractable address dirichlet basis extend effect admit efficient implementation kernel topic model generalise price increase modelling flexibility computational cubic number document cm pz pz south pi edge pi
element unconstraine project perform iteration reduction large fairly invariant seem neighbourhood fix quite closely sequential manner accuracy produce work fixing despite parameter estimate likelihood essentially identical seem advantage arise along move symmetry maxima explain responsible remove symmetry range standard deviation log estimate corrected value likelihood numerical multiply method choose six third intercept design intercept result perform smc gibbs step step case optimisation despite marginally comparison smc gibbs perform mean estimate one find standard value gibbs agreement distance value sampler automatically provide likelihood likelihood smc marginally comparison standard integration option still smc em probit monte truncate normal build student freedom produce smc sampler seem typical ideal smc evolve update avoid greatly particular performance example six fast since monte provide truncate student clear extend variable extension multinomial course complete em result generalise full examine identifiability model
share strength conversely fail bootstrap edge motivated development bootstrap subsampling setting problematic interesting investigate subsampling maintain robustness averaging implicitly error optimize estimator combine way average proof hz hz suggest notation view outer define set behave asymptotically though directly empirical subscript brownian furthermore pf almost surely desire state sequence process converge brownian noting class behave eq functional delta delta conjunction assumption b b j assume proof desire first support sample simply use computer yield consistent upon align capacity computer immediately method two trend attention regard become increasingly resource toward architecture provide hundred thousand processor distribute present capability storage inferential clear bring issue include issue exploratory visualization remain inferential uncertainty remain dataset frequently fit accurate assess efficient processing necessary achieve bring notably wide inferential bias uncertainty risk
dimensional spectra genome face novel approach fit unsupervised framework dimensional manifold relate base near regression optimize variable reconstruction problem neighborhood solve allow topology appropriate sort iterative variant experimentally dimensionality important play understanding
modeling rule provide formal framework p normalize parameter ignore distribution empirical evidence aspect make inference unobserved reasoning uncertainty become lack explicit characterize observation publication interest considerably simplify nuisance variable value paradigm principle statistical complete process contain expectation relative central challenge often include analytically demand use approach include mcmc sufficiently pool random underlie characterize intensive slow variational convenient potentially accurate analysis description uncertainty summary posterior sufficient obtain instance ignore interpretable summary point sufficient cost crucial outcome discussion learn thesis formal inference set incorporate paradigm view rational concept average analysis learn highlight often focus optimize estimate bayesian prior distinction comprehensive demand analytically often approximated analytical procedure mcmc variational summary summarize uncertainty result convenient available learn procedure challenge complex topology global optimum exhaustive intractable contain ultimately resource central thesis standard representation identically p fit maximize maximization likelihood estimate model operate function optima additional less posteriori additionally ml map ml case assume sample dominate become point ml role prior task needs take principled accelerate order stochastic mcmc exact solution computational potential acceptable give gain efficiency characterize uncertainty tractable distribution approximate potentially approximate easily maximize subsequently datum yield divergence q equal minimization analytically tractable kl maximize low variable factorize approximation infinite model marginal density variable optimize value expectation step evaluate analytically variational optimize expectation determine density update contrast marginal maximization step belong suitable convergence identification optimum guarantee incorporate prior avoid framework publication learn learning identify either globally optimize value method require accuracy objective assume topology set derivative optima estimate gradient direction towards desire gradually improve ascent characterize optimize manifold appropriate function bfgs method newton approach thesis probabilistic probability unobserved event characteristic generate observation characterize overfitte refer avoid overfitte overfitte avoid collect investigate accurately new learn test performance assess testing publication thesis widely probability underlying simulate variability observation space event resemble density multiple bootstrap estimate variation bootstrap assess publication flexible therefore simplicity impose soft penalty prefer solution publication theoretically principle
characterize number graph weak depend graph define expert reward reveal multi armed select reveal assume mean reward action change round case establish achievable combinatorial large clique partition number clique node deal constraint regret minimal clique find np hard clique handle graph change round versa regret direct graph efficiently graph regardless demonstrate approach analysis armed action alternative endow rich however paper armed bandit include action reward assume draw action property approach combinatorial bandit consider choose observe crucially reward separation obtain value partial
turn q co vector physical value physical phase time choose least disadvantage approach freedom location section suggest possible good belief belief panel relatively coarse exploration right exploitation blue bottom panel roughly forward cone bottom leave system section radial function initially equally large scale cover trivial I sample region intuitively belief suffice belief evaluation comprehensive discount control control
kl divergence permutation increase demonstrate kl place place c c kde anomaly pure false false test would receiver operate contaminate robustness check anomaly contaminate detection alarm pair well yield aim contour kde tb htb c kde employ psd view rkhs kde correspond quadratic employ robust robustness contamination training kernel estimate iteratively weight influence problem make assumption percentage contamination nominal problem interpretation influence
discuss disease prominent role non disease transition state sometimes disease depend duration influence medical decade state suffer disease specific duration show disease incidence high rate historical reason number disease henceforth rate depend time
sufficiently subgraph order near neighbor hierarchy subgraph approximate hierarchy form perhaps concrete approach context guarantee prune cluster carefully work pruning prune spurious guarantee set tree salient still estimator underlie cluster recover interestingly prune tie base rely identify remain along dense region cluster clean formalism propose seminal linkage empirical cluster unfortunately lead
tangent local vary analysis accurately track subspace bind yield behavior demonstrate trivial introduce principle quantify perturbation tangent matter need manifold noise concern indeed perturbation convolution clean point geometrically manifold help center manifold volume clean correction idea volume clean issue concern determination estimate tangent plane taylor series generalize dimensional exist represent principal direction system align point coordinate quadratic manifold first tangent inside ball sample subspace capture inside volume ball standard sampling linear subspace sample contaminate noise I store allow decomposition plane component vector q wish pca direct access work proxy span recover invariant refer local perform pca give density quantify define local neighborhood note align coordinate axis principal direction quantify invariant frobenius rotation axis align direction embedding rotation indicate proceed configuration position span note population covariance enforce mild density usually extra achieve implicitly consequence eigenvalue create instability prevent notation account center required realization realization copy version problem pose prevent directly perturb dominant proceeding review perturbation relevant reader familiar skip subspace respective orthogonal distance equal large angle angle use onto subspace may version present purpose arrange partition entry eigenvalue span possible tangent span column onto subspace entry variance space fact hermitian frobenius measure reader refer f u u hold quantify exist I map span invariant angle zero restriction normal tangent numerator measure quantify effect tangent component tangent contain tangent measure
compete finite interest among via particular analyze theorem class model computationally prohibitive crucial ingredient selection multivariate describe efficient combine select computable compare method one select illustrate analysis set machine cognitive collect discuss overview adaptive present reduce rank immediate component canonical cca perhaps literature recently refer historical reference application extension propose penalize nuclear penalty achieve adaptively minimax see calculation note remove side response regression model row transform univariate later reference optimal despite advance base practically ease comparison achievable estimation
predictor general abstraction actual implementation method pattern output region partition extract expert result must validate assess particular whole generalization achieve gate expert find gate resolve mlp network optical band also improve overall improve statistical refinement determination conclusion set compose respectively forward neural network represent mapping input variable output one mlp variable neuron arbitrary layer feed forward layer adaptive represent neuron feed network activation th layer feed forward input hide single layer kind define counting connection instead number perceptron weight connection th node activation sum include bias unit activation activation activation function set hidden layer output respectively th unit activation activation mapping logistic mlp connection good network kb method determination minimization back propagation bp simple important role view update paragraph task aim define way good regressor stand asymptotically regressor uncertainty sample directly extract approximate noise case convolution abstraction regressor reproduce accurately mlp learn space space correspond evaluate case involve determination mlp also throughout color turn since color depend statistically population kb vary region space kb define learn region network well
thank author thank david pointing associate comment recall fact chain chain obvious accord form factor draw rewrite second backward detailed known filter forward time backward sum message fig kx mx px
problem performance generalize principle pursuit literature requirement reader consider comparison show logarithmic mc emphasize close shall isometry however isometry theorem elegant david phrase detail notation also keep zero restrict isometry rip define rip infimum x suppose f rip check constraint show absolute coefficient define hand moreover result cs eq singular let whose constant xx ax lemma
chain ergodic rely provide necessary verify reflect beta construction bivariate chain evolve split x sx sx appropriate sx sx paper split
path total path length bin bin passing equally contain particular average neighboring symmetry path let length path pass first cauchy path note path number path edge eq thesis take least algebraic graph stand product q analogously matrix one last cc let without generality label j ij dm low notice q bind quantity ij z ts diagonal z z z da ij union variable apply hoeffding q eqs use eqs union dr j decompositions definition dm n let laplacian establish property spectral exist claim sum equivalently write lemma position node k manifold geometry base random graph precise noiseless reconstruct measurement distance
satisfy assumption wu infinity var u var u nu j end proof proposition nu nu n inequality note also ap n give statistic proposition n proposition eq supplementary proof result notion wu theorem wu wu wu depend j drop subscript jx qx cx cauchy schwarz k decrease lee lee critical sake brevity wu carry suffice j n cauchy inequality te observe n give assumption tu n sake brevity j proposition consequence hold p equality nr cauchy schwarz pr np markov four maxima np n nn pr three focus step approximate martingale counterpart lemma deal deviation lemma conclude explicitly deal residual proposition sigma define u q nd jt shall dependent version lemma hold jt cp give q j k u j wu substituting give jt independent suppose provide moment absolute third
quality fundamentally impact subsequent datum continue work much underlying aim gain insight nature datum contamination particular phenomenon mis highly desire formal give primary treat arrive contamination arbitrary though joint thus contamination attribute image contaminate mis pixel extent capture essence relationship amount mis shift certain impact highly scene e drastically consist forest form amount mis classification accuracy phenomenon mis contamination error error occur contamination contamination consistent contamination contamination amount discussion
similarity experiment multiple graph tune detection modularity maximization fast greedy graph layer truth adopt three criterion angle define intersection cluster respective entropy binary decision negative false negative show algorithm three result highlight bold font indeed easier much lead improve independently achieve perform present benchmark baseline combine result imagine regularize particularly come way compare maintain competitive computational significantly original modify namely term show difference two truth dataset unbalance cluster nature much attention characteristic work improvement confusion mit illustrative column row intend reveal data performance mit literature method exist widely powerful modeling pairwise object highly major
jk anti symmetry desire scenario e perfectly anti gp preference call readily pairwise number closely active perhaps design gps use eqn eqn calculate marginal observe decrease efficiently covariance motivate objective fundamentally carry update point yield update update filtering indicating procedure entropy model eqn use regression uncertainty confident perform classification repeatedly boundary observation uncertainty domain show mutual
mle abc mle great abc abc establish property procedure asymptotically hmm hmm transition original hmm likelihood immediately perturb perturb mapping boundedness convergence demand immediately hold hmms hmms next consider normality mle fisher invertible asymptotically asymptotic equal proof perturb dominate perturb invertible abc fisher general quantitative qualitative see strict sufficiently small mle tell estimator relative mle grow choose noisy mle ignore provide fisher see appendix doubly sequence variable difference law law law law asymptotic hmms perturb additive immediate corollary assume hold defer comment similar summary abc form fisher fisher hmms manner inherent lack smoothness estimator ball problematic try smc due abc likelihood smooth lebesgue estimate via
adopted argue often convergence mcmc bayesian optimization advantage exploration objective method exploit result optima consist phase sample learn policy explore phase procedure uniformly adaptation ergodicity adaptation increase shall soon detail require process adaptation storage need reason step consequence back conclude remark strategy detail mcmc auto value run chain entire observation function prior distribution noisy statistic select next refer reader review statistic acquisition function zero automatic determination ard eq surrogate intractable expectation respect invariant measurement hyper hyper typically proceed parameter experiment however good alternative quadrature integrate obtain extra indicate make
sensitivity specificity accuracy hc normalise ratio instead estimate parameter network nod one network trend size verification hc consistent sensitivity well consistency moreover substantial structure correctly hc successfully recover alarm successfully recover alarm attribute effect test specificity sensitivity combination network reach overall rapidly converge slow rate compare two likely consequence edge per alarm slow convergence also inherent limitation dense
show exhibit derive encourage result order second transition self enable consistent multiplier transition symmetry separation shannon contract statistics principle entropy additive regard analysis blind
km ff leibl trivial constraint interpretation observation pr need common assumption surely light continuity rate stochastic preliminary sequence assumption martingale assumption converge surely complete eigenvalue calculus reveal matrix write f f ff definite transpose jacobian irreducible reversible distribution unclear lyapunov main rate
column norm internal via parametrize often take thresholding independently function drive occur node activity whole function output case separable ensure guarantee specifically convergent fast cost differentiable activation invertible norm likely invertible thresholding relaxed calculate used appendix barrier solve know soft similarly program zero activation work focus interior include ls gradient projection homotopy entirely problem decrease tradeoff utilize recovery signal leverage hardware architecture burden unit processor gpu utilize perform fast improvement favorable property unclear sized architecture embed digital similar iteratively minimize apply thresholding enforce euler approximation analog basically digital incremental rather iteration digital linearize iteration show section demonstrate analog cs benefits analog architecture
powerful reservoir internal reservoir reservoir dynamic neuron architecture reservoir able reservoir depend past reservoir must operate threshold dynamical ensures gradually forget reservoir compute flexible detail reservoir regime threshold instability tune parameter flexibility reservoir computing amenable large realize water analog digital implementation report delay difference type non linearity period delay highlight isolated channel comparable digital implementation reservoir experiment almost magnitude order speed flexibility reservoir promise computation physical digital provide solution reservoir road build analog computer discussion reservoir computer key detailed treatment refer reader supplementary material task discrete internal reservoir reservoir mask note use mechanism subset state use build output layer combination state reservoir target training mse easily typical run reservoir fix minimum reservoir non delay feedback system widely nonlinear loop internal store delay loop entire internal process implementation instantaneous absence system simple nonlinearity implementation system evolve way convert discrete hold procedure loop regime delay act storing delay nonlinearity state
meaningful position pearson paradigm classifier solve q function series ii first identify classifier enforce counterpart positive fix give lipschitz satisfy q error check strong ii extremely would certainly achievable also subsection suffer degradation well achievable binary desirable result proposition fix surrogate far exist nonempty ensure surrogate continuous optimal mention proof uniformly consequence chance programming classifier candidate
nan ready decode almost property subspace hold suppose nonzero every vector range matrix satisfy kk solution apply triangle obtain eq property error decode error phenomenon isometry lagrange lagrange optimizer appearance energy clear later explicitly get direct estimate almost subspace subspace subspace satisfy almost property intersect mesh mesh subset distribute h follow half absolute standard lagrange duality find objective restrict still bound plug minimize minimize serve upper I complementary go infinity probability
difference define generality vector da result lem invertible define position section since favor fuse pt aggregate square combine difference use imply invertible aggregate invertible essentially technique lasso treat one deal present advantage lead impose partition overlap k among order sparsity small pattern assign cf outside sparsity obtain estimator define obtain group aggregate satisfie support overlap illustrate power oracle setup namely combine remark class estimator
n element wise diag
model quite lda dt low transaction one thus stable affected mainly rely fair target historical scenario may nevertheless whenever demonstrate cm cm cm trust management derive base past sufficient behavior reliably scale setting experience resort indirect experience assess agent history assess form trust relationship trust always drawback wrong recommendation affect derive trust iii trust nontrivial develop trust determine eigenvector trust rely design impose drawback trust group like member belong author control calculate author eigen eigen trust transaction management aggregate score stop reach advantage system trust service give rate rating value shape determine assess however cope may influence trust virtual beta compute attention issue evaluate potential I knowledge trust probability fall certain estimation rely experience trust ensure consider address inaccurate perform opinion represent successful adjust feedback inaccurate work initial trust profile accord aggregate member trust agent belong additionally
estimator function consistency simple simultaneous band weight compare achieve end future finite suppose real belong size probability unit pair draw simplicity subscript convention curve available discretization measurement center index necessarily temporal also random unit smoothing eq reconstruction perform use fan statistical convenience support finite constant classical thompson curve membership otherwise hold unbiased establish framework discuss study infinity recall notation constant whose may place study nk x ks kt normal deal design must fraction inclusion far fulfilled design old continuity mild regularity design essentially negligible compare
possible specification increase magnitude entry property example return volatility process separate vector infer outline account output account fix correlation introduce use gp process correlate prediction spatial set methodology multivariate volatility interest evaluate dynamic make prediction inference relie solely fix limit correlation structure matrix quality focus contrast scale need dimension conjecture wishart mainly limited dimension cholesky operation general operation
cover song negative network ideal centrality node sub threshold apply short mathematically th song index prototype community cover song second span previously closeness centrality result percentage music hypothesis song tb algorithm closeness centrality centrality nan p communities accuracy centrality discriminate become statistical significance assess test song tend centrality discriminate centrality valid dissimilarity measure represent think incorporate aspect audio content accuracy complicated song enhance coherence furthermore organization cover tendency song could query task audio song identification detecting piece extract raw audio important modern organization song nevertheless song music effort
tradeoff try avoid enough hand affect approximation available lemma remark di transfer reinforcement source rl adapt source target illustrative reinforcement rl speed rl underlie assumption source combination task useful transfer rl thorough focus transfer trajectory mdps use target particularly suited involve interaction sample available source case arbitrary reward finally index I available sample generate index available target source target e notice action generate l build lp access return case form iteration propagate transition kernel mdps weight determine I average bellman
n ordinary equation sequence tend ik ng ng ng ng ij exist away zero n expansion n fx pt nx nx nx compact converge atom rescale coefficient coefficient vanish j ij w vanish vanish contradict tend infinity conclusion converge converge vanish entail hand display must almost identifiability contradiction true subsequence equation contradiction term strategy triangular g w k two convolution sphere fourier function support fourier transform inverse fourier transform bound f g wasserstein distance thus note dx bound lemma line sign denote variation g constant
q give denominator component ic x iy ignore dependence approximate boost specialized leave tree forest thousand real uci market specialized classifier prediction probability moreover experimental uci artificial market recently implicit observe market never outperform market market outcome contract market win contract contract market possible outcome outcome exist price simplex r instance make market consist market participant budget importance tell market price price know describe price classifier feature show represent artificial prediction dot probability example base function logistic play potential feature market entropy contract price amount win mm mm set participant govern price achieve example involve market outcome example correct htb
size configuration extract used force identically vanish sparse datum inspire natural mean analytical formulae efficiently cross configuration equilibrium exist prohibitive ergodicity breaking affect quality question acknowledgment thank leibler thank center system contract pattern rewrite partition eq temperature considerably entropy partition configuration role dual entropy give entropy system measure expect measurement respect assume configuration pattern derivative calculate logarithm replica introduce expand I enforce definition lagrange multiplier give symmetric elementary gaussian saddle fact role pattern define self entropy center biology nj sup paris paris sup paris problem infer set frequency inference case ise space much negligible compare mechanic correction stress include attractive infer interaction criterion many attractive pattern noise configuration require good illustrate understanding fundamental various scientific model application small component pattern identify mechanic originally intend average power pattern linear correlation validate discuss issue configuration amplitude pattern plan define bayesian correction decide bar synthetic devoted real biological alignment domain reader interested
discrete dft field spectrum covariance via exponential expand spectrum truncation correspond adapt formal expand yield z feature coefficient summation field spectrum coefficient multiplying quantify complete orthonormal impossible value produce model mirror symmetry coefficient specification describe straightforward discretization utilize control mesh without generality fill refer indexing indexing map grid frequency wish grid maximal lag entry k h provide formula derive write rather produce take large produce compute determine exact important likelihood reverse device causal skew field skew field precisely causal causal may define move
already walk case find current vertex occur fix nearly discover unable continue self avoid past find total ht component class vertex walk account candidate vertex probability way drawback degree consume walk roughly since self class st see table st classes self singleton order observe walk
ordinal order set countable ordinal resp length resp quasi order naturally equivalence try explicitly p write suppose rx vx rx rx v v difficult quasi order ax rx vx aa accord lemma quasi order strong
next attempt physical application discuss ref reconstruct individual model procedure response direction reconstruct filter estimate angular power ref discuss sec bayesian diagonal facilitate via parameter reconstruct way coarse grain conduct ten middle still bayesian right iteration region dependence show pure estimator ten evident bayesian boost marginal time consumption main strength lie reach
compatible semi indicate duality mapping banach uniqueness product duality mapping difficulty uniformly differentiable banach convex uniform convexity duality uniqueness close uniformly fr differentiable ball uniformly unique compatible useful characterization banach space banach fr analogue representation banach space banach compatible semi duality exist unique compatible banach space compatible inner lemma homogeneous semi product however reduce variable namely ready value banach sometimes prescribe banach function norm sense banach consideration evaluation usually refer call
draw distribution eq define likelihood remark distinguish distribution eq maximum draw figure specify specify parameter via draw distinguish actual estimating maximum specifie sort great draw contain great draw bin contain great remaining find draw plot number draw require via maximum sort specifie mean distinguish integer estimate sort bin great draw bin remain contain great bin draw distinguish actual define sort specifie distinguish plot draw distinguish distinguish thank fan wolfe also thank many include point identity show hellinger version confidence significance significance set positive negative whereas figure probability false false negative remark figure rejection figure investigate draw say surely limit actual fast consider significance converge fast draw obtain accord simulation bar top bottom level remark estimate please draw increase plot trace converge straight line
leave path update parametric form indicate path involve unary potential potential cut variable define max product depend bottleneck capacity recover cut notably alpha location bottleneck bottleneck pairwise remain since valid expand beyond chain know potential cut expect example choice potential cut graph cut never unary potential modify mass cut fair back potential normalize leave thought view tree ascent block choice see however chain coordinate dual analyze chain subgraph optimum bottleneck optimum subgraph dual objective optimality way path setting conjecture mass cut side min marginal compute second recursively root mass differently pass theoretically analyze two guarantee assume converge guarantee update bind whether notable exist yield submodular belief optimize solution convergence message guarantee converge suboptimal possible via update somewhat temperature guarantee relaxation numerically implement even submodular energy belief propagation reveal message guarantee assignment submodular short path strong potential strength nonzero unary convergence could reach fix modify unary potential take interestingly modification mp become cut solution map notable closely relate block ascent dual interestingly optimal
discriminate action current state principle intuitive description formalize decision mdp find reinforcement definition necessary explanation decision formalism sequential process action state reflect relative use agent state apply agent obtain reward continue reach rl cumulative cumulative reward system goal reinforcement maximize reward
correspondence logical appear logical logical par inversion elimination product simplify sum true false another insight large identical member identify consider assignment argument truth assignment elimination truth group count truth efficiently count elimination par constrain choice counting elimination context rv fact eliminate par elimination exchangeability par par rather section may par appear par rv aggregate variables aggregation par extension consider alternative setting recognize computation first implicit specification presence second case identify share input value discover construct without rv graph later carry computation speed improve perform approximate extend algorithm two last computation simulate rv effect distant influence place bin close bp stage first graph compressed template represent group factor nod message super edge member weight super equal edge template bp modification message send super node weight super node counterpart except next algorithm target evidence group factor variable node par lack ensure type identical induction bp message evidence mostly useful learn bp template graph proceed stage bp determine evidence affect send initially variable super contain subsequent super factor node separate type factor super super refined type super process guarantee minimal e template obtain simplify description factor logical
newly add contribute active terminate step sequel greedy greedy node conditional negative loss likelihood threshold backward step factor inverse ji break k ji j j row symmetric zero positive follow appendix convexity notice estimation converge always suppose per
cc enyi penalize cc cycle enyi penalize logistic ise positive potential graph topology attractive nonzero uniformly entry uniformly ise gibbs knowledge employ criterion since truth possible measure good figure model model potential trend attractive note distance decay increase method decay graph regime rate alternatively fast theoretically regard run fast graph consideration global threshold expensive large unified paradigm ise estimation complexity condition succeed ensemble world separation acknowledgment author berkeley berkeley wu discussion graph author anonymous comment notation observation paper appear support uci award star fa dimensional ise propose structure independence conditional derive complexity consistent third novel require learn model
svm standard original raw pixel dimension pool anchor per anchor round anchor require anchor predictor state art thank dedicated dimension class multiclass hypothesis rely prove establish smoothness property first taylor hessian point hessian rewrite h maximum value form last imply eq equipped identify feature greedy iteration progress use denote index zero two rearrange inequality convexity hand conclude conclude claim ex multiclass classify multiclass predictor use linearly share multiclass predictor predictor fast set benefit success art classify relevant surface
probability side express intersection divide length due box simple straight line summation break show dash frame note sect projection complex two close position along calculate probability intersect box intersection line except come reference scan narrow box length well difference node position scan corresponding point correspond line come corner define measure intersect analytically evaluate break connect break corner easily change aligned corner calculate break adjacent break pair
discard remove early development game keep sophisticated preprocessing unnecessary reduction conventional focus reproduce rather dimensionality reduction preserve essential illustrate analyze dynamic complete construct win probability describe associated complex phenomena game dynamic play introduction complex high reaction bad coordinate fast configuration exhibit dynamic project close reaction propose base protein atom sample represent form optimize coordinate htbp
classification library
data fashion point clustering aggregate dimensional projection plane fig distance graph graph figure correspond variation clustering dark clustering highly band row order clustering close clustering significantly lack unsupervised clustering apply idea meta represent clustering connect inverse exist coupled clustering differ vertex meta dark around meta block corner large meta result depend expect enable handling topic averaged meta cluster cluster average clustering cluster traditional clustering return representative reduce clustering representative partitioning combine clustering equal co clustering never connect cluster representative clustering cluster representative clustering identify original nine clustering identify perfectly clustering individually clustering point set information representative nine clustering reconstruct two choose fourth clustering representative clustering clustering
nc ff domain optimization block implement local assignment carry could regularize iterate expectation nc c strategy time subsequently centroid update minimizer yield keep parameter fix eq solve solve simplify n scale update c interestingly show outlier outli hence step solve numerically interior root input cluster initialize update via robust spherical operation per large convergent minimum q n minimization proper semi imply empty non unique block block unique wise iteration term different coordinate wise converge cluster whereby assignment posteriori I outlier cluster sequence warm expect solve warm efficient suffice paradigm convex argue offer tight enhance
norm mu space far transpose stand norm vector magnitude fully tractable integer triple satisfy u u mn kn n km n positive integer regular recovery associate nothing triple augment appropriately speak triple well efficient synthesis extension choice consequently case achievable b h ta vanish complicated maintain validity intractable know imply gap system admissible computationally tractable necessity necessity assume norm assume choice implication nearly routine satisfying form v nd proposition condition nearly nd proposition
arbitrarily neighborhood uniformity corresponding assumption present section thresholding coincide adaptive soft thresholding coincide natural distributional adaptive thresholding insight simulation sample distribution respective simulate depend histogram ii adaptive lasso scaling component least normally partition block block equal factorization respectively comprise row three correlation correlation number either parameter adaptive imply estimator irrelevant outcome correspond proportion value simulate variable purpose scale derive red present compare adaptive soft thresholding remarkable agreement respective design design condition difference marginal turn counterpart situation somewhat strong figure ill matrix figure qualitatively h h h I subsequence may converge constant maintain assumption must since prove converse subsequence without hold choose opposite sign bound zero maintain contradiction complete part obvious n assumption eq first claim immediately claim q handle analogously subsequence suffice converge result support p n supremum prove prove observe p p ic select pointwise converge I se p uniform se se ps p ps complete eventually se se I
summary abc opposite value summary within three block use abc tolerance quantile previous experiment quite satisfactory highlight ability summary statistic statistic inconsistent choice thus bring answer abc true statistic wrong place fundamental choice practical setting statistic abc comparison mean statistic distinction differ classical summary addition testing summary range expectation summary statistic simulation model setting production pseudo quite preliminary neither final toolbox handling motivation description core feature particularly bayesian rely statistic thus result distance tolerance choice ideal carlo factor
policy class see relaxation could case extensive discover addition anonymous helpful comment support I fp ip sequential decision extend reinforcement exist none tight show near bayes current obtain robust reinforcement acting change obtain act instantaneous markov action equip suitable algebra
sphere cumulative lie proportion spherical total surface case mr r b xt b shannon calculus definition chain derivative encode sphere spherical gray bilinear encode exploit orthogonality intersection unit sphere subspace quantization
tree structure markov message unique bp r surely unique bp node potential message upper convergence vector guarantee conservative may element grow exponentially show conservative behave follow graph conservative address bp converge ordinary work log related technique condition message message oppose message recall normalized compatibility minimum row feasible involve graph suppose contraction consistent converge surely bp claim tree non mean error component dominant norm establish norm theorem expectation hand order include logarithmic factor part probability guarantee satisfy sample high exhibit bp contraction message geometrically quickly mean iteration accurate early bp message put piece ordinary operation compute comparison long desire computational become comparable graphical represent approximation reality theorem analyze guarantee convergence require message oppose relate message
check contain embed ask fix come figure suggest width line eps graphic graphic graphic http www ps width arise set demonstrate cox problem influence way small cause large bias bias variance
q lie unit eq right root determine stationary solution hull basically return ar q proof random attain vertex g calculate complexity interested diameter stability give tight prediction g care predict constant rate st daily
partially claim notice lemma replicate kernel center decompose influence note slope kp j correspond see small geometrically decrease use order geometrically finally discrepancy theorem discrepancy discrepancy choice simultaneously discrepancy range space accomplish vc distinct discrepancy change use slope size kernel kx kx nz ignore dimensional volume exist net slope
first suit perfectly comparable randomized initialization repeat time pca diag gmm gmm accord full performance full em dimension fact discriminate em matrix gmm diag gmm trend low diag gmm little bit increase tendency dimension performance fall poor performance explain fact subspace simulate fisher increase furthermore suggest succeed latent initialization model accord transformation simulate comparative purpose bic gmm diag gmm com gmm observe link different secondly select simulate right present datum leave axis separate accordance point bic conversely fisher discriminative gmm com gmm diag gmm pca bic leave dataset recent hand compare full gmm diag gmm gmm model mean version mean lda benchmark uci repository split specie describe dataset dataset characterize dataset describe detail family discriminate divide class accuracy deviation experiment
extra reward core converge priori keyword lyapunov tu von communication community essence tu comprise player thought distribute allocation benefit characteristic bound bound characteristic value mean expect coincide long dynamic core tu continuously player instantaneous subject know process bound generating find extra receive budget core drive priori convergence despite uncertain time vary tu arise value time vary see robustness latter work generate unknown line concept formalize worth mention common recent game interval similar generally optimistic
type mixture normal lie normal mix distribution arise exponentially alpha variable close recognize approximate issue overcome way package implement triangular density normalize mix conjunction work context proposal early implementation point complicated excellent supplement appear sampling exhibit novel turn mixture normal design nearly specifically representation result monotone triangle posterior mixing explain detail lead deal alpha joint difficulty work bridge gain show many likelihood penalty proper discuss facilitate bridge mixture normal completely distribution q exponential laplace alpha stable exploit conditionally conjugate mcmc
random minimal information determine estimation facebook graph show greatly outperform art future future plan problem review related technique remark california usa uci grant nsf award usa sample metric walk appropriately begin find weight sampler impractical equilibrium balance weight achieve weighted achieve collect facebook college simple random walk rw accuracy metric population weight c graph independence node blue node white naive approach rw irrelevant network web www measure desirable small representative hard information base rarely publicly available measurement element importance example depict world population china blue node assume median china take come china even sample china much budget people draw category sample much
r unnormalized probability condition candidate momentum position detailed select element must choose preserve use add jacobian unit r restrict treat unnormalized say require state slice must moment satisfie follow invariant r leave uniform must resample constitute gibbs sampling joint allow conditional density conditional condition r u imply free leave invariant specific r conceptually generative process building repeatedly size tree leave momentum tree single node initial direction take step height half tree balance binary height leaf position momentum visit trajectory illustrate accord likely reconstruct tree height leaf node leaf expand tree course expand make begin visit visit want expand simulation extremely discover large distribution density go easy stop leaf state nonnegative recommend moderately rule sampler neither detailed balance overhead whether stop height height stop leave subtree backward
row decoder lead expectation rip isometry property gaussian constant numerically decoder conventional known checking np rip measure quick concentration next proposition signal decoder expectation draw map rip lead mse hold vector distribution assume kb nk second equality rip expectation last since nan check subsampling decoder reconstruct different realization signal since coherence simulation dct basis carlo simulation rip figure typical increase small ratio fix slightly plot bernoulli vary little subsampling slowly linearly typical cc bernoulli sense approximation calculate decoder signal follow gaussian eigenvalue average upper bounded term distribution follow description lead inverse real score among introduce
normalize depend start connected component apply success value start point several point parametric point explicit preserving constant orthogonal orthogonal viewpoint expansion constant differential existence recurrence ode recurrence since require huge resource expansion preferable derive series integral invariance expansion change see unless odd change sign unless come odd express jacobian hence express integral find right laplace give gradient derivative hessian
recover base contaminate essential focus approximately low structure regularizer include sparse application gauss random field asymptotic frobenius relaxation decomposable regularizer specialized extension behave closely completion recent probability frobenius decomposable complementary remain explore acknowledgement three partially sn partially microsoft acknowledge international decomposition prove part satisfy property norm decomposable regularizer require parameter r special parameter sample regularizer nuclear regularizer separate also compute gradient property norm suffice suffice condition begin former suffice upper application triangle put piece since equality equation yield eq q since substitute equation substitute early appendix form corollary yield constant significantly scale corollary specialized norm careful term throughout choose split first function remain ii thereby remainder claim rate alternatively high see examine condition sufficient hold require analysis bind variable around conclude use extend interest constant since adequate careful make analysis expectation cauchy schwarz function gaussians consequently eq
linearity ensure reader detail consequence kernel kernel distance dimensionality space projection easier space neighbor tool collection indeed complicated shape represent rkh popularity machine context many consensus easily exploit linear distance reduce significantly within significant improvement motivation
area brain show micro property neuron example analyze neuron fire implicitly send clearly micro neuron concatenation sometimes superposition neuron hence successfully rate spike noise neuron unknown record neuron neighborhood constitute core problem signal consequently noise audio speech classify spike shape wave neuron still neuron depend neuron wave relation fire neuron wave vary neuron spike commonly know sort perform pca spike cluster signal spike observation eigen vector matrix capture since select variance neuron
main player round behind forecast auxiliary strategy give assess strategy group round round play mixed inclusion inclusion follow probability inclusion use boundedness simply repeat value game follow therein respective denote payoff player player action possibly accord end round player mapping h player monitoring game problem constraint armed notation useful finite ingredient unchanged ensure value payoff respect player random signal sufficient prove indicate contribution lie constructive soon efficiently auxiliary refined lead past essence use able rate third short layer notation internal monitoring auxiliary strategy condition component convex define set value condition sequel partial monitoring refer vector payoff satisfy reformulate subsequent constructive structure clear follow non concave strategy illustrate linearity efficiency game column force dark action payoff respective
know trivial many illustrate fractional diffusion type let let consequence definition property self every increment however positively exhibit sense older admit martingale term process integrable martingale quadratic hc gamma recently refer chapter add evy recently establish say iii iv old path variation sense
p b similarly b b b b b u u calculation result previous proof lack hence implicitly every go conclusion proof bound spend time define admit let spend result value impose fix proof follow z time spend case clearly similar reasoning low lead state begin update convergence proportion flat histogram
bring limitation discuss conclusion subgraph count programming develop package gmm fortunately package highly dynamic function network exist code capable attribute ideally suit computational implementation gmm essential structure special termination gmm simulate evolution base structure class place verify model gmm store store simulation dependency structure belief generate belief pmf previous third scientific computing require therefore package compound require evaluate subgraph include calculation rich illustrate quality scientific package unit test code website test software list index description view gmm simple gmm specify gmm base termination rule simulate study termination rule whereby terminate network growth implement graph implement pseudo gmm rule graph connect mechanism chain illustrate right panel direction cause growth several classic model method model exercise gmm highlight strength recover classic gmm model growth I specification attempt recover classic model early excellent gmm utility classic attempt network growth
variable drop constant use nc nc ij quantum spin acting represent spin spin interaction field act spin encodes ij ix jx sc iy nh jx calculate weak involve implementation device wave processor alternative weight problem methodology develop weak classifier correctness implement dependent interpolation hamiltonian eigenvector identity act interpolation satisfy condition time sufficiently slow manner minimize energy find optimal weight measure opt q opt opt classifier appear perform assign optimize group classifier together time ground assumption provide fidelity state quantum respect te instantaneous small quality overlap difference equal eigenvalue integer depend value depend boundary smoothness ref crucial gap shrink numerator mild grow problem exponential gap speedup speedup classical prescribe relaxed software amenable quantum speedup gap hamiltonian beyond speedup numerically fewer comparable give generalization error overlap adaboost quantum hard promising shall theoretically perfect majority vote strong weak construction shall weak classifier classifier relaxation shall weak vector eq accurate set three meet classify accurate construct entire correctly classifier every classification specification classifier agree correct correctness classifier stand weakly correct weakly receive vote weak classifier comprise classifier weak comprise majority
regard degree freedom second moment infer posterior stationary rescale degree drift assume transition sampler satisfy quantity fact get assumption bold line dot freedom illustrate proposition respect check attain ap decrease ap enough inequality computation show fulfil iff hand eq compare reveal consequently mean sequence martingale difference obviously examine bound fix bound gray solid assume obvious dash correspond equal actual root conclusion analogous table focus
match link pca square euclidean al multi relational relational couple analysis include couple tensor problem propose square summarize relative relative prescribed converge factor unit solve couple mode fitting I true convergence word cp simultaneously extend handle miss deal issue opt couple heterogeneous gradient analysis common tensor factorize factorize extract couple matrix objective solve optimization gradient first gradient rewrite I partial component combine matrix
basis depend polynomial well n fourier spline close embedding represent center give closely approximate kernel construct unary b spline corresponding follow b spline turn classifier spline b spline consist uniformly spaced knot truncate polynomial basis polynomial spline knot space repeat matrix p order regularization experiment cubic
parameter interval sensitive choose cluster experience reasonably change implicitly reliability estimate cluster hence limitation mean always extend eliminate issue extension isotropic cluster raise explore explore particularly configuration explore explore algorithm long overcome available account edge algorithm bound point outside induce measure window create integrate sample outside model account bias create around complete analysis note possibility include surface account example case tangent investigate e introduce spatial cluster along system curve galaxy finding facilitate aid line estimate permit various perform structure cluster often around lie boundary lie around family understand technique c pattern describe stanford introduce describe curve assign undirected orientation window produce
apply appropriately choose vector convexity linearity figure correspond exponentially weight apply dt tt coordinate last lie length upper conclude square adaptive introduction tune first cumulative regret indicate regime algorithm original regret b main however contrary alternatively without self confident weighting already adaptive algorithm limit initial square x setting get bound conjunction function section use adaptive modification lipschitz continuous refined bound corollary address loss high curvature slightly real forecaster e square high address proceed
use proposition pattern recover last component imply low satisfy optimality condition repeat sake ty ty ty c mean virtue support occur imply plan iv sec isometry plan result tx ty ty use orthogonality relation x tx ty tx tx ty root quadratic ty tx strategy confidence interval x x hence tx cauchy schwarz z tx z property begin tx ty tx z thus c r ty ty ty combine last equation tx ty x x obtain c ty support sign clear exactly strategy strong note imply satisfy optimality exactly sub sign mention conclusion
short path different delay show improve accuracy scenario message along short short respect delay exchange along short goal discover use delay topology equally well delay participant inherent ambiguity discovery point context latent tree eps minimal eps detail topology node end delay redundant hide graph minimal characterize minimal give participant hide graph iteration remove redundant initialize nh nh nn accordingly v ng topology delay random graph suffice reconstruct representation small participant delay edge variance le g discovery desirable original structure reconstruct unlabeled graph distance graph adjacency vertex graph keep permutation node unlabele goal small representation obtain delay graph asymptotically recover minimal definition assume infinity challenging set large sample topology large structure size network formalize set say goal require number discuss simple concept incorporate discovery set topology discovery base delay delay variance topology
replacement follow error support h h ta ta inequality hand ta ta yield desire clique ta hold corollary note ta cs j j contradict j clique clique satisfy solve clique kb ax prove exactly identify eq intersection satisfie column column disjoint need prove side n j eq need prove n verify writing exact wish construct respect intractable infer network describe time primal lagrangian assume n n multiplier polynomial constraint yield dual submatrix extract column original relaxed polynomially many index interior duality gap upper th sequentially call analytic new incorporating violate interior method
simply yield lot zero theorem least row index correct correctly show subspace indeed load loading matrix nonzero big coefficient eigenvector magnitude include magnitude fail pick medium obtaining include along achieve focus paper subspace interest eigenvalue separate spectrum theorem satisfy j hand connect sparse j restriction subspace ball display logarithmic near estimator adaptive minimax sense choose define set intend lead principal subspace gap note incorporate could step suppose hold satisfied sufficiently hold restrict jointly spike turn pt justification estimate consistently important need theoretical algorithm section conclusion use consider multiplication element come vector
subset j jt I lie possible ranking fix hoeffde less subset replace exponent query consider object thus r time query thus query determine rank index rank rank rank line mark around mark around ball center object object ball probability object object none position take jensen fourth conclude pm pm rank amongst label correspond vector assume fourth corollary university ranking comparison ranking sorting method pairwise comparison interested object may comparison euclidean ranking reflect preference human subject could certainly support embed could determine similarity audio interest comparison reference two
ratio evaluate biased coin eq q show nearly large interval little hard cumulative equal confidence interval obtain solution
top advance solve problem frequent suffer scalability mining frequency previous collection guarantee quality case provide contribution contain frequent sample result rarely transaction article convergence apply field field advance vc encounter field application database management aggregate vc related field database privacy show private concept database vc privacy bind vc query dataset although present goal technique inspire present detect failure balance short way prominent tight new time proof conference version numerous improve theoretical connection far experimental concern comment evident ar platform describe definition later transaction transaction transaction transaction transaction find decrease tie break arbitrarily discovery ratio intuitively transaction confidence association interpret transaction dataset transaction minimum resp resp triplet extracting
accord communication bundle online minibatch minibatch ns per hybrid maximize possible save machine involve small approach specifically bundle use implementation accurate per demonstrate thank second distinction take execution slow hand control carefully scheduling decision robustness slow batch elsewhere iterative machine evidence project logistic regression substantial also approach base run average empirically effective create pass briefly argue theoretically partition carry separate display site recognition datum example undesirable node node transfer require create predict constant body recent year focus predictor function method exploit node environment perhaps reduce dataset subsampling really good argue problem
regression estimate hypothesis assume differentiable obtain elsewhere density series expansion aspect c lx lx consider dx deviation result set ii iii see give chernoff estimate upon function constant whenever condition satisfy function ask condition satisfied proposition assume function bound state
guarantee constant allow finally tree marginal factorization forward compute marginal factorization message message message part message bp compute factorization follow algorithm property section es communications department mail com research visit university grant grant national mobile project intel mobile communication technology foundation wireless hybrid enhance mobile contract combine analysis method equation region free energy convergent message passing provide underlie addition factor correspond division message iterative decode variational decade physic mf area e visible pdf mf pdf e g factorize leibler minimize message pass factorize pdf additional
application solve solution previous consistently encode laplacian non zero encode bernoulli parameter mr zero encode video image bilinear identical laplacian encode sequence predict previous value residual encode code method simple part discretize laplacian look quantization negligible minimize encode magnitude element keep produce algorithm two b video left image sample contain remove video eigenvalue scale represent eigenvector bottom recover particular eigenvector need background correction leave people pass present base inherent use metric automatic per basis collaborative variant learn dictionary theoretic lead robust dictionary regularization formulation prior spatial modification box result user dictionary fit characterize model application work address establish theoretic result derive virtue principle allow incorporate markovian include area
structure index distance equal final distance consider component indeed read cluster indeed constraint computed eqs refer htbp cluster cluster figure mark symbol belong cluster area china version propose homogeneity component close object component cluster homogeneity similar contribution close component result component reach allow homogeneity medium table dispersion globally allow homogeneity cluster pressure mean wind speed relative cluster cluster cluster cluster cluster fig well partition two histogram datum adaptive wasserstein clustering optimize use wasserstein algorithm square wasserstein dispersion advantage ability standard mean find reach square minimum give
number uniformly integer assign else cell cell integer bivariate verify sis far enough count exact level failure success write statistic via sis table h l option reject table polynomial sample table marginal via sis rejection general sis may rejection rate table sis rejection whose
move play adversary constrain game play adversary adversary slowly change decision adversary move strongly sequence euclidean move game adversary conclude decision making played conclude restrict move throughout restriction notice force restriction distribution indeed force value formulation make information explicit minimization observe batch supervised come relevant strategy adversary call blind I blind analogue value general supervise proceed relationship online blind batch eq pass strategy proof learnable adversary sense know learnable specifically exist strategy use give rate rate decay convergent sequence learnable batch q thus learnable learnable distribution player formulation restriction learn supervise game pick adversary player suffer restriction define write define value blind supervise shorthand classical rademacher define setting say supervise learnable learnable blind sense
angle converge fix correspond prove smooth although stochastic gradient find identification filter constant process algorithm work decade stochastic online identification parameter set appropriate lead new deal general riemannian manifold introduce cast general framework algorithm intrinsic embed convergence almost establish never prove hold extend four online convergence immediately pca randomize intrinsic illustrate third detail concerned semi definite mahalanobis distance allow application g novel algorithm meaningful guarantee algorithm simulation indicate riemannian fast usual natural b brief preliminary z w differentiable cost measurable stochastic cost compute explicitly unknown instead access compute perform approximated cost I application hope almost g attract lot machine community filter matrix preference rating item book way overcome ambiguity constrain
netflix set propose max compare theoretical conceptual justification moderate ccc ccc test sampling estimate location marginal like via trace bound smoothed marginal theoretical disadvantage learn smoothed provide suggest marginal introduce well reconstruction smooth empirical smoothed distribution sx although smoothed trace smoothed marginal meaningful even e spectral deviation frequency marginal establish result turn set could guarantee
random hypothesis bayes unknown section present alternative particular say define introduce explain function surrogate satisfy pa aa ig posterior truth surrogate proportion equation even stand come requirement predictor asymptotically vanish support entail yield n equation generalize argument technical definition
fix task accord regular configuration worker second probability r iteration prove k r main theorem sensitive size leave run decay predict even requirement worker range rate suggest error depend effective right crowd affect efficient run comparable majority voting operation ensure run increase majority voting eigenvector suggest top sparse next make remark guarantee assumption distinguish task worker worker performance guarantee second generate need limitation additional cost per step allocation compare estimate agree amongst repeat stop even knowledge affect previous expectation maximization sensitive converge unique computing particular observe transition show significantly small majority surprisingly empirically exhibit fundamentally cf figure get regime get phase transition universal discussion limitation task master core concern reliability number want use degree factor necessary budget allow prove minimax optimal constant restriction assign worker simplify necessary adaptive iterative provide match minimax necessary adaptive bad case worker scenario approach minimax ask pay surprisingly condition algorithm adaptive query constant adaptive optimal large depend application regime restriction allow worker degree establishe achieve adaptive assignment scheme introduce probability show error quality worker need achieve practice assign worker comparison probability different regular run spectral use lead left approach detail voting performance value hand worker observe iterative inference start vote
theorem select loss bound select width histogram easily obtain approximation guarantee optima problem know obtain contain query contain tb column column number b subsection fix extremely width heavy dimension histogram minimum lead practice formulate use sparse transform wavelet haar wavelet orthonormal cardinality haar transform piecewise wavelet wavelet recovery compress describe estimate number zero compress setting obtain unfortunately random range query formal guarantee leave future particular omp omp greedy technique start empty omp large value omp eq note dynamic produce density quadratic attribute frequency modify provide code aa set residual ta rt obtain histogram subsection approach section next sparse general range q I note rectangular restrict histogram set analogously optimal incur compare histogram w bb bl rl detail tighter
analytic form result random mean figure histogram volume deviation histogram recover sampling notation show computed eqs strictly speak likelihood across quite obvious live find mode basically find unimodal live one probability live mode live exceed volume span small mode must actually analysis relate evidence function step evidence sense result note
equally target bias towards well target metric target heterogeneous general max parameter demand edge sample hold grid uniform become demand object exploit selection greedy particular demand proof appeal topology classic membership term adaptive oracle learn search take theorem object normalization bound sequence receive index containing mean rewrite q reach message occur phase link object since therefore dt dt v r tv phase stochastically dominate lie property message close edge object triangle object phase thus linearity thus
nu f satisfy bernstein cb moment gaussian inequality var fx I fx f ef ef ef cb rearrange e expectation v respect hold tv h corollary conjecture invoke imply smooth support respect motivate
purpose appropriately chain reversible exhibit gradient reversible metropolis hasting differential drive wiener correlation primary insight monte hilbert must approximate dimensional dimension order work herein mcmc explore step langevin motivation optimization minima gradient flow computation expensive stochastic partial carefully specify idea know dimensional context gradient flow develop noise go name simulate review reference noise construction method simulated anneal idea directly emphasize apply technique lead degenerate adjoint space gradient flow complex purpose describe walk combine metropolis hasting accept mechanism well idea anneal review reference paper develop dimensional adopt optimize viewpoint chapter dimensional technique diffusion construct basic building evaluate combine proportional finite algorithm respect quadratic form via lebesgue flow limit motivate chain observe
half external benchmark capability new scene semantic labeling tree associate year semantic labeling challenge broad community obvious etc early approach recently problem e learn patch specify field demonstrate success real one reason make difficult notion region demonstrate object inter understand challenge locality brief little part among contrast part label scene pixel pixel broad notable object recent year suit object abstract require part make call limitation image understanding propose global part object tree road tie pixel parameterization rich plausible graph learn complete relation paper manually jointly
hinge reward restrictive many maker allocation reward identically distribute reward arm arm formulation recent reference therein assumption value variant greedy horizon infinity whether notion case type address regret armed bandit underlie functional response mild smoothness encode function describe arm response arm prove first limited upper improve idea elimination static policy deal covariate explore build arm remain horizon tune regret investigate bound upon canonical paper elimination principle solve group similar covariate look bin bin view indexing aforementioned horizon restrict class difficult remain easy adaptively successive elimination severe limitation naive globally
subgraph order subgraph place around set triple disjoint x disjoint contraction implication weak say reverse implication property notice undirecte trivially satisfy hence compositional direct model separation probability variable space disjoint subset conditionally independent measurable almost independence cp independence probabilistic semi converse strictly regular compositional fact entry hence compositional multivariate say pt edge arcs edge line type mixed loop arc em em ki ki follow text let mix three em em ji em em inner
shall dimension subject recent year cluster problem paper point subspace subspace dimension consist partition accounting outlier unless special unit sphere orientation identify assign hide model always subspace affine goal leave introduce treat observe n n quite attention literature ssc algorithm expansion select vector motivate whenever belong subspace detection nonzero entry column subspace detection support subspace might construct vertex construct normalize gap eigenvalue estimate subspace cluster step subspace cluster matrix represent affinity exist involve restriction minimum purpose broad shall ssc correctly intersect principal vanishe prove ssc subspace grow almost linearly ambient sure allow grow root ambient modification ssc data corrupt exceed clean well provably capable handling improvement approach analyze cluster combine tool probability theory functional clear insight ssc successful viewpoint prove propose introduce section ssc give
easy argue enumeration convex feasible approximate correct satisfy hypothesis v ta tv ta f bounding conclude nonnegative hardness result arguably model nmf compute explain explanation hide take occurrence document nmf compute document express topic assertion focus question design algorithm resolve assume nmf whether polynomial geometry empty dominate distinct occur play determine algebraic depend quantity additionally compute successive set time naive constant would run heart algorithm theorem variable define semi algebraic factorization tool geometry obtain exponential time search heuristic maximization run time natural independent nmf express small hidden linearly lemma translate alternatively combination set section associate factorization sf exponential previous exponential system solve check complement use inspire exact nmf hold sf exact sf nmf problem turn nonnegative factorization separability segmentation identity matrix name easy search find row provably work
subject reflect development undirected graph generally region topological measure currently consensus compare mathematical essentially advanced suggest topological respect graph integration literature address integral monte method number estimate topological metric researcher tend involve short path include propose edge propose short
tune dt regret may grow match straightforward forecaster forecaster ds logarithmic satisfy intersection optimization quite small scenario ball reference article convex square aforementioned small proved intersection ball explain gradient satisfie regret differ infimum jx jx tc regret jx jx bound prove latter norm divergence regret look prove aforementioned specific ball contrary see prediction open sometimes essential derive sparsity pac review pac literature refer reader thorough introduction pac inequality recently sequence loss rely online true loss exp inequality automatically apply automatically tune thank regret imply hold solve besides priori latter since unknown adapt regret forecaster fully automatic setting model contain game
truncate expansion give reference side length less achieve obtain value determine evaluate reference develop hierarchical translation operator gauss reference consideration field compute far center first bottom reference expansion translation field shift q moment centroid center moment third entry wise field moment field moment iterate act clean routine approximate query must perform traversal shift centroid child local q center summation center centroid center expansion center node accumulation child wise scalar multi query reference approximation exhaustive computation high approximation exhaustive query reference encounter whether query node field moment accumulation constitute moment evaluate determine asymptotic description fast gauss key even none cost exhaustive method exhaustive query fine approximation p fc dc te reference query direct accumulation right query centroid main overview appendix kde routine maintain store pre
bound write ss lk rt cs il rt expectation arise thank tree calculation exercise prove calculation support fellowship award nsf dms ise introduce toy fact covariance symmetry correlation edge graph odd node adjacent number edge correlation matter explain node fix q instead case computing form edge belong figure eq quickly pp g g seen connect e make success care ss component q hold applicable name also proof proposition assume assumption proximity mean mind straightforward application inequality occur event q small enough verify upper bind ab bc e negative v cauchy v x k small k theorem easy probability bind outline note represent replica lemma j g gx application
flexible module feature implicit building feature datum ill define input important bring interesting idea self context put road formation careful feature drive complex principle genetic programming principle formation configuration approach decide promise hierarchy formation experience combine promise translate
change alarm anomaly frequency detection detection anomaly method frequency job link point st anomaly st st detection relate recent use alarm propose analysis aggregate seem fail detect change detection alarm point among user conference discovery link anomaly figure alarm change detection conference frequency table among twitter
monte former measurement discriminate source truly qualitative processor generators generator architecture hardware programming communication gpu execution code cpu gpu consist compatible device globally memory processor mp finally processor unit perform software provide language device share memory memory barrier barrier processor implicitly schedule problem execute gpu device
selection proposal example example scale mode show correlation obvious tail highly might acceptance ever know front though percentage amount draw demonstrate large class token describe model topic additional include discrete mixed bayesian difficulty lie large discrete difficult estimate discrete popular model close direct tractable advance parallelization use processing might integral practical al case log posterior smooth alternative augmentation kind mcmc approach suffer introduce augmentation weakly normally interested miss small could treat point require require hessian posterior restrict size dataset
cm stanford department science berkeley department berkeley author equally modern essential inspire ease scalable divide thorough characterize divide magnitude enjoy also present collaborative filter background demonstrate collaborative divide distribute randomized video surveillance pose science often entirely infeasible become challenge focus attention massive may satisfactory statistical strength research develop inferential task need write singular decomposition orthogonal onto frobenius matrix framework scalable interest matrix entry sparse outlier dense noise represent onto goal focus problem location zero save entry uniformly replacement combine expensive subproblem subproblem combine framework algorithm differ strategy submatrix submatrix parallel submatrix perform mf retain factor project onto subproblem form rank generalize input input step generate rank approximation use replacement column practice reconstruct rather maintain factor e
correlation cca direction common illustrate tendency cca score loading joint score closely good job distinguish panel signal reflect loading panel reflect loading group loading effect individual loading application object subset measure motivate fit sparse version exploratory pls cca penalty induce variable sparsity accomplish loading induce sparsity loading analogous loading small contribution penalty thresholde penalization loading component color panel color display nonzero loading pair color loading blue otherwise panel linkage correlation sparsity accomplish analogous minimize
trajectory event ms agent c ahead none ahead none avg life lose avg score life make lose ms look ahead ms pac increase considerably power dot deep ahead interesting ahead apart look ahead base could would thus markovian power decrease deep look ahead architecture relate rl algorithms evolutionary emphasize easy learn game branching factored may considerably exponent rl hard machine density correct decision dot dot low argue td decision reflect density parameter scenario however general unclear module module entity consider play module activity ball running involve body pac simple view ball challenge application body module properly ahead factored example rule analog ms illustrate information fig fig ms wrong dot ms one observer simple
anomaly background treat properly possible restriction novel consistent search physics semi supervise detection show successfully search physics systematic error learn extensively datum decade especially multivariate tool ratio search new physics signal background event classification obtain mc physics physics begin certain new event correspond choice generate
comparative competition organize provide blind performance state post new perform conclusion column transpose transpose expect represent calibration prediction provide descriptor make date round involve descriptor remain round ccccc nature round bind affinity affinity complex methodology propagate establish response
absolutely rotation scale family equal range q rotation invariance right side unchanged orthogonal matrix haar orthogonal range statement likewise jacobian claim f nu v dp v f absolute na sx sx transformation random df sx na ia proposition bayesian multivariate model characterize alternative matching interpret offer normal normal center parametrization cluster use cover entire normal model calculate simulation normality nonparametric bayes match integral statistical testing provide simplification nonparametric important parametric scientific hypothesis
optimizer k consider algorithm add remove active find long loss meet terminate backward contribute remove ensure round terminate loss guarantee recall definition restrict convexity specifically rsc restrict rsc parameter function smoothness give similarly loss bound require property loss gradient I capture loss parameter stop theorem
number rank play implementation assume bind meet solution terminology w x tp vb z z u u z w riemannian hessian compute formulae trust numerical complexity handle duality gap completion linear domain candidate final gap candidate trace next random generate accord mean fraction os ratio determine randomly priori mention remove scheme table trust illustration purpose also stop function fall stop duality gap convergence scheme show c incremental right solution right rank characterize look reconstruction reconstruction trace minimization experiment initialize report indeed trace time order compute entire regularization predictor grid illustration geometric fix various demonstrate advantage scheme table warm basis prediction
approximate positive continuous minimum normalizing density family compact exponential estimator take uniformly leibl exponential get small goodness fit fitting separately unconditional compete denote parametric choose goodness ideally goodness evaluated minimize kullback infeasible
guarantee side root pt pt pt pt whole tree select tree implement attain unbiased select true simulate test plan different branching keep paper formalize statistical value decision feasibility obtain approach ideal bind solve stage stage combination simple branch control program poor use valid extreme event scenario supervise section form stochastic scenario tree dataset approximation method statistical describe exploit feasible current describe simulation good statistical notation trivial value run vector single observed stage represent cost uniquely determined domain support
stable vs spatial ideally amongst likelihood criterion logical logical recent cast approximate abc consideration factor realistic set wider less trust leave without model selection might outperform short demonstrate compete exploratory describe statistical limited close log log likelihood yield normal likelihood method widely develop analyze spatial approximate extreme three implementation first third successful benefit show approximate compete composite dependence also bayesian naturally field computationally intensive independent identically distribute univariate q must member location parameter extreme limit fr tail bound extreme hold
obtain illustrate formula validity dt quantity compute expression eq mix asymptotic eq c c discuss mix approximately take representation see ensure desire mixing ap column monte verify approximation kullback optimal mix distribution however mix considerably less go e p accord ccc mix uniform monte side limited practical problem sequential test procedure build combination paper certain implication practical want stop soon either time variable take depend accept say sided threshold show attain alternative
handwritten modelling pose image reconstruction pixel code subset subset assign comparison point measure fraction misclassifie substantial confusion train significant target relative target relative top patch highlight result qualitatively setup color fig target background overlap patch ratio art detection fig tool nonparametric bayesian obtaining start suitably parametric tractable demonstrate argument amenable trick define one predictive evaluation
response enough relate regressor overcome regressor relevance regressor able concern regressor describe use contribution regressor calculate gradually select respective ratio ratio error regressor decrease response often begin gradually decrease remain regressor observe reveal redundancy regressor determine parsimonious suggest help either surface temporal capability analysis flexible build capability regressor principle regressor explain alone often response true local noise naturally cause improve robustness locally use stop regressor analyze converge variable iteration ratio begin uniformly observe examine regressor rich concern regressor dynamic change nonlinearity dimensionality carry depth information may regressor structure training important provide efficiency expansion experimentally order regressor alternatively
abuse terminology equivalence remain show finitely process go class symbol give class symbol stochastic claim ergodic alphabet history triple hmm irreducible hmms hmms definition sake brevity construction example formal definition idea process alphabet machine hmm irreducible symbol generator machine generate cover label right observer nontrivial uniquely even must odd sequence two state finitely exactly positive equivalence x generally generator generator machine finite word depict generate abc choose uniformly inspection generator irreducible shift history machine despite initial law limit limit odd converge initial equivalence state argument like generalize machine noisy depict generating period np alternate symbol state generator sequence probability odd step state even induce machine form grouping
price price storage occur price explain concept convenience yield implicit associate stock hold stock service contract fundamental consider economic relationship imply cost identify factor interest convenience positively stock one need asymmetric basis contract close limited economic effect movement price soon large price correlation near price contract decrease propose general long firstly allow incorporate stochastic discretized volatility option observation methodology advance sequential monte jointly volatility section explain methodology detail methodology rao version markov carlo methodology non typical differential e market price market order risk neutral neutral factor payoff reduce follow motion convenience treat yield fail price secondly extend study model motion convenience model
analysis decay people mutual link mean link seem decay precisely formal statement rise activity social predict decay window decay problem science originally study interval see snapshot longitudinal research link prediction focus evaluate predictive technique order address systematically characterize attribute extent fail attempt importance several identify characteristic evolution email extent structural closure contribute formation new edge people common edge similarly formation edge social thing role formation factor decay neither decay network allow validate social unclear edge effort close persistence mobile phone primary rely set well edge average degree source information record cell phone consistent duration type call text dataset primarily phone dyadic communication exclude exclude auto data phone call week period call information call call decay call phone vertex exist actor actor window criterion
algorithm see vector level initialize index thresholding technical need ax appear rewrite solution exact subtract get rearrange present thresholding algorithm sparse thresholding novel enough also tight thresholding experimental vector million face diverse area finance curse knowledge structure study accordingly huge recent machine signal devote modern processing deal measurement matrix recover see sense setting recover able isolated property restrict isometry rip long measurement rip true vector formulate linear efficiently reader say ensemble provide choose goal weak convexity linear typically suffice rip rip
stream predictor recursively average square summarize detail propose provide letter case letter transpose denote set denote empty denote sub vector form coefficient dimensional comprise observation sequence dimensional standard filter terminology reference filter problem recursively time q factor control
account phenomenon probability distribute response although generalise model amenable simulation likelihood increment increment experimental reaction bottom exclude chance outlier fast describe behaviour pair heterogeneity experimental accumulation rate experimental boundary share therefore prior parameter clear interpretation reaction normally less dominate quantity drift lie outside yield drift rate transform transform attempt abc difficulty choice summary summary experimental summary due inherent curse standard abc take small statistic abc ep notice condition iid employ describe section save go second adopt third stem site case algebra update marginal
quality pmf optimal norm norm time fast pmf implement whereas optimize implementation feature dynamically regression orthogonal consider minima pmf conclude preferable pmf strongly input moderately problem local principled local approximation remark pmf include extra flexibility use capture extension explore could problem improve sense analyze theoretically superiority suggest optimality penalty parallel eqs sequential updating idea gauss update thank code pmf file thank sharp discussion improve manuscript variate see variational feature region unique transition imply quadratic solution twice stroke
requirement estimator influence common influence equal potentially unbounded preferable assess robustness limit easier depend claim evaluate robustness uniformly technique show statistical dirac theory
exactly random effect variance record variance hope estimate bootstrapping effect find approximation around quantity coefficient ideally method expect dominate way resample distinction multinomial apply bootstrap regard factorial bootstrap might naive resample replacement naive tend infinity effect see netflix datum compute variance severe quantity treat bayesian degenerate parameter degenerate thus user need continuously distribute tie bayesian observation get weight bayesian distribution take weight match bag boost integer double nothing independently observation double nothing version remove sum section bootstrap z eq method taylor reliable proxy naive bootstrap bootstrap large datum greatly offer
bit accuracy experiment expand hash accuracy bit hashing per course surprising demonstrate magnitude large random projection issue relatively gb market regression dataset course age growth hashing method test bit solver improvement choose popular familiar validate generating solver without notice code unlike practitioner window experimental agree hashing substantially storage hash platform
learn unsupervised manner prototype dynamical shape field take sign probable automatically illustrate signal modern artificial ai device broad include recognition language medical mention intelligence ai essence able perform optimization make understand merely ai device pre ai device neural
response calculate binary w alternatively allow probit pseudo treat except consider base value consistency vector begin response restriction impose avoid place scenario describe processing section outline conclude predictor influence value matrix denote predictor model link deduce relationship particular wish toward write influence additive interact predictor interact interact predictor practice far simpler distinguished contribute whether demonstrate possibility binary continuous simplest influence category perform alternative consider detect marginal ccccc c ex g main pt useful thought problem pairwise method extension likelihood could often time
step player move player classic play opponent relationship performance drift toy respect drift first fix examine fix row two set depict opponent set affected increase result affect tracking opponent many poor approximation capture opponent decrease opponent depict high result big change small good strategy initial value scenario value whole real choose observe opponent strategy opponent depict red blue fix pre opponent prediction depict perform jump used action opponent action game second action depict opponent depict respectively opponent prediction red occur
marginal call rule bethe algorithm freedom message learn bp initialize run choose initial message stop previous learn approach begin bp bethe place minimum belief marginal possibly point however reason provide bethe bethe attempt learn marginal belief reach vary still reach equilibrium temporal equal marginal bethe equilibrium marginal match belief average bethe equation representation give average reproduce
achieve policy action achieve future sensitive policy linear bottleneck increase contextual expressive overcome sensitive instance policy return accord create weak rule weak learner solve contextual contextual bandit choose exp step set example know probability adversary hope assume context computational scale attempt get perturb computationally special mechanism application policy space oracle aside drawback worse improve adversarial possible adversarial set feedback delay round adversarial setting I previous use epoch regret scale analysis exist reward action lie continuous accord regret round extend vc special case new simultaneously small argument context reward
lebesgue manifold deconvolution receive attention deconvolution raise interesting hausdorff distance author contaminate noise deconvolution interest bound wasserstein distribution homology additive estimating homology principal surface approach manifold establish bound hausdorff noise denote open center set point euclidean supremum fold product eq denote measure fourier denote use dot product nc expression
finally summary quantity use characterize population element mean converse pdf number imply measure rate support support point wise great pdfs pdfs homogeneous pdfs imply homogeneity pdfs large great heterogeneity aggregate exist aggregate normalize general support relative population support quantify raw measurement individual population circumstance collection come identical per denote individual arrive imply element measure insight rate iii rate metric diversity characterize rely force homogeneity recall individual variance support simple instance minima maxima length intersection support differ support maxima length support characterize way maximum band population pdfs infimum pdfs union pdfs collection coincide collection population population substantially near zero supremum supremum near nearly method sensitive heterogeneity rest maximize second method diversity pdfs population quantify support union support pdfs maximize maximally orthogonal calculate l analysis pt quantify average quantify aggregated population quantify aggregated support support population aggregate homogeneity pdf estimator bias variety way bias individual permutation bias preserve relative quantify diversity quantify imply diverse cardinality homogeneity support diversity population diversity p population point individual rough individual interpretation split three broad perform preliminary interpretation iii understanding use pdfs yield understanding proportion algorithmic depicted circumstance useful population calculation variance distribution pdfs interpretation scientific circumstance purely
satisfy kolmogorov appendix basic definition kolmogorov shannon item process markov process class bernoulli variable outcome random class typical outcome kolmogorov form former code alphabet constrain index put last additive item take string string represent source take account encoding outcome outcome joint x x copy stochastic chain dependent whether stochastic
transform base introduce pointwise continuously statistic channel unnormalized haar wavelet remarkable orthogonal transform risk wavelet optimize dependent article organize first unbiased risk chi square arbitrary degree freedom apply pointwise transform threshold give chi unnormalize haar propose inter conduct mr evaluation method international conference independent draw chi degree recall see random characterize data q k order modify function kind chi understand transform first straightforward designing consider deterministic realization distribution requirement
decrease depend x I remain eq term positive plug obtained verify decrease material I lemma I clearly dominant utilize simultaneously lipschitz general present synthetic one highlight fit theoretical finding real world even early two difference iteration monotonic datum adjacent monotonic generally difference theoretical require outcome particular might try computationally quite subsection sometimes fit
light monotone call interval even bad interval interval light conditioning contribution bad note bind monotone interval side suffice sample probability conditioning equation follow pi pi pi pi turn clear contribution partition g p monotone light interval tv sample pi tell either accurate sample hypothesis distribution accurate draw sample pi I argument henceforth hypothesis output probability monotone interval consideration hold claim fact distance I c bound ki statement monotone tv k ii expression maximize final equality fact prof run simple perform increase motivate level dominant let modal time monotone know efficiently monotone learner separate clear hope efficiently mode natural try decompose modal level novel modal sample yes monotone monotone distribution stress modal essential whether monotone monotone care running parameter decompose monotone
offer scope interaction offer way interact average user thus increase hope meaningful projection pursuit exploratory previous narrow argue allow exploratory much insight class make dataset demonstrate task far demonstrate quite subtle spatial explore formal application easy straightforward histogram tool traditionally principal transform exploratory pursuit chance use question necessity exploratory pursuit replacement benchmark motivation mean benchmark notion thus really occur meaningful certain context really context usage year rather poor successive generator
handle aware restrict seem restrictive class action guarantee section restrict base cost meet requirement bound bandit trade reward number key use r dependent propose simply difference bad trading way bandit reasonable reward reward satisfy leave observe term p second consider relate function uncertainty outcome trading cost affect like come observation maker position outcome independent outcome independent bundle share exactly bind much perturbation change need argue
mix allow consistency I dimensionality mix nj ix balance consistency assumption uniformly sufficiently depend q coefficient decrease confirm theorem play fundamental role implement thresholding estimate threshold select minimizer function q observe time consecutive segment remain illustration procedure convergence oracle loss perform oracle justification adapt band choice let x w oracle estimate define empirical theorem satisfied c p generality justify coincide end white satisfying constant cross validation minimize oracle problem high subsection procedure moderate variable explanatory covariance thresholding generality standardize actually pearson correlation distribution population classic person correlation pearson two former
small looking interval cover simple plot long abstract automatic parametric kind important green article quantity svms stochastic error term several establish x iii three directly however time quantity interval special wavelet univariate nonparametric
expert many expert insight study expert rate likelihood mle fast divergence converge produce rate nonparametric exponential model I mixture tool density know covariate space adequate fall generalization classical constant covariate expert widely recognition audio finance distinct include among one conditioning variable parameter glm example binomial fall might lead discussion many simple new consider use interpret gain increase complexity direction
bioinformatic homogeneity performance mmd depend width mmd permutation investigate nan hypothesis hypothesis sample set perform homogeneity summarize show adaptive type situation two generate denominator thus numerator positive experimental summarize show adaptive tend outperform mmd term homogeneity propose outli detection outli another regular tend use wide support regard evaluation sample denominator numerator outlier outli outli outli detection outlier trade adopt auc bold face illustrate behave outli dataset let dimensionality setup regard sample table describe input auc get small outli detection become challenge result input dimensionality tendency sharp point ratio preferable density
rest globally discuss possible interior rest sake monotonically equal temperature straightforward check along possible rest depend ratio depict light grey region correspond ne intuitively correspond ne rest ne exploration vanish correspond anti game three rest light grey additional rest point sufficiently obtain characterization consider obtain quadratic equation reasoning intersection meet fig anti game illustrative original variable recover briefly interior single globally nature multiple rest jacobian eqs symmetric equilibria recall critical
piece foreground put overlap sake result specific lasso strong correlation lar independent simply simple converge apply problem separable scheme pixel resolution rgb channel add result test remove lack structural constraint encode neither group sparsity improve norm correspond reconstruct method foreground mask original detected overlap present foreground color aim linear patch adapt dictionary prove interested less multiply scalar inverse prove practice prevent induce norm element natural image another put encode decomposition relate arrange independent corresponding element model dependency natural like grid call dictionary norm dictionary grid group spatial neighborhood grid grid cyclic norm task consider project one perform orthogonal batch drawing signal mention recently hierarchical dictionary image dictionary group notation subtree dictionary idea propose prune irrelevant use additional decomposition remove w operate instance overlap group define alternate optimize respect protocol element improve denoise sparse hierarchical predefine dimension training dictionary impose handle
apply simple univariate strategy weakly dependent weakly dependent correlate identify calculate correlation pick correlation variable perform leave dependent dependent strongly dependent partition identify amount estimate direct procedure use learning identify dependent play lead role later reduce correlation accord size depict weakly dependent correlated term domain variable reflect value reflect optimize univariate less cost note gaussian base dependent identical weakly need course imagine define strongly optimize separate dependent one interesting consideration typically strong multivariate gaussian obtain project build impractical increase model sm flow variable sample project subspace extent trust reliable divide subspace project consider dependency among belong subspace probably offer feasible way growth grow validate section construct partition user define multivariate htb x cx x randomly subset multivariate gaussian gaussian subspace matrix approximate diagonal fig mean current capability sample user experience rest later sm perform every generation group interaction strategy also variable belong evaluation subspace partitioning simple straightforward sm indeed performance dimensional course need divide cluster coefficient treat however still disadvantage suffer curse size expect cluster comparison partition sm explicitly reduce sm accord population denote realization variable perform sm form univariate gaussian sm algorithm control need individual individual contribute replacement htbp initialize meet individual individual individual
ghz cpu ram discard thin picking obtain posterior chain converge comparison purpose run variational bayes second attain variational plot shape prevent clutter significantly towards accordance gibbs sampler note informed guess treating improve drastically formulation constitute behavior mixture normal connect broader namely arise inferior normal mixture normal particular guarantee result essential unnecessary signal oracle
develop low output scale fine code closely resemble hidden however subtle neural major difference weight learn independently node propagation minimize typical ensemble converge unless stationary difference seven bit file one size figure rectangle seven bit datum order magnitude mechanism input history fourth return th order context recent operation index mod set history also mixture expert mixture expert et backpropagation network interference effect lead learn naturally interference compose plus feedforward switch select expert expert expert development compression divide subset separate expert train expert train expert difference mixture al feedforward learn algorithm specific one area future learn distribution ideally partition however use mechanism expert govern unit recurrent example lstm hide gradient g deterministic mechanism intend prediction vanish computation recommend work rnn
linearize bregman journal science pp bregman minimization application compress sciences invariant rank vision pp mc low mc like know entry stands sample mc low feasible lead objective popular alternative summation rank exactly cardinality portion entry corrupt low portion entry corrupt sparse stand np exploit convex convex summation balance recover assumption exist solve problem large aforementioned approximation involve objective mc use solve contain analysis
design single maximize function appendix lemma relate x x strictly necessarily often convenient support set find interval reduce polynomial k well table notation design
round since loo stable shown result potentially binary potentially randomized uniform loo online exist potentially uniform loo stable c minimal need cover loo e g exist sequence round bound exist sequence om learnable loo particular playing round suppose must round apply setting diameter tt notion I learn adversary pick time always setting supervise input instantaneous regret z h instantaneous interested characterize condition set stability stable asymptotic minimizer output regularize mirror weight interpret thought empirical erm dataset erm stability albeit strong require guarantee type form stability slightly strong weak
solve simplex optimal finite set linear function piecewise uniqueness quantify metric task suppose simplex nr ir ir small object weight whenever later paper weight respective notation move restrict index formulate distance agree favor histogram histogram criterion balance favor metric idea pair formal perspective ordering criterion neighbor weighted neighbor sum set stand necessarily subset kn kk consider positively homogeneous unbounded solve issue restrict search variable upper piecewise consequence admit least cast convex
help approach problem directional sphere ray event observe set begin interested propose framework test directional datum spherical specific show perform situation competitive tend really priori spherical test incoming particle energy simple threshold implement joint directional energy numerous dependent multiscale use cr make public one tune use sophisticated drive method tune high although threshold approach threshold test power however possible give although compactly support window appropriate spatially concentrated function knowledge distribution ray spirit directional specific like fine coverage take benefit full significantly obtained investigate finally addition extension stress two theoretical experimental theory part acknowledge package interface package transform thank concern le fr ed make aspect thank anonymous associate suggestion improve proposition france paris france decade origin ground attempt discriminate try arrival energy suppose toward directional sphere nontrivial make turn detect form explore wavelet
base expensive computation algorithm score trajectory without computation complexity state practice number training example feature quadratic number able feature research sequentially classify step linear computation choose action action already acquire classify score contribution newly order feature equivalent constrain practice choose second couple minute minute hour category indeed name number binary breast cancer binary diabetes binary heart binary binary guide segment multiclass
run time repetition vary various linear weight age seminal predictor first standardized shrinkage try aic bss sub list h lp aic bss shrinkage estimator prediction ten error repeat l bias correct validation small base bss aic instance bss sub clearly shrinkage estimator utility positive practically may well correct shrinkage estimator less sensitive misspecification reduce maintain superiority misspecification behaviour detail monte carlo section call case percent
matrix element diagonal element converse let position let column agree index form generalization let satisfy equal non let element column goal case claim large subtracting expand positive definite together proved need conclusion maximal take denote consist establishe acknowledgement acknowledge arc centre mathematic via team corollary lemma comment mathematics university edu university france fr maximal determinant search design find optimal design
range increasingly complex grow automatic density adaptive nature box exploratory mcmc aim practitioner exploratory stage preliminary exploratory carlo fitting attention grow complexity tool tuned author publish acknowledgement support e support fellowship acknowledge analysis author helpful wang interact parallel chain mechanism comparison flat histogram bin split interact hence former chain state chain chain use present bin split desire new bin reduce desire experiment however iteration xx sequence typically update bias invariant parametrize proposal ad I extend necessary extend split bin bin flat histogram I I tx stepsize monte chapter recent ergodicity fail explain parallel version prof infinity precise statement impact upon require impact add proof consistency flat stepsize flat stepsize stay rely threshold result result flat histogram remain stochastic algorithm adaptation homogeneous hasting
q depend apply practical unknown validate heuristics oracle spectrum rank large corollary finally rise appear estimator slightly heuristic notice index auxiliary distribution yx lead lead conclusion quantile unconditional enyi representation law number e write entail collect conclude denote u k
include median minimum variance dissimilarity center dissimilarity method mention cluster specify dissimilarity agglomerative hierarchical dissimilarity method formula center diameter median centroid j j attribute name coefficient define recurrence formula agglomerative similar cluster center table dissimilarity must equivalence two point formula use q center distance expression readily verify agglomerative centroid variance store dissimilarity store examine dissimilarity two point cluster representative center treat remain object center method consideration since storage object linkage term refer theory single link dissimilarity couple store implementation look reciprocal mutual near use practice agglomerative hierarchical conclude agglomerative
normalize convenience homotopy prove view monotonic know need condition fact correlate natural mean full symmetric ii ii eq jj kronecker move right side dd matrix dd cone principal minor principal point cone absolute nonzero
weak hard extract rescale useful property formalize empty size margin produce loss generality orthogonal margin norm assume pick subsequence limit nan firstly margin extremely example contradiction secondly otherwise arbitrarily example attain arbitrarily contradiction margin everywhere nan space contradiction positive subset margin reduce empty sequence zero restrict inductive zero margin property combination margin within existence example suffice row f kf I kf corollary prove item set loss margin less upper contradiction suppose margin bound element present mx exist whose loss lemma achieve empty subset contain margin convex combination f false margin show arbitrarily large entry rate hypothesis doubly feature
strength part several restrict elastic limited past subject treat processing align e computation study comparison jointly energy careful convex combination f ds ds pp recommend analyze statistical use match disjoint change alignment may additional orientation interval composition parameterization domain space separately jointly solve problematic symmetric one enforce double solution however degenerate ensure symmetry symmetric eqn commonly function keep close suffer step seem problematic empirically conceptually template use different basis turn relate observation nice solving implicitly eqn geometric natural fundamental function shape parametrize train problematic eqn understand issue
trivially hash consumption avoid disk use task many value preprocesse gram response generate disk occur mask hashing note substantially low indeed concern cost accuracy continue increasingly resource partition compact representation sparse binary hashing hash positive definite naturally logistic leading bit variance random projection significantly interestingly bit hashing improvement speed department science ny edu microsoft research storage efficient progress among integrate logistic
row dot product although clearly let onto span matrix span rank extension statistical leverage quantify leverage row square influence square leverage score quantify leverage low quantity widely diagnostic large exploit example degree cluster coherence value customer network dense thousand hundred even traditional qr dna single nucleotide snp increasingly genetic snp possible previously confusion emphasize main contribution important concept statistic several random score crucial require qr partial fast existence fourth implication actual row arbitrary related streaming contribution rigorous understand practical constant realistic size numerical interested consider brief summary bottleneck section similar base approximation also random art hadamard algebra library
definition bregman eq expand gradient inequality consequence convexity give xt xt sum optimality old algebra assume eq desire may integrate take respect shorthand old apply valid control term inequality invoke combine inequality bind provide provide subgradient descent receive converge stationary modify subgradient mirror generalize recent incremental markov subgradient control couple autoregressive sensor wireless sensor attempt sequence statistical come dependent experiment example ram perform stochastic mirror descent step describe paragraph provably convergent procedure govern quantity radius lipschitz function familiar previous stochastic rate converge main mirror make expectation ergodic high show general remainder paper description expand corollary throughout explore provide fy fx x c describe description algorithm familiar literature mirror generalize address geometry prox
single follow object similarly distinct estimate n manner unbiased block derivation generic detail interpretation popular agglomerative mean spectral tp calculate c static cluster dissimilarity return flat evolutionary agglomerative interesting function agglomerative cluster merge agglomerative merge low cost iteration snapshot make merge dissimilarity temporal merge merge dissimilarity give see expand recursive membership begin static initialization employ speed reduce require evolutionary initialize time initialization evolutionary snapshot cost temporal dx modify measure fit past datum compute rather cost function static average spectral clustering perform optimize front snapshot simply term note case modify incorporate note easily accommodate history use exponentially factor factor cluster evolve ratio spectral form laplacian admit interpretation modify operate rather assume scenario object object may scenario proximity combine describe observe remove object handle add perform factor contribute result illustrate cluster
I incomplete run poorly optimum estimation offer advantage na I form respect em generally ascent property ensure reason algorithm accommodate high dimensional space increase many perform covariate example speedup discrete observation parallel speed issue graphic useful computational em algorithm evolution effort often ignore evolutionary relationship incomplete repeat separate categorical nucleotide composition equilibrium intend sophisticated model inform consideration costly maximum various show compete technique accelerate convergence finally synthetic mutation birth death likelihood markov death time negative particle one give birth particle decrease count popular biology characterize epidemic dynamic probabilistic alignment dna sequence traditionally birth death particle birth
whereas company provide day estimation france consumption measure long channel individual equip home able channel around day consumption people send determine advance profile relate economic sample behavior total people sample day aggregate measure observation belong spend representative really center draw great majority simulation minimum error optimum center automatically start draw order
simplicity distribute allocation problem static channel associate game equilibrium show distribute learn fact game converge extension sort achievable efficient constraint e g transmission power user importantly extended strategy constrain long suitably modify value allow recall mind convexity r star strictly star global weakly unique minimum star convex game star convex potential start employ gr evolutionary index name iff fact calculation dt h l qp lyapunov z z definition qp prove need k shorthand direction convex f q follow quadratic far k f z plugging noting near
horizon item find strategy particular ucb present use probabilistic support finite cardinality performance good ucb oracle describe indeed oracle slightly make reason appear expert request notation generic setting lead bind armed bandit time good miss quantify virtual aware time optimally draw assumption sense set infinity maintain precisely find go uniformly infinity present optimal allocation turn
change vast exist segmentation detection correspond continuous signal precisely different notably profile technology collect extent change translate increase empirically aware theoretical set sense correctly change towards center share change position zero th respect center correct infinite jump let characterize first select assume generality point group fuse tend give deduce find unweighted fuse lasso position unweighte fuse tend tend comment position monotonically boundary detect change detect consistently asymptotically profile relative identify close enough correct point find benefit unweighted suffer boundary fact tell middle group fused fuse correctly find position tend independently position increase problem detect allow change model th value identically result extend show discover unweighted condition
previously prove enforce descent challenge address introduce follow splitting model stepsize computational uniformly eq vanish q reasonable still nonconvex split monotonicity analysis ultimately gx inexact point subdifferential exercise equivalently equation splitting define inexact stationarity residual call prescribe satisfie iterate stepsize become correspondingly strictly
network difference particular bic size assess true parameter element probability network estimate independence provide situation black dot represent pearson permutation plot structure size structure learn test predict behaviour alarm interesting improve improvement sample
attractive variation poor censor associate ht contamination performance decrease adequate censor recommendation conduct extensive carlo several divergence theoretically efficiency robustness go beyond scope paper weight devote previously continue proceed technique largely make little nh consequently inequality expansion w probability analogous hand taylor series expansion write straightforward combine infer introduce algebra probability analyze rewrite keep use
strictly concept site positivity everywhere positivity though concept start useful regard consider event belief formally uninformative let
reader paper address discriminative view dimensionality representation two tend goal vice versa important overfitte objective simultaneously seek case name paper inherently include semi case drive learn combination cost dictionary clear dictionary share overfitte learn dictionary predefine fisher despite drawback form reconstruction consequently generalize incorporate classification cost loss cross validation misclassifie sensitive outli optimization
observe synthetic zero draw p k p kp kp marginal stop show curve artificial average consistently outperform accuracy summarize performance algorithm aspect value almost identical outperform thresholding kind nonzero numerical like emphasize dimension limitation expression measurement experimental
method propose specification calculate resample increase highly deviation close estimate evaluate calculate divide ols ol know enter tell traditional prediction prediction summarize rate decrease rapidly relative slowly observe correct set parameter deal regressor variance relative decrease extension adaptive literature method converge oracle enter model candidate enter sample size allow ols integrate another
would type work improper benchmark absence frequentist confidence interval value continuous bayes posterior posterior interesting widely variation confidence simultaneously work complete moderate plausible sufficient meet absence work bayesian posterior expense knowledge base hand allow make less dependent precise extent uncertain third moderate posterior confidence illustrate implication corollary variation px px entail assess strategy function set mass biology department road k
mixture dependent notice stick break straightforwardly rich stick break rest briefly discuss atom analyze feature stick specification stick multivariate atom otherwise measurable correspond case eventually law covariate eq mean mean th component reasonable essentially choice correlation measure discuss definition give subsection stick variable assume correlation simplicity shall every measurable exchangeable array indeed expectation joint array joint permutation observation practical exchangeability represent suitable measure marginally dirichlet process representation consider beta tractable procedure follow independent beta result independent beta product beta give independence follow specification independent beta r ik
top seven log scale capture simulation single show beyond decay reduce well significantly square wave include collect set learn drive uniform illustrative figure empirical involve propose balanced reduction general key knowledge literature reduction dimensionality reduction pca reproduce hilbert space rkh balanced functional analytical understanding ready approach case dynamic simulate trajectory emphasis broadly begin empirical estimate dimensional direction observable behave linearly linearity perform state therefore map feature situation closely separability linearly separable separate map nonlinear feature decision
parameter number find minimizer problem rest want nonzero want avoid minimizer zero key exceed accomplish suitable desirable minimizer empty cardinality satisfy minimizer contradict suppose nonzero contradict solution note know minimizer large minimizer
besides causal ordinary relate conditional obvious direct cause observe quantity intuitive mean direct cause correlation theoretic study causality lead direct extension generalization direct prediction source investigate causality artificial advanced particularly causality functional relationship stable mechanism together thus infer relationship term discussion ideal set observable another mechanism force take result
vector family task spherical candidate freedom also difficult leave investigation aspect idea previous estimate function unit sphere spherical proceed interested would second portion infeasible spherical result accord subsection step feature fix mixture spherical coordinate consider spherical spherical feature formally graph kf estimate equation share choose parameter carry optimize respect kl lagrange multipli minimized sample graph identify associate accord result multinomial distribution metropolis hasting outline
reward know arm reward status static propose logarithmic system status evolve accord show logarithmic policy achieve pre agreement rewrite regret markovian transition algebra reward stop show logarithmic sufficient term third term reason play arm show term logarithmic caused spend arm exploration epoch consequently cause epoch term playing expect arm exploitation
norm crucial heavily influence dictionary rank admit decomposition refer contrary component zero refer structure introduce pca decade alternative potentially notably nmf many support factor seem consider structure analysis essentially retrieve partly face seem meaningful priori induce enforce priori factorization support reflect pixel organize grid support factor localize respect expression pattern microarray involve group pathway neighbor protein base remark early readily explain respect priori relevant hand slight define define appear tool exploratory naturally analyze decomposition learn obviously exercise assessment face consist image pose resolution computational reason show dictionary pca nmf spatially pattern sparse segment meaningful
nx consider expect experiment hope exhibit experiment exhibit though goal mechanism suggest pool section viewpoint web feature relevance query irrelevant perfectly many hence may prediction fact know vector cluster insight datum distinct suppose diverse exhibit obvious unit discuss various mn tm denote least motivate suppose vector index iteratively estimate algorithm require guess initialize vector step eq
description aim build help traditionally could train could interest direct benefit technique visible light measure perform compare search common group together datum currently present object preliminary show concentrate spatially occur concentrate plot highly analyze frequency see range
cdf information focus share deep list c top mkl rank near column refer first among interesting classical start say top correct candidate list gene short list candidate half position mkl soon accept place gene candidate start list wish gene preferable correctly top mkl beneficial typically top set find gene soon interested list hundred candidate disease become share disease disease share ht curve curve mkl across number causal disease gene identify disease disease know gene require know disease one share disease tested context discover disease context contribution dirac vanish know disease disease disease association involve disease associate remove scoring association disease remove rank disease turn method similar result relative performance overall perform list
recently construct shall follow slightly large desirable use rkh discard automatically combine improvement estimate advantage reader strategy sampling applicable space norm different semi
eqn replace appear understand mean failure position algebra overall assume rescale failure hold w r ok feature compute follow row ok value preserve done call multiplication indeed matrix sign reading sign however sign therefore compute correspondence matrix partitioning running follow notice kk ok create projection run low running mean datum result low run recover subspace matrix drop increase frobenius k term eqn eqn statement triangle norm projection drop increase eqn replace appear give indicator statement eqn give failure union approximation project clustering result straightforward reduction distance preserve preserve clustering preserve way suffice preserve preserve
take note winner select testing improve average combination adopt equal mean note decade nn organize computational intelligence method challenge multi ahead miss time competition represent daily amount machine uk competition forecast value day method assess absolute relative denote h five forecasting nn competition know selection want forecasting configuration increase compose preprocesse step step way g two alternative model come configuration detail specificity series mean occur observation value corrupt gap adopt removal gap possess decide role forecasting adopt discuss week month forecast embed equation select central apply state review realization function select relevant past final dimensionality vector series step require estimate describe explore adopt dt
train namely original corresponding significance versus simultaneous cost reject nan size run training model inference training characterize environment gb ram core machine case analyze effect problem discuss section discuss create pick seven computing distance additionally priori cost second cost cost option problem decision equation pick distance solve solve cost box plot simultaneous process problem cost node generally hard computation second nm nm take iteration subproblem second time time among three use scalability initially find hypothesis generalization case think much constrain hypothesis predict probability generalization ml lf I lf capture space f small since failure lagrange multipli explicit constraint
strongly objective fail objective issue boost present meta combine hypothesis complex superior achieve weak marginally boost focused optimize meta perhaps adaboost specifically via together near look adaboost boost trivial recent attempt successful generalize problem boost multiclass analysis restrict multiclass utilize optimize boost analysis learner provide intuitive adaboost boost reformulate modify functional al additionally functional specific pac weak rigorously boost notion performance extend foundation
create thresholding obtaining thresholded covariance interest thresholded broken construction describe component solve exhaustive different scope summary per coordinate roughly problem complexity complexity possibly larger desire computation impractical role play solver thresholded total k op jj difference numerical example large property help fold speed section consider discuss real microarray example problem glasso code
successive cycle run enable concept concept concept equal multiply factor analogy discover average evaluation affect expansion quality keep expansion maximum possible expansion limit expansion knowledge basis mainly mit medium contribution people maintain graph relation use knowledge grouping word thing article useful mit interface database file stage plan add base
seem suit round pair correspond entry select reveal suffer goal minimize w reality pick measure predict collaborative filter remark mention trace learn predictor predictor stochastic mirror descent technique bound matrix regret scale get obtain computationally work predict see convex w expert remark perhaps surprising forecaster advance forecaster key namely necessarily bound forecaster entry forecaster know reveal convex forecaster consist bound
count similarly prior dag compose addition dag neighbor obtain single start add edge mode mass dag recursively lead greatest mode view discrete counterpart identical maximum recursion ascent manner differentiable implement md dags move proposal jump dag define otherwise need node pair common dag update limit work randomly choose modify dag dag reverse retain e bb add analogously lastly check cycle reverse graph parent follow application interested adjacency domain edge identify local mass adjacency node dag million representation truth independently please simulation sampler burn local adjacency adjacency true dag enumeration confirm indeed mode mode maximum graph report table md single mode demonstrate search recall detect burn mode global mode identify dataset observation confirm burn serve number calculate mse ignore probability report ccccc
random course non usual inequality past generalize provide let measurable couple immediate bound iid wherein need iid start functional deriving bad measurable suitable suppose deterministic say long
control fluctuation deterministic state procedure update avoid theorem state dual nonnegative convex respect smooth behave smoothing density affine hull constant respect almost fx f respect measure ensure f gaussians satisfy corollary follow subdifferential sample respect lebesgue subdifferential distribution ball corollary follow ensure use decrease f g note explicitly corollary state require suitable streaming result precede provide convergence bound g si achieve involve error mean reader smooth author corollary condition compact convergence identical decrease discuss corollary appropriate yield corollary optimality guarantee focus concrete algorithm choice corollary begin corollary provide uniform hold lipschitz assume b attain normal lipschitz parameter l remark deep corollary constant essentially whose smoothed version close necessary
high record average content nucleotide loo cv box plot utilize negative centroid plot utilize negative bar end rank name nucleotide pair information content whether rank centroid testing procedure sign centroid centroid correct method measure weight content follow relationship justify similarity centroid low average show test pair ic ic wang propose transform vector shift left alignment scoring distance measure consider st one see distance centroid compare measure propose tf median across general median give low rank tf well produce high tf loo cv present knowledge negative assume bind
conditional univariate take I logistic dimensional triangular logistic conditional logistic univariate logistic regression arbitrarily parametrization observe impact mind regression system typically instance section fit drawback close fitting discuss fitting construct conditional estimate fast work parsimonious weight di mean component precisely correspond logistic firstly really separation might conditional construct parsimonious regression identify component small significant acceptance reduce via iteration update formula iteratively least square quasi work h di maximizer complete complete separation proceed numerical option computationally might variance likelihood target grow beyond threshold logistic approach procedure elaborate add cause extra newton ensure decomposable procedure absence information naturally smooth parameter algorithm regression four move logistic qp justify simple parametric product p denote requirement likelihood hold trivially
empty intersection pairwise choice lie regularize interpolation prove equivalence minimal interpolation regularize fix acceptable minimizer bf b q functional compact attain space belong regularization functional choice vanish recover satisfie also satisfy linear interpolation nk restrict subsequence go shall x evaluation functional continuous arbitrary letting minimal characterization theorem learn minimal interpolation introduction x gx gb interpolation g k x xx prove ready minimal norm kx n add repeat sequence give since converge j j draw conclusion acceptable linear satisfie likewise regularize
increase appear distribution shape error already logarithm median essentially even error eventually
suffice polynomially grow stay almost set neighbor belong subspace rank space complete uniformly constant local neighbor issue form seed incomplete third neighbor us uniformity correct seed sufficient seed observation seed seed column neighborhood sample notation minimum distance support seed entire pattern support seed let seed binomial seed seed eq tt thin variable value pz desire distribution accomplish follow draw subset support remainder seed eliminate seed probability perfectly complete perfectly eq thus chernoff p
argue arbitrarily close exactly bayesian shrinkage frequentist posteriori observe estimate rely simple gibbs model computation excellent performance study double pareto shrinkage broad variety suggest close penalize likelihood prior generalized shrinkage modeling prove note right fu eq kkt optimality proof prove n n p u p u u fu normal prove acknowledgment award es national environmental health sciences content solely environmental sciences national lee national education science robust bayes multivariate normal statistical york heuristic instability estimating journal
however computation predictive propose drive fit scope burden learn knot knot hereafter write mt deal property factorize replace justify st place knot tend eq hence small news play provide probabilistic zero process process appendix constant knot dimensionality smoothness arbitrary entire bind value modification addition give pointwise cost equal apply continue process choose knot remain canonical knot possible adapt modern gaussian independent covariance loop
way regularity condition concern normality tend consistent asymptotically value subject divergence bayesian posteriori integration easier set order monte markov give compute construction confidence interval
toolbox publish correction mean classification instrumental importance double base analysis quantify exist variance remain consume black hybrid strategy introduce smooth counterpart accounting practice failure decide increase rare prove carlo intractable world output expensive black box finite remark frequent exhibit simulation run approach first replace known order reliability
outline model nonlinear might impose distribution scale idea matrix adequate define impose derive process consume hellinger proof adapt metric part summarize lemma cover intersection intersection sub pack information serious drawback depend raise see undesirable consequence optimal prior design want calculate allocate available sequential jeffreys jeffreys crucial surprisingly overcome current parametrization much informative expect intuitively analyse problem lead coverage shape
normality multivariate investigation residual auto residual residual hypothesis series series whether see weak test reject significance level whereas evidence run relationship test perform constant agree impulse response series show week response cause impulse series response response normalise compare impulse causal ordering follow order write orthogonal source variation cause variation alternatively could positively except response strongly much impulse response around slowly response affect affect affect
thank helpful lemma david find blind controller np np place science np nonetheless outline admit keyword partially observable markov bilinear complexity square root fractional decision prove conceptual tool ai include plan briefly
co justification matrix view semidefinite semidefinite engine cut ensemble body work please overview reference ensemble phase ensemble multiple instance aggregation separately instance instance bootstrap sample instance project clustering projection subspace contrast quality individual fashion rf performance essential ambient possible mean main co count time hyper solve cut problem add vertex meet another likelihood aggregation incorporate similarity thresholded improve clustering demonstrate study closely aggregation finding agrees assume ensemble applicable unsupervised forest rf distance metric randomization fundamentally analysis
policy bad would policy unfortunately seem consequence ergodic environment currently process negative computable discount strong asymptotically weak function deterministic computable discount asymptotically policy asymptotically weak policy say strong asymptotically computable discount computable discount policy discount somewhat asymptotic weak asymptotically necessarily real idea computable rule stochastically computable asymptotically discount function discount computable geometric discount analytically easy contradiction basic weak asymptotically asymptotically indistinguishable asymptotically reward omit correspond strongly asymptotically I
non fs fs ss define natural economic lack submodular decrease marginal marginal base fs fs fs fs fs fs necessarily combinatorial represent tree max leave location location set fs w j ji appear tree every permutation build marginal value role sum submodular call stand denote demand weight demand max leaf weight several unit demand tree also weight demand way ks k gs great informally g expensive g formal review far strict hierarchy separate class approximate learn approximate primarily input target hypothesis exist mostly pac model value allow unknown target polynomial number approximate everywhere fs fs definition framework
dictionary use pseudo appendix technique type mainly focus image signature dictionary well dictionary use dictionary classical problem vice descent dictionary prove efficient remain nonconvex method experimentally enough gradient material technique algorithm potentially require sometimes note updating matrix introduce
summarize follow matrix svd compute n operation interest immediate usual characterize bring set identity identity q solution algorithm common problem hundred note achievable reasonably sized dataset implication step order practically require computable eigenvalue singular represent additional storage requirement conclusion prove
rbm sampling perform manual expert short assess bayesian optimization manually tune cube manual intervention rbm adapting advantage approximation discuss apply space complementary technique adopt wang propose randomize mcmc bayesian objective exploration costly discrete densely graphical problematic optimize adaptation mechanism tend optima finite practice parameter bayesian stochastic
completely atom union two must location atom atom location ordinary poisson c also calculate atom give put together j proceed atom far location eq identify weight put together full propose distribution propose counting may integrate marginal prior since calculate calculation also work component atom number ordinary counting process atom atom atom ordinary equality increment poisson useful refer mean notation usual increment measure find q step integrate integration exactly case set intensity main measure tend posterior case merely note imply start define may measure prior may count distribution proof laplacian enough observation finite enough fact combine limit completely finite draw similarly binomial completely measure keep intensity height generate completely consist ordinary component jump less count eq establish henceforth shorthand quantity assumption choose chebyshev inequality write I follow note continuous c n yield every eq desire truncated case turn propose therefore integration choice approach exist atom bind bound next need part atom desire follow transform difference component share notational convenience new ni ni ci two together conditional conditional proportional otherwise univariate hasting slice sampling binomial conjugacy
determine agent action random size current randomly product mh mh three linear programming theoretic reinforcement long trajectory provide since sampling must convergence chain hybrid mh prior reward bernoulli mh sampler conditioning mh form mh suggest verify reinforcement learn actual estimate cumulative discount feature expectation appropriately discrete environment state expectation first transition robust performance
dt give proceed duality ix mx mx inequality present rademacher class convolution operator hilbert highlight generality apply specific case mixed norm recover bound remove see regularizer learning kernel regularizers linear associate exist admit novel countable certain
describe main scale sde dx bx z dt gx fast solution ergodic rapidly unique converge diffusion govern sde construct average define exist probability generate topology convergence begin general describe introduce seek reduce dimension present expansion paper probabilistic dimensional backward doubly transition bx z component slow n kf function assume measurable rapidly measure later assume dx aim calculate eq q brownian f ix z j z b transpose vector tell suitable sde suitably generator estimate slow marginal
mean allow local arm neighborhood exploit notably al process focus parametric curse query typically structural main request convex cost imply convexity natural function make polynomial note focus bandit stochastic vast emphasize imply bind detail jensen optimization converse necessarily optimization candidate suffer iteration setup prefer solution involve distinction free book sufficient descent concrete emphasis work nesterov schemes et evaluation function accelerate analyze set rate degradation regret plan attack query outline first every pass function order interest source regret opt give center device might act unlike discretization device solution clean intuitive develop version zero optimization vanish center device require center device conjunction ellipsoid motivate device slope value distinguish slope
root induce parent may imply combine variable combine child work entire tree decide subtree combine return nan test neighbor leave subroutine along current return variable return evidence subroutine subroutine determine merge parent child check parent child avoid merge recover empirical moment maximize exist return h u u return node leaf set root parent parent u x h h h x x grouping enjoy guarantee model vector use eq eq recursive return tree undirected consistency imply reveal solely spectral applicable setting complete aa award fa copy tail first tail type average independent provided define random copy tight exponential probability simplex copy probability least eq
convex let theorem corollary extension nesterov loss corollary mirror descent stability note eq extend mix corollary follow concrete gradient descent dual strong assume function corollary online svms regularize satisfy sharp guarantee begin martingale concentration martingale define previous random theorem give martingale sequence note bound ff f use deviation sequence define give precede inequality side q note union relate sum definition sum final complete turn mix application markov strongly great use bound
hamiltonian node weak enforce free saddle exactly used generation infer know determine assignment hamiltonian want correctly probable group marginalization maximize label configuration boltzmann correct size correct number ground minimum create group
vb deal section unknown iii describe symbol transmission iv combinatorial bayes symbol transmission numerical receiver current section vi implication remark future action redundancy know code partition message analog map alphabet
update accelerate proximal propose update proximity fast quadratic count value rate scalability large operation involve zero arbitrary tw u version influence nesterov recursion fista proximity define stage possible case wide solve next study estimation aim competitive art employ lasso choice optimization toolbox stop decrease proximity stop relative smaller study
account non calculate path reveal burden shift conditioning phase ode output treat input inefficient disadvantage want retain form stochastic model basis mention knowledge retrieve e
particular pose level al extend recently level mrfs shape g know advance initial pose shape product expert black therein mathematically high auxiliary overlap high potential use crf show gibbs expressive shape obviously complex graphical become learn potential consider application point view instance shape instance composite explore expressive first separately relation segment shape capability shape finally discrimination shape node image define neighbourhood I edge unary graph obviously subset composite
p x yx infimum x nx subset denote coupling independent identically empirical suppose couple attain concave likelihood empirical reduce follow term weak concave deduce another convergence nb hand notation eq even nb nb nb exchangeable deduce theorem almost surely converge compact strong lebesgue converge interior immediately convergence notation suffice continuous fix n definition lemma smoothed concave
word least sentence cover concave discount multiple time analogous submodular example illustrate analogous choice word importance heuristic highly occur sentence document bf x length hard achieve linear monotone empty relative objective relative add sentence terminate rule select I score greedy despite additional provide propose submodular scoring summary similarity allow pairwise feature cosine share whether
huge sometimes prohibitive integrate approximate variational technique kullback kl approximate improved distribution choose unfortunately log configuration lead involve smoothing presence optimal divergence end problem differ approximate derive estimation go back specific unsupervised classification collection family complete simulation study highlight average classification also public surveillance average average account evaluate compute field allow mainly bayesian
pointwise mapping frequent branch unlike branch outperform consider chance pointwise mode practically marginally sometimes demonstrate mode good potentially achieve lie selection mode information mask forward isolated g mask huge regression nonsmooth trajectory unimodal symmetric mode reconstruct trajectory contain branch reason wrong priori suboptimal candidate reconstruction correct appearance trajectory reconstruction constraint despite approach recommend trajectory reconstruct smooth trajectory reduce eliminate part original trajectory reconstruct global particular break ambiguity change candidate reconstruction vary pattern miss type spurious reconstruction shorter jump conditional become unimodal series change dramatically run reconstruct point shorter long appear nonsmooth pattern never vary pattern toy mask problem mask c robot arm mask component one toy toy nonsmooth density arm nearly mode location mixture superposition tendency develop log still represent qualitatively well mixture mode trajectory mode area reconstruction mode close achieve largely insensitive mask crowd reconstruct shorter due appear horizontal nearly vertical extent inside period scatter become trajectory geometry spurious appear distribution remain low usually get conditional spurious mode also full besides hard log small rate trajectory trajectory whose normal trajectory start trajectory particularly bad lack trajectory surprising give mask nonsmooth still trajectory long actual pointwise reconstruction short branch branch really constraint trajectory
alternative seek test statistic split three expand definition adjust reader appendix q n nn k section substitute limit equation integrate apply assumption limit suffice chebyshev last hold term jensen eq step degradation replace bs propose replace bs trivial statistic refined chen show achieve without explicit restriction du sd short demonstrate general satisfying moment allow function scaling paper set project average matrix working framework allow tend constant outperform test bs term efficiency conceptual differ past bs sd estimator analysis limited estimation capture effectively pool testing consider previously cl remainder intuition formally define devoted number performance condition achieve great relative roc curve comparison discrepancy test
van discussion theorem corollary phone phone author propose allow detect shape algorithm linear consistency algorithmic paper new technique quick image theory object noisy image primary diverse automate limitation essentially scope application object picture color colour intensity object thick colour application present paper rescale colour process digital image crucial section natural
identify quality new restaurant low restaurant review restaurant restaurant deal quality price quality restaurant low restaurant deal price customer restaurant restaurant high restaurant restaurant quality restaurant improve make potential true quality try deal price order part game chinese restaurant analyze part illustrate specific application chinese restaurant game cloud social spectrum cloud exact activity service platform deal tend signal affect paper chinese restaurant game social chinese consider chinese restaurant table customer label size restaurant restaurant restaurant customer customer distribution customer learn signal chinese restaurant game sequentially request request receive customer payoff determine rational maximize chinese restaurant game customer customer restaurant grouping
layer flat flat model combination figure recover flat close across model sampling model cccc flat use data set dash blue outer contour represent respectively solid black blue solid dash blue inner outer contour represent product peak figure plot prediction versus value validation sufficient see model number hide physical allow error final significantly specify flat range network large increase log log likelihood
boundedness hierarchical q right evidence exploit distribution evidence estimate independent hyperparameter implement treat variable thus derivation exposition efficiently subsection initialization value accord repeat guarantee subspace idea signal sensitivity measurement incorporate
complete regard complete whose edge expect slightly remove option agreement intuition ready lemma state concept properly slow graph deterministic compute average slow graph carefully need graph random outer eq vertice fast electrical circuit appendix grow low fast upper approach fast probability one ratio bound zero translate mutual subgraph concentrate section exact truncate noticed behavior walk behavior difference slow fast graph confirm experimentally expansion actually separation fast slow subgraph fuse graph experiment eigenvalue fuse envelope fast notice slow connection therefore slow slowly away display graph exhibit graph similarity fast os right appear decay dynamic control mix measure step necessary reduce distance justify fact spectral fast take walk fused graph walk begin fuse step probability find subgraph walk initialize subgraph impose slow subgraph decrease fuse confirm experimental fused exhibit thereby rate fuse graph join histogram experimentally transition fuse affect h slow fuse right subgraph slow subgraph
dimension show separation model particular publication paper substantial follow interactive mechanism query way query ask expense notion differential privacy mechanism extend rather extend arbitrary sensitivity differential privacy de achieve achieve set arrive online analyst pay query ask course query ask necessary interactive multiplicative weight interactive set offline version mechanism agnostic learner et evaluation unify mistake private release reduction achieve improve special query et al give error privacy time database universe variable guarantee laplace requires run proportional mechanism size block time al query prove low et synthetic giving count universe query improve run size database representation possible release answer size synthetic
least posterior probability mass posterior similar variance seem model look derive single posterior might suggest ratio proposal jump approximately mix improve jump determine pilot automatic propose automatic proposal mainly reduce trial identifiability center determine proposal move weak non choice value essentially simple efficient detail implemented know approximate expand taylor call center density order center
uci california repository detection case sequentially update real time uci dataset part training determine use stage combine framework assume arrive sequential uci learning repository free kind structure good response bad response sample element pass perceptron back neural test nn classifier use bad scale range accuracy sub algorithm intersection turn solution include scheme dataset get uci c c data success nn adaptive fusion vision concept framework compose number center represent weight projection onto describe sub apply
sign recover high nesterov quantity lasso x practitioner severe detailed proof rely view result lasso belong uniqueness continuity piecewise affine support sx tx tx tx covariate easily guarantee automatically
exponential theoretical guarantee independence discuss open future advance independence produce test reliable underlie statistical learn present assume learn discard structure error cascade error produce learn polynomial evidence proportional row execution compare without numerical markov strength guarantee complete ht sound use local test sound global reduce useful sound strategy design computing use useful domain comparable dynamic number test domain comparable respect cm quality correct use test design learn exact improve accuracy improve accuracy markov relate improve quality approach direct markov network require improve efficiency exploiting axiom infer unknown observe avoid redundant environment run design improve global efficiently structure improve efficiency subroutine order publish show independence reliable uncertainty outcome show accuracy independence exact present improve cost produce quality improvement cm
motion motion characteristic self diffusion calculate pe examine motion motion start flow profile ns correspond roughly profile flow examine plot velocity without profile presence use number equal contact contact vary intermediate jump good balance fluctuation within detect unfold order separate ensemble unfold ensemble copy classify trajectory unfold ensemble trajectory regardless reach trajectory reach return rather region show see single region entry list divide point list dedicate come come lower help ensemble force velocity store trajectory counter determine streaming unnecessary
unless approximation exploratory probability analysis furthermore exploratory abc software option software abc select genetic scenario shift abc approximation towards whole bayesian select assessment practitioner project grateful look since jacobian hence discrepancy connect lack consider sufficient satisfie illustrate behaviour operate fundamental integrate approximation carlo practical indeed unstable survey wide array application implementation argue abc bayesian discriminate insufficient lead fail entire infeasible due
attempt answer element conjunction provide challenge seek consequence phenomenon try decision uncertainty area science decision likely review statistical understanding conduct decade familiar wide array statistical player mention significantly especially run mean speed player fewer equal combination status able obtain walk evaluation player shorter argue first base type sum triple home double triple triple home triple home run perform analysis remain transformation indicate statistically trend state trend towards imply become recent seem reasonable assumption player try year occurrence dominate mention proportion score increase derive measure argue player already effective seek different lead evaluate markovian recurrence seem logical modification team team performance measure evaluate team team notice production model influence trading team rely production aspect game measure team serve team consequently contain availability
detail true cluster convex show observation model density unique optimum technical require gradient smooth make true construct requirement constant apply refined guarantee via direct definition singular u entry matrix similarly except plant spectral norm infinity easy probability completely determined distribution plant partial first result handle spirit j sign disagreement sign vanish paper optimum universal discussion appendix guarantee condition high existence desire dual lemma probability strong allow
merely form sample representative compute th patch index thresholding feed entry new matrix entry indice iii train classifier scoring cost effort large take hour often performance drop sample aim achieve reasonable accuracy co adopt propose paper extremely idea separate use classifier iteratively classify assign example illustration co natural redundancy relationship training loop however co superior coupling boundary naive margin proxy confidence margin define vote element list algorithmic description respectively label example membership f kf kf kx k co rarely obtain feature possibility constitute natural split subset acceptable independence label fortunately represent naturally property construct among
concave laplace cavity remain may induce numerical moment convergence report fractional improve initialize remain variance cavity problematic multimodal extra care discuss loop rigorous stationary loop parallel early stage approximation marginal improvement parallel update double necessary ep prove adopt propose modification double fast ep parallel double loop optimization moment matching loop marginal inner site update loop distribution message loop consideration direction extend modification check obtain additional evaluation every site integral update site cavity variance choose computationally modification illustration small tolerance modification result double loop optimization initialization iteration achieve distribution loop usually inner loop utilize objective determine respect cubic interpolation derivative site I efficiently comparison sensible hyperparameter achieve parallel double loop update ill many cavity utilize modification cost limit maximum inner loop additional size adjustment site outer loop evaluation rare may cavity though loop optimality
want previously financial index change consequence type drive change specific financial evaluate supervision concluding remark iv asset order volatility deviation specify information majority occur great
subroutine say within therefore meta procedure operate meta simplify explanation ignore constraint result fundamentally focus correctly identical meta recall iff universal uniform meta become case meta improve passive need large round behave follow thus contain vs put would vote infer request sufficiently high least step request x limited precision example necessary target function target z z type classifier one interval nonzero far interval split exclude total large number interval use require classifier represent particular among classifier represent separation vote request large vote favor particular decrease associate reasoning sufficiently satisfy label correctly separate point intuition operation result show return abstract follow include vc strength recall provide meta achieve complexity nontrivial target function know advance exist algorithm mention approach remove merely calculation actually somewhat general result meta achieve target corollary theorem class function passive algorithm achieve imply satisfie observation characterize possess vast array question passive describe question give learning efficiency specifically suppose label find return indicate concept method capability subroutine body store simply try possible additional check evaluation prohibitive easily calculate occurrence q calculate appendix determining total meta remain question polynomial efficiency subroutine actually elegant efficient specifically bind finding need infinite correct difficulty simply search number take restrict unlabele show state probability select modification unlabeled guarantee increase concrete corollary suffice use meta example result efficient running find previous always gap label complexity refine advantage characterize achieve gain address open problem similar explore say label p foundation upon begin gain achievable disagreement active gain serve improve quantify sophisticated disagreement subsection already publish slightly version meta disagreement learn result begin disagreement though refinement meta great aspect two meta request focus disagreement therefore region request request meta budget output request request else l procedure define satisfy inductive well thus infer meta stage version space passive return classifier request add guarantee passive achieve surprisingly meta meta claim universal however specifically improvement achievable step address passive label region small example request union f f consider request example meta positive say restrict small large disagreement reduce fraction request complexity dependent accounting observe positive observe positive request upon reach passive similar case union improvement observe interval negative region separate interval j z ji h difference interval disagreement meta improvement passive toward generalize example concept informally disagreement characterize label complexity disagreement wang disagreement include related variety satisfie condition explain implication go calculation toy take threshold example simplicity distribution uniform section z rr z rr h z consider r b b h p disagreement mean consider interval c set label region arbitrary location point separate gap ci extra remain mention disagreement complexity achievable disagreement reason request version disagreement request meta disagreement coefficient specifically essentially though vc class achieve learning modification instead formal detail implicit theorem essentially identical
measure cube variant endow pf ft uniform orthonormal indeed unique property concentrate conjecture follow endow measure universal confirm numerous implication imply influence tight clique currently conjecture would answer state formula polynomial fourier concentrate
lag matter technique lag lags forecast grouping remark group grouping grouping grouping select optimize forecast variable forecasting average forecasting segment pattern coefficient strong mention lasso detail universal grouping estimate whole combination modeling type underlie month suggest grouping estimate realization mainly grouping estimator depend problem produce appropriate proof closely upon consider ar equation regressor entry lag assume coefficient move fractional base variable say subset variable subset proper j spirit far state tp positive complement form vector w eigenvalue gram see cardinality remove column gram state
I quantitative ergodicity constant hasting knowledge setting kernel proposal metropolis hasting target density density vx f p fx vx readily px q qp vx choose vx pt projection finitely eventually stable ill frequently projection occur expand difference first focus allow random size analysis adaptive monte carlo generic expand set define stand algebra x practical mechanism include expand projection ensure feasible potentially adaptive particularly provide significantly well setting expense require convergence certain criterion step family smooth g include framework dependent particularly former share quantify ergodicity crucially projection present may situation
score see number redundant feature feature indicate score nonetheless give similar evaluation incremental classifier add irrelevant task increasingly reliable goal literature principle fundamental study known redundancy result rely illustrate amount redundancy evaluation reliable principled combination reliable assessment subset work extend rich continuous scoring repeat self def subset attribute motivate certain definition fundamental study control measure compute behaviour solution size relevance reliable relevance presence sophisticated matter study filter case error loose set actually feature benefit less
construction hypothesis reject identified reduce health physical activity week health option health good fair poor include indicator economic status formal education status origin parent product moreover information age gram per yes variable percent overall percent missing miss assess imputation available alternative surrogate imputation vision speak mass weak lack range head back activity getting accounting use major turn range phone call friend phone friend association variable category limitation index restriction activity formal education consumption gram day coefficient rate functional health social support connect five node background health false positive quite bound identical
operation reconstruction perform estimate node internal let fix satisfy leave pair vertex q child repeat role rely topology used subroutine building child think neighboring subtree weight leaf form standard bias technique calculation ready complete correctness fix proposition divide topology correctly reconstruct hide sample everything level four test proposition choice least reconstruct third account reconstruct ensure weight conclude state general outside polynomially q balance root say state mutual go
attractive fundamental terminology fisher mean would prefer statistic variation describe student variable acknowledge abstraction student many abstraction random toward statistical aim statistical article principle model presence inductive statistical analyze likely principle identify variation would probability capital sometimes kind call quantify principle principle evaluate procedure explain vary circumstance link connect specifically somewhat advanced elaborate describe detail train follow reasoning picture indicate
interest tag recommendation allocation hypergraph document pass model goal model topic broad tag characterize tag identify tag publish treat compose word manually label object interest multiple tag tag illustrate document tag author author annotate tag build document fig compose tag image link bipartite graph times corpus times build implicit link encourage topic document like motivate variant topic focus generation influence document document cite document labeling depict co citation document large parameter document distribution topic label generate limit relax relational topic hadamard product value generate citation adapt account use topic markov field multiple type model either proportion cite document regularization regularizer encourage labeling topic among nevertheless pairwise topic expressive insufficient document fig associate tag word uniform tag use
calculate entry code
partially projective algebra product topology sequential automatically either though arise unless countable product adequate projective construct constitute projective turn consist finitely additive set crucial set finitely additive call additive algebra countable additive countable helpful projective mapping define space product necessary algebra word measure projective projective projective limit countable sec consist measure definition facilitate projective specify sec start countable fix topology open ball rational since separable countable set disjoint q element partition algebra partial iff refinement index projective limit partition
exceed ill sample poor scenario variance belief generate model approach generative identical impose additional identity symmetric specify factor efficiently principal pca established involve ms last loading span top estimation illustrate conceptually obtain procedure thing study comparison estimation consider subset validation detail appendix pca unclear impose constraint number could motivate shall experiment constraint constraint efficiently straightforward together motivate follow residual variance pricing closely relate trace constrain problem address solution variance stand exist multiplier context practical long practically contribution
network node field laboratory identify node theoretic decide rest generative network connect network stress could location social web stage already decide joint far give block key mutual essential label correlate explore entire label effect node start explore central boundary alternate call agree agreement rest network network
appendix glm heart datum dimensionality range label class letter mnist letter normal gaussians uci learning repository benchmark repository feature range classification split mnist resolution reduce save prevent overfitte split letter perform split train testing letter lie letter time sensitive class heart mnist letter comparative study nn classifier large glm classifier cf distribution follow glm un consistent metric describe energy metric validation overall include number near neighbor interpolation glm margin classification display set dataset though good strategy local metric display split letter scale appendix mnist
svm indicate statistic dependency generally outperform prediction across indicate label elsewhere literature rapidly train relatively performance notably robust demonstrate dependency lda outperform svms rare dependency svms label except frequent dependency outperform document large dataset face large number extensively literature multi label text majority focus corpora relatively label label annotation label long tail rare number increasingly text attention future combine discriminative type generative use discriminative lda hybrid model etc etc text classification multi computer author anonymous journal helpful suggestion improve material upon work support foundation grant intelligence project via air contract fa office research grant microsoft award reproduce annotation contain herein interpret represent express imply new york annotated corpus available linguistic nearly publish york st manually human new york indexing service correspond subject article table numerous example document document descriptor remove article train remain article assign training result corpus remove word document occur set consistent common literature union dataset document descriptor type restrict dataset split equivalent form split first remove document descriptor appear fewer update corpus dataset classification etc consist assign label category original present set commonly convention restrict evaluation randomly select test score margin problematic manually automatically automate avoid automatically subset hierarchy automatically assign hierarchy expansion although sensible lead per relatively multi unique quite small single automatically although nothing inherently wrong lead able assign label statistic real world assume combination yahoo world power law yahoo split work training document remain label assign single label part collect pre wherein category yahoo keep actual add complete set setting motivate setting optimize result optimize cross svms show hyperparameter training experimental applicable lda incorporate indicate value generally total strength priori document large train
discussion permutation write refer distribution refer underlie collection translate poisson approximate binomial translate poisson binomial distribute translate iff write random fractional poisson translate poisson binomial give translate distribution eq description namely positive rational relatively integer sample provide ingredient use give implicitly explicitly cover cover construct size nk integer binomial tv cover theorem part extension algorithm figure modification give detailed description proper algorithm theorem hypothesis run hypothesis return output subroutine sparse sample access design find accurate unknown inside cover design imply close heavy finally hypothesis choose hypothesis subsection return point return choose description start unimodal inductive proof learn unimodal follow unknown unimodal bit operation follow list mass assign sparse bit output description form contain interval explicitly guarantee form cover constant lie level mass unimodal close unimodal sparse obtain list define return put point otherwise run unimodal return
differentiable every far eq rkh inner product dimension continuously differentiable xx eqs sec slight abuse identify operator take space comparison sec interpret gram xx x j x hadamard method propose apply handle strong output gradient method mean subspace span symmetry find various function avoid less
help eq derive starting introduce standard density lebesgue variation ensure please appendix class triple mle mle convex detector label factor small constant phenomenon detector know label achievable matter minimax extra assumption mainly influence slow label smaller slow statistical intuitively challenge passive sensible know density quantitative
old gp equip rescaling offer true old smooth square regression nonparametric model well add convergence hope reader deep fundamental extension try study gp remark estimation restrict define subset unit origin actual rectangle coordinate argument shift domain technical diagonal define measure identity matrix project axis one
ph university california california la currently electrical computer department endow wireless access interest digital digital communication human interaction endow wireless center wireless communication communication channel reveal advantage common measurement reason practical eeg considerable evidence give eeg frequency source exist direction indicate direction recovery numerous basic form impose algorithm algorithm algorithm mixed reweighte among algorithm much achieve bayesian greatly extend researcher example first later extend always much minima application contain structure exploit structure source focus relationship besides purpose theoretical assume source independent identically contradict rich structure exploit decade besides high source exploit design smoothness sparse temporal source structure learn often sufficient limited notice
difference wavelet replace classical space piecewise affine scale operator play classical wavelet show sec imply affine produce organize adapt expect application radius decomposition open j j kb j decomposition structure j notation dyadic cell generate algebra measurable algebra constant dyadic distribute variation map connect nearest kx ik weight construction partition dyadic tree publication computational cell dyadic regular compact embed manifold set property enough ambient therein h close large appear name manifold recent manifold learning complexity positive local pointwise tool build upon geometric especially harmonic therein estimation short account associate dyadic cell covariance identify prescribe value svd projection onto span column affine parallel pass orthogonal onto plane affine projection onto linear coarse geometric onto hausdorff define fact tangent albeit need go fix decompose
task automatically categorization science biology science engineering classify knowledge categorization complex unsupervised classification name task grouping processing analyze validity hypothesis cluster respective pattern matter recognition image bioinformatics region influence g analysis separate drawback cluster arbitrary division repeat reach good division identify community identification able drawback
posterior sampler sequence scale important evaluate integral I chain ie give suggest particle particle normalize unnormalized collection particle time allow explain hasting seem converge find update walk mix demand complexity calculation update intel approximately hour computation setting unify device parallel smc advantage likelihood particle hardware run around speedup run pick start trivial estimate effective computational smc rapidly density map initial value choose remove possible multimodal independence change effect
treatment model dynamical state dependent version modular decomposition conclude discuss direction connection broader distribute quantify coupling modularity shannon measurement q take entropy outcome logarithm entropy bit uncertainty measure provide random px uncertainty measure interpret knowledge hx yx capture marginal reach mutual case variable random x distinct constraint information often two importantly kl equal amount present formal modularity search
hold imply ia pa r case age show case pyramid duration age duration simple duration disease number depend incidence age
represent local
test condition subset sort size give acyclic undirected graph edge record adjacent cardinality order adjacent less triple adjacent adjacent adjacent direct edge orient complexity pc bound maximal degree vertex test pc algorithm hundred number node large even intractable due author grow shrinkage use phase markov dependent condition two phase sequentially real markov shrinkage condition try identify algorithm direct dependent give set great cycle put neither execute long exist gs markov relatively variable sparse complexity graph gs learn graphical minor modification possible try score enforce sparsity structure edge approach establish principle description refined offer sound criterion network approach intractable bayesian score optimize hard criterion score approach scalable combinatorial far away compare candidate structure bayesian involve compute undirected structure
b sx r yx cs disjoint whose intersect ct sx z z
rv set combination assume state contain signature massive period roughly day outer period rv euler cat rv date combine datum discuss analysis set receive bayesian b consist four signal parameter rv amplitude I date reference describe represent measurement analyse combine receive imply star confidence calculate assess denote receive description good model need combine set lc assume variation bias expand every consist
similarly q obtain integral obvious integral case conclude condition evident variable nd accord hold moreover check hence imply hold apply conclude proof fractional fractional simple immediately pp support modern mathematical shift away handle stochastic process develop dependence stochastic process application economic finance devote involve motion well long study
advantage life dataset systematic datum three computational artificial demonstrate smooth critical analysis deal event indicator event occur censor event end last observation cancer survival hazard cox ph cox ph build survival rate originally intend outcome tend covariate cox exceed observation machine cox one artificial sufficient often lead address chance certain interpretation individual early acceptable practitioner
atom distribute define density zero take hasting metropolis walk detailed sample indicator likelihood account possibility great numerically likelihood q slice sampler use sample discrete gibbs eq gibbs sampling discuss process poisson derive sampling atom observe atom round unobserved atom poisson poisson
confidence make useful large dataset social explanation study identify whether could explain despite test point algebraic specify nf directly problem give x algebraic set onto finite second necessary accord algebraic doubly intractable instrumental algebraic gr equality unfortunately limitation experiment small drawback complexity domain domain equality suffer drawback still insufficient graphical instrumental paper general produce inequality optimize
hasting either sampling call start iterate accept fairly weak stationary chain burn converge stationarity initial portion discard case metropolis hasting metropolis hasting metropolis hasting proposal strongly affect algorithm work fraction candidate reject cover important however prefer approximation proposal often limit walk sensitive target frequently nonetheless well true mcmc often consist highly sample therefore estimator posterior separate chain move rarely take stationarity inaccurate seem independent seed hope chain however run generate correctly chain reach mode depend mode mode concept move easy target sequence distribution multiple isolate together sampling idea independently interest
distributional density pf density choice write pz nz sum lebesgue dirichlet process difficulty employ measure advance fit dirichlet microarray nucleotide reach thousand fairly consume pr methodology enjoy mild misspecification proposal mix mixed resolve difficulty marginal mixing parameter output recursion density produce point labeling integrate distribution correspondence exact filtering additionally identically common minimum kullback leibler range weak limit regularize recursion methodology distribution density give assign typical
sampling calculation denote orthonormal dx lx x since q free exist ix I give analyze side ix second tt moreover evaluate equality last symmetric similarly x I early gauss equation work isometry finish rewrite theorem equivalent together together q translate unit vector relationship holds translate orthonormal basis x x kx ie kx kx express orthonormal come rewrite schmidt invariant since otherwise sum square apply away conclude remark demonstrate manifold focus I theorem imply approximation q identical independent hereafter expect equation deviation bind deviation constant term lx complete away inside boundary similar begin lk may plug note integral vanish apply taylor expansion orthonormal express order term py simplify notation expansion calculation q apply conclusion denominator appear numerator taylor expansion integral domain symmetric define slice basis x x xx reduce q next expansion become kernel numerator similarly denominator expand spectrum laplacian diffusion weight complex low space study graph development mathematical orthogonal ji iw optimally affinity actually digit two discrepancy even optimally perhaps connect edge set small patch e usually
program abstraction abstraction body program abstraction program pair abstraction compress compressed abstraction abstraction abstraction abstraction find abstraction match return call match primitive return define match abstraction unify abstraction body abstraction abstraction abstraction false unify si si abstraction abstraction var var abstraction abstraction abstraction illustrate expression anti create occurrence example replacement program begin define v procedure anti term begin f return argument function expression determine abstraction anti process create abstraction opposite list algorithm size return size expression return false return assignment pass place succeed repeat unified repeat unified pair primitive primitive primitive primitive map si si assignment abstraction assignment neither match assignment expression transformation repeat syntactic generate use incorporation syntactic pattern notion first lambda abstraction lambda calculus abstraction two anti via usually correspond incorporation program anti find possible small color abstraction color look abstraction f f f node f program abstraction f color size color color abstraction correspond even replace color draw multiple
one weight krige meet reliability surrogate reliability span span surrogate reliability reliability determine step size along current accuracy check design obtain converge criterion deterministic performance purpose surrogate refine reliability essence purpose validate algorithm long constant service load material characterize behavior modulus load euler allow involved consist mm deterministic design fashion allow service load satisfie limit reliability minimize area target reliability probabilistic multiplication state surface straightforward turn algebra failure denote random cross numerically refinement limit initialize sequentially accurate reliability depict initial design base satisfied design reach stable exact subsection mm approximation error higher optimal reliability thank krige
average explicitly easy take limit retrieval take temperature dynamic embed thus kind recall process explicitly recall asymmetric let embedded follow solution differential two show amplitude ht evolve ab respectively namely quantum pattern process leave panel find getting collapse
right step mean square steady enough accordingly steady random recursion approximate stability guarantee step steady state determine evaluate depend state size sufficient substitute return evaluate become eq approximate matrix shall notation euclidean kronecker u sufficiently size power eq block furthermore show recursion guarantee size accord become q expression proper weight able steady invertible resort choose proper square estimation computing zero elsewhere vector whose th one oppose explain assumption interested weight hold size enough consistent consider connected simulation sequence spatially consider regularization respectively popular choice non need denote go apply minimize manner
show pac theorem h nh bounding pac fix simultaneously eq tight formula relative subsequence first sequence martingale theorem simultaneously minimize grid geometric almost achieve would hoeffding assume theorem variational bernstein ki simultaneously value random take grid close value side almost weight sampling bind reciprocal round bernstein inequality satisfy inequality compare form inequality average
prior simplify choice summarize formulation provide denote inverse gamma numerical analysis parameter mcmc partially collapse integrate example datum bayesian short upon request estimation method denote frequentist require first resolve sign simulation single error seven quantile mcmc burn diagnosis mix start illustrate burn summarize
matrix equality q obtain stack stacking multiplication operation provide way represent equation wise multiplication generalize multiplication multiplication array r matrix multiplication review ib ib stack j n
augment lagrangian lagrangian converge saddle theorem close proper augment saddle factor marginal return problem q p f p value close solve augment special block furthermore well similarly separable fact enable admm marginal residual residual marginal iteration optimum optimization ise figure show
multiple base start introduce random also goal predictor risk introduction stage good consider separate coordinate align ideal eq underlie center k cx kk alignment kernel center state center center alignment effect alignment see underlie alignment center underlie center c k kk kk maximize label shall combination kernel continuously parametrize base kernel combination alignment target kernel come learn kernel omit yy yy k maximizer ascent additive optimize sequentially add previously direction neighborhood always stop iteration reach happen tb n text datum maximum stepsize k terminate eq material directional derivative maximize obtaining function optimize supplementary material use matlab point option final usual optimizer necessary achieve
let decrease amount establish since inequality prop rearrange map ready rank corollary hadamard submatrix particular thm map negative see may ask extend compression extend exist definite max last equality dimension recall since cover ground riemannian ref ref computation need cholesky suffice kernel many cat cat easy verify result establish rewrite relate quantity helpful tu obtain z briefly mention nonconvex similar obtain input nonnegative problem essentially investigate ignore whereas stationarity neither global strictly
lee way york new york usa department university york new york usa institute studies york york purpose sciences
infinity weak coupling correlation contrast net correlate net surprising experiment superior stand stand lasso correlation elastic net add squared term dedicate prior knowledge trace behave provide seek problem exhibit acknowledgement european technique obtain strictly leave singular vector eigenvalue equal associate closed
numerator denominator since lemma observe confidence sde domain technology gene protein chemical sde learn reaction reconstruct crucial model sde whereby drift many equilibrium special broad drift finite function drift chemical reaction ax
mc mesh topological mesh latter generally gauss complex require increasingly refined mesh interpolation gauss provide mc ensure statistical analysis test topological mix experimental formalism introduce subject specific topological fix subject specific specification eq back evaluate test simpler present mixed growth utilize cost integrate illustrative domain integration mc panel clear global domain integration diameter tend one variability cost testing effect global separately test integration integrate integrate statistic back topological metric impact experimental factor different domain yield systematic bias report factor df df cost influence importantly domain affect result large amount variability characterize large domain report statistic cost find influence separation cost topology cost topological metric illustrate interaction plot cost experimental reporting efficiency metric subset visual reach topology global vary investigate coefficient purpose simple unweighted recommendation topological population summarize main finding fix cutoff strength fix topological difference integrate entire successfully connectivity topology monotonic monotonic association iv topological metric weight cost association comparison analyse ease secondly systematically moreover difference per se informative relevance
detector device desirable deep biological engineering aspect problem observation signal consist least part purpose detector device detector device processor analysis detect signal determine finally arrive eq q detector incoming processor basis detector choose environment problem address processor obviously filter light intensity device biology filter transform incoming detector reality detector light brain come detector device fit biological moreover work bound detector device construction corollary lemma remark phone phone object pixel human stop step view
agnostic rely modeling filter filter transform parsimonious estimating coefficient identification polynomial dimensionality kronecker imply p multiply term multiply permutation retain discard redundant dimension exploit module order outline involve admit representation accomplish via due specification interpretability purpose special employ consist impulse cascade nonlinearity filter set redundant tuple length redundant nonzero location drop second wiener module immediately separable likewise sparse filter even case model high locate case aid sound environment adaptively impulse comprise undesirable cascade short wiener long approximately room usually follow channel represent impulse expansion also bioinformatic filter model ensemble spike record neuron probit selection neuron furthermore genome sparse determine phenotype human trait specie reveal multiplicative gene fact phenotype occurrence disease pose multilinear regression apart
bind together assumption bind exist event event general estimate bind mention every quantity role dimension justify hence conditionally inequality event bind let event trivially cube inequality convenience brevity subsample imply take unconditional true combine discuss sampling subroutine eq view run use notation resolution dyadic estimator always finite dyadic cube enough resolution find piecewise sort
part action take iteratively objective iterative nature well htp important uncertainty quantity bias bias gps straightforward mathematical incorporate aspect characteristic date specifically original discussion gp little relevance uncertainty g determine action objective problem aspect objective already subsection infinitely candidate action infinitely high carlo scheme problem domain computationally likewise increase seem restrictive spirit anneal nonconvex priori due costly method rely abundance arise give satisfactory answer multivariate compact admit random guarantee probability tight important note sampling nonzero measurement gp determinant newly point entropy multivariate entropy point maximized formulated computed matrix scalar summarize
n log markov give distribution innovation assumption asymptotic parameter stationary assumption c far autocorrelation dynamic innovation mention satisfie innovation process matrix symmetric eq could tractable seek n expectation recurrence ahead q one ahead q simply one additional ahead expression get parameter terminology cause cause gaussian var interpretation diagonal I cause cause triangle cause cause conversely causality innovation constrain innovation margin q causality constrain independence triangular causality denote conditional constrain model innovation
I n resample move metropolis hasting provide good instrumental parametric long easy histogram
contain elastic fista without outer augment lagrangian albeit magnitude truncate sparse range yield accuracy effectively cc fista solve linear avoid solve complement instead definite yield background separation marginally elastic net yield separation order fista work fista typical run fista take outer hand take iteration meet stop criterion percent elastic fista interestingly majority inner iteration test synthetic breast cancer mean behave like number inner step require second p fast linearization counterpart fista suggest reason fista whose gradient soft thresholding operation system show return varied setting tune contrast fairly rarely general well fista lasso without augment lagrangian fista take fista p solve believe evidence balancing demonstrate problem build unify
hilbert reproduce hilbert hilbert always functional natural space first thank space vector space understand secondly input many application orient usually provide systematic inclusion rkh concrete appear machine recent study embed rkhs requirement show rkhs small demand rkh embed rkh requirement outline inclusion devote investigation translation invariant schmidt reproduce relation translation relation require set refer input space reproduce hermitian
positive phenotype share target disease phenotype solution identify use phenotype cluster gene association validate finding throughput validate complex gene association help closely gene similarity disease phenotype phenotype future tool infer gene go weight straightforward acknowledgment support seed grant institute university tc cm cm zhang department engineering usa college correspondence gene identify throughput association gene phenotype fail reveal association phenotype exist annotation incomplete discover association set phenotype association rank gene disease phenotype rank phenotype formulate coherence disease gene association efficient coupling ridge propagation evaluate baseline leave task predict discover association experiment demonstrate ranking baseline apply identify disease disease recent dna rank candidate top rank list across author summary determination molecular
vector mn thus direction form j np nevertheless alternate maximization easy maximize fixing implement candidate candidate mention pair large reader correct addition section perform replacement attempt procedure previously utilize procedure use iteratively direction perform plot error rank small test medium start increase beyond avoid trace start overfitte
connect edge add connect vertex black white picture white lattice special lost sense overcome obstacle critical case satisfied get picture white black within predict formation shape relation explain formally split vertex consists belong vertex set call cluster go mention high form relatively black cluster region mathematical
finally multiple show multiple individual six doubly family table notation operational advanced measurement capital estimation become accept distributional lda process ii business line paper derive class doubly ability typical also compound aggregated compound particular develop scenario loss compound poisson arrival increment year loss increment dependent increment capture binomial success change couple comprise per derive analytic operational distributional doubly poisson process ii statistic email service pt operational take prominent occur ii requirement operational risk complex financial product information technology combine
mcmc technique markov diffusion numerical normality markov mcmc become tool ergodic great markov quantitative bound work fairly property satisfy bad iteration mcmc procedure category step early number satisfy performance procedure poor da mixture know performance da procedure
probability subsample dr building reward predict ridge estimation dr error rmse dr dr reduce rmse std confirm dr tend reward doubly always widely doubly give dr improve algorithm develop offset take acknowledgement doubly sensitive multiclass error specify vector policy give adjust aa work policy randomly perturb return learn try
square rsc basically define norm level increase particularly agreement furthermore dedicated discussion matrix complex otherwise brevity depend denote support zero vector denote hermitian matrix project gradient choose kk operator minimization q iteration outline broad sensing except solve impose whose great
volume volume fig analysis confirm web traffic informative trading reverse beyond causality power engine volume test check various hypothesis predict volume trading trading predict trading trading predict query volume trading query volume model hypothesis bootstrap sample bootstrap empirical p sort significance reject suggest reject find support third column cross trading direction compute clean large trading volume volume report small much direct value model regression square compare run clean significance outcome support four trading volume trading volume predict trading trading volume predict query volume volume use query consider compute model residual compute aim assess independent tend nan sample opposite bootstrap clean result testing search engine answer question user even user ask regular basis group user drive science statistical insight social human traffic request www track successful include car
open exact rate game separation er call build game remove split game action action game must armed bandit regret action change subtract column game subtract second one bandit armed bandit action split put half second recall dominate action accord loss split game recursive splitting binary strategy play internal outcome action conversely action optimal boundary share game matter restrict outcome hope know outcome reveal step idea action game stay large previously action start play happen switch fail boundary outcome optimal action action half game enough recover play avoid repeat problematic happen hence minimal generalize mention thus
book explanation black size great lying marked pixel black denote white black useful lattice orientation coordinate inequality limit finite lattice side symmetry arbitrary subset triangular lattice affect event measurable event decrease event ci ci e nn c moreover following let matching definition black rectangle vertex side event white path side black
almost mean unlike marginal beta formal beyond build model measure accord bernoulli place ibp drawing atom beta turn finally observation stick beta process stick length stick length segment break nonparametric mixture call chinese restaurant dirichlet factor latent specify advance allow flexible sequential group relational datum organize detail counterpart mixture latent limitation extension hope flexible study available software analysis grouping identity understand estimate population cognitive contribute produce behavioral generative g cognitive might rise worth fundamentally analysis behavioral attempt explain cause response recover distribution cause finite model observation generate rt rt mixture accommodate mixture condition
rl r dl anchor direction decide threshold nonzero along linear software http edu tw correlation basic continuous experimental study performance page size window kernel estimate equal sample replicate simulation bivariate normal distribution equal let sample variable diagnostic roc degree line cut hypothesis five statistic simulation versus increase true coefficient increase current multi variate deviations standard bootstrap deviation show standard table gold three cauchy three association correlation estimate roc index
issue stochastic bandit bandit usage introduce difficulty sample play weight handle hoeffding variance control variance action improve improvement bernstein evolve way analysis far improvement detail work emphasize although bandit
strength convexity highly dimensionality dimension subspace covariate dimensional gp c prior widely matlab package take run tolerance variance burn model extremely insensitive hyperparameter hyperplane set test dimension important likewise hyperparameter test little relationship proposal hyperparameter poor constraint region subspace general outperform outcome traditional problem range decision convex compute surface create maximizer result function unconstrained function solver noisy square place rather produce estimate posterior demonstrate objective produce stochastic randomly piecewise smooth contour
hence currently investigate employ transform arise challenge general nonparametric noise amount functional set estimator satisfy condition maximum quantile cover selector vast area arise situation neighboring signal gain average neighboring residual subproblem square orthogonal feasible hand demanding application illustrate estimation nonparametric regression image applicability algorithmic novel inherent behaviour consistency inequality extent area attempt value error gaussian discrete observation tend indicate cardinality get likely degenerate unless properly normalize utilize guarantee boundedness bound true regularity eq condition interest expect purpose future imaging application rather algorithmic framework select partition location concentration issue deconvolution step great explore f support support support department institute provide grateful associate anonymous helpful
fact additive borel e theorem correspondence definite sequence spline etc penalty easily literature work unknown thresholded validation method bayesian preserve study classical lie linear use datum replace angle also write unit kernel commonly circular von representation study taylor variance lie two derivative thus close give support illustrate take c increase increase kernel estimate density large region peak raise kind smooth adapt base spline work regard comparative also know calculate detailed explain also disjoint interpret deal circular deal outlier two major inherently detail boundary support presence edge lose adapt mixture presence knowledge able break compact adapt spline point detect exponential join general partition unity unity topological set unit interval neighbourhood function sum requirement partition unity often extend construction whole unity assume index say open space index index support compact compact satisfy construction use analytic category thus unity exist unity help cover localization treat reflect localization lie enable store structural adaptation basis especially fouri spline function result estimation use fourier spline give adapt various outli edge deal function partition unity real life datum discussion follow conclude probability absolutely continuous haar aim regard estimation circle bandwidth fouri estimate comparative perform usual procedure method usual incorporate penalty estimate trade simulation detail give fourier common integrable article variable represent suitable reconstruct fourier analytical heat fourier consider modern transform integrable transform integrable bar fouri science fourier transform fourier translate convolution integrable transform convolution transform definition transform factor operation product give respective fourier transform clearly ahead know calculate moment coefficient power conversely know coefficient correspondence periodic say converse circle transform negative solution circle belong class definite thus theorem definite definite sequence absolutely continuous rise spline integrable correspondingly definite sequence come actually fit minimize minimize empirical datum circle u situation density belong inverse transformation note write circle minimization give run moment coefficient absolutely respect haar fourier penalty existence fourier inversion theorem coefficient density say problem spline format problem correspondence thus spline technique solve minimization empirical kx n zero q coefficient nx value show define q value see literature reveal kernel kernel curve rate calculation mean penalty obtain inside expression thus smoothing expression mean eq go infinity situation similar variance trade find nd value clearly approximation methodology able simulate smoothing define eq density perform simulation follow comparative spline technique bandwidth usual kde technique usual kde mode take usual kde spline smoothing comparison repeat simulation perform matlab consider bi method kde kde nj centre accept reject nj note test reject information edge suitably mid region function control jump effect proceed edge test false vs proceed exponential mle experimentally numerator chi square degree percentile chi degree freedom reject report cumulative chi square detect step vs testing vs proceed way cumulative chi square detection contain edge break contain local edge estimate explain unity support region create cover base unity smooth outside support partition unity say concerned region possess region local estimate section thus start estimate local lie support fit ix say likelihood denote repeat possess local adapt spline lie circle treat empirical fourier coefficient become come region decrease belong region remain procedure get estimate q choice obtain minimize thus final estimation local proceed highlight show aspect density comparative come uniform mix proportion come triangular execute broken work exact leave little proceed detection region estimate region final density show figure integrate square calculate spline kde identify point effect allowing apply vertical feature finally position local kde actual density far justify spline small kde outlier come come normal truncate take whole repeat integrated squared table local kde support detect case figure outlier perfectly exponential presence outlier create kde fail motivated work able spline kde consider successful detecting come point c fourier usual spline perfectly successfully mainly fail explain modulus compare jump detection moderately see fail far kernel rao come methodology group consider group uniformity degree upper gray depict kde depict almost perfectly extreme region identify local kde estimate smoothing belong would see spline constraint influence use multiply unity detect notation cover support estimate multiply final proper function family give estimate comparative study give new circular smoothing circle density work smoothing density develop spline try circle get circle transform derivative fourier solve spline result density density comparative application novel working profile outli detection done consider perform consider exponential overall consider partition unity local exponential spline lastly life well estimate kernel estimate solve spline negative comparative local focus fourier paradigm complete
version set tune smc preliminary run vary option general facilitate efficient accurate strategy sample iteration ease exposition variable wish fact require invariant set auxiliary conditionally ki pi define target note scheme proposition mcmc kernel upon auxiliary auxiliary accept vi ki possible auxiliary accord possess notation induce everywhere algorithm adapt theoretically hold procedure addition admit auxiliary primarily use disadvantage posterior slow dependent many might find tune example value level close together illustrate use deal toy compare implementation deal model genetic another case adapt entry uniform proposal normal walk compare implement development computation order several extension firstly adapt implementation rare event examine framework quite framework multi level splitting smc might practitioner familiar could sampler article scheme sometimes focus clearly gibbs model methodology feasibility development methodology apply thank work education grant author support grant work complete author department college result adapted conditional expectation e denote convention n define normalize line construction regard validity metropolis hasting kernel density pi p k metropolis procedure similar one may sum equation admit part consequence similar proposition statement argument marginal write denote independence ki apply university sg mail electrical college mail article process stop first naturally posterior advanced mcmc particle carlo smc embed within mcmc smc stop author smc level formulae nest set resample intermediate naturally intermediate scheme multi level proposal propose proposal methodology stop article markov stop reach boundary wide population finance majority deal observe stop stop datum cope partial good previous direction fully area stop efficiently process start terminate return getting achieve importance overview smc prove smc achieve popular traditional sequential monte describe distribution density know wise normalize smc resample idea introduce density sequentially term particle proposal success lie incorporate resample operation whose fully stop process resample take particle likely particle start result particle approximate long reach early author later split resample reach intermediate sequence idea appear formally interpret interact multi formulae naturally intermediate towards level systematic exist deviation adaptively path process contribution address issue law stop process dataset context employ feasible difficulty trajectory stop inference successive bottleneck static chain construct spirit apply update e stop bring possibility appear within enhance flexible unbiased addition via adaptive adopt article structure formulate motivate multi stop multi smc approximation strategy level theoretical level give find measure write borel denote total ji possibly class dimensional everywhere else collection stop measurable throughout process evolution empty markov empty define positive direct evolution vector assume restriction simplify exposition omit un normalise constant subscript explicitly stop process f likelihood trajectory process throughout assume density dominate measure design mcmc normalize become clear framework rather motivate figure particular start split choose continue evolve move genetic stop mutation genetic observe forward split mutation point join compose begin default chain first individual change mutation event figure confusion present compose alternate tree could consecutive mutation stop individual terminal correspond match space write carlo inference reverse simulate data sampling adopt weighting simulate backward two tree define stop markov definition initial terminal appropriately modify convenience reverse previous bottom tree obtain respect likelihood brevity omit current mutation stop integrate remark crucial shall briefly introduce extensive refer reader ease exposition present shall drop smc simulate probability g possess respect dominate normalise density g choice wise require knowledge recursively properly importance resample part density possess normalizing path obtain smc approximations normalize compute resample n nu compute smc resample step particle path particle backward recursion view smc approximation measure induce simulated sequence obtain smc approximation introduce use establish complex extended target enhanced stop process smc seem general stop provably much introduce nest denote implementation level generic reach path remain multi smc smc ultimately target sequence distribution eq finite value stop spirit generic intermediate density r dominate natural density nz define sequence density dimension compare grow increment consequence sequence addition nh hx base particular generic firstly presentation multi convention importance herein write smc whereby reject assign weight resample reach smc multinomial define normalize eq n compute weight smc begin show proceed backward process indexing root level candidate expect quickly lead poor one construction th resample hard result smc empirically case level advanced method next median rank sample aim next vary mcmc importantly sequence monte mcmc algorithms variable proposal sample lie order target providing insight question valid smc level seem obvious strong propose extension introduce three particle metropolis hasting metropolis sampler paper level sample accept probability pi ki pi p pi pi pi
vector gpu implementations benchmark single computation rip compressed sense repeatedly start large objective way batch give possibility matrix significantly multiplication small enough size limitation core gb ram core machine ram gpu enough big code store column iterate letter cardinality loading outline solve formulation variance formulation fx ax ax equivalent form x ax x ax n p x p n ax n ax ax x propose via yx fx yx xy problem subproblem close subproblem list z sa aa aa u elementary correspond objective function region maximizer objective result four operator give vector integer vector problem subproblem give formulation write select constraint v stop satisfied necessary resp however normalize maximum reach happen generalize compact work maximize subgradient nontrivial relationship apply formulation apply start formulation iterate correspond iterate convex start solution subproblem formulation iterate iterate formulation note iterate x start produce ax k ty v ty precisely penalize formulation th iterate iterate start consider individually identical q iteration start iterate ax precisely vector iteration start compute formulation develop approach scheme load batch need perform parallel multiplication product single g case library parallelization lead compare idea operator start scope parallelization vector product parallelization start choose computation algebra case parallelization naive batch dynamic strategy informally converge quality sp run obtain explain poor amount useful run number experiment choose randomly generate find effect substantial nevertheless employ different start point naive solve start run start resource architecture penalize hence besides choice htp axis vertical strategy core computer benefit run range speedup plot speedup especially batch size nothing continue already converge minor negligible speedup dynamic predefine batch list scheduling replacement variant look compare right notably com implement parallelization strategy describe use optimization implement processor parallelization computing real numerical large corpus remark instance core see plot show core consistently number core across core random cpu result htp marker setup high marker code linearly core slowly code gpu code right gpu speedup reach several penalize problem matrix focus table matrix file take second take second iteration table depict take perform treat use elimination preprocesse able expand code size sp gb gb gb gb gb gb gb gb gb gb constrain medium article appear york times article publish article abstract clearly sparse approximately million million exploit sparse component pc fourth education united interpretations pcs pc nd pc pc child play company player million percent student us team stock united st rd pc th disease child human primary tumor multi implementation intel mkl implementations multi parallelization interface architecture intel mkl library serial nevertheless implementation fast intel mkl implementation gpu allocation gpu gpu core cluster mkl communication formulation maximization observe case suitable function formulation study notably constrain variance parallel aim core efficient multiple starting pc variance speedup point achieve implementation gb fully acknowledgement support mathematic digital resource centre grant ep g support simplicity partially start research grant theorem corollary com principal analysis aim explain combination paper formulation compute single loading vector factor employ measure l way constraint formulation propose maximization al formulation parallel implementation code parallelism aim computation time serial code deal gb maximization compute big tool area engineer finance encoding feature aim combination principal pc column center extract first suitable measuring loading vector pc
model model word unnormalized rate regularize across compute estimate corpus smooth differential usage proximity hierarchy ideal applicable corpora author annotation document challenge offer dirichlet multinomial conjugacy likelihood smoothed across membership parameter explicit distribution label result challenge conditional word membership poisson likelihood tractable divide member conditionally allow inference factorize hamiltonian conditional sampler efficiently space leverage hessian allow intractable walk sampler hyperparameter membership augmentation affinity hyperparameter marginally exact loading explore augmentation parameter find entire comprehensive annotate corpora topic aid factor scientific progress biology progress enable collection topic allow quantitative evaluation list frequent word new intuitively list news use analysis include twitter unsupervised infer orient around axis create prior ensure discover document latent conjecture problem pt pt collection content infer interpret frequent word limit interpretability exclusive characterize annotate collection tree nod poisson convolution analyze annotated infer semantic word exclusive experiment amazon demonstrate summary score interpretable frequency summary produce model also hamiltonian sampler allow scale keyword categorical inference challenge statistic interpretable summary simple analysis correlation recently powerful however interpretable statistical summary value aid qualitative dimensional quantity scientific practitioner design scientific interest mind balance interpretability live high live link scientific interest two low interpretability maintain direction future text analysis motivate count document create content represent length organize specific leave news label hundred inferential discover content time content low structure express approach arise commonly parameterize loading frequency usage useful onto word list interpretability stop sometimes appear incoherent post meet expectation instead differentially across exclusive summary less likely content rarely form look frequent word corpus content idea usage idea variable capture content perspective base topic vocabulary stable usage word within suggest popular usage arguably lack across topic trade strategy word versus tackle generative poisson count unnormalize whose regularize lead word extension code interpretability category create collection focus use topic build membership conditioning hierarchy hierarchical introduce hamiltonian efficient design interpretable restrict correspondence supervise lda covariate restrictive lead associated individual infer label term word exclusive topic automate domain topic specific predict hierarchical poisson convolution collection whose hierarchy topic label topic vocabulary document normalize topic vector assume word similarly present represent feature overall corpus topic draw parent control child branch greatly word close parent express similarly learn equivalent similarly otherwise hierarchy occurrence equivalence count combine topic model proportion allocation document membership topic count poisson give draw vocabulary get content document human extra content topic content allocate active generating define topic proportion element constraint non since implicitly model two source generate receive content vice latent affinity document associate membership document affinity active membership topic inactive restrict topic parameter simulation generative parameter feature draw hierarchy terminal draw k membership parameter document draw di fw outline vocabulary word simplicity presentation terminal child level word vocabulary positively content univariate semantic document membership nuisance content topic parameterize via across topic usage also child force toward explore empirically topic arguably distinguish related topic require qualitatively difference important content measure diverse dimensionality discovery construct measure able low alone necessarily harmonic toward score harmonic default cdf apply membership hyperparameter matrix word count length scalable critical offer advantage group posterior membership hyperparameter hyperparameter break independent interface computation draw block resource document core conditional direct walk inefficient uncorrelated hamiltonian hmc hessian distant acceptance hmc work unimodal relatively curvature implementation initialization follow scan block draw conditional posterior hyperparameter latent set posterior explain conjugacy variance parameter influence group hmc variance parameter poisson covariate exposure rate parent relationship use hmc speed sparse conjugacy specifically cardinality second posterior affinity rate document sample covariate regression covariate simplify generate corpus rate v discrimination independently convolution specifically topic back distribution hyperparameter put average every draw burn word vocabulary unlabeled count unknown predictive without condition contribute size affinity factor likely base evidence alone analyze model corpus large collection great word content give special corner word alone finally baseline model month period hierarchical category remove redundant allow distinguish assignment category modify annotation document child document token stem frequent stem stem million token level divide fine content divide high level business news market economic bag category grain broad node branch market split market topic market divide soft present panel compare child market panel cluster gain exclusive topic appeal aspect strongly parameter least frequent word rate least regularization prior parent topic science technology branch panel branch shape joint posterior rare unable exceed usage rate toward frequent even rate frequent exclusive great topic similar quantile frequent exclusive word alone science technology term american program similarly topic almost research ph cr gold say year nuclear million water nuclear say million deal frequency average table top topic alone topic model neighbor topic examine red set solid case since incorporate list word everywhere market understand word rank across right panel stop extreme frequency exclusive prevent highly base index regularize logistic york sample balanced sample across split fit set topic membership train micro weighted topic weighted logit micro micro macro macro york l logit micro micro recall macro compare average dominating corpus label lda display accuracy sample maximize interpretability summary term frequent exclusive neither lda may offer quantitative illustration statistical support vector machine interested reader analysis hypothesis summarie interpretability summary regularization rate occurrence estimate frequency affect variation turn translate summary create effect light plausible less word word amazon human outline issue experiment dirichlet score diverse summary summary arguably provide interpretable summary obtain lda metric quantify diversity summary across summary whether present diverse word summarie interest rank rank score estimate leverage lda lda summary topic lda cb summarie cc summary lda summary combination score frequency summary contrast proportion word drop length summary half distinct concept reflect interpretability fit lda estimate rank increase proportion gain diversity summary dominate summary topic summary occur next determine extent summary contain word interpretable analysis summary frequency summarie lda interpretability automate method human result randomize conduct amazon amazon execute comparative three different experiment participant interact extract coherent summary human amazon produce summary interpretability outline measure coherence summary belong topic another intuitively identify summary express coherence coherence topic picture summary summary ask coherent state preference p word coherence score word along topic present ask summary come number interest strategy topic summary response figure coherence provide summary summary randomize incoherent include interpretability include option topic size result relative summary rating summary preference summary word summary low probability space improve model topic equal topic detect probability summary estimate use result interpretability summary increase show coherence absolute summary strategy word regardless interestingly summarie strategy rank indistinguishable size smaller quickly drop increase lda summary display rating three summary preference topic summary increase question average quality degradation summary number grow topic trend average coherence summarie number topic summary vary coherence column right column summary grow
cl cl cl cl cl cl fractional cl cl c armed bandit homogeneous moderately diverse asymptotically fractional tight addition demonstrate fractional fractional mab theoretically performance efficiency budget approach outperform demonstrate fractional counterpart cost performance density significantly differ relaxation cost similar easy order approach behaviour cost diverse cost density order greedy fractional relaxation give fractional three homogeneous moderately cost extremely diverse cost homogeneous cost within diverse randomly set number arm represent see test performance divide see fractional differ typically low regret value fractional typically fractional unbounded problem fractional counterpart similar homogeneous behaviour moderately outperform counterpart improvement diverse apart also outperform budget particular counterpart limited logarithmic bound paper mab problem phase subsequent good agent reward plus interval take order algorithm determine fractional fractional counterpart good separate exploitation provide theoretical prove differ well finally simulation counterpart increase cost average implication fractional practice work consist tight bound extend mab distribution dynamically algorithm rely expect thus real proposition corollary budget multi bandit mab problem action constrain consequently exploitation optimal arm repeatedly agent within budget policy namely ii fractional former provide latter less expensive prove logarithmic asymptotically good multi bandit mab originally present trade arm must payoff receive sequence reward arm high play agent reward sample choose optimal arm mab trade ucb mab incomplete end variety study recently number arm model exploration budget limit arm reward subsequent exploitation phase exploitation phase address crucially limitation application wireless sensor network sensor action capacity setting perform limit exploration reward exploitation take phase budget mab mab therefore mab efficiently deal simple mab overall budget limit suffer value value exploration inefficient exploitation give particular budget limit cost typically poor policy budget limit regret function learn exploration exploitation adaptively arm reward detail provide high unbounded determine set however unbounded np hard literature occur well reward unbounde solve behind budget arm form arm fractional set highest estimate reward approximate arm instead use order solve unbounded fractional analyse counterpart devise asymptotically differ well possible one numerically demonstrate fractional increase give first mab algorithms regret algorithm logarithmic mab bound present counterpart mab arm step denote reward arm exceed operation total arm exceed typically world application denote reward agent receive arm arm budget initial must total agent budget formally give represent arm since total exceed total let q order thus optimum precisely denote regret sequence arm describe introduce fractional challenge algorithms arm similarity unbounde reward unbounded introduce unbounded capacity item weight item exceed capacity one either problem hard however detail put differently reward arm know budget limit reduce budget mab estimate frequently thus explore greedy computational calculate update estimate turn approximate unbounded within differ fractional unbounded crucially fractional fractional unbounded arm within form agent fractional relaxation solely high ratio fractional randomly arm tc I tc also see version ucb computation order fractional computational show fractional counterpart asymptotically focus regret fractional define end asymptotically begin simplify assumption define useful ease exposition reward arm cost appropriate mean density simplicity minimal denote cost high mean note finite operate average always budget estimate achieve regret main budget positive e bound follow chernoff range nx far devise time I e guarantee show case regret average state arm q number prove lemma arm arm unbounded arm tc tc arm high confidence total step budget arm imply inequality drop necessary add feasible show add arm budget time count inequality great small cost arm great inequality arm implies last obtain equation give possible conclude lemma already abuse drop notation explicitly number consider right sc statement hold apply since give obtain I combine together give conclude denote expect difference number lemma arm
grow locally discard information comprise pool form leaf likelihood leaf prior proceed follow time smc explain section criterion update associate shift likelihood time predictive change next dt absence active data pool tree grow active split assessment make discard prior information grow move move stay close counterpart stay move stochastically action mechanism prior discard likelihood information leave pool leaf suggest additive rule eq data discard sensible novel child proportion active share parent active parent preserve total preserve reversible operation prior bring discard future lack location actual therefore newly must immediately dominate grow partition candidate hierarchical inference prediction sensible data point go concept concept formulate choice discard ad regression separately require argue ad easy al enable easily update smc learning procedure sequential heuristic common heuristic active learn gps alm select average certain case demanding alm require integral numerically lead exploration sample change rapidly alm disadvantage cope search evaluate simplification ad grid need preferred integral evaluate location discard al leaf special data variance give design expression discard active location integrate rectangular general previously grow exponentially reasonable accuracy leaf tree key analytical integration observe remain near high partition surface fast active allow illustration learned sequentially sample initial show panel round proceed smc update implementation leaf regime surprising degree variance small final end response change indeed high absolute heuristic predictive surface class greedy near largely literature fortunately analog low datum discard tend interior pool discard divide update ad node ad leaf version update ad need unchanged posterior change propagate discrete tree change particle keep limited cm tell discard discard result good rate accumulation historical information eventually streaming mechanism evolve additional point update effectively effect strength update recently oppose old datum leaf recursive conjugate irrespective family show theoretic enable streaming refer historical discard perhaps obvious context contribution past become discard discard benefit behaviour discard heuristic mistake surprising retain clear pool heuristic resolve lie beyond scope historical henceforth smooth replacing allow vary smoothly control surface responsible complexity simulation ahead performance follow first generate rmse dynamic basis ht probability via bayesian conjugate shape indicate away extreme change distribution repeat change datum distribution peak figure discard model measure height vertical complexity drop dt without discard incorporate dt mild first pool size rise complexity deeply drop retain level adapt easily novel streaming henceforth assessment paradigm wherein allow first problem old become increasingly effect fuzzy display left plot old way discard useful unlikely hard illustrate introduce dt discard discriminant streaming context mind area newly prefer alternative auc discard explanation consider way evolves show visually full online interestingly latter bottleneck auc offline exhibit comprise minute identify four prohibitive transaction status determine detail time provide inferior example dominate example remain dataset lowest suggest essential maintain informative prior information much although classifier application experiment give use flexible tool fully potential updating smc model preserve flexibility enable availability allow devise active feature incur prior enable result algorithmic parametric classification tailor massive streaming context technique automatic factor streaming heuristic laboratory university department college b business il massive become wide variety setting vast offline database stream single pass streaming context remain change online evolve stream parametric technique suited increase take stream modern parametric streaming performance inference incorporate information become without algorithms remain operate stream arrival massive operational become area scale case estimation formulae conjugate dynamic modelling sample filter variable quite parametric cloud dynamic track parsimonious regression surface arrive sequentially speak online require history summary essential parametric modelling wherein allow maintain operational cost new necessarily discard historical help response real value one dynamic tree review section allow sequential local adaptation arrive fit simple discard require work concern discard whereby discard partially informative flexibility new require access consequently discard come cost surprising employ scheme discard active heuristic active discard lead performance streaming datum temporal concept learn exhibit evolve formulae require modification adaptive whereby contribution past smoothly historical suitably construct reproduce remain real show compare modern alternative fast smc yield surface increase available specification partition refer logical rule instance node three leaf belong model leaf parametric hard seminal generative split induce via leave employ tree py py sampling change grow long integrate regression leave multinomial leave choice incorporate see move embed describe old new arrive set partitioning set insight dt tree evolve transition ty allow move pruning grow occur leaf build area move grow move among location choose random complete stochastic dt specification inferential mechanic replacement py outline step involve require parent nevertheless great distance govern particle appeal division ensemble assess effect compare method like cost dt may access
repeat randomly predict metric biology evaluating feature first coefficient tune fold cross exist mse mean biology correlation e lasso lasso lasso propose propose method solution lasso negativity lagrangian furthermore redundant propose real feature show promise real world vision bioinformatics speech extend investigate acknowledgment thank provide spam dr valuable international ms yahoo st usa institute technology university find subset responsible lasso computationally feature selection input redundant strong statistical dependence measure hilbert schmidt optimal problem many microarray categorization domain could either structure sample original transpose goal responsible predict allow assumption dependency tend regression irrelevant lasso large package develop compute critical limitation lasso handle optimization I wise apply transformation wise represent k nk vector output transform linear transform th programming hessian number much big number dimensional instability irrespective linear limit flexibility capture non statistically particular problem compute efficiently scalable dimensional knowledge dimensional clear interpretation statistical regard method highly preferable practitioner hilbert schmidt spam selection ccccc dependency scalability structure feature highly linear scalable linear dual spam hessian guarantee convexity although validity select exist alternative feature frobenius norm I k negativity meaningful addition gram select incorporate structured structure formulation non negativity impose kernel combination lasso selection intelligence rewrite ignore statistically minimize regularizer relevant output lasso redundant feature redundant redundant strong idea preferable remove negativity select allow hard interpret interpretability thus non negativity constraint kernel laplace permit moreover delta thus kernel input gaussian delta normalize kernel regression scenario unit kernel categorical kernel scenario tends poorly rewrite plain feature high shown incorporate negativity dual formulation expensive overall computation practical large g memory dependency choose term redundant redundant strong dependence output deal high feature possible intractable backward elimination feature potential approximated window window mutual maximize advantage feature selection maximization selection backward elimination solution regard continuously relaxed version feature continuous optimization problem memory bfgs bfgs many initial expensive attempt base computational able try problem originally feature perform qp hessian singular spam feature spam kx k kx x spam closely relate employ induce spam optimize moreover spam spam additive spam thus spam optimization tend expensive spam spam deal output multi label section experimentally synthetic selection code report computationally expensive dimensional solver solve experiment experimentally x multi variate distribution mean e datum additive rank regression spam horizontal axis vertical scale lasso horizontal vertical figure selection function fraction lasso select spam work poorly assumption violate compare number run observe lasso respect compare spam trick deal trick compare image microarray microarray experiment sample repeat evaluate classification use regularize accuracy feature feature coefficient width parameter fold investigate red absolute red reason decide th select
I f gs fourth put similarly stable respect input bind expectation also e f bc e e sf en sf en sf e bc monotonicity symmetry give rademach average training conclude double lem lem corollary lem lem microsoft concentrated solely handle supervise also classification w supervised adapt technique learn effectiveness real ordinal ranking real goodness conjunction vector recovery generalization ordinal reduce effort limit reason notion artificial constraint like psd paper successfully technique albeit guarantee notable exception define goodness goodness provably accurate classifier yet classification view point margin instead target similar propose goodness definition space psd learn challenge bound learn specific b appropriate surrogate generalization restrictive showing definition admit psd additionally goodness capture influential recover require sampling test time predictor result issue construction typically ease practice try datum theoretically give experiment baseline kernel learn broad use convert psd project psd perform approach expensive operation apart inherently store operation notion notion goodness give enjoy advantage exist psd ability give bound psd model adopt third supervise goal similarity supervise predictor domain similarity enough predictor enjoy property natural learn good preliminary learning fraction inspire similarity tie definition call weighted neighbor define goodness propose definition incorporate practice loss final say loss goodness definition sure predictor utility sure commonly psd kernel unlabeled least hypothesis e kk learn put semi goodness definition existence guarantee outline choose unlabeled predictor definition second learn generalization prove utility dd l admit psd psd us correspond notion kernel margin psd psd kernel learn psd treat similarity rkh adapt modification value ordinal utility guarantee supplementary material space utility ndcg ex ex ordinal regression pt solver primal spaced dataset quality label ordinal measure compare sigmoid almost case sigmoid refer criterion provable show restrictive psd focus problem identify influential aim predictor formalize typically small fraction influential problem provably sparse predictor empirically benchmark ordinal recovery outperform direction would function real social notion amongst microsoft fellowship award microsoft ex throughout prove statement originally present certain proof step allow give statement bound third technical help fx dd allow project euclidean various exist margin appear always clarity standard hoeffding argument ff x probability expectation apply switch expectation apply inequality p f domain lf unify analysis predictor however analysis bound cover number strongly respect lipschitz c l weight psd kernels psd exist bound weight k unit similarity prove generalization use variational technique problem problem domain h optima example choose p p I weight discrete problem actually contain discrete variational analysis practice act require label use fall average guarantee elsewhere generic specialized make presentation easy f l sf k bf ik f b stable change embed predictor ss sl jk jk ik I argument easily extend incorporate respect stable
eq form method estimation fisher result error distinguish additional close true determine task theorem coefficient error constant size estimation iii relation function generalization unseen observable predictive dominant dependency appear approximation enable calculate observable variable unseen substitution latent numerical approximation exist discuss ii estimation summation error method estimation estimation type eq estimation type bayes variant criterion immediately asymptotic form conclude estimation maximum training affect analysis bayes type result subsection bayes fisher include true ii bayes straightforward n variant type depict rational get satisfy integer type remain ii equivalent change indicate estimation joint target change enable behavior estimation search determination ii iii expensive successfully em commonly search model maxima method expensive integral ignore prior reduce iii show case expensive vb computation form reduce many vb present formalize kullback leibler derive maximum mathematically determine case accuracy approximate cross approximation form foundation prove first hold error another expansion rewrite eq remainder number eq term w taylor expansion accord w consider end us taylor remainder converge apply end corollary positive triangular matrix fisher symmetric definite assumption prove corollary taylor remainder term correspond use complete end proof theorem taylor expansion state complete definite confirm right complete mm theorem statistical science datum important optimal active however accuracy quantitative latent model method probabilistic employ observable variable datum model employ observable label concern unsupervise learn analysis hide process analysis limit many criterion evaluate present vb latent theoretical play important observable selection simple analysis true generate recently redundant range latent conventional criterion validity necessary unsupervised study sufficiently singular observable estimation table summarize target target example observable regular conduct estimation goal provide error measure first simple attribute mathematical leave follow three fall manner error explain prediction observable form conclusion observable target observable represent latent us datum dependency dot observable gray target estimation prediction three estimation latent observable gray item refer bottom refer type ii marginal unseen panel unseen type ii include estimation section present maximum likelihood
predictor arise sample location sampling see instance realize scalar column measurement scalar predictor subject longitudinal outcome effect usual subset independent subject specific functional functional index time index somewhat role longitudinal densely vary time effect q prescribe independent term reduce baseline point independent situation function rewrite equation association model sampling approach impose directly combine subject n kk encourage preferred via operator penalty consider penalty heuristic informative dominant dominant eigenvector large construct penalty dominant eigenvector prefer belong treat preferred estimate prefer compare prefer penalize longitudinal estimate subspace longitudinal constrain tuning penalize minimizer estimate minimize expression obtain q explicitly term value decomposition component example continuous impose inform near return sampling penalty q scalar determine estimate analytical property penalize discuss appropriate minimize tuning b simply fitting equivalent whether randomness device truly follow conditional estimator unbiased trivial see conditional unconditional expression v v dt discretized version smoothing ratio derivation knowledge allow regression limited index sufficient flexible appropriate propose decide unknown point confidence band band drop non preferred subspace current grid select maximize aic restrict property study compare second simulation study influence size functional tune coverage band feature summarize subject visit flat pre white predictor instrumental noise generate location vary use select discretized function prefer estimate regression decompose trace square bias denote noiseless method estimate regression vary simulation regression use predictor generate table draw c simulation panel column penalty panel true method principal basis panel use finding table display exploit exploit limit use feature surprisingly relatively precise l c simulation average aic maximize mse aic increase decrease vary primary assess variance relative contribution generality indicate great emphasis estimation process section list realization simulation ny ny scenarios iii mse display plot mse mse evidence scenario increase mse increase result however beyond result almost unchanged plot column use define contain function represent increase small mse hand feature similar maximize aic mse mse guide set standard mse increase mse decrease mse panel penalty middle panel average decomposition penalty display bias location true b positively approach panel middle panel band coverage dot base subject display longitudinal horizontal mark nominal describe matrix use define panel middle bottom probability notable coverage band cause large lead shrinkage result solid line span line middle panel ridge penalty external process simulation information peak display figure ridge eigenvector highlight appendix contribution ratio contribution external shrinkage towards display panel larger minimal observe gender residual plot pointwise band panel panel pure investigate association obtain longitudinal stage patient association evolve elsewhere response mr comprise spectra background profile cr figure decomposition include age baseline gender choose covariate see appear gender respect vs pointwise different lead aic interpretability hard leading follow specific intercept distribute model formulation score residual display purpose model residual obvious indicate lack display band aid spectra display reveal pure cr part regression coincide profile cr significant peak location pure longitudinal suggest find several associated band suggest structure longitudinal available period propose longitudinal derive valuable extend longitudinal longitudinal scalar predictor advantage dependent regression incorporate equivalence advantage exploit inform penalty smoothness spline constraint probability band nominal naive band bias method still relative contribution space sample size absence penalty penalty weight onto generalize ridge regression formulation informative prior formulation linear use criterion insight convenient way derive possible two observe express functional model simplify outcome important arise status patient collect setting author thank manuscript support health grant mh tr extension two across multiply ridge situation without generalize put estimation gs interpret tune order gs vector respectively w l w expressed far express w kk l estimate obtain generalize model longitudinal plus plus minus abstract longitudinal association scalar collect consist vary exploit several apply patient collect phrase mixed model pt minus availability storage collection part time course longitudinal function analogue connect covariate study collect regression longitudinal approach longitudinal subject regression consequently suit association predictor evolve extend relate functional estimate fit generalize impose inform quadratic extension longitudinal allow priori structure procedure within implement analysis response follow square integrable model denote unique solve regularization require impose smoothness expand
way capture ts attract efficacy detailed discussion comparison mab achieve display news contextual ts delay ts search ad search thorough know ucb ts run however provide weak guarantee namely context version problem significant regard open include contextual bandit amenable mab challenge novel martingale demonstrate ts generalization ts near contextual bandit payoff bandit additionally ts solve problem bandit payoff provide upper implement long efficient arm paragraph discussion theoretic fact algorithm work gap theoretic question contextual technique discuss context context adaptive play history I unknown reward contextual bandit problem arm play context optimal tt rt appendix norm indicate norm assumption bound scale free regret regret appendix sampling algorithm precisely bt bt bt bt compute nt bt computation thompson distribution play maximize reward thompson contextual bandit algorithm completely reward play bt rt td algorithm sample arm td time analyze notational thompson variate problem vector arm time step thompson algorithm tt td thompson without require thompson achieve regret bind arm notational change contextual name reinforcement name problem correspond cube bind arm advance complicated subroutine regret infinite arm assume give bandit regret set give none arm point maintain np arm sampling propose optimize linear maximize convex even combinatorial pay regret away theoretic low achieve achieve good regret find computationally achieve heuristic thompson contribution thompson bandit despite attractive use useful multi bandit analysis ts basic seem applicable novel idea resolve play arm high mean problem reward regret play basis mab variance estimate arm small mab proportional play every arm know incur also arm quantify exploration exploitation tradeoff inverse suboptimal arm play arm arm standard estimate large regret arm bound improves play arm deviation small context arm able distinguish utilize play small step play probability group arm arm standard deviation arm property enough concentration arm useful bind regret irrespective arm play play union context probability arm establish super martingale adapt super inequality mention early introduce notation notation also appear notation begin material define distribution difference arm rv p ep te concentrate around respective precisely tt v tt always time keep vice versa observe determine history context include identity arm arm true proof appear appendix use state utilize prove state arm reward anti concentration mean standard deviation provide concentration around proof follow arm arm choose highest play great arm optimal play ct b tt ct f p tt g p g inequality apply define tt ie super regret super completely ie trivially ready apply along lemma see notation begin hold pp pg g rt alternate remark thompson stochastic contextual payoff resolve open thompson mab ts achieve martingale technique arguably technique ts amenable extension gaussian analysis anti prove happen anti concentration exceed deviation similar prove lemma regret remain tight remove desirable believe would provide bound contextual delay agnostic bandit payoff agnostic refer setting slightly modify version r bt bt concentration anti inequality super martingale random martingale hoeffding super measurable imply value measurable conditionally information reward conditionally mention make eq q tt bt inequality definite therefore r tt bt bt bt b tt b bt alternatively tt union tt tt tt tt b tt b tt tt tt f g p hoeffding last use line result imply r bt b derive along lemma observe f ti respect new apply hoeffding inequality bound condition simple hoeffding tm tm tc x reduce clutter unit greater equal exist subset event occur tm tx tm tm ty tx tu ty tu tx tc tc tm tm
bag frequency apply sentiment document bag gram well onto window sentiment learn gram sentiment try rbm nlp task effective explore sophisticated rbm rbms difficulty natural vocabulary size rbms linear issue chain monte carlo units yield demonstrate rbms million word gram previously rbms sentiment classification application boltzmann expand year rbms image bag movie rating although rbms originally include observation rbm observation issue natural word vocabulary impractical vocabulary typical nlp vocabulary gram window training scalability obstacle use rbm processing directly address associate softmax unit visible gibb visible carefully transition training rbms million window capture meaningful syntactic learn induce yield even extract art sentiment describe boltzmann observation vector layer unit parameterize weight convert simple conditional give train visible scale parameter q visible fortunately approximate replace second carlo rbm observe maintain parallel carlo step simulate step chain eqs mini procedure belong class requirement stable outcome rbm representation construct visible group use index observation encode unit contain rbms illustrate unit I bias weight unit ik visible probability softmax nonlinearity refer softmax equal size layer language vocabulary visible unit expensive update distribution dominate activity select mini batch negative computation since contain gradient rapidly compute barrier efficient multinomial rbms gibbs multinomial address binary tree leaf external consecutive word wish introduce undirected strategy conditional rbm rbm conditional rbm deal boltzmann large open nlp visible multinomial operator sparse satisfying approximation learning sampling conditional chain leave current ji v visible unnormalized separately efficiency operator move speedup sampling rbm v efficient correct possibility designing explore metropolis marginal word corpus I outcome use sample constant setup gibbs make proposal simulate metropolis operator distribution linear space bernoulli dependence practice although poorly mix good occur well examine mixing use corpus vocabulary describe imply initial accord computation require offline purpose hasting converge target six randomly gram corpus metric variation tv mcmc tv across group figure break dark green highlight mix mix slowly benefit h use neural use boltzmann machine model rbm vocabulary tie weight word bag tie single document sized multinomial distribution word condition softmax incur notably address computational burden observation rbms boltzmann machine could large softmax although motivate nlp softmax boltzmann natural language task representation task previously different mention conditional last gram window approach training network middle use base objective contrast rbm use fill within window single rbms nlp rbm windows parameterization rbm sentiment useful gram rbm separate position rbm gram window would prefer word position dependent rbm inspire neural learn possible word dimensional projection encoding energy position let vocabulary see value gram representation bias energy refer parameterization easily construction rather train metropolis hasting joint train use observation find helpful momentum updating representation additional window text english corpus text source extract window boundary two sentence also correct relate replace digit word character discard vocabulary consist frequent word plus token map word publicly available task feature crf f use window unit baseline comparable representation try
cart subtree select motivate extension concrete take uci repository seven coarse aggregate fine aggregate per day concrete height flow objective predictor cd cd intercept water sp fit multiple regression water coarse amount decrease water one may conclude water predictor see interpret hold main inclusion interaction bring interpretation challenge instead control effect similar goal partition restrict show predict terminal node water water flow produce easy nontrivial affect jointly flow water great intersection yield concrete model three response ideally predictive variable bias extend guide solve fit data node observation sign th two perform small exceed threshold depend characteristic categorical goal classify sign pattern residual work sign instead response residual residual pattern residual predictor residual pattern contingency table sign group row chi test independence specific method univariate guide except function sum split mean sign node categorical mean end categorical create contingency chi test mean singleton k predictor select small step possible ability detect pattern interval reduce guide page split find consecutive total deviation categorical form computationally quick approximate variable describe procedure pair water concrete group contingency form sign group water chi fold cross description multivariate guide refer bar concrete water coarse water concrete high combination predict concrete water split guide three unit apply one sum mean result close guide guide univariate uncorrelated flow weakly strength carry simulation experiment compare bias guide concrete randomly predictor select left probability guide trial standard error proportional variable water sp aggregate aggregate demonstrate select split predictor multinomial equal base panel seven variance regression univariate guide different utilize response variable structure challenge absence effect variable mutually generate regression cart model mean response estimate pt half terminal also terminal expect univariate guide scenario scenario take relationship multivariate guide detect high guide interaction step second multivariate zero mean covariance independent experiment result except scenario guide scenario univariate guide notably twice average splitting wrong effectiveness greatly predictor variable challenge guide use miss group carry chi allow datum technique pattern search split value categorical additional missing value split miss split consider missing mean usually miss great select approach cart maximize square bias toward sample concrete randomly brevity teacher applicable grid may restrictive broad applicability motivate explain longitudinal black white time varied section fit partly overcome partly model complete near force entry black school intercept subject denote time applicable interval user algorithm node datum great number restriction linearity fit split show curve node eight give tree terminal curve five terminal tree contrary finding page distinguished see tend rate decrease trend imply simulation predictor longitudinal spaced q effect fit case estimate package compound almost perfect interaction guide except simulation generate appropriate simulation obtain point error q record table simulation good model guide make assumption slightly approach predictor child daily presence stress child child week status education health ordinal people regression answer question child association stress cause child day pruning suggest variable health status education child interaction day significant two separate health figure plot group child health find curve day health cause child vice versa fitting predictor feedback fit simultaneously predict stress child interval stress health predictor plot tree confirm stress curve solid vary monotonically fair bad frequency stress child tend decrease study period significance specify specification parametric approach often possess important feature consistency estimate increase longitudinal datum response assume compact consist correspond constant setup treat longitudinal total collection terminal node partition node contain pt assume ij terminal ni n follow condition sufficient assume algorithm capable choose right curve regression g condition first hand independence identically density everywhere condition constrain standard estimate ensure correlation error small write u ik ik u ij ik ij n ik u u polynomial nh fu imply nh n nh fu k hence algorithm tree longitudinal cart bias covariance guide bias likelihood select contingency trajectory mean trajectory nonparametric split set selection pruning number normalize response implicitly residual trajectory pattern assumption node quite powerful situation satisfied assumption autoregressive justify satisfied within partition though robustness mean easily evident longitudinal smoothed mean terminal realistic summary parametric substitute parametric parametric model unknown may model construction base small powerful sample predictor tree quite interpretable serve purpose identifying subset implement guide grateful anonymous associate comment suggestion lead improvement cart statistical support nf national health grant previous construct tree cart consequently bias difficulty cart guide treat longitudinal use chi square tree besides predictor result square example compare effect generalize condition asymptotic estimate regression tree regression partitioning set value predictor cart split ordinal categorical sum deviation partitioning continue node test subtree lowest select extend longitudinal often use measure longitudinal symmetry em difficulty space extend cart exponential response response transform binary value function computational difficulty every node cart square deviation mahalanobis different treat longitudinal trajectory fit longitudinal low curve fit regression coefficient predict predict trajectory spline coefficient mention component multivariate
proof omit choose dominate probability one provide estimation compare term term drop subscript proof odd scalar theorem obtain regard derivative accord q n x n stand hadamard let denote index use imply contradiction operator positive semidefinite similar problem class argument density tt ii unique contradiction entry equality follow diagonal result side thus write zero positive definite consistency accord prove ni rewrite side rf definition theorem em widely nuclear circumstance basis completion various completion quantum recovery paper propose rank correct establish error importantly quantify nuclear reduction substantial provide interestingly highly concept foundation rank rank simultaneously recovery capture low structure keyword rank consistency without considerable interest quantum state idea address rank completion np nuclear convex envelope unit spectral nuclear remarkable norm property rip rip incoherence matrix recover noiseless sample later improve count employ technique develop extend coefficient relative analysis besides noiseless address plan noise study nuclear completion literature nuclear technique encourage efficiency circumstance certain high completion point recovery trace norm base column marginal prior penalize satisfied involve concrete density quantum quantum state semidefinite trace trace norm completely nuclear fact much relative room motivate possible norm seek completion support low correlation completion include special fix due observe sample high recovery propose correction nuclear minus initial estimator choice optimization scale stage statistical attractive thank lar overcome stage naturally occur particular stage enough desire efficiency lasso important broad overview interested reader refer natural reweighted trace minimize rank extend lasso however completion initial valid numerical reweighte nuclear approach matrix reweighte square minimization extension matrix transpose smoothness subproblem encourage hard constraint basis involve difficulty inspire matrix problem non frobenius adopt discuss impact correction recovery importantly mild rank correction nuclear square estimator asymptotic consistency sense ideal analyze gain insight limitation optimization key analysis interestingly recovery criterion construct correction consistency broad rank unless rank correction three improvement advantage propose apparent involve theoretical foundation set matrix optimization introduce completion basis bind correction step provide quantification improvement devoted report correct conclude relevant brief denote complex matrix symmetric semidefinite matrix nn hermitian positive endowed induce n mean n real case matrix orthonormal column short denote transpose notation adjoint cardinality I let denote euclidean denote norm nuclear almost sure respectively set basis nuclear norm solve problem relative recovered practical matrix assume reliable index basis need recover collection basis coefficient eq identically distribute control unless general weighted follow copy e notational slight express present example correlation real hermitian semidefinite entry entry zero ie eq completion integer hermitian semidefinite quantum rectangular rectangular aim n introduce operator frequently operator convenience rest arrange x n xu rv rv rx situation matrix efficiency sample model extreme nuclear completely nuclear propose generate pursuit rank estimator correction magnitude closed operator refer operator subsequent penalization nuclear bind restriction mild correlation boundedness hereafter call correction term rank step nuclear singular certainly directly serve practically lack convexity extent available achievable suitable substantially offset penalization nuclear expect low outperform penalize function theoretical support rank capture choose penalize suitably estimator generate adaptive nuclear semi penalize associate constrained optimization n x idea penalize version next iterate subgradient convex compare correction step least different noiseless noise constrain assume uniqueness though observation purpose precisely constrain correction bring similar norm correction impact rank argument use define quantity basic relation matrix estimate problem emphasize penalty sampling therefore base quantity probability easy extreme large constant universal case sample rectangular reveal derive error bind frobenius operator previously absolute choose correction control roughly degree freedom logarithmic recovery important error fix bind therefore recall nuclear theorem nuclear recovery rank case ensure simply reduce nuclear norm penalize large easy true account optimality table demonstrate substantial relate magnitude increase relate upper bound slightly decrease find small illustration relaxed attain q proportional important nuclear least c bc find validate numerical consider asymptotic step expect reveal chosen parameter true set matrix follow elegant behavior solution follow sufficient consistency recovery path sequel hermitian hermitian notational simplicity divide term extend argument least square case rectangular system solution r semidefinite nuclear simply reduce positive semidefinite condition hold consider eq assumption consistency necessary next theoretical guarantee system extensively eq meanwhile may define n adjoint rectangular rewrite semidefinite rewrite result either rectangular semidefinite self adjoint definite immediately constraint q correction step hold correction positive completion consider structure entry fully corresponding correction constraint problem completion special completion fix basis control trace constraint correction redundant thus constraint reduce automatically importantly uniformly class completion class correction rank focus correction section error meanwhile suggest choose semidefinite need replace decomposition eigenvalue conduct exactly good choice operator twice continuously differentiable correction let spectral constraint mf omit suggest ideal overlap error rank recovery error ideal guide contain rank regard divide perturbation confidence shape one need approach rank singular choose certain exchange recommendation choice particularly nuclear penalize remark semidefinite reweighte al reweighte surrogate meanwhile correction n however notable broadly speak rank correction reweighte trace norm recovery different problem adopt proximal solve admm reader sequel respectively stand least square correction short randomly diag bar take true randomly diagonal hereafter true pattern note correction consideration report whole diag diagonal point show consistency estimator possess result solution infer recovery unknown advance monitoring search different matrix diagonal entry gaussian set norm least four rank correction function plot figure decrease together attain exception observation test chance subsection covariance different initial r upper diagonal double produce produce penalty choose attain small penalty see result desire plot clearly observe initial always substantially subsection reveal good small attain search find actually consistency strategy question correction step quality report result covariance matrix rectangular report small different know estimator value take second correction function covariance c rd pt ratio e e e completion true subsection weight rank number non partial fix double among noisy observation reasonable sample low entry bring recovery completion r bar test trace besides frequently system strength row differ assumption noise randomness may quantum solution drop trace trace constraint one recovery among attain step report table besides recovery report square fidelity closeness fidelity relative htbp ratio fidelity fidelity fidelity pt e pt rectangular completion weight ml bar ml mr take dimension scheme sampling respectively word leave much top bottom partial un set large rectangular recovery ratio low meanwhile advantage remarkable rectangular rd pt ratio e e e e
model compute cp vanish obtain discard finally representative sample universe universe situation seem simple reliability base introduction share may decrease combine category spurious every eq note immediate recall end assume moment upon atomic transform class precede long provide validity model change category level change could identification prove last uniquely thus obtain prove obtain get q solve obvious formulae part formulae involve fluctuation lead thus expectation value formulae accord e c minimize note introduction base investigate accuracy remainder section vary inverse model quantile absolute range value parameter model define actual error range half determine proposition note exhibit affect affect range five belong category statistical distinction accuracy contrary rather significantly attribute fluctuation finally error deviation frequency expectation number influence five error item item rough beta reasonably low item email inter agreement account agreement chance exhibit tendency high low depend condition reliability researcher agree view accept reliability upon reliability upon rate nominal model inter item reliability uniquely relative frequency exhibit setting nominal like medical science content analysis simply measure adequate occur reliability cope prominent measure chance correct agreement vs maximal agreement way chance correction take correction compute give rise favor certain behavior explicitly assign category choice assignment whether assignment may extreme assume category distribution chance chance correction assignment thus marginally influence category population distribution hand agreement category considerable literature e hard reach consensus category certain distribution chance agreement exhibit usually produce marginal marginal distribution similar proof contrast inter contrast intra reliability conduct way group individual reliability prescribe another score category example justify reliability rare category argument problematic design typical approach define reliability common category assignment correct item reliability item inter side effect reliability state direct inter reliability simulate evaluate accuracy section author might statement acceptable close look impact set category classify use inter item independent mutually exclusive exhaustive exclusive existence usually call category implication category relative category category category knowledge sure case follow base category cf certainly assumption field e medical diagnosis upon kind item education vs formalize item true fail recognize identically item random q atomic convenience let model subscript refer choose assign item assign chance rating process choose agreement call category occur cf first model distribute seem discuss inter reliability contrast intra reliability use parameter characterize consideration actually arrive situation situation assume indeed dependent put item outcome reliability randomness setup would would subset attention belong independent imagine easily distinguished subset necessity value reliability cope return aspect inter parameter observable frequency category r expectation condition expectation
though occur finance class mean optimizer rate convergence property simulation extensively statistical statistical ambient concept area dimensional much tool literature procedure zhang penalization concave penalty well numerical property fan fan unified zhang focus dimension although appear information criterion bic provide forecaster apply often case variance non variance transform many transformation ignore heterogeneous aside estimating often important parametric explanatory finance control high explanatory positivity capable capture vary order magnitude main contribution propose iterative optimizer mean variance estimate mean simulation show outperform analyze vector form contain index denote submatrix column index norm usual extension notational matrix frobenius common variance penalize estimation procedure introduction correct probability tend literature variance jointly study extensively set solver new develop author acknowledge log study however discuss principled way resort ad hoc optimize convex estimate stationary however final optimizer estimating find maximizer solve form vector residual reweighte estimator principle likelihood generalize second cast result unknown likelihood alternate estimate mind usage result satisfy property sparsity continuity support satisfy mcp penalty produce particular scad replace iterate establish theoretical non trivial observe stop iteration selection candidate aic likelihood eq freedom compare bic simulation procedure scad scad grow relate ambient large size establish correct selection parameter satisfy far weighted assumption establish convergence second weighted estimator normal generalize least know true ccc bic bic bic bic assume conduct study sample identification precision precision variance ccc nd nd bic nd aic st nd aic st bic nd bic nd average data iid normal logarithm associate normal marginally illustrate scad iterate summarize toy perform correlation predictor already visible tends pick complex select standard previous number observe aic select complex size furthermore iteration demonstrate benefit prove important country base community country current international forecast form return country country year forecasting remove variable group category balance form ar initially lasso lars bic plot residual bin fit residual apparent variance third bin difference country log response equality variance justified variance tune select metric compare mse log table outperform mse score mse fold address high control mean exist perform solve
table temporal baseline still baseline centroid regularize visualization framework grouping temporal use past introduce version static counterpart demonstrate regularizer intend preserve area concern visualization dynamic thousand node scalability extremely significant large equip grouping human challenge expression compute algorithm gradient jacobian obtain evolutionary design dynamic object adjacency involve ordinary static smoothed quickly adjacency drop framework adaptively time minimize mse frobenius adjacency view characterize choice q affect replace variance sample static perform smoothed obtain membership estimate cluster membership refine iterate paper cut algorithm reader detail acknowledgement partially national research office grant nf xu award sciences engineering research structure evolve typically addition provide preserve consecutive human evolution network visualization set present create static belonging step penalty increase sequence preserve map dynamic multidimensional group interpretable dynamic topic recent year application science real node removal old edge development low rank detection community closely dynamic open attract visualization provide insight intuition static static typically graph representation create graph include direct scaling present aspect remain trend poor structure create visualization treat snapshot stack result visualization interpret present evolve time snapshot challenge preserve human undesirable drastically position frame may cause reference visualization involve create interpolation constrain successive way make real create snapshot static time often due static network cluster set visualization encourage preserve map draw explicitly employ regularization dynamic stability many ridge preference stability preserve map design set modify function static node belong knowledge group evolutionary grouping keep member close help preserve node belong group often evolve fashion temporal helps preserve map cause human side good framework visualization incorporate employ grouping appear work graph selection importance temporal interpretable time indicate square dynamic discrete sequence snapshot adjacency choose edge assume graph simplicity time row dimensional position visualization norm trace column summarize multidimensional operate graph snapshot multidimensional family variety refer interested reader cost stress give denote weight maintain refer confusion matrix stress adjacency distance metric calculate two weighted weight denote dissimilarity convert dissimilarity compute crucial kk force special stress stress minimize decompose independent optimize iterative iteratively stress optimize eq stress eq derivative upper solution solve note consequence stress translation remove solve cholesky convergence iterative laplacian optimize weighted dissimilarity spectral solution eq easily function aim edge short placing heavy edge remove place location view remove translation mean correspond scatter differ constraint variance vary undesirable would scale number node generalization give exclude abc easily often normalize problem differ replace degree dot optimization th small exclude constraint normalization static snapshot snapshot visualization visualization interpret regularize framework define optimize stress nod member group choose node far previous control quadratic form let define introduce grouping add map grouping representative decrease towards node group place knowledge membership position position often refer draw literature maintain preserve propose temporal decrease towards unlike assign step thus node act anchor measure measure goodness topology prevent adapt trade stability next grouping penalty give modify stress eq q stress temporal group optimize write sized similarly desire add row column I denote node stress write minimizer result solve sequentially dimension use iterate attain iterate take ordinary equation rank compute short compute show fig factorization follow back substitution position position fix invariance dominate initial subsequent back dynamic second term penalty compact define augment add representative laplacian two write final independent henceforth modify cost consider degree derive optimal zero orthogonality become denote eq center combine unnormalized problem replace obtain scale static consist close appendix get poor multiple initialization center form assign appendix initialization interior solve initialized consume solve cholesky previous dominated cholesky choose applicable graph force static ultimately decision type preference often prefer maintain entire series tune grouping encourage grouping treat rank group belong supervise optimize function grouping vary eigenvector u addition prevent eigenvector well preserve static method generate network software explicitly control stability insufficient node experiment enforce dynamic objective several criterion stability criterion stability position position spherical center probability constant scaling amplitude logarithm exclude grouping cost maximize preserve tend specific rule q journal update include localize optimize grouping way create stress minimization penalty mostly several penalty visualization modify term modification optimize without anchor view modification grouping come explicitly stability static eigenvector initialize node position second compute smoothed perform constrain method force direct achieve implementation emphasis improve scalability extremely initial apply improve applicable incorporate sort apply result support website priori compute baseline line use previous modification method compute smoothed standard propose grouping baseline regularization neither summary statistic section choice snapshot optimize static energy close group calculate centroid cost respect even learn centroid respect square node consecutive preserve mention stability goodness display depend require tolerance lie ensure choose minimize choose pt stress centroid sbm learn static mit static ccccc centroid mit spectral table one lower since grouping might method centroid significantly low centroid temporal position static employ less employ add regularizer experiment detail block model sbm sbm group stochastically equivalent I form node belong sbm specify probability cd represent probability form particular node sbm membership assign remain unchanged assign group simulate create static centroid cost method average simulation grouping regularization centroid expect group centroid although temporal static hold structure alone temporal decrease benefit average fig plot static centroid much generate cost grouping centroid slightly method combine perform centroid fig centroid temporal cost change beneficial penalty strong mask undesirable centroid temporal cost event exploratory long period respectively fig distribute decrease decrease sensible node move significantly representative temporal slightly increase centroid centroid place move representative grouping regularization centroid temporal indeed show always penalty penalty grouping towards representative increase centroid temporal cost fig part examine incoming transfer student week rank individual house private collect week break process data preferred convert graph edge convert dissimilarity divide priori affect plot use step membership time poor job topology temporal example mostly student switch blue around fall break back continuously change preference observation student large compare mean student fig respectively top row correspond
evident induce rather something layer introduce model moreover complexity mapping reduce typical toy real world set dimensionality first process obtain stack pca vary usual latent basis create level stack gps depict hierarchy top stack stack respectively dimensionality learn ard find correct hide signal one encouraging indicate truth confidence toy unsupervise learn describe observation input simple create toy stack two gaussian process exponential receive equally spaced kernel gp create sample present equally spaced randomly leave rest effect range become traditional gp layer layer trivially learn actual obtaining split deep figure suggest deep enough less assign deal control ht toy show towards five contain frame motion consist digits handwritten digit digit represent gp range bayesian hide evidence layer quality mistake decide number depth plot fig ard final hierarchy abstraction output encode whereas high encode circle many conversely dimension parent two vary output level abstract introduce bayesian process mapping approximately allow discovery hierarchy describe human motion handwritten digits variational handwritten digit experiment give evidence powerful enough encode abstract could validate improve something past effort combine gps deep pre present consider layer deep architecture experiment simultaneous allow principled model use top hierarchy lead model process future across task jumps gp set publication show variational formalism gps acknowledgement research state foundation section figure table chapter chapter chapter section gps belief gaussian model input gp variable variational marginalization model layer belief fully treatment selection justified modelling neural model popularity complicate associate abstract human inductive concept question question structure justify useful applicability around binary latent rbm stacking technique amount importance rbm deeply small seem machine core interesting approach modern much back implication model family vector almost introduce perceptron mlp certain unit relate deep learn autoencoder dimensionality expressive variant sophisticated gps complex structure powerful base lead principled truly associate rbm rbm model parent output input condition conditional role significantly rbm input likelihood dependent function nonlinear integrate analytically gp analytic integration propagation book consider deep marginalization treat might include rbm binary scale exponentially intractable similarly analytically solve maximize respect combine gp posteriori model algorithm available exploit advance stack go show variational gp stack truly rigorous allow selection applicability deep different hierarchy cascade layer layer layer place process mapping gp gp regular rbm latent likelihood automatic relevance determination ard prior place gp dynamical extend process obtain approximation latent output flexible gps gps process input store seek unobserve responsible prior draw operating input reflect covariance mapping quadratic l infinitely aforementioned normally analytically omit clarity gaussian latent elegant treating input latent employ prior gaussian process take form mapping likelihood analogously architecture three kind observe intermediate latent unobserved prior constitute input focus scenario intermediate depict generative take involved play extend deep segmentation already layer number layer crucial priori unlike allow automatic also add step automatic relevance determination ard gps weight latent switch drive introduce treatment nevertheless analytically procedure available regard e distribution node whole cascade prior step apply jensen bind variational appear expand integral expand gp auxiliary augment p clarity x able augment tractable variational free back replace joint version compute analytically break logarithm grouping fraction divergence expectation associate leave find bayesian gp output quantity see expectation involve involve output main calculation demonstrate hierarchy I
name field extensive survey find considerable develop cp still concern cp decomposition agnostic optimize outlier cope determine noisy despite existence paper calculate algebraic specifically algebraic determination rank due component avoid determine correct less sensitive noise superiority briefly basic tensor slice decomposition cp call tensor denote cp decomposition want cp noiseless find simultaneous svd determine n rw w k equivalent tensor simultaneous iii coefficient notational difference write column fix form diagonal slice component whether cp include terminology whether tensor tensor probably case retrieve notion appendix generic cp cp replace equivalent prove rescale generic element presentation reformulate imply lie span presentation correspond existence invertible correspond combination exactly one full state together component slice representation v converge calculate svd candidate mode switch mode simultaneous omit algorithm center component shift cluster link cluster center pair triple unnecessary mode g majority vote vote closeness decomposition output rank determine center belong return simulated toy input tensor slice uv uv independently independently normal normal covariance estimation detect htb numerical analyze increase figure synthetic ls emission rise emission thus suit assess physical priori knowledge show emission auto detect excellent agreement spectra fig lack negativity spectra attribute scatter thus well agreement shape identify present determine decomposition noisy tensor argue unstable component demonstrate competitive art advance outlier use intrinsic low tensor oppose method fit prefer whenever proper emphasize structure
implication work scale descent bfgs randomly variety bfgs nd fundamentally literature curvature minimum l bfgs cause variant fail curvature near shaped I plus transformation though ignore operational suited storage discard batch iteratively receive lx rx base simple rule tx subgradient prove regret remark loss estimate sublinear slow growth package implement online whole cost utilize collect file communication baseline learning step manner adaptive minimize generic square solve tx check fit newly represents forget previous emphasize datum indicate past equally correspond fit perfectly concatenation newton minimize function computation computing grow costly fortunately differently approximate analogue eq update finally q coincide standard return pass algorithm evaluate cumulative remark incorrect evaluate precisely case change set determine subsample old datum roughly previous point serve anchor fit batch expense global regret generalization learn recent subsample routine associate start store l bfgs hessian initialize descent routine summarize process check datum tm run learn batch aspect partition significantly speed gradient run parallel subsample parameter make subsample quick check finally remark reasonably acquire choice matter compute directly situation discard typical gradient descent priori bad bfgs consider ad necessary order appropriately rank ad click time display combine classify property display search query acquire engine publicly learn unique ad description display user ad position ad depth depth normalize click training click rate build vector go regard experiment compare bfgs technology randomly high randomly choose one compute auc portion million basic conjunction perform approximately winner win exclude future set relevant model auc bfgs implementation relevant learning quickly heavily optimize utilize robustness job failure gray ccccc cr bfgs seconds auc sa accelerate iterative bfgs sa bfgs induce derive large less speed set capture interaction query ad frequently well accuracy bfgs relevant technology bfgs sa amazon bfgs add measure second auc auc comparison feature cr model achieve bfgs cr time fast furthermore sa bfgs achieve auc likewise speed enable require compute note speed bfgs tie gain reduce reduce bfgs force expense specificity tp gray gray ccccc cr cr bfgs bfgs second rapid model use new l bfgs combine question ask model explore discover sa suited underlying behavior market pricing change term usage sa l bfgs dynamically batch datum variable size furthermore statistical aspect bfgs bfgs affect learn new care flexible technology context relevant company friend family work company list five united microsoft windows internet payment system lead expert political micro degree university master science engineering experience technology innovation engineering management team microsoft massive storage microsoft team microsoft principal microsoft microsoft ever window lead range science science recognize machine publish paper microsoft yahoo spend decade building system processing microsoft work understand yahoo focus ad relevance school focus recognition incorporate massive semi machine master electrical engineering investigate broad question people twitter social tie publish cover medium national public work microsoft master mit work medium laboratory medium receive applicable medical imaging principal experience performance load balance build optimizer computer mit present enable rapid training example million sa batch bfgs perform balance old iteration
mine practical widely theoretic express observe another categorical pa degree association one predict evaluate redundancy variable call feature selection correlation filter relationship variable variable e exclusive neither capable handling way desirable computationally cart recursively subset tree multi variable theoretic mention select tree select theoretic redundant result redundancy separate set decision split split redundancy redundancy problem tree eliminate redundancy new similar select eps eps number make return satisfy order incremental refer select first relationship limited categorical analysis limited categorical tree section splitting leaf limit node level depth position decision tree level node level child md denote decision condition md decision refer md md show md tree level md instance maximize node mutual calculate md tree feature add redundant information redundancy introduce recursively node g tree build belong unless coefficient produce large select splitting framework regularize sequentially predictive expect contain non redundant advantage svm penalize form interaction extract objective redundancy node feature tree subset conditionally terminology call practice usually unknown selection feature instance induction expect commonly subset preferred preferred criterion classifier strong classifier capture may subset uci empty select add nb feature accuracy classifier nb stop add feature rf continue feature indicate add nb contain validate classifier prefer capable high subset regard instance class instance auto c cc cc cc cc cc c c auto uci benchmark database vs implement regularize framework tree eps eps rf feature selection set accuracy rf pair subset replicate well feature level denote subset show trend evident regularize ensemble loss feature note present tree ensemble though ensemble rf rf tree capable capture interaction relatively differently rf accuracy competitive use rf predictive capable extract performance select next ensemble simplicity datum set equal rf versus backward elimination automatically rf feature randomness run rf rf competitive optimum still cutoff computational efficient second capability model forest quality subset strong computationally deal categorical numerical variable etc provide practical acknowledgement enable regularization gain previous forest tree easily tree tree select subset strong categorical variable interaction effective training set consist instance variable significant role curse interpretability framework splitting node produce gain subset single build wide forest handle categorical miss scale etc many background tree propose regularization regularize regularize section demonstrate efficiency extensive experiment work feature divide embed filter
ap proof make extensive proof proof theorem condition measurable lemma arbitrary sense proof c mh deal condition obtain recall hence geometric ergodicity spectral fold iterate mass every x arbitrarily small relax statement assumption follow independent cumulative associated suffice ar consequently take small two example g generality q g satisfied calculate lemma computation common scenario intractable markov chain facilitate fail geometrically ergodic consequence reliability phenomenon typically tolerance bound ergodicity condition alternative asymptotic involve pure jump sufficient condition variance bound reversible provide bound geometric ergodicity mixture reversible kernel refer branch parametrize computation approximate increasingly diverse likelihood computationally prohibitive standard prior metric artificial commonly approximate computation value interpret slightly abuse simplify draw general version standard sampling avoid carlo methodology whereby iteratively stationary sum start latter ergodic converge geometric total iterate mt property geometrically ergodic motivated ergodicity approximate reversible theorem bound integrable course reversible bound markov seek control bounding geometrically situation use partial problem identify negative reasonably mild geometric geometrically addition condition ensure property meet g prior everywhere tail quantitative example background bound geometrically ergodicity reversible markov spectrum operator restriction order bound equivalent ergodicity gap geometrically closely aforementione spectral gap variation eq accord common accept reject variable densitie common dominating g case pseudo kernel parameter model smc evolve involve fix kernel simulation readily output evolve denote pseudo auxiliary balance verify involve fix auxiliary variable kernel stop concern value behaviour demonstrate nevertheless fail propose move locally suggest may subsequent computational effort particular place successfully ergodicity hasting proposal many relation provide assumption however output output kernel share write represent extended g satisfie infinite collection proposal broad common proposal collection bound geometrically mt follow addition far get move away proposal tail decay exponentially mutually exclusive hc theorem simplify general reversible indicate geometric ergodicity coincide lack bound appendix concentrate reversible irreducible invariant kernel measurable bounding concern poor irrespective value former implication upper almost almost uniformly establish uniformly ergodic uniform ergodicity little motivate correspondingly exclusive computation indicate autoregressive cover kernel analogue far asymptotic systematically geometric condition order mh p condition proposition control manner different difference dramatically condition condition regular contour obtain corollary decay super tail decay exponentially everywhere satisfy euclidean geometric former already argument state q np ergodic corollary acceptance suggest cover countable collection proposal modification state could conjunction mh mh mh invariant metropolis hastings mh mh justify geometrically ergodic reversible invariant implementation qualitative effort unbounded iterate irreducible q quantity rejection sampler gap require much three proposal poor initialize little mass chain compact interesting value comment support improper success series give proposal need rejection sampler pn rna supplement qualitative ergodicity investigate number consider truncate explicit spectral figure spectral gap range gap exponentially gap even relatively reversible converge slowly reasonably considerably p figure stable former require simulation result obtain illustrate well behave p appropriately need region figure constant tail might infer tail indicate appropriate inference behaviour mean value kernel informative n implement parallel core device graphic processing hand exhibit biology approximate set pure markov q consumption death pure historical article straightforward simulate process inter jump exponential sophisticated stochastic observation discrete approximate use transformation kernel horizontal correspond two deviation add horizontal line estimate diag posterior sample tight strong give also run give indistinguishable inspection partial sum reveal figure estimate sampler percentile use sample sampler estimate seem correlate perturbation kernel sum practitioner estimate second use p rejection proposal one around average reasonably exhibit tail sum reveal difference figure illustrative produce chain practical determine easily geometrically common condition hold bound geometrically ergodic variance bound verify systematically course
avoid reasonable nr observe e variance ml implementation em treat nuisance major package ridge newton iterative iterative simultaneous ml restrict g information ml variance inverse estimator realization estimator denote replace say detailed section consider effect full u section variance square mse variance assumption trivially statistic exact q denote chi freedom estimate statistic empirical e version estimate covariance notice eqn consequently make simultaneous g prediction estimate statistic statistic generalization degree freedom df nan approximate fisher degree freedom denominator expression df given see degree df approximation kk variable unknown estimate natural estimator k taylor estimate estimate evaluated estimate component provide variance gradient estimate sr section q consequently denominator degree freedom eq indicator freedom unitary argue say say mse matrix source bias comprehensive solution consequently mse mse prediction simple take square get variance covariance mse get mse covariance matrix second derivative natural notice depend covariance argument blue justification however point effect additional mse expand expectation assume order ignore degree suggest estimator nan scale overview make statistical conventional improvement effect suggest effect notice derivation distribution unique solution modification consider expression simple practically environment present straightforward get covariance parametrization matrix approach development education science grant valuable feedback banach international discussion implementation assume matrix z inference leave side particular assume shall expression matrix rule inverse symmetric far whenever equal get explicit present mse linear predictor give I decompose sr ji ij base xx ii covariance property mse also component empirical covariance sr ip scale eq compare expectation estimate case general statistic however derivation suggest component consequently suggest version use variance sr ip present procedure implement iterative step solve system calculate iteratively dimensional diagonal stop ml estimation evaluate estimate asymptotic otherwise iteratively rank q log estimate fisher information estimate similarly final solution detail chapter al completeness component variance component g define system equation quadratic iv easily evaluate vector form use solution equation denote eq define uniquely unless inverse appropriate vector biased overview statistical hypothesis construction interval fix effect simultaneously mix fix mse rao research mention experiment meta analysis medical well device tolerance interval application uncertainty recognize sort mixed improper usage still problem brief fix effect conventional simple pointing appear improvement generalization detailed technical description solution mixed
broad application private add list choice initial publish know application student plan enter college year year year area mathematics computer general science year examine herein student ten preference university college college college university college college clinical speech human tr college college mi primary college college european college place allocate basis leave student seven nine subject leave six leave overall allocation degree solely basis point choice accept degree publicly leave subject minimum entry requirement score leave college receive medium part student leave fair especially gain leave explore choose coherent focus choice people pick basis factor behaviour example application table illustrate influence health science choice appear course application include degree first choice area close km year education science college application system report review recommendation concern research report examine exploratory technique without preference order profile bayesian subject student fitting paradigm cluster manner additional mining college combination select oppose college influence choice degree influence intrinsic matter look student college degree ultimately factor degree choice year college office ranking college degree herein appeal account offer preference degree discover similar bayesian paradigm infinite nonparametric operate nonparametric process item index suppose partial ranking item denote notational partial ranking construct yet pick interpretation item let arrival item let arrive ml ranking unfortunately alternative item see joint probability ranking latent exponentially result instance take joint km inference gibbs sampler algorithm occurrence ranking differ article author full ranking partial item gibb derive choice item among partial limit item observation degenerate define unobserved item sampler update item unobserve allow ranking ad hoc infinite limit capture underlie space choice item college gamma gamma marginals atomic gamma construct chapter distribution assume poisson intensity concentration know evy intensity homogeneous l evy intensity play gamma interpret correspond describe extend arrive atom arrival item constitute depict ranking appear ranking dirichlet ranking partial ranking rescale partial ranking inter arrival ranking variable give arrival g ranking associate jump evy via theorem word deal nonparametric infinite necessary condition strictly generative subtle prior simply index infinitely many natural atom location impossible priori mutually item infinite sum ill approach satisfy sure well atom purpose define explicitly observation variation normalise measure denote ranking list unobserve choice introduce auxiliary independent probability list auxiliary cf appear list appear denote unique item among list finally occurrence take partial ranking laplace moment completely formula decompose jump conjugate derive conjugate posterior process random mutually gamma k measurable denominator formula proof gibbs sampler conditional analytic integrate except leave latent versa simply joint partial ranking rescale keep proceed construction completely simple sampler generalise completely propose rank consist nonparametric augmentation derive school ranking preference dirichlet denote concentration stick breaking nonparametric describe th atomic specify atomic use draw process unfortunately share come atom result ranking mixture degenerate large consequence fine preference ranking motivate dp different atomic dependence gamma atomic connect leave marginally root gamma atomic form measure shall consist formula law mutually gamma gamma jk define degenerate number atom inspire coincide define jk jk jx jx jk addition neither since law marginally law compactly model observation interpret control indeed large large cluster ranking cluster different group construction induce atom number qualitative difference hierarchical dp law marginally gamma marginal law individual measure gamma process alternative induce atom marginal share measure control infinitely share purpose dependence component atomic dp well allow similarity hierarchical dp generalise normalised observe rank trivially develop observe occurrence item list cluster indicator similar observe atomic atom denote ranking process atom jk jk mutually process atom dependent probability mass poisson sample work sampler derive sampler mixing g scale total unobserved update backward recursion independent jk mass remain distribution allocation model finally dirichlet valid collapse gibbs however cost average scale college office year flat hyperparameter run gibbs order point posterior sampler decrease cluster matrix record show student student cluster member etc science business business outside outside business finance engineering music engineering business degree determinant student table cluster besides concerned degree heterogeneity area college see number location cluster correspond science outside inside college college science university college college business software college finance university college college university college biological chemical software computer software net university city computer application university city college science national college business college college information college science medium student pick rather area requirement determine available place phenomenon student pick law separately therefore consideration look variability student quantify normalise average normalised cluster normalise table indicate variability choice within normalised student popular teacher education member teacher education variability student student three popular matrix reveal reveal belong degree require select specify major observe fairly exhibit share name marginal respectively correlation rather college often appear top relate variability bayesian base random generative biased atom derive consist nonparametric model college cluster structure observation choose mainly gm generalise generalise distribution discrete whereas component rating flexibility capture strength generalise precision rank generalise preference cluster coherence term support primary factor reflect intrinsic matter aspect third location far degree education live home extension propose completely measure preference location however totally density leave careful carry probability gamma process formula e poisson apply iteratively use idea calculate numerator denominator already theorem numerator technique evy poisson process divide numerator denominator characteristic functional iw e iw see corresponding measure gamma note update atom locate w iw complete proof posterior sampler easily exposition process atom general evy satisfie lead infinitely law evy naturally homogeneous proof hold evy probability latent moment exponentially evy give law mutually evy intensity z z kn explore nonparametric generalised gamma beta generalise gamma form gaussian process include intensity form recover stable moment generalise gamma gamma generalise homogeneous recall ranking consist rating one partial ranking l evy intensity
close two geometric good utilize parameter cover achieve entire facilitate simple error computation proceed eliminate high peak dataset reduce input greedy minimizing use could linear harmonic currently length parameter histogram nonzero space nk x matrix tn construct entry construct transform inverse j symmetric part lead discrete cosine integer nx use analytical recurrence see appendix qx qx combine recurrence relation approximation refer chebyshev chebyshev polynomials quadrature order convert periodic making substitution serve purpose shannon kernel analytical straightforwardly present chebyshev apply rate qx c fourier absolutely qx schwarz analytic slow series also inferior convergence series still list chebyshev approximation another strategy obtain problem pca care need perform scale conjunction multiple extensively hereafter rf pca dimension learn memory multiple kernel operate convergence rate monte suffer curse dimensionality decrease generally make suppose another aspect pca bring unlabeled accuracy amount dimension pca discriminate strategy pca training fully need core high computing unable datum singular rf rather center loading time memory eigen transformation pca dimension large eigenvalue denote ki td convenient hessian regularization regression compute core td retain pca use ridge n order compute need additional involve read sophisticated perform separately l rf needs transform still term td retain load memory rf randomization square worth core square regression scale extremely well core inverse hessian need obtain scale need classification imagenet ridge baseline conduct challenge imagenet http benchmark subtle observe conduct medium compare imagenet approximation intel ghz gb core approximation cut match truth subsequently match plus segment segment well skewed carlo quite trial seed trial seed final approximating perform reduce dimensionality kernel type bag sift gradient feature via linear svm library empirically library produce context dense feature approximate optimal approximate case htb chi chi skewed chebyshev pca chebyshev direct kernels htb c chi skewed chebyshev direct chebyshev pca exp recognize image generate pixel ground truth pair art train class create database image tractable conduct multiple different descriptor scale sift foreground color foreground avoid fair processing step report high regression latter yet avoid gradient since tune convergence potentially bias average trial different htb chebyshev direct direct chebyshev chebyshev nystr om ap sift foreground show imagenet million author primarily descriptor spatial pyramid could calibrate calibration output score class htb chebyshev pca chebyshev direct nystr conclusion go conclusion go conclusion go conclusion conclusion go go conclusion go conclusion go conclusion go conclusion appendix recurrence concern symbolic software compute b observe symmetric odd argument get solve cl integration identity trick cl k z acknowledgment author thank author author like author author thank author text text text text text text text text text text text text text text text text text text geometrically rate lead segmentation besides core principal expense factor perform jointly unlabele testing datum improvement imagenet method histogram tool visual descriptor recognition utilize histogram descriptor extract use near machine descriptor score metric histogram paper derive pearson test utilize object performance current datum million thousand comparison often facilitate testing approach devise represent two top fourier kernel approximate transformation nystr subset comparison follow principal pca represent nystr hard functions regardless able nystr om partial rf metric paper elementary enjoy term error previously directly translate classification rf kernel analytical chebyshev slow linear chebyshev derivation paper record research rf generate theoretical carlo eigenvalue fisher kernel imagenet large classification influential dimension consume nystr hence crucial max code spatial informative descriptor whole image undesirable recent propose successful semantic
validate shaped spatial cluster detection empirically method spatial scan detect spatial point similarly straightforward limit want proportion boundary p zhang foundation international partially identify shaped ratio measure discovery spatial discover shape heavy computation large many detect intensity compare specifically collect location follow alternative region location example scan scan comprehensive multidimensional develop spatial scan scan scan windows location test window window cluster modify base spatial scan methodology powerful power circle ellipsoid approach issue control false discovery fdr shape scan every individual region spatially method extend adjusted discovery take spatial make test maintain detect shaped method internet detection idea form spatial novel variability test threshold help maintain specificity comparison fmri scan multiple testing fdr identify spatial organize derivation binomial poisson normal validity study approach fmri discuss variable hypothesis alternative objective success assume resolution detection expect region location maintain balance specificity sensitivity detailed formula statistic distribution justify set discuss idea location presentation column sequence I vector aggregation likelihood theory likelihood regularity unknown denominator instead derive parameter distribution binomial shape square circle square illustration calculation next independent straightforward introduce notation leave detail k ij easy straightforward adjustment eq sure small hypothesis median mean hypothesis hypothesis formulae approximate test eq alternative ij statistic fdr introduce location variability choose cell five point variability circle circle circle circle mm mm circle circle circle circle mm circle circle circle region close identify average test consecutive threshold monotone region increase region exclude variability similarly variability threshold average neighborhood high carry location test statistic measure arithmetic k ts illustrate certain condition surely region inner non boundary total spatial region boundary black fill gray rectangle fill gray black draw rectangle black fill gray black draw black draw draw fill gray black gray draw fill gray black gray presentation show normality signal strength proportion theorem give justification outline statistic relation remark statistic purely region large threshold substantial part one boundary boundary variability good chance scale large depend method tend show scale scale even paper simulation aggregate method region likely false region scale follow use window simulation single control fdr scan save manuscript binomial example provide specificity simulation times shape region shape shape datum cccc shape shape percentage paper real location survey application signal success method measurement include specificity percentage sensitivity percentage pixel average number show alternative increase sensitivity specificity variability increase variability specificity increase dramatically meanwhile sensitivity increase cc cc shape shape specificity sensitivity specificity specificity sensitivity specificity std std std std std std std cc shape c triangle specificity specificity sensitivity specificity specificity sensitivity std std std std std cc shape triangle specificity sensitivity specificity sensitivity specificity specificity std std std std std specificity select control discovery use false sensible level lead sensitivity show specificity decrease sensitivity alternative specificity scale specificity spatial triangle performance three shape sensitivity increase even shape specificity spatial scan complicated detect calculated frequency detect signal weak chance detect region probability map shape almost slight effect corner shape identify provide detect adjust shape correctly identify identify relatively small strong signal figure similar map detect scan default circle scan excellent shape bad shape shape subsection window adjust alternative success local variability adjustment
bring series na complex interaction achieve purely na case still probability empirical attribute dirichlet implement partition combination depth criterion identical attribute whole classifier decision letter splitting classifier member cause give vector tree bag show classifier replacement change index object include thus build subset binary cope one fit introduce reliability tree view non attribute split great comparison threshold attribute possible category cart include effectiveness usually measure retain stochastic implementation correspondingly attribute category except two contain generation split classifier gain split building way subtle interaction usage attribute beneficial accuracy score class become less contribution final increase imbalance reach point whole vote large develop assumption equal rare cope enforce divide count leaf object bag obviously misclassification might raw score voting package function require pass version explicitly pass predictor frame interface fit find class aside even object false model confusion prediction specie truth add raw score r bias test external reliably assess accuracy bagging sort internal train prediction build classifier object e subset idea originally use trivially bag random calculate object confusion forest access raw prediction data score appear bag prediction use ensemble also bag moreover importance information misclassifie importance attribute attribute also attribute evenly even marginally relevant attribute ensemble call process element model importance length width diabetes cccc forest forest cv random forest tree error deviation repetition testing build range next minimal verify subset compose classifier performance assess repeat time cross validate algorithm forest good optimisation target depth selection give lead suboptimal significant approximation forest yet show practice variability forest almost use attribute certain attribute importance cc speedup speedup importance mean repetition consecutive triplet region sequence align lie st describe signal model spectra thus attribute signal interval frequency band class peak importance obtain case expectation attribute site location score band interval differ qualitatively agreement forest model completely possibly high code repeat make training I collect time certain object scale object forest importance fast even slow complexity usually
euclidean together q proof lead desire distribution quasi minimax optimal contraction result greatly work focus scalar difficult naturally vector contraction situation dimension another interest restriction explanatory resp principle conditioning ill one conditional finitely mass continuously estimate observation analysis brevity direction future research note resolution quasi approach investigate bayesian typically frequentist implement usefulness bayesian able avoid finite frequentist analysis instrumental model nonconvex difficult obtain guarantee globally optima interest result nontrivial since estimate pose stochastic major author department economics mit suggestion constructive comment associate anonymous improve quality assumption aid aim develop quasi instrumental asymptotic generate induced gibbs prior slowly derive contraction bernstein von type result distribution greatly triplet scalar random instrumental much may unbounded instrumental variable structural interest form potentially form statistical challenge difficulty ahead attract interest structural ill pose equivalent operator identification though hilbert schmidt integrable value pose roughly method involve assumption however frequentist perspective theoretical bayes quasi bayesian posterior approach bayes neither put prior likelihood quasi moment estimate moment nonparametric quasi put value formally call gibbs proposition ahead quasi build interest finite grow dimension role ill choose basis space wavelet basis treat smoothness old space unified series setup condition contraction attain contraction state standard asymptotic normality view bernstein von bernstein von parametric quasi posterior expectation attain finally specific quasi bayes estimator attain minimax closely work form assume gaussian gaussian conditionally posterior study distribution clearly present largely differ fs speak tie tie slowly largely different present estimate comparable paper unified framework scope detail prior differ conditional restriction unconditional moment restriction restriction restriction approach importantly neither rate cr derive contraction comparable formally derive important distribution notably contraction structural contribution paper considerably asymptotic property bayesian bayesian main construction suitable ill alone sufficient contraction additional theorem analysis ill pose stream research topic mathematics literature however substantially finite error prior rate fs fs fs result prior hence contribution build upon bc b bernstein von cover account none paper inverse pointed condition relax identification approach organize quasi bayesian main contraction normality technical limitation contain scalar z distinguished population transpose ratio let denote usual lebesgue denote let function norm denote denote singular value outline quasi say older contain conditional equivalently robustness assume proper available instead use quasi let n candidate quasi maximize structural however unknown instead suitable series consider use quasi proper put shall prior grow consider another material basic say approximate space become prior subset span prior widely quasi call quasi distribution quasi proper contraction posterior intuitively posterior rate contraction correspond distributional normality framework broad statistical page foundation gibbs open line research interpretation quasi posterior borel suppose q divergence proposition show quasi minimizer posterior posterior optimally average measure kullback leibler constant typically see sake take positive reduce provide estimator quasi integral attempt restriction argue empirical interpret posterior quasi characterize purpose space concentrate estimator form basis begin state b generally add file fix sufficiently j wavelet appendix basis notational convenience convention span denote j property basis base modification however wavelet suit periodic p c g thereby quantify ill ill pose definition mild ill severe ill pose density analytic ii example adjoint eigen basis case ii trivially assumption ii self adjoint slack study convergence past denote satisfy minimax rate pose moment bind pt replace c determine class readily possible assertion borel construct put exist converge pt importantly contract rate page line hence sense fast possible ill pose pose ahead indeed contraction quasi baye fouri two assumption positive satisfy sufficiently cn ns nb nb I suitably bind counterpart ill nj abstract section g p ng b ns contraction arbitrarily satisfied constant ill attain attain contraction bernstein von type radius prior p contraction j r fully satisfactory appearance ball state page dimensional proof work effect n j suitably devise contraction w establish sufficient condition condition ii see quantile think process lebesgue measure pt w bernstein von result state quasi normal center type approximate reason add nature quasi model center coincide approximate bayes satisfy gx c file thm page typical goodness estimator hellinger uniformly bound need three typically relative indeed convergence n estimator attain rate contraction convergence isotropic prior choice notational think ill suppose isotropic lipschitz qx b rx rx e kk take product isotropic sequence n file class prior construct grow lead minimax contraction rate isotropic know arbitrarily case isotropic convergence proposition case take product isotropic appendix file proposition play role grow thereby abstract cover deal allow place prior shrink grow proposition method slowly grow upon request supplementary material approach sharp contraction lemma lemma preliminary contraction proposition notational convenience technical characterize variation center multivariate increase dimension j p ir definite ok eq quasi contraction rate take generate theorem dependent empirical sequence q understand context tb ib last independent posterior distribution eigenvalue likewise event construct lemma iv db c n n nn n
problem link collection relation role link limited link prediction reveal type connection tensor jointly address latent factorization bayesian overfitte chain monte conduct world demonstrate improvement exist art relational become problem analysis link concern predict unobserved link pair observe network link via explicit relation membership like share interest develop treat ignore dimensional interaction predict multiple relation relational network define overall object pattern object illustrate left figure social network among link multiple unobserved indicate task miss refer link pattern fine network relational capture improve include therefore factorization consider factorization address challenge ignore previous challenge factor characterize object feature latent factor relation capture correlation among reveal distinct quality sparsity usually cause sparse employ treatment placing handle moreover parameter conduct world relational prediction accuracy effectiveness follow briefly introduce problem fully carlo experiment conclusion link entirely network factor stochastic relational link extend factorization model probabilistic task concern strength link literature link task link address several enable jointly collection feature entity object type multiple impact relation prediction factorization attract lot attention mining community use web email communication tag tensor present propose multi infer factor object tensor multi pair another status involve type relation pair tensor observe information relate relational predict link show link link pattern factorization observation factorization factorization unobserved assign latent pair object relation learn three factorization modeling introduce generalized term part tensor mean cp tensor factorization incorporate logistic map miss obtain rewrite sigmoid somewhat prediction datum tensor usual bayesian impose independent gaussian latent factor type latent place place predictive unobserved link pattern condition relation interpretation modeling link pattern specific factor correspond different example company less interact type distribution average constant value hyperparameter model minimizing follow regularize error factor inference tr predict link simultaneously however limitation map choose average ignore parameter limitation manually validation treatment estimation large solution employ distribution hyperparameter predictive observed generalize miss model multivariate consideration hierarchical bayesian conjugate precision precision inverse latent still latent multivariate subsequent wishart precision wishart degree freedom relation w weight indicator contain representation follow denote conjugate specify tune hyperparameter change bayesian next predictive equation resort predictive draw whose proposal gibbs variable parametric gibbs partition sample iteratively initial apply derive predictive conjugate make efficient procedure rule condition observe sample follow gaussian factor distribution form latent hyperparameter k p ki nu u type influence object still parametric form q conjugate gamma hyperparameter simultaneously conjugate form wishart form latent mean denote hierarchical relation choice complexity efficiency eliminate adjustment distribution initialize sample desirable discover pattern real multi discussion examine propose behave world relational country relational social consist extract relation among relation evaluation construct tensor country dataset international multi relational sharing site interact video network construct propose link prediction bayesian probabilistic gibbs latent factor factor obtained compare art relational nonparametric relational object latent relational slice tensor implement report examine relational experiment e set use auc robust test link repeat report result model auc performance model dataset posterior sample good show
regularizer assign noiseless noisy scenario snr scenario denote height output initialize iteration block terminate threshold art basis inner loop besides omp noiseless situation pursuit index mean noisy rate rate define successful successful trial experiment trial experiment ram sparse situation show rate number identical vary zero block intra varied trial fm fm omp speed block five locate zero block generate uniformly snr estimate support fig although slightly advantage well fm fm even outperform fm fm intra block wireless body area network health branch compress raw binary compressed sense energy consumption algorithm ica fidelity clean decomposition iii sense column compare recover cosine dct accord basis reconstruct original raw accord measured fig fm slightly recovery fast speed fact fm clean decomposition ica experiment notice ica recover reason coefficient sufficiently sparse recover propose life application outperform close fast among algorithm fig liu fan fu technology recognition laboratory national china com sparse measurement structure signal exploit intra sparse show fast derive much scale sense sparse intra marginal likelihood recovery signal find rich structure structure widely structure refer case nonzero cluster location information block zhang rao sparse superior recover block fast implementation likelihood thank framework structure intra block correlation conduct synthetic exploit omp however much throughout bold symbol compute build follow mean suggest gaussian deterministic confidence block capture bind optimization extend bayesian compressive sensing index rewrite I rewrite ii rule relevance structural investigate eq unity note require noiseless global cost global correspond regularization converge minima largely reduce although regularization correlation recover signal transform via transform coefficient may zero block regularization fm ar
change dynamical iteration method utilize study converge see utilize change text eq eq static horizon nothing show dynamic evolve standard euler simplicity euler approximate taylor become graph compute initial k ht propose method expense product linear could use set sense euler simply method study change flexibility beyond brief beginning relationship time hour day implication time wikipedia hour hour forward euler convergence change hour essentially equivalent converge minute iteration power precede change interval base smooth jump average investigate time time give rise reference extract score instantaneous rank summary average experiment rank node occur intuition normal page large topic news popular time figure american score frequently help euler series trend omit wikipedia month twitter euler period unchanged wikipedia copy database page wikipedia page page page remove wikipedia page hour raw page wikipedia graph aside vertex indicate degree dynamic page degree page visit graph generate seed extract tweet tweet represent use tweet vector represent tweet month dataset demonstrate effectiveness adapt page external provide insight page help intersection top j operation identical wikipedia similarity static cumulative measure combine external appear produce something top page reveal ability change clear find actor medium result demonstrate effectiveness identify page external page magnitude cluster series page similar trend page share pattern amount popularity tweet temporal adapting dynamically perform step ahead move average specifically exponentially series twitter twitter validation predict percentage tweet predict dynamic evaluate informally time time shift bottom approximately stationary compare across time series information forecasting evolve external importance treat incorporate change utility use predict evolve graph external evolve adaptation capture change external interest influence node generalize converge external stop effectiveness evolve twitter external page important graph variety science bioinformatics many study node something execute parameter model pick node depend chain stationary govern graph variation instance study dynamic graph closely utilize quantity
paper mention derive e provide approximation conditional moreover mn n recursive formula matrix define k tt taylor expansion diffusion neighborhood symbol respectively satisfy eq one go reason asymptotic theorem realization sde satisfy additive reduce emphasize evaluate one computational viewpoint efficiently van alternatively subspace case de preserve alternatively formula equation make evaluation predictor determine discretization prescribed error tolerance performance estimator exact order estimator adaptive histogram limit observation different distance length equation estimate respectively estimate initial value addition van pool respectively multiplicative parameter first illustrate moment simulation error approximate moment exact one table rate realization euler linearization noise simulation time take way time allow explore period interval distinct estimator set exact conventional large parameter signal sampling reason drift provide estimator parameter period increase estimator shape c c l l ccc r r r average estimator nh respectively compute ii estimator calculate specify estimator histogram estimator compare figure decrease u negligible adaptive usefulness available c cc r cc r table order r cc r r r vi equation conventional example order h respectively figure histogram confidence example estimator negligible thin show bias approximate conventional unable illustrate usefulness shown happen insufficient number adaptive accept acceptable explanation accept step accept ten modification method estimation complete moment scheme time observation asymptotically linearization scheme performance simulation simulation order satisfactory stepsize discretization observation decrease higher adequate reduce complete distant estimator also sequential numerical simulation international centre physics thank support work diffusion theory inference time vary covariance weak differential equation correction diffusion process business statistic diffusion method http http continuous linearization time volatility financial market linearization bit discrete noise letter linearization space simulation differential equation linearization comparative equation normality system estimation overview review third identification application water nonlinear dynamical system linearization statistical process comparative stochastic process time nonlinear stochastic equation versus kalman taylor dependent multivariate example compute length c example compute example sampling period example period interval c period n axiom conclusion corollary criterion exercise theorem remark pc pc e mail modification conventional quasi method diffusion convergent moment result approximate likelihood asymptotically distribute bias mention enhance estimator intend situation observation distant process currently intensive basic except analytical decade method focus grow literature rao al deal estimation series simple estimator derive additive discrete typically euler rao linearization approximate number increase method observation enough estimator display performance due simplicity modification reduction useful base euler whereas taylor expansion result specific sde demand second affected expansion propose estimator approximation moment mention observation completely observation bias finite observation modification conventional diffusion orient converge two consecutive sample approximate diffusion decrease asymptotically approximate detail new approximate enhance estimation give distant typical practical example local linearization present section implementation increase right continuous family process function mm smoothness existence uniqueness bound process kt sde quasi denote diffusion asymptotically quasi maximum likelihood paper rao recently k k k kt pseudo prediction inferential consideration want contrary time impose completely difference quasi continuously growth equation approximation weak sense positive definite choose quasi sde et sense mh stepsize accord construct likelihood euler linearization numerical conditional quasi rao left side give pair discretization go clearly include particular since design term first conditional weak imply estimator asymptotic go next deal time sde realization sde sde k identity h h k h jt moment imply first sde sde follow underlie sde sde mh assertion exact zero control criterion clear assertion mh sde decrease deal sde observation sde maximum expectation probability realization sde sde trivially mh mh space sde sde assertion complete worth approximate convergence state order restriction order close time measurement restriction consideration designed quasi study eq generate sde sde index error loss minimize
use multiple target retrieve via pathway pathway classification reweighte recursive elimination select pathway associate phenotype cutoff improvement auc compare test confirm via hoc reveal auc far individual figure stability b stable comparable schmidt breast cancer interestingly consistently show assess et behavior compare network method retrieve signature subsequently relate drug target signature associate knowledge previously interpretable signature test much affected integrate disease restrict target candidate disease additionally integrate gene target signature show association surprising rank disease filter diagnostic google gene expression absolute score show often disease phenotype significantly classification signature stability dataset moreover yield signature clear additional integration disease gene could enhance nonetheless improve performance signature stability appear effective integration disease disease significant computational might candidate consume machine algorithm mechanism include school like provide gene area roc curve figure fraction cv stability si signatures disease gene know drug effect integrate protein signature gene overview dataset cancer year schmidt breast cancer year survival cancer non cm fr gene signature towards well decade purpose important obstacle make signature biological purpose grow integrate molecular protein selection discovery compare reveal well signature moreover disease information candidate disease target decade gain goal reliable molecular patient effect avoid reduce drug cost diagnostic signature copy entity signature gained take diagnostic signature group patient predict patient forest combination nn retrieve gene signature interpret recently try pathway annotation interaction review fr hope make signature stable signature diagnosis centrality protein interaction differential expression association disease cancer rule error svms rank construct conceptually previously hyperplane pathway pathway extremely signature investigate gene target disease gene candidate gene breast dataset repository cancer normalize normalization carry clinical consider breast cancer cancer clinical treatment residual except et assign describe al protein denote consist differential gene suggest protein web page link expression gene node apply filter rank mapping use rule upper cutoff margin disease gene disease compute employ candidate proximity annotation space test combination disease gene rank well go go information information light go propose disease annotation addition rank gene rank call major tf target associate fdr cutoff target conduct disease tf gene bind human sequence gene retrieve
different change prescribe normal estimate quantity unknown magnitude vary procedure estimate change stop detect time window length residual detection choose alarm characterize average time delay detailed accordance detection detection run upper eq mean unit pdf cdf normal accurate even exactly change target without derive statistic I ask ask good statistic use approximation numerical quite provide convergence challenge characterize aspect approach early multiscale closely geometric characterize favorable multiscale particular magnitude coefficient asymptotically datum lie well depend curvature leaf node approximation result suggest approximation number estimate appendix complete show projection close counterpart miss omit denote notation coherence basis white gaussian constant q bind bounding theorem theorem show first first demonstrate detect tracking intrinsic space vary change method average metric denote distance parameter figure horizontal vertical percentage well fraction fairly consider ease study observation tracking demonstrate able coarse tracking slowly vary intrinsic identical tracking http figure line union sign past sample color recent dynamic keep parsimonious curvature track maintain stable leaf meet study intrinsic ambient signal phase element correspond spaced change entry tracking compare intrinsic fig demonstrate dimension however small error significantly force use tolerance approximation assumes distribute gaussian close numerically verify generate mc trial track apply evaluate theory different comparison mc miss mc miss mc delay jump subspace delay magnitude delay threshold employ threshold detection delay track delay delay subspace subspace single subspace c single demonstrate http edu l display define minimum pixel streaming state anomaly figure background image shape make detection base background detect ease intensity consistent demonstrate detect present accomplish far even suited change detection around yield occur c peak statistic tracking less reliable significant change statistic automatic idea typical numerous detection examine competition person transaction transaction dimensional transaction bad dataset generic anomaly detection generally transaction particular track detect label exceed recently detect person transaction isolate transaction large spike show mark near threshold repeatedly repeatedly stationary paper describe novel method online tracking tracking perform fast important subset ever analyze multiscale heart approximation manifold update manifold estimate adapt change curvature dynamic potential play volume stream arise monitor traffic paper result approximation correspond dimensional many relate dynamical change alternative pose interesting problem restrict subspace cost recall assume estimate hence term bias condition expand write since equal zero since ab minimize optimal tend second tend online asymptotically restrict sample projection u identity hence asymptotically uv gaussian vector matrix consider datum projection u hence w denote du u eigenvalue xx dx dx u upper sufficiently noise statement ex email edu email electrical engineering describe point scale detect data conventional address challenge model dynamic lie dimensional ambient streaming use track measure deviation series statistic deviation detecting change sharp dimensional include subspace tracking multiscale point online analysis experiment highlight efficacy detect otherwise slowly dimensional manifold anomaly change accept change soon stationary away dimensional result challenge wish quickly change social surveillance video structure traditional change method deal scalar impractical high accurately delay false alarm poorly develop extract restrictive assumption address univariate change high lie close ambient model non test reliable multiscale efficient calculate element miss literature manifold remain challenge batch design observation lie lie accept sequentially change significant rapidly special appear parallel tracking least incomplete manifold negligible curvature union core basically track underlie observe generate kde naive variant quite computation poorly stream datum hoc face computational allow spatially bandwidth increase additive assume work applicable broad align manifold kernel open efficient kernel shape kernel adapt time connection reduce preserve challenge challenge setting vary merging learn precise gmm consider gmm impractical high additional assumption ill pose setting geometric resolution subset encode straightforward wide surveillance modern collect massive video stream analyze volume collect huge quantity motion multiple connection cause failure phase activity mind detailed expert essential reliably detect salient scene detect explicit anomalous surveillance track motivate alarm evolve network detection attack characteristic network traffic characterize rapid primary contribution present online tracking analysis experiment organize describe multiscale statistic change theoretical numerical suppose ambient measurement lie white random slowly time track vary allow tracking residual varie slowly change projection euclidean norm simultaneously without problem determined specifie might project ellipsoid projection numerical mahalanobis data quadratic covariance mahalanobis cx however distance lie curvature near collection well mahalanobis hybrid mahalanobis subspace let projection coefficient residual eigenvalue mahalanobis approximate u u motivate mahalanobi eq complete side approximation avoid numerical instability cause divide mahalanobis distance measure distance x definition distance mahalanobis distribution available subset update close virtual child preserve c calculate structure new minimum accord virtual calculate parent close virtual algorithm index calculate mm du turn virtual initialize virtual parent leaf virtual create mahalanobis distance sample alternatively close virtual near computationally attention greedy readily however nearest tree virtual center whether tree residual eq structure prescribe tolerance approach update subspace tracking reason alone shape fu orthonormal offset instantaneous derivation update size subset close estimate write ignore accomplished solution recursively denote row order recursively guarantee orthogonality onto approximation obtain schmidt require extra cost continuity frobenius subset lack continuity track fast
total time multiple regression regression set eq avg avg vector note avg via opt opt opt avg opt opt opt compute leave derivation opt opt avg avg opt avg avg opt avg opt follow regression still complete transformed block repeat copy identify important identify multiple minimize opt involve need somewhat costly sophisticated unconstrained frobenius rank opt opt rescale opt section theorem construct eqn need compute approximation ratio k bind matrix construct problem eqn svd low approximation lower achieve perhaps sophisticated agnostic approximation ratio need whereas carefully target also present heavy fact simple lemma opt approximation last property pseudo full opt conclude opt rescale proof theorem n run time svd svd compute svd opt rt svd svd satisfy second opt opt opt agnostic carefully guarantee agnostic agnostic appealing rescale line briefly agnostic construction construct rescale opt solve eqn run compute vector rescale opt opt hold agnostic column whose except sequence look algorithm probabilistic guarantee previous bad bind exist randomize randomized may depend agnostic depend agnostic construction integer orthonormal whose theorem appear select last r opt theorem ratio regard conclude success agnostic unconstrained multiple bound r satisfy kk whose k kk observe opt integer integer exhibit frobenius span truncate svd construct bind suppose give rescale eq opt open size guarantee simple linear conjecture tight certainly size zero give air force laboratory support nsf dms nsf decomposition svd contain contain singular pseudo inverse qr factorization frobenius spectral norm frequent fact n provide proof lemma result lemma optimize spectral property matrix slightly abuse sample consistent v r multiply technique select column describe convenient matrix time satisfy achieve exhaustive total r procedure nr exist deterministic svd svd svd q take inequality inequality deterministic procedure svd let take lemma prove corollary constrain regression square ask whether suffice construct room improvement linear area robustness error contain essential yes vector meaningful full complex solve significant construction multiple addition identify considerable huge incremental approach distribute costly essential constructing arbitrarily achieve performance regression fit response sophisticated community discuss point feature raw q positive opt x least hold datum possibly weight suppose minimize construct also provide paper switch equivalent formulation background similarly element effective c depend seek say various formulation linear c pt thm unconstraine response eqn pt unconstraine agnostic thm notation dimension row response ratio constrain single eqn eqn thm frobenius unconstraine frobenius thm unconstraine response agnostic ratio opt opt nk multiple four opt constrain describe construct size achieve constrain randomize arbitrarily bad despite logarithmic provide concentration concentration break response seek minimize series row seek exactly frobenius norm regression column response seek vector trade quality writing equivalent regression construction multi theorem describe size k response frobenius spectral norm case theorem present size note algorithm show construct
say feature feature detector detector high low entropy detector high low selective svd mlp see b mlp omit generator initialization architecture mlp detector generators appearance singular detector fact norm detector feature generator also feature detector feature generator detector diverse strong correlation feature detector full spectrum mlp rank basis random mlp able patch cc mlp variance image observation htbp code hide binary surprising activation prior application normally distribute create plausible support detector norm feed noise mlp layer activation indeed binary binary activity detector htbp htbp regard behavior mlp explain mlp observe figure feed compare apply input responsible responsible effect mlp denoise reduce thresholding domain wavelet operation typically leave unchanged fix value zero absolute effect denoise always trivially achieve question preserve feature version feature figure input detector feature detector value unit look create clean noise present layer eliminate repeat activity activity hide layer absolute small conclusion feature detector high activity one feature detector high summarize denoise image preserve high autoencoder denoise autoencoder stack multiple sequentially layer provide layer optimize performance initialization supervise suggest wise contradict suggest deep architecture deep net gradient difficult abundance generalize phase impose restriction stochastic local fall gradient descent activation almost completely agreement regularization force hide restriction regularization information input preserve input preserve virtue denoise reconstruct preserved question well classical achieve form indeed useful parameter weight activation classify vector activation mlp rbms deep net stack autoencoder feature detector rbm visible learn appearance learn cc activation pre training supervise real unsupervised learn real activation activation mlp logistic sparse train deep layer patch detector maximally feature many mostly location image sense high remove detector activation see activation fact explicitly denoise fit regularization denoise easily interpretable achieve output layer patch output identical denoise autoencoder generator study mlp architecture generator mlp notice generator look mlp single look perhaps seem difficult intuition learn layer mlp mlp hide detector generator look detector output detector still two mlp layer two mlp interpret lead denoise htbp detector htbp detector output mlp clear detector feature generator identical corresponding feature detector lose layer separate basis detector answer single unit pass mlp input mlp provide question answer detect activation effect mlp layer hide correspondence detector mlp effect mlp detector high look clearly generator basis detector activation c name db couple db db hill db db db db form mlp image well patch arbitrarily well difficult code allow answer try image form word try patch clean slide region patch noise see image perfectly though image slightly conclude last approximated related figure full rank upper spectrum flat mlp layer imply diverse image mlp mlp db db db couple db db db hill db db house db db db db db db db db db image typically refer pseudo problem omp patch denoise slide averaging overlap ask mlp combination sparse image mlp dimensionality column approach denoise approach dictionary hide serve mechanism create code pattern maximize patch absolute activation neuron third hide neuron third activation layer activation layer input cause activation activation ii evaluate activation activation maximization patch neuron save image contain maximization well large patch pattern mostly center intuitively make covered fall away patch filter layer layer observe output exhaustive patch case clear patch hide layer detector look feature detector useful look less answer behavior set detector remove used performance replace detector htbp detector detector iterative choose mean detector detector detector row detector yield good interpretable yield detector yield yield bad look detector generator train make observation strength noise detector generator htbp detector generators detector generators generator look similar detector detector cover patch away agreement strong patch result provide explanation unnecessary large patch explain achieve result small see detector generator detector feature generators detector feature generator figure generator respectively type horizontal noise detector feature detector output generator somewhat affect visible noise generator sometimes type learn autoencoder mlp block generators detector learn mlp patch patch correspond neighbor patch adjacent patch filter reference surprising update connect neuron neuron input update show state image paper possible procedure part gain insight unit train vary size observation make conclusion regard denoise hide iii go increase architecture finally fine lead deconvolution address might also benefit problem benefit briefly feature noisy copy patch activation observe mlp internal work maximize activation often remarkably similar output cause true rise rbms force representation interpretation denoise autoencoder binary representation give rise representation rely sparsity deep architecture recognition also representation require criterion make argue representation obtain representation criterion image rbms denoise autoencoder noise type noise mixed noise achieve avoid activation image denoise patch achieve denoise denoise way toward gap discuss paper explain technical achieve good patch make denoise problem difficult require consume modern somewhat neural especially gradient descent sometimes science exist training lead poorly attribute initialization especially time consume understand lead good box merely observe output logic usually inspection certain interpret human hide maximize procedure qualitative train varied set encountered training initially degradation phenomenon undesirable explanation phenomenon gain insight mlp difficult hide layer activation insight denoise mlp hide contain patch patch input patch input procedure mlp average refer whereas refer patch denoise show art denoise show tracking process mostly many day time modern small training set one use dataset train mlp architecture reason superior test refer opposed patch denoise denoise denoise area patch test still albeit slowly briefly become bad overfitte abundance zero issue use either size hide patch unit see add wide layer beneficial mlp patch hide train patch five four beginning decrease later mlp achieve degradation performance difficult landscape mlp layer effect figure deep narrow difficult network good ask question image train architecture set either gain image full training degradation figure layer patch increase lead patch bad patch patch still degradation still approximately later degradation explanation size difficulty patch stochastic may fail input patch ideal architecture patch difficult remove size expect bad question happen patch patch slightly architecture keep vary patch size point degradation patch input output explanation good show patch layer create degradation combine patch layer degradation patch layer quickly degradation however architecture patch degradation performance comparison conclude deep narrow optimize hide layer beneficial input patch may procedure layer hide patch attempt least fine procedure initially switch learn suppose encourage low suppose htbp show indeed fluctuation addition test turning obtain achieve bm patch patch layer size architecture four never bad approach bm fast fast slightly outperform bm slow explain training well achieve patch reason achieve large patch become strong weak indeed patch large cause difficult ideal influence strength patch block lead mlp search window four progress progress use block match plain particularly begin advantage plain evident perform size window well mlp block mlp use plain achieve compare progress win test particularly begin begin procedure procedure helps regular rather match procedure size block without match procedure block later stage superior present achieve plain mlp stage achieve noise level patch employ step bm achieve medium noise important avoid achieve ask question insight mlp work mlp
knowledge series chen rely specific algorithm splitting approach transform convenient reason generic spectral short specific unclear specify spline average sum plug infinite must error sum suppose decay slowly number explain fourier bound simply discard value memory follow directly process modification method idea correction il il il il il g du du I admit cubic rather interpolation interesting spline otherwise grid give essentially fourier range q simply decompose fouri two fourier correspond noise close fourier term differentiable since mean quadratic computed point aspect next idea common theory memory reason explain introduction posterior assign weight eq approximate certain spectral rely heavily parametric class true notation great generality certain statement term say go infinity implementation simplify ignore uncertainty simplifie observe particular converge fast feature quadratic denote coefficient toeplitz obtain evaluate quadratic explain typically rely expansion refine toeplitz one obtains easily adapt far reduce rewrite dataset fourier need value cost typically perform monte practical ignore operation spirit nearly independent rigorously speak establish valid true merely indistinguishable develop section speed rather respective langevin discuss carlo sampling approximate discussion see k step walk conditional birth reversible jump sampler death birth complete death walk sample birth constant k death step many basic strategy could consider variant two behaviour parameter scale walk tuning parameter shall manually principle form mcmc review past distribution however less learn address difficulty comment refer difficulty explore modal posterior tend mode may paper multimodal posterior propose spectral dot dash fit fourier hard distinguish except unless large multi mode may correspond true refer reader one mcmc spectral solid line line line coefficient describe generic smc sampler backward detail former must easy smc operation sample multinomial assign value markovian kernel anneal geometric bridge discuss conclusion alternative may sequential adjust dynamically coefficient solve eq take default equation sampler birth death advantage particle motivated hasting step perform move observe phenomenon increase lead improvement case repeat close variance correspond particle seem bring illustrate phenomenon seem obtain slightly equal formal suggest investigation context sampler instance one use reversible step currently particle sum acceptance birth death coefficient run algorithm cpu second correction star compare sampler mcmc simulate give time minute sequence smc run start set pilot run acceptance seem alternate mode seem poor trace middle plot visit satisfactory also reversible jump burn period proposal calibrate past posterior significantly chain auto correlation respect post decay posterior range one mcmc scatter add trace five nine mcmc bottom plot post burn posterior evolve grow goal property procedure effect correction step marginal resp confidence band light grey grey true one correction concern marginal vanish sample correction particle grey dark grey band spectral spectral cpu smc unless sampler expensive dark grey plot bottom row fortunately correction see prior smooth density inference give previous section spectral top plot marginal moderate impact spectral essentially nuisance critical inferential purpose interesting difficulty observe sampler figure mcmc trace band bin plot bottom prior preliminary methodology either inferential view seem detail author literature rarely g temperature seem additional work practical package record per time convenience dataset side although empirical biased true bias estimator quite frequentist parametric return suppose maximum likelihood return plot spectrum order see side find strong evidence long marginal left panel mention briefly thing correction negligible smc sampler little variability marginal weighted smc band dark grey corresponding order dash work say could adapt little parametric point sequentially course dependent distribution provide sequential sample could smc distribution likelihood lead cost reason far thus interested instead smc sampler acknowledgement comment j conditional coefficient distribution belong regard admit representation l enough q calculation n p tf idea show lf lf l lf f step n supplement f supplement n n p control lemma supplement fr n supplement lf nj j uniformly properly let n successively lf n term note supplement term n supplement second kn kk without loss generality positive covering ball nk k nj enough q k nk ok nj integral separately lemma supplement supplement q combine supplement together set go replace abuse notation ks n cn cn obtain precise controlling terminate theorem theorem often sparse nature propose recover importance show importance vanish go posterior modal sequential sampler annealing evaluate world semi spectral reasonable counter several assume observe toeplitz vary specification separate behaviour short certain choose semi run methodology take short expansion finally spectral density exist computational difficulty arise bayesian model expensive compute entry reliably quadrature feasible metropolis reasonable context since pilot tune process implement sampling green reversible hard tune spectral possibly memory vast parametric approach rely commonly use frequency paper hypothesis bayesian inference finite frequentist necessarily entirely satisfactory reason frequentist discard see parametric provide unstable inconsistent misspecification satisfactory provide frequentist propose address manner discuss likelihood hardware propose fast likelihood scheme motivate step monte carlo constant front intensive sampling however multiply front small put thing many discuss literature discussion importance discuss sequential
privacy basic place main begin impose certain family consider base differential corollary privacy proof result smoothness usual notation g illustrate multi median eq lipschitz respect belong classification loss set loss compute verify natural communication strategy amount maintain proceed review problem form outline essential stochastic information iteration receive appropriate mirror descent enjoy defer privacy subdifferential corollary information privacy apply assume notation definition take important precede paragraph optimally locally private contain ball radius geometry function perturb belong ball notation constant unique achieve loss proposition precede paragraph optimally private theorem yield minimax focus second result specialized analysis previous analysis necessary appear require restriction actually indeed domain establish consequence corollary ahead perturbation theorems stochastic mirror obtain section theorem sense mirror additional turn result suppose ball contain radius state result section alternative computing definition setting differential communication theorem quantity differentially interactive begin let collection differentially definition differential corollary differential applying mirror assume privacy result interactive privacy long optimally private guarantee minor error interactive conditionally eq infimum mechanism method work together find subject apply class lipschitz loss state minimax restrictive simple optimize privacy assume channel differentially private universal domain optimization loss restrict small captures functional corollary sharp factor note theorem corollary dimension substantial theorem inequality characterization private minimax tight relate set dependent query gradient depend obtain convex theorem method set open privacy play allow noise achieve necessary provide sampling precede privacy increase corollary roughly rate privacy comparison divergence scale roughly though theoretic privacy explore give sometimes sufficient compact distortion channel conditional privacy distribution wish capture locally private optimally differentially locally private attain begin characterization reasonable focus suitably index unitary set extreme minimax saddle polytope constraint unique uniform attain saddle brief remark somewhat deep understanding theorem attain saddle maximize still obtain privacy maximum theorem characterize minimax ball maximum entropy attain certain proposition discussion section binary almost surely coordinate slightly understand scaling concavity imply bind tight complicated theorem characterize saddle mutual ball variable support replace proposition somewhat sampling accomplish define mutual appendix lie result characterize saddle non trivial attention result generality constant almost fix constant satisfy equation q accord satisfies definition technical remark simplify matter show proposition point assign initial flip corollary proof classical exploit due consist begin appropriately finite member mean approximate property deduce low test obtain low inequality mutual detail outline begin reduction error assume proof construction collection representative discrepancy measure q separation optimality nature choose draw demonstrate q observation bound possible minimize risk testing uniformly procedure observe variable measurable satisfie contrast uniformly observe le inequality become apparent argument provide theoretic le step constitute argument technique satisfy definition definition loss lemma mutual interactive differentially scheme present pack conjunction step final lemma turn precede outline exhibit minimax separate risk separate collection relatively base generate standard define linear function final separation risk scheme domain sign multiply complete median loss loss pair cardinality q define sample functional construction whenever risk minimizer collection functional precede discrepancy subgradient outline proof private available mutual bound loss mutual independence single randomize careful calculation yield final somewhat accord proposition subgradient appendix accord proposition subgradient appendix family information apply prove le complete median begin state statement theorem pack construction median numerical couple dependence pack linear lemma le obtain variation appendix multiply side complete statement part eq eq construct analyze norm median recall packing cardinality choose among define hinge strategy risk radius minimizer risk risk immediate verify separation directly minimizer attain construction require careful guarantee provide gradient radius provide complete characterize proposition turn mirror use note low large probability similar eq focus identical universal minimizing satisfy low mirror privacy general utility statistical number interesting access perturb view convex question whether dataset could insight release privacy induce lie compact perhaps fast estimation preserve channel alternative might know care may question appear answer especially wish privacy technique herein hope work pt receive arbitrarily q oracle minimizer amount consequently guarantee devote mutual proving construct sequence element information support set mutual inspection must let denote minor giving apply convexity q subgradient independent define scheme uniformly choose scheme q define entropy concavity substitution imply mutual addition value fix component subgradient entropy minimize q proof concavity proof subgradient individual hinge loss equal eq shorthand therefore construction imply similarly parallel take thus use marginal condition via algebra seek bind mutual generality add claim coordinate similarly similar hold e expression note recall expansion must sum case probability remain representation mutual coordinate symmetry marginally concavity let fold product exploit concavity inequality achieve mechanism ingredient distribution subgradient differentially identical private also let sample independently probability set scheme differentially private calculation corollary sampling odd strategy eq strategy apply stochastic bind note regular probability appendix classical arbitrary inequality conjunction g must point precise address issue involve precise regular probability markov consistent transition probability construct appropriate chain extend proof state measurable though subset moreover topology clear measurable mapping accord additional drop indeed since theoretic turn contain slightly outside extreme represent regular define random construction measurable version sure set see turn analogue lemma private support bad markov without private discrete maximization lemma establish completeness p vector choose constraint exist may lagrangian domain kkt minimize dual conditional q since must constraint kkt broadly outline guarantee distribution mutual reduce entropy minimal mutual optimality begin consider extreme extreme define extreme generality extreme technique convexity symmetry constant determine upper attain extreme establish statement lemma denote minimize mutual problem extreme moreover convex inspection since satisfy equality uniqueness solve write shorthand normalization expectation constraint q satisfy tucker kkt g derivative shorthand symmetry note x px maximum rewrite entropy inspection mutual minimization problem maximum saddle claim extreme point maximize choose solve maximization additionally simplex w g remark immediately focus verify fix leave take multiplier problem infimum derivative identification mass coordinate say loss write plug perform logarithmic imply equality arbitrarily along support extreme radius mutual plan sum write lagrangian solving guarantee I symmetry normalize appropriately thus solve nearly side whose root belong desire algebraic statement outline proposition satisfy support extreme reduce optimally private simplify lp finally map radius ball choose privacy channel differential privacy level differentially channel implicitly converse view abuse hypercube conditional vector exist additionally small possible perturbation cast mass differentially private channel lp p optimally private lp use problem satisfy suppose sake contradiction matrix similarly satisfied symmetry information information give contradiction optimal differentially indeed entry corresponding eq characterize structure solution linear proceed large specify lemma uniqueness due begin lagrangian constraint negativity generalize constraint hold vector satisfy vector inspection kkt must vector eliminate negative homogeneous kkt indeed inspection see question remain whether definition seek q sum symmetry kkt argument remain define statement lemma eq strict apply objective previous lagrangian modify linear condition similarly definition suitably uniqueness completely set vector argument complete proof identical mutual follow invert mutual show define denote expansion f ff expansion yield tackle note f berkeley electrical department university california berkeley privacy model sharp procedure consequence exhibit tradeoff privacy arise whenever aggregate individual wish learner quantitative exploit freedom whenever paper statistical theoretic advantage foundation development decade goal goal thereby determine third theory permit abstraction detail specific procedure allow fourth loss allow optimization theory optimization practice theoretical decision risk randomization privacy measure minimization independently minimize risk provide access give denote explicitly history go suggest privacy census presentation orient privacy key hope comprehensive survey reference member belong dataset generate statistic literature limitation linkage yield maintain privacy currently standard privacy privacy formally privacy probability differentially guarantee adversary know privacy information adversarial attack break researcher algorithm empirical differentially private add estimator obtain privacy statistical estimator suitably asymptotic stability perturbation demonstrate maintain privacy providing though intuitively tradeoff precisely impossible call useful differential guarantee guarantee sample give amount necessary work relaxed differential privacy give additive failure also closeness low bound match theorem involve similarity somewhat goal count number output use geometrically optimal notion utility provide accurately statistic substantially minimize guarantee drive force focus keep private many individual classical privacy true set privacy interactive sense depend private locally natural population give trust medical application perhaps status develop access internet activity search provide utility web wish search show setting locally coincide concept complexity query powerful model rely count query interested precise polynomial quantity broad risk sharp privacy datum wish nature differential privacy explicit private e interactive turn theoretic recall version mutual random measure mutual theoretic idea security survey important standard mutual privacy notion mutual say generate possibly minimize characterization estimator privacy constraint collection function minimax estimation domain guarantee collection constant characterize e moreover turn differential able constant risk exist constant procedure ratio universal numerical establish quantify sharp amount effective maximally decrease roughly differentially
square penalty rest organize group effect give devoted criterion huber criterion en technical random mean indeed sequel existence vector unknown minimizing unnecessary introduce intercept appear already paper fact penalty tend group attempt solve group lasso procedure penalty impose predefine precisely consequently group level see choose group next naive minimize coefficient square suffice large reduction small consequently penalty act scale fan li oracle p weight effectively penalty en penalty make penalty know enjoy notice penalty behave correlation procedure rely sort coordinate zero achieve unique integer l consequently en elastic variable see spirit create group group group avoid remove group penalty scale coefficient smallest individually penalty tend keep large heavy outlier huber criterion introduce quadratic describe quadratic linear take huber criterion minimize respect propose huber estimation huber coordinate problem run determine constant huber usually j instance ordinary root form huber criterion penalty observe type least classical bic minimize huber criterion bic minimize eq non satisfy group high lead estimation property stability quantify group net net goal theorem grouping penalty huber task future us remark since initial effectively quantitative penalty ridge compare net provide sign natural happen adaptive net initial grouping elastic estimator initial estimator adaptive elastic avoid ridge avoid du grouping effect property various fair study involve normalize explicit calculation variable normalize upper property enjoy replace huber loss two denote matrix suppose consider need see huber criterion satisfy distribution note hold absolutely lebesgue strictly origin stand distribute random result hold matrix asymptotic huber combined ridge ridge ad en huber ad huber huber en huber ad compare present study simulation insight provide paragraph performance inspired involve group correlate model intercept way form ny tn matrix compose diagonal third block coordinate equal coefficient amongst highly correlate group intercept nature noise intercept block gaussian mixture gaussian deviation block group allow group penalty divide light tail whereas heavy error model quantify robust absence likelihood square performance performance measure design design center stick theoretical fix framework size provide associated selection report table report constitute procedure indicator absolute I vanish amongst count I case go overfitte zero least u report aim provide zero zero correctly zero recall model closely estimation illustrate estimation figure concern hyperparameter choice regularization penaltie bic criterion describe grid method linearly space space huber study report recommend huber penalty remark tuning ridge cross validation grid linearly space similar protocol pick relatively grid linearly fold validation selection en surprising en impose norm square close example en behavior penalty zero correct zero comparison fact type reduce huber contrary en method identify huber zero zero zero figure coefficient good observe penalty lead performance term ridge bias figure expect ordinary square well performance especially ad huber numerical come cancer early elastic net eight clinical logarithm logarithm percentage score predictor name divided part observation study set performance observe allow huber ad lasso slightly let observe great huber procedure stable except ol associate variable en penalty en solve optimization package program programming impose rapidly convert problem reject version convex self well huber huber satisfie kind lead characterization number trivial interval guide manner however expression compute represents begin alpha mu l l w compute multiply derivative kkt eq last index belong become second cauchy schwarz get belong become cauchy eq imply normality step adaptation concern treatment penalty notation difference normality u nu nu limit together p p tu suppose consequently selection part suffice n infinity claim infinity tends infinity tend sufficiently respect differentiable entail z z l l large q depending since involved sequence since possible tend infinity tend concern since converge infinity u entail concern j j differentiable tend bound entail second expansion j vi stronger hold treat b j ai ax ax p imply convergence involve finite ensure moreover tend numerator tight denominator part page notion weak net theorem eq show convergence convex u ji ii vi convergence prove f successively theorem iii vi previously page ensure imply since deterministic p u satisfy intermediate piece li semi note semi value low infinity recall ai j set I
conditional ic simulate conditional lasso neither automatically scheme variable thus credible infer let credible interval multimodal partition density great zero ideally zero firstly density current mean suitable modelling mean sample mean large integer credible define eq number sample fix active increase six dataset six rr produce validate dataset pair cancer normal validate find literature validate expression study interaction generate intersection prediction cancer researcher derive broad contain tumor fourth experimentally identify interaction search also interaction validate interaction assess initial interaction infer interaction coefficient rr control parameter determine integer large value shrinkage fold cross cv separate broad estimate part select sample gene training error finally test gene datum coefficient value gene subset coefficient estimate fixing algorithm select gene choose rr threshold mark gene method constraint perform describe rr lasso across dataset rr similar interaction sparse zero see gene example infer identify exactly zero lasso several detect less rr able gene credible active credible interaction infer interaction density look infer confident c ignore density infer select density peak around zero peak calculate credible interval significance gene credible statistical significance shown experimentally experimentally detect c estimate rr identify estimate bar plot involve information red bar plot involve expression rr produce protein form computed magnitude around zero vice versa unless rr approach credible show target eight seven target plot trace vice versa imply agree conclusion prove rr performance auc uncertainty interaction bayesian method automatically credible looking advance perform prefer method produce effect protein protein simulate competition target acknowledgement would thank david stage manuscript target compose protein gene ridge lasso algorithm produce ht credible highlight experimentally validate gene value fix roc rr threshold ht ht ht square ridge bar red ridge bayesian lasso detect experimentally validate infer list gene credible b draw bayesian put interact black red credible credible computed lasso gene great highlight experimentally significance credible credible negative bayesian lasso six identify significance greater show highlight experimentally c significance credible credible compute seven great highlight validate significance credible interval active eight significance experimentally credible interval significance credible b b credible list bold experimentally significance credible credible credible effect significance list bold validate target validate broad experimentally validate significance auc broad cm cm liu engineering university technology china mail cn abstract rna important target useful understanding role potentially disease point estimation interaction lasso explore infer target rr four sensitivity specificity bayesian meaningful provide manually choose meaningful matlab google express process primary combine rna induce protein reveal report important role development give sequence identify interaction target complementary seed region show throughput datum greatly potential target computational throughput mutual partial least approximation etc method probabilistic employ carlo require employ variational density regression approach propose lasso negative employ infer point parameter non identification target non employ rr validate dataset representation thresholding produce unable calculate credible uncertainty significance significance significance expression represent collection expression interaction collection model target matrix model bilinear could since illustration model direct advantage consider represent complex multiplication thus negative part negative degradation gene involve example activation involve target algorithm algorithm briefly
nonnegative corruption corrupt clean noise handle error moreover concerned partial corruption portion kind portion clean traditional follow obtain constraint control tradeoff dependent portion corrupt optimize norm effective convenient additive nonnegative need decompose clean nonnegative difficult find instead work nmf decrease reduce follow update ms decrease I I pe nu value correctness updating rule z tf z problem update rule need prove positive rescaling substitute updating additive result part gained keep auxiliary second important z ensure thresholde e updating cause objective always converge present experimental position reconstruction deal assume advance fortunately able position missing locate experiment patch average experiment face image vector face randomly pixel additive face image scan nonzero position evaluate precision total algorithm knowledge handle nmf nmf gain try detect appropriate performance sensitive try poor probably partial corruption nmf recall face recall face sample explore large enough long set set face middle speaking algorithm precision large little recall indicate detect precision present denoise first simulate additive reconstruct handle position use image cause face face generate denote similar since mask new face experiment face mean percent mean percent pixel faces see traditional noise large position partial corruption enable norm prefer pure nmf subsection denoise natural affected convert patch set reconstruct original denoising show image show nmf traditional space truth material application prevent nmf algorithm handle require advance miss solid justification application rank automatically adequate application nmf widely etc nmf many face motion segmentation etc dimensional nonnegative corrupted position variant nmf treat corrupt unknown application prevent usage traditional variant nonnegative corruption position clean nonnegative position noise iterative justification nonnegative factorization recognition etc nmf receive substantial attention practical improve incorporate nmf propose corruption propose pca recover propose robust
map basic discuss relate dynamical compare range piece extend kalman batch discover identify dynamical structure fourth dimensionality reduction discuss connection among show spectral localization motion orientation range robot throughout know range associate handle localization motion standard gauss unknown robot parameter online filter map new robot linear motion though sensor informative enough robot range application exact multimodal gaussian inaccurate linearization lot range attempt enough extended initialization delay initialization parameterization accurately multimodal show track multi hypothesis much expensive instead maintain statistically mode approximately sparse approximate identification attempt directly identification learn carefully observable regression originally algorithm almost identification observable tb history bottleneck rank method discovery spectral state appeal implement operation contrast em consistent typically impractical requirement range spectral discovery discuss measurement give property make carry spectral efficiency finite bound structure motion problem vision recover rotation sequence pose discovery represent frame camera factor position camera rotation state inspire range way identifiability geometric information beyond give dimensionality view possibly nonlinear like multidimensional example use approximated geodesic contrast inaccurate robot integrate additionally measurement force sec block resolve ambiguity popular classical sensor feature manifold lie reduction nonlinear dimensionality network location think suggest dimensionality unnecessary reduction well simplify strong portion generalize sec discuss robot sec discuss discuss datum online discuss motion measurement measure robot insight robot n position recover factorization unfortunately transform factorization hope metric estimate available four position simulate orthogonal transform translation constraint show set contrast camera orthogonal nonlinear tb environment svd recover transform robot position position information recover high frobenius receive pair robot environment robot successive velocity factor key observation write easy position contain contain robot value four singular learn make either target angle successive pair location th location collect top singular c matrix suggest alg thin discard remain randomly sample estimate measurement top include datum show number location robot robot position robot true detail recover factor presentation although zero noise finite time weak proportion gain argument sec detail bound svd learn nonzero eigenvalue covariance receive practice rarely satisfied receive range entirely often outline relatively miss em use penalty miss structural second divide subset robot build predict read datum range next finally average locally interpolation work much well practice drawback many extension straightforward sequentially need motion motion require want derive latent location learn motion identification sec robot idea complete receive next collect substantial computer gb ram take simulator place environment robot move step receive read range perturb sample equal solve coordinate accurately path investigate map increase generate environment spectral measurement exclude true datum collect fig collect use wide range find range mobile robot stationary environment place throughout robot coordinate ground ground truth truth path accord characteristic km pose receive turn pattern robot km pose measurement worst depicted qualitative spectral localization square show baseline robot standard report rmse worse particularly initial portion also gauss matlab range subject optima intensive follow suggestion minute evaluation nonlinear run computation time empirically optimization time use initialization batch optimal start likely certainly tb last good opt opt opt batch opt path last spectral path spectral batch last batch good full c solution differ substantially previous essence minimum free algorithm robot system state real acknowledgement support grant support nsf position case position know hope robot position orthogonal translation rotation turn metric row know column exactly exactly dimension analogous matrix learn robot coordinate position transform coordinate two matrix high fit surface scale surface surface step linearly transform approximately linear coefficient orthonormal coefficient column select right r bring quadratic spherical start transform coordinate quadratic diagonal set write r eq iv surface coordinate else guarantee unique take set pick software automatically quadratic get quadratic desire coefficient coordinate expand know first solve last equation bring coefficient eq leave one vi satisfy learn metric advantage correct coordinate identically since function identically svd top small singular remain provide estimate counterpart two part concentration hoeffding show element covariance value empirical multiple kronecker minus constant derive bound
exp exp identify see exp exp exp exp fail noiseless exp robust vertex hull perturb toward hull matrix index maximum well interior convex hull column exp perform ill conditioning matrix oppose exp ill perfectly extract well exp oppose exp fast algorithm computational c exp exp loose partly consider bad structured value show average guarantee guarantee exp exp exp algorithm see already perform solve solve interior third ten middle five twice add hull second solver lp mention solver much percentage correctly show exp exp exp although large percentage decrease fast noise increase percent correctly extract always c exp exp exp exp algorithm noise condition far still make heavily rely separability assumption perfect practice preferable analyze hyperspectral pure generalize hyperspectral framework explain algorithm robust direction bound one tight improve feedback improve lemma nonnegative separability column hyperspectral mix pure assumption prove small perturbation generalize exist hyperspectral hence provide performance hyperspectral pixel robustness hyperspectral consist acquire scene measure surface pixel hyperspectral material model spectral signature pixel linear combination call weight image band entry matrix equal signature pixel material whose signature abundance define equivalently give nonnegative hyperspectral abundance factorization ill pose assume material pixel identify hull call write cone contain sense mining interpret topic bag topic separability appear require topic discuss pure separability appear row topic pure sense actually part document reference hyperspectral mix pure nonnegative assumption handle situation sense comprehensive overview hyperspectral identifying normalize column prove work input separable perturb robust notable et program suited deal pixel several expensive world hyperspectral namely subset column balance importance extract al convex problem deal large fast incremental gradient tune impractical huge scale problem paper family separability noise fast requirement extremely priori tune factorization recursive section repeat separable identify convex hull volume section desirable approach guarantee recovery also need matrix practice satisfy always hyperspectral imaging abundance uniquely derive noise pure propose handle algorithms generalize successive automatic generation successive pure pixel justification base pure experimentally illustrate result synthetic denote position th entry x cc decrease identity denote discard nmf separability strongly function permutation theorem j corresponding maximize give step project column onto complement select number stop whenever residual last specify analyze separable small perturbation assume original matrix separable pure pixel hyperspectral imaging see implicitly otherwise general ill pose one make generality column normalize construction sum column entry correspond hyperspectral slightly image contain background pixel zero signature hyperspectral image intensity part angle camera scene hence although contain material signature noisy account signature matrix global notice whose whose continuous tell assumption recover set correspond column w k w k rw iw jj induction therefore column extract project onto complement condition step extracted say loss full unchanged correspond extract affect go assume perturb noiseless original satisfying sufficiently introduce kf satisfie convexity r I rx kx analyze bound imply convex parameter w give eq fact lipschitz continuity gradient prove induction work perturbation convexity column denote maximize satisfie contradiction extract perturb convexity strict vertex lemma equation obtain contradiction therefore condition large singular proportional algorithm memory sensitive hull discuss previous discard outlier simple outlier describe satisfy noiseless row use abundance column quadratic program separable contain cf j easy column unique index must hyperspectral outli abundance large perfectly reasonable assumption convexity cardinality step correspond column let row show identify extract precisely go prove bound row prove since step prove derivation exactly omit denote also projection column span column finally bf relate derivation whose good choice along hand worth qr perform respect column successive projection use show technique robust able refer generation empirically hyperspectral explanation explain theoretically analyze data dimensionality reduction successive successfully take hull maximum unless logarithmic achievable polynomial advantage algorithm separable identify whose hull robust clear experiment hx choice limit large potentially converge go infinity check choose check use choose depend condition derivation bound precisely ball locally continuous convexity investigate derivation use fy hold gradient gradient continuous respect apply guarantee noiseless hull column consider matrix select fail norm sensitive pixel average author perform numerical justify give synthetic highlight synthetic hyperspectral reader algorithm propose compare test variant robust comparing perform step compute summing element extract step way operation total approximately operation eventually impractical matrix minus reduce formula operation interested column test implementation experiment turn seven hyperspectral time less half residual matlab intel core minima attain generate function sphere identify maximize separability column attribute column correspond generate column large let number vertex computational robust noise vertex noisy soon perturb identify small perturbation confirm noiseless critical conditioning arbitrarily noisy column isolate column could high column extract example critical hyperspectral close pure moreover reason correspond column step principal component see notice preprocessing set
kernel value query hardness depend concentrated surface hardness depend characterize reverse closeness closeness concentration pi directional concentration project point set figure spread ball diameter point well possibly ball diameter number radius index onto notion directional concentration indexing indexing indexing space partition many indexing scheme medium hierarchical indexing scheme branch addition branch bind importantly build select efficient desire indexing tree metric align split tree use representation induce make tree prohibitive cover way quadratic construction multiple moreover analyze extensively cover tree requirement theorem index integer scale decrease node invariant cover dp q finite explain infinitely many level contain dataset node store tree child child supplement distance kernel evaluation construction construction would tree implicitly point modification branch provable support branch branch triangle absence obtain max subtree present search object subtree root node possible kernel query subtree notational convenience give rooted query suppose possible figure cauchy invariant cover tree node level bound recursively j ht surface correct loose root object maximum object q supplement r node retain far remove branch alg maximum kernel value current subtree possible subtree retain node correctness cover alg complete present supplement sketch I approximate focus exact max max absolute relative care take follow end iteration loop alg good approximate stop iteration r I stop point achieve technique supplement space make bound explicit search directional analysis tree depth required result child encounter whole pruning I ir r directional bounding bounding ball ball hence query scale price solve search experiment top candidate evaluation perform tree supplement fix length mnist uci machine repository netflix yahoo music dataset length representation protein reduce input sequence summarize close magnitude except mnist quite clear attribute directional constant property protein speedup order logarithmic sequence size require completely multiple load whole memory constructed distribute inefficient high branch method compare obvious exchange framework indexing scheme scan evenly distribute master node single master perform scan parallel match node master return match total good match important actual master runtime distribute system us scheme indexing contain build save scan single every tree match return master report phase query logarithmic time scale constant runtime enough small enough yet efficient achieve scale scan extension technique achieve comprehensive general date first rigorous hardness provably logarithmic query speed practice tight analysis without directional constant abstract object graph kernel detail tree single explicit kernel trick metric dp call current level tree recursive call subroutine represent splitting form completeness root object bound denote angle lie level angle angle origin simplify give cover consideration r kernel search query find ram al draw eq simplifying present always miss top query replacement want rate k tree subtree root return max operation follow improve actual construction parent impossible experimentally verify store reduce number evaluation computation gray proposition claim section wide max search focus begin hardness problem notion algorithm object feature provably search well order magnitude speedup extension search present accelerate object object similarity similarity scan set ht similarity class fix protein text music financial kernel kernel trick evaluate inner product require neighbor match computer vision million grow make scan prohibitive appear posteriori well widely successful obtain representation preference ever scale million cost retrieval recommendation find dna sequence set object various maximal alignment preference biological matching implicitly letter like p crucial lose normalization normalize kernel
seven gibbs bayesian iterate time keep size report result illustrate figure represent gibbs successfully outlier model contaminate table contain estimate outli occurrence size present remove detect outlier detect outlier variability outlier illustrate detect correctly see conditional obtain remove biased typical htb true consider motivate regard ip server university indicate outlier remove respectively autocorrelation partial autocorrelation function accordance estimate detect poisson affect identify require outli outli occurrence respectively effect cause depend position outlier methodology higher derive distribution implementation additional model work operational mathematic project mat de mathematics commonly problem additive outlier integer consider show methodology use keyword problem model poisson autoregressive series intervention often occur effect framework gaussian series detect outlier intervention investigate author intervention yet albeit relevance inference motivation stem adequate worth model intervention effect estimation contaminate period additive wrong motivate focus autoregressive independently real motivate concern address page construct concern page find author show infer improved outli bernoulli arrival additive outlier occur otherwise identically contaminate magnitude describe estimate outli contamination distribution conjugate ga distribution q marginal describe full posterior tt x first markovian affect fx I therefore conditional outlier give denote vector full conditional markov chain distribution case conditional log concave rejection metropolis arm gibbs
general evy process need monotonicity process rich parametric fa sd simulation introduce evy therein normally intensity realistic volatility determine drift brownian motion time brownian motion model infinite activity index respectively imply risk neutral measure base price throughout time negative put price option noise level error due bid market simplicity market contain bid note evy dimensional triplet infer price accurate estimation evy pricing characteristic finite imply martingale condition curve plug theoretical result observation additional smoothing spline two might improve interpolation spectral procedure rely identity look convenient option identity shift lead scale jump therefore use allow cut frequency parameter fa separately precise correction negativity jump latter right hand pricing apply inverse option benchmark minimize measure employ square approach correspond evy triplet pricing formula l evy plug efficiently numerical effort determine minimization consideration term mention include lead correction hence difference residual square impose price derivative least small minimize calibration drive choice quasi optimality study perform well setup follow version subscript onto quadratic sense domain particularly high frequency frequency smooth transition improve like weight outside determined reflect smoothness estimate scale version flat top kernel suffice transformation interpolation whole estimation finally choose differently quantity let simulation european option quantile fourier additive consist level choosing determine measure interpolation account use aspect mention quadratic smoothing gain high weight noise frequency reduce illustrate spline function whereby increase jump tp level iteration spline zero finer decomposable infinite activity u nonsmooth yield eq smoothness away zero analogously coefficient q determined lead solve substitute know jump one function q numerically compact large truncation reasonable decomposable monotonicity monotone arbitrary decomposable normality estimator fa setup decomposable let treat similarly nonparametric part choice cut trade construct negligible approximate u logarithm term linearize error construct variance linearize error nevertheless asymptotic approximation error negligible exact vanish analyze linearized observation quantile q incorporate may sample equivalence find fu fu asymptotic equivalence approximate tu integral yield finite variance linearize error isometry variance central distinguish finite shape sd scenario pointwise linearization kx version quantity evy representation l evy triplet latter cut density estimate point bandwidth interval give denote standard confidence cut wide interval delta estimator confidence interval gamma simulate price relative european call price linearly application interpolation cccc ccc level carlo oracle cut confidence percentage true value sufficiently confidence range fa fall bit picture confidence evy density pointwise interval evy density k confidence illustrate true evy pointwise everywhere contain confidence interval far plot confidence estimate negative peak cf method european option index therefore price financial market accord put respective price option two seven day price b unknown sd focus option price sd option calculate square fit especially seven minor trading day confirm sd coincide figure calibration l evy jump positivity correction might look interval model suggest neutral stock pure process decomposable reproduce option activity jump diffusion index equal contrast investigation estimate jump tp sd residual option pointwise option sd trading section stability moreover trading day procedure option estimation option significant decrease hand curve significantly slight respect axis volatility time market activity activity calibration option latter jump measure shape calibration unimodal minor differ reflect jump obvious trend pricing model consecutive market misspecification pricing however difficult improve activity fa evy model admit determine variance linearize estimator set precise error european option price fa fa volatility trend activity decrease long evy misspecification forward high available market interest whose risk activity interval base linearize tu tu finite variance estimator linearization write brevity mx purely linearize give decomposable compare differ characteristic tu finite version define u view linearize x theorem proposition thm grateful anonymous lead considerable improvement research economic observe price option evy evy activity well self decomposable l evy interval volatility pointwise jump probability infinite base option index calibration trading day european option nonlinear evy frequently price assume constant l triplet jump price account tail appropriately evy model reproduce volatility fact shorter adequate fact
relate express interval extract either equation provide place represent volume trading integrate choose location follow stop iteration optimum adapt impose element potential city aim identify high identify area would open additional evaluate maximum market volume city daily market characterize market volume around million daily copy far city concern commonly economic sale copy economic conditionally sale datum market potential justify daily represent available consist daily day locate sale total daily sale volume available around copy volume location circle measure measure type customer day copy fix effective suppose customer suppose attractive potential go network zero area covariate covariate represent density joint stock induce sale covariate distance car distance comparable rescale c estimate provide discuss interval original run test global likelihood particular application bad strong competition report approximated consider coefficient positive sign low control run retain pt average area city estimate suggest highly spatially competition nearby quite apart measure market potential location equation market city provide copy maxima area global note city market estimate potential copy place locate maxima represent location additional standard area city b copy city pt volume relate area market interval market daily copy increase total market volume optimize copy volume market volume actual light economic daily improve volume additional economic daily guarantee additional market daily phase potential prove essential statistical spatial market sale uncertainty include spatially unit potential market volume economic daily city daily sale focus city characterize potential include study temporal fluctuation relaxation worth outside sale city drug disease spatially available aggregated form g privacy reason precise usual multivariate variance estimate give equation evaluation derivative distance matrix vector eq acknowledgment provide preliminary analysis always recover market spatially sale tackle way measure location recover measure collect interact affected measuring maximization inferential apply market city analyze order area evaluate market city give market trading area sale expect spatial market low maximize issue potential already sale spatially available aim market sale spatial interaction interact sale volume affect presence potential ignoring regard spatial trading perspective spatial spatially market random ever potential instance recover realization spatially case interact interact location measurement nearby measure miss sale spatially market city aim area identify area total city respect organize background spatial datum interaction case technical estimation spatial term turn sale receive attention solve new interaction potential flow interested reader refer seminal store often attribute product store main drawback spatial market potential spatial location concept suppose beyond existence market also reach product attribute store affect spatial spatial potential drive interaction uncertainty critical application provide market answer question denote market potential generic spatial company trade company maxima store near company wants evaluate market presence actual suppose three mean covariate process w completely characterize reason later paper assume respect variable location realization interaction adopt f nonnegative binary continuous monotonically decrease distance parameter equal equation directly model line rewrite element explanation namely measure equal added function particular contrary reflect fact influence site worth exchangeable measure potential action measure measure suppose effectiveness case satisfied sense potential second equally simplify model measure measure typical potential namely suppose order supposed figure potential measure measure measure location place collect vector covariate partition site miss partition elimination permutation proper partition measurement partition sequel
cubic trend work mechanic penalty extensive comparative penalty tool often consume develop penalty interested reader toolbox site material derivative function list table lrr drop subscript henceforth penalty set derivative reflect compare path path power family thresholde objective soft scad penalize function solution objective mc path either smaller penalize display glm state art solution glm penalty algorithm warm setup predictor default show short ode major compute ode solver smoothly track whole definition institute nc david department university nc propose high dimensional remain routine lar behave highly combination perspective rapidly posteriori hyper instability validation procedure corresponding correspond prior mainly extend filtering scale efficiently large dimensional glm operating apply several keyword model glm lasso lar posteriori estimation convex penalty ordinary solution via prominent last decade find across variety norm lar lasso popularity excellent ordinary square unfortunately serious estimation bias motivate regularization properly large shrink signal non convex convex concave penalty estimation hyperplane iterate support regularization produces iterate estimation penalty category numerically implement sensitive setting sparse demand lar important expensive challenge cross incur criterion criterion tune minimize penalize shrinkage freedom often unclear extend mean bic lose criterion issue general twice differentiable subset scalar parameter penalty general penalization assume condition symmetric iii iv decreasing generality fold problem likelihood precision commonly aforementione assumption vary net vary regression elastic net thresholded oracle scad via partial derivative scad natural knot act scad mc derivative shrinkage function frequently use development list supplementary material general framework knowledge gps rigorous fundamentally gps path specific homotopy piecewise solution penalize efficiently consider lar ode approach fit piecewise adopt paper move improve impose great track least mc penalty empirical imply parameter path shrinkage power induce exponential power double pareto regularization correspond utilize engine appropriately take penalize generalize trend filter generalize regression produce less toolbox regression author site web site remainder organize path follow discuss various correspondingly z index current define hessian active path follow necessary denote article optimality satisfie furthermore semidefinite convex directional derivative order necessary satisfy track stationary along path slide regime semidefinite stationarity minimization check optimality claim stationarity directional function j scalar convexity h tt remark directional bivariate directional derivative negative local minimum passing local go hill abuse terminology ht first serve illustrate secondly orthogonal design building coordinate thresholding rely path non objective fashion list material derivation find assume standardize depict jumps minimum iteratively update take format covariate loss around iterate eq apply thresholding sequentially iterate precede difficulty regression non penalty enter vice cause contrast lasso piecewise make along lar strategy penalty descent find success net article explore descent penalty determine grid tuning advance larger likely devise path strategy track path smoothly path ode z nonzero penalize coefficient hessian restrict continuous differentiable vector write predictor calculate variable independent suggest solve segment promise path care stationarity condition inactive due may reliable inactive regression become thresholde formulae step cause may along whenever nonsmooth optimizer start coordinate strategy summarize determine penalize enter correspond predictor become inactive penalize coefficient inactive penalize jump segment remark path ode give reverse sign termination termination path specific stop predictor exceed subtle poisson separation complete likelihood occur surface behave linearly along dominate mc scad flat infinity penalize appropriate square prior include un penalize probability ap appropriate bayesian penalize proper marginal analytically fortunately manner informative information path also double pareto writing place double pareto un particular bayes penalty pareto otherwise log path restrict normalize jk h unknown laplace normalize q suggests easily compute family exponential unnormalized coefficient approximation bayes penalize h regression laplace laplace normalize plug numerous application path recent regularization penalize shape regression nonparametric density parsimonious sparse non pareto scad etc path hard readily path ease presentation equality assumption trick regularization amenable twice differentiable transformation matrix full lasso matrix eq rank filter fuse trend full row section cubic trend demonstrate financial highly structured compute use cholesky regular variable example illustrate procedure mechanic logistic regression prediction penalty third trend filter simulation small setting display I cpu gb ram example site first concern cancer seven predictor logarithm cancer volume age logarithm amount score set observation path nine penalty penalty continuity cause along power unbiased achieve allow model along path informative variation path upper panel plot penalty proportion solution scale explain scad brevity penalty advantage high explanatory power avoid display evaluate squared error test set path show low panel penalty achieve well highly concave penalty power achieve along lasso moderately quite achieve prediction admit logistic heart set measure heart split path train cancer regression lead bias along path plot versus panel power evaluate test figure penalty able concave one h third illustrate acquisition article constitute company become seven covariate company list long market return finance theory indicate otherwise market market return explore possibly nonlinear effect predictor say code discretized circle figure bin nonlinear monotonically book ratio log market cubic covariate demonstrate use convex log discretize specify bin covariate piecewise cubic two end linear spline regression trend location knot choose display unconstraine path dot bayes mostly match regularize middle correspond model bin formal significant cubic effect term index explain linear logistic finance company unlikely company flow unlikely meet burden associate company possess technology obvious estimate company exceed evaluate performance bayes glm replicate determined canonical model explore
incorrect ol adjustment still error adjust bias subsection ol adjustment sample improve incorrect regularity technical omit motivate experiment imagine slope regression correction choose ols slope fix slope variance sample slope strict naturally conjecture experiment treatment interaction improve asymptotic incorrect ol conjecture confirm unless uncorrelated efficient efficient unless group equal covariate finite infinite finite increase regularity condition sequence population still preserve influence would see regularity condition generalize covariate applicable covariate avoid extra subscript population th treatment whose finite invertible population limit finite least square analogously let covariance theorem corollary supplementary variable eq assume difference slope regression seem analogy help thing adjustment far add add eq ol association adjustment design treatment group derive treatment covariate interaction unless uncorrelated asymptotic variance I estimator asymptotic precision asymptotically iii square pool ols tends fall two group z b ab ab subtracting adjustment add time pool adjustment group covariate value population nine subject nine generally run observation asymptotically material equal van difference mean weighted square iii usual adjustment adjustment covariate sufficient either must similar treatment usual adjustment adjustment even pool single covariate interaction subject outcome independently assign subject remainder treatment adjustment potential outcome opposite sign remark panel theorem limit replace panel deviation bias simulation pt adjust adjusted ol ols ol precision approximately justify accurate solve asymptotic linearity prove finite population q regressor th huber white analog huber formula show consistent asymptotically conservative imply weight supplementary material prediction error material condition great probability eq equal difference generalize design size variance also conservative analogous pp basic related problem pp randomization high leverage substantial discuss university support service service estimate year evidence service alone alone improve likely contact simplify huge percent student service service percent percent available service service group financial service service service estimate close school difference treatment interaction include school school predictor first year usual adjustment appear social outcome lin al pp examine social find value outcome standardized outcome researcher prefer adjust improvement way confidence strong margin interval assumes normally distribute small conservative adjustment iii technical simulation constant effect heterogeneous estimator conservative interact asymptotically percent coverage sample heterogeneity ols estimate bias hc hc hc classic hc hc classic hc hc hc hc hc hc hc keep datum actual experiment service service unbiased slightly simulation treatment treatment improve adjustment panel hc regressors hc remark hc mac page residual formula factor appear panel way percent standard estimator pp use hc standard degrees freedom fourth panel average agree adjusted adjustment yield inference without one limitation bias treatment heterogeneous covariate lead minor potential center bias balance constant substitute school covariance college square center high service service limit remark would order magnitude table bias negligible replace lead pp useful interact validate analysis limitation shape extreme skewness group size serious service roughly simulation inference goal check sample coverage interval weak strong effect permutation asymptotically ol effect heterogeneous literature remain valid asymptotically nan hypothesis exploration approach adjustment adjustment summarize empty highlight extent encourage researcher probably difference specification encourage strength conclude always adjustment reporting always randomization traditional believe adjustment health agnostic perspective theorem major contribution discover property derive adjust assume treatment argument detail acknowledgment small valuable david associate read berkeley helpful discussion david help early education theorem ordinary adjustment estimate randomization conventional adjustment lead bias sample either minor easily ol interaction include interval huber approximation illustrate evaluation student reason randomize ideally average assign researcher adjust characteristic adjustment tend argument regression correct researcher conduct social adjustment adjustment many influential assume randomization effect subject linearity assignment variability treatment adjustment asymptotic inconsistent iii adjusted treatment randomization justify behind ol perspective agree general defer argue sufficiently minor regularity ol full interaction huber white whether include distinction traditional ol property assign interaction asymptotic omit example write random justify biased negative second provide intuitive property ols adjustment incorrect view depend assumption similar precision subject infinite interaction generally adjustment may adjustment covariate interaction page source randomness infinite pp purpose population need address major concern help understand balance adjustment main result
iterate empirical counterpart provide mistake unbounded iterate exist penalty far constrain norm precede paragraph manuscript problem split optimization piece thank piece unbounded form core linear linear regressor correspondingly track support core split hard core core nonempty produce simply unclear core countable existence hard core somewhat appear let follow property say grow optimization proceed hand either set region etc impose mistake occur loss within force core r h nh property transfer sample crucially bound quantify exist every restrict sample establish order manuscript measurable property probability lf f general q provide useful logistic determine precede statement draw sample weight suboptimal correspond unbounded portion eq unbounded bind bound differentiable available correspondingly strong albeit h suboptimal l almost assumption presence finitely manuscript class asymptotically grant probability measure hypothesis l h significance consistency class borel denote axis align split take family classical theorem compact measurable function similar differentiable exist subsequence sequence suboptimal f manuscript basically consistency course elegant apply powerful class scope manuscript measurable lastly everywhere measurable everywhere follow everywhere contradict since turn lastly bind probability loss truncation since bound subdifferential correspondingly every lastly desire let rademacher b choice every rearrange use finite convex notation eq define subdifferential subdifferential integral heavy result cf zero everywhere fact conjugacy formula next notice give since continuous set define continuity dominate term conjugacy result provide integral continuous mutually conjugate lastly derivation subdifferential result supremum attain supremum zero operator semi convexity question lower since everywhere fact let take optimum p np entail provide transpose conjugate thus desire appropriate note significance act constraint lastly banach fr meet follow desire goal duality expression statement consequently term next grant restrict always iff dual lastly present optimum iff part subdifferential desire mean structural crucial define set statement exist satisfie everywhere exist cl metric direct construction crucially subspace orthogonal exist h n moreover argue projection countable take nothing compact attain provide remainder desire say sequence every representation every contradiction representation satisfy unbounded e define eventually similar due low cf assume replace h also minimize lie within compact perhaps subsequence since continuous cf attain duality measurable primal minimum since everywhere finish construction satisfying order invoke result material minimizer although appear depend proceed demonstrate suffice mechanism proper may duality attain infimum employ show subgradient grant rate step size index finish attain measure weighting extract suboptimal representation lie lie minimizer finish attain infimum necessarily optimum I achieve particular infimum attain generate possess logistic finish let exponential early imply henceforth low suffice furthermore eq state next define countable support positive example margin go key max class maximum solution enough henceforth one positive sample point example maxima since margin solely eq mean rescale wrong follow employ let restriction existence core section property limit exist monotone convergence exist limit pointwise measurable meaning ip thank dominate countable q nonempty always supremum grant continuity attain supremum relationship core tie analogously possibly primal core contain follow satisfie close countable intersection start intersection every correspondingly integer due finish whereby countable family consider whereby continuity intersection sequence definition consequently exist subsequence primal core always contain minimize infimum finally finite grant continuity q attain infimum existence equivalence core core elsewhere p construct meaning follow uniform thus subsequence thus exist exist suboptimal b set property core exist core take provide definition core per automatically since exists misclassifie arbitrarily example every nn example fall within within consider line segment thus every deviation twice low return form grant r mh comment fix grant correspond lf inverse failure various final statement x depend interpret whereby statement chernoff basic mf henceforth exist cf portion measure linear space deviation version vc mistake plug bind suboptimal predictor particular since grant sequence c lie since combine piece precisely cf provide simplify precede bound refinement let core provide core set large refer corresponding strong ff event guarantee de meaning bound large proceed next allow consideration restrict approximate infimum attain exist correspondingly cf lastly q crucially pass compact verify borel separable metric compact hausdorff henceforth measurable compact satisfies continuity compact imply uniform since outside form half open interval correctly partition side vertex partition arbitrary finish form align lattice model modification stage straight choose necessary massive proof study minimization input pass treat consistency boost class hinge loss fit scale complexity class surely binary operate feature label iff interpretability popularity rather try pick mistake intractable attention clear zero penalty margin specifically specialized logistic analysis scheme path positive interpret perform search provide way analyze parameter analyse full certain
bernoulli bernoulli inequality leave organized loss generality optimal suboptimal integrate want forecaster forecaster mistake reason arm play reward random arm th respect case event algebra order behavior forecaster bandit shorthand modify optimal satisfie step observe strong positive eq deduce q recalling n extract last result prove reasonably restrict except section ucb argument allow whereas require improvement subtle condition author take ucb light theorem gap towards gap replace hoeffding bernstein derivation theoretical guarantee second approach replace neighborhood ucb neighborhood line start later also ucb attain finite kl ucb attain kl strictly reward tend incur idea easy regret shall modification ucb get attain small numerical small order regime attain minimax always improve plausible regret appear variant arm typically quantity probability close expectation dominate concentration ucb analyze detail fact around theorem horizon impossible concerned strategy concentration greedy strategy play empirical time grow practice first distribution thompson proceed past I recently mainly prior reference therein behavior thompson property first fact attain essentially interestingly reasoning frequentist moment generate tail distribution become provide moment surprisingly essentially refine basic ucb order satisfy interest robust variant armed bandit make generation reward gain step simultaneously player arm adversary arm player loss gain step regret gain version sense translate set whenever proof simple achieve sublinear round possible adversarial gain deterministic exist suffice q key randomization play forecaster adversary get stochastic goal respect randomization adversary task first regret upper argue pseudo regret adversary pseudo coincide point guarantee adversarial fundamental idea exp exploitation number trick estimator arm namely next play indeed forecaster weighting scheme standard science weighting provide pseudo assume forecaster version instead knowledge pseudo exp non exp divide verify particular imply idea reader recognize quantity arise analysis two step step third let fourth putting obtain term easy conditional eq simplify uniform probability unfortunately exp define adequate task exp weight distribution uniform variance cumulative order say exp idea estimate loss core gain gain trick estimate expectation last use exp uniform gain estimate gain arm describe sake describe time horizon exp eq exp first prove exp q first immediately g thus key however notation let induce exp mix use inequality last third summing obtain come lemma union bind consequence yield derive regret adversary regret expect regret pseudo case difficulty integrate exp p integrate deviation formula value random variable show section reward pseudo forecaster average probabilistic kullback leibler aware technique prove bandit convenient bernoulli arm supremum reward generation reward internal randomization forecaster r immediately regret forecaster time bernoulli bernoulli parameter whose forecaster formalize kullback inequality forecaster arm use low expectation reward parameter forecaster five statement max empirical play forecaster play round law distribution adversary recall inequality immediately j I computation forecaster deterministic sequence reward forecaster uniquely determine play law conditionally stochastic adversary law adversary kullback example law precisely fourth step conclusion concave proof deterministic observe result forecaster randomization k realization bit chapter originally forecaster observe complete history problem strategy p detail analyze chapter many analyse logarithmic exp bind construct strategy inf forecaster idea first exp forecaster inf translation precisely inf probability tc tp regret take exp correspond realize inf mirror significantly refer reader pseudo significantly free forecaster action moreover vector include adversary strength propose adversary bad loss chapter logarithmic ask minimax prove adversary exp pseudo regret log maximal reward gain explored building author attain exclude variation arm apply bandit ingredient reservoir slowly bandit describe combine minimax pseudo compete consistently play define regret might big natural chapter compete policy switch play simple attain switch prove viewpoint exp safe well match regret describe stochastic adversarial logarithmic guarantee ucb n n flexible sequence reward relevant interesting strategy logarithmic adversarial armed bandit variation full feedback signal loss feedback chapter describe propose end round decide ask loss current know set minimax regret fundamentally algorithmic idea besides forecaster select round reveal speak simple coin regret model bandit undirected vertex vertex arm encode loss neighbor equivalent graph full feedback minimax regret factor number naturally click plausible display example monitor incur stochastic customer dynamically adjust customer item price item pricing item signal incur loss reader include historical recent problem arm contextual optimality mapping arm well optimum typically desire context policy contextual information forecaster arm news recommendation pool candidate news user website arm click article activity may detail application bandit general presence create variation combine reward nature available remark mention aware basic example contextual round mark forecaster must context arbitrary exp introduce bind adversarial bandit section exist bandit contexts instance first theorem identity subsection extend several immediate adversarial theorem mark variant basic adversarial set policy randomize context bandit seem role decide include set bandit recommendation forecaster obtain arm randomize kkt jt depend pseudo adversarial bandit introduce pseudo bandit eq weight exploration exploitation number uniform get probability arm probability loss estimate expert exp chapter set give order contextual forecaster exp framework even exp contextual setting run exp arm expert maintain exp come exp expert exp show moment arm bound exp exp without apply analysis forecaster use pseudo loss inequality similarly inequalitie jensen get note draw use proof besides similarly exp modification basic scenario arm mark contexts eq contain forecaster set find variant bandit nontrivial combination examine exp particular scenario exp expert aspect expert combine learning context theorem give regret improve scale idea explore exp class combine us logarithmic albeit dependency intuitively expert immediately exp hold play draw requirement bandit game play arm distribution kt ki ni tt exp different tp obtain observe statement reader give contextual variant exp exp assignment assignment mix coefficient mix achieve pseudo exp mixing coefficient satisfie along modification expert past forecaster way exist forecaster set run forecaster mixing instance expert distribution exp instance joint k bandit inherent exp improve construction mapping context forecaster supervise standard context supervise contextual setting forecaster time research generation variant contextual context draw analogy learn theory evaluate risk class arm view counterpart adversarial regret characterize supervised rest policy theory follow dimension satisfy arm uniformly play simulate exp policy exclude logarithmic vc price bandit information essentially vc internal randomization realization element function state class binary finite randomization chernoff least internal randomization quantify agree point remain remain randomly position g realization sample desire viewpoint variant protocol sequentially protocol keep classifier step side notation online denote predict label online protocol learner observe associate use adjust observe prediction bandit online multiclass perceptron matrix rewrite update correctly generalization perceptron number mistake notion infimum denote hinge iy multiclass represent variant multiclass observe p tp operate protocol multiclass true available correct inspire bandit w expect prediction mistake make sequence mistake multiclass perceptron bandit prediction mistake q need bind condition prediction otherwise hence ready mistake follow perceptron derive w n du jensen inequality see iw eq low statement assume bandit study reward parametric class non metric lipschitz remarkably slowly change reward covariate finitely dependence reward study model capture dependence reward model subsection adversarial p correlation expert contextual arm finitely use contextual bandit model important loss function hand sublinear bound infinitely time arm naturally bandit source introduction good choosing path path optimization forecaster simultaneously adversary choose incur pseudo forecaster internal randomization similarly feedback forecaster observe loss note far assume incur vanish extra playing rather original approach computationally us idea chapter analyze without generality play play remark critical powerful subsequent scalar loss without rewrite element arm discretized show appropriately exp describe strategy first geometry concern ellipsoid minimal volume contact contact convex set ellipsoid contact contact position strategy hull set ellipsoid play detail transformation satisfy select build play outer product matrix invertible tp quantity exp algorithm exploration correspond playing probability parameter correspond exp exploration support contact point exp finite adapt analysis exp take whenever hand write v low small introduce mirror descent problem losse convex end forecaster observe analysis section online mirror apply obtains let subgradient convex convex closure admit partial definition bregman f fx fx fy understand act proof q map original primal divergence primal exactly bregman divergence resemble geometry square inequality convex moreover simple space primal back tw rewrite convex imply set describe description outside dual differentiable exponentially average similar chapter know strategy take powerful compact convex obtain w q divergence fx domain step compact effective use bandit information extensively gradient perturb way observe estimator present restrict loss simple specialized performance true mirror mirror descent compact set rate initialize perturbation observe fx g relate bandit strategy forecaster forecaster arm coefficient pseudo theorem assumption pseudo compact convex replace use obtain combinatorial loss many fall rank planning semi feedback playing observe td namely observe loss thus full still bandit basic key tackle kind random table view perturbation play surprisingly randomization perturbation loss loss unbiased unbiased indeed directly concrete factor subtle start entropy set perturb note computation give entropy continuously associate potential regret instead reduce entropy potential check f u taylor expansion ready bind run potential specific potential improvement adversarial multi run choosing first inequality sx qx particular old loss bandit section action belong loss time feedback receive scalar obtain order logarithmic show one obtain show chapter another strategy factor base follow perturbation interior random draw rademacher variable unbiased ready suitable euclidean ball feedback play incur convex composition euclidean open regret second need first algebra fact inequality soon since triangle conclude suffice online feedback uniform improve last exp mirror mirror originally seminal convex online community algorithm online mirror explicit see mirror application mirror self barrier unfortunately drawback suboptimal dependency regret precisely attain mirror bound extract mirror development take go back inspired topic extensively recent account view propose mirror strategy proximal see viewpoint finally mirror prediction several see semi series adopt last generalization bandit feedback done derive introduce inf strategy understand instance list omit important briefly review assumption feedback bandit case minimax regret semi order match exp remark exp provably set chapter focus pseudo chapter concern bound partial result direction exp strategy algorithm similarly restriction adversary prove logarithmic long polytope progress chapter scenario arm loss necessarily function precisely adversary select value define consequence loss perturbation order study chapter agreement initially set face nonlinear loss problem kind view dynamic evolve evaluate loss play choice evaluate point forecaster regret investigate section similarly arm return unlike mean lipschitz unimodal necessarily keep thing pseudo regret analyze point forecaster loss extra choice neighborhood estimate control reason euclidean result strategy descent parameter convex rate initialize estimate introduce technical point play two sx symmetry compute side preliminary unnormalized consequence random differentiable uniform q satisfy hence equivalence relate integral sphere ball generalize immediately conclude tx provide smoothed exploit loss smooth take lemma perturb estimate convex lipschitz convex loss process obtain gr second moment exploit ready gradient order point loss feedback set rx immediately since loss conclude proof pseudo point pay feedback rate estimate obviously tx
employ high layer triangle soft threshold feature encode image patch effect order similar approximation code sparse code solving close example video author take solve code identify predict dictionary dictionary directly code quantity surprisingly uncorrelated evaluation experiment outperform proximal structure proximal extension operate dual space outline encode chapter soft code proximal outline soft chapter usefulness conclude offer necessary support focus proximal scheme variant dual tractable throughout chapter directly code applicable broad problem presentation reader familiar lagrangian justification discussion support find standard concern give quadratic equation introduce access refer variant central chapter problem iterate step converge method scheme scheme pick inverse form proximal consider approximate approximate arbitrary algebra show update amount multiply step ensure selection scheme specialized consider primal connection form q forming require minimization split ascent converge unfortunately problem problematic unbounded ascent augment parametrize lagrangian problem nice augment ensure guarantee dual feasible write dual proximal ascent ascent derivation connection proximal iterative difficulty gauss block instead modification lead since problem code feature process many introduction allow reconstruction dimension reconstruction alone unique encoding code uniqueness introduction phase phase representation encoding phase dictionary provide external source since reasonable performance offer condition want explicit phase formally encode collect dictionary write regularization reconstruction fit chapter element add positive sparse code encoding phase together quadratic easily optimization successive focus lead since sparse coding refer together term refer equation non whereas non sparse typically imply treat fix problem first three proximal framework fourth solve encoding work iterative soft thresholding name thresholding arise threshold encoding refer statement iterative thresholding fista momentum fista fast reconstruction handling problem chapter framework fista set scheme development discuss full coding algorithm share namely fixed objective still iterate present chapter encoding name nonetheless encode form lead rather alternative cause mapping vary solution cast functional descent weak learner start regularizer set force arbitrarily equation optimize arbitrary case iterate adjust size fista feature proximal choose available suggest feature threshold multiplier operate space value choice essentially arbitrary value expression long single encoding admm soft threshold perform iteration admm force factorization order inversion choice feature proximal add get feature sparse admm smooth newton overcomplete thus rank replace inverse numerically admm ridge take interpret elastic admm inverse encourage briefly unlike one element lead magnitude mean step feature arbitrarily dense compare accommodate sequel writing feature omit iteration find sparse report cifar experimental framework pixel cifar feature examine accuracy encoding library patch subtract library centroid include dictionary directly normalized smoothed random feature dictionary appear build training image extract densely encode dictionary iterations pool encode patch involve patch separate encoding alone repeat number different algorithm encode rather non regular coding give select admm well order maximize c admm summarize cifar notable feature exact run iteration performance fista admm nearly high produce multiply long iteration different problem equation vector problem aggregated column replace library multiplication figure fista actually drop seem interpret implication carefully figure reason expect lead performance optimize optimize contribute drop fista unstable regularizer threshold comparison threshold cite work discussion fista l fista admm design relate relationship reconstruction previous construct describe result previous experiment library cifar encode patch encode previous experiment correlation reconstruction give consider bad another feature experiment lead optimizer reconstruction beyond confirm include run reconstruction produce inferior ht soft gradient descent sparse encoding surprisingly successful broad framework four feature sparse encoding approximate proximal encoding image benchmark objective around minimize reconstruct patch dictionary would expect reconstruction demonstrate intuition obvious extension would thorough exploration insight suggestion evaluate properly usefulness report use dictionary examine problem investigation indicator appear effective smoothing connection elastic net understanding regularizer empirically well different coding extend induce potential author investigate proximal method definition recently certain feature achieve image belief net factor rbms rbms autoencoder several moreover intuitive put forward remarkable performance justification main realize negative sparse variant several sort ignore problem localization predict location report common tool introduction neural model serve purpose tailor specific provide encode construct way vector classify accurately raw pixel neural network prove tool representation major barrier require image technique
ensure sample variance estimator default apply power randomly power eigenvalue power eigen leave towards direction type analogous shrinkage optimal orthonormal basis author assume target rank aim excellent even relatively noisy adaptively detail compute basis corresponding eigenvalue low compute project computational randomization multiplication runtime normalize runtime stable propose stability shrinkage describe rank stability guess e basis assume approximately enhance th eigenvector assign eigenvector th direction dominate expect sharp eigenvector signal partition estimate value optimal accuracy bi validation row partition respectively result submatrix block omit modify complement cross frobenius submatrix training stability suggest fix top become estimate bi three block optimize arrive final formulation generalize solution base reduction sir outline nonlinear assume characterize reduction correspond assign basis subspace sir span structural impose applicable unsupervised first statistical visualization predictive modeling goal latter quantity dimensional focus g r rx sir regression sir effective data global marginal projection manifold inverse sir sir matrix estimate linear sir assumption problematic linear manifold structure sir adapt localize inverse explanatory dimension structure around suggest compute construct local index neighbor consider response within slice belong sir reduce truncated reduction encode grouping classic typical cholesky back canonical approach progress projection span datum prohibitive approximate explicitly recover rank restrict span subspace contain consider sir slice matrix construction tx construction full reduce svd sir number slice substantial respectively entry near fix knn nj knn pairwise point knn knn lemma dimensionality projection hence fix n generate matrix eigenvalue eigenvector transform ambient estimator runtime randomize fast approximation information reliably solution truncate generalize add control improved sample begin contaminated pca method achieve art require runtime scale input propose applicable lastly adaptively coarse default uniformly sphere exponential increment iid es noise signal noise rank effect large strong use input average exponential sample variability cost multiplication ht c ccccc error absence randomization well established operate tend complexity typically product runtime rank relative approach base notice runtime approximately increase suggest order top p rank implementation p consecutive simulation replicate q percentage explain focus noise regime vary weak scenario direction covariate direction assign factor spherical criterion correlation report third predictive criterion use square onto subspace coefficient proportion set different n examine size sir slice ten neighbor subspace value randomize method result pn randomization beneficial irrespective set low randomization seem sir term adopt seem ccc ccc sir sir rand rand l ccc ccc low noise sir rand sir rand rand performance knn reduction supervise accuracy slice sir equal first contain digital handwritten grey scale digit extensively past contain ignore dependency neighbor remove become detail appendix reduction method reference fix value rank neighborhood accuracy training consists select identical fashion rank digit sir limitation allow number sir tend small sir pca increase digits train rand acc pdf digits train rand acc microarray purposes variation process classify origin expression cell individual population china feature pre neighborhood slice estimation classification report accuracy reduction estimate size upon baseline projection sir strong structure gene predictive produce comparable increase suggest potential advantage randomize parsimonious reduction model massive modern paper problem analyze tool recent estimator implement popular parametric supervise method sir regularization randomization quantify randomization contribute practical utility interesting relate acknowledgement sm acknowledge wu sm acknowledge support grant biology gm fa dms sg acknowledge slice nearest step fix instead basis matrix dimension rank describe set project onto axis digit mnist com mnist handwritten grey digits dimension large centered training file image microarray microarray expression replicate use version genome array input background correct intensity thorough annotation annotation processing pre process expression individual nd mark annotation remove technical improvement outline idea randomize text manifold focus localize locality laplacian method introduce allow reduction embedding seek datum embed ambient infer manifold represent neighborhood adjacency le preserve adjacency association laplacian decomposition eigenvalue nd notice certain laplace manifold motivation typically computational advantageous linear projection without locality projection approximation non laplacian reduction specify space locality preserve projection state bandwidth exclude trivial direction order eigenvalue neighborhood optimal require generalize last derivation entrie sparse neighborhood construct detail algorithm neighbor capture manifold subspace default power iteration value embed subspace p projection randomize complex generate underlying data scientific application increase feasibility reduction natural massive play role visualization predictive significant impact research review theoretical instrumental powerful provable stability classic efficiently dimension consideration error consideration runtime information apply estimator generalize mean scope work implement truncate randomized independent propose number infer svd reduction provide attractive accuracy argue due implicit impose approximation adaptive svd randomize estimator inverse sir localize illustrate methodology algorithmic extension three pca sir robust estimate direction randomize reduction low serve core engine reduction reduce truncate generalized setting input consist feature response sir apply widely case provide outline extension develop manifold focus locality
lead bt ct transform regression regressor collect observable reduce copy estimate large estimate quantile technique cutoff I bt give observe covariance sx similarly typically il simple notational estimate j integral estimate result slope function optimization transform solve plug choice cutoff discuss pca basis basis fourier basis become drawback discuss analysis paper reproduce quantile example plausible hard see theory also apply interpolation observation understand nonlinear correspond regression case limitation shall discuss nonlinear ill interval monotonically construct explain iii xu xu xu carry project adjacent convex xu xu show sense derive previous optimality j beyond scope dependent mild restriction moment exist c xy xy xy xy xy xu assumption smoothness require ensure identifiability eigenfunction thereby follow uniformly page role flat quantile essential define il xt impose discrete covariance twice continuously differentiable control discretization depend vice versa relation integral kernel reference condition precise case satisfy discretization speak old continuous xt compatible establish slope suppose satisfied inspection continuously observable assumption discretization negligible seem quite example seem mild functional analysis supplementary file establish average criterion tend small affect estimation work well cauchy work look one find perform dominate bic work figure increase become well essentially similar comment figure supplementary file deviation c normal cauchy cauchy theorem divide three subsection prove supplementary file notational uniformity line almost surely let b mu constant q soon mu mu conclusion straightforward hypothesis mu proof minimization u proof distribution absolutely surely however absolute conditional right mu define step event mu mu complete verification lemma pick define n separately order xu x xy pick fix independent rademacher apply conditional vc subgraph class universal constant envelope assign q universal q show taylor xu q xu xu xu xu order evaluate bound independent rademacher independent apply conditional regular conditional function h proposition appendix bind measurable c I constant necessary order conclude order evaluate mu mu mu bt jt jt jt j jt bt dt mu j j u j b u j mn complete acknowledgment author mit thank suggestion anonymous constructive theorem remark aid b quantile covariate subject per increase size give unit quantile index likewise function quantile covariate slope estimator plug plug necessarily monotonicity quantile index consider estimator rate norm rate minimax kernel slope initially seminal kb attractive make estimate quantile material quantile functional quantile scalar covariate quantile model covariate suppose differ subject number sample allow quantile vary interval slope two time quantile quantile quantile index covariate slope slope pca terms standard estimating estimate estimator construct necessarily monotonicity conditional quantile estimator slope plug estimator rate minimax smoothness cutoff empirically namely integrate aic choose cutoff criterion limit simulation although criterion work increasingly grid multivariate large grid high view take datum refer reader comprehensive treatment early study functional linear reference establish fundamental linear derive slope continuously expansion slope function early quantile function quantile model establish rate sharp nonparametric estimation quantile function consider indirect conditional quantile possibly adapt developed invert estimate regression conditional quantile parameter check conditional quantile indirect direct indirect flexible offer contain value conditional useful appear say occur growth height picture predictive height know mean response growth datum quantile linear benchmark quantile quantile vector slope empirical ill pose cutoff level ill pose slope function conceptually handle paper nature nonlinearity technical view challenging build upon asymptotic estimator g arise essentially regressor estimation careful moment calculation additionally bring technical account covariate technical formal remainder
penalization powerful role classical regression popular penalization technique net regression covariate direct penalization dimensionality exceed incorporate structural importantly fundamental parameter regression well low number see affect regression illustrative example x z usual size rest inner differ usual detection imaging infer adjust b criterion bic along solution suggest truth solution nuclear singular nuclear size display nuclear matrix nuclear norm achieve substantially lasso might argue fuse lasso might piecewise signal numerous fuse lasso article family covariate regularization formulation within generalize glm variety penalization lasso elastic net scad response outcome highly scalable scalable effective select parameter extension freedom development hand aim scientific receive attention involve tensor multidimensional array covariate idea cp introduce fit tensor thresholding covariate case contrast fix thresholde non logistic work research completion aim portion concern covariate organize follow optimization degree propose discussion potential future section notation vector order mapping instance model belong simplicity covariate associate straightforwardly univariate extension model generality regularization glm likelihood bf square parameter indexing tuning penalty lasso ridge penalty elastic penalty scad penalty scad spline knot act signal penalty lead regularize estimate mc scad beyond lasso shrinkage comment besides penalty depend scientific w q produce e cluster eigen convexity essential study necessary sufficient furthermore subdifferential singular v v lead optimality regularizer convex local minima regularize program global minimum strictly u loss convex nesterov minimize state building algorithm v share b singular thresholding nuclear regularization singular u singular matrix thresholding ready nesterov nesterov attract regularization resemble descent first utilize search algorithmic incur improve rate dramatically optimal wide nesterov predict gradient force iterate initialize h critical role update nesterov whereas also base loss search irrelevant optimization ridge act trust region next iterate loss denote differentiable u u update dynamically obtain value decomposition intermediate matrix singular top retrieve efficient solver rank row lasso part minimization know nesterov iterate nuclear regularization summarize omit reader refer continuously differentiable iterate monotonically remark decrease guarantee potential algorithmic essential objective step fail objective iterate fortunately rate nesterov cover commonly convex nesterov reweighted square nesterov penalize least convex way glm step expensive rough potentially cut twice differentiable continuously v glm systematic b x cauchy v parameter yield well along regularization path criterion commonly practice datum considerable attractive criterion report penalty power finding illustrative shape synthetic compare vary lastly motivate eeg mention employ examine variety shape follow aforementioned regularization square signal disk exact see version counterpart yield signal lasso lasso power power lasso power signal compare rank generate covariate covariate consist percentage entry entry level rest zero binomial response systematic normal binomial former bic mean independent validation tune set rmse mis common table highlight bold face since rmse array coefficient brevity sparsity method power power power lasso power lasso lasso response penaltie exception non zero element binomial rank superiority predict signal insensitive contrast propose version insensitive sparsity agree expectation coefficient finally penalty yield lasso perform well agree useful eeg regularization individual individual subject channel perform stimulus match one association pattern eeg randomly subject control horizontal vertical axis curve distinct analyze stimulus trial result covariate indicate th power specifically divide full training fold leave far fold test misclassification error report leave see regularize counterpart lasso achieve well leave fold remark glm method regularization employ ridge maximized classification well fair solely train report testing error optimistic principal pca employ variant pca reason preprocesse eeg apply potentially information layer choose version determine number goal datum analysis primarily counterpart require tune modern area regression regularization crucial role regression complex propose regularize spectrum upon signal approximate focus rather nonzero entry penalization original coordinate jj often eeg fouri direct interest regularize analog concentrate problem throughout fmri array tensor extend tensor regression tensor analogous fundamentally currently report elsewhere pt proof lemma develop matrix fan fan entry order low statement fan singular let function order conjugate f f value v subdifferential inequality x x simultaneously u f equal decomposition b b b proof calculus gradient multivariate jacobian hold partial derivative jacobian chain jacobian fit value freedom denote usual square estimate v tackle piece singular right singular eigenvectors b coincide singular rule b p u b jacobian b b p cyclic permutation trace u p u p p b b piece symmetry let value utilize matrix chain b v shows b v I u yield next ridge response piece show
region select good function introduction weight feature feature entropy addition feature speed viewpoint redundant list construct line help redundant viewpoint line example remove fail might compare exhaustive fortunately extreme happen fail construct decide decision system q case indicate would execute remain base similar hard expert matter always competition working specifie obtain maximal winner construct exponential specify competition strategy max purpose answer question select repository database list pt ht vote game decision vote name library test information correspond three distribution pareto simplicity test ds test cost ds ds backtrack three backtracking step size diagnosis optimal always third run time backtrack propose step min backtrack voting table show backtrack namely backtrack size backtrack tree dataset sometimes indicate sometimes sometimes maximal compare backtrack run algorithm take real backtracking algorithm time information extension computation concern among sensitive reduction define classification classification boundary classification address extension concern symbolic partition transform combine concern different computed loss major model semantic application rough indicate precision rough concern probabilistic rough rough set neighborhood rough region rough error range range test information entropy meet misclassification decision cost misclassification value likely select weak less coincide compute reliable parallel unlike positive conditional increase change relation similarity entropy change minimal total meaningful identify definition change issue combination exist extension involve extension area research viewpoint concern test cost constraint limited backtracking algorithm experimental efficiency backtrack one effectiveness competition heuristic note competition viewpoint rough selection natural generalization viewpoint help identify meaningful aspect input research trend concern beyond rough set natural foundation science foundation china j technology project china grant world cost include limited resource shall informative constraint backtrack efficient solving sized backtrack scalable heuristic heuristic find theoretic rough set viewpoint new research sensitive constraint backtrack theoretic set employ improve hundred attribute rough jointly individually decision often address region problem minimal simple representation often provide generalization problem aim maximal minimal store free application resource take like select number develop relate identify address numerical observational searching subset preserve information nevertheless resource limited necessary constraint resource serve problem coincide constraint condition fall namely input new simple viewpoint furthermore viewpoint show selection rough rough minimal aspect viewpoint insight trend concern broad viewpoint attribute systematic backtracking medium backtracking algorithm obtain however deal see e reason people problem exhaustive backtrack respect heuristic complexity dataset employ function prefer low stop improve algorithm employ competition value open algorithm experimental backtracking sized obtain backtrack time fast another backtrack competition rest organize problem propose backtrack uci california exist feature rough viewpoint interesting problem briefly conclude review rough classical rough concerned new selection test constraint fundamental ds q object universe decision information feature partition determine brevity pos iff pos sufficient individually necessary preserve context decision name necessity attribute reduction pos pos may exist relative simple minimal objective constraint namely necessity meet necessity feature minimal word simplify view hierarchy sensitive minimal output see problem input cost external sometimes issue respective ds constraint eq feature subset q set element constraint brevity constraint meet select ensure test minimize construct read problem cost input bind objective important primary objective problem simple phenomenon appropriate observe constraint region objective secondary primary minimal subject problem meet essentially redundant serve constraint objective consequently coincide backtrack backtracking produce problem heuristic subset backtrack rough people employ attribute reduction partly form backtrack invoke global statement backtrack execution store tb store backtracking continue constraint violate exception coincide subset backtrack backtrack discard coincide need devoted optimization serve implementation repeat positive note remove ensure correctness
contain projection code maximize entropy partitioning provide select projection equally candidate entropy consume thus center simplification reduce time calculation obtain entry sort top overall l definition adjacent group intercept sort create binary data dimensionality mean adjacent intercept alg compute obtain step alg clear scale stage need code sensitive hash high near million feature dim publicly collect extraction dim publicly sift dim publicly remain return close distance accord distance average precision recall average pt seven algorithm near neighbor locality hash projective randomly locality hashing generalize space use shift approximate invariant code principle hash direction projective pca publicly code author anchor construct anchor superior use author nearest publicly three control group adjacent selection lsh projection method regard lsh cccc c lsh cccc lsh sift set three lsh low consistently increase fail code increase decrease code length probably later use poorly discriminative geometric projection advantage method almost outperform sift large sift present bit time algorithm method dimension consider projection lsh computation relatively slow explore fast algorithm test lsh method long analytical eigenfunction involve sift sift alpha alpha alpha sift sift control number group performance bit vary achieve change increase reasonable balance consider efficiency sift vary adjacent consistent separate projection critical redundant decrease hash call hash neighbor base hash locality hashing guide projection hash empirical study three set scale hash let proposition thm li fundamental mining pattern recognition hashing locality sensitive hash lsh hashing find table hash require precision recall address propose hash hashing regard lsh explore geometric avoid purely projective function agree extensive experimental world set locality cluster neighbor nn efficient build structure propose neighbor unfortunately bad scan dimensionality high intrinsic difficulty propose search key datum preserve hash random preserve one locality lsh offer time query sub linear however lsh random preserve therefore order probability object hash lsh many hash long lead storage full hashing affinity item item similarity code success small code fail increase hash call effective dimensional near neighbor regard extension lsh hashing try utilize geometric guide projection hash adjacent group one select entropy hash art review hash experimental result art remark hash hashing intercept hashing find respect hash locality sensitive lsh lsh relative map hash code hash lead storage space limitation hash propose pca hashing simple choose direction exploit affinity binary spectral analysis affinity consume hashing analytical geometric ignore anchor hashing overcome generate affinity efficiently hashing hashing stack restrict rbm ph supervision learn hashing fail rp detailed sensitive hashing increase length framework lsh lsh generate guide selection present toy illustrate idea gaussians plane ask encode hash code lsh projection hash projective four gaussians hash generate code generate subsection quantization recently performance search
regret cumulative loss loss typically notion sequence evaluated online convex mirror consecutive bound mirror regularizer regularizers achieve work corner different corner expert sharing share achieve set simplex proper trajectory corner mirror generalize set simplex notion distance rely generalized notion include case regret project mirror fix essentially technique online improvement loss parameter simplicity show extended linearization may cast repeat game forecaster denote simplex forecaster choose forecaster accumulate g goal forecaster expert choose rich state goal describe generalize formulation algorithm parameterize past make serve generalized algorithm share rate dt e past weight g discount see regret discount factor instantaneous forecaster sequence vector variation expert regularity dd dd traditional obtain share algorithm perform supplementary material generalize optimize tune tune corollary share eq exactly share expert almost well vector eq correspond time fixed forecaster advance norm valid order optimize maximal illustrated omit vector optimally proof theorem multiply examine first update substitute substituting summing sequence loss forecaster regret specialize modification forecaster study drop forecaster corollary adaptive regret forecaster change shift notion tight exp forecaster enjoy guarantee forecaster update optimally mention obtain factor regret sequence remain r bind theorem introduce discount old horizon close game theoretic set matter recent close mostly monotonic sequence decrease non assume requirement get satisfie u natural quantity something claim whenever monotonic regularity monotonicity know advance arise tuning discount order fix via contrast guarantee therein trick fix forecaster gain manner next must horizon discount latter weight positive fix tuning time basically know advance forecaster easily focus loss tuning parameter regret forecaster improve expert focus share update run update sequence l l online technique refinement substitute analysis section replacement state introduce able switch time appear naturally sequence upper union support generalize update generalize simple satisfy exist improve theorem suppose share vector sequence corollary material tuned quantity horizon trick parameter extend focus run replace description loss number define way update performance sequence loss constraint provide illustration acknowledgment acknowledge national grant exploration exploitation resource allocation use body simplex simplex forecaster bind different beginning depend remain modified u I proceeding claim next left side rewrite I I entail dc I w nonnegative since maxima nonnegative e satisfied one q real well understand optimally choosing get numerator denominator straightforward need use ss bound
linear price vector budget bundle bundle structural immediate pair equivalently bundle good operate ratio sort optimal bundle spend item ratio high probability op price budget intuition bundle ratio bundle bundle need bind good preference well follow observe bundle class j pr j pr j bundle I example pair less bundle know preference bundle modify utility agent separable derivative concavity marginal item agent ia v ia ib derivative utility first adapt function bundle characterize p ix bx maximum bundle budget bundle intuition follow utility observation optimal bundle derivative need infer bundle k initially li kx li kx kx jx j kx kx new thresholds j b budget maintain ratio derivative distinct equally maintain ratio discretization convenience bound analogously maintain ratio concave maintain throughout pairwise ratio maintain pairwise ratio derivative infer item bind analogously update update appropriately completes use item property ki rest step represent comparison prefer section threshold sequence ratio threshold sequence threshold learner desire polynomial threshold include conclude threshold bundle optimum bundle discretization bundle admit separable introduce get strong feedback complexity require interact constraint linear bundle agent else return bundle either bundle almost receive linear program enough cut fraction polytope probability polynomial argument polytope begin model agent utility say bundle bundle accept agent bundle propose instead inequality optimality return pair two price domain distribution suboptimal bundle result exist inequality evidence suboptimal pair price value approximately bundle ip jx optimal return bundle object budget enough object suboptimal exchange unit object unit constraint pair vector without generality bundle object budget limit decrease make value increase unit unit rv I v completes claim maintain represent initially sample example sample uniformly body convex polynomial time end add program volume stop learn algorithm replace new body confusion final return end future explain example mechanism bundle either reject constraint end restrict form terminate hypothesis succeed predict valuable iteration example number phase terminate return utility function body bundle return bundle suboptimal subset suboptimal suboptimal probability return bundle return bundle body prove claim contradiction pair begin iteration compute end bundle suboptimal probability volume less volume probability feedback feedback suboptimal pair new add fraction case loop example hold property cca happen less reveal reveal preference course tight dimension reveal preference dimension ill suited unit without constraint exchange beneficial optimality case bundle identically exist prove claim I complete follow item item infer low bound infer find p p j p il il satisfie threshold threshold object maximum increase time result bundle range stop strictly budget unit threshold budget unit satisfy remain unit threshold want must try pair algorithm lemma object v jx j nk jx nk update low thing claim similar lemma lower less union accurate point threshold least jx nk happen therefore threshold infer imply property threshold threshold lemma find prove bundle property case ix bundle completely optimum might something else budget completely compare remain budget object bound least value fraction object increase budget optimum increase j optimum v budget comparison q kk construction update begin prove low iteration actual value vector vector loop constraint claim reveal preference perspective price budget draw bundle utility price budget utility hypothesis agent future algorithm agent linearly separable population merely suppose day day day attempt utility budget predict optimally behavior reveal nice preference focus determine whether classic utility sufficient explain past future increase piecewise restrict utility make inefficient computational utility accurately utility maximize access behavior restrict class linearly separable utility nd polynomial learn predictive polynomially agent select bundle goal learning correctly price utility generally utility bundle price price agent budget maximizer price
measurable dirichlet stick break process form thereby inference need inference monte carlo bayesian collapse unlike lead base base function parametrize draw define reproducing embed therefore important observation truncation stick already excellent approximation version almost sure truncation nonparametric sure hilbert inequality arbitrary kx f kx yield te sufficiently work much small choose effect truncation level would employ hilbert kx x kx cast prevent overfitte regularizer qp identity ij kx f similar thus reader perform use conceptually variational approximate moreover vb require access require value investigate understand relate method likelihood effect connection k answer method hilbert embed dirichlet stick break powerful avoid need intractable frequentist suffer comparison benefit efficient good dirichlet explain nice come source property heterogeneity parameter regard extensively community existence
derivative propose unconstrained multivariate kernel explore applicability convergence simple flexible finite behaviour new drive finally proof th multivariate density derivative derivative multivariate taylor compute th derivative involve partial variate derivative organize hessian hessian usual convention x however clear derivative adopt define product naturally h estimator symmetric symmetric definite f ik see context density dd dramatically draw definite moreover issue loss severe unconstraine bandwidth derivative improvement integrate error quantity integrate minimizer f stochastic detail integrate iv f rf expand show explicit x wise application approximation minimizer h expression analysis kernel involve strategy quantity optimal density derivative propose bandwidth datum asymptotic cross validation density univariate name cross cv study detail seminal paper either oriented consideration estimation part write smooth rf k cv coincide h mn sum equal third x useful bandwidth plug pi formula plug univariate density vector plug selector criterion analyze estimate f pilot possibly pilot bandwidth square square pilot bandwidth r g square optimal bandwidth overcome involve successive kernel estimation basis bandwidth minimize selector derivative counterpart derivative estimation explore gs criterion pilot selector lead g g analogously plug pilot selector selector build say rate denote indirect appendix rate eq bandwidth proof defer hold convergence selector plug selector smooth cv partial e previous unconstrained cv attain constant asymptotic expression seem possible pi bandwidth slightly special term achievable bandwidth general unconstraine slight loss optimal unconstrained see similarity note asymptotic pi estimation exhibit rate cv selector noting spirit cv selector different lower indicate slow tend rate sample size tend attain high provide realistic take account ignore explicit formula latter finite bivariate examine closely section c c cv pi pi pi kernel matrix short plug initialize th pilot selector sample stage pilot selector minimizer selector eq derivation previous facilitate execution selector modification pi selector include simulation minimizer nr cv pi smoothed cross develop exist library bivariate display definition find htp selector square conduct conclusion decide selector selector give mixture density selector nr selector previously available uniformly line simulation cv selector though former cv attention variability introduce oracle complicate different occur check try discover true datum consistently estimate slightly property pi correspondingly examine nr shift direction density produce ascent follow recursively density recognize variant algorithm employ advantage normalization illustrate accelerate sequence density attain unimodal profile decrease therefore g estimator eq ng understand unnormalized kernel term shift arrange note reasonable equation lead recursively define eq adapt cover unconstraine shift point successive gradient take mean bandwidth selector support result estimate relate derivative thus bandwidth goal identification mind contrast take variable convergence reach however task seek determination estimator engineering point way cluster local modal specify advance discover start partition apply shift mode grey point automatically nonparametric procedure via compare cluster technique impossible selection introduce shrink closely relate shift shift median pdf density identify normal multiply instance due include since recognize three compare normal label nr bandwidth cv plug bandwidth label comparison along five bivariate density situation shape different scale intended explore since explicit modification equally component break bivariate define mean normally matrix break weight half weight component radius indicator picture figure sample em adjust rand ari correct chance version partition prefer ari value membership membership estimate assignment cluster accord seven plus ari ari nr cv pi break none shift pi bandwidth performance choice rate second ari inferior performance mixture acceptable scale nr inferior break intensive provide quick initial follow especially difficult unable situation contrary powerful setup seem reasonably study ad hoc comparable shift exception surely careful bandwidth shift conjunction selection apply tends produce spurious mode tail enhance due phenomenon chapter sim real form singleton group outcome shift algorithm identify less singular mean bandwidth singular naturally newly requirement similar call merging rate henceforth group uci repository molecular biology university represent feature protein localization site observation om pp I cp find original seven five scale cluster correction pi merge group shift nr membership performance configuration nr pi group datum remarkably ari performance poor ari give reasonable ari introduce consist eight chemical sample divide nine area region cluster compositional reference chemical measurement transform place euclidean principal euclidean major try discover sub clearly recognize either grouping region ari division area nr grouping method remarkable ari ari bandwidth shift respect although region divide several shift bandwidth assignment smaller show ari shift pi lead pi always derivative negative set definite reject distribution adjustment testing modal maxima bandwidth significant curvature selector figure record depth west date distance surface depth depth pi estimation region three mode obtain bandwidth grant author author france institute sciences work henceforth symmetric x partial derivative integrable develop section integral suitable exchange taylor expansion well define lemma r r h r integrate thus h h theory lemma function x h k k term involve express r r h h x derivative read desire return asymptotic real moment pa
set diagonal complete consider projection solve subproblem corollary pass generalize describe spectral project gradient simplest suitable one iterate discuss important fairly invoke typically fast ordinary projection formally thereby bb global extraction subsequence theoretically depend compute obviously operator include norm must fact due treat inexact main aim separable say entire preferable simple realization replace stochastic result suitable additional account potential projection result norm show running method ease projection onto ball refer qp experimental fp experiment core bit ram setting measure table difference qp fp qp competitive fp consistently although fp twice qp fp method qp fp magnitude qp r qp fp qp fp qp e e e e seek column parameter row group row entire eliminate combine column overall norm notation yield long reduce ordinary make use mini row subset contribute w objective whereby upon iteration mn w become bottleneck counter may restrict subsequence implementation mini comparison qp fp solver solve h million million r projection reach spend method sgd use fp reduce science spread method initialize rapidly accuracy start eventually get interestingly long attribute difference begin take general either sgd yield rapidly problem prefer extended matrix duality theory mix norm especially special norm overall problem norm suggest algorithm gradient experiment lasso instance class mix study projection g bound extend method euclidean operator explore regularizer remark joint structural lasso efficient surprisingly seem present gradient projection onto open mixed norm norm stochastic sparsity encode permit widely therein literature reader sparsity constrain problem cast follow nonsmooth regularizer alternatively prefer perhaps latter latter primarily admit optimization algorithm constrain formulation attractive include nonconvex applicable separable inexact projection realistic type remain simple recently lead example enforce especially choice use lasso compress version letting differently unless impractical mixed norm hard since move projection ball batch mixed norm open develop basic aim efficiently algorithm iterate gradient constraint computation projection operator tie projection projection indicator compute whenever proximity exploit lagrangian dual duality primal optimal key equation nothing observe equal accurately solve useful let define exist decrease differ assumption claim fx yx finally differ change interval unique therein root iteration recommend use rather invoke combine interpolation fy fy fy fy fy ball generic projection upper prove dual conjugate dual partition p follow old invoke h u p conclude q u
treat theoretic plug link maximum hilbert schmidt rkhs depend bilinear form moreover definite definite operator deal look property empirical version note definition like calculus spectral gram construct infinitely relate spectrum I satisfy eigenvalue therefore eigenvector positive consequence relation proposition focus attention spectrum previously follow find compact adjoint separable hilbert compact let enumeration nonzero extend enumeration theorem operator normalize infinitely matrix hoeffding take hilbert schmidt combine yield definite consequence convergence constant gram seem extend conditional normalization q x similar follow report entropy hadamard gram experimental setup similar pick benchmark shift exponential student double multimodal unimodal asymmetric multimodal asymmetric gaussian asymmetric analysis theory need entropy product infinitely reproduce space dependent gram operator numerical usefulness want show metric positive symmetry cauchy schwarz relate calculus reasonable hermitian nd exposition space countable necessarily finite consider generalize generalized probability h strictly denote sequence entropy average define partial ordering measurable say shannon htp p concatenation union argument refinement orthogonality refinement possible refinement generalize cut simplex positive function respectively come possible belong turn fx f ff minimum state reproduce reproduce q definite respectively tensor definite x x j definite representation look theorem derive restriction restriction f gx hilbert space set definite class infinitely proposition corollary section rao department electrical computer engineering department university usa edu principled way among theoretic statistic pose challenge plug operator reproduce kernel infinitely functional entropy matrix axiom entropy result difference integral positive avoid estimation moreover mutual information independence art operational theory base law generation probability regard interest probability advance available finite theoretic call two quantity estimation case quality continuous plug rather employ choose assumption parametric tuning computationally possibility bring incorporate smoothing capacity control mechanism despite difficulty suitable version quantity entropy serve unsupervise variable density random equation entropy functional shannon entropy suffice monotonically plug estimator c potential show special potential density reproduce kernel space rkhs idea already rkh bring estimator go establish employ positive without independent input important mention recently minimum span consistently estimate nevertheless similar unbiased near graph introduce error despite nice objective conventional propose differentiable introduce shannon differential work estimate distribution functional positive event evaluate kernel pair reproduce differ scope matrix evaluation definite available datum drive act presence gram information property propose functional choice define show infinitely subject suited informally motivate measure plug base windows mechanic operator definite functional set axiom closely definition entropy hadamard product infinitely particularly entropy mutual sum entropy behaviour matrix statistical gram relation arise dimensionality finally carry independence art reproduce space idea however decade popular one deal provide kernel practical datum problem involve notice compute first statistic hilbert schmidt measure motivate descriptor datum concept algebra probability feature family xt xt completion deal difficult impossible result give deal bivariate define product hilbert reproduce kernel hilbert kernel eq call definite abstract space let simply allow relation element element verify gram fundamental method order base interpret reproduce convolution integral correspondence reproduce space implicit potential correspond law integrable square integrable bilinear uniformly integrable probability density square kernel define rkhs note x bound albeit set verify eq take window integrate information motivate information gram concept property entropy carry generalization argument quantity information monotonic theoretic also method derive serve mainly aim structured subsequently example probability product investigation information would reader theory definite provide set axiom measure axiom completeness provide axiom entropy employ nonnegative definite generalization extension matrix use spectral definite real aa definite satisfie condition monotonic take unitary trace invariant unitary continuity calculation thus power eigenvalue simultaneously matrix eigenvalue respectively yield notice provide eigenvalue lot operational quantum von entropy extension also functional gram obtain evaluation positive construct positive mapping induce key matrix definition hadamard product hadamard product definite extend entropy representation pair represent two realization ij hadamard compute x compatible individual entropy impose expect joint entropy component concept say order doubly easy important convex define eq concavity convexity concavity ready hadamard entropy individual entropy positive entry prove preserve simplex first consider order respective eigenvalue order eigenvector order accord respective inequality extreme shannon remain observe joint contrast multiple positive nonnegative unit quantity nonnegative mutual knowledge marginal gain entropy entropy analogy unit base divergence joint inequality need extra condition element notice element simple focus
svm obtain primal sub translate bad convergence paper ascent different proof comparable nevertheless dual analysis show gap achieve addition analyze version minimization dual inferior neither analyse logistic shall bound optimality interested duality sdca superior early randomization important cyclic dual coordinate order slow bound cyclic inferior analysis mean cyclic ascent inferior structural frank wolfe sdca hinge convergence rate sdca lipschitz relevant gradient sag recently analyze loss match regime sdca practical explain loss type primal frank dual sdca sgd dual describe code parameter good choice gap terminate sufficiently iw w return analyze simplify assume follow lipschitz duality suffice total optimality take choose type advantage average easier implement stop duality gap average hinge however inequality consider sdca duality gap suffice moreover requirement sgd namely suffer loss problem behave like regime smooth bind like practical observation smooth still dominate sgd sdca however sgd look sdca basically much sdca sdca may take size sdca help natural sdca epoch perform modify dual end sdca introduce convenient notation denote primal dual primal objective maximize dual objective end modify sgd modify prove sdca employ sdca require extra sdca procedure sdca sgd initialization modify sdca iid sdca sgd initialization duality stage suffice sdca duality ideally show sdca sgd ideal result show step sdca hinge analysis sgd versus advantage sdca explain sdca sgd try refine sdca asymptotically refined rely complicated precise interpretation result use explain sdca sgd loss although pass small gram matrix arbitrarily bad argument dependency gram smooth loss example hinge convex dual rely refine strong everywhere smooth loss variable eq therefore deviation convexity svm close away mean nearly point assumption may procedure iteration superior consider three ns os factor remove slightly convergence duality gap consider sdca eigenvalue define small superior complex case duality imply convergence specify sdca loss sgd may epoch sdca sdca permutation iw hinge use procedure sdca loss sdca close hinge significantly confirm ia loss sdca solution hinge hinge step sdca solution smoothed hinge worse confirm experiment sub gradient fact primal formulation proof theorem increase dual duality gap lower bound therefore upper duality theorem similar objective duality strongly iteration update dual write strongly combine rearrange obtain take obtain eq side conclude duality optimality last equip ready small require duality require prove first part eq either imply need turn lipschitz rely fix lipschitz ready theorem tell claim choose rearrange obtain q vector randomly choose iteration draw optimizer maximize dual derivation basic eq obtain proof need feasible display inequality follow nearly identical focus valid update objective obtain q equality I rearrange moreover fact n sum prove hinge take dual denote gap perform dataset different sparsity paper physics collection detail sparsity ph convergence indeed run sdca vertical straight linear original sdca minimization regularization show smoothed hinge apparent value linear still refine slow observe refined figure smoothness rate convergence dominant effect zero hinge error smoothed hinge provide hinge apparent hinge one repetition sdca sdca vs fix cyclic see cyclic rate slow cyclic dual ascent similar behavior cyclic ascent inferior sdca sdca particular sgd w iw x ii clear sdca calculate primal sdca sdca sdca regimes sgd theory sdca quickly smoothed hinge primal depict epoch ph divide set epoch gap depict ph horizontal divide corresponding epoch duality gap round sdca smoothed hinge plot axis one axis training duality achieve dual repetition choose random repetition sdca fix cyclic order depict sdca smoothed axis epoch optimality
approximation expect peak furthermore show evolution posterior sharp form evolution vertical colored enhance peak minimizers minimizer estimate minima median popular noise variance function minimum addition illustrate active surface apply acquisition criterion relatively initialize random surface grid evolution estimate sample contour select sample dot minima dots show median perform response surface slow due inaccurate estimation black dot one global fail minimum accurate accurate two minimizer c propose active criterion learn optimize transform process plan accuracy computer engineering usa mail edu minimizer criterion ignore uncertainty process uncertainty minimizer tractable optimization criterion explore unknown space search quite costly aim active carefully order reduce learning follow interest need surface query oracle potentially quickly minimizer mostly obtain function minimum providing consequence often discard minimizer could discard effective solution acquisition criterion balance tailor find problem densely potential minimizer minimizer maintain enable multiple minimizer target function furthermore unique noisy objective minimizer function often utilize focus contribute random variable minimizer functional next entropy additional straightforward intractable utilize widely posterior pointwise estimate minimum equality definition minimizer fx x upper q covariance normalize proxy px broad two minimizer minimizer natural modal multiple lead ambiguity independent remove covariance acquisition estimate
utilize experiment calculate conventional method include success summarize result ccccc specific success time size less compare utilize consider orthogonality pc utilize ground truth verify evaluate method assessment minimization maximization sparse pc loading reconstruct error pc reconstruct variance reconstruct utilize method introduce utilize evaluation method pc extract nonzero element nonzero table figure illumination methodology calculate pc pc pc include setting pc achieve lowest compete advantageous easy superiority extract pc component propose small propose maximization view consists extract dna micro array employ extract adopt pc loading try time besides carefully tune pc similar get ccccc easy low pca demonstrates extract pc advantage dominant propose usefulness exist nonnegative formulate covariance j schmidt design respectively success introduce list ccccc pca success nonnegative pca utilize experiment utilize nonnegative calculation adopt specify pc pc fair comparison pc evident dominate low pc furthermore advantageous extract propose nonnegative pca calculation far verify recognition together performance application mit mit resolution utilize loading face comparison cccc depict nonnegative pc clearly exhibit utilize face among show usefulness choose non mit training rest image extract utilize pca method respectively project pc obtain four respectively fitting lr directly fit original easy cccc face face potential novel methodology method decompose pca sub thus easily effectively new extended pca certain extensive current sparse explore model heavy correspondingly difficult adopt much problem although convergence know stochastic optimization anneal computation far improve thresholding correspond equal indicator function q feasible easy attain kk optimal complete denote permutation soft soft tf optimal thus solution exist feasible easy optimum deduce monotonic decrease tf k number complete deduce monotonic decreasing prove please lemma imply solution attain present w I follow discriminant solution express mm lemmas conclusion appendix prove equivalently solution first sign zeros one region v position nonnegative otherwise pick denote get w j w positive nonzero element new get first fact j j p p conclusion conclusion conduct c hold tp p methodology original pca simple sub recursively analytical solve sparse pca computation extend pca nonnegative method maximization face principal divide sparsity classical wide successful application throughout engineering pc maximally preserve always intrinsic feature conventional pca suffer component combination loading typically zero interpretation biology gene loading sparse asset pc loading especially expect nonzero loading transaction cost much attention recent decade topic develop attempt certain post e rotation loading pca enforce sparsity advance simultaneously pca loading al net et call pc semidefinite programming sdp series pc factorization include et issue concave maximization induce extract block simultaneously e solve recently briefly additionally set solution number coefficient etc pc sequentially pc properly extract pc pc utilize optimization sparse simultaneously attain pc requirement nonzero pc divide sparse make greedy approach method appropriate sparse pc iteratively pc way constraint pc constraint complexity dimensionality suit idea method introduce series scalar upper bold bold letter low letter denote maximized maximization viewpoint pca formulate trace denote empty imply exist denote I note decrease respect optimum main idea close form attain update implement iteratively recursively update stop summarize aforementioned algorithm pc solve pc loading briefly stop monotonically step terminate update reach computational evident algorithm essentially iteration e calculation operation involve require around sort computational approximately dimensionality input evaluate monotonic optimize column low convergent evaluate formulation indicator equivalent unconstrained solving set compact generate assumption regular hold condition theorem see advantage propose methodology easily need pc
stochastic difficult pair give result filter decomposition yield true backward filter h estimate likelihood generalize decomposition numerical suggest new sensitive point forward backward filter filter may beneficial forward explore introduce extend backward target upon w dx dx n n remainder mix backward follow formula zero property schwarz deduce convergence conclude subsequent notation define q series schwarz expectation note jensen inequality corollary paper adapt deal fix depend upon section mathematic college e mail department national mail hmms use sequential smc filter expectation marginal smoothing approximation two filter smoothing decomposition investigate likelihood description phenomena time process unobserve observe hide n dx xx dominate observation I x k n n dominating hmm often state static shall observe term interest popular collection hmms smc parallel combine sequence increase state introduction hmms normalizing constant representation understand smc recent design filter algorithm approximate appropriately meet smoothing relative standard likelihood decomposition via first perspective second follow investigate filter backward proof joint omit smc idea parallel reason sometimes particle adopt compute un j ai un j return joint distribution technique smc generalize two representation define smc meet density convention essentially efficiency normalize approximate sometimes index adopt notation un l ai ai un n return estimate normalize one calculation p tx x resample time marginal ny ny undesirable cost x x dx backward respectively time ny ny dx approximate estimate detailed omit establish divide e normalize obey filter backward intuition vice versa w joint calculation reveal e ny z q appendix notation lemma univariate idea compare variance filtering allow one representation within distribution section x n r calculate via observe affect recover distribution resample run nine plot variability estimate
condition identify offer penalize concentration matrix matrix suggest penalty penalty trace nuclear necessity could prohibitive efficient attractive penalty rank respectively course attempt penalty nuclear computational integer equal except replace modification reflect adapt deal clear call estimate glasso lin also hereafter refer method short ex pt infeasible interestingly though computation thank combination recent advance compute lasso latent clear observe estimator datum em iteratively penalize current lk nk k recall eq therefore q note replace penalty become glasso solved iteratively correspondingly maximize glasso purpose recover uniformly pair location location variable connect entry covariance ensure typical latent left panel figure along glasso use provide insensitive report little variation shall brevity play critical role fine recover pattern roc glasso close necessity account interesting perform penalization method
refer cell available cd cd focus variable cd seed summary output fit component fit model summary fit single skew mean skewness mixture component matrix freedom five fit fm posterior membership row evaluate argument aic bayes show fit multivariate component five rest cluster rest omit aic em user suppose specify list specify demand achieve illustration datum body record component skew height setting examine skew component iteration iteration matrix skewness component proportion compare restricted version distribution skew normal skew skew use component constrain label true label reveal fm high fm give fit skew group true false r nan nan eps fitting fm mt skewness fm nan statistic fm respectively asymptotically difference hypothesis alternatively value directly specify last consider examine fm high skew mixture skewness fm significantly c specify fm perform datum fm specification fm fm fit classify seed x train contours generate contour grid cluster level nan scatter heat contour cluster nan coordinate either five default include plot contour require option specify label specify percentile contour contour require quantile plot bivariate scatter option fit specific mixture mixture contour plot length take argument argument colour contour generate contour fit heat return scatter contour contour fit fm point cluster r model fit suppose interested simulated datum dx fit contour model plot intensity plot contour component plot provide contain intensity know marker cell patient parallel intensity surface protein involve identification population perform manually visually separate interest approach marker develop method sample three marker cd cd group fm increase dot accord expert consider b percentile contour display visualization device support user interactive viewpoint follow code fit contour fit level effectiveness label manual take label permutation cluster result report minimum note remove evaluate fm superior method fm accord true contour fm dataset present package fit finite heterogeneous asymmetric form fit fm visualization contour major demonstrate institute result via restrict skew distribution gave skew slow higher intensive calculation multivariate benefit acceleration strategy acknowledgment support grant research correction helpful package distribution skew distribution implement em ml fm visualization contour finite skew flow various multivariate skew step paper focus mixture iterative usefulness propose application bivariate unimodal distribution institute comparison achieve misclassification rate analysis skew heterogeneous contain mixture distribution skewed model datum intensive past decade application scientific bioinformatics processing survey volume paper year skew exploit tool model multimodal asymmetric introduction skew sn study extension multivariate skew automate model density mixture normal family skew impose component expectation involve skew skew mixture model implement restrict expectation multivariate truncate variate central package mixture distribution em user contour fit density interactive viewpoint paper organize section brief fm fm explanation visualize fm present fm cluster dimensional vector skew distribution skewness eq corresponding cumulative p tt skew skew recent noting version incorrect consist incorrectly update degree freedom eq simplify make update rarely preserve working consume algorithm implement quickly mean attempt perform log initialize sample component th th create extract systematically across search traditional stopping criterion size acceleration describe absolute less tolerance asymptotic give fm fit represent represent degree proportion htbp cccc dimension description matrix skewness degree proportion function evaluate fm sigma delta skewness parameter
multiple way interaction reach amount hierarchical polynomial regularize differently algorithm typically good matlab proximal used online mkl identify variable code error feature mkl uniformly nonlinear radial validation testing repetition mkl polynomial regression use minimum polynomial report tailor mkl less nonlinearity evident radial section estimate repetition benchmark census delta segmentation uci regression build training set draw breast algorithm subsection protocol sample neighbor since method polynomial degree mkl result selection stock select equivalently case similar nonetheless bring provide choice method among plot clearly second derive definition follow example note mention easily start immediately second q proof procedure consequence general inexact algorithm proximity fast condition say inexact tf tf exist since regularizer composition ensure generate inexact share follow matrix operator trivial derive adjoint eq observe extend rewrite proceed hypothesis f prove observe operator functional satisfie euler arbitrary subdifferential use euler contradict also theorem construction subdifferential plug kind subdifferential subdifferential subtract inequality start prove hold ff f n technology usa di mit edu di university di variable dependence depend nonparametric key make derivative intuition propose new notion regularization reproduce hilbert method correspond solve estimator term analysis common bioinformatics computer thousand thousand dimensional regime feasible quantity satisfy regularity idea behind call turn construction interpretable interpretable situation multivariate relevant detecting largely motivated recent advance extensively linearly naive try subset meaningful greedy convex relaxation basis algorithm therein guarantee therein situation latter understand approach mostly additive function relate encoding among assume function depend though provide interpretable size initial notion additive idea suggest way sparsity regularizer essentially question derive actual issue reliably high ensure drive stable reproduce kernel hilbert tool question linear functional representation computation regularizer ensure derivative sparsity correspond concept space since correspond functional suitable rely backward splitting proximal third statistical practice mean preliminary appear conference discussion derivation proof consistency augment experiment paper organized section consistency extensive future functional precisely statistical measure minimize prior paper interested estimate relevant sparsity requirement norm often norm studied extensively understand reference regularizer minimize norm univariate norm respectively multiple kernel consider limit variable interact partially consider simple triplet fx general essentially propose attempt tackle problem allow discussion function interest paper discuss detail reference paper follow variable basic idea locally linear window depend locally select depend one partial select risk criterion include local thresholding backward step wise dimensionality variable emphasis quantify optimal pointwise improve see discussion recently estimator prescribe study careful regime cardinality globally relative scheme classical spline modern variation utilize capture notion partial notion variable consider follow issue first need variable partial regularity need depend support restrict linear capture relaxation functional encode classic replace regularizer sufficiently ensure good poorly spirit manifold propose hilbert rkhs differentiable latter ensure make strongly rkh allow computation partial potentially devoted algorithm remark entry subset comparison remark difference regularizer regularizer gradient version classical uniform measure regularizer derivative regularizer give force propose utilize root group point penalty note consider derivative differently localize correspond partial trivially pointwise define complicated uniform interval moreover although relevant impose basically imply open rkh summarize contribution main derivation provably convergent begin extend dimensional coefficient forward backward admit closed form proximity operator overall procedure multiplication thresholding derivative universal depend selection estimate consistency hold extensive analysis simulate work rich space correctly solve outperform set interpretability select subset hierarchical reproduce associate reproduce kernel hilbert start discuss regularize observing make tf standard minimizer divide part allow function minimization prove x denote derivative main outcome coefficient provably clearly soon explicit expression also unitary consist two iteration describe add size priori ensure experiment accelerate simple warm procedure discuss principled way discuss induce smooth practical ultimately definite function q reproduce basic fact pointwise induce point follow return thank rkhs allow variable importantly derivative return partial write regularize adjoint matrix differentiable minimization nonetheless computation splitting algorithm belong proximal large paper nesterov optimal first order since become signal fista accurate regularize functional modulus differentiable convex class backward suitably proximity term simple penalty use additive proximity discuss next paragraph briefly basic previous iterate function provide provide sometimes iterative backtracking require computation share rate well strongly accelerate scheme minimizer set fista tackle minimization formulation proximity operator subdifferential describe practically start norm unitary ball convex sum smooth indicator v easily correspond proximity indicator mention convergence guarantee thank structure possible see compute convergent accelerate outer proximity procedure iteration v initialization inexact proximal approximately split proximity inexact accelerate inexact still computation proximity sufficiently inexact convergence accuracy operator suitably control adapt define f accelerate split outer remark evident positively influence crucial minimizer minimum far stop third remark inexact generate belonging discussing describe summarize proposition update update iterative operator reasoning make size second since quantity v accord outer iteration detect evaluate projection proposition algorithm inner stop depend inner iteration result consequence euler characterization subdifferential observe subdifferential belong subdifferential subdifferential control evaluate irrelevant discuss regularizer ideally result show consider let restriction force derivative guarantee motivate add square spirit manifold approximation allow sample counterpart deviation give study useful since belong always precisely convex continuous rkh norm asymptotic explicit learn relevant irrelevant sparsity context partial everywhere relevant goal variable correctly assumption regularization safe filter variable sufficiently large relevant converse inclusion give variable work implication additive model prove achieve probability sparsity much total considerably source possible challenge interesting structured sparsity operator adjoint operator sense one penalty divide partition penalty obtain vector restrict orthogonal th form rewrite infinite reproduce reproduce hilbert one operator range practice empirical indeed structure still operator penalty projection penalty q regularizer conclusion highlight typically would difficulty arise operator scope paper subject future divide tuning always consider step procedure variable apply fix regularization validation validation influence discuss solution principle parameter stand show propose pose ensure reason coordinate aim penalty systematically investigate enforce degenerate toy coincide
bind argument first graph dependency graph whose graph vertex edge vertex give pick vertex graph define bind codebook hold sufficiently er rao sufficiently symmetry code substituting since eq get share term code eq choose theorem consider show interval r right lower fast grow lower write choose number study combination dictionary polynomial ensemble achieve rate exponent source show robust design source source within distortion sparse cover property ensemble goal achieve distance encoding lie denote distortion govern grow compare mean happen standard solution approach realization ensemble go coefficient section design one use encoder asymptotically attain compression relate recovery linear recover position zero recovery priori connection investigation follow er sequel denote value gaussian q large source increase divide side grow grow hold g first strictly equal see indeed solve verify imply value attain write q therefore large similarly taylor minimum expansion attain get claim large author regression definition corollary distortion combination column call sparse superposition codebook analogous recently communication channel encoding attain shannon distortion exponent sense codebook design gaussian square ergodic variance sparse covering I term matrix distortion distortion exponent sparse ne problem information code compression general source rate shannon superposition code compression distortion criterion sparse block codebook communication sparse computationally rich minimum proceeding case random low vector normalize otherwise mutual gaussian interval distortion length nr quality measure q distortion criterion near source distortion distortion give achieve shannon codebook selection storage encoding codebook grow base code compression widely storage certain code distortion complexity recently distortion encoding decode code consider code main achievable exponent ergodic source moment error distortion special I source result imply distortion exponent bit per rate robustness ergodic distortion ergodic source codebook attain distortion function sparse good much codebook I consequence dependent deal literature know exponent refined indicator codebook encoding decode procedure describe exponent state distortion feature make I random refined distortion level exist code design distortion draw n nm theorem bit encode achievable region lemma particular source distortion yield distortion performance describe theorem shannon rate distortion coincide begin exponent distortion rate optimal exponent pair supremum sequence distortion give exponent describe error grow length gaussian criterion exponent kullback sequence within distortion decay consequently obtain sequence variance great exponent characterize exponent source eq equation sequence level great matrix draw mean satisfying achieve exponent attain I er deviation q exponentially eq get pick matrix codebook vector equal encoder code q great restrict trivially use codebook negligible ergodicity regression codebook correspond l uniquely section overlap position sum
energy maximize set consistent satisfy apparent carry step similarity belief propagation equation several adopt algorithm sbm several belief instance way assumption graphical loop decay treat graphical fully connect interaction edge weak put decay argue belief asymptotically probability prove rigorously probabilitie I ij assignment product recursively term message compare equation belief edge bp simplify term message update marginal use bp fast change find use bethe formula free bethe exact graphical many mf free set bethe derive equation order modular network modular approach section ensemble know true assignment follow permutation would successful mf right know I unique maximum seek practical situation estimate marginal overlap confidence quantity agree large undesirable give reconstruct ht mf bp approach network node generate group modular structure rr rs group note possible mf overlap region iteration effort maximal bp converges fix infeasible mf region mf large average probability use bp mf plot fig superior mf well agreement region overlap bp mf equivalent region overlap converge contain group parameter close impossible phase overlap group assignment mf confident part bp agree evaluate mf confidence considerably large overlap region mf depend overlap reach close explain bp mf time middle mf need bp converge different initialize message converge run mf overlap mf different mf quality mf conclusion bp mf reach problem bp concern give region superior notably suffer maximization maxima get fortunately indicator run small volume grow linear considerable initialization desire large correctly recover class label bp whereas spectral serve initialization em bp achieve modular example bp well exhibit panel fig core modularity overlap group adjacency cluster conclusion search original always spectral modularity test fig marginal improvement important make case clearly region clustering claim average opinion average degree think end start point improve overlap em result example network belief propagation traditional mean field recent discovery sharp block inference provably control heuristic speed reliably deep feasible sparse difficult method block subgraph community short serve quick claim optimality spectral network technique find burden spread fully connect structure belief expect formulation may field connect maximally sparse case outperform albeit finally could decomposition select reduce conjunction expectation wish thank discussion aspect dim france support fellowship science foundation inference classical commonly approach belief contribution perform comparative na accuracy tendency portion phenomena particle consequence capability neural apply system pattern far result phenomenon datum research understanding ultimately easy map explain allow pairwise bind protein protein link protein operate protein network sense structure make guide aim similar consideration social network online datum discover difference often important detail infer parameter undirecte unweighted discuss consequence capture fluctuation dataset theoretical technique sharp impossible briefly expectation maximization algorithm conjunction introduce em mention infeasible transition particularly highlight discuss finding consequence inference node unweighte enyi fall link generate degree characteristic small length unfortunately explain fail density may entity property protein protein form involve interior would soon entirely social gender education tie node simple stochastic assume node indicate link easily assign multinomial specify develop multivariate analyze embed behind adjacency measurement dimensional feature row apply multidimensional analysis find direction onto spirit remove symmetric mean say transform modularity via modularity row sparse since eigenvector large magnitude eigenvector recently aspect rank eigen decomposition well frobenius norm matrix order decrease eigenvalue rank approximation eq row anti parallel model comparison density mix modularity matrix instead
propose parameter interpret posteriori map justification lasso estimate credible interval markov mcmc mix model towards even evidence bayesian literature concave log prior reduce recent lee bayesian interpretations net median series pattern predictor irrelevant dynamic modelling important receive attention bayesian adapt elastic accommodate bayesian type include magnitude latent follow dynamic construction recently rely normal tailored monte carlo smc generalize numerous sparsity propose bayesian consider standard regression response identically distribute adopt p independent interpret laplace prior express admit hierarchical construction normal generalize density write pdf concentrate heavy tail make desirable distribution law laplace law show performance regression chain also consider superior parameter represent control behavior time identity priori independence value slowly evolve sparsity sparsity unchanged multivariate evolve smoothly pattern semi one q normal turn use vary introduce sparsity likely sparse zero decomposition multiple overlap joint distribution marginally follow define predictive normal latent model stationary pattern independence pattern iid sparsity successive appealing literature desired gamma define pattern go evolve extreme independent sparsity tune alternate period see induce minimum autocorrelation non way note sparsity lag share pattern clearly autocorrelation share pattern statistical uncorrelated autocorrelation share evolve depend stock stock trading period value diag carlo particle hastings sample replicate particle weight replicate particle particle compute run conduct generate artificial observation red circle truth solid additive figure show map ground truth parameter notice control correlation truth fit implication firstly create smoothness stability secondly slight sparsity aforementioned iteration particle metropolis hasting moderate depend circumstance flexibility practitioner unique extend bridge replicate see drop step replicate fully produce induce shrinkage stock derivative alternative micro micro trading asset consider news horizon day day variability stock order magnitude stream bad news increase trading activity macro correction market stock source yahoo stock derivative stock study mention early massive price next asset technology index observe activity market measure interest stock index constitute core begin stock technology volatility trend figure contrary attempt plot filter time range moderate panel early market close practical portfolio situation largely home specifically home aim portfolio correlation immediately leave appear highly correlate market induce derivative market conversely two positively might choose house stock directly might decomposable clique
seven recently object non capable capture inherent ambiguity localization compare demonstrate appearance variation capable capture incremental principal construct tracking track sparse principal decompose motion cover object appearance motion capability complicated appearance illumination change head pose video scenario illumination pose motion vary video scenario scenario video scenario partial body variation scenario head video clutter car pose eight highlighted bounding box color show fig figure video walk illumination head pose track break nd l fail able successfully face video scenario toy affect illumination l fail nd th lose tracking toy inaccurate track toy illumination change car dark road scene clutter vary condition st track car illumination clutter break frame keep car inaccurate tracking tracking car video variation head th frame head th nd frames l track head video person compete suffer severe track person frame achieve head vary body pose begin frame fail track frame stay far away track sequence player piece fail track face nd frames frame performance capture face throughout fig sized traffic scene track car th frame car video scenario try parallel car partially car achieve nd frame subsequently begin drift break th able track car tracking contrast car tracking throughout video ball ball l fail ball rd inaccurate tracking result fail another due severe contrast continuously severe video drastically person track face nd frame begin break influence head plane rotation inaccurate tracking result frame th track inaccurate result frame show car fail car st frames track car th track car th inaccurate rd frame car situation tracking result video sequence performance pixel tracking error well evaluate call tracking frame video sequence successfully criterion frame pixel location truth localization ground bounding box successful otherwise video use h sec influence sort exchange order neighbor video choice configuration within close dct sensitive supplementary tracking achieve equal track accuracy utilize state method representation improve tracking make refer dct particle dct slide clearly fig tracking inference show dct particle obtain accurate slide conclude dct object representation responsible enhanced tracking highlight color eight sequence compute standard location video report report total sequence propose sequence inferior difference e head appearance motion body appearance usually color appearance greatly plane circumstance haar color video sequence dct object representation appearance appearance sample dct reconstruction encode robustness complicate pose c video car car ball car seq seq simultaneous tracking dct compact construct dct energy spectrum component discard construct dct successive dct dct video dct dct newly well lead computational efficiency computing store cosine dct dct discriminative base contribute evaluate test belong foreground foreground background reconstruction discriminative appearance e rotation use discriminative conduct visual time video include illumination pose complicated change effectiveness acknowledgment arc discovery fig author theorem li van wang center visual school sciences intelligence key laboratory brain like system university china manuscript tracking model change illumination factor video appearance sample form basis limitation easily b challenge paper model discrete set cosine dct energy appearance discard signal reconstruction update representation dct dct successive cosine dct discrete cosine transform dct video incremental dct frame dct dimension computational dct algorithm belong foreground object discriminative time track cosine transform dct incremental template tracking move object fundamental computer vision include surveillance human despite effort challenging appearance variation pose etc crucial effective object challenge appearance change popular learn appearance variation appearance reflect appearance learn appearance compute reconstruction project subspace evaluate likelihood test belong foreground object approach basis correspond inspire track cosine e cosine basis propose projection dct incremental lead type enable long tool encoding image video many promise video dct lead correlated mean remove rapid often dct cosine fix cosine simple construct dct computationally incremental tracking represent calculate dct dct encode temporal redundancy correlation sample compression discard still relatively low sample cosine evaluation compact independent wavelet capture adopt dft amplitude haar aim detailed wavelet transform although conduct function work minor modification dct propose utilize compression power dct construct novel representation dense dct dct compact measure loss evaluate foreground give set efficiently update dct successive operation dct dct video newly add well dct along cosine advance reduce dct predict foreground discriminative criterion foreground background dct reconstruction likelihood capture adapt appearance focus learn compact discuss dct review relate tracking claim dct aim mutually uncorrelated cosine function express manner vision face recognition retrieval video object video localization application typically extraction construct coefficient geometry illumination focus effective dct object visual tracking field researcher variety object representation discriminative representation object include kernel density subspace covariance tracking base object review elaborate em appearance change appearance use mode kernel e framework incremental learning achieve real li compressive matching pursuit vector namely track video view frame n cosine define dct high encode high coefficient low relatively principal tracking build compact maintain control degree tracking dct obtain compact dct approximate reconstruct high form basis section cosine basis matrix dct employ efficiently dct dct I dct visual object representation dct object transform incremental dct aim current z formulate stage coefficient axis correspond computational increase normal dct grow much fast incremental dct accord dct unchanged visual tracking remain concatenation along algebra decompose update computing view dct e dct dct employ use fast dct dct summarize n frame traditional mode strategy dct computation become computational dct computational incremental dct dct incremental dct z n f collect negative object region pixel perform incremental dct dct matrix compute compact discard reconstruct reconstruction likelihood likelihood optimal refer update training maintain element keep last set module selection dct tracking module spatial region object select region draw make sort set final positive set middle fig negative around show fig generality evaluate candidate belong foreground object appearance candidate point locality locality versa locality constrain dct h compute nearest refer n left fig incremental dct
encounter smooth uncertainty principle bayesian gps offline series size matrix scale demand although experimentally scale poorly dimension equation integral integration practical gp necessarily smooth gp slow function evaluation ode cause eventually require principled smoothing gp gp computation exactly linearization sigma representation integrating model induce code publicly acknowledgement partially support grant intel b university usa propose bayesian system increase importance signal processing function modern finding estimate parametric analytic gp demonstrate fail nonlinear smoothing machine filter latent history methodology smoothing dynamic propose gps robust approximate gps contribution article derivation principle smooth gp compute close filter linearization evidence nonlinear kalman filter kalman refer ability background smoothing introduce explain gps detail system sec run prior purpose article gaussian approximations bc xt sufficient approximations eqs computing include extension etc differ covariance require eqs derive follow smooth relatively detail gp dynamic process practical application form increase gp control assume h infer value input fully eq specify respect square automatic relevance result square characteristic se function easier justify fix se zero prior common retain great inspire correspondence basis along universal writing axis white convolution function jointly convolution function specify distribution random prior require everywhere use complete eqs see universal make function se implicitly latent eq encode assumption give gps assume uncertain observe use collect maximization estimate evidence automatically avoid setup g notation hyper analytic dynamic extend filter analytic smoothing account linearization require sigma smooth standard frame eqs covariance smooth involve linearization deterministic joint distribute x ij q noise gp target role q desire definition law expectation kernel integration turn eq ac responsible definition unnormalize gaussian integral product obtain identity obtain q full completely eqs filter necessary provide smoothing analytically analyze state stochastic contain transition originally function uncertain broad eqs relatively small consider numerical eqs robustness filter sir resample particle filter particle gp gp ground closely filter compare trajectory evaluate filter make analyze filter sir performance mean mae sir pf average randomly start per setup develop indicate robust filter significantly outperform sir green particle amongst filter gibbs filter sir pf filter express difference sir pf near filter depict filter instead sir matching poor linearization filter sample small statistically even tb mapping panel sigma dots predictive show sub panel area solid gaussian track gp anti angular velocity constrain transition moment acceleration state ode hold w
get q positively definite uniquely normally euler sde expression independent wiener uniquely covariance divided advantage disease uncertainty sometimes calculate addition sometimes sde severe disease clinical despite heavily life ability rare incidence research patient duration account regard take sde age show pair path single path figure region path lie dotted indicate path mean quantile calculate additionally ht big difference gender incidence ratio age hazard estimate burden total number obtain statistic specific path figure path b becomes describe derive duration duration person divide total ever respectively age path factor unity disease differential equation decade transformation accomplish disease stochastic equation rare disease preferable deterministic incidence fluctuation strong age course obvious figure middle respectively empirical observed agreement describe equal incidence duration disease easily eq pyramid mean incidence close empirically observe per year relate uk incidence high person year datum way inconsistent duration basic contradict although difference duration noticed duration etc global european country publication mention incidence help burden disease especially limitation application patient publication find disease long duration applicability disease derive consistent indicate institute diabetes death disease incidence basic incidence model model also death transition mean disease consideration number intensity incidence rate depend also ht consider disease disease depend age disease specific disease duration reverse incidence inverse problem cause death
entire summarize appendix current amount perform build entire tree efficient tree sub recursively reduce single leaf join tree recovery tree divide correspond involve perturbation edge perturbation edge perturbation correctly marginal guarantee also successfully nuclear assume return tree subset return sample topology adapt nuclear base test divide simplicity ease guarantee translate tree building spectral representative nj liu nj proceed recursively close additive ij ij denote diag nj design true configuration need p k subroutine reconstruction still choice apply liu tree add neighbor algorithm tree mutual information determine nj four configuration hide observe start identity perturb formula pa u percentage varied perturbation randomly initialize compare spectral varie lot stay one especially hide method lead state achieve choose allow resemble indistinguishable even topology size experiment topology randomly recursively recursive stop group variable join recursion control partition versus close obtain skewed shape path close skewed balanced hide perturbation vary construct tree leave partition imply partition tree show large try resemble work perform well discover stock dataset stock relate stock www finance yahoo com daily price discretize value visualization learn tree topology discover figure htb stock accord subtree international group subtree subtree subtree subtree utility service energy ed subtree american express see subtree financial american express car branch financial operate segment services services company financial service real hide algorithm term hold randomized likelihood tree state simulate validation select present tensor automatically liu however produce discover structure model pre specify quantity unknown practice joint table variable tensor spectral design exponentially simulate usefulness discover latent property tensor multilinear algebra allow address arise pt edu discover structure important discover require hidden contribution characterization design relation use subroutine recover tree condition error sample size compare stock dataset meaningful fit discover observe task discover understand potentially heuristic need determine structure provable guarantee recursively nj algorithm distance variable variable nj provably consistent resolve subsequently good repeatedly singular value joint table require advance number rarely number base th insight reveal community concern focus state represent tensor miss translate propose nuclear rank advantage compute since tensor subroutine latent tree structure divide scalable mild use compare alternative among agnostic hide latter improvement cross expensive stock grouping stock lead well conditional observed variable furthermore letter letter latent tree way although px parent r lead throughout assume rank learning topology discover structure tree topology neighbor observe variable latent figure ht style mm black fill hide inner black hide circle sep circle inner mm draw hidden hide name name empty mm style dot empty empty width dot dotted width style dot h dot tensor th rank relate algebra rd tensor tensor denote along tensor ht mm characterize property test although propose unfold tensor subsequent computation provide conceptual structure tensor exactly grouping variable group grouping matlab column rao assume correct structure factorize mm mm mm mm factorization different interact column similarly middle whereas similarly discover assumption suggest attain nonzero correct grouping variable condition verify rank discover structure due matrix nearly deal rank rank nuclear summarize three way matrix nuclear algorithm work base base bind meaningful nuclear job concentrate fourth nuclear measure measure
solve constraint agree solution lagrange original community decade dual increasingly computer vision flexibility problem enforce tractable correlation zero write following decompose edge sub problem correlation cluster identical optimize configuration optimize energy set recover optimum since maximize set simplified entail optimize relaxation ascent non smooth tackle objective like optimize solution appear complicated early highlight inequality violate describe weight use cut successively approximate constraint constrain cut edge become long edge simplify edge weight independent edge objective amount objective also upper constraint impose constraint sensible wise decrease appendix explicitly exponential constraint possible partition solve lp cut successively violate collection perform constraint break cut add batch find produce solution e constraint set fs x optimize general tight cut cut constrained edge individual cut problem second term cut agree motivated start find subproblem partition edge pseudo display lp equation whose cut cone straightforward valid cut valid cut cut segment cut exactly constraint impose cut appendix solve contains produce result thresholded take cluster cone unit cube cone polynomial lp optimization considerable full cut cut delayed generation scheme dual cut correspond primal lp tight partition cone allow access partition without relative speedup logarithmic computation produce optima much demonstrate cluster run contour region merge average contour merge perform globally return perform slightly final performance move segmentation segmentation contour perform well edge negative average perform range threshold result visible report measure merge successively low objective expect segment break contour could practice seem one explanation proceed merge new contour form average average simple averaging outperform substantial possible explanation greedy truly successful correlation threshold fairly still solution suggest combination via structured performance note segment small benchmark outperform optimal segmentation basis apparent application local biological imaging quality clustering exist variety exploit decomposition subproblem dual acknowledgment google derivation optimization equation cut let write standard far simplify expression define cone cone yield objective main lp insight compactly cut plane optimize lp define add constraint cut primal add another dual lp delay column generation scheme dual lp keep grow optimum find additional affect formal decrease without optimize satisfy without low cut indicator polytope graphs polytope compactly inequality relax discrete polytope relax lp constraint side decomposition collection cycle optimizer dual cycle cycle minimum energy zero constraint th cycle cut subproblem ec implication weight edge case update edge quantity tight produce satisfy additional parameter update repeatedly remain denote ce establishes lemma claim optimizer optimizer satisfy uci edu find perfect energy demonstrate segmentation outperform competitive quality pixel image surface come bottom contrast recognition pixel develop bank detector formulate markov pixel quite valuable segment scene understand salient predefine label partition pairwise similarity rich technique partitioning appeal segment function parameter structure trivial graph partitioning np et hardness cut hard difficulty correlation et segmentation score cost integer programming branch cut et correlation potential edge define linear programming lp technique new specifically weight cut weight correlation low output always yield tight bound range criterion dissimilarity graph specify dissimilarity edge vertex correlation seek disjoint minimize specify edge cut edge within let configuration valid triangle enforce express cluster energy optimum partition objective etc weight specify segment place weight example vertex edge place valid objective appear standard pairwise convert mrf unary next call minimal color partition color g example partition useful term vertex equivalent
guarantee hold least positive index guarantee p bind bind square recovery exist intuitive multiply analysis show nevertheless quantify build satisfie require vanish modified natural analysis around suggest strategy know estimator gap regime confirm section follow instrumental instrumental interpret snr measure instrumental design column covariate independently key suppose dependence move previous main dimensional understand correct allow avoid optimization enjoy support guarantee expect correlation however sub everything else interestingly knowledge instrumental directly advantage previous support recovery course quantity add error unbiased omp element correspond large somewhat surprisingly estimator advantage support knowledge distributional effect know result recovery omp support recovery final recovering column square submatrix row note unlike omp assume adapt h input I output design mod omp identify provide mod omp identify omp covariate obtain non however proof implicitly residual hold unable apply result I thank mod identify reduce dimensional exactly previous sub mod omp mod omp identify correct recover clean covariate reduce omp concentration miss seem error sub mod use technical lemma concentration statement zero parameter union corollary sub p substitute w z e w w observe sub n n w z proposition apply sub u u I entrie great mod omp miss illustrate high empirical corollary theorem correct rescale normalize clear alignment actual result regime efficacy different noisy covariate demonstrate addition fast empirical term recovery result set omp problem low identify correct corollary corollary scaling look vector estimator build knowledge corollary scale roughly straight line origin align trial plot b simulation knowledge instrumental instrumental gaussian corollaries proportional control build recovery scaling versus go noise magnitude roughly trial additive versus control recovery versus corollary curve align additive trial turn become roughly line align future noise point trial subsection study mod dimensional setting gradient additive omp use omp outperform project ccc omp knowledge instrumental error control parameter circle gradient dot mod claim mod omp omp enjoy formal omp mod omp iv estimator plot error performance mod omp method gradient recovery quickly graph contrast estimator excellent recovery error project precisely throughout case figure mod omp independent believe omp project gradient performance comparison mod plot correlate dot correspond mod omp gradient mod omp trial concentration random repeat statement convenience union assume sub kn kn n h rescaling assume u u u last thm thm pt corollary engineering tx completely noise efficient omp deal set omp dimensional seek measurement set recover regressor noise also improve guarantee omp compressed deal projection physical measurement often noisy corrupted standard include popular know noisy surprisingly corrupted give recently turn scale significantly even far particular prediction hinge sparsity thus provably subset recovery large call computationally focus fast additional good well noisy set dimension regressor exhibit regime standard compressed sensing setup describe dimensional omp condition high regressor recovery corruption aware compressed sparsity low computational noisy corrupted estimate dimensional estimate set without issue convexity move covariate recently attract attention make important cope practical recent exist handle selector mu selector follow let covariate selector thereby adjust effect modify lasso minimize w xu similar computed program interestingly project descent optimum method bound minimax covariate guarantee recovery weak recent simulation seem critical recovery considerably require performance demand prove theoretically successful clean covariate greedy prove method clean complete omp measurement response receive noise paper paper result noise seem weak consider clear contribution greedy omp noisy noise receive sense sparsity specifically exceeds regressor simple covariate covariate covariate instrumental correlate covariate noisy guarantee far available case instrumental variable aware rigorous require column high dimensional scaling greatly give noisy regressor result interestingly noisy noise recovery simulation require corruption advantage covariate indicate result algorithm seem promise covariate measurement g meanwhile gap quite paper paper exceed regressor omp determine low omp regime section illustrate advantage regressor low setting regressor dimensional dimensional case assume covariate link via observe miss following seek unknown setting basic definition use gaussian gaussian row simplify subsequent notation generality lose section general result parameter analytical across define sub
simulation experiment validation widely statistic estimate statistical square estimator reference cross selection cross validation fold cross enjoy computational classical small second useful surprisingly result answer question least cross suboptimal stay distinguish value intuitively reduce validation estimator risk aim provide two valid variance computation would like common least particular fold validation fold penalty constant unbiased fold method knowledge asymptotic lead estimation framework rely new penalty proposition paper asymptotic variance computation understand cross penalization depend see remark say necessary getting confirm experiment synthetic integer bt infinity reference study estimate estimator measure risk want estimator comparable form large call front drive criterion oracle inequality penalization criterion criterion ideal orthogonal projection follow reference definition necessary main train test n fold cross one thank fold validation criterion study name penalization resample resample associate resample penalty distribution invariant permutation cost exchangeable exchangeable resample fold validation jj n resample focus cover resample base loo x loo exchangeable exchangeable resample factor framework prove fold fold cross leave validation prove square lemma family exchangeable aa cross unbiased loo penalize lemma penalization free front penalty study cross fold leave exchangeable take penalization bm lemma penalization picture oracle result hold constant front prove oracle satisfied leave concentration establish real value random piece satisfy mt mx proposition fold penalty deviation term help discriminate value histogram collection subset lebesgue countable histogram span precisely illustrate result every since show nd study interval equal state hypothesis family hereafter space real piece space either let least q section take fold classical assumption term inequality loss drive asymptotic bias fold penalization interpretation model therefore lead upper unless use correct cross show oracle constant remainder regular analyze correspond validation much large simplify presentation set previous less simplify message first asymptotic inequality prove penalization procedure concern validation oracle square framework inequality compare consider oracle inequality leave treat lemma recover concern fold penalization either square bound infinity valid remove particular remain loss ratio compare performance several fold method minimize make set goal understand validation develop apply ideally prove procedure inequality probability alternatively review result validation good square bad model limitation accord set distinguished leave certainly must go take account variance already challenge prove specific challenge compute precise seem heuristic depend second term square different partition satisfying case solve necessarily yield precisely unchanged translate random make desire often mistake behave lead remark quantity tool clearly hold dependence small claim reasonable least equally intuitively minimal magnitude drive instead decrease third much say really pair penalization come back piece two span model performance correct validation improve take section constant multiplicative factor eq difference lie one behave like ideal cover theorem conclusion decrease cross leave soon similarly derive leave formula component clearly believe show focus fold leave write major cover correct c former quantity dramatically matter variance increment compute section depend basis prove section section illustrate theoretical lebesgue density sx sx weight gaussians collection model consider every regular histogram bin dyadic histogram two n j piece set simple flexibility optimal small risk consider show help reduce risk even consider collection histogram collection procedure datum oracle model setting experiment less difference risk two well drive penalization vx suggest multiply penalty multiply procedure independent measure uncertainty measure empirical divide report setting p report penalty already know table performance fold method consider result decrease increase large l go fold confirm help picture fold minimize indeed increase simultaneously variance validation depend least estimation fact perform much contrary fold regression explain fold perform fold fact fold validation coincide fold penalization multiply fold penalization phenomenon discard setting difference comment fold influence unbiased satisfy distribution independent index ms conclusion decrease significant leave distinguish set l figure close confirm appear indeed mn proxy similarly heuristic decrease translate well distribution around explain observed influence comparable report lead conclusion b computational fold penalization validation density naive exposition fold criterion estimator square framework orthonormal describe precisely occur avoid risk even criterion correctness histogram algorithm close complexity particular histogram discuss fold selection discuss model square fold hold datum definition hold dramatically variability note fold follow unbiased fold either hold x prove similar theorem quantify criterion comparison fold similarly role split criterion proposition paper inequality least square penalization procedure lead constant inequality prove stein practice inequality suboptimal constant oracle overall none imply strictly mention propose evaluation regular contrast slightly modify compare well regular histogram likely present performance trade square since see well decrease increase remove completely fold penalization variance decrease reach article provide concern bias fold resp fold factor resp simulation factor even literature behave address penalization answer least fold cross penalization want would depend want thank careful reading author acknowledge de project bs project program use repeatedly proof number right side schwarz consider b converse true proof eq p cp follow us eq fold always leave hand use imply statement prove algebraic show cv hand cv minimizer coincide lemma variable v I otherwise density separable let e
observe model intractable connection computation finding although consequence previous uniformly whenever ergodic uniformly ergodicity soon pseudo direction nonzero right bound pseudo chain nonzero gap rely gap proposition ergodicity pseudo bound modification remark observe establish necessity existence ergodicity neighbourhood ergodicity pseudo marginal existence possess robustness perturbation marginal pseudo metropolis primary implementation uniformly rate rate scenario support imply geometric together mild additional integrable establish existence lyapunov satisfy sub lemma distribution possess power moment establish walk metropolis rwm conditions ergodicity rwm existence polynomially ergodic situation e allow tail remark intermediate tail vanish tail natural average counterpart one fact situation one marginal theorem support observation efficient version go converge show geometric prove practice implication central theorem possibility asymptotic variance pseudo introduce measure measurable two fx gx start play order concave jensen desire order facilitate marginal reversible lead acc marginal qx gx quantity ax rx g qx gx ax consequence truncation rx qx last follow generic add x qx q bound scale average reversible measurable denote whenever fx apply lem h specific gap auxiliary define nonzero bound also imply integrable respect dirichlet kernel operator spectral gap pf qx decomposition bind imply concentrated simplify lem appendix exist imply fx w fy qx fx qx fx let satisfy definition satisfy coincide marginally together imply observe fact kernel reversible repeat marginal next support theorem different statement explicit fix x w may imply necessity existence consequently geometric ergodicity existence uniformity mh walk second case existence bind necessary bounding ergodicity may boundedness pseudo geometrically ergodic modification unnecessary marginal exhibit define leave gap existence spectral ergodicity reversible positive establish ergodicity require record pseudo positivity spectral follow rwm write version f qx w fx z sum share indeed scenario u z multivariate ergodicity soon marginal algorithm admit nonzero spectral gap marginal geometrically ergodic able find completeness counter pseudo yield toward atom geometrically ergodic define otherwise choose gap family distribution index reflect relevant scenario particle clear hereafter pseudo invariant easy weakly invariant invariant marginal situation ergodic improve convergence situation take integrated autocorrelation see pn consequence corollary suppose kernel weight uniformly consequence corollary unbounded geometrically ergodic formulate technical return check lyapunov suppose autocorrelation time finite assume exist satisfy g autocorrelation inequality generality claim autocorrelation n k n corresponding coincide marginally couple set n nx show follow inequality k n qp n ng old cauchy two probability n nx nx imply exist uniform nx qx qx qx third therefore let eq q qx u additional find go sufficient imply condition quantitative bound behaviour marginal result rely establish algorithm drift keep presentation simple set vx gx v write w gx u n w therefore metropolis hasting ergodicity pseudo illustrative develop relevant analysis assume ratio w word x ergodicity pseudo rely inspire establish ergodicity explore sub x ax ax w ax bx bx w ax v bx x fast cf suggest marginal uniformly state compact uniformly integrable pseudo drift toward practically small page quantify geometric drift indicate consequently ergodicity q record implication exist result drift away pseudo chain away geometrically ergodic suppose acceptance marginal away vx w vx qx qx rx qx w convexity imply lemma exist claim often moment record highlight straightforward pseudo drift vx w b proposition toward imply geometric showing sub qx qx recursively nx b assume establish condition fact condition marginal geometrically walk rwm exponentially decay contour existence rate ergodicity marginal limit rely moment condition section may appear general cover condition admissible corollary extend result beyond case condition sake clarity detail unbounded distribution ergodicity possible perturbation indeed rwm establish exist almost every geometrically ergodic throughout region almost acceptance rejection r moment supremum lebesgue simple throughout hold also inequality observe w result respect lebesgue continuously differentiable tail exponentially decay regular contour moreover satisfy neighbourhood origin drift pseudo satisfy constant metropolis invariant density proposal define constant eq vx vx apply fix lemma define must least polynomially allow establish drift drift negative presence establish ergodicity seem however condition behaviour follow write w large drift suppose vx w z integral partition intersection acceptance rejection set notational straightforward x x w u w crucial unity grow q z x imply w w vx vx vx vx x deduce regime vx r x constant subset x term yield drift u u c consider condition z enough conclude replace moment moment tend infinity exist lebesgue z possible rejection metropolis implicit assumption exp whose help gain intuition metropolis increment satisfy constant polynomial drift furthermore quantitie lemma lem theorem quantity prove give sufficient additionally increment satisfied drift depend marginal choose choose condition satisfied compact set monotone decrease tail sufficiently large due imply define easy tail ray conclude hold condition denote b upon appearance denote condition observe x u u z similarly z w w u w may lem ensure increase sufficiently large vx vx vx vx w let exist vx u w z cx z cx cx decompose sub obvious abuse observe x z u x x hold z x z x z z cx u u claim verify decomposition section case bound central limit hold function specifically relate drift constant reference therein ergodic drift reader due well limit take
proxy use technique low rather quality undesirable request approach facebook platform al build tool request statistical approach match request pattern could facebook platform support friend application profile read preference access resource limited request application request request et facebook find request information approximately build version page website find page list yield link page feed select permutation ac market engine additional page rate rating et us facebook application rating limitation data request attempt explore collect secondary request might incorporate global rating popular play role request list frequently depict frequently facebook name email offline friend friend access communication view storage phone call read phone fine gps location control tool device tool start phone read contact control video tool retrieve run application globally price tail high evident almost decrease slope pricing application amount area decrease price extra rating user rating logarithm rating rating except peak rating follow center score star examine average see high tend quality rate additionally extreme score less indicate score alone insufficient cumulative price axis start cut rating score total dot use logarithmic rating discard rating rating respective htb infer request application request entry request request input want matrix encode frequently request share request pattern boolean matrix concept factorization request z employ factorization binary likelihood statistically outcome factorization representation originally role request mixture mining request request correspond request application assume follow sx boolean request assign model entry modify parameter flip request distribution reduce computational ultimately terminate predefined input I probability request request particular assignment request assign request request cluster group request simple pattern suffice could lead overfitte contrary complicated might pattern significant individual heuristic selection instability analysis instability pattern subset independent produce clustering clustering able precisely two subset lot I subset datum difficult classifier second I run multiple arbitrary htb request facebook suffice facebook depict five mining technique identify assign sort common second disjoint member application request multiple include reflect conditional conditional estimate whenever score facebook color end conditional detail request false ideally positive false false positive assign occur request cover define false incorrectly assign application otherwise false false please resolve discuss facebook positive thereby small application rate reflect great application positive negative partially attribute large place application request fully high training set statistical request application htb request help quality user high incorporate could request fit risk positive high regardless request false high negative indicate suggest request recommend request pattern application incorporate market group application category purpose book reference weather category roughly request strongly figure illustrate relationship category category classified many application communication fall indicate category contain request application kullback report pattern always find one category informative check simulated request want application request exhibit pattern pattern reflect chance request high pairwise conditional eq quantify one request test request request application request pair htb conditional probability request circle red request low pair figure dataset please axis logarithmic also histogram bin bin significantly please total average suggest provide potential direction user rank result receive enough review possibility request would notion need subject research technique suitable detect aim quality otherwise without receive favorable request find request alternative create good obvious drawback approach manually manual analysis consume hundred diverse available bad corpus significant application quality partially influence request exclude application specialized request interested user believe reason rating request significant find stability essence necessarily require fraction application request use request pattern facebook fit request low application request pattern application indicate request part risk relationship pattern application category intel national science facebook party access private ability call security restrict application privacy study often understand request request system cluster corpus facebook application pattern request factorization facebook request fit complex request request pattern application suggest request user application platform result availability hundred thousand click end potentially choice look decision security hardware facebook access user social operate access privacy relevant resource request example application text user complete intend privacy security unfortunately demonstrate pay major know combination simplify identify request application method
usefulness change uncertainty lead correlation treat interface I minus minus optimal middle maximum figure interface period maximum amplitude interface marginally interface well uncertain interface come three thing geometry secondly facilitate change elastic position introduce specify position correlation field specify depend idea rather inversion natural soft opinion may mail difference devote use change reader field denote element depend implicitly define position denote denote operator equivalent mixed operator convergent separate angle inversion discrete layer use system become specify prior flexible incorporation advantage markovian introduce huge differ additionally geodesic quantify multivariate spatial specify different prior multivariate correlation cross problem inversion primary inversion propagation equation solve include yield well inversion result adopt text convolution shape wave relative difference wave elastic relative pressure wave velocity velocity wave background discretize discretized operator novel system partial differential flexible region figure appear comparative colour scheme min individual make sense without inversion importance choose depend heuristic covariance covariance define change field widely inverse correlate correlation specify knowledge discretize eq correlation elastic typically possible transform computing covariance together lattice sensible require stationarity storage another computation requirement pseudo differential pseudo differential closely short transform approach storage statistical mat ern covariance firstly noise mat ern computation operator induced cholesky requirement vector relatively quickly address field alternatively manifold boundary merely namely box boundary direct interpretability term specify condition retain usually opinion stationary prior point space sound merely desirable letting correlation vary flat define direction eigenvector stationary flexibility bottom layer stationary interface interface direct interface almost isotropic relate usual model white noise inversion article obvious specifie way strength operator inversion answer obvious investigate alternative criterion new model replace symbol local operator state polynomial markov approximate rational jx suboptimal discretization suit candidate eq stationarity convenience ern choice define restrict hard correlation field matrix inversion phase layer phenomena potential drawback possible correct correlation situation actually interface expert incorrectly lead geodesic discuss handle quantify interface possibly require suppose say interface bottom geodesic grey cover interface properly fair interface section investigate purely possible combine interface increase complexity model estimate range location many modify demand flexibility domain simplification interpretability motivate behave hand kronecker carry case natural way cholesky locally square possible define way get minor concern practice long interpretability triangular inversion hyper well natural way hyper identifiable field well wavelet denote possible latent treat correlation interface straight geometry scope enforce interpretability namely multivariate field matrix free specify must great three apply depend different model stationary interface simulation simulate multivariate density unimodal quasi estimate unimodal identifiable estimate eq much difficult reflect broad distributional recover come may come smooth estimating density test essentially noisy reconstruct field fidelity follow subsection reconstruction base identity noise second give reconstruction iid noise three flat interface priori correlation structure level function flat optimisation one reconstruct field attribute sum prominent use smooth exhibit middle model seem lead range
continuity probabilistic order express conclude mm mm corollary corollary difference estimation mm ac university ac du technology ac song liu technology song ac institute technology ac address estimate density computing procedure well without regard step error shot procedure separately usefulness demonstrate experimentally density kullback quantity element approach two separately poorly stage carry regard incur stage big error appropriate without seminal separately estimate learn sufficient ratio two density separate explore probability shoot various purpose change change extraction detection segmentation shoot directly separately estimate derive stable equip cross systematically parametric setup optimal distance show accurate estimate compare experimentally usefulness semi change rest investigate approximated section shot density analyze formulate set identically dpp goal use kde give determined cross argue kde density good nature cause final tend estimator tend smoothed overcome shot estimate estimating difference loss parametric gaussian kernel center optimal computed replace add objective function parameter derivative function analytically eq finally density investigate parametric basis q matrix imply parametric setup setup width consider parametric rkh prove satisfie utilize dimensionality vector regularity non achieve naive kde rate kde regularity negative difficult implement always regularity superiority cv divide n denote validation average hold implementation available make expression replace approximate average follow replace estimator estimator eq approximation nevertheless generalize bias eliminate impose yield regularization lower replace reduce see eq plain essential generally popular divergence illustrate distance kl implication density draw depict kl distance whereas divergence infinity infinity outlier possess direct depict outli run show profile divergence value mm mm mm finally procedure conduct density dataset assign distance nan ranking depict rejection nan outli outlier whereas rejection base keep two presence test depict outli outlier keep low rejection tend nan reject nan p apply supervised change series real bias balance reflect test recognition classify unlabeled test set wise input mix illustration match uci randomly label unlabeled class misclassification weighted run result translate datum series subsequence single dependent incorporated subsequence measure plausibility distance record environment display true manually annotate indicate score next provide orientation take bottom score much interpretable method directly propose framework solution systematically optimize normality finite case convergence also bias cause experimentally outlier kullback paradigm useful topic estimation explore application acknowledgment comment support program du song liu define rkhs endowed width difference exist member integer modulus technique regression exist constant
think equation term negative recognize obtain bind treat constant lower call q indicate select complete complete latent maximize constraint easily introduce hide select posterior lagrange easily incorporate pz pz pz nd pz nd pz nd pz pz nd pz nd pz pz z pd nd pz pz pz pz pz w I thank zhang point error manuscript information medium social mining recommendation collaborative filter paper establish originally early technical report estimation estimation hierarchical factorization equivalent show well behind topic vocabulary represent probability topic secondly topic original specification multinomial appear document generative although subtle need token vocabulary distinct differ token place appear multiple time imagine want decide term token document position want step multinomial suppose choose term generation token summarize token probability token document appear formalism lagrange multinomial optimize summation parameter apply token position choose word token position token position document plug standard posterior hide di pr di di di di di obtain give hide similarly token position term simplify simplify subtle need indicate token satisfie element calculate still lead write token variable follow auxiliary auxiliary jensen lower bind goal clear left hand side possible q previous way formulate see equivalent pz pd whole q whole maximization guarantee likelihood maximize useful really maximize likelihood likelihood maximize show
architecture communication inter machine scheme simulate architecture microsoft windows platform speed virtual investigate parallelization call know parallelism mean satisfactory aim goal reduce need reach theoretical parallel aim dataset mod operator stand division operation sure prove belong stochastic gradient idea entity perform instance represent prototype technique denote investigate scheme rest organize provide bring satisfactory provide consequently bring finally asynchronous adaptation latter fit communication cloud computing test synthetic investigation start parallelization resource start apply sequential subset process processor synchronization share unit local comparison typical scheme implementation communication instantaneous resource bring entity investigation satisfactory series sequential rewrite synchronization iteration resource rewrite assume distortion consequently iteration drive speed choice sequence exploration trade respect share observer section process exploration lead suppose seek evolution sample accelerate merge share version average translation result typical display speed obtain resource acceleration great phase frequent could attract would slow compute parallelization deal communication cost update delay delay architecture share computing hardware issue subsection follow remove synchronization reduce phase realistic receive share entity synchronization unit share soon complete dedicate share barrier delay performance result
originally feed via agent interact environment uk partially observable decision widely provide making provide slightly call decision going join basically interaction agent light make observable markov make agent environment notable characteristic plan uncertainty mention mdps computer among mention demonstrate interaction markov implementation remark development call former one basically factor benefit state factor two different writing stand observable factorize eq music compound simple make decision receive concept follow environment fully state whereas observable suppose simplicity beyond interval eq q action agent stand make decomposition decomposition relatively partially observable intermediate make make possible value q finally factor second testing
marginal equal input system respectively function probable modelling phenomenon determination kernel describe literature complex kernel allow reflect htp take depend point view convenient inference conduct use determine hyperparameter value likelihood experiment take web execution environment service simulation request proper work iv request v service htp input execution output system quantity cart matlab cart toolbox occur demand size exponential express likelihood htp environment allow point iii universal time unit iii represented box mae comparison cart represent square perform slightly sample mae significance cart share variance reject value reject statistically cart htp htp process attribute service regression field learn service gaussian index namely process statistically cart show develop result high experiment sum paper try fill acknowledgment partially european technology web service performance system attribute execution service repository stream request index square perform statistically compare prediction service modern network method theoretical consideration explore web service consist layer service service design service paradigm application layer predict express service repository g service service dependency attribute stream request technique detection use resource allocation fact internal unknown execution system random probabilistic seem consider application parametric regression system application web service organize sect problem web sect gaussian environment draw input execution maintain execution execution attribute spend execution call simplicity target output however know responsible process dependency input hence historical consider need
ode initial converge uniformly note arrive iteration avoid randomness stable equilibrium proof g sf briefly update track n update sf one taylor g sf obtain surely theorem allow satisfy still sequence difficult tune appropriately analysis sf exclude verify claim hold alternative sf feed arrival process customer leave probability service complete customer service node n ii distribution service node individual update customer cost minimum nn optimal measure measure method average run value indicate use begin network control arrival leave system service simulation intel second relation trial gaussian simulation trial clearly fluctuation side cauchy sf last indicate flat ccccc cauchy side sf sf axis side sf axis sf axis iteration axis indicate sf sf suggest run sf sf attribute behavior quite obvious plot numerically achieve zero plot independent trial trial govern run hence multiple fluctuation reduction fluctuation prominent middle well smooth occur distant gradient become provide exploration nearby c whether control parameter step tuple run take entirely second concentrate overall case customer choose cost queue length experiment quite one table trend specific time service exponentially ensure well constant vary scenario expect reasonable performance since decay gradually v however theorem simulation increase drastically pt c sf c sf distribute service mark origin popularity notion build fact generalize equivalent smoothed functional two smoothing ode significance demonstrate smoothing improve algorithm appropriate modification develop adaptively conclude note idea hessian develop newton random identically uncorrelated hence possible gaussian generate box algorithm variate use correspondence student distribution duality observation state variate na functional sf scheme efficient kernel include gaussian cauchy distribution among new gain decade attribute ability generalize exist smoothing motivate sf constrain project set ode numerically numerical author computer institute ad objective expression common engineering financial encounter financial noisy optimize one commonly originally approach algorithm develop control track objective employ estimation limited applicability provide system parameter efficient technique base sf stochastic newton involve long estimate parameter perform long average constitute approximation sf performance sf gaussian sf significantly tune size improve perturbation sf distribution smooth paper propose smoothing aforementione extensively mechanic gaussian family distribution smoothing sf unify control control convolution thereby kernel kernel derive smoothed exhibit smoothing sf incorporate procedure algorithm neighborhood optimum differ early straightforward simulation sf counterpart sf present sf gaussians smooth gaussian sf provide conclude remark technique gaussian unique measure minimize existence continuous procedure optimize functional verification trivial instance translate impose turn process general lyapunov say conditional py see parameter control field compact nn one depend function idea functional call call smoothed sf control purpose indicate sf sf also functional behave behave retain sf aforementioned function ng ng g distribution dirac zero concentrated mean fluctuation estimate discuss gradient rule simple show gradient long single stage parameter tune simulation gradient side sf simulate parallel pt gaussian cauchy kernel search univariate dirac delta ab list column indicate retrieve clear motivation study gaussians suitable infinitely kernel continuous value evy super finance distribution shannon expectation coincide usual notion retrieve gaussian maximize shannon eq q generalization first moment usual form q well probability gaussian distribution discuss kernel also covariance variate normalize constant limit fact infer apart special table circle smoothed algorithm sf notion gaussian case imply uncorrelated standard convenience center functional gaussian apply sf ensure rest smoothed functional two sided property evident g ng ng g ng claim manner hence result sided nature control existence sf sf consider respect side smoothed I ji write sequel identically random ergodicity assumption govern similar manner two sf eq sequel obtain satisfy nb must note quantity positive number average inner principle generally typically generation consist uncorrelated random describe project vector step ht fix simulation govern z nz sf similar except update estimate simulation affect fix vector generate generate govern update nz prove technique compare present gaussians provide variate consequence result key uncorrelated zero variance zero axis case first use integral beta expand integer get I gamma function result similar equation however define easy limit qx subsequent analysis sf update fast g sf rewrite iteration use p pn disjoint p quantity convergence invariant satisfy martingale difference variance xt exist ode well unique globally stable differential equation ode xt rewrite
node potential guarantee decay ise degree absolute potential respect observe short depth trade depth identifiability impose degree low representation observe distance bound information impose requirement estimation edge characterize presence absence correlation extent enough edge impose impose edge potential path low potential need depth cycle graph determine build roughly depth ensure distortion additive test choose appropriately establish correctly node close absolute distance bound consistency tends supplementary locally high requirement logarithmic require complexity graph locally like match fully apply distance form local implement improve complexity extension learn tree establish like graph extend discrete notion acyclic subgraphs subgraph encounter induce denote consider di j j v j di access distance estimate algorithm relevant assumption graph marginal neighbor q converge locally bind q routine variable decay degree node identifiability tree local ise edge potential use bound require similarly b impose parameter local span constraint imply compare enable reconstruct bound concentration bound distance merge node latent condition structural q pac reconstructing model graph large graph latent tree model notable require requirement decay establish converge attractive ise model without analysis require local convergence applicable discrete decay accord gaussian mixture step correctness tree distance di concentration distortion presence briefly establish correctness graph distance cycle correctly latent correct local correctness guarantee reconstruction learn analysis performance ensemble degree constant depth straightforward union vanishing scale uniform scale far reconstruct notion graph inexact let unlabele without latent page decrease test indicate generalization also evaluate occurrence observe select version capture degree graphical edge lda dirichlet bic monotonically decrease bic indicate complexity bic general involve employ propagation exact configuration bic addition coherence frequently topic model pair score good measure human coherence external corpus contain article bag vocabulary article top word probability vector select method graph ground hand bic overall normalize output show node edge overfitte threshold l na na na employ control cycle tree long cycle find short effective discover intuitive relationship link computer see tend subgraph computer program context modeling contrast design hide input competitive coherence find short cycle intuitively unable topic coherence suggest however model well indicate use model lda top word disk video window memory windows graphic pc windows software version disk pc space disk window file graphic format disk window mac evidence human power outcome market observe threshold tree another score threshold certain learn figure capture particular degree various group home group together however relationship threshold decrease add first connect index cycle decrease company chemical connect stock stock return stock return density stock initialization building cycle add decrease neighborhood trend stock predictive predictive edge prediction evidence however reduce degradation predictive effectiveness discover line early reveal learn established method decay derive selection finding reveal add complexity certain regime tractable latent thank berkeley discussion begin algorithmic uci david uci anonymous substantially appear remarks award award award nf award graphical consider characterize tractable develop provable locally regime ise structural n node node depend potential ise structural derive requirement implement provide output learn discover latent variable extension many form tree effective evolutionary give rise day specie financial modeling recognition vision learn g dimension consistent much small moreover despite restrictive topic encode topic reality relationship assumption nontrivial challenge develop area active neighborhood cycle regime practical tree latent like correlation implication decay immediately subgraph cycle challenge subgraph estimate merging matching estimate clear locally like graph latent introduce cycle model consist add neighborhood precise structural consistency page general simplify ise random obtain number require near depend minimum potential homogeneous variable match fully graph select maximum degree respect propose attractive parallelization dataset provide flexibility cycle variable output incorporate score bic schwarz estimate fidelity control cycle belief propagation experiment discover intuitive compare well generalize allow multiple greedy computationally expensive algorithm guarantee em base extensively mainly thorough overview provable guarantee paper inspire work recent classified group convex latent decay make connection explicit relate limit learn graph analogous study provable guarantee discrete high gaussian theorem et al consider convex relaxation easily model incoherence method hard contrast graphical estimate poorly dimension accordance page graph discrete edge variable mass p graph clique configuration clique serve normalize correspond family special ising ise subset node discover presence hide learn observe variable observe general graphical model tractable construction bind graph recently although tree like general global specifically constrain constrain latent tree establish tractable condition limit uniqueness weak condition correlation configuration decay neighborhood provide distribution markov potential neighborhood let graph former refer edge short distance denote norm say exhibit variation decay decay maximum latent graphical know tree factorization distribution pairwise distribution root root parent say recover distribution satisfy counterpart equivalent positivity particular ise potential positivity configuration give graphical subset structure model extensively study topic method additive metric structured joint consider metric metric special ising model logarithm operate additive tree path refer tuple denote characterize tree term characterize distance refinement robust test meet search occur side output test relationship infer involve child weak merge use modify grouping routine locally node observe node j j v ac bc relationship far break forest hide length merging method term add add local neighborhood consider neighborhood set construct especially attractive reconstruct graph long liu grouping develop latent section main acyclic approach challenge piece presence metric address decay challenge merge
often correlate differ order magnitude narrow local proposal propose underlie intrinsic thereby principled proposal automatically tune move possible mix chain chemical reaction obtain paper method compare commonly hasting study chain monte behave next overview markov jump approximation section mcmc x process satisfy q conditional state problem transition depend time rate expensive exponentially initial simulate next state exponentially mf complete respect time transition occur transition specify suitable apply transition solution propose trajectory transition time lead construct iteration trajectory observe demand system observe poor mix many sufficiently result demanding suggest simulate use exact metropolis master provide simulation observe result purpose presentation formal derivation reader requirement approximations constant jump fluctuation example chemical capacity circuit langevin equation match dynamic satisfy condition transition poisson variable transition large imply approach limit compute variate langevin columns return wiener differ due intractable augmentation type transition unobserve euler follow langevin dividing get eq see fluctuation equation eq linear differ stochastic taylor differential nothing finally collect remain jacobian fluctuation state magnitude negligible coefficient deviation thorough supplementary convenient expression approximate linear integral sense integral simplify imply multiply time assume trajectory trajectory state continue kind encountered measurement reaction inference independence applicable available stem normal diagonal need notice still exploit markov likelihood invert variance brief work concept mcmc require introduction scope metropolis hasting algorithm move accept context inference density involve proposal mechanism proposal multivariate normal therefore replace tune fast mix high mix markov hand deviation posterior substantially correlation adaptive hasting reaction rate scale simulation metropolis scheme scale independently target manifold mala langevin employ euler solve integration th symbol coordinate mala stationary metropolis hasting correct proposal local gaussian metropolis contrast tensor avoid discuss simplify derive assume manifold depend see mala tensor identity correspond euclidean probability momentum potential hamiltonian manifold energy energy simulating require reversible volume preserve generalise simulate detail generalise path across iterate tuning integration simulate hamiltonian hasting mix e h tensor manifold hmc mcmc discuss metric natural tensor fisher multivariate normal general time point integration derivative quantity ode system ode system equation sensitivity derivative respect ji operator stacking model chemical differ order mcmc allow operate unbounded unconstrained parameter numerical simulation determinant jacobian derivative follow prior incorporate exist suitable informed knowledge design want restrict chemical become high since enough prior restrict introduce normal strictly moreover note fisher use add near simulation chemical consist unstable unstable convert reaction time ts ess h know make explicit infer constant simulate simulate state discard trajectory observation trajectory mass rather trajectory perform drawing index independent point paper similar derivation namely system approach transition ode system density identity condition include draw hasting sampler scale adapt burn phase acceptance follow simplified algorithm initial tune achieve sample adequate sample table compare ess different gradient fisher mix good achieve system sampler system poor normalised ess mix contrary report marginal posterior posterior figure mh sampler similar standard around significantly system chemical reaction transition matrix consist specie appear depend fail system concentration depend posterior reaction constant reaction demonstrate procedure parameter namely burn sampler converge variance obtain posterior show along constant fail illustrate applicability methodology system also involve expression protein adopt model specie dna chemical transition degradation degradation reaction notation dna dna system state transition rt rt rt rt gene section model stimulus towards gene protein concentration gene model hill make protein refer auto single expression probability simulate discrete unknown constant interval take time independent trajectory discard rest resemble experimental observation finally also trajectory system unknown figure infer b sampling independent point chain gene firstly see provide accurate parameter mala perform explain correlation parameter make sufficiently degradation protein affect rate expect heavily joint posterior exhibit correlation trace auto gene true hasting ess ess mala ess ess mean ess ess hasting ess ess manifold mala ess ess l ess ess panel expression dash dot posterior contour sample p red colour bayesian important show although inference autocorrelation limit applicability stem discrete simulation manner simulate ordinary differential fluctuation analytic sufficiently diffusion approximation research metropolis demonstrate exhibit metropolis sampler strong reaction gene system geometric proposal mala conceptually provide good trade auto believe
extent spectra near overlap sec time sensitive lie little spectral mode sensitivity examine accord criterion work nominal fouri periodic frequency show movie embed separability periodic doubly degenerate periodic mode fig movie mode generate anomaly amplitude mode exhibit amplitude domain movie e along variability california current frequency laplacian suggest one low mode power power shorter upper north typical movie prominent scale temperature develop west family gradually short peak short movie key temporal arise sharp year periodic counterpart nearly base integer multiple year result fouri spectrum peak skewness frequency mode fine spatial activate amplitude small aspect enhance mode propagation scale temperature anomaly spectrum include anomaly composite temperature anomaly consist lead periodic window near spectrum comparison scale parameter singular mode mode anomaly north see raw year field determine mode lead frequency mode describe degenerate describe variability dynamical pattern highlight account nonlinear high existence variability variability restrict temporal lie lead datum riemannian capture rare event occur behavior truncate expansion show norm dominate lead hand clearly moderate amplitude occur embed introduce qualitatively pattern illustrate spectral gap separate mode spectral evaluate via versus parameter display fig evaluate embed cf fig line preprocesse subtract filter overview fact lead degradation conceptually preprocesse nonlinear neighborhood relationship extract north study entropy temperature field temperature anomaly field structure mode sec characterized decay obvious peak periodic spectrum obvious turn mode spatial experiment dimension depend crucially dimension window temporal pattern conclusion qualitative scheme series take arise like utilize partial high distinct mode linear differ crucially square integrable nonlinear manifold comprise coarse grain space tailor nonlinear geometry riemannian increasingly behave orthonormal address theoretic method assess assimilation k overview online release initial prominent g statistical temperature identify regime nonlinear application v advanced w preserve decomposition compare laplacian low frequency variability qualitatively spatio learn representation diffusion map reaction g intrinsic system diffusion et regime shift definition h f volume geometry series detect volume page r geometric tool harmonic definition volume page determination reaction coordinate locally scale lee york wu nearest search j temperature california extract qualitative g reveal nonlinear analysis von w university et data assimilation range forecasting part I grain simple quantify predictive coarse grain analysis generalize spectrum complex represent pattern exhibit manifold constraint notion smoothness require temporal belong hilbert span lead eigenfunction eigenfunction evaluate ambient singular fine graph important rare dimension overfitte criterion north year run besides recover process carry variance capture dynamical keyword spectrum acquire large weather forecast seven organize spectrum variant physical interpretability algebra detect portion dynamical building work operator use differ crucially tailor hilbert square ambient employ eigenfunction imply relation cf partial capture laplacian capability rapid transition rare efficacy north version mode contain behavior domain characterize five year period amplitude spatially mode enhance region anomaly dynamic little raw order capture pay characterize minimum beyond particularly prevent overfitte machine hereafter define exist fulfil eventually theoretic step familiar system delay indistinguishable differential large manifold incomplete thought euclidean e tangent construct inner riemannian effectively constitute adapt exhibit regularity pattern projection rapid geometry nonlinear produce ingredient replace datum specifically temporal mode well behave k associate define theory theoretic basis scalar product solution problem orthonormal span lead element derivative
eigenvector dimensional isotropic center multiply unitary thus row assume isotropic take probabilistic close likelihood rotation generalize model semi pca observation formulate noise coordinate remain part linear pn q linear toy comparison spline case introduction generalize look verify eq finally eigenvalue covariance obtain pca order combination knot position function formula drawback equation matrix rotation original datum perform inverse step p intercept directly non linear select among candidate penalize criterion classical criterion certainly observation either bic popularity compare pi measure interest tr maximized method index parametrization refer topic pursuit problem index normality complex scatter dedicated outlier coefficient matrix entry otherwise could iff neighbor proximity generalize near projection axis eigenvector article additive b spline select coordinate sample deviation standard axis near original axis htb axis use criterion select number point htb c dim bic residual draw library give coordinate sample standard plan display library bic criterion residual variance factor expect axis dim residual spline function library consist dimensional classified network network terminology article model select largely control regression parameter step datum cloud pca display htb summary table visualize point dim htb predictor evolution coordinate visualization prove principal model simple truly interpretation depend highly projection linear library simulate display proposition example section corollary cover property propose set model experiment simulate pca pd affine scatter principal linearity assume cloud principal start technique curve principal surface belong neural non linear model projection adapt point approach estimate model give hereafter auto function dimension construct auto affine precisely write equation principal surface linear thus pca study view situation projection linear however difference appear get tractable noise assumption seem estimate estimate extend moderately high complicated simplify inspire perform present seem linear organize probabilistic
clustered hmms ms despite suffer memory hmms need store evaluating time replace output consequence result appropriate assignment similar spirit bregman base divergence difference form random whereas validate point discussion experimental density automatic benefit cluster sc high synthetic similarly annotation favor suggest robust experiment run suffer delay number hmms standard em considerably section hmms consist input hmms hmms hierarchy level hierarchy first level input hmms experiment input hmms simulation cluster hmms various summarize hmms input hmms learn explain build hmms hmms recursively h form cluster hmms group center virtual hard cluster virtual cluster calculate hmms hmms satisfactory allow gmm general experiment good affinity integrate extend construct k mean hmms virtual particular observation reduce consistent life goal series sequence input hmms hmms hence cluster cluster directly cluster model texture dt series hmm modeling stage fashion variable immediately level hierarchical together dynamic dt mixture analogous h time gauss discrete multimodal metric measure cluster assignment whether pair incorrectly assign well fit time likelihood hmm expect generate hmm assign sc hmms assign explicitly optimize similarity likelihood consequence optimize intra wise intra cluster tb walk b motion motion series represent hmm cluster hmms represent motion motion apply sc cluster motion capture action dataset motion span class jump marker spatially illustrate build hmms learn motion state emission level contain em dimension repeat action dataset sequence consist time coordinate body marker capture subject emission e h sc random initialization performance dataset hierarchical color indicate ground cluster motion motion class jump together portion probably dynamic action movement shoot hierarchy together indicate walk highest create desirable return considerably rest experiment sequence properly incorrectly quite furthermore high level ccc rand quantitative comparison rand vs level particular cluster sc level overall keep mind similarity favor h addition novel center motion center conclusion effect table generally rand em original motion sequence implicitly integrate virtual variation motion sequence robust roughly time sequence favor validate still closed expression virtual approximate statistic without center h large significantly long finally report consistently cluster dt note rand sc optimally cccc cccc rand present physical two rand e level level outperform high center sc ground truth rand index metric h sc cluster hmms hmms hmms hmm noisy generating estimating noise noisy version hmm noise vary collection hmms number always emission sequence original hmms state transition emission hmms emission distribution set hmms differ emission emission specify experimental b figure bottom differ variance emission sc hmms cc result clustering rand expect log likelihood center hmms assign average hmms experiment different variance produce superior line line color rank song retrieve metric tag least usage tag fold validation tag sc song gmm annotation f time sc algorithms annotation look overall runtime music runtime sc h song level efficiently song tag runtime sc correspond set run significant performance affect contrary em strongly list regularization leverage song sc considerable cluster affect result annotation model novel cluster retain annotation achieve considerable sc computational demonstrate hmms observe outperform sc suggest former accurate center fact cluster cluster maximum sc hybrid contrary separate cluster optimize log respectively center may affinity finally generative outperform audio metric base pair test annotation h base contrast hmms gmm non time experiment outperform perform similarly music audio short frame investigate character originally motion character segment hz force numerically smoothed training half testing write character hierarchical character relevant gmm character aggregate hmms component use virtual virtual hmms assignment hmms determine mix similar temperature hmms desirable hmms estimate character character write character example classified character retrieval character example rank character sc hmms learn since vary variation threshold varied retrieval performance average tag include intermediate hmms trial initialization stop retrieval classification accuracy em sc list retrieval various consistent annotation retrieval sc metric center h representative relevant intermediate hmms hence relevant classification training fast sc hmms character efficiently actual perform retrieval reduce number approximate suffer overfitte suggest h consistently improve metric converge suggest regularization average sample virtual actual positively elaborate annotation retrieval section character character song audio tag datum performance evaluate lead slow iteration character control character intermediate clustered summarize provide good candidate center conclusion face information select hmms cccc p generation virtual virtual annotation retrieval training tag song level varied retrieval section vary similarly virtual music annotation retrieval virtual song level improvement experimental virtual sequence short positively reduce quality variational cluster respect distribution represent demonstrate efficacy h summarize annotation improve model virtual prevent learn virtual performance virtual sample stage hierarchical intermediate particular partition song song ms execute song tag impact depend slightly hierarchical may suit cover inspire datum intermediate intermediate h ms final plan extend gmm universal emission hmm commonly speech move complexity estimating background inference g similar plan hmms emission extension background model song motion acknowledge wish acknowledge yahoo fellowship national science grant grant china support foundation research support part appendix derivation maximization carry independently follow terminate maxima step involve hold distribution substitute reduce mixture weight collect weight constraint maximize state independently hmm collect summary consider appendix give similarly hmm collect transition maximize emission function index gmm give gmm optimize optimize hide markov generative condition hmms base collection hmms characterize representative manner consistent cope intractable leverage variational demonstrate motion capture annotation music hand show method variational reduce improve robustness hide model markov hmm probabilistic model assume double hidden evolve markov encodes encode appearance condition current hmms music online hand write cluster hmms representative center appropriately represent application motivate hierarchical speech hmms indexing reduce hmms large semantic annotation video cluster hmms estimate work exist cluster hmms operate directly accord pairwise manifold application would solution propose construct cluster apply cluster prove successful group hmms generate center give hmms hmm maps close suboptimal hierarchical estimation hmm spectral sequence instead hmms algorithm construct output hmms derive hierarchical input hmms input hmms hmm guide estimation em main em observation algorithm probability mixture reduce gmm cluster dt image indexing represent I mixture semantic extend mixture hmms hidden markov mixture ms marginalization state process tractable hmms formulation leverage derive make tractable hmms hmms representative class graphical leverage approximation general suitable center manner memory requirement hmms mixture large dataset estimation run intermediate principle maximum estimation average possible step provide prevent robustness compare full costly operation pairwise fold hmms generate center evaluate involve iii hmms improve work originally propose organize markov present follow discussion section emission evolve take evolution encode matrix px generate accord emission emission mixture multivariate covariance matrix specify efficiently observation sequence sequence hmm joint observation finally obtain summation state parametrize parametrize collection reduce clutter hmms emission probability could cluster motion hmms indexing retrieval reduce mixture hmms annotate hmms efficiently subset compact hmms hmms hmm input weight generate input likelihood sample explicitly generate instead maximize likelihood way marginalization input model marginalization gaussians long hmms leveraging propose kl hmms present derive markov likelihood b likewise variable indexing distribute base component group cluster encode pz mixture reduce virtual manner require generate virtual r hmm gmm emission gmm py b I hmm hmm state gmm c hmm gmm emission mixture gmm emission sequence px rp py py hmm b l z py j hmm py gmm component clutter short emission denote indexing component gmm reduce denote reduce finally hmm base hand notation e b sequence base component imply addition expectation take variable specify short b summarize include name short notation table summarize variational variational subsequently set virtual base draw entire virtual base reason space different hidden cause problem virtual sample hide instead treat estimate likelihood virtual reduce use number virtual start different virtual virtual eventually maximize preserve otherwise general deal estimation present maximization view step family log optimize variational respect formulation readily extension replace practice tractable ms derive observation eq observation kullback leibler kl variational approximate reach compute lower let take expectation side follow jensen convexity family probably cut broad context expectation intuition compose cost successively arise result nest lower hmm expect likelihood average exactly forward calculate expectation essentially emission another gmm emission hmms variable variational I hmm essentially mixture low first hmm state state state respect hide assignment gmm emission therefore q py log likelihood gaussian virtual compose nest hmms emission variational variational alternate variational subsection variational maximize particular first gmm emission base maximize hmms statistic calculate first form variational well maximize low gmm variational represent computable bind chain variational explain substitute carry independently pair separate break substitute q assignment base py r ms
parent visit count action initialize throughout maintain backtracking h exploration simulation sample set reach leaf expand action mdp determine discount provide leaf action node value e return diagram simulation treat forward explore number effort nevertheless visit often eventually h ar h bs ar h h ar transition statistic fully observable mdps transition reduce planning give address expand lattice tree reduce offer tendency concentrate equivalent limited reduce path previous indeed tree costly transition require parametrize action additional latent p p parameter end chain rewrite require parameter stop parameter state instead necessary individually action performance improvement mdps single example policy locality reward work optimal free manner use interaction acting translate pure exploitation exploit instead greedy policy bias high provide complex policy overhead g state construct h h h confirm efficiently supplementary exist give list planning exploration rather bayesian rl maintain transition sample execute idea combine finite quite difficult variant provide criterion suffer tree node bellman value value child node wang search belief non trajectory sample take optimal node sample adaptation search mdps fact bellman monte bellman wang tree expand accord trajectory albeit action selection decision promise upper bind solve et al step step precise turn hard translate sophisticated c loop bayesian mix boltzmann first standard problem comparison popular task infinite location implement sample set exploration domain vary varied factor depth search set domain large algorithm mdp grid direction execute small design collect collect give since last visit step execute give quantify fair comparison prior criterion discount might rl dynamic domain run multinomial grid multinomial describe collapse scheme ba algorithms transition transition inside probabilistic general collapse scheme section domain plan increase code branching parameter specify performance ba learning l rl include different test task require exploration scale well plan opposite build merge optimistic explore planning generally obvious trade branching depth greatly naive example drawback forward sparse bound expand depth solve successful monte carlo towards trajectory grid average environment discount run generative prior correct prior gain performance report infinite associated otherwise agent know reward return oppose standard dynamic correlate transition transition posterior approximation conjugate supplementary planning solve mdp dp state ba deal must belief affect planning sampling full many planning time behavior drawback policy hide reward period algorithm treat node armed planning theory bandit nevertheless prior principled principle encode prior work explore structured prior match class agent base rl show tackle approach poorly provably carlo tree explore augment ba expensive belief tree root sample rl comments ba apply bayes adaptive equation ba ba effective define history lemma lemma induction root induction hypothesis definition match node induction sample node occur h induction ari apply ba mdp simulation ba result intuition unnecessary root update root evaluating mean illustrate empirically cell table act arm act accord guarantee bayes index arm mean discount step root root breaking execute result root execute estimate update get expand action select ucb obtain count green dotted prior employ video relatively block agent exploit explore b rate find might sub bayes agent sub likely plan qualitatively match construct hasting transition notation location reward column rough correction row account observe column due potentially since biased get proposal parameter could introduce enough keep link probability propose accepted row accept reject finally sample observe omit minus ex mm david reinforcement elegant exploitation ideal unfortunately find policy paper bayes planning outperform rl benchmark avoid bayes model belief qualitatively previous work exploration process mdps discount dynamic discount short potentially costly exploration reward long exploitation state agent acquire bayes
unweighted opposite approach right lebesgue ease rest paper continuous bound away shorthand metric metric bx r volume dataset accord connect vertex neighbor versa distance carry graph length weighted edge weight path case short path short let connect parameterize define know geodesic path geodesic distance particular inequality geodesic path monotonically pass work monotonically region paper geometric unweighted graph statement interior distance rescale distance unweighted consider unweighted two one condition stronger satisfied small see prove couple proposition ad imply imply dense assumption say exist exist geometric graph x bottom geodesic divide dense sampling vertex ball fu fu apply triangle fu geodesic vertex write low fx continuity boundedness approximate ball hold continuity lead geodesic connect path write lt qx lt dt qx quantity definition unweighte approximated near neighbor bound eq pr min pr max bound ps ds px ds follow kx ps k ps es binomial prop get p low finally sampling decide keep parameter maximize success sample prove exist density increase weight eq q edge assign interested limit respect go distance distance guarantee distance determining distinguish regime call prefer along distant ir formally prove contrast scaling factor simple set weighted consider I density increase edge fix two essence sketch step adapt graph adapt general nearby say probability probabilistic criterion geodesic connect near connect sum along adaptation fu fu graph path part segment get write fx call intuition take example straight prefer go rather big graph prove effect special aware regime study manifold case difference unweighted underlie low distance geodesic unweighted last unweighted graph short path unweighted high unweighted heavily region region uniform semi either implicitly laplacian alternatively take suggest way author omit estimation show simple family assign edge paper path unweighte even avoid result seem obvious machine practitioner unweighte robust unweighted behave degree graph density graph short short graph decision largely drive consideration implicit consequence choice carry machine learning make use structure benefit remark corollary conjecture exercise rgb weighted build short path tend infinity unweighted graph function shortest weighted distance fundamental machine geometry near assume go question arise behavior short measure give shortest distance study geodesic extend recent consider completely power little regard second
instance fit summation distance summation multiplicative representation find seek exist median integer projective admit general speaking point importance fitting shape denominator numerator sensitivity quantify importance mean cardinality dimension euclidean space size polynomially problem scheme range functional pick subset assign appropriately pf sensitivity pp f definition certain shape projective many area small thus construct small remove analysis article prove bound projective total total shape fitting shape set hyperplane sensitivity answer euclidean roughly intrinsic shape consist shape contain sensitivity projective problem shape tuple contain dimension tuple total tuple approach sensitivity projective sensitivity optimum fitting computation simple median distinct sensitivity greatly proof projective shape project contain subspace constant usually directly method translate computing compute approximately sensitivity bound ambient bind point sensitivity consider way every ambient dimension small exploit f f ps negative construction get positive get shape fitting run heuristic fit heuristic modify via original item interpretation issue broad projective paper organization article establish various clarity omit sensitivity methodology also stream article summarize relate low section upper approximation integer projective result study crucial define fitting hold fp fs literature requirement weight requirement satisfy instance shape problem sensitivity understanding denominator sensitivity fit somewhat read skip definition instance system r p projective cluster power distance vc set dimension fit sensitivity fit projective problem sensitivity projective upper projective clustering depend integer projective integer depend point order optimum total sensitivity fitting quantify shape sensitivity contain regardless ambient shape fp pf object sensitivity upper contain fitting minimize let relax p qp r us triangle p remark function derive upper shape p bound sensitivity contain turn problem sensitivity projective cluster fix dimension contain b projective phenomenon somewhat general sensitivity high projective clustering euclidean contain dimension basis projective tuple subspace derive one derive proof power distance projective shape total sensitivity correspond center p copy I cc remark result projective subset point use upper problem shape fitting tuple sensitivity line cluster problem total shape function alternatively recent sensitivity line lie sensitivity projective sensitivity give cluster total shape r I vertical horizontal vertical horizontal nj ip I nk tuple instance sensitivity projective theorem dependence sensitivity generalizing dimension readily accomplish manner size notion row th matrix condition column span satisfy ij u matrix rank z step use upper sensitivity fit hyperplane consider fitting hyperplane total sensitivity hyperplane u x h I denote whose u md md total dependent sensitivity shape total fitting sensitivity set hyperplane hyperplane arbitrary
solution however user gain guarantee service minimal goal convergent satisfactory solution numerical realize payoff basic banach eq order strategy iteratively htp distribute banach guess slot banach solution banach measurement get noisy invertible respect non alternative mean iterate without several remain without feedback start impact player fraction fully feedback moment iv speedup deterministic function stochastic question investigation iterative procedure stationary equilibria game space contraction banach iteration converge rate contraction propose behave lipschitz pseudo however acceleration cognitive convergence nice mean game without examine equilibria application financial speedup satisfactory illustrate game market biology cloud rich equilibrium little would allow scale game huge therein mean behavioral less scheme large player player action player aggregate form action player slot simplify drastically partially distribute player function capability observe previous play logit distribute dynamic would term action payoff fully payoff mathematical know gradient ascent directly player numerical realize payoff employ pattern payoff combine distribute payoff strategy algorithm estimate want gradient gradient observe payoff scheme estimation adjustment hierarchical three algorithm combination multidimensional address regime reduce learn partially distribute question framework partially know payoff capability observe field boltzmann field response fully scheme generic player numerical payoff noisy mean feedback information player player scheme payoff overview fully game fully reinforcement promise learn heuristic bandit difficulty extend game balance maintain exploit gain explore update dimensional adjust infinite action conduct claim recently selection dimensional even dimensionality idea continuous space slot standard continuous action widely correlate equilibrium compact equilibrium finite work game number framework behavior player decision manner create field contribution follow game player learn depend field learn drastically analysis interactive case acceleration time gap satisfactory player fully field scheme satisfactory game reverse speedup improve solution numerical illustrated games service section speedup player formulation game yet game aggregate game function player depend payoff game action j constitute shot game wide economic market market price etc influence total center server influence center serve share utility cost sharing bandwidth share wireless influence interference delay depend aggregate profile equilibrium deviation ask well e compact quasi respect possess least equilibrium development equilibria area characterization nash topological convexity detail examine admit equilibrium good mapping scheme scheme memory unique element map game follow simultaneous mean require observe common j compute htp slot user aggregate banach iterate consist contraction map unique fix contraction exist real satisfying banach fix unique suffer drawback continuous resource share player demand unit payoff simultaneous good derivative interior clearly contraction banach limit convergent modification banach action response evolve go least banach lipschitz may point attempt take decrease rate refer read htp sequence slot observe due clearly extend htp response initialize slot observe aggregate f j htb big rate mean field response cycle scheme well know convergent say converge fast class banach comparable banach reverse convergent point property asymptotic important engineering result limit compression asymptotic generic strict contraction follow gap regularity class set mapping subset empty combination follow get constant converge geometric minimize integral uniqueness ode ode aggregate continuous steady response nash equilibrium lyapunov equilibria game call lyapunov game turn first reach phase say trajectory explicitly convergence trajectory reach set e go fast field payoff single player field banach eq reduce banach mean field know mean mean field strict empty convex learning find field mapping iteration field strongly empty fix suitable condition approximate point major concern iterate exhibit slow speedup learn mechanism convergence trace measurement solution speedup compression regime go infinity asymptotic window bound aim non speedup learn algorithm aim pattern gain outperform traditional scheme problem convergent exhibit user aim point generic accelerate slowly converge limit consideration I quadratic cubic speedup extend multi speedup basic newton method iterate newton generate locally root cubic update speedup map player classical fix point iteration speedup derivative fix bind fix point guess differentiable fast locally learn arbitrary high method smooth systematically high quickly converge degree fix converge field learn great nonlinear avoid computation gap let convergent convergence range transformation transformation frequently computation present newton replace derivative speedup obtain note method write scheme two sequence speedup order htp method guess slot observe compute speedup obtain variant use htb speedup newton speedup newton model base technique non speedup however computational speed iteration slowly cost iteration iteration one derivative iteration may newton speedup consecutive sequence reproduce convergent speedup fully player algorithm perturbation observe r j j j taylor expansions proportional second hessian payoff player difficult convergence estimate pseudo hessian independent field learn second realize payoff j w tr generate estimate pseudo action mathematical feedback player action result false consensus belief feedback speedup initial iterate payoff j action analyze integral scheme social optimum payoff sum divide significantly reduce optimize technique conduct limit limit next mean game compare aim guess prefer initial analyze experiment reader choose way see guess third guess besides would guess still stick guess nash would stick game simultaneously choose number interval target player number close target payoff study experimentally game useful number iterate reasoning game illustrate never stochastically dominate player choice iterate iterate nash feasible profile modification interior equilibrium target interior equilibrium satisfy assume learn explanation feedback game
department mathematics universit du universit de cp nj usa abstract reduction frequently natural represent curve argue though technique benchmark adequate indeed essential serial inspire dynamic component propose study illustration improvement entail static procedure dimension functional functional functional technical storage phenomenon life improve record high frequency extract characteristic possibly high specification analysis recent prove consequently field research community realization curve observation cube surface deal surprisingly efficient role arguably technique analogy multivariate rely decade take early later influential book work asymptotic et smoothing et robustness application include linear usefulness also recognize scientific chemical et reference cite paper section background though serial serious numerous either space daily transaction daily pattern environmental etc realization time series daily observation day parameter time study ar refer ignore dependence series context inefficient serial quite problem estimate traditional traditional adequate fact seminal recognize operate dependent hence negligible instantaneous predictive besides failure produce optimal static uncorrelated exhibit motivate development dynamic idea transform say individual mutually lead lag autocorrelation allow thank orthogonality dynamic analyze analogy reconstruct low dynamic version expansion dynamic principal first suggest series purpose similar setup methodology domain rest organize section application describe proposition asymptotic feature simulation study contain proof appendix c paper aim objective essential difference develop main serve approximation study mathematically quite elegant increment disadvantage behavior eigenfunction contrary construct remark work technical leave consecutive curve level underlie show reconstruction panel panel expansion component color notable completely symmetry addition daily average extent curve trend spike considerably illustrative reconstruction much well application static mutually orthogonal mutually orthogonality lag component contrast treat illustrate superiority orthogonal auto break et al detecting mean sequence functional project proportion account enough pc th eigenvalue converge roughly statistic converge score become independent obtain separate statistic aggregate structure hold converge still need non dynamic principal long let get ensure aggregated principal feasible series response say depend regressor intercept natural frequency parameter scalar constitute functional analogy involve operator vector mild score cross greatly reveal frequency th consequence regressor assess individually quantity version one may hypothesis justify dynamic retain impact tool technical detail integrable unit within consider take integrable function stand modulus use serve equip inner define possess curve define ex h ex operator kernel well quite consistently mean center preprocesse element eigenfunction orthonormal eigenfunction loading approximate instantaneous static perform observation take serial likely exist globally speak base goal introduce density operator analogy classical concept operator denote sense hilbert schmidt see condition provide spectral operator create filter section build block construction define ty shall filter q addition write emphasize operator filter play density absolute coefficient hc pointwise general consider element I collection frobenius equality thus understand g consequence every adjoint hilbert schmidt analogy admit eigenfunction eigenfunction assume choose ty desirable discussion wish suppose measurable discuss eigenfunction standardize imply zero hold conclude filter lag functional principal zero value satisfy assumption th principal component call dynamic multiplicative devote dynamic elementary satisfying ex hermitian real coincide one ex uncorrelated lag ex matrix tell analogue static formula dynamic square dynamic mention score contrast draw analogy static replace however curve reconstruction involve replace alternative filter expansion dynamic theorem suggest well define score stationary mean series operator impose preliminary integrated square estimator functional state show estimator satisfy concept usual lag estimator eigenvector need identifiability th large finitely essentially common ensure eigenfunction sign remain provide sign situation slightly since setup multiplication way fix eigenfunction impose identify long assumption frequency denote exist choose almost consistency variable define hold mt practical record often necessarily datum require limit common error tend specification omit technical exercise cast proposition necessary carry represent basis spline wavelet survey preprocessing refer sequel matrix entry span dx linearly orthogonal linear follow let operator p ij matrix operator q iii linearity I eigenvalue dynamic task multivariate put throughout lag common choice tuning estimate coefficient ms numerical simple well available power substitute replace eq expression create observe dynamic variance estimate incur consider reduction q quantity datum half diameter ambient air transformation perform heavy observation remove traffic intensity business software transform discrete curve daily represent display figure fourier black represent curve center empirical traditional operator tuning still leave element away rapidly justify element ht filter focus dynamic explain latter static figure entirely idea static score course loading appear remarkably another total get insight static say tu deviation effect attribute regard dynamic interpret sequentially advantage fact approximation single dynamic expansion study impact triple eq mean panel impact score mean curve follow panel concentration curve highly correlate indeed panel first dynamic static interpret static day increase decrease data trend roughly consecutive compare middle panel dynamic expansion simulation performance static variety simulate result static dynamic expansion performance measure computation implement along linearity satisfy v var fourier process mutually choice combination follow methodology spectral density tuning calibration lead moderate variation fundamentally obtain perform integration chose impose time moderate fast experiment time deviation setting thus basically dynamic replication systematically procedure finally contrast exact numerical integration require truncation lag little deviation matter practice explain contribution higher visible slightly static static static deviation multiply value principal component take literature functional sequentially serial happen instance process static setup reduction paper dynamic serial functional case reduce static serial dependence static quite significantly serial strong outperform implementation ii toy air iii simulation study bring evidence dynamic functional small cast rigorous show mathematically rigorous methodology introduce adopt specialized setup throughout denote separable equip space since frequency domain nevertheless series space mapping pd pl hilbert product fourier denote limit almost fourier eigenfunction eigenvector scale unit additionally unit circle exclude impose way expand eigenfunction fouri sense line stationary hilbert schmidt operator mapping assume hilbert space sometimes hilbert schmidt hilbert impose possess convergence easily show self adjoint establish condition random call I applicable series purely moment dependence verify many impose follow result trace
problem opponent space addition state agree def unfortunately intractable agent belief expected utility arise payoff opponent advantage select opponent payoff sequence probability payoff gap second version expect gain bind prior indexing gain experiment parameter environment action stage norm first stage finally measure choose gain relate environment consequently guarantee two adversarial must strategy obtain stage exploitation hard reason execute belief execute calculate posterior well know reinforcement confidence alg reinforcement thompson alg choose exposition restrict attention alg use q optimistic evaluation efficiently mdp construction belief stage belief type thompson multi armed problem confidence p sign convenient payoff payoff stage policy regret stage follow shape policy begin two type slightly main terminate action take stage terminate set payoff payoff function sample uniformly opponent knowledge maintain payoff select payoff explain estimate belief stage problem quite case opponent fig greedy policy payoff empirical linearly grow slowly adversarial opponent nature fact payoff greedy r r r r r r r r greedy r sparse reward capture act environment previous theory multiple arguably purpose another purpose gain without action mind instance overall force payoff adversarial opponent sophisticated environment arm begin opponent importantly thing noisy signal hard consider herein bind discount well opponent example problem task hard agent close game approach bandit essentially involve policy relate search bandit actually action experimental show greedy policy strategy sequence encourage visit frequently naturally suffer change goal exploration agent measurable negativity easily execution agent explore environment necessary thus setting define reward stochastic agent opponent act accord apart formally describe link bandit factor mdps examine reward act environment arbitrary objective question act current knowledge might analogous motivate human behaviour formulate multi game opponent act markovian environment stage begin stage opponent determine agent agent act payoff opponent select problem reward process first stage want payoff stage second adversarial uncertain environment resource must spend important effort towards future may lead maker explain necessity contribution analyse opponent nature unknown stage gain exploration approximation adversarial confidence well greedy policy however nature greedy perform payoff force explore next introduce environment opponent nature adversarial sec act reinforcement confidence bound conclude link problem theory opponent payoff reveal regret utility maximum utility stage total require strategy stage remain environment perform stage stage act throughout stage markov tuple index shall use class text represent sequence kernel opponent reveal begin stage opponent encode task go rl map rl agent acting process mdp equip reward discount utility discount map ts apply payoff function experimental rl payoff maker choose policy use interact environment jointly select action
column select selection algorithm run nk nk reason call advantage avoid load whole memory none three step svd algorithm memory expensive fast algorithm maintain require whole section several run natural bag natural uci dataset biology bag dataset vocabulary million sift matrix upon large truncate svd experiment matrix dataset source image image matlab implicit subspace error bad conduct ghz much ratio pt pt experimental match run become efficient fast small row randomize problem fast scalable art e algorithm require row achieve subspace require subspace nk nk ok moreover enjoy loading make demonstrate effectiveness efficiency low proved problem work sake call algorithm nk x compute weight j kk I guarantee definite efficiently eigenvalue eigenvalue eigenvalue comparison moreover overall equivalently column convention adaptive give r ai ii bc ac ip c take symbol bold matrix distinguish vector ji accord lie r w w complete large singular column f c lie span singular along lemma nj first r r jj r prove lemma propose line lemma algorithm row give r prove randomize adaptive near target column r cost line total mr mn space thus assumption time algorithm stage nk nk nk cn li china important nystr om approximation approximate term advantage possess tight low complexity avoid maintain memory several improvement scale computation matrix stock web video bring modern effort understand factorization method informative facilitate principled truncate find svd little concrete difficult uncorrelated people claim interesting basis insight actual decomposition technique construct stage row selection implement widely recent work later sufficiently particularly ratio p divide method parallel require art subspace exactly relative cost svd numerical unstable impractical paper art bind list notation review section mainly section compare art defer let th row column frobenius norm svd write eq top u z discuss development algorithms additive section column error ratio hold aforementioned stage stage relative seek stage stage solve relative ratio select achieve ratio algorithm inefficient step svd exist column q expectation right input since time consume employ approximation speedup via approximate randomize svd target factorization optimal dual spectral nc ac optimal sampling present column define ai nc expectation r procedure give theoretical work three theorem
probable dense linkage linkage observer knowledge order convention black observer white zero unknown thin edge study formulate observer star topology utilize show facebook separation facebook number typical topology rapidly count heuristic neighbor far exactly study first contact topology obtain direct reliable raw detection community within dense linkage community linkage implement mathematically partition detection become maximize quality modularity adjacency edge modularity essence result reader lot clean amenable reason people start detection large truth validate quality recent year node associate school company person reality name put result ground truth reasonable observer local topology recommendation mean level application automate categorization cast label predict confusion show fig notation way label depth cc analyze reasonably kind possibility variation informative section computation conventional multiplication classifier framework assign score node score reflect one strict feature heuristic metric eq cut feature train probably pilot potential investigation observer compute adjacency matrix degree degree n inverse get also walk pr interpret distribution step pick walk node successful web axiom fit biased proximity visit node high proximity observer likely version informative however proximity observer score evaluate normalize biased instead stationary know put high walk w x node initialize portion pass neighbor use positive community connect connection rest node put capable distinguish numerically investigate matrix eigen provide irrelevant stationary bounded matrix multiplication separately total complexity computer asynchronous manner algorithm arbitrary portion share direct e call termination imply walk j since portion neighbor complexity multiplication complexity real also tight benefit structure graph currently user view list direct perspective consider dedicated friend friend observer eight run class rely user actually collect eight eight follow quantitative one observer denote manual review large community community interested ability approach name node default observer regard regard preprocessing cause grind impossible name list experiment observer friend positive practical look framework calculate rank graph roc approximately line distinguish informative human instead positive choose expect dominate curve everywhere experimentally effective run simulation plot curve partly focus compare light dark get get node return know statistic point typical user university threshold node reach drop study graph cut drop difference whether applicable current heuristic simply intersection observer intuition two likely intuitive bias reason information node connect contrary low community heuristic exactly many beyond node observer basic approach pr fig g high pure bad heuristic actually analysis user manually positive improvement already considerable bad concept put high observer vary alternative node fill positive put repeat round randomness scatter improve little I improvement besides manual labeling cause find cost degree vary scatter obvious plot heuristic perform application suggest v put node benefit friend recommendation already observer meaningful automate categorization randomness evaluate next evaluate heuristic plot increase node observer mostly l curve reach degree prohibitive degree heuristic note labeling node degree node recognize induce roc evaluation promising application realization multiplication observer section behavior use high two ghz cpu matrix multiplication section norm implementation stability denote choose fig exponentially second algorithm enough practical part fig also indicate approximation suppose require make upper converge proximity score use detect sharp detection leave detect constrain formulation work tackle locally author topology discover propose solve combine globally propagation community node combine locally detect community work go auc l roc extension current direct friend protocol help trust overlap community lot proposal available detection essence reveal seed observer seed reveal detecting seem people topology natural check friend list allocation observer manually degree directly however degree know label random node regardless paper community limited topology topology cluster traditional community collect establish find rank adapt justify applicability evaluation datum topology small discover commonly manually label boost fig service communication presence operating characteristic roc curve rate axiom department success service centralize lead serious concern operational robustness service party possess entire social service like friend community tackle detection impose page rank justify detection topology automate collect practice demonstrate adapt pr heuristic manually friend vector boost relative auc scale facebook part people daily life share platform role friend discovery community formation interest success service serious concern operational knowledge profile activity user obvious seek control overcome drawback decentralize implement party possess entire server super user even restrictive network book friend still effort design building adapt p design trust concern bt enough people within scope privacy monitoring attack difficult support critical discovery recommendation architecture many datum availability global require discover bag part observe acquire scenario community centralize one require localize centralized setting worth researcher tackle label lp switch
loss previous first relax mixed optimize relaxation utilize design input prediction classifier subset intermediate classifier determine tree distinguish circle black classifier child parent full base validation test node produce prediction two path deep handle threshold lead hard therefore input iv I sigmoid child exp aa reach child j node parent child naturally node risk probability likely risk classifier serve cumulative illustrate highlight input follow root denote along exactly analogous except weak classifier zero exactly weak feature weak classifier free reach terminal along marginal random reach l v use norm norm final naturally encourage combine penalty split cyclic v classifier traversal reach terminal depend overcome term relaxation prove minimize function show introduce alternate auxiliary perform conjugate latter follow guarantee decrease respect classifier jointly lack tree optimize help validation compute node removal decrease performance case even ndcg pruning relaxation term dimension mix classifier correct classifier optimize classifier terminal make final prediction final tree weight carefully specialize large yahoo rank set category feature feature correspond synthetic feature learner usage design identify perfect feature per test input along identify expensive mean task public yahoo challenge extraction weak query indicate match ndcg level relevance unless node hour tune classifier minute yahoo set coefficient classifier ndcg versus cost weak derivation insensitive fine early stop validation evaluate six sensitive curve simply compete improve evaluation reduce overall propose node discount correspond significantly improve accuracy curve limited evaluation compare beneficial ndcg long budget reduce achieve ndcg oppose tune ndcg yahoo input pass branch branch correctly rank low branch away share apart investigate fraction extract yahoo observe depth expensive deeply input precisely obtain high ndcg introduce novel time cpu principle fashion partition identify cost feature region allow accuracy fraction tree relax minimize classifier unnecessary consumption application far specialized case incorporate demand author thank suggestion world email spam filter cpu challenge balance test accuracy principled dominate drastically tree input individual optimize specific space tree match learn email spam test reduce unnecessary classifier imagine introduce email spam email web service daily day filter compute prohibitive introduce time classifier address principled manner reduce cascade incorporate dynamically path traversal expect cost tree single loss direct expect lead expensive necessary make framework relax mixed relaxation derive optimization synthetic effectively classifier search significantly current art sensitive web regularization feature extend cascade mostly classification cascade several classifier order center input predict cost cascade enforce feature reject later cope particularly effective skewed imbalance majority contain face often haar suit different significantly reduce never learn dynamically select feature average detection case fundamentally mind input terminal path prediction recent work speed vs evaluation differently feature extraction algorithmic mutual feature extraction tree possibly similar learn direct different adaptively learn motivation algorithmic different regard introduce formalize sensitive consist categorical limitation focus throughout linear decision trick particular first depth
fill f f v f label label v n v x b semantic assign factor indicate determine fix type template describe manual f gm size gm ex model gm gm minimizer minimizer size library allow structure prototype modularity tool file format potential exchange ex template define discrete wide art restriction impose factor factor handle efficiently function differently several parametric e metric dense interface graphical file format line tool modular graphical optimization equally standard tool rigorously c base definition operation include summation marginalization conjunction variety fig deal occur repeatedly store impose graph format extension automatically template elementary library property order grid contrast support drawback table prohibitive model factor focus impose restriction contrast library set publish cost slow optimize mrf twice fast modular corner height ex bp cut wang decomposition branch expansion swap subgradient bundle branch semantic graphical w operation determine semantic
truth original translation affinity cosine similarity two soft ml enforce help unable soft spectral translate surprising version enforce translate use translate original merely tradeoff integrate partition document word fmri fmri scan person consist scan node absolute correlation brain activate indicate certain instability scan subject thing scan result spectral spectral cluster fmri different min cut little insight activity subject transfer scan constrained c cut agree fig far california center cognitive individual cognitive scan rbf kernel enforce constraints fmri scan constrain cut similar cut great disagreement original vice constrain interpret fmri scan clearly study people disease width principled spectral cluster soft constraint flexibility framework side information metric transfer use dataset exist technique validate acknowledge automated discovery explanation event environment nsf grant nsf enhance usa university california usa cluster hierarchical constraint algorithmic setting one many develop contrast effort implicitly encode link modify laplacian principled explicitly encode optimization encode degree constraint constraint unconstrained remain empirical artificial encoding property extensively process community superior traditional deterministic polynomial ability shape cluster cut capture cluster fail fig validate spectral originally propose unlabeled information laplacian become insufficient toy cluster undesirable side large expert explicitly assign must link short solid line cl dash line spectral successfully recover neither complete demonstrate greatly improve result infer example document language context transfer laplacian domain side information transform pairwise ml cl constraint side belong graph cluster area constrain category incorporate exist mean study prune intractable build partition constrain spectral promise assign cluster inconsistent spectral develop category enforce directly graph modify graph second constraint category several limitation design binary real real belief yes many lead example could reasonable ignore small portion exchange natural interpretation either encode enforcing incorporate flexible address limitation benefit cl accommodate relaxed moreover constraint addition handling allow new entire alternative distance metric consider framework amount soft enforce threshold amount ignore exchange side raw consider objective explicitly create optimization unconstraine special turn produce interpret perspective embed standard benchmark transfer section apply namely fmri comprehensive task briefly survey previous constrain spectral formulation empirically conclude constrained extend form convert incorporate spectral group th affinity laplacian encode walk cl penalty affinity mean propagate affinity regularizer principled decide constraint guarantee subspace indicator accommodate inconsistent partial grouping enforce make sensitive extend propose spectral combine use incorporate pairwise constraint cut stress clustering well establish spectral incorporate spectral compare technique follow model spectral laplacian self reader familiar material skip formulation rest list symbol mean affinity degree unnormalize normalize constraint graph undirected vertex set matrix call l call unnormalized assume let one equal show eigenvector cut graph constraint principal cluster solely decide affinity laplacian extension incorporate side information reflect graph spectral pairwise soft new formulation constrain show generalize eigenvalue system encode traditionally link indicator belong ij increase sign soft believe belong magnitude belief value believe different assign similarly belong assignment constraint rather constraint individually substitute constraint unconstrained constrain graph constraint optimize objective solution unconstraine cover special eq encode trivially solve constrain tucker find set candidate satisfy condition choose force feasible lagrange accord kkt take constraint check solution complementary imply either trivial second eliminate unconstrained cluster focus hold kkt condition assign variable solve explicitly produce infinite feasible solution relaxed color side satisfy sign contribute positively sign color assign side cut case formulation joint numerical possible cut coordinate correspond axis line visualize unconstrained cut cut numerical randomly horizontal indicate solution cut threshold constrain show way unconstrained affinity relaxed assignment indicator practice partition side encode soft inconsistent algorithm labeling constraint always consistent algorithm par spectral big solution normally naturally way usually instead feasible preserve eigenvector associate eigenvalue mean embed specifically encode scheme eigenvector feasible routine discretization relaxed matrix due orthogonality step help moreover treat ideal cut cost column respective cost remove rest constrain scenario source edge target derive form soft affinity carry structural knowledge graph cluster knowledge target correspond trivial second large eigenvalue guarantee feasible eigenvector begin experiment soft fashion metric translate transfer fmri analysis study incorporate meaningful truth partition constraint technique list labeling metric knowledge arrange accordingly implement available segmentation pairwise outperform interpret human meaningful berkeley benchmark image image compress use segmentation set segmentation segmentation segment save irrelevant segment pixel close segment blue black belong white visualize reconstruct indicator unconstraine spectral partition pixel ground fail background correct introduce block bound corner block ml pixel cl pair pixel gradually help supervision successfully blue red note try aim unconstrained block fig right two part one red human spectral fail background fig spatial continuity block background intend assign background side cluster thus face achieve image alternatively block rest examine underlie truth claim extension cluster question ask converge cluster spectral draw sample truth constraint ground fig dense point unconstraine could ground ground truth adjust rand ari ari truth partition truth exactly constraint record ari different consistently ground constraint insufficient well lead perturbation constraint provide quickly robustness create double background although cluster correctly background uci constraint choose six breast cancer diagnostic optimal
axis time age period subject segment age entry may death study excellent extensive diagram disease axe age direction three system axis life line plane life triple age duration certain pointing illustrate line subject birth disease go direction disease henceforth life line death without whole life disease person record people affect person incidence person year often take usually divide subject spend event subject group risk spend group equation become diagram clear age product whose rectangle risk sum risk spend dimensional rectangular time risk volume subject spend voxel graphic field seminal present terminology rectangular volume call six face voxel adjacent plane voxel parallel union voxel face play voxel grid life voxel age rectangular voxel depth calculate subject starting point intersection voxel voxel arrange regular equivalent union voxel intersection us intersection plane operator formula define set occur intersection happen voxel order subject sort division voxel summing times yield period person year calculate intersection voxel faces million form implementation tune implementation ghz patient next method article patient population age assume study assume death censor aim transform age calculate f year show person
necessity adaptation infinitely eq index thus contradiction infinitely many combine eq thus necessity robustness establish metric almost attention distance form otherwise x value nonnegative lastly prove robustness geometric robustness say robust p subset equal subset z z z p z framework indeed example partition ensure framework pay convenient prove frobenius frobenius norm f tx tx tx tx x x special order derive however stability ability recent approach stability frobenius arbitrary use norm induce row c example proof version norm function cover kb b bb derive metric formulation use technique completeness consider bilinear mahalanobi simple bilinear triplet robustness generalization sparsity induce applicability framework advantage flexibility robustness tackle metric setting method adaptation promise robustness property input minimize maximum belonging concept optimization deal z n l l z z inequality proposition robustness bound p weakly robust n n p p p sample consist equality apply thus contradiction p contradiction value exist constant fx x approach proof satisfie z z tx tx x x u x product continuous w take frobenius solution subset b term training get consider obtain tx tx j solution triplet optimality derivation partition tx tx tx tx tx tx tx tx x c first follow cm attract lot interest decade robustness introduce xu metric illustrate result include automatically adjust similarity function training tailor improvement attract interest decade see survey rely resp metric generally mahalanobis parameterize psd seen find linear well function without psd result distance use performance despite practical success unseen datum metric independent identically iid indeed give extract consider near example diversity still assess convert result online generalization rely al uniform stability regularize induce propose pair constraint metric robustness xu allow generalization bound closeness input result etc triplet work assume build weak necessary generalize well fundamental illustrate deriving argument work vast regularizer without example simple organize algorithmic metric necessity illustrate applicability framework derive bound metric discuss related work norm similarity training build z label generalization loss bounding notion algorithmic xu classic learn base deviation two close formally subset every belong prove true latter ensure obtain notion space say convex quantity finite cover consider partition partitioned belong adaptation learn pair pair fall metric say formalize np jk quantify robustness robustness sample relaxed robustness easily extend triplet base iid triplet share interpretation z k z eq generalization bind present concentration help iid multinomial random huber generalization metric iid iid parameter z ic z ic c z j z z b n I bn bound lastly equation previous robustness give adapt straightforwardly triplet output theorem robustness property relax robustness z ks z associate robustness come iid draw metric
classification svm community room discuss technology specific separate convex optimal separation ht nonlinear disease disease goal support machine dimensional hyper plane separate space interaction certain gene roughly piece create technology achieve feed try ht gene linear polynomial sigmoid ht svm fold cross rbf sigmoid sample decrease svm literature transform kernel include spline histogram student etc modification hundred thousand kernel paper would new actually separate disease possibly case interaction meet challenge effort generate true modern statistic complicate huge able national maintain claim claim replicate condition finding medical survey journal come trial far examine claim american medical journal journal national cancer institute come observational likely false claim researcher relevant blind control field gene identification microarray category observational scientific gene gene cause flip coin consequently adjustment fdr gene show efficient detection taylor situation tool machine randomization technique laboratory point justify logistic logit versus link side simply boost adjust often single gene imagine gene solely responsible disease protein gene primary gene gene fortunately identification technique gene study differ discover induce tumor formation powerful feasible identify gene cause genome pool reasonably sized could examine rigorous approach microarray first identify disease study stochastic gradient identify pool gene interaction interaction situation possible cancer correlation pick technique important fdr least neural network decision problematic crucial distinction gene irrelevant false non discovery gene subsequent biological false gene lose exploration classification positive false specificity roc curve roc precision recall etc commonly produce accuracy select parameter currently growth tackle hard datum create thousand select variable know many process trying find black cat dark house investigation college student project field gene literature technology sample need select case pls disease moderate search platform beneficial top height pt researcher thousand tool variable responsible trait include education microarray brain imaging name focus investigation limitation variable et al produce stand artificial disease reliability tool widely variable responsible pre screen gene dimensional modern statistical international competition analysis bin predictor cancer research cancer predictor gene expression image processing snp area user predictive user furthermore probably involve satisfactory ask explain predict different la explanatory statistical extension find field microarray identification tool every cell express express cell cause trait compare cell allow comparative express cause formation hundred hundred point microarray need statistical size attract amount survey reveal whose group two category include correction discovery bayes mid platform parametric reference bar et etc vector machine neural near diagonal linear discriminant I bayes near rough pattern base fisher discriminant mahalanobis latent approximated microarray pathway neighborhood fuzzy mutual numerous reference instance et al wang bar et al lee et al reference category list variation instance analysis al eight representative test structural plus non lasso elastic nine toolbox generate vector least modification hundred suit hand cart variation literature include index genetic fill wrong path come neural variation cart radial basis rbf network architecture software package nine kind architecture function kind lot google neural marks day situation statistical et gene equally whether translate wang tool achieve false gene identification microarray vast reliable article learn analysis microarray generate simulated microarray employ identify differentially gene classify dataset result dataset motivated gene cancer patient outperform accuracy top may scenario datum may analyze microarray relationship significantly tool explore microarray large analysis performance model publish classify purpose fewer publish analysis subsequent cancer note classify gene rate ht n svm svm al lee bar analyze microarray leave validation leave one model run base train pls nine table always classify microarray pls variable square available dimensionality reliably thousand subsequent pls achieve always classify cancer elimination gene lowest eliminate without gene cancer however cut top gene slightly pls pls utilize cancer neutral networks nn pls neural analysis split validation cancer display regression pls neural pls leave pls regression conduct place pls gene select split compare neural pls notice pls gene desirable classify instance plausible use different classification rate explore gene method statistical three appear seven statistical pls statistical ht gene gene gene gene gene gene gene gene gene gene gene check ensure partition different seed split find place gene statistical take appear encourage pls error seven small standard corresponding chi square thus conclusion draw separation maximum run seed fix logistic parameter pls ht traditional cutoff significance standardize estimate outside fail normally distribute cutoff value estimate cutoff reliability estimate neural ht lc top cutoff elimination forward contain behave pls table say conclusion pls complete separation regression know pls machine model next provide patient analyze team medical institute institute technology imagine conduct microarray magnitude cost alternative scenario budget collect compare recent article van interaction reality expect build base purpose instead dataset generate microarray need mathematical equation select gene dataset design state state responsible disease complex nonlinear predictor eight simulate normal disease linearly contribute disease state gene gene normal gene disease disease disease disease dominant breast cancer attribute two breast cancer breast cancer genetic lot involve gene move cutoff function cutoff normal heavily skewed display disease disease remain disease disease present exact cutoff value disease simply cutoff disease different gene gene interaction gene contribute disease account represent scenario disease gene similar disease except nonlinear combination gene individually combination q gene interact single breaking et near neighbor obtain maintain patient outcome decision making regard current option rational patient high clinical pick disease summarize rate x boost pls x nn none x disease exception excellent job method contribute classify gene disease much disease disease level influential boost pick subsequently desirable identify biological method x nonlinear interaction important pls irrelevant gene none irrelevant gene irrelevant x turn disease identify table several statistical statistical correctly interaction gene column give pls lasso irrelevant raise achieve gene determine state compatible finding article et markers important use angle regression fold run encourage fortunately boost correctly though unable minor clean cut tree pick leave cross stand popular network irrelevant gene decision boost related representative basis center center create histogram cancer panel patient panel histogram unlikely happen disease cause popular accuracy believe disease ultimately statistical disease cancer patient balance reality try handle technique cancer patient equal patient work well give cancer patient work model interaction effect gene recall important disease gene interaction disease moderate accuracy show include nonlinear x nd rd rd interaction semi boost boost disease patient patient pick relevant minor summarize gradient find al compatible accuracy accuracy fdr gene fdr x gene gene fdr gene gradient boost procedure able pick microarray experiment year may many consequently handle highly turn gene gene lose recover nevertheless gene screen fdr fdr cut gene number fdr gradient well able nonlinear phenomenon
write also expression proposal r walk notice e early rejection actually albeit negligible factor density suppose prior early simplify verify early never take summary importance whereas essential development build close achieve determination automatic exploit classic hold loss choose therefore abc output e thank regression approach linear square optimality order measurement dy jj nj particularly summary addition lasso lasso regression coefficient return software prediction abc methodology multidimensional process previously introduce methodology handle general time vector artificial affected identification mean previously methodology knowledge abc pilot identify posterior mass suggest schedule automatic pilot abc region posterior use set parameter datum iii simulated artificial statistic iv statistic density knowledge informative prior prior region abc question prior determine distribution markov mix reasonably distribution course produce satisfactory enough augment prior mechanism iteration mcmc allow generation large avoid mind set truncate iteration metropolis increment attain long draw discard next section transition proposal density density easier simulate approximately sample exact posterior corresponding small acceptance posterior ease methodology package although find otherwise two inferential carlo strategy particle method introduction smc example simple necessary regard comparison tolerance sequential carlo particle fair abc drug literature devote longitudinal modelling follow although mixed drug concentration drug receive subject elimination rate drug intensity stochastic subject nine min min drug e reasonable sde min practice logarithm cl readily available therefore datum generation k cl perturb obtain however inferential algorithm typical sde euler divide abc multivariate regression exact inference denote start must obtain artificial separately abc truncate impose determine posteriori abc walk reasonable ability explore surface approximate rate acceptance rate discard burn th equal correlation subsequent save computer memory spirit plot bandwidth vs plot outside final draw panel limited draw mix well discuss large biased select allow accurate balance filtering give encourage ess part intel core ghz gb ram rejection produce acceleration true lin ess abc ess ess ess result carlo smc likelihood hasting allow inference ultimately call proved approximation algorithm particle filter resample particle randomization consider prior abc run particle acceptance case carefully code trajectory suit explain poor nine surprising due setup density regression order ess length chain consider obtain lasso statistic train use abc via report bandwidth retain outperform ess value mean fail often literature handle conclusion least methodology remarkable approximation sde limit sde fully measurement error define represent simplified auto protein code specific reaction represent via specie specie reaction negative integer specie value presence law degeneracy therefore redundant prior copy genome reasonable replace occurrence follow assume associated reaction occur continuous dna dna c rna rna I could elegant solve root definite trajectory simulate guarantee stay absolute square disadvantage preserve dimensionality increment unnecessary redundancy setup rna represent mathematical reformulate system partially observe system design define via rate time denote denote exposition dimension sampling equality hold inferential trajectory numerically euler stepsize different coordinate set setup ii perturbed vector ii denote vector object prior abc execute million equal correspond discard purpose report ccccc sd vs bandwidth evident range allow interval theoretically could inference practice safe imply plot sharp increase decrease result associate mixing region draw long mean look vary ultimately draw posterior notice excellent ratio ess eight ess use draw implementation affect measurement variability return estimate partially unobserve total available perturb independent consider variability uncertainty placed generate long draw acceptance retain result despite increase uncertainty setup initial residual variation compute ess ess ess partially error propose reduce experiment practical application methodology largely automate thank informative summary encouraging monte carlo multidimensional demand flexible limited modelling stochastic latent via multidimensional sde mcmc alternative monte notably excellent inferential possibility avoid one device difficulty try implementation smc hundred thousand parallelization deal multidimensional rather error available see paper sde model incorporate science grant tool author acknowledge comment associate quality work considerably model equation relevance grow already financial growth multidimensional challenge theoretically either prohibitive time sde limited package implement abc provide keyword rejection space equation chemical reaction likelihood popularity practical due would computationally considered review dynamical whose unobserved dynamic solution equation allow fluctuation behavior area model mathematic population dynamic model chemical inference work sde noisy moment issue bayesian multidimensional sde abc tackle difficulty stochastic represent state wish sde introduce scenario system unknown know corrupted generic reasonable conditionally independence illustration consider I development assumption infer complicated consider consideration correlate wish form carlo mcmc algorithm fail reason include explore chain surface difficulty sde trajectory mcmc proposal locate difficult sde transition underlie diffusion inferential sde available review sometimes prevent situation many presence considerably sde exploit recent computationally algorithm exploit abc computations stochastic chemical sde already potentially sde common compare simulated statistic role bandwidth consideration closeness trajectory block abc lf generalization additionally case via fact markov section initialization simulate start start start sim law conditional sim sim u sim sim sim recall marginal
scalable feature employ learn factor object inner lie ease excellent performance representative multiplicative receiver factor factor also representation focus capturing discover interaction relational useful absence capture interaction effect object redundant incorporate structure relational observational structural discover intrinsic property object precisely assign object block cluster contribute coupling factor model discover well local paper contribution novel function latent factor propose mm latent extensive conduct synthetic predictive introduce algorithm section describe experiment real provide work give represent miss use mean observe cluster datum denote latent membership object assignment ik membership relation predict among consider relational factor generative interaction hybrid object base relational bernoulli function define follow characterize variable encoding attribute user introduce latent matrix latent link exist object within denote object cluster within receiver inner object cluster actually model discover latent use type relational cluster fix also fix mean structure way adapt structure kind bias generally side information model hence interaction q represent object denote associate pre could assign observe benefit correspond block prediction interpretable blockmodel factorization style traditional process selection accurate latent preference certain successful generalization capable dense interpretable cluster assignment link pair cluster hence integration multiple generalization prediction impose latent factor follow description summarize build section latent maximize monte adopt cost expensive latent follow factor globally algorithm learn fix update instance learn feature one equation feature iterative rule newton logistic complexity respect latent cubic generalized function commonly style concave function step maximization mm factor derive aid auxiliary increase update q estimation maximize low learn feature fix auxiliary eq proof low optimize maximize transfer iteration need optimization function assignment relational latent log assignment value relation directly map term kronecker product matrix learn generalized model optimize obtain latent fix update latent derive update rule latent similarly latent factor employ kronecker rule function monotonically combine learning update em factor information learn latent latent require possibility handling demonstrate term nmf indicate nonnegative semantic membership blockmodel use relation latent latent factor latent relational task task prediction relational relation consider random train relational receiver operate characteristic curve check difference discover assignment well relational normalized way examine represent specifically two inter connect connected object cluster check relational propose different reveal structure nmf indicate model fair comparison wide selection number work check model repeat average roc link task well propose indicate efficiently cluster e g nmf due discover special integrate block conduct assignment label ground measure obtain directly latent assignment factor model feature label score observe well feature prove relational compare propose dataset task also latent vary latent increase however two link online social user consist social link structural choose relation entry time number model outperform model latent lead compare effect take dataset prove factorization respect model indicate effective specifically propose rather bernoulli may prediction moreover vary wide range high performance fix social effect vary real true label object evaluate latent assignment look auc respectively simultaneously construct paper citation experiment case genetic rule theory paper indicate truth g nmf dimension note dimension latent discover future evaluate table achieve reveal simultaneous latent adjust feature citation contain cluster content paper model dense one structure cluster feature relational feature varie extent experiment c c classified distribute prediction inner product lie ease continuous excellent performance representative latent generalize model capture equivalence latent structure dyadic block network example relational group membership dirichlet
variant evaluation corpus ng performance keep ng three compare asymmetric lda five burn optimize versus df df total ni corpus tend pattern many topic ng decrease investigation explain phenomenon table ni proportion ni ng compare characteristic word corpus document corpus tf ng examine word ng note systematic pattern ni alone sufficient class quantify df df show scatter versus word ni circle corner locate tend ni suggest one ni level show pattern word category ni ni windows ni file ni ni ni df category ni ni ni learn ni recurrent feature ni basic ni ng order ng neither order systematic addition stop topic corpus ni posteriori word likelihood result study lda hold likelihood higher comparable ni topic comparable word ni form contribute test stop informative likelihood likelihood combination preprocesse cccc cccc ccccc corpora comparable ni word comparable accuracy ng outperform five document linear one classification fold cross validation accuracy reduce exclude ng ng symmetric divergence capture several major inconsistent output token pose topic assignment token vi topic assignment token vi thereby minor propose metric maximum minimize kl generate consistent topic sum find topic attribute minor topic regular ni exclude clear pattern initialization mcmc run ni computing coefficient degree divide ni run represent run lda select well hoc preprocessing reduce explicitly non informative exclude vocabulary simultaneously topic combine advantage symmetric lda asymmetric prior apply size get vocabulary monotonically control effective vocabulary segmentation collaborative find topic show incorporate selection improve improve latent allocation multinomial distribution entire vocabulary contain topic adopt variable modeling lda excluding corpus robust discriminative prior document lda classification well asymmetric consistency comparison allocation corpus finite distribution vocabulary unique word frequency preprocesse contribute corpus systematic corpus relax must topic subset vocabulary study depth lda model achieve short identifie generate semantic topic topic specific aspect probably globally form topic word token word necessary form use preprocesse also selection dimension inference carlo demonstrate correctness model world compare lda show find topic robustness document classification lda occur frequently corpus contribute discover priori latent however exclude truly include lda selection conduct simultaneously latent combine search lda topic vocabulary preprocesse topic vocabulary topic mutually exclusive respectively prior per document obtain sample make selector observe informative informative token collapse update metropolis rely di word token depend assignment token update vocabulary fix varie di rely carlo integration q done burn correctness dataset synthetic corpus start ten topic topic probability word topic five word ten informative topic generate document random proportion draw token topic draw informative synthetic
define invariant preserve emphasize solver hmc generate eq equivalently express semi implicit interpretation carry size hmc hmc sample acceptance table moment express still term equation solution preserve would paper validity directly equation path computer finite resort hamiltonian justify attain acceptance set q hilbert chain chain rule third I calculation prove denote projection definition last corresponding modulus preserving formally preserve volume transition invariant assume next recall uniform bound need stress expectation integral subscript notice velocity validity hmc calculation forward volume jacobian analogue sense jacobian transform hilbert rwm mala hmc define advanced rwm move algorithm langevin sde drift calculation drift brownian increment sde interesting nonlinearity sde derive euler increment mala euler advanced mala advance mala hmc rwm derive term rwm metropolis hasting rwm advance rwm three sampler together h algorithm hmc mala x v x vx vx coincide mala section hmc appear mala derivative synthesis deterministic hmc avoid behaviour providing illustrate target hmc gain mala rwm mala gain complexity rwm derive bridge target general bridge complexity sampler length bridge analytical appear chain vanish acceptance grow connect proposition derivation thesis similar scaling decide completeness make modification follow derivation bridge expansion correspond bridge orthonormal corresponding eigenfunction co eigen specify expansion iid advanced mala rwm algorithms target probability current position stationarity hmc scale control step write equivalently express effect co power I modulus one power unstable require stability scheme hamiltonian derive probability acceptance constant arbitrarily long informally connect acceptance advance rwm mala hmc step give ignore nonlinear map synthesis effect offset proposal position arbitrarily advanced rwm show proposal distance position fix sub mix mala sde mala carry along continuous dynamic require propagate path rigorous beyond paper accommodate wide unknown conditionally mcmc sampler drive naturally rise due probabilistic brownian motion inferential issue relate applicability advance succeed parameter cover transform offer extension irreducible utilize transformation framework irreducible require ease exposition univariate however advanced hmc also section sde model treat continuously sde transform q brownian motion conditionally eq generic verify advanced density arise carry calculation assume time determine datum application sequel target integrable path reference motion boundary brownian bridge recall specification covariance space brownian motion bridge definition abuse proposition term integral practice hilbert space path specification correspond derivative eq appear specification continuous piece linear turn respectively necessarily lie hilbert space weak continuity hmc ess additional present e sde construct euler diffusion skeleton set relative value advance mala think hmc step indicate much step overall hmc perform ess particular remain level note substantial advanced mala hmc mala improvement rwm sampler rate drop relative rwm mala hmc n hmc relative rwm mala hmc rwm mala stochastic volatility analysis record daily frequency volatility hmc consecutive write poorly mala rwm nevertheless enough hmc reach advanced nearly time rwm mala acceptance mala version rwm mala hmc step hmc step numerical illustration simulated survival event drive hazard figure despite fluctuation hazard decrease could case follow trajectory draw exercise sample sde think behaviour keep ability shape hazard table calculation experiment associate extremely acceptance rate rwm poorly acceptance move mala well advanced hmc specifically hmc fast rwm mention figure diffusion process determine hazard display credible indicate shape hazard capture rwm mala briefly case sde drive one far omit parameter context equation sde coefficient indeed existence v ji k transform suggest wide scope advanced mcmc absolutely brownian motion relate briefly verify transform onto drive wiener sde simply write mapping driving imply sde remain calculate involve w driving noise see appear integral expression functional process path check differentiable condition advance develop apply driven diffusion validity establish relax diffusion scientific contain functional integral extend diffusion handle drive wiener direction algorithm facilitate updating either potentially sized block boost important posterior correlation path critical currently investigate extension advance hmc diffusion successfully fact support pseudo mcmc formulation diffusion see choice hamiltonian prior exploit mass location specific mass boost efficiency memory diffusion cover sampler block time lack beneficial hmc control try grateful comment greatly improve thank dr lot ep rwm mala relevant working result suffice point replace second high crucially grow provide acceptance analytically inequality grow slow retrieve rwm analytical inspection term clearly last vanish term analytical rescaling write bridge thus mala calculation indeed term eq come last h vanish use analytical expansion measure bridge get thus exploit recall calculation one operator eigen stand sequence expansion eigen examine eigenvalue verify calculation rest henceforth recall x eq prove illustrate proof sufficient proof calculation notation equation derivative mapping dirac delta immediately expression expression statement stochastic practice involve take much derivative denote consecutive instance q calculate term summation unless last overall operator covariance standard easily combine give line bridge immediately appear line consider difference construct covariance motion follow recursion derivative recursion calculation derivative notice approximation look equation k calculation get vector term q multiply obtain calculation assumption persistent advance available intensive advance familiar chain change law arise context drive wiener constitute emphasis advanced hybrid carlo make drive move analytical computational advantage regime latent model advanced version mix refinement inherently path finite one apply gaussian mixing stochastic carlo distribution application rapidly practitioner ideally suggest flexibility direction propose study advanced version performance advanced distribution change focus drive differential concrete brownian motion advance computationally demanding reproduce direct indirect ability realize drive mechanism importance behavior sde variable within sampler fast critical feasibility advanced method development take target measure hilbert exploit evolve advanced time projection use inherently sde difference commonly approximate infinite dimensional path mesh look advanced rwm adjust langevin mala hybrid mala hmc dynamic whereas rwm use proposal emphasis hmc employ hamiltonian sde computational advantage experimentally hilbert value rwm mala hmc require strict derivative paper simpler minimal wide range practical inherently specification concrete mathematical verification well infinite mesh mix practical run dimensional projection difference another contribution order avoid walk behaviour greatly outperform flexible phenomena biology molecular survival see application large class specify diffusion coefficient respectively form sometimes imply physical law langevin due quantity skeleton path study extensively observation regime appear pd bioinformatics stochastic observe error financial application model diffusion event involve observe unified consist diffusion complete conditionally diffusion proceed develop hmc inferential whereby sampler path augmentation mcmc address arise characteristic diffusion diffusion diffusion e conditionally dirac cause identify noting unless instance consider computer currently available sequel proposal path overall diffusion process typically brownian motion path candidate path case perform poorly many target one advance connect issue achieve consideration involve brownian bridge approach gaussian method advanced sampler issue advance hmc sampler provide powerful latent blind update mix material regard hilbert advanced mcmc sampler diffusion advantage family general coefficient regime section generalization collect material separable presentation later coincides preserve mathematically diffusion upon square integrable reference brownian boundary covariance gaussian operator bridge q brownian motion brownian specify brownian motion brownian involve eigen eigenfunction constitute orthonormal hilbert particular iid dynamic accept force target amongst alternative hmc synthesis reject test version hmc advanced target appear
achieve high forest due overlap always total combine fista combine tv include compare superiority compressive sense multi method cs validate forest conduct reconstruct mr however achieve good reconstruction fista measurement fista fista respectively contrast feasible way datum channel number fouri significantly fraction measurement bottleneck call literature final contain visual measurement cs sense configuration fig image similar forest compressive sense reconstruction recover fouri channel clinical diagnosis image discuss forest sparse fista solve recover run converge ratio snr fista fista cccc snr fista fista comprehensive algorithm increase improvement reconstruction fista performance sparsity gained combine tv however fista easily color capture optical camera represent blue three color color human observe color channel highly joint regularization gain snr regularization wavelet reasonably htbp color fista fista fista recover penalty different color hyperspectral band spectral utilize sense cost huge imaging compressive sense acquisition image reduce band scene band forest band hyperspectral image site show band band sparsity band joint sparse convenience wavelet band modeling achieve high propose compressive family numerous field sparse benefit theoretically validate compressive fast forest problem practical superiority forest sparsity tree sparsity computational convenience root prove paper dependent accordingly combination sparsity number first rip diagonal matrix tm n x diagonal probability indicate q choose prove c complete l b forest less tm li email investigate compressive channel hierarchical channel follow forest intra correlation family compressive sensing theory forest channel far sparsity theory multiple measurement share validate experiment demonstrate propose sparsity structure sense compressive tree technique become popular compressive sensing recover shannon sampling suppose matrix wavelet coefficient perfectly satisfy isometry property rip large find hard impractical basis pursuit prove cs lot efficient sparse classified group beyond sparsity structure come compressive standard sparsity zero datum arise cognitive arrival estimation compressive medical common joint joint solver bayesian contribute estimation structure already utilize image nature signal image approximately relationship tree zero channel sparsity hard implement exploit subtree hierarchical structure rely parent tradeoff often approximate parent assign joint study combination far compressive problem datum across channel channel differ fully guarantee practical researcher give intra ignore inter propose call bridge connect trees forest compressive forest measurement much tree sparsity structure forest imaging image parallel well forest application organize exist work tree review forest conduct compressive signal compression integrate sparse measurement measurement measurement restrict isometry rip subspace cs gaussian graph rip isometry isometry robust require measurement quantify rip lk rip probability intuitively subspace coincide intuition result prior utilize cs datum basis wavelet subtree kx subtree sparsity wavelet coefficient subspace number reduce proof rip recover measurement classical partially channel learn intra result sparsity sparsity sparsity especially exploit sparsity tree forest sparsity gaussian however data compressive follow rather dense system zero dense less analyze exist structured sparsity concentrate random extend block bind tm result dense entry depend sub energy concentrate e channel vary depend measure block diagonal matrix forest still experiment focus sparsity article structure group although sparsity model forest parent parent pair channel becomes overlap approximate assign example encourage zero encourage forest efficient iterative shrinkage thresholding fista fista convenient formulation composite forest sparsity article fista accelerate minimize object function smooth nonsmooth transpose n x closed form soft sparsity second close fista fista overlap introduce contain else coefficient row use constrain utilize alternate direction alternate formulation positive subproblem ta tb order fista fista fista benefit clearly ax whole b keep
achieved maintain u column update z jj parameter l update element outer hyperparameter maximize ml negative nlp cumulative gaussian finally distribution unseen q u effective vector site optimization f exp loss function stagewise basis add coefficient optimize coefficient exp al likelihood design stagewise additive show site select basis loss leave side inequality note monotonically scale training firstly unlike combination write separability function successive weak distribution keep previous optimize desirable log strict sense approximation secondly gp large get sparse along figure without reference value basis forward function notion stage iteration equivalent closely select updating site update simplify tt j jj j add basis j x u k site parameter approximation site previously add site fix relate predictive nothing training basis parameter stage coefficient summarize exclude see associate optimize next viewpoint propose select basis work min select vector cost useful achieve work adaptive accord add normalizing computed sampling along correctly predictive vector relatively misclassifie find violate difficult technique dataset move desire movement enough nlp improve wrong help getting alternatively get generalization minimize condition summary experiment benchmark test dataset large pick test add use reduction pick dataset conjugate specify outer optimize hyperparameter keep track good nlp loss evaluate nlp three constraint result site equivalently dataset figure minor difference effective estimate site certain aspect variance numerical due nature method loop optimize become slightly computation restrict evaluation therefore experiment change vector algorithm goal basis demonstrate effectiveness conduct experiment nlp loss panel sampling consistently perform dataset nlp make predictive distribution performance reduce working size see sampling nlp generalization see column gain basis ensure selection conduct performance partition compare nlp obtain partition observation nlp difficult relatively nlp r higher perform well base well gain w test error well method six dataset entropy gain conduct partition set nlp result hoc reveal significant difference measure gain nlp low overall p performance base measurement show take second dataset gb ram find fast matlab believe operation significantly expect speed improvement estimator viewpoint vector site selection complexity selection storage complexity several relatively set nlp loss dashed sampling dataset vector order yahoo com department institute designing generalize construct additive model boost effective stage new site vector basis achieve conjunction site complexity hyperparameter dataset aim full input memory basis far develop classifier propose sparse gp base informative particularly inspire filter hyperparameter site matching hyperparameter estimate optimize marginal predictive nlp gain validation experimental benchmark dataset showed particularly require vector achieve compare validation generalize expensive need efficient construction additive like boost introduce basis loss site coefficient effective cost reduction use selection experimental give entropy information gain section present design cover experimental generalize well represent latent
alternative detect value rejection zero remark relative rejection nominal large investigate instantaneous relationship fundamental proportional relationship price refer fair taylor concern link make study view macro u instantaneous causal period stock price us department bank series non series autoregressive test adapt time outcome residual implement study iteration instantaneous causality reject nonparametric nan period c c c see adjusted var deviation display c c c display two break point trade balance balance service see economic health country u balance service similarly account service service instantaneous causality relation analysis department frequency available site bank difference var adjust check suggest statistic display test quite view variance study value test correspond display instantaneous causality unconditional unconditional investigate kind instantaneous causality important unconditional vary propose test outcome set bootstrap framework structural proof introduce q give pd du tt v martingale central limit expression obtain appropriate replace e di theorem follow sake autoregressive conditionally process increment pointwise convergence proof hold similar pass economic survey directional influence reveal causality bootstrap guide chapter identification quantification causality investigate heterogeneous splitting bootstrapping box bootstrapping unit root regression heterogeneity causal stock interest university economic model linear causality stable multivariate autoregressive work I change multivariate regression volatility transaction work efficient estimation presence causal system unconditional variance financial work les shift series autoregressive variance cm corollary abstract instantaneous causality unconditional investigate test avoid precisely white autocorrelation consistent correction suffer severe test investigate instantaneous keyword var causality define financial lee others network et domain causality relationship account past improve information case causality relation var process test restriction innovation tool statistic correct covariance nonlinear pass model account unconditional autocorrelation correction unconditional structure van u investigate unconditional highlighted production sale break variance variance constant unconditional lead instantaneous causality unconditional numerous zhang test detect unconditional rao non unconditional variance xu reference unconditional variance stock constant restriction structure highlight standard instantaneous causality implement use provide suitable correction statistic severe situation precisely detect alternative periodic sign change note previous intend unconditional instantaneous unconditional variance distribution statistic variance asymptotic widely unconditional dependence restriction constant establish modify preferable test base constant unconditional plan variance causality var property test unconditional variance avoid take consideration test investigate finding autoregressive var triangular form subscript rescale approach measurable matrix assume gaussian retrieve tool develop suffer drawback test instantaneous causality piecewise smooth change allow unconditional variance xu reference therein propose use commonly instance structure view dynamic testing restriction variance follow h usual kronecker ols establish denote alternatively ol residual autoregressive important length instantaneous causality lag structure hand autoregressive propose fit particularly ol lemma may error unconditional constant drive markov pass detect unconditional weight statistic z choose statistic involve autoregressive order adjust uncorrelated surprising exclude statistic investigate asymptotic standard establish eigenvalue addition also standard hereafter consist instantaneous appear reject hypothesis causality resp test turn study situation converge strictly follow grow detect causality enough consider stand alternative relative st h c st st every error assume iid structure constant compare ability test instantaneous causality centrality w assumption suffer severe power consequence test intend account consider bivariate case r spurious unconditional variance avoid summary correction contrary addition non unconditional well test prefer time power bootstrap introduce ts consider h ts ct non fourth causality literature investigate var reader therein resample draw take iid give ts b generate motivated fact restriction test residual equivalent bootstrap replicate pattern variance weak
sensitive learn insensitive medical diagnosis business costly unbalanced situation cost rule insensitive insensitive highly make important develop extension technique current understanding extension machine architecture theoretic many extension svm insensitive hinge sensitive group category address adopt majority class due clear effect informative approach involve base transformation modify svm due cost unclear modify naive boundary movement bm shift adjust theory would svms accurately area aid sensitive calibration transformation receiver operate roc curve performance boundary movement separable sensitive optimality modification separate plane penalty bp svm introduce slack training transform bias obvious limited slack slack standard assigning svms fundamentally sensitive specify risk function classifier sensitive risk enable svm sensitive rule approximate sensitive bayes loss classic avoid method produce sensitive result cost closely appear change upper consider sensitive learn moreover sensitive minimum sensitive metric connection roc robust reflect tolerance briefly review generalize connection cs present cost examine classifier sensitive present demonstrate classification map class draw function negative assign minimize loss risk minimize risk predictor provide bayes decision minimize loss omit notational simplicity risk decision prove finally bayes boost hinge svms loss margin loss assign penalty positive encouraging loss enforce margin conditional minimize generative formula derivation consistent rely comparable goal find maximize maximal equality theorem hold appropriate margin invertible symmetry eq q design bayes invertible satisfy equation novel bayes consistent previous check derivative extend connection minimization false minimized predictor include associated classifier implement sensitive sensitive cost minimum risk associate sensitive minimum rate sensitive error generality equal highlight fundamental conditional maximum write intersect trivial thus prove note property concave extend paradigm predictor minimum conditional insensitive certain use g generative invertible ensure bayes optimality necessary lead apply bayes rule classifier approximate sensitive bayes cost bayes form arbitrarily choose close conditional loss equality say determine assign require enforce symmetry property satisfie property risk bayes speak sensitive classification algorithm predictor concave risk measure simple conditional reduce obtain I derive illustrate application cost I start recall adaboost cost sensitive easily satisfy note sensitive symmetry property eq boost extend bottom cost minimize hinge result sign sign replace insensitive cost counterpart sensitive conditional hinge loss condition close zero conditional criterion degree hinge hinge slope margin slope positive weight heavily leading select component result sensitive svm minimal insensitive svm class small margin slope assign higher enforce risk quadratic insensitive soft control parameter responsible sensitivity type impose example margin assign separable obviously define justify guarantee implement decision ii approximation cost risk guarantee heuristic solution intuition problem linearly separate green line arbitrary maximize margin equally distant near class blue irrelevant also maximum positive margin specify increase increase positive enable control limited level undesirable general regardless preferable attempt error bp necessarily cs allow choice positive independent unlike cs margin class clarity formulation svm bm regardless separability slack write dual norm vector svm dual dual solver act coefficient insensitive ci scale svm major ci dual relaxed modification nontrivial connect sensitivity sensitive ci dual allow extra subsequently cccc f effect norm dual consider positive classification class usually lead different intensity majority range training example linearly example imbalance solve apparent implie take low imbalance ratio decision class boundary margin ideal result cause value bm bp result implementation margin classification bm svm dataset imbalance non sparse highly sparse sparse induce norm regularizer lead balanced use choice zero result figure highly boundary choose regularization algorithm choice uci dataset ratio sparsity deal implicitly prevent movement discriminant class enforce margin class cost cs svm class equivalently extra make asymmetric favor cost sensitive k kk corresponding show cs dual coefficient regularization hyperparameter space minimizer dual depict problem formulation cs svm dual substitute appendix ig retrieve problem reveal duality primal decision vice regard expansion bp extra different cost sensitive number basis contribute part highly discriminant mostly dependent kernel basis majority training pursuit add majority cost sensitive implement computational medical diagnosis retrieval business decision make rise concept dependent ed main ed consider accord accord train example probability resample method suffer cause implement bp svm call ed hinge ed hinge dependent ed hinge ed benefit include asymmetric margin ed experimental sensitive svm base ed ed bp hinge svm evaluation prior respectively measure simplify call insensitive risk zero classifier produce induce component optimal theory find follow optimization equal cost sensitive zero one choose optimal risk minimum know classifier well cost roc robust performance measure auc evaluate area within true negative region tp tn roc true region respectively auc tn auc demonstrate cs ability sensitivity specificity bm bp svm group namely dependent sensitive class cost learn example dataset experiment far explain section create cs svm show specification experiment dataset point multi different imbalance ratio http www edu tw target type bad heart presence breast diagnostic breast cancer diabetes cancer rbf hyper perform auc fitting grid cross evaluate appear bold hyper train bm cs generality perform grid actually hyper grid implicitly cs svm asymmetric margin advantage cost http www finally tn auc know problem readily modify hyper test determined phase
function network transform function toeplitz unit bias constitute finally feature see patch descriptor scene weight share network share scale extract scale justify remove sharing predict pixel easier consider grouping neighborhood suit pixel formulate search adapt neighborhood minimize unique assume component minimal formally th partition explore subset methodology confidence good partition define pixel component let set pixel wish component explain cost set lattice overall component classical simply explore depth finding figure aim component kind leaf root minimal component optimal subtle class good root weighted two neighboring function region span region construction quasi linear naturally produce merge component horizontal hierarchy equivalent thresholding map equivalent cut hierarchy various one neither propose hierarchy produce produce classification dendrogram dissimilarity two ultrametric map contour contour constant thresholding map yield contour show simple dendrogram confidence component class available predict simply way achieve illustrate feature part mask component produce vector background representation max grid invariant relation attribute nice shape object handle dominant combine pool solid function recognize underlie object grid fall aggregated pixel fall cell descriptor encode relation experiment define number class paper predict distribution matrix note represent interested find cut optimize power edge qp energy equivalent approximately foreground fold sift flow liu compose split training image obtain semantic label test test range scene type network channel bank map pool dimension map map produce filter follow pool layer dimension feature component combination filter locally laplacian pyramid pyramid rescale output network produce feature map parallel find decay expand irrelevant include rotation paper hierarchy cover volume complete removal informative segment segment optimal include bad stanford system multiscale alone describe report complete example dataset show baseline multiscale network multiscale predictor classification aspect clearly suffer consistency hierarchy network input vector unit stanford sift sift dataset sampling class balance equal network balanced frequency discrimination recognition view balance give balance work poorly dataset large amount example extremely overfitte sift show demonstrate remarkably stanford server c multiscale net multiscale per accuracy report author modern core intel competition c liu multiscale net per pixel average per frequency multiscale cover multiscale per accuracy class accuracy multiscale balanced framework use operate raw mid convolutional supervise scene parse rely segmentation tree image segment contain instead cut cover extract consistent except production state art stanford pixel per compete single construct contain segmentation optimal improvement learn learn segmentation produce institute mathematical york new york ny universit paris si paris france parse consist belong involve parse segment dissimilarity simultaneously dense encode size multiscale convolutional raw segment cover node aggregated fed produce contain segment tree class distribution segment accuracy stanford class sift class order magnitude compete approach produce scene require recognition contextual integration simultaneously label pixel come distant category pixel indicate human nearby grey part road building cloud depict rely scale extraction multi scale dense vector around context convolutional apply multi laplacian pyramid convolutional fed raw tree graph pixel pixel connect near measure two segmentation merge span final segmentation subset wise feature segment encode grid aggregated grid component max feature center fall cell produce classifier apply aggregated feature classifier histogram cover segment define attempt segmentation pixel complexity linear produce full second adjustment threshold three key multi convolutional region class decide segment oppose object part object procedure optimize scene parse variety year rely mrfs account rely pre segment candidate extract individual segment neighboring segment consistent segment aggregate train scoring individual like train feature question scene parse make decision histogram look fine somewhat densely pyramid coarse hence scale encode decrease author candidate aggregate segment tree allow rely base cut finding cover system densely multiscale pyramid convnet feed raw mid big dense efficiently detection image segmentation particularly publish scene parse network feed scene parse base align label produce convnet boundary take segmentation representation pooling paper point seek image typically problem rarely contain object properly poor observation predict integrate grid finite enforce usually achieve
high x take optimistic costly proposal incorporate gram efficiently weight represent proposal corresponding take acoustic word max thus efficiently dynamic viterbi backward forward weight either max suppose produce edge draw unlikely thus reject produce account realistic trial experience sequence refine extend account stop process observe threshold refine stop process maximum evaluate retrieval possible word convert mobile phone assume english million language sentence remain length number retrieval experiment sampling latent token decode top plot viterbi fix sentence decode gram iteration reduce gram gram hmm gram gram c gram hmm token refine reach acceptance ar reflect gram input take iteration successful note trade variable resolve batch step note time find good iteration ar length move average trial os approach graphical loop function potential undirected unary function potential determine minimal bind span intuition gap outside improve upper correspond partitioning configuration write kx k potential unary potential eliminate loop particular equation neighbor replace condition different remove proposal refinement piecewise iteratively partition grain graph cardinality grow exact past decade rate refinement could refine proposal sample experiment show help stop refinement acceptance sample binary unary normal certain ise time elegant properly exactly target rely assumption typically attractive coupling strength make hard use policy take proposal conditioning pair policy addition use reject sample two reject refine contain conditioning variable reject strategy experiment select probable iv greedy policy total maintain queue triple condition ii refinement policy policy reject policy iteration conclude iv computation take trial estimate accept refinement unbiased current rate estimator decide expect trial distribution spend plot acceptance figure start decrease regime acceptance rate refinement see iv despite policy look optimal ii apply confirm reject adaptively region refine compute propose viewpoint rejection functional upper sampling refine refined decrease depend class max bound gram probability give decode hmms problem simple illustrate decode graphical reject refinement computational attractive model speedup motivate development extension two simple typical agreement free syntactic low level gram structure language improve limited refinement improve element partition example sampling far refine conditioning value region refine incorporation high n algorithm asymptotically produce os approach combine adaptive sampling refinement present inference hmms discrete elementary rejection produce sample produce propose os exact rejection space upper proposal dynamic rejection refine complex way implicit constraint obtain relevant acceptance never activate never explore region become explicitly gram latent hmm unlikely present never rarely explore treat consist proposal assess dynamic select interesting decoding construct os hmm attempt generalize sampling improve curve author propose curve rate technique continuous line convexity exploit tight tight piecewise rejection consider graphical introduce sampler increase variable recursively precede graphical similarity cascade sampler configuration dynamically proposal os unify algorithm sampling move background notation l px px unnormalize normalize define able distribution easily give rs follow unnormalized precisely ii dominate ratio accept rate measurable area illustrate illustration attempt say acceptance attempt space measure value q think roughly abuse function generalization avoid confusion say sense normalize dx f find informally sampling tend tend formal note connection mix mcmc hasting sense os almost measure unified os describe follow goal os sample refer cover density space os efficiently limit one refinement small natural control candidate optimization small costly one prefer simply good either certain ratio threshold
objective default inference interval match calibrate reference prior despite effort satisfactory inference goal seed plant formalize extend association unknown model clear equivalent I attempt accurately predict conditioning focus fully alone insufficient accurate therefore call amount association auxiliary set probabilistic start association auxiliary produce post probabilistic yet I three pair collection set valid combine set modeling probability without inferential true calibration dependence I probabilistic meaningful provide I well description calculation I output posterior belief poisson provide summary belief posterior belief consequence calibration property output control fundamental present give I solution marginalization nonetheless illustrate I conclude section website consist value depend induce variable equip unit cube lebesgue model observable observable scientific investigation formal describe produce realization single population familiar simulate satisfied interpret link variable surprising association many association triplet equal association help resolve uniqueness I sample practical prefer association prefer evident view moderate section role characterize observe unobserved quantity course never inference solve solution continuous could well mean continuous identify fix determine integration part reveal invert inference sense true inside information highlight accurately employ simple general description map measurable predictive draw good predict unobserved predictive design variate choice perform interest default predictive random set predictive subset description assume predictive satisfactory predicting available combine intuition set consistent unobserved random satisfactory predict capture assertion assertion act summarize subset association dependence singleton emphasis validity separate make remark first consider nan simplification induce ignore I modify fails discuss follow assess conclusion extreme convenient call singleton summarize report predictive singleton belief plausibility symmetry apparent plausible statistical singleton assertion function plausibility neighborhood poisson plausibility function singleton assertion treat observe argue observe plausibility similar predictive plausibility poisson belief function dependent familiar fact trivial drop measure recommend calculation precisely represent knowledge gain claim auxiliary familiar association primary determine auxiliary predictive certain aspect I clear hope accurately ordinary random association proceed space reasonable natural characterize summary evidence favor assertion predictive evaluate random section desirable assertion question argue meaningful indicate meaningful calibrate amount mean appropriately refer calibration definition predictive random ideally variable function precise predict unobserve stochastically q proportion possible mapping check related validity easily nest valid nest measurable predictive support definition outside set realization iff rich special random validity nest claim I I validity predictive essentially prove correspond I belief refer suppose assertion I belief I state plausibility I suppose I stochastically small turn hand complete proof key predictive I particular predictive valid whenever change validity parametrization include transformation forward correspond I addition provide plausibility frequentist I base frequentist valid type rejection rule therefore type error singleton counterpart frequentist eq plausibility plausibility coverage plausibility plausibility inequality classical plausibility display plausibility interval well plausibility certain exact interval show fisher I singleton sense correct singleton large inefficient validity shall assume ignore conditioning belief function predictive light shall predictive sense ab b u b u eq exceed call I base ratio unity denominator follow theorem theorem valid kind consideration assume predictive construct serve clearly nest bt ta iff monotonicity ax ax omit general reduction technique reduce I conceptual problem since idea prefer presentation sided assertion assertion one assertion I stochastically I stochastically predictive random optimality property light notion optimality assertion one side possible sided assertion suppose define resp optimal predictive nest case similar eq consequently stochastically optimality previously treat predictive leave sided assertion versus I x classical pearson strictly connection test example omit side difficult sided classical I assume assertion unity random set property corollary follow roughly speak stochastically loose test goal thing continuous parametrization argue certain property nest subset shall balance corresponding q take predictive set set measurable score last follow sense least must score balanced choose reasonable balanced random side side optimal balanced proving set satisfy start depend implicitly shall decrease function distribution satisfy analytically typically require condition verify b argument xt family natural condition score describe simple sketch idea balance around make small make score balance determine integral definition hence optimal absolutely assume relaxed thing although illustrate phenomenon thin gray density line convexity figure interval black case integral break take part make gray sense fails transform score score symmetric correspond support x minus plausibility conclude consistent intuition match discuss identify plausibility function show optimal balanced gray default plausibility interval horizontal line much shorter I one might consider nominal transformation short plausibility interval coverage set score balanced default gray vertical mark balance predictive well sake refer reader elsewhere suffice plausibility balanced I require need compute indicate show plot two plausibility balanced although shape vary balanced plausibility plausibility plausibility x black gray suppose inference standardize reduction sufficient sample mean variance justification though step auxiliary student degree non centrality dependent marginal slice space work q consideration thing central develop evaluate use entire integrating set plausibility plausibility interval invert usual frequentist interval mean also agree frequentist plug style marginalization whereas ignore I marginalization available discuss dimensional consist assertion simplify presentation emphasize I produce well product auxiliary vector value simplex association characterize case remark sort easy alternative perform sort component ignore partly partly assertion datum association belief vector greatly simplify plausibility illustration ratio plausibility predictive neighborhood power three test equal ratio old test power connection divergence two substantial due assertion new I default method entirely fair assertion drastically first experience knowledge fundamental science combine respective post probabilistic inference calibrate plausibility frequentist bayesian goal simultaneously I surely benefit I framework logical meaningful frequency calibrate probabilistic without latter property something inferential able final I depend user association particularly effort predictive particularly multi make ambiguity random frequentist choice neither argue uncertainty association therefore prefer framework feature attack depend assertion lead drastically output slight predictive construct variable relevant reduction connection theory statistic parameter developed bayesian theoretical I promise reference expect inferential framework frequency acknowledgment grateful effort partially foundation grant dms implicitly balanced express predictive let text balance b tb sign neighborhood iff balanced remainder vanishe term hence
broad community student arm serious imagine diagram familiar experimental property encounter broad scientific mostly irrelevant category winner anneal backtracking anneal perhaps familiar markov chain employ projection von feasible hyperplane version intersection family backtrack mathematic backtracking determine exist despite computationally problem backtrack alone examine alternative arrange fill cell follow three rule grid rectangle rectangle rectangle rectangle appear complexity incomplete belong exponential nonetheless exhaustive force matter execute simply option anneal alternate projection yield good approximate discussion student computational backtracking grow become partial begin backtrack block potential discard en backtrack construct cell cell order cell possess cardinality come cell list cardinality come cell character string alphabet cell limit treat backtrack grow grow element take backtrack string stre first another backtrack back string discard replace another grow backtrack depth search string string dictionary constitute tree traversal systematically move branch prune pruning branch implement backtrack backtrack virtue exist mechanism node infeasible grow parent draw child grow parent none child parent parent draw grow edge child node child infeasible child child node node attempt solution beyond partial eliminate consideration anneal attack combinatorial define state function quantify ideal sensible constraint zero annealing propose choose markovian current process history decrease accept early high stage temperature physics anneal broadly local integer integer appear nine anneal start feasible proposal projection projection fix theory intersection original complicated know projection projection rule namely projection onto ap simplex I fortunately purpose either alternate advantageous simplex whole intersection alternate project onto several smooth tucker involve multiplier solve search subset contrary small substitute function q denote th entry equal inequality dominate sort describe alternate projection relaxed imagine generating constitute solution reasonable constraint feasible tensor heuristic generating integer put probable integer unknown easy construct imputation relax solve specify constraint projection amount constraint requirement digit appear average constraint requirement digit amount contribute digit take pair projection cycle slightly code sake omit test imply medium hard allow project origin projection backtrack medium hard th projection sim anneal backtrack l medium l heuristic successfully method category computation ghz intel processor ram anneal backtrack implementation c show backtrack vast probably go hard simulate annealing find alternate turn hard rectangle rectangle anneal inspection reveal admit constraint show typical simulated anneal something cpu solution versus backtrack point line solve alternate tends backtrack student mathematical science combinatorial neutral starting stress poorly solve hard combinatorial certainly electrical engineering coding assume dominant science phenomenon computing continue execution drop second tackle large third certain notably communication imaging amount create modern dna sequencing still origin adopt
recovery choose cs pursuit denoise formulate bounding standard l q semidefinite sdp software etc similarly frequency recover fouri transform reformulate signal sparse multiple helpful operator balance transform sdp electrical produce long usually require wireless device signal domain signal quantization random multi balance sparsity l solve sdp solve l norm relaxed square l recovery signal recovery sparse signal static reconstruct signal distribution ls optimization newly propose l reconstruct signal balance measurement quantify root eq th simulation normalize simulation simulation choose fig three person patient patient time signal recovery performance length reach perfect constraint price noiseless available rmse fig reconstruction sub ratio profile optimization fig well bad optimization reason l compressive degree uncertainty l second second long optimization signal newly multi encourage multiple enhance develop hybrid method decrease bregman accelerate multi compressive linear sub investigation signal sparse domain improve multi programming formulate multiple linear constraint improve performance example result newly signal compressive sense cs attract employ preserve projection rather compress new recovery technique cs original measurement classical exploit zero analogue analogue digital operate cs sub obtain time aic convenience formulate matrix use
backpropagation enable perceptron know mlp time whenever simple linear value mlp learn series order computation question question model immediately network prune train large mlp remove connection heuristic introduce criterion prove surely bic layer result establish series generalize iid bic suppose suppose hypothesis likelihood converge toward square index class score establish ratio consistency bic application find suppose admit practical perceptron determination unit one unit keep mlp section mlp cite iid stationary model involve hereafter let markov finite space hybrid hmm e weight iid data peak financial mlp obvious reason compute shift inner see pointed give space generalization common object evolve function mlp neuron property mlp approximation statistical see also alternative mlp hierarchical mlp map som artificial som visualization map low grid string neuron codebook prototype prototype method som prototype point point vice versa away assign neuron prototype consequence visualization capability illustrate instance som design vector survey series build upon recursive version som tree cluster version som temporal code advance include symbol string som som component write represent slow correspond month year correspond month etc series slow year know month suppose series dual lead value view give vector future term would one ahead forecast however generally forecast nonlinear explored forecasting separately mean prediction profile som normalize transform profile hc profiles som lead prototype prototype assign predict standard forecasting scale match neuron instance day week predict profile average rescaled scale become naturally som naturally context normalization one calculation shape mainly another som segmentation technique represent instance som advanced norm visualization capability display globally survey categorical adaptation som way multiple analysis categorical bt contingency table contain multiple bt else bt previously transform individual som use bt som train provide categorical two som world economic field extension neural previous sense construct specific hand strength new technique present general som dissimilarity existence vector format frequent directly belong dissimilarity map negative minimal point dissimilarity readily set string string distance hypothesis minimal rule base choose lead version proceed observation dissimilarity minimizer distortion neuron modification know early version choose associate neuron som modify implementation som concern school transition identify generation sample people market nine category include education stream appear path lead stable situation west however north south region north region contract red contract west transition service education path north initial contract become south east exclude light dark color nine extension som dissimilarity propose avoid constrain deterministic relational number neuron dissimilarity function positivity kernel inner follow convenient come elementary operation datum rely batch som quite assignment distance translate solely kernel
would give calculation term first order induce correlation integral arise parameter residual deconvolution reconstruct contain problem statistical mechanic cell type sample turn ever optimally dataset base experimental issue leave acknowledge discussion research expression level typically type mixture proportion sample principle possible underlie result deconvolution complex support cell two dna set cell decade cell arise cancer proportion proportion concentration gene concentration gene call residual sample specific cell take concentration gene combination contain different mixture product constant cell type variation across sample fluctuation proportion basis reconstruct mix deconvolution reconstruct product mix sample mix fluctuation proportion induce positively fluctuation fraction vary outside molecular outcome example context face recognition analysis question paper fluctuation need algebra formulate deconvolution statistical physics accuracy trivial model suggest algorithm formulate deconvolution optimisation demonstrate drawback derive constraint algebra mn denote gene unknown right side equation gene level hundred thousand condition residual fluctuation type cell induce bold symbol measurement bayes probability marginal keep calculation tractable mean different choose contribution constraint mix proportion delta hamiltonian deviation third proportion pattern cell measurement physics proportion hamiltonian describe focused part phase set mechanic x aa aa aa aa aa tt numerically similarity cell average system thus replica saddle numerically case cell interested difference reconstruct mixture number normalize reconstruction surprising bring induce gene cell reconstruct finite sample reconstruction non value carlo reconstruct average target hamiltonian agreement analytical fig boltzmann sample deconvolution measurement reconstruction weight arbitrary configuration generate within positivity proportion new configuration accept acc configuration enter algorithm create accord mixture need distribution datum allow space high mixture mean configuration visit also deviation deviation quantify reconstruction limit serve reconstruction target deconvolution deconvolution start guess frobenius improved distance solution never guarantee reach artificial outperform accuracy algorithm right plot estimate solution reconstruction bayesian algorithm clearly reconstruction serve uncertainty
round cost size number round round stop meet cost exceed communication cost number k e party party protocol real dataset six party party gaussian carefully partition contain positive lie protocol real uci dataset separable repository protocol work perfectly separable limitation linear protocol design noiseless work communication claim noiseless perfectly entire present well bold require low cost compare time case frequently far desire specify circumstance well per player perfectly player separable synthetic aforementioned protocol perform classifier produce despite rather accumulate become algorithm exchange point c acc acc acc protocol dataset acc acc acc cost protocol dataset case yield classifier party show communication cost variation number finish increase multiply express word htbp party protocol table present party protocol party protocol party simulate player simulate second input work work storage back complete pass continue player arbitrary first completion pass work stream fix dimensional mean low point algorithm communication streaming look deep streaming room turn streaming pass need pass result solve distribute linear communication linear programming constraint streaming stream multiplicative weight allow programming general lp form value multiplicative solution hard constraint call solution approximation would achieve protocol search soft typically feasibility parameter I I xt xt describe party distribute protocol asymmetric value maintain construct feasible consist original feasible use update weight round merely compute soft program apply base algorithm minimization etc protocol hyperplane arbitrary framework distribute applicable wide distribute pt pt goal learning distribute multiple expensive prior propose separable distribute party classification protocol considerably demonstrate distribute across problem convex point streaming adapt multiplicative update set extend range major challenge learn minimize overhead communication carefully point node classifier extension player distinguish linear separability make node paper substantial classification present rely construction exploit multiplicative weight specifically desirable theoretical communication moreover player demonstrate stream weight convex problem task minimal party background protocol party present study section distribute focus infer accurate sub classifier individually improve protocol like admit partition dataset depend extract batch accumulate regret bound recently improve margin apply specific subgradient communication learn resource use demonstrate classification greatly communication major bottleneck many percentage minimize round motivate aspect algorithm consider distribute total agnostic agnostic draw total communication bound decision list proper extend preserve distributional resource process key primitive focus way general weight streaming distribute addition party possess party may negative example correctly assume classifier misclassifie pass pair communication word extend communication issue require count bit instance player cost goal protocol allow get dataset family vc random long exist probability party draw identically party henceforth protocol restrict size construct classifier well threshold axis align hyperplane one bit distribute hereafter way protocol back protocol use communication rely pruning whereby either acceptable classifier classify correctly contribution base communication sampling party player report sample party select union hypothesis vc exist way classification way communication general known word hypothesis vc clarity label protocol round construct weighting end set point perfect deterministic presentation bad two party mention deterministic version weight send initially weight multiplicative propose party misclassifie point correctly intuitively ensures misclassifie eventually weight future round replace line weight deterministic see protocol support point compare ab htbp randomly analysis multiplicative update resemble first key use collect ds although run time vc hold error could fraction mis sufficient specifically yield follow party succeed construction perspective slightly party round round weight construct misclassifie c c misclassifie end misclassifie point point relate inequality hence word communication suggest set lemma weight probability protocol need misclassification set apply failure protocol cost formalize party protocol constant round jointly use communication party protocol obtain classifier arbitrary say protocol player jointly player theorem parameter report fraction player failure round union lemma union party require communication round party terminate round communication correctness round theorem aside party party collect claim randomize achieve construction failure round party misclassifie result protocol linear communication classifier party party scenario amongst naive voting
predictor threshold z stage behave criterion satisfy collecting start decrease gradually maintain close cell analytic table benchmark cell generate build benchmark production maximize optimize medium contour x c ji constant zero prior gaussian kernel kx x function directly corollary theorems z e corollary entail select first locate center inside hyper sphere whose hyper sphere point inside value fc speedup speedup match cl benchmark denote experimental benchmark dimensional benchmark benchmark iteration dimensional benchmark note typical encounter recall theoretical suggest estimation expectation confirm estimation estimation observe good output experiment ei max ei algorithm expect improve experiment value min effectiveness matching approach batch set batch pure cell sample finish step experiment output constant add experiment repeat several way include demonstrate particular function justification theoretical indicate experiment selection meet approach benchmark sequential ei experiment setup table perform well exist practical less matching show constant competitive bad hybrid batch suggest stop criterion fact identify condition stop batch performance degradation ei investigate batch goal policy problem clarity focus maximize concave contribution firstly systematic limit universal bias simulation outcome estimate provide bound relate good experiment secondly propose behave optimally add dynamically early behave gradually move toward batch policy synthetic speedup theoretical light approach ei recall introduce conclude proof theorem conclude proof optimality get ei taylor expansion finally observation exactly addition quadratic root try introduce proposition aim optimize evaluate either sequentially evaluate multiple mode generally evaluation number iteration systematically hybrid dynamically batch size justify eight benchmark result achieve substantial compare try costly evaluate point optima black characterize example motivate enhance micro efficiency surface property problem run expensive consuming focus maximization consist function via select cycle repeat meet stop experiment policy take capability run experiment parallel motivate one select heuristic select experiment improvement ei select ei select add repeat batch speedup experiment sequential policy especially total experiment motivate dynamically batch paper simulate batch purpose select outcome pick batch policy pick upper bound proportional naturally rise hybrid sequential ei next outcome gp choose process go experiment small batch large appeal behave experiment stage prior sequentially evolve manner experimental speedup performance increase experiment speedup increase model probabilistic experiment ei criterion ei gp build outcome unknown new gp variable kx kx kx estimate output consider x location e variance new set characterize change variance z k ba b point enable previous scheme fast complexity actual expect heavily available td tell proportional parameter intuitively know return observation tell batch remark trivial counter intuitive gp optimistic practical bind would focus error oppose simply take corollary capture remark entail computable us criterion algorithm current estimation want accurate selection require suppose capability run speed without sequential
low permit tensor decomposition unified express observable sum rank parameter extract decomposition symmetric solve point variational orthogonal decomposable tensor perturbation variant decomposition subtle unlike decomposition robust use random decomposition tractable manner analysis I would accumulation avoid previously complexity finally address execute dimension require number entry combination robustness decomposition overall framework tensor decomposition model long history across decomposition idea gain numerous thorough application independent analysis decomposition separation statistically mixed observed ultimately projection contrast excess fourth method tensor particularly rigorously analyze bound decomposition estimator relate algebraic simultaneous markov via pair triple probability table later hmms topic lda context explicit estimator identifiability argument source develop tensor develop overcomplete source tensor long recognize algebraic work moment work concerned basic question identifiability structure work symmetric orthogonal tensor combination tensor tensor property decomposition although eigenvalue counterpart computationally symmetric orthogonal decomposition power tensor analyze fact imply general obtaining focus machine statistic far heuristic maximization require practitioner employ computer science machine address issue discussion hardness range mild degeneracy condition tensor observable notably observable propose successfully hmms subsequently generalize extend class low state update estimate emission organize basic definition model structure estimation extract certain decomposition tensor power analogue arise deal tensor euclidean restrict use array pi pa basis multilinear map follow matrix I identity observe basis tensor consider sometimes invariant permutation array I I check order small notion reveal singular decomposition similarly rank one term notion considerably instance clear tensor addition removal approximation norm operator denote tensor first exchangeable invariant exchangeable view conditionally identical simplified model document assume distinct document draw discrete variable draw independently document moment joint word estimate document encode vector conditional moment eq follow x structure reveal certain decomposition exchangeability triple resp form section yield tensor desire various moment mixture spherical covariance start covariance probabilistic mean choose component exchangeable vector spherical covariance differ draw topic document x I spherical covariance variance small eigenvalue eigenvector eigenvalue I common ica mix denote x fourth tensor derivative tensor x correspond excess furthermore increasingly simultaneously correspond mixture oppose dirichlet positive document hx independently specify topic iff word pseudo characterize concentration entry extreme behave interested independent mainly comprise x know assume distribution parameter style inner sep hide inner hide observed name name observe h observe inner sep minimum cm inner sep name hide observed h view call class observe conditionally independent similar require identical markov certain certain variable conditionally importantly may even mix moment k possess roughly speak recover easily handle value linearly independent define three observation concern model mixture product align require certain role incoherence explain distribution partition three possibly view achievable instance align incoherence entry assume rank large span hadamard basis cardinality incoherence condition degeneracy component isolate least partition dimension independently pick put entry put asymmetric component lead mean dimension slightly great base multi extract structure create three apply rotation three x dd invariance spherical check view mean column rank nd surely long imply choose kt ta e jj th large analogous rotation causes fraction large suffice discrete state observation initial chain whose markov give rank iii find decomposition discuss show decomposition estimate decomposition matrix v exist view two scaling fix linear u view eigenvalue uniqueness q maximize zero maximizer subject orthogonal eq maximizer efficient multilinear u tensor unique fortunately permit decomposition mild condition attention minor collection vector correspond scalar focus odd add convention follow generality since analogy two way view fix approximation generalization linear eigenvector satisfie assume eigenvector replace decomposable tensor subtle arise linearity multiple status eq proportional eigenvector spurious say unit repeat start map unit euclidean robust tensor orthogonal imply orthogonal order uniqueness sign orthogonal zero appendix readily orthogonality consideration robust whereas eigenvector large eigenvalue since sign robust fix map third orthogonal isolated maximizer local ica use order adaptive isolated implication two decomposable I function u tu u ii reliably identify iteration analogue moreover tensor model reduce orthogonal single k general form decomposable recover see discussion simultaneous degeneracy linearly scalar imply condition generally degeneracy tensor rigorously formulate smoothed first transformation matrix orthonormal define w identify eigenvector upon eigenvalue everything robust eigenvalue eigenvalue theorem discussion robust eigenvector take convention eigenvector eigenvector correctly along relative latent role applicability tensor characterization provide robust eigenvector latent maximizer objective interpret along tensor decomposition decomposable decomposable moment tensor unlike matrix extract approximate decomposition detail employ establishe convergence iteration decomposition determine convergent point subtract term element repeat power eigenvector ti k c perturbation analysis per preferable estimate eigenvalue tensor power update start symmetric perturbation symmetric empirical decomposable derive population moment throughout reduction whiten theorem return perturbation provide kt tensor orthogonal orthonormal iteratively time call tensor return estimate eigenvalue return least appendix one algorithm specific algorithm perturb decomposition use intuition eventually contraction eigenvector dominate bind note determine converge frobenius euclidean expect tensor perturbation stop initial tensor exhibit effectively tensor build address future research consideration may model topic distribution document primarily initialization document eigenvector application orient tensor variable practical concern arise tensor bag moment document document moment order triple length implicitly document contribution tensor check quantity triple contribution moment allow multiplication document corpus document matrix serious form multidimensional array value typically symmetry store third memory prohibitive dimensionality tensor procedure avoid linear single algebra computation efficiently span use compute whiten matrix product implicit access need explicitly form whiten third w implicitly via core computation perform finally order step non entry computational tensor power decomposable operation random eigenvector extract dimension k reveal value pure note recover eigenvalue unstable gap appendix worth differ roughly gap remove initialization work crucially population algebraic desire perturbation moment exploit convergence directly practical error observe perturbation power example discuss remain previous work accuracy orthogonal tensor method relative cite improve depend requirement contribute complexity far extract need suggest number perspective simultaneous consideration model robust guarantee experience tensor method exchangeable model misspecification indeed affect result reasonable nevertheless robustness advantageous suggest complete microsoft partly mt aa supported award award award theorem completeness decomposition converge repeat set lebesgue measure v exist lemma convergence claim claim first sufficiently thus eigenvector eigenvector point variational eigenvector isolate maximization unit stationary eigenvector pair characterize isolate enough local maximizer must point tu ti nu isolate orthogonal depend cardinality q u isolate maximizer note open pick small maximizer stationary maximizer suggesting constraint decomposition constraint isolate fix relax relaxed power projection analyze iteration complicated tensor initial rate soon dominate due appendix subtle due consider number implicit probability least random vector absolute integer least uniformly distribute sphere l unit separate k comprise normalization specifically least cumulative cdf j argument display inequality hold replace eq define tensor next lemma power method throughout define tv combining read overlap case first case split r sub claim last turn universal I consider three phase use ii iii remain phase iterate hence also hold iteration moreover I rate r claim rate iteration count satisfy imply induction quadratic therefore hold lemma replace check iteration argument
numerically calculation weight event histogram weight determine square candidate kernel could continuous central position practical usually significantly bandwidth subset provide description time find forward take determined minimize expression measure linear degree center evenly spaced start forward step fit consist try give large reduction find procedure stop otherwise include satisfy stop include fit fit positive take increase certain threshold backward step iterate reduce remove try forward threshold opinion stop band around bin integrate interval element singular bin histogram define physical compact error unfold result shape singular bin comparison span case expand unweighted fit indeed underlie unfold degree constitute theoretical properly ignore practical sure good vary function discuss scale quality residual estimate normalize residual quantile position kernel consider significantly width avoid number least bin leave unfold scale parameter function general lead bad fit observe narrow feature favor statistical fluctuation therefore try range overfitte region value satisfactory fit literature leave good method account uncertainty relate also experimentally resolution bias gaussian resolution function fig show measure simulate event distribution show determination monte weight event proportional position analysis scale normalize residual represented weight seven kernel list quality superposition kernel residual plot histogram measure histogram weight unfold illustrate unfold unfold true distribution range vary bad fit also significant quantile small range confirm simple bin measure calculating predict residual content calculate exclude bin unfold determination select unfold parameter minimal statistical error overfitte sum interval band p cm produce statistically independent histogram calculate unfold bin bin make bin independent bin bin q bin bin bin eq unfold total q root characteristic unfold bin gaussian kernel error unfold agree actual finding bin bin unfold result one observe bias distribution value become illustrate complementary grow tend rmse cm cm cm cm bar deviation histogram bin bin show strong region determination unfold datum unfold ill solve solution type smooth act regularization cross integral unfold source solution unfold criterion discuss unfold unfold example property validate execution unfold ghz spectra multidimensional handle properly determination grateful discussion careful reading manuscript ng physics box due resolution detector unfold know ill example smoothness positivity kernel act cross determine parameter numerical simulation statistical unfold problem pdf characteristic
guarantee kalman define unbiased positive partial order define define sampling period order otherwise force dynamic trajectory input solve everywhere time differentiable differentiable derivative mean twice qualitative mean modify filter scheme trajectory make propose modify scheme input scheme scheme piecewise take input piecewise regression improvement identically distribute bound indicate suppose measurement correspond even right parameter entry similarly diagonal side lastly ready unit zero everywhere else leave side filter advance first leave vary computed state estimate give sure try exceed filter pointwise polynomial r strictly speak result apply filter imply perform much discuss intuition measure filter preserve property state bound assumption filter weighted implementation computing choose filter empirical give compact replace leave sided derivative time derivative consume filter implementation compute filter coefficient compute filter provide know dynamic trajectory se mode exist nx nc triangle inequality first term give assumption preserve composition subsequent issue domain occur lie define use pr showing probability convenience far show give kernel c q simply term act ensure lx lx lx v n continuity whenever proceed analogously immediate noiseless intuition correlate instrumental nx pr variable k corollary pr another u pr nx I I uniformly converge convergence series setup often term define deterministic evolve time know se ergodicity hard general se ball center satisfie assume se know true approximate result noiseless se approximate arbitrarily nx g estimator nx iw I triangle setup match interior point cover consistent control second govern would always system explores formalize always exist construction limit subsequence visit limit exist correspond trajectory theorem control oracle converge se appropriate estimator note drop base upon research laboratory agreement air office research agreement fa provide identify rich model system technical proof reason measurement give proof concern simultaneous state dynamical ordinary ode different model prove nonparametric design deterministic numerical implementation distinct part section concern state ode concern control technique assume vector dynamic ode matrix appropriate describe dynamic control input generate piecewise appropriately design
construct confidence plausibility attention complement assertion optimal I something need proposition use desirable set accomplish iteratively intersection idea order equality collection nest equip admissible predictive since eq corresponding plausibility I event treat want break validity follow impose respect vanish include expectation rank suitably balanced remark accomplish numerical parametrization natural representation e respect substitute order choose high rank fact absolute unfortunately satisfy formal recursively permutation achieve optimal describe substitute expression negative contain element algorithm implement stop side positive experience occur tolerance initialize r e k r r r approximately hold close plot indeed realize fluctuation seem zero maximize fluctuation row plausibility belief plausibility classical testing versus size plausibility procedure somewhat less iff plausibility treat random look plot exceed demonstrate I plausibility function criterion thing look stochastic plausibility stochastically former panel I tend outperform approximation plausibility dominate gray line plausibility I plausibility plot plausibility frequentist plausibility various general normal plausibility peak describe plausibility plausibility confidence interval I plausibility unlike I plausibility coverage see confidence interval sort mis model xx solid gray plausibility solid black I plausibility variable challenging consider review introduce I frequentist develop count comprise independent establish mean fact ignore constraint positive I observe must lie consider intend case large mass c constraint belief case constraint validity elastic formulate detail boundary plausibility point imply move predictive recursively refer method balance plausibility black line gray plausibility interval short variety confidence adjustment ms plausibility interval function column winner nominal interval describe plausibility wide must frequentist procedure say remarkable plausibility htbp arise develop theoretical computational contribution construction approximately predictive develop handle challenging high herein part challenge problem binomial handle optimal focused primarily compare frequentist performance plausibility derive tool develop procedure probabilistic summary inferential meaningful singleton hypothese exist general satisfactory probabilistic point help topic identify speak dependent large predictive belief validity bias make I plausibility value difficulty interpretation however meaningful I interpret plausibility claim acknowledgment national science dms dms dms poisson fundamentally problem application physics challenge even large approach construct approximately poisson sort background belief predictive random score validity discrete datum fundamentally example problem physics datum frequentist method reference bayesian popular conceptual also suffer prior inferential challenging handling primary statistical knowledge consensus reach desirable inferential measure uncertainty observe logical way bayesian function plausible hand frequentist hypothesis procedure justify ii error pre assertion majority mathematical definition frequentist interpretability frequentist fail satisfy frequentist application take property carefully reference guarantee fisher theory calibration ii I build inferential reflect inferential ii continue regard strategy develop I review reasoning framework moreover output I poisson naive analysis brief introduction I mean namely side develop I construction recursive construction I side translate directly estimate poisson directly distinguish comparison compare frequentist mind tool procedure fisher build framework monte plausibility note predictive set optimal recommend predictive section particular belief function assertion I stochastically validity property description property value validity whenever predictive consequence I versus testing rule control frequentist plausibility unknown invert plausibility nominal plausibility frequentist plausibility meaningful free probabilistic claim sharp interpretation herein valid want validity belief subset agree assertion probability datum show admissible bound happen nest admissible bind say I result assertion exist admissible
dynamically mainly item frequency advanced enumeration cube monotonicity anti exploitation solver monotonicity property constraint significant previous relation frequency generalization enumeration option equal frequent mapping problem return maximal start frequent reach short maximal eq adopt expressive enumeration strategy anti maximal valid undesirable consequence search optimum potentially need particularly critical database maximal frequent maximal order relax frequent early type frequent search frequent happen clause find frequent repeat search next cm benefit guarantee maximal iteratively length low restriction maximal add form however solver support addition removal pseudo boolean add simple initially maximal frequent fine enumeration option encoding option different boolean encoding support removal clause additional deal cover encoding description additional clause vector search clause satisfied search item belong assumption solver solver removal closely maintain refer incremental clause encode although clause clause add clause grow encoding clause part reasoning outside solver clause maintain solver finds learn clause need satisfy clause strategy storage reasoning iteratively binary clause option item assumption control variable search option distinguish search search encode boolean solver removal clause clause implementation need adapt turning solver solver interface removal clause allow interface solver deal depict option introduce either adapt encoding result medium clause I adapt solver suggestion find set maximal fine suggestion variable wide variety factor dynamically attribute scope sensitive manner decay variable activity activity dynamically tune finally solver resolution adapt sensitive transaction item focus improvement must support addition flexible constraint evaluation option cp property observation retrieve support encode expressive subset cover enumeration option anti imply growth advanced search support introduce solver option one previous flexible interface efficiently solver synchronization parse low frequency solver comprise solver open cover option source simple iteration support assumption clause although solver try learn clause turn impractical adopt solver belong solution support uci repository transaction divide suffer scalability nature therefore intel ghz use operate implementation propose transition low high threshold fig fine solver natural option suggestion decay propagation remain locally maintain remove value domain never satisfy constraint solver deal summation summation constraint deal constraint weight summation discover early whether summation boolean target occur update item possible activate support simply summation branch frequency clause clause justify implication impact solution retrieve dataset nominal attribute label attribute threshold depend near case frequency exceed enumeration aim verify frequency formulate tune base enumeration strategy compact suggestion transaction orient adopt search strategy accord search adopt suggestion good frequent among transaction suggestion guarantee discover major obtain large frequent value frequent otherwise prefer although maximal translate generic translate implication distinguish cp type constraint stream apart simple illustrated ii pi p r r solution implementation large need transaction partition use integration frequent find voting constraint gain measure formulation technique stream pm cp user define novel combine constraint axis predefine expressive constraint variation briefly contribution expressive transaction involve build purpose pattern pattern direction account pattern produce pattern importance pattern devote adopt cp highlight come association due task cp encourage heuristic value extend adapt representative model frequent pattern use principle reduce redundancy maximal frequent pattern pattern variety concept conceptual cluster frequent I attribute stream formulate ip research boolean rough building classification verification accord solver example task multiple representation interval pair interval exist b database arbitrary frequency expression additionally probabilistic use mining expressive formulae option enumeration encoding important adopt association perform efficiency problem high fact clause understand tune solution extension evaluate impact efficiency discriminative exploitation limit intensive rule tune nature density item item transaction transaction exploitation adopt enumeration search level expressive constraint besides anti monotonicity extension dataset require discretization beyond vector boolean reasoning frequent structured new assessment necessarily rely base additional knowledge certain cluster wish relationship mining tree exploit cp verification monotonicity rule advanced bound rule prune speed turn cp option partition stream stream decade mining programming combine manner state formulate boolean frequent mining multiple task drive enumeration option although majority scenario appear non competitive cp cross pm large cp programming wherein variable state greedy pm artificial intelligence develop highly tailor towards specific cp generic high modeling language solver rather pm domain pm count shape minimum efficiency bottleneck cp solver deal option frequency though cp solution scalable pm expressive user drive search nature lead improvement wide diversity cp solver solver boolean evaluate although commonly adopt pm pm discover pm need efficient cp solver extensive cp behave pm section introduce enumeration search option describe implication review finally conclude remark research item call transaction transaction simplify language ti e p item product easily convert database pair item discretization convert row map transaction occur coverage cm consider database pm database frequent frequent place database fix core task pattern classification propose et pm powerful across variety application adapt traditional accommodate easy change support selection drive cluster background knowledge spurious interesting mining drive mining represent support variety expressive impose relaxation pattern bioinformatics cp like already support expressive cp cp task variable transaction assignment transaction transaction define set transaction cover frequent restrict assignment neither fix result different tuple database transaction mine coverage frequency constraint provide essence modeling language current art formulation rely adopt approach dedicate solve boolean scalable solver decade solve thousand clause potential formula clause handle solver encode enumeration alternative option cover map cp clause constraint adopt transaction encoding formula derive equation equivalence mm cm consider transaction binary clause clause cm encode extend clause cardinality need adapt focused transaction transaction encoding formula cardinality ti mapping formula boolean sequential diagram sort
output moreover avoid computation joint drawback formulation kde input overcome generalize allow kde input value step kde operator value rather scalar input exploit describe kde choose kernel recently deal link kde since step rkh space provide structured work base structure evaluate kde paper value joint space output kde data mapping structure output value learn input kde kernel kde formulations ridge learn drawback take dependency datum feature mapping retain output propose value kernel build output encode relationship output component output kernel allow kde kde take feature contrary formulation operator kernel kernel detail table summarize value kernel negative x j adjoint kx value kx value call reproduce input space scalar operator value technique function kde formulations kernel ridge substitute analytic block substitute step structured explicit quantity readily computable output compute input follow correspond operator inherent difficulty problem unknown kernel trick solve image problem thus trick trick implicit mapping material trick pre th express note form scalar formulation kde building suitable output correlation operator kde order advantage kde formulation dependency objective learning kernel functional functional consider value discrete continuous act scalar value operators rkhs receive considerable simple dependency successfully play kde empirical operator last scalar value operator value operate encode specify belong separable beyond separable value address propose conditional define operator kernel correlation effect supplementary material pre suitable tensor jx value product interesting work reduction work tackle operator main view believe reconstruction optical value kde second problem kde lines digit apply kde compare rbf input ridge try yielded performance perform digit select randomly handwritten pixel digit fold testing induce parameter kde rbf kde kde covariance kde conditional kde method value handwritten value promising kde explain fact kde kde kde formulation formulation dependencie contrary numerical output optical subset handwritten mit language character binary pixel representation consist word base handwritten character table experiment measure correctly recognize cross character polynomial third character propose feature mc vc vc py maximum length except easily separately character cm c kde kde experiment operator operate output output consider input allow study output learn dependency kde formulation value kde recovered kernel kde take interaction output input address issue introduce effect structure problem state art well observe conditional covariance covariance kernel another future base structured generalize l reproduce straightforward reproduce implicit feature adjoint symmetric sort increase almost eigenfunction orthogonal dot delta orthonormal basis detail kernel form compute e eq adjoint value yy kx yy yy define variable rkh l ii idea eq take form abc come conclude gram pre image computing compute large operation matrix incomplete reduce acknowledgment thank helpful suggestion de op situation mapping object characterize structure statistical synthesis algorithm structural becoming increasingly extend refined sophisticated need handle output difficulty arise increase depend class value euclidean structure date develop output output output deal scientific community little output recently effort
simple sequence density hasting alternative draw sample pdfs metropolis adaptive however though proposal adaptively proposal converge propose adaptive mh analytical whereas shape become finally eventually produce quickly draw pdfs technique available expense organize section alternative show section letter e letter denote unnormalized normalize proposal pdf whereas unnormalized proposal unnormalize general unnormalized pdfs adaptive rejection construction tailor appeal reject improve mean acceptance unfortunately pdfs unimodal improvement propose pdf normalization procedure time grow index piecewise construct possible construct tangent involve straight pass suitable linear density also piece functional straightforward well since piecewise easy draw reject e approximation one construct table build accept sort stop else back include indicate proposal discrepancy approach ratio close ensure cost htb tangent point line every reject hull construct well approximation close expect acceptance denote respect pdf define distance curve pdfs acceptance w tx vx px everywhere disadvantage density concave pdfs extension deal pdfs log concave metropolis mh rejection arm discard improve standard however sample accept another statistical guarantee accept distribute sample arm arm describe whereas pdfs step rs add point improve sample initially rs happen determine whether accept notice arm reject accept support rigorous mh omit see auxiliary observe link construct pdfs fact issue study procedure suitable markov build discard otherwise px k tm stop support straight line arm introduce piecewise form q htb important new point inside affect adjacent region therefore subset proposal practice pdf slow general target concave form drawing proposal require know straight see might suggest entail point contiguous straight construction without form piece use quadratic order proposal otherwise build well limitation arm technique univariate pdfs satisfied proposal stay target much procedure improve everywhere indeed classical pdf neighboring interval building allow improve even inside proposal interval drawback point eliminate show example remark guarantee except special structural limitation region inside x px clearly illustrate problem arms pdf multi modal show point tangent construction within c support contiguous vary within pass incorporate arbitrarily figure within figure b sequel construction depend straight pass pass look apparent point add inside contiguous interval c limit incorporate close locate line adjacent interval tangent show minimum tangent line straight tangent moreover tangent stay pass meaning even therefore standard arm entire pdf example discrepancy target trivial drawback arm unfortunately problem unbounded major pdf reason standard arm ensure keep cost low strategy improve arm two variant w arm proposal instead clarity technique idea possibility follow inside decrease evolve grow stop adaptation convenient simple draw sample tx tu tm tm tm tt remark point arm accept rs acceptance always accept never incorporate point reduce happen code implement unchanged calculate control step proposal pdfs difficult chain adaptation observe coincide issue chain current balance could good proposal remove final return follow fact technique firstly provide incorporate tends effectively vanish new proposal close close pdf arbitrarily algorithm support unfortunately guarantee although good simulation arm adaptive arm indicate build pdf use possibly account support convenient procedure e one draw tx tm kx tm tm lie proposal independent mh technique markov invariant procedure already calculate density improve arm density also subject restriction arm good support add computational cost easier inspire straight go extend straight line towards minus plus mathematically although look eqs simpler involve calculation one moreover procedure construct piecewise line mathematically proposal e uniform figure construction describe concave log since guarantee except first modify tx vx log target mode upper location depict obtain produce concave finally build structure completely arm domain pdf basic straight point piece tail unlike area indeed follow simple calculate pass tail construction use draw pdfs interval htb pdf domain procedure computational arm type remark could inefficient arm without due explain exception lead combination probably stay depend possible initial procedure describe avoid speed convergence avoid tail proposal initially construct attain proper proposal pdf add tail algorithm tail second arm incorporation support know kind tail reduce burn period proposal build instead straight arm build pdf arms eqs build pdf mixture q mean estimate distance proposal iteration form inside proper therefore inside note adaptive construction proposal slope tail scheme never table show chain build correlation consecutive always discrepancy pdfs std avg avg avg estimation second std estimate avg avg rs std avg avg support rs control table scheme decrease minus plotted mean propose arm perform h b figures eqs figure figure notice iteration two variant arm propose standard arm greatly improvement expense support moderate increase bind slight notice building strategy propose original third quickly procedure support great incorporate region piece pdfs curve shape procedure slowly second always support fourth arguably provide density efficiency proposal relationship approach proposal pdf see discrepancy speed convergence automatic pdfs good approximation issue complicated target approach building rejection arm important arm introduce allow draw concave metropolis log pdfs multivariate within univariate distribution gibbs highlight approach build limitation arm inside prevent converge alternative arm scheme target statistical direct inside adaptation one adaptive mh previous sample ensure measure computation pdfs numerical approach arm accurate consecutive pdf issue two guarantee discrepancy target pdf quickly imply proposal close mh parameter pdf ensure r increase due incorporation support inside moderate remain effort implement remain control arm evaluation proposal pdfs finally notice technique reduce construction slight test arm important fact dedicate introduce procedure piece plus pdfs tail standard arm efficient draw directly piecewise approximation pdf among partly author thank di iii de discussion
identify event put together event observe trace equality present eq keep use gradient generalize number neighbor I rare zero element similarly element correspond scan usually furthermore arrival enable arrival estimate employ last assign fig word reconstruct spectrum notation fire stop criterion meet receive electrical stanford work statistical signal message pass bs university physics theoretical post de th de sup sciences research institute berkeley usa centre de la stanford associate electrical award receive national science award research receive university electrical experience modification conventional pseudo trace overlap little currently employ hardware highly trace efficient massive computational expect domain mass mass ms family chemical application technology protein protein biological interest biological sciences medical development identify analyze protein chemical protein throughput include measure exploration technique configuration mass utility application consist three module possibly advance phase open analyze mass module common force broadly effectively stream toward mass particular stable mass act scan wide band module detect detector surface motion benefit alternative technique essentially mass rate advance range basic acceleration drift detector electrical acceleration effectively fire region drift proportional acceleration hold mean therefore reach detector proportional technique accelerate drift impact detector continuous sample discrete call noisy scan repeat hundred thousand many consecutive chemical reaction analyze precede automate fed automatically metric describe mass mass sensitivity speed mass specie specie mass refer detect acquisition per trade among metric speed accuracy sensitivity acquisition consecutive e scan detector fast acquisition eq drift region acquisition far apart increase factor detector characteristic speed reach mass remain improve mass length drift region obvious requirement sensitivity could restriction monitor chemical reaction use mass significant implication hundred thousand improve speed keeping improve automate drug simultaneous speed fundamental conventional choice mainly simplicity particular sophisticated technique resource generate mass power trying call subsequently accelerate drift detect hadamard reach detector frequency ideal shot output signal detector shift index long spectrum spectrum deconvolution implement efficiently hadamard inversion ignore require substantial modification hardware accelerate mass improvement hardware simulation real improvement scheme detector scan abuse scan process average obtain accurate let scan infinitely multiple overlap scan find vector transpose matter convention refer spectrum bin represent detector generate detector refer short rate response one bin describe without minor modification present simulate use algorithm single scan electrical spectrum expensive perform drift scan repetition n incorporate powerful scheme f fig require collection scan overlap subsequent trace avoid overlap consecutive relax random superposition q superposition overlap total firing consider adjacency matrix bipartite row bin scan bin scan trace refer version row case measurement observation bin reveal adjacency ls reason lie approximate shoot signal similar arise limited shoot limited regularize likelihood stochastic optimize regularize bin span algorithm log log attempt ff presence chemical iteration let cost equation right part negative calculate gradient follow firing constant estimate adjacency repeat criterion step worth note mention bin spectrum fig spectrum thought event cause problem terminate slow confident generalize instrumental spectrum chemical scan length interval average ten evaluation fire prescribe pass marker level indicate start marked event corrupt henceforth trace unless mention across electrical additive electrical detector mark high select potential potential impact event event exceed peak height event trace support valid preprocesse trace event whereby trace scale amplitude purpose identify enable evaluation constant event match negative match exist match say match negative rate tp tp rate fp fp ill false metric width pose state ratio acceleration four fast satisfactory accuracy false factor ten false proceed negative rate value discovery establish iteration fdr design firstly require discover truth rarely consider costly mistake miss line type fdr converge ground reconstruct inspection match vs fdr bottom admit peak peak ground truth difficult detect right plot list peak pick practice publicly available peak magnitude spectrum less tp peak pick estimate peak value slight abuse peak match within four curve acceleration acceleration acquisition acquisition pick truth spectrum list pick pick amplitude choose pick run pick peak ground truth significant peak peak ground estimate top peak achieve perfect reconstruction acceptable inferior acquisition outperform fast achieve decrease peak match fig top peak fdr since detect performance remain unchanged dash solid acquisition peak pick factor cdf intensity ratio peak peak understand achieve sample confidence show vs false acceleration ten construct nine ground interval standard obtain mapping spectrum adjacency simple trace equally cause curve conventional e obtain perform acquisition time long k
thank anonymous remark weak assumption fourth constant consider analogous b apply fix kl n provide high q sufficient schmidt calculation assumption conclude markov derive previous apply except show kl lead define r center x v functional translate design eq conclude proposition adapt regression supremum nr x inequality conclude bind derive cn cn markov derive lemma shall combine assumption derive k investigate take expectation upper derive observe derive n n first eq integer k g note freedom kx x x kx kx derive desire result finish denote freedom k kk k proof get apply since k eigenvalue france universit real combine test analysis interestingly drive knowledge slope level assess slope range numerical linear constant denote intercept represent slope function center random variable lack effect test widely literature generic procedure support address dimensional belong henceforth endow sake clarity thus center independent unknown vector stand essence testing general originally briefly review result minimization square penalize plausibility spline estimator thresholding hilbert functional principal span previous estimator reference therein property estimator rate convergence aforementione recently prescribe basis e spline estimation rely tuning quantity variance fact smoothness operator unknown literature test functional statistic functional drawback set arguably issue apply bootstrap asymptotically paper power viewpoint non test correspond projection power section mild additional sharp art principal rely perturbation theory result rely section power power close problem testing small distance powerful minimax achieve separation distance formalize minimax control functional wide class optimal suitably unknown choice quantity regularity regularity lead poor performance issue minimax simultaneously regularity testing viewpoint combine test test technique spirit require level power analyze viewpoint section test minimax distance involve procedure illustrate simulation discuss involve perturbation give technical refer hilbert euclidean product covariance trace schmidt basis increase eigenvalue correspond sequence eigenfunction sequel line expansion pca tool functional expansion uncorrelate exist center uncorrelated principal amount eigenfunction eigenvalue unknown empirical covariance sequel functional usually technique appeal variance core procedure seminal convergence assess asymptotic tuning besides plug create usually introduce dependence sequel hypothesis dimension expansion introduce define vector intuitively proxy reason stand orthogonal main classical projection kl j nj orthogonal instead behave statistic least prove numerator correspond norm statistic statistic et al consideration transform naturally unchanged replace crucial synthetic functional white exactly normally distribute admit fourth moment error noise fourth constant jj assumption since operator come behavior fourth moment truly mild brownian sequence exponential j restrictive restriction dimension projection classical eigenfunction dimension eigenfunction become difficult dimension exist constant proof show degree argument rely normal assume belong advance span test extension belong span capture onto probability regularity look precisely consequently convergence regularity power trade term assess optimality procedure assume regularity ellipsoid regular positive denote supremum ellipsoid number correspond minimal hypothesis ar tt separation infimum quantity minimax separation ellipsoid notion separation distance boundary ellipsoid exist process ellipsoid upper upper bind positive define minimax consequence separation multiplicative constant interestingly separation distance behavior increase regularity regularity quantity sequence separation test exponential decay achieve reliably estimate eigenfunction lead restriction art condition conclusion achieve optimal regularity take zero lead large good correspond term procedure achieve trade prior regularity dyadic priori discuss kl test accord one refer correspond hoc test approach vector independent conditionally conditionally simulate conditionally work carlo choice obtain appendix require take size small let admit n counterpart multiple counterpart constant proposition gaussian test power minimax optimality rejection pay replace term appendix us sequel stand integer logarithm constant direct minimax satisfying term ellipsoid minimax collection nest constant infinite ellipsoid consequence price pay simultaneously consider collection adaptation framework compare low decay separation proposition decay separation distance interestingly decay experiment brownian eigenfunction operator brownian use form evenly spaced testing experiment compute carlo quantile smoothness stand explicit fix smoothness dirac center trial rejection ie confidence procedure implement ghz intel processor cache gb cc c c cm pay correction furthermore become smooth perform third table correspond sequence become dirac test evaluation quantile computation second confidence cc c cm cm c cm cm cm h cc cc cm cm cm cc c cm cm cc cc cm cm cm two testing completely drive assess address focus extend testing v nj kl kl th behave p degree situation kl ph slight monte state describe nonparametric unknown well prescribe fourier decrease slowly project necessarily suit prescribed spline eigenfunction procedure fact randomness design prescribe also unknown combine procedure emphasize perturbation integer jj orthogonal projection onto span translate analysis I nk j correspond sx class operator numerator statistic square estimator generate n proposition adapt proposition conditionally size conditionally small respect quantity prove appendix operator replace definition use replace three behave distribution rely theorem tell prove appendix b kx hold statistic three n union rely condition enough appendix b deviation turn order conclude derive term assumption
alpha old split problem need monte carlo alg propose transform mdp policy dealing representation must map choice combine let sub original action binary mdps mdps denote define mdp knowing mdp vice versa original mdp environment binary operate mdps mdps binary light training try binary policy manner describe sub problem mdp increase problem complexity sub f except instead action sampling see choose action call action state decision different sense mdp equal noise correct split study present respective define spend multiclass one spend current spend monte cost classifier come multiclass cf cost reduce policy mdps since possible learn binary cf result number show addition illustrate complexity computation really assessment turn day computation car ht number x axis first line second rl vary discrete action experiment obtain set range performance reward episode length quickly second problem grid correspond agent mdp construct action set define contrary car notion consequence agent reward avoid reward small action intractable action learn particularly bad state problem number trajectory hinge perceptron iteration code random procedure alpha reward three figure action average reward first similarly strategy involve binary classifier performance indeed learn baseline intractable non trivial policy approach intractable car action consider explain performance set require last figure number r cccc car sim algorithm rl mdps monte phase costly couple lead theoretically assess efficiency method reinforcement generalize leverage deal space additionally behave impose binary search space mdp thus place solution value rely solely although dictionary key rely code actual hierarchy output mdps learn obtain classical implementation optimal policy work depend automatic adapt particularly theoretical plan relation fact deal thousand study international conference reinforcement learn scenario rl hundred sometimes rl obtain multiclass output code set reduce first classifier classifier dictionary mdp far set tractable finish experimentally demonstrate rl mdp rl learner environment action though understand rl face many dealing generalize large learning explore difficult regularity smoothness action great go planning problem action regularity knowledge regard consequence action sub draw idea multiclass assigning bit know real world problem propose computational leverage idea learn allow optimal mdps first reduce even brief overview rl section using explain mdp factorize accelerate analysis relate work cover key classification code tuple transition function choose expect immediate immediate article adaptation formalism multiclass integer binary code multiclass classifier predict transform supervised infer pass hamming look tree contribution part building begin show integrate around assign action one action binary action code code combine policy original want code action
co co et co co stage find propose approach fitting coefficient select tool refine improved efficiency description dependence application real interesting dynamic ar select fit bic discard ar explanation increase sufficiently inclusion similarly good bic illustrate construct coefficient dimensional var eq iid show stage fit summary define likely procedure use truly conduct correspond parameter execute modify ratio refinement truly exclude inclusion substantially discard pre specify replication refer modify bic refer bic panel compare compare procedure zero coefficient original one forecast bic fit include oracle bic raise implement var lasso var column stack ar matrix usually constraint matrix specify ar rank constraint ar var illustrate consider var express equal non ar estimator ar equation product l know coefficient parameter estimation therefore iteratively ar lasso fitting var ss var compact product l iid minus var ignore additive constant penalize choice ss log tuning parameter control var estimate respectively sum residual minus log likelihood lead method var derivative ss lasso function ar entry matrix lasso lasso var model ss since lasso fit efficiently angle lar descent ss var complicated involve iterative var lasso cast q eigenvalue decomposition define root target residual variable explanatory target var pt iterative var th ty fitting penalize var choose tuning cross use determine explanatory order determine drive manner take fit ss lasso var lasso var apply coordinate descent minimize aforementioned iterative ss lasso model parameter give minimum average ten give minimum average either lasso var ss acknowledgement would thank air science dms part support nsf grant google award section autoregressive var widely dependence series unstable overcome stage var many ar ar partial spectral together quantify far illustrate datum google trend levels air autoregressive sparsity coherence autoregressive model temporal multivariate series consist serial dependence series suit var ar moderate prediction interpret drawback sparse interpretable description popular penalize problem determination use lasso penalty tailor purpose lasso advantage selection estimation small setting tendency var formulate regression series treat variable explanatory theoretical temporal dependence response explanatory fitting stage screen conditionally conjunction information contain spurious far refine fit stage screen strategy ratio organize var section procedure var model stage causal lasso stage trend air contain supplementary pp interpret far pa without generality mean var matrix var want ar need dimension large prediction interpret description also sparse preferable fit sparse var ar develop fitting stage select ar coefficient correlate free correlation frequency specifically coherence introduce correlation effect removal effect result linear k opt another marginal correlation residual distinct conditionally uncorrelated lag lag residual cross density reflect correlation marginal spectral coherence distinct marginal scale residual series show computed spectral process via invert dimensional marginal challenging simultaneously f spectral coherence compute one spectral density encode pairwise conditional series selection sample e concern relationship zero distinct conditionally dimension entry replication spectral density remain conditionally use inverse pair series conditionally uncorrelated pair ar lag fit ar possibility use refine indicate correspond conditionally stage pairwise var particular ar belong lag rather individual connection ar approach achieve return zero identically correspond conditionally uncorrelated word need separate depend quantity summarize different modulus summary statistic supremum indicates conditionally correlate therefore rank summary high low way add base var select var select ar control ar coefficient use bic simultaneously bic eq maximize estimation var detail find appendix restrict specify respectively stage summarize stage distinct marginal inverting density construct distinct marginal ranking select sequence choose give shrinkage adopt obtain ar coefficient much fully parametrize var first stage execute group model examine employ determine introduce g examine multivariate question ar causality series causal relationship shrink e propose group purpose conjunction bic discover ar coefficient reader refer correlate marginal bic however spurious zero ar stage model pair group refine eliminate spurious ar rank absolute non zero constrain estimator column stack coefficient matrix estimator stage triplet absolute ratio high ar retain complexity ar retain bic contain pt zero coefficient triplet high low consider select coefficient top triplet execute appendix correspond bic bic numerical stage approach stage compete stage google trend concentration air lasso var formulate select ar ar loss choice square residual minus affect ss coefficient estimate mse mse ar ar estimation panel figure replication ar ar ar color shows correspond ar see panel belong series stage contain spurious ar coefficient position I circle panel c stage refinement implication first circle stage select true coefficient probability circle panel approach implication show stage correctly ar coefficient panel ar coefficient lasso ss medium circle circle ar unbiased legend circle approximate truly var spurious consequently interpret tendency select ar consistent bt true ar coefficient panel circle percent replication ar compare impact display ar coefficient approach well remain variability stage matrix therefore adjust change lasso var interesting change variability lasso lasso change panel increase circle first suggest increase ss increasingly temporal spurious method influence change variability ss ss result lasso address lasso benefit lasso ss bt bt stage method respectively horizontal ar stage past ideally coefficient stage lasso much ss increase estimator change bias significant estimator nearly vary systematic lasso datum view notice many researcher internet activity see researcher top google select produce google trend google trend center surveillance visit google data surveillance first trend report possibility forecast google google google trends fine report google trend throughout surveillance national fit trend week region north south due simplicity apply range stage parametrize bic stage coefficient far select ar var ten cross var non coefficient lasso x refer ar coefficient retain curve bic varie panel dash minimum compare structure discover ss ar interpretation region surveillance estimate diagonal reasonable activity previous week week region diagonal var signal var hard interpret lag dominant ar ss since description structure model discover activity ct nh indicate panel within moderately lag cross co mt la nm ok general lasso zero large absolute ar lag var three week week quantity ahead root square h forecast summarize forecast
site global vary specify conceptually specify comparison distribution option explore note record match status purpose consider count agreement individual sum within alternate specify choose repeat distribution draw posterior draw independently specify stop converge distribution decide record link review leave record record cut assignment group examine cutoff value potentially record equal small file size divide structure draw cutoff factor transform sample scale pair large match simply one draw multiply specify distribution latent class one beta could transform scale proportion approach implicitly exchangeable match specify field vary independently probability across block without strength block parameter separately insufficient matching agreement version allow correlation could restriction number across surely poor block match indicator simulate sampling conditional hyperparameter matrix distribution assign match initialization option randomly represent match per assign cutoff cycle draw converge target indicator value enforce constraint eq indicator comparison block hasting hasting hyperparameter draw hyperparameter hasting bernoulli stop converge converge suggestion make metropolis hasting step use hyperparameter calculate sm sm hyperparameter value follow step outline index replace parameter hasting outline index need tuning parameter accept approximately adapt tuning second tuning replication perform two condition across block block link together correctly estimate file record organize yield record match seven probability match increment agreement among information depend acceptance hasting step performance complete linkage reflect simulation area work interesting census health automate apply file would regard specification record linkage available linkage site site way decide site discount analyze site file challenge block study cycle gibbs sensitivity specification one indicator restrict future metropolis hasting new instead acknowledgment work contain report partially support grant selective author necessarily national security university else zhang linkage author north hill national center health census present record linkage w record linkage international exposition office management budget j base estimation record variation linkage factorial survey methodology match record linkage journal american statistical monitoring iterative simulation incomplete via journal american association b record linkage model record linkage bayesian survey method f calculate marginal association simulation h sequences stochastic gibbs image transaction intelligence hasting monte record linkage york science business media advance record methodology apply matching census journal american linkage health file link american association imputation record survey record match census record census comment american york city l new york hierarchical linkage department record linkage national death journal iterative record linkage american statistical association latent linkage link american research york discrimination dependency linkage model record linkage american statistical association research linkage census conference advance linkage american method match record linkage ed cox p new york pp record linkage rl file matching identify entity file include count survey study file identification file name find score status come comparison prior class rl key metropolis linkage rl name activity individual one rl file record correctly associate two survey frame survey longitudinal study file rl activity review subject file identification file error rl trivial unique one name address record one file comparison comparison record initial file requirement outside block much evidence favor pair could imagine take determine statistical supervise analysis logistic fit score status unsupervise model group initial rl hierarchical rl impact distribution current study vary across block hierarchical use inter heterogeneity development study rl error rate record linkage section report file file correspond file contain identification variable file similarity record pair agreement file base birth name house name age birth file name standardize accord address standardize separate day month pair consider field record equal agreement disagreement many typical informative location suggest match record determine evidence census file group record well reduce
estimate three ad quantile model high rmse consistent high percentage regard copula constrain quantile ht htbp quantile cccc cccc copula ht ht c ad patient drug blind clinical period laboratory raw data transformation heterogeneity dataset apply log consequently post adjust median quantile equivalent laplace zero scale find indicate association baseline whereas baseline model combination value low e baseline simulated baseline sample transform box cox relate modelling firstly implement univariate describe variable stem observe high post period trial pre period tail simulation incorporate result univariate find effect estimate quantile unconstraine proceed joint simulate post baseline population fit dependence assessment estimate via regard order make justified significance hypothesis conditional exceed threshold output nan order strong approximately order level quantile appear quantile figure order maximum majority order change quantile conditioning place state take survival early simulate post baseline subsequently constrain equation replace likelihood estimate produce simulate simulated survival uncertainty variable induce change survival ht behaviour tail order quantile standard laplace dash vertical correspond statistic row mm third column conditional quantile jx yx solid dash grey estimate model identify assess however insufficient duration clinical trial tail laboratory relationship attribute strong dependence methodology formally assess view behaviour test dependence baseline adjust formulation build extend order trial bound additional mainly asymptotically evidence tail behaviour predict high modelling attribute use survival indicate order constrain modelling primarily baseline quantile propose order effect come remain longitudinal order consecutive acknowledgment program ep mathematics discussion constructive comment constraint dot constrain explain find profile surface conditional small quasi highlight joint affect quantile area constraint join mostly quantile near line show bottom side join boundary constraint induce small quantile area induce quantile constraint section order large tail complement estimator composite additional maximum generalise distribution stage simulation necessary approximate derivation variable analogously outline conditional quantile limit copula second comparison independent coefficient model tail dependence constrain refer ht stochastic performance assess denote monte conditional
versa denote dags arc either dag arc lead variable variable arc arc model arc model network model choice class integrate smoothly arc bernoulli marginal among simultaneous multivariate specific specific find know expectation immediate covariance interesting basic minimum follow close q construction multivariate bernoulli multinomial effort property cover contingency analysis variable tt uniquely identify eq moment two bound prove either constrain segment axis maximum deviation equals reach maximum bound eigenvalue lemma family dag let appendix component probability associate derive distinguish correspond among mass possible configuration identify non informative posterior set useful identify clearly dags useful reference determine arc probability one configuration edge arising property display informative behaviour value derive section consider arise behaviour explain tendency represent relationship underlie almost surely exclude edge consider good closeness multivariate intuitive consider simplex point boundary origin intuitive entropy however model point comprise edge three edge brevity covariance position show entropy due increase difference eigenvalue fact nearly strong close axis distribution correlation clear multivariate similar bernoulli many reason acyclic arc dag direction influence arc even close form make distribution obtain reverse direction arc another every acyclic immediate include arc dag include arc dags direction arc arc arc arc arcs q expression arc arc arc belong arc additional replace undirected combination belief arcs direction uninformative case expect covariance arc converge arc equivalence direction associate way order arc drastically reduce free arc numerical arcs dag dag arc maximum quality examine possible dag value theorem compute dag size generate probability clearly figure estimate limitation easily via enumeration quantity evident absolute edge absolute interpret arc ij marginal arc remarkably graph dag node arcs approximated probability graph motivate case outcome contribution presence direction imply arc direction dag decision presence arc direction reach maximum covariance tight decomposition dependent form equation joint distribution dag arcs q appear tight light correlation compute figure bound dag loose value covariance dag dags non correlation vary dag modulus node represent apparent appendix dag size enumeration arc uncorrelated property dag support sparse proportion arcs converge assume conjecture proportion eq furthermore arc common strongly arcs modulus arc take value modulus value modulus dag conjecture two concerned range structure display little substantial eigenvalue graphical representation variability illustrate usually multivariate normality generalise value property bernoulli total easy bound eigenvalue generalise hadamard negative matrix bernoulli respective maxima generalise variance strictly equal zero convenient rank present option behaviour square frobenius arise case high unstable unique none correspond make interpretation seem arise let unique global maximum correspond matrix approximate normalise follow normalised vary associate display variability vary interpret goodness one total rewrite provide complex different present assumption algorithm datum result rewrite dependence edge arc distance use efficient furthermore possible commonly use literature rely variation ham knowledge possible include example restriction degree denote tuning measure learning algorithm increasingly hc towards never coherent lastly world attention reason parameter assign support drawback firstly effect presence arc conditionally secondly uniform assign discriminate network sparse preferred example arc order parent topological clearly variability prior challenge undirected acyclic graph make problematic propose alternative structure focus respective smoothly extend frequentist present literature behaviour individual reduce super exponential inferential task network moment easy interpret algebraic thank college comment suggestion author real real prof inequality inequality eigenvalue trace proof uniquely multivariate definition would assume theorem direction direct graph arc eq arcs arcs family hoeffding
delta quantity construct gradient procedure begin optimization parameterize kernel mean variance property normal boundedness satisfy importance section cumulative distribution normal interval iteration dimensional density multi gaussian kernel rr truncate dimensional multiplying correspond multiple pair learn cf demonstrate deal parametrize space learn constant multiple diagonal key issue datum square feature offset normalize run validation average run method maximum objective divide average examine speed method coordinate one per polynomial method sampling algorithm kernel parameter machine repository dataset htbp l consistently stochastic coefficient per difference naive slow algorithm representative mkl examine mkl generate regression dimension input choose free separate datum method learn decide cost linearly make keep exhibit consist build recall dd specifie add constant polynomial degree possible method training distinct fed keep gb run increase depend linearly kernel experiment variable large space learn predictive performance start kernel consider suited dimension among close aim mkl new algorithm kernel also product interaction feed kernel six uci method mkl kernel degree specification set overall non interaction linear bad believe observe kernel easily prevent overfitte assigning select purpose repeat confirm nd nd nd finite mkl time experiment fast almost since learning counterpart recall mkl multi slow dimensional infinite considerably fast dimensional since perform tb nd nd mkl finite breast diabetes heart exponentially predictor randomize mirror descent derive optimize variant trick careful unbiased keep combinatorial practically representative literature optimal solution kernel satisfy strong apply applicable beyond study combine combination gaussian remain acknowledgement support proof rate proximal proposition carry argument solve lemma strongly I hold one express divergence thus plug identity introduce part define convexity notice side linearize appear standard proximal learning convergence algorithm give analyze adapt martingale difference sequence satisfied boundary optimizer optimization lemma term yield q develop bind expect measurable together combine result rule bring thus k adapt martingale strong furthermore k old book sure bind sum place k seem strongly case substitute hoeffding martingale obtain union write primal lagrangian n multiplier readily primal equality constraint clearly feasible condition maximizer expression readily mild derivation quite argument find e specialize loss thank implicit provide case derivative manner combine implicit continuously differentiable list far shall readily verify cm ari science ab consider linearly combine predictor number large method whose scale intractable propose descent computationally variance propose base ideal proportional magnitude correspond surprising result coefficient combinatorial structure kernel allow sampling look computational predictor kernel mkl interested base come view nest convex first optimize weight minimize empirical risk randomization mention paper update argue keep variance actually low kernel make potentially use bound prescribe linearly mirror similar prediction considerable attention challenge interest lead overfitte infeasible avoid combine important induce follow approach call use range encourage group actual reason convenience trick see exhaustive survey reference multiple popular decide priori map appropriate task priori choice fed huge increase bring even infinitely kernel soon bottleneck linearly large mention infinitely lack guarantee consistency practically kernel new present specialized version give experiment formal index indexing define input x w consider solve optimization eq specify later nf nx penalize empirical know favorable pass sake simplicity abuse define mention mention call group introduce encourage sparsity handle come however reason convenience reason feature thus kernel penalty form jointly reproduce hilbert rkhs learning rkhs particular page equivalent lf see risk instance mkl therein allow run memory exploit finding slightly notation symbol convexity minimizer unique exploiting convexity use randomness propose randomized coordinate thus alternate point separately design proposal would alternate minimization investigation definition let nonempty barrier gradient correspond bregman kk mirror descent assumption note since infinity tb size subroutine k k g slightly form literature analysis respect norm time deterministic finally hold q q theorem strongly convex implementation project choice use two subsection choice make algorithm start first verify way expression calculate stop iteration include sake section ix nn I iw base f compute proper differentiable derivative write vector define sampling gradient index case whenever derive stay define k belong compact note latter condition empirical k k see efficiently may also store infeasible project project show simple result base calculation follow implement importance sampling tb randomly use algorithm presentation necessary drop gradient drop dominate integral countable back formulation counting measure form parametrize parameter come notation learn product base shall learn kernel combination set ix index permutation define statistical model interaction kernel cardinality depend write rest emphasize without keep write index draw description denote matrix high product hadamard overall would subsample empirical bring leave future exponential algebraic efficiently hard bound independently tb
suggest user web search cascade consider present select list reciprocal cascade observe click family straightforward via triple user choose nothing estimator probability define member order conditional theorem proposition reciprocal representation preference acyclic loss loss recover recall pairwise rewrite discussion consistent special recall structure ij derivative fisher inconsistent exhibit inconsistent employ consistent precisely one coefficient inconsistent simultaneously associate loss edge assumption consistency preference acyclic strategy solve convex procedure broad description presentation specific difficult grow fortunately leverage minimize minimize function argument zero idea et specialized composite objective maintain live iteration score secondary concern verify method insensitive aggregation estimate denote minimize loss display risk reference different pair broadly plot match consistently plot appear sufficient loss aggregation risk salient moderate large effect interestingly significant improvement consequence sampling attain risk asymptotic score display risk suggest minimization empirical risk feasibility design consistent consistency ranking ranking base inconsistent rank surrogate partial preference behaved empirical benefit surrogate experiment thus toward formulate rate rank risk interesting grow infinity least solution asymptotic normality robustness risk explore formulation supervise acknowledgment thank anonymous associate helpful comment valuable proposition national engineering research laboratory office contract query procedure consistency provide rank rarely preference pairwise comparison rank aim partial preference however raise serious many commonly loss comparison yield surprisingly motivation present rank base aggregation preference risk show parallel complement procedure task class year significant notably strong theoretical quantify rate qualitative aspect extension increasingly understand overall satisfactory world label classify real value rather list supervise complete provide treatment example retrieval goal user object probable disease recommendation system aim order accordance preference population document search engine must page statistical fall understand aim characterize statistical inference generate mechanism theoretic preference draw consist candidate item nature preference set retrieval natural preference select return discover provide specific ordering index retrieval need natural language assign score adopt theoretic perspective risk intractable machine select fail ranking reasonable generating mechanism remainder failure strategy aggregation begin loss propose literature discount ndcg ndcg require preference relevance preference makes asymptotically minimize ndcg loss preference collect trust biological evaluating involved click feedback even consideration collect complete highlight human difficulty relevance lead researcher suitable arise world patient treatment match pair store moreover show human reference predict item associate preference prefer associate zero loss intractable convex fisher link fisher consistency consistency hope ranking guarantee hope generally computationally fisher favorable surrogate consistency fail practical strategy satisfactory justification approach difficulty rank preference aggregate theory statistic preference aggregate theoretical establish discussion aggregation conclusion proof defer begin several way preference arise aggregation base match comparison may naturally generate practice preference preference direct preference order example response query engine order record user store track customer provide preference candidate example preference specify partial candidate candidate order preference item wish treatment aggregation us aggregation appropriate case pair relevance weight adjacency item entry nonzero aggregation strategy query represent user preference query adjacency matrix capture mean preference thereby item partial preference framework formalize hereafter specific existence series ks pairwise loss eq adjacency hereafter procedure sequence conditional law aside aggregation addition assumption loss ms ml assumption asymptotic place guide pointwise conditional risk predict closure subset query motivate minimize preference go follow face difficulty minimization nonsmooth typically minimization precise formulation demonstrate commonly use pairwise aggregation tractable procedure losse second consistent surrogate statistical procedure statistic law sufficient query countable pointwise bayes risk equality countable infimum qr f throughout exist consistency achieve risk draw uniform measure assertion rf completely condition fisher general analyze fisher consistent main surrogate fisher fisher pointwise fisher begin define measure classification surrogate suboptimal risk suboptimal conclude completeness provide material pointwise fisher consistency hold proposition clear consistency strong consistency situation connect weak surrogate fisher pointwise connection indeed space finitely multiclass classification setting assumption depend partitioning local often continuous q satisfy intuitively property surrogate score surrogate loss surrogate coincide fisher state give supplementary material hold fisher loss definition structure conditional theorem consistency uniform loss consistency coincide assumption restrictive practice sufficient condition hold weak satisfies fisher structure surrogate risk finite preliminary focused set preference demonstrate popular surrogate inconsistent pairwise loss preference work explore connection focus loss place function aggregation let direct dag direct impose separate distinguish avoid preference generalize disagreement describe et use number equivalently return bounded finite preference assumption let polynomial widely proposition loss convex function minimizing minimize fisher convex inconsistent pairwise show surprisingly surrogate inconsistent noise minimize preliminary loss surrogate function penalty surrogate preference aggregation risk dy surrogate convenient let edge low condition self reinforcement adjacency reverse equal definition graph global preference reasonable consistent nevertheless surrogate inconsistent low see form even noise margin encode margin preference base loss inconsistent noise weak theorem preliminary work noise set inherent development loss achievable review fisher preference typically section combine strategy procedure discount gain ndcg loss loss like score et general ndcg monotonically increase argument inspection standard ndcg criterion form measure relevance item form generalize precision assume surrogate ndcg family since permutation induce fisher ndcg recover result al surrogate preserve order integrate tend infimum satisfy zhang present example loss note differentiable contain example deep ndcg loss extend uniform set loss invariant shift dependency among order select satisfactory position let denote indicate empirically user rank product indeed value relevance independence regardless supplementary let permutation minimize preserve property fisher ndcg see variant corollary suitable condition conclude section space ndcg consist value think simply order direct acyclic transformation score place advantage finite exist give loss readily pairwise consistent surrogate fisher tractable aggregation partial datum list score make obvious procedure estimator law develop analogue population define belong query count develop surrogate expect risk converge uniformly analysis complete require knowledge surrogate develop broadly instead surrogate trading limit aggregated draw assumption loss loss might appear high datum substitution large final grant perturbation moreover choose obtain broad surrogate structure surrogate appearance query size expectation surrogate risk target suitable asymptotic procedure hereafter sequence scoring query convergence occur representative bind empirical risk argument hold proof material loss true probability query tail reasonable receive entirely day finite main concern behavior contain norm loss lipschitz continuous constant satisfied contain interior cover number assumption place give enable condition interaction aggregation datum point grow state shorthand exist sequence expect consider cover condition arbitrarily one relate growth directly sequence satisfy additionally satisfy eq function fix similarly argument n l n consist functional represent ball mean condition familiar necessity empirical loss condition
fair code regularization research topic comparison interior method group variation proximal employ publicly matlab implement coordinate descent refer implementation fista group fista instead refer observe coincide tolerance thus tune stop level range value approximation prox ip support prox concern label corrupt otherwise additive rescaling coincide variable sure working solution determine correspond variable small return nan solution build geometric order robust time ip prox ip prox c ip prox report entire regularization prox ip reason apply store il time know step fast benchmark fista familiar prox fista fista different projection project newton fista refer former latter protocol describe first correspond relevant index ten vary group space amount take thought correct loose iteration necessary fista cc c cc number iteration respectively computational fista fista comparable behavior generate coefficient subsection differently subsection problem cyclic cyclic yield guarantee comparison compute cyclic cp latter stop compute two repetition deviation computing plot cp much show thank apply overlap deal preprocesse selection present analyze gene pathway cross select fista two loop fold validation cross validation split entire fista comprise loading gene validation improve probably absence unstable overall computing overlap problem group advantage wise expand build belong scheme use optimization enable use selection high throughput acknowledgment absence thank carefully read paper appendix project need concave convex ht f usually employ k x px f di intelligence mit usa universit di mit consider sparsity induce allow overlap nonsmooth optimize version proximal nest accelerate proximity exploiting devise strategy thank pre empirically toy keyword regularization popular way arise process broad sense possibility write towards model example sparse minimize motivated explore regularize exploit pattern solution example partition group priori estimate know latter group euclidean achieve consider potentially goal union group common bioinformatics throughput datum gene mass characterize sufficient encode online database bioinformatics penalty admissible pattern arise apply expand paper direction coordinate proximal variable implementation overlap lasso generalize group loop base accelerated version name fista require deal large group exploit operator minus convex active group active active allow projection via cyclic accelerate reduce solve correspond via project newton dimensional extend conference contain presentation proof organized highlight short cast group overlap modify penalty compare extend describe scheme precisely recall proximal subsection proximity present projection active use projection norm perform performance study propose scheme run art conclude real improved preprocesse improve newton norm pp propose type group continuously differentiable simplify exposition penalty canonical identify belong adjoint lead support I e entry union subset see group group penalty study involve penalty complement complementary many algorithm study hand p overlap call allow case strategy potentially goal propose variable practical hand norm valid element belong sparsity instance empirical risk take form hold infinite dimensional kernel hilbert space readily due smoothness trivial moreover high use possible discard satisfied eq projection give proof rely allow proximity proximity conjugate homogeneous obtain minus convex derive convolution compose projection conjugate I indicator unitary ball contain convolution conjugate conjugate unitary ball rewrite indicator function basic projection restrict denote group solution number denote I g contradiction brevity thank hypothesis definition q every precisely cyclic therein propose exploit practice prove fast projection tolerance outer onto intersection cyclic projection modification cyclic ht trivial c lemma ball soft depend recall sort newton onto amount minimization problem dual advantageous much lemma theorem solve onto f lagrangian duality primal efficiently project newton report basic simplicity project newton thereby avoid overhead mention next prove recent provide proximal tolerance x I follow thank equivalent thus l equation use cp happen basic backward long guarantee unless strongly convergent exist enjoy algorithm minimum rule internal admissible projection report guarantee choice would ensure strong see proposition adapt comment subsection complete employ propose
know measurement variable estimate sense measurement question describe know question want optimal even example cm stand expectation estimator mathematical consider define mmse estimation world bias admit bias order mean random world appear aim measurement density property label example expectation mean find tell wiener filtering estimation even rough kind come specific straightforward signal simplify suppose assume gaussian absolutely information additive high frequency zero estimate low frequency gaussian hypothesis simple although asymptotically unbiased consider measurement independent unknown calculation pass certainly repeat however restrict case unlike two give explanation give let obeys freedom estimator value coefficient variable follow show distribution negligible consequence huge induce huge huge negligible divergence consequence give realization negligible huge express respect almost expectation value though measurement figure great non negligible mmse logarithmic estimator solution would course confidence obey freedom mean high confidence order determine suggest replace call fisher location pdf I transform characterize density imply constant probability density hand direct yet constant obtain biased world measurement lead linear unit u mean extend estimation mean definition log unbiased euler make origin event unbiased wrong induce try unbiased return light consideration variance ensure measurement available difference situation mmse estimator propose also construct minimum mmse mse infer product lead gaussian complex form case adequate verify efficient estimator measurement available irreducible biased direction give location ensure three estimator university e wiener filtering datum coefficient mean square error hypothesis noise cross vanish determination error filter replace lead mmse posteriori square coefficient relate biased q normalize obey monotonic give immediately measurement lead pure view calculate gray france email france
take reduction describe condition sometimes convenient plausibility space assume plausibility evaluate important I belief meaningful validity I proportion outlier hold singleton fact predictive plausibility familiar stochastically dominate set need measurable subset valid predictive contain measurable k set hold I however association statistic exist ta mild separability trivially nan case hold association hypothesis admissible plausibility generality reduce baseline I subset set follow p support consequence I notational consistency proceed plausibility evaluate justification right hand complete singleton point plausibility decompose piece plausibility general prior often focus probability value p I posterior interpret perhaps surprising give bayes view attractive approximate light argument concern value fail satisfy value explain respect posterior different prior plausibility describe situation arise mean I predictive set empty moreover sense I one literature uniformity variance goal quite plausible plot plausibility horizontal characterize plausibility keep plausibility develop p hypothesis I plausibility value light parameter clear modification I plausibility via method numerous part author recommend value odd interpretation strong suggest adjustment ever alternative familiar valuable offer connection plausibility p I I plausibility understand evidence plausibility useful I p correspondence probability I unified grateful suggestion associate two national grant dms dms dms demonstrate value plausibility nice ease use issue comes keep brief detailed description idea method find present statistic far trivial present I side albeit plausibility frequentist test goal predictive precise shall p stochastically stochastically value function binomial mass propose scheme carry binomial brevity paper show illustration plausibility function several determine plausibility case plausibility shorter similar hold hold statistic propose p plausibility inferential practical inferential plausibility notion plausibility practitioner use nan connection p trivial parameter inferential plausibility function statistic sort medium simple avoid reason frequent inconsistent people sense goal plausibility plausibility define within I build upon principle inference global mild choice statistic valid I plausibility understood plausibility highly plausibility fit way practitioner interpret p plausibility calculation avoid logic use interpretation summary prove beneficial organized notation p incorrect interpretation plausibility corresponding I bayes problem trivial two example binomial conclude remark observable sampling problem hypothesis mathematically subset write nan alternative whether thing occur relative chance occur false put suggest simplify tx intuitively compare outli conclude conversely sense acceptable reality adopt arguably interpretation go something observable actually sort program condition potentially incorrectly value support interval fairly medical effort confidence interval value free difficulty bayes factor simple valuable contribution inferential I exact prior assertion unknown accomplish explicit auxiliary random predict obtain random I exist confidence reference prior develop sort parameter invert target uncertainty probabilistic inferential assertion special
mathematical text section subsection paragraph eq number eps abstract
look opt gradient warm describe analysis algorithm return terminate solution always direction whenever check easy opt satisfied great therefore similar control practical I implement homotopy homotopy iv explicitly exploit linearity path verify modification code inner cm dataset full approximate path stop ill check experimentally verify correctness approximate duality gap check present complexity interestingly look example path full exhibit complexity also significantly path homotopy worst formally optimistic require number path approximate homotopy acknowledgment nsf grant dms dms regularization path popular optimistic always obtain every path duality analysis practical approximate either observation small problem induce penalty prove quality purpose formulation require value induce control regularization sequel follow classical optimality uniqueness rank subgradient optimality say otherwise uniqueness necessarily w j also reduce w hessian reduce strictly rank formally full uniqueness w sparsity consequence linear patterns path moreover limit easy always present homotopy initialization trivial compute path x record cm maintain optimality correct reasonable make work real become ill may typically truncate assume algorithm single jj enter even problematic view segment homotopy present issue show exponential see state segment therefore w w convex rank remark w optimality increase obtain contradiction proposition build regularization lasso path extra increase multiplicative adversarial design adversarial let span small bound two zero possible optimality inequality verify last last definition sparsity pattern characterization last mainly continuous continuity argument appear segment characterize easily show continuity argument path lasso linear lead segment recursively segment matlab numerical us small segment example quickly lead close publicly optimistic regularization path another refined path quite strong analysis tailor gap lagrangian minimize primal formulation function dual call optimality say solution build approximate perturbation condition condition satisfy dual duality cm w q satisfied path simple opt dual piecewise control machine upper contrast cm obtain guarantee
electrical row think additional v simply give give unit precise unit flow note identical message close message approximate ji ji lemma induction height message easy compare achieve ij I ij iv trace submatrix apply transformation ji step take precision ij ij message height p li il il l optimal argument let transformation ji ij ij induction correspond message satisfy ki ki satisfy induction hypothesis ki li li ji ji fact term complete approximate message ideal state set subtree ji obtain q ji ji induction height base leaf consider leave define subtree simply leaf happen notice ji q r constraint ki ik li il ij ij ik ki kt li il hypothesis height satisfy compare application claim compare eq ji ji along addition operation recurrence go message induction remain ji finally account trace eq introduce round tree unbalanced simple exponentially always balanced decomposition allow polynomial balanced theorem exist decomposition requirement construction diameter decomposition decomposition height decomposition element precision dynamic programming tree follow run ij term require tree produce transformation net leaf extract mi I mi zero matrix lemma produce pass support net passing claim construction edge scheme mrfs width semidefinite element mrfs whose arbitrary instead net net polynomially consequence algorithm mrfs precision impose net wise round precision description describe round mrfs mrf lemma corollary theorem pair observe usual w orient vertex vertex sequence child w I iw lx j tw j scale w positive diagonal dominant new connect obtain submatrix size row index let observe describe run ideal message pass construction net bound give integer support reason way factor ensure rank message ideal small iff round full q observation well eigenvalue within factor semidefinite eq follow imply next net rounding pass decomposition svd see round orthogonal net ingredient net set orthogonal obtain rotation second ingredient define ready net precision v v diagonal set clearly small precision eigenvalue message pass fine net next formally input orthogonal transformation satisfy p orthogonal statement round near e v computation u v p require unit sketch proof triangle u z triangle side schmidt orthogonal column apply schmidt round gram schmidt onto span along give gram schmidt apply triangle give net round message big ideal message error set observation approximate message big approximate message ji ij ij ij ij program subtree almost lemma analysis element round section due svd round error unchanged compare error wise rounding exponentially height whereas svd scale linearly hence provide brief omit depth identical message height ki ik li il ij ij ji il il ij ki li respectively successively application increase rounding inequality analysis induction term equation ik ki il remain obtain scale message message main mrf input mean decomposition requirement page use give net respectively empty extract approximate mi I mi height time construct pass per pass thm observation exercise remark corollary thm thm conjecture example random field observe total variable np supervise greedy graph joint markov gaussian mrfs give budget minimize particularly provable error main mrfs tree widely inference see class mrfs among computer state describe relate research denote denote row non occur matrix symmetric chapter order positive semidefinite positive matrix use eigenvalue rank large let index subscript clear fully scheme mrf matrix rank matrix markov clique assume mrf know w origin center gaussian e minimize I expect error equal conditional actual light express n gaussian decreasing follow state given find version exist small gaussian free chapter mrf fix q set density select laplacian define precision treat whose precision thought variable main finding np hard give greedy graph base message difficult formulate mrfs extend width require theorem programming mrfs graph expect error early process krige see unobserve krige variance solve krige I krige also extensive literature explore criterion entropy area quite similar problem e g overview objective aim provable goal average unobserved variable equivalent trace submatrix matrix experimental optimality minimize trace conditional experiment objective differ want prediction unobserve require unobserved budget formulate expression semidefinite nystr om nystr om approximation process first subset width mrfs exploit precision matrix mrfs strategy om greedy heuristic see provable bound moreover rather error tree mrfs motivation study exact extensive graphical mrfs chapter mrfs involve inversion subset hard show field hard g maximum framework second free mrf thought analog ise computer e discrete mrf uncertain adaptively rather ising prove solution feasible regression within special hardness regular graph nod regular entry q expression step harmonic iff iff budget hard regular graph node np bound set iff question average error since I e notion rather return b fa fs cover compute know minimization problem result know electrical see consider elimination decomposition survey decomposition tree width small yield assume width cluster give decomposition one trick like add empty exactly presentation edge split consist eq set separate variance depend variable mrfs happen joint precision intuitively induce prior markov property allow use message pass density matrix moreover v elimination cholesky row ji lower desire upper eigenvalue elimination cholesky take representation two transformation marginal transform marginal precision q transformation precision think operation first take principal inverse integrating intuition precise application small later discuss message follow fact principal generally matrix decrease invert submatrix invert submatrix large decrease submatrix small set lemma belong subtree note belong subtree define abuse actual need notation subtree subtree subtree q error go algorithm version message message produce approximate message split ji budget allocation mrf come gaussian mrf allow justified message proceed round round leaf receive exclude message otherwise express later notion maximum message message send happen leaf height argument il must observation describe compose cluster consider receive argument compose follow infimum easy describe
practically number psd section effort decomposition mainly influence calculation considerable reduction computational achieve psd paper di wind term simulate variate reasoning eq tm q approximate latter read store novel digital field develop spectral fractional transfer white time valid alternative case psd di extend use spectral two consist perform spectral decomposition psd eigenvectors multidimensional uncorrelated white unitary wave engineering design di calculus journal physics conference series fractional calculus probabilistic characterization pp g di use calculus di representation density fractional pp di r exact stationary colored fractional pp ed pp di digital wind field velocity pp di digital generation wind pp perspective record pp mathematic science multidimensional pp york modelling pp p wave moment di di ed universit di universit digital wind velocity field fractional calculus taylor digital velocity field fractional show construct coefficient fractional moment target superposition fractional white process simulation taylor practical digital wind velocity field need wind vast refer overview different attention simulation wind velocity field digital wind I superposition digital filtering commonly ar al et method wind field contrast filter equation fractional appear wind method slowly become engineering brevity sake fractional refer topic publish novel fractional differential extend investigate digital assign wind wind velocity psd organize multivariate wind velocity due assumption second characterize velocity field specification moment variate along remark straightforward theoretical concept wind reader comprehensive reference wind velocity think around stationary value locate velocity represent vector spectral psd read q spectrum velocity spectrum calculate flat neutral consider cite paper psd velocity longitudinal coefficient eq coefficient determine accordingly next section attention psd construct need reference sake wind field let matrix element hold density eq di di wind form normal increment bar complex conjugate kronecker decompose coherent elementary q eigenvalue form point di frequency pay consider combined start variate next extended gaussian stationary variate assume wind assign psd propose follow firstly psd term fractional spectral output psd show fractional moment linear transfer filter taylor contexts di al generalize taylor di variate topic z store finite use process solid ground simulate fractional stationary psd suffice ideal system white noise process zero power characterize impulse response namely transfer relation read characterized calculate fractional impulse transfer notation last integral eq transform fractional operator relevant psd reader must carry truncation discretization discretized counterpart superposition fractional integral white continue example simulate digital mean calculate form reasonable achieve function good firstly calculate return implementation require step integral gaussian white firstly fractional integral gr discrete fractional approximated become true also time partition amplitude realization gaussian gr approach efficiently transform p calculate decrease many suffice word calculate realization follow show target correlation implement see mean w strength component output impulse integral output linearity mean characterization linear output course linearity system next represent fractional transfer turn sum spectral assign fractional converge analogy eqs impulse transfer absolute eqs approximation truncation interval integral eqs
lead weight logistic g specification posterior generic coincide let independent model yield maximum likelihood result let fully data logarithm detail hypothesis indeed local coincide remark previous matter marginal density parameter likelihood analyze mix weight log nest fact consider depend empirical simulated equal proposition reduce step log function require ng calculate variable depend density respectively step likelihood maximization equivalent generalize contribute log stationary derivative algebra maximization reweighte ml maximization estimate density give closed g rest specify value whether sufficiently acceleration l kl log analysis algorithm stop introduce paper different classified type accord datum arise know numerical study consider criterion ml estimate rand rand rand comparison pairwise perfect make small difficult interpret ari chance possibility classify observation ari rand partitions original ari measure goodness fit linear detail goodness special pearson remove impact sample square goodness fit glm difference achievable dispersion form binomial call generalized q form divide dispersion respectively illustrate artificial aim analysis herein write environment package state nest show numerical concern accord poisson regression group group case three gaussian conditional accord random fit follow discrepancy q respectively poisson reveal bivariate group generation binomial gaussian estimate fit model scatter model display different joint binomial misclassification error model gaussian approach binomial mr group fit binomial binomial bivariate randomly binomial give ht give estimate display scatter list misclassification datum modeling like binomial outperform possible see ari poisson artificial size respectively poisson specify round fit poisson scatter plot ht fit ari obtain model poisson underlie ht ari datum number price run unconstrained poisson group large unconstrained result poisson outperform result value directly two class constrain poisson attain measure goodness substantially parameter confirm prove coefficient cccc poisson library consist application million application associate bic unconstrained poisson outperform result respectively outperform result attain value goodness group ht ht cccc concern length stay related display moreover b base plot scatter ari poisson clearly perfect separation contrary able evident poisson carry patient poisson outperform yield perfect separation class result ht fit study effectiveness comparison mixture regression result theoretical investigate reveal poisson regardless strongly outperform give gave outperform outperform outperform give comparable slightly outperform give comparable outperform show covariate figure highlight performance clear highlight well covariate range confirm obtain contrary covariate figure paper generalize categorical depend numerical covariate come mixture show mixture linear artificial comparison cite paper distribution extension student student estimate e currently extension issue concern first behaviour em suffer confirm g initial marginal distribution maxima finally come discrete specifie coincide get recognize generic specify sufficient complete since posterior reduce neither attain proof sufficient logarithm algebra estimate like prop prop prop remark di di via weighted modeling come heterogeneous family model exponential family link suitable mixture nest
last independence poisson process markov ergodic interested steady equilibrium probability neuron equilibrium satisfy supplementary neuron word neuron ergodic product marginal neuron use unknown variable external arrival neuron weight appropriate spike combinatorial system recurrent use application characteristic connection circuit connection cyclic cause history store internal state recurrent tool forecasting efficient gain form neuron receive recurrent linear produce output connection reservoir reservoir neurons reservoir reservoir circuit possibly reservoir expansion separability approach observation assumption linear often sufficient task know model input connection reservoir possibly integrate neuron activation layer allow connection input concentrate reservoir connection reservoir row contain update output vertical concatenation dynamical input produce series evolve accord spectral radius role reservoir two suggest intrinsic backpropagation etc simultaneously temporal weight compute regression ridge widely literature use research empirical reservoir respectively depend start performance preprocesse rescale offline ridge adjust autoregressive bt bt bt st traffic service european minute training neuron map configuration suggest neural topology trait traffic united education research internet traffic triple compose study validation sample ci last use reservoir zero use radius reservoir good performance improve neuron reservoir use criterion reservoir unit version present slight empty european significant performance accuracy obtain version largely reservoir important reservoir reservoir linearity necessity illustrate estimation reservoir interval begin day set figure instance type reservoir neural successfully temporal learn good relation reservoir performance significant property simplicity reservoir counter software aspect example reservoir reservoir decade computational name recurrent process recurrence neural circuit occur rapidly success neural drawback propose come design consist reservoir recurrent position example performance largely tool powerful complicated engineering many design machine come random mathematical network literature actually interpretation mathematical neuron queue spike among space internal neuron fire spike potential neuron increase supervise descent easy hardware implementation introduction nevertheless suffer feedforward topology refer rnn avoid use rnn network concern circuit topology recognize powerful machine traditional limitation come difficulty implement main drawback guarantee algorithmic sometimes train require learn relatively reservoir overcome drawback topology ten neural adapt connection output approach achieve begin section iii discuss also idea inspire experimental end conclusion discussion regard network propose concept neural context node neuron neuron receive one negative neuron neuron receive spike neuron receive spike decrease equal far strictly
whose zero formalize produce function respective source common expert similarity score strong recall take unless state shall empirically goal source attribute score interpret entity match magnitude achieve depict typical merge thresholded produce feature scoring processing matching produce clean cast release employ pair entity hash movie entity stop rare matching scoring runtime absolute difference cast entity goal formalize amount desire confidence recent community task complexity perform present approach transfer problem inter overview intuition transfer obtain learn set task net wish learn statistically separate structure share subspace depict task jointly take model difficult achieve transfer element transfer learn appropriately reflect task euclidean ball arise set source pair handle source approach function source treat task would source accurately score pair derive er produce appropriate allow fit available degree freedom amount consideration specify er problem model perform serve purpose store large statistical feature interaction capture act weight place score split half finally pair multi state baseline section allow result separate pair lead quadratic training could motivate transfer pair effectively example successfully use either learn separately great labeling expense represent extreme form none limited information characteristic adapt introduce capture movie generally weight account induce also program pairwise develop technique experimental proceeding recall index furthermore denote pair model solve convex estimate take moment encourage feature vector match pairwise function build alternative option term machine extensively encourage closely exist avoid overfitte weight due desirable encourage procedure act non parameter encourage capture part source represent assumption appear decomposition act sparsity allow away nominal source avoid overfitte tune complexity control select extend result popular number risk functional work practically linear learner svms proceed derive result derive tucker condition condition eq half difference subsequent choice remainder simple require characteristic common characteristic across solver source dimensionality must rely solve minimization proceed optimize algorithm estimate current soft whose maximum intuitively vector direction decrease case term truncation operation second solve problem dimension gradient smooth objective geometrically next art present gain world synthetic dataset scale source pairwise pool argue machine er assume pair impose example expensive consider share transfer model hence order example heterogeneous source denote function learner arbitrary vary source source identity pair encode flexibility individually implicitly take machine svm widely flexible mapping also kind success adaptively balance label svms regard er adapt however typically moreover may learn classification svm build general art implementation fair comparison experiment standard computational package library implement employ fold validation grid netflix investigate algorithm match vary produce potential label evaluate precision recall q recall positive negative employ six movie internet movie movie title alternate release attribute perform string stop cast absolute runtime year human source pair experimental movie learn scoring source precision order specific available pair movie conditioning subtract dividing place feature raw true attribute entity interval entity source level score attribute entity incorrect score inequality property er synthetic stress test approach pair movie compare pr deep understanding transfer method require labeling achieve four apply unobserved curve vary three pr curve figure summarize nine fix match vertical expense achieve total little training inferior characteristic behave manner available datum become adaptively dominate baseline recall trace endow structure significantly shift pr endow per source per trend observe figure apparent dominate recall row two assign another apparent width band poor movie focus learner vary transfer experiment take deep look source fine generating explore take individual difference flexible though freedom learn increment source increase intractable labeling compare observe far improve perform quadratic note require significant unable adapt increase method able example source quickly achieve test baseline method fast inferior test increase example prior vary consider er vary first highlight labeling barrier er multiple source transfer paradigm transfer numerous explore serious varying traditionally researcher pool vary level propose matching require evidence similar effect heterogeneous matching motivation paper via real weight grain transfer source simply learner comparison balance transfer task investigate selection heterogeneous acknowledge tailor experimental google quality digital library approach label combine thorough evaluation human effectiveness er work
clearly tradeoff z proposition lemma n plan online planning focus exploratory planning assess regret algorithms plan good effort regret introduce new carlo search exponential scheme part dedicated exploratory objective superior improve exponential bind exhibit attractive process model uncertainty agent action reward problem assume agent flexibility generative expressive type simulate execution mdp current act maximize accumulate reward use accumulate predefine handle mdps exponential lead researcher mdps plan mdp focus consist planning schedule external follow recommend action state action environment repeat action recommend step go assess sub closely capture take optimal policy follow begin recent mdps plan mdps sparse ng offer action discount time exponential discount sparse offer guarantee really setup mdps day mdps successfully partially adversarial relative planning ahead comes expect plan mdps barrier contribution carlo search guarantee choice space dedicate compete exploratory objective objective grow current reduction suitable reinforcement mdp benchmark art variant result factor upper attractive carlo search planning depict construction root issue store node phase recommend compatibility prior nod mdps distinguish subset action c ss actions monte concrete ground later describe core illustrate tree end state counter update action propose cumulative multi armed bandit mab select still sample transition sample popular appear expand dag expand node expand tree counter rs never adopt rule time select protocol root cumulative regret incur explore forecaster possible logarithmic cumulative seem planning reward bandit minimize cumulative simple objective polynomial rate reduction reward simple attempt recently plan general optimize online attempt rather barrier bad case reduction reduction regret plan achievable introduce state guarantee zero achieve exponential multi armed bandit stress bind cumulative minimization mab exploratory recommendation sampling mab action exponential reduction exploration exploitation situation mdps mdp answer exploratory mdps special intuition mdps complicated go act simple induce applicable optimal exception provide action pseudo need information alternative root objective identify optimal actual exploratory compete actually priori devoted confidence improve act reinforcement plan bound hyper exponential separate aforementioned exploratory investigation state mdp outcome horizon get separation exploratory concern mab mdps arm flat act state flat pair action specify act mdp step arm action encounter arm mab flat policy loop update obtain therefore arm mab recall mab actually compound policy policy outcome policy consistent note sampling systematically update stochastic uniform iteration arm sample iteration consistent arm hand q turn bound trivial h note well hyper regret rate transition help require reasoning concern planning introduce refer allow large extent family vary switch update node pair associate initialize value initialize mdp sample end either state reach phase policy accord policy expand update accord action follows call generation exploratory planning aim improve current candidate specific estimation recommendation tailor switching exploration choose sample accumulate accumulate reward depend subsequent overcome novel modification seminal hoeffding select exponent refer reader discussion mdp go applicable induce exploration along action reward horizon applicable lemma horizon hoeffde around serve induction horizon go action applicable online planning accumulate estimate select increase accordingly differently stage decrease put could course weight reasonable rate reduction regret perspective empirical differ point addition action reward responsible accord mdp reward similarly provide explicit expression constant recursive bring potential benefit horizon step line fact hoeffding inequality modify capture hoeffde sample bind q exponent multiplied pose tradeoff decreasing reduce term leaf go tend grow perspective formal guarantee appeal nevertheless try optimize optimize lead obviously would rough optimization valuable theory experience similar bias mdp mc plan original evaluation domain environment wind reach location neighbor move duration direction move diagonal take straight move wind wind fast tail wind assume mdp space action position wind mdp length terminal size path two consider area recommendation orient end state term problem grid branching depth different grid size show base modification replace policy motivation behind show slight modification updating switch identical four algorithm within suggest recent work parameter result respective original assess choose consistently outperform margin grid large relatively short planning allow attractive guarantee effective likewise practical parameter game person sum game go winner decide reward terminal assign terminal move tree move max act depend payoff aim possible objective min many player payoff branch tree domain convergence configuration average appear encourage quickly carlo online planning mdps guarantee interest guarantee goal remain future discount mdps infinite employ use account additive action value horizon infinite horizon mdps set inspection unlikely sophisticated sample another speed action good action towards mdps hope focus root recommendation planning sampling employ scheme previous difference play formal unclear reduction acknowledgement supported carry microsoft center partially support air scientific fa rely correctness well claim diagram depict overall central claim depict rectangular concentration binomial bernoulli equivalent choose state via action sequence apply action exploration end go function accord q pair consecutive iteration consecutive finish exploration action put consecutive exploration phase apply phase iteration finish apply
read tree variable binomial distribution derive parameterized tree range slight preference tree preference detail hyperparameter hyperparameter affect affect hyperparameter range range read nine per read binomial frequency figure structure likelihood although prefer vary pearson range setting range structure range place choice setting although read baseline slightly x due merging whose nearby distinguished note infer pearson recover read frequency consistent goal establish integrate order remove bias correctly sample simulated read frequency reliably recover representative good high dataset run combination simulation single report sequencing sequencing suggest single cell confirm provide truth show major single cell confirm frequency copy read count plot order partial via high parent sort parent ground clarity represent cluster place frequency clustering interpret plot indicate reference figure include single plot representation indicate largely consistent indeed posterior file reference list major estimate single appearance tree switch frequency two table variant allele count allele ci ci b ci ci ci ci ci ci ci ci systematic bias note support tumor cell agree e early root secondary uncertainty rest tree table branching lot allele suggest k cm x allele read count read depth allele ci ci ci ci b infer show figure stop five posterior relatively probability probability probability file ground legend tree largely perfectly reconstruct cell explain systematic deep sequencing nonetheless evolution sequence frequency quantify different point span patient identify sequence tumor label frequency reconstruct evolutionary cancer semi manual automatically allele allele patient reconstruct describe sample cancer share evolutionary history frequency read likely normal case appear sample tumor little tumor good tree agreement frequency figure tumor structure structure likelihood allele tumor sample indicate cluster dot plot form structure multiple frequency reduce evolutionary tumor reconstruct produce semi manual call tree visualization plot represent uniquely reconstruct profile tumor enforce structural population frequency able relationship tumor case detect tumor sample highly constrain drive semi define frequency estimate sequencing put read enough relax hard constraint large possibly phenotype clear ground gold cell sequence heterogeneous cell ensure potential magnitude manner minima represent determine reader extend recent whole genome copy attempt profile genome sequence site use extend full population change evaluate area innovation modeling allele replace binomial allow variability observe allele object follow place weight parameter draw parameterized dirichlet datum fix component stick provide point center e dp stick breaking view break stick location stick break large let stick break stick break mixture component parameter draw result eq every object take stick stick produce extended stick rooted agglomerative internal exploit sequence positive use index tree let denote zero indicate therefore denote child node stick break allocate respectively whereas determine sequence construction tree note concentration parameter model throughput sequence genetic population variant allele copy let allele denote whose probability population dirichlet variant position cell fraction cell population position count posterior appear inside summation compound single draw rewrite gamma dirichlet count useful population nonparametric prior avoid group evolutionary root tree stick breaking count tree structured process count stick indicate hyperparameter indicate latent include frequency population distribution root node evolutionary section crucial stick break infer chain monte gibbs involve subsampling sampling stick length stick break hyperparameter contribution frequency way proceed evolutionary subsampling procedure sampling frequency population node equal enforce constraint satisfy result use denote auxiliary singleton root node node denote parent population child node population appear parent tree current tree weight sample h root create queue q frequency weight non leaf hasting auxiliary frequency asymmetric dirichlet proposal markov chain converge sampling iteration posterior root initialize density extension cf multiple allow tree structure read match variant allele position let fraction population tumor probabilistic share stick break hyperparameter multiple frequency evolutionary constraint separately tumor move base tumor plot summarize visualize tree cluster direct fraction mcmc place aggregate information mcmc summary possibly quite uncertainty assignment input co difference number number assign cluster cluster cluster assign burn fix metropolis hasting dirichlet mcmc sampler initialization pick auto package sampler complete trace autocorrelation plot burn period figure file dataset input experiment observation additional file acknowledgement science award publication software publish manual high sequence quantification frequency single nucleotide variant heterogeneous tumor evolutionary population tumor reconstruct however reconstruction available condition reconstruction describe evolutionary history uniquely frequency tumor nonparametric prior major joint tree identify frequency high generate multiple consistent frequency tumor order real comprise tumor demonstrate branching inference agreement ground apply frequency mutation small partial order supplementary material background disease often characteristic genetic substantial effort genetic identify tumor contain reconstruct history tumor population quantify whole sequence tumor deep sequence tumor associate nucleotide frequency reconstruct evolutionary history multiple tumor sequence frequency methodology evolutionary infinite site tumor frequency reconstruct major tumor automatically perform reconstruction topological triplet consistent great branching contain population must sum
generative pd px conditional state proceed usual sampling indirect possible use auxiliary variable similar sampler contribution theory backward posterior treat duration single notation standard model message filter backward condition message transition modelling assumption unfortunately duration support lie backward fail duration code length propose behave like truncation define duration well sample result incorrectly admit entire sequence infinite hide markov truncation countable message filter sampler sampler state duration distribution space slightly modify application sequence generalise markov transition state duration speech speech synthesis suitable dirichlet hdp hmm allow address transition experiment three state distribution rate pd uninformative duration distribution inverse iw burn learn observation transition visit comparison per would forward computationally impractical second except separate reveal duration show indicate sampler hmm sequences truncation work explicit hmm available letter cardinality nonparametric variant develop black box hmms consist indicator direct duration duration truncation require perform hidden hmms tool variant formulation duration time specification duration priori geometrically hide semi allow explicit parameterization forward backward though short hard code run representation tractable lie within outside incorrect duration reflect allow pre duration hard duration exact incorrect develop hmm key hmms countable state duration countable box specifically estimate demonstrating technique finite duration duration review inference capture state duration observation consist distribution observation duration observation time segment markov chain
closed restriction resolution restriction inclusion axiom discuss straightforward fix deriving appear derive rule cut cut eliminate clause cut resolution restriction clause indicate dag resolution paragraph dag component original derivation node correspond node restriction formula subgraph long resolution use first explore associate clause add allow clause restriction instead simply require utilize clause appear demand utilize clause appear never clause clause derivation equivalent add restriction space remain restriction closed restriction close resolution assignment correspond construct derive clause derive subsequence establish carry construct subsequence step immediate add originally derive likewise derivation construction include therefore thus resolution proof appear elsewhere history alg completeness formula return otherwise either variable naturally convert return correspond limited kb partial draw process assignment valid note previously give find proof theorem proof size bind find reject accept clause precisely refer appear width naturally width proof exclude clause originally resolution generalize resolution e present general straightforward omit iff resolution run resolution resolution restriction close resolution assignment resolution proof take node new dag note step derivation derive dag latter therefore width obtain implicit learning resolution kb target either exist partial else valid respect run assignment length resolution naturally close proof utilize learn section search resolution restriction background briefly proof resolution formula clause formula introduce detail recall conjunction derive essentially formula infer given elimination simplicity derive elimination power include restriction axiom naturally hypothesis otherwise four since otherwise conjunction appear likewise elimination must evaluate otherwise elimination introduction new one introduction derived derive introduction original suppose apply first first rule apply possesse bound alg say proof width reject efficient iff derivation main conversely width run width take derivation formula input tuple derivation via check checking appear width first checking formula common derivation collect elimination width width length formula conjunction formula linear checking width formula note syntactic close let appear derivation formula simplify decision implicit kb target suppose either n assignment calculus simulate prove heuristic could computer algebra gr basis fulfil due heuristic practice nevertheless represent potentially furthermore calculus arbitrary present hypothesis calculus enable solution original correspondence boolean serve polynomial boolean formula product degree variable axiom infer multiplication derive derivation encode modify allow axiom course step calculus combination also note generality restrict great boolean axiom could multiply axiom formula original replace repeat trick time trick appear rest refer formula nothing lose look ahead focus restriction calculus formula system polynomial restriction restriction representation since vector express conjunction sc verify restriction polynomial arise apply effect equation evaluate kind nevertheless system equation np np interested solution calculus encode encoding undesirable recall correspondence require calculus simulate polynomial calculus extension polynomial calculus resolution introduce related axiom cut capture coefficient multiplication purpose formula assign whenever calculus calculus step restriction restriction axiom easily assign otherwise latter boolean axiom calculus axiom axiom simplify axiom inclusion axiom inference preserve say multiplication axiom clause consider formula representation formula degree polynomial fix multilinear ordering nonzero coefficient polynomial calculus appear used proof resolution simulate calculus calculus degree calculus construct lie f polynomial derivation otherwise bp decision pc theorem solve calculus resp run calculus work reader gr bad run paper case alg calculus need restriction system restriction easily polynomial calculus proof degree note restriction calculus valid calculus resp recall polynomial calculus restriction degree decrease restriction learn calculus list either polynomial least calculus resp else q integer interested cutting integer satisfied integer fractional current cutting cast et system cut natural furthermore cut formula encode cut plane syntactic analogue enable although cut connection work cut integer attention integer program boolean value axiom allow ix b multiply integer key division positive integer common crucially due integer round cut inequality name cut technical much note trivially derive cut allow ix ix n find convenient restriction evaluation intuitively end broken stage assignment formula induction else either induction evaluate false case immediate formula note construction matter formula must cut restriction close axiom restriction hypothesis likewise axiom assign simplify axiom assertion axiom remain rule evaluate derive true sum evaluate therefore partial addition simple ix ix division develop syntactic cutting decision main use appear formula sum naturally every appear also restriction cut norm cut appear remark resolution cut sparse bound intuitively cut norm easily contribute assume cutting generalize restriction know special case cut restriction size nevertheless cut sparse cut plane restriction let cut plane partial assignment proof restrict cut encoding bound sparse norm appear appear assume cut else reject accept analogue decision alg programming sparse bound cutting cutting solve cut run constant sparse algorithm width guarantee conversely step note could repeat axiom need otherwise table eventually formula remain integer nonzero choice cut plane formula consider formula formula check checking formula take arithmetic claim apply implicit cut sparse cutting far list cutting assignment process probability bounded cut derivation else run unit operation assignment introduction primarily motivate weak issue concern form problem include axiom informally normally nothing take aside problem assertion impossible reason fail change world axiom etc fully toy domain show algorithm plan early explicitly modal criterion planning planning approach solve system reasonable approximation may fail likewise fail circumstance discrete time markovian might reasonably expect axiom valid small course solution probabilistic plan rather valid pac semantics rule pac plan broad involve reasoning semantic limited decision classical semantic concrete suggestion result perform resolution proof satisfy work al certain choice resolution modern largely et explore work speak algorithm variable satisfy decision add search different variable rule consequence notably propagation false except improve semantic one might learn learn extend partial resolution depend ask actually decision actually arbitrary e number iteration feasible decision match nevertheless obvious tuning illustrate might world modify serve solver highly optimize easier sensible exist solver implementation simply assignment loop remain count bind validity crucially clause common solver expect sharing solver black box formula assignment another sophisticated reasoning pac semantics involve pursuit reasoning might possibility hide since mask independently underlie enable useful extend produce formula satisfied find resolution perfectly although goal one consider et learn even acknowledgement heavily influence appendix show implicit efficient broadly speak axiom formula axiom satisfy say assignment probability valid hypothesis validity proof algorithm whenever argument axiom collection distribution whether polynomial efficient serve formally implicit learn thm partial axiom system query hypothese assignment formula polynomial pair hold assignment distinguish case polynomial know likewise guarantee recognize assignment point precisely behavior pac simulate must decide first hold part checking time decide formula distinguish case usual axiom since check coincide formula formula derivation often optimal two space root induction must worth attain axiom formula root formula continue subtree label carry space subtree utilize space derivation derive clause label root much derivation subtree derivation order derivation root empty resolution step empty still assume derivation either include contain clause occur subtree overall derivation space subtree remain configuration derivation subtree clause clause appear actually et refer number correspond parameter knowledge old although heart recurrence proof exist convert normal resolution dag cut direct clause label contain along variable source cut space normal general initial eliminate introduce along encounter cut past towards finally replace lead proof cut node set cut internal label clause source cut step leave third eliminate subtree root close source replace subtree subtree label subtree mention still child subtree great subtree increase describe theorem find exist return none derivation clause step find path root choose labeling clause remain recursive call describe recurrence verify induction note check bound find proceed child subtree tree induction hypothesis carry derive root mit mit edu theorem lemma consider answer formula base represent partially weak semantic semantic version space calculus explicitly formula crucially explicit knowledge example implicitly consequence distribution essentially agnostic pac logical reasoning hand background axiom may perform logical collection semantic typical know unless visit seem infer explicit knowledge whether broadly pac attribute sentence introduce describe pac query background axiom collection partial cope target roughly efficiently use validity preserve introduction cope rule quite impose class assumption partial assignment restriction impose applicability remarkable approach discover completely would agnostic al et note agnostic distribution pac pac also theory hypothesis barrier less perfect seem useful perspective ai problem typically frame axiom nothing axiom useful therefore desirable learn utilize furth fundamental application approach propose know graphical logic programming inductive programming address kind task distinction datum demand require simply answer naturally work framework task efficiently much svms boost predict reason reasoning couple task state could distinction broadly speak prove aspect incorporate cope handle clause mention able probability agnostic inductive inherently kind valid intuitively quality pac semantics quality pac learn logic precisely target attribute attribute formula pac learn every binding agree valid may particular formula threshold partial guarantee evaluate infer conjunction sequence valid polynomially turn property semantic special sound sound pac semantics sense degree validity merely note without derivation al actually return detail mention sake interested syntactic restriction restriction wish contrast rule syntactic restriction follow sense restriction close say formula hypothese assignment phenomenon restrictive impose artificial difficulty source integrate extract answering
algorithm develop section integrate elastic net induce symmetric vision box fall box constrain adopt surprising box non absolute keep instead problem box easy prox bind dual thank closed function check cut consistent obtain piecewise high interested preserving learn manifold aim neighbor minimize distance nearest neighbor low preserve manifold regularize tradeoff regularize convex solve h adopt optimize objective equivalently eq closed piecewise solution trivial overcome initialize wherein minimum write presentation part second totally unique slope convenience slope slope piecewise strongly convex unique point change suppose wherein proof update one nesterov since nmf explicitly nmf incorporate inherently structured item belong pattern feature greatly improve sparse representation successfully e group objective follow index group sparsity usually wherein group modify smoothing g gs optimize index group overlap eq gs convex slightly project project onto define sign therefore solve project ball projection complete time nesterov adopt simultaneously eq optimizing neither slightly eq solved solve sequence wherein counter construct optimize approximation decrease efficiently solve fortunately answer rewrite proximity dual problem back problem normalization equivalent solve construct optimize combination historical point smooth proximity efficiently replace sentence modify leave proof work due limit converge multi effectiveness negative sparse group nonzero select introduce elastic effect elastic coefficient select correlate take sparse part group negative term extension induce euclidean smooth nesterov optimize solve update net induce negative similarly h convex wherein replace adopt trade role part spectral cluster laplacian laplacian specify graph data symmetric inspire cluster q neither convex smooth involve fortunately equivalently rewrite solved successively update variable discuss normalize cut ht section compare residual nesterov optimize synthetic effectiveness compare conduct illumination video sequence challenge face image compare robust analysis include stop consecutive stopping first decrease dramatically ht iteration nesterov practical compare synthetic study conduct start identical dense high obtain ten versus round cpu obtain cost fewer rapidly confirm scalable also real world I face face pixel seven matrix value cpu second observation summary suggest relatively nesterov capacity challenge face dataset traditional algorithm include seven individual eliminate effectiveness deviation train jx face obtain test project wherein pseudo inverse neighbor test efficient implement bring element aforementione suggest method representation sample learn cosine nmf percentage dx x jx five outlier laplace conduct clean contaminate evaluate robustness setting encourage ht extended image pose illumination manually align image contaminate type pca use baseline outperform robust cause study robustness figure outperform contaminate laplace heavy tailed show noise contaminate successfully effectiveness base different algorithm reduce successfully laplace contaminate interesting noise confirm observation heavy tailed extent ht face database contain individual illumination position simply illumination give accuracy deviation contaminated figure almost contaminate image mainly contaminate illumination successfully low reason contaminate poisson ht select dimensionality observation laplace noise laplace noise image contaminate e face image obtain capacity compare dataset wherein reconstruct reduce show face contaminate laplace successfully conduct simple type experiment evaluate representation conduct construct wherein evaluate ac mi selection repeat report ac figure even contaminate perform presence serious illumination outlier cut successfully image eigen image second eigenvector recursively partition pixel equivalent e g row correspond berkeley wherein edge node product feature node parameter partition laplacian wherein normalize similarity label segment reduce dimensionality segment figure ht correspond successfully separate object mostly g low find three successfully find last help mean segment perform simply set aim segmentation deduce illumination sparse capability compare video surveillance background challenge foreground video reveal background foreground evaluate capability surveillance video whose select frame low background foreground video figure successfully separate detail recognition illumination reduce dataset illumination estimate rank face capability illumination removal conduct extended dataset matrix keep consistent illumination removal result four remove illumination confirm obtain figure computer set inherently many type gradient view vision retrieval annotation beneficial learn multiple factorize structured sparsity learn space view share compose view learn dictionary group contrast respectively achieve great robust heavy noise sift multi group sparsity wherein keep ht row compare image collect object feature construct half image remain image set latent experiment aim compare construct classifier mean cc map reduce versus present basis second zero view different reduce learn pattern different view imply learn private view view table map outperform present heavy tailed laplacian low much account low recover comparable robust component illumination suited negativity keep objective neither propose include nesterov successively update fast make propose nesterov approximation nesterov smoothing improve develop elastic nesterov optimize inspire develop experimental several show comparable taylor taylor nmf approximate two extension kullback leibler present nmf tail decomposition robust estimate part thus contaminate extend various develop box constrain elastic induce symmetric major fast extension residual nesterov approximate row row develop iteratively solution flexible extension extension nesterov smoothing recursively set smoothing iteratively extension conduct experiment synthetic surveillance nesterov variant nmf non factorization nesterov smoothing rank decomposition nmf popular approach approximate negative rank factor account particularly video negative negativity factorization negativity constraint part support success video environmental propose greatly lee nmf achieve great nmf enhance computer discriminant fisher regularize nmf recently nmf distortion several liu mathematical viewpoint nmf variant minimize kullback distance poisson call short optimize multiplicative tailed image sift contain noise traditional nmf tail lie show distant illumination nine component space presence outlier knowledge illumination video traditional outlier knowledge sparse model matrix capability laplace tailed capability perform contaminate outlier extend develop box follow integrate grouping sparse develop elastic matrix try model nmf program slow scalable fast residual nesterov smoothing approximate outer iteratively scalable flexible objective another synthetic world scene surveillance dataset efficiency nesterov smoothing optimize face illumination residual optimize nesterov extension solve nesterov smoothing conduct efficiency nesterov variants vi fw optimal non smooth fortunately nesterov show approximate primal dual strongly prox project space feasible dual exist w however solve prox prox function prox control smoother bad lagrange multipli corresponding constraint median back form smoothed continuously differentiable gradient continuous w define use smoothed smooth naturally motivate optimize nesterov optimal construct auxiliary historical sequence minimize solution determine constant round historical achieve optimize point respectively combine check precision exceed
f universit scheme achieve comprehensive wireless enhance wireless device effectiveness current focused detect primary tv wireless wireless business system resource primary cognitive exploit tv band service challenge secondary device context desire sense device identify secondary wireless band group decision rule first base signal psd cyclic cp take classify psd band signal study detail cp pilot network identification signature signal scheme signature wireless system exist scheme uncertainty problem receiver sense receiver validate combine group algorithm parallel unique feature utilize detection detection utilize list spectrum receive signal false alarm detect tv signal practical pilot period symbol detection metric compose adjacent autocorrelation analyze noise pilot mode pre align dft dft length cp length cp dft length cp ds par ds psd utilize slide name cp autocorrelation lag dft total cp cyclic cp cp length cp symbol slot symbol accord frame frame probability form threshold mac behave structure detection compose summation probability peak ratio par estimate psd tv dft step segment rectangular achieve resolution par psd calculate since high par psd experience selective call par classify logic threshold receive process metric decide channel ds different unified algorithm algorithms mode classify long receiver frequency normally simple exclude psd ds cp sum autocorrelation autocorrelation performance band stop eliminate sensing performance frequency psd psd sense par ds notice threshold cp sum noise practically nu become inaccurate mis original metric require denominator dimension receive require call inherently nu configuration spectrum signal specification utilize classifier cp dft td dl mode mode td dl dl cp cycle fm evaluate white signal capture combine snr receive noise directly sense test use ability scale cp nu nu model limit calculate fig show algorithm notably nu detection become nu close ideal nu ms tv channel center acquire transmission
noise also consist norm marginal marginal compute fit optimization use good report error fit empirically weight norm norm compare collaborative movie dataset netflix rating c movie netflix max test max trace protocol root netflix norm reliably select evaluate max give rmse smoothed weighted netflix local max smoothed weighted give achieve compare smoothed empirically norm give known norm netflix achieve rmse improve trace introduce max norm generalizes examine family max different option weight max norm lie improved matrix reconstruction simulate movie rating suggest improve material prove sign generate independently complexity first xx x k smoothed use complexity smooth start prove u write l apply hull norm x apply result last norm norm ball finally return rademacher prove u u u want need eq mark fact sufficient sdp formulation sdp ba ba bx see quantity inside square minimize infimum attain minimize subject norm semidefinite program main norm special element bound see note sdp lemma equality calculate write show cc ga cholesky since therefore see q theorem exercise title title institute statistic introduce norm norm generalize exist max norm weight norm extensively unweighted conservative simulate netflix norm theoretical matrix reconstruct accurately many modern application collaborative netflix video norm low rank rank describe introduce generalize exist trace norm include lie trace max norm give improvement exist norm allow scale defer material take n analyze accord similarly column common reconstruction trace denote number column unbounde choose emphasize trace location observe guarantee recovery factorize definition intuitive q rescaled rescale norm still regularizer error row marginal improve place penalization represent weighted column use weight trace poor suggest smoothing avoid marginal smoothed trace smoothed column empirically weight yield vector uniform trace empirically supremum generalization norm norm max large standard trace empirically smoothed trace smoothed amount marginal smoothed line simplex weight smoothed trace norm simplex row th compare max minimal decomposable trace et balancing learn adversarial arbitrary affect supplementary material penalty convex determine depend impose low simplex intermediate max norm correspond rely equal rescaled norm take smooth two familiar situation want flexibility trace norm generalization correspond singleton exponent smoothed yield e hence column loose upper trace sdp easily supplementary material sdp write semi include mention sdp approach norm max factorize version norm dimensionality max applicable wise respectively special writing simplify use trace norm factorization last material prove look unobserved approximately ball gain norm think root square whenever ball hull extension trace statement hold trace convex corollary sketch u see convex version supplementary material weight dual inclusion max supplementary materials bound max smoothed norm
per hold fair nb also poisson likelihood th document link also nb base appropriate dirichlet dispersion list table sharing mechanism various algorithm lda nb nb process nonparametric automatically learn active fair comparison consider sampling process document topic weight tb sharing mechanism evident nb infer nb dispersion transition smoother order order last iteration corpus along axis either convenient document specific red per hold list lda refer lda concentration parameter optimize base crf hdp dataset vary nb lda nb hdp nb beta crf hdp gamma active percentage setting train partition table rank bottom proportion usage small nb alone direct generally zero well explain mark nb process mark combination would allowed responsible could indicate frequently document indicate heavily percentage confirm mark nb negative count naturally apply disjoint mixture model nb process completely dirichlet borel connection discover unique augmentation marginalization nb process able fundamental property theoretical structural advantage distinct sharing property nb make discrete model importance nb dispersion ratio acknowledgment anonymous constructive comment manuscript pmf joint count f lm p rf ml l sm pmf express let pmf l f pf p follow proceed thm theorem computer nb process employ rate normalization whose marginalization lead count model nb distinct reveal bivariate connect chinese derive augmentation gamma advantage nb variety nb beta mark gamma sharing construct apply make exist poisson factor importance infer dispersion chinese restaurant completely measure binomial process measure analysis predict claim disease document corpus count geometric originally model membership word member assign topic appear latent count variable mixture random assign count model multinomial dirichlet dirichlet conjugacy dirichlet mixture enjoy popularity group hierarchical share hdp conjugacy solve alternative construction chinese break distinct atom extra expressive tractable paper count count assign nb perform modeling nb process bivariate connect nb chinese restaurant table develop augmentation marginalization nb compound representation efficient bayesian nb employ rate normalization necessarily process mixture model marginalization since nb directly beta nb conjugacy group hierarchical employ share nb probability measure wide variety model provide efficient flexible construction part appear material conference paper provide logarithmic connect chinese table restaurant number nb gamma nb process process nb construction wide nb process nb thorough investigation property relationship key modeling nb dispersion empirically paper review commonly prior study nb process gamma nb discuss nb modeling example let dp aa ap k assign infinitely disjoint borel sign evy random measure l evy even poisson intensity nonnegative random machine continuous base evy poisson valid measure gamma ab evy cr analysis evy cp c base measure draw beta therefore invariant normalize process process base dirichlet completely correlate recover g predictive also chinese restaurant number nonempty chinese restaurant customer random chinese restaurant pmf variance equal due heterogeneity difference individual usually variance poisson restrictive gamma pmf dispersion nb coefficient term show generate compound poisson logarithmic pmf mr nb widely investigate numerous nb parameter nb conjugacy dispersion challenge use provide incorporate robustness bayesian able incorporate information bivariate distribution total number customer representation binomial restaurant nb various distribution poisson bivariate distribution pmf chinese nb joint logarithmic provide describe count equivalent circumstance customer customer represent augment rr augment connection various show corollary key paper allow derive posterior understand fundamental hdp nb connection process gamma reduce group completely random measure subset yy illustrate united count datum measurable disjoint define poisson share completely let equivalent let measure place gamma measure atom lead e evy nb derive draw consist finite almost surely nb critical distinct atom atom number work commonly use mixture model chinese restaurant customer count augment compound impose gamma conjugacy finite lx lk atom g kx ji ji rule chinese restaurant describe multinomial poisson equivalently without variable poisson dirichlet use mixture inference augmentation rigorous discrete process augment regardless whether base continuous analytic gamma dispersion modeling group nb process restrictive proportion group count distinct poisson model count group poisson compound poisson representation bivariate jointly count customer nb nb suit derive analytic posterior joint count group modeling sharing dispersion express express jk nb process augment jk normalization group gamma nb process distribution closely construction fig graphical negative gamma poisson compound binomial chinese restaurant construction construction corollary allow form x gamma exchangeable chinese crf process variable become thus augment hdp nb theoretically process random borel count normalization hdp marginally practically corollary conjugacy achieve analytic usually alternative construction crf stick break representation concentration nontrivial infer simply one may augmentation method practical become base may estimate especially share analytically update play role hdp readily mass analytic nb variation gamma model construction treat hdp group process log gaussian analytic posterior provide option purpose mixture compare beta priors gaussian nb share nb dispersion nb explore group nb dispersion group atom gamma nb hdp normalization follow beta member beta nb construction find beta conjugate apparent lack exploitation beta nb let express dispersion parameter note nb empirically beta nb beta dispersion parameter nb consider hdp motivate construction nb paper beta nb measure count normalization modeling hdp suitable mark nb nb dispersion beta process independent mark conjugacy tractable nb mark beta gamma process point mark govern nb nb draw marked draw construction focus normalization fully tractable link ibp beta spike various nb modeling corpus group document constitute group exchangeable index simply word nb modeling nb express express hierarchical equivalently fully exchangeable drawing insight model denote place dirichlet topic nb nb process since nb nb nb mark beta nb process nb word without document time variable count produce topic loading atom encoding relative importance importance atom tn n n kn jk would exchangeable topic unify connect model differ function kullback leibler kl divergence dirichlet dirichlet topic mixture algorithm nb gamma lda hdp hdp geometric geometric nb nb crf hdp nb nb count sum factor gamma nb parameterized construct variety nb topic count imply setting eight differently nb modeling improve model gamma close conditional posterior require nb lda tuning topic replace nb topic share statistical strength hdp nb hdp fix e comparable construct process process model replace draw bernoulli use explicitly count gamma comparable focus ibp compound nb infer fit gamma explore nb dispersion hdp
advantage generalization exhibit power nature nature tail kernel generalization far introduce similar laplacian kernel hilbert kernel demonstrate apply machine regression svms indicate counterpart kernel entropy generalize event logarithm case shannon functional additive nature kullback minimum discrimination theory shannon moment entropy existence second obtain normalize constant maximization lead doubly translate version write retrieve special multi dimensional gaussian laplacian gaussian inside define kernel l write kernel similarity decrease kernel increase rate decrease parameter similarity laplacian see decay rapid slow satisfy theory al exist space definite point positive first present require counterpart x kernel n lead positive laplacian certain range law behavior positive definite obtain retrieve kernel rational quadratic obtain laplacian define retrieve map higher dimensional reproduce kernel hilbert rkhs definite significance rkhs rkhs exist explicit gaussian taylor rkh continuous non rkhs fourier transform integrable lebesgue must note negativity corollary describe determine kernel q transform follow constant derive fourier transform similarly fourier expand exponential term component odd expression infinite series substitute gamma half integer observe summation observe result agree radial radial present correspond laplacian fourier provide kernel gaussian set perform breast mass heart auto quality red machine class svms similarity variant various separate hyperplane lead formulate c c laplacian illustrate hyperplane various kernel boundary say value compare svms multiclass polynomial kernel dx accuracy obtain use kernel varied type laplacian mark good kernel case require normalization set laplacian well counterpart justify flexibility achieved low law distinguish sometimes kernel rarely upon function datum fix linear optimization wave uniformly space sample c data c parameter laplacian gaussian various fold cross power law relatively behavior depend polynomial paper laplacian distribution retain law kernel counterpart kernel kernel positive rkhs propose law recognize law community far introduce lead theorem motivated importance
million hundred text provide classical problem explain variance pca less researcher hoc rotation technique thresholding include svd generalize converge local semidefinite hoc solving indicate compute principal pca achieve hard cardinality formulation describe safe method block ascent propose observe real datum allow reduction safe elimination pca become pca review base formulation highlight safe elimination pre pseudo extend semidefinite pca encourage sparsity loss generality cardinality cardinality write convex leverage safe elimination pattern correspond optimum relaxation variance feature less elimination look safe elimination huge third practice bring heuristic rigorous feature rate algorithm converge coordinate ascent well converge row row context sparse estimation precisely apply problem impose update imply element element entirely diagonal directly general lagrangian technique augment add objective optimize exploit come optimal apply strictly convex address solution problem column consider problem fix translate problem inverse problem relation valid fx ts ty z hz hz x relationship eq ts ts tv ts tu u stage first qp exploit reduce solve polynomial equation follow summarize denote remove column iterate constrained coordinate solve polynomial j apply ascent converge optimizer coordinate solve product fix size compare htb fig much sparse eigenvector publicly available news repository text collection record contain file gb give gb memory difficult technique present ascent pca variance drastically show fig elimination computation text hope principal corpora cardinality end equal cc fourth education mind repository label document level article website htb component pc nd pc rd pre processing search range implementation ghz processor gb top principal st nd pc words rd pc word pc th pc surprising safe elimination
learn course possible block system block gradient magnitude hessian bind second noise relative figure move recent memory component gradient wise adaptive learning tb gradient hessian want increase increment decay quickly step allow eliminate otherwise update correctly local curvature invariant obtaining estimate hessian estimate gauss diagonal exponential move curvature close infinity instability hessian necessary regularization average sample start algorithm keep exponential average robust near value initialization detail sensitivity empirically simple algorithm sgd whether good estimate block specific estimation tb use variance term operate architecture similar adopt connect regard fact overhead even trivially follow reconstruction variant regression understand behavior sgd function quadratic figure effect fix rate rate htp htp realistic curvature example rate schedule optimum cc ce ce cr mse adaptive memory appropriately initially quickly gradually gradient allow algorithm increase variance circumstance curvature vanish clear htb ccc cr obtain epoch ten initialization mark bold statistically significantly sgd observe full element almost significantly tune tune cr training average ten marked bold differ tune benchmark line total error set balanced statistically main outlier element lead overfitte compare training sgd network training sgd detail good hyper low average line show dataset digit small namely reconstruction use four feed forward softmax regression multi parameter mnist case second denote c layer perceptron cross denote use mnist fully connect layer perceptron hide hide couple reconstruction reconstruction tb formally processes sequentially apply affine transform element output feed loss delta loss mean number hide layer initialize cifar add term avoid numerical instability diagonal never htp circle drastically benchmark table table across per sgd main outlier lead overfitte training table hyper sgd mnist benchmark scatter obtain well tune cifar perspective learn initialization long learning optimize expect update expectation norm linear experimental completely need manual systematic make relate adaptive automatically adjust distribution period system fine dynamic change landscape drastically circumstance appear deep heterogeneous careful input structural classical hope enable truly box want cl helpful discussion title part grant descent quadratic classical base lyapunov stability positive super martingale since sum everywhere tb h give I global case detail learn hyper auxiliary initialization long surprising accurate average interesting quantity compute unstable safe benchmark careful rgb rgb rgb stochastic gradient descent learn decrease adjust error one variation make task sgd approach systematic effectively remove tuning scale linearly dataset recent interest besides fast sgd sometimes well get sgd require manual initial anneal schedule particularly stationary contribution novel rate possibly learn different minimize estimate loss every separable capture estimate practice common local lead variation none manual tuning variety convex rate systematic signal recent review practical sgd machine learn quantity gradient decrease loss without require anneal change schedule quadratic preserve convergence describe couple possible index datum contribute
come difficulty directly outside mistake simply vector hypothesis induce hypothesis equivalence perspective indistinguishable picking change perhaps weak simple tree leave mistake learner representative hypothesis adaboost classify scheme w different mistake row mistake arbitrary open tie break formally also important set closure often consider mistake effective important zero indeed imply adaboost update would arbitrary mistake certainly adaboost mistake adaboost initial pick point w calculate adaboost fact trajectory understand secondary converge study adaboost natural essential establish invariant section establish existence preserve decompose segment define w w w similarly suppose w clean inverse simplex space regardless decompose union w w w express secondary initial average round converge classifier goal adaboost carefully certainly converge round adaboost must grow classifier tx change prove formally discuss theoretically section converge condition kind something adaboost intuitively adaboost converge crucial understanding discuss state give apply secondary center notion able need reader detail existence context great ergodic capture say mean f two proposition state show ready measurable function adaboost exist like subset limit adaboost abuse define tw typical dynamical property adaboost set turn point mistake motivate theorem besides adaboost w w j w recall equation optimal adaboost eventually away adaboost condition instrumental prove measure satisfy formalize speak condition say long weight tie hold world dataset try practice conclude remark optimal eventually pair either condition become roughly condition mistake tie tie dimensional subspace implication exist thus adaboost weight positive converge example evidence condition variety high world behave equivalently remove mistake remove create dimensional removing reveal dominate mistake mistake remove mistake removal become empty update must least positive mistake every presentation avoid clutter repeat phrase explicitly technical mention state theorem condition proposition ergodic formal measure capture paper cover second quantity adaboost combine hold borel dynamical borel preserve way extend construction assign whether sense discuss give practical perspective next w adaboost w simplify denominator inside term w lebesgue measurable implicit lemma appendix technical average adaboost borel function adaboost tw statement tw recall form yield convergence adaboost almost statement repetition clutter presentation phrase state implicit secondary useful delay next let measure follow w obtain example state theorem notation mistake directly representative hypothesis unlike hypothesis need scale appropriately exact select confusion function adaboost build converge optimal adaboost let impose deduce whereby exist x tx take c converge theorem condition use low theorem tx tx tx row difficulty extend input correspond mistake mistake hx I equivalence perspective indistinguishable picking trajectory however learner hypothesis pick simple simplicity number regardless equivalence representative briefly whenever learner finite number selection candidate construct pruning remove never repeat mistake representative reflect bias tree simple complex mistake mistake mistake remove dominate mistake equivalently row set mistake sometimes well mistake mistake must correspond representative weak final adaboost sample extend theorem section prove extend instance whole outside mistake hypothesis mistake produce differently weak learner use put classifier representative one demonstrate adaboost build combine converge representative hypothesis tx tx f tx similarly arrive closely follow full replace yielding exist intuitively boundary hard think degenerate situation know borel almost correspond subspace think unless something special ensemble weak output adaboost likely borel measure adaboost converge borel reasonable relax discuss appendix assume phrase proof classifier implication adaboost subset adaboost exist tx tx f converging limit express dp space sense place exist dominate hx say h dp dp dynamic adaboost away impose outline proposition establish weight represent satisfy tool aspect converge training generalization probability devote fundamental dynamical tell invariant couple deal system formally call induced topological induce treat w directly proof open set convergent motivation additionally continuous map admit limit convergent second convergent must let sequence subsequence g g tie sequence subsequence contain compact subsequence w g close contain tie impossible tie equation clear q continuity clear equation construct whereby compact convergent converge satisfie convergent compact understanding continuity mention tell away set mistake inverse mistake adaboost away w w w w adaboost rearrange update use algebra w I iw weak suffice element case depend w manner exist hypothesis equivalently mistake example optimal adaboost round start initialization weight interior already assume make w continue template claim know w particular mathematical construction contradiction conclude reach finally compact space follow admit invariant borel state slightly tailor follow borel continuity condition follow state whereby mistake round decision heart cancer behavior depict plot suggest text look mistake ignore mistake adaboost draw cc margin sign round margin round evidence sign sign round bin positive adaboost train logarithmic sign margin example histogram sign zero discussion depict margin round boost cancer margin objective empirical adaboost favor tie condition result one adaboost begin potentially theoretical pac generalization adaboost empirical adaboost context author student supervision perform preliminary test attribute define implementation implication simplicity effectiveness arguably adaboost practice decision relatively small c mistake mistake inverse false show simple adaboost display mistake differently optimal adaboost suggest rule ever reduction repeat whole decision induce dominate lead constant realization good plot repeat perform split realization dominate hence include lead size realization density fit provide effective decision table number adaboost world publicly available uci repository note c classifiers train total dominate breast decision use percent classifier column total dominate respectively number validation test table like time opposite rest refer reader thorough cycle adaboost cycle minimum maximize margin boost previously interest find also inconsistent synthetic adaboost open problem select adaboost adaboost meta use learner adaboost classifier decision classifier adaboost round see plot important growth suggest adaboost ever repeat classifier logarithmic learner mistake finite unique representative base base representative logarithmic growth figure continue long run breast number adaboost dimensional suppose course adaboost open effective imply fact adaboost illustrate take weak respect adaboost tight course assume world uci dimension hypothesis dataset abuse notation style convergence account hypothesis tt adaboost h adaboost representative determine also respect exploit empirically also potentially draw application adaboost use commonly representative follow respect recall practice part plot expect round round reduce considerable certainly trivial dependent uniform error general curve convergence even still insufficient increase experimental require pair tie tie weight limit provide evidence evidence dataset representative purpose demonstrate validity available world many researcher adaboost decision heart breast cancer tracking good mistake optimal mistake mistake ignore mistake adaboost initial draw simplex traditionally weight mistake tend happen go minimal refer support term index example output always index negative mistake representative round cause row essentially part mistake tie ignore mistake cause jump round support stop appear tie zero suggest decision optimal adaboost boost cancer figure round little plot margin essentially view converge predict adaboost adaboost tree move increase stop sequence still question interface algorithmic behavior discussion statement contribution work present thorough enough stand technical establish thus reach prove validity tie future address current common ml seminal publish ml adaboost adaboost produce margin begin whether cycle open empirical carefully synthetic impose yet contribution title adaboost cyclic margin reasonable implication seminal partly prove tie result largely use assume synthetic example provide evidence favor tie real world significant establish important relationship tie condition employ adaboost suggest imply cycle observe tie evidence figure immediate tie adaboost generally imply reasonably logical evidence none inconsistent practice repeat far lack tie adaboost tie condition imply adaboost cycle similarly emphasize considerable empirical whether cycle adaboost world interested adaboost cycle evidence cyclic observation informally steady imply view cycle come broad main average class generalization evidence know explain adaboost converge evidence tie condition word favor sensible natural reasonable involve potential serious work quantity object leave proof adaboost condition obviously considerable favor improvement perhaps adaboost lie simple adaboost mapping state introduction optimal vary slow hand strong may generalization adaboost tree leave formal tie create conjecture seem believe result likely involve weak generating amount relative characteristic induce currently interact hence careful statement output minimum weight among respect weight believe extend tie case purpose avoid analysis set beyond non adaboost weak output may weight hypothesis dataset extend version adaboost tie tie would serve avoid weight adapt analysis require careful significant effort beyond scope regardless appeal tie relate phase fundamental implication condition impose adaboost adaboost risk bias deterministic classifier via function perspective intend converge amount something converge time logarithmic tight generalization see selecting seem distribute dataset use benchmark roughly drift state consist weight outline corollary adaboost secondary quantity main derive margin converge learner result limit would conclude adaboost boundary measure probability draw boundary converge manuscript conference except thesis science author part work technical report appear national early award cycle round average almost ergodic tie thus adaboost initial suppose cycle n f cycle denote without n f result ergodic theorem perspective alternative shorter thought equally simple appropriate adaboost tw convergence adaboost cycle paper leave formal statement cycle invariant measure skip suppose adaboost cycle borel cycles cycles brevity measure borel invariant measure give proof cycle adaboost cycle cycle adaboost w nc define collection disjoint pair algebra derivation order well transformation invariant singleton sake singleton contradict element cycle yield element hence either singleton empty present brief formal definition depth subject interior axiom call order pair topology implicit call contain pair satisfies negative triangle whenever call topological topological neighborhood center bx topological topological inherently perspective every topological limit general mathematical topological concept space denote set exist algebra set non set countable countable collection pairwise order pair another inverse e triple event law axiom measure measure note measure satisfy measurable transformation measurable pre measurable preserving measurable decision round adaboost un dominate attribute generalization split round boost vc exactly class split size parallel split induce axis vc dimension threshold apply node ensemble mm constitute consideration probability draw probability ready prove sketch idea specific normalize formal pt set integer substitute eq union obtain turn particular substitution ready provide follow space size specific weight constant simplicity k k k apply bind expression eq turn particular substitution another dependent bind classifier probabilistic account logarithmic output adaboost achieve select representative function hold draw theorem dataset I basic induced hierarchy base classifier hierarchy note normalize formal suppose k positive substituting expression turn particular substitution similarly half decision dependence decrease corollary practice considerably considerably small f measure stability depend h q follow take side equation dp dp h immediate infinite w equality lebesgue dominate third function lemma theorem conjecture condition theorem department science university ny usa department stanford university usa usa department computer ny usa stanford university usa department science usa adaboost learn practitioner mathematically elegant theoretically sound practice test researcher decade machine establish classifier margin round function weight adaboost dynamical provide condition employ tool theory previous work cycle actually evidence hold weak condition evidence real world formally analyses optimize burden adaboost popular implement practitioner mathematically elegant theoretically sound researcher point characteristic adaboost exhibit formal understanding well characteristic converge adaboost refer intuitive description reach call adaboost term adaboost learn weak base call begin boost output good hypothesis base classifier class relative argue subject stability round world decade intuition performance adaboost increase round previous frame adaboost dynamical sufficient existence invariant briefly implication community adaboost cycle employ tool ergodic adaboost impose call rather weak condition tie consider condition impose margin paper optimal adaboost always meet tie world try create big lead hundred perhaps thousand classification available differ bag bagging use learner bias adaboost use learner form evidence bias variance converge gradually hundred add work I five speech hold want broad impact adaboost perspective form present paper establish demonstrate implementation applicability success reduce theoretical hold artificial intelligence experimental clear last decade award year still widely theoretical essence popularity still state consider adaboost perform input training mt ti w ti tx f tx adaboost stand adaptive boost algorithm round sequentially fig algorithmic description output ti overview remain part remain introduction adaboost attempt explain view subject contribution significance work adaboost section round adaboost margin error mild substantial formal proof mathematical tool adaboost learner dataset substantial evidence favor tie summary discussion open problem adaboost add weak hypothesis intuition suggest combination increase long meanwhile performance ensemble improve stationary iteration go general accumulate appear adaboost generalization round appear reader evidence max theory put disease dataset heart disease dataset axis letter note time unlike previous canonical adaboost seem overfitte behavior vc dimension adaboost seem attempt context bound really explain generally inconsistent fundamental graph remarkably generalize combination complexity long meanwhile remain work adaboost raise previously adaboost try misclassification rate dimension margin th minus vote adaboost produce publish margin adaboost less accurate apparent drive force behind far generalization training independent provably reasonably tend margin effective adaboost generalization margin loose precise couple question different overfitte take appear question ensemble notation question I bayes asymptotic predictor inconsistent year long time thought answer yes claim behave like select adaboost strong law number certain learner adaboost version article mostly risk adaboost put put try indicate go generalization adaboost set adaboost instead behavior resemble produce behavior understand informally follow conjecture later conjecture ergodicity finite misclassification low weight increase complement decrease top understand infinity combine produce see evidence favor growth pg addition although optimal make ergodic ergodic dynamical paragraph turns establish adaboost suggest adaboost classifier condition turn characteristic learner adaboost exhibit tendency overfitte dataset heart disease plot pg overfitte datum stability stability help adaboost overfitte adaboost stable analog gauss minimize enough terminal part population learn paper similar sense tool real g invariant requirement tool e convergence round weight example margin show converge learner favor hold high ultimately emphasize ergodic cycle turn need reader discussion last technical adaboost real world dataset delay condition concept round tie weak representative hypothesis one minimum significantly express mathematically provide validity converge answer question go happen believe contribution area reasonably positive fundamental question state wide behavior reasonable generalization cyclic non cyclic mild extend remarkable adaboost explain converge us community evidence trivial trivial know exist dataset implication adaboost round come perspective emphasize space ideally goal classifier something mostly version forget nothing property adaboost boost effective establish convergence generalization state formal present paper need recognize usefulness important well understanding iterative size least decade asymptotic variant non similarly behavior convergence view despite know update interest ml fundamental area without work generalization condition empirical provide convergence rate deviation bound performance adaboost asymptotic result convergence among thing mild informally paragraph quite reasonable mainly adaboost tie without formal adaboost provide considerable evidence hold uci repository find practical adaboost believe concern ml article work publish adaboost quick clear address reason work actually discuss form variant understand go discuss believe relevant previous concern manuscript try put present work believe concern concern form aspect adaboost minimize type demonstrate adaboost iteratively exponential weak context section minimization understand exponentially fast show adaboost minimize provide prove enjoy achieve explore primal adaboost deal term attention complete version manuscript repository concern several perhaps meanwhile interest basic adaboost classifier average relate mostly within consider round adaboost let variant consistent stop example inherently concept distinct consistency adaboost error training say differently concern generalization important note adaboost machine complete version manuscript complete mostly deal convergence put close enough dynamical demonstrating cycle case prove strong asymptotic fully understand margin paper understand cyclic remove condition cyclic whether adaboost cycle date present relate version adaboost version I course like start regard art prove regard art paper follow behavior adaboost cycle adaboost support margin train typical mistake equal incorrectly element transpose mistake value mistake matrix hypothesis incorrectly semantic difference syntactic prove simplify adaboost cycle indexing mistake adaboost vector margin submatrix involve adaboost produce theorem start discussion previous paper something line several adaboost margin interest state adaboost turn good approach margin lead maximum margin support cycle also prove cycle tie tie adaboost cycle fact adaboost easy formal ergodic straightforward approach extend relatively via large typical select weak hypothesis relate condition selector characteristic adaboost convergence state state experience high real dataset condition formal formally speak adaboost face hypothesis w ti concrete selector great fundamental involve aspect get work begin kind adaboost classifier generalization minimize exponential iteratively loss procedure understand quick later enjoy rate achieve similar implicit meanwhile adaboost averaged margin concern adaboost zhang recently adaboost distinct consistent error limit set error iteration sample early adaboost early adaboost case application ergodic cover technical adaboost high technical formally roughly adaboost tie weak ti delay formal later concrete prove behavior cyclic case cyclic behavior higher dimensional always remain date arbitrary weight measurable remarkable explain converge
skewed database statistically plausible goodness fail event via imply law indistinguishable fact give treat event treatment theoretical single primarily drive perspective statistical physics population individual limitation apparent law remarkably robust vary somewhat year change international independent attack economic development country size age experience attack illustrate ci confidence interval observe sized tail log ratio ratio neither significant indicate law plausible alternative ambiguity illustrate difficulty identify tail likelihood rare event imply slight visual deviation empirical tail three past severe automatically minimize kolmogorov statistic tail truncate empirical provide thorough motivation ks law discrete tail appropriate count finally use yield city empirical leave large derive analyze covariate versus international country bootstrap accounting tail event bootstrap parameter agree historical probability interval event global historical record unknown jointly estimate difference concentrate produce tail cause slight curvature empirical mid arise aggregate event single curvature automatically wide ensemble correspondingly density confidence confidence law estimate robustness especially alternative different estimate form alternative x restrict appendix exponential heavy yield heavy decay asymptotically fast law parameter choice marginally traditionally negligible event comparison tail alternative yield ci ci chance power law consistently away outli bootstrap well free law none place failure law attribute roughly track different covariate illustrative factor covariate instance international different rand database exhibit appendix ci figure covariate exclude international event develop important tail specific power law exhibit heavy tail fewer significantly handle categorical covariate produce estimate examine hazard large contribution follow political generate principled statistical forecast event make stationarity estimating event rand calculate number past trend increase exclude significantly maxima severe remain latter trend decade expert new decade lead unless replace potentially specific prediction optimistic attack year return ii status remain refine forecast production use quantitative forecast estimation historical generating estimate facilitate alternative model table summarize forecast scenario status chance decade optimistic event forecast chance strongly popularity great likelihood progress move toward optimistic event place range statistically unlikely illustrate say calculation refine incorporate address treatment event several investigation stationary generation unlikely term technology exhibit event use identify subtle trend impact implicitly actually occur interpret condition historical effort extent policy impact accurate achievable policy actor similarly actor responsible incorporate capability say grained make country incorporate attack likely unclear estimate scale appear play perhaps independence scale central averaging birth rate population contingency unclear although event study control appearance small scientific true event whether classified historical development forecast provide principled naturally incorporate source uncertainty tendency skewed produce reference properly historical rare occur six time could probability observe sized world year choice much anomaly six large statistically unlikely historical record event conclusion frequency degree economic development country international type large international chemical rare use oppose datum historical hazard event change consider forecast next year comparable course treat phenomenon hold example micro extremely long deviation overall process true statistically justify global spatial long temporal scale apparent scale sized global event fundamentally event unclear likely generate event question global event expectation aim event incidence dramatically tail frequency severe suggest possibly political social effectively describe detailed trade benefit preference actor investigate global pattern offer complementary rational actor framework implication normalization tail likelihood attack integer I incomplete equation straightforward past show provide discrete take moderate value experiment percent automatic goodness ks method fall select fit value event several improve statistical analysis large alternative tail modify significantly easy present tail quantify regime analytically draw ensure generate fluctuation power law whose appear generative error mae ratio observation fairly error decay decay add outli small absolute mask example reveal mistake panel probability contaminate fairly sample method correctly relative probability present parametric model ii iii covariate refine historical large process I random distribution iv version use department report event historical sized slightly normalize constant value repeat event process approximate continuous event observe one simply value complementary cdf substitute rand database substitute one period calculation decay slowly power law model severe restrict least accurate repeating chance event event rough total main alternative global global database count evenly record fractional yield analyze rand law fit estimate binomial large inclusion highlight broader good etc round report figure drop cdf round power law number thus year ci repeat along law notably heavy rand center roughly agreement rand furthermore estimate dramatically rand support rand mainly international country international international character provide robustness versus international ii change decade pre event database heavy tailed international international ensemble visual empirical international rand event dash ci dash international large seem confidence fundamentally cause tailed international include international exhibit analyze condition estimate event economic operation development cover rand event tail decay slowly chance year period slightly unlikely event result fit place weight yield ci attack generalize algorithm covariate case event classify chemical biological include device iv sharp vi empirically least generalize carlo categorical original jointly fit covariate least zero apply ks technique tail univariate hazard hazard repeat bootstrap tracking probability drawing set bootstrap estimate cutoff size unlikely calculate event type great type chemical object estimate hazard value indicate dash historical frequency historical rand future exhibit relative frequency attack unknown exceed trial category w online discuss skewed political economic network six large event life modern accurately estimate upper tail present generic make combine parametric historical observing sized result condition global variation economic international datum drive next decade attack event modern history people attack sized decade answer trend global political rare determining generate event common would long expectation planning effort chemical nuclear event pose quantitative mechanism historical record contain event estimate mechanism large alone big fluctuation poor tail misspecification severe financial model event aggregate covariate example resort effort qualitative human actor attack quantitative adversarial treat event like outlier say little hazard algorithm event social particular make broad rare combine likelihood model quantitative uncertainty
variational guarantee define family ensure alternative motivate provide optima view proof college science bt united optima discrete unified description circumstance concave particular method sparse maximization vector differentiable prefer alternative relaxation wise popular discrete simple expectation distribution define space adjust lower tight enough mass restriction bind objective smoothness deviation objective dispersion increase give sufficient concave purpose way construct alternative make q bring differential lebesgue integrable ii integrable non discrete differentiable normally sufficient prevent form even quasi concave advantage concavity kullback leibler variational generalize linear distribution concave pf concave mean factor use triangular element component concave top figure lasso optimal lasso objective away closely match objective result optima differentiable section lasso objective bind problem choose approximation optimize bind implement surprisingly sophisticated main interest svm l bfgs alternative square encourage objective origin optimization variational gaussian expectation affine objective variational q cholesky cumulative hessian straightforward differentiable use alone z split term give maximal objective hence assume optimum find find optimum within first normal iy ny mean dominate solution sparsity iteration approximation inverting gradient whose numerically closely experience optimize set time reduce absolute current cm sd sd iterate sigmoid c large sd iterated ridge sigmoid test solver schmidt matlab relative since optima measure algorithm give mean deviation indicate capable optimum scale well smoothed approximate use sigmoid lasso subsequently effective optima also introduce component converge gradient vector underlie element subsequent create clean label label run rise impact vector package competitive state package nesterov method backtrack line shrink convergence relative cpu time algorithm implement support machine find hyperplane separate hyperplane dimensional separation possible map space separable machine add slack misclassifie datum cost soft solve lagrange multiplier convex refer preferred convenient deal primal remove primal problem objective minimize slack possible yield minimized slack loss misclassifie hinge whole hinge loss convex convex across hyperplane allow reproduce hilbert without without objective kernel k nm amenable choose gradient variational difference difference maximize condition tolerance hinge derive newton huber form huber tighter way give hessian dimension whether quadratic hinge loss small huber solution point lie portion offset except hessian show definite cubic inversion newton construct generating choose completely separate kernel positively label multivariate normal cost coefficient finding value matlab training set normal modification shrink huber dual bfgs package treat huber similarly analytically component iterate toolbox set matlab scale quadratic highly small finding fewer extremely fast surprising shrink roughly gradient opposite evaluation huber take inherent shrink schedule advantage shrink impact huber
code sd sd vc vc element proper appendix thus sd sd sd equivalently great sd j subset fr bound e p deviation close clearly ix code theorem point coarse x least theorem difference loss double deviation loss original loss deviation loss least eq consequently sufficient handle range eq proposition union stage theorem appendix lemma elementary sd p l remain prior across prior choice assign infinite dependence dictionary cardinality bad cover good good overcome require device justify unlabele yet switch rademacher average predictive level coding measure unlabeled simplicity guarantee bad case special disjoint conditional rademacher exploit proof lemma trivial non continue important role overcomplete set deterministic previous low deterministic immediate probability bad event occur last line treat subsection choice hence large deviation code denote mostly random sign definition q routine sign fix rademacher bounding rademacher average dictionary factorize u dimensional dictionary choice encoder gaussian behave standard normal fix proof use show difference perturbation solution linearly constrain program dependence index lemma supremum expectation supremum depend k rademacher dictionary factorize isometry rademacher complexity partition index index occurrence precise index note minimum taking intersection approximate disjoint union union arbitrary choose disjoint stability approximation disjoint useful rademacher complexity loss rademacher combine independent draw proposition result prove theorem bound elementary choose nearly bind fix unlabele eq sparsity margin label unlabeled sample digit versus atom level zero per dot code train digit versus atom sparsity code dot margin away separate employ task digit predictive coding per gradient normalize unit apparent sparsity indicate colored axis non trivial margin margin factor bound stability consequently infinite unclear encoder important open rely strong require turn true greatly would constant work upper sparse thing remain unclear one fundamental importance dependent provide small encourage code margin follow lemma key sparsity establishe perturb exploit old flow sparse code review optimality imply code operator likewise let observation symmetric argument second norm optimal close q consider dual definition follow optimal claim hence eq convenient roughly stability optimality combine eq rhs obeys newly bind rhs implying perturb sufficiently consequently q objective equivalent formulation z plug expansion incoherence result xx sufficient proceed characterize reformulate satisfying must consistency check optimality satisfied clearly sign check hold sign consistency satisfie replace side chebyshev loss assumption follow expand replace choose yield equivalent new related solve eq q choice assign q recall measure sample entire equivalently level u x x k apply program constraint denote equivalent optimal z z add yield cc zero become reduce operator z select similarly coordinate cover number radius space dictionary property one must careful cover proper cover proper cover cover simply subset ambient banach space banach cover cover number cover banach cover proper cover number name ball radius hypothesis name dictionary name name joint description name name return name description singular incoherence sd sm description second label description description w w description description average x ss theorem ex corollary coding learn hypothesis code recently variety establish bound predictive cover overcomplete exceed dimensionality setting stability sparse encoder measure fundamental stability lasso characterize stability code dictionary overcomplete decay infinite respect theory dependent dictionary architecture predictor theoretic many learn architecture base representation representation learn represent k z predictive loss coding seek dictionary evidence abstraction higher predictive intuitively atom generalize theoretical knowledge code previously generalization set extend introduce certain difficulty control perturbation predictive perturbation stability hard realize assumption interaction learn dictionary assumption hold newly provide ambient dimension infinite setting dimension free hold certain mild perturbation overcomplete incoherence system induce column bind absolutely stability dictionary perturbation collect measure product unit ball equal consider iid combination atom dictionary xx solve risk erm objective k space overcomplete handle overcomplete produce fast linear hypothesis predictive class hypothesis measure code objective formally code regularizer analyze erm objective know return certain sparsity stability holding present potentially additionally influence stability learn well apply define encoder induce regularizer design encoder perspective theoretical behave lack strict image unclear parameterized map drastically small perturbation hence perturbation central statement throughout sd value among guarantee code sense introduce property encoder point exploit sample property give point sample importance margin property flow intuitively inactive second sample hold label learn hypothesis highlight stability sparse dependent relate overcomplete minimal learn root unclear set introduction free dependence incoherence exhibit appear significant setting let match rate fraction open question represent dictionary reflect code perturbation quantify sparse stability theorem enough primarily extent extent dimensional previously factor unclear appear price encoder encoder stability error lasso stability stability readily use ensure result learn incoherence sparse perturbation since reach theorem pass stability drastically need avoid aspect margin hold margin away zero inactive atom margin integer small learn predictive learn incoherence trivial low mixture elastic fall guarantee scenario induce regularizer simple datum overcomplete bind infinite dimensional simple attain label iid drawn marginal specific cover cover dictionary metric essentially due main allow depend label completeness prove overcomplete deviation empirical deviation risk overcomplete lemma specialize
combinatorial graph perfect approximate generation generate approximately bipartite combinatorial function mapping relation arrive approximately satisfy assignment feasible uniform generation element speak reverse precisely sort object string uniformly output r randomized output previous paragraph input perfect assignment boolean uniform assignment satisfy assignment boolean hypercube sampler generate assignment proceed briefly statement definition stronger support put element uniform thought suggest impossible efficiently achieve even problem generation satisfying differ thus impossible meet another generation require object definition would approximate generation attempt intuitively appeal schema algorithm accuracy object interesting efficient inverse generation efficiently solve hardness version object define boolean term etc relation sample satisfy assignment high aware unsupervised specifically sort uniform satisfy assignment member know sort similarity since difference goal goal output significant demand hypercube hx set constant acceptable hypothesis function variation distance indeed inverse uniform additive aware boolean error discussion essentially directly yield boolean assignment element uniform access learn additive less algorithm draw failure positive uniformly finally reconstruction subsection inverse problem inverse hard inverse uniform reconstruct mrfs much make past decade concrete problem seem mrf reconstruction reconstruct need impose uniqueness random goal high case underlie give approximate generation note deal satisfying assignment assignment negative present present obtain approximate uniform technique count statistical call section roughly dense generating approximately possible uniform count need reason detail section technique function class construct online point know uniform generation counting generator random satisfying assignment carry stage obtain result check statement problem uniformly second study class full size formula boolean variable run output formula challenge formula time around view output hypothesis term observation since uniform distribution algorithm improve art learn open use query formula carry technique main class either reconstruct reconstruct efficient inverse generation approach hardness problem near computationally signature scheme public roughly speak simplification see full precise statement parsimonious construction signature inverse uniformly generate satisfying hardness positive lie quite boundary generation statement construction signature scheme time uniform problem either follow formula intersection signature hardness parsimonious type hardness approximate class hardness result generation particular obvious produce instance assume seminal np oracle approximately interesting ask require adaptive np whether hardness understand polynomially assignment satisfy assignment formula access simple elimination formula access assignment formula signature hardness hardness sometimes barrier along line statement technical version construction circuit verification result construction construction easy plausible assumption generation construction generation computationally uniform version plausible uniform hard uniform general organization section approximate inverse suggest probability point ds definition total ds dx dd function usually threshold function variable formula stress view take input dimension denote conditional distribution f familiar counting version precise notion approximate count boolean time output generation function efficient f let class variable randomize uniform u uniform dependence dd tv inverse need circuit bit bit clarity emphasize accord formally boolean approximate access test one thus need select pool candidate hypothesis c finite exist sampler evaluation oracle output call oracle perform arithmetic operation output certain crucial difference proposition explicitly carefully kind essentially detail full competition e algorithm failure describe analyze competition subroutine proposition accuracy confidence parameter kk om j could seem thing bit simple approximation thing since approximate totally competition iw variation purpose ideally oracle approximate turn similar ij iw particular proceed crucially multiplicative exact show eq similarly establish establish iw yield second inequality hence choose oracle provide subroutine oracle make draw perform arithmetic operation p j subroutine amount consequence confidence use ix empirically consider h I subroutine subroutine j ij follow decide membership give jx random sample query total claim perform competition set estimate ij jj return competition competition output sample arithmetic straightforwardly verify remain correctness union correctness hypothesis immediate follow competition winner intuitively likely winner competition winner winner r z ij r mutually follow chernoff j beyond stop winner competition ii winner otherwise paragraph competition winner correctness follow compute read sample read bit operation encoding set hence read need bit operation belong call oracle run dominate oracle call competition distinct exist output sample access oracle draw n dd j exist otherwise sample running bound choose hence never competition output ii j competition never variation probability failure assume element ii conditional condition easy proposition extend sampler invoke repeatedly event repetition ever failure generation conceptual heart speak algorithm construct essentially state roughly speak approximate uniform generation iii counting query algorithm suffice uniform generation statistical recall natural pac learning allow estimate access value v sometimes write state accuracy follow boolean randomized input oracle query maximum tolerance parameter ever say bound succeed independent say learnable independently sometimes write accuracy throughout independent pac label robustness useful nf minimum value tolerance ever execution complexity access simulate hypothesis confidence time introduce boolean give unknown output subset much smaller particular enough moderately dense take variable boolean input fx g g state conceptual follow maximum query minimum ever course inverse function analysis expression long uninformative inverse approximate generation work three main conceptual g generation support variation step hypothesis third notice seem introduction reconstruct object process standard consider fraction would kind run obtain good variation imply learn motivate possible accurately query positive draw technical learner issue conceptual seem precede motivation issue arise implement step concern algorithm close need able easy obstacle see version efficiently query claim issue handle statistical previous paragraph sufficiently accurate small provide query oracle succeed multiplicative full accurate function approximate counting accurate estimate value cover obtain distribution guarantee true variation hypothesis proposition find use distribution distribution generic count version theorem ready proof need query positive example sample algorithm efficiently valid e one oracle simple whenever simulate early access sample number computable tolerance use take range additive use empirical ix ix refer empirically lemma identity simple estimate empirically confidence way subroutine sample query fx f expectation dd need total subroutine run bind bind latter triangle simulate performance evaluate minimum provide course execution exist access efficiently accuracy dd om fx procedure query describe algorithm output fx hx fx simulate f obtain moreover confidence time certainly event query additive guarantee algorithm complete tell query access draw estimate sample even exactly sampler sufficiently distribution bound dx desire imply subroutine additive accurate estimate proposition distribution efficiently sample evaluation output subroutine circuit compute om precise subroutine simulate u failure gx hence get access count computable u take confidence bias run input return counter return claim argument assumption gx fx recall present succeed satisfy algorithm dd output outline subsection fx run n learner generator simulate run sampler rejection accord approximate generator hx ix output let exclude code approximate generator error fx hold sampler polynomial correctness step succeed confidence assume throughout definition satisfy property u fx event distribution definition generator probability call union bind failure call successful obtain goal call subroutine time give sample obtain finally obtain conditioning b successful fx hx union analysis hx observe hx x u hx ii remain item run establish I close total attempt draw output sample output default fx tv consider produce immediate triangle everything first probability code call approximate generator call fail least thus probability draw hypothesis fx hx imply therefore conclude desire big probability satisfy assignment failure tv happen lie happen draw identically tv claim u fact writing equivalent construct bind eq rhs proceed finite term rhs term zero property third property tv rhs turn bounding case guarantee learn numerator eq imply conclusion bound f g f h g imply third bound complete finish establish run verification run run regard algorithm claim function parameter sample union turn make draw approximate oracle subsection sampler close satisfy fx going set use need approach follow target available subsection construct detail randomize check high pass pass check give desire approximate evaluation g mx return fix consider run ng ig follow eq ng ig ic guarantee q draw I md probability none go forward hx multiplicative chernoff bind failure failure circuit value output construct simulate evaluate either evaluate step definition generator string behave consequently take lie combine recall desire string alg ready theorem complete nn right element negative ok correct iteration reach mt say mistake learn end section general give problem class boolean think uniform parameter run fast term n significant learning theorem provide count main technical contribution give fourth ingredient present analyze give algorithm exactly generation n approximate counting count tn fast know literature formula run boolean rather formula close intuitively provide target actually know learn accuracy explain e theorems class length boolean run evaluate query hypothesis moreover polynomial property let boolean x fx kk fx note pass learn target function intuition behind oracle hence target tolerance learn tolerance learn boolean regard theorem sc u gx sc c time abuse terminology function theorem step algorithm execution simulate algorithm theorem requirement succeed bound describe straightforwardly verify overall theorem target quite reasonably consecutive repeat high pool reasonably straightforwardly algorithm f output initialize ss repeat string claim start precise jx st term draw independent sample iteration least ss ready prove claim candidate iteration union every jx jx sf take term union term define gx item fx gx f bind least satisfy pm x approach term conjunction inverse generation class run kf kk counting formula formula boolean sum disjoint define express tree special first value get work satisfied assignment add say include high consider satisfy assignment certainly position exact choose satisfying assignment include repeat union hence boolean proving whenever hence clause prune pruning sample uniformly random statistical thing notice deal exact counting oppose chain one distance run via count really point expand space consist simply write trivial writing variable sign make I imply hence I whenever variable assume leave depth thing make since repeat round going want variable tree sum learn much let begin initialize repeat nf satisfied assignment say c ai nc ix assignments independent choose uniformly conclude assignment small remove less hardness generation class boolean learning theory base hardness hardness study hardness learn introduction potential inverse generation reconstruct sampling assignment object generation one constitute proof generation efficient specific step provably well effort get hardness come two signature provably hard hardness assumption assignment hardness code easy satisfy standard subsection hardness detail prove relate hardness speaking say signature hard generation intersection function begin public scheme simplicity suffice verification algorithm triple take message signature verification every signature first fix signature public valid sign message possible sign message signature range potential signature message g special sign sign message sm dd dm message similarly algorithm next notion attack signature holds choose need define notion hardness approximate uniform tn inverse algorithm generation invertible relation reduce binary hold furthermore class binary relation whenever say invertible corresponding relation relate inverse scheme boolean tn efficient generation invertible signature verification public sign point point generate translate sign generate new sign signature scheme formal contradiction generation close target key pair adversary signature v polynomial circuit invertible adversary adversary signature I xx follow claim random let correspond distribution statistical marginal coordinate special image apply likewise apply instance run time produce whose output definition probability adversary succeed produce adversary time adversary contradict security scheme literature requirement similar signature different generality state follow slight variant appear probabilistic eq mention state theory plausible sake hold hardness go hardness et signature use say signature signature message every message signature construct signature variant go back get special signature state special obvious special sake signature signature scheme function invertible generation constant depend hardness generation invertible conjunction clause standard invertible exist inverse corollary formula learnable uniform generation formula variable fact invertible reduction polynomial invertible say problem formula variable occur hereafter constant approximate invertible reduction time reduction follow show boolean inverse uniform intersection approximate integer np complete invertible outline boolean assignment show easily verify invertible see polynomial observe instance intersection hardness result directly prove np extension hardness uniform generation formula conjunction clause trivially begin one need follow hold instance circuit efficiently computable follow helpful invertible one let distribution obtain uniformly uniformly choose fact uniform extension assumption corollary assume inverse uniform close target signature security v v meaning adversary receive message signature choose satisfying assignment adversary denote output call close hence imply succeed signature pair ensure probability success run probability reduction reduction technique invertible reduction note invertible invertible reduction many virtue time invertible construct introduce every put vertex graph assignment every cover computable invertible assignment vertex cover assignment vertex vertex assignment create every vertex cloud polynomial formula map cover vertex cover cloud include else include vertex cover vertex size construct formula correspondence satisfy assignment cover far vertex map vertex cover hand assignment cover assignment map least satisfy assignment satisfy cover gx map conclude true hard approximate uniform absolute multilinear class variable degree generation absolute every monotone formula degree true hardness intuitively correspond function efficient generation reduction would np invertible make thus prove class algorithm code mac hardness generation class begin definition see triple property randomness verification message purpose hardness result special hardness specify generality mac say message exist likewise code say potential tag message tag r tm r tm r cardinality tag security attack message eq see hardness inverse uniform mac message inverse towards contradiction generation run output statistical security mac tag independently mac special uniformly tag message satisfy set taking compare definition output sampler satisfy suitable choice recall contradict mac unlike signature scheme cf hardness belong special construction intermediate verification construction hardness tend hardness approximate uniform intermediate year hardness noise version return independent sample study intensive fast take assumption let let henceforth assumption conjecture seem closely assumption whether formally perfect investigate run plausible strong dx vector input output state say henceforth similar consider show imply minor change security along line minor change proof subset problem hard proof lemma key assumption imply assumption get ready generation random string randomly verification tag operation mac theorem suitable mac assuming describe mac observe mac describe mac mac assumption provide description mac mac mac except exact come uniform remain verification generation easy towards class boolean n multiplication verification r say uniform generation algorithm crucial cardinality return randomly let draw ir procedure simply run distance give efficient approximate give quasi polynomial uniform crucially answer answer fact hard detail versus assignment show randomize polynomial formula assignment immediate even f tn get every element particular inverse approximate uniform generation draw trivial argument generation uniform hard inverse question function uniform generation polynomial approximate combinatorial object assignment let undirected symmetric generation relation vertex graph receive n easy permutation understand define graph unlikely decade effort fast strong claim establish uniform uniform relation randomize
asymptotic normality establish sufficient formulae literature asymptotic direct state checking avoid transformation model l semidefinite ccc nonzero assume contradiction vector uncorrelated follow since converge surely express converge weakly random uniformly uniformly bound surely infinity surely show investigation state model space except ergodicity satisfy short basic equivalence model mainly discrete estimator main time provide auxiliary result follow state discussion space condition develop finally introduce technical need derive process randomness increment almost evy evy mt gaussian l evy purpose enough evy l evy moment use time process l evy process differential dimension drive evy true general l process differentiable interpret consider suit type equation output multivariate causal easy negative part unique strictly second eq b immediate drive evy representation analytical allowing replace apply complex spectral density h relation prove tu satisfy ready q fact spectral density last h complete converse identifiability proof find rational f uniquely determine transformation orthogonal sample n nh partly space zero tu rational less notion explain origin terminology rational triple call realization algebraic realization convenient let realization realization important realization notion play definition mn realization n investigate realization minimal matrix tu assertion straightforward generalization triple differentiable show characteristic q side second infinitely distribution th evy analogue parametrization family continuous strongly normally strong consistency consequence four hold space identifiable multivariate satisfy logarithmic moment mix theoretical applicability practical research go notion confirm study bivariate present smoothness drive evy satisfie parametrize consistently normally distribute additionally evy hold unable verify analytically explicit check computational effort description literature canonical decomposition rational rational determinant polynomial satisfy polynomial root triple degree denominator polynomial kronecker index k small kronecker transfer process process unique structure block th vary concentrate kronecker rational invariance inherent restrict parametrization non description integer rational kronecker realization parametrize matrix th matrix order normalization one special h ij b h preserve multi structure matrix prescribe method inverse example cc cc ccc cc cccc cccc n cc cc cc z z z parameter present bivariate index drive evy gaussian stock return increment determine choose skewed covariance simulate apply eul stochastic making value realization compute numerical routine conjunction trust mean deviation report moreover entry asymptotic display c std est std driven horizon second report fourth deviation obtain one bias small accordance large deviation construction confidence uncertainty deviation desirable overall simulation study acknowledgement support international science universit grateful grant author acknowledge financial study process theorem theorem theorem eqs eqs process general space move result linear model many decade applicability detailed discrete precise formulation investigate second quasi maximum surprising mathematical case eq high noise sequence space aggregate asymptotic normality general satisfie restrictive satisfied sequence absolutely process recent deal weak make mixing model analogue familiar move introduce drive motion evy heavy tailed occurrence path characteristic many series finance multivariate give possibility time series interpret expression apparent process describe detail approach investigate reference therein typically digital observation become recent year variant maximum investigate second estimation multivariate second base spaced discrete coefficient restriction drive evy however autoregressive drive evy consider consider autoregressive letting infinity autoregressive move drive evy parameter frequency time horizon equivalence aim directly therefore condition induce turn paper sample finite moment mix ensure absolutely continuous component need appear rather univariate variance look behaviour function high organization develop general moment result strongly distribute infinity section establish multivariate grid relation continuous identifiability able main result consistency asymptotic sample final canonical demonstrate applicability stand transpose image identity rational indicator stand ccc ambiguity use respectively constant change space time consistency wide system result apply property estimator strong section linear characterize strictly stationary cc mean eq state output simplify considerably eigenvalue unity class aspect filter theoretic series output importance eq closure x immediately imply process uncorrelated absolute unity discrete steady kalman eq write process representation allow representation eq together collection stationary variance tr n logarithm always assume state fail difficulty history therefore approximation steady pseudo recursion one additionally kalman recursively advantageous burden kalman asymptotic deal paper kalman converge steady eq n main state strongly surely prove impose guarantee absolute unity matrix non hold suppose follow positive exist assertion consequence entry claim noise usual ergodicity chapter process via move consequence pseudo identifiability assumption parametrization state true estimate consistently quasi asymptotically term recall process f hold describe far parametric accord impose lie impose condition mapping continuously differentiable imply moment space sequence ergodic alone amount independence result sense impose condition strong mix strong mixing always satisfied restriction appear autoregressive equation exponential output impose condition whose distribution possess asymptotic normality parametrization derivative gaussian vector stack top asymptotic initial lf parametric state l covariance covariance deterministic actually fisher alternative task detailed scope work sense need concept limit estimating framework applicable l rely filter likelihood also hessian achieve kalman equation result pass nz r l ns n l z matrix regression claim il consistently estimate describe deviation technique exist bootstrapping technique extend considerably strong strong estimator sometimes work estimate turn moment reason gaussian know ergodic strongly consistent despite step kalman filter kalman theory obtain true pseudo observation function unique true divide surely function evaluate stationary pseudo kalman filter surely cn n exponentially positive positive c uniform iterate last almost claim observe infinite k complete imply moment one claim thus hold sequence view approximation result assertion ergodic consequence sequence compact analogous sake j surely j tc j j minimum difference element span definition orthogonal expectation dm dm dm remain argue inequality inequality strict alternative inequality equality converge l minimize observation imply complete suffice show neighbourhood lie every neighbourhood define equal normality assertion l idea say estimator central translate asymptotic estimator technical extend pseudo steady kalman derivative obtain certain covariance scalar different strongly bound use gradient quasi uniformly pseudo extended derivative step allow show asymptotically determine rescaled exist invertible taylor function divide number true combine strong consistency third term derivative control normality collect kalman
cut balance effect unbalanced balanced partition cut value lead balance term partition balance toward thm sec provide flexibility tradeoff constrain avoid cut size sometimes constrain draw separate hyperplane two volume volume contain exactly range matter scale regularity proper choice unweighted graph vary cut area notice minimum attain area technical connect point close unweighted compare graph small mode nearly near penalize unbalanced unbalanced way statistic sec rate compose shape cluster match sc fail reason sc cut balanced position cluster recognize region spurious along curve big sc outlier singleton recognize enable robust choice improvement focus unbalanced match rbf include graph sec rbf choose partition near parameter suggest nn meaningful point tb vs dim digit set total vary proportion fig graph adapt c vs vs vs rbf rbf rbf error vs vs rbf rbf matching rbf rbf rbf rbf several database dim letter dim recognition handwritten digits dim order even optimize illustrate network small detection small gaussian show cut average cut cluster leave cut fig cluster away decrease phenomenon curve correspond cut value minimize happen since represent impose flat attain deeply already point find say leave method relatively insensitive choice construction belong low encounter namely rarely point context anomaly detection complexity statistic construct spectral semi supervise unbalanced systematic neighborhood effectively robustness outlier degree framework cut detect multiple meaningful synthetic ability detect indicate utilize crucial step cluster rbf poor unbalanced spectral put balance cut unbalanced adaptively neighborhood tend neighborhood vary naturally cluster justify idea limit unsupervise semi set tool represent graph base learn identify critical unbalanced proximal unbalanced arise graph construction refer construction include nn graph link node graph rbf nn close versa recommend robustness propose suppose however unbalanced graph appear poorly conventional unbalanced proximal put importance balance size lead cut meaningful reason outline whereby objective proximal unbalanced section edge near explore section synthetic show significant construction investigate reason sc construction unbalanced justify let draw iid surface edge spectral denote variant cut unbalanced fundamental drawback unbalanced dataset illustrative draw iid proximal unbalanced density examine construction full rbf nn parameterized sc depict choice balance cut axis line axis sc seek achieve cut cut relatively proximal unbalanced relatively control balanced cut possible parameter account increase understand acceptable threshold outlier globally large small cut ratio discussion clear adaptively neighborhood nod plausible region adapt vary degree proximity comparison construction method learn calculate choose base distance dimensional anomaly detection sec range extreme construction close neighbor neighbor point follow optimize control minimum difficult uniform regardless degree around remain parameter third cluster algorithm alternate approach plus reader final step small cluster size optimize admissible small
sparse sparse component gauss sparsity variance component parameter treat unknown appendix se simplify output match rely principle match oracle em adaptation illustrate signal performance optimal form mse measurement completeness compute se recursion outperform adaptive reconstruction result characterize nonlinear poisson cascade response early pathway visual cascade field neuron amp combine estimation propose connectivity model gauss measurement pass vector q measurement channel nonlinearity also characterize ml section vector imply initially simplify iteration correct induction adaptive continue enough algorithm oracle know vector output asymptotic short mse nearly oracle know em adaptive essentially computationally scalar problem applicable provable learning learn possible extension implement estimation additive interpret product value scalar transform reduce update rule simplify relate need review vector element regard component pseudo lipschitz nature abuse notation q topological write write say topology follow along adaptation argument generate adaptive line z tp tp z precisely set prove non proving limit hold usual proving limit induction need argument hold thus derivation induction argument note limit respectively suppose since mean ab apply induction apply prove first end b h old k since satisfy limit also induction output conclude limit continuity prove argument proceed continuity continuity limit induction make equivalence p converge finite obtain equation similar prove limit two application assumption verify adaptation function fix scalar eq wish suffice convergent subsequence z z lipschitz show maxima maxima unique convergent limit condition satisfy analogous continuity manner immediately theorem part proof since already need need adaptation se adaptation definition hypothesis theorem false ed ed possibly non cascade model transform probabilistic possibly measurement call adaptive enable joint learning channel algorithm recently em posterior approximate message pass methodology apply prior identification nonlinear cascade dynamical system predict scalar perform method asymptotically consistent oracle know correct remarkably arbitrary systematic nonlinear provable random assume identically pass address distribution transform bayesian apply cascade dynamical response reasonably call pass belief propagation receive attention compress sense survey approximation relate exploit couple amp large performance condition optimality consider amp extend however although formulation attractive distribution limitation learn performance amp generalization algorithm general domain adaptive case amp adaptive scalar se converge coincide performance oracle know value remarkably result apply essentially unknown enable evaluation prior compress nonlinear cascade simulation method simple exact performance provable consistency em generic term mixture gm e step approximately parameter gm appear output simulation remarkably distribution sense argument present equation confirm numerically special particular choice contribution rigorous justification analysis however methodology way spatially open extended well alternate present source output call iteratively combine amp future work apply simultaneous unknown appeal conceptually strict alternate problem belong class maximal minimax approach recovery minimax may achieve oracle know em method due provably organize review non adaptive characterize demonstrate consistency demonstrating conclude conference appear paper proof description additional describe algorithm input output parametric see method along adaptation procedure case output set estimate describe two choice two variant bp approximations mmse vector function quadratic sum bp map equivalent mmse estimation scalar version mmse map generate random se empirically moment hold surely variable analyse remarkably arbitrary distribution function nonlinear letter compute similar physics I tp ti z j tr n tr standard algorithm consider adaptive show key modification two correspond output know datum understand via observe identically attempt empirical eq right attempt maximum ml depend role similarly joint component density ml estimate covariance briefly ml propose bayesian amp em sum amp output provide amp correctly distribution give implement via distribution every converge distribution give perform procedure particular choice adaptation rigorous point update simple family optimization search see similarly develop consistency expectation random variable output random ml adaptation convergence satisfy function weakly pseudo detail adaptation regard scalar evolution define similarly see weakly lipschitz functional uniformly open eq lipschitz continuously assumption c technical mild class functional function appendix continuity average q pseudo continuous continuous similar functional ml functional satisfy assumption vector output se equation empirically
code ccc combinatorial submodular middle disk axis row extreme middle fa consist list p mm mm equal trivial non explain cover fractional allow interpretation interpretation define small induce fractional optimal probably fractional weighted cover value minimum weighted cover combinatorial convex combinatorial positively ask exist large h wise extend combinatorial positively homogeneous coordinate decrease envelope minimal cover inequality homogeneity wise since prove second coordinate monotonicity homogeneity fw relaxation norm collection clearly submodular cone submodular cover norm integer note rather cover overlap group overlap w sum term sum situation subtle perhaps distinguished general sparsity behave correspond fractional weighted associate latter different still latent cover number one statement discuss support form core set allow intersection set actually must norm define weight submodular cover advantage need assess empirically remove instance three norm naturally relevant sparsity direct graph chain grid sparsity acyclic nice count worth present penalty calculation submodular thus envelope prior combinatorial submodular overlap natural function I norm include consider standard exclusive example formulation also much exception vector coefficient index combinatorial exclusive lasso sparsity impose group w otherwise explicitly pc j g j norm relaxation last would see result formulation w fa fa w easy verify fa exchange maximization eliminate dual actually norm norm particularly interesting since section operator form focus fa monotonicity thus example partition empty mention section group perspective study special norm submodular present submodular derive g extension fw extension extension vector set turn minimize e minimize sa submodular canonical submodular I q recognize extension simply negative function play crucial norm submodular particular norm say augment set strictly cardinality stable allow sparsity separable submodular soon set stable set terminology word sa fa core derive concentration inequality cardinality previously notion cover combinatorial show submodular extension submodular low combinatorial envelope converse envelope submodular retrieve minimal weighted cover context structured efficient gradient convex principle solve solution proximal proximal efficiently another submodular literature minimize separable submodular polytope dual fa sequence submodular w proximal problem maximize concave submodular divide take involve j ax ct ax ax see appendix allow support rate norm require weak submodular local small result sort reverse triangular involve gap complement separable consider response define study induce determine assume extend recovery condition proposition retrieve proposition extend support recovery normal jj extend condition let restrict eq concentration normal control via result diagonal stable either k overlap overlap surely specifically interval induce rectangular support p illustrate submodular group blue green red interval rectangular pattern lead possible combinatorial natural regardless support unfortunately envelope norm lose relaxation tight candidate axis make candidate interval design matrix norm elastic choose overlap w notation since unweighted example precede discussion regularizer recovery square permit optimal path hamming figure assess incidence generate support vary cosine report hamming figure outperform counterpart reasonably ham weight overlap although dominate hamming outperform neither well achieve vary lot indeed relaxation interpret prior unit relaxation display edge corner priori generally amplitude little vary encode regularization visible term similarly error relaxation likely corner value improve note relaxation slightly square first encode sub coefficient support grid constant combinatorial I cg distribution noise amplitude propose relaxation allow recover principle induce establish code combinatorial submodular establish priori question priori yield performance oracle specify priori acknowledge european project like thank provide fa p bb definition duality indicator function formulation show exclusive computation low envelope function g relaxation note illustrate derive let w positively remark fw opposite bound norm sum theoretical norm imply immediately therefore dual fa fa norm resp would three necessary sufficient condition characterize subsequent let minimizer set singleton component strictly w solution latter unique consequence convexity decrease respect equal contradiction order level j partition component eq eq stable statement fa fa p value j thus extension hence irrelevant formulation equal value I reasoning apply corresponding decomposition algorithm indeed denote I w proximal operator amount w last simplex decomposition minimizer q using obtain directly respectively proximal operator proximal section apply decomposition submodular section vector follow x mean covariance r r c jj covariance large invertible q jj jj jj jj p unique global simply show jj r c r j j result consistency let property decomposition condition thus z derive q z z concentration lipschitz bind expect minus plus minus pt plus pt team sup paris france proposition remark theorem axiom induce simultaneously vector norm obtain function moreover link representation lasso field identify express encoding support regularizers generalization lasso particular group formulation submodular convex reader overview relate detailed presentation given vector find appropriate combine enter convex control motivation stem regularizers penalty restriction unit model relaxation assume coefficient unit ball seem desirable assume continuously alternative paper combine penalty appropriate relaxation combinatorial function preserve capture envelope introduce relate convex group lasso exclusive discuss case analysis experiment section motivation follow part part parametrize encode approximation vector motivate surrogate code length naturally appropriate indexing denote shorthand element norm let would natural define cover denote penalty regularization positive since relaxation natural convex non arguably compute requirement regularizer ask formulation rescale instrumental give positively constant pointwise maxima call da h w hence positively homogeneous penalization gets obtain p w introduce support structure encode combinatorial envelope actually able capture intuition many number formalize next lemma minimal subset q prove redundant non redundant remove redundant recursively redundant still remove stop imply motivate upper combinatorial envelope compact capture canonical
mean natural derive embed simplicity assume underlie belong reproduce hilbert value operator exist h cauchy schwarz norm circumstance wish measure estimate embed sparse lasso rkh start multiplication closure pre hilbert pre take closure hilbert space reproduce know isometry idea I countable countable arbitrary exist k x v kx v kx first absolute kx kx norm continuity continuity operator fourth like show v I think I replace term x divide also operator direction one spectrum h independent value problem sure embedding rkhs kernel either non compact certainly new subtle first thing definition assumption measure etc normalize rkhs need hold xx iy I xx g ix show sup extend check regularity need measure reproduce problem reproduce assumption assume integrable x g g g contradiction together reproduce problem reproduce assume f iy switch mean eq fulfil nuclear sequence nuclear state element extend uniquely potentially kernel arbitrary forget lebesgue another incorporate guess could rkh actually want guess learn rkh learn powerful universal I constant rkh hold dim bandwidth control rkh bandwidth rkh allow old minima compare old paragraph couple assume hold pick fulfil issue cost contrary bad occur solution likely cost suggest balance surrogate tell roughly speak cost advantageous sg think people real see link easier need sg scale element scale use opinion make less theorem theorem claim example cs ac uk ac uk com com cs ac equivalence hilbert rkhs embedding distribution vector value regressor introduce function rkhs intuitive justification furthermore regression derive sparse version consider result value minimax state art valid assumption minimax upper rate rate reinforcement cholesky year framework reproduce space introduction one representation conditional rkhs appear naturally expectation ease rkh embedding application generalize expectation multivariate establish norm embedding behave hope obtain conditional expectation valuable incomplete prevent technique like resemble dimension access rich vector value apply result rkh embedding demonstrate connection provide embedding practical theoretical theoretical side establish novel significant improvement also show embedding resemble regressor year embed increasingly operator expectation naturally machine expectation advantage embedding integration reinforcement independence motivation space generalize expectation conditional embedding rkh norm conditional embedding behave would expectation feature condition valuable characterization since optimizer cross validation value loss value demonstrate provide embedding important side embedding give due measure require intuitive rate analysis side embedding resemble parameter demonstrate embed regression loss resemble apply embedding derive version embedding resemble apply minimax class low derive validation demonstrate conditional regressor connection embedding provide solid justification strict suggest regression specialize efficiently burden compute embedding somewhat choice place embedding connect establish idea investigate obtain version embedding introduction rkh reader give yx l map element thus apparent expand problem embedding define k h kx kx chosen suggest embed underlie section present terminology mean empirical ridge rearrange yet establish natural value draw value convergence derive sparse practical kind derivation estimate embedding rate alternative address key mean embedding value ordinary problem go back observe would regressor form label version vector establish regression draw empty measure one function vector take value analogy vx continuous reproduce exist evaluation mapping value relation reproduce property hold v unique sense unique isometry reproduce rkh limit isometry closure importantly perform replace estimate restrict rkh r thus arrive regularize q q hold fx tell prop value least estimate depend sample convergence th upper drawback fulfil embedding function rich restrict embedding conditional regressor hx complicated optimisation like restrict pick rkhs rkhs h v l operator kernel note function directly risk relate relation replace take loss add pose problem prevent hilbert section value rkh function satisfie embedding rescale embedding song attempt sec analyze immediate validation usual hold subsample j grid choose achieve good fold improve embedding present l ii hold separable vi fulfil examine risk objective x lp objective case r minimizer bad expectation subtle closely expectation originally function element acting function minimizer convention x couple worth hold choice approximation rkh decrease fulfil condition apart obvious word important thm divergence optimum suggest risk tell balance roughly speak allow function solution advantageous last comment estimate predict expectation value apply study rate considerably current state intuitive assumption obvious investigate detail section situation approximate sparse large want look norm objective equivalent optimum computation equivalent solve fista reach per ij zero otherwise q employ replace norm norm n ij encourage approximation output penalty ik link develop useful make use label embedding incomplete cholesky distribution reinforcement discrete done start direct angular policy useful dimension cosine angle angular cosine angle angular velocity approximation cholesky approximations generalization report value derive cross establish well low number problem valuable employ embed hilbert schmidt infinite schmidt result technique deeply embed sparsity equip certainly regularizer demonstrate fulfil thing separable space operator etc space infinite space compact compare element decay something thank ep union fp integrable particular fulfil fulfil x ii exist h x furthermore right q assume show measurable norm x uniqueness way h p e mx h x reproduce lem thm get eq reproduce proof like x follow q functional hence integrable consist furthermore borel algebra fix h nz h bn subspace thm iii sub converge weakly th closed iii weak hence x lx integrable function approximate simultaneously l compact pick product let ia lem represent integrable construction wise measurable measurable measurable gx gx value hence supremum p general weak certainly general measure borel topology finally similar x induce construction space algebra borel space exist lem space cb integrable fulfil group category summarize fulfil verify fulfil separable completely separable able certainly dimensional countable h closely separable finite respect rkh separable vx I dimensional schmidt simple borel algebra continuous continuous borel couple generating discuss detail want infimum h term rkh norm converge infimum infimum attain iff rkhs norm intuition fulfil appendix integrable h generating formally measure transformation induce concern generate fulfil follow appendix guarantee fulfil need lebesgue fulfil concentrated element integrable l furthermore integrable v h integrable integral v sequence converge infimum infimum intuition fulfil need sense optimize integrable norm definition find converge infimum infimum attain rkh rkhs hence weakly iii furthermore close weak thm move suitable equality dimension
rkh recognize explicit trick symmetric k finally kernel taylor expansion reveal correspond rkhs extend allow variety kernel necessarily common choice option variate express dimensional obtain restriction model mean I array note wise eq array implicitly level separable dimension exclude fix vector explanatory form independent mixed stage exclude array density z km variate b z multivariate obtain estimator explanatory model b k km j variate b kb test univariate kb l kb statistic lambda hypothesis available generalization curve array response place observation take design array mx x zero hessian maximize linear estimator attention know array along mixed estimate efficiently likelihood decomposition maximize kronecker product eigenvalue eigenvalue decomposition along estimate using criterion eigenvalue covariance real section show change proportion array array variate randomly cell implement cell proportion calculate independent replication improve htbp american combination trait protein date incomplete array datum selecting cell cell addition snp marker along dimension assume trait dimension trait observe estimate replication generate combined environment mapping trait height yield year condition marker fashion accuracy array entry trait correlation miss corresponding replication setting datum involve model sample increase stand array variate correlation response miss estimate know covariance htbp formulate array variate develop estimation possibly application way imputation estimate array part array value reason poor might dimension decrease extension model another array array suitable array variate response dimension allow make prediction combination separate section remark challenge deal high arrange array propose method variate partially observe develop imputation mixed effect model multi define algorithm spectral recommend simulation real life involve effect trait array variate array variate repeat array stacking array etc variate dimension array delta array kronecker array accomplish array multiply arrange form include video spatial measure array kronecker parsimonious variate model sample array array many main array development effect useful effect incorporate along mean variate model calculate likelihood method explanatory genetic marker model updating define semi array variate model study follow density delta give order matrix generalize multiplication multiplication array element wise tucker mode operation unfold element th stack dimensional level operator vector stacking array j jj j follow useful property array normal variable kronecker delta covariance remain assume root decomposition definite put overall matrix th evident context stand zero dimension miss em usually utilize go back adopt partition represent miss eq zeros eq vector lr rl covariance calculate assume write stacking therefore treat model variate flip prove attain parameter variate normal flip variate incomplete variate kronecker delta imputation modification flip repeat likelihood datum last update calculate
compare descent correction correction stop last correct coordinate descent simulation newton type correction descent correction invert big condition scenario coefficient solution make explicit find notice involve spurious terminate extreme predefined terminate pre application upper reason common restriction transaction cost put maximum select stock limit stop rule always point experience decrease set solution towards kkt hold otherwise coordinate check two depend choice mcp usually accomplish cross argument problematic glm bic add extra top bic subset pre estimator mcp concavity regularization mcp glm propose lasso convexity mcp penalty main glm condition hold mild regularity manifold dominate concavity convexity therefore small kkt valid global condition eq kkt long stay decrease mcp exist package mcp exploit active method appear later method mcp decrease stay activate lie estimator sign step experience point satisfy kkt later long convex path stable accordance size active active derivative respect intercept penalize sign enough order mcp hard enjoy adaptive rescaling similar replace diagonal coordinate algorithm update coordinate rescale together new need u kn k avoid mcp although long derivative correction avoid compute implicit derivative likelihood turn formulation coordinate stop lasso activate mcp inactive activate b mcp lasso lasso discuss mcp possess stay value interval tuning parameter conduct study package penalty highlight first use enough fast package handle penalty penalty different adaptation method logistic popular lasso mcp compare package report include cv fp tp mcp model sparsity level setting covariate zero four setting consider table repetition perform dot path mcp lasso path dot mcp respectively dash line dot mcp solid line dot correction panel line small get select correlation selection criterion tp method newton unstable quickly recommend hybrid newton path switch fp tp result table however provide mcp find overall compare job mcp provide increase mcp correction correction exceed square miss lasso conjecture modification subset active variable job mcp penalize particularly penalty entire example different take time dimension zero repetition solid line dot mcp dotted line solid line dash dot line path line line becomes select get near zero positive fp report repetition standard error c package fp tp cv cv mcp mcp cv mcp cv mcp cv cv mcp cv cv mcp mcp cv fp repetition error l c package tp lasso cv cv mcp cv cv mcp cv cv mcp cv mcp logistic lasso mcp flat level mcp square root jump lasso path cause correction path concavity continuous get smooth tend yield perform much well cv case mcp job fp logistic regression main reason penalization another interesting behavior mcp present mcp parameter previous work warm start find covariate change quadratic path warm affect repetition cpu l previously come microarray project gene profile free survival patient diagnosis subject available negative gene negative year small regularize regression suitable impose mcp accuracy yield package split set time test simplicity use tune mcp turn high mcp path little misclassifie subject notice test mcp test well mcp l error propose calculate penalize likelihood estimator real evidence update correction vector concavity mcp maintain stability newton public website x regression condition give mcp penalize kkt define rescale derivative eq iid poisson regression kkt active mcp penalize poisson kkt q thank associate constructive comment author thank na stroke york ny superior approximate penalize penalty concave mcp hybrid modify method coordinate simulation statistical include modeling miss first many promising bioinformatic finance throughput easy make handle mathematical attempt aic addition also unstable dimensional computationally prohibitive numerous attempt modify burden regression equivalently pursuit penalty convenient regardless penalty well mcp also linear glm dimensionality view interested dimensional likelihood estimator lasso regularization parameter vary penalize least lar homotopy entire path linear mcp local approximation yield optimize lar unbiased linear penalize quadratic spline penalty coordinate descent considerable attention include include linear derivative glm regularization calculate estimator glm small active lar also regularization generalization net exploit coordinate glm likelihood ordinary differential base quasi lar straightforwardly coordinate mcp scad glm quadratic approximation descent estimator mcp penalty adjust concavity penalty find minimizer less change glm rescaling range relate paper include glm different difficult explicit glm feasibility previous warm convex inspire rescale concavity adaptation mcp glm algorithms concave glm mainly mcp extended quadratic penalty detect
acc image simulate namely fig truth separate evaluate recognition contain common sub image randomly image image sub sub image simulate pair thin spline example test point perturb gaussian independently perturbation detector collect sift descriptor use correspondence true among score traditional statistical acc affinity acc gs tune ground eliminate outli elimination acc utilize information outli elimination vary outlier acc gs task challenging method design however outperform find acc gs ab ab c construction omit construct store initial cluster affinity pair cluster require complexity initialize cost find affinity cluster affinity cluster compute complexity matrix complexity loose algorithm reduce maintain store near cluster updating affinity ab ab ab ab ab neighbor affinity affinity neighbor neighbor therefore measure ce ce define overall mapping small ce result c link mnist region inspire difference connectivity sort consecutive score treat outlier acc gs method outlier thm com engineering university advanced technology chinese china simple agglomerative explore different role concept cluster average characterize around insight define affinity average affinity fundamental vision outperform computer involve cluster mean determine cluster agglomerative begin select affinity merge agglomerative conceptually produce informative agglomerative limitation computer different shape manifold conventional agglomerative usually directly pairwise manifold cluster sensitive noise tackle agglomerative various machine rarely agglomerative build nearest nn show lie cccc average linkage c linkage linkage use fundamental concept theory characterize affinity vertex dimensional vertex cluster reflect near density cluster density effect successfully wide web social show result vertex define via affinity inter vertex two truth synthetic affinity naturally affinity vertex advantage noisy multiscale cluster comparison linkage linkage affinity propagation ap sc direct cluster multiscale multiple greatly automatically extensive correspondence demonstrate superiority art fundamental matching suggest many affinity express product external library extensively employ cluster finally acceleration especially scale dedicated agglomerative linkage affinity distance capture global datum sensitive linkage method propose mining community satisfactory fail tackle dimensional sophisticated affinity observation several agglomerative min cluster toy suffer affinity min describe structure affinity inverse affinity slow sec segmentation besides agglomerative mean use mean density shape spectral handle greatly existence eigenvector laplacian affinity propagation explore intrinsic message perform manually spectral cluster graph task affinity keep algorithm utilize robust set vertex vertex edge manifold high space weight nk degree begin initial iteratively affinity merge nn graph vertex construct initial undirected subgraph path component undirecte direct undirected edge present detail initial v ccc cm affinity square fair close keep affinity affinity keep affinity agglomerative affinity affinity vertex vertex connectivity quantify concept vertex average ji w sec characterize nn size normalize favor cluster instead merge small connection normalize well unnormalized degree vertex merge cluster strongly connect mathematically vertex cluster strong otherwise intuition statistic affinity robust asymmetric summing q symmetric affinity affinity efficiently affinity index correspond easy lemma theorem use average linkage three conventional linkage although find linkage nn pairwise linkage linkage simply direct base linkage measure b experimental sec superiority linkage implementation accelerate time merge ab ab ab use ab store asymmetric formula ab row material formula present material maintain pair cluster set affinity cluster neighbor nearest cluster merge update neighbor probably among near create neighbor nearest b summarize material please time section demonstrate experiment run matlab ghz carry six publicly benchmark database write digit mnist database testing dataset image intensity feature euclidean pattern linkage link cut multiclass normalize cut sc spectral use handle distance distance fairly fix set dataset algorithm cluster performance quantify
irrespective deal deal outperform dependence clearly parameter depend run accordingly increase almost inefficient ylabel white north blue thick cd option solid dash bar chernoff inequality use easy obtain e start start end nk da ks kt ks end end bn k start start parameter output u theorem theorem axiom novel attribute time evaluation sparse graph sampling kronecker neither actual far enable rational design algorithm concern scalability become issue especially become million twitter regard kronecker graph attractive traditional latent scale thousand scale million recently realistic practice surprising clearly parametrize million order expressive recently name multiplicative argue matter attractive generalization ask part sampling address na expect I extend theoretically outperform knowledge address behind certain match classic accept reject address proposal compactly guarantee integer direct order target direct graph straightforwardly convenient adjacency exist I call edge allowed call either graph kronecker multiplication matrix define real kronecker product np kronecker product parametrize additional parameter kronecker call observe edge adjacency note matrix definition adopt convenience array expect view associate node digit verify edge write may resp denote absence assume attribute obtain bit node representation node need bernoulli variable additional j sampling entry individually computation graph edge normal chapter convert position ball drop coordinate solve employ divide graphical appendix code rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle yshift rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle edge thick rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle node scale rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle thick thick cell imply observe edge position divide weight fourth choose recursively location place pair generate multi poisson independent instead poisson good g poisson parameter taylor interested graph poisson modeling consequently generate sample analysis bernoulli non negativity enforce parameter bit entry efficiently reveal difference th map bit integer sampling matrix rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle scale rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle color easy verify chapter adjacency matrix sampling cc ba eq filter poisson adjust step remain summarize first generate convert generation pseudo reject target first generate like find remainder section devote behind construct proposal generate definition hold investigate complexity generate edge expectation overall complexity speak irrespective resolve suggest heuristic instead careful color color color behind var mean thus behave high rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle node rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle yshift cm rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle yshift rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle yshift cm xshift rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle concentrate highly probable relatively construct eq bb cover hand b cover frequent color notation follow prove discuss validity pseudo code overall generate calculate expect find many illustration legend pos south pt marker xlabel ylabel expect color black x marker thick dash blue thick dash km legend pos width height pt marker xlabel ylabel black marker marker dash blue marker thick I b overall k km e simplify equivalent least attain guarantee estimate select value empirically scalability design number edge algorithm write ghz title expect ylabel cycle legend pos north blue thick bar explicit error mark options solid error bar cd explicit mark scale title xlabel edge ylabel cycle black white legend pos west blue cd x mark thick dash cd xlabel ylabel name white legend pos north west bar cd explicit error table color red mark solid thick title xlabel edge ylabel legend pos blue mark bar cd explicit x mark option solid dash x xlabel ylabel black legend pos north bar red option solid cd x cluster scale xlabel run cycle white legend west blue error
mass pt component length put assumption hence put mass portion away support length portion maximize two together long k cover ball gx gx dx dp direct p b supplement p p follow effectiveness semi establish density estimator density appropriately sensitivity two show adapt cross validation excess ef rf f tv notational denote expectation everything ef nu inequality fx mm ef ef ef therefore cb inequality define rearrange use concentration probability respect validation follow sufficient achieve introduce series set use slightly roll show sample size repeatedly sample computed estimator size range supplement repetition error bar indicate size increase decrease beneficial provably outperform mainly distribution concentrate family control supervise semi supervised depend hence advantage extend density diffusion estimate density sensitive work broad two exist besides estimator report air support air force fa use label datum together invoke marginal ad hoc work example foundation analyzing method include mostly unlabele together label know motivate select huge much toy example unlabele challenge boundary label datum reasonable link unlabele estimate ss cluster call distribution develop always explicit outperform supervise precise improve inference paper improve manifold assumption parameter control assumption strong add deal outcome formalize define depend consistently supervise condition predictive depend control define denote datum unlabele estimator supervise conclude estimation label extension discuss condition discuss useful cover therein statistic include paper method strength classification covariate upper could smooth dimension free create boundary rate approach may version idea follow metric sensitive minimax discussion technical article let sample let denote work area region put mass precise pair curve unit everywhere curve eq supplement correspond euclidean definition formalize define scale space lf lx lp z n p ball cover cover function estimator mention showing concentrate small covering say eq short path connect regular covering cover cover ball regularity thus diameter use estimator detail supplement event depend unlabeled suppose every n since condition distance eq b h I simple page result np nm characterize much suffice intuitively
impose must family maximize natural parameter variational variational update gradient term independent variational connect therefore summation term term use h put together initialize pt update ie low q update precisely variational pass message conjugate factor unlike kullback problem consider might term univariate simplified case derivation calculus technique suppose denote write eq dd da ad express product simplify detailed manuscript cycle n ts f ty I I q I q negligible pt pt variational belong approximate posterior normality indicate perform discussion inverse message ic qx r j v say element update obtain formulae parameter conjugate vb update compute ne q simplify update connect evaluation gradient vector poisson evaluated analytically handle construct effect variance negative use quadrature compute gradient reduce high integral univariate give appendix run additional update experiment bind trend instability begin converge change initialization increase monitor change bound lower maximize variational tight useful traditionally computation bayes role prior odd favor q ratio marginal likelihood favor comparison likelihood review low place log obtain equally probable verify comparison cross selection mixture note bic straightforward degree criterion account estimation uncertainty default address issue see performance consider set initialize update cycle significant penalize quasi assess gold via use interface implement specify parametrization specify produce well use penalize quasi three discard chain apply deviation time update effect code dual processor windows pc ghz study poisson intercept logistic intercept intercept logistic intercept model set generate problem may matrix fix effect penalize initialization serve guess estimate penalize quasi algorithm different include l algorithm mcmc gold root c c penalize fix update partially update posterior effect center quite close deviation effect shorter lower attain parametrization update take converge produce parametrization update logistic centering center posterior mean deviation partially parametrization center well mean center take partially parametrization fast provide penalize quasi difficulty mcmc logistic posterior deviation fit mcmc second magnitude example produce fit close parametrization took attain high easy center perform big parametrization tuning although use partially update dash posterior estimate partially dash line figure mcmc could parametrization effect quasi fit previous real center partial centering centering tend consider variable status child time age intercept slope estimate different penalize quasi likelihood initialization method center short deviation improve upon improved deviation fast fast example hand straightforward aic analyze whether intensity sample call parent extra take treatment parent parent per model stage pt final th offset ij ij ij ie correlation intercept henceforth determine interaction retain model ij ij ij ij preferred model arrival drop ij ij ij prefer drop drop treatment turn drop e ij ij ij u indicate none time well effect add arrival c time e conclusion parametrization thus sufficient taken fit short likely cross deviation solid parametrization dashed center fit partially parametrization centering tune slightly close mcmc density good tuning performance pass recently focus poisson logistic longitudinal quadrature intractable integral parametrization center partial automatically toward center case partially parametrization upon well center fit close partially parametrization rapid center particularly slow partially parametrization center give tight parametrization variance improvement could parametrization offer fast alternative mcmc compare bind selection without x pd r bind p q poisson response I qx I logit link function tv ij evaluate quadrature see appendix p nr expression r bx nj ij univariate ij r bx b vector say q I gauss quadrature gauss quadrature integral correspond quadrature sample range recommend gauss quadrature ij sample appropriate deviation mode ij rx ij significant implement gauss package quadrature approximate integral quadrature example l partially support reservoir research thank constructive suggestion improve thank available simplified multivariate pass reading width effect computational center partial accelerate bayes vb implication examine partially model four variational message partially determine parametrization third accelerate accurate approximation center chain greatly choice parametrization improve partially hierarchical bayes deterministic interest vb intractable factorize factorize implementation algorithm message passing examine partially vb model four variational passing partially able adapt quantity center parametrization show partial approximation center produce generalize inclusion account observation applicability numerical quadrature mcmc integral intensive various likelihood approximation consider integrate nest focus logistic mixed model datum literature parametrization partial technique inspire van hierarchical center mixed slow old center complementary role neither partially parametrization lie center parametrization gibbs yu boost mcmc center expand vb method update speed vb parameter seek hierarchical show parametrization property gibbs sampler similarly center vb hierarchical motivation partial vb algorithm td w iw x iw x ty tx iw linear effect prior know let tuning parametrization interest leibler divergence qp py py py leibler minimization leibl maximization lead py expectation vb pt iterative iteration tune rapid convergence center true posterior recover partially suggest partially parametrization outperform vb rest organize section parametrization passing discuss briefly consider simulate conclude cluster denote th distribute canonical dispersion specific assume depend effect predictor q vector fix effect allow
inequality establish close v sa connection induce exactly incur encode close close quantization wasserstein past highlight link hold dx key show support close close combine behavior measure problem quantization quantization codebook situation depict draw intersection nan ambient absolutely jx decomposition atom would count atom double counting imply correctly quantization codebook clearly quantization cost measure absolutely part hold support unit constant note strong know convergence number formal manifold effectively force assumption absolutely describe several widely interpret output free pca performance quantity minimize set see reference connection problem respect average distance minimize datum behave question characterize dx performance quantization measure absolutely continuous inequality almost surely strict x dx k characterize measure vary choose widely deep include characterize empirical population fundamental speed law number mean measure propose diagram show decomposition prove use decomposition arrive optimal whereas label simple ht blue color triangle depict upper optimal quantization manifold technical bind optimize side probabilistic bind convergence absolutely measure intermediate optimal appropriate small output good bottom figure give absolutely let hold stem bind term c output k optimal suboptimal estimate introduce currently match low mean automatically institute technology mit edu problem metric support establish learn theory bound classic probability derive course probabilistic convergence unlike study random error probability typically analyze density closeness two see reference therein low particular estimator manifold pointwise convergence topic riemannian manifold wasserstein distribution approximate measure respect deep connection field quantization sec sequel widely unsupervised mean estimate wasserstein novel work field technical summarize estimate measure measure convergence number unlike probabilistic probabilistic bound measure end formulation well relate dimensional manifold inner product let measure th moment wasserstein law respectively guarantee metric space consider probability distance sequence iff evy point excellent wasserstein difficult combine wasserstein bind strong wasserstein distance use phenomenon algorithm unsupervise extend sec sec induce population measure useful h old take strong establish population convergence sense arbitrarily fast convergence upper absolutely recently propose prove optimal relate distance permutation matching
dx frequentist jeffreys normal arise formulae lead contradict full frequentist use really lie interval determined factor full determined formula jeffreys depend construction dx bayesian approach coincide frequentist frequentist determination confidence frequentist coincide jeffreys prior support investigate relation bayes frequentist determination coincide always coincide jeffreys value probability frequentist investigate frequentist frequentist x dx frequentist determination coincide frequentist coincide jeffreys ref relation credible interval find probability density probability random lie unique limit interval inside big outside determination confidence general depend determination level consider dx correspondingly eqs find frequentist
completely turn observation highlight conceptually simple equally many part detector split comprise asymmetric introduction result significant gain happen increase translate aspect fail provide nonetheless example object pose car g illustrate scheme try look encode homogeneity visual homogeneity within simplify lead well semantic heuristic directly appearance cluster insight detector refer involve annotation object heuristic work learn complex several computer viewpoint annotation associate separate right cluster group video simple instance cluster annotation space category significant cognitive seminal rely member closely relate train use belong positive classifier although promise overfitte emphasis place example explore intermediate spectrum detail detector analysis label bounding box positive instance wherein training model latent binary task hinge training parameter control separate hyperplane indicate latent using mention early step initialization initialization experiment find algorithm euclidean difficulty merge phase calibrate appropriately address transform output sigmoid yield score comparable possible specifically directly reliable one calibrate learn logistic overlap score box indicate predict box help detection experiment car cat table tv result part turn twice resolution challenge protocol row baseline row show visual detector baseline improve detector job cat class supplementary material category aspect ratio leave heuristic ratio rather part template twice relative template amongst high template detector pyramid fine pyramid visual par result align see supplementary thus model discriminative detector train visual detector seem visual actually suggest relatively visual part issue detector root resolution template eight part template detector part detector latent round detector preferable difficulty analyze tv plot variation gradually increase around key requirement success latent appearance initialization compare aspect aspect base lead performance drop notice initialization discriminative help mistake initialization first intra scene category exhibit visual viewpoint scene category visual expect aid scene scene scene database collection organize exhaustive scene well fine grained scene arrange leaf category node hierarchy original experimental human arrive e choice versus level classifier category level half training half classifier belong except serve negative instance positive ignore care tackle intra p feature representation descriptor bin contain patch filter acknowledge unlikely multiple performance use descriptor choose simple analysis focus generic see material category evident take close discover discover correspond level contain front car fine grain category category subsequently deep assign category category seek analyze benefit annotate run initialize initialization unsupervised interesting supervision create grain category may wherein imagenet basic category fine grain need annotation effort contrary belief part contribution need benefit performance simpler interpretable benefit scene human supervision grain contribution part detector orient part secondary vision include ordering
observable term physical physical model value model evaluate uncertainty variance model rather specific demonstrated example commonly illustration dimension indicate correlation use effort calibrate denote pdfs represent conditional note denote probability bayesian acyclic form conditional probability parent representation uncertain well herein relationship write construct chain direct need pdf experimental begin normally normal probability discrepancy process index deterministic deterministic conditionally deterministic variable exist conditionally variable derive observation set correspond similarly eq represent experimental input calibration base point practical series technique quantity available bring actual interval likelihood quantify likelihood develop normal half multiple interval available observe thus lie hypercube hence derive mixture value type challenge particular prediction system replicate may take characteristic g tt time discrepancy k likelihood series measurement time input choose equation complex pdf need make may need time theory matrix instead series replicate e repeat another attribute directly repeat assume variance repeat series series negligible treat condition consider become simpler simplify become real operation prediction kalman extend particle calibration etc implement interest determine identifiable infinite bayesian sign pdfs pdfs flat infinite non identifiability practical non identifiability structural identifiability redundant identifiable level identifiability insufficient bias noise due successful help redundancy detect identifiability overcome develop identifiability analytical explicit state solution analytical various analytical non whereas address second identifiability necessary identifiable rigorous identifiability analytical function theoretical profile likelihood effective detect identifiability preferable non identifiability become since evaluation require likelihood taylor identifiability model analytical detect identifiability insufficient likelihood thus demand detect taylor practical identifiability physics calibrate new form discrepancy measurement mean random expectation accord expansion show use number e experimentally observe input setting uncertainty system system infinite satisfy vector identifiable assume quality cause identifiability former suggest model either insufficient continue infer non identifiable order redundancy detect identifiable may eq retrieve checking dependency since column column column dependent identifiable use set identifiable derivative identifiable parameter remove th column remove model see physical corresponding identifiable analytical expression derivative forward give model rank whereas identifiable identifiable also identifiable calibration due reason function consume actual sparse may along simplified discrepancy another sequential step pdf represent probabilistic repeat mostly surrogate much surrogate krige gaussian polynomial svm machine surrogate uncertainty uncertainty random various source uncertainty natural variability datum input likelihood calibration even surrogate case approximate actual parameter surrogate instead function radial integration pdfs parameter efficiently quadrature widely computational effort conduct unnormalized mcmc slice etc multi physics individual analysis ideally calibrate experimental model method discuss employ variable versus versus share network model individually existence two connect full network option calibration present option represent except e posterior pdfs option expensive dimensional option calibration follow calibration present calibrate parameter combine option pdf pdfs base computational effort option option option flow use calibration pdfs whereas base bayesian calibrate option calibration existence basic bayesian method issue application calibration identifiability difficulty multiple physics model available parameter relate physics numerical implement multi physics rf device identifiability perform use first taylor series base identify rf physics unknown temperature six barrier height fp attempt frequency mass conduct nm mm combination repeat four current second available series observation replicate noise series time term gaussian example least square without consider difference prediction square correspond experimental versus rough appear exponentially significant varie adopt input exponential discrepancy addition variant trend covariance identifiability select since addition identifiability unknown corresponding suggest identifiable taylor form likelihood formulate observation construction require determinant inverse rise around determinant difficulty subset reflect closely precise convenience advance sparse account posterior use range kde posterior pdfs length corresponding pdf significant expect calibrate predict validate calibrate model map unknown fig density calibrate without calibrate model blue prediction observe fit prediction square standard probability cover prediction account take full pdfs demand square expect second later reflect wider early indicate discrepancy density device switch apply contact exceed threshold device close reliability device euler model simulate device take apply air input loading pressure air gap correspond force device geometry material boundary condition device long term loading calculate value cause contact due limitation currently measurement early collect device keep switch form study type device device relative type device construct hour discrepancy construct relate give common third option present option identifiability calibration check expansion method device directly half bayesian obtain rank e identifiable point also examine identifiability hour identifiable since measurement option bayesian prior pdfs pdfs posterior pdfs black fig note pdfs example except calibration pdf calibration calibrate posterior pdf fig marginal pdfs corresponding statistic calibrate pdfs statistic fig h posterior prior common pdfs calibrate grid computation fig option pdfs statistic calibration draw fig due pdfs first two pdfs calibration use give difference posterior pdfs insufficient reduce addition table third two calibration option give calibrate bayesian integrate calibration computational source available discuss calibration interval
write inference make intractable tractable fully factorize posterior nm n nm component turn factorize cluster instance unity note depend therefore bind maximize bind r keep update unity however part perform step inference estimation without class cluster visualize arrange distribute scenario arise target different location instance denote classification refine without share site careful look equation reveal server server site dataset locate site site cluster split multiple place always performance transmission one another summary updating update variational server helpful conceptual sharing need server place server computation carry illustrate fig respectively framework case class site avoid due space already precisely datum assess capability e store location semi benchmark create supervise ensemble target label remove logistic regression discriminant ensemble privacy preserving present accuracy post hoc show significant among accuracy motivate soft transfer acknowledgment support university usa privacy combine supervise site privacy aware computation instance target accuracy share extract knowledge centralize file database due variety recently emphasis source via simultaneously mining approach preserve mining technique restriction inference database ii record iii technique party party communication meanwhile notion privacy expand substantially focus privacy recent approach differential privacy notion impact large mining association also limit partition partition typically privacy true early datum algorithm site privacy aware bayesian combine effective address combination classifier prove cluster ensemble improve combine ensemble classifier cluster nice unsupervised classifying thereby capability motivation mind combine instance idea classification issue provide label base detail employ label instance target datum apply input consider produce consider comprise align label refined target label instance assign
temporal nonparametric predictive greatly hard nonparametric em like yet mixture pose identifiability suffer identifiability identifiable spatio dynamic simulate new one condition check simulate prediction unobserve principle intensive much plan imaging analyze forecast indexing variance delta become finitely gaussians conditional weight make forecasting hard parsimonious sufficient unique forecasting forecast point forecast view highlight benefit us past light cone configuration say mixed optimize h note theorem conclusion conjecture definition exercise lemma remark summary department nonlinear high spatio simulation extend hard non asymptotic greatly forecast limit implement available recently introduce reconstruction spatio temporal every spatio temporal associate prediction future single agglomerative state minimal capable soft often predict well robust allow introduce soft version optimize like naturally interpretation state em automatic corresponding prediction predict hard fix notation spatio process random tn dd optimally future u conditional x propagate therefore spatio neighborhood possibly past analogously event could influence dimensionality light figure illustration plausible find still spatio temporal pattern predictive x refer configuration predictive minimal invariance light cone regard relation among process oppose realization encode sm thus simplify estimate prevent use joint pdf satisfie completely distribution restrict applicability cone forecast alone spatio implicitly specify predictive I exactly correspondence mapping equivalence class predictive state formally keep distinction predictive hide soft mapping interpretation minimal state introduce component randomize version optimal constraint tell get useful fix give solve evaluate candidate moreover nonparametric direct nonparametric likelihood kernel approximate variable play role turn variable current likelihood come brevity x soft log update n matrix kde th get hard bandwidth r estimation normalize ij forest currently slow conditional simulation adequate start difficult bandwidth even increase feasible either would search cf split x train obtain predict iterate pairwise test k sample mse mix state variable since none estimate like weight update step although simulation henceforth usual stop em algorithm density test metric merge datum drive fitting model start reach optimum iid thus cv integrate mixture datum weight difference q combination mode simulation predictive mix practical evolve accord evolve observable integer word say state sample near site include become kx k start state nonparametric estimate top plot log realization vertical right discard trace alternate blue patch observe clearly residual fit residual mixed initial first half estimate drop mse reach optimum merging forecasting return optima prediction fig show mixed practically compare visually indistinguishable residual obvious forecast independent realization train future outline state low initialize initialization remain nine mse field mse mix
accord default comparison simulation candidate estimator value eq computed option estimator evaluate empirical square compute average number positive fp repetition explanatory control large three follow z g z g model give design table show performance minimize understand ideal minimize redundant large pl perform well pl nonzero additive correctly pl bad redundant component performance estimator candidate make pl cccc sl pl fp refer estimator sl estimator pl penalize refer cccc fp ideal select variable fp refer square fp refer select variable fp standard analysis restriction model denote positive restriction smoothness preliminary theorem concentration around variable primitive condition concentration see discussion component exclude fractional preliminary argument concern restriction grow section second although lasso result selection state j analogue infinite dictionary independent condition large case obtain reasonable believe long stochastically procedure theorem condition c depend eq possible exposition despite contrast characterize variable reflect ii reflects reflect behavior term collect property issue comment proposition lemma go bring subtle technical issue proposition lemma obtain growth dm ns dm allow allow slightly condition exist g c assume magnitude situation unfortunately cover inspection condition exist cover observe replace union proof state connection model additive scope review control doubly penalize general hilbert establish split double remove shrinkage cause simultaneously smoothness two proposal estimator modification replace use group lasso jk adaptive lasso post adaptive g besides notable group smoothness estimator notice analysis additive separate sense argue introduction adaptive oracle strict sense directly comparable investigate estimation dimensional additive especially generic overall help bias double penalization explore suggestion practical use c depend index argument define go event may c j q fact similarly invoke restriction ij jj n bind q c substituting wish j noting use p desire share dr author economics mit greatly grant aid lemma section proposition remark regression nonparametric potentially additive step implement penalize square penalty reasonably despite intuitive theoretical selection random generally contain redundant additive component derive implement group lasso additive identically situation larger sparse let denote without class positive q identification interest additive interest conditional despite redundant make sparse contain ii possesse nonparametric novel term event thereby enforce sparsity result control thereby overfitte result see progress theoretical show e penalty reproduce apply estimator regularity boundedness achieve risk minimax point double penalization selection bring correct choose tuning optimize shrinkage sparsity recognize parametric square select lasso remove observation step additive implement penalize devoted two generic situation variable apply speak stochastically redundant significant interpretation rate could know second term effect select redundant significant one plausible guarantee perfect hard group adaptive lasso need general situation perfect separate considerably fact side achievable make meaningful comparison exist estimator performance perfect first despite smooth primarily first selection hence refine additive recent theoretical apply estimation additive idea deal e approximate class complexity namely deal complexity regression expand procedure regularity estimator meaning achieve rate fail happen perfect enjoy adapt adapt believe result finite two alternative estimating work estimation post analysis extend note build paper important nonparametric ii step smoothness provide proof study rely omit case sequence notation exist independent unit sphere integer use supremum symmetric positive symmetric square root index agree describe appropriate second penalty result give j theoretical make show give lasso function except consider subject tuning control estimator give show give choice let convenient concentrate ig ij ig ig ig ig ig ig ig j dm g jk jk n euclidean know present estimate group lasso
simulation variable exact learn exact elimination feasible structure learn heuristic pc importance bn model evolution gene interval instead depend use distributional present important limitation global normal expression continuous skewed unless cox able power multinomial sequence ignore subsequent inference aware trend particular allele comprise outperform gene complicate case molecular least process perform impossible pathway network prior know selection present drawback separately accept task parallel intensive backtrack information speed possible increase positive false negative include markov merging gene proof correctness hundred observation raise problem correctly molecular extend model flexibility incorporate knowledge modern application field diverse bioinformatics customer survey weather forecast system biology among dimension raise curse challenge affect model associate used express graph principle separation two independence commonly undirected acyclic choice datum aim focus mostly associate conditional table consist conditional ideally coincide dependence structure identify difference theoretical approach conditional base fit al al al structure operate correspondence graph must dependence definition observable really unlikely structure implement rarely far common graphical define practical reason local distribution involve small apply modelling problem single regardless maximal non mass clique function accord chain rule clique property identify graph share child markov make system biology employ gene aim molecular mechanism commonly available gene presence code protein protein result nucleotide biological reason mostly vary nucleotide individual possible call interested grouping determine gene node indirect influence unobserve property completely molecular unable measurement direction causal pathway reflect pathway flow molecular consideration protein interaction limit cell interested one human production etc capture indirect genetic effect unless trait gene genome identifying strongly bayesian problematic care gene trait association expression association may hand gene locate likely together configuration occur less expect phenomenon strictly gene link undirected direct trait compose measure rna pattern probe intensity rna measure several intensity result study include meta difficult practice use furthermore within abundance systematically biased chemical collect grey control involve molecular nature gene investigate pearson robust gene relevance network construct correlation correlation order correlation rather presence determined corresponding genomic stein gene undirected scale direct graph microarray review low mean usually unable equally behave produce feature protein expression protein cell time involve perspective apply datum important note protein sometimes gene study network fundamentally gene expression datum genome rely indirect provide individual limitation gene state individual snps snp differ single pair possible variant combination aa model easier independent aa aa individual allele aa individual model perspective model snp discrete option multinomial receive literature approach variable model additive population structure often implementation regression mostly framework statistic genomic et al graphical systematic extend classic equation panel contribute trait significant lasso context model include bottom panel justify computational interaction snps even able completely hand picture capturing trait I implement analyse present crucially vary substantially depend
potentially dimensional yu segmentation euclidean ultrametric journal pages f journal theory practitioner l scientific wu grid classification international computing mind http f ultrametric model mind ii text content http lee statistical record van van sparse face xu clustering comprise value attribute imply ultrametric topology great powerful hierarchy proximity match address address th thus report grow decision planning include type text structure storing scale capability intensive support exploit datum activity big worth flow topology greatly involve challenge issue directly discovery well logarithmic constant dimensional exploit sparsity neighbor triangular metric space triplet ultrametric triplet ultrametric van range example triangle form triplet side represent among possibility semantic accomplish ultrametric convenient ultrametric dendrogram dendrogram root label precisely term ultrametric set ambient dimensionality define cardinality traversal terminal van terminal traversal ultrametric simple tree ultrametric distance computational need dendrogram terminal path integer let call traversal dendrogram dendrogram day construct map topology ultrametric dendrogram near write dendrogram terminal traversal dendrogram terminal dendrogram traversal dendrogram find ultrametric two terminal twice traversal dendrogram informally potentially common parent traversal dendrogram root dendrogram bad dendrogram structure integer agglomerative dendrogram favorable well agglomerative ultrametric carry one candidate check dendrogram neighbor find firstly low terminal follow cluster tend ultrametric far carry indeed endow ultrametric note carry computational structure ultrametric build balanced dendrogram logarithmic review ultrametric distances neighbor constructive computationally dimensional space retrieval form consider human computationally carry ultrametric retrieval massive require cope volume furthermore ultrametric induce hierarchy support arise directly distance ultrametric cluster store ultrametric tree retrieval easy member enhance inherent ultrametric relationship onto alternatively map onto express generalized ultrametric pairwise relationship partially lattice algorithm hierarchical direct reading scan discuss simultaneously ultrametric hashing bin precision hierarchical ultrametric consist common long sequence et al precision example precision length digits generality order cardinality measure attribute ultrametric boolean work number store effectively many theoretic algorithmic allow scope distance set proximity domain include demand typically domain agglomerative dissimilarity imply time number agglomerative reciprocal near cluster quadratic agglomerative neighbor quadratic agglomerative linear agglomerative algorithm characteristic algorithm often develop handle idea separate group within grid partition overlap cell calculate sort accord traversal neighbor grid follow
discuss dependent factorization method lead prediction time validation several choice misclassification misclassification rate stable minimize misclassification bin emphasize exhaustive lead improvement subsample validation average split train curve test set challenge one view paper mention respect challenge figure pair split bar split cc misclassification cc cc cc cc cc cc profile month period direct classify time namely rating appear useful distinguish member exhibit rate movie day week movie rate provide incorporate day week well rating cc consider week rating importantly pattern member well separated show frequency movie different day week label member tend movie day week mostly movie week repeat quantify empirical rating day define average variation p q tv pd rate movie day week possibly figure phenomenon bc tc tc bc week bc bc tc bc tc tc present three predictor member movie third exploit week good suggest predictor rate movie predictor assume movie assume independent everything subscript fix time movie per month bin occur everything else conditional occur user rate movie everything day week third take rating present rating account give generative movie user around prediction make low approximation user movie time center bin show residual movie well normal user agree overall b motivate user time normal distribution rating condition day rating previous tuple movie rule classifier generative evaluate section low particular ccc derive term misclassification correspond rating second variance residual error probability bin unconditional marginally come day week unconditional incorporate decrease misclassification rate condition week variance misclassification rate report detail provide statistically respectively finally excellent roc area rate product rate define average use original challenge study previous yield excellent improve contextual rating provide rating separation raise scalable incorporate multinomial multinomial vast reduce classification binary fix hereafter whenever characterize denote assume logit whereby feature fit assume attribute user maximize rating implement software standardize solver feature vector day week rating indicator hour implement feature vector week suggest adopt vector week day month adopt rely outperform unified framework describe bin length binary indicator rating scale shift reach test logistic assign regularization feature improve achieve misclassification estimate subsampling rating people fold validation vector challenge note bold proposition corollary conjecture remark usa department electrical engineering stanford stanford track aware movie comprise provide movie user comprises identify find rating significantly useful achieve preference know misclassification incorporation contextual play ever availability source information social pool view investigate relation social behavior recommendation result summarize table remainder describe challenge explain performance overview c size consist rating rating user movie provide movie rating throughout first user movie rating training composition belong tuple comprise movie rating user movie track rating train vast majority form user form consequence slightly misclassification size provide roc true rate false entry respectively entry assign minus false positive roc way total per consider user roc positive rate positive union class incorporate increase amount contextual effective tool movie hand latent large infer rate select movie vector generalize temporal variability movie latent first rating miss factor take temporal rating aforementioned rating provide misclassification roc curve throughout ti matrix rating column movie column regularize n kk ki j f j excellent performance performance prove suitable relaxation alternate update minimizing stop iteration quadratic inversion convenient instance ij e I define matrix read I proceed analogously rank prediction bin duration rate movie rating tensor represent rating predict miss section matrix
q provide follow subsection back research focused efficient estimator valid prediction set thorough efficiency lebesgue measure set level one lebesgue upper level prediction band normal center gray observe band quantile band optimal area validity require band refer joint type validity illustrate example validity band band q satisfying asymptotic efficiency band satisfy fact atom euclidean radius conditionally valid finite validity thus trivial validity shall instead band asymptotic finite asymptotically supremum validity consequence asymptotic efficiency marginal validity notion naturally validity partition band locally valid q sample valid prediction available thought become validity validity conditional validity strong validity whose elementary omit marginally valid validity one technical band mild regularity conditionally valid marginally valid prediction band straightforward event case marginal increasingly close therefore simplify construction kernel prediction develop sample suppose set q p statistic agree augment principle fit exchangeable p invert hypothesis reject density bandwidth mean high avoid augmentation validity order increasingly show appropriately characterization level version state construct slice marginally band measure marginally valid region band fix slice joint marginally let define band marginally band local band also simplicity presentation support side kernel bandwidth augment check band finite define finite locally marginally fix another independent conditioning optimize optimize effort interval approximation analogous finite local long satisfied density little inside validity must subset smooth approximate prediction argument choice cubic histogram bandwidth argument density counterpart definition solution existence guarantee contour smoothly estimate density roughly oracle first approximated set conditioning rank plug regularity quantify put conditional density old correspondingly kernel concept also old enable regularity xt cx notion exponent condition cut value set mention empty simply put optimal level difference validity know smooth give asymptotic efficiency prediction band satisfy validity rate minimax risk valid infimum follow result low error distribution fix hence proof use somewhat band note bin marginal consider allow bandwidth bin validity small measure procedure detailed preserve marginal validity drive split sample subsample bandwidth subsample prediction band split sized construct cx c work locally marginally bandwidth construction efficiency excess risk scope section centered performance band valid band equal sized whereas marginally band although locally valid coverage marginally band cover bandwidth panel car original car per acceleration use book linear linear per per power figure transformation sense intuition inverse per power panel prediction reasonably band wide narrow large panel band measure enhance smoothness construct ensure output without tuning procedure conventional band truly valid coverage sample construct free validity efficient achieve completely believe first band property rigorous bandwidth sketch argument exces study stability plug show excess band construct band exploit supplementary technical give old text book old class h old uniformly approximated polynomial integrable old class kernel convolution lemma total atom fix total mass uniform imply hence q q conditioning old consider assumption constant q consider fix theory universal constant q l xt fact fix suppose satisfy estimate satisfy pl c moreover modify argument assumption l l pl pl pl pl part second result eq lt direct measure estimate omit dependence set level obtain I easy inequality conclusion bernstein union ignore tuning generalize symmetric conditional density support take verify hold verify py x py note let uniform verify histogram cube pairwise separation density band l
weight eq section weight expression derive statistic reader test longitudinal explicit location subject multi reader roc large variance get marginal longitudinal also asymptotically take q estimator simulation estimator finite statistic simulate roc longitudinal reader importantly power expect optimal equal weight longitudinal repeatedly repeat measure subject study coefficient simulate statistic auc use show bias square square coverage close asymptotic show indicate violate well method surprising disease status curve base score roc monotonic new score empirical roc power equal let simulated dataset estimate weighted define simulated power weight datum take normal x choose ik ik define give within simulated auc square mean square coverage correlate nonparametric study diagnosis medical cell cavity see challenge exist include diagnostic history diagnosis outcome address issue investigate whether sequentially aid disease reference participant participant digital conduct conduct modality set participant correspond digital report question clinical diagnostic diagnosis original participant gold standard four weight weight estimate weight set indicate precise diagnosis report method apply compare diagnostic marker longitudinal illustrate study propose method conduct study finite set complex subject extend constructive suggestion award american act national security part american institute child human solely author view national institute health derivative finite partial set derivative normality marker therefore element element expansion side asymptotic normality combine device e htbp rmse rmse norm weight htb bias rmse bias coverage corollary conjecture condition department statistics usa division research national child health human development usa college public health human sciences medical imaging modality test marker longitudinal roc method weight area roc maximize detect imaging modality asymptotic power apply diagnosis newly image modality discriminate subject imaging modality distinguish diagnostic marker identify exist specificity correctly truly diagnostic characteristic roc tool compare marker curve accuracy marker apart threshold marker nonparametric measure auc distribution normal marker method perform assumption roc nonparametric robust compare empirical auc roc intersect roc curve marker interested range acceptable early cancer part nonparametric specificity classify non subject may thus desire marker range rest follow multi reader test roc longitudinal roc report performance roc apply example diagnosis notation suppose marker location marker normal subject total pairwise function marginal similarly define q iy roc obtain compare evaluate marker roc longitudinal roc effect among estimator covariance structure generalize estimating derive large estimator valid work reader subject imaging device one rating location subject denote reader approach compare image modality device reader special auc reader combination propose weight possibly empirically modalitie reader roc instance homogeneous rating equal weight datum introduce reader experience vary greatly biased auc estimate difference consistent detect modality calculate weight explicit expression estimate eq marker come longitudinal marker several time roc
directly comparable contour mae contour line dependency axis analogous rmse similar trend mae display indicate count density remain general example pmf sensitive count sparsity long valid level multivariate univariate prediction contour curve contour horizontal imply depend item item contour show count last summarize dependency trend absolute baseline insensitive name highly l variable user average dependent user w count item default item nmf pmf dependent pmf pmf strongly dependent cf count mae rank indicate dependency function density good varies non linearly count especially svd pmf perform analogous mae rmse function asymmetric undesirable word true rating specifically star belief difference recommend important issue recommendation error recommend preferable item high penalty predict bad high opposite recommendation loss mae rank rank recommend present consideration slope well good conclusion identify cf user average item base regularize matrix ii nmf individually slope cf display accuracy baseline svd slope pmf fair fair poor good good poor poor asymmetric poor fair ndcg poor fair fair poor fair good fair fair fair slow fair slow slow memory consumption high high well little short asymmetric asymmetric long use adjust large dense long repeat major conclusion svd pmf variation perform mae rmse except nmf asymmetric evaluation one simplicity vary accuracy user vary relationship dependence dependency appear influential trade factor low consumption accurate algorithm experimental help resolve specific hand efficiency important slope source software reproduce experiment cf conclusion describe practitioner implement recommendation examine novel art rapidly propose conduct study several collaborative filtering art variety context control criterion conclusion collaborative rapidly classic include method base around factorization include factorization despite considerable clarity central clarity method differ substantially user filter method perform setting collaborative compare investigate mention concentrate classic recent art control user comparative follow generally speak factorization nevertheless distinct accuracy depend user number relationship memory consumption crucial describe implementation study describe introduce recommendation collaborative broadly speak software recommender broad seen consider recommendation base recommendation information main system filter former explicit concern information gender location item recommendation category filter cf use user exception rating rating hold rating column typically example five star netflix recommendation rating implicitly activity web interpret positive rating extremely item hybrid content combine user content comparative study serious collaborative system task experimental content tie likely transfer rate different collaborative filter usually model recommendation rating rating recommendation predict rating user whose rating item item motivate user item rating estimate desire rating cf average rating vary considerably example averaging include neighborhood vote inverse recent neighborhood construct kernel estimator ranking predict model rating recommendation include slope successful factorization regularize profile miss nmf variation pmf pmf maximum nonlinear mae root rmse predict observe item range aside purpose test f argue metric conclusion concern relative cf however paper motivate criterion write survey cf extension concentrate hybrid art include netflix recent recommender system explore additional issue couple memory model cf focus comment computational issue mae netflix competition competition improve accuracy rmse netflix recommendation win dynamic achieve win linearly combine netflix competition use competition evaluate cf item represent entry old standard dataset rating item rating describe concern study conduct experiment facilitate implement list public update state art research elementary baseline average constant prediction prediction item rating use factorization category user memory default default regularize pmf pmf pmf slope rank cf experiment netflix benchmark cf literature recent benchmark measure sort rating matrix row realize specific pattern select subsampling require sparsity sort corner sort contain top item word subsample prescribe purpose time train conduct processor ghz gb memory investigate dependency user rate variability dependency determine start prediction quantity user multivariate accuracy variable row mae panel focus cf baseline item count rmse evaluation trend omit voting user cf variant work compare algorithm quantity top row slope intercept appear intercept algorithm mae slope indicate decay mae table method user sufficiently svd little difference simple item svd pmf sensitive baseline nmf insensitive sensitivity item cf extremely effective user dependency user count bad memory base pmf pmf pmf rank cf represent user count mae prediction loss analogy fix respectively middle mae category coefficient get overall svd factorization method simple
n show lemma regular estimator converge let matrix regular w nn n way already mathematically conduct c std firstly selection reduce denote dimensional normal matrix true determined element take sample generate sampling trial normal exchange metropolis eq six model criterion compare standard define set true experiment secondly use experiment reduce theoretical reduce case average c c c std applicable firstly let study qx px minimization leibl generalization define minimization important learning criterion well aic bic regular onto singular statistical v n p e w statistical prove regular model coincide pn singular study remark set singular respectively small model model property viewpoint simultaneously mean weak selection main purpose jeffreys recommend jeffreys however jeffreys theory model selection minimization secondly numerically free firstly asymptotic large expectation disadvantage huge cost present secondly sampling approximate arbitrary last whose analytically numerically evaluate strongly depend true statistical nan hypothesis procedure recursively nan asymptotic contain statistical table model determine training energy conventional recommend asymptotic importance lastly relation algebraic paper statistical prove affect negative value secondly set give statistical set conjecture relation lastly hold odd likelihood widely true expansion bayes threshold partially science aid scientific keyword fisher positive otherwise statistical minus asymptotically approximated schwarz energy invariant real parameter several discover methodology true present bayesian log mathematically expansion energy numerically information bic onto statistical otherwise singular model neural mixture reduce regression network markov layer rule singular word hide random phenomenon naturally normal neither issue theory denote independently loss minus log free energy understand minus logarithm marginal play evaluation bayes equivalent minimization bayes energy asymptotically likelihood schwarz bayesian normal energy approximate bic minimize kullback rational real firstly singular algebraic geometry behavior free algebraic fact machine artificial neural mixture regression mixture boltzmann hide statistical indicate quantitative singular depend know apply evaluation present estimate widely show inverse purpose mathematical theorem firstly temperature temperature satisfie secondly even eq asymptotic singular lastly regular regular statistical cost generalize version true section notation singular representation theorem corollary mathematically main section result name free loss posterior optimal log loss kullback leibler real threshold table variable name paper average loss true leibler eq exist minimize large unique arbitrary function set element equivalent eq define characterize purpose present prove definition say say hessian strictly regular mm mm fundamental open kernel define several open condition hessian definite satisfy value invariant statistical indicate present conjecture analytic function leibl prior nonzero compact local resolution local c c jacobian determinant b essential respectively even satisfied regular statistical canonical threshold theorem generalize true unknown case make even present temperature unique temperature therefore behavior mathematical fundamental log threshold law mm main theorem three important corollary model let canonical mm bic singular regular bic small constant regular general asymptotically distribution singular hence replacement appropriate corollary converge law nk r zero nk strictly respectively maximum w u ij j inequality determinant schwarz inequality mean complete proof evaluate integral integrable generality consequently coordinate note arbitrary understand random process fundamental integral asymptotic two loss essential denote dirac hence behavior value expansion measure
pdf bad estimate improve large method work far cycle mail estimate density estimation large local suited heuristic optimisation entropy anneal everywhere describe computationally form j pp j order maximum principle cc kf kf value estimate shannon entropy condition ease end divide smoothness give locality small condition neighbourhood anneal noise introduce pdf eliminated move final result determine experimentally reproduce well support assumption theoretical consider function center pdf neighbourhood replace perturb average assumption restrictive since taylor pdf total compute approximately distribute large number variable normal average est dx dx put taylor h x f want put numerically nevertheless relaxed term p third equation exactly c solution otherwise equation label solution another put increase vertical allow pdf moving carry dx important
iterate full rank imply establish set verify satisfying requirement broad undirected practitioner look heterogeneity model provide give dataset sufficiently empirically sparsity conservative practice compare literature first show exist force contrast deterministic focus sufficient existence failure polytope sequence nature sparsity avoid polytope evaluate nine size tie college division costly expect theorem variant degree correct blockmodel may establish national health research office office lemma early derivative bound change lemma omit three c p p ij f r r approximation x kl l kl kk l kl approximate verify hypothesis newton inequality l lemma bind lemma consequence corollary aim hypothesis simplification corollary obtain finally simple logistic implicit datum heterogeneity broad property rise likelihood regime practitioner choose context fit variety dataset implication approximate long recognize call assess goodness g facilitate g exploratory outli context take edge appear extent heterogeneity commonly practice node heterogeneity node direct therein take edge implicit depend model residual community associate linear since canonical analytical convenient take outer practitioner lack issue sparse adjacency regime wherein typically purpose essentially parameter log member broad irrespective node degree binary symmetric zero correspond graph node diagonal family probabilistic nan family let specify model introduction parameterize function complementary link logit undirected version exponential note degree case intuition equally likely log alternate parametrization degree loop see appear commonly generalize placing degree model high lead expect whenever log introduction approximate technique suffice gain claim likelihood intuitively behave mean rough bernoulli treat poisson poisson every correspondingly respective maximum include furthermore log likelihood evaluate pair absolute specification arise assumption component log smooth satisfy assumption edge result log evaluate analogously notably corollary total misspecification theorem specifically error ease reference existence likelihood newton
forward pass despite necessarily computational nature especially abc remark smc abc estimate normalize functional hmm scenario need already abc smc scenario density smc quantitie whether associate generate smc abuse x incremental smc incremental instead herein denote add subtract nx one follow grow conversely nx remove smc hmm dx dd dd state facilitate investigation obtain approximation answer smc numerical abc consider forward study elsewhere abc improvements hmm utility high investigation batch static worth smoothing estimate simulated parameter implementation forward particle approximate smc abc label abc dynamically drop smc abc approximation hidden accuracy abc abc discuss calculate nx distance smc hmm corresponding similarly average error smc abc allow understand impact describe preliminary become conclude numerical study smc abc smoothing respectively mean error display expect fall smc control outperform illustrate error small vast interestingly abc accurate smc counterpart smc stability therein smc run ne ne smoothing perform abc error distinction subscript display note abc comparable even marginal estimation mainly hmm smooth quite biased substantial h basis specify prefer perform would appear series forward necessarily functional improvement accuracy produce parameter bias smoothed link method quantity dependent see trend parameter investigate smoothing static hmms likelihood construct abc work batch estimation practical arrive online investigate methodology online consider article abc procedure acknowledgement author support grant acknowledge acknowledge platform self explicit consistent notation rely hx gx avoid notational independence additive assume expectation w abc n ng nx hx quantity consider induction one conclude deal dx p dx qx ng h qx particle convention filter exist proof ng n ng p add definition sum separately r clearly q second subtract function two bound n q similarly deduce ng n theorem follow therein latter depend upon repeatedly omit description abc auxiliary smc work nx dx backward remainder adopt decomposition use p depend depend elementary exist abc decomposition denominator apply use uniformity us algorithm remark school business south mail department apply national mail engineering cb pz uk mail bs tw mail mathematic college mail hmms density involve value precision quantify abc expectation regularity adopt form quantitative treat simulate static particle batch markov sequential unobserved dominate conditionally dominate depend functional approximate hmm expectation functional score see rarely smc rely ability pointwise conditional evaluation avoid evaluation simply close secondly evaluation technique interpret instance abc recent context hmms abc see abc note fix lack abc smc error provide objective theoretically empirically perform static hmms hmm incorporate value auxiliary approximation turn admit technique scheme control sample note adopt filter backward smc smoothed functional estimate expectation similarly smc expectation hmm study smc expectation hmm error control particle mcmc order hmm particle use smc generate state hmm particle hasting place context smc adopt adopt backward pass summary contribution nx henceforth quantify henceforth refer illustrate finding abc characterize result combine smc error numerical section article proof negative addition banach endow norm tv tv pa via advanced computational tool assume throughout associated completely idea facilitate compact exposition hmm bayesian even advanced tool deal tolerance follow represent statistic note data accurate focus return density probability particular abc maintain markovian help abc smoothing resort smc mcmc consistency grow intrinsic bias bias noisy article address potential hmms smoothed investigate concentrate particular facilitate smc abc minor modification adopt gx fx functional rather lie however typically employ smc perhaps discuss verify demand assumption establishe fast estimate grow linearly overall parameter necessarily dominate abc abc context smc recursively particle resample smc terminate sample whether stop else return choose implement smc resample perform increase commonly refer resample particle replacement I alternative degeneracy drop go former case degeneracy degeneracy bottleneck static latent central path degeneracy hmm particle variable smc back sequence whose element smc degeneracy effect want forward filter backward algorithm simulation eliminate smc nx px p note nx dx eq lead idea particle write perform eq apparent recursion v px assumption asymptotic linearly
available analytically consider control chain recursively family recursion rewrite recursion eq traditionally noisy sequence define I h closely relate effect negligible ergodicity sequence start various quantity crucial analysis assume traditionally modification recursion indeed major difficulty dynamic stability property rely good ergodicity vanish instability exist ergodicity ensure result aware numerous scenario follow significantly different developed dividing prove boundedness simpler use section sequence transition uniformly ergodicity trajectory accurately visit eventually provide deterministic stable establish sequence analysis ergodicity study reasonable state boundedness lyapunov away norm although symbol section lyapunov subset continuous compact introduce need describe ergodicity precisely stepsize account denote crucial j establish follow infinitely q eventually prove equation scenario soon ergodicity example focus condition combination exist conclude mcmc let convenience successive separate deduce imply l induction obtain x borel condition notice construction briefly size stepsize form robustness tracking application adaptive know impact convergence vanishe consist adaptively sequence due far give modification x h behind expect probability mean stepsize recurrent aforementioned neighborhood essence homogeneous recurrent scope establish section straightforwardly chain mcmc aim optimally chain invariant specifically walk increment algorithm concern situation cone definite circumstance implement algorithm theorem establishe section conclusion eventually compact boundedness already show sophisticated robust version tail situation stepsize sequence multivariate scenario research project symmetric density increment acceptance aim optimize result display broad tailed situation sequence case contrast scenario scale prove new stability prove scenario improvement thank early initialization increment stable x scaling rely establish establish establish algorithm n contour proposal standardize follow quantify ergodicity vanish eigenvalue become matrix choose statement apply assumption bind weak assumption vanish density previous walk notational piece instead piece assumption increment distribution away twice define r tail x increment proposal symmetric condition condition tail already argument presentation handle crucial difficult establishe scalar increment b vx rv rv vx r proposition vx vx vx vx vx b x vx vx inequality detail straightforwardly lemma z z z eq treat fashion state prove establishe imply differentiable z c exist j convenient x x q z x term bound address x thank know recurrence soon establish satisfied time homogeneous control resp w proof establish lyapunov result depend w inequality x denote x c b w yield w deduce jensen constant appropriate vx n vx b vx naturally let possible lemma theorem imply subsection proceed decay tail target cover stepsize stepsize q exist r infinitely proof subsection differentiable density moreover absolute first notice assume practically relevant rwm proof notational simplicity function establishe consider control valid obtain expectation yield assumption cv deduce algorithm u cx one lead notice I lyapunov exist q p x w cv cv w cv combine intermediate yield x c notice choice implie since pt introduce stop proceed claim stop would page care omit notation I I monotone thank obtain conclude result pt see sequence nonnegative number sequence change part cx cx negative setting constant exist consequently xt consequently deduce monotone statement deduce eq imply existence conclude xt x conclude vx vx q dy x almost sure acceptance pt thank sufficiently take therefore z z z z b expansion integral z x first indeed ensure vanishe obtain term bind choice conclude therefore case aim combination line establish x equivalently pt pt choose conclude let sufficiently ensure conclude pt z x bind taylor enough proof cover imply hand z hand x xx exist chebyshev deduce p lebesgue measure integral x z deduce statement cf r mathematics tw united capital support project letter al recurrence set control markov chain area lyapunov parametrize family transition combine lyapunov lead approach process crucial practical recurrence call adaptively statistic control numerous encountered control markov transition control markov consist define family mapping concern stability recurrence process relevant tool process depend process parameter particular interest open despite relevance discuss follow one invariant ergodicity lose organized recurrence drift stability characterize respectively drift characterize recurrence worth dynamic exhibit separation main behind recurrence clarity remain abstract practice main show familiar dependent drift characterize condition characterize abstract focus practically update cover subsequent find noisy aim section highlight central numerical early adaptive chain type optimize specifically result apply approach adopt throughout drift commonly potential lyapunov function drift lead follow together transition explicit exposition cover statistic lead dependent exist aa notice hereafter convenient mapping cb ec I neighborhood appear abstract motivated concrete situation simultaneous
derivation write zero view claim conclude particular recall decrease resolve second application formula back integer define turn supplement proof proposition provide concern like lemma state zero sub obeys nj square gaussian exponential inequality sub gaussian q absolute denote maximum sub unit eq entry concentration gram matrix lemma union matrix mean shall make use sequel entry population gram becomes accordingly combine lemma exist variance position lemma combine sn equip auxiliary let first restricted rely recent therein variable say parameter essentially simplify sufficient row pr eigenvalue set parameter satisfy eigenvalue c sub self paper consequently virtue exist claim event depend row suffice write transpose transpose decomposition unit vector equality eigenvector cauchy schwarz estimate moment third analogously proof scale expand square treat minimize separately read derive decompose remain aside constraint unconstrained minimizer coincide rule eventually read comment meet fact case condition shall comment provide zero minimizer substitute back turn obtain collect apply consequently depend make use yield exist suppose fulfil hold conditional depend ij x e two table report table l nn nn nn l l theorem conjecture lemma sketch support multimodal number paradigm regard necessity paradigm negativity regression property meet non regard hence support view noise vector throughout concerned number even hope recover sparsity constraint simplest support pixel intensity time histogram count rate negativity numerous deconvolution diverse field image reference excellent seem simplicity solve minimizer solid appear reference decade author permit reliable sparse signal study body negativity alone may suffice recovery unclear continue hold realistic consideration apart intuition specifically dimensional paradigm prevent adaptation noise enforce main contribution characterize certain tailor negativity thereby apparent understand empirical success non establish design achieve regard support recovery combine survey overview appeal decade computational popularity paper empirical limited performance arise responsible sub optimal rate burden grid since small literature recently level free extend conference publication set set self may certain equipped resemble available develop use bound section provide sup datum section look empirical deconvolution appendix contain apart supplement denote denote obtain correspond denote matrix matrix row likewise identity vary symbol respectively scalar simplex set positive differ line line positive symbol denote asymptotic understand w array x n state say position follow submatrix column result excess resemble contrast correspond certain extra design necessary self establish slow self decomposition square modify quadratic role coefficient may negativity square reformulate least square ill square contain fail hold many minimizer position condition interior cone negativity yield perfect fit light constitute pure concern overfitte quantify turn study estimator expression mixture zero conclude nx strong obviously half constraint turn design still overfitte satisfy order quantify define fulfil duality convex hull scale origin sec separate origin argue overfitte fulfil orthonormal fact comment proposition qualitative main illustrative purpose entry zero onto entry n fit cf bind consequence far rather mild additional tail noise comment explicit correspondence level monotonically function understand desirable estimation rise general prediction error correctly problem excess nearly constitute persistent spirit notion persistence distinguish consistency bind recommend roughly computational lasso design regard improvement slow lasso require parameter non context noise continue section elaborate property admit establish optimal guarantee relate selector similar support state condition share selector different I sub least follow assertion analyze condition regard estimation j constant weak several comprehensive random statement line addition eigenvalue lasso sparsity attain apart factor achieved support lasso less multiplicative combination property eigenvalue satisfied resolve apparent contradiction satisfy restrictive isometry rip condition constraint discuss result linear bind rate partly restrictive term compatibility place compatibility sufficient bound distinguish additionally thresholde geometry subsequently study thresholded sequel two purpose need give projection subspace span orthogonal context lemma contain let minimizer negative problem proof separately establish minimizer next require quantity nothing else introduce respect term hyperplane accordingly q duality highlight connection assumption submatrix invertible appearance quantity upper set summarize sufficient amount separation effective provide entry broad roughly os may imply level relative empirical discuss performance well understand light reasoning ordinary cf subsequent discussion necessary generating arbitrary least fulfil scope application remain design bind value scale ss singular value detail class call even imply may thresholde might recovery basis regression specification purpose predictor give rise ranking arrange easier equally order hold denote whose rank rather moderate setup need rank top speak operational replace variance dimensional continue advance minor efficiently qr decomposition removal accurate coefficient elaborate similarity regard result succeed argue general attain provide specific finally benefit negative negative condition negativity coefficient condition employ e support necessary point necessary recover absence condition require highlight conceptual connection note distinguish scheme non recovery low minimum coefficient suppose playing role considerably literature fulfil impose regard c need support entail appendix issue provide represent bad sup e mutual incoherence require fairly restrictive bound two list non thereby formally lasso already restrict specific design slow spirit design paragraph attain estimate sup room leave present design empirically regard estimation recovery crucially succeed recovery without little familiar set directly specify tune close look basic design uniform distribution distribution generally multinomial integer sequel instance theorem bound counterpart well recall self condition turn design light statement restrict x center entry eigenvalue counterpart random scale c center entry q np contour plot solid dash dotted display sample size complementary experiment fix gram matrix decomposition interior vary correlation mass small ij ij reason reader refer supplement brief confirm replication relative bound away zero dramatically short compress cs recover sparse vector adaptive setup linear contaminate additive fall model attribute rare disease strategy affect form testing discard repeat adaptive aggregate group retrieve assignment need proper fact number ensure linearly otherwise prior zero light paragraph measurement sparse recovery approach base discuss set adjacency distinguish since constraint therefore self characterize aspect relie provide sketch paragraph several range component replication consider vertical bar empirical distribution bottom surface quantile summarize panel configuration stay set sparsity shift toward display reason quantile top case consider surface plot form fix roughly hand performance sparse spike deconvolution appear location spike goal spike potentially signal demonstrate deconvolution candidate position densely noise eq think place densely mean may spike process fast theorem result adequate eigenvalue lr part vector return bar represent dot median respective position replication square prediction spike evaluation gaussian construction gaussians evaluate result different parameter ii choose validation tune cross validation spike panel replication inferior roughly substantially ridge remarkably concentrate near spike present investigate lasso drawing independently rescale gram matrix numerical figure set gram setup lead design regard deconvolution sparse fails presence place densely concern component order consider localize function center center draw uniform uniformly sufficient amount separation choose scale uniform turn yield perform parameter control magnitude design aspect fraction vary fix configuration across thresholded nn negative matching omp regard recovery additionally nn subsequent regard table contain supplement thresholded regard recovery I permit support recovery estimate determine manner fix slight advantage equal estimation negative lar obtain solution hold solution base constitute second advantage recover support usefulness thresholding nn obtain non check recover omp serve success h consideration thresholded version excellent recovery superior thresholded lasso difficult parameter regime strength high experiment reveal may competitive recovery select correct pointed thresholding stress regard lasso entail design thresholded threshold adaptively without ii occur apart competitive behind partially keep succeed situation omp success consequence cf success omp error accordance sparse close nn arise extreme l l nn l nn l nn nn thank helpful thank read provide valuable suggestion early comment gaussian parameter generate obeys tail combine fact use satisfie condition vector setting square decompose eq property claim presence h establish expand minimizer lower bind self property j obtain assertion would immediately already feasibility strong substitute add particular adopt analog divide side assume c trivially event apply latter assertion rest invoke along omit auxiliary kkt exist imply solution least equality constraint separate substitute estimator value must observation second minimize negative cover tie think considerably mean parameter first analog hold minimizer observed equip bind
unless clarity exposition regularizer penalization every regularizer virtue prove coordinate multi perform without generality permutation reformulate complement respect imply complement positive iterative suffice g positive e z descent learn convex subproblem section e k still still k subdifferential nk n k reduce remove hold depend sample equivalent regularize separable permutation k u reformulate connection structure solve sort time I quadratic quadratic q relate lagrangian partial solution feasible separable belong prove difference equivalent solution replace inequality equality boundary lagrangian tucker optimal finding sort straightforwardly search regularize remark quadratic trust solve extensively study mathematical optimization trust region arise behind approximation original quadratic inside circular trust usually eq therefore involve provide separable trust eq lagrangian case assignment unconstrained solution inside trust region relate hold region become prove solution theorem optimal block method detail careful time experiment polynomial infinite accomplish implement accomplished newton trust initialize perform newton nk formula iterate split sorting use regularize separable region section penalization unclear penalization lasso contrast penalized penalize reduce penalization solution namely precision task eq sequence regularize logarithmic subproblem z block definite precision additional learning reformulate let b enforce complement k k strictly constraint initialize element logarithmic separable one related lagrangian duality strong positive matrix change sort newton method continuous tucker optimality additionally inverse inverse produce similarly order permutation index b b b tc initialize newton case problem separable solve duality initialize iteration initialization lead k initialization synthetic test truth repetition topology undirected weight ensure closeness truth measure leibler minus fraction exclude specificity minus purpose lasso also show kullback leibler recover edge remarkably comparison bind produce kullback observe significant difference kullback leibl divergence versus diagonal kullback roc curve choose penalization task recover ground remarkably well task upper rule graphical regularization tr kullback leibler always method regularization real world brain subject condition subject acquire second spatial template voxel group please see region span entire effect side six fold subject figure task observe penalization bad therefore multi bind graphical selection tr well multi second idea graphical fmri remain testing procedure six method stable perform poorly multi method multi upper b value significance previously log likelihood technique report minus compete method multi different r tr well r r r mi method mark statistically method statistically subgraph learn structure subject interaction negative interaction red subject method c dataset world publicly state brain outside brain reference template space smoothing perform extract matter group please appendix span effect region side subject r r b site site task one third third validation remain six repetition make subject take turn validation report likelihood scale visualization log comparison moreover well upper bind result penalization well difference statistically mi di multi method task cs tr bind penalization produce subgraph learn site structure produce consistent collection site covariance blue task replace regularization provably show multi trust region recover topology truth cross high compete fmri lead ground experimentally well penalization believe negative way extend practice analysis sample hope trust problem fmri comment grant da leave leave inferior leave leave red xx edu multi graphical boundedness lead provably sequence minimization subproblem efficient experiment well dataset learn aim algorithmic challenge super measure model non inverse equivalent measure technique enforce precision sequence regression encourage structure allow imaging region interaction become setting fmri available consider generalize learn graphical replace regularization norm prior sparse contribution coordinate convergent multi learning learn solve penalization experimentally relate short conference paper general assume assume experimentally recover ground truth always discuss mrfs pairwise
permutation popularity avoid adversarial total priori packing generalization pack goal coming generalize choice present notice ratio despite pack lp recently obtain value state obtain competitive notice guarantee competitive actually various option choose option online learn classification explore see make suitably topic seem pac weak allocation column replacement replacement use independence substantially improve requirement low model behavior lp algorithm permutation still understand requirement column handle column keyword turn engine furthermore guarantee column understand case fundamentally latter difficulty call obtain competitive packing state depend bound connection online lp pac classify correspond consider family classification learn obtain goal whole classification refine bind covering consider group classification budget lp lie subspace large mutually bad classification column classification much small indicate capture lp column approximate lie induce classification classification subspace embed establish useful possibility face set pack lp column lie lp section follow section pricing find dual infeasible classify otherwise classify column homogeneous hyperplane motivation select column reduce solve lagrangian multiplier obtain lp appropriately scale column index output feasible quality solution ultimately lead dim contain column b index use completely determined simplify scenario dim contain x feasible least denote linear classification column property work decision make analyze complementary budget ix ti scenario ii reduce cost concentration inequality argue although chernoff type obtain learnable scenario either classification look sample put depend skewed compare usually equation unfortunately overlap scenario skew skewed introduce right expression classifications b ix ix bx significantly skew direction equivalence event serve skewed additionally reasonably concentration inequality ability term I similarly contain discussion set put chernoff bad size collection unfortunately enough allow give flexibility give unfortunately contain set find directly case inclusion lie subspace single take minimal lie suppose subspace contain assume hypothesis equivalently norm span sphere namely close definition homogeneity norm lemma think robust phase lp column fashion guarantee dim theorem fix solution dependence arrive improved generalization see combination lp denote aim lp describe find lp bs try partition bad classification index let u dim subspace column fix integer number tx bi order description integer think acting phase robust phase union column error notice online robust routine formally lp whether introduce work generalize allocation difficulty classification linear conjunction subspace seem strong enough classification geometric course dependence side competitive possibility permutation real interval random every give duration least modify eq lagrangian lp x sx optimum follow latter linearity fix duration proof let lp complementary loose complementary complementary definition since feasibility get complementary get sx sx simple helpful bounding event occur take iw iw iw iw proof alternatively column jt c analogous claim canonical point whenever limit negativity third inequality observation thus conclude suffice family follow budget set inequality v intersection hyperplane hyperplane l face cover empty face hyperplane inequality mm face page conclusion paragraph non empty let lp denote column hand inequality triangle lp eq lp suffice optimum optimum use argument easy feasible conclude analysis change tp ss slight possibly infeasible mean coordinate dim integer sa tx bi say sx sx ba sx ba appropriate prove scenario definition budget tell ix condition ix classification unlikely infeasible classification total budget follow eq event similarly replace expression event classification take let contain contain reasoning event mx construct set follow lemma j equivalently concatenation union check I element impose denote good scenario definition last combine lp hand side lp return solution jx jx verify return column dim linearity n employ summation bound pt packing lp arrive variable maximize lp
representation gene node interaction plot consider interaction ie isolated gene appear split cutoff significant per genetic pathway modify already know knowledge often aside believe nuisance role interaction example seem independent highly correlated ignore age correlate adapt deal nuisance resolve issue correlation assume potential compare remainder usual replace mean center give motivate appendix section show condition asymptotically interaction fdr rapidly result asymptotically base mean variance realization infinite covariate order furthermore r j sufficiently little bit straightforward assumption fairly trivial must correlation difference may reader might bound independent necessary tail share one relaxed tail converge consistently probability statistic converge thus fdr condition fdr proportion remain otherwise algebra become clarity carry scale little procedure nice effect give rescale permutation like clarity originally observation class class ie minor statistic large converge estimate technical denote mean let integer correlation covariate every k tn q notation particular one pn theorem may bound fix cutoff use n also note straightforward give permutation original statement already relaxed consistent discover permutation discuss permutation base method backward efficacy simulate convergence rate plan replace similar reason hypothesis group implicitly nan nuisance computationally project onto nuisance runtime permutation dominate nuisance regression nuisance runtime mle grow nuisance much technical theorem manuscript technical lemma imagine consistently correlation fisher change formalize order r j ki one rate begin value ki pairwise inner formalize variance distribution write eq first note complete rate random known correlation distribution small subgaussian bound variable lemma wide light tail agree lemma begin hoeffding type inequality j I tn quite inequality triangle jx moment generate know subgaussian random sub triangle sufficiently take far apply union sufficiently combine cutoff ie find begin slightly discussion th letting split range change simplify size simplify straightforward lemma theorem sup distance bind sufficiently sup correlation begin reasonably sufficiently correlation close matrix integer average pre class class probability symmetry begin explicitly quantity contribution eq similar large great lemma make tight bound lemma along fdr converge satisfying satisfied great lemma combine probability get violate complete helpful comment corollary interaction dimension issue model assumption nominal issue permutation testing search differ method simulated find significant false discovery especially presence effect real genomic although gold tell restrictive assumption logistic discovery rate many modern massive amount science throughput vast accumulation technique classical error situation label covariate vector class class mean instance patient belong class develop disease biology detect consider covariate fashion main approach many logistic regression post hoc nominal nice summary subtle effect fdr regression call contrast backward propose attack potentially continuous show straightforward insight fdr asymptotic go specify generative row nonzero index pair furthermore test generative standard past pairwise logistic normality calculate standard estimate fdr problem approach even misspecification cause anti fdr term feature try add move avoid permutation method joint nan main interest difficulty resolve main effect nice adjustment heavily issue back logistic correspondence generative model mention equality toward class simplify logistic term traditional interaction correspond particularly satisfy marginally permutation test logistic forward logistic include interaction match conditional interaction marginal logistic interaction nonzero diagonal entry standard interaction thing happen interaction characteristic single toward end nan interaction pair test property interaction necessarily find forward omit interesting approximate describe x n mean matrix argue test interaction correlation coefficient variance version variance q compare nan work interested difference high calculate calculate assess significance long distribution instead directly discovery fdr reject truly procedure cutoff short know hypothesis nan numerator nan scale denote calculate nan permutation standardized permutation class label estimate number cutoff significant cutoff choose significant statistic interaction correlation summarize center calculate correlation matrix class fisher u kt execute standardized use new class fdr interaction rank testing interaction variable restrict consider necessary first variance correlation backward permutation scenario serious similarly real usual poor job find interesting attempt simulate version biological protein gene act biological diagonal block correlate interpret
complexity pattern variable check pattern mind l h j j elementary pattern set extract census mainly old adjust datum education capital capital week country response indicate set contain ht age age service state pay work education college school th status service sale op house armed force relationship individual pac capital gain individual capital capital capital week work week less country individual country united states south china france pair elementary event intersection elementary data age age education three say pattern pattern disjoint covariate implication form meaning exceed threshold probability occurrence condition event support frequent simultaneously correlation event name association next subsection pair bernoulli multivariate categorical marginal random pattern binary characterize metric sensitivity specificity true predictive u target proportion three classifier since true increase value high attention decrease turn procedure follow highlight binary base binary liu definition else definition easily association sequel binary association statistical discretization numerical suggest choose sort continuous variable expand choose improve learn naive recursively find partitioning step use association widely prune exponentially growth follow rule satisfy condition pattern redundant practice add exact maximum redundant frequent redundant indicator sensitivity specificity predictive etc pattern look step set define rule pattern classify pattern among focus satisfy contain redundant profile profile low prune redundant moreover respect well classification classifier redundant final constitute contribute rule bring basic principle pruning generate end subset nest h u sort opposite pattern nest one worth generate high good criterion help redundant algorithm huge redundant proposition cn redundancy remove pattern j tell performance prefer short hypothesis discard j follow u l corollary h mention l statement equality pattern small ratio high misclassification associate weak hypothesis consider opposite discard accept l u x u nest nest generate short predictive small ratio point li anti monotonic property hypothesis opposite discard generate nest nan accept equality support profile propose stochastic sample randomized u try nest pattern whether hypothesis vs nest try test pattern vs define try equality false x nest observation pair random independent deduce binomial take uniformly distribute stochastic reject n kx kx use test summarize step pattern test positive nest pattern nest frequent frequent generally less pruning value reduce pruning eliminate nested pattern select relevant nest pattern nest general predictive logarithm positive patterns yu yu u pattern pattern asymptotic normality ratio bernoulli indicator identically distribute accord suppose central multivariate delta method nan hypothesis following select relevant select pattern quantile validation redundant n diag nc method three subset machine learn size number select technique generate draw empirical depend write whose summary replace observe replace unknown distribution original let sample denote g test hypothesis proceed simulate replacement denote value following select pattern select pattern shorter frequent bootstrap g diag b diag b datum come uci repository environment nominal h ccccc nominal breast cancer diabete unbalance conduct sup database unbalanced assess statistical start choose rare class select observation rare observation combination classifier specificity etc performance proceed roc fitting conditional discriminant network law raise issue select sensitivity specificity etc roc generally binary forest classifier compare step adjust word whose measure well design create example replacement sample segment class near perform near paper generation artificial smoothed bootstrap approach combine sampling estimate fitting logistic generate principle show specificity auc specificity boost cart boost glm specificity specificity auc boost cart boost error auc specificity boost cart glm specificity specificity auc boost cart boost specificity specificity auc boost cart boost glm specificity auc specificity random boost cart boost glm specificity auc auc boost glm specificity auc random boost glm cart c specificity auc specificity auc boost glm n auc specificity boost boost cart association well mine discover interesting relationship variable near missing automatically interaction building predictive classification critical learning interaction search combination deal
enable well discriminative embed call simplicity formulate equivalent regularization original domain kernel solution xy denote gram matrix yy xx yy way optimization generalize eigenvalue yy yy projection basis help kernel yy degenerate either invertible formulation general yy yy xx xx yy yy another laplacian tend estimate collection denote empirical laplacian yy yy xx xx yy xx yy yy extension locally non linear find embed nearby high exploit graph sample affinity sample gram laplacian minimize embed find nearby build preserve relation manifold gram x x column zero solve calculate next find n therefore extension cluster method sc normalize cut nc concern admit detail precede integrate extension self self pca formulate localize scatter matrix localize matrix scatter laplacian lb weighted laplacian gram matrix generalize pairwise provide unify easy methodology cover multivariate paper design new general method formulate scatter augment generic multivariate scatter gram matrix generalize expression compact highly include also several localization generalize eigenvalue extension methodology design desire multivariate methodology adopt template appropriately methodology specific massive text article video difficulty find intrinsic trend nature analysis traditional hide embed actually report annotation fisher multivariate cca least square pls formulate generalized eigenvalue scatter augment scatter tackle number small deal non cca formulate augment instead scatter need overfitte fit smoothly outlier locality fisher discriminant reduction lot major formulate introduce et al show generalized square mild la square design researcher exist seem good address develop tailor specifically view discussion multivariate analysis methodology exist template new method combine template appropriately characteristic yet knowledge calculate combine enable supervise one label unlabeled follow fundamental method section review analysis viewpoint review template design desire precede section various linear extension help dimensionality several clustering analysis method concern set dimension follow brevity suppose center co occur sample resp resp occur sample I first concentrate pair many scatter matrix quantity denominator ambiguity convert lagrange multiplier confirm transformation unitary implie embed uniquely determine arbitrary practically useful heuristic eigenvector eigenvalue deal convenient close xy scatter xy extension scatter let expression graph laplacian theory definite matrix independent yx yy cca vector cca mistake equivalence cca sample case projection direct viewpoint objective follow calculate derivative square independent direction minimize maximize xy xx suppose equation objective cca cca part already reveal property discussion use arise exploratory linear variable often variance whose top eigenvalue minimize substitute x obtain n description subsection least pls pls try find direction observable predict pls space pls formulate interpretability original formulated follow mean every cf discriminant analysis da marker regularization popular technique optimization include area statistic machine sometimes call popular utilize ridge norm regularization square xx xy equation objective ridge derive yx xx ridge addition several include form detail previous major technique preserving seek embed nearby space reduce affinity heuristic n th neighbor choice minimize derivation convert fisher supervise discriminant overcome original similarity obtain affinity matrix lb lb lb n local manner preserve contribute near neighbor search follow supervised discriminant
without appeal eq corollary q exactly prove simulation entire good general purpose decentralize design provably approximately publish main grateful pointing tracking decentralize learn measurement distribute protocol noisy measurement make distribute cope vary inter communication noisy connect agent system mobile physical environment network reach full uncertain environment require development protocol distribute persistent environment goal change environment subject failure hope contribute development environment protocol capable study protocol evaluate structure tracking chemical mobile sensor circle sensor initially sensor sensor sensor neighbor challenge protocol adapt neighbor node converge challenge far wherein lie mobile subject inter agent connectivity noisy speed protocol especially concerned identify salient inter communication edge exchange message neighbor convention scale connect connectivity condition graph maintain node noise use corrupt offset neighbor mean random update wireless relative measurement frame alternatively may measure measurement measurement vector assume node incorporate node making namely pass successive precisely let measurement positive motivate collection mobile sensor direction take protocol describe social interaction collective cost associate majority individual rely near update describe remainder neighbor node intuitively seeks align neighboring node node align motivated number advance neighboring metropolis within lyapunov stepsize recent agent stepsize crucial intuitively avoid response accomplish ensure decay stepsize analyze collective collective motion vary stepsize similar reduce social target converge estimate almost surely stepsize nonnegative initial remark proposition may early conclusion noise spirit early protocol think protocol possible variation quantitative rate take place particularly adopt introduce metropolis walk move whenever neighbor walk walk start node number small least time node decay variance eq general expect convergence alignment believe vary connected measurement possibly communication somewhat time decay decay depend limit large solely possible nearly decay examine every node analogy consensus convergence bound counter high connectivity explanation mathematical phenomenon low update asymptotic choice decay g stepsize good effect maximum various topology mention graph time considerably time inverse exception multiplicative factor time concrete possibly vary grid node fall scale dimension learn noise constant choose protocol moreover slowly decay period rigorous result network observation incorporation communication first agent indeed learn classic attract couple decade due part application multi system paper work game distribute attract attention reader mention survey much link failure reader representative spirit recent vary node consensus iteration certain lyapunov certain closely relate number recent paper detection protocol like mention consensus consensus idea literature phenomena estimation contribution tight choose limit static unknown compare speed setting outline comprises broken piece step essentially basic fact devote solely bound analyze proving conclude prove several subsection begin begin whose positive entry direct edge undirected standard th basis use th argument simply subsection lemma useful immediate corollary provide multiplication statement sum side row side suffice entry side th may use change multiplication kl square introduce connectivity nonnegative stochastic denote metropolis aware make name geometric picture entry hold keep measure much conjunction decrease multiplication symmetric require bind feature nonnegative denote small positive graph weakly diameter evident definition necessarily orientation immediately without loss generality problem index absolute path make else assumption absolute applying schwarz continue preliminary prove proceed inequality increase positive deriving current devoted analyze consequence corollary lemma prove product moreover logarithm definition use last bound prof yet inequality continue bind eq rewrite equality bind whenever integer indeed consequence lemma claim suggest provide suppose b rearrange need argue lemma occur q simplicity henceforth notation place turn main decay special lemma rely previously subsection suppose nonnegative adopt shorthand break j piece piece term quantity piece piece upper eq piece sum straightforwardly put turn extension later proceed lemma suppose nonnegative corollary subsection place protocol begin proof theorem bound step convenient omit bt symmetric measurement time dominant matrix measurement measurement variable else variable put together use kt kt every q observe satisfie involve sum nonnegative row sum square lt kt since rt inequality second consequence definition lemma main assumption apply component assume remainder convergence strategy repeatedly sure exist observe three fact large consequence four remains apply iterate expectation obtain assertion infimum undirected communication graph satisfy connectivity k lm k km k km lm v km complete last strictly argue infimum undirected communication connectivity speed nonzero determine expression infimum scaling conclude measure achieved give set infimum nonempty nonempty connectivity measurement former latter expression expression infimum function imply fact infimum since finitely graph measure length proof proposition split proof recall two begin analyze eigenvalue connect one time walk probabilistic namely transform node every observe introduce node way stochastic within initial transition probability q necessarily eigenvector prove proceed prove theorem namely apply time use lemma expectation obtain use eq algebra exactly let assume q far assume proof portion imply claim associate argue big association introduce abuse notation contain else say measurement measurement happen uniform nevertheless assumption measurement say
terminology discussion course take let identically selector perfectly intuitively least one perfectly calibrate tend average well proof notation bag removal bag sometimes bag take selector randomness equality conditioning bag reflect notation average weak predictor suffice selector perfect calibration argument replace left expectation element decide bag observation side become arithmetic become equivalence equivalence easy care probabilistic ignoring prediction predictive make object overfitte validity regularity small irrelevant imply property complicated map say order invariant predictor function change natural invariant iid invariant selector whenever iid perfectly calibrate belong selector satisfy calibrate definition rational bag stand matter size bag bag hold bag predictor requirement essential depend predictor perfectly calibrate impossible indeed perfectly calibrate union set test bag obtain test selector probabilistic length probability case calibration train fed test fix threshold alternatively apply attempt score fix scoring calibration follow compute training observation score unique easily algorithm predict direct find close tie method tie never happen experiment refer overfitte give score classifier predictor object scoring predictor unless bt behind scoring label different different scoring function score flexible apart matter score validity predictor show corresponding give proposition arithmetic reproduce distinct would like increase place recursive partition consist formally perhaps element ratio notation proof division cell leave partition assign repeat decrease terminate one lemma recursive general predictor computationally inefficient large training scoring q pre spirit inductive avoid inductive pre train predictor section follow proposition pre predictor corollary bag predictor predictor order nice demonstrating calibrate fold average fold compare loss pair probability extract simple preliminary describe function loss predict whereas fundamental loss equal predictor assign mode independent online cumulative predict true main e course outcome regret minimax intuition interpret unnormalized normalize get case equal version towards neutral typical never ever predictor remain interior produce average section discuss score omit scoring tool nz decision j tree bag bag lr na nb network calibrate function svm standard however inaccurate score input see role comparison publicly purpose nine label uci repository breast cancer diabetes voting vote vary well order practice calculate test mle infinite incorrect confident compare dataset experiment remain drawback score test label object score training scoring simplify set predictor therefore never suffer hand proposition predictor cf simplify observation marginal somewhat score high violate predictor however proof artificial hope nearly valid experiment w computational split set w loss accord mle table probability direct bold classifier short name name svm especially sigmoid method redundant table come logistic sometimes output close hardware improve bagging bag decision tree bag involve set instability bag log make datum calibrate bagging rmse also often prediction whereas square produce obvious suffer infinite prediction mention arbitrary calibration simplify worst indicate table notice four calibration combination classifier would breast become mle algorithm early poor interesting interesting equal number dataset rmse despite theoretical validity similar confirm study preferable computational efficiency perform mle well mle set introduce predictor thereby applicability experimental suggest potentially well calibrate probabilistic log yield improve calibration probabilistic predictor theoretical guarantee share interesting multiclass solve multiclass predictor predictor simplicity rest ask set value fix function yy ad yy ad thank helpful proposition
dimensional think sparsity concentration many plausible remove single practical aspect realistic software available many go implement imputation readily available software approximate well test concentration hide propose variable nuclear norm penalization hide achieve apply likelihood variable moment matrix joint q except matrix iteratively replace finds guarantee penalize stage save converge minimizer generality identity determine write statistic involve miss overlap result clique vs six left imputation representative graph case isolate display unstable bootstrap simulation edge result figure edge stability propose correlation identify ed alternative paper effectively replace nuclear penalization paper easier choose reasonable penalty trace could combine
expert eq key step satisfy maximize complete third inequality fourth inequality clearly end shall ucb restrictive derive qualitative statement easy uniform finite support ix assumption strategy reason appear later uniform I request generality investigate go way expert good ucb oracle strategy limit particularly interested limit section shall derive limit section request expert go infinity surely interesting replace denote index draw discover expert interesting expert step request expert step step interesting successively suffice show surely elementary enough take fourth borel permit almost surely infinity accord conclude study sampling cycle uniform request proposition make precise extent exact proof omit go ii asymptotic optimality good ucb assumption converge almost eq item exist q end proof ucb proportion display draw intuitively discover possibility correspondence quantity draw converge furthermore ucb algorithm denote denote geometric permit comparison ucb uniform unseen interesting item arithmetic ucb proportion unseen small arithmetic mean expect get large proportion item expert unbalanced item find time ucb solid scheme dot present size remove interesting chose plot former easy visualize number item also section clearly ucb moderate ucb outperform sampling simulation conservative round run actual miss course long remain illustrate good artificial probabilistic expert geometrically expectation interesting item number loop good display figure ucb perform sample preferable fact proportion tight concentration inequality mass acknowledgement especially anonymous suggest section proceed obvious playing policy respectively deterministic sake clarity everything go randomized choice item round observe round imply sequence independent one miss large I conclude observe f discover interesting item run history history expert also expert request define consider item unchanged force hand behavior play moreover prove conditionally obtain everything obtain proof hand imply case sum final loop oracle policy hypothese ii open request advance make interesting request go infinity almost almost define eq find good allocation convex solution hence eq instance obtain denote give conversely eq limit proportion item oracle converge proposition ex financial engineering edu department electrical engineering science ac universit fr power expert optimistic paradigm estimator prove expert restrictive optimality provide finding request finite probabilistic request distribution discover request issue amount identify indeed security system lead consequence country electrical identify occur note usually security every contingency identify unfortunately credible credible contingency usually available time security therefore frame propose address rapidly credible significant simulation point new probability strongly sometimes choice possible within therefore engineering build raise try answer able contingency instant result security application web finite set denote assume independent describe pick index observe horizon item request element choose accord time strategy interesting small expert strategy expert non support restrictive support cardinality description generic term good ucb rely lead ucb confidence armed bandit rely analysis ucb large capture property contrary armed bandit bound understand make future policy key grow analyze first paper ucb close precise emphasize bound explicit ucb probability interesting occur sample fraction emphasize denote modify impact least armed reference rely index ucb miss confidence ucb rely tp estimate accordingly ucb without assumption expert shall make assumption mass request significantly difficult hand assumption miss mass reference arm form ucb meet shall good ucb hereafter satisfied loop denote request expert next expectation number find crucially rely yield consist choose expert prove ucb arbitrary policy horizon grow problem indeed thank discuss alternative ucb obtain obtain obtain eq jensen cumulative well bound armed bandit regret completely
average third model average three component merge three component ii average I ari lead great average ari compare ari averaging probability ht average bank window measurement originally collect consist available data mixture bic bank either ignore model I merge within component component classification good ii inferior species ii result slightly inferior ari probability average window bic ii property species world r mixture use within window perform averaging average respectively breast case establish status package observation percent composition respect eight nine model mixture lie within choose component breast end accordingly model inside window equivalent report classification bic three scenario generate via generate scenario generate perfect datum consensus cluster outperform consider ht case bank propose paradigm dominate window probability merge ari take window approach one think use model model average probability average probability carry probability efficacy merge herein real performance issue three instead choose three seven seven slightly comparable approach component average perhaps great notably probability avoid component performance great may argue compete window preferable end user want model term carry usual base carry fact irrelevant average limit clustering consider penalize future herein perhaps begin corpus family cluster family illustrate non family window work explore lead weight focus herein also analogous fashion acknowledgement grateful member cluster comment suggestion computational award research innovation clustering model report good circumstance criterion often multiple thereby cluster result weight average component membership determine closeness paradigm merge method merge effectiveness model use mixture datum approach mechanism accounting model compete approach successfully cox regression comprehensive previously application try least square indicate voting algorithm similarity partitioning arise merge mixture often back model cluster social network expression cluster report well small case criterion component relevant argue criterion approach away necessarily matter reporting value paradigm within report average idea cluster mark significant accept consider herein average produce interpretable directly average probability remainder outline section describe average merging component conclude discussion suggestion work recently mixture parameter total introduce component impose covariance way quadratic spherical volume impose constraint eigen covariance g proportional eigenvalue impose parsimonious provide member member family carry detail model ht orientation na na axis axis pp pp pp range model associate classification report popular criterion maximize bic component wide nonetheless small necessarily well predict classification gaussian cluster might model merged different cluster hierarchical entropy fitting subset procedure criterion merge minimize misclassification procedure maximize adjust rand model recent work sharp although discuss merge fitting mixture skew skew merging give inferior flexible fitting mixture herein arise care would need mixture skew inference family criterion proceed without ignore difficulty practically impossible decide frequentist model take consideration average posterior model prior give performance implementation probability compute overcome choose fit long discard analogy average integral bic q bic equal herein compute describe merge window introduce suppose component model want merge produce component mixture another representation original mixture could combination gaussian case gives merge merge merging outcome ari rand rand compare data partition agreement divide correction ari perform ari agreement ari random ari herein merge component purpose merge reference case window reference component merge perform give number ii equivalent assume bic take regard merging illustrate inside four component partition correspond underlie merging call reference avoid confusion denote seven merging represent nd component go component go go component new component possibility put merging reference partition arise merge calculate row ari reference partition merging store merge combination choose correspond ari base cluster application g via classification either probability otherwise allocation merge merge simply merge within drop case component weight cf probability classification addition model time window estimate use predict averaging probability focus merge classification classification average come parameterize interpretable average careful might equivalent related problem switch mixture strategy match minimum mean generally naturally objective averaging approach situation model average reference model discard merge carry tend cluster averaging averaging merge averaging may reason component say averaging single inside mixture cluster arise average produce average real simulate ii successfully window component
copy journal library service sometimes refer time day copy volume include c data science institute scientific science citation social citation processing year text format format restrict quantity journal keyword exclude publish latter medical algorithmic exclude started note rapidly storage medium sense production additionally allow previous book cd explain retrieval thereby domain content support wide area system retrieval standard early come year either book cd term service supplement journal european north american branch largely branch north classification historical web site journal classification volume always date journal last cd due plan web completely access service www edu volume explain carry service journal appear service reader journal paper criterion employ paper classification journal obtain institute information service paper sciences citation social sciences citation index database service book profile contribution journal paper procedure paper profile service contain profile suggestion improve composition profile collect volume record number year year example cm author another publication publication book publish work business school field multidimensional application cover fu hill huber rand wishart wolfe increase see figure net nonetheless couple grow research production iv give follow economic start replace jointly uk university key table name pick appearance publication title title biology physics mathematics engineering literature economic service follow explanation change service cumulative user interface van books journal process long successively observable attribute observable firstly eigenvalue factor semantic term relationship project post salient hence profile source text label therefore analysis occurrence plane issue content axis dominate connect around north west factor year little north regard management typical later year regard look rather dot figure crowd project cite come follow euclidean correspondence full year min project passive hoc subsection project minimum euclidean distance aggregation hierarchical initially figure relate clear branch dendrogram mathematic management include cluster discuss association order home pattern label fisher hand rand fu wolfe hill cover huber cluster classical characterize dominant period mathematic certainly cite period come modern see management characterize cluster broadly pattern cluster tune role certainly characterize statistic would projection open search show repeatedly e author web wide set appear three record practically return display record display grey display imply e access lead shift role subsequently management trend massive proportion service continue challenge greatly reference http www dl survey special publication pp analysis link computer science engineering journal visualization open platform http security profile author journal book title recognition l tm pa pattern bs j ann fu ks syntactic pa fu ks syntactic pattern k discrimination hill huber ann maps multidimensional principle discriminant infer b rand multivariate string ed ss rr principle numerical j annealing manual wolfe ct automate search service citation service term produce decade distribute journal increase production post quantity couple approximately major analysis time mathematic recent time centre management different
filtering node weight assign set item rate edge indicate chance walk rating assignment depend rating graph I nod item content profile social assign walk weight chosen eq indicate walk move graph node vector walk back follow interpret node path recommendation directly rate likely user similar item rating recommendation prediction equation iteratively rank node recommendation rank represent separate category tag sort exclude user recommendation collaborative filtering similarity item rank rating item q rank weight predict rating rating item rating evolve recommendation change incremental estimate every scope recommendation similarly recommendation prediction recommendation system sparsity namely effectively recommendation na recommendation study social tend similar availability network offer extra social node recommendation recommendation user user connect recommend item user order experiment benchmark website post long review randomly item trust datum set consist movie profile gender rating rating user evaluation recommendation percentile size high percentile rank st recommendation consist item percentile prediction set rate relevant recommendation information art collaborative describe recall compare baseline warm start percentile warm start note considerably start h cf recommendation item content recommendation target improve start rating perform collaborative filtering support science science foundation technology grant research office grant nf dr wang valuable mr experiments department electrical engineering university nj filter build recommendation hybrid model walk system consist item content user profile social setting graph recommendation prediction algorithm evaluate start system hybrid last decade early system achieve recommendation serve component demand service amazon netflix recommendation intensive work improve recommendation assume user recommendation explicit rating item challenge recommendation preference information especially helpful give recommendation user little rate good integrate user social collaborative preference user user collaborative database prediction net dissimilarity bipartite link movie movie average build rate node random recommendation give rating edge user rating score unit user social make rate contribution hybrid collaborative incorporate user item user history recommendation construction edge assignment application recommendation multiple list list rate rating netflix implicit click user movie date item correspondingly profile gender denote user contain social
often jointly conditional sequentially chain initial note variable da one q jacobian determinant auto px da generic call markov leave sample z n n n da da px da slightly great case practical increase computational negligible benefit sampler quite simulation family sequel direct result px da term intermediate essentially implement kernel reversible marginal imply density chain generate law note auto correlation generate da algorithm trivial da remain mcmc scheme improper shall correct remain unchanged concern px da improper although translate almost everywhere markov transition density reversible integrable expectation expression markov initial distribution da px da address haar da transformation haar px da scale transformation haar px reader scale variance da proper haar px da indicate generate subscript proper haar da haar small asymptotic variance measure note augmentation counterpart subject proposition augmentation stress step kernel result suggest modification px da perform da eq translate specify optimal variable regard regard b conjugate gibbs meet follow conditional follow set v unchanged fix multiplicative important make hasting describe briefly mdp way multivariate probit set zero g however notation benefit exploit da arise joint abuse density admit gaussian come discussion put aside px e da conditional augment px da z z ig gamma appear da incorporate connection description generic px da abuse jacobian variable px da step hasting kernel proposal quite percent improper corresponding improper prior sample careful improper able proper may reward px da model iteration result truncate call step hasting joint verify z could implementation one structure detailed far impractical state setting may value onto discrete see mapping assume ix easily controller generation likelihood invariant scalar even px da transform application dependent study think identity noise mdp computer game henceforth henceforth mdp consist component block take angle move combination necessarily boundarie overlap state state occur map configuration configuration allow reach square bottom remove row irrespective termination reach subsequent influence state consist configuration total outline section control independently detail assume reward observe q case likelihood algorithm scaling function capture height square column square difference column automate system emphasis make prediction setup amount post burn da prediction basis end assess number draw subset prediction assess subset assessment action situation resort heuristic explore recover purpose qualitatively typical lead style region rarely second column third generation game immediately termination termination occur step full concatenation three assessment value px incorporate run run burn hasting relatively uninformative prior value improper improvement post histogram marginal significant autocorrelation da da indicate px da standard da observation observation increase qualitative characteristic prediction show block sequence state indicate predict action qualitatively obtain lastly move play inference observation termination step play well termination occur step aim unknown play subsequent assessment da experiment rate trace histogram density action use result even three information pressure allocate aim validate statistical record inference player action preference amount record player force play appearance player second record allow fall translation approximate posterior marginal pressure value indicate allocate preference panel term mode make make decision allocate difference computation action situation arrive gradually posterior monte carlo amenable kind sequential computation possible model paper microsoft indicate da predicting rank microsoft support choice da move comprise operation definition may routine integration integrable change line establish state procedure omit scalar decompose w denote cumulative several algorithm sample truncate marginal metropolis hasting proposal current issue approximation one latter quickly first multiply discard refine newton curvature mode acceptance program request implement un un problem controller statistical markov adopt unified framework new include essential good illustration apply controller observe action variety economic aim mechanism case assume arise specification state controller choose receive instantaneous specified state evolve action receive reward controller horizon controller noisy mdp involve specify reward aside difficulty specify heuristic observe adjust controller repeat simpler desire behavior human generating know controller intelligence biology economic aim purely statistical tractable behaviour statistical advantageous principled properly uncertainty controller mistake jointly future action model specific functional parametric maximize likelihood observe respect perspective augmentation make contribution adopt action mdp inspire statistical gibbs subsequent enhance new devise sampling infer expand da px da improve upon da simulation involve moving augment extra computationally da da px algorithm propose movement augment translation scaling also implement metropolis proposal augment illustrative controller apply game literature treat give player perform posterior player solely specify reward learning demonstrate model organization detail inference da px da assume discuss practical issue numerical various px da px present capital case realization short density mass random two jointly density marginal omit indicate correspond transpose comprise denote omit similarly otherwise mathematical denote concentrated point mdp comprise markov control process additional optimality control control state action probability evolution determine mapping value call reward consider criterion discount accumulate horizon discount ensure lead expectation finite set discount
work flat clutter object visually give consider symbolic object al consider place recently successful scene understand et propose dp learning object present idea specific mixture consist mix proportion mixture model finite model several independent model choose parameterized component product model cast define new combination total would prohibitive method draw component obtain observation effectively computation challenge optimize model jointly distribution mixture model informally present section refer model stick chinese restaurant process select everything assign exclude infinite model dp breaking process except sample number exclude conditional replace however assumption hold general control tune concentration construct smooth single two auxiliary distribution multiply l fx differ depend conjugate prior long prior multinomial sampling hasting present concrete section differ prior hard conjugate case lda employ hierarchical describe proportion symmetric multinomial whole vocabulary size lda allow share world type define fig assume probability model maximize integrate document inference follow classic q difference quantify goal provide supplementary iteratively infer document estimation challenge variational since lda term hard close expression convert eq value gradually violate bfgs quasi penalty one type separately pose assume therefore mixture assignment turn give use sample first behave document baseline hybrid object share infinite pixel represent group ground block diagonal draw I two draw wishart synthetic term density estimation pc z contour successfully identify correct specie specie average fm figure four corpus section appear http ac uk include remove word word article take article result fold learn topic article vs vision article one document leave obtain lda number significantly across lda trend present ability hierarchy sub tree topic proportion fact topic change number domain training dataset would right average different largely depend vs significant baseline consistent topic less multi parameter topic fig model obtains demonstrate different topic dimension contain keyword digit sp bottom topic set affect topic heterogeneity try asymmetric set worse lda hdp lda fm topic cccc vs cn vs force digits statistical realize algorithm cell rank list dimension different popular change object room location room scenario place test room place room fold location difference human pose object predict truth place object together due object share pose multidimensional membership mixture mixture multidimensional infinite membership finite build two model increase robustness sharing situation challenge sampling ability achieve model application pt pt minus pt cs membership membership generate jointly share unique variance nine require parameter fully describe hybrid mixture dirichlet mixture allocation combination topic well mixture ability complicate simple model vocabulary document build document analog word type let fig different five parsimonious scenario work multidimensional draw mixture covariance result parsimonious structure take topic combine specific control topic paper unnecessary identify structure multidimensional sharing parameter different mixture dimension effectively parameter different topic model hierarchical dp branching need accordingly hybrid finite employ name area topic development corpus latent variable statistical mixture indexing later propose corpus document share
observe slightly value large ise mrf mix close due dependency size number accuracy indeed set experiment show component compute size mrfs experiment ise mrf regular lattice sample require minute ghz intel run critical phase dependency mrf fig logarithmic logarithm logarithm field component significantly behavior statistically component decrease state indicate ising compute estimator small performance explain gibbs sampler explain degradation ml variable estimator accordance compute mrf chain generation minute ml similarly exchange close degradation letter depend intractable expectation mrf monte carlo frequently bound turn efficiency method process extension mrfs currently investigation future include derivation bayesian rao bound assign proposition letter consider rao field exact interest likelihood intractable derivative bind successfully ise monte carlo intractable estimation intractable receive recent computational statistic signal estimate markov mrf research unbiased mc letter addresses rao mrf knowing point view limit information carries specifically define practical perspective mean unbiased mrf likelihood letter address statistical propose express mc specific mrf widely namely ise remainder letter class work propose original method application methodology ise present conclusion whose eq mc normalize integration appropriate model important comprise laplace multivariate distribution frequently application intractable inherent evaluating integral establish lower existence weak regularity inverse q q rarely address letter propose exploit expectation approximated integration relate substitute whose element previously known physical property knowledge inference expression replace perspective alternative fundamentally expectation intractable efficiently approximated mc integration mc approximations suit number mc integration simulating chain carlo mrfs output algorithm markov summarize ergodicity sampler guarantee increase please invertible enough stable regardless simulate letter mrf use used z simulate specific ising wang ising mrf completeness discrete value mrf q
parameterize vertical positive isotropic hyperparameter align k km hyperparameter variant predict might resolution along little resolution parameter implement automatic relevance determination ard individual scaling adapt datum value variable important variable less variation coordinate zero coordinate could rely state benefit remove irrelevant decrease stay variant identify combination variable rotation coordinate system direction advance along axis tb calculated ard convenient establish identity matrix one compute derivative h ij x partial computational derivative completely automate set unnecessary state aspect particularly reason see consider formulate solve fit transition episode align ard ii good implementation ghz conjugate gradient give adequate visit trajectory differ start predict slightly known substantially state selection iii note iii choose dominant take look vary whereas along opposite diagonal ii along benefit hyperparameter likelihood predictive circle train insight gain look iii decrease fast consequence complexity capability help well indicate effective efficient small important employ sr online learn count keeping size essential good approximate sr ii iii center select approximation f subset detail reward except episode leave leave transition ii result without see however ii irrelevant unable weight consequence ii additional gain look cf likelihood moreover completely set rapidly even ignore task effective sr base approximation automatic feature generation primarily principle rl ultimately theoretical guarantee due less state action product policy optimal approximate policy iteration world batch setting online gain selection improve research artificial intelligence laboratory grant nsf fa class science edu rl value rl world application associate hyperparameter benefit solve evaluation problem fully allow turn enable relevant subspace eliminate variable substantial approximate value function approximation arise horizon rl problem satisfy function manually important require would could automatically adapt without resource factor easy example relevant input lie rl regularization promise instead specify individual specify key demonstrate selection rl sample transition automate hyperparameter likelihood loo minimization gp policy method benefit base search manual sophisticated concentrate effort part really matter improve generally space generalization despite rl explore variant stochastic hyperparameter use score online agent standard gps employ since approximation structured follow contain summarize benefit large problem sr approximation selection derive associate illustrate representation parameterized rl dimension directly target function guide either bellman reward unsupervised graph connectivity conceptually relate approach hyperparameter rbf basis location either descent bellman basis adapt individually overfitte place problem point use complex x r produce x value regard share gps traditional machine require inversion dense would regressor initially choose approximate take denote k mm mm motivated example nystr submatrix nm ij approximation nm nm plug eq b b similarly gain storage every prediction additionally evaluate sr produce degenerate predictive variance near zero far novel project approximation expression add term combinatorial find subset summarize forward step element well specific criterion general unsupervised target cholesky aim reduce incur equivalent schmidt every active span residual
exist latent scale regularize section scenario code writing intel ghz processor either dataset dimension computation initialize stop objective careful inspection reveal newton find quite apply generate tuning newton event safe stability panel display coordinate coordinate model non search dataset objective indicate objective panel figure plotted well low dense panel however small point suggest spike htbp three bt dotted cd plot different bt coordinate cd line initialize nonconvex guarantee global recommend initial algorithm experiment extreme case start avoid exact pick represent three run recorded point objective report thing value diagonal case try initialize bt get adjust tune safe careful initial difference diagonal initial surprising sparsity start column reach regardless coordinate good zero diag diag diag bt cd sparse bt bt sparse bt cd bt cd bt bt cd dense bt cd bt bt cpu cpu run element bt maximization cd much easy significantly numerically software package new author plus ex plus minus wang statistics sc edu apply lasso element marginal independence generate propose solve graphical coordinate attractive discuss descent basic covariance minimize matrix shrinkage norm graphical shrinkage element exposition method difficulty exactly zero point zeros encode marginal component concentration graphical concentration associate conditional objective impose challenge minimize complicated minimize solve current alternative investigate computational fast numerically test coordinate model variant framework find posteriori estimation regularize graphical map estimator know representation suggest consequently graphical expected inverse criterion remove term depend focus partition transformation drop fix rewrite conditional give conditional maximum every total cm step column algorithm partition repeat
exchange datum presence exchange drive scenario convergence cost pick stochastic run exchange assume regressor still covariance eq repeat validity either statement sufficiently justify ignore depend power square satisfied neighborhood covariance moreover rate two mean mean adaptive diffusion link introduce block quantity define perfect relate follow expression limit latter analogously devise adjust noisy exchange illustrate construction strategy weight update since simplify problem separate associate variance incorporate information exchange continue leave combination need motivate recursion node view manner similar steady satisfy recall r fourth factor ignore size moreover product comparison multiply instantaneous enable variance coefficient one q converge close replace equation adaptive construction several diffusion variation nod temporal strategy state highlight select network share combination add processing example solely current neighbor store estimate say smoothed enhance mean especially motivate strategy smoothing mechanism continue satisfy model sec continue scalar weight instant coefficient solution minimize still argument arrive version strategy incorporate intermediate estimate nonnegative need simple three filter smoothing illustrate letter label step sequence diffusion adaptation spatial combination temporal adaptation result variation filtering reduce variant processing reduce diffusion strategy extend argument sec doubly temporal processing adaptation spatial six variant satisfy note within operation moreover perform reduce reduce conclusion early outperform diffusion I u iw k u iw diffusion diffusion k j iw k u f relate reference start exchange add useful spatial occurring follow use project raw refer act project onto vector vector projection projection temporal plot table base use past current ki ki ki k p temporal processing tend across square enable least design distribute column length use instant vector model snapshot capture collect likewise history collect q estimate square represent hermitian represent hermitian common choice square time instant error occur heavily occur close diffusion develop rely solely compute node recursive square follow every matrix time node scalar nonnegative algorithm instant node incremental intermediate diffusion incremental step neighbor intermediate quantity matrix substantial resource matrix entry diffusion square start diffusion recursive equation stage introduce follow choice strategy variable q correspond illustrate special relate equation q approximation diffusion lead enhanced consensus update space smoothing briefly diffusion kalman filter evolve accord linearly objective track system interaction observe vector measurement measurement zero zero mean covariance matrix uncorrelated solely evaluate approximate filter denote block matrix unitary unitary appropriate next useful bound norm useful size argument nonnegative note equality induce establish bind attain left norm obviously q therefore establish dimensional equivalent norm exist nuclear define sum hermitian definite block norm absolute matrix specifically introduce block matrix matrix property maximum obvious vector vector establish maximum sum entry sign moreover combine next establishes block norm combination right matrix entry introduce matrix establish property maximum evolution error network hermitian hermitian hermitian spectral radius argument agree large radius block radius readily establish reverse entry large hermitian induce radius explain diagonal left stochastic block diagonal hermitian block matrix never exceed conclude matrix conclusion corollary stable lie unit propertie lemma conclusion stable possible appear lemma lead part stochastic prove converse choice must measurement possibly evaluate require processor node centralize operation suffer communication central processor problem fusion critical failure central processor fail need recursion elegant whereby interact locally assign node assume combination entry combination satisfy describe operate repeatedly neighbor iteratively state collect recursion express entry collect entry convergence condition tend initial consensus doubly condition state doubly fact verify induction validity likewise identity matrix iterate kronecker eigenvalue fact coefficient stable necessity must doubly doubly condition q rate since doubly know us term magnitude equal follow magnitude radius arrive condition successive iterate consensus strategy determine large worth doubly also satisfy condition combination doubly regular satisfie therefore ensure generate towards list connect consensus condition doubly satisfy condition connect associated nonzero combination weight include therefore towards namely multiplication entry power connected value path corollary consensus converge comparison strategy strategy employ quantity construction keep iterate converge consensus algorithm strategy quantity side process exchange update filter diffusion collect instant inherently keep streaming diffusion incorporate instant diffusion manner optimization iteration cf cf turn influence evolution interesting advantageous manner consensus strategy initially information exchange ease derive body optimize intermediate combine likewise neighborhood combine subsequently update also motivated consensus suggest e early instantaneous side equality diffusion strategy consensus influence help incorporate reflect likewise strategy rely desirable evolve recursion aggregate diffusion set matrix mean sensitive establish matrix size satisfy diffusion regardless whenever regardless contrast consensus individual stable diffusion relation consensus development application diffusion adaptation insight contribution student student laboratory http www student tu chen yu lee author also yu early chapter r work nsf grant author electrical engineering california usa email edu electrical university california suit decentralize various self behavior nature adaptive consist link connection topology solve inference relation streaming condition adaptation agent provide overview diffusion strategy adaptation network divide section motivation square distribute strategy descent diffusion diffusion noisy extension consideration appendix kronecker b laplacian block reference consider say q agent agent become neighbor enable optimum decentralize spatially rely solely continuous sharing neighbor localize article connect network separate disjoint subgraph network arbitrary connect node pass intermediate figure provide node able sharing directional particular node degree higher denote situation relation node flow nonnegative neighboring node receive reliability assign use subscript subscript denote send different exchange zero mean though node reach node likewise whereas situation use fig draw information control turn also reverse htb denote send datum send exchange illustrate neighbor leave admit interpretation example entry location location chemical localization identify band share communication medium agent albeit frequent among beneficial general important lead solve strategy overview area adaptation illustrative diffusion strategy enable adaptive useful consensus distribute nevertheless explain diffusion consensus strategy body present treatment distinguish quantity use random example letter quantity font letter need distinguish capital letter small letter scalar letter refer scalar variance distinguish vector scalar letter scalar thus refer time presentation letter row convenience symbol complex conjugate entry stack denote argument top vector maximum absolute among eigenvalue drop use article cl font capital letter letter scalar small letter scalar letter scalar entry diagonal entry top argument weight induced identity drop reader development optimization process square advanced skip read start cost utility algorithm nevertheless convex cost minimum choice dependent purpose concept underlie diffusion focus mse network beneficial argument extend mse cost individual cost already summary non solution agent would operate determine interaction derivation reveal converge tracking effectively relate noisy region observe ar scalar agent spatially nod eq far assume allow noise vary figure consist node spread space figure neighborhood parameter equation equivalent relation arise context process jointly sense order case noise profile allow depend index vary spatial definite convenience shall definite hence result follow recover multiply take obtain linear moment square mmse quantity mmse appear find quadratic respect find see therefore solve mmse coincide explain justify deal linear besides mmse noise level expression result square error attain find minimum recover expression attain mse location substitute simplify find expression mmse determine information successive become necessary devise measurement adaptive alternative node observation evaluate successive pass adaptive operate compute instant many accurate yet structure note discussion article structure process iterate instant compute exist appear constant positive discuss article size replace filter normalize satisfy slowly virtue size turn size environment reason shall ensure continuous learning equation quantitie deterministic interested nature step especially filter useful actually variable kind stationary change estimator tend towards virtue mean ideal variance filter loss adaptation gradient cause actual steady amount offset size smaller easy introduce priori measure assume instant variance priori mean excess quantifie offset adaptive far steady small step small well performance adaptation regression accordance variation noise perform however node model among mean neighboring share estimate network starting motivate node carry adaptation manner go achieve develop algorithm manner objective add individual note every verify global optimization desire scheme ahead square relate move finite impulse agent interest communication biology agent probe additive situation illustrate operation highlight inside diagram assume mean process assume white spatially scalar agent seek identify collect column input express likewise parameter equation continue expression therefore mmse attain location alternatively adaptive show limit perform worse small since observe underlie among achieve global cost adaptive manner ahead third relate source chemical localization application agent allow adaptive biological behave towards away e g tracking motivate situation target node agent spread assume aware point agent express inner product ok access noisy practice distance direction towards denote assume perturb continue either perturbation model deviation occur direction along direction perturb relative along amount spatially assume compare contribution assume independent find measurement since assume linear model q difference eq vector moment continue side compute however equation justify linearly estimation attain mse information iteratively adaptive implementation early performance therefore large distance direction towards location beneficial going way develop algorithm ahead role application help highlight advantage adaptation track change mobile close agent become dependent actual distance target vary well mobile time target close expect remark hold eq lead repeat lead variable eq nevertheless adaptive work directly reflect change statistical track non stationarity slow fourth relate sense cognitive type primary secondary avoid interference cognitive device detect band condition carry aggregated primary band communication environment consist primary user secondary spectrum facilitate secondary function say scalar expansion angular measure spectrum vertical therefore interval basis possible illustration purpose divide location control illustrate basis large class spectra h gaussian collect primary column collect basis express alternative coefficient secondary primary user spectrum determine stage involve repeat accommodate albeit primary secondary secondary highlight path location similar path primary primary power secondary user receiver collection coefficient primary vector contain primary user every power spectrum discrete frequency mean variance assume estimate every instant discuss ar localization implication vector scalar continue collect quantity eq subscript expression early difference scalar represent continue application secondary aggregate spectrum secondary amount vector secondary reconstruct spectrum kronecker verify moment secondary determine solve follow minimum error knowledge similar lead verify attain performance level level location alternatively similar secondary noise perform secondary secondary user observe arise natural expect among beneficial going performance develop mse tool frequently application sum assume article nevertheless focus column assume clear get square compactly view covariance eq covariance definite sensing focus derivation argument come beyond sec covariance ensure seek couple even determined describe lead estimate realization adaptive rely instantaneous powerful behavior change change reflect observed behavior construction strategy enable track act individually limited power perform well node process across consider development series approximate amenable distribute alternative motivate start introduce neighborhood mean relate row translate add denote say right use associate fig also eq weighting eq amount taylor series side consist quadratic cm along edge weight node local substitute second expression minimize follow relate equivalent newly side local cost correct cost minimizer value likewise neighbor suggest alternative cost lead powerful solution neighbor function node expression latter optimize weight matrix knowledge however non arise whose nonnegative coefficient node substitution property ensure constant un weight positive use hermitian characterization eigenvalue approximation theory newton former reveal select scalar set adjust far ahead replace argument modify global rewrite exception solely neighborhood soon fact node minimize evaluate start positive denote evaluate size parameter vary size sequence tend shall case develop return arrive iteration correction form implement step add correction update intermediate gradient actually try stand examine minimizer value actually perform step minimizer help throughout neighbor neighbor incorporate information approach widely item replace observe coefficient relate like treat weighting theorem collect translate one nonnegative satisfy intermediate scalar correspond weighting coefficient coefficient use first scale explain column row stand example illustration hand case correspond coefficient connect instant strategy combine update exist intermediate node perform exist estimate neighbor second aggregation combine neighbor step reason name adapt combine second adaptation follow reason combination real influence information exchange aggregation strategy reduce step locally node observe pass exchange gradient local strategy q significance applicable cost quadratic detailed involve neighbor label evaluate neighbor subsequently dot represent diffusion involve exchange exchange represent return reasoning step arrive nonnegative coefficient strategy step combine estimate intermediate perform aggregate estimate neighbor exchange node information intermediate simultaneously name adapt first involve step strategy combination reason network special exchange rely locally node rewrite cost follow involve first exchange structure fundamentally lie update combination adaptation step ease reference list derive follow early use dependent require reach consensus agreement likewise scheme exchange aggregation appear subsequently mean decaying adopt towards function l k du w k w strategy weight word set derivation comparison common strategy doubly row reveal converge common consensus strategy agreement end adaptation flexibility process tend square deviation manner require index vanish size property adaptation function adaptation combination topology mobile solve knowledge agent expectation become arrive adaptive coefficient adaptive usually start convenient clearly view approximation successive iterate iterate nevertheless shall continue use ease implementation data realization implementation practice approximation introduce update interpret involve cost affect diffusion close list htp mse individual adaptive l l w k w l operation diffusion time instant step first receive neighbor information use node combine intermediate neighbor node step exchange aggregation instant strategy step combine estimate aggregate information exchange node receive neighbor information intermediate estimate exchange exchange aggregation node run kronecker delta situation study implementation noise necessary sec rather useful descent general scalar set correspond implementation correspond become furthermore correspond reduce stand alone recursion exchange examine fast converge measure subtract relation relation compactly collect collect node vector represent likewise entry neighborhood denote neighborhood combination q reduce appear block likewise kronecker replace matrix entry matrix ease reference list sequel variable nm return error end purpose non node evolve special evolution vector distribute implementation examine convergence quantity block block extensively exposition reader review stated lemma establish descent establish end converge solution node optimize pick run diffusion estimate solution positive step nm stable meaning eigenvalue lie ti nm verify ensure matrix long weight matter influence nod actual classical stand alone bi directional stochastic combination stochastic employ non q row add early satisfied weight share strategy enhance step size optimize doubly satisfying consider distribute diffusion mode individually say condition decay establish satisfy stability verify also covariance e eq hermitian matrix coincide large jensen hermitian matrix nonnegative scalar choose radius r jensen decay zero ti nm likewise determine radius matrix note ti nm ki u k r nm individual spectral radius handle size assume covariance enhance set size stability condition case meet hold magnitude diffusion decay rapidly diffusion doubly stochastic ti nm strategy information exchange convergence enhance ci consider optimize stochastic situation mode algorithm operate descent satisfy meet decay zero ti nm r nm nm previous theorem highlight important fact role combination convergence diffusion influence stability matrix radius ti nm r dynamic error move adaptive implementation gradient noise steady perturbation average reason mean special resort measurement order become update adaptive scalar coefficient respectively matrix choice correspond adaptive correspond likewise correspond share become operation run classical stand alone performance diffusion exploit indicate related likewise analyze strategy network present early shall satisfy form white spatially far network turn result expression well simulation small independence stand filter whether estimate adaptive diffusion implementation towards introduce desire measure measure would term implementation capture priori second component variable confirm always least priori order quantify particular excess steady quantify offset square square indicate stand filter non section examine among node influence since operation network require nevertheless say expression performance interesting subtract relation recursion replace matrix instantaneous I compactly quantitie error likewise individual diagonal say matrix contrast r simplify combination eq time reduce column vector zero eq covariance relation end evolve operate individually non across would evolve appear replace independent early noise error recursion replace I would say asymptotically follow statement theorem analysis pick right network measure describe converge optimal k combination combination influence neighborhood lemma become classical result stand estimator node case converge bi diffusion choice convergence optimize pick network adaptive left theorems stochastic enhance satisfying network datum condition consider diffusion node operate case meet select decay rapidly enhanced uniform say independent far employ meet diffusion decay rapidly consider establish strategy enhance enhanced ci cost pick satisfy situation measure operation network run diffusion operate individually case choose require magnitude diffusion decay word diffusion highlight follow important matrix influence mean stability influence matrix matrix stability influence convergence since radius ti nm converge fluctuation examine evolve steady node evolution mean steady quantity datum square thus square evaluating quantity hermitian weighting say reveal energy block hermitian nonnegative stacking column write sequel convenient start compactly short note constant denote side q side evaluate expectation expectation compatible nonnegative define evaluation order continue sufficient exposition focus sufficiently term power size ignore let topology statistical profile expression transform follow compatible side coefficient matrix reasonable replace expand plus square whose effect term notation square convenience notation exploit simple rewrite appear relation independence impose sufficiently size power step ignore leave stochastic hermitian nonnegative definite actual recursion side equality relation transform recursion expand convenient exposition except ensure stability fact establish relation remains bound converge steady radius steady value adaptive square lie strictly unit satisfied satisfy define moreover rate determine compatible form note stability ensure also stability second order would stability require size stability stability node steady term error I n diffusion strategy satisfy hermitian expression average therefore order select q matrix early repeat substitute arrive diagonal obtain likewise diagonal except index whose network measure obviously implementation perform series express alternative filter expand desire simplify regression employ combination matrix doubly refer uniform variance noise across network sequel sec expression kronecker identity compatible likewise introduce noise simplify additional representation two separate dependent compare use steady iterating compare expression recursion relate measure evolve clear stability term grow unbounded obtain average node evolution select substituting arrive correspond follow recursion network table ahead summarize derive section doubly combination eq row auxiliary verify doubly part outperform information profile consider exchange scenario follow description expression derivation get performance exchange perform exchange measurement case would doubly stochastic eq without exchange exchange case end effect exchange correspond note follow difference performance tr u c network
bayes neural network propose neural network layer well go thus algorithm natural part role learn representation share neural outperform hierarchical n sentiment al answer wang name entity chen al extraction chen et parsing know still investigation recently learn work seem however instance small vector highly whether sentence encode segmentation module module stop word text come intuition word topic occur target word w word type encode position word order feature cross module preprocesse experiment window yield well use speech tag part speech tag word nn noun phrase vb dt feature encode word sentence follow seven na I neighbor logistic label na I bayes nb conditionally independent nb na module al neighbor choose close learning near take vote nn extremely use near neighbor principal pca eigenvalue problem covariance magnitude basis small et nn nonlinearity introduce trick map reproduce rkhs convenient nonlinearity implicit mapping compute use q center equation eigenvalue like pca basis square eigenvalue double dimension use polynomial regression technique probabilistic prediction prediction softmax et sgd rate sgd perceptron mlp feedforward artificial mlp layer layer view input try output classifier layer mlp layer stochastic gradient hide architecture perceptron find hyperplane programming control tradeoff width extend form optimization nonlinear rbf deep graphical hierarchical latent start learn feature layer field restrict rbm recursively parameter set fine backpropagation discriminative second adjust architecture deep belief sgd function hold cross validation tune iteration rate hide layer restrict boltzmann rbm another connect different make rbm h restrict boltzmann machine make visible unbiased rbm try reconstruct function bias visible layer respectively activation rbm achieve descent start take input rbm try h restrict boltzmann machine lot divergence cd introduce perform gibb gibbs come various relation rbm combination stack rbms complex label current backpropagation bp smooth possibility optimum backpropagation slight model task instance datum grain measure performance micro recall side frequent sense word task nn perform improve polynomial kernel rbf kernel principal component machine comparable among work good logistic perceptron mlp make nonlinearity work well outperform baseline learn algorithm rbf nb mlp rbf micro p lower nn well na nb strong bag binary bag sentence work mlp good feature outperform mlp nn na I nb feature among work mlp bad feature score score outperform logistic regression mlp outperform learn feature feature nb feature speech svm score higher improve little worse still outperform baseline much improvement alone basic make fairly belief word three compare art superiority art support machine outperform reduction pca I however regression perceptron significantly speech local much feature need help extract engineering deep nonlinearity instance lot cross show instance rate direction firstly representation helpful improve per secondly source one incorporate department cp ac intelligence laboratory institute technology ac intelligence laboratory institute pi ac rbm
important issue machine learning show bayes boost machine classification fundamental comprehensive boost svms prove label previous consistency pairwise surrogate class difference conditional consider expect distribution minimize challenge consistency surrogate loss formulate base new consistency whereas yield calibration necessary necessary insufficient hinge loss calibrate inconsistent consistency ranking quite setting instance consist accord yet label instance exponential loss hinge loss hinge inconsistent auc consistency surrogate logistic surrogate loss surrogate generalize calibration necessary yet insufficient consistency surrogate loss present exponential well general surrogate loss equivalence loss auc present denote underlying distribution identically otherwise maximize auc fx f risk infimum take measurable get hard practice optimize auc hinge infimum instance define define follow auc show sufficient error auc calibrate auc eqn commonly hinge hinge square expect pair instance different consistency instance minimize minimize study focus conditional risk hinge contradiction consideration loss simplicity hinge loss contrary prove calibration necessary auc consistency lemma surrogate loss partly motivated converse direction inconsistent auc hinge surrogate inconsistent detailed defer addition hinge absolute prove inconsistent auc loss surrogate f inconsistent auc hinge loss absolute convex calibrate lemma auc long lemma insufficient auc parallel binary classification error auc consistency whole present detailed defer differentiable present simple lee hinge provide condition bound later theorem surrogate prove consistent exponential surrogate x f f consistent consistent auc reformulate follow consistency weight loss x consistency hinge convex increase derive variant hinge consistent example consistency norm hinge loss hinge surrogate x immediate auc hinge general hinge eqn f auc hinge loss inconsistent auc loss loss hinge loss bound corollary consistent detailed logistic fix rf instance minimize norm hinge regret bound exponential focus fr partly theorem logistic hold detailed corollary rank bound q negative instance e x f hinge norm hinge loss hinge auc corollaries regret whereas provide regret analyze auc regret bound learn improve accuracy infimum take measurable popular formulation loss logistic etc risk f f infimum take measurable begin bind eq detailed theorem classifier optimize accuracy optimize auc classifier easy smooth score rank exponential x f section learn optimize exponential proper theorems rf f rf r pr f asymptotic f auc consistent threshold consequence auc infinite sample adaboost training consider show equivalence auc provide equivalence calibrate exist auc differentiable auc conditional assume convex eq case obvious g exist similarly eqn instance eqn give f f hinge construct eqn yet complete begin crucial differentiable increase definition contradiction similar introduce far derive give differentiable function therefore f eqn contrary eqn immediately complete e jensen c f expectation f c x f f get therefore x p inequality apply exponential logistic function immediately therefore proof eqn hold q easy yield surrogate x r r x pairwise loss auc e x x f complete area roc diverse convexity auc algorithms surrogate calibration yet insufficient auc e hinge calibrate inconsistent auc base asymptotic consistency approach surrogate loss least square etc prove derive obtain many surrogate finally bound equivalence accuracy straightforward adaboost limit infinite design pairwise propose require scan auc superior auc hinge theorem area roc criterion diverse imbalance optimize convexity pairwise etc
notation z q fix simplify hereafter ease gram lemma simplify q density starting matter appear I third calculation put full see sum case claim simplify combine row eq fix index actor let th actor domain operator abuse word row identically markov proof leave reader survey proof force start proposition fix fix treat recall fix claim empty f convergent subsequence consider arbitrary convergent subsequence observe algebraic real convergent subsequence real orthogonal real algebra fact take subsequence distinct diagonal generally see eigenvector hence algebraic eigenvalue orthogonal implicitly moreover eq fact convergent share common subsequence convergent activity doubly three develop infer actor latent stream actor process latent actor movement model govern dependence population actor position online actor value numerical scaling doubly classification g present ever challenge wide inferential contain message short message infer community asset actor word income summarize retrieve past transform detect mathematically streaming time actor represent actor message suitable deal notable cox hazard doubly also know although self sometimes cox hazard reference information transform actor multivariate take suitably statistical hazard activity actor model bernoulli family multiplicative actor time mild consistently participant explicit actor model intensity model spatio information series multi model among actor pairwise overlap interval actor mention thought capture underlie social empirical dramatically useful information doubly process drive latent process interaction actor proximity position close actor configuration message model actor approach representation formulation population position convention communication activity undirected mean know actor receiver probability probability proportional set k wise vector order product give column th entry slight vector entry number actor fold bold letter bold letter denote actor identity column one abuse notation indicator confusion actor big observe actor longitudinal actor feature frequency pairwise activitie illustration far position detailed diagram message generate actor community population member distribution proximity actor message distribution path eq value vector covariance positive smooth degenerate smooth degenerate diffusion taking th coordinate finite dx modeling example yet co exist dimensional motion stationary relative center ht b order actor population define begin give taylor value actor actor actor roughly speak influence actor space actor actor latent discussion motivation appendix actor deterministic path assume latent position assume let column vector transpose compute symmetric xt xt b message send actor actor message actor experiment actor possibility actor actor time interval jt ht paper continuous x formula update posterior present formula actor generalization case actor notational complexity nn ni jx ji ij dm ni ij hereafter develop actor dirac delta generalize formal adjoint u update projection latent systematic cf cf idea approximate weight discussion generality dissimilarity arbitrary solution mapping run dissimilarity relationship g read influence last section fix note away subspace lebesgue algorithm existence simulate property fact poisson ij u u ij ij ij jx x interval practically mean normal experiment hope detect possible position even though information estimate experiment environment end actor far away mode affect actor small eight actor illustrate eight actor able adapt five actor axis jump actor update discretized unit mixture initial actor sample distribution drop low simulate actor value near show behavior vary intensity produce simulation observe amongst actor change unit interval amount eight actor message equivalently actor hand eight actor tr vertical latent eight actor position eight actor experiment dash filter position actor position eq sufficiently interval height recall close connection model paper general setup experiment experiment comparable generally roughly amongst actor translate actor rough stochastic run q move average fix haar modify straightforward reader latent instead get large medium showing compare clarity suffer embed dissimilarity color clear lag accuracy clarity compare figure misclassifie indeed figure embed position true unobserved position bit intel core cpu ghz machine gb ram actor take red cluster gb monte replicate take slot replicate take ari maintain ari maintain x nearly prior show embed row position embed embed illustration actor activity complete cluster posterior present level level posterior online study often address stop illustrate simplify mild task structure crucial maker whether act confidence infer measure establish take explore robustness
exploit suboptimal example adaptive adaptive incorporate localization check play suboptimal nature abstract level subroutine return round sub particular often procedure exposition block block learner stay start start localize history move last learner suboptimal incorporate localize parameter guarantee relaxation x else round x admissible relaxation regret localize meta block block pp take advantage optimality situation literature conceptual online setup strong adversary meta either say next separate minimizer round localization update round next gradient recover follow style gradient specify admissible suppose play strongly specify turn study well unnormalized remain winner end localization q erm minimizer say minimizer non minimizer behind strategy next expert round gap suffer relaxation algorithm exist eq loss localization consider expert become winner round winner throughout cumulative throughout game suffer heavy dominate depend choice investigate full minimizer future history erm erm consider online lipschitz loss localization localization strong norm small eigenvalue exp loss dependent localization gradient pay adversary play adapting sequence play happen play sub adversary rate already adaptive localization remainder try introduction notion relaxation burden development like stress thin air rademacher obtain computationally relaxation something one would follow omit explicit sake constructive instance response relate protocol proper learner select simultaneously suffer improper learner side information pick nature simultaneously improper version easy may equivalently protocol learner necessarily mostly focus improper distinction difference improper write eq side achieve equal result specialized section relaxation extend term consider prediction complexity define contraction style future definition latter future term sequential rademacher attempt step rademacher condition conditional relaxation obtain relax finite proceed weight rise spirit attractive function function sign relaxation achieve yet lead admissible alternative method regret guarantee predict label correspondence proposition combinatorial analogue alternative remain trivial algorithm technique consider relaxation sequential rademacher involve coin computation couple expensive generally realize rather strategy draw sequence solve optimization estimate utility move random solid furthermore randomized strategy informally idea trace form mix term drawing verify make infimum expression inside outside example randomize tree variation regret walk related idea interestingly many turn rademacher complexity constant factor supremum draw erm rademacher complexity relate finite natural method perturb previously kind randomization rademacher new case style later trace norm completion expert one rademacher complexity complexity later verify consider x partially average randomized method relaxation forecaster see sequential rademacher equivalently bad classical rademacher ensure draw rademach argument regret rademacher particular setting expert specify game need appeal case distribution algorithm pick look closed randomized perturbed simplex claim especially attractive draw provide perturbation unit draw rademacher draw independently pick enjoy eq weight perturb uniform surface follow show draw likely produce direction ball unit previous address ball sphere consider randomize round draw pick enjoy importantly depend follow perturb adversary simplification perturbation begin game method also outcome play adversary contrast version side present pick observe exist x assumption relaxation assumption admissible loss case assumption rademacher complexity instead box access return bad depth round expect regret expectation whenever strategy g convex scenario static make outcome suffer lipschitz loss sequence turn expert sequential classical rademacher rademacher reason sequential rademacher general conditional classical rademacher average admissible absolute forecaster static expert plug value thus variant forecaster relaxation prediction supremum potentially preferable idea start note convexity solve online alternative game pick choice pick adversary indeed convex first game eq linearize relaxation admissible specify regret algorithm compute linearize note need evaluate expectation enough draw estimate close randomized draw one sequence show linearize already randomize idea expectation relaxation strategy specify expect employ alternative novel completion predict unknown collaborative pick shall predict relevant round entry know advance value algorithm regret bound depend order entry mild condition function regret entry game two alternative problem subsection alternative variant small trace regret guarantee moreover variant also computationally present equation hand spectral round matrix ask notice round efficiently subsection round second second loss equation trace position elsewhere ask fill inner respect entry first value problem yield involve trace constrain per randomized specify use situation constrain treat adversary illustration constrain within mention step selector define algorithmic step supremum satisfied expression feasible rademacher complexity relaxation upper complexity relaxation mirror step size problem mirror universal log factor exist whose conjugate q regularizer descent sequential relaxation arise relaxation adversary let relaxation whenever relaxation form mirror bound remarkable universal mirror algorithm algebra th admissible opponent repeat obtain remark leave side process recursively immediate consequence minimax q infimum expression sign supremum arrive equality effectively add swap follow admissible exponential infimum weight improve function rademacher relaxation show weight relaxation arise rademacher future tree last represent worst remove argument see general mirror admissible relaxation optimal one attain infimum optimal relaxation yy f g tf increase term hence concave rise eq point clear force fact value obtain value familiar update plug algorithm admissible derivation smoothness remove eq establish relaxation recurrence gd block block get gd interesting two block relaxation turn start initialize block length trick round round strategy pick apply also note case summation block continue become small block trick entire round regret block regret initial block sub gap minimizer round already play round block suffer playing rest thus suffer let think choice two whose sum introduce arrive bound justify gd zero bad depth let consider optimization eq write side rademacher substituting complete part finally relaxation sequential sign sign class take possible denote sequence projection relaxation q I factorize derivation randomized strategy propose match randomness arise eq theorem compact last draw draw multiply instance convexity minimax swap infimum supremum need non randomization extra throughout observe remain q coordinate j remain consider eq hand symmetric sign coordinate expectation argue ball ball rf hand ball h conclude round play randomize define randomized view view satisfied lemma show randomize specify admissible conclude statement randomize admissible additive hence sphere equivalently euclidean observe use omit symmetric prove geometric argue uniform decompose sphere align htbp direction vector length plane span angle deal dimensional span align coordinate read prove side mind four square cross square root completes yield assumption dropped expect specify choose start however calculation identical thus conclude need strategy propose rademacher tx convex first minimax eq minimax term prove analogously pick definition notation step minimax bound term expression write pass supremum upper shall start show admissible game pick adversary directly pick achieve equal last theorem pass case choice achieve relaxation convexity convex interval admissible bound w randomize jensen convexity pass variable pass strategy plug back q inside convex interval hence conclude yield increase I yield supremum two equally divide yield term distribution irrespective strategy round draw hence almost proof mirror let pick expression instead square derivative force conclude tree rademacher smooth case update specify follow proposition yield small pass must loss potential minimizer block admissible eq acknowledge nsf grant dms minimax bound constructive know derive one perturb forecaster emphasize rademacher complexity algorithm allow fast learn localize include randomize perturb method completion trace problem learn player observe distributional assumption past decade refer comprehensive sequential learning average number measure classical supremum rademacher concerned supremum martingale dyadic though complexity tool study value algorithm constructive literature study naturally certain sequential constructing corresponding method improve view constructive upper value surprisingly method perturb recent method show whenever sequential rademacher randomize certain development norm perturb prediction expert throughout paper appear air sequential rademacher understand inherent one aspect development average key also perform hand relaxation localize arrive complexity complexity lead fast equally localize present adaptive take move suboptimal study localize develop
categorization optical character recognition spam bioinformatics prediction carry standard scenario popular sequential baseline label propagation intend way global vs perceptron sequentially node test diameter imply affect prediction common underlie similarly intuitive fast algorithm majority predict time requirement need read approach sense node affect adjacent introduce method graph scalability comparative way span edge run give generation span span create via visit visit probability vertex adjacent visit yet span fast generate approximate spanning minimize sum tree short twice diameter short span root node random diameter check carry edge neighbor rule edge span near neighbor combine unweighted span nn take weight unitary span actual run span also classification span run one vote experiment choose world preprocesse normalization throughput protein dataset combination edge inter create exploit weight spam train split training remain unlabele create follow experimental euclidean j compute generate frequent category connect belong class functional depth classification database order associate task binary multiclass binary task combine try proportion size particular set macro average binary result contain table dataset resource run moreover inferior span dataset decide go refined tree table report seven span six split per tree well span purpose level achieve operate span see contain report span show algorithm span tree c rate take across use dataset span result relative among algorithm systematically run storage quadratic see near dense large situation real perform span tend original graph approximation actually practice fast way generate span inferior take take implementation actually rely span simple improvement reach introduce analyze algorithm term ignore linearization phase algorithm provably evaluation previously online predictor moreover combine span batch label scale analysis reveal loose tree slack second express low object well reflect tend graph generality prefer bias sense robust perturbation introduce ask robust number mistake weight question nice active classification support google ec publication reflect view reference main prove cluster sub start terminal cluster terminal empty close ii node adjacent node near mistake make iterate argument subsequent mistake mistake q define summing expression step jensen k obtain split predict split sub connect almost cost extra mistake edge suppose split node mistake make respectively pdf semi drop light increase mistake removal drop make coincide split sub cluster number mistake bound one mistake due let sometimes cluster overlap node fashion mistake argument mistake make conclude edge let linearization mapping edge pair start define visit sake visit terminate backtrack node path visit contain distinct construction pair create single traversal mapping remove edge eliminate transform l prove definition transform spurious create spurious edge create spurious free create spurious edge sum spurious elimination edge linearization let part immediately remain description pair gets replace edge establishe label free run line number mistake exhibit mapping edge union pre establish partition partition turn r last recall spurious edge correspond exploiting finally conclude proof lemma free hence associate giving turn lemma linearization span tree mistake equality lemma jensen concave corollary grow polynomially sake contradiction assume r n free edge give free contradict theorem proof part case adjacent flip cause terminal total sum weight weight twice therefore hence mistake node inverse polynomially mistake proof span label span span polynomially bind via summarize combination test macro set tree span perform thorough report four c split error split error class h c predictor average train split c measure l c c error rate train c f f theorem lemma universit di universit di universit di sequentially predict label arbitrary logarithmic random induce adversarial characterization achieve mistake polynomially random span tree time world compare perceptron propagation approach represent node item weight quantify similarity datum code input datum web spam categorization edge model relevant classifying order learner must predict label observe world application involve I role scale online edge advance start look kind graph unweighted online prediction mistake induce adversarial I mistake obviously lead span whose edge minus span mistake predict previous span maximally span tree predictive recall online suggests span prevent span edge adversarial assignment yield mistake unweighted term span tree unweighted prediction uncertain paper bound fill gap lead bound hardness weight weighted graph online must span exhibit mistake weight total weight spanning predict tree extremely like stress central issue involve context slow consider impractical operate near neighbor besides run linearization bring benefit turn perturbation label clearly practical empirical evidence compare literature graph prediction perceptron kernel version propagation baseline perceptron propagation method carry sized world dataset test span tree span tree aggregate majority vote comparison predictor baseline recall basic section survey literature relate mistake span tree mistake restrict match connect quantify provide constant devote let undirected labeling ng protocol predict possibly randomize adversary parameterized choose labeling node adaptively choose associated mistake occur learner total note depend randomization labeling learner use randomization adversarial choice requirement fact randomize increase mistake regularity regularity follow label free quantity I e weight define fix denote maintain tree q expectation choice rescaling probability thereby irrespective measure weight large dense unweighted large give big happen I disjoint path span see example ball graph edge contribution must contribution large bridge one independent path edge imply clique survey literature closely relate end learner perceptron embed canonical basis perceptron gd expect bind dense order investigate span mistake bind perceptron suggest algorithm span diameter reason behind span hand guarantee concentrate remain introduce show unweighted reduce span linearize first visit give line graph extent diameter inductive nn small ball diameter take cover recover unlike perceptron hold trick unweighted perceptron tree help diameter number perceptron like nn mistake speak way consider graph clique unweighte much scale cover clique hand large tend stay constant get close diameter become include tree effective strength flow square edge diameter clique sometimes yet fair algorithm logarithmic propose refined exploit paper application weight diameter diameter version speak select dual obtain logarithmic diameter unweighted select appropriately comparison present one paper scenario problem graph label energy minimization instance subject span draw long history g unweighted span tree sample graph reduce technique take due hardness reach walk connect light edge experiment test approximation building span tree finally generate conclude span tree reduce g result currently show node label deterministic mistake make pdf adversarial strategy graph consideration node node set contain small adversary node unique label previously use weight edge belong span edge node label matter mistake yield illustrative less label labeling connect mistake mistake unweighte set bound drop total mistake q mistake line free contain terminal terminal proof cluster split terminal node bind mistake near mistake must occur iterate get mistake k k jensen follow omit drop mistake mistake refine mistake solely let edge include eventually terminal terminal summing mistake bound mistake I w last mistake occur obtain number mistake minus contribution include bound mistake predict generate span span way mistake span tends reduce light theorem span mistake scale affect rescale w mistake bind every graph node substantially connect would uniform rescale follow polynomially connect fraction overall pick polynomially connect ratio total weight span corollary satisfied example nn contradict assumption connectivity need fact drop mistake extra mistake property bind elaborate compare algorithm perceptron section span tree distance depend weight reciprocal covering case unweighted whereas norm mistake approach namely diameter bind unweighted generally cover ball seem allow parameter diameter typical unweighted clique algorithm mistake corollary yet fair deterministic compute laplacian cubic quadratic tree interpolation initialization whose quantify comment drop run trial memory space constant linearize line initially terminal traversal make calculate constant top complete binary construct leave simplicity assume node build th maintain I subsequence leave reveal marked figure reveal dark grey link depict mark traversal operation predict traversal stop soon mark traversal begin go right determine look close locate start mark traversal mark path order find close mark go child node keep via distance determine trial visit trial place gets mark visit occur subsequent visit guarantee subsequent visit stop operation preprocesse linearization calculation distance trial show label graph significantly small running polynomially extend span tree labeling perturbation label explain beginning operate graph linearization whereas principle difference always couple edge unweighted moreover node label linearization step generate consequence linearization graph sign item hence star graph linearization tree free label regular labeling see quantifie mistake
nearest express interpret rule instead still neighborhood svm optimization finding recommendation author national national science foundation thank yahoo yahoo near amongst compare efficacy three none metric lead particularly improvement rbf introduce combine mahalanobis rbf capability nine benchmark difficulty outperform accuracy establish serious metric selection apply sake binary scenario restrict reason svms popular perfect furthermore maximum margin reliably generalization perhaps importantly svms highly boundary overhead trick map dimensional infeasible explicitly svms completely inner computed efficiently infeasible ij k svm entirely sake brevity interested reader description offset separate hyperplane hard margin ensure label hyperplane minimizing error solve equivalent separate hyperplane slack optimization simplify section svms well positive basis rbf dissimilarity must scale
dramatically variant demonstrate significance estimate vote actual opinion crowd take majority member collect vote crowd overall gold refer gold entire crowd initialization figure initialization gold gold accuracy gold improve initialize perfect expense budget gold performance substantially agree crowd gold standard reward vote yield crowd receive preferred crowd vote gold illustrate believe handle note achieve binomial early rather parameter trade would achieve illustrate point curve course increase help exploitation parameter example shrinkage uncertainty ask vote correctly highly seen correctly label make stable consider label mistake iteration mistake constitute vote choose recover cause mistake early one purpose reduce almost estimate approximately early stage majority vote prevent chance uncertainty vote explore quality gradually vote exploitation quality majority vote quality exploratory scheme high accuracy collect vote quality overall remove collect vote well approximate crowd crowd member vote discuss specifically exploration estimate exploitation crowd majority vote modular outline pool majority vote overall modular outline determine overall label entire baseline exploration exploitation idea behind crowd challenge approximate vote aim complicated distribution accuracy truth consider independence sub crowd crowd majority calculation crowd majority vote majority seem essence yet pool try like thank helpful project mit intelligence approximate crowd majority opinion crowd crowdsource present online dynamically exploration vote crowd opinion crowdsource useful vote quickly vote every member crowd may nature crowdsource vote crowd budget sampling comprise capability sample align opinion effectively crowd determine crowd crowd arrive require explore vote want vote likely decision vote align crowd limitation pay lot vote estimate align crowd majority decision pay vote suffer decision clearly exploitation tradeoff exploit mainly agreement crowd main modular template crowd include exploitation choice arise choice template dynamically determine vote make request uncertain iteratively quality estimate keep exploration exploitation iteration rate datum sufficiently experimental demonstrate identify member crowd approximate discussion representative crowd whose vote crowd vote hence vote informative approximate crowd vote voting wherein vote differential weight vote contrast vote wherein vote multiply scheme place emphasis vote high rest organize within modular approximate crowd baseline baseline specific choice conclude remark crowdsource increasingly lead resource amazon collect multiple generating label collection crowdsource clearly effective little disagreement approach develop reliability example crowdsource label treat demonstrate label preferable single label cost negligible pruning label less improve vote collect discard lower viewpoint effect researcher explore trust majority vote characteristic true sometimes difficulty effort prediction item quality however seek correct opposed turn know task precede effort collection simultaneously stream estimate exception reasoning label data accuracy classifier vote active labeling example pay price per vote reflect crowd majority vote sure repeatedly vote example look check whether potentially majority vote vote vote could bring criterion add candidate get follow quality add assign majority l c l l l conduct separate model crowd separate recommendation movie vote subset rate rate subset original rating vote boundary chemical document track define competition develop retrieve overlap list citation dataset article appear select document category divide test format half develop learn adaboost na I bayes svm decision several decision specific total combine half total baseline crowd uci repository collect census add noise add diversity strongly diversity issue aim varied prediction prevent due baseline accuracy half combine quality assessment vote build interval estimation ie refer item reward metric critical student sample vector majority vote select size subset exhibit criterion ask vote cause strict hand control use cost adjust spend compare demonstrate ability accurately vote modular view impact module
turn brownian distance distinguish marginal euclidean namely distinct independence measure informally speak independence easy problem homogeneity testing hypothesis borel pz w recover characterize statistic trace trace hold case diameter respect square integral operator xy operator apply induce distance characteristic briefly interpretation however interest completeness characteristic characteristic norm weight aspect kernel let translation kernel borel weight function integrable translation translation embedding one attention embedding eq discrepancy type ensure distance need proposition coincide whenever generate n z z inequality convex clear induction reverse triangle jensen z z imply z k k nn give natural interpretation w embed generate half addition finite also must mmd impose condition underlie type mmd energy conversely kernel small c gauss ii type type I type type extend analogy moment xy xy also kernel em x p yy xy schmidt yx x translation assess various multivariate microarray tumor event small type report trial high vary thm thm thm remark independence hand energy covariance literature embedding kernel establish energy negative kernel embedding rkhs family independence one member family statistical machine community space sample accomplish consistent random finite dimension moment community introduction discussion advantage certain expectation distance lead notion metric negative machine reproduce kernel test embedding associate schmidt criterion rkh far graph despite similarity base kernel link context statistic straightforward confirm integrable question rkh dependence formal extension brownian distance family induce kernel consequence large get sample bind third relation characteristic hold quite unlike characteristic kernel via structure definition rkhs theory relation rkhs distance relate quantity mmd schmidt respectively give quantity distinguish empirical test describe investigate variety strength family introduce concept reproduce kernel rkhs review relation space reproduce rkh associated rkh reproduce map say need triangle empty say iii say terminology type hilbert euclidean space use hilbert relation theory hilbert space exploit negative symmetric definite summarize nonempty set let definite call brevity hereafter simply abuse terminology work distance scale equivalently obtains express canonical rkh z z z imply every proposition clear positive kernel combine valid negative zero induce zero e z energy covariance measure demonstrate discrepancy measure criterion sign energy distance use characterize scalar coincide covariance imply negativity every mmd introduce measure dependence characteristic choice weight expectation pairwise euclidean recently establish replace type domain embedding kf embed obviously embedding measure discrepancy mmd characteristic characteristic characteristic characteristic interpretation thus immediate application distinguishing characterize satisfy additional property extend r finite p kernel embed well space say first q finite directly type measure finite moment check whether characteristic state although rkh hypothesis sample biased induce pairwise result k center statistic sample replace distance require nan chi yield bootstrap spectrum operator study context compute gram aggregated entry concatenation sample center nan estimate converge spectrum covariance respectively chi square analogous design attempt estimate distribution interpretation test independent variable conservative compute associate kernel approach section experiment kind synthetic multivariate second compare gaussians differ univariate second univariate frequency frequency plot design indistinguishable outperform bind conservative checking quadratic significantly test differ similarly differ kernel advantage similar cc cc mmd distance mmd various sample addition investigate value case perturbation kernel value yield high frequency result kernel advantageous distribution sensitive distribution fine differ high frequency appear real rotation rotation right use propose ica dependence fill pass random dependent uncorrelated dimension plot vary obtain wide range performance dataset case small rotation
forecast rule expectation scoring encourage careful probabilistic scoring definite nonnegative scoring proper yy score example class rank probability score score calculation cover apply score calculation dirac future observation incur bayes pattern theory break whenever metric continuous definite family member interesting open question equal cone generate asymmetric function near neighbor acknowledgement author grateful von foundation thank reference j harmonic protein bioinformatic I neighbor rule comment probabilistic pattern new york netflix challenge statistical learning introduction journal statistical statistic forecast journal american strictly rule prediction american association ari advance american association learn statistics j graphical journal definite space definite transaction american physics remark cover universit observation well perhaps surprisingly large refer cone metric negative slight explain performance measure success reasoning analogy loss metric definite prediction predict observation real value structured forecast population sample predictive performance mean loss forecast future observation information approximate cover misclassification risk twice elegant thought cover paper seek remarkable section satisfy consider prediction forecast predictive performance evaluate proper analogue dirac discussion relate neighbor let hausdorff space borel algebra measurable argument consist yy function cover contain predict past bayes risk belong class slight extension assume bayes measurable cover inequality desire argument play role harmonic arise structure increment ari example reference negative hold finite dd space mm qp dy qp mm special case summarize kernel apply reach generalization standard
sim w nn false w pick col nf w nf col col col nf col nf col w x mr r matrix exceed sim sim nu nu alpha n sim cn nu sim home sim home sim c nu sim home nu sim mm doubly matrix deviation stochastic projection doubly sum doubly scale non contain component say scalable number doubly zero scalable permutation eq thus approximate reduce corresponding subject generic computable since carlo sort computable formula doubly stochastic satisfied identically check scale connect independent lie additive row col maximum equal variability doubly gamma hold scale doubly sense close maximum apparent appear matrix accomplish linear proportional prefer provide moderate matrix doubly attain equal permutation extreme doubly stochastic easier doubly doubly stochastic permutation weight average helpful norm extreme doubly one helpful hilbert deviation large scale convex fix moderate deviation purpose deviation consequence though affect improvement approximation apply matrix obtain doubly value exact nevertheless decrease agreement unclear sequence moderate decrease block block associate doubly stochastic table error evidence suggest error moderate deviation n row doubly deviation value weakly strictly doubly associate assume moderate degree explicitly summation calculation nominal also least sense sum circumstance similar conclusion sum example restriction expansion odd term hold scalar bound degree restrict restricted expression r r r r r constraint value row index x express reduce form restrict sum arise speak sum sum row column index hold constant elsewhere one block element relate lattice write value may put together scalar restriction block reciprocal factorial partition q neither partition least upper less either three term symbol block pair series convergent spectral assumption expansion group degree six less express far n compute adequate gamma freedom e odd scalar chinese projection eigenvalue size matrix moderate occur moderate threshold comprise odd exist arbitrarily indicator corner approximation compute panel magnitude standardized residual log middle panel residual plot plot panel plot tend occurrence leave panel residual odd type constant moderate mean square approximately gaussian happen inequality rate structure matrix block structure slowly decrease course must depend generated positive deviation expect tend large doubly pr w adjustment sample table approximate polynomial journal approximate n matrix theory new york generalize relationship doubly ann alpha alpha return cyclic product else alpha alpha need paper else alpha alpha else alpha weight diagonal iterative fitting make col sum alpha alpha du n alpha sum alpha cycle u true log strictly positive rarely occur return alpha cat fail alpha alpha alpha alpha rarely alpha exp exp x x pi x mu mu else x nu nu
score shift unit orthogonal selecting index h aic rank usually unimodal dependence integral copula exponential estimator goodness fit test role understanding comprehensive speed issue ask give h scatter em section thm thm example modeling dependence united science many department statistic college set big size consider first value orthonormal mx x comparison enable expectation lp representation tail normal height keyword phrase comparison density lp co moment mid function orthonormal nonlinear correlation parametric modeling modeling quantile theoretic fisher algorithmic discuss researcher many include research quantile theoretic modeling seek comparison measure call tail decade extend discuss theorem describe routine specific pose involve method box intensive come applicable traditional plot exploratory polynomial score function discrete separately mid dependence continuous density divide fy discrete concept contingency ccc discrete give density denote connection major apply density innovation mid observe copula comparison comparison special iii score estimate regression quantile true construct relation x construct shift polynomial cosine regard mid observable construct schmidt power constant shape polynomial take copula influential product score function selection increase increase chi freedom drive chi discrete contingency dimensional moment traditional sample sum copula function integrable influential score maximize dependence sum influential exact combination compute vi display interpret fact lp integer satisfy x high tail polynomial lp score extension extensively develop skewness concept regression expectation apply naive regression quantile scatter notation compare display lp rx rx united aim bayes theorem fx fx interpreted formula probability unconditional marginal formula heart discrete f copula density continuous x express theorem odd du x x x express theorem equivalently linear logistic density exponential smoothing estimator
case approach show optimally incorporate even consistently robust nevertheless contrary exploit improvement proximity obtain gain match hmm access side side wrong naturally recover hmm partial turn state achievable moreover training expect support sequence hide access partial hide state derive accordingly considerably improve performance baseline training application speech processing bioinformatics language process stochastic generate unobserved state certain regard hmm particularly concentrate hmm alphabet sequence form chain observation hmm completely emission maximize expectation local equation ordinary observe sequence instant partial state sequence ordinary hmm along span observation ordinary generalize might ever say circumstance mathematical derivation em incorporate derivation state incorrect provide define simulation confidence match quality side ab fall category partially supervise label model suitable unlabele contain noisy may occur label date study relative frequency initial model feed ordinary hmm rigorously incorporate ordinary framework likelihood estimator know theoretically analyze mle maximization special set state derivation provide explicitly hide sequence furthermore state brief hmm derive iterative equation incorporate partial noisy iv remark briefly hmm sake notational finite derivation readily extend come outcome discrete time hmm alphabet hide unobserved set generate py py pz ib hmm characterize hmm without equation give maximization procedure recursion observe give q observe transition ease drop sum observation model incorporate proposition explicitly backward h marginalization use definition update markov reach q side update forward update backward find reflect ideally name accord unknown bring immediate set confidence accurate guess present confidence low confidence limit iv transition transition state relate forward backward definition q x j j n otherwise proof wherein markov condition observation side summation addition information likelihood maximize via auxiliary em provide incorporate definition convergent least let carry maximization split log division probability obtain maximization involve side derivation derivation indicator numerator denominator probability outer summation sequence observe divide estimate new incorporate possibly corrupt information next new scenario simulation test along relatively high emphasize exact hence side confidence side I match sensitivity side viterbi use average efficacy method compare state ordinary
cauchy also stable detail iid eq need act simultaneously include subspace challenge although correct behave classical affect distance histogram quite tail behave median distance preserve cauchy near subspace query obstacle qualitatively increase much heavy ii well behave tail projection increase distance distance good mis detect low sharp guarantee choose correctly significantly close need argument rigorously key sec need establish theorem argument proof routine technical lemma close unique long close p p lemma purpose appendix numerical fix cauchy bad subspace task projection come something strong denote hold bad subspace use unit sphere argument p associate close follow tail fix vector exponent unfortunately give power detail defer sharp w bounding cope heavy tail insufficient elegant argument rough basis analogue orthonormal preserve preserve cauchy appropriate value detailed calculation tie instead ask become fix nearest one rest near rest relative distance control become subspace compare target state theoretical demonstrate relevance success failure sample new project practice maintain subspace pool economic yield perform subspace search consider case employ augment alm numerical solver affect extend otherwise solver project handle solver subspace iid pool take randomly subspace induce reasonable simulate error divide magnitude add fraction growth corruption retrieve necessarily subspace pool report note distance small enjoy chance preserve near gap entail dimension htbp visual varied induce gap thing simple pick pool repetition refine fraction low half find corruption vary illumination well nine phenomenon physical cause low formulate recognition subspace norm face recognition recognition happen prefer save detailed discussion line nevertheless popular recognition extend face align face image acquisition randomly image moderately camera angle great angle face suppose closed world present evolution image stay ns achieve perfect ambient take nine experiment achieve imply search represent evolution success gap theorem need achieve already repetition significant extremely first htbp increase weak exponent become experimental result confirm specifically take high extreme achieve accuracy back refined description dimension recognition way htbp vary evident order recognition preserve recognition subset testing find corrupt original pixel resolution size location accordance gap take case effect gap gap gap drop rapidly corruption suggest accord significant corruption level level demonstrate norm ns variant htbp performance exhibit less beyond pay working dimension efficiency bad corruption corruption consistently applicability proposal recognition multi library library comprise toy pose take illumination direction take although nonconvex shape nine recognition corruption percentage corruption c corruption ns method tolerance variant report performance scheme recognition illumination choose datum perfectly corruption level beyond scheme remain gradually beyond burden expand rapidly associate predict lower large obvious run largely determine rt concrete task object instance straightforward exhaustive dimension cost search boost practically generate iid corruption fraction take htbp run bite matlab ram turn regression plot vs logarithm turn matlab scale see recognition run significant random experiment confirm projection figure dimension search empirically tie part provide essential fact rv stable rv constant strictly e distribution characteristic stable characteristic rv stable call stable characteristic distribution stable also stable exist virtue stable iid rv symmetric sequence complete distribution standard cauchy hc remarkable aspect standard cauchy invert characteristic scale control fact subsequent addition follow half cauchy rv fact iid sake page rv weakly converge f z g ax q half rv determine appear half cauchy expand series center plot convention subscript subscript figure htbp iid cauchy corresponding cauchy interest behavior sequence ds ax stable eq strictly eq note assume application k derive claim kk k obtain constant proceed condition let column independent sr rd existence subspace existence basis existence condition lemma well write last well condition whenever finish side relax apply markov take loss rv arrive claim expectation complete net q least choose bind notational eq intersection j side triangle obtain intersection remain ensure satisfied choice ensure choose eq tc simplify bounded constant ensure solve sparse record stable sparse c query similarly matrix theorem tm ok subset detect near subspace span canonical basis e projection parameter tuple subspace gap identify support assume partition I I subspace enough guarantee constant identify corresponding obeys put construction hence hypothesis proposition kt ic require start rr q sr p nontrivial probability tie submatrix rank foundation partially university zhang object collection linear high ambient determine naive exhaustive large burden significantly query dimensional distance linear program independent get back space exhaustive preserve nontrivial order multiply logarithm subspace dimensionality hence vision application investigate empirically subspace cauchy projection recognition w zhang space dimensional intrinsic structure central computer vision impact denoise object view match associate image say illumination recognition nearest put image prescribe paradigm face development sufficient achieve appropriate model depend encounter variation illumination capture linear model whereas variation alignment highly metric set appropriate observation perturb gaussian induce norm datum error due alternative model input would resource ambient application quantity could dimensional subspace well justified illumination variation build induce p z p x x p turn empirically count robustness current analysis likely justify image moderate cast unique convexity robustness subspace query determine objective interior method thing careful discussion expense guarantee achievable even aforementioned fit aforementioned tuning prohibitive object useful problem recent modeling error correction norm provable corrupted imply even nonzero recover present theory recovery strong recognition subspace precisely distinction cardinality hard find small agnostic solve purpose approximate problem concern individual regression heavily sampling e leverage score
word replace exist real world document collapse magnitude provable practical recovery machine estimation robust probabilistic latent allocation topic multinomial vocabulary topic distribution lda normal document begin document position assignment topic document matrix unknown stochastically never expect task topic case dirichlet lda maximum estimation distribution np result typically optimize markov provably parameter provide word topic separability word anchor word occur partially could document let topic topic learn document q learn additive unfortunately solve inversion make tolerance word topic two step identify recover anchor co supplementary material high learning follow recovery anchor purely combinatorial find anchor word recovery column anchor notation refer document outline set anchor tr inversion simplex original poorly likelihood recover anchor whereas besides ignore co estimate inaccurate probabilistic describe notation document assignment topic role anchor step word matrix normalize row store normalization anchor ar ta index anchor index special hull row anchor word fact pz k w negative hull mix easy co occurrence algorithm recover objective reason measure understand particular maximize observe count problem constrain gradient write k prior recover recall constrain least pseudo learn hyperparameter bayes calculate pz pz pz specify recovered find grid perform range material also polynomially word anchor anchor infinitely document hull simplex anchor number perturbation hull define simplex randomly subspace origin notation denote compute complement give anchor word solve whether anchor span find hope exactly anchor choice succeed anchor guarantee recall robust convex hull rest reasonable define polytope lda let hull write cover follow hull vertex appear perturbation cover vertex provide help infer topic separability material find argument volume simplex determinant vertex algorithm prevent exponentially proof run alg vertice phase point vertex simplex anchor word question variety also assumption hyperspectral clean whose prove infinite able give provable guarantee perturb g sample non negative factorization separability application topic application document provable guarantee even programming slow comparable three gibb average iteration burn train set evaluate performance correct document document dimensionality sparsity generate corpus generate document guarantee separable large york times article vocabulary length corpora ny document document topic corpora reconstruction true word matrix well align topic learn intractable reliable topic interpretable semantic metric coherence coherence use avoid occur well quality perfectly co frequently redundancy inter topic probable overlap expect ambiguity recover heavily linear size second second corpus second corpora semi corpora synthetic corpus corpus synthetic recover algorithm document bar show poorly noiseless infinite corpora recovery low variance reduce mcmc corpora recover corpora match comparable ny times corpus poorly corpora much recover zero error even noiseless lack separability semi synthetic corpus test separability add anchor word unique topic assign probable cause great reach exactly problem even correlate corpora ny article symmetric add block result fig recover bad case dirichlet corpus especially robust consistently corpora uncorrelated zero separability produce quantitative qualitative probability hold document top topic panel ny times split fail small ny recover produce worse hold token algorithm hold pair within range document method recover unique consistent sized corpora supplementary material ny topic shown observe tend specific gibbs file read internet email site www algorithms yet provable
sense hadamard means exist solution unique stable article organize review ode solution examine introduce prove consequence discuss relation incidence three state figure go transition density people consideration incidence disease individual number time assume homogeneity duration path figure henceforth homogeneity assume close birth furthermore continuous age member population number normal respectively q linear structure incidence combination normal age allow application ode unknown ode interestingly ode follow depend type ode ode ode analytical analytical equation depend type ode last par examine dimensional ode e system however obvious proven solution fulfil subsequent application derivation age equivalent costly ode solve information age relatively study application effect cause incidence interpret problem oppose infer incidence e cause ode ill pose sense sufficiently pose differential equip na pa pa ill pose study relation incidence link dimensional ode article show meaningful ode change type implication analytical exist important ode derivation specific proof ill distortion incidence disease consequence unclear ode ode incidence medical trend appropriate formulate partial subsequent disease disease diabetes diabetes duration u shape duration unlikely expect term I
reconstruct reconstruct explicit easily convert exact entry entry uniquely many position sufficient uniquely entry relation observation rank pattern special allow minor specific combinatorial position completion property completion algorithm separate relevant estimate miss entry allow estimation estimator completion whether term encoding position necessary theorem sharp apply determine minor minor experiment movie predict underlying matrix completion key idea set rank additionally fundamental algebraic observation question unobserve specific structural show numerical fundamental aspect algebraic set characterize exactly polynomial relation polynomial connect major ingredient independence independent row finite characterize perspective enable completion broadly speak two spectral j n rank generality follow variety r n algebraic vanishing vanish vanish view position entry vertice mask position bipartite tb every unobserved informally go interaction structure structure jacobian smooth jacobian j ji mn denote kronecker row entry order satisfy let n part nan parametrize see nan jacobian corresponding position row submatrix correspond zero entry purely combinatorial structure make boolean irreducible variety hausdorff kind call generic generic statement concerned proposition tell mean polynomial imply open topology surely respect show generic jacobian generic first compose critical attain maximum critical point prove uniquely reach first generic intersection devote mild answer position relate row jacobian show separate observe either real start precisely define mean imply observe position finitely fix infinity whole finitely whether position care issue nm kk finitely rank position position entry position finitely jacobian generic implication subset combinatorial independence generic language say finitely closure equal closure rank perspective prove combinatorial condition relate finite describe closure testing entry correctness continuous closure closure compute jacobian decomposition great residual return routine svd ram high rank detect minor uniformly elimination factor project neighborhood fm rank smoothness position finitely generic finally statement directly closure preserve generic finite every argument smoothness essential identifiability statement phenomenon explore imply similar jacobian show use parameterization another section show whether mild analogue well exactly one practical relevance mean analogue theorem finite entry uniquely true matrix analogue constant theorem proper overcome geometry defer statement state position position call uniquely nr maximal index uniquely finite check position analogue jacobian characterize general gr computationally impractical jacobian finite jacobian entry characterize space intuitively concept mathematically jacobian rank kernel stress call denote remove jacobian allow unique remove matrix maximal stress depend entry rational proof generic define generic stress uniquely stress theorem stress unique generic observe stress jj remain correspond position real number instead field rational substituting thus compute evaluate left correctness one imply similar consideration analogue remark algebraic irreducible cardinality depend whether section nm correct size stress hold keep dimension keep development finite define remove alternative probably statement pre image let stress inverse neighborhood e df leave definition stress prove determine row observe uniquely row recover column span connect reconstruct miss entry cover one call associate address reconstruction version wise actually answer general arbitrary one concept set circuit connection reconstruct circuit unique let circuit mask correspond scalar circuit rank fulfil circuit polynomial determinant generalization algebra matrix vanish call circuit scalar make circuit interpret let circuit circuit entry entry circuit observation respect property completion circuit position unobserve entry circuit polynomial entry general exponential circuit circuit correspond circuit minor closure mask neighbor denote intuitively iterate edge terminate edge add bipartite closure ki ni neighbor k jk e terminate say terminate e reconstruct minor uniquely crucial return subgraph efficient many vertex employ would one entry general strategy position observe fix construct position cause grow span tell position terminology dependent already spread sparse section basis maximally basis edge consist basis unless circuit I independent subgraph linear closure closure notion subgraph v subgraph position finitely basis part corollary rank set row column vertice q statement statement first sparse generality empty time edge induce treat symmetry situation need n maximize product n er hand rank sufficient independence bipartite mask sparse amount basis together along preserve infinitely circuit circuit jacobian vertex circuit imply core show rank section proof generic circuit stress circuit power see circuit circuit stress support column index zero stress hold concept core useful position circuit concept let denote subgraph minimum degree core circuit circuit follow need subgraph lie inside note thing closure circuit circuit rank circuit rank c circuit tangent stress uniquely identically remove would none coefficient addition polynomial vanish open stress mask equivalent bipartite random value interested bipartite graph enyi bipartite bipartite row random bipartite vertex vertex regular regular mask p n cn isolated vertex connectivity suppose depend p subgraph core graph span smoothly low completion rank generic incoherent perturbation show generic sampling powerful experimentally rest heuristic generic call conjecture threshold circuit circuit subgraph core edge value smoothness imply threshold thing together circuit inside core threshold reach give structural circuit core circuit yield threshold nonetheless moreover threshold establish rank span nonetheless completion explore experimentally also conjecture regular provide incoherent case regular mask dense enough start seem quite practical favorable entry add edge closure conjecture investigate finitely position phase transition case matrix quantitative investigate influence entry particular completed complete slowly entry occur entry various condition consider uniform sampling edge order edge matlab sequentially preserve property estimate success plot successful solve rx degree b chance success minor pass around closure nearly show minimum b exceeds require minor start minimization tb phase mask mask almost regular mask edge independently list list become th order way take check mask average make phase transition regularity experiment mask varied phase different plot regular plot bottom row regular transition sharp grow rather slowly likely column tb devote publish entry follow convention movie row grow core standard entry complete core size core grow vast majority entry majority column attain rank row core exponentially speed rank rapidly start empty core finitely identify check whether minor closure core theorem check process figure number determine way thing point correspond transition core start exponentially simultaneously experiment result reconstruction prediction one matrix choose entry independent figure three implementation competitive outperform low compare change actual predict priori actual error independently algorithm entry quantitative bin qualitative figure come measure calculate predict mask mask multiplicative entry priori variance mask affect known entry less come successfully entry structure mask pattern mask structure structure adjacent square mask depict red blue entry half implementation blue increase increment completed predict square code respective color error variance x axis complete entry logarithmic error
form instance observe situation ii x case substitute formulae situation forward x x formulae previously multiplication hmms maximize issue formulae multiplication operator common forward backward geometrically machine I architecture suggest solution keep track parameter store expression recommend initial become example u arithmetic em repeat iteratively formulae consider segment emission emission normal backward obtain heterogeneous ib ix segment emission segment transition identify transition hmm regardless occur observation equality follow base allow specific marginal regular heterogeneous chain change point cp contiguous homogeneous formulae occur plot estimate posterior mining set dot horizontal line state dash line dash r dot horizontal segment posterior probability solid line segment segment mining hmm emission segment greedy change state poisson initial transition change occur display probability change I segment contiguous state segment hmms almost historical perspective look literature year occur end though meanwhile coincide define end world posterior dot lrr horizontal dot line cp estimate breast cancer dot lrr line posterior marginal segment segment dot line segment point r line bt log reference genomic segment genomic hmm simultaneous point multiple simultaneously identify outcome implementation deal resolution current snp wide amount bioinformatic typically find contiguous extension fast rule smoothing technique model comparison find two modelling address computational issue allow hmms bayesian model variable dependency direct acyclic dag code edge dag factorization evidence obtain fashion bp sum product product message belief secondary shape structure call quantity distribution term typical criterion schwarz integrate complete assess exact bp forward hmms kind evidence bn framework permit extension realistic accounting variable straightforward aforementioned segment model segment segment specification two segment common segmentation advantage segmentation toy distribution probability state lambda transition pi pi reference change location pi lambda backward recursion probability evidence check ok b sum pi lambda posterior col blue col col j col col need large segment c parameter segment ref lambda lambda lambda lambda k lambda evidence consistency ok change cp matrix cp lambda lambda c segment col col col k change col blue col cell bt breast cancer lrr display probabilitie distribution change probability mean example transition segment r figure approach point begin end contiguous ir draw minimum proposition lemma heterogeneous finance biology characterize number change follow real value ideal non interval change emission observe segment index position segment two refer refer simplicity clear commonly use hmms bioinformatic widely financial series music brain image security hmm observation switch likely occur hmm inferential state quantity classical forward backward hmm change accomplished maximization method reversible jump mcmc recursive sampling characterize hmm mcmc summary inferential involve find chapter estimating include frequentist modify forward backward state hmm investigate penalization information adjust previously location present framework hmm change summary procedure hmm discussion property current efficient may change hmm permit use hmm variable simple dependency variable height text depth var node var var var edge thick edge thick thick thick thick thick hmms prior knowledge aspect hide markov introduce notion set outcome evidence evidence know unconstrained case contain hmm allow base level underlie appropriate share segment hmm transition observation evidence usually state homogeneous often segment overlap state segment transition increment segment segment particular effect arbitrary computation forward backward problem technical detail simple hmm find apply evidence evidence accomplish sum joint variable perform highly small cache avoid generating time tool factorization order factor factor depend place eliminate accord eliminate cache eliminate recursive backward message thus naive elimination consider elimination
sbm class let straightforward q space regularization free free maximizer within within diagonal close eq mle exactly mle substitute profile high set cluster estimate class incorrect assignment use divergence dimensional diagonal asymptotically decay additionally denote block tight high proportion population assume block define theorem require two expect high slowly specifically every equation identifiability class merged concerned ensure assumption make implicit block affect large mle block memberships blockmodel scenario particularly heterogeneous computing mle computationally intractable pseudo likelihood fit replace element without modification fit adjust remove block estimate meanwhile hundred remove mle return reduce nearly mle reduction optima initialization make seed follow motivate connect call neighborhood combine node pseudo spectral cluster laplacian eigenvector diagonal whose element grow everything else simulation sensitivity algorithm heterogeneous diagonal htbp keep population grow ten right panel block separate asymptotic expected connect eight noisy ten axis examine deviation model diagonal sbm maximize probability asymptotic identifiability node correspondingly demonstrate mle represent involve propose estimator incorporate insight gain straight giving definition outline expectation maximizer maximizer first step difference lemma building block establish union regularize likelihood bias tradeoff bias necessary concept refinement idea refinement connect choice similar make ab maximum aa argument take aa aa j concentration inequality asymptotic degree true om nc equality r pm leave maximize tractable concept refinement connect refinement regularize refinement log regularize refinement regularize refinement define partition index low triangular define notice assignment induce correspond straightforward refinement partition triple partition assignment large together process terminate pair choose routine connect refinement contain satisfie far essential refinement partition later previous subset result refinement analysis refinement partition remove define set block set group refinement refinement refinement refinement partition correspond refinement increase set taylor grant research dms pt mail edu mail edu fan mail edu pc pc pt theorem theorem sbm thank associate comment grant grant minus abstract blockmodel network maximum neither grow grow allow fast additional motivate empirical physical develop maximum likelihood estimator paper introduce regularization parametric word phrase consistency pt recent advance technology produce complex community highly actor identifying help question field depend interest interact element people computer cell communication network page provide community discussion topic network might contain activity relationship essence observe ability correctly understand lead blockmodel community actor belong connection node add rigorous likelihood blockmodel clustering blockmodel line author plant stochastic blockmodel spectral recover plant consistency plant paper blockmodel spectral high first statistical criterion though finding various brain find size brain human roughly brain suggest human maintain community roughly people refer similar human blockmodel asymptotically previous even result grow enough paper size slowly high ignore fast eventually contain roughly subgraph remain estimate stochastic blockmodel setting unified parameter set restrict space amount potentially technique lasso restrict space restriction diagnosis graph statistical propose restrict
ds overview come processing remarkably result history go name variable model develop fairly however representation issue namely mention argue solution algorithm near discussion bridge particular develop weighting lead somewhat aforementioned reason second wish argue general necessarily hold possibility datum design isometry coherence circumstance efficacy appropriate sparse update modify orthogonal pursuit omp stagewise f lasso selector view homotopy method take path direction forward stagewise fs prototype analysis note lar omp bp solution uncertainty homogeneity design pursuit argue uniform preferable growth argument selector lar ds validate repeat variance uncertainty lead prediction reduce lar formulate challenge pose importantly derive example illustrate sparse sometimes less motivate response imply interpret uncertainty arise source express variance absence admit application discuss repeat remainder variance challenge true solve ol expect bias towards depend signal noise new exact know explicitly error bring uncertainty residual access structure variance v system light great expect error hence become prominent track high briefly greedy sparse forward fs particular particularly understand main algebraic interpretation implication selector fs small initialize identify correlated update stop otherwise qualitatively find coordinate high minimal like implicitly solves often express attain increasingly main accumulate design equation unless scale uncertainty scale uniform contribution slight abuse modify noting solve specifically norm different comparable term say identical scaling associate uncertainty obvious correlation fs recall transform penalize scale lasso lasso penalty represent uncertainty recall penalization uncertainty uniformity uncertainty notation path geometrically uncertainty homogeneity uncertainty lead constant growth growth course random deterministic random carry expectation rather specific path scope brief moment connection selector tuning selector distinguish pursuit algorithm explicitly linear respect residual seem would property variable notice feasible result residual correlation expect explicit quantity fit panel identically lar scale line however associate less panel f right residual lar lar apply characterization datum without highlight challenge datum efficacy scale motivated candidate natural chemical different predictor molecular measure twice detector ratio spectrum peak measure allow pre processing peak may ratio relative control fraction infer nm visible light linear calibrate measurement sample biological run fraction proxy chemical composition mass sparse brief approach sparse correlation peak break redundancy cell hundred peak incorporate peak ensure noisy peak little consideration sample signature figure mass spectrum show height cross every peak among especially estimate variance normalization indicate proportional marker radius clearly peak variance attribute model nest fold outer validate choice lar lar chapter predictor another fitting lar peak validate matrix estimate detail improve validate lar ds lar figure consistency ds axis scale accurate associated plot figure noise scale number indicate lar solid line region uncertainty scale uncertainty slowly scale input almost provide graphical scale red peak indicate circle number sample increase accurate scaling remarkably lar ds function remarkably lar somewhat surprising expect selection utilize zero datum either neutral selection lar ds scale agreement lar peak ds alternatively total distinct peak lar ds combine common state front make formulate say reduction cv increase well agreement lar ds still practically ideal circumstance scale lar recognize interest peak less particle furthermore unknown peak small far unknown peak chemical mechanism argue uncertainty challenge knowledge focus term context uncertainty regardless enforce uncertainty selector lead residual reflect characterization application uncertainty improve lar peak addition one scaling
child denote way dag restrict find subtract way connect arc leave reconstruct drawing possibility leave incoming arc old result old receive arc reject arc term arc old fall drop draw arc old arc accept rejection highlight possibility constructive preferable draw parent invert arc number child enumeration recursively remove reach restrict avoid directly dags partition node partition dag easily restrict replace expression ratio section dags dag triangular influence bernoulli act uniform space dag underlie preferable practice dag rather dag arc basically recursive enumeration receive link link term binomial possibility arc exclude could theoretically arcs dag arcs possibility even add dag rational store integer limit reduce towards dag large term include model average parent necessary number mcmc treat dag analogously dag express split two reduce hybrid dags weight calculate perfect rational sampling mcmc offer remove bias operate wrong advantage adopt triangular adjacency produce markov avoid efficient perfectly uniform dag vertex compare dag chain dag recursive enumeration mean practical limit behaviour base dag belong alternative cost check approximately small markov adapt dag include restriction parent node idea could integrated scheme procedure combinatorial go add resample new connection move reversible one amongst alternatively underlie combinatorial sampling act space matrix dag partition produce avoid act structure sense combinatorial score order satisfactory possibility seem dag several set identify dag probability know markov equivalence dag although formula run dag correct essential implementation compare dag essential fraction move essential suggest dag little exact essential graph also dag importantly number essential graph dags dag corresponding essential actual might sampling uniform dag possibly essential sampler real label responsible background depend background satisfie result inequality frobenius square must exponentially fast constant maximum bound return irreducible uniformity eigenvalue one unity difference uniform discuss upper comparison enumeration study low weight evenly dag overall would weight amongst dag handle focus arrange empty dag still spectrum dag arcs transition q assume cover dag dag intuitively dag encode far store looking fill matrix evenly amongst dag high suggest matrix require move irreducible integer binary treat integer follow bit store grow multiply done perform multiply copy operation sum final multiplication add multiply calculate recursively without overhead recursively multiply calculated computing affect repeat bind complete step limited decay give simplified length representation numerically single dags step integer subtract number binary sum discuss sum sampling time seem complexity choose discuss complexity lead code precision integer package recursively store number sa n sa kb binomial readily recursively loop computational overhead r sample store value recursively generate loop km sn ki ki l ni l j randomly draw sample dag correspondingly column row define power bring replace since rgb rgb universit acyclic graph representation underlie network represent practical reverse engineering gene great require analyse acyclic uniformly enumeration consideration discuss enumeration provides actually acyclic us large idea enumeration various restriction include restriction enumeration hybrid markov chain graph acyclic acyclic graphs dag collection represent relationship short property independence apply relationship seminal paper tool infer structure model particularly important light drive dag class reconstruction review development dag discuss simulation study aim assess reconstruct graph datum crucial space relate remove currently rely chain limit dag carlo context graph uniform dag require neighbourhood expense extend dag recently pose negligible triangular adjacency matrix ensemble matrix provide dags point hill slowly uniformity small increase uniform sample essential evaluate certain count dag insight dag landscape assign structural therefore strategy base recursive enumeration dag combinatorial uniform dag enumeration note early report think impractical evaluation complexity show enumeration expensive establish practical enumeration exploit dag fast minimal triangular dag uniformly devise share easily restriction method parent certain perform triangular hybrid mcmc offer avoid bias vertex dag admit label dags asymptotic dag find dag entry vertex remain acyclic low triangular node eigenvalue dag easily sample triangular graph dag several permutation example dag arcs arcs non uniformity entirely start arcs proceed repeatedly pair direction otherwise long remain acyclic arc operations arc cycle stay exclude sampling reduce marginally chain path dags arc empty reversible ensure approximately long chain elegant dag apart remain acyclic order edge cycle applicability dag arc irreducible uniformity element moving consider go dag arc dag none remove arc add arc arrive arcs transition path move empty path order last move actually require probability dag uniformity derive triangular uniformity markov markov needs say accurately background restriction twice often uniform uniformity fine additional one twice entry ensure useful much express term importantly uniformity element probably element satisfied logarithm consideration dag easily find respectively remove arc improve verify uniformly dag away first step chain also discard like may check graph convergence start dag randomly uniform enumeration label dag incoming arc far classify label dag dag figure dag allow possibility arc old give way give dags arc turn inclusion dag finally order small integer multiply store many dag arc receive bold arc allow remain triangular bold case arc nod node arrive sample dag dag sequentially marginally well chain integer perfectly uniformly sample dag significantly chain require subsequent uniform dag rather quick integer still obtain run core intel processor gb ram method per machine dag perfectly recursive become irreducible recall take reasonably uniform chain hundred million thousand time enumeration hour give complexity thing bad enumeration account store memory computer several fact see next access memory reach space exact dags growth lie behind atom observable universe move dag extremely multiply exclude lr dag uniformity sampler sample limit dependent weight shift towards meaningful store small iterate long return enumeration sequence limit behaviour reconstruct order step dag sampling upper complexity therefore dag build dags procedure remove alternative dag sample vary triangular zero rare dag limit nearly uniformity return exact enumeration indistinguishable enumeration variation enumeration markov enumeration sequence order partition introduce partition accord dags dag account connect edge empty order partition integer represent correspond size large previous partition treating build split adjacent ensure th draw exclude step become irreducible metropolis hasting partition dag scheme dag digit single flip directly modification affect edge hence partition simplify split ratio convenience evaluate calculate weight take account partition allow dags preference move calculation chance move note suffice complexity completeness acceptance involve clearly large repeatedly draw soon unlikely event draw total binomial bernoulli effectively soon average move chain overhead
another implicit population crowdsource offer expert quality concern estimate expert labeling yet high base label ir direct considerable toward effectiveness relatively blind label via crowdsource regard pseudo gold label aggregate gold label build develop aggregate set classifier evaluation reliability tradeoff investigate crowdsource direct evaluation supervision method generality test establish rank correlation estimate figure depict conduct classifier crowdsource track investigate reliably classifier expert much crowd blind correlation evaluation result score see significant score benefit outperform blind complicated label yield though blind crowd likely common accurate broad concern human labeling relevance ensure label consistency decade system human evaluation system evaluation positively human system document vs retrieve wu exploit popularity frequency document retrieve fusion blind label al labels et pseudo generate evaluate demonstrate label exhaustive expert pseudo aggregate classifier consensus pseudo gold spam filter predefine applicable label effort effect direct performance label blind crowdsource class et evaluate pseudo score combine research integrate redundant produce area crowdsource aggregate human I method supervise consider majority vote expectation em variety calibrate naive glm adaboost also label expert evaluate classifier notation classifier example label example turn three human level gold estimation sampling arbitrarily reduce majority vote pick vote round decision q round vote tie break generally varied specificity definition relevant maximization em probability classify membership eq indicator receive iff unknown em classifier every independent probabilistic maximum summation become maximize miss crowdsource though rather way method naive nb support generalize glm adaboost class training expert set detect section unsupervise metric consider default nb supervision infer gold likelihood compute glm label glm fit example return response binomial learn plane decision learn support class adopt identifie classify dot meta iteratively weak respect adopt well adaboost method classifier begin describe dataset study use score difference score discuss test correlation achieve label per ex crowd train test crowdsource track comprise task collect crowd involve aggregate crowd task yield classifier output come distinct expert university final balanced track crowd label collapse single label majority across range classification positive negative metric fair perform operational pair difference compare preference induce ranking order hard tie question rank classifier preference minus pairwise correlation statistically significant vs vs obtain largely correlation score score rank adopt measure ranking swap pair closely tie measure reflect score outlier worst receive low whether estimate concerned determine ranking correlate nan ranking classifier rank ranking correlation exist tell directly detect one achieve involve triangle significance ranking expert vs familiar ir letting correlation score ranking hypothesis denote reference expert estimate ranking compute statistic freedom triple order evidence evaluate classifier crowd estimate achieve correlation expert begin section show produce classifier early establish diversity label score expert compare score supervise supervise present section place evaluation c c c c begin measure relevant diversity overlap coefficient intersection size union show standard study crucially classifier exhibit extremely low vs classifier bad suggest likely widely agreement represent fair chance exclude remain indicate moderate acc pre rank c tie rank statistical testing ht acc show tie rank significance confidence measure classifier accord actual table four tie tail pair indicate vs merely classifier classifier specificity actual test significance confidence result insight summary reliably crowd enable high pearson metric crowd significant improvement rank crowd low condition answer method correlation alternative testing label axis line acc metric typically correlation correlation four blind follow rr next supervise nb calibrate axis acc plot line detail achieve acc specificity lower significant see crowd blind evaluation correlation achieve similar simplify presentation correlation correlation improvement crowd observe crowd method wider across metric crowd blind range c score correlation swap blind acc acc acc rr glm measure ranking refer classifier achieve achieve statistically tie rank indicate reflect confidence rank tie achieve expert validation raw value reflect rank achieve statistically well tie rank indicate crowd direct correlation direct select evaluate cccc c cccc c c c swap type acc pre acc acc acc sampling nb alternative rank data blind crowd combine approach direct directly crowd significance difference prediction indicate confidence inform validation top gold blind evaluation correlation em absence correlation method consistent early crowd significantly rank offer potentially tackle aggregation gold gold well concern however pseudo gold labeling error derivative score rank quality secondary estimation four acc pre secondary primary combine swap negative metric hence correlate indicate c acc pre glm method predict score ranking measure pearson coefficient pseudo validate intuition correlate effectively predict axis axis indicate predict pseudo gold strength pearson expectation pseudo achieve coefficient correlation gold accuracy critical pseudo gold label evaluate classifier absence truth label effectively predict rank acc difference label correlation difference reasonably large metric swap acc pseudo evaluation pseudo gold gold gold vs expert grouping value metric label vs correlation value shown swap well difficulty uniqueness pool acc blind analyze classifier classifier test effectiveness worst achieve representative evaluation select predict rank score present row well rank
package code age f na na na na na na na na easily person site gender analyze key format omit item display ht option specifie contain expected one select fourth estimate option reflect typical item total moreover display fairly monotone behavior cover exploratory parametric currently analyse quite check reason apart software computer smooth prominent software year run within user perform several package handle believe enable pose plausible option extend package smoothing well serious statistical currently nonparametric program health international center california kernel smoothing implement option curve add subject choice mail alternative category must reflect accordance degree option option order naturally set item hereafter sake option often response refer comprise set ji jx assume select response reference mathematical underlie trait measure measure refer choice item typically proportion subject quantify discrimination account option characteristic mean universal among literature survey name aim analogy classic parametric structure estimation idea model vector interest item difficulty discrimination assume accordingly might approach unless although experience confirm communication appropriate display justify year recognize paradigm give item characteristic basic set statistical require package perform propose approach nonparametric smoothly keep ij consequence preferable weight amount trade continuity non argument far largely nevertheless common choice assume vary highlight subscript important different use replace nk consideration estimation process begin transform common summary alternatively come quantile ability denominator avoid infinity f standard equally value span ordinal ability consecutive start group n bandwidth plug former kernel generate estimator formulate framework deviation induce consider validation considerably high simple widely context let kernel nj remove estimate ordinal ability omit statistic nk vector denote suitable visual inspection graphical estimate pointwise point relevant extent consider moreover useful error argument ignore substitute complicated approach interval estimate several quantity compute take jx j score coincide refer option straightforward define kernel jx jx jx jx jt really respectively analogy single function call counterpart preferred substitution people display axis interpretation method fails affect extreme generic relative say differently summary addition characteristic describe tend step time step iterative refinement really reason consider package reference generic see simplex dimensional nonnegative coordinate vary move curve analysis simplex location convenient triangle unit length side opposite vertex unitary sum one triangle regular item option nevertheless three four option main create option item illustration capability number alternatively list weight item use simply correctly option preliminary item necessary vector option option scalar item analyze subject rank nominal item rank item vector equal complicated weighting specify help section default alternatively point default value quadratic select global computed user input numerical opt cross validation specify implement treat option add multinomial subject one class function return brief description variety select option evaluation point observe subject score mm subject quantile subject score quantile probability overall mm km option correspond weight matrix contain error mm mm ht method allow exploratory return correlation mm observe vector ii nj evaluate alternative return subject expect list package return return object return ht mm simplex plot high option display multiple plot group include contain response student item option analyze create key format produce smoothing default gaussian kernel measure performance correlation value create become available overall display produce display display blue red use dash fall item figure plot figure different item apart item item monotone problematic trait level option trait select measuring trait contrary item subject trait level item display power near subject great subject roughly select regardless total figure top student recognize option incorrect select incorrectly expect total constitute probable aside since expect select correct option consequently incorrect code display figure weight blue correct pointwise interval dash red illustrate via argument remove change specify wide total high due datum less precision plot average group include triangle simplex illustrate plot item naturally normalize experience people tend eliminate option characterize option item generates display figure ht curve medium red green blue value break three ordering level possible make consideration format correct term requirement toward important individual trait figure fairly item curve concentrate close correct student slight tendency wrong speed bad student item triangle figure see share lie option perform way simultaneously show define x paradigm consideration rank ks compute pca pca produce graphical principal represent item place component axis difficulty ability extremely principal discrimination item high slope case top respect component user identify discrimination possess strong clearly conclude principal plot tend summary composition dominant subject red line
formalize outli region locate region outlier cell outli poisson region poisson variable ax indicator region cell count name notion outli easily estimator exchangeable robust preferred procedure contingency region respect outlier look ensure full space call submatrix subset consider definition pattern cell full strictly coincide add minimal cell yield non singular pattern outli identifiable rule identifiable develop notation model correspond cell take yield indice detection omp define identify set large majority minimal ml x w w idea outli detection cell pattern outli region set minimal notice minimum identify base knowledge omp outli cell identify identify number minimal cut discriminate discuss ml j nj jx w enumeration minimal minimal potential outli free ml procedure open choice estimator call also derive categorical detect outlier adapt pearson least chi residual create subset minimize chi tuning generation subset code j ml j ml adjust perform difference outli detection base analysis estimator large table analyze pattern contingency design eq indicator design latter parametrization usual parametrization implement former proof many formula become handle cell table min case configuration submatrix hand produce strictly minimal first singular complete among case different algebraic investigate notion describe problem example nevertheless structure cycle cell exactly submatrix definition submatrix cycle case cycle ingredient cell cell independence cycle cell cell row design coefficient nan minimal pattern conversely submatrix nan submatrix indicator row belong coefficient cell negative therefore column coefficient cell choose cell positive otherwise iterate reasoning another cell certain column two definition corollary following produce strictly pattern table uniformly tuple cell cell produce cycle cell strictly select uniformly statement table empty cell row remain cycle must empty cell row column exclude remain minimal cell cell force constitute loss form cell cell strictly choose minimal need pattern stop two pattern section present outlier discuss situation conduct exclude omp poisson contingency table value contingency cell hence upper bind expect small bind corresponding contingency place row outli moderate outli contingency table analyze extensively create work three two type cell notably scenario time replace contingency affect simulation table similar finish generation contingency minimal constrain outlier type outlier outlier outlier identify analyze comment satisfactory table outlier outlier notably hence position within major placing classify outlier reduce considerably evident case outlier rise parameter estimate almost respect seem notably small scenario well scenario valid large though less regard outli rate outlier discuss method motivate cutoff consider case contingency nan plus outli cell table size proportion detect outli motivate compute proportion display good trade outli first discover outlier treat independence model become apparent detect social network go applicability general h ccccc within mi mi mi point omp outlier e outli look stay cell study cell sure outli cell result category contingency class yield contingency table omp identifies identify surprising hand model wrong seem plausible alternative offer satisfy final present behavior seven work class concern friend daily week week friend cover category whether daily identification classify outlier outlier yield eight definition pattern yield time omp one outli cell eight yield solution produce result detect outli time cell time detect rest cell time cell interested outlier cell outlier detect method omp contingency regard outlier get see co friend omp good contingency detect simulation real summarize consider provide detect table use increase outli simultaneously detection see sized big table procedure suggest omp outlier scenario detection however expect dimension scenario identify outlier perform
ranking return return importantly set depend concern social person loss application respectively incoming training neither differentiable development end consider set would edge latter optimize conventional hinge simply equation denote relation beneficial differentiable function edge knowledge forward relation restriction product mapping express pairwise kronecker feature pairwise dual choice rbf rkhs closely function compact bound rkhs dense universal define universal base interesting optimize universal apply kronecker pairwise expect low amount universal consistency kronecker consider valid relation approximate closely whenever expressive illustrate detail domain knowledge relation similarity relation relation domain object need learn symmetric relation binary call symmetric real analogously relation relation edge become undirected domain metric similarity symmetry incorporate framework modification kronecker symmetric kronecker use predict arbitrarily type relation symmetric relation rkh knowledge learn unnecessary expressive still let type relation call hold relation place reciprocal edge graph induce edge appropriate application arise bioinformatic represent preference win real high interpret interpret set instead incorporate kernel symmetric reciprocal also rkh reciprocal kronecker arbitrarily reciprocal let continuous relation dense kernel model reciprocal relation give detailed algorithm rank system equation relation powerful tool matrix correspond reciprocal matrix vector use omit consideration follow matrix skew armed definition corresponding reciprocal symmetric note cover encounter encountered consideration involve node set reciprocal kronecker ordinary kronecker immediate consequence kronecker claim pay entry evaluation edge entry contain skew straightforwardly contain evaluation kronecker p kronecker kernel represent follow node edge node enable drop present case application graph belong classification bioinformatic e feasible experimental imputation edge notation matrix prediction parameter whose represent dual solution contain per occur regularization multiplication thus minimize becomes set system kernel interpret simplified solution enough rule solve kronecker product let matrix matrix shift kronecker solve actual rule concern kronecker introduce namely eq q ss eq side leave write well multiplication time kernel definite nonnegative strictly therefore exist proposition continue reciprocal ordinary kronecker identity certain property furthermore prove multiply inverse result vanish orthogonality inversion identity calculus analogously inversion identity reciprocal shift ordinary kernel advantageous computational cut provide ensure inversion shift kronecker accelerate computation reciprocal inversion identity symmetric reciprocal ridge regression reciprocal kronecker ordinary kronecker learn symmetry encode kronecker kernel equivalent use analogous kronecker label ordinary kronecker equality fact couple suppose regressor determine vector kronecker result reciprocal show way reciprocal kronecker corresponding kronecker kernel use represent representation matrix entry center multiply entry quasi start contain node exactly form provide order compatible entry arrange analogously regression kronecker time ordinary kronecker write equation p invertible rewrite product nonnegative g involve semi definite call empirical e kronecker consideration reciprocal kronecker kernel similar close form corollary concern regression drop fortunately design take advantage closed form solution proceed entry node edge possible edge several edge adjacent write ordinary kronecker analogously reciprocal variable store occur sum solve become identity matrix system linear product cubic nonzero quasi diagonal multiply gradient show early learning process together regularization insight back ordinary learn model wise center correctly edge rather common algorithm relation utility value center help achieve utility form ordinary rank diagonal center make indicate regression eq inversion identity solution lemma provide block center conditional therein indicate regularize prediction minimization kernel level center conditional wise behaviour give compare pay attention corresponding construction rkh expressive wise able express help kernel next focus term wise latter promising predict value ordinary regression pairwise challenge consider task graph use prediction application retrieval preference induce document machine optimize closely rank regularize least square previously minimize equivalent article edge training graph enough train however exist limited training node furthermore mean edge learn need construct store less kronecker use node see assume wise solve avoid complexity present special product close regularize ranking generality first experiment reciprocal task win condition experiment consider retrieval document document summarize goal bipartite specie rank species capability generalize observe conditional minimize convex approximation minimize regression also train specie possible due memory requirement cost enforce reciprocal experiment kernel experiment thus measure variety approach characteristic task solver pairwise square loss experiment train software whose generation detail consist well pair play player player first win player differ move game player another meaningful sensible rank player variation set strategy play move one rank game adapt exist game represent start winner start thus product edge generate kronecker three algorithm directly score minimize edge pairwise hinge preference node preliminary notice regularization reach zero appear run zero rank present successful easier beneficial set pairwise rank approach yield rank surprisingly experiment reciprocal kernel experiment cc comprehensive plot performance repetition regularization behaviour rank range reach start increase implementation reach good even narrow suitable approach parameter highly model successfully enough exist use reciprocal kronecker regression approach pairwise rank c learn rank document document publicly consist document contain document feature similar message document document rank highest next last rank undirected graph relation conditional ranking document grow node nod experiment already million paper experiment training rank directly graph long experimental static rather document already simulate form contain message hardware window message graphic os ms windows mac thus form consist node second training test rank greatly outperform value relation investigate enforce final experiment compatible symmetric increase computational node task effect early discuss contain see quite outperform regression enforce symmetry increase loss rate curve monotonically decrease due scheme sometimes solution demonstrate show training consist million edge generality available improvement base enforce knowledge type kernel potential rank classification problem huge class happen dataset class correct prediction occur capable introduce article unknown rank normally define identify specie profile task profile profile profile scenario profile relational consist edge connect single bipartite detail version divide different form point large class randomly divide small entirely combine profile result also run kronecker experiment result rank rank close method point result specie test rank regression similarity query serve relationship r biological newly discover think try machine kernel similarity ec ec level detail ranking ec rank ground query count successive left start digit ec soon example ec query ec structure bind ec annotate purpose ec lead ec comprise member heat consider art label superposition generation equal part cross validation addition cross validation loop implement recommend hyperparameter power whole hyperparameter benchmark bioinformatic construct compute query appear end represent potential similarity rank query principle methodology cb fp global well supervised outperform kernel put traditional benefit supervise method account training phase retrieval ec conversely characterization ec supervise preserve ec support summarize set different correspond well correspondence truth different distinguish third improvement exploitation direct method indirect approach interpret indirect kernel entry noisy case similar kernel loss cb loss slight experiment prove trained kernel runtime conjugate early stop close rank train edge file software intel gb memory parameter limit conjugate efficiency solver sample fully vary number vary number consider scale behaviour cubic time edge thus observe beyond apply worse iterative cg allow efficient training share behaviour applicable solver kronecker product necessary edge stop cg prove perform experiment algorithm linear solver sample uci repository label point belong solver product feature even competitive e less feature low efficient inefficient infeasible long also beyond scalability large dense graph kernel solver million matter minute solver ranking condition rank loss function generalization motivate rank symmetric reciprocal treat important confirm rank optimize loss beneficial instead moreover boost performance discuss follow computationally present conjugate early matrix structure recommend become form solution proposition recommend method solve magnitude large rank exist acknowledgment thank comment grant research foundation european conference corollary definition domain one task goal consist rank particular object ranking type relational ranking object efficient theoretically regression prove symmetry relation efficiently illustrate symmetry improve present review conditional suppose play computer people player unfortunately game game game tend sense player great mathematically speak cyclic relationship rank supervise biological consist interaction highly confident interaction similarly conditional context predict rank protein likely conditional task differ ranking compute reciprocal relation ranking arise relational relation model social database object information retrieval interaction
ranking dataset os enyi rank yahoo movies division yahoo movie user consist represent movie difference movie user pairwise dataset construct comparison fisher randomly impact base schedule pairwise incomplete team b team team strength schedule among collect rank elimination expect team remain game possible collection interpret design schedule ranking context study schedule games informative spectral connect division argue improve scheduling conference continue construct synthetic estimate graph laplacian establish used section collection reduction synthesis experiment employ turn learn rank statistic schedule conference include extend comparison os enyi connectivity fisher additional study optimally feedback arc rank consider ordinal pairwise preference label specify work preference represent quantitative optimal alternative geometric ranking easily survey literature find design pose inverse e image optimal imaging tradeoff measurement reason cost collect often place acceptable reconstruction construct desirable informative equivalent variation schedule static scheduling schedule throughout scheduling elimination team next round elimination world dynamic scheduling gain view ranking past process dynamic game thus disadvantage scheduling method scheduling focus elimination agree rank particular team objective assume paper investigate round team schedule synthesis maximal study many algebraic review review optimality robustness network failure highly algebraic process determine fire game game connectivity load balance consensus mention rotation measurement algebraic edge proportional quality briefly eigenvalue algebraic extensive give recall incidence direct graph terminology arc direct arc tail arc define edge weight refer un laplacian interpret triple direct arc connect vertex degree define degree laplacian interval eigenvalue nonzero connect second eigenvalue characterize subject wise connect graph u w connectivity small tight incomplete set connectivity adjacent removal algebraic connectivity vertex algebraic bound inequality also perturbation weight increase indicator use obtain indicator respectively weight however unconstraine round greedy add maximize algebraic augment work refer greedy heuristic finding edge large second w alternative complete let arc incidence comparison denote comparison alternative comparison gauss state unbiased equation finding agree square sense square ranking prove empirically rank present datum laplacian square estimator since unbiased fisher collection definite ordering criterion function prop square bi eigenvalue laplacian expression prop criterion proposition datum collection criterion problem reduce synthesis add algebraic interpretation theorem tree condition effective circuit construct identify return comment ranking proposition structured graph laplacian graph compare algebraic os r enyi yahoo movie rating correspond division schedule construct via active accurate cycle mm fill dot dot edge dot edge dot thick style draw fill thick thick thick thick dot fill thick dot dot dot thick thick inner dot style blue fill blue dot dot dot thick dot dot edge thick alg diameter connectivity represent informative ranking computable graph node algebraic connectivity connectivity vertex connect unlabele algebraic connectivity number edge algebraic connectivity vary connectivity algebraic large node equal degree graph add algebraic connectivity study fig nontrivial maximally algebraic algebraic achieve connectivity panel connectivity remains describe distinct graph nod additional connectivity graph prop ranking review review movie yahoo yahoo movie movie movie define pair user movie write note expression arc yahoo movie user rating movie graph pairwise comparison l l movie day friend degree represent yahoo comparison histogram residual top rank number comparison black pairwise give increase movie relative estimator prop compare comparison experimental develop approximate matlab initialize use eigenvector lead eigenvalue laplacian maximum increase rate increase pairwise randomly movie tf yahoo database movie legend comparison color bottom collection square color graph visualization via pairwise top plot yahoo movie comparison improve system enhance generate spectral algorithm generate reasonable position movie movie use full dataset graph plot use figure top panel primary connect weakly advantageous designing schedule division schedule rating division divide decompose division college member opponent member member division game static ratio example american team primarily division within average strength among schedule consider team game per year team play ranking mathematically expert opinion aggregate ranking none none relatively schedule via game vertex indicate conference represent reveal conference indicate text use visualization method division play conference cluster conference algebraic cluster community toolbox algorithm optimal cluster centroid division division vertex team color conference membership bottom cluster interaction conference implication algebraic connectivity os enyi nearly representation schedule game edge game conference use primarily conference introduction note scalar three compare division os r let vertex graph laplacian comparable e harmonic rather logarithm determinant define remark division division record also fig division division table fig enyi take approximately time threshold connectivity connectivity similar scatter indicate graph nearly evaluate solid obtain finally blue table division os enyi graphs sense division division reveal primarily conference conference imply algebraic cut dash fig division schedule connectivity edge consist conference os enyi top primarily reduce ranking schedule flexible contain permutation previously optimal scheduling future game performance schedule completely begin play schedule os enyi division schedule solid line represent dot os enyi circle os enyi optimal os enyi division division os enyi comparison propose division truth rating distribute truth according ranking ranking ranking right algebraic connectivity dataset enhance dataset choose game comparison greedy selection collect initial collect plot rank determine collection strategy error collection thick minus strategy plot vs algebraic comparison red connectivity graph greedy random algebraic produce fidelity experiment design informative statistical ranking heart framework unbiased rank outer identify information square outer inner reduce optimality fisher weight find algebraic division optimality scalar strongly strategy remark optimality furthermore use via improved collection rank yahoo user rating consider show pairwise add contrast datum collection comparison add movie review review review review review improve ranking potential constraint collect extension employ accommodate incorporate penalization compute constraint handle weight least rank orthogonal free cycle harmonic cycle harmonic inherently inconsistent ranking lie graph give first improve remove connectivity alternative estimator method I mark nsf dms grant support nsf dms thm thm question vertice alternative rank agree develop collect least square experimental view problem maximize
eq final arise potentially statistic apply point whenever statistic impact allow issue value produce change detector shift detecting shift difference return find conditional analytically instead compute carlo thresholds stream find successively proportion exceed discard equal successively expensive hour needs perform computed store table overhead give seem reasonable since threshold bernoulli equal conservative table actual empirical variety take seem suitable stream far towards implementation world scenario resource spend calculate equivalent package optimize level correlation statistic sequence failure failure observation compute algebraic eq recursive formulation recall q use though hence smoothed still time far retain memory use efficiency store discard length recently window memory discard calculate whole contain computational naive old discard detector discard point entirely statistic maintain sum old time outside current carry lose give notably threshold post symmetry bernoulli c detector parameter run delay recall degree smoothing c comment occur suffer detect bad detect increase smoothing cause jump statistic partially detection performance demonstrate usefulness worse read relatively poor latter bernoulli everything far design complete change minimax parameter misspecification c detector realistic compares parameter misspecification likely happen early monitoring since insufficient allow estimate design base value correspond misspecification control assume post change design change misspecification occur substantially bad every appropriate detector exact sequence pre unknown device favorable chart change occur relatively stream comparable chart knowledge change bernoulli stream method poorly data optimal realistic investigate perform degree misspecification see drastically tool situation methodology available http website co uk research agent jointly engineer physical grant ep detector c c c detector eps widely approach violate observation occur approximation detection computationally suit sequential monitoring empirically datum find optimal assume detect change spaced datum assume realization integer constitute arise receive finance monitoring increase work hence terminology bernoulli change soon change sequential detect encounter find monitoring change begin reduce successive multiple simultaneously avoid single stream performance detector expect false positive delay detect denote denote false positive say occur play type important sequential assess reason literature bernoulli detection suffer several make monitoring computationally inefficient context require store memory although fix monitoring asymptotically assume substantially work come literature size treat binomial random chart tool chart individually commonly inspection detection treat failure approximately standard tool detect parameter sequence approach transformation chart also pre parameter know however situation sample always misspecification pre paper task parameter regard pre post computable monitoring point highly hence receive stream detect practice trivially side change detection tool design monitoring change extend distributional shift bernoulli implement available section end proceed begin bernoulli deal length develop sequential monitoring discussion follow performance detecting point length neither change assess occur hypothesis observation identically several test exist exactly rely like change detector number change point idea observation break nan sample identically number failure observation reason failure follow binomial coefficient fundamental probability dependency value hypothesis make test know introduction failure first probability failure observation observation identically convenience occur reject appropriately choose threshold change locate hypothesis
literature model autoregressive comparative baseline engineering order show return note make day volatility forecast evaluate mse model series historical windows contiguous historical constitute consistent volatility measure result observe significant improvement term employ metric evaluation return historical volatility horizon step c evaluation horizon step step evaluation c return historical volatility step evaluate covariance asset return repeat experimental copula employ similar input observation comprise available sample interval one comparative art volatility specifically ccc use asset result yield scenario yield one order magnitude yield comparable employ return average prediction ccc evaluation square ccc nonparametric financial model separate gaussian drive law capture model addition covariance asset return build top efficacy several asset observe improvement consider successful model financial return alternative widely field machine specifically novel nonparametric series impose component well capture model distribution tail skewness asset model gaussian mixture conditional volatility asset require tendency asymmetric generation financial market index tail asymmetric autoregressive address dependent volatility finance market approach commonly model exhibit time volatility follow period excellent define field last decade represent squared return variance computation prediction field machine function significant application domain gps several gp difficulty deal multi modal approach ensemble space issue work volatility financial approach effectively volatility introduction novel distribution constitute gp variance process gps model way novel approach nature asset procedure conditional bayesian technique especially process become popular year nonparametric realization seen give shape large highly indeed dirichlet prior selection value turn frequent dominate entire inspire advance switch mechanism prior derive examine efficacy consider financial remainder organize provide presentation present process prior summary regression propose copula jointly model mean conduct application deal model financial final summarize result dirichlet dp characterize referred innovation parameter dp dp subsequently draw joint exhibit sample draw allocation proportional previous allocation concentrate dp scheme innovation parameter role determine parameter induce tendency draw new base get contrary randomly discount innovation parameter yield rich rich assign draw likely effect rarely allow scale note case unconditional distribution stick break construction consider two random beta draw stick representation draw comprise unbounded component density proportion process completely specify define covariance real eq take concern large employ depend application eventually identically phenomenon scalar predict target information contain consist observable express superposition regard normality target observation yield identity conditioning training expression yield q optimization employ conduct type maximum maximization likelihood easy regression deal corpora model comprise derivation infinite innovation infinite reason break variational equal note full prior truncation impose tractable truncation prior impose hyperparameter function variational stand posterior kl nonnegative strict bind tight implicitly consider take variational read ignore constant term derivation posterior involve hold manner consecutive updating guarantee monotonically maximally denote posterior stick break innovation eq posterior regard process semi definite diagonal variational parameter q posterior element element estimate comprise hyperparameter prior maximization free resort bfgs model output derive derivation furth variable eq base variance one component largely essence model jointly various output return correlate give input approach gp aforementione exist learning entail optimization estimate hyperparameter kernel get bad local optima work issue financial devise novel way model tool copula seminal standard restriction example hypercube cdf exist variate copula cdf marginal variate copula unique conversely univariate hold inverse cdf model easy show copula widely assume cdf parameter copula model copula several frank enable capture dependence coupling copula copula model able skewness tail dependence copula rigorous effort motivate excellent discussion copula c covariance specifically vector cdf define cdf correspond give use copula value
make lemma perspective problem low rank similar low cx decomposition regression constrain multiple regression place elementary regression optimal thus multiple build improve previous well nd multiple except instead note simple equality approximation optimal optimality constrain b minimum point dimension randomize output subspace run section connect round follow fast round quickly basis generalize general easy start eq clear number conditioning pa relationship quantitie q recall da easy algorithm naturally connect round symmetric respect connection round deterministic compute rounding separation due convex round ellipsoid ellipsoid find problem algorithmic suppose describe oracle provide ellipsoid round result aware particular use construction find theorem round convex origin center subgradient matrix qr decomposition maintain conditioning column take construction well condition extend construction round leverage next improve run slightly condition run application find condition full compute pa storage work depend well norm condition leverage norm row norm estimate sample row together complexity due reduce modify obtain random construct specify diagonal solution section evaluation implement evaluate cauchy transform transform ct transform ideally evaluate distortion evaluate non convexity seem accurately transform basis approximate describe column well basis transform metric scale invariant condition fail bad input comparable much conditioning ct lead regime algorithm worse base increase perform cc third draw dash correspond difference inferior explore sampling fail completely fail sample ct instead next transform implement section compute norm instead error thus permit indicate leverage similar side entry independently simulate corruption regression regression conditioning instead determine tolerance accept size third correspond failure remove error fail completely fail slightly test certainly favorable error theory fact leverage approximated capability scale corrupt generate standard canonical measurement number entry million corrupt simulate noisy corrupt regression estimate regression certainly robust alternative separable response cluster core condition ct un time fast ct particular moreover store storage spend running implement ct ct reveal inherent ct condition sampling row solution simultaneously check measure error ccc ct randomize error ct well follow ct large third without conditioning corrupt normal treat overall entry draw wise clearly condition sampling ct uniformly entry wise ct maintain base ct measurement adjust towards summarize condition promise digit pass twice cauchy transform analog hadamard base demonstrate fast base improved running extension provide first evaluation evaluation rectangular precision solution acceptable base setting connection theoretically exploit scale david fast problem rescaled row polynomial improve previous fast round well condition basis round time conditioning provide algorithm compare thing asymptotic guide practical performance construction condition fast cauchy transform approximately preserve norm distortion constant cauchy transform coherence embed prove development dimensional importantly sense implement distribution mapping subspace embedding geometric quickly variant hadamard thing construct regression lead problem rank fast give q interested case although least absolute regression programming problem convex relevant work solve sampling basis embedding thereby improve cauchy analog variable slow problem subspace problem construction fast subspace interest call cauchy provide detailed essentially analog represent sense depend preserve distortion moreover actually relate construction rescale fast analog improve result construct well condition regression row probability row regression reduce regression consist rescale independent use black run exploit improvement well basis fast guarantee run improve run condition algorithm improve extension improve run fast computing basis problem condition basis round general conditioning alternate way particular round round much round time present basis round algorithm competitive develop evaluation construction slow cauchy transform core matrix evaluate latter nearly infeasible clearly paper brief use present basis leverage approximation include condition large conclusion presentation assume find minimize mostly set th row th frobenius norm ij p pp vector identity generic may trial due immediate first I cauchy stable distribute cauchy heavily sum note prove cauchy random variable dependent conditioning magnitude application lemma completeness tail necessarily mt logarithmic dependence cauchy assumption random substantially improve cauchy independent latter scale tail bernstein inequality suppose define cauchy transform analog actually present coherence rescale cauchy random hence main theorem quality construction first matrix cauchy random finally combination parameter failure uniformly algorithm fail suitably diagonal independently cauchy sg sn integer reader matrix hadamard may recursively hadamard transform hereafter spread comprise cauchy scale finally variable theorem find cauchy hold far constant since establish factor variant hadamard additional plus evaluation use tail concentration independently bound construction generic change parameter construct random informally slow transform multiply dimensionality slightly construction inequality dy fast mapping give distortion arise random whereas use requirement restriction proof originally appear strong prove correspondingly improve second easily well construction basis approximation least absolute algorithm basis condition adapt basis range u exist basis generally condition norm norm condition summarize run step product condition lead multiplication let projection construct construction section step factorization next theorem corollary combine proof condition construction condition fix construction result require rank case modification small sampling preserve discuss subsequent basically either subspace approximation motivate score let element name random regression rank regression ultimately follow define invariant basis score depend polynomial orthogonal span give lead leverage degree analysis goal vector sophisticated yield run prior fast permit arbitrary detail summarize well probability sample row quickly norm row scale leverage input carefully sample rescale problem theorem summarize improve n exponent multiplication purpose norm row norm condition cauchy construct diagonal solution minimize weight regression dimension construct remark preserve norm mention minor change natural right approximation subspace efficiency solve simple thereby replace expense size improve essentially accommodate detail
effort constant analysis recursion recursion f ff ts difficult q equally q plugging estimate form become integration calculation plug obtain hold essentially thm imply decay average polynomial decay scheme convex choose dataset run gradient extend mirror question individual iterate ht sgd iteration return common heuristic last well rate polynomial average last indicate return last iterate far whether especially implication probability last variability unclear whether iterate behaves comment proposition definition zhang edu nj usa descent simple popular stochastic theoretically decade usually smoothness machine paper investigate smoothness average convert sgd iterate optimization framework prove round iterate non non smooth first averaging averaging attain propose implement simple descent sgd unbiased learning minimize sgd simple highly scalable making scale proceed round describe line round estimate subgradient iterate suitably analysis decade instance reference therein perhaps gap understand look rate budget year attention devote e classical non continuity use solve machine standard smooth objective non non whenever smooth function smoothness smoothness carry see definition error general provably suboptimal averaging iterate average rate result leave significant averaging return iterate often say optimization iterate convergence aware well simple gradient know exist theoretically practical memory one know advance trick natural compare iterate rate average require contribution prove individual general convex iterate much bad pose knowledge sgd rely iterate iterate implicitly somewhat improve error average call enjoy new scheme easily compute study discuss emphasize algorithm convergence strongly focus paper widely bold letter convex strongly subgradient hold instead see wish loss close choose I subgradient individual subgradient error oracle bounds note without much generality size take strong function substitute subgradient assume optimize diameter begin consider individual iterate attempt constant tt convexity extract sum rearrange get convexity trick pick instead plug back iterate imply dividing repeatedly sum remain iterate analyze yield simplify also individual constant explicit convex c begin time instead substitute substitute simplify kf verify eq proof iterate inequality repeatedly sum remain term side use upper bounding plug simplify rate average rather individual iterate rate mainly since iterate first examine iterate assume optimization show tight flexible hold identical thm tf tf get tt obtain bound estimate theorem eq eq general
gradient iterate function sparse simplex constraint author true constraint problem project jointly consider regularize minimization rely low cardinality constraint easy plain letter represent scalar resp vector estimate p c dimension apparent subset without sort onto simplex onto large operator keep rest operation onto simplex euclidean projection onto isometry property rip k k rip assumption project invariant stepsize obtain via similarly rip sub entry ok case approximation guarantee statement task support support unique em w I definition threshold small introduce meet em c em delta delta obvious greedy project remarkably selector complexity sort w unfortunately obvious select grow mean adjust lambda selector overall symbol break tie hermitian semi almost construction call nuclear singular leverage algorithmic affine tractable amenable provably fail constraint relaxation obtain solution normalization meet constraint ingredient rip rigorous analysis unitary f apply reduction similarly eq ability rank jointly generate add follow snr measurement noiseless rank value freedom hence recover noiseless vary know experience estimate convex parameter rank normalize need exploit trace proximal sophisticated nesterov acceleration projection project stepsize acceleration algorithm try initialization x poorly noiseless theoretic high approach substantially convex solution solution neither advantage fewer perfect measurement convex highlight partial eigenvalue decomposition convex overall complicated factor dense solver approach iterative solver large discrepancy leave middle approach middle bottom right kernel relative gaussian additional nonzero tend small weight I x minimize result estimate negativity induce avoid overfitte interpretable one might sparsity context extend cardinality consider follow quadratic kernel priori find width depict individual mixture center around unfortunately program middle window interpretable cardinality enhance constrain order convex useful see profile pdf along position weight enforcing concentrate fully frequency illustrate pdf inaccurate mean portfolio return construct realistic adjust exist portfolio cost transaction naturally introduce cardinality mathematical portfolio selection seek optimization highlight synthetic solution rip goal refine art solver accommodate pursuit criterion solve use equality constraint matrix solver
advance fisher discriminant wireless take protocol extend propose capture relationship classification combine communication network interaction effort put tradeoff stage accuracy overfitte wireless surveillance environmental monitoring age application presence absence physical supervise result take length limited operational operational dominate communication wireless sensor sense popular operate wireless sensor part wireless sensor communication medium successive scenario measurement classification perform collect varie network supervise discriminant analysis algorithm train operational operational fundamentally character detection characterize operational classification accuracy training approximation network proportional throughput operational operational training two network rate reciprocal spend node likely consumption discuss operational back exhibit one regime quite usually encounter literature concern detection function sensor tend aspect model communication therein classification consider application mac focus case likelihood performance consider wireless measurement simplify wireless detection much reality algorithm estimate covariance training encode later adaptive back far less detailed network interference offer cover broad topology remainder organize sensor perspective operational accuracy balance act idea consist take measurement combine classification classification unseen focus fisher discriminant analysis covariance discriminant plug ratio covariance mean operational sensor classify performance specifically generalize unseen operational datum first specify statistical sensor spatial likelihood probability exposition one neighbor near sensor sensor graph element near neighbor graph encode remain element sensor highly accuracy rule known mahalanobis insufficient accurate simplifie decide wireless protocol represent node network set edge simultaneously ease presentation neighbor graph mechanism inactive transmission time unit neighbor inactive try back correspond incidence transition make measure refer throughput may consequently expect operational fraction dedicate operational throughout disjoint first node network simultaneously correspond interference empty simplify particular node state back notational active activity equal throughput operational consider prevent node correlate state use eq throughput choose shorter back position network throughput operational operational parameter spend interference communication node produce activity thus matter expression incomplete activity address include elaborate cost additional computation communication separately learn classifier operational setup associate compute overall accuracy generalization accuracy pattern set disjoint stationary throughput sample pattern sum sensor mahalanobis distance substitute expression four mahalanobis mahalanobis state state weight accuracy derive operational operational accuracy wireless sensor communication spend quantity comparison bayes optimal detector regime regime qualitatively plot operational function fix devoted training cc operational back rate mostly probable drop always cc sensor monotonically operational state negligible acquire state initially sensor approach examine function plot cc monotonically expect decrease local give phenomenon overfitte demonstrate decrease sensor number wireless network effect sensor change initial increase overfitte sensor fix fig examine accuracy comparison devote represent correspond range different part close corner plot extreme maximized increase contribute behavior turn three correlate sensor similar gd gd distance correlation independent fig
innovation whereas datum bias innovation extreme situation parameter decrease shown summarize iii u accept fail innovation iii average reason indistinguishable inaccurate integration error example innovation iv limit iv illustrate respectively realization solution linearization compute thin simulation interval period series I space innovation estimator u compute figure show histogram limit h innovation innovation negligible thin estimator provide biased innovation unable illustrate usefulness order innovation estimator estimator happen one exact innovation insufficient accept step acceptable first accept estimator accept ten estimator well r moment noise simple efficient moment covariance journal molecular nonlinear world scientific simplify differential http b http http continuous volatility market scheme stochastic equation additive bit local linearization comparative innovation diffusion series equation model equation level estimation differential overview review model finance prediction measurement mathematical processing third linearization filter application ed conference innovation j continuous st conference decision usa bold gaussian signal inf comparative likelihood nonlinear dynamical maximum versus kalman expansion graphical eeg model compute interval period interval length period interval compute c period period time length example period interval period sampling theorem conclusion theorem theorem exercise theorem lemma theorem la mail innovation method introduce two filter exact filter increase decrease conventional one enhance intend recurrent situation stochastic identify distant describe subject intensive simple joint unknown form diffusion contaminate noise observe extra arise typically reformulate continuous sde term discrete observation analytical simulate approximation decade see et deal noisy maximize function associate underlie state class inexact approximate like local linearization kalman et filter discretization mean numerical et innovation estimator way actual financial molecular et al et et al feature innovation exact approximate alternative innovation orient bias innovation minimum finite decrease increase normal bias decrease mention innovation local linearization algorithm simulation simulation innovation estimator enhance test reduce distant basic estimator innovation convergent linearization filter present section illustrate various differential differentiable compact q gaussian time increase sequence diffusion kt sde time specifically innovation give observation innovation k denote recursively filter ss first specify prediction normality conditional equation formula innovation approximation early paper like linearization filter extend filter hand filter discretization equation error innovation limitation estimator nevertheless innovation negligible actual variety financial molecular mention cox approximate innovation al similar comparative denote differentiable derivative n n discretization equation initial weak weak definite follow solution give instant z solution approximate innovation naturally continuous state ss innovation filter discretization kind converge approximate innovation euler linearization order might derive reduce conventional filter situation innovation innovation consider linearization scheme filter measure give consecutive state go include approximate weak innovation state decrease therefore convergence innovation innovation innovation innovation estimator q respect realization h identity h k k deal particular prediction filter k k finite imply imply assertion hand depend realization new expectation measure realization mh conclude assertion innovation zero weak criterion clear innovation exact filter decrease assertion error mh deal average estimator state partition innovation innovation give probability mh e underlie probability generating realization assertion conventional innovation section property theorems innovation restriction partition specific application innovation variety reduce close innovation account specification inference automatically concern simulation section approximate innovation relation minimum define generating remark depend index good predictor average loss regularity identifiability stationary ergodic diffusion definite definite ensure ss derivative let innovation mh theorem identity h r e k k er deal particular prediction state positive whereas account k moment addition imply finally together increase innovation decrease go zero exact innovation approximate theorem go innovation estimator order estimator neither restriction partition discretization end previous definition apply define observation twice differentiable transform high definition innovation far depend innovation deal particular denote reduce estimation innovation approximate innovation estimator estimator theorem concern property approximate converge innovation estimator viewpoint linearization reason prediction formula additive noise adequate well performance innovation estimator filter satisfy nz k h recursive computation prediction give prediction variance filter gain go innovation go zero innovation average reduce innovation innovation emphasize whereas evaluate conventional innovation estimator linear equation additive achieve prescribed relative tolerance useful performance four innovation estimator compute conventional order innovation histogram time period distinct observation moment filter formula exact state additive noise observation conditional filter van pool addition van pool observation value multiplicative two innovation four state previously table give example realization equation linearization equation compute thin precise time take state model compare sampling period exact conventional innovation
time trading share day news per record company record word yield start rough aggregate flow news topic individual stock order network flow news apply impact trading order iii peak trading explain news decompose news piece lda global corpus word document simple sparsity multinomial learn easier already remove stop news record set stock analyze record stock significantly fast volume appear indicator function day present time evolution news volume four topic topic support lda topic general every word specific though mixture word draw estimate topic highlight follow negative click top simply reflect repeat phrase repeat phrase corpu difficult would huge topic employ focus top eliminate phrase day symbol number short eliminate topic describe e stock market news suppose topic leave relative trading stock regression trading volume trade median trading move boundary regularize provide presence trading activity contribute determination ten fold perform divide entire ten measure ten cross stability estimate focus peak percentile day day pay attention study period divide overall month window time window peak day fraction explain attention peak day fraction volume article useful sensible correctly fig compare trading trading term yahoo discrepancy quantify quality explanatory peak day define define success trading peak day subtract day among peak volume successfully sense fraction peak explain peak day explain peak day yahoo peak day explain peak day exercise swap extract topic yahoo news yahoo explain trading fig modify decrease considerably substantial explanatory find confirm regression pruning record e g always report stock explain trade rarely stock index news news record stock stock find recall logic topic visualize similarity construct topic link broad list tend emphasis report important piece information influence financial market million news thompson trading landscape news affect automatically simple trading news piece piece trade able impact difficult way visualize whole landscape stock utilize network visualization technique topic careful reading news include confirm piece stock finding power news stock trading activity provide insight news market stock large volume trade explain news trading news novel provide reason success require effect sophisticated probably news source specifically news collect confirm tweet compare extraction activity financial market summarize major external influence financial market news trading activity explain pricing extended universe topic acknowledgment author grateful helpful discussion partially support student service organization support aid scientific trading red plot aggregate volume trading volume volume index news record company latent allocation different topic implement lasso step impact topic peak news methodology text lda associate volume production black trading volume stock dot day select method period line actual trading yahoo bp number yahoo bp comparison actual trading volume topic explain yahoo trading b yahoo try explain trading volume regression exactly peak explain news stock news record period blue circle restrict distribution manual read company course red triangle quantify degree associate quantify word six six topic link topic quantify company top frequent quantify topic proportional explain topic link topic sale drug panel business national budget summarize frequent support financial production percent price percent car car company ht create manual record extent column show stock analyze stock classification stock sale top business accounting trading trading sec recall stock energy bp medical east business plan environmental education business east cm cm management technology economics institute finance university school information school economic mail yahoo co abstract mutual relationship flow activity financial key regard understanding quantify news social financial economic affect trading pricing organize stock seek issue news provide thompson trade stock stock landscape stock price automatically summarize regression activity news decompose modeling use quantify news trading represent landscape stock representative topic able extract piece information stock namely trading explain news trading restrict time news relevant information financial price instantaneous change reflect news news type environmental social financial economic expectation flow project translate demand price imbalance demand towards fundamental present flow view market without feedback occur solely paradigm relate price news relate frequency failed recognize price volume flow concept past present create feedback loop dynamic issue understand news really incorporate price priori nature
claim na quite model practice compare time time slow greedy procedure slow general submodular quite slow perform give feature nb model accuracy result svm nb significantly outperform nb correspondingly case also selection submodular partition modular vs cost associate feature modular cost objective submodular bad approximately strongly evident provide justification heuristic make procedure selection submodular feature procedure moreover various combinatorial constraint may great yet concave extension easy rest submodular material support science foundation google microsoft award electrical university usa electrical usa section definition proposition function submodular algorithm guarantee monotonically reduce theoretically efficiently difference hardness difference submodular function bound minima show submodular function show validity area ever grow problem show submodular function submodular minimization np complete constant submodular submodular modular function wherein element big economic submodular define machine function problem possible model alternatively maximize mutual fix independent submodular submodular subject location property discount sensor may place sensor location g use mutual want suggest initial optimize produce submodular structure generative feature find submodular moreover example particular choose remain may group strategy fast transform appropriate model cost source feature ix hx submodular typical probabilistic desirable minimize probable factor tree bundle form solve programming hold approximate inference define vx width inference vision also submodular cut minimized efficiently address difference decomposition compute minimize submodular way probabilistic class rich potential stand possibly I say submodular contain potentially rich function particularly vision also minimize two inspire address equation minimize modular express submodular function minima question first describe tight modular submodular include describe submodular constructive procedure construction np certain class set find decomposition two monotonically converge note algorithm order moreover give exist guarantee heuristic however additive polynomial computable upper optima mutual near taylor series particular taylor series function q strict convexity tight polytope sub point ny fs fs f parameterized observe parameterize submodular characterize function submodular modular tight set modular submodular submodular review minimize henceforth ds suitable new combinatorial convex provide submodular express define submodular vx vx submodular complexity decomposition low difference gain low tb g every converge minima monotonically every iteration minima checking permutation describe iteration minimizer since modular improvement permutation minima fs fs minimization large reach costly algorithm practice ds monotonically objective minima briefly iteratively minimize modular every submodular tx step submodular maximization complete admit modular bound run procedure mean alternatively modular upper first maximize expression modular replace modular bound tx tx xt every modular minimum non two variant moreover decrease every converge monotonically increase permutation adjacent modular bound early observe different fs modular bound correspondingly choice g von permutation modular correspondingly modular local iteration experimentally quality gx might great h achieve permutation order gain former progress much heuristic indeed address question selection minimize subject minimize submodular cardinality np hard approximate since seem unclear constraint np hard admit constant correspondingly cardinality easily introduce negative function constraint utilize minimum span tree min polynomial non difference submodular function modular directly minimize negative modular subject constraint analyze approximation assume normalize correspondingly section theoretically hardness think justification case inspire algorithm procedure size application list general hence np hard surprising large randomized perform poorly easy np positive computable opt p give positive choose unknown define observe n notice guarantee solve contradiction exist hardness function optimal define g lemma issue unbalanced let clearly use unbalanced query unbalanced query balance regardless actual thus never achieve approximation say easily randomize polynomial factor theorem factor global optimum hardness hold even submodular monotone monotone monotone submodular function eq simple decomposition decompose modular totally modular add decreasing prove monotone decompose monotone decrease modular totally repeatedly low obtain guarantee correspondingly open minimizer ds function generalize local max cut trivially generalize cut polynomial monotone decrease guarantee optima local optima
cover run intersection edge clique constraints lagrangian lagrange derive lagrangian clique therefore dual use cardinality evaluate due sort edge htb entropy size output clique iteration initialize obtain solve c c dual optimization define iteration evaluate respectively correspond primal allow constrained step proportional dual moreover visit primal approximately feasible go feasibility illustrate violate violate observe reduce clique clique e element feasible I convex relaxation equivalent solve dual optimal ic hc clique define span polytope liu trees solution htb infeasible sequence clique feasible initialize sort fractional infeasible add adjacency elimination maximal perfect elimination connect graph average solution edge select edge clique clique run algorithm synthetic distribution several decomposable uniform identity value value normalize definite decomposable decomposable ensure relationship adjacency entropy covariance factor row column therefore multivariate decomposable graphical independent graph structure star decomposable matrix correlation algorithm decomposable generate ten represent multiply dual represent replace negative constraint key obtain tight relaxation primal represent fractional fourth primal represent project optimal fractional corresponding primal dual greedy add mutual empirically correlation simple greedy htb ccc dual htb primal primal learn propose produce order combinatorial liu liu variations pac pac pac jt traffic alarm performance alarm approximate decomposable graph traffic dataset traffic month california location discretize bin learn approximate decomposable graph entropy generate likelihood structure learn traffic dataset high likelihood figure illustrate gain early relaxation decomposable graph empirically currently explore design heuristic speed acknowledge european project dataset thank project true minus plus pt pt sup paris france l sup paris france theorem proposition undirected likelihood np consider involve search problem variable synthetic benchmark set tool collection define probability many domain vision language bioinformatic naturally problem situation might rich discovery relationship tractable often variable simple ensure graph however rich work simply possibility fit tractable number learn design approximate high indeed tractable variational one convex design fine trade apart learn direct constraint type definition independence maximize likelihood factorize context learn latter hard lead various algorithm base search independence test heuristic least incremental al maintain graph kl et convex decomposable give kl iterative cut et base perform independence derive decomposable programming use weak conditional tree use information criterion recursively small paper decomposable datum combinatorial section loop complexity round state method synthetic dataset gain ccc decomposable clique dot set maximal represent decomposable cast maximum combinatorial assume clique clique differentiable potential within clique say maximal clique I connect clique vertex clique contain decomposable time merely problem sufficient decomposable convex zero vertex clique contain number clique contain clique loop blue loop acyclic encode combinatorial minimize eq represent imply clique blue decomposable red clique decomposable graph red decomposable color provide convex relaxation combinatorial number clique intersection property already clique I e clique least select incidence hull thus new constraint crucial maximize linear polytope maximize weight span order select weight form long add restriction see polytope define polytope clique otherwise ideally hypergraph similar subgraph hypergraph hull incidence acyclic may applicable notion number contain trivially select graph forest hypergraph sub
goal motivated help goal exploration allow agent arise easily relate relevant capacity subsequent regard universal define utility find fully specify transition reward need option heuristic agent conversely default mode equip quickly target position middle regard automatically human identify even environment present continuous utility priori specification reward typically try goal choose wrong nothing would death natural balance discretized case learn initially instead interact environment gradient state agent balance wide condition would study priori contribution vector initially state approximately monte underlying imply estimate triplet interact iterated forecasting structure definition illustrate know finite domain main technical portion formal high integral model rather matter go beyond description reinforcement natural coincide way incorporate action loop action high state guide part discover explore without although formal definition motivate example informally logarithm effective number induce action measure extent influence general show left location apart agent want location r car state usually rl factor explicitly salient flat elementary action move indicate direction chance movement result movement agent pick correct car pick drop meet environment place state transition effective step action elementary state slice thing apparent location move want location corner car away situation episode agent stand heuristic agent task specify goal compute alone essentially freedom create salient environment imagine high sensible blind quickly salient state environment remainder typical design establish subset black section define give dynamic system interact space simplicity give interpret natural realize hardware denote go make system interaction action induce notational generality form exhaustive enumeration step every consider evolve step action regard abstraction allow treat simplify drop index instead irrespective loop model variable shannon channel respect possible length entropy conditional strictly speak entropy differential entropy density e entropy finite limited resolution n ip eqs proceed definition look simplify exposition integration replace summation environment label fully describe right transition entry want calculate aa ad da dd x logarithm x aa da dd natural probability among section mi take maximize various illustration mi mi em column illustrate full action maximal action action horizon make stay matter whereas stay important reason horizon increase step large action agent contribute mi zero dominate indistinguishable action action outcome outcome correctly let agent transition system system zero state action distinct outcome briefly discuss mutual channel comment consider controller action main one maximize achieve action fail resolve property distribution obvious action mutual action different state every outcome state mutual still action whereas distinct action highlight degree attain capacity transition action algorithm em denote th achieve since domain notation start obtain normalization ensure x give carry stop look regard assume simple gaussian consider approximated depend state compute input calculate transition fully np I ni p k k k z achieving associate n n n requirement computational neither readily triplet result infer proceeding iteratively predict accomplished regression gps mathematically elegant offer considerable gps exactly furthermore hence certain polynomial instead gps analytically close bayesian gps automate adjust predict perform gps action desire output difference variable independent gps univariate gps deal subset approximation subset greedy aim incur incomplete cholesky decomposition avoid project px th equation mean note set hyperparameter independently selection transition see multiple gps predict individual train corresponding turn sequence integrate unfortunately solve closed approximation numerically integral alternatively laplace prediction iteratively step instead step produce repeat horizon step np slightly different every produce ignore uncertainty precede prediction indicate intuitively appeal salient study continuous reinforcement latter learn need optimization use result behavior actually external subtle detail effect heuristic fundamentally oppose informed need horizon desire extent lead external act heuristic guide agent incorporate intuitively state lead high quantity far reduce consider start know nothing environment combine exploration choose batch gets predict accurate reinforcement optimize physical system describe usually control goal optimize determined drive reach aim drive operate characteristic system alone turn close enforce external intrinsic exactly wide action stay achieve stay certain problem one end depict force plane equation force equilibrium position enough back create difficult nonlinear control space angular velocity force depict model simplify move surface keep continue dynamic appendix roll angle roll angular inherently horizontal turn neutral position deal u propose bar depict around bar force nonlinear control informally degree body system balance dynamically stable task first link bar significantly difficult first link reach close unstable state system appendix equilibrium link deal continuous alone sufficient choose therefore primitive control appendix balance action actually balance choose assume state transition loop current possible describe use deterministic corresponding meaningful usually increase ahead single one action exhaustive enumeration agent number computation perform informally number step grow large cost prohibitive action action hold amount extend horizon go locally inform sufficient treat scenario show start follow action noise demonstrate alone happen straight without future current system controller determine optimal accumulate minimize behavior remarkable unlike value tie operate characteristic alone goal keep go side angle great happen stop matter step behavior run trial simulation show keep stable wide condition dot successfully keep second indicate impossible velocity pointing column angle outli keep stable ht successfully initial vertical bar state long avoid deeply thus demonstrate equilibrium makes actually translate bottom increase reach position vertical bar balance would sufficient phase control reveal high action cm discuss potential applicability space explore case state probability interact challenging space make avoid sample agent agent learner x step action selector action value state would evaluate approximately simplification prediction predict adjust face exploratory assign uncertainty assign uncertainty depend gps implement observe accordingly performance step loop initialize loop observe step use prediction provide adjust nc hyperparameter action easy compute episode start action selector compute learner update train ard automatic unlike programming find agent optimize want goal transition former gps region flat whole predict high thus agent create therefore discretization transition prediction grid attractive reach explore figure base exploration various grid division cell every cost function episode act steady act optimally minimization graph thing fine optimally grid reach optimal episode grid need episode episode episode behavior somewhat bad versus consider optimize performance explicitly agent fast explore model transition visit compare visit treat plot accurately transition go gp confident mean exploratory action know agent behave plot happen first visit cell much even goal desire behavior carry behaviour show hand create fail imagine instance state reader task attain nevertheless task arbitrary nothing specify balancing seem unbiased course outcome naturally arise arbitrary select actual dynamic approximate theoretic maximization place minimal guess agent place goal seem consider thing survival scenario stay move possible related free particularly concern system property discrete display somewhat scenario optima visible dominate state maximization task require picture open question system property seem continuous environment degree globally interact environment otherwise majority pick dropping degree freedom increase box due option horizon algorithm suggest
inductive value define predictor parameter score fold current proper calibration rank example fold separate eq fold give fold popular set encode encode output odd program program default help tree default value insufficient size fold region validity cross interesting package partially foundation discuss approach fisher valid statistic denote chi cross predictor define rather analogue obvious computed fold may theory testing generate testing kind naive tb mm cross predictor ac uk ac uk inductive validity empirically method unconditional confidence inductive inductive predictor typically produce compare predictor validation explores mainly consist ask element number ask predict classification always example unknown label label object label check corresponding level inductive split part value check calibration prediction proper training produce calibration calibrate calibration obtain inductive predictor much use proper develop use calibration prediction fold inductive set combine way combine fisher produce heavily main lead study involve datum predictor predictor measurable sequence stand list training non part measurable interested example choice rule label allow method appendix logit label q letter respectively proposition obvious proposition predictor exchangeability note
minute minute minute minute always dense shrinkage small need smooth preserve ht c efficacy datum one gene denote drug variance use select high variance compute table r dim cpu sp e optimal u kkt hold condition subproblem respect update e second summing obtain eq large equality combine complete algorithm start converge lemma get monotonically non thus subsequence get note imply satisfy kkt unique thus existence limit use identity hence convergence gray condition unobserve aim optimization direction alternate multiplier exploit take thus solve problem numerical synthetic expression datum two five time algorithm latent direction theoretical convergence compute variable graphical selection covariance dimensional joint translate gaussian capture system variable dimensional rapid advance high throughput microarray imaging regime develop penalization receive neighborhood regression scheme aggregation intersection propose jointly neighborhood regression selector constrain minimization call establish decompose graphical establish frobenius rate spectral absolute normal normal norm normal usually define convention graphical set realistic scenario suppose jointly multivariate concentration sparse marginal observe matrix base hold naturally convex relaxation introduce regularize eq theory concern recovery problem determinant cg inefficient special note rewrite dual max dual lasso programming problem interior requirement high thus impractical large problem develop block descent project variant nesterov recently show propose solve graphical selection multiplier solve problem problem proximal apply method block alternate subproblem take advantage problem different type include second completeness cg significantly cpu times paper organize proximal problem alternate direction multiplier gradient alternate method draw conclusion rewrite reduce function drop implicitly function involve operator split monotone operator study due success structure convex arise compressed machine e alternate easy proximal note proximal proximal shrinkage mapping discussion follow natural solve lagrangian multipli subproblem correspond proximal mapping convergence three block error require strong consensus problem respectively describe lagrange multipli subproblem solve subproblem mapping partitioning block second subproblem multipli thus eq substitute equality get admm solve block consensus discuss literature g another alternate direction type reduce block result group new admm multipli proximal mapping subproblem couple quadratic overcome difficulty second proximal note subproblem q proximal solve follow quadratic part proximal mapping r gs k incorporate alternate vary technique low optimization alternate direction method seek completeness include global convergence group implement grouping one find alternative yield practical absolute penalty basically modify q present data method admm matlab intel ghz gb ram admm set dynamically change multiply iteration solve numerical step proximal choose test use nonzero p kp matrix function inexact admm run admm objective run achieve admm iteration objective cc c admm e e e e e admm
extension centrality centrality subsection quantify pair therefore nod probability path bag short path visit indeed indicator equal path least otherwise path find interested give start end assume reach posteriori probability posteriori pair q node network derive formula node ik k z denominator ij j ik jk appear ji ik ij ij ik ik path q substituting element simply ensure normalize result quantity put form transpose numerator read assume normalize simply matrix path normalize numerator derivation form group see quantity neighbourhood within high node belong neighbor call cluster assumption derive numerator belong element element transpose observe express numerator want classify class find algorithm unlabele weight contain contain usually inverse binary indicator index mutually exclusive c hadamard product max supervise refer multiple introduce goal classify unlabeled medium partially graph medium compute network first describe subsection experimental third set class set classifier nine set label average laplacian rl regularize normalised walk discriminative walks walk comparative report follow nine fig set result significance labeling one side determine read indicate significantly well column frequency summarize equivalent labeling rating eight mean classification seven rating consider set total always competitive range set labeling come kernel achieve well suggest labeling rate notice largely notice walk labeling label score percentage eight position since nr drop significantly label surprising remain cm total take table five set seven ng variant ng ng ng ng label class since ng ten ng ng class ng class set record run run report competitive subsection two rl less class last slow rl competitive augmentation increase strongly come contrary system equation compute graph pre cross virtual bag novel describe sum posteriori draw accord sample boltzmann gradually probability path walk set outperform label competitive time include interesting provide feature feature assess experimentally label protein protein could protein path framework thank classification path bag tackle graph nod topology label node assign boltzmann distribution possible path pick path posteriori node path receive appear path connect network invert number group constrain node classify show outperform test state benefit mining centrality kernel semi supervise automatically predefine traditional pattern amount difficult effort reduce supervised mixture learn actually semi label accuracy semi classification receive grow recent year describe input numerous link categorization protein facebook internet dna readily available gene database large unlabele supervise infer effort label work capture path assume weight direct graph cost arc consider boltzmann cost low bag pick sampling start closed inversion quantify extent naturally posteriori intermediate lie n receive path connect define class unlabele show group within framework bag framework compare performance base technique classifier path paper mainly path bag path work semi bag us work discussion finally far background notation classification bag finally introduce direct vertex class class class since membership class otherwise zero node unlabele denote transition cost contain element transition proportional immediate apart building section instead virtual bag connect include graph path path low path also walk transition loop allow exploration pick detail derive relative path graph probability short large probability pick walk towards short path short one bag probability thank author show computed product entry ij define q bag behind bag path appear word node allow paths derivation path model show interpret reward agent along path path expect sub probability base subject intensive wide approach spectral method regularization framework svm alignment regularize rl generalize show large similarity alignment kernel laplacian previous normalize laplacian seem sensitive prior rl alignment sum similarity previous method start step comparative harmonic closely framework rl structural interpretation electrical potential rely walk markov precisely stationary distribution label unlabele interest discriminative walk walk see
take derivative product variable parameterize discrete imply ica show rescale respective distribution cone affine distribute coordinate iid invertible imply iid measurable linearity independent imply result elsewhere independence ica proportional nonnegative join product everything th power rule involve fall positive strictly power interior determinant jacobian row reduce determinant nonzero support gamma compute analog density vector give joint distribution part early separate permutation permutation sign ica sample gaussian introduce ambiguity respect rotation ambiguity ball symmetric different problem solve efficiently ica show satisfy efficiently obtain many question remain come particular iid sample v simplex tu algorithm rd within distance equally near sample pick canonical equally different equally likely algorithm output true process run randomness output optimize analysis simplex v mean pi pi invoke algorithm tu theorems orientation preserve nr discussion whitening mod ica simplex et distribution format choice algorithm mistake thing isotropic subroutine engineering microsoft research science university microsoft computer science engineering conjecture acknowledgment problem give simplex simplex arithmetic polynomial answer bound fourth moment show direct connection problem randomize base representation dimensional uniformly need approximately seem intuitively clear reconstruct require theoretic consideration turn approximate theoretic running know full fact body know affine study intersection half suppose space advantage proper mean output half perhaps approach mind volume contain simplex volume estimator applie situation hard hardness directly find idea ica ica recover give requirement full independence rigorous special generalization rigorous ask convex give simple rigorous popular algorithm use fourth briefly fourth variable input moment sample distribution minima function find maxima minima ica historical remark projection pursuit mid ica problem directly among kind dependency among appear learn intersection intersection space open sufficiently g intersection half pac access membership random effort simple therein towards goal intersection fall seem sample uniform intersection one work learn intersection half space reconstruct direction moment order require recent closely apply variable latent gaussian latent local optima closely showing algorithmic technique arithmetic division root distance access randomness vector simplex run polynomial mention early idea ica moment fourth version ica moment cube simplex useful fourth moment understand moment maxima simplex get ica dependence discussion take succeed error chernoff type simplex distance additive simplex simplex distance within thus algorithm ica substantial learn ica problem ica simplex independent representation ball cone measure representation problem affine ball ica estimation obvious ica implementation ica result one put position recover simplex simplex map full invertible affine translation simplex well efficiently choice simplex hull canonical call begin invertible affine map isotropic bring sample error final result rotation leave denote rotation keep fix recover rotation rotation simplex characterization origin orthogonal normalize result consider third along orthogonal descent sphere find highly maxima isotropic sample come simplex leave closeness depend accuracy mean simplex close total subroutine find subroutine succeed probability definition total variation subroutine succeed simplify case isotropic transformation plane kind need handle successful algorithm exact rd moment arithmetic gradient rd moment simplex rd real algorithm canonical isotropic access perturbation canonical nearly rd precision perturbation canonical limit third moment sec individual vertex position subroutine sec characterize maxima third learning ball ica hull affine hyperplane convenient work simplex canonical simplex origin euclidean ball clear isotropic distance express imply two set nd tv particular trial union p denote x distribution distribute distribute cone measure cone moreover variation distribution th moment derive formula integration newton identity direct computation divide standard simplex moment copy rotation invariant approximately simplex algorithm coordinate simplex simplex algorithm note introduction algorithm also idea rule maxima use first think fast state get solve coordinate simplex simplex rotation thus independently coordinate work u tu tu obvious way moment unbiased give bit coordinate extremely fast equation subroutine version convergence fast show growth overall subroutine copy notational iteration copy rotation leave output approximation pick normalize divide subroutine rr simplex pick algorithm likely vertex equally randomness multiple sample vertex different likelihood outline iteration except inequality quite loose gradient show handle coordinate simplex analysis clearly randomly coordinate value similar argument question happen coordinate happen angle volume happen vertex drop exactly choose evaluation least end chernoff omit thus true least starting coordinate great loss get mind let get last coordinate continue factor iteration r c c omit easy detail succeed overall claim lemma subroutine algorithm use map map isotropic simplex simplex orthonormal basis everything else qr definition let except set simplex distance vertex origin affine map isotropic tx cholesky decomposition rt simplex let invoke subroutine get hyperplane length simplex simplify pick isotropic core first put case carry practical prefer select loop vertex bound choice gradient variation equally likely least simplex polynomial input simplex invertible affine apply transformation simplex output equally transform remain ambiguity rotation remove end total affine transformation isotropic easy application chernoff choice moment imply st far ambiguity precisely moment subroutine fix point like iteration successive subroutine vertex euclidean
flow fix transition enter leave equilibrium detailed balance instead directly sum allow transition equation flip form require pixel intensity case integration momentum corruption sampler image return see technique present rapid dynamic style explore slowly otherwise would due momentum rejection cause back walk slow momentum accomplish maintain exchange probability opposite reduce exchange direction momentum normalization constant partition flip f momentum momentum x fp hamiltonian hamiltonian step stepsize property preserve perform r n draw uniform additionally depend
act like metric notably learn position rather implicit position position single position strong learn explicit metric capture evaluate differ distance image database type etc contain representation position form mahalanobis method detail paper plan future use diverse new mit set grateful part grant nsf nf mind w nf ap finding author reflect view ex minus ex plus ex minus ex ex minus ex minus ex plus minus ex ex plus minus ex make plausible label date base single mahalanobi handle accuracy metric single implicitly metric accomplish random forest base incorporate pairwise representation implement test type consistently fast bad euclidean convenient underlying application retrieval face verification present angle learn random forest emphasis capability space metric per vary space achieve function limitation illustrative elaborate dominate mahalanobis metric representative brief constraint sampling collect supervise group large find appropriate satisfy positively link link point separate expensive component learn mahalanobis explore minimize constraint quadratic develop global learn although single efficient capture mahalanobis point base need store divide subset generate possibility require metric specificity transform binary forest incorporate efficient pair encode absolute demonstrate importance incorporate underlie representation several reason forest yes vote forest vote correspond positively constrain constrain number yes vote relative forest scale powerful handle specific multi independent input fast state multi third forest inherently follow thorough forest inspire several advance learn learning learn adopt remove instance geometric consider interpretation throughout metric vary adapt instance multi pairwise link constraint set I ij specific minimize flexibility f j input function train ij mapping function implicitly mahalanobis term metric j denote mahalanobis encode general position concatenation represent take enforce symmetry metric contain two map region adapt heterogeneous reason metric incorporate absolute position incorporate add alternate example raise order yield arbitrary meaningful usefulness largely include particularly difficult significantly l dataset dim dim diabetes image handwritten digits feature handwritten interested reader forest tree leaf similar forest essentially q output dimensionality compare inherently nonlinear scalable flexible mahalanobis incorporation describe adapt different word select split value sub subsequent refine emphasis localize although negative sometimes construct appear problematic necessity vary feature triangle triangle triangle extensive uci image take position uci neighbor task mahalanobis position dependent former latter rely publish author mahalanobi follow finding section overall ten range table comparable art specific nine category benchmark metric theoretic near feature include position well position result heuristic metric simply obtain medium uci see positive select generally constraint tree test use neighbor task vary vary able gain global set identify neighbor enable linear perform globally nonlinear take linearity linearity become global strong implication specific nn highly retrieval figure show plot uci without position set act difficult diabetes dimensionality forest low ten value absolute position mean metric must address learn increase forest past certain yield possibly
group extensive usefulness drawback notion recognize like level level set parameterize content great probability proportion parametrization classify commonly exp right impossible univariate phenomenon group identifiable set three several coverage orient goal cluster graphic show cluster description methodology set arise independently notice reach maximum fairly maximizer provide restrict sensible detail make empirical level paper paper level note plug excess issue aforementione reference corresponding estimator surely induce modal admit illustrate thorough agree constitute contribute cluster introduce precise population rely answer process mutually positive content population ideal clustering reflect high density begin formalize illustrative exp exp node node right node node first split cm exp node node node partition cluster tree one set methodology identify depict correspond consist reach split branch two constitute partition part refer assign obtain divide assign either lead equivalent left branch branch core tree splitting cluster observe immediate observe precisely minima equivalent consist define minus attain solid circle involve constitute color sigma sigma exp pi exp grid major sample thick gray cs axis axis axis axis idea generalize proportion mixture distribution identity intuitive black line identifiable density function precise theory branch differential manifold critical differentiable reference book useful range represent goal indicate water flow extremely smooth call degenerate critical degeneracy number negative fairly neighborhood critical minus point figure minimum saddle maximum index white color blue function false grid domain sample title view major title width view sample title suggest unstable manifold explain next define minus satisfy integral minus vector descent curve properly speak trajectory water subject unstable critical analogously whose integral curve xt firstly manifold critical manifold unstable define population unstable gradient flow maxima water flow region peak describe look index unstable manifold manifold manifold rw w clusters nan definition univariate unstable manifold local maxima manifold unstable respective point minima moreover focus flow integral curve become respect xt notion domain mode follow ascent gradient flow cluster goal distinguish concept procedure however data procedure whole population immediately objective usually interest minimize natural due involve redundancy disjoint every twice account avoid redundancy measure cluster many add partition partition interpret penalization probability mass partition cycle thick node two show differ low probability cluster denote match depend less component match indicate match achieved match match note replace analogue call clustering explore differ correspond transfer distance minimal partition alternative clustering hausdorff population hausdorff ab two nan two clustering view hausdorff distance express every hausdorff regard distance set standard theory eq demand mean partition really close hausdorff distance check previous hausdorff clear picture wise hausdorff wise minima take copy distance compute add hausdorff instead solely obviously analogue measure lead clustering indicate understand procedure induce precise datum get increase formally mode stochastic clustering sensible plug base clustering naturally arise smooth estimator domain mode candidate kernel extent cluster dimension rule easy nevertheless development scale exp exp plot exp line completely discard step solid line pointwise kernel maxima function really clustering also seem easy clustering estimator surely theoretical consideration previous help compute clustering note know connect induced density use mode surely popular class shift algorithm introduce subsequently highlight engineer every location produce convergent local initial sequence notice multiple approximately ascent since make gradient shift algorithm recognize employ normalize accelerate convergence common point density replace allow produce moreover closed maximum correspond refer convergent j harmonic component definition partition normal mixture density show kernel h nk shift convergent proceed guarantee form explore expand direction nonparametric iterative explicitly denote shift restrict thus reduce per direction research exhaustive include variant mean shift oriented robustness consider shift mean shift previous median near neighbor concept would modify obtain proposal geometric median riemannian application respect shift algorithm third research paper deal subject procedure ideal ideal population notion interested cluster accurately make induced maxima appealing extend meaning point even resort differential mapping cover book hand case key role play subgradient generalize cluster measure distance clustering discuss section aim modal rely pilot density happen method shift surely identify shift locally adjustment suggests perhaps support project de well suggestion concern thought lemma theorem remark example group despite recognize aspect relatively surely classification quite difficult try paper aim nonparametric present branch activity note rigorous methodology author concern formal partitioning problem seem
restaurant restaurant section predict high hand movie recommendation member likely something thus prediction individual recommendation provide preference procedure recommendation likely opinion certain network decision user active node already make explicit statement preference target decide recommend item member influence power modify member social activity disagreement iteratively social influential recommendation individual recommendation take friend make recommendation group opinion future validate real com rating ideal collaborative recommendation setting threshold weight area movie restaurant department electrical university recommendation considerable focus collaborative filter people preference quality recommendation model group respectively individual improve cascade recommendation influence pattern disagreement group member introduce concept rate flexible essential social collaborative filtering study collaborative cf recommendation great success serve important demand amazon netflix recommendation book movie accord preference order recommendation mostly improve recommendation base assume user social people decision datum attack challenge recommendation system social social receive less attention dataset past network provide structure social improve recommendation recommendation traditionally system recommend item individual decision activitie user go restaurant recommendation desired recommendation nontrivial extension individual unlike influence preference individual recommendation way aggregate recommendation separately often mathematical past year pre exist social preference record datum linear threshold effect collaborative provide recommendation recommendation give recommendation reflect setting list user rate preference infer netflix click rate undirected node edge friend case user information organize influential recommendation group follow conclusion future recent paper recommendation matrix propose et network item node aggregate user estimate weight recommendation base network reduce limit target immediate friend consider friend recommendation rating friend rating identical good knowledge consider influence role decision effect result collaborative filtering addition recommendation merging g intersection aggregation particular recommend movie merge result recommendation yu group preference merge individual profile total virtual profile user recommendation opinion formation influence within collaborative system reality crucial people decision amazon choose read particular mind mention book book strongly recommend friend game theory scale rating state inactive respectively influence intuitively trust communication frequency lack neighbor user proceed neighbor must become active either state tendency adopt opinion associate g set rating process operate discrete inactive fraction influential intuition activate either equation user expectation easier opposite newly activate social decide subsequent potentially like activation note assume recommend short local view social social connect payoff c cm c playing game neighbor represent receive represent play neighbor payoff node w choose inactive solution rating active odd active equation threshold valuable minor modification influential user simulate social process binary show initially probability assign influential uniform influence average threshold fig even small portion network majority active newly activate node choose show fig social fast initial influence fig show realistic set node draw various activate half become eventually stay speed simulation result disagreement expect rating newly activate social network majority opinion attribute item recommend ignore inactive recommendation immediate distant friend supplement recommendation user group recommendation differ aggregate prediction merge preference profile couple opinion consideration
experiment potential speech corpus rnn gain source information core training contain select slight disadvantage many core test reduce speech preprocesse apply audio file bank emphasis coefficient compute coefficient hamming window interval create normalised mean rate sequence percentage record rate one record put quantity convert per lstm lstm hide input give counterpart momentum noise stop low network stop lowest weight range decode result lowest aware result recurrent network nonetheless prediction label oppose million performance almost target per much would hope alternatively jointly dataset hmm language corpora acoustic model speech corpora log bit epoch pass epoch rate calculate allow analyse dependency relationship speech recognition directly intermediate heat probability lattice input network binary vector encode character low short vertical common sequence th input bar graph indicate red prediction simultaneously predict correspond prediction output sum give heat sequence show sensitivity heat previous network dependency visible across dark horizontal heat correspond sensitivity space word jacobian conclusion acoustic linguistic speech speech experiment end look wide problem particularly example speech small input label translation due complex alignment acknowledgement suggestion institute research learn transformation sequence sequence speech recognition machine translation secondary prediction text speech name challenge sequential shrink rnn powerful prove capable rnn traditionally require sequence severe alignment many even introduce probabilistic system rnn principle input speech corpus ability crucial human intelligence everything reach everything interact world action important major apply output example transform signal ability identify despite apparent create speak language lexical rnns architecture general expressive conventional particular rnn well store early rnn recent capable state task recognition range perform character delay handwritten character rnn problem advance example rnn frame signal protein output length sequence purpose sequence output length advance prefer distribution output align ideally cover rnn layer long text speech sequence length jointly model similarity graph conditional rnns dependency mark potential close paradigm module often network transformation segmentation recognition define present speech conclude remark give belong input value would receive add omit previous standard add hidden layer feed layer rnns vector whole input rnns extent rnn compute hide tf tt iterating implement network input normalise simplify determine lattice possible correspond therefore similar k lattice output horizontal represent element black bottom node one start product pass red path calculation backward variable recursively initial condition define backward product lattice output sequence forward variable backward speech recognition forward image bottom map variable bottom lattice text sequence way log respect perform diffusion lattice u top q network
independent freedom indicate put everything ec coefficient arrive look cone side functional presence unknown analysis fmri subject receive neutral heat stimulus right rest repeat fmri model stimulus delay second heat fmri region stimulus stimulus delay shift taylor convolution stimulus stimulus regressor eq ingredient structure nature regressor shift plausible restriction specify coefficient illustrate result observation independent dependence iid fields course cone alternative angle intrinsic volume middle appropriately normalize course correlate add regressor neutral heat stimulus cubic scan leave effectively result whiten regressor nuisance calculate depend remarkably brain fix field isotropic volume term make fortunately estimator residual difference lattice take voxel inside course radius ready get p statistic large cone statistic almost identical cone note increase cone getting threshold detect heat stimulus statistic delay activation question powerful test range outside statistic statistic volume statistic cone increase indistinguishable twice f twice statistic volume intrinsic usual half surface area dimensional intrinsic simple intrinsic come formula concave corner lebesgue lebesgue ball boundary curvature volume hausdorff surface sphere intrinsic lebesgue gauss half zero part nsf dms pass field statistic fmri datum delay find euler ec test statistic result ec density formula class begin appear david work together intersection interest david smooth smooth field application last pass cancer author pass david main approach problem formula change preferred euler ec generalization ec volume formula ec densitie back david change concern maxima field field variety test statistic cone package fmri give use ec formula conclude subsection relate fmri purpose define gaussian unknown must mind generality testing unknown shall way likelihood ratio variance maximum cone case normalize span cone field mean embed onto span sided cone statistic field powerful random field outside cone side statistic acceptance concave dominate cone advantage admissible specific always powerful reason would denominator conservative strategy degree cone fmri shall likelihood ratio test pass principle point whole would standard reproducing argument function center statistic region thresholding suitably threshold choose threshold detect control outside something field degree freedom representation locally picture non arbitrary scalar linearly subspace actually compute field solve direct regression fit negative amongst fit fit alternatively separable warm location negative application geometric project could generic span event non negativity restriction satisfied achieve actually face surface region depend dimensionality q probability fit eq correspond depend large effectively generally ignore later ec interpretation limit approximation realization see inf field gray medium patch scalar patch freedom value curve define light field freedom term correlation field theory ec subscript ec paper devote evaluate ever wide field space simple next get ec build gaussian gaussian transformation domain field build discover take volume formula somewhat power ec seek radius rejection denote long omit far generality ec random rejection region ec field field union rejection union arrive ec show ec ec ec lead ec respectively part evaluate maximum generate intersection circular empty ec term intrinsic volume fail convex right ec represent dependence random field ec field ec sketch q face determine negative empty ec determine differentiable define ec differentiable though contain interior critical critical surely critical theoretic computation ec show ec actually sum subset ec field complicated replace decomposition argument complicate prefer complete counting result incorporate work ec elementary geometry remainder reflect origin fail content fail ec ec density field freedom u rejection red rejection region radius rejection cut ec density seek random lead expand appear sided f yield follow expression ec follow ec ec easier closely resemble tackle
segment parameter except hyperparameter vector ensure transition process finite crucially tie hdp limit tie via connection state encourage visit state elimination self generative similar process illustrate self transition define affect global hdp factorial factorial factorial rich duration greatly improve demonstrate outline factorial factorial hdp component eq factorial factorial factorial hdp replace chain semi evolve combination state factorial identifying bayesian uncertainty ambiguity demonstrate use factorial hdp home separation ill condition solve separation uncertainty motivate modeling duration two gibb begin hdp develop limit limit sampler hmm collapse hdp conjugacy hdp develop auxiliary augment recover conjugacy gibbs limit assignment trade integrate transition simplify duration however must sampler hdp resample pass discussion leverage side greatly accelerate perform inference construct duration follow iteratively appropriate respective easily reduce depend prior sample dirichlet diagonal tie resample require care see section introduce independent block distribution employ pass block state must duration message describe begins efficiently sample state conditioning initial draw label sequence limit hdp hmm construct hdp transition motivate fact infinite hdp encourage practically limit transition also represent greatly accelerate circumstance parameter control guarantee grow convenient approximation stick represent probability construct atom limit hdp greatly accelerate sequel long tie share introduce conjugacy hierarchical difficulty clean technique avoid priori sampling technique priori overall easily self link long conjugate resample necessarily easy conjugacy weak summarize generative component deterministic transition transition state extra transition proceed conjugacy general datum augmentation technique describe extend simplify us cycle produce focus namely depend extend row drop parameter convenience write portion generative state represent transition represent transition depict show auxiliary variable conjugacy iterate compare simplified consideration sampler compare sampler asymptotic unity augmentation auxiliary conjugate weak well numerical tp da insufficient marginalization estimating model mention direct assignment hdp hmm transition analytically observation parameter prior represent explicit sequence use simplicity additionally recover hdp conjugacy requirement correctness auxiliary finite immediately infinite dirichlet hdp hmm finite basic hdp hmm da sampling count portion predictive write exchangeability joint likelihood hdp hmm distinction new transition segment case hdp writing segment entire contiguous term segment may merge depend adjacent segment use first score correspond duration duration duration segment thorough affect predictive merged allow must deal conjugacy issue assignment sampler rather segment associate datum serve preserve conjugacy hdp via resample use I I simply self count self self super auxiliary variable way integrate entry auxiliary hdp hmm describe circumstance super may switch many obvious divide though super sampler bottleneck pass step asymptotic dramatically message similarly modify sampling element sequence block label adjacent hdp assignment limit well hdp assignment hdp assignment hdp weak synthetic hmms number hdp unable geometric demonstrate duration distribution geometric hdp hmm little factorial whole home device encode simple device advantageous side experiment matlab compare hdp hdp weak sampler sampler hdp hdp direct apply datum hmm limit direct weak sampler much execute great believe superior whenever choose priori geometric hdp direct assignment hdp hmm compare hdp weak sampler median dash percentile summarize hdp datum dimensional run hmm unable capture non markovian duration statistic state hdp equip poisson duration transition emission reduce hdp concentrate hdp hdp hdp hmm duration distribution geometric distribution one class negative denote analog gamma cover geometric place non well mode hdp hmm priors informative concentrated duration hdp hmm hdp hmm hdp application factorial unsupervise power signal device aggregate whole home consumption efficiency detailed power device significantly solve problem monitor application demonstrate duration speedup rich history variety specification factorial em distinct state mode device sampling model explore additional build factorial dataset frequency period top drawing device identify hour device least sequence rough level duration statistic device home hdp negative binomial power usually hdp accordingly specification duration parameter twice device device perform inference independently factorial hmm bias hmm present device separately signal identifiable jump term device reduce potential able state resample magnitude tp greatly accelerate true device consumption report emission mean run machine core detect second resample duration collect result summarize device fit device parameter incorporation duration hdp detailed comparison finally figure consumption select factorial factorial hdp c house em hmm hdp tp tp tp nonparametric modeling consumption mode user home level priori device provide strong providing duration hdp weak direct assignment sampler duration bayesian nonparametric allow learn markov demonstrate algorithm upon provide tool sequential simple hand free encode device prior mode use mode near specify hyperparameter uniformly parameter setting factorial table observation gaussian denote fix gaussian similarly latent success draw geometric factorial hdp hmm prior parameter encode duration prior extension elaborate hyperparameter experiment emphasize hdp device measure interest hierarchical hdp nonparametric hdp hmm strict undesirable wish encode hdp structure draw parametric set interpretable admit natural introduce explicit dirichlet hide also provide modular gibbs sampler hierarchical tool toolbox demonstrate utility inference synthetic semi markov set unsupervised aim distinguish single audio wish infer characteristic home individual device would knowledge device power sequential build flexible expressive encode appropriate prove datum real state priori infer data issue hmm hdp hmm infer hdp disadvantage markovian lead unnecessary rapid one avoid rapid switching hdp hmm introduce self improvement hdp hmm geometric duration structure duration induce markovian difficult construct sample find efficient important limitation improvement hdp hmm investigation explicit parametric idea hdp allow duration sampling hdp approach hmms difficulty result mix applicability model synthetic remainder duration message build bayesian brief hdp hmm assignment sampler hdp hmm improve demonstrate effectiveness hdp demonstrate quickly hmms unable necessarily require operation chain pass however duration length message express message often duration cause message increase message pass offset sampler experiment hdp treatment employ
recalling explicitly write calculate low gradient paper sgd cg task method gradient element md independence complex jj jj jj cholesky jj notice e mm jj equality jensen yield form optimal observe jj notice process exactly bind goal calculate distribution key prediction predictive gp coupling next calculate via need form fact separately obtain make eq evaluate md yielding calculate calculate j I calculate effect admit gp task sample share gps add variation inference gp hyper parameter model process mutually generative process perform variational previous hidden membership auxiliary induce effect nk j k j jj jj jj learn hyper marginal likelihood low complete data jk represent hidden make simplify derivation variational f q jk jj variational optimize variational em algorithm estimate derivative k variational rewrite thus set zero use obtain complete estimate variational find induce inside extract log term logarithm multivariate integral jk interest follow conditional eq form single write bind calculate derivative optimize base easy complexity key derive kk identity kk km therefore next calculate first calculate sequence operation calculated implementation stochastic coordinate perform coordinate g q index experimental effect white recognize gp combination gps effect newly input task kronecker implementation criterion standardize differ low use sample optimize uniformly task advance maximize mt sd sd single discard sample mt pp hyper parameter mt pp pp experiment kernel inducing obtain optimize likelihood scale equally spaced share whenever center iteration support control three also use gp finally hyperparameter specify initialize spaced randomly assign gradient small subset task time maximum entire experiment algorithm task artificial task interval function eq individual interval sample space draw realization prior function show top qualitative initial variable induce variable clear predictive showing show input recover indicate sufficient time plus propose intel core pc reconstruct profile developed case frequently among task realistic generate explicitly reconstruct profile commonly analyze denote action dirac delta four task previous multivariate normal real gaussian profile interval uniformly testing deal efficiently multi center center arbitrarily number center base significantly outperform method develop classify star star constitute densely sample star sparse effect task star shape inference approach equally dimensionality full h star normalize universal slide center shape variational mixed support evaluation demonstrate interesting online investigate induce feasible make partly nsf paper perform computer usa gp main solution avoid instead desirable group unknown individual problem idea gaussian handle datum validate outperform method outperform art sparse solution multi task many quite task aim predictive advantage successfully area medical diagnosis recommendation screening share parameter multi focus shared represent deviation share hyper portion difficulty number cubic improvement remain example case cardinality approach call induce variable sparse solution multi functional mixed effect variational efficiently hyper sparse sample approximation different point real datum validate approach show full sample gp first first perform organize mixed gp inference gp group section demonstrate use conclude summary mixed grouping model extend grouping give j aim take data effect perform mutually assumption kronecker function concatenation similarly see new extract sharing task lead inference scale though
random walk outcome coin valuable theorem claim coin give relate well use adaptive possible action also outcome first strategy bayesian employ markov problem application bioinformatics medical arm choose receive reward choose rich motivation identify bandit minimize number step regret choose reward arm exploitation exploration perform know address identify good arm bandit fix propose strategy focus reward show bind budget subject quality action address sample pure maximum expect minimizing need pac formulation introduce al et show total identify arm whose reward away arm correctness reward attempt address theoretic outcome exist minimize whose least pac appear theory decade field research decade introduce normally reward sequential variant application achieve arm special know strategy various relaxation independence normality know variance distribution address particular reward arm bernoulli coin adaptively rest chernoff trivial adaptive coin coin empirical coin bias coin yet strategy choose particular coin coin optimality employ tool number coin provably optimal coin probability heavy probability heavy necessarily identify coin heavy goal expect coin history outcome coin outcome coin coin whose strategy say main optimal employ adaptive heavy coin optimal infinite value fix least need general set probability bayesian exploit stage outcome coin p b optimality game game system cost start state cost system token token say pay state system terminate token denote token minimum cost say pure choice token entirely pure playing markov state game incur associated state reach state small play game state system unique pick token minimal straightforward infinite system system satisfy determine walk correctness coin straightforward ratio coin coin coin history coin log likelihood coin begin history likelihood identically zero outcome head outcome coin log coin coin great barrier since coin heavy boundary coin outcome system step equivalent system reach strategy associated dimensional log state real state state random oppose random uniform transition probability locally state coin heavy heavy decision conditioning log consider likelihood pure strategy strategy infinite choose child reach since show mixed label edge onto root leaf let child node adjacent pr xu pr maintain initialize root leaf choose play coin move system coin generate move yu pr yu yu bl xu bs gx use consider leaf leaf node node reach leaf leave incur r relate expect associate non leaf leaf node observe node non leaf leaf node pr xu let incur incur incur root node expect process mr xu g likelihood also coin coin markov random walk
sample exploit number edge induce sample b fundamentally state technique exploit sec accept rw challenge consecutive rw several rw stand technique rw problem issue estimator available sec simulation topology facebook require facebook state paper sample node l art exploit among node exploit edge sample repetition undirecte connect estimate calculate eq every neighbor depend distinguish random replacement collecting would trivial know alternatively rejection currently facebook impractical unless exploit proportional node sample presentation unweighte long collect refer technique approach prominent follow classic population split equal discard repetition discard valuable limit suited population specie biology node separate specie review page likelihood derive define number equation index across entire elegant extension consider node weight I give reduce outperform eq various depend special service broadly applicable ii service iii less prominent technique inefficient often layer widely interpret rough estimator take estimate device area fundamentally describe fig sample knowledge sampling node rarely study illustrate edge count every occurrence probability average facebook basic accept node extend rw graph density interpret choose form node assign weight divide term correct correction finally plug arbitrary possibly repetition replacement cross node consequently reality resemble except repetition lead derivation independently happen employ relatively additional cost list clearly consequently remain discard estimator accept arbitrary node explain category node make discard arbitrary simulation latter perform well especially skewed reason explicitly note discard vs consistently require correction edge rarely correction reason henceforth technique node www connect acyclic rw probability proportional demonstrate indeed arbitrarily say impractical fed directly rw sample poorly consecutive rw consist rw close rw increase rw reduction literature efficient l nn analogous rw take subsample fed sec find easily observe rather different exploit create problematic aggregate denominator avoid perform consistently henceforth yet sample main modify additional transformation consequently easy exclude lie within omit brevity implement note naturally strategy pair come walk result count node neighbor sample consequently number pair make contrast rw dependence sec sec pair node drop node markov rw sample typical happen hundred reduction straightforward implementation estimator lead complexity become sample million facebook node implementation fortunately sum thing especially auxiliary dedicate guarantee rw life topology facebook concrete rw eq eq rw topology ground truth topology come million summarize size grow example estimator sampling graph g email pa efficiency improve topology weight turn give already however primarily berkeley facebook k link k k email k amazon k google pa average highly topology rw gray dark gray perform grey region percentile percentile dotted life fully topology sample corner rw reduction technique medium grey dark grey grey percentile dotted line art rw ten large topology rw topology fix vary parameter margin weak rw systematic topology range acceptable variance effect many well say nature occur interpret contrast regime margin yield value much interpret exception topology rw fail probably lattice diameter consequently rw possibly sample absence rw date performance rw ten long regime rw close fig reveal rw highly node facebook entire topology us sample period rw rw cover user light gray dark gray grey cover percentile median set dot top facebook return facebook correspond translate rarely soon facebook exist continue remain fig rw facebook column estimate impossible contrast apply rw fig concentrated attempt compare entire rw represent grey rw dark grey improve interpret facebook contrast latter rw
q represent approximate value gene equivalence microarray conduct university aim investigate express day equivalence express day priori gene colour long treat ratio gene mean hyperparameter joint appendix equivalence case study approach interest equivalently day log mean distribution flexibility motivate histogram ratio day shape tail compare degree possibility distribution approximate normal therefore eq true mean log gene rearrange give calculation constant adaptation use equivalence region hyperparameter rather approach nine posterior employ em hyperparameter study behaviour observe express variance log value hyperparameter table ratio contain within lie outside increase prior demonstrate equivalence equivalence ratio asymmetric htbp scatter separate hyperparameter appendix htbp nucleotide nm nucleotide gene validate purpose increase equivalence adapt estimate note testing beyond scope conduct assume prior considerably mixture hyperparameter use em algorithm apply full observe unobserved complete calculated substituting algorithm ij give step replace numerical optimisation find I search successive interpolation interval choose hyperparameter proportion em mean consist small observation integer sample sample pair function initial component mixture proportion often method recommend choice algorithm iterate em algorithm iterate estimate achieve value normal dominate normal run algorithm limit initial produce log algorithm compare calculated order show consistent log initial maximum consecutive final monotonicity number b symmetry sufficient case let weighted observe average suppose decrease p odd school sciences sa edu school mathematical sa edu school sciences sa edu testing study define notion strength equivalence wide adjust evidence nan expression formally satisfy consistency requirement strength mean widely gene positive discovery inclusion translate equivalence posterior basic consistency credible equivalence propose analyse experiment keyword false discovery microarray equivalence assess treatment comparable pre interest potentially side treatment costly produce trial level statistical prescribe within provide sound application equivalence microarray bioinformatic study illustrate biological establish expression rank gene profile many profile gene remain part play central role system include disease however population ensure desirable establish gene naturally occur cell baseline rather find evidence appropriate identification stable future variety study population result aim scientific test gene test goal bioinformatics exploratory know rank inferential equivalence mistake test hypothesis hypothesis false dramatically increase adjust value gene discovery discovery adjustment applied incorrectly discover significant controlling example control family alternative derivation increase calculate equivalence describe probability basic strength rank motivate cell microarray conduct equivalently propose ratio equivalence employ algorithm hyperparameter finish brief conclusion credible evidence false outcome perform accord accept nan nan total alternative true suppose hypothesis statistic nest denote return consider unbiased statistic large evidence increase away statistic decrease value eq hypothesis u u se se se z se alternative derivation equivalence pair base significance possible rejection region observe associate recover strength evidence page demonstrate illustrated plot increase decrease equivalent observed confident value margin false fact lack monotonicity increase equal assign incorrect equivalence value highlight
estimation cone sufficiently proximal iterative shrinkage thresholding iterate satisfy contraction denote convergence importantly outside framework estimation either testing framework reader theoretical describe formulate section proof constitute primary mathematical concluding remark excellent exist modern numerous design broadly literature primal method directly primal dual problem optimize block penalize regression solve coordinate reduce lasso regularize iteratively solver graphical lasso implement unknown variant nesterov interest one stem convergence tolerance dual max study similar show sublinear block approach primal take block oppose decrease rate primal hence second locally around rich body numerical provide shrinkage algorithm technique gx hilbert associate differentiable convex semi necessarily often lipschitz gradient low convex operator example fx choice size iterate fx fx optimal say converge sublinear fy within satisfie p I approximate power discuss iterate minimum satisfy fix constant contraction give convex objective prove section constant follow directly lie specify iterate explicitly close convergence property rate I contraction closely relate condition soon smooth become effectively newton advantage bound solution global bring literature advantage element section compare time implement run intel gb ram zero simulate sparse somewhat finally matrix way definite condition percentage nonzero derive show heavily regularization generally numerical optimum duality result sparsity error convergence demonstrate dependence algorithm duality investigate publicly implementation record present far present three processor time time times processor important cholesky advantage dimensional cholesky cpu cpu apply gradient widely algorithm competitive example section competitive condition competitive many national science grant dms national dms dms grant department energy office thm remark statistics stanford stanford stanford stanford tx stanford stanford great community produce sparse inverse estimator gradient method present although numerous proximal attractive theoretical convergence reach tolerance give closely optimal convergence comparison
recovery edge phase transition correct nearly transition newton synthetic blue variable green dark blue plot recovery maximum census consist discrete year state status education level state state com supplement population evaluation set study variable size increasingly need sensible baseline compare multiclass regression condition rest evaluate performance evaluate regression multiclass logistic well objective similar separate regression since figure separate regression outperform benefit test conditional marginal regression education train use figure outperform generative small less size generative outperform conditional estimate computable originally evaluation find train evaluation model well outperform although suggest see specify synthetic acknowledgement helpful lee department engineering fellowship program science research fellowship stanford fellowship dms national health verify jointly check fu j tc sx jx tu compose fu convex iff establish cv cv complement iff inequality entry mrf large model exact difficulty mle gradient difference sufficient discrete computationally involve matrix inversion continuous difficulty method model method propagation reweighte belief surrogate case state discrete estimate inversion joint primary derivative expect use show efficient develop mrf independence q last entire py j indicator let p z st b w theorem conjecture axiom consider model continuous pairwise continuous amenable structure natural generalization line novel undirected regularizer sparsity take path learn value however population click gene expression gender consider continuous previous assume via graphical lasso several efficient find pl discrete markov field py derivative focus connect discrete previously well understand multiclass markov gaussian lead natural generalize graphical lasso block different calibrate fairly discuss conditional mixed view mixed response discuss discuss approach section discuss section calibrate discuss consistency discuss census population dataset synthetic pairwise model pairwise random discrete variable parametrize j rl important regression multiclass logistic regression simplify usual pairwise case critical absence conditionally read iff coefficient capture desirable property distribution property represent continuous eq discrete variable mix discrete pairwise reduce familiar multivariate parametrize discrete reduce pairwise second markov field eq although gaussian differ parameter standard calculation see variable gaussian common mean depend depend know mixed condition model pairwise simplify although exponentially vector homogeneous mixed allow make alternative limit via independence however marginally undirecte decomposable model regression applicable know primary learn parameter exact likelihood decomposable regression knowledge optimization procedure mix maximum obstacle since include special case difficult case well maximum product distribution take familiar variable state multinomial multiclass regression contribute effect take log twice negative appendix separate separate think asymmetric share conditional double expect outperform wise straightforward predict regression confirm separate regression outperform setting parameter dimension even dimension would eq compute factorize q weight uv section edge truth use extended estimator straightforward consistent regularizer adapt calibrate regularizer vector estimate estimate q likelihood edge type difficult result parameter identifiable popular fix issue drop identifiable asymmetric formulation treat differently q group regularizer optimization equation present consistency active inactive inactive orthogonal convexity q lipschitz imply convexity convenience exist convexity ensure uniquely estimate condition active dependent inactive form partition differentiable hessian lipschitz radius suppose give sample select large applie estimator maximum constant true thus determined theorem less useful proximal proximal convex penalty block coordinate descent coordinate close many smooth solve appropriate routine suit proximal accelerate proximal proximal compute consider familiar group overlap proximal simplify soft group soft thresholding accelerate proximal gradient work solve current iterate proximal prox determine line search theoretical property proximal accelerate proximal rate package allow proximal algorithm less proximal next newton nd analog incorporate minimize quadratic center subproblem
extension crp ibp dirichlet dependent crp partition restaurant process necessarily interpretation endowed notion dependent nonparametric exchangeable distribution support atom share across probability share base share atom base result hdp model point share variation population covariate nonparametric process survey number variational method model replace hierarchy variable conjunction dependent base collection atomic eq q simple assume share vary process stochastic make spatial replace create mixture extend incorporate interpret straightforward find beta main random induce dependency markov generalize employ vary across probability construct stick break wherein biased order atom repeatedly break random stick recover dirichlet commonly ibp dependent prior stick break marginal might obtain whose function construction elegant inference go way application related breaking process define measure bound recover stick break prior vary stick break process accordingly vary maintain dirichlet arbitrary result marginally non much easy perform inference hierarchical break appropriate structured jk matrix countable weight construct permutation row matrix process involve break follow construct random atom location permutation g v associate location distance share space covariate variable whose neighborhood e method stick break construction limit dependency multiple dp biased stick ordering size atom tend permutation distance maximum size covariate location tend large use class discuss induce dependency mixing assume I induce dependency covariate restaurant covariate marginally spatial dirichlet extension spatial replace multinomial correlate spatial field field mixture spatial process aim ensure value marginally dirichlet mixture collection field unit yx kx ix surfaces component enforce cluster explore break surface dirichlet break surface select marginally arbitrary stick break similar image author truncate technique employ ibp ibp element jointly element marginal recall conjugacy exist case analytically approach induce predictive induce n observation different exchangeable hierarchical hierarchical model document exchangeable arrive batch create observation crp maintain crp extend approach covariate covariate unlike elsewhere survey share model covariate lack generalize construct leverage combinatorial subsampling specifically assignment customer z leave restaurant scheme proportional say stay leave remain atom g describe sample assignment th crp customer step restaurant restaurant provide recurrent chinese restaurant special customer leave end atom step atom transition measure decay determine alternative crp say customer customer customer interpretation crp define table ie sequence customer distance crp utilize dissimilarity satisfy triangle customer monotonically decay customer assignment interpretation crp concentration customer person function window describe distance customer customer dynamic cluster predictive ibp create covariate customer customer dependent ibp first customer draw say analogously choose occur create dx dependent map poisson evy measure homogeneous obtain rather process distribution use customer covariate try probability popularity dependent segmentation interpolation model evy evy gamma gamma atom covariate contribute measure share atom close vice versa atom appear process window gamma placing allow recover obtain subsample measurable complete atom active center covariate covariate prohibitive create poisson carefully rate vary construct vary kernel kernel l utilize stochastic order use integral perform algorithm utilize metropolis hasting particle filtering infinite straight collection straight line plane pass set proportional set process intersection clearly intersection process marginally process correlation map induce collection poisson poisson covariate origin unit representation describe alternative look dirichlet process describe section covariate covariate restrict large covariate algebra disjoint algebra subset gamma process unnormalize gamma weight gamma gamma process weight normalize bivariate measure dirichlet autoregressive measure step distribute innovation measure marginally distribute marginal distribution depend dynamic hierarchical extend group hdp apply single covariate bayesian combination measure plus innovation eq approach nonparametric traditionally form arbitrary dependent focus dependent base process thorough covariate early community nonparametric process covariate dependent model dependent apply processing interpolation denoise provide goal remove pixel model common pre processing patch frequently patch treat bayesian vector stationary typically accord beta exchangeable well sharing achieve use achieve superior denoise house task sort achieve break achieve segmentation natural analyze song highly time evolution music music allow transition hmm segmentation obviously local correlation suggest use dependent hmm correlation topic model exchangeable collection specific distribution word text document latent finite hdp allow nonparametric topic corpus evolve news volume document exchangeable expect topic time dependent dirichlet create example hdp dp hdp turn use document since active topic appear topic popularity multiple addition vary vocabulary allow evolve recurrent chinese restaurant manner model apply finance volatility price measure volatility versa overview choice stochastic volatility simple return determine positive variance report indicate model fit length phenomena snapshot range nonparametric employ statistic community paper highlight measure hope develop well application hand similarity easier efficient history dependent relatively short modern field drive mostly machine learn community range application dependent analysis also variational introduce often less many meet know marginal certainly put restrictive marginal well rely covariate well sample question still applicable still play loose often lead start emphasis inference dependence appropriate restrictive rigorous analysis nonparametric community develop underlie machine provide development process complementary interest statistic machine dependent extend give collection typically space assumption nonparametric formalize statistic model understand help develop leverage recently paper develop dependent prior bayesian snapshot currently model traditional nonparametric chinese restaurant exchangeable exchangeability series proximal set may exchangeable relatively exchangeability arguably go seminal formally dependent nonparametric process exist prior prior collection member collection similar report nonparametric process range fall sometimes relatively group collection atom dependency introduce stick break locally sequence induce dependency collection locally exchangeable modify exploit collection model difference consider underlie related adapt numerous application point similarity hope aid understand current development new representation first nonparametric particular survey early construction grow rapidly focus process particular dependent stochastic process describe nonparametric prior reader reading model fit specification method satisfy conclusion body challenge currently discuss survey base nonparametric review prior random random measure notation convenience denote unnormalize arbitrary covariate covariate observe random covariate index notation extensively observe spaced observation draw variable useful representation process product let evy evy evy measure correspondence simulate simulate alternative often jump evy poisson
propose deal artificial characteristic movie make recommendation movie technique system improve across deal start rating apply induction hybrid idea behind many recent work exploit history trait social network yu wikipedia article ingredient know advantage content factorization hybrid content achieve improve produce recommendation insight otherwise interpretable recommendation ever desirable act recommendation big netflix diverse distinct even netflix content enhance ensemble proceed review collaborative content describe evaluate useful mention end brief summary review rating user rating form user rating principle indicate certain range indicate level rate highly sparse recall eq rating user item recommender predict would define rate despite often capture fair simple overall user effect obvious please common factorization experiment factorization algorithm factorization use predict factorization use know decompose another user rate user item simply imagine item item nearby long component mathematically factorization frobenius prevent turning view come likelihood penalty come spherical gaussian posteriori estimate bayes computational burden rather substantial result term improvement convenient term subscript bl stand baseline purpose bl convex solve descent move keep vice versa initialize small iteratively later cf factorization outline value netflix review speak perhaps factorization mathematic problem let p p factorization outline confirm netflix work orthogonality degeneracy question avoid practice elaborate matrix negative nmf negativity constraint nmf image reveal recent nmf though structure primary orthogonality svd g nmf sound mathematically somewhat ill focus remain community partly year long netflix stack attribute indicate whether item possess way incorporate slightly extra penalty selective regression constraint share call intuitive notion closeness model mathematically feature preference alignment problem attribute express solve size make alignment penalty size subscript ab biased idea descent apply alignment shrink centroid share page play central role introduce smoothed respective centroid square generalization alignment penalty proportional relation unity alignment smooth monotonic function ht u I become allow share attribute contribute share attribute attribute much enhance create tag exploit tag follow tag baseline optimization user history content apply item content information subscript stand original idea lead mathematical consist particular front multiply alignment respect become selective clearly reveal front centroid shrinkage measure attribute cosine intuitive amount interpretable incorporate information store attribute feature method group conference behave coefficient latent feature attribute feature vector correspond problem become replace regression technique fit sort site latent coordinate site similar likewise specie environment cca latent linear actual environmental site become extremely technique environmental content mf bl ab section table briefly summarize compare ht summary bl ab eq one early factor use allow mind bl recall sum multiply everything calibrate extra include rate least user movie set ht movie attribute lp lower integer indicator whether contain ingredient movie different serve learn rating p truncation movie mae cf mae discrete rating literature increasingly mae dominate netflix share attribute share ab share attribute generally regard ab overall metric mae rmse reasonable fairly opinion think strategy alignment activate certain sensible affect hand item certainly toward little sensible nature result result sensible smoothing behave like ab whereas eliminate alignment main fine provide descent hence initialization reasonably follow first miss regular svd p enough degeneracy somewhat ill pose explicit orthogonality prediction miss remove prediction take initialization would applicable bl extra natural invertible diagonal element practically pose subtle comparison force relative disadvantage explanation fair mixed svd sample initialize bl ab initial yield approximately algorithm geometric explanation initialization force disadvantage good factorization initialization ensure factorization way scale quantity remain fair use list increase contain give large movie e table bl measure could cc reach percent respective pre descent algorithm keep ensure practical finish large differ significantly relatively allow objective iteration figure summarize use content ab bl behind shrinkage error hold set data validation triangle emphasize overall fair section break vertical axis explicitly biased ab recommendation show dimension reason recommendation present recommender distinct e g category plot bl ab use two solution distortion together alignment bias key likewise movie close say child movie three produce alignment bias ab vector item lead bl alignment constrain similarity simple co occurrence merely drive content range similar highly result dimensional one calculate use cosine away meaningful pair example easy people like likewise like movie child favor action movie tell like like insight insight suggest familiar ht vector ingredient ingredient cosine black movie p cosine child action like briefly cf fill miss keep matrix behind minimization behind believe drive factor therefore rating differ explicit zero compress np recovery norm nuclear norm nuclear remarkably restrict rip rip involve minimization norm minimization reason attention limitation example maintain complete sdp matrix completion approach apply although devoted importantly mathematical bring
column ill condition unfold say column factor merge unfold helpful merge matrix highly condition one possible thereby mode reduce matrix method ordinary unfold tensor mode mode merge correctly estimate rao thank special rao product structure column cp high tensor assume knowledge suffice separability linear mixture bss unfold relationship bss respectively cp decomposition bss overcome little general incorporate transpose tensor lie estimation fundamental generally impossible additional corollary likely column rao factor thereby may tensor idea cp bss estimate unique may flexible cp corollary unconstrained ambiguity orthogonal bss employ ambiguity incorporate mild mode reduce perform rao product original rd tensor simple uniqueness mild section interested order tensor rd tensor paper mode involve unique detailed cp unfold via rao projection consider simple frequently mode local solution rd alternate apply order full rank correctly consider factor say column purpose keep unchanged matrix run tn efficient scale mode reduce one large unchanged correctly eq rank always column rank practice efficiency reduce want reduce row reduction technique employ consist lead let lead sometimes maintain essentially adjust sign matter related discussion row equivalent may svd multilinear worth method rd dimensionality often efficiency perform rao formulate matrix product specify apply sequentially original least eq example procedure jk optimal rank apply implementation mode unfold equivalent hence singular uniquely impose follow projection operation constraint element nonnegative repeat alternatively convergence analogy power derive straightforwardly tensor impose repeat achieve actually tensor involve operation extension repeat tensor exploit nature mode tensor convert one reduction algebraic proposition unique theoretically essentially corresponding solution unfold relationship unique reduce order thereby unique word uniqueness condition simplify suggest uniqueness rd important uniqueness point essentially lead important issue investigate uniqueness tensor extend th uniqueness th order unique rank simplicity uniqueness hereafter move without generality hereafter transpose change tensor likely generally increase hand uniqueness order unique th consequently also unique assumption proposition tensor corollary proposition let way th rd maintain assume factor tensor unique word mode possible show ill condition rank factor matrix mode rank way uniqueness relaxed uniqueness propose simplify may feature method may interested play task etc consider extract rao product actually th order consistent order uniqueness reduce mode high theoretically unique exact compression common approximate practice performance provide q obtain rank consequently able give essentially truncate cp proposition fortunately rao product rao conclude actually almost direct hold optimal use initialization different tensor unfold rd etc pi interference unit proper synthetic noiseless propose matlab intel cpu gb windows feature boundary add tensor combine cp include toolbox matlab set construct perform method perform way finally recover use monte run evaluate solution plot run minima give mode analyze unfold important method influence configuration randomly select tensor unfolding bad perform hierarchical alternate constraint replace algorithm perform achieve way final tensor meanwhile improper tensor result global limitation unfold achieve cp cp simulation improve feature technique normal component shift frequency jt configuration neighboring setting performance run fit correctly actually uniqueness monte c c cp real analysis image object angle simplicity object image scale perform influence center achieve cluster typical rao final see method slightly bad mode rao satisfactory able proposition c runtime accuracy iteration bottleneck paper reduction tensor efficiently incorporate unfold respect method overcome bottleneck cause component full loss structural essential uniqueness mode reduce tensor theoretically investigate confirm algebraic rao role tensor intrinsic one determine choice phenomenon develop future statement show statement show also regard note prove p procedure proof factor merge essential uniqueness lemma jt jt author thank anonymous suggestion original manuscript receive ph south china china laboratory advanced brain signal brain institute interest processing processing sc ph dr degrees electrical university institute electrical engineering system electrical technology title spend several electrical engineering von laboratory brain head advanced science technical translate chinese china china ph china technology china laboratory university two scientific paper conference research interest automatic control blind accelerate cp widely tensor square tensor frequently performance bottleneck overcome realize rao unfold mode promise bottleneck high mild convert rd theoretically order tensor essentially analyze presence show easily cp decomposition decomposition alternate tensor gain importance representation practical attempt find informative sparse tensor attractive account latent name extensively study decade blind separation unique mild useful factor cp matrix another form rao product remain specific
preprocesse extraction classification boundary segmentation knn ann classifiers improve state art classification six various promising recognition toy object realistic new progress aim experience recognition action video recognition intra class clutter object static problem handle bag feature support remain generalize experience spatio spatio bag weak approach outperform realistic show eight class include water read automatic automatically label dataset author location human event system section two foreground section solution define detect particular wide classification video condition main location compose four phase follow segment stream video capture candidate key separate background image video define description foreground foreground event result detail characteristic phase segment detect actual camera frame correlation shot sequence frame camera frame shoot detect similarity previously mention two depend camera movement instant movie instant video digital mobile camera shot video capture frame video convert frame convert frame original color frame calculate block mark new shot frame convert q red color separate object video play important decade still object hard adjacent spatially coherent demonstrate combine advanced computational model stream sensitive use initial object equation change drastically object background region color intensity correspond people experience vary color vary red back vary color vary component vary color become increasingly space feature pixel background moving object foreground statistical background color component background detect video summarize capture frame input convert color equation statistical pixel frame background pixel measure current frame pixel otherwise foreground pixel pixel follow q mean detector image isolate connect interesting template descriptor interesting area intensity orientation point detector computation matrix salient region step position extraction computationally expensive describe feature develop sift approximate multiple image limit intensity decision tree event track feature lx pyramid gaussian smoothed derivative q q sift descriptor mostly sparse event instance characterize training use parameter testing determine determine kind near neighbor knn classifier svm propose briefly give subsection neighbor neighbor find close set classified classified classified predefine query find number query point minimum near neighbor neighbor majority near neighbor query dimension advantage nearest robust noisy near neighbor determine neighbor query instance inspire system neuron work solve neuron artificial neuron separate negative class training vector label belong linearly separable geometrically find maximal solve weight call complexity superior capability function extensively past popular radial multi general independently discrete sa site seven see sa mean hold seven order water breast plain important significant gave hill try mark place move remain cc seven east day day fig category video resolution category video format system result record retrieve relevant record define relevant retrieve recall usually equation respectively video shoot shot identify accordingly c sa false precision knn ann svm performance system cc human spirit use web character video sophisticated tool action text discriminative representative however boost huge making applicable motion human action descriptor aggregate optical flow measurement spatio volume center channel optical template match spatio correlation motion energy create moment mahalanobis moment description avoid explicit rank constraint intensity spatio enforce spatio temporal video location several explore control recognition localization recently dataset manual annotation develop
cca retrieve visual database cca space combination tag cca retrieve modal embed tag directly search tag database tag scenario database variant keyword query retrieve set accurately go cca coherent tag decode g scope task similar neighbor tag return five tag tag accord diverse dataset local somewhat task protocol task subsequent tag annotation automatically v refer view baseline tag visual tag supervise semantic keyword unsupervise automatically computed tag I completeness view cca cca give embed model detail compare obtain intermediate visual tag baseline structural formulation tag tag predictor ridge task predict visual validation cca tag unlike cca cross retrieval sift sift h pca cca without learn weight classifier keyword feature modal retrieval stochastic function difference explicit use early sgd iteration pick dimensionality embed achieve good cca overall slow early stop use cifar conduct detailed study include feature tag purpose perform evaluation cifar dataset learn cca imagenet use quantitative imagenet validation split validation search query retrieve fraction imagenet search query tag exclude embed truth query examine percent closely ground example car car ground auto accurate image tag search cluster mean retrieval row initialization method cca image tag image search map nine tag section evaluate dimensionality reduction involve third view cca tag search visual apply slightly less significantly tag retrieval help possibly tag noisy reducing motivate give platform core ghz gb ram take cca versus minute nine minute get scale try likewise thousand completely infeasible platform point address finally report retrieval view cca fairly center image close tag present number tag view truth table method cca approximation tag discuss section compare solely cluster bad performance tag base cluster visual demonstrate view c visual bad cca v baseline tag cluster range nc method cca tends depend use tag joint example image visual coherence show tag though completeness figure still long correspondence frequent entire cluster confirm difficulty alone help explain incorporate cca l cc cca cca cca cca v scale cca v cca v multi view denote euclidean correlation cca nc clusters cca weight tag automatically interactive system user adjust concept query weight image quantitative annotation annotation neighbor imagenet cca space tag ask member research tag suggest mark tag image interface tag present tag correspond hard complete tag human voting tag gets mark three report tag frequency cca v lead cca v vs cca test wide randomly embed image sample query retrieve image use cluster nc end cifar surprising large semantic ten precision since ground keyword per keyword image keyword top I cca cca v structural refer query split around report ground annotation cca cca cca task v find weak model entirely surprising however tag mixed concept fewer well intuitively rich diverse truth annotation unlike cca strong cca especially noise cca flexible suitable tag image search cca c return compound query consist tag compare tag view cca cca tend retrieve compound annotation cca cca cifar three image result confirm tag produce explanation result image specifically either abstract object suggest tag landscape light appear somewhat try suggest specific wide cifar embed solution exploit multi important subject search explain information dataset give view supervise cca image database query query database mark tune nc report tag search dataset diverse absolute accuracy may database image annotate retrieve query quantitative evaluation cca still consistently wide work wide cca well cca tag noise qualitative search finally image tag present approach internet tag semantic start view cca work performance third ground truth search keyword cluster quantitative model cca cca consistently outperform extremely appear second simple concept cifar tag actually capable label even concept tag sensible concept attempt discover structure may connect visual tag latent image semantic view cluster indicator highly nonlinear second expressive cccc cca cca cca quantitative successfully visual consistency diverse flexible retrieval system capable visual semantic discover tag subsequent cca projection content internet like user retrieve query tag keyword manually adjust weight keyword serve basis discuss satisfactory develop sophisticated incorporate consistency name intermediate recognition parse retrieve embed tag retrieval return lead improve parse describe image tag sentence generation may become easy anonymous constructive comment implement manual auto support nsf ap microsoft fellowship website http www edu microsoft view department internet tag image tag tag annotation analysis cca popular latent capture semantic category mutually exclusive way supervise come truth unsupervise automatically tag ensure accuracy learning process multiple visual feature cca produce qualitative view task scale internet dataset keyword image associate internet order enable retrieval scenario search image annotation task must meet requirement big give extremely heterogeneous annotation million flexible cross retrieval tag enable tag without tag promise cca two visual common maximize modal embed vector treat image text retrieval principle handle cca cca cite use cca consider correlation feature improvement two semantic semantic high characterize image concrete figure three semantic view keyword think view visual tag stochastically tag vocabulary semantic may object category correlate etc alternatively semantic object attribute thus image annotate ground color shoot semantic give mention ground three high view cca embed embed three embed confirm derive capable considerably diverse ground truth keyword define ahead annotate realistic tag underlie fortunately clean truth still semantic explicitly informative signal search keyword retrieve tag know search ground category search effective view unsupervise tag inspire information semi reconstruct distinct modern method find necessary multiple nonlinear cca scheme improvement dataset contribution explicitly incorporate semantic function adapt cca retrieval scalable discriminative visual view reduction section ground truth annotation unsupervise tag vector confirm help improve perform comparative tag extensive evaluation image tag tag protocol vision community jointly image area exhaustive several important line work research focus occurrence image tag generative lack annotation image challenging contaminate internet tag tag frequently characteristic localize establish relationship conceptually compare attempt annotation unlike concern symmetric model correlation treat tag assign scalable inference test tag tag nearby cross tag tag recent canonical cca cca image modal retrieval modal approach relative importance word user annotation develop cross image embedding use domain unlike cca retrieval annotation approach add third semantic underlie basic cca pay price mahalanobi relate near optimize optimize neighbor annotation modal retrieval evaluating traditionally content base image tag image retrieval use query tag though directly contaminate purpose collection third interested evaluate automatic traditionally help sophisticated scheme annotation adopt experiment latent tag transfer annotation approach obtain relevance image retrieve unfortunately learn computationally expensive scale annotation consist tag dataset k descent simple image annotation annotation system use optimize dataset million learn feature model distinction tag image annotation dataset image single label hierarchy annotation occurrence different tag retrieve belong category plane tag query incoherent exploit annotation multi limit visual would impose multi decode outside scope paper image classification web baseline tag multi another way bank concept unlike produce embed supervision computationally intensive training view semantic scalable cca assume training image associate visual tag discuss stack number image may entry rest several keyword possibility indicate scenario possibly noisy annotation tag tag point learn rely instead formulate projection embedding project embed vector distance pair eq th result understand term align tag term view cca objective align tag formulation embedding formulation trick cca linear combination one must solve eigenvalue infeasible base reduce mapping use describe trick substitute cca eigenvalue w ij ix jx w iw add constant covariance view matrix space learn view without tag retrieval near neighbor implementation validation aside section range typically fall cca function distance embed dimension latent magnitude project objective reformulate view let view cca whose corresponding experimentally lead high retrieval euclidean section visual text mapping semantic unsupervised third appearance nine use different filter channel base dense corner patch mean build pyramid sift sift texture large codebook spatial pyramid dimensional joint bin feature compute set deviation descriptor adopt wise combine quite put respective mapping dimensionality reduction top approximate combine feature feature vector dimensionality balance efficiency origin construct frequent vocabulary summarize remove stop include etc characteristic etc tag tag tag feature two however tag require visual sparse show actually embed sparse compressed top sophisticated rank tag list early user salient image initially cca indicator straightforwardly
maxima different natural multivariate extreme weak convergence univariate reasonable apply affine transformation margin sequence normalize stable distribution th convergence take know strictly normalize maxima vector henceforth weak distribution margin sequence copulas maxima component give check joint margin maxima copula arise call copula call extreme value copula exist copula extreme copula arise copula copula say extreme extreme copula coincide copulas maxima max stable also conversely extreme stable tend fix limit max summary distribution appropriately maxima margins extreme max copula margin extreme margin copula former tail go infinity tend good copula characterization rise function denote generic one exceed percentile distribution copula originally involve maxima familiar univariate tail copula component exceed percentile simultaneously q govern formula dimension somewhat retrieve margin value copula tail copula even zero depict cc homogeneous eq similarly simplex w homogeneity frequently write union consequence q multiply let extreme must satisfy display attain independence perfect association max property copulas function law form fr reverse yield fr margin notation transformation disadvantage action neighbourhood identically geometric success return time return pareto find map evident dependence function take mode vanish origin intuitively grow profile degenerate equally depict exceed interpretation order statistic fairly max device represent tail measure parametric obvious work measure profile margin condition different fr independent vector calculate copy fix q j rescaling case margin compare identify da comparing give absolutely law profile profile variate distribution put variate stable profile dependence random profile encode dimensional face simplex empty let otherwise measure profile contain simplicity apply order parametric remainder work ensure perfect constant finite profile dependence spectral discrete atom suppose ensure dr w spectral profile discrete magnitude give max spectral call value factor outcome result type event random indicator b pair pair stable dependence asymmetric parameter extension equality constraint yield one indicator switch structure keep law equal empty way structure think logit indicator copula device copula associate model let tail dependence know dirichlet dirichlet model dependence mixture introduce transformation yield generally max stable eq variable count variable expectation exponential polynomial dependence bivariate random margin put stable dependence q bivariate standard put expectation stable function calculate yield author thank journal manuscript centre participant encourage discussion strongly author grant science contract max universit random component mathematical suggest stable univariate multivariate various stable device stable distribution device role yet fully exploit copula measure extreme value concern sample component bring univariate extreme ignore mathematical max
irrespective statistic accept homogeneity even hellinger large refer logarithm negative log likelihood generalization exact unlike one ratio facilitate comparison likelihood display entry homogeneous proportion give underlie rl hellinger frobenius please p value frobenius display entry display assumption table log hellinger please value display homogeneous correct rl log hellinger negative note frobenius distance value display display without optimally display party party b z lrr party divide square root party z treat lack recover another efficacy recover lack efficacy another root lack candidate lrr divide square root treat treat treat reaction reaction prior square corresponding reaction frobenius distance classical homogeneity paper typical similarly important circumstance classical statistic fan wolfe fellowship p foundation section remark homogeneity proportion contingency probabilistic process generate fixing row illustrate frobenius schmidt classical statistic log ratio hellinger distance divergence family hilbert schmidt consistency homogeneity chi fisher exact divergence root sn sn n sn categorical contingency table table provide comprehensive common fluctuation homogeneity proportion displayed consider homogeneity row construction therefore table assume homogeneity generate draw identically draw note rp count equal observe please construction exact confidence observe draw consistent homogeneity see discussion interpretation section value measure canonical possibility hellinger hilbert schmidt q take number member divergence member read advantage illustrate advantage statistic recommend use classical sequel define several section conclusion definition p involve compute probability carlo guaranteed specifically perform generate draw rp draw depend define draw row discrepancy simulate p equal discuss number compute similar procedure bound report present
ibp generating process ibp pre perform vocabulary ibp due restrict sample single atom posterior ibp use sample estimate predictive maximum binomial distribution train document classify ie label ibp uniformly explore way distributional exchangeable allow distribution flexibility future benefit distributional exchangeable column useful datum poorly suit propose exchangeable exist model set ibp relate gamma poisson exchangeable infinitely contain flexible model specify row distribution ibp impose distribution exhibit give ibp marginally appear want flexibility allow non marginal per ibp use markov see nonetheless ibp feature prior similarly wish exclude interesting arise exhibit perhaps team encode per tend parsimonious situation imply ibp well text tend suggest impose tail ibp point exhibit measure feature remove random unchanged far exchangeable important tell exchangeable distribution write probability exchangeable predictive px n measure may motivate exchangeability nonparametric ibp binary exchangeable row infinitely column measure process distribution bernoulli beta process bernoulli assign form parametrization beta commonly use ibp scalar cdf atom distribute accord piecewise countable jump ibp cumulative atom beta give think matrix infinitely binary result entry row row non entry column exhibit rich rich give time appear precede vary distribution degree ibp introduce extra beta limit result ibp feature point disjoint replace beta measure beta process include ibp exhibit feature law exchangeable gamma poisson process correspond column process ibp atom result negative integer matrix exchangeable distribute binomial exist able sharing feature ibp remove ibp power customer try distribute aspect elaborate marginally row ibp conditional distribution location trial sum restriction specify row priori bernoulli exchangeable mixture recover ibp ibp marginals poisson ibp see case construction explicitly condition draw de mix measure infinite construction exchangeable discard outside interest certainly find exchangeable sequence exchangeable restrict infinite process infinite previously assign count restaurant restrict sum chinese restaurant restrict chinese exchangeable sequence restrict predictive exchangeable restrict distribution process consist mass index red represent blue ball pick note color return ball sample color iff exchangeable trivial ibp restrict probability model give example exchangeable introduce normalize restrict previous broken exchangeability obtained directly restrict ibp section restrict ibp appeal exchangeable modify ibp scheme exchangeable refer ibp modify sampler crp distribute binary accord entry per location non th sampling especially hasting entire binomial recursive fourier skewed approximation metropolis hasting wish evaluate ibp unable analytically predictive use ibp predictive n equation describe exchangeable modify experiment appropriate datum design careful latent improve evaluating
find manifold window manifold enough bend local moment matching already match discretized consider density moment mild suffice chain converge convergence chain chain precisely point minus proportional sketch ergodicity convergence counterpart asymptotic distribution interesting note operator equation multivariate see kind specify component like density noisy approximation step small magnitude intuitive point neighborhood appropriate normalize define local mean scale take assume calibrate mean mean obtain chain ball define approximation definition result q mean step mean equal asymptotic moment version target similar observe term produce gaussian version construction make follow vanish desire neighbor moment minimize reconstruction estimator train contraction whole encoder equivalent criterion corruption set completely covariance encoder already set corresponding example go infinity go around taylor expansion neighborhood non infinity neighborhood dirac delta rewrite choose dirac expectation arise covariance get derivation happen force practical see relate asymptotically decrease jacobian parametric infinite perfect much magnitude indicate look increase singular relative direction practical want take non generalization parametrize perfectly generalize learn reconstruction equal even jacobian mathematical link situation control variance hyper optimize inspection introduce many non density window generator mix cover much true density window larger likelihood result auto encoder train generate likely look novel approach density moment auto justified chain local moment auto decade everything capture first present justification auto particular denoise interesting advantage criterion estimate difficult future follow try answer happen learner non consider compute future give rise sampling moment extend auto encoder experimentally give mathematical algorithm derivative dx mse x r q respect trace x obtain j starting x get j x x j c c solve limit note go goes plug recent encoder job unknown datum density contribute mathematical understanding phenomenon help justify auto previous covariance capture jacobian novel density call local matching propose extend auto encoder learn aspect generate learn manifold identifying set point configuration observe plausible attempt recent year feature auto encoder stage encoder decoder sample data feature stack deeply abstract much evidence suggest perform purpose classification generating target manifold concentrate locate manifold variation movement stay density remain concern many feature criterion implicitly learn whole aspect essence density formalize exploit question contribute propose motivate intuition jacobian encoder direction tangent plane direction preserve reconstruction auto reconstruction entropy loss easy success criterion parametrization tie weight force tie singular contraction singular least direction little variation encoder train denoise criterion corruption mostly corruption handle mathematically expansion rx rx x derivation auto contraction parametrization mm couple respectively
edge satisfactory merge intra benchmark four reveal art meanwhile possible intractable exclusive major approach hybrid evaluation major statistic score mutual mutual advantage cost however multiple another drawback usually require observation sake constrain constrain algorithm use conditional ci test reveal dag pc drawback ci variable parent besides orientation perform search scoring super get dag serve hybrid integrate constrain min hill superiority reconstruct skeleton network constrain hill search orientation devote overlap popular clique compute belong exist clique http overlap detection roughly detection subgraph detect overlap locally subgraph furthermore augment combination fuzzy define either node accord similarity besides cluster conduct agglomerative dendrogram partitioning overlap identify copy identification continue sufficiently undirected graph weight speak play unweighte one consist unweighted undirected edge community belong community isolate community could partitioned group hierarchical weighted partition number contain row column support node example tp node set partition weight exceed weight co model propose divide originally intractable task scale sampling section serve crucial drastically reduce run incorporate overlapping may ideally possess possess edge community coverage correct edge fine small build unweighted predefine map weight graph keep weight exceed generate partition htb discretization partition step outline step start unweighted step employ predefine weight set translate original unweighted various predefine mutual mutual entropy square entropy respectively mutual standard value edge coefficient standard correspondingly weight low threshold prune sake prefer community organize meanwhile sake wish maintain generate build partition v result satisfy criterion couple robust learn per community challenge mix eliminate bias irrelevant retrieve markov challenge step outline sub sub sc structure sub resolve find community node precise markov intra make problem localize solve markov parent child parent compose child parent word close topological closeness edge play ci measurement among require identification markov divide variable look node edge weight exceed mb xy w xy connect give justification association phase apply markov add heuristic denote mutual candidate node independence remove combine expand small intra member community expand community conduct reason contribute sampling build precise pick inner start neighbor denote bayesian large keep remove neighbor sub community sub community bayesian different constrain score interested confidence relate topology per expectation estimate classify exceed however dag overall dag average transform due current domain consideration remain markov dag assume prior possible order markov chain sequence approximately estimate sub community structure intra community resolve characteristic find major introduce try triplet however fails eliminate indirect interaction one edge triplet indicate interaction miss weak weight adjacent mutual small two first collect candidate triplet triplet node mutually triplet indirect unweighted edge triplet cluster graph subgraph hand indirect cluster direct interaction indirect eliminate correct group employ triplet create node community remove community merge add pool individually community resolve intuitively strategy large achieve expect edge continuously eliminate idea try strategy community perform greedy strategy combine community community community similarity coefficient community resolve triplet combine community hybrid community put community calculation value community initially calculation sum nk efficiency advance add would nn evaluation five close enough indicate learn loss learn alarm consist arc arc arc average network include arc degree benchmark besides weight mi entropy mutual entropy value normalization network weight transform weight certain bin lot possess ratio bin threshold threshold remain edge edge remain average short diameter dominate example mutual alarm short yet worst always belong never drastically average order path range alarm alarm alarm percent percent percent percent percent mi average l alarm link mi pearson second order denote htb comparison various weight l link mi pearson denote htb c alarm benchmark four learn namely pc greedy hill information extreme popular widely score serve hybrid prove art correctness greedy causal http www format default test greedy pc type choose weight equal greedy person threshold manually tradeoff true positive implementation name alarm five benchmark divide bayesian averaging se would could evaluation fair pc structure target keep everything unchanged regardless influence usage regard binary pair metric performance system edge edge classify negative number miss art pc search alarm table dataset recall reconstruct show versus counterpart avg intractable relevant pc superiority superiority serve harmonic global precision intra structure case even algorithm alarm htb evaluation alarm c recall recall score pc greedy avg na na htb evaluation c precision recall f score f pc avg na na na na evaluation c c global precision recall score avg na na htb dataset c c score greedy avg na na na htb link c global pc avg na na precision state cccc significance four cccc alarm normalize cccc greedy alarm present large large divide partitioning intra individually perform partition strategy verify
location location simulation table work spc knot result well knot especially spc simulation r knot knot spc c spc spc spc interval score mcmc simulate simulation table still work slightly spc suggest model process stationary knot model result eight fix knot world collect resolution south sense monitoring precision far costly scale gamma ray front smooth krige calibration hyperspectral ray krige improve ray spectra focus total integrated count unit count carry exploratory spatial prediction effect assess test region contain remain leave panel count colored dot location circle right mean deviation mean process see text region rectangle color count error tc count identify trend two another r knot knot spc spc spc hour parent spc parent covariance distance interval score design knot result low average square knot improve spc knot result accurate mixing slow knot knot slow simulated time slightly focus parameter assess base fair article spatial combine rank remainder process often encounter extend way parameterize ern parent spatial function second component function shape jump describe detail section indicate covariance mat ern closely indicate seem unlikely simple stationary simulation highly vary acknowledgment research advanced anonymous comment grateful taylor collection universit email modern high resolution measure monitoring analyse dataset spatial statistical technique feasible heterogeneous appropriate challenge combine component flexible via compactly address propose extension parameterize base mat ern smoothly model high measurement improvement covariance massive reversible jump sense environmental ground day large quantification uncertainty feasibility angle composite likelihood low compactly function produce contain computational feasibility appropriate achieve feasibility component spatial r small component model differ discretized convolution convolution author view fix orthogonal wavelet parameterize allow formula make clear general covariance fast computation range dependence inherent component rough I smooth short stein effort stein context process divide remainder component covariance compactly support feasible inference contribution specify mat ern parent smoothly linear combination function extension allow location refer knot allow fix priori treat achieve discretized convolution model spatially shape basis infeasible location assume shape special piecewise location spline contribution article parent truth rather way spatially dependent flexible hence preferable modeling real fairly involve large reversible jump take advantage operation strategy chain section deal extension approach use simulated conclusion denote spatial suppose namely simplicity identifiability assume practice origin process model model standard extensively standard computation likelihood modeling problem parent knot knot strongly prior flat improper point dependence write valid function feasibility compactly function great covariance quickly invertible feasibility give describe medium range range spatial account local zero gaussian covariance flat place z modification knot select location jump acceptance bayes fully automatic flat statistic inference true might include observe q first mcmc convergence q var p n explicitly p sect dense full update large employ formula obtain range nonzero order cholesky operation location indicate decomposition fix regular algorithm aside spend evaluate spatial obtain considerable extremely massive achieve feasibility current large spatial result prediction predictive approach knot mat ern comparison vary knot mat ern covariance spc spc special three study location simulation process together mean credible ci pm e without knot true create measurement examine medium model test miss design interval location location refer simulation observe measurement model mcmc plot pilot element per parameter covariance describe spatial trend intercept proposal knot uniform knot eight evenly space evenly spaced knot take exponential model knot basically pick consideration interval credible credible interval penalty contain goal small error average average group ht
observational estimate individual population length period credible result table run trajectory original interval gibbs augmentation detail discuss epidemic ode proceed euler approximate gibbs algorithm update euler time determine translate mix epidemic suitable differentiable sde standard vector write intractable accurately alternative suggest drive skeleton eq transform euler alternate step particle implement metropolis respectively accept formulation replace algorithm every ode part functional accept small variance epidemic simulate posterior draw draw run hold clearly reliability difference correspond formulation datum inaccurate term essential driven epidemic address problem helpful improvement error bias estimate particle filter particle posterior posterior interval cccc I quantile r shift shift shifted median shift shift median shift shift estimate euler discretization axis illustration contact record green observed incidence green contact panel line dark light interval level noise bottom trajectory estimate contact rate h implication dot line pointwise dark light blue interval respectively panel alternative explore diffusion right panel posterior density bm maximize mcmc particle gibbs statistic school economics political uk ac department centre school epidemic supplementary part illustrate various suggestion appendix kalman analysis give regard continuous implication associate provide present specify determine euler ps vector discretization encounter epidemic trajectory approximate euler ensure value choose quantity sufficiently show datum reasonable point theoretically regardless smooth particle particle smooth mcmc affect acceptance mind need get acceptance perhaps beyond observational decrease assess validity approximation extent avoid approximation month bm realistic epidemic reverse select google contact significantly decrease experiment methodology text incidence correspond figure fig moreover present credible interval estimate seem value credible aim assess robustness brownian motion sigmoid curve perform capture trajectory explanation part volatility h plot child additional alternative time conduct main approach integrate imply smooth differentiable path
formally current future cost current satisfie bellman strategy minimize cumulative contribute heavily prior knowledge environmental dynamic traffic advance therefore approach mdp require knowledge traffic specifically adopt algorithm name component actor illustrate select way transform actor action execute actor update td prefer smaller versa adopt actor generate store select policy useful traffic separately knowledge actor illustrate let begin meanwhile load state controller stochastic balance compete objective bs operation exploration exploitation result controller methodology probability call temperature indicate tendency stage exist possibility serve traffic stage however conventional scheme commonly controller bss hence controller meet load requirement transmission traffic load connect bs modify communication join cost bss determine intuitively simplify I minimization energy consumption simplify location prefer choose join bs transmission traffic load consumption transmission stage traffic bs transform meanwhile cost consequently td would k feed actor way update occurrence stage iv policy td action indicate execute stage one action higher execute infinitely limit greedy infinite converge address methodology exploit classical ac conduct bs switch effective energy end utilize knowledge strategy historical period bs switch basically indicate tendency converge tendency bs switch conduct system optimize policy period within attempt action traffic come might beginning exist come traffic high target hence target turn bss mode avoid decrease choose certain controller experience take consideration overall traffic load choose euclidean b additionally p kp actor initially dominate overall hence contribute optimal transfer continuously decreases take advantage learn also get propose connect bs transmission cost start scheme traffic td policy transfer update classical ac result one start several relate lemma boltzmann exploration thereby eq next policy track ordinary equation ode eq asymptotically track ode algorithm transfer transfer factor rl km km transmission macro bs micro bs w operational macro bs micro height macro bs micro channel interference factor arrival file constant discount transfer negligible turn mode due scheme consumption bss turn scheme bss start reasonable since exactly scheme controller find bs simplify adjust table firstly much static traffic load arrival bss turn homogeneous traffic offer correspond bss homogeneous traffic rate reflect static arrival expect energy decrease arrival bss stay bss utilize continue decrease controller understand traffic know well efficiency scheme traffic inferior scheme simulation feasibility save energy fig depict performance energy consumption delay scenario incur increase obvious put emphasis delay choose tradeoff performance improvement begin contribute fig depict task leibl divergence divergence transfer plot impact transfer generally speak expect traffic profile approximate approximate traffic load arrival scheme ac k detailed sensitivity analyse c various reflect file consumption controller new action controller try controller would action large result energy consumption fortunately certainly undesirable action classical ac demonstrate effect file similar arrival constant consumption proportion cost turning utilize bs make difference save red region bss exhibit scheme different bs switch operation vary decision besides adopt actor method reinforcement learn switch energy exploit traffic transfer actor improve learn period propose provably arise extensive knowledge transfer propose approach apply scenario map less straightforward bs dedicate meaningful yet challenging knowledge author l anonymous kind author suggestion improve paper quality national program china green cb chinese education key r china grant laboratory without stage state occur stage simplicity k tt tt jt tt define depict z z piecewise integral jump z negligible hence convergent see equivalent ode claim chen zhang china email edu cn box chen sup france email fr universit sup de la france validate energy access turn bss switching operation load dynamic still forecast firstly minimize consumption reinforcement learn bs switching actor utilize historical neighboring provably simulation various contribute demonstrate feasibility expense delay communication energy reinforcement actor popularity load demand massive consumption huge emission speak account overall emission besides operator doubly five china mobile meanwhile account significant improve energy consumption access reason behind largely bs peak stay irrespective heavily body load aware bs adaptation validate possibility energy dynamically adjust status predict traffic reliably challenge bs besides find turn bss bs mobile mt moreover subsequent user association traffic load bss bs switch bs switch influence consumption word scheme minimize consumption concern effect bs operation present combine bs association load solve instead predict traffic traffic actor reinforcement rl necessity knowledge traffic bss centralize centralize bs operation controller network generation evolution rather result bs switch controller estimate traffic load variation experience possible consumption bs switching take experience repeat know switch bss traffic load profile bs switch rl convergent hence rl bs switch controller hundred bss fortunately traffic exhibit make load aware bs switch moment region relevant utilize conceptual transfer mostly recognize one previous novel appeal lead research activity learn bs historical neighboring region bs switch operation enhance incorporate actor ac namely transfer actor propose previous work firstly save consumption moreover assume already extend rl thereby contribute field especially ac traffic mdp energy scheme conventional focus incorporation idea rl section evaluate scheme present validity mdp state subscript td error occurrence stage occurrence discrete shift arrival file consumption delay summarize run bss traffic bss bss depict exist bs switch traffic bss correspondingly status bs stage centralize beyond bss meanwhile transmission request location arrive poisson point arrival per location therefore traffic example high arrival file bss turn traffic bs ix ix bs versa otherwise variation bs coverage partition indicator describe subsequently whole region traffic load variation constitute state chain transmission user bs convenience change
approximate overhead subproblem suitable issue return optimization adopt method I k k center radius example consist determine index approximate solver enough rapidly point strictly indice fw originally design solve iterate frank wolfe consist local k fw update iterate stop sufficiently suitable proximity search direction towards vertex feasibility assume prove exhibit tendency intuitively lie boundary often problem get close direction reach face sublinear frank wolfe later improvement stop practice tendency fw lead resource desirable towards aim enhance introduce direction know move vertex ascent follow find fw algorithm minimize fw k k iterate feasible direction stationary correspond face lower reach whenever away stepsize impose constraint stepsize continuity concavity strict problem assumption aforementione problem particular ask concavity force perform happen behave unconstrained optimization linearly objective obtain explicit procedure maximize carry compare argue whole associate I k k immediately follow update component introduction sequence make structure output preserve formula increase criterion bc identify procedure differ bc square value get optimal round line join round triangle pt circle circle pt circle bc fw derivation procedure compute r j k ji k apparent identify small step optimal stepsize determined correspond lie remove ball remove hard perform step whenever step ascent fw fw standard expression easily insight away search htb join round cm cycle pattern color black black color node node linear result algorithm substantially need remove correctly optimum possess eliminate get near originally motivated advance lead efficient train use satisfy impose obtain fw objective concave constraint regard addition strict optimum make normalize kernel add objective normalize apparent need reformulate long fw procedure large similarly explore iterate still propose immediately evident notation explicitly target qp j whose column constant g hard write follow k stopping hand concave eq addition virtue feasibility objective function classification acceptable frank wolfe build considerably small fast software evident classifier time discuss several issue several include scalability increase assess capability frank wolfe study significance analyze penalty experiment capability wider highlight purpose paragraph summarize conclusion detail cover number like dataset original number dimension classifier tuning category adopt one beyond binary case cm cm cm feature brief description classic handwritten united states service capability digit create experimentally collect optical character objective black rectangular pixel display capital letter english file bioinformatic regard concern neural international conference network handwritten come national connection record detect type one categorization document appear document extract census original aim predict individual us year series task appear minimal instance subsection svms rbf reason kernel know choice svm capability fw handle polynomial satisfy small svms use among pattern logarithmic set consist randomly computed determine fine parameter tune significant subsection effect fw adopt method require evaluation quickly possibility scale sampling technique call overcome cardinality major previous generally technique rely equal least choice make adopt lrr c dataset computer core ram implement source code report concern accuracy support size number fw respect training algorithm fw contrast considerably pattern great fw importantly speed monotonically range fast time significant training set obtain dataset create scalability pattern fw confirm propose tend dataset actually fw case reach fw moderate exhibit test accuracy fig correspond accuracy speed svms price slightly fw significant grow fw particular peak show case tend however exhibit around ht l l cm acc acc std acc std e letter e e e baseline acc std repetition experiment protein train std std e e e letter mnist protein time second algorithm baseline correspond deviation std dataset accuracy run use fig pt cc paragraph significance perform reject conduct performance adopt method test safe parametric binary propose apparent advantage accurate fw run test cm fw p cm fw rejection obtain book p sign preferred tie employ default correction suggest cm statistic fw accuracy w sign rank test respectively cm binary fw vs fw fw time fw vs run low significance level low summarize conclusion hypothesis confirm time run section run magnitude well conclude training statistically conclude extension fw reject reject fw reject accuracy test significance use pattern conduct experiment study effect fig time accuracy change confirm fast svm h c solve svms capability propose even satisfy conduct homogeneous squared use meta note however determine optimal consider capability fw wide thus great flexibility building summarize result test accuracy svm select experiment fw demonstrate ht cm cm dataset accuracy std times std fw fw fw fw fw fw fw time statistic repetition experiment detail competitive tendency favor speed sometimes expense fw among single subsection several binary subproblem great subproblem subproblem quite run bc reasonably traditional constitute fw instead capability handle show testing offer fw say fw large order probably lack size solve finally fw advantageous offer considerably run comment understand performance gradually commonly scalability regard appear encouraging fw outperform collection cpu speedup increase reach order fw algorithm accuracy clear fw probably explain considerable dataset evident become advantage start optimal support algorithm try number progress towards remove approximation imply weight vanish drop consider possesse remove directly enjoy base computational geometry svm algorithm base frank wolfe scheme analytical formula learn solution compare present compare testing accuracy confirm conclusion reach wolfe build prove software conclusion statistically assess non parametric preliminary handle wide kernel allow flexibility variation theorem quadratic programming qp become expensive traditional mainly restriction adopt condition within svms core task building embed propose novel method svms frank wolfe increasingly complex step case sometimes price binary classification wide support machine problem improve technique svms optimize optimization usually formulate qp require complexity major effort scale dataset structure traditional svm address small svms essentially training call prominent minimal disadvantage method exhibit slow close get slowly sensitive size active like use avoid scale adapt qp still handle efficiently look new propose small contain adopt impose recent independently point build core demonstrate new method software start solve small subset look point outside define large new current approximate prescribe sequence optimization problem use efficient warm avoid full gram ref employ end variant algorithm formalism need wolfe optimization define simplex solution considerably idea nest linearization exact line linearization consequently external numerical solver discover support look separate hyperplane maximize misclassifie unseen deal noisy example compute margin svms address svms see g degeneracy address
spline output consistent evenly spaced covariate visually credible posterior gibbs sampler plot surface generalise mixed spline give parametric basis degree block parametric effective degree freedom draw way density trace goodness number describe effect schwarz residual autocorrelation autocorrelation remain matrix chain u remain residual integral predictive observe forecast forecast ideal forecast calculate routine autocorrelation beyond used may uniformity choose predictive forecast credible distribute normally credible divide half give performance methodology ability univariate bivariate interaction model significant amount illustrate simulate covariate uniform fit trend evenly spaced interval univariate second spline order bivariate consist product two basis six spline order penalty prior direction intercept weakly ar residual area include planning environment portion particle linearly week measure flexible motivated generalise spline generalise average record record different autocorrelation fit spline motivate estimate capture maxima occur trend represent significance parameter type size covariance nd week year spline wind spline nd penalty wind speed temperature nd penalty spline nd penalty nd penalty fit thin spline wind speed effect order polynomial tensor wind attempt spline wind spline undesirable final fitting table wind wind term replace spline spline fourth daily trend replace trend change year exclude inclusion fourth daily trend fit trend daily trend trend basis mesh copy year residual lag hour hour lag capture hour hour fitting daily variation fit daily year year estimation conduct draw discard burn periodic interaction interval periodic similar uniform covariate htbp capture curvature function trace posterior posterior converge unimodal whose figure trace distribute credible interval distinct parameter maximum posterior occur still approximately long parameter coefficient coefficient autoregressive ar residual final distribute marginally htbp contour fit spline figure spline log wind either remove nearby trend year daily trend vice daily exhibit pm trend day daily daily peak pm show posterior residual exhibit autocorrelation semi regression daily trend temporal variability explicitly residual significantly examine model particle air middle fairly peak dependence peak traffic increase hour dependence wind direction mark direction peak occur around effect trend wind proxy good result present effect temperature wind generally wind effect estimate credible see lag credible interval represent specify regression autoregressive daily trend peak occur pm express specification quite daily trend autoregressive daily trend trend autoregressive contiguous model residual select residual find intercept intercept freedom freedom spline around predictor agree spline sd intercept daily trend wind direction wind wind entire forecast modelling base reality weather forecast estimate model observe forecasting rather combine spline autoregressive method underlie smooth autoregressive b spline small none smooth temporal trend result function exhibit fourier despite basis sampler ensure first sample discard sample stable density spline count basis wind reduce indicate dependent basis flexibility univariate basis basis qualitatively model residual show lag daily trend autocorrelation model capture smooth trend explain trend model autocorrelation residual methodology outline spline rather allow knot vary reversible jump use knot necessary smoothing fit choose covariate ensure spline typically knot model product spline investigation particular functional guess random glm generalise analogy generalise product spline identity stay include desirable come argument setup basis form tensor spline present provide fitting non focus cyclic autocorrelation residual model much remove wish division sciences institute health innovation school sciences science author wish feedback suggestion development comment thick observational exhibit cyclic trend autocorrelation realistic forecast combine spline modelling model auto error simulate efficacy particle autoregressive generalise particle continuously series exhibit temporal dependence covariate motivate able take account specify smooth surface time covariate establish spline extend penalty periodic basis basis generalise additive model spline periodic trend spline coefficient evolve admit error forecast fitting assumption independent identically residual modelling autocorrelation realistic traditional distribute error specification move model lag term series inclusion account remove lag interpolation trend adjust examine temporal forecasting forecast generalise model b despite development spline incorporate approach error cubic spline spline spline function smooth combine spline parametric modelling error propose define autoregressive ar incorporate model outline review generalise spline outline penalty identifiability spline spline autoregressive error discuss metropolis forecasting discuss include section series dimensional covariate show model trend error covariate forecast apply record section review light suggestion make extension model present parameter belief posterior belief mathematically condition posterior parameter specify response identically additive covariate discuss semi methodology matlab fit whose omit generalised treat equivalent generalise spline alternatively augment say glm express spline glm glm spline many equivalent spline non knot linear adjacent knot basis overlap desire covariate grid knot spline constant knot basis create recurrence cubic spline three piecewise knot knot e zero interval flexible response basis spline spline affect speed mcmc knot choice particular spline basis make spline include spline order b spline piecewise piecewise individual grey scheme across thick grey thick black show b spline zero covariate boundary preserve impose fit spline introduce spline frequentist spline coefficient derivative spline element spline equal row operator analogy sum penalty conjugate multivariate coefficient spline parameter small concentrate improper penalty overcome add spline analogue periodic trend spline evolve accord daily treat daily evolve accord day obvious factor continuous sense spline term covariate tensor form product identity size element penalty spline formulation penalty term two covariate matrix block corresponding penalty covariate indexing spline univariate reflect different amount smooth dimensional precision formulation amount direction multiple address evolve recover walk spline penalty parametric polynomial model outline periodic spline briefly choice basis parametric modelling periodic fourier series term ensure spline identifiability already treat e equation fourier weakly informative structure diagonal fourier series trend include function high spurious coefficient zero use periodic spline adopt merely point basis term effect basis covariate element simple assume factor precision knot knot space smooth spline basis hour treat lag describe autocorrelation structure residual multivariate prior censor absolute strictly order prior precision weakly variance autoregressive error conjugate forecasting realistic successive within residual mcmc forecast readily important dependent forecast ignore uncertainty forecast would produce formulation respective weakly informative prior respectively autoregressive precision informative block smoothness spline hyperparameter prior treat similarly consist distribute spline penalty coefficient model htb lag lag include autoregressive similarly contain column residual lag euclidean distribute autoregressive condition vice versa comprehensive acyclic marginals hasting fill thick minimum black thick mu edge beta edge eps eps delta edge lambda draw black corner inner
parallel computation provably converge nested problem complement backpropagation setting serial computation reach cloud become device etc thank availability area computer speech compute mac many choice optimize step nsf award input deep net net simplify ignore bias form follow nonlinearity sigmoid per hidden sometimes notational particular constraint single feasible mac qp loss nest subject qp bound derivative us compute c z f nk gradient suffice look gradient r short construct eq kk problem rather qp define problem minimizer nest give qp satisfie nest qp stationary n satisfy eqs point problem remark since constraint infeasible converse true stationary define eqs hold q mac constrain minimizer cm thm prop thm corollary thm conjecture thm wang california sound nonlinear layer sometimes inspire offer way ever example computer front joint consider numerical optimization execution suboptimal system strategy learn extent replace original involve augment without alternate coordinate mac provable derivative competitive serial providing model year practical applicability ever powerful machine translate natural physical limitation serial suggest scalable thousand processor cloud hierarchical neural originally brain successful face define hierarchical feedforward mapping dataset numerically transform output linear sigmoid net map wide net unit encode net cascade scene understand process classification reduction architecture share ideal nest layer objective probably occur stream visual composition produces inherently usually backpropagation compute gradient sgd feed optimization convergence criterion satisfy suffer layer higher slowly slow researcher practice net around architecture net recently initialization much computer architecture mapping decade remain good strategy architecture net front combinatorial search trial costly effort solution often irrespective cascade lb lb lb lb optimization nest call partly parallelization optimize describe give illustrate advantage mac give input deep net ignore k mapping nonlinearity scalar sigmoid output fully connect share term backpropagation auxiliary per unit equality see intermediate hide problem minimizer complicate involve small ill cause may solve mac quadratic qp mild converge minimum original follow path break unfolding every square involve equally condition gradient gradient qp single unit unit k n nh kk separate coordinate derivative step exist new however proximal operator mac complex couple step auxiliary dataset potential operate variable qp objective function iteration achieve along coordinate device algorithm nested architecture convergent nest architecture elimination architecture need feedforward net coordinate semi net hide auxiliary nest step lagrangian rather depend gradient newton inexact step mac maximization alternate multiplier optimization area illustrate mac deep autoencoder technique need mac qp decrease reach mac serial remarkable minima unnecessary perform inexact qp term common net error validation nest sophisticated take maintain mac qp fast post auxiliary coordinate simply kn kk kn weight except fit one prove value another advantage heterogeneous architecture layer perform specialized quantization recognition layer radial weight layer jointly minimal something easy mac autoencoder encoder decoder code net remain mac qp large decrease iteration advantage mac value architecture architecture traditional architecture architecture criterion picking involve suboptimal hand select mac achieve layer let coordinate e ce function aic km choice mac step separate require testing layer still costly parallel alternate run multiple iteration guarantee decrease near mac reduce exponential complex architecture mac rbf autoencoder try architecture early mac mac formulation full generality equivalent optimize jointly auxiliary quadratic exist several line qp give set principled way net nest unit past early recent sparse activation net go early day neural net backpropagation good representation important deal activation objective activation intend distribute representation nest develop net nearly focus reveal parallel processing aspect truly deep extract overcomplete dictionary one nest work layer backpropagation limit nest recent greedy weight sequentially optimize weight converge problem machine homogeneity dimensionality mapping dimensional mapping minimize nonlinear spline neural net radial nonparametric function drive separate infinitely hoc prevent dimensionality supervise dimensionality mapping rbf see version single layer therefore result nest biased summary work solve representation encourage mac auxiliary purely mathematical construct parallelism good none heterogeneous layer criterion separate mac introduce however term functional negative contrast mac new break able thank introduce subproblem mac characteristic deep autoencoder heterogeneous architecture autoencoder speedup mac alternating square nh gauss approximate hessian search search backtrack practice minimizing separate formally enter objective nearly gauss mac gauss reasonable result intend efficient augment lagrangian combine inexact warm unconstrained alternate explore topic parallel mac qp yet processor nearly parallel toolbox programming effort matlab processor run processor report matlab toolbox parallel model cache dataset handwritten autoencoder architecture map dimensional try reconstruct mac qp introduce auxiliary learn curve machine image handwritten digit training equally digit autoencoder logistic layer weight layer dimension fan initial adjusting attain give gradient random weight mostly give initial mac qp run newton gauss since lead local optimize inexact nest evaluate tolerance term tolerance classical stochastic descent conjugate gradient sgd carefully ensure minibatch learning cg regard cubic interpolation backtrack long minibatch batch mode work plot objective initialization validation closely train iteration mac cg epoch red circle mac processor also show right sgd cg iteration error mac qp beginning reach mac qp serial remarkable linear processor objective runtime mac mac c ex nest marker iteration mac epochs sgd mac qp quadratic indicate red solid processor parallel mac curve processor memory toolbox runtime hour mac opt mac c nest marker mac optimization reconstruction sample initialization embed algorithm mac encoder decoder rather rbf train mac qp introduce benchmark sequence degree thus close manifold pick half cat leave autoencoder bottleneck layer dimensional visualize reduce decoder decoder radial hide image decoder concatenation layer total million weight problem minimize plus quadratic layer weight rbf stage encoder one train center mean fix input obtain preferred fully center achieve net strategy network beyond example vector machine slice wish train deep autoencoder construct rbf center alternate mac approach alternate step mac objective embed decoder slow mac mac problem step encoder decoder separately
primitive infection cascade result np another paper cast albeit good knowledge crucial cascade learn mrfs classic cascade epidemic assume parent child child edge reverse happen initially become seed every try child active correspondingly time node become inactive spread infection infect sir epidemic remain epidemic never reach inactive sample path cascade sample cascade four around square around active inactive respectively inactive time child turn inactive step epidemic stop epidemic cascade observe seed infect infection times cascade cascade refer complexity cascade super prior network direct super node parent course super infection find node cascade cascade cascade thus cascade infected speak graph far seed far incoming edge follow infection time node infected node infect presence provide infected attempt edge activation sample number need graph meaningful interpretation need time infect cascade via theoretic knowledge estimation infection find identity associate infection ml value maximize likelihood crucial particularly nice enable particular j set parameter parent every finally sample infection vector seed concave please sum depend variable algorithmic proposition small subset solve parallelization speedup fast super infection analytically cascade reliably optimality program particular complementary condition program neighborhood formally involve optimizer define prop eq analytical cascade needs reliably node consider super graph strength parent cascade great ml algorithm e enough remark appropriate derive corollary depend entire neighborhood I infect reliably error scale value sample great set need entire scale system parent seed reasonable seed next slot make hard neighborhood counter indeed degree node power learn ease restrict algorithmic cascade specifically parent amount generalization enable implementation formally denote child infect recover less information diagram series q entropy e conditioning inequality apply need low lemma cascade decay corollary ensemble graph information bound set use time infected need order specialized case number sample particular degree subgraph thus parent super allow corresponding must big exact remove ml graph ensemble choose result sir cascade establish connection classic formalism mrfs e g introduction sir direct graph define parent e direction arise comment causality result sir sir infect discrete causal govern parent include encode history neighbor history cascade node neither cascade random mrfs classic formalism enable central notion therein collection every condition purpose graph variable clique direct graph connect either child relationship common child undirecte direct present child infection generate sir epidemic direct graph graph epidemic evolution graph imagine divide assume specific mrfs ise discrete pairwise require knowledge enable free test know typically causality generate apply sample leverage example phenomenon alternative formulation acyclic undirected cycle causality empirical evaluation ml greedy synthetic graph pick mention structure govern number infect set super leave infected grid cascade grid graph vary much presence cascade regular ml scenario regular graph see extent reduction sample complexity fact effect super direct graph twitter tweet feed put direct choose person edge weight parent simulate cascade use greedy super axis neighborhood h scatter number infect ml explain treat graph learn super example linearly linear cycle graph epidemic cascade time get infect require establish theoretic paper epidemic cascade extension infect weak infect cascade much ml result assumption reconstruct entire trajectory weak observation fall class propagation focus one decay cascade reach cascade seed reach would efficient case well interesting performance greedy greedy epidemic give ex pt lemma epidemic cascade infect offline online analytical impossible cascade cascade sir establish cascade require ml greedy reliably graph tree theoretic tight super cascade sir epidemic cascade infection keyword cascade field epidemic successive spread failure phenomenon nod large disease cascade recently example include network understand consumption medium e video news article spread twitter facebook keyword epidemic cascade optimize security reliability epidemic cascade spread complex epidemic protocol file fashion basis popular content streaming network learn vast majority graph small etc affect spread cascade cascade spread infer learn first find influential online network facebook access graph user link priori spread offline social practice seem twitter survey drive cascade structure learn cascade primitive investigate summarize many reliable recovery main characterize characterize observation need
sign impose reduce use allow detection dependency furthermore feature weight interaction simultaneously deal redundant redundant increase implicitly deal issue avoid redundant weighted help execute tend systematically solve perform constraint firstly assume feature procedure start algorithm step search eq middle find update accordingly find reach subset feasible rr time subset find sort k close head one estimation form product eq overlap kernel classification otherwise reduce simply gradient positive onto ball carry practice sophisticated e newton super gradient linearly take general method convergence rate rely per iteration line search find sophisticated solver actually decide efficiency instead every iteration significantly iteration approximately section report experimental result backward replace redundancy relevance state select mutual relevant solve problem obtain program trade low redundancy categorical correlation categorical experiment recommend weight solve varied another discriminate experiment toy binary bernoulli take probability chance bit flip characteristic redundancy dependency binary function characteristic interaction candidate selection candidate adaptively scale median available f adaptively distance convexity time randomly method f report correctly select feature measure none tb dataset lasso pc rank redundancy feature redundant forward necessary simultaneously f unstable instability incorrect heuristic become non inclusion many obviously make b since start feature well detect dependency feature sometimes redundant feature keep one eliminate strategy pairwise feature interaction therefore surprising fail information reveal dependency since feature interaction drawback redundancy consider hence dataset pc consider feature simultaneous feature interaction regularization combination detect dependency correctly choose usage attempt maximize manner flip inclusion feature illustration feature problem evident high inclusion drop extreme table considerably value cccc cccc demonstrate conduct experiment real summarize classification dataset cover wide include speech bioinformatics dataset task class per per class uci repository second international evaluating house repeat time trial low entire dimensionality kernel use availability selection competitive especially class pc multi toy experiment pc take redundancy among exception would problem pc dataset component component redundancy consider redundancy see suggest ignore redundancy many unstable another bold face method pair test significance mark problem large computation perform number time number choose alternative interact non interact explanatory detect interact c segment speech dimensionality help facilitate learn predictive cause difficulty include redundancy feature sparse importance keep allow interact dependency implicitly redundancy obtain usefulness method valuable comment international mm institute technology mm department institute technology feature technique less important feature exist algorithm redundancy select key characteristic world attention attempt interaction propose maximize variant output numerical handling redundancy feature mutual supervise number e learning could lead interpretability useful aim remove unnecessary target many show insight ability removal feature relevant explain redundant likely output redundant another feature two context explanatory characteristic interact consider reveal attention focus redundancy study consideration square information output experimentally art artificial real handling detect interaction formulate describe sect several measure argue list regularization weight account balance consideration mutual information possess desirable argument sect propose refer artificial sect sect formal realization feature attempt identify index cardinality index feature accuracy classifier yield classification interpretation wide applicability practice feature amount challenge np guarantee exhaustive since impractical quality issue sect quality sect strategy feature complexity optimization exhaustive optimization find strategy fast feature redundancy feature give feature form joint operator hilbert schmidt could xy k universal kernel gaussian dependence property criterion heuristic set theory dependency extraction mutual information kullback divergence measure vanishe density difficult avoid information accurate computationally expensive existence another square mutual pearson information log kullback like information go goal close estimation formulate least square minimize since find tractable learn admits find optimal purpose q respect call possesse selection fold nk k hold
consensus step accelerate convergence distribute average differentiable communication substitute induction constant induction compare side constant lemma mit edu distribute method private objective topology objective distinct share favorable suitable computation proximal communication network nesterov acceleration multiple per round distribute superior verify distribute enable storage agent number network objective local private information agent maintain optimum updating iteratively team explore regularize square fit distribute learn sample variation order gradient subgradient function computationally naturally implementation typically agent carry subgradient average exception distribute gradient communication converge additive distribute vary maintain update step take step negative function neighbor consist communication agent estimate current neighbor consensus stage nesterov stage bring estimate agent proximal step allow inexact centralized use centralize establish exploit objective proximal accelerate nesterov acceleration fast cite paper method global problem use parallel computation relate consensus grow optimization decentralize subgradient direction multiplier build seminal paper centralize proximal organize gradient establish rate verify effectiveness finally conclude open scalar centralized write average denote centralized standard product ij stochastic exist number hold concept establish key subsequent give section inexact characterize respect minimization optimization attain map optimality proximal operator basic proximal operator scalar view inexact proximal control iteration inexact centralize characterize inexact attain accelerate inexact proximal iterate eq indicate q sequence k inexact gradient verify inexact multi proposition allow omit sequence th order integer every therefore make q objective private function g ix hx n follow continuously differentiable gradient scalar exist subgradient ix first propose proximal problem initial agent k iy I k stepsize scalar represent vector agent communication agent exchange value value step consensus shall estimate serve method agent update estimate along gradient differentiable part multi consensus proximal respect objective follow nesterov type acceleration stage perform us error inexact centralized method step vary condition double doubly scalar say receive neighbor interval exist part agent equal update estimate current receive pair limit establish let shall decrease geometrically respect communication inexact centralized proximal inexact centralized method satisfy q lipschitz centralize inexact simple expansion inequality hz fact optimizer z hz combine desire bound term control step consensus inexact exhibit optimal section polynomial geometric establishe iterate sequence k x l give recursive let scalar polynomial geometric polynomial geometric proposition therefore polynomial geometric sequence distribute step consensus generate kkt take communication iteration number require execute communication great equivalently state proximal operator clear communication sequence error sequence address step consensus wish small guarantee become result equivalent nonnegative hold since
minimizer last follow specific let datum probability summary generative dataset refer list intended admit low dimensional model totally structure contaminate outlier I ambient dimensional ambient increase outli start chance scale energy dimensional parameter result ratio contain information behavior dimensional stability number draw except appear restriction comprehensive statement constant statistic outlier contain recover perfectly stability sufficient condition outlier seem fix ambient dimension l subspace number increment increase increment specific invariant heat identify outli ambient leave find dot regime capture qualitative determine identify experiment repeat probability fit theoretical bind empirical understand behave control definition statistic plain residual l square subspace exceed claim q noise stable regime stability suggest experiment basic perform linear subspace dimension space increment increment model determine close statistic projection give repeat heat blue heat map begin regime pca heat mean top valuable tool robust semidefinite interior problem favorable numerical reweighte whose weight evolve proceed svd exhibit type directly motivate estimate eq seem plausible minimizer computation water water appear box label correctness dataset solve matrix pt heuristic motivate counter weight solve influence increment box guarantee converge pt observation regularization stop pt satisfie initialize iteration counter initial increment optimal optimal objective fail convergence satisfie q establish schema summarize cost read keep typically point somewhat solve subproblem computation calculation arithmetic operation matrix spectral calculation arithmetic operation achieve calculation direct compute thin approach also iteration accelerate randomize page dimension curve associate choice ratio algorithm sequence detail experiment assumption iterate amount omit analysis indicate empirically experiment ambient outlier mark single unique run tolerance compare iterate package precision measure frobenius seem experiment usually much leverage iteration proof take machine algorithm terminate iterate error practice precise result degree reasonable much database numerical experiment involve formulation solving summarize recommend procedure algorithm pt pt center datum dataset dominant assume center center observation center q incorporate center modify problem omit detail second formulation address point solve reference program denote transform experimental idea pca finally proxy small exceed solve huge proposal limit attention formulation polynomial programming success largely consequence hard way standard pca spherical pca observation sphere pca perform et recommend reliable observation solve consist outlier comparison sparsity decomposition consider effective design several generalize single face database image convert pixel face subtract median apply spherical pca subspace choice experiment display project compute nine center every onto meanwhile face describe pca capture well different database original project tune noiseless xu al underlie ratio dimension model appear standard draw subgaussian method alternate high numerical experiment significantly fast strategy readily different situation observation scenario semidefinite relaxation work zhang minimum obvious equivalent indeed together implicit observation yield claim subspace specific tight relaxation find dimensional point analogous inverse determination dimension eigenvalue present know incorporate require requirement convergence idea broad perspective nonconvex problem establish combinatorial relaxation work relaxation proof maintain page statement proof section compare perturb objective see triangle eq justify third reach residual dominate become apparent restriction lemma insight behind involve side derivative direction confident take directional recalling perturb sum separately term involve l twice outlier page outli therefore q take limit determine produce low directional low bind component bind section alignment substitute expression directional bind finish relation draw norm involve consideration explain argument ingredient simple express comment obtain homogeneity thin value dl normalization unitary euclidean side concave construction extreme reach finally state block equivalence norm inequality equality third coincide trace exact model probabilistic statistic statistic follow detailed fix parameter let dataset accord page satisfie except verify demonstrate simplified collect numerical old reach probability bound alignment bound gaussian develop next except add side obtain second right invoke calculation rademacher sequence draw sum unit rest jensen expectation close dominate centering probability f frobenius complete develop alignment matrix know see whose standard normal cumulative column sphere argument distribute sphere gaussian variable degree freedom jensen normal cdf proposition complementary algebra use simple arrange recall ratio begin invariance imply next statistic independent center probability bound statistic satisfie complete contain argue near optimum solve problem correctness statement range eigenvalue q construct via quantity whose either enforce convention proof satisfie optimizer point simultaneously nonzero among therefore objective perturbation expand particular condition ensure derivative choice side continuity solve index eigenvector argument observe semidefinite negative perturbation optimal phase iterate matrix fact point achieve huber highlight interpretation continuously strict objective generalize auxiliary fact argument prove iterate innovation handle definition collect term next verify relate gm relate b equivalent monotonicity eq motivate stop criterion imply f require information bilinear inner strict imply span impose constraint set argument necessary constrain minimum function convex iterate line however characterize global nearly identification define dx x latter reach respective objective zhang support nsf grant dms dms award anonymous remark improve manuscript rgb rgb theorem theorem thm thm definition thm thm fact thm e computing sciences institute technology mc california ca zhang institute mathematics se mn explain along outlier describe reliably low use relaxation orthogonal formulation document confirm find lie near dimensional describe model huge array highlight bioinformatics pt illumination nine body lie dimension camera involve rank concern topic dimensional snp use correlate model difference substantial outlier principal linear sensitive consequence scientific year researcher alternative principal favorable low approximate convex proxy discussion dimensional formulation convex demonstrate cope outlier optimization seek norm matrix frobenius refer refer trace semidefinite symmetric dd trace range great exceed integer respectively extend transform motivate research begin principal explain residual approximation subspace classical method fit minimize residual elsewhere sum dataset appear pca linear algebra convex right nevertheless analytic datum column arrange svd extract principal imagine contain hope another challenging pca linear model less possibility residual sometimes orthogonal absolute book extensive way mathematical contrast tractable although many none minimum proposal linear involve intractable option analyze rigorous linear optimization propose replace straightforward equal eigenvalue enforce eigenvalue convex set observation dimension attempt tight formulation restrict argument effective rank provide good hull feasible impossible constraint reduce ultimately immediately clear auxiliary q norm solution dominant invariant precisely spectral entry weakly straightforward good fit even outlier solve issue accept original believe solve involve subspace two hull relaxation character robust type theoretical numerical indicate type model manuscript appear apply orthogonality approach suggest relaxation close outline develop describe dataset outlier simple appear section evidence effective setup locate outlier ambient property result gives summarize leave remain lie ambient structure contaminate l dimension space dimensional subspace dataset outlier model locate near hope assumption geometric check find subspace evidence outlier exhibit structure unsupervise justified finding let discussion imagine point must sure summary statistic design requirement direction statistic direction along approximate residual pca outlier major linear exhibit signature
cl bivariate margin explain either cl n frank std deviation create multipli car std cl h cl normal copulas lr lr lr c margin margin multivariate normal variate margin bivariate c lr paragraph copula std deviation create multipli car std prop prop prop prop lemma prop detect distributional change series often lack alternative change sequential copula contrast attempt rank computed respect consequence multiplier resample serial account scheme finite sample involve financial present give alternative involve f multivariate regard capture empirical appear little margin unchanged copula evidence detection particularly change practical change structure good copula obviously copula change test functional copula power aim powerful alternative copula crucial lie computation rank whereas rank rank way accurately copula detect quickly phenomenon empirical copula copula another illustration dependence organize follow asymptotic nan contain description multiplier resample validity result simulation defer rest equip uniform describe statistic difference ij l dependence tie process rank compute continue adopt convention write ns test copula aggregate er von aggregate thought find powerful er von nan value distribution limit multipli reject change joint could concern marginal copula change statistic differ copulas copula rank subsample compute process compare use process one obtain analogue carlo usually powerful copula intuitively copula copula nc empirical bivariate empirical course cite serial write sided stationary margin side link statistic difference eq focus briefly denote mix integer say side empirical define strong continuity absence dependent actually pointwise continue weak actually transform convention ns u aa state asymptotically uniformly turn trajectory probability one consequence continuous argument draw continuous margin satisfy corollary covariance compute resample back suffice resample partial observation spirit resample frequently compare behavior various resample result state later case multiplier seminal multiplier extend multipli bootstrap sequential key idea replace multiplier suitably serial dependence h way sequence appendix copy multiplier define multipli resample rescale rank arithmetic convention subsample x ns ms partial simple vary slightly upon follow canonical form effect ns nan reject estimate significance statistic multipli multipli multipli sequence condition copy multiplier strong copy multipli margin satisfy also strongly automatically appendix margin keep assume break assume change copula frank version obtain device hypothesis single occur copula one copula copula marginal rescale rank rank fast base sample package sake brevity table provide percentage alternative involve change copula copula serial independence change copula value margin occur besides conclusion reasonably serial table minor fluctuation statistic copula copula serial conservative size consideration base powerful alternative involve table power low distinguish copula basis low fact change point make involve detect copula another reading regard robust behave rather illustration apply computed index bivariate multipli explain value provide evidence course early decide basis nd compute daily composite former take package th approximate extreme occur period report sensitivity constant change achieve limit statistic resample sequence reject rather concern margin one change margin intensive rank calculation next speed grateful mark anonymous pointing might center nonlinear project foundation contract de e grant p science integer let margin quantile collect convention quantity replace decomposition supremum small ij complete shall case strongly mix statement map put j du jj continuous second side converge assertion boundedness third converge zero u tu assertion observe without ns ns assumption existence q verify asymptotically finally display desire convention sum zero decomposition definition respectively take different value observe exceed specify stationarity markov large corollary show n ms probability exist set u n sufficiently uniformly exist use fact ns
condition truth goal objective capture match experimental adopt fitness experimental free maximize focus art choose going define follow protein logic compatible logical give hypergraph weight finally combinatorial several condition protein logic experimental compatible want total number output compatible implementation systematically generate comparison logical perform identify compatible middle scale case dataset predict whole consequence see equivalent minimal compute together obtain minimal solve middle minimal second take second equal show score second enumeration take boolean show genetic material optimization hour predict whole dataset evaluate depict perfect depict compatible perfect appear number compatible therefore suboptimal run large genetic minimal phenomenon dataset optimization approach minimal recall find global clear nonetheless percentage axis experimental computation plot evolution experimental I contain maximum time score go minute hour outperform issue test conclusion formal boolean rule paradigm rely powerful free open source software main conclusion reasoning solution moreover outperform order opinion optimal issue robustness shall suboptimal gain whether biological relevance loop hope offer author project b partially support protein protein integer program logical steady biology mathematical logic become approach datum introduce heuristic solve paradigm encode logical program answer solution improvement efficiency guarantee well application cell via effect pathway gene define decade biological pathway exist public pathway sign oriented retrieve sign orient inside event signal molecular network vast concern type starting generate high throughput correlate protein protein include control modification key condition design insight control throughput establish comparison public resource integrate knowledge exist recently introduce implement www genetic logic compatible realistic suffer lack guarantee intrinsic train experimental equally optimal experimental regard explanatory related event reverse engineering assessment www project encodes distribute general public offer boolean constraint technology search logic program modern complex preference optimization global reasoning complete search obtain go towards prediction combinatorial logic represent graphical representation logic give motivate formal formally function logic relationship capture graph two protein encode graph either logic gate logical network represent adopt formalism example reader cite literature compatible logic formula logic sign conjunction green activation green node nod drug specie measure toy b logic circle whereas multiple informally logic good explain criterion quite intuitive appropriate way clear give base convenient protein logic clause three input sake loop mechanism transmission pathway include responsible signal input sign acyclic direct measure moreover subset mutually sign happen extract one database compress experiment correspond force tool experimental experimental function rv rv proteins active inactive great low approach future compare formal dataset appear definition recall domain property define example look hypergraph hypergraph capture evidence property logical positively resp formula redundant minimize model reduce formula redundancy logical conjunction
upper evaluate task audio corpus sub sample atom initialize extension use code algorithm number iteration result optimum spectra train dictionary reference line spectrum figure coherence inactive decrease decrease svd result middle increasingly unable consistently self coherence svd dictionary self enable maximize sparsity code atom efficiently dictionary coherence joint atom atom achieve objective code approximate demonstrate benefit coherence acknowledgment excellent implementation valuable feedback draw code learn dictionary successful denoise separation solve dictionary dictionary iteratively factorization dictionary sparse code dictionary coherence coherence class code self residual code goal high coherence self seek trade depend application dictionary effective self train enable trade sparsity approximate frame coherence particular code dictionary place densely region redundancy similarity dictionary atom angle atom self permit well admissible coherence increase avoid atom degeneracy enable span objective self coherence dictionary support generalization orthonormal contain unit span recover natural signal suitably achieve dictionary atom atom densely feature however redundant long unique need omp orthogonality atom maximum note definition self product magnitude therefore consider sec self dictionary magnitude dictionary formally exist minimum coherence self state sparse code recover norm curve omp al control gram approximate expert appropriate family provide adapt characteristic advance therefore dictionary suited source atom atom another atom lie atom representation atom coherent multiple atom practice dictionary self coherence fall self use independently atom algorithm pair inner bind increase coherence atom pairwise iterate step grow present self atom step sum multiplier possible trade sparsity code maximize dictionary signal parametric necessary furthermore make incoherent empirically task training incoherent improve unseen approximation sparsity jointly propose alternate minimization convergence optimum follow svd atom newly update replace large incoherent atom one atom replace atom therefore self coherence update fall atom well enforce constraint introduce atom optimize self minimize thm lagrange multipli trade
supervise account uncertainty unsupervise consider support national science foundation fa cs model online transition specifically achieve complexity unknown balance exploration exploitation able rapidly past rl allow mainly learner capability require interpolation process learn highly accurate provide determine implement control method domain accumulate maximize one rl learn g obtain physical system interact environment real able quickly amount robot interact environment good optimal learn interested state smooth dynamic typical primary keep complexity rl spirit extend continuous space gp part model dynamic environment transition prediction accurate amount implement non parametric gps give enhanced modeling flexibility automatic determination gps us implement exploration way bellman via concept key estimate bellman behind transition simple transition inclusion often sharp bellman actual bellman point gain efficiency compete advantage transition estimate conceptually closely relate fit interpolation primary contribution gps trade practical unlike could implementation learning approach make always dimensionality space grid grid bellman scale exponentially advanced grid allow somewhat increase curse alternatively nonlinear unclear approach really application curse still smooth state close close interpolation work fundamental gps observation small bellman operator rather learn comparable assumption justify criterion specify something generate sometimes elsewhere make amenable tb transition cm mdps reward justify control discretize assume discretize reward action rx x addition continuity fulfil domain violate e car else despite case still action accumulate maximize decision find bellman yield value bellman operator well number bellman uniform order solution affine function grid write grid cell interpolation invariant perform apply obtain apply action respect grid z vector associate point grid slightly discretize bellman contraction point control approximate posteriori residual grid cell modulus continuity continuous way rl problem agent environment sample observe behavior goal keep possible done use place sketch plan interact action discount planning interact current choose execute observe store evaluating criterion plan step exact determine else explain detail functional module essence theory regression particularly flexibility automatically principle via optimization marginal automatic determination relevant determine guide gps target estimate triplet perform dimensional change state variable independently univariate gps prediction respective subset e lack sketch subset regressor approximation element subset greedy aim minimize incur decomposition avoid degenerate variance gps gps rl also study offline gps functional parameterize tv constitute adapt relevance determination projection whereby variance formula find uncertain stacking together per uncertainty ensure receive scalar maximally maximally uncertain discretization planning stage plug let uncertainty reward solve discretize bellman repeat maximum reach necessary gr une case update empirically make agent space uncertain employ bellman suggest show become uncertainty either explore unknown explore exploitation take process discretization keep final call model stage plan summary model planning discount give see update reached examine various well comparative car hill car powerful hill necessary hill opposite car velocity possible reward hill reach maximal episode discount velocity single link car provide need energy create angle angular discretized remaining discount next consider balance state roll angle roll angle bar angular velocity inherently vertical turn usually experimental setup comparison start recovery impossible discount goal link robot height since problem setup state initial episode episode end state pass discount four negligible offline efficiency grid car balance tolerance full planning second problem min large spend planning module offline transition possible domain step balance total gp tolerance hyperparameter would computationally perform single optimal
share availability secondary result scheme propose algorithm asymmetric planning approach cognitive justify important channel policy primary rely algorithm suffer number channel regret suffer end consider play q refer one channel good channel play upper high index round inversion two sum second equality read slot equal number equality us share index channel play substitute combine previous lead end secondary primary limit every channel secondary select access primary rely algorithm expect number coarse verify least inclusion last inequality ease assertion write implicit km mb write second concentration hoeffding inequality exist km implicit avoid sub channel belong proof event km n mb consequently hoeffding km finally fr cs paper secondary exploit primary secondary environment need learn availability channel sense argue mechanism exploitation formulate mab share integrate interference limit learn ucb resource primary user definition channel spectrum access type primary user secondary user access resource dedicate service pool user resource area refer exploitation network challenge one observation thus substantial communication discard unable activity resource consequently interference quality resource realize secondary address key concept order implement communication user knowledge channel machine secondary among multi mab gain impact explore channel element cognitive scheme enable user quickly resource learn interference among resource interference entity single allocate channel deal uncertain environment first formulate network arm mab derive ucb issue share learn iteration unique per interference rely algorithm modification fair third fundamental result collaborative secondary network scenario yet learn sensor environment simulation rest paper organize work find consider mechanism instance job assignment detail discuss describe mechanism empirically evaluate section conclude strategy resource brief relate extensive allocation present various multiple secondary mab perfect channel acquire suggest channel multi user tackle lead nash equilibrium know since several paper suggest mab modeling tackle author context tackle model framework analyze draw bernoulli contexts modification take frequency approach decentralize error fundamental realistic scenario require sensor converge show hand alarm detection signal band free lead miss hand slow rely author complex empirical estimate instance contribution find user compete communication yet algorithm handle mention environment fact provide heterogeneous worth specific framework rather expect resource consequently mab framework refer mab rely close explicitly error homogeneous latter address detail primary channel appear channel channel suppose tackle time divide channel slot n distribution realization availability availability subsection generic index choose channel trial per see refer channel centralize decentralized detection denote sense ta k refer channel slot sense accuracy characterize usually probability detection observation channel finally policy policy scope depend outcome access access I interference primary small although send transmission interference occur transmission secondary fail channel channel receive numerical feedback transmission fail share reward share interface discuss aim promise resource reward adversarial exploit learn firstly environment resource index rely computation usual form index confidence bias resource q resource iteration design among round rest consider suggest method job mainly matrix opposite depend worker combination mainly invert solution refer discrete sampling time refer weight maker let refer operator division refer maker assign resource consider subsection making say optimization decision context network formalize resource ensure consider refer discrete define weight use row matrix iteration maker need fail moreover scenario worker solely relate usually detection suggest base resource allocation compute section share enable quality theoretically network among rely previously notation throughput slot equal free observe consequently achievable channel equal eq job context usually throughput expect slot achievable expect throughput implement channel slot symmetric symmetry channel slot every combine every common index slot symmetric optimal compose channel round describe fair collaborative symmetric bound symmetric secondary user primary channel assume secondary access suffer notation matter expect state upper one hand refer behavior report suggest verify selection environment optimal fundamental allocation mechanism environment tight exploration parameter notice converge one previous scale heterogeneous environment piece answer information paper communication interface cognitive share channel receiver hand purpose assume without thus purpose adaptation receiver band communication configuration solely receiver outcome receiver transmission message receiver receive slot communication dedicate compute reward consequence fast rely outcome band aim herein mechanism describe consider scenario result secondary analysis numerical value false impact analyze resource tackle user secondary user divide set symmetric user spectrum secondary mechanism channel inefficient learn condition conduct generalized scenario average horizon regret four illustrate collaborative iteration theorem vector illustrate matter base slot collaborative perform individual regret symmetric learning quality enable observe figure handle notice figure quickly
show evaluation multimodal black however black function train big unit without duration cloud make actual unit desirable understand concept cost second parallel multiple core optimization advantage good learning argue fully parameter critical hyperparameter examine default exponential core experiment kind interested find take bayesian evaluation local hessian find evaluation determine evaluation perform easy justify extra computation formalism review discuss contribution two choice make perform must express optimize choose must utility determine evaluate prior gp property induce elegant marginalization conditional impact section draw observation variance induce acquisition denote evaluate proxy depend observation proceed denote good standard intuitive gp analytically alternatively ei current development idea exploit consider construct course acquisition exploitation show behave unlike upper require improvement function several machine unclear consuming vary take core parallelism onto modern solely non degenerate correspond function determination ard ard mat ern differentiable quasi newton require squared behavior bayesian problem interest hyperparameter estimate result input hyperparameter alone compute observation estimate integrate sample acquire slice chain computationally assume sensible ultimately find quickly possible acquisition expect improvement evaluation execution rate improve optimize good quickly cost resource know use independence duration ask generally batch parallelism however evaluate clearly repeat experiment roll acquisition policy appropriately balanced exploitation roll intractable monte different complete yield ideally new acquisition evaluation simply whose covariance hyperparameter straightforward acquisition select would operate note attention monte highly effective compare machine gp ei optimize hyperparameter ei opt ei ei time gp ei ei tree common bayesian define compare classification task hyperparameter gradient batch epoch run error report integrate ei outperform logistic grid dirichlet allocation lda direct document generate learn variational minibatch word minibatch exhaustive search hyperparameter make online wikipedia article article split validation word count document must specify weight follow analysis lda generally take five hour approximately processor compare strategy expensive restrict repeat time pick report figure achieve compare number time evaluate duration optimization run choose optimize expected improvement integrate hyperparameter ei evaluation ei find significantly figure ei well run fraction min svms svms dna sequence hide term svms consume grid search report hyperparameter avoid find computationally however entropy parameterize outperform expense protein tolerance grid possible split figure grid search ei mcmc version constrain term time optimization strategy status mcmc superior ei gp ei learn strict exploring parameter ei least compare ei problem optimization error selection critical infinite function commonly restrictive tuning multi thorough beneficial demonstrate often prohibitive explore numerous hyperparameter remain nine layer network cifar code tune human competitive publish cifar epoch softmax scale layer optimize nine strategy validation five separate run achieve well find ei error expert cifar bayesian hyperparameter
distinct iterative aggregation detail centrality score item comparison graph centrality efficacy popular multinomial logit comparison associate determine probabilistic comparison sample laplacian evaluation synthetic estimator popular aggregation task range contexts recommendation system item wish order item player player book book display big collection book pairwise movie compare preference wish score result engine assign online outcome use score computationally work comparison datum reasonable aware question rating rely interest assumption research recommendation matrix arguably provide individual user inconsistent lead aggregate rating design aggregation challenging item study potentially existence set axiom paper interested outcome item consider want score indicate intensity set imagine ranking permutation item time generate permutation become hard stop scientific practical designing fall work call model logit research design price select winner round winner assign player centrality research global ranking item line research refer survey choice pair lead eigenvector slight accounting multiple spectral propose centrality rank eigenvector form construct markov similar ranking build classical novel showing near popular formally popular open exception axiom satisfied axiom make spirit centrality novel ranking justification work ranking comparison several distribution difficult know ranking provide generalize partial expectation scheme comparison provably variation pairwise might comparison additional margin win team score team lead score score generalize traditional distance comparison whether exist meet criterion count comparison entire perfect pair wise marginal limit author order adaptively pair underlying observe comparison true rank centrality produces propose nice intuitive explanation item player random likely go vertex ever depend lose random give immediate neighbor effect comparison capture walk node wide graph notable include version web page share network assign trust use nice rigorous property establish choose unknown underlie centrality note even produce probability centrality may comparison object comparison arbitrarily os enyi strictly positive high compare option show choice summary centrality always remark analytic study walk chain like scenario relate chain comparison use dirichlet form characterize reversible adjoint operator expansion graph result order position position remaining choose verify ordering rank notation comparison pairwise graph connect comparison pair weight edge object fraction weight set pair walk independent entry non negative valid walk maximum node rescaling ensure sum one finally ensure row exactly add node define reversible reversible ideal I proportional chance item item ideal act markov hence least self unique adjoint behavior set due surrogate start distribution large eigenvector stationary top makes formally assign call centrality rank centrality rank distribution suggest alternative receive high object preferred remain since state fix markov chain irreducible interestingly meaningful complexity derive find solution implie per laplacian identify play ability score special compare os enyi graph bound require scale optimal complexity bound rescale euclidean underlie stationary quantify prefer present simulation ranking day international centrality rao effectively comparison characterize centrality notation node min degree laplacian thus think reversible jump neighbor probability random normalize laplacian real eigenvalue denote shall eq eigenvalue walk symmetric laplacian establish graph outcome produce hold probability enyi logarithmic guarantee specifically comparison centrality score comparison everything choose pair compare comparison produce b constant remark immediately go os number thus centrality range resolve score consider regime order great preferred consideration graph centrality effectively centrality require ignore performance centrality reversible random matrix mix walk end play centrality choose pair result gap play goal compare oppose preference wish n metric ordering order iw indicator natural unweighted loss normalize connect begin e representative depict show
replace jeffreys jensen shannon discriminate density class employ shannon gm divergence generalization jeffreys divergence multiple good maximum discrimination naturally binary comparative text dimensional divergence jensen categorization broadly divide generative conditional class generative classifier include discriminant lda directly without function minimize error term classification discriminative classification example support generative classifier small counterpart require training contrast tend incomplete take care entire incorporate allow close datum generating knowledge incorporate easily lastly understand discriminative counterpart method take divergence minimum proved language processing curse dimensionality maximum entropy model generative obtain early number tends prefer discriminative also allow incorporate classification criterion call entropy basic present study binary extended class several binary among unique classification main jensen divergence divergence well arithmetic study jeffreys study accuracie discriminative portion notation label binary space propose conditional form function individual observation q form I presence maximum moment function statistic mean sample draw unknown iterative like descent iterative scaling version jeffreys simply divergence give context problem law divergence divergence jensen appear relatively recently characteristic divergence combination negative symmetric interpretation connection relate seminal entropy text use data label maximize naive propose maximum entropy feature notational assume notation us surface hyperplane strategy feature scenario select maximize class ideally would address classifier bayes label quantity role hyperplane former hyperplane class using indicate I train separate term get divergence distribution similarly instance class classifier indicates given subset I use particularly number high set et feature thereby reduce divergence especially since naive classifier optimality loss simplification greedy extent divergence class class write set p divergence discrimination step deal classification invoke natural distribution problem suitable interpret multi classification similarity class datum mix mutual prior classification basically deal give device among divergence turn measure discrimination divergence property equivalent case become greedy maximize among information latter complexity feature linear text theoretical need also linearity divergence discrimination discriminative measure drawback motivate form notion average however similar basic divergence replace interpretation discuss weight symmetric observation usefulness divergence stem multi divergence help difficulty extra exploit nice follow easily equivalence replace weighted class form feature I divergence calculate rank divergence decision assign density bayes scenario natural performance discuss c testing rank approach multiclass multiclass approach svm discriminative use I cross moment global popular commonly summary well introduce refer algorithm special computational algorithms complexity various feature vector choose rank svm assume svm implement mention technique affect dimensionality addition quadratic complexity indicate I distribution appear unstable skip algorithm determination involve considerable require numerical calculation worth datum hence estimate twice following thereby make twice estimation use moment variety gene number use divergence obtain fold randomly distribution I cross highlight observe distinguish disease prune strategy away gene validation feature os windows window mac mac category experiment class group way different multiclass remove frequency less much corpus jt number corpus represents contribute document table classification propose use fold main
accelerated newton classic adjust curvature step iteration method store full demand task problem ease burden hessian difference iterate use lead dynamically curvature storage acceleration mm employ comparison acceleration scheme direct reader early review instance broad term unconstrained minimize unconstrained f auxiliary function identify g alternatively unconstraine satisfy satisfy sequential unconstrained generate iterative barrier method backward reference mm unconstraine obvious meet speak closely iteration nonetheless since must iterative derive iterative ultimately convergent surrogate globally algorithm ambiguity algorithm highlight mm fortunately carry broad set finding idea well know projection attribute close simultaneous projection enjoy advantage invoke operator find np p two intersect derivation simultaneous projection bregman divergence generate strictly bregman generate bregman bregman onto bregman projection equivalently function c brief denote hessian reader thorough treatment proximity set consider intersection minimize optimal minimize contraction coincide point stationarity point way euclidean euclidean w suffer instability rate accelerate mm systematic quasi newton acceleration reduce number sake describe dual nesterov dual solve dual dual consist I subject primal reduce I maximize sake clarity novel derivation iterate primal respectively derivation nesterov fista appear projection regard project onto doubly nonnegative numerical project symmetric intersection matrix example entry projection onto set project nonnegative accomplish matrix accomplish decomposition outer generate identically project simulate onto quasi acceleration fista decision switch update track progress algorithm eigenvalue newton path reflect switch point obviously amount vary method record include second frobenius fit equally version acc shape criterion readily amenable projection cone accomplish pool arc component project component fitting pair space geometrically penalty constants newton acceleration next result fit track progress constraint solution rule compare fit improvement accelerate version fast acc square convex predictor vector weight seek residual view convexity preserve define concave impose interpolation accomplish minima maxima reduce semidefinite inequality feasible region nontrivial w jk j jk ip jk c jk admit w w nn sum mm reduce p jk jk nn n display point least geometrically took five quasi acceleration second total objective maximal vector discriminant predictor svm classifiers penalization f constraint pass intercept negativity yield iterate except entry report svm set geometrically sequence iteration maximal consider handle optimization choice include cholesky ij essentially cholesky cost huge trivial cholesky factor fast eigen example distance distance indeed city occur rectangular guarantee short spread city norm treatment arbitrary hard calculate fortunately side component I objective minimize separate side sum underlie projection point euclidean distance broader relax requirement widely robust convex convergence prove strong comment algorithm subproblem mm monotone instrumental proving set compact minimize continuously meet scenario assumption eigenvalue hessian take rule practice know convergence unknown nonetheless determine classic point mm converge unique iterate argue turn entail quadratic expansion inequality g convexity g single minimizer proceed check proposition easy verify map take arbitrary descent f stationarity c continuously differentiable limit g strict property uniqueness continuously cluster sequence fix conclusion iterate converge stationary possesse represent converge require sharp ascent ascent sequence penalization parameter expect subproblem possess minimizer subject constraint one justify converge minimizer boundedness feasible principle distance formula parallelization intersection variant competitive program raise question rough insight encounter formula several nonetheless devise area thorough constraint employ advantageous introduce change qualitative conclusion may improve useful parameter estimate although convexity long theory show relax extend target feedback thank associate constructive suggestion comment unconstraine minimization attention connection mm research partially united states public service grant gm appendix ascent algorithm iterative dual project convex reader reference proof background low relation indicator close convex convex thus function bound zero convex attained lipschitz ensure associated function convex convex denote recover gradient important algorithm class ready consider slightly general convex intersection minimize rise ic iff f therefore since separate proximal denote size simplify p combine identity lipschitz thus actually require convergence convex early n mm fail quadratic bind algorithm employ figure depict global fail algorithm geometrically iterate convex reduce descent step size lipschitz circumstance condition maxima minima maxima remain hence meet helpful
proof generate tend certainly constraint merely conditional constraint method causal relationship iv constraint constraint possibly form understand difficulty contain control effect directly generalize control effect yx let vertex appendix always distribution two directly bound compatibility exclude rv style draw minimum mm rv style xshift edge corollary case joint z kp jk unlike strong condition iv program possible constructive crucial determine test graph specifically give symmetric likely graphical approach dag separation derive fourier elimination size infeasible doubly exponential elimination check theorem search intensive case could highly advantageous intensive search benefit much easier intuitive human user iv take state find practice theorem dag observe dd element condition valid conditional density dag remove edge vertex property say pa dd cp unchanged give compatibility pa give strong conditioning factorization clearly pd expression maximize sum take minimize tight analysis lem lem lem lem lem lem graphical acyclic dag neither adjacent latent parent generalize instrumental inequality effect characterize term separation provide acyclic graph dag commonly intuitive display dag identify conditional possibly bias remain marginalization understand merely term conditional independence causal identify hope intensive spirit use elimination determine construct dag relate terminology new derive instrumental applie order say set parent denote adjacent path towards direct path say direct vertex associate vertex admit convenience variable bold letter refer dag say joint v internal vertex call vertex whenever criterion imply dag interpretation dag extra system intervention edge parent dag implication margin latent observable identifiable underlie impossible presence assumption space latent discrete state obey iv inequality specific clear derivation interpretation see adapt provide value discrete obey iv p z p p z instrumental suppose iv effect break regardless factorization z equivalent satisfying quantity equality give dag give instrumental insufficient elimination become moderately sized rv thick minimum inner sep control possible effect give strength shorthand generalization construct eq py py z include instrumental next provide graphical graphical criterion dag vertex dag disjoint
set assumption class metric limit criterion see low distribution property asymptotic claim specific hypothesis learnable binary hypothesis distribution hypothesis class limit hypothesis distribution tight fully characterize thus tight significant desire chance statistical asymptotic tight depend moderate machine margin achievable error write multi refer element misclassification margin receive return classify object size distribution margin define follow excess distribution specific fix require general provide regardless specific sometimes omit simply separable stand real stand denote unit w homogeneous x x w uniqueness sort fix vector matrix eigenvalue use follow constant stand constant rademacher independent variable rademacher respect hx get desire upper margin appendix classic let function dimension class class pseudo linear classifier bad dimensionality nonetheless neither unbounded dimension arbitrarily depend converse dimension arbitrarily norm bound loose tight direction dimension sum dimension average direction capture let integer sub x margin minimum drop x minimum statistically independent first covariance generate small latter quite former ease carry dimensional hilbert eigenvalue imply unbounde complexity cover space cover covering follow class choice sum require notion hausdorff eq minimal f provide mapping hausdorff provide hausdorff distance lemma let let domain dimension dimension follow subsequent corollary integer value consider therefore generality henceforth write x w projection dimension x like class equal w f hausdorff cover pseudo ff theorem eq q theorem bound equation adjust fact side note bs upper definition conclude second right one want complexity matrix eigenvalue example moreover estimate covariance matrix necessary tight dimension answer need relate bind relate sub gaussian prove family closely evident formulation hypothesis class distribution hypothesis linear sample least prove simplicity otherwise replace set addition point hypothesis misclassification show h whenever return h argument last simply half least hypothesis domain uniform label origin return alone suffice hard homogeneous high present simple characterization condition eigenvalue gram require auxiliary lemma state exact margin lemma provide representation pseudo rw w theorem rescale sufficient necessary invertible invertible necessary origin hold exist invertible ready let matrix generalize linearly probable matrix formulate let dm theorem generalize linearly hope extend margin sample bind say sub variable relative also family distribution quite sub inequality gaussian moment psd random useful sub moment moment random define distribution moment orthonormal relative matrix multivariate corner hyper full rectangle low constant depend regard small gram assumption dimension approach example series matrix finite fourth asymptotic use random fourth moment draw equality control upper separately eigenvalue product proof appendix state dd orthonormal direction define natural w draw let low norm control sub moment equation distribution theorem show upper product bound moment margin dimension fully complexity follow two identical sub gaussian direction let gaussian moment moment projection moment point provide sub relative formally depend result characterize adapted dimension conclusion equation spherical alone logarithmic logarithmic stem margin might logarithmic equation characterize behavior improve gap irrelevant feature regularization regularization bound establish one ng assume use low coordinate bound low sample gap discriminative spherical mean v looking calculate generative spherical classify estimate ensure learn abundance unlabele costly active need decide current without abundance estimate distribution adapt dimension calculate easily unlabeled label bind far would information show true margin adapt characterize comparison characterize complexity matrix great conclude lemma since g g sg exist vector yx ki eq dimension prove induction inductive denote projection z r z r denote moment dy tx x expression ds hold fact ms x x prove squared moment psd b therefore eq method matrix fourth lemma work assume eq column far center psd value let ij I ij ik ik index sub matrix row j substitute brevity constant later since inequality l row column change moment lemma
enkf enkf solution enkf first figure right stage start ensemble forward twice dot enkf look apparent enkf linear tradeoff moderately enkf quick turn prior produce fully inverse wishart spurious specification function give produce update ensemble finally note bootstrap useful accounting run carry describe make use enkf explore enkf determine take suggest involve aid computationally demand perhaps simple g scale universe background dark composition fluctuation universe combine digital survey give map galaxy body evolve matter history begin big year physical produce frame slice map galaxy spatial galaxy predict matter universe require effort set scale carry simulation dark matter background force middle frame simulation spatial physical observation digital digital body right matter datum periodic galaxy censor snapshot structure dark spatial matter scale matter periodic cubic lattice since difficulty geometry bias challenge publish line diagonal deviation frame run array table spectrum draw prior application enkf cccc upper spectral galaxy amplitude dark prescribe spectrum gray line frame physical parameter incidence select correspond physical along ensemble enkf describe section vector frame figure presence skewness fit uncertainty give right figure distribution triangle member circle black dot contour black posterior pointwise credible spectrum density universe light line ensemble enkf produce far elaborate statistical posterior similar estimate gp application come effort measurement well ice ice rectangular location ice distance exchange use approximately depth measurement guarantee configuration minimal determinant multiple anneal enkf community sample dimensional range apparent physical cloud ice cloud particle cloud air consumption mb cloud density warm average relative threshold warm ice air temperature ice heat radius cloud environmental air cloud cloud ice fall cloud cloud ice produce could output explored record two air temperature observation directly observe orthogonal long pilot air temperature show green model dot basis element output combination summarize correspond pilot variation output scale basis actual physical light green prediction dark average compute observe simulated field basis produce long pilot black dot solid black variation compute pilot run dash due discrepancy light sample run dark line update scale standardized air temperature field pilot least variance discrepancy precision discrepancy give dotted line discrepancy air temperature govern prior dimensional model define get plug reduction basis pilot vector air figure ensemble sample uniformly rectangle depict figure ensemble rectangle formulation discrepancy proportional physical result enkf ensemble run finally point enkf collect ten successive synthetic highlight enkf show variety enkf use mapping make easy ensemble often gp well deal output helpful enkf enkf start fairly little depend specification relatively exist literature tail clear analysis simple largely implicitly use enkf see ice compare ability enkf quickly rough ready demanding task problem computer kalman sciences laboratory laboratory sciences group national laboratory sciences laboratory institute physics division laboratory high energy physics division laboratory price laboratory fig ensemble enkf effective quantify estimation assimilation weather successively aid forward next recently enkf prove effective estimating describe reservoir history match challenge model calibration involve well computationally demand forward calibration combine observation explore enkf calibration consider keyword computer validation assimilation quantification process ensemble filter enkf prove quantify weather model assimilation dimensional successively physical aid demand enkf ensemble kalman produce update ensemble affect enkf involve describe reservoir production time history especially since involve number demand forward model assimilation static evolve explore enkf collection constrain go enkf address end strength enkf model inverse denote map physical observation noisy horizontal denote result inverse order inverse inverse forward produce physical normally specify posterior trivial deal range demand forward experience range second monte mcmc remain popular explore vector forward model inspire recent focus effort simplify multiple parameter produce conditioning physical four run exploration process gps computer produce infer forward incorporate model function function typically product require dimension take gp conditioning forward example fix variant enkf run result posterior draw paired simulation ensemble covariance setting draw simulator enkf normal motivate calculation variant enkf calibration leave figure vector eq simple problem vector nd write element simulator output normal rewrite commonly filtering gain computation effectively assume relationship induce enkf kalman enkf depict symbol distribution depict gray marginal density ensemble dot draw second basically enkf time member order member whose match posterior enkf update simulator simulator
terminal either load color package graphic explanation terminal graphics macro ltb lt lt lt lt lt ltb lt lt lt lt bp ltb enyi match right generalised random graph cutoff gp walk show rather subtle correlated limit reach cycle large threshold kernel reach walk typical globally variance complicated manner structure undesirable prior normalise deep understanding apply approximation fail regime outline cavity method approximate average global show vector output match obey component independent identically distribute prediction vector n conjugate case effective vector teacher differently normalise change matrix weighting prediction simulation curve list large diagonal plot generally end region spike vertex figure normalise globally normalise student globally normalise teacher locally normalise student clearly large spike edge dy typical enyi match red globally normalise teacher normalise pt dy cm normalise student normalise teacher kernel similarity space graph counter use variance euclidean undesirable modelling walk normalise variance analyse consequence study curve prediction curve obtain depend belief propagation approximation gaussian process cavity belief propagation field wide attribute mainly nature ease kernel space relationship encode ease gps set gaussian explicitly rule require error trace average error example study variety understand parametric parametric gps euclidean graph connection relation include financial space graph become understand expand curve gps output graph rest walk particular correlate proceed walk gp section looking function local extend accurate asymptotic regime curve core development normalise belief propagation act warm locally normalise compare numerical simulation find agreement globally locally normalise qualitatively normalise gp normalised versa result curve prior arise fundamentally different monotonic discussing complicated everywhere jj fundamentally accurate broad range parameter last surprising continuous like correlation space calculation correlation avoid trick inner kernel machine result develop wide kernel normalise graph vertex generic encode connection vertex connect otherwise exclude denote known matrix construct conjunction load graphic need graphic macro ltb lt lt lt lt ltb lt lt lt r ltb ltb ltb ltb ltb ltb r ltb ltb ltb ltb ltb r ltb ltb ltb kernel graph calculate conclusion kernel value short apart remain limit cycle focus regular qualitative vertex study walk completeness gps gps machine direct posterior input correspond observe choose covariance matrix focus gps output corrupt independent identically noise target teacher noise cx cx cx simple loss preferred correspondingly kernel belief smoothness expect amplitude predict one desire achieve location kernel spatially graph connectivity vertex along among usually undesirable model unless justify return depend dependence general imagine show variance walk globally unity ci gps unity vertice single instance present give poisson distribution figure generalise power generalise graph edge assign one generate superposition p dpp know large enyi locally relatively prior even give specific example spike vertex spike edge subgraph vertex additional increase weight opposite subgraph spike large even step would occur centre star illustrate alone degree constrain vertex end probability theoretical prediction large limit fine peak agree linear rather broad tail power feature variance similar unity spike subgraph slow exhibit significantly large accordingly value two constitute vertex retain spread latter complicated ij ii ij kernel euclidean vertice variation computational unity plot figure gp remainder describe terminal option explanation load graphic see graphic lt lt lt ltb lt lt lt lt lt r degree limit generalised exponent cutoff performance non method study square student teacher prediction give teacher teacher average simplicity assume input vertex graph average still specific consider average ensemble degree specify degree equivalently degree degree mean analysis therefore broad mention regular graph enyi power law generalise typical curve teacher distribution likewise assume teacher noise simplify student variance value note eliminate output average general calculate input enter see situation learn prediction euclidean space case gp predict graph broadly ensemble extend discrete graph see fully belief sketch derivation include treatment cavity mechanic eq input enter count average logarithm partition cavity discuss exist graph cavity fully replica power replica index independently independent poisson average average evaluate eventually equation variational justify exponential truncate hand fluctuation issue prominent order unity fluctuation example nearby posterior vertex posterior fluctuation mathematically proportional close contribute compute normalise walk dot centre generalise right compare show line range noise approximation justify large expect package color conjunction terminal explanation load color package graphic explanation terminal graphics macro ltb lt lt lt ltb lt lt ltb ltb ltb r solid numerically graph dash cavity note visually indistinguishable simulation centre law generalise exponent cutoff move cavity look curve focus large regular trivial large stay important conclude naive bayes unity reduction v away vertex volume get distance proportional large summing grow detailed large proportional characteristic cutoff distance decay decay eigenvalue graph identical regular tree explicitly normalise overall color option explanation explanation terminal graphics macro ltb lt lt lt lt lt ltb lt lt lt lt lt lt ltb ltb ltb ltb ltb r ltb ltb ltb ltb curve master curve slow curve numerically far eigenvalue numerically simulate fluctuation graph detail average indeed cavity rely local nearly indistinguishable cavity curve must function relate vertex create factor arbitrary eliminate cavity begin ij eliminate transform integrate term respect coupling link vertex remain interaction binomial walk variable pc pc sake uniformity instead interaction hide definition additional represent via dirac delta fourier conjugate get adjacency interaction graphical apply belief propagation focus globally rescale interaction adjacency product respect prescribe order eliminate subgraph locally tree root approximately cavity create iteratively subgraph belief propagation cavity message vertex factor cavity self consistently character cavity explicitly correspond matrix rather translate equation ultimately large degree individual cavity edge pick random edge degree vertex pick incoming cavity distribute sample cavity equation vertex label omit index example vertex simply also replica element cavity result allow vertex exact worth learn prediction specific regression gp applicable level briefly isolate vertex careful avoid division end find consequence vertex datum give globally normalise cavity eigenvalue regular enyi generalise greatly outperform confirm cavity already graph vertex cavity globally normalise kernel cavity fraction bayes dynamic predict posterior become variance cavity without prediction distribution go cavity numerically distribution prior cavity distribution cavity provide particular detailed tail agreement fine peak shift rare cycle also cycle cavity cavity normalise walk argue probabilistic define normalise challenging calculate ensemble normalised scenario account block cavity update set bayes trick well behave expression parameter determine way predict really want normalise analogue dropping write graphical iterate equation cavity cavity form define equation specifically update one calculate large outcome cavity cavity covariance cavity normalise couple look joint replica method fix equation globally normalise second covariance marginal covariance subject marginal constraint approximated globally normalise reflect cavity message first delta cavity cavity may enter represent cavity send cavity belief set cavity message reach set apparent reason full cavity case look curve normalise kernel normalise cavity show figure cavity enyi random graph law generalise centre globally normalise cavity accurate even capture aspect curve discuss detail package conjunction terminal explanation either use package graphic graphic ltb lt lt ltb lt lt n ltb ltb ltb r random line simulate curve mostly visually indistinguishable result show single line centre generalise exponent curve globally solid line normalise random right law cavity locally normalise indistinguishable curve simplification drop consistency requirement look cavity prediction value curve package explanation color graphic explanation graphic ltb lt lt lt lt lt lt ltb lt lt lt lt lt r enyi vertice cavity prediction prediction curve gp regression global plausible avoid variability prior trivially relate local justify compare kernel well normalise match case kernel kernel bayes reflect answer kernel error try
lemma q regard difference operator simple derivative refer convenience define p consider quantity taylor dt use u u conclude equation apply taylor equation exist dt v dt dt result u use bind w difference ti jt jt jt jt jt jt jt v jt jt jt jt jt jt jt jt jt v jt v exploiting jt exploit also v curve repeat argument recall know hold particular bernstein let independent eq adaptation use coefficient follow condition independent true conditioning behaviour x k v rt w v fix random variable satisfy constant twice focus fix last line hypothesis continue bound integral recognize incomplete l dt e l dt dt dt e standard piece together draw satisfy also lemma processing consist model signal atom paradigm range support particular rely minima analyze paper admit local study dictionary noisy extend previous limit dictionary key quantity coherence modelling signal field framework signal design dictionary prominent effort dedicate g wavelet representation notably many processing decomposition simply therein although sometimes formulate submodular widely definition coding bring sparse dictionary instance quantify differ non uniform later explore consist characterize minima non formulation recover identify important direction signal possibly corrupt outlier compose atom knowledge carry signal straightforwardly noise point discuss local code presence make contribution within probabilistic derive low local reference understand around reference hope completeness result integer extensively frobenius ij extract conversely n square diagonal vector matrix sphere v set pp atom learn ik aim reconstruct atom introduce signal lasso code level sparsity paper processing however context topic atom simplex characterize minima signal accord draw precisely regularize neighborhood subject signal already blind permutation instead affect issue soon carry dictionary unit turn consider unit column parameterized stand v leave drop dependence correspond tangent frobenius w parametrization scalar proof sec curve onto appendix describe require lipschitz minimization since non moreover however close favorable leverage key denote p property appealing make possible form remark sign q sf nf n ingredient exploit sign least understand reference therein ic machine control support square reasonably reference dictionary quantity eq measure correlation column deterministic coherence dictionary coherence light transfer proof rip weak coherence assumption however face ic use rip coherence rip equip present signal dictionary sparse build generation draw replacement define generation yield support whose I background variable g simplicity case formally eventually signal x model prove admit local size detail main local minimum model k follow hold admit minimum highlight distribution keep main dictionary incoherent signal guarantee ball resolution coherence condition see impose slightly hadamard dirac addition handle regime resolution minimum term rely recovery worth note dictionary coherence follow generative reference orthogonal minimum sufficiently relation previous remove unchanged limit mention obtain noiseless result noiseless analysis noisy work differ linearization equality technique easily noisy building structure proposition control many concentrate high correspond bilinear form concentrate uniformity net collection coincide number large recovery signal get seek well rather appendix cr exact recovery dictionary sufficient w event turn question comes study regularize square problem associate see conclusion turn go away noise depend statement proposition supplementary material previous place v conclude highlight experiment zero coefficient uniformly invariance see auxiliary hand know sign close frobenius assignment absolute solve fix epoch random dictionary begin hadamard dirac fix k noise primarily limit plotted fig hadamard mp speak decompose function residual discuss surrogate function building expect concentrate uniformly bound assume second uniformly x instead event around relax expectation seek rather equal show dt coincide definition x function coincide uniformly zero hand indeed r cr exact sign dictionary v control proposition exact recovery consider v modify handle simplify noiseless recovery perturb noiseless signal satisfie almost dt residual find small signal proposition set recall induce ball local minimum recall sphere positive accord prove hence formalize lemma give vice w v result case column uniquely previous vector handle thank convention define j j reciprocal v j come column product term w j p consequence conclude exist finally observe conclusion general c stand universal make thank sufficient k constant keep hidden success theorem even conceptually noiseless regime follow q noise handle jj function particular imply stems read eq c positive radius tt display surely main consequence simplify residual surrogate coincide almost code radius depend simplify moreover ask lower bind positive p universal explicit choice c jj second c define onto w jt jt j jt c dt dt q jt next norm isometry set matrix z z dt matrix control coherence dictionary k q similarly matrix briefly prove introduce positive definite term I norm obtain stand one four corollary computation normalize finally normalize column pi simply v v p I pp overall approach concentration model lipschitz showing lipschitz control together estimate size signal proposition six denote denote w x observe union bind conclude average w lemma exploit cover p respect background cover reader net euclidean resort sphere second dimension net conclusion metric ct check hence n mp mp mp mp mp mp mp mp recall ct k obtain ct l ct ct l ct observe moreover dt estimate vanish
identify convenience sup define closure sup thereby ensure complete classical fr inferential approach relate empty separability sufficient non empty observe event trivially certain prefer value neither existence sequel sample fr space equip allow space metric identically mode mean sometimes denote linearity lebesgue probability assume integrable key make generalization take metric space need class separable whereby real value every every fx fu say indicate rao p value separable dominate continuous integrable ii exist albeit closure within due fr every fr mean quantity pair fr converge q definition tr inequality tr q number maximum prove proof construction choose mention invoke continuity fr hence boundedness sufficient guarantee assume definition mean belong point locate respect square fr locate interval author highlight fr interest boundedness family find prove restrict fr original argument thereby uniform appropriate weak assumption aforementione theorem remain valid fr mean result variable commonly define consideration group pt remark proposition axiom sample mean support air office scientific research whose grant fellowship uk national institute health center mh south foundation college conduct like useful fr mean idea space fr graph fr necessary sequence sure fr mean value pseudo metric exploit fact metric bound fr order generalize metric square summary analog expect value abstract show specification space interest fr minimize notion abstract physics riemannian manifold sample fr cumulative produce convex metric space negative curvature mean interested simple separability mild topological boundedness metric condition modern statistical likely bound bioinformatics dna naturally family commonly similarly bound space albeit metric lead node fr fr sample pseudo compact metric two publish conference probable particularly proper bound subset riemannian manifold make apply manifold mild fr sample non take riemannian manifold take metric boundedness strong fr mean space indeed space difficulty arise special issue also space metric consider several metric hard restrict fr greatly simply importantly fr show fact consideration limit fr difficulty consideration function power set become end resort sample mean asymptotic main paper necessarily manifold play outer fr constitute extension mean point proper emphasis value study sequence mean show outer adequate consistency fr devote fr font scale circle circle font circle circle circle font circle circle v v v ig may graph say multiple loop edge graph statistically type fr interest hamming simple realization element distance element graph fr graph obtain result somewhat average concentrated mean hamming sequel consider separable bound random special popular choice font node circle circle pt font circle circle circle v v font scale circle circle font v font circle font circle circle hamming space endow assume every measurable particularly construct base remain behaved term value element approach marker thus true commonly fr fr exist sometimes refer fr non set observe space endow inner unique random variable fr version fr attain ambiguity refer sequel respectively denote fr albeit fr close clearly x expectation infimum lead result q argument mean particular note assume underlie fr thereby fr convergence define observe treat subset sense ns outer subset however type fr define abstract fr fill gray corner font west anchor east anchor east anchor north anchor node east node south anchor west anchor east anchor anchor north anchor north anchor east south panel close equip fr inference conduct take median panel coincide arithmetic fr evaluate set fr
topic associate topic correspond make efficient scalable become popular parametric specify grow chinese restaurant crp dirichlet enter restaurant denote cluster assignment customer customer give customer rich get rich table customer high get table crp independently topic vocabulary post topic choose multinomial sequential crp exchangeable chinese restaurant process widely medium since aspect network value relationship accommodate chinese restaurant chinese restaurant point context medium say post relationship imagine hyper element element share nu distribution table assign customer neighbor already look social medium crp wide entity post death capture unique label along relation hyper relationship one factor content preference movie evidence topic empty table capture crp post friend case cardinality nz friend influence influence trend national lot country capture construct location country location profile user cardinality maintain explicit relation statistic time capture medium way crp crp define remain individual world user relationship reality act determine content specific affected differently multi relational chinese restaurant aggregate influence label post associate additional take indicate influence label multinomial iid dirichlet interpret influence imagine given assignment define post equation aggregated influence capture r w reflect topic two word post iid pz create help capture factor medium analyze dependency one relationship set world user relation country create across user contrast distinct friend couple post friend three flow user relationship preference relationship distinct dependency create user couple user medium factor distinguish medium dynamic capture aspect fall count extend aspect evolve old topic b topic wide network preference user topic evolve enter vocabulary go chinese account temporal reality change epoch label take epoch hour week dynamic dependency describe separately start popular preference influential evolve need change epoch popularity dynamic epoch make early equation epoch nr nr u k neighbor epoch exponentially nr k epoch define conditional well may friend early epoch sample iid temporal add u time influence relationship epoch distribution evolve capture dynamic component depend epoch epoch define present posterior influence involve posterior distribution couple solve resort collapse gibbs traditional repeatedly sample size sequential old infeasible parametric model scale expense sub begin medium first describe require online set influence factor condition influence user look additionally vocabulary vocabulary topic baseline discover trend exist algorithm experiment collection million tweet extract publicly profile default setting baseline lda generally assign topic available twitter parametric topic also dynamic unseen ability exp n dd log indicate ability different consideration tweet month facebook evaluate record lda topic topic facebook baseline capture relationship acc model facebook index lda lda assign topic post indicate find vocabulary word topic indicate modeling way compare reasonable gold topic task significant trend qualitatively knowledge post accord assignment gold twitter indicator know indicator post www etc label normalize rand measure model twitter dataset correctly cluster virtue input come knowledge well inspection find label propose movie comparison split grain distinction gold twitter facebook additionally indirect classifier topic medium gold model twitter facebook twitter na predict author twitter facebook last month create twitter facebook positive negative equal random user topic near neighbor topic post user post set twitter facebook number accuracy topic baseline well usefulness topic infer consider temporal lda lda hash tag significantly outperform topic task involve compare label post conjunction user epoch user epoch aggregation subset different segment major dominant break label event wide popularity aggregate major wide event discover include sep google wave google popularity aggregate death actor uk verify wide event top search interval specific page microsoft twitter page summary enable discover trend major aspect indicate post accuracy infer factor aggregate perform rich insight factor influence epoch aggregate factor epoch trend trend anonymous plot aggregate user subset heat epoch color probability world trend wide user specific day period trend heat color color interest observe wide trend largely influence happen expense preference happen news death discard gradually user preference wide sep expense popular google wave usually influence mostly event trend aggregate user specific trend uk china heat trend uk largely influence epoch around uk around strength influence somewhat weak china strength various stay relatively stable character trend variety look trend topic character aggregate user distribution influence topic epoch character move world color wide google wave preference preference wide summary wide influence lead insight beyond capability employ core ram employ child read batch plot micro post version sequential process increase demonstrate scalability twitter facebook twitter facebook analyze final performance addition model attempt study analyze medium crp incorporate influence process dynamic essence million tweet extensive demonstrate trend model superior baseline beyond capability model study influence various factor social challenge various type influence social medium network act influence multi relational chinese account user evolve design extensive twitter extract prediction baseline valuable trend trend capability facebook extremely user share view short understand medium become important identify segment security lead major distinguishing feature influence variety four preference event world factor affect user different influence secondly medium datum inherently trend interest influential sometimes lead popularity slowly individual evolve factor intrinsic analysis social exist fall address isolated interaction scalable parametric medium specifically augmentation chinese restaurant relational restaurant multiple relationship assign relationship new parametric process version evolution topic trend rich capture crucially multi perform dynamic million twitter dataset facebook demonstrate qualitatively goodness topic discover user use baseline importantly discover interesting user mostly influence global network influence preference analyse effectively medium discuss media section discuss light work parametric model medium dirichlet dp infinite mixture dp restaurant provide generative description algorithm permutation probable dependent crp dd
crucial since compact unlike countable countable countable algebra borel one correspondence measure separate interpretation separate interpretation hence set interpretation restrict show interpretation give separate alphabet separate interpretations borel establish note rs first clearly separate closed ir I remark precede paragraph eq proposition separate interpretation intend belief interpretation member separate interpretation alphabet countable separate interpretation algebra let sentence interpretation separate interpretation close term type enumeration r interpretation immediately measurable proposition show proposition countable alphabet concept strongly interpretation correspond probability strongly imply let borel algebra relate iff set interpretation sentence set interpretation sentence also separate interpretation probability separate clearly interpretation algebra probability relate sentence interpretation suppose sentence conversely probability set sentence separate probability discrete countable real call borel algebra probability alphabet countable enumeration model discrete interpretation sum separate interpretation interpretation countable interpretation separate discrete construct separate characterize axiom also sentence assign countable exist probability sentence enumeration countable choose interpretation assign mass define countable define sentence illustrative concern sentence alphabet alphabet interpretation standard close close term separate countable alphabet probability example non separate interpretation strongly choose construct form enumeration sentence put separate sentence correspond strongly alphabet exist sentence separate model put mass separate sentence alphabet logical two map suitable alphabet element logical interpretation everything separate interpretation everything far except separate disjoint alphabet sentence domain separate confirm note choose sentence empty form enumeration sentence put mass alphabet sentence alphabet non logical term separate interpretation standard interpretation continue logical add logic constant interpretation still axiom later induction interpretation modify interpretation expand nn n ni everywhere separate therefore axiom either replace subset choose absence operator cause constant select singleton non number enumeration interpretation avoid axiom hand logic formula replace something construct good section distribution logical axiom enumeration sentence iff construction problem infinitely constraint build measure illustrate model combination zero odd I theorem characterization sentence sequence sentence finite infinite complete left depth depict conjunction sentence root ex bl bl tree let sentence sentence pairwise disjoint state necessary sufficient well yet characterization probability alphabet countable enumeration sentence sentence resp iff separate resp follow condition addition resp part demand separate enumeration type item explanation enumeration satisfy appropriate intuition I require axiom put set hence would branch enumeration sentence nothing probably prefer keep thing keep enumeration branch formally branch proposition immediately strongly strongly trivial direction imply thus case induction complete induction argument prove summary imply great conjunction contain conjunction together condition separate r axiom hence allow neither large sm unfortunately item item e separate something else interpretation enumeration sentence separate sentence extended sentence ask meet idea uninformative consistent unfortunately possible concept kullback sentence necessary extend pairwise valid eq finite sentence valid sentence illustrate sentence htbp result sentence depth sentence let sentence sentence implie imply empty sentence sentence extend alphabet countable extend definition equation trivially equation finally fourth trivially induction suppose depth assume hierarchical sentence depth restrict equation sentence pairwise disjoint see q show htbp simplify sentence depth index depth sentence expand let index sentence path valid full valid sentence equation replace furthermore equation sentence hence sentence depth inside proposition conclude brief reasoning discuss inference logical implication knowledge base observe exception construct usual forecasting index belief word intend logical axiom uniquely satisfy optimisation else solution logical case expect logical important kl intuitively sure separate forced sentence separate evidence even model problem black problem integer sentence th uninformative let logical degree true happen apply bi bi observe approach course weak black bn want sentence range give extended bi bn bm bn bi bn hypothesis absolutely sure bi bn black counter ever conclusion qualitatively free free hypothesis quantify inference like type probability x generalize follow possible separate crucially exploit term type long break could confirm asymptotically determine separate whether sentence separate determine separating calculus satisfy sentence separate finitely sentence increase appropriately tree assign probability procedure inductive choice make every inductive reasoning operation approximation assume endow etc expressive order logic convenient definition achieve maintain consistent knowledge g outline proposition remark agent past kind ever observe e agent need action regularity time formalize type intend black immediate agent black formally whether allow inductive system basis process result inform belief derive ensure limit belief reasonable reasoning difficulty get right game three nothing game select one ask preferred behind reveal constraint open nothing behind either originally switch good switch use three predicate define set sentence important symmetric predicate true branch requirement uncorrelated rather scale style circle mm predicate location swap swap swap swap swap j swap swap generality locate allow right root add predicate predicate predicate branches circle fill k swap node swap node g swap swap n swap node r n swap branch two force player leave hand branch player likely outcome research history go back main far mathematical bernoulli historical idea sentence back contain reference important appear theory find reason artificial intelligence third machine artificial intelligence typical useful technical distinction approach external probability sentence logic incorporate mix appear uncertainty careful logic particular logic alone expressive extend extension simply higher crucial high logic exploit admit take return property modelling concept directly view integrate logic outside researcher logic build early sentence bayesian network constrain interpretation system approach example provide integrate inside outside sentence lie discovery currently partly dp architecture agent department communication digital centre axiom theorem ex ex ex school school ng national university ex reasoning difficulty develop system integration logic expressive language order ideally reasoning structured knowledge rather main true sentence wish consistent procedure logic iv inductive reasoning vi quantify hypothesis sentence wish list criterion construction counter finitely sentence suitable sentence bias brief theory might reasoning consistent satisfactory logic formal insight generally expressive knowledge reasoning suitable logic admit high result probabilistic logic write answer computational sometimes logical answer probabilistic consideration work quite idea prior general concentrate develop sentence logic sentence use high employ make interpretation sentence order introduce interpretation connection connect probability quantify sentence limit separate interpretation accept circle countable justification real experience statement relevance first statement furthermore guide less equally conversely short mean probabilistic coin one event sentence possible satisfy I strong sentence condition something prove false chance intend non belief non inconsistent weak version plain strong construct interpretation sentence model characterization general probability interpretation countable domain example important value sentence sentence necessary determine probability constrain probability sentence coin head fact logical axiom number require interpretation prove specification complete proposition hierarchical sentence determine choose biased relative informative consistent prior proposition brief end discussion broad motivation well outline model logic direction extension straightforward nevertheless useful material proof technical wide attention hope reference logic discussion history give logic mathematics situation description logic foundation mathematic several advantage furthermore logic logical formalism area hardware verification order science logic present minor way omit operator logic omit logic terminology along type definition type function convention logical type logical type alphabet constant term term together mm type term type closed term formula formula logical countable truth quantification axiom logical axiom truth x g indicate indicate axiom axiom axiom axiom axiom reduction substitution replace occurrence occurrence provide occurrence immediately logic basis logic simplify notation omit type always x application frame frame call truth member call individual frame type variable assignment map variable element define pair interpretation variable term type mm mm pair interpretation every interpretation call crucial allow logic formula mm satisfy valid interpretation valid logical consequence consequence theory need separate interpretation term separate separate close alphabet separate close close thus distinct concept separate play completeness sentence complete say closed interpretation provide separate sentence suppose completeness proof sound separate expand alphabet separate expand alphabet separate alphabet interpretation alphabet alphabet consistent expand separate sentence fact development carry logic version logic practical extension extension deep extension include tuple logic probability probability connection conventional interpretation make sentence sentence alphabet negative valid valid sentence probability sentence follow intend degree disjoint sentence sentence valid valid valid probability include since valid iff induction induction assume imply valid conversely imply thus valid I term close let sentence range nx range set term type finite close hold eq suppose alphabet alphabet countable range closed type enumeration alphabet countable set countable hence close sufficiently large appear enumeration enumeration type rh supremum conversely sup include monotone replacement class necessary term countable alphabet alphabet equivalent
label approach determine acquire crowd amazon allow collection retrieve area g multiple svms supervision task include amount label amount notably closely computer shoot single training leverage category different source field adaptation class model success verification vision method gaussian trivially transfer discard letting source constitute background modal rather time sample modality provide overview input variable denote denote probability let seek joint px represent available label source tp u tp n generally assume classification matrix column domain da summarize predict unsupervise da introduction relatively label application domain language computer vision reason categorization weight da utilize source across domain make sec survey method multi modal relaxation da supervise model space empirical da arrive formulation source minimization already really discussion consider relaxation imbalance relax imbalance population drift training sense change landscape assumption account need explore make class relaxation target margin classifier situation problem assume datum regardless covariate become fit minimize dominate since simplify biased metric distribution distance prior sense discriminative three distribution target common source standard logistic parameter specifically find solve adaptation adapt entropy argue suggest ensemble portion fig unlabele adaptation adaptation adapt model successfully also illustration adapt gmm gmm figure gmm background together train right show survey support domain train subsequent source thus sec machine basic decision learn source act regularizer existence model domain let classifier attain svm similar enforce proximity support vector close introduce constraint old classified old target specifically vector sign close survey da perhaps domain adaptation create map finding might equal depend entropy feature likely good alignment take account straight forward way propose remove approximated source minimize de u another towards goal mean discrepancy mmd mean two domain reproduce hilbert rkhs integrate function separate parameterized mixture function relate representation correspondence unlabele behave required noun help noun algorithm maximize correspondence recent vision variation mapping label mahalanobis algorithmic domain state ensure decompose mapping change regularizer frobenius encode mahalanobis new become improvement table compare recently goal source transform combination following relax rank constraint lagrange multiplier alm propose mapping motivate incremental create intermediate representation create representation final stack location cm p cm naive amazon naive train image category domain mark bold sometimes call multi learning da transfer prior da formally think transfer task view feature space augmentation name recognition text classification class share formulation author concept assume assume available goal transformation vice versa common begin work popular method project projection unsupervise agnostic project discrimination find minimize scatter class scatter lda direction give solve note eq eigenvalue cca analysis though cca projection onto mutually maximized sec cca formally find projection solving dimensionality write constrain eigenvalue solve eigenvalue problem detail excellent
minibatch sample rather sum subset condition field benchmark closely detail include train accomplish step consist maximize likelihood rbm consist rbms maximize far include discriminative ability somewhat mean exist closely phase criterion code epoch base last start increase entire mnist training last obtain mnist trained rbm follow variational obtain jointly train correctly optimization result number likely introduce boltzmann jointly learn train machine time deep model call visible unit always train evaluation usually apply represent label code discrete label available contain organized unit neighboring layer allow gibbs entire likewise fast variational q summation fortunately structure every element conditionally visible likelihood intractable overcome layer procedure thus layer simple consisting value repeatedly apply field essentially run beyond admit simultaneous excellent invariant version mnist handwritten digit knowledge allow conjunction train boltzmann machine layer rbm rbms train enable jointly train procedure suboptimal deep general optimal influence deeply training procedure procedure optimistic layer able current boltzmann machine deep architecture pool factor multilinear interaction unit layer use hessian without greedy never view difficult show hessian poor parameterize intractable great expense approximate costly search early mcmc take make markov chain
rapidly occur reflect target elliptical multivariate call random field elliptical advantage mix induce easier apply tuning parameter elliptical take invariance namely draw eq q marginally nevertheless previously proposal latent gaussian elliptical slice sampling rejection elliptical transition consider pass current induce purely gaussian prior choose elliptical slice sampling auxiliary sample sample angle choose slice procedure specify current state satisfy slice intuitively slice compare depict elliptical slice illustrate produce elliptical slice reflect full elliptical slice depict elliptical slice log distribution x elliptical slice arbitrary refer need target sample elliptical slice approach put residual density note correct enable use elliptical slice sampling elliptical mix practice difficulty tail rapidly illustrate get back toward hand low unlikely origin iteration elliptical allow approximation gaussian serve auxiliary residual residual possibility combination point equation view augment fully choice drop component elliptical slice update markov transition operator focus simulate heavy simple convenient location density real value parameter observe conjugate parameter update elliptical define current perform elliptical slice new contour sample red new point make way choose maximum laplace regardless section parallelism dynamically require exploratory create markov chain immediately obvious discuss fail way current chain lot shape operator elliptical slice additionally stationary core generalized elliptical slice desire property begin markov chain use determine multivariate parameter apply x individually update chain introduce detailed balance create joint updating multivariate parallel however factor efficiently problem maintain multivariate denote chain practice simulate leave invariant generate chain two part transition use determine use operator composition idea maintain chain state markov chain chain see description choose appendix subroutine return maximum likelihood subroutine elliptical slice update x chain algorithm choose may converge set estimate describe fit poorly appear well k ad r r fit multivariate construct dimensional space identity avoid degenerate distribution intuition emphasize nature important devise sophisticated idea high merely simple seem work choice least likelihood parameter diagonal covariance parameter nature establish aggregate transition wish preserve x stationary desire chain elliptical slice nonzero nonzero transition chain therefore nonzero transition irreducible ensure invariant distribution multivariate approximation fit iterative empirically parallel chain challenge construct manner cost evaluate density function overhead applicable promise complementary introduce technical software requirement addition population parallelization could apply mix compare scale core number ec algorithms environment parallelism empirically mix mcmc sampler seven mix compare chain effective effective effective result metric sample carlo experiment unless burn iteration five variability omit aggregate diagnostic elliptical several sampling direction slice coordinate wise slice sampling variant slice direction direction align mh gaussian current tuning adjust acceptance ratio optimal hasting chain also sampler carlo hmc combine procedure automatically evaluation term include comparison hmc though differentiable principle access manual complicated subroutine instance require image run sample distribution temperature regular interval target produce often practice geometric view entirely principled choose goal box restriction require describe ten shape normally nine condition remain initialize spherical component gaussian spherical component distribute uniformly hypercube breast cancer explanatory breast cancer consist mean prior initialize chain spherical explanatory repository place prior variance spherical origin stochastic volatility simple fit synthetic burn iteration chain take absolute parameter constrain length scale place burn iteration spherical snp gene eight dimension actual genomic sequence burn initialize spherical center origin take leave five bar rescale effective mix accord metric attribute unlike chain hmc although fail snp arise reflect mostly reason rapid mixing even vast away successive highly multivariate build shape skew transition mix arise result length parameterization frequently approximation greatly parallelism b x wish function core available chain triple make sense core chain section equal triple origin degree use model initialize spherical gaussian second half discard second used estimate vector marginal triple five average squared aggregate chain would set additional enable effectively chain enable sampler effect column fail fail burn however increase core provide square respectively converge one chain mh converge elliptical slice approximate parallelism gaussian shape transition chain particular hundred core mcmc elliptical variety evidence number work reduce overhead share chain speed chain obviously chain run slice sampling perform per update location chain narrow evaluation chain rapid chain leave imagine perhaps chain machine gain gain parallelism mcmc able mix area extend take core magnitude multivariate core give feature location mode use mixture accurately gaussians explore leave explore algorithm detail parameter likelihood q update parameter function probabilistic conceptually find practical effectiveness approximate carlo step rapidly mix leave regard multi speed paper distribution take advantage hundred core information chain elliptical slice create chain mix require curvature computation markov carlo parallelism slice elliptical slice probabilistic fundamental machine coherent formulation representation unobserve estimate typically integral rich rapidly expectation carlo simulate chain equilibrium principle gold inference mild limit simulation often expert method space generalize tune multiple core parallel detail arise world dependencie parameterization directly primary efficient markov reflect structure imagine thin region high necessary length metropolis mh proposal monte carlo behavior introduce momentum method efficient move even approximation target elliptical shape require curvature word encode us information oppose hamiltonian valid preserve guarantee limitation expert difficulty stem tune transition mix metropolis appropriate proposal carlo step probabilistic widely develop black box attempt mcmc theoretically understand narrow transition operator hamiltonian slice local search acceptable avoid potential adaptation significant computational
define estimate percentage high simulate axis vertical difference consecutive consecutive horizontal represent difference show must rule class read well multiclass ccc horizontal kernel namely project pca mahalanobis mahalanobis kernel pca mahalanobis previously mahalanobis correspond extreme simulate construct spectral library hyperspectral spectral noise adjust figure present spectra number train repeat hyperparameter term accuracy report vs classifier intrinsic present estimation try fix value lead drastically configuration class increase decrease estimating important another estimate correct live experiment axis represent value classification propose kernel performance result provide mahalanobis kernel kernel pca mahalanobis confirm poor generalization mahalanobis deal conventional sensitive datum redundant class sample could l mahalanobis mahalanobis average correspond mahalanobis load computed set first program matlab core ghz low eigenvalue demand hyperparameter computation eigenvector fast second demand mahalanobi regard eigenvector demand mahalanobis perform fast clear winner process computational viewpoint less demanding mahalanobi might nevertheless mahalanobis demand pca mahalanobis mahalanobis mahalanobis adapt propose parsimonious mahalanobis base space original space split signal subspace accurately purpose demonstrate case accuracy increase superior simply datum mahalanobis regard computational load mahalanobi demand besides consider tune optimization hyperparameter demand computation strategy concern development model conventional mixture kernel could mahalanobis kernel summation intrinsic author tc university definition article exploit computation mahalanobis inversion condition inversion unstable parsimonious signal noise subspace inverse stable use hyperparameter provide conventional kernel dimensional commonly automatic huge number variable hyperspectral hundred pixel gene thousand gene customer web service associate noise cluster filter literature moderate suit pose need specifically exhibit intuitive statistical property space summarize tendency corner tend concentration difficult analyze unfortunately discriminative also suffer tend distant clear near fail distance assess similarity affect neural locally potentially phenomenon consequence space phenomenon follow bellman reflect difficult separability comparison hyperspectral hundred spectral sense former contain hyperspectral conventional accuracy several dr dr reduce mapping onto space meaningful dr dr algorithms without exploit project accord variance independence improve discriminant analysis dr however maximize class scatter matrix scatter condition limit effectiveness popular dr laplacian one dr purpose map onto global discrimination maximize dr recently specific subspace original processing class normally embed white property without discard model dimensional dimensionality discriminative conventional generative local depend neighbor mostly local kernel choose approach subspace mahalanobis inversion whole associated matrix diagonal purpose covariance sample mahalanobis project apply euclidean space would specific adapt data subspace ensure parsimonious characterization possible explicit parsimonious mahalanobi construct mahalanobis several subspace hyperparameter optimize classifier accuracy subspace select hyperparameter section detail signal simulate dimensional report conclusion space curse statistical estimation I restrict covariance reader find clustered cluster generally follow identical similar intrinsic whereas class last understand decompose noise contain noise ci sample eigenvector correspond eigenvector computation usually opposite dot major drastically covariance estimate need furthermore stability eigenvector mahalanobis rely significant discard dimension theoretical advantage two far space seem discard noise even worst consider ccc represent contour b mahalanobi hyperparameter optimize tuned hyperparameter reason know direction optimal classification maximize variation direction discriminative modify discrimination regularize mahalanobis induce hyperparameter analyze section function map onto space evaluation input live riemannian metric equally task compress right
raw exponentially straightforward ad ad independent observed achieve self ad degeneracy query equally ad tuple ad b calibrate ad self calibrate ad calibrate calibrate prop prop prop prop thm nice thm prop sec prop prop sec prop nice guarantee nice prediction calibration expectation condition selection selection define self calibrate sure state query bid raw previous weak define invariance raw map calibration show ad raw prop prop weak z insufficient property behavior occur mechanism prop prop equivalent prop prop nice bid prop fortunately prop prediction f p calibrate ad element also event selection break ad ad calibrate assumption consider prop query candidate tuple share selection ad must ad prop prop imply prop query two c f prop imply nice four equally ad scale ad whenever prop impact thus map efficiency show half bid query add system normally reasonable query ad seem unlikely grain bid running experiment ad prediction nevertheless current calibration iid assumption question efficiency lose randomize deterministic maximize calibrate lemma calibration prediction system statistic calibration adjust prediction click prediction ad want maximize notion calibration impossible detail impossible calibrate prediction give achievable maximize roughly speak already capture calibration event predictor happen fraction occur event problem prediction make decision engine past decade business role engine select candidate ad query keyword reasonably candidate rank ad bid bid click bid show ad top ad show ad score threshold nice prediction value generate equivalently produce mechanism combine value reflect predict face example volume rather new output order prediction consider informally state efficiency computationally map efficiency formalize question mechanism different mechanism impossible prediction calibrate ad iterative fail since ad rate ad np sufficient fairly system bid query selection condition fact calibration predictor calibrate fairly make example easier calibrate classification calibrate example good predictor excellent method adversary easy classifier prediction calibrate sequence predict interaction calibration care theoretically key begin define unit ad select provide raw concern raw prediction establish study interaction engine fix string like car ad click bid pay click prediction prediction drop sometimes ad tuple query ad query drop ad consider ad ad achieve high pick ad model ad raw position ad theoretically change change ad respect ad candidate ad ad formalize distribution ad proportional ic choosing suppose two query candidate candidate think ad occur ad q break tie thus ad f calibrate ad multiple concern finding prediction q able issue assume exactly addition choice mechanism show take selection bid reflect click justified pricing per cost incur query incur engine cost vary ad would notational burden focus simplest generate transform efficiency prediction calibrate say exist first relate offline problem datum efficiency prediction order ad problem nice calibrate map exist maximize calibrate concerned start prediction serve like train large train prediction ad calibrate ad batch may ad prediction map assume calculate exactly ask ad potentially answer iterative calibration cycle start finitely ad map permutation cycle address general restriction select ad ad purpose counter section prediction candidate bid cumulative ad candidate ad three ad tuple show ad ad ad generate near ad occur generate negative ad exponentially far shorthand pick candidate fact ad figure distinct candidate query consider single query ad observe click self calibrate ad click rate show ad start cycle efficiency example calibrate even efficiency access calculate time sort bid must list exactly ad query show ad compute contrast practice possible accurately simply grain estimate select ad query raw trivial answer become show nice answer show ad bid guarantee z tie break answer q
replacement design design practice design require frame list may complex design design use appropriate distribute natural design entail detailed presentation reference substitution extend functional assigning unity elsewhere namely dirac total mass depend respect sample membership indicator assign weight elsewhere thompson weight reader lie information mention nonparametric treat elsewhere rest functional expression base straight obtain argument implicit thompson well directly formula give thompson variable fr measure jacobian identity two strictly unique eq fr respect respect influence exist tm tm dirac definition functional usual definition robust survey mass unknown order von write tn tm tm use tm finite influence nonlinear linearize artificial variable linearize suggest estimate give linearize role exactly design linearize respect nevertheless put know curve discretize describe employ product quadrature view multidimensional vector kt kt kt kt estimate linearize k linearize volume treat france greatly fact de france million request store analyse challenge huge consume desirable consider median curve design population minute consumption aim curve week consumption record week auxiliary population kt week consumption curve peak peak day decrease time measurement effect figure consider strategy fix distinguish kind auxiliary auxiliary opposite sample realize classical survey update practice simple design put chance sample perform survey draw list median unique equal kt kt kt kt kt basic inclusion median obtain sampling inefficient systematic sample efficiency frame correlate adjacent element frame accord week discretization point simplicity systematic appropriate really frame advance solve eq may substantially compare replacement sampling homogeneous worth improve median linearize indeed linearize asymptotic variance sum estimator within usually know strongly computed week linearize compute linearize consumption week use propose split week distance linearize cluster prop allocation allocation figure week within accounting allocation kind build consumption induce consumption consumption week well allocation prop allocation allocation plot second week correspond consumption linearize week population linearize design efficient proportional auxiliary study information cope suggest discretization consumption week inclusion give thompson distinct element replacement use advantage sum consider one simple improve median partition call describe practical consideration may favor perhaps costly whole frame belong membership impossible perform group know improve estimator obtain thompson auxiliary group stage weight size median remark zero group whole suggest aggregate calibrate estimator auxiliary membership argument remain group agree allocation proportional allocation multivariate g linearize variable drawback reduce inefficient improve efficiency week size thompson study post curve four design design replication study equally space discretized quadrature linearize variable cluster space term divide compare simple result proportional allocation design week consumption surprising report design perform estimating consumption fail median believe encounter small need prop also simulation week consumption table c prop remark surprising construct account consumption variable optimal allocation compute minimize variance mean estimator allocation usually survey consumption linearize design need consumption costly obtain impossible storage contract consistency analyze design quality estimator use estimation sampling give deviation design h median prop dash allocation give observe sampling divide ten compare proportional example optimal survey appeal consequence consumption curve confirm rule lead reduction estimator get well nevertheless rather complex future concern auxiliary information concentrated median thompson complex develop general regression constructive manuscript spatial b b estimator median banach efficient geometric median space average gradient component e asymptotic band estimation outli p geometric notion quantile pp storage complex statistic residual methodology estimator survey principal di ai multivariate center area population journal american association influence role robust population g thompson replacement universe b median banach quantile york nd york zhang multivariate usa von differentiable english france de universit email fr fr widely consumption currently de france profile unfortunately outlier high consumption load median avoid whole median test linearize compare area future highlight thompson substitution france load calculation load use future load customer load load curve profile use consumption customer nevertheless highly sensitive presence customer infinity datum force extreme customer view context join company one leave important amount demand load profile demand customer class synthetic describe situation profile profile besides analyze population median univariate central way univariate advantage survey first motivated mainly census spatial median among definition median spatial study detail zhang company device individual consumption new accurate point work census united concern center population median explicitly drawback transformation goes deal variable isometry require invariance share lie
tuning reason tuning parameter purpose except tune unchanged conditional chapter tell respectively sde diffusion sde brownian motion remark nn homogeneous initial degenerate distribution discretize find replace uniquely see limit governed wiener limit n derivation trace norm sde langevin univariate adaptive mcmc properly behave langevin little metropolis adjust langevin mala langevin mala general lipschitz uniqueness lie b sde ds ds take integral ds ds ds use e cauchy ds ds e ds multiplying integrate side ds ds ds ds u ds u ds du f f du equation bind prove sde comparison discrete call try tail compare mh tune fix proposal cauchy generate discard efficiency perform kolmogorov ks remain asymptotic measuring empirical burn high value great choice well adaptive suggest optimal lie simulation extent naive mh heavy lie interval adaptive although compare solution euler discretization choice euler z euler h sde euler mcmc give target compute kolmogorov value table note part sde move leave modified accordingly well almost peak average dominate differ interest quite heavy address adaptive figure plot value e e compare asymptotic generate adaptive e e e e asymptotic use target cauchy cp cp adaptive cp value non adaptive h ht cauchy pdf diffusion many iteration invariant approximation chain limit target limit metropolis early adaptive attention example however adaptive mcmc early scope mcmc continuous fine detail suggest mathematics valuable comment cm cm markov carlo mcmc proposal chain mc stationary diffusion standard limit trivial diffusion plot comparison adaptive mcmc sde monte carlo mcmc consist markov unique invariant parameter possibly infinite mh measure choose proposal probability form generate accept show chain drawback proposal distribution make stationary dispersion interest turn value mcmc quite main concern definition grows follow algorithm algorithm select move proposal multidimensional mh detail technique mapping continuous limit metric endow algebra variable nh homogeneous probability let almost surely specify drift coefficient e k dispersion
branch replace step similar finally replace constraint hand side start cell outer sep minimum fill gray none draw none none fill none draw none none circle fill circle white fill cell outer sep circle fill gray sep cm sep draw none fill draw none fill fill white draw circle draw fill draw node circle source target record linkage record tuple record linkage computation ratio record pair match record subset match vice versa record subset record possibility record record tuple subset weight record tuple membership record belong denote record tuple assign tuple however regardless decision appendix record tuple membership record record order weight determine cutoff weight multiple procedure maximize assign right subject tuple belong maximum record possibility whether record tuple possible increase indexing subscript record membership j p p finally record tuple configuration belong record decision optimal availability assign wrong keeping subject admissible level assign record subset linkage practice matching evidence actual error rate cutoff linkage expect resource linkage perform three record census three gender outline follow find record triplet suitable triplet every field train divide set within triplet belong order method gender triplet pattern classify remain triplet date age explore option construct three categorical variable date create comparison new construct category category date period use age record gap record new think record assign help identify replace triplet multinomial pt c comparison category table linkage option age date obtain hand thought usual misclassification group control linkage carefully scenario misclassification error indicate multiple record linkage comparison date treat date way agreement three correspond agree unit unit agree small age category misclassifie triplet misclassification indicate use properly naturally performance specific link implement bipartite record third approach assignment nominal level triplet misclassification error try decision record triplet assign decision triplet record linkage record triplet one classify linkage provide decision uncertainty something record practice common variability size dependence record field existence replicate record explore propose method affect model generate categorical record measure give represent value field unobserved model record field distribute field generate generate around value value important linkage inefficient need good adequate linkage product census three accord date measurement triplet record linkage option error compare result method city simulation measure panel b misclassification specific effect close subset measurement true triple create option low explore generate contain common five field contain equal also variable scenario category scenario database field category scenario known quality across four database triplet triplet database linkage correspond exclude made assignment use performance report mean assign mean meaningful linkage unbalanced subset extremely small show misclassification panel solid dash error grey represent method quality extra panel large large amount impact misclassification class finally scenario inclusion low extra integration common modern decision common match tuple bipartite incorporate linkage posterior model belong go beyond dependency naive integration believe linkage census census decade linkage enumeration post enumeration census block source use american survey various file incorporation linkage describe sampling census order ps ps ps logit order define decision record tuple assign record tuple subset keep rule record tuple record tuple admissible rule construction divide wrong minimized linkage linkage identify unique available building upon application record linkage merge post enumeration survey census census health care database name integration use ad linkage decision thus suppose record linkage linkage record file match truly match could occur measurement record record refer another another individual bipartite linkage pair file resolve matching record ad hoc resolve multiple bipartite appear survey implementation accurate census evaluation census adjustment require census evaluation match survey response census dependent joint inclusion heterogeneous survey evaluation process list homogeneity attention multiple linkage exception event whenever capture diversity source coverage assessment record system clear record maintain census de de number record conceptual difference whereas national record information daily census occur evaluation census record system linkage source linkage generalize approach linkage supervise record product link union possible agreement record record kind fitting present decide tuple record study different scenario exposition population denote record exist record generating record member vector record tuple entry correspond describe record tuple information different individual I individual record tuple classify record tuple individual order formally set associate tuple partition group entrie tuple pattern common record tuple agreement certain associate tuple set describe match consideration grouping agree similar idea record tuple us record agree agree agree since match partition equivalence record tuple equal one agreement field vector field represent agreement present represent belong connect need link three agreement fr full triplet field probability kp p p pg j pg note entry tuple complete bipartite linkage take state independence multinomial indicator define obtain arrange k k conditionally independent give tuple membership baseline produce
outperform fact filter ds remarkably close length accurate forecast svm preferable may outperform historical evident experiment ds test initial add five multiply figure value interesting condition look around mse begin consistent repetition phenomenon create repetition parameter converge converge wrong poorly repetition result two green red specify pattern location misspecification forecast misspecification factor lead incomplete knowledge system historical validation procedure stop cross pick dimension well range dimension allow pick determined dimension hyper pair l entire sort experience svms ls dimension ds ds fit filter laplace diagonal choice degeneracy calculation use possibility approximation posterior previously create particle particle degenerate employ addition encourage parameter state perhaps forecast mind state state equation propagate specifically one step dynamic practitioner intuition perfect knowledge yield forecasting system know incorrectly forecast accurate system fail highly system paper forecasting historical preferable exhibit misspecification extensive misspecification examine circumstance historical datum predict traffic flow week nuclear forecasting forecasting however vary drastically historical traffic volume predict flow may physics historical forecast forecasting forecast use perhaps city traffic flow considerable traffic flow united lack world record infer ice core preferred perfect one gain arbitrarily system knowledge historical become limited nonlinear include system forecast become inaccurate particularly prominent field connect science chemical determine strength market physical formalism usually equation system ordinary euler approach base approach forecast rely solely next year management give file rough origin test testing conference one somewhat parametric linear autoregressive parametric vector assumption relationship note combination well exposition wish knowledge method describe form various level misspecification method organization discuss historical datum parametric tool approach remove effect misspecification study simulation conclude future may parametric parametric rely describe datum relationship autoregressive historical comparing build class svms latter specifically begin describe support machine ls end square free parameter kernel uniquely hilbert rkhs e consist map suppose sample l svm find unique solution optimization problem hilbert sample diagonal require operation infeasible find maximize usually discuss early process differential initialize forecast adjust current noisy assimilation gaussian noise choice kalman filter class dynamic system forecaster process process assume write linearly additive noise order frame recursive interested posterior merely often filter density know include forecast able filter analytic covariance simply calculate however dynamic kalman provide alternative method dynamic methodology kalman nonlinear kalman linearization initial estimate wrong incorrectly jacobian kalman filter approximate transform variable instead taylor expansion kalman mean transform recover carlo filter importance distribution particle associate importance unnormalize typically sequentially take markov nature measurement x z recursive weight formula choice proposal simple proposal particle filter simply generate evolve forward proposal calculate weight degeneracy resample step remove multiply high weight filtering concerned dynamic information noisy state space state representation equation iid run state filter parameter state filter many detail volume purpose misspecification broadly include traditionally term firstly complete complexity model misspecification incorrect alternatively might attempt misspecification understand term lack availability complex require considerable amount rough elaborate gain mechanism misspecification cause forecasting method forecast distant forecasting misspecification unknown look svm filtering early study learn may really limit specifically dimensional system forecaster mention later svm consecutive observation forecaster complete dynamical system forecaster dynamic forecaster addition entirely base specifically system explore wide length historical fold increment repeat forecaster repetition historical historical repetition filter laplace specify laplace general setup generate trajectory dynamical historical set svm random describe index sample save index examine historical misspecification fully drastically describe system stochastic observation ds none unknown none ds aspect ds ds contain various attempt ds contain detail forecast appendix ahead forecasting rmse
sphere study index tree greedy strong tree margin investigate success evaluation algorithm population evaluation exhibit assume procedure copula fit exhibit sphere bivariate characteristic bivariate explain well summation optimum fraction sum select reflect include provide appropriate situation point interesting explanation simple interaction combine non copula summation linear summation model multivariate well copula copula similar present copula construction exhibit robust multivariate product level copula simplify construction determine selection study example compare tree aic determine tree summarize evaluation I truncation tree select appropriate tree technique aic find pairwise show crucial copula arise propose benchmark correlation seek copula copula product copula copulas copula construction product copula consequence underlie normal bivariate univariate normal copula function bivariate student copula student univariate student distribution degree derivation bivariate copula dependence bivariate transformation degree bivariate copula control dependence denote expression bivariate copula bivariate copula controlling dependence close form numerically derivation formula bivariate bivariate represent dependence bivariate copula transformation copula copula control bivariate copula institute physics email institute physics email institute mathematics physics email study copula build copula copula study dependence aspect crucial success show copula appropriate distribution explicit probabilistic sampling build solution multivariate derive evidence copula separate statistical structure fit rich univariate model dependence model copula limitation copula strength dependence coefficient parameter characterize pair alternative problem multivariate copula powerful copulas decomposition family literature author behavior problem indeed copula identify model build product copula empirically show well endowed variable notion copula respectively notion terminology conclusion consider random distribution every multivariate distribution function marginal copula separate effect dependence margin provide immediate random copula copula give independence margin support besides base reformulate copula normal margin copula multivariate normal univariate normal normal case dependence preserve margin normal inversion estimator ij matrix positive correction margin normal margin I select population individual margin distribution third fourth limitation available copula bivariate parameter appropriate construction approach result decomposition multivariate copula bivariate copula building construction nest call regular copula construction regular rich bivariate copula dependence bivariate copula set node regular constitute edge tree share instance regular graphical four model specific particular denote scale rgb rgb rgb joint consist density copula construction q copula distribution component remain recursive evaluation expression special univariate reduce joint facilitate function define expression bivariate copulas simulation develop implementation specific copula imply way random heuristic heuristic assume weight appropriate variable maximize node root greedy maximize weight original instance add node edge node heuristic structure completely copula decomposition tree require might step conditionally c yy consist fit tree detect truncation level bic expand value truncate tree copula type goodness copula way repeat copula accomplish way bivariate empirical copula copula significance reject copula combine copula student normal low student copula dependent copula copulas tail copula copula likelihood become almost indistinguishable normal copula distribution method w x j expression although present crucial statistical package know function definition summation maximization optimum ensure fair find algorithm optimum optimum precision evaluation truncation population sample interval value take global locate middle interval interval experiment sphere summation asymmetric interval summation joint asymmetric interval ccccc population c ccccc evaluation well large normal margin interval margin global optima capture
short time sampler st lattice site express take integer graph exhibit phase start update algorithm periodic boundary metropolis heat gibbs iterative sampler alg compare confirm convergence gibbs ergodic fast quantity factor acceleration locally rejection transition gap markov st autocorrelation conventional autocorrelation variance autocorrelation bin autocorrelation metropolis heat iterative gibb investigate exponent temperature exponent gain metropolis see st geometric allocation autocorrelation kind mention introduce gibbs review show set well mention update applicable heat usually inverse cumulative inversion apply candidate choose proposal reject autocorrelation explain reduce rejection general inversion reject ratio canonical mcmc candidate achieve rejection picture simple metropolis generate configuration stochastically detailed keep reference rate state px configuration isotropic bivariate x proposal propose candidate current position account broken proposal avoid configuration current configuration proposal use proposal probability state fig process rejection minimize st indeed proposal get short candidate construct several ergodicity extend efficient future review ergodicity assess conventional applicable efficient need analytically grateful valuable discussion present university code library support grant aid scientific program support markov physics university south wide variety tool chain carlo curse care candidate configuration determination kernel construction monte dimensional wide tool interesting topic physics phase transition physics curse effectively suffer configuration update sampling decrease former quantify variation transition assessment latter decrease autocorrelation observable th independent autocorrelation effective roughly total number simulation central care key ensemble replica method apply protein spin wang overcome growth graph many physical model hamiltonian move candidate momentum third determination mention chain construction rejection transition matrix detail however chain metropolis type apply algebraic transition conventional ergodicity monte balance impose method detailed balance impose simulation balance elementary force actual simulation metropolis algorithm heat sampler performance seminal analytically simple reversible optimization transition finite state transition element follow statement ref suppose irreducible matrix invariant call gibbs usual gibbs accept reject metropolis scheme modify gibbs sampler sampler say iterative version gibbs sampler rate diagonal minimized see transition balance condition invariance target meanwhile modification additional axis version axis apply momentum hybrid partly chain generate history discuss asymmetric sphere see net flow hybrid carlo detailed balance recover break paper chain reversible metropolis however al apply solve algebraic algorithm discrete continuous space probability balance visually allocation optimal solution introduce allocation update huge implicitly construct elementary configuration proportion quantity raw flow average rate metropolis proposal distribution give heat find minimize sense visually allocation entire box keep fig ref describe alg satisfie allocate rejection maximum rejection rate certainly reduce candidate rejection discuss tb geometric propose rejection figure weight candidate choose matter box remain box remainder allocate iii ii continue likewise iv step iii box discuss ergodicity previous irreducible condition ergodicity although update interesting ergodicity color tuple state assume unnormalized proportion state update alg configuration allocation arbitrary fix calculate reduce update irreducible define determined alg transition make alg irreducible consecutive update candidate multiplication certainly cyclic loop end visit b monte update accord lemma similarly choose show space irreducible ergodic rejection exist
possibly necessarily identical I graph separate path separation condition separation mit coupling process mit source process mit solely convolution exist mit parent parent parent mit additionally distribution far simplify parent mit mit proof give remark modification mit autoregressive coupling lag connectivity generalize analytical lag mit mit yy shannon communication couple communication channel mi shannon read bandwidth coupling capacity alone interpret strength mit mit might leave exclude parent parent parent modify mit call stand modify mit stand separate mit hold coupling strength link thus mit coupling coupling strength make mit solely couple filter mit numerical next section couple strength numerically mi te mit investigate process independent white process variance b dependency process dependency appendix functional form neighbor lead bias high variance mit coupling strength link depend mi mit mis entropy mis input mi approach analytical exclude exclude different always equal mit bottom panel fig mit red line te gray dotted link series green dot te much mit compare lead te analytically input apart te link dependent consideration mit show input due estimation mit aim measure distinguish couple property equitability parent expect mi take broadly mit coupling distinguished analytical variance add total plot te suffer negative mi mit solid related conditionally line almost nn seem bias small measure te relative mit theoretic mit thus measure coupling strength use te possibly also mit intuitive well interpretable te coupling ingredient equitability interpret coupling assess mit mit x large mit coupling surface area correspond strong mit large increase surface air apparent spurious coupling mit measure strength reasonable picture interaction preliminary analytically mi te coupling strength overcome step lag link link multivariate theoretic mit lag strength interpretable experimentally set couple strength process coupling attribute suggest modification mit mit practically besides extract causal direct indirect connectivity assess coupling acknowledgment national foundation research education research research environmental thank comment appendix proof mit discussion regard couple strength linearity condition mit mit x stationary series article condition parent hold coupling include ti separation time x mutual parent already path towards chain twice covariance variance auto q kronecker delta else entropy derivation markov entropy simple form top toeplitz e toeplitz toeplitz te toeplitz utilize relate geometric toeplitz wiener absolutely toeplitz te integral contour integration te since block zero everywhere simplified complement block since zero infinite te depend strength rather exploit infer involve mit mit source conditional firstly process correspondingly I secondly arbitrary entropy source generally depend parent invariance simplify arrive since independent past add simplify g therefore extend arrive parent remain parent mit additionally ic mit condition entropy first invariance entropy know know dependency parent communication graphical science operator new york method new york j causality reasoning g meet mm mm conjecture united assess strength meaningful focus strength theoretic demonstrate known instead certain delay mutual mit association coupling computable root mit graphical mit component graphical admit low reliable mutual mit practically mit strength compare interpretable case solely interaction particular mit many able exclude theoretic formalize idea general illustrate mit produce grow abundance hypothesis statistical measure key reflect heuristic notion give dependency idea multivariate reconstruct field propose zero imply causal strength alone remain understand interact bivariate coupling strength influence correspond keep solely measure impact realization regard ingredient demand require somewhat transfer due property reconstruct interaction link attribute delay mechanism give already denote assume stationarity te aggregated lag density te choice truncation lag affect via number process influence te reliability causal couple delay influence coupling dimensionality overcome next transfer derive enable estimate remainder another truncation describe somewhat arbitrary lag avoid drastically still quite simple lag lag non lag dimensionality causality sect te strength property mit past parent series infer introduce supplementary detailed detection positive determination coupling strength detailed causality link mit infer link parent certain key stationary time yield realization uncertainty observation realization non process noise always observe mit entropy diagram attribute mit closely couple show link mit correspondingly due mit mit mit sensitive mit use causality non also parent drop source mit causality graph entropy reach measure situation interested matter hold sect mit already estimation truncation analytical property series graph mit website http mit intend different analytical interpretability motivate couple tractable process series graph parent dimensional determinant evaluate derivation formula te te entropy past easy yield entropy innovation term hand treat process apart infinite toeplitz theorem another block thus
unlikely region obtain ucb complement colored panel denote task shrink search space simply algorithm back lattice lie stage relevant inside practice even exponential course improvement begin show explore location residual specifically hull set hilbert projection interpolation hilbert bound well reproduce kernel hilbert rkhs norm bound tp p tuple minimum norm interpolation consequently satisfy assumption lemma follow x location derivative hull pairwise distance diagonal gaussian process subset standard variance vanish point exponential vanish branch theorem state prove supplementary material gp appear interior twice vanish alternatively vanish lattice decrease exponential main use bind utilize together picture expression ucb finding assume compact convention subset give differentiable continuous fx x xx bx df two q specify branch bind like remark theorem ern hessian co dimension function chance lattice fine interior important decay lattice choose sake rate follow branch algorithm cumulative regret ucb monotonically factor shrink necessarily lead regret point large ucb sample every lattice propose modification ucb address key regret evaluation word speak case far discard piece unlikely compare show additional step improvement branch cumulative obtain noisy unbounded search region less go free achieve identify noisy reflect achievable depend suggest noisy limited exclude branch lie outside perhaps cut would encountered guarantee contextual bandit parameter context v stack functional provide remain standard adjoint arbitrary equality calculate claim prove see project follow correspondingly interpolation tight variance inversion second without generality minimum consider exploit need dimensionality large latter prove rkh lemma derivative vanish bind bound sequence sample covering banach set define ball banach space entirely first show step neighbourhood enough neighbourhood densely envelope around tight eventually neighbourhood please graphical compact continuous unique suppose contrary radius exist bx fx tb theorem densely densely inside reason localize neighbourhood regret exponentially proceed twice densely shrink claim prove number follow carry lattice sample iteration begin f sense combine give x neighbourhood precise form irrelevant maximum argument goes depict tb need govern equation differential separable integrate side integral use integer illustrate somewhat satisfied overall expression x derivative indeed cs process observation ucb gps gaussian great prove vanishe rate complement attack attain much fast regularity function near global maximum sake maxima might free could feed consume algorithm global observe sensor deterministic configuration automatic entire architecture hardware uncertainty characteristic elsewhere mention search impose objective vary require change point point tight near elsewhere relax fast discard bound convexity global force reach search guarantee produce function lipschitz give imply methodology interested result stop ideal regret hessian behave cf therein improvement methodology function possible
underlie among give variable assumption sparse try column use penalty copy block condition induce dependency machine partial tool condition input indicator annotation associate feature graphical tag identify dependency tag interaction ignore impose consequently block order estimate impose assumption sparse affect moreover estimate inefficient unfortunately dimensionality document bag web dimensionality order category usually much small formulation precision estimate one consider matrix contribution graphical model point marginal precision dense expression quantitative trait gene gene take effect gene paper simultaneously matrix drop consequently reformulate efficient statistically connection conditional quite partial graphical different impose lead convex formulation lead summary estimate contrast global minimum directly scalability formulation conditional assume sparsity establish numerous precision recent estimation popular precision matrix guarantee investigate formulation know estimate support nonzero separate neighborhood estimation aggregation hide dimensional dimensional write complement exhibit sparse dimensionality low minimization standard advantage realistic many closely analysis precision likelihood convex solution solution precision rate precision approach precision via significantly procedure statistical model estimation study regard multivariate matrix text entry vector row column ij ij ax j remain graphical analyze multivariate unknown discuss paper distribute e parameterize give vertex order pair normal convex sparsity particular precision formulation aim estimate matrix confusion idea eliminate respect follow allow allow convergence sparsity introduce indicate decompose l proposition behind optimization high penalize define separate formulation sparsity induce penalty respectively induce penalty full model formulation significantly assumption analogous situation conditional field unnecessary relatively compare yy yy yx yx yy yx yy yx concept strong q guarantee present probability proper hold compressed follow sense assumption obtain let frobenius minimizer wise penalty interpret element assume absolute c imply may depend least algorithm subproblem respectively objective procedure global q proximal utilize smooth next problem eq subproblem term subproblem inverse product gradient subproblem product evaluate therefore high per precision dimensional standard significant show formulation noise matrix estimation set sub gaussian discussion maximum noise setup true covariance matrix follow remain algebra write rewrite equivalently express multivariate yy generate condition add generate training goal increase compare recover conditional glasso glasso modify recover support via modify selection method adopt study method precision frobenius evaluate retrieval concept stand positive fp stand positive negative recall practice score metric evaluate f support recovery glasso expect glasso select parameter exclude figure since glasso inferior due penalization figure speedup glasso glasso glasso consistently form norm observation consistent frobenius performance stock price dataset parameter sample summarize describe become standard benchmark image retrieval image annotation contain around manually annotate keyword keyword set visual feature publicly construct evaluation feature tag label joint allow available high score annotate make tag average word per word along extract sample training graphical set contain scheme utilize remove transform set category subset sample frequency purpose vocabulary keyword price stock stock stock daily price trade first adjust return point choose sample goal precision condition size test value sec training glasso glasso glasso experiments glasso glasso precision component formulation skip regularization ground truth evaluate recall definition training since construct two link regard connect category training observation small value glasso precision graph identify glasso glasso stock correlate well approximated construct link identify threshold show graph tend glasso potential link induce tend slightly construct method illustrate detail top present estimate advantage conditional include interpretation term dependency variable without sparsity assumption show rate convergence depend partial competitive current derive normally distribute relaxed normality easily set useful undirected believe extension application follow determinant algebra eq q parametrization pointwise objective convexity proposition hold yy yx yy yy yy imply absolute contain th q combining display therefore complete singular last desire bind q verify q yy yy yy yy yy yy yy yy
rule classify every category binary variable represent say instance th otherwise label category simplify general naive analogy classic category joint along feature true instance might seem basically perform naive show neither intuitive realistic clearly clear jx cm j p jx formula probability instance estimate instance train binary assign last give probability would know assignment shall make approximate notice really variable apply equally approximation remove threshold component classifier compute capable deal input classifier propose content binary capable give instance obtain category category compute score thresholded hard categorization width text corner minimum minimum height text height distance text width block classifier cloud classifier load apart contain say usually part classify binary label fast computation needs summarize see neither issue classifier methodology second heavily suitable content classifier possibilitie classifier exhaustive give aware collection combination different categorization corpus together preprocesse news use split name document test assign test medical every assign mesh category mesh huge often category disease category subtree category discard corpus follow relatively news document final document term name split give remove reduce first english apply mark removed occur less document document preprocesse consider base measure shall along brief measure certain label subset measure hamming compute normalize label hamming label range test subset loss case predict set average measure make case hard categorization assign suitable task recall stand true positive false negative categorization harmonic macro micro average version denote classifier tuning aim make result implement evaluate rank instance partially rank label top list relevant baseline secondly obtain well multinomial naive bayes nn use normalize svm choose recall could perform perform gain max combination selection note implementation use contain except alternative label choose base classifier category correct category real usual dot learn maximize criterion generally select instead proposal reason deal fast work environment accurate regression include approximation accurate vector probabilistic vector value classifier value probability simple svm modification canonical reasonably outperform select classifier effort naive machine refer classifier machine respectively distinguish correspond explain also purpose micro macro baseline least one baseline reach macro respect remarkable present run couple classifier micro micro sign perform macro choose kind nevertheless notice specifically true positive negative positive also macro account number improve sometimes counter intuitive significant macro micro second hypothesis say show table sign resp plus bad baseline sign indicate difference time bad significantly bad ccc ccc nb nb nn nn svm svm nb nb nb nn nn k svm ccc nb nb nn nn nn nn svm svm baseline micro macro methodology follow fact technique improve macro improve baseline micro also especially baseline around classifier bad micro compare bad regardless remove capture base assign probability assume example methodology seem benefit lot macro category heavily whereas micro micro quite baseline sophisticated one table value bad nb nb nb knn knn knn knn knn svm ccc subset error nb nb nb knn knn knn knn knn ccc loss nb nb nb knn knn knn knn svm hamming subset alone pattern behave really collection improve baseline version work naive perhaps improve baseline show present measure great explain method great increment macro micro fact ham false positive negative regardless predict baseline show methodology explicitly take account classifier way assign training content build provide document collection content baseline classifier naive classifier experiment tend known technique question first output threshold question explore relaxed environment categorization natural form acknowledgement support pt la machine unlike choose complexity exponential alternative common use take methodology improve obtain train occurrence exhaustive standard corpus probabilistic base classification text categorization category instead nature corpora article belong collection article multiple document scientific paper associate keyword vocabulary subject internet example furthermore tag sometimes tag instead category scene categorization protein medical diagnosis although solve ignore concentrate obtain I
use attribute test set role business attribute parameter evaluate result business entropy indicate guess assignment user good configuration entropy small determine configuration attribute role dispersion resemble must business attribute uncertainty reason business mutual heuristic score generalization experiment organization unit dash generalization split train mix user role role term compute evidence pz numerator distribution b rewrite beta make must normalize consequence bernoulli derivation substitute back eq analytic evidence assume bernoulli bit bit use outcome noise indicator role negative auxiliary variable li solution I derive update energy introduce derivative derivative first equation rich department mine base control give access mining candidate small minimize paper mining combinatorial need represent derive probabilistic model likely generative reflect configuration additionally assignment configuration assignment experiment real role wide variety popular access assign resource access role role decompose role assign role role conceptually easy assignment user experience well recognize may change within different must aspect develop two kind top role top business security policy exist assignment engineering incur pose security simplify automatic bottom engineering numerous role develop notably unfortunately mining algorithm suffer artificial role undesirable link principle mining goal deviation role deviation goal compression role role configuration call attribute user assign limit among make configuration difficult maintain join business within role fit close possible user assignment attempt make business mining process business relevance role control matrix necessarily role arise role mining reflect setting practice role maximally fit configuration access definition algorithm prediction compression address wrong problem argue mining quality solve contribution mining definition role role likely motivate involve probabilistic configuration likely control configuration role role hold generalization role mining compete experimentally role incorporate business mining thereby improve interpretability role follow inference role problem appropriate evaluation class model user world control show business user mine hybrid role mining algorithm attribute draw conclusion explain underlying mining business relevant account pure without special assess mining rectangle text center height em text minimum rectangle text width em text em node auto input nd assignment assign user role assign fit residual matrix encode mining term user di entity induced p heart mining role say search assignment assume principle organized find access therefore perspective business policy reflect business encode policy enable business knowledge security policy business business attribute principle name reflect mining structure distinguish mining hybrid role might contain unknown perturbation control exception fully system error discriminate exception expert rank procedure substantial manually assumption instance fraction determining constitute role problem mining infer assumption view role differ availability hybrid part influence bottom role case remain solve bottom well hybrid assumption influence pure mining influence availability c dependency involve mining indicate generation grey pure mining role mining compression deviation role alternative problem either contrast quantity involve determine mining algorithm little optimal create closeness fit treat mining require obvious distance metric compare configuration hamming role usually know practice create knowledge evaluate configuration generalization use assess learn dataset generalization unsupervise role conceptually general unclear transfer role hold relationship role employ cost along validation mining configuration neighbor user role near mean create matrix row row correspond neighbor hamming divide ratio assignment behind configuration assignment mining generalization configuration mining fail structure thereby perfect mining inferior adapt compute describe agnostic role mining particular probabilistic posterior user discover tailor employ work method role rule core two role assignment mac role relationship instance particularly relevant deterministic configuration convert probabilistic version observe user matrix hand matrix assignment determine source parameter bernoulli distribution ignore moment treat treat random describe entity mean circle variable circle observable user assignment empty hide statistical dependency entity entity rectangle realization entity exist index assignment explanation text px boolean f term must outcome see go value reflect exploit property probability current deterministic role eliminate model parameter observation appendix px chance assign role user role take px ik eq accord entry conditionally complete user extend core introduce role core model depth level role introduce depth role layer super role generic repeat depth see flat section propose variant level hierarchy motivate consideration assign user column group resource department assign similarity simplicity matrix second group boolean assignment motivated similarity resource grant access object orient group execute alternatively group denote user call business role technical assignment track introduce boolean role hierarchy use rule understand boolean partially make business link assignment logical user assignment user assign path indicate assign multiple path easier express union path I appendix expression assignment business role technical role resemble one role switch condition alternative generic strategy arbitrary treat assignment substantially likelihood derivation avoid probability take constraint assignment group turn already domain arise particular decompose role lack one role assignment convert decomposition freedom fit role flat role layer hierarchy constraint template template specialized environment instance binary variable relevant model instance later generation contain l condition become collapse technical role serve assignment tb even constraint group limit belong belong user partition disjoint business role disjoint decomposition reduce hierarchy user role explain determine model repeatedly algorithm optimize cross validation generalization level hierarchy explicitly role role algorithm role provide role role new plus probability reflect favorable assign exist create enter rich influence role configuration role large likely configuration role model dirichlet number role nonnegative role assignment except event role create role exactly role accord kl account assignment active beta derive equivalent infinite propose hill repeatedly keep track figure assignment calculation hyperparameter htb prior circle denote principle add role applicable could underlie assign necessarily assignment replicate user account bit assignment add bit noise bit generate generation assignment flat generate coin flip let binary indicating bit observe depict explicit threshold denoise way denoise role mining show cutoff htb generative boolean mac indicate noise bit probabilistic model role mining require access em gibbs algorithm update current centroid assignment mac role introduce temperature uniform role set low make expectation assignment sum equal iterate modify negative result minima apparent early minima guarantee effect anneal scheme regime ultimately user role assignment benefit decision row initialize row initialize stop user assignment gibbs assignment keep fix exploit exchangeability cluster involve iterate assignment stop several predefine reach posteriori report book keep score compute average entire configuration restriction practical ultimately choose configuration experimentally investigate model artificial mac dataset compare mac world dataset core differ layer additional encode assignment create mac flat second business role disjoint term user role assignment business role role constraint role assign role difference mac implicitly make encode dirichlet variant suit solve experiment control accord mac assign outcome fair coin flip role distribution repeatedly role connect kind infer mac assumption operate role require additional generalization describe section quality difficulty vary level noise user run generalization hold infer decomposition depict right mac trend type level mac generalize mac even slightly bad mac mac range expect assumption behavior mac mac mac violate though extra mac layer instance assign technical flat configuration role overlap term model variant mac user mac inferior mac algorithm result appear mechanism finding role step gibbs scheme annealing perform beta bernoulli improvement unnecessary extend mac general generalization provide criterion select role assumption world focus explain give input mining assume hide noiseless noisy infer input probabilistic approximately reconstruction assignment investigate question conservative configuration reconstruct user assignment provide question wrong wrong input fraction randomize vary mac constant negative false positive towards repeat old error introduce sum old error mac rate negative false bit htb error distinguish negative positive trend configuration illustrate true value reconstruct learn strong introduce addition configuration result uncertainty practitioner mac factorization technique large matrix business attribute business attribute mine publicly dataset system scenario control server customer access configuration create compare boolean matrix different binary component learn fit learn boolean factorization mac noisy dirichlet capable learning factor role cost solver find boolean matrix decomposition minimize boolean decomposition false differently compare decomposition successively create large role select minimized subsample training user hold user model partitioning mac repeatedly validation select role threshold low error train customer user error run run mac user c min run min mac table number discover median customer generalize equally inferior mac mac winner largely significantly fast time author different criterion impossible fair comparison time mac increase generalization grid le illustrated mac generalize good bit bad tb hybrid business mining remain unchanged assignment business property approach hybrid mining fit configuration assignment business modify set role role assign business cost influence business make likelihood business special whereby optimize minima incorporate business toward minima business attribute input configuration meaningful satisfy business predicate attribute also ideally identical account
solution therefore q net norm ball convex hull unit immediately norm illustrate support u support norm differ sp hold norm show generality yield strict inequality strict efficiently solve composite particular accelerate exhibit proximal proximity map sp lx correctness sign sign hence require order sp fista fista elastic net corresponding complexity generalization guarantee expect huge whether tight support gain experimental mean total way contain validation point net lasso elastic net validation set report table oracle v gain show learn absolute elastic learn use response absence heart predictor training select three method version repository name remove word appear randomly split report accuracy datum regularization give heart mse se se elastic relaxation show elastic exactly light relaxation motivate tight relaxation demonstrate well prediction induce evident unit tradeoff population predictor predictor predictive presence beneficial correlate feature elastic net yield less solution direction yield sense course refined handle within question lemma remark case novel correspond sparsity tight relaxation elastic good elastic net support light justification expect justified envelope zero vector bind vector convex impose accurate magnitude convex convex outer convex hull yield zero empirical inside np scale perhaps good sparsity expect bound assumption broadly aid robustness motivate sample use constraint scale identify support regularization predictor standard get tight convex equivalently elastic net alternative paper relaxation whether tight well possible outer bind associate support norm tight convex convex elastic elastic net elastic two ignore ridge three select elastic correlate large may correspond consider hull clearly nest norm ball dx variational denote subset cardinality imply overlap traditionally application case tractable depict notice elastic cardinality differentiable net vector elastic norm support vector surprisingly entry support invariant derive formula every let unique integer satisfy cardinality combine shrinkage large
adaptive sequence k k tt u v u tu reasoning u k check case even odd obtain contradiction consequence several primal primal proportional case least iterate guarantee alternatively fix candidate thus primal average primal prop mirror note apply logarithmic applie focus primarily iterate interesting distributional worth investigation primal conditional algorithm cut acknowledgement partially european author like mark schmidt lipschitz b convex domain denote minimizer duality relationship x mirror recursion average candidate appropriate similar reasoning condition dx hx hx lead get integration part upper bound gap gap fa ax obtain gap pt plus minus pt plus minus pt pc proposition mirror generalize duality regularize regularizer dual interpretation lead mirror primal dual many problem cast situation simple potentially preferred method interior fewer expensive objective certain decomposable part classical subgradient extension sometimes refer frank subgradient adapt algorithm tailor hard show fact b recover previously c dual iterate review primal dual converge precisely minimization hx make follow assumption constant note conjugate bound situation tend boundedness allow explicit define differentiable gradient include cm quantity compute maximizer duality gap run converge rate convergence rate primal continuous strongly hx hx gradient pair primal candidate dual pair optimality subgradient equivalence mirror descent generalize strongly format original typical often regularizer strongly convex convex generally many barrier fit square although continuous entropy exp e proximal step decay extend general max formulation object vertex hypercube rather lead domain associate relaxation combinatorial linear maximize variation interpret formulation square regularizer lead descent hx proposition consider mirror strongly denote minimizer mirror recursion tt u follow adapt strongly x x x hx hx cm convexity rewrite smooth summing function average size sum weight additional averaging come guarantee close convex strictly include mirror recursion prop classical project classical mirror need equivalence assume still optimality prop review correspond maximization domain equal change conditional follow order towards small correspond linearization later search hx f taylor expansion part maximizer dissimilarity add extra proximal may u reformulate f f mirror equivalence mirror gradient lipschitz b mirror descent start
reduce present complexity rule language think complex learn order apply rule permutation great number begin universe vast reduce block addition improve order rule rule give else block reason linearly order rule beyond need less know great give rule gradually rule rule set first need short need thereby ultimately division complexity example rule step purpose rule however hundred seem child significantly exponentially rule relatively quickly number step time fast start end rule adapt rule give exponential middle rule linearly apply middle else proper new rule hence rule order sequence rule less pick give binary search logarithm number integer counting step different factorial n equation disadvantage impractical give exponential rule rule need less small child day learn rule year learn order
nonparametric estimation nonparametric regression show local procedure develop cubic periodic essentially intrinsic difference key assumed section value fix reproduce certain envelope need simultaneous denote eq strong crucial theorem away infinity existence bias result address show explicit bias specifically asymptotically vanish strong condition uniform boundedness amount choose suboptimal demonstrate validity rely simplicity find bt know simplify remove example consistent therefore coverage increase bandwidth pointwise ci exclude cubic periodic share surprising spline structure z thus imply ci periodic spline cubic calculation rely regression weighted value c value periodic belong th proportional show minimum base meanwhile maintain fourier satisfy minimum bandwidth construction otherwise boundary point make band thin hence likelihood require tune vast dealing nonparametric stand technical smoothing applicable composite hypothesis nan identify chi degree freedom nan limit show minimax well sample smoothing testing hoc thorough either eq q derive remark nan nice remark likelihood satisfy n r nh e cc copy direct reveal asymptotically approach distribute specification global next equivalent condition h reveal scale theory regression possibly change bootstrap reasonable value find testing whether belong space integer nan ny decompose l setup show modify theorem testing specify limit negligible minimax general find supplementary restrictive write pt hold positive achieve exist g ap cr detect local alternative turn minimax see testing establish lebesgue however separation norm minimax whenever equivalently local detect satisfy detect alternative since example model z package see case trivially useful identifying quantity pointwise ci become I smooth l z k dash dash length equally spaced covariate pointwise ci true periodic spline figure coverage cp ci space cp computed proportion replication cp smooth report indicate construct coverage band reasonably meanwhile area cover grow band area remark detail conclude significance setup except test provide local polynomial smoothing proportion correct nonlinear show moderate advantage especially intuitive rapidly vanish example test c cubic repeat procedure document summarize band factor tested significance true nonparametric gamma lead choose conduct similarly logistic binary logistic relationship length simulation straightforward give z find approximate eigenvalue significance numerical eigenvalue value simplify analogous consider true z length space test value power various significance level c plot title spaced validity specifically power cp desire level grow validity grateful discussion careful du author thank co comment lead improvement dms award dms study local spline unify first technical tool call functional traditional parametric equip develop I pointwise confidence ii local likelihood test simultaneous band prove point short arise ratio applicable likelihood carefully discover periodic tool smoothing spline variety literature primarily interest conduct together rigorously derive smoothing nonparametric assume construct interval band interval test local simultaneous confidence global property good little systematic rigorous partly important main smooth function cover aforementione relate tool asymptotic property square equivalent approximate reproduce become extremely complicated smoothing spline effective employ process develop new tool directly broad nonparametric immediate asymptotic normality spline lead construction pointwise confidence theoretically valid point length next testing function nan limit chi constant converge smoothness discover phenomenon nuisance arise nonparametric useful band global literature effort estimate see hilbert applicable validity ever nonparametric demonstrate optimal width achieve assessment aspect inference fan al testing namely generalize call penalize chi freedom locally test complicate periodic smoothing spline mild remark reveal inference mainly study however issue currently spatially peak drive future research reduction quantify need see necessary benefit asymptotic apparent mention extension complicated multivariate smoothing conceptually approach rest notation present local several local inference together discuss give concrete illustrate theory numerical periodic spline suppose copy nonparametric unknown cover assume model instead assume specify conditional value nonparametric second quasi modeling overlap coincide combination fa abuse may also refer homogeneous periodic smoothing use rather simplify existence uniqueness guarantee theorem denote order w tail condition continuously concave open positive z z slow rate impose boundedness information trivially order g achieve linearization technique quadratic device technique hence surprising condition cox assume weak ready present technical tool develop practice construct representation naturally apply hold nh h pt z pointwise asymptotic direct equivalent regression result global hold bias true obtain detail explicit pay obtaining could therefore alternative imply envelope introduce normalize eigenfunction satisfy iii boundary condition follow spline weight involve furthermore remove bias exist literature global mean corollary mainly focus conclude global infer construct pointwise local example corollary delta proposition pointwise ci remove confidence interval corollary asymptotically vanish nh z ci propose asymptotic ci aware rigorously pointwise spline know ci coverage exactly asymptotically furthermore ci uniformity across evaluate point however ci valid even short perform comparison case equivalent version heuristic ci nh proposing show see ci wide meanwhile turn correct ci remove bias inclusion limit ii introduce consistency short consideration furthermore demonstrate superior spline ratio testing chi
black whereas similarity dominant stationary blue motivation subject impossible exist subspace subject g measure table summarize datum improve however subject focus stationarity estimation surprising significantly improve subject shift like toy bottom individually word obtain filter parameter apply method combine statistically pair permutation estimate permutation obtain subject actual value cm ccccc audio competition iii subject std ss analyse stationarity activity investigate gain detail one highly reflect relation experimental mode stimulus mainly area responsible visual test minimize presentation stimulus change increase correspond filter feature little subsection show protocol may subject invariant train crucial training system use consist five subject right calibration experiment second indicate record day scheme band pass day subject mean increase effect lower prominent indicate variability less sub sub prominent change non similarity subspace span one change part question score span measure similarity generate actual lie significantly discriminative datum project discriminative direction method non relate noise reduce need show dominant change discriminant show prominent change experimental condition subject subspaces stationarity interpretable meaningful reduce non robust perturbation subspace weight investigate transfer learn covariate shift imaging modality foundation education project university national foundation education frank mail tu de frank tu de tu department cognitive university subject change computer challenge great importance subject reliably use user construct subject aim common discriminative information conceptual difference subject matrix shrink towards reduce shift paper multi eeg meaningful common spatial stationarity transfer learning process gain brain computer reduce construct classifier approach average order quality especially promising setting transfer spatial pattern subject optimization thus incorporate subject construct note strong namely process stationary satisfied common challenge two signal characteristic method classifier regularize wrong direction careful opposite change induce assumption thus remove unlike reduce shift discriminative subspace advantage characteristic spatial filter regularize towards dimensional assumption true discriminative compare unlikely remove towards discriminative subspace effectively complement subspace different characteristic reduce one stimulus visual test phase system increase increase computing filter lead feature stable pattern extract discriminative user prominent complementary successfully promising transfer novel common organize art introduce analyse toy prominent shift phase popularity domain stationary basically strategy belong second category limit review extraction discriminate state induce band filter eeg pass band power condition stationarity original adapt author trade extract feature filter lead stationary ensure stability consequently compute training incorporate ensure remove stationary preprocessing step neither consider instance learn datum discriminant subject brain utilize collect previous kullback leibler subject largely dimensional write covariance matrix control incorporate user restrictive namely subject author recognize violate subject variability thus subject propose similarity filter different subject goal eq apply learn trade value force opposite force vector filter perform high newton kk extract filter spatial filter restrictive function formulate generalize cluster tackle inter introduce way brain information tackle stationarity like kullback adaptation multi main utilize subject principal subject summarize filter invariant briefly extract invariant utilize datum user prominent capture suboptimal toy example see stationary subspace relatively direction behaviour change explain difference discriminative stationary contrast extract represent real experimental discriminative eeg visual since pattern performance switch different stimulus remove stationary classification subject subject impact classification moving generate five effect subspace sum stationary call contribution source contribution source thus toy normally mutually subspace span source variance discriminative stationary source subject trial point later extract per classifier determine classification leave experiment toy datum repetition increase amount make remain rotation aa add simulate dissimilarity discriminative subspace artificial error dissimilarity plot figure subject namely rate however become affect dissimilarity mix performance stationary remain experiment opposite namely fig stable improvement subject show decrease non stationary subspace drop important transfer goes increase although non become increase average behaviour discriminative stationary subspace course subject experiment advantage meaningful contrary discriminative may regularize stay gain stationarity severe consist calibration without perform two specifically indicate stimulus visually appear calibration locate ms band hz filter training hz densely filter three filter linear discriminant competition iii consist eeg subject right behind randomly move indicate presentation period subject relax eeg signal hz hz trial available subject manually densely cover division coincide competition trial trial run subject band pass hz addition spatial
use ni configuration time estimate run quite time simulate draw estimate panel full panel note curve b ni mostly well less effort ni air region daily pm matter diameter index early calculation air health health range minor environmental air take proper map air daily european human health ec region recently spatio dynamic pm concentration use daily pm daily probability european ec daily year map consider probability produce make limit daily pm measure monitoring web denote spatially uncorrelated latent unobserved air assume covariate spatio field daily maximum daily temperature km km spatio temporal assume autoregressive dynamic spatio mat ern give estimate marginal lc lc lc lc lc lc lc lc lc lc lc lc lc lc lc lc lc lc probability level posterior see avoid result show km section family configuration integration right fourth possibly level expect divide forest basis proportion thus noise area pixel km km lc lc lc lc lc lc lc lc lc lc lc lc lc lc lc lc lc lc lc lc lc trend green model method south region show deviation reasonable semi bottom positive significant trend green conclude period conclusion draw seem spatial control uncertainty region contour stochastic precise definition introduce calculate behind family sequential practice simulate even problem extension fall broad category thresholding compare threshold disadvantage many interesting accuracy interesting contour curve quality contour map present family initial comparison complicate far fairly verify situation family complicate family set contiguous could incorporate matlab see material acknowledgement observe system system datum center pm provide web grateful valuable contour curve uncertainty need thorough walk scale version distribution give determined simplify mcmc still computationally cholesky entire set predefine value several well related find contour sequential accuracy environmental area air region exceed daily european health region early key word contour curve interested area study exceed level typical air level scientific imaging spatio might common science want region common specify location marginal exceed hypothesis act control confident information family hence quantify instead want modify solution marginal value define differ basically category discovery thresholding thresholding euler characteristic threshold latent field focus latent measure mean measurement priori motivate problem example go unlikely base parametric importance parametric extended parametric family region contour curve contour curve estimate computational integral test example application consider air north spatially remark formulate hence one exceeds given quantify uncertainty contour curve bound well area notation contour set contour curve point incorporate continuous contour neighborhood complement union interior negative contour contour interior set follow empty contour respect include occur environmental field essential treat contour region close region exceed certain region process level give similarly negative also formulate large another probability interested idea uncertainty might example simultaneous regression interested significantly avoid set avoid non overlap non reformulate case similarly contour set contour contour set let avoid region contour interpretation important uncertainty region union interval level somewhat find satisfying satisfying requirement report wants know exceed might something probability interpret function contour visual tool answer question function contour function member retrieve function interpretation function location close set value member calculate set practice popular practical application setup covariance parameter far contain location prediction field let main contour uncertainty set contour calculate region probability method shape algorithm calculate use computationally demanding involve propose minimize outline simplest complicated base family routine calculation integration node parameter sequentially soon fall large want candidate na I method large probability run integration retrieve retrieve extending set extend optimally discuss section method calculate full vector considerable effort approximate year approximate mc fortunately number technique integral integration transformation transform unit hyper cube transformation separation sequentially quasi mc replace mc see integration problem integral show technique applicable parametric model note motivating reduce prediction use property main matrix difficult inverting matrix advantage sparsity auto cholesky eq depend element matrix integral iterate integral band efficient particle simulator integral last density sequentially importance sampling first l nh dx dx b importance like conditional target update set select resample meet size technique evaluate computationally optimization family marginal calculate simple one base family quantile q marginal simultaneous non stationary example quantile set set fix dependency therefore set seem reasonable grey posterior mean curve contiguous center smoothed smoothed probability circular radius q general disk smoother equal modification result demand method calculate family family choose set parameter increase sequentially order step probability fall optimization use search uncertainty region avoid parametric level pair family family avoid uncertainty cholesky precision sparse reduce point cholesky improved bound marginal perfectly marginal level obtain correction method improve eq large variable perfectly class lower contain node contain node example add node domain integrate sparsity monte computational practice computation section directly unless laplace good alternative estimate posterior distribution use affect appropriate estimate posterior calculate probability ignore conditionally estimate quantile follow calculate b numerical numerically approximate scale parameter measurement prediction krige location estimate together grey cl cl tc tc ct ct ct ct ct ct tc tc ct ct ct ct ct ct ct bc tc ct ct ct ct ct ct bc tc tc ct ct ct ct ct ct xx correctly curve result contour set show red b grey show dark show grey set grey red panel want correct denote requirement difference calculate twice base mostly monte carlo nothing accuracy correct panel b contour uncertainty krige show black level complement union large contour set field mat ern parameter kind spatial center lattice us calculation expansion set field location noise give posterior estimate krige contour gain definition set parameter special contour two small family arguably computational family although large b lc lc lc
contain power normality empirical level test notice significance small true cn ct cl lr lr lr lr lr mp mp mp mp mp mp mp mp mp mp ex g overall close notice test mp rejection vanish multipli procedure bootstrap investigate specifically rate rejection reasonably nominal scenario simulation fit procedure multivariate five absolutely continuous family state continuous copula copula family three nc multivariate normal normal multivariate copula five parameter copula nine margin copula seven margin copula generation margin n nc expectation dispersion order copula use nc margins copulas f nc extension package unlike multipli carry situation distribution matrix decrease multipli procedure bootstrap powerful multipli dimension rejection vanish mp reveal powerful lr lr lr mp nc nc ex nc look table multipli conservative agreement significance level improve strong nc easier notice finally difficult distinguish table make multipli looking bivariate overall empirical comparison also reveal rejection two suggest difference practical perspective power reach scenario lr nc nc ex ex mention early rely package evaluation finite experiment default indeed quasi carlo random number generation therefore second analytically multivariate approach close significantly nine extremely multipli simulation improve change routine latter carry study multipli datum bivariate fit notice could procedure multivariate would solve lr univariate bivariate experiment suggest computational multiplier analyze consist year daily intel microsoft stock sample univariate goodness intel return simulation execution execution multipli bootstrap order heavily ii multipli numerically bivariate goodness n intel stock bootstrap time multipli procedure minute processor compute nc returns third multiplier even instance execution minute hour parametric mp approach lr mp mp nc ex nc execution expense analytical could horizontal little evidence basis plausible financial would thank associate constructive comment proof random belong brownian follow map p respect equip metric probability follow k similarly proposition converge mapping assumption obtain immediate proposition weakly probability proceeding proof turn supremum proposition consequence center respect except compute jx center dispersion correlation matrix formula multivariate discuss except jx obtain whose otherwise obtain corollary compare cumulative function estimate cumulative employ carry goodness parametric easy implement expensive alternative technique fast empirically outcome generic computationally efficient multipli goodness fit procedure sample power model contain nine gain multipli procedure product fast procedure degree phrase central cumulative integer base computed parametric assumption extensively goodness two statistic er von minor variation important goodness concern computation critical value regularity state weak drift goodness theory free several statistic martingale transform boundary review location er use distribution statistic weight explain generic implement resample carry test fix idea let repeat stand version realization convention rejection ns n procedure generation consume consist reference therein technique recently assess nan large parametric reduce minute aim investigate empirically bootstrap sense determine parametric bootstrap roughly power small powerful acceptable recommended become practice multipli appear extend establish context instance jointly weakly speak copy new approximate hypothesis f explicit sensible tend verification assumption family illustration purpose er von reformulate adopt procedure compute n nz z statistic regularity indeed jointly copy limit compute tend advantage multipli parametric bootstrap introduction roughly speak generation fit generate generation multiplier typically fit respectively fit resp multipli realization indicator concern smoothness discuss instance smooth continuously candidate square integrable relate estimation prove hold
flow treat conditionally imply multivariate mean come different slightly align system action due characteristic robot signal different modality follow methodology delay maximize alignment signal delay manually basically gmm approximate desirable compression affect likelihood hundred multivariate case threshold need spurious one likelihood equation basically covariance weight contribute proportion update mixture inverse learn feed sensor optical optical flow prediction mixture rule map probable identify component depict time show optical v mixture capture information check predict application predictor sensor robot new robot active frame activation component event follow active optical optical flow highly likely around second obstacle experiment core ghz processor ram gpu optical object aim robot action perform human decide decision situation stress acquire knowledge action robot five forward velocity angular threshold correspond distribute look optical flow field angular unstable magnitude nearly zero unless ground sequence absolute normalize na I predictor basically expect horizon predict predict optical predict flow vector choose compatible I derivative optical flow confident prediction predict test could predict account show log ratio na I predictor decide experiment platform information optical axis flow present row big flow black row encode shape forward well condition mostly assumption I ahead present mode usually information valuable segment alignment stream align significant reduction na I error e without use reduce incorporate half almost independently gmm action consideration sensitive h logarithm na I gmm test gmm I learn obstacle later object significant horizon second detect robot obstacle binary optical flow change middle optical flow graph optical flow mobile advantage improve dynamic extract accurately predict application model mechanism build robot flow technique particle work principled extension spurious refine underlie ram department electrical engineering college bt email event agent need make term uncertainty internal external environment acquire mobile learn optical flow image learn optical time horizon reinforcement modal simple one interact environment consequence effect agent sensor acquire capability optical scene move movement body encode scene appearance benefit aware action learn optical action capture task perspective mainly motion effect scene appearance discriminate change optical flow primitive day old behaviour mechanism enable term optical dynamically analyse optical flow optical flow prediction consistency predictor state online task build optical flow mobile robot predictor axis interested term option sensor value decrease unfortunately handle optical flow grid field optical variable robot encode angular extract define angular velocity optical current hypothesis optical na I predictor assume decide analyse sparse backward modality use present indicate region show learn useful period line method propose joint optical flow previous action optical flow example flow environment optical optical
scope article successive time process speed count limited gps user nevertheless traffic huge data position traffic compose day et france et france flow road france flow regular weather weather soft medium traffic high due several gps accuracy wrong incorrect variation individual traffic eliminate outlier flow speed free flow datum record reference link enough record link confident already consistent france treatment main issue build weather build weather understand weather consequence road behavior associate traffic weather datum weather propagate get pair find rule forecast weather velocity indeed build include range road yet provide estimation road weather method forecast scope work road weather depend road road weather go fast speed vx vx tt vx speed call adaptive spline k hinge point suffer lack indeed homogeneous among may weather fit drive bad weather variant well thresholde bad weather traffic critical speed decrease wish traffic weather condition following extract neighborhood neighborhood practice arbitrarily narrow enough time find fact thus decide relax traffic stationarity day existence stationarity belong stationarity want weather e e two represent weather section good link statistical modeling nevertheless turn link task residual selection established present match france day also weather france basically link highlight thresholded classical choose one compare good classical selection datum contain sample remain validate select estimate rmse goodness calculate q obtain rmse thresholded fit well thresholded moreover thresholded constrain model fit model fig mean conclude thresholded far advantage homogeneous detailed h apply correction weather take separately road link weather global justified generalized interest link weather condition structure complex different instance three link another network build kind road matter fact road weather may level behavior one h sum link deal linear thresholded linear thresholded model build pair homogeneous among link show parameter km km highly wish network dependence depend correctly highly normalization fig joint concentrate h remain parameter e compare rmse aggregated significant link nevertheless construction road correct thresholde road easily homogeneous thresholded remain eq interesting interpretable road really illustrate fig speed proportion proportion flow speed represent condition let km flow km km road limit decrease thresholded normalizing speed also need quantity calculate speed speed network obtain loss link compare predict weather version calibrate get basically proportion flow speed threshold value difference speed thus correct information weather learn overcome stationarity mix change weather traffic desire decision extend road quantitative major forecast weather contribute forecast company study weather actually thresholded think weather flow global build weather nothing free speed consider weather thank provide respectively weather road road weather france mail understand weather modify traffic dynamic change change thresholded road road network provide speed term forecast linear thresholded road traffic spatial weather accept weather modify significantly traffic actually bad
controller methodology determine whether quantification tractable many temperature temperature quantification j amount solid default controller mark solid line characteristic controller substantial future direction speak experiment energy wide modality berkeley edu master berkeley edu cs berkeley control improve energy efficiency air system potential economic build wide hybrid update model effect air temperature controller large usage justify present energy per p confidence interval united air systems drive research consumption ensure building respect modality methodology scale design difficult weather design operate efficiently setting require measurement moreover consumption controller reduction usage temperature water outside air energy controller controller air single room warm day day mathematical dynamic experimentally usage weather wide building control along distinguish describe controller able begin procedure identify next describe hybrid adaptive vary differ change furthermore robust nominal measure provide experimentally controller controller methodology use nonparametric energy part berkeley efficiency platform building laboratory seven office building measure protocol interface water air air water air throughout operate design modify operation box modification general default controller loop box keep air temperature control box air minute e denote control air send air dynamic nonlinear simplify typically adjacency eq adjacent additional vary principle make affect bilinear finite value mode I unknown scalar function system week identify hybrid system change fidelity modify modeling approach amount hybrid two hour hour consecutive hour reason pick experiment roughly constant construct discuss modeling load change purpose load short within quick span serves provide additional measurement suppose prior coefficient ib j ic j constrain least mode system ensure fact reflect capability box relatively across lastly window air amount temperature box controller controller controller restrictive portion k temperature amount model fitting point comparison point air eq minimum air flow allow building scatter amount controller largely leave approach several air explicitly implicitly vary n consequently simulate minimize cost optimization fortunately greatly change hour change hour combination far reduce specify mode hour four combination low computer mention early need variable would specifically quadratic program reasonably formulation form piece usage consumption water air usage usually subtle energy energy reason usage physical stress fine usage source cost energy really get simplifie model usage energy energy controller behind system nominal use maintain improve rate minute intuition represent correction provide adaptation inherent controller load due weather control air
structure output space yy hinge surrogate loss apply prox sdca note strongly norm indeed maximum definition therefore objective overall imply prox sdca bind option fact v prox sdca maintain w x iw tt maintain explicitly maintain vector efficiently find sgd structure prox sdca output gap eq problem instead imply corollary replace match main duality loss approximate point coordinate ascent frank wolfe match handle regularizer involve notation prove theorem run sdca careful option choose side choice simplification option iv employ simplification convenience list primal dual formulation key lemma update expectation obtain lemma third strongly n dd duality randomly next rely fix achieve maximal dual suppose lipschitz gradient combine lemma tell imply next inductive suppose turn rearrange choose obtain sufficient condition n imply remark version numerous regularize rate sometimes improve follow generic associate regularization structure svm analyze runtime coordinate solve u z u associate optimize dual dual immediately duality gap regard sub version round optimize analyze sdca lipschitz smooth functions lipschitz differentiable convex define smooth generic prox sdca idea describe maximal allow step let sufficient primal lead w v write gradient method sgd show strong result prove prox sdca loss motivate sdca sdca choose maximize rely directly maximize dual general propose choose step increase objective htbp prox sdca scalar pick option large option u option definition lipschitz non iv return prox sdca expect everywhere superior sgd one numerous interest categorization bag short choose approximate base sdca goal solve linear require strongly calculate therefore iv iw w I hand set prox target accuracy prox sdca runtime obtain theory factor sdca sgd require intermediate iteration relevant shrinkage fista approach batch fista enjoy however fista rather runtime q contrast sdca happen choose need regime prox sdca fista runtime
news manual setting already balanced file document actual description typically section subsection sub subsection paragraph example throughout article environment indicate paragraph character indicate phrase code part subsection care begin character english character three display text formula simply notice equation style look slightly display style display equation set equation environment environment structure somewhat differently enter able article conference book article occur text automatically several citation cite short reference uniquely identify document author title identify item file scope detail form citation format across page top table environment wide table figure page bottom page figure detail figure table environment forget file article logical like axiom corollary proof form distinguish document environment create gb ga environment logarithm e list construct use form construct environment gx fx gx gx contradict complete environment source code primarily intend create camera copy omit problematic paragraph acknowledgment make exception book article nothing location acknowledge file rule discuss indicate start b title include body add file run resolve create file file file comment file helpful comment moderately expert read please usa format lars rv rv rv email email alternate tight design file write
parameterize convolution sub reconstruction form q notational brevity combine convolution sum matrix convert map combination reconstruction part integrate straightforward also feature pool operation direct eqn fix reconstruction eqn eqn impose pooling become employ constraint fourth importantly directly pool employ multi model stack manner channel property solely intermediate weight convolutional operation convolution layer derivative repeatedly version filter key section respectively ensure invariant small change pool otherwise able perfectly pooling would generalize axis mean precision introduce weight sum give unit gaussian location show illustration parameterization brevity neighborhood rewrite several advantage sum pool gaussian select allow variation max change invariance small adjust variance continuous occur pool illustration gradient analytic activation enforce motivated factor notion intensity parameterization weight represent individual negative third negative brain finally negativity reduce flexibility improve utilize enforce constraint optimize sparsity two though latter lower toward descent neighborhood layer variable eqn input problem adapt scheme set element shrinkage projection shrink project onto order employ estimation technique advantage layer significantly account architecture manually good inferior per mini repeat minima proceed epoch overcome set feature epoch perform demonstrate explain stage mini datum cause new map turn reconstruction remain continue get gradient early therefore suffer map epoch reconstruct optimal element continue wish intermediate layer optimize assume variable combine layer update pooling variable reconstruction error chain parameter neighborhood q q coordinate eqn pool however estimate likely pooling coefficient map pooling size epoch reconstruct propagate I loop low eqn cg project length output map filter update term bottom map index update filter plane practice parameter infer project feature pool optimize infer shift layer pooling layer filter initialize work well second jointly take trade filter capture detail filter become dot avoid fix learn layer layer layer nice part optimize learn filter initialize random project begin inference start create feedforward forward pooling use moment extract pooling parameter provide initialization pooling filter initialization evaluation mnist handwritten digit dataset instance per compare interpretation image input digits architecture filter map contain connect randomly amenable gpu processing layer configuration input one motivation make top level activation infer input patch pool analogous sift show allow operate high dimensional classification patch map multiple layer patch hyperparameter mnist classification layer epoch pass dataset epoch map test improve optimize inference group connect layer visualization layer model learn fig demonstrate reconstruction b display raw filter helpful input sensitive search infer input activate fig representative selection feature activation activation inf biased view large feature pooling correspond image activation stage visualization examine shift analyze separate connect understand visualize second project pixel pixel projection activation activation layer alternate pooling feature decomposition pixel reconstruction color depict high assignment pixel color indicate color feature color purpose understand also activation utilize show operation map box one second starting show reconstruct layer benefit smoothly non overlap pooling layer long range structure group common max pooling max smooth transition overlap use convolution pool activation ie maintain adjust confirm break regularization display comparison pool significantly max optimize max despite train smoothly throughout tune property pooling stack max cm test encode fundamental drawback procedure improve performance layer separately map pool directly differentiable pooling variable layer pooling inference need examine depth combination inference find training separate phase work training optimize pooling filter combination filter make allow filter move need updating pooling table optimize optimize joint key differentiable pooling mm infer phase update update use model phase update update respectively mm discovery evaluation classification run also reduce compare iteration classification happen removal high make discriminative change level order reconstruct feature pool suit due summation negative pool enforce positivity gradient negative pooling filter feature fig many rd nd map encourage filter similarly layer due notice nd improve varied help cm cm mm previously show encourage hyper laplacian apply train run regularize training control various sparsity
round care use exchangeable exchangeable ti proceed tell sequence identically apply sequence number partition argument exchangeable partition appear draw rest use generation argument belong apply condition frequency partition contain imply every partition exchangeable partition exchangeability draw induce partition construction think partition convention interval break yield exchangeable consistent case collection follow feature exchangeability allocation belong specify frequency summation partition independent ensure summation unity stick breaking obtain break unit return generate stick proportion stick remain length stick v unity convenient condition independent rapidly simulate stick finite fall stick crp chinese exchangeability value mixing frequency outcome bernoulli find break chinese restaurant divide customer crp rest color former collection gray white color gray begin white customer crp whereby start gray white draw replace gray ball white ball number check define crp gray customer sequence ball ball ball number round exist customer customer equivalently customer crp construction customer color gray customer table color denote therefore play gray customer white customer customer scheme gray ball initial white ball ball thus customer table outline recursively independent customer identically table table v identically frequency process well define since frequency sum crp recover stick length choose contain consider feature p sort encounter ex stick crp gray ball ball feature draw likewise round ibp model point initial gray ball white ball iterate across function partition stick length stick length sampler mix impossible partition block feature frequency stick crp around examine finite crp stick use approximation create truncation stick break size exist technique avoid slice thus far posterior length stick different particular use approximate minimize family potential stick breaking demonstrate number stick length ibp example cover build ibp examine slice stick ibp posterior stick allow underlying stick length exchangeable collection appearance random sequence exchangeability sequence assignment give point sampling exchangeability suggest exchangeability stick length random label assign label block belong feature allocation exchangeable continuous open interval correspond jump form partition stick construction partition label measure unit may sum way increase stochastic one correspondence mean choose distribution weight due summation independent stationary measure follow nonnegative left increment purpose wherein ultimately scale assume range interest stationary increment poisson measure countable subset process characterize disjoint treatment drift jump component result write eq countable process evy eq jump frequency jump may frequency hand side jump interval identical e jump stationary transform evy eq drift nonnegative call drift evy particularly since jump process methodology stick generate membership take bernoulli jump success jump every associate jump eventually jump appearance iteration let choose recursively jump see round particular see early derivation stick stick length ibp beta stick connection zero evy parameter parameter allocation l evy index appearance sequence ibp stick poisson suppose point value point also process define measure eq evy assumption distribute beta without atom prove theorem distribution atom weight stick length check recursion assumption note choose total mass poisson desire cf atom rate desire recursion note remain bernoulli minus measure next generate process stick jump however normalize measure finite mass sufficient must stick length partition jump jump exchangeable laplace derivation distribution jump find normalizing jumps exponent exchangeable partition exponent chinese restaurant start gamma exponent correspond crp concentration jump crp derivative laplace calculate partition generate normalize jump derivative always straightforward calculate note general integration yield substitute equation pn b kn b n n line form beta crp desire final laplace exponent know find partition normalize stick must distribution normalize jump feature appearance jump correspond recursively none jump chinese restaurant jump crp stick example notation sum jump minus j j show lemma evy jump density lebesgue lebesgue dt dt change variable transformation former variable latter derivation term notational clutter end evaluate ft gamma determined e k second read ahead imply assignment object random uniformly partition label special general straightforward gibbs stick ibp illustrate find exchangeable usefulness stick representation crp stick discussion jump unnormalize jump label unit uniqueness exchangeability sequence label uniqueness moreover wish associate point height vary randomly specie likewise collection correspond assignment equation book person appearance block similarly appearance label complete let nz nx ny atom completely disjoint jump us size point intensity point yield tuple intensity tuple completely completely random gamma labeling partition specifically come point process parameter along point axis atom plane dp normalize gamma atom length dirichlet form induced partition partition restaurant process feature block I measure distribution beta choice uniform line segment beta value atom measure bottom accord process induce allocation index model first outline context feature structure generalization beyond option remain section infer underlie infer generative block handle dependency parameter particular depend block dirichlet mcmc provide use beta augmentation simple partition exchangeable stick encoding frequency occurrence block random stick length likelihood focus show successive restaurant process specify induced partition form draw dirichlet completely specify induced form I I draw weight extension idea lie scope extension crp form stick ibp beta explore generally demonstrate alternative partition stick length measure expand suggest structure might provide allow feature permutation research national science foundation material foundation award office contract grant point group block notably formal analogous call integer feature topic combinatorial representation analogous beta thereby bring previously achieve process allow collection often move bayesian paradigm rich express belief thus field dominate dirichlet flexibility arguably aim broad kind useful toolbox bayesian mathematical structure useful specification combinatorial mathematical field focus area version classical combinatorial combinatorial analysis establish speak statistic combination latent compositional modeling essential calculation serves exhibit bayesian combinatorial one well dirichlet expand upon notion stick dirichlet process factor dirichlet statistical idea associate underlying factor interaction rise need counterpart characterize combinatorial model connection stochastic model stick break nonparametric adapt expressive specification essential inferential tractable reason popularity dirichlet stick break yield inference algorithm analogous discuss representation associate beta process consequence inference remainder organize follow introduce model allocation exchangeable know crp correspond ibp choice rich stick frequency associate label crp ibp ex stick ibp marginalization crp ibp mention stochastic study bayes suggestion development intuitive idea constitutes formalize idea proceed underlie combinatorial datum index mutually exclusive exhaustive nonempty partition similarly partition exclusive nonempty example n introduce mutually exclusive exhaustive restriction allocation nonempty belong block belong infinitely coincide index consider array feature represent may picture impossible display infinite pixel allocation block block partition block allocation note converse general continue development special partition allocation feature exchangeability modeling rigorously block allocation apply f nf imagine similarly agree randomness say allocation allocation q consistency feature allocation characterize restriction sequence space allocation sigma algebra sigma index restriction circle customer table middle customer customer useful feature emphasis exchangeable partition exchangeable exchangeability block chinese restaurant restaurant partition increase belong chinese restaurant customer chinese restaurant customer restaurant belong customer table else restaurant recursively restaurant table latter next available exist possibility distribution table customer shown summarize choose n consist customer customer try customer gray sample indicate customer exactly index generative model allocation discover enumeration crp poisson number feature factor associate density let choice k n index restaurant finally process allocation generate multiplying factor f f n h n k quantitie ibp exchangeable recursive exchangeability equation ibp see form specify index yield exchangeable crp example crp ibp refer conversely partition label index partition sequence result call give sequence partition interpret contain label feature cardinality definition allocation unique collection share label similarly collection cardinality form induce feature allocation feature reduce allocation concern label feature appearance assign labels st index block uniformly recursively see label use scheme appropriate partition find appearance know induce completeness conditional z probability underlie z n k chinese chinese restaurant chinese crp find rule imagine must
template extract capture think score model database involve face modality fusion arrange vector performance svm experiment arithmetic give svm give equivalent combine previously work various fusion architecture base score fusion issue fusion automatically fusion solve order allow reproduce database first create available database modal tuple tuple modality present subsection present summary two tuple intra capture tuple tuple score recognition time second database create combine template ar database compose individual illumination difficult database reason involve ask time capture compute inter capture subset produce help correspond score empirically choose g validation main difference benchmark modality performance intra important private tuple intra score instead database score system private template adapt physical access building modal adapt logical access web modal tb intra tuple tuple subsection fusion genetic score need normalize normalize operate indeed score come classifier necessarily evolve normalize equation present normalization score capture user call represent stable impossible paper procedure genetic fusion genetic programming quite genetic population evolve create evaluate program represent mainly take child leave kind case ordinary genetic choose tree replace another computer program function build step repeat termination criterion fitness reach reach number fitness measure program fitness operation new generation operation previously individual program create randomly select program example provide mutation new program randomly provide population winner tb genetic reference field list symbolic economic biology bioinformatics compression seem apply find genetic use genetic meet maximal reach half score tuple first tuple tuple program basic constant fitness case thank library fitness generate global computation roc read genetic root function terminal avoid unimodal set function function child depth set avoid stay mutation individual evolve quantity individual much interesting half validation mutation mutation individual scheme generation first generate keep generate program benchmark rule return minimum return score score return sum genetic fitness genetic engine present configuration genetic fitness search validation performance present performance database system fusion mechanism concern art performance see good operator outperform method fusion c fusion far function far gp present performance operator private fusion point give global really compare system fusion scale quite programming give global curve inferior overfitte well state weight fitness criterion meet always reach generation fitness evolution run database scale easy fitness evolution database fitness appear bad deviation fitness huge explain generation say well automatically return score tradeoff pattern already way really train pattern empirically normalize fusion inherent genetic probable simplify another parameter fitness program interpret present depend complex compare subtree tree modality terminal set genetic fusion give genetic weighted time genetic important two method genetic need ten less propose paper approach automatic generation decide automatically fusion program genetic concern design base inspire return threshold heavily database improvement hope fusion system thank genetic genetic engine performance metric add genetic template fusion thank library problem database region financial suffer drawback general capture extraction modality state svm performance genetic give well performance database score classical validate propose method art score performance new modality example classify even literature biological dna analysis signal individual perform dynamic handwritten computer way drive pattern face recognition user method bad whereas performance instant accept several accepted security less well modality correlate generic different sensor sensor acquisition capture texture recognition modality instance right per video compose previous interested fusion combine score compare need evaluation evaluation realize curve information regard quantify device physical attack service attack fusion depend modality metric represent ratio user overall small globally system
contradict incorrect overcomplete solution solution necessity general lebesgue measure zero subspace one large notice instead objective therefore lemma true optimisation formulation multiple cause fact dictionary indeed make much map r r nk sufficient uniqueness particular optimisation unique contradict standard cause pattern need uniqueness boundedness solution bar mention coefficient generate matrix reduce subspace practical robustness successful distance embed sensitive restrict definition isometry measurement proportional total function propose comparison option choose also sparse position subspace quantify number subspace q concept theory binary l lf depend figure plot bound plotted ratio new sparsity ratio small boost recovery see optimisation technique powerful project admissible norm schmidt frobenius norm relate quadratic onto keep large onto keep absolute letting intersection set provide necessarily projection onto alternate projection projection statement projection element zero simply point statement unique choose projection iteratively update current negative gradient direction onto solve f find select selection objective half optimal descent constrain size thus objective increase parameter initial sparse change gradient algorithm check update dictionary selection type dictionary may greedy sparse technique however reference sample signal stable close set subset accumulation h admissible black dot zero dot horizontal line gray horizontal line plot area zero unit normalise norm vector randomly select distribution magnitude sign descent base superiority coefficient recover panel sparsity admissible figure explain panel show clear set correctly recover keep repeat plot transition joint constraint colour high area exact recovery large add demonstrate new dictionary dictionary times overcomplete want large dictionary difficult solve need keep dictionary joint overcomplete environment ghz take another test select dictionary representation learn joint joint well reconstruct reconstruct reconstruct setup audio signal propose dictionary dictionary learning method record mostly select eight hour record overcomplete two dct dirac transform incorporate audio combine atom subset appearance figure low frequency dct atom atom although delta dirac atom window close atom audio database get plot overcomplete dct reference solid line snr dictionary overcomplete run learn sample select learning similarity fast sparse necessary dictionary use mod time simulation show dot dictionary aim dictionary dictionary ghz fast ns implement datum ns sparse expensive part bottom right new dictionary selection atom signal dictionary reformulate approximation sparse coefficient overcomplete joint generally infinitely solution sparsity active show overcomplete joint approximation problem dictionary gradient approximately synthetic support namely simulation select subset overcomplete dct dirac dictionary handle vast need keep dictionary vector multiplication audio small overcomplete seem extension investigate left investigation exact work paris university langevin paris france ed fr representation describe sparsity processing rely simplify unknown domain knowledge signal paper call reformulate consider representation subspace synthetic realistic sparse successful low application elementary plus noise atom set call overcomplete extra failure redundant main lot redundant already use knowledge example combine r select dictionary fast dictionary product give atom complexity search learn learn avoid atom signal represent reader ask dictionary sparse complexity success sparse structure space property affect dictionary cause atom atom similar similarity indeed atom equip line search monotonic investigate test large formulate dictionary overcomplete introduce iterative show section lk find objective decomposition efficient
already show fact explain expect poor performance single expert unlike active indeed incur active poor performance convex weight cause specialize helpful mostly conclude benchmark rule worth good second compound therefore fix operational consist produce forecast hour forecast expert run rule access instance expert different prediction similarity estimation fair expert form expert expert fair weight weight equal expert regret operational section describe weight usual sum real statement figure basically need run round two share deal expert day ahead period convex n f share update correspond name whenever share update multiple empty extension share aggregation operational forecasting regret statement extension coincide instance weight differ loss one quantification appendix note form interesting fair get performance comparison note sensitive initial allocation well theoretically one rule aggregation indicate summarize turn depend weight fair sequel compound excellent significant performance aggregation rule track expert improvement respect aggregation rule obtain prior fair potential black important operational forecasting side review know expert aggregation rule hoc accommodate specialize exponentially average indicate operational forecast rule datum consumption parameter adaptive rule tune theoretically hand improve accuracy note improve finally term aggregation expert come rule anonymous valuable drastically exposition completing write partially support grant adaptive grant induction define normalization expert round expert entail e w rewrite thus appear get note proof straightforward share note rule implementation choose j prior indeed efficient implementation indicate part sequel sequence expert denote define introduce ti interpretation switch inactive stay current active slightly switch current expert inactive switch expert compound distinguish whether prove immediate bound definition loose second induce replace recall distribution update definition transition function rewrite j ti w last fix compound vector tt sequence index recall trick bound note show plus ratio weight use enough also magnitude state entail paris france sup paris france france sup paris france paris en france specialized expert relevant contribution fix one consumption concern general methodology state parameter demonstrate improvement square behavior large error specialize expert arbitrary round forecast past outcome error aggregation expert interest aggregation rule expert bad sequential ahead forecast consumption take variant basic expert call specialized round expert output scenario specialize expect forecast mostly external consumption specialized work specialized expert rather reference formalize specialized expert one paper context specialized pass another theory many field mention forecasting consumption study aggregation rule already load review framework expert discuss take important theoretical bound online put rule provide france market organize standardized historical result rule tune sequentially section also behavior occur aggregation rule bound predict element number forecast henceforth available one forecast active know expert time assume empty produce forecast weight prediction linearly expert precisely reveal start round loss output thus loss forecast sequential ensure combination expert expert shift possible square percentage instance equal dirac weight average aggregation rely denote choose convex put perform exponentially follow loss denote quantity theoretically uniform advance like trick cope limitation concern see different rule sequel replace depend family simple sequel content next introduce statement regret heavily specific loss hand work square contrast compact solely lemma rule two statement exhibit little learning predict j ty rule convex belong forecast proof theoretically choice lead comment compare follow formally generalize trick convex subgradient product subset gradient replace definition regret figure aggregation imply theorem subgradient uniformly supremum vary sequence expert observation third regret performance expert sequence expert constraint expert specialized formally sequence instance call compound expert put vector compound expert compound weight minus one correspond expert convex compound compound shift might rule index weight control aggregation loss efficient exponentially weight spirit uniform fix share specialized depend adaptation set forecaster mixing rate predict ty ti convention denote fix rule use section sake completeness update bound belong varie vary expert forecast theoretically theorem define desire soon theoretical choice section gradient trick replace loss loss obtain variant denote bind loss subgradient sequence forecast aggregation rule automatic advance rule parameter almost optimal indicate online tune rate exponentially bound thus poorly illustration different set result come poorly remark theoretically satisfactory consideration sake completeness first symmetric tuning rule tune equally performance report table theoretical practical theory tune thin exponentially need far concern aggregation set base rule aggregation report elsewhere family rely possibly value past forecast instance weight aggregation rule denote consider equal member family meta sequel offer guarantee term underlie family need parallel instance together meta rule impossible minimization instead whole seem somewhat second meta article report application prediction cluster choose theoretically optimal sometimes losse see improve slightly datum respect sequential target expert good example article application management tradeoff energy consumption wireless tracking traffic also online e vast average expert set enough method sensitive topic forecast performance latter already standardize outline treatment datum section expert strategie choice operational automatic comment robustness assessment strategy uniform forecast expert strategy convex finally pick good forecast expert forecast good element form prediction expert observation period absolutely merely black heavily hour day large enough hour day ahead arbitrarily hour interval characteristic hour hour consider report error root aggregation consumption characteristic lc number day interval expert cc graphical datum leave expert scatter relate frequency activity pair first choose differ one weight instance compound expert explain fact good predicting instance active compound expert require difficult cope active give partition active expert eq active corresponding choice partition choice even relatively choice versus achieve compound achieve possibly benchmark serve procedure benchmark aggregation compound j element accord expert convex aggregation introduce rule summarize tune outperform oracle type rule choice far preliminary exploit concentrate limit carefully weight remain change converge vector meta get somewhat second grid around grid evenly grid evenly grid choice preserve meta grid choice need summarize table comment performance sequential aggregation rule choice obtain aggregation meta grid middle grid meta grid constant p cc convex cc tuning parameter choose meta select rule cc rule use department importantly length period weather cloud wind pattern summarize characteristic detail divide use expert forecast
schwarz differentiable density build exponential need choose auxiliary laplace transform ensure many distribution multinomial von wishart start retrieve canonical usual canonical include parameterization family parameterize coordinate moment parameterization arise see case strictly function conjugate maximum l strictly differentiable compute functional inverse gradient strictly leibl exponential bregman follow moment often divergence use mixed coordinate geometry system orthogonal since mixture single component distribute report unique maximum hessian consistent asymptotic normal covariance mle report duality bregman square euclidean multinomial leibler divergence divergence wishart use bregman duality prove maximize average sided bregman coincide center follow geometry average likelihood reach shannon x reduce empirical bregman center obtain diversity report finite probability exponential latent variable get otherwise mathematically rewrite family mathematical equivalence average write mixture maximizing eq gives hide since per g parameter convention mean sense bind fix function bregman define conjugate bregman decrease monotonically reach correspond local bregman greedy swap heuristic swap take geometric show assignment reach prescribed reach local denote proportion assign th jensen diversity minimize complete shannon simplex entropy weight complete q summarize identically independently family characterize moment exponential family cluster far later I I depend parameter unless reach weight unless local reach parameterized system assignment weight single mle correspond bregman maximization describe straightforwardly code e hard mixture von log compute function invertible approximately proper initialization assignment tx reach per cluster degenerate em identify convergence em gmm way initialize define partition diagram precisely joint gaussian weight ix gmm xy cell explicitly weight bregman diagram design partition assignment mle em bregman tree mean heuristic greedy optimally global swap mle tx yield von need reciprocal gradient focus von remain problem properly step way soft convenience mle properly initialization bregman guarantee initialization far minimize design far et mle may mle infinity see discussion penalization boundedness mixture complexity exponential future compare model modal family determine appropriate many criterion criterion aic problem exhibit make selection unstable interpret nk mixture simplification polynomial complexity practice entropy acknowledgment early prototype discussion center popular iterative fact swap per cluster additive mass cluster generator eq could exponential family split bregman loss center monotonically decrease terminate finite optimum bregman generator minimal diversity cluster rewrite assignment step close cluster since center iterate inequality since function local optimum I I iy ic I I w ni loss minimize cluster variance bregman quadratic induce generator identical et rewrite centroid cluster center within cluster bregman mathematically generator arbitrary minimizer assignment mean monotonically base global bad exponential plane let dual natural consider parameterized matrix usual cyclic product sufficient auxiliary parameter log symmetric eq moment parameterization leibler leibler decompose bregman write explicit usual number ix initialize seed mahalanobis let ix diagram update unless reach reach mle initialization report seed uniformly probability c c eq add seed initialization soft bregman em write deduce conjugate reciprocal mixture cluster parameterization px soft calculate moment parameterization bregman divergence conjugate generator meet decrease complete weighting iteration initialization build mle seed seed tx kx exponential parameterize coordinate versus log parameterization information matrix conjugate canonical natural natural moment usual usual space density parameterization use parameterization convention shannon mixture component cluster hide sufficient statistic maximum weight parameter partition cluster center cluster proportion bregman divergence generator b jensen diversity fp n n fx e f j respect additive bregman keyword frank computer mail frank family traditionally maximization monotonically increase incomplete expect assign likely update duality bregman divergence local complete mle bregman heuristic greedy swap cross entropy proportion update successively careful possible family bregman mean mean denote mixture summarize mixture often mixture parameterize positive definite mahalanobis positive definite draw variate model variate cholesky variate drawing sampling mixture doubly essence multinomial gmm component learn set sample gmm frequency information image figure gmm image model point color gmm representation retrieval ir paper mixture belong bernoulli multinomial poisson mixture model wishart mixture gmm model depict hard
study system various entropy thank small suffer overfitte insufficient approach overfitte prior response drawback likelihood place center key principle principle idea inference combination framework special introduce inference free parameter make size available day day amount datum variable information theoretic inequality capital letter capital letter variable capital letter denote set bold capital shannon equality therefore preferable many require low entropy many system estimation probability combinatorial event significant role regard know overfitte parameter often suffer overfitte viewpoint avoid bayesian background datum increasingly natural add frequency knowledge avoid generation g usually always distribution interpret prior knowledge probability parameter regard generalize although widely use accept axiom probability improper still inference another keep characteristic ml estimator knowledge regard accordingly insufficient small becomes consider regard regard increase principle optimally method physics method utilize ml modify regard essential principle estimator effective I principle within constraint far principle log contrast I low high preferable case I principle necessary devise entropy I principle balance optimal contrary principle branch analogy select achieve balance minimum energy analogy play temperature represent hyperparameter entirely hyperparameter propose method interpretation estimate nevertheless free include treat fix regard therefore potential free energy extract determine still organize theory joint experiment estimation conclude size utilize entropy concept accordingly energy constructing hereafter multivariate knowledge assume I discrete probability joint entropy quantity accordingly come author aspect mechanism internal kullback leibler kullback leibl distribution manner cross according define way empirical random empirical eq relative denote energy internal leibl estimate ml also divergence hereafter minimizing define define practically ml q indicate data temperature often hereafter regard introduce statistical available size new temperature unit uniform denote I distortion divergence divergence temperature datum temperature random define omit recognize temperature assume see get viewpoint select explain I principle similar relationship internal principle probability estimation formalize manner define energy define empirical replace relative binomial principle estimator minimize solve shannon energy alternative energy express temperature estimate distribution physics determine size estimator denote range note propose I temperature limit low kl cross entropy x formula px x px partition mechanic mechanic statistical mechanic probability shannon assume statistical mechanic feature state denote canonical distribution value right log function single value equation meanwhile estimator true consistency need equation accordingly asymptotic limit approach preferable ml insufficient definition adequate free dominate large dominate new count occurrence event count occurrence regard event state uncertainty consist degree uncertainty principle quality quantity regard obtain optimally denote canonical distribution characteristic property mass temperature canonical canonical follow cross principle prove statistical mechanic fluctuation denote distribution define equation equation usual call fisher canonical fisher similarity mechanic relationship relation obtain equation energy differential capacity temperature incorporate belief bayesian count event sum prior bayesian maximize lead canonical express equation temperature calculate replace I dirichlet jeffreys variety entropy px px px hx px hx I dirichlet jeffreys posterior viewpoint estimation kullback divergence average value identical ml inferior overfitte superiority I opposite ml poor tend entropy example entropy one increase entropy entropy tendency view relative I sample size entropy even improper point
well increase overlap little dependency increase criterion result ht variation mse among b outperform criterion case merging component equal however difference mse merge focus simulation compare simulation bottom dependency bold correspond well calculate b b b estimation value estimation explain fact reliable cluster show seem outperform merging probability estimation term confirm study number merge throughout remainder markovian method estimation try markovian dependency make b value regard beneficial interest accounting stand overlap emission neither accord cluster nest correspond distance see display distance spatial well misclassifie dependency two separate consideration account require array experiment array biological protein protein involve genomic computational time section ht method start cluster bottom grey leave diagonal red contain whereas cluster green density project axis unimodal result biological expectation know element name easily interpretable knowledge flexibility model hmm emission flexible provide hierarchical lead local criterion simulation highlight component real illustrate biological describe approach component brief enough link computation require backward dramatically pruning criterion approximation lead algorithm algorithm easily criteria update step get close optimum group paris france paris cr france cr france hmm account neighborhood emission suppose belong parametric distribution distribution flexibility classical combine likelihood criterion select combine criterion derive carry determine merging selection accuracy hmm constitute longitudinal hmm process neighbourhood dependency unobserved emission state parametric class gaussian poisson lead poor fit distribution effort capable skewed tailed multimodal emission cluster latent great interest mixture emission distribution mixture hierarchical review merge good approach limit hmm achieve component hmm hierarchical consider criterion simulation good criterion eventually classification experiment describe chain transition conditionally emission assume emission mixture denote denote transition specify finally emission collection paper devote parameter hmm existence px stand expectation parameter aim x ps pz ps pz conditional step first markovian emission density complete log get cluster complete like call hard observation entropy xu clearly dedicated purpose uncertain simply bic complete mixture behaviour model couple decompose different entropy measure hide uncertainty among may relevant refers class focus integrate complete derive penalization display mixture remain always hide dimension hmm differ criterion go merge initialization note g l simulation performance aim good selection criterion advantage account markovian design dependency four emission mixture six impact markovian ps aa ps dependency decrease
general precise identical expectation effectively reduce expectation avoid usefulness class map lp constraint operate bind experimental performance nonempty whose union relation fine briefly empty binary operation exist element operation composition name play na permutation group assign composition group define iff group call induce fine equivalence refine give action preserve lead vertex preserve g name denote clear cardinality symmetry node equivalent act let edge also arc problem np program graph subset compactly factorize ki ji must argument track argument factored encode graph pairwise feature discrete family indicator overcomplete overcomplete mean marginal overcomplete representation letting parameter overcomplete family tu x family symmetry exponential permutation equivalently straightforward act action remainder hold x x x x characterize factorize tt permutation group symmetric pa pa consequence lemma permutation action variational quickly depend constant map domain size yy yx yx design simplify call model yield strong two partition fully find six method reference domain size logical case dramatically runtime domain suffer expect graph partition partition virtue work observe domain current implementation curve finally cycle illustrate change cut plane size cut plane point outer variational observe substantially slowly affect much need work network converge early observe grain lead slow notably work constraint improvement optimal solution model present new introduce symmetry construction device cycle demonstrate constraint art obtaining extend approach full approximate condition pick express factorize reduce must argument j k x kt kt satisfy pick kx j hypergraph represent structure exponential family model iff throughout change inner product next integrate lebesgue integration establish result linearly lebesgue integral log hx hx hx hx equality also px hold act essentially partition subset consequence summation sequence since jx jx observe linear consequence write denote define contain main clearly induce proceed step overcomplete via action together x indeed u loss v group elaborate mean depend intuition purely formally indicator give assign index obtain I cycle every cycle sized form first connect arbitrary node use straight forward strong induction exist path must pass constraint constraint pass bi node must pair permutation
ridge collect reservoir reservoir analog reservoir design reservoir weight reservoir retrieve straightforward require optical implementation encode light intensity summation time depict panel figure reservoir state intensity optical reservoir generator provide pass send keep constant intensity input output light intensity multiply reservoir weight output analog proportional output exclude reservoir across discretized follow detailed multiplication negative realize follow balanced state reservoir periodic reservoir pair intensity eq light intensity come reservoir arbitrary wave filter choose appropriately output gain drive multiply coefficient multiplication state intensity weight choose summation achieve summation equation give reservoir minus side figure provide integration eq constant significantly single instant begin encode write time integrate equation time multiply would multiply residual value reservoir measure reference node whole b output balanced panel integration equation indicate dot discretize take analog simple encode reservoir define equation effectively would therefore several magnitude undesirable require precise behavior adapt reservoir state desired keep weight minimum force generate hardware implement analog start experimental depict maximum sampling feed balanced response read national digital acquisition rate experimental end circuit simulate circuit apart circuit every external output behind choice implementation generator issue architecture analog wireless bilinear filtering subsequently capabilitie reservoir task become benchmark reservoir wireless channel presence find reservoir symbol wireless analog reservoir input reservoir measure triangle digital square reservoir ideal circle reservoir analog simulate star experimental amount three quantity reservoir digital triangle ideal equation red square experimental third reservoir analog reservoir analog integration circle performance analog fairly ideal significantly digital reservoir report benchmark handle performance node reason loop time hardware increase leaves noise incorrect discretization network nod concept show simulation bad effect circuit network complicate rule may infer contribution node effectively term large capacity nf reservoir parameter reservoir expect extend dataset far fine tuning amount ridge procedure good one lead architecture quite modular add reservoir whether circuit complicated summation increase analog reservoir computer leverage main characteristic operation indeed need slow challenge field million nonlinear node digital recover reservoir output second setup retrieve symbol analog remove rate symbol order reservoir analog use reservoir input successive technique support office grant channel detailed description use reservoir reconstruct symbol symbol together reach path time create gaussian ratio sequence fed reservoir close value measure rate symbol service universit u f cp universit b reservoir hardware hardware reservoir computer digital implementation reach system bottleneck slow digital design analog reservoir computer work real time build test benchmark performance reservoir room improvement thereby major limitation development reservoir computer reservoir computing machine idea computation train external powerful technique range significantly easy neural network easily hardware implementation comparable implementation almost recurrent nonlinear topology network nonlinear requirement rich dynamic essentially dynamic exploration come input reservoir computer appear optical reservoir particularly optical inherent high parallelism without interaction reservoir construct weight reservoir bias actual conventional recurrent network make unit reservoir time step regularize reservoir computer obstacle design nonlinear unit
possibly constraint model clearly fw tw fw motivate minimize observe stream sequentially model observation online measure regret gap accumulate step cost allow good online achievable rest cost e convex unique example setup svm stream hinge formally type stochastic descent achieve scheme online problem computer computer message undirected serial minimize wide objective incur naturally extend cost distribute decrease rate similar serial hold describe endowed consensus assume doubly generalization stochastic detailed propose update update point processor round main within accumulate variable incorporate neighboring gradient accumulate merge constant reduce half per round dual subgradient maintain accumulate achieve perform exponentially step proper kt I kt kt I kt w kt let suppose doubly produce minimize obeys execute node maintain graph topology consensus fast dependence go worse dependence spectral characterize performance process order point available advance minimization account available observe manner hold doubly evolve derive back recursive algebraic bound accumulate triangle term invoke sequence collect far q step depend node iterate describe towards optimum recall recursive obtain use prove arrive eq bind must control round eq consequence view repeatedly convexity theorem integer update need restriction exposition assume select need algorithm round solve constant conclude present previous section easily variance gradient moreover old jt theorem past q hold collect expectation still room optimization processor illustrate simulate arrange geometric input generate classify relative position minimize fast dual simulated boost yield still also version fast end optimization random expect denominator unable online propose batch convex convex optimal iteration batch serial logarithmic every preliminary investigation perform communication explicitly account time require optimization doubly node eq eigenvalue consensus doubly stochastic moreover frobenius show demand less obtain eq since fact arrive true definition lot effort motivated dataset interest towards present distribute computer converge continuous iteration reveal time batch access start rate corrupt additive form formulation scenario sort point tend infinity rely smoothness trivial mechanism batch optimization set processor project type converge reveal sequentially rate lipschitz reference aim processor similar arrange graph gap e optimal processor
les dans un la les de l dans est dans par une le du dans prototype un le centre des es par le dans la des des la assign object par des des le en dans par les la pour un de le si ce une une les optimisation le maximal dans objective une l de la par la base de http www des des pour si de est des dans des de cluster dans le les r un la du la la distance dans les dans un sup par l de est dans une en sa les la par la pr des dans k pour des les les international conference discovery mining usa method pattern intelligence extension pour de des method overlap international conference recognition page consensus overlap microarray bioinformatics international conference mining usa c international conference intelligence cm les sup de de tn overlapping learning issue mutually exclusive may belongs present method overlap separability overlap overlap le de un dans les es pr de dans en pour la mod la les exp et les classification dans en un la en en th en pour les de classification un en les affect es une les ce en et des des g en des r par des des des th des pour al dans la des des les une les la pour la pour des des la la pour des partition le il des
life reinforcement hamiltonian systems energy balance actor storage successfully electrical circuit key exploit structural endowed hamiltonian ph ph widely geometric differential pde concern simplify pde search stability rely technique supervised solve stochastic control explicitly equation controller receive immediate process maximize cumulative reward actor policy actor reinforcement action space disadvantage progress incorporate partial address ph framework property parameterization particular type energy balance pde split part parameterize actor reinforcement part pde see synthesis seek controller instead online structural bring allow control fashion system simple consider reward else counterpart specify desire hamiltonian bring intrinsic energy consumption offer robustness ph learning control point view incorporate yield controller interpretability solution combine advantage aforementione control trend synthesis ph system root stability aim ph experimental show control input ph system solve base present interesting controller objective aim semi oppose supervise learn network fuzzy specification output instead rl consider alternative fitness analogous reward ph system parameterization actor reinforcement introduce experimental conclude natural represent system energy exchange ph introduce formalize ph eq store matrix assume paper denote system call semi bound satisfy call passive closed loop equilibrium desire control loop passive storage add energy feedback balance hold desire energy rank eq solve pde I passive kx eq reinforcement control environment rl mdp mdp action transition process control state apply apply return scalar maximize cumulative expect reward paper discount reward discount policy factor deal continuous necessary actor actor policy approximate improve actor direction usually beneficial actor ac approximate approximated parameterization actor temporal use trace speed include visit state decay rl exploration visit new find policy towards away lead actor pde split loop hamiltonian simultaneously parameterize loop actor rl actor first pde term close energy eq clear call loop energy assignment energy form desire closed automatically guarantee relation storage basis choose grow unbounded constrain parameterize basis impose obstacle stability demonstrate analysis section knowledge energy hamiltonian prevent total grow unbounded avoid possible set control rewrite stack version mean bilinear oppose find actor actor rl condition observe practice fast fast free rl system never unstable additionally policy model approximate rl balancing suffer obstacle application system even position hamiltonian energy hold split shape widely closed read desire energy update zero long validate control physical fig commonly benchmark motion angular momentum full measurable state model p furthermore htbp mass read potential shape one define hence two actor desire two actor ease incorporate symmetry ability problem fourier topology define basis possible integer frequency contain combination order dimension scale several advantage periodic prevent value small topology learn policy third beneficial periodic value momentum restrict adjust actor table low slow result goal entail critical hamiltonian rl position pointing equilibrium position top able back build momentum equilibrium reward unstable mapping e order basis p rd done randomly distribute white incorporate define take policy zero run system trial approximately begin initial position estimate learning curve simulation give table htbp trial trial sample trace max dots average average minute policy zero initialization value optimistic explore space lot learn state back momentum eventually hamiltonian acquire learn minima desire undesirable shaped try undesirable fig extract negative region initial necessary momentum disadvantage control basis also initial constant initialize otherwise stay set overcome perturbation system less present hence conclude local stability calculate opposite negative x look fig gray equilibrium around stability indicate positive white gray region x dot infer naturally function see stability similar behaviour show fig experiment fig algorithm slightly convergence minute near fig attribute system performance attribute optimistic htbp systematically law extra control make actor reinforcement landscape
draw popularity pair illustrate text ideal popularity adjust vote position determine mixture vector text ideal popularity understand suppose everywhere position issue adjust ideal adjust adjustment capture ideal point algorithm vote scan ahead show posterior vote vote bottom involve issue multiple yet issue political data issue ideal classical ideal equal zero model relative depend text different figure e adjust interpretable multidimensional point yes vote dimension separate subsequent capture kind pattern vote researcher leave right divide classical dimension readily nothing tie concrete issue ideal vote bring issue multidimensional ideal explicitly tie issue determine vote unlike multidimensional vote issue relate deviation voting pattern cccc education attack york serve history attack percent community child national substitute college frequent label lda describe issue language proportion dirichlet associate political issue exhibit treat issue encode e collection collection bag unsupervise topic fit use exist scheme document assign heavily influence give interpretable tag readily fully unsupervised case study political history tag provide service service provide pass code phrase cover may label illustrate top perform frequent mixed draw exhibit proportion posterior tie political modeling portion complete give roll text adjust ideal ideal point turn central problem roll encoding prediction ideal position issue bayesian tractable must variational tend gibbs update next section roll select simplify posterior adjust fully factorize distribution family specify latent endow capture specific marginal conservative variational fit prediction excellent variational proceed fully factorize successively update kl divergence reformulate jensen bind closed optimize ascent coordinate ascent adjust close form effective many algebraic derivation take alternative carlo integration approximation give easy rewrite integral exchange order integration apply rule q monte distribution note replace discuss material estimate find optimize stochastic optimization gradient step alone achieve convergence convergence minimize eliminate improvement briefly mean hidden random vote achieve infer expectation induce behavior adjust roll united house evaluate fitness roll issue responsible qualitative u issue preference demonstrate rich explore political house roll vote hold office attack th mark gain take house roll call vote record one want vote useful roll serve record roll www vote receive roll call summarize ccccc ccccc house separately two period vote house treatment house completely fit represented phrase count phrase bag commonly vocabulary phrase omit vocabulary gram far use phrase section interpretation identification ideal arbitrary multimodal address point adjust ideal traditional explore justify first outline issue adjust traditional report quantitative adjusted ideal table ideal adjust ten show ideal bottom row vote separate vote roll call place cut vote improve vote vote education trust provide program community service provide provide develop house ideal model issue ask differ ideal house ideal panel compare ideal axis bottom ideal parallel compare separate line different treatment traditional ideal surprising able explain fact political alone check discriminant separate party along illustrate discriminant bottom normalize political political party additional dimension extent entire house fit vote contrast analysis fold axis logarithm weight vote issue issue log increase relate vote issue issue political single explain ideal demonstrate number explore information estimate position adjust expect adjusted illustrate issue figure inspection relatively house blue red vote political party th house across denote issue show issue great variation issue preference become ideal understand point issue explicitly control fit effect ideal offset observe residual share offset vote ideal issue moderate ideal adjusted right increase unit leave corrected identify preference nonparametric describe label I vector correct issue adjustment issue find expect extreme absolute counterpart several house house correct issue focus vote full vote house correct issue issue preference replication house motivate analysis issue adjust vote increase place improve adjust improvement restrict significant tend vote vote recognition house frequently house local majority party house largely symbolic topic house business national national effort production symbolic vote evident post symbolic vote yet point alone recognize voting behavior issue explain house way sharp voting vote vote house party species power party party house ensure file member decision vote support systematic systematically issue systematically position adjust ideal finally improve vote vote apart position roll capture vary issue issue large history vote well illustrate exploratory tool several way issue differ vote issue activity outside voting issue area modeling position issue change series provide additional detail begin variational ascent iterate lower specify threshold traditionally symbolic expand solution expand stochastic repeat converge upon variational generality perform variational parameter motivate taylor taylor variational notational convenience estimate describe next paragraph taylor monte variational current estimate variational posterior mean current coefficient straightforward eq increase change instead monte sample decrease space pass cdf variational sample note estimate bias sample method paragraph enough quasi select paragraph markov current marginal factorize gaussian form practical implement happen issue adjust parameter little posterior drop pass drop perform second order describe section definite enough also limited step reason issue regularization share inference considerable expense yield adjustment estimate report effect house hold vote calculate b approximate drop summarize well range represent vote good log drop house accuracy house issue issue e service range five consideration result fairly well political table law east business requirement service health day service local child human finance resource p management care force security resource history name elementary secondary education force education international education house house rule vocabulary convert form part phrase word eliminate phrase phrase result gram processing feature characterize classify whether use vocabulary feature number external appear article feature list phrase negative word figure phrase vocabulary phrase contain wikipedia appear wikipedia containing contain phrase phrase phrase phrase phrase phrase term term phrase term phrase appear expect chance david nj cs david science computer department nj fa grant develop specific voting exploratory language correlate political approximate base across inherently multi key variational voting vote vote capture row roll vote item r roll political ideal binary roll give political call vote two voting favor roll point recover familiar division example vote point incorrect vote adjust service point behind classic black line mark vote incorrectly capture broad illustrate vote act h colored vote classic vote four eight vote model incorrectly sometimes vote incorrectly predict circumstance often vote explain consistent political get closely error differ come policy classical political position political content propose political position voting expect develop capture place spectrum inference encode issue unlike political incorporating vote conservative conservative adjust tell hasting issue vote following describe develop algorithm usual mcmc fast year house vote give ideal exploratory tool analyze item decade political science overview historical perspective political ideal dimension vote yes explain explain party issue careful study roll neighbor former make inference assume different point inference political variety incorporate text text focused issue evidence answer political science method behavior toward text vote receive vote predictor vote individual vote well ideal affinity toward type adjust model conceptually content recommendation article already user web specifically design political work item employ orientation orientation spectrum consideration political roll ideal discuss account specific ideal point quantitative political member overview voting record point place interpretable political historical dimensional characterized discrimination often difficulty popularity yes probit underlie empirical popularity vote popularity vote vote yes ideal standard prior vote ideal
norm sdp solver first optimization trace thresholding available sec problem write provide optimization complex selection publish basis pursuit order nesterov proximity gradually use smoothed also idea linearization iteration fista depend know advance indicator zero inside infinite positive definite matrix smooth play central role carefully present envelope well summarize proper function x left side property lemma lipschitz gradient require reason nesterov smoothing derivation somewhat refer propose diameter convex convex conjugate diameter focus identity I envelope conjugate slightly lemma derive follow nesterov formulation somewhat different relationship operator gradient lipschitz control quality approximation kx l fy k l accelerate method track update formula satisfie iterate possibility give iteration share limitation choice additional converge optimum original smoothed objective optimum elementary b b dx assertion follow constant price pay advance logarithmic factor report convergence rate term adaptive version accelerate backward lemma thm eq thm apply use obtain x lemma change parameter denote use obtain apply fista guarantee nesterov smoothed method important provide explicit lipschitz discuss sec nesterov rely bound fix advance set guarantee unbounded practice motivate optimization advance share domain primal minimize problem translate objective discussion point clear constant different direction multiplier lagrangian associated updating approach satisfied make ill suit problem admm linearization alm alternate projection augment lagrangian applicable decompose fista smoothing share another applicable differ nesterov part affine applicability pursuit similar attain boundedness combine benefit max unbounded domain priori number iteration advance max u max regularizer collaborative nice commonly nuclear recently max commonly lack recently way clear far aware non proximal rely regularize demonstrate entry entry max problem eq entry max operator correspond thresholding movie entry among rank matlab stop less small eigenvalue increase proceed projection produce procedure big speed kind strategy semidefinite programming parameter good consider rough run list highly optimize rely method interior runtime increase run memory could run large try hand iteration increase iteration decrease range first yield runtime hour generic sdp convex fast reach hour regardless able reach minimum objective value attain plotted gap sdp practical require significant parameter produce rough solution c c cost datum given solve choice tr surveillance stack correspond sequence frame matrix represent move foreground frame since implementation alm solve method alm strategy beneficial iteration analogously static strategy video since roughly order reduce hardware result number iteration theorem art alm static alm sparse see therein q give
surprisingly device grow popularity mobile device pc combine security require dedicate sufficiently mostly simple short scheme dedicate explicit security attention analyze operate reliable device contribution framework serve behavioral behavioral feature mobile device task discriminative design decision usage continuous insight operational online characteristic identification verification active category behavioral hand feature behavioral aim activity early behavioral pressure identify base input work average depend datum gain popularity security survey modal mobile enable bind sign verification high signature fusion yet highly read enter scheme explicit user particular literature aim implicit instance scheme monitor user digit particularly behavioral study high instance action click work inspire experimental visible one continuously clicks carry move position reduce system flip resemble line signature verification surface geometric signature subject pressure feature achieve differ signature verification signature sophisticated main kind implicit always signature verification paper probably contribution try historical behavioral classifier support unless carry use author two contribution entry implicit second interact like trajectory goal understand would reliably continuously record enough behavioral perform read coordinate stroke addition phone record time cover available phone various extract system profile profile phase classify challenge action frequent primitive perform system slide usually page slide move email web distinguish horizontal within action across extended complex focus appropriate continuous monitoring exhibit discriminative rely action feature continue converge equilibrium point device device long switch mode device begins phase continuously track make consecutive result precision individual influence precision need choice proportional provide detail affect decision usage scenario suffice reliable likely identify cause get conventional modify security configuration phone hour usage standard mechanism gps device scheme operate feasible framework challenge record interact phone try collect realistic analyze distinguish make week investigation aim decision technical feasibility must compare main motivate user produce ask read question reading inform would subject phone mobile minimal many improve assign participant ask start document ask difference image stop user minute pair participant ask week later one participant collect across different interact protocol enable interact phone various document image allow entry document image primary application wikipedia article see panel free go often new article orientation device user could switch landscape mode record sample variable raw event code absolute ms device orientation pressure orientation orientation provide standard reading usually take minute trial minute participant specification appendix possible concept study outcome generally speak appropriately freedom variance operate device time cause condition issue try avoid know serve analyze behavior reflect interact behave use reveal purpose collect limited artificial limitation intra set aside phone user phone possibly interaction regard moreover round record one part document influence user therefore protocol adaptation user improve ability device behavior relatively amount snapshot study term one give device user use aware play ask experience device user world two long stable phase phone phone ideally must phone phone would record acquire record phone take care phone enable describe record report extraction record begin end trajectory encode n na phone landscape record list believe relevant trajectory notice tend device stroke vary trajectory able lift velocity stay latter gets accelerate even distinct direct stroke consecutive scale straight angle ensemble angular dispersion straight constitute project distinguish end leave important time read detect informative reading discard method keep store near neighbor accelerate training k distance odd interestingly nn hard user decision powerful classifier two divide hyperplane margin svms maximally hyper generalize datum individual outlier within margin control trade minimize class linearly feature involve hyperplane kernel transfer space radial basis parameterize tune feature usage scenario way explain detail normalize rejection require fraction recognize fraction reject classifier quantifie resort temporal user misclassification rate far increase detect security equal rate equal fold cross tune stroke individually user estimation several consecutive classifying instead classify majority project thresholded score depend far near put count resolve individual report carry feasibility experimental analysis reliability classifier output analyze influence several vary number per decision stroke lower increase classification stay another affect affect interact device give attack carry please influence turn device decision slide precede adapt classifier early intervention odd could come make second take decision decision might provide phone home restaurant enough increase might insufficient exclusive security mechanism device implementation trade security accuracy experimentally constitute difficulty different experimental inter week assume train stay pick device train week round acquisition record user day every put phone pick record day device every overall already make device user directly device device detect phone imagine extend minute scenario draw htb datum result outcome color black nn colored blue percentile individually median range usage intra outlier seem user reach reach depend quantifie security reduce rate distance interestingly inter week classifier rate inter interpret round round text find text subject round week short user phone way interaction prefer way hold phone intra class record yet week experiment horizontal scenario user way image comparison distinguish way depend chance correct accepted rate future equal accuracy scheme highlight critical account categorical take value read email write email read music operate scenario probably behave use video heavily temporal instability current serve exclusive device satisfy security trade achievable contiguous extend time minute adaptive false favor gps available network computer finding computer claim would differences computer user believe help content collection collection phone classifier get large user long without might consider introduce degree affect variable bias classifier test principle binary variability user clearly affect influence subject repeat one inter via number ten time subject repetition test equal user fluctuation range apparent demonstrate size influence influence phone phone risk might record necessarily slightly dot convert normalize device signature device aware minimize condition much possible datum experimental protocol try strictly protocol check device carry device carry device total phone phone constrain user record device respectively result illustrate phone user collect unclear signature help device responsible relation phone differ nn equally suggest phone responsible however rate inter phone suggest introduce signature alternative explanation number phone collect phone alone reliably detect play role estimate finding attack attack base try random attack sophisticated try observe stroke imagine pressure etc look successful also attack place application device pattern attack success chance argue device lose attack situation construction investigate question serve behavioral importantly
expect achieve misclassification plot zero block include lexical syntactic review amazon website review review review per identify feature uci repository neural use author report selection separately multinomial show utilize sparse gain efficiency normalization center evident k problem plot estimate rna datum patient various unbalanced normalize lasso datum available validation run trend lasso significantly sparse reasonably outperform investigate primarily interested trend error cross validation trend depend restrict multiclass center low interval distribution typically confirm center determine cross generalization sparse give trend configuration specific one degenerate location zero construct site choose singular primary quantity set interested problem non false false negative true negative rate method non zero entry predictive zero central distribution show area average configuration center configuration center accord model thin configuration error configuration result case choice dense area dash randomly thin configuration lasso sensitive look group low except configuration bad configuration group applicable require twice continuously require around bound requirement ensure range nest loop middle loop coordinate descent outer middle inner develop allow part improving middle provide group separable sense separability primary ensure straightforward subgradient b n nx sufficient proposition define closed proposition denote hold sufficiently jj deal hence equation finding point descent provide descent accord cyclic rule compute eq jx jx line search jt gradient need aim cost compute block sufficient consideration piecewise cyclic compute algorithm beneficial broad cauchy multinomial without three time amazon software package interface multinomial logistic sparse group template library template generic sparse package template library external library template library performance utilize boost library boost review library template library introduction library package library list current set fitting group lasso group amazon k see discussion section penalize negative log multinomial distribution primary multiclass classification implement algorithm template library regression present multiclass classification real example lasso example lasso comparable performance acquisition objective would optimize interpretation theoretical lasso different descent interested unconstraine coordinate naturally search denote projection onto block mi mx function say around rule I scheme make twice differentiable scheme rule differentiable sequence cyclic outline block coordinate coordinate minimizer consequence minimizer differentiable minimizer last scheme additional continuously give optimal guarantee convergent immediate descent use differentiable fp cluster generate stationary point point coordinate outline except minimizer corollary minimizer lemma contradiction point otherwise continuity implication choose ax ax sparse sparse real outperform multinomial classification sparse implementation multinomial package implementation high classification amount package group classification gradient descent lasso regularization lasso descent optimization generalize gradient descent consider logistic cox combine optimization descent coordinate modified application multinomial optimization multinomial group group efficient straight forward completely robust minor treat type optimize penalize approximation raise another upper sufficient whether attain approach enable update hessian reduce provide hessian reduce time set multinomial purpose multiclass lasso rate good multinomial lasso achieve example text amazon review improvement rate sparse template multinomial logistic available lasso require multinomial size twice solution define lasso decompose dimension write n group include infimum computable grouping multinomial multinomial multinomial grouping reflect group multinomial problem purpose set profile type classify primary site profile amazon review classification rna disease classify rna characteristic simulate problem assume categorical use parametrization multinomial intercept together identifiability parametrization penalize group parameter grouping part cancer site expression amazon review disease expression k set lasso two preprocessing entail center scaling center scaling feature design column column technical biological normalization procedure
trivial scheme verify use mcmc static metropolis utilize mixture ergodicity unbounded see metropolis combine augmentation augment copula specify distribution new update chain exist choice proposal covariance choose specify update define history j j theoretical motivation recommend choice next mcmc proposal manifold definite matrix property symmetric definite marginal consequence closure convolution wishart proposal matching detail wishart covariance note also riemannian symmetric positive matrix wishart fix freedom select average variate marginal chain adapting proposal ensure restrict matrix combination remain method present table second hyperparameter claim development incorporate estimation posterior gaussian copula augment iv predictive dependence compare incur versus independence claim payment payment copula include statistic perform metropolis iv adaptive metropolis use proposal heat riemannian manifold metropolis posterior convert matrix convert top left table summary payment per payment statistic provide component year decompose year quantify residual currently development constant year constant per year repeat incur principal variation payment development proportion easily distribution leading year incur complete development weight large eigenvalue posterior summary result distribution present involve contour posterior construct use adaptive development factor payment distribution present posterior copula development triangle payment incur copula model full partial estimate quantile leave measure worth note augmentation specify example parsimonious specification allow whether payment respectively present claim mixture via hierarchical copula full payment find agreement attribute bayesian model structure consider additionally model extend combine information present within payment incur allow prediction incorporate achieve develop hierarchical incorporate wishart conjugacy log form copula approach hierarchical capture potential tail dependence payment incur demonstrate incorporate regard augmentation incorporate challenging methodology inference monte incorporate efficiently marginal dependence advanced challenging claim model extended dependence incorporate recently chain payment incur model partial several different payment introduce discovery dp incur copula std applicable e marginal gaussian copula payment incur independent copula full post hierarchical bayesian prior variance independent bayesian scale payment report development cumulative payment marginal mean variate say variate matrix satisfied variate chapter pm rp tx cp partition matrix sub matrix variate distribution degree multivariate function wishart partition matrix denote follow chapter detail variate inverse variate wishart degree denote generator cv cc dimensions function copula family manuscript lemma function copula dependence tail dependence copula dependence copula dependence frank tail frank admit recursive expression q two variate copula u ic u copula weight secondly show b condition satisfied notation copula definition give q domain box odd consider n k copula expand volume box corollary remark proof article recently claim loss coherent particular hierarchical incur claim combine incorporate dependence augment incur usefulness incorporate payment incur result payment payment iii lag year gaussian payment incorporate transformation augment incur carlo data augmentation copula payment triangle adaptation component deal matrix restrict manifold dependence incur claim augmentation department statistical university college uk email ac uk university mathematics statistics school business predict ratio past come variety contribution develop claim however structure formal extend generate copula incur loss year row article significantly extend dependence problem variate inverse prior chain manifold treat unobserve miss likelihood financial security company predict claim account parameter claim combine claim incur loss available chain aim prediction claim incur datum adjust reduce gap drawback log normal model future claim triangle payment incur major loss quantify model applicable loss extend capture additional structure payment incur payment incur datum commonly exist ratio development impact development development period appeal claim practice payment plus projection incur find structure payment incur loss development period period life fully actual different period payment copula construct specify triangular structure loss permutation restricted copula across development period augmentation methodology involve mixture copula feature incur loss structure development lag loss diagonal comprise contain row previous bayesian hyperparameter development variate preserve conjugacy independence copula conjugate model sampler copula dependence non make develop iterate form class mcmc grow recognize inference interest utilize adaptive mcmc facilitate adopt employ scale metropolis algorithm financial reference therein involve adaptive definite create variate proposal mixture modify copula base challenge intractable arise triangle argue dependence payment incur challenge marginal year give payment incur loss integral copula bayesian augmentation overcome model involve claim payment second incorporate claim amount loss know report claim impose develop construct vertical development claim row claim incur year year incur loss moreover assume claim probability ultimately claim value p j diagonal appear diagonal block corner right corner clear definition iterate define dirac vi stacking create derive dependence framework assumption independent recursion l follow component payment incur component claim incur j note assumption make applie coincide claim incur assume tail beyond incorporate incur base particular result cumulative satisfy parameter summarize payment development year loss year conditionally eq conditional moment chain conditional upon consecutive incur loss see conditional payment incur give aspect extend develop independence incur ratio incorporate subsequently article develop extend convenient conjugacy often lose present generalize static assumption use variate wishart distribution develop statistical relevant distributional payment ratio copula log payment ratio wishart property positive accord variate gaussian possible extension second structure payment loss incur loss prior wishart give inference development consequence dependence payment log incur conditional k distribution q degree freedom matrix eq properties wishart covariance conjugacy payment incur loss convenient marginally result random payment lemma find th year claim payment incur specify unobserved year triangle incur specify consider multivariate covariance define family operator permutation permutation dependence specification admit conjugacy tuple tuple tuple multivariate lemma generalization model case year payment loss year unobserve payment incur loss payment incur loss consider year tuple give transform payment incur I payment incur variable payment incur denote I jj follow jj I jj formulate joint incur claim upon parameter accord normal jj j variate allow j either payment delay loss increment assume positively claim increment however structure especially party status change term medical substantially loss lag lag year always realistic feature enhance correlation maintain parsimonious consider alternative structure motivate practically appeal claim estimation claim recognize claim incur month month portfolio claim period claim benefit lag normally follow subsequent correlation equally change claim huge claim cost period replace stream period mainly capture dependence year propose lag assumption dependent lag normal statistical payment year analogously incur inverse wishart define inverse payment hence payment incur loss datum jj element random payment incur loss permutation index characterize distribution covariance payment loss incur loss prior hyper another give multivariate copula form conjugacy gibbs block ii matrix consider vector initial vector give rotation transformation obtain decomposition accord diagonal follow j u u incur loss j covariance vector detail consequence conjugacy directly scheme full estimation ii conditional posterior give independent posterior component payment loss gaussian specify wishart section form ii inverse q study copula framework copula copula article dependence capability remain framework however introduction enable modify distribution model comprise statistic alternative modelling dependence copula model copula g detail method know statistic augmentation combine allow copula model variation comparable present fundamental member family mixture copula characteristic copula member reference variate particularly copula construction analytic dependence parsimonious generator member family copulas frank copula copula denote copula density parameter frank copulas dependence tail upper tail copula dependence class copula payment log model comprise copula copula distributional variate distributional member mixture copula component admit tractable function denote proof provide augmentation situation bayesian intractable consider generic evaluate point wise setting partition admit augmentation due posterior dimensional augmented detail give notation augmentation invertible transformation consider log incur loss give I decomposition observe log payment loss payment loss incur th year introduction notation make payment triangle incur loss un observe quantity triangular loss auxiliary
require stage study current health classic current status study member period disease case age due contaminate technical reason I trial participant randomize treatment make treatment treatment period intend covariate treatment treatment treatment I measure measurement omit reflect fact measurement case set causal systematic inference account encounter world medical stage conclusion causal observational relationship estimate incomplete directly graph project despite effect carry systematic calculus distinction allow description miss presentation causal graphical specify hand explicit parametric relationship presentation effect identifiable linearity effect implication tie causality new possibility graphical second show key format clarity speed communication analysis scientific scientific knowledge anonymous comment description derive population population become explicit factorize accord distribution variable notation distribution argument unless otherwise integral clinical trial integral unknown variable need nested variable q cm abstract assumption design scientific research design miss structure direct causal calculus node causal diagram time observational incomplete make offer causal clinical trial commonly set typically acyclic graph represent direction effect discuss relationship mathematically object causal systematic carefully design sufficient effect specify objective collect collection take account analysis population pressure measurement design causal effect design need accurate reporting causal estimate describe clinical nest causal provide implication cancer effect unknown causal causal calculation assume assumption variable subscript index close variable describe determined symbol population directly circle instead measure circle belong available design consistently design control selection factor status cancer case status control analysis control figure situation difference causal effect apply calculus design rely causal causal extended reflect inference causal benefit calculus directly question relate selection mechanism probabilistic structural causal triple variable determine factor call model respective domain form probabilistic causal pair causal diagram correspond direct member toward design extension probabilistic causal causal follow attribute value determine selection parent parent parent miss node type observe type open open observational corresponding setup determine know principle lose determine determine unknown later population miss record identify truncation selection conceptual union population differ causal conceptual allow area sampling may differ area member priori unique social security member enter study induce consist individual effect also item causal value individual select value definition univariate extended axis may special indicate miss relationship node variable health study life health causal variable time link causal observational relative node visualization observational stage section causal version unobserved unless kind benchmark measurement subsample correlate addition sample design formally determine make measurement undirecte miss concern design description many general cancer operator intervention observation represent represent node income remove identifiable experimental front criterion use formula automate calculus step must accord factorize model notation refer define respect unless specify ignore response ignore note parent parametrization purely vast topic directly applicable likelihood value mention parametrization causal write link parametrize variable collect case design frequency number assume cancer become summation shorthand maximum carry htb cc illustration sum cancer control select covariate measurement probability table non select population difference third round aim causal essential experimental study illustrative design causal make causal missing describe realistic remove name design study two stage collection graph present applicable design design analysis factorize causal design offer point write likelihood need illustrate project aim classic genetic disease genetic project select underlie certain age typically individual examine factor follow disease genetic risk health status understand pressure baseline variable need classic classic baseline health status baseline conclusion make causal calculus causal disease correspond effect disease health causal classic health baseline gene health status baseline observe effect investigate birth unobserved individual make status whether mechanism eq consequently effect genetic disease accounting tell conditioning baseline situation qualitatively kind status project risk account potentially different participant participant must take classic
mc dx operator I lasso produce hard mc line value choose mc select correct estimated type penalty researcher algorithm entire solution e g algorithm algorithm descent explicit maximization whereas lead formula derive formula penalize utilize maximize maximization quadratic coordinate wise equation loading suppose variance update log old old old old old old old ii parameter maximize express penalty include descent vector update equivalent penalize close variety penalty update maximize likelihood diagonal column zero decrease describe column loading loading compute introduce entire path value give penalty mc path produce improve smooth surface initial loading unique variance might value loading matter choose stationary function say value loading nonzero element factor loading value factor loading estimate ij column loading exist element see stay may nonzero zero nonzero produce column therefore adopt newly value loading approximately lasso large monotonically employ traditional apply mse mse mse pm indicate proportion loading zero obtain follow increase often detect e mc much mc detect lasso penalization produce rotation mc yield mse select dense bic lasso penalty rotation mc mc mc mc lasso illustrate usefulness available evaluate add percentage loading approximately three technique penalize via mc analysis pca reconstruct via posterior mean reconstruction mse without reconstruct digit reconstruct loading right figure factor produce smooth image sparse mc yield loading loading yield mse produce traditional rotation descent entire investigate propose procedure produce handwritten show mc load sufficiently datum hour entire path would propose algorithm freedom lasso however topic apply em common regard miss penalize log likelihood take ni te tr f old old ne old e old ni old old e old old old old old old old old old old il proof contradiction element proposition section remark school engineering science rotation utilize loading even rotation produce introduce loading methodology rotation technique via em along descent path permit wide penalty investigate model strategy example give nonconvex rotation useful covariance observable variable exploratory analysis use maximum use rotation technique utilize loading however well unstable e mention furthermore even estimate rotation produce sufficiently case penalize lasso attention tool sparse generalized support penalization method nonconvex nonconvex method smoothly scad minimax mc elastic several path coordinate researcher discuss procedure penalization obtain sparse loading numerically show penalization traditional procedure yet although ordinary analysis early nonconvex yet penalization via nonconvex penalty penalize view I maximum estimate technique methodology solution estimate loading em descent introduce permit nonconvex include package r implement http package traditional estimation procedure likelihood nonconvex penalize method rotation obtain path conclude remark observable vector g unique factor mm pi unique observable distribute covariance estimate close form likelihood g loading generate rotation meaningful orthogonal criterion minimize loading note construct model loading q loading produce loading possess loss recover
carried processing moment demand impossible exception filter analytic problem transformation nonlinear kalman transformation approximate transformation taylor mild linearization acceptable linearization measure quantify linearization deterministic technique instead covariance nonlinear statistical possible linearization ii mean capture linearization characterize term covariance linearization correspond dirac affine calculate year choose example selection paper briefly simulation sec calculation sigma n factor depend scheme transform cholesky scaling hold root factor solve expression note adaptive linearization well desire demand single global linearization especially weight covariance filter local benefit decrease increase nonlinearity perform locally linearization linearization provide linearization assess linearization apply geometrically trace proportional ellipsoid linearization I linearization affine linearization nonlinear mass avoids split irrelevant together select splitting linearization interpolation linearization focus linearization error consider split calculate analytically component choose gaussian formulate replace gaussian vector large reduce degree concern preserve simplify splitting numerically reduce univariate moment preserve mixture require determine symmetry place hold library preserve univariate preserve constraint lead free equation dynamically linearization throughout simplicity determine component additional capture order like skewness split subsequent linearization split linearization control growth univariate q covariance axis thus eigenvector replace eigenvector choose splitting univariate mixture shift scale multiply transform splitting covariance calculation preserve splitting gaussian formulae eigenvector splitting eigenvector large cause linearization merely eigenvalue idea evaluate deviation transformation linearize along split gaussian deviation subsequent linearization desire split l integral along eigenvector eigenvector choose nonlinear integral efficient mean dirac mixture discretization linearization reduction stop linearization delay right node start delay north pos near line pos delay line pos west north linearization sec filter key idea dynamically increase mixture linearization filtering limit memory follow description operation prediction step perform linearize l linearization nonlinearity cause severe linearization large nonlinear linearize splitting linearization newly linearization component affect splitting gaussians subsequent linearization criterion combine criterion drop grow beyond original remain integral tracking keep avoid linearization stop let split base locally linearize system kalman predictor linearization component grow due subsequent step exploit redundancy component reduce algorithm year see typically much employ computational demand reduction predict filter almost identical prediction linearization filter linearization correspond linearize joint equation apply gaussian locally linearize measurement give rise component predict measurement linearization k k reduction posterior splitting max nonlinear consider recursively mixture threshold estimator linearization numerical integration weight linearization error rather select splitting base splitting perform large table leibl divergence approximation large eigenvalue scheme component account gaussian always since split inferior quality splitting importance splitting quality tb kk angle choose k model unimodal exploit filter strong heavily tail deviation threshold filtering threshold filter employ simple weight criterion consider example linearization fair scaling e split reach since exploit linearization error filter pf residual resample threshold simulation root rmse runtime depict good tracking high runtime conversely fast selects exhibit perform large eigenvalue track inferior less thank split besides direction nonlinearity split analogously filter represent split runtime see consume operation reduction already novel adaptive statistical linearization quantify linearization error linearization covariance splitting direction split linearization criterion reliably strong keep number split level linearization corner draw text circle thick fill fill dash sep anchor corner
rich stochastic production line consider measure server server buffer infinite capacity free keep service service go customer node place buffer service customer buffer service customer random mechanism represent visit customer let matrix refer table table firstly describe consider problem criterion produce representation notation arrival time service may note identity coincide customer service wish performance criterion follow network function every service occur node customer third arrive therefore point obtain representation reason extend node performance consider customer average per number time queue denote performance optimize random operation great necessary may represent indicator provide procedure deterministic note sample expectation network performance possess reliability recursive require proof possibility representation obvious deterministic procedure technical order simplify formulae introduce easy value traditional axiom fulfil axiom minimum small element equal denote first subset obviously table deterministic procedure consider network write examine arrival customer customer leave go node coincide one clearly customer provide customer denote happen join queue node account time I j representation finally representation superposition representation theorem require conclude take represent generality suppose entry introduce additional fulfil induction statement assume operation lemma axiom successively axiom statement optimize formulae estimating obtain gradient consider estimate random realization differ estimate difference formulae computation additional function simulation difficult perturbation method sample simulation dynamic combine calculate gradient network provide derivative one simulation compare require alone concern unbiased estimate mse order short considerable algebraic examine easy interpret practical lipschitz family etc various suitable ordinary change verify condition fulfil next technical operation break provide respectively variable w verify instance examine f operation multiplication w one satisfied firstly possible derivative vice versa neighborhood lemma may occur therefore h inequalitie subset obviously q g hold e condition analogous borel lebesgue measure neighbourhood violate nevertheless every hold p condition corollary continuous hold every neighbourhood remain closeness example show define eq although inequality lebesgue follow see nevertheless case fulfil lemma neighbourhood w neighbourhood consequently fulfil belong lemma fulfil neighborhood reasoning minimum index consequence importance obviously stable addition clear corollary deduce example continuity essential important verify fulfil nevertheless example definition algebra operation generate show example family k satisfy conclude fulfil contrast traditional exponential satisfy random integrable network describe gradient algebraic probability begin may mean completion normally case realization algorithm fix activity stop go step complete correctness reliability apply denote sample initial failure element exclude node able well arc set hold save go step rather deterministic mechanism conclude continuous estimate unbiased total fulfil customer hold follow boundedness indicator identity q arrival service previous server contrary arrival customer completion customer th arrive w expression define algorithm incorporate initial value completion add go iii visit customer change key role calculate gradient include function u initial upon add g determine visit customer server free completion quite note gradient give use value gradient measure optimization acknowledgement grateful discussion axiom example class measure expect analytically function give analytical study estimate unbiased rather meet study network perturbation management biology system help analytical method computer behaviour usually main aim performance performance evaluate sensitivity gradient generally estimate procedure
student institute research statistical institute email control along generalization chart prove robust generalize multivariate univariate chart establish efficacy methodology depth bootstrappe directional statistic mid ease chart prefer high speed production relevant analogue basis chart multivariate analogue robust dispersion drawback shift median univariate interpretability difficult liu rank datum datum rank rank moderate change reflect suggest use distribution central tendency performance chart presence chart clearly absence recommend development section program propose average simplification take deviation statistic robust multivariate mean choose depth accept multivariate observation dispersion observation four depth depth depth liu liu desirable invariance monotonicity background correspond normality line limit chart distribution limit chart chart propose control limit distributional transformation distribution chart follow value consider mean suitable choice chart distributional chart instead plug limit choice l say p ensure stay control time treat besides consume multivariate pp test statistic define depth resort chart extend variate th element simplicity mean mechanism bring procedure rational fix distribution construct phase distribution plot size chart size chart convenient standard preferred percentage take percentile table propose chart chart chart scenario shift shift outlier significance previously depth compare chart obviously simulate bivariate normal get select process chart univariate chart statistic bootstrapping control replacement dispersion dispersion adopt compute ti variable choose percentile control statistic significantly recommend multivariate chart shift shift outlier compare liu rank test comparative efficacy shift table liu shift presence adapt poor shift presence strength obvious shift chart chart recommend chart around fluctuation depth cut lead highly control chart long stay serve distribution point huge function limitation application depth depth reliable bivariate dimension depth performance change value evident circumstance liu bootstrappe surveillance change
repeat characteristic represent make life unstable group connect line rank like play likely opinion novel likely situation analytical refine go alternate view player expert give style mark four use position determine dimension reasonably demonstrate level connection indicate seem merely suggest section nn sec sec reference expert reference rescale comparison pattern pca reduce dimension fold measurement include show column examine classifier part already significantly generate normalizing explore investigate way relationship thorough internal player argue adjust condition color limit opponent might differently game stage pattern carefully move game game strength investigate extract summary pattern information various correspondence summarie player play play style player proposal characteristic remain work demonstrate exist practically information summary application inference direction future newly go theory specific go comment methodology mr acknowledge major paper information go games corpus student physics university also ik student move pattern record game propose extract summary several within correspondence traditional go feature player strength style accurate classifier attribute rank internet study go program theoretical playing approximation computer go focus create play position research game record human play game well player information play black assume basic use computer internet review game computer record enable collection far move position base go feature belong class intermediate go present play devise player information way play strength playing move characteristic player reliably game apart practical investigate style extraction game corpus explain apply finding game finally interpretation research direction large collection record format analyze particular player look effect process play move count encounter globally mapping generate describe need specificity tradeoff description various attribute description game sample difference significant inspire candidate pair position feature move last play move edge distance spatial configuration played maintain symmetry configuration produce go square match contain capture move opponent elsewhere shape move corner far local diverse object pattern thus describe multiple intuitive solution scale default process show fig mostly large diameter game pattern always try modify raw count post processing method beneficial normalization dimension knowledge inspection ultimately classifier conjecture frequently style aspect implement use feature go rating stream produce string per move course move play influence playing analyze several basic technique purely correlation vector component produce represent next significant axis dataset lie little pattern pca preprocesse sensitive component hand produce player project map connection assign pattern output represent infer assessment play strength meta country calibrate knowledge pattern output divide difference approximate trivial approximation representation pca dimension define interpretation play aside classifier vector vector artificial output generalize unknown network pattern relation nn naive infer membership evidence couple analysis together technique well pearson product correlation coefficient measure strength compare data validation randomly divide reduce assume inter dependency vector dimension center produce vector eigenvector eigenvalue eigenvector projection follow describe r n compute eigenvalue decrease eigenvalue mechanism relationship player distance preserve consider dataset reflect estimate quantify scalar probability high therefore preserve visualize quantify order member position subject descent know vector player similarity output average illustrate prove study pn network ability classification iteratively reasonably call neuron get output activity activation previous activation call neuron typical sigmoid feed training w n n exist construct approximate player separately cover sized interval dimension fit value distribution element feed estimate deviation element probable c mining method open source framework available majority implement programming library notably mdp library naive bayes around module playing strength play receive rating scale list scale rank game review game player strength game force discard various represent positive representing rank number represent etc create perfect correspondence measure dimension pearson see satisfy imply extremely trivial group strength confirm correct accurately strength confirm suggestion examine pattern position suggest pattern alone certainly thorough reason play pca strength nn classifier classifier play player perform reduce explore game determine accuracy small pattern compare fold measure standard approximated file select column describe game square classifier obtain albeit network classifier network due network sample small list mainly comparison perform nn use three architecture comprise c train take explore look vector attempt pattern style source game collection subset player choose within player play play frame analysis traditionally expert allow predict similarity pattern discover move ask mark four give style safe style base experience expert terminology go player much room confusion concept extra style accurately reflect diversity dimension research experimentally pattern year play r r year player complete answer reliable though beyond aspect pearson manual consistency game game age reasonably expert decision dataset either se li chen deviation receive three playing sort game go play several distinct diversity analysis outlier high common extreme style prominent balanced careful player regard ten pattern player set three pearson pca dimension correlation year
consider random error satisfy parameter become case error denote element specification norm norm page k combine thus become specification practice popular fold validation little contaminate performance ols pure obtain identify small contaminate pure testing contaminate merge penalize method identify minimize step deviation positive section integer large number infinity covariate estimation penalize lemma still stop provide supplement critical properly case covariate follow specification difference formation change highlight continue conclusion valid analytic property additional cause state theoretical assumption frobenius fourth consistency hold depend fix estimator infinity appropriate integer eigenvalue bound properly scale asymptotic assumption via extension theorem suppose estimator consistent supplement theorem hold asymptotic level estimate x plugging theorem require handle theorem condition q estimator simulation parameter specification p cp estimator know zero least square hard pls hard square pls soft penalize square hard pls pls soft ts oracle iy refer thresholding squared rmse penalize examine pdf realize realize leave rmse horizontal line rmse minimal rmse rmse panel soft parameter rmse symmetry rmse rmse forms rmse around soft cause rmse optimally rmse pls pls hard hard come pls hard soft pls pls perform significantly ol depict minimal rmse contribute rmse ts generate figure rmse similar fix increase bad estimator much ol penalize hard pls vary rmse around pls hard pls grow ols four rmse p set eight vary simulation report regularization step similar rmse step drive pls interval pls hard pls small size rmse drive penalize panel pls pls pls pls tell closely phenotype thousand snps three chinese false discovery proportion discovery discovery population filter deviation regression loading factor snps biased number proportion filter clear large filter filter slightly filter filter optimistic ccccc discovery number consider structural regression model exploit parameter consistent contrary possess partial selection thus structural consistently asymptotically previous show square continue significantly naive least square advantage interval verify genome association testing equipped drive penalize false discovery proportion estimate illustrate regression model high structural fast sparse phenomenon appear penalty impose structural parameter save proof ps ps p ps I ns p proceed theorem notation x xx assumption cauchy schwarz adopt related theorem matrix deterministic bound hold nd proof condition ds dt p n dd proof explicitly eq condition dd next show special q conclusion n condition condition q thus therefore estimator lemma supplement copy denote bound notation sufficient bound lemma p lemma dd n notation next desire result apply eq times v thus q q assumption dd v v p definition ip dt theorem converge denote p note thus nr theorem since notation lemma estimator supplement occur assumption long lemma assumption condition consistent occur desire need theorem specifically result theorem eq converge first n lemma note q lemma dd dd long corollary notation sufficient q next zero eq note lemma dd dd material section true thm thm conjecture rgb question pc partial sparse penalize principle extensively structural structural high dimensional estimate derive consistency possess partial selection consistency interesting partial phenomenon structural consistently estimator improve procedure challenge suitable region dimension slow drive simulation real analyze material paper methodology almost namely hold overview sense contrast another model different depend model arise arbitrary statistic factor statistic z dependence statistic unobserve critical discovery estimation parameter replace model critical possesse reflect instance nonzero measurement response contaminate response outlier several seminal mle presence stop work exploration assume come common elimination nuisance conditioning solve principle economic remain write kk condition parameter detail supplement mainly possess consistency consistently estimate penalize one step estimator estimator asymptotic thus construct efficiency step parameter penalize method rigorously penalize derive two estimator propose regularization extend covariate analyze proof positive random copy variance exist positive mean kind large understood penalty soft lasso hard general penalty function minimizer simplify minimizer soft penalty penalize estimator interpret minimizer subdifferential calculus theorem necessary ordinary integer simplify derivation keep message simplicity statement stop stop every algorithm stop simulation suppose limit corresponding limit estimator solution follow nonlinear soft eq necessary lemma must conceptual confusion minimizer interestingly huber q huber equivalence penalize huber indicate naturally formal least indicate datum huber sparse compare paper dimensional high review robust impose problem obtain estimator notation expansion n index I similarly operation set properly great greatly asymptotic index analytic expression derive property theoretical specification regularization addition result
class dissimilarity happen performance identify way eliminate improve collect identify graph edge connect node root refer graph extend supervise learn et al fundamentally content output node put pressure sign content feature weak address purely work semi solution lead propose mixed ir converge exist even relational many ir vote relational harmonic square use devise although straight forward form similarity dissimilarity hyperparameter two graph losse graph relate approximately call easy usefulness quickly parameter demonstrate et mixed systematically usefulness web citation link page page poor organize extension ir discuss hyperparameter notation use represent associated edge connect belong class either weight correspond label node distribution interest dimensional unlabeled namely scenario edge belong divergence kullback leibler shannon divergence etc shannon smoothed version divergence term information datum match regularization underlie graph regularization fitting regularization node connect term edge violate suffer objective q measurement distribution help regularization constraint find fashion iterative kl regularization ir clean often useful method mix graph way normalize w j w practice pure achieve improved node initialize j z ji jt formulate collective classifier relaxation label rl relaxation labeling component collective keep probability instant relational classifier compute unlabele sum neighbor may sometimes perform mixed modify ir method label k characteristic role key graph characteristic study researcher method method study characteristic class graph coefficient base graph represent weight weight ia ib respectively usefulness use graph negative factor type graph method degree way similar note pure description benchmark experiment binary people cs generate equal document categorie window categorization non link dataset page link page follow construct relational specifically weight knn relational purely derive comprise computer paper relational relationship seven class neural etc convert seven refer movie movie office versus movie share production company edge production two dataset available next dataset comment dataset percentage forming realization one labeling experimental give table vast performance good intermediate ir mixed set dataset small original graph coefficient good range increase move lowest extent positive variation occur somewhat away attribute fact toward fall similar conduct pair significance compare ir dataset significance graph intermediate subsequently graph ir use identify beyond scope occur graph demonstrate application next dotted black performance ir ir cv ir ir cv cv ir ir propose signature page http com refer http www com refer graph edge page signature match range put score signature accurate consider binary first page rest page table since varied evaluate auc ir method graph set cv technique quality grid interval cross validation performance inferior graph see performance instance improvement significant cross validation choice good performance slightly inferior estimate since auc around graph method restrict base perform ir ir cv obviously three competitive tune section computational et al ir fast time also provide competitive mixed get improve al
vice likelihood regard nuisance function regularity obtain property smoothing estimate curve q k I predictor I hand profile log need ii k km k obtain newton involve four ik tm repeat convenient ii alternative see chapter al category specific incorporate account order would need vary application vary increase curse dimensionality structure modelling computationally could profile context integration could h efficient fitting parametric incorporate gender region age flexible model fit irrelevant alternative parametric odd effect gender region age party support surprising significantly relatively category east substantially raise lp party strong represent decrease support support irrelevant conservative decide amongst within argument party obvious perhaps past g computationally implement context nest adequate framework category flexible case value confirm problematic political party say party enter type datum additive wrong conclusion average derive likewise extent peak lp low implies type potentially probability support low increase find level level support group party work across high drop away class confirm party mostly student constitute strong prominent role great background involve student anti nuclear movement surface interpret exponentially display party lp versus thereby scale difference clearly origin drive factor lp recognize group small party notable support lp old typically involve parametric section age function reveal monotonic led number covariate political party lead comprehensive profile model inherent assumption relation response explanatory overcome modelling model separability sophisticated modelling additive modelling design arguably interpretability age model misspecification extend discuss response mathematical property consistency efficiency estimator remain parametric profile logit give detailed subsample particularly case nonparametric approach due precede political composition interaction age highly shape party likelihood notably smoothed field keyword multiple investigate influence possibly categorical response behaviour demand provide mean factor support political great policy designing useful outcome opinion fisher et however conventional adequate complex pattern profile manuscript political party usefulness basic parametric thereby demonstrate superiority identify profile gender nonlinearity covariate interaction different model multidimensional explore thus usefulness flexible explanatory little utility recently nonparametric propose special generalize qualitative cf respect modify bayesian approach manuscript kernel specifically profile estimation spirit extend inter model multinomial response motivate introduce formulate kernel procedure fitting parametric party section benchmark result conclude remark motivate political party aim dominant political multi party collection interpretations broad party historical background party comprise political party form conservative party party party lp green party five consideration govern diverse group green local green nuclear movement movement movement form east unity party although system call party social throughout manuscript extract economic variable political party party commonly literature economic log e logarithm region origin gender region origin whether person
study n array investigate channel array difference baseline interpretable practice describe agglomerative approximation search ht implementation crucial interact gradually merge fig dependent reveal response improve however increase tends increase overfitte sample effect information complexity log size gene merge minimize l q agglomerative initialize univariate potential pair direct singleton merge network newly terminate continue merge distinct mixture neighboring demand task gaussian process variational type co mixture activate condition overall correlation gene profile fluctuation covariance component however heavily complexity leave difference response generative mixture maximum limited gene adapt gene sample per previously use pathway pathway base pathway www implement knowledge pathway tool consider pathway provide pathway set format interaction consider remove normal post gene measurement degradation minimize pass include biological replicate platform finding investigate human expression platform biological replicate available root temporal multiple definition probe genome would provide expression plus array identical compare set technical rely annotation alternative use available probe com validate pathway interaction network sample body compare approach wide response genome interaction network response step response step detect gene expression find connected subgraphs subgraph gene response detect list gene algorithm randomly keep association original assess influence prior gene include finding investigate significance detail detect response expect difference group datum test condition response qualitatively similar pearson detect characterize centroid response characterize response within gene datum consider validation corresponding predict response supplementary report readily applicable modeling wide implication finding investigation coherent response highlight functional outperform comparison finding identify supplementary median distinct response supplementary one distinct significant difference observed validation response qualitatively similar supplementary central fourth level group pathway alternative response may share mechanism may reflect reflect overlap pathway start pathway remarkably pathway pathway association provide supplementary file general signal system network cell growth could six pathway pathway interact pathway diverse vary detection detect expression array individual visualization mean baseline ht selective activation association response response grouping condition accord occurrence expression share suggest highlight functional connectivity share differentially gene response co expect individual overall connectivity signature co occurrence reveal investigation reveal supplementary detect fraction differential median gene response compare alternative detect amount response coherence association mutual information class detect statistically association condition average label supplementary high compare confirm detect describe obtain median detect significantly comparison detect response seem however finding biological gene affect analysis could specialize activate share functional help formulate context define support validity detect large response finding highly establish finding response biological network include pathway protein connect characterized process provide readily interpretable biased study bring prior information increase however collection pathway hundred pathway affect predefine demonstrate account aspect pathway pathway pathway predefine module association multiple pathway description consist module procedure rigorously identify coherent module interact gene response increase account measurement principled interaction factor bind protein base pathway database interaction limitation connect principled collection availability contain valuable condition specific experiment enhance directly exploratory task provide disease genome scale key comparison systematic investigation response unknown motivation subset process model focus interpretability since set potentially disease differential network subset patient many collection update extension datum topic another treatment interval remove criterion response without significance majority response verify datum biological sensitivity could seek constraint impose related cluster agglomerative model base selection connection feature global optimum model exhaustive combinatorial network problematic solution link genome interaction solution achieve compare interaction power agglomerative pair find merge affect considerably mixture run application computer ghz supplementary investigation interaction network group interact gene differ highlight gene measure activation characterize wide quantitative validate identification characterization scale diverse wide cell activation potentially help novel hypothesis gene previously context european research institute school science information technology department computer science box version manuscript biological process interaction associate response reveal unique share yet novel interaction method search local know interaction gene guide human pathway response functional condition context gene availability available contact molecular induce change level produce huge body cell public gene pathway model protein context activate reflect expression although indirect wide availability provide investigate detection response
guarantee worst running provide generalization inspire concept game theory game contribution technical contribution behavioral formally identifiability game concept trivial game game bind analogous vc dimension number equilibria datum game equilibria baseline approximation completeness search game well player show convex produce game player brief delay compare without reader aspect constitute attempt visualization similarity previous model behavioral game community ai nf nf n n n n n nf ce discussion graphical game form game payoff learn equilibria pure nash equilibria ce equilibria equilibria number agent relatively payoff learn game provide time learn game extract temporal dynamic branch thus vary probabilistic vs achieve utility exception assume action noisy set differ focus past behavior model method payoff agent reach vote vote availability lead use conditioning neighbor also validation player researcher payoff explicitly potential payoff e pure degree reader payoff determine assume joint observable non game furthermore validation game graphical datum mention direct cyclic parameter maximum describe although game direct cyclic structure factorization player probabilistic differ nature approach well suit mostly consequence keep mind competition graphical except behavior aim seek ai community game variety include identification popular social computer community excellent area social discuss depth discuss self study diffusion influential individual game type read game either behavioral start article couple year estimate influence consider lead influence g take theoretic approach static game approach strictly theoretic within static game framework make assumption choice assume conclusion player game see g e player game theory rational act independently subject irrespective core concept x prescribe equilibrium notation game theoretic model form direct player arcs payoff player set parent neighbor context graphical game payoff influence weight threshold f ia player define quantify player play graphical main contrast perhaps still influence modern view network compact uncertainty analogous intuitive description interpretation structure game become paper influence influence voting game people present influence unlikely ground influence u ever respect encoding game joint capture property really certainly direct influence type learn depict type proper rigorous scientific structure applicable shot generalize generalization require effort simplicity reference section discuss main later misclassification likelihood result causal take strongly indeed game lowest differently problem priori one find equilibrium would context game maximize low behavioral play cognitive social principle ml tradeoff complexity performance apply joint choose uniformly uniformly complement mixture stable outcome game formally pmf pmf need enforce also enforce mixture game completely mixture induce pmf action identical drop context parameter q q game equilibrium greater trivial game mixture p valid tuple q part definition depend prove contradiction q q contradiction definition game call refer tuple hypothesis consist tuple identifiable similarly game game brevity trivial pmf game hypothesis space trivial identifiable proposition completeness trivial trivial identifiable game identify discuss nash equilibria response equilibrium discuss realistic mathematically tractable bit concept human behavior social book behavioral address nash reason understand illustrated chapter propose nash behavioral theory superior setting payoff g logit interpret temperature available payoff propose payoff indeed technique hard graphical game mle payoff unclear tractable apply sophisticated extension scope paper unclear logistic regression logit variant behavioral model account cognitive reasoning effort usage assume payoff fair human experiment behavioral theory scalability game unclear view vote individual hope take one study concentrate vote reason scientific process sometimes alone concentrate publicly hold decision voting record decision case record information pre final work within model population justification make unclear gain randomization strategy introduce mixed pure realization general game measurable process complex expression thus appendix furth equilibrium could easily natural expense produce generative less amenable form see know priori action equilibria pmf bernoulli parameterize derive equilibria mle probabilistic definition q n prove mixture property kl remark proposition case induce pmf note low non game set mixture give furthermore trivial identifiable game give infer state payoff induce generative exclusive different property joint note game mixed equilibrium state previously work concept equivalent mle game reflect generative hence select equivalent seek attempt create model objective unknown assume truth learn I nature essentially translate ground game illustrate several parameter ability know selection core ml choose ml learn precisely elegant multiple invoke generalization practice pac support standard measure via invoke toward well know exactly prefer reason exploration interpretability everything else equal explanatory measure likelihood preliminary work algorithm heuristic operate among alone would prefer former show generalization maximum upper vc maximum achievable generative mixture e last technical include vc allow expectation denote game game maximum probability least objective b elaborate result clarity presentation generalization number player logarithm pure equilibrium game bound network layer unit input function unit term hide term weight quadratic respectively term hide weight define neural equilibrium oppose output instead dimension mixture section equilibria baseline exhaustive us negative show nash complete general learn player probabilistic loose allow obtain tight bound expense hinge primal program program ij ij regular derivation add slack duality logistic I behind decrease choose latter logistic show bounding generate strictly upper bounding ii generate claim iii logistic identity e strict show exponential function positive prove claim suffice set I simultaneous minimization become bfgs expand onto negative enforce justify minimization structure game absolutely proportion equilibria assume player large connectivity analyze structure present utility restrict assume enyi connectivity define player produce player proportion equilibrium player concentrate ingredient proportion equilibria simultaneous logistic game simultaneous svms degenerate case x lc c l l subdifferential simplify I I svms simplify j e degenerate solution j l c l smooth lc x claim termination b point careful reader subdifferential vanish proof still concentrate ingredient equilibrium show player interestingly ingredient bounding proportion equilibria main absolutely I x condition normalization axiom I b nx I I hypercube cover hyperplane recall note hyperplane zero half important surface I except hence state probability lebesgue still bind depend norm instance arbitrary covariance additionally requirement uniform subset entry allow graph equilibrium absolutely absolutely random draw I I I equilibria furthermore statement let I n f cc use hoeffde simultaneous svm additionally exponential exhaustive baseline proportion equilibrium sigmoid plot ise green class drawing probability component view nature arc every node truth p threshold generative move nature purely data validation respectively respective regularization ise model use describe repetition kl respective theoretic ise considerably purely probabilistic across nature seem learn model purely kl value much purely ise end bad relative probabilistic vs outcome summarize figure help still learn show superiority game ise player gradient ise np use propagation statistically latter value select value report leibl precision minus equilibrium minus exclude equilibrium closeness recover model truth world report synthetic equilibria proportion equilibrium significant avoid bar clarity marker simultaneous svm il sl simultaneous regression kl search equilibria model equilibria equilibria equilibrium equilibrium resemble truth game exhaustive method ground nash equilibria accord set mixture truth repetition lower exhaustive proportion equilibrium ise exhaustive exact likelihood suffer consequently convex simultaneous equilibria equilibria grind truth recall additionally proportion equilibrium resemble mixture ground slightly second synthetic interaction nash equilibrium accord repetition ise logistic low loss equilibrium equilibrium additionally equilibria parameter experiment minimization improve increase slightly repetition real almost move interesting arise concentrate decision course please understand leave diversity influence induce interesting nash equilibria magnitude sign b truth validation convex outperform maximum model synthetic proportion equilibrium remarkably well equilibria ne low kl experiment two approximation remove true equilibria maximum equilibria loss loss player contain repetition require bias number convex outperform maximum proportion equilibrium observe high density obtain low inspection ground truth usually seem equilibrium maximum computationally feasible game learn leave th row influence learn weight value top corner bottom right corner produce c directly influential least regularization voting loss publicly vote vote vote researcher g vote equilibrium np sampling randomly split repetition turn train maximum proportion equilibrium convex simultaneous logistic low kl equilibria equilibria learn game counterpart normalize influential high respectively influential th list aggregate party strong party structure game vote cast interestingly decrease voting adequate argue log action action action single agent work regard game produce order address identification influential refer behavioral individual conditional inherently without produce regression condition heuristic please area manner learn study plus get technical need common player utility despite similar estimate generalization standard also generative theoretic equilibrium well evidence support theoretic long extensive benefit find contribution successfully graphical model game theoretic model research payoff utility assume player play simply condition extend broad class boolean action binary feature player player non version still extension analysis linearity additionally new vc extend parameter regularizer future sophisticated process upper bound hinge account variation influence voting difference term influence bias topic preference grateful improve discussion topic behind suggestion improve presentation motivating causal comment question wider grateful share master thesis reference thank anonymous anonymous reference overview composite likelihood appear ph nsf award end dynamic discuss manuscript certainly prediction knowledge publicly vote vote seem sensible modeling perspective nature vote detail indeed go availability process considerable burden something world motivation view temporal provide wrong often inherently express separate differently treat individual example invoke page intermediate need general temporal dynamic model poor really care stand interest recognize significance social behavioral science economic high abstraction going decision reach growth physical international shift engineering entity goal course computational reasonably end state model extension mixture player probability action parameterize otherwise complement parameterized pmf behaviors q datum likelihood keep constant step hard keep constant combinatorial tractable provable distribution maximize change generative parameterize player action probability pmf behaviors q pmf alternate method step change jensen nature likelihood step yet another changing pmf hard formally surely pick computer university ny usa behavioral class parametric game stable influential social task also cast estimation goodness complexity capture equilibrium control number equilibrium unobserve generalization mle probability world voting record briefly applicability game modeling ai formally study behavior book core solution serve outcome system self interact setting role infer causal outcome therein say computation significant computational game community ai compact game decade game large encounter game graphical ai player determine player neighbor neighboring decade constitute arguably theory considerable progress ne see therein play prominent role ne games example non game theoretic motivate current work influence identification
auc gene intensitie roc foundation european community agreement review manuscript interest u bottom fully scalable fully spike presentation score value full material method abundance solid dash dotted line u classify cell exclude quantify forest validation mm pt title short microarray running department laboratory university european bioinformatics genome uk department material engineering de es mathematics centre european bioinformatics trust european bioinformatics genome uk department box email keyword online robust probabilistic average accumulation standardize collection function scalability current form bottleneck microarray collection constitute major source genome wide scalable measurement calculate training overcome limitation scalable online large microarray include thousand array alternative readily applicable preprocesse probe update batch novel take full microarray collection moreover collection probe effect affect various provide tool control available http www package accumulation house public create microarray standardize overcome inherent bias analyses measurement hundred genome disease size portion microarray collection gene million array database able combine analyse array challenge preprocesse central multi array combine array probe effect performance multi array technique limited requirement million probe bottleneck comprehensive meta collection variant standard preprocessing develop tackle rely probe term restrict microarray applicability microarray probe platform platform independent extract probe collection scalable introduce scalable probe online limitation extend averaging framework estimate online hyperparameter allow rigorous microarray ordinary consecutive batch requirement respect probe collection readily short microarray moreover analysis probe performance collection probe platform provide collection step preprocesse normalization probe probe array preprocesse applicability collection due memory reference extract complete requirement time hyperparameter appropriate microarray quality background probe apply collection step store disk preprocessing step operate quantile basis estimation quantile normalization average probe implement scalable quantile calculate distribution batch jointly normalizing array use parallel calculated averaging hyperparameter approach introduce online consecutive probe hyperparameter update correct normalise transform finally batch initialize equal prior probe batch provide batch final probe level batch ideally expect yield batch probe last probe summarize yield final summarize probe model normalize level gaussian probe probe gaussian affinity ij model additional assumption probe affinity estimate variance analysis randomly affinity posterior give version develop validate full advantage form update prior prior probe hyperparameter posterior hyperparameter interest estimate probe nuisance analytical available marginalization slow point fast approximation iterative optimization newton sample size follow gamma yield readily give equal j information update value final retrieve update hyperparameter next give hyperparameter complete yield probe probe probe probe issue microarray preprocesse assumption formulate linearly hour preprocesse file four processor core preprocesse classify traditional preprocessing multi array preprocesse perform array combine probe accuracy array incorporate preprocesse scalable preprocesse large scale microarray collection standardize database probe array probe however applicability currently limit variant additional incorporate utilize batch available heterogeneous microarray collection requirement procedure contrast fully readily preprocesse reference http edu microarray preprocessing spike quantify focus website design moderately sized scalable however applicable spike probe microarray comparison probe method field array single figure summarize complete available outperform u u set outperform bias fig false difference salient test interestingly although outperform fail spike outperform scalability scope summary preprocesse standard wide scalable collection type probe set tree split exclude comparison outperform pair batch outperform considerably limited scope batch array contain multiple test summary technical data pearson share package correlation batch pair supplementary requirement array applicability sized data probe preprocesse probe reference form bottleneck meta analyse microarray collection take advantage nearly million array array key introduce online individual probe effect large collection scalability limitation current preprocesse sequential hyperparameter batch microarray collection extend probabilistic alternative fully readily probe term hour run file optimize efficient processor notice yield result additionally probe affinity omit speed computation preprocesse analogy allow advantage method preprocesse array provide array diagnostic purpose suggest probe rna degradation snp annotation modeling yield source probe remain poorly model probe scalable comprehensive collection short array bias tool microarray probe single model variance recent probe comparable outperform utilize outperform model microarray collection primary scalable algorithms effect separate scope finally applicability algorithm currently limited plus spike microarray outperform scalability online currently fully scalable preprocessing applicable
strong organization ice event patient stay wind bin california area gene black fix produce moderate support alternative plausible stay city moderate conduct fix yield exponential cutoff law cutoff spatial ultimately distribution surface invariance truncate exhibit large systematic indicate alternative likelihood nest cutoff give ratio support law indicate probably law power moderate power law fit alternative remain plausible alternative complex often suggestion argue identify quantify straight doubly logarithmic plot straight law furthermore problem test extend researcher reliably investigate law hypothesis collect lose properly claim extend objective characterize generate question power whether follow pose largely scientific goal law fundamentally tail law part challenge face make phenomena law interesting reach modern hope statistical present aid contribution share available http www support institute phenomena law accurate law complicate fluctuation empirical tail lose adapt statistically include hypothesis kolmogorov goodness fit compare effectiveness quantify datum heavy tailed law attract broad consequence appearance phenomenon span physics economics sciences quantity law say exhibit invariance empirical presence underlie process network know law generative mechanism facilitate statistical event branch specie unit diameter rise two diameter diameter forest ad leave know test similar tree forest scale consider another interest intensity wind speed knot impact modeling frequency process datum rely upper frequency reliably tail follow existence upper fluctuation information lose convert count tail unable distinguish power heavy tail normal present principled answer statistically power typical u census city united york times extensive law find review obey draw pattern hold law researcher critical start detail statistically law hypothesis discrete goodness fit characterize fit plausibility range exhaustive principled consider law rather narrow adapting popular quantify impact impractical impossible histogram measurement receive due inference tool currently present power testing plausibility compare make assumption logarithmic bin synthetic accurate size amount lose statistical investigate hypothesis bin fit law consider plausible hypothesis power tail alternative model replace principled minimum description length real world exhibit tailed power finally highlight continuous paper material derivation indicate law constant function formulae simple distribution analysis case method count original discard retain range bin count let bins boundary bin convention bin count fall bin density subsequently external source otherwise raw exponent small bin study law first fit imply straight doubly straight seem approach parameter case tend sampling fluctuation pareto th early th researcher na I generate give mistake see fitting law generate na I estimate scaling parameter choice assume provably estimate maximum multinomial sample material section focus elsewhere symbol symbol power sample scheme space must numerically maximize em complicated section bin boundary successive analytic obtain bin hill estimator associate positively bias material role equation tail explore detail demonstrate experiment draw practical priori truly power choose fit model section sec alternatives law variety ten power result bin technique produce ordinary complementary regression bin operate doubly yield significantly value dramatically especially poor linearly f tail count induce significant ordinary complementary call frequency tail f complementary c bar sometimes dramatically accurate tail equal hill mle robust square regression mle quantity tail body aim simple value power discard fit uncertainty prefer conservative avoid maximum always use choose inspection log datum avoid method choose distributional synthetic principled none accept choose high reason require goodness fit options example pearson practice like find ks superior fit bin ks empirical bin reach law ks law poor ks distance high effect illustrate method methodology rt index minimize bin boundary idea sample minimum sample smoothing parameter size law logarithmic variety slope slowly numerical size logarithmic one threshold across second bin boundary characterize vary bin versus estimate bin true figure scheme dash show reference reliably slightly ks moderately rt accurately slight deviation law bias versus logarithmic target bin induce substantial second true slight rt rt work large reduce slight method logarithmic scheme kind number bin tail bins bias power vary sample bin decrease implication researcher must around method generate draw bin bin c logarithmic absolute bias decrease fit say constant perspective theory use asymptotically start dominate visually hill beyond maxima yield ks hypothesis power dominate correctly nontrivial makes inherently fail power law systematic deviation range power follow law quantity sample model imply mechanism law imply care empirical law actually measure describe allow model plausible generating observe bin wide normal power histogram axis despite one law power hypothesis sample dash law notably determine reject end plausible plausible alternative answer likelihood additionally impact power test count count would know plausible give goodness quantitative question turn represent likelihood deviation close attribute fluctuation reject view reject consider model law step law choose remain case empirical statistic bootstrap law distribution call law times fraction count generate law increment bin increment bin proportional count repeat desire ks statistic fit correctly original datum subsequent precisely estimate instead yield bias synthetic answer datum accuracy know true value know generate decide reject relatively conservative power law threshold avoid fact chance power value correctness arise well law test necessary cover number bin shape yield underlie happen goodness hard rule little bin fit analyze tail require frequency statistical power ks effectiveness goodness fit test various sized synthetic log alternative strong wide size bin reasonably law plot size law draw model expect rejection reject note however rejection power law require coarse provide follow law tail produce datum even demonstrate bad argument several compare law validation bayesian construct loss reduces interpret far commonly alternative theoretical mechanism effort application expert constitute reasonable normal power cutoff table bin likelihood pt cutoff pair logarithm natural make distribution indistinguishable event subject fluctuation direction unless require log chance log use fit fit bin law small say sign sign reliable favor technical likelihood supplementary material hypothesis answer datum sign without regard power power cutoff true yield must material draw law normal use boundary power dash performance hypothesis log range general hypothesis favor hypothesis difficult datum also power imply generative implication illustrate second log bin sample compare likelihood supplementary compare law well law law correct figure grow scheme interestingly reliably decision favor law illustrate alternative size already fit power hypothesis supplementary material step scheme quantify scheme power law hypothesis illustrate information pose question scheme logarithmic power achieve limit equal fisher scheme approximate equality supplementary material vary scheme fix show scheme estimation arise variation span source bin arise asymptotic agreement come dashed line log datum power axis make correct step framework axis sufficiently nearly eight option experimental phase fine grain collect maximize accuracy hypothesis log law plausibility computing law log normal reject law experiment scheme e log
follow likely occur likely likely occurrence event likely occurrence fundamental reasoning belief law probable limited knowledge say imply logic binary therefore knowledge possible universal applicability belief cox sentence consider strictly however coherent reasoning identify value act abc formalism determine complement happen form abstract naturally calculus measurable show partial belief represent show order course term proposition reject sentence accept necessarily sentence accept value belief notice value great value satisfy requirement aforementioned value dimension remark bayesian include abc formalism objective belief subset hence fill gap know minimal readily c case knowledge hypothesis experiment namely evidence favor value whenever decision derive loss issue future work minimal feature apply hypothesis note variable value differ drastically maximum likelihood estimate therefore logical reasoning proportional nan hypothesis sx take say induce compare difference value extensive review refer oppose posterior distribution requirement conclude reflect probability long posterior zero sharp directly latter section procedure produce readily classical alternative want specify computed compute respective computational principle main approach likelihood principle violate nan specify propose otherwise interpret aforementione treat undesirable nest hypothesis nest give evidence principle actual great report word distribution among many thing undesirable aspect evaluate critical value decision decision possibility choose critical threshold vary correct critical internal undesirable value incorporate scientific elaborate discuss know statistical scientific significance reader employ however internal undesirable certainly bring problem implement without logical open value next deal article rigorous mathematical treatment likelihood concave compare confidence region carlo require e actual criterion decision acceptance replacement value measure acknowledgement I acknowledge wish suggestion write manuscript attention report motivate student think contribution statistic anonymous helpful suggestion lead improved paper definition conjecture mat email corresponding value regard value scientific ratio measure evidence logical requirement meet e measure study abstract belief calculus formalism use establish end short problem abstract calculus measure base nested science discuss limitation well alternative parameter value alternative propose objective evidence connection calculus status proposal specific observed capital letter denote frequentist paradigm test cox least statistical common informally define small probabilistic specify common set advance threshold reject interpretation feed drive think statistic point serious logical argue highly nan conclusion arise regard implication frequentist detail parameter lie interpretation explain basically attempt possible properly understand statistic avoid decide family p measure restrict namely value family p p happen asymptotically probability way topology complement pearson significance value ratio testing account nan hypothese consider side computed likelihood reader asymptotic mean reason health g genetic study one frequency two know homogeneity frequency homogeneity mild logical requirement claim evidence widely hypothesis nest within logical reasoning datum word compute respectively dimension hypothesis compute metric may happen may evidence reject limiting happen exact consideration identically random identity example suppose ratio statistic hypothesis sample size show evidence ratio value statistic logical contradiction frequentist job define scientific take decision problem conclusion value decision hypothesis present produce surprising conclusion form regression usual non say generalize section plausibility therefore regard plausibility fail hold value short value final great closure inside way least point closure gradually confidence dot dash dash confidence general fashion measure relation region hand specifie specify test confidence readily compose see procedure consistent hope draw community conduct interval fuzzy compatible recommend author confidence level fuzzy whole interval membership goal paper author naturally depend consider connect formulation quantify yield nan prove essential evidence consider metric asymptotic statement confidence region interval level law weight evidence hypothesis definition although paper point relate monotone transformation transformation work version reader extensive review method interior hold theorem hypothesis fix evidence invariance confidence assume prove show ab plus concave increase alternatively ss basically strictly connect proposal author value boundary confidence notice converse typically approximated degree strictly upon
linearly study es function linear es increase invariant basis strictly l situation parent far optimum optimum undesirable situation handle increase search algorithm handle sa es es characteristic section study section variance result search normal zero standard multivariate zero ni tt selection child select f define adapt size path multiply approach easy front distribute path length length update decrease short determine simplification update consider rule analyse throughout linear determined distribute es vector generally non linear path alone would es geometrically log step size change formula immediately sum case constant prove es geometrically recognize thank quantity number investigate es e equal prove lemma number I surely deduce integrable show strong law obtain q sign determine infinity independent see idea normal invariant g gx dx recurrence recurrence vanish three converge size es surely expectation prove sure convergence satisfie expectation c analyse conclusion get remove use development formula equal separate side go c right summing expectation rhs positive increase geometric divergence step geometric mean affine geometrically geometric geometric divergence dimension pressure double future show hand increase increment suggest likely equal furthermore c equal I development prevent ex curve quantile set precisely quantile median figure evolution run variation deviation plot curve bottom deviation divide relative must agreement constant three careful variance relative critical make deviation investigate es affine compose rigorously prove behaviour contrast sa increment deviation increment go mean signal ratio go give stability confirm partially grant research mm cumulative adaptation adapted measure call path
bind noiseless nuclear nuclear spectrum net since parallel sum penalty penalty provide meanwhile roughly correct call calibrate net develop replace scale soft decomposition result level smaller calibrate net perform outperform method calibrate bind use sample extra appear bound appearance sample seem paper describe spectrum net convergence derive calibrate frobenius mapping letter op ie step rank minimization thresholding solve matrix tie need complete equivalent n lead e compute convergence result proof iteration may em calibrate e nonzero projection uv p net proof proper lie proper still infinity calibrate modify provide target imply support consider matrix ratio normalize suppose coherence hold let satisfying imply rd constant noise frobenius magnitude simulation use element equivalently existence spectrum net penalty n c remove recovery noiseless sufficient rd right shall independence requirement aspect symmetry set theorem replace root size requirement remove factor requirement root aspect ratio square block satisfy coherence provide random matrix iid projection noise htbp calibrate lasso modification estimator level penalty solution correspond solution calibrate spectrum always ny training error rank f report three level proportion result average replication net spectrum modify calibrate spectrum net spectrum true spectrum lasso estimator analytic unclear control approximation taylor expansion nuclear control tangent spectrum e omit uv tm p uv uv uv spectrum member differential h uv inequality n uv p uv uv h h p p inequality singular error follow derivation find uv uv uv p f acknowledgment research grant dms dms dms grant slightly provide l converge iii lemma remark department statistic paper concern calibrate frobenius penalty incomplete current guess scale soft thresholding singular result converge cause frobenius coherence condition penalty nearly order unified analysis noisy noiseless completion proposal
rs snp associate significantly manuscript analysis central genetic approach phenotype binary phenotype account serial longitudinal estimation novel center efficiently phenotype selection bayes simulation demonstrate utility application genome wide phenotype relate diabetes keyword carlo sampling occur genetic influence phenotype many genetic study trait diabetes disease study disease common phenotype conceptual phenotype disease phenotype response mutually correlate trait take increase jointly many collection type diabetes exploration sequencing sample longitudinal measure phenotype body mass american diabetes trial longitudinal diabetes longitudinal combine longitudinal study study phenotype measure genetic factor trait phenotype longitudinal family trivial correlation phenotype repeat phenotype serial complexity phenotype discrete b joint effect phenotype difficult serial factor apply conduct quantitative trait equation methodology study correlate phenotype formulation latent rely observed phenotype latent conceptual disease status widely field economic become increasingly attractive genetic trait sub common genetic trait study outcome covariate longitudinal consist part relationship characterize genetic marker account serial direct part longitudinal continuous desirable genetic raise challenge sample inefficient algorithm center expansion computational organize section discuss consequence na ignore structure section mcmc design phenotype selection selection bayesian extensive simulation study apply method genetic section conclude recommendation discussion outcome phenotype measure individual individual repeat measurement outcome u represent conceptual phenotype trait via mixed q covariate direct effect direct effect factor loading phenotype reduce capture mutually binary mixed link probit logit gain known representation probit response recover dimensional marker possibly clinical latent mixed indirect covariate effect phenotype interest effect model genetic significant specific represent subject ci modification nonzero model phenotype avoid assume covariate part individual status practice analytic burden assume independence exist ignore cause incorrect phenotype response individual index sample decompose j ci genetic marker phenotype first simulation longitudinal equation ignore conclusion longitudinal covariate efficiency popular rely modern computational markov dependent paradigm offer principle incorporate genetic contain outcome direct covariate indirect interest algorithm statistical unfortunately mix along sample chain next specification modification da sampling relatively phenotype extend phenotype phenotype gibbs sg due vector mixing improvement center hc hc move move hierarchy overcome px px auxiliary couple load update likely update vice versa hc ad ci original expand parameter transformation gibbs sampler induce distribution hc apply since assign prior prior analysis instance add flexibility improve behaviour suggest use prior loading set conjugate expand model parameter n specify prior hyperparameter degree freedom df prior respectively phenotype selection specify spike hyperparameter notice induce spike spike conjugate expand apply gibbs obtain appendix step involve involve suppose trait first phenotype one concern mcmc mix continuous specific gibb expand notice strong apply layer da scheme leave prior parameterization unchanged call doubly expand center hc summarize gibbs j conditional k j remain continuous part draw illustrate autocorrelation expansion medical interest determine phenotype study truly status loading phenotype phenotype loading conservative assess loading spike prior spike loading prior belief recommend correspond phenotype priori latent small determination phenotype posterior discuss introduce approximated mcmc threshold loading include practical discovery fdr phenotype factor central assume two factor support support calculation challenge dimensional arithmetic path specifically unnormalized via calculate identity expectation density py g py choose estimate grid calculate context bayes factor different remain concerned interested factor loading phenotype remain unchanged binary phenotype obtain calculate show bayes df bayes df another grid size approximation suggest grid big grid due distribution close suggest small grid observe good important inferential genetic study marker covariate component clinical marker interest two compete link calculate practically study method ignore method introduce detail simulation model phenotype specification family allele families segregation serial ar autocorrelation phenotype five time unless specify phenotype run discard first burn five phenotype interest table replication study ac hc px da scheme propose reduce increase efficiency improvement evident standard illustrate square error rmse result rmse response continuous phenotype discrepancy factor however phenotype serial estimate correlation incorrect serial correlation cause quantifie phenotype variable clinical covariate spike prior four phenotype binary phenotype choose phenotype strong association phenotype replicate phenotype disease status replication phenotype moderately associate phenotype specifically consider also phenotype via df phenotype truly bayes correct phenotype truly associate factor criterion correctly nan estimate criterion little identify weakly inferior explore selection purpose indirect genetic assume marker one standardize continuous covariate standardized effect part phenotype compare assume association genetic marker criterion covariate respectively covariate estimate consistently
depend dimension estimate word word may require sample situation near say mass contain frequent translate guarantee svd svd procedure simplified take v u provide fast accurate outline perturbation detail detail sample obtain whiten understand complexity dimension rather depend let focus analysis issue x w column deterministic perhaps surprising unit unit unit notable error skewness consideration dimensional notion sparsity decrease often burden explicit sparsity helpful skewness whiten complexity model skewness tend surprisingly beyond requirement read david zhang thank insight preliminary claim lemma section condition definition example topic model generalization cluster observation latent document increase power come cost challenging unsupervised distribution parameter wide probability contain term fourth via value scalable operation topic rather agreement powerful incorporate broad interest observe occurrence word document document sparse determine occurrence dirichlet dirichlet allocation permit multiple correspond hidden finding estimate either method maximization variational factorization modeling count document specifie topic specify word topic match observe efficiently observed decomposition low moment fourth exchangeability availability hide excess knowledge exchangeable principal pca two decomposition first data svd utilize high order find direction exhibit moment perform latent factor scalable applicable wide model include exchangeable model exchangeable model poisson generate analogous decomposition base order recover third moment suffice document finally multi view draw independently available topic present discrete model recover simple eigenvector presentation focus utilize moment correctness plug plug section matrix efficacy appendix topic document provide guarantee albeit topic essentially overlap condition separation separation condition learn statistically document topic separation utilize multiple document provide provable certain condition separation utilize anchor topic provably use negative procedure problem motivation limited provide result progress algorithm take certain technique know setting idea utilize evolution multinomial extend mixture single key ability multinomial factor method quite general hmms body algebraic problem blind approach decomposition see tailor source separation usually much without algebraic utilize idea column noiseless algorithm insight independently hide exhibit rather solution suffice estimation simplify eigenvector long necessary approach hmms approach exchangeability permit rather additive key technical contribution dirichlet careful gap topic approach exploit work separate variant canonical correlation permit procedure guarantee recovery h k hide th l first analysis four mild identifiability goal particular assume rest example gaussian consider component somewhat classic mixture gaussian permit responsible generate state component responsible impose noise linearity count sequence well understand separation without exchangeable independent factor case skew consider dirichlet present exact decomposition svd high third algorithm extension topic factor model throughout pseudo column allow square moment matrix leave singular q return pairwise project permutation identifiable column form transformation sign central third moment skewness skewness false positive return subset sample sphere consequence canonical assumption uk whiten possible w uniquely determine singular every singular nonzero reconstruction w entry standard positive skewness direction skewness practical freedom multiple find singular easy determine skewness straight estimate skewness suppose singular step skewness singular matrix find matrix u v set singular reconstruct eq subspace fourth moment tensor central fourth denote excess hope order factor subset uniformly sample sphere return canonical skewed note incorrectly skewness provide distinction remainder argument product simplex modification parameter manner prior denote modify second modify dirichlet equal rest extreme central eq expand exchangeability x moment find remark find identity set svd leave reconstruct normalize set pseudo show recover permutation column restrictive tuning also latent allocation positive return column random vector sphere return vector consequence rescale variance normalize algorithm finally claim hold functional form limit skewness eigenvector base show skewness appear skewness modify implication note effective skewness skew lda conjecture fourth moment utilize alternative model dirichlet application natural come different intensity prior
multivariate total identical mutual multivariate extension information information mutual combinatorial conditional mutual information upon mutual information eq similarly entropy mutual mutual information follow introduce lattice integrate statistical introduce notation define set multivariate lattice expansion information joint special lattice variable special give tb derive three basic rule mutual identity view equation equation lemma joint conditional show three rule mutual inclusion evident multivariate hold mutual n h x mutual decompose hx equation chain apply repeatedly x n n entropy variable partial equation total decompose explicitly mutual information dual follow probabilistic variable entropy conditional redundant entropy mean equation derive rule decomposition decomposition conditional entropy residual nh h entropy mutual spatio temporal variable essential let multivariate condition zero otherwise lattice delta transfer come rate free variable come entropy follow prove go flow entropy come lattice difference non go flow go flow inequality minimum lattice depict bivariate information cube cube plane indicate entropy depict I k also depict colored outline outline circle code specifically transfer red outline transfer variable theorem less come sum negative information blue circle outline indicate go go transfer circle residual entropy entropy total correlation x visually node lattice reflect mutual respectively indicate green flow cm cm cm corollary support mathematical proof theorem flow stand lemma mutual information information lemma joint mutual lemma neither stationary depend past state conditional mathematical variant convention cover except
match consistent denoise matching arbitrarily expect want conclusion equivalence equivalent contraction parametrize per force correspond jacobian minus jacobian energy interpretation concentrate near score nearly density tell direction increase ridge direction direction decrease direction manifold return mind derivation particular ridge expect reconstruction move manifold near eigenvalue direction manifold besides derivative property immediate recover integral path stay approximate pick line use hasting reject acceptance compute train capacity process concentrate train exercise along require qx show hyperparameter explore work train learn draw model mind original fall region come would certainly maxima kind lead ratio property metropolis hasting embed plot pair read pair leave concern spurious auto encoder capacity end fail properly desire reconstruction towards density case work region close dot thing outside noise small enough example find right sketch illustrate situation properly reflect accurately stable manifold concentrated conceptually spurious assume case end completely involve corruption capture recover generate denoise form closely estimate property second derivative contradict reconstruction derivative hold simply virtue reconstruction reconstruction alternative maximum likelihood unsupervise need rbms boltzmann machine analogously score matching estimator density minimize confirm mcmc approximately sample metropolis hasting question since heavily notion notion already norm show trace rewrite motivate continuously define expression applicable point fix part leave make notation denote first term implicit lagrange calculus reader either component optimize separately expression px satisfied optimum become hypothesis vanish linear obtain eq substitute side get substitution situation try series like dr kx term show involve get eq eq could would need study could possibly local appeal precise denoise auto capture local function second derivative move justify procedure motivate moment auto encoder continuous function ball product ball pdf ball around stick ball everything rewrite formula first local base associated obtain estimate direction q dx px x dx x dx hx come come probability term transform anti odd x z look front reason px hx remainder main exact expression make number ball origin integer absence absolute put negative theorem corollary drop odd become integration unit get determinant jacobian let radius dot decomposable corollary let show yield integral ball understand integral rewrite h xy h observe multiply lot power term yield odd function power dx dy reasoning write u ji integral function previously unit ball continue conclude translate give match auto auto encoder good job capture manifold regularize reconstruction characterize auto capture derivative contradict interpretation theorem generic depend parametrization encoder encoder would train similar auto encoder training criterion corruption contraction similarly consider criterion maximum likelihood involve show sample confirm aspect many focus region point variable plausible representation discovery feature latent generate base reconstruction auto variant encoder representation argument empirical well train counterpart generate quality restrict unknown concentrated region density core manifold identify question concern error criterion implicitly whole aspect essence target formalize link answer establish implicit define statistic contribute proof link criterion rest contribution auto generate estimate increase correspond hessian second access approximate mcmc achieve hasting approximate auto auto encoder recover distribution compare project visualization order presentation differ order gap name review exposition capture training thank reconstruction map maps back reconstruction made view encoder regularize denoise regularization regularizer basically attempt constant possible away precision make neighbor high value otherwise possible correctly reconstruct derivative direction remain direction manifold case direction around direction auto choose ideally tangent direction encoder linear encoder capture manifold extreme want nearly special obtaining function flat near yield towards density point mm large identity obtain regularizer point towards peak encoder corruption loss actually auto encoder auto encoder encoder whose contraction encoder regularize minimize regularize reconstruction element use easy strongly term tie variation direction fig denoise stochastic source mostly noise corruption easy mathematically many proof serve train consider reconstruction result make reader understand study effect equation convolution involve neighbourhood view noise neighbourhood total quantity include weighted numerator simple additive try pre give equation strongly minimize expected thing expansion go good reconstruction something equation allow asymptotic go family distribution mass connection auto taylor encoder expect corruption x loss take expansion similar encoder contraction impose alone motivate loss instead plus penalty analytic auto encoder contraction denote penalty coefficient involve notion notation see alternative way behavior continuously differentiable differentiable twice minimize derivative expansion understand function expansion wise behavior use calculus allow recover score know nothing minimize score subsection practical loss draw alternatively auto encoder train online gradient update stream auto encoder minimize generate auto encoder auto minimize empirical loss available capacity minimization ideal drop information train sufficient model capable function work round numerically evaluate setup describe pick quantity purpose give example happen model minimize train fashion discretize score get arbitrarily illustrate score ex fit divide evenly discretized loss free thing discrete score htb visual expect predict extend manifold one visualize field parametric encoder numerically along denoise auto corruption noise encoder decoder optimize bfgs use corruption bfgs run bfgs convexity solution corruption also final outcome reasonably may undesirable training select visually outcome find along field towards density close play vector place reconstruction implicit high manifold role function regularize auto small example maxima place whenever path region median region median clearly visible
incur take action loss compare action rest present slot loss pick environment action library marginal additionally convert gain library store action action similarly train idea control little gain sensitive multi environment slot action fill evaluate formalize calculate slot carry formulation advantage control instead loss regressor slot estimate marginal computed regressor slot per feature benefit regressor produce pick target environment establish sensitive slot sequence distribution environment greedy allow algorithm cost sensitive fs nr hypothesis fs define cost sensitive classifier nn thm nm thm misclassification marginal x dependent multi classifier hypothesis minimize additive n budget control take execute make correspond pick additive sequence state fs regret class class square slot reduction action fs classifier reduction regressor fs class sensitive planning recent show hard avoid soft penalty lead trajectory suitable often seed straight environment suffer local provide seed chance minima seed contextual overhead evaluate library negligible rely plan arise research difficult circumstance initialization position distance field seed leave target seed library environment performance train regressor without seed note regressor subsequent regressor also include remain action difference initialization library order order sort regressor sort regressor maximize absolute benefit slot failure execution trajectory execution robot short seed rank performance fail free plus trajectory execution time long computation environment summarize straight unable free result execution single regressor choose primitive provide seed benefit pick trajectory seed fail sequence failure execution second compare initialization seed find environment avoid obstacle enable produce planning environment top slot failure rate fast trend bottom execution column straight heuristic planning mobile library feasible trajectory low traversal portion repeat trajectory place challenge challenge generate tx robot maximum trajectory sequence cluster center compute whole overhead run histogram use online slack sensitive slot report traversal sequence trajectory choose appearance step fall empty around generate trajectory sequence try avoid theorem yu liu university pa edu order library optimization static account within description goal efficient reduction order repeatedly classifier regressor approach formal sequence cost prediction contextual optimize control library mobile web ordering admit interpretation item interpretation real requirement relevance diversity member library context library refer rank choice action trajectory mobile optimizer dimensional trajectory control library collection trajectory mobile demonstrate trajectory robot al expert dynamically trajectorie store control span feasible library specify action library mobile robot task traversal path usually constrain motion model dynamic library robot library every traversal robot goal minimize trajectory sensitive seed bad initialization suboptimal even action trajectory act predict library feature fail initial seed early fail library feasible time find naive closure fail similar principled reduce diversity art either library ad action past success fail g unable back heuristic code plan predict robust action predict label number difficulty selection predictor within expensive explicitly multiple regressor environment performance guarantee selection module sensor system leverage classifier map action action contextual loop et outline good attempt submodular submodular et al contextual section planning contribution near move predict make arise domain search library demonstrate efficacy robot plan mobile robot generate sequence either rate success etc diversity library formally monotone submodular sequence sequence fs concatenation increase monotonicity addition sequence sub modularity library optimization attempt optimize cost well sequence cost environment high cost dependent environment evaluate et monotone
entropy panel version namely stepsize search weight vector value value plot difference plot plot conclusion converge maximum entropy goal align hope well goal ar discussion support european project city paris paris theorem convex optimization conditional minimizing moment discrepancy enable result optimization consider approximate integral reproduce space learn approximate algorithm entropy attractive method field estimate mrf entropy requires learn mrf answer query empirical follow recursion observation statistic empirical frequentist mrfs parameter update motivate perspective mrfs mrfs explore update effective decrease regularity decrease fast collection representative super contribution frank wolfe another explicit minimize yield gradient active algorithm exist infinite algorithm bad sample property satisfy interpret integral hilbert rkhs mapping rkh identify associated kernel update f rx illustration figure constant self cm cm cm perspective corresponding moment match approximate approximate similarly fx schwarz control less rademacher estimate generalizing relate bring line integral two clearly outline present provide input compact quadrature aim compute combination certain problem thorough mention definite compute lead affine moreover quasi sequence error function opposed sequence quasi continuously frank wolfe linear segment e optimal upper rate gradient size subgradient ascent outline hull set finitely finite still converge follow gradient optimization change certain obtain weighted line lead set unconstrained happen uniform kernel interpret update minimize problematic choice mrfs extreme common convex originally algorithmic perform subgradient ascent temperature frank wolfe might interpretation establish connection indeed max operation cm differentiable natural ascent equal equivalent therefore surprising subgradient ascent reflect change bregman divergence would lead function problem size cm diameter polytope lead integral two conditional see section however affine hull center relative choice step size random search search relative interior satisfied satisfied definition hull orthogonal orthogonality distance cm continuous kernel true cm finite get positively homogeneous construction one show function imply imply compact subspace infinite exist apply projection orthonormal eigenvalue know orthonormal f kx function kx hence cm rate know justification cm context integral infinite expectation kernel choose general example graphical practice possibility minimization exhaustive sample could cm aim assess relationship cm section space th px x expectation basis consist interval exhaustive kernel density middle right randomly em gradient original weight search lead quasi density inverse cumulative min denote compute optimal follow conclusion cm perform good search perform bad infinite set extra projection significantly outperform regular least empirically achieve present explain theoretically eigenvalue cm
validity agree valid agree negativity relaxation negativity absolute avoid certainly importantly objective care much close tend encourage objective deal gradient previous optimize convex valid discuss matrix max row internal norm elsewhere relaxation eq alternatively whether recover cluster cluster problem assume case cluster binary matrix constraint clustering noise match tight relaxation introduce main c v maximum definition inspire notice cluster inside outside follow certainly ensure recovery affinity combinatorial region vs cluster lemma cluster combinatorial small affinity matrix lagrange multipli norm c I unique consider balanced unbalanced motivate clear theorem plot wise tight unbalanced two affinity binary relaxation relaxation section enhance norm constrain optimization norm linkage perform absolute problem linkage power hierarchy clustering node similarity cluster pair cluster similarity fig exhaustive search find separate hierarchy separate check recover solution tight expect max provide trace max balanced max unbalanced scale trace require proportional size guarantee proportional huge advantage provide max two diagonal entry large one entry compare algorithm ideal ideal ideal gradually graph although binary affinity matrix absolute unbalanced fractional reveal absolute objective advantage affinity recover max proxy formulate solver small sdp condition slow scalable change problem convex large reasonably rank update operate unchanged otherwise formulation optimal solution zero gradient likely gradient gradient however optimization problem solve loss iteratively operate update cause numerical precision gradually emphasis close introduce lagrange multipli saddle iteratively fix optimize optimal fix eq q soft ij ij check initialize differently avoid stop positive semi double equivalence two size every factorization norm update trade dual provide sparsity add linkage although tight relaxation trace go introduce relaxation relaxation norm indicate suggest intuition matrix node cluster strict spectral cluster relaxation exist matrix optimize constraint cluster secondly integer run round fix clustering pick affinity replace cluster clustering enhance trace pick experiment explain summarize enhanced outperform competitive like investigate level clustering objective trace counterpart spectral balanced unbalanced clustering describe indicate norm far underlie norm spectral cluster mnist enhanced spectral point construct explain report time complexitie matlab top rr rr rr r closed see equivalence rr exceed contradiction equivalence since sub relation trivial counter strict gray cloud clique link connect cloud construct recover cluster alternative cluster alternative step characterize optimality condition dual construct equivalent max wise residual adjacency guarantee gradient distinguish zeros zero space projection define define yu sharing column space contraction definition theorem eq tu define contain eigenvector repetition variational norm definite matrix sub eq suffice condition condition follow characterize sufficient optimality notice construction entry corner cluster take standard duality norm alternate infinite sum geometrically wise division easy show last inequality projection consider last attain optimize lemma I q proof straight forward conclude proof effectiveness cluster approach partition similarity different cut receive lot normalize symmetric cluster minimize total term disagreement capture member cluster decide unfortunately disagreement np hard formulate matrix relaxation constraint recently nuclear minimize penalty provide propose max tight
element use estimate coefficient vanish toy identify network network find turn element detail max min clique infer independently certain condition basis realistic true grow correspond true consistently unnecessary candidate element subject constrain em lead k ij poisson deconvolution truncate maximize involve linear normalize develop theory result extend unclear writing proof begin give adjacency induce subgraph otherwise expansion satisfied expansion consequence uniquely decompose encode binary symmetric matrix basis constructive practice robust algorithm use basis non basis nonnegative entry encode clique bernoulli maximum encode clique clique include network element basis matrix network asymptotic bind element redundancy never true redundancy scenario redundancy imply regime limit behavior node assume capacity independent basis tend include analysis facebook empirically recover matrix quantify accuracy consider approximation characterize specify basis element exclude reconstruction basis element low theoretical order statistic give basis recursion network red use simulate blue weighted estimation binary without enable comparative pattern facebook us historical datum quantification concrete illustration idea whether simulate parameter basis distribution sample network run th parameter estimate permutation normalize distance reconstruct count clique size quadratic reconstruct incorrectly clique reconstruct table summarize reconstruction last table support pose reconstruction accuracy edge specify basis choice storage comparative decomposition prediction interesting pattern facebook facebook period student analyze r r r college edge american mit berkeley forest summary result salient college include report basis runtime second error incur element range compression substantial decomposition element reduce recently analyze association record history sc record general union network set find non association power associate directly result suggest tight interested developing holds explain spatial organization operate build analysis involve member meta activity association reveal organization show association identify summary exploratory tool capture multi organization historical record take quantify explore quantify clique scale possibly os poisson random contain edge baseline adjacency integer encoding svd summarize interpretable quantify provide overview row network row row box svd cumulative utilize slowly svd poisson consistently achievable svd rapidly reconstruction achievable achievable top support able notion social information social exploratory simple amenable weighted parsimonious orthogonality constraint attain interpretability basis social social summary always achieve element suggest interest parsimonious complex suggest leverage social facebook high compression ratio utilize desirable sensitive miss properly maintain outline power network weight long window element message window period author wish science grant award office grant ex edu weight social scalable combine fashion parameter sample include redundancy expect synthetic facebook association keyword deconvolution massive parallel year media facebook collaborative spirit wikipedia services service include cnn com popular com service momentum social science study pattern interaction tool collect interaction network pair arguably adjacency integer diagonal zero modeling decompose exchangeable value number binary decomposition popular orthogonality constraint attain interpretability multi choose enable linearly inferential utilize infer multivariate decomposition key community multiple scale come avoid adjacency matrix infer basis idea poisson edge weight rate matrix interpret latent induce social interpretation clique identify
precede important interpretation response previously analyze response response integrate spike compute potential impulse process model spike generator impulse threshold however goal nonlinear usually repeat response introduce several nice form avoid yet additional three addition implementation importantly coefficient aspect circuit fit associated response interpret neuron type neuron drug affect processing assess drug prediction spike paper prediction accuracy build sequence aim happen future benchmark high curve high rest organize describe motivate future protocol train light nm code independent ms ms indicate presence spike spike spike ms model behavior hz choose spike potential simplify analysis output spike train consecutive spike next next number roughly neuron spike spike divide part first relate interpretation spline sequence sequence start status mean interval f tf denote otherwise predictor immediately interval form model assume useful fire predictive spike simplify datum datum stimulus validate assumption first function depend spike spike assumption coincide neuron input hypothesis fit contain limited integrate data stimulus special form logit intercept logit model assumption additive simplify well moreover logarithm function formation additive include stimulus transform understand sharp fire leave hand side sharp sharp similar stimulus critical method atom contain component describe characterize spike expensive importantly compare function construct explicitly parametric amount procedure check parametric order three logarithm logarithm transform log relative empirically also response interpretation straightforward last spike pf cumulative cf fit spread sf response spike plot top panel sequence red spike bar solid circle grey vertical grey vertical spike times note spike status design predict framework follow sf sf achieve either sf achieve st f implication intuitive function equal conversely achieve answer fundamental care sf measure include interaction order term predictor high order examine possibility cf sf backward aic history development generalize parametric semi straight recommend plot pf fit value panel validate first time build adopt receiver operate roc unified approximate fire mechanism neuron believe spike comparison design sf r rf matlab svm instead potential stimulus history neural firing ms substitute show model specificity curve obviously advantageous prediction orient auc interpretation limited complexity stimulus filter spike among difficult spike whether spike encourage spike hand provide answer auc behavior provide interpretability simulation rf interpretation model comparison example integrate employ follow potential assume occur fix process spike spike time constant spike thresholding potential conventional box stimulus represent simulate ms simulating purpose sequence spike length randomly ms bins box shape stimulus yield time replication level coefficient confirm capture classical pf half may indicate highly power extract coefficient pf sf frequency coefficient entry scientific insight fit replicate vary cause neuron consider scenario plot fit pf sf respectively fraction plot scenario especially pf drop pf pf sf vary illustrate relationship clear pf increase spike probability due increase sf closely plot pf sf plot sf level spike spike spike remaining use different use measure auc different outperform small se rf svm good performance highlight bold time need prediction cpu ghz gb advantageous fraction identical fourth fast fast c c rf fitting prediction total se well highlight bold modeling draw probabilistic introduce response sparse advantageous interpretation computation discuss capability capture history spike equation reverse exist spike decode glm encoding model involve neuron extend predictor neuron status fit coefficient term neuron network estimation joint likelihood criterion interested explore additional component pf coefficient spike occur address mean model predict spike provide important direction analyze datum interesting provide fix intra follow state hold seek inter point rgb circuit circuit circuit challenge frequency spike high light propose new relationship employ logit provide nonlinear link input output response interpretable insight neural validate spike within ms advantage efficient computation simulation integrate spline approach output specific pattern demonstrate datum summary parameter circuit disease generalize suggest great brain specific brain leave neuron goal genetic light light application circuit input neuron circuit early stage circuit precise circuit I train neuron encode circuit brain cognitive circuit make information thought affect brain slice recurrent circuit light neuron randomly spike neuron record neuron directly activate neuron circuit producing continue stimulus show light output spike neuron run experiment scientific question
active arm km km km last element bandit identification event reasoning make phase make arm active arm eq suppose arm assume active arm happen e induction objective illustrate arm identification accord arithmetic allocation budget report experiment bad mean arm geometric arm experiment arithmetic experiment bad interesting sr arm identification even performance uniform involve single arm fundamentally always outperform tune provable research university edu wang department mathematics edu department science rely successive bad successive one algorithmic contribution tackle particular good face unknown choose reveal agent identify multi maximal note budget possible formulation model want minimize attain accuracy correctness latter history go budget set budget logarithmic notion characterize hardness identify well intuitively good arm sr evaluation paper suggest generalize analysis top contribution solve distribution successive h v nu sr find experiment sr perform involve problem fundamentally one solve author face arm single identification evaluation bandit construct identify arm nu nm km arm identification numerous reader previously cite terminology bandit face arm probability sequential evaluation observe arm select denote correspond mean assume mean make measure mean evaluate fine argue quantity arm easy see logarithmic represent hardness identification useful surrogate ccc arm n introduce multi bandit face identification problem sake problem deal good arm arm quantity problem mh mi bandit arm forecaster sequential form evaluation select agent arm high mean sort weak gap conjecture replace good identification paper shall derive section build strategy tune much gap time round sm sm quantities identification sr successive good feature confident among arm informally end high low decide rely gap precisely accept end phase empirical among empirical empirical good arm compute well sr order gap ki successive arm identification good eq hoeffding probability complementary event bound eq suffice event empirical within respective active arm
hessian product hessian guarantee shrink coefficient guarantee decrease kl divergence heuristic parallel update heuristic modification nature opposite sign sensible prevent case mutually drive mode parallel update result include effect e produce away effect individually optimal assign evaluate usefulness supervise color ten example per cifar patch test arrange overlap train svm pool post process pool image feature compare size vector cifar achieve grid accuracy coding encoding sparse achieve enhance modify seem require patch achieve close cifar semi svm learn cifar train advantageous medium label aspect set yet cause transfer competition competition color unlabele test label class public competition label cifar competition unsupervise demonstrate effective discovery supervise clean dark work figure page span always immediately leave line figure separate extra white page figure draw table table table title one table title table terminal cell acknowledgement discussion computation done conduct computer consider code learn spike rbm sparse code intractable ability dramatically demonstrate approach capability cifar learn challenge transfer latent variable visible generate q sigmoid bias spike respectively precision respective conditional element product constrain restrict variable hide rather understand gate subsequent motivate discovery describe avoid apply discovery coding show achieve cifar object recognition dataset normally factorial prior cauchy choose posterior factorial spike drawback sparse variable merely remain even kind undesirable laplace active little reason believe realistic setting model avoid control unit via determine active control coding control integrate extraction generative boltzmann inference approximation scheme map inference make suited activation feature relevant rbm rbm work think overcomplete small advantage discriminative capability factorial poor generative purpose discovery spike feature supervise e appear contribute kind classification powerful discovery turn exact mechanism learning turn intractable maximize step rather admits close find online work goal variational maximize variable minimize kullback leibler family ensure tractable step step sparse code difference posterior
vertex problem hence clearly arm conclude acknowledge grant dms mm take oppose specifically sequence bound worst additionally achieve way paradigm partial side online include compete regret assumption feedback promise research stock variety field game level prediction irrespective often naturally develop regular successful appear prediction expert theoretic regularity set restriction possible move adversary constructive author provide method focus set linear optimization information carry online reader nature round learner observe regular fix may view nature suffer process word plus adversarial ideally pay theoretically key make constructive game instead close side tight make precise allow deviation game constrain sequence depend short serve motivation statement value find purely well even ahead total advance either one obtain eq discuss interest regret previous move proxy move tend go alternatively one past occurrence get predictive bound provide predict regret property arbitrary write far full linear reveal reconstruct dependence learner require generally nothing prevent appropriate learner bandit yet actual learner study information might contain complete delay set consider optimization take optimistic incorporating missing bandit appear scenario differ feedback delay fall trend advance optimally employ completeness set dependent slightly bad represent sequence also divergence begin modification follow regularize regularizer simplicity next mirror paper minimization variation style perturb present generality provide learner outside compact set loss local shorthand consider initialize observe update notice regularize see incorporate turn correct endowed barrier optimistic expense additive small boundary self modification md respect bregman require consider strongly initialize update recently convex dual convex optimistic mirror descent yield q mention usual optimistic md gradient sequence mirror respectively completeness exhibit norm simplex case feedback serve explore letting mirror correspond term norm hessian optimistic mirror simplex enjoy turn receive simply incur suppose include move external source present simply availability opposite decide follow randomized full f present update observe adversarial appropriate step self let eigenvector eigenvalue uniformly analysis simplify generalize ball regret statistic x sm ti f put aside nothing investigation far see low raise good go formalize process indexing strategy us predict optimally however scenario comparable set example occur goal set stock expert option learner access step stock day regret know loss stock optimally achieve initialize initialize q rely loss unit banach nature optimistic let different set expert forecast give one stock expert expert would guarantee among stock measure see minimize idea run secondary regret replace online full setting get compute scenario number section access compute initialize regret term loss radius round get useful stock provide loss stock day trading day service forecast day provide correspond bind initialize initialize f g tf bandit loss arm limited information improve analogue set banach ball optimistic md process variant set suffer partial information setting early self arm draw update contain radius regret perturb unit norm object notion attain trend present deviation advance learner add shall constrained constraint admissible recover unconstrained admissible relaxation guarantee irrespective easy search sequential computationally attractive eq q adversary small deviation key since supremum expectation might able difficult I verify satisfy may take rademacher norm expect vector view perturbation final process rewrite form randomize ball ball take simple indicator perturb output regret converse implication satisfying draw respectively rademacher variable output expect recognize perturb cumulative couple example set give consist feedback move sm immediate argument immediate bandit delay feedback draw mean full norm write term exactly quadratic expansion obtain couple transition argument work e follow immediately bind positivity use multi armed bandit loss attractive tight whenever thank process barrier set step state case bind interesting gain term good negative gain barrier couple function round arm observe f h ti q q regret completeness trick extend let stand receive introduction let contiguous interval make monotonicity f take fix constant priori regret q box broken phase define eq generality follow assumption regret calculate deal period suffer regret may forget learn function trick full bound instance enjoy factor lemma computable trick optimize unobserved quantity clearly us eq quantity computable lemma imply plug inequality get final optimize argument tight lemma bandit regularize observe hold add side conclude inductive newton f follow self function local e hence prove yield eq convexity tf together technique let project normalize eq diagonal use probability view random
produce table simply indicate cluster c cluster j ht set cluster apparent norm provide cluster well experimentally norm probably favor weighting measure diameter height weight weight pair determined omit age variable dataset aim age contain size unbalance various number value range ht contain characteristic width overall store find ccc ccc cluster cluster document file automatic collect speech normalization entropy individual breast obtain dr observation uniformity predict variable perform well though superior c principal make look gap interpretation argue although partitioning explore variant cluster research tolerance effectively control cause split ignore inclusion identification especially document extract web document extract look along look secondary like thank nsf possible grateful repository website breast detail http uci ml breast cancer matlab document herein variety social business retrieval refer grouping human classify human main work know develop involve eigenvector decomposition matrix center name step word replace word natural gap understand detailed algebra geometry support column trend way component principal spread another define trend orthogonal deviation minimal concept deviation sake development present trace tn vector frobenius norm center lx see among orthogonal onto j locate finding characterize minimum deviation line datum yield trace regardless minimize observe know minimum way line orthogonal line directional direction line line minimal total deviation unless state hereafter understand principal two disjoint line depict front depict side ht distinction front back determine jj center q principal singular compute immediately sign need determine respective evident determined right matrix sign negative sign column assign option partition option examine scatter maximal principal trend separate three disjoint continue produce disjoint project split principal take gap motivate column partition appeal easily superior look gap cluster three distinct gap unbalanced contain address figure create gap human likely depend whether point goal cluster ideal line create tolerance balance ignore project experiment percent particular separate tolerance small data tolerance percent identical aside data project principal split gap maximal entire collection center gap right begin sort step create take document traditionally store mn row correspond individual extract filter word remove matrix text mining term effect normalize document frequency inverse frequency weight scale text variant variant document term q normalization normalize unit euclidean length note discuss scientific document normalization commonly scientific unnecessary stay range evaluation validation important cluster relate important accuracy evaluation measure break
regression hide follow mcmc covariate sampler rwm states x perform rwm update simulate disease x rwm x require adaptive rwm perform disease disease covariate estimate run remain uncorrelated sensible disease proposal accept within take approximately hour start trace chain thousand adaptive algorithm learn shape trace six three estimate plot iteration final run fit examine see choose six disease disease capture mark provide arise bernoulli trial six disease b interest disease whether presence absence affect logistic coefficient whether old affect infection affect correspond via reversible jump infeasible probability effect individual might raise probability indicate probable interaction positivity posterior parameter firstly clearly probability specie evidence reverse interaction presence chance statistic plot specie less infect previous exposure old infection infection decrease perhaps evidence interaction evidence change infection recovery exposure prevent arbitrary disease estimate prior markov dataset describe create check draw interpolation growth year provide disease independent rough correspondence parameter likelihood nine parameter summarie posterior probability positivity interval parameter markov disease always baseline c c main none posterior change affect one effect important analyse longitudinal study record six model offer exist approach approximately methodology cycle disease disease sample regression probability walk part gibbs motivation recursion directly would provide parameter conjugacy conditional mh tuning allow gibbs hide mh efficiency mixing current mh sample states disease covariate disease therefore avoid entirely update logistic parameter disease hide conditional state however lead poorly mixing discusse examine couple hmms consider article figure allow similar article simplify collection practice consider apply regression assume separability computational complexity state gibbs qualitatively allow major briefly biological insight offer old complete indicate last month previous infection specie specie effective acquire date population suggest vary specie interestingly infection appear infection infect recovery next month evidence current infection month find covariate covariate predict finding mention state disease disease treat apparent dependency sufficient without infection group certainly could enough disease appear consider disease state scenario subject might model methodology give state minor west project university support natural gr trust z section definition uk environmental sciences uk institute university specie absence repeatedly regular different incomplete profile discrete disease dependent logistic regression disease time point influence specie probability specie via cycle disease metropolis covariate hidden disease disease parameter evidence pair acquire keyword forward population infect successively e intensity infection impact involve complex allow prediction outcome interaction order longitudinal sequence infection four spatially population specie aside community human disease capture mark study set lead incomplete profile subject contain profile give disease miss response incomplete datum examine occur month first influence covariate logistic realistic methodology investigate disease disease dataset infer covariate disease covariate five disease analyse english cut site forest site near site live comprise day description outcome tag identify site factor lm capture integer weight gram identify tag capture diagnostic infection variable contain least informative transition capture initial chain contain substantial almost half capture month even observe twice month miss derive status despite table table summary capture response c investigate potential six study wish presence disease perhaps infection first month affect applicable disease probability model time disease disease state structure length infection geometrically infect month phase phase disease semi first expense influence dynamic knowledge presence absence disease must lie markov hmms disease e relationship chain capture forward backward consider disease interact couple hmms couple disease chain backward e size size forward backward hmm states take operation naive time equivalently deal certainly chain sparse approximately justify f l hide markov arise reflect covariate parameter simplify presentation hmm disease hide simplification disease chain likelihood likelihood summation hide simplification couple chain disease disease likelihood multiply may useful tool time point detail covariate clearly record unobserve dynamically could perhaps adopt simple whereby weight sensible investigate remainder brief disease dependent covariate dependent drop reference dependence detailed description resource specie human covariate recover third fourth month month majority one month first month likely particular specie might recovery past disease specie model infection infection old infection infection either status hide time govern justify similarly effect relate prevent example logistic month elsewhere disease infect infection say infection recover disease depend infection month use infection infection infection vector connect observation transition relate include disease disease human dataset infected chain transition separate regression relate hide infection infection remain present
volatility standard forecasting demonstrate aid behave mis specification keyword valid non tool characterize fundamentally function plausible alternative nearly iid extend economic financial forecasting rigorously accuracy mis finite allow immediate asymptotic generate popular selection tool aic imagine iid bad respect distribution prediction use natural criterion actually calculate risk generate single neither infeasible call assumption uniformly distribution series expect poor sample model bind assume mild condition existence provide general series satisfy mild average model amount use way beyond traditionally empirical quantitative inspection application really prediction especially mis validation partial exception economic long recognize difficulty pseudo prediction performance remainder procedure approximate bias toward overfitte giving tend true three reason test compete despite already instance recent financial enhance ability c reflect phenomena period often rare robust lead typically employ generalization bound rigorous reliable generate finite broad confidence normality understand report policy interested forecast specification wrong economic case still predict future derive apply setting finance economic engineering science loss make form rarely time dependence therefore ar past assume instance state view show stationarity ar lead worse phenomena plain provide way assess good really scientific explanation evaluation economic policy ask another approximate reality correctness success fit wrong directly past matching remainder background give focus concentration idea complexity leverage powerful idea generalize state prove result conclude toward generalize elaborate goal control datum reader sketch classical setting make poor throughout solely denote independently generalization q word expectation perfectly function rarely plausible index call one pick choose particular indirect ad hoc minimizing tuning fit risk risk big match end reproduce apparent depend fluctuation learn theory control tight quantity unknown allow add accuracy risk hand size observation form variance trade quantify result fit also claim piece bad choose forecast expect risk show standard result proof begin fix hoeffding minimum substitution rf nf powerful say observe training fact function want hold function wish nf achieve extension number function take result model quantify prediction choose functional analytic covering rademacher vc starts collection infinite every every latter vc vc subset one growth subset form growth effectively distinct go definition dimension finite show vc capacity straightforward kn immediate risk inside pre include fit apply explicitly need conceptually right hand fit nothing I infinitely vc illustrate count concentration show average concentrate expectation generalize class must handle dependent information know observe iid empirical expectation exponentially effective move iid series forecasting modification observe observe predict value process wish predict iid sort dependent first reader strict strong stationarity integer stationarity imply unconditional constant limit separate observation restriction convergence occur arbitrarily slowly bound vc give describe sort restriction restriction regularity coefficient process mix many mix definition approach product individual probability overview know mix iid correction background piece mix find prediction bind modify risk forecast preferable tighter risk increase event risk exceed solid black event probability force upper right curve desirable negative slope within budget put form move depend divide odd block block let block sequence mean empirical risk base sequence nf nf statement slight get obstacle vc calculate class ar equally bayesian conservative class forecasting model model linearize place past notation prove grow datum grow memory satisfie triangle inequality fix condition mild satisfied sub multiplication value prediction norm likewise absolute approximation become apply trade effective whereas depend desirable consequence memory q apply state model calculate demonstrate behave well unfortunately linearize stationarity eigenvalue lie unit circle stationarity ensure forecast one conditional like replace maximum forecast form space model simple compute output theorem quantify volatility calculate method standard stochastic volatility observation show investigate long kalman match let log remove return day kalman log minimize actually calculate point upper vc stochastic volatility vc dimension large much error translate forecast produce intercept reduce predict effective spurious volatility fit risk methodology forecast business cycle model forecasting use bring four consumption hour work economic estimation form list unobserved variable map nonlinear set return uninformative parameter penalize rather filter produce grow nontrivial bound finite lag vc dimension deal result course suggest effective calculate order magnitude suggest error risk surprising try get idea address show bind standard forecasting course raise address selection training penalize minimizer risk truth loss describe minimization work iid iid inefficient slightly suppose exactly constant exist erm nf nf k ignore constant iid I retrieve setting pay upper slowly iid case tight since general rule fast characterize ignore choose frequently risk many different minimizer forecaster collection minimizer small course complexity risk minimize bic rely asymptotic penalty risk natural predictor bound allow selection principle knowledge penalty account complexity error prediction volatility forecasting exercise model function vc control vc choose risk procedure demonstrate control generalization forecasting otherwise linearize dynamic equilibrium linear derive hold generate unlike standard selection penalty aic inherent forecast term economic applicability forecasting whose forecast would decay derive attempt nonparametric fully version possible circular review bootstrappe fit sensible coverage fit quickly let alone require obstacle resample method step theoretic forecast exploration gain picture forecasting algorithm want low risk forecaster target forecast well forecast actually realize remarkably try make ignore stationarity
minimizer theorem origin admit minimized inequality conclude remain inferior fact family admit differentiable level broad functional regularization loss q minimize functional admit theorem satisfied sufficient functional form instance uniquely minimized family functional define hilbert regularizer radial characterization dimensional family functional say admit every admit characterization general admit non report continuity regularization methodology ill estimation focus hilbert space endow product enough regularization square machine principal variety kernel space rkhs value empirical desirable solution include framework machine concern situation available space objective functional minimize case dependence definition functional cast regularization regularize regression choice control give label interpret corresponding recover principal monotonically constrained unitary formulation broad class deconvolution evaluate convolution noisy form many classical appropriate finite measure algebra give input regularization functional regularization take problem functional let functional say admit choice reproduce easy therefore obtain admit appear rkh functional early condition also necessary existence existence exploit author relax admit merely satisfy existence minimizer radial functional present radial functional hilbert space hilbert space convex star shape respect point shape radial space hilbert let hold conversely fix generic eq last also pass follow shape origin semi close ball center align unitary subspace orthogonality addition angle span positively sufficiently form
conditional variable symmetric matrix self loop typically notably focus expect degree within commonly encounter practice remove ij satisfy identifiability discrete degree regular community derivation good optimize np bayesian chain method study generally scale million node propagation comparable slow profile likelihood trivially plug speed profile search thousand propose model block case involve occurrence approximation profile give label block correct version consistency estimate truth converge grow fast misclassifie converge one need study belief propagation analyze cavity physics transition establish impossible unless one sparse correlate paper purpose consistency degree grow practice find graph quite contribution likelihood dependency make tractable adjacency symmetry block compression divide look likelihood sum accurate allow million truth overlap purely cluster likelihood connect perform present numerical demonstrate political contain maximize optimize possible assignment convenience true idea group group give quantity along give row th column nr base mutually approximately poisson long ignore among latent write pseudo maximize standard em probability node initial label repeat algorithmic summarize lk lk lk lk l lk lk current label label value return converge return lk il fit valid identifiability stationary label skip empirically fast update result hand important substantial community divide degree design cope situation correct extra degree node likelihood conditional whether matter fit reflect observation multinomial node multinomial pseudo parameter obtain iteration update converge unconditional replace probability turn initialize note full function modularity modal arbitrary option interest way dimensional cluster degree work identifiable distribution pair laplacian diagonal collect absolute eigenvalue omit mean block poorly due many simple surprisingly belong add add matrix find empirically constant value rest forming mean perform need pt know act tx ax constant increase edge grow balanced extension unbalanced balanced naturally call node prove model section undirected block introduce direct adjacency adjacency matrix confusion introduce case adjacency draw loop edge analysis effect direct natural extension model practical direction link social model context direct edge restriction parametrization particular make comparable undirecte allow carry consistency undirected expect mild long condition satisfy soon sublinear growth start label label estimate obtain confusion either key analysis overlap amount overlap important naturally data small fraction community preliminary formally know match value community uniform consistency particular initial algorithm operating depend turn implement plug initial whereas label direct permutation account label community counterpart undirecte denote notion discuss function let adjacency probability consistent think hard balanced simple start update majority intuitively enough initial figure illustrate key grow primary may strong plug direct obtain maximum estimate estimate parameter local maximum truth even normality example undirected probability addition proof theorem condition meet choose recover label positively truth give move positively label vanishing extend unbalanced direct block supplementary initial include overlap initial balanced supplementary material directed investigate unconditional perturbation simulate scenario correct presence node conditional label probability draw enforce regular control determine relative degree degree degree relevant information diagonal otherwise overall expect mutual one think column sum r ij ij always reference matching dc perturbation unconditional likelihood dc initial outer specify figure smaller easy uniform moderate amount serve value pseudo perform poorly limitation scenario apart serve value show pseudo achieve gain poor start give surprisingly uninformative exception unconditional accommodate degree already room well initialize appear good value limitation effectively spectral sc take outperform dc might dc memory consume excellent complexity variety brief belief propagation different time scale rate slow constant bad likelihood political node focus manually conservative treat community direction analyze connected pseudo produce close expect degree result correct block unconditional put low unconditional fitting model correct model restrictive go go match estimated vice direct compression imply define normalization ie focus concerned slight abuse since case start classical bernstein sum ij b complete symmetry note variable rh see supplementary pick follow
I cholesky supervise variable greedy pursuit technique applicable candidate must start advance context propose include process unsupervised time error incur exceed sufficiently current contribution later ignore visualize add basis column k k ti remaining go overall long costly operation reduction forward augmentation exploit addition contribution try subset unnecessary ensure otherwise resource make necessity present square evaluation approximate automatic recursive represent underlie take reward tuple policy problem eq reward indicate problem state w tm h tm tm tm w fix r w derivative square state obtain eqs compactly k mm solve eq mm solve b recursively update currently update reconstruction contribute solution problem decrease add give threshold increase line least square propagate identity recursively compatible add second outline implementation unnecessary repetition equation minor symbol estimate far current cross weight execute observe loop execute simulate check basis additionally either step true currently consider least clarity matrix tm tm mm solution assume tm tm previous computation new observe update tm h tm tm tm step complexity k inverse reveal undesirable go away merely need access past example online tm h eq h already precede r show computed computation obtain via expression update need computation store square residual projecting span current inclusion contribute reduce well represent aside criterion usefulness usefulness take I together carry article use publicly procedure policy reflect transition function represent make actor maintain regularization actor use learn previous prediction evaluate propose ever grow irrespective reflect real amount computation function agent example complete evaluate actor iteration example policy state sometimes greedy recommend action pick greedy random among action observe differ associated value interval guide increase initial experiment show tuple employ suggest action take delta action state choose rbf length factor rl usefulness try effect supervise start unsupervised tolerance least reduction runtime simulation hour roughly hour pc try combination rl set report I scale rbf select basis function real time pc behavior interact agent roughly configuration run curve plot example additionally horizontal indicate code plot hand code consider latter considerable manual par second second code need discover supervise one basis function regard conjecture strongly depend I vs actor specific opposite observation least rl key point basis subset rl effectiveness benchmark overall rl possible superior grid fairly want function require considerable manual carefully specific manually suitable algorithm use online selection basis oppose reduce relevant policy evaluation bellman residual apply rl problem deterministic unified deal use respective demonstrate simulate pose dimensional control arm state rl vs deal state transition rl anonymous comment suggestion let action reward represent add element execute tm tm grow supervise normal tm tm grow tm tm tm z h tm r tm tm tm tm tm tm check reinforcement vs key like mean online require framework evaluation underlie family approximation supervise selection relevant indicate approach obtain square iteration widely accept common platform intelligence consider problem maintain ball region field team try gain reinforcement rl individually ball team team play dimensionality observe rl second noise agent imply environment need model server necessary learn successfully underlie fashion throughout state exponentially exploit knowledge various interact one also state simplify represent solution choose adapt try visit growth solve optimal control square policy ideally suit efficient paper formulation regressor subset select subset employ greedy candidate improvement supervise error incur kernel complexity ensure resource way learn necessity structure brief introduction describe derive recursive third problem long defer end regularization network dynamic time finite maker control set admissible distribution action sum reward maximize denote decision evaluate q discount reward choose proceeding accord discount policy policy greedy choosing achieve good obtain well trivial otherwise function obeys relation bellman solve linear transition reward situation infinite approximation unknown one figure depict parameterized mean practice approximate sound bound infer problem reinforcement carry approximation transition difference td initially prominent linearly parametrize td evaluation converge fast least base descent
control bellman lyapunov case highlight towards develop drive dynamical strategy give concept reproduce space nonlinear rkhs theory tool nonlinear system material define tc minimal show q lyapunov energy pair define make evident follow problem linear norm solve precede question give system define ellipsoid output compute stochastically force stochastically control system input give corresponding dimensional motion sde density evolution density differential operator refer context find covariance uniquely transition mean satisfie dt xx xx xx bb steady bb exactly iff fact steady suggest linear approximation obtain approximation equation explicit lyapunov find solution subsequently reduction system hamiltonian hilbert connection system define introduce rkh give brief reproduce hilbert heavily develop theory rkh hilbert inner product say reproduce kx kernel positive kernel reproduce unique x definite positive compact hilbert property rkhs oppose map symmetric symmetric operator practice typically rkh theorem guarantee reproduce rkh immediately nonlinear algorithm product instead look look eq inner reproduce variant implement rkh balance dynamical briefly however compute lyapunov equation directly prohibitive multiplication implement primal adjoint alternatively linear take simulation extend nonlinear proceed coordinate system orthonormal let response xt see matrix respective response finite interval assume measure response respectively matrix thought matrix observation nonlinear assume linearization origin observable stable rkh quantity map sample sample separately implicit rkh scale respectively operator w refer rkh version integrate refer empirical whose scale map similarly build express important index ill condition svd section introduce empirical energy control system estimate assumption suitable space reproduce rich capture strong space assumption set energy satisfy origin stable origin smooth equilibrium would solve explicitly solution simulate develop unknown nonlinear drive gaussian noise compact former balance approximately datum projection approximation event need impose precede overfitte limit q towards derive computable define operator adjoint associate operator mc check ms c cx dimensional column product let output kx vector product sample collect result kernel energy energy estimator consistent approximation making open make rkhs induce reproduce compact define rkhs furthermore space function separable introduce briefly recovery regular rkh introduce notation rkhs form control involve denote preliminary schmidt covariance reproduce hilbert schmidt establish consistency method adopt fix eq norm calculus make part adjoint expand orthonormal converge lastly iii part requirement fact enough e note precede estimate observable control definition energy function replace obtain sample general validation ellipsoid observable characterize definition ergodic nonlinear key provide great control nonlinear sde however system steady existence steady back formula solution often conservative balanced truncation context play key sde energy theory hilbert estimator measure aware technique rkh control systems parametric invariant density give behave map rkh may control energy measure system nonlinear system reasonably infinite sde dimensional rkh ergodic stochastically nonlinear value linear self adjoint wiener invariant along invariant absolutely respect proceed derive invariant interpret heuristic nature mild sde exist pg trace give condition zero back establish
orient detector different color encode orientation invariance encode orientation interpretation complementary pool feature return pool along pool across interpret strategy pooling bilinear visible structure course possible interaction architecture imagine overlap specify variable interaction believe overlap potentially consider topology exploit high possess building deep interaction intractable variable interact analytically resort approximation conditional standard choose minimize equivalently variational bind maximized take bind variational approximate factor fix depend three stage strategy parallel parallel maximize parameter evident parameter contain negative derive adopt strategy positive gibbs sampler variable outline training latent particularly learn motivate block wise organization structure allow remain local interact interact neighborhood procedure complexity interaction adapt block tractable super hidden interaction structure consequence relatively block extensive consider stack multiple factor local gradually toward local strongly attempt variation effort direction also variable multilinear critical attempt factor variation method interpret attempt extend subspace variation bilinear essentially model state factor factor element tensor thought generalization unsupervise consider separate develop algorithm demonstrate style font content later develop bilinear code additional observation factor develop multilinear simply compose together multilinear ica model face factor illumination orientation image identity people extract pose recent high interaction interaction boltzmann dramatically rely rank approximation employ highly connection interact structured norm interact interaction learn use ability variation synthetic fig compose object vary color color rbm negative bilinear factor color configuration fig bottom learn encoding color would enforce hidden generate perfect deep pool pool row tie evaluate rbm measuring usefulness evaluate usefulness use large spike spike pooling spike iii order arrange standard training performing unsupervise unlabele learn cross label various spike fold valid ss rbm number factor representation product representation representation form consider multiply factor outperform confirm hypothesis successfully learn low factored classifier deep achieve present extension spike restrict boltzmann factor three interact interaction act binary interpret previously future gradually stack architecture datum base high interaction see latent train regard latent datum generative involve complex interaction interaction challenge individual pixel showing expression well identity expression dominate image space seem appearance individual faces importantly interact combine easily affine often raw cope reality variation provide feature appropriate face capable effort cope movement computer vision toward engineering set common source motivation inclusion convolutional pooling induce pool powerful notion filter pool together purely unsupervised feature situation variation expense lose filter pool spike include high perspective interaction variable factor rise conversely see attempt interact factor combine generative unsupervised factor research direction information responsible principle organization learning train subset relevant consideration lead factor discard little information motivation behind spike htb g precision additional gate group extra subscript term visible model progress variation boltzmann spike boltzmann previously invariant pooling spike add variable spike variable constraint model
dynamic programming improve bound tight frequency call methodology represent simplify uncertainty attempt model uncertainty robust uncertainty mdps robust organized mdps function term optimization involve concentration use bound leverage exist mathematical programming benchmark offline paper generate advance function identical section concept require general symbol respectively symbol appropriate element part algebra expectation trivially action sa sake action avoid idea action denote state assign probability randomized policy map choose randomized pair ap ar value policy always approximate formulation mdps mdp solution discount remain state take action denote define value equivalent frequency action measure closely section characterize basic e part hold policy basic bellman frequencie greedy action maximize return respect tie mdp specific mdp solution derive tight bound transition also property bound concentration transition formalize describe analyze sampling simply derive identically programming subset return mdps simplify restrict policy minimize loss return follow rely value simultaneously restrict policy approximate frequency element represent combination column approximation definition follow without reference feature follow show importance suppose represent assume ready paper gain motivate primal slack formulation mdp value upper approximate penalty return invertible ignore formulation particularly frequency equivalently define use representation assume duality show combine separately remainder assume policy generalize state solve program suppose imply equivalent bound readily show subtracting factor involve undesirable rely purely derive follow admissible suppose tight policy mdps hold derivation assumption inequality bound rely require feasible conservative solution well solve np easy practice np completeness feature action mdp favorable property solve mixed formulation solver develop formulation bs computed policy restrict assume take state policy represent linear program np suppose solution deterministic greedy respect equivalence optimal dual separately contradiction solution treat low optimality positivity th element q trivially feasible optimal choose bilinear separable bilinear formulate generic impractical relaxation instead existence bilinear upper optimal assume simplify individual whenever show discuss practical must bilinear similarly identically sec sample estimate reward nonzero zero action section experimentally synthetic chain three offline feature unbounded restrict balance cart either action represent force cart uniform govern differential benchmark arrange grid center term require balancing step average run figure show bar indicate compute poor feature space decrease become conservative optimization become quality factor good single benchmark generalize chance moving feature orthogonal polynomial show randomly outperform propose base formulation significantly strong theoretical guarantee improve solution additional state small encouraging mdps discussion inspire anonymous detailed comment greedy com programming popular approximate curse turn derive analyze theoretical translate performance problem reinforcement programming many study empirical many must carefully tune offer insufficient
supremum depend geometry upper specific algorithm prove adaptive query function class large infimum oracle supremum take depend geometry independent turn applie typical additive gaussian convergence bad dependence restrictive setting function polynomial focus feedback build idea optimization obtain question fundamentally somewhat describe convex point probe plus change time noise one theorem markovian consider priori belong simple easy original minimum query require desire minimized associated query let p kullback leibler ip take simply refer explicitly semi point specify select select define minimum fx proof ham denote element elementary radius ix ix jx differently bound differently accord oracle comparison markovian assumption conditionally conditional note underlie kl bernoulli f f ready apply want choose side definition satisfie apply since use slightly different unit define c iy ix exactly bound section recall consideration corresponding evaluation underlie algorithm depend kl eq compute claim attain oracle pick uniformly exploit guarantee expectation oracle error approximate line accomplish however confident comparison sketch leave supplementary material define choose random zero except analyze direction coordinate sphere line guarantee fx q last gradient lipschitz take respect strongly define lead within comparison pairwise comparison q wish concern minimize wish make operate maintain iterate iterate regardless away close fast gradient let make pairwise algorithm subset also rely heavily unclear assumption bound suffer contribution relate oracle bind x fy fy random fy fx fy fx density equal true unimodal laplace side distribute fy fy fy loose pairwise oracle summarize pick uniformly exploit gradient argument decrease make line accomplish binary line probably correct robust repeating query confident pairwise comparison active line position query estimate pairwise comparison taylor hand convex minimize q recall chosen eq respect law iterate unique minimizer convexity calculation complete proof wish fx pairwise concerned minimize single variable indexing search produce indexing index present assume comparison noise iteration eliminate iteration modification great fraction removal search provide fx fx initial estimate straightforward fx k side fx fx bring request straightforward terminate pairwise loop argument bring request comparison oracle iterate bring comparison error repeatedly comparison active ability adapt unknown mention way want implement efficient subroutine fx fx fy identify request pairwise convenient could result impossible low lemma algorithm ensure see algorithm begin loop repeatedly compare soon repeat sampling terminate strategy proceed noiseless location loop compare repeatedly noiseless noiseless iteration iteration noiseless algorithm unlike request distance evaluation small distance probe consider bad comparison straightforward k subroutine subroutine request call subroutine must union comparison search bring call query subroutine n rate remark definition university usa usa lower rate free noisy gradient situation algorithm comparison evaluation wider pair comparison example show boolean value optimize large tuning evaluate unclear influence thus free rate noiseless method equivalence gradient otherwise function function contrast scale strongly rate evaluation new boolean comparison decide interesting subject pair comparison convergence ambient new achieve formalize strongly convex gradient gradient define minimum query way optimization
n quantitative measure noise ratio widely image learn similarly slightly well interpolation measure purpose perform amazon http www com average win e g bp vs amazon web interface follow people experiment intelligence etc reliability explain measure reliability sampler train online vb super base superior natural ask visually assess reconstruction likely bp vs decision quality ground algorithmic fail select number vb iteration see vb complete mini reach optima vb batch vb find may local optima mini hour gibb burn collect sample distribution take approximately matlab implementation collecting reduce hold db finding super nonparametric training scale code traditional evaluation variational case slightly human speed important online regard analysis ratio capture run human show necessarily build time series generate process super via bilinear n lemma proposition super resolution resolution super beta element determine benchmark natural human assess visual gibbs sampling approximate large datum bayes dictionary need factor variational stochastic optimization resolution denoise compressive sense datum image represent combination pattern may beneficial build accurate superior denoising model derive efficient image sr lr surveillance imaging variety super general possible solution regularization researcher machine train ground image relationship reconstruct image lr patch use neighbor search resolve propose kernel regression another sr algorithms texture match know image super one use scale image via coding regularize dictionary element across position specify dictionary variance difficult assess provide nonparametric method latent provide infer otherwise posterior dictionary many image architecture segmentation multiscale representation paper super feasible super algorithm latter approximate enable evaluation quantitative success analysis necessarily devise human evaluation nonparametric model sr variance couple devise scale experiment assess sr give scalable describe super resolution parametric section present detail use natural analysis denoise model learn e patch resolution build factor analysis lr lr perform super correspond detailed forming depict preprocesse extraction first weighting function interpolation sized patch location lr couple train model element loading local use dictionary resolution lr dictionary dimensionality resolution element use resolution dictionary produce key property frame super inference assignment mean distribution binary encode element activate elaborate place gamma weight level share represent element represent patch illustrate sr phase dictionary lr expectation reconstruct lr distribution weight patch inference discuss assignment use bp binary ibp encodes activate one correspond distinguish characteristic specify priori condition binary ibp restaurant infinite try dictionary element ibp customer enter restaurant sequentially choose first beginning take th customer begin proportion popularity previous customer quantify took consider customer choose record indicate infinite indicate joint final customer restaurant exchangeability observation bernoulli trial draw beta ki observation represent usage inference scalar tend bernoulli approach ibp enough exhibit component pair super lr training variable computation posterior dictionary image pair rewrite couple writing fully reveal dictionary single scale amount posterior patch dictionary lead share dictionary lr reconstruct patches weight score lr image patch dictionary fit determine strength control score precisely set hold patch posterior dictionary step estimate lr stage determine patch correspond wise overlap reconstruction match lr image solve anti follow sampling prior conjugate exponential gibbs observation equation provide material sampler patch sample develop alternative sr scale streaming inference mcmc replace family minimize track see typical variational coordinate iteratively hold sampling replace subsample datum adjust subsample rather scale exploit develop ascent algorithm derive variational easily perspective bayes base field learn become factorize govern optimize observation divergence maximize bind variational dimensionality twice big scale function hold parameter go algorithmic online update parameter identity matrix q variational parametrize ascent update ascent equation variational coordinate free precision gamma variational variational gamma coordinate ascent update variational use nk compute k update develop divide variational image weight optimize per optimize find basic repeatedly satisfy optimum optimum sn new vb noisy estimate write uniformly sample thus noisy channel right element pair locate consist subsample variational sample local setting parameter variational copy image local equivalent fit cost optimize global versus ascent parameter early speed global full online vb mini mini initialize vb useful help begin recent near draw whereas algorithm anneal jump optimum train web berkeley image image various community evaluate sr
empty bin respectively lot careful basically process jointly really want available fact denominator normalize become equivalent set highly unbalanced expect would significantly reduce occurrence bin mention one conduct permutation long bottleneck use permutation chance empty bin decrease increase already review strategy integrate hashing modification bin sec bin e empty zero strategy procedure matrix depend empty bin th believe deal empty bin convenient produce artificial expand always long preserve random slightly accuracy permutation scheme robustness permutation news interestingly code perform extremely accuracy comparison random strategy actually original hashing provide scheme testing convenience logistic small permutation scheme one outperform achieve accuracy permutation reach compare bin code course news h merely test one strategy application dataset difference permutation scheme permutation difference length understand would connect count min sketch length divide evenly bin scheme grouping entry bin permutation connect min cm sketch count cm estimate remove subtracting sketch idea bias multiply whose probability recent show sketch equivalent projection hashing conduct showed believe fix scheme difference major bin empty actually bind bin difference ff possible potentially perhaps variable massive hashing algorithm require accuracy preprocesse hash context estimate similarity binary text method hashing linear learn logistic sublinear near drawback hashing require time expensive e process true preprocessing come efficient conceptually massive evenly bin reveal permutation hash scheme accurate replacement scheme test robustness also merely outperform news summary merely preprocesse efficiently bit hash low bit near search text major drawback hash hash expensive permutation text binary extremely dimension page contiguous mean need substantially increase english word super word practice total dimensionality convenience occur document note image vision either feature dimensional example distinct dataset potentially hashing design equivalently two measure inner product compute prohibitive consumption propose efficient computing permutation permutation probability permutation binary dimension q training regression store low bit dimensionality expand idea million achieve original building hash table sublinear sec example inner product permutation preprocesse hash loading dataset sample format take format second preprocesse document process preprocesse cost hashing nonzero use view vector divide evenly bin vector divide evenly bin small bin empty bin rarely concern set bin bin obtain bin representation example indexing original take modern show empty occur reduce permutation permutation much efficient perspective energy highly especially software implementation process permutation may meet interactive easy number generation cost gb realistic resort universal hashing permutation universal theoretically always hash much easier generate permutation one hash one permutation nonzero datum verify work scheme name basically sample estimation however align unified fashion scheme evenly nonzero align interestingly hash permutation author realize hence sublinear hash table product bit hashing projection work show use extremely sparse small merely projection scheme convenient count min sketch hashing briefly review original permutation large search numerous etc research modern practice often fall general framework lsh performance lsh solely underlie idea directly bit hash hash near sublinear scan permutation bit signature create point signature match phrase permutation bit signature many neighbor use permutation hash query retrieve table example permutation table hash permutation store bit signature low bit signature place apply signature become signature search near neighbor much confirm strategy perform logistic popular package sgd give regularize logistic solve regularization solve permutation binary vector store lowest merely run expand exactly illustrate digit expand fed permutation apply zero never permutation hash encode empty bin strategy later refer confirm search learn permutation figure practical although occur rarely unless vector interesting probability analysis provide rigorous foundation consider empty bin bin define bin see appendix binomial analog scenario often sparse lemma bind approximation expect probably empty significantly impact practical assumption datum distribution q en n q appendix confirm lemma eq show good see exact already derive seem note always original completely empty zero var see appendix probably scheme may slightly merely preprocesse replacement essentially vector appear web l contact web university business book job thorough figure result base repetition verify theoretical curve empty practically hash substantially would usually see strategy bin
applicability complex secondly modular graph simple graph consist node link node link conditional variable concentration determine independence mle concentration modern application observe observation thousand product genetic many behave term point involve trade cost benefit e error error incorrect subset forward backward procedure select predictor regression forward extensively adapt constrain precision penalize solve estimate absolute scad penalize approach alternative base theoretical property penalize derive well establish tend impose symmetry precision matrix reduce introduce symmetry important likelihood restrictive particularly need relatively paper symmetry modelling impose precision model useful random come use idea precision motivating time microarray cell experiment introduce graph factorial correlation detail problem address graphical aic bic adapt factorial encourage simulation edu system understand interaction protein gene develop collect gene concentration rna produce snapshot expression temporal evolve dynamically internal genomic continuously expression response thereby flow genetic dna snapshot profile reveal determine differentially microarray determine functional order infer interaction gene expression aim variable relationship variable motivate describe describe find describe item collect gene level analysis cell produce cell cell line laboratory reach collect hour contain investigation complementary second microarray experiment array investigation compose collect cell cell another set cell assume technical replicate independent replicate replicate two dataset replicate sampling conclusion cell step conduct obtain across firstly normalization systematic due describe dimensional subsection variance illustrative across compute select base partition matrix square matrix summation interpret variance predict behaviour lag self self represent conditional dependent give gene temporal lag cell gene gene lag element last partition graph lag concentration replicate gene across gene measure gene lower read gene education longitudinal teacher across study conduct student secondary school student illustrative purpose control influence proximity score behaviour score measure teacher compatible correlation class student show triangular covariance upper score measure experiment four measure illustration subject prox prox prox empirical matrix precision mean divide root range measure equivalent correlation pure influence indicate independence conditional dependency conditional relation undirecte correspond set gaussian vertex dataset whereas course dataset establish gene measure collect hour hour speak vertex order natural order convenience notation dynamic graph assume conditionally interaction term equation different type third represent line refer dynamic order measure cell gene across change correlation diagonal moreover think time self self across two graph triplet link induce partition type line line colour continuous line represent colour dot colour let partition indicate stand subset vertex partition partition consider lag graph time lag induce firstly indicate colour secondly edge colour across indicate differently tv tv j l f substitute specific natural induce partition I v jt jt ic g partition calculate colour edge resemble eq firstly follow natural partition create couple couple simplify secondly induce subset colour create bring graphical model graph denote ij tv ij couple satisfy property relation undirected connect factorial dynamic factorial triplet natural partition n factorial correlation factorial figure variable f sub partition natural impose design connect element concentration partition associate uniquely mn self partition induce parametrization induce specific equal restriction follow factorial description n diagonal empty empty estimate replicate induce subsection colour time gene generally comparable scale interpretable conditional matrix rescaling show rescale scale precision rescale correlation rescale e factorial structured restrict diagonal partial identical colour class structure graphical model conditional obtain conditional element conditional correlation constrain follow factorial estimate conditional element conditional triangular correlation equal triangular element partition partition total note I c c subject prox prox prox reach considerable interpret vertex connect graph tend density follow edge density could test selection parameter would bring instability penalty concentration induce convex hence variable hence feasible design produce operator necessary impose linear every induce describe order p frobenius want sparsity assumption absolute constraint equality introduce slack variable equal diagonal penalize constraint impose structured priori minimize e standard determinant allow impose factorial optimization solution employ proximal conjugate develop matlab linear graphical package analysis open source code package virtual connection within apply algorithm penalize restriction example represent constraint constraint guess code consider smoothing parameter framework several factorial choose well find false reach stability selection stability bootstrap idea smooth aic bic compare factorial factorial degree number zero e element partition chance fitting smoothing
asymptotically assumption description set obtain discrimination limitation besides discrimination generalization discriminant analysis analysis random help system decade wide speech music recognition asymptotic optimality classical actually go scatter covariance scatter projection thank expect acceptable asymptotic bayes optimality limitation infinity provide quantitative especially address superior increase dimensionality second examine respect w mild discrimination insight severe ability population power eliminate influence influence proportional worth result besides discrimination acceptable estimate theory main understanding draw originally motivated nuclear physics mathematic engineering communication law spectral wishart almost almost extreme singular random formulate proposition whose independently e surely deterministic letting sample notation bold letter space matrix compose first column sphere locate th sort descent denote denote column space distribution class follow scatter class nonzero eigenvector obtain fisher discrimination reduce scatter denote generalization discrimination training infinity power population counterpart however increase infinity discrimination dnn regard linear reduction loss generality eq cumulative cdf replace population counterpart error give discrimination dnn dd unit length sphere independent eigenvector v td estimation ready main total recognition handwritten digits handwritten digit dataset dataset get population discrimination power different discrimination horizontal population minus discrimination properly bound hold high dataset major considerably small problem distribution ratio generalization properly randomly select perform evaluation set classifier ratio result result generalization upper dominate lemma simultaneous theorem semidefinite clearly ii eigenvector must v submatrix u imply therefore I fact arbitrary nonnegative rescale rewrite unit independent unit must unit uniformly sphere give proof distribution wherein fact invariant orthogonal similarity transformation complete statement converge spectral distribution f complete uniformly distribute entry sample uniformly number divide side q first argument know r third nonzero examine hold eq let partial denote integral substitute thus proof lemma scatter wherein e c n x e ei tc q proposition
vector term example list chain loop count singleton singleton chain singleton singleton third singleton combine tail work normal scatter tail find spectral wishart distribution chi square individually collective eigenvalue top eigenvalue generating supremum b eq eq adjoint dimension bernstein semi definite q
decide lower decide similarly full recovery may follow problem page least coincide apply coordinate j detect ideal choice since elementary omit diagram popular include g expensive local way graphical structure guide utilize setting set tuning convenient second easy preliminary situation believe conduct long respectively convenience pe depend tune minimum strength tune insensitive point bivariate high screening use order screening memory need need change choose degree performance much slow case identify first p j case change pe take threshold take depend scad shape mc penalty integer take tune lasso scad ideally depend slightly inferior scad mc comparison fair unclear tuning mc range report average hamming consistently especially three lasso scad perform error conclusion save lasso scad model cd scad scad mc detail p bayesian report ham independent repetition use well cd consider post filtering model simplicity mean penalization post filter end show distance loss difference soft naive parameter theoretically optimal signal equal case address gram graph insight post filter live case stage clean candidate stage overcome employ technique develop focus regime rare achieve rate wide situation limited mc selector well even successfully problem time case covariance screen effective marginal closely relate attempt optimality recent work different motivated compressive sense genetic largely dna variation long series different ill pose filter paper study memory time change problem reason primarily broad coordinate without continue adapt validation post correspondingly modify tune slightly tf pe pe core could long series result largely many rare weak two main long live small isolated connect focus change post decay extend setting post matrix modify accordingly length still extension necessary specific pe paper closely recent dna save leave notion mean understanding operation compressive genome wide sense already motivate change analysis operation try study along apply design gram future acknowledgment david lemma condition example support nsf grant dms sciences national grants gm gm support part award dms major article complete engineering model sparse main separate nonzero fan primarily gram finite focus regime screening induce graph conduct interact decompose separate filter newly introduce give stage fan song candidate remove positive selection measure minimax hamming situation gram successfully apply memory change nonzero nonzero one deviation know generality gram eq often difference signal rare gram ill pose challenge signal well concept signal important largely focus regime signal scientific signal hard find easy partially explain publish relatively element filter removal gram play role application area drive dna copy receive exist major interest piece wise jump relatively parameter jump gram adjacent row display adjacent observed process exchange rate reformulate gram say toeplitz matrix norm gram matrix adjacent row example jump asset example sum example adjacent rank removal rare variable also investigate reveal give consider mild equation fact full rank solution see spirit ground look motivate penalization look noiseless penalization fundamentally intractable decade limit lasso mc penalization method four rare strong recovery penalization fundamentally unfortunately rare first fundamental phenomenon indistinguishable signal weak principle hamming distance appropriate fundamentally signal rare example rare regime tuning penalization detail penalization method imply penalization scad propose show achieve condition clean heart marginal face call relatively see effect motivate application example linear matrix small penalization motivate new multiplying one largely exploit free cause serious issue post filter regular regression model face helpful procedure correlation incorporate essentially sparsity deal cause innovation create rich graphical structure lasso heart screen reduction big date extend model major variate screen remove noise exclude result strong screening overcome challenge covariance screening screen retain almost signal construct connect exclude sub generic screening efficient fundamentally regime sufficiently setting discuss component theoretic three fold develop theoretic framework rare asymptotic minimax partition call criterion assess procedure phase diagram space different diagram rare challenge ill regime practical perspective paper rare regime term distance find tight procedure exact entail derive diagram section need usually associate especially challenging introduce optimality simulation denote contrast hamming shall symmetric arrange method introduce rare weak signal model introduce start filter explain filtering help interesting interaction cause call discuss section discuss complexity feasible asymptotic framework hamming function hamming hamming section apply explicit convergent formula derive diagram section find primary interest weak signal location randomly primarily relatively signal rare weak rare rw remark theory develop tie rare signal extend signal g ising mention primarily gram view reason necessarily filter say memory time change point induce strong filter eq lose reduce matrix h compare easy track naturally motivate three call definition parameter choice sparse exceed unclear sparsity help variable sparse path unclear remove try surprisingly answer lie sparsity support subgraph justify support realization p case many know component one view separate subproblem insight attribute sparsity latter attribute linear filtering tie broad ise largely filter induce discuss linear cause frequently denote form restrict row denote sub form vector explain matrix correlation continue must filter moreover di definition compare imagine fully never validate phenomenon content neighborhood add di capture fisher matrix eq orthonormal negligible phenomenon fisher substantially patch fisher information model although ratio cause case screen information technique summary filter q observe induce sparsity construct signal cause information overcome selection pe candidate signal carefully achieve candidate remove false positive step integer ps ps pe pe step suppose connected denote arrange size tie break toy example subgraph triplet sequentially decide test screen bivariate screening variable examine screen new far additional similarly never variable clearly multivariate screening exclude formally retain stage convention sub step th set fix tuning node node step random thought follow update node td terminate write accept sequentially accept stay course could spirit regression use collection threshold threshold test central circumstance arise appropriate misspecification alternative continue help provide ss property say subgraph subgraph expand introduce ss enable illustrate retain index example retain sure false positive compare reduce original parallel one node view expand recall retain end step graph subgraph component fix subgraph subgraph fix nonzero pe pe pe pe dense address moderate select fast graphical challenge desirable adapt variate desire coordinate test exhaustive force fashion marginal exclude obtain innovation variate subgraph connect subgraph great display variate tuple total tuple variate tuple subgraph subgraph step consist part obtaining degree exceed solve context exceed provide properly study therefore conservative computational complexity section set show term hamming distance section memory time add rare driving increasingly successful impossible relatively critical recalling fix rare weak strength signal reason remove future view prefer properly large neither adapt gram small small sub subset v constraints elementary omit discuss broad minimax risk selection hamming expectation take vector denote correspondingly hamming hamming minimax hamming long hold many assess hamming assess optimality study demand local hamming hamming bound error fix hamming risk geodesic local hamming characterize th exponent notation frequently multi provide index aggregation local low hamming toward recall favorable theorem similar r p p gd example include primary eq section broad properly case prescribe lb achieve focus right side lemma lb hold root constant adjacent satisfied mild condition number series certain singular value relax gram belong nevertheless slight continue use tune pe unless signal sufficiently constant principle depend unknown suggest choice relatively flexible section theorem appropriately subset
dataset report report naturally cast estimation group causality see reference therein dimensional infer adjacency unknown implie give operational causality node past series link causal causality insight knowledge selective make technique focus involve nonlinear appropriate problem l set component index history belong vector set dictionary I sum eqn identify subset node dynamic structure induce note eqn recover temporal output associate source dependency causal life simultaneously stage extract group gene group represent gene insight within gene conduct four dictionary kernel output show four functional input nonlinear predictive imply great imply causal graph analyze approach centrality key activity recognize contrast relate bind bind factor bind activity consistent biological knowledge capture connect protein bind successfully fail recognize relevance likely link suggested estimation provide insight show protein bind estimation separable predefine output optimize coordinate descent method computationally costly work scalar eigen gauss solve system quadratic solver inexact provide rademacher value analogous scalar multiple learn literature also causality high class functional theorem york usa h technology c research new usa lemma proposition framework high dimensional allow broad regularizer induce reproduce inexact solver simultaneously framework high causal cast lead graphical development analysis endow suitable reproduce formalism value far yet value extension popularity responsible complicated value turn polynomial default require associate greater value solver cubic multiply scalable learning become necessity broad family regularizer induce serve optimally combine value kernel algorithm simplicity separable kernel definite formally define full kernel estimate effort one address scalability carefully inexact solver subproblem value hypothesis extend result case enable causality believe state main body multivariate amongst naturally motivate algorithm treat independently reproduce rkh attention value challenging selection problem certainly scalar case contribution kernel technical element exist smoothness select elastic net penalty conjunction square scalable block gradient fista learning solver overall turn multiple classic var traditionally time vector regularize forecasting collection time small induce subset endow causality interpretation benchmark least follow eq reader value rkhs principle overview supplementary value evaluate input nn sense pair solution dimensional sized gram comprise ji identity compatible development familiar general linear value improve though costly scalar comparable input matrix semi definite system eqn eqn solver solver eqn method e solve completeness though point cubic solver diag diag output recent elegant extension eqn separable kernel correspondingly language denote trace denote frobenius denote cone stationary globally propose block fix show large dataset take day standard corresponding vector reproduce reproduce f f j eqn eqn joint scale denote gram individual scalar version reformulate strategy minimization keep minimization termination return eqn subproblem conjugate solution eqn dense prohibitive repeatedly outer cg solver massive cg advantage special kronecker eqn warm previous termination never cg q cg fast matrix fourier l never explicitly cg rapid iteration strong jointly trace eqn possible formalize convergence cg solver current iterate warm start dictionary gaussian scalar strength give material weight value oppose essentially reason mkl immediately close j elastic net type penalty also discussion regularizer structured sdp solver general differentiable successfully completion iteration optimize linearization result eq curvature eigenvector small eigenvalue appeal iteration note unit trace meaningful optimize gradient diag matrix search search direction k adapt eqn iterate g additional small block descent convergent discuss chapter proposition offer rate associate inexact subproblem quickly stock multivariate output first autoregressive var week eqn
handle work particular stochastic obey admm value quite online refer concerned design stochastic admm modification stochastic class function might directly update nonsmooth deterministic call solver consume contribution algorithm structural good feasibility contrast differentiable notation deterministic simplicity clarity denote stack vector tuple semidefinite vector product euclidean ambiguity often assume quantity frequently present algorithm list statement nontrivial constraint augment lagrangian minimize augment least easy alg gauss minimizing alternatively follow lagrangian initialize k variant name linearize admm alg semidefinite alg linearize proximal point note linearization help line alg close solution alg share linearize one line replace approximation mirror stochastic nonsmooth smooth linearize prox scale stepsize stepsize admm alg exhibit convergence first variation utilize probability adapt place able obtain deterministic admm exist literature finding aspect admm prove feasibility admm direct reach convex convex take alg assumption b magnitude scenario function rate besides assumption admm applicable broad subproblem minimize augment structural convex nesterov help achieve rate lemma divergence prox convex scalar bregman proof convexity definition alg eq handle last condition alg convexity alg inequality result operator derive feasible second k k apply therefore order claim adopt implement follow strong last bound
lie subspace might apply vanish limit become utilize input vector small small approximation superior example kernel hilbert semidefinite allow empirical imbalance replace weighted trace weight task theorem hold reduce equal rademacher average sum understand number benefit transfer give excess use risk lot observe approximated assumption see reference therein somewhat consider column marginal without know unclear convert reveal learn sparsity derive consider regularizer closely fall short introduce general elegant regularizer apply dominant read observe empirical task picture expectation convert dominant eq observation different disadvantage simultaneous letter real space map adjoint call unit bound compact finite linear map compact call adjoint nonnegative self adjoint cone operator linear nonnegative operator adjoint number ie operator eigenvector sequel self adjoint eigenvector adjoint operator eigenvalue whenever monotonicity use operator coincide invariant complement without map compact theorem apply represent restrict subspace tr tr x ax ax ax ax e basis case independent random satisfy q k r get kk pass eigenvalue combine inequality prove conclusion follow give slightly almost iii q previous subspace contain range realization contrast span argument lemma show rademacher span contain distribute k implication together jensen divide elementary algebra q line lemma iii excess risk thus total map inequality definition third hoeffding technique bound q q rademacher eliminate loss b tr ss operator old jensen tn v v td q central range expectation verify integer occur rademacher imply cauchy schwarz j b x nj b conclusion since b q together use jensen bounding term help subspace range rank invoke jensen lb together theorem algorithm axiom conclusion condition conjecture summary em height depth ne ne pt pt skip h ne abstract chapter head head head head head head ex compatibility conjecture compatibility compatibility compatibility compatibility false false mu size false mu mu mu th mu mu end align align end end end end end align environment environment environment style array allow false true tag tag true department university college popular excess explicit independent space infinite sum semidefinite fundamental limitation incur sample potential multi sample size small collection sample individual sample machine try theoretical preferable seminal bound trace time consider input probability pair observe risk minimization model z assumption remove replace attribute intuitive
hyperplane separate r coordinate xu coordinate coordinate yu xu xu coordinate away coordinate xu yu yu xu xu yu cycle baseline coordinate xu coordinate yu intersection coordinate away intersection xu xu coordinate away away cycle partition right model fair coin elementary gamble indicator gamble gamble gamble second gamble gamble ff assess acceptable assessment interpretation gamble ff acceptable decompose decompose wish minimal happen happen whenever intersect sided gamble gamble reformulate intersect acceptable cf proposition interpretation point interpretation restriction agent might make second background face assessment answer assessment coherent lead compatible coherent put correspondence remainder devote answer question relation interpretation turns uniquely associate coherent dominate coherent maximal uniquely mean background model compatible whole assessment gamble uniformly dominate gamble use equally subset interesting arise discuss well treat coherent real price acceptable gamble see infimum price conjugate linear conjugate coherent subset gamble ff weak thing previous concern compatible carry satisfy assessment associate criterion avoid confusion assessment purely statement model see notice abstraction purely statement statement still know direction acceptable price translate background correspond preserve resolve add pt scale coordinate yu intersection xu xu yu coordinate yu xu yu coordinate intersection away xu away cycle coordinate xu yu xu xu yu yu xu yu intersection r away coordinate away statement never less resolve coherent reject gamble relate gamble set back tool lower dominate natural extension coherent incoherent avoid sure gamble encode gamble possibility property assume composition positive gamble associate experiment agent transformation rise gamble assessment avoid confusion e distinction background cf gamble gamble ft background gamble status gamble special obtain overview idea paper exchangeability f linear invariant gamble gamble sequence state gamble exchangeability mention finite sequence atom turn expectation extend infinite started allow character directly accept statement preference counterpart even input transform reasonable requirement whether contain impossible basic numerous practical tool inference core need space large part deriving formulate expect bit essentially carry literature gamble build framework know closed set closure deduce polytope theoretic duality know know topological object able illustrative acceptable gamble gamble empty restriction exposition possible formulate counterpart proof may dominate existence finish pair coherent generalise scale take probability also framework aspect correspond accept thank version paper discussion three confusion mu equivalent assume equivalently former gamble change claim linear difference positively contradiction apply claim one q latter lemma scalar hull union latter equivalent effectively former prove lemma identity c give hand side extension follow corollary claim confusion proposition us inequality fact cone positively scale intersection g g cone use claim show confusion preserve non empty intersection empty still assessment set acceptable gamble family note mu maximal closure proposition prove dominate intersection closure operator nature dominate claim dominate dominate mu assessment dominate confusion mu mu onto wise cone mu gamble already furthermore mu proposition direction separately definition f know thank closure acceptable assessment confusion trivially satisfied equivalent therefore whereby confusion whereby confusion becomes I complete equivalent f equivalent f f f scaling reasoning coincide mean g g f section theorem proof preserve empty intersection empty prove equality claim preserve mu mu lemma guarantee axiom dimensional necessity proposition preserve lemma operator cone axiom maximal proposition imply closure proposition follow condition counterpart lemma side mu mu infer existence equality contradiction thank proposition verify confusion thank proposition accept accept gamble avoid lemma repeat add write condition state lemma explicitly go calculate claim lead theorem final identical converse equivalent theorem change substitution equality pp mean confusion contradiction coherent gamble intersect gamble linearity pf pf p assessment confusion positively scale remove cone fact pg ff create confusion proof avoid confusion avoid confusion avoid confusion equivalent prove expression follow distinction separately prove coherence gain proposition homogeneity inclusion separately reject left hand element f f coherent homogeneity f ff prove claim sequentially explicit derivation closure topology claim ff equivalent coherence tell confusion empty empty section reject uncertainty reject preference inner outer fill thick dash ex em axiom modelling accept reject gamble preference relation bridge simplify variant provide modelling statement reject preference tool deal lack phenomenon quantification reasoning uncertainty assessment event conclusion optimal mathematical framework around gamble value uncertain acceptable reject one speak gamble expectation expectation reject framework start consist gamble expert restrict unconditional theory utility express uncertainty gamble useful application arise symmetry character strength type express course expressive already mention representation far work event gamble closely speak gamble expectation experience working gamble attention theory simplify application basic bring centre focus develop uncertainty statement representation gamble incorporate much work theory specify probability expectation introduction concept give clear interpretation available unique precise possesse deal probability correspond preference order event gamble probability gamble may whose partial basic towards sometimes strict preference relation associate preference neither basic separately way preference axiom order preference gamble coherent uncertainty set gamble although use representation exposition prominent set acceptable keep type gamble present strict gamble discuss contribution propose axiom theory add allow strict preference axiom define axiom associate axiom call axiom avoid sure extension probability natural extension school want mathematical applicable analogously g opinion transform discuss construct goal basic conceptual set mathematical course useful conclude investigation section small section effectively symmetry improve give description nature gamble subsequent main criterion confusion consequence scale gamble whose reject axiom extend assessment full incorporate requirement axiom conceptual central assessment obtain rise answer question perform resolve encode accept contain answer important gamble section framework important assume accept shift away framework work relationship coherent probabilistic reader preference framework translation concept relation application theory simplified preserve express separate preference gamble encounter outcome construct exclusive outcome guarantee possibility apart axiom infinite space topology intuition finite formally name pay uncertain outcome pay express precise gamble gamble multiplication constitute space origin agent gamble coordinate xu fy yu fy possibility gamble format example fundamentally negative connect exist concept prove convenient concern operation gamble h secondly gamble value gamble associate subset section object axiom rise room priori agent ask gamble remain agent pay statement statement reject agent force lack experiment pay rejection gamble acceptable set similarly denote xu coordinate yu coordinate xu yu xu coordinate coordinate yu finite format accept gamble reject one extend discussion accept reject gamble thing indicate choice reject gamble mean loss base category accept reject reject accept reject gamble accept reject similarly non acceptable gamble mean rejection confusion avoid axiom confusion confusion although confusion ideally investigate remove confusion number confusion remove gamble accept gamble gamble statement gamble either accept reject assessment wise interpretation give statement agent gamble terminology moreover confusion course gamble neither acceptable operational gamble illustrate later statement make gamble gamble gamble gamble gamble though call accept determined pay express linear precise utility acceptance statement generate acceptable positive utility reject impact axiom dc express ps g extension never remove statement remove confusion assessment assessment mu feasibility assessment finite statement feasible reduce plain example possible remove closed flexibility modify suggest ensure assessment still confusion remove gamble gamble accept take formally gamble part cone either cone couple close gamble acceptable status gamble set cone cone solve optimisation decomposition sum like use complement space sum r yu intersection xu xu yu intersection yu yu cycle coordinate yu intersection xu xu xu yu background xu xu yu yu yu away away away cycle apply three right grey gamble include apparent work gamble amount thing close close necessarily general e even non gamble affect reject remove confusion close one remove rejection statement closure impact apparent I make assessment interested increase confusion close assessment exhibit increase confusion close assessment confusion gamble gamble reject yet thing reject confusion avoid tell closure gamble gamble consider acceptable confusion statement assessment satisfy confusion hold simplify closed confusion f start assessment gamble lead axiom definition increase create closed mean assessment close avoid confusion extension without assessment confusion without confusion result assessment assessment confusion wants remove confusion case confusion confusion set reject gamble gamble something deal something confusion coordinate xu yu background xu yu coordinate yu xu coordinate away xu xu yu coordinate yu xu yu away away xu coordinate yu background intersection xu xu yu away yu xu yu r away away cycle intersection xu coordinate xu yu yu xu yu intersection xu coordinate away extension depict grey gamble dash line example reject gamble agent gamble certain small arguably last gamble abstraction gamble visible accept gamble gamble acceptable indicate gamble example ray ray ray gamble reject gamble support material pair wise inclusion say former component resolve terminology gamble gamble exploit provide terminology component assessment acceptable gamble effectively transaction assessment gamble include restrictive restrict potential scale xu coordinate coordinate yu xu yu background xu xu yu intersection xu away intersection yu coordinate away r intersection coordinate yu xu yu intersection xu xu yu away yu xu yu away intersection xu xu yu away intersection yu yu away r away away cycle coordinate xu coordinate yu xu yu xu xu yu yu away away yu cycle r away xu coordinate cycle away illustrate neither right consequence one union operator play role infimum north xshift south west right ex east yshift west rectangle yshift east leave assessment confusion gamble nine class whether gamble acceptable illustrate label gamble visible gamble gamble symmetry proposition maximal gamble accept reject maximal class empty pt sep ex outer sep ex sep ap xshift ex east xshift ex south west yshift west rectangle yshift ex east xshift ex south west xshift ex north east yshift ex west north east node shift west rectangle shift south regular side outer ex ex sep ap rp east xshift ex south west node right yshift south west rectangle yshift north xshift south west xshift yshift west rectangle yshift north east shift ex ex north south east background stay empty useful assessment category encounter gamble invariant symmetry second show gamble maximal summary concept notation quick reading remainder gamble reject axiom confusion bold plain derive axiom gamble must acceptable assessment assessment close axiom gamble confusion acceptable reject assessment confusion element rejected gamble assessment assessment coincide axiom expression accept reject status accept framework former read despite imply nature relation follow axiom accept confusion closure status translation axiom theorem equivalent status gamble reject accept accept independence reject f vector link two useful two gamble exchange vice say exchange corollary preference f h h preference reject ideally suited decision make gamble accept exchange accept moreover stress gamble gamble arguably gamble latter cause statement instead relation conceptual moving equation pt scale coordinate xu coordinate yu intersection yu xu yu yu xu yu coordinate away cycle yu away xu away fm fm leave fm fm ex coordinate xu yu yu xu yu xu away xu away away intersection away relation illustrate acceptable reject accept acceptable reject reject background assessment go associate impact pair status coherent gamble monotonicity reject need look simplified statement allow framework restrict term allow framework model gamble side inner sep column sep sep ex ap rp xshift north rectangle xshift south node right yshift ex south west ex east xshift ex south yshift ex south west rectangle yshift ex give illustration reject acceptable gamble view statement therefore specify assessment provide set use requirement example gamble loss create g subset satisfy closed restrict attention useful cf accept assessment furthermore intersection structure closure counterpart accept care proposition preserve non intersection intersection proposition set cf coincide close accept confusion mu proposition intersection structure closure satisfy condition lead like dominate purpose accept proposition without status therefore accept status closure accept compact status accept attention gamble assessment accept status background status acceptable gamble cone deal restrict statement reject gamble acceptable end status confusion status would lead reject exclude compatible regular ex outer ap xshift north xshift south west yshift east node xshift south west rectangle east yshift west yshift north east illustration simplify accept restrict either statement statement impose gamble statement something assessment set set situation also remain restrict specific version cf mu yu yu coordinate away intersection xu away away xu yu away away away coordinate xu yu background yu yu coordinate intersection xu away yu away away baseline coordinate xu coordinate coordinate yu intersection yu intersection xu yu yu coordinate intersection xu away yu cycle away cycle shown graphical gamble gamble either conclusion still assessment status framework attention accept gamble theorem assessment model background status gamble form cone gamble linear deal form restriction look theorems extension give gamble background status side size ex none rp xshift north east xshift west yshift north east node south west xshift east south west yshift north east illustration partition work instead general highlight model namely focus lie gamble status f gamble gamble form
depict historical record identify contain set ip ip fs ip ip ip fs ip fs ip fs ip fs fs ip base attribute second variant variant ht c ec ec ec ip attribute ip time period ip table historical table ip unit email count aggregate definition feature machine ip address roughly distinguish high threshold large historical list operation resemble heuristic heuristic machine applying threshold historical another predict train set ip period ip slightly mainly second predict mean ip nearest future whether close heuristic historical window change ip predict ip threshold ip add list auc score consistently examine configuration compare ml set difficulty set tool aforementioned ground past window spam oppose currently black design filtering module list list mechanism incoming instance step cause email accept instance accept pass step order reject processing list contain ip final list heuristic receive ignore never reach space contrast continuous non ip pass operate activate predefine controller address address ht setup l machine history n auc bl I n heuristic min min evaluate use email receive online comprise label un receive online list dataset automatic device claim spam rate partition send set validation mutually exclusive vice mechanism roc curve metric enough load mail potential customer classifier tp tn false negative roc rate versus tn auc equal unity positive classification bad binary example auc auc consider performance discrimination experiment ip time amount resource spend email black translate without inspection white less spend spam filtering within learn mode low reject high result heuristic whereas among update auc auc metric well superiority white drop sensitive stay subsection suggest aggregate behavior create try address least list update investigate window construct construct window induce net instance income email black list per history list window sec bl performance historical time window figure historical improve historical auc score change setting w w respective performance experiment windows window rate window surprisingly decrease time window explain decrease dimension problem k dimensionality notice decrease historical window increase window probably reduce ip contrast black increase window trend interestingly window average since auc window window great nine overall window list increase aggregate email test heuristic email log art effective nine window similar approach another interesting base roughly classification relate g base particular believe multiple boost mechanism base set enough motivate effectiveness clear spam filter frequent execution less power inefficient period tendency spam short explain compare email service activate influence length general window certain result increase efficiency list decrease entire filter system grow auc occur probably dimensionality play mail history historical window small historical increase behavioral select currently due record machine construction end requirement suggest I bayes spam usually list list frequently minimal list content base slow demanding filter list list increase process process ratio error increase update black list white spam email list content filter black entire energy email filtering partially thank useful remark er email service spam continue gain vast send automatic distribute world spam manner propose new aggregated datum encode mail transfer agent historical receive learn build predict mail transfer near update white list internet service spam less effectiveness rely addition black white eliminate content inspection incoming result reduction filter load survey show email spam mail account spam directly exploit computational service mail box spam filter spam computationally sometimes speed spam complement rough incoming maintain gain spam attack flexibility important email receive reach refine strategy learn mail focus extract feature communication elaborate mechanism help spam attack manner analyze try pattern indicate address exploit especially require quick detection mail transfer white list address maintain email target suggest add black list email service email service historical machine create list empirically email setup pass describe encode filtering spam pass positive list incoming discuss section spam l level application spatio temporal three prominent approach spam work focus spam roughly divide three content filtering real refer file produce feature upon make email high open model content lot however internet service perspective classify incoming email heavy content filtering costly later disadvantage continuously plain spam spam ip list ip regard accept reject require email computational resource systematically spam keep track report email entirely ip inaccurate heuristic lower allow spam filter mainly low resource spam compute email base extract network social et al system change behavioral reject gain detection change address repeatedly spam usually accurate address email long exploit due high large service create address detect change address quick reaction require email service massive spam attack attack spam finally internet spam know ip range heuristic dynamic currently configuration dynamic address arrange range easy necessity address address ip heuristic furthermore large spam aggregated spatio application extract incoming email share similar domain work base email et classify email ip abuse tend ip another level domain reject email cluster report data spam organization filter al spam new ba imbalance spam spam spam mail attribute compute computationally mechanism positive could unbalanced use ba classify spam spam receiver show precision rate high communication filter trust system network example receiver likely list j al email book degree social email header construct single classifier email white black report white list test set black list list spam mail spam belong network white spam mail example spam supervise collaborative approach network email user assign high close friend assign friend network create white recommendation low positive several email domain email email record establish received investigate accuracy mechanism spam fully receive spam less another focus solely geodesic ip email receiver comparable ip liu assign return low volume accurately spam maximize west introduce name spatio potential spam email author identify spam pass positive work rate scenario week email email service hour email email maintain spam white email pass list set way address implicit ip inter notation ip ip ip nr relational represent line mail nr address error spend process email mail classification spam present property example email cm spam ip ip ip ip ip machine ml base see play significant fact able address pass email nr pt rank instance spam respective list list roughly art log attribute target attribute spam spam instance frame hour apply directly email heuristic base solely send past continuous exclusive ip denote ip start spam send ip window ip ip email window behavior history behavior ip time window describe objective black white list manner behavioral
control communication sensor rare optimality transmission observation especially large analyze suggest sensor positive message communication accumulate time suggest positive integer eq rewrite message fusion whether transmission fusion fusion fusion information likelihood ratio suggest detection continuous treat separately however structure design asymptotic suppose continuous close exactly sensor center value center kt corresponding eq fusion piecewise time centralized recursion fusion frequency transmission sensor moreover possible communication sensor threshold attain target value specification simply previous clear preferable correspond centralize practical excellent additional centralized test proof threshold sensor asymptotically preserve want transmission interesting claim walk increment n triplet order setup set need threshold j suggest specification resp interval resp clearly threshold computation follow unbounded establish centralized moment bad centralized change equality exact could time fully recover sensor small number bit main remainder design sensor communication j j admit expression simulation rare overcome k find j efficiently ratio account unobserved fusion fusion receive center approximate clearly quantify delay actual k lemma way fusion likelihood prove connect alarm period play role establish result consequently asymptotic ready communication order u divergence sensor come value latter quantity write k desire additional go infinity rate bit transmission order asymptotic achieve bit transmission low communication imply asymptotically optimal optimize magnitude emphasize alphabet bound away first optimality communication accumulation quantization error source alphabet large alphabet elaborate depend sensor minor become statistic measure contrary quantization period bit period I become obvious ask bit fusion center preserve quantization degradation quantization scheme increase reduce quantization almost illustrate conclusion simulation study take independent normally observation variance every k r bit per message period compare rule use bit communication communication centralize require transmission sensor curve centralize optimality operate asymptotically scheme infinite communication correspond period operate characteristic parallel optimal use two per observe similar gain sensor fig novel decentralize detection order performance bound rate show order optimality preserve decentralized detection independent need centralized decentralize assumption sensor indeed guarantee form adapt path sensor imply sensor correlate dynamic long indeed case every wiener square invertible b b log likelihood decentralized model sensor observation appendix subsection stop use choose ct eq sum consequently increase pair divergence equality proof define time due threshold following see write numerator kullback leibler sensor variance local ratio line bind observe furthermore classical walk whereas inequality u alarm numerator change jensen inequality maximal average take k proof goal connect threshold alarm period order elegant proof center indeed message fusion center fusion fusion correspond identity message receive message sensor simultaneous message label fusion fusion receive stop stop time relate physical term message fusion denote instant fusion center connect follow indeed number message every instant sensor message center suffice write component clearly epoch message justify observe v iterate expectation random variable ratio martingale obtain clear change outcome repeatedly proof change whereas j one conditional jensen vanish I u complete eq k z third expression imply moment order asynchronous eq consequently suffice upper clearly borel triplet identically distribute integrable stop prove take eq summing obtain inequality relationship lemma science decentralized sensor fusion bandwidth energy center responsible detect change possible novel communication bit parallel order scheme loss use observation asymptotically false go fix communication sensor optimal finally superiority decentralize rely class suppose area number fusion sensor occurrence sensor fusion exhaustive review refer however rule application mobile wireless communication surveillance system low device characterize order preserve necessary limit load transmission activity sensor constraint center sensor fusion sampling decide constraint rule contrast observation paper decentralize literature e sensor communication shot sensor fusion bit decentralize detection enjoy optimality asymptotically suboptimal asymptotically decentralize rule sensor fusion center communication summarize sufficient message decentralize testing call time establish order false go optimal induce low fusion storage capacity require additionally k transmission communication sensor fusion center sensor possibility impose decentralized detection feedback rule detection rule intractable make simplify compare decentralize rule attain
office grant de acquisition mapping present mapping object consist presentation together target word context exponentially presentation human discrimination limitation acquisition child learn fix two event human capability viewpoint theory child mind although demonstrate exhibit remarkable difficult account book review mechanism child issue human statistical principal mechanism necessary endow capacity object statistical term observational idea learner determine place cross mechanism build coherent mean confirm evidence whereas essentially counting occurrence object albeit much sophisticated counting contribution derivation explicit mathematical characterize learner performance study strategy work minimal word must object context refer occurrence object store count time occur word pick great co frequently paper expand et offer analytical expression minimal scheme finding effort monte compare performance human learn performance algorithm discrimination capability introduction law human somewhat act law eventually match organize sect introduce scheme counting independently learn complete word sect subject understand storage capability describe finally sect finding conclude remark word object mapping represent object distinct without name word object represent integer mind entity event say distinct object form present learner guess refer multiple ambiguity word list wrong association decrease exponentially learn subject never experimental mechanism deterministic selection stage may frequently selection guarantee word trial error mean utility dictionary basic point represent integer entry yield zero object appear confidence increase trial update consider correspond word simply select recall correct mapping trial fraction sect reciprocal easily detect intersection realization minimal immediately adapt incorporate magnitude ideal scenario salient minimal learning learn update aside rescale sect together word associate word sect whole event object state may incorrectly always equal trial possibility unchanged appear reasoning trial know trial q chain yield absence object word pick comprise hyper since transition main correspond transition leave unchanged eigenvalue recall write term produce analytical et derive analytical episode choose maintain notation mention curve result circle carlo simulation lower choose first time n total deterministic easily introduce integer deterministic variable rate rate single accordance intuition task complete fact learn monotonic reach increase sampling scheme lead non whereas realistic situation context linearly large find much random sampling produce result simplicity minimal analyze previous contain powerful regardless second perfect always large confidence regardless closeness say relax perfect ratio magnitude accordingly select object probability differ among identical confidence decay implement subtract word may difficulty rise responsible context relaxation perfect sampling analytical frequency compare aim algorithms map training episode episode comprise presentation word ref condition word divide word appear time condition divide time time time allow episode carry run modify result show fig straight line deviation discuss fig subset fig minimal quantity excellent agreement get excellent agreement model intuitively appear frequently outcome actually depend fig find frequency experiment subject three may considerable improvement direct consequence discrimination discrimination select trial imply discrimination large course large show fluctuation consequence ad hoc procedure summarize fig storage discrimination capability human introduction confidence bring sake minimal limit sect nearly impossible subject attention focus sophisticated thing attention learner describe briefly confidence adjust accord index run entropy word weight entropy ref explanation comparison experimental performance mapping simplify sect I circle summarize finding selection al reproduce symbol independent slope large insensitive minimal realistic conclusion vast extensively original contribution conclude although study determination scenario quantify learn learn author various inexact recover non pass introduction effect capability minimal previously object episode fix learn except word dependent
pi ji easy calculate mass expectation consider expression follow suggest require expectation outer calculate availability collection miss context target observation em need infeasible cardinality resort know em section p monte carlo particle iteration update involve step weight know stochastic em satisfy choice calculate calculate average particle smc em respectively da proposal available author smc obtain approximation z sequentially weight alternatively resample accord weight degeneracy contain index I particle resample particle proposal connect th path carlo step smc good sequential proposal show implement monte complete situation update receive alternative smc linear point statistic compute require expectation see memory time handle showing evaluate expectation functional also allow density technique intermediate eq additionally possibly dp b smoothing recursion follow else calculate smoothing treat separately perform show intermediate quadratic scalar ie obtain ij ij sufficient appendix calculate dy notation simplification show calculate extend target newly target time detection association ti tb step observe follow convention irrelevant multiplied handle statistic recursion target target target recursion k z mx recursion write subscript ij r ti variable initial conditional sufficient calculate variable filter expectation sufficient conditional sufficient respectively calculate store target terminate write function hence online rule implement short represent q forward recursion ti ti x birth prediction th th detect update recursion ti ti z tm recursion save expectation include completeness availability manner online average update estimate online update z modification reflect illustrate recursion k expectation calculate finally calculate add stability notice smc em implement z nj stochastic obtain b nm z list expectation conditional monte recursion incremental statistic th expectation target th target target tm compare velocity velocity assume estimate velocity target life expect around target per method batch smc implementation smc assignment sample association set smc appendix mcmc run statistic smc implementation use estimate use use follow x mcmc em respectively smc mcmc iteration estimate generate comparison execute result need give density I see smc em around mcmc mcmc move fix dominate average target cost smc dominate good assignment one smc far slow figure true approximately iteration mcmc slowly induce step preferable careful choose influence smc smc often without may smc online vs pass start smc cost pass roughly though introduce concatenation e target target free surveillance survival parameter depend crucially however little em run ii smc large target em use good assignment complexity em execute comparison create window surveillance approximately target time mcmc em bayesian em well worth stationary em converge iteration axis demonstrate first create scenario number target surveillance estimate slightly target stationary assume velocity trajectory create length target except use burn execute note target effect target thus online mle parameter initial negligible parameter distribution estimate batch online target estimate converge value particle estimate online estimate compare leave suggest target check comparison initial robust k bold plot experiment velocity target generate length take figure show around estimate quickly set true know birth death death indicate horizontal value bias namely illustration monte dash illustrative every precisely poorly run em birth drop indicate bias birth death unknown target hmm therefore birth death generally speak smc birth death tracking per finite integer variable assignment bias birth track assignment detail expect smc good obviously accuracy smc raise particle online em unknown address issue smc simulate smc target bad else stop compare birth death track present infer comparison offline implementation prefer smc disadvantage slow convergence smc batch cause boundary smc target clutter time budget smc em easier restrict smc parameter birth death accurately particle verify recommend consider use track estimate linear gaussian extended mle merge target target per need allow bernoulli measurement gaussian gaussian distribution target centre surveillance region type non monte sampling state provide require useful intermediate q smooth update online modify update multiply rest mainly section present exclude particle calculate perform calculate prediction assignment produce assignment score assignment infer eq q simplicity true smc filtering particle constant track average simplicity approximately batch smc trajectory sufficient statistic bit smooth expect batch apply data size us length particle track close expect time term batch iteration online use calculation whose stationarity cost smc stationarity solid example offline static target present expectation algorithm monte carlo assess simulated scenario estimation provide concern move observe extract motion traditionally surveillance problem biological framework comprise independent move surveillance region fashion arrival target birth initial target target observation record generate say spurious observation generate target collection generate record without target target extract motion trajectory record highly challenging problem large algorithm track move target handle association origin record tracking variant association monte methodology carlo smc mcmc smc implementation filter huge developing largely rarely case know include derive poisson target central filter additionally likelihood static target gaussian henceforth batch linear model exact novel development tracking stress virtue false measurement target generate false follow distribution mean value uniform superposition
shoot similarity use guide knowledge visually object however category contextual likely contextual exploit category object detector boost update round pixel work separate first step property non detector feature different category crf segmentation select previous thesis support exception early classifier task task hyperplane select feature perform logistic task likelihood adapt heterogeneous leave optimize cluster area visual huge underlie basic idea transfer important small allow quick thesis topic visually recognize rough suggest know approximately day learn appearance new visual performance machine method especially recognition thousand transfer learn technique try still gap human vision illustrate analyze category possess therefore number annotation common expect category illustrate plot phenomenon scientific current art detector detect want extract rich representation image try car etc likely rely category recognition weak logarithmic law similar database problem real expensive impossible arise identity person allow security obtain hundred person especially illumination consume hold verification handwritten recognition another scenario preference rating generalize rating high suggest probable application solve example simply improve rating competition lead amount relate case explain section prominent application vision quality production piece check manual control need localization arise obtain kind use example underlie challenge example common traditional machine assumption insufficient notion task database visual task classification cope clutter illumination diversity category intra class variance number representative learn high task number intuitively polynomial severe require require broader deeply theoretical bound possibly infinite hypothesis achieve refer sample task valid follow example sufficient achieve regarded complexity measure hypothese vc close regression vc variate polynomial exactly nearly indirect manner view say inherently try solve pose possible role reduce possible advance important want visual recognition often regularization smooth solution mostly possibility utilize unlabeled datum estimate manifold number unlabeled datum thesis machine vision amount preprocesse reduction classifier manually software visual classification task automatic reader detection increase expert prior knowledge decrease need however manual goal transfer knowledge new task automate advantage traditional build new manual effort obtain concentrate label example transfer tb concept recognition concept domain transfer linguistic interference quickly learn new thesis conceptual difference transfer contrast transfer single task rather emphasize rather class distinguish even though exploit transfer example recognize quality image digital relate body try handle lot superior example manual knowledge domain training concentrate principle transfer section survey perspective journal paper comprehensive machine also cover broad early development task couple incorporate answer application transformation manual supervision face benefit e illumination rotation try gray histogram target approach text estimation important directly apply without align face image current important relate metric piece early technique metric find metric minimize single maximize distance category apply use near learn mahalanobis identification task find distinguish car work classifier instance obtain training database adaptation apply gaussian decision appearance term shape transfer share base modeling combine target classifier learner boost transfer learner achieve preferred specific concept propose classifier extension try learner lead reduction slide approach evaluation grow logarithmic category benefit difficult assumption task valid estimate text news subsequent eigenvalue incorporate latent framework allow use approach transfer visual object inspire lot common target prior appearance share instance part lot manner connect prior distribution bayesian last regularization share task relate assumption share feature transformation regularization instead work idea propose optimization jointly utilize machine allow modeling want hyperparameter marginal idea categorization task powerful perform parameterized product compare
unary marginal extended even calculation partition bipartite interaction homogeneous sake simplicity bipartite discuss previous assume order first contribute factor reduce unary computed recursively graph partition invertible unary yield value bipartite vertex unary partition complexity show partition sum efficiently field complete homogeneous similarly partition sum homogeneous pairwise expect computer vision expect evaluate compute good class field unary factor outer width unary marginal thorough improve original manuscript author wants express special valuable strict motivating technical receive degree physics research interest probabilistic article accept publication issue journal note attention ise homogeneous pairwise potential unary potential probability bipartite homogeneous potential class large field provide markov field unary inference pattern vision markov gibbs restriction underlie structure marginal probability ise estimation propose new improved calculation class field ise homogeneous pairwise potential unary potential similarly polynomial complete bipartite provide homogeneous potential equal sum contribution unary factor homogeneity pairwise unary recursively sub label either vertex label remaining label dynamic programming size mapping unary probability calculate unary generalise field give denote notice assumed vector respective value kronecker contribution
comparable bioinformatics system science perhaps exponential possible match suffer place mass empty complete graph propose modification degeneracy non density applicable relatively small modification degeneracy describe family constrain around principle discuss distribution network moderate density model via outline possible future exponential distribution describe contribution variable base exponential I canonical open assume denote ff dirac family satisfy exist unique provide poor family find mle modeling assign little mass family approach function place around distribution heavy tailed approach use degeneracy case choose black red line non exponential tail use combine family add symmetric positive indicator version indicator hypercube center side kde g kde match accord empirical kde bandwidth dimension preserve determine multidimensional assign smoothed tuning canonical augment parameter ne constraint point know induce allow moment satisfy set aware capability turn learn undirected vertex loop complicated commonly model degeneracy place graph empty away modification parametric family degeneracy statistic feature motivated graph hull degeneracy mle find place mostly complete graph place c tb degenerate red bar right blue mass low address degeneracy information degeneracy attempt address degeneracy specific edge exponential suggest approximation truncation degeneracy interior due modify geometry towards degeneracy modify geometry space category modify uniform neighborhood graph family probability preserve function eq concave several challenge fit computed form mle sg however graph computationally expensive equal result modal close mode predefine illustrate constraint normal df simulate vary sample sample parametric assume schedule kde determine choice obtain exponential choice kernel kde kde estimating kde gmm density quickly consider model show sample increase similar moment schedule estimate mix sparse bad enjoy parameter whereas schedule schedule statistic non zero estimate normal number distribution zero normal zero estimate mixed compare discussion fit make commonly fit measure degree proportion share proportion edge common geodesic proportion connect pair minimum tb kp business datum set ad datum kp fa ad unique max dash line red dash line line close black line red triangle first consider tuple compute mass function put large vary since gibb use package mle sample generate whereas graph goodness test sample run markov chain graph tune scale find need variance able sample sample count triangle need state graph suggest considerable family capable fall empirically increase arbitrary continuous density control sparsity fall estimator global constraint require mle optimization raise challenge constraint adapt graph degeneracy bandwidth acceleration acknowledgement help proof nsf theorem augment augment eq augment statistic respectively prove convergence half kde expect kde uniformly positive kde kde f density identical draw ne assuming family parameter nf n n ne family chapter w e exist sure additional constraint match constraint small reality rarely condition though constrain regular canonical closed subset space pointwise parametric inequalities kde ne kde ne kde ne kde kde kde density kde continuity density n ne satisfy continuous combine lemma corollary
cast video subspace include subspace particular ssc datum lie union challenge unlike unified framework corruption membership subspace spectral cluster contaminate noise point perfectly sparse nonzero error free perfectly lie solution entry sparse sparse precisely vector entry linear magnitude square root dimension subspace program collect objective function program hence programming tool optimization hand corrupt eliminate deal noise consider datum program similarity infer cluster incorporate corrupt separate correct datum incomplete fraction incomplete cast fill datum follow apply cluster graph coefficient drawback fact know case cast miss complete see define index denote ambient column ssc keep row complete base nonempty address problem problem lie motion segmentation involve subspace naive way deal case cluster subspace fact include origin drawback increase origin indistinguishable deal fact subspace solution lie subspace comparison subspace subspace well also affine ssc representation nonzero datum point representation datum subspace direct operator figure leave space span every intersect origin show subspace intersect independence converse characterize two define principal union subspace underlie cluster minimization specifically n recover subspace without price subspace disjoint subspace appropriate recover consider investigate intersection restrict also solution restrict subspace except supplementary ssc succeed subspace intersection strictly norm precisely result minimization recover subspace sparse successful explicitly show program depend angle establish datum minimization successful sparse hold subspace angle polytope nearly degenerate reach I n I I I hold nonzero recover subspace speak sufficient principal certain subspace recovery norm subspace segmentation datum apply ssc norm subspace subspace condition always recover closely property difference optimal instead feasible violate feasible solution still optimal successful result subspace number relationship column hull polytope reach interpretation nonzero small polytope recovery middle figure sparse reach degenerate close orthogonal hold note sufficient translate subspace angle section study program recover sparse result point lie get synthetic always hence component verify subspace subspace great odd graph subspace right increase row common similarity correspond graph program connectivity subspace principal angle subspace angle merge add indicator hence number row choose common sparse np consider connectivity point trade add term representation demonstrate term regularizer objective subspace angle well solution recover uniquely show figure connect feasible representation hence minimize sparsity regularizer show success principal angle consider embed dimensional ambient subspace reconstruct linear generate basis two subspace addition subspace angle equal verify change sparse data increase probability normalize point subspace program error denote sparse I inside summation norm subspace correspond choose subspace choose sparse coefficient spectral ambient subspace trial predict decrease also success minimization recover representation note small increase undesirable spectral also subspace error imply exactly component ssc real face ssc art lrr ssc direction multiplier whose supplementary material variation optimization constraint choose experiment face experiment use program general solver coefficient spectral art dimension face dimension subspace segmentation lrr face note lrr accord ssc processing similarity effectiveness sparse rank objective investigate lrr lrr processing method motion segmentation alm variant subspace priori laplacian noisy set comparison subspace segmentation sequence video cluster b face subject subject lie describe present experimental present first pair motion segmentation face subject small principal certain range subspace dataset angle principal angle always principal angle degree subspace compute percentage show subspace dataset show subspace b near near plot principal principal angle challenge subspace lrr h median motion segmentation refer multiple region scene track nf vector stack q segmentation refer trajectory affine subspace lie trajectory motion segmentation lrr lrr ssc median median ssc problem consist video video frames trajectory frame singular dimensionality underlie video perfectly lie linear subspace dimension apply trajectory subspace table conclusion case ssc separation subspace angle feature success program number inside cluster ssc without step normalization help cluster result ssc perform post lrr error post try find representation function error dimensional pca projection close trajectory onto dimensional structure close error table effect ssc dataset ssc result ssc coincide random addition datum subject acquire pose vary illumination consider fig subject illumination subspace image well extend face face image acquire reduce memory treat plot value value dimensionality face singular showing corrupt image corrupt lie cluster algorithm devise divide subject group first three subject consider choice apply svd plot corrupt correspond cast image model sparse deal validate corruption face due error importance remove face subject subject experiment performance datum subject remove sparse conclusion ssc zero suggest ssc deal face ssc angle small ssc lrr show lrr relatively show post process low always improve lrr cluster number subject neighborhood addition neighbor subject number c lrr lrr ssc median median mean median apply collection point cluster make ssc low ssc obtain subject respectively come bring common low angle subspace increase respect table lrr lrr ssc median median median algorithm process conclusion subject respectively ssc incorporate corruption model lrr deal corrupt large increase clear corruption lrr general hand post step help large ssc try find rank lrr deal corrupted neighbor subject right lrr median time subject equivalently drastically high complexity lrr fast time supplementary material close union technique ssc build obtain show succeed recover datum ability deal affine experiment face video show effectiveness superiority research investigate theoretical sparse miss datum extensive real point form similarity graph point probabilistic optimization applicable future author like support proposition optimization theoretical optimization solve optimization program q side inequality achieve solution program equal equivalently theorem representation I prove write substitute right independent intersect contradict optimality obtain desire proof subspace also minimization restrict point every strictly succeed recover n I recover contradiction lie vector solution optimal imply second strict inequality sufficient contradict optimality optimization program prove contradiction exist nonzero intersection program select subspace contradict subspace condition norm full column rank thus establish bind norm multiply leave recall definition subspace angle column subspace except rewrite optimization optimization solve solver solver typically implementation alternate first eliminate optimization equivalently overall lagrangian respect primal maximize abuse vector entry admm set terminate update auxiliary whose help penalty vanish make variable introduce lagrange multiplier lagrangian admm consist follow kk kk system large conjugate gradient method thresholde act element give return return fix gradient ascent lagrange multiplier repeat iteration iteration achieve denote dual summary implementation program h ssc subject computational different b function code author mention lrr ssc fast lrr make lin r liu linearize rank fast propose zhang compressive sensing scientific corollary many world collection video text web microarray dimensional several category union dimensional among point point subspace subspace solve sparse optimization program succeed recover desire propose point near intersection subspace entry miss incorporate program demonstrate motion dimensionality programming spectral angle motion segmentation face area image bioinformatic video million web document affect effect insufficient respect ambient commonly refer curse often lie uniformly ambient recover low reduce computational memory improve inference recognition category trajectory video subject vary illumination write digit rotation ambient therein problem separate numerous image processing compression computer vision segmentation illustrate arbitrarily centroid spatial take structure algebraic spectral subspace alternate assign fitting subspace drawback generally require dimension initialization factorization algebraic segmentation thresholding provably independent assumption violate principal fit contain mixture probabilistic cluster subspace expectation maximization drawback sensitive dimension choose enough subspace noise know however must exponentially theoretic
although similar analysis conduct interact colour line mixed computational seem work good explore relationship cs ac collective classification attempt instance tend pattern interaction make instance class pattern interaction blockmodel link alone simultaneously provide interpretable well understand datum explore network long focus predict attribute leverage traditionally modern shift exploit improve performance attempt assume e stochastic blockmodel instance stochastic blockmodel efficient update relative investigate understand introduction new collapse inference avoid long diagnosis sampling update expensive social analysis refer zero occur adjacency interaction role belong respect role role interaction assignment usually accord attribute create role link paradigm blockmodel role infer attribute stochastic unsupervised role role assume homogeneous linkage pattern behave dirichlet name extension model lda dirichlet corpus topic blockmodel derive identify distinction role possible heterogeneous linkage pattern role membership supervise examine standard blockmodel sbm assume role interact interaction indicate absence application blockmodel role thing blockmodel sbm network draw role role interaction interact role draw draw absence link mixed membership blockmodel previously extend blockmodel role draw node interaction role interaction receiver draw role interaction provide py use inference sampling convergence inference method variational introduce approximate role field evidence conjugacy bernoulli model integrate collapse posterior tight bind implementation collapse variational expensive practice taylor collapse inference first order approximation implement standard stochastic blockmodel updating role count receiver collapse remove count role reflect inference class occurrence oppose sbm occur receiver assign rather assign role interaction receiver role involve receiver represent softmax conjugate conjugate gradient predict unlabeled rest label involve generate four examine citation web citation popular collective classification comprise represent paper represent frequently occur novel link network corpus corpus american english category occur noun result link classification type namely primary investigate maximum role varied sbm macro measure f tp fp tp fp negative harmonic mean recall macro use f remove
current target evolve system new position position stability depend sample evolve locate matter center case inside trajectory apart stability extreme small hypercube length center lyapunov hypercube approximately linearize pass well eq lyapunov eq trajectory although converge point stay around example suppose dynamical define high value dynamical however two situation stay rare case trajectory stay eventually could jump yet stay htbp converge stable hypercube strong situation small small q close thus evolve stability fix experimentally converge text especially demonstrate question connection neuron indicate text neural instead flow certain pass boltzmann machine message pass field thus structure necessarily algorithm directly encode neuron propagation satisfy point encode inference encode neural inference semantic learn exhibit parameter joint generalize backpropagation boltzmann machine preserve symmetric correct semantic context local uniquely determined term conditional methodology neuron variable neural network must way conditional boltzmann mapping situation text imply neural without machine neuron binary allow hidden neuron represent support information happen rigorously even though continuous neural preferable boltzmann machine question continuous encode efficiently approximated distribution encode number digit representation digits evaluation approximation inefficient look eigenfunction relationship fourier eigenfunction unify investigate binary preliminary question neuron bayesian model external introduce explicitly neuron correspond correspond observation choose part parametrization I I term hand contain two term recurrent neural roughly speak integral deterministic counterpart get well close match deterministic
limit something usually data collection tweet km radius uk twitter search able retrieve recently publish tweet km location english twitter feed format collect content feed library store request retrieve tweet database collection minute location retrieve project aim collect daily basis obviously short period tweet collect twitter user reach able collect tweet per sampling uk st publish user twitter rough number tweet day daily increase together publish reader plot volume collect tweet cumulative equivalent collect acceleration collect million day side increase tweet maintain representation twitter corpus tend tweet see method library basic field ir machine project deal content library engine index tweet word frequency document create index base fact database library source code create type indice another useful project software handle large scale implement machine algebra operation singular key paradigm operation computer operate retrieve implement comprise tf tf stop rarely cosine similarity need least software library text perform stop space representation well type like college practitioner weather uk weather read uk whereas use present chapter start characteristic twitter service serve main project twitter attract human massive platform opinion mining twitter retrieve tweet provide report collect tweet cumulative refer tool give version grind throughout project mm tracking epidemic reduce impact help plan various employ monitoring report monitor capable disease population twitter hundred thousand tweet search automatically identify turn test uk score obtain preliminary completely independent commonly hence provide epidemic extend version tracking monitor social medium detect infer monitoring epidemic population population insight health intensity epidemic existence demand affected give phone call network drawback delay engine health query engine query extend monitor content web tool twitter micro website option status phone device character mobile text twitter uk quantify various region country twitter reveal stream tweet hour delay furthermore entirely method early stage one twitter life provide evidence social marker marker word form phrase gram topic pick social could health datum validate daily marker appear tweet otherwise contain marker daily twitter corpus tweet divide tweet eq day tweet divide daily retrieve score truth report provide statistic base scheme metric express number diagnosis uk region retrieve equal twitter expand former period assign day week expand expand smoothed peak reflect epidemic uk marker temperature infection score time point move express tendency twitter score day linear coefficient twitter series correlation respective present whereas investigate correlation move smooth locally smoothed ol across implementation present smoothing window produce correlation move average well ccccc south e plot twitter high two notice actually build twitter weight marker tweet q number weight twitter daily compute tweet day marker twitter content twitter marker retrieve twitter unweighted vector unweighted day smoothed move ol time smoothed expand learn weight infer remain region method time indicator retrieve correlation train linear rate region region similarly report various smoothing interestingly smoothing correlation window day smoothing decrease perform produce fact correlation differ point produce interpretable tendency perform slightly well leave investigation avg point assess aggregated score epidemic form use correlation p test fold randomly decide correlation case score unseen train period train together experiment marker relate region uk obviously choice marker deal concept select enable effort make marker extract marker select keyword correlation form create candidate informative create marker relate use reference discuss preprocesse stop removal extract marker topic formation choice marker discuss justified section form candidate feature daily twitter day twitter represent tr day array total move smoothed rate order advantage produce sparse solution discard candidate prove redundant estimating least subject optimisation task shrinkage time region validation percentage region five choice lar apply table possible choice high correlation setting region comparison infer choice non extract marker majority spread heart mention page home ill counter check water confirm phone cancer train avg remain three whereas aggregated day period region respectively day axis day read follow page check home member exist cancer spread ill far aggregate data form percentage remain data value retrieve infer point marker automatically infer candidate signal correlation target unseen automatically marker uk experiment choose smooth marker move tendency unseen use overfitte chance lasso especially inconsistent try issue propose formation gram corpus index reference choice partly justify day period gram stop word appear twitter corpus retrieve gram smoothed move truth expand smoothed ccc epidemic movie movie epidemic movie epidemic movie get quick datum correlation ground correlate marker correlation statistically word derive series word something ccccc media market player school live record death public level bank wave award know region top twitter movie music popular international possibly popular name something signal behaviour target also select day cross weight excellent every term cross something another safe conclude set effect matrix estimate shrinkage bound expect euclidean nonzero proportional sparsity show bound denote loose sample shrinkage norm intuitively result solution holding assume conclusion increase well e far rate turn satisfactory dimension solution reason chapter present epidemic uk twitter give early various mostly plan base micro calibrate extend message send mobile device besides privacy concern character form keyword phrase normalise manually score discover actual ol retrieve one next achieve regressor initial use twitter usually approach ir form feature target wikipedia page health orient nevertheless majority choice justify firstly secondly understand error affect characteristic learn marker scoring function propose truth specific generic present methodology methodology limitation infer phenomenon explore rich amount service twitter study consist benchmark location infer tweet rate effort detect epidemic build lasso amount study show significant different learner core investigate chapter extended web learn mm already chapter detect twitter content core inconsistent variable close gram bad feature english language character improvement use gram chapter hybrid combine short span smoothing something improve application important lastly event epidemic allow finding commonly economic inference regard magnitude refer recent past reference consider magnitude event variable entire content observable proportion life web aim observable detect infer magnitude event tp event entity include web see learn observable logic web web twitter make content concentrate web content type paper paragraph manually relate related keyword application sentiment predefine phrase map sentiment report rate pick similar query furthermore sentiment apply effort extract vote office daily average exploit location tweet detect approach feature method apart reduce human minimum tend great later track component average subset query keyword high value preliminary chapter twitter content automatic rate incorporate detector tool infer besides retrieval distinction lie principle regressor handle candidate feature search methodology abstract summary use observation datum general chapter vocabulary gram marker marker extract web candidate target connection consider gram gram combine propose unseen flow web static behaviour document tweet facebook combination aforementioned comprise know gram indicate extract stream gram depend gram least subset inference gram take candidate marker suppose user stream raw count marker limitation tweet character twitter special setting value marker score marker represent occurrence stream interpret tweet marker tweet score candidate marker keep eq matter depend duration day hold interval restriction time retrieve variable candidate bootstrap attempt shrinkage solution shrinkage lar inversion either continuous range explain look algorithm feature space vocabulary array perform ol scalar bias term strict application feature go soft feature fraction refer consensus ct strict ct decide computational phase include ct retrieve ols regression ct optimal ct definition sample define iw naturally definition essentially try plus square outlier investigate literature research square comprehensive metric present result ct suppose threshold validation denote index testing phase take consideration rate infer zero perform filter testing ct track well go truth ct explore stop reach assume form perform lar entire set lar set uniformly lar end nonzero select ct next decide ct change lie fold cross preferable feature aim evidence characteristic therefore shall compute ct correspond ct feature train p train mi b mi jj train p class candidate investigate gram hybrid gram gram gram interpretation topic base context gram gram consist piece tweet character daily close zero tp v train I exploit advantage form combine try approach form hybrid consensus gram gram hybrid gram class denote tp p q I j q hybrid gram explore gram gram b hybrid combination proceed note gram gram follow class perform something class gram gram entire run unified name bad hybrid dimensionality without training perform feature mainly differ via describe training feature pearson coefficient candidate feature training rank retrieve compute fit incremental top correlate performance select evaluate disjoint set train train train twitter infer daily availability daily weather since piece available majority twitter various tweet easy due non marker study weather relate web wikipedia page language course weather vocabulary weather terminology marker probably good semantic marker twitter study gram keep candidate likewise gram year twitter datum form location million collect size bootstrap complete soon one stop criterion meet essential execution amount cross start month validation fold half marker five location ct principle marker ct batch retrieve inference selection select round validation present ct select turn able rate large number select month month tweet discuss day create weather capture evaluation table interpretation numerical rate equal range outperform total achieving comparing indicate feature interestingly feature denote validation testing fold cm present result validation remain cross validation month select feature gram outperform cloud comprehensive majority gram table semantic connection without direct connection act weather use weather orient positively select gram semantic weight particular multiply cc pour multiply cc cc air light stop wind weather cc cc air weather pour light look cross round inference term marker frequency location unlikely gram cloud font weight word content infer uk base uk region truth information result diagnosis accord baseline daily report interpolation rate consecutive compute factor day within duration week several reference web health service follow general case extract gram gram count tweet involve time collect twitter data period rate epidemic period feature assess datum randomly apply region evaluation tweet period tweet million previous study validation day round fold train fold day ct rest testing contiguous randomly day contiguous wise round fold study feature either significant period gram topic tp ccccc cm cm hold cross comprehensive interpretation equal deviation range class term method improve approach multiply cc cc cc spike huge public remain rough team behavior member perform wave health wikipedia tp multiply cc need site health fit head visit health care loss worker home channel take care multiply cc gram gram gram stage attempt web care check code check site rough health health care worker health home visit wave item wikipedia font proportional weight negative present round class gram body water solution large surprisingly gram relate large feature mind significant generic one tp figure fold cross correlation provide additional significance present carry contiguous day datum test remain formation epidemic period inference outcome smoothed inference move induce trend class experimental method inference public web medium distribute phenomena property uk rather unstable daily whereas evolve smoothly figure picture hard discuss weather precede forecast especially weather example bad day location exact average day gram day one propose overcome select figure high truth randomly turn major base experimental target applicable event draw similar extraction ir technique imply entire instead focused reference event justify lasso section span limit train risk avoid focused event candidate constrain dimensionality training issue nevertheless small word daily tweet location feature gram proportion properly contribute experimental process make manual gram rare stable ct operate adaptation application strict ct would apply ct tp methodology ct validation event twitter linear end predictor weight value choose either function motivation behind fold firstly want metric secondly investigate whether task cart tree nonlinear apply cart cart mostly latter intermediate comprehensive series tweet candidate learner cross validation identical measure performance month fold training month fold either pruning cart ensemble half tp three consider gram gram concatenation gram gram experiment threshold percentage level prune equal cart retain full insight infer cart produce datum gram gram word topic weather derivation replacement tree go ensemble decide validation ensemble average prediction feature present pruning cart cart elaborate bootstrap rmse gram gram combine improve error fold optimal tree use ensemble keep tp level tp tp c min perform variable every subtree importance give importance ensemble table important investigate select bold feature weather identical irrelevant gram bold act bold cc cc week wind like start ic weather meet al feature bold cc cc series uk rate fold day training day datum decide rest experimental apply pruning tp min max tp cm method rmse table cart case ensemble ensemble gram gram drop rmse bold cc cc low school warm live bold cc confirm check light switch care contract bold cc cc live feature feature class marker topic epidemic important ensemble learner conclude function feature far target concern reliable even hybrid ensemble irrelevant might outperform nevertheless select gram seem correlation initial amount must define manually preferable feature ensemble outperform incorporate affect ground section examine probability general limitation infer spread region operate vice versa inferences bootstrap see standard deviation cart divide ci compute multiplying se quantile normal tend quite three similarly model base mean advantage framework inference regression ability input covariance spatial characteristic event chapter methodology capable uk identify theoretical framework already initial insight smooth truth score different location respectively mm median mm likewise deviation previous rate unstable rate time zero rate never rate per inactive tend notable short period try derive likely occur pdf rate histogram well fit pdfs gamma pdf mean logarithm see event normal event frequency tend small therefore inactive inactive active precisely short peak expect topic strong chapter general methodology infer exploring web methodology specific procedure well claim gram gram hybrid combination gram gram give good whereas gram semantic event alone naturally comparison even variant learn interest message propose count frequency disease name well easier exist optimal furthermore obvious actual bias prevent study actual twitter population twitter oriented lastly insight characteristic behaviour generic characteristic look extract norm content twitter focus well daily characteristic daily correlate significant life partially detect uk twitter v uk exact major medium micro website twitter various way range already propose track epidemic infer seem nonetheless explore variation real large via twitter permit variation accept notion certain phenomenon even capture twitter content volume affect day contrary clinical concept load early confirm study show pa rather increase daily effort explore variation user assess period uk tweet within centre bias daily centre text divide bin hour day hour assess level activity namely message text list retrieve priori build select filter way keep preprocesse affect apply extract hour type base interval number twitter corpus normalise interval count derive frequency averaging frequency final scoring time w I iw divide hence interval weighted final figure hour pattern aggregated figure include multiply se distribution interval consider linear correlation high point deviation follow test stability daily sample pattern base day pattern permutation number correlation test eq consider value aggregated aggregated statistically instability consider weight extract scoring investigate stability describe depict first drop start reach apart reach hour much peak start increase reach peak tp produce pattern present drop start reach level something peak steady reach high hour whereas peak slightly reach peak happen level hour tend peak decrease start reach core high p hour tp linear correlation pattern aggregate pattern positively statistically show correlation expect derive apply scoring something table scoring type correlation tp cm tp cm tp section investigate previous section show occur peak unclear extract peak day form histogram peak day depict difference peak frequency tp figure behaviour period occur unstable amongst investigate rarely reach quite step autocorrelation figure scheme lag consecutive hour logic something high affect previous autocorrelation become evident type daily pattern strong significance statistic vector length day compute autocorrelation compute test lag autocorrelation periodic display unstable behaviour term affect apart interesting result justify score overcome bias create life tweet merge na pa pa patterns plot na strong count hour also begin dominate pa slightly affected negativity tp content try pattern level medium uk two main amount time reduce statistical stable result elaborate scheme positively correlate signal clear peak also peak flip peak early combine show peak pa also peak hour na peak take na evolve pa hour signal across fluctuation early peak something na na strong pa pa express express moment monotonicity reach finding state pa states behaviour day recent pattern pa na across day week finding uk one derive uk pa retrieve peak scoring period day scoring extract limitation study people therefore drop collect exclude content bias usually datum characteristic extract limitation arise due general twitter create present claim uk investigate daily similarly focus norm section uk partly affect tweet process million apply interval daily time interval extract daily basis section score count twitter corpus tweet likewise day extract treat equally daily score smoothed version allow retrieve peak smooth time series reveal period affect base norm life turn significant event sense identify time series period explain death people uk reach record security peak international phone period rd possibly day apply yet uncorrelated period month take place uk rest peak happen nd th scheme equivalent output moment signal happen uk day death rd co occur attack side period figure figure produce identify least throughout year significant list year day st day peak much event apart peak uk receive positively event international death possibly weather condition moment occur mark bin death rd death occur attack speed short peak series derive table peak rd period date match characteristic significant usually average minus pa day twitter user together event base peak level negative exchange movement death positive day death across observe follow exist high contrary happen smoothed reveal similarity see correlation might norm set twitter cm cm day lag correlation positive negative bound significant autocorrelation experimental periodic type interpret consecutive day lag consider week indeed large extract one exception autocorrelation happen retrieve include strong day belong principal component scheme see clear separate evident close might also day task cluster day day figure expect moment relate bin death attack uk cluster separate rest new day group also speed observe day probably day uk entirely separate rest place opposite space close form date uk together attack cluster majority twitter stream uk international occur recall define scheme significant event concern uk affect twitter scheme death severe scoring give across scoring day lag lag increase decrease pointing might day day pass perhaps due event interestingly lag equal periodic affected week autocorrelation nd solely cluster daily discover scheme consecutive tend common seem day behaviour identifiable area cluster face importantly acknowledge twitter large bias actual ground fundamental come partly justify moment signal daily affect twitter e one extract whereas second remove firstly twitter content publish uk partly agreement similar finding acquire bias pa vice versa hour show hour length day support affect user figure track event real uk combined death twitter track event day day scoring result scoring scheme nevertheless show two scoring scheme combine identify finally show investigate lag day day likely dependency day week cluster score consecutive day exception negative identifiable preliminary nlp way content uk form similarity show stable form briefly examine twitter slightly something use feature preliminary aim voting inference input medium uk still result serve investigate twitter influence respective tweet content window significant formation share location tweet twitter uk carry already describe centre location tweet area correspond conduct setting document tweet day since datum set comprise day span also reduce day month day document day location proximity twitter answer question pairwise location respective similarity content approximate cosine document tweet retrieve tf entire document stop denote cosine similarity publish total pair km end cosine cosine experimental smoothed obvious content experiment increase relatively experiment seem figure km p window pattern however cosine primary question location necessarily twitter look global pattern twitter uk twitter quickly place regardless proximity country unitary apply multidimensional scaling location similarity well pattern trial datum retrieve computed represent monotonic point satisfactory level minute location nearby location distant might observation uk space whereas location centre try understand content reveal influence similarity within country effort content relation twitter share among propose preliminary pre set average similarity already previous period degree content average cosine pair average cosine ranked decrease top one location infer decide participant else numerical expect comprised month depict proportional degree location something happen depict visible directly observe share read place far likewise embed month minute show average picture network consider month central j degree comparable experiment one month infer quite examine short content similarity location balanced manner nearby include network see significantly increase shared content additionally reveal connection connection appear day score range equal numerical observable dissimilarity form network location network stable therefore well description share introduce form metric consecutive similarity edge dissimilarity use q similarity base distribution original replace consecutive network test switch original ss nan divide justify first month examine whether month minute investigate network plot respectively daily instance always switch might score first low last datum investigate network minute figure draw period second week similarity smoothing move extract hour minute pattern rise indicate periodic twitter user importantly news medium twitter minute line smoothed twitter million tweet volume tweet hour aggregate figure time p work hour p hour twitter hour become therefore activity might activitie stability day set compute correlation also linear say operation day significance value pattern present paragraph divide one twitter day twitter patterns behaviour trend indeed twitter volume peak deviation hour possibly day week hour volume repeat previous test stable respectively least another whether cluster day feature twitter volume interval figure figure day week pattern observe consecutive place position interestingly interesting offer social medium tool majority perhaps easy preliminary twitter finding united uk recent publish discuss discuss three major uk party party extract positive sentiment sentiment voting day vote percentage per political party dense publish day tweet million section tweet tweet political party retrieve political party see per party party name party search gram insensitive latter look match search tweet contain gram search keyword base mainly entity could also create automatic manner repository human approach build elaborate approach sentiment part speech noun positivity negativity negative sentiment weight might appear stem positive negative tweet retrieve sentiment weight note tweet list tweet sentiment ignore pos use motivation behind pos twitter pos might inaccurate however case probably tweet principle pos map particular sentiment pos tweet carry stanford pos log pos sum sentiment tweet term retrieve sentiment score incorporate automatically list list content tweet tweet word list tweet sentiment tweet sum pos identify extend tweet tweet reduce add enhance semantic orientation english achieve pseudo sentiment great tweet method one previous negative sentiment tweet extract keyword political party represent vote party describe remove tweet tweet unclear help later finding experimental setting remove entire also tweet sentiment score tweet subtract score vector ground weight voting percentage bias ol receive value reduce freedom study three major normalise order sum inference thresholded take unless take existence political people vote create comparable normalise vote three major much voting inference unseen overlap learn calibration sentiment score party sentiment multiply example conservative party triplet represent normalise inference voting party denote thresholded sentiment tweet many sentiment vice sentiment tweet sentiment number tweet negative sentiment refer sentiment measure mean absolute infer voting metric easy interpretation read mae unit base voting measure triplet voting error infer incorrect ranking difference correct position since one incorrect triplet make triplet error triplet correct triplet equal range computed method extract tweet sentiment come tweet sentiment remove top unclear tweet retrieve figure table thresholding tend performance mae fail voting properly performance thresholded figure depict good inference ground mae leave sentiment tweet cc ccc one size focus testing tweet retrieve search increase day part close retrieve randomly come test process time count give value significance sentiment tweet ccc present pos tweet good fairly poor performance mae significant thresholding reach mae ccc c c quite publish provide twitter political propose predict word count semantic tool negative introduce averaging like keyword party name orient propose traffic party present tracking voting tweet deal bivariate e two nevertheless indicate specific proven chance predict tool google interesting conduct triplet consistent content medium author firstly prediction aware characteristic justification base try figure form three scheme extract sentiment tweet try twitter particularly tweet enhance similarly aforementione work sentiment try show poor statistically significant sentiment remove contribution thresholding improves begin thresholde far modelling preliminary aspect future research come proof capability choice keyword argue influence automatic mechanism bias sentiment tool also tool expression extract medium content political opinion extend tweet probably vote quite exceed mae derive therefore formation consistently truth issue resolve rich twitter investigate show country capable form uk prove time content uk secondly extract twitter twitter tweet evolve peak occur hour temporal characteristic day etc explain form cluster cluster exception infer voting twitter something topic tweet sentiment apply consider speech incorporate stanford pos third sentiment voting percentage indicate effort public infer several uk display four type set region publication tracking mm monitor important school delay demonstrate web provide epidemic work social web medium predictive infer region uk twitter chapter validation complete automate tool interface use uk twitter data train validate behind million tweet km uk rate ground detail give summary feature gram extraction comprise gram gram gram decide ct select gram gram n gram ol finally weight daily twitter content region daily since realistic large equal inference behaviour also inference maintain focus tweet km get tweet centre retrieve atom format database describe perform offline inference basis website apart three region south central display entire uk detector website chart website infer infer day display separate box patient rate exceed page medium predict stock market implement finding compute display already namely see scoring importance display raw score score divide remainder standard raw score plot figure indicate deviation large interpretation stand negative multiply score negative constant maintain dynamic type day automatically peak carry skeleton website amount maintain region twitter datum different implement basis show chart include type page website entire uk page day display display historical datum interesting discuss peak day mm chapter outcome conclusion research interesting behind scientific primarily amongst chapter operate give primary firstly attribute drive prove store twitter ground truth track epidemic diffusion exploit content twitter manually prove correlated rate propose fully automate regressor lasso extract gram semantic topic accurately rate uk epidemic limitation mainly learner gram possible experimental infer occurrence phenomenon rich amount method bootstrap consensus ensemble core methodology gram gram investigate combination rate usefulness rate hard test datum source twitter content problem extraction pattern assign importance weight analogous text stream remove bias marker investigate type infer pattern partly confirm one pa show vice daily gain support life twitter example observable twitter uk daily period length week additional pattern study twitter could country twitter depend location form uk infer secondly focus extract twitter distinguishing seem confirm match norm report address inference twitter uk general lastly online tool serve dynamic theoretical derivation uk display uk region tool twitter content thesis web text stream reflect part life social medium increase reflect thesis way type web service focused extract life trend inference event inactive discussion could quantify show stream general periodic norm similarity reveal share among nlp twitter vote therefore verification truth compare unsupervised acquire fact abstract infer voting web conclusion become life public effort develop possible side applicable content investigate study acquire target able several signal position platform course avoid twitter early success improve gram increase select combination gram gram improve performance selection bootstrapping lastly content sentiment paper idea short paragraph social stream conference journal recent book newly topic extraction voting sentiment tweet detection environmental twitter topic could extended direction grow something project scientific medium research technique application sophisticated linear learner new information key function hold could concern gram show semantic gram regular aim look aspect incorporation sentiment rational life medium already need quantification detect concentrate english language method language apply framework interest go challenge successfully signal stream challenge combination vast social answer question result path issue impose development carefully consideration propose computer closely quality speech arithmetic sized sample define deviation uncorrelated tend increase make sample identically distribute sample standard deviation transform suppose variable denote way mse broken variance bias average around something unless bias complexity extend reference vector size cosine simply cosine angle sim range point odd number equation move plot move average recover original nan hypothesis hypothesis nature nan trial unlikely contradict hypothesis might extraction principal analysis observation orthogonal space generally direction derive eigenvector component define eigenvalue twitter uk centre set belong list belong read hull york extract manually form marker gram describe particular ccc back infection track retrieve affect gram gram term ir alarm belong care concern favor good great heart like warm weak h bad blue dark tweet political uk character empty space twitter conservative party term cccc conservative conservative party david grant david david vote vote cccc ed vote vote david david david cccc mp mark david david vote pc thesis requirement code research award thesis author signature date ph old matter go grateful person properly primitive another ph thesis course significant scale ph good mind internal ph progress useful remark progress also sc study grateful interesting build sound people negative grateful mit technology special database person project I cover financial support ground computer department university year ph amongst thing ph book thank interact indirect aspect platform company spend write life east east east east art art core google web form unique web approximately time
input cope meaningful propose sparse principal exploit sparsity probabilistic base setting geometric distance neighboring dimension embed manifold volume dimensional performance affect author suggest base smoothed author property technique neighboring point exploit graph adopt either geodesic minimal span arc distance efficient base euclidean usually work exponentially curse reason dimensionality practically available insufficient acceptable manifold raise dimensionality increase edge behavior neighborhood propose empirical correction procedure produce generate fitting correction use local give extract estimate bi compose fit horizontal whose author describe base comparison distance neighborhood distance manifold high locally map manifold smooth mathematical drive spherical neighborhood origin radius restrict describe ball uniformly ball uniformly distribute neighborhood estimator information unit property draw sketch angle consistent locally nonlinear map sample independent uniform nearest j I normalize compute employ translation k q parameter section draw digit version letter alphabet circuit face database three consist gray test represent actually know digit range synthetic point particle motion nine series delay collect value letter twice training group set study compose realization circuit delay contain employ toolbox generator create point unbiased achieved execute contain number resample trial iteration shape distribution hyper table configuration summarize relax selection perform run result report summarize c estimation obtain manifold able embed manifold consideration confirm affect notice manifold embed dimensional space table percentage report percentage error manifold consider indicator good c obtain quality poor geometric approach test affected noise correctly dimensionality well perform strongly correction precision mean promise valuable tool finally w r employ reproduce average curve compose range combination obtain robustness novel call angle method compute synthetic aim employ employ overall really strongly really capability ii linearly embed iii effectiveness propose width dimensionality concentration dimensionality last decade dataset gain considerable deal work devote dataset point lie embed high propose intrinsic exploit nearest neighbor angle neighboring provide close form leibl respective distribution synthetic highlight effectiveness compare art decade great deal lie low manifold embed estimate term element lie entirely information follow use curse representation projection dimension minimal retain maximum useful furthermore suggest reasonable neuron capability depend empirical classification relate recently series crucial structure geometry research focus development present investigate dataset embed high space highlight report fail kind precisely note
comparison substantial effect evaluation recognize bioinformatics inference face objective letter highlight aware future develop protein natural science foundation china interest cn letter comment claim specify argument indeed conduct evaluation inference letter search parameter estimate point learn bioinformatics meanwhile unbiased research approach real application closely relate separate assessment final selection protein problem fig selection protein bipartite input produce protein protein assessment procedure evaluate assessment result optimistic analogous bias search calculate area auc performance ground description source code score allow positive ideally calibrate use ground true label assessment select final index assessment highly check search optimistic conduct procedure
code yet infinite learn learner even odd code notational learn version great result learn book anonymous feedback lemma corollary conjecture ask anomalous distinguished finite hypothesis infinitely often permit thereby answer pose strong learnable learnable failure produce family natural collection subset denote computable give code computable fix enumeration finite letter string text theory string natural depend initial either segment switch order list set use switch wish specify cardinality write mean symmetric computable enumeration machine code learn enumeration n mf sm I j identify enumeration set sm e inspection weak sense learnable learnable might j ci n theorem conclude remark make relationship notion anomalous search string learner code string exist construction string include enumeration never produce infinite difference index learner produce learnable begin need prevent family course string finite collection meet essence string hypothesis extension equal verify require natural thus sequence string extend construct sequence string next string sequence symbol use string give algorithm stage string yet empty possibility string least set process terminate end observe furthermore converge define number define conjunction statement substitute position return suppose
class randomly sample nonparametric specificity specificity sensitivity transformation correctly classify threshold score object particularly aggregate equivalent integrating unfortunately compare example score object curve incoherent relationship total freedom measure reduction way instance weight give proportion population proportion misclassifie statistic proportional misclassifie misclassifie equally etc requirement specify integrate misclassification classifier property misclassification classifier give problem reformulate calibrate score reference cost misclassification brief measure refer object represent total incur total yield rather normalised advance requirement exactly require different classifier classifier although researcher problem entirely distribution tackle distinct one universal response experience researcher wide universal class unbalanced another would rare symmetric would little would everything transaction detection transaction class size unbalance transaction phone call plus call bank likely less cost transaction easily run recommendation researcher another alternative class misclassifie incur object none misclassifie misclassification object class cost size unbalance small detection transaction transaction order magnitude attribute transaction transaction essence principle pick relative unbalanced situation serious mode several way leave open reasonable unimodal extreme result fully nevertheless suffer disadvantage treat understand undesirable form prior switch beta specify value wide mode sensible default universal default respective shape employ reflect parameter higher clearly guess normalise cost place mode cost context expert opinion type misclassification severe class guess invert single place guess misclassification burden specify encourage whenever expressive standard fact early classification solely threshold fundamentally incoherent different rule differently let propose measure choose objective belief consequence kind misclassification give work report belief suggest experience correspondence researcher world asymmetric cost distribution unbalanced c mathematics south college area roc measure fundamentally incoherent treat differently different overcome classifier extend propose match
error similar use correction notice ii minimum quality decrease e fine tune advantageous impose correction summarize base role adjust tuning correction small increase value improve know efficiency algorithm select surface factor surface value parameter neighbor correction structure precision apply mini cf neighbor correction mini r highlight highlight mini surface similarity rmse usage surface quality mostly image annotation sparsity yu absolute penalty group pp j overlap graph fuse svm pp joint subspace selection pp trust region mix simultaneous pp pp pp classification schmidt linear potential pp discrete r plus mit induce norm hierarchical f pp k representation learn invariant map chen compress pp pp plus scientific collaborative filter pp g neighbor netflix pp e email web pa email web international conference separation structure code dictionary learn novel filtering system numerical experiment present outperform several advantage structured dictionary collaborative online service alternative daily decision recommender rs recommend item among recommendation movie book news recommender system underlie rating item cf try rating make user recommender collaborative advances cf dictionary assume representation behind user product usually factorization include analysis one considerably problem observation enough minimal exist prominent approach enforce covariate structure tree sparse application ease sense number life benefit structure annotation group micro array iii transfer vi excellent sparsity code already representation find novel appeal transformation extraction ii background corpus face extend collaborative cf appear online typical reason appear time motivate advantage offline amount case small available rating cope use novel call online dictionary overlap non induce iv briefly numerical draw vector diagonal coordinate coordinate stand coordinate pp gx respectively treat idea structured observation assume give know observable column bound group call hypergraph vector available otherwise belong triple sparse code eq structure responsible quality coordinate similarly average dictionary factor td sphere child hierarchical cost eq manner actual sample estimate representation dictionary variational property introduce z g j z step minimize quadratic convex mean stability smoothing optimize keep fix minimum solving project diagonal b x otherwise update tc j c numerical approximation estimation worth improve update batch cf correction improve estimation cf task optimize group user task miss coordinate rating accomplished remove code representation neighbor may estimation neighbor approach assumption item movie rate cf detail idea individual rating user k prediction observable item error simple modification illustration benchmark detail prefer item evaluate element evaluate two rate remain rate user per similarity row quantity item mae quality popular mae measure square rate efficiency use mae section rmse sake completeness report rmse rmse item neighbor regression neighbor capability focus follow structure purpose sparse similarity correction affect numerical choose weighting indicator
directional spectral address value rigorously valid directional derivative rectangular repeat assertion prove full rectangular singular divergence matrix rank take convex relaxation np minimization norm proximal soft derivative take theorem strictly map everywhere derivative corollary simple provide sure completion encounter system netflix therefore low entry choose approximately rapidly depict function range single gain core proximal nuclear norm use recursively recovery regularization also automatically regularization derive mapping mapping set singular lemma locally write taylor expansion use product sort impose sign constraint directional j hermitian n v denote particular along apply system invertible inverse yx implicit svd neighborhood desire distinct composition function derive use anti fr paris france france paper recursively risk toward spirit sure derivative divergence solution close splitting iterate second challenge potential applicability automatically consider bound operator entail problem ill pose various research ill side minimizer non proper impose low remain largely typically want minimize quadratic risk value stein weak jacobian sure solely way contribution derivative split algorithm estimate finding proximal smooth yy study value svd singular value rectangular matrix main unitary left vector definition symmetric extend subdifferential spectral x u value argument number observe proximity operator spectral matrix spectral follow provide closed value value quantity
propagation iterative message pass compute marginal gain popularity paper new belief usual graph markov want underlie link form node variable together factor graph undirecte bipartite say mrf exhibit deterministic behavior exact procedure generally exponential resort computer procedure graph cycle apply several bethe minima bethe question bp series work mrf unique however multiple fix parameter bethe consist viewpoint looking condition single local quantify know bp normalization section exhibit case message provide bp paper propagation message belief include belief idea factor product contribution come message probability sum belief ensure constant converge follow compatibility often normalize define read worth convenient shorthand ax ib beliefs convergent practice refer express obviously aim policy pointed message belief strictly positive concern lead belief write moreover preserve belief resp resp imply indeed vector conclude look correspond invariance natural plain mapping span vector belief simply convergent iff converge space normalization normalization play normalize belief attractive jacobian modulus unstable modulus orient orient second ingredient node row unnormalized representation jacobian read jacobian plain bp pair element analogous jacobian encounter interesting depend graph simplify irreducible long always spectral cycle graph less positively homogeneous share look differentiable jacobian fix belief read summarize matrix presence message jacobian possible obtain belief indeed admit point generality restrict pair associate combine reference eigenvalue matrix fix associate stable I combine effect local give uniform degree correlation correlation variable stable fix order part consider local q w convenience somewhat I b reversible implie conclude proof deal express average stochastic ai jx ai ai contribution individual eq triangle inequality project ai ai ai ai ai end homogeneous respective level stochastic encode statistical connectivity average
quantile different relate also fail situation exclude covariate quantile consideration penalization special answering available censor therefore novel quantile major challenge censor regression quantile set sequential nature interact fundamentally censor iterative course generalization convergence additionally individual considerably scope applicability paper section realization way censor quantile result finding proof technical censor predictor censor covariate censor instead observe identically aim regard covariate study predictor quantile function combine idea reference coefficient iterative spaced define piecewise censor quantile estimator quantile censor computation directional satisfie present consist wide dependency structure provide point base see relate generalize solution equation definition distribution order negligible view estimate possible case censor version share start behave control survival version reason seem censor accuracy identification correspond vanish devoted penalization vector allowed depend lasso detailed penalization lead rate alternative way penalization avoid section state penalization vary zero technical ft px z ft corresponding generating process intercept maximum norm finite f uniformly denote impose consider possibly relaxed introduce additional leave restriction condition characterization quantile identifiable censor discussion refer contrast approach define converge center follow case setting violate note z denote interpret index vc chapter argument law function overview recent cite therein particular imply soon ergodic impose precisely imply soon converge towards process check establish literature class function dependent case dc mix time soon ready main hold eq equip supremum ball sigma algebra process uniform derive test censor discuss interesting theory q infinity arbitrarily order quantile g condition proof uniform censor regression converge censoring note tu correspond quantile classical aspect penalization process notation maximum non vanishing correspond mapping coordinate remain coordinate penalization power additionally number discussion give particular provide happen suffer impact quantile tf tf special structure eigenvalue statement view index vc subgraph continue c omit sake brevity ready main enjoy property tend coefficient remain coefficient p hold ns integral supremum see denote v asymptotic limit complicated special identity remainder oracle similar estimator technical omit brevity penalization statement theorem satisfy preliminary penalization adaptive lasso investigate condition result give partial answer question show optimal precisely turn exactly dependence sake brevity n basically reflect th enough affected penalization asymptotically coefficient tend imply intermediate assumption define aa hold preliminary consistent penalty encounter traditional avoid consequence quantile analysis quantile covariate impact exclude take use highly flexible adapt conduct presence censor weight weight value note weight weight censor function divide denote index kx minimizer basic idea procedure estimator weight consistent might quantile quantile curve total sum change sense invariance move censor lasso well average quantile finding estimate obtain penalization estimator behave penalization systematically component systematically intercept ht average row correspond censor roughly regression view wise penalization slow probability coefficient observe slight systematic advantage penalization quantile slow shrinking reveal picture suboptimal superiority penalization become ht local proof brief main prove result general coefficient major role aforementione coincide derive ingredient representation give detailed fact dc denote lebesgue sketch j imply constant minimize j kn j simplify minimizer find part observe yield p kn kn kn second part w make begin part simplify ni b kn last moreover part c assumption begin eq combine derive p n remain assertion exist would also minimizer component choose contradiction line kp kn dominate proof uniformly consistent quantity b b c r c n p result point additionally complete proof array random q assumption hold minimize let assumption hold obtain proof point difference major step note prove eq estimator supremum consider grid hold follow classical argument yield omit brevity condition finally front iii iii hold c n yield I n n algebra yield show complete iii combination establish lemma n n nc mc nc mc b nc h nc mc j db obtain j yield v r cb w nu integral denote value ji tw tw assertion induction suitable next summation u tw ji tw w end cb dc yield ji u u v u v nu u k cb finally u proof convergence yield ns v eq p taylor combined algebra statement matrix u v uniformly prove satisfie th moreover separately finitely interval without solution base prove estimator uniformly argument uniform additional consider exist consecutive contain large tend n careful inspection show continue eq quantity c n n note large ns lb nc n nc assertion establish let hold follow iterate ns yield computation nc pn assumption argument whole definition universit censor case quantile traditional penalization adaptive yield regression cross penalization introduce deal censor yield assumption kind keyword phrase weak censor
setting behave misclassification pl di sim former require therefore di slow setting describe detail bernoulli simulation outperform di sim algorithm almost conclusion profile h ccc pl b l ccc km b b describe microarray activation gene distinguish dataset microarray datum contain measure different dataset belong age study relationship expression residual log gene error gene independent zero gene block conclusion hold residual negligible light consider exploratory perform use profile preliminary appear representation result fourth fairly gene visually appear expression gene group high gene group find cluster suggest interaction gene behavior well new individual since vary product identify product application apply result split period individual rate movie sufficient base likelihood apply dense comparison variety situation procedure theoretical operate assumption row simplify selection future reveal finding find high level result review behavior individual movie period sim negative normalize example simulate block row column membership chen class ij ccc c km simulate cluster number column membership matrix row ccc km b n b standardized student parameter conditional column ij h ccc km ds b profile log criterion perform three example verification finding data review movie user rate movie rate five rating release movie gender code retain customer service movie individual already movie rate likelihood bernoulli determine vary movie seven qualitatively movie group seem data assignment cluster report movie within suggest rating explain alone release order movie median release date gender effect mirror face air brain fan extreme group day attack reservoir water group contact return future english patient toy air force roughly speak movie group activity popularity consistently inactive movie rate movie group movie discover suggest recommend review movie individual recommend engine corollary variable popular effective tool apply successfully source range review website currently practice sufficient condition consistency infinity apply broad bernoulli microarray collaborative filtering suppose row example gene patient seek patient profile find group similar activation alternatively indicate review goal simultaneously movie term cluster block co broad help collaborative diverse difficult purely base biological science one gene apply method drug identify association medical several well comprehensive survey goal reference hoc lack notion generalize collect lead different report formalize profile consistent tend infinity assumption notably handle general treatment recent binary cluster consistency network nod belong cluster individual specific effect cluster spectral develop extremely generalized simplify knowledge main text organize setup next finding study microarray remark additional technical result formalize follow determined solely column vector membership unknown refer undirecte goal assign true analogously profile assume bernoulli propose likelihood general simple misspecification element draw family conditional entry respect likelihood deviation derivation scale terminology deviation literature refer though require rate misspecification technical maximizer column constant exist row column class membership vector surely multinomial ambiguity subscript mean depend membership tend avoid confusion proportion entry similarly nontrivial neighborhood bound satisfied long handle element infinity word heterogeneous place restriction consistency variant away moment certain bernoulli data zero setup maximizer true row label f limit proportion row probability label multiple permutation maximize profile log furthermore result specify model misspecification datum could poisson make motivate distributional assumption method example outline constant hold neighborhood confusion sequel version neighborhood around true class proof literature symmetric symmetric binary network work whereas type paper work criterion potentially unbounded lastly criterion continue section notation establish uniform assumption matrix version version sum next locally lipschitz appendix maximize confusion close permutation independent fix inequality choice imply permutation misclassifie set g proportion n permutation study profile bernoulli standardize misspecification immediately obvious poisson function poisson sum equal maximize identifiable try simulation maximize though
jt jt I jx I I case hmm I forward x x x k potential running intersection hence u j although prove theorem x x jt eq partition eq achieve hmm exactly j I I follow k x k x next dd dd dd dd dd dd dd x x dd x dd dd roughly dd dd dd x dd dd dd dd dd hand hmm forward iy I forward since case jt direction subtle call message opposite message compute remain see involve recursion recursion another back jt perform recursion possible message process backward recursion x recursion possibility refer suppose matrix leave establish introduce message center without complete extension hide evolutionary loop sum propagation replace sum max pi lambda lambda lambda x col blue x col forward lambda e tf pi f col col col col sampling ps sample ss ss ss pi ss ss I I ss pf c pf x pf x pf pf k x x x x x x x k x x digits digit distribution x p p k x p p k x p x p x x x cp cp cp cp cp size cp cp sample cp cp cp cp exact propagation achieve provide contrast variant define forward derive introduce message recursive message consider along formalism code provide probabilistic graphical powerful like hidden hmms inference quantity use multiply variant basically compute point message hmms hmms message derive exact message implicitly algorithm objective forward hmms introduce paper follow hmm illustrate approach illustrative result derive code mm simplification pressure denote pressure day transition poisson eq typical trend pressure phenomenon pressure problem convention prove solid I b backward compute eq prove forward leave right term recursion simplify produce quite reference fig posterior l l l h l l l l h l backward conditionally heterogeneous direction backward forward direction numerator denominator use corollary discrete acyclic note thanks acyclic call ex var var var node var node var var edge thick thick thick edge thick thick thick dag parent set generic fig represent relationship loop cycle allele non allele indistinguishable parent allele general population text depth var var node var var var var x edge edge edge x thick thick introduce subset empty evidence hmm evidence disease disease individual affect affected affected disease status individual evidence dd dd dd dd dd dd consider jt connect cover u jt
significance show six code lee filter result rest lee edge consider situation c figure lee look universal quality present ground metric j display filter e sensor band nominal look al look homogeneous lee filter image smoother highlight almost corner visible apply lee stand filter hellinger windows level table assessment filter hellinger order index lr lr lee lee lr h h lr filter noise protocol carlo showing application real present use behave filter yet assessment proposal com paper filter distance pixel compare use filter vary look change heterogeneity window compare lee criteria quantify employ line edge assess optical noise information ta si model development plausible image statistical propose literature datum use multiplicative intensity employ describe et lee filter protocol carlo filter edge index pearson filter quality filter draw pixel outcome characterize specified deal homogeneous intensity image gamma look gamma obtain employ figure filter nine denote central estimate eight account homogeneous maximum look heterogeneous area estimator namely distance test modify level significance whole derive distance include due flexibility look vary kl r n decision hellinger since filter checking come block set reject local filter central around filter assessment performance par technique preserve follow assess two datum version quality filter et sim u corrupt image quality compute assess analyze
demonstrate value go distance two function minima apart shall fix oracle choose learner query domain function apart identify wrong horizon proof lower bind probability prominent active proof convexity state ff f unique return whose sc hence ensure depend ensure part convex maintain interest constant enhance return ix whole distribution correspond divergence p condition identity half substitute give jensen bound large conclude recover exist low strongly setting tight appendix batch proof immediately give tight bound derivative free optimization gradient exponent conclude strong help show satisfy fx fx derivative yy tp tp x f initialize tt tx x et tight query f f may variance keep use radius condition convex derive diameter g fx x fx g fx f appropriate probability use induction first induction probability correspondingly epoch event c e induction like condition probability c trivially g strongly round query alternatively use assumption prove convex class behaviour strength smooth hierarchy natural imply convexity correspond low classic demonstrate proof novel free generalize rate support see behaviour noise jump sign corrupt identify optimum extremely tight epoch gd extremely optimize lot value achieve correct number step noisy substitute strongly recover order proof bound condition point every small large use strong unknown value function old grow elsewhere behaviour bind active give hope intersection active optimization polynomially try hard problem boundary level interested get subroutine generic reduction set give problem conceptual field research support grant thank input satisfy convexity uniform say positive taylor grow go grow least slow need uniformly fx rearrange since eq eq result random variable characterize random around want integral rectangle small height tt px te tight inequality suffice return work divide grid interval x modify initially align ie mx du elaborate avoid grid align complicated variant grid exposition I firstly lipschitz norm bound x growth convex dimension constant ix fx h first q convex special argue rhs argue ty ty fy query complexity determine growth quantify like specifically least strongly function attain tight bound complexity interestingly rate characterize active connection rely drive oracle generality everywhere always deal furthermore convex diameter give function oracle unit let later analogously function repeatedly query estimate optimum question way formally convex small condition mean quadratic everywhere everywhere strongly geometric question arise minimize work determine intuitively connect central concept tight possibly equally point claim determine minimax rate growth optimum strength rate around optimum everywhere later form hierarchy sc state decision relationship seed function precisely rate hierarchy tight algorithm reproduce get free bound stochastic generalization say equivalent weak grow margin literature reproduce version ie decision satisfy exponent
acknowledgment thank manner multiplicative univariate look inverse law reference attractive describe space modify inverse moment moment belong multiplicative describe product variable outcome complex wishart rejection normal deviation correlation verify whose generate transform boundary detection active contour region spline specify contour boundary model area synthetic prove among mention map update ice monitoring among due sensor day illumination sense device weather extent devote application frequency imaging band sir able receive vertical capability provides value require specialized follow description see feature observation alarm treat design work propose al perform em detection maximum image segmentation law approach base description every edge ideal area mechanism property one index harmonic knowledge combine contour operate whole considerable virtue spline curve allow noisy image base spline contour criterion et techniques spline boundary tailor present attractive proposal begin manual specification region spline curve radial around point belong contour give assess increase capability statistical specifie assess edge detection real conclusion algorithm random design receive vertical wave receive ideal subscript indicate subscript indicate signal processing enhance complex q denote look summation hermitian real paradigm product random unitary variability heterogeneity random mean corresponding look exhibit multivariate complex complex wishart present density determinant matrix function integral compute q consider random unitary mean harmonic channel law provide law model distribution close issue impose parameter moment diagonal eq nz ig ig univariate th return total error I necessary relate correspond hermitian display font lb lb lb lb lb lb diagonal account I I feature intensity immediate area heterogeneous forest grow homogeneous area type regard texture intensity correlation three check describe look image exhibit look generate every look intensity target estimate forest target frequency allow htb value derive number look preserve fit excellent figure show synthetic image show background figure assess contour et method real adapt spline convenient representation spline function curve parameter control reduce effort polynomial arbitrarily relate smoothness control control control individually curve spline representation contour work sophisticated could description length present begin rough segmentation manual automatic scene boundary want curve fit spline develop initial specify automatic initialization perform ns homogeneity language text threshold correspond consider region also natural manner block array complexity big window evenly target mark else consider discard form convex compute automatic determination connect whose spline alternatively user search determine centroid proceed belong change candidate angle consecutive centroid interior object contour outside segment discretization array array sp font pt lb segment parameter segment slide mask consider method build curve candidate boundary determine segment rectangular window around segment detect set return spline methodology employ assess edge detection image establish true propose image segmentation algorithm surrogate truth manual mathematical boundary manually create possible option measure local use amplitude point image show right show vertical white dot situation replication situation estimate distance boundary th relative error pixel denote replication big algorithm simulate image find eq situation area model besides indicate horizontal consider three present three channel greatly capability accordance result exception situation fail fluctuation content individual notice three channel easily
several important fact implication theorem unlabele algorithm request example round unlabele unlabeled round learn sufficient passive consider nearly log nearly remove isotropic distribution admissible mm unit p c proof without admissible arbitrary active label passive example fx concave log distribution whose matrix whose prove isotropic concave tail disagreement claim start isotropic log within exponentially oppose argue tail log concave concave ec c u hx whose hx e x fact property desire sufficiently non constant isotropic light tail project isotropic isotropic well isotropic concave center constant small apply whiten see show generalization except since isotropic associate small log passive example bound complexity passive active theoretic procedure appendix small whose covariance center passive matrix effectively essentially consider closely capacity notion use begin definition p x w dr cd disagreement coefficient appendix small constant theorem immediately concrete result obtain dr dd request condition minimize generalize margin constant label example excess use condition passive improve dependence bound improve far appear sample achieve use amount computation unbounded seem hypothesis repeatedly unlabele example evenly split candidate eliminate roughly candidate would label constrain conceptually erm concave localization literature useful part nsf grant microsoft fellowship technology microsoft concern passive pac passive building computationally unit tight infinite hypothesis significant progress separable agnostic nearly concave erm low noise condition agnostic challenge long seminal accuracy pac time well lower improve polynomial open distribution resolve belong include convex area research classification exponentially passive learn answer active exponential improvement remarkably passive connection active case linearly specifically provide new widely disagreement coefficient case form passive label instance label unknown classifier new study play crucial providing algorithm tight especially imply learn example lower explicitly pose providing polynomial polynomial erm intensive local really bad hard rich bind far however suboptimal complexity ignore eq yield complexity polynomial specifically algorithm concave provide interesting non hypothesis function constant complexity gap ask unlabele hope classifier past year mainly availability large amount raw modern application development principle call active paradigm analyse conservative strategy amount consideration label far manner versus analyze active prove request answer passive learning improvement distribution uniform ball even multiplicative dependence inefficient splitting novel characterization disagreement constant region disagreement theorem fact use margin active round learn passive least hypothesis interestingly quite classic analysis erm conceptually run algorithm hypothesis tool active likely show would use preliminary failure later stage presence extend receive passive bound optimal bound label request passive example generality selective learning request previously change upper selective section make p xu throughout nearly log concave play decade area include integration theory extension nearly concave concave isotropic mean origin covariance broad uniform concave summarize fact isotropic upper density mm f ad isotropic isotropic fact universal constant two homogeneous time angle vector project disagreement implicit completeness include isotropic analyze provide disagreement log choose enough hold let imply imply turn ball carefully upper bind include integral eq exploit rescale probability distribution let volume return c complete weak prove unit ball addition tight passive analyze prove fewer passive somewhat ellipsoid except instead update multiple instead one c maintain instead constraint new target old small ball previously analyze ball oracle sequence cut draw put mm hypothesis datum reject label put isotropic lemma
inf inf inf inf inf inf inf inf os inf inf inf inf windows inf inf inf inf inf inf inf inf mac os inf inf inf inf window mac os window window os herein assess table beta except binomial compute binomial poisson student quantile lr em inf inf inf lr lr lr em lr em e lr lr lr inf inf inf inf inf c lr lr lr e e e lr em em e r lr lr em e inf inf inf offer analysis divide level two two seven explicit em r lr em lr em five numerical computing standard test dataset lag quantity none test bad platform respect variation note conclusion turn subtract instability perturbation original input change deal student also cumulative poisson matlab better version al platform produce binomial law normal matlab produce acceptable deal matlab fail return message serious use statistical test six consider matlab windows operational system platform safe result make determinant ill condition operate acceptable logical comparison careful equality interest involve assessment worse sensitive eigenvalue double test latter balance bipartite spectral double comparison equivalent situation km com com com use computational matlab architecture operating system os set function platform conduct verify include data national institute technology protocol include computation basic univariate lag correlation assessment compare digits secondary spectral statistical mathematical modelling rigorous computer model diverse area bioinformatic rely pde ordinary ode number effort final often finite mesh use operation rarely check approximate consider reasonable bound tight computational library function huge structure partitioning parallel need correctness either method assess algebraic graph mesh mesh dual element common associated graph partition eigenvector determinant decision offer library calculation problem little attention assess diversity operational accuracy institute dataset besides statistical employ assess viewpoint numerical scientific operating window sp mac os whenever hardware precision organize discuss assess subsection subsection conclude simulation arise model error computable solution discretization error instability availability limited operational system accuracy author instance measure provide obtain national institute consider measure assess mean deviation autocorrelation quantile base logarithm absolute value error approximately match assessment convention place correct digit inf perfect match platform compute platform numerical check stability propose platform deviation lag compute dataset obtain original platform produce error observe belong second deal first computation propose assess q accuracy logical platform assessment another assessment laplacian graph finite without loop vertex note eigenvalue connect call consider algebraic bipartite two say vertex subset connected cardinality laplacian eigenvalue assessment balance almost example size assessment denote percentage eigenvalue equal correct answer window os mac univariate assess lag nine classified dataset difficulty come world difficulty value precision arithmetic digit compute std st deviation inform return divide deviation n
regression example individual association gene genetic association snps genome association attempt relate variation many genetic variant snp association modeling phenotype explain available phenotype despite opposed genetic variant affect phenotype regression selection assume small affect phenotype performance application expect vary true genetic phenotype genetic architecture unclear prefer sparse mixed hybrid model case capable genetic idea use genomic prediction trait trait jointly compare detail model particularly emphasize benefit parameter provide exploit algebra avoid ad hoc approximation sometimes reliable result thousand thousand marker involve naturally potential use association specifically phenotype marker e genome wide potential underlying phenotype could ultimately application datum successfully combine variety setting outperform model phenotype method motivation begin consider phenotype individual individual genetic marker genetic marker effect vector code allele marker column see detailed marker accurately kind modeling assumption genetic become effect equivalence compute brevity paper restrictive compare usual definition commonly refer statistic combine commonly unbiased predictor assumption dimensional come mass proportion convention variable regression commonly phenotype phenotype distribution put mass variant assume effect assume large include effect mixture two normal assume proportion effect normal regression hyper include assumption flexible special obviously still normal mixture normal include alphabet model name genomic selection summarize specifically effect use component answer effect way model practice term treatment term greatly affect example make flexible flexible among certainly consider advantage although hyper issue tool less turn description specification focus linear induce prior effect terminology terminology refer effect instead emphasize mixture prior discussion specification refer imply equivalent symbol whereas effect application estimate although estimate genetic effect direct interest relate effect component specifically small marker environmental phenotype correlate affect addition environmental measure issue relevant give observation normal flexible natural modifying come mixture point modification change specification specification summarize phenotype control zero control expect magnitude non zero estimate framework chain carlo approximate imagine bayesian incorporation external e individually effect take inference variance q gamma arise limit improper posterior property measurement phenotype seem desirable intend specification extend marker upper limit marker reflect uncertainty put number marker correspond place event place probability prior follow easy term interpretable quantity variance effect phenotype genetic interval reflect predict phenotype snps together well predict alone function depend brief choose one reasonable relationship hyper make ratio expectation ratio variance marker mean matrix text derivation marker effect marker marker genetic term relative error properly scale provide bridge would near datum mass summary term prior distribution rather specify prior use uniform incorporate contrast seem hard directly specify reasonable default prior course note approximation prior estimate directly use definition computation approach introduce coefficient point write sub corresponding use mcmc detail use marginal marginal association sample sample value evaluating burden involve determinant calculation individual incur hyper parameter consequently impractical ad hoc commonly burden phenotype exploiting develop eigen unitary eigen vector phenotype make transform follow extend evaluate efficiently burden study analyze software implement software result assume snps small proportion snps genetic include hypothesis perform real snps phenotypes scenario simulate effect come genetic architecture simulate group causal group number plus snps represent realistic scenario phenotype create replicate variant marker either exclude mean estimate method estimate perform best rmse scenario figure poorly strong bias conversely method although approximately unbiased wrong robust wide true model advance make setting appealing recommend despite meet simulation phenotype phenotype brief mean square phenotype negative accuracy compare qualitatively interestingly phenotype differ scenario scenario close number causal scenario involve small capture equivalently due capture actual distribution effect pattern simulation comparison exclude snps potential explanation much phenotype effect reliably individual assumption poor tend effect reduce phenotype trait trait datum datum contain snps contain include density tc snps narrow trait isolate population tc three trait estimate almost trait explain narrow trait suggest trait substantially consistent substantial attempt precisely additional snps specifically come distribution also environmental estimate generally trait range tc suggest contribution trait disease turn trust case assess include disease share snps seven disease disease diabetes diabetes seven disease split individual perform split estimate treat binary quantitative taylor binary phenotype set auc seven perform disease performance result study strategy expect well number association qualitatively although well disease unlikely human clinical control generally disease relevant quantitative phenotype heterogeneous stock performance compare phenotype small human vary slightly wide five propose phenotype bayesian modification perform poorly simple greatly improve general fix online distribution component obtain computational burden fit similar lee posterior text focus treat control status interpret phenotype prediction one might modify probit probit find slightly bad treat binary outcome quantitative phenotype partly due considerably nonetheless burden remain heavy store substantial computational resource somewhat large currently collect fit moderately hope also help highlight principle characterize certainly rather important often interact conceptually characterize easy predict tend importance size assume normal flexible point phenotype method flexible well less flexible distribution importance estimate hyper parameter phenotype illustrate benefit estimate hyper integrated rather two procedure hyper focus proportion phenotype explain genetic component notation could helpful method use answer distribution underlie sufficiently flexible extent distributional mixture normal mixture degree freedom produce gain seem likely use therefore preferable expense effective flexible component effect directly unclear necessarily computational expense integrate ensure fast difficult predict size ultimately provide nonetheless appeal flexibility performance thank height thank software thank helpful comment manuscript research pi fellowship science program use full project trust award contain height measurement individual datum set phenotype age effect normal quality snps exclude snps allele imputation consist consist individual study phenotype present high tc phenotype quantile normal correct mass index status quantile snp array snps minor allele third trust phenotype data case seven disease case diabetes diabetes well control obtain control result snps minor depend split come heterogeneous stock consist individual family eight phenotype performance phenotype measurement set phenotype percentage cd cd phenotype previously method narrow cd median phenotype correct year quantile phenotype available individual allele frequency phenotype human height simulated phenotype effect scenario phenotype datum causal come fixed causal size distribution size come causal snps snps small medium effect size leave causal snps size addition medium effect explain proportion snps effect error mainly rescale call predictive predicting error covariance snp practice snps linkage snps snps I rescale apply formulae ensure result account random correlation also assess value observation depend q formulae estimate center measured marker marker code allele simplify center throughout unit center affect column assumption marker relevant throughout copy reference marker center matrix matrix center effect generally multiply side result semidefinite expression take parameter extension slight expression simplification expectation approximate numerator denominator rough expectation expression conditional matrix derivation assume center plug approximation prior solve give independent induce marginally decay another nice marker marker effect increase estimation previous use generalization restricted respect evaluate despite derivative approximate q correctness also bayes give identical phenotype estimate e effect multivariate theory conditional predict phenotype observation correlation correlation size present give give observe rao text add method fit fix example software double coefficient denote parameter gamma rate study scale stand degree freedom posterior similar key difference treat pre whereas specify prior greatly sample modify available online software fit fixing avoid update approximation study burden intuitively fix effect size prediction snps base except run due server million assume markov carlo explore marginal n perform decomposition eigen transform phenotype matrix multiply eigen previously calculation determinant iteration mcmc hasting posterior hyper marginal walk proposal particular snp standard rank snps small put snps rank high snp test distribution truncate denote add covariate switch pick covariate step switch hyper add new boundary reflect back proposal use improve mcmc move obtain h dimensional instead sample never normal diagonal simulation formulation iteration sample rao mean q end scheme fit comparable complexity practice maximal number causal text binary refer link trait row cumulative cdf auxiliary vector specification hyper mcmc text posterior posterior sample slightly q hyper identical calculation transformation need iteration probit set transform quantitative trait assign individual higher quantitative half consider approach probit view probit counterpart score exact probit calculate figure difference trait trait treat trait use probit partly reflect burden factor provide alternative derive notation proportion cdf pdf
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image follow another rbm family introduce binary binary representation include layer form binary binary layer refer encoding conditional factorial give advantage layer close match value binary deep ph training train second either cd good train cd generate get inference exactly discuss pass possesse sharing patch texture structure feature field layer hide map across map layer associate layer weight field mm texture texture texture dataset stochastic gaussian rbm high fidelity remain challenging model deep ccccc synthesis bi tm multi tm texture unit synthesis relate model fair protocol rescale original either texture divide bottom half testing report layer task synthesis model size minibatch deep always cd high mix negative relatively begin persistent chain especially possibility advantageous mixing model follow texture synthesis texture layer cd cd inductive impact unchanged filter field imply difficult task handle add convolutional share layer limit show result texture measure texture measure present texture synthesis inspection impact depth inductive versus cd texture help explain improvement mix gibbs chain improves already argue performance train negative sample improvement mix autocorrelation spectrum markov plot sample chain find layer texture specifically worse obtain consistent regard advantage vs consistent sub train divergence visible stochastic something cd cd help make sure information rbms job avoid spurious rbm mm deep texture heterogeneous train layer rbm sec filter convolutional rbm layer generate model apparent sample particularly structure rbm label easier share texture single texture amenable cd suit even effective interestingly find cd low deep integration development finally high apply boltzmann machine modeling achieve art texture synthesis parametric rbm layer layer result deep belief powerful generative capable model single resolution texture human image certain extent progress natural progress area graphic although capture property probabilistic model model model human vision boltzmann previously demonstrate sample preserve would texture model share weight filter convolution particularly choice texture model texture capacity texture concentrate evaluation texture exhibit strong largely consist regular repeating right wide wide decompose use pyramid interact spatial convolutional generative architecture depth part belief texture layer concentrate find training cd maximum persistent offer important similar model amenable cd contribution texture model model competitive art texture synthesis novel rbm stack belief train strategy layer third able encode dependency cd train translate texture model exhibit stationarity help introduce capable label constructing mode texture belief capable purely unsupervised community decade probably popular texture synthesis currently base method typically seed transform patch texture unlikely readily applicable natural statistical track scope rbm quadratic visible activation unit configuration change several student visible unit hide mean hide work multi texture boltzmann tm single oppose multiple texture deep texture impact layer model help texture feature recent stack rbm layer top convolutional ability globally coherent finding motivated attempt coherence three set visible random spike value unit work concern spike contribution energy joint interesting despite maintain bipartite boltzmann machine th consist share conditional restricted use block conditional mm covariance represent logistic h nh root efficiently block gibb also persistent cd involve negative phase likelihood phase maintain approximate negative gibbs approximate regard block gibbs mark important distinction monte hmc draw likely vector amenable gibbs ability efficiently sample amenable unlike
collection index composite term analogue give modify shorthand oracle correspond minimum output procedure furthermore inspection logarithmic corollary application discuss process satisfie like theorem follow establish analogue proposition coarse extend prove grid shorthand sample definition shorthand arbitrary event technical lemma choice replace constant specifically select event somewhat event hold hold strategy use lemma fail union lemma choice analogue grid inequality eq transfer entire suffice substitute theorem nest collection collection consider model forest family collection dictionary wavelet polynomial obviously challenge limit understanding outperform relate model impossible class least must allocate finite approach trade class class selection one quantum time linear box constant f assumption difference penalty successively choose iteration balance compete optimistic intuition behind definition class encourage optimistic formal begin preliminary optimistic budget example n index goal selection excess attain limit choose f gain I strategy non sequel requirement follow allocate quickly proof alg round round definition result fraction budget allocate procedure budget allocate rate vc dimension satisfied constant bind multi armed nearly general see connection multi bandit reward suboptimal plus always intuitive class try distinguish amongst visit suboptimal good oracle quantify risk remainder assume dependent result aggregate function appendix choose round alg define function vc clear factor ignore selection arm set reward force excess must scale split computational budget class roughly small key distinction logarithmic small improvement I split budget across class computational outperform I online nest case apply grid replace theorem similar particular setup risk outperform allocation multi armed bandit start give choose receive small recall notational q q lemma class error occur much risk high risk assumption three sub index choose need thing number alg interpret implication invoke obtain concern indicator markov examine modify multiplier new framework idea main attain computational selection nest take space competitive extension complexity problem exploration obtain optimal certainly raise address dependent penalty adapt instance somewhat would another question ask intermediate selection correspond understanding broadly believe number available performance broad hope risk even computational hope extend affect quite acknowledgement cl version read argument perform support microsoft fellowship support engineering fellowship acknowledge award dms arc award start recall shorthand monotonicity increase inequality monotonicity establish simple establish shorthand usual q finally proof suffice probability eq event occur q choose minimize right eq follow event apply union final empirical second part non combine event complete event far apply early bind inequality occur since define apply guarantee find rearrange lemma let hold recall grid induction inductive q assumption event hold claim claim condition inspection two possibility apply inductive use monotonicity assumption inductive noting complete occur break amenable analysis control event happen three must happen shorthand relationship none eq string event bad consequence q assumption imply probability particular remain recall satisfied solve immediately occur complete proof proof risk e fraction budget allocate model class argument drawback select even model theorem averaging argument frequently online define divide union penalty class differ get note proposition theorem technical somewhat minimization objective multiplier lagrangian infimum derivative see I I px proceed shorthand clutter definition minimizer see previous make drop probability jensen replace sum bound I immediately complete average risk department sciences california berkeley usa penalize empirical minimization practical model effort require compute minimizer budget satisfy setting ff minimal tradeoff class q excess error error common address tradeoff express class choose tradeoff standard approach early compete objective empirical class approximation error rise example therein complexity penalty perform penalty decrease follow output selection author penalty give roughly possible incomplete include paper give complexity trade error optimally drawback limitation feasible low prohibitive modern form key inferential consider rather quantity effect specifically difficult assume expensive computational model rich strategy criterion class allow receive thus overfitte class address issue make consideration provide many different result address order competitive know logarithmic extend rate condition guarantee concentration risk oracle refine regularize risk consideration class share structure novel exploit base prove select computationally paper organize estimator oracle model refine rate selection model technical argument need main corollary work setting class inclusion increase give natural proceeding computationally first perhaps example aspect computation inclusion natural minimizer say computational many estimator estimator formalize give penalty budget formalize notion problem model computational class part mainly result rule degenerate case upper sequel shorthand clear certainly many study penalty e extensively satisfied example loss unless model selection grid particular penalty continue hence large inequality least class hierarchy j class place model main explain furthermore unless trivial must computation brief remark ask oracle access could optimal entire budget satisfie compare hand know optimal budget logarithmic computational budget require priori g accordingly round computational update guess suppose round budget round get budget last round length oracle replace bad ease set theorem arbitrary may turn setting inequality specify process output sample process computation output algorithm satisfy proof slight generalization satisfy notice suffice additional notation generalization choose
metric agreement weight kullback area metric base metric area metric validation mean predict absolute error method reliability relatively less section uncertain two variable random probability discussion derive experimental relatively sample assume validation test eqs test model reliability letting decision decision acceptable within directional modify interval bayesian interval original interval half fail less case quantity predict deterministic zero case model metric propose al attractive capability fully pool validate experimental cdf observe q statistic probability standard match cdf cdf uniform variable small correspondingly observation deterministic applicable reflect directional due cdf gaussian random gaussian small variable distribute area cdf reliability interpretation decide reject accept disadvantage aforementioned prediction quantity predict coefficient uncertainty characterize compare device rf require reliability device rigorous quantification observe reliable device framework quantification strongly affect switch micro available regime micro illustration study major prediction failure mode variability quantification simulation surrogate krige r combine failure failure system integration denominator integration numerator although different test factor testing failure test reason pressure pa pressure pa failure percentage equality hypothesis table observe performance pressure locate pressure pa interval calculate reliability failure compare percentage failure pressure existence directional pressure pa vs failure failure r vs number four computed eqs result pa perform bad directional mean reflect pressure pa surrogate rf switch performance test middle two pressure pa pa whereas low pressure pa failure reliability reliability base metric failure base test bayes reflect bayesian testing base directional bad classical testing equality testing hypothesis reliability validate polynomial predict micro coefficient rf standard directional testing validate entire model formulation test fully characterize experiment hypothesis subject minimize average use mathematically relate reliability prediction reliability reliability incorporate failure reliability device explicitly account direct testing method validation paper upon partly support national nuclear security award na provide quantitative investigate bayesian hypothesis testing validate interval hypothesis consider measure report measure testing properly choose avoid error reliability existence may consistently specific reliability metric mathematically metric conduct frequency surrogate run fully characterize testing process real perspective comparison prediction widely supplement systematically effort application validation development prediction experimental discussion base method unclear validation fully characterize characterize directional threshold validation result characterize experiment input interval expert range partially characterize partially uncertainty input characterize limited practitioner input assume validation term input experimental thing physical variable experiment diagnostic quality input component measurement represent previous study validation experimental explore validation existence prediction physical predict literature deterministic value variable case study validation directional bias directional direction remain vary explore validation account existence directional fourth usually validation datum show condition mathematically rejection various quantitative include test hypothesis deviation entire two characterize characterize bayesian modify account directional classical reliability method present quantitative characterized quantity predict uncertain directional mathematical validation example switch generalize polynomial replace expensive micro simulation device predict quantity comparison experimental response prediction base relate type available partially experiment characterize measure physical quantity uncertainty source uncertainty modulus variation micro structure output input stochastic input report uncertainty lack knowledge probabilistic paper uncertainty value uncertainty uncertainty note quality validity discussion due uncertainty measure experimental limited section h predict deterministic partially interval equality reliability area treat select computing decide accept prediction metric fig validation involve plausibility nan alternative hypothesis could whereas coefficient prediction model coefficient pass exercise hypothesis reject incorrect validation matter necessary hypothesis especially formulation acceptance model prediction physical classical explain detail statistic facilitate hypothesis statistic formulate distribution derive theoretically statistic outside nan good test statistic statistic outside information decision whether make acceptable maker define exceed acceptable correspondingly accept alternative value reject note acceptable significance sample value reject level interpretation testing test equal equal hypothesis come kolmogorov test relatively thus predict normal random normal mean observation datum random freedom model equal mean end obtain cumulative distribution degree freedom reject fail estimate instead standard deviation standard nan tail hypothesis commonly similarly note test directly compare input form interval unconditional unconditional dependent unconditional model uncertainty reveal relation occurrence probability occurrence event occurrence q true calculate probability represent prior validity collect conditional original bayes support factor distribution formulate differently propose base nan hypothese formulate deviation within existence bias formulation distribution parameter either deterministic applicable standard deterministic formulation case two formulation characterize tail deviation may validity moment standard purpose observation random hypothesis derive practical issue underlie therefore calculate bayesian testing divide straightforward deterministic directional set hypothesis fail test directional increase chance combine illustrate combine test use overall datum first interval union support predict formulate correspondingly predict quantity e case observe pdf alternative assume informative pdf uniform affect estimate factor proportional eq parameter bayesian testing pdf partially pdf base report calculate unconditional prediction characterize partially separately equality
occur eq determine far achieve predefine common false run chart occur change positive desire determine classifier predict whether prediction stream stream view detect bernoulli although slight far may situation since affected drift far throughout concept drift increase error rate work subscript standard deviation estimator several modification streaming detection know whereas classification must stream estimator weight less sensitive therefore occur quickly new whenever distance exceed standard deviation standard q implication choice desirable detection positive false likely occur point give deviation keep false positive vary limit recommendation choose post advance drift chart stage limit change decide procedure inverse base carlo method conduct carlo choice carry monitoring begin overhead unknown positive control limit time vary add subscript would whenever update detection chart adapt detector approximation single monitoring begin overhead concept required various stream begin extremely fast approximation follow various monte adequate several derive suppose positive give use misclassification threshold table detect drift call drift stream classification choose classifier value critical sequentially time predict label correct incorrect table current concept modify take depend rest discard begin classifier classifier correctly classify incorrect presentation classifier whenever stream change change store train store practice store order threshold occur stream retain drift classifier later drop conclude false slight equal reasonable occur concept take detect suppose drift detect soon relatively value since pre change distribution recent pre magnitude concept drift generally detect magnitude class distribution switch change pick synthetic drift pair evaluate stream discriminant lda chosen write streaming choose utilize complex boundary knn recursive predict observation knn stream detector detect memory retain change rest discard detector user evaluate detector give performance prefer find give generate accuracy concept drift detector investigate value value emphasis place omit space reason sensitive choose rest gauss gauss compare pc realization change detector concept detector performance without positive occur fast occur less positive change detector highlight importance detector change algorithm method threshold incorporate improve possible performance difference detector detector affect affect difference significant follow use algorithm due simulation less gauss gauss lda lda pc knn knn knn pc knn gauss gauss lda knn knn pc knn previous change however cause change change investigate encountered majority drift change drift move concept drift static perform suitable modify gauss stream initially static drift period previous modification equivalent change point allow drift occur increase moderate c pc knn knn competitive drift outperform suggest choice concept may take drift limitation real world image advance concept drift stream incremental real drift detector serious artificial set desired however change control pc detector key limitation evaluate detector wide setting give good practice varied detector range pc detector free price collect south price market vector demand state label price increase take hour decrease price movement available give sufficient classified drift incorporate increase knn great accuracy obtain knn suggest identical assume setting allowed vary accuracy lda knn classifier accuracy gradually drop critical reason frequently varied drop amount classification perform move slide average accuracy whether contain point detect whether detect accumulation pc knn knn knn knn accurate video contribute early early diagnosis cancer obtain image preprocesse grey level pixel label tumor pixel optimal varied parameter detector set correspond accuracy lda gradually drop result performance suggest frequently appear broken segment change although require verify present streaming problem exponentially move chart stream stream feedback receive memory allow drift experimental performance possible direction research extend monitor entry confusion separately stream method cope known drift detect move chart misclassification stream modular detection overhead fully memory exist rate control time classification stream consist point receive stream meet follow criterion pass process discard acceptable store small repeat store constant increase computationally process stream must observe streaming version unknown predict learn assign literature streaming reveal immediately make classifier receive immediate whether accurate key difference stream conventional offline streaming change dynamic distinction make case gradually change remainder classifier stream significant deal drift fall category classifier automatically adapt stay date classification concept drift concept detector design occur useful adapt concept give occur take thought change concerned detect drift limitation streaming observation receive frequently control concept drift serious case desirable concept drift really drift detection stream contain change detect drift occur detector know positive occur
markov forward backward outcome hide variable adapt backward formulae backward heterogeneous consequence standard marginalization quantity distinct set consequence contribution formulae reduce q conditionally j backward forward recursively recursively forward quantity moreover complexity x forward practical influence real consist model hmm gaussian hmm obtain hidden transition year appear posterior influential look protocol j temperature depict empty dot validate effect five influential segmentation observation remove remove take account period characterize segment negative influential basically segmentation remove influential one bottom solid I point segmentation interpret either particularly outlier underlie statistical application detection linear argue influence effective model outli influence differ posterior condition illustration support supplementary material detection change assume random time outlier sample probability hence average outlier statistic maximum absolute normalize three outli score axis detail supplementary material assess result depict statistic axis individual meaningful play consideration investigate influential protein structure structural point influential measure might reveal property receive ph paris dr interest include propagation computational receive ph university france include network hmms hmm different outlier hmm hmms framework author suggest outli viterbi search follow outlier hmms influence outli hmms heterogeneous model hmms instance hmms sensitive presence sense outli point explain main paper significantly temperature fig estimate function outli peak interpret answer manually add outlier depict peak two outlier top two outlier triangle semi simulation temperature original simulation outlier original outlier obtain sampling average test contain estimated modeling cluster standard deviation cluster base score calculate inverse neighbor densitie dataset outlier sort depend distance parameter integer standardize rescale axis score compute assess auc auc surface receiver operate roc qualitatively interpret fail fair good auc proposition kullback leibler namely sequence hide influence illustration application meaningful datum forward outli local outlier process biology typical markov letter measure start fix sake consider hmms parameter fully set density omit write explicitly observable evidence except measure influence observation influential leibl entropy application distance suggest state distribution speech recognition bioinformatic letter kl one good
media communication generative community detection split stochastically block membership highly flexible represent structure combination asymptotic phase transition tool theory despite block impose restriction network notably block make model many network community fit stochastic low degree community conservative political degree avoid allow history generative edge attribute membership block allow block extend pattern ordinary model selection largely base ratio ordinary correct variable ratio likelihood must likelihood propagation bethe give deal likelihood hand turn usual ratio nature degree correct derive asymptotic recover classic large dense substantial correction correction confirm validity theory network represent want split community priori elsewhere traditionally stochastic entry adjacency version poisson ignore loop former indicating belong assignment multinomial parameterized node thus assign block generates make poisson draw block involve connect twice product q assignment jointly physics ground minimize energy find graph wish total summing distribution logarithm minus energy latent approximate estimate average monte gibbs however review marginal distribution belief efficiently approximate marginal typical rapidly certainly stochastic moreover degree sum poisson degree block lead within block skew heterogeneity block node assign block get number connect recover block provide force total correct drop node belief propagation refer belief likelihood purpose belief expectation belief propagation extend idea marginal update light message node take markov field weight complete opposed network track sparse scalability node simplification distinct reach entire estimate maximization set propagation propagation holding observe bethe energy log chain carlo typically propagation seem propagation numerical selection homogeneous block correct extra heterogeneous without prior thus distribution need pick natural approach nest correct really superior degree converge least stochastic correct model define usual favor elaborate alternative exceed turn nan nan bootstrappe large however helpful bootstrappe nan constraint must must already impose model well central datum functional dependence fail specific dense effective grow correct asymptotic grow recover limit dense correction even shall consequence ratio characterize distribution concentrate assignment major stochastic underlie energy term ground substitute kullback leibler apply k xy axis marginal ordinary degree model correlation appendix show asymptotic limit give analysis small somewhat asymptotic figure simulation mean well fit formula moment simple quantile fig indeed fit free imply concentrate around interestingly fig concentration theory remarkably illustration replicate formula observe ground fundamentally follow really relevant observation likelihood analysis apply graph expect say differently case least extra really ordinary asymptotic see add predict fluctuation test go asymptotic ratio eventually big nominal pass happen would incorrectly must derive distribution calculation consist member make heavily former division classic degree nonetheless bootstrappe asymptotic analyst think twice force block imply mild responsible correct low extent bootstrappe fit reasonably theoretical marked line value theoretical political theoretical gaussian well reject favor less clear political focus political either correction greatly political observe completely agree bootstrap theoretical deviation essentially extreme model network extra capture justify several theoretical
presentation help generalization useful temporal period trial choose trial learner rely memory learn learner fuzzy system represent past fashion naturally generalization delay profile match available trial learner seem natural resemble fuzzy system natural world behavioral human event curve observe invariant human ask reproduce discriminate scale invariance characteristic human wide finding specie suggest world many scale fuzzy memory generic connectivity perturbation instantaneous stable spectral magnitude one connectivity view reservoir node efficiently tune neuron reservoir past fact dynamical reservoir require define reservoir precision input reconstruct net area past reservoir depend connectivity step beyond generic connectivity smoothly exponentially sometimes law recurrent network turn memory tractable least special reservoir net memory increase unless tailor orthogonal feed grow slowly trajectory connect network memory low dense fuzzy memory view reservoir tailor connectivity linear weight input white decay power law net grow relevant interest predictive fluctuation basic essential world learn ignore sake time vary serious multidimensional separately shift storage resource shift sufficient storage general encode technique bottleneck slow strategy fashion complementary feature impose principle low component time fuzzy fashion shift potentially slowly varying activity issue ignore serious application involve ability svm elegant strategy svms prediction slide window shift slide recurrent correspond improve gradually memory step shift svms svms along fuzzy input shift long temporal contain need fuzzy directly svms aim build happen fashion processing datum agent memory resource learn mechanism rely tailor act memory could potentially find signal rely series construct free way information redundancy information fuzzy number correlation representation quantify expression mutual information share neighboring signal uncorrelated long correlation redundancy spread activity simply average integral generated define redundancy calculate correlation correlate node small construct integrating function artificial correlation exist activity turn q decay exponentially even short emphasize shift moment pass without integrate activity reflect temporal activity instant activity response input shift make across mutual share calculated activity individual variance quantity high mutual share vanish redundancy require redundancy need mutual neighboring neighbor happen arrange input long lead correlation k note integral come denominator temporal large shift instantaneous correlation turn summation yield proportional approximately node behave correlation shift mutual neighboring node scale possible eq acknowledgement acknowledge national science air force office fa predict vary maintain recent signal memory shift extend storage resource learner priori many occur scale resource availability fuzzy accuracy free information illustrative fuzzy system time available resource limited purpose learner continuously recent basic arise much maintain past evolutionary pressure minimize resource consume grow generally example occur commonly prediction relevant essentially unbounded learner long range natural signal resource argue reflect scale free uncertainty example second second resource yield grow accuracy free learner way fluctuation mind represent sufficiently evolve rely instantaneous external storage resource inaccurate memory long optimally maximally property optimally temporal scale maximally preserve memory sufficient construct self free past mathematically laplace approximate node lead sufficient fuzzy fuzzy forecasting example ability shift past ability prediction algorithm learn modular system memory use forecasting need value correlation past time present stationary ignore existence long predict axis unknown future shift denote buffer wherein store unique forward store node enter self sufficiently represent shift optimal way relevant prediction maximally preserve buffer fuzzy average linearly center bin bin past represent bin attain represent show prediction information fuzzy buffer evolve self lose compression bin moment correctly buffer buffer actually save resource series though buffer sufficient extremely buffer maximally preserve relevant storage resource linearly skip ahead quantify contain review range correlate since actually behind white integrated generate series long range hence auto word past instantaneous white specifie coefficient term two power series correlate behavior euler formula gamma large sake analytic relevance predict predictive content taylor expansion show restrict linear clear sum quantity entire accurately represent infinite shift contain memory buffer turn shift portion predictive correlate buffer rather single shift theoretic consideration information bottleneck multi variable systematically compress transformation maximize series moment correlation maximize eq immediately single buffer moment buffer statistic simply uniform bin bin examine averaging individually would error extent give functional bin memory summation perform approximate integral bin equal bin turn straightforward pick successive bin completely past buffer fuzzy buffer range fuzzy buffer node buffer bin power fuzzy buffer lot predictive buffer wherein overlap ensure optimally free fashion ideal fuzzy buffer evolve possess fuzzy buffer overlap non average bin extent fuzzy fuzzy buffer mathematical recent free fashion internal critical consideration implement past fuzzy buffer temporal represent scale unlike buffer require resource construct distribute free fashion node estimate decay constant denote gets activate instant gradually decay decay drive functional functional value fuzzy instant ss make resource mathematical infinite mathematical come history laplace transform reconstruction history moment past distant temporal plotted top activity distribute illustrate equation delta input whose width area scale invariance invariant quantify estimate width peak index delta column delta delta input invariance linearly term combination bin spread activity desire bin buffer value nonetheless evolve input node evolve self noise across long constant white noise snr large drop zero signal appropriate delta signal average spread signal standard deviation signal snr positive generate spike activity node multiply delta snr stationary snr long node truly buffer redundancy scale buffer share neighboring node buffer free mutual buffer node accord redundancy spread spread redundancy analyze delta buffer fig activity buffer point value buffer imply redundancy panel buffer activity instead activity whole translate invariance activity pattern pass explain neighboring node invariance activity analytically value eq say pattern inside law buffer except happen whenever integer infinitely condition approximately hold invariance pattern buffer choice accordance redundancy equally distribute buffer redundancy reduce redundancy buffer large moment buffer node redundancy loss moment two buffer formalize delta input past current input successive buffer value min delta minimize redundancy th th almost activity middle vertical line represent dot represent activity correspond activity substantially different imply redundancy imply lose redundancy activity th activity th activity delta past n activity delta input activity value range rough estimate eq require high b attain focus two node close node close intermediate represented activity minimum minimum exist activity great simultaneously information redundancy minimum fuzzy minimize redundancy compare shift forecast goal difference shift self fuzzy learn forecast consider series property generate integrate noise library http com temperature year measure series plot row plot correlation reveal difference like correlation short time short autocorrelation balanced series correlation average year year row row show row self memory red use denote sequentially feed sufficient fuzzy value shifted discuss instant hold exactly self value record coefficient extract standard lm open absolute intrinsic intrinsic variance quantity correlate construct correlation decay decay serie generate reasonable see error forecast reduce error see fig blue shift square white memory shift temperature fig seem small reflect reliable knowing year help temperature subsequent year seem correlation long scale could exploit additional
simulation underlie belong belong thus group tn causal relationship among direct acyclic equation give dimension contain direct mutually group define rewrite block acyclic also include non noisy restriction grouping know merely estimate causal among group observation arrange correspond row difference algorithm explicitly build allowing effect iterate generate group formally group assume acyclic group group generalize find directly state regression follow j ji appendix apply vector combine fisher one hilbert schmidt dependency nonlinear correlation p modify measure infer two scalar underlie point likelihood rx lx ml j meet assumption error within group correctness term measure advantage term section pair meet j consistent total group n lk ix k take obtain group accord adjust third base term among group causality independently base assumption error need measure matrix matrix estimate connection ol estimate direction et direction close suggest trace group infer minimize equation causal set formally state correspond combination group ji residual lemma section formalize causal arrange datum regression approach use connection parameter particular approach apply data variable calculate pick group minimize eq whole set value individual bold legend bold pdf fig black toolbox simulation variant hoc available follow randomly create connection average nonzero additionally complete mix random sample gaussian finally randomly causal order variant ad hoc modify search permutation matrix block yield low triangular finding causal secondly approach distinguish markov graph simulation complete datum generate show size exception fair design dimension perform follow hoc perform dash ica perform seem probably dependent ica replace apply algorithm large mistake make find strategy
simplification take theoretic leibl predictive obtain support take expectation result outcome understand gain divergence decrease amount expect utility typically expect close predictive must bayes outside logarithm introduce nest estimator nj mf approximately term maximize optimal objective need describe stochastic sample require gradient monte simplicity essential idea complexity solve design variable unbiased objective word sequence satisfy one appropriate scaling diffusion section choose technical solution view sensitive acknowledge choose logical converge iterate relatively criterion change fall tolerance successive maximum experiment stochastic fixing throughout entire practice dependent transform random moving example distribution realization transformation practice fundamentally really always shall generality design value application compute large attain low finally run optimal remain within converge respective stochastic true optimality unbiased value across replicate w difference optimality formula optimality decide run approximate still optimality tolerance replicate perform algorithm large compute w optimality post bfgs optimization robustness ease implementation update approximation evaluation bfgs backtracking search find stepsize satisfy first wolfe many commonly fall tolerance variable evaluation reach bfgs taylor l design dimension become start termination initialize termination meet direction search update define u tu kk challenge apply optimal lack gradient introduce simple expansion replace intensive first analytically come word forward expect gradient estimator analytical derivation introduce repeat forward evaluate stochastic enter experimental prediction describe function map discrepancy take intensive drawing evaluating evaluate calculation tractable one would replace surrogate accurate space option spectral expansion approach include recently loop bayesian experimental excellent unlike stochastic simultaneous series orthogonality convergent sense sum truncate common truncation retain certainly polynomial impractical construct pc expansion jointly affine uniquely realization design endow convergent pc expansion computation via approach need difficulty latter prohibitive compute project quantity g onto method treat box solve functional trajectory functional smoothly state employ orthogonality simply coefficient th component model evaluation expensive integration dimension employ quadrature care must avoid significant quadrature polynomial indeed full tensor approximation quadrature rule construct substitute approximation enable gain discuss applie unbiased use finite expect obtain simply unbiased idea availability unbiased estimator optimize instead support new tradeoff course realization monte requirement comprise condition unbiased estimator unbiased require parameterize continuous perspective product noise support case use forward distribution observational meet requirement preserve must term original simple variable stochastic perspective assume diagonal dependence standard normal example translate kronecker delta replace function illustrative extract variable second design incorporate substitute draw realize draw multiplying output therefore monte carlo approximation realization either likelihood expansion derivative derivation gradient estimator orthogonal expansion experimental design tool two place govern square space concentration parameterize center impose along spatial prescribed source location ultimately sensor spaced sample I mean mutually contribution concentration profile realization nd spatial backward integration model parameter use truncation evaluation surrogate g evaluation relative current formulation seek possible accord likelihood five concentration great information gain result numerical explore gain show interpret monte approximations carlo approximations become grow become value flat flat encourage higher solve sa effectively signal ratio gain suggest albeit point lie interior locate corner corner center measurement orientation distribution constant radius sensor sensor minimize area problem happen corner show density source true directly spatial note posterior moreover reason perturb relatively significant construct discretization pde pde simulate treatment understanding impact direction place corner tight sensor center keep realization depend source imagine sensor prior source location result assess behavior method compare ascent maximize figure path ascent direction gradient eventually naturally good gradient increase show instead intuition mechanic set realization find bfgs small subset large realization objective extremely near different objective none encounter maxima corner nonetheless subproblem low table bfgs conduct start location run uniform schedule termination criterion lead similar time histogram major represent result major bfgs different value immediately corner corner around high improve quality bad balance need choose sample size outer overall bfgs intermediate slight bfgs place flat near optimum need close optimum robustness design keep run information variance estimator observe histogram reflect design corner design mode upper sample enough large size shape expect high table show histogram gap bfgs deal sign upper bind low size run bind optimality gap decrease since able corner monotonically consequently bound tight toward increase gap spread indeed objective flat consider occur corner translate sensitivity change expect increase large intuition obtain consider change upper low value increase corner mention early well indicate contour figure choose value certainly indicator surface gap value gap appear another stochastic run table low bfgs take terminate maximum difference reflect bfgs histogram transfer number fold lead account sublinear resource substantial bfgs require take high freedom user stop attractive evaluation expensive integrate quality final design true optimum study take mse reflect plus difference via relate effort computational run symbol four confirm previously encounter solution quality balanced large inferior curve highlight optimal light monte estimator reflect bfgs achieve small mse give effort generalize experimental design paper stochastic employ formulate approach sample couple bfgs case extract gradient gain rather nest evaluate challenge forward polynomial expansion perturbation sensor cast design design perform matrix loop sample size impact gradient estimate efficiency optimization result suggest improve sample optimization size outer loop instance yield little quality consistent necessarily algorithm superior broad difference sa algorithm essentially optimization flexibility surface behave exploit source inversion gradient gradient may appropriate optimality replicate solution use adaptively adjust assess employ stream estimate interval method stochastic outer loop reduce relatively frequent fine terminate without termination present surrogate product method derivative free prefer surrogate increasingly small optimum similar idea additional potential gain point focus open experimental design collect rigorous design stochastic objective continue mathematics air office national science foundation diffusion equation five design vertical range vary htb surface figure corresponding value htb surface htb surface position htb cccc htb c cccc represent frequency cm c htb versus time optimization inner outer loop highlight red bfgs light derivation estimator form method present gradient form respect form constant vector mutually compose component become normal datum derivative condition summation expression model case quantity available
k concern miss feature conditionally word object observe assumption correlation variable generality suppose probability soft class object rating uniformity define use rating object rx co co miss feature motivate parameterization r rx vector similarity make learn metric feature provide version hybrid discuss generate vector class estimate parameter describe rx hybrid semidefinite aforementioned ordinal convex optimization analysis experiment generate data experiment consist ct along rating part project seek develop retrieval detail worth consider hybrid free use instance medical hybrid new explain generative vector detail generative c scan include semantic annotation vocabulary haar extract image simplify remove one feature study rating figure demonstrate image introduce note pair appear lc since cross validation c j co result distance retrieve ndcg retrieve list figure ndcg margin htp co give error hybrid learn measure class element datum fit rating try synthetic purpose medical retrieval substantial gain think generative growing explore combination hybrid broad detail randomly partition validation set ascent compute range determine trial choose optima rarely data experiment model mixture gaussian procedure produce matrix iid nx nz pm theorem proposition van xu distance hybrid learn assign assign base provide encode well real medical image leverage retrieval significantly system search object require assess learn similarity rating representative user kind object distance available serve search retrieval classify provide useful relate rating metric attract attention year propose label refer method category ordinal discrete convex metric classify similar distant method share distant minimize class aforementione literature kind stage similarity metric classifier distance optimization similarity rating medical retrieval problem object object rating comprise consist object feature similarity rating dissimilarity respectively class triplet class reason object give object sample metric result accurately reflect variety discussion choice metric retrieve similar discuss three exist ordinal offer learn ordinal assume object rating obey level together coefficient metric nonnegative rating approach compute rx
partition ng result note likelihood dependent conjugacy gps simply nest fact procedure motivating replicate trajectory allow trial trial parent structure hierarchical motivate application trial relax pt trial replicate trial share parent alternative structure consider independently location replicate exposition compactly denote covariance share show qualitative raw correlation similarity ccccc sim sim sim est sim trial recursive normalize set infer conjugacy q likewise predictive pt marginal decompose easily far assume base importance hasting sample hierarchical solely partition recall tree join neighbor contiguous region think partition balanced tree graph build randomly select uniform contrast setup straightforward point level uniformly undesirable lead highly unbalanced partition function adaptively use search tend correlation partition much space think location correlation partition correspond cut edge weight depict seek cut give set measure connectivity avoid small fig select cost absolute location straightforwardly propose partition normalize cut simply cut generate partition distribute extension distribution weight draw amenable efficient size match alternatively straightforward algorithm iteratively propose accepted proposal normalize cut dramatically improve proposal acceptance decrease discover especially many tree large benefit efficiency proposal first cut adapt encourage search shift towards great balance search particle proposal gp dynamic cut gp computation extension similarity gps collection gps node leave cart partition assume hierarchical inherent perspective instead motivation partition gp aim partition parent partition datum covariance value trait distance common tree define branching application internal gps gps approach fundamentally necessarily spatially define multiscale history cf variable define level process high level share vary level differ gps contiguous gaussian vector vary fashion combine formulate focus gps address inherently another fundamentally latent take mapping finally hierarchical associated tree closely task standard tree partitioning randomize unbalanced tailor pt ccccc sim sim sim sim map trace log likelihood versus chain minimize error assess ability infer propose replicate level set specific simulation demonstrate prior chain run pure search hierarchical estimate especially average region result consistently capture feature sensor replicate visual much well track jump stimulus break display expect see key occur finding processing length visual accurately standard response also compare predictive functional become approach trajectory share information trial limitation restriction grid enable direct word independently trial interpretability exceed pt generative characterize structure examine capture certain accurately previous model provide locally ii global iii change enable theoretical gps focus univariate formulation amenable multivariate naturally accommodate share parent possibly model interesting relate specification represent particular parameterization alternative hyperparameter mix poorly future exploration broadly new framework stationary analysis especially possible extension thank acquisition preprocesse useful suggestion support grant science es marginal compactly discuss therefore c k py pf derive assume additionally drive hyperparameter perform comparison scenario take take fix parameter optimize grid training sensor hierarchical gp assume single bandwidth parameter particular shared function jointly gp covariance constrain study bandwidth parameter gp assign trial setup determine jointly optimize example ccccc account significant variation account trial trial drop fairly rapidly level indicate note trial variability instead sensitive hyperparameter fairly robust reasonably setting marginal sec word sensor word cut amount record smoothed strength baseline prior information expert run prior identical aggregated fig demonstrate dominate related display example test datum sensor full ccc fig mu pp trial fig pp trial fig pp trial fig mu pp trial fig pp clarity slightly large interval observation mean subject give write protocol review record use device acquire pass filter hz filter hz horizontal participant position indicator place separation remove activity begin location filter hz remove contribution line contamination ms stimulus end ms stimulus second subtract signal amplitude stimulus multiply precision replicate iii proposal py detail search root select location hyperparameter previous partition point initialize structure proposal normalize proposal pt z py pt hyperparameter previous partition initialize proposal previous remove normalized proposal w proposal assumption example section definition david statistical science nc propose capture dependency gps nest capture top level partition inherent conjugacy analytically partition allow employ theoretic analysis key challenge insufficient one address challenge employ band covariance gps underlie mat enable smooth function assume adapt vary capture span memory extension smooth shift invariance fundamentally different gps direct interpretability local stationarity grid information across motivate activity stimulus snr key stimulus although similarity replicate key difference phenomenon well correlation ability share single trial long potential gps nested inherent conjugacy gps gps conditional encodes enable correlation generate distance encode stationarity importance science vision sampler global move inference integrate partition allowing
stop dependency increase demonstrate dependency capability capture moreover source reliability expect source reliability extent list assess true risk dependent approach jointly group true source capture dependency collective experimental exist edu collective intelligence low become barrier effective collective intelligence address issue assess reliability observe often source influence share high intelligence redundant possibly reveal latent source aggregate prevent collective dominate source reveal reliability result real respect intelligence collect task participant fairly extensive set crowd article amazon involve wikipedia claim fact object collective infer fact collective intelligence people source may treat aggregate dependency among involved group reliability incorporation reduce observation especially dependent source dominate collective intelligence especially reliable moreover equally reliable may reveal reliability group negative reliability reliability two closely reliability group aggregate individual reliability distinct generally reliable likely reliable individual object vice versa overfitte object reliability consider remainder formally define source sense evaluate section tp define multi sense description intelligence source object take source report infer true source discrete value many crowdsource binary assertion correctness test value however extend topic paper five object object category object alternatively book book value broad g present answer claim notational convenience claim bold subset meanwhile among membership source inherently dotted influence tend object unseen infer knowledge number determine bayesian break construction impact group reliability reliability meanwhile specify object reliability dependency infer generative sensing eq denote discrete generate bernoulli stick break latent membership pg l stick break need group determine capture degree construction determine dependency accord likely assign dependent share observation degree probability belong group extreme yield complete dependency reliability soft specify priori sample specify reliability high general reliability sample generally reliable reliable object sense general risk reliability adopt uniform distribution determine posteriori process observation source conjugate depend reliability detail specification adopt categorical generate different setting group object value hand object randomly object soft count false indistinguishable member source guess ii tend value contribute observation observation process wish family factorize factor determine u coordinate descent update sequentially update form still dirichlet em element vector paper short still q find sum reliability object intuition reliability reliability object object reliability logarithmic beta update line reflect reliability affect estimation reliability reliable group vice versa overfitte estimating simultaneously update source expectation l define index thus sum distribution exist demonstrate effectiveness infer reliability perform book book online web site book author data book science book com book com book extract book total book source book object author object book categorical domain claim name preserve name ignore middle name author book manually book cover list test one
energy use range energy non submodular underlie regular optimize compare multiscale swap energy state tailor energy multiscale framework state family energy apply multiscale optimize challenge metric energy diverse mrf energy energy well optimum opposed move global optimum able multiscale table fig multiscale energy valid integrate multiscale put track running time energy mm swap single show relative close energy cc c swap expand multiscale drastically visible swap numerical swap close energy cc unable cope step unable propagate far fill hand framework step gap numerical single multiscale scale sec sec second multiscale runtime construct pyramid take optimize coarse time serial construct effective multiscale scheme experiment compare energy measure sec three unary min unary unary agnostic within multiscale four representative energy percent different energy unary wise percent lower close bar energy aware consistently label prove example energy pyramid take principled swap energy energy label denoise sec sec use variable unchanged percent achieve relative well time unified multiscale multiscale energy multiscale landscape involve level involve method energy interpolation multiscale energy gap multiscale algebraic multiscale pde solver g allow method develop solver multiscale optimization minimization problem l l n look j unary put together I diagonal represent unary term energy long zero coarse interaction unary easy add unary whenever zero diagonal chapter energy pyramid energy pair energy restrict simplicity energy wise write q entirely wise differ energy energy pair general define unchanged energy agreement neighbor fine denote entry interpolation indicate fine affect ij energy q interpolation issue derive future however yield fairly straight energy variable label energy beyond interaction energy refer energy energy variable q high way order parameterize energy interpolation energy estimate agreement neighbor use interpolation entry scale variable coarse hyper interpolation trivial representation interaction label leave derive interpolation matrix coarse parameterize label mm derive multiscale energy pyramid unify different different energy multiscale framework minimization segmentation denoise apart submodular binary energy develop challenge energy etc discrete energy submodular regular energy submodular energy submodular energy encourage term energy reflect piece popular application vision focus energy encourage hardness energy energy consider energy optimize contrast energy contrast encourage energy parameter learn g functional optimize energy non energy explore optimization method energy minimization method primal act formulate provide approximate solution solution optimum labeling lower bind discrete hard one representative instance energy easy optimize energy show tend approximation tight discrete minimization exploration exponentially make task way toy effective method looking phenomenon scale fine local high multiscale landscape multiscale phenomenon encourage coarse apparent gradually search fine decade vision focus pyramid experience unify directly pyramid moreover energie multiscale empirical evaluation energy come optimize tend well approximation method multiscale space multiscale framework optimization process exist multiscale landscape fast multiscale problem denoise correlation cluster multiscale aggregation take underlie landscape label well pyramid multiscale framework energy include example submodular energy non energy superiority primal method multiscale scale increase optimization challenge multiscale directly prominent coarse pyramid work regular grid energy work propose scale aim unified analyze suggest landscape energy improve optimization method property restricted energy training address principled build energy pyramid decrease number label assumption consider discrete minimization many computer vision cast see restrict submodular energy build pyramid freedom key pyramid interpolation map solution level pyramid principled aware interpolation energy pyramid multiscale landscape energy apparent however counter intuitive semantic correspond label variable label allow discrete solution representation basis multiscale entry find energy label parameterize pyramid description scale wish generate representation approximate variable coarse assignment assignment yield approximate fine correlation construct desirable variable neighbor general may interpolation grouping formula unary dominate correlation naturally account influence unary pair inspire energy energy regular grid generate assignment neighboring variable iterate mode obtain energy assignment assignment choose yield condition random initialization locally assignment give together preserve energy smoothness preserve term allow explicit encode get algebraic select representative every variable correlate variable either consider strongly correlate c add yet select index fine coarse index fine interpolation row leave entry refine compute interpolation coarse eq energy describe multiscale put together multiscale energy fine matrix pyramid explore coarse fine fine scale serve two coarse solution binary round set intensive module framework empirical estimation scale refine solution fairly module restrict pyramid energy aggregation pyramid coarse refine main goal first submodular energy advantage primal method minimization unified improved exist primal diversity energy energy denoise measure correlation sec energy improve swap representative move make overcome non energy follow lower compare energy bind close multiscale scheme consistent multiscale dual discrete energy behind move make algorithm challenge energy multiscale give significant boost swap expand scale percent relative
erm tight online factor length notation mistake guarantee separable establish lower bind imply question bind study integer define number online batch achieve one loss rate hinge loss establish agnostic desire weight vector integer positive universal integer denote predictor associate half hinge margin denote q subscript omit clear threshold monotone formula generally denote case fractional negative refer unknown pair sample expect simply subscript minimizer ds subscript denote rademacher complexity respect draw average rademacher follow variation bernstein draw eq cases rhs eq least range b therefore consider fraction apply find mp unnormalized gradient unnormalize unnormalized theorem vector constant obtain assumption obtain rearrange definition desire sequence unnormalize list batch algorithm expectation x case x r I r three observe set detailed return average set canonical mirror set batch rate achievable next accord empirical w w theorem slow tight hold rate guarantee provide combine propose tight convergence analysis tailor positive label bound rademacher conclude dm dimension remove therefore j obtain inequality choose maximizer consider case otherwise q rademacher denote obtained side lemma define easy mu mu therefore get finally use uniform negative loss constant draw exactly lipschitz translation w combine use euclidean norm cover easy u w rademacher complexity entropy state member draw label follow combine rate give adapt fraction negative theorem case bind loss trivially prove loss eq bound induce get second provide constant trivially hold eq note bernstein prop become hand second since statement erm theorem prop eq light loss predictor factor via standard convergence provide convergence result show even domain restrict discrete pose complexity learn output example construction respect predictor norm construction loss adapt construction hold theorem hadamard matrix row exist construct hadamard exist hadamard eq q transform let every ki one coordinate label z term desire z z zero vector equal rewrite concatenation concatenation n finally h yx bound satisfie show existence algorithm respect statement construction h output verify contradiction show uniform learn theorem rate asymptotically indicate need erm exist prove lemma first class similar distribution denote thus leave define change cause change therefore inequality gs mt constant define appear hand difference cause fs fix walk chebyshev therefore check probability two ready low universal domain suffice domain vector lemma exist z w suffice define since microsoft school science engineering ram institute il science york ny department statistics nj non norm monotone batch minimization rate tight grow classifier empirical minimization low instance change extension binary classifier binary threshold formula formula specify study formula mistake formula analysis general case separable show mistake applying consider fractional note study separable order move vector therefore negative target necessarily separate fractional weight explicitly consider misclassified margin guarantee rely half upper misclassification obtaining translate hinge gradient online md bind statistical setting use rely regardless threshold yield mistake complexity relevant range agree md result mistake distinction lose md erm guarantee
impossible certain mixing consistent obtain select homogeneity upper length choose average linkage sample svm describe parameter definition distance artificial dependent marginal finite countable construct right remove integer time single distribution length min applicability choose brain interface competition subject letter originally one split dimension sequence minute method dynamic base estimation unsupervise svm interest svm unsupervise error time subject education project european program er de concern series cluster homogeneity address algorithms consistency prove experiment real problem algorithm perhaps binary conceptually simple try building algorithm different reducing many start problem formulate underlie theoretical datum hold exhibit range sample may well homogeneity testing binary result algorithm assumption stationary statistic homogeneity problem distribution distribution infinite evaluate consistent building finite dimensional distance apply homogeneity testing change far work time build stationary ergodic measure marginal sum infinitely marginal indeed fact algorithm classification setting choose concern interface present algorithm stationary ergodic distance trick involve reduction explain distance devote respectively condition address homogeneity testing cluster guarantee present experimental measurable space induce sequence time separable limit ergodic I express one study kolmogorov note mild particular separability sufficient metric rest easy apparent consist case slight abuse write ph true generate verify directly trivially set euclidean check high dimension space polynomial series two take decrease two distance e metric marginal q separable vc generate stationary ergodic finite vc stationary ergodic n I consistently find last arbitrary statement problem suggest calculate calculate eq indicator calculate follow dimensional attain minimize conceptually sample time distribution whereas generate ergodic mix memory assumption dependent stationary distributional statement stationary vc dimension generate straightforward instead distribution give sample thus algorithm distribution accord whether cluster asymptotically length point separable vc stationary strict separation separation cluster asymptotically algorithm theorem linkage work distance close assign follow linkage clustering consistent marginal ergodic spirit number situation strong assumption ergodic consider necessity distribution involve general result advance strong turn establish ergodic sample guarantee strong another reason statistical homogeneity provably ergodic shown analyse homogeneity sequence sequence mix process algebra mix ergodicity much generate indicator vc fix choice guarantee use similar bound finite term exposition sake rest letting whose whose decrease eq imply statement respectively homogeneity sample asymptotically stationary ergodic series consistent
galaxy correspond fig mostly galaxy autocorrelation galaxy red appear initial value random change successive factor pf show follow pf log band galaxy hierarchical galaxy formation model explain elliptical exist galaxy step mass formation spectrum average galaxy author thank anonymous impact foundation national foundation department office iii web site http www iii iii laboratory group de de berkeley national laboratory max institute university york university university university university university physics md usa edu universit fr ed france variety galaxies digital affine aim curve curve cloud principal galaxy galaxy measure find letter shape turn define branch galaxy population control low old variation important galaxy variate fit arc power log fit arc length galaxy increase autocorrelation power arc correlation arc length show galaxy decrease allow galaxy cloud red galaxy dominate disk ratio understand trend encode available curve dimensionality reduction support relevant physical scenario merge formation massive galaxy arise formation galaxy blue red constrain drive galaxy common property galaxy galaxy survey galaxy available constrain accurate level estimate wide survey survey million galaxy turning become feature keep increase amount physical want important galaxy dimensionality principal uncorrelated orthonormal base spectra wide galaxy property equivalent emission mix galaxy population show population galaxy enable contain correlation embed galaxy spectra take surface preserve local geometry review see practice span high curve value begin complexity multi rank associated arc classify rich galaxy principal curve belong low galaxy limited galaxy galaxy behavior statistic much volume keep galaxy assign p galaxy physical galaxy galaxy remarkably local universe detail galaxy pca method pca detail sec paper flat galaxy dr ms server galaxy constitute band apparent cut distribution cover resolution selection cut primary appear calibrate imaging field image interpolation galaxy complicate define area cover fractional area galaxy reduce galaxy statistic close great galaxy center correct apparent lower arise contain avoid slight variation limit magnitude galaxy limit subset magnitude limited subsample use study absolute leave galaxy galaxy variety relevant create within color shape spectra age color since color highly correlate computing color correct correction frame correction color color spectrum negative spectra good spectrum computed spectra provide cloud decide include partitioning cloud introduce desire artificial cut magnitude neither strongly feature galaxy elliptical galaxy concentrate profile galaxy dependent band also property degeneracy dim use k apparent magnitude star formation star formation mass mass dr galaxy derive band cover part galaxy correct spectral equivalent full galaxy include build cloud apart deal pd reduction description section component transform dimensionality transformation rotation new projection uncorrelate select highest project onto order low describe span rank pc variance svd allow variance eigenvector respective thus expansion transformation ji ji weight weighted schema point assign one calculation covariance general property might made point divide surface surface go ahead pl l arc compose segment connect projection datum project onto self series expectation equal curve j il practice compute weight cubic regression calculate knot close segment iteration cumulative change p valid first contain galaxy equally survey volume use weighted galaxy galaxy fail galaxy galaxy assign volume galaxy galaxy limit apparent magnitude limit give volume iteratively value directly inside database library curve volume density histogram measure visually outlier wrong property measure simplify calculation avoid galaxy initial ht galaxy orthonormal eigenvector combination property high property pc since ht point order arc line circle text curve present turn mark colored direction galaxy project onto pc separate main cloud galaxy sec galaxy notice contain component combination th th pc pc magnitude expansion galaxy share evenly important correlation high old population opposite sign galaxy surface star formation opposite next mostly expect star form galaxy color probably concentrate star galaxy variance pc furthermore pc obvious galaxy nn increase line population denote galaxy mark group mark boundary color turn value principal unity dimension example extreme change spline unique equally spaced quantile arc length turn across straight show maxima place along red galaxy population principal curve resemble letter regime branch turn galaxy place fix arc galaxy group value onto place galaxy amount branch choose arc branch branch branch branch table galaxy group radial like fine branch follow remain visible see many spectral hand evident nd branch branch although line visible arc turn right hand straight circle within bar span quantile bar quantile red galaxy log na h red red group sec evolution galaxy arc feature present great present shape behavior individual initial arc branch turn curve fig great leverage pc length scatter turning behave differently emission degree arc also galaxy spectra function property strongly shape emission belong proxy interestingly average dispersion make old population arc length present turn fact na inter consequence present high star formation galaxy index determine light profile galaxy disk elliptical emission ratio diagram galaxy galaxy present emission line well measure fractional velocity dispersion km cut reduce go length none emission line location group line cover star form galaxy branch region interestingly branch happen turn feature powerful describe beyond construction dispersion bar include colored dot random group dash pure galaxy composite increase arc blue triangle aggregated sample circle red galaxy ccccc unit group evolution arc computed method explain estimate lf galaxy parameter choose double law fit index define reach power cutoff q whole change curve st tail behave branch branch slope fit shape log st branch mostly constant mostly surface st see function fig give slope expect star form dramatically start continuously start rd branch slope turn st branch long end sharp cutoff end well fit double law since cut rd turning present law section flat fit drop fit since shape belong track eps diagonal solid black auto correlation represent dark left show correlation expect galaxy sample dark second galaxy arc length quantify explore galaxy cross parallel generalize two consider measure cross brief generate use galaxy draw correct sensitive bar circle survey volume sample sec follow galaxy remain group auto correlation solid plot high amplitude large scale color instance amplitude increase object universe small sample power scale suggest compose galaxy scope paper highlight correlation autocorrelation cross galaxy dark information pair galaxy dark matter trend context close arc length expect correlation galaxy distant particular mean less galaxy perfect mix galaxy arc signal galaxy distant dark matter show galaxy mix suggest two galaxy dark matter aa function analysis galaxy group stand aside galaxy sample pay galaxy apart cloud another galaxy whose appear eps curve panel small disk galaxy red fig find galaxy cluster cloud separate build galaxy appear arc precisely dominate opposite cut galaxy image average spectrum accord compose mostly minimal formation component exponential disk concentration galaxy lie interestingly size uncertainty value estimate arise spectra component galaxy around galaxy group member ram pressure mechanism disk red galaxy double power group study galaxy explain choose curve radial partition galaxy datum cloud high core sequence galaxy average show useful able support physical scenario principal curve power curve galaxy arc big
call select intel ram method write misclassification partition operate able fast fail complete memory time versus proportion take unable train hence fast scalability scalability kernel significantly exist mkl implementation result mention goal add technique kernel line kernel scalability support nsf grant thank geometric kernel mkl problem search interpretation combine structural formulation allow mkl optimization routine provable guarantee set empirical significantly fast unweighted principled kernel wide learn jointly optimize exploit heart elegant approach require exist albeit accurate large present perspective mkl view mkl formulation multiplicative primal show primal reveal extremely light sdp call library optimization base provably extremely light optimize mkl technique engineering mkl view work focus add already detailed evaluation nystr om approximations apply heuristic merely baseline mkl nystr om additional scalable baseline since determine early simply train well combination semidefinite efficiency researcher start alternate alternate update mkl cutting plane semi program mkl improve combine cut plane level variant lasso point speed heuristic work mkl study gaussian family regularization base dynamic briefly present learn good classifier recent stage kernel stage stage meta example meta svm solve highly inefficient scaling example scale scale past propose limited scale bad letter bold letter htbp cl semidefinite likewise duality margin short exist convex combination row close point hull express collect define expand familiar prove interpretation stand discrepancy simply search short distance formulation later solver approach store kernel exploit find short fix merely adjust iteration exploit sdp primal mkl forward step rather solve merely backward computation decomposition speedup update allow due convert mkl margin place general constraint soft appear simply ir mr rewrite canonical sdp apply brief overview thesis guess correct attempt either primal guess guess base constraint start end else lack solve program primal guide primal assignment form adjust paragraph backward incur traditional deriving sdp gap role important guarantee sdp explain assign solve denote ti convenient explain backward matrix symmetric unit eigenvector orthonormal equal rest call eigenvector clearly correspond part eigenvalue equality form orthonormal quickly cause rapidly double precision fortunately converge completely multiply certain result also factor suffice merely significance note avoid store ab store n program explain r shall algorithm iteration lemma subroutine p qp mn step wish template prove guess infeasible particular I trace become simply collection geometrically examine point choose correspond positive pseudo process max I highlight consequence produce sparse never need compute I expensive root explicitly find mkl eliminate close multiply side complementary maximize constraint need search eliminate start observe I complementary recall combine I summarize parameter objective loose via sdp much absolute ti eigenvalue absolute always ir require require run bound mn fail result small dataset baseline weight nystr hereafter mkl datasets uci repository medium classification scalability medium scalability constraint subset amongst kernel weight trace par beyond figure entire matrix memory poor employ nystr observation learning perform thought simple algorithm feature nystr om
chart observe lr chart reference choose chart lr scheme optimal value delay moreover change lr sr chart choice average delay attain equal decrease chart exception bad delay observe illustrate chart well small bad chart turn arise control gaussian assume change reference suitably control simulation compare control assume ar criterion chart small depend reference control value equal change lr chart chart dramatically analyze use except chart always chart change lr chart change expect begin small delay author management european discussion comment remark european box derive use sequential generalize give process extensive study scheme variance delay ratio process observe process appropriate tool monitor control cf en many public health economic environmental sciences model time directly series propose transform possibility chart extension exponentially weight move chart dependent overview topic deal mean behavior observe surveillance variance series instance economic detection asset control time introduce stationary process focus independent disadvantage procedure lr sequential deriving suitably drawback generalize lr sr great advantage reference process white explain control briefly new autoregressive process chart derive residual control chart section control section lr generalization approach generalization modify sr obtain scheme autoregressive average sr chart reference reference true sr chart small value medium lr chart chart dramatically generalized delay equal length limit average delay chart must prefer later spc structural examine consider target sequential detect occurrence course target detection increase variance problem arise economic increase asset variance reflect production process since bad stationary process observe process position observe exactly scale control chart introduce page control chart memory point previous former chart value mainly normally expectation likelihood time calculate recursively dramatically simplify practical calculation control design practice fix practice replace fix determine control target practitioner recursion stand equal control choose scheme determination target formula available via assume cf determine gaussian obtain minimize quantity denote chart likelihood assume likelihood occur control q indicator thus j j chart n cm autoregressive moving process follow x x j describe part lead chart recursively describe approach apply chart recursion apply residual chart chart normalize independent recursive presentation statistic calculation delay markov depend thus limit consequently simplify application sr paper change sr procedure prior sr asymptotically sr recently several variable independent normally distribute density univariate normal distribution notation sr statistic know quantity lead derive apply independent length chart j j jx replace chart causal get n n iv statistic change reference lead disadvantage available procedure must section likelihood size change receive spc cf section denote control let hold q obtain q test gaussian mean condition else n chart give causal ar v consequently rule nf cm nf x ccc chart run assume n normally distribute expression value exponential reason eq logarithm increase arithmetic independent variable iid nf x determine concave function attain n iid iid iid r n iid u iid chart get function position run scheme control detect optimality know aim give chart apply situation control choose limit performance explicit formula criterion extensive study run delay determine repetition presentation average delay consume
standardized fan scad penalty zhang mcp mcp big impact see discussion choice variable group coefficient top show estimate norm group panel show see characteristic path quite group mcp path bias toward path solid plot group dot understanding characteristic nonconvex group case group j lin lasso via soft unchanged mcp scad verify group mcp expression group group mcp scad norm mcp scale soft threshold scad similar mcp unbiased threshold complicate mcp pt mcp operator operator scad hard scad family close expression building describe descent fitting kind propose lin way path use fu wu originally scad discuss behind extension group iteratively reach optimize function particularly scad mcp expression group coordinate partially optimize coefficient define l j exclude group ordinary equal residual penalty consist f choose attain maximum group mcp norm bridge composite penalty possibility penalty encourage bridge mcp bi one mcp bridge penaltie mcp limit extent variable dominate norm remain bridge variable whereby draw prevent group bridge achieve individual norm mcp group bridge composite mcp group generate group line represent equal composite mcp three underlie model equal line equal reveal group penalty nonzero coefficient magnitude solid enter phenomenon composite mcp grouping composite mcp zero eliminate mcp perform penalization piecewise unlike although penalty idea still main solve first li direction equivalent threshold operator solution penalty whereby guarantee decrease every com group composite mcp achieve penalty group wu possibility convergence model fit coordinate long procedure preserve transform back propose portion replace likewise penalty replace mcp knowledge explore work performance genetic important age relate control analyze marker suggest relate group mcp marker gene belong penalize logistic fit additive marker dominant analyze single univariate logistic marker cutoff select marker time group composite mcp table penalization penalization gene age chance demonstrate different three clearly see although misclassification error gene marker genetic bi method achieve bridge identify promise snps look mcp global minimizer take mcp penalty condition estimator justification selection reasonable group j index group least denote empty first mcp chi given suppose proof oracle group criterion strictly tucker mcp behave oracle sufficient yu due mcp extension present therefore say identically selection consistency mcp situation example strong mcp strong convexity extra effort descent concave penalty largely equal close zhang zhang helpful give selection method model regression learn genetic genomic dimensional covariate intercept unobserve lin operator procedure reproduce convergence additive high zhang lin van propose variable closely nonparametric distance specify threshold compatibility penalize infinity form condition group lasso lasso spline consequently lasso underlie adaptive lasso select correctly achieve linear mutually exclusive index linear priori property method al rarely covariate nonlinear effect zhang liu determine component regularization closely relate show method consistent special four parameter difficult pursuit approximate spline expansion use mcp show consistent structure correctly time interval error investigate effect covariate measure repeatedly know longitudinal important smooth chen li estimation apply selection selection author li group regularity function great level select important probability vector identically regression efficiency simultaneously multi task author criterion general th variable regression task select drop study selection et matrix group method throughput clinical phenotype status phenotype result pathway gene pathway treat incorporate pathway select pathway gene study marker identify suffer perhaps relatively size effective pool regression arise gene song important identify gene complex hundred sample often control utilize hundred snp human genome powerful select simultaneously separately group genetic discuss selective group progress area important issue well variable determine even include bic schwarz freedom lin freedom least estimator feasible systematically estimate analysis group selection b choose applicable discuss furthermore estimate degree group selection prediction important insight group range whether penalty validation clearly lasso method consider mcp relevant solution descent group group overlap pathway belong multiple pathway extend lasso variable without group liu yu overlap et bridge assumption group appendix chi freedom chi distribution kkt intersection let x use recall j ji jj n define orthogonal projection mn j complete define j along zhang omit event reduce reduce condition prove acknowledgment thank anonymous helpful comment extremely grateful point lead improvement research grant nsf grant conjecture grouping include lasso article give selective concern theoretical pay bi selection additive regression genomic datum analysis wide association highlight consider naturally model write matrix th predictor around desirable unit grouping estimate select variable many statistical algorithm lin group coefficient extension fan studied discuss wavelet coefficient van b logistic yu penalty selection special zhang simultaneous individual bi bridge propose general coordinate grouping structure reason rise goal example group indicator effect basis also introduce advantage meaningful expression gene belong biological pathway group genetic genetic marker gene desirable account grouping
document calculate power well vb give train thorough lda prior hardness prevent infer solution main reason almost corpora fact converge quickly despite loose application reach convergence average quality even much efficiently empirical analysis second prior thm lemma thm integral part non challenge want yield document article framework probabilistic topic flexible easily prior sparsity goodness develop infer latent topic model significant interest recent nature integral np vb collapse variational collapse gibbs converge slow vb fast often development remain common limitation quality second latent representation huge storage lie document lie gibbs limitation goodness inference extend sparsity probability process induce gain suffer severe drawback meanwhile change non smooth inference find expensive common latent directly inherently ignore document meanwhile require consume nonetheless retrieval computer inference sparse representation significance contribution resolve problem way framework model enjoy follow rate incorporate sparsity representation would exist representation add play may solution always provably decide hence sparsity approach theoretical existence fast linear mf interestingly knowledge first mf tr work intractable discuss definition introduce framework equivalent map briefly lda section practical go vocabulary vocabulary appear document term consist document kk k assume corpus mixture lda variant model proportion indicate proportion ml topic problem maximize topic inference proportion maximize necessary infer topic contribute emission document nevertheless unnecessary many leave popular objective situation differ discriminative belong serve propose framework continuously differentiable frank wolfe topic proportion frank wolfe proven moreover find solution face simplex let continuously f frank wolfe find face select objective continuously concave frank wolfe vertex note rate provably crucial mostly explicit face lie approximate provably trade basically flexible replace forward basis selection flexibility objective difficult suitable serve give principle turn technique widely fact vb next subsection property corollary define iteration error frank wolfe replace forward basis want appropriate nonetheless extension open future research exactly frank wolfe next discuss proportion inference endow naturally map besides imply inference model topic reformulate maximization contribute pz j j task maximize need topic assume far density express reformulate p constant would complete proof reveal map exactly function choice topic document h form complete influential topic model us connection concave reformulate problem combine document exist solution allow proportion objective simplex frank wolfe modify consideration lda proportion lda problem function objective come dirichlet topic induce function cause hard require document latent sense lda modification objective log inference lda proportion task logistic correlation note conjugacy deriving inference algorithm inference slow contrary model employ proportion various correspond vector identifiability use recover key reason mostly correlation importantly inference correlation document far use topic proportion reformulate concave task concavity second derivative diag diag diag element diagonal diag semidefinite combine feasible concave interior consequence maximization maximization specify fortunately interior convergence basically exist belief believe derive note inference correlation slight modification objective imply inference exist converge rate objective well define unit simplex slight frank wolfe slight modification original maximize believe easily enable investigate framework sparsity infer proportion analysis library write reduce design library general enough topic flexibility successfully demonstrate work attractive deal application employ design effective supervise analysis
formulae root label different symbol arbitrary possibly generality tree follow symbol label adjacent sequel variable usually leave path root internal subtree root root leaf contain subtree devote relevance hypercube propose basis use denote often treat depend boolean relevance dimension induce assignment relevance exist indicate relevance obstacle correctness relevance hypercube set relevance hypercube subset sized relevance hypercube iff contain least hypercube kind read follow fill sized first subset relevance hypercube hypercube exist otherwise valid relevance sense know agree relevance know relevance hypercube one recursively reconstruct skeleton tree represent equal relevance hypercube check row sophisticated reconstruct use paper assumption subsequent include material arbitrary read relevance hypercube check read vector take read firstly root label symbol aid focus irrelevant need main proving correctness remain aid induction represent section assign colour definition relate function present need sequel projection colour leave subtree colour label symbol colour remove replace projection adjacent node symbol remain root support subset internal reason necessarily boolean variable relevance hypercube every hypercube size internal label symbol hypercube restriction symbol read function denote iff stable existence relevance say hypercube partitioning conservative branching inductive partial subset subset extension take composition need conclude induce read lemma relate structure property reflect variable colour leave subtree colour arbitrary variable conservative since read agree hypercube restriction hypercube therefore one internal colour colour leave colour repeat reasoning place reveal colour colour unique index e take colour root leave colour contradiction partition refinement node great colour leave different leave leave colour symbol colour define subtree short root let induction nothing colour claim set conservative cardinality restriction relevance input label leave colour take except one subtree construct relevance hypercube need definition tree read different depth subtree inductive colour subtree belong put state conservative agree hypercube root leaf label internal colour colour contradict complete read hypercube constitute relevance alternative read agree make tree leave follow mi rule put contain subset subtree colour conservative stable lemma leave lemma agree relevance restriction subtree depend restriction variable relevance set relevance hypercube regard projection recursively end formula read give form classic read reformulate correspondence internal independently transform table circuit tree represent boolean node reveal verify agree sized input concern issue polynomial acknowledgement author useful discussion grant md read variable distinguish read show check contain relevant improved reconstruct variable strong classic representation boolean function read appear check iff read function check iff distinguish unlike irrelevant denote read checking basis small vector check test regard basis hold relevance contribution method correct basis arbitrary checking cardinality previous proof point relate exact firstly constant secondly check regular projection may turn thing consequence learning know implement unknown basis query turn identity basically specify function
htbp absence record number tweet user tweet help recommendation accept item hence divide user small balance precision computational successful recommendation reflect user interest analysis recommender consist popularity rank discovery recommend item acceptance rejection select category reach precise specific group item organize dr indicate popularity possibility acceptance little recommend popular item item function return rank similarly whole normalize user accept item interest active keyword interest class analysis mapping suppose item compute item category satisfy candidate include distance normalization directly storage computational enough collaborative interest search interest let amount search interest return indirect merge adopt comment could happen link obtain sigmoid correspondingly target category include modify increase interest coincide choice affect recommender training supervised present initialize stochastically termination initialize recommendation epoch weight search round eq momentum factor typical update training epoch training control apply instead process discuss performance enhance system experiment record stochastically divide omit update comment computing train process user interaction inactive inactive prefer similar active popularity item train system recommend recall item low due difficulty interest adjust help inactive ccccc item active inactive enhance recommendation user interact preference group accept item possibility base similarity frequently update platform interest interest behavior stochastically gradually track keyword user find target category indicator order respect user initialization good accelerate reduce search competition competition thank wang platform university zhang precise recommendation help twitter design hybrid solution track task user might consist user extraction recommendation performance lead service considerable growth observe dominant since chinese helps result attractive recommendation crucial recommender system influential unfortunately consider little fidelity difficulty stable overcome hybrid recommender generate item mining platform discuss problem hybrid discovery interest order experimental popularity twitter services china lead platform attract lot million million daily dominant china instant service carefully platform nice growth group service embed user platform service write comment website party group china time public widely frequently comment message matter website active user twitter platform recommender real interest accept recommendation recommend item list present prior introduce recommender overcome main help interest association rule huge involve search parallel adopt mining association property support necessity frequent frequency keyword class user keyword database
discuss road represent direct road intersection segment intersection road segment along road segment account segment date trajectory discover sub trajectory trajectory cluster cluster possible proceed graph trajectory bag comparison segment segment checked individually take come individual segment spread naturally adjacent road segment therefore sufficient bag segment implicitly trajectory road segment fashion spatial analogously tf length cardinality second inverse frequency trajectory calculate similarity weight similarity calculation traffic monitoring portion pass undirected trajectory node two assign weight edge choice natural also put emphasis nothing never put cluster since analyze number edge trajectory one tend vertex degree reason modularity community modularity measure cluster modularity community randomly modularity graph perform modularity node discover validate retain proceed isolate sub nest level induce modularity synthetic dataset generator use edge direction start different similarity compare use td weighting achieve weighting vs classic hierarchy top low cluster produce third choose expand sub arrival trajectory along visit road visualization separate produce cluster understand visualization traffic one cluster expand agglomerative calculated adjacency agglomerative linkage comparison conduct hierarchy cut number cluster quality end overlap trajectory assess road segment trajectory share result number trajectory lack table trajectory cluster share road classic overlap achieve linkage come fact unbalanced trajectory linkage linkage modularity behave c overlap mod mod attract proposal include mainly base suppose move hand movement start path like configuration furthermore whole path trajectory might path contrary modularity trajectory road trajectory base cosine define conduct modularity optimization trajectory propose relevant hierarchical future direction exploitation relevance decision make traffic
gradually dimension underlying estimation size estimation neighborhood estimation depict estimation obtain estimation technique technique capable respectively rp precise estimation could cope neighbor value result rp see fig group magnitude obtain estimator among competitive computation also computation computation time slow time see fig reduce rp advantageous work directly rp dimensionality reflect view european implementation cm mathematic ph ph os year award award ph student image transaction method collaborative sc solid molecular sc physics technology os european artificial intelligence review journal review conference paper lead illustration test dataset neighbor method size column size size size column neighbor knn knn rp digital processing original publication tv os department mutual various exhibit computationally intensive consistent distance embedding rp technique method image efficiency demonstrate theoretical distribute problem rp classification near manifold geodesic stream reservoir review relate compressed sensing show recently rp potential patch vote patch naive promise correlation norm theoretical exhibit modal paper apply neighbor color intensity filter texture descriptor may dimensional formulate mutual accomplish randomly straight histogram bin image considerably efficiency theoretical shannon multidimensional differential efficiently purpose rp evaluation approach exploit formulate condition knowledge rp rp conclusion draw task section follow embedding projection transformation transformation describe parameter possible produce close measure measure among thing similarity pixel simple choose neighborhood pixel value color vector q similarity information concept joint shannon differential entropy quantity enyi similarity projection purpose distortion set preserve approximately construction notably property probability random strict cost introduce draw rp length construction rademacher image characteristic compare directly estimation cite reference efficient note pairwise sample rp decrease divide sample group call estimation multidimensional entropy pairwise approximate embed address rp base divide fp ab ng drop estimate entropy rp ed group enable quantitative version green channel version filter version fig choose evaluate objective angle interval ideal degree choose around feature rp draw construction size neighborhood dimension feature randomly project rp outside interval outlier estimate k partition neighbor minimum spanning type
rr outperform tree significant tree previously parameter poor mainly variant right combination straightforward performance project process random forest good appendix show benchmark training configuration project process closely forest variant rr train number perform poorly regardless amount instance unseen combination instance configuration configuration run set split random permutation combination configuration surprisingly benchmark permutation randomly instance l sp pp rt rf rr nn pp rr bound rmse data provide additional pearson coefficient sp nn rt sp nn rt rf inf e e unseen quantitative figure generalize section respectively heterogeneous set poorly outlier ridge case order quite experiment different region space heterogeneous sometimes runtime robust prediction instance observe ccccc ridge neural network prediction vary select correlation remarkably well high hour execute second user execute build ridge variant heterogeneous otherwise forest perform across heterogeneous benchmark remarkably low yield datum interestingly important give instance decision configuration well unseen instance already pure unseen instance rr pp rt rf rr sp nn pp rt rf test test ccccc runtime ridge rr instance long configuration good instance plot benchmark predict visually indistinguishable random forest closely forest ridge ccccc rf configuration figure sort hardness configuration value runtime combination represent advantage forest well heterogeneous appendix instance configuration visually assess work true indistinguishable random forest challenge instance configuration interaction parameter configuration hand regression neural network capture instance hardness distinguish even training instance configuration prediction yield capture instance hardness configuration simple forest instance configuration nd evident finally depend focus forest forest salient improve qualitative capture difference overall increase set yield return entire runtime practice comparison far second second issue forest run unknown censor usual censor runtime censor indicator typical strategy deal censor datum fast really bias bias mostly call building unbiased censor need people people use handle censor survival analysis handle censor regression make portfolio algorithm selection good method apply exist candidate consideration process one likelihood right censor forest describe context handle censor base configuration pdf cdf runtime gaussian maximization rf I denote predictive rf good rf confident censor runtime censor rf variant implementation detail potentially prediction know maximal runtime keep censor subtracting sample absence censor major mechanism need observe experimentally procedure baseline ignore censor treat censor runtime unseen censor different censoring threshold instance specific runtime instance represents generate experimental study section instance configuration prediction strategy measure rest estimate drop censor vary benchmark qualitative treating censor dropping censor yield consistent treat censor yield beyond case vary enable well l drop slack half drop drop censor treat one censor quantify show drop yield improved variant yield top threshold drop yield uncertainty value censor close yield half vary variant yield uncertainty prediction censor poorly figure varied brief respect yield low good rmse censor fix threshold bottom threshold censor benchmark advance combinatorial new building focus parameterize np problem aware algorithm compare forest process whether unseen parameterized previously unseen instance demonstrate setting coefficient small hundred forest far handle run wide variety researcher seek schedule portfolio automate datum source reproduce experimental online thank early thank benchmark benchmark clause preprocesse preprocesse benchmark comprise instance clause respective standard maxima clause comprise instance average variable clause deviation respective maxima clause benchmark comprise category instance contain clause respective deviation respective maxima clause union bound check comprise verification library clause deviation maxima clause encode software verification static verification filter window os clause respective maxima clause win previous mix publicly mixed benchmark instance instance deviation maxima cm comprise constraint deviation comprise spread red conditional decision random generator parameter instance comprise encode winner determination combinatorial select ratio deviation maxima comprise euclidean generator select deviation comprise clustered dimensional instance generator deviation set prominent repository code complete em em predict previously input machine application analysis portfolio parameterize decade much thorough treatment empirical kind span range instance overall new substantially runtime approach algorithm simultaneously machine prediction empirical model parameterize mixed np ai input arise realistic size conversely take amount theoretical objective runtime greedy competition reflect scope include variety practical classic selecting predict sequential setting algorithm algorithm setting basis benchmark generator assess impact forward identify five explain algorithm review past work describe contribution sophisticated forest approximate gaussian arise setting section new novel yield comprehensive advance perform comprehensive runtime date evaluate method data instance parameter section literature analysis handle improvement perform forest extension censor conference first evaluate work gain insight hardness ai plan three planning depend size run long community predict runtime various planning decide runtime specifically success runtime programming winner determination show runtime solver regression include lasso multivariate spline regression end relatively ridge regression subset expansion response polynomial yield runtime ai planning make connection literature survival censor work dynamic configuration publish characterize hardness combinatorial classification article neural also classification yield worse make online fast run good contrast predict meta compute stop act scheduling distribute performance analysis discuss show subsequent refined approach incorporate note runtime correlate mostly prominent fitness correlation autocorrelation exception expensive predict model describe rather number careful necessary issue literature design input generalize core automate literature consideration running instance continuous aware relax categorical regression instance analysis detect use linear quadratic note characteristic control call model much local problem runtime tackle employ suffice yield five feature yield identification mix previously quantify measure interactive use see provide thus detail list computable piece usually domain problem low parameterize configuration real second principle run runtime quality consumption communication configuration combination entire allow nevertheless review configuration set record vector performance transformation easy focus effectively predict log experience find important variation hard discard input across datum normalize deviation relatively rarely handle miss normalization miss respective ridge regression ignore multiply validate mean rmse simple frequently grid fit input simplicity conceptual interpretability combine competitive performance feature predict f optimizer scale gradient minimize error default hide neuron gaussian gps dominant ridge albeit great expense similarity choice input idea correlation general expect take variable subset tradeoff specify observation optimize improve book contrast hyperparameter hyperparameter grow integral gp maximize optimizer optimizer make big limited bfgs gp expensive inverting hyperparameter optimization yield prediction time know handle input tree disjoint predict follow point region zero point square mr regression procedure start binary scalar data point otherwise split sum square region equation continue find training tend branch contribute little fold see predict response variable split point continue child categorical select error split categorical good loss efficient sort predictor datum thus implement fitting tree dimensionality time variable continuous split split categorical partition consecutive split building depend balanced lead good case recurrence experience perfectly balance close bad point tree regression merely need propagate query tree node point categorical store mask member take balanced primary improve prediction parameterize advanced handle categorical prediction partially space new model family base potentially forest regression tree limit input prediction value input section value categorical input extend arbitrary handle encodes categorical domain binary input indicator column column point one rest column treat encode categorical gps dimension
political event prominent expect political sentiment remain sample crucially estimate discrepancy experiment demonstrate algorithm regression thank discussion paper derivation rademacher bound rademacher adapt proof jensen verify associate unchanged sign equivalent analysis discrepancy prove rademacher discrepancy scenario tighter substantially improve upon previous one generalization line batch combination exploit qp demonstrating keyword environment domain pac draw spam financial condition environment devote line scenario allow study label distribution consecutive step generalization scenario long show consecutive generalization achievable improvement assumption constant setting allow intermediate base relationship recently analyze consecutive nearby assign limited paper deal drift pac bound unlike bound generalize context adaptation previous admit favorable scenario use accurately complexity contrast discrepancy tight rademacher complexity vc simple standard batch term discrepancy measure line lead meta combination discrepancy section qp preliminary demonstrating finally discuss natural section definition input output space loss function l adopt complexity sequential rademacher element random draw coincide use even favorable efficiently loss adopt provide measure dissimilarity distribution take consideration loss function set discrepancy introduce adaptation fix consecutive hypothesis distribution define discrepancy already mention accurately finite rademacher h h discrepancy definition symmetric triangle inequality base discrepancy discrepancy straightforwardly advantage fact discrepancy unlabele two learn pac loss scenario environment oppose generalize discrepancy define sample return mistake mistake scenario carry upper risk also scenario pac agnostic emphasize expect depend x first commonly empirical bound loss th side yield rademacher complexity consider additive next good expect return empirical erm combine bind indicate trade value small grow increasingly challenge minimization limit last instead algorithm pac example exactly select rademacher vc fix mc minimize hand exactly many vc discrepancy divergence lead tight informative mh last integral rewrite mm corollary vc track track however satisfied tight base rademacher complexity importantly fine target accurately finite analysis empirical risk bound arbitrarily close discrepancy bound favorable absence distance would uninformative vertex uniform measure measure function disagreement also hypothesis present dramatically favorable illustrative greatly case sequentially assume adversarial scenario distributional return line hypothesis strong guarantee regret minimization combination need main draw form loss bound process probability inequality hold discrepancy independent hoeffding q follow sum inequality statement theorem bound argument hypothesis return process hypothesis least q scenario I sum vanish straightforwardly rhs th statement claim
class ad balance scale cancer rating block digits diabetes tumor segment vote breast variation classifier test difference variation gm multivariate attribute dimensionality uci dataset variation lead classification c c gm ad ad gm ad paper anomaly ad detect various anomaly content document challenge extract transaction automatic extraction content address unique transform algorithm since feature preserve prominent detection association perform poorly dimension comprise univariate model superiority algorithm detect anomaly transaction ad transaction filter anomaly prevent attack machine school science ac il transaction interact transaction attack potentially interact regarded anomaly transaction machine technique anomaly automatic method extract feature transform dimensionality ad utilize central novel four transaction capture system detection anomaly security machine anomaly detection interact transaction may fall attack mistake medium regardless origin especially schema exploit interact information anomalous transaction detect anomalous mean security end security protocol transaction signature mostly take protocol document process language definition language describe document tag format transfer share internet file schema schema anomaly action attack mistake technical error describe prominent generator attack interact web service attack attack exploit mechanism server prominent attack buffer field attack site schema attack attack attack produce anomaly attack stre transaction collection inherently modern arise cause attack attack simple human transaction lead simple way regular simple scheme file anomaly attack document application anomaly detect anomaly document attack anomaly opinion important behind approach anomaly nature would work detect anomaly require elaborate domain indicator attack document anomaly anomaly aim discover majority anomaly spectrum security financial surveillance name anomaly range cluster neighbor theory supervise impractical supervised classifier example pattern moreover easier obtain anomalous domain semi supervise anomaly detection semi anomaly detection transaction assume ht classify label general denote score close document e sx x dx pairwise distance mutual feature able document document anomaly anomaly detection plausible inherently curse dimensionality consequently dimension grow dataset concept another similarity use unable new incur allow localization anomaly pattern detect detection function domain detect far anomalous paper outline detection framework transaction framework section present several evaluation experiment summarize document anomaly follow detection method take supervise al propose occur relationship association quasi functional dependency anomaly previously association information represent outlier rule database rule incremental handle database anomaly describe relationship item occurrence chi attribute anomaly knowledge incorporate et classification anomaly extract accord attribute feature represent introduce liu detect frequent anomaly value arithmetic detect anomaly extract namely adjust software computer original extract structural transaction practical anomaly propose detecting stage extraction dataset generation machine learning transaction define transaction transaction feature extract definition put extraction extract feature element generation transaction aggregate arrange tuple contain transaction tuple add train final stage anomaly stage extraction start acquisition parse definition file numerical define visible contain file carry type etc enumeration range handle process single transaction comprise element store complex occurrence occurrence measurement correspond relate htb produce object string traditionally machine require regard input scalar feature transaction contain contain different item met overcome process consequently rectangular gain determine due aggregate tuple separately complex denote tuple aggregated since aggregate tuple derive rectangular aggregate aggregate aggregate feature however minimize loss aggregate function simple summing possible enumeration dependent enumeration string produce simple word minimal transaction compute document frequency lastly tf prominent word choose extract tf transaction text contextual mention dimensionality scalar effective detect anomaly process essential property translate element standard thousand feature anomaly feature anomaly detection instance represent class inherently compare unsupervised anomaly detection detection capable none cope dataset dimensionality high propose later unsupervised anomaly transform see fact regard source attack propagation localization complete operational meaning anomaly goal anomaly detect anomaly univariate classifier full regard normality dimension anomalous transaction identify low likelihood additionally order section method attack transaction use svm prominent branch svm algorithms algorithm specify seven experiment classification adaptation identify anomaly anomaly near neighbor although ranking problem classify compute neighbor classify point anomaly percentile density near compute compare ball ensure neighbor least one anomaly achieve l c ad mean gm ad gm main classification receiver characteristic roc plot specificity vs sensitivity equivalently fraction true positive positive select model discard one distribution strongly diagnostic use decade auc cross procedure subset utilize second subset role process repeat fold implement experiment pair level difference auc statistically follow hypothesis follow evaluate transaction collection collection transaction real transaction relate anomaly contain transaction capture university transaction actual attack transaction follow standard transaction item ensure collection comprise transaction consist transaction transaction process extraction transaction regardless transaction stand attack computer university attack capture convert duration attack minute tool network traffic computer file collection put process collection transaction differ aggregation evaluate ad experiment transaction document however document contain anomaly system entire anomaly ai module transaction site text attack perform transaction attack represent string encode attack encode attack transaction lastly attack element sentence child book attack ai module transaction element target transaction construct anomalous transaction module one mention attack class transaction avoid ai module attack schema mention attack introduce anomaly rare content anomaly affect system transaction stress train anomaly detection label document anomalous transaction focus determine approach document machine suit regard produce additionally examine capability algorithm examine accord detect anomalous transaction classifier transaction level attack evaluate ad ability detect attack multivariate determine anomaly belong whereas gm classifiers roc depict possible obtain rate gm cn result test distinguish anomalous transaction conclusion process preserve detect anomalous examine gm average achieve ad classifier perform auc close result aggregation dataset dimensionality relatively arithmetic aggregation multivariate svm poorly transaction distance detect anomaly transaction previous substantially well classifier univariate classifier inherently classifier deal detection well produce detection examine experiment classifier system new gm use geometric aggregation denote system indicate classification mu stand ct anomaly detector transaction classify real avoid bottleneck transaction time ad gm auc ct auc ct cn transform multi univariate increase effectiveness improvement mu mu respectively version enough experiment high auc classifier time
follow seminal intuitive advanced via idea behind measure system equilibrium u convenient conceptual parameter possibly convex element whole entropy consider possibly variable respectively parameter support eq strictly increase employ hellinger distance hellinger complexity intensity shannon gray proportional remarkable closely relate e discriminate various hellinger distance universal linear corner summarize provide value discrimination target stem analysis process os platform excellent reveal information model homogeneous heterogeneous appear promise include analytical hellinger sample generalization de col de generalize complexity et functional capture order distance former latter divergence scene corrupt amplitude non require model describe among law validate expressive tractable target assume appear little shannon hellinger complexity expressive identifying type synthetic prominent carry conventional sensor optical spectrum active sense carry illumination source operate operate spectrum price pay automatic visual use image analysis call assess dimensional signal able regime statistical nature law entropy divergence complexity stem feature framework describe stem image formation intensity format every outcome variable truth intrinsic unitary law return completely multiplicative single distribution amplitude format accommodate later description multiplicative reduce distribution law gamma adequate homogeneous eq denote model heterogeneous area law develop texture illumination absolute distinct order distribution system via pdf describe observable series primarily characterize give shannon one degree shannon logarithmic physical restriction represent system micro predict probability
price design reason study table code fitting compute choose cpu implementation use dataset slight genetic optimizing reduce variation result average design choose c table profile log discrepancy implementation precision double matlab far suggest precision arithmetic yield optima stress neighbourhood good optimum double precision run cpu refine large runtime cpu gpu somewhat short double might involve include software package matlab similarly library gpu accelerate possible four gpu cutting currently cpu gpu network like fourth fast world chinese develop execute numerous processor frequently heterogeneous technique popular intensive computing computing architecture graphic unit powerful cost capable modern central great deal intensive statistical dataset demonstrate implement gp gpu gpu acceleration suggest drive graphic offer computing processor complex parallelism modern execute task parallel execution time magnitude extremely purpose graphic unit intensive cpu gpu traditional fitting gp computer consume time model determinant numerous spatial cost determinant prohibitive moderate large moderate dataset thousand run fast speed leverage intend demonstrate heterogeneous computing platform lead remainder organize review basic formulation model intensive implementation gpu cpu discussion study suggest intensive computation gp model terminology simulator simulator model although several choice exponential parameter good power exponential correlation stable alternatively suitably avoid another like mat ern numerically cpu gpu likelihood parameter q log likelihood q hyper denote determinant expensive increase numerical stability qr cholesky system solve implementation viewpoint log likelihood back solve log likelihood evaluation expensive surface optima maximize challenging contour input generate hypercube simulator dimensional example simulator chosen contain minima near though region near evaluation evolutionary like commonly gp point minimum sufficiently involve matter search fitting computing ga require cholesky decomposition seven cholesky ga cholesky solve gp solve fitting use reduce burden heterogeneous gpu moderately intel core cpu gpu intel cpu capable core per core capable per cycle core per cycle intel processor gb matlab hardware slightly old class ram twice core computer multi core processor end fx core dual core cpu write matlab linear systems link performance inverse linear matlab cpu specialized parallelization core platform adopt study intel core core cpu development process matlab programming enable gpu free library step gp gpu library algebra operation routine source gpu block similar available gpu evaluation correlation kernel library include back transfer gpu mutually note parallelization value perform likelihood numerous simultaneously parallelism execute time computationally intensive
biological biological relationship gene biological represent relationship activity graphical distribute element effort focus example problem remain et likelihood develop nesterov method solve al ascent widely glasso call box alternate problem lin solve interior et newton estimate derive condition solution single subject associate block decompose small massive gain formulation observation draw single may work multiple matrix jointly penalty introduce learn et estimate fused grouping penalty admm require eigen decomposition propose node graphical even recently graph decomposable necessary fused modeling fuse penalty estimate graph paper fuse work formulate sequential estimate graphical penalize regularization encourage adjacent similar graph consider order common motivating disease emission graphical nc mild cognitive patient ad expect common identical expect evolve time order nc ad fail key contribution sufficient fuse diagonal duality precision screen computational employ order solve project shrink conduct synthetic effectiveness paper matrix denote bold format tr clear eigenvalue denote form diagonal share rest paper fuse screening present result sample identically variate covariance many conditionally precision notational simplicity covariance take eq furthermore usually dense induce sparse employ fuse simultaneously lead sparse fused encourage mathematically solve model fuse graphical assume throughout diagonal strictly say ensure assumption solution establish statement problem optimal solution sub f region nonempty one observe associate p diag tr kl tr hold u solution feasible compact fact solution order optimality subdifferential convex convenience objective unique strict optimal hence set nonempty diag diag j relation addition addition notice relation together every bound thus conclude hard compact ready prove theorem observe equivalent also problem determinant involve many precision reduce first block precision whole necessary fuse graphical remain derive necessary sufficient solution fuse multiple block graph first demonstrate decomposable derive say block feature exist permutation simplicity presentation throughout suppose k kl l demonstrate diagonal decompose sized latter problem problem address block decomposition scheme section satisfy describe adjacency satisfy otherwise adjacency matlab component partition scale decompose latter let system eq satisfy inequality solution lagrangian dual ready sake inverse order ij ij k ij k k diagonal solution block block ij ij ij ij lemma ij matrix kl kk optimality fact optimal conclusion screen capable partitioning feature size large sized block need compute solving replace problem simplicity simplicity tt iteration optimization iterate quadratic obtain newton suitably monotone show zhang locally computational global rate proximal decompose set sized addition search length ensure reduction positive prove exist let associated solution reduction one convexity adopt backtrack length hold cholesky positive associate term cholesky give advantageous minimize space issue identify single lasso allow update shrink single size quickly shrink improve technique also successfully shrink fuse graphical gradient iteration ki ij follow optimization reformulate ij j fix optimality solution shrink scheme shrink coordinate free fix addition quadratic result summarize determine free backtrack screen rule image experiment perform intel ghz gb memory screen following include comparison admm screen admm admm screen matlab involve involve double loop implement routine fair block ground block structure number matrix draw fuse penalty fix total solution terminate error admm stop admm great speedup time parameter c iteration admm admm experiment way draw reverse edge edge edge distribution precision vary replication fix accuracy glasso high accuracy demonstrate graphical ht attention affect school cost united states project fmri typically child research detail website dataset develop graph region obtain computationally intensive datum screening curve fast admm second utilize demonstrate superiority include h block rough cost block compare high rule identify identify block image ad mild cognitive control nc disease database region brain volume volume automate derive corpus examine whether information common percent group stable edge glasso say nc glasso
road cluster cluster meaningful level hierarchy cluster hand cluster top level top segment sake road classic agglomerative linkage visualization result omit due limitation overcome community participant positively modularity need loose produce two study trajectory similarity measure trajectory dynamic subsequence real ignore movement apply context road network unsupervised proposition trajectory exist proposition framework tf aforementione similarity suffer drawback sensitive parameter ii trajectory rather homogeneous rarely discuss author describe discover path frequently road resemble segment aspect produce flat parametrization trajectory road network measure shortest calculation agglomerative efficient trajectory unlike computationally road network require together trajectory underlie road focus trajectory deal road segment segment often visit compute road segment community present exploration various situation flat clustering produce however sensitive trajectory rarely thorough segment produce index like real trajectory dataset impact community cluster phase universit e paris paris even trajectory attract move euclidean presence underlie road evaluate trajectory two constrain trajectory approach clusters road orient approach experimental synthetic proposition device store thank device become dedicated move light challenge motivated management analysis move database mining road collect along deduce understand dynamic dedicate sensor expensive precisely study trajectory interested discover road segment aim retrieve road segment frequently useful context management planning road trajectory give road basis predict situation guide planning object movement impose address concern proximity road propose interaction modularity cluster discover nest exploration analogy road trajectory segment cluster organize follow present trajectory cluster road experimental finally conclude road represent intersection direct road segment edge road link node trajectory sequence segment e segment connect collect apply produce segment give trajectory road trajectory trajectory aim road trajectory appear segment appear segment belong different unlikely appear framework introduce measure trajectory road graph trajectory partition community algorithm discover adopt segment trajectory segment individually account segment simplification justify underlie direct graph consequently direction implicitly visit occur isolated spread adjacent consider segment totally reflect absence fundamentally exist similarity trajectory especially unconstraine along separate close trajectory road since similarity cluster conduct massive tend especially modularity adopt express modularity edge modularity generate community community discover clique trajectory group heavily intend segment modularity account node occur trajectory great well code graph high significance evaluate modularity randomly valid present clustering apply recursively cluster edge separately stop partition step complete hierarchy trajectory explore detail maximal detection phase rarely practice complexity interest discover proceed analogy previous comparison similarity discover road represent direct frequently empty subset large segment concept segment segment compare road segment therefore equivalent compare visit claim advance road segment considerable road segment versa contribution characterize road road segment eq contribution segment ratio whole
dropout finally globally layer feed test patch image four corner patch horizontal though pass imagenet dropout globally connect serious without dropout art performance validation find describe fortunately main dropout significant help net joint many year non layer lead thing bad cm feedforward typically poorly hold greatly detector prevent complex co feature detector helpful several detector neuron learn detect context operate give big set record artificial adapt connection detector enable correct input correct typically perfectly weight worse test datum data detector prevent co train hide randomly omit rely unit dropout good train computationally train huge network almost certainly network presentation case share present descent normally use grow large length bind update unit division prevent grow large allow thorough start contain unit weight layer compute equivalent network guarantee assign log assign network square network always network effectiveness mnist write test greatly extract without without publish result feedforward error reduce use dropout incoming weight unit reduce drop random figure dropout also generative feature detector discover available train deep describe fine tuned unit code use pre train boltzmann backpropagation error hide enhance apply widely system hmms need acoustic frame fit state deep pre train neural network hmm outperform gaussian test benchmark belong input net advance ms connect hidden layer softmax subsequently merge dropout significantly architecture rate probability neural frame decoder know transition hmm sequence hmm without recognition dropout improve record identity core benchmark hide unit improve cifar object image object use transform data error use neural three pooling layer convolutional layer connect layer imagenet consist thousand image thousand subset roughly per competition six state achieve comparable performance convolutional hide layer final softmax incoming dropout see b recognition decision hold validation assess dropout class contain exclusive document vector count stop feedforward network hide gets reduce try various dropout performance network fully dropout dropout worse throughout input dropout help often adapt individual dropout probability unit performance unit require regime probably improve create statistically expert get fraction implement model give complicated feedforward typically carlo assume eventually equal importance make easy dropout net pass efficient averaging bag different bag often tree quick quick dropout allow apply feedforward neural dropout bagging sharing regularizer standard parameter towards familiar extreme naive bayes input feature predictive multiply together little train feature context finally similarity dropout recent role interpretation theory gene achieve co nearly robust achieve perhaps optimally way gene end fitness co number gene environment decrease fitness learn thank h discussion google microsoft advanced consist digit classify digit architecture sensitivity dropout net architecture dropout visible unit stochastic entropy exponentially decay start apply minibatch multiply epoch income weight update length scale draw momentum momentum linearly stay multiply update minibatch epoch tp f f neural dropout consistently since input pre train rbm bias initialize number sample visible divergence momentum momentum start linearly multiply epoch layer rbms use activation rbm rbms neural dropout backpropagation momentum epoch minibatch hyperparameter mnist needs run backpropagation keep classification objective validation architecture dropout achieve overfitte net train remove momentum corpus volume corpus cover economic social market market account market production service class split category category remove cover divide test frequent dropout backpropagation backpropagation architecture hyperparameter dropout epoch net train dropout improvement smoothly stop million color image web search image search english come noun cifar image among ten class cifar filtering image incorrect cifar varied canonical viewpoint scale image contain instance belong indicate imagenet image category amazon crowd tool roughly class recognition competition object imagenet image spirit cifar resolution difference imagenet imagenet number imagenet feed convolutional network cnns feed layer neuron output bias add filter activation neuron pass filter refer differ ordinary organize bank organization neuron apply filter local extent center neuron decrease pixel object input examine neighborhood neuron bank different location roughly statistic image kind treat equally bank apply cnn neuron become channel pixel determine convolution operation large imply neuron bank unless share reduction neuron apply reduce advantageous convolutional cnns pooling summarize patch neuron convolutional convolutional pooling consist convolutional output bank pooling layer call layer space least apart pool convolutional pooling pooling unit aid pool convolutional unit layer translation pool activity cell exhibit invariance include response layer layer encourage competition activation bank organization neuron order course arbitrary begin real utilize nonlinearity neuron neuron bias nonlinearity advantage traditional time reach nonlinearity processing activity scale take pixel network raw across label initialize positive layer max nonlinearity take derivative input initialize sufficiently simply try initialization attempt neuron help ground reason use stochastic batch size variable objective respect gpu cifar take minute imagenet four equal layer whose power typically
determine sm statistically significant conduct test degree trial suggest sm statistically sm feature virtue encoding collection albeit million report sm feature highly movie baseline purely thick positively dash highlight font color highlight green movie colored topic popularity user interest mean account moderate edge show attribute pair popularity introduction pose question facebook pt interact content interest user thing interest movie distinct interest exhibit sm interest friend social user interest hope sm interest visualization say visualization sm learn subset parameter infer various fact dataset give towards various reveal friend learn topic across interest cosine middle commonly user interest next search topic high count normalize popularity popular show corner friend positively friend specificity valuable exception strong yet positively movie anomaly occur friend presence something fail statistically much popularity significant proportion separate life implication fit unlikely movie actual positively play connect bar south final concern topic notice user positively topic country ac movie within yet possess highly notice topic word highly activity contain modal connectivity speech serve anchor common meaningful inter interest term topic observation policy grow network increase modality distinguish exist model network either outcome text topic network link except asymmetric link crucially infeasible zero link model complexity also work facebook used implicitly sum feature neighboring cast topic text link neither outcome equal result link influence salient interest facebook million system model sm learn aggregate user interest million sm closely combine address challenge scalability phase phase parameter likely likely scale user facebook facebook title video social decade aggregate hundred million broad open problem user interest health social classic gues appropriate content potentially interest provide content user friend increase content consumption game player activity oppose play social interest facebook facebook interact graph content occur interest interest say movie group pattern facebook tool visualize summarize salient aggregated population critical enable individual detail social network turn policy aim network enable growth interest however mostly incomplete text act flow user attain complete view medium ignore user modality view link categorical principled enable capability yet produce address facebook hundred million diverse modality profile status name linearly datum fail presence structure facebook information simple multimodal require treat potentially sharp change require challenge mind present scalable visualize interest million facebook scale hundred key datum early successful certain modality mix blockmodel citation mix sm multimodal integrate user subset millions feature initial integrated feature pass information principled manner collection topic usually coherent assign intuitive g report label vote would topic positively correlate label value label facebook know positively interest movie contrast two motivation friend naturally interest interest degree mutual recommendation perhaps highly mutual driving potential drive prefer friend drive could latent figure label single proceed application four interest movie answer interest justify analysis interest bag facebook context movie status update link denote interest movie movie intuitively want capture concept word interest actor correlate interest movie friend actor relationship network insight nature facebook text pose user instead use learn label sm sm stage pt sm subset network learn text user label tendency towards topic user find topic classifier consist topic represent anchor interest topic topic user frequent give name american american self american friend label user topic might american contain word term user vocabulary desire generative background vocabulary draw dim word dim draw topic user text foreground word ik user topic user draw label shall circle red learn salient value text every dim probability label salient topic high word link friend american coefficient correlation represent status title compare form facebook document hence document exactly draw notable topic tailor paper facebook keyword irrelevant topic generic topic distribution per foreground background draw assign separate class distribution friend let matrix arise topic outcome generate word specifically draw upper triangular bernoulli draw essentially describe topic facebook sparse variable put pp music include learn positively interest user indicator topic sm proceed parameter follow explanation form right detail simplify reduce latent integration link beta binomial conjugacy random remain also amount ensure scalability direct maximization hybrid motivated gibbs sampler like easier implement easily optimize parameter dimensionality resort direct condition derive use background word user function prior term compound distribution integrate word notice importantly time sampling background topic ignore integrating otherwise come integrate background word simple count ji ij clause take care term integrate compound bernoulli integrate algorithm require metropolis hasting draw draw stochastic gibbs expectation whose th gibbs sampler ensure gibbs initialize eqs prediction input parameter randomly initialize user friend predict feature assignment gibbs observe learn sm learn topic likely friend discover facebook facebook user movie relation facebook page interest facebook select broad scope page sufficiently base collect complete million million explicitly mention collect follow text document status status one facebook title document typical nlp removal identification user symmetric list collection across document sm latent th iteration cumulative increase
relax prior family message loop reach loop exact penalize kl w relaxation easy choice part scale old message relax cause side product simplify overhead negligible practice choose expensive kl relaxation relax kl new propagation q w w energy detail update however update ep difficult matching demand exactly match ep ignore reduce final accuracy min max include special max ep bethe relax moment far along approximation rather identically sample input noisy latent gp projection encode smoothness nonlinearity factor factor relaxation share bf initialize converge loop posterior multiple minimize relaxed divergence cumulative obtain release implementation publication ep ep fractional variational case outlier ep increase algorithmic stability change divergence minimization necessary step message size reduce speed furthermore guarantee ep ep speed ep excellent inference gp ep thesis use update possibly improve quality figure belong reflect labeling x classifier dimensional input approximation importance sampling posterior boundary figure outli boundary bayesian decision boundary close decision perfectly power examine exact covariance align covariance ep outli force ep relatively small wide range consistently ep nonlinear gaussian red blue label vary apply tune decision boundary ep dataset set power relax ep clearly ep decision give well ep shape finally decision illustrate reflect curve time sample test iteration convergence run reach ep frequently always converge accuracy flip introduce labeling error low error test five uci benchmark diabetes spam detect heart time diabetes medical history uci diabetes two experiment patient five year breast contain patient randomly accuracy split outperform detect spam partition training test experiment demonstrate various additional error ep method increase ep relax classification demonstrate avoid give ep primary energy q main text update guarantee relaxed experiment ep give outlier relax ep function kl duality condition constraint dual classification divergence optimize line gp ep ep cumulative ep cs cs edu school west expense powerful propagation expectation ep efficiently exact nevertheless ep sensitive suffer inference propagation requirement propagation relaxation exact moment presence outlier relax moment outlier greatly quality uci benchmark demonstrate significant ep ep algorithmic stability inference relaxed expectation process bayesian principled make prediction learn exact expensive address challenge variety speed computation representative approximate generalize approximation discrete continuous posterior maintain gp predictive despite bethe examine model ep good experimental benchmark consistently power performance distribution probabilistic normalization could obtain
differ slightly optimize e constitute sized interior method idea convergence l bad big coordinate increase lipschitz logarithmic precision hence role feasible concentrate projection converge contrast devote obtain different infeasible sequence generate primal dual saddle formulation primal polytope relaxation pair cast saddle lagrangian iteratively dual primal feasible gap l one decomposition subgradient size apply relaxed imply convergence guarantee q none please operation take directly gap reconstruct primal method gradient vector converge optimum reweighte approach smoothed optimize mapping expression optimize unconstrained reweighted free energy optimization tool speed projection preferable approximate smooth optimize benefit lemma analogous converge primal optimization approach solution auxiliary see augment lagrangian base apply problem lagrangian combine descent versa scheme infeasible project optimize projection describe aware reconstruct primal relaxed non descent scheme relax convergence practice concentrate reconstruct feasible qualitative objective point conference benchmark implementation primal algorithm dual decomposition base subgradient ascent sg average sg nesterov accelerate gradient ascent smooth dual library optimize polytope elegant source author site primal potential lp map problem deal infeasible subgradient sg fast optimize exceed gradient require ghz primal significantly contrast integer relaxed problem converge primal bind sg plot line primal dual problem provide generate lp know generation unary potential assign dataset ran project eq suitably function purely potential significant contribution dual infeasible estimate lipschitz proportional plot l energy differ infinite energy correspond energy infinity projection primal one contrary efficient certain separability limitation however particularly overcome construct maintain feasible address support foundation application grant mm subsection corollary subsection definition subsection example end propose problem separability infeasible produce dual often feasible complexity obtain feasibility optimum property apply polytope inference problem markov demonstrate superiority relaxation appear vision often million thousand optimization strong primal primal sub infeasible slow size situation feasible polytope relaxation mrf problem relaxation integer application complementary heuristic search converge guarantee vanish duality determine sound ii basis iii adaptive optimization duality gap selection subgradient smoothing procedure problem cut plane scalable method infeasible separable construct feasible point optimum problem infeasible counterpart theoretically empirically polytope mrf formulate problem separable separable programming special illustrate method avoid numerous detail refer proof special generalization cone integer index subset matrix dimension program standard feasible ni main please due contrary euclidean compute solve assume scalable simplex due size additionally show speed assume polytope unary binary max unary potential feasible primal infeasible energy problem posteriori marginalization require relaxation hull set polytope local polytope approximation non theoretically algorithmic polytope relaxation dramatically simple optimization objective relaxed function maintain infeasible primal feasibility cut discuss message pass propagation optimum obtain infeasible vast conference aware formulate accuracy attain end primal auxiliary attain case grow size obtain slow get closely introduction section case relate mrfs allow reader familiar marginalization mrf specifie optimize construct dual algorithmic scheme dual primal estimate feasibility estimate optimize euclidean space paper notion optimize simplification optimize infeasible get big soon possess projection make computation additionally tree reweighte posteriori primal dual formulation separability construct optimize undirected finite labeling collection uv uv uv unary potential potential label minimize express denote uv uv r uv x relax programming form polytope later denote slightly briefly correspond satisfy iff else vector become collection similar suitably select write split subproblem use optimize local polytope collection primal optimize define simplex sized study example optimize quite grow assign euclidean quite bad demonstrate depict optimize actual value neither projection euclidean pairwise factor control infinite thick red line assume infinitely primal assign corresponding pairwise factor equal primal high relaxation model high local straightforward optimization split representation relax since unary one degeneracy u read follow explicit consist follow coordinate uv possess separability fix latter split uv unary potential improve denote reweighte unary potential constitute possible space optimize equation theorem optimize projection energy influence construct correspond polytope reweighte optimize formulate lagrangian base marginalization section reconstruct primal estimate dual technique base jointly solve
medium although preferable nearby compare one rating also stable characteristic mean less often ordinal number rating bit metric process address scalability inference predict measurement formulate partially complete contain node structural spatial long various ordinal completion great acquisition similarity recommender well applicability recommender simple factorization produce focus network example network contrast completion ordinal recommender describe network result application give end heart metric serve various example measure service indicate rate concerned streaming acquisition metric effort lead tool acquisition suffer accuracy bandwidth path paper metric bandwidth determine choose metric go addition rating reflect experience end performance define rating ordinal paper qualitatively good acquire determine metric belong illustrate evenly requirement need certain metric acquisition versus particularly significant internet objective bandwidth site medium service achieve goal use knowledge find globally allow rating matrix rank entry practice matrix often component negligible approximate completion entry preserve entry turn word try find approximate rank difficult constrain neither continuous rank adopt compact look class factorization mf contrast factorization appealing formulate node network come internet overlap heavily induce enable problem evaluate characteristic value original decrease hinge pt incorporate additional negativity besides divergence value close build prediction machine perform unseen reduce variance prediction combine final mf model mf different scheme adopt stochastic mf pick reduce sgd demand process locally interested reader decentralize ex path system randomly select neighbor sequel path choose trading increase always accuracy measure overhead lead pt ex mf construct complete proper give search optimal empirically one enough keep hand require choose limited evaluation among dynamic hour static adopt common use root square q small rmse dataset partition evenly dataset ms ms ms unbalanced rating learning rate equal regularization equal tune impossible decentralize empirically mf number regardless mf mf parameter predictor impractical mf mf centralize use mf strategy particularly generally perform nmf mf ensemble well mf marginal worth extra I nevertheless adopt sequel c mf c ensemble netflix netflix show adopt achieve three rating row represent rating thus mis rating rating class bad value connect choose optimality performing prediction class rating random baseline optimality perform good rating rating path qualitative cost metric address end rating recommender netflix factorization adopt benefit rating internet application ac li scale acquisition end distinct contribution ordinal rating completion former measurement
fold provide list alternative shortest discover close deal length node direct adopt short probabilistic uncertain relationship directly view probabilistic feature recommender experimental obtain decomposition represent method problem case conjecture exercise proposition world domain mean edge quantify likelihood existence represent likely language node connection two weight link instance link label link l regression predict unobserved real collaborative filtering well adopt last uncertainty lead usually edge quantify likelihood existence strength link reliability pair node connect graph probabilistic probabilistic along play network road concept whose task formalize link likely unobserve link graph important ingredient classical link prediction probabilistic need want whether two interested node paper likely probabilistic adopt tool connection may probability link particular label output correspond adopt learn model link choose recommender predict rating user world achieve induce svd kind probabilistic propose section work edge probabilistic system undirecte assign assign assign represent absence imagine instance discrete sample accord probability treat mutually indicate belong discrete graph probabilistic length edge path follow iff language language simple free path path path formally path randomly sample language otherwise give possibility free discrete path belong existence intensive intractable check every existence path sample discrete accord basic estimator eq check path iterative depth path existence visit sample adjacent stack iterative stop node stack non language path label element link belong train way classification unobserved class eq label link prediction language particular language eq query language consider language particular language real observe x x represent problem huge weight l regression belong solve unconstraine w I tc binary rest nonlinear quasi efficient minimize w score hessian element identity newton validate recommender probabilistic observable domain similarity graph similarity relationship measure heterogeneous entity item relationship assign goal user rate past predict unknown exploit user similarity user orient rating rating similar item orient rating rating let rating preference user item strong preference rate approach predict unobserved follow represent stand similarity user pearson item base formula neighbourhood connect item structure particular entity recently prove technique leverage mining item indicate similarity node kind similar user adopt pearson rated connect v pearson correlation rating rate user v connect item item compute pearson corresponding j rate item score add element belong solve constrain path path user belong language particular predict rate pl belong construct complex query connection belong user classification set computing item rate know map training link classification tool deal probabilistic research university international heterogeneity version range report rate movie user gender code site seven month rating consider fold rating test rating ccccc r testing create rating specific testing adopt procedure resp procedure sampling path language language neighbourhood probabilistic
recent book deal propose present overview context stein estimation elliptical estimation test along estimator deal mse estimator risk conclude square ls test hypothesis special situation elliptical q likelihood statistic hypothesis statistic let sn sn moreover maximum testing mixing get produce calculation probability significance call statistic cdf distribution follow stand insight lemma give possible elliptical see result measurable ease propose elliptical proof expectation see g far obtain addition proceed estimator estimator convex combination indicator percentile disadvantage extreme outcome test stein se se disadvantage become negative bias quadratic estimator consider matrix form determine bias risk evaluate quadratic risk estimator first expression estimator obtain make centrality risk define partitioning qp obtain rewrite obtain r pt f n f n conditionally te n te f provide equivalently eigenvalue easily zero thus estimator dominate square whenever pt use risk vice h pt vice versa superiority difference dominate shrinkage robust model error belong dominate quadratic get thus dominate whole away outside origin dominate expectation outside around origin dominate hypothesis always determine information order stein h dirac shrinkage inverse student remarkably behavior elliptical symmetry theory phenomenon estimator substantial robust elliptical could elliptical important c discussion regard tend regularity ii sp asymptotic distributional bias q pr ratio gd g p n n q r se difference e md dt see acknowledgment like anonymous suggestion put pointing grant ms ed pt maximum criterion stein type estimator inf stein stochastic constraint multivariate regression k transform new york ng distributions york k univariate york implication stein estimator north spherical location aspect new york md preliminary york useful statistics york stein elliptical preliminary estimation error ph thesis pc corollary md mathematics technology box school mathematics department mathematical
follow percentage apart percentage except error depict assess figure heterogeneous present leave draw independent outcome g I region enhance visualization easier hard summarize main look significantly exhibit competitive behave consistently bad mainly rate large peak shift top bottom hold employ error detect area look experimental compare perform edge separate area relatively look become detector display result respect execution many method speed unable mostly account technique replace achieve similar precision lower latter approximately distance derive acknowledge grant thank comment suggestion universit pe issue propose nonparametric detection numerically display datum planning environmental management detection coherent synthetic platform extend surface employ effect obtain direction platform motion remain moment movement throughout platform trajectory energy direction measure intensity send return surface illumination intensity presence acquisition region cloud resolution complementary optical selection band angle receive complex nature store amplitude image possible cover area surface little texture water ice example extremely area surface texture area texture cell employ sensor texture correspond cell heterogeneous resource operate l band sense band object group principle segmentation segmentation identification edge think characteristic occur image degradation illumination signal image degradation characteristic employ make local image viewpoint stochastic law pointwise analyze use information approach attractive perform area homogeneity intensity index look interest many parameter relate power reflect present target characteristic heterogeneous forest area method raw maximum smoothed propose outperform date maximum discuss image present use direction physics formation describe observation field coherent illumination look format model small high expense look exhibit degree homogeneity reciprocal tractable law denote density random l look independent correlate field moment distribution attractive mathematical linear tail display large lead dot nine eight look column image mean right readily find hard task local variation wish approach common define equation moment counterpart correspond arrive system follow numerically show severe numerical analyze related likelihood correction ml estimator effectiveness resample correction show ml show presence corner source contamination image estimator asymmetric motivated outperform summarize emphasis explicitly employ edge statistic additive show usefulness sample image visually detection nonparametric pixel neighborhood test local grey ratio describe detector robust detect noisy whether sub detection variety technique present compete acceptable contour spline parameter control smoothness use amplitude square image homogeneity value follow present feature final belongs exhibit value centroid angle successive around segment illustrate partition candidate small red dot come namely dark area estimation estimator upon consider segment lie lie transition maximize assume along straight population statistical nan identical intuitive two rank regardless possible rank assign population tend small population hypothesis consist population assign rank assign tie population within sample scale ordinal wish difference nan follow tie rank population tie divide different population hypothesis rank consist sample possibly different arrange cccc sample let assign tie follow sample sample respective mutual amongst ordinal either identical else yield large remain alternative sometimes population simplify small moderate equation preferred rank assess sample random sample rank usual equal tie assign tie require state respective population independence rank assign population divide get square range
surrogate truly appealing would need hide variable behave nearly well behave nearly theoretical one suffer thus present reach scale modulus continuity typically scale like accurate minimum nonzero singular component entry variable concentration magnitude scale dimensional well type condition interestingly instance approach robust analysis corrupt corruption totally nuclear suppose completion want norm need precise sample incoherent matrix work establishe recover even fraction corrupt reliable regardless corruption occur essentially error instead minimal require clean corrupted ls model move l survey circumstance powerful begin rich make range latent subtle video surveillance computer application know competitive loss robustness various application application flexibility robustness computer ls apply although low may relaxation practically model perform surveillance important foreground stack frame pixel hard imagine change foreground offer model texture texture name recover low superposition l work reconstruction recover align image component acquisition spirit ls acquisition large compressive sensing rely sparsity explain start two concrete efficient acquisition sequence interest pixel hyper frame thought index sequence important concern image clearly video time scene nearby correlate tensor trace nearly low roughly innovation frame moving object foreground could imagine might acquisition ever suggest image variation image imaging movement quality ct precise imaging variable context static correspond imaging recent potential since support substantial computer concern np size vertex factor easy graph select clique add clique find clique possible formulation difficult modern fundamental area machine pattern wide applicability mention new connection show optimal player game clique think plant clique decomposition represent clique interestingly apply problem clique sparse clique thus recover able break barrier investigate tight relaxation research find superposition introduce across follow reflect low decomposition extension
computationally feasible overview idea encode lattice compact representation need provably distortion codebook follow heuristic explanation attain distortion characterization encoding regard tradeoff distortion successive refinement distortion letter random letter denote denote inner vector unless otherwise mention rate measure n entry gaussian integer whose specified fig compose property property compression rate give find encode mapping encoder decoder produce proportional recover shannon storage construction choose dictionary sparse code satisfying offer trade error source generate ergodic encoder determine pick non zero give choose section section succeed distortion sequence distortion formal contain encoding consist stage imply source sequential codebook length matrix memory source encode module compute source column module compute residual module consist multipli fashion delay module simultaneously encoder computational memory automatically encoder bit indicate position decoder receive bit reconstruct rigorous analysis number law number precisely projection mutually component collection sphere product random deterministic q normalize norm list begin true independent list terminate pick distortion distortion make rigorous bounding distortion stage value sequence variance let define encoding satisfie I rate encoder distortion decay exponentially length enough interval constant choose I chernoff event bl ml np equality hence distortion probability excess decay exponentially large source ergodicity need theorem encoder distortion source line encode source attain small growing grow polynomially storage operation gap distortion illustrative choosing yield gap govern next complexity low convergence shannon propose algorithm essentially exponential gap dominate distortion whose grow polynomially length encoder distortion complexity grow polynomially propose encoding interpret code think codebook act sequence minimize distortion distortion distortion achievable codebook rate distortion attain codebook I codebook follow typical close distortion distortion fall source whose second moment ps redundancy code distortion function bind redundancy successive refinement linearly stage excess distortion tight determine redundancy encoder successive inherent codebook important encode bb focus high rate dash high rate distortion entropy code scalar encoder source sequence brief step selection almost amenable theoretical encoder variance dictionary obtain plot increase keep eq length graph come expense distortion column give well rule bit modify due deviation norm take gain shape source nearly distortion code ec distortion gap theorem excess distortion bit distortion summary include ec gain empirical distortion error exponent interesting work essence value residual step introduce normalized euclidean norm distortion value express statistic I armed goal distortion give hold whose hold satisfy appendix follow first follow due trivially side therefore since q prove towards induction sufficiently obtain apply proof ensemble compression design polynomial block result codebook encoder computational grow polynomially attain distortion rate variance source satisfy deviation excess distortion distortion exponentially block length may successively source emphasize successive unique inherent codebook coefficient choose encoding power choose distance analyze value design matrix design efficient length optimal among code fast expense high complexity simple successive refinement section distortion encoder greedy across sequentially pick successful recovery may approach distortion section explore small storage find distortion establish theoretical performance open communication performance binary codebook encoding analyze al compression communication rate shannon theoretic demonstrate channel superposition key scheme terminal channel framework joint condition collection joint q recall bind p moment similarly substitute distribution brevity column integral equality maximum variable eq generate use eq integral standard evaluate q verify maximum attain exponent dominate claim bound bind attain large go use eq author like thank discussion anonymous comment define linear combination column design successively source
mean termination guarantee check stop condition purpose efficiency sequence accomplish mean infinitely stage number positive n x appendix eventually stop termination n establish rule eventually stop termination sampling method purpose principle construction accomplish apply size sequel construction since variable interval use z x n appendix notation construct n take sequel compute limit give check sufficient sufficient decrease notation x b b I truth virtue follow fact true w w true check critical subroutine low st st limit adaptively interval truth confidence give truth width w I sufficient follow truth statement check virtue fact statement n w w w check efficient follow st let return adaptively interval check method sample confidence variable ready construct sampling purpose sampling stage term inequality inclusion termination take result confidence region mean variance subset c w point solve w method x n x point solve equation w would like point technique interval sequential procedure bound tail bernstein inclusion method geometric poisson sequential bound make information estimation guarantee prescribe rule notation lemma eq lem z hence r z z lem simplicity z z z prove lem nn imply definition n sm n n x nn l n n lem b n nn imply see n b u sm nn n n nn u u u n n x q complete show suffice r computation thus lemma z result imply eq combine lemma u x consequence lemma stopping ensure desire lemma know continue u sn stop n claim n x x terminate theorem preliminary corollary lemma lem lem r notation check inequality z prove lem g notation check z z z lem need since imply l sm n n n complete lem n g n show existence define u g sm u nn x u n nn complete x sm sm eq combining complete r show lemma simplicity r r consequence prove make lemma follow n n x lemma stopping ensure coverage stop equivalent sn continue show fact sampling terminate de p prove lemma pos r r therefore suffice notation r p r complete lem pos lem lem pos n need well existence define x x n n nn n l lem need together see x n p u pos n x lemma pos result define n nu n complete show p z suppose follow x x last chernoff position stopping ensure coverage rule show fact n sampling terminate de f preliminary r notation note p w z z r r r r definition definition lemma l n nu l u x p inequality z claim prove rule know stop rule continue continue need follow n u x eventually terminate z z z z z z z proof define u l x u u n region v follow region u similar prove inequality unimodal respect complete sequential estimation estimation construct sequential contrast rigorously guarantee specify issue estimate overcome difficulty advance evaluate stop accordance pre area estimation wide scheme prescribe level unfortunately nature pre specify level zero average infinity method overcome limitation shall develop virtue inclusion variety sequential estimation cast prescribe coverage use confidence sequence interval must termination situation impose sequential except probability specific principle scheme process include sequential inclusion justify interval variable assume theorem sequential inclusion principle bound
stock stock real construct take car difference graphical model markov network network fire conventional single firing process network manifold fire responsible system geometry joint joint call manifold follow flat flat manifold I replace replace marginal part skip save space bregman strictly denote property manifold role determine kullback algorithms difficulty design close decomposable intractable variable difficulty firstly propose fire information fire introduce algorithm geometry experimental appropriately fire network herein network fire node mean sample assign fire fire direct number fire fire one method cyclic call chain fire subsequence homogeneous matrix random manner markov firing process homogeneous fire empirical distribution firing ergodicity sequential firing ergodicity equally homogeneous converge distribution fire markov network fire process follow fire equivalent gibbs sample give fire aspect fire sequential fire stationary distribution firing note rigorous rigorous lose determine show work certain condition count sample fire fire sequential gibbs fire fig notion geometry fire onto fig gibbs converge incomplete thus converge however thus provide theoretical treat fire part paper define conditional part manifold kl bregman manifold close firing limit illustrate information subtract conditional already determine form good model role machine requirement construct complexity low avoid overfitte show fire trade requirement minimum treat empirical situation table denote define similarly depend distribution section cause rise continue break forward describe dominate source dash goal intersect intersect reasonable independently node situation construct computation form addition node evaluation subtract rarely node eq whole computation numerically transition therefore large learning markov carlo compute distribution numerically draw perform firing node separate part variable
likelihood coefficient newton fisher square replace information matrix continue criterion modern extension framework etc adaptively allow tuning penalize randomize widely freedom calculate nonzero multivariate logistic algorithm optimize handle know million fast smoothing popularity linear importantly generalize enable researcher complex increasingly computer describe member formulate smoothing spline consider multivariate restrict reproduce hilbert reproduce flexible formulate parameterization adopt really ratio project onto spline reproduce utilize smooth iterative square computation multivariate exponential way structure ise capable estimate importantly via significance multivariate property equivalent marginal bernoulli furthermore include statistical inference hypothesis interval extension multivariate logistic technique candidate smoothing spline linear effect clique proof proposition py py py condition distribution regard py follow bernoulli examine expand formulation bivariate bernoulli eq function separable formula conversely term assertion log combine correspond product expand exponent zero appear numerator scenario apply negative formula verify trivial numerator theorem take generating density rewrite separate moment separable assume expand hold decompose c independence multivariate moment dependent moment function node bernoulli form author nsf grant dms theorem proposition corollary consider distribution family regarding demonstrate importantly pairwise interaction among model interaction ise bernoulli distribution conditional bernoulli utilize canonical covariate node structure bernoulli linear predictor undirecte devoted resource study undirecte allow describe thus conditionally edge capture categorical pairwise correlation consider sufficient structure year area focus matrix example lasso determine covariance valid pairwise true bernoulli discuss represent third clique call physics gain popularity several discrete include ise structure study multivariate community extend clique order ise model assume interpretation interaction want variable potentially consider spline model node high graph multivariate covariate section start simple mathematical multivariate clique discuss optimization result bernoulli logistic section provide deferred case extend widely bernoulli bernoulli explore outcome consider take density valid probability simplify inverse exponential marginal bivariate bernoulli bernoulli bernoulli bivariate bernoulli hand bivariate bernoulli vector define log bivariate bernoulli separable give deal variable independent bernoulli distribution among component bernoulli component separability outcome follow defer wise importance component ensure assertion researcher multivariate bernoulli generality block proof defer graph researcher conditional multivariate bernoulli conditional distribution bernoulli state random generating bernoulli hence solely parameter hessian fisher optimization possible indice cardinality say definition reformulate form partial generating derivative respect bivariate bernoulli second order natural gaussian main undirected multivariate ise random symmetric positive define bernoulli ise pairwise interaction clique convert introduction retain markovian continuous ise consider interaction log normalize multivariate kind similarity independence structure ise
q row valid observation identity combination least argument obvious topic every row entry every constraint solve quadratic give define nonnegative projection onto k cx fx z theorem remark subsection depth sciences department sciences california technology paper describe programming computing idea drive factorization salient datum identify constraint select demonstrate similar recent contrast early propose lead experiment minute keyword nonnegative programming computing matrix factorization mining package include heuristic nmf theoretical heuristic correct develop rigorous nmf stem challenge hardness consequence additional hope practice provably nmf meaningful answer potential preprocessing impact practice machine scalable robust nmf problem appropriate contribution theoretical show succeed modeling algorithm substantially may make pose believe independent adapt class major experimental research system solver extremely sgd order implementation minute nmf drive factorization express row appear qr adapt row nonnegative indexing indexing nmf seek feature matrix sum may np whether motivate algorithmic nmf condition uniqueness nonnegative result I separable nonnegative jk pt circle hull circle nmf distance hull definition uniqueness result row hull row nmf call row permutation identify set distinguished row allow justified variety text reconstruct count word localize predict audio bin remain admit factorization well factorization nonnegative constant nonnegative nonnegative factorization robust particular compute although guarantee run undesirable priori knowledge distance third row row lie finally link norm drawback separable matrix key admit matrix matrix element accomplished extract construct matrix extract column approach construct factorization theorem solve single program extract nonnegative nmf matrix find unique row nonnegative sum row factorization suppose state paragraph find correctly identify topic assuming consist topic reasonably far away construct guarantee know show early version contain establish theoretical remain develop solve lp may suffice scale prohibitive describe algorithm large column subset popular qr localization use outlier et preprocesse recovery noiseless improve aspect enable bound rely incremental depth aim minimize proceed lagrangian multiplier set ascent solve lagrangian subgradient minimizer update make little difference minimize project descent index nonzero incremental choose subgradient project iterate sort plus drop result negative item solely break find incremental repeat epoch variable identify diagonal row minimization subgradient separable incremental nonnegative primal nk line parallelism cache critical technique cache intel incremental respect store contiguous encourage hardware choice row cache row similar classical loop join relational curve identical x core ram scale synthetic programming run couple solver stepsize fit incremental gradient combination uniformly add run run convenient performance compare metric profile particular performance curve outperform algorithm give convenient algorithm visually ccc profile correspond experiment linear slow must fed run fast noise achieve low matrix qualitatively code principle occurrence place dataset normalize recognize c feature gb e figure speed serial parallel exhibit hardware cache datum able correctly rmse middle quickly demonstrate dimensionality language svm extract misclassification versus topic right topic misclassification compare speedup versus number horizontal achieve analyze problem pose work factorization theoretical author like support nsf award fellowship
restrict rational coordinate actually sequence fact use addition treat define new matrix write v trace product create analogue formula moment coordinate map marginalization diagram ct cm x cm ct swap ct cm rt node node rt rt swap node stochastic end motivated observation v I e empty truncation index inclusion summarize auto ct cn distance distance right distance co cn co cn swap distance node swap diagram exhibit map direct inclusion map trace denote entry trace map inclusion relate hmms new e diagram auto co node right co swap node swap spc spc swap begin map trace q q write diagram co spc cn co node swap cn spc co generator check may map follow diagram cm co distance cm co cm spc cm swap swap spc node swap swap spc spc swap imply diagram auto cn swap cn node cm cm node swap map factor first take map map marginalization set open w applying complete diagram dominant cm cn cn node distance cm cn cn swap co cn swap co cn generator question hmms turn cut map point analysis reveal negative point light nonnegative test distribution membership reduce ask determine contradiction reduction lie respective induce observe formulae vanish square simultaneous mind stochastic end implicitly describe characterization rational closure assume irreducible rational parameter rational field application extent observational alone different notion rational theoretically identifiable theoretic rational identifiable theoretic identifiable identifiable answer question introduce rational fraction field subset dense open thus stable division show algebraic statistical question parameter coordinate simply q rational since field extension algebraic closure hence extension extension identifiable parametrize model hmm variety transition emission sum hence large exclude complete formula equilibrium turn yield reveal geometry consider readily e visible identification process emission triple define apply yield vanish sign hide alphabet emission identify obtain polynomial one hope state theorem thm theorem lem question conv nb explicit recover membership comprise gr basis coordinate parameter identifiable sense closure projective variety dimension gr considerably hmms selection problem hope hmms primarily identifiability algebraic geometry begin hmms binary hope eventually precise attain provide insight relate branch model series paper e begin hmm use speech well since gene dna biological alignment build measurement kind time algebraic algebraic geometry fully explore hence early investigate algebraic especially attribute interest manuscript gr walk subsequent include algebraic ps dms geometry statistical selection analogue model polynomial equal e encode equation analogous use hidden markov model bm exhibit fact actually modify non verify ideal hmm degenerate hmm verify make new coordinate hmm short reader sense equation generate fine membership hmm consecutive node algebraic analogue generic identify parameterization excellent context causal effect ode parametrization hmm composition dominant explicit hide component identifiable combination recover particular triangular generator gr basis ideal simplicity complicated look like parametrization variety invariant theory triple inside trace element find finally explore result find geometry equilibrium hmms visible thank lin suggestion note hide visible reduce throughout markov distinct hide binary hide denote respectively call hide index node row specify specifie transition probability state emission emission formula read precise set h v compatible distribution binary jointly binary process accord see give familiar model alone chain mind node unobserve vary sometimes allow vary set process closure geometry purely stochastic consist triple cube call triple arbitrary sum complex complex replace convenience sometimes algebraic function write make identification length write complex subscript intend likewise denote generator model simply word frequently reasonable usage image stochastic observe continuous cube markov closure equivalently closure ideal homogeneous vanish homogeneous type closure whose reflect algebraic ideal turn computation simplified become feasible compact indicator function thus confusion write specify subset avoid usually treat ideal moment coordinate projective cut observe elementary even hide alternate writing occur generate moment function respectively change formulae taylor relevant term vanish ideal conversely formulae describe computation run intel core ghz cpu gb ram light understand geometrically homogeneous ideal patch reduce continue long expression arithmetic trying probability coordinate instead extraction generator subsequent run generator generator order generator degree eliminate redundant generator reverse second minimal homogeneous set generator generator low degree save hour generator turn omit apply generator minute minimal generate set polynomial degree generator homogeneous ideal minimal inclusion homogeneous set moment run memory gb compute author place memory try compute kernel dominant uniquely follow c ct ct node swap p rational visible moment exhibit use explore observe swap permutation q generalize permutation essential use parametrization subscript letter string hope context avoid confusion make term fact coordinate act q word sign act matrix factor map introduce interpret consequence eq write diagram dominant distance cm auto cn cn node cm cn cn ct swap ct cn
draw take probability euclidean hellinger leibler bins draw even nan hypothesis repetition probability regard random experimental occur realization realization accuracy remark p number monte simulation conduct p follow trial trial simulation whose corresponding course fraction exact calculate discrete parameterized specifie meaning exist distribution desirable actual experimental assume calculate realization repeat experiment calculate value realization realization repetition procedure parametric bootstrap family discrete distribution highly repeat calculate realization maximum likelihood distribution number somewhat remark favorable indeed point generalization goodness fit generally much classical position remark goodness toy example use draw bin unlikely arise specify exactly goodness detect plot test whether via evaluate simulation draw extremely find bin increase draw bin demonstrate root statistic nearly report statistic bin sure support matter arbitrarily value bin classical goodness fit reliably independently power law behave agree draw bin bin bin unlikely well various fit plot arise simulation empirical square consistently classical little agree draw bin bin draw draw refer text page source english assess goodness consist english repetition come news randomly corpus bins figure word sort frequency statistic significance permutation sort frequency choose bin great number choose bin great among great bin via sort know similarly know bin plot value distinct must list word whose display must substantially large assumption respect goodness fit priori fortunately root digits value follow define contrast p fact one without know dictionary classical analyze analogous list english word repetition list word corpus sort plot compute monte carlo involve distinct word zero please note display though substantially figure remark list list whereas root square introduce great draw truncate law corpus randomly nonnegative method determine fit bin bin contribute aside bin total draw fall three frequently occur maximum turn model calculate via root statistic truncate law fit great frequency follow classic decay poisson report whether datum bin accord distribution truncate bin total depend draw demonstrate experimental bin strongly influence classical statistic simulation bin statistic fit reasonably agreement classical statistic tail whereas insensitive particle bin interval second dot report display plot whose poisson calculate monte carlo simulate four report poisson reasonably good ccc bin square dot poisson line suitably proportion pair expect random proportion nine naturally proportion genome individual give suitably random entail goodness high confidence confident suitably table via simulation negative root blind discrepancy please note classical nan fortunately irrelevant mean estimate inaccurate potentially inaccurate sensitive bin log section refer statistic simply logarithm generalization log facilitate rather good relative table associate log root mean square c match american number match report health generate rating rating value root issue small symmetry investigation modification provide testing root powerful detect root square statistic much powerful detecting detecting mean bin whose associate recommend ratio excellent fair poor excellent good fair poor ccccc fair poor excellent poor ccccc fair good fair report identify exclude experimental figure collect specie sort build bin since sort subsection sort permutation integer method obtain sort frequency please truncate geometric omit complicated discrepancy solely fluctuation log ratio unable root determine significant number goodness detect model quantify statistic success remark actual generate draw p simulation compute present p example subsection step remark cost subsection consideration draw statistic value estimate accurate present equivalent statistic calculate parameter root mean sensitive bin bin desirable recommend square variation simulation draw arise generate define root classical often simulation plot remark square classical furthermore draw root square next specify distribution q require increasingly root square exhibit opposite require distribution specify section specify require model distinguish bin specify draw draw distinguish actual specify distinguish mean powerful begin truncate draw actual distribution remark specify specify eq consider draw draw distinguish model specifie distinguish see involve specify distribution via plot define specify method distinguish define maximum distinguish poisson maximum draw require specifie distinguish q estimate eq distinguish specify draw distinguish actual model specifie distinguish q likelihood method bin great among choose great draw contain remain bin finally draw sort specifie distinguish specify draw distinguish maximum specifie acknowledgement would like thank fan w stein many pointing show hellinger version root nsf grant nsf fellowship fellowship nsf fellowship mark goodness distance test standard magnitude goodness article discuss numerous fit practical euclidean goodness interpretable black program rapidly calculate precise significance square identically model may countable accordance terminology discrete category draw use root statistic distribution rao root root square statistic confident draw quantify confident denote root square fact parameterize correspond draw arise model square avoid number pearson root square average produce classic statistic average involve division zero divide power round error arise thesis classic long appropriate superior computer illustrate conjunction goodness fit statistic preferable classic statistic division somewhat kolmogorov root certain circumstance kolmogorov root way complementary largely easy understand computing mean large continuous maximum underlie unknown value parameter clear devise none standard
mix essence graph number define approximate sampling accomplish state simple call neighbor otherwise rapidly maximum markov sample independent irreducible chain relate chain whenever respective locate appendix maximum moreover independent permutation exhibit chain figure depict grid since grid rapid mix identical follow fact whenever group rapidly couple probability locate refined moreover capture power coupling merely consider indeed strength chain presence hamming locate chain rapidly vertex runtime number leave detection markov people whether grid motivate lattice numerous core processor gb list scale quadratic set generator permutation w markov chain state art discrete implement file dedicated topology degree sampler bottom dimensional depict partition size edge topology center generating partition vertex depict instance generate b I need generator topology run start set require ram create accumulate topology sampler accumulate plot variation gibbs fast gibbs chain computational overhead observable topology summary chain result sampler reader recognize state bi count limitation chain combine generate highly advantage avoid intractable markov exhibit ise markov chain sampler numerous expect improve problem exhibit relational language candidate chain algorithms marginal symmetry detection use break group model conditional design stage markov easily incorporate permutation van discussion remark system markov uniformly state element chain remain hence statement accomplish omit state since state last follow chain p hamming element path argument distance degree five random hx g hx hx hx thm corollary thm corollary thm proposition thm remark remark thm thm example convention convention thm conjecture thm thm thm open thm thm construction com present novel detecting utilize two contribution computing chain family markov leverage establish connection rapid mixing chain probabilistic result effectiveness efficiency algorithm reduce first exhaustive exploration search answer integer utilize work relational formalism logic avoid instead notable propagation exception somewhat principle since design applicability contribute deep symmetry construction color graphical model consideration main first inference compact set permutation group link path coupling make couple chain respective locate coupling argument chain possibility investigate lead mix analytical insight empirically markov art chain applicable begin recall concept brief overview work utilize design logical probabilistic discrete structure preserve group binary element element form group act permutation cycle understand map relation equivalence denote f define give chain transition step chain start chain irreducible converge let markov chain leverage challenge exist breaking predicate symmetry symmetry symmetry put answer programming programming notion probabilistic relational work algorithm belief bi relational contrast applicable large formula large graphical colored undirected formula network weight model counting framework represent partially formula colored graph clause color construction belief bp contain receive message colored permutation acting feature partition variable message receive message notion exchangeability de finite partial state act whenever exchangeability contain every exchangeable draw exchangeability feasible compactly represent generator utilize go marginal via message pass applicability turn inspire previous introduce markov exist chain presence chain range transition original chain require generate act see generator set markov stationary integer state random markov chain chain chain state element element nearly uniform polynomial generating replacement initialize pseudo random perform multiplication verify overhead negligible irreducible base gibbs commonly perform chain follow step time uniformly conditional chain identity permutation equivalent gibbs sampler major property space irreducible irreducible stationary distribution original compatible capture
inequality central banach space banach let absolute explicitly specify state front example electrical engineering edu mathematics air institute edu group responsible tumor focus selection high exceed work variable response restrictive comprehensive low avoid limitation term predictor paper relate fundamental problem relationship sample tumor health problem many great time change observe variable per sample impossible collect imagine hundred clinical world stay paper inference small case response mathematically model sample call predictor compactly term comprise tumor classification correspond express tumor would tumor despite continue inferential focus high linear predictor responsible variation small gene design variable per x exist implication predictor imply model gene biological pathway situation problem group linear denote predictor regression underlie explain one contribution time estimate design nonzero computable establish predictor indice group contribution response basic researcher notable direction despite high report total body restrictive kronecker incomplete restrictive scaling predictor nonzero finally group study paper analyze variable selection addition limitation influential predictor discuss early incorrectly nonzero group coefficient predictor response assume understanding carry manner letter scalar constant denote shorthand transpose spectral finally norm qp qr mathematically formulate bad group result main numerical conclude attention relate comprise column since norm relax error combination group group third performance selection procedure term estimate sort marginal norm sort focus seek characterize performance group impose metric goodness metric time group check suited group total response thresholde dimensional arbitrary coefficient paper coherence property design coherence finally offer price correlation theorem marginal give return guarantee predictor early proportional group expand predictor estimate come affect confirm section develop notation facilitate group correlation yx predictor coefficient respectively marginal correlation require behavior two toward leverage proceed take subset group dimensional vector x similarly dimensional kx x z banach begin note condition concentration proposition upper construct martingale k martingale element eq u v order every consequently nature induce primarily need bind uniform coherence construction martingale banach appendix algebraic fact together km ec appendix begin element follow involve resort argument primarily utilize straightforward construction martingale proposition use together claim union distribute correlation correlation complement recall statement trivially I therefore establish lemma claim condition establish claim probability regard trivially note theorem create generate matrix entry draw schmidt stack orthonormal confirm empty specifically plot four figure solid see figure
upper previous claim incorporate claim claim marginal precisely marginal bivariate covariate marginal denote exposure covariate technique q q maximum likelihood eq general maximize gauss copula copula optimize margin step estimate marginal claim via dispersion exposure time wise vector poisson numerically time compare initial performance confirm finding likelihood solution asymptotically maximization maximum denote hessian partial explicitly bfgs matrix via derivative approximation assume fix copula family nest copula obtain restriction model copula pointwise family parameter difference variance prefer copula standard eq prefer otherwise adjust via joint step claim loss policy positive variable loss mean asymptotic total replace estimate joint numerically group gender car contain number constant offset consist marginal model intercept correspond covariate age draw uniformly last car car c car intercept car claim intercept car car claim coefficient variation copula family parameter fit copula repeat follow regression coefficient estimate r r freedom independence model prefer aic claim center policy bottom aic center display narrow display result run display error bar upper display square policy square leave significantly relative square low copula base value display value joint e ignore panel display total dash respective systematically loss confirm conclusion three family confirm make display company car year covariate exposure categorical analyze ccc description gender drive age year car analyze average claim poisson significance interested asymptotically normally interval addition adjust claim covariate age year marginal respective covariate fit joint copula family table result display copula indicate select select c gauss frank gauss frank gauss frank frank frank indicate model model conclude copula preferred family remainder continue family aic model comparison base moderate claim number claim equal copula considerable investigate impact copula total loss eq already dependency indicate conservative take appropriate rating company size claim tend skew theoretically incorporation estimation incorporate claim allow flexible class family extend study copula extend family overview appropriate marginal dimensional mixture continuous random copula construction see e average size claim portfolio severe us car policy covariate marginal moderate avoids portfolio claim bivariate family relationship cumulative cumulative univariate far kx dt gauss u gauss v u v display gauss frank number center leave aic center display narrow claim policy loss leave score center display width display narrow average size center right bottom estimate aic score display standard narrow joint copula claim accommodate restrictive skewed multi independence show efficiently test optimal copula usefulness car keyword claim size portfolio pricing base compound model independently quantitie restrictive effect portfolio propose dependency average claim combine paper derivation policy often skewed policy mean usefulness copula popular year book bivariate triangle management claim covariate overview claim separately claim flexibility extend copula generalize family gaus copula propose extensive study loss reliable total confirm bivariate copula bivariate distribute importance theorem state bivariate copula copula conversely marginal separately define copula monotone measure association measure denote joint x xx yy copula parameter relationship truncate gauss frank parameter illustrate copula family result claim claim copula claim size equal display mass gauss probability mass value due conditioning average panel copula tail dependent shift compare size
fmri order clinical score level propose linearity simulation formulation capture recovery brain pattern fmri formulation bold capital dot target paper vector prediction value brain map visualization pattern matrix form design volume target word voxel stimulus presentation lead regularization parameter penalty fmri linearity construct pairwise operate pair predict sign index restrict image associate pairwise hinge use successfully information retrieval loss pairwise extension might know close misclassification distant discuss implement regression solver input library via learn library tend appropriately major logistic simulated volume consist voxel real cubic size choose mean coefficient leave model regression cross validate square equivalent ridge non linearity correlation times linear mse dimensionality second correlation coefficient unlike increase paper result pairwise linearity pairwise assume unlike zero pair case equivalent adjacent coefficient pairwise logistic ridge unweighted ridge appropriately major set mse dataset describe sentence complexity simple sentence high sentence consist informative validation cross choose class adjacent computed score pair image keep superior temporal tp inferior order estimate project regularize parametric weighted linearity vary shape region exhibit investigation however trend decrease bold tp effect well p function effect region projection region apart investigate pairwise improve pattern learn fmri benefit fmri presence brain choice loss extension brain via grant ga team le de france france france bt france team paris paris infer functional specificity image fmri general glm remain
therefore cauchy lemma argument follow number compact ball center origin remark detailed every dt dl constant depend consequence get every deduce yield simple proof turn interest let step subgradient subgradient cauchy schwarz use twice thus would combine get proof fx q unit e divide side let directional directional eq dx assume notice derivative equal convexity right satisfie vc satisfie imply f dx dx work derivative right proof indeed replace cover l deduce existence whenever remark corollary section cover number convex function lower cover term constant uniformly summarize cover provide alternate implication study optimal numerous convexity constrain estimation risk minimization hausdorff kolmogorov metric pack metric entropy space play central area information theory nonparametric cover uniformly dimension relevant know covering function uniformly convexity characterize optimal convergence rate empirical implication regard convergence numerous convexity nonparametric study study crucially concavity estimate receive machine motivation prove section cover relate independent convex cube specifically value function uniformly bound absolute theorem sufficiently logarithm b convexity estimation usually uniform function difficulty cover number convex function deal class cover number metric space totally metric motivated number recall metric function work cover cover simple loss dx fy b cover b ingredient direction specifically real value x b euclidean constant inequality result bound ingredient section metric constant denote p f f also p rectangular clearly restriction function belong existence depend positive obtain set coverage cardinality suppose integer p r r coverage restriction belong v cover cardinality small observe cover cardinality prove exactly induction induction involve interval however elaborate control simple exist dimension every scale rest prove sufficiently pack cardinality subset whenever positive integer satisfying v uk vi vi k di ds vi h let fix vi j vi j vi let let property last distance dx sx integral ii ii vi vi j alone deduce
think least easier think regularize situation clearly reference example computation pruning remove section take improve exploit generation add know practitioner use speed effect truncate model well precision shrink iterative stop neural gradient addition ill pose problem pose distinction sharp divide line effectively assume away modify add smoothness involve modify objective box modify step prune preprocesse inside noise preprocesse empirically randomization include approximation small vector iterative heuristic lead sort computed truncation three via three way problem underlie theory exist difference interested compute smooth regular answer generally database graph many web eigenvector interest mix direction partitioning task importance node eigenvector theoretical characterization let weight graph strong riemannian manifold ad weight appropriate graph variability walk diffusion graph one whose reason lead nontrivial eigenvector lx yy interest compute quality scale medium box solver n I na improve entire ram sophisticated eigenvalue instance look vector preferable perhaps ranking page extensively computation surprising well advantage multiplication laplacian multiplication take advantage distribute indeed relate necessary consideration design constraint much perform rank root assign positive negative probability node evolve dynamic canonical evolve heat seed evolve move neighbor evolve input stay neighbor evolve power time seed walk control quickly choose seed carefully seed could randomly draw vector seed prevent state run value assumption early depend seed set justification typically expensive answer result well formalize regularization approximation procedure actually exactly nontrivial something source code follow program stand trace operation relaxation optimization distribution program versa solution sdp question shown arise regularize entropy determinant certain conversely run act ridge respectively three dynamic reader issue assumption implicitly formally result characterization algorithm compute lead exactly optimize partition objective cut cut balance set weight balanced formulation article interest cut variant say formulation study wide load balance vision theoretical primitive algorithm social network interested find meaningful community empirical social large scale noise property structure analogous cut spectral region locally class atom generally show short length spectral base light size node social scatter plot cluster flow spectral flow except nontrivial gap low axis correspond spectral cluster interest explicitly perform regularization explicit procedure figure present axis cluster interested axis clearly cluster reasonably aside tradeoff basically except two explicitly term algorithm essentially space approximation locally biased observe sense partition heterogeneity empirical function formalize intractable thus approximate flow implicit locally bias near nearly every small certain find cluster nearly linear sort modify usual nice property original formalize locally bias eigenvector use biased graph partitioning modify locality vector seed set parameter version usual nice exact quickly running perform bias like guarantee result seed find achieve good node volume great contain objective medium approach biased wants find cluster operational sort procedure either procedure use prove seed locality diffusion base nearby seed walk approximate modify heat small simply maintain thereby number need probability concentrate change take fast time output base web characterize cluster million though disadvantage complicate interpretability actually procedure problem thing seed root consideration relationship informally provide similarity thresholding commonly terminate equal manner application simply operational interactive practice characterize cluster body use diffusion scale dynamically evolve conclude general discussion modern theory completeness theory guide qualitative computation guide etc thus success problem status neither think algorithm near analogously bound often practice bound qualitatively analogously qualitative insight useful practice realistic reason combinatorial input emphasize apart application nearby reliable geometry like meaningful construction encode question describe involve go worst address question heart perspective heart notion entirely central algorithm computation implicitly either science practitioner implicitly suggest treat consideration rather much secondary practice analysis inferential algorithm implicitly violate exploit bad interest database database practice domain computer term statistical adopt scientific computer learner term datum aspect lie heart output property nearly every apply empirically fact implicitly lead exploit way scale database also inferential predictive several overview computer science adopt roughly area scientific specific make effect reliability thesis application domain share common force extremely way view algorithmic consideration box quite medium perspective scale understand exploit term statistical property bad genetic note couple computational datum seem particularly appropriate generally emphasis detail gap modern massive set analysis store variant traditional database relatively note thus way become increasingly resource complementary ability analyst understand analyze insight work noisy poorly hard imagine quite heavy extent big massive application preliminary main place analyst play test hypothesis unfortunately thing traditional database issue lie algorithmic perspective notion traditional intuitive idea basically objective optimize constraint measure analyst reason apply objective regularize thing interest nearly regularization form apply approximation perspective case scalable statistical inferential non computation either approximation computer sense heuristic decision early stop practitioner implement real often sort implicitly different interested give intractable approximation mean depend approximate amongst practitioner experience surprising practitioner algorithmic perspective perspective datum I three case illustrate phenomenon way involve lead nontrivial eigenvector laplacian second involve compute partitioning third computing approximation locally bias version problem implicitly smoother regular characterize sort analysis purely perspective scalable algorithm ignore likely challenging research front practitioner datum obvious reader perspective rich put look set help financial transaction hyperspectral sense microarray nucleotide web engine video etc implicitly root structure hardware input generation etc consideration computation interest appropriate algorithmic typically keep close enough process flexibility flexible enough intractable inference problematic several way view two member column member array another variant model science machine well scientific view continuous problem consist sort entity pairwise entity geodesic distance permit theory alternatively theory eigenvector notion pair vertex value encode correlation area scientific interest science rather array transformation space dot product orthogonal semantic table relational structure way dna model string matrix graph relevant discussion researcher probably familiar relational various extension rule logical suited datum see therein database application automate record keep system correctness reliability sophisticated rich ever either original noisy poorly dense obvious graph arise arise numerical algebra look good problem fairly easily almost algorithmic tool ram reasonably ram run reasonable sized datum detail access etc graph pose substantial science construct make cpu digital natural science extent area economic science provide source although typically involve something interest crucial secondary motivating roughly pose exist solution depend topology especially numerical sometimes amount input well condition pose ill pose answer draw occur split field compute numerical business tool versus logic lead two algorithmic mathematic use turn work condition problem form solve exactly poorly presence introduce truncation error error represent
multiple similarity unify representation modal technique establish improve knn classifier challenge question investigate knn grateful constructive comment suggestion support correspond author axiom attract relatively little method particular analysis rademacher sum specific derive similarity estimate complexity similarity norm statistic machine g neighborhood mean neighbor depend mean algorithm distance measurement retrieval rely devote automatically learn focus square similarity successfully various problem verification although devote metric learning address strongly frobenius offset similarity deal general metric reduce rademacher sum sample refer complexity similarity complexity similarity method rademacher similarity technique review metric learn term conclude paper notation let trace time denote frobenius dual input output space denote pseudo large bilinear similarity similarly report pair point require main term appropriate overfitte performance expect offset equal optimisation way overcome upper hinge upper order overfitte enforce term restrict emphasize general linear putting error lead metric formulation instance favor norm encourage wise trace analogy consider similarity frobenius encourage rank detailed similarity novel analysis form q analysis true metric z similarity sequel since similarity exactly follow brief comment modify offset firstly example label z problem desire label j x b z j jx j consequently imply rademacher average sum sample related reason refer complexity worth rademacher complexity divided let z b apply inequality hand equation technique b mx z rademacher q contraction property average see lemma inequality side inequality put estimation consequently put similarity replace exactly follow formulation metric without loss focus popular norm frobenius frobenius norm rademacher solution frobenius complexity estimation back dual singular value frobenius norm mention hold norm mixed follow obtain average x b norm estimate right hand side put back let put complete mixed norm mix side inequality apply lemma yield put combine main give key mainly less see x estimation dl mix deal high secondly similarity different frobenius estimation norm simplicity omit paper regularize learn rademacher sum analysis norm input technique rademacher analysis mention question firstly
relaxation lack symmetry outline optimality feasible feasible exist feasible solution tucker condition see vertex compose solution let sample plant multiplier explicit multiplier use satisfy theorem multiplier block solution multiplier provide explicit multipli complementary satisfied condition satisfy desire multiplier multiplier block explicit multipli block vector perturbation know feasibility optimal semidefinite extremely establish show dominate diagonal block high block imply semidefinite sum compute satisfied rearrange use see desire apply choose next delta otherwise c linear perturbation linear know perturb obtain use establish nonnegative symmetry c c column sum satisfy equation equation span nonzero satisfie unique sa si yield r scalar q define provide solution density disjoint clique clique subgraph vertex matrix sample plant accord choose nonnegative exist clique subgraph disjoint subgraph nonnegative show positive fix decompose rest index similarly index optimal sn kx unique contrary lie fact imply w q solution mm plant nonnegative tend sufficiently begin derive entry repeatedly hoeffde variable hoeffde independent distribute satisfy upper holding tend scalar r p apply obtain show substitute yield c r combine proof consequence tend large exponentially approach q eq sufficiently tend exponentially exist scalar combine bound show scalar union mm tend feasible tend lemma state give also sr combine show positive uniqueness tend next ensure feasibility derive upper impose I k c provide scalar satisfied fix equation define system form eq next show choice bind hold satisfy rearrange satisfied small since bound finally also identical remain bind follow solution c consequently combine n bound scalar depend nonnegative nz unique u u establish uniqueness sufficiently q tend exponentially random independent identically I accord distribution choose q correction block invoke rectangular random distribute sample c probability tending exponentially verify heuristic generate symmetric plant distribution entry plant partition similarly compare solution planted sample plant solve multiplier admm comprehensive manuscript recent survey represent apply x minimize augment lagrangian yx successively dual multiplier via seem work subproblem current subproblem let eigenvalue decomposition unitary subproblem admit see admit apply solution moreover solve efficiently stop relative duality gap particular admm augment lagrangian convex program k solve augmented admit simplex projection update apply project duality gap desire variety follow repeat rw model bernoulli variable success probability stop tolerance subproblem solve tolerance admm block successfully recover solution figure average sample plant cluster accord sharp perfect conservative observe trial satisfy repeat bipartite draw plant weight plant return construct successful behaviour heuristic predict theoretical guarantee institute mathematics science research grateful david suggestion especially grateful implement admm trial suggest relevant reference early thank anonymous presentation organization true theorem claim identify significant wide cluster clique disjoint clique densitie subgraph clique ensure clique solution semidefinite consist semidefinite seek simultaneously partitioning graph density result bipartite subgraph disjoint clique empirical goal group similar cluster fundamental play wide retrieval biology optimal depend fitness clustering pose intractable heuristic cluster practical unfortunately evidence usefulness heuristic ensure separate establish ensure certain approach graph similarity two object weight correspond equal similarity representation clique connect clique different obtain identify sense subgraph sum edge subgraph induce clique unfortunately np sum subgraph maximizing relax semidefinite program employ paper although establish relaxation certain input show relaxation include concentrated collection subgraph establish object co aim simultaneously feature call exhibit obtain seek respect expression across gene grouping topic customer preference recommender overview partition bipartite graph dense subgraph feature bipartite disjoint bipartite subgraph establish semidefinite instance correct recover special disjoint bipartite subgraph input give set feature build regard paper heuristic cluster optimization recent paper yu paper appear release sufficiently cluster relaxation matrix construct concentrate heavily disjoint subgraph datum consider early result spirit relaxation hard establish certain convex sum nuclear relaxation result generalize nuclear norm transform sufficiently subgraph matrix matrix clique adjacent every pair node clique subgraph disjoint subgraph vertex subgraph clique disjoint concern subgraph clique subgraph compose clique densitie subgraph equal n subgraph maximize find partition minimize minimize potential partition np hard note disjoint clique subgraph subgraph outlier column matrix characteristic disjoint normalize slight column complement use notation formulate quadratic combinatorial constraint constrain semidefinite replace variable component equal thus vertex subgraph column program q nonconvex program far ignore nonconvex constraint clique define feasible objective equal feasible clique plant disjoint clique ignore sake efficiency due entry unique suppose sum equal therefore point semidefinite approximate partition although think nuclear relaxation indeed semidefinite matrix x semidefinite obtain ignore nuclear feasible set equation minimum nuclear norm prove analogous relaxation clique subgraph ideally identify cluster correspond graph clique subgraph weight connect node within low clique focus clique disjoint subgraph vertex compose disjoint clique entry independently one entry satisfy ji independently satisfy c variable distribution plant clique mean say matrix plant plant model plant disjoint subgraph stochastic add within plant dense subgraph independently add edge clique plant disjoint clique subgraph simply bernoulli follow yield draw plant plant disjoint clique clique subgraph find probability disjoint graph random symmetric plant distribution delta exist scalar maximum clique
importantly construction see orthonormal great column row bound small outline row column infer discussion column permutation admissible compression matrix matrix block permutation respectively row column matrix diagonal still uniformly thus toward ensemble bound condition recovers dimension feasibility focus simplicity exposition condition rise utilize next corollary accord accord generate orthogonal probability suffice exact guarantee course outline throughout presence compression sparsity mainly dominant cf limit increase column recover deal solve non convex optimization accelerate originally success fista estimator stable offer attractive notably addition dimensional arise build eq square zero lipschitz f optimize p quadratic algorithm iterate aspect form convergence rate accelerate result stem possibility subproblem employ initialize decrease geometrically reach terminate drop tolerance detail conclude section respectively p generate f tt augment lagrangian especially suit prove tackle task directly subproblem challenge common auxiliary formulate equivalent tackle associate next positive splitting entail iterative comprise implement block augment lagrangian variable minimization singular thresholding likewise update via unconstraine program whose closed termination inverting fortunately perform line reduce inversion svd compression matrix I ad optimum iterate converge p converge k k b f c x trade convergence ad attain need appropriate predict extensive test considerably choice g ad tuning suit resort component bilinear draw entry column entry demonstrate solve wide range cf c c l suitable depict recover namely value apparent succeed sufficiently observe interestingly hope recover omit avoid unnecessary performance list less accordingly significantly challenging last though therein still possible l superposition feed poorly mainly due inaccurate flow experience change result service term external network failure attack service anomaly engineering network traffic challenge since load superposition next consider represent graph cardinality number layer flow large single intend accordingly connect carry time compactly traffic flow across flow traffic flow periodic period relative measurement suppose instant anomaly column link matrix identical whereas measurement central protocol overhead wireless network power constraint centralize fashion raise robustness since specific isolated motivate anomaly whereby carry rely directly subject algorithmic put general regularize cc line marker estimate anomaly agent place less see comprise flow candidate reference I collect internet internet usa record week internet internet comprise flow traffic multiplication internet acquire trace superposition traffic truth therefore apply ground dominant low anomaly compare whereas anomaly anomaly free unable anomalous flow scope anomalous slot framework enable identify anomalous flow instant assess anomaly across roc curve depict apparent false case illustrate across depicted internet pca effectiveness flow deal matrix via task traffic monitor video surveillance principal optimization local sufficient exact intuitively require incoherent compression behave isometry operate sparse condition matrix ensemble algorithm nonsmooth globally complexity real data traffic anomaly flow several extension provide new challenging research impose room obtain study datum naturally broad e pick satisfy nonzero must rearrange obtain value r l express eq w w upon arrive linearly suffice nonzero triangle b fact assumption value obtain nonzero follow establish property hand side fact inverse b orthonormal substituting bind suppose upper complement term summation e r ignore arrive bound r j p r j plug move rewrite sequel element inside th similar deduce index summation put together r suppose begin desire presence compression stage proof emphasis distinct bind norm r former establish lemma parameter u hold higher less build eq addition recall plug readily note apply sampling distribute bind j eq utilize inequality constant b argument completion omit put infer complete cm cm ann edu pt superposition deterministic low fundamental arise traffic anomaly recovery encounter leverage ability nuclear sparse program confirm say isometry ensemble exact high first nonsmooth provable traffic flow time outperform identifiability traffic let sparse observation deal anomalous flow network vary foreground identify region fmri fundamental plus decomposition absence one signal principal component refer pca sufficient special superposition matrix challenge identifiability presence stable early deal recovery lead guarantee therein cs variant corrupt noise last noisy nothing else pca arguably main contribution recover solve nonsmooth parameter nuclear aforementione norm surrogate respectively albeit natural np optimize compressive put assess capable real practice line first identifiable notion incoherence component restrict isometry constant ensure succeed small sufficiently subset recovery cs condition result duality valid challenge dual procedure distinct show sparse compression draw solve accelerate direction ad claim effectiveness traffic anomaly interesting defer bold letter set operator pseudo inverse spectral singular cardinality scalar th zeros n issue address identifiability matrix space still problematic sparse span space denote singular svd subspace l w notational henceforth element help solution admissible assumption put unique identifiability uniquely nonzero perturbation l belong contradict zero uniqueness h h locally intersect denote simply orthogonal onto row complement subspace addition identity throughout build derive ensure identifiability incoherence holds achieved readily arrive give u exactly high arrive matrix exact sufficient condition state next element exceed context row together last condition guarantee recover solution determine reference therein essence express deal lagrangian multiplier constraint subdifferential nuclear optimality f iii may contain dual r p deal construction dual tight existence special construction compression matrix lt express b arrive b find incoherence guarantee bound follow instrumental assume contain nonzero b v attractive norm substitute yield notational brevity substitute q hold tight back note upper eq cauchy schwarz come column exceed r become upon hold straightforward condition necessary requirement equivalently
fast aspect toolbox inform choice scheme informative take pursuit gp input induce regressor give fully independent approximate attractive conditional recommend development prediction substitute original equation substitution k index subset solve show set training input use toolbox mix match adapt prediction reduce basic without point outside close smooth surface example physical simulation important method recursive cluster random onto split sized recursively median median test split concern hyperparameter cluster cluster joint cluster likely implement toolbox modification sum method cg multiply dense argue method alone cg iteration preliminary ad hoc termination rule see section necessarily examine test termination recommend exploiting iterative provide fast certainly scale firstly linear construct memory hard dataset disk reproduce computation expensive storage computation dense implementation concern many possible regime identify piecewise provide meaningful method truncate series apply machine implementation matlab software open firstly dense program automatically secondly usually eq review time complexity fast compare mean predictor effect compare point lie fixed training comparison hybrid hyperparameter phase final phase argument expect superior explicitly explore experimentally sec hybrid mean compare full evaluate gold could computational alternatively small predictive option square relevant definition time count mind visualize help understand difference unless synthetic alternative compare approximate approximate freedom hyperparameter e approximate likelihood although hyperparameter second base implementation matlab suit efficient matlab hybrid implementation toolbox comparison problem approximation use usually variance dimension require accurately input space dimensionality irrelevant detect equip automatic relevance ard another noise level datum underlie noise average variation much large distance check level hyperparameter model isotropic irrelevant dimension ard parameterization appear fast cg cg dd mm horizontal residual diagnostic iteration reproduce however long measure later directly iteration test plot add insufficient iterative true cg direct cholesky either cg heavily ghz core machine multiple increase matlab intel cholesky multiply focus provide isotropic square share dimension randomly show known indeed standardize regression width direct implementation accelerate mm mm kernel ard power range power small memory intensive local range five hyperparameter leave four four make selection choice dataset individually learn although run overhead matlab always slow four dataset account plot local look plot leave column monotonically test represent predictor turn well improvement take optimize hyperparameter result inferior pattern monotonically approximation trend often operate comment specific function fairly obtain good turn fashion make error difficult slow function select randomly superior converge convergence local detail slightly select induce randomly joint separate training give report result joint consistency give random mm ccc cm mm dataset hyperparameter setting learn fix although learn bad small setting fix outperform produce generative well prediction computer implementation suffer behaviour keep track thousand careful code might reduce problem combine tend dependence could investigate dataset attention control optimize induce point improve expense potential area balance spend select move develop context challenge use require phase future approximation code gp compare try difficult run run give test behave monotonically varied comparison iterative cg dd comparable make provide dataset assume hyperparameter dominant simple hope act gp approximation approximate require subset partition work dependence scale partition scheme small recommend anonymous thank many discussion european community publication reflect evaluate gaussian regression rgb computation neural california institute technology usa ed ac ed ac school st ab uk even straightforward process approximation assess prediction compute baseline subset empirically problem comparison parametric component task unsupervise dependent gaussian algorithm understand recommendation quality obtain varied different approximation baseline subset set dominate publish code encourage baseline structure follow specific practice section issue select approximation direction task dimensional input function test
question tangent space norm point necessity fisher base condition graphical experimental describing raise consistent space identifiability condition geometric piece lie intersection underlie fundamentally identifiable quantify identifiable piece ask interpretability nature relate condition bound incoherence component row latter low completion choice regularization parameter method emphasize respect qualitatively experimental estimator stable observe ideally regard experimental end expressive synthetic experiment graphical reason tie useful upon marginalization g connect observe graphical intuition synthetic hand setup seem generate maximum average degree randomly latent may rank decomposition beyond observe direct spirit condition variable broadly similar domain beyond optical decomposition subsequent wish highlight practitioner sophisticated package develop package invoke convolution scale essence matrix also atomic viewpoint could decompose two suggest fundamentally composite highlight role broadly regularizer trace model gaussian interest algebraic major challenge specifically marginalization categorical understand intractable decomposition describe quantify effect careful reading would like thank contribute statistic attention several across processing comment also challenge variable latent gaussian specifically represent disjoint represent possible note observe variable sparse marginal latent gaussian graphical effectively uniquely individual study algebraic sufficient identifiability give statistical interpretation decompose sample graphical observe variable induce induce penalty induce rank important fisher formulation discussion constrain seem promise certainly investigation formulation develop provable computational drawback existence formulation potentially optima behavior rigorously nonconvex one property go solver well solver scale involve scale several analyze rank apply step follow trace sparse follow component roughly two cycle low piece also reader must vanish ensure pure rank away thresholding sign entry contrast component low rank rank reason vanish broadly analyze possible minimum entry component norm residual norm norm selector selector contain selector relation unclear might suitable selector study bring consistency rank spectral population purely low nuisance application structure quantity imagine quantify deviation low via weak may estimate rule careful investigation suggest component minimum magnitude nonzero entry consistent typically build bind one regularizer require issue polynomially note probability polynomially whereby desirable search sparse graphical consequently hypothesis affect may far refine fast factorization regularizer apply joint induce viewpoint sum
comment provide separately condition result low bound constructive prove sm sm si f function give f tm design sparse estimation relate signal leibler alternative density I make proceed distance distinguish choose interval subsequence pass subsequence arbitrarily high estimate eq kullback leibler lie contradict adaptive sense present difficulty instead bound accuracy kullback equal argument proceed mass allow ok n function q finally adaptive sense single sequence passing contradict denote accord k k contradiction prove apply context j k n make similar argument set j j loss k k n decrease slowly follow c sm similarly attain involve lemma choose discrepancy target let constant include dyadic index r discrepancy since would would since remain true design operation noise subject p region deterministic j k j thresholded k eq conclude little effect coefficient constant k j show term wavelet series lie support say n f parent fashion also make two estimate accurately deterministic depend result j e nj cn eq separately part may effective nj hold part argue f coefficient parent place design jt wavelet proceed induction j c eq n inductive obtain nj prove induction requirement choice conclude attain spatially point choose f may noise nj thresholded fall two eq bound suffice time contribution definition pick instead obtain instead r uniform condition tend old point time hold consider coefficient estimate satisfy nj k n effective quantity k nj depend cauchy schwarz thus constant f may lemma constructive proof b old class deterministic tend n holds tend sm proof wavelet thresholding ac uk g secondary sense design adaptation spatially far restrict paper sense applicable problem spatially improve spatially function likewise design typically advance design sequentially describe adaptive combination adaptive sense literature regression classification f author convergence describe adaptive standard intuition place heuristic justification know regression argue lie rather consider spatially hard seminal kind problem scale estimator function obtain improvement region rough overall describe show obtain rate contain class adaptively class adaptive find standard insufficient achieve might often assumption estimate gaussian sense adaptively significant thus sense nonparametric unknown spatially sense implementation finally describe algorithm varying design move design wavelet give need compactly wavelet wavelet vanish width l term wavelet f estimate wavelet distinct observation yx suppose scale coefficient change apply wavelet ik wavelet design consider literature transformation simultaneously accuracy choice note estimate depend yx k therefore coefficient index nj ease notation j able converge adaptive thresholding hard f hard otherwise fix adaptively select initial stage point space design density choose estimate ensure later point unknown shape thresholde decrease factor least jk wavelet know coefficient accurately fix constant split p exist n simplify point x design require design effective nominal least n grid describe density design regular effective approach require pick calculate tm sm thus wavelet lie must scale however establish indeed similarly locally old follow topology sm discuss begin result sense old sense ny together ny spatially attain sm upon sense f sm sensing sm two feature first rough place commonly measure old benefit achieve r factor sense uniformly f benefit enough rough behind class tm interested f function smoothness least restrict quasi locally old likewise restrict require fundamentally however obtain achieve eq log n bind design rough otherwise improvement easy locally derivative sparsity achieve spatial adaptation sense confidence even spatially asymptotic sm choose simplicity wavelet penalty wavelet set finitely many convert back nx evaluate ik I post thresholding transform uniform predict point describe standard always make nj tend error build make look resolve slightly different definition observation
generators generator extreme subset anchor influence simply I need residual point cone expand depend behave norm support hyperplane cone eqn variant rand variation may expand residual henceforth implement simultaneous matching pursuit greedy response variable explanatory solve r norm count nmf explanatory variable relax solve penalize variant sparse note greedy guarantee well concerned optimize albeit refinement selection separable refined alternate frobenius typically build one hold alternatively introduce anchor extensive scale benchmark parallel propose alternate code respective author tend cause perform less compare remove similar baseline perform guess nonnegative combination show image possible closely separability assumption minor image contain region factorization non learn version noisy select image add pixel normalize recover correspond unnormalized recover control noise separable entry generate dirichlet choose zero std correctly average noise range good robustness anchor perfectly resolve perform competitive turn competitive expand crucial h evaluate human label commonly tf representation document normalize column henceforth identify anchor unnormalized correspond cluster cluster significantly report sake figure perform almost classification obtain document term restrict select dataset inner dimension ten local search mean accuracy feature rest testing scenario induce representation base unlabele classifier select separable greedy outperform dataset initialization rapidly advantage minima audio checking cart absence economic absolutely gold activity approach backward benefit birth propose convert tuned clustering assign cluster refine solution iteration figure converge optimum initialization error bar indicate propose require run traditional nonnegative square consist similar nmf tf popular empirical task previously separable nmf conduct effect anchor column svm setup early show normalization tf require separable nmf predictive quality none white house site house inspection united team anti constitute contact matter united checking grow hour country constitute tune space weather gb gb c e anchor tend table word topic extract eps share distribute parallel parallelism run multi machine machine parallelism use library em parallelism parallelism message performance core intel gb ram run experiment co twitter relate present report greedy depict detect second sequential complete factorization core core believe speedup amongst dense architecture omit brevity state run head characteristic example per epoch execute iteration nonetheless dataset run shorter comparison separable nmf offer scalable minima streaming medium content acknowledgment provide thank condition edu com university college usa ny separability tractable find extreme hull new empirically several favorable relation scalability share say admit matrix nmf provide appeal signal topic geometry comprise non column cone geometry know np hard np hard g treat convex heuristic nmf elegant approach develop certain separability nmf geometrically separability cone small informally modeling problem association separability existence anchor identify across usage separability investigate show uniqueness nmf scale place right nmf assume normalize norm find hull attempt feasibility lp involve scalable robust knowledge estimate formulate exactly nmf robustness general specialized algorithm incremental descent parallel primal dual size asymptotically relate qr noise cast multivariate derive sensitive perform certain preprocessing scalable produce correct separable exactly require closely hull procedure simultaneous matching pursuit sparse variant quite point correctly cone multiple residual demonstrate superior experimental emphasis scalability robustness implementation correctness work short background relevant concept notation convex linear ray generate ray ray generator combination finitely element point see state point finitely possess extreme hull extreme generator vector express nmf contain pointed generator uniquely index non admit separable nmf dimension extreme cone support generate matrix I cone solve problem e residual th vector consist
issue individual estimate multiplied call mean relative variance individual factor simplify result page next two piece grow schedule approximation sample case schedule schedule interested primarily huber integral area distribution run approximation worse issue issue deal involve analyze far possible add assumption change gibbs difficult drawing describe show schedule piece algorithm piece always nonnegative early various gibb together generate return form process operate page different way output combine fractional keep know simplify page initially get enough chance even value value exponential enough concentrate balanced interval independently pair follow let half length interval eq q similarly calculation calculation show involve fall final concentrated tight concentration relative long balanced schedule repeatedly draw copy concentrate pair product oracle generate output time sort successive value hx iv iw complex value algorithm run time schedule well third part produce failure poisson run concern least poisson come bound run repetition always make failure get exactly exponential number k union z make chance least fail schedule next value relative I control interval come fashion increase th slope schedule regardless hx hx last case proof analyze w e w w chebyshev say chance successfully create schedule give union mean z complete average variable sample average concavity jensen inequality
distribute sample approximate bayesian recent filter image fusion segmentation mcmc paper generate posterior result adapt mcmc assume gibbs generate according successively draw conditional associated interest resort metropolis hasting generate proposal reject generate sampler metropolis gibbs sampler mcmc base see accord generate whose voxel quantity compute mrf gibbs finite use loop accord dependent discuss generally coordinate gibbs mh move sampler depend acceptance previous term present require possible intractable likelihood introduce carefully tractable sufficient target auxiliary discrete state tractable statistic generate introduce variable mh proposal replace simulation discrete auxiliary despite explicitly exact likelihood free mh acceptance candidate vector addition sufficient limitation address abc mh henceforth abc mh abc mh use statistic restrictive tolerance distribution discrete generate accord mh auxiliary constitute limitation perfect sampling costly million alternative gibbs state infinite perfect regardless small explanation choice candidate approximation around improved expense result abc correctly even small explained previously define statistic function generally fortunately mrf belong statistic neighbor voxel sufficient exact work euclidean distance crucial drift center eq period strategy walk medium pixel become sharp walk summarize know lastly incorrectly observe fix incorrectly either htb htb correct c c true mmse mmse mmse htb conditional associate th order assign fig map burn period correctly label jointly column fix ease interpretation second scenario highlight blue observe incorrectly considerably show mmse propose good agreement show map simulation actual class label mrf propose fig incorrectly classification obtain htb c c classification htb mmse mmse mmse apply acquire depict acquire resolution sum mrfs extensively model mixture detail experiment class result empty noise preserve contour markov burn period remove interpretation observe optical clear boundary classification inspection topic depth problem use couple mrf class report mode image acquire probe contain outline rectangle vector mmse result fig show contain layer produce boundary within half observe obtain fix corrupt hand spatial yield fig viewpoint surface tumor cut towards deep htb gray light gray fig fig c gibb jointly unknown bayesian image set cross validation standard require normalize within method abc likelihood hasting intractable rejection apply experimental obtain synthetic unknown incorrect value successfully relax reversible jump include texture field project region acknowledge optical grateful parameter jointly mcmc perform normalize constant estimation hasting obtain synthetic unknown value know incorrect successfully apply estimation sampler image processing field mrf recognize tool capture mrf classification segmentation generalize vector segmentation precisely define whose hereafter generally process fully unsupervised monte carlo mcmc powerful tool square mmse analytically mcmc mcmc method directly indeed compute design recently expectation avoid possibility eliminate define posterior although analytically convenient lead poor notice normalizing implicitly depend paradigm base analytical development strategy combination state bethe approximations sampling langevin mcmc recently expression compute analytically recursive mrfs base integration expense estimation arise apply monte integration approximate generally recent precisely technique recent statistic hasting introduce random auxiliary distribute hasting require computing admit unfortunately suffer acceptance ratio application auxiliary several sampler however substantial costly image application alternative auxiliary monte carlo sampler arises provide possible normalize free substitute intractable likelihood metropolis hasting auxiliary auxiliary step require distribution although abc study density increasingly regard processing contribution propose abc included particularly adapt show easily exist previously remainder hybrid gibbs posterior bayesian synthetic vi report th voxel n make associated stationary class fully follow characterize map observation work enhance relate amount spatial adjacent introduce coefficient exist segmentation priori consider jointly framework conditionally associate characteristic mrfs mrfs conditionally pixel pixel conditionally properly pixel voxel neighbor neighborhood structure neighborhood eight near represent neighborhood near voxel neighborhood consider clique horizontal neighborhood mrf variable indicate voxel whole random obtain pixel nn result
loss determine regression loss nonsmooth differently work guarantee nonsmooth satisfy necessarily magnitude objective convert objective bregman indirect approach regularize relation vector draw minimize strictly full feature scenario linearly separable case achieve small tend direction drawback usually candidate thereby aforementione nevertheless explicit sparsity addition regularizer logistic point denote sparse remain matrix self suppose assumption every none matrix henceforth imply adjoint index obtain furthermore thus possible choice union r regularize constant probability implication grow achieve provide include depend knowledge extreme average independent psd matrix well achieve glm error go eq hoeffding variance previous subsection also probability interestingly obey synthetic algorithm sparsity constrain induce one though optimize difficult characteristic model accuracy dimensional datum independent autoregressive parameter datum adjust data pursuit logit effect effect additional logistic elastic mixed available produce fair net measure label furthermore net regularization label logit omp include average evaluate parameter furthermore evaluate see correspond indicate net elastic logit omp convex method nevertheless set step application involve type identification gene interested cause greedy convergence guarantee base notion restricted smooth cost stable restrict linearization cost generalize square known isometry recovery square concrete requirement logistic function lipschitz incorporate broad medium scale optimization whose hand especially running dominate grow deterministic ordinary improvement gain coordinate introduce beyond scope analyze establish series operate lemma lemma turn main eigenvalue large symmetric definite respectively concave convex jensen yield q imply symmetric index thereby q tt operator jensen introduce q coordinate apply inequality proposition q use ft invertible multiplying c ss easy b therefore yield recursively part basically deduce proposition inequality divergence prove lemma prove proposition l inequality definition bregman divergence obtain bound eq yield inequality subtract q expand obtain side always inequality subtract sum discriminant desire denote corollary write eq current q proof vector involve th algorithm corollary remark constrain applicability learn processing feature compressive vast constrain application estimation linear fidelity relatively effort sparsity involve quadratic pursuit approximate minima linearization paper produce approach generalize known arise compressive sparse compressed demand decade bioinformatics finance datum often use infer circumstance inference ill significant structure exploit regression acquisition regime structure usually cast formulate requirement computationally optimization extensively receive area norm regularizer induce nonlinear elastic frame analyze provide work restrict function domain emphasize sufficient condition generally elaborate coordinate descent minimization formulation increase stand alone operator nonlinear consider residual error show hard formulation scope apply forward impose target multipli never multipli stop condition multipli magnitude pursuit sparse arise nonlinear prove characterize sparse canonical hessian analogous analysis guarantee smooth smooth derive function symmetric hessian prove unless complement letter v nonzero depend indicate equal except coordinate e capital transpose restriction identity indicate one outline overview work formulation present provide guarantee prove example algorithm apply logistic present gain attention title pn measurement consistent absence entry every solver np solvers proxy approximate solver require equivalence hold measurement high ideally measurement necessary interior method scale several another category cs convex orthogonal omp pursuit iterative subspace pursuit name greedy constrain least square residual iteration successively position zero exhibit convex though requirement accuracy meet restrict rip understand satisfy rip small number cs show rip accurate reader guarantee algorithms cs provide incur acquire variety desirable broad everywhere rely induce sparsity show et statistical decomposable glm shown establish rsc basically curvature curvature descent constrain guarantee different rsc provide majority regularization bound albeit implicitly logistic regression recognize sufficiently short desirable assume glm restriction restrict estimation perspective mention generally perhaps corresponding mention impossible desire convexity property know rsc quadratic location may investigation rely proxy fidelity appropriate acquisition application right significant advance prediction generic cost replace desirable approximate pursuit generalize broad course even simple combinatorial np cs obey certain accurate guarantee analogous rip framework smooth introduce notion restrict hessian smooth cost introduce linearization bregman divergence restrict canonical subspace somewhat restrictive requirement compute c algorithm maintain estimate first evaluate nonsmooth large choose index merge current sf c terminate change specifically gradient reduce proxy step reduce step form study minimize invertible step relax restrict approach large descent thresholde square variant characterize restrict function linearization rip standard basically require function stable restrict continuously differentiable q sparse say stable kf twice continuously condition number cost square condition sparse vector special condition equivalent rip function canonical subspace everywhere determine eigenvalue across therefore cx k tx diagonal norm ct everywhere convex notion nonsmooth function subgradient restrict subgradient
convergence rate method step accelerate sg default inner successive gradient stay constant convergence remain sublinear sg gradually transform achieve convergence specialized weight strongly numerically treat pass increase sg batch iteration rate oppose sag identical sag cyclic rather sampling consequence convergence convex derive treat pass proof involve lyapunov experiment show sag robustness suitably fast also emphasize sag perform show assume gradient lipschitz fairly average function strong regularization term transform strong convexity exist initialize result depend gradient norm optimum denote finally convergence consider internal constant sag proof sag achieve sg method albeit advantageous sg pass example large sag use sag assume iteration sg subsequent sag iterate contrast sag sg experiment sg rather minor sag appear analyze performance convergence know example basic fast sag much number indicate sag substantially proposition primal wise case rate sag reduce excess cost problem depend rate true may sag step size far indicate use lead sag iterate lipschitz describe sag requirement prohibitive form since store scalar rather full reduce cost sag update parallelism sg example mini batch sg iteration mini mini batch sag dense storage requirement since batch mini batch reduction storage early iteration sag uninformative far modification analyze sag outperform sg sag objective like rather approximate recursion regularizer proposition satisfy conjecture size start lipschitz instability allow high smoothness multiply extensive sg dataset dependent tune em full cubic step satisfy strong wolfe condition hand nesterov method base publicly limited newton method linear art sg projection contain regularize dual method sg sg average strongly sg nearly constant describe either sag estimate logistic section sag ls sag ls initialize theoretical suggest strategy sg sg method pass several sag pass l bfgs indeed website causality website website although differentiable regularize half testing add regularize standardize one testing effective pass datum measure divide practical pass cm cm colour experiment sg carefully sag compare basic sg accelerate multiply size stochastic describe use size iterate gradient cyclic average among power method bfgs sag show select step size cm bottom view colour observe trend performance carefully size substantially pass sensitive steady bfgs eventually pass sg sag seem make steady progress sg method sag reach cost often translate reach line search fix perform similarly method show surprising sag deterministic large size sag learn datum equal derivative optimal regularization asymptotically negligible regular batch enough high covariate aware satisfy parametric contribution though setting speed decrease step size pass unlikely testing cost sag sag iterate reach sg sg slow terminate step paper sag iteration sag advantageous termination sag termination specify sequence sag sufficiently le schmidt european fellowship science engineering proof proposition compare coordinate sag regularize pass proof convex function gradient denote unique stochastic k probability nf ix k definition n concatenation block diagonal proposition prove step shall lyapunov rate shall dominate case proposition show stochastic desire proof denote I generate choose affect pass datum stochastic gradient run sag section provide descent sag one use sag variance denote k take average convexity quadratic term cc denote compose diagonal diagonal diagonal throughout moreover side third side equal equal last expectation summing term n b rewrite conclude proposition sag size lyapunov lyapunov quadratic n n bound term positivity guarantee convexity third gradient line stem negative vector use condition convexity convexity give since function dominate x complement verify dominate eq eq lyapunov rate sag shall lyapunov goal express term positivity guarantee convexity property q get complete square convexity follow finally get finish positivity previous gradient prove minimize ng x stochastic use descent constant gradient initialize truly reflect apply sg sag use eq formula lipschitz maximum square column similarly dual convexity argument large fast speed determine basic size q fast determine independent primal prefer yield dual primal dual size iteration could apply cost coordinate apply descent happen variable sufficiently cost coordinate dual depend sag sag rate coordinate slow determined sag highly sag l pt pt plus minus pt schmidt sup paris sum convex incorporate quickly problem machine redundancy example class structure sg method allow learn sg sample average focus ix b ia example binary smooth extensive include smooth use minimizer cost fraction fast rate method
advantage importantly invariant vary component allow across task learn contrast change different state sample additionally free distinguished rkh make operator scalability note linearly mdps problem applicable discuss formulation path infinite horizon control section integral wiener system control act subspace good respect terminal expectation start require scalar give path dynamic consequence express make challenge point admit finite specifically use take dynamic cost hierarchy evaluation express term necessarily treatment basic concept function positive definite kernel embed constitute direct commonly consideration interest lie give allow express rkh although conditional embed generalization specifically variable q rkhs product scope paper satisfie argument conditioning directly proceed introduce delta treat formulation analytical immediately practical discuss supplementary function use e rkh embedding apparent convenient allow pre computed evaluation require evaluation kernel term expectation practical regularized kernel turn joint distribution representation empirical obtain denote hadamard take hence representation importantly note weight recursive use fine furthermore function material transform lead poor characteristic also overcome laplace mode small basic drawback relatively inversion subsequently per additionally policy overcome alternative weighted supplementary employ gram schmidt reduce novel choose discussion address opinion efficiency instance repeatedly optimize single reaching movement interaction change generally monte method require sample trivial state sampling case allow efficient repeat necessity evaluate infeasible estimator base ideally arbitrary set particular change importance estimation g reinforcement incur transition expensive explicit expect turn distribution sample agnostic often appear concentrate guide context incorporate amongst task execute task situation invariant relate cost look stay balanced path integral expectation path invariant component stay induced well objective abstract appear framework explicitly implicit cost double demonstrate monte close enough highlight task concern move velocity coordinate affect position aim position final avoid dimensional weight consider I true reinforcement learn access sharing time sample start low supplementary exponential kernel median comparison firstly monte reinforcement see b approach carlo compute knowledge true policy starting position monte capture optimal lead without approximation similarly new starting would affect dependence sample refer highlight recursive sample sharing equivalent embed yield empirical estimate specifically apply finite representation rkh
neighborhood filter filter pixel neighborhood nine parameter eight estimate likelihood looks give base within filtering whose si tie ti hermitian definite increase choice divergence divergence simple scale good prop er sample respectively eight test al derive distance yielding express transformation possible visualize color color decomposition green channel laboratory evaluating generate look counterpart use figure figure original figure window whole albeit instance present hellinger fine detail preserve employ directional instance structure within forest maintain image selective produce strong fine linear texture proposal decomposition assess iterate filter close equivalent homogeneous com sense complete produce consist hermitian matrix suffer noise literature preserve analogously cone color propose describe type type smoothing technique visualization detail visualization sense achieve prominent imaging coherent provide image interpretation segmentation analyse lee principle preserve signature filter average neighboring ii filtering execute element homogeneous neighborhood image good efficiently simulation statistical covariance organize smoothing distance complex wishart visualization kind imaging result intensity relative phase datum possibly vertical wave signal characterized medium complex multiplicative circular module determinant transpose besides hermitian definite characterize
ideal assignment sequence redundancy probabilistic generating quantity extra bit would code assume ideal exist statistic diagram perform average partition average distribution form employ average temporal partition know complexity propose partition still guarantee redundancy member method reduce temporal partition partition utilize advantage kind partition begin partition although restrictive temporal partition temporal partition possess later exploit parameter define define map tree show partition notice weight datum number associate depth weighting application special contribution specific computation equation clearly equation lemma run data main overhead bold tree update kind traversal need summarize stack significance many base memory describe space compute routine change context bit number bit bit representation differ effectively run keep dx depth model ni tw ti weighting limitation minor state omit base almost partition arbitrary model almost partition complete temporal particular precise terminology end segment begin partition say start segment abuse binary construction add ba bit q verify base source redundancy piecewise class whose redundancy negative monotonically non redundancy source stationary furthermore combine see partition contain segment redundancy definition concave jensen gb ga aa b complete part partition half tree leave segment decide include result require choose restriction n I overhead due advance appendix time depth tree extra straightforwardly modify idea size one copy relevant boundary alternatively justify weighting overview consider successive trial denote observe one kt use weight code n use next stationary source accord segment generate redundancy estimator base upper show technique perform segment relative complexity synthetic average iii heuristic exponentially decay non illustrate number effect report average uniformly point segment get slightly linear algorithm runtime already time book paper replacement compression binary source performance context bit consistently improvement file already considerably slow method also alternate set memory model empty switching model switch compose index map sequentially add string define convention symbol boundary bound memory bit possible switching define state redundancy switch x x number logarithmic tracking generating difference method prior towards small point prior arguably job ensure together however line remark tree unfortunately find would require follow derive hold partition tree weight efficient automatically piecewise extension closely redundancy match literature thank helpful technology note trivially equation depth dx j x j x x third step use inductive use prof allow derive low note ax dx x dx take negative ex corollary example introduce weighting algorithm piecewise source bayesian average large segment prior compression code distribution competitive redundancy exhibit superior trade
limited available logarithmic regret corollary di understand reward moment order surprisingly moment refine empirical achievable armed introduce agent action select one agent observe independently past survey variation vast majority assume gaussian moment generate variable factor reward hoeffde similarly assume ucb variant basic strategy satisfie guarantee ucb eq reward sub order know bound distribution weak sub suffice disadvantage gap become sub tail regret heavy tail weak may even infinite moment logarithmic though gets refine robust construct upper confidence strategy factor dependency availability guarantee various rough behind ucb sum estimate interval sub common factor arm q order accuracy know significantly wide strategy appendix property heavy tail handle tail replace estimator need like show mean precisely need let finite suppose also factor interestingly may recall estimator subsection satisfy shall requirement mean define reward arm ucb provide distribution assumption exhibit mean suppose distribution estimator regret least interesting v n assumption suboptimal precisely step prove notation round equivalent either three inequality v assumption union bind n false conclude hand old next regret heavy tail truncated estimator mean moment q bernstein computation conclude follow distribution satisfy regret assumption distribution bind even worse robust ucb find remarkable impose strong moment still regret replace armed satisfy suboptimal problem furthermore fix exist satisfy take bandit distribution bandit two bernoulli formally follow kullback divergence directly along computation arm run bernoulli straightforward satisfactory invariant strategy shift reason center quite around raw indeed possible truncate possibility mean estimator section next section divide data one standard median empirical show certain block size property robust ucb let let computed empirical distribution binomial next consequence rather moment distribution satisfy ucb median satisfy elegant well optimal however good term unique prove additional requirement framework robust strategy upper large distribution proof arm reward maximize policy satisfy modify satisfie well median validity heavy tailed bandit reward arm mean
phase synchronization coupling identification include phase shift couple identify synchronization synchronization research phase synchronization weakly relationship long history back synchronization decade interact study context nonlinear dynamical type synchronization self weakly couple uncorrelated coupling phase whole synchronization dynamical synchronization experimentally electrical circuit biological phase synchronization synchronization regard couple single activity couple activity cell area abstract year single eeg study human demonstrate phase synchronization memory synchronization phase brain support memory memory act communication mechanism gamma synchronization grouping synchronization investigate discover phase activity within pattern synchronization pattern review synchronization synchronization frequency prescribe also ignore work sound mathematical theoretical treatment tend form theoretically couple model dynamic phase test synchronization coupling identification include argue theory good investigate drive key describe physics transform spectral transform try directly process synchronization theoretical synchronization insight research ground process concept discovery aim trend consumption short run exchange level market introduction develop researcher become develop theory diffusion give heuristic elementary phase ti hilbert phase error white drawback adequate external keep phase increment simple case iid drift substantially phase evolve brownian motion stay relatively model exactly concept remain heuristic formalize appendix phase synchronization focus synchronization interact stochastic view brief describe statistical phase definition synchronization show aspect want address may proceed model example mainly start synchronization start synchronization traditionally synchronization define hilbert rotation center phase alternatively one system periodic represent amplitude latter signal determine transform hilbert obtain hilbert cauchy principal amplitude method estimation transform model particle filtering next couple phase shift usually know practical behind definition rescale stay move synchronization synchronization synchronization coherence close indicate synchronization often synchronization several article advance distribution present directional statistic phase measure find directional surrogate mutual phase couple drive mention many directional type synchronization process provide introduction large population interact phase white discretized diag mention fulfil specific us insight synchronization network namely obviously synchronization sometimes load phase synchronization synchronization testing representation matrix full synchronization intensity equilibrium lead test addition vector drift equilibrium phase transform meaning contain example section come linearize synchronization relation eigenvector synchronization stochastic structure eigenvalue synchronization almost synchronization process situation phase synchronization non test look fit addition parameter synchronization relation coupling etc synchronization difference replacement difference employ phase employ phase zero phase happen section see usual paper autoregressive identically distribute random restrict var e call process integrate stationary univariate thorough specify univariate synchronization long straightforward calculation decompose full stationary representation multiply stationary show move equilibrium force illustrative another move l rd r representation process component component process side walk multiply stochastic trend drive representation back occur illustrated detail synchronization belong correction eq stationary obtain x obtain true relation next back drift process deterministic contain use phase intuitive dimensional walk discuss simple intuitive restrict take identifiable synchronization know synchronization mean reduce suffice describe joint example confirm since increment usually correlate note one phase relationship test distinguish relationship synchronization synchronization relation inspection system usually factorization synchronization relation shift represent direction couple meaning section lead likelihood theory ratio theory test determination decomposition multivariate situation much challenging classical ratio integrate lead example strongly recommend reading presentation read reader keep mind inference lead drift role drift term least consider maximum system essentially first step remove eq lead view canonical denote equilibrium correlation increment thus belong synchronization zero belong synchronization split belong discussion drive force way determine call recursion call hypothesis significance level stop test hypothesis section test test standard know disadvantage test common restriction power alternative significance device construct test asymptotic superior inferior strongly p uncorrelated prefer test test also single conditional assumption solely error unknown follow important residual lagrange multipli test trend finally test next use synchronization idea context phase synchronization describe stationarity propose stochastic difference always back stress discuss model signal phase increment phase integrate sometimes stochastic trend occur trend occur see integrated synchronization phase process linearly non identifiability phase synchronization also rank relation phase integrate term usually drift increment var process still phase pt case synchronization synchronization give may relation may linearly example recall process stochastically error correction phase synchronization relation uniquely reflect r synchronization span testing view phase space difficult equilibrium process provide drive remain nothing know priori phase must produce phase cause external mathematically trend I intercept trend stationary regard phase cosine cause daily specific higher sufficient conclude additional fit var intercept trend test lr generate trend phase shift know testing phase synchronization collect approach present phase instead phase regard adequate regard process phase example often test lead stationarity instead consequence finding author maximal synchronization perfectly proceed phase ignore phase phase mean model unobserved phase model challenging consuming give original phase use phase prior specific conduct lr prefer bottom situation synchronization one hypothesis synchronization particular result slightly however lr test perform bad estimate significance significant calculate difficult testing coefficient software synchronization direction coupling investigate adjustment drive force synchronization force mean ratio whether hypothesis adjustment chapter software appendix detail additional system coupling strength corrupt couple inclusion couple integrate step burn point subsequent strength make realistic stationary estimate trajectory plane rotation symmetry plane rotation center exist figure phase apply window estimation plane coupling strength b coupling e corrupt case noise system I phase plot minus plot stochastic h indicate fact slightly ascent ar part coordinate line corrupt trend phase bottom ar simplicity discussion reject lr rejected accept thus reveal phase eigenvector system test c synchronization index coupling strength dash statistic critical test hypothesis reject plot determined value nearly indicate plot explain turn see appendix agree apart jump difference lr test penalize highlight plot whether black red blue procedure specify synchronization decide synchronization line dot critical coincide remarkable test tailor penalization mention include lr little synchronization relation dimension tendency prefer lr given analyze directional coupling std significant except agree strongly reliably identify coupling residual purely correction coupling parameter informative discussion modify system illustration theoretical set nice obtain calculation value obtain representation representation meaning get active soon equilibrium case process coupling correctly implicitly ahead mean phase mainly direction precise phase identical significantly deterministic part trend one keep mind hilbert smoother investigate vary frequency point connection theory process phase detail coherence process equilibrium relation shift couple couple synchronization network phase synchronization system regard essential coupling new come enhance coherence triplet investigate confirm potential synchronization perhaps synchronization point understand fit towards equilibrium synchronization little seem interesting important phase cover phase consequence couple calculate whose move regard contrary recognize inclusion structural shift occur close guess situation occur set topic topic economic framework detail able chapter iii possibly mutually simple amplitude vector meaningful phase synchronization prefer write form see shift trend full phase transform term smoother estimate technique course effect hilbert transform significance need investigate scope everything smooth one become conditional rescaled asymptotically argument regular stochastic unit lr heuristic inspection proof test stochastic term trend remain estimate phase hilbert transform original unobserved consider heuristic synchronization relation remain group different phase synchronization check stay transform obvious integrate stay integrate phase stationary process transform hilbert constant cycle reflect trend hilbert emphasize argument mathematical stay highly derive proper synchronization likelihood ratio hypothesis clear derive strong particle filter base furthermore misspecification generalization space signal periodic function typically unknown need may amplitude var nearly clear increment positive increment happen mainly overcome case autoregressive duration extended bivariate generalization lag smooth aspect module http www com www net package project web package index http project package em phase hilbert pt type none pt chapter correspond use appendix pt vs beta beta type drift specification correspond statistic quantile give quantile correspond lag ct beta ct lag lag em critical segment plot remain seem aspect simple successfully phase synchronization reason modeling choose aic synchronization bic test bic reasonably synchronization situation investigate develop least present highly situation true accordingly bic lag nee high approximation
rule soft rule accuracy compare predictive soft quantitative trait molecular provide prediction phenotype numerous genomic community ensemble genetic interaction marker highly contain short day line available variation website second evaluate display r r size l soft l soft rule rule output variable might use two combine lasso combine input unsupervise weight least obtain kernel rule model develop ensemble interaction importance dependence tool soft example rule importance rule involve interaction naturally ensemble big adjust scheme thousand loading time acknowledgement award section property article soft importance soft logistic rule soft boundary various soft ensemble hard approach challenge view provide simultaneously focus stable ensemble whole influential bag bootstrap aggregating involve sampling model produce although model use classification response tree soft ensemble attack problem rest post explain convert section soft conclude comment discussion ask predict variable restrict model consider framework additive base approach arrive involve obtain use combine code eq bag adaboost boost special generation procedure bag value boost fashion takes recommend mn w decrease include final learner precede section article indicator region variable tree terminal input say r lk lk subset categorical space terminal many terminal maximum tree rule logic rule call boolean logic statement involve construct logic cx logic construct input logic logic rule show logic express ensemble weight l ensemble random forest bag boost hard value rule include interaction variable explicitly model good subset term minimize perfect response explanatory likelihood finite deal separation approach recommend produce bias correct jeffreys fisher logistic ni diag g g iterative utilize linear product build model structure smooth model use calculate hierarchical child terminal function fuzzy decision perhaps model utilize tree build cart rule soft approximation rule n l estimate lead final soft ensemble prediction ensemble hard soft make slow however fast fast programming language part accomplished summarize soft rx convert rule soft sx soft additional soft hard set line recommendation several algorithm recent solution minimize rule final
detail start simulator hypercube design take al optimal vary simulator sequentially add ei evaluation ei vary complexity e estimate minimum simulator output scale lee benefit ei uncertainty appear guide sequential design utilize search trial point global point place uncertainty e global shape repeat time add point ei figure present ei ei ei performance realization h expect outperform shoot slightly outperform ei clearly gp ei run estimate global yx global minimum find replicate optimum yet function spike make densely spike sequential add ei median realization run global outperform naive shot performance ei ei result mean ei rate ei ei dominate ei sometimes wrong spike mean paragraph prediction encourage ei detect discovery region add follow ei summarize ei method median curves global attain replicate well suit negligible ei candidate engine situation localize surrogate still competitive optimum run simulator procedure configuration impact et suggest depend complexity simulator run design long notice piecewise utilize tree example employ average effective continuous experience effective carry run course possibly nearby simulator possibility response effective engine evaluate ei candidate since computationally identify relate sake current allow input comparison explore alternate way ei find provide support student award discovery centre hypercube package h additive c e spline bayesian application computer model lee case model computer optimization experiment guide stable gaussian mit multimodal multiple global optima th international conference optimize sc thesis mathematics ns computer mathematical represent physical computer experimental virtual phenomenon think consider goal simulator feature simulator simulator minimum utilize version simulator ill application tree improvement universe sufficiently complex scientific impossible impractical constraint phenomenon enable study place electrical observe mathematical computer take input consume input situation relationship simulator model model predict surrogate consist world phenomena simulator simulator run value result tool minimize sort collect sequentially accurate location popular overview response closely model krige interpolation quite rather strong stationary response entire limitation lead extension lee recursively partition rectangular model upon refer tree give capture relationship also deterministic noisy monte turn enable computation guide design identification organize surrogate discuss experiment briefly outline surrogate power improvement et al review adapt playing conclude future brief detail also discuss lee assume simulator value error output vector shorthand terminal return different tree terminal rectangular denote terminal flexible use variable model capable incorporate adaptively individual tree place split point area allow nearby say interpretation place gps place interesting receive mass gps way biased low continuity making response spatial correlation gp imply stationarity enable curvature locally model inferential tool importantly uncertainty ingredient design wide real considerable noise simulator error choice emphasis modification surrogate modelling simulator follow general distribution specification priori ii node generic computer believe quite observed tree give point accomplish scaling recommend distinct within shrinkage g output less prior mass accomplished chi recommend percentile default percentile specification response mcmc consider motivation lee inclusion residual approximate gps estimate specification parameter ensemble allow fine grid axis discard first quick observe cover involve put mass terminal large tree later make base briefly specification replacement offset instead terminal lee terminal node accommodate model computer modification setting default r package c indicate predictive stop specifie model comparison package r describe power modelling description give wang basic predict produce km wide high generate approximately power placing collection suggest energy consider simplified simulator place control simulator location along scale shape considerable flow location area thus optimization problem display panel simplify study exercise pre evident spike panel panel display prediction mean power display rapid fluctuation deal consistently observe fit case model unable consistently introduction consider kind optimum computer simulator switch maximization let improvement unobserve less maximum form expression normal first local seek improve currently identify minimum place uncertainty nearby variety similar involve predictive unobserved response account parameter employ idea mind calculate framework unobserve incorporate ei calculate ei directly straightforward get posterior ei incorporate uncertainty strategy calculate ei new ei maximize possible simultaneously hypercube point
description comparison simultaneous development microarray techniques genomic bioinformatics microarray gene expression mass simultaneously study thousand biological feature protein nucleotide snps challenge determination whether biological differentially reference term biological address use throughput include david review far modular concrete precisely whether biological category reference exist approximate test test test follow gene differentially express gene microarray go category statistical representation total apply value prevent positive incorrectly reject hypothesis gene control fdr value biological category reliably category application well false biased estimator application microarray gene expression microarray thousand concern category appropriate reliable adapt nevertheless broadly protein biological pathway section arrange concept gene cancer conclusion recommendation preliminary describe state de go category binomial proportion go category nuisance unconditional consideration represent go total go nuisance probability mass nuisance conditional justification concern nuisance interest statistic nuisance parameter alone contain variation independent minimal nuisance therefore statistic little parameter false discovery go q let ii multiply odd precisely category go category alternative bayes significance category represent differential go modification go tend maximization likelihood usual microarray probability parametric let mass observed category satisfie odd go category equation go eq mle normalize equation go next develop stand base odd hypothesis alternative eq mle member denote minimize regret mass define estimate combine equation guess category ratio bayes mle breast applying cell human breast cancer contain u file microarray gene gene microarray criterion de de gene variance gene estimate theoretical hypothesis bias normality expression presence absence hour exposure define go go molecular least de thereby go sided mle display bayes factor indicate tie h nan hypothesis alternative component component grey comparison grey commonly threshold evidence simulation study de conduct separate study behind assess symmetry odd ratio equation odd go category asymmetric randomly fisher exact odd ratio sequence time performance simulation estimate low moreover bias affect whether odd furthermore attain equivalently category differentially bias conclusion go category thousand gene study medium number go go category adapt mle medium go cancer sensitive component medium go breast cancer mle formula show sensitivity left plot component plot component nevertheless figure bias even mle number go category estimate category equivalently component freedom recommend number go interest mle simulation category finally recommend
benefit life challenge arise challenge utilize code hundred target follow assess introduce validate estimator estimator demonstrate policy robot constitute artificial intelligence stream generate interact period big diverse stream approach acquisition verification reinforcement life mobile expressive represent interaction al represent reward pseudo termination condition policy semantic intervention scalability computational represent massive mobile robot several operating form consequence policy great learn parallel learn sequential long learning challenge policy scale life sampling td size computation per memory available life long robot arguably make life recent highlight big et policy robot robot hundred prediction policy policy require place computable progress policy target time robot scale formulate agent model discrete system observe characterize note access environment current environmental step transition produce conventional predict discount special receive formally particular policy control discount factor predict action select reinforcement estimate common use action policy policy value behaviour behaviour policy td become unstable fewer reliably td policy function incremental td except secondary complexity td projection onto weighted bellman operator discount whose policy thing policy capture architecture pose generalize predict th scale predictive question happen behaviour different behaviour effect velocity action substantially dramatically learn million distinct easily sensor architecture consist triangle update prediction set sparse previous parallel mm provide parallel architecture et desirable characteristic run linear scalable distribute ability learn modular specification behaviour depict enable compositional td question support scale prediction physical build mobile robot see sensor detect entity ambient field internal status acceleration velocity robot mobile operate raw coding produce binary constant comprise pair sensor code produce always white fair evaluation present policy evaluate period direct evaluate robot update behaviour five reverse counter stop new execution behaviour execute action normal occur select one action policy five complete spend second centre behaviour robot spend learn generate reverse stop sensor question discount robot pseudo termination scale value specification step result learner parallel correspond match identical total computation ms entire ram dedicate wireless present prediction hz axis stroke profile return yield percentage non shape test evaluate online performance precisely execution truncate return test normalize configuration normalize square use weighted represent percentage question illustrate policy prediction physical robot scale share identical divergence substantial return learn important demonstrate policy experiment test considerable mass infinite estimate evaluate place scale policy frequency execute trade change step frequency ensure closely approximation propose measure relative ignore question never go quality representation common technical converge provide performance evaluation rewrite expectation use stationary expected approximation expect backward view trace additionally target agree incremental sampling current exponential trace update second storing first compare chain end episode begin middle transition incur determine scalar expensive mdp incremental sample estimate well experiment behaviour move measure provide simple compare change learn episode secondary trace online change simple chain provide match magnitude episode primary weight vector random online estimate online aggregate curve average task robot chain experiment run exactly prediction six hour time step effectively effect every question reaction error measure quickly vector time exhibit trend bellman storage predictive illustrate important validate policy behaviour estimate slowly occur also incorrect affect long quickly scalar estimate significant learning efficiently estimate consume twice estimate limitation prediction question number policy magnitude policy maintain action action parametrize policy copy otherwise offset random random assigning independently test architecture scale policy robot experience question policy correspond second evaluate learn cycle ms ram requirement result many learn strongly availability prediction environmental square bellman heavy denote estimate curve learn progress result provide policy online performance measure literature idea prediction along option temporal abstraction policy approximation develop et exhibit slow learn progress policy active relate al real sensor
common identify combination subject throughout describe study motivate use identify quantify upon cd cd cell present recognize recognize activation future constitute total cd cd production activation match label cd cd cell define must collect ensure rare subsequently cell maker threshold phenotype difference subject generate subject difference real might detect analyze hoc rule method test contingency test subject share observation develop name cell explicitly model binomial across place unknown proportion binomial multinomial dirichlet mixture sharing help count specificity multivariate model help detect cell organize alternative method model multivariate use expression observe subject two un cell negative measure cell classify positive combination count un cell vector define gene expression use set generate part trial dna modify boost boost goal assess multiple analyze consist il measure envelope ease cd sample peak subject cell isolate flow subject tm measure array pair control present vs handle multiple correction present analyze combination count combination univariate cell alone measure everything loss case one case summarize contingency un depict positive proportion call concerned identify expression follow sample proportion un pair respectively subject compete un proportion cell alternative proportion cell alternative parametrization describe sided proportion indicator subject share whereas marginally jointly two treat miss expectation maximization also hyperparameter via markov section propose em mcmc greatly calculation utilize nan likelihood conjugacy likelihood available form give eq beta miss define complete likelihood side specification involve calculation normalize calculation web log unfortunately form implement start em step convergence initialize exact realization mcmc except give short pilot run present iteration burn leave margin result compare performance compare expect false discovery false incorrectly analysis specificity side compare sided fisher exact base observation negative positive potentially day sensitivity specificity exact ratio detect comparison fold change alone give comparable compete panel il il consistent web cm x north west font b bar bar east bar west font font west font cell use side detect model figure specific difference proportion gene inter pattern evident probability preserve difference proportion fisher fdr adjust reveal identify distance south east west font right bar bar east bar west font south east north font examine performance unconstrained mixture primary event ten independent realization evaluated sensitivity specificity identify roc fisher ratio log fold side examine nominal fdr properly fdr unconstrained compete include specificity b additionally estimate reflect fdr compete web figure auto anchor south west south north right west bar bar south bar north font west east north west font anchor west south north west font proportion maximum web beta multinomial beta replace si unknown proportion respectively share subject exchangeable proportion dirichlet algorithm estimation mcmc binomial simplify use close nan function similarly marginal likelihood estimation procedure base dirichlet except initialize positivity call test em greatly although slightly alternative mcmc appendix b cell express expression increase upon approach sum cell il would appropriate change count different population decrease cell difference apparent condition combination advantage perform require adjustment power auto south east north font bar south east bar north look eight assess fisher table multivariate significantly power test auto east north west b bar bar south font eight compare fisher roc curve discovery color article mass quantitative routine sequence single become development become fisher result inference quite small importantly bayes microarray addition nature generation use multinomial count experimental condition share across non beta increase fisher underlie assumption violate figure univariate base beta binomial allow constrain proportion cell sample proportion match sided motivate example type cell set ignore express cell pre filter ability demonstrate broad importantly demonstrate cell gene c cell identify change cell population protein homogeneous cell population correlate disease context study increase cell detect express differential count functional iterative perform positive univariate satisfactory test might preferable detect true select combination side population differential expression sensitivity specificity compete condition multivariate figure c provide experimental design limit
om motivate introduce formally error appropriately nystr om suggest adopt study nystr om good rank use frobenius implication closely error frobenius find application pca low manifold improve norm understand application om domain stand spectral frobenius compare bound bind term scenario frobenius remove diabetes rbf blue vary overall indicate large explain dependence examine rank legend varie observe combine nystr om motivate error nystr om analysis nystr om measure exploit learn incorporate nystr development nystr om world well nystr om submatrix select row usually computationally expensive submatrix need let nk I reproduce space rkh endow eigenvector approximation nystr compute nystr om b already large inequality integral eigenvalue integral eigenfunction eq easy eq integral schmidt concentration integral operator hilbert sure probability l result last base two additional adjoint h n bind key large normalize quantity theory perturbation result foundation theorem eigenvector consistent unique eigenvector allow assume found indicate sufficiently imply eigenfunction compute approximation eigenfunction q appendix approximation nystr norm last hold obvious l besides nystr approximation approximation try bridge gap effectiveness nystr om nystr om base concentration plan nystr om proof j simplify nd eigenvector therefore decide relationship matrix matrix define canonical
joint usual positivity getting map scalar ensure usual recursive message relative column denote linear belief update output preserve non simplex immediate consequence basis wise negative fact instead track negative belief output prediction via relative column form basis express form right term estimate form inverse invertible substitute statistic estimate observation representation convenient maintain appropriate mapping note box method hmm spectral hide mit document markov though slight notational original maintain usual map necessary update limit analysis order give denote relative basis indicate
decision reject classifier binary towards biased two define risk surrogate solution solution framework margin stage without feature acquisition subject closely cost sensitive infer take acquisition cost study decision cost propose increase utility classifier regressor discrete deal consist million develop learn formulate multi reduce cost stage reject rejection decision rejection classification stage reject classifier investigate reject stage framework introduce classification class posterior rule context machine reject lie small separate hyperplane criterion implementation stage reject stage medical margin always meaningful illustrate degradation happen measurement margin meaningful reject cost cascade multi stage reject cascade much cascade cascade introduce reduce classification cascade detection alarm partial decision pass stage whether belong detection cascade architecture cascade primarily concern binary classification problem decision classification decision stage conceptually distinction fundamentally cascade unbalanced negative goal stage admit false negligible associated problem well rejection decision problem medical diagnosis detection argue sec item performance detection cascade stage average misclassification example stage sensor modality reason develop cascade nevertheless goal cascade reference therein perform cascade sensitive detection system decision see learn possible ensemble base classifier size ensemble current cascade distinction set partial stage computation classifier full label extract acquire th truncate th fix reject classify delay modality standard system classifying indicate active misclassifie incur analyze fundamental simplify minimize risk arbitrary decision minimize appeal remarkably risk risk stage optimization simplify derivation c account conditional optimally dynamic program possible conditional expect minimization rewrite optimal eq reject strategy class implication ignore stage minimize single multi optimization modify remarkably modify replace stage meaningful risk meaningful reject htb classifier reject option classification clear expression express two reject modify term agreement disagreement namely claim thm binary classifiers nf reject classifier eq minimizer inspection illustration compactly use derive derive decision assume task n wise parametrization stage go try learn wise empirical go training boundary cost recursion eq optimize function constrain decision minimum equal stage simplify due stage stage three system refer appendix program ensure terminate multiplicative step stop n correspond direction hypothesis minimize maximize classifier margin employ two system consist q thm reject subsample f sf proof approach complete detail please refer appendix compactly st stage nd margins stage interesting reach nd reject st second goal large early stage life stage confident reject base reject binary margin infinite case reject use measure attribute acquire attribute acquire work normalized margin measurement utility denote estimate measurement parametrize limitation compare error unclear set cost stage possible reject vary reject system nd reject datum stage classifier classify sign split averaged iteration subproblem surrogate outer c stage dim nd dim rating diabetes test bin image ir level synthetic dimension nd dimension uci machine attribute extract stage rate discrete employ opinion demonstrate utility approach possibly estimate htb three dataset diabetes dataset uci consist body mass index two expensive hyper attribute intensity spaced fine acquisition stage coarse course various device image modality ir wave extract many image training carry either clean fast modality slow c name centralized mix diabetes global margin marginally point reject nature classifier utilize nd multi imply half expensive opinion without medical resolution ir whether require person slow medical diagnosis penalty easily adapt risk reject roc high reject high incur lastly three algorithm generalize way cost stage ir vary rd color fig map correspond make decision horizontal example modality point system ir achieve reject rate one demonstrate modality every vertical ir avoid together achieve centralize ir htb ir strategy achieve error rate general sequential parametric principle erm optimize remarkably subproblem turn practical minimize surrogate several formulation multi fraction security ed term average drawing choose express equation size expression chernoff bind second nx px define rewrite note iid use collect two system minimize keep constant term get keep minimize sensing modality cost unnecessary classify majority reduction datum seek average acquisition cost risk minimization erm multi reject wherein stage classifier measurement next acquire cost erm show reject parameterization global surrogate framework generalization medical demonstrate substantial achievable classification sequential learn security medical diagnosis compose stage sense less informative sensor fast sensor expensive measurement practice allow modality make scenario attempt classify consume th modality make example detector could use slow expensive wave consume inspection stage ct scan procedure many example salient sensor measurement order sensor sensor modality stage consume situation sense modality case sense action become methodology primarily fix stage modality justify account come across consist sensor sense modality
vertex add boundary add integer order write size find order nearby insight thin minima must maxima order order give increase find component leave order right zero htb order thin width find globally thin see minima order property note flat consist width flat greater contain respectively flat write minus vertex eq let adjacency unweighted calculate sum index go subtract sum particular non locally maximal locally minima note locally integer shift weakly maximum ordering loss shift remove add vertex remove slope add shift could increase imply substitute find maxima ordering irreducible shift rearrange maxima shift helpful seek minimize finitely strictly guarantee reduce one maxima strongly irreducible come strongly irreducible cluster cluster prove terminology proof slightly add small add large boundary sequence remove create strictly boundary cluster vice versa increase complement remove portion irreducible order contradiction cluster e fail concave fail replace minimum flat define find look index locally minimal slope adjacency vertex slope negative slope vertex calculate respect remove vertex maximal slope find terminate every vertex negative slope define terminate follow sequence add increase decrease concave intersection sequence increase boundary add decrease boundary complete proof lemma vertex slope respect remove increase final convex sequence vertex increase final vertex vertex slope slope add increase boundary w reduce boundary contradict conclude convex repeat convex therefore terminate irreducible contain trivial discover correct increase useful irreducible stop irreducible choose irreducible ordering order irreducible repeatedly find strongly irreducible initial shift consecutive vertex minimum vertex order shift leave repeat argument shift irreducible middle irreducible order fact exists give repeatedly remove vertex zero none cluster non empty core say core proof lemma terminate cluster every core core contain assume contradiction remove sequentially first contain slope core slope strictly negative slope strictly remove remain thm thm translate subgraph thin knot dimensional manifold mining search large set dimensional point measure point geometrically fall category determine make cluster book algorithm underlie gaussian center way assume restrictive less effective euclidean less hierarchical building place tree flexibility final construct partitioning translate point reflect point define subgraph separate whole close cluster encoding generalization knot manifold thin surface thin position translate quite sense begin vertex define width look improvement criterion first start order cluster initial ordering improve repeatedly examine future
naturally period behavior take place short important behavior carry relate time period description integrate access pattern page indeed dynamic nature change influence week factor event economic volume profile start take author date involve investigate semantic recently outline discover episode partial periodic temporal majority period consequently datum interesting may short period domain period case web behavior evolve consideration give rise study analysis concern summary carry evolution objective change stream usage article organize propose include benchmark external criterion well report suggestion future modelling cluster evolve evolutionary dynamic splitting entire period sub period use year sub discover reveal cluster sub specification store datum prototype reflect individual subsection describe strategy correspond traditional order cluster contain sub period month year put partition period period us partition period process contain cluster prototype belong time cluster repeat period common completely allocation run request repeat request percentage repetition duration average duration request number request page site percentage request site semantic duration request adapt contain row real column except obtained present analyse apply external implie evaluate cluster match gold cluster f partition cluster second partition cluster combine cluster contingency measure formula element element correct rand cr two partition cr agreement cr sensitive number cluster cr indicate object rand index range indicate perfect agreement near confirm correct rand index near indicate partition obtain month figure value reveal local cluster cr mean partition word versus dependent local detect also confirm notice result conclusion local dependent use refine appear cluster dependent detect cluster local value surprising clustering similar whether carry suppose reveal summarize occur domain issue highlight adapt extract knowledge evolution devote handle evolve article propose divide regard scale offer change nature use change split set regard hardware overcome cluster acknowledgement author grateful cluster usage domain suffer classification detect occurrence change distribution ignore concept concept make inconsistent regular necessary usage necessity evolve solution article summary evolutionary sub period validate propose occurrence change mining appear end use order extract web
summary reference satisfactory result describe summarize reference automatically reference color achieve perform channel illustrate process detail subsection representation comprise hidden patch patch size much small goal gaussian size reference fix densely overlap associate patch image coordinate independently mapping pixel patch map possible interpret patch behave traditional map mapping take suppose mapping eq indicator equal true maximize function patch graphical maximize follow e respect mapping get parameter eq coordinate alternate reference applicable train sift convert channel represent dense sift patch divide orientation calculate grid dense vector train channel sift channel dense sift channel sift represent sift sift learn graphical densely cover patch mapping patch essentially take channel channel color channel channel patch specify patch denote overlap averaged patch channel square half reference densely horizontal figure result image convert image parameter color sift compare color reference level matching result produce continuity whole image contrary learnt contain sufficient color patch respectively top much spatially transfer target natural result user intervention segmentation exploit color information appearance result effectiveness htb htb department electrical computer engineering institute department electrical national university color image lack segmentation eliminate develop automatic use appearance effective train reference image image contain information scientific image illustrative color part manual consume suitable manually image automatically automatic reference target interaction selection approach biological human human
hierarchical interest represent scenario maximally away maker internet link parameter link depend moreover grow slowly increase child height grow go infinity aggregate majority non bayesian contribution derive bound decay break break consistent rate bad achieve achieve fast rule breaking rate mean therefore gap almost achieve rate certain pass message explicit convergence majority alternative majority explicit span binary organized leaf agent circle agent independent true measurement agent decision measurement forward agent next agent exception agent receive parent final hypothesis receive leaf represent tree infinity agent hypothesis alarm ii error answer question type ii error explicit function yes way aggregate binary agent binary break aggregate information daily life fusion explicit probability towards root derive probability turn characterize know error probability associate new binary use locally sense bind probability majority rule use rule derive explicit asymptotic divide odd type ii agent wise kind type type root degree branching agent tu binary fusion parent majority odd suppose message type ii ii binary message probability pair decision pair fusion induction suffice bound h mx km mh claim monotone monotone binomial mx jx moreover monotone monotone decrease proof analyze wise analysis error root odd majority rule know moreover consequently increase proposition odd integer rule fusion moreover brevity note addition explicit error substituting suppose majority integer upper bernoulli show assumption relax g type use suffice apply majority achieve derive break suppose broken bernoulli distribution bound proposition low level tree suppose root useful balanced binary remain provide analysis binary ratio agent agent characterize unit ratio globally reduction notice odd unified simply derive low root majority rule provide total underlie hypothesis easy type root rule bound majority recursion maximum type ii probability leaf agent test level majority identical majority characterize explicit error evident majority achieve gap describe negligible branching tree total change convergence rate test rule break agent level eq integer recursion omit c suppose brevity type alternative total n p decay surprising ii rule decay sufficiently difference small likelihood rule break break repeat consecutive decay combination contrast strategy involve strategy asymptotic alternative fig decay purpose alternative strategy large majority random breaking gap htbp section allow agent rich pass message alphabet alphabet tree investigate decay integer agent th level message u tm k leaf message alphabet agent subtree root agent leaf message leaf parent sum receive immediate child agent integer message number subtree alphabet know subtree leave star leaf message directly star decay height I hold fig leaf measurement intermediate sum message agent e see message alphabet enough agent abuse height strictly scheme know subtree leave connect subtree subtree ignore fusion rule aggregate subtree leave message agent henceforth simply rule throughout decay easy leaf connect agent recursive agent connect agent directly equal regime decay decay simplification agent tree tree rate upon simplification desire result notice mean theorem alphabet decay quickly change decay depend furthermore sensitive compare achieve almost exponent test convergence n strategy majority apply argument make proof corollary proof omit brevity message pass agent message leaf agent binary interesting scheme strategy suffice message height message agent agent size bit q large convergence rate average bit still small fig show message size become tight study error characterize convergence majority majority break suboptimal pass convergence probability alphabet many question remain usually topology branch different agent complex question involve conditionally another question communication agent reliable communication ideal ratio know inequality q proposition p network consist agent root aggregate binary make measurement hypothesis leaf message member summary agent type provide scheme involve binary characterize exponent error bayesian learn decentralized hypothesis testing attempt jointly consist agent measurement underlie past neighboring agent hypothesis decision fast respect network answer depend structure structure primarily feedforward agent private decision restaurant movie go popular appear behave market private decision agent structure common political structure online social e sensor network decentralize concern sensor message measurement compression maker goal typically compression associated intractable complicated decentralize focus process economic biology physics review relevant aforementione make equal make measurement agent recursively decision agent decision belief agent decision rest copy ignore phenomenon decision asymptotically agent unbounded private ratio decision problem change sequentially observe immediate agent identically bind belief derive observe decision randomly agent relevant situation network complicate wherein decision study agent structure largely network exhibit hierarchical structure concept hierarchy extensively structure network human
get p bit mathematically instead use qx qx qx qx qx likelihood row ensure one google gram gram vocabulary word vocabulary token take minimum small definition draw blue shape red thick thick red thick co occurrence frequency triple fast gibbs sampling reduce generate vocabulary elementary accuracy estimate either bound put condition useful drawn vocabulary widely often either slow optima zhang show hmms theory close onto value decomposition svd adjacent project observation state perhaps surprisingly co pair triple observation sufficient accurately course observe hide make precise generate require gibbs sampling observation count return result know advantage et mapping size alternate replace vocabulary introduce consider emission give observation joint vector length create everywhere else observation operator theory predictive state representation effectively distribution state directly certain exist fully observable operator fully orthonormal mapping moment derive several way show consist singular value property discuss detail estimate observe conditional probability observation recursive key take current produce map et al b observation equation dimensional reduce easily relate also invertible correspond singular detail material observation edge observation edge et edge h edge x terminal way reduce estimate achieve sample transition check instead focus relative know compute state thin lose estimating main reduce reduce term generate estimate define empirical third namely iy iy iy yy variance word tensor likewise version ij small relative estimating quantify sufficiently zero ability accurately invertible one infer require become identify hidden problem estimate fact close absolute relative large fortunately discuss shift rescaling term accurately basic scale bound usual hold everything enough state triple proceed two corollary correspond theorem proof theorem bind proof write x product write scalar b scalar state prove accuracy product thought relative allow give error generic show complexity rely know observable convert complexity hmm triple hold state lemma basically say element accurately use accurately give hold happen know hence since exactly bind hand involve know know interpretation carefully discuss different hard contrast address couple way improve rescale increase estimate word word basically would accurately challenge compute unstable basically significantly improve ratio probability could fact help change relative improved fact htbp vs vocabulary slope slope show internet corpus gram frequent gram gram occur figure find supplementary material increase singular small decrease straight complexity idea direction hmm learn discretization provide analogous transform rr learn observable rr extension preserve tensor transformation recent replace contain contrast reduce proof simple simple discrete discrete hide approach way present improve hmms tensor observe tensor
utilize q rhs u beyond simplex euclidean denote operation define generalize respect set function consider denote cube give bind low calibration note inequality define also hardness fix follow remain strong polynomial relevant complexity strong calibration desire consideration thm thm microsoft research new existence efficient calibration calibration existence nash equilibria unlikely weather forecaster calibrate every predict day approach probability forecast numerous application first randomize subsequently existence calibration think round low thought achieve critical calibration calibration forecasting henceforth calibration outcome forecasting explicitly pose net calibration widely utilize namely implication main suppose exist run ram polynomial cumulative notion randomize nash game widely round inherently concern compare sense statistical distance rather norm outcome outcome outcome forecaster distribution interior simplex constitutes forecast strong calibration least random use motivated primarily convenience restrict minor distinction literature standard forecaster strongly consider grid normalization satisfy cover one weak weak cover manner partition set intersect common face lie simplex simplex specify associate uniquely write neighboring test weight uniquely q say diameter hull test cover manner continuous opposed indicator weak note normalization tx define ram main base use calibration equilibrium game let nash bi matrix player payoff respectively mixed strategy refer simply equilibrium ne abuse notation payoff definition equilibrium mix approximation entrie particular r player game along outcome denote distribution smooth response uniformly sample law
prove guarantee computation fact often multiclass coding code develop show possible method interestingly naturally extension generalize large loss rest follow section discuss simplex analyze respectively supplementary copy describe semantic belong bx bx classification rule erm bx ir base section multiclass binary modern algorithm consider relaxation erm indicator surrogate consider often suffice special suggest misclassification label side step sometimes margin exponential machine square surrogate effective risk functional section replace misclassification effectively denote rule pay relaxation generally misclassification risk relaxation quantify requirement surely suitable depend random possibly possibly misclassification risk margin decrease case particular loss suitable x q polynomially multiclass multiclass mention interpret kind relaxation interestingly scheme multiclass seem experimentally performance sophisticated general large multiclass notably extension inconsistent classification rule apply map function consider problematic make come paper study fisher give impose constraint section natural function explicit cm label label right b thick thick dash red red dash red code strategy turn label decode return note implicitly treat naturally strategy simplex c correspond separate simplex figure natural geometric map vector hence close code propose binary follow simplex non bx yx bf f satisfying definition svm function several function classification naturally extend multiple code focus extension generalize misclassification induce one consider expect x svm ls l loss fx df sc l long theorem sc respectively square notion generalize noise say satisfy arbitrarily exponent simplex l tradeoff exponent constant constant svm ls enhance separable inequality derive misclassification risk min max technique svm loss could know work computational hilbert brief introduction kernel span choice discuss particular kernel induce kx set version expression consider loss notation simplex functional n classical value computing close loo loo loo hadamard leave df eigen standard see qp kronecker delta derive omit observe convex especially formulation fact programming iy formulation train bad consistency optimization simplex first sub finite offer neither matrix fit memory descent point ix iw computed x decomposition interested computing note convenient perform system explicit feature simplex versus simplex solver complexity complexity omit constraint path algorithm simplex essentially particular class conduct performance online uci list simplex boost uci raw hierarchical map selection leave loo l small eigenvalue width pairwise distinct point collect classification accuracy letter sc l ls simplex consistent omit sc achieve see good among method include simplex boosting compare rbf ls rbf I mit mit edu ai mit multiclass define strategy simplex code relaxation relaxation develop constraint derive provably class tool
compositional paradigm risk intuitive play maximum deviation zero indicate expect understand affect case classical candidate reduce one e outli risk task one thus task argument approach maximum minimax focus replace example consider formulation hinge new h task trace norm nuclear formulation paradigm time convenient constrain fair comparison ep involve task ep education analysis digit pairwise minimax ep regularize solve line multi attain various problem mm deviation relative mode draw uniformly sphere radius parameter task isotropic take task draw normal iid center normal mean task ball radius mean wise risk train tradeoff minimax maximum risk risk homogeneity plot respect risk solid minimax minimax visually school appear predict score student root mean rmse normalize point learner moderate share capacity minimax objective outperform specific capacity minimax stopping risk improve stopping build normalize mean relaxation capacity good capacity relaxation overfitte fold fold red minimax green compose rate computer training price etc maximum rmse objective rmse norm overfitte obtain low rmse objective minimax outperform rmse regularize norm differ mnist task classifier use component class training intuitively path node minimax figure confirm minimax capacity minimax competitive establish formulation newly formulate lie relaxed formulation introduce compositional paradigm operate risk induce empirical outperform build overfitte relative make formulation proper effort study even area minimax college technology ga usa ny usa task wise empirical risk loss compositional include spectrum minimax formulation minimax loss compositional vector relaxation case tends newly test encourage essence exploit observe learn inductive learn task connection include collaborative multi admit admit provably task random task know present challenge store future sensible ensure learn scenario music company business star rating would assign training company learn predict rating company limit quickly user rule company preference song ask user rating company leverage learn predictor produce rating task minimize risk task classical assume test notably em user fix user user rating adversarial preference company minimize negative feedback business rather mean whereas much locally outcome notion cast spectrum multi end minimax hardness full classical compositional paradigm equally goal admit hypothesis future handle exception theoretic contribution relaxation compositional include third empirically task learn criterion finally core empirical across training task level learner theoretically show future likely maximum risk short restrict attention subsection iid let empirical h hx focus observe risk newly bound emphasis bound expect expect markov show risk newly decay risk control certain space control tail cardinality theorem argument iid cf observe task high decay goal minimize task motivate empirical risk oppose singleton risk I test h union least decay probability minimizer take uniformly particular high probability risk minimizer empirical risk b maximum next section introduce compositional paradigm task benefit tx additionally define task loss compositional paradigm minimization typically convex yield
difficulty meet vary depend approach approximation error closely appear random identity fx fx identity role compact depend g g fx assumption old desire yield generalization h f minimizer diversity candidate function e var hx h follow algorithm q theorem var f e tell var h h right term cause small large quantity f decompose h f v eq estimate technical possibility h f h immediate follow error ratio probability almost bind get sharp involve almost constant almost eq see z u g fu u apply tw h old inequality fx f q fx power e q fx position f know whenever satisfie bind prof apply cover know small solution write understand part prove constant complete proof g fx gx n f jx bernstein cover h bounding side e enyi entropy scale parameter infinity explain value tune however entropy approximately estimator converge consistency far know consistency open question analysis instance consistent shannon error setting identical sampling process question require research topic understand relate minimize empirical differ extra vanish analysis mean page random r enyi explanatory variable random predictor entropy shannon entropy enyi erm variable power decay number projection interval generalization least generalization fx x ft remark minimum criterion empirical enyi present explicit novel entropy overcome technical difficulty arise square error analyze related keyword enyi theoretical inspire introduce machine paradigm framework several attention processing engineering datum systematic area find therein minimum entropy principle supervise maximally blind deconvolution topic idea extract much entropy entropy average shannon enyi explanatory become predictor use principle searching contain entropy variable principle classical minimize error put might still principle since moment order error account entropy enyi neither explanatory explicitly supervise reflect explanatory relation entropy approximate gaussian approximation shannon entropy enyi empirical computable principle decade effective foundation mathematical yet scaling parameter large enough algorithm tune convergence converge impose algorithms barrier analysis possibility detail consistency require effectiveness application minimize empirical enyi focus set input learn marginal explanatory identically aim r enyi appropriately erm learn let continuous define compact space computational purpose scale erm adjustment approximate requirement adjustment translate boundedness decay study weak constant large error approximation space regression pair hypothesis measure number exist radius center shall assume satisfy reproduce associate smooth cover number together concentration scope simplicity adopt cover prove section assume cover almost confidence explicitly bound small simply get cover small enough error bound cause method know adjustment theoretically well approximated uniformly bound heavy computable quantitative proved tell imply replace estimator put power index appear
derive respective fourier relationship identical derive close cross stationary kernel author smoothing function sound latter powerful develop gp model gps key insight identify smoothing convolution smoothing expression smoothing lead point express separable e g basis function identify form separation say universal smoothing function consideration derive kernel demonstrate auto covariance equation give equation expression auto equation gps scale dimension eq dl respective base gps dimension input term exclude refer length exclude task incorporate covariance provide flexibility task gp learn hyperparameter system learn length scale learn hyperparameter adapt filter experiment conduct datum make real sensor x chemical composition apart hundred deep east depth element element correlate capture metric fusion perform evaluation experiment gp conventional quantify compare kernel prove aim ten fold motivated fold validation estimation world set trial performance paper gp find appropriate subset identical point auto auto covariance gps optimize context resource modeling comparison vs independent vs cross block version idea rather block robustness point uniform block collection represent fold fold testing fold together small fold e follow estimate parameter result fold point e empirically implication cross fold number fold min max cross away block c comment fold min x cross x error prediction x prediction least high various mean value fold represent mse popular metric paper variance var outcome se var suggest confident inaccurate estimate outcome correspondingly var inaccurate uncertain prediction extent include gp gray mm ms se var nlp se var nlp se var se nlp mean mean mean std std std std std std std std std gp x gp c size method kernel mm ms kernel se var nlp nlp var nlp nlp mean std std std std std std std std std std std gp nn kernel mm ms se nlp nlp se nlp nlp mean mean std std std std std attempt hour ms attempt hour nn attempt attempt iteration hour individual reasonable converge well nn test figures nn produce se estimate test competitive marginally small block consider size lower confident se nn nlp nn base nn nlp remain mm rise nn cope incomplete ms competitive mm nlp kernel three gps optimize gps fusion figure figure derive independent se nlp demonstrate benefit heterogeneous source table gp small intermediate size x x gp gp improvement model correlate data ms figure prove independently se nlp perspective ms gp gp prove trust option meet nlp gp result attribute inferior element perform well nlp metric prediction se inferior figure e figure equivalent se nlp se ms se gp inferior uncertainty nlp trend element ms kernel gp nlp se small produce se prediction basically confident outcome nlp block prediction nlp trend large variance nlp overall poor limiting case mm use poor poor minima investigation stationary metric poor confident attribute correlation profile decrease increase support point interest trend able cope large support across block se take analysis se metric provides describe important understand nlp metric confident poor prediction increase bad outcome meet study answer fundamental model kernel I mean universal cut substitute statistically develop past domain model numerous sophisticated gp beyond scope develop experiment good question instance could hyperparameter metric ensure check question confident uncertainty derive gp availability ideally se significant I prediction net nlp metric test check metric behave would optimize gp datum hand purely base domain way treat modeling optimize good information multi gp multi representative g model kernel network kernel gp depend model suited competitive local minima sample reasonably gap modality paper resource individual gaussian gps demonstrate individual modeling problem effectively integrate heterogeneous benefit integration use resource modeling quantify validation gps neural competitive robust size acknowledgement centre paper evaluate fusion resource modeling demonstrate information superior estimate model provide powerful simultaneous modeling multiple consideration experiment perform fusion resource exploration resource flexible high fidelity challenge deal problem capability application limit applicability collect datum expensive collect consequently kind characterize numerous resource handle spatially correlation homogeneous model also gp ideally suit spatially use multiple correlation totally different quantity together quantify quantity evaluation understand simultaneous fusion independently kernel effective gaussian gps powerful handle gaussian list gps produce multi continuous domain desire gps manner spatially interest basically perform work large scale stationary kernel standard interpolation method applicable alternative stationary find superior stationary least work build address heterogeneous modeling usage correlation improve prediction process context presence attribute uncertain set entity model preliminary address former demonstrate model output available learn attempt generalize random source fuse scope integrate heterogeneous protein fold representation separate gp use idea composite soft human source domain kind information kind information sum encoding represent common heterogeneous demonstrate source heterogeneous model heterogeneous qualitative heterogeneous separate combine sum fusion gps derivation covariance heterogeneous example multiple former use gps incorporate surface spectra spectra map environment use implicit surface incorporate modality prior inside demonstrate context gps gps address sensor quantity extend heterogeneous idea refer gps one improve cross covariance idea dependent gp terminology analysis fusion focus resource picture relate practical heterogeneous together work study work provide resource use perform gp gps independently gps quantify benefit correlation quantity paper fusion modeling modeling present recent survey discuss multi regularization perspective focus review specifically address process convolution present efficacy broad perspective tie together process gps wherein jointly characterized specify resource modeling coordinate model necessary appropriately relationship random variable stationary square diag north l depth quickly change east direction east nn augment vector augment diag east east north constitute nn layer input use show universal tend kernel kernel smoothing equation east north depth model east north east depth kernel test free hereafter yield estimate eq nn kx n covariance evaluate likewise learn equation hyperparameter learn various approach posteriori maximize latter likelihood
offer useful representation well operator order number order th element denote clear integer absolutely median normally x obtain function max operation take function examine fourier method regard apply conclude combinatorial optimization programming consist discrete programming e question discussion minimize always difficult express graph surface produce illustration objective minima clear practical determining conclude problem subset condition minimum hyperplane intersection correspond always suitable produce sketch express e firstly function arbitrary local clear consist index solution coincide include express illustration independently minima active successively difficult represent minima outline problem produce clearly objective low solution define formally exhaustive consider actually reduce depth search provide examine point exact time dimension within reasonable branch design problem search subset would draw valuable discussion kind I paper arise method replace equivalent residual give local local dimension observation algorithm recently robust estimate regression follow optimization ip ij regression parameter optimization minima efficient design problem
define classifier convex wise maximum simple convex conjugate interested finally keep put appendix nonparametric model could discrimination margin latent dirichlet lda task project probability express model later constraint model example whose graphical document unobserve latent build mix index average mle posterior large principle correspond minimize constraint minimize insensitive loss n learn intractable variational auxiliary distribution q nm nm nm nm optimum z optimum although problem develop additional primal iteratively solve alternatively approximate solve approximate present mt discussion unknown fully framework ready present model ideas ibp margin machine svm simultaneous feature parametric automatically latent choose define likelihood task basic setup latent binary feature multiply specify dimension resort nonparametric bayesian let ibp review ibp successfully break construction efficient column sample independently stick generate decrease ibp property row ibp mass number column harmonic almost finite ibp task discriminant stack zero bayesian profile matrix need discriminant widely discriminant explore inference post want infer include place define arise expectation move well also zero ibp prior essential dot reader generic banach unnormalized practice problem finite level see truncation increase expectation note ensure fy finite impose latent gaussian super principle row deal vector denote definition fy fy fy ny penalty minimize hinge averaging prediction surrogate squared clarity cost predict label set besides perform may linear value loading replicate together form p arrive generic divergence kl divergence unnormalize although infinite finite ibp property prior decrease increase exponentially make feature truncation level detail show truncation marginal distribution exponentially hard deal duality develop algorithm solve second operator th ny g proof defer conjugate optimum distribution distribution conjugate make together impose true label datum likelihood prediction generalize share ibp cast impose problem hard resolve label inductive set latent feature train standard typically single task statistical strength jointly develop task particular representation prove relationship infinite mt brevity task let mn multi easily na task mt introduce discriminant one dimension projection view let couple share projection alternate learn pre latent unbounded projection use multiple task mt type pass transformation act input project latent representation projection view fully factorize discriminant possibly rule naturally impose define likelihood mn illustrate conjugate case similar minimize hinge rule equivalently derive mt deferred conjugate mt mt z similar detail duality strategy inference test margin test datum parameter priori appendix discuss perform mt formulation obvious basically method perform bayesian dual infer distribution however primal dual intractable mt hardness mutual desire method field dependency form impose constraint apply mcmc sampling infer iteratively estimate dual approximate e expectation sample mt idea apply easy stick ibp furthermore truncate truncation duality constrain duality problem lagrangian margin lp lagrangian step appendix corpus output given solve minimize theory adopt alternate solve margin update equation constraint large margin infer latent optimum scene closely stick break augment infinite e nonparametric mt ibp use ibp share among multiple mt mt ibp inference detail comparison macro micro dataset perform nonparametric mt ibp separate hamming loss actually equal mt art evaluate cccc cccc ibp mt ibp mt school compare bayesian multi learning report mt mt ibp concatenation mt mt ibp svm mt mt well study mt mt ibp separate classifier feature input boost mt ibp mt change parameter constant mt insensitive school insensitive affect mt training see mt achieve run mt change initially estimate maximize objective explain school explain present regularize rich formulate theorem induce concentrate develop learn task explore margin principle constraint latent automatically appear bayesian bayesian work plan constraint represent regularization nonparametric bayesian contexts social collaborative preliminary investigate direction mrfs hard plan investigation room far nontrivial normally inference truncation resource important adaptively determine feature preliminary progress along extend deal acknowledgement thank anonymous manuscript nc national projects cb cb national natural china china science foundation nc support fa fellowship appendix beyond bayesian network inference implicitly direct undirected general undirected model boltzmann could mle frequently practitioner figure need presence instance may mle lda choose parameter unknown inference converge specifically optimum bayes mle objective holds direct many formulation distribution illustrate scenario subset subset connect via graph connect observe direct property chain graph know factorization mrf hybrid boltzmann insight undirecte problem formula h put together generally problem formulation latter unconstraine normalization constraint discrimination dirichlet dirichlet result discuss loss unobserved topic topic build generate document proportion nm n index average nm expectation principle quality minimize insensitive use introduce nm posterior replace nm constraint nm optimum conjugate approximate primal formulation margin iteratively solve inference mt unknown dirichlet interpretation move proof appendix adjoint ex cc dual respectively regularity inequality take infimum infimum conjugate duality attain supremum infimum get q put appendix proof definition first easily eq therefore moreover together claim appendix easy infimum equality eqn eqn conjugate prof appendix similar operator function conjugate eq q conjugate appendix variational conjugate optimum posterior distribution normalization expectation conjugate constraint dependent algorithm mt mt consist truncated mn mn whose individual term q write finally effective discriminant easily adopt variational belong simplex normalization denote replace q tb input data mn mn mn iteration binary svm learner eq outline mt lagrangian constraint constraint specifically update duality theory solution although assume factorization form normal obtain solve solver light priori objective fix mt develop inference stick ibp outline constraint feasible truncation nk unconstraine duality dual let effective discriminant n computational bind get kl procedure solve computed mean equation cm testing datum absence solve duality normal dual efficiently update primal form update hyper g mt hyperparameter easily chen bayesian rely prior knowledge discover improve prior affect posterior bayes impose arguably direct natural regularize posterior post flexible procedure network undirecte markov network formulation chain induce concrete infinite margin present efficient report study benchmark dataset contribute interface largely keyword decade bayesian remarkable popularity field partly desirable wide heuristic practice determine unknown gaussian process beta restaurant crp bp describe ibp nonparametric grow traditional parametric recent relax homogeneity exchangeability dependent process exchangeability correlation structure structure dependency common share rely define case encode posterior accord bayes know principle offer arguably richer flexible rich structural etc hard capture prior denote parameter define mle latent post necessarily bayesian induce rule regularizer certain latent variable posterior drive type obtain augment mle parameter interest going regularize entropy discrimination discrimination dirichlet allocation give rise knowledge make impose nonparametric data nonparametric formalism bayesian apparent formalism formalism learn posterior ibp style constrain formulation objective enable regularize formulation desire kl minimization solve feasible appropriately perform parametric interesting impose slack formal description normally define formalism wide graphical chain operator solve allow regularization distribution source datum remark facilitate integration complementary largely treat disjoint core machine maximum discrimination extension markov maximum discrimination lead successful outcome flexibility bayesian handle develop machine infinite feature readily master large employ ibp unbounded feature infer pre specify procedure high optimization technique discuss relate present duality section preliminary finally discuss research arise scientific domain popular book mainly recently bayesian attract many develop much rich datum subject constraint estimation label development generalize discriminative entropy appropriate label empirical expectation normalize unnormalized expectation criterion alone usually maximum conditional likelihood al expectation supervise auxiliary propose measurement label model prediction satisfy auxiliary maximum entropy principle expectation moment match minimization special case later general banach draw paper develop regularize bayesian duality generalize present extension preliminary max nonparametric example infinite assign dimensional space multi feature share task however regularization mt popular theoretic atom endow borel algebra measurable absolutely continuous exist assume dominate exist reason introduce variational formulation density background assumption clarity optimum q q function distribution soon additional post sequel distinguish posterior post omit post posterior imply project bayes special mainly interested inference define model model variable input datum feature ibp feature input datum e capture knowledge effort optimization rule straightforward generalize rich replace normality constraint unconstraine regularize generalized constrain problem regularization impose subspace satisfy constraint besides normality rewrite master u kl unnormalize soft penalty type expectation expectation possibly dependent post make solve nonnegative interpret slack soft cover hard indicator figure feasible convex illustrate lead higher shall unconstraine u assume induce function infimum versa well many mention domain expert discrete natural expectation expectation sx yx general use unnormalized expectation supervise learn choice indicator please summary
freedom classification change planted confirm validity eq show assessment condition fig indicate mode mode imply plant mode solution plant free vanish become offer great hold change appear number linear perturbation limit dominant begin mean plant solution learnable negligible allow plot phase represents plant learnable dl region straight even letter nonetheless assess correctly plant dl use replica early learning plant square per sufficiently measurement show characterize rs dl solution replica symmetry break correct critical energy assess rs pure keep vanish promising framework refinement ratio work aid energy formula assess mathematically rigorous difficult replica evaluate limit take analytic precisely evaluate valid perform come evaluation identity constant mention nd eqs stand whose q saddle lead identity auxiliary saddle x ax exactly lead restrict candidate dominant saddle analytic xx follow replacement convenient handling saddle eqs rescale offer temperature intelligence science computer technology observation engineering material science major relevance number shannon sampling measurement recover arbitrary band signal match find attention signal fourier component amplitude wavelet basis property exploit sense cs enable cs rely considerably basis look therefore signal basis primary signal learn denote represent dl sparse dl characteristic trend compactly represent superposition strength specify dl efficient processing dl storage database standard fail patch respectively dl dl uniquely dictionary dictionary answer algebra plant learn sign permutation element zero column size dl supposed enable substantially respect consequently relevant assess central carry systematically replica keep replica compose pure free give x ph ph ph density transpose mean performance dl variable mean linearly represent effective body concern element statistically randomness effectively th active inactive zero active fig b thresholding base accurately sufficiently feature divergence fig increase degree ph suppose combinatorial
autocorrelation dash seed figure optimally curve marginal figure modify version online meaning perform topology illustrative figure restrict region component marginal restrict triangle marginal distribution distribution autocorrelation pdf pdf ordering pdf pdf consider build walk empirical variance precisely balance incorporate acceptance restriction area represent end run agreement solid certainly happen pdf pdf ordering suggest skewed roughly center respectively also distortion non marginal efficient chain ht pdf significance make resolve non adapt diagonal explore severe permutation require lead consider tractable mcmc target mix efficient key proposal proportional display obtain run example topology examine indicate criterion look seed autocorrelation figure autocorrelation tune walk seed gaussian perfect adaptation covariance restrict achieve result examine become nonlinear figure yield much nonlinearity seed selection change see select form remark pdf multimodal multimodal ready suitable region marginal mean moment restrict target prove stable density lebesgue compact invariant group organize behavior large number ergodicity theorem proof find cover ask lebesgue endow product tb tb euclidean real toward convergence stable give family adaptive approximation adaptive follow value history tend condition transition invariant stable satisfied soon almost resort approximation step design establish procedure rely existence lyapunov sequence make technical mcmc respectively markov compactly support prove eq also show algorithm one hold conditionally h q expectation covariance ingredient lyapunov lemma differentiable satisfie w lyapunov stable bound one mechanism make da x term sequence stable almost surely projection surely convergence neither experiment pointwise convergence accumulation large compact assumption almost except group simultaneously hold lemma lyapunov give field w exist symmetric consequence suffice continuous lebesgue dominate x show first term continuously w upon w th negative iff conclude symmetric exist matrix ib conclude note assumption denote away lemma process let prove measurable measurable define construction projection mechanism guarantee lemma rhs qx x dy similarly detailed balance definition separately hold conclude poisson equation p exist solve poisson assumption lebesgue cc h l l l pl tx p x tv characterize require satisfy three u hold u u set case tv lebesgue simplify coordinate remain generalize coordinate borel thus consequence circular line outer assume without enough proof regularity transition kernel rhs dy dy dy enough establish rhs repeating yield dy v cx x dy establish let result exist rhs lemma respectively prove counter respectively enough exist function martingale increment eqn term cc consider hx conclude q wu wu wu u compact see uniformly choose small conclude differentiable u set compact finite pick small finally taylor wu wu dt yield item remark lemma almost surely proof contradiction projection assume define prove induction hold addition induction eqs imply conclude induction contradiction item theorem along line thus check measurable exist tend almost lemma exist surely conclude limit consequence lemma note use almost surely permutation measurable tf continuity finally eq come projection recurrence f eq adapt stable replace research proposition lemma invariant permutation monte face marginal mixture adaptive address switch associate strategy devote consistent variant cope switch step base adapt metropolis online compare compactly distribution ergodicity consistency secondary generic explore many inference common face serious target know permutation difficulty algorithm visit inferential particularly usually situation model invariant component favor specific ordering component additive model represent superposition recover know reversible visit hard due vary address switch posterior conjugacy specialized assume permutation physics computation commonly defer code permutation also restrict finite know kernel space value distribution past internal along iteration reach proposal research last year paper adaptive metropolis hereafter variant aim identify hasting suggest tend minimize matrix unimodal proposal target empirically section difficulty obvious switching shape also solution switch identifiability post satisfactory post simulated trajectory adaptive improper issue algorithm manual prior target manual poor purpose provably variant cope idea idea step adapt proposal require loop small permutation modification heuristic scope evidence couple establish mode beneficial distortion identifiable restriction posterior marginal distinct recovered permutation problem early previously online quantification presence differentiable variant showing marginally identifiable alternative illustrative example convergence detail artificial define px explain mechanism stable sample also denote mcmc point center implement permutation permutation minimize draw uniformly minimum achieve possible usual hasting sa term toward penalty
rate spike train dependent method drawback dependent effort train dissimilarity code vary spike train spike complementary real variant essential difference fire spike distance track spike code spike distance relative spike respectively obtain distance piecewise dissimilarity profile value arbitrarily maximum whole spike train code spike material supplementary figure www user spike spike train profile instantaneous accord ratio identical train distance absolute deviation equally since rely thus knowledge spike causal real dissimilarity temporal analogous coupling train spike train flow unit patient experimental independently medical clinical criterion verify computer eight typically micro diameter mn micro ground channel digital filter hz continuously unity phase sort pass hz useful thank european project regard joint laboratory foundation award human grant education science fm foundation spike train measure rely instantaneous spike dissimilarity lead instantaneous like fire improvement allow localized train selective averaging comparison define instantaneous dissimilarity past train extract valuable monitoring train applicability model eeg synchronization train spike spike train tool area potential application evaluate role fire pairwise within quantify role synchronization bind spike train determine spike train spike adaptive complementary quantifie dissimilarity fire profile spike track difference spike kind circuit spike move certain compressed compare successive hand application resolution neural track understand fire code research pattern involve phase generation termination van distance spike spike capability spike train temporal instant van calculate yet spike temporal correctly trend move spurious instantaneous consider spike spike separate difference precede spike spike spike situation record spike train promise rapid online signal patient medical treatment aim train distance overall spike lack even causal instantaneous train spike spike future modify instantaneous train causal new three improve spike variant generate train measure able synchronization within spike train reliability visualize instantaneous follow pattern train selective subsequently train activity record distance fire involved termination finally train lack capable address application variant time preprocesse eeg series synchronization hoc mixing distance profile spike train give instantaneous measure dissimilarity train define temporal average profile e bivariate spike train show derive dissimilarity spike spike bivariate neuron case calculate instantaneous instantaneous result identical train profile bivariate subscript improve henceforth replace three spike take locally average labeling time spike train spike instant fig precede spike ambiguity interval distance precede spike spike spike spike precede spike instantaneous base precede spike divide instantaneous achieve normalization scale invariance train bivariate quantity dissimilarity profile improve dissimilarity eqs order average spike close dominate previous spike neuron weight locally make profile train average distance reliable neuron simulate allow reflect change already reliable slightly frequency spike train gradually half train spike spike close spike expect monotonic instantaneous follow monotonic decrease trend recognize short high spurious value background half evenly fire event without noise spurious spike still spike precede small spike spike spike among spike dissimilarity profile train spurious high firing reflect spike bivariate train space consist background start spike correct compare spike spike rewrite sake brevity omit dependencie j j give expand definition precede spike first spike spike precede second spike train wrong happen restrict spike resolve restriction flexibility corner spike train define analogously see dissimilarity four replace furthermore previous spike interval separately weight spike read analogously spike train contribution difference corner spike spike take care train difference fire spike train locally lead improve definition precede train precede spike spike lead spurious spike spike precede vice versa instantaneous dissimilarity profile modification spurious desire dissimilarity monotonic instantaneous monotonic decrease actual change match train bivariate presence less less drop half evenly spaced fire precision reflect rather decrease instantaneous reach ms vertical dash fire introduce eqs desirable consequence spike spike train fire spike spike information information spike position length interval like variant difference corner spike spike restrict spike precede spike spike local weighting divide corner spike interval precede indicator spike spike train gradually spike spike relevant go back precede go back spike half difference period decrease instantaneous desire sign common slope precede difference regular average calculation causal dissimilarity exhibit move average show reflect spike interval match regular trace depict move green dissimilarity profile attain maximum spike decay well move average reflect trend profile period high original causal spurious multivariate precede spike spike train perfect ms drop reflect delta spike successively event short interval spike start increase spike quick spike train narrow reflect decrease peak denote become prominent fluctuation eliminate move decrease increase reliability peak reliable spurious event first ms neuron correctly reflect average dissimilarity available train turn reliable spike train reflect rapid decrease accordingly decrease contrast causal peak spurious yet reliable identical spike train except omit second part necessarily neuron recently second evident neuron rapid firing reliability decrease spike piecewise linear one spike pool spike piecewise constant localize visualization per pool train even decrease interval actually small imagine example angle pool spike spike three step spike spike train instant spike train distance spike normalize closely correctly reflect dissimilarity selective temporal averaging spike train right selective average marked line average average certain idea significantly something obtain train external influence occurrence internal statistic exception neuron five correspondingly spike train spike spike train represent spike train spike spike train spike train fire five spike train mark horizontal red follow denote triangle time spike train regular selective whole tree external generate spike train neuron noisy half external stimulus vary stimulus amplitude horizontal green green triangle dissimilarity depict bottom f train stimulus level task neuron via visual inspection unable identify spike dissimilarity right c construct spike train identify link shaped line height mutual close repeat iteratively require minimum train obtain average separate dendrogram modify spike train train easily external stimulus relate spike tool code neuron stimulus stimulus representative neuron reliability minima order half trial stimulus varie spike half stimulus dendrogram decrease stimulus train consist spike train reliability supplementary train instantaneous selective individual interval example see spike group train manually assign correspond dendrogram respective dissimilarity average group spike train movie see supplementary train b dissimilarity average two separate regular spike train patient previous train spike spike knowledge university patient monitor prior description train patient later confirm formation brain variability exhibit fluctuation high amplitude fig remarkably time subtle continuous increase among neuron role spike variant multi patient formation spike train dissimilarity profile spike period offset marked vertical brain line la accordingly right selective temporal averaging depict selective interval horizontal bottom line obtain brain spike distance reflect record spike distance indicate brain fire neuron finding thus mechanism termination level provide understanding spatio functional spike region may insight variant aim series reliably instantaneous dissimilarity instant time discrete arise data dissimilarity signal resolution beneficial synchronization scale example observable eeg transform event characteristic potential spike shape carry minimal response spike maintain spike eeg time maxima make apply profile profile causal calculation average spike eeg duration transition perfectly minima mark red color dissimilarity profile apply trace trace move top ten signal generate channel mix mix identical eeg spike variant respectively black average box validate measure synchronization continuous internet eeg series www science university hz introduce independent maintain channel identical although decrease adjust maintain appearance regular eeg signal illustration application bivariate eeg use short channel depict profile alternate variation mix towards capable monitor zero identical transition observe easy identical signal deviation independent three contribution dissimilarity profile spurious obtain like fire add extend continuous eeg instantaneous reliability elimination capability track synchronization spike train fig possible visualize cluster supplementary evolve pattern spike train instantaneous pairwise spike dissimilarity instant continuous fig external non absence channel resolution neuron internal certain detect converge fire code localize stimulus reliability another possibility average application population response whether train successively reliable spike train possible spike train measure aim measure similarity control situation protocol stimulus repeat trial spike similarity approach trial select appropriate spike similarity response evolve extent train regular spike time spike past future spike spurious reflect uncertainty demand monitoring upon latter half regular spike capability track train possibility several role tool rapid decode brain device monitoring ensemble could lead furthermore
determine optimal quantization due oracle inequality co exchangeable array accordance co blockmodel blockmodel estimator dy b eq misspecification blockmodel serve proxy approximation stochastic blockmodel profile minimize divergence limit value increase specify blockmodel profile polynomial logarithmic growth generative long necessarily blockmodel leave open consistently sparse simulation suggest behavior blockmodel qualitatively similar across model theorem essence array yield underlie admit author furthermore time algorithms generative blockmodel suggesting exist clustering recently universal replace objective assume quickly absolutely respect implication accord blockmodel fit yield main technical enabling interpret clustering may clustering generative formally optimize yield appear relate co clustering bipartite adjacency partition empirical co piecewise define index proportion edge span subset mapping us encode use index co clustering partition binary array analogously resp partition subset k introduce co clustering admissible def nonempty zero objective mn mn mn line establishes least profile ab closeness least square risk profile likelihood average kullback leibler divergence equip serve establish exchangeable array accordance exist fix hyperplane support point closure convex combination point functional equivalently triple functional formally follow geometric hausdorff distance frobenius I schmidt metric base metric measure short subset nonempty hausdorff natural recall norm see h k since imply hold h also h result relate dense broadly speak analyze term hausdorff metric detail estimator require fast convergence estimator involve optimization turn hull optimize function interpretable exist yield connectivity criterion profile l yield x yx xy b ab ab dy ab ab ba kl ab ab expanding ab ab accordance choose apply blockmodel even misspecification identification cluster array co cluster equal size connectivity theorem attain polynomial regularity strongly dense small regular rademacher adapt ranking allow improve describe cover support fixing exist claim result upper lipschitz inspection measurable analogously count section st conjunction bind generate come rademacher devoted proof f nu clearly introduce rademacher complexity generalization sec analogy q serve define co blockmodel partition prove lipschitz condition bound random whose element without replacement lemma q construction comprise expectation invertible range exist analogously choose ready define complement complement compare well approximate set g expectation eq due hoeffding sided interval set satisfie rademach arbitrary generalization quantity similarly frequency lemma n auxiliary mapping close metric expand upper statement q quantity mn lipschitz iv claim e mn h f fp le h k least claim brief misspecification misspecification form parameterize monotone curve normalization maintain area regime introduce outer product separable stochastic blockmodel top percent relative excess risk decay zero row kullback leibler decay toward optimum grey horizontal figure separable array graph column regime suggest blockmodel despite misspecification array blockmodel profile algorithmic initialize coincide blockmodel class blockmodel contain excess percentage toward normalized divergence decay toward represent quantity obtain expansion blockmodel behavior square omit brevity address group class relations blockmodel blockmodel achieve limit rademacher nonparametric pairwise ranking set important seek thought covariate describe represent covariate take never effectively co misspecification rather assume real value value co utilize discrete otherwise technical consistently case blockmodel know number derive elegant proof implies give instance blockmodel setting good blockmodel able convergence prove first respectively lem prove support directly apply theorem satisfy ab ab l h ij ab ij satisfy dx dy ab dx f ab ab dx dy k dx dy follow similar mn b ab similarly dy ab dx dy h provide support use establish section state generalize definition arbitrary induce generalization rademacher lemma u k proposition support lemma lipschitz chernoff q result conjunction union recall mn h h let statement proof lemma st st tw ij lipschitz st st differ term sum st w therefore union statement recall deterministic variable respectively imply uv u recall definition uv r expectation side argument q nonnegative identity define expectation show observe tt tt side substitute yield statement first pt ax bx ax bx informally either group attractive latter analogously symmetric follow define follow induce likewise otherwise increase may interpret hoeffding I uv convexity mn pt equally convexity linearity expectation mn independent argument mn r I mn expectation induce fix apply conjunction vary result partition vary analogously partition hence dt mn hold rademacher hoeffde apply condition argument theorem theorem prop tr f lipschitz imply h proposition prop hyperplane prop choose k c excess quantity l lemma establish plus triple blockmodel show involve maximization learn ab establish analogous argument let maximizer exist assign quantile class quantity pt line third definition corollary office award nf award w nf uk sciences establish ep ep fp ga european establishes address co partition array exchangeability provide rate correspond profile minimization blockmodel nonparametric asymptotically connectivity underlie generative class tool wide rapidly grow use
pointwise poor alternative average choose minimize leave right table improve adopt determine regularization ridge although perform bad variability result highlight make expect perform draw intractable loose criterion mean neural bic well outperform entropy winner critical influential analysis comparative review one order qualitative dimension comparison relative illustrate compute relative example bic square outperform affect choice summary requirement suitably hyperparameter sensitivity uninformative understand abc perform efficient abc carlo carlo sampler implement practical decision dimension potentially bad sampling derive example chain monte carlo base arguably necessity evaluate evaluation diagnostic leave procedure serve comparative acknowledgment discovery dp national width width comparison observe summary implementation summary statistic rather full central question summary observe minimal article provide split exclusive subset regularization subset selection ridge regression reduction three typically update refers family computationally intractable feasible become statistical research see distribution first ps ps summary p informative ps intractable approximation construct standard density argument explicit intractable rejection sampling term proceed draw summary straightforward ps ps intractable posterior avoid weight suffer dimensionality help algorithmic approximation weight consider sampler chain monte carlo practice typically smooth small may balanced much effect allow small turn burden markov sequential adjustment aim explicitly simulate summary achieve trade summary highly dimension statistic abc choice comparison reduction characterize exclusive ii projection iii reduction within bayesian regularization involve evolutionary fitness mutation drug production article section classify review reduction outline comparative section conclude abc informative candidate loss dimension information analysis information vary exception establish statistic implement data summary dimension broadly classify exclusive class evaluate criterion contribute criterion criterion argument subset demonstrate importance choose final consider combination statistic layer abc framework whereby response transform predictor include feed expect consideration abc use however summary shrink regression toward uninformative contribution discuss idea adjustment strategy abc suffer curse dramatically increase adjustment aim avoid explicitly describe regression adjustment notational simplicity assume adjustment use eq draw prior zero assume estimate ed ed adjustment control variance trade reduce effective accept flexible conditional variance residual ms estimate flexibility produce adjust bayes alternative adjustment summary dependent variable subset simple summary statistic exhaustive enumeration infeasible especially monte result greedy principled reduction statistic contain note relative score content justify practice implement conceptually whereby posterior differ threshold specify particular procedure various follow definition summary statistic acceptable determine statistic choose sensitivity aside obtain statistic observe statistic unlikely generate considerable especially continuous discrete acceptable restricted reduction entropy shannon w minimize subset entropy posterior sample write denote empirical remainder posterior w work weighted minimum entropy summary stage assess author root eq compare component scale component wise naturally parameter unknown summary treat leave minimize likely vary observe minimize simulated set summary statistic minimize subset summary directly posterior know truth computational expense procedure stage example occur distributional tail informative exclude suit stage argument consider entropy easily alternative technique reduction abc criteria suffer require technique combine nonlinear transformation order informative advantage well number statistic handle uninformative addition accounting disadvantage project interpretability method projection seek explanatory high correlation dimension explanatory suitably cox response optimally correlate summary choose square leave validation response inspection minimum argument disadvantage square aim global relationship draw distribution orthogonal component truncate ps pi region pilot feed nonlinear generalization regression propose network learning dimension regression unit q first transform statistic combine neural neural neural network possibility hide unit rather dynamically hide specify parameter infer minimize least neural term decay neural network idea regularization criterion use theoretic previous reduction summary ensure p require good parameter assume model prior noise vector potentially use function consist power fit base truncate prior ps pi significant posterior pilot sophisticated alternative regression summary summary inference considerably regularization approach overfitte abc adjust uninformative summary regularization could greatly inclusion ridge adjustment alternative dimension abc method include comparative adopt summary abc specification computationally tractable idea likelihood spirit use estimate pilot regression adjustment call seek good bic evidence attractive contain bayesian choose develop genetic weight contribute equally simulation uninformative ideally negligible finally recent find way accurately full avoid find conditioning potentially joint density likelihood weight variance component depend induce marginal practically useful writing construct py dy computation ap conditional simulation analysis last hide iterative analysis processing importance whose claim curse abc method derive aic bic construct second modification fit available information goodness statistical maximized likelihood measure py determination equation aic number define effective simulation arbitrary insensitive favor uninformative aic subset summary kernel residual size aic replace involve could g adjustment element vector aic seek summary minimize variance adjust criterion section select however construction include predictor requirement informative range outside uninformative criterion section linear weight describe univariate notational clarity exposition approach weight least adjustment adjust uninformative summary avoid implicit reduction regularize square regression adjustment toward impose magnitude note consider alternative regularization procedure additional pt ridge weight minimization pt deal generalize average comparative reduction study analyse literature include evaluation mutation drug production clean simulation draw performance set evaluate value divide parameter standard first comparative ease summary adjustment score within abc euclidean statistic simulation nonzero choose summary size keep slightly adjust follow exploratory analysis fit use take pointwise obtain summary median function start choose chi model previously propose strategy focus scale mutation scale dna sequence individual site mutation mean distinct examine abc dimension reduction six genetic table site show regression regression method pt c perform regression adjustment substantially bad summary summary exception pure next reduction relative reduction regression adjustment contain obtain adjustment supplementary parameter combination reduction aic criterion partial pls networks loss ridge ridge obtain adjustment summary good row c c pt third example integration achieve statistic substantially uniformly improve partial net adjustment six population improvement standard adjustment reduction aic partial loose dimension compute adjustment relative follow adjustment bic net perform stage entropy gain evolutionary plausible mutation fitness transmission evolutionary incidence marker transmission relative parameter transmission transmission drug mutation stochastic statistic number diversity proportion case replicate reduction technique univariate abc well analysis particular proportion marker mutation distinct regression adjustment inferential performance argument favor marginal whereby replace estimate precisely margin reduction aic bic substantially adjustment depend aic result increase reduce clear improvement almost aic regularization mostly comparable improvement partial square produce result adjustment partial ridge neural posterior loss
event diagonal get put complete subsection take define detection assume sequence test discriminate converge pt price pay multiply minimum gap remain rate therefore tight sdp simple large inspection proof hold arise dual indeed control show proposition statistic pt employ hard quantile dataset behave sdp one strictly diagonal support problem seem qualitatively rank unnecessary indicate plant large sdp parameter section original functional sparsity notion order enough zero exclude h make notion sparse detection hold test old let vector exist unit vector eq generality otherwise v p p appropriate base let approximation proposition since statistic achieve context relate examine maximize quadratic argue solve quadratic form choice hold follow step discriminate define value sub coefficient I nx condition p replacing hold probability similarly two observe rely fact weak set attain optimal assumption constant initial situation necessarily probability n low take one discriminate test difference phenomenon method proof limitation hardness rip attract lot complexity theoretic impossible statistic arbitrarily clearly suffice achieve reason first polynomial approximate input hereafter develop another plant clique problem inspection level prove argue unlikely let q hinge optimal rate contradiction bound polynomial plant clique hereafter argue algorithm unlikely vertex place connect pair edge plant clique problem call vertex goal h consists planted traditionally plant surely appear even algorithmic relaxation fail provably hard see bring toward hardness note capable finding plant currently tensor yield polynomial lead researcher hardness plant clique include dependence equilibrium vector adjacency rademacher random take construction vector long qualitatively manner subsection illustrate method argument complexity begin show plant even plant cardinality surely otherwise begin side variable n display side lem inequality would detect presence empirical valid soon replace one clique probability k k pc hypothesis subsection propose approximation sdp tolerance recall plant clique control appropriately soon constant clique consequence way one clique intrinsic sdp reach level intrinsic polynomial computable control element sdp achieve dual however polynomial prohibitive statistic latter problem grid purpose illustrate empirical compare x x vs unit dimension yield matrix density hard distinguish particular statistic discriminate set almost discriminate discriminate long bring evidence consider tight factor need discriminate soon minimax behavior illustrate phase test different ii close predict reciprocal setting exhibit suboptimal scaling exist hold actually transition parameter sharp choice q vs vs distribute left level compare demonstrate critical indicate scale existence pay technical appendix various concentration inequality ce variable generalize sum sub gaussian chernoff tail follow bound expansion chernoff define chernoff e st st display support wu fellowship support grant dms level sparse principal high covariance optimal know np alternative prove detect principal theoretical science bring evidence inherent sample set sparsity belief explore high pca classical pca produce inconsistent namely relie explain assume identity sparse assumption contribution word may ask principal explain statistical answering consist construct detect associate variance ii prove test level lot vector propose detect shift sub encode two focus aspect problem detection close shift detect diagonal plant extend direction sparse precisely minimax usually notable exception unlike extended linear refined exhibit qualitative light important size testing capture although construct optimal raise prove topic consist overcome reference therein solution statistical address introduce call programming sdp drawback direction semidefinite matrix eigenvector notable allow simple near importantly bring support evidence evidence build conjecture plant clique well conjecture vector robust model discuss notion sparsity distance covariance sup follow discuss link probabilistic spectrum wishart detection particular test spectral principal probability prove sometimes become computational relaxation true whenever th row denote define norm form define radius trace rank functional denote usual identity submatrix real number copy random objective h covariance sparse euclidean large two imply perturbation weak critical minimum coherent hypothesis testing existence direction exploit submatrix affect perturbation matrix define eigenvalue equivalently behavior hypothesis govern nan begin statistic enough alternative probability unit sparsity define use matrix general satisfy condition indeed support eigenvalue wishart adapt desire bound q net sphere easily q since yield subset subset cardinality fix observe cardinality get complete sufficient lambda whenever follow previous subsection find h regime take provide discriminate converge goal exist joint hold infimum find distribution let proof get determinant moreover formula substitution display q eigenvalue together equality substitution desire turn proof yield inequality expectation
however per rest denote hoeffding follow principle rational multi armed high establish use action high must pure exploration arm bandit stationary work well rational policy heuristic policy instance armed bandit reward arm multiplicative ucb aware sampling computer particularly engine extend aware ensure aware average per engine engine still player advanced phenomenon win vs node outperform work suggest monte policy outperform pure largely aware estimate use policy differ decision principle david il monte carlo decision ucb policy armed mab minimize cumulative search differ mab arm move reward value empirical evaluation comparison ucb mab instance go appear although method successful appear successful past job trading exploration exploitation statement ucb simple principle scheme outperform adversarial search nature good computer go show improve monte suggest approximately optimal decision mdp explore trajectory termination cutoff multi armed bandit problem mab mab ucb near state art search get reward policy mab
problem cast shown adaptively rate sparse set general tend precision belong liu lin good conditional cv theoretical unclear contrast procedure minimax begin later attain precision norm loss rate observe nonzero need consistently bound rate convergence expectation constant p n theorem z p z z I depend analogous lemma hold depend thus prove condition similar way theorem mainly rate wise wise graphical rate norm theorem hold norm p pp cm p precision define theorem attain collection indeed establish minimax estimate discuss technical outline section particularly suit treat directional matrix generalization le tensor r lower alone parameter like bound maximum estimating belong need state density dominate measure variation affinity r rd vary establish non zero matrix observation projection r aa aa vary remain equation apply sharp major norm detail difficulty essentially finally calculate factor leave sec shall major yield precision satisfie make clear component sum less equal later associate equation immediately equation p lemma separately lemma comparison define q sparse collection put put theorems collection ball ball define w graphical enyi random random uniform generate end additional select choose sample obtain parameter select minimize cross liu various setting table loss compare tune three ccc band ar ar ar norm tend risk sparse covariance cf important distinction depend difficulty precision mention introduction equivalent undirected independence contains order indicate additional depend imply threshold recover context liu adaptive dependent thresholding recover sharp var much difficult gaussian liu transformation apply graphical neighborhood selector adjust correlation detailed leave prove key technical begin collect technical technical lemma px n suffice great ik jj jj jj j jj jj ij I ij classical general matrix proof contain set yield w p belong feasible p pn lemma belong n prove theorem np cm pn cn pn prove estimator minimax estimator projection inequality follow establish lemma p ph w n yield without assume affinity affinity without case prove constant affinity chi calculation triangular matrix possibly leave single normal r normal precision form nonzero element submatrix observe number row value upper triangular p place prove chi proof straightforward omit precision simple total immediately determinant subset j simplify notation r ready prove recall overlap equation imply equation last establish enough equation equation equation show v special simple first eq eigenvalues jk n k write nonzero equation drop upper triangular thus n p nk equation follow definition wide multivariate high convergence adaptive minimization propose parameter space implement step establish derivation sharp directional technique minimax low optimal convergence sparse adaptively minimax collection space simultaneously minimization graphical primary secondary fundamental many high knowledge analysis furthermore broad application portfolio lin precision tool relationship scientific recover equivalent liu liu distribution recover draw recent attention b matrix introduced estimate lin study also et fan et selector precision liu rates spectral many various unclear estimator estimating term serve fundamental benchmark goal present establish matrix norm parameter sample variate matrix ij order consistently zero graph symmetric spectral equal matrix model ball away collection range particular low liu thresholding variability entry paper introduce detail study estimate precision establish match minimax rate compare frobenius discuss connection difference proof minimization minimax establish together adaptively begin notation vector
differentiable neighborhood statement twice em satisfy let continuous q lipschitz subdifferential exist bring great deal design nonsmooth continuous obtain nonsmooth locally lipschitz computable certain arbitrarily hold every lipschitz subdifferential everywhere imply next divide optimal value relation conclude hold hence rule lipschitz continuous everywhere eq recall relation immediately iii every statement subdifferential em nonsmooth lipschitz arbitrarily eq statement hold lipschitz f lipschitz fact imply statement iii iii em nice find computable let order stationary moreover nonzero let contradiction definition contradict substitute multiplying use immediately second minimizer statement differentiable hold low immediately ii let minimizer notice order satisfy proof local solve also variant subproblem moreover unify variant subsection reweighte propose sequence vector q close provide unified analysis suitably solve apply minimization choose arbitrary weighted accumulation point first point accumulation hard q k kf x k f side definition stationary em reweighte whose computable variant subproblem simple form g minimization choose minimization k go go set outer inner criterion iteration th modulus x relation inequality imply outer k next accumulation order problem suppose accumulation stationary subsequence continuity f conclude optimality side x stationary generate accumulation multiplying conclusion present variant subproblem solve method minimization method scalar apply end scalar accumulation accumulation exist subsequence side equality problem iteration variant close moreover unified present em ik next establish accumulation suppose accumulation ks argument k ks increase combine inequality accumulation subsequence definition g p fx eq complement claim indeed see imply k kx yield observe equality stationary em reweighte computable subproblem simple form choose I step go iteration termination criterion inner iteration strong modulus inequality g kx kf kx kx whenever em show accumulation generate stationary problem accumulation fx fx converge outer argument let subsequence optimality take limit equality I proof study parameter dynamically adjust approach whether iterative reweighte share adjust address variant subsection locally computable certain stationary view iteration remarkably establish accumulation choose minimization ik point accumulation moreover nonzero satisfy bind n gx q accumulation subsequence monotonicity gx limit side recall imply substitute obtain stationary monotonicity result order rest sequence computable arbitrary arbitrarily weight go outer inner inner termination let th outer iteration modulus last due yield f sf similar accumulation point satisfie accumulation order hold nonzero accumulation point subsequence monotonicity denote value outer iteration proof condition limit rest conduct numerical datum generate convenience matlab mac spectral gradient choose supremum terminate accord integer one denote generate vector zero deviation choose list value column second give mention cpu objective former cpu c rr c c generate instance cpu report point objective instance method similar cpu much rr cpu c minimization minimization derive bound entry order stationary local variant unify addition lipschitz develop threshold value remarkable reweighte dynamically computational new generally stable cpu li propose minimization solution regularizers method discuss reweighte regularizer pt corollary support grant study minimization nonzero order iterative reweighte minimization solve subproblem unify addition continuous develop show accumulation remarkable iterative reweighted dynamically update demonstrate exist cpu key iterative reweighte minimization reweighted finding system regularize nonlinear programming observe minimization iterative hard penalty widely function one
scheme towards towards towards give short summary two stage stage stage lead trivial corollary prove np prove np hardness trust relaxation lagrangian quadratic give scheme well consequently well scheme decrease towards bound towards dispersion relaxation scheme stochastic framework successful uncertainty environment context interact efficient strategy design policy maximize reward bad adversarial receive observable generalization implicitly return consider compatible deterministic mostly environment aim policy strategy suppose know optimal occur work uncertainty robust studied formalize state discrete whose element denote element finite action horizon instantaneous r sequence action return optimal action far mode dynamic action transition set transition learn transition sequence u originally exist finite action lipschitz coincide system return intuitively action initial u subject bold min generalization aim bad return lead focus design resolution set action generalization problem assume e horizon step two case relate want safe clinical clinical subproblem section f u p x u matter drop argument refer bind two subproblem r subject obtain solve correspond stage except action trivially prove provide u drop first relax constraint u solution u intersection size also lie exist k depend triangle inequality replace satisfy index replace satisfy focus subproblem x p focus resolution element u intersection lemma cardinality l maximize u constraint low order side equal solve constraint prove introduce problem determine np feasibility feasibility n z px np reduction exactly similarly belong compute equality prove linear exist q point hyperplane choosing obtain suppose satisfie imply prove hyperplane dy prove ball tx furthermore inequality prove hardness choice space follow hard hard np hard equal time optimally solve general stage aim approximately obtain want propose tractable drop problem show provide relaxation constraint interior prove scheme relaxation scheme prove computed relaxation converge dispersion section two subproblem solve straightforwardly cf theorem problem drop suggest constraint drop index relaxation give u tr u radius minimization determine hand side literature value lie lie distance equal family relaxation consider combination relax relaxation make constraint trust tr tr f trust region provide exact original optimization tr way lagrangian section second multiply constraint variable constraint dual value decompose square observe quadratic fix soon interested never unless follow inequality never convex closed formula rest proof useful come write convex trivial write ml observe notation linear know formulate ml compute propose address return p l bind compare case relaxation first two k region bind always great u definition possible k tr construction equal prove due k bound two corollary let x tr f u tr always preliminary p u u u k tr tr f tr relaxation transition k u u x l x definition lagrangian relaxation bind u u section tr b consequence theorems lagrangian derive problem introduce action bound dispersion decrease zero dispersion equation dispersion intuitively see visit lagrangian dispersion u u tr f u u tr end f denote trust maximal maximal tr tr b also sequence action u u tr tr action lead return dispersion trivial u u tr us u sequence j bind satisfie relationship j b c u f end notice scheme dispersion suffer curse action actual lead state trust lagrangian bound benchmark x reward u denote th lipschitz state c u tr transition follow f u maximal maximal trust relaxation transition f tr lagrangian relaxation great trust equal ia ia computed transition optimal action cardinality order quality protocol cardinality space f kb maximal tr resp transition tr c u u tr sample transition observe lagrangian much trust bind close sequence policy show problem relaxation extensively perform min continuity reinforcement imagine environment continuous would interesting scheme problem space acknowledgement scientific research present system office author nesterov point relaxation author trivial trust tr r x u tr f f quantity
jx functional right hand homogeneous compare fx jx jx continuity exist open vx x jx jx jx jx let mx x mx vx x p vx un ni ni closure mx un I vx cover define around jx jx kx j sx ds gp may projection orthogonal g vx vx p fx vx x compatibility fx fx vx sufficiently kx jx x jx p kf mx mx hence kx take jx I c jx exist jx together property dimensional contradict cone sign place continuous kx b depend sum kx kx b kx twice interior since qx dt much jx c jx ds satisfy denote exist bx constant distinct satisfied constant distinct number positive constant x lemma fx fx n n thank p na fx ii replace lemma sufficient formulation worth state toward distinct number j j j j lemma p p p n x n dt n fx x dt x l fx dt note fx fy fx fx c n complete fx fx dt fy fy dt c n sufficiently n q complete admits support sequence compact set continuous observe proof lemma sufficiently representation replace n change line k bn ii give proof consequently conclude condition iii replace implicitly proposition proposition replace supporting imply c c x x x sufficiently way large exist constant x c constant p x exist c x n n event fx lemma original integration keep order exist help dimensional type limit inequality answer field connect newly manifold proof theorem obviously satisfied shall constant diameter may sn kn sn sn k sn sn sn sn subspace process tb wiener increment wiener theorem use brownian sn integral aid continuity boundedness old inequality yield bn sn sn hand term x x u ss qx sx sx sx sx eq consequently process likelihood examine asymptotic maximum value bayes estimator estimator independent deviation good bayes maximum estimator c mean c example previous standard deviation maximum type estimator compute likelihood estimator behavior furthermore small theoretical point view estimator performance mean mean criterion derivative shall remark qx c jj lipschitz x p satisfy theorem open ball open may call sufficiently sufficiently neighborhood moreover continuity contradict sufficiently intersect fx cn large remark theorem standardize write du nu later du rigorous time n dt eq complete dt cn dt u ds sx lemma terminal discrete martingale I n h every u h h every eq j easy proof check regularity satisfying yield complete every exist nu nu q nu obtain desire use consequently q note z nu nu u nu calculation yield dt dt b formula dt one decomposition second enough f k since k ds j n k nj ds theorem show bound random depend u show finitely show family r see nu nu complete proof satisfied condition setting complete checking existence obviously support support neighborhood necessary logic depend location increment choose continuously topological support besides side necessary effectively together theorem general derivative include start move quickly possible derivative fx bx sequence pure york diffusion coefficient estimation multidimensional diffusion ann coefficient normality diffusion r posteriori york conditional de observations yu rao exposition identification dynamical yu inference spatial york yu statistical inference ergodic le l asymptotic york le asymptotics york observe process journal statistic quasi jump appear statistical process rao asymptotic square process estimation jump discrete observation infer asymptotic differential dynamical estimation martingale ann related process n inequality quasi ann deviation institute mathematics deviation u definition lem example university sciences university mathematic quasi type deviation become index nature make na I require quasi estimate key phrase normality convergence observation deviation classification secondary study paradigm analysis p normalize factor tend tend continuous simplify condition p estimator since sup u sup u pp asymptotic mixed likelihood convergence separation moment derive process include serious process prove convergence exponential type place serious likelihood statistical field connect stochastic example mix drift convergence dimensional wiener respectively value dependent course finance asymptotic theory estimation unknown volatility frequency study refer rao ergodicity diffusion perturb limit score become distribution mixed quasi likelihood scheme highlight field appearing find prove wide applicability see limit correction divergence distribution obviously necessary standardized provide kind high statistical large deviation enable valid high estimate field quasi develop analysis quasi quasi likelihood bayesian give type inequality locally carry process le le notion setting ergodic diffusion difficulty besides exact expression calculus complicated jump involve rate associate drive evy positive index moment likelihood diffusion process jump deviation sampling polynomial field meet question difficult na I far quantitative author part differential equation dimensional wiener measurable process bound locally boundary lipschitz true asymptotic mix normality convergence contrast bayes unknown volatility type main context lx n set admit extension qx sx sx process admit ds take wiener time admits support function exist vx j vx jx satisfy qx fc jx relax strong condition degeneracy unless random thank exact available inference quasi satisfy bayes estimator dt standard article fulfil growth fulfil polynomial growth following suggest degeneracy easily satisfy differential dx qx fx case thus ii obvious w stop necessity introduce x satisfy stochastic wiener value differential dx sx sx degenerate neighborhood sx sx fx function x difficult fix time I however na I devote proof likelihood random
robust describe fista solve cs extension efficiency finally vi conclude paper matlab reproduce website sense cs compress compressive setting recovery pose establish reliably strength various solve cs recovery cs cs induce cs recovery hold normal cs inefficient cs underlie additive achieve quadratic g ir huber penalty derivative huber penalty outlier detail linear actual solve trivial cs option literature mr reweighte outer involve iteratively double loop limitation one loop powerful framework suitable thresholde fista optimization effectively smooth sense share fista variant minimization update involve historical couple easy solve smooth decompose univariate analytical trick fista variable approximation historical update subsequently loss ensure proper second convergence original fista readily loss see remain compute constant domain imply mixed quadratic part negative cost alternate multiplier admm simple powerful optimization large arise develop advanced compute different recently unified framework robust technical challenge couple make extra easy tackle variable element wise group use know admm framework term smooth tie affine cs objective augment lagrangian augmentation improve numerical augment update dual variable concerned treat thus optimality lagrangian iteratively particular operation due iterative burden robust update difference admm algorithm solution admm cs essence replace previously major oppose iteratively modify quadratic recognize fista choice converge see inversion hence inversion update subsequent case state admm update approximate converge discuss stop update affect residual residual dual vice versa penalty trade admm generally emphasis primal residual admm error primal dual due work case examine detail terminate primal small please fista present tackle slightly angle fista simple admm regularization fista admm split quadratic requirement numerical experience tolerance fast fista realize need cs make fista extension simply case would model priori cs lagrangian equality solved previously replace primal sufficiently residual e requirement stop wish regularization may realistic quadratic loss g elastic robust formulation treat special recognize convert solve many efficient trivial modify admm indeed update step fista quadratic fista need induce operation likewise case generalize solve slight modification mix propose admm huber select suitable framework restrict huber indeed type note fista easily differentiable overcome difficulty non apply framework introduce rewrite thus scale rewrite easily principle solve yield exact recognize stop vector include k k cs reveal basic sparsity exploit far knowledge exploitation sparsity effectively requirement comparable recovery compare slight cs haar wavelet image could example video circumstance similarity consist background move illustration sequence bar later exist cs likely task perform denote recover cs task cs formulate g l loss along reflect prior correspond wise respect single quadratic formulation extend fista soft result prove solution lie point satisfie e intersection minimize shrinkage generalize I I write solution meanwhile rewrite thus decompose row immediately step resort loss admm update k task fundamental cs formulation framework example affine set investigation practice robust recovery constraint construct residual exist become residual meet happen robust cs robust combine coarse fine search find robust bar admm variant inversion image inversion directly much algebra exploitation reduce dimension cholesky accord direct fact lemma note cholesky factorization whole regularization cs recovery bar admm compare nest nature double loop nest robust cs cs solver loop select solver speed know expensive numerical experience show inner solve high solver robust cs case see iteration inner outer compare approximately computation shrinkage examine take former insight iteration respect solution iteration take threshold solver inner cs termination tolerance within loop residual nest cs algorithms matlab roughly bar db noise gaussian time versus plot achieve initialize indicate cs considerably respect loop converge normally clearly profile iteration accuracy admm fista need number actual take tolerance advantage algorithm nest interested tolerance admm fista algorithms nest fista robust bar image top haar wavelet show method minor different robust next examine much improvement cs affine robust cs formulation formulation know examine robust bar example select error take admm robust take reach minor affine formulation random bar recover behavior affine cs formulation huber noise random noise huber new remain behavior admm cs show behavior right observe slow accuracy compare huber loss admm theory next examine cauchy fig cs use nest respectively completely fail meaningful recover nested maintain recovery quality regularize admm cs computational admm cs behaves model use previous show formulation provide db huber thus despite less favorable cs well loss admm robust cs demonstrate usefulness robust separately multi cs formulation share bar frame bar move block across frame obviously wavelet bar distinguish clearly illustrate image haar wavelet bar image common admm multi task show fig cs cs recovery cs recovery every clearly robust cs formulation
validate moderate modification amount noise motivation component modification could nonlinear difficulty component robustness measure different different noise curvature validation base nonlinear term nonlinear input explicitly model find set weight attempt auto nonlinear preserve validation network problem non nonlinear ability pca model model miss approach adapt neural pca nonlinear pca complexity lead error validate control new datum validation exist method validate optimal validate curvature even nonlinear pca perform supervised architecture unsupervise validate nonlinear pca able component fit decrease distance effect use function plus standard error datum large fit clearly specific unsupervised label target correct onto curve error architecture compressed generation locate curve pca use perceptron mlp auto encoder auto force square extraction generation hide layer nonlinear mapping layer middle hierarchical validation pca done generation mapping lose input nonlinear control decay add coefficient change component validation fail evaluate nonlinear missing specific nonlinear validate estimation sample reject nonlinear decay get nonlinear pca repeatedly initialization network validation miss demonstrate ensure embed nonlinear component circle projection blue dot complexity high consist lie manifold loop embed factor apply network optimize weight parameter loop miss incomplete per standard test validation repeatedly newly median missing run compare approach able minimum validation small complex fit contrary supervise fit validation optimal setting coefficient weight decay force nonlinear linear force solution well impose pca whether unimodal multimodal e ni nonlinear incorrectly detect inherently unimodal point illustrate obtain important show validation miss validation plot set nonlinear pca test contrast optimum absence test fit produce incorrectly distribution unimodal miss validation dimensional unimodal multimodal nonlinear set nonlinear give increasingly complex validation lack class complex pca high describe complicated cover space complete moderate onto restriction pure validation contrast curve remove target predict test datum validation remove unsupervised validation validation auto pca behind miss unsupervised remove validate correctly availability software pca auto neural nonlinear pca neural
constant nx kernel define fx nx constant k asymptotic confidence hence investigate band procedure profile elliptical galaxy base simulation simulate bivariate unimodal include dataset convolution correspond choice slightly bandwidth pt nominal nominal computational feasibility scenario preliminary motivated bandwidth band normalize figure illustrate decrease normalize average unimodal correspond small coverage band wide band decrease significantly figure bivariate band set increase precision band often spread use methodology profile elliptical galaxy pixel galaxy image model convolution sharp optical detector digital couple device pixel space dimensional elliptical galaxy band nonparametric estimator narrow centre galaxy centre normalization uniquely determine control curvature profile model coincide de detail already classify galaxy consideration galaxy analyse give probably point source middle advance bandwidth choose band unknown profile gradient see due partly cause bandwidth whole order region galaxy restrict check band satisfied parameter extensively dd jj dimensional direction j vector contain pt l one axis give example pt b b alternate b regard increment rise assertion decompose sum part intersection intersection axis auxiliary throughout convention nk hx proof weak brownian detail precise n da limit independent variable rt l assertion accomplish purpose introduce g notation brownian next consist replacement process quantity regard increment u step pt q pt negligible pt assertion representation straightforward nx da df df z integral r n n z nx ff hx r fx x x r n f h h da da n f embed observe convolution integrable f proposition application taylor h nh kernel hold h inspection show omit brevity replace precise calculation n rewrite increment sum express increment give n da increment accord account least dx rewrite sum index sum n application version recall obtain imply existence x du nx h k n du nx n nh nh nh nh nh jx bb bt bt bt db bound part wiener rescale brownian u nh x nh pt nh nh pt increment correspond pt nh u nh nh nh g nh obvious manner modulus brownian pt n bs bt dominate argument nh modulus continuity obvious sufficiently nh nh nh nh da nh law logarithm nh nh nh nh v uniformly negligible brownian nh nh nh nh nh u brownian nh nh nh nh u lemma contribution quantity obtain nh k u give k definition yield x p b p lemma da limit depend repeat lemma b df note pz df set z df px u minimum maximum observation yield n desire argument proof error partial sum replacement fold away helpful stein parts manuscript technical support center foundation style graphic remark asymptotic band multivariate convolution method increase set particular confidence band feasibility elliptical galaxy provide band application impossible observe inference problem type medical physics biology express well emission involve heat reconstruction imaging device closely operator study deterministic physics example numerical reproducing space investigation importance point view field construction convolution square recovery device biology observe image slightly physical propagation light surface application true mathematical observe image approximately convolution real call problem convolution regard problem become nonparametric band develop band dimensional motivated dimensional typical reconstruction also spatial behavior spectra like variable star depend refer predictor univariate investigation weak sup appropriately construct band periodic reconstruction biological device often extend construct band independent supremum center derive bootstrap confidence band band band local bootstrap van deconvolution band asymptotic confidence band asymptotic cut estimator indirect limited estimation univariate regression author band band standard extend straightforward manner multivariate multivariate case inverse band require present band convolution predictor originally motivate asymptotic band small strategy long sense restriction whenever constant kind convolution distinguished convolution determine degree class moderately pose decay precise subsequent existence b specify understand difference fast enough relevant structure result remain decompose two sufficiently fast completely illustrate h h kx definition f ff kx example condition factor polynomial imply odd expansion obvious even situation h taylor h give x nx straightforward dimensional exponentially tail assumption transform square assumption fourier transform weakly differentiable integrable derivative satisfy moreover
walk reversible add weak connect vertex direct particularly become diffusion graph place diagonal orthonormal eigenvalue place order diffusion state euclidean distance embedding time approximate depend spectrum top computed case align eigenvector respective distinct alignment extend union embedding define allow thorough policy choose piece low cut spectrum direct determine cut sequence cut establish partitioning sophisticated also free version truncate walk evolve restrict find eigenvector eigenvector small trivial cut range small eigenvector give side minimum identify state state side cut bottleneck give stop meet algorithm subgraph cut recursive application bottleneck partition associate spatial scale determine scale scale discover mdps pre compressed mdp transition matrix strongly type diffusion policy region light structure mdp accord goal strong multiscale multiscale suitable mdp mdp coarse mdp original event mdps relative indeed think inherent rather mdp view coarse fine coarse describe carry coarse coarse mdps learn detail bottleneck explain mdp priori simply choose suggest determine mdp coarse compatible fine coarse scale mdp tuple brief primary mdp subsection scale section bottleneck bottleneck discount collect trajectory reward along trajectory chain optimal fine coarse compressed respect scale vice type allow instance related part may always accomplish computationally simulation compute computation parallel prescribe cluster compute interesting relevant computation eventually multiple independent comparable section analytical proof multiscale read skip multiscale mdps obtain construction fine policy small diffusion everywhere regularization partially follow initially towards coarse iterative process fine transition guarantee except situation cluster boundary bottleneck relevant subgraph induce restriction p incorrect task error easily lead compressed boundary equal policy cluster correspond system discount sx ss compressed mdp refer restrict run execute policy mdp large circle bold fine vertex intersection undirected coarse dark gray cluster fine policy state execute execute feasible reach back top hand execute reach compress dependent discount even problem cluster loop path start reach compress another fine neighbor well blue draw left loop edge fill draw node draw fill rgb grid fill height loop edge loop loop edge leave edge two well thin two loop show per coarse coarse reward action see detail refer action trajectory restrict determined state cluster action reach either carlo compute analytically trivially advantage parallelism box simulator need slow develop light quickly separately solution form consider bottleneck respectively compressed may obtain non solution computing action discussion appendix derive reward discount reach define define expect discount reward trajectory bottleneck state cluster associate cluster repeat process example scale mdp depend action compress mdp coarse computation involve give accumulate variable start bottleneck concern accumulate reward discount reward solve scale action correspond denote characterize p aa h q system course discussion consequence length apply calculation negativity compress mdp still insight involved path necessary characterize hierarchical discount coarse scale experience consistency precede coarse fine depend length coarse mdp coarse discount correct discount path length transition partially correct simulate fine impose coarse uniform discount discount incorporate coarse compatible policy accelerate procedure policy cumulative discount tt proposition discount negativity coarse correspond denote markov chain boundary discount scale let compute non linear discount fine pair start coarse jensen imply ts loose connection length discount potentially context simulation suggest calculate proposition average previous subsection solution course value law terminal mm bottom font skip thick black every join style corner style style terminal solve coarse sp xshift sp row xshift xshift row point join p sp join sp join sp join branch sp join join join join join join sp join join sp join join sp join join loop join branch loop join join inner mm inner multiscale mdp mdp obtain several suggest flow fine update fine coarse update coarse coarse bottleneck collection independent iteration fine fine scale solution may update represent loop latter outer loop pass update instance bellman appear asynchronous iteration compression fine policy solving successive fine coarse compression iterate allow move fine therefore repeat transfer pt mdp policy coarse save fine bottleneck rest fine global policy set criterion meet state repeat average bottleneck criterion meet way recursive application particularly top bottom reach bottom compatible mdp path goal enforce coarse fine coarse update information update local readily handle successive describe localize fine bottleneck give comprise fine compressed absence specific reliable policy coarse action agent involve coarse cluster bottleneck vary depend provide additional action coarse mdp algorithm mdp convergence coarse bottleneck fine local cluster determination step think boundary cluster bottleneck interior explain solve require interior interior new policy updating bottleneck state use globally combination interior bottleneck averaging boundary execute convergence guarantee value modify monotonic condition note find policy iteration cluster bottleneck rapid hierarchy distinct piece efficient algorithm fewer primarily reason multiscale start coarse fine good otherwise iteration multiscale treatment offer since give policy constrain mix time cluster slow within rs context solve hierarchy coarse fine fall compatible range conjunction coarse realize precisely optimal fine local fine compression coarse mdp likely close coarse summarize cluster behind coarse get fine bottleneck cluster large reward outside may advantageous visible locally thus policy cover alternative cluster r denote geodesic bottleneck law original modify terminal state rr mdp truncate modification discount original mdp rr range find give rise r action scale bottleneck adjacent trivially cluster across within involve mdp fine mdp fix coarse cluster solver inside outside illustrative policy initial least problem boundary poisson problem x bound boundary seek solve system system capture label interior label fix interior state given solve unchanged policy bottleneck state update block towards determination cluster need another iteration completely converge tolerance cluster reach along value require bottleneck quickly function process value bottleneck may update asynchronous cluster update step comprise outline bottleneck iteration modify iteration variant analogously determination characterize need partitioning capture bottleneck interior state fix known assign system ideal virtue likely product furthermore connection likely solve likely asynchronous policy scale policy scale pass per bottleneck satisfy interior policy unique first value interior state fine may determination infinite challenge assumption modify asynchronous converge unique corresponding initial coarse algorithm coarse mdp zero everywhere fix shift asynchronous policy find discount operator direct reference transition define section recursion repeat substitution fix large ensure optimality convergence action bound reasoning policy length l satisfy determination partitioning compression analysis assume step general use approximate stationary iteration however eigenvector sparse solution cost far example compute randomize approximate compute find sub graph accelerate substantially compression make compression local space requirement assess complexity complicate negative transition iterative square guarantee generally reach newton et al competitive previously classic routine practice unconstraine linear system appear proposition negative complexity per cluster coarse transition probability coarse coarse solve constraint necessarily negative involve time complexity naive case iterative linear example reduce solve multiple side side system determine complexity reflect quantity element complexity solving solve mdps dynamic iteration iteration number solve entirely assume significant coarse bad solving compare bottleneck markov chain fast successive scale everywhere step cluster construction update policy interior bottleneck state assume since dominate collection solve multiscale determine complexity strongly dependent gain reasonable good multiscale structure ease scale equal scale scale say scale iteration require scale level multiscale assumption require relative relatively attention challenge substantial impact broadly problem second well policy refer depend type transfer several form systematically knowledge argue novel systematic mdps multiscale discuss hierarchy hope meta transition part appropriate former imagine problem sequence distinct identify part already solve global remain part solve conceptual distinction transfer policy value reflect global structure problem easily translate much applicable easy consider assess section compute occur coarse scale single element quickly compute candidate dynamic sense potential respective advantage reward globally transfer systematic identify transfer general proceed problem whether transfer potential operator step detail section give multiscale mdp compute hierarchy select cluster hierarchy transfer match pair algorithm globally locally match pair cluster proceed matched transfer already remove possibility correspondence section determine operator solve problem retain policy solve algorithm variant start respectively section respective transition discount cluster index quantitie truncation restriction context interior boundary question otherwise refer appropriate throughout information case special case within scale consist explain scale see single within transfer diverse restrict belong match address match situation correspondence focus attention transfer cluster cluster boundary receive regard policy bottleneck transition cluster bottleneck form involve decide keep entire coarse scale simultaneously correspondence close metric natural average wise embedding underlie state respective map embedding sign alignment problem match cluster denote weight subgraph e occur scale auto draw width minimum height width cm ap ap bp edge ap bp draw ap draw bp r ap edge bend node bp w match subset sub exactly database transfer possibility policy knowledge compute mapping either take denote subset resp j aspect transfer action action action policy action policy entry uniform previous result used summary transfer everywhere compute local coarse hierarchy instance derive policy function apply initial coarse transfer correspond either diffusion policy apply previous scale stop return suppose coarse scale operator compressed discount exposition entire scale moment correspondence development idea equation sub scale value j define system collect reward expectation apply back compute reward operator action reduce situation unlikely determination operator advantage subtle initially result compression guess compatible coarse determination knowledge eliminate transfer potential determine obey correct scale markov provide warm optimal policy pair match cluster detect transfer policy whether diffusion current policy use warm correspondence check support transfer give current uv function compare describe equation improvement relative computation transfer underlie take heuristic warm cluster boundary assess assume dynamic collect transition dynamic transfer reward transfer proper boundary policy coarse remain reward probability possibility problem operator current usual dynamic discount reward determine comparison quality use establish correspondence transfer graph state edge graph instance goal graph establish role play state example terminal state desirable able match terminal state terminal sense representation coordinate could seek abstract specific match similar role across several section computationally otherwise actually require matching solution match restrict attention correspondence detect poor choice transfer coarse scale collection compute involve build median apply match research several min cost give diffusion either locally cluster match illustrate dimensional problem base domain cart move cart balancing position finally carry sequence various setup compression reason dramatically per propose multiscale iteration small global several different boundary update iteration algorithm update summarize multiscale algorithm name interior boundary convergence boundary interior possibility either convergence iterate fall per make comparison fair iterate cluster local iteration cluster every local mdp boundary regardless interior construction transfer algorithm well suit whether initial question initial fine initial initially scale type iterate boundary information allow propagate throughout interior value modify interior solve coarse mdp compressed coarse iterate interior immediately solve coarse diffusion policy interior iteration otherwise propagate slow solution considerably initial coarse initial impose fine scale fine exist pool augment coarse scale ignore current fine action initial coarse increase action result coarse reward proceed coarse scale involve compression diffusion policy discard coarse policy amount diffusion cm range fine coarse zero mdp transition link state direct terminal mark correspondence transfer agent grey grey circle terminal action movement reversible otherwise agent obstacle fail reward state reward bottleneck detection compressed guess policy figure detect marked character adjacency compressed transition original successively mdp graph plot cluster successively compress determine cluster policy result coarse depict direct path state right right policy reward goal multiscale cluster though fine scenario fine section advantageous discuss transfer problem correspondence cluster source partition correspondence color gray connect cluster pair match exception orient detection safe thing matching pair general priori identify problem share orientation pre solve store database solution scale involve detect valid particularly match transfer candidate cluster coincide early visual intuition source map mapping procedure receive transfer give action action mapping action identical original transfer without modification figure place cluster corner grid transfer transfer enumeration state scan right create cluster impose optimal policy constant likely reason transfer alignment transfer relatively mix cluster various transfer comparison multiscale fine initial initial coarse mdp pool policy appear impose convention plot value give indicator progress plot description algorithm label transfer choose action various multiscale fine policy transfer scale vertical warm coarse solve mdp initial coarse compare curve clear coarse initial iterate cluster boundary trace ms trace update ms trace corresponding policy ms ms involve transfer multiscale multiscale figure correspond figure compare iteration difference start multiscale transfer multiscale able multiscale transfer curve iteration transfer reflect improvement due multiscale multiscale algorithms optimality ms non monotonicity multiscale give compressed multiscale fast policy iteration rather time reason iteration policy entirely fair multiscale section intel matlab release parallelization optimization call evident multiscale iteration ratio general identify cluster cut cart cart track balanced force cart cart location hold balanced must balanced cart force cart subsequently step additive deviation three state angle vertical along span end reach note reward task default move receive cart hold cart move otherwise able move balanced start balanced reach goal transfer task position simulation end region cart opposite carry balance transfer ability balance apply task multiscale achieve continuous control furthermore domain partition graph matching circle terminal see detail mdps simulation diffusion ignore episode hz input cluster net approximately pool state discrete mdp reach simulation separately cluster terminal boundary clearly discretize apply separately terminal default transfer plot show coordinate dark circle sample cluster define reward statistic claim state fine mdps boundary sample terminal near step boundary translation coarse geometry helpful approach different natural partitioning balancing without place default transfer state cluster mark gray circle interface bottleneck partition cluster compress uniform discount next fine fine across already partition match place fine matching assume across form correspondence coordinate map interval define coordinate across correspondence distance neighbor fall match coordinate fine policy default match match multiscale scale policy multiscale list text transfer fine policy policy experiment transfer choose force cart multiscale coarse coarse respect diffusion appear plot euclidean axis iteration optimal figure multiscale pi fine scale plot compare warm furthermore attribute slow policy improvement convergence still small multiscale figure evident error give improvement transfer visible compare policy converge slowly relative confirm near origin bottleneck fact converge suggest either fine initial datum contain initial policy pool simulation show give performance non bottleneck near gradient object room ball music switch action succeed marker object look music flip light switch order latter action must unless note modification period switch light stay state object currently domain illustrate compression pair task remain goal gain environment towards solve another build mdp simulate episode trial reach action small sample transition spectral stop potentially cut multiple subgraph assume fine task transfer coarse default sense compress policy optimal pair section light turn period switch marker alternatively flip default playing must music music place marker ball ball music turn switch begin marker random object flip light switch music goal look flip light switch action switch agent look music place look flip switch default transfer text visualization two appear nearly difficult mark ordinary terminal bottleneck colored cluster although embedding diffusion task cluster seven scale transfer default operator correspondence determine step canonical scale execute action map execute fine cluster match transfer fine multiscale list table match matlab confirm match match list default across algorithm list appear update non transfer coarse mdp policy natural correspondence error fine action compare multiscale canonical iteration vertical label error plot detail description algorithm without compare transfer warm start multiscale transfer transfer reasonably assume reason coarse iterate ms trace algorithm boundary update ms trace involve ms exhibit ms family since coarse value fine scale transfer setting conclude would expect initial coarse entirely potential good good absence ms involve although ms trace policy reach multiscale policy mention multiscale involve discussion figure ms without scale fine scale plot demonstrate coarse transfer transfer confirm fast fewer ms maximally comparison multiscale see policy problem look light marker turning differ task goal music play music look marker look ball light switch flip switch position music playing must music turn music place marker look switch flip light switch episode begin look random object random object default leave although pair lead compare previous scale match map two mark ordinary goal see default state vice versa direct match coarse scale assume hierarchy transfer possibility hope transfer portion fine scale policy deal music sequence music immediate goal confirm possibility color spectral case music provide illustrative purpose plot next section music music cluster music music intersection plot point magnitude next algorithm separately correspondence role problem assess blue trace label inside axis plot clearly expect everywhere follow music state suggest apply policy problematic goal either problem conclude transfer cluster absence transfer helpful state early correspondence relevant underlie map mapping generally priori change post transfer place marker policy state thus policy state default task policy match compression text compare plot correspond euclidean intermediate iteration true optimal trace pi transfer trace label pi correspond fine policy pi transfer pi plot except trace appear obtain fine transfer policy purely greedy policy condition compression pool diffusion specify collection pool respect value pool multiscale iterate coarse opt boundary several conclusion draw impact conjunction coarse fig policy condition robust policy multiscale fine even transfer multiscale compression fast emphasize true problem policy iteration suffer take compression figure impact initial coarse bottleneck figure optimality ignore improvement multiscale come improvement lead contact comprehensive comparison similarity several much literature multiscale principle approach strong may treat depth many ultimately require option generalize beyond coarse fine separate mdp notion multiscale impose improve plan reduce planning structure combine macro multiscale planning multiscale fundamental result transfer knowledge potential policy generality detection compression locally policy solve mdps analytically model estimate specific expense conceptual follow flat review address challenge hierarchy process essential literature divide similar complexity throughout biology abstraction coarse macro broadly extend primitive action framework reinforcement learn pre collection macro closely model formalism option problem hierarchy discovery employ option discuss option hierarchical flat abstraction consider policy specification option end termination element option important difference distinguish option option approach decision aspect substantial option framework macro action specifically objective mind option primarily plan macro exploration scheme macro action receive considerable attention yet coarse macro fundamentally take view couple macro planning action couple conditioning macro time fast mix coarse localize multiscale decomposition globally distant multiscale scale solve contrast option introduction additional necessarily add plan indeed macro sequence scale particular language option bottleneck terminate whenever markov termination coarse lead option always remain markov option direct terminate one option option problem branching avoid loop desire leave direction execute bottleneck direction action context able guide store transfer option query closely way goal expense ignore want differ decomposition able broad transfer enter picture option macro possibly option robot room constrain termination policy coarse scale carry scale bottleneck drastically partition time accelerate problem mdp problem solution policy policy scale arguably core option option state execute option scale planning may describe top problematic reason expert specify determine multiscale option accomplish hierarchy option option across scale lose define flat user avoid either burden burden repeatedly difficulty example embed markov termination could clear problem lack necessarily branching problem reason level contrast strongly multiscale impose action multiscale organization invoke coarse solution scale transition combine transition macro trajectory suffice option interest single abstraction transfer learn construction coarse transition discount advantageous length represent keep operator probability sub preserve mdp mdp trajectory rather constant concern distinction reward termination aggregate reward pre average possible serious start state path span coarse contain little fine definition multiscale sense keep track approximate analytically termination macro multiscale consistency policy layer specify recursive policy optimal overall hierarchy impose consider global optimum markov policy policy scale recent explore learn paper speed interesting learn partially observable learning macro emphasis macro open sequence coarse action loop state important literature mean interpretation must provided assumption provide paper automatically goal exploit locality create transfer primarily algorithm researcher discovery characterization top framework option approach overlap work literature automatic macro organize three aggregate state simulation bottleneck define state visit heuristic frequently visit state algorithm search policy trajectory occur approach intensive automatically create analyze relevant compact representation specialized maintain principle consistent intuitive notion bottleneck spirit definition appear although emphasize around graph identify online employ former advantage cut neighboring reach option separately reward similar consider successively explore without global option correspond graph cut incorrect learning option prescribe bottleneck state may base node define must high alternative technique give substantially graph mdp compressed third represent discovery reference coarse decompose piece solve guide accelerate abstraction domain reward salient outcome serve option learn policy layer abstraction manual specification event salient differ sub task learn tailor development nothing objective store method seek particularly query paradigm closely problem solve goal around previous consider constraint coarse bottleneck terminal terminal scale chain localize geometry localize function likely capture lead geometric tree level abstraction goal however isolate efficiently locally consistent several reward bottleneck goal principled way learn partition boundary may clear dependent scale finally mdps approach hierarchy share like type coarse deterministic decompose discuss solve maintain product coarse algorithm similar bad multiscale self mdps trading
p max sub band b pair level allow interference channel gain sub band spectrum represent realization constraint low want achieve goal controller know network require decentralized information formulation normal joint discuss function monotonically decrease consumption player achieve minimum utility ne state empty refer player profile player game reasonable mean link formulation nash equilibrium follow ne theoretical formulation modelling scenario player individual game follow equilibrium se due use k define efficient equilibrium player may effort minimize effort trial te characterize system scenario te implement state triplet reaction action content action new decide evaluate otherwise benchmark eq opt player increment observe e contrary stage step turn utility noisy opt notation pure nash pure nash equilibrium approach play player employ te nash maximize sum equilibrium play te converge action profile player te number player minimize employ k te corollary channel snr te te state study approximate allow expect converge ne ne se spend state player simplify define sec approximations restrictive player action satisfy optimal w r study te interested frequency ne ne player individually transition probability list description omit ne bounded euler proportional freedom nonetheless ne pd pd pd pd c shorter ne se fig se level achieve state theorem se bound ne thus increase converge speed validate second validate general channel power employ two two experiment summarize figure channel model prove precise formulation second converging analytical increasing bring possibility note convergence ne compose te fraction player curve employ player reach player satisfy power employ scenario inefficient configuration study decentralize able work receiver consumption knowledge bit feedback realization analytically expect simulation general channel work carry theorem proposition conjecture count minimization device operate signal interference introduce decentralized trial statistically meet exist local stable use framework form converge point correspond equilibrium nash similarly provide trial nash sharing receiver bandwidth divide several orthogonal subject mutual interference device service plus device design achieve stable operating operating achieve device power transmission communication centralize neither information current device manual inequality centralize game nash equilibrium interference water noting work
point box dash line kernel ascent find reasonably corner error rbf kernel maximum interest tb p local show hinge approximation compute clean report completeness black circle effectiveness attack mnist digit focus distinct vs vs nature handwritten digit provide attack digit mnist properly normalize pixel consider regularization validation respectively retain class digit tb plot class point attack plot display attack appearance attack reveal prototype appearance final segment segment become round increase show due nonetheless cause classification rise initial error attack iteration cause svm attack point multiple extended point randomly set size respectively steady attack effectiveness attack point variance toward security attack arguably surprisingly svm present also reveal assess carry function previous algorithm realization strategy implication future address method restriction interesting large solution simultaneous point attack sequential attack individually optimize attack simultaneous every optimize effect use attack selection heuristic allow improve regardless optimal attack significantly practical limitation label human user message ground although arbitrary message guarantee attack attack satisfy side work understand side attack incorporate problem desire handwritten digit world image many spam mapping smooth solve attack crp r scientific innovation acknowledge von financial carry solely necessarily attack machine attack motivation attack datum come natural behave adversary extent ability datum attack enables experimentally demonstrate ascent identify good maxima convex technique tool infer hide complicated adapt behavior help security problem behavior fact solution security task spam unfortunately generally adversarial adversary achieve goal adapt spam detector generally adversarial learn explicitly attack attack attack exist database often collect example attack family attack security machine assume know algorithm underlie learner real setting instead surrogate capability construct accuracy base solution svm technique depend smoothly program attack attack formulate optimization performance subject retain although surface reliably maxima surface gradient dot point attack latter strong disadvantage since practical ground impact indice contribution product gradient maintain introduce kkt q attack cause smooth restriction composition remain us solution eqs component rewrite use substitute objective attack gradient particular expression gradient common rbf dependence avoid substitute provided enable extension section present attack initialize class principle point margin adjust attack point may progress tb label attack attack final c incremental svm p crucially preserve classical search
act panel fig problem mind fig signal reconstruct reliably case domain normal illustrative package make domain signal segment periodic condition early one consequence overcome extend correlation leave unchanged reconstruction influence influence reconstruction rectangular pixel scale pixel deal scale bin appear formula fourier whose bin reconstruct bin rectangular spectrum choose reconstruction example exhibit reconstruction good agreement region far fig reconstruction relevance interpret manifold sphere parameter power spectrum angular spectrum often last case replace part effectively around resemble field version panel show gap structure infer fig wise fractional approximated region seen around observation expect neighboring infer infer normal wide use field power extend simultaneous reconstruction field develop apply field past field logarithmic power suit add formula implement associate prior stress derivation formula depend spectral expect satisfactory signal field reconstruct demonstrate reconstruction spherical pointed observational represent additive incomplete demonstrate information field matrix represent incomplete convolution make applicable mind reconstruction galaxy thank anonymous helpful perform medium correlation structure examine smoothness broken power shape logarithmic behave like tend toward double derivative change logarithmic value prevent prior sigma event break special case pure power arise motion function arise example transform space spectrum double scenario study remainder field statistic regard spatially calculate spectrum drop employ spectrum reconstruct reality therefore choose spectrum become support example triangular stationary spatially field transform power spectrum power double case correlate correlation keep finite add introduce signal case addition smoothness signal variation finite pixel sec spectral field potentially problematic discuss smoothness logarithmic derivative spurious variation true function latter spurious correlation feature present spectral smoothness prominent spectral line usage smoothness grid two exponent spectral exclude approximate derivative represent entry opt lin develop infer random affected normal fluctuation amplitude order use formalism free signal reconstruct gibbs bayes maximum posteriori power smoothness scenario reconstruction demonstrate validate degree reconstruct continuous field branch physic develop reconstruct logarithm arbitrary log normal suit physical positive order magnitude spatial correlation consist discrete often arise field cox field finance study often log field density universe simple matter galaxy per volume approximate well underlie motivation intensity come magnitude spatially log description natural bring observational uncertainty regard number apart deterministic relationship log normal accommodate observational setting field lead fouri transform field correlation description aid knowledge field region observation field homogeneous correlation describe spectrum wants simultaneously infer several technique formalism employed tackle infer log normal field gibbs spectrum equation noise field power uncertainty arise reconstruct spectra turn redundancy spectra sharp drop spectrum occur actually neighbor interact one prominent ad hoc smoothing follow idea show spectral smoothness introduce appropriate spectra feasibility gaussian field discuss applicability smoothness estimation spectrum formalism smoothness prior derive formula way easily additional demonstrate smoothness formalism formula log smoothness reconstruction case sec assume analyze signal field discrete continuous response instrumental instrumental response fourier contexts sec case linear arise incorporation signal restrict notational straightforwardly field statistic necessarily symbol quantity limit argument regard least field straightforwardly space realization call formula serve desire continuous reconstruction physical covariance priori problem reconstruct signal principle unknown overcome formalism energy formula signal formalism briefly sec homogeneity unknown harmonic fourier sphere mode harmonic dimensional space sphere angular stand signal spectrum symmetry formula q wiener see sec detail subsection regard unknown formalism reconstruction eqs iterate hoc part usefulness smoothing case signal sphere periodic spectrum form power realization field pixel homogeneous uncorrelated noise formalism realization fig reconstruction iterate eqs smooth fluctuation reconstruct scale reconstruct power chance reconstruct conjunction option load package graphic explanation terminal graphic macro ltb lt lt lt lt lt lt lt lt bp spectra dimensional black solid dash draw green power dot package color conjunction terminal option explanation use load package package graphic explanation terminal graphic macro ltb lt lt lt lt lt ltb lt lt lt lt r r solid show datum smoothing dot wiener show illustrate especially interested wiener filter power spectrum see residual reconstruction spectrum constrain point way property signal spectrum power exhibit fluctuation turn position rapidly power thus reciprocal length enforce formalism present incorporate spectral eqs posterior spectrum posteriori solution eq solution consider posterior define spectrum accordance spectral location value tune priori turn inverse jeffreys flat logarithmic always limit filter straightforwardly derive calculate signal datum spectrum negative logarithm hamiltonian definition power homogeneous isotropic hamiltonian make derive mean maximum posteriori power effectively delta procedure bayes note formalism spectrum rough hessian curvature regard complex formalism develop gamma dot horizontal line level power scheme top spectral green hoc power dot line spectrum dash terminal option either load color graphic terminal graphic ltb lt lt lt ltb lt lt lt lt lt bp power spectra signal top black solid line power spectra dash line realization draw reconstruct spectra smoothness power spectrum hessian dot conjunction option explanation load graphic graphic macro ltb lt lt lt lt lt lt lt lt lt bp r r correspond top panel signal correlation bottom show triangular signal realization solid reconstruction green sigma uncertainty area bottom plot different add realization calculate reconstructed fig plot dominate power scale dominate reconstruct spectrum increase dominate scale reconstruction wiener reconstruction slight excess residual reconstruction ad power spectrum window width ad hoc reconstruction outperform smoothness discuss signal triangular eqs power spectra serious practice figs reconstruction spectra respectively divide pixel power spectra reconstruct reconstruct scale power rapidly reconstruct stay flat triangular say spectrum drop zero see dash line stay reconstruct panel small represent reconstructed spectrum well thus accurate power magnitude signal dominate irrespective reconstruction perform even power spectra power priori case mild noise smooth neither priori significant feature problem reconstruct logarithm log simplicity interpret pointwise linearity posterior highly even signal add treat formalism free energy gaussian mean covariance calculate gaussian sense leibler obtain gibbs last definition covariance term approximate energy internal energy posterior hamiltonian temperature give around remove posteriori reader depth discussion temperature assume calculate expand logarithm appear expectation expansion ensure vanish simplification calculate gibbs confusion integral appear derivative respect yield right hand appear eq together last posterior self estimate correspond many uncorrelated response purely equation simplify somewhat eq sec reconstruction spectrum power maximize interesting employ formalism reconstruction eqs regard assumption viewpoint inclusion spectrum power accord derive package explanation load package graphic explanation graphic ltb lt lt lt lt lt lt lt lt lt lt lt lt lt bp r r r dimensional low realization draw panel field blue solid
generalized bregman divergence detailed maximization converge finite two variant divergence use divergence divergence different parametrization fix bregman partition select df intensity raw poor preliminary performing give bregman notice mean significant iterate lb lb lb lb denoise collection collection collection noisy image lt noisy lt two variant poisson initialize draw standard normal normalize euclidean norm euclidean rule constrain axis initialize many stop iterate algorithm long u add dictionary since coefficient whole remain information among instance average also investigate variant aggregate noisy poisson bilinear interpolation level significantly reduce computation patch computation deal count comparison bm method real image summarize result follow visual metric house bridge simulation fig find comparison variety way bit improve upon poisson multiscale low light interest tend transform classical pca section patch test generalization image cube comparison ease mae cluster patch suit count quality provide channel square intensity different homogeneity across spectral patch respect third dimension practice level legend legend I legend legend I bm legend I legend I bm I I legend I legend I legend I legend I bm I legend legend legend I bm I legend I I I legend I I legend legend I I legend I bm legend I legend I legend I legend legend I bm I legend I legend I legend I bm legend I legend I I I legend I bm I legend bm legend I legend lc c house bm bm bm bm bm bm c peak bm bm bm bm bm patch channel reconstruct adapt generalization denoise image find patch space logarithmic natural logarithmic might ask several see work alternative pca power else improvement theoretical question minima interesting reduce acknowledge award fa nsf award thank provide spectral anonymous propose straightforward hessian case origin l gradient computation use approach method numerical one possibly ill condition step inverse hessian inverting transform row introduce write hessian matrix u multiply lead follow usa electrical computer university nc universit france limited imaging sensor array limitation image vision inherent removal yield introduce denoise limited combine dictionary method employ adaptation poisson develop sparsity method help reveal conceptual base highly light regime broad detector certain imaging available limitation arise environment spectral produce cube location spatio count wide variety tool particularly challenge limited available intensity poisson algorithm challenge count designing bin create resolution spatial resolution contrast exhibit pixel generally whether detector conventional fail transform variance transform count suffer demonstrate advance dimensional modeling sparse poisson resolution combine poisson principal poisson pca sparse intensity non nature state art particularly principal introduction denoise approach extensive combine pca approach white extension image directly use data fidelity singular direct suffer recall relevant property exponential iteratively computational conclude acquisition device poisson mean intensity accurately represent concatenation patch combination one interpretation self describe setting denote overlap patch similarly intensity patch many method represent collection space spirit pca framework pca aim represent row element element wise uv different similar assume allow issue relate goal assume restrict act representation family though use poisson gaussian idea analogous comparison purpose space equip dominate measurable exponential r parametrize call independent know parameter gaussian possibly poisson distribute identically wise eq counting parametrization usually parametrize proximity analysis exponential kullback leibl divergence write function define gaussian unit zero divergence bregman bregman divergence non square size observe let patch patch weight dictionary element patch approximate th patch factorization exponential criterion bregman divergence amount matrix minimizer intensity pca remainder solve intensity estimate dictionary solution atom ensure keep follow jointly fix newton algebra entry represent diagonal appendix update propose introduce transform precise definition update appendix updating rule row tv
policy bandit agent ask produce reward great immediate exploitation action immediate great paper bandit reward thompson allocation bandit thompson attention optimality achieve give choose observe receive possibly choose past observation minimize expectation arm prove e must suboptimal leibler divergence reward policy satisfy equality asymptotically efficient ucb et al efficient good use confidence past reward optimistic arm choose action high suboptimal imply contrary kl ucb confidence asymptotically confidence successful thompson policy idea model yet fundamentally frequentist regret thompson choose action thompson reward recent investigate optimistic version propose provide theoretical guarantee extensive scope generalize without consequently major scale like meanwhile strategy policy optimistic sample analysis ucb thompson corresponding quantile bind thompson draw precisely thompson asymptotically experiment thompson optimal policy ucb present assume thompson bernoulli uniform arm arm explicitly cdf resp binomial link beta q binomial trick several stage resp ucb index quantile distribution computation inspire analysis index policy policy arm round index reward aim draw suboptimal arm arm suboptimal optimistic k ucb argument ucb also optimistic ucb directly apply analyse thompson since optimistic confidence close convention regret analysis sample proof explore sampling exist tell distribution concentrate inverting convexity last inequality lemma draw let occurrence optimal arm play suboptimal arm extract range play play concentrated around fact say action e lf interval eq fact hence condition also sample interval proof fall mean adapt j suboptimal arm arm last comes draw small hold suboptimal arm depend conclude inductive hypothesis complete induction show event begin arm interval deal range arm arm bind size get f induction complete summing hypothesis I link mention give exact lead bs bs bs j bs therefore median binomial inspire rearrange write affine function negative whenever clearly constant conclude thompson bernoulli thompson ucb reward different gap thompson ucb well ucb small figure display cumulative trial ucb ucb close thompson bernoulli incorporate reward index sometimes thompson implement kl ucb costly produce posterior regret various dark show central asymptotic bandit policy simulation show thompson outperform policy idea control rather valuable control
scheme despite simplicity mark application rational tree open empirically develop well non root meta reasoning principle helpful current complex address future successful go pure extend specific select current move really fair dominate nevertheless adapt aware acknowledgment science foundation computer sciences center production management theorem david computer university ac state games process ucb multi mab cumulative differ actual move rather oppose regret arm bandit ucb stage optimize although theory apply trivial theory propose aware achieve outperform formula numerous although past job exploration exploitation correct bandit ucb principle issue game nature action exploitation action close pure exploration selection problem cumulative small cumulative regret begin ucb propose scheme sampling principle suggest another paper generate scheme improve monte search initially suggest policy decision process mdp mdp reward explore termination condition cutoff reward arm associate action reward set literature bandit reward ucb scheme maximize time far search describe algorithm mdp adversarial get reward policy problem reward arm great minimize simple upper exponentially respective ucb polynomially theorem yield regret sampling control principle bound description rational maintain current root gain cost factor ideally select intractable simplify action basic non concave utility algorithm frequently multiple many case insufficient change current get rational define goal achieve scheme amenable analysis concept examine polynomially upper bound bound uniform confirm greedy scheme q substitute ucb aim finding move outcome mdps adversarial choice move optimize root deep tree move choice internal optimization far mark suggest combine rest select ucb stage select step line select statistic update root realization sr cr employ node depth node node next next node reward expect stage less tuning exploration ucb need step rest achieve choose maximize infeasible current incur change equation gain reward minus current exponential action sample one denominator aware maximum stopping leaving appear amenable analysis section demonstrate low result empirically verify armed bandit tree experiment sr cr scheme comparison bandit return search compare reward sampling vs average ucb dominate ucb large b vs perform sampling scheme child anti symmetric cause uniform incorrectly give tree exploration reward exhibit dominate everywhere advantage greedy ucb tree arm
explain automatically attribute complicated mechanism provide explanation empirically cutoff explain alone quantification within population increase replication series widely mechanism increase respectively regularize incomplete gamma last let acknowledgement office award fellowship ep fellowship award ep uk award fp integration grant european union theorem derive product enables realize network large approximation form enable quantify across population achievable expect need attribute complicated mechanism considerable study neighbor nod connection statistical construction exhibit expectation quantify variation remain important realize effect observe count select node sequentially valid edge loop lose requirement replace edge bernoulli trial success probability nonnegative normalization degree th equal p connect assign edge assign statistical insight recognize degree sample condition near later eq neither loop ij poisson fully graph study enyi reduce determine describe variation introduction parameterization understanding parameterization range weight increase regime sample property sequence network proposition degree edge trial ij proposition parameter strong realize degree expectation behaves scale thus nd th decay directly distinct maximum variance dispersion whenever poisson variate dispersion aggregate go dispersion cauchy schwarz quantify difference eq dispersion must theorem establishe obtain element accordance edge trial comprise degree exchangeable conditionally degree discuss natural treat deterministic decaying change relate inverse large choice characteristic mean consider node binomial counterpart indeed iid yield provide moment degree degree express dd dispersion compare network dispersion depend dispersion match unity dispersion conditional manner illustrate notion variability variability ignore network binomial heterogeneous either source ignore correlation degree arise specify special case equivalent classical homogeneous study lead binomial simplify early direction entire map analysis quantify laplace theorem act integral regularize beta concentration toward step transition admit lebesgue weakly whenever later series entire significant quantify incomplete variate imply law standard normal verify apply tail hoeffding quickly whenever choose relate standard describe survival dash transition region analogy survival observe fig censoring follow near examine simplify covariance counterpart natural begin decay decay power law relative magnitude grow small ni treat regime realize network polynomial decay variability distribution significant literature characterization proportion degree sequence law constant value converge individual drive growth growth argument euler law variate order chebyshev sufficient central behave variate grow degree nd become describe degree decay main specify relax form overall polynomial deviation control hold introduce redundant variation overall decay constraint via variability essence retain permit deduce total convergence law suitably whenever nd power law whenever I apply establishe claim ii numerator balance ii converge prove expansion gives constrain grow consequently element value er multivariate outside power law set I possibility section vary accordance estimator exploratory refine suitable provide toward understanding goodness fit network important open challenge highlight realization wish pareto restrict amount correspondence nn degree form pareto pareto uniform multiply k dt k expand f reflect n incomplete restrict degree within well unity accordance visible behave region dominate decay censor law panel average f b panel show taylor exponential cutoff effect investigate censor hand fig empirical rapid censoring power decay hand panel scale match censor cutoff motivate explicitly effect exponential cutoff law frequency infer n k due purely derive valid must generally enough admit second binomial dense expect degree linearly eventually edge essentially determine distribution quantifie later extend twice differentiable random strictly attain taylor expansion carefully choose multiply side recognize truncate establish hypothesis may write form taylor remainder substituting mean n hence q part identity implication follow concentration lemma expansion essence kf nn tail choice reveal via fine survival come taylor formalize split integral mixed binomial reveal essence multiplicative degree recall discussion mechanism impose beyond decay accordance lemma network substantial heterogeneity heterogeneity explore theorem variability marginal variability degree generate regime variability well understand smooth reverse occur limit recover apparent behavior achieve behavior total network law study regime studying realize concentrated heterogeneous degree variability require direct control turn sparsity smoothly network notion scale affine degree corollary parameterized constant mean behave property regime shift scale map admit limit mean converge grow recover set importantly achieve dispersion product characterize dispersion network instead estimate realization mechanism size expression natural rescale enable set rescale degree grow overall magnitude grow collective heterogeneity depend refine simplify complementary rescaling extend yield simplify expression every eq fixing thus recover leave effective normalize exceed toward begin shift regime simplification follow set corollary restrict distribution variable assumption likewise n first lagrange yield multiplicative understanding n unity latter formulation censor prominent corollary smooth beta gamma especially unity corollary highlight statement incomplete censor degree characterize binomial survival extreme serve heterogeneity limit must consider previous per q expand multiplicative exist eq
wind linearize equation become cart cost first far state reflect seem simple specialized capture application could robot receive human plan action try plan issue future wind control use issue commonly minimize square historical result forecast predictive control room improvement observe particular beneficial sometimes empirical minimization compute minimize historical historical certain technical infinite available future work main contribution fit regression direct large improvement controller aforementioned relation direct series regression forecasting control build present control static decision work become unstable whereas static efficient employ new devise slide cross validation evolve minimize period definite gx u simplify problem control horizon time subscript action observe horizon formulate time new solve produce process order expect future simplify objective become dynamic govern involve linear quadratic generate control forecast idea extend generate forecast becomes arbitrarily grow nonlinear drive scalar useful predict form noise conditional straightforward forecast event produce forecast policy generally discuss coefficient l note sum begin compute linear square coefficient observe time forecast natural forecasting model capture generate assume principle whereby empirical problem compute minimize historical action start easy ergodic historical optimize appear generally however study experience initialize detailed implementation appendix challenging parameter mainly evaluate gradient require linearization typically include reason sum practical significantly past decay influence future nonlinear one dominant motivate refer coefficient typical iteratively forecast fix allow set validation ht algorithms kind hz employ around derive physical therein keep minimize parameterized hence period ta b experiment policy accordingly ensemble series follow sample ar five coefficient tend five variation sample generative model helpful forecasting objective differ generative another choose control naive dominate select forecast five feature approximate extract period forecast perfectly generative context recognize sort misspecification practical sequence zero mc give mc clearly generate aforementioned five cost form generate focus presentation excess subtract influence cost choice control plot excess qualitatively rather small suffer prevent l outperform interesting gain substantial compare quantity interpretation cart system reach cart angle certain threshold twice root simulation period initialize state reach generate policy consideration mc ls failure summarize result much close mc l ht direct coefficient predictive version namely forecast methodology transform problem linearize slide cross validation technique applicability paper forecasting hope observation work involve broad consider paper multidimensional system involve relate idea perspective algorithm controller model control transition aside autoregressive conceptual fact ls analogous study data method hand discriminative provide combine generalize offer approach broadly combine method adopt weight square put emphasis critical component series develop forecast use decision line misspecification method well gauss newton carry backtrack search determine take iterate whenever hessian add spirit
standard ccccc low p low residual technique spectrum spectrum respectively root function sd innovation standard subsection calculation produce unit provide red empirical implement interval whether root outside incorporated root ar polynomials lie disk stationarity whether class standardized alpha residual decay function behavior typical asymptotic label plot formula converge make asymptotic normal measure sample exercise mobile forecasting forecast three function alternative two sided contrast estimator begin summarize difference vs mle mle three consider value significance mle value increment horizon parameter model tend zero tend increase prediction test execute package addition provide base evaluate evaluate property capability diagnostic forecasting would improved root long series grant em em depth autoregressive move cl mm de cat de mat cl mm exhibit persistence lead development account decay autoregressive integrated move package implement contain ability impulse life key memory variance impulse package long introduce key event package example package memory produce estimation density noise mle describe time series model forecast package offer forecast estimate package aforementione unfortunately computational implementation severe calculate variance forecast impulse response discuss develop function fitting via asymptotic aim paper package exist development package finding devote parameter impulse response estimation provide assess apart describe package also illustrate life memory beyond walk root autoregressive account long general process define autoregressive operator root expansion mean innovation asymptotic usual gamma function process unit solution stationary singular decomposition unique purely express measure purely absolutely lebesgue spectral write polynomial mle calculation likelihood normally give associated parameter likelihood calculate two indicate version minimization minimization newton type condition distribute da estimator assess ar plot include figure behavior consequently tend large function especially near inverse tend near discuss covariance useful tool make inference approximate likelihood calculation alternative computation hessian base explicit obtain spectral partial derivative sequence kf true consider density identically covariance spectral observe yield I j p analogously impulse commonly find outside disk invertible write expansion coefficient relationship expansion dy give assume illustrate general consider parameterization function term process autocorrelation dd root one cd correct absence ar formula reduce eq behavior mean asymptotic exact autocorrelation approach prediction square rmse forecasting presents rmse statistically difference paradigm prediction carry diagnostic base observation nan difference prediction test horizon windows estimator obtain simple autocorrelation consistent function life http area display persistence analyze illustrated tree series specifically h usage package fit covariance hessian mle study function r na option implement allow
hand side random q numerator eq follow current status censor mle smoothed status censor theory expect reasonable plug locally estimator purely mle actually hold mle indicate large mle mle bivariate censor bivariate status bivariate censor independent maximize formulation first second observation look rectangle tucker df generators mle maximal mle compute mass study status censor mle discussion mle datum put treat minimax low estimation consistency put determine mass computing somewhat energy spend intersection mle corner corner really effort mass bottleneck censor determination intersection mle put phenomenon simulation place prohibitive want behavior computer ultimately determine insight mle section attain determine smoothed asymptotically observation study accordance extend censor end conclude rate rate basically current status monotone diagram observation interval derivative simple model sometimes represent variable leave deterministic derivative derivative bivariate distribution diagram seem process incorporate duality condition optimization analogously status duality square hide point mass deal denote process coincide necessarily status must function fact must dimensional status complicate deal equality mle bivariate current status status surprise preserve get estimator gets argue attain construct purely converge locally remainder appendix define twice suppose normal point rectangular way necessarily negative turn nevertheless build leave get correspond seem seem research picture plug nonnegative support treat boundary representation conjecture order unless happen bias compare behavior distribution r indicator ht value mle mle plug mle sample letting coordinate coordinate randomly uniform permutation integer mle place mass put picture different plug put inspection intersection put similar rather mle reduce ht asymptotic theorem value mle r mle plug bias time standard deviation sample suggest vanish detect present theory confirm get active still usually partial vanish uniform achieve uniform distribution bandwidth plug correction bias bandwidth actually attain mle extent suggest censor censoring model eq define similarly maximize q bivariate probably theory measure introduce belong unbounded finite vector consist zero otherwise nonnegative optimization treatment analogous aspect experiment quantum frequency belong observation frequency change bound give upper put mle put apply package c mass facilitate literature preliminary follow mle convention mass mle right corner close correspondence apart package bivariate censor picture mle picture figure mean occurs mac see picture df mle ridge coordinate level I
equivalent imply rr preference specify agent alternative concave concavity log concavity hard normal log make location normal specifically p concavity p technique maxima concave follow necessary maxima bound alternative subset least sufficient maxima family suppose hold unbounded alternative increase simultaneously alternative suppose alternative edge path conversely follow edge alternative follow path log concavity bound estimate correspond unique maxima strict contradiction maxima contradict never reveal mle belong ef motivate domain determine iteratively proceed iteration previous compute conditional latent maximize log simplified w bx e aware analytical approximation sampling sampler approximate gibb choose accord tail gibbs rao jx ms tx tx l k x rao reduce tx j x k j e step x tx j bx I exact mc uniform mc e however application long remain large safe gibbs sampling sub ratio factor suitable ratio similar gibbs number rao make arbitrarily mc datum namely set preference order simulate use utility equal method agent iteration infer order middle well increase get case axis middle panel mn access obtain full interval method public collect partial candidate non rank rank alternative experiment alternative entire truth since find adopt compare data sub variance subset case fair great public kind mc parameter synthetic variance guarantee variance part parametrization variance rather variance fit model criterion log likelihood predictive aic bic value whereas negative model fit predictive likelihood likelihood sense data compute randomly choose vote standard deviation summarize htp c aic aic significant compare fitness fit well log outperform mc linearly agent intel ghz gibbs acknowledgment grant compute anonymous suggestion property sketch alternative draw alternative rank alternative case receive attention general enable mc concave global maxima world simulate scalability capability metric aggregation social part aggregated g crowd rank ordinal consist rank single select formulate true rank alternative report view noisy regard preference alternative agree rank otherwise maximize mle pairwise comparison agent preference pairwise preference winner random economic utility alternative denote naturally c illustrate systematic economic seminal economic alternative adopt cyclic preference correspond strength preference parameter alternative l term shape likelihood function analytical extensively apply recently em recently propagation p variant social p model use closely model quite characterize strict preference inferential particular preference set order agent voting preference profile win ranking tie win include singleton mle social unobserved truth ranking agent independently mle maximize preference report vote estimate ground truth determine win preference ground truth win rank mle return parameterization belong ef ef format eq e x x
assignment link determine entry q respect flexibility represent type structure link affinity core effect ccc connect membership topic belong close membership modeling flexibility feature section task nod network membership discover member link member member member likely affinity member member opposite behavior link likely non core affinity c membership model rich belong predict derive evaluate data friend citation document play increasingly modern machine provide friend scientific focus link decompose kronecker product structure though powerful network ignore node text document represent connection along complementary use simultaneously exploit network link node powerful link traditional provide predictive node predict node profile new link predict link keyword predict connection reach belong multiple group membership art assign node latent belong membership share group membership series truly belong multiple contrast membership necessarily membership link link link combination membership group entry capture link affinity link affinity understanding formalize illustrate value membership beta parameterize multiple membership node affect limit
evolve describe characteristic function functional function function instance make robust easy interpret extract raw functional extract give approach maximal inversion regression drawback extract easy link original feature feature original approximate piecewise functional solution us interval set term functional feature extraction consist feature context consist original unconstrained functional pca feature interpret support analyse elsewhere basis wavelet unsupervised basis I discard different version select spline orthonormal compactly dependency localize easy drawback wavelet compactly consequence function interval spline suffer length lead representation large function without among il norm support get approximated index partition satisfie order belong also minimize square quality find agglomerative dynamic lead propose bellman single iteratively quality good class initialization iterate minimize index keep backtrack class piecewise prototype loop use compute search precisely interested store array accord recursively j compute recursively cost storage scheme use quality measure rather problem restriction piecewise prevent continuity condition search continuous spline expect piecewise one measure evaluate need exceed reach particular leave quite noisy rely overfitte basis leave give constant segment leave maximal variant basis readily accord l leverage basis reasonable could solution likely select behavior htbp illustrate dataset mid range cm spectra along spectral consist support example solution unable pick detail peak spectra approach
challenge receive attention field include recently solve equation previous recover compare major approximation consider noiseless case recover exactly noiseless sharp omp real hence complex complex description model reformulate reconstruct organize classical omp solve omp geometric parametric set conclusion section obtain omp iteratively atom greedy fashion current add omp assume atom normalize e nonzero pick atom atom otherwise slight notation use atom submatrix omp detail e residual atom counter span element stop achieved go otherwise go reach iterative ls return algorithm key stop rule noise noiseless natural stop achieve construct maintain expand atom atom choose maximally approximate construct fall algorithm terminate require two effective omp non l omp select atom step among omp note gain insight omp main behind proof provide technical light properly noiseless meanwhile omp noise case well gaussian extend meanwhile propose restrict isometry rip omp signal beyond pose success omp equation nm guarantee generality solution entry begin decrease computed error eq utilize step first thus must h h substitute hand right side exploit mutual definition hand derivation sparsity imply decomposition atom I omp residual onto atom atom current operator span element residual clear omp variable necessary imply lemma u select denote omp recover correct atom coefficient satisfy lemma due complex b noise gaussian distribution real part part directly vector entry bound complex theorem suppose meanwhile suppose k I rule obvious easy however imply suppose support give cardinality besides success mainly vector dictionary fundamental unchanged application complex widely literature description image ideal center characterize center light correspond stochastic measurement measurement obviously problem carlo via application subsection range ghz ghz band start ghz five scatter locate contaminate noiseless respectively mutual incoherence situation atom small determine support omp far calculate present noiseless finally result recovery r sparse noiseless remark relative understand fig noiseless scatter recover sparse dictionary incoherence among support omp meanwhile also matter inter atom incoherence interval support mutual incoherence obviously incoherence condition fig paper omp approximation dictionary present incoherence atom interference provide sufficient omp theory omp importantly new complete omp make omp confirm omp consider yet see fig fact adjacent case second scatter
area high anomaly simulation look anomalous phenomena anomaly iteratively refine highlight regression value functional sample nonparametric edge fail geometry edge sample nearby embed nearby place first randomly learn skewness set display predict enyi rmse covariance drawing element obtain matrix rotation rotation outline repeat lebesgue weakly divide indeed accordance consistent grow whenever case estimator special rearrange factor ensure lemma conditionally limit th power vanish thank empirical law couple serious gap lebesgue imply handle strong concept need uniform lebesgue put density slight cm corollary algorithm treat individual object machine operate suggest treat employ define pair set projection cone semi enable machine anomaly dimensional embed numerical simulate extend classical area economics bioinformatic consist especially wish use quantity quite algorithm consider function density observe directly finite treat approach space represent point application representation traditional machine technique problematic wish survey certain group difficult similarly collection collection develop classification map set svm define machine anomaly detection focus individual joint goal detect finally develop handle problem distribution map generalize generalize machine estimator nonparametric consistent publicly generalization kernel exist kind drawback product family fit inner product base fisher family rare belong method inner product contrast provably certain nuisance divergence intractable optimization space reproduce sample sample convex programming distance appropriate functional computation demand fix small able kernel define affinity evaluation embed kernel hence generalize operate either set deviation show kernel svms closely euclidean appropriate speed empirical vision domain image problem collection local method descriptor represent image composite kernel pyramid feature histogram use intersection histogram curse bin problem near distance bayes integrate bag result improvement comparison class principle define possibly object functional improve limited available approach expansion b inference recent use scalar functional functional value functional vector although develop functional classification directly input density sample need product sufficient general machine generalize consistently divergence include enyi hellinger estimator problem however inner classifier divergence estimator investigate inner divergence consistently extend classification image show anomaly formally investigate unsupervised let supervised seek ideally review discuss perform multiclass vs classifier vote final experiment approach inner product map form eq gram matrix tool quadratic available idea class svm anomalous consider sample around elsewhere supervise set scalar functional mutual point reliable intensive alternatively size use problem look transform hilbert perform nonlinear preserve dimensional generalize distribution map local geometry distance kp p j kp j local characterize reconstruction vector reconstruct locally dimensional nonlinear algorithm include method require distribution estimate positive semi definite gaussian try r divergence divergence limit enyi divergence satisfy lead definite gram show ccc distance hellinger enyi set tool estimation enyi divergence euclidean neighbor th nearest neighbor provably consistent condition dd ball estimator thus plug formulae consistent estimator therefore increase however definite r enyi divergence instead euclidean gaussian asymmetric therefore gram project frobenius discard similar eigenvalue amount flip negative instead view noisy approximation jointly support find small discard separate euclidean associate analog tucker provide yield well well projection example kernel theorem might valid gram crucial solver approach predict kernel although computationally experiment see inductive approach label p calculate discard calculate predict applicability experiment mention publicly consist class point handwritten digit error algorithm achieve raw classification even distance easily dataset however euclidean distance become become necessary normalize intensity sample image take grid penalty degree try times kernel pixel perform use divergence near neighbor accuracy evaluation rate enyi indistinguishable hellinger polynomial slightly bad accuracy raw mean ht inductive raw raw enyi enyi enyi hellinger visually structure image multidimensional scaling digit preserve letter raw euclidean uninformative help explain turn kernel particular whole object extend set word might visual correspond dependency visual ignore unique give vocabulary patch feature category area extract concept category vocabulary patch represent category within image histogram chi square histogram discount root transformation extract patch stop bag extraction extract dense technique descriptor grid toward scale range extraction feature feature location produce use normalize unit sift evaluated image tuning applicable perform project enyi hellinger estimator extremely poor test enyi approximate twice kernel mixture fit gaussians monte previously chi quantization consider pyramid vocabulary pyramid kernel variant recommend computer vision group I two let wise match kx tune way pair get computation good match dominate see basic performance individual image angle angle image view color bin fix pixel invariance reduce preserve therefore test cross result show enyi well properly tune std std pair perform worse bad may hard divergence precisely show enyi near divergence performance r category dataset horizontal show scene method nonparametric kernel method dataset total show goal classify color location patch mean local allow location image
although direct reality assume case optimal access give access cope polynomial time fashion calculate dynamic simulation show good behavior link setup uniform sampling among case difference product find bound behavior set context thm thm remark examine various active sampling analyze normal distribution expectation setup address estimation although solution functional efficient structured functional induce yet obtain probe delay along efficiently delay accurately follow eq network assume move link possible trace trace reverse trace trace draw frequently trace since probe equivalently surprisingly trace result survey work experiment section formulate discuss section structure walks walk setup recent bandit chapter suggestion research similar pose study overview grow solution sdp insufficient complexity depend size recognize functional particular field concern resemble multiple focused estimating parameter al include weighted square similarity several difference entirely formulation focus efficient al exploration restrict although great interest assess address sampling issue effort solve specific present concern network deal overview al field find suggest parameter field effort find fine rather probe minimize another examine possible multiplication add subtract possibly infinite estimation logical define complex entirely clear new throughout situation occur since multivariate minimum diag diag take inverse information matrix mse like option believe appropriate option discrete time setup ease I addition shall restrict denote simplex formulated find optimal combination achieve formulate different sample setup per functional abuse notation row fit convex sdp programming minor variation restrict budget formulation affect solve nevertheless tend contain inner motivation describe sampling part start case subset variable natural functional functional one solution functional xx nn mb km diagonal entry tm nc diag generalize yield well kk solution show small suggest k combination least optimal functional show theorem entry unit functional dag direct acyclic node order normally estimate scheme delay delay actual path characteristic consider dag sample w exhibit instead functional middle far many path go counter suggest solution semi program impractical grid path large concern example grid distance subtract key problem although suggestion deal mse solver cope relaxed efficiently programming simulation show quite relaxed introduce product leave specifically denote leave easy recall distribution source path give product path optimal produce effort drastically order efficient solution calculate without calculate theorem use programming first time path sequentially finish node follow equation likewise start employ trajectory pass distinct leave probability minimize sdp specific entire even compute globally
separable constraint wolfe context structural svms subgradient wolfe convergence practitioner size need proper criterion size pass small pass review svms goal structured tag map information pair dependent slack surrogate loss th slack rescale unfortunately combinatorial nature one define structure replace structured hinge input efficiently many paper subproblem call non unconstrained formulation maximization subgradient respect maximizer decode subproblem write dual associated column tucker optimality ta cf finally b objective frank wolfe algorithm optimize thus wide iteration corner find picture next size start corner previously representation suitable case exponential second linearization yet compute linearization feasible special explain readily current convergence importantly choose theoretically sound stop know duality convergence hold appendix affine measure yield see formal apply dual structural due section frank wolfe main subproblem frank wolfe augment decode dual since potentially iterate frank sparse previously visit need maintain previously solution augment subproblem keep size use kernel maintain maintain structural primal obtain frank corner augment subproblem start linearize duality lagrangian problem gap compute maintain duality gap stop simply restrict give algorithms difference feature wolfe appendix obtain svm duality oracle since prove imply svm primal accuracy surprisingly frank wolfe equivalent primal frank choice search equivalence notice step subgradient wolfe batch step apply seem wolfe subgradient quadratic identity cut frank wolfe augment decode dual towards corner frank wolfe cutting optimize task program exactly visit result frank inclusion plane simplify ki disadvantage frank pass call oracle generalization frank wolfe maintains appeal wolfe algorithm block idea method affect method write pick uniformly block frank wolfe interpret nesterov svms need subproblem k k theorem obtain duality appendix partial assumption appendix compute structural svm classical wolfe curvature algorithm either predefine f h expectation block duality k apply frank structural dual maintaining see observe significantly vector analogous wolfe formalize svm bad new block specific search theorem overall rate approximate structural duality gap single oracle call require constant number duality gap cut plane subgradient frank allow compute iteration pass duality gap terminate unlike cut call use interestingly hold minimizer subproblem minimizer give accuracy duality gap theorem generalization improve applicability scale decode decode possible kernel maintain combination cut support xlabel effective ylabel line legend legend pos north east index header col comma product ls style thick mark triangle mark header col sep lambda product thick mark repeat option index header col comma lambda opt txt header sep comma lambda product txt densely thick mark option solid col comma lambda product ls header col comma datum dataset lambda confidence color solid repeat index header col sep comma include txt thick repeat index comma include lambda txt index header true comma lambda txt densely dot thick mark repeat header sep lambda table col comma color gray solid thick header sep comma txt densely style mark repeat mark option index header col sep comma lambda densely style mark repeat mark option solid header false comma txt xlabel pass ylabel area line legend legend pos north table index header comma lambda l txt color solid thick mark repeat header col sep comma txt solid style thick triangle repeat header col comma opt txt index col sep comma datum product l txt densely dot style thick square repeat mark option x index header true col sep comma dataset lambda l txt header col sep comma dataset lambda txt color style thick mark comma lambda style mark index col sep comma dataset lambda header col sep comma include data dataset lambda densely dot style mark repeat col comma lambda header col comma include lambda txt mark repeat header lambda color densely dash option table sep comma lambda txt densely style mark square mark mark index header col comma include txt xlabel pass ylabel false normal legend north sep include l confidence txt solid style mark triangle header col comma lambda product style mark mark repeat option solid header col sep data lambda opt txt index index col sep comma lambda product l txt densely mark repeat mark option header true sep include dataset lambda product l txt header col sep comma lambda txt style mark mark header sep comma include dataset lambda color header comma data lambda txt index col sep comma lambda txt densely dot style thick mark index index header sep comma header col comma lambda txt mark col comma lambda densely dash style mark repeat option table header col comma densely style thick mark repeat mark header comma include dataset txt xlabel pass ylabel false style log legend legend pos south true col sep comma include lambda color solid index index header col comma include l txt mark mark header col comma dataset lambda product opt header col sep lambda product txt densely dot thick square mark table header col sep comma product l txt table index header txt color solid style thick repeat index header col comma lambda color header sep comma txt header true col comma lambda txt densely style repeat index header col lambda index index header true col comma lambda color thick mark repeat index header col sep comma lambda txt densely style mark mark repeat option false sep comma include lambda txt color densely thick option header sep comma lambda txt xlabel pass ylabel legend legend pos col comma lambda l thick header comma lambda l txt solid style triangle mark option header sep comma dataset lambda opt txt table header col comma include lambda l densely mark mark header col sep comma lambda header col lambda txt color green thick mark repeat header col comma lambda txt color densely dot thick repeat header true comma lambda txt index index col include lambda densely style index header sep comma header col comma lambda txt color gray solid thick mark repeat header comma lambda densely mark mark x sep comma include lambda densely repeat option header xlabel pass ylabel primal e area legend west index header col sep comma lambda l blue thick mark triangle mark repeat header sep comma lambda l txt thick triangle mark option header col comma product ls opt txt header comma txt densely square option solid header col comma data ls index index header true lambda confidence color green thick repeat index true sep comma lambda txt mark mark repeat table header col comma dataset match lambda txt header sep lambda txt color densely style mark mark repeat header col comma lambda txt header col comma lambda color mark repeat header col comma match lambda txt style mark option index header col sep matching lambda l densely thick mark option solid table header false comma match lambda txt achieved randomized solver parameter solver batch objective setting pass result less e though surprisingly compare task frank exist labeling augment decode viterbi third sentence different structure bipartite marginal label decode flow frank wolfe option stochastic subgradient also kk iterate call recently converge instead analogously iterate method efficiently also provable convergence different several criterion dominate superiority especially practice term average sometimes slightly despite amongst investigation optimization objective reach specific accuracy measure gap regularization minor ignore cm cccc saddle gap bregman yes primal primal slack duality subgradient yes primal frank primal dual duality coordinate svms sequential structural svms output space estimate method oracle factor parameterization optimize obtain expectation degenerate wolfe actually dual exact accomplish search appear descent applicable loss include svm despite motivated perspective option exact direction provide rate method implement method summarize reach svms regret effect cut plane approximate unclear cutting plane frank wolfe oracle appropriately bound propose randomize frank block potentially iteration duality wolfe structural svm online issue address need optimal closed cost pass frank wolfe duality gap experiment structural svm realistic setting pass possible exact oracle guarantees computable gap available maximization oracle although expect frank wolfe helpful national ms partly support ms fellowship constant appendix frank structural svm provide appendix self frank wolfe main linearization duality interpret additional experimental approximation correspond affine form upper lipschitz diameter affine ambient frank wolfe lipschitz invariant wolfe generalize coordinate curvature expansion partial time diameter global block frank curvature structural maximal length proof twice plug taylor hessian two identical product upper time column formally curvature norm svm domain define maximal feature e block curvature block restrict block notation augment block analog whereas zero taylor word vector tight worst indeed exactly frank wolfe problem exact frank loop product simplex reduce corner solve decode vertex variable vector show simple linearization equivalent duality gap lipschitz continuous wolfe algorithms primal original meaning error difference n argument eq define structural convergence fw obtain iteration cost frank wolfe g paragraph iterate either predefine step use satisfie f furthermore algorithm iterate duality gap different block follow structural obtain approximate duality gap cost duality gap guarantee predefine step write convergence error f analogously plug dual detailed lemma bound predefine show stay valid plug additional part argument make duality add requirement restrictive structural need around comment practical aspect structural additional e wolfe svm sometimes get fix usually frank section duality criterion fast partial gap single pass necessary obtain efficiency reason soon convergence wolfe structural frank wolfe e block recall contribution denominator maintain mention subproblem duality gap duality gap paragraph instead primal combination number upper dot search I size implicitly dot product suitable term begin self contain wolfe prove two part fast line appearance know rate method good scalability large result optimization product domain I e converge let pick scale lipschitz domain structural nice example big difference variable subproblem loss simple variant solve approximately multiplicative coordinate pick single random leave unchanged wolfe gradient converge minimizer quality candidate additive error parameter default determine candidate still attain th error internal oracle together predefine step step candidate line maintain convention analysis average iterate duality iterate call appear getting prove fix say storing fraction iterate know stop alternative mention maintain uniform round trick round towards round motivate weight average prove irrespective frank wolfe paragraph example use coordinate descent block difficult prove strongly convex cyclic already use frank block could prove frank crucial duality namely sum domain linearization notation product curvature sum block show obtains iterate exact algorithm q approximate variant eq predefine search block iteration crucially lemma let move within direction towards obtain expectation choice block duality approximation take block condition write simplify curvature convex duality eq definition lead good therefore claim improvement random gap exactly follow idea variant condition hold line use directly duality gap deterministic inequality expectation side choice previous block q appear wolfe convergence difference expect similar argument slightly base immediately definition bind particular f induction simply rearrange analogous primitive quality exactly argument occurrence moreover combine easily additive accordingly convergence hold frank variant additive remove independent precisely give case match frank wolfe define curvature constant base get well dependence search solution duality frank wolfe look one true gap without iterate duality duality gap scheme iterate example quadratic dual concave affine function concave imply primal either predefine line iterate duality gap approximation define use average simplify expect duality gap start improvement variant begin take expectation lemma duality gap idea handle combination bind upper existence duality iterate combination coefficient use see master convex example consider predefined size master master proof iterate last proof primal induction line weak thank start convergence iterate line subproblem multiplicative quality primal decrease primal k subproblem additive quality replace variant decrease soon fall leave enter subsequent soon equivalently always plug recurrence recurrence appear solve recurrence monotonically decrease differential equation side q complete proof multiplicative rate finish hypothesis definition induction crucial simplicity structure need search eq even size yield recurrence could induction advantage schedule know frank wolfe step schedule case know batch ensure proceed frank wolfe primal line search get theorem duality gap duality average make use theorem master gap appear valid master iterate otherwise choice master fast amount conclude k follow argument similarly part upper integrable ft n nk work complete sign strictly decrease b complete original convergence predefine shift note fully weight average regime primal refined speed forget case begin gap use optimization crucial differentiable linearization give arbitrary subgradient position ff immediately crucial duality gap differentiable function subgradient duality additionally interpret duality subset give f indicator conjugate directly assume inequality become equality choose subgradient last subdifferential f f detailed explanation refer reader summarize simple linearization duality gap restrict particular position slack lagrange multiplier scale multiplier variable multiply correspond primal attain finite saddle simplify also wolfe w set q expression lagrangian lagrange problem negative claim provide experimental frank wolfe method compare simple col sep comma lambda product l txt header col sep include lambda confidence txt color thick mark repeat solid header sep comma lambda l txt header confidence txt green thick mark index header col sep comma include lambda txt densely thick mark mark index header col comma lambda txt index true comma include txt densely dot thick mark repeat header col comma datum dataset lambda index header col sep comma lambda confidence color black x header col comma include lambda avg header col comma lambda txt gray thick col sep comma lambda xlabel pass ylabel legend pos north east col sep comma lambda confidence solid mark repeat header true col sep comma include lambda txt table index header comma l txt style triangle repeat table header sep lambda txt color densely thick mark triangle mark mark header col include lambda opt txt header col sep product l txt color densely style thick mark index header col sep comma lambda product l col sep dataset lambda confidence txt solid style repeat header col comma include lambda color densely thick mark col sep comma dataset lambda header sep comma lambda confidence txt color densely dot style mark repeat header col comma include lambda index header true col sep comma lambda avg color black thick mark mark repeat header include dataset avg index col include lambda confidence color thick header col comma lambda txt xlabel pass ylabel header col lambda txt color mark repeat x true dataset lambda txt index header col comma lambda product txt solid mark mark repeat header comma lambda txt color densely thick mark option solid header col sep comma l opt txt x index header true col sep comma dataset lambda ls color densely style header col sep comma lambda l txt table header col comma include confidence txt solid thick mark repeat comma include lambda txt color densely dash mark mark col comma dataset lambda txt header col sep comma lambda txt densely mark table index header comma lambda x col sep comma lambda avg confidence mark repeat table header col comma data lambda avg txt header col sep comma confidence color gray solid style thick repeat index col sep lambda txt xlabel effective pass ylabel primal legend pos east index header col sep comma confidence txt thick col sep comma lambda product true col sep comma lambda color blue thick mark repeat x header sep comma lambda product txt dash style triangle mark solid header col dataset lambda product opt index index header sep comma lambda color densely dot style thick repeat option index header col comma include lambda l header comma lambda green style index header comma include lambda color densely mark header col comma txt index header col lambda densely dot thick mark table index header true col sep comma lambda txt header include lambda avg black solid thick mark comma include lambda txt table col comma lambda txt gray mark mark index header col comma data dataset xlabel ylabel false area line legend header col comma red style square mark repeat index header col comma lambda txt index index header sep comma confidence txt color solid thick mark none mark mark col sep comma include lambda txt densely thick mark repeat mark option solid index header comma lambda ls opt txt header col comma product densely dot style mark mark repeat option index header col comma include lambda product l sep comma lambda solid mark header comma txt densely dash style true comma lambda txt index header col comma lambda densely dotted thick mark mark true col comma lambda x index header col comma lambda txt color solid thick mark col comma dataset lambda avg txt col comma lambda txt gray mark repeat true comma reach increase thousand early xlabel pass ylabel log legend pos north east index col sep comma lambda txt color thick square mark repeat header col comma data lambda header col comma txt color mark triangle mark repeat header true comma l densely dash mark mark solid header sep comma lambda product l opt txt table index header col sep comma include lambda txt color densely thick mark option index col sep comma txt header comma include lambda txt color mark repeat header col comma dataset densely style thick col sep comma lambda header comma include lambda txt dot repeat header col comma dataset txt index header col sep comma lambda avg txt thick mark x col sep comma lambda txt comma lambda gray solid mark col comma include lambda xlabel ylabel area legend index header col comma lambda txt solid mark repeat col sep comma dataset lambda txt col sep comma lambda l txt blue style mark none triangle repeat index header comma lambda txt color densely dash mark table header col sep comma include lambda l opt table index header col comma l txt color densely mark square mark header col lambda ls txt header col comma include lambda confidence txt solid style thick mark mark repeat dataset lambda txt color densely thick mark table header col sep comma txt table x index header comma lambda txt color densely dot style repeat index true comma datum lambda txt x header col sep comma avg confidence black style thick index header true col sep comma lambda index index header col dataset txt color solid thick mark repeat index header true comma txt xlabel pass ylabel area style legend legend north east index header comma lambda red thick index header col comma lambda header true col comma dataset lambda confidence txt color solid mark repeat header comma lambda product txt densely style thick triangle mark option solid col comma opt txt header true col sep include product txt densely dot thick mark option solid index true col comma lambda l txt header col dataset lambda confidence color style mark header col sep comma lambda txt densely thick mark repeat x index comma lambda txt header col comma lambda txt densely style mark repeat header true col comma header comma lambda avg txt mark mark repeat comma lambda txt col sep lambda txt color solid style mark repeat x col comma lambda xlabel pass ylabel legend header comma red thick repeat col lambda product txt header col sep comma include lambda confidence none triangle table index header true sep include txt dash style mark mark option comma lambda col comma include lambda l txt densely dot repeat mark solid col sep lambda txt header comma lambda confidence txt green solid mark mark repeat col densely mark header comma data lambda col include lambda txt densely mark header col sep comma include lambda txt header true comma avg style thick mark header col sep comma include lambda col lambda solid thick mark mark repeat index header col comma lambda xlabel pass ylabel legend legend north east index header col sep comma include lambda red style thick mark x index header col comma product txt table index header sep comma txt color solid style mark mark header col comma lambda product densely style mark repeat option index header comma lambda opt col include dataset ls txt densely mark solid col sep comma lambda txt index header sep comma dataset lambda txt green solid repeat index header col comma lambda txt densely dash sep comma lambda index sep txt dot repeat header comma lambda txt header sep data avg color style thick mark header col sep comma include lambda avg index true comma include lambda txt repeat header sep comma lambda txt scale xlabel pass ylabel false area legend header col sep include thick mark repeat index header col sep comma include lambda table index header true col comma include none triangle mark col comma include l densely style thick triangle repeat option header lambda l txt sep comma color densely dot thick option comma include lambda txt index index col sep include lambda txt color thick mark repeat lambda densely style thick mark repeat col sep comma include dataset header col lambda densely dot style mark mark repeat index header col txt comma lambda thick index header col sep comma lambda txt comma confidence gray thick mark mark header lambda xlabel pass ylabel primal log legend north east index header comma dataset lambda red mark header col sep comma lambda product txt header comma include l txt style thick triangle table true sep comma include l txt color densely dash thick option table header true col sep comma lambda txt table header col comma lambda txt densely dot style thick mark option solid table header col lambda header sep comma match lambda txt solid thick mark repeat table header comma lambda color densely dash style mark mark repeat x header true comma lambda txt index header col comma lambda densely mark header sep lambda sep comma lambda color thick mark index header col comma match lambda avg header true sep data lambda txt thick mark repeat index header col comma match lambda txt xlabel effective pass ylabel log area legend index header comma match lambda solid thick mark repeat header comma matching txt index header col comma lambda l txt color thick mark none mark index col sep comma match lambda l densely dash style thick repeat mark option table header col comma include dataset match lambda product l opt txt index true col comma lambda txt color densely dot style square mark sep comma include match lambda l header col sep comma include color solid style thick mark repeat header col comma dataset densely dash style thick mark col sep comma index index header comma match lambda txt color densely dot mark index col sep comma include lambda txt comma include avg txt mark mark repeat header comma col sep matching txt color gray repeat col comma include lambda xlabel pass ylabel style log legend legend east index col sep comma lambda red mark square col comma include lambda col sep include datum lambda product txt blue repeat header col comma lambda l txt densely thick mark mark option header col comma lambda l header col comma dataset match lambda txt color densely dot mark mark header col sep comma dataset match lambda l true col sep include match lambda confidence txt color solid style thick repeat header col comma match lambda densely style mark mark repeat header col comma lambda txt table index header col comma include match color densely dot mark header col comma lambda txt header col sep comma lambda color mark repeat header lambda avg sep comma match lambda txt gray mark table index header col comma matching lambda xlabel pass ylabel area comma match lambda txt red style thick square repeat index header col include dataset match lambda product txt header col comma matching l
domain domain unlabele formulate knowledge statistical relation bad include remove audio comparison propose multi modal relationship application access domain task word recognition greatly benefit availability audio verification much accurately modality major multiple label domain example wish audio gender label label audio set limited subject although straight train image alone clear good modality pair audio collect nonetheless predictor even practical availability aid machine suppose age audio audio training unlabele audio visual help solely audio domain unlabele domain unlabeled set use design predictor alone account label case set view tend agree unlabele construct setting observe label I I make view framework assume availability mapping assume exist implication lack multi infinite however suffice determine give mean error mmse x situation label predictor domain fall unlabeled example domain available suited view extract handle via framework nevertheless setting pair unlabeled transfer scenario data modality audio video perform one construct label want classifier operate audio audio audio share construct alone instance want train visual unlabeled audio instance cross recently link instrumental also highlight show cross representation learn example one label statistical assume mse estimator use domain domain unlabele every label multi course unknown bad set problem available insight multiple domain particular help underlie illustrate audio visual word application demonstrate convert neutral pair neutral face recognize available recognize audio access preferable paper regression capital bold letter xx mathematical expectation px inequality wise xy several give access label training independent assume approximated nonparametric label zero relation label associated mmse say accurately term test treat ask predict base solely include label datum several fig double circle correspond continuous line multi share also label adopt generalize statistical l independently practice parametric try rich accurate come bias sequel estimator one moment yx xx sample correspond form predefine arbitrary set k ki j quantity linear reference trivially borel measurable case zero set label degree even number set determine joint suffice compute mmse uncertainty able specifically determine predictor accordance note predictor one valid example predictor many unlabele label estimate depicted assume nf illustrative suppose linear pdf restriction valid infinite mixture eq density requirement consistent construct predictor domain tackle domain estimator give alone access domain set possibility include two predictor alone worse minimize mse close joint triplet know belong subspace function mse attain expectation since depend concept estimation interested next find mse distribution solution need mmse estimator among possess simple form demonstrate utility mse approach performance sometimes area bad difference mmse depend bad minimization mse insight characterization namely show multi minimax solution fig consequently solution show come unseen bad perspective predictor distribution perform stand contrast basic help agree context view amount suffice identify predictor collection consequently imply moment unlabele training similarly determine apparent however orthogonality respectively carry knowledge formulation case suffice situation k orthogonality suppose determine domain good predictor mse particular twice obtain stage technique unlabele u l www yx yx yx case base minimize rely concept innovation innovation mmse estimator denote make regard minimax appendix innovation optimal consequently denote x conclude approximated training domain second linearly interesting improved operate proximity joint seem help improve accuracy however narrow distribution design belong optimize single regret coincide minimize bad determine minimax estimator hard bad remain goal determine next describe single domain minimax explanation sense measurement alone interesting example illustrate fig intuition boost solely label framework set situation domain minimax imply cross unless cross learning classify isolate unlabele visual fail boost audio bad nothing illustrate fig second naive feed predictor base rather mmse approximate unlabeled minimax predictor single predictor case generalize coincide good predictor case yx yx yx strategy suppose numerous x x coincide u innovation highlight come train unobserved help recall term vanish q rich need meaning already determine mmse example jointly linear consequently moreover jointly imply coincide indicate another switching role estimator label training second vanishe happen example concrete example function x therefore x x x mmse therefore satisfied domain example form rather concrete side approximated covariance approach derive result illustrative preprocessing aim remove observed database may include variation illumination demonstrate produce neutral face straight forward appear twice neutral unfortunately collect access appear relation neutral face impossible obstacle view example pair manually several predefine location image database triplet annotation neutral may unlabele annotated face neutral employ design predictor construct third image neutral face apply regression discuss depict manually annotate neutral annotation template appear automatically unity kernel tune constant root correspond image show result neutral fourth result set face neutral training comprise neutral perform side information equation rmse attain low set lead additional training indirect seem worse apparent sort average joint example neutral ever able know relate point annotation relate neutral half people neutral set direct nonlinear share claim output digit video corpus structure sentence sentence isolate distinct set label visual
inspire problem require allow variational applicable close approximate mixture mean approximation principle resemble perform allow introduce variational approximation solve new look variational posterior represent idea make generally applicable extension separate discusse improve section despite generality competitive conclude effect observe upon bayes distribution proper kullback leibler kl divergence almost divergence posterior analytic monte choose factorize often solve call factorize make optimization restrictive conjugate specification assume posterior independence block parameter factorize parameter shape bayes usually member vector statistic normalization base often call approach optimization parametric analytically entropy analytically determine derivative respect posterior often factorize able evaluate restrictive generally allow factorize next draw parallel problem variational bayes linear regression limitation notational write assume vector family distribution rescale unnormalized px qx respect expression minimum implicitly identifiable exponential family insight maximum dependent matrix maximum associate unnormalized consider analogy analogy notational base measure analogy continue residual base model right depend constitute optimization section require computable allow extend impose simplicity mixture approximation variational regression carlo update px equation iteration unnormalize approximate initialize guess match prior approximation initialize draw approximation px tc n inspire research start seminal relatively update step size taylor give comparison divergence first pre since stochastic usual pre close algorithm family analytically suggest approximated analytically first seem approximate condition turn approximate draw dramatically reason optimal use randomness term particularly example approximation exactly gradient vanish finite sample exact exact functional still deep contrary literature algorithm statistic asymptotically gradient infinity conclusion e decay fast choose either take long reach rate put order vanish final algorithm grow number remarkably average match optimization excellent average half averaging rather remove use previously use mc give mean early expect guess define negative pre converge unlikely good help pick good guess help quickly difficult guess rough choose curvature guess default prior increase sufficiently stable variational divergence unimodal optimization optimum multimodal unimodal variational mode desirable guarantee minimum kl approximate variational functional normalize variational precision strategy analytically bad run option even option option toy approximate functional dash use give serve kullback leibl divergence give likelihood depend measure leibl essential performing model comparison present final kullback follow intercept recognize vb integrate marginal eq convergence jensen inequality tell log everywhere lower often kl however often comparison oppose marginal evaluate monte carlo show approximate marginal something use regression perform relatively log plus capture approximate kl approximately equal relationship come square exactly highly dependent curvature comparable scale curvature posterior therefore across set interpretable application mostly moment therefore construct relatively moment tail kl divergence good propose bind calculate approximation quality different approximation efficient extension discuss example conclude approximate expectation draw work remarkably well explain form certainly approximation often well addition problem version probit classic literature exist gaussian cdf gaussian performance method major much wide model simulate make use expectation calculate analytically compare message excellent performance classification implement inference approximation negative otherwise variational use thought truncate univariate posterior prior parameter often extra correlated reduce variance approximation implementation stochastic dependent g language look since synthetic performance quantify root rmse mean score posterior finding estimate present well see rmse assumption make introduce rmse indicate approximation give significantly average variational indicate rmse score identical digit give factor hessian transformation efficiency linear section slow convergence term run draw ep converge small ep rather infer matlab section introduce implementation probit regression additive factor optimality condition regression equivalently regression conjugate example know analytically approximate importantly separate often regression low factor projection contain property correlation control subset remain sufficient zero course omit low regression approximation store probit depend write link normal approximate distribution py f control approximate particular transformation use variational new parameter transformation jacobian make combination hessian usual parameterization stochastic approximation approximation working parameterization express gradient average algorithm appendix storage new unnormalize twice posterior total iteration initialize guess matching prior initialize step calculate gradient storing invert induce dimension parameterization hessian store invert even probit structure also twice still log log additive unbiased learning step since calculate large tradeoff subsample many successful gradient probit implement divide process update minibatch reach converge require pass much approximate relax approximation typically analytically approximation often still moment mcmc indirect identify strategy two block posterior choose analytically easily freedom capture practice conditionally tractable block type bayesian hierarchical suggest exponential joint normalize conditional precede conditional optimality condition still variational slight modification derivative setting sufficient leave far intercept normalize approximated straightforwardly carlo still perform separate conditional section alternatively approximation mixture allow dependency precede approximation exponential signal vary variance extremely finance vs model govern autoregressive specification hierarchical prior q exponential q choose form examine notation prior normal therefore multivariate conditional approximate approximate form statistic block optimize approximation make fact respect kalman direct indirect effect case indirect little application give expectation calculate analytically kalman sampling element belong posterior inverting precision expensive still matlab ghz processor magnitude mcmc figure similarly posterior latent show indistinguishable ht stochastic volatility characteristic importance auxiliary regression infer mcmc method able leverage hessian posterior approximation optimize volatility approach ability posterior need significantly approach construct flexible family letting analytically tractable approximation mixture enough mixture arbitrarily close note multimodal suffer kl q requirement integrate auxiliary joint density fortunately approximation mixture normal apparent section equation auxiliary thus posterior practical normal consider problem estimate death cancer pair city cancer occur city binomial model follow beta binomial precision give improper prior skew eq bivariate gaussians auxiliary assign categorical outcome approximate adapting include discrete approximation fit rao expectations advantage statistic equation mixture posterior therefore use expression update introduce extra expand apart different mixture use examine quadrature tail density gaussian mixture apparent consist square clearly advantage much rich true quite especially unimodal optima good global optima approximate arbitrarily py qx lower height solid line interpret newly propose section thus usual inspire regression sufficient approximating since unnormalized density distribution show nature software package along net would considerably wide range automatic beneficial automatic selection appeal despite applicability demonstrate problem hessian rare without factor present extension important use hierarchical practice mcmc chain option mixture tune trade mixture family analysis model variational usually conjugate collapse integrate adapt work collapse complex model methodology collapse mix conjugate conjugate closely variational message algorithm infer net conjugate maintain convenient pass formalism specifie perform variational approximate integral expectation quadrature secondary unlike monte open acknowledge anonymous thank organization scientific support college microsoft stanford center cancer unnormalize condition q clearly normalize recalling give tx tx tx tx px solution divergence point notational variational conceptually mean kl divergence regression q usual mean parameterization parameter differentiable identity q parameter stochastically point author perspective sampling convenient distribution share specifically choose variance result evaluate sampling importance practice process stability condition identical fitting posterior important tail variational posterior unless recommend might directly instead suffer offer way combine monte aforementione recent literature collapse aspect
institute mathematics natural university institute consistent kernel machine generate finite basis sample fourier carlo induce fourier distribution correspond parameter domain identify band prediction correspond kernel scalable learning methodology multiple produce fast apply problem radial different linear combination complicate practical vision shape texture descriptor role category principle easy represent descriptor base combination category learn scalability combine kernel limited consequence capable insufficient massive million image imagenet article principle costly preserve non embed span infinite simplify notable kernel learn massive seminal radial focus kernel histogram empirical kernel largely multiple combination produce conduct recognition difficult object order learn linear operate broadly classify single kernel combination kernel prove learn definite involve attempt track conceptual model kernel semi definite programming reformulate norm regularization cone medium scale kernel combination direct search perform difficulty multiple relatively example scale become formulation within framework carry scalability wide applicability large attractive kernel offer substantial central dot point become infinite feature mapping visual popular trick quadratic example direct usage impractical dataset methodology linearly random methodology fourier transform every transform define j frequency carlo linear explicit feature two monte dimension dataset hundred thousand example train motivate interest class approximation randomize dimension htbp c u methodology achieve goal need criterion parameter fast due rely procedure kernel optimize parameter hold validation training overfitte sequel denote input wise organize target drop map kernel ridge cost hinge logistic etc eq optimize square rr q regard expression expansion regard th kernel kernel identify band useful prediction effectively expectation parameter change although conceptual difficulty become parametrize far away start resample potentially nevertheless convergence belong q fix product throughout entire process generate automatically feature obtain respect smooth norm formula parameter fouri optimizer parameter similarity introduce besides usage embed advantage kernel number invert connection interesting practical far section fourier methodology opposite lift group formulation initially fouri lasso prove approach feature well store organize wise target example concatenation would component apply overlap formulation write feature quadratic insensitive regression perform solver standard present build gram define set element primal associate theorem drop simplicity rewrite trick multiple shown observe group lasso formulation set equality happen well restriction order insensitive il standard insensitive function logistic group assume algorithm maximum solver dominate computing matrix approximate become constant could slower thousand kernel learn compare random fourier methodology counterparts accuracy challenge image image without follow
plant community lin first optimum run heat iteration refine ever possible initialize stability try isolated community focus community plant keep community assignment make concrete block vertex label label correctly label fairly assigning classify degree probably connection near equally initial correct unable degree well assignment structure help somewhat separate degree vertex degree even broad degree exponent work randomly initialize either assignment kl follow generate perform non correct work undirected network block degree distribute orient plant asymmetric choose work simply grow correct edge block versa network precisely generate degree classify vertex since distinguish block network run step vertex separate consist common novel corpus american english many corpus news contain network ever version occur size size news accuracy correctly label everything noun tailed degree asymmetric english often roughly block noun david news rs r likely block e compare model apply ignore direct start random lin heuristic find optimum run heat naive heuristic label noun correct assignment david news independent execute network bold sbm dc david fairly moreover mistake zero since away correctly block well fail bad degree separately edge full case thing even broadly slightly news despite likely assignment connect news edge suggest though kind look try give model assignment kl news take base degree refine low use random help stay accurate optimum condition follow step news sbm nh counterparts distribution heavy moreover vice degree assignment r r block noun table let improvement slight well alone degree generation benefit optimization let good kl step oppose degree give assignment generation converge avoid kl degree block deal network sense help block allow heavy orient partial total degree degree assume generate prior unlike direct able capture account degree put unlikely degree high direct subroutine impose community illustrate small block generation appear handle without benefit kl depend heavily know correct form degree truth assignment difficult variant correct degree infer structure network bad strength kind kind select meet grateful foundation constraint q r log likelihood partial give orient correct undirected number direct edge poisson check impose r rr ignore rs rs recover rather eq bayes line denominator however prior analytic call likelihood degree generate distribute proportional product gamma natural gamma gamma hyperparameter block multiply family uninformative reduce integrate focus estimate continue posterior simply posterior calculate integral algebra numerator denominator hyperparameter optimize empirical approximation integrate long degree result integral impose degree likelihood derivative setting get com tool infer community world degree degree correct block accommodate degree vertex vertex generalization direct tail english text high correct connection social daily contain link political speech vertex way feed relationship functional crucial highly generative block vertex block let capture many community community independently membership distribute solely membership block degree lead model heavy degree distribution community conservative degree undirected parameter vertex control degree accommodate degree community remove tendency degree vertex correct classify vertex explain may recognize separate dc direct generalization block two parameter degree english rarely vice versa strongly membership leverage type strength correct block able utilize degree asymmetric include stochastic treat law whose likelihood assignment interaction community structure community degree separate community experiment well network community enough vertex classify vertex block relative community notion degree strongly asymmetric partially capable take call try version direct point direction generate undirected degree correct oriented writing giving orientation direct inter possible substitute combine log add obtain poisson special case force edge way utilize domain exponent us membership tail degree help vertex unlike maintain degree generate specifically independently depend block
connect paper represent clique node degree increase rank capture paper characteristic observation degree connect connect share explain mean link connect connect biased connect connect plus connect branch implication observation use take difference solve generality consider eq normalise evaluate recursively ambiguity dependent tend unique evaluate rank scheme reflect evaluation entropy minor neutral network average obtain circle neutral line see statistical fluctuation evaluation also degree nan evaluate show two quantity z j link nan notice score closely link degree degree nan nan degree entropy degree rich many tend correlation good model closely reproduce nan rich connectivity lagrangian network rank sort decrease order useful comment suggestion restriction rich two share link describe nan approximate network topological pattern behaviour occurrence mechanism perhaps responsible complex statistical method surrogate reference nan purpose intrinsic preserve connection node basic procedure surrogate sequence pair link create degree procedure create sampling bias network self loop many example bias take consideration generate random technique order nan shannon widely describe attack shannon case entropy approach way maximally maximal entropy attractive nan probabilistic measure use model degree correlation community use explore walk equal law method structural like degree difficulty share degree lack heavy measurement insufficient limitation node degree classify limitation correlation classifying network limitation nan ability could method construct rich coefficient connection rich correlation accurately distinguish node degree node degree rank scheme node divide node notice enough allow multiple rich rich probability connect loop impose link formulate represent solution formulate normalise interaction interact via maximal probability maximal certain common use via link give constraint show network five network label degree link give map relationship inside node label inside read probability link correspond pair constraint sum lagrangian multiplier nn mf q equation define p
proposition noisy exclude therefore verification lasso replace still valid w I mu tm central limit probability tend normality tucker kkt j j shown verify note tm n constant verify slightly modify th replace replace right still tend end iii fan show consistent assumption verification assumption scad suffice event satisfy scad convergence assumption x es ij ik ik jk ik jk lx lk ik iw lx lk lk jk fact jk chebyshev eq together jk meet validate condition pattern q side tend specifically p cn sign th j addition uniformly tend independent satisfying end end verification definition wang wang supplementary material fourth scad assumption fan I suffice follow inequality follow finite chebyshev prove lemma I regression adaptive scad slight modification existence verify subsample magnitude element accord c select active
compare horizontal scale multiplication precise indicator tc tc bc k bc bc tc bc k bc objective zero conduct see iterate convergence slightly due conditioning reach precision overall gradient hand intermediate require loose suffer slow sublinear begin behavior vary keep big variation affect stage suffer sparsity rough last vector interesting implie still mc bb tc tc tc tc bc bc introduction homotopy study l reconstruction separable use strategy reduce iteration initialize satisfied say accord method able exploit restrict acceptance stop homotopy either monotone employ call bb option bb precision numerical figure demonstrate number stage number bb result loose option reflect figure use intermediate overall pg bb tc tc bc bc inner figure nonzero convergence monotone version bb low bb set bb method look high homotopy see homotopy counterpart instead mc bb tc bc pursuit experiment choose generate independently I transpose transpose soft modulus bp ls use bb remarkable converge requirement become small always reach loose homotopy recovery confirm stage default reduce bb stage proportional regularization stop constant homotopy path advantage homotopy homotopy recovery multiplication homotopy method need track break multiplication homotopy important strongly hence show iterate homotopy method structure objective become convex thus geometric homotopy accelerate automatically exploit convexity discuss convexity obtain convexity ls accelerate come explicit nesterov give suggestion direction strategy periodic investigation lemma assumption zhang square iterative slow nevertheless exhibit fast local homotopy l approximate stage warm although assumption homotopy iterate homotopy function effectively find solution interpret rate convergence solve l vector term learn statistic receive interest year build representation recover measurement l recover measurement relate regularization great investigate rip constant recovery denote nonzero element depend rip achieve closely paper numerical recovery focus sparse sufficiently assumption method provable complexity algorithm follow nonnegative appropriate choice however parameter directly nevertheless efficiently lagrangian root find similar nice survey practical briefly summarize complexity relevant ls minimum among use ls bottleneck approach normal cholesky factorization prohibitive large solver method conjugate multiplication involve computational cost multiplication conduct l problem shorthand iteration search procedure rule include appropriate iteration complexity establish strongly unfortunately sparse submatrix rip fast stage g nesterov minimize l accelerate typically statistic complete exploit piece linearity examine condition submatrix break quite general bound recovery homotopy idea gradually reach employ adequate approximate serve homotopy use e result mostly hoc sequence method provable along algorithmic geometric value precision precision nesterov parameter gradient target sufficiently final solution sparse satisfie like exhibit sufficient universal satisfie overall imply convergence homotopy proximal gradient stage always start parameter path stage rip imply along homotopy path effectively geometric nesterov homotopy track scale much small target number homotopy method phenomenon complexity also confirm empirical compare iteration term theoretical homotopy strategy path square condition near provable solution interior insensitive important free compressed l sufficiently special paper parameter homotopy analyze extremely difficult free pursuit technique demonstrate proximal homotopy path significantly simpler necessary homotopy main devoted proof map key nesterov ls condition nesterov close solution call proximal quadratic lipschitz see q satisfy composite mapping directional eq therefore presentation x b x correspondingly add subscript specify fy fy call ls problem gradient lm l k x composite develop variant proximal gradient accelerate primal need choose optimistic line try lipschitz iterate algorithm line ensure lemma objective unless mapping case criterion optimality l additional depend differ need line search although repetition nesterov search let lemma example nesterov complexity find strongly convergence complexity nice function whenever case objective strongly complexity though convergence observe nevertheless homotopy strategy enforce iterate rip homotopy geometric explain condition characterize throughout practical purpose constant message without special attention optimize natural pick automatically satisfy condition concern basically start precision geometric simplify presentation symbol denote restrict satisfie roughly proximal choose q condition step k hold eq terminate total output satisfie measure optimality precision reach let let achieve gap global restrict global plus last globally particularly interesting sense bp algorithm gap informative convergence suppose outer outer global geometric bp proof subsection iterate algorithm prove theorem appropriately homotopy lead iteration complexity direct sparse eq let subgradient achieve approximate ta ax z ax break part fact combine gives q rearrange arrive obtain x inequality prove since subgradient result much suppose assumption proximal optimality monotonic objective establish iterate small expand ax ax ax ax therefore ax splitting x drop side claim inequality q hold argument imply last eq mean much sparse operator eq nonzero truncation part inequality last exist eq take side inequality since achieve eq side prove recall take form monotone iterate satisfie search terminate imply lx smoothness termination search accumulation point accumulation accumulation restrict x fx accumulation accumulation especially stop procedure x second relax restrict segment eq homotopy proximal approximate eq k fx prove desire ready give estimate homotopy method within optimality km kx smoothness restrict property theorem obtain g k criteria eq prove homotopy outer performance follow moreover outer instead method terminate apply desire bind numerical support first property method implement follow solve pg nesterov line search propose described nesterov accelerate method pg replace dimension entry choose dense uniform choose
receive node receive recursively certain construct node imply consist ratio threshold error equilibrium fast rate rate slowly slowly respect backward search network integer node asymptotic law fast scale scale let network converge assumption distribution private proposition derive q much rate probability follow situation p j c scheme length least cn n kk consider situation node observe backward scheme length away converge moreover converge converge sequential testing model channel recall symmetric channel complement k probability flip symmetry section easily general know probability decision suppose exist exist strategy observe immediate decision immediate k xx ratio rewrite linearly without henceforth obvious become type error bound regime appendix error presence probability converge condition probability converge describe way suffice type I strategy map corruption away ensure contain true mean decision increasingly uninformative move recall denote belief belief eq easy derivative monotone follow ratio martingale converge surely expectation martingale almost public belief almost implication public belief completely wrong public bound away public everywhere imply almost everywhere suppose statement independence eq continuous contradict almost surely prove converge convergence proof generalize scenario away variance bound convergence generality generalize denote belief converge surely property assume exist loss assume characterize polynomial tail density public fast rate establish bind converge rate sufficiently type I decay probability decay substitute evolve characterize non proof away non constant belief fast outcome surely sufficiently small almost error probability decay linearly decision asymptotically uninformative maximally completely uninformative belief condition public belief evolve characterize sufficient negative logarithm exist subsequence show show contradiction eq public converge limit represent decision among public limit surely evident nonzero away truth hence provide learn nonzero belief collective arrival explain let event decision arrive collective false phenomenon use occur lead hope nonzero expect depend relate scale law condition surely almost surely almost surely provide error guarantee necessary condition nonzero space sequence infinitely proof wrong decision e sufficiently public belief convergence uninformative fast signal public infinitely slow belief exist lead exponent expansion density theorem necessary density assumption tail derive explicit establishes law belief uninformative tail probability almost asymptotically uninformative probability surely k kb k decay ii assume eq decay type failure bound decision case go relationship observe decision converge necessary characterize relationship convergence derive event nonzero multiple similar go decision want generalize technique network topology moreover failure expect scenario finite snr martingale convergence snr go go infinity signal snr error probability node observe decision threshold combination suffice ratio decision equal decision always small ratio exclude hypothesis testing node decision ii xx p xx xx j k see treat recursion ode ode exist two public belief plug establishe claim iv ode establish satisfie rate fast outcome result probability condition q conditional decay recursion public belief know decay condition event positive q iv solution turn thank anonymous reading manuscript constructive comment presentation node sequentially make give hypothesis observe decision two failure node subject interested ratio decision number bound immediate decision show grow number decision converge probability converge consist test node previous decision probability channel away locally convergence error converge decision locally underlie truth explicitly relationship probability learn decentralized channel social sequentially decision node measurement receive immediate decision true converge sequential single sequentially collect fix pre maker stop hypothesis decentralized sensor network case spatially hypothesis decentralize network resource sensor aggregate sensor sensor failure case fail message moreover channel sensor noisy message subject aggregate spatially truth sensor increase another agent case underlie also state learn action decision agent use objective locally minimize bayesian ratio social feedforward customer decide whether restaurant previous feedforward private customer receive represent reveal might decision decision converge rate e converge converge explicitly convergence error use denote nonzero denote denote node scene decision hypothesis decision corrupted carry make decision immediate find sequence making tend e proceeding definition assumption algebra assume history mutually probability algebra absolutely ratio
product evaluate whenever term follow constant occurrence notation although statement asymptotic defer ne intuitive become replace proof exponential tail sum concentrate around essentially guarantee term hence contribute term chance confusion us range affect verify product convention member order pair countable define let q application produce ready give functional brevity break proof original product part form denote first term apply give low matching rhs choice eq q lemma appear product second lemma get first apply rhs bind let proceeding bind miss equality desire remain delay r accord correspond edge break two e break edge total category obtain product extend sense term bound technique remove index describe put piece asymptotic behavior throughout conditional convention appear condition definition r tn intermediate nothing dependence equivalence collection q irrespective whether n ne en en prove theorem regard asymptotic behavior far bound eq long bound polynomial say n u ec obtain nothing boundedness take result proof elaborate proceed step various time shorthand notation introduce drop en event hold c bind long boundedness state large sum e break follow move couple throughout drop implicitly lemma statement n n I en ne en statement boundedness let eq prove lemma definition proof ne ne result log e u ne ne ne p ne ne p p note sum sum u ne u establish early see note lemma move introduce e sum rule take term apply ne recalling term en since lemma u r u get graphical stop change functional subset thresholde message delay direction change restrictive simplify place impact phenomenon delay detection rather interesting analysis approximate message pass algorithm computational scale constant fast approximate message theoretical publication remark I decay seem likelihood break complicated simple piece irrespective tail behavior whether one derive minimax consider node argument show independence priori depend notation convention product divide obtain hold definition one drop term nonnegative rhs version enumeration version introduce version let eq dropping separate start prove pick throughout pm pm knn kn hold shorthand kn multiply b n b b side interval nb nb bb b probability imply b fix change thus enough small integer satisfy previous assumption c generality denote assumption polynomial moment lemma remark ram discussion support part grant sequential detection present detection rule functional delay graphical formalism rule protocol linearly size demonstrate classical detecting time decentralized data rule constrain take paper interest change may detection traffic concern spatially treat change sequential accounting reduce alarm probability detection delay propose formulation multiple change point network adopt estimate globally locally across statistical site associate point another detection message pass network computation interestingly express leibl divergence define simulation rich sequential tend variable formulation sequential diagnosis associate take across array interesting distribute line impose severe constraint site use usually understand denote integer denote link sensor variable represent time sensor fail interested point take endowed central ingredient formalism specify linkage observe collect cf vertex associate share distinction aggregate observation denote nm nm ccc graphical depend hence often density density specification summarize I change dependent despite independent priori primary change eq algebra detection stop detection rule trade false delay pearson false alarm minimum delay ingredient formalism represent rule aggregate primarily fashion pass message one neighbor graph conceptually communication play two coincide obtain multiple crucially rule geometrically highlight case simplify asymptotic delay delay eq delay see delay decrease counterpart independently result delay regardless whether edge asymptotic basis give basis posterior counter result even moderately access one geometric exponential allow tailed brief state ingredient heavily message pass mp single star node pair point together false alarm tolerance depict node sequence post prior geometric mp posterior node private single change ij limit expect carlo change take single relatively alarm though slope suggest mp single access pair mp mp high guess regard nonzero observe establishe concentration ratio term cf marginalization asymptotic graph complete
decade progress make design efficient undirected graph dimensional observational method penalize regression work feasible excellent theoretical property normality copula copula replace extend normal transform smooth almost study represent relationship encode I conditionally invertible towards zero vector independent property pair construct random define kl kl also appear free explicit easy compute formula become easy procedure change compute code calculate energy construct pair ordinary invertible invertible computationally singular construct correlation p computation package code language graph tuning parameter os enyi graph os random graph construct relationship manner os enyi structures node recover edge section huge implement huge estimating copula stock huge price stock day market stock select instead force graph
grant exploitation low final gradient q filter term account second gradient know easily weight weight class difference convention author tool useful discrete also divide interest range real algorithm algorithm filter superior near knn learn support maintain conditional retrieve proxy demonstrate pair find sense return process retrieve visible tool map ice help water road ice pixel pixel instance pick produce require retrieval inversion classification vector know could count might correspond surface forest road water etc input represent normally likelihood thus near knn skip domain large necessity real important feature include filter describe advanced retrieve water retrieval comprise million fast first classify surface extremely ability confidence absence knowledge true value unnecessary probability estimate super section gradient knn estimate density perform classification work pick determine voting refinement probability th coefficient justification carlo except multiply essentially arbitrary assume roughly constant isotropic distance simple metric q q region tail region sample weight unity filter density fall wish filter substitute correctly select produce width valid integral quantity knn scheme bandwidth estimator filter width varied pointwise give width conditional advantage scheme knn produce desirable discrimination make classification class sample adaptive coordinate test derivative class search discrimination time necessary giving describe class size lie simplify probability behave assume pdfs roughly filter iteratively coefficient filter iterate square th width obey q call obviously take filter width target weight part twice contrast need computation member large marked removal member add tree mark removal operation member filter apply find follow paragraph must calculate least arrange element large one repeatedly large unbalanced tree actual great element entire new tree tend two belong solve locate find dimension considerable consider two evaluate sign least aid convergence estimate estimate repeat true within certain tolerance combine newton hence library use solve cubic solve rank linear synthetic consist class illustrate triangle centre minor axis dimensional knn implementation cost second compose member class rather class sample compare compare advantage quantify return conditional knn codebook search classification algorithm represent fitting ccccc classification correlation knn n show trial uncertainty affect true retrieve class quantify column visual figure figure compare main thing much four important need datum phase much fast unfortunately necessary measure accuracy classification prior retrieve come exact region codebook design codebook coverage high surface map seven channel simulate colour use automate discrimination statistical dense complex one use image manually le surface water le whose landscape resemble training six resolution minute minute minute minute fairly winner codebook imply section give speed since codebook utility water mis choose correct perfect job classify lie water world monotonic inaccurate fortunately rarely case figure equation right side classification accurate direct knn compete single whole efficiency algorithms classification time generally increase increase weak actually include bottleneck algorithm select although overhead calculate efficiency coefficient svm training read output phase issue codebook return equivalent svm number phase bit redundant appear bit well sample specialized extract classify surface type general statistical get local try discrimination resolution sample minima optimal neural classification generate value overcome variable retrieve dividing range govern range threshold class define follow conditional fine retrieve compare probability value water retrieval seen replace tangent robustness impossible result see normally expectation moment side formulation characteristic robust estimator
generative correspond reasonable validate consider boltzmann unit top initial bias visible ab initial bias handwritten threshold medium gray size handwritten digit feed binary persistent keep track particle gibbs collect tend shape iteration alternate datum independent likelihood rate minibatch particle persistent divergence epoch variant average stochastic reduce estimate k randomly representation build sampler take measured residual curve see produce gaussian perform estimate estimate use update update see center generative advantage strong generative simply discard highlight center fast stable center eps eps eps eps eps epoch mm eps eps eps eps eps eps mm top layer epochs eps eps l eps eps eps eps eps figure layer filter learn tend varied learn suggest rich center top form manifold may still contain digit digit lot useful discriminative balance modification boltzmann center involve center optimization criterion modification boltzmann machine center show machine learn fast center addition center deep boltzmann discriminative training layer still require many understanding come despite initial conditioning hessian exclude dynamically solution within behave investigate tu de machine boltzmann principle extract datum unfortunately layer without greedy largely initially activation zero learn hierarchy abstract datum powerful extract unfortunately jointly argue great reason mapping activity sigmoid default function center offset condition estimate boltzmann gibbs invariant train center deep boltzmann top useful learn fast stable center backpropagation centered center boltzmann boltzmann following sigmoid denote parameter operation element boltzmann machine connection unit contain bias associate denominator function center boltzmann associate unit center boltzmann derive unit x sub sub sub sub sub sub show argue center hessian center vector random equal project dependent independent absence dependent conditioning therefore datum due cause visible center think well perturbation behave perturbation show get limit show stability quantify ratio eigenvalue interpretation hessian project condition hessian number boltzmann unit initial offset ccc conditioning sigmoid conditioning machine ii iii technical reason introduce restrict unit typical unit deep architecture specific role unit layer structure easy efficient alternate gibbs sampler boltzmann figure associate take group sample alternate state persistent center cx w bx x x mb present representation evolve layer layer layer think aim approximate implicit measure kernel match obtain number sample close error counterpart represent respectively q build kernel sort good representation certain measure task lead lead kernel principal component empirical obtain residual task curve curve determine necessary relate shoot ask hand label information representation rich encode subtle purpose represent compare architecture analysis
minimax thus kullback leibler observation discrepancy outline rest supremum latter hold whose th column define chebyshev last find suitable f ip ip side lemma need hypothesis infimum test procedure center coordinate point connection dimension large satisfie convention condition ensure lie sphere inside construction let coordinate case element depend appear simplify moreover point equality apply crucially large lie exactly show f enumeration q nonzero coordinate nh nh hold q n mr argument use nh follow familiar previous non coordinate big satisfy relaxed r set cardinality ensure also ensures state collection ex shannon follow finish fix take observe n low symmetric symmetric r l r ng require lemma concern relate devote square remainder derive view quantity kronecker conclude also q entry tv tn expectation since product vanish verify calculation together conclude next argument convention define l since proof theorem select suitable nb one utilize eigen submatrix ensure involve perturbation eigenvector possibly aspect require establish step follow take theorem careful expansion show term expansion accordingly lemma together q entry rest enough since appear rhs fact upper uncorrelated na appear upper bound computation deduce q event use inequality complete three probabilistic deviation follow nc suppose follow role small pp variant implicitly ta sa bound I discrepancy eq log give respectively contain kullback leibler fx x rhs orthonormal exactly I nonzero cardinality shannon entropy set satisfy I fix demand coordinate nonzero coordinate last lemma expansion involve involve though explicit proof uniformly suppose b n sequences asymptotic principal large old result universit paris functional principal penalize dr temperature north estimation smoothed principal component estimator wang functional f active f operator random matrix banach space c stein l unconditional estimation bayes estimation l normal convergence principal functional regression pca break principal chi mathematical statistic book manuscript principal stanford university nonparametric similar component l financial correlation l stanford stanford university principal component stanford I component stanford university w york comprehensive identification cell cycle microarray statistical multi channel wireless communication e component root multiple root j recent sensor determination minimax convergence longitudinal form cite incorporate dimensional covariance observation rate sparsity constraint pca estimator regime usual principal widely technique reduce traditional pca applicable reasonably two observation distributional eigenvalue various among eigen covariance fact sample approximate large however advance acquisition dimensionality individual nearly magnitude increasingly article collection face typically e student contain taylor outline class analyze act think example relate number size study sometimes thousand datum collect factor hundred stock theory optimal discuss several consider aspect indicator monitor commonly refer dr give extensive treatment theory communication measure point consist person subject study time patient cell gene gene researcher interesting eigenvalue population covariance certain type wireless deal estimating grow problem eigenvector characterize consider cell two pca systematic identify pattern structure smooth corrupted element vector often practical scientific interest advantage time address develop estimation theoretic standard reference optimal model give spline eigenvector eigenvector viewpoint observation gaussian assumption though essential make risk mostly nature even effort make explicit assume triangular individual partly motivate similar analytical estimation mean loss eigenvector relevance sort chapter describe behavior scheme show class constrain parameter suitable regularity lead eigenvalue population eigenvector framework subspace scope suppose triangular assume observation covariance matrix identity vector notice among sign notice identifiability rank mean infinity think equivalently describe term invariant notice relate convergence main focus exposition eigen matrix asymptotic refer ba problem strictly appropriate interval suitable consistency still probability goal assess task specify model invariant sign specify loss invariant specify eq norm useful loss denote l two approximately bound remain assume l identifiability condition big eigenvalue l possible relax condition issue rate growth risk address question investigate gaussian among deal measurement uniform note result circumstance rate analysis restrictive moment consider eigenvector restriction eigenvector parametrization notion later theoretic behavior space positive satisfy sparse formalize demand belong think wavelet sufficient membership coefficient basis refer treat motivation impose impose restrict ball appropriate radius need reverse vector space appear large sphere center fit inside space q natural kk low minimax state condition applicable infimum estimator hold take take hold flexibility hyperparameter appearing notable aspect become clear bind bad suggest strategy able extract coordinate rate convergence subject possibly regularity principle c consistent part hold eigenvector th reveal contribution expansion certainly weak whether sufficient model assume propose henceforth practice datum entry serve slightly biased true sparse rescaled multiply observation slight notation eigenvalue eigenvalue motivate follow scheme sample variance coordinate index submatrix correspond eigen eigenvector n choose good choice convergence single scheme scheme view appropriately assumption structure extract lot focus diagonal suboptimal minimax point covariance case view one coordinate way former contain small preliminary among extract stage refer final stage eigen follow constant later n ok n om tt thresholding normalize vector except derive risk
express generalize usual optimality continuity lipschitz two monotone drop obtain lipschitz symmetric curvature proximal subproblem many approximation reduce usage generally strategie hessian function adapt choose proximal positive adapt handle hessian modification multiple subproblem proximal cause become skip update express proximal type search direction newton composite let scale mapping proximal exist unique strongly convex subdifferential scale proximal mapping cauchy schwarz h express direction composite mapping use deduce newton search satisfy newton type like composite subgradient minimizer direction substitute positive definite zero minimizer minimizer directional derive close newton search usually must resort subproblem exploit iterate decrease accept direction strategy arc arc relative lie low arc drawback trial lipschitz sufficient descent condition lipschitz choose substitute choose subproblem search gx td kx proximal subproblem approximately inexact counterpart indeed implementation proximal newton subproblem efficiency reliability implementation use heuristic inexact affect describe adaptive analyze inexact type conduct condition commonly adaptive function require side generalize composite require approximate near desirable perform exactly optimal solution near preserve convergence newton subproblem solve accurately direction implicit subgradient connection inexact newton stop condition k fx inexact proximal stop condition sufficient descent subproblem inexact converge unit length linearly readily composite generic result separability generic implementation state sufficient descent inexact newton converge linearly generic proximal newton show newton converge globally subproblem show proximal show inexact linearly choose composite lipschitz uniform convexity convexity strong smoothness constant proximal optimal solution neither general proof assume closed function uniformly method e exist length cf exactly decrease exist eq sequence must decay length sufficiently attain descent decay search cf converge newton smooth type twice usually relaxed constant purpose assumption relax proximal exact converge strongly constant auxiliary result add strongly decay descent since mapping lipschitz proximal newton twice continuously differentiable convex constant require quasi newton unity satisfy sufficiently proximal satisfy criterion step appendix definite bound search direction q side side eigenvalue eq x satisfy substitute q proximal quasi newton converge assumption quasi converge assumption lemma length many eq denote proximal thus substitute expression substitute expression rarely solve analyze adaptive stop inexact proximal linearly term small continuously ii assume iii iv eventually prove auxiliary kx non eq combine next gx express multiply obtain substitute semidefinite deduce inexact newton linearly depend sufficiently inexact method step length converge linearly decay inexact proximal unit length monotone cauchy inequality apply substitute deduce converge sufficiently converge decay finally accord inexact proximal newton inexact suppose accord inexact show substitute expression proximal converge explore inexact direction behavior proximal avoid subproblem proximal newton statistical show suit expensive smooth evaluation suppose seek covariance q wise avoid goodness fit probe collect patient gene convert match solve problem proximal bfgs method update bfgs would computationally expensive rule decide condition solve subproblem stop fista solve subproblem plot versus figure met empirically figure proximal bfgs characteristic quasi newton dataset fast ignore expense condition per fast adaptive like third stop yield convergence convergence affect condition number slow logit goodness handwritten digits feature news scale match value proximal fista nesterov problem relative figure figure proximal bfgs fista bfgs versus fista expensive exp log evaluation nonzero entry dominate bfgs method expense shift subproblem whose function nonzero entry evaluation make total bfgs optimizer en opt publicly available laboratory share interface software compatible generator website popularity function composite analyze proximal newton rapidly produce solution level set proximal convergence stationary point disadvantage cost subproblem fast subproblem hope result proximal minimize composite acknowledgement thank lin lin ed schmidt anonymous suggestion lee support fellowship stanford fellowship advanced program er national institute medical national health award gm
summary pa usa em introduce novel reduction find multivariate orthogonal financial macro economic successfully discover informative forecasting well available dimension simplicity transform transform dr dr orthogonal uncorrelated signal keep ica recover autocorrelation forecast accurate efficient dr forecast bottom daily return tree package thus forecast sake finance economic even value interpret return propose move dr technique find provably eigenvector dr find low subspace relate review computation simulation autocorrelation daily return stock efficiency growth correlate intuitively month highly temperature warm warm warm zero uncorrelated k k k stationary transform since non negative function white spectrum example temperature right peak represent year qualitatively vice recover inverse fourier distribution spectrum spectral density inherently eqs uncertainty deterministic shannon thus future entropy logarithm support flat indicate flat spectrum white noise contrary processing literature require forecast characteristic perhaps function q eq leave indeed return guide estimate normalize scale fourier frequency consistent package along multiplicative n gx sample fourier fourier density see future classic neither plug intuitive property estimate iff perfect yield expect qualitatively estimator differential future rely capture temporal dependence similarly rarely linear multivariate make property seem trivial intuitively combine general make simple side efficiently equal contrary univariate complex ib reduction k ts ns semi return eight financial series important structure discovery portfolio stock portfolio bi plot pca almost equally movement gold pc fourth energy though pc water almost twice water water vs gold vs china mining financial autocorrelation lag overall wrong fig sf fast large component latter still reveal white low indeed day month simple study nd omit lag lag autocorrelation spectra absolute growth last growth region baseline state economic help times scale know business well affect global display statistic autocorrelation relate correlation analogously lag lag forecasting compare portfolio density intuitive contrary drive cycle much easy year fit adjusted cluster space interpretable unlikely lag business cycle clearly visible rate similarly maximize lag correlation cycle face find important forecasting pc show period whereas look somewhat associated loading quite loading separate series literature analysis fit var recently another dr separate extent moment bss average forecast bss especially ica entropy fit contrary principled measure furthermore quickly use technique point differential entropy equal proportional coincide g
subtree document advance document allocate topic subtree select subtree stick construction document child subtree construction dp select variational optimization requirement maximize fix objective considerable freedom regard greedy variational update outline dp stick dp j kk switch probability greedy variational subtree sequentially currently activate activate contain subtree subtree determine add great increase document specific beta distribution prior variational distribution topic indicator remain question activate subtree great increase objective reason restrict distribution calculation iteration parameter subtree candidate fall formal description atom dp meaning define subtree document starting constitute activate add determine create subtree include point subtree include pa increment subtree note two greedy though stick break document dp atom advantage atom stick break subtree atom candidate atom change incorporate subtree select document optimize variational subtree structure start value update familiar beta though form independence closed update beta variable j level break proportion statistic pass second parameter document path transition stick construction switch stick break slightly statistic expect topic terminate select take global break data batch discussion section scale corpus size old new document reflect increase parameter datum value batch use sub batch see old variational wang nest topic batch result three set verify multi improvement nest crp stochastic perform million rare giving initial level dp hyperparameter slightly encouraging continue greedy subtree stop low fall continue iterate fractional distribution fall hold process top variational test percent split variational portion form document portion comparison evaluate hdp adaptively probable probable predictive unseen number document processed see hdp give document topic equally contain show word topic use level per three allow entire adaptively figure node probable accord four child relationship meaningful hierarchical subtree branch issue incorporate document learn distinguished topic datum single computer set topic wikipedia comparable figures improvement hdp increase figure topic word would york times though ultimately sensitivity topic wikipedia corpus base typically similarly dp impact posterior sensitivity hold fairly robust structure prior quantitative quality model well relative believe reasonable setting nest chinese restaurant observation follow stick construction new specific path hierarchy entire content variational inference scalable compare stochastic hdp engineering university berkeley university ph electrical focus develop involve wang capital management project department receive thesis award research focus david associate receive university focus image electrical california berkeley research focus graphical model statistical processing computational biology member science member engineering member american american association name international chinese restaurant node share document express derive massive collection demonstrate million million document bayesian process thing natural store might along decide well option end focus develop representation topic topic corpus inferential topic close become tree build chinese restaurant bayesian select limitation practical drawback document drawback example consider com compare article journal type word area yet structure relevant document rather analogy document result need multiple place subtree perhaps child topic dominate statistical compact prior perform restrict path drawback develop nonparametric objective access entire document specific make refer dirichlet allocation mechanism whereby define base collection dirichlet let hdp area separate topic thus corpus significant development inference lda model continue development ability efficiently corpora amount section dirichlet chinese restaurant process hierarchical present review scale well massive document compare hdp dirichlet build bayesian nest chinese restaurant process constructive process foundation nonparametric model rely model set statistical trait effective trait mixture process eq q respective parameter potentially infinite induce partition atom dirichlet briefly probability measurable distribution application representation chinese restaurant crp avoid integrate dependent unique display property give insight evident unique limit crp analogy chinese restaurant table select proportional customer select chinese restaurant work stick construct draw construct broken stick index expectation ig write mean inference crp chinese restaurant tree extension crp crp analogy accord crp restaurant uniquely indicate arrival customer act crp restaurant table occur potentially table tree number start child probability crp constructive construction dirichlet rise stick develop construction later nested tree li paths stick break child equal stick construction child index index child sequence atom topic document leave nest crp generate corpus start nested document produce prior stick break dp standard document drawing topic draw drawback therefore corpus specificity create appear many possibility topic document learn situation document three topic insufficient multiple occur bayesian document corpus path key path allow paths goal hierarchical hdp hdp dirichlet base continuous dp place probability mass atom base atom advance hdp dp use great topic nonparametric lda explicit representation hdp representation rely stick break identical difference framework document access still coherent ideally document main toward word path topic path share tree nest hierarchical first path path document share global stick break construction dirichlet transition dirichlet follow index dirichlet document dirichlet atom tree represent dp stick breaking eq eq lead dp randomly mass place one tree node since probability allow document place atom first proportion high probability leave probability document high node delta recover generate document draw base draw beta topic path mean allow atom path stick breaking determine stick break specific act currently determine topic node continue select breaking imply document imply equal path select independent aid development nature stick break evident along subtree tree al share cluster within summarize text corpora large slow currently index million article last year fast essential bayesian idea inference field inference variational variational exploit variable share brief stochastic work splitting group local one group update
network break opinion certainly affect opinion social agent element adapt environment self separate self interact political influence international effect point simplify control change illustrate figure simplify description htb political political opinion political political support exchange member interaction strength characterize interaction property number strength independent evaluate extreme left extreme simplify reality quantify depend distance environment average effect party way respect opinion interaction equilibrium mechanic temperature enhance interaction party party clearly equilibrium system environment interaction differently right political leave political viewpoint keep spectrum effect essential opinion political party viewpoint attractive character represent party opinion party party age two party opinion splitting way proportional interaction party party large merge age composite party merge split two mean party stay keep original agent one concern belong belong party contain graph opinion gaussian uniform political probability agent evolve belong party randomly another different party agent belong party depend party randomly agent belong proportional respective maintain contact agent opinion political party party party change opinion opinion effect take interaction replace dynamic time ratio characterize difference dynamic opinion correspond agent isolate allow party political opinion uniform set evolve stable range political rule interaction whose hence middle political merge decrease merge party mainly middle range accordance reality empirical study mainly european political spectrum party website datum locate political confirm influence decrease quasi stable state show straight line mean evolution relaxation quasi political long reasonable large evolution opinion whose number reason remain go splitting figure exponent initial large fix later certainly party union enhance since small observe sharp record empirical collect website empirical dividing find law exponent close heavy tail influence merge party link add opinion dynamic quantity party party agent link agent agent typical evolution indicate formation political character opinion maximum quick increase around sharp happen time period party sharp period period determined large initial value political increase agent formation political period agent almost evolution party degree belong party divide degree agent maximum average change despite formation political average degree result link evolution belong formation link link respective political time semi delta distribution formation political consider interaction strength evolution agent opinion social role opinion formation environmental force temperature temperature small average opinion large yield large strength show opinion result without opinion initially shift however always end opinion opinion leave number final number remain party accordance result remain political remain locate side political jump happen agreement effect interaction shift right political weak exponential small close temperature relaxation time work propose opinion compose interact mechanism opinion merge split link interaction political contact environmental influence dual opinion take party exponentially initial alternative period initial exponent verify state regime parameter exponent political regime place temperature opinion temperature numerous remain end political matter mainly strength dynamic close middle political spectrum agent accordance agent political topological structure political formation character consequence merge splitting dynamic extremely period political characterize degree effort realistic opinion network agent opinion party influence feature dynamic strong weak consensus dynamic accordance phenomena party exponential phenomena
compare covariance select note mean tend pattern sharp within band interval drive series wide band dependence market main index stock tool allow synthesis various stock underlie trend market specifically index weight stock weight equal market date date application stock index select behavior motivate logarithmic return national stock index distribution return heavy trend financial recent typical financial obtain well characterization market discrete induce latent perform rescaling sufficiently rapid figure miss solely burn trace show rapid dynamic volatility occur world financial european rejection budget usa jt accommodate heavy tail slow rapid ability capture economic finance line quantile usa european without dot quantile correlation usa failure mae mac financial financial save peak european rejection grow financial instability correlation national posterior immediately clear economic world european systematically south showing confirm consideration exhibit economic structure colored place problem update covariance predict update ahead return correspond future unconditional b mean step ahead previous gibbs return national country become I compare improvement analyze ability obtain characterization play crucial formula calculate present multivariate dynamic simple conjugate tractable moderately model heavy tail key enable fast useful study highlight flexibility respect multivariate show dynamic capture economic financial market demonstrate analysis financial reference heavy mean sharp iv update data vi possible explicitly incorporate prior dynamic detail sampler l equation simulation apply factor recall property state joint reproduce dictionary element equation relate recall lk I vector variance equation smooth latent state space finally b I kt k I I I ji I combine model lead eq online update distribution posterior lk I h smooth initialize state te similarly g te ii conjugate usual I important allow certain rapid inference prediction across smooth slow mis calibration interval substantially narrow wide time continuous allow smoothness covariance dictionary evolve linearly usual online algorithm bayesian assess simulation locally long multivariate nest stochastic analyze collect application monitor key characterize dynamic covariance cycle period rapid change model smoothness single restrict inference rapid event multivariate particular interest vary smoothness meaning couple smoothness construction locally factor formulation vector generalization autoregressive var polynomial spline spline implicitly locally local include spline perform well location knot selection via mcmc computationally moderately flexible kernel separate estimating time evolve weight move see use extension accommodate vary maintain flexibility covariance generalization variant demand step time formulation however uncorrelated evolution fall flexible vary lag missing tend application inconsistent varied join machine effort propose improve wishart process scalability continuous generalise wishart regression via trajectory covariance behavior process sharp wishart motivate rich dimensionality base model apply approach dynamic reference cite therein recent multivariate loading evolve dynamically sparsity latent lead improved portfolio utilize vary autoregressive result computationally proceed validation emphasis develop process accommodate covariance regression loading factor combination dictionary miss data scale moderately motivate assume stationary period sharp difficulty nest process reduce involve locally smoothness latent computationally accommodate specification explore updating compare study finally stock market across examine focus take probability variate realization stochastic assign algebra subset propose process broadly inference article focus equally spaced miss simplify accommodate substantially observation modeling rely dimensional successful advantage frequentist characterization loading residual recent article large cite interested vector comprise lk lk restrict attention generic process dictionary smoothness derivative expect center represent induce vary gaussian noise tw otherwise induce smoothness base markovian imply key advantage extend along instantaneous lk lk element represent nest kt kt kt crucial aspect firstly grid relate discrete time represent I scalar time factor kk time loading sparse linear set play crucial reduce model computationally tractable process time loading loading evolve closely refer loading characterize accounting loading vary independently nonparametric induce fundamentally carefully modify unique potentially constrain enforce identifiability undesirable dependence response follow avoid identifiability ensure identifiability induce characterization hierarchical approach maintain allow mean respectively lk inverse gamma k b great infinity shrink towards lead element inverse prior independently simple smooth sampling parametric outline draw instantaneous mean variance recall shrinkage hyperparameter realization state smoothing consideration mean minimizer ty ct respectively lead less suggest hyperparameter allow variance strong discussion structure smoother require state respect associate smoothing technique allow smoothed series estimator procedure choice recursive formulation approximate posterior available computationally tractable online initialize ahead moment latent computational conditional state sample I overcome problem previous initial drive initialization use online study multivariate specifically pc assess whether extent accommodate sharp vary flexible setting locally technique structure smoothness process subsection covariance time simulate shrinkage posterior use truncation higher find concentrated around parameter element proper initial place accord via similarly prior state balance rough mean also hyperparameter set heuristic outline run discard burn sample implement choose minimize mse estimate covariance formulation correct explicitly process directly therefore spline reproduce mainly simulation specifically truncation reproduce smoother without iteration reach discard burn analyze procedure splitting
value median bound vary highlight near box concentration four absence shall axis limit link period curve around day day vary fourth value peak day introduce step incremental continuously offline unlike synthesis functional streaming series european study abstract summarize content abstract appear user read chapter unless particular series book file chapter template file manuscript plain appear stream gain lot sensor use monitor physical variable traffic necessary potentially infinite flow store computational nature require development incremental exploratory processing cluster method deal identify stream stream usually cluster stream second stream homogeneous cluster observation moment aim find usually method accomplish identify centroid synthesis homogeneous since process synthesis point development stream cluster approach strategy name micro stream keep micro macro reveal final micro algorithm basic come sensor record variance group multidimensional extend introduce streaming allow fine streaming keep account five third median variable site grid propose incremental description track dynamic evolution incremental sense maintain datum receive stream overlap window approximated method micro splitting incoming overlap window associate window update appropriate micro macro cluster micro first incoming overlap window frame subsequence subsequence form subsequence jt jt residual zero summary streaming jt jt w jt jt jt min jt max w jt bind functional analog vary band modify band order band functional statistic envelope region analog band central region envelope represent box classical box envelope dataset vertical formally sample subsequence th band depth curve curve integer third micro functional jt jt f l jt jt dt min min jt jt h max dt allocation unitary increment micro cluster centroid keep micro allocate micro structure however depend choice high involve micro contrary bring generate micro deal value threshold micro allow propose compute micro cluster centroid micro much exceed alternatively discard micro merge micro update store parameter micro micro long reveal stream micro line get predefine snapshot micro snapshot collect update instant order user slot identify snapshot close upper snapshot remove micro effect occur snapshot micro allocate functional removing slot snapshot snapshot store centroid allocate item become process provide output micro macro summary interval heterogeneity macro iterate attribute cluster whose minimal centroid representation macro mean result available record daily make corresponding weather locate period weather way daily local record hour reason observation unable observation record accumulation since see high day could day second daily stream overall trend phenomenon help behavior along observation period represent functional assessment
c approximation term however universal require non result performance use order action verify effect design observation functional could show rmse converge smoothly exp become less exp rely every preferred automatic exp another implementation assess automatically exp move sophisticated partitioning become strategy look ahead practical setting evaluation partitioning space strategy behave dimension great idea across core decompose vast streaming thank lee discussion shape gray increase parallelism set particle filter second popular instead analytical inference paper proposal implement ahead decentralized implementation ahead outperform particle filter bandit enable particle filter problem example paper rao particle filter one two group condition compute group analytically decomposition analytical solve relevance however happen analytical derive approximate iii instead group group state automatically answer ii filter name decentralized effectively group particle use particle conditional parallelization parallel filter graphic gate array resample chen filter stream chen approximation performance obtain carlo ahead resample exact rao ahead describe significantly well pf domain context look bit involve monte experiment ahead standard question pose continue appear future paper paper involve online bandit lead future pose ahead strategy la paper state decompose x n z govern noise equivalently nested subproblem subproblem interact depict diagram implement filter nest except filter algorithm must employ subproblem use pf particle nest subproblem state derive look ahead reader mathematical involve derive pz ty jj n j n evaluate j time importance approximation eq importance weight markov process numerator express quantity numerator marginal replace approximation importance importance sample mechanism note require achieve expression rule proposal mechanism yield application use distribution marginalization approximation detail compute step careful keep track index aside filter common proposal intuitive complexity derivation considerably take prior proposal take importance optimal simplification importance predictive expand draw importance suffer major require ability new calculation importance step finite jump carlo approximation next expand follow hope particle require moreover small resample pf gain mention possible optimal swap resampling enable intuitive benefit illustrate look px pz multiply discard j pz decision decompose system play role subproblem decomposition group large execution change reason choose adopt state group consider choose possible action probability proportional reward action repeatedly yield gain different closeness observation predict real action jt initialize get select tt jt tt drawback try assume reward introduce overhead especially exp stand exploration exploitation information try call subroutine exp receive ensure associate exp simulate vector fill try code vanish regret conduct experiment compare performance ahead algorithm meaningful benchmark adopt multi stationarity kalman filter fail study bandit z ty e experiment standard pf simulation interval
could prune alternatively sequence contain pruning pruning let shorthand max score prune q th good position condition eliminate position fact immediately eliminate sequence definition max possible pass max true sequence edge pruning path label position equation suppose prune standard framework recursively label definition max vertex main pruning method apply give pruning shorthand average thus prune edge pruning position transition hmm reduce q per similarly sequence pruning wish perform update define eq sparse inefficient instead marginal reach decompose three piece piece piece prune logic decompose along hmm require position pruning modify definition pruning separability condition mistake simply prune average marginal achieve pruning vertex pruning pruning possibly suggest enforce choose threshold give follow pruning always base least strict pruning function pruning represent dataset pruning train pruning eliminate reduce vertex marginal position pruning pruning reduce volume respectively observe training development observe training lead set volume development example gram gram new test reduce go reason lattice pruning go run order baseline send cascade ms pos state n
consider monotonicity stress recommend implement overhead section theoretically section serial give precise solve list separable new paper devote h set block grow speedup towards situation grow partially speedup various summarize depend structure block choice framework encode c comment gradient serial serial random parallel new distribute continuously serial give deal partial block euclidean column small knowledge strategy fix uniformly name assumption treatment hence much beyond scope processor compute update doubly uniform nice binomial see definition three monotonic appropriate section later text select update ht bb bb bb l thm bb thm bb p thm en prove deterministic separable believe interest decomposition stochastic program derive method property name speedup situation well paper analyze coordinate algorithm set hence algorithm somewhat variant restrict value moreover lastly serial depend hard compute involve computation huge scale express easily computable quantity turn use hence necessary complicated allow update unfortunately coarse theoretical serial special update iteration uniformly involve node core achieve speedup variant control small set linear svms illustrate theory match reality open efficient implementation publish http code com iteration formally realization subset brevity characterize assign subset technical block proper sampling proper sampling block far say proper rise refine doubly generate equal whenever definition different uniformity uniformity let uniform whenever du characterize nu set nu partition note nice du processor core iteration block block assign block processor block independent ic ik processors core processor choose block random processor block two processor block parallel fast approximation nice du probability independently parallel iteration equal processor nice processor assign processor available iteration easy result binomial sampling nu update block du nu iteration block intersection du nu thin precisely du nu serial nu du cardinality nonempty doubly must n serial consequence sampling statement next sampling separable positive semidefinite w five identity let doubly uniform nice yield eq trivial doubly happen precisely eq vector choice recall choice explain need give answer step update fx k k one eq choice use minimization update could separable separable easily form replace w vector conditioning far note indeed separable recall h block update bound natural set monotonic separable proper admit indeed inequality step explicitly analyze monotonic sometimes behave monotonically enforce derive useful monotonicity separable form h fx fx analogous write intersection argument last combination weight vector j h plug convexity besides usefulness derive reason also say respect q j h separable argument formulate doubly proper doubly uniform let fx fx n could alternatively write combination nice nice sampling give serial prove descent composite monotonic result convex cover monotonicity restriction auxiliary hx fy eq convexity hx fy w fx w inequality ii prefer use finish complexity let nonnegative lastly property constant choose extend aid update deal inexact proper choose level accuracy eq fx random iterate otherwise since monotonic message serial see expression parallelization speedup du expression ht parallelization factor doubly binomial serial compare end fully parallel partial separability especially answer question speedup separability speedup sr ss processors illustration processor separable wish stress indeed huge scale ht choose target level accuracy letting since take finally could proof able hence convex force resort situation term w strongly parallelization speedup serial factor approximately average update parallel quantity separability show core problem hour sense tight predict speedup generate scale instance row enable particular exactly separability ht e e e e e e e e e e e e solve nice utilizing take around gb implementation version core iterate iteration reading proceed another current update soon compute second good nice allow processor pick block show table need coordinate total number coordinate coordinate serial coordinate would summary represent choice one conjecture phrase work turn roughly magnitude trivial find old coordinate slow slow choice ht progress coordinate update e move last nonempty show remarkable degree though method nonzero place local come parallel utilize core serial consequence observation core turn excellent figure b follow utilize roughly dimension separable demonstrate parallelization necessary reach equal measure iteration nice sampling value core core solid line marker seen figure theoretical reality remarkable partial degree separability useful replace far deep phenomenon experiment svm hinge coordinate coordinate nice sampling coordinate minimize box dual dataset gb storage separability e sharing feature theory nice add level processor nearly mean primal observe gap training decrease increase test small beyond say processor expect parallelization h ccc experiment dataset contain divide part training set testing cca gb depend small use nice depict regularize marker correspond epoch processor less achieve nearly offer processor similar work number however utilize processor job approximately speedup accuracy stop increase beyond rarely method h epoch ff section sec sec prop prop block associate block constant vector weight n x w x w update iteration update algorithm parameter central fx h ix fx tx ix kk theorem claim exercise author grant ep ep author centre numerical software grant ep publish http code google com dc parallelization apply speedup serial iteration simple processor separability separability case processor mode processor show involve hour core coordinate partial separability huge expect interest suitable optimization topology sense usually grow capacity grow dramatically decade fact currently challenge aim task algorithm big domain broadly strategy single block coordinate drastically reduce memory arithmetic certain iteration amount multiplication necessary usual classical smooth problem tolerance variable gradient balance observe author serial compete point truly huge necessary ever availability build root massive parallelization combine scalability coordinate topic answer observation minimize variable simplicity assume trivially processor task solve parallel processor parallel coordinate serial processor use separable would amenable acceleration parallelization acceleration reduction separability one message explicit simple formulae speedup many naturally mean promise big regularization unconstraine simple individual variable alternatively certain solution account reader simplicity next decompose precise assume throughout nonempty depend develop analyze need give many objective encounter big elsewhere block e ht logistic depend separability th function th partially separable section reader example nonsmooth partially function arise cut paper recently complexity various brief reader cyclic attempt complexity method update analyze independently regularize smooth unified refine paper separable develop linearly constraint include inexact analyze separable independently study writing aware paper appear various aspect topic building include describe block problem establish subsequently detail parallel contribution selection update development select develop expect separable tool promise computational instance gb core hour objective value iterate conduct problem come formalize summarize
form order optima denominator close close bound line quadratic behave distinguish hard bandit convex characterization setting result difference focus dimension extremely function provable bandit derivative free open bound focus strong convexity performance strongly smooth exist upper function table exist upper seem achievable quadratic function derivative free noise bound message algorithmic generic assumption still specific setting concrete classic set ridge sample derivative setting think query give draw equal set consider noise attain good useful round jensen round contradiction identical thm quadratic q letting get convexity fact substitute result term substitute expression rearrange simplify quadratic root eq root know nonnegative recall average regret remain take side specify let later expect deterministic strategy explain existence expect globally minimize smooth w w calculate substitute simplify get q exercise verify expression require item fraction verify use identity finally substitute require upper kl noise see q coordinate pick ball global ball satisfy thm microsoft institute science convex optimization bandit feedback gradient year upper low literature bandit derivative free available precise characterization imply trivial low moreover set require mild gradient rate despite imply contrary convex free convex consider ability query realization optimize important query free among solve unconstrained framework due datum learn kind study armed bandit optimization decision setting repeatedly choose realization error goal roughly value well bandit problem simplex refer discuss attain regret gap study mild convex query inherent free exception multi bandit query common non armed bandit away bound linear theoretic prove rely construct deal function much originally year seminal optimization pg quite strongly function enjoy known bound respectively bad dimension investigate complexity bandit free focus contribution prove strongly convex three question performance sharp bandit scale optimization linear provide convex analyze function attain error sharp show fast free may quadratic explain later establish low impose result fix domain show bound prove average derivative bandit distributional assumption stand armed free minimizer boundary domain however argue restrictive shall clarity exposition begin low tool tackle smooth set summary performance consider natural additional proof appendix quadratic tt combine table dependence strongly intuitively everywhere curvature subgradient intuitively fix curvature minimizer prevent consider lie proceed round pick realization bound wish domain free case uniformly noiseless pick noiseless set goal minimize namely possible derivative simply get return inequality bandit hard round interesting research extend quadratic definite away behaved assume spectral easily rescale see smooth appear provide insight case convex derivative knowledge free nonlinear oppose however achieve mild least one attain boundary query strongly common case actually situation assumption crucially strong convexity assumption must always optimum discuss free actually decay hold natural utilize gradient value function whereas query interest function structure query relatively far away use known point query simplicity input convexity initialize query quantifie return return iterate oppose average family derivative free instance repeatedly estimate feasible region since generally rate fast lemma appear algorithm perform bound average averaging plugging bound obtain tight namely bad derivative order besides strongly scale case function provable round possibly satisfy eq optimum must ball despite domain allow e bind appear sense see exhibit quadratic expectation attain imply uniformly gaussian random strategy deterministic deterministic generality randomization hold randomization informative value measure kullback determine kl divergence hard distinguish large none coordinate support return represent two use upper
yield eq together whenever l f g inner iteration iteration hence conclusion em kkt g g g hold constraint cx xx kkt monotonically bound monotonicity bound hold replace use obtain replace imply continuity lk arbitrarily find step hold accumulation kkt point notice k k k k pass subsequence assume side kkt convenience u I together modular lx q inequality due boundedness addition due conclusion statement dual view subgradient approximation concavity author attention pt section proposition remark assumption support grant establish iteration programming suitable accumulation generate kkt lipschitz associated result mention establish programming consider nonempty set convex necessarily throughout make continuous nesterov convex solve al order necessarily convex et al another special lipschitz ball recently convex smooth dc difference see program widely view q nonempty close scad ht ht l reformulate observe comprehensive order program suitable assumption accumulation generate kkt variant exact function establish outline paper establish finally inexact inexact solve convergence point normal directional along convex subdifferential establish constraint q eq satisfy condition observe contradiction inequality second xx every hence feasible point sufficiently minimizer contradiction assumption positive eq eq indeed suppose convexity hold define eq q convexity eq sequence notice contradict minimizer definition satisfy finitely cone differentiable convex convex exist let propose convex also variant introduce exact arbitrarily end em accumulation kkt proceed several lemma smooth continuous close differentiable lemma provide inequality close nonempty closed exist cx I cx iy x I iy iy iy I cx cx g convex em accumulation sequence generate kkt k I k exact method hold assume cx xx kkt since cx conclude cx I I k relation k I w w w immediately imply large relation yield verify let z relation continuity px x hz k side since imply kkt accumulation I v generalize em let hold
turn post provably separable near allow oppose robustness analyze dataset focus admit sufficiently permutation imply proposition prove proposition theorem close paper follow factorization permutation first proposition broad nmf hold q necessary apply theorem tight multiplicative condition multiplicative finally hold well synthetic post processing deal matrix th transpose matrix zero identity denote context diagonal denote trace cardinality sufficient column nonnegative imply sum constraint linearity simple would bound could improve stick formulation admit let combine equation eq fact distinct noiseless allow let one column dataset solution solution index imply show complete imply algorithm able reconstruct corresponding must process remove constraint discard since feasible relaxed program correctly tight bound multiplicative believe possible match theorem multiplicative able research apply rank nmf trivial admit separable factorization bad column belong section necessary construct matrix extract permutation investigate variant see provably entrie feasible large post noise bind prove section recall aim identify nmf give actually distinct distance extract twice desirable moreover nmf approximately let large optimal solution sufficiently large column extract construction contain construction theorem take extract extract matrix solution discriminate separable return matrix four extract original reason algorithm construction prove optimal separable construction condition et al column algorithm r loop entry distinct exactly equal correspond take checked column entry enter loop multiplicative multiplicative storing point negligible storing necessary hold vector distinct typically loop perform processing keep compute far design sophisticated extract relaxed variant processing use pre point noiseless robustness hold th datum neighbor discard processing et input topic algorithm inside assign show proportion claim open robust theoretical th let separable eq et bounds bind dominate differ arbitrarily column ill condition dominate factor et however know efficient perform synthetic conclude algorithm provably experiment superiority near run ghz lp develop construction theorem three avoid towards objective separable display column correctly extract extract one column even algorithm large level corner identify identify extract process computational post negligible needed propose provably prove tight also design computationally seem polynomial noise input section bind input impractical seem acknowledgment author would point also belong hull program larger small although simple let program parameter depend feasible feasible optimal solution linearly interval note since one large lemma imply show clearly q verify let let result construction let let optimal first x j r give another equal z last solution therefore problem subdifferential subdifferential slope minimal show equal q mm proposition corollary question institute universit de la nmf recently separability span column problem particular r nonnegative program programming refer new general provably process separability programming one express nonnegative combination nonnegative space aim weight np show separable separable column nonnegative trivial aim small cone generate mx mr reference therein separability sense situation correspond document approximate correspond bag discuss practice separability row row require see discussion separability use imaging refer separable easy vertex see robustness author rather paper general provably variant noisy also noisy exist nonnegative matrix submatrix permutation notice column column et propose separability prefer identify distinct read weight equation use column column entry interesting always
empty two sample respectively empty wherein linearly two column one row option row z z w option wherein apply empty state q apart active part reduce mention complexity complexity find one compute I move accordingly active maintain remain thus become newly modify place replace previous doubly sample consistent maximization hard margin elimination criterion combine lower publish related formulation misclassification additionally feature elimination soft sense utilize radius sense whereby therein whose minima small leave combine radius low light devise qp qp publish method novel herein outperform attain low several feature promise especially one employ tune herein herein information support vector svms interested reader brief manuscript label f w hyperplane act sign decision boundary separate optimization svm linear optimization svm solution associate lagrange multiplier efficiently since svm discriminant express solely kernel trick svm concept maximization hard sense call pick margin reduce retain step actual index leave margin anchor alternative second counterpart margin pick elimination objective elimination every classified elimination elimination hybrid slack little hard sense basic parameterization pose standard elimination decision herein refer solve intuitive lie right intersection point two feasible region lie thin dash accordingly region cone bound slope slope incorrect region define cone slope maximum slope label e generalization curve svm fewer particular elimination feature comparison note author current manuscript include fact discussion currently place result somewhat explanatory isolated explicitly specialized active provide feature feature elimination randomly split select fold validation trial trial average hundred highly probable separable feature hundred herein eliminate separability separability lose half qp feature hyperparameter select candidate denote classifier perform elimination jointly incorporate hyperparameter n mention specific highly form many zero discuss page great e optimization refer give assumption work qp mathematically utilize solve fulfil row need column discuss restriction lemma introduction enter picture highlight point intuitive description central point doubly keep doubly doubly move along active complete movement whereby less movement case movement doubly likewise doubly like single direction summarize movement possible call decrease objective sample movement movement switch movement take place switch margin active represent wherein pair row top half respectively elaborate row block two top category type sample category doubly property elaborate restriction raise nontrivial constraint initial initialization recall piece multiplier classifier ii generator explicitly identify particular would doubly generator assign albeit scalar multiplier set scenario brings assign indicate set wherein via multiplier less sample margin unfortunately within reliable determine extra measure determination find generator provide classifier multiplier discriminant sample positive normalization measure whereby utilize scale large sample paragraph possible provide active numerical make proceed describe essentially switch I move along current doubly movement new constraint require least constraint requirement address fulfil theorem ii qp fewer long summarize part qp qp extend elaborate introduce index row half bottom note contain index plus sample index specify sample index whose constraint e mean active denote three structure cc nm row linearly final row appear row placing
convergence epoch basic weak form locally restrict strong convexity rsc condition allow pool convexity rsc note namely low nontrivial pair rsc strong direction relatively sparse statement whenever application condition convenience namely simplify f choice prox generalize allow length apply dual epoch length regularization suppose expect satisfie epoch length cardinality q theorem return corollary apply iterate dual bind state epoch suppose parameter universal cardinality valid relatively average central proof optimal note proposition equation choice addition epoch future reference note make elementary inequality simplify version rsc condition condition average simplify equip broken case perform satisfied control I proposition recursive epoch hold epoch epoch upon substitute set simplify eq ii feasibility throughout run appeal simplify recall bind error cauchy simplify equation bind epoch algebra bind simplify result net epoch summing series give order convert error epoch let complete computing epoch set inverting allow deduce error bound term geometric inverting algebra observe bind stay epoch thereby bound long feasibility epoch epoch since large hold set early epoch epoch feasible algorithm epoch lemma epoch prox length equipped specifically least observe thereby corollary set yield calculation minimize order must demonstrate rsc notational simplicity shorthand yield corollary know consequently rsc suffice quantity matrix study result appendix rsc complete corollary main theorem lipschitz calculation since check satisfied plug early bind epoch iteration epoch early iteration epoch additional development prove argument ability error epoch epoch may continue epoch become enough encountered theorem error increase treatment epoch simple fix epoch challenge behavior epoch feasible first epoch lemma know constant lemma apply setup epoch epoch complete substitute recall exponent straightforward duality norm perform algebra refer behavior epoch well optimality rsc lower combine bind yield far simplify rearrange yield establish allow translate pair since lemma write q triangle substitute eq rearrange tolerance final use inequality equip begin shorthand result prox function form al upper bind simplifying require provide appropriate assumption size eq control start guarantee valid assumption bind gradient lemma control random early q inequality complete convert bind simplify rsc minimizer hold feasible minimize combine yield recall shorthand close statement application conjunction result convert cone result observe consequently piece universal substituting notation invoke rsc apply rsc yield rearrange recall notation complete inequality equip ready inequality lemma rsc inequality hold explicitly rearrange combine involve term upper triangle inequality yield identical early different finally prox center control error assumption recall meet epoch length suitably complete proposition state straightforward algebra remain result exploit martingale tail particular et sequence start lemma suffice recall plug statement complete turn moreover measurable deterministic satisfied h old use update condition satisfied plug state suppose condition proof result combination lemma feasibility epoch schwarz argument second upper obtain previous rsc second proof straightforward epoch fact consequently update repeat obtain assumption note epoch center construction guarantee recall substitute recall establish completeness establish universal ensure establish condition appeal define eq substitute rearrange case result assume ensure epoch calculation finally inductive claim inductive identical argument complete inequality universal epoch thus obtain bind accordingly rsc condition translate rsc assumption recall shorthand second f combine yield use inequality combine condition translate analog yield respectively eq simplify error least useful theorem lemma simplify simplify rsc technical main ccc department department institute technology university california berkeley loss optimum approximately yield objective dimension successively nesterov establish iteration sparsity locally include square loss statistical result rate effectiveness confirm baseline square stochastic desirable feature accordingly intensive several reference therein efficiency provide sharp function rate strongly optima sparsity precisely enjoy extremely sparse result significant convexity boost square type approximate prove useful many application overview paper reference many mild logarithmic precisely solution entry stochastic mirror descent converge loss linear improve rate attractive slow opposed rate strongly encounter exhibit convex optimum approximated enjoy type rate mild optimum use meaning substantially build multi new sparse optima objective regularization quite natural sample later stage effective appropriately close many example summary development structured simulation confirm regard convergence superiority compare algorithm direct method inferior regularization critical keep presentation restrict multi step note similar result mirror accelerate optimization focus extend structure subset optimization stochastic access vector goal design suitable optimum zero sparsity inequality upper involve optimum well choose appropriately contribution involve epoch sequence number specify length specify q epoch update geometrically upon termination r averaging averaging operate I prox describe average prox choice prox used previously see g example space prox feasible appendix schedule initial prox prox initialize worth subgradient inspire nesterov composite minimization prox norm compute prox mapping prox mapping prohibitive dimension discuss stochastic begin lipschitz instance sequel gradient sense goal fast locally strongly step local convexity concern local convexity refer strong statistical theoretic analysis sparse concern produce stochastic subgradient tail sub gaussian gradient constant component hold sub tail satisfy previously help apply scenario loss consist pair covariate predict decision rule minimize appropriately choose common hinge distribution either strategy illustrate assumption satisfy thus bind exponential marginal zero let minimum taylor expansion sampling pool convexity bound relatively simple verification instead boundedness sub tail maxima problem least covariate condition proceed condition local covariance exploit old inequality semidefinite via likely hold tail hold rsc condition establish sub need assumption tail vector obtain sub analogous calculation position property result problem rsc radius globally hinge logistic example apply square somewhat treatment dual averaging algorithm state epoch base choice suitably radius met outline use result algorithm iteration convexity lipschitz involve subset q measure sparsity result length work objective predict overall apart scale sparsity second concrete optimum bind yield compare perform stochastic gradient assumption choose bound find suffice choose similarly converge scale fails exploit sparsity descent regularize strong key exploit sparsity convexity mirror prox break epoch fails exploit strong obtain inferior convergence close spirit approach decrease schedule consequence overall guarantee procedure understand nesterov use fixed set initial tend reduce factor improve slowly regularization note work zhang assumption allow follow section set cardinality corollary specify high recall simplify setup corollary least directly note result generalize factor develop assumption approximately notion approximate sparsity formalize enforce magnitude small ball vector assumption simplify corollary capture update constant range range rather interesting show precise convergence sparsity seem rate obtain rate leverage phenomenon exactly match dimension sparse optimum convexity translate
theorem usual nonparametric framework construct rich packing construction independently risk norm norm tail bound important correspond tail random variety upper potential eigenvalue need nsf science fellowship dms nsf grant anonymous helpful comment relate canonical angle addition identity immediately bound principal lead eigenvector belong ball bound sharp obtain constrain pca exceed statistical increase necessity goal intermediate either size affect principal pca perhaps situation tend consistent situation sparsity eigenvector perform application error bound lead eigenvector wish look uncorrelated dimensional subspace orthogonal projection eq optimally reduce reduce pca spectral sample eigenvector analogously reduce pca eigenvector pca inconsistent beyond without addition enhance interpretability sparsity notion effect statistical inference research work add maximization relaxation form pca iterative fashion sort reason lead eigenvector ball provide appeal notion concrete correspond hard sparsity number entry vector small soft realistic nonzero statistical difficulty feasibility eigenvector fundamental statistical thus gap tractable main consideration usually constraint formally sphere large satisfying ingredient estimation distance unique frobenius non uniqueness eigenvector ambiguity turn euclidean asymptotic effect dimensionality highlight low minimax use base construct constrain maximization estimator solution feasible active constrain ordinary since replace optimum interesting point coincide difficult subsequent efficient pca analyze semidefinite sdp eigenvector magnitude matrix estimate notion imply wide covariance imply consistent covariance eigenvector condition imply minimax ball remarkably sequence model inspire aware provide bound ball attain allow optimal condition guarantee non proof lemma mainly technical proof
easy task relation treat uniformly closure inductive database also embed turn true programming language possess usual construct input way embed require language comparison method text categorization page computer task category student figure diagram relationship web page represent make totally even feature domain flexibility soft match occurrence page count text categorization problem additional signature common course page contain string follow page leave setup table research course f p course course research course task essentially encode domain exploit atom signature enforce mutually exclusive scale conjugate gradient describe wide result report mc collective result slightly low identical extract capture contextual lack page page page anchor link report table worse attain although accuracy comparable requirement term cpu core intel core take iteration attain internet cast working picture focus predict box office task interpretation directly necessarily interpretation notion allow overcome ground interpretation disjoint endowed total natural movie production e associate atom entity give reasonable training use ground ti ti ti similarly ground atom create movie outside discard appearance movie datum summarize model signature movie actor production additionally signature count movie involve movie year test curve summarize comparative three software ground introduce together hard query year design capture actor false case use exactly signature definition discriminative implement obtain fact fast complete phase follow consistently high year language logical inductive logic kernel underlie inductive logic programming sense variation background probabilistic entity al signature play role programming combine r provide diagram graph prove helpful hand framework signature predicate need relation inside inductive logic system relate relational logic logic logic program relational input partial interpretation prediction statistical relational implicitly logic program process network diagram statistical distribution learn case result tie particular occur template match statistical learn former really knowledge feature use feature relational commonly collective combination structure iterative incorporate em another inference try also kernel see distinguish graph obtain relational rich symbolic kind represent regard define represent specify collective learn relational relational approach relational graph explicitly employ derive resemble several mining employ graph relational encode bipartite whose connect atom atom usual employ learning mining typically label relationship node approach single representation three logical language base address language processing perform extraction exploit arbitrarily connect field dependency sentence dependency indirect logical logical relational kernel method constitute principle statistical relational rather use relational learner perform label original formulate wide learning task state art statistical also programming step relational aspect possibility perform collective semantic follow predicate collective amount important case signature complicated system example dependency regular structure programming exactly conditional field collective graph search kernel eventually inefficient develop purpose interesting open collective produce provide therein set another potential make use define concept neighborhood nearby combine generate consequence informative dense induce case kernel bias library integration therefore important future though implementation implementation issue similar employ develop vision acknowledgment analysis partially sf research partially start grant anonymous associate constructive comment code augment keyword signature header one clause clause signature connected header predicate mean domain additionally predicate measurement comma period signature signature signature header header level role level sake closely notation use call variable vertex bipartite vertex every root adjacent indicate radius edge whose subgraph neighborhood radius vertex repetition symbol denote map edge denote preserve preserve graph edge invariant follow give x semidefinite semidefinite eigenvalue kx converse reasonable always kernel call extension valid kernel kernel sum iff part denote yield demonstrate instance decompose q valid convolution way kernel part instance root neighborhood construct apply present overview hope efficient non graph assign hash graph hash sort edge hash hash hash hash sort vertex novel create compact molecular tool space present hashing mainly efficiency gain introduce vertex edge vertex sort sort list root vertex list root edge label edge label assign edge triplet uv encode vertex plus label graph resort construction list integer pair vertex note space hash naturally tradeoff space hash remark di di novel approach expressive logical relational specify build interpretation entity logic programming database access rich technique first graph entity diagram mix numerical symbolic program inductive logic system apply tackle popular collective logical learning learn database relational fairly representation alphabet difference logical distribution interpretation machine addition probabilistic type method learn vector amongst popular notable relational receive lot commonly accept markov probabilistic programming logical language wide task ultimately fill introduce logical contribution relational similar inductive logic base kernel call space eventually whole discuss relationship logical learn specify type logical relational interpretation entity object relationship naturally employ constitute start logical relational unlike difference instead feature immediately define formulae construct graph entity relation background formulae comparable much rich learn level specify logical relational description describe structure system interpretation transform equivalent knowledge system also conceptually represent graphical turn lead logic tie together capture important flexible specification entity completely graph employ subgraph reader keep mind incorporate similarly mainly variant learner situation section discussion main domain formalize assume interpretation background statistical position context system logic mn formalize problem section report finally relationship system detail illustrate real demonstrating capability anonymous science entity include student paper specify e activitie certain task student e binary student come atom logic function allow atom tuple relational interpretation logical ground interpretation correspond different interpretation false person post post person person person person person person person person person person disjoint database domain entity diagram two unary diagram diagram frame number leave position p student example signature see signature bias system annotate list explicitly file clause name list argument entity signature numerical categorical atom regard connect atom constant procedure signature signature introduces entity signature powerful database may argue co work together activity signature signature associate signature frame student paper student student signature predicate signature effectively paper student publish together relational method similarity map ground atom bipartite undirected graph node connect entity former interpretation figure predicate signature e base use procedure give section respectively person clarity still signature predicate e predicate accept signature specify learn statistical interpretation logical set atom interpretation unknown natural statistical direct indirect inference sake throughout case think consist statistic response framework distinction reflect ground interpretation partition atom query goal supervise associate interpretation e fit statistically motivated measure may interpretation cover number relational binary categorical naive share model yx actually optimize hinge maximize conditional output svm hinge bayes cite conjugate rich relational naive hmms logistic extend sequence hmms setting output simple three every pair feature discussion move suggest alphabet mention begin natural extension hmms stochastic logic expressive relational logic generalization context free investigate discriminative linear relation markov logic loss mn cope adopt rich contribute language generate relational feature relations logistic crf discriminative svm svm hmm table arrange share embed represent learn signature predicate signature g program specifie call library predicate specify formalize object atom name tuple learning expression name signature contribute every atom predicate semantic way take logic specify predicate clause introduce knowledge inductive logic hence ability specify translate maintain fashion opinion key reason relate logic order ensure well subsequent introduce assumption perhaps clear keep separate directly build function relational consist atom distinguish key distinguished relation introduce relation relation job relation signature must entity column represent domain see relation ground ground atom partition dependency predicate interpretation partial consist atom require interpretation output accuracy ground several situation learn minimal ability analysis simple implicitly predicate attribute interpretation construct notational atom every every atom ground appear uniquely denote atom map assumption degree degree vertex unbounde grow e web diagram diagram template atom template expand ground graphical network markov logic semantic quite perform interpretation requirement meet practice efficiency size phase flexible bias variety interpretation graph several exist literature reference therein interpretation tuple entity additionally subgraph kernel suitable sparse discrete edge propose deal soft match large graph whose tuple mix discrete introduce extension kind decomposition kernel part pair detail see integer let root let also introduce identifie pair neighborhood whose root pair neighborhood radius root family denote indicator root neighborhood center vertex section reason consider extension impose upper kernel furthermore g distance build countable illustration neighborhood subgraph bottom neighborhood subgraph fix radius neighborhood application dataset potentially induce section maximally discrete tuple allow case atom maximally obtain concatenation signature attribute follow denote feature transform correspondence subroutine problem whether algorithm bad degree hard high entity play relationship prefer efficiency distance spirit idea function likely number pseudo count neighborhood subgraph match express vertex exhibit likelihood neighborhood match quickly yield dominant overfitte well relax type match subgraph match base match cost give subgraph multinomial structural subgraph replace vertex pair close vertex either neighborhood subgraph dot histogram illustration soft generate kernel represent assumption vertex label vertex tuple extend allow match tuple discrete mix extension mainly focus categorical characterize define tuple subgraph write map vertex name set atom ensure signature tuple six discrete choose property treat atom vertex contain structure match jointly match concatenation tuple formally labeling receive uniquely identify summation correspond finally tuple successful match match tuple match product tuple fashion match tuple label value vertex product tuple collection index analogous labeling canonical identify neighborhood label vertice identity real tuple standard dot convenient efficiency reason knowledge select subgraph signature equation like vertex kernel neighborhood vertice kernel interpretation table conjunction plain kernel machine svm move involve entity tuple entity induce alternatively convert subproblem simplicity job relational ground atom intuitively target query specific entity web page section tuple link viewpoint highlight define figure illustration define graph center finally
first integrate laplace determine covariance discrete integral marginal formally equally give interesting illustrate consider case marginal pass diagonal procedure entropy product marginal constrain marginal easy lagrange multiplier constrain integrate independently express turn eq find actually I another integrate write lagrange multiplier correct covariance note form condition thus conversely end somewhat first trivial entropy marginal nontrivial correlate gaussian marginal index drop notational gaussian easy correlation distribution concavity maximize unique satisfying requirement prescribe marginal establish upon integration generalization expression joint marginal perturbation expansion power rather write power write correct average take guarantee positivity highlight corresponding maximum distribution expression immediately conditional thus result second order perturbation hold assume prescribe marginal construct distribution prescribe marginal entropy straight couple result hard exception gaussian among encode case marginal scheme moment perturbation present grateful valuable discussion grant determine specify marginal consider condition unique choice correspond maximum entropy maximum entropy complicate around product marginal moment give far know assume marginal arise range financial economic eeg system mechanic name finance become marginal copula describe two random variable frequently measurable physics ill infinitely distribution none marginal lift joint maximize
onto use line adaptively prevent maximum simple modification constrain specific project onto technique onto tf project orthogonal complement find basis subtract original rewrite decomposition projection q position constraint un tf f tf line convex program easily subproblem column approach individually subproblem program original value nature difficulty b inside conventional similar direction set brief total derive formulation bregman split tailor z augment lagrangian solve duality coincide method concave g b fix soft formula inversion computationally expensive inversion significantly simplify identity iterate parameter also particular x x k synthetic part concerned almost reference ask care exact true operator effectiveness imagine reference explain certain detect obviously learn mean variance need project onto make project randomly component reference normalise chose corpus project subgradient iterate time check operator reference trial plot practically start point far reference check equation investigate role simulation set population simulation show locally identify less distant way demonstrate consist optimisation program alternatively respectively whether satisfied repeat pseudo plot respectively column row many locally base haar imaging community figure select dc pseudo select constant choose enforce operator learn seem operator choose overcomplete haar operator haar learn objective base algorithm optimum problem use provide generating checking indicate learn synthesis use average dct dictionary row demonstrate synthesis well dictionary svd lagrange multiplier use previous average trial synthesis dictionary slight db dictionary framework marginally behind synthesis promise field denoise find reason leave row right middle ht learn synthesis omp paper analysis operator fact want select appropriate introduce canonical constraint suitable practically show relevance demonstrate recover synthetic analysis operator enough give two image class piecewise operator similarity finite harmonic type observation select part toolbox program optima program avoid check minima recover operator close neighbourhood rewrite optimisation admissible provide pair solution actually check optimality achieve tangent constraint subsection local condition optimality local zero objective tangent constraint see definition positivity violate easy contradict optimality complete lemma condition proof upper hand note x local lemma similarity separately term parameter separable may framework analysis unknown system operator proposition proposition dictionary learn synthesis operator identifiable follow note act identifiability use variational identifiability let n l v x constraint r nontrivial constraint training st statement iff follow positivity x c secondly identifiable identifiability cardinality produce pattern give variable theorem sufficient identifiability optimisation two constraint reformulate right multiply al presentation mi collection approach sample synthesis counterpart transform use overcomplete optimisation l optimisation optimisation exclude trivial solution answer conventional constraint normalise frame project demonstrate ground training set image noisy signal realistic optimisation often two practically identifiability learn representation low model dictionary learn appropriate actually involve describe signal example include music speech edge video low dimensional modelling signal overcome decade many achieve natural crucial dimensional imagine signal class dimensional possible map restrict admissible simple option map ask familiar structure synthesis set overcomplete call index cardinality processing perspective easy task live union subspace subspace one look satisfy come computational aspect look match operator zero row compatible interpretation structural synthesis sparsity signal work synthesis play ask signal design reader refer survey dictionary show dictionary perform dictionary wavelet learning method also norm preferred formulate admissible learn coefficient reason exclude trivial solution signal modelling scale dictionary atom frobenius norm ball make optimisation different sensible often alternate objective upon designing transform decade many harmonic wavelet fast transform perfect reconstruction signal relatively adaptation model current signal application learn counterpart formulate optimisation perhaps surprisingly analysis problem even scale report exclude compatible operator wave atom give optimisation subgradient implement practical open cope give local propose optimisation nature operator develop idea primarily remarkably concept pointed synthesis approach already learn optimisation trivial solution become trivial recent k operator promise specificity explicit optimisation expression analysis clean bold letter vector bold capital present list correspond subscript frobenius abuse notation absolute operator canonical norm space denote whose subscript iteration cardinality representation similarly cardinality x subscript original parameter elsewhere bar complement c c pf signal synthesis operator operator optimisation briefly constraint introduce optimisation algorithm simulation scenario synthetic operator appendix local optimality analysis purpose I dimension problem solve new adaptation problem optimisation use introduction future unconstraine exclude solution admissible assume thus formulate prefer lagrangian multipli simplify reformulate problem noiseless candidate suitable exclude solution space operator signal simplicity constraint individually exclude solution combine normalised frame subsequently constraint smooth differentiable optimisation find optimum repeat enough full operator close closure admit sequence function complete resolve ill conditioning geometrically separate let case ai admissible constraint admissible optimisation constraint actually construct call manifold tight constraint trivial empirical orthonormal zero cause element signal without insight operator operator bring compare motivate apply choose uniformly normalise row yield rest intersection normalise frame manifold frame manifold guarantee intersection two manifold dimension fact optimisation feasible global optima
snps mapping ad gene map identify multivariate trait phenotype measure arrange response phenotype arrange minor allele count snps record minor allele count snp arrange additionally phenotype snp square square include square eq give p q find reason singular thus invertible uniquely equivalently predictor snp regression ideally like exploit estimation boost limitation address restrict rewrite relate phenotype capture th predictor row vector back dimension represent set predictor set model set phenotype rewrite q vector relate phenotype obtain reflect response optimisation exploit response image derive structural correlation phenotype calculation computationally instead simplify approximation group map snps pathway begin pathway snp coefficient denote snps map pathway observe minor allele count snps corresponding snp coefficient small snps sense effect causal snps functional group assumption illustrate fig causal mark grey causal pathway pathway generate sparsity snps causal seek identify impose constraint snp vector htbp causal phenotype box grey causal occur pathway particular snps vary phenotype coefficient impose respectively apart benefit enforce also pathway sparse model correspond penalty whose weighting group depend penalty setting pathway snp coefficient enforce pathway expand note solution eq optimisation amenable coordinate hold additional response pathway group lasso tailor variable group accommodate coordinate descent descent grouped obtain successive estimate pathway pathway constant snp descent within group use h w increase pathway box estimate snp coefficient tend box full algorithm box group p pathway wise concatenation p require pathway tendency lasso pathway snps phenotype bias arise example variation snps snps tuned pathway select pathway factor iterative weight pathway unbiased biased begin phenotype weight adjustment maximum reduce pathway frequency continue even gene phenotype expect pathway snps gene pathway pathway pathway selection presence snps mean pathway selection probability reflect causal snps spurious association nan snps capture one tune substantial gain specificity causal rank capture association phenotype factor repeat factor rank need typically large snps pathway pathway method parameter determine variable choose drawback approach focuses select vary across establish importance alternative resample bootstrappe sense select adopt calculate selection frequency repeatedly relatively insensitive subsample snp pathway pathway measure across q indicator pathway selection snp voxel constitute phenotype euclidean subject voxel dimensional verify voxel discriminate ad classifier gaussian covariance figure accuracy sensitivity specificity optimisation primary concern figure cite use associate ad longitudinal phenotype pathway rank order pointwise pathway min l name gene pathway ad gene snps pathway pathway infection cycle disease fc gamma disease drug identify gene pathway snps rank snp describe lasso pathway lasso subsample accord size map range maximum sd pathway subsample max sd r snp snp map rs rs rs rs rs rs rs rs rs rs rs rs rs rs rs pathway rank phenotype pathway select frequency explain subsample selection pathway pathway ranking aid interpretation pathway ranking pathway rank select rank phenotype study extent pathway nevertheless interesting pathway calculate rank ad take average pathway gene ad gene sum ad gene gene score pathway ad tend compare empirically obtain pathway ranking empirically compute empirically chance permutation htbp pathway trait modelling show sparse modelling approach detect pathway approach begin model snp phenotype pathway biological pathway image phenotype identify model disease signature disease investigation especially case phenotype poorly ratio advantage extract image highly disease pathway function pathway link aspect gene drive identify snps gene phenotype ranking select irrespective capture signal salient pathway drive pathway detect highlight lasso activate driving selection pathway rank link expression formation ad brain aside validate ad rank gene occur ad particular amongst pathway suggest pathway drive small effect interestingly pathway single subsample turn snp ranking table gene gene identify major ad high gene phenotype allele study pathway pathway pathway disease select allele snp allele ad phenotype disease pathway rank presence pathway higher ranking snp snp also pathway evidence interaction ad fact snps interact simplify uncorrelated reality phenotype correlation association assumption gain multivariate phenotype finally tend pathway effect conventional approach demonstrate limitation pathway know pathway snps exclude pathway annotation reflect gene interact annotation different pathway rapid pathway suffer benchmark fail complexity pathway take snapshot biological acknowledgement ms collection project health national national institute drug discovery life company company la company development scale company clinical site national www organization california institute research education california laboratory california grants foundation present associate quantitative trait image characteristic longitudinal change disease pathway genome single nucleotide pathway pathway rank resample exploit finite application study whole genome mr snps map pathway voxel image signature characteristic ad change rank ad associate include secondary driving pathway identify validate ad gene disease imaging pathway regression grow genetic variant allele identify great date study augment phenotype signature effect image genetic phenotype highly common identify gene genetic molecular pathway rather highly multiple variant act pathway association reveal genetic otherwise variant individually help biological association example understand important biology patient care first able accommodate phenotype pathway genome single nucleotide snp voxel longitudinal change characteristic ad pathway collection functional pathway represent molecular interaction reaction accommodate source group gene network rely snp test univariate quantitative phenotype snps assign within snp pathway pathway account pathway linkage disadvantage snp consider snp snp consider snp effect accurately identification false identify pathway trait sparse lasso grouped demonstrate snp pathway show high sensitivity specificity detection pathway great gain previous accommodate phenotype incorporate sparsity regression identification pathway reduce incorporate phenotype single account voxel snps follow begin description voxel use map characteristic ad discriminate ad map pathway describe section pathway resample discuss strategy address pathway gene rank section imaging use study edu institute institute private year public primary emission biological marker clinical cognitive early marker early ad intend aid effectiveness well time clinical trial longitudinal public database serial ad individual control subject follow standardized mp protocol acquisition ms te ti ms flip angle acquisition yield mm isotropic voxel et et linear align longitudinal mutually align international brain exclude brain generate extraction create structural brain change time globally scale scan scan inverse optimize cost code jacobian create brain spatially across jacobian template comparison analysis conduct clinical subject inform participant experimental cognitive screening ad year ad cn detect causal pathway phenotype brain characteristic interest extract univariate quantitative signature voxel multivariate signature individual cn longitudinal exclude voxel intercept dependent change screen slope change voxel voxel ad stage wish clear imaging derive signature ad select discriminative voxel final set phenotype correspond voxel voxel age subject study database snps detail snps variant original
vector adaptive projective half wave pd projective angle consider practical adaptive ease update quantum rotation regard issue experimental scheme consider quantum concerned state reduce optimality see appendix measurement high need must deal possible constraint candidate discrete much simple candidate estimation could probable open consider experimental design measurement optimality state rank projective estimation trial perform first trial conditional include trial fisher include state unconditional calculate unconditional even converge essential unconditional design experiment conditional fisher view lie heart adaptive dependence direction radius solid black dotted spaced line dash light spaced dotted radius peak plot expect six six make thing easy average two thing less intercept effect combine create error quantum adaptively measurement outcome measurement determine optimality general nonlinear analytic projective reduce numerically show optimality criterion standard successful experimental implementation quantum protocol state confirm theoretical perform obtain involved statistically assume finite copy quantum operation mathematical always error quantum key classical precise quantum quantum estimator use context identically choice raise trial trial setting outcome design potentially performance adaptive design criteria fisher calculations shannon report propose experimental perform measurement large state state decrease update however infeasible whose sec terminology experimental brief review criterion analytic update analytic updating use numerically precise quantum sec experimentally appear adopt sometimes subtle formalism terminology apply survey interest include quantum measurement hilbert acting space know try lie include pure state measurement trial copy distribution quantum measurement quantum outcome trace operation represent sequential copy sequence use trial outcome measurement th th trial previously obtain datum let denote trial correspond choice maps nd likelihood specify estimation quantum sketch quantum evaluate introduce minus fidelity random variable density call true density latter least eliminate namely find combination average rao optimality space denote density nd nh operation respect rao estimate eq exchangeability limit derivative converge quantity converge interpret lower square known regularity choose explanation stand describe wish consider approximation parameter estimation must criterion necessarily I measurement adaptive require computationally sum measurement th probability trial consider relationship conditional unconditional estimator appendix optimality q nonlinear analytic state identify rank measurement trial denote projective axis whose parametrization projective measurement hilbert schmidt schmidt trace distance system measure calculate measurement behave hand behave maximum behave indeed make outperform briefly measurement terminology subsection update every trial simple step require measurement trial include trial calculate choose prove mathematically weighted mean squared result adaptation criterion er rao optimality criterion use cost disadvantage incomplete experiment explain paper projective first shannon distribution simulation pure projective maximum estimator perform set measurement behavior expect fidelity estimation pure state projective measurement average expect fidelity show criterion bayesian shannon propose distribution projective numerically analyze evaluation criterion eqs involve cost integral cost mixed projective state unbiased basis separable average expect scheme precise numerical optimality even least projective measurement analytic projective rank projective rewrite analytic sequence minimal equivalent basis invertible scheme choice arbitrary perform projective projective matrix rank minimal span vector dimensional space span third measurement formulae schmidt eqs schmidt eqs perform simulation design detail schmidt selection square schmidt distance adaptive optimality subsection measurement trial projective measurement trial projective measurement scheme selection randomly accord haar three fix projective fourth choose design know minimize estimation computation estimator use likelihood point sphere choose subsection schmidt sec expect black schmidt line blue dotted nu space include scheme measurement sequence carlo integration analyse routine statistical expectation approximated figure expect loss function number axis logarithmic integrate integrate via integrated four schmidt optimality large pointwise hand gap adaptive gradient begin behave consistent result asymptotic design explain sec average criterion estimate fig state time least expect nu n horizontal axis scale schmidt distance expect three state respectively measurement calculation analyse error pointwise number trial vertical axis logarithmic scale schmidt expect depict become plot schmidt square hilbert schmidt
produce fraction decrease eigenvalue covariance used similar test never measure move short autocorrelation value sample must take measure efficiency integrate autocorrelation practice performance reason long autocorrelation must time affine invariant reasonable calculate module module implement author project useful understand key goal metropolis hasting home build reason write tuning build read build package say make similarly piece sampler start spread reasonable another ball expect close practice find effective chance get low mode modal landscape expand fill autocorrelation time third go continuously increase anneal autocorrelation mcmc short acceptance range principle fraction low reduce mean acceptance fraction modal mode expensive human disjoint single small burn number require thought accomplish merge big one mcmc produce ensemble sample recommend hundred matter initialize obtain mistake run run couple way plot less autocorrelation try autocorrelation run time sure huge dynamic contribution think step long modal autocorrelation short mode mode time ratio autocorrelation target modal become happen long good proposal direction acceptance quickly fairly general application actually important move move implicitly linear true parameter value equivalent linear avoid case useful might package chen mit price helpful contribution grant er david address center particle new york ny usa pa nj usa institute university st york ny united states test affine invariant carlo mcmc open already project literature advantage excellent call implementation parallelism ensemble permit user core without extra effort mit package appendix visit page past decade gain come method example problem experience directly often expensive procedure find often understand pdf detail approximation space application probe arguably important advantage bayesian nuisance process otherwise little marginalization process integrate wish nuisance marginalization list nuisance value addition marginalization result regime valuable many evaluation statistically pdf present efficiency slight modification introduce chain center position normally something tuning dimension sensitive proof burn short chain tune extra call computationally look highly density regime principle tune hyperparameter h make sampling calculating get bad dimension transform much sampling isotropic motivate invariance equally insensitive covariance extend invariant sampling hyperparameter implement effort already project summarize decision outline complete discussion beyond interested reader classic like key concept draw necessarily quickly compute model important expensive compute available histogram subspace span implie value sample trivial process analytic representative simple commonly iterative position proposal position distribution sample proposal parameterization multivariate center tune worth step accept previous h converge level implementation need efficient short autocorrelation draw xt expensive informally move shorter autocorrelation clarity involve evolve ensemble current space position position clear
eps alpha vs eps pt pt eps clear experiment sgd converge fast average strongly prediction accuracy error still three exception fig accuracy reason inexact serve second stable rate discuss pt pt vs eps pt vs h vs eps alpha introduce composite nonsmooth propose convex propose batch well nesterov technique possibility exploit link promise negativity proceed clarity smoothly composite expectation smoothly approximate left inequality second bregman appear identity let scalar denote bregman divergence ready lemma smooth convexity subtract side inequality inequality due denote rewrite line easy inequality line alg term alg verify q definition substitute take l want minimizing analytic simply satisfied q ignore bind separately follow q college institute technology minimization nonsmooth convex central machine algorithm exploit common nonsmooth include svms achieve minimize loss confirm subgradient include issue computation nonsmooth nonsmooth sparse support vector regressor make nonsmooth absolute norm widely reconstruction nonsmooth theoretically difficult optimize nonsmooth stochastic learn fast nesterov deal nonsmooth smooth function apply nonsmooth function strongly paper nesterov smoothing propose stochastic nonsmooth combine analyze new algorithm name ml method minimize nonsmooth averaging converge nonsmooth still numerical dataset compare algorithm perturbation nonsmooth complexity serial cast function proceed paper along notation assumption first access stochastic subgradient basic propose objective drive gain variant recently composite convex nonsmooth model proximal stepsize composite idea rely nuclear norm machine loss nonsmooth become nonsmooth separability hinge absolute insensitive tackle study stochastic composite setting lipschitz smooth clarity focus generality drop analysis simply stem exploit propose smooth stochastic approximation attain optimal convergence analyze choose numerical present proof paper nonsmooth nesterov structures nesterov minimize formulate equivalent point form map convex non obtain smooth approximation nonsmooth crucial property continuously differentiable continuous drawback method present subsection dominate still affect take g keep mind nonsmooth indicate initially solution change eventually nonsmooth example sec turn alg denote retain rate without one fairly small without rate call dominate online typically theory batch call batch many employ contrary free effort convex bound e rhs term bound achieve achieve one retain perspective b inferior optimality nonsmooth stochastically approximate use hinge convex surrogate express objective g take hinge enough compare pt pt binary suppose ik alternative popular square regression express maximization look huber green curve fig available sim alpha classification regression dataset sim alpha use rest sec approximate hinge classification absolute regression compare
partition size sketch partition intuitive give motivate partition function group moreover g result immediate corollary lemma consistent consistent consistent example sized consistent partition figure group go probability go find instance problem rational encoding previous size group rational variable every j encode state stochastic define encode show correctness termination target teacher goal learner maintain proceed round teacher update reason mention introduction whether positive tree consistent simulate receive every make consistent receive sense unfortunately sketch unknown describe adversarial teacher keep teacher long hypothesis target teacher enough behind teacher execution mapping teacher seen return never converge able implement teacher interesting execution mapping fail teacher teacher every teacher execution long positive already become negative teacher pseudo treat line transition return least sized first partition make active terminate condition teacher line sketch iteration loop induce execution mapping sub structure least sized future new return hence current lift add finitely many terminate active learning teacher sometimes desirable learn hypothesis need impose every state learn algorithm guarantee condition teacher teacher keep return transition positive self loop teacher return always always partition terminate least correctness original style simulation reasoning verification large increase scalability check environment reasoning guarantee use reasoning plan investigate relation result compositional verification check investigate new checking david answer question suggestion union union simulation clearly induction height tree leaf define height maximum height leaf trivially height let induction therefore conclude induction strong without generality partition bound exist disjoint strong strong simulation arbitrary definition therefore strong let suffice definition choose transition also create initialize induction imply define ds g l ls l sr simulation mention begin furthermore imply say define furthermore induction define distribution way follow definition uniquely eq simulation therefore impossible target adversarial teacher begin round loop teacher act adversary far teacher return fail return fail teacher label compute update see teacher hypothesis make show round add definition add hypothesis label let state argue contradiction construction contradiction conclude step intuitively whenever beginning learner keeps never converge unknown arbitrary def learner def return positive example impossible teacher begin active describe teacher keep generate start initially teacher transition clearly tree return teacher return execution mapping teacher strategy keep transition state dirac loop latter finite return teacher figure force consistent teacher round state return action teacher proceed round initial negative every execution return except hypothesis b round force keep tree pp consider deterministic strong traditional partitioning use teacher query teacher fail latter problem conjecture semi intermediate assumption automate guarantee tree study label transition guarantee compositional verification yield stochastic shape appear compositional verification promise successfully check simulation work language automatically compositional reasoning exist acceptance tree language motivate language tree deterministic consistency verification want learn good space partitioning result partition stochastic partitioning result number algorithm framework active teacher teacher answer query membership target create transition probability assume teacher teacher equivalence target return check assumption teacher fail restriction conjecture restriction also turn algorithms teacher pa probability arbitrary verification system difficult readily intermediate guarantee style framework fully automate compositional verification moreover safe preserve compositional propose check probabilistic trace variant deterministic use sound terminate complete guarantee terminate lead complete rule guarantee termination using infer previous generation separate trace check guarantee separate make query equivalence query probabilistic state study algorithm stochastic acceptance trace say approach discrete probability specify explicitly number dirac e non label tuple distinguished action use circle figure transition give restrict dirac classical thus notion deterministic state finite alphabet finitely branch label system belong simulation belong pair belong pair support belong distribution relate state probability transition one match match possible general notion matching match possible follow split achieve check compute appropriate equal definition explain bipartite giving follow iff exist match iff iff simulation strongly iff simulation simulation definition immediate simulation simulation simulation check great simulation initialize iteratively pair terminate show start state remove exist simplicity characterization discuss exist def height state strong make definition conclude problem learner diagnostic two denote iff trivially l l useful check moreover preferable simple serve tree state make example structure simple execution mapping execution mapping state tree mapping execution condition impose teacher regard execution mapping finite stochastic say constitute sample goal say tree transition trivially tree trivially hence exist consistent checking reason guarantee infer trace state partitioning learn insufficient unknown enable obtain traditional partition equivalence state drop always positive map partition every iff consistent show iff consistent reduce always sketch simulation tree generality induce iff upper lemma size c way general merging na I find sized consistent partition rational arithmetic exhaustive I equivalence boolean say partition start encode consistency want introduce quantification say every fortunately boolean action expression transition add encode sl encode use iff nest quantification variable weight encode subset encode hold image detail encode constraint every correct
prove fx integer magnitude weight may integer weight nearly almost constructive approximate stress quantitative result dramatically previous self elementary algorithm heavily spectral correctness careful rather fairly theorem spectral rely turn required argument proof main theorem relate simple approximation boolean linear type contribution detail subsection give fix form domain need generalization gx simple finitely fact straightforwardly x rational f nf induce function e contrast familiar g fx gx familiar hamming basic structural converse function main may view statement sort bound relationship say require proof also improvement remove dependence carefully extend critical recent whether constant conclusion possible inspection argument theorem paper already ingredient construct construct construct approach substantially find bound structural apply find closely convert small main first define bound function exist gx state algorithmic main vector v proof polynomial imply e much simple algorithmic roughly work identify candidate first inefficient correctness sophisticated result make approach complicated novel much dependence limitation note boolean return approximate apply close order close distance accuracy contrast require weight linear optimal idea use idea distribution learn polynomial formulae fouri like similarly less throughout give section main structural algorithmic ingredient put prove main result present conclude paper let ess explicit ess independent denote cumulative cdf cdf say anti denote I elementary inequalities b ab b arithmetic geometric inequality obtain say closed combination affine letter hyperplane whenever explicitly dimension denote span space contain element regard exact version present time completeness write x first translate separate gx fx gx first second identity use fact strictly start point unlikely solved even check correctness intractable interesting naive variable one requires improve albeit problem target goal function constraint encode gx aforementione since linear linear programming truth obtain straightforwardly time solve provide overview main proof proceed explanation subsection clarity restrict attention boolean suitable weighting near theorem statement prove theorem use good anti extremely roughly moderately good radius original crucial small anti subject seem quantitative improvement instead closely approach use perturbation measure closeness ham direct high explanation modify statement hyperplane relate boolean hypercube statement hyperplane geometric statement hamming function euclidean center differ euclidean distance mass classify correctly closeness key geometric depend dependence heart geometric critical statement completely roughly analogue geometric lie euclidean hypercube stronger bind weak guarantee pass point arm careful iterative another prove theorem hyperplane euclidean recall show euclidean hyperplane hyperplane neighborhood element provide subset eq lie appear fact strict quantitative improvement theorem application lemma depend independent necessarily something require later technical reason give detailed lemma underlie proceed reduce follow prove later instead lie reduction critical apply define definition coordinate effect space tail case tail property index contradiction index ess behave like deviation norm entire anti concentration ignore tail early view refine generalize function various detailed proof key ingredient index implicitly explicitly define regularity fix regular helpful regular ess tell distribute particular anti gaussian intuitively regular quantity inequality tail weight critical apply inequality repeatedly technical use ensure continue lie use define idea coordinate tail tail coordinate tail tail norm thus essentially ignore tail n conceptual clarity continue k n w w cardinality least union bind also x q hence h deduce hyperplane h condition apply absolute ensure recall plug verify use side bound thus sufficiently indeed hyperplane h head tail suppose sake contradiction ess theorem get q consequently w contradict existence statement rest proceed hoeffding turn w x h w w deduce alternate v I true notation sufficiently sense upper ii claim prove case conclude essentially refine two modification generalize allow key thus quantity concentrate x inequality dt ready assume n fx gx generality satisfied give record proof characterize exclude degree bind low proceeding projection line orientation set whenever balanced balanced opposite side x x inequality suffice term write second first recall otherwise ni hence separate contain unit hypercube irrelevant orientation line p lower suppose bound project projection separate balanced x satisfied n obtain hyperplane satisfied proof use j beginning imply j hypercube lie side orthogonal parallel eq analogous say obvious j v set define x eq x plugging establish claim construct dimension nu jx jx jx nj get w n w ny j j j j n contradiction proof theorem convenience formal main algorithmic every boolean probability output intuitive overview motivated reasoning since function course parameter quite try close operation almost exactly current necessary modification ensure correctness proceed namely difference g boolean potential measuring arise add vector overcome counting argument result potential generalize follow process accuracy value tn vector coefficient eq add prove g tf claim proof observe ti operation tx part tx tx tx tx tx g tx g tx g tx tx x tx g g tx tx g tx claim stop algorithm time ensure therefore find g ti tx ti tt claim threshold markov v iw integer weight absolute boost close reverse distance ham everything polynomial together start give ff bit operation output representation proof run appropriately definition polynomial get write n x complete simple theorem produce proof note define integer satisfy iw weight input simple structural w w iw f satisfy gx complete proof say whether conclusion conclusion strong v f nf contradiction integer must application david restrict learner access uniform set receive label specifie bit ask sufficient integer weight satisfy accuracy early computationally give first result give learn running consequence much bit properly section show main briefly review agnostic hx agnostic learning give label arbitrary agnostic guarantee boolean give access representation probability operation describe correctness estimating algorithm theorem run f triangle dominate reconstruct approximately degree context paper give provably nearly existence open complexity believe intractable conjecture exact intractable attain upper believe show integer argument imply indeed output weight improvement improve run quite theory boost approach generalize precisely fouri degree specify degree boolean even bound algorithmic straightforwardly immediately hyperplane pass point give fix pass assume integer x dx easy also observe whose two final hyperplane pass prove prove element prove claim let denote span denote show first obtain th subtracting belong let position sign belong subtract previous imply suffice exhibit easy claim eq whose alternate subtract odd position subtract prove claim technical require extension lie hyperplane conclusion extension need deal find define da w w convenience reader proof towards contradiction greedy integer argument prove homogeneous nn w rescaling define comprise solution w w system system integer new
machine generator suppose ideal device disk make bit desire come resort physical process probabilistic physical device unnecessary attention rather string scientific community procedure possible obtain make bit familiar generate string distinguish determine binary string actually statistical numerical physical generators b generators deterministic mathematical resort physical generate binary string binary test randomness binary string pass criterion use string random string familiar ambiguity digit string digit either occur string generate entirely algorithmic however digit absolutely digit suppose example digit digit vast person string simple explain make digit mathematically generator number string mathematical whose number random binary string root technique cubic root number hundred thousand use mathematically root whole root amount digit discard proceeding justify discard digit digit root digit digit digit digit digit comparison specify figure either generate course neither two contribute binary appear h string figure compose order two number number consider increase appear consider number permutation permutation refer leave call subscript indicate permutation permutation letter first refer number second subscript use indicate number belong refer element order belong permutation subscript third subscript indicate belong order op long op p place long binary string desire result first op concatenation operation place exist finally string make last previously obtain accord successively continue carry root root carry precede permutation obtain permutation regard carry root pair specify permutation permutation different express clearly concatenation specific binary summation specify add sequence specify factor multiply concatenation indicate string product sequence may give triple likewise concatenation number consequence permutation respective characterize assume subscript number subscript indicate belong successively reach want describe order next operation binary already operation express last string base generate sequence discard first course random string tendency four different tendency present refer likewise expression string fulfil string tendency eight tendency seven possible two etc consecutive bit stre random choose suppose string tendency three transition comprise length calculate perspective probable different transition actually string naturally chi square determine give numerical theoretical consideration use significance difference show level case transition string length form use generator pass fail pass test fail fail pass fail fail pass fail fail pass fail test fail test excellent distribution generator string quantity curve string dot solid approximation assess let standard respectively normal area curve specify h area interval binomial previous article index randomness string string string feasible probable randomness string string none tendency generate string correspond aforementioned string index binary carry randomness randomness index probable average index randomness binary string potentially tendency tendency mention string compute average find randomness compute precisely likely generator string string follow carry computed string calculate already table randomness index generator quality ccccc max avg max precede generator binary string obtain generator previously article answer question notion kolmogorov randomness string generate length string comprise develop result algorithmic make possible sequence digit accept string digit generation program exist less bit string whether randomness currently suitable computer approach regular regard h approach vary string string computable close accept truly string two large comprise comprise string randomness string compute index randomness obvious string vice versa class index randomness high randomness string procedure digits root digit discard specify different permutation apply belong want simulation string describe ambiguity generator find mathematics www quality string statistical section generation document publish institute technology find test develop randomness long hardware software generator
approach marginalization process combinatorial category coherent calibrate model computational joint tree method separately particular one alignment infer alignment inferential alignment recursion recursion numerous infer iterate tree theoretical understanding getting calibrate problem finally analyze time considerably simplify disadvantage base joint approach substitution surprisingly relatively major equivalent tree exploit notably poisson computational computation obtain number description local treat fundamental description refer new light evolutionary incorporate poisson value space generation literature lexical character string remainder present formulation exact empirical view subsequent stay unchanged interval mutation substitution sequence achieve exponential correspond mutation character character position string exponential small winner substitution exponential event mutation multinomial variable substitution rate determine character multinomial determine character parameter stre tree visit single root stationary conceptually along infinitely long reversible useful arbitrary make description relate poisson additional continuous point equal branching set edge length whether branch write rooted subtree root dropping long model position character character identically length type mutation aspect aspect character string consist step atomic mass hence root atomic mass population branch black poisson root sequence deterministic two visit point x visit direct subtree root location character whose substitution path single character define single character along set substitution symbol homology path homology see construct symbol thereby observe comprise homology leave character character character point root probability generate homology random description subsection fact description string state characterize distribution substitution denote rate reversible substitution local describe section coincide appendix depend character follow establish reversible poisson advantage geometrically distribute stationary protein life adequate consequence process characterization allow discussion likelihood marginal likelihood condition homology homology unknown number analytically follow capture homology second way pick observed homology character symbol leaf drop exchangeable simplification internal appendix formula alignment partition computation depend locate column precisely look common character correspond occur location simplify fact chapter denote compute slight recursion subtree root derivation appendix column get total time system bayesian use benefit relative separate benefit infer tree infer benefit infer datum assess reconstruction reconstruction platform infer produce inference implementation evaluate frequentist precisely estimate frequentist study four type potential resample compare resample resample increase tree fix produce resample quality infer sample branch exponential two exp yes yes yes sp edge measure quality reconstruction alignment measure quality reconstruction partition difference metric metric quality difference term f report relative improvement relative relative improvement baseline joint full tree see test system generate baseline baseline large versus f substantial tree note simple metropolis suffer fortunately work literature potential sequential monte exclusive driving force sequence atomic segment also prominent consequence bias biological bias undesirable length gap relate poorly align sequence homology evolve substitution permit poisson exponential poisson role pure substitution knowledge representation vary vary sequence align similar length freedom indeed even note process genomic constant actually also acknowledge neither accurate biology inference notably account topology nonetheless useful hope motivate goal specie also view biology inferential example significant model capture arise model regard wish poisson extension superposition make non local poisson approach inference indeed superposition follow second superposition variable design irreducible integrating create bridge pair parameterization character handle assumption replace quasi stationary cox process large variation site branch acknowledgment would thank comment suggestion partially grant office grant health grant kp section proposition condition degenerate leaf length homology ix n exponential therefore conclude proof equivalence edge description rate distribution location fall global description hypothesis base construction character global measure poisson assign parent hypothesis well establish hypothesis mutation give yx item follow item standard random n pn dependency simplify find nucleotide nonzero weight leave c v compute use standard programming survival joint carlo object topology branch proposal auxiliary partition two use pairwise hasting irreducible possible link group move proposal brief involve
small time backward couple minimum histogram htp histogram acceptance eq way accept class ii accept pt discrete continuous extend hasting perfect broad applicable purely work motivated give decade appearance seminal perfect community immediately variation extension appear area statistical physics spatial operation powerful iterate transition law perfect simulate motivated mixed allow possibility become impose mixed result mix mixed adapt perfect somewhat relax assumption partitioning state describe single class remain possibility member second decompose space metropolis hasting algorithm extension example mcmc enable value create shorthand quantity involve usually simulate directly behind perfect law converge come meet couple zero path since path forward refer achieve perfect interested metropolis hasting algorithm way construct transition limit satisfying generate potential time evolve transition hasting metropolis simulator candidate transition evolve density verify sense borel stationary hasting unique limit distribution ratio run even generally metropolis rate study extensively hasting perfect need generate density metropolis hasting perfect ratio leave write ordering attain role move accept move state accept couple description start lower path accept move probability accept move upper move time otherwise state continue point hasting time time monotonicity candidate consequently move upper path accept candidate identify illustrate htp grey arcs grey represent start solid path ultimately perfect state inclusion transition candidate eq candidate partition sample density dirac delta concentrate hyperparameter draw density replace draw make hypothesis report interpret detail simulate hasting long regular proceed approximate start perhaps arbitrary q move arrive draw turn perfect backward care random path meet figure non htp path couple dash line potential path series possible acceptance depict circle image indicate achieve start depict coupling propose non meet note realization fail coupling htp open circle transition dash actual realize candidate value sample time accept stay accept achieve box fail starting determine must acceptance number negative path accept density since minimized likelihood mle include early serve reading coupling depict couple r achieve figure go two forward use exist path start accomplished set target backward path start stop reached extend algorithm point class point two overlap case modify candidate value include point support example propose support propose candidate ii density backward candidate point single atom hyperparameter know priori independent simulate candidate class accept class always ii occur sample path least step store compute coupling rx n k ix rx k require accept candidate
behaviour distribution inferential key aspect duality geometric relationship duality relationship consequence relationship existence family call parameterization affine across useful behaviour give vector set q dimensional change exponential image special thus advantage strong tool limit family multinomial possible polytope converge family converge boundary zero family limit geometric family dimensional define dimensional family base point embed span consider direction give one boundary first first change impact determine see redundant consequence fig eps family clear four ambient via problem htbp logistic full lie simplex design dimensional change explanatory inside space explanatory convenience distribution pass direction htbp plot clearly global geometry immediate sequence list also turn instead datum go infinity explanatory reach form identify effect uniform simplex high geometry region continuous geometry geometry derive geometrically suited use asymptotic formulation perhaps formulae rather approach ideal course deal two fundamental issue stability matrix formulae secondly boundary careful understanding multinomial use first like logistic accurate expansion act diagnostic considerably point contour widely example variable sample bin enough key understanding information control moment bin uniformly support continuously uniformly bound geometry exponential information loss inference two family make family uniformly measurable partition choice bin label geometry approximate particular norm satisfy skewness tensor corollary likelihood arbitrarily fine fig base base partition application affine unlike mixture example geometry numerically concern survival day diagnosis illustrative purpose censor give distribution family embed htbp geometry censor censor value interest mean width dot panel raw censor exponential inferential show skewness asymptotic full example fisher variability datum simplex paper mixture may mix polytope exist convex existence low exploit use positivity hull open generic tangent regularity family representation apparent contradiction curve lie subspace space discussion purpose motivated idea local approximate identification issue additionally usually space show considerable hull hull triangle within convex hull measure quality prefer far norm likelihood maximum turning point condition small likelihood euclidean unbounded change hull determine positivity directional turn appropriate finite far convex respectively simplex appendix evaluate couple examine hull hull decide singular segment consecutive theorem mixture model laboratory show relative variance fit binomial mixture good fit circle mixture proportion directional lie hull binomial node simplex joint observe easy include six opposite edge opposite face model dimensional unlike hull affine multimodal structure aid correspond model h bi convex polytope maximum structure maxima internal see construct approximate surface surface boundary opposite surface slice surface hull convex slice paper focus multinomial area geometry geometry allow numerically geometry geometry support finally build act paper look dimensional careful discuss selection thank ep decomposition notational convenience bin omit without loss vanish trivial eigen eigen denote order span eigenvalue expand claim much close approximated positive indeed proof theorem loss partition thm exist partition away uniform measurable far follow direct follow finally equation q image share preserve eq general suffice positivity expansion account directional zero hull directional finite cone directional case directional accordingly change log directional non taylor n small likelihood least hull maximum satisfie lemma draw statistical extend manifold fundamental thereby support start computational proxy space investigate selection model uncertainty vary far briefly indicate geometry different differential geometry largely focus multivariate exponential differential geometry reference include include consideration important demand modelling highlight insight fundamental inference design approach geometric contingency hierarchical family connection wide algebraic well review paper objective tool distribution framework family geometry home numerically implement information traditional start act use selection conceptual goal detailed development find implement confusion name topic learn practice statistical handle naturally varied illustrate aspect development example geometric issue issue full family look geometry body key model sample space distribution dimensional space space three describe variable accordingly possible variable simplex together vertex structure act identify extend work comprise working represent geometry geometry important nontrivial many consider expansion curvature dimension reference briefly describe author one provide type datum look neighbourhood multinomial space place neighbourhood computational algorithm censor look continuous response censor survival section consider theorem curvature reduction illustrate model response direct fig internal hide member model apparent likelihood inherently appropriate developed case mathematically excellent foundation world make precision arguably fundamentally categorical object treat thus carry treat categorical fig add select bin day plot inferential choice explicit geometry relationship information discuss look understand closure embed limit estimate numerical discuss use expansion look geometry issue geometry proof discussion key inferential construct affine linear duality fisher affine call family pointwise construct geometry introduce construction introduce categorical addition subspace affine geometry derive characterize computationally family dual affine hard subset mixed parameterization furthermore inverse bounding shown extend first sparse multinomial embed maximize shape boundary third order rarely across boundary simplex extensively graphical multinomial geometry geometry fact impractical explicit close simplex panel probability vector line terminology geometry immediate boundary line lie direction show panel natural geodesic panel show relative straight strict positivity strictly simplex parallel line everywhere orthogonal panel line find base geometry limit parallel connect simplex curve closure limit parameter triangle explicitly structure geometry multinomial
shannon uniquely finite countable redundancy finite must exist cx qx evaluate positive sided exist redundancy redundancy eq example express adopt convention infinite minimax theorem conjecture parameter symmetry enyi divergence exist divergence eq hold see term behave evaluate obtain understood finite eq infimum reverse trivial capacity achieve require denote achieve redundancy theorem enyi order read may extend enyi regard order seem nevertheless order carry negative people may avoid negative order property avoid skew symmetry identity follow tend remain identity skew order application symmetry enyi semi addition processing infimum property notably remain true enyi divergence order exceed remainder proof skew symmetry enyi continuous p enyi divergence nonnegative follow symmetry pd theorem divergence enyi divergence enyi divergence number let convexity argument imply contradiction logarithm divergence construct converge topology enyi enyi divergence locally behave root triangle square review include continuity infinite order identity particular enyi leibler capacity redundancy channel order acknowledgment gr anonymous useful comment author van grant van originally perform degree position obtain grant scientific research universit paris france interest statistic include minimum learning divergence state sufficient thesis es es receive mathematic art science university work hold various position mathematics visit network business college journal theorem conjecture remark r r enyi kullback leibler shannon satisfie axiom kullback depend enyi equal leibl review property enyi kullback divergence limit order channel redundancy continuous channel leibler divergence kullback fundamental success many concept numerous never code enyi divergence es gr operational characterization enyi code compress ar operational testing divergence crucial recognize computation throughout hellinger enyi use enyi enyi entropy study enyi throughout literature establish treat divergence detail kullback leibler preliminary version independently publish work finite another read adopt space density sum member long set positive case definition enyi enyi shannon kullback leibler relation whenever interval enyi densitie enyi r divergence long compact relate enyi r denote let tending tend information recover enyi wide value enyi instead enyi divergence kullback leibl order define enyi distribution chain square np enyi divergence imply enyi consistent normal kullback extend r enyi divergence continuous space enyi via definition show enyi extend behaviour section convexity property enyi generalize contain several treat chernoff hypothesis application channel minimax redundancy order order divergence violate enyi enyi divergence present power presentation corollary often skew symmetry often p jointly jointly quasi let p transition markov lower upper semi semi continuous pair outcome separation eq chernoff side conjecture capacity redundancy dp q achieve countable distribution side carry often continuous measurable write algebra may interpret event respect measure absolutely may absolutely continuous common none result mixture countable restriction capital letter g refer take px qx definition order formula order motivate definition arbitrary formula define order sample generalize space follow simple define divergence variance definition ensure relation square distance distance hold order may change integration depend dominate case whereas respect may subtle consequence count eq direct argue ability enyi divergence eq inequality hold stem theorem form conditional result process mass induced verify consequently enyi absolutely proposition expectation theorem enyi divergence countable let supremum would enyi extend use inequality converse inequality bin let supremum countable enough supremum finite end suppose countable find inequality continuity enyi divergence define order divergence divergence enyi original always present exist enyi enyi aa equivalent turn order result follow verify express closed form extend verify sufficiently case exceed dominate convexity since remains p p also kullback leibler q doubly standard way continuous opt limit leibler proof q q hold hence alternatively restriction convexity derivative qp monotone convergence remain argument imply q q monotone together random px countable essential supremum ordinary supremum finite extend range finite aa q completes imply divergence continuity extended order enyi extend fix r enyi varied relation supremum partition convexity order finally various positivity suppose jensen equality extend qp inequality study property divergence order turn continuity equipped topology event topology topology strong sense variation topology countable enyi divergence function topology simple p q continuous semi continuous continuous semi simple imply semi property extend divergence topology enyi skew particular symmetric inequality symmetry hellinger distance equivalent total variation relation enyi enyi inequality x yx enyi divergence q qp theorem enyi divergence total space topology convergence expression enyi topology discuss weak topology continuity mean metric measure metric equivalent metric topology weakly need interior consist algebra apply proof book imply topology q consequently set compact convexity quasi enyi measure continuity relatively compact since denote restriction processing tend depend finite partition imply analogous sequence partition theoretic theorem may hold member part evy proof without probability let q np semi continuity enyi evy family imply similarly md theorem extend nx q jensen expectation follow last processing theorem nx p evy q uniform relate integral continuity probability term r enyi divergence absolutely mutually continuity mathematical tool absolute continuity equivalent enyi version absolute hold space event subsequence entire separation relate continuity mutual theorems theorem equivalent p p continue relate theorems infinite algebra np p call divergence sequence nn special enyi countable pair countable give eq extend observe theorem product distribution proof symmetry letting tend countable theorem raise let arbitrary question space n consequently repeat argument also state hellinger responsible hellinger integral r observation contain p parametric know regular taylor interior argue property aware exact technical probabilistic explain equal relate leibler involve q convention density define infimum uniquely interpretation enyi divergence kullback leibler divergence formulate already identity heart ar equivalently hence infimum uniquely secondly suppose consequently infimum equal either reader hence inequality follow converse infimum corollary concave wise infimum extend continuity concave monotonicity enyi convention addition introduction version survey distribution eq continuity hand lemma continuity
consider turn increment classic give remove fix instead cholesky generality inverse cholesky motivation appropriate detail notation estimate column notation kalman incorporate dependent front see role minimize barrier would like generalized composite function objective step objective composite wish way rewrite nn nn formulation argument propose apply matrix specifically cholesky factor give explicit setting diagonal matrix case factor nn eq block v twice continuously vx nn nn strictly entry write composite nn gauss newton form difference function criterion search whenever approximation sure component great derivation gauss subproblem assumption well provide estimate statement subdifferential order stationary point ignore rewrite portion sequence rewrite q exist eq subproblem rapidly motivate gauss input termination criterion step follow counter gauss newton terminate set return algorithm terminate finitely point point sequence subproblem dependent direction burden addition require always closed form q block algorithms recall equation q advantage kalman present simulate denote model main innovation smooth take cholesky measurement full period time discrete equally spaced noise true state vary situation phenomenon sensor report particular easily belief cholesky function estimate take behavior simulation extend kalman smoother thick red dot ground show kalman filter thin kalman pick last important magnitude make kalman fail know paper formulation model know variance control modeling rise structure gauss newton repeatedly extend kkt optimality preserve smooth inversion inversion definite immediate useful lemma corollary algorithm every require invert corollary terminate eq finitely necessarily contrary let triple condition multiply condition third condition find first q since subsequence generality q work ff take contradiction lem lem conjecture proposition definition medical trend filtering require know covariance matrix framework depend estimate gauss inference apply implement efficiency kalman approach synthetic optimization perspective include smooth nonlinear kalman smoothing system know angle interest kalman modeling error dependent bar choice tracking depend turn turn rate state noise portion idea motivate extension measurement variance
infinitely converge true sparse fraction force sampling population outcome period coincide sparse policy equivalence e observe appear theorem prove result period yield least true supremum bm minimization denote period force possibility q outcome period force order rewrite eq q law case exist flexibility construction consistent sampling collection force section refine notion affect furthermore sensitivity perform simulation convergence criterion almost sure convergence assumption unknown purpose outcome absolutely I e imply outcome population period period appropriately ensure overlapping exponent length obtain period slow intermediate difference period follow force frequent desirable population soon optimal population also frequently force outcome long period force rare converge linear solution intermediate preferable well estimation value avoid non population evident seem address comparison base region correspond scenario narrow figure accurate valid expect value long another issue arise slowly entire duration actually preferable average policy outcome complete policy incomplete information infinite horizon period exceed population outcome could use switch constrain policy horizon reward however constraint period behavior allow specifically neither consistent high outcome converge appropriate amount sampling budget long something impose cost constraint unknown asymptotic cost development sparse force period unknown period employ estimate instead compare convergence sense weak since exceed albeit importantly currently another towards instead assume distinct independent I markovian decision case importantly efficient policy extend technology research program author thank discussion remark gr independent population outcome upper distribution construct policy true mean compare policy independent objective maximize period infinite period exceed introduction introduce dimension tradeoff control incomplete population maker population well versus effort view problem incorporate mathematical programming allocation linear programming advance maker policy ensure learning area armed sequentially population order maximize horizon generalize construct incomplete simpler base policy normal horizon armed bandit model constraint bayesian propose heuristic estimating benefit computational analysis bandit bernoulli beta must allocate approach adopt approximation idea population randomization adaptively modify observe outcome period type population outcome markovian construction family per period converge information mean sense organize reduced cost degenerate basic optimal cost express linear combination define solution cost constraint optimal population characterization policy randomization rational specifically set satisfy incomplete actual admissible restrict attention policy depend outcome specifically let outcome period observation history dependent eq number sample period eq reward desirable policy feasibility feasible requirement first long exceed outcome per period
discuss de impose define us positivity ab plug begin bind k get count follow constitute upper principle simple generate bound look put together nan probability dimensional subspace satisfied useful section rule causal st test fail causal look explain direct vice versa action history influence formation mechanism take range tend prefer attribute formation step attribute influence however st reproduce coefficient influence st time consider node observe estimate causal give due calculate try binomial excess coin extreme comprehensive distribution plug test base apply confidence like hoeffding want check empirical sampling central verify expect average empirical less would fail reject htbp fig trait circle state repeatedly pick node appear tendency become explain static square red generating identify sampling fig longitudinal study type neighbor co worker identical section heart pair survey actor way median mention test include unbounded identify causal effect use rule relevant observe sequence material day combine analysis future observational social mark begin address response center modeling sensitivity central latent pose barrier st identify causal effect bound seem primary practitioner paper take tool geometry possibility long convex relaxation algebraic geometry make approach lp define lead optimization address ultimately allow causal presence come human every reason important obtain causal strength effect could situation whether model unbounded find law restriction public act cause correlation outline way statistically causal social study dynamic correlation common validity future effect heart equivalence exchangeability partially initial count call markov chain partially depend state transition average arbitrary transition matrix property preserve chain pe requirement chain prove property nan ask chain see notion exchangeability differ pe pe possibility sequence back see ccc satisfy long type solve exponential theorem long may another nice condition eliminate sec nonzero remain check differ typical conditional independent perspective order markov look solve influence otherwise think instant influence choice time influence previously delay model equality list read pick maximally violate particular delay acknowledgement grateful provide access thank definition test social trait demonstrate bound causal effect observational study unobserve trait assumption associate effectiveness previously methodology preliminary heart tie year constitute observational social review latent act impossible non necessary identify social intervention impractical even formation action measure central study social method bound limit well additional actor attribute time step formation attribute refer latent edge formation could actor attribute result rule employ shorthand capital letter ambiguity arise pa pa ab principle action surprisingly attribute presence mean observe pick determine term unique sequence lp bind become successively tight increase size lp representation compact g sx ix negative integer rhs quantity ensure positive require impractical become algebraic proceed optimization
initial low purpose gram threshold use refined simplicity still refine resolve gs reason follow generate vector nx screen refined repeat record hamming step gs gs use experiment due critical parameter satisfy study unknown open problem model fortunately study experiment specify amount procedure reason experiment assume know iterative screening tuning iteration refine iterative screening estimate comparison hamming contain goal experiment gs lasso investigate fix fix design block wise block pair iid sign equal repetition gs behave behave lasso recovery hamming error yield lasso prominent similar comparison gs interesting due become increasingly less strong gs become prominent screening screening screen gs investigate fix gs differently depend part experiment alternate adjacent index index equally setting block say ai j I average hamming gs outperform additionally gs effect grow gs prominent less include investigate experiment comparison use sub block combination choice coordinate hamming ratio procedure c c strength equal screening experiment generate way experiment performance use sign alternate across average hamming repetition diagonal first sub sub hamming repetition experiment generate function entry pre specify non value multiply draw hamming ratio suggest consistently htb experiment gs tuning experiment reasonably investigate mis specification whole experiment experiment use experiment experiment experiment independent repetition sub gs know rr gs experiment replace counterpart result suggest mis specification effect reasonably experiment mis issue use experiment result summarize experiment robustness mis specification experiment except sub experiment predictor presence correspondingly experiment assume iid sample df hamming ratio repetition suggest reasonably robust binomial v definition right recalling follow let introduce follow prove section corollary short elementary show corollary subset equivalent claim trivially condition corollary odd diagonal note em odd argument case monotone give argument hold equivalently case monotonicity inequality show case use second coordinate combine odd definition consider similarly j eq write remain strength remove generality assume definition quadratic rewrite similar achieve must satisfy v v claim show event p subgraph form support unique maximal subgraph event depend note I sufficient triplet eq basic realization I I p ty ty dominate basic non recall direct calculation em stage constant retain subgraph lemma hold remain omit hard notation connect integer let I exactly partition gs screening design empty singular score ty I basic direct ty iy j differently iv I definition relatively easier induce constant begin accept algebra I I second w magnitude algebra show coordinate finite towards end definition path connect path subgraph connect connect separate effect matrix path connect sparse g wise calculation attention apply subset subproblem gram matrix tune ss lemma suppose apply aforementioned subproblem ham r q ss last proof reason omit proof em symmetry need see hamming short region plane c calculation bad q equality rv v r j em acknowledgment thank ji nsf award dms partially grant dms dms grant research resource theorem theorem xx xx zhang zhang zhang row large regime selection challenge gram relatively naturally induce sparsity decompose subgraphs clean mle main innovation guide selection procedure hamming optimality minimax compare support demand hamming naturally error broad hamming procedure penalization utilize convergence simple tuning http www software gs matlab graph gs weak signal clean sparsity design vector th motivate big datum though take small proportion nonzero nonzero entry identify selection regime regime signal find genome generation sequence despite regime rare individually focus rare regime appropriate rare evaluate rare weak signal weak appropriate focus regime hamming throughout gram approximately factor row simultaneously find reconstruct gaussian often induce network gene normal concentration available security edge strongly key insight signal subgraph size decompose emphasis paper case lemma related discussion motivate screen gs clean screening identify let subgraph form large subgraph split many remove objective fundamentally correct rare paradigm gs diagram especially rare weak popular idea reduce noiseless model equation mild solution must viewpoint frequently variable selection design arguably vector idea penalization fundamentally computationally intractable selection provide signal signal decade tractable approximate penalization scad say upon component truth penalization fundamentally unfortunately framework signal rare weak fundamental uniqueness noiseless long valid signal many vector perturbation indistinguishable long regime principle recovery consider function unclear penalization correct rare even wise tuning ideally since benchmark development imply optimality univariate call sure well column univariate variable correlation inner corrupted happen snr reason univariate effect refinement screening paradigm scenario positively association test bernoulli signal said mostly screen stage challenge expect overcome screen straightforward integer consist series phase increasingly phase variable retain significant candidate screen unnecessary threshold test order control positive candidate order screen challenge computationally screen gs screening major carry gram node generic fix gs clean method consist graphical screening step step screening retain end gs decompose separately dimensional gs sophisticated note counterpart gs discuss gs step gs screening able aforementione sparse definition hand side subgraph arrange subgraph one tie hold certain ise recognize gs efficient screening first design gs overcome little situation long small moderately computationally feasible moderate explain first frequently span set matrix restrict row sub rewrite model key orthogonality test near orthogonality happen negligible effect gs able retain way overcome gs sub whole already retain differently discuss node gs step g j moreover carefully tune gs counterpart size complexity stage moderate organized follow gs achieve hamming rare diagram visualize optimality gs gs attribute optimality separable two main discuss exist extension gs proof notation use denote occurrence occurrence vector hadamard denote spectral submatrix form restrict column sorted write fix graph strong notation occurrence denote graph involve maximal call great gs section gs rare diagram gs penalization section gs gs step initial sub let list contain broken thought subgraph list step index accept currently update keep continue finish retain index keep forward principle depend subgraph order give different difference usually alternatively example step finish screening theoretic gs would produce result reason skip gs sophisticated choice screen screen tune properly sure screen negligible fraction signal say integer regression small solve unique case whole minimize eq vector gs gs gs paragraph list p k u p u sometimes design matrix gs main exclude obtaining contain gs properly choose two separately step computation subgraph fact least size subgraph graph subgraph computational gs contain part break component part broad design tuning choose probability total great complexity gs moderately use univariate screening complexity gs implement screen subgraph complexity latter multi gs model hadamard product see primarily weak constraint mainly technical reason need discussion let fix range range cover successful impossible recovery correlate fix design presentation translate design notation see gs fix introduce row unknown assume call compressive security fixing model increasingly necessary successful selection rare call many loss weak hamming asymptotic rare hamming respect misclassifie component misclassifie aggregate bind global configuration resolve exploit new favorable estimating geodesic distance j well favorable risk th coordinate define denote hadamard reject type ii negligible survival risk v shorthand stand occurrence denote q risk favorable overlap graph pick appear uniquely non empty intersection p claim hold model hamming calculation say one grow large derive conservative improve neighboring discussion holds replace elementary calculus v especially broad favorable neither gs need hold sub constant coordinate approximately equal gs subset choose step set fix consider model model model gs gs gs screening set h gs p interesting range properly h gs say gs achieve optimal adaptively note influence convergence remark narrow start influence choice replace base procedure accept parameter hard gs need pay primarily interested skip turn form strength coordinate model q approximately number selection small successful universal extend tight across suppose additionally j pi pi corollary rather relaxed surprisingly rate satisfying condition red red panel red middle part illustrative complicated part dash panel illustrate corollary together corollary implication diagram partition whole different region difficult almost full signal recover recovery minimax general last illustrate figure blue line get strong end integer follow p p side long hold partition corollary view assess weak phase partition region inference diagram region rare selection full recovery rare achievable high sense incorrectly region infeasible signal nothing sparse find boundary non recovery region effect area deal issue deep insight variable broad penalization select functional aic bic good theoretic computationally computational relaxation limit scad mc take select penalization penalization problem surprisingly optimality lack adapt design optimal subset selection save subset scad mc selector mathematical block eq convenience broadly consider optimal continue relax represent subset parameter ideally bad ideal depend exchangeable hamming scenario form p p follow model gs set subset set ideally h strict exponent expectation gs outperform selection representative prominent increase gs phase plotted phase partition region exact almost region interestingly separate last separate vary calculations elementary boundary lasso hamming much recovery slow subset selection optimal region subset adapt matrix correspondingly one subset separate ideally explain optimality penalization lasso remain rate save leave gs key innovation guide force computation gs exclude overhead obtain utilize sparsity design
write database program may likewise program characterize input endow database phrase characterization privacy differentially achieve whenever privacy important relation require take simplifie rarely mention quite randomized definition discrete subset topology completion field small open value sensitivity bound privacy mild sensitivity mahalanobis distance demonstrate privacy explicit argument dp let ds ds ds second third finite dominate hold bound privacy via output definite symmetric satisfie output dp lebesgue dominating consider eq exceed multiply leave zero say examine pz tm py c define final proved give privacy mild sensitivity global sensitivity nothing sensitivity mahalanobis previously definition provide sense almost private private procedure remain unable analog case differential adversary database note observe private adversary determine database set private comprise exclude concentrate obtain analog let achieve dp note measurable obeys constraint implication sense incorrectly reject release raise section measure function space essence countable deal dominate exist space differential privacy measure finite family cube dimension randomize algorithm nature describe set borel take prescribe b operation collection field page closure intersection namely contain differential hold sense eq argument approximate dp release differential priori release satisfie claim guarantee union extend countable due theorem countable set hold family satisfy form set towards every obtain conclude dc principle computer would privacy guarantee computer point continuous description borel field therefore function differential lead throughout summary every projection differential privacy every countable explore achieve privacy entail suitable demonstrate differential privacy conceptually appeal remain challenge may sense normalization time consume bring maintain consist subset hilbert k kx x correspond I closure present form require finite distinct release field mean reproduce private rkh density gaussian differ gaussian upper rkhs trivial process zero differential demonstrate assumption bandwidth rate normal respectively evenly spaced grid remain peak hold kernel kernel case positive require employ fix privacy establish sensitivity process usual depend rather order necessary differentially private composition typical way employ leave choose choose maximizer private citation compact priori rule determine private technique private algorithm range work rkhs live function space amenable demonstrate broadly use whenever rkh detail determine level necessary privacy rkh take fold tensor see detail reproduce completion set form obtain substitute complete define derivative gaussian isotropic dimension x dx dx dx choose lead technique attain ad grow high estimation privacy developed use desirable determining among error bound require differential privacy vector privacy kind take classification interval place prevent confusion admissible call whenever lipschitz demonstrate minimizer adjacent norm swap rearrange minimizer functional inequality use together combine turn loss function term plus find svm transaction alternative former request evaluation batch online online typically machine batch nothing multivariate finite evaluation condition q track request inversion answer request general time proportional problematic consider always span evident take position algebra whenever fall kernel mean sort increase sort doubly link list link list constant differential privacy preserve functional analysis theoretical address specifically ask release differentially noise preserve privacy address real count value technique case mutually absolutely bound complicated quantity kl matrix assume operator term nothing statement restriction subspace thm axiom thm pa usa pa department usa differential framework summary database discrete develop appropriate yield privacy rkhs rkhs kernel machine reproduce hilbert suppose consist measurement release without privacy rigorously basic thought induce due coin datum define single vast privacy develop boost parameter logistic svm possibly consist differential privacy measure norm sensitivity rkhs rkhs motivation value fold growth
various primarily minimax precisely magnitude feedback supremum infimum allow strategy feedback definition adversary interested mirror idea mirror discover descent idea see recover upper bandit armed bandit much one question exponentially forecaster suboptimal minimax magnitude bound bound introduce paper discuss average suboptimal online general bregman strategy appropriately choose section magnitude regret obtain bandit case discuss combinatorial simple combinatorial independent minimize perhaps forecaster strategy exp exp correspond expand bandit exp bandit strategy regret derive combinatorial derive see author mirror section obtain latter parallel expert question improve exp provably suboptimal exp defer coordinate bandit instantaneous bandit game ai ta bandit notation expect online mirror mirror descent mirror descent update formulation origin idea perform descent descent tailor properly strategy thorough treatment subject convex differentiable fx depend norm bregman say moreover understand space bregman divergence example study presentation suggest respect switch barrier convex hull section detail study case specific semi bandit describe inspire computational complexity polytope indeed step jointly get interesting selection ranking span acyclic graph also however note role moreover gradient loss detail conv np aa feedback feedback project weight improve satisfied unbiased satisfie bregman divergence w fa divergence fx obtains conclude invertible third technical difficulty bandit discuss prove regret word ti ti x define computation use fact one obtain note second already appear basic armed strategy expert inf forecaster logarithmic notion potential potential every define paper restrict potential simply potential regret hold pseudo bound order direct inf coincide basis negative recommend optimal direct u desire non fact let old combinatorial optimization feedback bandit sublinear exploration component algorithm exp play distribution exploration play exploration contact point polytope ellipsoid polytope feedback exp q next less argue conjecture lower correct idea limitation exp exist bandit infimum strategy adversary information conjecture regret bandit low conjecture contrary far self barrier polytope admit barrier self exist open problem sake multiple hypercube q choose disjoint second half coordinate parameter exist exp adversary regret lower define adversary q put coordinate choose interval interval begin odd vertex pick expert uniformly expect even select precisely select loss exactly add fix remain proportional consider different adversary adversary half matter round identity big sum low important conceptual contain heart argument bandit class leibl divergence prove bound kullback leibler simplify multiple identify word play game attention strategy arbitrary definition time adversary follow word adversary small expectation action denote player face distribution adversary play adversary loss coordinate adversary denote kullback leibler moreover symmetry concavity square q third I since deterministic empirical conditionally probability forecaster play adversary respectively chain kullback iteratively losse sum bernoulli distribution plus see sum conclude proof player plug fourth let denote randomization thank realization randomization previous apply probability c k numerator denominator differ show denominator use hence q numerically verify decrease come fact imply bernoulli usual abuse notation kullback leibl distribution easy consequence chain leibler jensen inequality introduce nonnegative ec grant paris est imagine fr financial university edu action maker difference realize pick magnitude regret assumption maker model bandit exponentially average provably suboptimal combine mirror inf forecaster able discuss lower bind optimal framework setup maker player
reconstruct transform autoencoder htb mi v one regard representation static contain transformation form assess multi capture hide temporal taylor three frame sample epoch equal present include generate frame result dimension dataset see repetition outperform identical give temporal representation achievable dimension along reconstruction single architecture mean frame delay frame delay frame natural movie investigate compare implementation pixel frame normalise unit initially data convergence full temporal learn temporal projection past visible activation plot run complicated unit unit visible layer unit past projection delay whereby delay starting point unit unit choose unit activation unit repeat map hide x layer define trace display likely delay represent represent delay temporal filter field model display row temporal see learn different unit delay force allow weight learn filter show forward select active transformation translation motion call motion movie evolution learn temporal interpretable help well understand effort adapt represent propose learn temporal track employ visual interesting generative support would thank universit department intelligence technology bernstein computational characterize deep enforce constraint little investigate might domain investigate temporal suggest temporal feature usually fashion barrier feature think yield advance unsupervise feature extraction center machine train large train yield unsupervise even set autoencoder boltzmann rbms feature learn shape system explain shape primary visual property natural redundancy rbms structure supervision structure back field find brain vision focus find natural understand filter develop restrict show able movie dataset machine rbms autoencoder prominent variety machine learning field paper briefly two consist visible visible representation represent activation activation autoencoder deterministic weight hide visible layer respectively perform layer mean reconstruction evaluate layer evaluate activation additional abstract representation rbm boltzmann conditional boltzmann restrict boltzmann rbm whereby feed include hide visible layer visible visible manner rbm learn dynamic box see denote energy q given denote delay amenable stacking rbms boltzmann contain hide connection visible step hide architecture rbm likely rbm model dynamical motion capture data model rbm temporal connection denoise autoencoder temporal mark able outperform deep natural gain insight movie possible
version auxiliary preserve probability new topic possible state represent allow choose empty count assign go return pool topic parametric author topic root guarantee topic topic j q assign random top level sample eq one work lda author represent yet necessary propose choose consideration wide serve prior article author present non parametric extension available possible overhead interest author attract model community seminal modification feature set little work article complementary represent topic much valuable extension allocation lda available two document via gamma word author seminal author unknown topic condition condition assignment author topic collapse markov x transition iteratively exclude variable di ki
time competitive dark gray gray image internal meet image represent pixel normalize range near neighbor local point random display randomly fidelity point top white identify perfectly average li segment perfectly report run pre density approach tb datum set handwritten multiclass interface set automatically handwritten project approach require graph scaling implement iteration per fidelity obtain four fidelity confusion good correspond digit computing large mistake try distinguish digit unsupervised laplacian ratio version normalize cut sophisticated setup eigenfunction self neighbor report upon require r interface method obtain local scaling construct adequate exploit multiclass ordering fidelity method condition configuration produce method fidelity representative hold rely less choosing label work investigate interface conjecture variational type exact nature functional support air office scientific pt pt institute mathematical sciences usa air physics sciences china rely quantify meaningful framework desire category devote cluster binary problem alternative interface equation phenomena functional generalize graph minimization cut sense graph diffusion function segmentation cast label combination ranking iii recursive partitioning consist successively cluster desire number reach datum considerable contrast propose interface simultaneous multiclass modify remove value smooth binary interface multiclass setup expression locate graph integer represent multiclass classification smoothness kullback expression involve interface binary multiclass base functional quality segmentation characterize state let scalar denote double minima gradient term variation potential hand adopt discrete label cluster minimize towards region double potential segmentation term potential interface goal transition interface norm slow interface approximate variation tv formulation produce interface transition tractable minimize approximate tv norm furthermore calculus tv interface graph undirected neighborhood represent segment vertex weight symmetry neighborhood set close feature vertex follow calculate I I express graph denote component connect among neighbor represent convergence limit allow label potential li integer periodic multiclass laplacian equation effective calculation class state small multiclass framework contiguous class label label compose compose jump analogous vertical jump interface reason interface undesirable manner require symmetry class determine thresholding boundary class integer correspond unstable well near half difference difference vertice state half whereas function correspond speak nevertheless regardless correspond used fidelity include little effect functional give represent ht dt rand nu u u half integer use detect change minimize term fractional gradient descent update preserve vertex represent fractional term summary multiclass interface segmentation consider real three class construct analogous binary half circle center bottom center circle
dms grant regularize cox existing linear model empirical risk function censor survival neither lipschitz negative partial likelihood iid derive asymptotic inequality lasso cox tackle iid model receive attention regression oracle inequalitie nonparametric include example provide g classification hinge loss regularize survival censor sequence iid covariate largely parallel material let covariate cox cox eq hazard likelihood log non ease theoretical intermediate replacement unknown population negative partial view loss cox excess desirable non inequality cox generalized nonzero iid intermediate lipschitz similar boundedness tackle pointwise type error negative additional discuss assumption assumption identical assumption assumption regression replace censor strictly subset k denote typical constant q k denote version cox base rademacher sequence fix supremum find calculation cox f b contraction follow yx f yx e k k n satisfy triangular inequality obtain f k pz jt nx jt directly pn e assumption e te ii ix ie u ix te xu thus theorem r constant let xx argument k te f ix te ix r mp n te r satisfy event te nr te mu nr lemma show lasso pointwise let assumption f impose ii f meet hold exactly meet pi bb w n triangular obtain remain follow lemma van de meet I pi suppose meet since meet exactly slightly oracle
avoid find consider consist nan repeating thereby seek exist find find common orthogonal extraction present n kk control equivalently component extract basically signal simulation briefly optimize optimal truncate svd page q fact tv converge cca sense form basis require actually restrictive datum high low dimensionality state end section subsequent reduce reduction indeed extensively column basis completely principal independent distribution otherwise robust pca estimate canonical give vector maximize high specify center end bound low high cca cca give wave standard normal distribution mix red extract project cca sense cca another component high extract state propose highly satisfy due relationship cca pls principle basically seek principal span similar whereas pca seek quite useful relevant signal alternate truncate frequent datum hence consume method intuitive large use big issue extremely large significantly random solve follow first use case may lie fortunately rarely examine orthogonality orthogonal want onto desire property uniqueness source bss bss bss mix permutation denote bss bss recovered mixture remain permutation ambiguity knowledge mix attractive extraction pattern source consequently mixture bss actually version obtain desire sparsity independence impose penalty bss link bss bss multi block link perform multi high correction require extract group correction word bss contrast link bss ordinary bss discover case require nonnegative run component common low nonnegative low nonnegative example rule wise division see detailed convergence feature helpful top analysis bss method however method process reduction task visualize low discuss discriminative neighbor etc much unify careful stage critical purpose extraction flow diagram common belong category denote extract matching share th evaluate accuracy achieve although improve version pca close see extract almost achieve separation particularly accurately efficiency individual specify actually second converge however consume instance consumption detect bound component intuitive nf different term observation multiply tend justify base extremely ray accurate detection energy ray image diagnostic task early unfortunately subtle separation diagnosis ray extract separate soft four source respectively component interference different uniform highly separate ica random ordinary nmf algorithm mixture run soft realization component extract satisfied extract desire propose nonnegative equivalently pca c information avg c c avg face human face datum face expression database gray distinct ten expression close dark subject position take different pose illumination expression process face image randomly repeat first split form face image different feature common number method influence time widely accuracy information compare improved cut justify completely due original cluster detailed clustering significantly investigate change stable parameter fortunately cause correlation become individual small propose describe two different individual analyze scale contain view space equally category category share feature different make widely adopt evaluate classify see object category compare three classifier knn classifier neighbor svm fold validation search interval interval database run require sufficiently ensure covariance train training feature routine split two image score classify sample accuracy derivation plot robust well among naturally extract orthogonal whether extraction investigate propose exploit common simulation show able real show promising clustering task study analysis remain feature important manually discover group also quite achieve common task object tailor task incorporate feature high extend feature promise direction restrictive flexible beyond block link often common feature time collect propose scheme common individual block discover common accord feature two algorithm extraction perform common incorporate dimensionality blind separation propose encouraging synthetic correlation link source separation increasingly science tool purpose interpretation retrieve multi attract increase attention encounter take subject various subject design collect individual device diverse naturally link share common due collect possess link devoted promising block canonical correlation cca propose maximize variable datum later cca datum blind separation extraction square pls maximize image framework tensor tucker decomposition therein method name variation simultaneously good knowledge fully study individual analysis multi block main include common detailed method cca discuss interpret high pca propose separation bss nonnegative nmf space separately desire block extract individual able rest organize extraction cca method discuss justify validity finally conclude remark suggestion set follow consist necessity assumption pca component ica method treat naturally component
summarize care occur horizontal axis logistic explanation h limit h rbf kernel eliminate cause huge selection prediction svm experiment cross validation fold cross accuracy validation illustrate th cross validation accuracy accuracy figure validation point one validation ten validation accuracy table lr std prediction regression respectively point model validation well result work actual contrary serious big hence prefer interval mixed analysis percentage composition mainly method svm lr expression expression stable real keep compare apply collect predict application estimate normalize select practical select variable lr one interval concentration respectively computational expand end concentration situation validation numerical get applicability svm lr practical security special purpose situation kind prediction true application induce lead consequence meet practical security like thank zhang comment company support cb chemical primary problem essential machine logistic lr lr get range meanwhile effect occurrence chemical process bring people life property occurrence chemical become lin classify year cause effect prevent occurrence include existence source control principle people naturally source prevent occurrence past since investigation come cause people get static remove turn prevent controlling reach certain range lot give fast mixed storage production influence factor pressure density h development intelligence chemical technique like artificial network genetic failure since happen machine lr efficient tackle decide intelligence interference classification accuracy logistic regression consideration lr clear explicit interval structure basic penalty comparison briefly give preliminary company essential control tend prevent lot medium concentration occur reach establish mining base prediction suitable know pressure special pressure room device concentration experiment density different point pressure concentration pressure whether occur transform another experiment several get pressure p average h c c pressure prediction svm lr incomplete variable introduce incomplete remove datum monitoring scale normalize avoid attribute object summary quality subsection chemical correlation variable quadratic variable original kind feature algorithm extraction transform original dimension extraction reduce transform feature subset variable variable slight characteristic effective knowledge chemical nearly consider serious reaction pressure take end procedure algorithm svm technique classification acknowledge quadratic generalization capability optimize margin briefly speak decision class kernel space view decision bias correspondingly hyperplane slack user penalty error change separation
multiclass adaboost mh mr work gibbs believe multiclass confusion matrix error error positive good generalization confusion pac heavily rely sum introduce bound obtain pac framework sec multiclass basic briefly main contribution bayes confusion future work present consider output class unknown distribution family set find make prior pac kullback leibler otherwise class transpose identity concentration self adjoint matrix case adjoint give symbol call maximum preserve hold element pac classification label consider learner aim choose bayes true empirical bayes classifier risk classifier predict draw accord return true bind pac theorem gibbs eq ab b bayes consider matrix say multiclass learn act sample example draw I example class context confusion consider confusion build upon inherent desirable f correspond confusion example zero outside recall objective learn generalization guarantee give conditional correct consider kind discard element confusion misclassifie confusion every sample confusion confusion equal control confusion classifier task precisely aim confusion small small main multiclass prediction confusion difference remark might confusion p inequality norm op risk formally relate confusion space family example belong defer q inequality fix estimation gibbs classifier upper risk gibbs depend minimal close empirical confusion matrix risk gibbs bayes way proposition well multiclass measure risk inequality defer imply confusion bayes relate defer appendix theorem deduce confusion matrix classic generalize carry adjoint central namely confusion rarely scalar almost surely scalar self adjoint matrix preserve eq sequence fix adjoint refer I ia invoke confusion matrix confusion iy confusion every naturally p equivalent clarity give example one verify notation demonstrate thank note value random negative q second give recall leibler let last complete theorem lemma jensen f calculation leading simplification equation right obtain remain substitute focus give multi class boost mh adaboost mr adaboost take theorem confusion may risk bind instance depend weight vote similar would algorithm sound would probably besides might might kind extend possibly also propose new pac bayesian classification confusion couple confusion confusion concentration self adjoint matrix self adjoint empirical corollary risk empirical moreover classifier interesting belong e confusion gibbs true matrix
eq total mass em theorem form ranking I order example top book publish week york ranking application voting discussion rating product player ranking stage item list appear select proportional denominator item item sensible pool infinite list build work pool rating parameter probability say case formalize representation atomic measure note item probability normalize draw first remove pick basically partial biased problem particular gamma process atomic marginal introduction suitable law give accord simple gibb develop effective time aforementioned start describe take infinite nonparametric operate throughout paper ranking simplicity length item pick th stage interpretation follow item arrival arrive em apply derive ml multiple ranking difficult directly alternative parameterization item shown factorize k factorize gamma conjugate w km carry vb algorithm update occurrence item appear ranking distribution instead previously update rating one model appear ad hoc sampler structure infinite object show gamma choice item random atomic measurable atom iid base density poisson concentration parametrization view easily extend nonparametric partial item arrival item arrive arrive give arrival joint gives mutually exponentially depict visualize top list visualization sort order consider partial ranking consist atom list observe item express law gamma model auxiliary inter arrival times list f generative item unique ranking construct marginally suppose atomic random measure law suppose law gamma conditionally conditional law coincide conditional law tc maintain reverse atom interpret strength dependence measure examine mass mass atom never atom base atomic atom give recurrence trivially extend partial ranking size extend sampler item atom atom propagate time posterior item item define write fix atom atom contiguous interval outside variable gibbs proceed update condition latent note extra integrate directly along parameter previous constant desirable evolve measure measure distribution define able except advantage remarkably discrete section parameter applicable apply discrete york category book list correlation assign account publication date book burn popular book category book appear book book list represent enable category characterize quantify pn estimate respectively atomic generative exactly characterized stage generalization process also continuous york useful density insufficient list path book prior publication evolve event foundation totally neighborhood instead notational work
age true order random nonparametric aim simplest suitable diagnostic simply explore exhaustive nonparametric conceptual simplicity practical popular exist beta roughly speak although truncated age set age counterpart bandwidth discrete beta parameterize accord kernel bias commonly offer neither smoothing technique note kind find commonly cite exploratory tool number time smoothing regression organize estimator adaptive bandwidth aspect preliminary logit computation pointwise interval section relevance package datum section kernel age hereafter since age equally spaced mean move hx define parameter particular dirac delta uniform effect smooth spurious fine get large speak beta possess characteristic firstly automatically change graphical display match age assign outside order near property counterpart discrete restrict vary reliability measure convenient way thus reliability closely reflect reliability small smoothly old characterize depend reliability sensitive reliability number adopt reliability local extreme reliability old case ignore reliability give estimator calculate age age beta vary mode exposure risk death age allow likelihood reliability variability relative variability x take exposure bandwidth regard bandwidth drive vast without simple validation simultaneously fit compare omit description validation minimize xx age remove difference commonly write notational compact rate parameter consist last simplify qx qx perform available plot obtain par par par par par par par old interest cross validation video default six observe rate fit plot rate cross validation residual prominent visible know literature especially probably activitie notable simultaneous representation statistic residual residual display r arise fitting rate minimize cross statistic residual inspection add say need create ex alpha produce plot bandwidth pointwise pass code argument default specify confidence argument allow pointwise interval naturally alpha produce plot note manually pointwise interval manual specification pointwise confidence useful take code bar display great exposure usefulness change visible range bandwidth ex alpha par par par par follow cross residual specification estimate interval code ci observe one one logit bandwidth vc alpha par par par par par par par par par par par flexibility result ht residual apply confidence compute preliminary logit specification joint logit beta e e summarize quantity package package conceptually easy nevertheless several option among formulation validation score minimize may residual plot either confidence interval
characteristic cut regular see follow bound connectivity scan scan statistic bounded let spectral scan symmetric solution fact supremum convenient derive single component alternative refined hypothesis scope article indistinguishable rely square supremum heavily reduce expect recall generic intersection intuition behind spectrum phenomena statement concentration result univariate asymptotic spectrum topology property asymptotically distinguish attain previous logarithmic attribute generic performance estimator naive detector reject reject distinguish examine take result stem distinguished contain balanced result log factor analyze detail topology lattice kronecker scan depth constant signal least cut binary tree spectral scan signal snr strong versus give estimator figure subtree level size gap regime structure improve power dramatically lattice vertex volume laplacian root bound low statistic test acknowledgement support grant nsf grant statistic sufficient normalize constant nan eq statistic function set proof optimal pearson lemma alternative parameter snr indistinguishable versus consider error vanish testing testing remark result distinguish fix let theorem write contain eigenvalue vector use let show intersection ellipsoid unit ellipsoid supremum supremum simpler able hold recall supplement reduce q distance orthogonal dimensional subspace orthogonal fix centrality symmetric minimal alternative square tail proposition test corollary study spectra really notably particularly give algebraic balanced depth small give characterization characteristic characteristic satisfy general balanced kn h completely characterize kronecker graph cut cut cut edge cut enough control spectrum kronecker base sum eigenvalue stochastically bound choose geometric change piecewise analyze generalized likelihood ratio statistic relate find laplacian call statistic depend topology theoretically scan outperform naive thresholding concern fundamental noisy observe activate comprise induce highly variety scientific area surveillance inherent difficulty specific condition algorithm anomalous scan statistic intensive entail individually anomalous graph understand determine computationally tractable cut believe realistic objective cut np mind relaxation call spectral laplacian importantly derive performance scan combinatorial estimator main tree give logarithm spectral scan balanced trees model verify scan dominate simple contribution define cut reflect way topological show cut develop tractable statistic scan statistic performance scan explicit scan notable topology superiority detector thresholde formulate problem realistic scenario involve composite literature subject relate fundamental statistic portion area incorporate life problem normal mean problem generalize test graph unclear extent topology theoretical mention feasibility fail match subspace problem cast nuisance structured statistic formalize change observation possibly nod eq constant within formalize write thought nuisance independent noise snr true cluster bi partition easily formally c interested problem gap regardless value cast structured composite q join alternative meaningful detection separation hypothesis analyze asymptotic testing sense far though explicit establish snr spectrum hypothesis distinguish limit asymptotically indistinguishable exist notation terminology theory central combinatorial laplacian adjacency matrix diagonal degree denote take eigenvector algebraic connectivity study asymptotic testing unbounded nuisance comprise composite hypothesis index feature apart
player expectation take internal adversary move end performance player dual play information loss specific bandit feedback start expand exp geometric expand applie assign weight action use armed bandit loss vector mix exp choose uniform action idea geometry finite exploration attain factor moreover hypercube exp linear bandit expert without action regret say provably suboptimal strategy order address class study mirror paper grow rapidly set observe understanding mirror adapt suggest basic universal picture semi feedback bandit feedback see mirror optimal strong fundamental bandit successfully apply mirror descent seminal compact descent barrier barrier hypercube suboptimal ellipsoid note implement mirror polynomial contribution canonical correspond euclidean euclidean efficient ball euclidean ball euclidean ball study paper exp stochastic descent regret exp show corresponding discuss briefly section show computationally hypercube briefly template exp mix np bandit scheme let round play z bandit feedback exp several equivalent way write usually implicit regret recall bregman write action differentiable proof randomness induce easy exercise example indeed differentiable differentiable mapping computation bregman exploration propose first online feedback use geometry ellipsoid minimal scalar exist contact simplex preprocesse action follow first combination space work ellipsoid ellipsoid minimal x translate everything product loss play ellipsoid ball xu drop ellipsoid product slightly account product eq also estimate outer mapping valid tp moreover clearly unbiased exp product easy bind p ta remain let thus conclude ta proof small eigenvalue one conclude use discretization argument exp use compact set ellipsoid basis specify factor ellipsoid replace computed efficient implementation lead programming consider bandit expert exp suffice turn preprocesse build correspond straightforward detail omit time observe loss contextual notable special armed suggest corner algorithm achievable simple bandit expert restrict attention obtain weight distribution section regularizer random perturbation minimax interior sign rademacher check perturbation optimization regularizer exchangeable hessian precisely follow differentiable thank theorem term involve bregman divergence computation obtains prove fact inequality satisfied v consideration identity elementary conclude j ti ti e p tp remain small inequality restrict attention action exp problem one attain regret order precisely motivation regularizer come perturbation interior bernoulli else easy check perturbation modify key figure
q diabetes version appear ds relatively ds lag thresholding property mild prove lag consistent lie lag discover control noise proof theorem parameter play effectiveness lag lag adaptive provide well one let variable effort mainly divide stream procedure aic show nice property impractical shrinkage algorithm lar computationally favorable introduce gap discrete selection attractive lag application lag improve group lag path thm thm mathematical stanford partially support wang fellowship call selector estimate nevertheless program control weight lag enjoy attractive lag thresholde property asymptotically consistent discover lag identify predictor computationally simplex lag superior compare soft mean gaussian tail predictor initial bias enhance explanatory play ideally complexity coefficient eq coefficient omit keep coefficient one zero discrete process subset limit large convex also zero substitute net lasso selector relate soft processing hard soft select model spread attempt scad penaltie eq sparsity concavity show enjoy variable relaxation optimal short least selector discrete influence program q dependent surprisingly geometrically analytically demonstrate penalty choose hard cancer rest organize present motivation lag connect lag highlight selector provide section discuss lag demonstrates understand design orthonormal ordinary ol notice operator jump soft p coefficient hard regression interval change interval pseudo function motivated lag matter selector write equivalently another hence expect choose name lag gradient zero mean substitute heuristic mean explanatory variable less otherwise abuse pseudo hard ol suggest ols detail section lag formally redundant column furthermore condition c condition actually covariance us define eq c lag consistent hard property proof role effectiveness pseudo interpret lag ty iy flat break otherwise direction break case still minimizer even illustrate toy minimize break similar increase exponentially provide discover true lag move break another breaking fortunately lag possible selector ds similarities lar numerical inferior penalize pc nonzero coefficient significantly stand indicate uniformly nonempty condition lag identify probability ol accuracy lag quantify lag program condition b analogously take section tx upon contradict b imply hard thresholding eq therefore w lag normal hence enough prove py denote cc tx c mean two satisfy subgradient q lag convex lot interior selector lag reformulate solve lag equivalent linear constraint linear program relatively recommended routine absolute lag pass routine treat response lag need criterion validation solution warm justification technique simplex try adjacent current one new solution find thresholding property cancer weight age logarithm amount seminal percentage score set profile lag quite predictor exclude almost discrete lag penalty indicate jump jump include segment include unchanged profile role cause however role right lag piecewise constant demonstrate trend though prediction lag contrary variability diabetes predictor fit lag compare lag lasso lag lasso fold table lag variability parsimonious lag nonzero product lag effort notable magnitude nonzero lag lasso relax relevant consequence predictor weight select relatively argument verify
exponent ensure exponent measure affect high measure affect major guarantee high measure alternative obtain sequential estimation I measure firstly constraint problem estimator max I extreme dependence root square cube size come property denote constrain magnitude support strong percentage exponent htbp cb c integrate square deviation benefit unconstrained improve exponent bivariate conclude limit explain estimator exponent measure constraint beneficial max superior especially moderate dependence large improvement high measure promise acknowledgement financial definition proposition extreme link dependence margin inequality order exponent measure measure subsequently exponent impose estimator max stable exponent inequality estimator stable arise appropriately scale maxima identically throughout algebra vector fr margins max simplex satisfy margin due loss fr margin place structure copula function invariant transformation practice random dependence stable value back max stable positively imply additionally max strong dependence pair non review property max stable distribution although aforementioned dependence class distribution implement paper incorporate max constraints essence dependence stable inequality result nonparametric exponent satisfy inequalities simulation study conduct assess performance arises appropriately scale consider identically fr margin representation exponent extreme w jj exponent measure max stable completely eq measure describe completely margin exponent measure additionally homogeneous I homogeneity property dependence quantity define maximum measure term complete dependence trivially interpretation coefficient inequality bound set stable terminology refer respectively set yield satisfy ease notation set positively dependence property max tight bound high coefficient easily inequality analogously terminology introduce subset ms new exponent negative uniquely function set exponent negative line simpler consistent consistent modelling model limit extreme value identically distribute vector unit fr margins normalise maxima cumulative fr dependence estimator
name allele family identify assign classification tree describe name name recover insight drb assign drb assign give broad type type assign drb broad type medical biological assign solely dr dr dr dr dr st st st st assign name allele drb confirm reasonable check cluster make st allele cluster tree contain drb drb group drb drb allele dr broad dr dr exactly drb broad five dr dr dr drb drb dr drb drb drb drb classify drb drb allele table classified st drb drb drb dr consist split dr dr dr drb drb dr dr broad split dr dr st consist st drb drb drb drb consist drb st exactly drb st small drb drb consist dr broad assign confirm st st st st classify nine classification structural experimentally bind binding overlap classification group propose synthetic panel et seven drb drb drb drb drb bind specificity group nine algorithm result classification work classify drb et seven similarity study dr solely datum drb drb drb drb drb dr drb dr drb drb drb dr drb drb drb dr drb u drb drb drb drb drb drb drb dr drb drb drb drb dr drb drb drb drb dr drb drb drb drb dr drb drb dr dr drb drb drb drb drb dr drb drb drb dr drb drb drb drb dr drb drb drb drb drb drb drb drb drb drb drb drb drb drb drb drb drb drb drb drb drb drb drb drb drb dr drb drb drb drb drb drb drb dr drb drb drb drb drb dr drb drb drb drb dr drb drb drb drb drb drb drb dr drb drb dr drb drb drb drb drb drb drb drb drb drb drb dr drb drb drb dr drb drb dr drb drb drb drb drb dr drb drb drb drb dr drb drb dr drb drb drb drb drb drb drb drb drb drb drb drb drb drb drb drb drb drb drb drb drb drb drb drb dr drb drb dr drb drb drb drb drb drb drb drb drb drb drb drb drb drb drb drb drb drb drb dr drb drb dr u drb drb drb drb drb drb drb drb drb drb drb drb dr drb drb dr drb drb drb drb drb drb drb drb drb dr drb drb u drb drb drb drb drb drb dr drb drb drb drb dr drb drb drb drb drb dr drb dr drb drb dr drb drb drb drb drb drb drb drb drb drb drb drb drb type column remark stand marked expert indicate allele shorter c dr dr dr drb drb dr drb dr dr drb dr drb drb dr drb dr dr drb dr drb dr drb drb drb dr drb dr drb dr dr drb dr answer final statement question bind otherwise study look comparative scheme n extend framework protein suggest hope think chain author drb contact drb sequence marker suggestion helpful evaluate influence bind adjust topic future improvement explain capital h lc l see recover vector obtain coordinate precisely www toolbox composition mm mm mm proposition edu live com edu edu kernel substitution use machine allele allele dr allele classification relate chain biological offer simple powerful scientific effort certain string bind classification major complex place substitution simplicity find gap help bind bind bind hadamard power square contrast machine former emphasis describe chain inspire local vision begin finite may think positive definite let basic life protein string definition substitution align represent think symmetric addition check let next index x still recursively string q string kernel chain define chain abuse count occurrence occurrence kernel normalize correlation basic sometimes work place alphabet penalty even gap numerical experiment indicate context reason one limit choose protein protein realization individual protein set related ii drb simply drb subset play central bind body chain allele strength bind string substitution cross also span contain define vector refer ii available operating curve roc auc binding refer contribution nn auc drb drb drb drb drb drb drb drb drb drb drb allele square list auc use thresholding bind non auc mark weighted weighting size set contribution sequential generalization bind allele allow detailed drb dr groups important coverage gene variant overlap bind group accordingly bind specificity health organization assignment base work throughout world international exchange et assignment especially certain drb drb sequence exclude marker drb allele allele locate marker location occurrence location last occurrence reason marker constitute drb encode bind site reason position allele occur range allele transform normal different order collect normal form list since may impose call derive sequel family exclude drb family cluster construction clustering separately drb drb drb base order base instead cluster st st st st st st clusters play neural network sequence type check application string motivation relate string function bind work similarity powerful bind affinity value available output function bind allele take string symmetric call definite positive definite kernel subset kernel also x iii product follow normalization function space euclidean product linearly general kernel refer finite reproduce reproduce hilbert notion usual define example alphabet name semantic collect align find region greatly block block occurrence indicate frequently another eventually normalize indicate occurrence construct qx take logarithm round simply obtain analogue spectrum hadamard matrix positive entry hadamard hadamard logarithm value symmetric hadamard hadamard logarithm conditionally hadamard logarithm define fact follow iii imply verify hence index k semi index index count explain sum give cl definite elsewhere since know k positive discriminative string stand review write space call represent linearly coefficient see underlie sense run guarantee fold validation suppose partitioned assume paper fold one train division division compare define special refer leave cross tune affinity strength ic score ic ic determine bind bind bind ic bioinformatics usually score introduce ambiguity sequel ic bind affinity benchmark publish cover drb representation bind allele compare state art nn align allele allele step leave partition five fold merge training leave cross validation every geometric optimal predict bind aa pp since affinity label datum divide bind allele define allele value bind matter hence measure bind auc table relate reflect modulus dp p list cc allele allele drb drb drb drb drb drb drb drb drb modulus continuity big neighbourhood bind motivation allele bind huge phenomenon refer job experimentally allele put bind datum datum allele help sequence use step introduction sequel parameter define allele kernel bind bind affinity choose allele reduce allele allele kernel allele drb bind affinity whole divide follow contain datum suggest index define five validation division merge part testing leave cross validation employ except test auc affinity transform show l allele rmse drb drb drb drb drb drb drb drb drb drb drb drb drb drb drb drb drb drb drb drb weighted auc row mark bold use table table allele rmse auc allele rmse auc drb drb drb drb
basis spline find short severe reduction burden five level difficulty task compose essentially experience distance go minor member help effort compose task assign percentage difficulty difficulty illustrate shape discuss base posterior quantile plot covariate equation gender find negative imply bad condition line attribute disease posterior estimating count age use easily report severe cv quantile summarie entire aggregate age age group assign reason variation specific depend course age region difference population two influence evaluate classified fourth difference probability extreme curve age group l age call cv cv obtain motivate estimate population survey latent addition use unit latent proportional odd mix whole together covariate adopt estimate spline allow propose basis inclusion number knot p keep low avoid recommend spline simple people categorical person accord survey design reliable quantifying amount challenge adopt base estimation latent belong change influence penalize basis show mcmc posterior estimation spline base health survey task responsible health organization planning classify monitor people medical investigation medical person request provide classify population people public department important alternative hoc survey expensive survey statistical proportion european source health condition survey national institute every year recent employ account international world health organization evaluate activity survey person exclude degree difficulty person even aid item maker perspective definition loose tv burden health mainly plan status dependency approach measure score equally contribute measure may obtain functional national survey classify different classify extend tackle locate center large proportion region health department responsible health organization planning classifying level health age label probable latent within health exploit variable range population count population recent rao challenge latent hide area develop item summarize datum probable membership dependent value latent covariate error propose tackle classify get area hierarchical status capture form influence therefore mixed covariate spline spline area use cubic spline radial spline basis consequence provide via spline spline random mixed representation spline basis property behave chain time datum spline specification final procedure selection checking provide conclude population complex follow consist large self consist secondary divide population proportional systematic sampling member survey live involve divide health group age domain make technique account international organization conceptual health loss restriction lack ability perform manner disadvantage prevent depend large difficulty person aid item type accord difficulty movement difficulty one home intend people difficulty show problem rest bend something difficulty daily concerned independence basic get tv volume aid recognize ordinal difficulty group four categorization category home go capable effort lot effort capable category activity get category effort take category category volume capable capable speak use tool perform assess item latent conduct line nine provide particular remove raise merge cognitive addition redundant discrimination latent reason employ reduce nine item aforementione first continuous logit increase probability equal interval effect age functional assume smooth compare parametric approach functional form relationship estimate relevant application interest specific despite numerous area model inclusion area incorporate connection spline spline mix incorporate nonparametric mean unit spline function cubic spline radial spline knot desire make little property p spline framework thin spline avoid coefficient penalize shrink accomplish thin mixing eliminate posterior step implement ij td let triangular invertible cholesky decomposition symmetric transpose ease arrange fashion nature identify orthogonality order intersect axis transformation eq q w rewrite w c rr need b item select level consider prior informative computationally convenient flat real assume health effect independence effect representation thin equation normality hierarchical use spline effect task critical sensitive influence smoothness root component choice along exercise area obtain software count different age combine draw count specifically count small index run subset case deviation quantile summary proceeding choice specification multinomial logit vs purpose nonetheless model problematic consider index membership estimating count draw frequentist cdf cdf information use left gain emphasize propose tool people characterize level age health goodness fit clear interpretability pattern count estimator careful consideration selecting reason suggest class adopt lead class interpretable class approximately class interpret sample lead unstable count small area comparison odd specification model number class odd exclude status covariate give contribution slope parameter around knot spline keep model place value sensitivity exercise interest rule smoothness spline sensitivity exercise section proportional odd class therefore age three popular see detail parameter
consider kolmogorov type statistic xt asymptotic take schwarz c hx formula simple sequence research author grant grant definition xt goodness fit test statistic characterization family belong deviation asymptotic nan efficiency parametric goodness one problem distribution function family inverse pareto often g service period system economic pricing reliability goodness test far test base completely way introduce monotonic characterization characterization find family state goodness general alternative also introduce resemble statistic describe limit efficiency certain need rough large deviation asymptotic refer efficiency kolmogorov hence applicable describe value kullback may n statistic change non statistic degenerate first give theorem deviation degenerate get sufficiently pr generally alternative three contamination alternative density need kullback nan consider nan composite establish general dy elementary calculations eq eq efficiency moderate maximum equal slope admit kullback leibler third obtain therefore integral locally kolmogorov statistic depend calculation get projection point kernel degenerate limit develop show complicated distribution random find statistical modelling statistic bound use deviation degenerate sufficiently eq q calculate local f deduce formula representation eq kullback large admit asymptotic leibler local alternative slope statistic admit know information efficiency alternative see goodness fit testing section local sequence maximal local
rating aspect accurate sentence simple explanation aspect highly correlate I star refer unlikely rate rating text relationship aspect depict conditioning encode eq encodes penalty star vote co occur star vote prevent unlikely possibility occur eqs minimize used multiclass dependency rating section aspect corpus require choose review accuracy fraction rating scaling range large experiment hour randomly test corpus label algorithm test try baseline middle rating prediction high square show seven semi supervise upon learn fully supervise possibly due unsupervised english simple merely english supervision review address seed supervision baseline train correspondence sense occur user overall toy review acknowledge aware million require hour train dataset outperform supervise label unlabeled explanation semi optimize likelihood fully optimize certainly technique supervise extend lda topic lda unsupervised supervise like unsupervised achieve close five dataset consider aspect rating aspect rating task review naturally evaluation train using predict rating rating prediction rating rating per sentence review rating inaccurate capable predict rating text fact model outperform aspect predict rating text fail explicitly largely address combine rating text decrease baseline combine rating level supervision impact rating surprising affect significantly case rating prediction aspect label inaccurate unsupervised learn rating ultimately predict must rating mean rating rating task intervention aspect sentiment model learn match aspect high sentiment parameter confirm sentiment expect role different study old warm discusse interact decision sentiment separately aspect explain star normally ingredient mass account review surprising among introduce corpus million source system provide rating multiple product sentiment determine part review rate aspect review highly sentiment readily corpora yu intel fellowship microsoft fellowship online consist plain text together understand aspect contribute well know help product paper build rating system separate aspect introduce corpora review rate six prediction task rate aspect sentence rating rating dataset recover rating art scale moreover content automatically content well aspect task product sentiment make mean evaluate describe feature naturally answer different website opinion influence may body body refer aspect answer rating aspect aspect provide rating opinion figure tree dark light head excellent dark light thick body dark finish presence actually nice ccccc consist text one example six sentence look overall rating form supervision sentence discuss rate aspect medium thick warm may describe choose review well explain rate third rating content goal corpora million review interpretability modeling approach aspect separately describe interpretable sentiment new name short introduce corpora million review rate three handle dataset training predict supervision aspect supervision manually label addition unlabele supervision label human ten corpus find benefit supervision aspect sentiment discover separate review aspect recover find require explicitly similar author aspect term aspect paper discuss though sentence multiple review aspect rate recover rating discuss unlike topic topic aspect aspect sentiment review body describe aspect describe aspect manual intervention domain critical due complex five multi ten manually annotate million review period time adapt throughput highly aspect highly review I exist source much simple approach lie model contrast learn neutral word aspect simultaneously learn sentiment aspect contain consist plain feedback score many application include discovery sentiment among understanding demonstrate well task nothing good user preference nothing corpora approach topic frequently interpretable representative attempt way aspect multi aspect rating corpus identify highly correlated vote assign aspect sentence deal examine explicit aspect rating aspect sentiment assign sentiment corpus review manually specify sentence sentence publicly rating many system rating rating model model relationship work unlike whose aspect word distribution separately word sentiment aspect review considerable manual intervention learn automatically language review finding consistent aspect look amazon audio rating user aspect rating aspect rating review rate price service product review quality category user feedback rating product author dataset cc aspect price service briefly provide rate obtain use review label aspect rating aspect determine human sentiment label use sentiment label negative neutral index brevity wish label manually label review sentence irrelevant annotation crowdsource amazon review sentence coefficient agreement unfortunately annotation correspond random sentence evaluation address crowdsource service individual expert question approximately hour expert score score review expert review corpus review corpus annotate corpora expert publication model aspect word appear sentence simultaneously learn rating might might word appear discuss high word sentiment intuition closely corpus review divide rating overall aspect though aspect sentence sentiment encode discuss aspect index aspect aspect aspect aspect word star rating use expressive power alone separate critical would find sentiment word beneficial sentiment aspect aspect sentence independently entire learn aspect maximize learn scheme level supervision increase supervision must match aspect optimal highlighted segmentation task five node five ensure remain unconstrained include aspect aspect aspect rating unsupervise latent aspect assignment ascent optimize merely consist maximize sentence concave proceed square ascent sensitive aspect initially assign aspect name aspect parameter initialize randomly high among subtract constraint interpretable issue encounter aspect assign single notice word aspect regularizer perfect word reduce reduce address need prediction encode aspect enforce aspect subject aspect discuss application example encode must
setting however slow arrive finite certain assume many fall see quickly past major direct estimation space among aim direction collect give value temporal difference bellman bellman temporal basis function neighbourhood dimensionality technique construct td bellman angular provably approximation extraction online extract binary function approximation keep candidate combination td generate small td simplicity regardless bellman feature space like assumption space bellman assume linearity bellman derivation projection projection norm preserve class type projection scale project appropriate target exponentially immediate bias induce sparse space probability ok linearity help error fit linear value mild approximation bellman angular estimating bellman discuss reduce control proper return thus prediction error much compressed light result propose simplify variance simplify version compress weight eqn error determine subject projection guarantee algorithm memory discuss fast robust respect gradually complexity adding stop point easy set generation validation start provide tight work bind suffice bellman error td compress eqn let collect td bellman linear detailed appendix sketch bellman control constant transition third simplify concentration norm expect projection preserve bind due variance set trade compressed discuss arbitrarily bellman policy estimate careful benefit state contraction measure bellman satisfy add either challenging goal optimal rp far direct regression original future poorly small importance type sparsity randomize extraction e projection projection compress apply state size compress rp method choose consistency initial guess fail exclude address rp rp rp error rp among fix vary number iteratively minimize method small compare fit solution seem two minute work half million sample space prove projection preserve linearity generate projection reduction analysis projection memory behaviour happen common empirical analysis confirm extraction projection avoid car regularization rl complex course iteration validation selection tight theoretical help analytical closed answer question linearity bellman avoid linearity simplify concentration rapidly mix give base time markov measurable hide need characterization fast past transition kernel ia operator dependence concentration bound large value small chain markov uniformly past major result consider generalization inequality random homogeneous follow application proper constant write td bellman error term series martingale point bellman wise td bellman regression lemma term theorem prove lemma probability define x sum vector chapter satisfy union equation line set substitute simplification probability bound inequality base variance exist element element taylor expansion inversion inverse column see thus orthonormal basis union line second apply lemma line finish proof finish lemma bellman contraction contraction due corollary cs automatic bellman function evaluation function rate logarithmic error parametrize estimating generation reinforcement lot aim bellman current estimate bellman exact unlike fit iterative space feature generation bellman method rl bellman error fair many fail use project exponentially difference help determine size discretized code feature many feature iteration logarithmic time provide minimal code type also need
problematic markovian model trajectory backward simulation section detail novel backward backward development truncation non markovian generic applicable study severe accuracy simplify slightly extend version unnormalize assume would particle time independently proposal kernel implicit write adjustment auxiliary sampler weight sampler construct kernel validity assess mcmc sampler extend smc auxiliary variable target construction admit density indexing point notation x sequential procedure sampling appear know form smc index maintain furthermore sampler straightforwardly never sampler incomplete gibbs ergodic chosen however observe especially poor degeneracy cause collection new fundamental gibbs value draw verify correspond collapse leave plug numerator realize difference introduce break strong set ta tm mx x suggest far explore explicit pass discuss problematic markovian distinct forward sequence step pg backward simulation via problem markovian show compute backward computationally prohibitive backward sampler markovian model make progress markovian markovian hence possible replace follow backward backward fx sl c appendix compute backward weight result quite even accurate inferential know truncation weight evaluate start discrete tv truncation level exponentially decay move remove requirement specify introduce parameter easy change whereas stop cost computational trade well illustrate typically evolves figure example right adaptive early still line overlap dash markovian efficiency sampler state apply smc sampler rao approach rao since marginal process nonlinear density fix trajectory kalman backward particle smooth model evaluate backward weight thus kalman filter complexity employ truncation computational backward generate dynamical naturally model state admit dominating consider smc sampler straightforwardly apply problematic address smoothing particle reason backward degenerate normally simulation degenerate markovian see static appropriate definition degenerate markovian exploiting discuss deterministic markovian backward method see markovian smc inference work sampler markovian application tree node smc fashion agglomerative smc generative markovian observation give markov employ section exact run smooth state first rao particle th system e run coarse approximation discard state make five sampler accurate magnitude less would decrease suggesting cause dominate even simulator smooth forward particle backward computational sampler serious particle term iteration computational sampler affect truncation evaluate use function toolbox take order simulate step system noise sampler iteration use particle consider level system well posterior mean discard sample rmse relative mean box tend increase probability truncation representative th setting run track smc great success g see scenario beneficial instead problem target surveillance behaviour apply state v standard g acceleration straightforwardly parameter state system assume affected acceleration noise use matrix arise discretization state target uninformative arise visual tracking initialize process singular care apply state suggest space u sampler hasting random walk deviation density sampler target worth point sampler complicated sampler model natural implementation sampler particle conditionally similarly sufficient backward state backward discard truncation truncation employ sampler particle discard first smoothed sampler provide inferential despite weight problem tune
form task multivariate experiment outline h initialize construct nonparametric accord marginal minimize solve system equation kernel equation close former optimizer advance summarize base prediction jx classification new want separate class likelihood portion shift series accord learn via basic dirichlet suppose need infer assume parametric parameter belief limitation scope type inference approach dp base positive gb almost hierarchical specification g n fx parameterized generate partition interpret grow new limit mixture dirichlet prior dirichlet dirichlet concentration previous addition calculate variable approximate choose find distribution minimize leibl current factorize analytic truncate stick fix treat follow variational following probabilistic minimize graphical distribution z term expand lemma denoting z e recalling observe see distribution tv e hz hz j z j z j equality j couple observation observation yield hz j calculate hz previous identical z j previous log derivation exactly derivation summarize except calculate calculate equation implementation make extend implement function function fit stop difference demonstrate process st jt jt sample vector sample randomly finally task amount model fix mixture group expectation propose individual variation learn jt jt square qualitatively trial see task poorly st well st area estimate close figure recover set local yield similar st show rmse draw conclusion work st increase converge dp truncation dirichlet dp concrete motivate research star category research number star survey dramatically collect hundred classification star survey base behavior periodic especially study periodic star notice characteristic time light share shape series periodic star experiment explore dataset addition orthogonal therefore context experiment time filter sample point interpolation normalize standard series pre mixture near series slide series point take separate order cv result see densely gmm series aside effect vector gmm kernel hold place flat effect minimize j algorithm set learn consider observation modify covariance estimating put together form equation compare mean vanishe covariance accordingly view extension phase similarity share feature separate investigate truncation dp show bic data dp bic equivalent performance reduce bic many choose learn correspond develop apply star part preprocessing reject million approximate insufficient propose provide process performance propose time start simulate survey feasible rely series normalize linearly nn gmm rbf use order moreover classification close centroid series per sample perform additional discussion performance dense explain obtain interpolation sample notice plus inside improve third sample increase gradually become plus achieve densely sample difference use expressive getting gmm non reason attribute share gmm gmm constraint setting model come summarize densely sample couple perform excellent dp potential three label test assign initialization accordance run dp dp obtain accuracy result order focus point pure without label discovery time attract year range potential example diagnosis diagnosis speech common choose time fourier wavelet coefficient feature fall category base generate model closely detail framework across algorithm parametric mixed task task furthermore effect along different process mean allow gaussian second mixture share kernel focus stationarity regression divide infinite network different closely approach substitute curve series parametric incorporate shift shift multidimensional also treat e curve alignment parametric cluster hidden admit parametric regression handle series differ model shift estimate handle full parametric hand grid variation allow treat nonparametric task model sum group prior extend number variational learn experiment discovery particularly sample several work survey system therefore issue develop appropriate extend context drawback avoid perform cholesky wang wang share among shift periodic novel capture group task learn periodic lead infinite selection experiment use particularly sample multi variational world task study drug find good regression instead measurement leverage patient population individually multi intuition simultaneously attract approach medical diagnosis framework formulate multi value reproduce hilbert exist statistic share learn formalize share build extend include aspect modal second task shift original shift invariant group mixed alternatively shift random regression unlike exist extension proportion adaptively rather explicitly inference propose posteriori map parameter insight hide variable identity solve phase addition dp show complexity alternative gaussian mixture series interest primarily regression utility several synthetic series yield superior gaussian bic introduction interpretation main assumption model report relate conclude idea throughout scalar vector function extend fx fx n reproduce hilbert keep simple generalize framework convex relate large term complexity assume square function term time sample center normally shape thereby variation arbitrary known require section shift individual variation one hidden prefer get concrete train l separately given provide explanatory application excellent classification j analytically intractable couple sum distribution problem resort iterative algorithm iterate step local maximum stand last difficulty couple
come get irreducible reversible state ji reward player remain successful without player pick arm successfully frame sample player n jt max need prove reward player property match occur I j jt jt jt jt jt take eq index know frame decentralize markovian reward per md mn arbitrarily sequence arbitrarily skip bound event theorem display false p max max md put together bind desire previous refer bipartite one distribute bipartite arm know matching maximize match note goal optimal preferred bid player determine player price high arm high find reward player arm less divide frame frame allocate bipartite matching algorithm run phase length describe phase decision time determine channel match change change counter need phase phase figure bid channel user know bid allocation determine bid run return go successful transmission slot frame channel bit indicate bid precision frame match b successive reward channel markovian user bernoulli run show cumulative bold curve upper curve well even markovian case reward generate chain reward e performance algorithm ii much well decentralize achieve usually difficult progress markov state algebra fx department electrical engineering university california usa support grant fa nsf distribute player arm bandit mab pick pick arm get reward I model unknown markovian model arm model irreducible reversible unknown stationary arm reward player neither dedicate control player costly add regret index achieve expect motivation come secondary user cognitive wherein must pick wireless channel different user introduce armed essence player environment player learn suppose choose arm reward density find minimize type policy expect grow order arm regret asymptotically reward come much index policy asymptotically unlike reward come propose markovian irreducible represent single arm evolve probability play passive play liu extend problem trivial show policy chain continue played show space hard policy long high reward employ notion regret compare reward always arm high reward policy priori knowledge propose policy propose deterministic exploration author arm bandit partly spectrum access channel yet know nothing channel statistic bad channel expect channel learn explore formulate channel must channel show must decentralize setting I policy regret markovian weak assume underlying chain polynomially surprisingly regret decentralize armed discover propose index heart operate round price bid value impose communication must nevertheless growth expect achieve know organize section single mab markovian decentralize reward decentralize case reward match decentralize numerically wireless arm correspond channel want pick arm dedicated control among player potentially pick arm regard player play reward function generality mean unknown player statistic play player reward get reward share manner get time play player tn namely time maximize reward time match bipartite match reward pick policy policie decentralized player formulation reward markovian player model irreducible represent x j assume generality player use emphasis order unless otherwise first bayesian bandit later player player pick identically play time denote reward play jt x jt ucb pick instant useful analysis subsequent index every epoch switch space together yield variation schedule unknown decentralize mab might question various affected resolution high index pick square pick positive monotone computation algorithm computation give ucb positive ucb slowly asymptotically slight modification due precision player arm become c follow since p also last equation know get choose regret grow trivially bind regret scenario reward arm come reward model reversible markov finite matrix x distribution ergodic max xx max jt playing satisfy small contain follow derive markov min xx jt jt ts jt nx jt jt jt jt take jt jt mt concentration chains bandit markovian cost ucb sequence consider suboptimal start start jt event j event arm number observe event j jt j max inequality skip fx jt jt c fact similarly event summation eq jt jt max exponent mp yield computation formally section choose expect armed bandit reward precise computation choose monotone omit decentralized multi armed reward wherein player pick arm neither get bipartite bipartite matching however distribute bipartite incur computation exchange time bipartite matching frame kind decision index bipartite determine phase exchange player change frame frame length decentralize define successful play pick arm denote frames time player yield bipartite expect reward initialization least counter match use signal player change increment counter decision player matching
parameterization modify complete may interest pmf amount spend failure formula pmf state state cf calculate ft ft ft rv ft discrete computation ft dft expectation variance ccc desire pmf extremely quickly accuracy handle probability computational dft function dft author thank david pmf hold sample transition distribution pi pi fourier transform f st f pmf proof remark em height em depth interest field analysis find use characteristic inverting calculate unstable general lattice quickly calculate discrete fourier extremely fast able useful inversion characteristic rely existence central rely property general seem approximation calculation partly difficulty case science make fourier preference draw area process another way representation branch represent state label characteristic detail see know calculate challenge cf transform dft dft support term negative binomial fouri inversion secondary easy calculation dft dft deal lattice dft code software package q transform fouri transform understand dft fast transform dft fourier control dft dft rv discretize evenly spaced point adjacent dft similarly ft evenly interval transform domain eq dft lattice nonnegative pmf lattice dft dft approximation ft dependent dft pdfs lattice able essentially information variable include dft sampling rv pdf class lattice essence little census locate allow dft accurately ft sufficiently lattice properly lattice integer argue nonnegative slightly derivation lattice need transform let transform control pointwise dft less ft great dft accurately transform difficult transform demonstrate inverse dft forward dft dft fourier rv evaluate pointwise dft less couple notational help integer negative large imply therefore express forward dft assume approximated dft pmf ft nonnegative bind replace thus error introduce fourier primary estimate division occur address additionally also part primary dft one pmf overcome one bind moment primary case turn transition exist find impractical another approach
straightforward verify hold large fast solution q stem involve supremum diameter tuple see combine condition term within agree obvious correspondence appendix derive bound theorem constant far completely term situation invoke scaling term proof hold modify theorem parallel analog column eq column dominate selection orthonormal select error problem process unlike matrix radius dr proof theorem slightly weak sparse match corollary result optimal reason enough eigenvalue recover individual eigenvector dependence low shall desire remain focus assume appendix proof bind packing set particular packing give k covariance fold lemma minimax gaussian principal remainder correspond packing consist metric packing set reflect post two bind careful maximizer ingredient bind maxima traditional purpose elementary tool functional curvature let span eigenvalue alternative perturbation distance use inequality variational lem function g span satisfy condition suit ideally constrain combining recover frobenius parameter solution corollary sharp nontrivial symmetric obtain thus argument probably detail eq hold let index quadratic form sharp recent advance corollary standard gaussian see proposition hence subspace regression correspondence sparse pca far final minimax theory let penalize play consider penalize subspace analogous u entry row penalize simultaneously enter multivariate encourage sparse go beyond rate independently condition achieve constrain straightforward rate bound theoretical appear intractable rather correspond optimization challenge although minimax noise knowledge modify technique exchange along case whether exist without either thresholding seem front restrictive possible outside one approach follow estimate principal penalize detail future consequence state essential separately capture essential subspace vary subspace give theorem row satisfy eq universal constant every otherwise hold see let exist every imply sparse satisfy constant estimator setup subspace prove theorem tool minimax low generalized measure measurable leibler measurable satisfies calculation multivariate let consider subspace kl exact fold divergence divergence find pack small diameter subspace frobenius prove method construct packing let allow packing diameter conjunction follow generic minimax estimating define define derive allow analyze subset satisfy absolute v lemma relation fix unique verify guarantee thus substitute packing set pack packing choose element manifold representative pack subspace pack manifold satisfying exist lemma proposition convert bind frobenius invoke let necessary constraint eq side substitute low bound f c n capture hypercube hypercube orthonormal support combine pack packing approach however pack product may differ pack require packing without final packing achieve apply round code th coordinate code specify place column packing differ pack set column modification substitution brevity outline major step definition I satisfy small letting every invariant estimator remainder high order term proof ignore dominate orthogonal write quadratic term cross term lower singular spectral norm universal bounding norm v v q v v eq tail controlling involve universal eq thing together condition q angle subspace span ii eq inequality precede conclude write determinant thus f proof lem subset satisfy ii right belong hand argument brevity u since e apply conclude orthogonal symmetry w e f e ef ef dedicated cross proof variable satisfy constant matrix net duality exist schwarz inequality u j number g q side inequality minimal choose satisfy v recall u entry constant follow remain norm thus inequality minimal net constant acknowledgment article complete university thank comment support nsf fellowship dms nsf grant sparse principal analysis high number estimate subspace span eigenvector sparsity bound sparse subspace class estimate optimal without restrictive interestingly appropriately employ early arguably know use dimension numerous diverse much early serious difficulty inconsistent lead conclusion correspond estimation assumption matrix development focus methodology concept eigenvector include development limited result lead establish individual eigenvector open pca span facilitate assumption variation mathematically convenient imposing conceptual principal unlike obvious present complementary notion sparsity norm row sparsity orthonormal intuitively mean subset crucial existence frequent rotation practitioner help principal component principal estimation sparse subspace constructive wide nearly illustration row measure section allow intuitive explanation subspace variable corresponding basis advance subspace sparse estimator iterative thresholding assume perturbation showed nearly achieve eigenvector track dimension subspace work covariance model two reason simplify analysis enable exploitation gaussian possibility equal variable covariance case technical ingredient subspace novel variational corollary useful recent advanced bound framework construction pack develop key distribution similarly satisfy estimator subspace generalize angle subspace canonical angle projection additional angle subspace orthogonal orthogonal denote familiar frobenius subspace singular singular nonzero singular twice identity convention relate subspace ordinary euclidean orthonormal minimal orthonormal achieve constant additional subspace convenient manifold matrix compatible show problem maximizer subspace define essentially lasso estimator idea chen sufficient know convex replace convex
prior event c p bind paragraph right combine proof n start cc cf hellinger dc nc bound n verify condition display vanish equation constant mb nj ng extreme union kb g j nj due theorem last j contraction n proof accordingly involve construction volume hausdorff general position vertex hull lie adjacent distance vertex g rest multiplying depend vertex g kp p g q display distribution v mp g cn kp cn side display employ part use kp g kp n proceed polytope pg mn lower remove regularity induce ki second exploit k ki g k k n conclude proof l kl note dx gd g ray angle ray edge bound intersection ray resp intersection contain support lie intersection segment x x x b c repeat extreme g segment cone b dx g gx h proof fix satisfie soon say general either lie form face among hyperplane center map stand pg c show pg continue bind face hyperplane contain contain omit key ingredient establish existence measurable hausdorff two lemma test highlight hellinger begin existence density vary overcome packing consider pack nh nh tp n g consider inequality dd g existence suppose every every exist packing proof maximal set tt take take way p due monotonicity every probability lem satisfied hold n display thank display term vanish third rule bind denominator I p last bound acknowledgement support grant author motivate work style graphic g mm nm pt distribution population polytope also contraction population kn n population polytope mild contraction hausdorff distance support extreme nm quite interesting rate root density datum entropy layer continue fall effect additional datum set sequel additional contraction contraction compare exponent differ algorithm recover population nature extreme polynomial exist establish consistency contrast concern specific contraction paper asymptotic geometric technique exist proof tool asymptotic continue useful estimation section formulate abstract contraction main endow hausdorff metric opposed endowed hellinger quantity indeed hellinger fundamental analysis tie amount hierarchy paper devote feed hausdorff divergence functional kullback leibler induce relate integrate technique geometry come derivation remainder main basic geometric assumption consequence abstract contraction whose verify present provide leibl neighborhood theorem technical radius g diameter throughout affine polytope denote leibler hellinger variation define independent convex call distribute accord use dirichlet kp detail equation write multinomial specify I ij px I give relevant variable dropping indexing individual n count joint mp product draw nz z early amenable significance work g polytope behavior infinity number population polytope serve bind purpose parameterization consider contraction behavior access contraction distribution g give population metric minimum matching hausdorff mild metric family concern behavior boundary p regularity dirichlet characterize contraction relevant section satisfy j j priori symmetric let hausdorff consequence relatively widely symmetric technical require kullback hausdorff fact replace statement relatively dirichlet difficult try use establish technical asymptotic primarily establish point far endow contraction exposition contraction suffer even total amount appearance root kl divergence distance appearance proof viewpoint highlight provide issue discuss manner knowledge order rate kullback leibler situation lemma estimation technique always posterior contraction rate gp incur independent since spirit regard g hellinger rate size generally derivation still yield rate remark iv remove trivial role exponent intuitively large mass near boundary locate boundary equation enable variational distance hausdorff extra exponent convex polytope hausdorff entail extreme moreover share rate hausdorff convergence positive constant convex generality spherical ball convex polytope lie vertice p part broad polytope arbitrary polytope moreover either note obtain asymmetric role hold state dependent abstract contraction hierarchical whose detail specification irrelevant integrate notion hausdorff ball eq useful hellinger hausdorff metric hellinger metric simplify hold kp g choose around leibler fix suppose scalar define scalar g certain condition defer note applicable model choice hausdorff remainder devoted condition hellinger low prior define kullback utilizing describe total variation metric hellinger population hausdorff metric minimum matching hellinger inequality let contain spherical gp g convex polytope polytope either g gr satisfy tighter allow consequence lemma lem respectively main test order distinguish organize nx ii ease measurable probability b outer gd g g c ii pa ga hold pa g regularity ga pg px define ga gb ga p n conclusion proceed way extreme suitable existence transformation contain polytope obtain hyperplane away extreme solid radius share sharp corner pair two hold ga g g g extreme pa iii latter proof proceed support neighborhood alpha regularity dimensional hausdorff definition reduce l kk rank variable chapter induce dimensional hausdorff jacobian map admit hausdorff kl l dirichlet give density imply regularity concern behavior near weak kl divergence endow structure
play crucial role medical trivial incomplete consider complete thresholding noisy measurement exceed projection matrix correspond fourier describe image speaking setting less ambient dimension acquire impose physical resource recovery high essential cs pseudo projection examine extensively essential typically compressive compressive measurement perform comprise extension design measurement analyze complexity set eq q candidate design dyadic risk surrogate risk idea behind formulation excess concern estimate compressive measurement proceed manner spirit proxy rather comprise careful reader detail examine level set compressive measurement quickly entail tv effectiveness compressive remainder optimization briefly tv fast shrinkage simulation section estimation total tv initially denoise tv minimization collect projection acquire representation operator either isotropic function give tradeoff fidelity measurement quantify tv fr g pose term tv take estimation slightly less value procedure optimization gx fista approach rely quantity fast gradient dimension termination specify terminate successive sufficiently www mapping en wikipedia show test bit format take random I several setting deviation outline target approach algorithmic evaluation excess fig excess measurement estimate art approach proxy define noise achieve noisy measurement figure provide estimate level set several different estimate parameter excess even small ambient reconstruction correspond reconstruction
prove convergence part decision tool section error bayes loss lemma q derive optimal defer closed center solution include eq norm respectively sample satisfying covering supremum norm eq defer appendix constant present hold moreover infinity dataset show error classifier figure replace tend assume risk sufficient existence proof first assumption first continuously differentiable second continuously open interval function calibrate existence lemma increase z condition note loss existence confirm truncate quadratic easy order differentiable condition exponential hence use similar correspond hence yield note monotonicity convex depict bayes consistency general numerical algorithm devote linear regularization outperform hence method examine benchmark variant simplicity subset unbiased train function clearly unbiased outperform unbalanced function parameter regularization evaluation prediction suppose identity way negative define distribution show test estimator svm unbiased unbalanced estimation setup balanced datum slightly well comparable ccccc unbiased real uci benchmark diabetes heart see show dim test denote sample test average performance respectively unbiased ellipsoid covariance correspond exist take loss uncertainty input uncertainty set benchmark deviation function unbiased bold letter test small opponent except breast cancer result close indicate compare achieve unbiased unbiased superior covariate covariate uncertainty capture distribution point appropriately achieve unbiased dim test breast heart conduct unbiased bold opponent significance svm heart paper binary show conjugate property parametrize uncertainty present margin parametrize prove statistical consistency uncertainty proof exclude nice regularization investigate hinge loss modeling optimization theory frequently apply datum see work acknowledgment grant aid support aid ts prove existence accord argument restrict function form problem column property vector kx shall subset compact infinity tend negative compact therefore next since slack also satisfie inactive ensure max gap lemma ps negativity satisfy derive function subdifferential subdifferential present hold convex exist satisfy attain real subdifferential hence corollary hold clear marginal ensure first satisfied reproduce negativity optimality subdifferential derivative monotonicity negativity subdifferential bind mean number subdifferential less define lead hence choose satisfy high choose appropriate positive hoeffding inequality lead direct exist eq exist large say boundedness property lead hold second calibrate sufficiently inequality respect fix fx b equal indeed q hold respect depend vc number higher choose independent uniform respect probability condition observation inequalitie q hold high hence tend follow infinity positivity condition uniquely determined fix continuously differentiable derivative convexity decrease negativity convexity non negativity positive satisfy inverse fix choose straightforward confirm satisfied cm corollary classification uncertainty loss vector machine represent classifier loss property use hard margin mini set employ regarded margin study application field learn study two conjugate relation learn problem goal input training sample label sign algorithm logistic support property base understand intensive another statistical label case picture convex subset hull learn hull label hyperplane hull point paper term optimization minimax machine set study concern set learn focus function uncertainty conjugate uncertainty learn apply uncertainty accurate uncertainty specify margin uncertainty set recover consistency exist relation uncertainty devote illustrate way uncertainty nice kernel consistency proof notation equal otherwise vector space describe face euclidean hull element denote drop subscript measurable function set short supremum estimate binary associated input article sign refer binary decision evaluate e expect measurable low denote q subscript drop hard learn use computation tractable hinge adaboost minimizer surrogate avoid empirical surrogate loss classifier adjust loss computational retain base robust uncertainty determine probably solve optimization application uncertainty design sample high label confidence resp consist convex sample resp hull hard margin svm robust bad uncertainty solution decision illustrate problem appear hard svm algorithm briefly minimum minimax probability propose decision play minimum margin term decision estimate line estimate boundary achieve maximum uncertainty accord margin uncertainty direction margin follow set introduce accord investigate relation euclidean trick obtain rich constant point hinge loss formulation negativity confirm negativity last inequality come dual minimum see lagrange index equality change resp introduce uncertainty represent uncertainty estimator tool analyze uncertainty expression theoretical tool understand property uncertainty uncertainty parametrize section uncertainty exist kind parametrize level set convex proper q uncertainty z replace base popular learning decision constant consider primal derive give representation uncertainty derive correspond via gap primal primal training empirical thus tool consistency learning study algorithm develop primal function keep conjugate define condition resp eq uncertainty define apply shift reason description follow theorem suppose equality equality prove lead statement projection statement impose unchanged formula h p uncertainty set q empirical property optimal tool primal show uncertainty function tc definite matrix matrix hold uncertainty solve vector illustrate panel right uncertainty q uncertainty estimation set phase slightly brief uncertainty define depict hold quadratic positive misclassification decision boundary original present variant use necessarily let reproduce space endow decision
well less measurement number bind suggest indicate evident compare model part l competitive knowledge confidence argument need simulate know need exactly confidence compressed measurement formulation objective restrict quantization error interval finally utility formulation previously along variation attempt acknowledgment foundation grant dms foundation fall competition award grateful constructive comment version lead develop high estimate assumption define yes appear mistake something inconsistent check ok yes assumption least define similarly compressive lemma find completeness proof give set complementary without intersection indicate entry feasibility suppose satisfie ij inequality eq yes checked factor factor yes complete noise prove inequality fact consider leave claim inequality complete subgradient existence prove assume inequality hence prove claim hold satisfied hold take error measurement matrix entry draw submatrix row define choose threshold satisfy large note similarly minimal prove probability large note inequality working ensure exponent dominate conclude derive discussion iv quantity dimension since definition claim check problem could counter counter lem counter counter reconstruction finite thus quantization near outside account explicitly quantization lagrangian consistency appear date show extensive formulation variant compressive reconstruction quantization consider sense cs represent define quantization measurement round near record quantization round represent lie beyond represent namely setup compressive sense hardware architecture remove sense matrix unknown without noise record vector part correspond recorded matrix quantization particularly quantization interval choice weak quantization variable course robust replace add quantization region viewpoint play least square appropriately clear inclusion rather apply reconstruction hold art place vanish eliminate place require code cs norm sign two recover though recover magnitude reliably norm denote mention nonzero use submatrix discuss order notation positive write dimension write denote depend zero partition matrix norm eq quantity place admm solve discuss property bad comparison formulation conclusion claim describe simple combine constraint way specify write lagrange multipliers lagrangian primal turn gradient increase proceed ax ax ax break update conjunction many algorithm accelerate fista primal admm section property obtain difference true solution certain assumption sense formalize quantization error distribute quantization asymptotically variance selector assume recall rip estimate would I vi hold converge constraint place happen specifically p formulation formulation compare five variant formulation try variant constraint omit recovery performance lasso remain alternative appear objective x hence comment discussion notation discussion need I page keep report actually constraint consistent bind ok formulation enforce actually enforce add constraint l define signal satisfy certain familiar selector notice result synthetic entry location draw independently repeat choose yes essentially admit tradeoff confident since tight determine tight simulation elsewhere quantization error note make proceed generate numerous empirically similar seek dimension bit confidence percentage figure value average trial recover extreme formulation give performance lasso tie continue plot
spatial dependency inspire advance motivate different towards predictor hyperparameter stick control proximity identification compare break remainder organize stick break datum dependency section nonparametric cluster dependency relation method efficient derive dp dp characterize scalar refer innovation dp draw subsequently independently draw random joint show exhibit sample new exist draw allocation proportional previous draw allocation concentrate let draw sample variable discount innovation distribution expression g dp rich rich sample assign assign draw particular value scale dirichlet number unique grow characterization unconditional random draw break two collection beta distribution regard stick breaking notion introduction dependent prescribe position measurement arrange dimensional lattice sequential naturally associate lattice depict temporal e point case location prior comprise predictor observable location eq select spatial close location become typical kernel kernel aim cluster relatively feature far location space nonparametric goal sample draw distribute value value take location assign th innovation random kernel stick break construction discussion location innovation valid random purpose follow three summation independent share common idea stick bound unique instead consider beta prior stick employ discount bound kernel location unique cluster contrary great stick stick hence predictor appear bayesian typically mean variational monte impose gamma innovation q hide th impose q also impose conjugate formalism consist family variational innovation inference fix c n c parameter prior hyperparameter comprise hyperparameter impose innovation derivation approximate maximization iterative free consider exponential configuration take functional read let eq q obtain regard hyperparameter maximization determination location assign energy latter
recent exploit similarity nevertheless dictionary unlike dictionary adapt image approach penalty well indeed promise process svd error intensity pixel scale present denoise penalty bottom well bold natural develop well undirected dag breast cancer consist gene breast cancer goal versus argue graph regularization objective improve use identify gene disease lead yield well prediction task interpretation necessary drug penalty regularization statistically significant improvement asset function improve help medium solve time internal procedure gene gene obtain interpretable connect component pair link arc approach require transform dag treat direct choose arc arcs dag arc denote pre prevent penalty goal formulation negative intercept gene expression include ridge elastic penalty link arc variant equation use place norm arc structure penalty problematic vary cope issue try strategy every penalty describe unable select prove penalization number belong logarithmic elastic net penalty select internal set repeat report average table start answer penalty sparse significantly improve interpretable trust preprocesse acyclic report processing run observe concern seem produce connect similar sparse well assess difficult number result unable test statistical rigorously without experimental neither seem experiment try graph structure table conclusion task hand none statistically prediction ridge ridge select form sparsity accord asset aspect stability sparsity method introduce strong regularization structure provide stable solution half experimental whereas believe claim penalty without encourage select improve stability solution biological knowledge connectivity test connect component pair dag random balanced deviation internal fold sparsity gene connect answer question proximal relatively fast ghz cpu intel core slow small value solve proximal reasonably fast conduct reasonable amount proximal structured subgraph feasibility link flow penalty variant flow framework problem encourage connect able non penalty valuable context find convexity helpful prior knowledge approach fast interestingly penaltie dag remove like exploit induce extend lasso allow encourage group line encourage group link arc clear experimentally set task dag exist group every vertex dag define vertex similarly contain encourage intersection subgraph dag number contain connect unable make useful flexible suffer belong concrete arbitrary reason penalty structure proximal compute involve graph group inclusion relation different terminology encourage sparsity pattern predefine decompose latent support encourage desire regularization give show equal introduce define equivalently g g g addition index j g strong decrease rewrite r g g satisfy p interpretation interpret theoretic code inequality cover well uniquely inequality show weight lead interesting interpretation source code length path define unique vice versa denote bit indicate start bit indicate cost weight uv path matrix walk prove definition equation associate flow path cost cost flow vertex minimum achievable flow constraint satisfied flow path flow arc integer classical say cost flow arcs integer tell decompose flow path flow unit cost flow keep capacity therefore flow equal use proximal pattern proximal therefore rewrite uv flow constraint exist flow solution loss generality integer replace constraint fix flow rewrite uv flow piecewise minimum equation without loss generality easy see sign accord proximal p j uv yield j u w w program place define rewrite identify path show converge correct ensure zero moreover unique give g l g update solve operation w w convergence easy solve condition lemma p denote stop select want stop trivially stop p check observe h old inequality yu yu fr bin yu berkeley department university california berkeley usa embed gene gene knowledge easy new computationally feasible select connect direct acyclic sparsity dag path penaltie exist interaction penalty tractable path experimentally show connected subgraph network flow much priori inducing well nesterov supervise available assume predictor gene regularization automatically identify subgraph connect component group gene involve two connectivity improve connect easier involve feature arc risk formulate order choose cardinality encourage subgraph good penalty connectivity graph category involve pairwise interaction vertex arc encourage simultaneously usually tractable connect penalty approximation problem penalty long graph interest group induce regularization encourage sparsity pattern penalty similar strongly two go beyond sparsity new challenging example define subgraph solution address greedy vertex encourage long range interaction define connected subgraph subgraph subgraph soon large connect naturally direct acyclic build upon penalty pay allow function penalty go beyond interaction long form subgraph number path path dag even though dag flow thick draw blue minimum fill blue mm minimum thick draw black fill red minimum thick gray rectangle fill thick dotted line width auto yshift mm yshift mm xshift mm yshift xshift mm xshift v yshift xshift yshift xshift xshift yshift xshift xshift mm yshift mm yshift xshift mm xshift mm yshift xshift mm v yshift xshift yshift mm xshift leave mm xshift yshift xshift v yshift xshift mm right yshift xshift v yshift xshift yshift yshift xshift v mm yshift xshift v v v v v sparsity form subgraph represent cover bold development flow technique chen structure regularization relate flow penalty terminology overlap group complementary encourage encourage connectivity link arc however obvious effect discuss question summarize penalty feature penalty involve optimization deal flow tool implicitly exponential allow penalty operator polynomial long interaction tool brief flow penalty optimization technique devoted experiment genomic scalability conclude penalty flow since concept brief topic section proximal gradient popular define arcs satisfy arc except figure flow remark appropriate give admit variant go arc vertex distance group v yshift yshift mm group xshift source xshift node node node bend angle yshift group yshift xshift mm leave xshift yshift mm xshift yshift mm yshift xshift yshift mm v yshift mm yshift node v cm bend angle auto right xshift xshift node bend auto yshift mm group yshift v xshift source xshift yshift xshift yshift xshift v yshift yshift xshift yshift dag send along path flow flow send along along cycle flow attract attention engineering physics exploit structure prove minimum cost interpretation always flow unit send cycle unit flow cycle figure flow build upon interpretation flow e flow locality neighbor vertex graph locality exploit flow capacity penalty respectively iterative thresholding similar mirror descent insight problem classical w interpret algorithm deal nonsmooth efficiently solve formally able compute regularization denote efficiently represent bit select complexity count code count encode zero selection easy interpret selection isolate sparsity group consider nevertheless equation np g otherwise boolean abuse denote consider relaxation problem addition penalty union group define norm introduce link norm need variant penalty detail large much send every path assume configuration equation flow uv uv enable flow turn equivalent flow solve hold cycle flow acyclic mapping proposition formulation capacity handle flow solver vertex equivalently vertex link arc carry main proposition proof compute consider define flow capacity constraint strongly definition penalty challenge general hard variable exponential graph difficulty convex define obtain define unit path select context methodology define cost flow constraint relaxation address optimize regularize operator flow proximal cost flow cost u ff optimum equation establish detail minimum piecewise equivalently define cost g u cost flow cost cost solve flow often instead choose indeed weakly polynomial require transform cost cost flow problem integer round denote worst appeal heuristic possible even proximal algorithm appropriate recently inexact proximal increase deal cost complicate fortunately relaxation modify handle cost keep describe relaxation convergence come refer issue constraint classical obtain uv uv formulation interpret algorithms network perform update dual primal present instead chapter algorithm active adapt efficiently define additional length nonnegative prove g g l acyclic correctness complexity op appendix complexity loose empirical complexity compute norm useful problem l adapt experimentally allow solve medium instance subgraph call observe time close subproblem easy solve check update accordingly ensure use warm l shortest acyclic u subgraph update solve subproblem stop relaxed could duality guarantee enough behave practice interface make available open software package proximal fista convex available duality gap stop duality gap experiment typically choose grid solution initialization warm follow decrease objective strategy far penalty consider sensitive fine validation validation offer connectivity tune prevent overfitte find choice good coarse parameter detail experimental experiment control since network different partition edge mail arc accord order structured compare ability regularization design less entry remove component strategy sequel perfectly recover snr choose different penalty criterion denoise outperform ordinary ol ol change ols shrinkage penalty improve regime snr also parameter solution path penalty exhaustive testing
successful completely free away major year reader experiment correlation perfect anti perfect correlation exactly bring argument mechanic mechanic statistical theory physic explain individual instance actually happen happen cause driven science spin would influence particle place outcome measure direction particle measure spin correlation merely cause origin common source locality quantum mechanic strong outcome physical merely reveal act quantum mechanic describe aggregate exist nothing physics mechanic instead mechanic deriving turn head assume principle quantum mechanic agree though actually measure run exist may locate one imagine discuss observe take assume locality bring freedom allow analyse tool base like assumption free true way one justify another understand randomness well coin pseudo generator use kind clinical design really want observe come physical mechanism coin another jointly deep otherwise unnecessary never value mechanism completely unknown ensure know measure really know three negative think tell discard super mean discard observational everything explain nothing freedom possibility reject coin click past invoke instantaneous process extremely subtle effect quantum seem phenomenon surface prediction mechanic causality difficulty I force please actually admit outcome like limitation experience physics universe notion evidence cognitive birth framework data experience experiment module algebra module notion interestingly module causality agent continuous agent act together I build create every physic causality quantum physics causality physic quantum imply say problem also quantum mechanic isolate normalised length evolve quantum outside laboratory equal square system together much evolve reality outcome lead framework mechanic opinion von device respect quantum mechanic hilbert quantum unitary inside look operator hilbert space space past copy picture change observable family special exist value definite unitary evolution determine joint fix quantum notion restrict back abstract quantum field theory hilbert traditional unitary evolution mathematically continuous measurement initially invariant yet problem reader work two cite solve world protocol source distant randomly implement long time almost call particle know perform pair huge many event experiment detection correspond thought detector easily overcome necessary explain experiment mention pair pair detection event classical particle leave correlation source agree another like measure decide desire pair generate outcome draw advance choose go probability successful setting particle identical aim advance want experimentally without statistical miss context put history result take call outcome vice versa detector turn sharp efficiency suppose particle particle amount set detector give particle correlation sharp ab ab everything else perfect limit allow quantum go publish journal fair close albeit experiment actually simply analyse good question close indeed never perfect repetition actually logical experiment nothing quantum mechanic rely logic move ball nature take necessarily find implicitly business statistically inequality section clear argue picture inequality worth measurement party outcome unbalance role study case hyperplane polytope everything correspond hidden vertex mix mechanic joint quantum party etc take measurement party outcome elementary specific whether generate table define space sum probability moreover theory quantum mechanic sense marginal marginalization local also list restrict positivity positivity constraint euclidean space euclidean sub create polytope model quantum contain set successively large combine let special class constraint satisfied imply party outcome prescribe correspond thus mixture distribution measurement party joint distribution pick model convex polytope mixture also describe intersection hyperplane model describe affine space full nonempty polytope form strictly slowly picture small within outer square quantum circle boundary picture vertex extreme quantum body generalise polytope sub exclude corresponding positivity boundary hyperplane nontrivial polytope nontrivial hyperplane correspond inequality outcome indeed increase new turn turning mean old measurement grouping outcome seem exercise try classify nontrivial generalised actually violate mechanic pose open question probably counter see generalised inequality turn statistical connection design much material cover excellent author second specific challenge researcher make forward correlation find hard get become serious paper publish use local classical computer protocol past computer measurement message coin course request output setting outcome collect computer computer contain huge table number share course generator randomness exponential like computer I enable challenge two us enforce discover program look inside computer thing difficulty computer detect challenge quantum http www alpha challenge science phenomenon see wikipedia cut necessity communication verification long write post internet news spread prefer style detector computer produce result reasonably run make save program past must accept pair successively neither correct seed exploit force know pair actually pick row seed could run seed calculation model everything claim correct section correlation get single seed style local program simply seed internet draw internet devote discussion quantum interested repeatedly create classical physical system systematically inequality thereby stop round replicate experiment physics local program create instead communication neutral learn think kind rating carry test demand price interested ensure implement generate pair keep run repeat time publish internet appendix theorem inequality red ball red ball number equal probability row ab ab n ab ab ab ab ball replacement contain ball rewrite similarly rewrite exceed three average last exchange role everything eq possibly q want make choose home translate trivially restriction acknowledgment grateful especially thank conjecture suppose establish physics experiment aspect work theorem issue causality understand thereby also conjunction principle refer locality causality match direction influence need propagate spatially connected causality reasoning statistical observational study assumption properly match statistical unobserved issue challenge discuss locality quantum mechanic fundamental physics short name locality world supposed demonstrate mechanic consequence force reject principle hinge outcome measurement spatially separate physical outcome precisely principle outcome outcome actually choose measurement super form implement freedom assumption design experimental actually outcome existence actually experiment existence mathematical phenomenon refer model reality somewhat position adequate physical reality outcome experiment demand object kind physical constraint quantum natural physical locality often consider principle call imply mathematical local phenomenon important thing classical even statistical causality note time statistical observe unobserved correspond independently use coin setting outcome measurement also observe represent grey unobserved might arbitrarily validity causal place restriction instance imagine fundamental familiar life keep mechanic locality randomness become irreducible world primitive merely position cat causality principle mechanic claim interpretation quantum opinion entail reality actual path turn reality wave reality cat observe average purely label column label name denote fair row outcome either one four product product equal former case latter possibility ab b original expect average average sample expect inequality binomial distribution completely traditional limit mean value conclude original four identically suggest conjecture come back quantum mechanic locality want mechanic must hold theory least principle purely certain quantum physics detail mechanic reject freedom almost freedom matter change locality argue ask go positive assertion randomness randomness dependence initial variation explanation randomness quantum randomness irreducible reality purpose need understand mechanic behind quantum call model refer predict spin spin half quantum fortunately understand two distant measure real dimensional space unit spin away direction outcome time imagine repeatedly binary setting description quantum prediction separately pair particle state quantum mechanic correlate product angle outcomes location pt
exchangeable assume replace exchangeable I understand idea testing imagine capital place outcome never risk reflect acquire capital unlikely exchangeability martingale idea fit previously measure one rather deal neighbor many case pp way example example different euclidean accord example symbol line calculation produce tb value follow resp exchangeability produce exchangeable uniformly independently value exchangeability calculating focus martingale calculate sure indeed check martingale define construct denote tb martingale grow shape grow might reject exchangeability use martingale martingale avoid function implement density since unit poor point boundary get boundary point p reflect kernel calculate normalise integrate rule conclusion plug martingale update recursively take evaluating take plug martingale life dataset current plug martingale asymptotically growth rate infinite sequence sequence borel say thing exist us eq inequality contradict investigate martingale compare martingale life test exchangeability tb tb service example handwritten life code attribute gray scale exchangeable reasonable reject experiment merge testing exchangeability sure example set mixture final plug martingale show plug power martingale plug martingale power final advantage level difference old suffer great flexibility example pixel picture spectral band indicate classification label exclude describe reject exchangeability arrive martingale plug martingale plug martingale martingale figure close second peak flexible end old assumption commonly threshold mixture martingale plug martingale would construct testing exchangeability stable martingale life dataset approximately generate introduce idea lack exchangeability look therefore small regime start situation exchangeability p observe ideal shape several kind predict kind value exchangeability martingale past plug plug involve prior integrate efficiently yes whether ac uk cs mm machine learn exchangeability assume distribution independently devoted exchangeability arrive would like measure degree exchangeability construct exchangeability finally investigate testing benchmark former know satisfactory new flexible become necessary many deal algorithm standard exchangeability assumption although violate satisfy popular independent meaning testing exchangeability exchangeable permutation exchangeable distribution exchangeability traditional challenge previous employ theory calculate exchangeability testing proceed two implement generate mode previous exchangeability deviation martingale grow rule exchangeability reject propose construct second exchangeability martingale use martingale assumption martingale martingale namely former grow exchangeability
depend despite convergence ordinary form incorporate choose suitable distribution b gaussian easy result integrate eq mode determine model put b p good constraint see prior vanish asymmetric z q posterior one minimize quantile consider novel form analytical uniform prior pp informative derivative order improper pdf proper improper lambda p seed sort add sr mx sr mx mx mx mx save txt save save txt save txt save se add set sr mx sr mx mx mx mx seed bad proper plot lambda fx ms add mx growth sr mx uncorrelated mx mx label proper growth sr mx improper prop improper proper improper lambda seed sort add set sr mx sr mx mx mx sx mean save txt save save save save save txt se sort add sr mx sr mx mx mx colour typical prior place dot thin dash black value medium sized iid proper correlated grow thick solid proper uncorrelated size thick dotted prior thin improper thin red thin dotted spectrum phase kk anti correlate natural prior spectrum use derivative uncorrelated improper fig concentrated frequency kx k g spectrum f typical unbounded growth super decay spectrum dynamical gaussian decay f law decay get f improper predictive likewise predictive taylor approximately assume fulfilled suggest fulfil depend become eq treat jeffreys prior transpose w transpose demonstrate basically fig iid spaced dot derivative use improper whose trajectory much basically reconstruction delay reconstruction deal compare colour true true derivative cubic spline noisy thick true thin line black dot simple gaussian polynomial alternatively low behaviour plausible although seem also sum fouri linear derivative open source software implement acknowledgement education environmental author thank estimate several unknown real function correspond error correlation find posterior move error special case observation analysis value several real argument noisy occur scientific task classify parametric regression krige regression nearest distance weighting although usually rigorous theory guarantee function belong justify available various form heuristic easily interpret reliability several measurement error also identically iii argument derivative estimate amount correlation minimal diagram estimate assumption contain precise plausible error apparent uncertain however highly measurement knows basically shift direction influence dash line uncertain uncorrelated right diagram retain three error dash show amount error colour effect error article estimate reasoning exist special limit move interest taylor data belief update uncertainty become measurement mixture gaussian derive function simple argument see reason prior article covariance value derivative derivative parameter large want distribution approximate interpolation regular relate also move interest bayesian motivated approximation plausible generalization bayesian weight covariance interpret take account estimate estimator method locally weight smoothing estimate infinitely infinitely differentiable least either measurement infinitely smooth depend smoothly distance method nonsmooth nonsmooth change involve also counter intuitive argument g suppose result argument involve reflect instead four measurement slope slope switch measurement distance uncertain seem use rely correctness rank near measurement extreme might increase robustness might exist tailed use smoothly decay distance variant special common include kernel method near interpolation suggest lie outside maxima consequence weight individual overfitte prior derivative fast estimate vary resolve reliably overfitte derivative order continuously convention component subscript index symbol denote x ii estimate magnitude measurement error belief variability make later I involve plus order derivative taylor polynomial convenient integer derivative relationship note many choice vector matrix linear model estimating error regressor interest determine usually evaluate variance translation multiply constant unchanged multiply ki ic input case get online interpolation lagrange local cubic polynomial uncorrelated intermediate rational vanishing become dirac delta free rational remainder derive eqs improper case quadratic formally regression compute yx w improper also trend proper absolute derivative mean absolute increase vanish q derive substitute care numerical nonzero exact I k addition I vx local j g dot method away though weight approximate behaviour choose prior proper variability one negligible bayesian although truly addition possibly measurement error regression ns yy xy xy estimate square study already symmetric exchange variable absolute vanishe least total give independently cubic eq bs xy show total function vice versa linear cubic one independent b symmetry get I polynomial case minimal choice prior measure weight kf give distance weighting value error occurrence eq smoothing note rational square exactly show directly iw dash dependency vanish word interpolation exponent give interpolation power implicitly assume derivative vanish vanish plausible example feature coincide derivative estimate slope sharp order spectrum certain obtain rp j j simple constant function argument vanish since j hand mean
direction contain residual stress depth one stress illustrate stress profile residual setting component output krige desirable krige analyzing caused especially collect intensive principal instead simple create single computational krige datum irrespective naive extension krige response include liu west experiment conduct nine stress difficulty correlation krige determinant correlation total computationally intensive numerically unstable experiment inversion determinant calculation iteration optimization extremely consume overcome issue krige extension liu collect location run grid sometimes practice dynamic force model tool profile collect different tool feed engine report liu al profiles truncate point functional profile also degradation profile fail especially organized krige modeling illustrate experiment summary conclude remark krige computer response collect note run differently location krige point mean separable tt ir tt case space output universal krige r tn nr I krige ij et eq q et al algorithm employ hundred make prohibitive krige response krige stage profile mean initial interaction compute location collect define location define ij response functional model selection forward blind krige function ix tt td initial stage implement collect grid output regular location liu dramatically reduce result krige x complexity reduce al grid equally spaced functional response collect intensive loss functional location evaluation predictor degree smoothness correlation write evaluation predictor grid response necessarily kronecker problem miss datum grid develop extensively create miss overcome moreover maximization introduce carefully framework combine I view collect negative datum kk e efficiently expression construct direct application overcome monte carlo em main conditioning complete th sample datum distribution al tt I ii result q eq q proof appendix mean three profile except th weight determine evaluate conditional efficient involve observation calculate also easily simple result extend z attain convergence simplify speed conditional iterate expectation z expectation worth note I update simplification complexity per argument proposition n ik indicate row miss assumption recall I side decrease suggest accurately miss simplification make equivalent implement update herein collect regular grid em procedure estimate miss add prediction krige miss z k g approximation convergence guarantee wu residual stress experiment originally al apply optimize turn challenge al cut force turn residual functional output branch hypercube nine cutting stand respectively angle level consider cut edge categorical isotropic al rest factor experimental residual sophisticated element software wave range degree cut angle tool mm cut speed min depth run residual stress originally collect procedure observe assumption side percentage truncation profile theoretically residual stress towards nonlinear profile fit average stress profile least transform original functional fitting h htbp fit krige al apply suggest variable krige ready move build second iteration reasonably substantial em take ghz pc naive krige extension four htbp confidence pp interval illustrate select predict near evaluate matrix lee seven hour ghz computer concern evaluate construct augmentation perform I widely naive krige four basis identify eigen profile entire propose krige krige augmentation marginal propose krige method prediction surprising augmentation completely lose severe quite optimize objective optimization minimize objective maximization large predict stress experiment regular predict predict stress illustrate solid regular sensitivity effect interaction et effect feed cut shape cut stress clarity residual stress surface stress increase feed speed stress profile positive explain physics attribute increase cutting conclude remark computer experiment functional scientific output extend pose article krige challenge successfully stage building procedure incorporate product application sampling run run illustrate hard turn response stress factor setting quantification possible fully bayesian incorporate uncertainty application fully infeasible inversion tackle stationarity assumption relax al additional effort due recursive leaf acknowledgment valuable comment grant dms also laboratory office contract w nf proposition objective q normally
consider estimation component proper approach per monte carlo estimate bilinear measure define include evaluation section among conclusion analysis partition extend factorial tensor domain attribute historical use orthonormal product haar together term hoeffding represent subset clarity complementary disjoint recursively subtract sub df jj l quantitie exclusive complement satisfie easily introduce way importance generally subset introduce measure denote index index count subset important investigate sensitivity index mostly unnormalized work publish translate computation hybrid share component jj jj special result f I nf n I show get I asymptotically small bias important q default nf n require unbiased type relation cube method quasi monte use plain effective expensive approach attractive way another importance drop interaction set black box component dominate version behave differently affect superposition fs easy superposition effective offer well instance identical follow index evaluation fixing subset uv product subset specify integral vanishe vanish vanish equation describe dimensional function generalize square negative compactly mostly zero compute bilinear rapidly computable element element bilinear call combine pair write bilinear low combination one column convenient row equal linear combination variance write representation pair di derive expression dominate evaluation require evaluation product f function evaluation row column sum unbiased nonnegative nonnegative index square square express measure version among classical mixed every expect contribution variance measurement error square coefficient next equal vanish lead degenerate result sum index twice bilinear evaluation begin variable helpful illustrate omit follow simple bilinear require function evaluation evaluation presence evaluation treat variable bilinear use per integer evaluation require let choose put alternate sum otherwise zero bilinear argument theorem bilinear estimator nonempty contrast disjoint w equal coefficient vanishe otherwise equal therefore estimator mean solve carefully evaluation reduce u index along index remainder present value next second contrast q eq therefore way evaluation respectively evolve value nearly value initial set strategy apply index great index empty variance component combine yield truncation contrast measure extent distant lag contribute also base pair segment evaluation per require function per biased estimator proposition bias correct suppose nn first put di much great biased require additional need sum bilinear calculation interval average space estimate desire combination original simple asymptotic estimator matrix importance estimate bilinear ideally variance fourth unknown hard component series papers rao comprehensive version quadratic proxy variance component effect two effect interaction however set write eq proxy choose minimize original former cost proxy argument original estimator q third fourth moment evaluate example non minimum function evaluation answer specific set bilinear formula dominate function symmetry compare large sample estimating error bilinear half standard bilinear analogously bilinear ff estimator modification ff correction asymptotically negligible difference require evaluation per base combine available I involves thus expect small looking subset pair range size report superiority contrast neither always hence proxy reliably specific product deviation proportion last contrast time ratio deviation square little square theorem square advantage way interaction bilinear advantage
approach difference neighboring enforce assumption frame frame apart achieve sparse smooth code advantage primary classification availability label label unlabeled semi supervise without respectively motivation datum visual pattern hence high level domain propose useful motivation supervise useful base well low task impose code use simultaneously collaborative modeling hold datum via weight experiment semi dataset face dictionary described dictionary dictionary sparse code supervision dictionary compare code two conclude describe data ssc incorporate space smooth traditional substantially measure modify code stage add enforce incoherent speedup order scale framework computation temporal supervise transfer number generalization way approach extend kernel bandwidth interesting generalization bound analyze regression iterative alternative scale experiment challenge large recognition variation illumination use extreme conduct consist range various experiment sift patch vary medium resolution follow experimental human perform record consist category challenge video illumination resolution etc randomly densely video sample extract smoothing experiment conduct smoothed person dataset similar technique flexible incorporate norm furthermore significantly dictionary propose speed use improve code code popular paradigm standard code independently dictionary propose smooth incorporate user sample principle construct traditional sparse coding express parametric framework type kernel encode image space incorporate temporal relationship unlabele learn setting fall alternate scalability remain novel code regression traditional alternating univariate regression step speedup traditional code order statistical show proposing smoothing incorporate coding provide sparse code efficient successful various lead improve computational speedup local smoothing lasso temporal achieve locality difference dictionary along regression propose use speedup technique computationally notation correspond vector dimension vector use correspond frobenius notation dimensional explanation code solving correspond denote column object patch sift scheme representation encoding every encode sparse vector structure sift feature class close sift class neighbor apart propose mechanism smooth convenient function capture smooth traditional sparse coding encode encode code square smoothed empirical common kernel gaussian cross validation choice kernel input advantage bandwidth may standard distance may express code code spatio univariate bandwidth video kernel frame feature alternatively fuse approach datum smooth sparse computational fused penalty capture frame may immediately situation sparse handle regression least constraint marginal regression obtain regression proceed calculate least threshold operation provide algorithm ii dictionary code learn dictionary analyze smooth coding provable global prove learn code main obtaining begin represent smooth let random q empirical population true relaxed constraint also unit dimensional ball bound cover every element subset function cover prove concern covering cover generalization code derivation length differ term incorporate factor demonstrate approach speed code description experiment synthetic examine regression regression relative different relative reconstruction report regression significantly less perform method spatial sec ssc ssc mr conduct several demonstrating setting face static code obtaining locality perform experiment recognition video space image generate sparse densely sample pixel step sift code max process repeat report per together base cross validation select smooth result report previous result unsupervise code scene ssc size scene demonstrate scalability classification increase dictionary
aggregation denote coordinate pseudo product function finally exponential define hold arise natural paper design gaussian regression paper concern exponential extra averaging suboptimal exponential weight regression optimality heavily case particular weight natural ms aggregation problem take call aggregate onto hull difficult design high coarse fine would oracle expectation investigate term heavily use fact random design j jx f jx n font throughout rest weight write n first nm later moreover jj scale employ use homogeneity may simplicity treat choose event temperature small enough complement j j note yield eq rely construction ensure nm jj projection aggregate hold large note homogeneity observe n I remain probability old get since choose aggregate correct aggregate automatically minimize interpolation weight family estimator ii put weight fit function equal identity know leibl weight call estimator note weight choose deviation restrictive condition noise variance denote eigenvalue variance proxy directly imply hull interested aggregation simplex aggregate unlike aggregate explicitly solve ms aggregation restrict infimum nonetheless pointing focus aggregate estimator satisfy probability employ uniform may original aggregate aggregate estimator see moderate purpose approximately aggregation aggregate appeal simplicity exact algorithms term quantity go generalization valid minimizer approximate hereafter convention pt variable gaussian choose q satisfie moreover type entropy correction paper nevertheless uniform order aggregate vanish simplex design term optimize range overview simplex promise especially become method sparse case sequel focus greedy take selection property aggregate make explicit appropriately design greedy achieve dictionary simplex statistic greedy type aggregate option algorithm th algorithm add iteration output iteration resp reduce resp refer former use frank wolfe literature minimize value purpose ms appear give approximate solution classical frank wolfe two unable achieve produce j optimal aggregation therefore achieve disadvantage result imply bound dictionary proxy eq aggregate simplicity although completeness prop since relatively fully counterpart advantage convergence fully additional essential update optimize subroutine description order uniform prior solve ms aggregation optimally assumption bound importantly deviation order obtain later accurately achieve remove boundedness variable aggregate q theorem imply deviation bound decrease indicate critical careful inspection aggregate require case easily sufficient achieve ms constant beneficial run confirm experiment stage greedy aggregation paper star stage ms theorem beyond construct iteration constant usual allow bound avoid boundedness problem vanish technique problem greedy suggest increase reduced constant configuration identify define standard identity regression define draw oracle om temperature tune remark random due mis good scenario lk experiment repeat replication order avoid detailed regret exp star indicate aggregation star surprising star regard averaging estimator keep consistent good stage although regret replication noise model nearly good beneficial procedure aggregate ms aggregation analysis illustrate ms aggregation compare greedy bad small theoretical variant sparsity hard display yield eq replace expansion n jensen plug get take follow allow high f successively jensen f observe complete prove chernoff bind q combining identity f j eq use convexity yield statement high proof taylor expansion expand side apply plug q induction second inequality complete assumption conclude vanish simplex similarly j f therefore theorem complete lk c replication suggest example section section proposition notation part dms e function aggregation regression model problem aggregation surprisingly sub limitation lead sharp oracle minimax design produce aggregation performance class statistical learning considerable past
light tail random path cost edge moment path replace hoeffding bind represent consist establish classic sequence evenly exploitation select estimate set compact efficiently path subset dimension consist path construct path belong exploration past observation integer define l dt da hold independent exploration cause th exploitation contiguous denote th period exploitation show hoeffding hoeffding arrive exploitation sequence time mean property select exploitation sequence prove hoeffding define minimum constant exploration well increase cardinality sequence amount lead arbitrarily theorem follow regret exploration difficult value consider increase replace enough proof distribution moment exist cost draw construct exploration sequence base sufficient exploitation exploitation arrive eq path mean regret general special dimension action minimize expect xt identify short repeatedly choose gradually action run epoch epoch epoch epoch epoch cost choose policy point arbitrarily stochastically state tail link propose problem I evolution stochastic lemma property california email edu adaptive short wireless unknown stochastically problem aim optimize path randomness uncertainty dynamic quality vary quality path delay quality observable formulate armed bandit dependent arm arm dependency maintain regret policy ignore furthermore include find ad hoc wireless quality stochastically vary model reward evolve source communication end link policy total model armed bandit mab mab liu player play offer cost draw unknown arm cost player since grows treat share dependency path policy mab naive growing linearly preserve optimal logarithmic specifically propose algorithm light tail horizon show arbitrarily allow performance tradeoff network state stochastically mab address logarithmic asymptotically regret cost light tail light tail tail sublinear tail linearly incorporate exploitation arm term preserve generalize assume fully observable link cost choose adaptive achieve link paper study bandit adversary deterministic achieve sublinear formulation version support regret nontrivial algorithm achieve term cost achieve online bad address stochastically consist
distribution invariant sampler element formally proceed next repeat write sum exceed threshold entire chain rare large increment big jump increment average mix fast big jump really invariant invariant goal update step preserve stationarity conditional invariant clear preserve stationarity step th borel observe produce ergodic ergodicity follow borel number particular order expression display bound complete integrate borel distributional step random rare event essential become section assume tail sequence vary tailed big heuristic step q note view correction factor variance vanishing complete cover heavy tailed elementary sharp algorithm long e prove efficiency computing interest queue er company density integer value sum problem additional difficulty update ease description draw step suppose k nk ty updating step proceed thus j markov chain invariant step possibly preserve stationarity write ta write last order summation expression equal b equal equal eq definition equal desire complete uniformly ergodic q require modification consider distributional rare event property therefore ease reasoning impose generate sufficient instance vary density n vanish normalize denote depend cover wide range random study present efficient algorithm complete section literature include algorithms et monte li find appear al estimator label construction simulation mcmc single variable carlo step get fair computer pareto batch record consistently event asymptotic become well event become rare geometrically batch also different estimate observe indicator well runtime batch consist importance sampling carlo mc approximation avg std avg mc avg est e avg c mcmc est std avg mcmc avg est std avg time mc avg mc avg e std e avg est e std avg avg est std avg batch consist simulation est avg per avg est avg e std avg est std avg avg est e avg est std avg generate mcmc draw lemma corollary theorem research g foundation rare underlie rare invariant generality heavy tailed walk exceed reciprocal whose vanish sum illustrate numerically importance algorithm heavy tail primary secondary carlo mcmc computing rare basic use mcmc algorithm probability outline full heavy tailed walk rare sample repeatedly simulation sample assume rare consist satisfactory unbiased estimator useful inefficient consider relative popular instead original importance exist unbiased depend distribution distribution distribution zero unknown serve select easily sample likely distribution unlikely proving error propose sample unbiased construct part ergodic roughly speak event control ergodic ratio normalize however well study context event simulation methodology problem compute heavy receive paper present markov estimator form suggest conditional belong elementary complete sharp restrictive assumption tail outline efficiency computing event describe computing contain efficiency computing present mcmc exist mcmc well importance sample rare markov heuristic fashion general computing section precise efficiency measure ignore upper lead decay emphasis normalize small choice mcmc determine dependence markov dependence high computational irreducible chain fundamental irreducible chain condition mcmc sampler ergodicity sampler study metropolis hasting xt markov chain geometric uniformly ergodic exist q rigorous markov ergodicity sampler condition ergodicity condition theorem mcmc reference therein walk section gibbs ergodic argument lead desire property estimator unbiased estimator approximate conditional choice appropriate gibbs proposal metropolis hasting geometrically ergodic problem compute somewhat assumption lebesgue integrable special density analogue distribution eq
loss context analogous coverage criterion criterion one well minimize length subject never value scalar interval utilize uncertain prior confidence period statistic theory nd theoretic motivation york w journal american statistical mm definition decision theoretic interval mathematics interval assess coverage apply find good interval interval combine bayes confidence loss combine bayes usual interval I behaviour keyword rule interval estimator author la university mail pmf pdf also good point estimator define expectation pmf pdf pdf improper finding estimator much criterion namely volume decision directly loss include pmf pdf expect yield however point lead poor confidence set set behaviour introduction prior pdf choose solve consider modification prior however generalization contexts iid let also usual tn use new section rule usual since u denote expectation distribution expect distribution posterior distribution marginal subject minimize denote substitute minimizing
convention classification semi version base classification gaussian model amongst early multivariate paper multivariate include skew normal lin lin lee distribution recent paper development gaussian clustering nevertheless certainly introduce skewness also location mathematically elegant straightforward methodology maximization outline illustrate approach work variable inverse function property dimensional asymmetric laplace density skewness distribution list prohibitive application address asymmetric density mahalanobis theorem f density shift laplace recall cf mixture distribution th parameter g incomplete computation latent step step maximize parameter iterate attain common em heavily dependent overcome deterministic annealing deterministic conjunction cf give type algorithm fit observation likelihood covariance formulae maximize specifically mix n ig ig e iterate update parameter update consist cf convergence probability cluster classification anneal increase sequence transform likelihood surface improve find determine many iteration anneal anneal ten high acceleration log estimate log value analysis herein iterate must handle specifically happen term create computational overcome search value proceed take overcome restrict acknowledge quite thorough exploration suppose membership use membership estimate model k nk membership therefore value classification analogous fashion maximize observation herein prefer specifically ig ig reflect group assess true group membership adjust rand index rand introduce partition plus apart total multivariate component skewness shift mixture perfect run ari give relatively poor ari component four component ht ari gaussian ari htb depict typical correct instead comprise argue inspection resolve old national skewness use many skewed fit classification sensible group classification associate contour skew also fit contour htb htb repository localization site development result expert illustration span alm content distinguish two protein terminal cf component mixture model choose three performance ari superiority merge gaussian ht htb would perform well criterion poor produce classification ari I compare within membership know table outperform counterpart I model site take known raise around efficacy flexible merge anneal select base clustering illustrate give whereas gaussian consistently furthermore notable model component old give membership contour reveal model capture shape far counterpart protein present difficult illustrate good performance ari outperform counterpart ari force modelling give worse expect ari model give excellent ari surprising gaussian location know question efficacy gaussian datum merge decade case substantial paradigm skew elegant computationally away gaussian mixture type mixture mixture however poor merging section introduction describe mixture g g g explore cluster conduct selection compare cluster skew university statistic
update message step exponential compute log normalize constant update repeatedly ep ep moment context cavity return moment posterior message ep compute log solve form measurement dynamical nonlinear predictive covariance respectively nonlinear dependency intractable approximate relationship functional effectively use accord explicitly linearization otherwise inconsistent backward partition gaussian z cavity obtain influence derivative I implicit linearization guarantee describe message outline require update division partition measurement make intractable solve uncertain input long however gp cavity linearization explicitly implicitly ep update remain backward take account coupling lose respectively integration inner matching instance approximation implicitly analytically approximation similarly forward involve partition take see propose directly message true approximated instead suboptimal ep context filter tp series update moment via log update dynamical write z new pass ep generalization exist g kalman message log function update dynamical also kalman smoothing prediction linearization compute influence derivative moment cavity pass formulation general solely prediction propose synthetic set art specifically linearization gp generalization ep ep evaluate measurement measurement infer latent absolute mae norm covariance subsequent ep ground truth dash quickly reveal ep matching iteratively closely average space track control angular velocity mass order control covariance measure z train trajectory length trajectory start ccc mae ep ep various ep ep across especially ep iterations linearization ep numerical typical cause exclude moment numerically coherent approximation capture remove observation process trial learn state datum ground label focus tb trial ep occur region red embed period acceleration summarize trial iterate backward infer tight ep bit ep generally pose capture test trial ep trial present dimensional inference motion example moment matching linearization posterior ep trade approximate improve generalize forward improved prediction comprise inference dynamical system inference include investigate alternative linearization computing message acknowledgement receive european grant agreement institute advance thank wang review approximation analytically iterate expectation target dimension eq law iterate target additional dimension term I take determined see conclude match kl divergence distribution conservative mass alternative way approximate posterior equivalent linearization linearize apply gp diagonal evaluated linearization approximation optimality kl lose linearization beneficial largely due simplified college uk complex series system financial market video phenomena diverse require gaussian appropriate message pass backward smoothing thus accurate latent structure result improve compare hence pass filter challenging application economic series often dynamical dimensional noisy approach advance art estimation develop novel inference parametric generalization allow time series gp dynamical broad interest develop contribution pass inference message recover dynamical backward inference system evolve measurement central model explicitly gps mean input distribute gp non restrictive assumption compare dynamical transition measurement train target function function determination covariance function account gaussian dynamical task analyze recently bayesian smoothing analytically intractable previous state
boundary origin triangle intersection triangle contain prove help reason triangle instance follow lemma proof completeness arbitrary must consider origin angle contradiction exactly I tangent suppose parallel tangent angle contradiction edge hull row conversely row intersection ray entry unit hull triangle row map nonnegative nonnegative rank question characterize submatrix satisfy condition acknowledgement thank discussion thank comment stage lemma theorem corollary question question meta conjecture definition definition theorem support part nsf dms nsf innovation fellowship solution inequality notably task inequality algorithm motivate algebraic implication need decision nonnegative yield exponentially yield establish form algebraic entry additionally large submatrix nonnegative fundamental arise admit equivalent nonnegative write nonnegative nonnegative hull nonnegative refer inner application machine complexity machine factorization factorization representative entry compute nonnegative factorization express matrix retrieval exhaustive nonnegative combinatorial polytope subject rank give polytope construct slack column slack prove slack exactly extension quantum remarkable polytope polynomial formulation conjecture deterministic communication complexity polynomially equivalent formulation nonnegative boolean boolean relationship computation nonnegative biology economic process range dynamic historical name modeling resolution priori observe equivalently system polynomial treat entry entry variable valid factorization exactly nonnegative inequality well known finding appeal say run exponential time np hard run crucial reader polynomial polynomial maximum run sense number explicit analogy lemma inequality give exponential run run exponential time run exponentially doubly exponential algorithm proof perhaps system inequality remarkably expressive believe polynomial reduce drastically many polynomial inequality probably make nonnegative rank think result nonnegative inner dimension rational bit r notice exponential exponential normal nonnegative matrix dimension crucially inner dimension determine submatrix submatrix row column row row exponential transformation apply reason previous run doubly number variable dominate transformation exponentially transformation algebraic among transformation particular ratio share normal number also nonnegative submatrix submatrix play crucial admit characterization subset netflix like yet nonnegative differently nonnegative submatrix nonnegative thought though equivalently system infeasible constraint contrast inequality infeasible denote submatrix column submatrix row small respectively call affine contain recall call notion nonnegative requirement want admissible admissible subset row nonnegative always stable preserve inner dimension nonnegative factorization inner approach lemma update update preserve throughout process terminate column ever accord order phase update admissible nonnegative satisfie hence end update analogously throughout procedure maintain dimension support monotonically decrease update row column strictly decrease order nonnegative inner dimension throughout demonstrate need furthermore list row invertible linearly set show column vector break part otherwise could strict would stability set row restrict row identity support next prove main two zero outside u u u combination w j vector support yet identical early early admissible contradict update support early linearly role encode variable attempt ensemble factorization conversely valid dimension ensemble apply vector immediate description completeness boolean vector I u c j compute nonnegative tie output ensemble stable immediately establish uniqueness vector vector nonnegative dimension factorization combine variable nonnegative guess guess semi boolean function semi empty run bind sign configuration need configuration call boolean polynomial degree large vertex cross need linear transformation hull doubly exponential semi stability somewhat exponential reduction number algebraic dependence ensemble column invertible matrix row boolean sign degree determine q algebraic polynomial output matrix compute algebraic algebraic non empty moreover lemma factorization inequality describe expense extra give polynomial inequality run extend also return approximate formulae algorithm algorithms oracle boolean maximum bit apply r
base give bind scheme point study work employ column base bound work recently pick bb vector hadamard eq proof give high rr bb observe follow lemma case vector rademacher hadamard mi mi subgaussian lemma possibly subgaussian lemma union statement choice together substitute obtain combine acknowledgement thank pointing error reference note give randomize form subgaussian mb straightforward prohibitive product specify property
plan allocation mass discuss elegant roughly vice quantity instance parameterization l evy say reach multiply time return expectation know usually large time lt calculate long hold however difficulty equation pass calculation theory illustrate movement patient heart attack patient patient movement planning resource quite analogous inferential movement patient bayesian method focus patient assume nine state care post intensive care unit care home represent quantity q quantity euler integration calculate euler give column demonstrate many computer begin calculation consume calculation time calculation discard way code spaced hour state probability line color patient home calculation obtain hazard denominator conditional reach show conditional hazard process ht markov consider interesting likely state visit show care take define choose calculation interested base uncertainty estimate full summary sample frequentist uncertainty account repeat band alternatively bayesian solve obtain wise credible interval apply parametric datum substitution transform show bellman bellman numerical laplace american network space model thesis university inversion report national laboratory nm partition implementation new york survey comparison journal physics introduction laplace inversion laplace transform relate journal inversion application transaction introduction ii york york health monitoring report national nm semi markov finance york movement volume york evy international mathematics semi process reliability j flow graph institute preliminary markov finitely mathematical r theorem process mathematical existence uniqueness york inversion advance apply sciences numerical fourier inversion concern stochastic process mat time problem transaction american transition department statistic air force base edu david national laboratory markov rich implement capability quite develop exception simple solution numerical practitioner demonstrate implement patient rich model rigorous theory semi provide rich process chain process process birth death process name survival dna real require reader list closely laplace transform property make useful pdfs go theoretic detail make state roughly transition state constitute chain time depend origin contrast state exponential continuous time state may make process time I explicitly depend instantaneous state transition transform self state never return one survival reliability analysis death process recurrent state distinguish start find state count transition occur denote notational also define operation product computation move simplify solve opinion necessary transform unfortunately numerically support indeed book bellman inversion ill inverting transform much discuss experience confirm inversion transform numerically tune polynomial need close numerical numerically euler euler method q lt without integration contour contour emphasize avoid detail exist always uniquely fundamental usefulness process convolution multiplication lt solve transform extensively paper price lt however obstacle way numerically term area research improvement paper euler euler inverting probability smooth cdf two smooth possibility euler euler approximate lt inversion integral associate euler coefficient accuracy terminology move semi process reliability might quantity time reach average spend question application probability probability process find probability demonstrate application amount state period reliability figure interest lt integral easily therefore spend item distribution
matrix know classic reconstruction task carefully orient several dl belong track track use classification discriminative inspire meta learn et classification orient incorporate redundancy noise information recognition computation sparse bottleneck focus al method contain class dictionary sub dictionary et al sub share common visual use reconstruct identify incoherence dictionary independent incoherence term I see atom different independent incoherent derive incoherence n jj incoherence dictionary feature repeat almost exactly dictionary different reconstruction absolute detect represent product atom ignore reconstruction track track discrimination track discrimination track need learn dictionary present dl propose dl logistic conventional dictionary logistic enjoy prevent wherein bilinear wherein k zhang li discriminative achieve desire support discrimination add conventional dl position element scalar control two fuse term drop original final fast image label lc method discriminative dictionary q suitable occur input share term discriminative representation encourage linear mechanism fast discrimination dictionary atom correspondence structured specific associate denote training solve derive discriminative fidelity discrimination coefficient c ii jj nearly fidelity sample code coefficient fisher criterion maximize intuitively bf unstable elastic incorporate issue relate convexity discuss utilize track review representative dl classification track track sophisticated conventional dl dictionary dl discrimination lagrange scalar balance weight mean matrix propagate make track omit note example indirect besides concern seem trade classification consuming extend research ex l l l pt minus pt minus minus edu cn artificial intelligence college technology china present representative dictionary dl sophisticated framework dl pyramid rather concentrate dl classification deal meaningful representative roughly divide category dictionary future learn dl sparse signal aim view signal sparsity dictionary classification code face example track lc dl expect extension c face
reader refer optimization application criterion aic criterion degree freedom estimate specifically bic denote freedom estimate generalize least regularize degree freedom justify local glm collect regularize demonstrate path third implement ode glm conditional completely know scale normal logistic quasi glm assume variance include case appropriately reader refer classical text quasi know predictor general form fuse trend many denote matrix matrix glm formula simplify penalty impose coefficient introduce penalize show standardize leave bic predictor enter match detail reveal monotone effect book nonlinear predictor market estimate criterion naturally fuse lasso trend five predictor penalize linear assign define coefficient simply tool monotone impose order point translate uniform I concavity translate inequality computational theoretical complexity generalize propose linearly constrain derivative loss list provide estimate constrain availability predictor market estimate undirected lasso likelihood variance negative mle precision correspond propose determinant concave apply recent attempt make primitive predictor approximate ode symmetry part column exactly nonzero calculus omit brevity derivative graphical initialize penalize objective cone explicitly incorporate path solution minimize cone symmetric path constitute penalize imply path algorithm student score mechanic algebra display path choose part attract much year involve nontrivial demonstrate applicability maximum likelihood univariate concave estimation estimation elsewhere recently active besides provide solver offer whole solution solution log estimate middle prediction error flexibility nonparametric attractive include gamma density recent review log concave logarithm concave give iid unknown support nonparametric mle continuous piecewise linear knot outside obtain consistency mle prove pointwise mle notation objective derivative derive recurrence facilitate recurrence recognize recurrence symmetry unconstraine estimate article generic implement matlab provide whole solver linearly arise application regression nonparametric extension twice objective huber robust estimation quantile generalization regularization require formulation regularization propose penalty penalty convex pose difficulty continuity fortunately enter promise constrain corollary machine prevent overfitte problem term encourage constraint lasso method develop method penalty exact solver ordinary penalty regression path along goodness fit practice couple aic bic tune generalize regression nonparametric keyword concave density equation quasi restrict framework lasso generalize linear ease tuning yet extension nontrivial propose efficient solver smooth dimensionality denote sum fold learn broad application recover encourage fuse smoothness late equality incorporate properly second encourage estimate require nonnegative regression achieve complicated constraint occur shape nonparametric regression example finite consequently path unconstrained end special regularization homotopy angle lar linear illustrate goodness sparsity sufficient linear expand wider devise generalize least generality quadratic concern attempt long piecewise propose predictor ordinary exact penalize derive loss regularize separable restriction penalty important encounter generalize aspect convex loss equality equality fuse example restrict log last path acquisition constitute whether company seven market book return record company intensive study effect company target exploratory regression explore nonlinear effect quantitative covariate vary adopt predictor say use discretized covariate chance monotonically book log utilize neighboring illustrate flow impose monotonicity market covariate enforce concavity market covariate regression covariate regularization specify equality constraint estimate increase cubic end natural cubic trend regression filter contrast bandwidth semi regression parameter tuning location knot fashion coefficient gradually become monotone covariate book covariate large enough line unconstraine constrain estimate dot line availability solution easy choose show criterion constrain path within second revealed finance company unlikely target hard meet heavy burden company high flow possess hard minimum unique third stay turn piecewise article affine e lead optimization formulate constraint residual principle convex beyond scope loss strictly convex relaxed residual along segment configuration imply continuity coefficient path lemma throughout call inactive ode segment interior segment index increase amount difference active keep active segment ease notational burden lead correspond multipli c cc solution f result path segment configuration solution ode side constant quadratic recover study two inactive vice versa detect constraint differential equation detect type event happen track coefficient active boundary relaxed segment admit current segment active solution stationarity multiplying side current active readily end set inactive set initialize ode inactive constraint p summarize proposition segment extremely use ode ode matlab mathematics numerical notably path algorithm specific solve correspond ode explicitly path ode burden develop path regularization repeatedly evaluate multiplication cost available cardinality computation avoid organize inverse suppose th diagonal new matrix kk k ij ij kk operation
uncorrelated mean study introduce mcmc initial function increase prior momentum operator hence eigenvalue operator posterior observational noise parameterization sampling parameterization target observational pick evenly inverse observational precision method alternate explicit sample show precision consistency distribution wave fouri expansion momentum figure observational increasingly numerical arise condition article paper space demonstrate standard langevin bridge ia method figure purely reference measure mcmc accept stochastic dynamical highlight langevin hybrid carlo metropolis method problem mean modification exist mcmc demonstrating efficacy method incorporation idea body theoretical desirable property highlight technology applicability wide suggest possibility numerous development modify desirable prior arise acknowledgement support fp grant st college support grant grateful financial article support skip section assumption department united result probe bayesian condition diffusion process mcmc typically mesh modify target speed mesh value algorithmic truncation prior part model flexible modelling tool range example assimilation mechanic design principle formulate preserve lead modification exist range process field prior statistical bayesian theory function stem recent problem add grow use concrete directly generate field evaluate draw probability efficiently variety purpose focus expansion draw simply exploit eigenfunction series introduce characterize either application covariance advantage readily prescribe distribution specification particular advantage interpretability g advantage context create difficulty since project pose major challenge discretization apply involve credible law express equation naturally reason I computational ii notable convenient adopt specification specification markov tool simulate application prior approach condition diffusion study generalization prior arise prevent overfitte determine informative bayesian naturally evaluate posterior almost kind mathematically infinite model become curse aim devise strategy devise dimensional possess early within modification monte substantial algorithmic speed approximate furthermore interesting construction detail prior likely absolutely continuous respect typically dominate derivative likelihood dominate natural absolutely continuous end idea introduce random differential employ proposal metropolis equation preserve reference reject lead know major algorithmic algorithm later slight walk outline end random pick variable reject pair metropolis hasting reversible u standard differ slightly type modify clear proposal reversible speed discretize natural accurately infinite robust walk field assimilation mesh dimension b show acceptance appear standard modify walk imagine acceptance curve mesh refined mean mesh refine probability curve mesh refined obtain acceptance mesh implication difference acceptance new independent mesh use represent old grows rapidly mix become great refined providing key proposal carefully measure approach new speed apply space possess measure gaussian set bayesian inverse condition diffusion goal range require respect explain principle derivation lead illustrate give deep throughout paper assumption part model detail mcmc generalization walk mala sampler metropolis hmc action section estimation arise shape problem brief denote euclidean define range give rise measure density field thus evaluate numerical method refer increase tie finite certain common property algorithmic arise frequently unnormalized form specify mainly methodology variable positivity forecasting frequently pde dynamical gain namely equation field nonlinear pair relate velocity later weather forecasting determine inverse encounter equation thus flow head water height require positivity solve pde domain measurement interest observation database inverse second dynamical match approach inversion describe outline methodology curve orientation preserve constrain curve family field choose length appropriately hilbert dynamic define solve dynamical euler lagrange dynamical initial momentum scenario j noise thus form precede concern methodology employ readily extend reference field diffusion process find solve brownian motion end arise ii brownian signal eq arise zero drift three describe primarily field denote gaussian covariance make way truncation result sum mesh building motion prior operator mutually exclusive problem possible representation naturally tool develop numerical approximated similarly naturally concern link efficient inversion develop highlight possibility eigenvalue form orthonormal draw sequence trace conceptually think sequence coefficient thus grid efficient computing exact truncation refer gaussian prior draw indicator surely formula gaussian useful conceptually think make coefficient formulation express dimension basis switch write eq consider sequence conceptually practically unknown random infinite vector gaussian coefficient prior mesh refinement demonstrate generalize walk langevin sampler hmc preserve potential reader take state arise discretization generalize langevin incorporate parameter choice algorithm introduce idea part section specify expansion briefly describe interested define space adopt transition measure take precede equivalent sense reversible walk langevin reference operator potential motion identity may square root via refer rapidly function intuitively mathematically continuity either condition generalize case application target paper show assimilation finally mention derivation limit proposal drift lead known walk certainly fine mesh lead singular move reject return method alternative irreducible possibility drift discretization spatial rearrange cn find operator also form algorithm target g prior mathematically fact look number note absence suggest far familiar argument motivate proposal many way specification efficiently proposal via write form clearly dimensional numerical show proposal improve upon naive method similar cn proposal show explain designing formula cn reference cn acceptance probability sense cn define via accept entirely happens walk truncation parametric contain sde behind metropolis adjust mala reference invariant measure function acceptance require ia define u u proposal draw measure special introduce choice proposal give proposal simply prior probability mala give useful proposal context emphasize proposal section random write hence cn proposal generalise accord function thus eq view formula metropolis sampler partition formula robust mesh within sampler base behave mesh alternate updating word alternate give consider base independent u pi precede metropolis hasting method propose walk satisfy detailed balance respect acceptance monotonic simple random purpose equation posterior measure modify gibbs conditional acceptance move slightly remain pt move mode proposal discretization measure hmc hamiltonian introduce extra momentum velocity appropriate add result remain break walk type behaviour method hamiltonian flow reason nonparametric various sampling employ example introduce highlight standard technique compare walk goal advantage algorithms partition second goal model algorithmic truncate gibbs sampler density recall indicator approximately chance two gaussians dimensional density sampler computational graphical case reader comparison quantify well experiment truncate former use covariance independent draw integrable facilitate fair tune proposal acceptance around require probability refer accept proposal mcmc importance acceptance pt correlation function integrate auto correlation significant treat function decay chain integral determine variance path average integrate determine integrated autocorrelation mix mesh mesh comment prior reduce runtime table improvement primarily cause reduction number due adaptive proceed problem nontrivial arise assimilation distribution start size behaviour proposal aim velocity lagrangian observation posterior observation observational case time restrict divergence spatial follow note adopt basis power easily eq lagrangian scenario hence surely figure term observational use velocity solve fourier find lagrangian integrate particle initial particle lagrangian evenly space evenly lagrangian condition comprise lagrangian observe evenly spaced volume
effect edge highly shrinkage shrinkage impose enjoy benefit spike prior formulate hyper pairwise plus log bias enough act uninformative represent actual distribution equation hierarchical insensitive simple setting parameter sparsity parameter learn necessity validation unlike structure edge detection mrfs due mcmc langevin jointly reversible method mcmc parameter posterior mrf fix drawing fix use mrf mcmc among langevin carlo brief noisy posterior step langevin hmc matrix isotropic discard usually accept reject detailed decay step mrf f langevin run step langevin algorithm modification markov motivate persistent change slowly sampler approximately stationary allow step every scale size hessian prior average burn period sample persistent chain help suitable momentum langevin dynamic know inefficient random draw momentum update momentum partially preserve thereby similar fashion hmc multiple correlation significantly improve mix figure mean however value update correct hasting tb pdf fig pdf momentum typical parameter langevin change reversible conditional add edge exclude degenerate proposal edge restrict explain jacobian accepted metropolis represent likelihood distribution ratio however ratio partition approximate approximation origin reduce problem estimate computing derivative expansion ratio partition log centralized persistent chain unbiased estimate variance consequently plug equation unbiased logarithmic domain unfortunately domain transformation taylor unbiased estimate ns nf consider replace variance estimate acceptance cause acceptance small jump grow set acceptance proposal truncate add estimate since demand sample enough parameter change accept move much sampler jump parallel set indicator inference partial momentum proposal initialize momentum draw draw assess real ise bias convert boltzmann exact sample two consider equally group within group strong positively couple truth ground datum mnist digit convert gray pixel thresholding value pick image pixel necessary compete bias result always neighbourhood output max min implement lasso mle use specify one bayesian pick single posterior bayes pm insensitive use use momentum proposal set subsample exact accept reject decision method sake consider vary induce evaluate validity exact marginal show sample four pick title approximate well bayes exact level value component parameter exact tb validity deviation set task recovery precision quality evaluated hold compute general intractable instead validation randomly group grid lattice I train bayesian remove edge typical precision recall tune tb fig lattice tb lattice fig edge percentage edge fully chain edge suggest sample size tendency dense range sparsity almost peak contrast data mle design mrf parameter estimation model inference generate exact bayesian model regularization sparse result dense globally fitting prior instead another automatically selective spike discuss bayes bayes phenomenon density deviation fix include real edge figure decrease however return bayesian release hierarchical improper automatically vertical fit level harmonic high corner lattice exist ground truth learn dense model method bayes pm robustness although sufficiently model e get quality without produce qualitatively compete tb fig horizontal pdf limitation experiment bayesian able parameter need hyper performance computational complexity grow clique mrfs turn regularization model cost bayesian mrfs prior achieve langevin reversible attempt mrf structure prior work present bayesian mrf search selective shrinkage mrf fitting fitting set strength bayesian insensitive hyper automate choose sparsity method upon department university california california automatically regularize model expensive parameter suboptimal regularization induce bayesian spike prior regularization suffer langevin dynamics reversible conduct hyper induce highly graphical know mrfs large variety domain social subset automate relevant become sensor attribute increase help overfitte
background recognize color able thing use good create location construction detail store location maximize usefulness percentage corruption address address indicate location corruption update location signal describe address far identify need percentage corruption create could surely satisfactory performance original reach strength hamming distance reach strength hamming wave percentage test demonstrate graph figure increase counter counter increase decrease hamming flexible storage necessity location explain previous justified wave generate input say corrupted background retrieve white corrupt white background remove algorithm identical summing pattern test radius pattern q intend give test pattern corrupt original static straight percentage corruption new see visible important emphasize number address pattern retrieval self address retrieval little pattern use retrieval go retrieve pattern percentage bit error signal retrieve highly test well decay recognize highly corrupt pattern purpose come get biology trying say way enhance memory recognize rotation signal overcome inspired human brain recognition dynamic hard location process article replicate human moreover capability promise reference neural chapter university sparse memory fully memory suit connection institute di alphabet short long architecture permit binary pattern retrieve partially match efficient non capability recognize approach purpose create signal try memory dimensional consist
geometry gaussian model intuitive determinant suitable meaning grow recall undirected model markov field family case factorization translate precision random say graph precision pair consequently sparsity inverse encode selection determine edge sparsity interest norm sense recent entry well scale et al challenging observe random need complement second least rank consequently latent graphical cast involve insight negative log proxy rank method program attractive guarantee incoherence guarantee recover sign support associate optimum recovery probability challenge potential decomposition see author overcome sparse hand quite relate result concrete incoherence selection assumption return precision norm significantly small small discrepancy dimension raise requirement possibly rank direction develop theoretic exploit model impose nonzero nonzero scale incoherence impose ensure identifiability particular intrinsic graph recover determinant neighborhood whereas difference demonstrate aspect incoherence relaxation cardinality incoherence intrinsic population decomposition identifiability could weak notion concrete begin pair imagine perturb l identifiable decomposition sparse relax requirement long proportional guarantee matrix q observation form sparse robust enforce incoherence way radius low ratio fp matrix range achieve e mass precisely plus sparse low limit thereby also nuclear estimating involve arise possibly partial accordingly seem use error bound induce second component scaling
technical difficulty spirit sample estimator find condition expansion though early involve correction necessary begin lemma couple rewrite yield claim combine expansion sample I eq see equality analogous lemma claim focus use schwarz order may entirely analogous result lemma eq sample algebraic combine obtain desire eq run sgd onto sharp rate iterate choose bind assumption eq application triangle inequality give apply h old conjugate choice markov lemma turn data point algebra eq q taylor lagrange q since old jensen iii bound inductive q define shorthand imply inductive defining indeed inductive hypothesis bound instead analogue bind statement conclude hold prove strongly conclusion argue locally gradient minimizer similar analysis optimality globally chapter note strict inequality inequality inequality divide complete appendix two moment combine proof lemma variant separable banach say matrix independent distribute apply jensen inequality see upper involve ik argument definition complete application success event joint hold side except remainder joint event occur notational derive recursive p shorthand long drop define remain jj th definition schwarz n equality indeed product similarly apply schwarz second apply linearity indeed proof final lemma completely final finish exactly reasoning remainder early zhang zhang em ex em engineering computer department university california berkeley berkeley usa berkeley communication statistical setting involve scale sample evenly perform average show mse decay guarantee appropriate decay amount parallelization attain square decay expense potentially slow mse provide investigate scale efficiently solve prediction chinese search engine logistic distribute optimization subsampling procedure define solving scale centralized minimization among infeasible keep study distribute empirical recent distribute scale survey paper therein contain relevant purely optimization explicit benefit arise statistical computational family computer must high distribute estimation limited synchronization associate simple term size machine machine certain extremely communication failure synchronization yield essentially good knowledge however rigorously generally provide sharp showing rate naive provide error decay match centralize access likelihood statistical programming attain scale bad contribution appropriate form level evenly among processor computer instead return processor estimate correct estimate decay match centralize gold first small empirical procedure section normal model relative baseline split datum investigate sensitivity amount favorable gold access sample experiment search engine click experiment enough involve sample dimension storage essential resampling give substantial naive averaging begin real integrable estimate population quantity population minimization impose parameter risk instantaneous classical estimator deal space compact convex radius addition amount curvature term differentiable exist denote semidefinite condition require method consistently sample sample evenly processor collection processor objective notation describe processor local empirical minimizer involve subsampling subset uniformly replacement local processor compute minimizer average compute standard subsampling roughly estimator argue understand condition sense centralize risk euclidean differentiable subgradient define derivative relation indicator true comparison assumption regularity function constant continuous meaning require insight analysis type smoothness condition method work necessity illustrate necessity indicator w v population strongly second unique given establishe necessity give problem logistic long rank suitable provide associate independent averaged estimate population square easy step I vector establish decay pre depend grow gradient interpretable upper loose original bind vector multiplication perform type regularity condition chapter neighborhood le addition parametric theorem eq linearity trace except bind obtain rate without calculate calculate attain inspection proof expense reduce make even note introduction certainly expect unbiased reduce variance sense distributional estimator behave average reduce desirable bias introduce difficulty note contrast classical asymptotic finite explicit square lastly tie distribution machine relatively processor close lipschitz continuity concern motivate development introduce subsample smoothness condition euclidean smooth third derivative strong mean eq constant easy also non g covariate finite moment establish bound q bind eliminate elimination subsampling subsampling minimize averaging affect select paragraph condition compare grow polynomially hand dimension local per limitation intuitive low curvature population mean risk effect per machine course allow grow total cross model leave open question compute multiple reduce opposed replacement minimizer need order minimizer sketch argument minimizer achieve statistical error provide argument argument analogous iteration thereby obtain view minimizer output triangle elementary bind iteration obtain condition minimizer share convergence minimizer one require perform particular subsection descent descent yield minimizer radius local convexity descent local strong enjoy excellent especially baseline evident degradation much high grow somewhat convexity condition satisfy comparison distinguish gap parallelization figure show grow result suffer proportional see machine expense loss square error cc machine regression cc machine estimator develop intuition benefit drawback describe remark yield aim mis experiment sample begin feature index method offer note error mean error setting subsample correction plot square curve square sample accord regression least even oracle estimate roughly agree somewhat benefit increase case choose reasonably experiment compare mis specify generate specifically dimensional choose close mis improve number machine r description token appear gender gender user word token appear title age user position page ad occurrence ad query ad title click click click ad search predict search engine click business section study search com retrieval book search present user response ad dataset user transform regressor datum description mean title bag encoding encoding assign possible title corresponding index set age interval per feature interval fall bin entry corresponding value unknown categorical also indicate combination dimension incorporate goal predict click negative loss ridge regularization strong suggest practice regularization parameter click split sgd baseline pass entire evaluate square hold specifically five fold use study compute computer consequently full pass pass sgd rough baseline attain sequential pass sdca enjoy rate hold loss error versus subsampling plot well proxy substantial even split give performance pass enjoy gradient since pass gradient give minimax ranking may wish direct prediction end area curve auc auc bipartite broadly show parallel split sampling explore split plot hold set versus subsampling ratio split com mis specify appear inference challenge solving grow grow fast speed storage capability computer performance oracle able access interesting remain problem datum far generally communication study environment informative concentration thank point mistake statement relate feedback fellowship facebook office smoothness present attain processor inspection odd subtract normally binomial calculation somewhat eq allow control moment argument local objective processor event begin inequality relate bit algebra give eq q average statistically remainder intuitively convex begin hold guarantee closeness rough event behave population close guarantee ball u guarantee three follow lemma previously rely relate minimizer chapter care locally appendix careful expansion via initial
justify layer generative model provably close optimal justified provide encoder fine training deep model highlight rich model contrary much hard look possibility test auto complete rbm create outperform state art stack rbms layer future layer selection algorithm low extraction part display color color display color remark architectures generative propose optimistic proxy interpret auto generative stack rbms improve highlight importance hide rich generative mistake comment tag comment tag deep architecture layer network object help representation consume require expert difficulty whole network procedure justification fall somewhat short expectation cite log add reflect validity wise deep new criterion condition optimize deep derive ever intractable complex simpler wise optimistic train subsequent training successful relation restrict boltzmann auto generative hide generative scheme auto approach deep synthetic real auto auto stack auto architecture much rich latent rich inference model basic architecture traditional optimizing model probability generative learning estimate deep architecture distribution latent variable separate parameter deep recursively interested interest observe layer quickly present frequently architecture stack rbms leibl optimum obvious tackle parameter impractical able deep architecture wise bottom distribution variable reproduce recursively replace surrogate objective architecture stack rbms rbm maximize ignore moreover approximation layer improve follow question parameter latter convenient layer reduce hyper search aim layer optimistic section stack rbms auto p ascent space deep ascent require influence perform display training successively train conditional variable bottom part tractable infer bottom optimistic assumption cf provide go realize value obtain enough guarantee suppose trained think color think new top train reproduce perfectly optimal used difference optimum kullback leibler strongly suggest original recursively top fine tuning fail global layer wise training distribution priori version optimization subproblem solve sequentially layer train successful auto keep layer indeed layer layer unchanged suggest rich conditional contrary practice auto expressive prop solve optimization part generative rich reach use overfitte come family relevance representation learn irrelevant experimental evaluate bottom layer whole able produce max likelihood bottom subsequent may converge max display really redundant redundant coincide color remove really redundant really statement show low coincide theorem marginal thus bottom one optimistic assess bottom optimistic display ok ok consequently maxima argument particular writing may help incorporate crucially rely practical optimize architecture already optimize possible probability see optimize conditional represent proof bottom latent assume layer use display ok ok whole take proposition display remove train reproduce perfectly case reproduce perfectly prop propose perfect color ok prop bind prop perfect give reproduce keep fails reproduce global optimum importantly optimize heuristic use may relevance compare optimistic want dp hold reverse optimum express leibl performance difference kullback leibler divergence appear ok nice argument color ok nice precisely loss respect abuse understand obtain log sum I exactly characterize conclude display ok ok ok check ok stack rbms stack boltzmann rbms stack rbms wise rbm learn target layer distribution generative top top bottom rbms bias bottom rbms never train maximize generative generative weight tie first layer different future comparison ascent ascent generative layer ignore tie rbm use top rbm initialization rbm new deep still tie different low layer associate upper account likelihood might actually test see suggest tie part optimality guarantee deep auto credible stack rbms train stack train backpropagation layer reproduce auto p auto training layer error encoder train model auto encoder manifold scope possibility learn top rbm generative concern theoretical stack deep model auto rbms expansion likelihood keep sense stack maximize follow commonly sigmoid give let keep op perspective consider intermediate wish f g training backpropagation raw optimize particular conclude statement keep justified approximation justification underlie situation perform choose form could possibility explore impose wise stack lead tie layer determine mutually follow sense maximize training wise try train stack rbms wise hide rbm know optimal maximize encoder weight criterion rbms coincide tie retain stack rbm train approximate rbm distribution rbm stack rbm tie weight encoder see full rbm clear extent criterion yield optimization criterion train consistent though seem unlikely layer match perfectly nonetheless wise consistency rbm layer likelihood layer fine tuning layer case target distribution non require upper deep model layer incur upper able subsequent consequently layer respect procedure fine backpropagation confirm expect recover principle limit auto encoder layer wise deal issue local yes ok transform yes ok yes except include different term version qp maximum sufficient w vanish variation quantity display let take quantity maximize since ok calculus variation ok variation statement proposition general ok optimum ok optimum prove incorporation display equivalence critical likelihood point constraint follow critical likelihood constraint log vanish elementary multiplier qp n critical point incorporation display look ok detail ok sure empirically deep make evaluation latent new log yet enough dataset indeed try assess modified auto explore bind wise give future log dataset give reasonable picture sure deep spirit rbms evaluate likelihood train stack rbms cd equal consistently single rbms hypothesis deep learn namely architecture capable represent compactly architecture layer epoch give obvious head number l rbm layer layer hide layer cd rate bp epoch epoch dataset dataset probability baseline model dataset scheme give equal independent pixel bernoulli log rbms rbms confirm deep check deep likelihood rbms distinct test part image lie amount capacity one give display air color train propose bernoulli adjust validation sample concatenation log validation stack rbms confirm check deep validation rbms deep auto layer auto use display keep mind possibility bernoulli bernoulli kind new rbm stack study model focus equal although increase rich layer procedure circumstance optimum deep gradient ascent ideal deep remark increase model could greatly positive exploit modify auto use usual auto hide tie depth stack rbms ordinary rbms auto rbm sake generality backpropagation rbm train likelihood validation set distinct comparison generative find optimum propose approximation really layer reproduce auto study approximation provide comparison stack cd cd general training auto depict figure train adaptation bind backpropagation tie generative weight encoder rbm hyper stack rbms bp learning epochs ann standard initialization training figure backpropagation cross loss maximize auto encoder upper weight tie auto model rbm train size deep auto comparison figure pareto front display dataset distribution describe average validation log pareto generative dataset generative deep auto baseline independent single rbm evidence deep auto low rbms compare optima auto outperform rbms auto stack rbms consistently arguably auto framework universal significantly improve rich auto us rich auto generative exactly value likelihood raise whether indicator approximation w model many though enough unless otherwise turn likelihood estimation various measure maximization procedure globally experiment result actual definition assumption wise ideally idea architecture use layer validity maximization possible learn approximation conditional second training relationship un optimistic tight part reproduce practical generally low situation refer optimization might reproduce perfectly layer introduce could high validation distribution bind ideal definition optimistic final obtain perfect upper either class poorly actual layer upper several effect go affect really predict context architecture hyper selection involve generative intractable sometimes visual rare method train criterion maximize parameter exponential evaluate layer otherwise hide always color hyper evaluate layer always become rbms intractable prevent dataset evaluate summary wise hyper layer upper evaluate sample distribution layer experiment robustness hyper mention representation irrelevant train generative representation lower log assess quantitative discuss rbms compare actual model training checking result instead
weight decrease singleton entire removal increase consist maintain singleton usage clear context reward record object terminate backward low cost remove explain low element currently singleton singleton add backward current object correspond row backward forward execute thus backward pair decrease convergence backward forward share threshold backward factor singleton singleton reward q find record break stop estimate support find break backward new share true element correct interpretation main quantification share state scenario randomness gaussian restrict eigenvalue constant minimum task gradient specify towards magnitude large entry entry magnitude constant bind noiseless yield small fast algorithm fewer backward however need thus range complexity contrary zero time row optimize row distinguished atomic setting fall algorithm weak assumption hold proof b inspire j c see exact never consequence stop previous backward fail go lemma bound complete em divide eq inequality imply converse ii setting stopping comparison suppose share portion support location value entry noise draw substantially interestingly tend phenomenon around stable pick oppose single row single different define rescaled version parameter plot success outperform less know theoretical sharp sp sp model sharp suggest problem design sharp ss I gaussian greedy substantial conjecture sharp threshold size show sharp conjecture precision open average classification error handwritten digit optical handwritten digit handwritten task collection digit different people digit otherwise disjoint report set classification find average distribute task change get error terminate forward fail go entail loss function separable next lemma bound estimate parameter stop parameter optimize rhs stop carefully choose would arrive hence q conclude terminate backward fail next consequence stop j hold stop reach lemma arrive contradiction assumption forward step support begin rhs conclude error support forward identical step assumption lemma immediate backward separable column fix jk entail notice always large provide element rr omit b super sub forward optimizer equation latter term first I j claim c theoretical term sake replace hold provide lemma appendix satisfied sample assumption noisy measurement backward operate distinct object iterative addition removal support exist algorithm complexity interestingly extend greedy structural infer observe accord design column interested infer inference recovery potentially small number feature arise recovery recognize regression range support handwritten character course indicate infer jointly often task learning attempt iteratively drop algorithm
loop generalize predicate atomic predicate try infer loop invariant predicate consider abstract incorrect conjecture form x observe variable share state characterize denote I induction versa induction hypothesis formula I statement induction also x atomic predicate find formulae atomic predicate inductive follow show see characterize loop annotate proposition condition loop incorrect conjecture teacher return abstract otherwise applicable loop annotate loop inductive condition nevertheless pair atomic predicate incorrect assume incorrect satisfie necessary collect atomic predicate predicate inductive corollary mi abstraction abstract section try boolean resolve check weak teacher abstract intend equivalence teacher teacher abstract truth abstract abstract abstraction give another chance refine abstraction inconsistent atomic predicate inductive generate atomic predicate abstract formulae recall inconsistent inconsistent predicate loop generation algorithm predicate annotate formulae equivalence first atomic predicate section initial set loop invariant exception predicate invariant case find predicate learn find contradict abstraction distinguish predicate find loop invariant exception query generate coarse abstraction predicate start predicate number boolean support incremental improve overall threshold atomic predicate abstract equivalence resolution box teacher abstract return conclude compare conjecture approximation atomic predicate fall approximation possibility set predicate sufficient iteration infer solution predicate insufficient predicate arise learn algorithm invariant abstract exceed threshold atomic predicate abstraction refinement intuitively predicate insufficient random abstract exception atomic predicate approximate space solver query example add extract translate annotate loop manually average collect ghz intel cpu r case atomic predicate choose atomic predicate pre loop statement automatic interestingly predicate suffice loop fail due ill predicate infer loop invariant example atomic predicate thank small example take order give abstraction addition preprocesse outperform three case tie atomic predicate generate always item buffer keep buffer item buffer copy item atomic predicate program text express invariant specification successfully loop invariant invariant atomic predicate text generation find loop benchmark atomic predicate predicate atomic predicate program text loop invariant find invariant contain predicate suggest redundant predicate predicate make loop easy follow loop summarize implication summarize true atomic predicate execution range chebyshev loop probability atomic predicate invariant algorithm device figure generation perform predicate atomic loop surely atomic atomic predicate atomic needs query loop generation present technique apply atomic predicate imply text efficiency code report need realistic example implicit predicate text additionally loop quantify quantify gray ac com national version publish support center education science national award dual address generation loop invariant abstraction refinement interpolation program text effectiveness learn way annotated post annotate loop post specify intend behavior annotate loop specification verification tool whether annotated specification loop require intelligence recently automate technique base abstraction atomic predicate annotate atomic predicate teacher able loop construct technique abstraction atomic predicate crucial learning extract atomic predicate text atomic predicate express invariant infer loop invariant heuristic atomic predicate redundant predicate algorithm generate atomic predicate technique atomic predicate interpolation logic formulae formula inconsistent logical symbol occur interpolation order formulae inconsistent many software interpolation atomic predicate refinement interpolation atomic predicate loop algorithm add new predicate execution learn new generation effectiveness efficiency loop atomic predicate loop loop body decrease become loop iteration eventually loop explicitly atomic predicate express establish specification atomic predicate exploit pre loop inconsistent extract predicate x interpolation loop specification introduce technique algorithmic technique quantify loop invariant atomic predicate address free recently technique termination technique loop termination combine algorithmic paper author design transition predicate interesting adapt invariant inference interpolation implementation base refinement software check abstract may abstract logic quantify generate predicate template review learn base loop interpolation generation automatic conclude denote free logic equality inequality rational number formula free write evaluate satisfied formula return inconsistent logical symbol occur formulae third condition make generation symbol observe condition specify loop formulae loop particularly atomic predicate invariant annotated loop atomic predicate enough predicate least one propose algorithmic technique abstraction decision teacher answer abstraction engine combination predicate state software check base section review invariant atomic predicate formulae free loop adopt abstraction relate boolean formulae teacher guide boolean formula inference show level view loop invariant framework loop teacher teacher course loop invariant try answer information program text teacher employ suffice framework traditional formulae boolean interact teacher query abstract query teacher otherwise equivalence teacher exclusive abstract abstraction boolean remain teacher guide abstraction loop free loop annotated formula weak loop membership teacher whether approximation inconsistent simply membership yes randomly approximation loop membership query give query give accurate exploiting well learn orthogonal static analysis answer equivalence indeed pre post solver resolution return find
classify validation classification include major challenge accuracy assign everything total accuracy fail capture diversity vote require vote maximum weight grid finally forest use information differ classifier cell examine neighboring cell predict cell question majority smooth noisy classification second pass weight city demand class reduce overall display incorrectly classify however spatial intra relatively total accuracy com matrix com com tendency consideration leave share remain four display result large account mix use correctly classify sub classifier exclude mixed incorrectly classify onto without affect heavily classification accuracy time classify nature actual differ classification accuracy increase heavily area plausible change finally analysis reveal fundamentally different mobile phone activity heterogeneity cdr potential infer act guide suggest mobile phone activity measure heterogeneity simple broad classification update traditional well reflect activity planning large expand aim relatively low aid fraction city high resolution additional balance interest ground may mobile phone measurement cdr hope location public private resource develop apply novel mit fellowship national research fellowship collection analysis manuscript understand people city crucial planning create currently sensor gps device collect massive amount communication million record information utilize mobile phone relationship population course week cluster mobile phone show mobile phone capable useful database application database depend turn city live location understand individual efficient planning choice influence determine demand maximize popular location maximize access city city office kind usage business office hour relatively different might differ somewhat intend information area note part dedicated relation use human traditionally via survey survey require subject record move day whole week survey method expensive limit give survey thousand capture period fortunately past decade nearly country currently phone purpose understand particular call cdr provide location mobile send obtain cost aggregated area level risk information question arise whether mobile usage measurement advantageous result share usage whereas usage usage monitoring allow shift development work apply aggregated cdr infer dynamic I area city supervise region cdr use classify reasonable normalization application forest mobile human first utilize device mobile phone activity university student regular daily moreover pattern level student upon extent et nearly anonymous mobile phone reveal persistent human human song et mobile mobile phone use al mobile phone link mobile phone mobile phone activity km km activity qualitatively city decompose activity usage et cdr mobile phone location similar qualitative similar technique analyze profile cdr proven detect movement census call across area attempt associate source learn date exist employ traditional mobile phone activity partitioning region region profile active order identify pattern characteristic specific correspond two datum mobile phone region cdr mobile phone strength unlike cdr record location location mobile phone provide accuracy continuously across space locate phone activity count call text cdr roughly home million mobile phone mobile phone activity obtain classification office actual proxy actual impose obstacle study phone phone activity record pair partition phone population rarely influence due transform cell lattice size test coarse level enough mix reduce phone compute hour mobile phone certain analysis give single classification area large heterogeneity densely characteristic reflect census phone activity block display actual frequency grid vast percentage examine mobile phone activity macro city mobile phone activity average cell count classification differ greatly maximum hour cell activitie huge order typical mobile phone normalize unit profile remarkably city fall user also partly phone across normalize activity residual interpret mobile phone activity mobile phone city hour mathematically cell time average pattern average notable relationship residual activity behavior higher visible area early activity level reflect subtle phone hour suggest area city mobile phone link high level treat phone activity proxy spatial people give pattern concentration people work shift behavior visible residual activity volume day affect mobile phone persistent phone hour week display activity top activity plot absolute activity density order city logarithm city strongly dominate usage spatial nevertheless dominate difference different perspective region km spatial activity rich early hour activity locate activity become heavily concentrated later activity area suggest activity residual
refer show well leave simplicity value important fold give fluctuation run transform regression since perform high transform change dramatically advantage well across decrease whether l normalization frequency good range classifier combine performance transform keep refer transform transform project smaller omit generalize dataset omit bayes classifier feature reduction transform run bottom axis refer dimensionality effect lda names drug dictionary protein drug logistic lda count treat lda omit outperform binomial omit count binary incorporate via occurrence weight count feature produce document cross weighting tool run except lda binary occurrence significantly datum feature run transform apart run drug drug also several transformation normalization number principal reach investigation thus enable support drug interaction prediction mining patient record drug drug interaction deal drug mechanism efficacy drug aid kind extract publish clinical database drug inclusion still preliminary area benefit literature mining mechanism significance classifier identify relevant document identify causal mechanism drug drug interaction important linear classifier dimensionality investigate benefit publicly dictionary find distinguish relevant give well alone normalization adjust improve classification linear proper dimensionality large help classification drug medical rate refer incidence include aspect drug g medical record database drug drug detect signal database become methodology scale extraction information domain database ultimately research genomic drug collective database especially relationship complementary conduct independently research automate literature method whose mechanism clinical conjunction previously gap parameter probe ki ic text mining approach may particularly causal behind complement mining patient reporting bias previously show automatic literature work orient toward perspective first relevant contain evidence goal automate report evidence towards integration mining li working develop goal report manually wide logistic regression support machine binomial previously find well mining transformation normalization technique use name describe corpus section deal cover discussion li automatic extraction retrieve article manually classify drug irrelevant article contain one four class study clinical clinical drug initially one extensive well detail request field author title subject mesh field si latter contain code biological entity entry process certain convert less character occurred omit token document occurrence combination run classifier simplify angle version threshold proportion occur proportion document otherwise full version protein include additional account entity occurrence dimensionality linear svm interface cross validate logistic regression interface validate naive bayes naive validation discriminant lda covariance toward shrinkage validation validate shrinkage toward equivalent naive multivariate follow occurrence rise occurrence occurrence inverse document feature term apply document total minimize difference feature normalization projection onto commonly precision
parameter relationship hold transformation involve parametrize present w great deal lead discuss fall multiplication wrong definite movement suboptimal use make rarely thing help deal remove dependency inherent topic precisely remain head around provide review ill analogy widely concept whitening problem connection whitening begin poorly mean datum negative parameter much sensitive change step instead large nearly movement get via slow show illustrative parameterization effect parameterization dominate remove descent path blue section initialize fisher notice descent quickly gradient describe descent converge directly value difference dependency assign metric define choose magnitude representative change result uniquely correct objective well g whiten analogue covariance signal remove dimension remove dimension whiten quick covariance inverse mahalanobis whiten symmetric zero whitening refer rescale axis unit inverse rotation signal orientation unitary transformation linear mahalanobis variable whiten common preprocessing signal processing perform direction find write update gradient illustrate function direction parameterization effect formula specific minimized learn take natural metric even metric accelerate fisher require instead empirical alternative approximate field gradient practically simply ignore work surprisingly greatly square problem compute however gradient convergence application evaluate may find condition eigenvalue eigenvalue tend worse dominate deal subject rather unstable give plug play call robust approximation small behaved ridge regression useful gradient gradient frequently cause gradient pass true space natural
recover equation rise compressed sense paradigm signal admit ambient measurement denote compress possible nonzero entry recover quickly instead relaxation also es stable measurement take alternate minimization weak regard recover incorporate algorithm weight x recently sequence minimization belong update performance iterative call two weight converge update algorithm solve subproblem overview sequence subproblem detail preliminary demonstrate recover signal incomplete scope result leave index complement refer van find efficiently spectral determine l pareto trade least square one initialized point hermitian transpose proceed prove pareto curve continuously solution l rise guarantee root method generate ls expression example pareto root solve weighted lasso subproblem arrive update subproblem l exactly lasso ls find generate contain large vector weight subproblem different iterate lie curve pareto switch r problem coincide signal pareto l solved switch path weight apply support path subproblem note axis one oracle curve support oracle ambient varied still recovery clear far show less signal every subproblem allow improve test compare minimization recover synthetic
ref problem introduce demonstrate building consider decomposable ise spin vertex together publish broad transfer discuss distribution inductive summarize conclude hierarchical work string candidate string accord candidate promising learn solution candidate encode diversity new candidate incorporate use restrict termination global optimum number reach represent solution structure acyclic specify direct conditional specify value variable string represent probability parent use internal variable variable leaf assignment node unconditional test split repeat goodness knowledge metric relevant combination favor simple exponentially description complex probabilistic probabilistic crucial building probabilistic population small population learn experience address basic experience examine bias search instance transfer building focus identify bias structural future analyze build implement one sure collect solve work pair classify describe classified predefine fitness q string may additive typically prefer difficulty subproblem subproblem even subproblem order paper variable create denote path variable q subproblem distance length variable mainly subproblem additive metric correspond locate interact confirm numerous spin dimensional lattice ex describe inspire al run apply process introduce start model split compute dependency split recall quality probabilistic contain population prior statistic ex split normalization constant contribution experiment done know evolutionary three ise spin consider periodic boundary condition unique minimum cover node ratio unique instance map create combine graph nearly regular lattice half refer reader preliminary copy ensure population minimum optimum independent run hill hc incorporate cover vertex cover ref use use define effect mean random instance equally sized subset subset run subset analyze remain round instance small run experiment perform across computer configuration base case always run computational node two therefore cpu execution speedup respect execution run speedup multiplicative factor execution improve bias base execution speedup indicate base percentage speedup addition ability base prior apply run instance examine finally examine combination building delay suggest carry building source multiply due requirement effect ex ex confirm ref examine bias yield obtain minimum vertex improved improvement observe much majority instance substantial instance size instance problem vary argue nearly multiplicative distance cc cpu speedup cc cpu cpu c cc cpu speedup c cpu cccc speedup paper extend optimization derive substantial applicable execute combine technique great several topic central technique linkage
relate orthogonal aspect interpretability human view label visual interpretability closely member semantic interpretability view function projection axis attribute experiment visual interpretability show dataset ask participant aim investigate view aim vary level generic order interpretability overview interpretability label address detail human view dataset group relate automate human expression automate discussion visually interpretable view manual view impractical degree apart visualization lee al exploratory pursuit fisher discriminant method et al search projection dataset evaluate neighbor user et propose two measure centroid base entropy class author measure alignment finding view good participant al separation compare measure suggest combination investigate oppose visual interpretability label axis address interpretability propose simplify coefficient et dimensionality reduction arithmetic assess author contain feature less interpretable however report related interaction investigate human develop human computer complexity develop system mathematical expression visually presentation notion human infer experimental participant expression expression human derive calculus represent automatically experiment without intervention detail design execution along automate visual interpretability complete degree field physics biology account participant ask fill course experience participant mine commonly mine visual table dataset shape affect relationship automate measure diagnostic breast cancer dataset characterize chemical three contain seven kind path uci learn repository eight nine country diagnostic breast cancer order view automate choose visual design assess usefulness extract classification assess mention use measure quality include three visual propose measure user present list automate utilize name section vector index consistency measure attribute experiment value automate ensure choose diverse automate measure create width bin bin frequently range across automate measure upon experiment label build observe class unseen item two category generative generative infer mapping regardless common feature classifier usefulness base method feature view choose common algorithm characteristic decision boundary generate dataset machine tree generate boundary decision boundary characteristic separation near neighbor label weighted measure generally algorithm utilize assess choose create partition algorithm build internal split dataset respect partition decision boundary attribute tree bayes algorithm simplify despite bayes algorithm outperform variety machine technique search separate two class utilize minimal problem item quantify cluster algorithms validity indice interpretability validity aim separation class scale rate five view separation automate interface build rate user calibration pre select display randomized outlier median user comparison automate automate strong good automate relationship human table c support naive k near index histogram measure lda value support bayes lda near histogram naive support consistency lda rmse naive k index machine lda indicate fit rating base automate without consideration view near individual notice long match might member class overall lda human seem extent match automated shape member therefore human derive measure individual ten automate median cast linear automate leave six ten see combine human report composite winner composite measure dataset observation estimation naive exclude favor df interpretability understand user understand mathematical transformation characterize projection automatically intervention use interface connection visualization take experiment start participant interface expression consist five variable logarithm root participant expression expression include analysis expression depth size e l tree avg block time spend write z x log u participant inform mathematical expression possible numerical square power expression display second expression back ask rate easy understand interpret difficult show first calibration participant linearize specifically choose display division fraction create easy expression multiplication record spend rating understand disk manual inspection study response correctness summarize rating median take rate number examine expression rate participant frequently correctly expression incorrectly rate participant pearson participant consistent observe behavior answer incorrectly meaningful rating participant much hard interpret present utilize tree expression compose depth block indicate expression participant rate expression easy rating rating operator total rate expression rate participant c attribute operator total tree avg cast learn mapping interpretability participant leave q predictive rating number block df human long expression interpret block increase relationship human automate measure interpretability exploration interpretability interpretability concern easy looking argue validity machine assess view besides compare automate human indicate single outperform extent linear combination automate correlate well interpretability human original investigate human would expression operator participant rate rate long
form overlap segment incorporate nominal prototype separate approximately extent separation similar fix dictionary basis background approach dct basis dictionary initial column frequency audio separation separation whose amplitude symmetry retain separation identify nmf estimate separation qualitatively propose aim separate source online generally separation domain overlap spectra signal separate note hand learning implicitly basis overcomplete amplitude spectra still furth time experimental investigation blind application domain edu blind source propose sparse recent effort representation key background separation partially source problem domain demonstrate separation separation representation blind separation separate signal comprise superposition source bss arise call audio perhaps know independent ica gaussian separation factorization appropriate factorization nmf sparse source separation mixture separation knowledge source manner aim partially background source novel unknown source describe source tool task domain effort audio processing law device utilize device load encounter device qualitatively low audio periodic approximately load audio separated subsequently classify separation would facilitate accurate remainder work description motivate aforementioned audio conclude remark fix decompose partially motivate audio application comprise underlie continuous time consider small prototype observation base approximation generality local inherent let evenly equivalently length segment goal effort essence entail leverage contribution partial column dictionary column express broad approximation recent describe effort decomposition effort put contribution represent linear separation employ simple variable low approximation particular via optimization square matrix interference cause accuracy svd comprise amplitude noise may numerous traditional setting effort aim denote nuclear relaxation non sum absolute essentially count pca possess representation source imply represent cosine transform suppose sparse may directly approach implicitly priori restrictive dictionary column aim base dictionary accomplish respectively comprise column pursuit pursuit denoise form priori represent background approach approach semi try represent semi blind modify decomposition column learn column form assume express superposition possess estimate estimate base alternate minimization coefficient know type approach comparable pursuit omp iterative fashion outline follow subsection lack make depend initialization strategy datum unknown strategy coefficient essentially learn identify estimate dictionary word update extract repeat iterate demonstrate
video video rank knn transformation fr mmd change problematic capture set knn mmd fr video subject assume mobile phone datum set walk individual feature magnitude raw approximately represent whole gender walk retrieve high precision mid early point early recall reflect flexibility similar large goal find group group variability group provide sensitivity score reflect allow flexibility quantify estimate neighbor neighbor point describe capture distribution distributional distance work include provide give significance score dependent dimension domain statistical gain bootstrapping addition interpretation attractive lastly insight discrepancy area represent hypothesis hypothesis test significance normalize apply hold reject error bind differ result reject even though grow value get decrease need ensure rate alternative similarity reject test may principal rejection area change projection projection aid distinguish define value let mapping let distribution projection expansion follow denote ks inequality apply combine conclude proof far surely projection probability least denote notice let fs source randomization projection equality eq clearly depend q therefore infer whether test type level significance sample unit ps ps corollary assume threshold moreover ks theorem consistency bound projection decay power projection clarity thm suppose I let prove totally presentation space define volume box intersect cover mass ba z discretized version distribution discretization turn discrete lemma relation structure two initial refinement discretization discretization split equal difference discretization refinement norm let variable proof lemma yield recall number discretization lemma combine union combine union least get proof belong element ba ta da exist z add bipartite graph case decrease decrease hand discretization x neighborhood bin result histogram may cardinality large claim discretization useful may remove absolute feasible optimal sufficient solution let constraint fix show exist obey stage know result difference bind solution describe obtain constraint constraint optimize plan discretization k bb cardinality feasibility feasible feasible z ij k leave side problem q whose existence td td infeasible feasible next show solution feasibility equality hold conclude difference discretization refinement substitute assignment inequality element sum hold solution procedure optimal feasible solution inequality lemma hyper size definition spherical point unit sphere let neighbor neighbor union bind cardinality neighborhood match q equation department electrical title proof lemma theorem definition distribution sample much research determine score optimally score hypothesis set come simulation detect example mining vision scenario adaptation da intuitive input yet whether generate work similarity statistical procedure score however score statistical testing similarity equality sample generating test one equality equality transform similarity see work similarity predefine physical expect represent similar distribution apply measurement change people put name discrepancy two distribution distribution difference difference specific figure blind follow put distribution minimize cx wasserstein problem mass function rewrite dx plan mass distribution perturbation area equally costly due function triangle quantify intuition similarity consider similarity tv characterization plan costly measure dirac delta perturbation therefore wasserstein tv explain sensitivity optimize plan allow perturbation aspect contribute problem unify neighborhood point mass neighboring non depict scalar objective plan perturbation plan scale width pt plot coordinate optimization identical account difference optimal integral whole solve appropriate let link link assign edge version program assignment typical problem demand example complexity bipartite cardinality score may test notion mmd capture rkhs distance perfectly equality mmd rbf advance highly dependent clear domain accordingly capture case rate let cardinality disjoint cover n rewrite metric change theorem exploit use dependency box inherent namely curse theorem sphere least identical I theorem converge expectation combine difference exponential distribution intuitively dependency bootstrapping bias present different possibility project appendix material type complementary dissimilarity hypothesis p relax distinction similarity flip role nan equality test supplementary material inference bootstrappe ci bootstrappe approximation resample replacement computation
simulate iid sample replication four sample observe plausibility suffer coverage length final ability plausibility function coefficient goal select suitably collection plausibility immediate profile distribute wrong subscript evaluate singleton zero plausibility coefficient type strategy choose numerical analyze diabetes average pressure six tc possible make much figure select comparison four bayesian diabetes section point relative likelihood unbounded although depend potentially quantity respect integrable denote hellinger define metric constant denote universal assumption precede paragraph constant large omit space last outer center hellinger bound exist na application give plausibility vanish plausibility singleton consistency mass vanish hold extra plausibility region rather abstract comment family ii second control theory transformation transformation chi square converge pointwise support conjecture stochastically large check numerically claim I remark frequentist program inferential framework frequentist plausibility plausibility finite sample justification extension plausibility carlo pearson program frequentist error rate despite strategy exact inferential derive asymptotic normality inaccurate challenge require even preferred arguably perhaps issue claim correspond make fail somewhat numerical method exact frequentist property desirable approach construction frequentist assign plausibility datum plausibility use whenever plausibility plausibility function follow hypothesis confidence plausibility control rate sample large sample justify efficiency plausibility plausibility intuitive implement frequentist result problem generality make method appear previously interval certainly new critical version plausibility paper paper nuisance plausibility interest exact sampling plausibility function structure discuss demonstrate efficiency region test theory plausibility method good practically effect model propose exact bootstrap conclude remark give section unknown say e function indicate reasonably sort square negative minimizer relative choice q often difficulty define function act plausibility claim singleton I plausibility variety problem test plausibility base q intuition plausible outside plausibility test control plausibility construct confidence plausibility plausible value connection show region nominal level sampling plausibility unify minimizer plausibility empty likelihood contain plausibility plausibility precisely outside parameter value plausibility plausibility compare asymptotic ask plausibility region plausibility general neither figure plausibility single portion around plausibility understand complicated plausibility properly explain phenomenon play however suppose convexity concavity chi limit one expect plausibility size sufficiently see besides tool frequentist plausibility potentially deep plausibility function inferential employ plausibility continuity statement make sense stochastically large monotonicity may continuous jump know stochastically term plausibility achieve immediate singleton important estimation singleton continuous stochastically general rest go coverage probability plausibility plausibility frequentist furthermore moreover corollary bootstrap analytical validity suitably function particularly efficiently wrong case mild mean plausibility percentile appropriate chi display region similarity plausibility display asymptotic efficiency conclusion region precise plausibility I rigorous plausibility valid fix size motivation investigation let iid version decrease law probability vanish probability atom continuous plausibility correctly distinguish plausibility far difference strictly tool empirical also require quantile propose derive rare need plausibility remark number unless example herein conservative evaluate plausibility equation approximation interesting depend indeed monte substantial transformation transformation tie together model transformation result group transformation certain depend check loss invariant special result suppose dominating respect hold depend establish multipli immediately success fundamental statistic widely substantially coverage likelihood give numerically mass plausibility plausibility plausibility unity plausibility binomial gray plausibility monte binomial almost indistinguishable coverage plot coverage claim simulation interval particularly yy exponential plausibility normality parametric asymptotic normality bootstrap plausibility interval iid unknown gamma literature e evaluate plausibility illustration present survival time certain plot plausibility show jeffreys along elliptical plausibility elliptical roughly region plausibility coverage mark normality maximum likelihood gray jeffreys bayesian posterior consider covariate probit iy ip distribution coefficient plausibility function illustration real relationship exposure death otherwise show plausibility comparison normality plausibility confidence region indistinguishable likely region region nuisance parameter component case kind easier interpret plausibility nuisance example negative profile replace global maximizer obvious carry plausibility turn marginal plausibility q carlo check nuisance straightforward property theorem carry exactly provide plausibility corollary hold rare write corollary structure interest model transformation namely composite give follow invariant marginal plausibility composite see case iid chi reach weak effect turn suggest plausibility convenient justification provide one different might less choice likelihood sign convergence suggest quantity use conjunction bootstrap scheme limit special particularly illustrative unknown relative profile usual residual monotone squared see plausibility interval efficiency plausibility independent nonparametric empirical probability th show quantile plausibility essentially binomial problem bivariate distribution correlation nuisance calculation correlation transformation gaussian convenient illustration replicate probability plausibility display fisher normality approximate normality approximate sign log work parametric percentile bootstrap digit reasonably suffer plausibility range take last correspond plausibility base straightforward evaluate check nuisance shape I negligible iid size robust fixing reasonable
two sale volume namely sale bipartite graph product degree bipartite sale output prediction predict cross sale co sale aggregated category relevant collect book week user category time sale category category setup type include feature involve descriptor matrix contain past sale vertex growth degree market relevant underlie factor instance cluster proxy useful quantity infer future connection simplicity assume node ii indicator eigenvector contain velocity acceleration history average quantity accounting representation fouri polynomial include task completion link vertex asset future appropriate lead graph completion aim prediction relative wish predict top sale volume component assign label magnitude sale volume specific compete compare baseline ridge address time prediction hypothesis completion dedicate link prediction denoise ignore feature version aim singular uv ta uv order convex standard objective denoise separately emphasize decrease appropriately compete baseline observe rank advance book category sale volume good graph f benefit joint regularization represent relation goal advantage although identify case contain optimal explore greatly method determine within domain key contribution lead thm link predict application node formulate hybrid graph predict joint empirically synthetic real node feature forecast system collaborative market bioinformatic rating activity technique apply multivariate multiple autoregressive develop approach enhance assume encoding dependence graph inference graph latter interesting se part recommender appropriate completion realistic setup evolve matrix adapt account dynamic graph simultaneously vertice network cause forecast sale market reliable user product evolve history expect item sale sale induce corresponding item situation arise financial reflect dependency stock predict stock price dependence show available efficiency multivariate enhance broad scope objective predictor adjacency mention factor create set discuss implementation performance objective simultaneous feature experiment synthetic object interest nonnegative unweighted one time represent vector depend function qx tt goal predict future adjacency introduce descriptor encode matrix value qx ff estimate past th equip n indicate value edge large value prediction vary feature local change reasonable evolution govern unobserved evolve smoothly account choice factor measure adjacency coefficient good indicator popularity refer centrality trend read degree singular adjacency formulation set underlie mechanism rigorous history factor assume partly contain assumption collect feature satisfy assumption rank rank factor growth growth exhibit regularity frobenius regularity close predict consequence stationarity mechanism graph correspond base reflect eigenvalue define regularization sum per allow adapt separate correspond task rkhs static reflect predict term reflect assumption matrix minimizer subproblem simplicity rkhs represent regularity degree q w frobenius work adjacency inter intra degree projection statistic length interestingly improve optimization norm term instead vertice issue discuss learn standard gradient project challenge ensure desire domain happen minimization functional degenerate eigenvectors iff define add define trivial henceforth take slack q schwarz basic eq cauchy q let w choose replace nonsmooth nuclear smooth spirit eq lower bind approach envelope unit ball spectral differentiable case relate multivariate vector center also eq randomly assumption linearity make laplacian non book sale volume item aggregate category test book aim predict sale book predict sale co set zero user define product node co product sale volume book target sale volume user algorithm output prediction predict sale sale category relevant book category week period entry item sale time include adjacency involve past realization feature direct volume sale popularity market measure factor structure cluster proxy infer future assume contain indicator several cluster eigenvector descriptor contain quantity make recent average account fouri prediction could criterion optimization evaluation focus completion link task asset least square well use relative graph completion aim report pattern predict instance item market sale volume component product magnitude sale volume address graph completion one direct compete compare consider baseline x account future graph choice low hypothesis dynamic graph shrinkage dedicated denoise ignore hypothesis obtain singular value uv td uv implement cross objective denoise synthetic lagrange coefficient efficiency minimize objective validation set appropriately outperform compete item nuclear free well understand due high efficiency predict future graph benefit simultaneously time series represent relation investigate feasibility convex greatly method condition global attain work approach problem future methodology far predict link dynamic solution reflect paper formulate learn structure predict benefit prediction evolution challenge collaborative filter market bioinformatics response movie rating stock price statistical technique multivariate either multiple regression develop enhance frequently assume encoding correlation discovery unobserved graph per se occur application recommender tool among evolve completion adapt account dynamic goal simultaneously predict evolve one make effect cause sale market semantic reliable evolve history expect sale sale induce corresponding financial graph reflect dependency stock aim predict price infer enhanced tackle broad regularize risk objective formulation adjacency separately convex synthetic mention factor create set discuss implementation issue objective multivariate simultaneous dependence structure joint relevant synthetic notation paper undirecte weight edge unweighted edge th row represent feature node series series n qx recall operator graph standard computation follow eq w frobenius tensor node inter intra onto specific cluster statistic later choose
method add run noise level require impact quality impractical moderate give approximate produce mechanism account dimensional bias towards pca chain carlo simulations subspace produce variance algorithm privacy understand well lower differentially private show sample differentially require nearly theoretical demonstrate experiment many depend minimal differentially private pca grow several question suggest computational privacy interact place mcmc use implement investigate set guarantee angle develop theoretical general challenge provide notion extend use finally many choose look differential privacy selection differentially private manner learn database survey find survey strong guarantee attack definition privacy syntactic definition attack individual database several differentially approximation perturbation desire computation perturbation add exponential base measure output variety mining task differential paper deal differentially prior calculate decomposition mechanism subspace toward private exponential run datum unclear may implementation differentially private reconstruction provide hold plus additional term depend privacy power calculating differential release transformation preserve privacy measure utility example preserve differential privacy privacy base create privacy reference preserve mining set study singular publication differ privacy release pca actual datum private individual let moment frobenius reduction low rank matrix much dimension schmidt decomposition area science suppose top subspace large refer quality measure maximize result top close top preserve privacy privacy quantify take privacy supremum measurable interpreted numerator take privacy privacy privacy privacy differentially individual private close well individual guarantee privacy relatively perturbation therefore characterize scale differentially modify query private pca private privacy differentially privacy top apply privacy differential correspond eigenvector pca easy differential privacy gaussian necessarily eigenvalue entry output dimensional differential utility sample matrix privacy focus special output mechanism mechanism private expensive computationally implement recently propose gibbs gibbs popular assess burn add change induce copy copy copy sensitivity median smooth sensitivity difficult function private solution problem perturbation require property namely convexity norm pca sample differentially show scaling require despite privacy even algorithm pca challenge pca manifold private differentially fact differentially follow calculation complexity privacy correlation gap hold satisfy pca differentially certain technique q differentially private utility eigenvalue gap show scale private scaling match contrast number certain level favorable bind satisfy notice dependence reduction space dimension suggest well application guarantee differentially private match showing nearly privacy column column step mechanism geometric lemma unit sphere bound probability union spherical complement output good boundary maximize minimize sphere fact bind simplicity q private collection correlation collection differentially private eigenvector computation characteristic polynomial also follow product give expect utility average database differentially private radius disjoint hold inequality norm product database database differentially private basis product eigenvector small coordinate packing sphere span weak bind eigenvalue requirement copy copy moment q construction obtain upper bind eq q follow plug put thus differentially consist copy copy database eigenvalue eigenvector q additional guarantee differentially distance eigenvector complexity illustration tradeoff calculation vector achievable orthogonal first loose still tail set ij j yield rearrange differential formula particularly expression differentially provide true top eigenvector privacy parameter space pseudo construct set matrix matrix algorithm input input use pack minimax utility bind private element top covariance expected lemma show small loose set yield low symmetry q density lemma vector inner euclidean satisfie condition theorem low impose condition yield large dominate set eigenvector produce horizontal axis four correspond plot lead plot figure order correlation logarithmic range experiment limitation particularly subspace present good dimension rich regime favorable little algebra q concern large eq get involve mcmc burn must reach stationary theoretically chain reach must interaction differential privacy rich report choose four domain connection medical instance individual activitie dataset contain mix feature discrete preprocesse dimension respectively normalize row maximum normalize utility account four fairly c implement gibbs sampler gibb widely scientific machine finding mcmc still open area much time weak practical user iteration employ diagnostic empirically chain stationarity provide appropriate burn distribution perform qualitative develop characterization gibbs sampler procedure come execution privacy implementation guarantee note suffer privacy try burn confident performance affect burn run copy trace chain initial uniformly column iteration ht ti data gibbs dataset dataset illustrate behavior plot show datum start column rapidly case number sample sampler thus sampler dataset chain good column fact distribution private try generate utility maximize subspace reflect subspace although hold practice datum run several randomized result utility size subsample non private nearly illustrate blue utility subspace well produce subspace utility par choose exception produce furthermore run burn suggest differentially result immediately average bar dash blue utility green indistinguishable utility projection respectively randomness bar standard mean pca algorithm dash line indistinguishable datum onto datum summarize svm private predict majority label classification half onto base use repeat projection projection dimensional algorithm well random projection accuracy compare private pca subspace classification application understand utility subspace point draw
matter select epoch bit datum hashing substantially loading present accuracy sgd datum hash achieve accuracy present loading table datum bit hashing loading dominate achieve accuracy h epoch transfer gpu result make bottleneck similarly dramatically resource requirement online due reduction loading epoch use hash reduce incoming various query classification etc communication hashing approximate similarity bit hashing critical must bit application especially bit preprocessing compute permutation require concern parallelization scheme substantially load beneficial g page hash increase online major hashing substantially reduce amount memory batch become bit improvement hash dramatically epoch significant often epoch reach impossible space study bit hash g u hash family hash technique similarity context matching redundancy file management spam etc substantial bit regression binary task enable massive expensive show million disk gb format crucial apply bit mainly focus view set q permutation practice store storage prohibitive low bit convenience theorem permutation hash vector new point consist expand binary run day inherently example discuss distinct beneficial hashing inner linear naturally suitable application big impact operate linear time linear package dataset regularize linear solve eq regression solve penalty elaborate major issue bit hashing scale bit hash step costly value permutation large concern bit hashing accord classification appear signature signature page still hardware improvement signature reflect expensive preprocesse change data user testing affect new incoming process signature computation graphical offer compare current parallelism access gpu many especially benefit gpu characteristic suit gpu main parallelism hashing typically hash minima speed applicable hashing mention online become increasingly web loading avoid overhead store large memory architecture demonstrate hashing also beneficial substantially load dominate training cost especially epoch overall hashing serve scheme text storage researcher develop loading e loading long g million hash basically matlab verify permutation order alone impractical resort simple hash universal hashing function randomness hash study report hashing reasonably hash reduction unless fairly limited memory beneficial reliably replace massive hash large storage difficulty perfect common standard universal u number choose increase randomness universal u hash storage store hash hash gram gram gram binary feature iterate section describe graphic processor sketch far hash suit implementation purpose recent primarily high cost offer increase computation graphic limit datum memory suitable consist number cycle processor execute locality consecutive access access gpu bit read set disk gpu compute minima gpu hash minima minima back process transfer block memory main within gpu consecutive gpu internal oppose access perform time access parallelism inherent gpu limited capacity even impractical store crucial simple hash function operation expensive trick simplicity note hash odd much integer evaluating compute p else integer evaluate bit trick v summarize datasets gb study pairwise combination product product expand dataset testing l gpu platform processor unit memory bandwidth gb intel processor overhead cpu implementation break load memory use function mod hash bit convert operation bit operation note report mod table preprocesse u loading take magnitude u hash hashing substantially operation avoid million feature permutation actually algebra hash storage impractical gpu processing demonstrate second second fold observe version gpu implementation fold cpu gpu preprocessing substantially load achieve preprocessing become h loading gpu gpu bit figure hash operation separate overhead spend memory gpu minima back memory leave panel adopt batch gpu reading disk gpu appropriately affect gpu report range cost affect reasonably e may practitioner spend gpu speed vary significantly spend order spend process gpu key cpu implementation resort hash hash practitioner validate hash impact hashing bit addition provide another bit hash hash experiment long hash estimate fully random permutation section list sparse dataset gb million choose conduct cpu ghz gb ram windows u scheme svm interval experiment repeat experiment whenever page experimental reporting validate conclusion accuracy average case correspond practice scheme slightly simple u hashing scheme appear well domain whereas hashing show variance provide hashing practitioner please randomness accuracy affect fully implementation hash meaning hashing early limit expand training experiment figure time exceed merely accuracy accuracy accuracy achieve accuracy next hashing algorithm influential hashing bit hashing conduct two large gb use hash train training gb svm logistic solid dash curve bit hash range accuracy value hash substantially storage substantially storage bit hashing compare bit plot bit select h hashing bit hashing see even bit hashing course
image average improvement almost summarize compare result bm average berkeley varied step bm train also outperform bm mlp mlp medium outperform art level good recent tend equally raise inherent denoise db outperform respectively capability patch noisy assign patch bound structure synthetic denoise bm already close rich geometric rich estimate expect yet db mlp rich assumption quality db db db db db bm db db db db db db mlp db htbp db bm db db db bm db value dependent db bm derive perform patch patch tight version theoretically achievable use db bm db bm bind test train bm summarize table noisy version image bm result note quite db relatively small agreement report much bm achieve par achieve bm mlp achieve visually improvement patch patch unable tight bound large size reduce clean bm bm bm use patch step effectively support patch see estimate bound step mean db patch bm difference achievable bm suggest quality achievable patch suggest bound lie db achieve noise level remain bm furthermore patch suffer return particularly become patch observation area figure statement denoise return available increasingly suggest patch denoise mostly flat area mlp area denoise corrupt independent imaging corrupt shoot denoise assume transform imaging handle db db db however difficult transform specifically design effectively denoise simulated noise effort adapt procedure general yield good image whereas also violate poor adjacent canonical mlp train column correlate image corrupt high bit result remove filter filter boundary bm vary db train million outperform middle problem except assume position pixel pixel truth filter db mlp quantization filtering transform dct transform thresholded bm value hyper procedure table mlp match patch omit match plain mlp achieve mlp plain mlp mlp mlp outperform mlp plain mlp large house approximately plain mlp db repeat find house far superior matching mlp ccccc image bm mlp bm db db db db db db db couple db db db db db hill db db house db db db db db db db htbp mlp plain average dataset large improvement dataset mlp db match plain mlp average htbp image berkeley dataset mlp match plain db image lot mlp db mlp explain fact patch blind surface average achieve mlp plain mlp plain surface block repeat matching allow outperform matching plain still achieve average matching learn feed architecture toolbox cpu add noise mlp run produce start load denoise noisy clean image perform I noisy take contain step achieve denoise test image image level perform house outperform bm approximately bm bm method repeat house advantage method low still db bm level exceed bound cluster addition suggest room improvement bm see approach image bayesian patch result superior assume medium yet method great improvement estimate patch infinite size progress toward reach approach reach half gain bm agree room improvement complex see merely achieve result quantization poisson two case seem competitive state art improve little complicated long useful kind addition plain vision clear block encounter worth specific knowledge engineering denoise mlp take approximately cpu gpu bm fast hour cpu architecture layer patch surprisingly explanation operating often box behavior understand extent bm db db advantage sometimes bad art happen lot regular attempt successful ask result combine denoise good plain mlp image outperform achieve theoretical toward gap approach extensively study poisson excellent well combine certain training inner mechanism layer denoise poisson clean count corrupt accord capture digital usually white procedure noisy image approach smooth part noisy rely wavelet conceptually image patch bm successful rely patch elsewhere bm similar image noisy patch noisy patch rely natural achieve bm procedure purely free noisy clean instance noise clean map complexity practice image noisy patch clean patch give patch patch affect quality denoise patch clean potential noisy patch patch invertible almost denoise noise clean explanation least denoise patch clean easily create patch patch denoise choice influence complicated high whereas difficulty capacity usually require model patch model bad potentially denoise possible state art perceptron mlp map patch onto capacity mlp mean sufficiently unit choose patch contain recover free agreement finding enough high capacity modern graphic base plain improve thorough poisson discuss strength work limit two hard limit make step reach third achieve thorough experiment remove extensively image diverse denoise aim smoothing class exploit patch statistic set berkeley exploit image involve wavelet coefficient patch learn neural network belong image image category type cnns various task digit traffic sign cnns exhibit design compare plain result amount layer potentially cnn think cnns kind report strong layer wavelet attempt incorporate auto auto neural pair add art result performance rather representation noise auto become standard deep apply patch learn however learn level correspond overview result base natural image art obtain assumption rely pure define denoise problem noisy patch patch different dataset mlp nonlinear via mlp hidden layer mlp operate wise mlp hidden layer layer mlp hide I hidden layer could imagine potentially train noisy image remove patch patch white feed noisy mlp represent patch mlp parameter update map patch clean patch error maximize backpropagation transform close pixel multiply weight use normalize uniform eq neurons output combine trick ensure part reach division divide modify rate discuss layer hide capacity layer need approximate function provide sufficient neuron compactly would require practice unit noisy overlap normalize patch subtract dividing multiply add place counterpart overlap region weight window patch third slide window patch decrease factor able approximately slow bm intensive operation mlp multiplication operation suit efficiently implement mlp gpu used factor core cpu speed factor allow image unlikely identical amount effectively could image exploit channel publication imagenet imagenet hierarchy image category transform grey scale define six algorithm set select select imagenet set use become diverse lot contain structure choose thorough well compete choose five image performance method choice train extract noise resemble mlp form detail result follow dictionary noisy noisy approximate gaussian novel image sometimes bm state denoise bm prior rather fact similarity rely block patch state exploit use approach bm also excellent choose bm usually least result achieve art introduction learning approach rely bm bm ccccc mlp db db db db db db db db db couple db db f db db db hill db house db db db db db db db db db db ccc bm mlp db clean bm mlp mlp denoise mlp architecture detail mlp denoise image clearly inferior bm house contain ideally suit like bm adapt noisy outperform image suit type note every image cc htbp approach bm five image imagenet dataset db berkeley db perhaps berkeley imagenet plausible imagenet contain regularity bm outperform bm dataset db berkeley db highlight row create mlp complicated structure mlp advantage bm comparison db improvement berkeley dataset db small improvement db initial train outperform art denoise consistent tend bm contain perform patch frequency blind frequency structure match bm patch image art noise level low high extremely describe architecture patch various level ccccc mlp db db db db db db db db couple db db db db db hill db db db house db db db db db db db db
goal good construct graph bp walk addition knowledge exception restrict class learn complete overview mention discuss present precede deal road scale real specification notice without adjustment propagation lead relate bethe heuristic new one organize review mainly base perturbation expansion develop refine ise explores refine expansion coupling expansion dual propagation complementary particularly graph give give gradient reduce tune field sake adapt standard induce solution inverse comparison make section consider spin historical mean resp maximum impose field multiplier associate entropy minimizer log likelihood lead ij note entropy distance empirical ise mean evaluate make transform impose expectation duality e yield solution connect matrix relation gibbs small coupling expansion correspond contribution expansion successive letting subscript expectation successive derivative eq expansion quantity eq convention approximation invert coupling require need mean field inverse vanishing coupling bethe inverse parametrization q relating give ise model neighbor notation neighbor give explicitly invert response result expression albeit suitable discuss ise appear hand side existence check directly give relation non vanish invert full equation nevertheless restrict way correspond construction simply bp equation heuristic proceed lead coincide introduction correspond ise temperature refer state physic introduced originally auto ise sum outer spin inference interpret axis introduce pattern axis coefficient highest plain bp bethe modal coincide attractive basically temperature topology four one factor conditionally tree yield recurrence notice tree tree possibly order term elementary building block coefficient notation generalize point relate basic eq along intersect case distinguish align intersect vertex corresponding lead leave aside introduce name continuous aim construct identify conditionally covariance variate precision matrix pair selection empirical derive easy conduct statistical goal discover conditionally penalize solve eq determinant regularization orient since positive definite thus definite construct norm know element cardinality matrix intuitive infeasible solve norm penalty practice use continuous solve exact optimum guarantee enough firstly square name lasso penalty encourage variable far basic norm allow entry weight constrain penalty diagonal avoid unnecessary introduce structure exact scalar see penalty play envelope norm convexity convex penalty sound also underlie corresponding regularization perform linearly increase result w issue penalty element suitable assumption bethe approximate check sparse either bethe direct graph belief fix belief satisfy constraint meaningful propagation unique allow approximated procedure amount select mutual find span link indirect invert potentially visible correlation indicate well weak assume equation iteratively propagation justify assertion like pairwise variable joint link optimally multiply loop good mutual read good maximum weight relevant classical inference dependence tree start want add produce corresponding since derivative attain correction log sort link quantity current yield distant target contingency table estimation mrf compare bit rapidly inaccurate link incremental propose construct indeed factor read pair precision reference form complete zero add variation give therefore determinant add modify way iff definite one ij ij individually deduce give finally fulfil degenerate desirable remove link desire remove factor assume precede likelihood read rearrange property observe compatibility able use strict connectivity precise condition validity sufficient definite aa modulus wish increase modify link assume sufficient impose express determinant ji ji simply impose track remove wise elementary link strong constraint easy compatible present remain equip define graph correspond link compatible wish repeat reach restrict several intermediate update covariance link coefficient select highest compatible respectively modification repeat convergence one absence simply walk satisfied implicitly slope want backtrack allow converge connectivity adapt dynamically adapt let carry toward connectivity concave pareto front boltzmann trivial point order yield order either bethe give explicitly empirically following consider individual expectation let ise fulfil ise correction implicitly set reference start bethe coupling model expansion expand assume follow follow free response bethe point gibbs energy moment bethe bethe coefficient log give quadratic contain hessian likelihood parameter else parameter associate convex find solve resort mutual partition decide global perturbation form discuss soon ultimately max parameter information read q interpretation direction exact fisher computed way edge link connect group link bethe approximation links information define coupling field consequence drop implicitly maintain impose reference suppose single temperature practice correlation capture belief modal average intra precede follow bethe equation proper counterpart inter cluster single intra cluster former bethe co formula yield superposition higher superposition along indicate natural field case obtain spin variable temperature regime coupling use correspond term spin configuration path closed express last run subgraph empty denote edge loop propagation correction generalize loop subgraph reduce loop cycle span loop convention edge superposition figure therefore weight dual primal lattice duality allow express j ise spin ht loop cycle combination case via disjoint cycle basis still basis link cycle link cycle read loop level constitute systematic take cycle general link cycle factor interaction cycle involve expect strong pairwise dual pairwise graph wise dual response read read arise part involve order precede solve eq contain restrict basic cycle cycle might break rapidly cycle start sized remain pass restrict exist cycle actually sign possibly proceed ordinary belief define corresponding message message eq update obtain cycle result belief function belief interact connect complete shown figure bipartite graph indeed case propose literature decomposition linear cycle far determination concerned cycle propagation precede suitable relevant proper might propagation weight possibly approximate graph edge share cycle dual denote edge dual factor cycle replace product formula read interest generalize expression weight message message factor read adapt expression generalize concern function unchanged generalize degree factor combinatorial burden come back write separate odd eq cycle extend cycle interaction concern lead derive share modify read factor represent dual factor order distinguish send cycle send cycle come weight dual partition read q measure cycle extend fix response ht extend care dual coupling see figure cycle cycle take last pseudo variable measure consider node absence free read involved example possible take formula give latter use joint present context exact weight form determination limitation possibly expression arbitrarily limit present doubly lagrange programming convex split introduce augment minimize function obtain proper initialization optimize jointly respect usually optimize alternatively give lagrangian successive result scale augment lagrangian log determinant programming inverse problem challenge differentiable make furthermore determinant feasible approximation definite determinant term give continuous domain speed procedure hereafter symmetric matrix last iteration auxiliary variable decompose mainly determinant part separate penalty single descent version penalize attack challenge cause exact make name approximate calculate explicitly order optimality unbiased ol estimator square order taylor penalty intrinsic relation explain perform definite impose accomplished perform calculate order optimality eigenvalue optimization clear guarantee contrast definite control margin increasingly iteration make converge gradually reduce margin definite sparsity constraint geometrically predefine upper optimizing reduce gradually simultaneously penalize square regression besides analytical compact functional clear intrinsic robust superior penalization make taylor expansion penalty within good initialization taylor read eq scalar constant solve w j wise weight zero configuration influence self pruning entry magnitude free unnecessary robust induce outli non magnitude design robust stable correlation fact weight expansion robust robustness entry compare step condition therefore cubic define entry cubic rapidly within besides decomposition multipli penalty pareto curve optimization different procedure converge validate penalization norm conditional jointly use cost search optimum avoid vector complexity improve efficiency solution learn name work compress equality penalty penalty design satisfie restrict isometry restrict isometry sometimes necessary norm likely inverse ise ab width cd various rmse plot finite zero b solution assess find actual center determine bp bethe propagation bethe precise field show use method conjunction bp bp bp co beliefs conjunction inversion yield absence field less local bp belief case field vanish bp compatibility approach propose build link schema mrf norm link axis version constraint explain compatibility marginally little maximum span schema bethe costly schema difficult validity bethe control instead favorable compute show datum produce simulator inference correspond traffic consist link extract large albeit simplified network paris link snapshot give link gaussian heavy tail take individually cdf center study actually mrf task display various one solution comparable greedy full pareto figure level
reconstruction e reconstruction solve clearly interestingly estimate almost generalized variance set correlated error generalization immediately multivariate dimension still residual addition improve incorporate nuisance parameter post quantification posteriori application robust formulation obtain portion derive formulation term model parametric posteriori formulation constitute portion student density give smoothly heavy near gaussian also kalman smoothing meta work fix show inverse treat nuisance parameter score estimate freedom residual q dimensional solve routine box source place vertical consist pick pick motivate inversion wave model receiver simply velocity ray velocity arrive ray geometry perturbation interest velocity ray operator transform medical imaging figure source inversion invert primary grid frequency output stage know avoid minima result starting recover important square approach design recover optimization reference cite reduce project project design arrive computed evaluating need update example represent weight inversion might possible find many involve primary significant calibration great inverse dynamic quantification overall optimization square carefully name idea entire corollary immediate ability modify fit nuisance practice c inversion variance dataset gauss cg calibration source bfgs match development present alternative derivation covariance interesting note estimate freedom fix match mass library knowledge degree freedom taylor inversion freedom view engineering discovery collaborative grant research project b cc residual allow home ignore cc lemma corollary nuisance direct primary parameter strategy projection nonlinear square idea broad ml map freedom nuisance large unconstrained inverse variety include optimal design improvement primary compatible minor modification primary parameter secondary degree freedom many example separable square extensively year class parametrize form class major insight reduce long note fast approach descent instance inverse provide detail include degree freedom automatic calibration robust paper section theory nuisance variance freedom formulation important detail see variance student formulation illustrate contaminate automatic illustrate image dependent weight application discuss conclusion easily rather work reduce follow adapt continuously differentiable continuously differentiable function gx differentiable condition exist smoothness hypothesis gx suggest natural design minimize whole consider method specifically derivative gauss type bfgs bfgs allow exploit project modify enter computation next constrain additional constraint normal minimizer x g first unconstraine systematically minimize problem avoid modify provide work formulate framework via forward reflect discrepancy variance affect true come source parametric student measurement freedom variance unknown nuisance estimate propose remainder provide source student estimate important drug approach gauss numerical importance parameter error share eq q modeling operator ml
concept name answer decide deterministic hybrid kb kb dl extend dl safe dl overcome limit dl safe keep scheme still semantic stable semantic predicate interpret predicate reduce analogously answer reduce statement show answer boolean problem currently expressive logic intersection lp lp inferential mechanism induction prominent however distinguish concept individual presence depend classify orthogonal dimension discrimination clause unit clause aim induce classification positive finding explain observation definite clause cover entail interpretation hypothesis cover concept traditionally view partially ii searching clause clause clause partial pair clause usefulness among chapter definite clause standardize apart program substitution c order syntactic admit structured hypothesis refinement function compute generalization search refinement refinement couple bounding clause iii concern syntactic clause search link link link link occur link link occur occur tight scheme attract attention build upon comparative attempt reader closely integrate arise many sort language dl refinement chapter discover frequent pattern dl safe rule rule focus discriminant induction interpretation hypothesis non head play coverage relation one interpretation hybrid generality relation hypothesis generalize procedure coverage relation relation processing enable rule hypothesis represent clause link range clause language name check constrain resolution coverage relation interpretation query answer oppose precisely variant mining pattern conceptual order description pattern unary refinement rule framework hypothesis inspire operator top bottom analogously differently assume coverage generality query answer reformulate show value dl induce dl head flexible integration dl cl safe evolution l kb kb rule hypothesis language recursive rule target interpretation interpretation description extension answer answer ref yes see concept change business trend usage consistent evolution preserve consistency distinguish conceptual specification representation consider conceptual due fact become consider role fact know need together already available crucial requirement follow definition induce concept role kb building rule cover part name kb adapt upon p mm mm r concern generation rule p predicate l admissible restriction head guarantee satisfied concept rule belong represent hypothesis contain tuple occur tuple incomplete reference hand need equip mapping example extended background theory note reduce answering kb reformulate cover nm reference example cover nm satisfy cover cover cover definition choose deal adapt characterization result generality reasoning derive standardize apart kb substitution say denote exist ground substitution variable immediately answer easily prove also quasi refinement top language build disjoint u intuitively semantic fulfil specialized note refinement rule apply mm produce mean refinement h h e onto previous definition rule classifier loop sequential covering learn remove covered rule space perform iteration increase instance view hypothesis begin specific general cover positive inner perform fine grain determine search conjunction hill begin rule promise step consider choose cover degree iff iff increase degree favor confidence cover degree take classified concept require deal unlabele outer start refine specialized cover hypothesis stop cover assume cover rule pose researcher cl community area acquisition bottleneck severe open demand task rule cost address relational framework match art research rule viewpoint suggestion framework implement intend specific section augment expressive thank evolution role operation kb comparative framework review common trivial semantic clause framework framework deal differently expressive dl cl language recursive integration scheme head requirement hypothesis take account course may turn case nature intend impact process rule differ greatly finally consider loose theoretical devoted concern expressive hybrid framework dl many web specify w work group theorem di e mail complement refer rule onto dl language support relational within logic programming database demand engineering automate though inductive machine programming relational adapt onto sake specific onto relational definition dl concept inductive logic programming rule language rule representation read sentence rule field logic lp database play architecture format activity perspective integration source scalable manner core standard standard extension unify expressive follow extension logic monotonic around top aside extend indeed rule mean frequently inference constraint discover new fact expressive see nm likely hybrid system integrate rule interest chapter shall specific language integration rule basis kb constitute vocabulary relate simple demand project produce approximately hour hour rule feasible partial task ml introduction even guarantee correct critical relational rule ml seem particularly promise reason ml lp recognize major approach distinguish form prior logical theory induction chapter proposal learn relational proposal difficulty program atom correct query answer ground query logical consequence answer denote consider advanced handle semantic program presence inherently conclusion alternative accept semantic extension semantic integration research system two deal kb reasoning motivation develop namely crucial semantic loose semantic kb cl share particular dl cl hybrid kb semantic requirement know respect property hybrid
regime graphical relaxation method separate model undirecte observe equivalence dag member technique conjunction standard refer norm matrix form row column position zero th row cardinality vector element diagonal product observe circle minimum cm draw fill black draw observed name observe name joint distribution know de imply exchangeable conditionally see illustration modeling latent occur total topic view generation realization represent topic word draw topic topic occurring draw encoding moreover allow basis framework variable moment recover natural degeneracy topic full degeneracy hope distinguish topic become moment word hope word unchanged correspondingly identifiable appropriately hope canonical follow transformation unchanged sign degeneracy second moment col identifiability col provide term condition topic proceed key establish expansion form otherwise sd ns degree term close necessary weak multiplicative property last generic assume mild perturb independently fix variable prove expansion word describe column column exhaustive search provide state need j represent word child row gap permutation word least column n iv correlation column differ definition set arbitrary section essentially novel traditionally lp inequality often fast projection iterative thresholding accord learn topic hide degeneracy assumption one bayesian model include topic topic bayesian incorporate causal direct acyclic graphs applicability intelligence social sciences structural economic parameterize specifically ss dag dag hide hide dag topic generality hide variance respectively moment variable convenient triple inner consider dag variable skewness identifiable prove skewness chi topic dag independent similar ica also dirichlet allocation lda suitably adjust need impose topic since directly proof skewness third moment observable column topological find pair uv sphere let sake direction vector power iteration describe orthonormal repeat framework combine statistic suffice moment I e accomplish word denote multi tensor hide dag observe provide remark oracle topological ordering need topological dag node result order dag suppose topological ordering induce dag node permutation dag identifiable framework identifiability topic expansion framework recall topic linearity exchangeable degenerate view word unlike topic single observe view random n uncorrelated model hide uncorrelated independent furthermore matrix non let matrix setting application sound source assume topic presence remove linear relationship linear structural progress identifiability paper contribution discuss remove noise fix third looking represent rank extract noise observe decomposition partition return c ab da fix return defer low diagonal diagonal low part I I u ib partition full show fixed let thin orthonormal column fix choose uniformly random put row submatrix fix full partitioning least denoise e recover noise matrix appropriate rank procedure section expansion generic coefficient matrix consist layer formally linear level within level node concern model specifically suppose induced condition hierarchical level induce model satisfie observe e entire node arbitrary relationship theorem section view third present proof emphasize validity general bring require reliably concrete example hierarchical model node require validate consider contain coefficient represent among construct accord gaussian I bernoulli entry indicate make expansion assume positive sake simplicity consider gap gap keep entry unchanged complexity algorithm similar select include chi equally denote chi slight variant make robust essentially er er present find partitioning row coefficient observe moment lead partition estimate coefficient partition due finite necessarily realization realization scaling estimating coefficient return remove ambiguity ambiguity since precision recall characterize support retrieve true edge retrieve summarize result ccccc recall recall sample around line far learn node hide level induce level relationship node observe triangular entry normal coefficient matrix describe relationship experiment gap maximum maximum row previous uniformly family chi describe summarize cccc recall acknowledgement david discussion support award nsf award award fa award nf complete microsoft research observe lead contradiction combination follow multiple linearly different canonical sign aa mm scale row n iv proof equivalent variable l observe aim notice inequality solution optimality claim b z z z twice older moreover put strict contradict optimality scale th ready hold least loop choose linearly hence let ai k lemma decompose part diagonal since pair part decompose low svd kn key determine singular singular sphere q rotation sphere wu ai demonstrate theorem expansion return column permutation inverse ai ordering dag direct triangular topological proceed column get way correspond dag exist exactly zero triangular version row row correspond parent remove node dag equivalently eliminate different triangular topological denote form associate hide write lemma decompose rank argument proof column inverse notice recover moment permutation column entire identifiable within contain fully submatrix ready hold every henceforth observe fully clear subset correspond submatrix nan space conclude let ic dense one dense vector consequently complete vector entry diagonal separately recover leave put correspond definition topological order topological order hide topological triangular matrix entry zero triangular require invertible invertible invertible identity distinct I ib ab thin indicator variable ie I moreover least complete proof chernoff independent topic pairwise correlation iterative column permutation let return denote l w l pt plus pt minus plus corollary proposition numerous statistic estimate broad topic linear order condition identifiability weak expansion constraint direct variable approach topic latent study recognize incorporate model latent datum effect far predict concept human affect abstract belief organize incorporate model incorporate incorporate set causal acyclic dag applicability intelligence sciences economics datum discovery hidden estimation dag suffer model criterion observe third dag roughly speak property influence identifiability dag probabilistic correspond observed model allocation topic dirichlet approach lda low order correlation g topic establish identifiability provable e word topic determine arbitrary impose expansion constraint word support different topic guarantee identifiability topic moment model identifiability degenerate bipartite graph impose word neighboring require intuitively number contrast note subset topic degeneracy identifiability necessary identifiability model degree identifiability expansion word span establish word strong expansion call parametric degeneracy learn topic matrix moment obey gaussian characterize topic however impose modeling via incorporate propagation show sparse graph topic model spectral propose correlate linear discuss rely blind deconvolution ica assume general impose linear view component cross moment topic diagonal prove relax condition strong incoherence condition set column full column decompose employ decomposition apply novel third order connectivity focus correctness plug address difficult reliably technique involve precise deal I matrix tensor future study propose year overview recently provable overview approach theoretical
visualize type cluster low cluster ht discuss later eigenvector could expensive algorithm appear first summarize experimentally second cluster modern business like company company replace present method separate object near method aim square intra distance contain index denote vector locally chapter repeat set point mean cluster toward true perform shape incorrectly perform apparent search pdf point datum end entry similarity induce complete object partition wish identify objective cut way guarantee simply minimize function cut perhaps trivial cut divide sum association normalization one bias pdf minimize np originally due relaxation eigenvector similarity matrix define n degree argue small relaxed assigning index eigenvector eigenvector one formal ht eigenvector eigenvector second small eigenvalue index computationally computation general drawback use method denote close neighbor spectral intel core ghz mb cache gb run intel core machine core mb cache run core mb cache gb run dataset run half tangent spectral accurate dataset cubic practical obstacle approximation develop laplacian popular cluster small representative spectral small remain matrix small submatrix basis nystr om remainder article discuss display numerical conclude finding fundamental behind approximate approach rely algorithmic step ensure sense wish compare computed spectral remainder course magnitude perturbation validate context laplacian perturb eigenvector respectively eq angle perturb projection analyze tradeoff efficiency theoretically correctly spectral mis certain nd eigenvectors perturbation eq yield perturbation et al major spectral preprocessing construct large identifies representative seem representative perform dataset close mean correspondence correspondence recover cluster representative step remain cost runtime course cubic choose enough quantify precisely tradeoff accuracy theory utilize mis mis fast norm demonstrate mis cluster control perturbation laplacian incur cluster small initially cluster name representative assign original dataset cluster closeness dms run cluster representative assignment find neighbor assign cluster majority many way representative spectral clustering utilize identify representative sample centroid coincide fast spectral randomly dataset reference assign proportional column accurate reasonably size cluster subsample similarity submatrix submatrix approximate motivation behind part wish similarity matrix represent relationship column compute nystr om eigenvector unfortunately eigenvector property cluster semidefinite eigenvector efficiently eigenvectors eigenvectors nystr om algorithm nm compute spectral similarity tuning guarantee fact yield worse empirically manually via nystr om nystr exact necessary eigenvector runtime spectral nystr om submatrix entire similarity aim select budget store remain entry enforce approximation index give accord efficiently budget budget denote large perturb laplacian respectively eigenvalue relation factor either eigenvector cluster note preserve closeness budget result run set clearly aim convention percentage centroid utilize experiment machine vary specification dataset machine experiment intel ghz machines mb gb intel core ghz mb cache gb tangent intel ghz machine mb cache gb value nystr tune table display depict rate fast decrease small representative likely portion get dataset spectral gaussian dataset assume perform nystr om fail sample budget perform bad error budget budget algorithm small nystr om sample size change increase substantially consist low might nystr om well problem cluster l om time half depict om ideal small time trend effective c nystr om error display cluster budget inaccurate identify l c nystr om error analog surprising result note perfectly large l c budget nystr om c range figure tangent depict exact second hour second approximation nystr om ht demonstrate dataset l budget om segmentation identification application company contain list age year company child child etc distinguish company stay company resource appropriately effort maintain expense analyze solve spectral method high difficult quantify long overcome challenge utilize historical teacher appropriate center education survey teacher survey entirely take former teacher take teacher survey question survey break interval status code child rating six recognize never child none school teacher cluster yield likely proportion tail cluster easily either teacher valuable teacher age give vary away old child close age tend term work apply yield mix leave child mix teacher contain run spectral express label cluster consist tailed support similar status l status cluster cc work predict group express child cluster yielded tail less first go display characteristic bad consider finding similar child child recommend neighbor bandwidth neighbor cluster neighbor near neighbor newly find cluster neighbor group neighbor reflect appear teacher statistically likely child
write whenever compute play role probability easy status q put together eq expression arrive immediately expression symmetric denote mn ip sg I mp mp mp j mp mp intuitively possess concerned really probability individual individual proposition proposition cause considerable confusion match match indeed come far chance focused quantity unique q odd similarly ratio evidence odd ratio differ case replace evidence database people determination really individual interested therefore odd retrieve see odd probability effectiveness select individual pe ps comprise individual come union contain separately odd size database odd effectiveness increase odd database large give match explain intuitively add many people unlikely match individual add database unique actually unique match decrease database match increase cf likelihood match increase great small database specify probability cf odd long match database always increase contain match unique match increase unique unique match decrease illustrate obtain example choose cast example provide one type perform database allele estimate allele obtain account discussion choice contain actual seem seven frequency reasonable start homogeneous frequency posterior equal effect heterogeneous population dna equally probably member member rise profile ratio ratio odd odd odd belong take relative everything else condition inverse odd get posterior odd belong careful whether take posterior serious finally odd calculate x illustrate obtain probability profile uncertainty parameter varied keep assume consequence know region country relatively past difficult give belong choice compare would ignore namely call naive fix summarize p pg notice greater naive considerable ss x ss sg much however course naive likely ss decrease grow considerable profile strongly probability let strongly choice population population whole exceed naive pg uniform situation three whose priori sc I meaning probability approximately set biased search note section odd decrease probable negligible also procedure become less raise odd unique population pc read monotonically go derive possess population exclude status since solution population situation quite suppose may individual base estimate database odd odd match six match obtain thus dna match search likely match happen ten double search match almost expand database put contain dna contain probability odd unique database minimal odd true database match rapidly grow slowly grow odd pc database match decrease idea database course database yield match database theorem thm thm example give account heterogeneous population correlation effect conduct return conditioning importance wants quantify weight evidence likelihood apply might relevant ratio prior discuss apply identification weight evidence rule couple trial people california match description estimate million match description return abstraction follow terminology population quite extensively detailed population alternatively effect stop generalization paper albeit somewhat article knowledge appear apart duality paper distinction purely mathematical view viewpoint elaborate issue text focus transform evidence posterior odd evidence reason likelihood view quantity unable allow odd already datum evidence background likelihood ratio regard background interested prove point background information issue invariant choice hypothesis purely view concentrate thing early suppose involve probability article case classical expert rest seem ratio prefer one address soon ratio correlation outline review biased protocol type dependencie strongly frequency homogeneous population addition ratio deal find match database include conclusion conclusion recognize text hope specify measure characteristic value characteristic number primarily interested often state call odd similar evidence use evidence way namely side odd odd background information via odd odd transform odd suppose evidence prior evidence evidence classical case uniformly regard member equally probably base however allow way independence frequency incorporate outcome sake ratio equal easily follow really matter viewpoint take likelihood prior clear uniform prior acceptable start subsection express expectation si k p n kp kp distribution e write posterior inverse intuitively equally expect therefore se understand way replacement careful supposed subsection situation suppose repeatedly find population detail question informally else explain else express characteristic count evidence belong characteristic knowing seem unlikely answer depend system distinction strength ideally strength evidence draw conclusion strength discuss expert look apparent likelihood prior probability seem generally admissible justify suffer irrespective must distinction account need expert pointing conceptually property distinction evidence point scenario find dna profile sense point expert procedure g profile could report reduce plausible alternative frequent prior whereas see would determine immediately clear expert knowledge difference variant prefer population reflect justify come focus candidate influence frequency population involve ip analogue check analogue probability ps ps eq weight one value matter whether interpret evidence preferable conditioning average factor weight distribution assume take independent variable whenever odd imply note denominator readily eq inverse concentrate classical population likely happen select extreme follow np also want variance equality remark
verification assignment trial assignment exist alg problem clause indeed include clause exactly keep assignment satisfie also exist one always decrease stop never evaluate algorithm initially assignment oracle trial computation query assignment every time compare solve algorithm formula fix algorithm query assignment medical input diagnostic ask find hidden help find relax requirement cluster formula satisfy within hard finding separates learn randomized trial satisfy assignment formula formula formula input try answer formula success algorithm smaller lower bind trial bind need trial eliminate process clause adapt decide clause show proof slight least satisfied drop half return combine chernoff clause clause assignment satisfy therefore principle clause force subset assignment cut small turn clause case family formula clause verification maintain triple contain triple formula randomized algorithm carefully clause divide clause assignment block clause block enforce assignment block block clause clause index randomized query satisfy part reveal position block two block denote verification imply candidate remove element also obtain query satisfying continue namely least triple either assignment unique thus output yet possibility formula claim unique namely assignment satisfy satisfie b ix ix initially query divide block set formula clause exactly clause clause tuple block satisfying clause assignment reduce give group precisely give group table multiplication report long open structure group version constraint unknown group verification apply restrict group natural attempt solve keep propose consistent whenever add restriction three rule avoid triple computation polynomial trial arise oracle unknown triple exploit table surprisingly hardness stand runtime solve time group hardness shown employ hamiltonian hamiltonian cycle size hamiltonian cycle hardness hamiltonian achievable cyclic shift label hamiltonian denote group cyclic generator translate cycle immediately arise time hamiltonian cycle thus construct multiplication around employ come interaction verification answer require idea efficiently something simulator correctness correct hamiltonian cycle wrong hamiltonian control long care code vertex ask hamiltonian hamiltonian arbitrary way hamiltonian first cycle pick arbitrary cyclic shift group define multiplication module cyclic basically run mapping identify verification table simulator verification correct find cycle suppose cycle shift cycle group multiplication make verification oracle simulate answer answer hamiltonian x hamiltonian cycle several define input hamiltonian time define run trial whenever simulate verification raise design verification oracle really map roughly return outer statement eq emphasize multiplication group next vertex hamiltonian cycle roughly non whole return output output clique cost take query time non claim time immediate give language easily strong include elementary completeness inclusion formula next algorithm statement completeness perfect completeness protocol round complete basically mention paragraph problem come query protocol public coin protocol deterministic send know pair graph currently oracle suppose follow answer polynomial verification decide check answer output perfect payoff utility probability mix equilibrium admit mixed nash polynomial player equilibria number nash normal version function mixed strategy equilibrium player violate trial fast find centralize quite investigated focus dynamic nash game nash equilibrium ellipsoid see matrix nash equilibrium verification oracle return separate claim ellipsoid problem ellipsoid handle perturbation positive volume explicitly employ develop gr formal theorem totally payoff matrix oracle see ellipsoid problem separation consider game respect utility verification nash unknown verification nash matrix without verification return player nash equilibrium entry serve separate gr ellipsoid solve problem separation whether single separation oracle argue strong oracle thank oracle solving identify two utility find find nash comment volume ellipsoid volume handle applicable apply find contain fortunately interestingly verification fit briefly feasible verification ask cover entire center ellipsoid violate verification establish ellipsoid remarkable ellipsoid know violate oracle suffice continue ellipsoid must system solution positive ellipsoid get polynomial conclude dimensional lie extra constraint reduce solve hyperplane solve gr ellipsoid ellipsoid must thin direction short parallel ellipsoid hyperplane hyperplane round number rational number size hyperplane although claim apply game approach obtain note hope nash potential game strategy game algorithm however game admit graphical nash polynomial even aware game efficiently achieve remain verification internet equilibrium identify similar game competitive equilibrium market omit share agent basically connect function classic cost function special case bound rational present numerator denominator violate set precisely input additive unique solution core implication able verification determine trial core sharing inequality cs core subset violated handle unknown get strong oracle describe apply ellipsoid conclude version u propose verification trial instance integer integer always large unique partition solution query oracle big size derive query set sum subset know bad query investigate input lack level polynomially viewpoint meanwhile leave gap examine obvious specific hidden phenomenon work try computational complexity general framework systematic complexity trial tradeoff complexity note argument entropy directly allow pair comparison polynomial note would probably also polynomial deterministic polynomially deterministic complexity polynomial computation verification consider verification ask problem acknowledgment grateful comment point cm thm fact chen zhang nature thing motivated economic computer investigation model search problem goal find adaptively confirm valid solution return violate trial complexity summarize follow despite little provide efficient nash unknown version input considerably theory ellipsoid strong complexity substantially input exist graph simulator computationally lack difficulty algebraic structure exhibit serve supplement computation computer particular focus understand power limit central computing explicitly game usually explicitly circumstance player payoff explore new environment know alone instance business company lack customer sided preference precise example job market job market know precisely like information job company mistake systematic fitting organization event diagnostic serious search collection e way otherwise reaction severe everything composition diagnostic largely effectively could try dna long identify diagnostic input lack input database acquisition extensively speak proceed adaptively accomplished otherwise object goal one consideration trial also approach economic adopt adjust self converge individual involve position management encourage enhance conduct clinical side effect observe analyze feedback future diagnostic ingredient trial employ error propose future formally algorithmic perspective aim investigate broad input investigate lack viewpoint trial question input numerous arise variety application study nash multi search assignment formula individual preference side addition motivate early broad space domain solution define constraint polynomial exponential infinite intensive theoretical artificial intelligence research exploration practical paper input input contain preference list list task namely propose chemical trial candidate one violate return return make complexity study drug verification carry thus truly reveal violate exact constraint I player motivate surprisingly assumption verification efficient verification newly historical return difficulty compare input overall verification oracle invoke either oracle unit unknown input input computational complexity complexity class verification oracle occur extra translate omit become query need standard latter assume free verification query deterministic trial complexity u investigate trial rigorous hardness trial note diagnostic development expense motivate goal protocol trial time complexity lack deferred section equilibrium side preference satisfy also ellipsoid target contain input point core contain volume applicable unknown sophisticated work turn verification crucially use nash structure medical student practical significance complexity trial clause min principle insufficient query instance distinguish employ characterize partial case height correct bind speed examine another whose completely somewhat surprising knowing violate admit input add extra difficulty version considerably version whether hardness u group know computation oracle polynomial compare multiplication multiplication table difficulty trial phenomenon hardness classic complete hamiltonian cycle hamiltonian cycle existence completeness group find cycle verification know issue crucial know design verification perfectly question however favorable incorrect hamiltonian may already use find hamiltonian cycle certain low variable efficient u completely resort nash equilibrium I existence yield illustrate complexity lack imply distinct specific level elimination exist learn combinatorial structure shrink space e explore relation space design unknown environment ellipsoid model model strong connection fundamental difference essence identify unknown object concept sometimes quite attempt solution aforementioned development diagnostic exact input learn case may impossible learn relax requirement formula assignment input finding contrast exact specific application largely available scenario verification situation orient advantage supplement exist context object unknown verification detailed relationship relevant concept task survey give collection row concept row query column answer element map equivalence query cast membership equivalence query framework search class let solution query thus propose solution hide membership gain extra violate necessarily efficient satisfie become equivalence propose correspond set induce table know e need put possible constraint would identity column position identify merely row satisfie hide difference problem however unique lie concept measure exponential learning oracle instance unlabele set iterative give interact fundamentally oracle correct answer label constraint achieve accuracy prediction unlabele use label instance objective pac probably correct pac domain instance concept sample complexity objective approximate concept classic pac although allow rather solution many learning reinforcement reference therein discussion level sample different trial solution address environment exploration explore entire planning objective desire path specific location either space reader refer topic feature oppose robot planning look difficult scenario underlie geometric structure ellipsoid trial search g separate return ellipsoid elegant polynomial expression trial model much pure perspective ellipsoid trial indeed herein crucially ellipsoid base trial time complexity ellipsoid calculation u program make function query input allow view extension traditional query query instance proof interact although clearly would sided market complete preference simplicity exposition list prefer contain namely match label match computed acceptance list propose candidate solution verification oracle pair reveal first underlie result unknown another set unknown discover order indeed verification return otherwise return sort complexity u actually u solve use run make meanwhile solve u trial upper fast order distinction standard comparison technique straightforwardly carry turn measure allow height sufficient degree randomize relation set whenever element denote uniformly among linear extension order thus form order
learn sometimes partial processing device additional shot useful characterization hamiltonian hamiltonian estimating post process reference therein outcome measurement broad carlo bayesian design wide also measurement adaptive quantum idea utilize quantum dynamical improve hamiltonian suitable parameterization hamiltonian estimating purpose hamiltonian expect high concept hamiltonian organize formalism bayesian discuss rao carlo section discuss numerical key element quantum theory statistic interest suppose ultimately achieve encode batch control process offline collect processing choose control past experiment formalize various way natural future control via remainder notation subscript expectation experimental usefulness quantify yield motivated optimize two choice discuss detail outcome choice subsequent utility scientific maximize value distribution utility minimize minimize protocol overview show feedback iteration block background set collect produce evaluate experiment design interest analyze particular rao incur methodology design average explicitly minimize bayes case risk mse minimize maximize bayes question achievable variant experiment standard loss rao bind singular singular fisher information arise unfortunately assumption meet avoid integration iterative algorithm various interest mean rao give give space specifically space carlo back wide scientific application name monte particle importance filter smc behind bayes update rule difficult evaluation costly issue approximate approximate calculate ensure always simplify arbitrarily particle every feed support particle carry distribution without generality initial make particle approximation accord particle location covariance particle weight particle function particle location update unnormalized normalize weight return monte require careful effort introduce error limit first weight effectively fewer use amount resource eventually techniques generally adaptively change location simple choose location shall effective ratio threshold resample explicitly behind incorporate randomness volume parameter apply particle thus particle spread formally location come particle sample perturbation resample particle resample use suggest covariance q involve particle particle location update perturb particle address regard thing since algorithm tailor say something issue simulation quantum interact thus call wish minimize time involve high particle appear utility reduce reduce ideal since routine call average idea formalize particle location semi utility dc find weight scalar u dd np h dp dd particle location particle element introduce sort combine complete adaptively design smc conjugate conjugate nan optimizer evaluate choice specify choice experiment control variable experiment optimize I pick experiment highest well carry provide able task maximize region algorithm describe quantum design point volume lie run record record credible record region speak credible confidence credible obtain kind region explore credible estimate amenable turn particular region compute region least intuition function region large current smc state achieve property choose geometric hull minimum satisfy collect reasonable approximately normally accord information describe mass inside parameter transform variate invert generally normally posterior satisfying particle good estimate covariance matrix dimensional normal meet cdf actual rao use hamiltonian proceed generalization address consistent experiment hyperparameter describe hamiltonian hyperparameter integrate consider realization denote hamiltonian avoid subtle conceptual difference hyperparameter way yield convert estimate drawback likelihood function need ensemble experimental take large may require error approach straight gaussian dynamical hyperparameter develop region hyperparameter obtain region attention assess fluctuation performance especially unknown develop understand either discuss significantly figure build intuition movement note experiment sec describe center likelihood dot red randomly blue measurement monte particle see line indicate indicate initial rao error indicate curve numerical precision choose experimental occur averaged randomly choose figure error see remain optimization whereas sufficient yielded optimally particle smc scale know previous contain guess per high performance introduce new guess choose guess little heuristic reduce possible pick show small approximation cause relatively large value take guess shall provide benefit trial region set surprising confidence vary skewed provide uninformative force informative room search informative guess heuristic heuristic poor guess optimize also median square tend trial randomly bad optimization median substantial contribution true describe simplify make approximately justify yet find consider appear model distribution expect within volume use within choose standard deviation case approximate provide favor region improve collect experiment insufficient experiment model mass averaged guess heuristic choose guess normal challenge calculation distribution use figure absence improve even characterize much contrast qualitative continue increase implie guess heuristic unlikely informative fix landscape sufficiently find mse guess suggest landscape local optimization guess find substantially substantially subtle mse estimate mse three quantity suggest also mse expect optimize model section particle without optimization dash trial single initial guess use experiment indicate solid experiment average trial guess trial region trial guess solid guess trial datum indicate region unknown initial indicate dash line correspond local solid line trial optimize average trial error curve estimate measurement sequential numerical dotted particle mean solid numerically variance quadratic loss region algorithm presence hamiltonian hyperparameter parameter volume mass hyperparameter recall admit provide hamiltonian score gaussian probability lie agreement within estimate probability yield mass unlike ratio tail non tail perhaps surprisingly interest calculate compare experiment grow see infer hyperparameter systematically estimate bias vanish conclude hyperparameter unknown particle another assess infer hamiltonian learn call make previous minimized choosing show sophisticated reduce experimental classical tradeoff increasingly quantum grow quantum particle cost asymptotically perform experiment computational rather experimental time relative heuristic number call call second computer lead computational hz utility lose rate approximately comparable need guess yield experiment although cause cause intersect curve surface seem indicate expensive heuristic may get well percentile strategy continue improved regime consider choose mse suggest approximation ratio become stable require em choose incur show right percentile great show require trivially little communication additionally improve circumstance quantum simulation expensive graphical processing unit field gate array latter device processing provide applie learn parameter collect infer hamiltonian processing store recover even advantage hamiltonian parameter
also show extended perspective noise threshold solve subproblem minimizer coincide minimizer question v point use newton solve appropriately strategy evaluate simply detailed function linearity violate nonetheless solves estimate fairly quickly approximation experiment iteration weight verify computational iteration find framework toolbox purpose velocity perturbation grid frequency hz composite lead procedure outline run multiply dividing therefore normalize source reconstruction reference show source signature outline paper depict normalize depict source convergence propose nuisance regularize problem source imaging draw projection nuisance exist solver experiment demonstrate wavelet successfully figure wavelet figure nonetheless lot effort devote develop solve difficult come always pick solve several iteration exactly obtain c red blue source weight show processing recover give auxiliary signature unfortunately practice signal technique present novel methodology auxiliary propose projection contribution combine projection deconvolution imaging image sparse prove inverse sparse decay data measurement compressive obtain guarantee general inverse guarantee regularization particularly useful linearize several placing reflect array datum represent fouri record series frequency operator record velocity measurement typically unknown wavelet bilinear perturbation frame
aggregation yield energy however agreement energy aggregate together interpolation agreement aggregate coarse hard fine variable influence mm estimate generate relatively label locally temperature perform random variable determine set coarse interpolation agreement coarse select agreement coarse sequentially start add yet small result coarse index fine leave throughout energy compute interpolation construct pyramid consecutive level less energy solution coarse fine fine serve initialization step interpolation refinement coarse energy aware interpolation role multiscale make interpolation coarse sense discrete gauss multiscale multiscale cluster energy compare performance energy result close well show percent energy close pair hard percent baseline art b energy multiscale synthetic energy connect grid label unary term ij pair wise energy optimize see result energy contrast average coarse fine optimization energy single cluster matching frame co minimization adaptively adjust energy regular grid energy compare discrete multiscale tailor relaxation improve art challenge energy energy also algebraic derivation find acknowledgment remark discussion thank rgb rgb rgb rgb rgb art come contrast energy suggest multiscale derive algebraic us pair interpolation principled algebraic propose aware efficiently multiscale landscape energy coarse fine optimization energy improvement method energy weight edge discrete preserve assign label preserve energy energie large dd preserve energy yield energy poor contrast scale energy suggest derivation novel yield substitution vector yield scale wish map fine assignment fine approximated coarse assignment write coarse parameterize fine
good strategy half iterate across method weight iterate outperform also optimize objective continuous present arguably analyze trick use technique last half average propose epoch gd geometrically sized average scheme whereas scheme general iterate iterate scheme therein strongly size finite like update see fw hz w lipschitz respect svm setup regularity b obtain show prove induction subgradient w z w first base hypothesis yield w complete plus minus plus minus pc team sup paris france mm lemma technique weight iterate easy compare exist strong constant follow scenario project subgradient precisely field measurable subgradient unique support pair identically fw w lipschitz unconstraine w show appendix use method x standard proof w f thus w fw fw line sum inequality similar term stay sum q weight averaging scheme implement average uniform average cm bottom view colour empirical benchmark set website causality website website dense standardize regularization although sensitive whole strategy average iterate iterate average since average w weight
sec low glasso mse ff thm act number term frobenius estimate mse guide regularization relate publication li mse consider compare li order weak obtain tight outline report introduce notation throughout report introduce limit mse ff present superiority glasso sparse kronecker present empirically I j define tp f symmetric let submatrix j fs set psd definite ds close simply close cone nc constants number kronecker gaussian likelihood belong minimization definite unique iterative block coordinate descent develop zero well log jointly motivate flip adapt q p flip compute sparsity imply flip lead factor proportion definite satisfy nj k glasso computational glasso section provide prove formulate objective assume argument assume definite subproblem consider eq subproblem follow maximize similar nj I subproblem converge define converge see consider set fold product optimization without vector q k reader appendix interest positive automatically take care assume descent mm descent logarithmic barrier limit minimize minima assume f descent theorem converge objective flip ff achieve optimal statistical thm establish propose improve mse absolute b ff flip flip condition accurate covariance kronecker minimal error ff three ff quickly specify fast computational ff kronecker ff simultaneously achieve ht reflect visually ff consistency establish compression constitute satisfy nk f offer strict generalize thm exploit kronecker structure significantly low estimation ff kronecker need thm inverse hold I inverse three iteration although thm matrix hold ball blue ff rate ff glasso note glasso algorithm would yield inferior ht magnitude constant plot reflect error minimal depict fig region mse go grow rate control grow accurate ff glasso naive ht establish deviation n pf pf w establish chernoff diagonal tight dependence convergence use lemma respect norm section validate rate section iteration penalize step glasso algorithm glasso monte simulation use positive definite base enyi square assessment frobenius estimate calculate trial regularization parameter select nf constant front regularization experimentally performance show perturbation figure compare precision outperform glasso ff ff algorithm suffer outperform ff regime since kronecker htp kronecker lasso ff flip glasso fig lasso flip glasso dense similar see nonzero matrix glasso rmse expect glasso ff ff suffer sample regime outperform ff regime exploit kronecker db reduction covariance ff htp factor matrix nonzero place place ff account sparsity structure ff ff estimate fair comparison ff precision db matrix ff rmse instead ff db reduction reduction ff approximately significant htp center around rmse ff ff bar around ff ff remark reduce address large mse curve intersection ff mse match mse well rate case thm ff illustrate predict top curve htp identity thm shown predict convergent ff kronecker glasso exploit kronecker structure sparsity derive glasso ff validate ff exploit kronecker increase thresholded ff account kronecker thank deviation report let triangle eq sum argument I j equality achieve minimax invoke kkt dual use observe ij evident equivalent verify saddle formulation duality trivially follow positive glasso simple combine kronecker product establish optimize definite penalize flip objective kronecker low primal mle bound sample convergence monotonically iterate glasso kronecker glasso review definition subdifferential reflect geometry proper saddle lemma continuously continuous subset j jx jx bound e convex block hold j ij optimal primal j hold exercise c sum define k imply important ii apply every j convergent ji j j subsequence continuity yield j j show j j critical rule maxima rigorously exist maintain without generality contradict maxima maxima point minimum j contradiction saddle exist nonempty descent jj strict descent induction iteration fm symmetric j denote operator define sequence permutation e k j add I formula method bind u f good infinity constant k tail difference tail bound markov property p lp f inside ty lt elementary mt note mu mu u mu plug tail taylor mt mt proceeding conclude np random n nc follow modification thm due present iteration flip assumption growth sample matrix intermediate max define c probability let cauchy schwarz inequality expand c together union absolute kronecker product precision ff
universit paris de paris optimize extension quantization dissimilarity method refinement non numerous need dissimilarity dissimilarity observation datum neighbor usual hierarchical technique suffer three difficulty firstly classical linkage global quality naive implementation scale large greedy technique suboptimal order efficient prototype dissimilarity technique linkage quantization dissimilarity come cluster suboptimal equip di di di partition measure thank size see favor cluster intermediate arise twice dissimilarity merging increase linkage generalize naive proceed merge small linkage denote compose partition c recurrence devote agglomerative old complexity demonstrate author benchmark linkage previous efficiency close pair linkage naive maintain queue cluster candidate store sort cluster true neighbor cluster small candidate low merge queue long near neighbor computation search remain cluster calculation fast important drawback minima hierarchical heuristic previous section single refinement operate time operate level intermediate move individual fact local minima heuristic dense item happen individual move measure avoid error operate dendrogram multiply level upon consider size analyst give idea apply consider move entire update implementation heuristic give near graph dissimilarity linkage approach hierarchical proposal dissimilarity rely quantization dissimilarity phase method dissimilarity cat database g relational standard naive implementation describe start configuration choose approximately computer implementation cluster cat solid dash cat relational correspond configuration hierarchical quantization hierarchical bottom multi refinement achieve refinement
function bayesian key posterior arbitrary criterion unobserved process gp gp unobserved point normal posterior selection criterion selection optimize area area promise maximum expect improvement ei successful example focus sequentially selection reveal mild leverage distinct phase phase exploitation bandit exploitation introduce select eliminate maximizer hence try algorithm point close benchmark outperform ei art ei whether exploration boost ei help observe ei exploration eliminate region ei select motivate introduce insight aspect experimental section motivate key ei ei hence ei represent deviation pdf respectively observe evaluation variance ei widely study concern balance ei concern though asymptotic ei condition ei try information potentially request lot optimum experiment attempt literature vary degree briefly discuss definition ei researcher make ei replace much success show begin make little difference ei surrogate objective try improve exploit decrease area exploration method change setup impossible proposal exploration ei ei method budget select ei follow ei sample investigation understand ei ei explore necessary introduce section reveal never help ei since regret monotonically hence skewness c literature empirical investigation ei know operate achieve ei answer able exploration exploitation ei especially query budget exploitation ei exploration observe output r leave region product lipschitz hope infinitely strong optimize change radius point sphere assumption violate satisfie hoeffde replace high try volume boundary might case sphere sphere overlap point select pick sphere pseudo next value might negative happens far prevent sure mean exploration need sure affect e pick small start gradually side limited sample implement volume point inside point fall previously newly explore newly explore deterministic direct direct optimize lipschitz inside might slow inaccurate describe exploitation gain optimal suppose explored find would whose small small enough mean similarly replace entail code maximum constant observe output qx r nothing algorithm exploitation gp unknown ei rely lipschitz exploration gp optimize scale deal exploitation phase case complex enough budget width fit budget take width exploitation ei achieve role lipschitz hence budget achieve certain chance sphere well choose true safe derivative long prevent become certain function look cc cell v ij constant x ei scenario different six synthetic function mathematical shown use real contour benchmark output cell adjust output output cell sample benchmark maximize production ph growth medium benchmark except sake case budget exploitation ei ei ei first like different ei ei sample budget ei version ei ei instead take infinity behavior posterior ei light interested used exploitation ei exploration help ei set ei measure benchmark estimate ei benchmark exploration ei benchmark ei consistently benchmark ei ei slightly performance due ei width also note ei improvement ei ei selection em cell like exploitation exploration fail use ei summarize follow regret
probability broad plausible sampling total span curve amplitude use gamma prior scale time I gamma decay duration long assign gaussian equal former calculate loo cv likelihood robust prior nm light curve likelihood value second row use em describe apply describe model deviation uncertainty loo cv look first difference loo cv construction low two model inspection pdfs partition peak amplitude partition relatively broad amplitude phase peak half partition broad inspection extra offset offset reflect posterior mean zero confident relative evidence non formally true term explain reduce loo large relative stochastic explain stochastic shape deviation essentially bar top large happen model component likelihood deviation accurately miss whether require light receive low star variability light reveal variability occur hour star rest integrate light spatially otherwise rotation period observable variability magnitude light curve year apart curve previously hereafter hour long curve use nan mean calculate curve light long widely context reference therein term observe e reject nan accept implicit final generally per se tell important adequate data substitute assessment compare five use although negligible finite duration integration minute section top approximate delta calculation analytic calculation calculate process l nm column curve e ten variability inconsistent test reject p metric sometimes correctly limit know motivate converse indicate fitting identify good curve difference offset need surprising light eight light loo significantly gaussian variability error difference high model therefore none light light light discriminate turn confident seven significantly bad discount interesting light year apart explanation behaviour different coverage interpret forget summarize base loo likelihood curve much model exception equally plausible ten light curve describe model require without seem light curve next remain seven clear winner three seven explain datum four light curve plausible large gaussian variance light curve well bar something plausible multiple stage loo event sample pdfs per examine posterior curve associate nm curve line loo sensitive change loo cv report arrive loo cv likelihood well likely likely examine prior unity change something loo evidence observe behaviour light prior factor ten nonetheless remain issue always light setting cf prior canonical article probabilistic method temporal accommodate stochastic variation many series axis temporal offer total error average partition average sensitive pdfs evidence something confirm take series sample theoretical recurrence derive elsewhere main purpose exposition comprehensive series analysis elsewhere present demonstrate main curve nan variability fluctuation deviation bar well might interpret confidence theoretical curve stochastic probable long term trend periodic posterior early significant curve describe stochastic component component comment remain plausible model exist explain focus practical problem inclusion loo sensitivity comment interested answer question explain prediction probabilistic modelling logical introduce comparison principle measurement signal uncertainty generally write could combine variation top
nb learn significantly beta usage positively use mark nb diverse responsible example large mean confirm beta beta nb across usage might helpful model alone experiment direct yield result zero well explain beta ibp gamma nb hdp viewpoint fix natural count viewpoint result confirm importance together examine adjustment variation potentially fit tb share evident nb compare infer smooth document topic number value convenient variety negative binomial nb process count group propose process completely borel set oppose hierarchical hdp disjoint borel discover compound representation mixture derivation demonstrate nb share nb dispersion hdp structural computational advantage distinct sharing mechanism connection result show dispersion acknowledgment program summation independent bernoulli become sm poisson vocabulary topic th topic crf hdp model crf restaurant table restaurant infeasible sample approximately result enough sample implementation convenience parameter experiment result obviously lower hold thus augmentation one allow infer dispersion eq hdp special gamma nb nb gibbs construct express gamma express thm thm electrical computer university nc develop augmentation binomial disjoint count fundamental derive efficient gibbs unique nb sharing model exist showing infer nb interest count binomial nb nb originally model hdp substantial suffer nb mixture distinct united fitting construction neither explore inference concern nb dispersion infer metropolis hasting paper fail connection nb hdp compare perform joint simple construct computation nb process count property inference construct gamma nb analyze hdp structural advantage hdp nb distinct select completely random modeling typical nb dispersion introduce nb distinct united poisson denote borel give group partition jointly variable measure expression poisson independent count model measurable conditioning normalize prior continuous process dp gamma representation construct equal dispersion assumption count model construction poisson gamma proportion treat motivate layer nb compound mass pmf r p rp rp investigate numerous scientific nb augment p scale compound r p logarithmic abuse add place lemma since gamma conjugacy assume use repeatedly suggest nb process within conditional analytical pmf table customer chinese restaurant concentration chinese restaurant count pmf nr poisson compound key count modeling examine understand fundamental nb nonparametric bayesian number compound l pl pmf summation random w sm sm sm r generate compound lead compound sharing nb dispersion nb count gamma gap represent poisson gamma gamma three inference nb conjugacy ga x ta ga lemma base atom discrete gamma conjugacy exchangeable conditioning equivalence gamma denote hdp normalize hdp hdp gamma nb theoretically disjoint practically nb exploit conjugacy latent inference hdp alternative construction crf stick analytical posterior nontrivial simply crf apply method practical base biased gamma whether shape govern continuous equation nb variation gamma define evy cp dp cb construct jk construction nb nb dispersion nb nb share beta process mark variable thus process k x j jk jk conjugate tractable zero nb process introduce bernoulli process b jk jk jk link inference tractable link ibp beta spike nb parametric show variety differ dispersion share deeply model imply typical group word six differently construct process factor nb tuning document automatically learn topic strength document close gibbs inference even lda topic model nb count framework set hdp normalization hdp construct whose nb introduce process process gamma focused topic ibp compound dp nevertheless likelihood improve allow well nb explore share hdp allow hdp share beta nb explore probability group improve binomial mark distinction update construction nb process table poisson th loading sum score link nonnegative factorization factor proportion count mixture jk corpus typical relate nb hdp hdp nb crf hdp nb modeling variety crf fair comparison implement base atom nb lda nb parameter proportion topic toolbox last sample nb topic measure vocabulary smoothing location ji ji v uci edu toolbox restrict occur corpus term randomly learn dependent term f
relate fitness agent evolve capability learn large lead add bit agent create next accumulate generation median evaluation generation half q small take fitness evaluation cost capability small thesis evolutionary start pre encode gain evolutionary machine learn beyond learn boundary evolutionary possibility ai development ai recursively ai span thereby intelligence evolution cost learn principle valuable suggestion access vast people build present thesis goal achieve complex evolutionary create require thesis efficient go learn blind search environment give inefficient capability work evolutionary learn goal complex need order discuss limitation imply machine study supervised pass phase example field example propagation machine probably computational discover pattern achieve machine study unsupervise learn structure study unknown machine receive environment external agent henceforth goal sensor body achieve goal environment capability specification thesis present goal capability identical machine processing goal agent run platform process incoming control alphabet agent agent modify denote number start memory emphasize use expression size finite use memory arbitrary thesis state achieve run platform control agent body capability state physical pair physical body platform let agent achieve agent achieve goal sequence path energy environmental time possibly agent repeat randomness amount execute achieve thesis limit computational period result dependence thesis present theoretically speed physical computing device energy achieve know achieve phrase agent repeat trial identical start agent achieve learn effectively even achieve start well cost vary repeat identical agent effectively effectively achieve trial agent goal body unbounded agent would large environmental environment integer say grow unbounded phrase short effectively multiple discuss allow arbitrary agent must environment call henceforth able achieve goal string evolutionary bring high fitness landscape phrase efficiently define learn evolutionary information evolutionary change pre calculate fitness benefit target change environment physical case probability success description string exponentially principle state agent string denote agent would inefficient change miss learn value efficiently learn extend agent state efficiently learn g environmental efficient evolutionary cost agent difference ham efficiently principle bit agent zero us denote median
markov chain type delay draw mean model run aid clarity combine rapidly peak decay slowly production peak monotonically know possible reach unlikely rare logic ii plot equation iterate random trace need space value sufficient reliable initial convergence parameter path quality course clearly subsequent note rare assume variance true suggest importance sampling use model address probabilistic checking construct rather representation smc intractable exhaustive smc offer pose monte simulation reduce rare construction exploration importance provide optimum significant applicable kind space system biology methodology importance sampling e probability markov restrict logic precise sampling automatic description syntactic describe produce robustness chapter adapt improved exponential growth state give confidence rare challenge objective circumstance level establish maintain check find motivated simple contrast low avoid transition magnitude improvement simulation efficacy methodology reliability engineering provide prediction behaviour complex system increasingly become compact system becoming increasingly create correspondingly correctly user expect high ensure correctness system testing outcome highlight discover incorporate sophisticated despite testing cause fact unlikely cover mode remain series test check formal system logic scenario boolean answer checking quantify numerical alternatively offer accurate explore state apply protocol precision satisfy property interest exponential limit applicability universe circumstance system impossible compositional reasoning reduction size heterogeneity infeasible also successful match precision quantitative logic ever solve simulation approach become increasingly availability hardware check simplicity checking execution trace individually whether variable advanced technique combine whether satisfie know confidence tight checking offer quantify complex system evidence check checking establish statistical checking cope large key challenge cpu trace cpu core grids purpose processor production despite model necessary rare unlikely problem base since probability rare property temporal rare state rare mathematical applicability work iterative refinement must produce trace present arise syntactic constitute low comparison unique optimum highlight area future error rare bounding formulae chernoff hoeffding property become error useful monte unbounded increase mean importance simulate rare simulating monte originally material sampling good sampling I must simulation must paths use term zero importance sampling fact satisfy estimator meet necessarily address ii scheme fall broad state state dependent sampling individually transition probability former greatest infeasible tractable kind potentially affect differently function indicate path property trace monte carlo random absolutely eq ratio importance goal rare result importance rare property frequently optimal condition rare produces trace rare lead check specification explicitly specify class multiply rate absolutely markov latter density particular vector rewrite rewrite function cross alternatively kullback effective direct regard optimum importance cross entropy directly importance construct successively estimate eq f function eq scalar purpose occur ratio form substitute substitute partially almost hessian k nd note furthermore need positively hessian semi fact unique optimum newton quasi newton force hessian individual sum parameter individual transition equation imply circumstance improve useful side hence formulae base unbiased biased optimum equation previous scalar work reduce distance successive distribution explicitly converge trace establish heuristic trial effective iterate suitable observed increase obviously rate help along failure property express temporal finding system trace successive trace trace satisfy
graph complete even graph complete consequently transition generate operator contain chain state complete time markov markov completed complete irreducible satisfying equation irreducible exist stationary reversible irreducible lem moreover chain operator score search equivalence class direction validity chain operator design reversible easy stationary distribution subset class discuss property need reversible explain markov markov direct chain markov class complete denote property reversible irreducible perfect show transition two complete complete operator operator confusion apply accord complete probability completed complete complete operator denote obtain property operator complete direct transform direct sampling operator uniformly generate uniformly random transition denote property right sure irreducible operator complete sequentially irreducible irreducible operator complete result reversible pair complete q define probability reversible stationary irreducible stationary complete sm irreducible reversible unique stationary construct concrete perfect carry equivalence complete set describe interest compute literature equivalence equivalently direct formula approximate define equivalence proportion number chain probability ergodic complete set perfect markov show equation operator sparsity run reversible equivalence edge complete perfect operator construction complete say complete perfect operator complete efficiently perfect operator operator completed complete six edge six addition guarantee operator introduce sure reversible notation distinct neighbor consist vertex child represent operator complete number six operator occur undirected yx direct edge occur complete subgraph x z cx z xy constraint five operator complete type operator lemma condition define perfect operator perfect set dd key without reversible supplementary material proof precede toy supplementary operator irreducible next accelerated version generate base perfect sketch show step explain subsequent e construct randomly operator complete set operator chain dominate implement time obtaining provide accelerate hundred fast arrange implement discuss complexity acceleration speed step construct algorithm include possible operator check condition te te tp accord tx vx tx vx tx x check result complete getting complete number avoid result complete material three classical whether clique path undirecte two vertex check trivial small complete condition whether undirected path vertex check source look undirected path within accelerate accelerate time number edge structure subgraph construct step equivalently possible less operator operator condition consume path depth know maxima evenly divide consequently fortunately complete experiment less approximate implement experiment cubic notice obtain irreducible reversible markov checking accelerated generate irreducible chain idea operator accelerate complete type edge reach bind easily possible edge version tm nt operator e te determine check set check replacement operator back operator uniformly step complete operator irreducible reversible algorithm provide om estimator sampling approximate normal distribution let variable distribute describe replace define show accelerate nearly roughly conduct experiment reversible equivalence estimation propose true efficient estimation quickly length increase complete constraint edge direct ii maximum measure vertex around ten iii chain grow know assumption latent selection variable inference observational need infer undirected complete underlie complete equivalence causal infer direction markov proportion direct component equivalence sparsity difference study asymptotic estimator pp markov equivalence get true size markov size calculate size chain chain calculate result table close standard deviation second chain ccc implement propose method without acceleration run computer ghz take hour ghz proportion markov size worth estimate neighbor concept characterize equivalence characterization proof essential acyclic occur lemma validity use validity validity condition type two valid equivalently hold contain operator equivalently hold adjacent valid equivalently dd clique undirected path adjacent equivalently undirected path contain operator prove operator reversible irreducible theorem validity valid complete contain complete equation complete adjacent pass vertex vertice parent reversible parent give reversible parent complete chain undirecte skeleton direct structure structure edge edge direct parent parent parent structure acyclic contain cycle contain acyclic child must include parent denote condition rv path contain path thus contain get acyclic complete complete complete except complete operator completed complete contain complete thus show complete result theorem need show define operator reversible operator reversible operator reversible operator operator denote reversible operator six several complete adjacent know graph direct complete vertex common strongly b parent denote result become undirected adjacent contradiction either occur hold must occur lemma occur iterate get direct graph circle stop step eventually lemma complete result extended let set complete whether denote obtain cycle cycle contradiction chain cycle cycle two clique cycle length great undirected cycle chain occur induce subgraph direct induce induced subgraph edge strongly remove ii modify cycle must contain vertex adjacent lemma partially cycle would induce like contradiction undirecte must undirected path vertex undirecte suppose subgraph occur induce subgraph subgraph contradiction yield direct direct complete clique vertex clique hold vertex length subgraph operator satisfy complete result complete clearly hold valid undirected occur occur adjacent occur configuration adjacent adjacent undirected condition valid lemma occur occur possible adjacent adjacent path must adjacent yield contradiction clique valid definition modify show equivalently skeleton common child structure exist structure structure parent clearly parent direct occur hold introduce lemma vertex direct short let partially direct edge undirected occur direct path complete adjacent short must obtained complete direct figure involve thus direct strongly yield contraction lemma proof undirected become parent partially direct child parent contradiction partially direct short partially direct denote direct undirected path direct occur shortest adjacent get exist cycle must partially subgraph structure clique parent parent neighbor vertice parent parent parent modify complete equivalently skeleton skeleton must condition occur structure must child clearly structure occur hold skeleton remain parent direct occur undirected thus occur occur undirected undirected undirected case occur adjacent neighbor become parent parent parent parent condition hold parent must parent otherwise occur also edge hold undirected path undirecte change get proof lemma complete undirected complete lemma contain edge direct exist least vertex parent whose direct find exist go since acyclic finite end parent need reversible reversible operator complete procedure step lemma undirected edge new skeleton subgraph skeleton repeat procedure completed direct whose parent complete empty structure complete edge complete whose skeleton subgraph skeleton repeat remove complete subgraph skeleton contain undirected repeatedly sequence finally partly department berkeley wu visit berkeley grateful manuscript co associate comment suggestion lemma corollary program cb nsf grant dms dms dms nf agreement research school science center economic education education microsoft popular tool represent system statistically say belong equivalent understand vertex design reversible irreducible propose operator markov stationary closed construct operator class introduce operator accelerate provide sparse acyclic vertex experimentally markov equivalence dag edge subgraph subgraph linearly base direct acyclic graphs dag denote dependent scientific bioinformatics business causality dag encode dag observational equivalence represent dag encode dependency equivalence uniquely represent complete acyclic complete short direct undirected correspondence completed call maximally orient modeling task discover understanding equivalence important dag searching complete search base dag bayesian edge direction need active hard set markov complete approximate markov equivalence define dag pe na proportion contain graph vertex year popular tool introduce restriction edge paper markov equivalence operator transition stationary interest class pt section short representation equivalence edge denote edge call partially least edge direct undirecte direct undirected vertex dag direct call acyclic direct cycle consist dag probability conditional imply joint read encode skeleton arbitrary regardless characterization markov class dag skeleton among dag preserve involve consequently uniquely dag
number moment solve efficiently cast decomposition demonstrate theoretic barrier precisely one mixture gaussian separate statistical exponentially imply grow component work bad degeneracy work finally algebraic develop model component analysis ica ica handle mixture model issue section describe gaussian main observable observable small eigenvalue covariance denote coordinate special spherical eigenvector covariance well k w strictly strict separation fact imply k observe vanish h hold crucially derivation claim relationship simple moment give random I recover ks km distinct therefore eigenvalue matrix geometric zero eigenvector sign claim regard plug derive restrict share spherical complexity problem mix worth inverse accuracy moment population central theorem bind exist theorem singular pick satisfie q easy accuracy either w become finally also efficient orthogonal tensor extract parameter robust recommend practical noise zero multivariate gaussian arbitrary straightforward recover definition open handling estimator axis align mild require moment partition conditional x kt describe recover component spherical rotation let x td invariance gaussian spherical conditional x kx dd almost split three view dimension guarantee somewhat conjecture sufficient share similarity independent unobserved signal independent generalize mixture value spherical spectral decomposition approach describe theorem completeness ica non eigenvalue geometric eigenvector choose surely deterministic algebraic recent sample previous moment mixture consider view setting long conditionally view moment map work view suffice estimator signal spherical unnecessary contribute correlation separate view degeneracy virtue tractable previous tie additional largely unnecessary drawback prevent automatic satisfie lemma condition provide availability deal separation randomization matrix denote large norm third eq analogue ny u essentially implement two moment w nx furthermore define moment moment half sample half th eigenvalue average w w w e w w w tt w recall basic special structure moment subsection w proportion inequality bernstein inequality inequality follow per moment third separately first third throughout thin orthonormal freedom thus q schwarz tail term combine handle check handle per ir w shorthand old empirical definite per observe I I I triangle g r r claim recall orthonormal singular analogously u assumption definite column uv uv second u third observe u whiten iw third claim u positive first third first claim w w kt wu w wu write e w e orthogonality I structure first third claim let sphere I take e kt eigenvalue eigenvalue arrange increase ii trial I sign simplify notation sort increase last therefore radius eigenvalue contain inequality simplify permutation range range term guarantee wu since indeed fourth use satisfy lemma exist sample lemma define therefore condition least furthermore inequality eq tail use freedom let vector random inequality term summation zero odd time odd non negative odd integer step standard normal follow cover cover cardinality exist u fact j require example remark microsoft computationally statistically mixture mean moment efficient mixture position connection relate study widely important covariance matrix probabilistic spherical gaussians let component define probability variable multivariate coordinate vector isotropic far k h hz observable mixture indicate component mention accurately recover component mix work efficiently degeneracy component span strictly entrie estimator plug moment moment empirical
norm get reach expansion point last appear discussion expression high initial discuss situation converge simulated initially anomalous introduce distortion moment rx certain moment author optimize implement implement significantly despite fast dimension standard code dimensionality addition add dimension dimension add transform give power situation grateful discussion transform classical problem encounter research importance year come intend suppose include possible recognize purpose trend discovery anomalous idea covariance generalization among widely hyperspectral recent year appear generalization tucker decomposition generalization rx normalize spread convenient go normalize well example slightly gram expansion present completely solve moment practically case exist compute step value choose normalize unity performing obtain suppose pdf unity would normalize moment normalize eventually discover restrict moment third note moment multivariate tensor moment tensor hard treat tensor lack analog procedure despite existence tucker decomposition kind normalize highly nonlinear expect third quadratic logic rx sense rx determine reproduce second carry normalize distribution transformation argument distribution possesse moment moment implement spread space lift want assign point certain satisfy possess requirement lift completely compute coordinate requirement overall normalization fig initial abuse lift lift rotation particular orthogonal rotation third tensor rotation minimize recall still freedom define lift ambiguity coordinate independent amount dimension describe follow condition generalize I normalization normalize rx space third onto compute follow use tucker order employ modification algorithm rotation matrix dimension moment run first whereas carry onto orthogonal rotation orthogonality tensor rotation mix dimension rotation upper corner corner projection
retain section fit logistic involve group mcp scad tend point global local minima mcp scad logistic likelihood refer concern certainly terminate explain write pseudo response descent affect updating remain spherical update equation present mcp generalize family typically unbounded iteration guarantee possess nevertheless well test penalize poisson study extensively obtain either criterion vary minimum beyond strictly convex continuously fact mcp scad present residual previous value next value warm comment regardless choose computationally intensive step calculation operation pass operation fit depend number depend broadly speak factor nonzero require group fix consequently spend end compare publicly package fit increasingly datum package package appear mcp scad tend tend lasso mcp scad mcp scad lasso worth note package problem purpose choose attention entire implementation terminate prevent present scad require compare scad illustrate mcp methodology allow flexible modeling association involve categorical variable term predictor covariate whose true study cross regularization scad fix group default recommendation suggest group case section rmse fit root prediction minus irreducible gaussian mcp coefficient normal vary towards allow enter occur model four amount theoretically gray regularization similarly increase mcp theoretically group perform poorly group mcp scad small model fast lasso far many scad nearly selection thus two behave similarly mcp seem somewhat well panel rmse oracle know advance mean zero coefficient variable oracle nonzero nonzero old cause genetic multiple probe reliable probe influence gene cubic group mcp describe table norm probe group group group symbol mcp scad nr c select scad fairly shrink nearly mcp select probe plot mcp group scad top probe aspect standard rest observation responsible qualitatively gene mcp select gene fit explain scad group correlate gene advantage possibly correlate somewhat predictive versus approach mcp parsimonious group lasso single potentially measure add note mcp fashion aspect adjust mcp make recall special mcp parsimonious considerably course flexibility tuning parameter important parameter group age consist control may disease section represent snp carry snp cross misclassification snps remark improvement misclassification mcp slightly superior group group mcp scad considerably parsimonious without compare snps select correlate cause pathway mcp scad powerful grouped selection lack lack publicly software development prove implementation package acknowledgment grant grant dms dms author thank genetic association anonymous helpful remark lead considerable proposition convexity mcp convex objective group group mcp differentiable everywhere strict infimum algebra quantity consequence updating consist along square loss continuously differentiable thereby establish point stationary note guarantee unique suppose group algorithm possess limit involve regularize strictly function scad mcp proceeding previous infimum proof proposition claim stationary differentiable exist segment inequality matrix algorithm minimize convergence note property stay compact possesse point apply conclude stationary furthermore also eq eq abstract attractive possess grouping group propose problem group nonconvex mcp also extend selection scad group mcp modeling explanatory group categorical flexible analyst scientific understanding take grouping gain variable become build early group covariate lin lasso lasso suffer drawback namely change magnitude biased tend coefficient smoothly scad minimax penalty mcp simultaneous asymptotic imply identity truly coefficient group scad group mcp scad mcp largely due lack publicly algorithm fit publish article scad fit estimate computational provide combine large make inefficient fit scad demonstrate superior approximation upon demonstrate coordinate scad mcp efficient show extend scad mcp fast publicly potential group mcp group explanatory overlap linear portion design form regression let member matrix covariate covariate think discrepancy encourage prevent penalty index regularization tradeoff collection regularity often outcome wish include selection penalization regression loss take logistic binomial loss variable ignore originally orthonormal unclear apply orthonormal explore author group norm demonstrate variable uniformly powerful article term penalize greatly reduce computational lasso straightforward extension group mcp importantly accomplish generality solution penalization software singular group diagonal eigenvector follow j may transform original compute note procedure rank eigenvector denote invertible possess rank transform back original transformation experience model case naturally member identical coefficient article orthonormal group orthogonal consequently subsequent section norm explore penalty group normalization group coefficient group describe estimator originally propose minimize behind penalty euclidean thereby encourage variable select residual large presence thereby normalize across group roughly group optimization group obtain lin generalization describe algorithm clearly illustrate descent algorithm employ coordinate descent considerably presentation fit scad mcp following section impact square center ol solution path orthonormal note subdifferential q square solution design exclude implication equation multivariate optimize problem lie direction thereby enable solution determine furthermore determine abuse definition vector act amount less way thresholde soft single descent penalize replace essentially modify replace updating refer recently update execution loop update refer consist expression repeatedly residual efficiency clear complicated arithmetic group update step lasso complicated view lasso lead clear penalty focus smoothly deviation penalty mcp penalty motivation penaltie p mcp scad originally penaltie group substitute discuss refer regression scad mcp behind understand consider mcp scad begin rate penalization relax aim lasso among coefficient mcp penalization scad move mcp penalty plot penalty middle panel derivative estimate light identity differentiable behind consider least solution simple note mcp scad toward depend mcp scad scale plotted illustrate essential describe mcp thresholding mcp shrinkage hence scad involve scad modify thresholding solution limit thresholding multivariate mcp scad
convolutional formulation color multi modal feature connection map map reconstruction compute inverse connect layer feature unsupervise b perform unsupervised stage complete stage order next stage absolute normalization sampling operation wise feature normalization pooling contrast one exact map index experiment average size feature extract predictor hierarchical x sample I ig I I I ds display since invertible use local contrast normalization figure form perform stochastic datum map constitute convnet classifier component stage nd stage feature traffic sign house number object detection multi traffic house number strictly feed take layer subsample branching produces extract contrary output transformation pooling subsampling convert subsample uv keep uv channel value normalization subsample uv absolute subsampling produce extraction feature map randomly connection mapping break transform value normalization sample fed stage magnitude great improvement detection minimal gain number classification less bootstrapping typically setting multiple extract answer add bootstrappe answer image bootstrapping pass I e total accepted overlap bounding box box overlap modify replace contain avoid instead modify bootstrapping include body part positive bootstrapping detect big window reliably however demonstrate multi stage convnet convnet convnet f stage convnet ms convnet ms windows width context pixel correspond image supplementary fair majority train train table system represent figure along rate show contribution convnet convnet combination convnet ms compare supervise without multi stage exhibit compare convnet reach competitive error rate result publish system clarity plot show convnet datasets datasets convnet competitive datasets convnet sensitive work convnet likely large auc mean less convnet dot line reasonable plot report convnet c fix auc auc pixel pixel auc perform among highlight auc positives convnet convnet ms competitive multiple measure measure report supplementary contrary popular level use extended feature performance benefit art publicly available improved future resolution successfully real hardware combine highly optimize graphic performance roc exp scale roc scale medium oppose auc discrete auc convnet auc exp get convnet ms exp exp roc scale roc exp report dataset curve software available dataset ranking convnet f continuous observe convnet rank oppose rank convnet ms convnet benchmark c c call auc large auc near medium auc auc auc car highlight bold row auc percentage curve plot positive small auc great false positive em auc discrete auc fix medium discrete auc partial auc heavy discrete auc discrete table except auc detection list successful deep vision report convolutional model stage skip integrate unsupervised convolutional stage surveillance variety pose variation classifier forest hand good mid level combine help sort multi feature little demonstrate usefulness unsupervise end technique stack stack stack decoder detector system also learn complicated corner convnet contribution convnet consistently competitive result major use train relatively end fine tune integrate additionally connection enable stage detector detection great enable real without entirely avoid introduce applicable gpu focus feature show algorithm produce successful supervise many find adequate datum design domain alternative unsupervise recently generic recognition train deep achieved actually follow layer machine image algorithm output location bottom row tend aggregate horizontal towards bottom extraction feature perform filter generic parametrize input gradually abstract convolutional code predictor contain code unsupervised multi train architecture follow tune many field code overcomplete sparse coding
distribution balance explore distribution distribution focus get look wish promise reject early main sample increase thing arm armed approach speed mention datum collect predictor generalization attempt step start configuration loop page learn first pointwise estimate configuration accordingly drop line test sequential applying algorithm conceptual overview depict additionally package name via contain learner cs p ip n break np p accord precede top tb sr tb calculate indicator robust trace filter sequential drop stop case complete configuration information whether top configuration rely test store pointwise good performance compare pointwise performance subset remain order sort regard want find configuration similar behavior remain current behind performance build configuration small relatively small available versa apply directly individual exploit pooling outlier affected overall testing affect effect helpful overall good perform configuration look outcome configuration subsequently pointwise test experiment significant difference configuration essence test whether show significant difference case behave differently term performance yet estimator check fine rest pointwise row yield whether significantly submatrix significance configuration increment one test index since indicate configuration configuration must configuration one configuration different configuration remain incremental thus correction actual calculation incremental e add increase speed improvement first collect show step procedure iff amongst configuration step generate step trace summarize learn trace record fashion whether robust binary random configuration top transform error scale top scheme overall represent binary top configuration course want estimate observe behavior binomial mean win configuration variable originally context quality assessment production process biological stop focus approach give deal potential win sequential observe bernoulli want denote accord success bernoulli variable significance acceptance control user meta test hypothesis likelihood meta geometric representation binary sum accept red stay red another fix since sequential eliminate allow evidence configuration want eliminate configuration acceptance mark use approximation expect take real success winner solve equip check trace detailed statistically exceed decision top violated transform winner configuration thereby behavior top approach would change success probability violate accommodate act drop control choose configuration remain stable drop configuration adapt sequential potential switch independence implication please sequential contain necessary need implement algorithm early configuration past submatrix trace apply test submatrix illustrate configuration configuration mark red one top configuration configuration mark gray drop picture configuration keep go heterogeneous behavior configuration mix red perform black early stop tb win pick step configuration determine way course restrict optimal increase model pick configuration suggestion significance suggest usual significance winner suggest setup fast really sure accept winner procedure exploit regime stable regime time future research win probability feed subset capable deal certain investigate suitable subset first property test employ come process give top respectively step drop deferred consequence drop guide stable regime configuration see used guide control observe drop configuration discriminate configuration win adjust act probability step drop insufficient notion learner show real performance configuration behave reasonable strong extensive experimental speed small impact compare heuristic configuration certain inside exhibit act step global win configuration configuration look bad configuration mark superiority global win enough marked winning process process tb visual step execution win configuration straight zero fast end express recurrence recurrence intersection lie decision boundary number split definition configuration line zero construction diagonal reach straight one actual recurrence decision number plus current reach option therefore recurrence path recurrence prove global configuration suppose win trace winner eq calculate configuration correspond outcome binomial probability path point survival path lemma sum conclude early indeed go maximal probability mass maximally spread decision early tb size fit dotted line bind success probability depict observe fast increase step nearly converge optimum rectangular grid impose fluctuation overall small impact balance conservative configuration control dropping impact overall probability analysis assume happen change win lie occur often datum set drop rate point simulate switch configuration change success choose constant change behavior switch turn winner tb indicate relative configuration plot show specific setting switch configuration two false positive similarly theoretical confirm study change variant test remove depend configuration result range later change occur false increase negative significantly low even room even drop configuration box come situation optimal cross amount process explore grid time adjust constraint model evaluation budget available step parameter lead near available budget idea configuration rough estimate total need parameter configuration configuration drop need calculation configuration drop consumption depict correspond difficulty hand need calculate maximal number use fix bind configuration sample appendix really ensure kernel name incorporate without training get regime cross stable essential configuration aware could learner yet world often might nevertheless analyze misspecification indicator configuration show sense grid happen mask normally configuration chance certain redundancy configuration redundancy similarity underlie redundancy theoretically new way configuration notion interesting incorporate function test dimension configuration scheme explain deal fact clear configuration potential preliminary learning evaluation version least dependency overlap set dependency evaluation form behavior throughout property yield task tailor cross choice support gaussian leave right task pose severe stop suboptimal intrinsic task use range devise eq plot point time parameter contain configuration interpretation apart mse learner validation speed gain encode misclassification difference mse measurement impact experiment difference mse well normal low noise normal suboptimal configuration increase tendency increase yield distribution use seem picture speed gain svm show high seem direct consequence inner perform speed gain svm configuration tendency drop rate steady increase kept spread high speed gain transition algorithm intrinsic algorithm choose suboptimal configuration cross gain range direct intrinsic show overall variance much leave datum correct configuration configuration setting extract configuration correct much demonstrate investigate choice benchmark repository entry classification panel difference square error never exceed show although pick problem mse classification task pick suboptimal error always relatively learner impact correspond combination high cross validation performance datum cross speed gain diverse vary picture speed higher task seem solve fast task trace keep regression ratio bank svm bank svm svm top configuration crucial test recall iterative testing pair compare section estimation test compare instance replace test non version result point replace classification rank benchmark get reliable statistic report mse look pair runtime procedure significantly configuration pointwise time bank bank block svm repeat time get sound report mse cross annotated thing negative indicating find configuration impact across always pick perform impact runtime simple evaluation huge choice configuration impact choose setting time outperform like validation combination superior heuristic trade speed simple robust configuration speed cross promise candidate model tb aspect potential module look configuration capture point stop depict top step solely configuration early stop act global determine stop focus landscape module look configuration essential evaluation show behavior drop benchmark easily square use information datum utilize fit less yet similarity come affect alternative calculate sequential configuration accord configuration behavior compare configuration strict drop outli behavior configuration residual residual check calculate raw outli similar specific configuration one see nature configuration obviously speed drop roughly observe datum conservative outlier keep low base ratio impact speed shift keep configuration conclusion fold section modular construction exchange turn extremely devise instance outli flexibility multi structure flexibility comparison observer increase expense set compare accuracy outlier compare conservative sequence hypothesis accept contrary accept neither decision exceed prove test potentially leave lead development kind close algorithm full cubic tb speed close compare conservative alternative increase step rapid close sequential tb indicate negative switch configuration setup behavior start enough turn winner reveal come price drop advantage procedure identify clearly configuration promise candidate large size advantage heuristic effect systematic understand statistically argue one error fast insight lead introduction sequential configuration time without combine bandit problem furthermore get size setting future research performance converge parameter tend infinity discuss minimization refer book therein interpret set ng gx I consequently order vc feed node activation vc dimension rkhs induce converge condition ridge optimization rkh attain regularization hypothesis discussion care correct scaling parameter datum package name version whole intrinsic dimensionality execute run early stop follow trace error remain configuration win configuration tool cast notation deal matrix configuration ccccc configuration configuration kind act use determine whether set configuration perform configuration behave compare trace remain test task test test configuration equally effectiveness configuration least significantly calculate denote total reject degree freedom long entry suffice exact calculate explicitly rank tie configuration position point tie reject denote degree significance section treatment sequential publication prove equation eq yield decision test minimal therefore maximal construction graphical intersection step configuration set learner observe proportion configuration parameter drop small give fact obvious solve substitute rt sake large mild condition power sum mention fulfil since furthermore constraint note condition assume term monotone decrease asymptotic behavior start negative grouping value sum negative finding consume propose candidate method computation power procedure cross validation testing cross standard tune supervise
give voxel voxel exactly voxel contain voxel treat write indicate canonical voxel spherical often voxel voxel size voxel contain show voxel particular identify response pattern extend single voxel voxel place voxel voxel voxel less voxel contain voxel vice versa contain contain multiple voxel voxel simultaneously less voxel voxel show absolute single voxel increase radius radius center one radius voxel spherical distance voxel two maximally distant voxel contain radius namely follow voxel contain voxel radius contain irrespective voxel frequency radius group let since voxel radius radius since one radius contain hold theorems radius voxel adjacent voxel lemma radius within voxel imply least contain consequently voxel establish voxel bias influence detect particular radius increase monotonically radius every voxel informative contain radius since necessarily adjacent logic center voxel within fully assumption center voxel radius radius increase voxel multivariate difference simulation concrete intuition implication ease single slice direction voxel voxel radius systematically take value mm mm contain voxel voxel voxel voxel first voxel voxel scatter text location voxel slice voxel exist voxel exhibit difference condition identify recall voxel contain signal voxel center signal voxel simulate response illustrate response voxel response voxel requirement draw position voxel figure voxel voxel decomposition testing implemented evaluate accurately membership use support machine leave loo cross panel map across slice radius go right expand high cluster accuracy reference cluster information show accuracy voxel slice dot horizontal obtain thresholded circle square separate voxel voxel separation radius mm voxel radius contain simulate manner section voxel map center voxel voxel weak difference organization voxel observe cluster several voxel accuracy red voxel map correspond voxel less separation map map radius mm top map voxel voxel voxel horizontal voxel voxel evident smoothing grow radius furthermore increase htbp panel voxel voxel separation voxel separation panel plot profile information map center horizontal slice voxel voxel voxel dot upper indicate motivate produce difference consistent theorem informative identify manner previous demonstrate effect inherent monotonic increase informative irrespective actual task voxel group monotonic size plausible informative recently study map remarkably map whole radius establish previous ask arise chance suitably question cover voxel brain cubic volume voxel direction voxel contain voxel signal voxel central voxel would informative informative single voxel informative true htbp informative voxel volume proxy spherical volume side voxel contain sphere would simplification cover volume voxel divide volume cube voxel voxel voxel voxel minimum number volume cover take cube voxel voxel rapid size would mm total spaced voxel spherical volume state single relevant voxel voxel voxel enable distinguish due presence random map draw actual voxel informative would information irrelevant informative counting inclusion identify task relevant voxel combine across require informative voxel could classifier compute voxel response assign dependent specific machine assumption appropriate allow technique make analytical alternate interpretation sensitivity count number informative sensitivity trivial explanation geometric property underlie organization sophisticated optimistic assume sensitive satisfying support office collaborative contract w nf lemma popular pattern analysis construct spherical voxel brain despite map challenge challenge size signature issue formally examine geometric mapping geometric spatial produce radius informative brain complete multivariate property capability pattern fmri cognitive provide systematically value cognitive voxel relevance criterion basis inference neural cognitive technical technical motivation mapping concern region response cognitive might voxels relevant though individual conventional univariate restrict voxel explicit concern simple enable sophisticated information irrespective whether informative response univariate unit voxel spherical neighborhood voxel response group base statistic center voxel map central employ dy soon bl kx fm pf sf ph jt vb gd popularity pose map single voxel map mapping protocol irrespective voxel information coarse location unique central voxel voxel voxel neighborhood property map ask reliably pattern signature study qualitative require interpretation map intuition voxel systematically continuous location voxel neighborhood priori informative virtue sharing group signature exactly voxel neighborhood deduce informative signature formal prove voxel signature information brain increase multivariate pattern importantly voxel brain voxel define define voxel voxel convenience treat voxel voxel principal direction assume path connect voxel voxel simplify intend emphasize principle ignore special boundary truncate white voxel latter agnostic surface ad chen abstraction subset voxel voxel voxel indexing indexing voxel voxel radius indexing conjunction structure voxel member convenience write result q clarity restrict entity tb voxel gray vary depend dark gray statistic compute map back correspond information map text indexing voxel difference experimental generally measure whether contain specific compute bl statistic response informative corresponding overall radius geometric voxel voxel informative decomposition virtue regularity sample since choose shape unlikely voxel necessarily relevant response alternatively accurately dependent combine voxel contain task make procedure restrict common procedure geometric voxel unless state symbol requirement sound requirement
quantify cardinality technical make analogous wang ensure neighborhood ef logarithm small meet provide theorem corollary provide risk also ideal yield fast illustrate theory k verify ef ei fx verify positive assume e multiclass estimation compare baseline classification tree multiclass wu et quantile method z pairwise optimize employ tuning refined burden example probability distance measure compute set p x denote cardinality avoid computing correction comparison next covariate freedom covariate example generate covariate testing size example different get large become difficult example additional covariance example error standard superior numerical free relatively cut boundary get intensive satisfactory multiclass apply dataset publicly university california repository http uci ml length width observation white attribute range score make focus score white white attribute measurement different class scenario misclassification set predict p x q replication table evident small except performance due burden propose free conditional estimate construct distributional assumption efficient propose establish experiment simulate example demonstrate advantage regard naturally density technical py pr kx k k x ib ib inequality bound bound let x x p mp mp k mb k establish mp version li p pr pr pr k pr multiclass estimation discriminate logistic regression distributional formulate cumulative function convert exponentially number class study relatively interval multiclass estimation tune statistical machine class probability strength information hazard multiclass discrete know liu zhang wu opposed class estimating probability distributional fisher homogeneous baseline assume ratio generally verify distributional violate gain popularity practitioner classification tree sensitive suffer instability wang propose model estimation conditional binary classifier various weight weight contain obtain multiclass attempt wu lin develop couple multiclass estimating wu zhang liu extend wang liu search vertex contain require intensive computational binary classification vertex exponentially propose multiclass via conditional cumulative series quantile compare free burden importantly establish show highly illustration nonparametric li liu reproduce rkhs induce specify associated li liu rf computational remark asymptotic behavior estimation space employ grid along direction whereas complexity wu efficient cross become inconsistent order suboptimal wu liu wang wu finally performance choice appropriately multiclass indicate dependency tuning estimate conditional k comparative loss eq indicator model wang liu searching distance candidate dependent
finite rate model independent eigenvalue detect dependent result would eigenvalue select also population counterpart strategy determine eigenvector accurate decay polynomially accurate estimation showing section stochastic covariance trajectory discrete corrupted noise study however context perfectly trajectory without bootstrap select sample eigenfunction counterpart parametric observe trajectory establish instance fix eigenvalue eigenfunction bridge gap covariance spectra decay brownian motion connection jump spectrum operator eigenvalue latter evaluate observation instrumental analysis already denote induced let j section sample approximation eigenfunction eigenvector turn develop eigen present eigenvector theoretical operator euclidean square also hold multiplicative constant result sub vector constant x x additional assumption zero constant moment interest meet sub assumption vector distribution sub constant jx provide sharp bound term frobenius operator sharp either state constant specifically proposition depend matrix accurately offer level assess satisfie eq satisfie furthermore continue singular establish independently unbounded employ rest include good rate bound logarithmic derivation norm low frobenius derive order theorem keep minimax rank bound estimator frobenius operator matrix little gain thresholding shrinking would performance simulation conduct situation theorem relative motivate introduction class covariance introduction study low population operator estimation matrix effective rank also appropriately rate sample operator frobenius note guarantee large less specifically small imply growth induce bind make bind non result offer satisfy probability least high q scale norm irrespective show decide believe small upper discuss consistent separate separate relative q notational clutter constant appear eq recall quantity need regarded jump estimate condition make index quantity threshold note implicitly thus well order high index enough large exist call jump spectrum relative triplet order concrete threshold notation constant let hold eq fine jump minimal noise threshold estimate pr specialized assumption spectrum show condition appear naturally brownian specific assumption belong irrespective assumption differ analysis minimal jump note differ satisfy notice index exist immediate exists imply theoretical spectrum occur large noise level follow theorem assumption technical holds exist iii prior illustrate spectral population theorem detect offer dependent threshold manner suppose analysis trajectory ks xt discretize trajectory assumption continuous positive theorem decrease eigenfunction orthonormal moreover hold brownian motion mapping distribution j q scale covariance trajectory focus employ densely smooth estimator appropriate defer detail estimating jump operator accurate illustrate analysis address eigenvalue order need form evaluate whose us spectra close show moreover eigenvector vector evaluate eigenfunction establish modification feature plot apply slight abuse denote eigenvector eigenvector denote assumption fix point hold positive give appendix whereas immediate consequence crucially eigenfunction finite dimensional decaying automatically scale rank jump occur theorem detect jump size spectra construction detect suppose process technical index regime employ translation noise employ recall motion detect covariance apart existence term quantify hold upper suppose assumption directly evaluate eigenvalue eigenvector subject recall reasoning eigenvalue separation analogue hold define immediate identical theorem estimate keep usage brownian motion reasoning thresholde accuracy eigenvector brownian eigenvector relative proof generic u u generality immediately justify orthonormal argument employ transform matrix evaluate either frobenius sub eq let q need non random equality hold independent gaussian constant u sub f combine proposition combine proposition average material need prove proposition next subsection begin instrumental proof exponential use use exist furthermore straightforward last let inequality prove article proposition frobenius smooth change inequality prove let eq let ix n z expectation article proposition adapt completeness result value zero tn special therein matrix singular frobenius f xx xx present proposition sequence independent exist two psd semi definite integer markov ax ax ax sketch note operator therefore inequality appropriately prove inequality sec section easy proposition k k proposition sec proof notice integral kt finite integral equality easily derivation constant otherwise small nd hence upper q hence ii ir eq ij ij c lead q first invoke derivation eigenvector give complete mc full rank acknowledgement david grant dms foundation dms foundation national institute imaging ns national institute stroke proposition class reduce effective class decay spectrum measure complexity define class reduce estimator perform detailed empirical necessarily goal sample eigenvalue population counterpart plot level provide checking eigenvector analysis population polynomially decay extend operator polynomially decay spectra estimation receive amount attention last largely motivated covariance consistent regime understand decade whenever instance seminal rest effective hope year depend type sparsity entry wise wise diagonal decay shown adapt see many matrix
function term regret balance wide range composite special case proper binary necessary sufficient let strongly direction let c proper strongly thus characterization loss regular proper strongly loss tool establish function recall follow tie p note follow rank estimate completeness let f strictly link yy p p p regret several composite property loss exponential exponential proper loss concave strongly proper logistic loss proper c regular eq concave composite apply square hinge loss square probability predict coincide proper associated composite learn apply appropriately obtain loss pt yy yy pt canonical exponential pt spherical yy function regularity condition loss accord spherical proper composite essence exploit proper proper link scoring denote marginal distance rank risk treat show certain rank plug ranking inspire xt adapt rank risk condition version proper yy take noted restriction give obtain hand exponent bipartite rank composite term composite loss include widely characterization proper loss elsewhere concern necessity strong loss term concavity bayes concavity regularity spherical loss upper regret pairwise via natural pairwise standard adaboost standard logistic hope establish study future discussion thank lee conference mining hold ann work support fellowship department science observe trivially x similarly instance negative scoring mis instance equivalently area dominant framework pairwise classification bound et al show regret rank term balance logistic exponential loss bound bipartite broad proper composite simple proper hide balancing variety strongly logistic loss tight condition cl problem arise variety range retrieval drug discovery last several variety ranking rank problem label negative goal minimize mis area algorithmic theoretical indeed enjoy since bipartite suitable follow summary nevertheless algorithm adaboost logistic svms exponential loss yield loss surprising effectively also far quantify score surrogate loss show bipartite ranking exponential loss build analysis bipartite analysis exponential balanced loss nature bound bipartite rank proper composite proper hide balancing variety proper exponential square squared special case set bipartite instance problem definition background composite reduction ranking characterize loss bipartite term together brief open question background binary proper loss assign specifically scoring simply receive tie break simply bound score binary function belong briefly notion regret proper composite give prediction binary yy x concave background material material simplify compare include important start estimation space say minimizer unique proper proper basic strictly strictly decrease proper binary regular proper risk loss strictly several way attribute twice result proper strictly strict convexity equivalently give third contain principle helpful loss proper loss direction concave strictly conversely assume strictly proper strictly concave notion composition say invertible loss widely note composite describe composite form form etc characterize composite bipartite note bipartite review result build pairwise distribution x scoring r
pair verify therefore derive I gaussian matrix universal exist e eq inequality implementation alternate solve admm lie express q admm consist denote solution admm pre constant specifically least eqs via efficiently eq admits add solve iteration intermediate iteration x f problem via system linear equation obtain verify admit rv rr admit analytical plus minus theorem section lemma regression scheme assume assume matrix admit program regularize accelerate gradient multiplier develop efficient key admm scheme optimal program performance composite estimate predictive observation receive area machine estimation assumption set effective base use trace become popular tool trace encourage singular dense application predictive function lead interpretable rank cardinality encourage sparsity motivate use trace induce structure investigate extensively recent develop program recovery trace establish trace norm study trace norm minimization collaborative filter convex paper multiple function basis set function assume coefficient formulate trace norm employ composite regularization induce coefficient simultaneous low structure incoherent propose solve multiplier also component conduct convex basic property lemma assumption basis prescribe regularization conduct effectiveness propose trace sum singular sum prescribe dy set th practice basis sparse low induce sparsity couple coherent low parameter cross use sparse rank globally many technique alternate multiplier consider accelerate multiplier non develop algorithm admm attract deal scheme th search intermediate specifie increment commonly refer operator efficient efficiently dual min verify duality substitute dual convex coordinate cd compute globally solution optimize two fix optimize rate fast iteration optimize via compute unit summarize optimal eq therefore w x f material assumption derive performance trace norm formulation restrict eq less restrictive condition denominator one rsc assumption matrix imply condition trace bind x dy entry eq lemma material last similarly substitute eqs set inequality choose refine bind proportional refined n note tight evaluate benchmark set study admm implementation material trace multi label comparison macro micro classification benchmark business health yahoo multi experiment experimental stop value determine double validation I business health auc micro compete algorithm benchmark set business result demonstrate effectiveness multi outperform benchmark high data numerical study convergence observe admm much slow focus observe trend admm convergence stop change stop attain curve successive curve present plot figure converge fast left plot plot dual alternate algorithm operator similarly alternate successive curve figure efficiency predictive specify coefficient matrix sparse structure formulate problem
monotonically monotonic monotonically fast monotonic unbounded fast propose theorem strictly set sample generalization development rv depend ease point within back fundamental simulation boundary since also contour rectangular depict rejection whether need analytical expression boundary necessary know relationship describe contour enough rs obtain dot dash line solid transform htb standard region standard pdf condition function also increase limit eqs entail follow since rewrite form monotonic rewrite recall hence fast region generated bound monotonic verify vanish vanish condition condition indeed density relationship rv pdf although unnormalized study variable monotonic q denote rv see distribute sequel sample density method obtain connect look region pdf uniformly recall eq write indicate extension theorem simulation htb density pdf point distribute area depict draw equation connect rv density use formalize rv monotonic transform transformation point choose region distribute moreover devoted method transform variable describe investigate connection recall define eq simplicity treatment eq lack decreasing also decrease since recall therefore trivial express increase consideration develop monotonic pdf unbounded rewrite q finally increase monotonic b non monotonic locate mode locate consideration case transform variable eq moreover region bound section remark namely satisfy apply transformation random pdf give rv coincide area transform rejection unbounded b display region depict proposition show inverse du hx px provide rv moreover distribute generalize inverse moreover rv area obtain proposition extend approach whereas generate extension fundamental fundamental link pdfs clearly decrease exactly rectangular circle triangle useful rectangular allow infer function indeed rv consider instance inversion rectangular rv unnormalized rv region p yy v inverting obtain completely inequality rectangle second end contain assume alternative area region second second formulation technique sample eq replace vertical formulation choose pdf summarize relationships pdf dash line set consideration remark assume decrease also multiply obtain area p far minimal equivalent rectangular rectangular region simplicity consider decrease localize mode mode transformation distribute pdf decreasing vertical rv indicate choose later obtaining pdf write rewrite q exactly strictly relax condition need relax study possibility formulation indeed know apply rv apply rv rv transformation htb correspond second slight depict increase way follow combine assume section observation distribute sequel suitable unbounded scheme uniformly acceptance rectangle htb region region axis accept second possibility I uniformly eq illustrate htb uniformly boundary show combination rejection specifically completely pdf fundamental simulation relationship random whereas realization rv pdf function obtain need achieve bound transformed pdf see density monotonic transformation pdfs monotonic piece moreover rv pdf namely rv work see section design deal pdfs consideration aspect rectangular partially science innovation project ref program ref transformation rv uniformly write pdf rv jacobian inverse namely substitute yield integrate marginal pdf rv calculations trivial turn pdf integrated pdf express term rewrite integrate rv transformation rv u gx consideration easily infer light strictly pdf uniform yield expression proportional pdf light decrease decrease draw constant define take important possible symmetric axis axis decreasing rewrite moreover jointly transformation clearly come back decrease define summarize expression rv express decrease target eq easily observe indeed area p htb unnormalized target yy depict solid area derivative cdf calculate use notation cumulative generalize iid ratio rs rejection tr representation generalize density symbol rv distribute unnormalized unnormalized normalize pdf target rv rv meaning unnormalized rv alternatively frequently indicate constant indicate constant rv q close close closed side rv value arbitrary unnormalized rv instead unnormalized instead pdf commonly start type interval consider associate imply lebesgue length sample rs algorithm refer pdf constant distribute sequel rv rv rv pdfs half gaussians inverse target rv pdf give imply target pdfs unnormalized target invert proper unnormalized however rv rv multiplying normalization hence perform different whereas region region lebesgue identical support inverse pdf give region unnormalize generalize lebesgue inverse imply common support paper never invertible use convert rv rv rv rv indicate rv rv target rv transform target rv invertible pdf compactly rv great obtaining convert inverse rv unbounded rv inverse target rv obtain tr invertible transformation inverse rv case write simulation realization transform rv tr rv invertible compactly transform rv strictly q region inside sample must unnormalized target pdf unnormalized pdf pdf formally draw target pdf formally unnormalized target differentiable use inside uniformly define use unnormalized pdf eq let note become pair distribute inside case due simulation tr rv derivative vertical vertical I denote vertical vertical vertical I point vertical department communications la es circuit engineer de km mail david es transform rejection generalize ratio monotonic pdf equivalently transform rejection target classical showing completely monotonic concentrate monotonic pdfs decompose monotonically case transformation inverse handle generalized ratio technique carlo mc often nuclear sequentially smc filter use chain core efficiently question remain rejection exact inversion approximation ratio present efficient monotonic discuss technique case section appendix consider technique often monotonic unimodal pdfs monotonic denote pdf unnormalized pdf able feasibility drawing extend monotonic multidimensional moreover relationship see vertical rejection another discard sample favorable rs occur pdf bound domain density proposal calculate bind ratio find easy task scenario sophisticated rejection high etc unbounded pdf critical region area region straightforward theoretical sampling transformation make transformation rv unnormalized draw convert sample invert obviously restriction unnormalized target pdf furthermore suitable function similar cumulative cdf transform rv achieve imply draw sample sometimes technique ensure v pdf interest us sample draw uniformly unfortunately tail fulfil consequently pdf uniformly inside region u introduce separately explore far goal relationship approach prove region pdf moreover introduce consider monotonic unnormalized pdf transform rv unnormalized pdf provide monotonic pdf monotonic pdfs technique unbounded important consideration notation fundamental discuss rejection technique background paper provide focus possible sampling introduce ratio consideration section monotonic pdfs devote unbounded consideration conclusion sequel unnormalize pdfs mean integrate whole domain consider normalize pdf denote unnormalize furthermore unnormalized monotonic pdfs pdfs pdf unnormalized multiplying variable instead attain inverse pdf inverse pdf usually perform also obtain target rv issue clearly half pdf identify unnormalized pdf normalize logarithm proper normalize unbounded pdf similarly unbounded kp clearly target important remark normalization unnormalize generality attain formulate pdfs although approach pdfs proportional even carlo rejection etc base result sequel density constant equivalent inside region area whereas play auxiliary many monte theorem jointly discard univariate marginally unnormalized depict unnormalized method density apply sampling htb unnormalized target inverse monotonic unimodal density extend pdfs unnormalized target pdf inverse unnormalized describe illustrate alternatively eq depict way generate draw interval e figure always unnormalized inverse non monotonic interpretation generalized pdf straightforward distribute fundamental first coordinate unnormalized whereas pdf able draw second yx generate inside inside e draw e uniformly monotonic proper monotonic obtain feasibility unnormalize pdf unnormalized pdf practical variable since pdf draw pdf inside accord method form random decrease unnormalized random unnormalized pdf call alternatively relationship technique clearly apply rejection rejection rs unnormalized know unnormalized proposal pdf work accept discard repeat step pdf rs remark connection fundamental inside accept whenever fall inside repeat step rejection green target pdf inside locate define eq rejection complement q rs pdf happen dark accept belong red region happen point indicate dark circle discard express condition belong green I red demonstrate equivalence description rs whereas red indicate rejection region locate define sample draw pdf dark green dark red circle fundamental rejection sampler candidate lebesgue tight make crucial performance bound pdf bind convert problem unnormalize easy rejection improve scheme technique unbounded proposal proposal close illustrate pdf support pdf bounded pdf show figure section devote deal problematic transform inside region transform tr htb unbounde finite last proposal simple scenario several author line invertible rv define inside b figure conceptually inside unbounded unbounded target pdf support rv pdf rv different suitable rv support unbounded appropriate rv unbounded unbounded finite support imply infinite unbounded discuss always generate distribute hand invertible apply rv rv unnormalized sample distribute result rv unnormalized target pdf section apply rv sequel monotonic either function inside I inside domain rv imply inside interest rv unnormalized make g monotonic pdfs unbounded pdfs rv sampling technique obtain literature rejection also call close unbounded consider monotonic transformation rv unnormalized unnormalize idea draw invert drawing choose adequate unbounded transformation close look notice term assume unbounded monotonic vertical imply limit situation monotonic unnormalized target transformation increase decrease htb monotonically monotonically corresponding transformation vertical conclusion unnormalized transformation dx f limit tend alternatively order support bi interval extend infinity support towards infinity whether
variable n property imply independent vanish zero tensor tensor constant vanish gaussian variance first variable ica assume latent random via additive assume last assumption serve remove ambiguity otherwise convenience first orthogonal term whiten sign noisy main additive affect covariance whitening could orthogonal without whereas scale follow matrix main clarity presentation constant time division sample nn simple component independent length method already ica modify reasonably way whiten popular include high ignore class additive gaussian validity fourth second ica presence statement restrict exist list maxima th canonical list minima sphere canonical course coordinate maximize restrict ignore rescale tr summarize whiten without mostly algorithm provide moment observation recall fourth operation proceed lead worth first multiplication take role describe transform primarily equivalently b bb idea quasi whiten noiseless quasi whiten matrix construction quasi whitening know multi equation essence however several demonstrate first cauchy variable recursively I arise limit summation generate certainly intersect fraction finally tool proceed use binomial odd dominant particular term come q get follow chebyshev lk j lc choose apply tensor whitening throughout quasi whiten canonical act independent remain demonstrate cosine orthogonality section bound suffice demonstrate fast shall statistic denote max I give define get investigation propagation tensor use growth tensor operation place goal portion useful norm result theorem split precede lemma use propagate tensor bind demonstrate angular correct sample later n I follow equation show ad lemma apply equation theorem apply yield restriction except restriction meet suffice basis canonical approximate whiten canonical stay apply rr yy give equation case q complete thank discussion nsf grant abstract give quick emphasize spherical remove notation consistent abstract proof reference final bf define eq engineering university science engineer party people different room recover mathematically sample whose random ica address propose presence necessarily additive distribution independent routine property tensor ica get provably presence propagation blind set room simultaneously superposition produce variable superposition linear transformation give recover need name range address extensive literature processing community applicability speech various biological comprehensive introduction book ica remarkable fact reach consider correspond maxima orthonormal whose element compute fourth independent apply pca transform principal direction transformation rescaling absolute sphere independent recover direction maxima sufficient sample coordinate paper separation contaminate underlie signal valid analysis still minor address signal polynomial compatible invariant pca special case underlie method base pca sample compatible year blind signal separation ica concentrate provide run analysis analysis al address learn equivalent ica subspace blind
package develop originally integrate crucial ability way arbitrarily development organize calculation facilitate flow sect class implement inference sect sect sect begin terminology commonly measure pdf multiple unity estimate constrain particle uncertain shape nuisance principal build concept density integral handle collection pdf handle easily multiplication representation fitting generation study inspection great modularity principal factor start field distribution background constrain observe event factor number event specify nuisance specification specification idea provide obvious element prior frequentist calculation pseudo limit complete option carlo iteration test conceptual school data limit frequency various outcome broadly permit probability typically interpret frequentist require inference obey brief method available refer detail example test credible class nuisance credible return depend return ability lie within specify retrieve find value side convention value implement likelihood numerator denominator absolute regularity condition demonstrate shape deviation two sided invariance ratio valid physics community program asymptotically variate perform hypothesis characterize systematic systematic uncertainty hypothesis implement estimate interval return object dimensional contour hypothesis visualize interval newly develop inspection profile relate hypothesis give inversion achieve function probability characterize typically nuisance product parameter unity credible interval calculation nuisance remove marginalization integrate perform current interest use posterior algorithm root numerical integration include integration metropolis number step construct markov default proposal mode invariant class visualize input bayesian implement release interface development integrate confidence guarantee fully specify probability implement derive return concrete specification order define observation add desire introduction another decide assume asymptotic mc nuisance nuisance profile nuisance toy carlo nuisance point parameter consequently accord inside class user specify use give statistic obtain toy helpful application importance implement frequentist approach hypothesis frequentist method marginalization nuisance technique often refer define alternate hypothesis along order exclude choose statistic possible outcome functional form statistic monte approximate lie cl uncertainty marginalization distribution generate toy net presence uncertainty hypothesis hypothesis nuisance result consists associate computationally permit merge vary signal side implement test finding level interval limit condition define upper alternate drive project would useful without object file object file complex multiple easy via save file fashion convenience publication newly develop utility permit intuitive string create store allow pdf one create observable parameter class two prefer aim c physics specific single file describe line fact need would everything addition add include comment easy combination channel collection class template base analysis require knowledge instead file model user histogram template event channel efficiency affect systematic uncertainty gamma log systematic channel identical across channel list mention implement technique theorem incorporate systematic pdf combination
context erm costly must optimization paper cs propose provable guarantee secondly aware distinction regard estimating paper principle know accomplish pursuit directly propose derive dimension extend soft section randomize essential estimating minimax finally simulation write norm product b present bp important motivation well result describe draw suitable p constant fundamental condition apply signal additional aim must invariant signal determine determine large hold eq notable feature xx level little measurement recover structure derive technique area deal cs area seem fully usually role technique set randomized approach estimate generate vector characteristic function form tv refer example cauchy write stable set noiseless well study univariate however corrupt show moderately affect impact noise choice control measurement leave first generate measurement cauchy cauchy variance define confidence noise gaussian coverage width confidence vary take measurement mild standard algorithm basis pursuit expect perform lastly growth well assume hold depend dimension sparsity fix confirm simulation lastly small order high follow taylor naturally extend recover year many researcher refer paper application theoretical role sensitive perturbation matrix straightforward attention quantity recovery order explicit section robust motivated quantity quantity elsewhere less follow randomness noiseless deterministic estimating noiseless sx validate consequence decay setting image dependence relative error grow moderately lastly reconstruct consider set alternatively implementation pursuit illustrate coordinate plot agreement three indicate take signal accurate parameter figure average sx according always right middle signal profile profile choose curve probability reasonably reconstruction bp instance choice plot plot reflect theorem implication hand prove implication vector define fact q define measurement version fact imply detail find chi freedom follow delta prove limit statement way depend limit allow conclude relate define consequently may eq derive term complete give asymptotic bound relation almost omit reason theorem analogous essential difficult case idea measurement indistinguishable yet theorem let satisfy old column orthonormal let standard realization define amount show event equivalent accomplished variance gaussian respect fact obtain portion version inequality long j w verify finish make expectation finally definition prove need begin first optimize side enough take infimum without third enough replace supremum two px ax aim eq construct bayes also prior define two put mass rule return ax third california berkeley theory compress parameter unknown due aspect depend know drive practical concern unstable value coordinate paper sparsity sharp rely little width dependence naturally estimate guarantee randomize essential accomplish deterministic sense cs linear q measurement small signal draw ill pose reliably pn draw line commonly algorithm play issue recognize theory relatively quantify conceptual hand estimate assumption sparsity simple estimate emphasize aspect depend know several valuable address assumption likewise check active area recognition image natural device theoretic accurate reliably recover measurement deal sequentially case determine recover take additional may discuss enough characteristic rip guarantee closely rip grow body devote whether treat already yet meaningful consistent tuning orthogonal pursuit omp appropriate examine omp estimate reduce restrict ensure coordinate decrease vector one dyadic red dyadic color code triangle bottom axis indicate number despite cs practical unstable particular equal description effective
count source vice versa criterion define degree relatively external directional zero alternative cut however concentrate explore asymmetric component propose preserve directional divide accord connectivity distinction terminal develop identify directional community sim use node direct laplacian graph define degree matrix remark graph laplacian strong connectivity community di sim nod two way k leave sim stochastically receiver co relaxed sim discover huge direct pair terminal require number scalable search community time community division propose regularize balance lead cluster observation bi adjacency svd find good directional community reason two first directional community balanced division graph division expensive second global np hard undirected regard size addition define us parameter determine trade svd minimization proposition directional community vector result relaxation membership replace interestingly induce penalty sparsity induce helps introduce directional component structure adjacency undirected know component laplacian connect component relationship extend directional length represent terminal directional denote obtain replace singular direct directional singular span moreover directional follow directional community subgraph strict directional q directional directional directional community svd use multiplication hard thresholde local solution find similar exploiting linearity definition h thresholding treat maximize proposition constant tie determine sort entry sequentially large small meet hard direct solution svd search alternatively function monotonically local detail initialize l similarity algorithm laplacian step detect node principal find regularize svd tight community solve combine inspire induce penalty type penalty elastic sparsity control non size discrete report significantly regularize solution net svd local thresholding regularize svd fix become seek time entry search verify search method number define maximize follow threshold regularize svd replace search threshold algorithm find describe appendix scalable elastic regularize extract next extraction repeatedly sequentially advantage identify world web link exploit property devise community regularize vector initialize community repeatedly net value magnitude vector different source modification future level propose among community achieve change obtain change regularize svd smoothly level initial contiguous change huge fraction name elastic vector initialize initialization specify vector level pick discover community search community stop early reach simple stop value time minimum previously detect candidate pre l desire stop search rule burden keep early stop rule tight community direct apply idea modularity first elastic net penalty identify submatrix zero say edge typically network directional edge call procedure take sparse singular singular singular submatrix require directional component keep whereas detect community membership source edge concern regard initialization driving motivation scalability massive investigate computational requirement specification edge know drop potentially community relatively explore locally regularize smoothly believe tackle massive modern setting hand algorithms di include generate benchmark generate directional part consist node asymmetric community terminal node generate law size terminal control aspect community community degree degree information criterion measure configuration simulation result repetition reference directional valid practical emphasize directional community general accuracy di sim community big community repetition error directional community three directional community cm di di sim well di sim give accuracy even strong community accuracy algorithm community however di seem community experiment detect directional community contrast direct citation form science cs paper manually category represent major field computer sub remove self citation symmetric average degree terminal un run candidate cover node sparsity take en take value decrease help community minute minute discover en cover part yet cite comparison consistent perform average degree highlight asymmetric directional community di sim partition require algorithm detect di sim citation arrange arrange remain row citation sim membership column arrange terminal part remain node end internal appear diagonal meanwhile dot dot internal membership sim adjacency separate partition community detect reveal citation yield community reveal correspondence di sim treat manually citation information validate quality category paper paper intelligence value list paper major paper theory database hardware computer system program directional l db ec ir net os correspondence detect community manually assign significantly low community large significant major operating os intelligence ai major strongly connect big community find show field embed field example community relate ai category ai community meet expectation regard manual detect paper part within manual direction massive million scalable detection search community social highly asymmetric micro website china direct contain zero symmetric hour hour algorithm run six ghz gb check quality community detect large community small directional community additionally verify small scatter scatter plot directional community community look similarity node community community asymmetric community popular terminal source highlight directional community integrate role critical characterize community notion community capable svd sequentially identify linearly meanwhile small directional community enable asymmetric direct real matrix multiplication parallel acknowledgement grant dms dr dr dr wang discussion present search find directional find dc article proof negative edge notational convenience cs c adjacency remove row without denote node node whose zero back vector direct terminal undirected connect connect undirected dc examine connectivity connectivity connect terminal connect member node node g node member contradict node directional dc directional component obtain step apply eigenvalue matrix normalize define row say equal component graph span k indicator th component eigenvalue follow eigenvalue l eigenvalue eigenvalue q value principal eigenvalue vector break entry q find exclusive span expression component submatrix directional directional generate corollary submatrix equal include representation submatrix selecting row row principal value eq principal tell enough include directional component row column start direction place entry belong span span include include directional membership vector obtain set row decompose maximal connect within principal singular community connect connect contradict maximize directional component subgraph subgraph prove cs q condition community panel figure size randomly subset submatrix laplacian principal singular sub directional include external add three left panel adjacency directional scatter submatrix laplacian direct directional component matrix external plot graph derive direct external edge right panel pair submatrix component mark intercept maximize small directional describe directional directional encountered connect small directional community directional exact directional node limit small illustrate external merge together pair pair principal value directional external slope still directional approximately regularize embed directional entry objective write notice monotonically keep go keep thresholde large absolute resembles express objective multipli tucker solution boundary happen unless zero satisfy kkt satisfie threshold become satisfie z setting solution obtain z come inequality come way obtain approximate increase monotonicity z increase find lemma z initialize n second already therefore quadratic quadratic formula know di sim singular mean singular vector within centroid distance result directional community separate partition match terminal large common use implementation available default except option repetition v terminal value range determine community size k grid linear community early continue community reach leave citation network four manual category di sim table paper manually assign category h summary citation source terminal node stand c ht l ai db ec ir net os ht paper directional en category ec ir os ht source di sim
way difficult square covariance model slice easy step size tuning size lead computation change metropolis sampler slow paper slice latent highly correlate slice sampler metropolis sampler simple log model dominate covariance gp update cholesky decomposition th latent require decomposition almost costly cholesky decomposition slightly complicated consist gps hyperparameter covariance change secondary unchanged change unchanged change require major discuss table std operation operation operation sampler latent distribution q parameter great triangular cholesky prior standard normal transition reversible leave proposal ratio sampling obtain new update hyperparameter slice secondary hyperparameter metropolis tuning scheme highly sample use possible notice hence component update update model residual gaussian distribution gaussian uniformly true contaminate residual residual covariate method dataset u randomly seed dataset observation drop sample burn predictive distribution compute mse probability response hyperparameter mcmc give gp gp notice gaussian small residual winner giving numerical sampler latent metropolis slice sampler metropolis update sampler computation list adjust sampler efficient slice width adjust deviation acceptance around get c slice efficiency modify five run run bottom trace slice middle initial iteration sampler adjust time plot modify least mix appear metropolis roughly conclude mcmc fast reach equilibrium hyperparameter series main gp elliptical slice observation elliptical wang department university department residual arise propose serve since change derivative unobserved covariate dependent input residual model synthetic deal handle carlo addition residual non introduction gaussian gp text perhaps flexible function degree smoothness additive covariance function regression residual though many variance depend residual include unobserved variance change residual gp std method synthetic section association covariate pair length gaussian could specify zero reflect negative covariance hyperparameter covariance covariance suitably choose constant sometimes residual parameter length covariate lead ard use exponential covariance observe prior logarithm hyperparameter compute integrate let variance gp model expansion q denote derivative second e depend gaussian residual variance ignore consider chi illustrate translate purpose curve axis produce plot observe change look big around notice residual strong quantity although exist see trick produce exist unobserve often gp treatment response secondary logarithm noise residual variance gp prior
sample distribution accomplish call moment measure covariance major impact compact representation certain work euclidean space embedding space discrepancy mmd measure independence hilbert schmidt criterion characteristic rkh despite test question discussion context independence use definition covariance page confirm integrable show rkh dependence measure precisely formal function translation variant kernel two sample demonstrate distance maximum arise consequence arise derive arise kernel sensitive second obtain characteristic arise quite via bootstrap perform distance discrepancy know probability hellinger divergence family measure metric wasserstein metric mmd equality energy show mmd establish total discrepancy divergence equivalence implication practitioner use test member broad alternative one second readily general topological space indeed readily structure euclidean text string structure negative distance negative review mean mmd hilbert schmidt independence correspondence definite mmd give estimate quantity investigate variety conference publication technical proof omit independence type nonempty triangle arise also nonempty say terminology satisfy say function page page hilbert map second type take satisfy square root obeys assume topological borel measure finite sign borel measure borel condition ensure expectation twice von notion generalize exist ready see sufficient expectation namely need come remark energy whereby covariance random distance covariance testing measure dependence weight characteristic joint particular expectation generalization let space energy distance moment integral view characterize independence xy satisfy additional term discussion suggest closely whether xy question corollary understand reproduce space mmd schmidt begin reproduce hilbert z reproduce rkh page rkhs value say notion map embedding finite borel measure chapter let rkhs kf alternatively paper unbounded finite sign restrict later consider finite clearly theorem borel probability measure introduce notion borel hilbert distance embedding q square mmd useful characteristic characteristic characteristic entire iff characteristic alternative see mmd employ variable et topological space q product xy hilbert schmidt joint follow show hilbert operator xy see cauchy schwarz x dp pt dp dp xx yy dp yy identify tensor k p negative type section definite proving equivalence mmd equivalence definite closely adapted nonempty z consequence valid convenience induced kernel say pt brevity drop induce abuse kernel strictly would suffice center distance express type z z proposition valid kernel center covariance fractional brownian page metric type embedding relation kernel hilbert kernel define whenever say generate clear condition equivalent require shift kernel f inner product proposition special obtain pp positive definite illustrate kernel characteristic introduce definition clearly test literature consistency notion relate coincide proposition interpret space w generate n z z clear jensen z z note finite z able sufficient w condition namely generate mmd marginal moment approach approach two independence energy generate kernel generate generate denote page link rkh base mmd rise possibly merely metric induced isometry z z simplicity bound endow note finite moment otherwise energy expectation infinite satisfied finite discrepancy addition must equal invoke asymptotic maximum xy yy dx x dx dx xy use form marginal result use embedding metric addition immediate existence covariance strong marginal p k p hilbert schmidt relation energy covariance remark space x xy xy distance naturally analogy moment appendix distance term review interpretation measure first include completeness characteristic covariance choice nonnegative borel clear correspondence weight integrable continuous invariant test majority obtain testing translation invariant kernel rkhs extend topological unclear two sample characteristic function independence satisfy term review establish interpretation kernel k q quantity exactly immediate negative checking check appropriate borel embed terminology embedding rkhs hilbert characteristic embedding turn distinguish strong kernel characterize equality informally distinguish outline empirical distance mmd define moment generate suffice establish test strong z I statistic recall generate distance key role characterize nan degenerate associate page l p desire class operator hilbert schmidt rkh trace requirement hilbert schmidt kernel pn precisely hilbert case independence xx center analogous chi square product p p class operator remark xy k p asymptotic type quantile distribution yield bootstrap operator extensively compute gram aggregated matrix concatenation center show square estimation threshold bootstrap provide indistinguishable performance obtain empirical page establish converge sum chi present page work design attempt besides bootstrap propose test distance bind independent remark sensitive gram associate kernel assess performance rkh test kernel synthetic investigate kind two differ mean dimension benchmark one univariate hard plot bootstrap indistinguishable outperform test far conservative average I list test pt also kernel set median
allow discrimination character thus remove alphabet contain character character contain character implication development text regular modern form character word chinese text exploit capability recognize regular visual computer instance approach capture denoise model regular wavelet image go one explicitly pattern major pattern position ideally probabilistic generative straight salient appropriately character regular share character recovery corrupt document task robust heavily corrupt page character yes course yes amount self corrupt relatively character character entirely unknown aim optical character character method character generative patch corrupt text static mechanism discriminate character representation contrast allow varying pattern account introduce character discrimination representation truncate probabilistic pixel position patch feature thought patch denote mix patch choose uniform entire patch shape pattern model latent mask mask draw bernoulli mask patch outside assign pm illustration graphical definition background require feature document different background observe appropriately probability histogram model individually histogram b channel compute across patch include potentially nevertheless compute give pattern entry pattern position cyclic positions equation generative histogram show patch class choose pattern mask variable class eqn pattern mask translate combined eqn parameter likelihood frequently parameter em optimize summation joint rule pointwise multiplication computationally computation combination latent simplify observe decompose posterior individual binary q denominator efficiently make principle complexity pattern patch size hundred thousand still exceed improve efficiency therefore pattern position class apply suit factored variational approximation zero approximate point patch class obtain define function position scoring selection give feature pattern define high mask space position mask approximation reliably consider area large value proportional pc py experiment find simultaneously relatively feature low character document evaluate curve histogram blue course heat map generate rgb patch five different character choose colored character mask fig background mixture modal apply class observe blue fig compare variance mask course show converge organize pattern g matrix color individually successfully experiment generate apply document display character b manually corrupt line create resolution scan document automatically cut scan patch interval instead filter process structure patch entry third pixel array cut segment pattern position way increase large character take visualization course represent character mean represent character algorithm pattern pattern document character mask b much character high class patch representation e representation character illustration character detection character document detect identify document leave patch character patch assign match report match identify type character define template result match match match match binary mask mask mask probability mask maximally reliable maximally pattern match instance position close reliable equal zero one scale match quality patch assign input match poor match perfect match use representation character match remove corrupt globally good bound store fig use good reconstruct document match match quality use match visible patch onto document illustrate procedure visible patch match prevent reconstruct one match character reconstruct reconstruction reconstruct character bound competition find reconstruct terminate match accept sufficient successfully document perfectly reconstruction character character character supplement however reconstruction part error supplement random manual supplement decrease unchanged example supplement pose partly extend segmentation processing comparison result baseline apply standard experiment document recognize character essentially corruption cause correctly performance document non character fig respectively approach fp character evidence character improvement training label however fair intend versa generate different
expectation gradient let formulate call equilibrium solution nash j nash equilibrium dependent nash assume realize payoff unknown node equilibrium node keep equilibria different optima bad maximizer capture price suboptimal stability whole system j mention clarity decision state whole uncertain payoff stochastic call robust nash point game state difficult state definition equilibrium find observe amplitude perturbation signal represent payoff instant update perturbation node get realization dynamic environment time repeat sure vanish learn appropriate initialize perform observe learn subsection discretize present payoff function clarity try function rate limitation fine vanishing coincide ode contribution step vanish discrete uniformly existence maximizer j hessian dominant imply hessian hessian integrable ode system trajectory differential equation ode recent development find limit write rewrite jk law zero martingale rate solution ode kt te lt vanish solution ode ode window specify define helpful discrete ode ode need chapter important result ode assumption converge trajectory order ode convergence constant however notion weak vanish learning sort almost sure time ode isolated stability gap sketch theorem theorem first vanishe exponentially bound amplitude ode exponentially independent nash equilibrium payoff deviation nash profile euclidean nash equilibrium close equilibrium nash precision payoff nash equilibrium nash constant nash hold ode reach equilibrium corollary follow ode inequality hold inequality together result constant depend player stochastic ode state dependent payoff assume ergodic drift ode ode almost sure surely stochastic ergodic wireless illustrate contribution interference channel receiver show receiver receiver back payoff correspond receiver interference onto payoff interference wireless channel payoff receiver contain power receive local payoff ascent htb general application utility time power time gain receiver draw dash black thick minimum height em thick fill black scale node black fill sum em black bend right east color thick bend north bend east black em thick bend north north thick bend east black thick bend west thick bend right east node west interference numerically run payoff analytically nash choose analytically represent function transmission structure payoff h jj please payoff follow j assume identically variance assume mean follow wireless tune slow fast tradeoff discussion omit initialized interference bandwidth dotted plot converge htb type payoff bit bit communication channel physical interference source expression receiver receiver numerical advanced etc without channel interference particularly scenario reward tune global optimizer system optimal respective satisfied scalar vector able equal nash minima state sample ode error close numerical generic wireless illustration convergence step size work introduce payoff wireless area amplitude vanish subgradient consider include et local focus nash equilibrium deterministic scalar scalar action wireless example consider user reward surely j follow lipschitz continuity jt j rate step j l j j proof l jensen r r r complete compact need shall remark maximum I impossible imply continuously differentiable gradient absolutely integrable detail condition please q use imply finally check condition martingale k k surely martingale integrable realization value j j j bound show l z j j z sure ode interpolation get line chapter equation vanishe length helpful obtain compact te lt lc te adaptation dt r j trajectory trajectory ode time converge vanish start process concatenation interpolation k gradient payoff martingale ft k mt f j ds exist k weak vanish follow form form invertible combination gain invertible gain almost invertible control transaction liu stochastic application mathematical introduction basic theory algorithm asynchronous optimization automatic transaction subgradient journal sensor rd international sensor incremental subgradient optimization mit mobile american conference noise application mobile equilibria mobile sensor automatic dynamical viewpoint http www liu nash equilibrium general payoff nash person sciences equilibria game economic study pp method pp j pp mathematics sciences york university continuous wireless network wireless communications international pp conference focus proof receive european grant agreement rs france france fr attention system equilibrium distribute become application equilibrium extend payoff develop algorithm nash trajectory define vanish provide conduct stochastic payoff distribute consist node model distribute node interact another payoff node system reward change node k contain action interact reward interact existence access reward maximized scenario interactive game iterative nash draw thick
dataset denote sample miss set least denote corresponding contain miss element variable keep mind mm mm I sub inverse keep instance currently represent sub part context directly miss gaussian maximization compute probability clarity iteration compute eq gaussians fill equal expectation observe assume denote equal q covariance result imputation value sake clarity miss miss regularization diagonal extend high cost burden somewhat nearby analyze computational cost em present require inversion operation pattern miss operation clear miss real missing since lin suffer fast pattern number miss barrier problem motivate reduce cost variable quantity determine another pattern miss observe pattern differ nearby miss compute miss consecutive want cost cholesky show pattern another two optimal minimize result summarize argue fast numerically stable cholesky low triangular triangular compute pattern question update pattern find remove row variable variable find row column add dimension minimize remove remove consecutive miss pattern find order discuss em computation conditional rely cholesky need order advantage inverse partition covariance part matrix conditional partition inverse go go miss miss follow inverse miss sub per em miss small dominate left remove add xy yy yx yy term remove miss update order suppose order order pattern compute cholesky miss pattern miss fast present parent miss pattern matrix visit miss pattern tree fashion thus span note constrain miss update simplicity use number true cholesky varie add dimension argue try sophisticated decrease virtual observed update set find al np scheme provide huge stability computation incremental accumulate incremental link depth may grow bad case never lead significantly exact solve interest summarize section sketch em pattern span pattern span miss pattern initialize covariance ignore increase expensive matrix computation naive em mixture gaussians handwritten optimize gaussian miss remove pixel e mixture rest sample select namely fraction principal keep random initialization ensure computer ghz architecture ghz memory algebra library realize train row grey imputation pixel capture sensible value uci repository validation behave systematically normalizing compare compare imputation expectation value learn imputation value neighbor alternatively obvious way nearest presence miss allow neighborhood base original obtain neighbor one report predictor feed discriminant optimize feedforward descent hide among weight among initial among ridge hyper decay kernel bandwidth dot three imputation mechanism regression imputation gaussian global imputation near imputation try keep set seem reliable miss value go neighbor illustrate generative mixture learn even mixture regressor greatly improve supervised htbp database financial learn company number preprocesse obtain contain record dimension purpose comparison validation test discriminant value imputation mean ridge discriminant ridge imputation imputation mechanism report statistically paper training mixture context imputation computation update span hybrid gaussian discriminant model significant imputation datum matrix problematic learning
condition relation statement addition statement statement establish outer pd solving problem suitable accumulation moreover accumulation minimizer suppose fx compact statement accumulation problem condition bound order affine minimizer view choice together boundedness accumulation exist recall index bound subsequence ki k j statement indeed let imply know use claim contradiction pass subsequence otherwise divide side take limit relation sufficiently since fact identity subsequence accumulation subsequence definition immediately lead relation hold subsection method establish reformulate associate quadratic ready pd let define arbitrary apply subproblem solve l yx k k satisfy step termination subsection method mention enhance solve subproblem b establish omit solve accordingly replace sake brevity omit point affine let also relation definition addition affine together sequence accumulation saddle function convex minimizer follow accumulation moreover observe bound together implie hence observation q limit continuity addition subsequence equality relation limit saddle statement due establish problem q strongly rely subproblem addition reformulate outer suitable generate nonconvex accumulation generate pd fx bound accumulation hold accumulation first convex statement statement let rest proof conclusion statement theorem conduct performance section apply sense code method matlab www perform matlab b intel cpu ghz gb ram subsection subsection vision bioinformatics neural processing see logistic sparse logistic integer solve aim therefore suitably solve effort project terminate termination criterion pd apply set termination pd next conduct pd compare quality approximate first solver parameter default first pd small medium data uci repository tumor internet magnitude discard apply bind pd sparse outcome predict predict detail column column ratio column cardinality average logistic second pd report six solution generally achieve low sparsity c pd c pd method three different size second equal manner sample outcome sample normal draw apply solve five let approximate pd result approximate pd slow logistic demonstrate quality approximate pd surprising c c pd c c pd subsection inverse real recognition see conditionally sparse covariance selection formulate integer follow eq subsection let clearly thus suitably pd eq computation pd lies give slightly pd solve subproblem arise step q solution v tc arise address termination pd outer termination accuracy random next conduct pd routine full eigenvalue symmetric routine parameter first experiment compare generate manner describe particular generate prescribe letting pseudo draw uniform covariance positive empty apply four pd covariance define pd instance four objective normalize give six respectively observe fast instance quality loss pd c time pd c pd c time randomly nonzero diagonal equal sample covariance ij j share diagonal pd method apply pd observe pd completely pattern solution see pd large pd performance pd method widely pre describe entry pd sparse set modify normalize name fourth loss pd last pd pd outperform normalize entropy summary experiment c datum pd pd solve problem noise cs formulate approach solve see apply pd thus pd suitably effort pd method subproblem arise address initialization termination pd obtain matlab set penalty termination criteria pd solve random obtain solver experiment observation let value apply pd solve adopt criterion mean error successfully detail recover cpu column two respectively observe pd large also second experiment except randomly orthonormal computational also pd c pd cs formulate integer control popular approximate solve study subsection form pd subsection suitably pd solving subproblem unconstraine solve conjugate method termination pd randomly initial outer termination criterion pd set associated conduct solution algorithm find solver increase accordingly pd mention warm start approximate pd resp apply plot detail residual average accumulate cpu cardinality second graph observe residual pd almost substantially method conduct generate orthonormal plot pd slow trade general minimization coordinate pd nonconvex accumulation compress sparse generally lagrangian simply replace pd augment lagrangian pd provide recover arbitrarily denote easy recover regularization relative fairly true remark paper problem optimality problem pd method solve subproblem solve accumulation sequence generate pd satisfie optimality condition nonconvex accumulation minimizer saddle moreover accumulation subproblem performance pd apply inverse compressed computational generally minimization compress sparse concerned formulate idea recently selection discover independence popular approach inverse covariance likelihood propose promise minimize logistic formulate solution continuously entry solve iterative square arise deal application compressed relaxation replace fail penalty decomposition problem accumulation sequence first convex show accumulation minimizer point saddle subproblem apply logistic inverse sensing demonstrate outperform organize minimization x solution
dimension formula density challenge even evaluate numerically stable introduction overview nest copulas tackle first copula family evaluation density three nest copula briefly address convenience defer copula dimensional later copula copulas copula responsible dependency several package way follow nest copula copula possibly argument bernstein transform sufficient proper copula completely monotone u frank see generator laplace property copula family family allow example power nest copula reference copula copula possibly condition frank fulfil generator belong proper composition completely generator eq integral compute theoretically especially complexity theoretically copulas child challenge leave generator compute integrate solve di polynomial prove back fa di fa formula q q come frequently denote denote recurrence align leave kx ns ks nx sign inner generator copulas fa di generators generators time derivative cauchy appear correctly allow compute expectation derivative solve monotone align leave coefficient cauchy align coefficient evaluate challenge log evaluate adjusted derivative quantitie part child copula exist theorem nest copula turn generator transformation transformation generator generator form proposition generator theorem generator note u slight copula outer inner generator copula concern former concern plug simplify term power q furth eq nest copula type formula plug kt simplify generator nest copula type follow family generator calculation show hence argument part generator derivative kt ks derive nest component generator generator inner generator coefficient polynomial derivative inner frank generator one inner generator appropriately shift part copula show hold copulas root copula copula refer copula case plug provide clear part basically piece one density numerical typically compute logarithm logarithm face density briefly nest sign term theorem st double typically correspond computed implementation package use kb precise package l two nest copula equal graphical insight copula copula copula dimension degenerate copula st product accordingly display plot copula sample minima display child plot easy copula behave see likelihood copula surprise marginal copula package directly fit nest copula life set figure nest copula htbp u section analogous nest copula briefly address working turn convenient consider level convenient denote copula root copula cd section transform scale grow south level style mm style mm level style level draw mm style draw style dot thick child child child child child node mm mm mm mm mm mm mm mm mm mm replace thus denote respect key challenge align find integrate challenge di formula suitable block di yield special strong lead partial moreover exist function position slightly l q summation write solve similarly eq pattern nest copula previous heuristic useful copulas trees representation branch tree branch branch polynomial polynomial vector equation term stand polynomial apply lb q appear author would like thank feedback align x l n
open since vanish branch use denote lebesgue denote gauss study prove side parallel axis characterize limit modern terminology absolutely continuous measure rectangle equal circle identify point thus random variable way result rectangle side axis deduce also measurable borel b bb main interested q informally negative observe factor agree characteristic variable write independence integral product q true elementary representation split variable put interval interval follow half definition use polynomial use integral follow euler ib proposition entire convergent proposition since product suitable even nonnegative recurrence expand power desired define verify reverse get recurrence simplify recurrence recurrence polynomial writing recurrence integer recurrence convergence series number determine also integer recurrence coefficient positivity correspond sum substitute recurrence sequence zero recurrence give recurrence appear satisfie sequence remarkable identity number polynomial number omit limitation l assume result remark integral representation expect confirm corollary make lemma multiply last integral representation proof proposition real thus dominate dominate uniformity q consistent analytic extensive numerical ib let immediate corollary q order numerically compute sum express compute inversion analytic half need illustrate technique fail fact origin fortunately sufficiently large corollary zero disk practice choose somewhat large proposition evaluate return evaluate equation arithmetic varied dynamically example sum alternative compute advance recurrence evaluation expense number depend interval recall sum value therefore may van root root constant measure detail omit q recall result evaluate function make periodic period function dt characteristic quadrature stepsize measure finite get simple odd computation eqn similarly course analogue proposition evaluate constant computation give difficulty extend motivation analytic independently implementation recurrence directly agreement statistical error carlo region latter van place sum show slow illustrated slowly computation critical suggest plausible seem strong imply pt precise consequently generalise cover dirichlet character euler product would sum modulus would course nevertheless sign mathematical sciences institute national van van pt explicit characteristic accurate numerical practical expression obtain accurate gram positive real sense critical assume
program traditional programming within calculate asynchronous parallelism boolean boolean logical fuzzy logic use form value explore collective behaviour delay representation competitive exploit brief give behaviour detail present require exploitation memory dynamical continuous discusse section use continuous production present discrete action final dynamical array cell lattice set update parallelism extensively genetic network exist extension connection influence body explore value differential wherein logical investigate fuzzy background fuzzy logic lead logical fuzzy life generate toward fuzzy behaviour form herein base connect include read write connection execute parallelism cycle finite machine programming graph structure evolution recently traditional symbolic expressive power represent purely rule direct token stack label string language token translate similarly program node label subsequently stack token specify matching condition action graph stack string token non form fuzzy representation radial condition hybrid system explore base contribution determine extent action node ga date rbf explore hybrid originally introduce aspect genetic since area quick typically boolean generalization machine dynamic affect behaviour wherein behaviour order consist regime around trajectory neither per analytical reach closely describe must play game wherein mutation change connectivity boolean typical high fitness increase approach enable heterogeneity type feedforward value value great rapid change regime occur change update step asynchronous possible asynchronous form wherein suggest realistic underlying discrete network certainly development fashion asynchronous device less towards asynchronous tolerance particularly delay insensitive scheme may hardware asynchronous see asynchronous feature significantly thereby aid evolve exhibit behaviour equilibrium asynchronous change asynchronous network significantly lower expect asynchronous lack complexity interaction transform reduce state asynchronous loose term take alphabet string reward unit predict fitness symbol environment condition input receive match logical match generate classifier environmental covering match current action present action represent prediction make weighted propose covering match current relevant alternating employ wherein exploration conduct otherwise exploitation occur compose execute environment feedback receive update ga ga great ga parent fitness produce usually mutation payoff error fitness average parent include parent update ga oppose step problem loop instance maintain weight input extra maintain environmental classifier weight reflect prediction rule correction control correction subsequently prediction enable accurate payoff piecewise constant boolean fast optimality see asynchronous rule node initially many connection input connection assign way current update etc cover apply l table connection node node input cycle typically implement cycle overall occur explore eight action extra node enable logical regardless join exploit within neural decide last update cycle node would randomly cycle status output create match mutation maintain mutation mutation locally evolve search mutation hand evolutionary algorithm mutation truth represent string connectivity maintain string form entry evolve string mutation adapt evolution connectivity map mutation initially uniform pass mutation rate within truth connection new connect either add node remove occur rule increment initially scheme whenever mutation mutation short term buffer input buffer past clear require present duration define pattern require length distinguish position temporal whereas memory state index maintain internal instead explore soft operation within explore hypothesis inherent content due asynchronous maintain input output cycle significant adjust computing addition short number network place cycle receive environmental input place trajectory different locally therefore environmental fall affect environment require resolve see environment binary bit furthermore task simply short start mechanism employ whereby goal f cell action environment contain non require action solve optimally internal state performance memory linkage recurrent direct np capable represent twice figure figure bit classifier figure mutation rate also rapidly around trial performance also rapid subsequently fewer adjacent increase complexity would three memory resolve evolve bit situation thus large memory optimally bit achieve step prior classifier internal increase observe trial slow inherent selection policy activity unlike number population converge mutation rate reach lowest htbp fuzzy logical fuzzy thus generalize continuous generalize fuzzy topology able dynamic network cycle give set time update q randomly fuzzy logical decide input logical membership membership update simultaneously fuzzy logic min commonly fuzzy mention potentially leave appropriate htbp fuzzy min asynchronous scheme fuzzy benefit figure result experiment increase change state cycle value less cycle due tendency fuzzy I increase initial change fall respective asynchronous update scheme change asynchronous converge around whereas aspect use adopt create assign connection input first message connection assign random cycle population initially empty cover generate rule match regardless join neural fashion binary whereby calculate represent integer reference operation upon input see fuzzy function list integer represent mutation rate evolution select fuzzy within rule along connectivity wherein agent within attempt short locate corner otherwise action would outside move whereby trial agent choose north south east west step state combine
always suggest function eq reference point reference point exist candidate contain generation however unnecessary density reference candidate proposal jointly importance procedure figure candidate work important consist depend need importance use pdf walk distance vary possibility delay rejection trade thank comment manuscript work partially project ref ref process ref ref try classical hasting choose set extension upon flexibility balance scientific implementation family core monte sample powerful markov carlo generate stationary coincide density pdf typically able mh try metropolis weight chain select advantage portion decrease acceptance domain carlo good performance interact adaptive basic modify rule analytic specify extension reversible candidate result asymptotic strategy consideration weight differently delay augment auxiliary stress remark upon flexibility rule within mix building pdfs present construction acceptance scheme theoretically possible design drawing algorithm affect detailed organize explain balance acceptance rule introduce comparison draw mh possible draw pdf movement balance correlate good proposal density whereas author mix together extended drawing candidate proposal analytic arbitrarily function draw proportional state chain scalar treatment draw independent pdfs therefore sample algorithm joint draw calculate weight normalize draw let satisfie detailed condition eq proposal express use choice finally scheme fair generate candidate moreover equation simplicity maintain sequel multiple mh approach weight acceptance indeed choose k finally fulfil e equation separately mh similar function provide summarize draw proposal pdfs normalize draw otherwise probability go back markov method generate condition write delta previous one exchangeable write total candidate correspond integral recall definition eqs expression symmetric vary assume generic balance suitable acceptance separately yx instance I possibilitie non function reference q obtain acceptance need possible previous consideration design reference point author consider drawback scheme since draw balance avoid sample draw calculate draw compute difference position balance show seem evident great say jump increase surprisingly decrease get bad consideration explain reference special fulfil kernel express candidate show reference integral obtain eq rewrite straightforward vary proposal pdfs choose therefore case necessary draw discuss depict pdfs proportional pdf proportional simple instance fig acceptance importance numerical compare pdfs drawing acceptance average markov chain table rw rw one proposal pdf proposal pdfs proposal pdfs table walk proposal densitie independent proposal important proposal weight observation extremely design good density scheme heavy call evy normalizing tail evy goal normalize constant via carlo chain apply three try choose suitable two proposal rw heavy feature average summarize c std rw consider p note possible combination function section different candidate acceptance movement next acceptance illustrate use great acceptance try almost invariant try acceptance rate correlation acceptance rewrite indeed obtain power increase complicated dimension mh provide pdf pdfs shape density shaped kind compare shape pdf performance sampler show htb face remain sampler iteration remain repeat standard mh sample markov draw apply walk proposal pdf I sampler mh standard deviation proposal table provide acceptance probability state average jump etc correlation rate c try mode jump rate table clearly mh since grow decrease jump rate increase acceptance obviously jump movement acceptance mode jump standard scaling candidate since movement namely evident vanishe grow great jump mode quickly face target clearly explore large portion sample study flexibility introduce analytic density draw propose acceptance provide example need reference satisfy balance theoretical infer observation consideration standard mh pdf candidate quickly consideration computational suitable apply kind target unbounded tail moreover advantage mh scheme black box algorithm simulation try independently choice proposal proposal
sx collection take uniform reduce computation notice calculation sx sx cardinality change one permutation state sx formula sx take q case ii site dash shape draw draw circle dash dash dash subset note connected neighborhood site measure probability x compute joint x treat later show herein covariance match choice example neighbor match covariance thick match circle shape draw circle draw shape shape thick thick thick herein field construct satisfie x g ray imaging herein space neighborhood analogue formulate size qx radius n nm I list pixel image pixel I restriction ensures directly generate verify abstract connect natural application meanwhile j site example necessarily albeit necessarily follow mm right dash dash dash dash dash dash dash dash verify connect abstract enough illustrate setup I cardinality denote I I simulate I nearby site x k I uniform random I notational convenience valid site compute x generation target generate automate public computer apart prevent intended abuse automate agent appear character digit easy difficult wu propose name specify marginal pixel embed quantity conditional formula real yet generate markov end generate include detect coin flip unit coin tail classify coin flip either person generate coin coin flip sequence count approximate total period expect way behaviour average covariance flip follow flip simplified algorithm coin flip head analogue wu problem hide still forest surveillance equip capture camera gap correlate either correlation forest observation overhead pixel small area generation use forest target function filter resort specific generate regularity right consistency permutation guarantee follow sufficient regularity singleton proposition neighbor property property site site neighbor site zero type relate marginal singleton valid condition constraint therefore proper equivalent appendix side sx verify tx x sx require singleton simple corollary condition apply also necessary assume neighbor probability x satisfie lemma give form assumption I n repeat quick field definition fact ix ix permutation neighbor rule hence simulated order simulate matter site site site neighbor sufficient permutation give u u use u thing u let I ix adjust component u I triple integer formulate equation triple contain n hereafter write cyclic site neighbor solution canonical permutation n pmf x sufficient permutation n necessity h g satisfy distinct solution canonical linear I without guarantee I prove generator I permutation x point albeit n site neighbor condition joint x imply I ix loose degenerate structure equation canonical structure exist dimensional j k distinct construct satisfy multiplication triple apply n similar say l u separately u u u k l j equation verify also one summarize nonzero solution nonzero formulate satisfied investigation guarantee verify contrary obtain equality n site neighbor hold u application site neighbor site ix I x I let site determine condition I permutation x follow follow x form pmf directly lem necessary formula uncorrelated uncorrelated statement follow immediately formula assume x produce field remarkable concept extend system site ss uncorrelated independent regardless rest state exactly field definition field condition sequence I site formula permutation choose site I compute I treat recursively repeat compute u x simulate value x generation field x make site gain new permutation extra condition actually field site markov field uncorrelated n prescribe dynamically take guarantee site behave every much treat employ reason site multiplication rule property break concern multiplication big neighborhood alternative h nonempty proper pick otherwise subset I site h consist site nonempty since induction ib general nonempty exclusive subset j mb connect nonempty way choose nonempty b start site b pick within b stop repeat neighbor b procedure must stop nonempty exclusive replace third repeat paragraph mb mutually exclusive exclusive b component contain site henceforth nonempty first exclusive I choose site proposition distribution probability ease subscript fourth fix mass therefore q follow section corollary section section herein new quick rely pass convergent method mass site base quite maintain site successfully object limitation impose covariance property field markov derive simulate field field phenomenon use statistical mechanic application field tv image image functional imaging web extraction etc reader researcher synthesis classification optical optical character match recognition recognition video surveillance sparse language chinese interested inverse motion precisely formulate mathematical certain various estimator need simulate simulation smoothing technique chapter technique random field typical certainly exceed modern computer simulate field resort impose discrete field need gibbs sampler briefly speak realization site local configuration sequentially update hundred produce deal propose new incorporate give function site covariance nearby mind configuration discrete simulation far correlate carlo simulation impractical establish covariance field maintain ease proposition site portion random sequentially feasible precisely simulate conditional portion construction actually construct pass dramatically random object field generate condition base singleton particular investigate permutation property case want covariance pair site form auxiliary collection ensure field regardless order component really setup simulate unknown portion discrete valid multiplication rule configuration probability formula use kolmogorov permutation condition condition kolmogorov extension existence vector satisfy n condition site site setup I portion x way k consistency mass list follow permutation g I k h kx x happen x order
integrate plane away derivative vanishing know result likely near useful mnist dataset intersection present behavior embed cause integration disk case intersection involve subtle manifold near neighbor onto onto decompose exactly denote ball center tf follow eqn eqn eqn projection du derivation fourth relax result second eqn third replace replace intuition behind derivation slightly whose projection follow mm derivative intersection e boundary intersection point point boundary boundary effect manifold produce proof smooth nonempty boundary smooth continuous rt angle normal cumulative edge limit close move normal boundary equivalent effect manifold intersect co actually regard piece manifold together advantage view allow intersection continuous case manifold boundary corollary let smooth manifold boundary f eigenfunction intersection continuous within piece manifold useful probability constant inequality appendix asymptotic arbitrary change singular within rescale seem counter intuitive require inside estimate differential operator play likely subtle case implication result place put converge necessary derivative exist impact numerical sample infinite near x see mean eigenfunction laplacian automatically boundary experimental direction hence directional limit surprising reason imagine eigenfunction laplacian near singular structure preserve intrinsic distance edge laplace numerical phenomenon give implication eigenfunction clear investigation eigenfunction situation analogous manifold neighborhood least minimize quantity orthogonality contribution bound department computer science usa laplacian construct considerable exist interest argue boundary aspect geometry realistic boundary process bounding manifold intersect smoothly near boundary manifold together direction laplacian near manifold phenomenon somewhat interior laplacian laplace exhibit implication volume difference large contribution behavior distinct comprehensive understanding toward lead significant deal non key year become intuition surface situation riemannian reflect boundary basic arguably intersection manifold come happen right computer graphic edge surface whenever phase transition boundary occur bound constrain range motion ambient negative intensity paper laplacian near inside gibbs operator different scaling behavior ignore globally despite relatively locate contribute fine believe comprehensive understanding complex design gain acceptance task include supervise mathematical many construct datum laplace heat obtained study infinite reason edge type somewhat related development considerable recent aim algebraic term guarantee special manifold recent interest reconstruct topological manifold line relate provide singular appropriately scale section bandwidth reference infinity appropriate differential manifold exhibit near approximated scaling counter relative inside eigenfunction number application subtle proof smooth answer brief boundary boundary limit eigenfunction smooth one paper intersection intersection seem effect eigenfunction straightforwardly walk normalize graph discuss popular derivative expression involve boundary condition restrict manifold simplify exposition intrinsic boundary smoothness way associate intersect interior point exactly smooth intersection type exposition intersection happen interior edge j technical reason singular manifold moreover intersection tangent tangent piece union notice manifold smooth kf ic px sample build vertex assign q gaussian euclidean analysis weight unnormalize laplacian laplacian primarily involve bandwidth limit various analysis regular see version point limit bandwidth analyze
uniformly cdf univariate cdf note univariate density write express satisfie j f jx sample I estimation function inverse covariance matrix transform light natural candidate large dimensional relative alternative use tradeoff choosing estimate sample standard estimator jk replace respective normal transform truncation tune amenable main covariance entry theory jk jk positive constant detail give show frobenius j nonzero precision precision probability particular rate rate graphical however recent equally flexible approach normality restrict tree nonparametric lebesgue measure identically sample range f bivariate density univariate family forest forest density univariate density log minimize entropy forest satisfy forest hand constant forest connect node find span setting way stage add connect yet visit stop edge replace plug bivariate univariate liu edge degree automatic need adopt split set marginal carry evaluation j univariate estimate cube numerically evaluate kernel point choose th approximated make concern care need truncation use ensure liu span k cycle k j k f full overfitte edge induce liu forest validate f evaluate solely hold hold forest construct estimate carry iterate edge obtain compute maximum span hold estimate property type density old exponent error kernel yield kernel density kernel choice family density access liu omit technical main hold notation follow fact increase increase rate excess still note estimation state order value long integrate slow technique correct work investigate achieve minimax protein study gene analysis assume know sample discussion pre gene pathway treat glasso graph wide parameter suggest lead different regularization compare path regularization evenly space similar subtle nonzero compare value column regularization glasso graph close fit glasso vice gene typically large consequence regularization lasso step gaussian sparsity obtain glasso well surprising univariate unable held likelihood hold log likelihood glasso maxima inferior kernel figure graph graphical automatically hold clearly big co yahoo amazon com collect stock yahoo finance finance yahoo daily price trading denote stock day consider replicate series share compare stock classification stock stock stock stock stock stock stock stock stock show correspond stock stock clustered stock interact indeed example two yahoo target forest graph glasso forest color glasso span show edge estimate glasso forest glasso node good graph regularization edge neighbor remain node point density full sensitive exploit estimate graph stock make dimensional density cluster outlier describe forest stock result display difference common estimate glasso difference forest graph draw hard conclusion effectiveness independence relation fully surprisingly graphical high piece relate spline decompose one identifiable dimension still tractable see markov function equip penalize induce solving require dimensional efficiently log analyze another conduct markov factorize clique neighboring clique exact expensive strategy structure global theoretical paper undirected graphical least simplify tractable consider efficiency rely function distribution acyclic bivariate together clearly possibility nonparametric potential work function accurately support method estimate selection approach discrete normalize strategy variable framework conditioning collection cart go cart estimate conditionally model idea build partition cart classification leaf glasso go cart reduce vary variable explanatory variable simplify straightforward mixture forest read assume joint distribution assume forest forest forest extend handle extension graphical approximation yet experience show incorrect estimate wrong graphical draw finite already essentially clique number discuss flexible nonparametric use estimation graph construct graphical discrete alphabet log linear clique parametric distributional yet alternative particularly build two allow arbitrary distributional restriction copula extension kernel graph forest nonparametric expense two
become active stepsize value small cause constraint become entry support small critical sign accordingly homotopy jump one critical value updating solution homotopy come compute element homotopy homotopy close multiplication experiment propose homotopy algorithm update homotopy cost homotopy solve minimization adaptively every homotopy step accord homotopy add element shrink homotopy act homotopy attempt desire adaptively shrink homotopy path sequence homotopy homotopy value encourage active reduce fast achieve inactive weight change direction homotopy predefine summary solve homotopy problem adjust initialize want homotopy index active inactive follow select fast initialize correspond denote weight select select rate reduce form solution homotopy inactive equal value active illustrate various homotopy choice block length accord setup sec homotopy way homotopy size become small rest select select homotopy select toward methodology optimality support toward along straight move direction maintain respective subtract alternatively inactive become maintain optimality suggest maintain weight value priori scheme adjust setting stepsize cause support select inactive large set repeat update direction stepsize updating homotopy termination main homotopy system compute multiplication element homotopy update rank cost every step multiplication homotopy homotopy performance h accuracy high quality signal cost comparable h quickly expense h exist solver present solver fast sufficiently warm start old warm outperform h recover type random accord generate signal wavelet transform block toolbox interval disjoint region add uniformly transform piecewise signal piecewise signal haar present result haar transform nonzero number wavelet element signal length signal signal jump select amplitude divide three add transform generate jumps transform present fig decrease near generate entry select entry reconstruct algorithm describe example block haar perturb sparse wavelet measurement although weight accord make instead rule good manner homotopy http edu homotopy reproduce weight follow priori select adaptively denote previous homotopy shrink large dense norm proxy support every multiplication identify change support rank update use inversion update every accord outline iteration homotopy solve equivalent iteration weight fix main involve update inversion alternate problem detail solver package use solve old warm solver tolerance order use shrinkage see h iteratively solve use use old warm termination tolerance accommodate weight solve constrain initial weight warm default optimality multiplication summarize select use update matlab implementation reconstruct db reconstruct performance h recover simulate procedure record figure figure signal signal plot first weight superior performance plot row display although iterative h solution h top solution five iteration top row haar wavelet transform randomly perturb wavelet signal measure gaussian gaussian reweighte unweighted perform reweighted row iteration row top multiplication fig fig signal application plot count initial depict since present count first compare matrix product least application count appear count count count third count count iteration count iteration h appear block signal row utilize row iteration runtime display nevertheless runtime small third first reweighte unweighted five reweighte row reweighte third row iteration reweighte unweighte iteration third row brief observe recover well homotopy problem computational iterative scheme iterative warm homotopy efficiently algorithm iteration quickly expense sec homotopy adaptively homotopy method iterative term computational homotopy embed thresholde iterative solve shrinkage generate solution previous determine stepsize entry respect soft thresholding shrinkage shrinkage algorithm iteration adaptively accord solution offer parameter value code method gray compressive measurement setup model generate sparse wavelet wavelet noise entry three presence db snr original peak application experiment third reconstruct warm employ last reconstruct solve code value observe modification cost simple scheme exist enhance scenario detailed regard scope east image north south east north west rectangle south east north west rectangle original inside update reconstruct count average experiment north south east north image south east west rectangle symmetric extension inside box reconstruct image count grant foundation often solve many replace common update two homotopy reweighte present change homotopy replace one second propose algorithm solve weight adaptively select signal algorithm along homotopy change support single homotopy compare solver method yield reconstruction quality fundamental recover measurement observe signal noise obey incoherence program solution term keep commonly pursuit denoise yield enhance weight adjust coefficient original nonzero location elsewhere since location nonzero priori critical perform iteratively solution next appropriate value use compute iteration arise solve solver homotopy homotopy homotopy change linear homotopy path homotopy cost homotopy standard homotopy trace support homotopy respective follow homotopy present weight homotopy update solution change view strict elsewhere solution maintain optimality must keep direction violate indicate add cause
localize multiscale conclude brief graph oppose graph prove particularly visualization numerous science area expansion transform definition enable important classical ne separate signal application way connect thresholded physical vertex describe especially supervised connect distance e signal represent unweighted one associate connect original graph combinatorial laplacian element weight edge laplacian difference neighborhood eigenvector real eigenvalue consider assume fourier expansion function eigenfunction expansion eigenvector laplacian classical analysis carry frequency slowly whereas far associate eigenfunction rapidly provide notion connect eigenvector vary I eigenvector associate rapidly value demonstrate eigenvector eigenvector signal vertex vertex b eigenvector sensor positive come zero non normalize laplacian sensor graph latter laplacian associated laplacian fourier give equivalently different vertex domain useful spectral domain refer signal heat analogously analog fourier decay closely approximated fourier coefficient e way b road represent blue positive negative come vertex signal domain graph domain plot take emphasize geometry incorporate differentiable manifold definition differential operator operate calculus mathematical precision intrinsic examine normalize normalize manifold vertex value note quadratic norm neighboring vertex large return laplacian tell also minimizer laplacian smooth interpretation laplacian carry notion frequency connectivity laplacian transform eigenvector box demonstrate content depend mm signal vertex edge show signal bottom spectral domain smoothness graph signal structure least respect intrinsic see visually ii laplacian low frequency fourier transform popular option define graph partition every connect eigenvector n spectrum also carry high eigenvalue unlike normalize laplacian connect may graphs vertex vertex walk converge go infinity walk asymmetric laplacian identity laplacian due detail basis clear normalize nice contain fact eigenvalue useful extend dc review develop multiscale transform filter start extend filter localize vertex domain classical frequency combination complex transform domain correspond convolution domain notion also generalize theory write filtering discrete version many arise solution variational problem discrete reference relation discrete filter arise differential application image processing mesh example particular application mm uncorrelated additive wish enforce clean view example noisy pass x pixel connect horizontal vertical set neighboring pixel similarity noisy weight take perform diffusion smoothing display row image comprise filter smooth area image classical smooth laplacian via filter signal domain signal neighborhood constant vertex domain relate graph filtering filter also interpret domain yet short distance vertex number comprise connect filter polynomial filter relate filter vertex definition convolution one define product graph convolution multiplication graph translation define change generalize translation delta center weak way convolution center vertex remark act define translate apply second normalizing ensure preserve control localization vertex decay distance increase property translate heat translation operator operator graph eigenvector h translate c heat figure rely frequency content represent eq q define generalized laplacian graph discrete nature however graph domain localize analog set instead transform setting assume unlike generalized generalize require entire diffusion operator example discussion wavelets intuitively apply different power heat describe flow heat proportional weight signal heat vertex heat vertex variable entry far apart small e justification structure increase nice illustration heat notation q initialize dyadic power heat diffusion interpret original show k e kernel dyadic power dyadic power importance wavelet section multiscale transforms signal graph require successively intrinsic geometric connectivity spectral sparsity transform give fine reduce vertex preserve process separate closely reduce assigning connect new additional first often refer special bipartite connect subset bipartite notion bipartite technique algebra mention lee weights walk transition greedy recursive repeatedly part sign subgraph minimize connect generally bipartite graph subset large refer reader therein review literature connection domain closely extend concept signal graph show class reconstructed reduce localize transform design wavelet analyze computer wavelet wavelet graph dependent wavelet tree graph wavelet channel wavelet filter fouri design filter domain wavelet transform thus exploit transform graph signal graph locality spread principle trade signal open recent spread define center vertex interpret mass pmf signal geodesic signal affect purpose example trade exist graph wavelet normalize laplacian one wavelet reconstruction expansion resolution decomposition remainder design simple example broadly type vertex domain design transform base graph vertex localize transform filter vertex node transform transform wavelet wavelet transform unweighted graph compute node neighborhood weight neighborhood outside guarantee wavelet function localize location minimum wavelet transform wavelet originally signal approach set odd node compute neighbor prediction neighboring wavelet balance hierarchical tree partition define level modify graph wavelet base eigenvector wavelet quadrature mirror idea graph construct diffusion wavelet operator basis level schmidt scheme translate version discuss spectral bank synthesis filter prototype graph bipartite intuition present design wavelet short center constant wavelet across vertex wavelet vertex depend exactly wavelet interval take wavelet graph wavelet function vertex wavelet low pass filter wavelet generalize pass satisfy q transform spatial two transform classical invariant necessarily wavelet spread transform scale wavelet scale spatial transform change location take average different graph regular wavelet wavelet axis axis wavelet localize localize spatially wavelet understand spatial signal graph htb ability graph wavelet transform piecewise signal unweighted road color represent well kernel design toolbox scale coefficient respectively magnitude concentrate mostly near pass signal fourier transform f f piecewise smooth severe unweighted scale wavelet cluster area review way generalize elementary set core processing localize multiscale transform generalized operator section multiscale transform review process fairly frequency extend transform surprising property moreover thus quite challenge ahead briefly mention extension method paper incorporate structure way yet construction affect localize transform signal mention clear graph domain short diffusion analyze transform operator useful high adjoint scale confirm fast transform computational signal transform involve computation require eigenvector normalize graph area approximate operator open include fast fourier transform datum deep class wavelet see open field signal localization generalize design transform may heart localization vertex appropriate notion non unlike classical euclidean
subject reconstruct robust face b represents argue solve hence necessary effect consider aim well sparsity block minimization pose tradeoff counting hard work develop tractable relaxation group replace relaxation mix derive entry main understanding relaxation give surrogate however relaxation surrogate depend raise whether preferable relaxation minimization question duality derive relaxation introduce derive lagrangian original relaxation minimization problem importantly lagrangian respectively program gap correspond lagrangian relaxation minimization provide relaxation sparsity lagrangian minimization optimal primal objective relaxation lagrangian generalize lagrangian value choose value describe introduce sparsity otherwise block contain non zero express introduce matrix entry belong otherwise definition reformulate enforce aforementioned original equation constraint g ensure indicator entry lagrangian lagrangian lx unbounded coefficient consider entry lagrangian dual simplified program make going eliminate min operator verify lagrangian give notice going value relaxed real relaxed real constraint verify wise lagrangian hard primal duality gap lagrangian dual dual lagrangian primal bound dual problem np duality gap primal moreover increase conservative primal unchanged duality may analyze effect conservative estimate dropping drop box see conservative reduce entry sparsity lagrangian entry problem importantly solve wise solving precisely lagrangian minimization solution desire conclude e duality problem bind substitute group minimization lagrangian wise conservative relaxation entire surrogate sparsity interesting choice sparsity norm consider norm non block minimization mx unclear finite minimize entry mixed datum face minimization top edge indicate percentile mark box potential explore corollary experiment experiment analyze bound explain tight duality reduce lower grow linearly function trend loose increase nonetheless ability solution pre condition norm minimization able obtain duality hence error image model image system minimize dominant optimization special problem conservative evaluate ar manually align subject contribute un training un image whose great present group primal mixed sparsity entry get consider assign block subject classification minor notice image classification state art correct correct un present minimization allow interpret several relaxation np hard primal lagrangian equivalent derive relaxation sparsity ability per contrast either condition perfect relaxation hard verify importantly condition pre correctness hope prove important instance verification specifically explore tight
regression mle ie adversary solve first square attack attack estimate column dimensional column c polynomially apply produces scale high recover pt pt theorem corollary conjecture attack wide range attack far want release attribute database relate code gender partly arise medical attack record contingency study like classifier risk minimization include transform linear format amenable algorithm surprising solve nonlinear formulation attack distributional consider statistic attribute relate boolean global sensitive record vast statistic formal privacy guarantee security attack seminal rigorous study tradeoff utility access information require roughly datum little recently design private algorithm typical evaluate seek notion privacy bound differential rule notion privacy reconstruct attack sensitive want sensitive relate gender partly arise medical database individual sensitive assume database concatenation construct attack private consistently estimate attribute change error randomize mechanism attribute adversary attack run instead simply reconstruction class natural reconstruction attack construct namely typical minimizing least decode lp attack answer per attack query answer simple adversary coefficient component seem database counting query database inner attribute hash example adversary ask sum database hash statistic unlikely publication paper attack attack appear analyze release marginal contingency marginal distribution subset per attribute private attack fraction arbitrarily remain noise generalize recently attack greatly expand applicability attack setting specifically reconstruction attack release give record boolean multilinear representation table error estimator broad estimator record estimator linear release attack statistic like obtain highly nonlinear like release sensitive attack distributional assumption statistic statistic attribute total boolean subset size submatrix entry evaluate statistic degenerate boolean add per entry attribute mild condition case bound base square lp decode handle fraction entry arbitrarily insight hope broadly release record fact linear know arbitrary value depend public affine variety obviously linear attack classifier see observation optimization every record function must lead equation form first technique add mention attack depend attack give draw underlie analyze geometry arise attack relate product refer matrix show product matrix entry row product matrix satisfy introduce square lp noisy degenerate boolean release attack either lp analysis analyze matrix arise associated clarity discuss attack estimator hamming distance letter vector denote row concatenation eigenvalue arrange value z ns subscript non early often dependence variety processing estimate output summarize sequel entry regression entire attack let orthogonal orthogonal define use sa affect norm entry map som privacy decode name stem lp attack slow attack considerably strong attack geometry euclidean say cauchy schwarz hold say decode bind operation bind describe entry lp upper euclidean nd time bound non degenerate boolean standard background represent boolean multilinear polynomial represent fx x degenerate multilinear non degenerate degenerate constitute degenerate depth evaluate database entry allow repeat let restrict index attribute degenerate attack section degenerate mechanism add entry attack achieve privacy every database attack privacy run transpose reduce database reconstruction boolean degenerate boolean kf boolean polynomial multilinear multilinear multilinear represent degree degree multilinear strictly less aid function jk correspond change position row affect singular row matrix tt j ki define row correspond follow contain vector approximation noisy adversary try I adversary set adversary know goal iterate logarithm part depend mechanism non run outline equation database analyze need follow constant satisfie exponentially probability show exponentially recover noise translate decode solve slightly equation euclidean euclidean make k decode attack matrix appendix notation f decomposition establish definition matrix define attack part boolean depend database add attack decode outline establish section multilinear polynomial high repeating least exponentially euclidean adversary recover attribute analysis bit translate bind reconstruction attack rely geometric property certain matrix conjunction build upon summarize establish product consist row product kt j product dimension order uniformly uniformly random call iterate induction main establishe x matrix independent dimension nc random wise appear independent random refer multilinear step behind simple multilinear multilinear represent representation multilinear multilinear ki ji I ji variable multilinear p repeat coordinate get ni ji ji j hold q dd multilinear constant depend x j triangle last ki thus equation note alone inequality singular independent therefore exist order theorem schwarz right theorem estimator k assign differentiable nm estimator estimator similar natural may
acquire uniform uninformative commonly problem informative open causality consider constraint belief first equivalence maximal constraint network incorporate path markov chain mcmc neither deal dependent incoherent briefly bn dag graph connect equation p iv call drop index equation ignore triple vertex b essential equivalent skeleton reverse connect set imply dependent take logarithm method k log ignore maximization become pair path connect important devise path mutually exclusive knowledge v single belief table path bn cause associate belief causal relation prior mass probability avoid pair belief incoherent bottom induce belief part dag configuration belief describe computation configuration determine p graph dag equation stem mention computing factor preference assign score first employ appropriate uninformative prior belief uninformative uniform prior property factor count intuitive reason everything else prior graph desirable drop score place satisfy dag solve good dags certain count dag grow exponentially option number specifically avoid expensive method configuration never may even variable estimate arbitrary order huge dag upon developed exploit configuration variable become hold effectively path become z yu yu yu u sufficiently negligible formally clear contain node keep add possibility create appear path pair yx yy u belief appearing appear j node configuration influence negligible path directly affect compute graph show variable nod dag random factorize estimate number dag recommend laplace kl sample provide determine approximate node size path r belief sample dags laplace correction divergence measure distance claim belief node relatively l dag cost time memory set relatively approximation infeasible number run small step complete dag constant measure divergence average small reason relatively small variable approximate compute value jj marginal equal belief belief path semantic detect discuss specification impose eqs satisfy axiom belief coherent incoherent equation typical uninformative certain configuration prior preference uninformative prefer information minimize formulate optimization problem efficiently iterative proportional fitting case incoherent equal marginal belief coherent belief ignore seek marginal close input incoherent belief amount measure call provide conduct reasonable comparison belief belief value bottom close belief show configuration score configuration z g yx cost solve dominate practice solve efficiently memory belief obvious would way upon seem however uninformative respect result also u I large path belief practice detect however belief give preference relation example correct suggest belief incoherent probability axiom implicitly reduce belief example px px belief thus imply definition c configuration simple configuration common case method space dag dag hill dag search extend dag configuration closure store closure compute visit fast complex graph trivial dags total overhead straight dynamically closure closure query reduce advantage extra belief improvement belief operator also change configuration application allow swap belief equivalence result describe proportional repeat randomly number dataset dataset scoring score dag uninformative prior equivalent sample ess result plot true network orient high prior perhaps notice informative skeleton tend association tends run correct uniform expect belief identify true effect exactly belief prior role generate belief component appear coherent incoherent path overlap contain result total consider large path randomly assign true remain probability uninformative way remain repeat component incoherent dag search ess start uninformative informative prior swap operator case informative structural hamming introduce markov dag use correctly orient varied path belief varied within incoherent dataset incoherent belief belief similar incoherent small uninformative informative reason score prior ignore usually consider maxima swap happen whole counter intuitive increase size suggest ess parameter reason behavior
j gaussian pdfs I f read respectively intractable message occur message also vector eq pass I pass nx ni code equivalent decode belief proportional gaussian reduce bp rule output gaussian identity specifically bp gaussian message coefficient update manner rest mf f split tractable exploit correlation representation mf message n f compute belief I nx n code part back mf channel mf dirac basically include message pass scheduling start correspond message correspond algorithm pass variable message factor node back message pass end decision belief wireless parameter carlo simulation derive coefficient encounter ep instability ep require multiplication pdfs division pdfs message channel compute versus combine mf bp receiver employ pilot receiver algorithm show pilot essentially bp mf soft channel notice snr value would account bp mf receiver active evenly spaced pilot pilot convolutional coherence channel decode message pass bp include particular derive four message apply employ consider computational stability conclude receiver bp mf em variant effective receiver project support project contract mobile national advanced foundation process wireless mobile project grant grant within national research university university technology iterative receiver introduce inference belief bp field mf include propagation embed bp mf modification consider algorithm probabilistic four pass receiver numerical wireless receiver mf candidate algorithm design advanced receiver requirement motivate successful application decode devoted receiver module design soft well inference prove useful tool detection belief decode probabilistic bp mf usually pass mf bp propagation mf pdfs refer ep belief pdfs exponential attempt unified region energy hybrid message pass combine bp decoding purpose either framework ep receiver wireless ensure belief obtain bp expression reduce message computation bp mf similar member specific exponential belief ep modify message mf belief constrain dirac delta mf maximize unconstrained graphical representation establish baseline derivation message receiver represent r alphabet symbol j alphabet finally aggregate symbol x send channel output vector h channel inter
reward draw rely tu result mab suggest order time quantify performance empirical need currently investigation answer regard chernoff mean sequence x il continuous initially propose lemma large deviation follow ex event notice finally apply combination beginning conclude occurrence anomaly tt deal reward provide since inequality decrease negative write sufficient prove analyze positive ht g term step subsection tu upper occurrence anomaly play equal anomaly appendix mild include end introduce armed bandit mab learning paradigm popular sense allocation reward channel name utility index highest associate sample collect arm confidence assumption sequential face exploration choice decision maker herein focus maker discrete value reward quantifie maker behave problem multi armed mab problem exploitation mab value arm utility past quantify interest index index ucb index provide optimistic ensure select decision maker build policy selecting bias deal remain challenge tackle matter inspire aforementioned multiplicative additive expression main contribution paper analyze multiplicative make available arm collect choose optimistic consider low non restrictive rest refer suggest policy outline independence reward suppose instant select arm instant shall refer seek regret expect playing refer play instant instant present contribution multiplicative index sample machine ex scaling factor index round last lead index machine index play adopt convention history index analysis focus determine sub consistency armed say good policy matter ensure expect expression remark suboptimal upper expect regret k draw prove
control feature channel kullback skew pp feature spatial rate gps sequence tc york rank variance analysis b inversion overlap base method mm toward understanding b g mm team statistical http www co l aid interpretation validation accumulation measure r mm e prototype f h gps mm comparison rank pt proof definition correlation high de email mat cl di universit associate recent nonparametric estimation form area probability nonparametric kernel us frequency spatially nonparametric estimation short nonparametric assessment occur achieve km al show region appear patch locate part secondary et et probability poor area et heterogeneity appear indicate potential great well aim work area area require choice methodology area identification region improvement adopt available science e social sciences standard book focus mainly distance multivariate variable multivariate associate density assume probability option adopt datum california link correspond identify retain parametric construction association cluster mode formulation concentrate kernel type conceptually normal function symmetric outcome parameter simple different cluster separate low refer describe briefly continuous unknown concerned denote position observe form move range cause move accordingly regard increment cluster early vary idea additional specification language r nonparametric adopt outcome bandwidth step toward bandwidth influential vary quite notion build geometry line sequence segment path include give step span admissible mode determine end early allocate sample mode subset core belong aggregated distribution represent density core allocation q option depend whether core represent refer diagnostic technique representation object spatially allocate cluster allocate cluster distance adequate method explicit diagnostic play allocate allocate refer index maximum partition diagnostic index time correct figure tree diagnostic lack surprising give proximity visible variant take exactly examine implication exploration log quantity meaningful outcome adopt quantity measure adopt produce association event examine way regular option evaluate observed choice computation grid region lead non rectangular grid comprise area cover event last refer figure scatter label drop explain early htb selection grid overall provide provide evidence consideration cluster association last focus refer form point feature grey cluster associate involve produce city locate maximum blue locate gray figure association increments htb diagnostic us rank testing classical whether assume population normal tie value make use produce measurement outcome provide group assume nan exist difference non level repeat correction significance regard htb min green gray red cluster represent red gray city low practically green estimation day involve result necessary number day moment great cluster versus index incorporation activity mainly mechanism process event magnitude magnitude comparable magnitude region occur km sharp apparent
domain disagreement source error network discovery publication view reference proposition universit france universit france da arise differ generate source label imply task seek vote da da seek take trade majority kullback leibler capacity majority structural source mm target task mp source ia tr low label domain infeasible hypothesis domain david follow marginal quantify equation capability guarantee usual vc trade traditionally vote hypothesis consist lead generalization essence order error pac expectation error major empirical sg kl domain pm restrict distribution specialized bayesian weight precisely spherical centered expected gibbs give prior kl theory learn weight express pac contribution pac da structural marginal h h suggest instead focus nevertheless derive follow propose trade error classifier p r joint minimize da da pac pac refer independence rewrite expectation g restrict classifier posterior loss center result pac linear need develop rely equation da consequently minimize side gradient evaluate
sufficient necessary omp recently show recovery necessary drawback stand require propose easier read dictionary inner product atom definition ols imply sufficient ol wang sense mention necessary omp ol fail satisfied ols observation vector necessarily omp ol select atom omp numerical simpler strong base coherence necessary ol atom follow refer stand linearly dimension index submatrix column index value x dimension omp vector must say omp ols understand procedure search ol ol select omp pick atom correlation residual sequel different formulation however confusion straightforward omp ol throughout use omp notation whose linear combination eq atom hereafter ensure success use follow recovery succeed weak sense point exist end support number iteration omp omp include ols present perform delay wrong decision failure exact derivation condition narrow algorithm specific condition whereas select iteration constitute necessary condition omp omp recover constitute bad omp ol drawback evaluation carry operation ensuring support easier mainly distinguish type restrict coherence condition exact step omp prove omp succeed tight dictionary term omp select wrong atom us virtue theorem result ol derive sufficient coherence dictionary condition ensure success iteration also wang show dictionary recover summarize succeed exist wrong first coherence success select atom term iteration exist term select iteration atom specifically resp sufficient omp resp ols last iteration differ omp ol omp derive upper restricted isometry project ol connection coherence section prove proof omp sufficient state depend state derive therefore prove theorem characterize appear implementation generalize isometry rip project project rip rip rip asymmetric restrict isometry asymmetric isometry note since many property isometry remain proposition depend coherence dictionary rip result report ready right side proposition calculate indeed full imply submatrix full finally consequence coherence project coherence project apply useful function coherence report state ol combination result meet recently linear combination representation exist dictionary context select bad condition atom first exhibit scenario atom representation result bad condition dictionary dictionary elsewhere play gram eq unitary check distinct eigenvector sort corner zero eigenvalue span proceeding define concept subset subset say subset ol dictionary reach omp exist two representation disjoint dictionary exhibit result input apply representation project respective select q success complete read coherence wang corresponding bad cardinality dedicate condition since evaluate pair mutual coherence cumulative intermediate mutual coherence inner dictionary atom involve type evolve investigate contribution interesting proof proposition technical denote eigenvalue eigenvalue express lemma proposition hold first norm see fourth rip q inequality inequality relationship recursive obviously decomposition moreover full family turn unit norm jj q appendix proof denote change confusion prove technical recall read take statement dictionary reach satisfied ols virtue latter verify dictionary result without atom hereafter remain matrix arbitrary recursive construction omp successively select iteration selection yield
equivalently batch conceptually one location adjustment distance discrimination batch hyperplane approach assume variation observe variation batch factor addition subsequently poor factor correlated batch effect contain remove large process center normalizing deviation array know remove effect dataset large multivariate biological contribution remove variation factor good factor context difficulty arise regression term turn ordinary least square square form deal difficult dedicate random term way adapt dedicated major difficulty covariance amount expression gene way replicate array variation involve control study different know cause help variation third subject allow explicit variation model address unobserved random first factor equivalently know discusse improve section performance method gender use evaluate finish discussion na ab b f removal another fix non effect address factor influence sake clarity unsupervise control affect interest negative gene gene intuitive two w reach singular svd singular value back I maximize plug use step expression simple call simply project datum orthogonal column regression maximize plug algorithm correct direction expect orthogonal factor estimator sensitive variation supervise remove include naive present lead smooth correction estimate unobserved model introduce random normal j know estimator version appropriate correction scaling amount signal direction posterior formulation serve highlight relationship solve close form unsupervised product make convex unless third discrete optimization problem finally discuss difficulty typical cluster center constrain dictionary norm constrain penalize could equation solve dedicate generalize purpose use dedicated discuss fix idea denote matrix spherical convex iterate minimize classical simply solve general fix row couple close iteratively solve observe empirically fix objective sensitive instead us expense correction w hand side minimization hand conditional side appendix identity right side need unsupervised clustering suggest observe unobserved still close involve dedicate numerically general various estimator amount poorly thing use propose available maximize unsupervise generally possible maximize step optimization ridge step unsupervise maximum yield naive hand side joint fixing naive however similar naive setting maximize procedure naive ridge benefit naive obvious replace ordinary dot regular variation interest project sample remove lot center naive projecting remove come lot gene amount result spherical identify naive account information variation along right panel take ridge remove signal uv control remove provide remove leave association summarize fix naive joint iterate procedure alternative arbitrarily equivalent row covariance supervise maximum incorporate encode shrink positively towards enforce separation additional posteriori detailed deal noise gene would lead penalty shrink regular loss assume encode need covariance usual square regression option feasible least ordinary square estimate covariance matrix mention try ml discuss procedure case naive model regression consider supervise show lead supervise estimate correlate canonical column correlation association gene lose direction variation estimation benefit direction estimator first obtaining replicate two say platform profile replicate influence variation replicate replicate difference replicate row gene control fashion new control assume noise maximize likelihood singular svd large value plug regression require constitute unsupervise discuss first gene first variation control difference replicate gene happen influence solely gene association whereas difference replicate affected replicate variation thing control influence simplify describe c develop w asymptotic behavior wishart gene regardless control dot depend gene large enough estimate ignore implement rw da easy db replicate factor remove estimate replicate perfectly replicate two c c leave unchanged combine replicate term amount covariance pair replicate combine lead factor already observe common ordinary compute covariance residual iterate correct amount could factor datum use mean center procedure yield lead know expression affect lead whereas effect may quadratic allow non correction fit affect evaluate correction microarray expression unsupervise typically explain datum away gender centering remove gender remove choice cluster kept replicate correction solid non iterative similar replicate perfect clustering gender correction use lead variation observe replicate signal dotted figure iterative iterative panel replicate method replicate separate shrink clustering send lead high pair gene green line correction control gene result reasonably separation space leave panel naive replicate advantage come naive replicate correction good cause variation even probably well gender gene influence gender particular gene little interest dot green base correction plus error try improvement non iterative two lead well correction rely improvement replicate gray estimator also repeat experiment center replicate correction filter gene since replicate replicate something vary identify array effect performance array measure berkeley laboratory array filter base variability within merge identify restricted neural study three across identify core study allow leave one aside recover correct mean setting first build keep array array replicate potentially replicate select present correction platform completely platform replicate correction good correction perform design full platform orthogonal expect correction easy platform cluster platform centering replicate iteration table show recall deferred appendix datum maximal presence figure platform full reason center platform well remove platform remove replicate available center replicate platform improve regular center presence disadvantage amount platform whereas possibly benchmark becomes center give extremely form replicate naive replicate full platform principal fourth principal remove two platform effect remove fourth platform remove two component lead perfect contain platform allow used presence full reasonably gender one remove platform lead naive design cause value first represent strong batch effect iterate ridge far improves replicate base correction gender quality correction give respectively retain consistently absence correct show come correction remove platform effect gender correction explain design naive replicate performance remove enough platform correction remove array replicate performance correction lead explain behavior estimate remove correction check variance qualitative gene expression assess drug three combination test total sample gene array measure array array array array experiment retain drug suppose effective early time lead array restrict gene replicate platform replicate gene convert design obvious purpose array influence drug numerous variation array array drug hard gender one drug may gender clustering behave h center naive replicate correction display try partition cluster principal center platform array platform like gender cluster drug see figure lead improvement organization drug still clean replicate well performance run different lead clustering objective error clustering correction well organization fail correct replicate performance platform small remove platform drug iterative poor naive replicate replicate procedure work change far gene little allow introduce three benchmark iterative gene performance control gene gene cm cm gender control replicate combined gene dataset overall affect control gender factor rely heavily correction one replicate replicate introduce combine less solely gene even gender replicate variation replicate robust affect gene gender dataset good gene figure interestingly gender control ii control gene gender eigen eigenvector gender eigen gene small control less gender good gene available help remove variation replicate control individual replicate amplitude correction propose unobserved factor interest dataset affect introduce idea negative control affect replicate affected replicate gene control cluster interest affect replicate remove replicate variation sound platform use know explain factor factor addition replicate suggest available remove variation scenario variation remove interest actually rank technique give benchmark difficult technique variation never benefit appropriate surrogate hyperparameter could stand cancer dt author chen discussion r nx n plus additional orthogonal orthogonal regression denote particular equivalent regular ol identifiable normal map equation plug equality carry lead california berkeley usa bioinformatics large contaminate batch variation account analyze spurious association variation account correlate unobserve former carefully gene replicate expression generally remove without compare state correction last microarray gene help like study several involve several
activate frequency task individual individual pattern individual eeg exponential generative eq gamma design basis design test test kind model simple speed guarantee variational gap nmf typical inverse gaussian gives likelihood expand approximated jensen second term low give optimize inference variational parameter eeg group performance hyper common comprise eeg record duration task letter preprocesse raw eeg psd eeg channel psd frequency bin constitute subject desirable intra variability separate kind side propose basis basis share subject whereas common basis pattern individual additionally competition indicate able concentrate well subject expect concentrated region subject concentrate individual basis column achieve comparable often time perform basis subject show capture well limitation show individual present generative analyzing find common class subject subject variability seem capture pattern seem less size limitation individual variability well pattern advanced institute south model eeg eeg variational art competition eeg multivariate electrical potential among neuron high temporal brain brain interface application mobile eeg svm factorization nmf nmf determine useful spectral eeg testing pilot pattern reflect intra variability generative give
remain probabilistic difficulty straightforward persistent divergence minima fail present greedy wise overcome layer rescale seem importantly necessity difficult organization influence organization like would influence layer layer wise simultaneously influence layer deep boltzmann fall vector hide add difference norm principle previously form rbm proportion application somewhat explicitly activation encourage hide effectiveness boltzmann stochastic unit group unit dependency unit interaction specify encode visible visible visible visible connection connection parametrization boltzmann function ensure general rest evaluate q restrict interaction rbm restriction rbm possess distribution unit rbm readily posterior immediately rbm extend imply however still resort much comparison boltzmann typically maximize likelihood accomplish gradient ascent rbm full persistent divergence mcmc update unit filter term vast layer filter successfully detector characteristic mnist superior lower prevent poor regularizer encourage unit layer empirically layer boltzmann contribution demonstrate simultaneously layer well layer future would many simultaneously
limit correction correction monte carlo number replicate non zero interval correct nominal practice however corollary precision function bias univariate parameter multivariate adjust impact nuisance control consistency mean size theorem however asymptotically quality estimate analysis moderate gold great scope provide adjust credible potential credible implement amount exchange alternative would assume correction make minimal grateful use matlab abc financial research provide university south write l x first g l ii l x equality first order taylor represent simultaneous confidence fan method nominal probability distributional frequentist estimate interval correction wide double improve coverage keyword interval correction approximate mm intend credible many problem credible asymptotically sample many problem composite adjusting generate simulated parameter statistic derive draw distribution pseudo credible determined frequentist adjust interval similarity double involve demand demand chain obtain interval estimate confidence base bootstrap bootstrap expansion quantile estimate bootstrap function iterate produce correct interval bootstrap bootstrap coverage double adjust bootstrap compute autoregressive replication interval bootstrap fail interval procedure credible intuitive coverage frequentist reflect posterior distribution calibrate software technique bayesian describe give theoretical relate involve know narrow conclude comment tail interval parameter seek write interval compute quantile abuse notation bayesian case interval bound estimate produce approximate coverage theoretical result come simulate simulate replicate value good estimate population amount large consequently example occur whose see frequentist setting likelihood bayesian posterior case less accurate limit suppose denote th c correct coverage interval write th observe estimate coverage replicate simulated limit manner value l simulate g provide correction nominal asymptotically step obtain confidence interval step upper limit x random example example assume suppose location normal estimator illustration interval coverage amount usual frequentist confidence x z interval x c display correction confidence interval replicate illustrate correct limit correction adjustment horizontal qualitatively limit irrespective correction choose result arise change location adjustment correct location correct plot correct limit leave plot plot interval cp retain property however monte adjustment systematic interval adjustment correct coverage retain replicate second row adjust interval correct coverage confidence cb double bootstrap bootstrap double db correction bootstrap cb correction provide broadly comparable calculation package compute nominal e correction different specification level auxiliary specification comparison bootstrap correction possess structure record broadly double whereas procedure code broadly comparable interval complex composite technique composite likelihood bias narrow approximate abc modelling mechanism behind model abc analyse parameter quantile parameter interval nominal coverage pairwise specifically composite specify bivariate spatial location provide normally specifically score vector ordinary set max spatial standard composite likelihood obtain analytic estimate readily employ narrow compare composite matrix spatial model stable degree covariance spatial model extra marginal parameter cccc coverage nominal confidence interval replicate analysis standard correspond correction procedure correction adjustment clearly coverage base hessian take modify achieve setting algebraic representation available moderate computational ccc hc follow procedure abc likelihood estimate density control approximation require credible analysis daily exchange return type model quantile function control location use particle generation correction draw dependent quantile table credible kernel
model mixture gaussian kernel admit store save store representation second nonlinear distribution clarity key reveal important loss fx f ix instead iy fx risk fx I compute universal simplification preserve distribution nevertheless simplify fx fx variance lipschitz lipschitz second fx fy variable around behave continuous fx risk turn linear train svm x gx gx important differ need furthermore gx consequently virtue kernel place distribution place rbf large bandwidth generative define via example idea surrogate kernel also k kernel previously treat probability restrict robust instance svm incorporate maximization margin cone generalize svm reflect classify relate specifie distribution center miss explicitly involved nonlinear comparable svm expect extensive quadratic qp learning baseline base apply firstly fundamental binary problem gaussian virtual distribution rbf boundary region density region heavy treat implicitly incorporate trained choose nevertheless secondly different combination embed two class wishart freedom consist classify digit digit digit digit initial virtual aforementioned virtual third original example exclude virtual virtual reduce pca dataset svm rbf kernel depict equivalence svm virtual term gb illustrate benefit scene bag represent patch vocabulary codebook encode occurrence detect codebook image generate base framework result four room office therefore focus separate image training codebook training firstly randomly detect local patch detection dim collection cluster local patch define center represent histogram sift svm nonlinear use ic sift vector rbf adopt ic parameter perform choose adopt pairwise voting benefit understanding employ method elegant high natural image propose kernel probability rkh simple nonlinear choose suitable place illustrate benefit pool compare pool example world km thank discussion virtue proposition bound j result loss equality consequently minimizer kx distribution lipschitz function continuous follow accord fm x similarly yield complete bound gx equivalent gx svm minimize gx minimize admit complete proof lemma system rely learn distribution represent embedding hilbert straightforward fashion machine analyse insight svms base insight svm place experimental real world demonstrate effectiveness learn collection problem arguably distribution reason preferable uncertain application gene microarray source reduce unfortunately feasibility microarray cope uncertainty amount array across may give abundance throughput amount challenge space besides potentially incorporate collective point learn distribution kernel generalize closely kernel leibler kl semi object semi recently introduce jensen shannon application specifically domain make first prove theorem regularization kernel probability measure input location particularly reduce flexible svm give iy setting preserve permit hilbert rkhs rkhs endow reproduce preserve see embed associate kx kx second solution combination minimize risk admit indicate contribute usual regularization recover case mx regularization limit infinitely minimize fm x something optimize g cc endowed topology borel algebra
optimization project comparable q probability notation simple full multiplication let margin problem use eq eqn achieve svm clear eqn thus optimality eqn right rewrite w opt opt follow get geometric margin accuracy eqn lemma proof project radius construction minimum ball space denote radius point b b b b b b b b radius exist b project minimal radius radius clearly similar theorem prove construction use ready bind finally regression solve regression rs matlab version run varied run refer fast medium partition ten fold cross estimate repeat ten ten fold dataset cross randomness construction random ten ten matrix classification experiment report random projection need random projection svms run square square average ten choice matrix evaluation namely collection document dataset genome diversity datum correspond classification report use solver synthetic dimension rs rs family use four matrix validation experiment ten family synthetic trial normalize w ij generate family contain family dataset experiment three family grow projection small svms figure running projection svms nearly projection obvious combine svms dimensionality instance tp datum family consist document datum comprehensive maintain matrix binary collect document word column diverse set project versus full rs four table second quantity ten ten cross validation experiment ten random projection set reason large less accuracy present text task set contain feature generate contain point task ten multi vector default setting binary class project datum observe close training training ratio project except zero predict take small dataset datum tp rs data c rs rs rs data pca summarize center compute known pca error principal performance rank pca retain experimental matlab pca keep matrix principal matrix refer run experiment entire denote pca set experiment sample dimensional though projection svms sensitive projection svm greatly like therefore random rank standard quite varied svms randomize hadamard see projection svms svms full compute pca projection compute bottleneck svms rank svms second former pca principal second principal desire sensitive take long matrix dimension pca c project pca dataset average matrix pca second depend indicate deviation ten ten experimental world namely gene convert label classification general full consist image center leave etc regression set time decrease increase c project indicate correlation mean standard ten fold ten four consist cancer cell line ten correlation influence square dimension rs rs show depend explanation projection useful handle predict achieve approximation radius ball relative excellent generalization value svms feature despite full rank improve random method well dense rs random projection method work medium expect excellent indicate experiment benefit projection combine time zero oppose svm compare popular dimensionality see random fast pca extend approximately low open investigation vector point construct soft develop prove margin preserve relative ensuring comparable margin extensive design programming short international conference intelligence short detail svm fast support advanced project air force laboratory fa nsf nsf address computer science institute edu cs mathematical department com respective form hyperplane maximize distance hyperplane separate soft dual lagrangian formulation unknown lagrange part measure classifier ball consist hyperplane width error monotonic soft insensitive dual insensitive loss formulate multiplier preserve subspace preserve I margin subspace preserve hyperplane comparable vc svm regression b n contain contain singular spectral n lagrange multiplier determine solve eqn imply separate hyperplane e appear vector hyperplane generalization svm worth lagrange multiplier determine solve eqn entry separate hyperplane give point support reduction transformation reduce regression svm present construction fast hadamard take address propose ground breaking construction datum depend classification regression memory serve theoretical relative generalization preserve relative randomized hadamard running apply randomize hadamard transform construct need transform run need equal random randomized hadamard three transform randomize hadamard eqn margin dimensional radius row state construction preserve relate technique insensitive respectively practice show problem random importantly comparable manifold regardless margin result dramatically improve need projection run separable show margin separable running projection margin preserve show error free preserve binary meet angle angle well angle inner angle preserve two result additive whereas relative big margin small moreover analyze separable analyze non weight relate analyze claim regularize weight approximately preserve approximation hash kernel datum random projection increase datum sparsity show hash
invariant markov addition first part find ensure mix marginal continuous f omit polynomially ergodic hence basic begin observation hx hx borel similarly proof theorem suppose note corollary eq let transition eq necessary rs principle hasting denote right side jump proposal condition establish accept jump q unnormalized density together suggest choice piece sense choice middle course depend walk normally qx trial error deviation remark statistics california edu school university statistics college university edu quantile establish asymptotic construction asymptotically valid estimator recommendation practitioner quantile absolutely notice bayesian quantile typically monte simulation report notion approximate produce estimate method practitioner rigorously increase reliability inference mcmc entail simulate gx gx valuable carlo finding assess mixing much weak constant serial difficult interval student quantile interval assess reliability explicitly implement batch subsampling simulation rs require previously begin brief require chain section bm sbm rs illustrate conclude recommendation essential preliminary material borel algebra ergodic irreducible recurrent mean geometrically equivalent characterization chain ergodic measure establish gx strong number omit assess approximate sampling monte scenario markov refinement description chain proof markov polynomially order consider monte sense expectation find satisfie q bound work area practically ergodic case easy q marginally student degree variable sampler update metropolis marginal invariant markov chain estimate marginal ensure satisfy every v example replication absolute median distribution carlo appendix polynomially need method substitute consider estimate study chain process estimator estimator familiar corollary continuous consistently mean bm n iy n q putting standard estimating quantile error substantial amount use bootstrap b experience method computationally focus block subsample order subsample sbm note avoid give normal simulate augment markov apply derive alternative estimator recall give simulate otherwise thing apparent represent use simulate residual height simulate pt receive considerable gibbs employ establish practically frequently easy draw chain v hx split straightforward markov geometrically polynomially ergodic exist sequel develop base gx set exist consider polynomially ergodic require follow polynomially exactly need recognize variance moreover consistently investigate finite sbm rs example conduct common replication order interest heavy tailed say normalize slowly vary quantile freedom walk jump proposal draw proposal kernel finite tune scale order autocorrelation result result acceptance varied tail bottom sd length interval rate first agreement nominal median quantile average sbm great case couple appear bm variability agreement slightly well rs show half empirical coverage least huge variability standard tailed bm interval slightly slightly wide demonstrating length sd jump proposal ccc metropolis bm sbm rs bm sbm bm sbm rs ccc distribution bm sbm method bm sbm rs distribution bm sbm report concerned occurrence indicator disease normal bayesian credible distribution geometrically ergodic attention coverage estimate da quantile truth rs example setting replication simulation table bm sbm probability nominal investigate nominal level rs nominal mean bm notice across rs average bm coverage probability ccc ccc bm mm rs sbm mm bm rs sbm know analyze consist average player first player represent reasonable hierarchical specifically flat gamma proper gibbs simulating markov implement interested transform study correspond bm rs draw
ls likelihood nlp normal like example measure label negative example via confusion give table tp correctly false fp proportion incorrectly classify positive rate precision let effort false positive negative worst thus address positive negative actual actual preferable us detail measure optimize case maximize recall choose depend much want combine recall criterion harmonic prevent class q experiment write true maximize positive tend positive f positive f summarize classifier play evaluation binary optimize hyperparameter validation situation separate loo situation next estimate otherwise one value estimate auc applicable probabilistic classifier sense auc however compute estimate smooth nonlinear want gp define smoothed indicator sigmoid make criterion hyperparameter loo combine loo predict positive rewrite q smoothed fashion quantity loo estimate smoothed loo show useful use version classification classifier becoming optimize hyperparameter early gp trading compare maximize minimize overall loo define criteria loo loo cv various criteria cumulative write refer help value optimize approximation compute loo ep use full gradient calculation hyperparameter expectation type optimization gradient implicit site iterative ep cv hyperparameter ep loo cv gradient sequence mean covariance derivative site work use call ep cv optimize f optimization proceed dataset smoothed monotonically case exhibit behavior smooth difference smoothed base take depend problem increase true case car dataset clearly trend case right th always optimization cv depends ard give problem equation method storage complexity recall discuss loo cv relation derive field binary gps originally physics loo loo cv ep loo cv count indicator rough loo error ep ard hyperparameter work classifier feature associate sequential update outcome though optimize evidence prevent overfitte loo choose count work gp classifier optimize include ard directly various loo include gradient loo cv sake treat interpretation imply approach give pass sigmoid loo loo gp treat hyperparameter ill optimize l within ep optimize ep ep optimize ease nlp table l predictive maximization ep optimize ease fm ep optimize method diabetes ep diabetes heart method ep diabetes c dataset car diabetes dataset fm nlp car diabetes conduct summary experiment ep routine matlab code http www code hyperparameter hyperparameter benchmark dataset web project benchmark summarize first available web code conduct correspond post hoc multiple dataset mean point partition dataset performance get rank significantly nan hypothesis rank nlp ep ls statistic reject nan nan reject post hoc pairwise reveal ep ls post hoc significant ep set rank method l method improve test significance detect significant well method significance thus l table nlp performance l inferior performance kind see nlp score breast diabete observe nlp close compare dataset dataset heart difficulty nlp increase score wrong classify example ep score l ml ep estimate relatively poor result poor nlp hyperparameter variance value except difficult dataset apart pattern seem car multi classification consider example treat dataset web edu machine partition class ep cv second nlp smoothed criterion step describe early hyperparameter observe smoothed breast cancer diabetes remain width analysis give method believe arise example carry significance use rank significance nan hoc reveal nlp method smooth f nlp rank observe conduct sign statistically significance demonstrate usefulness smoothed consider gaussian loo cv optimization loo criterion smooth hyperparameter specifically apart nlp usefulness measure useful handle distribution propagation ep expression useful ep cv real benchmark nlp nlp demonstrate smoothed performance ep excellent gp classifier loo optimization nlp predictive py nlp give n py loo predictive loo ep analytical eqn derivative predictive distribution ease z nz give approximation ep approximation ep k avoid inversion note entrie k classifier loo cross criteria practical loo predictive nlp weight useful unlike loo approximate loo distribution expectation propagation conduct several benchmark criterion optimize nlp approach measure f excellent choice criterion cross predictive error precision integrate require analytically various expression two important function integrate within approximation integrate hybrid hyperparameter gibb integrate hmc integrate hyperparameter variable integrate hyperparameter choose focus address model loo cv predictive nlp classifier use integrate hyperparameter maximize laplace ep method integrate optimize information em ep approach loo cv address problem loo rough hyperparameter ep cavity distribution loo select automatic determination ard hyperparameter loo least nlp classifier measure likelihood measure measure like useful image understand example work hyperparameter literature regularize weighted loo curve auc criterion handle regression optimize f general loo experimental compose target pair true label gps model prior covariance k across dimension automatic ard covariance ard latent binary value label w py hyperparameter characterize assumption latent
approach select analyse outcome break avoid bayes amount good actual structural thought principled method classifier bayes underlie set almost certainly extent error true term currently consider domain learn construct theory say provide enough create appear accurate classifier valuable still whether drop heuristic relate decompose inherent variability arise bias considerable insight performance regression adapt bias adopt considerable definition bias negative variance bias variance quantity guess set guess around cost cardinality term bayes specific number reflect classifier form decision boundary elliptical never classifier use study make drop remain sum inherent achievable note many assume hereafter adopt bias accurate value statistical model unbounded distribution integer potential iid underlie practice evaluate training success predict component raise methodology give example partially rely distinguish go misclassifie misclassifie type decomposition use preliminary examine sample argue fundamentally partly instability derive result show stability design degree essence fold cv ensure classified time sample please evaluation purpose ten characteristic naive nn bag adaboost ensemble adaboost bag decision treat entity decomposition classifier cart cart knn naive bayes implement library use require datum artificial five visual inspection problem create image good bad act generator datum inspection cd label operator label inspection describe augment descriptor add total different range cardinality derive artificial cd derive image cd label uci repository observation regression estimate error make unseen pool independent dependent procedure measure simple independent quality prediction close give way process single iterate clarity variance combine show predictive combine change extremely model various field learn sufficient provide achievable height height height scatter sample marker indicate axis correspondence plot constitute combination vertical point predict build word show observe final majority former apparent considerable plot total lie final marker size fall fairly evenly thus bias seem end scale line would consideration take taylor expansion value dataset size single visual inspection fall line bias error account proportion perhaps expansion even variance size even follow intuitively absence training region probability major point simple quantity take far try model confirm original bb model increase determination bias relate either use sum final variance well show correspondingly examine build combination able unseen dataset evaluate cart logistic knn library default final make without build classifier seven dataset see estimate predict near actual poor bias inaccurate original pointing decompose show coefficient determination initial unseen classifier segment f perform pair relative I bias size marker logarithmic different scale somewhat visually trend towards cause trend error decomposition apparent trend show height width different algorithm concern represent mean variance algorithm fairly survey variance ambiguity page negative handle knowledge ensemble take classifier however approach treat entire classifier definition next predictor limit behaviour potentially universe limited future finite predict value complete treat ensemble entity boost create ensemble train set train create training predict fy subset train incorrectly fy possibly item training look depend follow well fold counterpart state refine take average turn attention class far constrain output decision partitioning contribute bias correct x fx contribute give ensemble bias provide dependent furthermore always non quantity constitute strict rate treat entity section build predict attain ensemble classifier require run fold time computationally heterogeneous ensemble pool build model relate bias different training treat regression evidence training decay bayes error decomposition experiment bind derive minimum achievable feed fit predict use feature space cd artificial cart decision tree bayes decision combine discount cross prediction item datum clarity hereafter increase use fit form prediction well coefficient low relate value sample situation size predict dominate observe use predict lower also train perform e across form achievable also different ensemble estimate analogous model deviation law change discount constant constant modelling ensemble ensemble make sample build illustrate cd artificial show build entire accurately ensemble would first show secondary standard deviation curve artificial observe fall standard error overlap regression nevertheless standard predict entire depict investigate early scenario ability approach although component progress way training increase qualitatively similar across different combination create statistical technique examine relationship perhaps surprisingly provide accurately behaviour different component confirm prediction finding validate range dataset model also examine reliably new range slightly datum component suggest complicated depend final bias covariance available function ensemble straightforward methodology ensemble achievable ensemble law error direction combine previous theoretical finding suggest worth law period predict useful major work definition bias investigate whether definition accuracy european contract author view machine system topic establish predict set cost lead accuracy algorithm training composition observe decompose bias although differently behaviour hypothesis take create limited predict full result range analyse various prediction case predict unseen widely understand theoretically empirically e g unseen example validation average fitness run evaluate repeat user change increase effectively achievable thus accuracy acceptable come insight successful place management sufficiently order learn pose develop user management example store come however failure stem descriptor whole must user costly classify field diagnostic inspection medical frequently sufficient store necessary process hoc collecting database significant quickly accurately predict go successful importantly viewpoint valuable save paper future give still learn final achievable focus descriptor limited later predict relationship train theoretic loose motivation investigate approach reliably observe hypothesis use source
simultaneous spread recent tail exponential separately conventional reduce blind paper consider alternative appropriate impose image empirical bayes blind deconvolution exploit blind tune bayesian estimation image decomposition perturb nominal specifie log bayes relate wise gaussian basis tune strategy reconstruction expensive propose estimate precisely nominal perturbation nominal relative truncate reduction improvement full markov simulation comparison quantify advantage real cover bayesian propose result unknown positive recovered reconstruct measurement convolution mean observation kernel model imaging device vertical configuration perturbation nominal experimental field mass probe etc approximation derive error true direct nonlinear direct expansion deviation deviation know basis coefficient influence denote notation rewrite convolution sparsity measurement noise mass single w conditional prior interesting enforce natural furthermore ensure positivity uniformly define e improper informative jeffreys noise equivalent informative jeffreys variable associate hierarchical hyperparameter prior definition hyperparameter express conjugacy integrate hyperparameter posterior yield section present generate describe metropolis distribute sampling simultaneously result consistently output subsection noise pixel parameter pdf distribute w conditional therefore sample achieve posterior stand component stand truncate generation require generation truncate appendix enforce condition image summarize order detailed procedure appendix algorithm transformation costly strategy propose gamma result deconvolution algorithm blind deconvolution nominal mathematical mcmc sec mmse ensemble average burn period sample maximize h amplitude external slice field perform describe image propose assume physical physical tuned table radius mis specify fig image basis span subspace set carefully tune produce ellipsoid specify center nominal pca residual allow axis basis determine empirically fig eigenfunction perturbation basis depict estimate coefficient show agreement correspond eigenfunction true blind show blind nominal hereafter accord appear outperform preserve h quantitative trial six fig histogram error criterion indicate alternate minimization reconstruction notably unknown blind method reveal investigate criterion stage blind comparison fig direct comparison several reconstruction initialize fair comparison modification variation tv suggest estimate poor deconvolution sharp iterative approach fine adapt method term present use basis image prior apply suggest show fig fig blind deconvolution motion incorporate penalization poorly trivial poor due image blind camera motion shrinkage effectively bayes several second method fig modification extension pixel add voxel scalability benefit impractical slow additive paragraph basis obtain select use domain axis grid fine along high bayes interpolation semi blind presence experimental synthetic pass noise semi blind reconstruction initial blind experimental neither ground truth image fig nominal estimate estimate validate difference gibbs give residual nearly experimental blind show algorithm experimental intensity variance estimate region p measure indicator voxel provide confidence significance reconstruction bayesian reconstruction identifiability common deconvolution ambiguity prove reasonable solution trivial restriction estimate reasonably solution beyond scope image reconstruction extend blind transform assign exponential laplacian distribution use pixel hyperparameter image replace reflect nominal metropolis adaptive evaluate uncertainty demonstrate blind blind several criterion method include enforce eigenfunction use algorithm selective future acknowledge dr provide thank comment paper generate exploit describe require without evaluate decomposition procedure coordinate coordinate vector I iw accord I moreover recursively evaluate follow overall expect pixel quantile previously exact require quantile tw computationally especially restrict burn period save selective sampling iteration compare mcmc blind fix blind need careful selection step
correct intervention supervision establish unsupervised shrink transformation variance feasible instability question arise overcome supervision mathematically possible question impose force pde form function convergence supervision assumption pde supervise cluster converge general pde decompose part pde satisfy pde green green satisfy pde replace normalize proper density fact original function pde supervision correct assertion theorem adaptive current historical density leave shrink algorithm design deal law stable due particle speed supervision interesting future aim fill critical gap mining application lack stability employ physics partial absence supervision cluster correct unless transform normal lyapunov anti nature cluster without proper supervision preferred ensure incorporate formulation mean cluster pt corollary convergent anti shift wang wu york analytic equation propose mathematical law successive underlie stability algorithm show supervised shift possibility correct transform multivariate convergent adopt supervision mechanism dynamic set accord cluster set study normally partition small homogeneous excellent review cluster emphasis wu discussion many cluster optimize strength partition merge et determination cluster challenge problem cluster shape boundary difficult issue specify ease normal introduce year algorithm dynamic algorithm treat include wang zero gradually towards cluster law category cluster arguably shift receive flexible generalize operation idea find transform toward variation wang design bias center movement resemble tracking center employ robust recognition excellent image analysis segmentation practitioner apply area underlie understood initial transformation chen point shift point statement also dynamic intuitively behavior major difficulty function transformation setting dynamical dynamic evolution impose shift framework useful criterion reliability general identifie suggest improvement consideration successive transformation general law characterize evolution form dynamic pde analytic differential broad converge normal density anti diffusion lyapunov increase anti uniquely instability phenomenon convergence happen nature shift place supervision dynamic potentially supervision rest follow presents establish brief view sample clustering mean consequently view particle field law require cluster pattern self organization reaction order mean algorithm framework form dimensional density function element surface element denote side minus gauss take inside component derivation discussion law associate differential form supervise influence certain impose supervision lead law framework many dynamic process keep partition process give unsupervised approach functional gradient derivative variant method often govern law move less adjusted interest mathematically show belong category category formulation wang et al center locate wang show force trajectory seek mode movement function long area combine eqn correspond one anti diffusion move region anti unique apply reaction transformation detailed derivation fourier transformation f dx fx fx dx f also follow convergence I uniquely evolution deterministic status unique path lead lead naturally normal respect time converge speed location eq conclusion occur spatial variation time contraction prove result cluster center generalized anti dynamic shrink boundary fourier I fourier side eq transformation integration simplify way analytic function
severe sampling rate ml fit tb plot posterior credible dash square dotted posterior credible indicate small observation level naive lead severe set focus energy interval uniformly define short point restrictive exponential solve energy set constrain energy frequency ern function derive interval appropriate derive uniform length reasonable prior bound obvious fitting naive mis effectively way noise value expression variance level processing rna sequencing apply g variance multiple replicate approximate low total fitting cause bad length synthetic ex tb ex ex tb ex avoid reasonably sec create data sampling equally space fit unconstrained fitting result scale fraction drop gp coincide square point space calculate square true mse large likelihood indicator mse likelihood indicate opposite frequency instance small fit large four present support together model drastically fig clearly length make model sensible bound c htb bin time series use version scenario unconstraine fix processing differential model tf set show moderate fitting vary drive especially fit propose example tb slightly model tf gene discover target without model fit list demonstrating bound bound improve gp instance rigorous bind sensible length exponential mat ern spectrum reconstruct justify collect place prior help fit synthetic usual fit functional freedom plausible essentially model highly constrain ode usually one incorporate delay severe cause case length method base ingredient model short biology carefully justify prior future beyond ode suitable suitably lead fitting failure mode institute technology technique multiple independently gene severe model depend avoid gp focus energy frequency fitting avoid informative prior allow reliably automatically instance illustrate real process gps widely apply nature differently size easily gps work gps gaussian identically widely gps learning square
sufficient data region manifold question remain simple exist system number point estimator answer question develop algorithm lemma unless rearrange j rx x rx j dx multiplying vanish multiply eq whenever apply lemma j j would schmidt many comment manuscript version mm corollary message mml link compression close kolmogorov mml np common approximation strict minimum mml calculate apply rule boundary rule heuristic dimensional calculate dimensional cut estimator prove result certain defer section example address issue idea briefly estimator exponential family open subset pdf bayesian pdf give elsewhere make technical moment n code x code length constant lattice variable estimator cut estimator probability fx parametrization standard result family differentiable page eq say mass say exist discuss informally construction family determine general transform simply ni ni ia ai vanish cut point therefore solve give corresponding derivative denominator step let lemma since never simple numerically behave method j j equation minima estimator point increase global minimum combine though say surprising satisfied estimator make agreement exist boundary seem symmetric g cut odd minus positive place reverse code length number cut code cut point cut estimator due fact difference hence impact minimum large idea minima cut point continuity guarantee corollary exponential exponential though exponential distribution parameterize pdf pareto moment additionally prior case gamma local minima outside range extremely small briefly discuss simple cause cx rx appropriately term many section behave right side numerically
dedicate error message send pilot user vector bit datum symbol ki k complex alphabet pilot draw alphabet aggregate channel ki interference lk h receiver li sample interference ratio receiver consider briefly unify pass combine mf arbitrary aa group factor set factorization index argument similarly index mf approximate pz belief combine mf framework graph distribute collect exchange augment depend could exchange symbol code bit focus share construct augmented pdf scenario replace variable original keeping pdf read next define set factor pdfs f lk lk lk lk lk lk lk lx account form set factor stand code operation lm lm bit probabilistic graph depict structure receiver subgraph factor link receiver process belief pilot symbol channel channel user external iteratively pass overhead adjust depict connect bit receiver make arbitrary factor subsection mf focus mf statistic q define mf therefore belief lk lk lk lk lk lk message lk lk lk lk lk lk obtain select pdf l pdfs non bp eq message l send bp message binary lm lm lm mn c lc lk input decode output equality therefore receive receiver obtain processing message pass main stage receiver obtain initial iterative pilot position include pilot successively repeat li li successively time decode perform bp part initialize equal bit receiver receive compute receiver use successively repeat process decode define denote number stage iteration take go step take belief lm lm lm consider noise interference e table evaluate monte snr almost achieve exchange receiver benefit improve weight precision present decode vice number number pilot evenly spaced symbol user receiver indice db pass design distribute receiver interference wireless unify combine bp mf jointly perform power sharing show remarkable compare system performance share could design schedule exchange approach accommodate value become
document illustrate inference use involve generate class statistical class elaborate step generate generate coherent define extended inter co dissimilarity article similarity extend article extend union article co article article dissimilarity create proximity extend dissimilarity article distance common vocabulary text text mathematically text vocabulary frequency vocabulary frequency construct article give contain inter agglomerative cluster proximity agglomerative amazon amazon user amazon crowdsource amazon difficult use computer human survey question user survey retrieval confirm article success user survey theory member article worker agree member bundle overall agreement independent semantic c comparative ask worker two extend aim comparative check commonly semantic employ worker two bundle appropriate name ask point name diverse bundle result comparative table contain overall user co similarity similarity h c result promise believe structure well conduct towards effectiveness semantic extend year keyword etc automatic scientific article characteristic interesting digital retrieval system properly organize scientific retrieval etc generating article article expensive report automatic extraction article scientific article extraction dirichlet lda make hierarchical agglomerative technique validate semantic make use crowdsource amazon amazon carry setup retrieval effective popular work relevant dedicate services scientific google mention retrieve article proper input query modeling result article google indexing rank page group result coherent task study generalize effective precise definition closeness pair notion closeness closeness similarity dissimilarity closeness item scheme start create bundle scientific second agglomerative inter similarity article validate article document suggest lda base dimensionality reduction important consider independent
dd explain maps hypercube hypercube employ choose angle insensitive exponential leave dd excellent concern normal dd well study dd perform well outli alternative family speed quantify time shift location concern variate normal various consist training base phase distribution classify computation time processor I physical exhibit computation deviation setting dd phase classify calculation computation grow slow datum hull thus depth outside point fast c concern section uci repository contrast dd well know literature detailed description point train number dataset table table diabetes diabetes table synthetic performance term dd respective benchmark datum apply treat several procedure c nn dm dd dd table bad classifier case presence treatment procedure point mh point correspondingly treat random fact calculate mh comparison randomly obviously assignment subsequent benchmark random mahalanobis author classifier contrast md par handle neighbor rule either mahalanobis moment alternatively estimate maximum employ moment nn restrict five rule treat remain exhibit dd depth classifier bottom deviation dd md md md md pd pd ss ss ss classifier nn specify size train test dd classifier class small form constitute testing sample see dd except handle method depend mahalanobis moment perform perform treat mahalanobis euclidean c dd c nn mahalanobis md tr present dd dd perform however depend dd properly robust mahalanobis distance perform handle good machine solver classification use dd depend svm dd two optimally systematically range correspond range interval small arise benchmark fair parameter four employ sequel diabete follow similarly choose big dd calculated use radial basis interface leave employ result twice preprocesse dd inspection plot l c mahalanobis opt time train test pre technique dd bad svm optimally optimize considerably dd svm varied report table classify svm many computational burden phase dd regard take second determine approximate parameter diabete similarly classification completely dd transform variate depth space transformation first fast testing setting compete classifier rather nonparametric approach good alternative certain family consider perform binary knn svm tune dd derive operate elliptical nevertheless dd rather behavior robustness point hypercube insensitive outlier hull must treat classify assign dd classify properly maximum depth rule arise vary specify even classifier mostly large dd dd skewed participant robust discussion suggestion pt plus ex definition corollary acknowledgement classify completely dimensional efficient discrimination depth depth classification perform discuss dd real new rate fast alpha dd recognition misclassification theory recently new employ depth liu point assign label depth centrality indicate employ supervised classification liu al depth transformation analysis dd multivariate regard way classify maximum depth recently li dd depth representation differ notion extension depth depth employ depth represent literature answer ad combinatorial span rather calculate mahalanobi use calculate reflect form mahalanobis depth remain still insensitive depth procedure heavy depth calculate li classification problem plot separate unit validation class point rule firstly classification restrict secondly often assign ad employ calculate excellent concrete application robustness issue outli space operate efficiently space like depth employ dd plot class depth class point classification plus whole dd simulate dimension employ fast multivariate classify introduce transform depth first discussion depth present theoretical dd elliptical mirror extensive simulation comparison benchmark section svm restriction invariant convex level vanishing sometimes impose survey restriction special notion classified class depth mapping mention indicate closeness represent closeness translate closeness e classify depth rule rule al construct degree separate depth several mahalanobi one map origin depth correctly ignore principal allow classify proportion origin classify near neighbor method properly mahalanobis mahalanobis mahalanobis depth depth properly beyond hull sequel either polynomial yield basic include terminate result point object classify accord vector procedure replace long feature leave separate assign order dd transfer set depth properly probability include invariant way matrix measure restriction depth detail cite survey nonparametric depth interest nonparametric spherical elliptical distribution play parametric uniformly sphere random support operate elliptical simple insight elliptical elliptical ellipsoid unimodal elliptical strictly increase claim hold spherical surface also ball around strictly hull unimodal elliptical mixing probability increase strictly prove include projection limit dd depth estimator particularly projection mahalanobi hyperplane mirror respect half independent dd line plot rule correspond requirement satisfied mirror unimodal mirror symmetry dd plot well let unimodal elliptical dd dd common affine transformation dd pc explore classifier procedure dd dd quantify simplify supervise two
fact dual dpp operator solution enhance dpp rule able inactive dpp speedup gain one generally need commonly use cross validation involve lasso consume address dpp briefly inactive idea screen effective discard inactive art dpp rule strong discard strong version version apply rule unclear sequential rest organize dpp idea screening rule set inactive exist screening rule efficiency solver enhance high set screen solver conclude remark dpp rule first geometric briefly safe property basic version dpp exploit geometric lasso dpp enhanced dpp outperform dpp inactive feature form contain derivation dual variable notational convenience recall view kkt potentially make inactive problem apply identify inactive inspire safe first region relax eq find screen rule detect inactive view accurate inactive screening geometric interpretation dual look notational convenience problem space projection mathematically set solution indeed nice illustrated lead enough immediately interior p conclude naturally exist well rest section derivation screening divide contain optimal find step develop geometric dual illustrate fundamentally important dpp screen dpp discard inactive relie operator follow screening contain able notice imply another ingredient feasible nonempty closed operator nonempty convenience convex hilbert projection lasso problem center ball center radius develop dpp upper different inside notation screen rule express easily solve plug bind statement applicable lasso please dpp safe safe st dpp respectively lead estimate commonly approach cross involve idea dpp rule rule corollary inactive reduce problem theorem inactive feature repeat sequential dpp rule sequential dpp lasso sequence k know corollary dpp dpp discard inactive small assume large illustration basic dpp dpp rule easily paper propose screen dpp correspond dpp present careful operator step fundamentally develop dpp imply accurate result screening rule discard inactive dpp operator dpp discard inactive propose approach accurate estimation idea improve dual develop enhanced dpp section screening rule screen rule inside nonempty close hilbert converse nonempty hilbert projection ray projection ray e operator please define dual lemma case technical view easy let define q ray direction lemma inequality distance hold conclude view statement b note reduce briefly review technical convex nonempty cone elegant nonempty closed subset hilbert nonempty close convex subset view key prove statement need x easy arbitrary follow ready view clearly need please note easily complete proof complete theorem inside center develop inactive optimal omit rule improvement parameter screening estimation effective discard inactive dpp rule properly rule discard inactive dpp estimation dpp onto nonempty close nonempty hilbert inequality converse true projection parameter value complete trivial vice accurate dual dpp dpp half know theorem screening easy see discard inactive dpp dual solution see screening rule inactive dpp result screening enhance discard inactive several able inactive screening rule subsequent generalization evaluation develop rule three step optimal lasso recall expand side rearrange follow hand side bound ball follow omit similar easy radius discard improvement dpp family dpp e dpp data b operator screen rule lasso applicable lasso kkt eq generally inactive group coefficient therefore find contain relaxed follow screen word also solution result effective screen discard interpretation let projection onto feasible close notice projection operator theorem rule moreover case lasso specific group problem e subsequent dual nonempty please refer operator theorem develop lasso problem know definition easy immediately need check see eq plug moreover result easy lasso lasso give follow indeed follow immediately easy q inequality due discard parameter measure rejection speedup please safe discard section safe strong version safe notice safe safe strong safe unclear exist sequential favor sequential lasso assume I feature aware safe point group compare basic version safe rule version safe briefly speak version rule make rejection ratio safe along spaced length safe normalize safe structure section cancer face mnist handwritten gene every yield image per object trial randomly image description face mnist handwritten digit report ratio basic version safe five six cancer cancer mnist face provide pointed strength screening stem reason usually method aforementioned counterpart section compare version strong screen real datum perform sparse synthetic datum I generate truth generate rule equally spaced trial screen safe fig ratio safe discard inactive speedup provide screen active nonzero ensure correctness screening discard guarantee run strong twice fig present rule observe pattern variation em c safe strong solver safe solver report combine four section safe strong six real equally spaced datum breast cancer cancer face mnist handwritten digit house rejection ratio safe report solver screen h breast cancer contain tumor gene matrix set microarray containing contain testing response one performance setting image datum mnist digit solver safe solver solver safe strong without solver column second comparable safe rule inactive speedup strong check kkt correctness screen speedup gain size e breast cancer set gain slightly high mnist strong observe speedup e breast gain mnist speedup gain magnitude screening need hour lasso need screen especially strong rule safe discard introduction experiment solver least angle regression lar experiment setting report run lar solve two substantial speedup gain low see discard inactive see em lar breast cancer mnist solver without combine second screen evaluate strong number group entry generate denote size discard group behind solution accurate notice average discard inactive group robust c solver solver strong report screening screen report time second screen far demonstrate effectiveness solver improve solver respectively screen operator screen inactive moreover dpp dpp discard inactive dpp family inactive extensive demonstrate mention dpp rule solver speedup tool plan idea formulation consist structured penalty detailed derivation dual formulation assume completeness constraint therefore problem new dual variable get eq lagrangian primal variable q objective subgradient attain optimum equivalent v optimum denote rewrite q everything dual kkt affine condition problem duality denote primal variable lagrangian kkt exist eq q eq group problem become dual eq eq follow denote eq split problem subproblems smooth let satisfy clearly due get conclude function eq everything
problem variety instance time distinguish metric dense instance assumption stability solve polynomial improve previously primary infeasible solve instance practice realistic fuzzy solve vast completely formal yet interest meaningful way instance efficient partition every many practitioner opinion set find intuition solid study continuous two criterion optimal remain perturbation distinguish another function candidate solution cut partition cut perturb distinguished cut least sum weight degree notion machine correctly metric notion stability stability perturbation modify algorithm distinguish locally stable substantially result regular graph stable instance weak impractical assumption stability find terminology complete graph symmetric bipartite explanatory separate edge separate resp side cut wu weight cut abuse etc minimal average degree potentially follow dictionary useful stand external internal stable locally switch vertex definition intuition optimal concrete definition iff maximal cut locally necessarily resp hold instance unique bipartite stability quite every instance hand numerous cut tend easy stable cut computational perspective hard even arbitrarily stability whereas efficiently decide stable decide cut stable simplify stable instance unique maximal switch decrease define distinguish maximal cut local distinction namely distinction vs cut distinguished highly distinguished stable side cut slightly motivate notion expand expand expand however expand see distinguished distinction expansion derive instance metric dense hard adapt correctly instance algorithm vertex test seed h cut prove part metric dense preserve local instance metric metric instance practically applicable polynomial seek practical reasonable locally good analyze computational significance method simplex solve almost polynomial instance problem improve admit cluster plant instance edge resp resp add edge drop certain consequence plant instance efficiently go adversary modify input improve optimal take forward solve instance dense analysis dense stable cut distribute c mx x x cut polynomially yield randomized instance locally cut cut uniformly stable cut instance vertex map instance prove cut cut map stability cut cut locally cut stable cut map cut construction dense instance let instance consider let easy v v locally metric locally stable cut stable cut stability l z r locally cut either ball l local stability proposition assumption claim stable found consider ball sense show metric stable neither ball even express ball distance identify denote finally locally stable w n conclusion prove psd orthogonal semi cut cut cut ba psd thus show bipartite cut w cut w laplacian iid prove distinguished
consistency satisfied choose exactly part need satisfied important oracle e part existence tight concave satisfy proper choice parameter choose cross validation cross regression also serve paper work study modify observe moderately fall analyze data bic dimensional real high cluster analysis membership adjust rand classification agreement agreement isotropic run member family choose get fail dimension choose associate ari confirm capture bic high dimension simulate model ari ari ccc bic parsimonious behaviour select ari correctly select bic component setting herein one careful modification take preserve asymptotic accounting simulate consistency asymptotic computational convexity prefer concave course problem penalty stay initial mode penalty adaptive lead oracle shall author wish numerous collaborative research science aid collaborative award innovation derive derivation become law large number hold taylor expansion derivative penalize likelihood within p integral laplace apply large usual bic second number penalty mean first line conclude therefore conclude part part time n order tend true decompose efficacy family base clustering depend parsimonious tendency drastically high merge impossible clustering introduce overcome extension factor match model likelihood multivariate gaussian assume fitting decide review mixture focus gaussian review work cluster discriminant mixture eigen mixture actually subset parsimonious family component mixture parsimonious select factor amongst salient family like covariance linear every select mixture away maximum bic theoretically justified number author bic consistently choose observation bic drawback approximation influenced parameter correlation cause laplace addition method prior version mode parameter avoid flat prior fitting case perhaps minimize residual constraint coefficient thus interpretable model behaviour provide penalize gaussian follow model select parameter condition especially mixture though selection criterion note regard author tune estimate extend bic high bic proportional instead consistent interestingly author penalize maximize conventional mainly applicable mixture model author point infeasible possible herein upon mathematical clustering modify suited limitation criterion work property inconsistent ideal criterion analytically work herein propose penalize bic selection likelihood maximize maximize penalize element use penalty penalty scad penalty satisfy asymptotically hard scad penalty prefer lasso easy criterion bic paper estimation property simulate exhibit result compare well property g maximize go function origin locally successive suppose penalty element marginal mle maximization stage estimate variable show membership belong component first stage likelihood z ig complex analytic
I auxiliary representation refer problem mean correlation I modification plausibility outperform drive asymptotically I variance benchmark observable take interest I auxiliary space probability together characterize random inference sampling I tie fundamental quantity predict conditioning invert close measurable serve support sort admissible predictive set admissible set mention admissible unique primarily combine I unique empty predict observe admissible specify eq assertion alternative tend inferential belief convenient report I start singleton I distribution singleton subtle interpretation probability observe parameter original one regard despite start ultimately make explain inference hand datum probability explain valid without common dominating measurable sufficient condition simple ignore let shape I function step singleton simple shall characterize draw finally plausibility assertion singleton q plausibility gamma assertion plausibility plausibility function belief I belief I I false relatively show validity admissible I valid plausibility formulation validity theorem help objective belief interpret unlike example common frequency validity validity output plausibility plausibility plausibility construct frequentist procedure interpretation example collection credible region nothing plausibility auxiliary variable easy however rarely admit scalar association vector first look seem challenge reduce desirable first step reduce efficient auxiliary price auxiliary shall look normal independent step predictive auxiliary singleton assertion c plausibility eq alternative note trying predict uncertainty step predictive singleton plausibility inference base plot simulate sample validity stochastically tend mean wide plausibility I predictive preferred difference efficiency necessary dimensional quantile plausibility correspond remainder give efficiency reduce function since conditioning inference statistic consideration I whereby simultaneous information dimension overall gain efficiency unobserve characteristic need predict unobserve observe help predict unobserve information accumulate characteristic least mostly define relate unobserved auxiliary familiar relate familiar formal decompose distribution take refer understand remark dimension often dimension sort variable reduction advantage I aspect predictive ability section construction I simplify associate fix unobserved see get belief use plausibility depend predict auxiliary conditional association ask loss suppose baseline admit decomposition set association proof say baseline sense obtain former bad baseline I however random auxiliary efficiency construct valid prevent make case validity set setup random valid I efficient baseline I condition decomposition fit conditional representation insight conditional I level sort reduction focus dimension dimension variable conditional case reduction end section appear important I reduce far provide see connection aggregate across sense solution generic distribution induce determine I hard belief set I predictive I consistent posterior I possible develop conditional partial example available incorporate slight previous remark application conditional extend validity result I context obstacle handle collection contain either predictive extension admissible time remark general allow validity identify depend since holds translate statement plausibility plausibility desirable property I conditional plausibility tx tx mind meaningful probability aspect focus relevant subset though conditional validity validity random set tx tx I corollary specify dependence observe could happen depend find conditional definition family obtain probably course problem similarly consideration desirable exponential family condition association method something fact nice check differential minimal invariance etc experience familiar thing satisfactory parameter case intuition insensitive baseline association clear partial e correspond choice solution derivative association formal existence solution powerful tool far track step base student degree freedom somewhat I consideration suggest likelihood observe I build theoretical justification use plausibility assertion plausibility illustration obtain center plausibility I asymptotic normality likelihood credible simulation summarize indistinguishable favor plausibility interval guarantee coverage normality prior alternative naive indistinguishable arguably drive suggest seem suppose independent sample namely available goal come application maximum conditioning recommend consideration reduction step efficiency gain employ differential sx sx reveal conditioning differential familiar classical statistic jointly statistic modify simplify q complete p plausibility readily evaluate illustration display plausibility function derive ignore plausibility sample conditional I naive conditional opposite variability reflect plausibility conditional I plausibility gray gray curve I scale association dimensional sufficient independent familiar default square plausibility singleton assertion evaluate carlo illustration simulated figure display jeffreys prior display asymptotic I plausibility besides guarantee coverage capture elliptical large jeffreys estimator I plausibility I baseline association exist may distribution correlation natural towards independent baseline reduction free decomposition requirement meet I alternative elaborate via type conditioning regard justify localization describe separability extend relax direction start fix section pair association distribution suppose compute point set like produce plausibility refer local validity property local certain theorem local conditional I depend suppose validity localization imply conditional plausibility confirm simulation plausibility plausibility depend place I structural I assertion development solve normal fix construct I shall modify I real vanishe take expression evaluate differential give expression eq denote conditional plausibility local plausibility nominal coverage illustration consider plausibility table conditional frequentist due summarize credible jeffreys message interval interval average moderate large hope well jeffreys local I result along length interval normal respectively I review jeffreys bayes consider number replication treatment unbalanced interest component vector stack useful variability come source nuisance eliminate mixed function fix covariate handle marginalization setup case variance component consider elsewhere orthogonal diagonal fall correspond minimal equation unbalanced model plus sample parameter employ connection note sum familiar characteristic logarithmic since vanish satisfy eq orthogonal association define association chi square numerically random detail I elsewhere brief simulate group size moderate box I plausibility region give walk random contour indicate plausibility region shape roughly consistent work plausibility contour I develop auxiliary strategy simultaneously goal argue remark section fisher case condition auxiliary dimension give satisfactory reduction addition section even predictive conditional standard I set case dimension identify auxiliary predict validity fact could locality assertion singleton way one develop conditional focus thus extend latter extension special case idea association necessarily formulation idea focus framework g regular family I describe herein difficulty new tool come conditioning need dimensional nuisance sort deal marginalization point acknowledgment partially national foundation grant dms dms dms author helpful suggestion associate association r hx hx hx hx distribution hx define distribution hx hx x belief conditional issue fix admissible section like theorem parameter next characterize right turn imply eq supremum prove proof since free tx take supremum validity unbalanced give rescale one assume relatively simple partly fact whether boundary none I full sided credible mean unknown weight framework marginalization present consider association observable something describe look I I pde relationship derivative identify derivative respect simple calculation therefore solution association association omit rewrite obstacle case negative association state word efficient fortunately modification assertion I side suggest get effectively ignore interest care need plausibility size perform test indistinguishable sign something validity locally choose maintain expect arguably standard conditional I gold suggest lose focus choose one function one one function assume assume observe likelihood familiar many exponential family decompose association generate equivalently fit assume independence determine carry summarize factor baseline decompose directly minimal sufficient replace base require result dimension basic basic conditional claim full may statistic great equation technique see baseline association decomposition association determine take independent work analogue mapping factor association I work shall subsection group onto admit association correspond invariant work maximal tie vanish therefore solution equation consequently association free take statistic summarize mapping group decomposition invariant consider association group residual invariant take step proceed baseline association unit maximal I induce distribution expression transformation maximum likelihood quantity maximal condition determine formula general suppose familiar basically find basic insensitive association construction fully depend solution determine decomposition summarize give association one differentiable differential pde system
follow way estimate write explicitly euler c prove rejection state case vx vx c similarly prove upper mala ball require act lyapunov mala kernel suppose assumption observe hold indeed definite explicit exist eq upper q f assertion satisfy chain initial estimate construct lyapunov choose jensen stop process note provide assertion length combine metropolis finally deterministic fix lemma hx h acceptance proposition fix since h h assertion acceptance proof assertion theorem metropolis hasting wasserstein induce distance hence r borel coupling measure assertion w dr nh nh dr nh nh acknowledgment discussion prop prop prop prop prop remark prop prop adjust langevin mala hastings contraction wasserstein mala euler probability density reference measure regular depend contraction rate langevin hasting chain result fact mala proposal metropolis hasting state attract grow show particular product measure acceptance algorithm rwm metropolis langevin mala converge go zero case diffusion scaling maximize speed limit diffusion optimal acceptance rwm mala target sufficiently w point corresponding perturbation measure euler rwm mala replace semi implicit euler proposal metropolis realize infinite arise chain langevin langevin well convergence wasserstein quantify strictly diffusion might expect metropolis hasting chain heuristic wrong sense huge quantify cf chain remarkable exception well work chain uniform measure concave quantify wasserstein euler precise implicit euler require restrictive proposal diffusion result geometric ergodicity particular establish equilibrium wasserstein metropolis hasting chain log concavity context measure space develop thesis fact hasting wasserstein quantify rejection implicit euler proposal bind corresponding ball proof bound measure density normal decompose measurable constant absolutely degenerate absolutely continuous form important cf brownian bound law process abc consider wiener evy number interpolation path dyadic fix let consist component expansion image brownian distribution sampling return absolutely transition strictly let kernel markov monte integration choose accept homogeneous markov transition eq q metropolis hasting reversible hasting standard prove hasting wasserstein look adequate small satisfie straightforward balance case simple kernel give growth detailed balance diffusion solve langevin equation adjoint dense cf algorithm corresponding mh obtain sde mala mala proposal euler discretization detail view euler discretization step euler scheme langevin time kernel stationary cf scheme implicit part substitute semi implicit mala langevin proposal kernel give proposition correction vanish goes fix sufficiently polynomially constant approximation surely finite measure dimension path directional assume q minus norm correspond convenient come end path norm one cf path restrict ball hold constant depend r euler analogue bounds mh assumption exist polynomial r k r r proposition semi implicit euler proposal upper respectively acceptance euler fy determine depend depend euler make state proposition norm e section bound hold kernel wasserstein borel algebra marginal coupling variable define joint derive bound generally couple mala setting coupling mala suppose exist constant moment depend explicitly stationary wasserstein mala chain satisfied explicitly estimate stein step mala require provide minimal step hold polynomial error satisfied radius small take metric borel infimum marginal distance measure bound yx yx qx yx couple markovian coupling markov kernel markovian coupling couple markov apply equilibrium markovian let markov transition initial respectively couple since chain fact stay wasserstein finite assume markovian coupling distribution metropolis product yx yx dy acceptance transition chain variable yx yx yx u yx u yx x markovian coupling hasting coupling accept vice versa mh kernel triangle inequality obtain equality indeed x always simplify diameter x proposal efficiently estimate old hasting q hold wasserstein necessarily close minus consider constant hx dt xy x continuous sufficient guarantee equivalent q derivative consequence mr mr derive mh rejection probability mala probability w r semi euler state euler proposal step size h h vx vx constant vx vx vx similarly implicit euler proposal obtain hx vx h h h sx vx h vx vx hx vx vx upper compute mala rejection vx upper bound rd vx vx ty dt expansion ty vx dt x vx ds ds averaging equation vx vx ty dt ds sg ds ds dt vx vx vx bind semi implicit euler explicit proposal one vanish general valid bind proof vx vx vx vx vx vx vx vx h hx hx hx n p consequence inequality follow old combine order proof hx vx hx vx l hx hx vx proposition semi euler proposal lemma polynomial semi euler proposal pt c proposal
analyse simulation million histogram figure display marginal semi automatic solid represent obtain parameter spherical reproduce correctly reproduce posterior unable reduce spherical marginal posterior regression use semi model semi close mcmc albeit far overhead appear broadly marginal obtain wide especially would construction compare simple bayesian direct argue method toolbox density challenge reliability flexibility enhance focus give estimate al indicate approach automatic expert mixture expert density histogram draw marginal solid density posterior density summary statistic middle semi row quantile plot estimate solid mcmc approach histogram posterior bottom mm mm corollary approximate computation regression mm fan abstract mm approximate difficult calculate active topic current abc common strategy several advantage first easy approximation reference analyse third sensitivity several prior need approximate likelihood build adaptation extend reference frequentist via keyword copula estimation regression density interest initially biological science wide range abc simulate involve simulated weighting accordingly receive see weight practically match impractical post abc posterior g dimension replace statistic reader summary alternative abc construct direct prior summary simulate attractive way firstly posterior secondly likelihood purely analyse maximum researcher familiar inference prior prior bayesian credible base contour bayes posterior perturbation prior expression useful reference prior approximate intractable wave consider approximation approach generalise summary covariance chain mcmc choice summary relate multivariate normal simulate informative statistic appropriate localization summary statistic help available respect auxiliary marginal posterior estimate inference still process pointwise estimate stand functional could serve goodness fit diagnostic abc build recent flexible estimation easy estimating adapt multivariate precisely individually implement density intractable function connection conclude goal obtain intractable datum draw carlo summary determine must necessarily convenient pilot broadly density region define consider flexible marginal mixture expert regression mix vary express component fast parsimonious fit propose copulas margin cumulative density denote normal multivariate normal sample model normal mean conditional component density note exactly another albeit additional experience acceptable approximation enforce marginal summary simple primary flexible construct greatly selection summary statistic orthogonality transformation ij pt ki ik sample derive combine final frequentist provide estimate posterior multivariate informative conditioning mixture provide good parametric technique alternatively condition approximation normal distribution estimating likelihood provide discuss performance normal difficult marginal estimator normal joint benefit simple normal great flexibility copula transform flexible expert wide vector draw range easy linearly reduction via pilot moderately abc adjustment focus production clean require particle observe consist measurement inclusion spherical univariate generalise pareto scale take equally spaced statistic quantile highly instead region density square euclidean large moderately parameter manner methodology summary statistic addition perform fit expert adopt variational number spherical work summary statistic relationship automatic observe display transformation greatly simplify model likely estimate spherical evaluate
belong category simplify problem normalize choose multi represent situation label dimensional ht width compare four train number instance portion instance unlabeled one portion gradually step report micro deviation real world explore regardless pool mml well confidence lack usefulness iteratively learn spectral embed connect high dimensional dataset uci benchmark mml three standard minimal understand also mml two mml mml simply feature mml mml implementation happen rbf j training zero mean square latent model view variance overfitte gradient limit mml decide success consist select table extract color word histogram feature outperform type thus desirable descriptor learner description color color histogram successful descriptor sift accurately image suffer carry ignore perform learn sift preprocesse datum construct color color range visual histogram vocabulary cluster sift description image visual codebook bag task name vs task vs vs vs vs vs ball vs machine vs video predefine use set deviation see mml compete statistically level view confirm transfer knowledge description discriminative performance sift dramatically ball vs reliable sift name sift significantly color necessity multi transfer difficult user mml place multiple view normalization ht c svm sift svm mml mml mml task task task namely census datum record diabetes patient sensitive signal obtain various angle single emission image choose six algorithm view dataset disjoint uci benchmark summarize c feature diabetes heart mml error rate six uci benchmark trial table outperform view technique evaluate mml statistically performance propose framework view demonstrate simultaneous learning feature description one confirm conclusion ht sift mml mml mml diabetes heart conclude mml advantageous multiple description instance multiple task relate mining area multiple attribute investigate together total learn learner use simultaneously one study task dirichlet although many world usefulness practice propose measure success benefit task reject comprise learn together multiple space disagreement branch disagreement classifier semidefinite program conjunction regularization reformulate problem standard implementation applicable although suffer theoretically rarely hold real machine kernel scheme sometimes induce mml require view nonlinear various generally unsupervise traditional classification exclusive multi correlate specialized capture usefulness dramatically large annotation hypergraph concatenation attempt dimension label learn utilize motivated unlabeled datum various learning attract amount interest introduce field propose classify structure unlabeled induce label propose unlabele encourage many propose demonstrate unlabeled survey mining move towards demand supervision view low paper framework handle task view setting simultaneously solve miss identify label apart reliable rate explicit success challenge make purpose use beyond complete supervision dimensional view possibly address problem separately mml optimization reduction set design handle multi many information view combine reconstruction formulation mechanism near neighbor fundamental question task multi present quantify view mml outperform many state semi supervise reduction challenge datum well suit high dimensional datum separately supervise benefit challenge formulation efficiently interpretable reduction mml consider experiment independently collect people shall project face supervise semi supervise fig show five denote people indicate associated face stand green blue people surprising make label marginally infer simultaneously dimension reduction produce belong person associate aggregate gradually image apart width width width subsequent learn technique environment successful elegant framework structural well mechanism view clearly label view examine structure reconstruction create many view set show place single suboptimal raise difficult view fundamentally indicator issue multi multi view supervise construct show mention learn simultaneously quantification successful examine paper section perform structure determine experimental particularly promising discuss issue experimental compete experiment compare compete usefulness learn monotonically improve view significantly benchmark work additional conference extension framework setting facilitate labeling propose quantify view claim approach iterative simple prevent compare mml mml place equivalent mml present early aim simultaneously inference attempt discover structure datum exploiting instance feature abuse refer function explore semi partially formulate function eq near measure near neighbor determine bias advantageous immediately enforce label label matrix minimize function express label multiple label gender involve contain unlabeled problem task relate multi task finite binary classifying contain label otherwise point task description aim asymmetric represent avoid self reinforcement diagonal degree th vector table task classify task label instance instance degree node regularizer vector unlabele identity carry related partially task intuition nearby tend adopt nearby learn encode propagate write something enforce sparsity formulate description classify task third penalty share tuning description decide control section guide error view partially chance less ignore favor class introduce matrix regularizer balance label calculate denote multiplication definition satisfy matrix propagation diffusion balance even label framework task multi learn mml benefit paper experimentally multi multi multi discover aspect description hand weight partially mechanism enable take knowledge mml handle reformulate online learning specifie discuss issue linear fashion mml kind fusion enable detailed important contribution determine obtain zero alternate current iteration predict small value tolerance alternatively make confident treat prior cost confident large derivative confident locate decide find entry entry newly regularizer label apply take advantage complementary carry multiple description optima alternating show recover covariance multipli specifically prediction reduction processing step visualize work datum weight completely intrinsic relation step unnecessary desire vector q reduce vector avoid solution eigen eq recover eigenvector discard eigenvector eigenvector minimize multi apply many problem solve avoid essential approach transfer view empirically usefulness mml perform propagation mechanism instance step build upon carry learn task view wise sum use walk quantify success multi propagation cp positive step propagate matrix interpret diagonal report transfer task occur view value imply view successful occur view algorithm measure world gene dataset collect accuracy value normalize dna synthesis task two color diagonal naturally view different result quantify trial curve mml black proportional relation approximately parameterization lead well thus high outline believe easy motivate multi improve exploit contain obtain highly irrelevant intuitively view suppose description hand multiple description view still knowledge moderate fit task fashion view task source task view weight encode respectively view indicate task l c k c md k j k uk show propose weight avoid propagation sensitive controller label avoid naturally fit preferred choose contrary fit preferred parameter wish greatly minor change parameter empirically respect pick highly dna task micro balance task choose box ignore definition histogram influence view see surface relatively though open question
obtain identity prove constraint display minimum end solution image denoise enhance remove differ example path suppose ij represent record gray across true level denoise serve reconstructed achieve replace isotropic penalty penalty focus path follow convex loss many image poisson ray denoise circumstance square view vector penalty sufficiently reduce goal quantitative notable advance include site summarize recent progress reality iteration tune path attractive option recover minimize linearly dependent violate constraint equation uniquely difficulty four fourth well redundancy neighbor pixel illustrate corner pixel transform segment along path path path accelerate constrain build differ homotopy path introduce effectively handle tracking explore knowledge first make method enjoy simplicity rich numerical matlab regardless slow exist optimization insight predictor interact quadratic piecewise permit denoise suffer side problem reliable separate local current solution development rely strict path denoise role enforce work expand list grouping straightforward remove restriction exact method equally necessarily unclear one behind modern interior broad issue researcher corollary theorem grant gm kl grant hz great stress tend constrain penalty recover penalty examine strategy consistent constant trace constant thus start solution programming piecewise take jump piecewise operate differential segment segment projection onto quadratic programming semidefinite mechanic denoise demonstrate path penalty decrease optimization penalty program ordinary differential constrain regularization device problem penalty operate barrier strength barrier method gradually barrier gradually send either solution optimization barrier approach program application control method central reliably quickly minimum nonetheless penalty augment lagrangian potentially competitive method penalty avoid problem disadvantage penalty quadratic penalty lack objective argue path unconstrained solution penalty increase along constraint path boundary advance lagrange multiplier special programming path one entire segment theory homotopy year exploration method goal assess path follow constrain performance method probably later practically orient paper experience rich numerical matlab include solver review exact derive path particular attention constraint algorithm elaborate demonstrate particular denoise problem apply mathematics penalty discuss function subject affine equality constraint differentiable differential partial transpose second partial constraint meaningful lagrangian l capture lagrangian satisfy nonnegative complementary one take choice create favorable circumstance minimize twice differentiable lagrange rule local minimum inequality program constraint condition impose quadratic minimum prove quadratic minimize surrogate complicated increase lagrange unknown application follow property surrogate increase convex likewise finally assertion generally finite strictly combination strictly two point strictly strict inequality multiply b third restriction result direction sequence scalar tend e n obvious section devise path strategy start program regularize problem primary find solution path mind area path order characterize minimize one strictly minimize subdifferential equation rule calculus book know path uniqueness continuity uniqueness existence subtle solution continuous linearly near convex consider fix point inequality solution fail exist tend subsequence take limit e e unique verification claim defer say remain subdifferential smooth path solve equation ode path algorithm segment determine sake simplicity begin occur plan attack lagrange constraint multiplier satisfy g constraint lagrange multiplier unique consequence continuity hand side observation stationarity constraint equation write equation theorem require dependent equality left implicit importantly implicit following summarize finding strictly coefficient occur subdifferential solution lagrange multiplier satisfy differential segment become active constraint boundary subdifferential determine segment inactive occur move keep constraint active constraint occur inactive comment move continue simplify considerably penalize affine vanish segment satisfy previous highlight generality constraint surrogate along differential two row inverse amount scale inverse sequentially organized operator computational every burden plus compute slow algorithm problem problem invert suppose full satisfy active furthermore f equation computational side improvement cost balanced fortunately practice solution constraint leave sequentially ode loss number regression ode interested reader refer book versus employ euler update ode euler inaccurate connect amount replace easy probably ode package ode intend mechanic comparison comparison feasible point program design project convex algorithm consider toy project close close h path start movement path toward c ccc toward origin toward path c toward axis plot vector point ode solver evaluate derivative converge square nonnegative principle useful modeling nonnegative observational article estimation statistical matrix nonnegative entry imaging range pixel rank minimization nontrivial enjoy property guarantee converge even exhibit alternating solve separate denote column correspond separate path algorithm solve subproblem end problem straightforwardly project typical next path roughly unconstrained programming example minimize affine amount positive semidefinite path equal bivariate radius start unconstraine along circle end ode solver evaluate time programming contour contour branch stand application chemical equilibrium structural mechanic digit circuit process subject geometric leave equivalent fractional power instance constraint programming
must satisfy primal feasibility iv dual feasibility stationary candidate primal plug primal vanish ii hold last two meet give rise decomposable simple strictly sub admit eliminate respectively establishe initial lagrange n redundant identity plug multipli become identity sequel move across lagrangian update obtain I identity penalty upon nk follow similar lead complete algorithm convergent recursion upon imply consensus statement proposition go define limit upon iteration comprise tucker r limit arrive n nk thresholding yield follow ii iv f add iv readily check identity obtain last role optimal equivalent b cm cm signal advance wireless communication edu superposition plus recovery completion rank minimization nuclear surrogate count minimization centralize processing separability overcome limitation characterization nuclear adopt yet minimize alternate multiplier reduce message pass among interestingly attain centralized counterpart regardless initialization outline highlight impact include traffic anomaly rf wireless centralized sparsity nuclear norm low spc considerably measurement observe miss one introduce set unchanged compactly fit square error entry pseudo g describe unfortunately rank norm np nuclear surrogate close accordingly rank control convex appeal special instance attain traffic exactly absence compression term robust available special offer completion stable recovery completion early effort work assume jointly process solve challenge interest wireless preference share fashion raise central processor represent isolated point failure scalability driving motivate distribute propose medium singular aforementioned reason motivate minimization recently though applicable topology also distribute absolute centralized decentralize sparse base small corrupt spatially distribute measurement task separable nuclear norm amenable minimization nuclear separable convex cost multiplier ad complexity per pass single interestingly convergence centralize initialization propose algorithmic four namely traffic volume anomalie ii robust matrix completion iv cognitive cr networks centralized benchmark defer bold letter letter operator hadamard product cardinality scalar matrix matrix top return inverse entry semidefinite trace inner mi mn n adopted consider capable computation message neighbor abstract g monitor internet cr communications network model link agent agent henceforth agent eventually outline incomplete corrupted n l r develop regularize processing locally exhibit centralized estimator keep overhead inter agent communication neighborhood facilitate requirement distribute argue value internet traffic analysis origin intrinsic addition small reduce adopt p obtain convex bilinear lt efficiently handle amenable non coupling address minimum bilinear leveraging provide obtain adopt frobenius enough one globally p ensure less stationary optimum interestingly show relatively globally optimum point globally sufficiently becomes violate sufficiently expect long addition variance certainly decompose couple global variable cf auxiliary estimate utilize minimization become later variable norm cost regularization cf additional nothing importantly understand estimate coincide even consensus impose whole connect desire consensus constraint eventually eliminate tackle constrain minimization problem associate lagrange splitting associate additional positive split notational n constraint n fashion multiplier ad adopt ad iterative lagrangian well parallel tackle task g entail comprise iteration implement coordinate variable lagrangian treat ad generally cycle group ad special section sufficient subgradient iteration implementation augment lagrangian decomposable separability come group agent turn highly simplify four specifically initialize algorithm present l thresholding entry denote r k k k f n k f simplification redundant inherently redundant auxiliary multiplier keep track redundant multiplier communication burden stem program carry agent neighbor communication matrix efficiently observe encourage algorithmic accommodate penalty maintain agent complexity n k one need unconstraine strictly quadratic admit anomaly term operation n n k n b flow invert inversion need addition reduce inversion employ n rank computational ridge large element operational main burden communication identical incur traffic measurement allow datum analysis science g site web surveillance massive volume datum result low dimensional nominal ls pca remarkable presence large related reduce sensor flow see corrupt local low agent want form special discussion agent estimating row constraint n per variable one primal variable close minimize soft neighbor n c n l matrix cost neighbor localization wireless recommender give noise observe entry corrupt rely processing aim complete form exploit nuclear norm whereby feed programming rank brief operator operator linear symmetric characterize introduce whose equal otherwise n general component remain agent suffice primal iteration obtain property subproblem admit solution receive neighbor I k l n agent obtain inversion typically acquire prescribed computational iteration iteration transmission overhead small value observe regression l retain consistency however fail parsimonious adopt regularization agent reason available distribute fashion absence sense cognitive cr whereby sense across frequency map enable spectrum spatially expansion sense form narrow spectrum locate operational variable selection motivate collect location amount solve protocol accomplish accordingly readily update multiplier select fold validation numerical convergence convergence follow find penalty lasso k n k n k r c k convergence agent consider realization agent randomly place distance prescribe flow bilinear draw collect internet internet refer flow record internet assess agent flow end flow collect operation internet comprise flow control condition interval respectively short flow attain experimentally fast convergence evolution agent depict representative metric interest error compare n left agent accelerate collect per processing centralize right depict evolution level distribute centralized counterpart give flow measurement begin internet though management protocol trace flow superposition clean anomalous truth precise exhibit dominant singular receiver characteristic roc highlight anomaly low rate e accurately consistently flow illustrate estimate anomaly flow aid centralize obtain use ad agent pca assume apparent converge
show impose hierarchy section notion hierarchy remove though build remark removal technique column overlap favorable discuss role hierarchy aspect sec primary straightforwardly logistic material modification primary predicting base concentration size c six classify measure top method sparsity pick good level incorporate predictive bottom provide indicate next effect constraint unbiased degree freedom freedom valuable primarily prefer tie advantage characterize optimality tucker effect hierarchy simple take function jk residual predictor linear kkt thresholding c write toward examine statement strong method strong hierarchical loss satisfy sx sx supplementary correspond weak insight prop reveal overall form identical certain effect shrinkage certain constraint loose weak condition become involve lasso th loose match coefficient hierarchy would naturally I dual increase weak identical loose show hierarchy hold show particular jk j jx j analogous get exact establish study elastic term modification ensures note suppose absolutely drop hierarchy weak jk jk absolutely quadratic freedom refer dimension space fit measure notion procedure see thorough discussion produce fit degree freedom ridge depend pattern constraint tight evaluation p replicate strong along estimate df circular bar estimate plot datum therefore useful unbiased amount pt difficult quantity make j j jk pair df strong hierarchical fit main force constraint accommodate interaction pay effect make sense could advantageous make nonzero visually indistinguishable considerable fitting overview restriction discuss iteratively consider remove add consider elimination enforce recent version selection first without include hierarchy post modify lar hierarchy procedure interaction come viewpoint interaction hierarchical selection indicate whether coefficient normal get write similar series modify add inequality enforce hierarchy sense approach method make describe composite penalty broad penalty group achieve hierarchical put induce interaction framework structured literature note remark close vanish rewrite penalty although class involve sum norm combine involve main study advantage restrict true generating consider truth hierarchical case element ii iii submatrix ratio snr snr effectiveness refer sequence add variable choose forward modification modification add give main approach lasso pt method tune select note simulation know avoid biased favor simulation panel color effect respectively solid forward respectively jk prediction assume hierarchy well rest receive succeed panel predict truth expect winner situation surely anti identify main get interaction variable hierarchy even interaction constitute truth present well able interaction inferior scenario hierarchy dominate relevant interaction identify interaction since spurious hierarchy pair favor scenario ii iii well correctly far sensitive joint act contrast select effect regard six mutation drug drug location six site occur focus weak lasso addition standard effect dependence result average six case abc achieve well test error concern rmse dominate several main appear option fast solver rely amount iteratively coordinate minimum convex specifically hierarchy symmetry couple mean get separable block discuss weak discuss lasso solution descent elastic tx r involve lasso gradient convex generalized gradient descent suitably step since replace state descent guarantee fact look differentiable subproblem p solution elastic net remark observation solve along sequence large warm start value symmetry tie method multiplier admm widely applicable apart subproblem problem three separate part serve result practice affect admm hierarchy symmetry involve result update throughout algorithm solve version admm call replace argument propose lasso strong closely tie exploit admit characterization impose hierarchical implication demand hierarchy introduce distinction measure variable hierarchical interaction measurement consume provide implementation weak method gaussian gene prove solution ease equivalently notational write pt change strictly objective amount possible treatment lagrangian variable kkt primal dual pair ty l necessarily term follow equivalent ty result pt characterization assumption freedom write l inequality describe ten row pt ptc ptc follow proof strong hierarchy property include elastic fy pt solve design lemma part vector show imply zero except consider te automatically specify contain begin show union already jk jk j leave begin tu tu u jk jk jk jk jk jk jk u np u u u u jj j ie ie pe np e j set long lebesgue weak hierarchy nearly section jk jk establish weak hierarchy fit kkt get q satisfie ty write continue fy satisfies optimality ty fy h iy fy lf since lb fy fy fy hold plug leave argue ny ny prove nearly situation piecewise degree bind estimate interpretable linearly precisely linearly r span row clearly lie span r jk add show row p p give drop kkt condition subgradient function involve stationarity involve via dual step knot max useful associate fellowship dms dms health contract add set convex interaction interaction include marginally precise characterization hierarchy hierarchy degree reveal fitting hierarchy constraint distinguish nonzero raw prediction hierarchy focus closely tie concern available empirical insufficient outcome medical diagnosis occurrence confident patient either situation I status additive moderate measured variable idea develop order predictor eq refer part interaction summation consequence notational interaction take throughout everything carry restriction provide net training select main estimate practice among interaction effect restriction name call jk argue hierarchy sensible reason special position origin must include hierarchy position origin regression must go take nonzero model strongly norm
reproduce kernel hilbert rkhs prescribe hilbert function functional rkhs possesse reproduce characterized property reproduce kernel uniquely determine rkh thus rkh reproduce kernel emphasize hilbert function vanish everywhere wiener wiener eq depend reconstruction approximation application consider reconstruction error norm borel banach space borel measurable function throughout continuous reproduce property equip borel optimal performance reconstruction eq concerned point shall try remove end algorithm choice finally minimize minimal interpolation point reconstruction reconstruction reproduce kernel determine observation hope another distinct reproduce positive strictly definite furthermore eq reproduce every equation get result orthogonal projection prove corollary minimize quantity calculation tell complicated form discuss space reproduce radial norm define reproduce kernel must monotone measure minimizer minimal mid point unlike nonlinearity exceed example two elementary let careful elementary measuring seem natural dominate multivariate hilbert save computation effort consideration restrict subspace minimize eigenfunction eigenvalue principal machine course point span eigenfunction minimization computing eigenfunction exist transform purpose span infinite algebra consist subset measurable f approximate transform question determine satisfie eigenfunction functional operator moreover imply bound ball weakly word shall j h tu equation imply lebesgue turning theorem orthonormal maximize part transform eigenfunction eigenvalue return positive borel continuous explicit form orthonormal span kx give stand eigenfunction operator different orthonormal eigenfunction eigenvalue k h le ix practically concern cardinality rank notation eigenfunction costly shall eigenfunction efficiently standard follow shall first eigenfunction eigenfunction assume minimizer problem wish close subspace bound projection onto subspace supremum identity kt p kt k kt kt accord lemma distance projection onto accordingly shall show section singular exist span use subspace yield explicitly characterization equation rewrite u kf h u nc k k nu kf get equation introduce matrix matrix eq get hold h h k lemma give problem solve search sampling remark favorable orthonormal search candidate point efficiently occur inverse computation illustrate point reconstruct system throughout kernel matrix therefore experiment consider q two use spaced kernel target sample reconstruct error calculate present plot pair comparison follow lebesgue sampling mark star equally mark eps standard marked eps approximation large situation point perform space comparable error equally sample usage sampling bring relative superior equally spaced lebesgue optimal marked star equally marked improvement figure error mark star eps relative instance space obtain spaced improvement experiment dimensional lebesgue mark spaced eps mark circle star rather lie could equally space example lebesgue distribution spaced eps marked star eps pair relative close equally spaced point consequence error comparable present equally space mark equally spaced point algorithm eps eq mark marked star error instance improvement experiment superior experiment algorithm one uniform support equally spaced point mark circle eps mean improvement figure mark circle mark eps pair except outlier improve conclude sampling yield significantly equally space true corollary thm
assumption meaningful way fix ap minimum iff th differ exist minimizer constant prevent generalize assumption trivially minimum frobenius unique th large differ identity establish estimator next satisfied solution trivial assumption remain require frobenius unique norm objective demonstrate see solved replace invariant norm consider frobenius unique optimal resp spectral objective minimum generalize far rank ex equivalent constrained version take admissible well admissible even solution tune constant desirable optimally usually replace suitable norm counterpart norm strong j sd problem couple considerably invariant norm definition invariant constrain norm nearly prefer constrain norm adapt version moreover suitable constant yield theorem obviously remain different transform trivially iv thin svd norm trace recover thresholding correctness proof characterization subdifferential trace avoid deep form trace thresholding solution hadamard subspace next corollary let thin either form classic point hence let obtain form way derive pp classic result recover look verify recover weak surprisingly appear new subspace clustering consider set subspace estimate ill pose subspace solve subspace cluster widely though refer reader survey successfully membership reconstruction obtain minimize instead independence think turn proved thought recall either point come show unique note shape sim know surprising aspect noiseless nothing frobenius close practice corrupted consider discrepancy choice norm norm typical regularizer frobenius consider program solve multiplier matter choose appropriately one thin svd function remark heuristic computer vision literature provide justification show optimization purpose intuitively amount moderate sim singular noise sim shall shrinkage interaction discrete might instability preferable thin call shrinkage shape interaction purpose also call shape measure regularizer equivalent due equality remark therefore matter result solution sim close derive latter subspace variant lrr lrr trace case performance directly except sim tune sim affinity apply segment cluster misclassifie point follow subspace construct randomly noiseless achieve expect level sim noise lrr range probably discrepancy lrr start trace norm regularizer regularizer sim form single lrr generally require converge svd cost lrr order slow threshold method level sim motion segmentation subspace regard manually independent fair comparison apply though result sim contain obeys subspace row obtain sequence good regularization constant well close art select individually clear lrr significantly slow method include sim lrr lrr provide interesting regularize apply cluster result subspace clustering first body use norm large fan case frobenius three norm fan fan follow fact invariant norm easily remove clear extend iff th large minimizer proof fan value lemma admissible feasible admissible k suitable integer definition note complete thin projection let orthogonal ap k unitary apparent need consider k frobenius minimal choice simplify uniqueness j sd unitary invariance norm diagonal argue objective due xu c follow use everywhere zero solution unitary invariance ba easily verify invariant restrict singular matrix main body norm see value guarantee frobenius step need minimizer also optimality conclude frobenius former principal reduce datum characterize close solution model obtain insight promise typically subspace manifold sample provably correct identifying characterize likely union subspace would subset unfortunately normally membership subspace therefore course difficulty segmentation
construct stick construction paper process allow al deal introduce background completely sampling introduce technique poisson introduce section dependency operator section poisson process random measure completely measure poisson process measure measure along kind work slice describe illustration basic domain process countable infinite set dropping line picture class admit summation part deterministic purely atom jump atom restrict pure jump jump measurable measurable kind construct process take poisson surely number denote call poisson product evy construct measure measure construction completely mean arbitrary disjoint measurable space finite finite always jump bound get point finite without loss generality evy represent probability l evy used concentration parameter sampling logic sampling jump later go evy mt base corresponding go l l evy deal evy generating jump homogeneous note important name exponent exponent guarantee jump lead interpret functional evy equation jump sampling block jump discuss normalize unnormalized construct normalize increase additive term random increment prove distribution also normalize measure take measure survey kind transformation evy use mass usually sample hyper jump depend gamma process construct shoot cox process evy form use normalise l factor normalise normalise parameter evy similar different induced mass parameterize law compare dp cluster create total familiar stochastic special limiting normalize generalized gamma process process normalize process n stable gamma formula l l regularize incomplete mathematical library evaluated necessarily trick eliminate term relative mass variable follow replace introduce variable idea variable future slice method behind deal jump mixture I belong slice auxiliary introduce description follow mixture auxiliary return otherwise actually compound mean expression give integral turn incomplete gamma function large size component slice variable density density evy consider sampling go eq shape separately conditional jump interval sampling introduce dependency operation review transform construct dependent completely superposition poisson superposition poisson superposition superposition poisson subsampling sampling select poisson acceptance transition poisson independently operation describe move integrable measure aa poisson superposition give superposition subsampling give transform new operation generalize superposition measure subsample subsampling define transition atom base superposition subsampling point superposition subsample underlie posterior version condition version recursive simplify marginal weight various integral introducing prior make carry result upper incomplete integral recursion otherwise term may unstable poisson process well ease major issue poisson generalise discount vary concentration might significant computational burden develop simplify marginal variable predictive posterior also present adapt normalise u jointly jointly via transform obtain adapt integrate slice bind jump observe jump value follow n k j k hx k unique count jump index match conditional hierarchical reasoning jump jump retain apply covariance property laplace exponent case dependency involve significantly mass long jump normalize however correlation normalize q dp correspond upper gamma superposition underlie l evy dependency via evy dependency transition measure ba disjoint operator appropriately covariance superposition subsample straightforward extension posterior superposition superposition nx xu jump subsampling subsampling evy q subsampling evy give dependency operation composition operator two operation operation acceptance superposition superposition constant composition transition superposition subsampling apply operation lastly operation form subsample acceptance subsampling point superposition operation rule q remain top remove lemma ready theorem three equivalence subsampling point poisson dependent measure evy evy denote transition sampling subsample posterior subsampling completely bernoulli acceptance nj j terminology pz pz z department communication digital research centre variable rearrange evy formula since something something hold result lemma allocation indicator slice jump auxiliary q introduce jump threshold evy evy proof come posterior proposition simplify yield concentration posterior datum distribution equivalent surely proof lemma equation expand expansion power incomplete expand recursion k n therefore chain expansion hold bt arrive expand inside definition denominator kn u posterior discard theorem proof come end prior see true simplify mass introduce make change tu
much star systematic surface star good although dispersion bias black dot star show namely respectively value indicate datum h model magnitude range g ap average start bp begin decrease detector plot figure introduction improve accuracy magnitude ap range reality represent upper ap star upper assign star ap value equal datum training strictly distribution figure fit limited star prior g star star explain high prior datum half residual whole much star may star vs interval posterior posterior pdfs star circle circle green dot axis ap model ap estimation magnitude range star ever pdf generally width important distinction precision residual top panel project precision red difference low rp spectrum precision accuracy say figure pdf eight star parameter reduce star early ap also ap estimation produce uncertainty pdf bad simply see g magnitude fractional error star distant build star high completeness drop surprisingly low contamination svm true star range ccccc select completeness select contamination select contamination summary star performance inferior science give nothing estimate extreme recognize investigate parameter datum algorithm system star close investigation result select sample star issue ap high star snr something find despite biased e fig less sensitive suffer different svm perform produce regular svm motivated forward fit grid ap table small well ap small mr less bias always ap test naturally report ap uncertainty characterize use also introduce considerably however magnitude possible ap considerably slow work accelerate feasible star observe year allow post release expect include rp spectra available website estimation introduction processing g star emission star spectrum star present star room improve grid improvement grid need star make spectral wide estimate preliminary alpha abundance difficult want fix use improvement modification apparent acknowledgment thank thought ap something support grant well contract innovation obtain source yield star scientific reveal formation analysis rp spectrum three different algorithm effective star wide range account diagram improve estimation broad star spectrum absolute always large amount estimate star across priori performance magnitude estimate well star method survey fundamental star diagram detail accurate census star ever course position proper star source visual addition five also radial velocity star magnitude datum detail statistically part composition formation evolution anonymous star equip together distribution target nm resolution main name bp rp galaxy star physical reliable central understand galaxy fundamental property star age composition star primarily temperature surface estimate ideally star absolute magnitude apparent magnitude development describe refer e bp system estimation approximately object vast module galaxy module classify emission line star module spectra radial velocity star rp also much achieve noise ratio cluster exist try network kernel quite successful many magnitude confusion observe broad partly algorithm take parameter compose separately logic behind collective rarely article confirm work experience survey svm bp rp spectrum uncertainty complete star estimate report final inference account ultimately make user e library observation process place learn datum quality behaviour evolve l environment article effort correctly galaxy noise datum complicate indeed part develop library simulate differ algorithm describe science available version analyse additional prediction notation summarize unit unit ga empirically fit apparent band k logarithm rp arbitrary comprise literature description far algorithm code originally perform implicitly transform nonlinear original normally tolerance svm scale tune search training process identify star tolerance define determine newly star present work discrete magnitude train near build separate ap model iterative initial phase multidimensional forward model newton ap generate observe whereas spectral band ap full hereafter report early bp rp essentially forward infer divided weak thin spline model regular addition covariance fit probability measure star contain information band define hereafter basic spectrum pm g g later pdf ap ap advantage method pdf estimate take account uncertainty magnitude ap possible include compare ap spectrum band gaussians likewise depend bp rp include estimate four predict everything iteratively jacobian find model spectrum resemble spectrum determine apparent consume forward datum even give training although lot observation attempt sufficiently dense broad ap number result heterogeneity limit estimation grid rp spectra arbitrary final grid datum calibrate determine via resolution ground spectra increase vertical band spectra fix lower vertical convert rp spectra simulator library broad spread resolution line response detector like overlap medium band bp cover nm range spectra blue resolution instance difference middle difficulty degeneracy using spectra due see reason former strong star difficult quantify plot bp source course primarily time nominal source noise source accommodate random error systematic currently band high star actual assume magnitude would apply deviation range around colour star star median large nearby star black rectangular value regular spectra distinct grid interval rectangular range step calculate grid locally high resolution I physical combination star really rare star star initial diagram source colour reflect population expect period formation typical white point discrepancy star train test order call grid ap speak ap accuracy strongly investigate datum grid star star star magnitude magnitude set distribution representative obvious varie train svm test set processing mix star magnitude star magnitude send svm near magnitude kind matching adopt train datum ap star data different ap grid sensitive rp spectrum individually normalize divide spectrum remove apparent three reason could arbitrary star select initial assign library construct affect use spectra limit grid may infer rely self consistency gm place derive library star constant make certain star star star star star star f star star star star star star star star star star star star ap svm star star star star star star star star star star star star star star star star indicate dot green dash model solid accurate residual ap true ap summarize strongly plot various ap table permit quick summarize ap star range four spectral type range star star star star star star statistic star residual possible square outlier summarize g systematic mean mr reader appear almost high show residual skew systematic accurately star star prominent spectra explicit line bp rp still third p look bottom show leave star symbol method circle vertical scale panel dependence error significantly bin star phenomenon grid weak effect consequence model rather increase nonetheless temperature star star star systematic part think correct unfortunately rarely plot residual residual use see green line red line ap range plot degeneracy top mean star open circle green closed circle dash vertical type scale figure encourage science star course law large turn somewhat star mr total error residual imply degeneracy svm accommodate observe degeneracy lack sensitive affected try solve
make conjugacy simple exist decrease spend space disjoint measure beta gamma process evy product evy represent tell exchangeable describe mix exchangeable distribution exchangeable binary matrix infinite collection bernoulli result exchangeable matrix beta distribution cluster estimation example distribution dirichlet normalize stable directly hazard pp auxiliary space random evy theory herein evy evy measure atom mapping give q family measure collection bernoulli value atom control dependence measure mapping vary form simplify take unimodal dispersion interpret atom location dispersion unimodal multiple location marginal indicator evy form smooth intuitive measure nearby covariate use measure evy unchanged conjugacy put original allow specific normalize collection normalize completely dependent construct large dependent normalize sub covariate marginally random probability measure remainder leave section describe two hierarchical base simple covariate dependency choose unimodal away covariate dependent model example decay atom correspond model go contribute covariate unimodal arbitrary might random transform sigmoid every value consider construction covariate dependent process homogeneous evy choose location kernel location covariate location fix auxiliary decompose gamma choose resulting imply valid pointwise construct covariate variable weight point select weight atom dependent corresponding give superposition generative sample covariate simplify gaussian ibp combine instance combine k model decompose text finite vocabulary simple topic latent draw word specific extended way topic topic drift allow topic time topic topic localize formulate poisson usage beta topic localize force parametric model activation allow modal disadvantage force treat twitter sense address happen use popularity vocabulary let dx dd db dp distribution present atom rate global topic baseline wang auxiliary dx xt kt expectation exactly form beta programming language inference poisson process derivation easier simply mark poisson poisson beta normalize construct x normalize process snp dirichlet exist covariate incorporate snp model could benefit scheme correspond well space thus naive poorly merge metropolis snp possibility explicitly yield scalable efficient implementation easier employ paper location allow write independent independent inference sampling grow covariate increase design inference represent originally tailor allow kernel manner step let g dependent probability evy ta dp unfortunately easily space however treat mixture show measure marginal pick framework addition represent poisson poisson define process normalize biased fit markovian dependency modeling consider probit dependent latent prior capture likelihood intend highlight ease dependency gain covariate yield ny n observe truncate beta beta ref probit carry gibbs truncate computational case atom converge conjugate update auxiliary iff sample joint distribution scheme lastly sample derive conjugacy supplement covariate dependent synthetic bag item covariate line eight pixel varying imply result location usage location use eight perform describe learn scale kernel depict learn dimensionality probability depict sometimes feature explain evaluate un dependent ibp collection transform gaussian test country observe country process country deviation compare exchangeable feature beta beta obtain rmse exchangeable indicate incorporate modeling result surprising use flexible structure add covariate conjugacy increase cost exchangeable bp run hour day account run simple latent dirichlet allocation ng document word distribution draw word basic way drift topic usage time apply topic usage time assumption allow localize formulate factor assume document observe dependence adjacent instance topic usage beta localize topic unimodal non parametric extend dirichlet allow topic disadvantage force quantity allow sense twitter document arrive address denote document vocabulary word vocabulary use model popularity pn x beta topic effectively learn poisson decompose pn x pn multinomial eq denote topic generate document baseline define control realization topic particular model gibbs sample supplement dependent qualitatively union dataset full text cover year break document result document keep occurrence corpus quantile comparable result topic vice versa qualitative evaluation hold hold word supplement static version binomial see obtain superior show incorporate version much
geometric convex half determine representation setup system study theorem statement share common let follow equivalent setup big class rise hyperplane ask every answer exist hyperplane family answer chapter involve undirecte unlike concept demonstrate demonstrate henceforth undirecte orientation enable orientation orientation else orient way direct graph undirected subgraph follow subgraphs undirected cycle cycle orientation orient orientation mean direction cycle sort linear remain orientation cyclic follow see strict observed cycle yield direct cycle yield yield cyclic subgraph vc dimension vc dimension vc dual vc cardinality strongly easily equality number component g ds se statement implication position orientation agree obtain orientation edge connect agree path particular vertex lie orientation theorem tree short path equal subset separate e ask closed intersection hard demonstrate system example sort subsection lemma get intersection system class intersection theorem theorem convention master thesis dr dr dr vc start numerous application rich elaborate discover researcher set definition give system trace al first strongly characterize strongly al discover characterization equivalence characterization phenomenon different discover field pure mathematic connection hope group enhance functional geometry system extend maximum geometry find new definition differ known duality transpose phenomenon duality via translate claim claim unary notation literature operator preserve property maximum work duality demonstrate stem boolean algebra moreover reveal concern sort reach presentation relate theory sort operator operator simple weaker weak naturally geometry combinatorial orient introduce convex study prove certain concern distinct direct subgraph advantage demonstrate application work study finite thus structure standard clear prefer system call system mean agree cube cube equivalence equivalence relation agree cover follow statement sx merging agree say eq definition identical except highlight duality observe example straightforward calculus statement similarity former replace strongly natural variant partition statement se section statement se first statement strong duality certain english text proof transform text lemma proof symbol pair pair sometimes dual true true valid phenomenon easy verify valid chapter strong system dimension dimension denote generalization accumulate work independently several discusse prove trivial useful let equality remark true prove conclusion consider assume hold equality f lemma let object pair restriction associate system induction pick restriction enough lemma induction dimension easy remains accomplish gives namely demonstrate duality transformation nature transformation theorem hard find claim dual claim case dual present proof transformation take care suggest follow equivalent prove every otherwise pick restriction lemma whose valid wrong system follow statement eq q lemma finish property mention serve prefer pair restriction se se se divide part conclude first valid system se euclidean discuss se maximum se seem possess property theorem cube accord terminology refer pre pre clearly translate restrict follow normalizing cube mapping naturally ns nf system system restriction maximum example operation complement satisfy boolean follow variant unary operator way system operator lemma array order row sort sort preserve sort fact easily array phrase member appear immediate following let sort follow author let se henceforth mean study define regard boolean operator symbol maximum se statement et case system verify lemma easy cube conclusion operator begin permutation intersection appear begin sequence permutation sequence lemma immediately imply theorem picture leave boolean next straightforward lemma every hold yy characterization follow system two system verify hold every cube eq easy q x term sequence member system definition object vx distinct replace every vx lemma desire finish proof system argument seq chapter introduce natural via concept discuss basic operator second generalization theorem regard form system partition correspond cube informally depend surprisingly operator become formal later extend cube cube cube original terminology extend definition operator x call follow hold reader might simplicity note cube x thus determine behaviour operator let henceforth extend x compute locality operator meaningful satisfy sequence boolean meaningful composition operator theorem prove regard remain true obtain meaningful operator similarly operator let cube follow concern cube theorem refine shift statement equivalent hyperplane euclidean half determine hyperplane cut chapter h section follow hyperplane independent addition note plane first amount intersection demonstrate h cell discuss generalization system question give line side htb linear hyperplane usual simply translate nonzero vector note normal hyperplane half associate orient hyperplane e
ern poisson parent cluster cluster sample distribution independently inside disk similar mat ern generate poisson process parent intensity replace parent sample poisson contrast mat ern position point isotropic center deviation potential scenario problem regularity clutter clutter consider regular density unit apart denote generalize incorporate control interaction exhibit regularity computational spatial environment extend inside case sampling clutter generate clutter density top toward bottom target locate window result mat ern whose spatial pattern realization desire six sample realization instance poisson use point average however point clutter discard realization benefit traversal clutter source clutter spatial clutter note rejection achieve actually point conditional poisson point distribution window however crucial observation interaction unconditional one purpose illustrate clutter environment choose point point realization average exactly clutter goal place true obstacle clutter probabilistic disk disk boundary limit disk sampling window poisson obstacle disk window fact obstacle line perhaps traversal obstacle disk know second perhaps operational point view obstacle disk center intensity inside respective total obstacle disk clutter window linear window corner w shape corner coordinate obstacle clutter name p name resp corner shape four corner top six corner point wise leave corner w ten corner point w shift unit axis obstacle sampling window shape place traversal target place along straight horizontal obstacle relatively detecting assess impact obstacle window condition specific point fact many window realization obstacle sampling center window figure obstacle point realization obstacle within clutter ccc disk center denote circle obstacle disk solid show field walks take ard total traversal walk avoid traversal happen traversal length long walk outcome traversal unit b gray reflect color indicate experimental treatment clutter obstacle window treatment level statistically significant traversal length window clutter treatment combination run consist pattern clutter ard walk runtime average simulation core processor speed clutter realization clutter exclude variability clutter realization clutter clutter clutter treatment combination total clutter realization obstacle combination obstacle monte realization sample obstacle obstacle background clutter type ip ern distribution distribution convenience presentation obstacle sometimes label v stand obstacle window corner coordinate obstacle short notation window window shape window obstacle p l l obstacle mention early precision obstacle obstacle combination traversal traversal clutter obstacle obstacle level hc obstacle realization consecutive level highly perhaps positively obstacle linear shape shaped distance magnitude take correlation account traversal difference treatment possibly traditionally repeat situation subject realization clutter type obstacle window obstacle number combination need obstacle clutter assumption among add repeat homogeneity covariance repeat correlation repeat besides plot residual present distribution satisfied setup clutter obstacle number obstacle level dependent try compete structure addition compound symmetry much benefit compare repeat test increase measure setup usual pilot study repeat clutter obstacle setup clutter realization repeat clutter realization setup clutter clutter realization obstacle traversal obstacle stand degree numerator denominator ii analyze obstacle obstacle factor repeat level usual setup clutter obstacle set setup variance autoregressive compound cs cs variance treatment covariance treatment cs setup imply usual structure combination unique ar assume distant obstacle obstacle factor treatment measurement treatment combination autoregressive experimental detail autoregressive heterogeneous heterogeneous combination obstacle var structure comparison var structure selection criterion criterion overall section traversal clutter obstacle form give perform obstacle clutter investigate interaction treatment combination interaction treatment obstacle ignore clutter consider obstacle clutter neither obstacle form clutter mean trend length obstacle clutter obstacle type different clutter type likewise traversal length obstacle form average obstacle mat ern cluster tend shorter traversal regular clutter tend long traversal length clutter tend traversal clutter traversal obstacle average order p shape shaped obstacle I obstacle obstacle interaction obstacle type obstacle number obstacle obstacle trend plot figure significantly reasonable traversal length level compare obstacle p shape form traversal length obstacle shape obstacle form trend increase peak shaped length concave disk obstacle clutter decide boundary often reduce traversal length avoid cost concave obstacle occur shape obstacle traversal length occur obstacle form traversal occur form obstacle short traversal obstacle obstacle ignore clutter type clutter plot compare value clutter traversal tend increase obstacle increase length obstacle traversal traversal length mat ern clutter type traversal length present short p p occur hc initial overall compare length pair profile plot discussion defer pt pt treatment w p hc ern w v hc hc shape v hc hc hc w hc hc hc hc investigate test obstacle type obstacle background clutter background clutter significant interaction obstacle number obstacle obstacle test effect obstacle form clutter shape form length increase shape obstacle concave trend obstacle clutter v shape length traversal length combination present traversal length traversal occur treatment type traversal traversal length obstacle traversal occur shape obstacle short traversal length occur form traversal length occur traversal treatment short traversal occur shape traversal obstacle traversal length obstacle traversal occur shape obstacle traversal occur p traversal length traversal occur obstacle form occur shaped obstacle obstacle number short traversal obstacle form short occur w traversal traversal obstacle traversal length shape obstacle obstacle short traversal treatment traversal short length occur obstacle form occur obstacle short traversal shape short traversal length obstacle traversal length occur shaped obstacle form short traversal traversal treatment traversal length obstacle traversal shaped obstacle form obstacle obstacle clutter type obstacle type obstacle obstacle clutter obstacle p obstacle obstacle clutter level test effect obstacle level traversal length different type clutter obstacle level obstacle obstacle obstacle clutter type obstacle type clutter type obstacle interaction clutter level clutter type level obstacle interaction obstacle type clutter compare main obstacle type clutter traversal length clutter obstacle interaction clutter obstacle reasonable interaction obstacle obstacle clutter effect obstacle type obstacle number obstacle clutter traversal length different obstacle obstacle level clutter traversal length clutter type obstacle type obstacle traversal length obstacle short treatment hc traversal length tend increase obstacle short traversal clutter type traversal occur mat ern clutter obstacle clutter obstacle traversal length tend shorter cluster clutter treatment length v treatment shape obstacle form short traversal length occur ern clutter shape short traversal length mat clutter type obstacle type traversal length clutter traversal length clutter sort shape form occur length hc treatment type linear obstacle short traversal length occur mat ern shape short traversal length clutter shape obstacle form short traversal obstacle traversal occur clutter traversal linear although clutter tend traversal mat ern poisson shape obstacle traversal length sort p short obstacle traversal length clutter p shape obstacle form short clutter short traversal length mat ern clutter shape obstacle short traversal occur clutter obstacle traversal length mat ern shape obstacle traversal length shape obstacle form traversal shaped obstacle type except clutter mean linear shape clutter mean traversal sort order desirable length reach clutter know clutter type actual clutter clutter type determine obstacle traversal refer well henceforth good type clutter shape length clutter form overall shape ern clutter shape pt clutter shape v overall shape p p shape obstacle obstacle table traversal length clutter background var obstacle form shaped obstacle form treatment obstacle obstacle type compound symmetry un ar autoregressive perform clutter poisson clutter mat clutter clutter clutter assume var mat ern aic clutter cs agree clutter var good yield aic likelihood ratio significant follow pick parameter difference type freedom criterion aic likelihood compound symmetry autoregressive ar autoregressive un small aic clutter un ar pt clutter un ern clutter type cs ar type un ar clutter type cs un pt un ar ptc pt clutter clutter ptc pt traversal mark bold type traversal length significant intersect vertical indicate clutter describe shape shape obstacle shape w shape shaped recommend v obstacle obstacle slight advantage otherwise low cost recommend length obstacle type recommend clutter shape obstacle obstacle recommend length decrease shaped shaped obstacle length v shape well significantly restriction slight employ shape obstacle form recommend well decrease shape obstacle form shape obstacle recommend l l v v mat ern v cross clutter obstacle mat ern clutter traversal occur obstacle pattern clutter mat ern obstacle obstacle obstacle scheme localization considerable scientific engineering community cite operational wherein spectral examine location potential algorithm call appear disk shape clutter coordinate scale shift clutter disk center inside take disk simulation environment inspire ard data traversal circle see cc inspection clutter clutter consider instead look realization clutter concentrate away investigate place use obstacle place scheme long traversal length actual obstacle obstacle traversal length traversal occur obstacle traversal length obstacle traversal length placing scheme compare realization replication traversal clutter realization use compound symmetry length obstacle obstacle length obstacle traversal significantly obstacle obstacle shape traversal length obstacle correspond mean traversal length shape obstacle obstacle w obstacle significantly different shape shaped form shape obstacle form shape range insight clutter pattern type combination obstacle obstacle number stand traversal obstacle type obstacle interaction obstacle type obstacle obstacle level trend plot traversal type number level obstacle form instead obstacle c linear shape w shape p trend traversal length shape obstacle form traversal length obstacle number shape traversal trend window shortest v shaped window length occur occur traversal length occur shape obstacle traversal occur p w treatment length v investigate obstacle obstacle obstacle well profile obstacle obstacle obstacle form obstacle stand shape w obstacle traversal length obstacle obstacle obstacle level length test obstacle obstacle find obstacle obstacle level trend significantly reasonable effect obstacle obstacle interaction obstacle type obstacle figure significantly compare effect level traversal length obstacle traversal length treatment present short traversal length tend increase obstacle short occur type obstacle length length obstacle decrease mean traversal slightly however reach short occur type shape pattern decrease traversal length stay stable short occur type length occur traversal stay exhibit concave concave type obstacle obstacle level traversal present traversal treatment short traversal occur w treatment occur type l treatment short w length occur treatment occur treatment occur treatment length occur treatment type length occur obstacle traversal v shaped obstacle furthermore traversal length concave trend overall obstacle multiple obstacle obstacle traversal length obstacle overall shape w shape traversal length obstacle obstacle difference family plot notice linear obstacle form shape shape furthermore shaped shaped obstacle wherein place clutter traversal theoretic specific obstacle place know clutter exact clutter investigate pattern clutter explore number traversal clutter extensive world system setup three measure treatment factor obstacle traversal choose furthermore measure flexibility different correlation homogeneous poisson mat ern hard core point total obstacle sample clutter shape extensive analysis cluster traversal long short obstacle tend reach increase traversal concave trend traversal increase disk optimum avoid window become obstacle obstacle form short occur p obstacle traversal length shape shaped obstacle tend especially large clutter obstacle traversal v shaped traversal obstacle obstacle form close start occur mat ern clutter clutter obstacle traversal tend linear obstacle get among traversal long window small linear obstacle window close obstacle e well shape window close conclusion valid different clutter obstacle window mark cost clutter type nonetheless environment clutter actual real world clutter clutter obstacle scheme world conclusion obstacle length obstacle form close moderate obstacle traversal occur shaped form point follow provide brief discussion several inherent know distribution clutter whereas clutter update disk mark accordingly assign mark fit overall clutter assign conjunction framework incorporation clutter account clutter mark sample obstacle mark obstacle disk center poisson obstacle window overlap argue ideal strategy sensible true obstacle disk reason choose poisson obstacle disk obstacle marks disk true obstacle information actual status crucial ard currently mark dependency instead poisson give disadvantage leave ard dependency obstacle disk center ard currently art presence capability heuristic guarantee yet challenge computational traversal attribute short ard algorithm benefit investigation ard leave future extended type obstacle problem know background place place oppose randomly inside likely term know sensor technology specific area obstacle disk location obstacle disk challenge would false traversal valuable comment suggestion flow article matlab code university obstacle wherein traversal traversal disk shape disk obstacle disk obstacle place traversal clutter traversal theoretic know clutter exact clutter traversal repeat obstacle obstacle place clutter spatial combination clutter becomes cluster short hand follow case datum application vision problem problem introduce graph theoretic considerable attention xu article consider modified version sized need start target disk shape refer henceforth brevity place another clutter clutter piece etc place clutter traversal disk disk obstacle traversal option disk obstacle disk disk clutter execute cost overall traversal assume obstacle location obstacle clutter disk en minimize total traversal give clutter maximize traversal markov computable show nonetheless efficient rd efficient heuristic provably possible complete spatial true clutter broad detect spectral field characteristic point distribution knowledge placing maximize traversal article study comprehensive limited investigation obstacle scheme clutter type various insight obstacle traversal address two question obstacle maximize might optimal way place clutter type analysis measure variance setup measure treatment clutter disk center homogeneous process ern clustered experiment obstacle process clutter window shape cost radius obstacle clutter efficiency adjacency discretization adaptation rd lattice rest manuscript organize formally ard clutter I clutter simulation clutter clutter obstacle place disk obstacle place obstacle equip assign traversal disk disk open fix radius short arc stand complement agent option cost pass
even accept accept unit steady sample score sample high track graph proceed accept consistent graph iteration generate new node e change prior effectively characterize relationship causal relationship pair prior existence scoring follow equation edge multiply influence graph thus large probability graph sample meet interface user unlikely edge bias convenient iff iff satisfy around empirically impact mention follow cubic plot clear gpu refer gpu figure gpu implements part remain part algorithm handle cpu cpu gpu score gpu gpu blocks grid access share global gpu gpu architecture architecture provide cache modify stream sm contain core feature core integer per memory interface support gb sm call sm unit execute gpu implement assign evenly block compare number assign set evenly block local score compare maximal score need assign parent predict local score convert subset indexing problem index number regular consider limit subset index subset subset index subset indexing gpu functions recursive parent parent composed choose update combination give without count straightforward combination one see begin get obtain large combination base parent set gpu store global storing suppose ram gpu ram perform gpu operating maximal implementation gpu scoring speedup score implementation gpu different gpu implementation significant detailed gpu implementation together acceleration rate list acceleration acceleration switch gpu gpu per gpu speedup sec gpu apply cell alarm c preprocesse runtime runtime graph sec gpu sec gpu sec gpu cpu mention gpu base implementation still runtime gpu runtime node part subroutine primary preprocessing runtime runtime runtime parent second parent second partial parent second also compare implementation generate parent implementation generate implementation possible consume runtime acceleration almost generate speedup time empirically accuracy receiver characteristic roc introduce measure roc true rate fp rate fraction positive give observe negative point accurate try curve close corner highly accurate small figure leave generate knowledge edge point assign interface remove without knowledge third add generating assigning remove add prior knowledge use generate fourth become generate h realistic contain noise state flip realistic show roc good acceptable noise tolerance network bn gpu traditional gpu speedup iteration gpu accelerate bn fold demonstrate highly one propose method scoring add enhance accuracy score evenly gpu implementation take parallelism scoring accelerate part work accelerate preprocesse gpu like thank wu author underlie gene pathway combinatorial nature carlo combination still mcmc purpose processor purpose graphic processing implement novel assign acceleration serial use incorporate prior component scoring search able inference big current gpu mcmc priors bayesian causal relationship direct work characterize relationship matching graph use score still accurate maintaining require datum part prior enhance accuracy significantly search add aim knowledge another integrate bn novel easily add existence edge nevertheless intensive demand hardware field gate array unit gpu gpu highly exploit parallelism novel gpu gpu background learn bayesian performance conclude paper set via acyclic node direct causal represent conditionally independent parent point condition variable write node influence parent instance figure parent determine figure common model model bayesian variable expression expression gene mechanism continuous structure experimental type observational observe experimental individually observational require sample complete due super network huge sampling explore small sampling topological dag node dag parent example topological node possible graph table list due reduce combination compare sampler advantage order learning aim explain graph prior q serve complex hyperparameter dirichlet refer parent refer maximal parent plot preprocesse well accept strategy randomly select order order subroutine section start preprocesse include initialization generation score part heavily score use accord consume need score parent store hash node parent later parent need strategy fold speedup experimental node parent change change comparison well
reference marginal able recover statistic observe isolated fix marginal recover tree w reference conditionally provide lemma invertible observable exact matrix version eq perturbation uniformly relate eq rr lemma r lemma apply previous node event component bound fix least ba pd give bind u define k provide statistic g h r inner ai di matrix uniformly simplify pairwise liu consider consider convention choose g improve improve fix substitute result bind mixture relate mutual relate entropy ix hx recall function bound mutual decomposition bound py iy liu see tree liu previous exact denote neighboring follow upon approximate separation function path vertex convenience py markov limit parameter remove claim correspond singular great subgraph fact similarly thus v simplicity isolate define b base u invertible u root close regime proof imply absolute distortion observable desire version slightly bound correspond induce pa aa large xx u u real k ki ia real eigenvalue r k ki unit column eigenvalue distinct j satisfie I I invertible property conjecture claim observation microsoft research new microsoft mixture discrete mixture variable markov propose mixture mixture vertex observe mixture mixture prove correctly maximum complexity scale locally correlation decay offer represent structural quantitative relationship represent natural vision financial amenable inference propagation recent computational requirement see brief simultaneously analyze model think fix set depend applicability change influence recent provable variety paper combine incorporate relationship parsimonious perform propagation graphical model base expectation maximization scale poorly dimension issue guarantee graphical offer include mixture graph product distribution section restrictive since direct significant incorporate mixture aim mixture offer tradeoff fit inferential tree since inference reduce mixture graphical tractable observe learning model graphical base stage correspond union propose union independence efficient node hold mixture sample marginal mixture hide component develop mixture observe conditionally variable adapt triplet suitable estimate marginal mixture degeneracy tree estimate component marginals liu liu span mutual recover component mixture complexity scale pair union component graph extend family include tree graph graph local significantly scope mixture depend proof correctness analysis careful use spectral success setting incorporate local dimension principle node another limitation however impose learn distribution know singular rank least hard another variable latent however mixture model observe variable well provable trivial product significantly advance scope selection outline variety recently focus deal estimation mixture separate recently recovery separation constraint complexity component call complexity scale polynomially impose outline method employ discrete product general triplet pairwise thus applicable selection seminal finding model establish span mainly efficient provable g graphical approach classify convex mixture mostly work tree mixture direct acyclic term use asymptotic decomposable recently learn condition think accord realization setting require component order test distinguish roughly impose although overlap moreover allow mixture allow variable latent estimation mixture operate significantly graphical finite cardinality say pmf local markov include say global property disjoint p property positivity condition henceforth model respect markov positivity markov iff clique z clique clique serve normalize otherwise allow potential discrete graphical observe variable mixture vector mix draw variate unknown know draw decompose draw stage graph accomplish series test special give graph respective mixture via spectral liu mixture extension v sample u satisfy consideration order property union yet respect observe union property recall node together perform test group graph crucial depend relax neighbor instance marginally independent value approximate separation set separation finite recover graph union component markov result graphical graphical output least test graphical work base alternative addition previous procedure graph component graph graphical propose test later recently product r separate variable independent py w form set proceed consider triplet full denote matrix u u estimate eigenvalue v h py I prove recover mixing learn perform triplet remain triplet hide variable triplet share node triplet order triplet decomposition learn order adapt simple observation consider three node removal line w g configuration successful estimate fix describe triplet obstacle triplet decomposition product previously fix triplet additional graphical conditioning unchanged h imply hold operate triplet generalization distribution require place lemma pair triplet dimension see marginal scale see addition follow success subset probability exist node isolate e matrix uniformly manifold r impose also learn require alignment label decomposition discuss component marginal marginal align estimate various characterize provide mixture component least appendix variable op p scaling note special case distribution well approximation estimate marginal mixture impose standard degeneracy existence unique approximation liu exact iy iy connect number n condition mutual liu model represent replace span denote error likely make exponent liu correctly mutual within liu structure marginal spectral succeed recover complexity recover tree liu discuss op approximation graphical technique previously product provable method establish polynomial complexity variable guarantee mixture continuous mixture acknowledgement author uci award fa pairwise reference isolated estimate algorithm carry triplet neighborhood separate fix upon spectral marginal liu
drive ergodicity volatility volatility stock return asymmetric volatility effect ex current commonly investigate leverage stress leverage exhibit asymmetric considerable conditional specification capture asymmetric exponential manner vary family volatility framework volatility allow volatility volatility process originally develop white al framework innovation price basic stationary uncorrelated gaussian time taylor asymmetric model suggest capture return asymmetric behaviour absolute incorporate li model autoregressive depend sign volatility excess residual basic normal skew distribution skewness heavy specification skewness return volatility initially interpret return future return decrease develop martingale property heavy tailed leverage use volatility generalise financial fact include excess volatility sn skew st non specification account leverage asymmetric central hereafter asymmetric hereafter asymmetric hereafter sn finally asymmetric skew hereafter st excess skewness multivariate instantaneous however technique inefficient base specification multivariate estimation volatility set solution normality hasting algorithm provide reliable mcmc associate importance ergodicity essential geometrically addition rapid ergodicity existence limit monte geometric applicability theorem aim ergodicity general incorporate et nonlinear measurable mutually identically distribute observed trend volatility skewness nonparametric series finite identically random mutually adopt strictly irreducible observation past function compact set iid restrict follow assumption iid variance moreover fourth conclude assumption process chain
minimizer inactive active e minimizer inactive duality non whose computed threshold follow operation unfortunately case directly characterize relation equation give median finding treat unknown approach guess say great method design triple q optimal follow lasso projection result supplement satisfy monotonicity employ description modify optimal restricted accelerate first observation element less secondly decomposable involve routine approach compute thus defer supplement brief advantage explain interpretable feature group interpretable nonzero group advantageous especially interpretation vanish incorporate prior amount large may simplify certain permit tuning counterpart note although model propose ideal theoretically moreover computation efficient perform selection conduct pc cpu memory operate dc programming accelerate propose nonconvex evaluating algorithm projection direction adapt formulation supplement generate radius ball fair record run admm terminate iteration stand admm summarize average algorithm replication ccccc admm next demonstrate toward end toolbox generating report distance euclidean convex unique termination relative difference consecutive less terminate k accurate problem great burden one method yield accurate projection one evident follow I standard partition truth possess nonzero group enhance nonzero accord test lasso choose validation conduct choose parameter metric error estimator capability recover structure table well well group glasso eeg genetic eeg electrical fluctuation place technology widely diagnosis genetic encode eeg electrical activity region utilize potential stability place hz therefore naturally divided group lasso adapt validation select lasso logarithmic sample scale select set table although underlie reveal fact accuracy lasso group selection theoretical analyze dc accelerate efficacy propose validate synthetic investigate extend modal multi task promise direction pt pt minus supplementary material sparse group nonconvex inequality likelihood ratio g last towards c pn constant j multinomial theorem j j j c g ig utilize suppose differentiable derivative differentiable addition gradient proof appendix infimum complete restrict c vs get gradient linearization define l x couple augment utilize minimization optimal possesse amount solve projection multiplier summarize note value implementation whenever fast u w x intersection carry projection alternatively square projection p cm plus minus engineering evolutionary institute university demonstrate parameter paradigm motivated application contribution statistically nonconvex sparse reconstruct oracle parameter efficient propose nonconvex compare synthetic achieve high past decade sparse extensively investigate statistical property possess certain explore feature propose ultimately detect site justify sparse encourage feature simultaneously computation suboptimal study nonconvex truncate potential superior formulation preferable theoretical nonconvex ideally wish follow condition define hellinger density dominate b space conclude regard global
point score dataset rapidly c next base area roc curve auc true rate false positive time change strategy peak regard specifically alarm true alarm remove alarm filter change point peak gradually illustrate curve random seed describe auc good test significance level tend perform value plot specific consistently well accord unchanged moreover reason medium evaluate dataset activity sense evaluate consecutive singular eigenvector reasonable evaluate target sequences subspace column span series value ar auxiliary ar ar change score log ar criterion descriptor signal set median distance heuristic kernel indicate outlier activity human activity collect change segment behavior arbitrarily decide orientation axis series change score plot score trend behavior change time change recognize regard roc curve describe tend next national institute environment speech offer segmentation manually annotate annotation final evaluation experiment signal score still clear speech roc curve auc significantly outperform si std c ar std apply twitter twitter interface track popularity keyword occur twitter frequency perform change detection hope correlation change keyword change evaluation wikipedia entry world change occurrence development news platform soon reach point score peak visit stock one year low change spike cut peak formulate two consecutive comprehensive art ratio key block various analytically possess optimal numerical novel divergence paradigm recently possess parametric convergence artificial real world dataset activity speech demonstrate promise density two observe margin reason decide model segment affect hyper discover challenge investigate show however advantage density translate performance detection clear intuition score point even attempt dimensionality change practical usefulness compare expensive validation procedure analytic method iterative detailed comparison improve focused point represent testing provide threshold determine recent often consume setup bottleneck recent report world event show twitter challenge along include interest acknowledgement sl program ms support project mm mm mm change series objective point discover lie behind time series sample method divergence accurately efficiently ratio human twitter message propose method keyword change kernel change series call attract researcher community decade depend delay point target application immediate response robot reaction period accurate detection certain delay change genetic analysis demonstrate change compare interval move typical alarm becoming significantly outli attract pre trajectory past interval dissimilarity subspace span column model lead successful detect south explain rely auto parametric accurate know density vice versa know density density estimation ratio develop g report outperform change point ratio promise advance fold idea square notable advantage analytically achieve optimal robustness experiment base change second improve change employ ratio ratio unbounde basic smooth bound possess convergence plain imply base change compare rest describe review artificial human activity sense twitter conclusion future formulate subsequence subsequence represent transpose treat series sample information incorporate start let segment strategy certain plausibility point specifically dissimilarity likely point change rest review section algorithm kl kernel cross divergence minimize ignore irrelevant second term constraint purpose constraint come non negativity density unique global solution gradient achieve estimator divergence kl estimator apply recently call pe different ratio fit squared specifically minimize constant substituting state approximate average parameter element easy confirm estimator give pe construct pe divergence divergence low conjugate pe notable possess stability robust experimentally compare infinity convergence govern sup overcome q p relative reduce plain tends smooth one confirm plain ratio unbounded contribute use divergence eq separately approximate parameter square estimate ratio approximate expectation density
bind perform confidence choosing choose os mix chain diameter chapter useful periodic taking generalize periodic transition give obtain distinct state converge respective mix aggregate state period theorem latter adapt period try reach arm cycle periodic force learner deal overall claim horizon state space arm exploration determine thus execute episode many sample action hold rs show mdp plausible mdps lemma argument assume diameter first state repeat ks ks sect k ks state term ks rs ks ks ks get color let ps split cf reward contribution pair concern second show episode analogously episode sect obtain evaluate sum episode bound k sequel ac evolve learner action know arm show index necessarily decide arm produces accumulate follow traditionally reward produce probability reward high setting measured extension set govern trivial set often natural transition name bandit bandit cognitive e bandit channel available sense arm simple sense also example illustration much arm two remarkably notably well difficult non bandit regret measure respect bandit take arm regret arm obtain useful compare regret allow regret optimally nontrivial depend diameter bias vector express obtain well good diameter try turn far regret bound problem latter consider armed hoc optimal policy regret unclear exploration exploitation finally bound exp hardness reward arbitrarily chain mdp horizon arm state diameter eliminate however chain arm low improve arm irreducible state know discuss periodic deal space know neither transition probability reward choose observe receive transition matrix learner compete know mean observe minimize average select selecting set eq somewhat next nature demonstrate immediate arm index arm bandit close armed state cc typical look average optimal policy otherwise close natural interpretation cognitive either device channel want channel typical policy reward stay immediate armed bernoulli reward know arm obtain arm switch arm reward choose arm observe reward arm sometimes appeal form maintain policy sample arm index seem independent seem evaluate independently work I case ucb motivate two arm index policy markov also intuitive index suboptimal index right start dash reward reward r observe optimal choose l give give reward probability subsequently arm miss reward l clearly behave consider bound eliminate periodic bad detail arm total depend arm clear intuitively wrong cost mdp step chain evolve diameter mdp recall number time step mdp j consider distinct action mdp markov power transition iff arm mdp correspondingly consist regret measure optimal mdp problem structural close arm tb observe initial episode let action visit prior number time color visit estimate plausible mdps colors mdp value optimistic tr could aggregate aggregation replace reward probability transition state whenever transition probability aggregate helpful parameter know aggregate guarantee mdps colored generalize structural structured set color function ps aa identity mdp aggregate structured mdps modification confidence interval plausible episode optimistic average colored action respectively adapt episode termination basically episode end color color tb confidence execute colored parameter mdp paragraph structure I j j
even specific inequality root commonly subsample behave uniformly provide quantile quantile q understand two side consider replace respectively differ hold pointwise notion way suitable well ensure compatible sufficient closeness ensure closeness discussion closeness metric ensure closeness quantile heavily relate closeness metric coverage usual argument asymptotic subsampling rely show tend construction precisely subsample use construct consistent construct contrast confidence pointwise sense less infinitely often likewise satisfying desirable analogous reason test fail finite course point nontrivial confidence sufficiently rich reason restrict test instance weak also relate complicated setting shrinkage ill develop independently discussion page whether asymptotic relationship subsample result verification moreover converse requirement relationship fail nominal coverage statement fail essentially bootstrap asymptotic univariate cumulative arise autoregressive subsampling inequality statistic discussion statistic uniform asymptotic validity differ subsample fail pointwise valid multiple testing despite area appear asymptotically valid proof result theorem supplementary material result uniform law number aforementione value valid describe subsample integer tend satisfy th generally toward feasible explain remark confidence tend infinity statement true iii bx nx nx bx bx nx deduce strong hold argument validity tend infeasible without feasible region case test nan hypothesis construct apply part conclusion worth though state root applicable especially hypothesis next feasible estimator root nx integer tend satisfy theorem replace tend root q sequence normalize limit integer statement theorem root computationally show obtain result correction factor assume establishe converse tend infinity bx nx p bx bx nx pt let variable root goal approach lie necessary require random true iii nx nx nx n hold satisfying deduce replace replace verify argument pointwise bootstrap construct consistent distribution q metric compatible yield word measure metric lemma whenever nevertheless coverage fail provide equal whenever event p n similar establishe iii kolmogorov coverage proceeding frequently correlation usual diagonal standardized expression variable reflects center normalize deviation nonparametric construct rectangular region root eq form suppose positive integer tending replace suitable restriction generalize root real suppose replace moreover continuously theorem need distribute proof requirement nonparametric consider root differ one sense hold fail bx nx central lem limit hand generality corollary define replace inequality generality hypothesis sequence p problem recently receive consistent level sense tending argument theorem establish test multiple hypothesis distribution sequence hypothesis control accord eq eq stop reject integer possible extend analysis straightforward nan empirical random one confidence natural true replace variable denote natural suppose eq q integer tend remain nonparametric construct example counterpart let generalize way generalize continuous z moment inequality propose testing alternative level adjust quasi follow understand dimensional matrix construct refine moment illustrative simple theorem distribution root generalize straightforward fashion statistic theorem way construct behave large versus hypothesis q bootstrap counterpart extend sided testing cumulative
development dna dependent specification domain bind bind bind activity mkl anti rna mkl mkl mkl protein bind activity protein bind bind dna activity bind nucleotide bind dna direct activity activity bind bind bind bind nucleotide bind dna transition interaction synthesis dna dna dna break end cell dna synthesis dna nucleotide process cell cycle dna cycle environmental indicator rate ie distribution inverse hierarchical conditional number initialize hyperparameter universit de france france environment act large number weak effect way identify correlation environmental integrate signature adaptation algorithm mix probabilistic unobserved genetic variation infer background structure evidence factor number development gradient adaptation selection role fundamental biology intensity environment effect e adaptive cause evolve trait advantage environmental involve achieve wide dna specific signature selection population spatially adaptive detect compare among marker genomic alternative way investigate signature especially beneficial identify environmental population quantitative trait selective act individual phenotype reflect population evidence detect environmental background genomic variation basis environmental adaptation gene drift correction consider association environmental positive study allele account covariance assess adaptation allele environmental explain neutral build face need identify neutral genomic imply lack reject correlation environmental genetic effect factor background structure perform extend recent statistical approach deal thousand rapid control population autocorrelation estimate detect adaptation human allele genomic size simplicity nucleotide allele environmental could environmental association environmental background level response vector deviation prior zero variance gaussian prior environmental variable model factor separate neutral variation explain environmental factorization estimate identify score loading pca factorization probabilistic factorization analyze genetic environmental factorization row decompose data decomposition vector minimize loading simulation choice application estimate population genetic essentially estimation score loading gibbs sampler product speed scale snp addition algorithm environmental retain score absolute preliminary experiment define equation find effect quickly individual fold recover latent factor alternate use deterministic gibbs check consideration incorporate population proportion principal study approach program start individual q score equal identity consider equivalent equivalent vector depend empirical point main conservative principal component exist individual series generate replicate generative test set report distribution conservative moderate number test produce false association represent high genetic generate replicate error model rank generate regression first matrix environmental effect check similar distinct initialization report quantile absolute error square indicate pc regression hide absolute algorithms pc quantile error shift fold poor performance increase series method lm glm pc model enable ms ten adjacent include individual test implement find run compute genetic neutral lm factor range test lm test produce result latent pc value much find lead heat stress genomic dna individual population refer genome diversity centre genome cell panel array filter remove include file weather year period temperature maximum month variable score factor additional total snps great disease trait association rs associate significantly correlate notable disease synthesis activation production gene heart involve rbm development base factorization unify effect environmental genetic environmental allele include genetic environmental genetic specie environmental statistical gene adaptation development summary environmental association detect species genome however implementation example result positive association mix regression estimate suggest utilize separate neutral variation selection phenotype marker neutral sometimes distinction explain extremely difficult neutral background use actually correlation environmental allele frequency genetic much fast analyze computational program outcome take validation computationally theory select find lead conservative estimate still restrict approximately suggest cluster specie test directly fine motivated trade accuracy future development develop numerical base algorithm confirm discover link activity heat stress association gradient example identify gene color associate contain snps height synthesis disease snps involve heart brain confirm correlate evolutionary example result support soft ever genetic throughput sequencing model contribute toolbox landscape new environment association source code program available web work support nsf lm cccc fp lm glm evolve specific response stress heat temperature stress protein disease response stress bind probable bind dna dna cm rs trait color vs disease black red cancer h disease rs rs rs rs rs aggregation cr rs rs pr height height rs brain p cl heat protein cl dna heat bind probable protein alpha cl alpha sf cl nuclear bind protein group cl bind protein domain contain dna h temperature temperature month temperature month temperature temperature bp bind binding
som extend som correspondence approach consist standard computation extend som dissimilarity drawback increase propose however increase stay algorithm relies express som make kernel space version batch kernel design handle string dissimilarity combination classical version som dissimilarity approach name idea som organization describe arbitrary input dissimilarity non grid neighborhood prototype randomly random combination matrix iterate assign close prototype accord distance carry straightforwardly computable distance step accord equation dissimilarity batch call batch relational som introduce long suppose som line som close prototype non position decrease neighborhood whole neighborhood neuron vanish relational som som basis som possess several drawback organization bad visualization unbalanced batch online scan significantly batch summarize som som illustration uniform batch relational som version relational som identical initialization available batch relational som iterations grid organization iteration initialization som converge organization iteration initialization organize minute batch minute processor ram relational som presents application line relational dataset deal numerical surface categorical first relational som simulate roll performance figure simulated distance run different algorithm uniformly first geodesic algorithm perform map unfold som som dissimilarity geodesic important unfolding square relational som project roll separate moreover roll completely rather grid heavily test rectangular grid result clearly batch som grid line som median som som use dna dna comprise molecular bioinformatic aim identify biological assigning specie accord publish dna hand specie discover species neighbor construct dendrogram large become rapidly sensitive unsupervised learn dna although use helpful cluster sequence hundred thousand site specific distance dissimilarity euclidean dissimilarity median som test amongst median som provide encourage main drawback organization highly among allow empty existence group acknowledge som mixing allow detect main specie labeling unsupervise address issue distance nearest cell distance neighbor vertice two distance vertex cell specie diversity radius proportional cluster use graph book us amazon com edge co book accord political orientation neutral extract relational som short dissimilarity figure provide network display direct color classify simplified represent density second organization graph group connect whereas
pair canonical transform fouri relation signal q unit norm u triangle q q follow supremum spin recover restrict isometry relevant full incoherence specify operator easily basis retain spin true spin sparse let mutual spin recover component spin conceptually separate mixture incoherent basis sparsity quantity minimization similar tight therefore spin yield careful spin manifold incoherent basis direction spin case dictionary comprise manifold powerful flexible conceptual image ensemble consider ensemble vary nonlinear manifold include acoustic vary frequency represent disk black disk image solid variable pose six correspond location orientation manifold manifold class construct preserve geometry randomize operator rip constant volume space ambient isometry projection operator reconstruct compressive disk component generalize signal manifold fix template define image denote unknown problem equivalently compressive demonstrate image demonstrate spin consideration recovery signal canonical spin provide incoherent rip manifold shift element corrupt spike spin projection match template simply well assume spin observe spin measurement constitute ambient figure measurement vs reconstruction plot db spin far corollary relationship interest hybrid signal rigorously analyze projection incoherent manifold recovery pair signal spin geometric criterion two disjoint incoherent operator isometry manifold computational spin present demonstrate utility spin thorough study spin future gradient count relate step isometry constant preliminary size give result choice size thresholding scenario rarely ambient concept approximate projection rigorously demonstrate spin indicate spin inaccurate clarity brevity focus attention signal belong spin conceptually sum spin require manifold incoherent operator isometry component manifold spin component matrix reconstruct affine rank attract attention key rank matrix incoherent two manifold matrix vice phenomenon quite challenging need lemma approximation cs nsf n nf nf thank valuable early manuscript department electrical computer engineering signal signal manifold ambient incoherent spin order project method nature recovery measurement spin provably recover manifold incoherent operator certain restrict isometry spin low dimensional match exceed art observation problem instance limited possess effort instance advance separation compressive affine principal write differentiable manifold give linear measurement measurement objective signal also numerous arise identifiability simple measurement noiseless operator observe fundamentally ill pose manifold unique signal situation operator possess nontrivial particularly ambient issue order two issue nonconvex non numerical descent successfully convex optimization design linear type signal prior successive incoherent manifold spin nonconvex possibility spin provably recover true require incoherent restricted rip formally statement recovery incoherent satisfie rip observe b b k b propose iteration z manifold onto component manifold play stability manifold stable operator detail spin arithmetic essence extensively hard projection spin spin project algorithm pursuit generalize mixture nonlinear manifold spin component manifold spin exhibit strong comparable art despite nonlinear nonconvex nature certain special sophisticated higher strong stability guarantee pass amp lagrangian recovery appeal spin conceptual simplicity plus generalize nonlinear manifold manifold paper convention vector quantity appear unless product interested manifold informally manifold signal applicable identify capture signal locally continuous possibly signal manifold riemannian manifold example define include signal excellent refer analysis core orthogonality isometry measurement availability incoherence element generalization normalize simply incoherent manifold direction incoherence manifold orthogonal inequality direct decompose strict uniqueness b b incoherence unnormalize last arithmetic henceforth gm impossible unless em direct sum lie incoherent manifold incoherent b rearrange desire address restrict isometry manifold matrix isometry property rip belong generalization approximation sense matrix rip traditionally generalize manifold isometry certain manifold construction rip dimension ambient discuss section define informally closeness term euclidean nonconvex manifold ease exposition henceforth signal operator crucial signal precision arithmetic uniquely define nonconvex manifold admit operator example signal length canonical subspace projection approximation efficiently simple thresholding describe onto incoherent spin view generalization recovery key spin formulate measurement b return demonstrate spin possesse uniform comparable approximation broad signal theoretical describe spin incoherent measurement isometry manifold observe noisy measurement spin moderately sized explicit spin output z spin true iteration positive paper informally recovery fine precision spin availability operator approximate multiple mechanism projection spin z note implication signal isometry constant use spin spin iterative recover manifold match guarantee automatically mild condition unique full spin proof analyze sparse define error spin incur th current iteration z k k b z k k b z take adjoint inequality
ti remark scale laplacian definition ti desire focus convergence obtain establish consensus agent average dynamic obtain boundedness iterate agent successive refinement I correspond centralized action elsewhere tc tn ni j jt j satisfy q construction contradiction argument action pair evolve scale version obtain construction conclude action event imply conclude hold often together establish action inequality derive q hypothesis lemma suffice following proceed instant induction hypothesis eq property induction obtain q establishe reverse direction hypothesis establish learning establish update sequence agent iterate merge establish reach consensus evolve rewrite whose adapt evolve quantity adapt nk kk state asymptotic average iterate see agent innovation take generate pair trajectory depict solid centralized factor uniformly agent distribute reach consensus readily reasonably importantly centralized asymptotically limit centralized trend equivalence asymptotic convergence loss centralize attribute consensus enable agent essentially track investigate reinforcement setup sensor building entity compute differently environmental setup collaborative competitive learn stationary discount focus approach processing mean consensus strategy assumption communication formulation action set mixed stochastic markovian state general applicable broad memory low dimensional state indicate centralize argue per distribute scheme asymptotically asymptotic consensus convergence state impose action approach simulate centralize practically motivate concern perfectly observable agent instead act responsible proposition result htb department engineering pa mail edu consider multi agent process agent differently instantaneous cost control distribute reinforcement setup state consist horizon discount propose distribute agent sparse cost agent yield network agent analytical interactive dynamic uncertain markov control influence random instantaneous incur process collaborative agent network minimize horizon discount instance resemble control environmental dynamic spatial temperature sensor application building reference minimize measure sense desire important scope agent correspond social entity dynamic interest pattern policy controller shape economic growth scope formulation limit scenario management agent motivate learn instance valuable practically involve information include agent agent instantaneous cost bellman generate sequential trajectory rely trajectory implement control direct various exploration technique would cost instantaneous stage cost locally agent cost central resource bit communication medium fully distribute agent computation extensive multi survey formulation range stochastic call investigate viewpoint formulation somewhat setup optimization unique stage local agent cost key additional instantaneous realization stage global cost observable specifically instant instantaneous cost whereas often formulation decentralize stage cost time comparable often decentralize controller emphasize require literature locally cost mutual neighborhood communication pre decentralize specifically perfectly access contrast decentralize controller control action perfectly agent limitation agent simultaneously receive instantaneous potential consensus communication one cost appropriately communication consensus one agent functional inter exact instantaneous appropriately potential suitably design optimal performance network agent consensus control minimal generic lead scale markovian analysis technique broad class problem viewpoint centralize explicit consensus distribute scenario problem network literature optimization scenario broadly network goal static objective agent aware view extension environmental dynamic model static agent obtain minimize contrast scenario formulation transition state sequentially cost rest set sequel learning setup formulate present propose formalize inter agent communication intermediate distribute present learn section centralize section finally conclude component wise denote use correspond cone denote denote definite matrix denote vector zero symbol use dimension operator standard euclidean matrix denote existence object correspond respectively inequality object interpret surely state inter agent communication denote agent communication pair exist loop graph li pair edge matrix diagonal definition positive laplacian matrix order eigenvalue eigenvalue algebraic connectivity value detail control markov denote generic note bold symbol govern satisfy agent often reduce former control proper augmentation apply satisfie evolve dependent modify policy horizon discount factor global agent mdp concern horizon associate provide centralized programming denote bellman readily see contraction imply start obtain form basis classical successive value know reinforcement method lack information bellman class method stochastic call action often recover trajectory oppose rely implement direct adaptive offline simulate response far trajectory time dependency due relax context rely centralized instantaneous cost centralized obtaining expectation instantaneous cost agent location feasible due agent medium motivate learn collaborative local communication scheme agent learn counterpart state process agent learn scheme base stage formalize agent impose requirement characterize locally conditional require condition random adapt formal trajectory control markov accordance obvious measurable give instant sequel characterize agent locally message formalize message obtain possibly agent slot algebra agent network available local agent information reward consist locally reward information agent view require across formalism readily join I e instant induce moreover inclusion strict usually agent communication inter strict global explain fundamental exchange run lead wide eventually obtain accurate successful involve communication inter generate neighborhood satisfy distributional failure failure link failure spatially network failure time wireless motivate interference wireless failure connect stay capture assume capture broad class asynchronous random asynchronous protocol analyze fall hand event communication link exchange system instantaneous action eventually instant adopt show pair note agent simulate often impose note centralized condition u n collaborative distribute weight reflect sequence update instant refer transition reach realize slot stay maintain adapt serve weight w identification eq send process update agreement consensus innovation incorporation sense result algorithm trading impose sequence follow consensus innovation potential persistent sequence guarantee innovation weight consensus dominate innovation comment sequence associate innovation potential consensus innovation condition far effect randomness local asymptotic mixed scale
conclusion value optimisation scatter indicate probably case mean optimisation remain r orthogonality display bad negligible surprising design orthogonality criterion design exhibit term ml design slightly orthogonal orthogonality criterion unique w design criterion slightly restriction orthogonality restriction unclear design good orthogonality lead orthogonality improvement ml design slightly bad good improved combination visually plot nearest choose design worst minimal bad property crucial projective sa negligible impact response parameter omit surface onto original none project projective call situation small corresponding simulation hypercube hence projective property measure quality projective property minimum project domain use redundant design present onto histogram represent redundant easier redundant list cn show superiority redundant design bad decision number call strategy fine add coarse grain batch one produce one shot fine grain sequential possibility generation sample sufficient advantage non orthogonality course optimisation may display coarse grain sequentially produce one design shoot orthogonality property searching hypercube latter test sort optimisation orthogonality purely factorial design integration grid generation bad orthogonality last group employ modelling place suppose error sensitivity hard confidence broad design equal level discrete new sequential randomness free point figure plot distance aim show strategy start design preserve column row present design within sensitivity although bad generating show importance monotonic relate impact evaluate engineering majority monotonic quality sampling monotonic consider discrete discrete plot together correspond full design involve mutual correlation design store difference parameter plot per associated apply design accord restriction model multiply cc cc ml cn max max max max cc cc c cn max max max mean overall sa ml optimal design term variance free slightly bad design free suffer large average compare variant classify hand criterion contrary evaluate suffer sensitivity aim analytical devote illustrative benchmark optimisation represent ten bar area bar benchmark continuous discrete together cross area discrete p material lb loading concern bar material loading give thank symmetry bar group hence variable cm p material lb modulus f model maximal restriction restriction automatically specify design feasible bar bar design h anneal iteration correlation compare use consist statistic parameter independently maximal correlation multiply list ccc cc max max bar bar sa structural model cn ten bar design predict three significantly improve well ten bar criterion design balanced able among bar obviously parameter eight hence bad eight comparison ease mutual usage base sa analytical structural conclusion cn present aspect make optimisation tendency design subsequent analytical structural error difficulty optimisation nature tendency even restriction bad design suffer high variance criterion regard optimisation advantage simple computation good orthogonality prediction criterion generally restriction design ml design orthogonality sa sa bar projective criterion concern optimisation besides drawback lie necessity bayesian ml criterion common winner comparison course consume growth computational exploration task basic investigate input simulation e point design call aim available suitable sensitivity orthogonality hypercube sensitivity sa tool investigate essential part response individual system present contribution particularly aim able reveal nonlinear monotonic relationship output sa expensive exhaustive perform relate contribution present review several generation generation aspect discrete continuous possibility continuous domain beyond review literature include method generation present difficulty arise devote mutual compare projective improvement section present assessment design usage sa respectively conclude criterion assess preferred preferred need evaluation admissible input value orthogonality necessary assess parameter may preferable orthogonality nevertheless base employ one propose potential distance probably know minimal two maximal minimal use quadrature normalise dimension value space discrepancy consume ml work field regression determinant dispersion inversion negative point order become eq design illustration fix four corner domain coordinate corner value colour instance optimisation criterion idea high response add overcome problem modification imply add b need keep mind add preserve c two define manual two orthogonality cn commonly design center scale define eigenvalue generator discrepancy design review design heuristic annealing employ design concern briefly possibility optimisation particular sake clarity present four five four corner evaluate last shape way whole row grey colour colour corresponding ii detailed surface accord domain imply several concern optimisation optimisation space optimisation situation present criterion poorly another negative demonstrate criterion cn extreme corner pointing character cn evaluate corner undesirable smoothness appear optimisation criterion much anneal feature concern r corner optimal design well test quality ml
result college political book case correct block variant employ monte result obtain obtain block degree actor node connect cast member reflect actor always something impossible modularity besides block temporal line actor correlate datum enables fully scale thank point correction american b universit module length principle seek prescribed block structure maximum block simplicity yield efficient monte carlo application network actor bipartite module problem literature systems blockmodel attention drastically majority maximization derive structure inference blockmodel communities unknown infer predicate choice amount blockmodel ref generalize accommodate arbitrarily arbitrary community structure penalty monte arbitrary structure blockmodel compose block node twice far impose direct analogously becoming fix degree respectively ensemble entropy entropy blockmodel ensemble correct belong node edge analogous material overview block maximize network compatible henceforth entropy number block otherwise matrix principle necessary blockmodel eq upper necessary observer one datum loop simply multiply number take derive ref b without restriction must exactly restriction show correct direct replace equation infer unit width dl mm mi vs benchmark width mi vs pdf prescribe together infer vertical line mark c true detail pp impose diagonal structure otherwise original common express pp value criterion useful albeit recover discard serve structure mi vs pdf partition pp different grey line ref difference impose detect convex global give easy even prescribe partition block possess impose criterion ref blockmodel variant show recover fast infer hence compute resolution modularity interpretation simply modularity entirely avoid properly modify modularity knowledge improve infer structure eqs variant closed width unit mm width profile b blockmodel american network correction describe political book infer red match infer correct blockmodel circle block actor
underlie number tend hypothesis indicate l evy ad tends clearly indicate propose skewness difference fusion device analyze fluctuation phenomenon mode h mode rapid drop device operate device investigation physics diagnostic tool characteristic potential probe array behavior mode like transition reveal radial three typical fluctuation radial observe obvious amplitude position set constitute consecutive examine part evy examine tail set whereas examine datum datum statistic depict follow confidence consider tend whereas value confirm one propose indicate analysis evy ad hypothesis distribution examine skewness parameter corresponding interval add possible prove normality active research determine develop able close show law essential identify alpha simulate stable distribution work might detect evy develop procedure assess stable fluctuation measure fluctuation concern observe evy fluctuation measure stay evy index detect stable sufficiently heavy tailed pdfs low turn qualitative change important scope check device well discuss alpha l evy available law clearly combine visual fourth check method furthermore density transition occur transition gaussian statistic stable alpha stable evy stable central gaussian independent reason evy stable naturally appear determine characterize evy index stable law slowly decay law asymptotic process general four l skewness location deviation comprehensive alpha stable evy distribution quantity evy exactly motion arrival find evy central brownian walk jump heavy tailed review evy light medium evy evy field stochastic electrical engineering biology economic lot analysis evy parameter l evy stable quantile quantile evy stable law consideration evy fast come hill log log focus assume parametric whole section v address issue evy index shape pdf gaussian log scale hill evy example fig ref especially biology contain evy belief analysis inspection l test relatively rarely physical biological effectiveness application analyze fluctuation device detect l evy device sec pdfs simulate evy close sure almost visible plot moment simulate demonstrate evy outline l evy demonstrate one large number theoretical evy index number evy visible empirical totally skewed evy distribution symmetric asymptotic evy contrast difference pdfs demonstrate technique evy htb algorithm recognize evy detail statistical test goodness check idea check probable reaching analytical cumulative piecewise zero jump point usually measure either supremum quadratic well kolmogorov ks distances analytical incorporate define suitable ad test kolmogorov ks exhibit poor sensitivity ad fit tail crucial stable law testing propose ks ad goodness test match normal skewness converge skewness deviation skewness l evy stable implement various package g first stable general structured whereas favor outcome statistic reject ks ad test propose
certain diffusion chain marginal information imply rest part introduce unsupervised couple dx np xx unsupervise unsupervised form principled labeling build I partition misclassification learn train c text method search partition minimum scheme correspond labeling mention rather study misclassification misclassification error scheme misclassification error quality estimate average generalization cs conditional density prior let measurable far old underlie isotropic bandwidth one assumption prove kernel density uniformly converge bandwidth assumption probability kx dx indicator almost sure n n min f note indicate unsupervised connection nn hard give nn cost function neighbourhood q misclassification misclassification give show nn facilitate introduce cover rp r misclassification nn I em nz l x x r l eq x moreover n h h f iff cx approach nn far asymptotic misclassification soft lm lm lx n rhs nn c plug kernel density suppose eq sure e cover point z g r dd rhs suppose g min g generalize almost min kx dx vc function envelope exist open min b min g vc obtain classifier lm em class indicator bandwidth satisfy old h old follow equality convolution h follow verify misclassifie almost volume cluster boundary region separate low density separation cluster prove slight proof plug weighted lemma bandwidth rhs follow rf pi lm plug volume boundary pi ds vertex lie edge determine induced volume misclassifie plug suggest plug normalize minimum solve computing laplacian hx forward laplace backward operator moreover forward operator operator normalize capture riemannian consistent similarity reflect geometric volume misclassifie classification practical method misclassification near plug connection error plug volume cluster similarity classifier close relationship remark edu classification unsupervise manner exist misclassification error except classifier misclassification error popular near classifier unsupervise bound similarity prove recover type map close classification normalize laplacian similarity misclassification induce volume misclassifie classifier laplace set representative include dissimilarity cluster identify cluster complex lie low parametric learn classification learn max manner far
token rgb rgb rgb rgb rgb relations consist ignore computing tool graph high system signature relation positively propose spatial secondly force interpretation mechanic devise embed multidimensional rely assumption dependent capture short property dimensionality ability preserve local produce maintain meaningful neighborhood gradient trajectory sensor result superiority propose embed various reduction extract complex set popularity decade back principal component classical multidimensional recently local fisher discriminant discriminant modern enable capability reduction technique meet limitation structure hyperspectral acquisition object pose challenge set employ variance unfold embed subject good point laplacian draw correspondence laplacian laplace heat devise geometrically construct sample high widely method extend property classical nonlinear embedding extension dimensionality probabilistic widely separate lead embed distribution embed stochastic neighbor graph capture embed include distance non kullback leibler divergence nonlinear dimensionality compression visualization reduction represent manifold comparative study manifold hyperspectral reduction community problem tendency embed result increase discriminative boundary form within develop reduction visualization incorporate benefit unify framework nonlinear dimensionality form high predefine draw interpretation suggest design dependent novel embed enforce pairwise range force towards dominate force dominate range environment neighbor field importance together similar act barrier generate force map reveal meaningful structure similarity construct second local kernel signature space image characterize channel pose challenge visually exhibit overlap signature structure review force formulation formulation multidimensional field framework connection popular nonlinear field unbounded setup various force interpretation relate study biology discover pattern force analogous physical force early propose interaction maintain stable developed discuss later power law individual strong short compare parameter constant inter law artificial system carry complex rely fact basis rule aggregation short planning biology control study seek address relate maintain stable obstacle grow importance force field area remain concerned draw collective behavior devise predefine formation manifold give force field loop element property compression extension pair force embed vertex map particle motion edge force law govern model configuration map pairwise map heart mechanic embed present imagine particle move velocity centroid denote position move aware position correspond force edge graph individual centroid inform dependent force field interaction velocity describe symmetric interaction th map map attract attract whereas represent superposition define interactive insight section force dominate dominate unique balance manner barrier force embed function follow ij ij odd symmetric origin u ij artificial term whereas term establish alignment interaction force act along potential explicitly instead neighbor superposition field enable cope change rewrite reflect force map along occur around define motion potential neighborhood let describe dimensional map map embed adapt platform reduction many nonlinear visualization though field preprocesse building visualization task ask illustration popular neighbor le stochastic neighbor benefit various functional combine insight create embed preserving assume neighbor presentation focus version weight use binary method ensure effective neighbor low gaussian pair weight map proceed minimize kullback distortion neighborhood distribution dimensional neighborhood far demonstrate superiority include locally optimization unstable lead experimentally attain meaningful expansion dimensional work reveal term nonlinearity force field form vertex space attractive force force interpretation field magnitude dominate cause establish force describe stochastic neighbor well model modification lead improvement student embed kullback divergence correspond expand cost dynamic force field term vertex map attractive force square law large force describe formation establish force short field spherical variant embedding assume surface probable value similarly desirable unfold many spatially drive probable control map implicit unit yield motion contrary describe manifold turn define sphere neighborhood configuration magnitude unbounde pairwise distance also make spherical apart force field I unbounde inverse distance exhibit capability split unbounded configuration unstable technique present proximity embed dimensional embedding obtain idea minimize magnitude obtaining whereas uncertainty uniqueness embed constraint compute solve eigenvector problem connect field identify artificial whose motion map incorporate cauchy interpretation step insight force field energy component involve choose incorporate neighborhood high technique equation however area hyperspectral suitable parametric neighborhood extension pixel tend drive therefore fully address image remove smooth enhance result feature improve technique partial differential equation hyperspectral demonstrate significant improvement devise pixel graph spectral artificial unbounded vertex potential suppose force map I potential corresponding force short range distance force dominant observation model formation pair whereas set exist table force function exhibit behavior c n ne ce nz nz objective nonlinear require stability learn meta rate adaptation embed even fast minimum configuration description termination configuration rate optimization world initialize parameter neighborhood randomly distribution new u iterative optimization equation equilibrium distance motion guarantee minimum configuration invariant visualization consider energy negative motion graph continue decrease invariance principle configuration converge conclude result embed force field property discuss termination embed embed neighborhood neighborhood weight dimensional optimal embedding establish another comparison enforce map coordinate learn nine natural acquire delta signature remove water overlap band band classification carry hyperspectral acquire national visible sensor resolution water remove leave class signature subtle visualization low coordinate scene north area west west consist tree image resolution gray select testing euclidean compute classification class scene hyperspectral comprise pixel locate six original scene california cover comprise water discard band green min heavy center c water c short water evaluation le take neighborhood obtain principal component reduce construct neighborhood value ensure neither unstable observation observation implement gaussian determined generate graph principal establish spherical magnitude set norm termination solution algorithm iteration terminate embed map admits namely pick lead le pick eigenvector dynamic serve describe gradient field map minimum configuration trace affect cost corresponding scheme field present hyperspectral class interior water field formation high random change cause pair map smooth optimization iterate design dependent range force generate magnitude contrast initialize seed display trajectory direction field indicate severe optimization point negative direction gradient water demonstrate gradient point reduction demonstrate performance problem manifold tendency onto characteristic increase class representation effort visualization complex figure reference le superior map method compute coordinate seem separate seem spread imply tight demonstrating class separate effect significant le produce interpretation image reference embed embed construct spatially embed maintain disjoint truth boundary appear existence water class little ten class presence le little meaningful interpretation embedding show figure generate embed information disjoint neighborhood establish neighborhood interactive field generate relation many maintain embed straightforward denote low euclidean respectively model local representation highlight coordinate continue frobenius turn preserve force tendency map small separate property useful suitable task close include entail establish automated image object class label determine nearest angle cover dimensional carry pixel several visualization table trend indicate outperform enable accuracy observe mean ks classify label belong percentage provide interpretation consistent seem lead accuracy however lowest achieve embed representation separate achieve ability make label distance force field optimal metric k near always class term separate term formulate map classify trend provide representation low embed achieve visualization correspond category tree spectral signature mix subtle graph incorporate boost category advantage pairwise embedding separable suffer algorithmic optimization nonlinear objective misclassification error embed dimension reduction rely call embed estimate neighborhood low allow regular automate property acknowledge open research deep separate study include figure channel choose low drop display nn classification le ks class c c ks embed demonstrate nonlinear category resolution technique framework valuable couple dimensionality reduction
illustrated drive formulate comprise markovian suppose originally access e heavy drug widely several health development understand comprehensive branching markovian alternative base constitute strategy assess reach generate random traditionally associate existence priori learn specific characteristic population research factor account propose population collect participant seed participant ask ask participant drive quantity associate association within tend categorical contingency value row status status pearson contingency check observe naive ai ai ai ia ai size individual however account structure friend population author confidence interval coverage confidence build naive characteristic drive sample characteristic risk incorporate risk risk logit characteristic individual contact know latent markov network individual hyper parameter diagonal prior region laplace de et al comprehensive population comprise comprehensive criterion ii another age old participant study long comprise equilibrium reach start seven seed start slow six seed eight seven participant month study treat seed subsequent deal local instead besides heterogeneity pool weight former single city comprise country evident structural secondary structural drug progress another carry confirm finding much probably social difference large take underlying reason choose site unfortunately participant major open scene structural drug attempt researcher health worker knowledge parametrize study contingency table pearson independence test nan hypothesis suggest dependence analogously contingency table participant pearson test nan suggest evidence therefore structure population provide use confidence evidence obtain function logistic regression logistic effect well agree although coefficient interpret odd participant three ci receive material participant odd school odd ci participant college education participant odd ci summarize suggest participant reduction remove connection lead dependency factor relate odd ci participant time odd participant live live participant time infection
component unit equation repeat run burn apply experience behave augment produce sample unobserve turn enable etc remark specify metropolis hasting gaussian independent exponential element draw otherwise save element algorithm posterior initialize expand run produce augment posterior unobserved produce predictive full unit size step complete pmf hence sample level package front include parametrize population unit overall mathematical size naive dispersion tail option package focus enable degree improper prior size binomial poisson log effectively common typically thin tail prior sample proportion sample size size simple prior infinite moment flexible beta proportion translate discrete assigning uniform size mapping close easily numerically median amenable used section modeling design add great see amenable pg pg design proportional condition adaptive full equation worker drive hard reach
subsequence hence x k x statement em suppose modulus hold whenever let eq imply immediately let suppose obtain consider convex constant dual cone cone lagrangian lagrange multipli lagrange multipli q pair relation eq lagrange multipli box constrain convex hard aim study throughout function gradient solve thresholde eq subproblem close solution study proceed subsequently follow establish operator subsequently lx lx lx x follow show magnitude component method suppose define second relation argument conclusion lemma em minimizer moreover let local minimizer problem continuous moreover hence minimizer addition know minimizer establish proceed strongly modulus follow change local number sum monotonicity hence ii first arbitrarily depend monotonicity yield convexity apply obtain one q hence contradiction satisfy next establish upper summing project apply fact obtain l fx fx imply iteration establish local solution method apply perturbation modulus finding define strongly arbitrarily method number satisfying pt satisfy observe hence local solution use conservative improve performance dynamically result variant solve subproblem satisfied go go go end em method need number finite termination satisfy outer iteration similar argument derive together method local moreover final similar argument replacing imply convex programming relaxation minimizer penalty dynamically point smooth convex assumption lagrange multiplier feasible exist observe let approximate approximate variant quadratic moreover suitably assumption strongly convex establish apply find minimizer give define accord replace modulus let find local minimizer c generate local minimizer lagrange multiplier inequality assumption local find convex convex variant strongly perturbation associate modulus moreover clearly establish replacing lagrange generate find minimizer theorem iteration generate approximate eq together approximate em conservative practical update dynamically result present proceed project variant method sequence multiplier problem go show satisfy iteration method apply loss variant follow lagrange multiplier inequality apply know imply em inner terminate outer iteration accumulation minimizer accumulation local satisfie exist subsequence subsequence necessary addition pass subsequence necessary upon take limit know hence local cone programming propose box constrain programming sequence converge minimizer solution solve regularize programming apply quadratic establish approximate local dynamically show accumulation extend work solve author thank suggestion substantially remark assumption support discovery author leave department engineering university thank visit thresholde constrain converge local solution method cone apply relaxation approximate local propose penalty dynamically accumulation minimizer pt key iterative
procedure sense flexibility knot hope initial spline procedure knot enough least hand fan li xx basis may variance q able operation operation classic xx square estimate truncate important scad estimate estimation criterion suggest generalized cross fan li let eq q hence modify generalize validation criterion factor due lot basis select adaptively fan fu predictor estimation criterion also aic bayesian schwarz suggest penalize spline estimation criterion wu model classic strong result rao wu rao wu agree lot wu rao wu procedure well two slightly smoothing spline spatial initial knot two vary knot method insensitive though cause knot depict figure knot select non spline frequency example histogram scad every scad scad tune tune knots eps eps width examine examine impact factor simplicity spline method sensitive factor factor method rao wu rao knot may affect cm knot cccc cccc knot cccc knot factor multivariate chapter model outline suppose th define impose univariate knot spline non penalize concave penalty replace release impose concave univariate additive concave still make accurately select dimension corollary section true cm true cm plus spline find non spline optimal knot insensitive knot method simulation compare smoothing e concave spline concave knot much attention attract penalize spline smoothing spline regression simplify knot spline penalize spline penalty quadratic penalty penalize spline van propose regard variation kind regression spline theory penalize chapter smoothness position knot firstly variable adaptively optimal knot lot work direction adaptive backward approach establish spline remain estimate procedure insensitive avoid involve high dimension like spline efficiency penalize optimal knot selection still penalize little spline spline easily multivariate estimation penalize penalty avoid select knot knot simultaneously insensitive knot enable study chapter penalize compare procedure fan li non concave penalize traditional property high spline extend linear basic idea concave penalize take penalty threshold spline basis coefficient knot penalize avoid procedure effect shrinkage fan select claim spline sensitive knot knot spline green approach trade flexibility regularize insensitive knot trade flexibility control regularize property validation validation green chapter spline various chapter optimize likelihood spline chapter newton fan convex spline penalize spline simulation discussion nonparametric regression knot q jump knot dimension truncate power basis hence classic power spline truncate spline minimizer weight function scale sake interpretability penalty fan three bias result automatically reduce model instability whereby continuous fan li satisfy continuity three principle useful spline generally penalize spline cause number knot penalty produce rough penalty smoothing smoothing reduce knot adaptively keep smooth spline knot fan li smoothly non follow fan scad scad discussion refer chapter
subsequent yield perfect classification bic value datum model true c c group plot value ht line yield analysis purpose consist species datum consist original measured age purpose evaluate clustering consider ari display cluster cluster cccc est est cccc est clustering approach parsimonious describe package parsimonious mixture gaussian distribution estimate via package classify correctly two misclassifie select group color factor evaluate bic bic among six characterize remain four top choose covariate hence scatter presence observe red species green indicate versus support loading though informative unique constraint loading ensure uniquely load fit result huge computational burden approximation useful reduce accordance formulae inverse inverse partition utilize start term ig variable explain section follow load iii isotropic th term g k g g yield g attain k g accord n g k simplify g g estimate loading solve manner without consider last update update g g computed matrix update g g g compute accord get moreover estimate matrix p load unique datum example attention matrix weighted constitute vector constitute response explanatory assume impose variance introduce alternate expectation maximization algorithm maximum estimation parameter family artificial model direct mixture model component sub group describe cluster base mixture group ideal modelling framework manner mixture regression model joint marginal well approach gaussian mode parsimonious model gaussian density tool purpose applicability space remain challenge issue gaussian lead matrix gaussian matrix principle mixture develop work start family component parsimonious clustering paradigm linear basic maximization address aspect discuss evaluation artificial suggestion step introduction next suppose partition weight mixture usually mean vector conditional become linear leading see assume independent triplet distribution importantly mathematical become factor recall conditionally generic shall herein extend across whether isotropic parsimonious variance loading variance isotropic unconstraine unconstraine unconstraine unconstraine unconstraine unconstraine unconstrained constrain unconstraine unconstrained unconstrained constrain unconstraine unconstraine unconstraine unconstrained constrain unconstraine unconstrained constrain unconstraine unconstrained constrain constrain unconstrained constrain unconstrained cccc four letter whether impose constraint covariance variance letter apply load isotropic observation unlabele scenario draw know small proportion ig indicator observation notational prefer cluster cf classification substitute dynamic section specification miss stage generic incomplete ig z ig otherwise ig say easy consist cycle cm correspond partition iterate section illustrate cycle detail family give miss n eq st complete current ig ig k g j step maximization ig miss factor conditionally ig cycle st calculate follow z ig ig give maximize eq computing guess st carry set g g g start standard select natural opinion parsimonious procedure group multinomial first constrain cccc second start membership initialize ig initialize initialization continue accord scheme display cm cm cm decomposition detail value eigenvalue th element eigenvector acceleration acceleration iteration l k analysis stop l analysis herein characterize relevant classification integrate complete
vc q unlabeled request return er em specify adopt likely case aside indicate multiplying factor improvement know occur agree prior regard slight depend generally equal dependence undesirable application easily follow factor whether always achieve tight access value case sometimes specifically condition plug corollary bounding summation arrive theorem sketch include calibrate ba log appropriate unlabeled label request er examine asymptotic unlabeled sample log log log request indicate theorem multiply bound second interesting indicate surrogate access prefer use somewhat surprising ignore case calibrate loss indicate size ever competitive method optimize reflect indicate extent otherwise complexity result dependence typically directly log log log tight dependence apply fix almost everywhere survey class marginal fact density condition exhibit scenario particular choice pf w f p f p p f pf know plug probability sufficiently budget universal contrast k log indicate say active assumption method turn certain represent xy q xy condition result van calibrate q erm er analogously brevity condition calibrate unlabele request function factor indicate multiplying case strong distinguish case corollary reasoning proof universal calibrate satisfie finitely argument appropriate request return er sufficient multiplying particularly case indicate erm slightly specifically f j j xy specifically corollary reasoning analogously follow universal constant classification calibrate satisfied let label request probability er sometimes theorem small erm include vc vc major analogously brevity leave f hx logarithmic derive erm achieve vc excess rate vc major contain special define subgraph envelope hull vc conjunction adaboost hull vc van note vc hull envelope vc hull envelope function derive vc hull calibrate xy condition theorem satisfy least corollary derive analogously erm active since analysis significantly convenient offer unify algorithm inefficient study relax loss class vc van uniform classic jensen next f pf satisfie satisfie combine monotonicity easy proof would little algebra take suffice make right side suffice define monotonicity j xy suffice plug complete proof appropriate universal calculus reveal u u j choice j u u require condition definition strictly j summation let sum n log n convention note appropriately success least note therefore imply state sketch follow analogously integer log ba substitution place theorem bit calculus choose identical remain show satisfy necessarily toward end note expression sum n appropriately constant sketch analogously theorem modification u j follow definition constant proof theorem proof constant u j reasoning include proof lead prove vc subgraph application condition throughout adopt notational except apply variant specify specify chernoff law event determine determined completeness define sf inductive claim trivially inductive step take event mf furthermore inductive eq mh mf fact monotonicity xy xy universal side definition e mc brevity let right complete inductive prove hold j er furthermore sm chernoff least side suffice complete note discussion expense logarithmic next arbitrarily even prove exist every mm serve base inductive fix v universal f g mf follow derivation localize risk least sf v claim h v implication simply eq v sf principle induction establish j q appropriate specify definition prove every I serve inductive take inductive exist I sf j chernoff bind law least xy imply j j xy xy imply j jj hand induction hand fact e summation sum right inequality take suffice satisfied imply er note union great theorem complete sequential design supervise algorithm sequentially request select instance pool unlabeled label request label work use active specifically surrogate label classifier return give active extent passive additional passive accurate go subtle topic namely certainly active develop computationally condition e say many yet reach level practitioner turn heuristic practical attempt problem common perform convexity within execution guarantee loss formulation adaboost indeed come understanding use surrogate learning actually primarily surrogate sequential design learn large pool unlabele sequentially request pool active learning produce small achieve already specifically minimize make produce relatively passive surrogate bounding via inequality error technique zhang active optimize guarantee passive might seem possibility surrogate way less direct help may guarantee even surrogate insight truly identify optimize location focus request elsewhere construct active optimize surrogate increasingly number request achieve bound analogous passive minimizing find passive extent tool develop thesis conjunction localize rademacher work surrogate relevant zhang develop excess surrogate risk conclusion insight study plug rule loss value smoothness study analogous learn obtain remarkably obtain method work method optimal term surrogate loss loss strong generally lie zhang extent active active develop several maintain plausible candidate request make mistake label request technique number mistake derive request method surrogate behave nearly previously study analyze request achieve excess risk base complexity surrogate method result determine excess risk excess evaluating bound label request method immediately number request excess constructing algorithm make generally active passive method instead propose surrogate optimize increasingly space thereby strong space g denote usual simplify event measurable van discuss xy py xx xy ph certain conditional follow hx h hx hx hx contexts function write equivalently f variable refer arbitrary protocol initially unlabele may select observe another request protocol conditionally independent ni ki throughout primarily interested satisfy condition discuss statement convenient xy interested optimize characterize z represent xx subject error necessarily minimizer reasonable calibrate though convenient infimum actually z necessarily instance interested modify natural handle general substitute risk f xy xx p rely combine stand formal assumption loss choose function surrogate necessarily note significantly restrict care surrogate add essence analysis relation h however leave open important learn passive generalization significantly h r h xy xy h related define ph g h ph g ph radius radius xy h r transform guarantee excess surrogate abstract transformation define xy calculate context also arbitrarily subject subtle relationship excess hold calibrate argument calibrate satisfie context fix r f though modification replace family originally convexity quantify specifically condition calibrate take discuss many calibrate loss specify context machine optimize exponential vector machine term hinge hinge condition require loss learn estimator binary classifier quadratic loss quadratic exhibit gx iy subsequence reason g g h r passive serve derivation motivate bound excess erm mu erm h h return think kind erm erm h h minimizer erm erm point function erm erm erm mu erm follow quantity q interested quantity however toward define follow take h quantity variant mh claim hold roughly within factor follow localize sample quantity completeness xy erm h typical calculation derive sometimes envelope function proceed p entropy tend complexity calculate much van explicit conjunction relax p relaxation source slack however case condition tight convenient explicit later section make benefit allow abstract enough capture specific measure conjunction toward definition h p satisfied instance make see explicit example quantity quantity p p p mf h p crucial h p p pf invert bounding express abstract h xy xy perhaps simplest make surrogate know basic comment potential drawback context optimize number achieve excess specifically modify fix xy er mention several optimize loss many learn excess surrogate passive specifically problem z equivalently state constraint x f slight imply minimax optimal active achieve specifically positive decrease convex twice variety calibrate loss neighborhood know loss satisfy produce provable passive general describe request implication improvement active sufficient merely surrogate produce algorithm interest interested help computational maintain surrogate propose relaxation unlabele budget mm request label v behind rate informative therefore focus label request uncertainty update step empirical sampling maintain shrink region practice maintain implicitly keep track step define step check one likewise find problem function convex long efficiently quantity present internal previously define appeal interpret way compare empirical risk conditional compare empirical risk original sometimes problematic computational challenge considerable class continue derivation description relaxation case mild dependence noise condition represent main request er briefly idea rough outline maintain also round latter upon reach index r theorem algorithm request label label request index chernoff statement indice label budget formal keep minor request indicate often achieve certain type class vanishe calculate alternative calculation measurable immediately fix u theorem label request er modify way analyze analogously convenient interpretation analogous restriction modification constant event argument argument way lead interesting possibility update update occur point update could choose factor modification restriction allow additionally check another update substitute abstract result
style conference wide http www file pdf contain illustrate various satisfy rectangle paper title bold horizontal rule top bottom thick title page start page author name list leave name address follow author side side please pay regard figure level word level first proper line apply regardless within text order end section style format style acceptable consistently blind publish use paper widely e review anonymous author name citation anonymous text bottom appear horizontal clean dark place figure figure may clean hand table appear title line title proper terminal terminal body cell aspect style file modify width font perhaps please page file letter letter file cause year file file reader available box machine generate file acceptable please http www pdf generating file figure otherwise please pdf letter ps ps check pdf file shape implement shape try figure file program eps eps simple clean figure achieve window microsoft pdf office http www microsoft ed save office os via pdf click drop box save window via ps file create computer file http www com pc take ps click click advanced font font outline select click ok file ps create pdf file ps file option file contain embed ask margin come figure always specify figure width eps graphic graphic graphic bundle http www graphics ps properly please acknowledgment acknowledgment reference acknowledgment level citation consistent reduce font reference long cite reference mit book realistic neural neural journal assumption laboratory mit institute institute random constant direction reconstruction manifold reconstruction bound higher previously k novel tool motivate suitable one close low proved practice well amenable theoretical lead work set variety semi embed ambient approximate suitable sense geometry manifold typically support survey quality approximate measure reference therein minimize associated encoding crucially work focus important widely k algorithm see approximated extension induce collection possibly result manifold resemble locally tangent mean support provide k mean algorithm thus combination fact novel develop tool rest begin discuss reconstruction mean present space endow compact drawn respect goal learn approximate reconstruction reconstruction easy respect word increase mind give belong effectively constraint negative space typically definition space setting continuous distribution euclidean domain lebesgue size see equation induce subscript virtue minimize set mass locally number though closeness global randomize global minimum respect randomization collection affine analogously k mean aim minimize minimizer relaxation flat often cluster interpretation rewrite sum pairwise distance consider k mean quantization error interestingly coincide precise sense problem quantization distribution excellent generating support point analogously euclidean decomposition effectively produce euclidean problem quantization manifold define constant approximation support perspective interesting dictionary interested dictionary associate reconstruct maximally reference crucially quantity interest question characterize piecewise affine manifold manifold bundle sec k yet currently mean approximate sense increase ultimately interested understand quantization approximate minimize thus population infinite increase derive rate technical depend order hilbert study keep report train small increase experiment think concentrate around analyze factor derivation somewhat surprising trade error order entirely might expect reconstruction really result could tight tight remain e derive whether bound tight importantly aware point exact distortion justify heuristic choosing perform show trade indeed regime trade easily e setup unit sphere nearly orthogonal clearly indeed origin complex setting direction embed hilbert complexity rate arise solution generalize derive rate result theoretical either widely past involve throughout follow smooth metric absolutely density consider access solution grow equation vanish convergence compute depend px dc ed k empirical equation may bind follow ns ns consider total difference measure tend become large note discrete variable probabilistic performance choose complexity explicit error novel quantization quite transmission distribution finite moment measure absolutely therefore unit cube clearly make formula result sequel manifold know appendix let absolutely replace clear restricting mean equation error uniformly widely recent year particular sn restrictive condition moment mean whose solution practice mean original relaxation provide closeness global equation practical approximation proof result let choose equation argument multiplicative incur third multiply equation probability respect prove next section problem begin introduce uniform expect affine combine result expect smooth metric affine space positive second grow smooth hilbert manifold embed separable constant eq dx b begin begin upper rademacher subspace projection onto independent unit schmidt inner lemma finally desire consider simple analysis metric tangent absolute value fundamental tangent measure choose dd intermediate clear everywhere associate since absolutely therefore minimize among fourth quantization metric definition match geodesic follow adapt characterize quantization riemannian packing result must diagram triangle follow minimize region diameter measured distance tangent begin establish tangent manifold geodesic appear give neighborhood tangent plane geodesic form clear dominate point satisfy equation qx imply collection neighborhood hold consider point equation lemma contain neighborhood clearly compact admit lebesgue every diameter set since hold inside need kf theorem absolutely definition definition mention exposition absolutely
eigenvalue correspond instead operation rank prior diagonal consist corresponding eigenvector mode determinant determinant column one diagonal determinant inversion lemma draw posterior covariance basis rewrite fast low triangular cholesky drawing form avoid optimize induce require choose location induce expressive automatically location input challenge gp preserve small input trivial overfitte kronecker product separable respect test reduce suitable density covariate target region space contribution contain associate th slice grid similarly consist th slice contain grid non distribution laplace factorize total slice grid denote consider closely derivation present issue become computationally dense grid large limit form mcmc limit measure posterior use inspection factor initial one la approximation remove burn density estimate grid size second examine approximation logistic simulate la mcmc dirichlet gaussians mixture variance assume mixture allow component gibbs compare advanced gp default mixture model exclude using would toolbox test mat ern skew datum grid truncate mixture truncate less sensitive measure time estimation sample student la grid size take core ghz multiplication take option second second comparison simulate student x compute estimate la illustrate low show la take mcmc minute test set student mixture gaussian measure divergence grid times grid size equation reduce exclude kl la performance sense large prior density density set demonstrate scheme estimate dr mcmc dr dr difference dr illustrate directly la fourth process la close art laplace avoid posterior la grid take time fine paper cubic demonstrate predictor laplace approximation speed variational bounding approximation ep laplace variational binary however poisson slightly laplace process well ep could performance la diagonal quadrature free ep way gp preliminary turn slow improve correction correction latent correction could la mcmc plot toolbox matlab acknowledgment author like thank de importance sampling anonymous acknowledge resource provide ki department prior model analytically inference grid integrate type sufficiently interactive reduced speed transformation gaussian modelling density parameterize smoothness control covariance challenge intractable construct focus ensure normalization describe property establish grid leibl infinite approximation consideration bound extended unbounded interval transform integrated metropolis set latter automatically location analytic make fast fine laplace grid quick practical focus way relate literature posterior involve mode penalize maximum consider marginal enable posteriori map covariance alternatively use derive moment process approximation evaluate likelihood hyperparameter grid newton mcmc guarantee guarantee hyperparameter dominate covariance matrix straightforward computational interactive figure speed stationary covariance avoid dense grid exploit reduce exact prior grid exponentially suitable reduce computation impractical normalizing term avoid elaborate rejection conditioning algorithm place make space easy however scale construction quick combine idea integrate value slow tailor la rejection importance hyperparameter result additional inference several experiment independently focus unknown maximize pd limit delta locate belief unknown obtain realistic equal employ logistic density transform unconstrained smooth estimate density covariance gp gaussian widely use smoothness scale decrease magnitude combine placing gp polynomial lead density tail go eventually basis demonstrate illustrative mixture gaussian latent fourth show hyperparameter value exponential collect center prior latent vector associate contain basis regression fix tail towards negativity posterior zero rejection discretization belong th denote count prior bayes due non approximate integrate implementation laplace gp section efficient mode computation likelihood far obtain multiplication reduce gp brief mcmc resemble laplace approximation intensity estimation expansion approximation lead full diag element matrix implementation form avoid use multiclass positive maximum mode stability inversion preferable conjugate evenly lead toeplitz covariance embed enable toeplitz speed multiplication frequency domain multiplication convolution single embed matrix multiplication fast two form la ii hyperparameter marginal determinant evaluate intensity evaluation exploit low intensity require evaluation marginal newton typically fast implementation default reduce compute respect compute derivative magnitude informative freedom prior equal addition estimate integration composite design la scheme joint predictive monte
propose fix minimal complexity dominate whose bind atom assign alignment protein share variation similarity compare variation atom score refer respectively sup sup invariant translation semi valid k deal metric area auc author bind classification map problem decide auc nevertheless provide supplementary metric rmse determination coefficient suited paper nest validation rmse bind outer fold validation fold union fold test since average square nest denote fold affinity coefficient error always return eq therefore predictor give decrease report sign rank database protein along protein sequence bind energy force field diversity structural protein domain database computational structural dissimilarity database structural protein bank refer unique base bind energy modification approach bind datum bind bind affinity fold minimize overlap fold avoid fold since predefine fold nevertheless conduct method contain dr binding per allele transform ic learn allele purpose useful pseudo highly position potentially contact specificity author method use pseudo compose dr respect chain consist also conduct well design quantitative recently pls networks ann biological activity gp find protocol thus tend cluster use split select small dataset kernel attempt predict bind consider major challenge consequence la protein yield secondary structure validate try prediction protein protein assign great protein alignment aspect protein interaction interaction surface protein bind protein share bind motivated sup binding sophisticated rbf vary kernel motivation design gs descriptor sup sup sup bs bs gs gs l table bind affinity protein well gs simple bs sup sup bind sup sup benchmark atom provide relevant figure illustration sup sup absolute binding maintain bind drug discovery ultimately serve drug rational gs ability build perfect predictor quality responsible ultimately achievable biological dataset method experiment predict bind energy specific bind train gs validation validation predefine fold provide predictor generate three common metric root square rmse roc auc rmse auc find see significantly inferior dataset rmse cc c c drb drb drb drb drb drb drb drb drb drb drb h ia far potential gs author cross allele training use determine bind allele bind specificity use beta chain experiment sequence yield slightly suboptimal gs allele obtain allele assess auc supplementary appendix rmse individually yield allele show outperform value ic calculation require predict author indicate globally cc drb drb drb drb drb drb drb drb drb drb drb drb average cross result extend rbf gs kernel database gs use gs likely rbf kernel table rbf method gs rbf kernel ridge slight support hyperparameter tune insensitive require short gs outperform rbf datum limit consider method gs l design pseudo bind gs elegant eight kernel ridge bind kernel first capable accurately bind target ii bind state task well quantitative affinity database benchmark would substantial tool community predictor still major machine hard computer expand author gs ad conduct provide biological insight supervision final manuscript acknowledgement foundation innovation sciences university work nature pr grant mm say alphabet symmetric convolution string alphabet ai lx I finite alphabet semi ff string note line rewrite string string contain put real fact l suppose calculate roc curve auc problem transform case aim distinguish achieve allow require predict bind bind affinity threshold convert technique calculate auc convert bind energy bind bind value range latter auc drb drb drb drb drb drb drb drb drb drb ia ia calculate explain discriminate drb drb drb drb drb drb drb drb drb drb drb cm cm theorem protein physical protein template choice protein interesting display activity drug drug furthermore base possibly affinity method potential accelerate low drug discovery select potential biological validation specialize sequence bind incorporate chemical generalize eight radial basis programming computation ridge bind propose kernel relevant good prediction bind affinity specific major benchmark dataset quantitative model benchmark dataset conclusion benchmark p art predict protein bind apply predict reliable bind improve model interaction pathway lastly research accelerate drug vast majority protein protein biological process protein protein interaction control interference chemical discovery novel pathway agent interaction surface secondary essential bind interaction secondary furthermore activity drug drug serve bind region specific process target identify utility throughput screen million test biological however chemical library generate false negative challenge reasonable accelerate provide increase aa recognition complex capable affinity response bind affinity protein model pathway high binding bind affinity predictor affinity huge set candidate stochastic traditional classification bind bind bind quantitative obstacle database bind protein family bind protein machine biology database contain bind great success simpler bind significantly allele target reasonable require allele accuracy multi reasonable bind allele allele know propose capable multi prediction kernel bind information protein propose gs kernel generalizes currently weight also inverse elimination also protein one predict bind energy vector bind energy protein consequently equation thus kernel extremely successful decade choice good subsection design choose protein biology protein homology detection kernel design small like exploit exploit unify manner call pair string string string position encode vector component encodes possibly similar encoding string length encode comparison gs contribution rapidly position control contribution differ measure square vector gs sum comparison parameter distance dirac delta spectrum rbf radial rbf rbf string string great string weight kernel gs eight list free approach gs free measure accuracy elaborate
netflix contain million user movie book music recommendation graph employ similarity movie movie build movie wikipedia user whether certain movie netflix retain music book wikipedia set netflix contain wikipedia record movie shorthand recommendation collaborative filter netflix partition part movie identical part rating movie density serve source world source book movie set target different item construct netflix share movie netflix extract share movie netflix rating rating netflix knowledge movie task item totally task music source movie utilize wikipedia record graph movie perform rating netflix movie predict rating rating c c dataset target selective selective pmf c simulate selective utilize baseline pmf show work non adopt technique recent prove uniformly weighted utilize domain baseline selective framework performance result collaborative task target pmf fail give especially help selective source domain prediction selective transfer outperform selective truly helpful issue selective observe selective significant netflix source world case handle source factor affect table find movie much improvement simulated entity movie rating netflix try system recommendation domain target rare source domain user graph knowledge movie domain heterogeneous utilize source target domain selective level domain selective globally noisy contain inconsistent observe transfer level transfer eliminate irrelevant source highly target even variance weight fix one adjust adjust order fix prediction learner movie base reflect training topic training suffer overfitte selective selective number latent topic compare fast go slightly large decrease obvious clearly overfitte fine grain knowledge selective selective learning etc pmf etc selective transfer relate collaborative summarize knowledge work learn ever grain consistency source collaborative filtering recommender gain probabilistic machine introduce concept selective transfer well overfitte utilize domain build application domain work transfer collaborative involve manifold alignment jointly build focus auxiliary recommender distinguish user preference align framework transfer knowledge target domain recently researcher propose via weight source individual boost base well limit task systematically filtering besides perform selective transfer variance empirical would introduce useful source embed criterion boost transfer source domain target experimental world selective transfer method long robust overfitte pt filter user historical user many model prediction recently several work knowledge manually part target novel criterion empirical cf setting consequently boost perform selective several state selective improve rate real world recommendation collaborative cross recommendation recommendation attempt recommend movie tv book page recommendation aim rating item historical user preference record system although item often usually small available rating extremely sparse service limit recent via collaborative filtering source domain transfer previous trust similar world cross music web site rating music international rating web site rating good rating inaccurate music site obviously international site data tackle cf challenge source work adopt cf target item get truly word inconsistent dominate happen domain give careful user expert criterion far variance domain like preference transfer observation consistency source implementation criterion selective collaborative instance help build contribution follow domain filter influence factor novel selective transfer collaborative extension boost issue cf base implementation probability semantic solve various wish prediction regular recommendation illustration user task observe recommendation example contain adopt commonly collaborative either appear source domain derivation target domain item co collaborative base probabilistic analysis filter probabilistic integrate selective transfer learn observe careful reader generative base introduce topic item rating mean variance mixture estimation minimize extend cross domain transfer filter knowledge knowledge source domain assume user source world approach jointly model domain target relative could domain motivated domain expectation statistical estimate derivation tb weight boost initialize generate minimize weak fitness weight source update hypothesis target domain propose selective novel empirical selective mutual like transfer knowledge record reflect preference record empirical variance factor boost care mis predict instance domain
opposite estimate separately necessary construction propose auto probit actor network probit maximization employ treat bayesian description estimation actor type question vector preference covariate coefficient term responsible covariance structure structure mechanism exist basis tie describe act structural membership mutually exclusive effect correspond compete structure group actor embed term model describes account integrate unobserved matrix pose significant since treat description regime autocorrelation first estimate expect unobserve complete calculate expectation respect replace estimate possible analytically estimator mode see produce approximation find mode expectation degenerate solution bayesian observe choice decide unobserved chain generate series distribution summarize appendix choice truncate unit distribution multivariate follow gamma distribution language mechanism ensure generate please use close choose prior assess algorithm ideal pre distribution temporal narrow show temporal consideration available suggest high effective sample high posterior car compare original consist decision mid car otherwise car price interest preference among network measure location live explanatory actor information age education information car car actor location construct home code thus structure membership contrast separate coefficient show explanatory network term equivalence calculate simple distance structural equivalence undirecte weighted adjacency distance number individual otherwise node inverse plus positive equivalence element adjacency connection respectively addition avoid denominator equivalence leave right deviation interval thing second structural get increase first autocorrelation equivalence variance autocorrelation investigate phone around call back phone take soon back automatically phone company raw phone record month period phone information age user pruning algorithm imply equal relationship call include several explanatory gender phone connection structural assume phone call party eventually adopt technology call drastically normalize number element structural adjacency obvious mechanism impact relate relationship trace autocorrelation coefficient second autocorrelation structural variance term log show autocorrelation opposite model auto actor specify fit e hierarchical find solution solution use validate quantile accurate structural recover variability vary number observe capture affect get get transform side decompose likelihood parameter get estimate expect first analytical equation first right q analytical generate interest follow truncate distribution identity generate n matrix corresponding bs generate metropolis increment one get software develop greatly chance validate return estimation bayesian software implementation history program quantile validate simulation base generate software code true credible interval bayesian distribution quantile value series software test number draw posterior estimate quantile draw software true software quantile replication expect simulation run want distribution mcmc simulate replication validate hierarchical generate network generate keep draw parameter count software correctly write randomly estimate replication pool quantile sorted show replication roughly confirm mcmc draw burn listed figure autocorrelation autocorrelation term equivalence autocorrelation plot right bottom coefficient variable red randomly show valid th randomly n show red otherwise bivariate apparent come htbp r invertible finite coefficient matrix represent weight rise characteristics one obstacle measure closeness type actually decision difficult conceptually address probit behavior regime auto sensitivity impact nature interest include govern connection explain topology make network embed topic technology past network hundred people collection enable access technology handle scope aspect complexity individual decision product technology solely attribute age education though due mechanism handling indeed development associate unobserved share tendency ability predict person beyond characteristic produce correlate member phenomenon autocorrelation outcome actor distinct network family outcome explanatory term term whose specified interest autocorrelation estimate autocorrelation accommodate actor friend research autocorrelation
uniformly iteration chain stop randomly sample square error respectively correlation fig depict function function increase close goal bivariate target use mh black box simplicity gaussian set stop pdfs form pdf mh pdf parameters gaussian location weight gaussian quickly remain mh modal multi target build extend proposal update gaussian lemma corollary carlo widely communication introduce modal density vector previously two chain monte mixture hasting chain problem scientific digital communication learn etc mcmc normalize simple proposal produce mcmc hasting mh drawback mh method high sample mean extremely depend discrepancy target target several extension propose reduce burn among mh adaptive proposal technique usually parameter capability technique least unfortunately problem adaptation proposal lose chain mh carefully recursive empirical sample covariance whereas jump algorithm mixture obtain fully mh complicated iteration introduce mh gaussians use recursive formula adapt mh multi modal target always improve mh organized mh describe box usage finally conclude multi modal density mh covariance variate vector approach fully adaptive mh update since adaptation adaptation stop use recursive formula initialization initial n n identity gaussian pdfs accept otherwise set parameter find index add parameter denote er set update definite pdf mixture expression update technique show sensitive information initial box way gaussians great dimension select cover support diagonal big value train period parameter big great numerical suggest could great gaussian generate adaptation mh irrelevant vanish therefore cost control adaptation gaussian locate target discard important refined improve toy univariate target pdf clearly pdfs proposal uniformly respectively I stop
rule db I refer adaptive evaluate two mean excess ii exist distribute adaptive filtering subsection special rule propose fitness filtering fitness I equivalent rule fitness selection fitness give fitness fitness c degree n ik jk j jk metropolis hasting correspond fitness utility define hasting rule fitness definition exist summarize fitness definition graphical distribute fitness ii filter topology node error aware weight instantaneous denote local exchange instantaneous neighbor fitness note fitness form use I update propose directly adapt environment adjust accordingly base instantaneous mse aware power implement performance lower verify noise poor principal adaptive signal whole enhance analyze dynamic expression regressor graphical game structure eventually model reveal good signal node concept signal probability filter probability show center node node center possible steady variance steady adopt strategy information noise interact one instant nod instant user specific node neighbor expression intuition adopt beneficial adopt node well therefore utility follow sufficiently calculate accord compare therefore hold conclude signal whole filter adopt percentage adjacent similar either imply follow dynamic I subsection dynamic understand update include strategy fitness q fitness normalize include fitness fitness fitness among neighbor node fitness strategy probability replace strategy meanwhile I select show among phenomenon fitness q fitness percentage meanwhile taylor weak go parameter dot dynamic variation within limited player derive strength limit long biology help close strategy combine besides reflect node signal negative signal good closely good subsection begin equilibrium rate selection e rapidly converge accord therefore characterize focus diffusion characterize regular degree suppose edge probability good signal db I equivalent undirected adaptive filter characterize good connection node update update diffusion I update rule diffusion diffusion consider adopt favorable state even node use ess game theory ess ensure another obvious good favorable ess check whether stable ess follow distribute network characterize ess node node equally respectively scenario majority adopt fraction ess evolutionary stable left hand ess adaptive graph incomplete graph ess incomplete graph characterize regular degree ess strategy approximate rp rp u u I since three compare adaptive ess network leave respectively regressor simulation average independent conduct kind aware power propose aware exponential aware update use projection first comparison variance node hasting kind variance propose well adaptive perform degree algorithm relative degree steady kind adaptive filtering algorithm steady besides performance steady performance six convergence algorithm fair network compare variance degradation significant hasting since rely less information clearly power achieve relative variance similar adaptive power advantage notice performance fitness function b b iv simulation generate degree type trace regressor set node I iii update either reach reach calculate run reach simulated theoretical gap approximation derivation good along noise variance ess I rule node solid represent setting network node strategy denote color use process strategy propose exist adaptive algorithm game node always unlike bottom theoretic provide important distribute offer unify design future give substitute define increment case approximate finally adopt update backward kolmogorov satisfie follow differential weak approximate bad connection substitute theorem expression initial term state whether increase decrease percentage shrink favorable receive engineering ph high visit group electrical computer china dr currently department ray liu post interest game theory wireless social dr distinguished award national fellowship distinguish receive technology china college associate electrical engineering college currently origin wireless communication information interest social signal dr chen receive fellowship chinese award university distinguish fellowship mention school research award ray name distinguished teacher college technology lead research broad communication cognitive communication security green communication technology include signal processing award distinguish year award award award school cite author process vice journal advance signal processing chen ray liu processing estimation network distribute filtering algorithm regardless nature characteristic study theoretic distribute graphical evolutionary formulation node regard selection framework two evolutionary game adaptive network stable strategy verify adaptive game recently adaptive filtering interest wireless hoc localization distribute sense cognitive classical centralized distribute robust node network fusion diffusion process adaptive filter instant satisfie follow vector measurement covariance white independent combine incremental incremental allow node neighbor adaptive summarize project mobile traditional mainly design rule topology statistical degree relative variance incorporate node intuitively sort adaptive exist offer combination unified reveal rule fundamental answer essence parameter similarly evolutionary game general bottom focus framework fundamental summarize evolutionary game node regard local combination different neighbor regard evolutionary exist special case propose aware distribute filter complexity evolutionary theory analyze derive diffusion information good work detail adaptive filtering problem graphical game information section iv section simulation vi finally always adopt ess player rational take equilibrium strategy ess evolutionary game utility matrix whose versus population fitness fitness fisher ess eq fisher population sufficiently strategy strategy process update adaptive evolutionary adaptive classical evolutionary consider complete scenario player location incomplete evolutionary game structure player relationship valid replace
albeit claim formalize algorithm condition online stability play adversary step online regret condition assume lipschitz equality regret picture regard follow let adversary regret stable regret intuitively proceed average batch loss unstable strategy stability condition formally new set batch batch end average batch process picture lipschitz lipschitz element stability pt prove bound last batch maximally regret stability regret condition forward regret fairly general convexity major simplify exist technical brief generic sketch stability exploit lipschitz regularizer l non bound optimality regularizer divergence evaluate update minimize would commonly obtain iterative sublinear dual although solve successful maintain intermediate sparsity however impossible every behaviour exact give regularization strongly add approximate eq stability q triangle successive iterate eq know couple stability write simplification use rhs bound bound sublinear stability light crucial concept relate ability minimize online remarkably simplify extent arbitrary stability regret arise solve approximately compare offer per like perturbation stability iid unfortunately gap exist stability class existence concept batch set think major empirical risk concept online stability way fundamentally batch learn ex mm property definition conjecture axiom claim ex ex ex ex microsoft stability implication bound introduce novel look bound forward also bound general non stability regret online restrict regret regret simple exist tight approximate version follow stability ability recognize connection stability generalization fair say stability adversarial show minimizer erm apart insight stability potentially help design algorithm example setting generalization concept erm online adversarial online sequential two player adversary loss player adversary player control stability derive critical area privacy connection iid online set erm serve hypothesis sufficient analyze erm set unfortunately canonical scheme make connection stability difficulty study connection regret concept sense end stability essentially leave also uniform alone guarantee example move bound force end forward incur move adversary move fundamental online bound regret regret equivalent regret forward like stress general convexity contrast equivalence illustrate learn follow regularize mirror demonstrate arguably simple approximate version solve important practice precision provide version framework section usefulness analyze exist online recall bregman find notion key behind online strictly interior nonempty bregman rr convex modulus convexity refer strongly present characterize optima particular bold letter denote dot product refer arbitrary norm unless compact norm describe play point suffer measure player perform compare move know move advance goal minimize regardless sequence online linear game set introduction description stability set last md type online progress connection unlike extremely generic function prove exist family fact connection consider algorithm iid iid fashion define new notion provide connect move sample general follow choose minimize play surprisingly simple achieve adversary play add follow give strongly norm tradeoff another describe bregman divergence update unconstrained minimizer call mirror try minimize current loss similar general interesting note special mirror descent gradient descent look fundamentally spectrum algorithm mirror corresponding counterpart
successful build bayesian linear datum exact prohibitive present variable though use novel provide efficient implementation lin cut plane implementation cutting couple heuristic remainder organize dependent section metric scoring decomposable compete bic minimum description order decomposable find empty go permutation define know program operation solve dynamic see variable minimize arise treat however ability bayesian may fact equally htb permutation list history dependent receive mathematic community decade list city find cycle city city np develop overview reader popular pick removal add replacement underlie city asymmetric replace opt opt step note increasingly extend naturally edge add randomly acceptance rejection replacement opt opt implementation find ignore history opt opt implementation due integrate history dependent part effort improve approach dataset publicly uci repository census several attribute work week country education status attribute unfortunately miss value individual discard datum break capital continuous attribute number possible bayesian greedy hill h education capital hour week country learn hill capture study work week education status hour work week hour simple arithmetic accurately input order random variable predict significantly comparison ratio check predict individual less square expression p case find approach predict apply ni machine repository consist try answer relate state discrete clean dataset entire entry entry highly fig dependency air air temperature surface air temperature find dependency surface dependency seem failed quantity quantity suggest c air discretized prediction concentrate wind eqn wind wind predict learn hill new structure order expert space hill technique structure lin hill
allocation simple hierarchical flexible discover topic use efficiently massive set encode variable factorize estimate hide want many prominent strategy monte carlo variational sampling posterior run chain equilibrium collect distribution set parameter member close posterior inference optimization variational kind describe researcher speed tailor specific correctness optimization optimization maximum objective optimization provably optimum particularly term independently set subsample amenable datum estimate subsample subsample independently expectation detail idea attractive point variational implement repeat require repeatedly set analyze graphical probabilistic mid seminal statistical parallel originally publish lead mixture understand lead automated allow write inference review early whose set much wider amenable coordinate field approximate service alternative chain monte despite popularity focus develop mcmc strong guarantee gradient langevin advantage develop variational variational derive variational algorithm probabilistic massive set divide part apply hide within review hide variational ascent algorithm objective gradient coordinate stochastic technique variational estimate variational objective repeatedly subsample build inference datum nx correspond global global may involve observation collection vector partly random keep global local hide variable distinction dependency local conditionally hide q notation refer frequently arise endowed gaussian global mixture variance local hide complete conditional hide parameter shorthand convention conditional complete conditional global complete determine local e equation conditional relationship imply specific conditional hyperparameter dimension equation exponential parameter derive stochastic discussion conjugacy family conditional contain learn bayesian latent allocation kalman variant factorial hierarchical probit analysis factorization nonparametric keep use explore prediction future list modern turn field inference root strategy variational optimization introduce family variable index free find member close interest closeness divergence use distribution call field variational variational field ascent inference kullback leibler kl maximize marginal negative hide jensen fy fy q term entropy variational depend variable family try member maximize solve member close kl divergence pz depend simple family govern global local factorization variational family distribution lead ascent optimal without conditional equation likewise substitute mean advantage q respect expectation computational gradient variational ascent ascent hold variational conditional coordinate coordinate eq involve third constant quantity depend hidden quantity depend equation derive likelihood separate leave substitute form identity simplify derive set global natural complete holding optimize play thank local depend identical depend th computational scalable section coordinate inference iterate update global guarantee optimum computing expectation direct graphical conditional update tractable many aside field inference expectation maximization across global reveal begin reflect complete must analyze expect something parameter variational inference detail component variational classical finally global old intermediate improve global calculation arise coordinate gradient account traditional discuss riemannian distribution classical function gradient ascent equation away direction reasonable direction space might set parameterized dissimilarity univariate indistinguishable euclidean distribution overlap reflect distance dissimilarity distribution parameterize transformation kl method q ascent ascent space kl complicate riemannian define transformation euclidean nearby matrix plug equation ignore term derivative identity variational gradient mixture address variational respect fisher metric fisher fisher reveal follow analogous go closely coordinate consider variational subtract classical natural coordinate reasoning around importantly easy compute classical parameter many compute ascent inefficient point stochastic inference use fit parameter form follow variational conditional natural easy discuss noisy gradient decrease algorithm optima complex write subsample optimum overview optimization overview equal type optimize realization iteration convex whose resulting suggest fisher replace stochastic noisy gradient function result figure current optimize variational variational ascent repeat weighted intermediate ascent variational variational maximize write return local optimum parameter local natural hold reason jacobian stochastic optimize maximized subsampling decompose term local choose variational natural gradient objective noisy suppose objective thus natural replicate compute replicate natural whole consider gradient noisy optimize comprise gradient noisy size update parameter sample appropriately compute global datum intermediate weight step control old information iteration delay rate way satisfy iterative optimum describe basic inference algorithm stability bayes method estimation sample stochastic optimization datum finally gradient gradient gradient expectation global use expense help optima converge basis datum see per may want hyperparameter maximize empirical maximize fit figure increase variational maximization optimize currently scalable derive variational lda nonparametric counterpart hierarchical dirichlet process use latent encode model collection aid extend apply many word arise share proportion represent document exhibit health document business business topic depend assignment dimensional q lda collection document variational collection complete conditional derive ascent form coordinate natural variational inference topic iterate document local coordinate ascent routine iterate update assignment proportion update complete conditional equation back parameterization assignment expectation proportion document variational depend document batch inefficient collection document topic complete analyze document topic initialize randomly schedule appropriately document variational global dirichlet variational per topic multinomial follow phase optimal iterate equation batch document use variational intermediate equation contain replicate document next iteration topic inference assign topic corpus document per lda root scale variational fast batch variational inference reasonably size subset lda collection document limitation lda researcher cross practical nonparametric nonparametric variant lda hdp membership text hdp assume collection document posterior hide determine topic need describe hdp flexible unseen broadly inference bayesian us model mixed membership model grow expand prohibitive search specific structure validation use nonparametric topic scalable correlation section dirichlet breaking hdp topic stochastic variational place context place flexible prior distribution draw datum draw flexible potentially mass variety review see dirichlet parameterize non negative atom closeness scale small place atom look spread around atom draw representation gamma marginalization chinese restaurant stick break stick explicitly define draw discrete atom q atom stick stick breaking use infinite recall following combine infinite imagine stick call aside break proportion aside stick result stick length collection realize return stick th combine g draw place mass draw tend describe formally see dp stick break intuition stick break atom stick encourage draw atom break location tend later break location dp dp dirichlet vocabulary topic draw dp collection proportion construct hdp stick break construction corpus construction construction chinese restaurant alternative stick breaking mention hdp generative topic corpus breaking proportion document break proportion draw draw corpus define probability topic break b draw document stick length membership unknown advance unbounde however posterior advantage complete conditional exponential family separate global global variable break document break stochastic hdp begin indicator interaction level indicator proportion th account index variable indicator conditional topic sum document allocate statistic keep index conditional break proportion stick break di di conditional distribution generative family main parametric model optimize variational breaking allow level breaking proportion topic enough topic necessarily topic truncate variational topic easily correct stick truncation large corpus expect exhibit subset conditional update variational batch global intermediate sample local stochastic hdp level breaking proportion beta optimize conditional multinomial di di di di k di z w hdp summary initialize set set size sample uniformly intermediate illustrate collection variational inference fast well per hdp model corpora variational well batch subset study effectiveness stochastic allocation lda hdp compare collection investigate mini stochastic traditional document collection vocabulary stop rare nature span year vocabulary york times document span vocabulary wikipedia wikipedia processing observe word vocabulary collection aside document fitness set fit corpus topic topic predictive vocabulary assign divide hold word disjoint approximate imply distribution avoid compare bound form evaluation dirichlet parameter topic use variational topic proportion approximation depend metric evaluate hold hdp via stick breaking truncation stochastic inference introduce schedule equation control old early although stationary speed size minibatch hdp hdp concentration equal level truncation unique hdp inference large model give large number lda datum modeling hdp stay robust overfitting overfitte fitness hdp lda regardless corpus proportion give hdp certain priori likely appear topic l lda batch document sensitive consistently stochastic hold batch slow preferred hdp hold varied batch enough turn sensitivity hdp present fix explore rate mini figure corpora ten big sensitive corpus sensitive rate hold size fix varied rate prefer inference hold varied big variational massive stochastic variational objective noisy arises repeatedly subsample datum illustrate two latent dirichlet dirichlet model stochastic collection million document generalize many setting develop way apply membership blockmodel communities social non uniformly adjust stochastic mcmc update modeling update direction conjugate exponential expressive rich use expense mathematical expand probabilistic tool example capture topic change present develop non conjugate scaled optimization develop update another recent advance close fully factorize posterior collapse variational trading simple another structure variational relax field well arise connect inference sample uniformly
cause assumption incorrect time nonlinear instantaneous model assume direct suppose extension regard time series substantial causality structure finite broad class exist provide assumption causality wrong causal conclusion simulate iid try structure acyclic graph joint say constraint causal dag reconstruct graph g recover equivalence distinguish different vx jointly independent drawing element acyclic yx xx stand lead extension post nonlinear additive discrete bivariate provide section principle time value full contain series vertice full address problem causal summary causality causality article cause past help predict translate phrase help multivariate assume var causality g tx ix tm ta tx residual follow significance p p causality nonlinear chen bivariate noise test whether degree correspond short relaxed already say infer causal structure independence effect linear extend ts instantaneous effect relationship describe suffer causality intrinsic effect predict causality instantaneous summary still identifiable instantaneous causality conditioning causality structure fail exp bad conditional test desirable partially perform violate conclusion lemma fitting checking residual check implicitly simple although causality series extension causality furthermore testing suffice section describe identifiability consider pdf pmf say set tn tt node appear require acyclic assume identifiable model hold assume kx exhibit acyclic full identifiable condition regard feasible result stationarity ergodicity var nf I I rp ip f ip linear special causal hard drop iid ts fail causality find output dag estimate principle output correspond know intractable section concentrate series without feedback loop additive recover dag modify regard independence atomic user input tuning causal occur independence cause causal relevant structure independence test scientific discovery relationship rather hypothesis lead causal reject independence number useful develop heuristic size generate code mat experiment lag x see figure g infer large contain wrong causality draw circle draw circle circle circle circle circle draw dash draw x z z z z gaussian effect length causality structure lin lin ts nonlinear instantaneous simulate n z ground causality thus causality wrong answer wrong gp specific nonlinearity causality ts wrong r dag lin correct simulate tn figure show linear mainly gp sample one effect gp accurate answer due causality ts answer causality show bad causal conclusion linear experiment opposite tn correctly identify causal discovery length tn ta line discovery dag wrong causality ts answer circle dash b inner sep draw circle draw w instantaneous effect make fit true ts causality lin correctly wrong lead false old duration next old interval whereas causality g causality experiment temperature minute expect ts causality infer remain insufficient temperature cause time result diameter ts probably relationship price decide whereas lag store exp experiment give wrong decide causal benefit framework causality compare substantial practical ability multivariate discover structure complex model preprocesse trend case instantaneous feedback loop fit force although promise lie scope present conference satisfy variable parent noise crucial series believe hold technical full
prove well product start arithmetic bind proven note subsequent q value definite write nonnegative product form positive complete hermitian software find decomposition sum hermitian construct discover proposition arithmetic necessary considerably inequality actually arithmetic semidefinite reader comprehensive concern technique specialized inequality vary arithmetic mean inequality positive list arithmetic definite derive resembles mean mean nonlinear map tuple geodesic flow riemannian however average matrix arithmetic consequence scalar sum arithmetic geometric matrix mutually high also product satisfy arithmetic contrast without replacement large replacement collection k unit follow dominate operator inequality upper final order property deterministic product construct frame odd identity inner factor harmonic frame cast validity conjecture frame k additionally arithmetic mean harmonic treat use effectively fouri appropriately group coefficient generate combinatorial prove broad expectation independent identically symmetric explore identity hold variance theorem would infinite matrix analyze let without arithmetic since iid power low final contribution random demonstrate replacement record fourth entry verify calculation rr rr way lower use linearity couple v u j since distinct odd zero must write apply since index equal expression expectation prove commonly detail extensive fourth moment surely replacement grow scale exponential replacement expectation square wishart matrix wishart demonstrate kk also analysis problem look sample independently depend expand since concave hermitian jensen replacement reasonably mild condition subgaussian moment inequality sampling least mean similar reveal replacement sampling bad numerical evidence replacement randomize article learn completeness example demonstrate six comparison randomize define third row row generate haar sphere replacement rate column harmonic frame running row run haar I piece open conjecture demonstrate frame quickly sort combinatorial exploit arise employ prove beyond conjecture conjecture conjecture frame list version certainly inequality argument follow could analyze arithmetic matrix product incremental increment reach seem suppose minimize amount minimum path simple within success extend result incremental descent coordinate modify randomization tool jensen replacement nonlinear acknowledgement suggestion support award n nsf award cr support air research laboratory contract fa nsf award award research google finding recommendation express necessarily view include wishart geometric self loop complete arithmetic fix lower matrix angle slightly goal fourier nn recurrence pattern computation kind odd term induction fix follow write remove assume place permutation choose appear permutation element occur factor complete case care cyclic unity harmonic shorthand compute use geometric invariant frame tell since sum fix conclude remainder alone claim equal unity root generate eq equality unity union line line unique yx inspection else hence possible series eq lemma conjecture subsection subsection height pt pt sciences university randomize base decision pool progress theoretically replacement focus least formulate expect convergence replacement demonstrate inequality hold many well provide gap discrepancy explore prove consequence descent keyword definite randomized nesterov free replacement sampling quite
increase factor per equation application health proposition necessary accurately give scale disease would surveys health status health care measure make incidence observe compact continuous smoothness side question term approach inverse exponential although approximated affine interpolation quite well appear quite line inverse pose generalize publish relation incidence three death former theoretical example disease treat incidence death model model incidence disease prove death sometimes refer normal people death transition intensity henceforth denote incidence intensity age duration book duration follow differential equation describe system intensity figure system look relatively heavy easy imply eq population although population merely value could ode importance incidence age patient uniquely respectively clearly force incidence uniquely qualitatively quantitative infer cause force ode age profile know directly incidence cause allow study incidence study need recently inverse ill pose article organize generalize allow central partial pde ode case inverse pde disease section finally death henceforth age assume duration additionally proportion introduce ct partially hold incidence hence important overall negative disease stay result death absence necessary affected initial cauchy hand pde year solve problem follow show disease show disease address age time want age arise course incidence change cause formulate straight problem design period birth death disease place uniform equation denote positive term approximate life incidence death constant time simulation piece integer year birth diagnosis person ill age person three correspond identification counter birth year year decide non person year ill compete risk simulation des accomplish cumulative inversion decide death occur first case age disease death get death age exactly people year transform diagram extraction person event assume course advantageous context try measure incidence furthermore later age incidence addition show age incidence dash red blue comparison solid initial affine numerically ode ht visually agreement curve actually age specific predict percent point age disease measure incidence much direct function know profile assume incidence information assume incidence express product limit stem incidence non low
interaction original prediction evaluate membership protein unweighted interaction result prediction table measure analysis unweighte substantially primarily real value pair protein affect wide edge statistical protein transform network generate much result network unable observation original benchmark edge network class show perform original transform measure perform almost metric produce high increase able large major well neighborhood configuration two protein protein mean auc auc decrease c auc class auc change increase increase decrease auc decrease experiment protein original confidence utility content protein extensive investigation explain subsection result investigation protein even connect original versa configuration consequence transform improvement measure attribute change network part focus prominent change identify change scatter plot figure remain transform produce indicate tendency network focus assign accurate reliability interaction lead substantial difference transform include well formula denominator numerator assign protein protein degree network bottom corner low numerator high high indeed high network retain original connected neighbor point corner figure denominator high high numerator equation node original form neighbor configuration lead natural structure drop responsible study case subsection similarity filter interaction protein spurious interaction number connect benefit analyze difficult methodology edge original give noisy analysis study extent noisy version measure average auc noisy worse encouraging able dot even network interestingly function robustness transform original serve important improvement drop protein operation address protein study able prediction advantage measure discussion quantify protein measure advantage value reliability score able substantially improve prediction continuous version produce change original especially structural factor likely spurious introduce related protein researcher protein direction extension validation functional direction would measure perform characteristic presence weight edge since measure combine well support nsf grant fellowship school confidence measure mean max auc classes auc class decrease auc decrease h c c auc max classes auc auc auc apply text interaction cover protein bp member none network outperform metric capable extract rich refined protein interaction change quantify introduce transformation examine dropping measure record drop transform eliminate lead elimination examine comprehensive procedure two prune original transform collect non original edge network respectively respectively figure show small percentage edge drop drop noisy indicate indeed effective valid interaction percentage drop transform network result major original process change namely global coherence edge drop add overall avg avg study examine change base transformation influence transform edge three original drop keep approximately average share protein analysis along trend observe although small fraction retain drop add drop coherent edge add variation type determine functional answer last connect protein transform encouraging note transform transform network coherence network match preserve original drop coherent coherent although add coherent fraction transform small transformed institute multiscale department genomic new york ny usa science university mn usa mail abstract protein interaction study complex biological despite rich face quality challenge similarity form address measure interaction graph convert interaction transform corresponding effectiveness estimate transform original find transform original particularly reveal improvement measure link biological address disease protein module function protein since protein tend highly base protein despite rich embed interaction challenge affect prominent primarily positive interaction presence network affect another interaction interaction lack completeness mainly specific protein annotation biological insight gain datum present positive major challenge rich study local interaction issue interaction traditional approach connect however addition direct association protein association idea two protein direct interaction protein similarity network module module discover original fs similarity protein show performance utilize measure context handle protein interaction protein robust protein likely benefit performance measure context understand interaction firstly comparison difficult furthermore measure functional module discovery set use comparison hard attempt fill extensive comparative context unweighted interaction follow systematic graph network measure estimate compare quality original biological several transform accurate obtain contribution change due investigation ability measure noisy important well prediction effective novel association performance efficacy handle interaction protein detail assess protein source microarray since focus complementary combination accurate material evaluation protocol annotation interaction set include interaction protein unweighte study include database go analyse vote member protein interaction similarity define use notation direct similarity one measure two coefficient follow assume form unweighted incorporate presence interaction propose probabilistic measure significance neighborhood configuration two name chance binomial non unable take however significance measure protein section attempt fs functional fs measure common neighborhood protein interaction unweighted measure refer avg uv avg protein network protein least one protein score reliable assume protein direct separate similarity protein probability neighborhood conditional similar generalization name interaction unweighted weighted version note protein topological overlap co expression network association common neighborhood unweighted straightforward respectively numerator denominator inclusion desirable sensible contribute generalization zhang co network measure transform protein hc demonstrate application originally design
dynamic statistic process switch nonlinear continuous allow analogue biological us perform rescale dynamical describe across force position model offline discriminate learn approach change derive base switch propose monte constant diffusion dynamic approximate dynamic maintain present detail switch representation implement winner multidimensional sigmoid strength external behaviour winner implement determine centre determine identity dynamic implicitly one one lyapunov stable provide remain switching element lie interpolation observation parameter set dynamical system ensure stable fix switching logistic sigmoid choose increase logistic maintain evolve dynamic behaviour point point switch external switch movement represent gain describe learn dynamical limit improves govern implement motion around cycle position implement strength thus position dynamical implement position combination normalise circular von cycle evenly ad hoc consider learn position phase suggest allow actual model assume equation w covariance diag process switch induce switching provide flexibility reliably datum aim infer switch filtering solve optimally filter dynamical linear dynamical resort extend kalman filter trade version suggest filter integrate sigma transform sigma factor prescribe wiener standard I choice scalar correspond manually select facilitate switch choose moderate discrimination performance otherwise likely switch choice experiment describe synthetic human motion filter long datum possible infer fraction response trial identity choose typical motion capture dot connect illustration purpose show c line error grey switch output determine switch true green period begin high b noisy infer state noisy reliably inference dot trajectory plane form overlap depict position shift thus dot dynamically hide period dot dynamic generate uncertainty unless otherwise trial simulate switch current external dynamic online base dynamic variable start switch fix trajectory trial trial switch impulse quickly switch typical movement sequence increase step plane introduce noise average twice time prescribe fig nevertheless still movement gain free value performance cf half cycle response input r true walks walk noise walk noise motion trial identify correct dark blue red second switching dynamic motion example aim principle capture similar switching present motion walk important walk remove capture marker data dot see frame walk span dot highly walk principle walk capture normalise mean trajectory close normalise walk four normalise another represent infer stand normalise original marker position marker challenge test model original generate walk switch introduce jump total walk uncertainty trial dimensionality true achieve response addition add gaussian noise marker deviation equal marker position yet standard give furthermore misspecification uncertainty model nonlinear kalman dynamic process online hide synthetic plane light represent switch achieve stable dynamic process environment switching perform misspecification represent goal motion present dynamic cycle successful
take variation tail balance negative motivation loss net negative yield balance hill establish cdf normality characterize model equivalent cdf essentially statistic hill substantial considerable value represent issue tail reference square rmse enjoy estimator fortunately bias reduce exploit censor couple variation define equation eq reduce quantile establish author define eq replace get estimator establish normality carry section conclude remark section appendix optimal fraction function quite function determine known order unfortunately second regular asymptotic estimator variation specify rate constant sign near extract fraction asymptotically k fr cdf quantile taylor identify example fr view realistic show indeed assumption asymptotic follow give term normality thing normality meet need major approximate brownian integer define corollary sample ccccc ks ccccc ks ccccc ks ccccc ucb r r c r r c r r panel panel eps shape eps population parameter horizontal represent size new rmse moreover panel case tail show range tail panel pass normality test panel probability cdf brownian amongst state order eq cdf formula make representation statistic rewrite show theorem use application calculus us behavior integral elementary k rewrite eq make expansion observe may q n use approximation dt show remain converge ks complete theorem normality us integral formula simplify let verify first rewrite nc nc easy eq let show recently eq nc combine normality element analogue calculus finally calculation bias tailed bias high newly confidence easily estimator perform mm mm remark mathematics quantile introduce normality goodness square interval reduction extreme value heavy distribution hill I negative v tail infinity tail index e cdf tail model large price return etc construction rewrite quantile
namely modify summation start inequality density parameter equation determination coincide generalize modified formulae equation formulae introduction q definition approach prior signal read nonzero appearance denominator formula background interpret really flat lie conditional provide determined formula coincide naive eq drawback interval unity background contradict full lie infinity must unity drawback correspond bayes inequality inequality read derive coincide satisfy evident equality look q frequentist eq result modify frequentist show support grant modify frequentist definition namely pn definition prior propose modify physics determination observe due parameter probability determine lie except evident popular option interval inside confidence number event determine value determination identity determine parameter lie modify frequentist equivalent
present phenomena stochastic interaction interaction particular community naturally label chemical may movie collaborative rating email main contribution generalization describe identifiable context label block transition propagation threshold label conjecture validate belief propagation detection draw address label reconstruction regime et spectra grow high et al belief determine regime initially al tree simple census threshold survey understanding threshold still coincide et al community contrast detection label explicitly consider follow node split block namely block node refer belong node relate one observe draw al relative type infeasible feasible degree make un several support conjecture couple see edge consider follow starting node consider branching characteristic birth child poisson distribution poisson parent draw everything else consider depth denote subtree root together bayes entail recursion uniform leave uniform constitute robustness belief ratio less belief root belief I variable point say insensitive ready denote average branching tree quantity insensitive prove implication un label infeasible insensitive classifying correctly nod conjecture technical branching process parent branching endow I moment sn sum path let er transform er determine behaviour deviation weight branch sure similarly branch q expectation summation read large summation borel lemma finitely thus branch indeed nd positive sure tend tend infinity lower bind minimize desire linearization formula read absolute suitably label l weight derive laplace point supremum necessarily convex duality case equivalently equivalent strictly sensitivity perturbation perturbation parent child type parent distribute tree non get exactly type vertice child ij type vertex throughout child opposite label correctly reconstruct clearly infinite growth realization distribution adapting infinite label reconstruct type th level derive bind effective electrical upper maximum background notion follow child bl ij r r ba tend infinity hence computation ij also uv uv g eps ab cm eps numerically label stochastic model symmetric I among node type type reconstruction value validate conjecture great lead characterize success reconstruction use introduce denote assignment community setup assign take overlap metric range assign randomly label graph belief base vary value seed line curve attribute fig accordingly belief propagation label correspond curve shift fail threshold overlap achieve analysis context label conjecture symmetric community affect rich community potentially characterize namely sensitivity coincide label tree main conjecture characterize theorem
span span proof find light qr iterate triangular obtain qr give define kb tu u tu notational simplicity break lemma state complete proof satisfy rip norm th spectral rip matrix lemma u k v show recover use measurement prove observe constitute sense satisfy approach incoherence done exhibit similarity sense incoherence perturb completion iterate step partitioning set prove establish subspace w h incoherence see sense alternate enough incoherent maintain guarantee lemma define also iterate pt u start alternate use proof general let q similarity update sense essentially special incoherent induction distance incoherent factor step incoherence incoherence incoherence step incoherence w multiply constant similarly multiply q assume argument prove lemma prove stress incoherence also incoherent incoherent directly run sample total large step stagewise one x recover nm significantly information modify call stage stage solve stage recover top step k also k noisy sense goal I formally present proof base f hold th step initial stage ok show initial stage formalize proof f satisfied iterate iv alternate minimization empirically appeal solve main motivation result theoretical would aspect result matrix rip incoherence algorithm iterate computationally fast optima high statistical minimization rank problem perturb perturbation extent rip incoherence demonstrate rigorous result minimization pca light aspect initialization easy iterate show succeed initial orthogonal subspace suggest iterate preferable stagewise adaptation modification alternate fact mostly definition microsoft com edu university mail edu alternate widely applicable method efficient form major win entry netflix step convex paper theoretical performance minimization completion problem pose tractable certain minimization succeed compare exist minimization guarantee particular find empirical represent bi form reason crucial several practical application recommender several estimating million matrix load memory hard disk optimize recommendation may rating million hundred thousand store rating final rank efficiently optimize modeling impose constraint matrix impose pca look due bi finding minimize result bi linearity correspondingly popular overall solve efficiently usage date almost theoretical work motivated practice completion complete observe popularity primary come user become appeal provide fast parametrization refer measurement recover back study completion entry completion element measurement without sense ill moreover svd involve manifold need manifold intensive exponential several robust additional relaxation rigorous minimization establish recovery norm assume problem recover give additive scalability problem completion provide scalable respective exist albeit drawback rip theorem level decrease base observation rip hold alternate view analyze perturb see detail proof rank span orthonormal basis space depend span column distance subspace decreases iterate angle distance ready correctness inequality follow rip follow
dimensionality improve furthermore signal less crucial hyperspectral scene fractional linearly mixed simplex abundance rely library usually laboratory term dictionary library different library observe constrained observe spectral infer signature implicit fractional I hyperspectral implement simultaneously linear baseline describe identification class technique fractional devote devote sparse contextual mathematical summarize obtain experiment plausible distinct negligible partition illustrated spectrum approximate linear mixture fractional measurement denote fractional abundance g give fractional name fractional area th fractional abundance vector sum abundance abundance researcher sometimes abundance account consider error signature cover hull green red vertex correspond simplex mix green identify simplex exploit algorithm adopt direction show fig noise vertex correspond hand side pixel contain material spectral datum middle infer fitting yet seminal work several hand correspond near resort adopt suitable linear fractional signature tend condition sensitive yield characterize expect value snr snr sd couple ratio snr must acceptable significantly corrupted fig mix fractional simplex db united survey fractional well column band remove noise matrix code read snr indicate signal subspace snr singular decay high signature nevertheless big shape slowly band assume good low linear vector yield gain snr usually advantageous necessary operate subspace require extraction suggest exploit objective minimize sum projection empirical maximize additive power power mathematically sequence transform first transform snr sd identity differ optical develop laboratory dimensionality extraction module identify pixel variability scene collect pixel angle metric create identification signal criterion minimum come mind criterion adopt approach turn develop pearson theory method determine hyperspectral base detector build sample term whiten infer estimate select mix dimensional linear dimensional subject topology method estimate example analysis manifold linear projection projection pursuit also refer orthonormal orthogonal projection spectral onto replacing project storage snr gain direct consequence explain latter assume covariance power project denote power signal subspace estimate side noisy spectra mixing interval fractional db project datum noisy project spectra identify db colored additive db top noisy scatter eigen blue dot final although projection often remove percentage change value eigen possible attack contextual leave I filter bm structure scatter plot noisy dot green dot eigen image bottom figure simplicity notation project original product simplification reality variability treat primarily invariance shape fairly signature affect positive factor pixel instead entire spectra pixel observe simplex rather illustrated transformation match reality mapping transformation impossible identify affine orthogonal onto dark accord set projection spectral angle affine choice issue illustrate fig publicly hyperspectral cube united hyperspectral line band hand side angle high angle vector usually area figure side project vector discard project vector pp volume approach pure pixel assume mean spectral vector simplex enable efficient view strong pixel probably often use hyperspectral conceptual mean pixel dimensionality snr random correspond vector pixel one fact simplex pixel find define large iterative implement vertex project orthogonal span already determine iterate simplex algorithm iteratively grow simplex finding vertex angle cone cone represent spectral vector start identify make cone making angle terminate tolerance maximize cyclic fashion quite concern project subspace span determine consider complete datum nonnegative abundance lattice value lattice operation construct max spectral contain appropriate use notion al seek mix volume simplex define pixel result fig simplex minimum datum true mixing simplex project onto simplex usually consider recall volume convex hull give origin simplex seminal identify apply projection iteratively simplex minimize simplex v identification lagrangian introduce allow positivity relevance depict perturbation true lead dash original positivity project onto theoretically conditioning aim solve linearly stand number follow maximize minimize b approach constraint hard optimize convex lagrange computationally aim hard positivity cyclic linear pure pixel version effect chance act soft constraint fractional chance volume mix alternate involve solver factorization solve original set minimize square simplex volume nmf implement alternate programming latter robustness h algorithm aim solve distance simplex vertex volume regularizer ambient non exactly recently ice implement alternate ice regularizer former minimization square nmf ice ice incorporate proportion prior large initialization proportion proportion zero discard ice sense datum correct independent source dependent constrain solution meaningful range pose adopt posterior abundance assume priori bayes paradigm stand probability observation model unknown popular joint matrix clear ice nmf classify minimize play consists conversely assign abundance respectively convenient ensure inherent observation mean gaussian autoregressive iterative respect abundance linear distribution interest conjugate physical informative joint posterior close estimate design remain impossible chain propose generate sample parameter mainly unknown chemical spectral mix distribution assign abundance efficient operational respectively instead estimate spectra hyperspectral estimate identify dimension bayesian approach depict toy illustrative compose pure fulfilled maximize volume correctly contrary statistical hypothesis star independent uniform admissible abundance equivalent seem span characteristic detailed paragraph simplex suggest recover conduct choose volume simplex implementation bayesian rely identically gaussian lead matrix vector note colored covariance handle many low band projection subspace largely spectral abundance compute em estimate ica attack assumption fractional hyperspectral fractional infer mix mixture dirichlet density enforce constraint abundance cyclic infer augment signature investigate interesting normal compositional stage stage respective gaussian second stage pixel abundance multinomial dirichlet abundance set quite consume joint maximum estimate joint sampler inherently theoretically correct speed dominate fig nmf algorithm generate library fractional mixture dirichlet mode dirichlet randomly initialize reflect situation region scene tune correspond scenario poorly contrary yield useful hand side dominate dominate image circle identify although really sure true reasonable conclude example fig bayesian refer link spectral observe signature linear pure signature advance ground amount optimal signature library mixed pixel scene combinatorial efficient regularizer mixed grow availability library area strong angle basis pursuit pursuit matching select say available amount find library pixel scene fractional library regression system satisfy linearly unique least noiseless pursuit omp replace norm term approach pursuit constrain denoise term multiplier interpretable perhaps totally fractional reconstruct solving provide incoherent sparse signature tend limit impose hyperspectral point simplex add sum depend optimization convert constrained square feasibility problem case hyperspectral library admit sufficiently sparse unique act induce regularizer point rarely reason noise run experiment hyperspectral angle spectral signature abundance dirichlet yield beyond reach evidence impact coherence minimum signature introduce optimal dirichlet mixture stand abundance respectively degradation perfect perfect curve challenge db useful notice plot material snr value correct selection material crucially availability hyperspectral library acquisition library consume library consideration adapt vice way library directly information idea term signal processing reference al attack modify exist learn representation learn signature scene approximate material criterion generally hyperspectral implement spectral usually algebraic positivity fitting term likelihood direct ignore contextual great hyperspectral hyperspectral contain image pixel process individually hyperspectral cube take advantage follow integration contextual information guide abundance step spatial attempt spatial pixel abundance dependency particularly adapt image abundance pixel partition abundance markov field labeling variable conditionally pixel individually generalize region must choose extension base nonparametric markov several exploit spatial information designing criterion addition classical positivity include structure autocorrelation abundance measure vary smoothly information incorporate within include account automatic extraction spatial singular value decomposition use spectral extraction preprocesse vector scene intend preprocesse extraction mention exploit contextual assume deconvolution variation regularizer spatial enhance nonnegative factorization replace regularizer neighboring abundance limitation regression signature add variation term individual band follow collaborative add pixel decade research spatial resolution environment signature material spatial instantaneous imaging hyperspectral resolution component development recent development paper development hyperspectral cover mixing versus nonlinear identification integration describe physical many algorithm high activity limit many address manuscript however combine provide snapshot challenge hyperspectral field include signal modeling algebra regard trend image general particularly application monitor tracking g type chemical contamination application response decision near different heavily consider spectral hardware architecture gate graphic unit implementation together discovery correct method gibbs sampler advance multi development offer possibility hyperspectral piece accurately process practical yet several mention proper distribution become structured library area process become cover different area investigate software quantitative perform statistical acknowledge green team make hyperspectral united states publicly library signature acknowledge center available community superior mail lx image instantaneous thousand channel higher refer high spectral resolution identification require identify material multiple spectra measure material pixel material signature ill pose condition variability many search tractable describe contribute potential small spatial spectral resolution monitoring region spectrum band range record light locate illustrate measure organize form cube plane correspond acquire band vector acquire spectral hyperspectral refer process pixel spectra image collection spectral signature per material present represent material hyperspectral spectra grow suppose percentage cover want care distinguish signature remove interested present scene obviously application percentage scene pixel states mixture author laboratory set calibration hyperspectral accurate linear material cross area highly dominate dark object inaccurate amount accurate estimate contribution material pixel regardless large hyperspectral ten year currently
optimisation modal slice specify energy boltzmann draw slice identify global mode project methodology include multi mode dominate boltzmann mcmc pose well functional optimisation mode boltzmann potential exploit slice develop chain mode avoid associated optimisation modal methodology optimisation literature build seminal hasting mh boltzmann optimisation additive functional dimension boltzmann distribution analysis mcmc local generally require mixing slice remarkably mix slice case energy mcmc popular simulation evolutionary mcmc liu al lee lee gray particle solely energy focus stochastic level wang algorithm rest describe base optimisation sampler conclude problem minima min optimisation multi modal minima simulation simulate exploit optimisation find energy z partition clearly minima mode simulation explicit knowledge boltzmann limiting case boltzmann distribution interest limit mode boltzmann minima original extend simulate draw recurrence start ergodic estimate modal however dynamic equilibrium distribution issue chain insight slice back chain converge high project lower describe development slice suppose wish high un density let auxiliary slice normalise provide slice define gibbs sampler hence draw wang slice product suppose auxiliary slice x u conditional I interest additive boltzmann slice extend slice pick set classic optimization inside long narrow flat get minimum need choose last nonlinear mcmc use collapse van conditional conditional invert slice sensitivity burn minima local maximum boltzmann quadratic slice augmentation slice inequality follow combine argue similar fashion constraint conditional sensitivity slice long chain level equal contour underlie function region distribution slice invertible set result conditional truncate normal slice figure show burn boltzmann mode straightforward domain conditional auxiliary slice uniformly run gibbs conditional defines sampler sensitivity namely slice boltzmann efficient optimisation slice sampling modal flexible enough additive multiplicative fx x minimum straightforward example quadratic minimum immediately identify simulation simulate anneal mixed optimisation sampling alternative ny liu wang application analysis york annealing b york simulate annealing york university thompson anneal lee optimisation categorical input sensitivity importance variant statistical mechanic chen sampler liu computation local liu slice sampler van partially collapse sampler dynamic problem certain comment et g g geometric
I jx seem tree definite exist asynchronous algorithm plug root r dependent computation leave two theory laboratory f ed electrical university ct belief multivariate gaussian equivalently minimum guarantee convergence fail converge quadratic function failure mode understand via graph parameterize tree remain positive demonstrate iterative quadratic gauss always converge work study reweighte minimize equivalent positive covariance definite gaussian belief propagation message pass variance provide variance sum area dominant covariance scale tree condition necessary definite computation tree definite matrix arbitrary tree occur dominant loading load matrix subroutine produce feedback produce repeat decentralized amount load feedback quick provably convergent min algorithm convergent sum diffusion belief reweighte propagation coordinate maximization pass guarantee correctness plausible search convergent message first graph cover combinatorial characterization characterization allow conclude convergent message pass correctness correct assignment outside walk investigate reweighted belief algorithms undesirable typically answer diagonal may definite although similar loading parameter reweighte variance definite converge exist parameter extend idea general convex outline reweighted generalization cover characterize examine variance reweighte algorithm quadratic examine reweighte min sum finally section summarize main sum factor proof derive min minimizing factorize self edge minimize factorization bipartite graph correspond omit single reduce graph figure typically omit clarity message pass execution min pass back pass jx neighbor difference understanding update correct central min arbitrary variable affects locate constant avoid numerical issue may execution think vector index edge pass value message typical assumption choose message order min marginal message set belief approximate ix j ix min solution construct equivalently uniquely locally global contain arbitrary graphical min version message pass min sum need belief correctness convergence algorithm graph single demonstrate tree min depth tree root length backtrack walk backtrack successively vertice node root recursively leaf neighbor copy node operate copy potential construction represent circle minimum none edge potential subscript message time receive message add one message tree terminate lemma belief algorithm correspond message initial example computation view change update belief small say algorithm converge real objective always equation theorem min guarantee guarantee alternative message pass suffer drawback effort convergent message pass result algorithm obtain standard choose call weight problem choice surprisingly weight guarantee correctness cause incorrect solution message message analogously x c ix ij jx x vector message belief message factorization correspond marginal belief exercise draw scale node label node node right label right edge x x edge x node label factor self adjacent variable reduce computation tree produce pass order however pass multiply potential tree multiply summarize message pass time tree computation leaf computation create connect node tree root contain backtrack length start length computation computation root multiply branch multiply concrete associate weight correspond pass new potential along standard node belief computation tree min generality tx tx tx tx pairwise factorization tx ix reweighte though appropriate reweighte step update parameterize quadratic eq update valid message analysis hold asynchronous belief locally always minimum assignment locally belief tree converge minimum completeness proof lemma locally minimize quadratic consequence definite reweighted solve dominant convergence computation denote entry graph cover iterative message pass address great strength two precise neighborhood copy cover study relation local cover pair cover cover original connected graph cover finite cover graph cover cover universal cover associate potential base sequel specify object objective pass reweighted sum distinguish graph identical node node message receive send message send copy local pass assignment copy lift belief define belief variable lift assignment assignment factor entry function nk identity quadratic notably critical cover critical similarly definition hx critical point vector fix hx lemma critical eigenvector eigenvector lift problem point cover cc negative eigenvalue unfortunately via minimization may figure cover iterative reweighted base reweighte reweighte converge correct assignment correspond cover characterize equivalent definite definite implication proof use detail consequence walk intuitive explanation condition message min theorem tree product correctness correct positive walk belief minimize early belief lift cover locally belief message reader reweighte message pass whenever semidefinite correctness belief edge example produce pass scheme quadratic correctness correct original dominant dominant may eventually possess non positive happen infinity course correct understand affect reweighte reweighted remain course belief belief determine begin consider ij valid ji c ji follow j induction hypothesis exhibit weak suppose computation c require tree positive sufficient convergence variance computation positive see ensure element force computation cause behave almost choice computation generate exploit computation scale dominant case message result suppose trivially satisfy c ij induction symmetric diagonal tree ta tc ij eigenvalue tree bound way long monotonic decrease remain definite limit belief definite estimate converge belief must minimize see message pass order update perform order message asynchronous asynchronous computation extend asynchronous asynchronous allow update cover via vector order variable message copy belong already bipartite kronecker double disjoint algorithm asynchronous kronecker double cover asynchronous alternate message label label node label asynchronous two specifically double early analysis kronecker double bad might solution reasoning message pass iterative minimization definite equivalent system study elimination cholesky system gauss show use algorithm positive minimize cyclic gauss produce iterate gauss symmetric ordering follow variable analog perform n algorithm produce cycle gauss gauss semidefinite matrix strictly diagonal gauss exist immediately semidefinite matrix converge semidefinite gauss extension may point correspond system system first construct j ki ij ji ij converge
pattern recognition tag machine learning supervise hard ml pg feature extraction work supervise learning include pg popular learn engine select technique pca vector like pca recommend deal want ml pg library provide pg library feature extraction mechanism proof library file library file level library transform file internal implement list default ml pg library perform library first question need increase library reason pg librarie lemma library introduce illustrate pg user small library pg list run table efficient inefficient lemma prove pg interface machine pg table group proof agree one possible lemma fundamental natural operation split keep frequency click name split description show mode pg library cluster current may technology aid interactive case cluster library pg interface two figure incomplete development library lemma ml pg pattern side figure pg statistic take extraction machine interface suggestion ml pg vector pg discover series sum odd different correlation fact sum current suggestion pg brief description flexibility modularity pg come free light backward pg implement translate form cluster name external concern external tool system external even section highlight ml light handle handle pg convenient environment divide learn input object example high serve may cluster example happen certain find run determine ml pg handling program cluster pg generate handle output statistic program three frequency explain cluster interactive mining may proof pg choice search pattern refine extreme avoid value produce library find big value cluster determine experimentally ml mainly learn heuristic optimal cluster pg interactive interactive consideration size library auxiliary cluster user ml stand produce cc c c frequency library frequency parameter lemma table pg lemma proof n induction note desirable ml ten suggestion relate lemma ml see figure proof finish n notice goal inductive rule imply way suggest pattern high similar lemma potentially pg analyse output pg actually double criterion close cluster range distant indicating indicate probably assign wrong whose ignore pg fix interface access effect proof property provide therefore frequency trivial initially frequency trivially pg time frequency cluster discard low significant pg item frequency serve proof hand acceptable differ recognition ml pg allow user vary threshold parameter show come experience library line modular approach pg wide choice keep ml pg different highlight c c matlab gaussian indicate pg frequency threshold cluster lemma cluster irrespective choose lemma result effect level ml pg library value frequency parameter proof split smaller notably see simplification cluster figure parameter discard typical usually produce big big split interactive cluster form choose goal example highlight form belong cluster highlight form become frequency increase increase effect increase demonstrate mining library goal level bring level focus example whereas focus relate finish pg ml pg transform file format consequence ml pg file node whose child lemma belong file machine process way depend pg convert list pair contain name list goal dependent cluster search list current pg high frequency display overview automate interactive proof background introduction general interface probably different technique generic planning interactive theorem interface recommender mining implement could method prove termination implement theory discovery mechanism derive program symbolic tailor certain library certain proof properly deal library hand statistical machine library problem discover ml belong category successful provide efficient integrate statement relevant result successful library since proof infeasible receive tackle library send syntactic predict whether proof library hierarchy figure approach rectangle naive anchor anchor north white difference tool handle grow pg achieve extraction compact feature ml pg also library interact stage library big well tool aim increase goal user feature extract order formula proof vector sparse svms bayesian mean gaussian big interface library present pg machine interface prove statistical result non highlight pg flexible proof ml pg interaction learn help analyse interpret avoid pg benefit knowledge determine frequency frequency easily extend modular tool regard level statistical design allow environment learning mode addition library notation level plan feature extraction method plain pg detect pattern extraction implement pg way first consider five integrated pg whereas proof varied patch partially proof library spread library tackle close lemma pg pg combine easy mechanism implement search pg pattern find move symbolic family ml plan integrate machine help tool track fail machine strategy experience could also discover various include interaction extraction mechanism interaction pg cluster library think server user useful especially program purpose pg name notation grateful anonymous comment suggestion individual research interactive prove fm project participant rgb rgb thm lemma thm remark definition remark figure pg extension allow goal structure library interactive write matlab pg automate matlab result pg interactive development interactive prove decade prove solver become increasingly efficient type environment conceptual mainly advance enforce employ art offer user generate lot challenge inherently especially output order external order proof suggest valid mention improve perspective statistical automate interactive interactive successful mainly improve issue statistical challenge rich finding proof structure apply notion regard traditionally proof nature aspect recognition statistical interactive challenge consume challenging find understand guide interactive interested include fail guide tool experiment interactive lack interactive mining extraction proof semi automate inherently interactive user important range among experiment build upon maintain strong interface trend statistical interpret result uniform environment matlab underlie programming language learn interface source interface wide variety hand translation use view meet technology improve user behaviour feed back proof primary accomplish become interface rectangle control anchor north fill white white rectangle build machine matlab pay attention address vision call pg ml pg manual pg interactive relate proof development goal tree challenge issue pg automatically aspect machine length challenge ml pg must enable range machine suitable assume want amount pre processing pg question ml pg automatically connect interface collect analyse interpret output stage challenge backward machine tool proof less demanding seek statistical arise backward finally ml interact relation proof detect library even challenge aspect development interactive potential recognition aspect benchmark statistical pattern recognition prove pg go useful option experience survey work extension possible run kind statistical help expect ml pg library lemma list notably simplification goal induction goal trivial library list user pg problem previously pg lemma library namely old give table go proof operation lemma shape statistical goal trivial goal n mining shape lemma pattern case general similar pattern also may apparent formula type abstraction automatic interactive column table pattern sequence apparent always serve important ml pg sensitive effect one several proof step immediate transformation proof single throughout dimensional array shown allow induction trivial goal top lemma feature feature correlation n successive proof step consider goal rewrite move prop rewrite prop prop move prop move rewrite prop l prop table odd pg name stand big index big seq odd odd add big library highlight bold contrary odd pg learn patterns style chosen ml pg feature associate level extraction worth level base application difference concrete concrete plain package implement impose pg plain row represent almost ml currently consist extract close correlation trivial trivial row table encode property plain type argument hypothesis none pg goal apply extraction bold table lemma odd none prop l l prop move prop prop move move prop bold sum odd pg big n fact big add add library ml pg feature table flow use odd tree move move rewrite prop move rewrite big argument inductive name add add big lemma library column
find ccc ccc bernoulli horizon c min bt learn policy bold report policy consider training horizon already bernoulli prove gaussian make tuning sometimes happen horizon learn policy systematically outperform policy numerical nature policy extremely hard interpret understand relate symbolic policy box enable performance symbolic interpretable strategy interpretability performance tradeoff several field worth equivalence strategy resp symbolic horizon policy good training horizon prove well policy propose paper knowledge exploration test armed bandit horizon policy outperform publish robustness wrong highlighted evaluate armed bandit opinion direction improve policy overfitte occur call technique along idea could identify candidate behave good policy certain expect aim certainly relevant policy alternatively see adapt well bound perhaps identify policy give small expect regret good performance armed exploitation scheme principle investigation study successful ac exploitation many bandit formalize canonical form knowledge lack approach incorporate address class e propose step I target ii strategy performance parameterize symbolic appropriate bernoulli various play meta learn e outperform strategy greedy robustness learn strategy truncate many science artificial intelligence finance step reward machine response characterize distribution rational risk neutral play expected variance reward reward distribution decide play goal consist try use current decide play effort essence difficult impose play theoretical armed focus design provably asymptotic assume reward distribution bound play confidence reward arm round high strategy index typically involve performance usually report manually tune share similarity problem trial simulation good policy play armed exploit bandit armed number training consist search e yield performance tune hyper index policy within much broad prior two compose learn generate symbolic formula empirically play horizon fully significantly wide previously policy careful test learn truncate distribution idea armed policy policy identification pure exploration formally armed index policy state symbolic report armed policy arm bandit play process arm play refer optimal denote bandit minimize maximize ideally tb kb bandit bandit policy arm value response arm play play machine subsequent policy score k arm tie break random bandit parameter note sum give exploration playing rely prior e bandit many desire strategy exploit knowledge family whose policy fully give play horizon since cumulative regret value perform training make hypothesis optimization solve section consider strategy define parametric family candidate feature rely describe feature define product operator policy history play combination feature rich e strategy propose possibility history perform four root logarithm inverse arm multiplied way give combination degree resp policy parameterization change bandit episode global optimization yet rely region sample solution repeat population model probabilistic candidate train problem sample candidate probabilistic adopt prove quite policy gaussian policy literature three sub propose small formula build upon formula close formula advantage formally formula unary unary atomic variable constant provide cardinality four elementary operator contain absolute opposite figure formula occur k policy index formula discuss symbolic parameterization good policy multiple time propose partitioned equivalence formula equivalence typically far trivial specific rule perform involve advanced static believe difficult solution consist formula return formula random formally follow respective realization ff f lead cause division logarithm discard minimal minimal cardinality formula naive finding implement inefficient contain formula relatively bad relatively formula formalize multi associate arm select episode index policy reward quantity episode arm reward try formula identify bandit arm arm multi armed index pure exploration bandit work bandit algorithm reward instance depend time play far select bandit clearly correspond bandit rely armed technique problem hundred benefit compare previously propose policy case exploitation strategy test policy learn former policy policy default hyper tune scenario priori parameter care issue bandit problem bandit
role increase amount behavioral naturally minimize contribution definition iteratively membership snapshot nmf matrix max membership role structural evolve role less important active inactive role currently probability shift occur role future structural represent analyze behavioral role basic center star drift evolves analyze dynamic role mass shift role active expect role would denote list max snapshot node feature iteratively membership give nmf membership consecutive inactive take role time role active role inactive whereas lot role role increase dynamic refer evolution structural structural behavioral node behavioral drastically communication stay structure therefore structural behavior center star incoming bridge connect network consistently behavior drastically role notion decrease trend g role versus versus home besides track detect change occur similarity role membership across node membership detect anomaly minute r ex dynamic analyze ip trace dynamic consist million node real citation propose model quadratic unable model investigate sized node unable realistic million exploratory approach node minute second dataset million approach take linearity method apply intel core ghz gb framework also trivially behavioral role parallelization make attractive pattern naturally apply stream fashion monitoring error demonstrate utility track individual dynamic result indicate behavior entire important communication domain explore dynamic specific behavioral pose particular type seek question characteristic learn behavioral role day change day behavioral role normal another node change lot trend evolve behavior across role drift axis importance structural dynamic ip communication role structural property role behavioral axis first analyze last dynamic university email email university email account trace edge across hour activity feature result behavioral trace address communication communication begin minute interpretation dynamic role show interpretation role role whereas role combination structural characteristic major evolutionary particular consistently property also cluster slight role many interesting certain role cycle trend structural behavior node additionally nod pattern activity become active inactive represent white structural world email network structural pattern significant trend pattern use role interpretation stable role period surprising membership day email communication role period represent inconsistent inconsistent behavior anomalous activity two day role node dominate role individual role clearly node identify dynamical dynamic email communication ip trace often dominate role nevertheless network dynamic nod human behavioral fluctuation day lot scalable vary completely capable handling handle minute capture particular topology coefficient biological miss perhaps evolve property may long pattern traditional connectivity pattern understand frequent actually instead fully dynamic paper imagine type select difficult connectivity understand pattern make manual impossible knowledge biological network representative truly property also compute edge manually tune interpretability extensive manually costly time inaccurate novel pattern furthermore pattern stationary system could completely automatic capture main disadvantage approach however analytical tool pattern widely plan sophisticated analyze connectivity nevertheless time anomaly goal detect node link anomalous interpretation pattern anomaly pattern anomaly fail capture novel explore therefore representative generalize evolve capable capture anomaly suitable attack may plan detect take g entire network change future structural dynamic track level connectivity require edge whole node use basis sophisticated future transition apply task plan nod structural dynamic acknowledgment laboratory contract de ac nsf number air office scientific national science engineering fellowship reproduce purpose conclusion herein interpret necessarily policy either imply nsf network biological behavioral role represent connectivity parametric automatically learn role behavioral center star bridge connect community novel arbitrary track particular dynamic non membership perhaps indicate anomaly trend among network pattern evolve time overall effectiveness tracking dynamic network dynamic representation building tool fast explore database biological network nod attribute dynamical change considerably induce pattern automate far observe network necessarily considerably usually significant continuously automatic identify track tracking analyze structural fast completely automatic manner approach capture linear heart build sophisticated tool dynamic individual activity instance ip ip want behavioral role allow characterize behavior detect begin dynamic behavior feature time track membership dependencie role track behavioral role behavioral role role connectivity pattern role similar network recursively automatically therefore node share analyze dynamic dynamic whole dynamic present evolve importance pattern entirely example behavioral dynamic initial privacy twitter mobile device etc fluctuation change entirely contrast dynamic structural structural behavioral instance hour serve work email activity behavior star large incoming connect community follow explore serve tool structural specify pattern explore perhaps suitable world network clearly nod network dynamic pattern communication agree human intuition increase predict temporal work mine one importance node work exploratory discover structural automatically therefore social communication biological applicable domain structural variety pattern become connect network period structural dynamic learn diversity adjust behavioral sequence graph graph generator feature behavioral internet topology generator topology generator internet learn facebook role analyze google govern e network
feature within conclusion induction accord signal without suppose relation take set iterate x note combine l q choose obtain signal eq induction hold index magnitude u n n last use decay eq contradiction definition matching pursuit natural extension omp atom per omp add atom orthogonal matching pursuit rip show rip absolute within slowly decay particular recover slowly within pursuit omp sparse commonly compress sense q sense sampling omp isometry rip recover compress eq q sparse eq I analysis omp primarily direction coherence rip analyze rip condition improve rip rip omp recover uniformly omp allow omp iteration omp atom least square direction rip omp sparse type algorithm base omp orthogonal pursuit sp variant extension omp matching pursuit omp select omp atom procedure omp omp study name iteration index feature z r x x rip recover iteration main give rip recover signal recover numerical us conjecture recover slowly decay within rip recover set case answer suppose matrix rip initial recover zhang omp make result extend paper additive strictly liu rip rip depend theorem answer question interest find matrix whose see rip make rip one comparison performance give parameter sample set distribution subset generate sparse omp repeat succeed record illustrate omp omp theoretical whereas right depict decay sparse naturally want signal less experiment decay run step decay omp firstly definition decay without decay rip decay omp improve rip decay omp iteration right rip rip lemma denote vector index generalize jt similarly inequalitie q part u cauchy schwarz imply n eq space furthermore imply similarly since x fourth follow accord na n noting combine last rip since p imply u nu indice u u h u x n lemma eq put arrive lemma result
red width pt south near color line south node start color red width south north regularization apparent certain present correspond top regularization worth series contiguous traffic describe accommodate dependent since flow satisfactory dl technique complexity suit monitoring wide count measurement sample collect purpose phase link slot assess entry link count tune choose show reconstruction comparison fix diffusion entropy penalize l solve ls regularizer encourage traffic volume stochastically dl base compete approach furthermore decrease remarkably actual traffic monitoring delay connect path delay operator assessment plan diagnosis challenge primarily grow delay measure partial current delay p tool environmental krige scheme delay measurement introduce therein topology delay build idea capable spatio put kalman filter variation delay topology krige predictor path comprise due contribution link model delay collect depend traffic temporal periodic well drive delay spatially path model link sec spatio generally refer random propagation channel spatio entail treat ls find namely recently diffusion wavelet delay correlation network operation minute delay network bottom wavelet bottom right delay several around change delay map summarize operational spatio temporal early efficiently past measurement towards kalman employ yield equation term gain final give covariance delay yield noise kalman allow prediction well second namely measured turn design monotonic greedy algorithm readily increase framework admit problem formulation capable measure art tracking path selection heuristic note parameter adapt include delay collect month ip square left observe far improve traffic anomaly across flow traffic engineering traffic physical cardinality operational traffic flow associate specific adopt mean multiple connect pair path discrete horizon otherwise estimate topology also physical traffic temporal traffic addition periodic respectively thus validate temporal flow since normalize rapidly flow experience due failure behavior g attack aim service amount account anomalous f traffic change difficulty anomalie anomalous spike flow traffic interference flow stem link measurement operational reality traffic indirect traffic measurement tuple e link measurement compact q argument keep unchanged flow traffic term nominal level anomaly occur short time possibly flows suppose anomalous instant anomaly traffic row give level measurement critical monitoring aim anomaly argue sparsity property instrumental network th flow one addition affect magnitude event examine link anomalous flow subsequently contingency heart seminal anomaly absence miss component analysis decompose anomalous model residual dominant nominal span dominant anomalous orthogonal operational phase exceed amenable priori traffic effectively rank alternative bilinear leverage provide obtain non bilinear term r partition table adopt frobenius optimality could considerably may point globally however globally violate unless implicitly cost inside introduce auxiliary copy per copy yield connect impose neighborhood agree framework regularize put adopt alternate direction multiplier admm unconstrained quadratic refine subspace thresholding anomaly map exchange estimate directly connect overhead problem far admm offer literature empirical especially convex favorable linearly bi potentially extensive indeed problem attain consensus optimality centralized monitoring computer addition change miss challenge identification effect load balancing cause e anomaly slowly table dynamic partially general change load traffic completely link live low subspace may true update frequent network scale acquisition monitoring task datum network reveal week thus safe lie exploit spatio anomaly top previous acquire recursively historical naturally place possible counterpart exponentially weight l distant anomaly environment formulation coincide provably convergent base anomaly traffic critical nesterov acceleration anomalous flow robust rank subspace corrupt incomplete namely incremental descent subspace projection tracking miss problem issue relate model instance outline sense physical wireless cr well completion collect link level protocol anomaly standard flow leverage low solve aim monitoring cm end internet application file prefer establish connection close resource quantify choice delay unfortunately pairwise infeasible measurement distance pair matrix strong delay multiple intuitively correlation nearby node belong overlap common link low distribute requirement network motivate factorization algorithm require symmetric avoid overhead leverage observe traffic reference collect observe total network miss typically column internet structure good delay typically imputation interest sec either path delay observe spatial cr rf fashion global power psd capture channel gain cg propagation medium per frequency map enable identification band localization tracking activity reader treatment introduce build psd frequency spatially frequencie unknown virtual spatial grid cast problem vector control narrow band nature operational due nonzero correspond map active motivate lasso basis sparse ls cope matrix arise inaccurate channel estimator capture propagation psd band plan include bottom span band estimate right recover total center utilize transmission level location dynamic dynamically heterogeneous key full yet delay well accurate identification partial corrupted look scale sp collaborative monitoring objective management include adapt environment operate ii development scalable tool track purpose management iii ensure robustness face attack develop adaptation design categorization cm cm cm edu communication specialize transmission system device become orient heterogeneous service streaming thank advance speed capacity demand experience entail high failure attack research vision enable robust operation management dynamically evolve landscape human intervention analytical relevance sp monitoring sp construct scalable tailor heterogeneous network service internet device volume day wireless connectivity dynamic rely mobile united capability make device cognitive cr ensure service operator comprehensive landscape key management risk analysis security come challenge traffic volume interest instantaneous network management capacity plan traffic count readily acquire management instance link approach series square complexity internet traffic effort toward network monitoring delay great interest service affect experience number origin impractical even base inherent trait task operational paradigm current entail central monitoring interaction prefer consideration couple additional limitation first resource heavy tend network enough separate reference therein parsimonious descriptor monitor management state incorporate traffic volume delay activity intend traditional signature increasingly raw dimensional anomaly reliable manner diagnosis statistical instrumental maintain experience dynamic environment ensure network security monitor management complex complementary online construction map network anomaly flow recent advance statistical processing sp kalman dynamical minimization matrix semi optimization multiplier sp dynamic monitoring enable health lead enhanced robustness deal incomplete measurement domain measurement stationary traffic delay krige internet comprise node flow count collect incomplete due hardware matrix synchronization count simple measurement processing software rely interpolation series accuracy miss entry series none true real stationarity contiguous individually origin traffic flow relate term carry difficult link count goal estimate flow approach flow gaussian logit ultimately mechanism regularizer link count contiguous time miss historical measurement leverage structural regularity semi dl approach dl dictionaries representation mean capture success compressive cs signal lead advance audio motivated link count linear column dictionary admit predefine dictionary fourier respectively exhibit traffic
france lot attention devoted kind scheme combine tackle elegant quadratic program come theory low misclassification evidence naturally fusion add preserve improve behavior fusion real fusion rank combine multimodal issue lot research effort see survey fusion input bring source semantic audio event modality correspond generally different view classical commonly tf bag texture sift spatio temporal descriptor able discriminate scheme generally fusion merge unimodal classification heterogeneous sometimes score available classification fusion fusion outperform significantly tend give often refer stack extra multimodal fusion stack associate classical look combination usually weight majority vote cost machine assess diversity adaboost multimodal fusion method adaboost introduce weak another risk completely fully classifier since output propose majority vote score account majority aim fusion able positive additional ranking pairwise conclude empirically section vote pac dimension I real value otherwise aim lead majority vote risk majority auto moreover consider opposite restrictive represent quasi elegant trick usual core majority vote n I moment margin generalization propose minimize minimize risk vote account quasi find minimize uniformity weighted majority vote pac rate vote appear classifier fusion modality randomly subset I achieve low risk application document retrieval relate vote adaptation improve measure real value ss j sm example score map example top behind preference positive method measure notion order preserve multiclass hinge order positive j h j h force hinge previous eq hinge quasi term hinge consider additional slack abuse highlight difference simply drawback make hard propose negative force positive score vote lead process cv lead map empirically extension fusion stacking implement base goal example independently draw keep test objective good benchmark
robust asymptotic design phrase asymptotic optimality sequential minus identically distribute I vector density degenerate testing versus common sequential select measurable terminal rule stop accept test control test sequential test maximal type sequential hypothesis simple probability seminal statistic parameter see uniformly asymptotically replace statistic f apart moreover approximate often overcome adaptive observation approach initially sequential lead delayed estimator bayesian cost density q integer hypothesis two serve replace subset mention statistic ratio always easily discretization hypothesis loss efficiency advantage many application second wide present area detect signal sensor take density resp present sequential parametrize stop call weighted generalize ratio ratio replace sense attain asymmetric channel jx asymptotic setup want answer select far select correction addition kullback leibler accumulate attain open favorable prior take observation expansion operate select lead induce especially channel expansion essentially design remainder paper preliminary asymptotic operating test whereas establish optimality section compare specification test use quantify kl order divergence situation attain situation every trivial coincide throughout paper time respectively transform connect number number quantity important great testing moment z size say emphasis parametrize arbitrary follow also role asymptotic operating characteristic section expect size rely decomposition hold weight eq follow precise slowly change page decomposition random walk imply allow random walk approximation theory specifically define vector covariance matrix follow lemma prove b distribution apply mu asymptotic additionally define argument lead along positive finally inequality first remain similar way clear b correct approximation expect sample size rest kk mt necessarily moment moreover select belong theorem bayesian upon resp resp define h h I sum ct sd sd integrate stop problem sequential structure seminal state sequential follow specifically mi mi cb rely order threshold theorem go write simply mi similarly threshold establish sequential test thus threshold clear right suffice q negligible specifically sequential fix doe inequality due recall cm ct second rearrange eq inequality become similar asymptotic select strongly believe way away zero allow asymmetric integrable thus word kl accumulate follow accumulate independent depend additionally select pd pd imply consequently kl follow additionally select pd threshold select pd follow weight scenario test density quantify similarly q uncertainty alternative require testing probability eq side expression determine cardinality error every I ie choice would favorable particular prior rank sense I assign relatively ratio prior setup assume embed simplicity resp performance channel use set channel strong present channel high strength check accuracy compare realistic setup table emphasis signal set choose threshold whereas c three column compare two experiment level whose go indicate particular since also sharp type probability h c remain figure probability dash line triangle circle see asymptotic moreover identical present case probability scale dash line asymptotic whereas refer triangle resp circle resp work detail simple composite discrete asymptotically appropriate optimality divergence favorable simulation experiment
number consequently expression proof di nonparametric prior derive letter sample science page institute mathematical california de poisson stable ann cm secondary appear parameter rely approximation central unconditional approximate estimation specie inverse prior introduction exchangeable infinite exchangeable integer exchangeable function backward identify arise extreme namely conditional evy stable stable q partition exchangeable gibbs correspond specie al typically lie diversity specie unknown specie et al assume realization exchangeable belong predictive conditioning stand huge amount nonparametric implementation mix stable obtain explicitly mostly explicit alternative generalize mixing hence exponent rewrite change combination burden implicit prior additionally predictive specie ratio term increase look therefore possibility gibbs coefficient rely non central number approximation posterior rely apply finding approximate gamma approximation easy task aim generalize number combinatorial coefficient term hence number correspond partition sum generalize number block exchangeable gibbs arise poisson eq generalize number limit stable partition almost limit example proposition agree approximation enough notice pd h polynomial stable correspond q introduction predictive bayesian nonparametric species weight specie number specie exchangeable namely number et central generalize number convolution replicate previous section look exploit central follow approximation hold central number central rewrite gamma exchangeable gibbs stable proposition drive generality box new block additionally et eq poisson square new specie conditionally basic observe intermediate even formula poisson generalize large size would approximate generalize idea reduction report formula nonparametric prior exponentially letter diversity gibbs partition proceed conference computation complex bayesian prior st neutral species exchangeable triangle unify generalize control reinforcement non nonparametric
numerical paths ridge concept implementation work purpose rule example manifold circle although circle example aware cloud find pt web clutter difficult reach point four phenomenon bias homotopy apart piece suggest great require reach intersection corner violate far component ridge converge relate currently investigate question first establish procedure adapt mode theory work allow corner intersection develop extension sequentially remove shift investigate adapt investigate run result purpose lemma recall hessian onto q let denote dimensional normal enough induce distribution density lebesgue measure bt da define quantity bx x lemma p bx p x xx z xx x e w dt finally z ok x ok part loss coordinate span span xu z x ok b xu h l result turn claim dx dx calculation order restrict change dt similarly acknowledgement suggestion improve thank suggest simplified theorem support nsf national grant dms air grant fa problem ridge cloud density manifold numerical accurate many problem intrinsic low existence cluster inference research commonly unfortunately manifold even assumption homogeneous meaning attain offer define set essential underlie open space estimate density case density dimensional manifold finding cloud much like goal hyper ideally ridge focus point cloud definition ridge define hessian projection local maximizer direction hessian yet think analogy hessian hessian space direction show circle smooth ridge coincide convolution fact bias uniform ridge indeed mode main well estimate polynomial estimate paper many propose make definition concentrate near ridge provide understand hyper ridge kernel paper informally together ridge hausdorff distance define hausdorff boundary ridge ridge treat practical density instead assume estimate finite algorithm accordingly extension call ridge apply dimensional ridge cluster manifold relate reference therein field work include paper visualization exploratory issue history global locally definition relate appear literature also structure sensitive method paper ridge rather use suit study generally vast shape representative throughout value may section hyper section start point cloud five derivative density ridge far subset convolution common assumption logarithmic valuable ridge deconvolution datum draw additional clutter point noisy consist thought rely hessian q eigenvalue eigenvalue write column write vx vx vx space call local normal span l lx tangent space flow line curve flow intuition pass toward unlike path toward path project algorithm discrete step interpolation approximate flow map respect flow gx ridge curve satisfy first critical critical one regular point call definition ridge thus lie dimensional ridge record main denote radius exist jk space ridge intuition mode consistently derivative instead positive analogous imply direction size ridge background version symmetric eigenvalue eigenvector correspond similarly symmetric square symmetric q examine show convenient gradient path parameterization often subscript tf derivative write formula derivative arrange derivative integral unique unique path condition suffice show show differentiable rule whose th near establish ridge formalize notion function drop along move away ridge want emphasize point immediate ds ridge side hence l l g eq l il sg l l g l sl l sl sl sl se ds l h follow etc fx hold r decomposition small clearly enough gx gx gx lx lx lx gx gx estimate hide manifold bandwidth ridge define assume second derivative vanish boundary kernel density usual result law result prove error prove use law g jx kx kx h h assume q suffice purpose theorem next section let role take tend rate without boundary twice usual way chart away add subscript want ridge surrogate show subset neighborhood property lose corner gaussian fact let surrogate ok k mr would hausdorff near neighborhood manifold gaussian paper mixture see nearby mode highly mode think fact mixture refer ridge function around several surrogate neighborhood whereas preliminary
row signal model observe zero support far average signal denote theorem support notation dependency recovery together constant capture tradeoff understand quantity mild linearly indicate benefit new meanwhile characterize role limit measurement cause drastically special signal identical merely however properly certain condition support wireless communication block contribute non theorem instantaneous capacity conduct random block diversity representative trial trial matrix choose noiseless successful trial study intra algorithm size block block zero entry model coefficient assume range value adaptively intra completely ignore result clearly show correlation exploit exploit performance unchanged phenomenon correlation exploit correlation perform exploit correlation degradation conduct noisy experiment treat time column increase start column nonzero zeros nonzero ar snr treat exploit smoothness short smoothing inter l true variance learn vary concatenation multiple measurement vector exploit correlation compare correlation non review intra model intra measurement framework incorporate level limit application involve endowed exploiting improve performance nsf grant correlate phenomenon exploit intra exploit intra exploit correlation properly exploit essentially operate operate exist reweighte estimate previous iteration algorithm let suggest strategy intra block correlation iterative reweighted assume variant framework framework regularization extensive computer performance two framework ability high exist domain break bottleneck wireless make correlation latter inter become properly exploit family performance degradation use approach iterative reweighted type detail lemma work among value entry intra vector correlation sparse inter correlation context measurement model sparse present incorporate limit recovery discuss impact intra inter limit attention development measurement mn measurement many application structure uniformly dependency application exploit view concatenation block block model attention application general nonzero motivated appear detail problem typical vector row impose mentioned structure correlate challenge consider case call modeling zero algorithm correlation intra intra block basic assume multivariate distribution tend due relevance thus block capture correlation principal scalar estimate type maximum procedure likelihood hyperparameter exploit well partition th block consist issue structure stack thus kronecker overall th capture inter capture time natural vary assume stationary situation abundance option adaptive kalman give signal vary algorithm localization event stationary segment series certain duration model multivariate signal certain statistical signal deterministic assume dynamical
number ever case unique ji ji recall active concept derivation supplementary material share among corpus ibp exchangeability current allow write factor histogram concept unique concept follow et proposal material sampling share e particular exchangeability p ji ji ji ji straightforward mm induce nest compare concept learn hdp hdp collection number distribution vocabulary corpus record health care mark two year either house statement day select document correspond active measure removal name entity noun token appear leave vocabulary augment corpus take sentence co occurrence project approximately sampler choose comparison select probable sample introduction concept find coherent focused hdp word evident histogram fewer illustrative incorporate induce call drug benefit order quantitative conduct user idea cloud hdp choose pair coherent ask ten vocabulary salient generate remove word question identity participant hdp correspond advantage model example coherent concept hdp prefer concept hdp topic preference subtle understand american participant cloud consider participant claim follow closely understand preference study illustrate birth death proposal ibp concept birth call draw unique concept force birth proposal j respectively birth otherwise maintain hasting rewrite definition plug simplify propose acceptance replace therefore value sample proportional concept aim concept specify distribute gamma construct purpose work directly dirichlet maintain infinite multinomial concept assignment frequency conjugacy multinomial abuse notation utility dirichlet along draw j imply random variable ji document observation conjugacy conditional distribution corpus semantic derive material simplify follow document ji ji follow sample ji ji ji ji f denote dimensionality semantic normal wishart matrix covariance respectively diagonal independently write likelihood analytically likelihood standard normal wishart conjugacy yield unbounded restrict attention noise although concept equivalent simplify across document presence indicate absence gray identifiability illustrate generate underlie infer definition concept reconstruct additional vocabulary recover concept filter user participant sure health care participant ten question result favorable vote hdp ask many participant prefer treat sample tie loss binomial binomial test initialize sampler semantic feature concept add remove hyperparameter term representative concept concept sparse structure associate infer inherently concept standard hdp direct feature demonstrate utility representation nearly member researcher million scientific article try read information ir evolve move search remain representation ultimately atomic consequence task set document representation must expressive fine grain name mr distinct name entity th united may end coarse grain representation topic problem individual estimate interest concept seek appropriate represent computer vision community concept model model interpretation concept incorporation ir system vocabulary occur underlie semantic concept differently document set address property rely strict concept uncertainty encourage instance say assume document create refer topic inform undesirable option expressive inherent specifically concept nest beta vocabulary benefit source example vector sentence information text lose bag formulation additionally vector information word feedback weight word assignment concept corpus encode semantic similarity membership mm model topic vocabulary traditional example top hierarchical hdp salient use health care package additionally concept obtain approach contribution use beta concept incorporate concept formation mcmc include study record showing efficacy approach wish unbounded concept probabilistic prior mixture cluster concept coin particular attribute particular flip coin assign concept formally know coin bias concept document exploit know write base mass finite expect realization formally beta define realization poisson rate base improper distribution measure contain prior active conjugacy predictive assignment concentration ibp ibp draw bernoulli infinitely concept document customer select customer share concept absence concept level document lead ibp customer concept word happen customer previous select come vocabulary ji z z z x k nx mm coin bias process draw level beta process assignment word concept indicate concept idea discuss note nest weakly combination entity encourage concept component close correspond sec avoid incorporate sensitivity way maintain exchangeability computational document collection word concept entail identify concept indicate ji binary nest beta feature assume discrete come vocabulary presence actual text document multinomial distribution hadamard product word multinomial distribution importance base provide issue sec concept contain word address semantic describe cf illustration fundamental semantic identifiability explicitly vocabulary allow help guide representation provide efficiency much want concept co occurrence count g sentence basis work study distributional know company keep semantic sec semantic often occur semantic explain concept consider correspond semantic likewise column per simultaneously car value feature concept weight j ji set inclusion responsible meaning incorporate independence
write equality bound operator order v x vs therefore lemma prove claim follow multiplicative lemma establish eigenvector perturbation eigenvalue k ki ia ki eigenvalue aa notational ia k c triangle inequality give cauchy schwarz follow third norm bound eigenvalue simultaneously subsequently column k matrix obtain column matrix invertible satisfies lemma claim define I inequality spectral r rearrange claim multiplicative j v project vector vector distribute obtain bind proof establish gaussian clear statistic true magnitude simplify norm use property eq rule verify mixture satisfy nk note handle idea coordinate dimension remain check matrix km ks hadamard large axis roughly speak non vertical block satisfied partitioning coordinate induce roughly row value call partitioning view preserve distribution view similar cca require separation contrast put least matrix selecting row index together union submatrix e ie chernoff let u eq require concentration multi clear conditionally recall independent therefore effectively bound statistic subgaussian definite standard theory decomposition decomposition invertible invertible n fact together see case matrix l x show mathematical identify result document identical case simple satisfie rise triple differ lemma theorem fundamental treat current practice search scale broad dimensional include gaussians gaussian model method rigorous simplicity offer tool apply machine treating generate identify important include mixture maximum em drawback practitioner alternative maximum approach method back pearson order several equal adapt variety intuitive guarantee mild high dimensional determine moment accurately broad class high dimensional model result estimator implement algebra eigenvalue decomposition estimate low involve explicit indirect variable determine view hmm present future view axis align gaussians coordinate non redundant mild work relevant due variant offer em mixture vast thorough find text advance science decade comprise work learn subsequent provably gaussians roughly deviation gaussian separation expect application spectral projection enhance separation develop grow dependence show related thought modern literature method technique decomposition pearson root computational present notable exception situation evolution mixture polynomially mixture component success transition matrix work remain space approach mixture hmms component dimensional second moment know discrete moment demonstrate moment moment method problem availability exploit mixture separate view mixture separation direction projection availability view remove separation entirely gaussians study degeneracy view text video linguistic datum paragraph predictor semantic example warm simpler wise wise exploit work view method moment mixture document explicit algorithm correctness efficiency view mixture gaussians hmm appendix inner integer context suppose partition document word independently multinomial topic corpus vocabulary document short accord multinomial vector draw independently accord specified encoding become conditional depict degeneracy document prevent distribution whose tensor triple identification coordinate expectation view di view involve parameter triple via operator assume matrix invertible invertible probability distinct every topic exactly geometric observable xx algebraic transform vector observe notable consequence perform transformation xu mb xt eigenvalue equal non zero probability every exactly observable suggest approach base relate focus estimate estimate handle estimator context general frequency triple arise moment concern triple v matrix moment tensor lemma straightforward generalization matrix invertible observable kb scale vector nevertheless carry possible reconstruct kb eigenvector whose entry system element th polynomial observable template decomposition crucial distinction polynomial involve involve lemma suggest estimation b present average independent copy similarly matrix orthonormal corresponding vector pick invertible km euclidean u ki u view estimator care depend abstract scalar technical splitting discrete technique hold gaussians present exist pick rotation matrix manifold permutation addition multi view framework hmms technique parameterize observe multivariate covariance mixture covariance j various dd j axis align diagonal partition view randomly hold require v column cluster degeneracy show separation efficient comparison minimum recover concern condition slight recover pair matrix finally gaussian still concatenation condition requirement matrix covariance full meet even component markov latent model state form state conditionally consider homogeneous vary describe hide state readily handle handle restriction hmm identify independent give u therefore recover observation matrix recover scale zhang discussion comment david anonymous pointing perturbation appendix also illustrative modify implementation empirical copy frobenius norm euclidean claim let union mu invertible imply let permutation j bound feasibility modify begin third middle end article stop instead single extract eigenvector automatic heuristic leverage score c bit save run
view actually conditional hmm consecutive recover emission spectral view state come hidden organize section short view view state simulated illustrate correctness effectiveness view section short summary view vector target variable take nlp view word topic mention property word connect dimensional response view view view htbp pair view linearity assumption lot fall example widely nlp hmm observation gaussian edge edge h often word english vocabulary lot easy internet getting lot human observation lead briefly go though simplicity mean since view refer conclusion view identity introduction cca svd correlation canonical theorem claim aspect view together drop uncorrelated group predictor hilbert predictor onto assumption span covariance span span projection onto therefore optimal predictor dimensional cca regard dimensional order view feature uncorrelated help view hide get hide view illustrate content help classify find combine hide new available assume view cca square optimal dimensional optimally everything subspace span predictor predictor canonical random canonical component predictor subspace indirect cca actually cca rotation I variable moreover know cca first last subspace easy svd column reconstruct find follow except tx th tx space lie pick canonical direction I denote moreover q canonical lie direction cca last direction hence satisfy rotation last canonical similarly covariance canonical convenience q full rank column step step rotation matrix compute find optimal dimensional subspace dimension view cca view reduce algorithm reduce cca dimensional huge contain useful information reduction space feature cca normal normal view predict three dimensional three predictor label linear one time rotation trial square learn large square large h second sample size view amount experiment square loss average well sample square loss experiment advantage view label linear decompose view moreover variance due predict amount size different square plot right label view feature become small cca multi view assume carry achieve dimension view need disjoint part act view
let past l l bn remove parent minimal joint parent minimal negativity kl separately parametric generative graphical direct process node might exist causal strictly causal satisfy positivity violate graph unique process process edge equally minimal generative unique causality identify causal influence adopt wiener apart past use formulation work direct name causality condition recall statement causal prediction cumulative loss sequentially predict thus direct influence process process direct iff graph immediately graph case exist arguably bad ambiguity generative graphical direct minimal model equivalent make correspond pair search first help correctness satisfy parent bi bi iff equation parent accurately learn bi ai identify parent require determine satisfie potential parent require exponential direct alternative motivated chain trivial sequentially remove condition generative proof appendix alternative direct graph separate test entail direct condition process condition direct ai ai ai algorithm recover algorithm element line execute parallel number graph dimensional dimensional statistic demonstrate involve recovery next direct algorithm network identify parent pairwise test etc small degree relationship exclusive counter process ai ai ai return proof increment even degree recover minimal dimensional minimal bound degree result graph optimal show direct involve sufficient identify parent show parent intersection parent formally sized subset degree ki ai return return parent trivial algorithm parent information test q question output show modify return validity marginal exact parent divergence form approximation direct direct sum direct along specify degree let follow case ki specify however use exact direct information interval recover point ai reliable ai develop find good estimation confidence practical simultaneous rectangular multidimensional denote ai scenario value ai ai parent error characterize I approximation well given perform poorly different scenario attain minimax regret input b mb b b lb mb lb b return confidence identify robust approximation ai thus construction alphabet plug second parametric jointly result jointly direct process jointly stationary markov finite markov pairwise network jointly direct couple network irreducible notational simplicity indexing start history assumption stationarity definition jointly plug j constant event examine sample characterize order compute empirical direct estimate time sample appear biology field complexity network p estimate mle analogous interior p hessian q pg twice maximizer iii extend parameter vector jacobian entry singular multivariate delta specifically information pair continuously jacobian dependency different direct occur nonetheless joint matrix sample unknown calculate confidence separately calculate sample confidence markov coefficient use trial trial parent non scale magnitude distribution pg identify direct parent child compare parent infer dynamic capture parent estimate least square fit entropy avoid used length number complexity n initially parent resolve except parent parent test parent respectively show algorithm identical degradation dynamic misclassifie edge present computed version setup section degree draw normal stationarity show proportion dynamic increase monotonically percentage edge identify concave peak parent parent parent parent identify identify next plug simulated network binary value node limit bias respectively pick bias round example bias color correspond figure perform almost game game amount use use resample parent replacement mean direct information width normality interval perform algorithm percentage infer game bar random would thus quickly suggest robust utility infer news user online micro covered event middle east analyze precision correctly proportion influence proportion randomly influence depict roc show average algorithm approximately non influence tn baseline user single even parent influence correctly tn variation amongst reflect monotonic decrease precision truth graph c acc alg alg alg alg visually depict axis overall influence precision baseline network significantly research social sciences economic biology widely meaningful interact underlie case reliable demonstrate utility news influence direction future research estimation technique direct information several rich well characterize another graphical estimation assume likewise assumption depend past biological network graphical handle topology beneficial assumption strict causality relevant observe section strictly causal influence influence need causal nonetheless non x linearity iterate expectation divergence joint natural strict causality minimal instead direct parent algorithm correctly recover direct repeat causal causality corollary ensure hold state direct model corollary parent p strictly closely p chain correlation mutual notation correlation condition close lastly identify statistical dependency causal use linearity rule mutual strict causality direct propose causality direct form causality preliminary set causality predictor conditional causality direct far beyond strong causality conditional sequential setting conditionally help thus want strong anchor capture causality causality setting term side help sequential consequently precisely quantify causal predict sequential sequentially form full past specifie know except simplicity mt l help decision combine time simplex hellinger could discount manner prediction measure form causality causality logarithmic cumulative expect cumulative minimal test compare model error linearity expectation focus show condition direct information j continue past give direct hand non negativity use notion reduction tx x predict cumulative loss causal side predictor probability interpret quantify strength sequential demonstrate sequential special axiom alphabet direct I prove causal pc property lc satisfy pairwise property likewise lc lc generative lc invoke extend condition place turn induce direct information contradiction j ig mean I model direct yield non chain chain way bi zero consequently bi I occur equality use chain add zero reverse valid generative parent minimal ai ai ai ai recursion parent set return contain set suppose current inductive base hypothesis trivially inductive hypothesis rule lemma set true holds remove remove algorithm parent show direct involve let process process exclusive note argument three return parent first contradiction parent evaluate necessarily j j ia ai ai execute line case evaluate parent chain parent loose maximal ib j ki bb jj lemma thus appear ba empirical hoeffding entropy translate entropy direct estimate measure error direct event realization hoeffding four realization order pair next correspond denote concentration l evaluate entropy maximize increase bound entropy j x analytic bind attain aa increase maintain probability increase several condition check define information assumption alphabet q evaluate rule move linearity plug proof condition specifically simplify addition normality assumption form sequence follow derivative finite definite iii assumption include matrix th r linearly hold column orthonormal matrix inverse column direct error wise analogous variable denote normalization normalize volume respective integrate normal rectangular specifie corner positive volume eigenvalue volume constant two coefficient repeat form imply grow parent parent maximize bring scenario independently hold rectangular irrelevant optimizing hold identify parent regret apply ks line parent lb j either add side return thank inf j computational science fellowship de er support grant fa fa nsf grant propose represent generative base distribution information causality quantify sequential develop bound use event valid near known certain criterion analogous structure undirected use direct characterize plug direct information confidence point estimate lastly analysis real twitter network influence analyze causality generative direct economic social involve interact agent neuron stock computer expert seek computer infect time activity price traffic work influence agent question whether influence agent want user micro twitter many several news wants identify strong influence decide pay activity want influence reliable compute indirect influence distinguish influence user sure influence influence issue propose graphical agent depict influence show information generalize causality one influence direct connect principle beyond strong causality use independence network nod million graphical prove correctness infer graph degree optimal valid return bound adaptive early instantaneous influence influence identify reliable estimation interval plug empirical estimator propose identify influence activity news account middle knowledge accurately infer influence organize follow discuss make algorithm identify exact estimator demonstrate algorithm contain direct graphical direct communication channel direct
lr test denote likelihood parameter table two model repeat item group implement package item lc restrictive item equivalently dendrogram show lc obviously cut dendrogram several value correspond lr suitable multidimensional lc section software follow speed analysis aggregate record pattern record configuration distinct label index provide configuration among distinct multidimensional trait list datum apply original code response k latent start deterministic start user point parameter lc specify link difficulty item difficulty row equal index item measure correspond dimension latent class estimate vector item array response class lk aic bic index outline two procedure pair perform item dim class item est est multi specification dim specify multidimensional model multi specify multidimensional default multidimensional structure dim obtain est output est lr df lr merge item collapse list sort represent hierarchy dendrogram statistic lk maximum result aggregation np list hierarchical merge height use package application illustrate item consider ordinal illustrate choosing class logit cluster mathematic testing service national progress replicate example aggregate record class pl link display dendrogram figure detailed object height first collapse identify statistic aggregation aggregation concern define group item item item aggregate visualize also type lc former detect global logit link carry lr item response structure response structure item test dim firstly point link dim model freedom test link dim constrain unconstrained degree freedom coherent perform adopt cluster grouping item outline presence threshold item application four response type properly parameterization difficulty difficulty know constrain difficulty p parameterization account logit parameterization basis sake likelihood bic provide est start rs est multi rs vs lk df np df p lk df comparison lk df lk p vs np rs lk lr rs analysis rs fit four response free taking lr parameterization dimensionality accounting criterion indeed select logit link constrain free dimensionality logit logit c free constrain constrain constrain patient suffer mostly first belong severe patient condition outline selection binary logit link difficulty initially class extend trait trait introduction latent detect notable simplification case trait characterized multidimensional treat formulate treat ordinal item different link provide order maximize contribute specific lc base likelihood ratio tool useful nest multidimensional lc availability application collect service implement cluster concern scale ordinal subject class explain type reject favor measure provide package multidimensional class package deal name traditional trait also trait depend type model estimate package discuss selection illustrate concern refer aggregate describe fit keyword trait deriving make trait case trait unfortunately restrictive traditional flexible extension take one trait among main contribution example multidimensional wide see overview topic another advance literature latent characteristic context introduce certain number patient treatment secondly trait maximization way multidimensional characterize assume multidimensional comparison traditional formulate discretized variant class represent mixed ordinal mixture concern mention extension multidimensional lc trait latent trait random subject individual moreover parameterization item diagnostic rather interesting multidimensional base continuous latent trait perform goodness fit parameter require aim lc ordinal package issue trait rely link allow response cumulative local category link function presence possibility possibility discriminate differently different difficulty special level category basis item extension traditional introduce trait procedure aim type parameterization dimension allocation model output package estimating model neither trait among package multidimensional trait propose package application collect service within assessment progress binary homogeneous group trait concern ordinal patient characteristic latent trait follow propose multidimensional lc devoted devoted illustrate implement analysis model specification constraint response item item latent trait latent trait realization discrete latent trait latent trait category express category parameter identify difficulty suitable belong specification three link base local first compare category moreover category previous category model see item coincide global typically trait contrary intermediate level award reach intermediate model model base item may discriminate discriminate item category share degenerate response also observe logistic pl refer parameterization difficulty item unconstraine ii special constrain difficulty category rating parameterization case category item difficulty category combine obtain specification member eqn free rating scale cm item level cm accord multidimensional derive latent trait know traditional multidimensional lc multidimensional lc rating extension equation multidimensional pl multidimensional lc due independence contain measure trait denote identifiability constraint latent trait element moreover free item rating identifiability lc obtain probability free parameter item depend type parametrization difficulty class probability equal ability however difficulty unconstraine parameterization parameterization unconstrained ht difficulty free express category extension item response zero zero zero element equal indicate concern way moreover logit link logit vector suitable find ability single taking difficulty explain unconstrained column constrain constraint accordingly suitable column scale parameterization remove constrain accordingly remove deal likelihood deal aspect mainly concern observe formulate propose may express free
discretized spline rw smoothing rw rw rw derive solve stochastic sde resolution spline equally good computational aim extend rw carefully smooth sde representation explicitly exist method adaptive smoothing convenient understand continuous satisfactory adaptive intuition build correspondingly rw knot spline equivalent improper sde dt wiener refer sde ft du improper ft e sensible straightforwardly carry method unfortunately prior intensive dense sde note provide good overall product denote choose element ft let common choice piecewise linear h jt j field interior determine continuously finite substitute linear dt weak sde neighboring basis give modification dense computationally expensive show change replace give straightforward result cubic smoothing spline expansion approximate show approximation depend enough function describe actually sde cubic derivative however aware global function theoretical sde spline flexibility straightforward spline control smooth present sde formulation sde td ft dt smoothing function compare global spline second smoothness spatially smooth spline ny ft dt piecewise derive reproduce hilbert computationally intensive reproduce dense case achieve dt leave side proof side th h discretization boundary affect leave corner h h h note involve sde dt instantaneous quick decrease adopt formulation minimize ny ft dt also dt write define appendix mean h h n n take smoothing assume differentiable proper smoothing intuitive sum previous link sde dt use dt dt leave right remain variable full approximation numerically integrate laplace work hyperparameter case use reduce rank vary sharp peak highly gaussian add true internal knot add example spaced noise ft adaptive spline estimate ordinary smoothing spline square eq sde fit second sde knot knots median square table slightly estimate slowly smooth significantly peak spatially different yield sde offer inference mcmc number well inferential knot capture efficacy software reason space error acceleration draw minimum illustrative concentrate general gamma regard shape scale heavy sharp peak spline adaptive technique cauchy fit credible cauchy cauchy adaptive sde mcmc performance give drop back model attractive adaptive smoothing smooth drop compare cauchy offer credible wide capture opinion cauchy error well overall cauchy paper unify smoothing spline base smoothing spline sde easily adapt element smoothing spline demonstrate effectiveness application proof use adaptive sde hand entry dt tt tt dt th tt dt dt dt dt tt dt I tt I I nonzero entry happen dt letting write dt dt sde linear system jt nonzero tt dt tt tt tt tt similarly dt dt I previous I dt tt nonzero dt dt n dt entry approximately boundary diagonal matrix define spline popular evidence support partly elegant firstly become prohibitive number roughly secondly amount smoothing function class adaptive smoothing certain differential drive smoothness markovian make study example demonstrate keyword adaptive monte spline partly support partly elegant mathematical I I function index spline degree smooth sum derivative cubic smoothing frequentist view reproduce hilbert bayesian yield partially improper restrict spline become computationally obstacle stem smooth underlie spline set typically impose point knot spline unfortunately mark penalize p spline require spline note distinguished basis penalty penalty compete truncate ridge penalty spline gain statistic implementation formulation
confusion dictionary advantage sc report qualitative sc independent hand advantage indicate sc outperform sparsity become transfer employ pixel handwritten character digit capital follow experimental instance possibility train task learn target task approach dictionary task set vs task case instance character per tune hyperparameter create process describe task associate train target run multiclass accuracy task character character learn possibility learn experiment assume digit compare show dictionary miss pixel per image employ pixel explore code widely process justification insight bound hilbert depend quantity intrinsic datum code promising improvement compete would valuable vector could arise structured norm lasso g family regularizer mind qr k encourage code group code divide dictionary hilbert task express acknowledgment part grant ep international joint dictionary combination transfer principled bound generalization setting real dataset advantage grouping extension code approximated space quantity principle bound numerical real dense decade development linear guarantee performance reference therein atom dictionary free principled choice exist representation underlie benefit individual justification new task automatically task environment task contain learner task demonstrate give single concern subject considerable attention idea meta dictionary support predictor atom dictionary sufficiently task dictionary together justify analogy sparse coding unsupervised replace observable correspond combine coding list incomplete transfer therein common perform generalize present first coding provide learn learn present connection present practical also address problem learn demonstrate utility set learning apply thereby hilbert organized section present bound numerical experiment conclude turn technical exposition notation way hilbert integer dimensional dictionary canonical regard code implement assumption set could readily ti ti value tm tm represent predict label lipschitz value atom vector minimum section present experiment generative model standard distribution well minimization always specify rule assign I propose frobenius use employ use place depend bind denote redundant eliminate restrict learn learn setting interpret context task training ti input upon return minimize th task error minimal analogous risk bind input state implication appear average square suppose surface dominant extreme datum bad sense little dominate dictionary atom could bind noiseless generative number minor increase suppose task generate model list dictionary extend disjoint correct total relate cluster sl ball radius bind sl apart sl outperform utilize correspondingly task sparse superior theory employ crucial end regularization scheme frobenius place quality may poorly perform set environment context transfer risk simply draw sample test read loss incur predictor linear adjoint observation task draw simplicity fix sample large independent datum retain dictionary environment quantify compare well minimize achievable result ts implication interpretation remark imply behaviour dominant denominator excess theorem possibility task possibility appendix risk task say term plausible discuss example linear model vector input assume uniform sphere haar measure task fully ball draw combination square reduce code quantity choice order bad code experimentally section propose code multi rr base line versus
stays correct apply mean th line c stay graphic
graph assignment run side part propagation offer would belief allow recall comprise additive multiplicative satisfying multiplication identity multiplication multiplication requirement meet usual arithmetic identity operation backward property preserve rise useful call next far carefully allow exponentially follow four tuple property weight multiplication observe component multiplication leave result fourth component thus belief propagation second perform need perform message important remarkably suitable rise tractable normalize marginal efficiency belief configuration graphical generalize inference sufficient normalize time pass compute diversity q assignment pass run compute though note pass perform second second order significantly modern processor tuple use follow operation easy computation scalar normalize marginal structure alternative marginal structure single assignment ask assignment efficiently message pass form therefore computable marginal simply separate runtime single factor run pass obtain everything compute require make sampling practical familiar marginal probability run take pass marginal offer efficient sampling show naive iterate conditional marginal efficient integrate part return belief propagation assignment singleton create new message assignment since see unnormalize assignment example assignment go handle maintain assignment hold naive algorithm run might expensive run extend input factor run arrange belief propagation pass proceed backward receive message conversely message message backward pass independently execution factor variable belief suffice compute part send full unconstrained forward right assign forward leave message likewise send variable unchanged backward computed illustration mr thick dash yshift yshift cm row sep sep minimum circle draw minimum cm draw minimum minimum cm circle minimum minimum circle f minimum cm minimum edge edge edge bend bend bend bend leave bend bend right swap b bend bend bend bend right swap b bend swap bend right swap swap message need circle assignment propagation loop essence complete forward backward pass linear message assignment send hold imagine root subtree root distance draw circle child node child node child c child child child child thick fill subtree anchor sep subtree north east include disjoint immediate tree neighbor weight node message neighbor incoming message message assignment subgraph claim message neighbor belief propagation assignment go proceed part sample incoming message set current message message take account neighbor claim visit visit know walk entirely assignment hypothesis neighbor know assignment conditional complete walk start proceed depth message thus procedure second traversal fixing assignment final piece replicate dpp number eigenvector first asymptotically dot tb b leave paper per total similarity cite paper importance weight cosine dot normalize tf define eq document filter remove common appear rare appear filter word score stationary thresholded cosine plus term centrality score design diversity cosine zero imply correspondingly dot feature dimension behavior sample offer type examine illustration cover topic apart visually salient locally control discover multiple tc thing explanation mobile database mobile server organization handle mobile wireless challenge management length subset project tf centroid word high paper two news comprise york times collect part corpus article article corpus list number discard article deviation six month period article article cosine article six cosine furthermore go enforce document feature feature control make one model maximize half project report provide resolution neighborhood article annotated reflect position article publish digital read likely online fig visualization article article entire million place figure depth exceed modern instead upon display visually ranging leave sample color compare baseline method task month period sized slice similarity cluster basis article slice slice average cosine coherence repeat clustering exhibit article successive slice naturally align attempt smooth publicly fit set slice topic topic randomly produce baseline obvious distinction always span nearly period select one dpp tend continuous news explicit relation collection risk health raise ability home though ill world choice meet may message name salient single breast disease patient benefit heart breast patient health black breast patient news topic model half salient word quantitative evaluation generate baseline news summary summary approximately daily news summary france corpus distinguished tag summary cover news dataset gold summary month tf evaluate obtain vector cosine percent hyperparameter metric validation describe automatic variant six produce summary cccc sim summarie bold high verify bootstrapping rating average score average article identify bold distinction baseline former orient choosing approach orient article distinction see online complete task ask first article totally single clear rate average column rate coherent mean poorly since time slice ask implicitly ensure enter display article ask thought remove improve coherent average right task tf advance second include baseline fitting projection matrix sampling core summary mean offer possibility wide practical application suffer correlation arise conclude briefly mention question shannon dpp q strongly currently useful quantity remain future asymmetric encode dpp dpp inference constraint set might learn dpp quality approximately use parse translation corollary theorem conjecture chapter section proposition conjecture probabilistic arise physics traditional model hard negative efficient marginalization conditioning focus extension learn community apply informative summary sentence pose automatically important news modeling learn analyze discover process lead space goal make introduction combinatorial impractical tree approximate interaction intractable offer arise physics theory elegant correlation offer marginalization inference extensively rise deep new aim focus extension dpp fix result database dpp see correlate make inclusion item strength correlation similarity item co occur assign item dpp prefer distinct query focus salient diversity place work diverse information retrieval unlike diversity context kind interpretation query topic alternatively item exhibit quantum particle dpp dpp serve metric weak large pose dpp final application summarie news choose diverse sentence diversity image return google pose task improve human image incorporate bias toward non overlap news task automatically extract news balance intra coherence inter background along modeling extension theoretical result aim enable material begin introduction tailor interest focus discrete simplify proof description efficient describe decomposition dpp fundamental tradeoff quality diversity compare expressive representation show still provably requirement efficient quality diversity news allow item practical expressive dpp dpp experimentally structure number doubly diversity inference pass structured toy pose process process quantum anti effect year give matrix attract mathematic make formal combinatorial probabilistic accept survey overview relevant process pattern interval record output characterize spike spike tend occur might tend spread correlation finite point without extend continuous simple distinction document simply measure empty unnecessary nonnegative eigenvalue use sufficient dpp marginal observation singleton element dpp diagonal element tend co demonstrate think measurement element unlikely perfectly similar conversely independently element co occur correlation draw dpp sample process set plane relate spread independently paper focus world receive much section briefly example remarkable whole technical uniformly say location less process previous probably thus likely position decrease independent symmetric walk integer walk let walk begin position fact intersect dpp walk intersect far apart edge span spanning distribute edge begin orient arbitrarily nearby likely uniformly demonstrate assign zero cardinality particular case minus spanning issue subset dpp hermitian draw entry normal unitary dpp shape square half square color gray suppose known subset b cover vertical subset distribute dpp data symmetric whereas atomic every semidefinite eigenvalue less normalization follow zero everywhere else trivially hold cardinality less give split partition write column whose inductive term give shorthand write write measure diversity prefer q reasoning marginal intuition l kernel obtain zero except eq rescale conversely dpp l ensemble invert inverse dpp inverse equivalent give restriction reason nonzero rare cardinality dpp limit ensemble describe offer alternative l ensemble atomic offer appeal furthermore need effort interpretation always square intuitively describe measure use product say volume view span magnitude probability similarity increase intuition verify probable vector orthogonal span volume feature define else magnitude feature appear multiply span volume decompose kernel direction magnitude section dpp distribute dpp kernel dpp kernel may seem complement encourage dpp assign marginal probability hold constant define take dpp element bernoulli immediate generally expect plot ensemble include infinite appropriate uncertain image situation may cardinality front ten desire goal achieve advantage realization review associate estimate large interactive memory gb already partition write matrix well go multiplication use practically computer interactive storage remain extreme marginal set item compute probability see probability configuration conditional dpp ensemble marginalization dominant inversion gauss elimination kernel sample latter practical working complete ten dpp row element obtain simply drop element also dpp one diagonal entry zero everywhere else inversion appear combine dpp appear closed formula allow arbitrary marginal conditional dpp note dpp marginal partial eq q dpp expensive dominant inversion although appearance inversion conditioning expensive marginalization dpp theorem tb input orthonormal loop phase eigenvector produce second previously cardinality random express elementary phase elementary dpp mix elementary dpp phase dpp elementary dpp kernel elementary orthonormal subspace span due elementary arbitrary expand vanish dpp arbitrary definition hand expression reverse marginal agree fix draw dpp whenever almost theorem eigenvector loop select elementary dpp mixture loop sample geometric introduce span span generality span height formula projection onto proceed iteration begin update basis probability choose exactly argument identically select generate visualize influence dpp grid together initially successive select shift avoid choose dpp unit fig particle circle color dpp probability offer cluster eigenvector eigenvalue strength cluster perform choose select identify accounting overlap expensive operation schmidt expensive cardinality potentially save often bottleneck require interactive large minute instance eigenvector since choose kernel multiple desire recently sample significantly sometimes refer posteriori hardness closely entropy determinant covariance find find principal covariance submatrix maximum dpp mode semidefinite input dpp mode hard dpp reduce cover element decide cover constant intersect semidefinite reduction require time value must suppose base product column correspond cover orthogonal optimal dpp hard dpp hard even branch mode heuristic set find optimal hour modern computer order application greedy algorithm achieve approximation poor may nonetheless whenever yield return maximize submodular subset discuss kind monotone approximately maximize polynomial search grow variety way show rise wide involve diverse item physical diversity take sort class probabilistic particularly appeal machine section briefly wide world process perhaps right poisson item come coin setting mean process disjoint generalize least likely since poisson real world address various modification poisson introduce correlation construction make induce embedded euclidean remove radius item space apart ii mat ern design keep begin uniformly small remove item remove lead dynamically early radius inference ern computationally moment compute iii process expensive ern enforce radius choose issue mat ern spirit surfaces cell surface arrive random bind remain like mat ern process core pack much achievable restrict find initially select ern intuitive seem plausible analytically provide framework item energy course without constraint assumption instance common interaction argument lie local term markov markov pairwise process potential piecewise radius otherwise cluster continuous integrable discount general core typical pairwise depend control diversity call union center little item diverse fall twice radius decompose individual similarly however item might appear item interact potential mrf element discuss expressive possibility mrfs simply mrfs intractable even normalizing process carlo approximations nonetheless moment rarely dpp entry sum permutation dpp submatrix appearance index natural generalization rise special case irreducible theoretic character partition element determinant single describe opposite dpp diverse recent paper property detail furthermore process induce computationally matrix work class efficiently computable leverage perform inference beyond computing determinant replace count determinant modeling advantage third originally index operate set additional modeling preserve setting however randomness study estimate practice finite draw poor concern set generator randomness ensure appear even speaking set diverse evenly origin point discrepancy define discrepancy box set cover unit cube deterministic uniformity property unit q discrepancy lead quasi contrast monte stochastic discrepancy offer uniformity generate efficiently seem plausible uniformity characteristic set tool work generating prediction learn possibility point process deep exactly characterize theoretical promise exhibit make intuitive computationally variety fundamental serve extension later intuitive unary decomposition splitting quality comparative expressive power show qualitatively characteristic despite advantage impose super dpp show dpp inference simple projection dramatically bound consider finally formula ensemble analytically ratio concern interpretability practitioner understand dpp kernel totally similarity primary practical want diversity balance preference diversity dpp gram quality term diversity decompose dpp arbitrary high dimensional entry think goodness item sign similarity main potentially simplify independently diversity diversity diversity model choose quality tend quality item quality diverse fail quality combine balanced return determinant volume span magnitude previous decompose objective diversity go diversity intuitive setup exist turn dpp geometry square volume span probability item increase probability contain model offer already task like essentially practical make advantage global negative traditional elaborate expressive compare take representative random field mrf whose purpose mrf variable value denote bold assignment set encode dependence might fact tend co mrfs field parameterize depend mrf clique nonnegative clique constant mrf think characteristic define independent neighbor converse clique mrfs offer intuitive interact encode independence unique clique limit clique mrfs large clique potential clique node potential mrfs clique unbounde inference inference also hard problem likewise probabilistic hard approximate constant mrfs identify mrfs submodular tractable mrfs depend clique encourage take formally least depend pairwise mrf eliminate ij parameterization sometimes visible boltzmann mrf whenever mrfs encourage potential necessary build mrfs diversity opposite potential graph indeed potential cycle inference failure improve algorithm wide informally approximate effectively potential outline mrfs familiar potential directly relationship similarly mrfs come expressive dpp relate binary negative negative correlation model dpp mrf difference individually semidefinite kernel mrf actually mrf induce virtue disagreement see prevent dpp form dpp therefore obey imply guarantee semidefinite trying intuition mrfs dimension difference equivalent correlation start apparent dpp mrf unnormalize per potential unnormalize single see dpp node mrf item dpp ccccc mrfs potential show subset edge mrf dpp set visualize manifold slice four slice dpp slices mrf origin appear gray surface rise qualitatively similar surface mrfs describe anti anti impossible improve constrain slice constraint away mrf slice plot express grey dpp blue mrf surface primarily similarity dpp distant item rather far look distant induce mrfs constrain model data mrf exclude item generally depend express say item naturally restriction inference rely kernel inversion may efficient construction semidefinite diversity become database furthermore identical q thus dl apply quite fact include normalization marginalization sampling dpp normalization equal determinant eigenvalue nonzero dual dpp marginalization course require time q dot product marginal give arbitrary pairwise marginal obtain determinant implement represent orthonormal relationship tb select iv seen allow setting effective case also diversity language reduce maintain dpp random extremely nonetheless distance classic randomly onto dimension preserve volume set point connection volume projection dimensionality dramatically maintain result projection dimension sample probability least span project effectiveness projection restrict portion restriction application relatively formally seem bit since imply conditioning dpp apply work dpp dpp quality dimensional condition dpp project projection say dpp volume conditional must adapt reason low sign definition inequality directly follow combine representation exponentially expand expression nice volume normalization denominator simple matrix dimension analytically derivative likelihood plug identity formula advantage use diagonal positive semidefinite everywhere else identity matrix expression determinant way complement dpp similar asymmetric offer appeal geometric inference kernel prior knowledge dealing set dpp diversity introduce dpp choose student think student tend merely entire article sentence advance separate great even importantly learn nothing generate summary unseen article time place alternatively sentence appear article single address rarely repeat depend input enable share article denote imply input g sentence article conditional dpp conditional assign every dpp efficiently quality quality diversity receive identically conditional kernel parameterize base prediction unseen input learn log observe optimize dpp learn unlikely dpp estimate ascent bfgs must exist optimize fundamentally begin diversity fix support automatically consist desire even long proper score use feature distinct parameter advance ease go single drop extend parameterization show able negative exp expression concave modeling empirical count dpp assign higher diverse compare example diverse attention overcome bias impose determinant see practice sum instead switch summation eq q marginal item appear dpp count per inversion idea efficiently note need diagonal exploit multiplication entire multiplication unfortunately asymptotically irrelevant still dual fast along line recently currently tb instance demonstrate dpp quality document news summarize news news article selection sentence relevant sentence diverse fit document article preprocesse try dpp summary construct place sentence policy unlikely perform justify automatic like sentence order understand cluster use collection approximately article ap cover sentence word use training summary evaluation summary character space depict set human reference summary performance follow automatic evaluation overlap human reference summary sub simple show well f primary development recall turn removal actual recall setup require unfortunately reference summary high summary summary human simple round human normalize add summary precision reference remove reference count character oracle summaries summary reference automatic competition well reference human probably human could summary optimally nature compare believe summary training score summarie human use diversity stop sentence word document article finally per entry word proportional similarity cosine cosine word thus salient feature score good cosine confident remain fix throughout experiment hand training find list feature cosine distance tf produce series bin boundaries determine bin evenly spaced bin quantile current summary small sentence bin character five global bins position document position plus indicate position appear cluster similarity compute cosine feature salient occur frequently cluster raw ten local eigenvector row cosine raw score global bin count may feature include previously unseen task apply inference posteriori high reason primary metric impose character summary thus goal summary budget length simple tb cluster character limit iy u discuss formal generally nonetheless seem bayes decode automatic speech decode specific loss function realization evaluate since exponentially summary expect efficient inference sample satisfy impose sample character outside inference report show time require processor decode randomize lr summary cluster parallelization bfgs optimization parameter system simple merely since consist news article entirely many identical similarity include advantage dpp take diversity contribution impractical properly nearly long short inference employ baseline document perhaps simple relevance similarity measure sentence logistic regression sentence score sentence optimize sentence add refer submodular implementation rely include method test originally result update version highlight stochastic improvement dpp outperform significant boost logistic inference run produce relative dpp perform report contribution performance drop significant intuitively two essentially centrality important removed assign probability dpp position team tends spread team exactly five likely exist elementary fix however achieve focus equally aspect notion fundamentally serious limitation express dpp uniform cardinality dpp may cardinality dpp series trial characterize certain might look may tend diverse position warm might impose cardinality offer search engine diverse mobile user dpp real world conditioning dpp content size simply expressive limited correlation define time diverse image query dpp subset dpp size dpp concern content dpp obtain dpp set cardinality dpp give semidefinite standard restriction normalization dpp cardinality subtle attempt difficulty likewise valid marginal eigenvalue matrix full easy special item expressive quite dpp condition case depend flexibility practical advantage assign possible since effectively situation size hundred nice know expressive increase expressive whether convenient useful case close seem simplify dpp polynomial polynomial examine characteristic property recall dpp apply dpp elementary last two eigenvector elementary polynomial give every eigenvalue formally compute identity difference essentially summation fast numerically stable rely precise number eigenvalue kn inner iterate elementary symmetric polynomial thus every advance since dpp repeatedly efficient two phase subset eigenvector reject sample begin rejection eigenvector formalize intuition dpp whenever decompose elementary dpp elementary dpp dpp dpp accord mixture like task recursive elementary tb input e l nj desire return loop single index induction hypothesis loop begin suppose added loop occur immediate inductive probability return nothing begin observe since begin inductive otherwise iterate overall dpp generate sampling assume expensive sampling dpp normalization dpp normalizing simplify observe side dpp normalization conditioning marginal polynomial whose thus probability standard derive know singleton use item elementary dpp elementary except marginal scale probability eigenvector compute marginal require polynomial fashion would tree leave correspond represent associate elementary symmetric polynomial auto fill black child draw black node child node draw child draw child fill child child child child node background line pt leave interior node represent leave leave combine leaf constant interior use remove eigenvalue polynomial along leave represent leave root tree take necessary elementary polynomial singleton marginal inclusion dpp dpp exclude offer dpp easy still cardinality cover greedy approximation like demonstrate image motivation run search engine goal image user unfortunately search look city perhaps furthermore user specifically shoot expect search small argue least diverse respect another want maximize search return satisfy maximize want diverse natural course evaluate task amazon establish diverse comprise comparative image prefer use whenever correct set task seem work practice give expert advance optimize idea combination receive loss make mistake logistic alternate step project projection standard algorithm create query retrieve top restrict search pass safe average image city great paris category spread evenly across order compare directly baseline method probability full differ single classification candidate redundant set use sift candidate uniformly instance obviously redundant choose usual order decide actually diverse diversity amazon worker label reason candidate result offer candidate order instance category fig label example five candidate right correct calibration instance belong level inherently difficult keep check five instance four agree end half image form block kernel image kernel normalize ensure assume image equally relevant partly justified fact come actual google search relevant function image variant pixel color space color sort align bin either dimension sift variant toolbox sift descriptor descriptor category combine cluster normalize near cluster descriptor process use five center half kernel create kernel every vector flexibility act diversity increase derive training combination tune query run result candidate dpp kernel apply test kernel optimize mixture minimize technique diverse idea set add round weighted combination relevance detail merely small give prediction dpp good apply attempt learn replace metric substitution optimization non dpp practice optimum easily find provide advantage random split significance bootstrapping regardless outperform achieve decision make number rather human obviously significant measure improvement percentage versus kernel actual use dpp mixture category dpp majority bold fig dpp exhibit significant diverse might sift center sift color color center color center center sift high category try cover image draw whether select satisfie virtual generally simply similarity expect well method high similarity average single virtual whose desire dpp dpp indicate result average perform good virtual mixture select ten subsequent drawn fraction fraction virtual user model gold measure expert combination expert result average virtual dpp job cover result significantly see section offer inference ground exponential naive would imagine linear position obvious move
coincide gaussian probabilitie posterior component I coincide associate generate write r I kx kx aim highlight finite regression wrong although well separate polynomial consist cubic report generate ht name visualize beginning represent base gray dash allow dedicated marginal histogram component scatter display group regression separately generate joint via display ht suppose forget classification regression start parameter confirm value adjust rand index ht contrary shall obtain differently mixture estimate corresponding component mm display ari value regression ari impact possible remain scheme range randomly indicate use discrete set generate replication classify ari replication conditional ari improve remain extension mixture polynomial name polynomial justify carry within em framework bic use membership know polynomial apply artificial datum excellent clustering compare future several extension initially concern characterize outlier finally evaluate theoretically separately conduct indirect acknowledgement helpful comment suggestion di mail present cluster bivariate dependency consider base base framework estimate use use bic integrate complete condition posterior probability membership coincide excellent artificial mixture polynomial weight employ statistical purpose indirect nonparametric see hand application original datum biology mark overview focus direct indirect represent bivariate hereafter vector constitute suppose linear conditional regardless shape mixture regression mixture adopt latter use unfortunately linear capture polynomial regard regression highlight indirect view implicitly affect classification bivariate cluster furthermore elliptical approximated well mixture several like helpful modeling purpose application group elliptical density consist elliptical fitting rarely satisfied become difficult grow propose elliptical represent correctly elliptical create provide alternative difficulty contrary bivariate allow increase applicable purpose easily interpret presence model detail component frame artificial datum suggestion mixture parametric respect associate th component mixture implicitly assume density model constant hereafter denote n paradigm represent way functional component integer degree model become polynomial factorize denote eq free model aim classify unknown membership unlabele general scenario estimate fit adopt unlabele correspond know proportion improve define zero component origin label observe x indicate l polynomial suppose em ml gaussian step detailed th follow unlabele th independently regard maximization subject constraint augment set equation equal yield eq maximization I nz ij qx ij nz nz qx ij nz finally regard maximization algebra ij nz nz ij ij ij nz nz nz nz nz nz ij nz ij nz ij r nz nz nz nz I nz nz ij obtain em respectively say unlabele via map analysis adopt classification algorithm although maximize almost except step one partition em section environment value constitute model specify strategy detail mixture regression maximize log likelihood run draw acceleration use iteration base decision make regard whether reached log k iteration l analyse algorithm k parameter determine optimizer particular provide method option among logarithm quantity purpose consist convenient selection range couple choose couple good exist criterion integrate reference choice mixture bic use classification model mixture selection base bayes eq present component represent component attempt cluster approximate
maximum algebraic contain many bind check feasibility iteration initialize g hence bind terminate analogous consider far observe relatively determine principle every situation theoretical practice analysis lipschitz consistent set however search feasible constant supremum behave alone scenario entirely possible robustness thereby lipschitz control value identify detailed one stand notably concentration hoeffding turn influence rescale cover calculation first numerically second impact find implementation response model exactly implement necessary follow form associate provide representation refer x store compressed form elsewhere convert numerically represent come constant regard add class calculation whether integral overall calculation outer loop outer inner loop apply candidate ever scenario cx impose outer loop describe expand solver interface solver termination size population objective value improvement great consecutive outer loop objective optimizer generate position direction optimizer also constraint algebraic impose outer optimizer necessary scenario cx valid constraint equation evaluate optimizer standard multiplier objective evaluate optimizer fx px approach explicit build object optimizer generate first constraint c underlie discrete solver impose shift coordinate optimizer pass c scenario use impose unlike constraint impose feasibility scenario little discussion scenario load collection subject area interest dominate numerical convergence show bind indeed datum ignore find experiment reduce determine discover decrease compute markov difference confirm run consecutive long mass locate worth significant algorithmic computational result distinct much observation approximate illustrate point discrete cube uncertainty velocity dominant phenomenon natural try calculation always valid greatly find relatively consider automate product dimensional event simplify improvement future show column position column coordinate bottom effectively approach great generalization admissible independence independence moment include use optimally propagate uncertainty hierarchy acyclic output relationship figure g il bound precisely suppose respect value play pair geometrically cone must remain close whereas note confidence interval nature application real easily define admissible scenario admissible also analogue problem induce q measure validity similar uniform norm many application may strong therefore room metric lie distance inverse system system neither even observe uniquely partially assign subset notion continuity space neighbourhood equivalently hausdorff response lipschitz extension define single need acknowledgement us award fc na california institute technology material california institute center science finally anonymous helpful comment thm thm thm thm j quantification convex r universit provide quantification rigorous carry objective pose setting output solution trivially monotonically upon furthermore extreme carry analogue simplex efficient high system suggest optimal maximally finance rigorous uncertainty practical concern uncertainty quantification address cope operation paper thereby presence uncertainty pose scenario input uncertainty uncertainty concern unknown partially distribution infinite reduce type physical know method variant system variability diameter optimally bound bound study extensively discussion lipschitz locate hand smoothness point calculate without evaluation couple concentration produce last optimal datum know motivated show unknown lipschitz approach large calculation optimal see survey historical remark topic structure exploit greatly burden shown propagate bound bound determine propose use offer rigorous motivating easier less terminate number addition extended objective next hence maximally informative induce great change optimization upon assumption interest note hoeffding often reality assumption toy detail example variable consider constitute failure give value failure use set information alone trivial impact piece q lebesgue distribution gx g z information generally improvement possible say measure set unless take evaluating pose lie close contour notably non monotone establish problem treat determination function upper failure treat directly optimally bound failure least consistent still provide non discuss concern redundant general remark section implementation many redundant large quasi lipschitz short differ component lipschitz lipschitz constant argument borel measure kf f index sensitivity bound short diameter diameter place value independent finite independence relaxed control event system complementary provide rigorous computational physical interest determination diameter great failure safe know sufficient truth value know exactly gd question address constant probability failure problem determining evaluate maximally treat bound uncertainty quantification verification method applicable stand physical quantify bound suitably space place unified framework paper question use calibration difficulty good use bayesian area attention condition response globally lipschitz attention globally lipschitz broad old use modify suited requirement inequality constrain hold inequality example constraint suitable constraint must though value isolate elsewhere type pointwise evaluation finitely order upper obtain available state constant space let exist apply old value modulus continuity preserve modulus language metric say subtle hilbert lipschitz banach consider scalar optimization negative lipschitz variable intersection double illustrated note close cone contain empty say four dot dot note feasible lx kx x g ask bind rely denote lipschitz value sense denote gd short g ss sd kx xx gx complete proof although l say may mutually short also upper upper give g solve least kk candidate mutually run constraint mathematically formulation identical differ addition optimality number large metric part arbitrary eq q may necessarily member estimate lx lx gx constrained entail feasible compact extension constant restrict even simple suggest consider fix great l note quantity finite differ defined easily k observation corner cube size constant affine determine everywhere half vanish section spirit emphasis provide diameter failure necessarily short input simply give couple numerically may seem next extreme find search structure simplify structure correlation use remark let cube opposite corner ham cube separable borel probability regular countable dense borel measurable hausdorff separable compact simple inner topology right mild space extreme obtain combination dirac support matter convert product eq indexing upon make easy easily search finite feasible extreme value optimality space eq assertion discrete cube failure final equality contradiction extension short contradiction establish similar omit infinite dimensional value follow write assume distinct redundant bind see p dot location observation grey dot location dot feasible assign error maximization basis possibility q sharp next observation unit failure depend maximizer event sufficiently consider threshold must otherwise observation lipschitz give five case contour plot neither boundary critical conclusion infer attain satisfy unique dotted line various dot dot failure impossible least upper application whole input space partition lie lie lie outside reasonable one obvious formulation objective notion information content calculate entropy make notion optimization constraint determine extreme value behaviour rather solution elimination redundant section notion redundancy
classify beyond replicate reach three curve explain classified entirely level level vary display smooth author exceed usual relate thought status patient differ hour line period rapid within appear increase final period slowly curve greatly reach within positive curve low deal variation age sample provide henceforth distinction class alternate find sense programming know employ attribute machine svms applicable technique classification include handwritten digit recognition operate represent find separate separation hyperplane ideally far concept refer perform accord gap motivate formulation accumulation decade progress learn upon discussion improve observation pair form classify spam filter email spam longitudinal patient identify patient seek hyperplane separate observation contain vector observation lie half hyperplane lie half space find side eq convention break arbitrarily hyperplane exist training set separate hyperplane several identify hyperplane chapter near point hyperplane give iw I margin hyperplane via optimisation note objective positively homogeneous equality refer hyperplane support remove act optimisation space separation consider penalization approach slack variable measure th meet relax slack lead optimisation classify misclassifie penalty observation hyperplane reach increase objective hard correspond linearly separable least must zero agree force remove slack recover original margin two class criterion sometimes desirable reduce contribute separability unnecessary express desirable know encourage encourage interested clearly recover increase interestingly proper goal control huge regularize optimization descent serial randomized performance alternate sensitivity positive specificity already give correctly separate together refer reliable classification svm paper robustness quantify work portion remain provide useful feedback overfitte training training set classify hyperplane otherwise fail specificity gain false positive negative classifier create robustness separable specificity support remove hyperplane upper cross bound negative negative bind positive perspective understand support vector act test sensitivity specificity pseudo sensitivity specificity pseudo consequently specificity equal pseudo sensitivity pseudo specificity optimally test non fraction sensitivity specificity value else know refer number classify train subscript vector machine raw attempt find svm optimal employ nan read hour hour observe equally hour average threshold decide adjusted margin separability hyperplane test oppose calculate recovered svm specificity rather hour read hour maximum margin performance calibrate optimality guarantee latter great distinction two suggest high reliable validation classify differently remove optimally indicate classify precede read read margin svms separability classification error margin measurement specificity sensitivity extremely time see margin take hour perhaps entirely rt svms exclude present figure actually due threshold towards negative clearly contribute interesting error increase remove likely cause stability inclusion dominate train big parent svm dominate freedom greatly remove parent svm remove pseudo inclusion validation accommodate performance figure around hour cause reduce inclusion increase average argue exclude margin still hour margin hour argue inclusion reflect couple remove subsequent accurate svms lot choose hour positive four negative high read svms false negative average svms test take hour confident probably rt hour notable reading hour read hour average winner exception hour choose positive occur fact result penalty increase svm obtain specific sum consideration specificity svms justify training improves arguably adjusting consider far potential test machine observe help distinguish class variety namely extract information difference arise order replicate must wrong meaningful four identical arbitrarily replicate replicate within replicate flat replicate replicate show aggregation distinction lose average sensible three flat show increase flat would closer negative sensible replicate great reading hour replicate replicate increase occur replicate significant plot heuristic maximum increase train length specific validation replicate associate pseudo specificity fit optimally robust full svm suitable replicate weak robustness absolutely relative reading dimension consequently surprising dimension training practically separability seek separability compare plausible overfitte induce separation curve random also general present attempt separate hyperplane confirm separability dimension poor contain simply stack svm stacking vector consequently lose measure within independently seek induction try eliminate essential information discriminate class recall nan take fit curve function least space attribute dimension provide noise unlikely clear curve aspect occur unlikely reveal approximate piecewise major profile approximate stage three connect simplification define gradient location allow choose highly clear curve figure curve decide think derivative approximation sample degree way appropriate improve fit useful keep proxy intercept conclude information piecewise help find curve train svms try poor lower frequently unstable figure display curve fail trend sigmoid function growth positive description plausible follow trend model case sigmoid flexibility accommodate many express identical therefore aspect stack dimension curvature locally regular analysis involve stack dimension stacking assume present extract certain value feature svm training point apart variety difference define central follow stable divide second derivative approximation hour must multiple hour interval measurement minimum gradient time approximation calculate significant noise cause suggest issue partly great certainly appear flat guarantee sufficient preliminary enough sufficient trend noise employ identify trend longitudinal smoothed easy implement contiguous fix longitudinal central odd length enough large capture visually compare smoothed raw smoothed selection sample visual clutter plot first approximation smoothed difference value brief highlight value firstly information gain reflect rise trend derivative zero increase reduce trend maximum rise rather increase appear stop second derivative close trend suggest assume curve perhaps behaviour mean examine vector stack derivative overfitte robustness measure selection svms robustness consistently ccc case omit clarity feature classification assess work appropriate optimisation infeasible imply separability separability dimension stack meaningful appropriately sized ensure separability performance present colour graph tendency see consequently light robustness tend separability overfitte fit explanation apparent interpretation figure another smoothing derivative svms explanation investigate induce observe svm induction actually stack svms long frame instead curve difference suggest real error roughly balance shift positive less toward dominant select feature fit robustness difference either maximum feature inferior measure correlate curve high similarly svm extra positive negative improve robustness inclusion achievable via analysis rt analysis patient age date lp duration procedure plot display figure firstly plot age segregation conclusion consequently additional impact axis additional svm clear apparent profile positive horizontal axis difference prove induction merely see measure distinct patient age old old old patient characteristic correlation exploit characteristic improve probabilistic classifier report case curve time derivative predictor show predictor separability reducing omit rt clinical case rt diagnosis case class need multi class construct separate svm one class classification assigning class give great hyperplane soon apparent reliability type svms maximum separable freedom validation confirm exploitation omit stack end next try derivative information construct difference way meaningful find separability case robustness highly absolutely reliably patient discriminate age date lp procedure far available divide duration disease observe disease patient duration alone influence partially explain profile diagnosis
efficiency feature class therefore concern simple knn network since contextual requirement great show categorization store amount prototype cardinality calculate document centroid measure similarity regard use final hierarchical classification assess cite centroid comparison hierarchical hierarchical big flat method confidence main concern label result boost accuracy obtain tc superior grant pn ii te organization new categorization retrieval mining content spam filter mail categorization web characteristic typically categorization attract field machine mining pattern categorization totally class run hierarchy single category variety categorization take benefit goal massive far hierarchical category structure top upper used concept specific propagation major disadvantage misclassification recover propose error limit classifier per parent node iii classifier benefit node classifier great hierarchy classifier clear strategy method strategy return reliability goal confidence assignment weight goal add related occur accept reject candidate label consider reliability score hierarchy experimental categorization reliability score reliability validate text conclude literature hierarchy root typical child process start proceed class positive c c c separately strategy membership node fact classifier correspond leave unlikely membership avoid force level strategy commonly assume evaluation start go root level base local next classifier permit stop internal hierarchy certain class predict classifier pass phenomenon whenever threshold score node test originally force leaf worth point leaf node combination top prevent stop false negative propose label evaluation try ensure reliability hierarchy assign classifier idea sample option classification process manual account hierarchical parent relationship level high rate weight candidate use formula label generate accept answer classifier threshold rate lead equal false testing apply hierarchy hierarchy accord node select plot histogram truly reliability specify equal rate fa false rejection fr equal regard
identify positive positive run model fact mixture negative solving sentence admit negative sum adjust sentence equation element rank assertion discuss mle us start case function sum let vector sum variety non maximal interior critical global maximum try ran sample polytope converge one eight local lead precisely solution maxima lie negative q discover maxima carefully geometry em case comparison em global project acknowledgment thank comment national science dms dms dms example corollary problem fundamental statistic matrix rank mixture distribution two likelihood numerical geometry point discovery likelihood complementary maximum fundamental typical encounter discrete respectively write thus non closed I sample negative rational projective divide problem find rational point algebraic compute point reliably find maxima among determination bold number small compute symbolic fail beyond author solve use degree finding interested topology algebraic degree sign euler characteristic open suitably ultimately topological table know rank result n x x factor sign euler organize constraint mixture identically numerical geometry practitioner refine computation might theorem proof computation statistical closure fold independent algebraic distinguish equation affine system equation regular point subspace complement inner subspace positive logarithm likelihood critical contain entry solve express orthogonal make exclude strictly singular point exclude degeneracy jacobian defining format jacobian format homogeneous row format similarly write follow column rank jacobian third translate requirement derive formulation elegant geometry serious namely solution little row impose replace first row far sufficient ml get replace let kernel hadamard format linearly column either suffice explain span matrix equal p product invertible indeed equation note redundant far variable concern way reduce row replace critical still simplification lead computational result recall matrix matrix imply provide give u equation see consist many sum simplify column diagonal last entry column nine nine nine equation solution rational nine nine solution generic algebraic ml arise product rr nm homogeneous coefficient expression refinement fact upon variable root exploit various aforementioned table build balance tight computational linear product discuss impose variety independence state equal entry theorem entry multiply similar symmetric multiplied diagonal system consist local column sum hence fact diagonal symmetric arise entry eq equation take system six theorem suggest ml degree symmetric first version later document advance project use numerical algebraic address global numerical significant symbolic equation generic computation fast numerical homotopy critical fix homotopy change methodology wide likelihood treat agree statement homotopy substantial commonly statistic option preprocesse formulation generic first homotopy build discuss notably homotopy build root count option intersect notably essential algebraic preprocesse subsequent solve case processor perform summarize separate processor distribute processor second minute attribute represent alignment dna table take solve subsequent second integer become degenerate preprocesse polynomial appear clearly use small shall preprocesse introduction numerical homotopy book homotopy compute set complex isolated root compute point approximate solution software construct family contain isolate sufficiently track solution path isolate generic count degree generic root know root isolate root track ml root connect along segment general position contain avoid gamma trick trick arc critical issue due choice local write suitable matrix give affine random homotopy quickly critical preprocesse namely computation serial double duality root root homotopy read solve homotopy system arise generic construct table root return homotopy root use implement construct advantageous significantly less degree intersection advantageous intersection build discuss idea solve system intersection build product polynomial variable define system arise fact ic algebraic yield finitely hyperplane isolated point solve univariate hyperplane homotopy compute apply homotopy computation additional confirm trace centroid hyperplane linearly centroid critical matrix linearly test randomly generic theorem hold critical analyze hessian lagrangian polynomial critical maximizer form tangent remainder symmetric critical likelihood maxima local minima six define extension instance coordinate proposition irreducible coordinate hadamard point illustrate distinct complex critical real consider precisely seven six maxima respectively expect real positive number maxima maxima equal real maxima algebraic geometry include ml duality regard ml variety equation verify whose grow refined statement entry equal statement conjecture maximum likelihood duality likelihood preserve reality positivity critical perspective result exactly estimation matrix vice versa refined formulation duality speed complementary trivially ml degree illustrate theorem specific already literature conjecture maximum datum datum follow scale normalize occur illustration ht compute critical express rational remain point fall four symmetry calculus remain distinct pairwise smooth depicted tracking homotopy path function analyze arise member family pairwise arise establishe performing path using solution remain distinct retain critical size q sort point
compute first patch average stage compute eigenvector reconstruct average patch mm snr figure signal different level level moderate noise achieve patch nevertheless quite pc pc different cause much image lose general practice stage eigenvector general level patch contain patch extract dimensionality patch texture extract region patch image low spatial proximity image proximity conversely contain experimental instance patch classify accord compose class extract region texture etc author observe patch distance note proximity texture patch pixel texture patch distance patch gradient patch much texture patch low smooth eigenvector able reconstruct patch reconstruction patch texture eigenvector patch graph texture figure snr eigenvectors patch eigenvectors snr initially reach index adapt incoherent need reconstruction figure eigenvector pass see evaluate denoise gold pass evaluation image periodic texture add image pc pc pc pc level moderate middle follow denoise provide provide k svd display reconstruct original outperform squared visual wavelet yield consistently bad noise svd l reconstruction wavelet svd yield mean smooth texture even stage estimate rely eigenvector random patch smooth organize along low structure patch reconstruct denoise outperform work raise address compute eigenvector zero method implement option propose eigenvalue fast recent indicate method still computation patch clearly coarse processing eigenvector patch reference therein algorithm currently existence extension collection lee patch optical organized construct show lie image sub manifold provide eigenvector similar pc pc pc pc pc pc pc pc pc pc pc perturb analysis perturbation graph laplacian acknowledge perturbation acknowledgment grant dms award eigenfunction computation influence eigenvector diffusion operator perturbation change connection patch organize patch reconstruct demonstrate gold image denoise goal experimentally perturbation laplacian novel eigenvector patch graph image computer universal transform replace perspective indeed recently image shape contour patch image use patch reference taylor dataset patch aggregate eigenvector provide expand patch image intensity obviously smooth graph laplacian yield patch often corrupt eigenvector graph problem become eigenvector result note non eigenvector perturbation addition bound predict image angle translate effect perturbation encode eigenvector image original eigenvector aim understand low laplacian geometry geometry quantify eigenfunction laplace equip experimental understanding robustness laplacian jointly eigenvector experiment extend image location define notion patch block odd integer patch set patch start explore patch ignore imagine intensity dimensional though patch form collect lattice horizontal vertical lattice pair translation coordinate patch discretized argue move object form sub differentiable lack acquisition device smooth provide illustration patch extract structure shape surface encode move cone fig explore different generator cone orientation effect orientation inside encode patch always edge patch manifold review patch interpretation manifold patch denoise construct local point measurement tangent global dataset reconstruct tangent patch coordinate observable remove delay therein estimation coordinate tangent plane plane basis plane therein project noisy onto tangent plane ideas patch ball project tangent manifold orthonormal multiscale recently worker compute construct global diffusion manifold try around comprehensive denoise denoise geodesic manifold replace mean explicitly graph various interpretation idea pde depend local neighborhood graphic community implement discrete version surface therein propose diffusion discrete operator apply diffusion kernel patch compute eigenvector laplacian diffusion eigenvector spectral decomposition similar fouri seminal work perform remove image diffusion estimate noisy expect eigenvector different perturbation eigenvector propose originally think regard vertex structure patch connect construct call patch graph weight patch near neighbor patch encode region domain parameter drop rapidly zero patch noise fully characterize patch weight entry symmetric define laplacian opposite laplacian manifold ball radius around volume ball wrong sign share small similarly eigenvector kn write vertex eigenvector inner product define image left encodes intensity irrespective take horizontal derivative eigenvector fourier analysis intrinsic patch propose eigenvector encode feature texture etc graph patch replace regular lattice image patch reduce single pixel noisy denoise clean set access clean appear circular stability laplacian study experimentally add weight iterative recover eigenvector corrupt noise eq image e graph neighbor vice cause accord set edge near perturb define laplacian expect eigenvector reconstruct clean perturbation eigenvector depend separation predict problem separation small large limitation invariant subspace translate term perturb approximate effect employ coordinate plane orientation laplacian balance texture similar perturb row clean comparison visual observation analysis paragraph appear reasonably eigenvector pc pc pc pc pc mostly capture texture original instance texture fig quantitative perturbation quantitative comparison transform practice research analyse radial angular frequency etc orientation integrate orientation thin sampling two within perturbation individually dyadic eq dyadic eigenvector motivate observation eigenvector dyadic similar fouri inside eigenvector show index scale index radial total contribution group perturb eigenvector energy clean red perturb start perturb energy clean distribution radial create confirm try capture increase h il respectively radial bottom radial obviously suitable denoise topology topological change create removal significantly graph laplacian modification eigenvector verify experimentally entry location topology perturb weight perturbation topology local perturb eigenvector experiment factor preserve guarantee experiment base noise accord neighbor graph perturb constitute previous paragraph word perturb display perturb visually eigenvector radial eigenvector remarkably plane change image spectral geometry encode metric phenomenon pc pc k image pc blue red radial frequency image
simplex certain ideal matrix parametrize family simplex definite perspective package allow collection statement compute vanishing case analysis acyclic compute associate ideal vanishing follow package graph create contain entry normal store hence take direct acyclic know pd vi definite parametrization imply generate vanish satisfy I follow j j j graphical variable explanation independence translate matrix briefly explain construction generate variable whose simplex p primitive quantity create use p let statement translate certain probability follow generator p p p ideal degree three correspond undirected independence local statement direct undirected context statement n capability graphical acyclic satisfy recursive factorization parent map vanish q ordering vertex acyclic vanish graphical binary variable vanish generate ideal vanish ideal path previous ideal independence global statement ideal compute vanish gaussian model mix mixed ideal parametrization triangular direct edge otherwise symmetric zero entry vanish ideal ideal principal ideal generate ideal g problem consist solve graphical mixed example identifiable index contain rational acknowledgment people lin david partially national science institute mathematics anonymous helpful comment suggestion research sp grant fa air force scientific advanced research project ss support foundation dms david foundation thm thm thm thm edu edu edu package undirecte vanishing ideal
three flip reading comment cosine similarity pearson correlation tool preserve put correlate maximally apart transform fall second transformation derive metric distance pearson coefficient cosine use denote already know metric angular equivalently anti object maximally apart correlate anti correlate angular pearson coefficient cosine similarity strength pearson commonly deviation coefficient pearson sample define cosine standard information retrieval cosine angle transformation pearson cosine evident terminology model solely interval henceforth pearson indicate applicable dissimilarity satisfy triangular informally always short distance allow building accelerate skip take distance distinct distance object satisfie commonly state distance distance preserve shall metric namely increase concave refer state exceed essentially mean relate g important ease reference preserve preserve first inequality rewrite fa ba scalar rewrite fa ba ba proof quick determine concave twice concave understand acceleration imply calculus twice preserve transform dissimilarity transformation generic principle ax metric currently distance triple derive embed equal distance vector identity page second hence find preserve obtain trait angular certain interest trait angular preserve counterpart single object use ax ax ax angular ax ax angular distance
monte replicate randomly total accuracy plot via bootstrap resample solid circle data remain curve plus cca get f dot nn result pair dimensional classifying mi consider simultaneously classify test dotted plus curve pair dash indicate incorporate domain via term classification correlation reduce additional regularize cca choose properly trivial noisy keep figure regularize cca result cca regularize counterpart noisy dimension c non c regularize cca cca tf mi tf mi mi setting training relation figure superior cca investigate choice combination class regularize yield setting document embed early regard indexing outline table pair example classifier relation indicate choice class superior cca classification cca tf tf inference relative monotonic give inherent nonetheless characteristic regularize version match reduce apply wikipedia datum domain efficiency result analysis improve regard text document canonical generalization improve finally increase available document improvement performance amount matching work identify embedding multiple enable fusion source canonical generalization investigation cca document canonical efficiency object domain example translate language domain classify representation task ik need common training classify separate different learn represent relation datum consider canonical common cca training investigate near relation datum investigation discuss manifold well setup present conclusion survey kind explore view domain testing language later paper dictionary latent etc translation involve classification translate match whole divide follow space domain learn multidimensional euclidean cca generalize cca common e combine generalize firstly low manifold cca paper focus learn investigate dissimilarity space manifold matching show mapping manifold dissimilarity denote kind fidelity fidelity well dissimilaritie fidelity q measure define multidimensional dissimilarity dissimilarity ik jk ik ik jk ik ik multidimensional mapping column orthonormal requirement similarly imply subject language write differ english article neighborhood article english view case point training document manually topic topic people date thing respectively document class relation total document train classification start dissimilarity matrix dissimilarity dissimilarity graph topology dissimilarity matrix contain dissimilarity wikipedia connect document short document english neighborhood document document connect text dissimilarity ti ik jk ik jk indexing wikipedia description common pick via scale embed dimension joint model fix throughout dissimilarity training document problem different large preserve noise second indicate total c nearest nn assign close
status study disease death consist depict symbol figure incidence scale age duration disease simulate population population accomplish disease model compete birth cumulative failure failure diagnosis death measure birth person know event death disease person failure integral eqs calculate question random variable available store piece date birth person diagnosis person age death person person contract age diagnosis na follow h simulate na file file store person store text file dot file devote analytical relation common measure measure duration disease diagnosis lose life year disease obviously age diagnosis simulation typical question mean diagnosis subject another interesting incidence status typically current status incidence estimating incidence status decade useful integral eqs describe integrate integration give event co system axis age sometimes refer age entry entry death clinical trial diagram second subject without life end concept hand eqs integral failure time associated life choose sure occur year calculate life life field graphic exist calculate intersection volume voxel start calculation parametrization ideally suit approximate integral rule j necessary calculate f voxel grid calculation interpolation voxel j transform would well obtain affine consecutive j similar compute intersection grid accordingly death disease interpolation section present year birth death incidence total contract disease important age death disease year death duration ill compare disease age black several group confidence analytically agree quite article simulate population death consist disease birth subject without subject life diagram locate part life direction change allow modelling disease duration play disease follow
convnet ms convnet accuracy demonstrate clear dataset point inferior pooling pool art stage give slight increase performance unsupervise learn method mean auto run attribute supervision high seem exhibit address problem introduce edu house convolutional learning inspire unlike popular design automatically optimize traditional convnet stage state art dataset improvement analyze stage sf character recognition document handwritten hard like behind human mainly de image illumination recently dataset extract dataset digits input big label contain character digit image template matching way superiority superiority traffic sign challenge accuracy art convnet different pooling implement open cm cm convnet compose stack feature stage contain convolution module module module pooling module convnet lp oppose stage stage oppose h pool pool rich representation compare feature add fine lose improved work work however gain sign likely explanation observation gain texture multi characteristic channel set extra easy train information sampling order yield allow measure put emphasis process contrast channel contrast normalization channel invariance overall convnet feature two convolution filter filter output also global classifier hide unit rate gradient dataset pooling represent report epoch
directly c rest write sign consistent vector n probability lemma write iy eq n w I put ready proposition n utilize c since combine q show lemma w c suffice complete satisfy pn I I p proposition hold probability proposition hold stand theorem ise binary markov apply range scientific engineering infer represent edge equivalent dependency rest graph focus draw network type graphical ise structure equivalent diagonal precision paper inverse covariance appear recent year paper establish many fast glasso recently ise constant make demand spirit fitting assume sample model node study life covariate genetic study role tumor development since interest tumor observe various clinical phenotype tumor mutation status factor include motivated situation model binary incorporate covariate connection covariate mind impose allow select covariate paper way quantitative fit separate subject interpretation covariate concern nearby region covariate ise lead covariate strength subject organize ise establish asymptotic evaluate instability conclude physic q binary z probability sum equal otherwise specify related parameter independent suppose covariate ise jk conditional odd way imply conditionally jk jk jk jk model express vector instead depend eq choice linear parametrization jk logistic regression parametrization straightforward relationship depend among desirable formulation convexity negative log likelihood maximize likelihood spirit maximize give conditional likelihood q network interpretation suggest good sparse encourage propose conditional likelihood regularize regression guarantee estimate compare magnitude initial separate regression jk jk jk jk min conservative sense turn separate often conservative identify detail regularize entire logistic careful rearrange obvious treat separate fit model omit regularize derive fashion regression covariate slight notation drop intercept irrelevant true without additional j j j hold regression exist bind effective covariate rest assumption c q uniqueness jj j j establish proof vary signal strength covariate roc zero estimate zero specificity curve replication sample adjacency randomly exception intercept term generate stop estimation min separate max joint combination first fit augment uninformative total remain zero false curve generally increase particularly dominate joint method tumor play tumor development complicated relationship time vary tumor dna profiles microarray cancer patient advance breast cancer collect another base copy profile dna array copy event bin band sample status cover group patient retain association among event association clinical characteristic include mutation status variable er status variable tumor stage ordinal denote array datum I jj contain phenotype covariate matrix ise covariate fit selection infer stable repeatedly replacement tuning record non pair note correspond interaction covariate measure finally rank list depend heavily primarily pair gene locate likely depend column relate name third record tp status status tumor generate hypothesis experiment molecular mechanism breast pairwise selection tumor tp associate genome instability tumor tumor association interesting association group find breast region several tumor patient tp contribute tp pathway study suggest positive breast cancer previous finding find association tp role tumor gene cancer htbp c main tp status gene gene gene gene p q p p p p p p p q highly node role pathway covariate selection subsample node finally across stability covariate list interestingly rank role breast cancer study region tumor region breast cancer report candidate tumor transform confidence selection change tp status frequency finding lead discovery tumor breast allele survival association role gene
leave aim residual empirical partition test baseline scalar ridge ridge integral ridge evenly weight kernel well combine identity residual enhance use constraint ridge outperform kernel functional response induce lc value simultaneously operator deal regression method predict movement regression performance work mkl mkl collaborative minimizer consider tool guarantee generalize corollary banach suppose proper bounded space attain functional give present convergence infinite reproducing let kx q strictly convex directly strict convexity us eq ridge rkhs converge number subsequence convergence value convergent analogue convergent subsequence space subsequence obtain proposition hilbert operator state subsequence convergence rkh operator reproduce subsequence minimization fix converge subsequence whose minimizer pair minimizer convergent subsequence converge bound de analyse des de hilbert deal case value function considerable learn entity typical include brain interface design precisely movement measure electrical activity give period instant whether move amplitude signal amplitude clear formalize functional point benefit multiple value movement movement fix several value reproduce rkh whose map target work draw rkh value kernel recently reproduce build kernel try scalar kernel seminal effort carry theoretically analyze problem motivation propose framework finite value kernel mkl without difficulty arise space point work coordinate descent multiple although combination kernel value reproduce correspondence positive definite value kernel traditional scalar kernel value cm reproduce hilbert value furthermore hermitian adjoint definite hermitian w rkhs reproduce hilbert reading value kernel lagrange multipliers lagrangian lagrange multiplier banach finite lagrange multiplier suitable infinite constrain lagrange multiplier minimize respect ridge belong rkh similarly scalar convention cast derive admit supplementary material line problem follow block kkt kx present devise scalar mkl coordinate descent closely gauss operator typically analytically iteratively solve initialize non simple system still fix rewrite close optimization operator basic matrix value make detailed supplementary showing generate continuity boundedness argument minimizer scalar space reproduce technical norm cm cm f common build scalar carry set positive operator integral fact operator provide interesting extend structured identity finite kernel encounter kernel infinite space solve value form value kernel operator product analytic solution invert gauss method initial cm cm kernel value inverting matrix gauss iterative iteratively convergence satisfied expression positive operator value variational n jk split role indeed problem vector deal constraint derive multiplier equality constraint kernel easily gauss highlight operator real involve brain interface address related movement focus direct scope prove aim fourth dataset competition grid place depend unknown record band pass hz signal
well screen fs fs often perform combine study draw conclusion relatively fs prefer fs converge yield well simulation cr cr cr sis sis fs sis fs fs cr cr cr sis sis sis fs fs ct dataset frank author ct ct slice histogram describe structure histogram location air body histogram form axis construct manually ct volume know detailed dataset et square fs fs want improve fs fs fs least error number fs fs highly procedure fs fs term classical screening enough word subset call fitting well rule property model initial virtue screen initial estimator screen subset include subset screen hence fs wang well well screen fs addition satisfactory fs modification also fs fs fs choose wang fs fs bring whether rule believe valuable topic national natural china grateful laboratory science variable inequality omit ax ax note v ax ax superior subset hard complete complete unique proposition ft generality pa b c b b c b c boost prediction fitting van de york california school science http uci ml ct images hill company york http corollary mathematics science method screening include residual square rule well fitting regression algorithm propose search subset yield property popular screening screen show competitive high subset combinatorial reduction em orthogonal number candidate variable wide variety scientific become coefficient loss generality denote cardinality often eliminate important exhibit variable model actually sparse fan two estimate sparse screening stage guarantee effectiveness screening possess sure fan fan fan wu fan song li aim screen much relationship fitting square sum square good fitting say interestingly variable pick fitting well fitting screening make fitting tell sum asymptotically size one likely include denote number nonzero component equivalent optimization square algorithm reach sum screen obtain exhaustive branch later moderate subset search infeasible forward optimal solution e basic idea possess e square put fitting screening well call fast real underlying actually desirable include yx x ax pa eigenvalue matrix hold rank column eq x ax make design wu lin identically sure screen fs literature fitting tend include small mt ratio dm theorem indicate superior subset state sure subset point call power see normal nx q minimize subject constraint fs track path flexible minima relatively point final write monotonicity ft proposition fit improvement asymptotic mt p mt k square stop take p correspond subset well asymptotic screening subset corollary connection point sure independence sis sis take similar yu well combine successively whereas keep besides virtue boost paragraph convergence monotonic converge study counter special case stop sum decrease tool wu certain recall theorem effective nonconvex solution neighborhood practice neighborhood provide direction sufficiently know consistent obtain way scad regularity condition van fan like disadvantage least replace fast subset screen ft ft monotonicity converge simulation study design kronecker product wu hadamard configuration cr cr sis
could discrete distribution give dependence serve tool evaluate specification compare parametric nonparametric kolmogorov integral former develop datum integral strong bring uniform guarantee transform nice follow add apply transformation use extension copula test uniform von kolmogorov transform necessary estimation make prove parametric bootstrap correct arise ml idea project transformation ng projection test require projection approach extensive static study base moment comparison parametric nonparametric despite stress situation could provide specification contribution dynamic asymptotic asymptotic bootstrap valid rest introduce specification experiment conclude ty dimensional parameter space cdf use subscript nan correct conditional order would family distribution nonnegative integer fy strictly version strictly increase typical p ff matter result distribution copula marginal invariant realization realization fix realization respective denote property fact u u motivate know either uniformly function von kolmogorov statistics q test lag independence correspond obtain generalization kolmogorov discrete distribution alternative deviation box correlation lag note variable nan correctly specify specification good goodness fit alternatively residual eq normality addition square capture alternative example misspecification well test distribution bootstrap parametric statistic recursively bootstrap repeat percentile prove limit bootstrapping derive simple know fix alternative denote ba r fr f ff smoothness tr linear nan e dynamic probit logit introduction adjust satisfy close variable study effect expansion estimator smoothness continuity uniformly parameter although satisfied estimator know assumption mean account asymptotic expansion study limit eq q cr df define alternative equal nan local assumption hold random around may nonzero stand appear project effect suppose assumption hold v test justify bootstrap procedure e prove triangular array tt x tt similar assumption require hold investigate finite sample exercise autoregressive dynamic specification specification set try hypothesis static dynamic interaction probit check dynamic marginals logit alternative sizes table bootstrap base parameter lag von kolmogorov denote residual normality residual lag additional omit nan probit probit static probit probit probit interaction logit static probit static chi static static logit dynamic probit dynamic probit static chi interaction probit dynamic logit interaction static interaction probit pt traditional almost slightly improve fast rate ks size static logit static probit hand static probit test static well case dynamic misspecification add logit improve add dynamic alternative dynamic probit logit even well high lag account versus lag summarize dynamic misspecification statistic misspecification marginal distinguish statistic possibly test additional noise effect since test residual dynamic misspecification indirect effect power attribute develop introduce still wide check goodness many financial parametric asymptotic test experiment base
covariance know correct formulae krige proof enable interpretation krige key though formulae krige weight valid square integrable good coincide case case ordinary krige write covariance integrable bayesian construction improper coefficient equation use equation take come counter q early regular krige equation diagonal happen enable decompose weighted krige krige formulae incorrect formulae covariance krige would conditional formulae enable avoid co sup e attention weight david support david lot effort computation krige update formulae enable krige variance avoid costly inversion already available addition traditional formulae new formulae formulae sequential correct establish correct expression variance covariance parallel counter first notation recall formulae real krige krige formulae krige formulae argument counter sake simplicity case wiener even krige case stationary kernel initial weight lead krige krige lead expression
robust loss omit design consist c find condition rather strong exact improve yet maintain restrictive tucker kkt moreover gram design say condition meet link q ps compare bring lipschitz respectively one constant square logistic let constant effective arrive furthermore take depend fix keep away stay zero formulate differently imply hold appear appropriate choice estimator error likelihood oracle np definite matrix condition enough say compatibility prediction argument argument population I order compatibility another refer paper selection also small heavily convexity loss unbounde priori bound result loss mixed effect regime pa theorem example section kkt consider theoretical extension result quasi likelihood robust positive high co much literature eq orthogonal lasso extension model least loss therein concern lasso say good oracle state equal true result compatibility neighborhood stability equivalent hard strong latter imply compatibility concern oracle inequality early paper context orthogonal consider along later proof developed remark technique penalty design hinge penalty design compatibility eigenvalue penalty consider generalized possibly compatibility cover case lipschitz compatibility one spirit condition concern oracle model extend beyond paper generalization robust high discuss concern selection modification lasso procedure scad introduce quantile aspect study apart theory description datum case huber loss loss numerically present finding prediction compatibility dependence linear situation large assume assume vector quasi eq quasi describe lipschitz assume quasi robust likelihood section handle let abuse vector f large correspond regularization mean shrinkage expression square e usual call estimator either euclidean quasi present condition selection paper map allow turn sup nonzero denote small call approximate elaborate issue brief outline see active well term coefficient zero refer appropriate sufficient result state facilitate give formulation next robust definition inequality likelihood address similar compatibility strictly depend tuning paper except see condition become involve much sparsity compatibility line connect sparsity alternatively square suppose lasso error jj regularization allow see rather say sparsity compatibility constant stay estimator constant know measure term include say error ahead idea approximation neighborhood compatibility sparsity least estimate immediately bound abuse terminology prediction behavior level sup inequality allow neighborhood link far note quasi measure linearity term choose true constant serve normalization albeit principle say exist exist q exist impose link hold logistic canonical come appear result let use square describe effect choose eq neighborhood arbitrarily reasonable take comparable canonical compatibility eigenvalue condition error compatibility know empirical risk minimization penalty coefficient prediction know true actually generalize truth sparse one sparsity compatibility trade arbitrary thus condition constant regularity
refer matrix factorization visualization interpretation temporal network membership represent binary capability attribute covariate link prediction latent space distinct technique much graph remain I nodes vary characteristic domain single non metric rich evaluate future work may wish well discussion focus base link however link form use trace page visit walk discuss task relational link event type assign stochastic modeling prediction predict presence static frequency occurrence link prune away link facebook noisy add used remove link indeed assign could weight weight link often possible effective discuss link weight link interpretation jointly link construct weight link somewhat assign link thus link link except assign link negative relationship flow link generate continuous instance might word appear link feature view summarize weight feature summarize process collective toward compute much supervised learn weighting link construction exist link represent importance link discuss weight accomplish apply technique simply link perform apply decomposition singular similar technique discuss result unweighted graph unlike design exist graph technique unseen know additional sensible link simple weight aggregate property link aggregate phone communication network case interaction link like aggregated generate weight weight link facebook identify user friend facebook predict link facebook datum participant strength link large network via tree find perform alternative two education level interaction find interaction feature helpful day communication kind feature attribute topological feature communication take normalization send friend prediction predictive accuracy parameterize possible evaluated training approach manually prediction two dataset interaction propose link capture coordinate optimization predict strength strength task directly evaluate strength autocorrelation gender status researcher event occur interaction metric link base separately incoming message impose decay base old metric implicit represents demonstrate predict however basic task weight heavily connect frequently recently alternatively appear summarize link ever link new recently past classifier exponential decay kernel weight yield handle serve incorporate v link relationship facebook create represent enable subsequent classification link label text link unsupervised technique dirichlet analysis traditionally technique use topic document collection document topic form manual reveal sensible concept relation however semantic obvious infer topic link aid topic represent kind technique mind software document lda technique content associate link go impact vary topic run lda one significantly temporal topic describe label essentially bayesian dependency associated art message author message infer make use role email network demonstrate simple technique network label predict label learn relational link prediction page utilize feature anchor possible link link performance separate inference often significant challenge limit consider approach learn entire graph classifier treat link negative create sign nearby link indicate relationship edge interestingly decrease train vs status partially explain ability technique section work text general sign opinion mining sentiment reformulate predict associate kind label design logic extract text yield node represent object produce link analysis example group intend graph two people yes directly cluster people description vote topic implicit link add discover construction link construction many considerably computation feature construction discuss summarize feature describe datum aggregate average common link precisely aggregation collect feature center subgraph target positive sign adjacent link node way aggregation link summarize kind link figure source go link input rather bottom show four subgraph link subgraph link subgraph vary amount display use input link solely link construct value message link feature count might compute form compute aggregate phone call link weight depend topology link coefficient link extent discuss feature link use link nearby link instance p work identify link target link form take label use link feature work graph sign figure feature sign target link path finally relational feature close close feature label node distant new base friend two people node predict construct node consider distinct discover email imply message graph supervise paper certainly add node knowledge base relational existence focus prediction node community characteristic social process kind facebook newly discover node represent node refer latent depend link include many node become link close share impact reduce influence propagate collective exploit node group node side may associate affinity group discover technique create leave alternative may represent unknown dataset potentially simple relational disadvantage allow propagate influence newly large liu large link collective unnecessary good clustering depend discuss assume create originally create link kind discuss relational node discusse relational attribute type algorithm agglomerative algorithm self map since study relational attribute desire link group original finding grouping kind two topology table latent belong value pair value instance detect extend group short path go along link link connect assign correspond reveal remove high relational addition link cluster coefficient describe metric node walk reach node nod euclidean link modular often lead useful simple module find modularity simple identify kind similarity compute every node link remove link place weight intuition vary form particular leave large form cluster original remove link reveal community group identification transform row algorithm interesting matrix variant involve compute original motivation identify cut connect identify node walk overview describe relational metric adjacency describe identify group enable ultimately yield compare ignore also number learn supervise cluster clustering liu metric web node prediction instance represent prominent normally prominent community search secondary treat link represent detect pattern add technique topology graph technique attribute produce consider simple similarity combine single similarity mention algorithm instance weight attribute information iff link control importance attribute attribute similarity entity resolution attribute incorporate hoc set logical probabilistic system logic component primarily node person primarily describe connection people principle approach kind model link instance model belong member link membership belong search state chart hill membership well use estimate news generative sophisticated treat membership belief group membership particular round round soft assignment membership membership demonstrate focus simultaneously multiple g connect also depend group membership group connect group assign cluster area complex facebook might way might status multi node topic form predicate consider distinct individual cluster baseline alternatively node represent real world object case entity cluster representation create link representation yield efficiently section use enable inference collapse super yield logic see interpretation weight feature node seek node labeling seek discrete represent likewise systematically reduction subgraph weighting labeling general use label practice technique tend label rarely nonetheless interpretation substantial actually label vs discuss section represent weight social discover prediction whether attribute topology construct attribute weight contain represent sale sophisticated strategy indexing concept corpus rank connection quantify rank extensively elsewhere ignore structure topology topology search example web conceptually implement systematically compute describe eigenvector algorithm rank identify influential variety centrality local global characterize metric coefficient centrality addition ranking ranking particularly relative ranking short path g addition alternatively formulate technique extend define notion temporal local notion closeness metric understand dynamic accurately identify concern interaction communication people could interaction node node join apply link link weight also instance various approach weight seek vector add anchor high approach probabilistic discuss far relevant transformation recently adversarial air moderate spam web technique topology necessarily kind attribute discuss propagate trust site try spam air widely suffer human site favor assign label label classification labeling consider end facebook predict political desire anomalous detection indicate ever connect estimate label enable subsequently node enable learn accurate labeling final goal label change stack classification label relational classifier relational collective classification cc analyze training inference stack bias estimate single pass cc extend idea generate training simulate classification approach outperform stack cc accomplish aware describe collective done knn logistic simply exploit assign supervise assign content communication traditional abstraction recent technique lda link incorporate link discover describe attribute topology topic specific systematic instance classify likely represent information neighbor construction construction discuss feature input operator discretization automatic relational link node value feature value possible include link node relational describe consist four type relational relational value feature consider relational value relational node might construct dimensionality several feature value thresholding etc compute exist node topological shown count adjacent short pass relational link new kind aggregation mode link value link target could relational feature construction away value adjacent instance value mode alternatively feature number count adjacent feature apply recursively topology combination recursive classification network might compute might hybrid feature topology aspect relational feature potential collective meta depend neighborhood contrast applicability independent occur semi co kind describe summarize case aggregation operator relational input walk topology color indicate walk conjunction input rarely ever discuss aggregation refer return another frequent compute fraction meet c operator may threshold node base complex relational via join compare performance alternative aggregate topology count adjacent link carefully handle temporal aggregation raw define importance recent temporal discusse notion distance modify node closeness traditional domain operator intersection represent presence contain word intersection collective represent union node adjacent value adjacent node e indicate draw relational outperform count potential probabilistic relational without compute specific dependency naturally accommodate vary neighbor sense choose aggregation still different clique link appear later clique enable random co citation link subgraph pattern pattern adjacent produce true exist simple pattern link involve star node link star node three know direct possible argue triangle share common side help avoid degeneracy arise generation subgraph sign positive relationship pattern left compute feature dimensionality reduction kn information original capture accord dimensionality component ica reduction adjacency create explain collective classification relational demonstrate apply pca substantial mention operator elsewhere walk classifier sampling topology easily imagine type discuss technique prediction weight weight computation easily lead refer however feature discuss newly discover create datum similarity graph instance type also discussion feature link text node whereas link relational link aggregation accomplish computation include node target link operator well suit vs select manually prior experience trial error situation automatic non relational study select receive search evaluate summarize search search relational possible specify raw consider friend divide easy operator possibility effort expert step appropriate usually specify operator strategy consider feature evaluation usefulness strategy selection evaluate way determine retain instance candidate evaluate improve immediately feature scoring feature retain decrease new continue correlation mutual strategy bic many frequently must metric summarize automatically describe system search relational tree extension relational involve topology g predict class measure yield value discard remain choose inclusion adjust bias relational sum exhaustive argue tree allow complex alternative single exhaustive operator base logarithmic recursive random add temporal spatial feature exhaustive pre add remain system system candidate view create base predict include recall pr feature continue extend feature add ability dynamically object different demonstrate helpful system develop independently describe impact system weighted formula systematically candidate clause empty testing clause top inefficient proposal clause response markov learner template guide construction approach learn long clause consistent pattern create enable structure parametrize attribute improve justification propose extend clause demonstrate fast well baseline seek learn explain whole political force subsequently clause conjunction appropriate smaller useful logical technique approach logical exhaustive decision feature relational link apply construct relational use refinement equality refinement possible proceeding refinement seem combine specific bic include reduce notion aggregation aggregate relational confidence discovery cd possibility instance clause metric evaluating clause primarily relational instance build transform labeling section transformation transformation instance way avoid labeling might propose perform link labeling entity merging solve task simultaneously notion inter relational relational feature task task compare perform technique across instance link focus recent associate link kind kind input text text document link document link link connect email message list prominent kind input type perform node link discuss document link first associate instance mention discover typical weighting use weight topic discover use represent kind perform seek topic word contrast design lda imply text document likely topic simultaneously semantic extraction discuss text second type add lda prediction use weighting generative document topic multinomial link link directly lda link lda extension link alternative lda replace link link model model bernoulli condition link link lda model document word document document work divide link incoming argue limitation largely overcome incoming much scalable lda likelihood lda link comparable pairwise slow change use approach encode lead introduce topic compare lda argue force topic pairwise link provide accurate suggestion pairwise lda baseline change model type link likely topic consider author create new document outperform lda link text link email message relate protein protein interaction several previously link model author art extend labeling strength topic assign node art link model allow role lda idea specifically share information share model block label link dataset email outperform lda baseline task protein category group model link prediction discover document surprisingly approach demonstrate link prediction discover link connect also learn discover topic discover graph connect topic connect frequently document represent topic add discuss additional relational transformation highlight representation transformation subsequent task whether accomplish goal guide provide ground link particular evaluation rank document last link classification run change surrogate change labeling prediction naturally problem weight relevance direct metric related autocorrelation presence sensible collective success strength autocorrelation social likewise attribute could assess naturally appropriate vary technique different upon ideally transformation execute selection surrogate evaluate retain improve even desire walk section use link final accurate supervision however supervision always helpful final program linear link lead via algorithm ensure particular transformation task remain suitable refer effect relationship e cause cancer challenge causal statistical take circumstance situation provide control discover discover different demonstrate understand media system remain extend broad range relational article task also subgraph frequent informative classify subgraph semi technique classification subgraph weight generation many node subgraph generation attract use prominent product method global potentially attribute classifier graph survey additional statistical prominent dependency structural logic transformation regardless kind statistical subsequently interact kind link connection relevant publish interaction domain multiple incorporate temporal represent examine temporal network specify manner simplification remove transaction efficiency support update comparison need purpose specialized maintain kind survey relational mining alignment involve pattern similarity useful overlap private publicly support preserve privacy transform way minimize information e prevent naive operate replace individual unique adversarial true identity user discover particular adversarial attribute identity user within early sensitive relationship kind describe cluster approach remove address attack adversary able recently study privacy issue goal topology create model generalize describe drastically change oppose make change allow variety analysis perform available resource attack must addition obtain related graph challenge need enable release importance relational article relational present section primary interpretation accomplish simultaneously technique work section highlight task link survey suggest develop instance reformulate traditional link likewise topic discovery use labeling node vice versa much discuss link together range transformation wide range weight decrease challenge technique come social retrieval inductive statistical relational relational resource evaluating change consume challenging understanding affect characteristic interact combination characteristic thank anonymous majority research research laboratory support fellowship make air office scientific engineering fellowship relational relational artificial intelligence transformation relational transformation transformation link label link construction node weight gray em minus width em depth transform statistical relational david department west usa department usa apply research artificial intelligence laboratory code dc usa cs edu david representation become due relational domain article examine issue relational choice relational feature affect capability introduce representation relational domain incorporate link include predict existence systematically motivate detailed compare transform highlight challenge remain research assume identically violate encode business person task involve associate person generally relational describe type link relational internet world scientific citation bioinformatic association communication location mechanism implicit relationship relational address reason relational high characteristic across link instance friend political people inference relational reasoning identify issue heart intelligence decade important example whether feature add high decision ai domain large complex way model performance relational datum use chapter relational relational algorithm relational represent attribute g logic focus address grow application people place likewise relation link content representation choice range aggregation kind path researcher recognize importance separate study examine weighting prediction survey assess relational give link subsequent representation transform feature apply transformation vary upon intend sometimes substantially add node add represent underlie simple increase dotted predict represent predict link link link produce discuss focus subsequent focus examine relational figure pre increase output valuable prediction insight protein survey multiple purpose applicability g suggestion great follow prediction survey many could g side facilitate transformation collective running analysis classification create possibility label useful pre stack wide likewise survey issue knowledge representation dependency link structural logic network briefly focus modify modify feature change logic program node addition program closely analogous logical could logic anomalous transform transformation understand appropriate intuitive representation identify specifically transformation task predict iii estimate iv link consist feature examine necessary comparison organize next relational label weight section consider summarize transform discuss method evaluate transformation challenge conclude datum facebook relational introduce relational aid define type consider facebook www popular social link facebook user gender relationship status school movie preference likewise link formation send link formation one political moderate assume construct label every representation sometimes focus useful pre process subsequent analysis political correlated degree connection type kind relationship instance indicate people join facebook addition facebook weak person often weight notational add component symbol transformation predict might combine measure similarity instance recommender implicitly link two rating movie book case cosine commonly metric link problem transform train approach machine
function lasso estimate test number govern predictor average compute pool together level sensible feature thus represent precision run run problem tend within precise fast high ex contrast affect measure fall specify scale stop stop burden reach affect obviously precise difficult descent setup ill condition system surprising proximal experimentally roughly comparison believe handling explain optimize greedy overall highly competitive accurate much rough dominate agree competitive variable conversely sensitive optimize second method competitive arise difference protocol accuracy stop criterion step occur early slow considerable obtain optimize soon sensible problem figure implementation highly fast improvement rough fista homotopy fast alternative large condition true restrictive fulfil obviously require rough either removal irrelevant relevant recovery high need hundred medium sized generate set determination average correlation medium sufficiently say evaluate compare lasso return various argument default correspond report htbp error mse mean bottom expect bottom obvious well level htbp cccc ms accuracy opt focus performance slow quadratic solver become precision par ten time slow viewpoint sparsity induce optimization viewpoint penalty compute low minimum objective function provide assessment test state art implementation compare mixed possibly ridge penalty lead known net derive viewpoint address non efficient wide accommodate penalty symmetric ridge provide structure elastic net wide range inducing fuse irrelevant bound stem triangular side reach choice vector side slightly proposition state prove define relate penalty infeasible inequality stem lemma trivially conclude define compute hand stem trivially proof generic problem trivially tight gap guess near value complement current vector function much compute gap nan compute meanwhile monitoring could comprise zero optimality violate compute meanwhile easily optimization france et universit france universit france france interpretation elastic net beyond viewpoint unify efficient medium software solve get rough compete aim solve viewpoint provide novel interpretation machine robust framework location affect closely concern show assume penalty assume corrupted suppose response adversarial according induce recover formalism regression simple unconstrained quadratic strategy iterative plain strategy solve approximation medium several computer variant uncertainty thereby detail derivation lasso ridge penalty net general purpose formalism section optimality resolution scheme demonstrate solver exist motivating linearly variable sparse perturbation variable datum consist source affect design corrupt strong though set option lead equation relationship pattern neutral perturbation denote response form sum contribution input input neutral noise discuss support amenable unbounded deviation confidence level neutral confidence noise scenario valid estimation perturbation belong without specify derive robust regression eq robust easily follow robust reveal case corruption minimizer ordinary amenable propose know two follow dual spurious regularity condition express convex define convex finite possible perturbation induce elastic net implement strongly subsequently prescribe magnitude activate group derivation problem suggest unified iterative resolution assess bind gap current solving series small increase involve guess solve respect active currently penalize submatrix comprise index adversarial pick move along descent direction update adversarial check coherent value bad gradient respect choose minimize current picking problem variable active provide behavior net measure computed plot large display hand meanwhile optimization monitoring well subgradient comprise situation violate eq compute meanwhile obviously objective objective trivially tight optimality complete complement propose magnitude gradient solution bind low size duality check gap nan compare performance efficiency assess value computing time return obviously package compare comparison language library calculus library algebra second package bias simulate post genomic signal processing domain predictor exceed dimensional setup active bad subspace thing local govern affect heavily run characteristic reach rather determination conditioning variable coefficient true ill three art coordinate name elastic net admits embed routine approximately differ regard active resolution descent implementation approximate optimality reach end precision warm routine resolution nine situation parameter medium predictor low medium medium high medium level medium medium
sf sum prove together cdf binomial q early sf sf j prove microsoft com microsoft bayesian idea study empirical state provide novel thompson simultaneously open regret prove et novel conceptually beta bandit armed exploration trade inherent study mab clinical trial unknown arrive sequentially decide treatment patient choice reasonable treatment patient time treatment bandit many generalization study basic stochastic multi available bind ucb algorithm discount reward bandit problem randomize regret basic probability arm ts ts idea independently emphasize bayesian free arm see interpret assume bayesian knowledge bound thompson directly comparable ucb attract attention thompson provide technique general setting favorable comparable low display news competitive popular delay delay play delay require decision arm play ts randomize unlikely delay microsoft search ad idea thompson competitive ts namely provide bind bound e bound logarithmic reward problem exist far lower bind pose thompson optimal bound thompson analysis technique conceptually arguably extend distribution beta contextual substantially state ts formally mab give slot machine play yield reward obtain play repeatedly reward play mab decide arm reward goal maximize expect play amount arm formally define introduce notation play regret performance measure guarantee provide thompson reward reward thompson sampling closely extend bandit bandit maintain bayesian convenient prior bernoulli reward useful observe bernoulli trial trial success sampling algorithm initially arm time observe play play accord summarize thompson arm play arm observe else expect first matter state course optimal arm armed stochastic bandit thompson algorithm big bind armed bandit thompson big notation contrast bound dependent bound depend prove regret precise give asymptotically achieve bind bayes ucb quantile low sampling also achieve independent regret ts exist imply suboptimal imply additive dependent appear involve problem bind ucb give theorem follow step essentially condition history arm lemma core decrease arm desire merely introduce early introduce notation denote cdf mass binomial number play time arm threshold bind event event intuitively later hold time step play play reward define probability assume probability leave hand trivially exceed suboptimal hold q last equality event involve hence f f true side inequality trivially suffice exceed suboptimal arm prove inequality give involve essentially chernoff hoeffding appendix observation concentrate mean happen I careful binomial sum play lemma mark observation change play give give eq rt notation absolute constant obtain independent x dx e rt
empirical work example approach practice orthogonal series kde non estimation datum inference population sample solution continuously twice integrable derivative essentially smoothing degenerate integrate hx kx z x base usual spline minimization problem model smoothing spline twice q control fidelity datum converge square typical spline part spline criterion fix remain spline write element penalize square write minimize give suitable contain function unknown different give ref basis nu nu kde sequence cauchy schwarz hand uniqueness inverse fourier hilbert space hilbert space basis fourier smoothness choice hilbert drive series also estimate nice
simple case adaptive directly use multinomial case equally version take hour complete run intel cpu result allow sampler result pdf mode non clear adaptive well n autocorrelation plot acceptance ratio respectively allow one brief interested forward let composed concatenation genetic type go mutation x zero probability mutation mutation diagonal generate number reach reverse sampling forward previous mutation uniform concentrate infer shape equal adopt approximately level proposal walk c take hour smc unit gpu left plot trace sampler result encourage practitioner adaptive ht ht bottom autocorrelation plot acceptance respectively perform inference stop process level auxiliary recent
lead normally load pc usually variable sparse loading vector enhance store loading vector incorporate induce mechanism formulation via induce two norm denote optimization formulation extract loading compute arise modeling induce si cardinality two constraint version stop effect choose enforce encourage sparsity si usage x ax x ax ax ax p x ax ax formulation involve previously formulation version four propose formulation constrain directly cardinality load introduce additional variable
hierarchy document label multiple redundant membership overall percent document content two branch market economic social percent label branch example market market market branch document multiple level topic trading market market branch market therefore completely membership corpus topic table policy account comment c capital issue rating asset production service market market c c competition management management e economic economic price price finance sale finance e e output capacity trade balance trade start lead mm cb lx cross branch name european community ec internal market ec ec policy g ec economic ec ec ec ec ec international world health issue issue interest science weather service market market market market market soft trading market mix cb lx branch mm exclusive topic parent expect usage candidate highly exclusive child child topic rate greatly parent child mean branch consequence although model regularize fit figure differential usage across
survey among simple approach greedy item type sort operation next density second identify many feasible round another fractional version fractional solution fractional ii item solution integer fractional relaxation choose j cost determine high type depict budget arm budget small low still estimate set greedy receive upper limit drop subscript since np hard use near optimal arm confidence arm indicate arm next arm budget repeatedly draw equal reward unbounded current imply converge solution unbounded budget form
store leave correspondingly statistic integral calculation classification propagate step particle change require first update occur new evaluating stay dynamic swap ad leaf computational cost incorporate leave explore benefit ad heuristic discard estimator datum discard full may discard enable operate stream eventually become use massive synthetic illustrate multivariate modern batch x online keep intend represent I budget full comprise fashion leaf ht cm full reveal even nearly full predictive second respectively sized data cost gap grow roughly online stay classification consider uci machine repository classification attribute predictor train fold full fold two stream
size specifically depend difference normalize implementation advance table refer compute gram rewrite n ns require memory small would run instead regularization less amenable regularizers elastic regularizer well feature easily incorporate negativity closely stage center alignment propose selection problem perform however hessian positive non feature suit review relation approach eq joint window
sp ex e e e kernel e e ex e e e ex value benchmark source compare form eq baseline psd ordinal regression sigmoid sp insensitive sp use sparse implement forward greedy surrogate mse accuracy size significantly learn able guarantee sp offer mse accuracy sigmoid superior seem sigmoid dataset method baseline along give
iii form iii theorems ii asymptotically imply asymptotic type iii combine maximum c type ii summarize definition maximum appendix relation show advantage respective error type estimation type supplementary difference type supplementary accurate corollary maximum type performance method g discuss type assignment
approach functional combine spline eigenfunction represent basis eigenfunction predictor project onto function instead incorporate penalty eigenvector penalize estimate arise expansion term function determine empirical structure see scope longitudinal setting allow vary concern uniqueness instability arise lack predictor infinite estimation subspace uniqueness mr spectra plot amplitude involve repeat subject subject collect value predictor specific effect random similar slope longitudinal assume decompose several invariant mr collected stage scalar mr spectra interest association mr evolve time mr show figure transform frequency pattern spectrum spectra
tc event least de know u grid close mapping construct fix set exist close remark corollary thompson heuristic multi bandit problem idea significant demonstrate well however many design thompson sampling contextual multi armed payoff context study version contextual thompson prove efficient theoretic low armed bandit mab exploration trade inherent armed bandit bandit round arm arm make choice arm play dimensional arm reward arm play play round learner aim relate likely good reward
phrase movie movie learn appear near though analysis train gram positive sentiment avoid specific gram movie set use difference energy gram sentiment sentiment explore training pair well yield examine gain energy class bag average energy linear classify document result give notably model bag bag bag science china sciences economics engineering probably slowly retain whether effectively describe rbms softmax unit efficient train rbms
simulation trial variation guide probability three importantly sharp missing value multidimensional simply miss zhang require miss observe equally fit autoregressive compound miss care user split illustrate idea survey health vs health vs fair poor assess separately parent teacher absence code response child teacher child teacher report two response generalize equation simultaneously fit child health fair otherwise page cd intercept status health teacher single parent poor health guide figure split health status split parent child health terminal terminal report behavior teacher interaction parent report teacher report child parent single parent teacher report frequent child teacher report parent reason
calculation inequality obtain substitute prove operator I definition adjoint convergence minimizer regard sequence pt minimizer applicable optimization problem need convert set sequence random feasible problem function converge pointwise prove pointwise convergence distribution theorem converge pr rank perturbation let satisfy function accord limit sufficiently sufficiently nuclear encouraging decay establish transformation directional q accord sense tangent cone set argument unique feasible result explicit characterization n r subdifferential lead together conversely easy kkt necessity n n tend characterization subdifferential equation eq since exist uniqueness proof similar easy b tangent assumption n proof necessary consistency explicit f note r yield desire conversely easy satisfied necessary part theorem due necessity condition consistency proceed theorem multiplier simultaneous know tend exist argument tend complete first rectangular self
finally easily solve substitution part unstable virtue uniquely expectation triple expectation rating applicable raise whether uniquely even proposition expectation value ce c e c invariant trivially q satisfie imply assume conclude complete square prove imply q nn precede fall reliability uniquely range straight line parameter category case estimate popular triple uniquely situation involve typically
iterative maximize scad standard minimize coordinate detailed defer analyze standard literature statistical fan assumption independent satisfy subgaussian subgaussian variable constant need constant regard assumption np satisfie appear due nature establish oracle sense support asymptotic ol estimator ol demonstrate correctly mean
create supervised learning task visualization ad create different yield insight particularly belong group place result low within group clearly experiment propose design line available entire advance line become past snapshot snapshot propose utilize place belong together place position neighbor temporal context previously summarize grouping propose consider three force intra force inter force intra force force force different group force correspond force group subject inter force meta force belong close utilize force direct intra force meta force da tu add virtual virtual group virtual virtual differ square grouping cost propose da static graph scale graph contain temporal apply characteristic dr attempt preserve dr grouping dr pose close space pose version optimize denote group membership point notice group regularization control two grouping grouping group ordinal
character lead total hierarchy conditionally layer close ard layer share correlate ard bottom fig discover common intermediate layer ard set non two subject perform opposite ard weight automatically model latent method plot encode information interact subject constrain space two projection latent dynamic encode top different whole first small dimension ard blue wide red ht abstraction analytic evaluate quality overall hierarchy mnist utility digit set
uniqueness statement slice mode belong cp tensor routine component slice span true equal step randomize span repeat attain non singular principal reach step mode
potentially run distribute accordingly acquire want answer completely fast method utilize distribute extreme update neither appealing bit completeness relate lie minimization widely fast force learn bfgs offer robustness relatively fail situation acquire previously situation old certainly leave increasingly similarly reasonably regret reach error descent combine advantage number bfgs sized batch forget change significantly outline problem interest widely outline algorithm
redundant tree available randomly random forest recursively split tree focus detail framework regularize tree way select splitting belong belong enter need upon gain select criterion tree consist tree tree build relevant regularize simplicity single regularize ensemble indeed form tree ensemble regularization feature need measure minimal markov minimal
derivative reversible distribution let first since reversible satisfie reversible reversible invariant satisfie show upon define proof expect upon draw two geometric irreducible chain number setting indicate lack ergodicity utilize proposal fall could proposal bias towards centre global end define q autoregressive similarly lack r b iy recover target attempt improve markov kernel concentrate markov analogue extension hold irreducible admit invariant mild proposition factorize x follow artificial dp decay suffice x
informally unbiased formal justification taylor bias parametrization like equal adjusted mse inference linear function effect empirical blue suggest blue suggest effect test construct fix simultaneously matrix explicit section accordance approximate scale distribution estimate analogy derivation estimator denominator degree trace estimate derivative detail
irreducible appear add jointly jk jk jk w accept probability q remain variable slice build truncation completeness briefly recall admit stick break slice latent give allocation dirichlet q update augmentation technique w accept acknowledgment thank pr helpful work european intra fellowship preference college european intra european fellowship rank develop nonparametric popular random specify effective simulation model apply preference degree amongst existence preference establish determine subject matter datum consist order list object ranking arise shall top school consist private application chapter provide detailed item assign represent assume item item among item choose select proportional overall partial select many collection unknown sensible item assume possibility item appear I sampler available infinite unobserved group outline section gibbs limit mixture whose directly
rate level nystr om question translate version pca add little overhead rf combined square loss g group pca classification approximate unlabele covariance turn select organize summarize describe elaborate test analytical chebyshev imagenet comparison kernel visual system good performance among histogram kernel propose cross different bin histogram include also hellinger shannon invariant kernel approximations rf nystr om sub operate kernel long slow speed propose good learner embed deep code prove
cluster shape conjecture method one another worth investigate fully derivation proposition statistic relatively derivation test assumption calculate distribution let likelihood simplify could statistic derivation statistic likelihood ratio derivation statistic normal x lead lead proposition collect region notation statistic define note region show case signal consideration region region noise show c straightforward similarly bn stochastically eq boundary
much fast especially dataset quickly depth assessment require acknowledgement work national centre grant perform grant scene allow one video classifier still ensemble computer application image action reality attention aim bring wide generalise package briefly bayesian version list variant discuss
function use basis fourier natural structural move posterior q integer wavelet inverse probability event nc j n ng nb ng nb j later two q correspondingly quasi write log quadratic result map quasi borel integral soon gx relation soon integral hand hence practically expectation relax keep essential establish technical since theoretical frequentist allow use level approximate basic theorem I usual minor primitive iii used identification assume identification conditional reader discussion completeness domain ball substantially relax however analysis inverse continuously invert plausible know value decomposition see wavelet system suitable quantify ill basis quantity ill least l ill pose
type prediction dataset relation type relational incorporate relation fraction type link prediction observe help boost model capture correlation among reveal influence link prediction performance likely vice versa relation impact performance follow paper propose new social introduce specific addition object provide treatment derive conduct dataset relation reveal distinct network evolutionary link scale human association provide paris
intra block observation sensing sense noise lead cost together covariance th semi th block parameter cast specialized variant block exploit structural beneficial block class compound auto etc cost type maximization author
formulate evolve intuitively change human page dynamic available spirit research place graph row probability manuscript utilize walk diagonal degree none column linear system table section rx computation decay time vast idea incorporate update related et vector change new
improve sample period precisely summarize estimator observe difference become indistinguishable significant way length accept fail step therefore adaptive good contrary accept step adaptive note result finding r ccc r r ccc r iii difference average exact l l l approximate estimator conventional histogram limit conventional time available whereas period sampling period bias decrease shown summarize iii h tt accept fail step shown accept big iv exact iii iv result realization linearization euler thin partition guarantee simulation realization two
patient report breast year respectively survival shorter report event table interaction comprehensive network pathway interaction remove pathway r package node undirected edge protein protein type direct interaction package microarray probe accordingly microarray map ignore network approach
follow goal mind tradeoff tracking residual thereby track manifold vary old cost online estimation impractical address cost f form subset sequence previous illustrate dash red union correspond past detail apply sequence residual assume point I detect soon method streaming setting change incorrectly change stream alarm time point detection false long alarm multiscale online subset subset use multiscale multiscale harmonic literature subset parent child roughly cover child tree index subset leaf use index tree notation transpose matrix matrix offset hyperplane origin hyperplane parameter specify ellipsoid capture tree virtual maintain fine leaf instantaneous need explain tree subset use determine
constant fraction weak tight see agnostic approximation agnostic construction agnostic thm thm agnostic regression thm formulation cl pt agnostic agnostic thm thm second deterministic opt opt opt opt nk whose correspond point response table opt agnostic probabilistic fail probabilistic algorithm away event choice sense agnostic success unknown approximation agnostic construct without bad vector fact exist bad
mlp output perfectly detector fact never combination feature input generator generator cause high activation hide neuron maximization base technique normal mean norm initial patch find feature activation maximization strong generator neuron htbp correspondence input pattern discover maximization deep demonstrate correspondence mlp four detector inferior detector mlp single detector visual appearance detector therefore denoise high capacity seem operate principle layer detect noisy patch weighted patch consider see mlp difference mlp previous one output patch somewhat generator figure detector generator patch generator look similar feature detector look seem focus input look detector focus center input explain patch input patch patch correspond center region patch correlation fall pixel outer denoise center activation binary mlp single lie essentially binary dictionary hide zero layer information unit hide entropy train mlp mlp mlp unit layer unit detector generators generators unit low generator high feature detector high entropy feature low latter seem
elaborate proposal transform correspond learn particle strategy however user reasonable indicate resp describe simulated figure modelling determine persistence really characteristic range behaviour properly must understand spectral quantity coefficient belong particular towards consistency size prior independently outside prior fact consistency strong indeed simulation ghz resort language library program author web page smc final approximate posterior correction particle cpu minute variability run top plot see plot reveal rate birth death may successfully particle go axis grow slowly roughly smc sampler step time hour minute satisfactory result box expectation approximate ten run identically histogram bivariate sake top plot repeat run smc versus colour proportional
following provide sampling use construct specify similar except apply pack recall consequently choose equivalently somewhat theorem though emphasize choose loss since pack consequently complicate privacy begin state let accord support proof appendix binomial differential combination channel f mutual information combination channel lemma thus obtain choose q property differential require index establish bound amount underlying bound locally differentially private packing element induce mixture summarize previous paragraph hold locally differentially private interactive place proceed establish establish require formal proof proof corollary identical nearly paragraph uniformly interactive differentially private channel immediate low inequality separation b note le identical argument pack conditional contraction proposition construction le obtain proof outline exhibit exhibit special packing hypercube packing packing usual conditional pairwise separation pack discrepancy theorem hinge loss since distance bind indeed pack hypercube sample long channel locally differentially locally differentially previous paragraph hold choose note universal result apply repeat completely imply bind noting complete low reasoning choose result mirror obtain match second statement mutual follow uniqueness high mutual allow level allow must must must must multiplier apply technique convergence
form constraint q since q computation definition bind acknowledgement part work network p project grateful constructive discussion mm mm lemma group effect france universit abstract huber regression operator satisfactory method huber call elastic coefficient cause ridge ordinary huber enjoy approach compare elastic satisfactory illustration purpose keyword elastic huber oracle property implement http fr subject outlier encounter predictor eventually response heavy error ordinary square ol estimator huber type estimator hand linear analysis sparse enhance fit easy interpretation prefer simple put light relationship covariate tune suffer satisfy nontrivial consistent differently
identify validate current gene credible significance great dataset non constraint experimentally validate lasso routine significance threshold roc various value varied threshold set roc dataset plot show purpose sensitivity specificity reason lasso coefficient use false auc rr bayesian large term auc note meaningful select auc perform constraint infer protein target substantially base apply broad discuss gene detail interact
preserve local structure dimensional manifold propose maintain nmf performance application sample partially g image face kind corruption corruption treat noise estimation nmf kind underlie recent deal partial usually corruption give ahead ignore assume position
robot essential maintain physical coordinate frame sensitive error error derive identification explicitly rewrite feature follow look kronecker notation element dynamic pick form give transform repeat representation convenience discovery system simply interpretable track kalman need approximation plug kalman h gps state collect observation u u x coordinate compute step motion pa simultaneous localization mapping algorithm identification desirable include consistency optima optimization multiple track low requirement track extended transition severe non gaussian posterior encounter finite demonstrate world theoretically justify comparison good attempt estimate location hypothesis filter optimization likelihood
w diagonal see implie theorem residual induction lemma second approximately satisfy assumption let algorithm permutation let projection extract column condition say generality projection reduce first inequality projection orthogonal orthogonal projection last w k actually empty imply extract th ip ik al recover column order compare bind theorem one separable matrix without separable let recall matrix hull add separable et identifie see equal column eq theorem matrix tighter particular extract imply large well bound expensive least et al rather difficult trial separability sense separability satisfied summarize simple growth hyperspectral comparison depend solve program algorithm
application accelerate contribution characterize hardness value direction directional index require explicit representation branch max exact exist technique space shift radial algorithm set posteriori shift applicable shift invariant equivalent input solve study shift embed inner product approximate value follow representation locality sensitive lsh locality sensitive hashing lsh normalize kernel without normalize similarity rigorous technique branch bind restrict runtime suggest preprocesse kernel example kernel require un kernel recommendation normalize inaccurate item
roughly impulse additive provide lead outli bayesian detect describe additive explain illustrate methodology simulate well set concern ip address server department university conclude poisson identically variable
calibration misspecification confidence exponential evy range pricing calibration focus parametric cf therein l evy approach parametrization misspecification activity evy construct confidence calibration neutral measure option price calibration l evy process assume evy mass sd evy component second set mass assume evy fa sd modification improvement square quite fast fourier whereas focus theory realistic sample size jump finite confidence derive confidence activity set sample self decomposable scenario sd well coverage simulation variance introduce use self decomposable option good residual datum calibrate model option behave
company even sale market assess seem interaction estimate finite solve krige stationary field mutually normally spatial however conditional interaction term flow paragraph problem address paper develop spatial though refer location collect measurement aspect sense general differ suppose eq q measure measuring say h ultimately act measure measurement distinction value measure location n paragraph estimate main statistical analysis arise follow spatial h parametrize set pattern pattern assume
estimate terminate whenever may happen design small implementation allow predictor stop convenient complexity ode solver repeatedly evaluate derivative path stop event check derivative semidefinite inactive penalize along optimizer coordinate number segment ode evaluate hessian glm loss take detect inactive thresholding per evaluation optimizer utilize warm fast adaptively choose size simple software ode ode package matlab differential equation user fulfil knot scad mc solve ode partial due scad penalty knot mc ode rarely numerical role lead parameter path usual implication next follow certain avoid procedure aic variant frequently recall bic laplace fisher observation maximum posteriori likelihood plug aic derive operating necessary fall path somewhat pick
improve confidence hc bias wu preferred group mean robust conventional estimator unclear pool variance analysis become economic adjust question ols adjustment rapidly unit standard brief remark actual substantial imbalance cox inference covariate imbalance adjustment perhaps either consequence occurrence randomization population mean member assign biased finite context equation bias square analogy lead tend depend largely importance omit quadratic covariate omit first result determine randomization adjustment suggest focus social illustrative conduct service financial college student except school randomly treatment service third service group
logistic adaboost without attain analysis produce output iterate must focus suffice mean deviation cause issue apply choice apply generalize limitation present boost tree manuscript organization primary force iterate finite break piece core complement albeit hard core direct establishe effectively carry instance significance prove sample quantity risk display simply many bound consistency appear simply guarantee share arbitrary structural tree constraint split meet risk note support variety associate mention related generality sufficient boost work map weak learner function convenience operator abstract convenience search classification instance boost throughout make function notation hinge origin requirement three precede loss arbitrary
go bandit derive ucb chapter bandit nature process markovian three distinct playing effectively ucb randomize call markovian case stochastic adversarial refer survey markovian bandit player forecaster implement bandit performance horizon play arm sequence arm study select receive reward play horizon forecaster forecaster allow averaged definition expectation forecaster pseudo weak since compare expectation regret build arm select arm possibly round forecaster reward independently past reveal forecaster pseudo write seminal bound chapter describe naturally stochastic bandit game quite recognize instance motivate theoretic machine step gain would go forecaster select gain still minimize regret adversary mechanism gain mechanism independent adversary forecaster behaviour adversary instance clearly adversary player randomize pick gain game simulate presence adversary gain depend randomize player round consistently choose gain non connection equilibrium observe theoretic equilibrium behave differently opponent change behaviour interestingly minimization round forecaster choose possibly external randomization adversary gain vector possibly external randomization observe arm adversarial bound forecaster opponent irrespective opponent adversary start since hold opponent randomization forecaster play opponent playing topic david seminal payoff observe opponent move consider game round observe turn equivalent adversarial non adversary simple playing recently propose approximately discover science apparent connection bandit bandit adversarial arm process probability state change markovian fashion reveal player go bandit process compete resource project get resource matrix nature seminal provide optimal efficiently notable special markovian bandit assume state change update reward observe markovian bandit economic different adversarial
especially mean advantage disadvantage soft feature view proximal minimization present chapter demonstrate connection iterate start appropriately descent negative regularization dictionary chapter eq q separable element therefore unconstraine optimum set soft literature slight split encoding obtain proposition appear threshold feature separate compete entity triangle code treat proposition important insight proposition nice code study feature code proposition tell even approximate code even typically see single sufficient show discriminative three question possible encoding lead extent investigate experimentally examine soft feature variant proximal code present chapter fast soft first fista
datum study ability estimation dimensionality large cause rank small direction difficult distinguish projection tend separation panel estimation min rank min max min demonstrate reduction axis align noise contain investigate project onto direction reduction method cluster consecutive within sir implement whenever randomization projection include scale sir evident rich scenario randomization perform regime improvement scenario ridge h tb rand ci rand ci reports ambient panel report bar standard dimension set assumption strongly estimator randomization latter sir randomization pca denote randomize implement regression behind covariate response latent decomposition bs independent noise principal q latent factor
notational convenience estimator corollary give section fast slope take conditional quantile bound previous convention apply address tight restriction rate least nonempty side indeed prop file three criterion investigate simulation singleton xu define likelihood asymmetric eq aic see may analogue bound bic du du integral replace summation monte finite case repetition website quantile correspond spaced point quantile simulation summarize figure table criterion error error show
aic often cross normal glm simplicity drop covariate vector zero denote response normal aic fit square degree freedom stein unbiased take degree angle lasso freedom nuclear normal derive unbiased comparable non orthonormal b convention expression freedom regularize fit freedom several aspect least degree third surely rank degree freedom qp exactly plot well I count equal freedom na I count search number effect freedom theorem limit require existence exist give conduct first regularize covariate second classical regularization regression nuclear lasso covariate penalty power regularization simply correspond penalty coefficient tune instead examine nearly unbiased scad thus
compared show advantage viewpoint convenience exhaustive run exhaustive moreover efficiency deal exhaustive compare choose competition look fig show find feature although set hard specify ht depict follow find specify acceptable find feature discuss third competition well case user know viewpoint rough view aspect optimization analyze aspect first extension concern since require rough form accord cover rough deal neighborhood system rough well might fuzzy example rank directly
generate lsh thresholding thus lsh define hyperplane different opposite hash point match cosine specifically point lsh guarantee within time optimal empirical study lsh efficient hierarchical decomposition vision extension lsh lsh probe lsh propose lsh lsh address limitation locality hashing shift hashing mapping invariant convergence guarantee well relatively base need preserve pairwise distance database
sequence round sequence consider proof eq forecaster share simplex extension forecaster share kullback leibler divergence p generalize theorem bregman kullback bind rewrite summing apply I I universit di sup paris within team classic paris sup paris paris france lemma definition mirror regularizer achieve carefully share equivalent generalize discount capture
polynomially relaxed stage bundle improved complexity preference economic begin modern explanatory utility finitely price observation piecewise concave reveal excellent survey utility finite view come hypothesis length relate give think future decision hypothesis produce polynomially circuit intractable utility reveal preference computational goal produce monotone utility non guarantee agent utility jump continue function efficient polynomial complexity nice work consider reveal set rather reveal bundle select value bundle bundle arbitrary agent bundle pair price bundle
probabilistic long modelling mixture component source dp process I speak interpret whose measure strength
mixture recent advantage parameter significance illustrate real keyword adjust rand index validation significance nonparametric integrate shift plug choice full note seminal concerned unimodal cluster shift cite preserve density crucial significant dimension extend scale digital visualization new problem closely gradient finding application medical imaging sense rigorously analyze one curve embed ascent path gradient concentrate estimation useful tool density arbitrary formally crucial bandwidth multivariate setting bandwidth consist definite bandwidth constrain degree along bandwidth identity mean smoothing coordinate direction note bandwidth validity previously check unconstrained counterpart reason smoothing encounter analysis detailed analysis derivative simple lead substantial become bandwidth selector develop sophisticated drive choice applicability
e r qp fp qp fp qp r qp fp qp e e ball next display project ball varied reveal ii value time approximately large moreover run apparent code scale bar still nontrivial projection relatively currently medium problem htbp group feature separate important less important information task mix usually matrix
unimodal multimodal gaussian unimodal asymmetric multimodal mixture unimodal rotation angle gaussian add extra rotation cause dependent across vary angle dimensionality gap gram component break permutation quantile type hypothesis accept reject experiment rate dimension solid acceptance rate test span dot reject line dot figure statistic kernel dotted line statistic base joint entropy span graph propose notice correction configuration small acceptable type statistic estimator method notice method angle determine hypothesis propose independence gram size therefore gap monotonically phenomenon relate entropy notice smoothing associate slowly characterization resemble quantum
convex bias term regularize absolute obtain set runtime optimization find stochastic gradient descent sgd sgd find runtime favorable sgd approach stop tend begin especially reach fast convergence become interested solution coordinate solve conjugate namely associate optimize dual variable optimal know immediately gap sub optimality version sdca round theoretical understanding gap sdca loss paper smooth equivalent smooth main finding smooth hinge precise several experiment large sdca sgd sdca understanding rate svm basic ascent establish mean achieve
evaluation devise prohibitive typically accord acquisition acquisition lie estimate point estimate apply functional statistical minimizer relate highly hence intractable
pc loading second project one pc loading project attain loading model enforce orthogonality unnecessary task sparse enforce orthogonal pc involve pc include etc spc etc method minimization perform variance objective function sparse equivalently formulate eq separate respect column follow small divide solve construct construct convenience rewrite form dimensional closed form I present equal omit element th equal proof q large element
forward smc f n filter acknowledgement project initialize input thank associate enhanced article un dx x px dx dx p eq semi q q define backward normalize td td dx n n dx group tf dx
recover graphical rigorously method compare maximization step support part nsf award fine piece hereafter refer address graphical
version simple skew simplified advantage expectation calculate e reader different form skew efficient th matrix degree shall fm finite distribution fm component fm generate adopt denote shape respectively sample fm implement admit mle yy framework complete likelihood unknown fm implementation alternate change log use maximize iteration calculation expectation bayes membership component current expectation positive hyperplane evaluate express central recently truncate multivariate correspond
preferable h suffer limit parametric semi parametric model paper possible parametric context derivative rkhs rkhs provably iterative study exploit universal propose safe discard variable prediction selection nonlinear toy towards role differ previous local scheme computationally algorithmic solution general distribution lebesgue differently explore analyze selection view computation consider towards natural improvement possibility consider supervise naturally suggest generally plan differential operator selection beyond lr absence would discussion suggest proof sm discussion science mathematics mit biological institute brain department artificial intelligence laboratory integrate project health child grant national foundation nsf di support usa science technology b foundation x x define thank empirical property quantity proposition meet operator continuous derivative schmidt partial derivative schmidt base hilbert schmidt operator respectively proposition extend henceforth satisfie follow write f problem
exponent brief proof entry length whose specified think compose column later property mapping encoder q mapping produce column compression q style codebook column exponential construction choose column develop automatically decode encoder binary binary indicate zero bit involve share common code two distortion encoder decoder achievable
probability comprise specific together membership estimator identifiable contribution assume correct situation primary possibly circumstance toolbox circumstance usually infer unobserve preference show sufficient contrast conversely reduce component belong impossible sharp region provably accuracy high theoretical grow contrast consider small amount vanish impossible inference however network parameter highlight difference impossible inference come introduce infer justify fit reduce approach deal search common way interaction far apart originally
represent pattern share b evolve share middle model rely evolution vary put alternatively therefore algorithms provide latent might estimation successively follow w em scale mixture follow slide maximize accord reduce convex previous conduct carlo conduct bayesian requirement prevent implement particle employ hasting latent x
time fast discriminative object representation maximize inter object region boost instance drive etc object representation design adaboost robustness appearance cause rotation illumination tracking liu yu present base mechanism boost lead tracking build online weak pixel wise employ object localization instead boost image patch object attract attention line tracking distinguish classifier adaptively discriminative supervise base constructs review cosine signal first dct section dct matrix dct compact dct dct goal dct express discrete digital mutually cosine frequency information dct dct n reflect energy e signal texture discrete discrete dct whose mutually uncorrelated furthermore dct sparse compression tracking dct cosine form write formulate dct decompose dct operation z terminology algebra terminology tensor algebra dct n accordingly dct special dct
take marginal filtering sec mean cross integrate source uncertainty law total expectation expectation take xt rewrite fm se center input predict xt dimension gp across na f ji integration serve test gp tractable choice combination integral signal dimension see compute law
dependency relate number report publication eq play population furthermore incidence plus age profile patient uniquely force incidence uniquely qualitatively forward force
g way connect ht ccc observe inner style circle sep name z scale sep style inner sep name h z h observe circle sep hide circle inner name name name name consistent topology capture recover motivated design way connect join together tree variable relation total taking resolve reconstruct set suffice make method new main contribution approach method practice test make order
berkeley segmentation graph orient generalize boundary detector propose pair connect parameter segment edge segmentation output range simplify optimization perfect code solver tolerance add constraint work describe recursive contour early low bound lp relaxation term potential neighboring relaxation unary potential cycle collection cycle enforce consistency fast house implementation branch remove solution inconsistent namely within component add enforce consistency edge bound comparable tight low give batch cut upper solution histogram comparative
w pn op pn ij v z ni z row lemma z z piece n corollary applying theorem induction mod pick similar mod mod identify identify order may define z follow substituting term lemma I c w p x op x op z cn ki op union bind w c op I op w op term I iw I conclude w pn combine pn argument ix h z pn small h h complete variable argument show ix ix ix I apply I
penalization variance hold penalization penalty equal concentration start analogous particular proof p p eq bernstein depend constant notation definition eq tm lemma eq deduce computation train set compute j risk train mn cost penalization criterion j eq v yield initialize datum unique j iid random value measurable common distribution q provide simulation result analogous illustrate extended version compare setting standard deviation datum drive separately loo cv sample deviation drive consider knowledge l loo cv cv complete test validity without sum mm right provide mn mm setting figure b mn v b b mn b mm theorem corollary definition remark study fold square goal theoretical choosing minimize prove non
show p continue px qx n px qx x exist chain monte mr author mr mr mr mr mr author mr mr avoid mr mr mr mr mr author carlo markov mr mr mr mr algorithms automatic author carlo mr metropolis mr author mr ed sampling chain mr mr mr mr mr mr unbounde mr graphic theorem support advanced research fellowship capital award project markov chain monte marginal chain spectral normalise pseudo chain many hold gap equivalent ergodicity unbounde recover case independent hasting super contour imply asymptotic pseudo marginal accuracy estimator increase interested defined achieve scenario carlo metropolis hasting hasting call proposal implement wise either unbiased nonnegative estimate paper
differ take instability define I cluster factor cluster fair clustering instability cluster item assignment figure depict instability request instability request pattern number instability two median instability bar instability minimized request pattern instability pattern randomly exist division pattern overfitte phase extra structure result instability instability increase realization subset analysis select pattern fit request facebook application methodology request request differ find request technique request facebook application must review rating application facebook facebook application perform instability outcome facebook fit tb entry sort decrease color conditional htb
come contain information correlation region multivariate correlation spatial transition see correlation random prior geometry geometric incorporate pay attention field change bottom layer look layer interface interface field positively interface smooth key consideration remain thing certain transition manifold reasonable interpretation leibl divergence completeness account definition need exposition boltzmann entropy arbitrary positive definite matrix line q nice length curve minimal
non thus correct estimator empirical take well use define numerical artificial dataset mm kde c kde kde denote normal mean kde behave depict sample kde show accurate kde depict next kde distances distance distance kde estimator cause fact kde tend density difference tendency kde base typical reasonably true
nd introduce hidden variable background nd pz w pz bayes pz pz constraint version pz derivative nd pz pz nd nd pz obtain nd pz nd
performance sequential derive process parallelization asynchronous update share latter suited communication compute machine cm paris france paris paris france universit paris paris france
contour see contour action pair state couple fully observable obviously help
draw body demand universal generate demand uniform generate demand distribution go round execution demand time execute execute demand environment universal demand determine output use mae square simulation environment
discuss resolve convergence gradient contrast reduce gaussian smoothing see minimum subsequent less sf sf comparable case sf considerably pair sf achieve achieve sf exhibit sf experiment perform sf trend sf pt c c sf sf sf sf sf nb segment discussion list retrieve essentially choice mean wide previously applicable restrict sf cauchy search similar table also trend small v particular quite may prefer sf nature table relate behavior sf justified analyze gradient ideally tune difficult pair theoretically
isolate adjacency permutation unlabeled node keep word minimum permutation unlabele correspond provide g node count argument supplementary state fraction node ise homogeneous degree scale eq nearly match frequently trade complexity datum fitting criterion schwarz stock return software code ise model depth potential keep randomly potential employ latent graphical relationship word occur latent graphical word constrain keyword select indicate appearance divide equal group purpose financial modeling dependency consist stock list sp dataset performance graphical simplify interesting sophisticated kernel complexity fit well avoid overfitte propose method lda module control neighborhood present range recovery theoretically guarantee h learner criterion threshold threshold thereby output choose value reference dataset experiment stock return choose previously discuss
chemical specie solution specify reaction pair reaction reaction place current number must chemical assume reaction population ax reaction together state reaction specify jump occurrence poisson reaction specify initial exact relatively partial physical discrete parameter property kolmogorov equation master evolution probability master intractable observe process incorporate show markov posterior computationally strong require carlo approximations systematic obtain approximation physical yield fluctuation extensively see applicable fluctuation system limit describe set state dependent master likelihood brownian increment thus limit applicability methodology transformation likelihood less inference linearity non
space unity mode define act scaled unit temporal dimension embed illustrate temporal closely resemble fig embed evident representation mode recover sec embed prominent datum mixing find place temporal mode useful physical available dataset influence factor non markovian initial compatible factor dynamic coherent candidate embed extent describe degenerate mode evolve quadrature instance capture prominent low frequency involve organize situation information separate verify mode periodic frequency window
arbitrary result restrict maximize say try equivalently variance common approach also criteria point disadvantage absence density measure probabilistic model desirable permit comparison may extend model use class density classification posterior membership compute thus appear estimation
exception rand interesting gap believe available sc affect suggest hmms instead automatic annotation music digit estimate annotation auto ms gmm emission long model therefore annotation retrieval account consist provide annotation range acoustic song audio coefficient window audio instantaneous song audio approximately second automatic tag modeling audio tag database tag database processed learn song tag song level ms pool together h algorithm tag tag tag song extract feature song tag song annotate retrieve tag rank tag tag sc write retrieval tag h tag use necessary implement sc approach song ms single hmm poorly hour times annotation yield sc song hmms centers hmms method sc hmm cluster center estimate hmms assign cluster hybrid song hmms hmm form h mixture tag proportional hmms cluster tag audio empirically ram h overlap audio evenly subsample sequence actual song non overlap believe em roughly similar still slow report running song require processor oppose would extremely intensive considerable maintain mixture bag tag feature far consecutive tag tag annotation automatically song precision recall score tag measure tag retrieve positive ranking
learn often large tractable exploit extend recent advance carlo search avoid agent belief mdp episode outcome simulation many future obtain belief innovation considerably representative competitive algorithm consistently algorithm sampling belief avoid bayes expensive simple mdp suit planning domain reinforcement illustrate benefit infinite state challenge intractable generic make unknown mdp mdp sa discount component mdp tuple mdp use unknown distribute observe history belief ph dynamic current inside augment space tuple
generalization cover construct ball center ball center call mass ball small ball disjoint ball concentration ball pr lemma constant complete constant finally question extend density give converge length partition fine chapter suggest many application plug distance estimating present simplify go approximately similarly consider
integer expense adopt anonymous reduction project let r qp projection definition relax p f pp p f pp p opt statement theorem conference title shape total projective include median cluster projective total reduction type greatly early result total sensitivity positively projective projective problem high article shape fitting shape refer b
feasible sequence banach converge belong satisfactory solution sufficient existence feasibility condition banach rate banach satisfactory time advantage important satisfactory see global mu mu j mu mu optimum remarkable distribute banach range aim possible fast rate speedup technique speedup satisfactory reverse choose rate big converge banach learn convex use speedup speedup quasi static lead learning find satisfactory realize context express generic satisfactory payoff generic reverse rate banach iteration target background noise load satisfactory convergence satisfactory field banach reverse speedup summarize banach banach iteration get clearly error iteration satisfactory solution speedup la acceptable tolerance study additive simplify equation per class requirement speed acceleration partially player numerical payoff happen hold observe realize observation
hold background refer appear sequel value possess expectation satisfy provide operator e operator specific play covariance operator definite self adjoint depend process result space banach proposition need cauchy statement conclude fourier eq statement orthonormal denote lebesgue since turn imply fix negative adjoint orthonormal trace v stationarity verify negative adjoint triangular kronecker latter two imply converge continuity adjoint negative get e cauchy schwarz dominate complete g p eigenvalue continuity fact e c observe complex conjugate eigenvalue spectral limit transfer operator sl filter spectral cauchy sequence prove appendix l yy routine argument iii dependent z r h cauchy inequality furthermore deduce sum independence conclude e e de office research support science policy office cm cm
type stage stage transition start state select take unknown receive immediately terminate problem define transition bandit sec radius policy maintain distribution stage observe payoff norm multinomial employ solve augment mdp action additionally act optimistic probability policy maintain dirichlet see distribution reward draw sample mdps parameter individual reason correspondence
relative algorithm propose highly compute additive eq also randomly time simultaneously row square norm row sampling subspace sampling ok expectation intersection dominate svd choose least seek devise mild requirement column optimal selection column aim choose construct achieve minimum constructing well hard polynomial particularly bound
address problem partial fundamental service automate focus fully decentralized page localize community localize partitioning available leverage decentralize execute acquire node need exist exchange among scalable comparable exist collect china topological already direct friend topology user many community contribution formulate new constraint page justify applicability user design community evaluate use scale social organize survey work formulate topology social proposal commonly vector possibly subset community satisfy intra community inter linkage include maximization cut minimization popular combinatorial np hard researcher eigen approach several batch focus detect community pure community researcher try incorporate enhance algorithm combine topology
fails successfully classify feature strictly require correct extend cascade mostly cascade several stage classifier either reject input pass prediction input cascade enforce reject classified input later however structure skewed class imbalance generic detection reject response haar cascade suit expert optimize building cascade weak note heavily stage regression boost work learner gain feature introduce additional learn tree reinforcement
one fill table set encode function user manual call function optima member semi
drop assign define ratio substitute equivalently notice give substitution let leave multiply side mean exact implementation specify hermitian matrix eigenvalue remove normalize trivial automatically step eigenvalue number feasible choose minimize say n eigenvector eigenvalue eigenvector need appropriately combined feasible matrix consequently feasible eigenvector increase make small small eigenvalue definite generalize eigenvalue feasible trivial drop normally great satisfying whenever care toy follow affinity cut constraint want group cluster happen make distinct feasible mean want thus cut plot fig group nod group indice entry optimal relax bias toward unconstrained finding similarly formulation clustering interpret colored way node color assign node cut graph entry interpret relaxed sense minimize interpret
case disease duration duration exposure population risk usually formula go describe risk expense institute diabetes year broadly incidence need determination sometimes duration exposition
triplet approach rather allow counterpart bound convergence rate framework cover establish accordance robustness bound introduce propose assumption specific form stability induce regularizer limitation parallel analysis tight tackle regularizers derivation involve compute limited accommodate robustness tackle triplet constraint illustrate following propose ability metric notion robustness originally notion characterize metric robustness
dataset patient sample pre statistical selection run cross dataset pls pls double check ht cancer cancer cancer normal posterior extremely excellent pls use pls extraction also notice widely g encouraging hold water problem experiment depict select pls pls pls common collect use pls find gene confirm pls gene believe believe patient analyze cancer simulate cancer gene forward expression generate formulae generate identically q correlation simulation facilitate gene gene interaction scale mean value gene x control patient perform classify simulate normal advantage approach know dataset responsible classification rigorously examine gene gene interaction heterogeneity interact gene assume dominant allele allele capture potential mode generate train dataset interest thousand gene greatly microarray nearly gene perspective gene interaction disease discuss different interaction associate computational article search level seem impractical genome point drive far home
approach carlo smc likelihood free therein concerned sde contaminate system dimensional statistic real represent increment standard motion matrix assume regularity directly might conditionally inferential method error e tx tt nj system concept paradigm methodology letter hasting mh regularity convenience q simplify likelihood generate numerically sufficiently accurate approximation scheme dimensional simulate draw stepsize numerical euler sde law gets simplify whereas free methodology attractive simulate function previously mention typical sde trajectory posterior large sample abc useful section difficult generalize multidimensional methodology propose sufficiently close summary tolerance abc simulate states observation sim embed abc mcmc chain sim abc kernel
effect membership discover globally predictive intrinsic property also transfer cluster datum dataset suggest outperform state art behavioral relational become object citation co many relation object discover among relational predict unobserve share common service relational make statistical challenging correlation give various structural relational characteristic imply divide within relation object form dense take social dense circle group relational generate proportion rare largely observation contain relational side
word topic informative assume indicator specify simplex dimension informative know whether incorporate word generative include divide entire vocabulary informative word determine word token step ib di b di di di corpus corpus token dirichlet multinomial prior number token th vocabulary token dot represent token ni simulation study generate third lda
guarantee skeleton times eq operate independent observation write derivative brownian used involve stochastic independent diffusion work give brownian motion volatility stochastic popular modelling financial asset interest financial derivative asset diffusion volatility diffusion sde reflect ease exposition stochastic volatility leverage still paper volatility diffusion denote form pair multiply volatility normalise brownian latent survival survival target aim hazard reflect event diffusion parametric formulation assume positive diffusion motivation consider occurrence event give eq obeys exposition unit although observe hence brownian motion censor refer simulation diffusion volatility survival aim efficiency unit cpu environment carry latent diffusion survival language equilibrium suitable burn period sequence amount serial correlation autocorrelation truncation sampler time burn period match variance obtain independent ess relative sampler mcmc application volatility hybrid univariate trajectory context assess monitoring draw report ess ess reflect percentage draw consist propose brownian experience literature aim rwm mala performance
article review sparse measurement sparsity different channel joint sparse joint tree former utilize utilize channel bind joint practical happen mr image tree naturally b channel physical object nature human tend boundary several connect like forest forest mr form unfortunately exist recover channel separately exist result well htbp previous channel support subtree kind forest kx ki similarly forest coefficient channel zero zero forest forest structure solution intuitively tm tn measurement satisfy
performance experimental construct set site constrain non negative turn I unconstraine p basis vector moment experiment section describe entropy choose inclusion gain kullback posterior inclusion take differ second quantity compute maintain loop hyperparameter construct minimize choose computation inclusion hyperparameter site site alternate estimate parameter conjunction selection validation except summation happen view nlp viewpoint
lag adjust dimensional upper determine instantaneous preliminary notation cm shall instantaneous unconditional test nevertheless remark eq instantaneous hypothesis lemma clear gaussian note expression simplify notation optimal date base one step predictor instantaneous causality non triangular instantaneous causality pp test case instantaneous obvious test standard q block usually probability vary interpret instantaneous causality interesting entail build unconditional consequence variance test spurious stationary general iid consistent
dependent svm ed bm bp ed ed kernels ed cs exhibit among ed rank complicated ad hoc cost cost bb winner rule ed cs ed ed bp svm kernel ed bb svm ci ed bp ed bp bm svm svm predict recently view sensitive function cost unlike separable enforce large slack offer optimality implement rule cs svm sensitive theory metric classifier sensitive cs cost evidence confirm superior method convex r nz conjugate duality induction eq q primal dual regularizer decision I formulation introduce hinge q v conjugate ci hinge bp cs hinge q moreover rgb rgb rgb hinge sensitive draw connection probability cost manner
tendency favor tree surface cut biased cut leave node optimal tree edge parent edge receive weight cut parent cut node entropy map section train independence rest maximally construct classifier well figure two section step feature output network input image multiscale pyramid ideally linear categorization pixel achieve target pixel elementary deviation multiclass location distribution pixel training maximally hard target discard available produce descriptor kl component label assign cost stanford background evaluate semantic dataset public
produce much low leading although gain much going everywhere extreme obtaining take space discrete accept generic set move improve acceptance much possible move slightly much large concrete optimization os illustrate fig one try maximize indicate curve whole easy check immediately due maximum reject refine refinement max max find small predefine threshold usually completely task view range consider
test precisely mean I frequentist good I near distinct assertion interest predictive ratio specifically limit r lebesgue optimal assertion respect sense set plausibility ratio favor iff determine pearson statistic construct free conclude side I classical notion optimality restrict class set light familiar I exactly illustrate powerful unbiased random plausibility optimal poisson choose unbiased integer critical test x construct predictive random depend comparison show plausibility plausibility xx black gray line general set dominate group show mention choose particular simple dimensional equally trade make simplicity optimality currently define simplicity theory function default date satisfactory picture present unknown automatic long run calibration identification auxiliary present I calibration belief mild theory develop help resolve illustrate belief predictive validity convert
cell preserve count choose select cell non cell typical associated current temperature whether accept neighboring drawing uniformly accept less likely make temperature move ingredient schedule general temperature favorable neutral slow method rely one close quantify euclidean projection take p give collection close nonempty intersection alternate convex recursively nn nn odd closed method alternate
measurement signal domain take consideration multiple l encourage domain enhance mean coefficient equal traditionally synthesis linear predefine call outcome operator representative element
sophisticated structure advance decade focus universit paris machine relate raw limitation limited handle description rich relation network etc quite world aspect observation hypothesis justify mathematical learning efficient develop analyze context statistically independent decade numerous survey solution multi perceptron mlp som mlp artificial neural network point consist real bound transfer introduce linearity
deconvolution uncertainty reconstruction deconvolution give measurement statistic measurement generate proportion target aim system target compute call result reasonable parameter weight probability generative integral saddle scale concrete also saddle equation p distance p nm
party classifier htbp acc acc party party communication cost versus size communication cost versus dimensionality protocol baseline htbp protocol protocol well technique perform perform synthetic dataset outperform baseline usually fail yield classifier particular protocol achieve far result particularly support node good partition formulate act constraint svm player natural goal early solve communication efficient distribute take input working space look past streaming pass sublinear work look stream protocol streaming word storage pass prove indicate sublinear input exposition first streaming work let player
calculate clearly experiment cl interestingly consider similar comparable ei speedup observe speedup achieve fully three inspection reveal speedup explain select ei close quantity easy consider two jointly set exactly set expect speedup investigation generally increase go forward change significantly compare lot observation likely meet stage experimental batch batch experiment ei batch practically intractable therefore select
eigenvalue decomposition generality basis contain u obtain projection perturbation iii concluding show triangle q mean final conclusion add stop satisfied inequality schwarz fact two tensor imply stop condition sketch guarantee somewhat form appropriate termination terminate universal condition frobenius condition outline relative stop stop invoke iteration eq consideration condition remainder essentially another simultaneously examine w cast simultaneous linearly independent permutation rescale simple eq chosen give sample parameter include diagonal separate perturbation sample guarantee loose instability contrast mild choice quite simultaneous beneficial uniqueness row requirement uniqueness translate perturbation fact simultaneous simultaneously collection cc find unitary matrix simultaneously interesting question yield improve computational section section tensor learn latent consider statistically efficient dirichlet exploit extract orthogonal symmetric value decomposition compute decomposition efficiently similar analysis establish analogue vector computationally variable decomposition model topic moment statistic prove basic paradigm
finite bin parametric unfold determine experimental unfolding solve priori directly priori remainder paper organize illustrate numerical given solve unfold problem use offset plus pdfs non location width kernel use common generalized vary form parameter unfold find infinitely discretization perform g discretization histogram quantization error follow
valid se generic design estimator proof nonparametric se require exist trivially measured continuous mapping parameter enter square continuous se nonlinear explicitly assumption minimizer least continuity uniqueness law use generic take l individually probability stochastically close nx pr n w nx ps
posterior inferential let parameter mind start I variable take value space measure association together characterize association I I treat unobserved tie quantity turn success framework predictive start measurable index generic collection predictive predictive nest define distribution measure construct admissible restrict predictive support equip three I associate empty predict unobserved associated association specify assertion interest inferential belief e c case often
decrease frequency item primary tumor vote kp j primary tumor kp heart option dense threshold common problem range association fall illustrative example advantageous performance frequency threshold range answer useful fig dataset j resolution frequency key management good imply extensive analysis solver different axis main detailed cp finite logic retrieve cm observation enumeration option significant level impact ability early discovery interestingly preferred expressive iteration remove subset redundancy search search overall good choice search search acceptable frequency threshold multiple search cm medium threshold quickly search light discover search low maximal frequent encoding orient prefer threshold minimal significantly option suggestion adequate medium towards large suggestion indicate frequent option solver frequency threshold constrain resolution tune
value kernel eqs gram operator operator associate vector kronecker operator important towards main value kde generalize take try perform incorporate regularization kde conditional value way take account regression moreover rapidly dimensional know fix operator kde formulation succeed effect explanatory input another take account output scalar function similarity reformulate kernel joint induce see map function joint output value literature recover
carlo universal indeed generate independent correlated mh drawing proposal unfortunately situation limitation drawback rs generate reject adequate pdfs rs analytically ratio pdfs best pdf proposal correlation chain correlate mean converge practice establish burn ensure great mcmc focus extension point propose technique rejection rs acceptance function procedure build proposal exponential inside associate newly reject always refined proposal proposal discard sample decrease cost due unimodal generalization handle target jointly indeed instance technique approach attempt overcome limitation metropolis tackle multimodal proposal remain metropolis accept properly pdf mh initially accept also step determine whether accept rs sample rs support build pdf accept mh support mean pdf procedure point improve never add guarantee pdf multimodal
impact trace extend realistic span detector scan scan specie enter however slight chance pass detector impact mass detector arbitrary thousand acquisition chemical acquisition area normalize round close empirical result poisson poisson variance tie variation detector cumulative distribute conditioning event cumulative normalize reveal satisfactory datum poisson adequate model variable describe detector response dirac delta pmf q account spurious detector eq indicator define function otherwise modify drop log strictly remarkable give follow strictly log concave transformation log likelihood remark variable irrelevant great operational scenario complicate additional appear multiplicative specie output original
nr k nr ar nr k nr nr ar space uncorrelated call component euclidean k kx k apply proof explain covariance define j towards define z j note base type uniformly random definition previous inequality w x degree freedom centrality non conditionally centrality normal straightforwardly fix q ji eq derive conclude j lemma straightforward consequence lemma nz n hold eigenvalue inside circle eigenvalue inside circle event consequence include denote lie contour circle eigenvalue eigenvector j nz n z z lead n j j nz nz nz take contradict straightforward assumption classical us expectation
action restrict situation set action vary drawback mapping state mdp discount function rl policy rl environment problem mdp provide policy improving use simulation approximate form estimate policy repeat properly action action classifier without q binary action carlo mdp optimal generate classifier state label algorithm rl multiclass supervise
result lasso multivariate noise var square lead equal issue choose address literature residual possibility minus simulation paper explicitly implicitly apply method lasso detail simultaneously perform parameter approach simulation simultaneous model var var method ss tendency var phenomenon specify complexity increase square ar coefficient estimate var stage correct ar efficiency two compete lasso method result ss perform among three stage lasso ss model dimensional iid specifie series change three compare three method five metric coefficient ki ki coefficient coefficient k ki ki ki jk first reflect estimate method fold sample comparison stage ar coefficient number lasso ss mean spurious ar compare produce coefficient non coefficient var reflect variance ar ss large stage stage small mse var method notable marginal variability ss
vary seven match select agreement another file record linkage class choose third sample fourth fit mixture probability agreement match likely tend deviation approximately beta sd narrow percent block match status know comparison produce hierarchical necessary tuning prior distribution formulation covariance correlation transform across block hoc suggest logit size logit scale limit range log scale four log hoc follow probability range value repeat agreement constant initially
marginal logistic copula bivariate copula section pair imply expression algorithm simulation bivariate residual limit laplace conditional threshold simulate independently limit set I constant bivariate extreme logistic bivariate simulate preserve stochastic ordering dependence decrease survival simulate observation asymptotically logistic
imply first follow rewrite graph dimension super exponentially choose computationally paper approach distribution graphical presence investigate probability edge identify pair achieve model edge concern super inferential paper likewise reduce basic prior derive mainly property variability section inferential appendix exact computed graph statistical intuitive representation compose describe different graph separation conditional two commonly find undirected direct acyclic graph well determine probabilistic low element adjacent node conditional link arc dag indistinguishable
modify degree currently compute scheme add early iteration add kernel kernel want discuss update kernel suppose effect less negative suggest instead keep operator combination make value version optimal perhaps run end suggestion operator slightly weight enforce low kernel mkl sum polynomial kernel term degree first kernel among weight roughly zero value experiment achieve experiment three uci regularization improvement conclude specific ensure pick choose dimensional along coordinate run dimensional run discretization geometric fashion finally uniform specification give ridge tune kernel low threshold however image dataset finite mkl dataset table
norm understand parameter convex step size take index choose iteration assume choice q neither rate apply query uniformly index mean minimize computing gradient axis estimate gap leave vary aggregation fix vary sample work risk experiment next optimality iteration nf true evaluate unbiased risk see experiment confidence interval iteration attain lie perform experimental microsoft web search microsoft engine retrieval benefit aggregation partial preference ndcg three surrogate aggregation inconsistent fisher consistent recall ndcg prediction vector ndcg increase give query relevance observation set pair score average log preferred structural ndcg addition corollary minimizer square least square logistic surrogate loss previous loss completeness fisher surrogate score loss access score aggregation strategy empirical risk order sample minimize minimize risk horizontal axis aggregation statistic vertical ndcg plot loss regularization goal empirical minimizer aggregate varied extent
interior equation newton first inspire nesterov proximal map accurate robust sense fine furthermore computationally process convex convex smooth gradient proximal shrinkage choice respect computation combination consecutive fista choice proximity p q proximity compute available show error accelerate maintain advantage one fact fista recently criteria proximity penalty admissible minimization scheme special proximity evaluate reduce wise thresholding subsection proximity write compute initial lemma allow efficiently proximity operator consequence computation amount operator onto intersection theoretically show project intersection
amount case great assess statistical behavior frequency well less basic understanding statistical simple giving measurement note part suppose successively unknown process either location location variance call relevant location go definition recall whose
maintain validity plausibility assertion range alternative model unknown inference trivial case function simple produce construct p I side assertion uniformly favor equip determined produce plausibility value side binomial unknown goal nuisance conditioning determine minimal sufficient auxiliary goal scalar leave write testing versus ft predictive plausibility exactly p
forget
pattern precisely either cm enough pattern exactly behavior path first tell inactive associate variable side variable instead close switch segment rigorously regularization rewrite optimize obtain equivalent r solution therefore span necessarily true denote segment
equivalently end g sensor mutual node necessary detail instead analogy network note variable condition process nod weight lemma imply give budget immediately algorithm question budget set lemma exist compute function increase keep mind q bad algorithm prove need g set lemma principle see e chapter state flow flow flow flow unit flow exist imply finish proof flow composition prove program together flow feasible iff affect cut see follow ai ai us minimum non iff negative scale assignment define flow cut include side flow total flow cut side give approximately precise mrfs width graph bind entry degenerate full assume approximation mrf time pass decomposition
integral calculated step di sm fractional moment one psd fractional keep terminology fractional calculus calculus complex keep density cite introduce integral derivative show psd integral valid belong interval integrable generalize function treat paper di eqs path integration locate eqs application previous software calculate apply thus whole eqs fundamental equation choose reconstruct di approximated application
gp recently reformulate showing consider gaussian obtain robustness linear multivariate start generalize polynomial categorical depend heterogeneous effectiveness estimate effectiveness group user take account characteristic stay education framework generalize paper canonical particular belong product show generalize variable useful common simultaneously rather use principle real problem pattern really histogram probability state remainder show likelihood approach introduce
keeping introduce reservoir dynamic network reservoir compose unit toward reservoir reservoir design reservoir network identify external spike traditionally neuron expression precisely neuron behave load activity rate stable reservoir solve linear equation reservoir concept need output dynamical parametric function use system compute reservoir network
precision address challenge new far superior train optimization behind scoring source incur runtime maintain recall labeling number source artificial pattern challenge joint multiple source intractable cost source significant focused er mining community typically precision recall reference discussion er achieve recall combine expert overall comprehensive offer effort label numerous science motivate source suffer label easy matching source master learn heterogeneity explicitly requirement amazon cost privacy sensitive trade negative precision labeling er moderate heterogeneous address transfer algorithm score source adaptively fast er movie major internet engine movie entity
maximize iteration immediately improve locality satisfy formal guarantee illustration comment sampling template guarantee case minor purely uniform policy parametrize empirically respect exponential uniqueness sophisticated give formal denote accumulate depth go uniformly choose auxiliary notation number key result proof outline issue high claim call horizon probability bound statement establish horizon choice branching partly go mdps cumulative eqs mdps period period try lemma constitute prove theorem stem mdp finite horizon parameter step particular induction verify chernoff inequality hold actual chernoff hoeffding directly inside biased stem numerous root
branch tumor frequency tumor generative implement infer frequency infer uncertainty measurement dirichlet major namely tumor derive advantage begin expand subsequent fitness survival subsequently frequency tumor snapshot evolutionary contain multiple major advantageous appear appear back illustrate circumstance highly allele frequency frequency tumor tumor tumor evolution generalize model three frequency panel consistent panel may change take branching simplify population cell discuss estimate frequency sequence copy per consider population allele available copy change specific whole genome sequencing site occur contain cell population greater equal tumor population different branching topological branch circumstance already establish plus
countable sampling apply largely without modification introduce sum pass finite allow particular important follow give duration graphical otherwise conditional indicate range new state whose derive full slice
knowledge reasoning formula valid valid construct place assignment easy claim subsequently boolean give formula logic form variable basis formula threshold list formula semantic logic atomic formula focus formula polynomial atomic tractable everything essentially carry usual formula insight logic offer independently main formula reasoning semantic guarantee high formula true good estimate validity formula know valid whether interesting develop partial assignment whenever draw mask property value assignment process allow know problem hard restrict learn something evaluate way assignment high formula kind certainly essentially true partial assignment evaluate iff false partial evaluate iff true bc regardless formula partial false precisely motivate likewise simplification evaluation restrict partial assignment recursively follow represent evaluate suppose restriction formula state say axiom proof hypothese triple formula system triple hold triple say check impose formal object sake reader aware expectation fulfil application although least preserve classical semantic fall know preserve restriction system system close partial satisfactory subsequence axiom system restriction derivation formula derivation partial step consist formula variable logic sense mean extract case restriction formula ahead typical proof essentially interested version give proof limit tractable limited hypothesis either else limited formula resolution
capital scalar full one subscript column th respectively subscript respectively non vector norm frobenius division recursively optimize objective stop optimize differentiable contain fortunately update optimize alternate optimization variable row except follow term residual wherein look carefully piecewise w l change slope sort wherein sort accordingly remove piece slope increase change sign sign slope w l w slope three range account negativity solution ht summarize successively stop satisfied successfully previous warm accelerate convergence n h r I jj end stop end main spend sentence sort piecewise cost worst piecewise th slope change stop output bad complexity find however scalable optimize equal column th problem negativity constraint deviation presentation much square l especially contaminate outlier keep negativity capability inner r prox dual e e
detect detect equal test favorable detect observation far present five signal result result validate effectiveness feasibility show cause degradation ideal observation time signal scheme wireless eliminate influence apart consider experimental simulation successfully feasibility
result fix want clear infimum apply reasoning plug everything get restrict attention treat x b imply q desire side e take define conclude prove since hold reasoning plug prof follow inclusion since uv u trivial extend take factorization marked replace v u uv eq bind
document sensitive topic toolbox crf hdp dataset word small crf hdp crf become parameter crf hdp base fix optima nb lda document weakly couple result large large ratio usage nb improve tune appropriate parametric nb outperform hdp percentage nb get nb count nb sharp far j kp particularly mean rarely ten infer nb corpus variation compare curve geometric beta nb gamma nb transition smooth gradually kp topic model large allow rarely confirm gamma learn topic beta nb mean positively correlate correlate hdp process viewpoint fix normalization viewpoint count interesting examine view concentration adjustment variation proportion fitting crf hdp nb predict hold model model crf nb crf hdp binomial distribution count shown find crf hdp crf fig indicate across document use
pareto later power behavior various physics biology hence law though first exponent expand include tail decay gaussian pareto measure begin birth shannon distribution lead increase application point theory measure introduce mechanic entropy axiom shannon entropy information
original size easier safe elimination problem life method work especially often interpretability text pca promise corollary electrical computer sciences university california berkeley berkeley provide small number maximize sparse apparent interpretability generally computationally
increase year ever increase theoretical optimality result generalization scale use practitioner deal setting parameter test sgd decrease originally recently asymptotic understand like affect convergence speed researcher make per overview early adapt descent multiplicative approach update fisher estimate simple natural newton combine covariance second approach retain hyper tune another disadvantage start problematic landscape continuously contribution maximally decrease expected formula
art leave adaboost cycle conjecture tie adaboost leave particular computational community emphasis statistical characteristic believe capture essence consistency statistic scenario ml hand round reason round ml rarely amount data essence practical size ml truth often recover care learn perform whether recover ground datum come statistic certainly concept extremely useful ml fundamental play include core cycle cycle also provide often low dimensional also exhibit cyclic randomly dataset adaboost cycle real alone vary technical e matter carry infinite realistic g randomly seek something cycle adaboost margin interest previously maximize adaboost turn come science machine perspective nature study running rate forget rate something mostly function version nothing original optimal adaboost recognize usefulness asymptotic adaboost really adaboost whether adaboost error reasonable know move rate say recognize significant iterative dataset decade asymptotic variant study convergent property often round nature manuscript within focus old learn fundamental central many weak frame adaboost sufficient tool ergodic margin adaboost classifier practically function generalization stable show dash line select log weak base find round pair example input note round error logarithmic growth select suggest real dataset numerical error calculation result open whether adaboost adaboost dynamical invariant dynamical system average adaboost converge ultimately stability hope open htb axis letter dataset round plot originally appear max put forward believe formal heart disease round axis converge letter behavior typically adaboost margin adaboost quality loose explain couple question margin remarkably seem stationary may community adaboost show margin tend yet little significance result ml community care surely learn classifier sense per margin tend limit believe useful dataset implication justification converge consistently formal stable practitioner formal guarantee adaboost classifier generalization similarly unstable behave require establish establish considerably certainly trivial concentrate adaboost generalization important without establish concrete adaboost seem fundamental interesting guess read relevant summary community emphasis deal asymptotic happen round hand asymptotic study question seem asymptotic consider quite iterative ml adaboost article recent literature manuscript hundred year literature thus scope mention likely paper begin
period nontrivial historical highly direct right skewed heavy tailed event attack record even statistically unlikely significantly historical close discuss estimate tail fitting parameter distributional large distinct maxima peak finance several uncertainty upper frequency replace identify define tail severe true uncertainty discard statistically one principle average example aid return finally tail simulate empirical inherent variability likelihood bootstrap extreme technique principled provide tail instance negligible statistically unlikely plausible away likely remain least great mark although mark relaxed incorporate uncertainty principle choose practice large empirical meaningful historical remove describe univariate covariate straightforward denote particular denote event value tail variable tail region binomial variable deterministic event least big bootstrap event integrate drawing sequence bootstrap one event event qx fx event correct jointly
computable minimize gradually around either discrete optimization provide case free energy broad iteration generate determine fp tw fw w fw bioinformatic compute applicable normally objective non provide solution soft guarantee solution convexity objective interesting since bind straightforward apparent analytically case analytically version readily description also make relation
atom residual ss section result fundamental lasso critical motivating may interest stability margin retain perturbation mean inactive atom perturbation stability perturbation dictionary inactive perturbation depend incoherence high contain qp algebra view difficult strategy four problem objective value consequence closeness q duality show complement residual perturb consequence considerably strong satisfy q support atom solution demand former one nevertheless hold considerably theorem theorem behave lead desire optimality solution original perturb govern system aid subsequent lipschitz sparse index induce operate similarly empirical measure associate operate compose return conditioning speak overcomplete set overcomplete individual overcomplete overcomplete error problem implicitly
give algorithm independent output sampler ig pair initially run testing use draw use oracle way phrase sampler prop say interpret mean get sampler sampler oracle rejection take affect correctness probability happen whole thing exposition perhaps basically actual goodness issue issue issue sample even still mention leave real come fix thing assignment tv f hold output simulate requirement precisely index desire tv f claim call conclude linear generation three four count technical giving ingredient recall record approximate count uniform algorithm arbitrary run probability give whose exactly opposed representation parameter output hx algorithm provide al algorithm time query tolerance ingredient proper nn intuition unknown fx satisfy f equivalent note u intuitive henceforth assume e constraint natural first approach program variable turn valuable present note relatively suppose feasible base representation solve simple condition fx union feasible solution satisfie satisfy example feasible along line negative contain constraint prove hence fact course infeasible algorithm problem naive one ellipsoid one intuitively one algorithm one intuition actual invoke stage learner small mistake guarantee terminate small number stage force mistake recall notion learn boolean online unlabele ask identify minimize mistake mistake mistake example essential class class mistake mn current maintain base receive reduce online ellipsoid mistake proceed follow previously basic idea execute learner stage sure solution decide counter output terminate generator learner satisfying assignment output generator continue stage generate upper stage stage terminate online put output b n execute current hypothesis give run ix go current hypothesis satisfy section satisfy condition condition event event call generator successful union bind least successful h ix x condition conditioning certainly p succeed hypothesis observe point give identical point execution fx bind stage complete correctness running involve
minimal definite tu un sufficient assertion observable key ingredient minimal particular triple principal spectra tu q prove thus sufficient show imply strictly negative spectrum unit disk process equivalently state drive autoregressive move average polynomial describe hence l evy process assumption ensure model describe stationary output analogue strictly negative part algebraic degree triple shall inference irrespective identifiable impose strong collection positive definite root ib tu calculus triple rational conjunction realization invertible representation conjunction imply imply contradict theory develop equivalently multivariate process see b apply choose way hold previous specify parametrization continuous impose rank imply observation tu eigenvalue matrix entry number eigenvalue equal observation eigenvalue imply state sequence able impose interior positive hold follow b tu infinitely
large density nn unweighted nn behave near rbf behave smooth insensitive outlier contrast reject minimum cluster size vary unbalanced varying vary proximity partition partition unbalanced inherent cluster balanced partition partition cut cut ratio ratio partition unbalanced unbalanced partition cut unbalanced unbalanced cluster curve cut small cut rbf choose cut unbalanced volume depend account insufficient
scalar problem transform scalar nonlinear operation follow transform similar proximal review equation well equation computational generality key motivation characterize behavior se amp review particular se realization component component appendix precise assume output let denote lipschitz se adaptation sequence value gaussian distribution analysis mild define adaptive remove assumption true conditional thus vector first represent component output output dimension asymptotic state nature constant deterministic constant depend output choice recursively parameter well describe repeat write equation ignore value pre
constant depend lead positively homogeneous ff envelope w w fa fa equal result function corollary relaxation bind penalty illustrate graph penalty blue combinatorial potentially sparsity obviously encode relaxation allow case simple cardinality always norm relaxation w literature fact coincide show detail provide see extend consequence tight necessarily suggest tight characterize extend true couple concept capture intersection relaxation reflect immediate unit f combinatorial envelope q construction decrease envelope extension submodular get tight relaxation f immediate corollary fa always small contradiction satisfie imply consequence construction relaxation figure illustrate value remove formalize illustrate low envelope specify combinatorial range enable answer small large motivation
look put value problem start compare statement follow address embedding asymptotic convergence algorithm space rkh necessary subset measure measure detail moment affect rate assume follow schedule upon follow schedule every exist value n tell achieve logarithmic factor embed map rkh value song al bound thing rkh expectation applicable converge coupling minimal main likely deep detail ensure statement theoretic fulfil subset rkhs last integrable intuition fulfil need sense optimize cb spectral correspond value operator ax l complexity namely essence assumption measure finite yx analogy song need schmidt covariance operator mean give rise although still translate property regard value due two schmidt operator compact general compact operator main rich associated finite fulfil condition q invertible yx schmidt priori unclear fulfil interesting measure rate
dimension array consider hadamard operator easy idea source analytic auto compound toeplitz finally array array however computationally demand delta covariance value repeat array useful statistical recognition compression pattern dimension product original random covariance eigenvalue column call th calculate covariance covariance least one zero kronecker covariance eigenvalue vector r I sample base
repetition correspond method fp tp lasso mcp cv mcp cv mcp cv cv mcp cv cv cv mcp cv mcp cv cv mcp cv cv mcp cv cv different correlation selection tp loss repetition package loss cv mcp cv mcp cv cv cv mcp cv mcp compare paths mcp smooth path stay simulation mcp sparse near exceed newton notice path correction stability newton path coordinate path identical mcp concavity path although path stay optimal easy mcp path concavity get flat short mcp path path lasso whole path see two
combination attribute effective affinity measure effective method agglomerative direct fundamental concept widely receive much attention role data power linkage superiority cluster believe work powerful vision graph partially support research research grant foundation china support introduce team key author thank read li acc detect connect initial cluster n cn v ab ab bn ab sample vector n weakly initial create neighbor cluster initialize near two pair clusters b ab ab bn ab ab
cd table cluster title xlabel ylabel list name black legend pos mark x thick error bar mark x red mark mark solid thick dash explicit cluster title xlabel ylabel run name black white legend pos north west error bar mark mark option dash cd mark table title xlabel ylabel run black legend pos mark thick mark mark bar efficient primarily work well investigation improve correspond graph graph become remain number probability mass write marginal follow poisson function hand define multinomial n algebra furthermore poisson apply multiplicative hoeffding chernoff inequality chapter form
estimator hence condition outperform hand reflect sample change develop high must get ball radius define short connecting bound large symbol generic expression follow need compact density high concentrate near precisely notational drop
converge depend rule convergence burn addition show inference sensitive count consider analyze many clinical trial anti drug precede week base logarithm age treat count successive visit second bound model iv code record visit otherwise ii intercept mod iv poisson intercept slope ij age poorly decide covariate age deviation fit initialization quasi time second take variational especially application parametrization effect center parametrization fit center parametrization fit center give marginal emphasize relevant partially parametrization dash parametrization center general know center automatically choose optimal parametrization improve marginal dash line density variance compete infection information patient seven patient seven total patient arrive variable severe
suggest carry step exist parametrize sample measure must n kx mean point store cast lemma measure close reduce find near reduce find optimal finding non x n always close increase however point sec problem minimize
find coincide jeffreys dx poisson formulae fx jeffreys proportional
split visual instance viewpoint partition truth viewpoint annotation ground category semantic aspect split heuristic obtain bottom category notice huge variation appearance pose camera instance direction different close build slide window detector accommodate diversity amongst recently recent detector et entire vision towards challenge detector perform well state detector discriminative behind root
determine parameter identifiable e calibrate follow probabilistic discrepancy account gaussian process available combination model available small limited example device model physics use device calibration physics multi physics reliability device require solve several accounting calibration computational source view adjust comparison calibration square bayesian inference account various computer uncertainty uncertainty due insufficient bayesian various source discrepancy account calibrate framework effort devote engineering application issue system calibration point interval identifiability use computer calibration common calibration amount aim provide potentially issue network system node observation incorporate facilitate node calibrate account pdfs note focus calibration develop moment paper calibration use calibration
access accordingly respective label instance cluster update variational n class cluster label server eq break q second site separately send server send server site difficult become retrieve label server get present site make
brain space valuable nsf grant grant ns entire margin gray equal product simplify notation assign number iterate fix margin spatio temporal field field x horizon fully mark red extend fully red gray grow stay constant truncate red lie lie furthermore k limit another happen may simple base familiar
literature especially could triple guarantee property sake completeness appendix restriction restriction basis z om g g basic material spline normalization concern lasso think spline knot g ig ig ig ig appear well behave cone dominant penalization statistical eigenvalue restrict originally eigenvalue restrict eigenvalue ig j md play j sparse subset dm dm preliminary proposition statement bind order condition value iii iv concern
ordinal perform trend among alternative independence test statistic asymptotic homogeneity decrease advantage illustrate assume linear force effect snps centre trait hand hypothesis dominant exhibit independence network specific particular dependence expect modern trait molecular part curse gene trait redundant system either processing gene marker study concerned markov literature learn one bayesian include curse markov influence
electrical engineering dc computer fp overview white paper np tree ed group v cluster center f multidimensional survey discrete mathematic structure clustering retrieval analysis management house p massive ultrametric embed journal scientific ultrametric ultrametric analysis ed computer pp fast cluster search concept ultrametric f ultrametric
vector year achieve misclassification method cf crucial user behavior indeed single user compare rating across day week distribution almost use week discuss generative incorporate rating rank approximation section propose binary classification contextual module regularize regression replace composite several aspect investigation confirm claim early importance accounting present allow evidence precise form extremely day context recommendation note music track dataset tend rate recommendation identification three deal classification evaluation rate filter rank rating training
smooth choosing choice theorem pt pt pt band notion coverage guarantee combine idea optimize prediction finite regularity converge minimax fast bandwidth simulate desirable interval automatic want goal produce prediction iid next fall level set observe shall set fall nonparametric prediction regression construct relax usually parametric nonparametric smooth behavior remain construct adapt band sense band investigate study finite desirable guarantee band give weak validity good solution achievable propose sample infinity valid band
th area curve cdf assign vary specificity substituting nonparametric obtain marker eq q auc statistic form compare location normal location statistic possible pair normal assign location normal otherwise average location within subject view statistic auc form range fall sort measurement statistic great statistic compare screen number marker low rather interested sensitivity th marker particular measure
importance top list half life exponential formally associate item list item assume utility user test evaluation ndcg contour plot asymmetric count item count contour line differ pmf equal contour figure user contour line performance largely affect score sensitive svd outperform comment nmf well mention ndcg trend time issue appropriate cf list arbitrarily scenario realistic within since netflix bigger much limit minute severe scenario realistic exceed well perform cf mae time observation apply nmf pmf pmf regularize work constraint svd one color good work pmf pmf exclude
true regular singular condition let constant constant integral outside affect theorem prove general kind make form assume sufficiently small manifold ax infinitely satisfied fx du index one equal mm say use analytic function compact set function recursive enable resolution fix resolution system local log canonical concept algebraic follow definition local say essential equation j k coordinate odd model say odd mm
dx dx taylor series perturbation respect become eq condition reduce lagrange multiplier multipli variation interval
ia paper subject centrality amongst researcher theory experiment among high show analyze summarize nine range empirical supremum euclidean take conservative suggest less known computationally convenient conceptual monotone transformation see parametric centrality arise asymptotic expression avoid matrix
smc abc hmm variability seem slightly allow grow smoothing criterion preferable forward time smc obvious increase associate linearly approximation appear grow less obvious red static throughout datum smoothing smoothing forward smoothing allow dynamic resample smc hmm smc abc hide proposal run burn addition computational cost smc similar computational smc procedure error result observe accuracy estimation smooth use smc update forward exact roughly well forward smoothing smc mechanism observe abc hmm reasonable consider effect evident smoothing particle mix likely
proof hereafter w show lyapunov lyapunov function individual drift condition characterize involve definition individual worth point drift vanish vanish practically situation increment vanish may role accommodate update numerous later scenario exist constant eq establishes set establish strong return weak guarantee existence time return contribution rescale lyapunov respective drift compare scale control define surely jensen concavity identity I deduce obtain iw x iv x iw conclude notice comment lyapunov degree find establish region resp notice concave could lyapunov scenario whereas note practically relevant encountered practice establish strong satisfied scenario dependent drift condition previous abstract add drift
main state subject combination eigenvalue extent fulfil design negativity tailor class comprise design arise paragraph design identify evaluate must condition fulfil mention motivate matrix study noiseless broad class I imply negativity constraint cf apply regard hyperplane consequence hard continue present satisfied design non negativity regression gram lower follow I partition corresponding principal bound equality simplex figure apply matrix entry function spline kernel function traditionally point evenly effectively band I u h mention remarkably deconvolution intensity measure model dirac arise limit device paragraph deconvolution spike train bivariate denoise study design population gram population limit design gram design eigenvector scale uniformly cardinality uniformity consider specific scaling structure structure gram cardinality event well assumption noise show appropriate result remain valid distribution gaussian include cardinality extra depend let negative distribution trial probability cf shift form increase sequence I larger decrease extra arbitrarily quantile virtue various monte generate standard bind active dot quantile frequency solution problem consider whose gram correlate random paragraph design gram possesse involve non design
recover fmri validation considerably experimentally validate problem connection section detail discuss penalization experimental n n mn na mn na normal covariance model simply reflect precision concept estimation selection number zero fit overcome dense matrix precision regularize encourage among term log technique box dual via determinant inequality constraint perform regularizer graphical enforce know assignment unknown assignment change regularization scale free variable reinforcement structure model mrfs heuristic conditional block ise
induce define induce get work subspace imply fix classification column formal cover mb grouping classification classification classification empty inclusion small element element serve add appropriate meaningful x v vb mb set size although classification ambient sketch basis determine whole similarly belong contain element capture definition hyperplane conclude cell nonempty give lemma prove lemma apply lemma would like belong dim perturb lp precisely column unchanged lp slight matrix lp simplify integral constructing
release r permutation estimate permutation calculate calculate assignment eq approximate estimate slight conservative especially small nan alternative increase however conservative slight general large practice covariate otherwise change increase permutation numerator theoretical approximate simple robust permutation truly proceed potential variable basic gaussian matrix nuisance unknown however standardized version space partial mean scale variable interest onto nuisance ready correlation proceed correlation condition covariate however approach nuisance variable adapt original algorithm deal
min redundant eliminate profile extract redundant classifier follow pattern corner white b c I corner thick white b rare amount available association rule pattern class short occurrence specify appear preprocesse perform evaluate scale application real world proposition class possible method overcome short come advantage allow identification correlate mining deal database discovery relationship variable association rule point derive perform deal target occurrence classification tree etc possible effective ignore class devote past evaluate contribution
semi definite definite follow call expression fundamental large also row column claim concentrate prove third positive semi definite resp obtain relationship c deal notable analysis method reveal combination addition describe section section review multivariate method viewpoint give pca xx b w class scatter typical example term typical canonical correlation cca formally cca coordinate maximize either yy take lagrangian obtain xy xx
broad broadly consistent inverse scaling complete star capture correct complete graph contrast inferior take reach roughly star reach iteration threshold line graph star cc value row correspond graph star last case star vertex simulation walk property subgraph sampling bad simulate iteration plot function exactly complete simulation emphasize depend precise location sampling choice appropriately agent protocol quantitative learn crucially agent polynomially fast connectivity play turn relevant primitive point number question protocol achieve depend polynomially appear speed loose several could potentially speed finally develop decentralized handling situation complex learn vary arrival group learn strategy deal situation try action
latter wide classification paper automatically theoretical guarantee method improve probabilistic prediction prediction adaptation regression goal calibrate reporting possibility prediction poor theorem predictor predictor probabilistic diameter reflect uncertainty explore efficiency empirically use loss popular prediction minimax way former version nine set uci repository datum set usually work original interestingly slightly score classifier example simplify seven nine confirm wide study prefer achieve predictive empirical recommend output carry valuable
interesting give figure stock return example entry conditional concentration identical method
ii iii grow infinity maintain constant ucb sense ucb quantity limit assess tend prove interesting item ucb iii technical detailed discussion number prove term term ucb strategy discover request adapt miss simply know reinforcement entail prefer miss ucb simply request expert high upper brief ucb prove find item completely satisfactory main non step obtain interesting item request I good efforts ii subsection independently define every interesting see exactly idea total
run ari similar table performance consider preferable ari small ari std scenario generate add bic within component merge average slight classification former ht ii bic iii simulate triangle rejection point triangle component generate merge within analysis data mixture four model window ari perfect averaging notably perfect nice average merging cluster value bic good section consensus cluster form clustering object clustering heuristic clustering suitable value available http correspond window clustering cluster five set consider average probability one probability approach average apply give outperform consensus
day probably cd year profile change format week inform available totally email see format keyword engineering design h k sciences ai stand profile entry title multiscale de france author van h p management science multi keyword spatial aggregation landscape keyword management risk f profile year observable attribute marginal alternatively empirical frequency occurrence correspondence sort
walk item accord target preference combine trust collaborative recommendation take walk rating propose collaborative walk filtering incorporate social show approach prediction social create graph interpret recommendation similar connect item record unfortunately construct clear author weight capture preference effectively item
infer merely time estimate noise third space burden burden increase great difficulty clearly impractical sake completeness account matrix constraint normalize sample wishart new scaling augment infer turn metropolis hasting possible sample remain component complete cycle suffer case da da half discard burn prior choose fig show auto augment isolated improvement alg improper prior px restrict value depend variance law fig autocorrelation value improper scaling demonstrate da da comprise action prior improper mean px da translation discard px da translation practically da shown figure plot use initialize half fact quite far true finally translation autocorrelation compare da appear beneficial popular
need infer latent membership observe make formulate three infinite finite upon process allocation finite infinite modeling document modeling compare lda dp corpora achieve document baseline extract dimension reflect orthogonality learn scene understand recognition place govern I human potential human pose mixture associate human versa mixture joint pose object govern pose preference explain pose front certain therefore object parameter
model compare previously study ml ise mrf observation generate ml compute intractable ml finally variance estimator abc ml ml gibbs coordinate implement auxiliary tolerance mrfs mrf ml markov whose minute intel core matlab
rely predefine incremental approach bellman successively reduce construct orthogonal basis bellman derive function eigenvector induce gps reward p mdps rl ergodic introduce terminal episode sequence state reward train discount terminal terminal case state episode function short v eq scalar parameter collect ordinary rl observe define term define n unknown capture mdps remainder solely consider e function
minimize update hold generality focus row matrix covariance last variable inversion block involve lead remove term solved coordinate direction vary soft iterate follow column descent block summarize start let solve repeat utilize representation
manner satisfy instant incremental symbol assignment scalar nonnegative satisfy instant node intermediate cycle incorporate represent perform incremental neighboring node intermediate update p retain standard covariance representation filter require consider satisfy every instant node follow incremental kalman start time implementation arrive order facilitate comparison equivalently rewrite degree neighborhood diffusion write observe employ estimator generally employ enhanced performance update adjust enhance discuss convergence kalman filtering detail along diffusion smoothing early mechanism global assumed diffusion involve reference strategy step satisfy leave regard scheme coefficient nonnegative corresponding behavior diffusion noiseless strategy statement condition adjust complex noiseless let individual practice situation network attain chemical physical identifying minimizer would converge pareto solution global individual cost real individual constant generate towards desire global norm gradient restrictive differentiable condition relax allow require norm hessian would exclude unbounded body chapter adaptation update exact use version weight error realization gradient become condition past noise noise variant long grow fast requirement furthermore condition require variance satisfy eq verify relative noise vector let steady state noisy consider assume minimizer data strategy employ zero development diffusion adaptation insight student laboratory http www edu students chen yu lee di tu j chen yu early article ease reference kronecker product compatible kronecker denote replace scale consist combination property kronecker bc ec te ec ec consist node connect node consider edge connect loop exclude self loop set still loop loop path denote consist connect addition integer since denote entry eq term location associate incidence define every connect column display one entry entry deal simply assign index negative high index node incidence consider example column incidence manner observe laplacian incidence row small algebraic connectivity zero subgraphs connectivity nonzero algebraic identity add b zero network separate say laplacian matrix laplacian generally subgraph algebraic connectivity obvious must would diagonal algebraic contradiction converse statement graph connect eigenvector e already b verify individual imply connect entry desire add leave add doubly add arise frequently list right leave doubly leave doubly one add conclude spectral radius unity eigenvalue equal integer strictly strong characterization eigen regular eigenvalue circle eigenvalue correspond eigenvector right eigenvalue entry part follow frobenius regular doubly matrix hold doubly eigenvalue nonnegative definite hermitian b likewise symmetric eigenvalue real nonnegative follow part matrix orthogonal entry therefore end scale justify
refer guess mean document identify appropriate word fundamental task translation ir corpus paper corpus base employ learn feature text word correct manually speech consequently competition consider english lexical task unseen see na I bayes nb near neighbors nn machine svm learn algorithm learn consist nonlinearity model behavior algorithm create simple lot feature input composition manner abstraction
adaboost auc area roc curve adopt diverse sensitive imbalance traditional criterion etc convexity even optimize auc yield hard avoid pairwise surrogate usually adopt g exponential loss hinge square etc surrogate lead improve actually expect surrogate converge risk also call bayes optimize yield limit sample optimization surrogate calibration loss calibration necessary yet insufficient auc inconsistent pairwise surrogate risk whole provide sufficient auc minimizing find exponential auc exponential provide
probability member choice haar wavelet multivariate basis task develop use appendix implement ks cs r differential entry application complete posterior distribution instead posterior object object euclidean cluster uninformative would conclude actor actor situation multidimensional obtain achieve represent actor decrease take value example possibility choose solution probability solution transformation ensure continuous like aforementioned uniqueness show embedding classical return dissimilarity dissimilarity provide rank discussion concern rank least dissimilarity begin scale diagonal entry matrix whose th entry dependence need general remain
demonstrate norm arise section present superior know often inner function denote rademacher depth mapping learner nature protocol round minimize study space set able algorithmic idea game distribution solves henceforth omit understood range moreover partial recursive game eq minimax specify mix recursive appear tool yield array constructive translate minimax interpret take present value serve regularizer online exponential follow relaxation first yet tight upper sequence x admissible minimize relaxation meta however need valid relaxation relaxation tx irrespective adversary hoeffde admissible deterministic eq recognize potential know loss player difference potential extract potential origin author conceptual arise conditional characterize view sequential rademacher shall refer already future whenever rademacher admissible appear prediction arise relaxation conditional rademacher tight rademacher admissible version chernoff inequality tell proof softmax maximal weight realization finite probabilistic concentration deep exponential relaxation optimize dimensional cost schedule loss suppose upper admissible furthermore important absence automatically mirror aim relaxation arise step algebra example
within edge term combine equality let p way twice put everything together label graph transform list neighbor method label line technique scope significantly experimental confirm often beneficial pdf span span linearize traversal arbitrary simplicity assume traversal soon visit get line backtrack produce visit eliminate elimination display bottom show bottom elimination eliminate eliminate adjacent drop directly adjacent right eliminate connect node weight without obtain edge weight gray predict remain neighbor prediction shown initially create together suitably weighted order create starting perform backtrack edge current backtrack one traversal eliminate soon encounter edge iterate elimination happen adjacent eliminate among eliminate let algorithm predict operate metric label di j v v e path I mistake free drop edge define use shorthand point please proof mistake loose ii diameter extremely argument change rescale value make compare label edge concern eliminate contribution g exponentially light mistake refined way equal mistake subset imply add extra mistake mistake label mistake logarithmic advantage high make intuition rule close combine rescale edge insensitive appear
appropriate weight majority margin thank soon motivate material base national health grant foundation grant
baseline algorithm know aside naturally approximate crowd appear accomplished process typically much population characteristic nothing closely crowd predictor let crowd l denote arrive vote number time far consistent quality collect smoothing shrinkage crowd toward valuable whose certain time opposite vote threshold ml ks l loop v v tt new pool high estimate another uniformly pool enable balance exploitation estimate vote
embedding measure restrict measure discuss borel probability satisfy reflect random consider test appear appendix underlie negative describe relation energy theorem prove appendix depend choice link family kernel accord may generate rise possibly subset merely provide triangle schmidt pair jointly hilbert schmidt mean discrepancy distribution kernel mmd k k k xy x x last k show demonstrate link covariance prove xy similar make yield rkh equivalence lead result
natural near nonparametric predictive valid conditioning integrate empirically exceed suggest distinct prediction analogue forecast interesting many risk access netflix root broad cover understanding
seq cn lambda exp exp k sum n exp mu I exp n x col col shape break cat value sim shape n alpha ta sim sim alpha alpha alpha alpha alpha col c col alpha col alpha x x x tx alpha alpha alpha alpha alpha alpha alpha alpha alpha tx n alpha alpha alpha alpha l alpha
directional lp x x selection index plot convenient formula contingency xt concept apply non continuous variance regression multiple logistic discrete dependence coherence trace yy yx regard coherence quantile mid copula em nonparametric relative theory
summation state give definition estimate hmm maximization iterative algorithm hmm cf reference therein incorporate section iterative em hide observation access state partial access happen ease notation ever model labeling framework b order define backward equation variable z actually observe noisy observe backward
careful approach work space query subspace cauchy variable say prove embed sufficiently e polynomial require vs recognition point solver reliable speedup fold rely price computational profile failure quantify use repeat end maintain regression recognition complexity observe improvement example query strategy basic building block database hard subspace problem near neighbor sublinear point randomized locality become sublinear algorithm et lift space near neighbor several hash hyperplane sublinear computer science limitation intrinsic sublinear near hyperplane hypothesis boolean version variant variant difficulty sublinear unlikely well motivate cauchy projection choose cauchy family remain cauchy exploit previous level obvious heavy yield preserve df logarithm certain non distortion result notably distortion incur nevertheless elegant turn
fraction implementation work support scale proposition proposition fact restriction constraint question question remark fs mm grant support nsf grant support david nsf innovation fellowship support dms nsf innovation fellowship david google award grant wu method exploratory corpus inference exist provable practical inefficient inference provable implementation run popular collection without supervision topic model vocabulary token document specific distribution intractable result researcher
differential prove side correspondence equation ode ode exercise calculus solution meaningful obvious type age population true restriction
finitely noisy circuit completion single set observe position find complete circuit circuit entry return average appropriately polynomial per take variance circuit high try decide except contain imagine bayesian explicit yield entry illustration statement circuit graph simple cycle bipartite arbitrary disjoint number disjoint circuit occur moreover simplification elementary become algorithm take least regressor complete circuit observation efficient computation polynomial complete efficiently ht completion denoise entry position observation estimate complete circuit circuit write set weighted mean determined sign circuit locality circuit also reconstruction say complete circuit r solve side suitable estimate plus wise error local completion variance logarithmic single entry observe position observation variance variance estimate return standard error actual reconstruct section finitely closure uniquely associate bipartite corollary bipartite vertex isolate position grind convention isolate care indicate ground set want maximize closure
change point copy interest sort accord position observe emission distribution point segment square share level observation figure level occur change cp maximum segment segment approach location peak dot lrr line change r change curve include second slightly change due bioinformatics genetic pointing phenotype disease detection copy characterization dna genetic treatment hide map likely extension hmm procedure change specify transition improve extension include reversible markov monte carlo account
expect per expect top axis ensure expect gamma probability heterogeneous implicit reference exponential mle leave submatrix adjacency simulate clearly plot except would identical block break adjacency deviation model level htbp regularize maximum likelihood asymptotic membership high population grow slowly stochastic blockmodel edge decay probability decay block stay bound away otherwise induce second implication small size number diagonal grow linearly
classic measurement error regression independent design motivate measurement enhance algorithm scale scale view validate characterization angle selector predictor uncertainty perturbation matrix source theoretical point view interested theoretically argue variance efficacy point validate empirically characterization algorithm lar ds motivated characterization experiment national energy laboratory repeat spectral value domain small predict fraction variance interested mean understanding indicate research algorithm way familiar lar variant tailor parallel evolution
discuss explicit bernoulli digits string represent string sum equal final dag small one large dag recursively sum integer bit integer entire computationally resample integer integer numerator repeat dag nan triangular recursion end remain stay zero connect newly add connect hence bernoulli reject alternatively direct binomial sampling exclude add complete order since integer dag rather equally likely permutation procedure mapping dag uniquely identify edge map simple ignore mapping draw compatible edge node permutation sample total dag detailed number illustrate list integer big far leave small dag divide way dag fact node integer would next arc also least arc node last far bold bold may draw receive arc merely permutation node label draw reconstruct permutation adjacency sample inside line denote draw bold old adjacent matrix permutation label arc next step ensure receive
actual classifier predict difference utilize originally investigate familiar establish ir community correlation three consider crowd score correlation crowd evaluation outperform blind round typically effective outperform accurate though reasonably regard outlier though still fairly base expert crowd approximation additional likely collect future comparative finding consider induce system impact measure school consider classifier share expert traditional evaluating limited expensive classifier scalable yet preserve high evaluation label crowdsource scalability raise serious regard blind investigate label aggregate label output label performance additional crowd direct classifier supervision assess establish rank correlation classifier vs crowdsource expert expert classifier rank accordingly score tie ranking score
entire finally item already herein vertex option option create addition different code gr gender r vertical blue quantile plot score performance nearly diagonal line plot strong test obtain r display item ht compare evident nearly overlap also display code point individual actually attention confirm produce result add gr site format gr omit produce separate plot ht plot dash diagonal situation vertical quantile answering highlight among group distribution slight people toward people large people figure figure note score density item testing efficacy plot presence confirm environment within along well application
f analysis appear remainder author outlier contingency among true table treat couple connection residual test detection find context contingency deal count contingency sample cell count great identification recognize outli contingency outlier goodness test apply algebraic statistic approach towards outli contingency go back outlier minimal set contain enough cell full rank remain although base minimal minimal table derive pattern independence example nevertheless order applicability notion run outlier organize briefly base estimator
contain ranking object conditioning conditioning phase object phase set ranking predict rank neither object rank observe phase four different conventional framework use correspond imputation link know solely exploit link incorporate feature information label rank new setting fix able conditioning object set realized rank encounter learn rank literature retrieval document rank new predicting capture previous use joint query possibly encode typical kind particularly design retrieval bm match task require human expert information always construct perform access possibility representation experiment example addition case object object domain enforce relational symmetry consider explicit one nonlinear kernel know similarity aim object advantage kernel rank kernel building pairwise capture similarity edge predict kronecker generating modelling consider structured define input usefulness kronecker remain challenge require computation processing usage overcome computing training advantage kronecker traditionally solve g reference therein relate link set computational kronecker exhibit solely prediction body g similarity path kernel kronecker sense infer similarity path walk could use related pairwise kronecker kernel domain kronecker application bioinformatics task target knowledge concern
dataset represent informative consideration connectivity unable small maximal algebraic os r enyi graph contain edge result graph laplacian os r enyi algebraic os enyi random connectivity know algebraic use concentration connectivity os enyi inequality hoeffding pm bn pn pn least match side red green os enyi value obtain indicate circle initial take black value figure nearly indeed algebraic connectivity heuristic produce optimal also algebraic connectivity optimal average os enyi graph blue solid finally give os r circle formulate yahoo movie rating rating yahoo movie rating movie nonzero rating movie rate review rate movie review make movie give movie imply comparison complete majority movie receive movie occurrence movie receive ranking discard leave movie review remove remain
discard mean information lose loss also old window detect performance chart detector assumption delay nothing inferior benchmark define misspecification detection set equal delay change order choose method generality knowledge require learn change affect occur detect consider early moderately case contain correspond locate proceed performance chart follow observation know reference choose specify approximation note previously serious control limit
interesting gp constitute constant variance correlation return noise clearly approach finance allow capture effect volatility vast majority financial return capture address bayesian paradigm obtain expression covariance let modeling th consider uniquely describe possibly infinite impose latent gaussian gps rather effect mix introduce function relate q n cx eq latent impose accommodate fact definition cx innovation also impose gamma complete conduct bayesian bayes g prefer variational due well scalability
condition hence ct require essentially limit empirical evaluation apply large comparison size problem subtle motivation basis induce entry gaussian result entire randomized use hold list worst show performance trade compare ct advantage ct base transform relatively implement straightforward theoretical algorithm advantage set ill condition slightly randomly ill heterogeneous leverage illustrate linearly spaced gaussian linearly space way ill due due dd rest low leverage ill condition submatrix snp snp genome description lead submatrix create rgb convert intensity result implement evaluation closely seem use run third cccc e third size superiority condition close worst bind ct cccc conditioning algorithm choose conditioning well condition algorithms test increase decrease expectation algorithm interestingly reasonably reason
sgd machine take draw respect polynomial averaging set besides polynomial decay run iterate report log repetition iteration omit indicate polynomial decay well achieve averaging early amenable significantly average require performance particular simple iterate attain well theoretically also extend sample individual iterate smoothness finally bound
simplex hyperplane empirically demonstrate substantial accuracy exploit refine enforce convex portfolio illustrate degree european future proof sciences de paris corollary definition challenge relaxation lead norm application benefit feature constraint efficient sparse projection onto simplex use
independent monotonically jump infinity constrain physical limitation communication bound drop jump infinity cc cc monotonicity make analytic approximate expression detection allow overfitte operational stage state active risk overfitte high active research successive correlation sense temporal interesting phenomenon correlation mention layer access keep track detail place eliminate sensor probability set similar use geometric certainly com wireless sensor
v innovation conventional r vi innovation order innovation estimator noise innovation innovation maximum likelihood state four simulation interested identification problem innovation introduce noisy convergent innovation filter exact variance also demonstrate approximate distribute decrease innovation linearization filter algorithm performance simulation thin innovation satisfactory innovation innovation estimator maximum discretization decrease innovation innovation much exact innovation less adequate tolerance adaptive order innovation automatic approximation effectiveness innovation identification reduce observation distant easily measurement miss within international centre physics author partial accord tt functions taylor expansion drift matrix function symbol viewpoint formula exponential depend method van alternatively suggest computation moment formula
distribution proportional label frequent quantify company fix size edge software show topic stock microsoft yahoo report reflect clearly potential deal yahoo stock microsoft agreement yahoo difficulty useful impact whole topic stock isolate rate financial major observe company name confirm successfully economic information drug national product deal global regression news topic meaningful among news click market imbalance extract news record sometimes information news record top news exclude blue exclude overall support trust contain topic stock reflect report
use first modular modular notice perform maximization guarantee ensure next increase monotonicity follow submodular reduce iteration maxima modular reach optima modular next value local minima submodular maxima optimum gx tm gx f tm fx j modular optima gx gx j gx gx gx gx gx vx vx ensure tight approximation time submodular question form bi randomize importantly practical greedy bi randomize pick amongst randomize combination lastly note local heuristic submodular entirely step heuristic via procedure submodular modular
denote random belonging minus alternative set hypergraph say acyclic connect acyclic graph acyclic greedy algorithms section use lead exact greedy clique decomposable decomposable graph approximate find target entropy sx sx sum selection consider width bounded parameter problem decomposable graph entropy since clique size clique characterize selection incidence incidence minimize equivalent form decomposable
velocity angle first angular nm angular restrict cm symbol mass eqs solver keep xt discretize continuous however alone keep position unstable introduce complex balance yield physical controller yield result stay inside valid work intend produce meaningful close balancing roll roll measure axis angular velocity control center mass nm dynamic model interval reach angular computed physical front radius vertical mass horizontal distance mass mass mass roll angular velocity keep roll angle restrict roll suppose terminal define terminal state stay matter go forward use keep xt us compute finite possible action discretize cm pt department cs usa school science college ab united continuous agent generalization loop drive transition influence influence early identify salient dynamic act intrinsic reward require external reward state know initially address carlo address prediction iterate induced application exploration model keyword dynamical self title continuous enable
source plot visually tb mm see stability much give e mean confidence mean confidence seed validity inductive region significance proposition inductive
e c dim sp sp cpu generate need hour minute solve minute obvious generate objective value always much hour minute take minute dim latent large expression fast cg thank consensus research national foundation fellowship institute mathematics application part science office proof sense completeness introduce define function rewrite
group diffusion topic originally implementation http www ac research relate topic nine topic nine sample list extraction set web page four set manually page link co undirected composition collaborative internet movie movie recommendation movie focus movie whether office contains link whenever share production company edge production two movie report node know label remain phase use assessment phase labeling run run fold cross validation external fold successive rotation specific fold moreover fold external cross validation fold internal tune classifier
component section recover transformation component component certain distribution ica state certain intrinsic state ica randomize simplex simplex learn presentation ignore ball transform linearly ball measure section ball map sample use linear scaling turn find axis align alignment ica ica somewhat rescale distribution fact even isotropic ica justify algorithm denote routine vector coordinate square isotropic distribute entry p nx n p isotropic ica routine state explicit simplex hull generate scalar obtain approximately separate inverse multiply matrix remove row isotropic v permutation diagonal entry sign correct orientation sample output every let invoke obtain separate work iid coordinate accord isotropic ica
state metropolis acceptance eq eq one flip p momentum
comprise texture color feature edge detector texture original image filter decomposition texture euclidean distance absolute position vary tree fold cross retrieve counting retrieve fold achieve good accuracy category par baseline fail global distance less angle forest incorporate relative position pair implicitly capability speed benchmark learn feature function form position clear question remain
assign give clustering comprehensive apart hamming proposal concern partition ideal goal introduce distance partition difference clustering clustering moment number cluster introduce distance surely distance practice hausdorff distance measure hausdorff compact compact metric try capture proximity set symmetric nan identify hausdorff first rely involved measure seem ideal clustering incorrectly represent significant cluster result clustering two clustering distance permutation metric partition moreover usually combinatorial represent particular comprehensive understand set resemble possibility logical add equally possible well name bottleneck
put wish book decision informative comment via relationship social filter social influential detail filter recommendation know preference user prediction user variety adopt relate item rate user rate item rating collaborative predict rating target certain collaborative filtering prediction recommendation display rating item unfortunately ignore social user take effect consideration unfold epidemic make recommendation start explain model
typically encode continuous space tractable differentiable nothing determine alignment collapse onto remove one refer output recurrent layer layer length sequence encode vector length unit compute iterate input hide matrix bias term rnn application lstm exploit lstm composite vector obvious mean hide gate gate state gate gate
intersection face random independently cone marginal weight mixture recall argument empty inside well similar limit restrictive want interior patch numerator allow isolated intersect patch really closure isolated patch avoid closure entire field case difficult exactly maximum isotropic ec dimensional volume ec high ec ec approximate maximum giving field volume essentially ec light gray region threshold expect ec ec surface diameter two parallel tangent field field necessary intrinsic
hdp whether hmm hmms bayesian hdp hmm hide chain collect transition form integrate sequence generally expense expensive treat prior approach model focus model duration explicit distribution bayesian perspective literature parameter estimate procedure particularly constitute basic underlying formalism generative standard draw duration new probability random duration graphical picture see transition super super state length segment observe symbol super transition sd state specific duration defining must end segment boundary final right censor algorithm modify censor chain similar alpha beta dynamic hmms inference message pass message symbol message hdp symbol message duration segment begin observation duration indicate begin sum future boundary expression constitute contribution segment provide censor survival duration subroutine sampler hmm expressive model
integrate k last line evaluate complex effect jk kk jj jk cholesky j k jj jj jk kk kk k newly start derive task calculate predictive partition predictive group derivation computation calculate bayesian task preliminary give newly task predict procedure variational keep particularly much random effect yield task gp multi task task also model multiple output work derive differ
heavy coin unique minimum occur hence finally proof dimensional walk walk less heavy coin respectively heavy gets let heavy coin respectively expect algorithm follow inequality achieve heavy minimum cost measure future strategy good outcome history modify initialization optimality action major heavy heavy interesting devise coin condition set coin
edge efficient independence graph exploit sample estimator dependence sample finally combine stand rw graph simulation topology facebook require least facebook graph sampling walk measurement important number practical user user month business case user strong report compare facebook different case network scalability reliability obtain architecture system protocol www instant rich content million network track predict medium obtain potentially thus extent estimate population major technique currently context variation drive widely public health
thing buffer expression observed lie equivalence gene total lie neighbourhood equivalence lie gene posterior great gene microarray explain equivalence indicate identify correspond specified profile profile would htbp nm nm nm nm bc nm nm nm nm probability equivalence day day posterior equivalence decrease value gene equivalently day gene equivalent small htb show set requirement evidence particular
dimensional one continuous lemma restrict meet condition thresholding finally backtrack procedure terminate specify duality attain author satisfy work well practice note accomplish cholesky cholesky j u j require section criterion however approximate hessian take step safe
work extend wise regression generalize specify model poisson specify specify conditional conditional regression arrive joint cx procedure mixed categorical regression closely baseline generalization pairwise edge unfortunately incorporate likelihood procedure graphical representation probability absence conditional maximize penalization graphical model pairwise edge either variable conditionally variable two absence edge entire mixed motivate non sparsity penalty relaxation scalar regression irrespective balance way group treat equally fully factorize px py r base
entry customer customer must exist customer covariate latent section construct poisson give poisson operation yield rate allow varie covariate define dependent normalization process dx three poisson construct superposition transition restriction rate superposition superposition poisson countable poisson superposition poisson transition tx tx tt dx measurable associate atom qx dx qx respective associate special treatment partially construction bivariate poisson represent baseline superposition process bivariate take form partially exchangeable additionally distribution hazard lin superposition subsample chain poisson chain dirichlet addition transform use kernel gamma gamma operation atom subsampling affect location size lin generate process wide variant gpu need lin mcmc chinese chen slice drawback construction application ad validation alternative construction drawback use kernel process use wider dependent spatial poisson
filter recommendation accuracy insight recommendation interpretable svd internet service internet market vast maintain customer recommender strategy example suggest base recommender system cf content base item match cf approach netflix user rate preference overall item rate extensive review discussion refer reader recommendation boltzmann broad netflix neighbor gain popularity result netflix perhaps netflix prediction ensemble many take predictive table netflix base label knn error rmse factorization svd table drop rmse achievable combine together also netflix project focus fundamentally one focus propose add value paper extra capable item without cause general user relatively
column separation bss mode lead cp cp useful allow bss notation bold letter bold unfold tensor I pn ny j na denote rao wise hadamard product reader notation q outer b exactly cp ni nj illustration rd outer tensor regard e letting shorthand reason hereafter alternate widely standard update use quite attractive essentially permutation matrix number suffer slow respect effort improve reduce accelerate preliminary tensor tensor transpose tensor accordingly example transpose permutation transpose mode e transpose unfold thank rao replace rao successive vector simultaneously
multiple pre separate background foreground scale sift identify feature use identify sift major stage detection localization orientation assignment invariant generate successively gaussian laplacian filtering localization accurate contrast discard absolute pyramid detect small lie discrete neighboring pixel determinant orientation histogram region histogram gradient magnitude circular window orientation bin descriptor vector sift represent pyramid lx equation calculate determinant equation eliminate orientation sparse sift descriptor smoothing
supervise keyword method v cca v cca cca v structural learning tag protocol truth query retrieve v visual pca reduce nc structural refer deviation query nearest retrieval cca compare plain euclidean distance scale component consistently adopt propose evaluate view tag section performance class baseline feature pca get precision term poorly baseline interested keyword dataset tag vocabulary improve embed project tag coherence image e query also image cca third keyword retrieve retrieve search incorporate view three target version view view nc clusters almost identically automatically supervision list performance cca v replace lower high cca significantly though noisy view help tag view report comparison structural cca v reason discrimination batch batch optimization sgd beyond scope design sgd neither produce tag unlike cca baseline suitable retrieval query show tag note advantage view tag fact cifar imagenet lack tag retrieve tag main ten keyword particular retrieve complex query multiple keyword tag give minor form tag intuitively cca objective influence distortion tag rare one observe effect accurate tag
neighbourhood average cloud dependence function least component simultaneously component dominate one put thought profile measure borel weak compact converge profile equivalent copula indeed homogeneity homogeneity eq consequence intensity integrable indicator computation representation unit find total measure profile vector must law appear max profile vertex asymptotically asymptotic perfect dependence
homogeneous p distribution rl log likelihood hellinger likelihood frobenius example statistic produce arise hellinger appropriate associate entry contingency cross result call calculate accurately always extra principal advantage frobenius frobenius
poisson exchangeable homogeneous random ibp remove family distribution joint marginal later cause sure one imply direct familiar subset truncate restrict contiguous line restrict subset restrict exchangeable distribution exchangeable distribution de representation n example ibp zero per condition latent ibp accord restrict arbitrary infinite trial probability
encoder denoise encoder justify apply denoise well local matching auto perform achieve local derivative target implicit one gaussian location result method base partition function denoise encoder perturbation contraction magnitude applicable extend auto denoise second argument successfully auto encoder small chain local reconstruction jacobian respectively factor function finally empirically verify include experimental criterion auto train
isolate employ whole network pc search comparable summary scalable structure complete information serve meaningful biological protein gene serve central area underlie relationship bring understand graphical acyclic dag flexible proper result integrate model bayesian average grow enumeration impractical overall structure network variable parent network range protein average support logical attempt beyond number thus employ manual yet influence result knowledge domain exploited distinguish closely variable guide partition frequently similar pattern neither practical collect special second quantify bias result lead structure result attempt correlation information recursively network multiple size intra community
parent covariance parameterize mat ern correlation modify function kind spatially vary mahalanobi describe write diag rotation angle ern mat ern deviation geometric rotation angle feasibility vary spatially accord combination determine generic transformation ht l l smoothness rotation angle cumulative standard normal distribution mean see smoothness order determine parameter smoothness solely integrate n full furth probability density variate mean describe jump markov monte gibbs metropolis emphasize
generate accounting perform compare expectation well correspond bias tradeoff smoothing sm sim reduction mse associate hour motivate approximated initialize present detail section assess validity deal series experiment conduct validity trajectory capture experiment epidemic observational epidemic
policy action include indicate index introduce shift prove shift around must ode sufficiently large theorem come addition introduce ode define region rely establish meet strategy derive policy give ode p track solution ode value operation thereby v v n n claim lyapunov sufficiently explicit account hold definition regardless transfer sufficiently direct establishe track solution ode second ode strict lyapunov validate energy propose extensive km km exist macro bss micro file transmission request arrival file size ease process traffic load realistic historical indicate bss e w micro bss transmission operational consumption operational power macro bs micro w channel modify influence fast noise
propose endow working side exhibit linear rate research originally motivated framework computational optimization possibility kernel equivalence popular polynomial since kernel application capability training recently kernel improvement second minor acceptable actual run statistical assess significance conclusion addition confirm article introduce machine treat feature space call instance infer prediction term correct category candidate assess ability correctly map call hypothesis induction problem multiple category possible separately rest train binary svms versus classifier organize direct acyclic frequently performance method decision model space dot embed mapping decision represent feature input precisely computationally avoid belong value pass sign classification label prediction mechanism decision predict well unseen minimize implementation induction principle classification problem address build reliable pattern misclassifie margin pattern attain mechanism full implement regularity margin instance without
contour fit spline covariate correspond credible wide region edge space maxima minima indicate fit spline plot quantile grey grey figure sample band lag apart model expand lag suggest modelling residual identically u average degree freedom require trend element eight effective freedom five sd intercept forecast day day advance forecast store throughout bar year condition triangle slight model indicate provide variance joint daily trend provide also lowest fit goodness fit forecast subsequent analysis univariate daily trend credible mean quantile fit temporal trend vary year day peak around peak period daily peak pm daily trend day trend contain peak trend decrease effect day account autoregressive autocorrelation lag capture model wind wind linearly wind weak wind observable interaction wind speed direction wind angle approximately correspond
constraint mac correspondence prove nest unique mac constrain versa local minimizer exist nonempty kn nest exist nonempty neighborhood mac constrain problem mac nonempty local nest constraint obviously exchange everywhere non strict minimizer max mac formulation correspondence well manifold minimizer mac vertical necessary tucker mac point simplicity exposition special layer analogously r omit kkt nest mac constrain problem equivalent nest necessary condition minimizer minima maxima saddle mac eq kkt problem conversely eqs kkt hence correspondence stationary kkt saddle correspondence minimizer saddle mac first qp penalty sequence qp exact global sequence limit point kkt multipli theorem theorem involve function local minima applicable derivative weight differentiable mac qp nest positive qp find minimizer k multipli noting turn sequence mac
integer proposition ji ji infection denote infection model version generalize vector log likelihood number node please course follow parent cascade infected node pick large remove cascade consideration proceed remain cascade cascade u I result suppose suppose ed least greedy neighborhood attention establish cascade clearly could tailor instead standard information ensemble cascade need approximately low ensemble corollary cascade epidemic ensemble infection collection cascade infection observation say approximately recover graph singleton graph probability randomness generation infection recovery define fail theoretic entropy mutual
notable feature scheme laplacian simple evaluating take rank popular sequential backward search work f x reduce potential nature search different backward search incremental backward never consider likewise feature find feature diagonal place solution tend simplex select reveal non may give sequential incorporate necessarily nest characteristic useful multiple feature subset size indicate feature counterpart optimization correlation statistical measure covariance imply converse capable linear would quadratic dependence give positively pearson schmidt detect dependency require formal hilbert map
nesterov acceleration technique convergence number superior generalize function distinct secondly condition accelerate single consensus useful convergence proof let note bound also sum next ij k substitute omit increment start use convexity x j turn result back eliminate k j reasoning give lemma proceed hold suffice show due second positive
onto nine dimensional computed modeling face versus image last row show projection face explain qualitative subspace order face computed show model generalize decade intractable researcher optimization admit robust focus aspect book comprehensive robust overview major challenge notion quantify recall point arbitrarily place estimator bad quantify recovery stability approach residual early orthogonal consider method hybrid modeling aware tractable aside receive attention square residual book generally computationally tractable data maximum principle often formal subspace fail subspace compute covariance aware review pursuit pp construct direction scale component repeat proposal appear aware provably maximize pp remove offer algorithm provably correctness criterion tailor randomize iterative consist eventually identify project sphere pca recommend method practical guarantee researcher start effective technique variety algorithm guarantee assumption split plus corruption first problem rank norm regularization tradeoff goal recommend norm return appropriate outli contrast formulation possess appropriate discussion
strictly multipli one could dependent multiplier loss monte base use multiplier equivalent usually independence sensible work multipli nan propositions eq define independent copy finally copy result theorem adapt ii conclusion replace somewhat claim proposition obtain monte study detect question single change cross dependence margin alternative happen margin hypothesis combine factor representative question via resample distinguish size per autoregressive exponential autoregressive normal multiplier observation move
reach hour boolean go model minimal find go due perfect main advantage compute minimal allow meanwhile exhibit develop minimal question biological relevance discriminate precise pathway nonetheless exist fit reduce compatible induce perfect strong performance efficiency experiment describe perfect size range minimal model criterion parsimonious fitting observation however mention least
available frame hour intel figure show increase approximate flat setting result spectra atom ccc dictionary self constraint indicate residual code cardinality train dictionary term trade code code omp report result residual norm residual curve identical
state produce bottom actual circle induce uncertainty panel model rapidly whole lead exploration optimal transition episode already counter uncertainty cell approximate function state similar continuous domain gp value separate problem interpolation transition learn advantage situation limitation bellman globally space advanced discretization grid grid curse limit minor concern simplify make transition know conceptual limitation simplify make address acknowledgment take agent artificial intelligence laboratory university research
positive heterogeneous homogeneous step allocate channel access iteration channel step sense outcome round mechanism mechanism introduce fair resource allocation round argue subsection multi assignment job framework latter adequate relate let refer assign unique one resource formalize set maximized cost minimize logical logical logical resource aim interference resource advanced technique interference user performance division division division name agent virtual entity plan decision every coherence worker exploit primary moreover characteristic quantify channel availability ratio sense characterize f resource observe implicit functional relationship relate primary resource state resource among maximize secondary network observe
expert select competitive cifar state art grant amazon web services universit algorithms tune often rule force appeal idea automatic hand automatic within bayesian gp tractable induce gp enable choice parameter impact machine algorithm variable cost duration experiment leverage multiple core previous reach set dirichlet svms convolutional machine rarely regularizer generative
consequence exist comparison mle art analytic centrality centrality seem room remainder centrality main centrality dataset international centrality maximum exist analytic parameter centrality mle bind imply rao conclude remainder line transpose euclidean operator sequence event go grow comparison various analyze centrality algorithm comparative interest assume I n iw ji model ij independent invariant scaling equivalence w outcome equivalence representation onto define equivalent upper distance estimate order object
rise I class standard prior rule correspond denote I divergence use divergence interesting px evaluate divergence marginal decrease practice approach classifier incorporate basic modification follow class capacity feature marginal class easily provide natural q average list dataset denote feature use rank consider classification assign density extension although inefficient require additional I
shape restrict fit onto great progress newton quasi extension successful constraint constraint correspondingly room unified approach smooth attractive onto intersection write unconstrained introduce challenge take denote progress make parameter region example appeal convex distance closely tie c stationarity analytically due projection resort principle rely call subproblem fortunately close euclidean rectangle c set matrix g ball analytic last organize follow place iterative illustrate distance five different set close machine advance present theory algorithmic concluding limitation distance enjoy great convert smooth principle function current combination
unobserve observable also arise distribution see next variable parent rv style thick minimum mm u xshift mm b causal represent arise trial take outcome factor affect interest na I estimator effect biased encode amongst assignment treatment assess effect restriction make observable however finite
c combine conjecture microsoft usa university large introduce upper governed margin hold rich family feature characterize tool hardness property linear classifier use rule effectiveness complexity tight classifier learning focus instance free vc rule learn classifier maximal excess understand positive understand rule restrict require predict upper tight low instance vc cover well upper entire characterize identify distribution focus select classifier margin input treat origin correctly predict specific margin vc class classifier complexity optimal error
ice parameter heat bt output ice ice height base fourth span year ice prescribe ice year configuration important measurement ice measure measure rectangular spatial model ice large ensemble run show ice plot control heat ice mark indicate log ice take location set location index incidence measurement design consideration determine column enkf describe give compare determined measurement standard identity output parameter element behavior mean variance spatial sample covariance spatial correlation take maximizer treating maximize gain shannon minimize course number need compute define write computation relatively system
transform introduce vertex rewrite graphical cavity belief locally normalise require cavity serve find lattice cavity finally qualitatively show qualitatively correct consider cause teacher student lastly bayes gps locally normalise locally normalise error variation see individual vertex globally normalise spread prior change example cavity case teacher cavity gps enable look ahead extend cavity graph student teacher limit curve graph heat normalise exactly kernel give heat kernel method relate lattice discuss section eigenvalue find call consider p simply modify tree limit contribute expand everywhere similarly calculate decay explain understand behave outline previous obtain linearly break suggest look integral integration part lc l cutoff unity state visually cutoff version remove decay expect find numerical suitably normalise plot clear dash may surprising cutoff appear understand intuitively take let evolve diffusion process break diffusion scale diffusion reproduce quantitative scaling adapt walk normalise x mul exp mul mul understanding leave mean function vertex write nx independent
orthogonal consider sufficiently radius display versus oracle initializations curve imply appear fact account agreement prove interestingly would expect left plot radius much dependency coherence initialization potential minima conduct presence noiseless local minimum develop assumption lead also generative spike believe total remain plan formal future appropriate use relaxation technique useful acknowledgement european project fp project appendix statement proof simplify core appropriate problem dimension number possible local neighborhood control version present minimum reference quantity q universal provide c regularization minimum great success decompose contribution concentration surrogate term us generative dictionary noise introduce provide condition find theorem heavily concentrate constitute indeed corollary consequence discuss bind strictly exploit quantity rip noise level parameter define
metric trivially verify adopt argument assume aspect somewhat quantity formulate class function invoke law definition moreover see imply rx exists dominate empirical iid taking every observe suffice verify dx proceed number equivalence cl closure one subsequence md kn subsequence take negative minimal respect nr suffice equation extend originally interest fr value computationally hard fr element variable practice iid realization employ instead infimum necessarily
epoch count static remove historical relationship count forward online single infeasible data sampling base parallel hierarchical receive data processor execute independently parallelization parametric create explicitly master end describe master first post iterate new read joint receive label iterate label update child master receive iterate post master phase master iterate master computation master many new topic begin child happen across child create topic maintain count back master child master sample help quite similar section experiment carry medium aspect model goodness unseen label topic qualitatively trend major estimate insight find trend scalability strength ability million evaluate importance factor able combine factor usefulness factor medium would able perform array medium available truth quantitative
leibler form scale rescaling constant natural introduce uninformative probability measurable absolutely respect respect call x reference entropy sentence sentence countable alphabet sentence entropy equality general conceptually refinement limit always enumeration sentence equivalence require justification entropy sentence monotonicity routine order enumeration limit equality enumeration sentence sn let elementary algebra tail show borel generate borel algebra ni measurable p form martingale z consider z existence show definition measure find measure relative practice scale meet constraint rescaling integral converge measure ix uniquely ix n jx construct end relative choose indicator elsewhere tell function piecewise constant sentence proposition lead follow constrain expression informative contribute define relation indeed factor possible informative measure theoretic first finite convenient sum restrict term elementary expression minimize constrain minimization separately non strictly finite compact lagrange multiplier derive uniquely finite elementary necessary condition probabilistic coherent finite determine probability sentence answer alphabet alphabet probability meet remark informative equation since pairwise proposition finally proposition conversely equation put well suppose valid eq valid finally separate guarantee first sentence sentence valid condition
lda cca versa needs introduce regression cca outperform par p cm name correspondence face contain pose item text instance tag office object amazon department science engineering california assumption train underlie unfortunately instead label exist target improve classification domain method domain vision apply datum make sometimes impossible sometimes transfer leveraging hereafter unseen domain case
field discriminative augment discriminative discriminative value greatly generalization operation next fed perceptron resembles initialize respectively treat constrain remain transpose learn
share chain chain approximation elliptical amount chain take different time may slower one update period share validity fraction spend measure overhead point operator transition preserve indicate round long multiple time need approximation randomness compute algorithm maintain collection advantage core core make use collection seem good collection chain collection burn motivate distribution section convergence amount summarize amount approximation iteration every varied approximation change four work update slice straight point elliptical slice along use step affine invariant make differ elliptical slice perform variety scale adjust member parallel population parallel another parallel involve sample separate chain encourage chain auxiliary hamiltonian burden user combine stack support practical
conventional accuracy subspace two mahalanobis mahalanobis confirm mahalanobis compose class sample therefore small zero accuracy comparison achieve pca mahalanobis kernel perform bad mahalanobis high gaussian kernel increase observed subspace estimate classification signal classify mahalanobis mahalanobis class sensor spectral pixel feature nine define principal increase conventional kernel pca mahalanobis equally accuracie influence accuracy classify report mahalanobis mahalanobis test variance influence compare mahalanobis kernel estimate
prop suppose combination none q prop fix raw distinct candidate bid treat either occur ad ad etc say matter ad ad ad query reflect inductive result p map selection prop maximizing maximize eq since one suffice single decompose ad bid raw I respective ad expect ad since ad negative ci quantity ad hard selection mechanism prop consider sort ad bid show efficiency hard
date consumption distribution variable consumption realization multivariate amount almost customer collect expensive minute total consumption customer storage get estimate profile reasonable select sample population compare compression population conclusion situation rather simple design survey property sampling treat far know nothing framework investigate estimator use apply functional arise domain functional response time index structure section concern mean linearization curve variance without replacement proportional replacement linearize sampling adapt stage thompson population functional element trajectory straight generalization univariate customer company want location interpretation centroid three
say x go arbitrary let tune accept accept explore reject see tune accept proportion accept history article approximation sa sa adaptation condition p tv verify rate convergence therefore aim stochastic common diverse field model branching process advantage discrete apply try
clear record element nonempty recurrence relation convention subset tuple record contain record belong tuple record record classify tuple appropriate record notation product function tuple inefficient problem bipartite record linkage block thereby reliable categorical code quickly record code link discussion link record tuple assign subset agreement link file record decide remain assign record tuple tuple assign see practice tuple classify partial subset record tuple fine equal record subset potentially present file illustrate gray become record match census subset census subset pair link record assign link record linkage classify belong direct implication obtain tuple subset determine
explore misspecification forecasting initialization adequate historical observed must care take training datum forecaster take misspecification base outperform adequate extensive study demonstrate use dynamic forecasting method little system forecast likely poorly demonstrate system surface long svm determine width end dimension correspond set pick grid number training speak actually factor use cross approximately grid geometrically validation pair grid validation performance l one repeat change validation
strategy exhibit tree sphere enough summation preferable expand determined procedure cutting expand appropriate preferable opposite summation less copula aic function never great algorithms construction nine bivariate appropriate ordering copula sphere variable first order I success good evaluation sphere algorithm exhibit well resemble indeed exhibit random cause efficient evaluation conclusion careful copula decomposition respectively copula flexible product copula rich confirm correlate always necessary
rejection bottleneck mention nevertheless balance mention tb allocation sec start procedure cumulative shift shifted assign shift obvious amount allocation unique construct overlap position order explain shift example update cumulative gibbs determine uniformly detailed known good way make negative generation order candidate order position update uniformly symbol nothing satisfy detailed net flow globally autocorrelation method candidate sampler surely explain significantly rejection case minimize create
limitation observable process causality require exclude whole universe impose source entropy face process set able source solely graphical applicable estimation bias effect estimation skewed come positive link material coupling attribute coefficient mit filter influence parent mit strength link regression regard equitability property coupling couple partial correlation approximately analytical mit adapt air temperature anomaly heat towards height km coupling pressure pressure mit panel significance white estimate use separately surface significance average e lag parent spatial lag mit leave panel peak indicate month mi significant lag couple delay difficult
figure discard search find high place slope checking cf gps job provide low upper bound limit vary costly would avoid optimum easy surrogate aid optimization restrict gps enable evaluate optimize reader general surrogate utilize gp confidence ucb deviation gp choose study rely heavily idea surrogate key trivial notation follow set
regard verify log regularize problem estimate regularize yx formulation convex convex desirable section high formulation formulation impose natural dependency interpretation special univariate reduce high regression replace row formulation element penalty arrive regularize maximum linear penalize also study sparse regression unknown univariate variant neighborhood write entry row entry row matrix formulation due quantity regression know multivariate regard generalization graphical precision author suggest estimate selector mean contrast estimator convex framework investigate performance monte employ sub
instance section collection categorization validity light previously point classification deal state compose assign learn infer word assign subset unlabele cope classical consists divide binary decide category naive baseline relevance collection object assign tag due absence another tag motivation look result binary assignment capture improve utilize assumption relevance still categorization year basically transform adapt adaptation contribution algorithm field variation cope intra dependency improvement deal probable category document belong category characterize
demonstrate increase hybrid mining attribute fortunately increase generalization generalization role indicate optimum mark circle define attribute hybrid able researcher recognize importance explain top bottom top description resource instance analyze task enable work manually pure manually carry method thereby task bottom role algorithm role mining different instance candidate role role cover role role select role differ proposal cost role role mining variant paper problem definition discover role upon prior role assignment rule mac focus role select pattern unsupervise hybrid generalize particular connection concept cover approach method solution motivate analyze world usually realistic prior mac one core model highlight role role assignment theoretical comparison sound confidence assignment modify assignment mac user mining prediction compression problem role configuration definition mining inference analyze variant access control robust generalization ability ability role mining convert joint joint depend outcome realization vector individual contribution ik sum pick half factor term sum sum range modified set bit except successively product sum probability two start exclusive event p substituting yield necessary gibbs collect distribution ik ik beta beta euler integral new
beneficial also lasso despite number simply provide support zero prefer might gain predictive include correlated variable drawback elastic q response column view trade ridge depend relative equivalent method observe outperform elastic net measure second handle sum orthogonal trace penalty
loss minimization cast separable many proximal regularizer norm interesting naturally smooth term square typically subgradient norm efficiently common situation machine processing regularizer atomic norm norm decomposition compact translate correspond may choose constraint I differentiable interior gradient converge g bregman always see project start recursion x
computer build south road qx email ac website http com language thought language thought permutation require begin
certain regularity fa fa g tail boundedness mild quasi assumption could regression endow property far adjoint q w g exist sequence bilinear eigenvalue assumption hereafter satisfy sum sup sequence delta convergence enable expression g theoretical derivation underlie assumption know polynomial say marginal view rely ode bound infinity eigenfunction finally use notation fr z ng w fr derivative ds g explicitly write k z g ng sg ds f identity operator develop key asymptotic rigorous theoretical develop present norm exist constant choice process possibly continuity condition proper necessarily state type dominate q
change adaptively subject threshold aggregate eigenvector subject give grow extract projection without invariant subject towards define penalty denominator perspective subject rank aim remove set subject want dimensionality subject interest training determine validate manner type subject discriminative scenario important valuable datum illustrate domain discriminative subspace datum discriminative stationary subject user subject give similarity discriminative non stationary subspace task furthermore common perform subject subject reliably subject present bad affect please amount maximal remove limited subject important advantage avoid utilize vary discriminative subspace
location main area region desirable fine km resolution lc lc lc lc lc lc lc lc lc lc lc lc lc lc lc lc lc marginal level limit area km cover see blue set take complement indicate grey panel indicate region certainly ni method pair level avoid uncertainty cover uncertainty contour curve contour lc lc lc lc lc lc lc lc lc lc lc lc lc lc lc leave complement grey contour green show contour trend cover related feedback surface rapid receive much attention period individual occur significance testing field thus find simultaneous trend year prior measurement determine trend restrict spatially vary evaluate distribution choose correspond measurement phenomenon would entire assume measurement one pixel
bivariate nine devoted illustration financial proof code test study framework theory empirical probability assign mass elsewhere measurable evaluate function mean sequence weakly notational weak denote indicator lower recover version state advantage general low may goodness consideration choice half paper van satisfy x x open measurable fr map np asymptotically linear q proposition prove appendix subsection exist brownian limit state proposition class depend family respect proposition
unless provide author forward step show behave effectively sensor datum author expert recurrent neural rnn environments low sensor datum environment process error filter schema rnn also take application view forward optical flow detect forward acquire obstacle free compare expectation novel incremental generate capability purely optical spatial enabling predict locate time phase optical field capability robot follow
evolution traffic study involve affect weather prevent provide traffic flow weather weather direct whose variability quantify impact weather traffic pair datum traffic different weather road variability road condition occurrence traffic study obvious heterogeneity datum weather road conduct nevertheless previous modification enable velocity furthermore enable different weather change conduct therein achieve build select observation traffic different
deviation building intuition amount proportional amount temperature controller control day span day provide use implicitly variation usage due building keep use use notational reason default controller energy usage day controller day estimate day statistically significant estimate characteristic statistically difference enough exclude controller day statistically suggest default mark
gap total obtain proximal sdca one pass general unlike batch gradient accelerate gradient relatively even significant loss convergence prox prox sdca gap nonsmooth advantage duality check criterion discuss convergence nonsmooth nearly
properly horizontal material follow sentence input file english name wider take width page live area environment table location immediately sentence file comment author alignment author enumeration table
generative discriminative pre layer mm method improve positivity encourage amount use finally avoid feature finding applicable cs optimize single global contrast exist scheme heuristic model store information appearance decomposition signal although pool incorporate range demonstrate secondary issue within present detailed machine representation recent feed attempt generative deep belief network compositional mechanism representation give invariance perturbation level large common preferred pooling many hierarchical directly pool region global pooling max max averaging sum joint hierarchy something build layer hold fix optimal joint notably machine
assignment customer generative specify customer customer form index partition table customer restaurant table configuration customer factor notation factorial similarly except numerator factor find numerator customer customer restaurant collect configuration block customer exchangeable construction exchangeable turn particular demonstrate instance bernoulli represent allocation collect successful draw feature allocation latter case feature distinguished frequency order order reason suggest random random achieve maintain across associate assign uniform natural follow unique augmentation order random factor number block nearly equation argument construct size partition case implicitly summation condition long hold argument explicit necessarily one partition exchangeable consistent allocation whose order feature frequency represent identically form appear feature order choose exist argument call assign index block ibp recursively chinese restaurant like crp form customer partition feature crp ibp start choose sample customer yet restaurant recursively choose sample customer customer sample equal customer try customer belong feature allocation
recognition operate genetic evolution generation individual leave system condition genetic programming fusion return system order make decision rejection none equal system low far good accordingly reason use evaluate lowest generate low increment high step far comparison inter intra score roc far mean couple fitness various genetic present evolutionary produce division minimum number return mean mm score score distribute linearly score linearly tree depth em individual depth limited mutation selection fitness inferior
accurately exact would noise noise atom finally code indexing significance grow combine atom consider combine dictionary dictionary learn multiplication matrix interpret dictionary investigate similar finding dictionary reference dictionary allow significantly problem fast goal describe selection formulate submodular canonical derive neighbourhood small alternative submodular formulation mention generalised form author select support non help quadratic objective include boundedness dictionary matrix optimisation practically complexity iterative
fair fair sequential aggregation rule obtain rule grid report fair prior adaptive run weight allocation select sequentially good pair maximal grid eventually construct cover span implement extend grid user start say value test different try impact bounded place grid possible grid table fully character meta limited performance would period allow enough base rule symbol well good single expert p benchmark grey black quantile residual group half move measure residual want come aggregated hour subset aggregation discuss study aggregation meta benchmark expert overall aggregation benchmark performance hour good combination forecast share aggregation slightly bad seem excellent term probably benefit intermediate around intra depict third value residual distribution concentrate benchmark conclude aggregation never prediction expert expert error strongly favor
global order sufficient full initialize cluster dx I poisson isotropic etc initialization mle good give diagram dual cluster al cluster bregman bregman interpretation efficiently weight maximize complete likelihood amount minimize weight ratio mean standard bregman bregman choose q fc b fc seed optimal bregman factor explain triangle bregman bregman interpret remainder taylor consider proximity property zero mixture weight increment combination rise weight al em mixture bregman duality bregman divergence bregman decrease complete converge monotonically decrease expect doubly
positivity temperature large temperature grow mathematical formula energy significant role free define shannon entropy free energy cross instead multivariate define side energy introduce statistical positivity differentiable monotonically increase energy coefficient partial follow reasonable accordance accordance I principle finite accordance data principle mass energy state part interpretation shannon adopt select accord free principle maximum entropy difference arise ground basis shannon entropy energy estimate finite
td td step follow step combine base criterion adapt hmm obtain merging th term merge merging criterion cluster obtain close local iteration hierarchical em choose framework view maximize non informative uniform satisfie integrate approximation bic maximum bic consistently devote classification global
project read air research laboratory contract fa conclusion recommendation express material view u release thm thm corollary remark family notion symmetry probabilistic framework exponential family equivalent class marginal deal variable usefulness general framework variational map lp relaxation lp bind result art function relatively efficient
need reservoir computer good performance operate drawback precede absence reservoir process limit analog bottleneck point modular computing hardware parallel analog possibility back output apply reservoir computing category generation intensity propose drastically optical experimentally reservoir report although reservoir reservoir computer usually neuron typically neighbor linear combination instantaneous coefficient evolution reservoir discrete total reservoir input describe topology favorable dynamic treat
batch devoted therein intermediate definition assumption direct convex strong convexity combine fw fw furthermore fw fw average evolve accord next convex optimization serial begin sequentially repeat recursive typical descent characterize appendix serial algorithm immediately regret standard study distribute discrepancy state
de un ensemble de class la action dans ce ensemble le du age des la la distribution axis les des en dans un en un une de la
accommodate input position obtain term setting enforce benefit become ambiguity improve ready introduce equation denote discrete time policy actor reinforcement algorithm parameterize generic ms valid exploration term actor update adjust choose distribution gradient lack gradient traditional matrix outside type actor update gradient desire hamiltonian parameter algorithm balance system action k r x actor x balance actor raise stability balance lose effect preserve passive definite hamiltonian immediately specification rl guarantee exploration necessary guarantee
coefficient classical separate political alone vote drive political traditional ideal adjusted figure stand become overall traditional adjusted interaction specific depend vote name popularity ideal adjust also assign advantage include popularity extreme increase issue qualitative difference ideal quantitative issue well predictive vote divide vote individual pair fold vote vote evaluate hold vote assign probability hold several ideal label describe study issue ideal rather topic lda topic make issue datum variational mcmc ideal issue remove contain maintain mixture improvement traditional topic change five ideal issue adjust label issue ideal lda adjust lda adjust issue summarize table issue adjust vote permutation vote issue adjust issue adjust ideal demonstrate issue give well fit roll point validate exploratory tool traditional ideal characterize call datum demonstrate approximate collection vote adjust fit discuss several vote party line explain preference like issue model measure log adjust relative formalize define vote log vote adjust likelihood point issue improvement vote log likelihood measure house
whose h breast cancer alm admm alm nonsmooth area splitting method way accelerate distinguish wide applicability problem covariance show code implementation handle prescribe application sample interesting advantage varied definition motivate problem completion cluster robust pca sparse simple simple non advance domain optimization proximal iterative smooth fy ng
click directional click feature mutual mutual information entropy determine user must convert bin span quantile make bin mid cover velocity mid pressure stop direction velocity stop duration direct distance pairwise velocity velocity velocity acc end trajectory large end acc acceleration end line acc phone stroke orientation leave coefficient analysis percentile pressure trajectory informative position adjust position orientation orientation insensitive please rank rank constitute gain complement hold correlate percentile percentile end segment direction end connection understand feature redundant color green encodes color plot highly selection still understand discard thereby speed merely decision distance highly almost thing constitutes ignore trajectory classification due discard orientation end velocity feature feature stroke exhibit motivate stroke framework neighbor rbf kernel various reason work stroke observation close observation classifier merely observation huge limitation
infimum quadratic coordinate descent paper simple newton descent minimizer approximation descent minimizer find line repeat meet algorithm search necessary ensure q equal q reduce optimization gradient xt algorithm imply definite computationally beneficial g middle loop penalty block separable differentiable separable separable twice differentiable quadratic decompose follow symmetry minimizer consideration cyclic j j j last term penalty due part ordinary descent block descent claim follow minimizer descent
gaussian development affect affected reflect copula payment incur significant shape quantify contour sample contour vs scatter plot third eigenvalue eigen combination variation payment incur year ht plot marginal box plot box box posterior distribution present presents copula log observational figure structure copula plot copula copula posterior tail payment incur uniform copula contour mixture copula year incur assumption year posterior surface posterior dependence year incur mean scatter correlation correlation versus payment incur mixture box distribution box plot box marginal posterior box posterior distribution section detail able payment incur give integral analytic distribution incorporation significantly posterior development payment incur clearly shape dependence hierarchical mixture admit analytic augmentation stage sample cumulative payment loss year note base mcmc interest sample complete model dependence construct normal approximation locally factor precision covariance gaussian may copula augmentation alternative distribution total give payment incorporate development precision development good agreement dependence ht posterior predictive box kernel loss estimate mcmc denote depend
theoretical bias causal first ensure estimator relation utilize collapse graph e node parent graph causal diagram parent edge node edge direct applicable selection mark parent collapse transform contain model conceptual population model collapse selection diagram follow diagram selection diagram conceptual identify ii consist causal causal include identifiability surrogate design identifiability causal effect identify rule calculus effect integrate effect allow rule however express estimate identifiable specific instrumental clinical trial identify even causal identifiable form may
across operator nonconvex penalty mc parameter realize model procedure evaluation problem degree freedom degree freedom selection introduce df bic df p df penalty freedom well turn refer improper improper converge slow unstable issue penalty tr tr prevent solution select value whereas generalize bayesian criterion penalty prevent occurrence loading dimensional diag penalize likelihood traditional rotation
discretization state facilitate arbitrary applicable increase exponentially space exhibit kalman gaussian filter try adapt filter assume state represent kalman apply series offer employ linearization density true property approach approximate accuracy depend dynamically linearize linearization mixture induce linearization quantify covariance base moment preserve splitting introduce direction nonlinear linearize linearization linearization applicable formulate sec brief introduction linearization novel splitting derive sec describe component conclude remark time
technical representation briefly estimate gradient main introduce state conclusion prove condition fulfil section present network criterion normally value behaviour analytical formula normally realization carlo function form variable offer potential analytical study begin management scheduling project project activity activity time perform activity perform normally interested describe oriented graph arcs correspond project belong must formulae di f activity describe duration random represent involve completion express
pz chart plot z difficult distribution resort chart chart give mean chart control comparative adopt outli outlier outli number largely shift place put choosing outli take take outli carry simulation order compare
minimize effect run validation analysis different suitable nn classifier network approximate per classifier neural network nn nn bayes class variance measure cell variant choose minimize mse network classifier comprising train due take average discretization ideally marks go play aim aid classify style dimension many recommendation game various base currently promise world rating play especially play similarly go quickly human opponent game auto provide pattern provide experience hope look analysis pattern especially hope insight various shape improve investigate challenge extract game play classification highlight room improvement record discover classification origin game aware previous
rmse pls hard pls pls hard significantly drive pls ols large show rmse pls pls soft slow rmse rmse least pls pls drive promise finite rmse four next sample penalize step drive variant plot approximation close left reveal size informative panel influential plot obviously line coincide demonstrate construct pdf c comparison suggest adopt interval plug obtain specify drive choice rmse suitable confidence first four index pure cc cr rate setting drive interval previous except coverage close nominal proportion coverage rate decrease sensitive example implement penalize study expression gene
graph pure therefore idea good algorithm time label node nod benchmark example respectively g window benchmark form web page web performance construct graph gain pure combination performance naturally site method al label consist similar pair tell construct pick oracle know node consist unlabele unlike percentage construct even connect edge connect label unlabeled take construct take pair filter edge
party I local standard deviation age presentation deviation visual assessment alternatively one parametric region list model gender region g display probability support function age package surface bivariate bottom indicate surface gender region surface probability similar htb party sign change frequently obvious play profile adequate appropriately reflect evident aim precisely group
screen target ht response model expression signature signature prove clinical reliable disease signature coherent signature routine clinical practice establish signature design classification signature association underlie enhance derive signature network validate wide detect model local condition independent predefine classification gene extend approach predefine reveal unknown know related modify applicability datum identification coherent component predefine pathway collection entity complicated gene suggest gene predefine predefine small module signature search set miss purely orient biological bring connect difference relate approach single additionally specific expressive module signature specific
perspective continuous concept remain constant bounding equilibrium denote convexity vanish exhaustive iterate nash equilibria exponential tb sort uk note constrain equilibria fashion observe candidate nash equilibria maximize increase equilibrium nash equilibria joint frequently observe maximize aside constrain equilibrium equilibria linearity constraint thus fit subsection game check whether nash operator account equilibrium make satisfie np empty complement equilibria search regard refine hypercube function possible linearly separable first number tight bound player conclude vc vc neural easily unfortunately weight I find induce experimentally find estimation game equilibria rely mild space likelihood trivial sensible variety base may look would range fact broadly objective want behavior need tend try convex upper find slope upper maximum bind kl bound informative since compare red bound equivalently next maximize equilibrium game sufficiently hypothesis identifiable kl kl kl kl kl kl prove third approach sigmoid equilibria additional ascent maxima gradient ascent hard proportion equilibria implementation regularizer encourage attempt lower control approximation maximize equilibrium minimize equilibrium maximize proportion equilibrium loss far introduce note avoid obtain speak make hinge require minimize develop efficient solving hinge encourage attempt fit loose simplify player independent regularize joint novel output logistic equilibrium kl
identifiable approach correctly probe zero propose keep probe deviation high variance j yield noisy prior expectation probe support probe content element probe affinity estimation probe prefer accommodate potentially probe ij ij maximum j affinity estimate across yield final specification probe level j j weight inverse ij learn far validate comparison preprocesse standard datum ideally batch confirm compare regular moderately sample estimate pearson version batch memory resource sample correspondence batch confirm convergence probe probe convergence later indicate array typically case online file
cutoff law cutoff law alternative fit normal exponential cd cd cd pt branch volume moderate none stay moderate wind speed none none wind speed moderate cutoff city moderate cutoff moderate none rare gene analyze city intensity come similar analyze counterpart intensity consequence choice large slight raise compare furthermore poor power law cutoff heavily contrast fail reject cutoff lose obtain intensity case conclusion direct implication illustrative work branching argue law imply branch forest law distribution branch entire collection branch analyze law cite support claim statistical west result match diameter branch diameter branch prediction instance stay well fit power particular hazard probability worth hazard leave decrease stay heavy tailed investigation covariate predict hazard article principle conclusion coarse power sound make method practitioner variety field behavior datum regardless number field quantity compatible hypothesis tail distribution explanation effort
however sometimes great discard hypothesis general reject logical reasoning evidence almost expect shall may occur let main x x p notice eq evidence build come acceptance hypothesis practitioner become summary frequentist general computing statistic statistic choose depend value give frequentist metric dimension hypothesis arise nested hypothesis new regard logical paper present evidence frequentist belief show final remark evidence discuss refer review evidence manifold evidence measure write notation nan item hold nan least coherent plausibility
I es expectation logarithm unbiased walk precisely know get geometrically rate tw accord unbiased geometrically investigate es contrary geometric chain irreducible borel start addition one precisely suppose
scheme approximate iid uniformly km km proof theorem outline coherence r km rs matrix omit proof would adjustment advantage theorem error achieve small recovery noise imply minimize large error
application consider phenotype interest clinical trait simulate snps phenotype rs rs detect bayesian association study account serial structure variable represent trait characterize phenotype significance strength phenotype unified approach challenge met adopt show variable spike detect weak application genetic diabetes demonstrate relate measure limited phenotype continuous phenotype association marker interest effect phenotype impractical genetic variant advance parallel partially less moderate propose uncertainty however must science engineering ls health lx disease receive genome institute national diabetes disease institute diabetes disease national health diabetes diabetes study support institute health grant
matrix linearity may responsible variant hide specify dirichlet distribution think probability simplex denote distribution q pseudo uniformity pure one one rest allocation represent independently fall represent vocabulary multi exchangeable model random dimension full rank long share exchangeable rich view factorial series hide abuse product factorial make mean shift make linearity furthermore recover transition furth topic underlie hope recover column extreme good recover multivariate see transform invertible indistinguishable statistic issue
entropy entropy uncertainty past transfer right side lemma information follow entropy non conditional entropy equation mutual negative later equation equivalence come go flow state three lemma multivariate
successfully equivalent match discrete namely generally implementation corruption implementation fundamentally get would limitation ability discrete continuous review would interesting generalize besides seem qualitative pattern illustrate arise force mostly parametrization energy experimentally mathematically formulation assess score inconsistent gain constrain chen acknowledge support research use quadratic corruption behave first equation auxiliary rewrite put quantity focus quantity optimum read rewrite term equation taylor expansion expansion inside represent term odd reason expectation vanish px px use geometric expansion q result study asymptotic would would say deal pointwise asymptotic expansion problematic contribute use quadratic corruption rewrite loss index mind consider family taylor involve take expectation substitute taylor expansion express
library initialization seed feasible seed early initialization include slow attempt environment minimize trajectory environment position target initialization trajectory move exploration goal exploration original straight line point goal seed short left configuration configuration environmental configuration library notice straight initialization obstacle therefore difficult initial seed mark seed find also execution blue end initialization generate default order evaluate library free initialization seed fall slow contextual role quickly evaluate trajectory result
rational middle plot entropy rational mean line procedure converge entropy state consider empirical conclusion conjecture rational converge cm cm rational certain moment compare version obtain exponential lead lead direction therefore assess ability outperform algorithm originally another whose entropy suggest fail far efficient moment suggest entropy suggest question well
single post recovery advantage rely subject constraint optimization constraint graphical trace regularize max formulate fairly relaxation relaxation max generally cut norm fact cut negative classic differ several way problem multiple cluster integer sdp number cluster round technique projection typically employ classic relaxation linkage work aware cite focus bad factor guarantee guarantee need cluster hand clear translate affinity enough underlie true
estimation recover element error negligible propagate reconstruction quantify element provide achieve column minus element accuracy instance empirical accuracy simulate node difference clique instance basis oppose accuracy c error reconstruct evaluate term versus network model coefficient bernoulli corresponding network method runtime accuracy comparative degree fit model denote blockmodel mixed membership value summarize comparative consistently second desire
spline basis additive implement discover cf sf fitting cf sf smoothing pf extension name summarize extend integrate apply first illustrate validate rf predictor pf sf spike explain model aic pf sf strongly associate estimate intercept pf sf df spline basis pf cf sf pf partial describe contribution except drop linear pf represent time channel thereby notably coefficient sf space light fit confirm sf end increase discount possible interpretation may fast input sensitive observation fit interaction seem minor deviation improve prediction much somewhat study
conduct multi bandit strategy evenly budget arm identification sr plain successive design arm budget return arm identification idea extension bandit arm figure require propose sake attention permutation gap arm high arm gap cluster
successful obvious variational achieving commonly use discovery rbm energy energy rearrange energy model energy identically comparison reveal energy turn inclusion undirecte direct undirected describe immediate consequence transition modeling tractable training rbm guarantee factorial due away rbms
least negative vertex relaxation equation enjoy prove unchanged every instance think simplex attain regret solution statement begin note look ball pick theorem effectively adapt simplex randomize enjoy since bind adversary since bf randomize simplex bound assume however tend conclude interested bandit barrier bandit play simplex convert armed bandit develop self solve bandit pick adversary bandit pick pick f f conclude step pick bandit simplex use armed note choice output round sampling pick bind bandit simplex observe reality observe unbiased estimate
partition partition affine hyperplane shift hyperplane gap cluster project onto principal trend measure gap adjacent unbalanced size figure two world singular plot depict algorithm gap ht ht division situation two separate ht aside trend sort right singular sort permutation v eq obstacle implementation something gap end gap take separate
fraction close gibbs hmm incorporation framework via link therefore state hmm covariate hmms require six wish wish state disease disease change gibbs sampler employ forward metropolis true hmms covariate describe model use disease covariate monte sensitivity hmm influence process hmms subscript notation disease state observation eq conditionally independent l chain month take missing time example disease give covariate disease deterministic forward fitting model month month covariate influence month covariate covariate month site disease detail description ht dd pc pt pt pt l
state economic subsequence assumption stationary show mix coefficient single process stationarity deal try joint extend ahead set notion function p expectation take therefore may prediction recent I series somewhat subscript within forecast memory forecasting model grow method use make sequence series use say ar obvious meaning memory allow control forecasting loss assumption unbounded loss retain control allow memory allow grow implication considerable training specify tool aic asymptotic class increase uniformly w explanation inverse cf worse note expect supremum event error expect concerned discuss way understand visualize confidence effective eq minimize thereby ensure small outcome sample risk take take movement toward
functional minimizer compute independently functional admit theorem widely also technique machine therein theorem hilbert space rkhs functional exist wise functional coincide section start
mixture consistency value purely arbitrary long speaking degree equal unique neighborhood need converge theoretical unbalanced material block like label truth instead weak assumption label plan far guarantee practice initial option empirically perturbation initialize pseudo empirically topic appendix tail consistency bernoulli let apply result rhs chernoff optimize yield let thank mathematic discussion unbalanced remark support nsf focus group dms grant dms algorithm fitting scale network fail sparse new pseudo range include political perturbation work spectral fail mild condition
framework top standard agent entity receive act distance object agent sensor well ball team learn hand code however learn individually world happen step macro ball must decide hold ball automatically receive ball continue ball leave rl duration episode immediate pass individual call act work consider state vs result large space make benchmark finally approach optimistic use implement try actor transition result outcome pair optimistic learning meaning execute policy changing estimate q optimistic iteration online become available interact paired actor
use approximate nonlinear estimator linear nonlinear control system map infinite act reasonable dynamical nonlinear dynamical hilbert leave lyapunov operator nonlinear thank thank receive international incoming fellowship nsf thm definition remark remark quantity arise nonlinear nonlinear system exist readily extended system success mapped develop computable approximate induce approximate ergodic stochastically force nonlinear easy system study analytically analysis yet necessary analyze dynamical basis previous reduction
subspace invariant decomposition boltzmann machine filter together group cause window filter filter subspace direction invariant unsupervise inherent use representation pooling somewhat datum representation away procedure level exhibit control lose subspace pool invariant sensitivity direction goal extraction many situation return pooling expression subspace subject expression form associated appearance lose feature lose successfully task irrelevant unfortunately ultimately
except greedy notation assume randomized apply state construct individually simply implicit include assumption indicate policy necessary consider policy common approach section focus type make easy theoretically show bind iteration policy optimal upper choosing otherwise highlight advantage bind small limit right instead bad programming bound policy
error line comparison response accordance essentially line oracle except comparison confident comparison active know mention practice fy fx identify request comparison sample opposed ensure intuitively strong ensure equally spaced line difference away bound coin whole exceed probably correct boolean comparison evaluation class possibly error decay strongly convex tight resolve gap due world desirable extend
data decision reliable provide number win bp bp nearest binary method approach lr moreover need seem similarly well bp human bp reconstruction bp statistically significant bp statistically human assessment figure present train via resolution mix super estimation section demonstrate nonparametric image super beta bp assignment activate binary assignment cell distinguish characteristic examine posterior matrix element illustrate upper sensitive sufficiently use equal reconstruction histogram bp batch size vb set algorithm batch present subsampling patch dictionary segment call mini evolution hold natural patch learn dictionary patch represent
without feature see conduct svm hashing convenience slightly regularization svm hash performance achievable reader able easily accuracie linear panel regression panel hashing compare well accuracy permutation e second merely absolute already present permutation hash less original hashing occurrence empty phenomenon bring additional reduce advantage reduction show code code superior coding add new expand desirable algorithm encourage hashing work permutation scheme conduct empty occur sample vector permutation coding work let well accuracy permutation without without matrix sample
precision graph follow keep independent keep number replicate point replicate number link fp false false discovery rr aic bic aic bic well average graphical neighbourhood lasso rr glasso glasso see graph necessary dynamic graph human cell table c aic graph model describe accord scheme summarize characteristic lag constrain temporal lag constrain across self interaction couple lag interaction sparse offer straightforward interpretation I particular part zero priori orient parameter estimate real describe finance consideration determine system big molecular gene structure penalize lack specific consist
lemma lemma lemma complete substitute discrimination power generalization discrimination ratio note moderate characteristic inherently fit evaluation generate population covariance discrimination scatter discrimination scatter since expect generalization bound side figure power repository image segmentation constant class
chain assume singleton must contain
strength range average hamming outperform thresholding post nontrivial sophisticated signal strength investigate half case pattern different experiment report inferior ideally result repetition table suggest outperform various two pt c focus memory accord way tuning parameter u pe pe need take case lasso scad shape mc range vector tuning know scad mc set hamming assume argument comparison report hamming repetition suggest outperform comparable mc gram process cd c scad scad mc signal severe b end except generate appear adjacent opposite ba repetition suggest outperform mc mc signal major signal cd c scad mc scad mc scad p population covariance tune therefore misspecification affect appear adjacent parameter mis report misspecification instead apply tuning comparison tuning ideally experiment insensitive misspecification outperform misspecification adjacent panel adjacent triplet panel pattern panel lasso severe compare pattern signal vary block fraction sign pattern block sign block hamming base repetition report pattern negligible inferior sign outperform theoretic insight
limit goal seek expand eqn simultaneously approximate inexact solver execute dictionary separable kernel denote infimum minimum supplementary write notational mind impose value trace denote cone definite whose sparsity regularizer admit representation broad extensively sparse allow norm norm admits close norm admit optimize specialized solver involve partial involve follow quick bound trace conjugate cg iterative rapid numerically start regularization rademacher complexity give give material scalar concept
get engineering institute science technology corollary stochastic setting optimization nonsmooth linear multiplier nonsmooth closed solution minimize augment directly demonstrate algorithm structural function strongly function
let product notation save covariance uniformly mutually iii q pa ii b pa b ds pa take side lemma ii optimize conclusion sum finite dimensional range weak strongly dependent concavity finite dimensional suit self adjoint dimensional ii eq I ax get
interesting nature mu empty intersection infimum play infimum bottom multiple maximal respectively form mu explicit maximal gamble confusion coincide confusion ex xu yu background yu coordinate xu away yu scale baseline ex coordinate xu yu background xu xu yu yu xu yu cycle intersection xu yu yu xu yu away cycle baseline xu coordinate coordinate yu intersection xu xu yu intersection yu xu yu close second nevertheless top mmd md cp pos dash pos cp pos md md md pos sep white dash mmd mmd relationship set diagram set go top diagram indicate dash line set maximal element move correspond assessment confusion step far move assessment model viewpoint closure typically bottom confusion diagram simplify non trivial assessment correct extensive criterion give intersection closure closure effect generate least dominate assessment dominate assessment structure encounter closure proposition mu mu mu closure apply operator form interest use indicate one agent assessment impossible conservative closure confusion interested turn observation mu dominate counterpart envelope coherent combination dominate assessment closure applicable whenever extreme extreme point set constitute instance style belief priori assumption gamble capture replace small context valid appeal use negative gamble condition one model de section set desirable gamble restriction event issue treat model section assessment share theorem lead natural assessment formally mu assessment combination assessment natural extension gamble interest coherent interpretation accept statement call acceptable therefore axiom status dominate status simplify closed assessment confusion status confusion possible coherent background background ar reject status gamble cone result relation allow representation variant regular east color size column rp xshift ex east west yshift south west rectangle yshift ex
environment figure implement classifier white list order receive validation threshold fix choose false relatively true empty begin black white valuable phase focus address list important whether identical informative derive feature produce correspond white list test arrival reliability mechanism machine machine na I decision bayes contain email randomly select w minute history present heuristic baseline window address threshold ml good ip behind place heuristic interestingly algorithm seem much sensitive compare work produce apply auc score highest whereas low ip configuration achieve comparable performance notice base match discuss continuous email high resource consumption learn white classify incoming email placing handle incoming mode update black activate save computational resource activation amount spam list spam execute batch update execute minute minute minute list update time address update figure superiority
simulation suggest strong property decentralize decentralized detection scheme analyze scheme discuss extension correlate support stochastic observe kt deterministic specify locally space coincide restrict process find time quantify delay formulation bad case conditional expect history word delay history know random cumulative alarm much rich class adopt idea detection delay kullback indeed measure q u optimality imply solve proportional brownian motion index drift exponent case imply detect brownian motion centralize apply decentralized communication constraint account detection center detection sensor center stop measurable transmission
recall immediately single trial illustrate semi reveal lead vanish contribution namely learn indicate fast explanation actually decrease turn original word learning randomly choose integer circles monte panel leave time course trial associate single mind write expect sum provide expression evaluate number trial double vanish expression term dominate decay already scenario interestingly exhibit monotone decrease selection increase fix increasing must increase maximized simulation effect word guarantee
target birth death implement target state association observation quickly resort approximate monte carlo mcmc different offline estimation tracking scenario prefer implement infer target approximation beneficial explore target tracking smc step assignment sequential proposal filter appear previously context da tracking essentially like implement mle technique formulate static contain discussion appendix smc mapping capital small letter density w denote write law distribute joint distribution expectation tracking surveillance noisy observation markov density paper specific hmm linear specify mean covariance x state measurement target surveillance target surveillance evolve transition density element target new target target create mean state superposition newly observation happen point addition
visual image transfer visual new category right available head body four similar texture idea answer support target common paragraph transfer task depend model random forest helpful task give answer question course expect incorporate always learn unseen task concern learner fail lead event happen life friend sometimes word thing occur tb besides learn learn classification symmetric jointly refer combine estimation helpful especially task couple relevant classifier collaborative filtering mention
computed programming computing field binary graph notice factor partition pairwise without form unary factor appropriate without set accordingly due
monotone lemma ne satisfying continuous ne triangle train n n p ne rgb rgb dark lemma conjecture note school electrical engineering department statistic university exponential family density statistic mild result approximate family propose exponential distribution address degeneracy typical family estimate family prominent role estimation family unbiase convex know readily non kernel convenient alternative number vector relative
handle subspace subspace find parameter addition proof agglomerative local affinity manifold local good build dealing subspace sensitive cluster resolve similarity similarity deal assume subspace complexity exponentially dimension subspace hence computational advance sparse low ssc rank lrr low build algorithm handle noise know principle paper study ssc lie union subspace represent combination infinitely express combination point ideally motivate infer overcome cluster choose neighborhood deal pick close solve hard minimization recover extend representation point unlike recovery problem basis basis subspace datum challenge program convex miss class affine incorporate corruption optimization experimental outperform world segmentation fig ssc union linear generalize condition connectivity regularization increase ssc motion cluster paper order last lie sparsity increase lie ssc motivate perfectly subspace generalize deal entry affine linear subspace lie subspace permutation assume refer number address consist find belong
show possible picture class role class corpus role usually frequently co occur role role see appear come noun come appear role relationship see together instance relational simple case blockmodel sbm role heterogeneity complex blockmodel single membership blockmodel
stochastic stability lyapunov deterministic force exhibit upon rigorous yet detail achieved proceed analyze detail discretize make poisson non ignore issue start reason indicate conclusion
uk negative media finding journal analyse see background fundamental usually refer collection reader social network develop content twitter previously present try resolve limitation generalise capacity rate social pattern extraction daily way able work pattern discovery task twitter content location method form second time tool thesis mm chapter fundamental theoretical notion start general research common learn short well bag present field retrieval vector extraction say statement notion learn characteristic experience previous probably make statement primarily human could machine computer definition past year broadly develop program play intelligence without address general machine respect measure experience machine quantify abstract improve performance several learning task one try identify primarily root machine quite year basic category reinforcement learning scenario also target response target trying discover content reinforcement instance discover try mainly unsupervised effort description theory use variable variable case break task independent know value dimension one intercept observation mapping function hold target know ordinary ol residual sum eq intercept derivative equal derive ordinary square ol solution interpretation square affect appendix ol generate predictor something ol equation regression know control dependent shrinkage aim main many correlated correlated shrink coefficient impose variable sum sparse still exist predictor hybrid approach disadvantage discrete quite often reduce ols define refer norm ols ridge close accommodate identity top diagonal way add process norm encourage sparsity offer interpretable maintain solution retrieve know lasso optimisation eq control shrink easily control transform ol nan moderate choice initial regression regression programming estimation efficient way achieve enforce modification coefficient coefficient direction joint coefficient current predictor become zero variable square direction iterate predictor compute residual explore predictor direction predictor variable active correlate predictor continue along least fourth propose drop arrive nonzero value assume predictor perfectly probability sign mean sign correlation relevant irrelevant make weight infer pattern differ fix relaxed bootstrapping categorical recall target represented category label label representation structure primarily later extend regression sort top label classification depend figure media content web high score big risk epidemic actual tree whether epidemic simplified instance middle tree classify yes likewise classified assess manually practice mean supervision order make application text mining computational failure diagnosis medical scientific intelligence construct automatically tree past tool chi automatic detector quick many briefly chapter interpretability self explanatory easily understand handle miss tree rely make create decision np apart optimality instability minor section cart sophisticated program et cart cart capable great established exist software package implement tp four terminal entire specific making similarly previous tree number perform recursively splitting form cart construct binary tree divide divide variable observe use variable create branch core cart subset set subset subset decide optimisation response terminal regression control denote square loss overfitte set decide version assess improve sample accurate replacement original bootstrap consist member appear appear appear bootstrap predictor primary sample element bootstrap deviation predictor aggregate well predictor replicate predict bag average version prediction operation bag bagging improve significant impact unstable predict bagging reduce problem test classification bagging interestingly necessarily tp give respectively unseen variable voting decide aim create dimensionality extraction method manner dimensionality find bioinformatics text speech processing aim effort dimensionality interpretability infer speed learn follow extraction selection specify basic selection analysis correspond primitive divided product indicate perfect anti absence rank important represent research version name modification effort cart instability cart able text stream use main collection material usually ir automate algorithm perform computer section go basic ir apply throughout corpus collection collection gram convert something structure receive form gram document weight equal exist alternatively count occurrence gram treat gram possible semantic random reason usually bag word inverse approximate actual similarly present tf bag count frequency normalise additional reference vocabulary result derive vocabulary
next mean equation call estimator norm concentration time algorithm ensure converge theorem represents summarize setting algorithm experiment synthetic report dataset employ extend draw follow multiply concatenation compose dataset sample c sphere affine concentrate nonlinear nonlinear roll uniformly hypercube sample hypercube band isotropic nonlinear face dataset
protein one incorrect illustrated fig ground problem already absence model
segment enumeration member guess least enumeration relationship may specific either eventually enumeration output hypothesis unbounde finite enumeration initial segment enumeration even code family
machine pattern terminology classification construct score univariate object assign know perfect score great support way measure rule measure complement discussion complicated classifier crucially affect classification tend object tend class receiver
mini achieve neighbor size mini batch lead mae approach outperform art randomly select mae bad demonstrate test surface surface well mae correction structure b minus l scalable collaborative large pp signal decomposition canonical pp em multidimensional pp lee negative matrix factorization regression shrinkage via zhang benefit pp v compressive inf zhang yu automatic
arise g iterate provably consist split sake clarity without generality backward close proximity adjoint operator dependency iterate drop formula allow compute iteratively
come exhibit equivalent normalization say satisfy obtain norm actually norm amount aim ai belief idea express message term belief start rewrite ai ai ai ai conclude whenever belief bp address series viewpoint look message belief
sp finite interval lebesgue c n later become infinity place restriction coefficient approach continuously differentiable th vanish slope allow run zero smoothly correspond since assume hold penalization theorem assertion uniform arbitrarily slow obtain bind inferior property procedure expect subsequent enter estimating equation small rate might b finding demonstrate use scad encounter quantile section implement adaptive sub perform wise might desirable rather keep certain range quantile preferable quantile introduce kind adaptive far disjoint preliminary converge uniformly bound wants set different quantile different interesting present fashion affect problematic assume exist
give robustness independent positive number subset suppose fix consequently apply inequality bind inequality go subset lemma zero two bound identical element nonzero set taylor expansion hold jensen define lemma side sample assignment maximize likelihood close finite text variance bernstein bind finite population assumption locally assumption detail maximize label small lemma place detail note large must rewrite remainder combine f follow additional empirical result standardize student misclassification rate size deviation normalization di
x x give divide sample either direction previous jt subtle recursively r convention recursion sampling achieve x x x use conditional recursion subtree message surprisingly indeed look bp message formula algorithms ex however center approach several firstly sketch hmms natural obvious benefit secondly provide compact parallelism tree
area characterize si tie ti q increase every convex choice divergence symmetric hz hz scale prop tie test likelihood respectively mild work al eight
lipschitz x low bind arise growth closely resemble determine modify proof originally lower active satisfied decision translate rate optimum think around optimum role sequential connection demonstrate complexity active dimension first optimization precisely classification boundary dimensional agree intuition boundary detail find boundary minimizer feedback minimizer continue exponentially passive minimax know upper strong convexity popular theory decay
q equation rewrite order moment parametrization grow curve various value grow grow dash function distribution modify third f wishart come homogeneous area return heterogeneous heterogeneous complex I component respectively block vector follow complex c give covariance multivariate return area wishart use model nm center wishart degree freedom analysis describe order define area different degree generalize characteristic al extremely heterogeneous datum reason tractable harmonic
constant passive marginal concave improve bound provide capacity closely disagreement characterize active immediately imply concrete complexity literature purely agnostic noise active deal concave broader mixture separate deriving might interest match nearly unit nearly concave allow paper label machine draw space also class linear keep notation classifier goal hypothesis xy consider protocol passive label hypothesis polynomial label also access sequence make request label hope active
os os em em em lr lr inf inf inf inf inf inf inf inf upon decision graph analysis em inf inf inf inf inf inf inf inf inf inf os lr em inf inf inf inf inf inf inf inf inf inf inf inf inf inf inf regard statistic pose little difficulty exception present small five surprisingly
phenotype phenotype narrow sense list quantity three trait method agree suggest enough datum half training individual closely different family study split intra individual randomly roughly inter family intra intra share inter covariate effect size root square rmse bad inter family split phenotype summarize rmse compare measure give trait measure snp large poorly trait cd consistently outperform seem bayes perform similarly trend tend improve capture large genetic effect small environmental genetic background tail capture either insight perform illustrate benefit approximation superior performance illustrate benefit fix pre fit prior robust specification show difference method reflect rather fundamental evaluation suffice nonetheless competitive speed implementation effectively effect addition snps effect substantially additional snps compare causal snps scenario order fast speed consistent hybrid behave result considerable speed statistical hybrid allow small balance infer computationally tractable moderately implementation moderately equip modern perform wide range setting accurate trait phenotype consistently outperform two modeling despite burden believe general problem phenotype example control one treat status correction estimate derivation text phenotype
integer adapt state join standard build select matching whose range usual similar fitness place ga execute exploitation high multiply receive input prevent rule l logic fuzzy action continuous adapting es great trial performance apply fuzzy action ga continuous achieve discretized trial continuous give receive action greater achieve minimal none drawback predict enable true reinforcement component evaluation
autocorrelation monte carlo texture three model layer cd except unconstraine model generative deep projection visible usual expectation collect layer quantitative texture sample texture patch within test patch compare quantitative comparison model competitive constrain texture cut texture texture image set zero result frame fed gibbs texture seed texture evaluate index compare truth compare texture method consideration fairly competitive texture multi texture layer explore generative
researcher constant budget concrete example illustrate ball covariate ball r risk often hinge logistic let uniform value complexity set stochastic computation output satisfy see give remain classification increase sequence structure capture order scenario design incoherence selection however inequality selection form exhaustive simple nest oracle assumption complexity arise case may hierarchy time provide classification aspect similar expansion wavelet instance satisfying computationally begin class budget modify quite inequality incur setting whether nest hierarchy across hierarchy recall constant see notation give characterize growth penalty budget coarse presentation result coarse present uniformly amongst budget indeed class attention present tractable look assumption unnecessary hierarchy assumption
interpolation support etc use target orthogonal wiener coefficient coefficient value jump asymptotically variance estimate vector matrix gaussian n section validation predict distribution switch construct gap height major source include four pa pa correspondingly construct use polynomial basis validity validation assess surrogate experiment device device record point combination h comparison experimental aggregate respect pressure bottom correspondingly marker prediction combination graphical comparison pressure systematic pressure mean prediction always experimental experimental combination value calculate dash represent fail rigorous make individual failure four pressure pa failure discuss reflect existence directional parameter requirement illustration device consider calculated pressure failure bayes number failure
acceptable wide positive detect drift positive increase whenever change stream discard classifier decide two drift detector affect artificial detector give detector change occur polynomial stress polynomial require contain adapt control define similarly two ii obvious without fair choose give false example pc lda gauss dataset difficult point algorithm low overhead calculation overhead situation artificial contain benchmark literature point identity draw gaussian classification datum feature uniformly curve time stream detect estimate take gauss gauss denote gauss occur stream stream change
three compute three assess importance influential observation suggest kullback leibl distribution hide focus parameter sensitivity suitable problem speech bioinformatics contribution forward backward triplet extend complexity configuration show outlier anomaly must influential context prove efficient detection appropriate rescale influence
form leave leave identically testing block expect block expression block probable large graph correlation degree begin observe large suggest correction approximate taylor computing hard use let put large expand yield simplify set keep term give plot block model raise challenge globally challenge way consider divide pattern link tool belong impose homogeneous degree block
sufficiently self sufficiently evolve real activity node value memory system accordance fuzzy buffer nod fuzzy yield buffer associate bin weighting since neighbor overlap convenient bin associate neighboring bin imply represent show subsection tuning redundancy spread fashion pick buffer ultimately give correspondence ks implement discretize derivative notational activity consider discretized non entry read respectively value calculate correspondence self evolve discretized discretization derivative discretization distance neighboring turn relative construction far small hoc low requirement closely spaced compute activity process numerical differential numerically evolve differential equation discretization propagate boundary computed discretization propagate derivative induce compute th derivative fact free coarse th discretization construct grain th derivative free grain weighting match discretize limit fuzzy system linearity equation uncorrelate small turn choice node uncorrelated standard representation neighboring generate delta magnitude function signal neighbor remain however node accord snr make node
collection focus applicable model performance causal even relatively infer causal among actual consist interested multiple approach applicable lose output connection replace respective mean aggregate denote important exploit full oppose towards approach design
family avoid need representation derivative univariate polynomial basis function available recurrence polynomial polynomial division difficulty fall robust alternative require evaluation three recurrence preferable conclude analytical mit edu technology become optimization space gain formulate stochastic ii deterministic quasi polynomial model objective conduct partial sample estimator quantify solution robustness experimental crucial development understanding across careful translate financial resource traditional factorial composite largely explore response experimental guide particular parameter inference discrimination extensive endow gaussian extension linearization approximation analytical impractical power nonlinear design perspective approach foundation noisy indirect constraint heterogeneous focus experiment design experiment parameter inference design objective theoretic leibler complicated must use objective available design many approach simultaneous perturbation approximation gradient great broadly involve average latter must invoke deterministic employ gradient involve nest differential box adjoint evaluate prohibitive address surrogate expansion
represent rating set j x exhaustive co dataset result every retrieve close object retrieve position ndcg ndcg base distance th similar object rating ndcg criterion assess repeat generative size grow rating interpretation bar deviation set image ct
minute ghz intel core level level minimal level fine effect gp kernel set variance match gp splitting predictive fig analyze distribute location head spatial per brain field outside ratio typically collect subject describe noun trial concrete fall information single key capturing vary still model trial similarly maintain smoothness trial model trial gps norm mu pp training visual likelihood analysis independently sensor test run independent chain global search discard first burn result hierarchical simulate sec cut point cut point temporal gp optimize hyperparameter predictive condition time straightforwardly
individual claim book test set accuracy compute book source accuracy book set explain well share com tag box car cat child purpose tag tag manually tag annotate user aim error yield follow book precision tag among compare algorithm conduct intel ghz cpu gb physical windows operate compare converge reliability group reliability
allow recursively pyramid label reduction well look perspective reduce especially move degradation principled scale fine scale look interpolation act coarse label notation emphasize type contribution trick form directly exploit move multiscale completely specific application practically wide diverse family energy multiscale effectiveness multiscale interpolation poorly interpolation fail landscape principle aware compute variable variable soft influence fine variable hard aggregation case variable influence coarse label energy pyramid approximate decrease degree exclude solution scale exclude interpolation exclude energy interpret one aggregate assign low however energy aggregating allow interpolation strongly correlation variable estimation
size draw get size draw dependent guarantee hinge divide side allow every bernstein prop apply obtain x lemma appropriate draw draw erm able negative class idea spirit online parameterize convert refinement state let consider properly cover denote properly cover follow bound cover spirit eq eq clearly maximize note need unnormalized dividing use jensen q unnormalize two value unnormalize
indeed risk obviously minimize long find mild require prefer high soft svms remarkable bayes obviously minimizer much achieve finite vc dimension word classification asymptotic asymptotic bound statement approximation generate choice kernel entirely need remark distributional short result consistency hold replace practically reason fast higher enough
localize galaxy arc measure show quadratic root compare length blue peak peak turn closely little rd sequence galaxy green point map scatter shape principal pc branch turn scatter length scatter apply scatter next mostly ht galaxies arc mean root distance curve spatial change equal order arc alone principal along galaxy property along w shape principal branch galaxy eps panel representative galaxy shape galaxy arc increase bar spectrum normalize perform galaxy marked emission line line galaxy spectra evident slope spectra emission line etc increase band reach arc galaxy middle curve red dominate agree st b subtle arc length branch connect turn arc galaxy distinction galaxy galaxy appear st branch principal identify green galaxy galaxy st turn strong second branch nd happen rd branch dominate form green turn galaxy dominate th emission line galaxy st branch line transition interestingly strong reach nd turning become th
dataset proportion algorithm competitive htbp example well mkl scalable kernel scale fit plot per million htbp perform range example run transformation sample approximation reduce fit reduction help kernel time outperform column demand memory improve without technique scale well beyond thousand htbp minute hour computation scalable mkl indicate deal limit direct head comparison make make show achieve time achieve kernel implement
see autoregressive process assume independent normally distribute root lie outside ar quantity consume derive recursion v quite show decision recursively length put variable variance chart independent cf lie let causal ar thus eq consequently x possible calculate quantity recursively b x identically expectation stop put moreover chart point zero continue chart run ratio sequential process random know
selection represent artificial member gene genetic gene marker indicator categorical individual depend address bi distinction goal make orthonormal bi derive result develop selective development theoretical computational bi selection genome association issue least square regression extended attempt relevant definite group local tucker condition ordinary square uniquely define definition orthonormal lin however scale proportional equivalent triangular cholesky criterion become q transformation j x x jx jx standard take gram predictor recommend lasso study idea lasso van de reference therein show consistent variant condition consistency error matrix estimation group sparse zhang property group estimate group formulate considerable progress lasso version
inference lda inference objective different likelihood inferior topic soft topic I cluster cluster document topic document cluster document visualize project vb separate clearly phenomenon meanwhile document focus document explicitly often place one document cluster document separate document explicitly include extremely topic principle trade time sometimes frank wolfe provably decide limit quick reach ap small learned lda reach find case average problem stable
relevance hypercube observe stable stable dy u ingredient function agree relevance hypercube choose relevance hypercube hypercube sized contain relevance stable agree conclude variable structure conservative iff label contain understand restriction relevance hypercube tree restriction label discard far occur conservative conservative branching contain leave branch nothing prove branch branching express formula conservative contain hypercube constant
substitute compute inactive also apply extraction recommend pick candidate sort generate final recommendation htbp mapping match interest obtain recommendation similarity popularity item suppose user specify domain q previously category active inactive mention value possibility preference eq acceptance proportion especially receive observe recommendation preference eq popularity
paris universit paris paris france novel move trajectory build optimization cluster profile study superiority propose give insight cluster traffic become activity basis result serious delay environmental collect information road g rate
heavily illustrate accord rp reliable estimation slowly feature precise estimation simultaneously open massive solution projection rp five demonstrate order ii dimensional parallel fit core architecture include diverse signal advance acknowledgment european union social provide grant research european ec fp conclusion recommendation material
prior coefficient see effect numerical improve rank bottleneck ridge regression dimension inversion cubic runtime approximate polynomial log sense create light number pairwise input even generalization performance uninformative instance suffer perform exist selection review ridge elimination redundant unfortunately run infeasible nevertheless cross rr rr completeness include rr rr half decade book build add cross stop expansion unnormalize normalize feature deviation carry forward expand reason prevent ever million selection exploit inverse update previous cubic fix basis million million parameter default describe predict runtime ridge backward lasso give contrast rr employ cubic essentially pass forward add backward phase specifically construct consist expansion bound candidate forward backward add reduction square rmse remove rmse terminate add finally outer choose phase iterate parameter well processing perceptron output unit input hide single combine single use mlp runtime neural matlab matlab neural package support weight parameter determine
training create instance give w line determine test final sample take classifier experiment discrepancy regular average average average last perceptron figure phenomenon average explain increase answer found consider thus nevertheless still outperform algorithm worth drastically improve previous proof simple notion divergence work motivate specific
algorithm inspire estimation parametric generative window supervise learn responsible attribute permit anomaly univariate per attribute normality score combine yet univariate discuss formal pdf scale draw convenient bag attribute attribute draw classified task model calculate likelihood propose hold x dd anomaly aggregation harmonic normality respect set normality instance aggregate normality score especially computer security cover anomalous example anomalous rule learn responsible wise normality score rbf normalization distance function similarity smoothness attribute normality weight weight technique equation attribute entropy approximate density weight technique range lastly employ leave training use training first normality technique likelihood obtain score accord instance score univariate technique algorithm anomalous respect approximate approximate value need thresholded case identify system anomalous anomaly particularly anomaly measure attack expert
hellinger area analyze model distance tend closely observe law look immediate estimation discuss parameter present experimental intensity employ acquire nominal look scene dark area forest
increase cpu implementation minute roughly simulator output six allow fit use cpu example average cpu hour thus perform table implementation computation gpu parallelization burden large minute suggest complete implementation gp gpu although focus hardware gp ad modify give plausible gpu correlation substantially reduce chance want alternatively
therefore approach may proceeding lemma em let respectively statement unbounded unique extreme unbounded ii linearly inequalitie exist linearly inequality together immediately point unbounded statement extreme g moreover independent total entry satisfy divide rest active active total inequality number active independent due observe observe inequality one case similarly virtue relation similar argument independent inequality inequality combine conclude integer j follow let fy fy ji fy unbounded equal finitely cone follow fact immediately
basis intuitive intuitively segment segment majority trajectory relevant contrary portion trajectory play key formation contain devise adapt tf case trajectory frequency occurrence trajectory rarely visit length trajectory measure segment whole trajectory total trajectory contain segment trajectory attribute road finally compare trajectory cosine trajectory trajectory road depict put emphasis fact common road segment
give high decay significant boost irrespective search good weight learning rate decay able small size high momentum scenario stochastic apart dropout dropout give standard take network belief fine tuning backpropagation dropout dropout unit unit learn rate use constraint impose length incoming hyper epoch propagation decrease take boltzmann machine backpropagation field activation deep whereas dropout major encourage useful rely specific effect feature learn figure dropout simple look like whereas backpropagation difficult discriminative less speech corpus dataset evaluation automatic speech recognition american english read phone speech convert give signal need extract output target open library bank extract ms normalize
highlight model sm interest jointly shall observation four interest middle facebook two facebook name page status common update popularity four user interest mild interest four interest likely contrast interest dominate popular four interest page social page notably four internet topic topic start popular correlate differ topic dominate truth school especially house carry sentiment interest seem parent normally interest high topic contain self converse put facebook interest proportion range whereas intuitively coherent friend topic inter topic rarely interact whole inter weak art topic visual whereas topic political support though interest positively former fine city phrase like star
stem ep message kullback leibler kl divergence old belief refine distribute fashion approximate well message pass suffer divergence approximation force ep converge slow message poor function propose relaxed propagation relaxation kl penalization current relaxation message pass moment constraint datum understand also primal differ equivalent
partial partial feasible hold already hence trivial call feasible indeed apply provide euclidean optimizing denote call lipschitz specify property continuity optimal additionally tight follow px combine continuity basically state optimum moreover convergence preserve please idea give bound projection constant optimize continuity lemma lipschitz al l three lipschitz cauchy place continuous derivative monotone hence I lf lf ff obtain switch section follow inference mrfs polytope formulation optimize dual estimate devote formulation introduce notion
phenomenon strong predict music book movie sense rating preference node contact motivation recommender netflix team win integrated similarity similarity focus paper applicability leave future mf mf integrate solve second prevent value ex ordinal entry round value ordinal rating require matter long practice constraint compare
path distance find probabilistic transform investigate give aim distance special considering recursively discrete group infeasible large focused compute important database efficient query traditional top query score probabilistic formalize unified ranking database graph uncertain edge direct visit edge concatenation visit go node propose generalization path system framework probabilistic link adopt method relationship particular
purpose consider vector response coefficient error belong say generator negative probably sometimes difficult three measure al satisfied generator elliptical density reader exist compare model dt n thus nonnegative elliptical interpreted mixture normal sub class
heterogeneity bottom present execution execution spend decrease detection time execution herein figure method never fast follow note discrimination similar focus table consistently reach look window pixel field band sensor display enhance purpose main dark leave heterogeneous due edge detect five agree dot provide compare eliminate extract information use use assessment aim turn determined heterogeneous region heterogeneous five carlo carry criterion
background foreground success also video video extract relevant feature high vision limited success extract class object image regular texture corner texture speak surface texture projection camera low imagine rank
codebook term trade rate well codebook grow exponentially code superposition code square error computationally efficient codebook construct recently dictionary low enable base distortion code efficient distortion successive trade complexity parameter per encoding choice excess source decay fast decay computationally decode encoding rate source compress distortion letting rate successively asymptotically distortion rate encode excess exponentially successive distortion gap typical realize distortion distortion rate designing rate question give excellent exponent minimum code computationally code source
sampling terminate th stage n nn n w prove lemma consequence equivalent continue n sn justify suffice simplicity virtue q cr r cr prove establish definition finally proof combine continuity eq continuity imply q prove nu n imply law z z prove z contradiction show claim b bind b prove desire coverage lemma stop continue sampling rule continue n
work fire model control h university em lattice series parallel application carlo chapter journal pp model mit existence limit many book assumption however example small limit distribution unique class expand b probability
explore refer denote univariate discuss bernoulli possess dimension construct bernoulli realization py k simplify denote quantity bernoulli correspondingly bernoulli realization interaction order interaction bivariate interaction multivariate vector log formulation bernoulli member bernoulli fundamental link j j among addition natural parameter multivariate bernoulli exponential interaction define determine gaussian independence multivariate univariate bernoulli
follow matrix row diagonal concentrated row ensure topic otherwise identify distinct remain row represent distinguished minimizer topic observe unit perturbation matrix nonnegative one admit rank robust sum decomposition quickly verify factorization construct factorization constraint eq member solve feasible suppose let twice fact sum matrix margin ensure th satisfie ii r trace row identify identify topic simply factorization
variety closure ideal vanish model rigorous anti equivalence affine scheme variety encode property vanish polynomial vanish hilbert finitely many generate ideal follow affine ideal closure closure hyperplane closure cube dense require apply hmms visible state trick visible visible emission let emission trick homogeneous homogeneous homogeneous find cut applicable consecutive process thus linear map write vanish use obtain fast derivation use parametrization parametrization describe moment coordinate generator motivation generator give short algebraic geometry cut show generator costly operation generating find instead make moment moment particular binary algebraic subscript index view probability zero provide
comparison mean coincide please involve statistic carlo programming language page complementary multiply sure problem subtle particularly example subsection black art goodness fit moreover require extra work principal require mean example draw small please although natural draw bin bin root draw bin probably useful determine discrepancy large fluctuation outside physical science exactly observe due due arise necessarily test suppose exactly fluctuation remarkably extensive introduce modern bootstrap test separate arise related homogeneity contingency goodness square log latter three member divergence bin bin probability draw respective bin actual draw fully notation classical pearson statistic g hellinger draw number draw distribution differ carlo draw testing hypothesis parameterize
use involved however extend partially partially clause otherwise permutation variable clause clause respectively map clause color add variable versa color clause occur weight add node edge occur incorporate distinct cv weight connect node depict result colored generate f f state colored colored construct clause class set bound degree specialize color remarkable million set position exist message pass algorithm belief propagation operate belief operate approach contain receive
usage hypothesis year one paper computable namely coherence case group coherence define upper pointing bad coherence early central thesis paper coherence particular address measure satisfy coherence straightforward definition verify finally define ready state result group nc large brief significance
family parametric namely depend copula definition rotation extend overview density refer derive formula copula derivative partial gauss copula definition xx yy xx yy xx f yy f yy present subsection hold general pair discrete paper gamma dispersion claim count model distribute f yy approach appropriate claim zero binomial claim accordingly copula average claim copula truncate claim joint parameter gamma
make incorporate improvement accounting use compare property linear fmri decode superiority fmri fmri supervise decode rank behavioral cognitive assess specificity certain cognitive kind implement fit fmri activation variable brain
depend every rectangle I meaningful special give dependence ready use class thought expansion constraint remove time accomplish induction argument focus removal idea constant involve also verify exactly infinity every exhibit cover
weight end complement adjacency es exactly combinatorial heuristic strength understand theory exactly approximately cut thing correspond dimensional euclidean space tight duality reason span distortion minimize lot third return graph cut originally riemannian manifold parameterize independent node finally spectral method method deep cut connection diffusion see biased compute several procedure effectively metric third prove partition cut return big graph space distortion clearly graph structural finally flow consequence average pair thus spectral flow complementary relax different may flow connection know fitting connection discuss remain fail complementary base path cut guarantee reason approximation partitioning provide good atom understand algorithmic flow implicitly regularize analogous lead nontrivial eigenvector laplacian regularize accomplish evidence spectral filter metric place explicit impose nice quality approximation challenging suited algorithm phenomenon observe work highlight base
contradict minimizer analogy special since minimizer x b z set prove contradiction b contradict x minimizer eq recall definition dual complete sample easily formulation belong subsection remark firstly restrict attention loss widely use function state mx b ij x
disjoint clique compose clique clique tend exponentially follow scalar tend exponentially establish hand lemma subgraph give disjoint subgraph consist decompose repeat follow entry probability tend exponentially exist tend exponentially k provide sketch technical establish satisfy eq vertex disjoint bipartite vertex sample plant scalar establish multiplier assumption construct block block index consist one block equal implie assume semidefinite matrix complementary equivalent require orthogonal rearrange system formula force row orthogonal nonnegative choice exist system apply orthogonal ks semidefinite decompose condition feasible correspond scalar unique optimal subgraph tend remainder consist establish disjoint subgraph tend exponentially construction moreover series nonnegative tend exponentially therefore semidefinite tend decompose
assume cosine become tight prove instrumental incoherence among isometry extent subset column behave like isometry satisfy normalization subspaces two disjoint small satisfie r assume contain nontrivial meaning nonzero rely equation w lt tr u ft rl w rl c lt w lt tr ft measure role condition I e x e row issue define small column row contain identifiability span column simplicity singular say projection th allocate direction need orthonormal row r hold program exactly recover pair upon observe function sparsity increase nonzero element column sufficient indeed note nonzero role sufficiently spread moreover restriction place position affect one multiply side theorem coherence
simplicity ard widely finally numerically b termination criterion difficulty difficult reflect real wide range hyperparameter carefully work well synthetic dataset dimension input draw gp se variance dataset derive simulation relate energy dimensionality datum report dataset concern dimension test split dataset available accuracy prediction set standardized log mean square normalize mse always predict predict ghz core least likelihood routine toolbox code section investigate efficacy iterative investigate utility compare prediction hyperparameter hyperparameter synthetic attempt dd
nonzero singular component less comment context potentially improve propose concentration graphical lasso build selector neighborhood b several neighborhood deal begin neighborhood aspect marginalization latent procedure state differently require preference impose wise global constraint require certain size vanish estimator latent believe selector additional selector
leave repeat effective thus ensure spatially variety l l begin convergence follow series thus membership class primarily old h old sm time differentiable derivative old class sm require condition interval wavelet one fundamentally affect follow sm class class sm coincide provide still correspond function r p define pm r variation unchanged contain meaning whose expansion h class coefficient wavelet often always small height jk axis j argue spatially rate rough wavelet also approximate describe tm sm I correspond f
noise dataset world separable technique metric highly nmf local minima associated initialization issue target easy model keep memory datum distribute scale share distribute tweet factorize less exist lead unlike require near propose approach new identify correspond expand cone ray entire eventually figure next extreme ray pick outside cone project vector call residual separate ray find maximize call intuitively pick cone ray ray h geometric algorithm applicable nmf provide datum separability exactly
graphic section family normalize gibb similar complicated algorithm class central idea monte mean use broad come gibbs finite function deal ise false gibbs graph vast devoted gibbs instance
b q conduct assess estimate hyperparameter fixing advantage p perform label field turn posterior unknown couple assuming assess compare value present different separate finally comparison art report consider gamma model frequently pixel extensively apply gamma paper prior distribution fig density gamma note overlap experiment map single markov chain correctly jointly fix ease interpretation scenario highlight blue propose perfectly table mmse simulation column table display generate simulation scenario report depict realization mrf synthetic fig perfectly
ratio regularization ratio vary neighborhood prune ratio begin decrease become unlike omp perform ratio higher appear loss never drop although stable behaviour low interestingly become low norm stability come cost loss ratio nevertheless test ratio close value effect low ratio penalty elastic net logit omp include illustrate denote estimate surprisingly none algorithm attain small error parameter affect algorithm regularization high provide comparable versus regularization mid ratio outperform logit mid unstable behavior sampling ratio change change dramatically regularization relative compare ordinary test sampling ratio sampling ratio appear logit omp almost sparse logistic indicate
variable computation hope achieve preserve cost sg new call sag randomize incremental combine cost sg sag incorporate gradient sg sag iteration like access recent gradient example possible sg pass problem sag unseen convergence translate testing achieve see training context sg apply focus sag typically cost sg review literature attempt aspect sg despite year sg focus aware sg preserve sg give technical applie sufficiently discuss include depend numerical comparison base sag sg pass datum available accelerate sg review outside scope comment relationship closely sg momentum sg momentum
impractical accord wish improve follow approach specifically two easy obtain bound capture due eq use induction song al operator schmidt reproduce hilbert consistent embed lead parametric calculation admit consequently much allow across example test illustrate general reinforcement challenge work identify integral
test wishart distribution employ statistical homogeneous et derive distance wishart work present stochastic test distribution beyond geometric al use technique intensity
stationary various kind stationary particularly kind model piecewise stationarity receive considerable attention graphical influential source universal possible partition powerful modeling piecewise efficient live code partition result binary stationary provable redundancy guarantee recently parametrize prediction change environment predictor directly compression weight computationally efficient weighting segment distinguish towards contain long
sample bandit I denote bind one let hand union follow chebyshev x h old inequality entail always easy see indeed restrict computation tight factor clearly property dependency term estimator empirical show indeed deviation remark basis approach fix tail obtain
phase synchronization nature review w j specification journal analysis th ed read understanding transform wavelet synchronization essentially physical journal asymptotic partial journal business economic w vector analysis vector journal dynamic testing efficiency equation journal autoregressive var brain ph thesis university eeg base generalized synchronization var york n integration integrate synchronization synchronization u synchronization dynamical physical review economic population couple theoretical physics york chemical york schmidt stationarity journal synchronization introduction series ni bivariate generalization present journal conference david coherence application eeg g physical direction couple interact theoretic physical avoid physical review dynamical concept polynomial synchronization review correction equilibrium synchronization universal university phase drive integrated york study force numerical ed relationship synchronization physical synchronization study physical review synchronization physical coupling interaction physical physics p reversible transition synchronization physical letter explore synchronization population couple detection application review j synchronization integration gr heart national united biological behavior population couple r interaction synchronization power vector synchronization physics report autoregressive time p var shift time thm proposition pc lead powerful decade theory couple lead physics area
hard soft machine repository classify normal variable individual area hard compare table datum point vertex five hypercube roc right tree perform hard mean element independently actually use constitute error sample soft replication another describe constitute error summarize accuracy
fast ei inconsistent performance ei outperform ei simulator al jx p contour global show multiple global easy versus global criterion force follow jump neighbourhood precisely ei try minimize prediction uncertainty consequently large trial estimate global implementation point use ei criterion evaluate median global clear well ei gp spike axis benefit smoothness response suit slow attribute inconsistent ei select ei play role candidate simulator function give narrow dimensional detection would volume e exclude function simulator small recommend leave
bayes determine whether differentially estimate factor almost simultaneously go acknowledgment thank useful discussion innovation university institute systems biology road school foundation road china mail author express researcher conduct biological gene go sense representation commonly use rate g strictly control family rate prefer approach control false discovery
learn pp residual machine partially observable dynamical association intelligence parallel online repository kalman journal engineering search machine point td advance system process pp h gradient temporal thesis j reinforcement robot advance information pp advance pp predict temporal reinforcement learn introduction mit mdps temporal abstraction reinforcement artificial intelligence temporal advance processing system ari gradient r scalable time architecture knowledge interaction agent make prediction model journal artificial intelligence mit artificial intelligence use reinforcement represent wide fact dynamic may
soon project material c web figure acknowledgment work institute disease grant grant grant foundation discovery trial laboratory support ai public grant ai ai thank trust distinct accurately cell crucial system biological study rely protein cell common problem datum identify differentially present bayesian beta cell response strength across subject distribution propose expectation algorithm frequentist include basic change experimental gene single method specificity additional robust misspecification combination dirichlet cell population particularly truly homogeneous individual difference cell lose mixture provide single cell protein development flow since numerous cell
theorem accord q f r compute rewrite edu nystr om base impact approximation concentration integral improve error nystr om frobenius nystr om method approximate large
answer hidden state deal easier avoid optima em usual
iterate solve minimize q solve fix q binary usual erm minimization replace many boost strong weak classifier finite subset classifier operate sigmoid boost boost solve stage find eq express solve keep term stage ease indicate reject term close agree expression go simplify variable reject go stage last stage binary initialize hard classify reject use nominal classifier alternate accord sufficient estimate update state optimize follow way stage pass stage pass convergence input f boost subproblem boost subproblem allow surrogate equation enable follow thm surrogate simply smooth descent learner
generality fail concave vertex boundary remove argument vertex particular edge vertex edge vertex new minimal flat shift put reduce case increase step assume large maximum note ks j irreducible define order locally minimal flat tell consist width local though flat multiple cut graph
use website site static page site web usage methodology define come analyse representative trace usage long request total duration second assume human duration request request exclude come web elimination justify usage rather site filter status fail request percentage
human source color fall two category user whole require neighboring intervention texture segment reference method perform intervention pixel value statistic require case image voting confidence whole treat pixel intervention
hence asymptotic useful many answer find rate convergence scale decay positive write exist write corollary derive type majority analyze small specifically corollary rate configuration height tree short tree recursion decay tree let suppose apply fusion probability similar decay ratio fusion turn characterize total bayesian fusion rule majority easy obtain ratio sense minimize use derive follow immediately tree fusion case bayesian least good fusion consider convergence combination tree agent mention tree branch odd majority test since fusion ratio locally total achieve globally interest decentralize globally tree test globally exponent assume message threshold
qx rule define c also assume hidden version item u q u q x hmm triple inequality eigenvalue go identical hoeffding increase obtain figure utilize internet capture google gram popular token token thin svd vocabulary gram per obtain multiply
coordinate within distance coordinate conclude ne approximate fix joint outcome distribution good vice versa first hence prove intuition p accurate approximate fix proof sized complete prove
class simplex exponential function boost hinge consider half loss constraint assume latter considerably slow computation hinge multiclass svm relaxation require far loss hinge loss simplex code fx v introduce notation fx fx loss introduce certain notion geometry coding derive theorem moreover linearly class loss h consider simplex code decode misclassification dx replace
outcome risk recall formally motivate theoretically introduce continuously parametrize minimax compositional paradigm present evaluation formulation model discussion basic live input typical typically addition apply inductive bias select set present precede goal transfer learn good probability task task measure index couple notably equivalent natural classical argue relevant expect yield supremum simplex supremum assume attain simplex empirical motivate
f fx fx true inequality uniform ratio inequality literature q surely constant v surely take cover c j v apply satisfy j u part lemma e v f f f last relation bound f give h replace error assumption almost surely h v z together apply eq q identity
std std std std std x gp gp rapidly prediction area typically area view region low htb e input datum predict view uncertainty around region prediction area correspond low figure low predict view uncertainty around rapidly area view region region high concentration figure e entire region section view prediction constitute gps neural resource mm ms numerous summarize main trend figure follow figure result reduce may whereas result increase deviation ten fold cross generally block well behavior robustness nn kernel gp could apply attempt obtain simulated annealing quasi bfgs update datum approximate marginal comprise training computational resource optimization consume part attempt code matlab core processor machine core optimization interest across analytical gradient significantly reduce initial attempt attempt time hour
determine conversely vector satisfie solution minima must great estimate improve replace rough consider local minima providing number minima residual
mn optimum parameter conjugate function lemma task appear classification extensively latent consist key frame belong category include scene scene weather scene dimension dataset image include cat dimension real value histogram moment define discriminant latent split cc ibp f show ibp discover learn svm discover latent feature learning model svd weight svd initial finite f score achieve table illustrate use either increase bad ibp randomly run appendix detail fold cross hyperparameter mt comparable nearly much ibp winner ibp dataset interesting discover latent show value per overall category different example discriminative value per category deviation probability like process lead feature inactive semantic discover dataset rank interpretable f feature category f different latent mt real scene uci repository treat label assignment treat task dataset task per dataset education student secondary publicly evaluate various define score student school gender band school percentage student school percentage student school gender setup describe variable feature form student school school student school student school ccc ccc acc acc f micro mt ibp svm acc micro f mt ibp mt mt ibp svm mt
provide cost dl complexity replica method analysis plant solution learnable element sufficiently support encourage dl strategy non convenience simplicity plant constraint consider differ impossible main study critical constitute expect property realize energy unity stand
bound easily permutation function surely let careful reading show quantization term distortion vector quantization penalty see target look already stochastic ensure asymptotically use lyapunov lyapunov stable require strong illustrate adaptive algorithm law number hold sampler penalty robust robustness toy alpha alpha left proposal make sound characterization user show quantization introduce approximation examine direction arise would make post counterpart prohibitive model work concentrate modification probabilistic reversible jump mcmc switch inferential study stable throughout lemma supplementary extensively cover zero r lebesgue furthermore v p iff linear
latter dataset spike train lead spike train segment auxiliary spike huge dataset demand segment train whole load spike train matlab ghz intel processor calculation took take less matlab code spike variant www application spurious gain instant instantaneous matrix train e spike train intra inter dissimilarity limit instant instantaneous cluster spike train train ms dash matrix pairwise four mark green regular top spike association indicate train cluster ms ms eight last ms contain regular real spike bottom instantaneous cluster figure fall overall spike spike train change ms spike train four cluster b reach form four time reflect pairwise past distinguish sensitive th differ considerably fourth spike instant event compare column time fourth contrast regular spike time interval average future interval interval relevant past difference interval spike dissimilarity precede balanced spike reflect precede spike although spike normalization dissimilarity spike train spike show instant average certain interval continuous separate individual interval interval linear superposition matrix whole interval column spike train frequently cluster third spike
co establish recently requirement exchangeability exchangeability imply blockmodel piecewise also provide simplified approximation nonparametric inequalities blockmodel identify practitioner actual blockmodel main statistical enable misspecification effort devote community draw community find flexible nonparametric generative understand exploratory organize inequality base blockmodel fitting quantify collection clustering main statistical theory study recent sec proof technical fitting blockmodel involve partitioning block bipartite graph vertex represent person represent exchangeability exchangeability array permutation permutation finite set column bipartite broad class indeed unlabele adjacency discussion separate exchangeability one require result representation class separately exchangeable binary array array fix generate exchangeable bipartite generate ij yx separately preserve
example average adjustment dimension reduction ridge network aic minimum adjustment alone statistic substantial adjustment majority analysis gain performance obtain adjustment technique partial sometimes rank disadvantage procedure aic bic evaluation potential able optimum example summary implement expensive computationally square adjustment raise benefit expensive form stage comparative likely rank expect outperform dimension production clean may alternative adjustment work naturally usual variance trade neural high analysis original square linear zhang presence cause degradation perform adjustment figure beneficial quite suboptimal third produce improve network weight large accordingly weight avoid overfitte approach produce superior general primarily attribute dependent summary extra determine appropriate regression example complexity relatively parameterized manner primary directly give model estimate statistic determine quantity derive naturally statement make advantage summary example fully dimension substantially analyse considerably curse exist dimension apparent reduction
sparse pca introduce invariant rotation simplify reason clarity relative express ratio general center say find indicator desire test value throughout assume determined see eigenvector candidate behavior statistic nan key assumption relevant arbitrarily hypothesis discriminate spectral lot attention perspective rest section argue moderate dimension discriminate infinity consistency entry large eigenvalue discriminate alternative asymptotically high large behavior hypothesis quite accordance hold reference establish fourth condition almost intrinsic limitation fluctuation discriminate hypothesis make formal spectral moderate phase behavior regime phenomenon phenomenon subsequently qualitatively critical spectrum exhibit eigenvalue
action even moreover many case regret background definition arm mab present suggest mab policy evaluate arm
eigenvalue respectively analysis establish rate shall introduce call drive implement norm loss upper bound risk major establish minimax rate convergence derivation sharp design scalar vector spectral present directional low estimating along direction le method column direction match thus minimax convergence estimate norm mean minimax satisfie view drive ball optimally pattern easily investigate estimator perform paper closely connect grow literature covariance study sparse covariance rate see fan zhang convergence introduce simultaneously
study q recover iterative reweighted also solution np converge minimizer chen derive minimizer hybrid pursuit smoothing propose wang q chen quadratic smoothing trust newton solve nonsmooth special suitable al interior nonconvex minimization problem box suitably minimization problem regularize derive minimizer extend solve provide propose novel lipschitz locally develop solve problem accumulation sequence computable minimization dynamically update outline notation low nonzero stationary minimizer
r f x l x x l k x x r l since x f r r tr end policy relaxation relaxation drop polynomial programming also theoretically relaxation reinforcement generalization computational j min mode rl aim design interact maximize reward signal develop researcher make problem many field finance engineering end several subproblem rl high environment batch rl batch dominant generalizing usually generalize contain area poorly cover property nearby cover low area cover explain policy need system
bn bn k j k use repeatedly kn order iii arbitrary continuity observe sn sn kn p sn sn sn sn kn cn sn sn sn kn increment existence increment initial stopping imply diffusion matter verify second stochastic x diffusion integral wiener value value satisfy assume hold well n x x p db u ba u exist satisfy ds ds cn vx ds p n constant p x mt mt n note du sufficiently exist constant c proof sketch th trial project increment bx n consecutive trial surely dimensional sx jx bound jx c writing sx dx dx sx sx dx dx dx dx
improve cs effective extension demonstrate algorithm provide well original extension cs imaging compress cs powerful theory acquisition recovery attention signal field theory unknown inherently reconstruct lower need exist processing several imaging hyper spectral imaging acquisition sense hardware play role inherent year advance application collect repository also variety specialized solver complexity focus recovery wherein emphasis robustness originally recognize scheme statistically inefficient corruption model corruption bit error transmission pixel buffer image processing work address cs combine exist cs formulation convergent cs formulation solve problem computationally recovery overcome limitation robust cs robust loop thresholding share nesterov alternating
validation nonlinear nonlinear nonlinear generalization principal component pca straight hence describe detect important frequently process complexity pca fitting cause often limit pca sample pca find flexibility little flexibility follow trajectory flexible relevant datum illustrate fig neither
discuss provide supremum increase system result generalized supremum denote convolution basic paper predictor equally space na term variable formula fluctuation estimator cause instability consequence ill pose require regularization exclude value multiply fouri transform smooth function compact satisfy exact assumption analogue express h analogue classical manner reason asymptotically kernel example theorem consist approximation field region approach small high derivative thus bandwidth justification choice useful constructing interval refer deconvolution kernel advantage carry call rapidly corollary sake parameter application
represent expansion factor neural bottleneck depend lead occur fine amenable dp adopt representation broad transfer possibility basis locally transfer careful accommodate change extension change reward change problem mdps observable belief decompose solve solve mdps extend mdp framework could efficient tractable coarse fine vs acknowledgment support grant fa u sub nsf grant dms author acknowledge helpful discussion markov first expectation obtain equation coarse chain transition matrix recall state denote interior state restrict compatible state fact chain mx measurable suitable markov property stopping time probability law right hand third equality equality px sx partitioning bottleneck bottleneck state non follow mdp unique negative transition bottleneck start p enforce whenever define future keep track sx give equation harmonic strong markov h sx process time conditional expect reward ultimately assume stop discount reward x nothing homogeneity prove proof use second discount reward boundary coarse h k ap h h ks ap p ks ap ks r h equation bottleneck potentially bottleneck define another graph induce second reward calculation discount factor derive discount discuss stop well lemma solve define give depend q equality fact compress discount find depend appear side compute compress previously compress vice versa solution discount negative separate square still possibility edu edu mathematics many decision often multiscale together goal infer leverage hierarchical abstraction multiscale repeatedly mdps wherein sub problem mdps representation task within sub globally problem significant aspect multiscale decomposition yield task amenable localize transfer policy potential operator compression illustrative include involve reinforcement transfer leverage sequential hierarchical generally suggest ideally layer abstraction broad occur task divide dramatically collection small computation super divide approach hierarchical subproblem subproblem discovery multiscale fundamentally multiscale planning relate concept multiscale procedure partitioning repeatedly markov contribution consists incorporate within multiscale multiscale geometry reward play prominent role wide regardless accomplish multiscale bottleneck control geometry partition one result hierarchy distinct sub efficiently propose perfectly compressed mdp subset multiscale hierarchy fine compressed transition analytically macro may different original problem scale function fine successively mdp multiscale way computation restrict conditioning time fine scale globally coarse scale condition mdp behind contribute coarse pair coarse scale repeatedly localize modify asynchronous assumption per iteration state converge globally sub systematic scale plan reinforcement domain decompose distinct part part appropriate transfer sufficiently mdps framework transfer proceed match various scale appropriate solve partial exploratory preliminary definition brief overview policy dependent discount describe multiscale mdps concern consideration introduce multiscale domain comment open subsection well definition notation formally mdp see tuple consist action set tensor discount
proportion equilibria always take consideration selection issue performance proof constraint game theoretical formulation sec algorithm state te ne represent network consumption theorems ne te action profile ne meet reach player satisfy consumption minimize generally ne large satisfied
attack adversarial I nx denote dual corresponding depend margin point sequel low letter part submatrix attack whose maximally decrease accuracy attack fix proceed validation section develop affect convex ascent iteratively optimize initial attack attack current vector attack attack gradient hinge differentiable
smoothness choose enforce prior maximize law deviation parameter shape could contain spectra change mean smoothness prior possible discuss specific smoothness write operator form calculation hamiltonian get differ extra term denominator calculate inverse hessian hamiltonian hessian reconstruction use spectra moderate strict show reconstruction much shape strict turn smooth resemble color conjunction terminal option load package graphic graphic macro ltb lt lt lt lt ltb lt lt lt lt bp power spectra smoothness solid power spectrum blue dash reconstruct smoothness dotted reconstruction region dot line sigma hamiltonian dot hamiltonian give translate interval mode visible shown reconstruct hamiltonian sigma component spectra interval spectrum calculate note uncertainty hamiltonian case scale rough uncertainty power spectrum typically example intermediate package color option load package graphic graphic macro ltb lt lt lt lt lt lt lt lt lt bp residual reconstruction realization solid eq
outperform bm visually mae compete ray ever provide information present early stage deep implication per voxel channel cube remove spurious propose bm approximation reasonably intensity provide importance fully transform bregman divergence variable poisson implementation classical optimize square update equation illustration improvement direct simple simulation gap appear form transformation transform refer method algorithm poisson refined transform refined multiscale partitioning bm refined bins bm mention patch patch length dyadic multiscale adapting consider inspection qualitative
general result regret analysis suboptimal constant mention exceed introduce q inequality follow present q sum arm link beta binomial display deal binomial trial draw proposition care ts sequence bernoulli q let notation hoeffde bounded sequence nb rewrite ease
every greedy scheme probability arm apply chernoff hoeffding exist non optimal arm obtain tight eq great mean henceforth rest arm arm accordance cumulative choose well often intuitively scheme ucb current sublinear average time arm exhibit polynomially decrease every ucb outline chernoff
heterogeneity degree diversity rich natural step realistic fact multiplicative model moving effect require rich class property degree correct blockmodel network amenable understand affect property observe variability nontrivial variability summary model summary multiplicative marginal network move statistical setting establish p root function derivative whenever converge approximation lead convergent convergent write side expansion convergent expansion yield asymptotic form induction nk statement recognize power thus prove double index iterate application relationship denote regularize incomplete part establish beta substituting q appeal lemma upper bind integer incomplete gamma gamma apply expansion lagrange expansion bound recall eq integer respectively low gamma q denote
eq linear see flexibility linearity refer linearize efficiently programming ls regression problem control benefit computational extension suitable series naive produce coefficient convex validation albeit intuitive intermediate disadvantage linearize combination slide cross window sequence boundary separate window control performance incur average select set close integer ls illustrate slide window
ir argument lag ir function normal computation obtain dotted line plot option nan innovation alternatively autocorrelation discuss function reduce considerably calculated calculated h prediction benchmark observation bold horizon west evaluate series consider relevant implement test produce forecasting ahead default mention matrix empirical variance name name model illustrate performance
special observation treat second reduce treat near near similar example bias near attain interior point simplicity support want generalization obvious symmetric moreover integrated kernel approach interior close boundary smooth define prevent support upper boundary density next hide score eq notation function
little beyond either efficient normally ef extend ef concave global maxima result choose cauchy application mc adopt adopt approximation establish error affect long carefully suggest parallelization agent alternative rao scalability generally order comment extend preference synthetic world preference experimental suggest approach flexibility world aic observe fit predictive aic criterion p dataset
link affinity develop tie feature belong multiple feature logistic base membership node affinity entry base share derive task
segment basis optimize accord avoid basis full summarize approximation original clearly illustrate htbp get principle thank anonymous constructive comment cluster functional axis france objects analyse spectra
mean stop rule span stop early suppose stop denote denote prove loss element vector identical real also far independence technical corollary procedure substitute result lemma pursuit omp redundant complex complex noise recover representation extend case validity discuss illustration transpose
learn entropy skewness locally embed subroutine version zero denote rotation angle run preserve geometry sample distribution rotation angle object dataset convert object perform detection detect proximity image detect euclidean detect fig successful preserve geometry image pose perform group machine sample unknown nonparametric supervise state several well image show promising estimator estimator operate derive divergence distribution denote euclidean distance th neighbor dimensional unit define estimator q claim estimator apply almost equivalent rx dr theorems point interior eq away zero uniformly similarly away condition consistent state lead unbiased p continuous uniformly let bound
programming path vertex similarly pass allow delay adversarial sampling show depend low eigenvalue form x symmetric example binary solution consider product show associate edge nj nk ni j ni kf ni equation write distribution simulate sub exist might graph path relaxed exploit programming solution
predefine predefine indicate frank wolfe dual predefine bad batch frank wolfe early additional solver several value include average version appendix trick uniformly sometimes widely recommend mention text objective value exclude substantially solver optimal htb pass ylabel false area legend north east header col comma lambda l txt style thick mark mark col sep comma col sep include lambda confidence densely mark repeat header col dataset txt index comma dataset lambda ls txt densely dash thick mark solid header col sep comma lambda index header col comma dataset lambda txt densely dot style mark solid col sep comma lambda txt index header col comma lambda txt color mark mark index sep comma include txt densely style mark index false sep comma include dataset lambda txt scale xlabel pass ylabel primal area style legend legend pos north east header true col comma include dataset lambda ls txt style thick mark mark header true col sep comma lambda product ls header col comma lambda txt densely dot style thick repeat index header col sep header col comma lambda txt color densely dash mark mark option solid true col sep comma lambda txt index header comma dataset lambda txt color densely mark mark header col comma dataset lambda txt header sep comma lambda txt index include lambda txt densely thick mark mark option header col comma xlabel pass ylabel style legend legend pos north east index index header col comma lambda l txt color blue style thick repeat index header sep comma lambda ls txt index header true col comma include lambda densely style thick mark repeat header comma include lambda col dataset confidence txt color densely mark option header col sep comma include lambda header comma lambda txt densely thick solid table header sep comma lambda txt col comma txt green solid header col comma lambda txt color black densely dash mark mark mark option x header false col comma lambda simpler predefine original htb xlabel pass ylabel primal area log pos east col comma lambda product txt solid style mark x header true comma lambda header sep comma lambda ls confidence triangle table x header col sep comma include lambda densely thick repeat mark option solid table x header col comma lambda opt txt col sep comma data lambda confidence txt color densely thick repeat solid table index col sep comma data l txt header true col comma lambda style thick header col comma lambda txt color densely dash style thick repeat header col comma dataset x true sep comma include confidence txt style mark col comma txt table header true col comma solid mark mark table index header col comma dataset header true comma lambda color style mark mark table header sep comma txt ylabel legend header sep comma lambda txt style thick square true sep comma lambda txt table header col comma lambda product ls txt style thick mark mark triangle repeat header col dataset lambda txt densely dash thick mark mark index header
see fix term depend proof lastly optimal estimator contain orthogonality see optimal estimate argument deduce claim imply two rv see orthogonality principle hold course estimate observation result optimal equal optimal estimate estimate optimal principle prove attain expression since respect eq expression nonnegative minimal lower bind bad attain find complete simultaneously auxiliary rv study eq specifically set e x prove set estimator fix result maximize within comprise therefore last establish estimator attain minimax optimal complete rv use orthogonality principle orthogonality equality result orthogonal rv mmse rv form office research office advance national foundation science foundation grant foundation google research technology ac edu
unnormalized univariate optimize factor rather single regression regression computationally statistically factorize implementation need far draw achieve offer far equivalently univariate approximate implementation equally statistic rewrite condition normalize appendix property know approximated easy long give seed monte approximations beneficial variance reduction provide gradient update factor propose assume regression figure probit large empirically gradient algorithm present combine rejection sampler approximation sampler correct proceeding might univariate calculate transform sampler parameter cdf matter probit z might equivalently proceed something entirely mostly although offer insight second approximation option approximate twice differentiable hessian relationship use identity stochastic combine classical square explanatory variable alternative form efficient classical regression explanatory directly equivalently transform explanatory variable principle rewrite invertible may variational thus regression carlo sample case regression give finite functional hold immediately obvious general choice statistically observation implement choice
gd report evaluate kernel chi square matrix create representative group lasso recommend segment scalability group standard present class follow weight perform sift sift sift color sift norm see feature preserve fourier powerful approximation carry scalability extend approximated g square leave multiple learn accurate plan explore alternative basis li
landscape provide inference correct variant well heavy tailed edge vertex illustrate use correct present variation orient throughout denote edge number block block subtle problem model bernoulli belong replace connect result analyze control loop degree correct block multiply impose normalization vertex constraint connect block direct network call direct number edge impose rr ignore constant log parameter degree completely degree fit block generate community perform quite poorly
property degree rich entropy internet entropy internet tend form fully connect mesh clique impose restriction network rich internet network link internet variation node ambiguity rank rank average consider internet http www network evaluation degree link agreement
minimize j pm j tm verify approach normality j sufficient mu mp approach replace bind verify
complement restriction various literature rip analyze l problem slight integer say satisfie restrict property equivalent know convex still possible sparse strongly segment connect moreover characterize smoothness namely much restrict instead global constant vector verify desire inequality key whole convex behave grow situation along homotopy convergence constant key idea gradient homotopy solve regularization decrease reach nesterov adequate warm pg list algorithm make presentation regularization since l optimality condition control reach solve absolute introduction ls type relatively rip turn precisely exist l whenever parameter appropriately condition iterate path argument state convergence sufficiently dominate adequate sparse recovery pick practical purpose recovery closely assume exist rip satisfied
formalize notion line asset comprehensive exposition decentralize vast game biology physic asymptotic structure research begin seminal cover observe e structure extensively measure private surely use error likelihood ratio unbounded private private signal derive error rate learn previous node ratio convergence extreme decision learn ratio author decision tree right unbounded ratio private error study show truth structure review perfect generate sequence test know failure decision model observe certain channel decision node learn bound constant slow agent learn convergence memory error node immediate learn strategy
large scale linearly distribute pass consider stop alarm verify asymptotically optimal detecting pass mp implementation stop establish model exact provide otherwise issue adapt infinite thank property pass protocol neighbor node chain message pass message ji ji n ij readily available message use nk k let pass posterior root initialize message one recursively reverse direction edge repeat reach step base normalize let base n nk ij pt following produce value time contain asymptotic stop share loop divergence run I j
subtract adjust variance output estimate package plot different department university structure unknown propose encode markov much
major interface class define spline final sample determine analytic analytic segment define offset gaussian width test effectiveness classification knn briefly knn one knn zero else filter building codebook match label test near perform codebook label codebook codebook svm good classification dot class fit minimize obviously straight line synthetic poor expand fit nonlinear polynomial transform independent trick apply expand calculate square dot
systematically layer may ultimately center ccc auc error offset center layer configuration offset top level simply discard one generative discriminative wise evolution wise evolution see epoch center stable clearly discriminative inaccurate become eps eps eps eps eps eps eps eps eps eps eps eps eps eps eps eps eps pt
determination asymptotic aforementioned expansion fix symmetric eigenvalue complementary set refer coordinate either second q submatrix index corresponding expression convenience several stage jointly refer stage coordinate define relation select coordinate combine ambiguity choice set ng depend convention correspond large eigenvalue let follow ng c define consistent reference denote establish ba valid observe imply follow generic algebra equality inclusion I n technical argument appropriate large except negligible also note deal involve analyze set n na ab task section expand isolate finally complete section establish c nh g nn useful probability rhs going fix carry important c next q
sufficiently add side add side criterion substitute fx remark remark newton minimize sum nonsmooth proximal newton convergence smooth computed method tailor bioinformatics processing case result nonsmooth proximal proximal newton many bioinformatics loss necessarily penalty regularizer trace describe type convergence interpret gradient curvature method composite art problem relate project constrained type behavior application drop composite function denote nonsmooth part trust region accelerate idea nesterov family iterative implement use statistic processing generalize successively curvature exploit quadratic cost
scale property shall uncorrelate observe eq estimate r package eq q discretize difficult analytic close em optimize iteratively quadratic semi analytically eigenvector nu daily return bottom omit converge minimum rv alphabet remain p p low optima repeat start position select x usefulness find tool
million stochastic model stochastic hdp use break posterior allow word come topic beyond truncation something reasonably use something small node three specialized nonparametric nonparametric process tree greatly quickly center contain relationship briefly document k measure partition child stick break set think capture distribution obtain subtract negative level vector second level second coherent sub result cluster initialize tree variational v randomness equal document compare inference lda hdp give entire versus topic gibbs papers corpora experiment year vocabulary size per three corpus use truncation truncation number increase corpora child stop root share document five fold validation predictive outperform variational also corpora benefit document share big omit present stochastic algorithm million article page somewhat consider vocabulary remove stop
thank program support des de la france grant university bs wang present opinion agent network political opinion party mechanism sub form isolate external agent add opinion influence social political opinion compose numerous locate axis mechanism splitting take account decaying end quasi remain study vote carry keyword opinion political network approach understand complex considerable modeling base political
two capture market bottom formula normal european plays role china european bottom european subject quantile time posterior mean black flexibility vary smoothness characterization dynamic major financial plot regime occur correspondence immediately market show clear international financial agreement financial level end begin capture correlation economic lead rapid correspondence budget recent grow instability period financial save notice peak represent dependence financial change correlation economic peak represent begin possibility prediction new obtain quantitative financial updating present gibbs observation simulation smooth show observe log quantile red conditional trace performance characterization together improvement analyze usa bottom unconditional online ahead use prediction predictive step ahead merely prediction
micro tuple assume role centroid allocate functional micro micro summarize adapt keep choose much window construct allocate allocation centroid micro correspond start window centroid allocation appropriate mean window consider w jt f jt f jt jt b max w w functional alignment
assess adapt swap transition action low adapt adaptation choose change stability filter significantly practical hence well adaptation plot rmse incur action policy expect exp well bad action adapt attain rmse rmse action exp change action dot switch action comparison exp algorithm great improvement ahead decomposition ahead perform remarkably though parallelization show ahead pf even serial
write pruning consider example pos sentence tag real vector classifier output correct pruning value linear pruning find goal minimize proportion answer
appear lipschitz replace note need separable pick exactly immediate consequence one eq establish easily vary procedure arise via let j ji writing ii follow depend doubly remark otherwise reduce arise combination clearly doubly combination establish nice doubly nice regard combination satisfy sense h fx fx fx j fx equip j jx jx h fx j smooth proper let replace j expectation last establish proper uniform monotonic establish minimize upper update hx fx ix establish uniform monotonic call conservative make nice nice sampling thing translate monotonicity objective merely applicable monotonic let uniform let serial parallel monotonic fx h establish remain monotonicity fx fx point nice fx fx fx show adopt expectation happen identity without generality indeed inequality plug ready bootstrapping
consider suppose deterministic variable assume whenever realization general convex slight variant constructing query away always handle depend uniformly rr upper describe may regression instance follow f td remark even set regret since emphasize result formalize plug improve calculation calculation subgradient recall also matrix back eq value require hold consider without term corresponding eq equal assumption distribution bound back eq generality play repeatedly choose
continuity fx fx v k indeed lemma argument give k w k fact recall k z hx hz side also condition optimality immediately em continuity set study inexact subproblem constraint subproblem kkt subproblem satisfy kkt inexact inexact programming
input update accordingly separable conditioning around feasible define q go feasible give column assign let rest j hence diagonal entry instead pick cluster ht rank initialize nx centroid large rw w k index identify identify cluster remain index I I j contradiction observation correspond individually follow must moreover contain normalize equation therefore iteration eventually indice small also
feasible either iteration algorithm empty solution single doubly discuss utilize ensure e discriminant function equal doubly active scaling accordingly adjust remain coordinate calculate straightforward computationally show reduce multiplication give initial identify lagrange correctly classify doubly active scaling necessary place accordingly sample place either c c modification follow contain tn take input e must like costly costly scalar
various illustrate aspect focus specifically co pick sign member differential evaluate baseline knowledge terminate epoch p favorable baseline dual average regularize regularization prox convergence method exploit strong square impose constraint converge trial observe baseline prediction ht ccc eps eps versus iteration tailor baseline remark keep henceforth maintain fair algorithm epoch square measure version epoch length epoch henceforth refer show large epoch quite rapidly phenomenon bad solution decrease rapidly though knowledge epoch slow iteration though exploit practical epoch remain relatively importance decrease ccc eps eps trial versus simple implement optima minimax optimal optima logistic attention idea naturally extend interest study development mirror method lead multi step convergence acknowledgement aa google fellowship sn support yahoo award appendix recall prox lagrangian lagrangian conjugate
pca belong satisfie assumption hold tool control ball agree factor case square fact distribution sharp notation proof dimension compatible define leibler kl measure sharp dimension greatly exceed size lead
relational predict property interpretation entity entity properly categorical mutual automatically interesting property situation occur consist relation job however recognize job necessarily capable correlation g principle language separate job look believe concern permit great job relation binary interpretation entity multiclass interpretation attribute interpretation currently database library predicate specify graph signature auxiliary predicate database read interpretation library except generation c incorporate interface solver set scenario grow task independent interpretation simple learn structured output domain naturally model e distinguish finally distinguish interpretation concrete extended task value biological activity water partition structure activity relationship formulate case handle introduce relation signature trivially sophisticated amongst task collective interpretation illustrate classic page four interpretation relation text r model relation represent classify possibility unary entity possible may subtle perfectly interpretation page atom page part connect category output study literature page independent domain movie movie entity relationship unary partial movie movie movie produce informative occur available algorithm space compute explicitly space explicit advantage deal framework undirected function directly kernel intermediate technique logical relational attribute possibly attribute tuple feature transform relational format kernel representation graph literature kernel
distribution marginal copula complete cumulative approach treat statistic marginal say tool extremely powerful relate test originally far complete description correlation
plot experiment face image image piecewise smooth signal pseudo admissible use normalised mean zero use learn noiseless aware iterate iterate loop iterate plot initial apply arbitrary learn show coefficient sample figure use noise formulation great suggest operator perform explore detail aim experiment denoise use operator operator tv keep corrupted face two setting bottom row visually successfully corrupt smooth copy table get learn instead db initial signal horizontal patch operator bottom operator horizontal line learn synthesis promising experiment comparative framework image previous patch reference overcomplete dct sparsity synthesis omp recovery convex optimum omp run additive
gene contain establish snps gene amongst set pathway subsample snps selection pathway model perform lasso matrix pathway select snp coefficient satisfy denote snps select snps map map pathway snp snps gene rank frequency open programming use module execution nonetheless considerable burden processor implement design increase computational taylor penalty avoid intensive addition identify pathway reduce follow final ensure place efficiency pathway major bottleneck generate entirely fit parallel strategy computer server node parallel server due study time exclude whereas job signature characteristic describe longitudinal voxel variation increase marked patient volume expand average per coefficient slope brain per plot voxel structural time point correct age discriminate ad cn voxel image showing extent voxel able discriminate ad top fig voxel extract final voxel value exceed voxel discriminative ad highlight bottom
considerably error compare average pointwise schmidt four square schmidt optimality projective projective numerical precise expect thank discussion numerical work project develop innovation education introduce let invertible matrix matrix substitute eq obtain introduce minimize eigenvector minimal eigenvalue eqs rao assume order definition probability py finite mix sphere obtain maximization rhs maximize maximal value multiply semidefinite take trace py explain unconditional reason call fisher know divergence everywhere part likelihood condition fisher unconditional divergence unconditional fisher however
combine region properly scope end multiple mode probability proxy expect jump move mean distance chain autocorrelation tend affine prefer autocorrelation practice evaluate short chain hundred principle go large issue twice compute return go particular almost improve acceptance fraction disadvantage large burn initial burn autocorrelation time burn together
smoothness follow extend smoothing stochastic stochastic vary nonsmooth part composite realization max stochastic smoothness parameter observation relate approximated ready algorithm nesterov smooth surrogate original analysis easier identify proper series achieve notation throughout center smoothed line alg apply nonsmooth rhs scalar could arbitrarily hence retain side
small assumption satisfy completeness trivially replace essential guarantee termination propose attempt reasoning restrict checking property incomplete may terminate present section learn intermediate termination teacher implement two check learner make teacher true fail teacher generate execution mapping thus teacher return learner fails return negative execution contribution towards composition enable maintain learner return teacher conclusion real succeed otherwise practical way negative execution terminate algorithm terminate learner use semi analyze reasoning describe partition directly max support distribution distribution total candidate state iteration complexity bad computing checking assumption well address leave work deterministic sample traditional partitioning discovery compositional investigate teacher partition
dx inductive ks euclidean infer inductive affine inductive hypothesis affine hyperplane else exist affine denote lie construction trivially generality assume pass origin thus x nz j rx jx jx dx dx j x terminate hyperplane satisfying follow dx nn hyperplane multiple every must integral actually upper claim theorem remark pt award california research fellowship california berkeley nsf university function degree since exact uniquely specify recently nothing efficient approximately exact approximation representation run integer previous weight also strong conceptually simple structural showing threshold close take take decade field machine theory weight boolean weighted majority game theory shall coefficient paper expectation respect may provide least surprising elegant completely constructive rise rough statement motivation briefly see extensive account motivated electrical engineering early suggest later researcher vote
digit base sequence string successively mention made order generate digit string produce digit string c next pair digit purpose string pass statistical test string familiar digit everything place choose imply equal obtain experimentally frequency frequency hypothesis consecutive order f f f f f f case case table equal frequency table show main confirm generator string statistical test string expression consecutive bit select string expression string fulfil reason
branch evolutionary time comprise substitution rise problem computation exponential poisson linear treatment global poisson computation inference inference model separate inference transform rapid availability sequence inferential cope inference increasingly procedure bottleneck modern molecular dataset issue align evolutionary figure depict evolutionary string string evolve stochastically branch tree computation branch make branch v string substitution star denote nucleotide circle substitution research paper markov string stochastic branch evolutionary tree likelihood far realization generalization broad class finite know dynamic simplification restrict case
draw posterior coupling time eq error test unknown gamma component possibly handle algorithm acceptance point ii probability occur restrict ii accept class mle run unknown prior denote common hyperparameter choose acceptance probability ratio class ii restrict similarly ii accept candidate q inverse gamma prior hyperparameter transition
might high hand panel censor solid embed contour contour full visually curvature family inferential example reduction effectiveness idea approximation maximum important tool manner problem tie understand relationship rigorous numerical understanding describe make asymptotic well exponential consider show level parameterization dimensional plotted implement figure become parallel fact describe linearity system describe geometry simplex extend advantage compute understanding variability htbp determine look model e distinct arise likelihood define hull non concave simplex work concern observe distinct probability lie euclidean projection pre image maximum likelihood completely likelihood likelihood map geometry see simplex space advantage explore exploit simplex exploit give simplex geometry structure
algebra theorem order finite processing achieve range simplex three set contain figure enyi argument well enyi equality first alternatively follow zero jensen whether equality partial enyi distribution equality note old sufficient extend let secondly r enyi divergence jointly order enyi jointly convex argument pair imply strict imply hellinger convexity imply ordinary hellinger hellinger integral imply essentially convex important result theory inequality kullback inequality distribution strong sense enyi ordinary set define simply generalize let q normalize normalizing normalize p bf fa jensen inequality side equal mixture element define eq cover kullback leibler example remain prove evaluate monotone q trivially imply put everything therefore converse dividing
find newton optimality kalman kkt newton method newton operation system scheme addition need extend smooth red dot blue dot smoother green outside axis specify previous kalman smoothing demonstrate efficiency kalman smooth function definition write explicitly follow form gauss subproblem decrease
follow incomplete framework adaptive policy prove explore policy conclude adaptive population successive follow respect uniquely determined assume period infinite exceed upper infeasible constraint redundant model programming randomize maximize expect standard e collection pdf program g basic feasible degenerate correspond correspond basic correspond degenerate degenerate include let optimality
unique causal strength causal additionally satisfy non treatment treatment treatment refer bound identify set causal function value function observable take must lie like offer little understanding develop inequality find violate contain model otherwise treatment see even demonstrating explain determine def seem parameter infinite see distribution satisfy condition begin model look satisfied constitute bound associated optimization
continue good performance experiment model generate flip coin retain coin let retain ratio repetition robust however proportion miss gs continue outperform additive linear flip coin evenly jx j repetition figure reasonably presence nonlinearity gs continue degree moderate say h h idea utilize lasso attempt exploit emphasis rare careful reveal subtle transition properly able exploit phenomenon ji paper framework similar design way technical devise current different gram matrix pick variable weak term challenge signal difficulty heart innovation gs screen overcome challenge objective consequently main use hamming optimality imply vice space corollary must note get arbitrary large gram paper rd state since ratio derivation match penalize optimality gs sure screening screen property gs line random ap iid signal independent design matrix model bernoulli gs really except negligible exceed integer constant p condition claim subgraph sparse ise another interesting model relatively degree generalize test proof simplicity connect subgraph must size connect combine em write short generality start update repeating terminate favorable configuration see arbitrarily follow index consider alternative keeping except perturb know know small minimum type ii hand ii
note discretization value vector piece conceptually appeal remain whether output regard discretization process reason connection measure space additional illustrate move differentially private vector function examine function subset covariance dimensional require family distribution differentially space finally result dimension sample database differentially field finite multivariate distribution demonstrate imply finally note borel privacy statement carry function lie reproducing correspond covariance gaussian simple basic definition first rkh closure combination correspond reproduce
randomization eventually randomization observe adversary end player observe require address adversary coordinate action information semi bandit observe coordinate observe variant optimization rigorously combinatorial repeat adversary choose draw adversary choose bandit measurable si measurable player incur loss observe full coordinate bandit try cumulative online time action possibilitie bandit bandit version adversarial armed communication represent choose suffer delay delay traffic edge observe delay delay
h activation visible reconstruct visible example distance bias visible stochastic cost boltzmann connectivity visible layer figure seek structure activation rbms train divergence idea represent optimally practice via visible hide correlation presentation successive gibbs proxy strategy use early stop rbms stack deep belief
second interest common term beta z beta alpha word document z alpha
large symmetric datum carry formulation implement task fidelity specification information point setup supervise real characterize smoothness fidelity weight datum vertex know understand example fidelity consequence attractive class goal address potential fidelity minimization attempt interface region absence fidelity term trivial diffusion final result well potential interface flow label class require
lemma I lead result summarize case pa pa n n b pa b pa b w n r argue exactly tail add remark section
q choice cca linear lda limitation cluster task object belong widely data retrieval bioinformatic unsupervise learn need sample many sample cluster certainly dissimilarity example human extend location remove similar show incorporate extract face manually face observe profile face individual helpful feature split construct remove common feature let perform cluster object detail reduction extraction consider link bss simulation total ten four column benchmark name mat six standard snr db first extract run latent signal good simulation simulation sir interface sir
prediction although adjust svm effect opinion error pay type svm first meanwhile important try control adjust appropriate control experiment denote radial type stepsize test fix type low prefer coefficient error fact error rate control second error component mainly predict whether
base pac inequality multi idea mix statistical give belief large information essence associate predict specialized appropriate pac bayes characterize deal upon majority classifier put lot enough bind classifier informative majority vote give support closely majority enter play normal different estimator
among list occurrence item finally define model expectation give marginal laplace l evy another application derive observation gamma atom random mutually distribution simple derive conditional form integrate total mass leave derive proceeding nonparametric ranking smoothly change ranking ranking gamma distribute random markov enable
row contain weight allocate smoother matrix name return tractable strength rate use mathematically demanding exist g analysis logit transform xy smooth back transform guarantee provide motivation model relative success transformation depend set interval relevant section eventually rate require substitute use package analyze
guarantee corollary exploiting focus observe phenomenon across notable obeys found diameter satisfy platform graph vertice graph vertex multi indicate multiply edge graph choice multiplier ensure easy cut coarse previous kronecker vertex distinguish cut scale strong explanation demonstrate figure graph triangle subgraph change noise end develop scan suggest feasible graph spectrum develop balanced kronecker graph statistically
q fact u u v u note order prove soon consideration since triangle equation u u conclude along straightforward acknowledgement optimal discussion regard van mm financial engineering university edu di universit di microsoft statistics microsoft com bandit contribution algorithm action instantaneous work show expert basic expert
argue property ds yx discuss ds threshold lag weight predictor conjecture ds lag behavior impose enhance discuss suggest orthonormal adaptive towards perspective double exponentially likelihood coarse lag address frequently nevertheless ridge enhance ridge lag ridge choice generality consider scenario group identical predictor reason obtain desirable bias predictor shrinkage sense follow claim lag
fr size estimate performance report carlo hypercube monte carlo mid numerical analogously replace expression histogram logistic v extreme logistic centre right case estimator high although bivariate exponent constrain estimate change regard well
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model couple area via multinomial age limit population level predictor presence disease health spline specification provide area within age denote unit variable latent area h refer area indicator latent small area pattern unit area weight belong class assume within probability take model multinomial one category status category enter aspect finally mean denote unit include variation explain include intra area covariate prior assume prior specification since mixed effect set fix effect estimate reference item fit describe parsimonious consider assume order class odd become equation
alternative sign integral put infimum attain gx dx right side slope alternative density assume integral sign alternative denote statistic limit see j hz
fully entropy semi supervised precision semi fully knowledge publicly available task consider seed study aid significantly supervision take rating rating adapt competitive sophisticated require jointly text datum competitive sophisticated world dataset star word star sentiment overall size aspect center star weight star sentiment publication identify sentence user aspect rating specifically review sentence maximize score rating setup motivated user prefer summary discuss aspect merely show increase supervision semi supervise improve segmentation high different user variety segmentation user primarily easy classify aspect correlate outperform lda lda incorrectly aspect sort rank cast aspect sake reproduce baseline figure aspect rank worse supervise classifier similar outperform
rl agent apply electrical collect slice model rl observation select step occur challenge state code convert variable encode policy place grid hz domain error monte carlo set size carlo sum rp rp number sample size projection iteration rp error projection result rp avoid include rp number projection interesting though place minimum error size relatively robust
method particularly magnitude truncation example rao particle degenerate report slightly paper space dynamical originally space decade research carlo monte carlo lead substantial challenge algorithm q static arise scenario model via marginalization non markovian particle family inferential construct proposal sampler sequence interested gibbs would sample procedure carry conjugate prior replace step metropolis smoothing difficult address smoothing final weight
apply smooth penalty single share regularization improve rkh term value rkhs multi task solution f k give functional extension substitute task give prior give j j distinct example ff df n furthermore optimizer express zero capture variance connection q data effect model task jx jx assume effect sufficient capture mixture mode like gmm maintain amount add group zero mean predefine kernel relate point z j know kernel relate shift periodic j j focus stand circular dirac periodic period shift mutually mixed model similar prior realistic cluster define function characterize j additionally denote
establish time fact next algorithm computation costly algorithm cost combinatorial np hard cost decentralize wherein decentralized indeed shall see decentralized armed problem abstract communication formulate bound unable computation let regret argue alternative define arm increment counter chernoff variable armed bandit problem per prove suboptimal suboptimal arm event make arm transition jt I time arm play replace chernoff get last expectation perform index allocation allocation update suboptimal allocation arm transition suboptimal bound player pick equation jt jt mab player select suboptimal arm jt
quantity calculate pmf less windows pc ghz dual processor pmf look like equal equation machine round indicate first roughly apply claim pointwise random monotonically decrease pmf support work trial error large reduce find lattice variable dft compute
technique packing packing follow careful formally framework provide consequence intuition conclude practical concern prove low major proof identically distribute pca look mutually uncorrelated coordinate maximal find manifold dimensional orthogonal least spectral orthonormal associate eigenvector must pca replace decomposition sample estimate lead eigenvector eigenvector additional structural necessary estimation notion appeal successfully context value parameter sparsity inherently coordinate coordinate span manifold notion term norm span wise orthonormal belong radius speak sparsity generate orthonormal notion orthonormal column coordinate orthonormal norm orthonormal unlike orthonormal ensure dimensional sub tail intuitively estimate turn
slow convergence qualitatively close additional impose either hold distance hausdorff next qualitatively nonparametric like either satisfy eq regularity equation hold parameterization need simplify follow give population n ij estimate assumption replace lower infimum take oppose notably dependent partial justification like posterior rate allow differ like surprising exactly extreme implication impose pairwise extreme point polytope interesting determine datum level hierarchy minimax optimal estimating support due elementary space hausdorff obtain state unbounded extreme within ambient irrelevant replace thick property corner support hyperplane angle form vertex lemma probably absence direct reference g lemma sequence
reconstruction projection level compressive simulation effectiveness approach compressive variety performance arise edu arise field image digital measurement signal estimate incomplete additionally corrupt constrain tv
convex hull subset hull sample identical solve minimum uncertainty intuitive consider uncertainty decrease sample solve constraint regularization depend regard define compare confirm way derive variable lagrangian lagrange optimality negativity lead constraint conjugate min rigorous min theorem gap label parametrize set let solution point min satisfy kkt hence solution condition nonempty nonempty z svm function know form estimating bias term boundary line segment vector base learn learn elaborate parametrize set way function set svm hull addition infeasible distance conjugate invertible truncate prove x yield though learning method conjugate q kullback weight derive parametrize associated uncertainty
uniformly think might preferable value yes appear find mistake cause inequality something inconsistent check ok yes arise ok phrase appear ji bit appear appear old think relate ok think current ok constraint support term sort restrict rip require effect least square term compare entry draw gaussian iii together q theorem bound regime
uniform proximity location predictor unsupervise bayesian modeling year nonparametric estimation infinite component limit posterior density infer mixture represent model modeling distribution popularity popularity generalization dirichlet process couple segmentation datum
combinatorial relaxation obtain interest automatically select connected subgraph direct acyclic coding encourage tractable formulation choose sequel natural parameter encouraging encourage sparsity function encourage connectivity negligible count cover support thereby encourage regardless framework handle us acyclic node node formally graph link every vertex arc path weight component present next figure cost choice whenever experimentally path start illustrate cost arc arc encourage path connect path code walk connection simplicity appendix follow issue objective question graph thick blue fill size draw fill mm draw fill red thick thick dot bend yshift mm xshift mm right xshift leave yshift mm xshift xshift mm yshift node node edge yshift mm xshift node yshift xshift yshift xshift v v near v xshift bend yshift xshift xshift yshift xshift v xshift yshift node yshift mm xshift yshift edge yshift xshift node yshift v edge xshift node yshift mm edge xshift dag along cost weight state sketch main first key component transform flow flow arc correspond along path equivalent cost look every arc decide
lattice detection ideally rate decrease thereby statistical significance datum confirm claim claim lead researcher publish reporting experiment fully various protocol system mechanic prevent ever quantum relation prevent simultaneously mention possibility read continuous toward critical quantum correlation would unlikely able go make experimental choosing help distant galaxy soon device still certainly subsequent soon possible memory opinion ensure freedom choose setting cascade classical randomness pseudo etc never super make proof ghz done implement claim experiment prove quantum outcome yet exhibit bar follow line local event another quantum mechanic arrange event contradiction probability observable event event experimental design assume exist coin course view super everything ever go world generalize outcome two particle possible mechanic mechanic refer different accord etc measure interested interestingly requires maximally amount quantify notion far formalism maximal quantum isolate example even repeat never like strong
exist positive function deterministic function particularly interested sequence function want say growth martingale mapping sequence stable continuous eq never reasonable rewrite growth estimate martingale growth rate suggest prove density kullback inequality theorem contrary statement satisfy obtain
next term I treat variable lebesgue integrable measurement belief lead convention symbol occur density density also interested equality argument argument argument measurement use tendency mode measure dispersion prior become simple gaussian rest prior equation square index behave dirac delta give eq assume instead general also understand diagonal ji j ir p power distance measurement precision weight general distribution eq choice generic choice possible whose measure decrease rational interest matrix degree practice instead systems note two uniquely discuss usually small avoid integral
nest v blind krige develop weight computer experiment calibration computer model liu bayesian liu dynamic computer k l model computer functional response element model w nd ed york efficient multivariate deterministic function computer combine computer wu f l equation evaluation computationally uncertainty uncertainty confidence bootstrap augment augment regular department nj school ga school institute ga conduct optimize residual krige profile dimensionality functional severe krige regular grid simplify complex gibbs regular kronecker employ efficiently
investigate computer combination carlo review relate well moving bring cost goal physical may least one center sum experimental interest function identical component fix pair basic kind measure natural consist function design correspond independently fractional factorial replicate second estimate involve difference three former great convenience measure integration estimate necessary interaction sum integrate kind integral product reveal internal exhibit moment combination
magnitude lasso note coding loop dictionary computational lasso smooth term top eq use cross validation iteration regularizer example regression marginal datum context incoherent way use bound enforce eq direction represent constraint use lagrangian update q incoherence constraint normalization least square update correspond converge provable yet nevertheless idea speed exist bi representation kernel similarity zero
probabilistic note inequality truly size oracle produce notable produce hold high later selection establish small small possible understand terminology see selector remainder small suboptimal selection successfully inequalitie precisely average convex combination carefully flat simplex define estimator refer theoretical call estimator lead choice choice successfully average irrelevant implement comparison inequality term suffer gaussian perhaps importantly limitation may technique actually inherent exponential consequently since expectation also paper
time large distribution sublinear achieve sublinear time time stochastic problem compact subset linear tail sublinear achieve action order adaptive cognitive exploit primary availability link dynamically communication nearby delay link model hoc
compute probability markov whose implement hasting estimator constant design choice q burn stationarity negligible alternatively period discard another estimate avoid simply essential determine influence concern control take normalize sampler control thereby convergence speed markov fast control control normalize variance ask standard estimator size normalize moment dx fx order close density play role choice let approximation explicitly close emphasis form chain
ab solve nd new york box c york l inference c theoretic interval utilize letter confidence
pair rand agreement index undesirable ari ari take classification ari also happen classification bad simulated site assess draw gaussian illustrate face mixture empty would expect mixture indeed merge classification merge mixture skew accommodate skew appear mixture apparent lin lee analyse anneal start mixture distribution treat remove
consider dynamical train dynamic gps point model synthetic ep ep mae report trajectory mae average kalman smooth ep iterate know iterate backward smoothing ep improve poor criterion uncertainty track period show iterate backward smoothing ep posterior follow indicate blue bar measure ep smoothing small infer track ep require iteration solution mean posterior match ground tb blue gray
throughout section triangle triangle lemma conclude boundary furthermore triangle unique edge triangle edge close origin argue interior ray namely contain note slight modification need involve rescale rescale instance consider strictly correspond intermediate lemma triangle contain vertex lemma exactly two edge triangle triangle consider column simplex problem arise modification issue plane plane nonnegative origin denote nonnegative vertex cone minor invertible nonnegative instance coordinate triangle vector plane nonnegative
orthogonal q bind side matrix copy q bernstein
likely visit visit state visited count unlikely patient day expected visit patient low fairly trivial case probability begin add reach infinity capability elegant transition time computational time dramatically parameterize asymptotic probability interest semi time extension scope applicability many state appear problem state methodology performance element find numerically number may program
recent researcher dl performance denoise merely reconstruction orient fashion explore information representative dl method directly discriminative coefficient discriminative simultaneously classifier discrimination name final track I face dl supervise dl discrimination dl organization classification use
reason obvious estimate company log unlikely large interested reader illustrative demonstrate apply complicated monotonicity constraint concavity constraint application present section illustrate potential section review method derive operator necessarily strictly convex discuss concern various discuss future generalization consider convex twice row partial transpose hessian penalty exact method way constrain optimization problem calculus sufficient coefficient make sense uniqueness continuity vary concern continuity foundation uniqueness unique minimizer gradient linearly coefficient path uniqueness sequence e continuity uniquely solve stationarity therefore continuity check sufficient convex function continuous strictly square solution
analogous article demonstrate effectiveness problem arise processing b show hmc variant mala diffusion space hmc superior demonstrate walk behaviour behave nontrivial acceptance dimension proposal summarize relate viewpoint many inverse assumption flow approximate satisfied image range condition item discretization noise markov mala langevin proposal straightforward fashion rearrange apply term proposal explain measure metropolis well choice pt desirable preserve sde invariant remain standard absolutely continuous carry proposal eq derivative satisfying advantage space scaling limit gap use limit recently hybrid monte concerned distribution conclusion relate proposal central see discretization mesh refinement demonstrate modification describe scale limit proposal variance stable mesh refinement gap idea gap closely resemble dimension via gap mcmc arise dimension demonstrate wide range lead respect field reference designing problem produce insight design
tree inference conduct various propose literature neighbourhood joint mle induce require gibbs popular enjoy mle strength provide credible penalty follow mrf framework offer characterization underlie gaussian prior ideal exhibit well unlike laplace unimodal apply context expectation propagation unfortunately mrf hard feature even intractable cause double nevertheless mcmc explore bayesian spike mrfs devise langevin sampler experiment show actual well
space location counter bit store always address pattern storage string intend different bit center pattern every increase counter retrieval summing address retrieval
necessary exact guarantee low extent relaxed graph broadly might exact guarantee either variant absolute estimate threshold selection analysis incoherence note relate
descent know converge geometric error require sufficient overall conclude naive processor previous involve combination stochastic ease limited requirement case machine approximate descent begin stochastic choose step stepsize project point convergence reader deep alternative convexity assumption require moment globally entire similarly hessian require nonetheless assumption hold common hold domain average gradient base sample result attain bad early circumstance difference th assumption correction base correction significant term arise bias stochastic trivial work trivial machine number split one specify generation choose similarly set correctly mis estimator variate independent experiment loss minimize specifically setting subsample stepsize give performance I split oracle cc machine across simulation square accord oracle line use plot set average number machine grow regression estimate return sample uniformly plot infer vector versus split plot red centralize gold apply batch make
good reach training tuning tune help note tune wise intermediate stage fine apply experiment layer bottom mention rbm train generative fine work practical optimizing simultaneously global hand architecture correspondence easy directly incorporate aspect distribution simple well formalize argument let fix marginal maximize likelihood might define turn know coincide former arbitrary improvement yield incorporation good point transformation fix point incorporation concavity maxima unique fix incorporation display concavity critical maxima invariant distribution concavity maxima incorporation incorporation coincide maximization em justification construct variable inference auto display remove ahead incorporation even generic map close optimum look look search sense simple already respect justification know thus describe arbitrary ahead incorporation though sense make close look look sense already good property respect model know feed maximization em increase assume approximate put optimize particular tuple sum algorithm transform optimize ok recognize em color ok recognize argument algorithm tuple independent eq display argument
infer individually turn depends actually result mode regularize level contribution novel greedy design target structure sparse matrix guarantee performance support subtle oppose sometimes statistically efficient convex programming case huge restrict directly recent via share approach paper regularize encourage conceptually constant multiplicative factor greedy
set symbol inconsistent formulae consist predicate interpolation inconsistent exist generate inconsistent domain p predicate abstract boolean boolean relate formulae abstraction formula map boolean formula map free formula specify boolean canonical normal lemma useful abstraction function predicate cube direction predicate canonical atomic proposition quantify canonical canonical conversely assume proposition canonical observe atomic proposition boolean since formulae query teacher teacher responsible two query ask formula answer query boolean polynomial query size target language basic term assign formula formulae loop sx ss occur positively give annotate loop formula
many classified area fully develop thus city com com com goal algorithm correct actual use incorrectly area mit clearly area usage political lead error may incomplete mistake examine prediction closely display activity cell ii cell incorrectly incorrectly predict activity dominant activity activity pattern find group support phone activity residual cell incorrectly group display algorithm identify share activity characteristic class reality examine potential cdr predict usage demonstrate show potential activity activity dominate life reveal subtle five category may aid resolution capability fine supervise
lda full dimensional reduction classifier dimensionality average describe previously interestingly tends keep principal lda diagonal lda account occurrence feature extracted name entity protein interaction task occurrence entity protein document extraction tool count name identify document feature occurrence feature publicly tool entity chemical name name clinical identify name entity dictionary provide li
prevent scale follow parameterization wish metric identity mahalanobis identity symmetric system step illustrate gradient always problem parameterization white everywhere constant express j many fashion bfgs rather rapidly
average realization experimental outperform recover moreover approach reweighte require dominate basis problem decade outperform find modification spectral subproblem replace subproblem constant estimate
service experiment maintain st resource bioinformatics conclusion recommendation material necessarily reflect foundation mm pt height laboratory mathematics science st laboratory science one university st di di automate practitioner collect statistic probabilistic previous run bias future purpose technique np spin vertex demonstrate even run yield multiplicative inductive experience
validity cluster pairwise distance sum small pairwise across whole value compact class measure separation inter index separation propose define separate experiment introduce report match amongst four study include index base introduce exploratory pursuit classification centroid group indicate structure closeness centroid view place close centroid score contain projection centroid separation histogram sum bin factor indicate separation choice bin size influence score bin dataset participant dataset task direction goodness point color name name ask
contribution row list dct choose well aforementione utilize obtain component even list achievable procedure snr well classical illustrative comparison depict histogram measure block primary motivate audio classify periodic prototype signal px c dct spectral comment broadly audio nmf examine blind
simple find assignment matching assign match illustration see obtain solve problem measure j ij dx amount step distance neighbor average bipartite many testing nonparametric consider equality asymptotic equality dimension minimal span statistic portion near testing wasserstein relation histogram histogram signature center converge choice signature capture wasserstein continuous distribution notion signature manner capture fine tv notion tv distance single similarity remain namely would partition last establish machine et base discrepancy choose mmd rkhs
order asymptotic reference could efficiency computation empirical improvement composite likelihood herein promise alternative bootstrap investigation considerable effort free include interpretation herein imply calibrate meaningful herein sort inferential framework shall elsewhere monte carlo approximations frequentist inferential procedure knowledge computational pay simplicity plausibility evaluation carlo estimate consume parallel cost principle problem nuisance infinite marginalization something propose make inference grateful comment liu anonymous associate relative technique maximal likely property plausibility emphasize establish plausibility consistency plausibility outside property imply chance make incorrect decision plausibility function assertion closure eq thing control sufficiently overcome process handle admit
tt adjacency introduce descriptor encode information matrix dimension prediction f qx ff estimate past row prediction equip norm n element likely edge evolve slowly change assume govern evolve smoothly network factor graph coefficient capture popularity refer centrality trend degree reflect factor set slow evolution factor unobserve rigorous statistical specific history formal contain factor approach study simultaneously dynamic collect feature exhibit regularity adjacency feature keep infer stationarity predictor neighboring graph function assumption notation trace root rank prediction problem square loss loss error different adapt separate x prediction denoise reflect predict assumption author minimizer subproblem presentation addition regularity laplacian
careful finding normalizing framework analyze interest broadly machine acknowledgement author detail manuscript ks would like thank research support ad california technology relatively refinement nf f eq technical basic randomly fix consequence shannon unit inequality packing sphere parameter satisfy eq vector distinct surface equivalently inner product end jj union inequality imply pack work eq product product bound question sometimes refer perfectly noise construction scale theorem theorem ad institute ks science engineering experimental support grant ad institute principal tool contain algorithm operate risk privacy paper differentially private propose new explicitly optimize output differ nearly furthermore illustrate dimensionality tool datum hundred reduce intrinsic dimension discover relationship transform greatly
red bit solid hashing loading average require hash lrr loading ratio course write inform sgd code provide dataset panel accuracy versus panel accuracy improvement sgd nevertheless epoch perhaps approach accuracy bit continue time recently overhead energy order learn truly frequently occur matrix due prohibitive storage signature resource successfully apply learn fairly efficacy become increasingly context search show normally highly document hashing demonstrate permutation bit hashing permutation increase threshold bit find similar address bit hashing task match accuracy random hashing number challenge able hash current gpu massive parallelism impact operation bandwidth
remove characteristics loss clarity quality common enhance compressed shift compression back achieve db compare state mlp noise sa achieve db peak peak bm db mlp corrupt mixed gaussian imaging imaging corrupt observation come distribute free follow take peak bm variance ii denoise bm example design design specifically potentially lead well result lc cccc bm mlp db db db db db db db db db db db corrupt poisson gaussian peak mlp peak example peak divided compare bm pure outperform state art many denoise notably bm patch reference patch denoise exploit patch clean image effect two ask image perform rather bm repeat structure answer train patch image sometimes average previously patch patch noisy patch size output one perform imagine provide patch architecture mlp hidden size training procedure note block neighbors bm choose patch distance threshold maximum bm denoise step merely neighbor directly noisy high employ
belief prediction detail practice state median value eq cdf invertible datum consider one fix reflect none overcome multiply mf bethe modify dependency global connectivity additional link selection calibrate fair ht width width ab comparison mrf stable favor well display approximate one project position set belief ise ideally calibration ise locate center help optimally curve coincide one hide generative compare generate traffic simulator fall gaussian version greedy reach level ise obtain encode full follow poor ise encode paper many case bethe start develop solution gaussian sparse compatible compatible simplify natural ise evaluation bethe regime hence keep compatible certainly basis schedule gradient efficient broad class simple extension local experimentally work support national research grant prop prop prop prop pairwise mrf problem inverse ise somewhat relate inverse mrf belief propagation certain approach bethe field solution pairwise mutual perturbation different way idea start bethe computed bethe expansion choose fundamental cycle basis loop characterize connect propagation pass problem refine regularization
perturbation velocity interpolation may design long evaluation derivative gauss update subproblem cg iteration cg student histogram final corresponding figure clearly inversion home ignore eq fitting modify lead interpret parametrized signature signature formulate index link eq student solve penalty refer velocity depict source percent stage use
relational meta inductive conversely extension elegant relational chapter devoted lp database survey powerful proposal conclude remark future programming root know logic form complex symbol mutually either quantify omit notation implication clause admit call implication implication call body intend quantify definite clause theoretic semantic symbol assign clause syntactic semantic know formalize proof logic inference derive resolution sound clause prove clause body semantic logic program accord semantic logic alternative none correspond view reality lp orient towards clause prominent database clause database treat restriction distinction predicate divide two fact involve clause reason
hide document view require additional assumption single recovery certain rank observe vector propose technique technique describe latent condition technique hierarchical model apply deep fig useful property structure include satisfie rank illustration latent variable tree path illustration effective central include independent chernoff expansion rely tool decomposition area cross moment word satisfy topic find matrix indeed impose natural degeneracy denote column establish column thus prove leverage dictionary counterpart ingredient establish word neighboring satisfy condition identifiability word exhaustive propose recover additional latent linear bayesian moment specifically equation acyclic dag correspond third excess framework reduce analysis spectral hide hide noise variable reduce ica latent observe original term analysis moment decomposition third direction exhibit moment recover expansion extract high depict view hide mixture conditional
alternatively acceptable om adequate tradeoff ht ht l budget nystr time cluster frequently accurate dataset easily cluster nystr om often quickly large dataset fast inaccurate efficiency run apparent face approximately spectral take three reach sample size increase preferable datum addition right use teacher run useful quite teacher cut minute would likely factor fine acknowledgement thank thank pure mathematic lastly nsf corollary theorem separate within cluster cluster utilize spectrum datum solving problem develop summarize application employ cluster likely leave company light cluster applicability
write situation probability publish search sometimes incorrectly problem subtle case q whether odd odd relax variable q necessarily search also selection likely write search uniformly random middle account search odd take course odd make I e last describe odd remove conditioning contain odd equivalently symmetry middle appear odd evidence everything odd situation see correction denominator fact bias case correlate contain write satisfy success independent give rewrite whether large role default population equal q default question weight contrary weight evidence consider background choice prior odd arrive reciprocal careful one want divide computation expert one face
correct come branch statement inner outer non combine claim hamiltonian claim know answer trial find terminate correct care even completeness use serve cycle far output hamiltonian answer finish proof cycle really correct infer return element multiply hamiltonian runtime give problem incorporate bound check accept hamiltonian cycle reject complete remark polynomial create simulator simulator notion interaction simulator cycle generalize invoke invoke simulate conclusion become time oracle verification likely polynomial oracle weak search give undirected ask exist otherwise u two unknown exactly want new chemical know chemical compound either verification pair may define strong piece distinction matter weak hardness result strong graph bind one low try propose knowledge imply bipartite node index vertex empty efficient try propose newly piece add edge decrease initially isolate finally stop trial efficient solve u come address question oracle list oracle look quite nature collection avoid exist whether process wrong follow necessity solve verification oracle solve solve graph clique exist isolated specific verification oracle return violate must polynomial instance clique clique isolate return thus find clique contain look output clique verification oracle returns edge first decision yet return design happen edge already clique contain clique least oracle
increase change drawback avoid analytically marginalization section marginalization analytically tractable converted expectation intermediate hyperparameter covariance slightly single hyperparameter translate estimator hyperparameter peak good derivation estimate mean parameter decrease approach region assess performance number design performance setting build complexity detail reference second generalize exist substantial unknown hamiltonian control evolve internal hamiltonian want experiment hamiltonian record unknown protocol provably protocol slightly generalize decay know treat normal exponentially whereas finite reach go remain convenience unknown generalization fisher rao utilize asymptotic approximation normal bayesian rao way smc numerically estimate way admit similar mean condition intermediate variable point entirely remove leave normal example consider hyperparameter identical formalize fashion dependence
popular particularly useful imaging nuisance must estimate imaging modification nuisance proceed wavelet nuisance parameter idea exist extend present imaging velocity smooth conduct
energie relaxation dd tight far multiscale energy coarse exploration allow avoid energy principled derivation coarse coarse fine aware interpolation multiscale landscape discrete preserve energy energy suggest hoc heuristic coarse
proof become nice note zhang propose weighted decay average averaging scheme whereas iterate paragraph exact proof much tight last
large structure penalize covariance interest many economic many lead parameter greatly exceed address frequently impose lin et impose al kronecker matrix miss optimization flip ff sparse call kronecker glasso dimensional assume whose separable kronecker channel wireless communication receive matrix arise recommendation netflix kronecker product graphical know time community variate normal study matrix normal n variate collaborative filtering kronecker factorization generalize fold measurement likelihood ml estimator ml alternate flip ff model order report know measurement structure graphical glasso glasso estimator idea glasso vary
efficient hierarchy set analyst cut dendrogram adapt heuristic frequently suboptimal construction merging decision lead refinement move cluster simple approach heuristic select switch lead error long move continue quite
clearly step whole exploitation walk entire fairly well regret well exploitation different explore property evaluate budget phase exploitation find maximizer evaluation property try remove many search search close ei distribute normal variable simply point write q point costly optimization find optimizer carefully maintain perfect exploitation area budget leveraging follow exploitation aim
way something frequentist course series method pose periodic fourier period could fitting parametrize however method one pose limited model type take uncertainty limit equally spaced parameter restrictive present von powerful considerably pay hoc suboptimal length cover whereas comparison mathematic relatively allow integral summarize rest paper early claim interesting possible summarize method hereafter arrival
nb explicitly become irrelevant construction treat nuisance interpretation fixing hdp group gamma process prior analytical posterior dr e cr poisson dr dr dr measure g continuous group component distribution express nb discrete marginally equivalently jk ji j n jk insight unite framework direct assignment crf likelihood would analytical normalize gamma model finite hdp may discard useful begin nb share nb dispersion alternative share construction distinct nb dependent scale thus draw
estimate value million measure evaluation unit capability year environment average human year simultaneous agent agent generation get local generation evolution divide start million efficiently start zero environmental year use evolutionary possibly upon capable small learn large efficient learning start
make ht extent system motivate feature moderately large numerical method model chain comprise respectively component fail evolution begin fail rate model investigation component failure express logic u failure show dash solid line highlight effect circle relative experience decay division step converge parameter type solid line make less agree failure fact include intermediate plot path close optimum course agree well exclude estimate variance efficacy
set markov ratio complete complete proportion direct number number report material length acc acceleration proportion mark solid box st rd direct complete box proportion near ratio near box box indicate rp box box respectively plot window window relationship number distribution maximum chain complete window approximately show chain increase solid circle chain complete size component complete box box accelerate algorithm accelerated accelerate fast estimate estimate four edge number size distribution complete line point window difference panel display second display maximum number structure three via plot accelerate window difference window top bottom component maximum component generate chain suggest speed approach nearly complete via study discrete show estimator chain second pp algorithm equivalence via proportion edge cumulative show reversible irreducible class markov equivalence equivalence class reveal undirected even equivalence thousand important equivalence class appropriately markov nearly equivalence widely calculate vertex potentially complete vertex complete vertex appendix approach proof definition notation contain cycle every cycle equal possesse occur reversible corresponding parent
alternatively span mean number markov use view estimator estimation computationally insight thank regard straightforward show assumption hessian diagonal noise alternatively orthonormal eigenvector multiply aa scale preserve product include special case mixture axis covariance align coordinate group view matrix kt theorem satisfy incoherence th large diagonal span basis coherence property partition formal gaussian apply coordinate partitioning orthogonal spherical multiplying cause coherence rank uniformly orthogonal
triangle become symmetric anomalous normalized distribution circular norm describe datum lift add dimensional quadratic initial firstly author neither aware make vanish describe section situation analytic although converge rotation
rmse theoretically covariate variable flexible equation f effect outcome maximum domain thus six visually effect nonzero represent b model group certainly lasso achieve parsimonious model mcp scad group mcp recall true carry design involve nucleotide snps version snp depend minor effect phenotype study mechanism yet detect important snps snps subject phenotype dominant effect include version variable ignore group snp simulation mcp false broad conclusion reach previous group variable either account grouping grouping ad hoc fashion mcp scad term mcp produce scad group gene expression genetic association determine snps associated fix mcp scad select validation analogously evaluate cross validate briefly microarray gene week
horizontal dataset variation scale ratio determine recognition accuracy equal extract yield unsupervised initial bootstrapping htb exp scale roc convnet report positive sort top bottom area measure bootstrapping phase sample scale desirable box file publish website version effort accurate evaluation curve discrete point instead compute curve sum area interpolation curve annotation dataset miss add modify code report fix auc yield supplementary material impact modification avoid
evaluation ease collect notational fold lp ng gx nc pp take speed validation reliable small subset setting motivate understand effect make estimate output ip overall error expect finite possible configuration approach predictor consider predictor parameter ignore minima test location test cross systematic configuration performance predictor configuration condition converge appendix counter neighbor prediction compare say section configuration full configuration hand suboptimal parameter correspond introduce fine grain see b indicate solid dash dash drive force approximation sure quickly uniform concrete see uniform convergence convergence minima typical train support subset training parameter sake simplicity see minimum rather one drive force still converge helpful continue discussion concept empirical denote error would difference risk class result coarse reach dash assume stay small extensively study theory know link dimension dimension hilbert mean small configuration complexity error error increase tend true become choose complex complexity ignore fast unfortunately make tight approximation realistic paper prove future work mechanism fast plausible look concrete example instead possible ensure sequential analysis learn body mostly focus evaluation already available reduce
upper going increase red colored center cluster panel expand cluster cluster voxel increase figure voxel diameter consistent accuracy exhibit degradation profile would smoothing multiple voxel exhibit experimental neither voxel contain informative alone informative accord voxel voxel voxel task response voxel take figure randomly deviation dot line voxel response dot voxel mean indicate vertical voxel condition correlate voxel denote voxel condition simulate response voxel uncorrelate response voxel simulation signal form response distinguish weakly response htbp scatter response voxel gray dot voxel spatial location voxel indicate single voxel separation either voxel text response voxel dot
organized computational tuning parameter example application devote present method multiclass computational multiclass conditional cumulative distribution furthermore regression function th conditional quantile define value various al chen add follow increase importantly k x estimation estimating specifically regression method ng li liu
meet assumption jump investigate appropriate threshold estimate population discuss eigenvalue theorem particularly informative combine corollary either term order rate still kk eigenvalue requirement take jump derive unnecessary would consistent eigenvalue jump occur identify dependent close fix assumption n eigenvector proposition depend gap eigenvalue recall denote counterpart select delta follow result reasoning make applied spectrum retain close counterpart estimate gap consecutive eigenvalue let define need hold depend constant usage plot outline eigenvalue retain separation population theorem eigenvector suggest small furthermore assumption plot
composite regularity satisfy eqs strongly r follow therefore proper loss canonical link composite comprise strictly factor correspond link multiplied function eqs satisfy proper composite desirable property convexity reader note loss canonical note link scale link appropriately proper canonical link r algebra proper strongly alternatively start scale proper define link verify strongly composite therefore purpose canonical
existence right value l duality subgradient subdifferential large value li li li leave singular completes give interpret projection ball admits follow multiplication ascent superior multiplier implementation detail admm implementation material estimation sparse low building block optimization dy eq respectively universal operator lemma material previous random value
mode localize sign indeed since consider always see general equivalent deduce rv consider must connection inverse analyze monotonic pdfs recall generic pdfs pdf lebesgue inverse version generate sample point accord note strictly clearly unimodal yield unbounded minimum pdf segment figure depend monotonic piece monotonic part increase mode pdf unimodal decrease combine inequality depend interpret localize switching region observe define write rv lebesgue subset unimodal localize generalized write target application unimodal loss region axis region describe observe figure b rv pdf region density compose rv simulate generate accord accept distribute distribute consider pdfs region represent note boundary eqs finally represent axes switching axis picture domain pdf partition form disjoint continuous even generate illustrate recall q hence note form monotonic piece region g introduce region target transformation bivariate relationship transform region also interpret pdfs piece monotonic transform region distribute rv transformation rv another must bound technique pdfs observation refer unbounded decrease vertical first pdf unbounded produce obtained evaluate general pdfs rv situation increase pdfs random pdf pdf high order
entry simplify property last come independence univariate rearrange factorization quasi whiten apply lemma give apply yield positive number diagonal positive root bb ti tb whitening b need fourth invariant distinct several tensor index fourth statistic statistic let I ic multiplicative imply estimate directly purpose analysis approach estimate fourth lemma
keep fix user statistical leave namely statistical recommendation method different statistical still discussion conference acknowledgement member statistic idea thank organization rich useful conference present progress team collection software tool implement modeling implement map within overview hypothesis testing estimation discuss development member root team exist extend root tool analysis establish readily tool compare
relaxation relationship analogous sharp invariant inequality iff eigenvalue stable eigenvalue separate set semidefinite exploit measurement next collect n refined measurement detail retain construct parallel valuable width depend signal ratio order construct rip since select ask randomization essential lead think game nature namely nature choose maximize choose freedom fix value take bad sx aim noiseless measurement rule large always choice state
design capture range grid en early corollary dms one link incorporate direction link reveal regard terminal connect adjacency exploit svd propose linearly efficient huge network benchmark network problem effectively interaction refer insight one interesting search module typically characterize link connect within practical direct wide web citation depth community community restrictive structure assume member violate main source citation present among paper computer citation restriction make symmetric community reveals completely turn allow different role detail htb comparison community asymmetric citation network implicitly express asymmetric social asymmetric community context web wikipedia drive play formulate pair community node connectivity quality directional community aspect capable directional aspect detection scalability huge network requirement regularize value edge community direct community propose theoretical link denote vertex source point community exist importance terminal community treat role community edge concept community ideal situation component undirecte
covariate random give give I constant real residual reason sometimes nearly covariance add diagonal change secondly produce add away gp covariance ard stationary hence achieve instead hyperparameter denote multivariate covariance prediction response depend response though variance residual taylor
reproduce kernel view measure choose design test strategy tool exploratory appropriate domain family arise chi square describe extend pearson straightforward schmidt clear whenever whenever invariant kernel broad dependence generalize exploratory take structured domain remark notion universal space say universal dense endowed universal laplacian universal due kernel universal direct universal also characteristic universal characteristic equivalent mean include kernel characteristic learn kernel notion translation acknowledgment b acknowledge carry university equally lemma institute mathematics provide statistic test covariance statistic maximum mmd reproduce space rkh establish energy compute mmd exactly
letter constrain would self natural several page character page character page alphabet letter exclude character would page consider character english page execute procedure improve character character linguistic predict context probabilistic generative sophisticated side recognize pattern patch simultaneously image invariance sift character handle add dramatically font separate clean text document heavily corrupt far improve sentence foundation early education research grant institute strongly manual remove letter character representation supervision character
circle node edge thin black em em sum em em em bend right east r west color black thick bend north thick bend r west north north thick bend east west color black em bend north em cl interact environment gradient optimum case concavity payoff gradient ascent ascent along extend j component ascent k knowledge state iii payoff state action equilibrium access unknown ascent feedback reward advance player project asynchronous differentiable sensor interested reader survey subgradient convex present application mobile greatly
many missing focus gaussians covariance expectation suit component non parametric e window put training make maximization training miss provide imputation keep em miss variable always like mention sensible absence gaussians attractive low analytical sampling gaussian train learn least gaussian use imputation observation generative distribution useful regard help discriminant reach accuracy contribution em novel inspection provide
subsection throughout develop method conduct experiment test inverse present conclude remark symbol dimensional respectively nonnegative maximization operate real cardinality number nonzero denote arrange entry likewise submatrix closed let normal tangent real denote identity positive semidefinite resp definite cone resp vector denote diagonal matrix element study order nonconvex necessary optimality minimizer follow condition assumption also local minimizer assume follow hold define minimizer condition problem nonconvex convex point fx fx x affine feasible exist local virtue know minimizer point minimizer second sufficient solution develop minimization show feasible denote follow definition imply conclusion
recently gain since copula investigate copula inner generator appear nest tractable copula arbitrary dimension family nest copula generator derivative inference interest able one nest copula monotone q completely monotone generator denote follow copula refer intuitive example joint say measure association furthermore construction explicit copula censor complicate argument possibly copula desirable parameter goodness transform see copula
also hence denote half consider real half obtain contribute part take simplify ib laplace infinitely zero term expansion zero infinitely real immediate expansion give real consider show eq constant ib xu xu x reduce ask
around compact showing provide average static mutation trial converge increase htb example update asynchronous parallelism topology recursive perform adaptive open scheme allow topology meet problem discrete ensemble asynchronous reinforcement evolve retrieve memory randomly update network match cycle environmental control network self complexity introduce dynamic necessary exploit solve mechanism adaptive adjust memory limit input topology term increase maximum content scheme boolean fuzzy logical collective ensemble asynchronous solve input reinforcement value grid previously research explore general complex problem wherein additional beneficial system investigation discrete dynamical asynchronous boolean represent discrete asynchronous fuzzy logic possible adaptive open number test
exception draw generate depend apply candidate deviation exactly mh summarize probability movement correlation number rw rw rw rw correlation mh rw rw without indicator indicate correlation point pdf true shape grow draw reference totally whereas outperform mh suggest generate generation reference always scheme independent proposal pdfs consider gaussian random walk try weight acceptance weight importance weight yield small produce standard mh drawn try target density independent pdfs try scheme
site addition simulate general site extend site neighbor site covariance herein flexible algorithm generate site field random site field proposition remainder next summarize simulation sufficient regularity singleton proposition effective generating proposition appendix recall notation state random follow site site denote nonempty neighborhood markov collection subset satisfy neighborhood q respectively concept definition nonempty generalize neighborhood subset site nonempty set subset subset contain one empty b nonempty subset exclusive detection two part herein denote bad unknown part know part nonempty site associate site connect neighborhood connect site site order unique subset exclusive choose I prove neither site connect rather henceforth site site connect site
replace finite sample type paper rx toward directional complicated manifold intersect near neighbor direction let distance fx fx type edge depend manifold dominate operator laplace away volume approximately volume depend scaling near singular tend infinity effect fourier vanish precise ccc reflect somewhat simplify think boundary quite operator example directional type singular usual operator expect negative think difference
analog function assumption approximated spirit nonparametric estimation seek impose graph call estimate rich still encode precision force graphical thus relax normality restrict undirected graph figure summarize family model think model regression marginal copula estimate transform jointly normal univariate fully model covariance glasso refer log model back least iterative optimization package implement inverse level multivariate forest acyclic estimate light yield nonparametric family acyclic two marginal sparsity tractable family
slightly small homotopy procedure utilize number homotopy step homotopy homotopy algorithm value adaptively follow update every homotopy adjust change support toward elsewhere predefine homotopy instead iterative observe adaptive reconstruction assign adaptive encourage remain homotopy step require entire recently level shrinkage reduce adopt selection explicit exploit regularization homotopy homotopy build homotopy numerical compare propose accuracy old warm well homotopy solve path decrease homotopy lasso homotopy toward final piecewise path length direction segment sign sequence analyze kkt optimality zero critical easy homotopy homotopy critical value desire every homotopy direction move calculate matrix
weighted extension direct vertex new analytic decade application technique vast hope come decade fs fs height pt pt fs mid fs scale rgb ed signal processing laboratory california institute david naturally field graph theoretic concept harmonic overview way domain highlight incorporate structure processing signal review translation multiscale transform brief open possible extension form domain numerous weight connect either proportional vertex graph signal bar bar example engineering datum describe disease census datum human pattern g stock brain image connectivity connectivity graph thus image view represent vision text popular supervise help label method build image proximity process reference method account graph signal way either visually dimensional storage use operator digital process question field
minimization minimization problem enforce per instance desire relaxation sparsity last decade problem entry aim find entry intend sense cs linear namely relation numerical competitive area cs error compressive imaging recognition enforce attract increase attention year group block kx non
since singular exponentially add attack exponentially attack attack attack produce estimate follow since hold lp decode attack exponentially fraction attack bit estimator logistic column solve I attack produce solve theorem error scale privacy mechanism add noise least attack recover entry attack decode attack mechanism add private compare theorem reduce full establish distortion attribute decompose k multipli adversary adversary every define solve analysis use discussion loss xy every attribute let get adversary q zero loss bound away zero imply
address regard general exponential bad complexity form desirable incorporate strength association interesting partially fp reaction would thank anonymous suggestion science university institute advantage search score bayesian informative belief network possible causal pair arise among novel operator advantage belief skeleton advantage former naturally prior since node way belief path correspond relation network derive experimental stem observational amount exercise week occurrence experimental amount exercise belief cause incorporate belief strength
denote cardinality complement letter represent matrix hermitian hadamard j k denote pdf dirac delta begin combine bp mf approach briefly algorithm slight modification unified framework arbitrary pdf grouped index denote index part refer denote bp mf marginal pz
present notion multi armed bandit exponentially reward prove optimality suggest conclude machine base slot bandit problem tuple reward reward associate objective reward crucial face payoff payoff payoff machine
chance category forecast observation forecast index nf nf nf I range perfect score total forecast category discriminant range score except fraction forecast random chance unbiased forecast notice rand correct correct rand adjust pair expect maximum index r nf maximum number extract magnitude observation locate team al comprise density function display cluster
notably elegant relevant allow tight cm green grey target rotation angle cm cm cm cm da cm european
apply prove meet notice v representation support mail fr universit sciences f france counter recovery orthogonal pursuit omp ols noiseless omp ol good first iteration sufficient condition dictionary ols knowledge pursuit least take overcomplete particular noiseless usually solution support suboptimal tractable approach one basis algorithm mp pursuit omp raise question ensure paper novel answer omp ols widely recent analysis
way assess correction often partition adopt turn consider course group close adopt effect correction cluster addition consider absence center variation factor procedure factor center variation course factor may correct effect unobserved correction replicate dictionary sparse dictionary minimize jointly convex obtain minimizer discuss many iterative iteration relaxation restrict method think paper require rank strength application new try heuristic three benchmark choice discuss robustness hyperparameter choose replicate former use choose model model ridge acting choose positive care iterative counterpart estimator minimize correction small iterative use experiment author r interface gender find patient assess gender benchmark affected technical microarray brain patient involve study send university ann array share miss lead control gene identical use non iid dependent discussion could reasonable result process center array type line iteration dot line since gene affect sensitive cluster gender assess therefore try different correct retain center avoid plot retain keep panel clear variance main
trait eeg signal brain consist subject pattern approach literature feature occur pilot always new deal nmf modify cost function standard
cifar also op universit deep machine probabilistic date simultaneous joint largely encourage training deep machine model view
normal specify x set regular limit u degree lower replicate leave illustrate correct confidence range fix bias respectively adjust limit occur contrast correction procedure adjust limit necessary assumption hence correction correctly enough require correction vary error although eliminate increase limit correction replicate analysis p cp cb db cb db correct cp cb bootstrap correct cb double db replicate empirical probability interval correct
distribution class plot accuracy use rbf leave lin lin experiment include kernel lin unnormalized rbf normalize rbf validation perform rbf kernel repetition rbf rbf degree figure value function embed embed kernel tend impact level coincide depict equivalence image invariant consider handwritten equivalence class transformation parametrize factor along parametrize adopt distribution virtual sampling task digit digit
try match margin time run averaged choice margin need projection small follow times rs theory optimize input lose svms apply combine run fast twice increase projection two vertical ten ten fold ten matrix four individual genome individual fine population individual correspond nucleotide variation across genome allele allele miss population constitute tp projection classification four level notice vertical since run sample broad experiment population classification nearly identical method level close strongly support main fit combine projection region population task fast follow rs dense seem outperform number
eq need background reader pick subset class subgraph fs subgraph subgraph index converge supremum satisfy begin measurable subgraph function stationary polynomially ergodic uniformly let g page page let satisfy continuously since define finish initial use proposition result proceed polynomially ergodic denote respectively equation yield composition q e corollary q eq eq
learn kernel nlp k fold tune validation function smoothed fold cv aim address two method hyperparameter standard non different cv probabilistic ls need design loo cv base measure function classifier optimize usage loo cv propagation approximation actually however problem ep laplace approximation ep conduct criterion logarithm predictive nlp smoothed nlp ep loo cv probabilistic ls classifier ml use method l real benchmark show l quite competitive method ep optimize nlp optimize hyperparameter nlp optimize bias experimental inferior ep give brief gaussian ml criterion loo cv criteria illustrate loo cv optimization aspect specific
capability previously unseen combination unseen set repeat variance use create final either directly via actual perform predict make effect decompose procedure illustrate variance figure predict seven previously cd size space investigate model accuracy model uci repository feature alignment e adaboost provide action tendency variance prediction relatively dense variance variance apparent explanation suggest future whether would create model height quantify ten coefficient determination predict value progress curve monotonically rise reach exception segment correlation via except consistently prediction direct prediction accuracy attain full ten diverse seven determination predict great predictive treat variance huge normal therefore pair test confirm level deviation deviation predict height new one five
function perturbation exploit unlike development blind deconvolution pixel sparsity gibbs sample blind alternate blind illustrate real sparse principle atomic scale demonstrate first spatial less imaging map atomic spin formulation spin force reconstruct force rely wiener whereas least square however parameter physics probe probe external unfortunately tune circumstance image reconstruction suffer image partially usually deconvolution paper problem hierarchical bayesian
mechanic study collective behavior present well comprehensive utilize differential equation diffusion mechanic except opposite justification number accuracy point represent continuous available underlie represent since examine evolution density movement transformation identify backward shift total merge perfectly simple dimensional location cluster flow law hence dynamic yield law law dynamic provide analytical evolution
develop book variational area naive computational respect small instance gps reliable important rank gene great instead lack fitting model thousand impractical fix problematic constraint phenomenon early detailed derivation gp stochastic completely matrix multivariate
close take cut alone chain rule jacobian e g formula apply set likely estimator mx hx show various exact column agree say generate procedure
pass technical distribute receiver interference wireless within probabilistic share information predefine propose probabilistic message message determine overhead interference wireless potential additionally technique extremely design receiver architecture could beneficial propagation decoding assume channel joint decentralize
intelligence task intelligence user user work call self question filter six select user content independent feedback semantic comment generate worker ask article bundle feedback article bundle
random classification near mahalanobis depth moment estimate separate multi regression vector alternative current substantially feature extend I rather procedure yield rather stable solution procedure class two dd modify procedure base validation linearize x x I exponent feature equal discrimination feature bottom approach successively new basic subspace straight subspace discrimination discrimination separate dd classify discrimination depth h restriction imply straight discrimination calculate two coordinate pair characterize angle indicator minimize pair next geometrically value straight discrimination accord far angle minimum angle attain calculate etc stop basic angle
image handwritten digit screen two rejection discard feature e solver screening screen run solver screen equally spaced speedup solver screen lasso rule solver dpp solver solver solver solver mnist run second screen cancer obtain mass index ratio protein patient patient cancer therefore patient image face image pose trial image trial digit data grey handwritten testing dimension randomly digit get matrix randomly trial screen improvement discard inactive dpp compare observe improvement discard row show speedup moreover improvement discard inactive improvement ratio discard inactive result gain rule mnist speedup gain mnist screening solver substantial four family dpp rule compare screen computational cost family dpp compare art screening rule flexibility family dpp see dual problem share similarity look projection onto close entirely
separate say nu wu wu find vertex nu wu w wu z nu w z wu wu wu wu together practically instance hard problem feasible toward computational even many stable instance seek dependency corollary practically locally stable instance theorem remark counter theorem corollary conjecture computational traditionally quantify lead
use ari bic superior array collect interval remove max min reduce gene select gene start ari choose factor factor ari great choose bic choose lrr propose parsimonious gaussian mainly intend term rather penalize thereby likelihood mean absolute coefficient scenario specify mathematical require cross
base relevant pde seem complicated answer obtain heuristic jeffreys simple algebra function pde clear extent approximation exactly problem idea immediately care regularity exponential family requirement drive pde connection describe lose case equally fisher build already instance familiar thing like similarity dimension reduction feature immediately choice bad may conditional focus know statistic general differential drive function I analogue statistic carry challenging section home statistic important space turn notion framework auxiliary variable dimensional subspace observe I take modification denote accord association normalize simplification helpful corresponding density I predictive giving example q plausibility corollary proposition definition remark step inferential free focus unobserved efficient I feature observe case simultaneous fisher differential drive validity prove admit I flexible variance bayes set validity fisher view middle bayesian approach influence idea current frequentist example datum dependent prior promise new I observe free fundamental idea uncertainty fully
proposal distribute correspondingly define couple hasting kernel center consequence wasserstein mh proposal explicit give wasserstein hold ball value contraction proposal wasserstein hasting easily wasserstein hold mh proposal disadvantage second condition euler proposal positive exist restrictive often satisfied minimum state ball r enough coincide moreover sufficient hold q consequence proposition wasserstein metric implicit transition mr theorem exist path generally modify suitable neighborhood wasserstein base correspond two satisfied h w chain proposal joint wasserstein r probability ball base lyapunov function implicit quantitative implicit
matter likelihood inference provide numerical mean marginal various model analyse estimate base analytical function frequentist estimate counterpart uninformative implementation abc analyse tail slightly short posterior wider obtain approach choice original incorporate inclusion unlikely would result affect conclusion posterior spherical inclusion additionally normal sample transformation conditioning automatic unstable density numerically importance transformation
whether task constructive avoid view development task view propagate along neighbor natural neighbor mml limitation potential sensitivity choose iterative algorithm great chance view formulation thereby success demonstrate algorithms concern multiple setting task view multi multiple feature description motivation performance task multi performance correlate contrary understand propagation task successful propagation idea behind identify success two problem optimally close alternate mml optima provide function reconstruct dependent weight independently reduce function denote function description refer lagrange multipli enforce sum matrix parameter feature respectively control sparsity construct place notice empirically indicate indicate contribute indicate goal miss fix weight learning relax classify rewrite format carry regularizer operation row attempt optimal method closed form inversion prediction obtain form matrix inversion identity final prediction unlabele instance
add entry determine obviously gray multiplication importantly diagonal position nonzero adjacency matrix twice entry within stack row equation execute backward substitution extraction matrix nonzero experience computational scale forward substitution compute second nonzero suggest format truncation non even display pattern image size similar image coincide sketch work objective affine minimization path move initial gray determine minimize call first begin parameterization variable path equation drastically simplify nan standard denoise ode equation principle segment segment differential solution solve involve yield nonzero path close time z early straightforward model operation end cost path approximately comparable
level scalar operator term encourage sparsity split ad experimental literature support ad especially exhibit favorable structure instance bi subproblem close per convergence extensive test indeed formal appendix consensus q generate converge benefit sparsity minimization far call application scale traffic flow change service user source network failure service attack service anomaly engineering network traffic challenge however load flow pair flow link traffic flow connect carry traffic detail traffic carry link measure compactly clean traffic flow anomaly load indicate flow order link counterpart traffic flow occur addition flow suppose anomaly give measurement goal fashion primary define low cf sparse pursuit applicable adopt anomaly p readily simplify general residual
hierarchical argument occur special hierarchy favor hierarchy word main likely small corresponding word focus main effect final useful distinguish call coefficient practical restriction difference substantial hierarchical split run top misclassification versus practical lasso node nonzero edge take argument extreme hierarchy develop weak hierarchy name think strong impose principle multivariate spline vector produce vector response lasso optimization one control relative importance square extension apply column deviation predictor center centering notion strong hierarchy situation want exclude grow focus generalization induce q onto structured choose nest penalty specialized penalty suggest penalty generalize propose lasso strong constraint interpretable remark penalty literature admit demand sparsity pay attention
convert handle computer viewpoint reconstruction equally space elegant motivate research mathematic brief introduction mathematically evaluate evaluation continuous endowed arise area term hilbert space reproduce product shannon band rkhs reproducing search shannon formula reproduce frame formulae reproduce banach frame banach space semi formulae enable continuous go countable remarkable however countable infinite computer infinitely raise question shannon lead exponentially decay often come
correct ssc correct subspace however computationally extensive develop generalize foundation prove generalized theorem invariant norm norm mathematical completeness arise popularity norm closed rank regularize connection previous prove minimize yield reveal true invariant choose suitable remain provably obey classic factorization demonstrate devise art computationally use although generally frobenius norm transpose pseudo state unitary natural small know fan frobenius three belong
form variable point correspond corresponding definition subsample induction subsampling process measure poisson another measure normalize measure measure evy derivation move thus subsample decade distance base take superposition induce operation definition complete configuration jump impossible z k furthermore normalizing denote argument subsample write subsampling equivalent divide use patch nm bernoulli evy mm mm mm chen mm normalize mm national act act nan computer university theory instance dependent topic model introduce background measure process slice
far sequence star principle accommodate expect poor fractional error significant rely full benefit magnitude realize p insight ap svm overall ap estimation accuracy spectra achieve surveys survey ultimately method spectra processing snr spectra bp rp dispersion make rough snr snr bp rp much large cover wide range unlike dominate far ap averaging trend restrict range address science structure identify star analyse three case mix turn rather rather ap science sample select completeness galaxy sample separation mix magnitude select estimate effective true range completeness fraction false high completeness contamination achieve preferable contamination star truly star accuracy provide ap contamination star contamination science k star star build proceeding table star conversely cost arguably true ccccc svm model star concern select star disk thing star define star four list separate statistic
apply completely exist dependent random framework construct covariate motivated flexible model minimize structure datum bayesian model attention learning statistic community us component feature model exchangeable justify believe structure grow challenge assumption maintain property original nonparametric process nonparametric measure index space know process nonparametric find literature disjoint subset normalize gamma process ibp known beta binomial positive exponential completely represent space dependent operation offer great dependency conjugacy dependent beta normalize relationship aid understand development covariate
operator notation work note notation stem boolean algebra page de law de several theorem phrase restriction theorem prove easy hence hold phrase phrase prove induction rather important analogue system se se se surprising establish follow easily disjoint set symbol q operator prefer sort characteristic transform change bit cube contain refer set partial note sort operation every pair derivative xx sort restriction sometimes translate classical sort close observation edge sort eq sort system apply shift sort require make sort
clutter type short occur occur type clutter traversal obstacle mat ern clutter length occur short length traversal obstacle level short traversal length mat ern clutter obstacle traversal occur clutter type form length cluster clutter long regular clutter obstacle window locate traversal occur windows length obstacle length clutter obstacle clutter length decrease obstacle clutter tends decrease obstacle short occur occur length window length length length clutter occur type occur short length short traversal length occur mat ern clutter obstacle short traversal type traversal clutter shape obstacle traversal shorter cluster clutter regular trend tend flat obstacle level increase obstacle increase tend obstacle number decrease shape window v window shape obstacle length trend similar cluster clutter likewise regular clutter ern cluster length short length clutter clutter window occur clutter v window shortest clutter window shortest occur window length occur hc w average traversal length tend obstacle increase short traversal short traversal length occur mat ern clutter short traversal poisson clutter type obstacle traversal length clutter shape obstacle traversal shorter cluster type obstacle obstacle type w shape window tend flat obstacle trend similar clutter w clutter poisson w windows clutter window ern clutter close obstacle significant interaction clutter reasonable interaction background clutter effect obstacle type traversal background clutter obstacle find interaction obstacle clutter main effect obstacle type clutter obstacle clutter obstacle form traversal length type obstacle short traversal length treatment hc obstacle length clutter short mat ern traversal clutter tend long traversal sort shaped shaped obstacle clutter type short occur treatment occur hc traversal occur ern clutter traversal similar clutter tend clutter traversal sort shape v shaped mean traversal shaped obstacle length
parent parent combinatorial fast overhead b mb parent proper responsible requirement versus candidate parent complete score memory need score parent store share efficiently reduce keep parent pick score search efficiency correctness illustration high high score reduction move record right half example entry assign large involved keep track original reduction memory store current involve entry entry among obtain original high parent algorithm ghz intel e processor
e removal u graph observation independence exploit variable rank property satisfie u choice thus neighbor find order criterion neighbor variable sufficient correctly recover impose find eq mixture tree separate separate degree component overlap model include establish sequel complexity exponentially relax separation satisfy property include propose effective neighbor neighbor involve computational complexity number test perform node conditioning involve assumption make dimension neighboring eq relax strict separation local separation decay refer bind local rank condition great satisfie arrange n relate allow grow cardinality sample also enough consider
continuity infimum drift iii conclusion derive sufficient ergodicity asymmetric volatility ask kind ergodicity ergodicity skewed purpose work ms support capital grant remark section
partition indicator complete group aspect estimation enable treat scale combination surrogate truncate approximate tuning dc decomposition nonconvex say replace approximate subproblem restrict provide update iterate indicate local dc critical routine detail section propose estimator achieve presentation utilize entire feasible group consistent oracle
determine computing setup reason take hypothesis approach extend problem address dissimilarity datum issue section dissimilarity estimate dissimilarity measure dissimilarity distribution dissimilarity measure divergence method dissimilarity contribute decide measure include popular divergence unknown cope plug definition divergence plug reliable novel basic direct density learn density intuitive know two know know necessarily estimate substantially easy density promise representative demonstrate usefulness principle avoid solve recognition density principle
exploration phase one obtain unknown large regret diameter necessary markov chain treat I next memory section stationary ergodic case asymptotically sublinear acknowledgment project european fp grant author currently science ks regret set plausible mdps lemma random identically fix state recall transition color choose transition state hoeffding union hoeffding color note action pair color write
fx fx gx reasoning gx note p inequality iv prove vi follow similar finally vi analogous variable define use bind establishe define k bx pp pp assertion vi ii note lem assumption bx j claim sequence iii exactly iii whenever vi corollary example provide quantile root behave construct behave uniformly coverage tend
figure detect environmental compare environmental create environmental previously neutral simulated frequency slope include correlated environmental dataset set percentage false base lm glm neutral fp contrast produce low power reject neutral simulated return bayes rank association positive number negative score parameter assess measure receiver auc replicate whereas linear performance value factor analyze genomic throughout great forest snps express sequence est individual environmental component describe describe fall environmental score great environmental snp great confirm snps snps functional annotation discover protein protein stress stress
original labeling book probably relational som combine advantage som organization computation som handle describe performance project either categorical paris fr example pairwise dissimilarity variant generalize whereas som rough relational som suffer version provide result bad article line justify several
limit arbitrary template binary background white disk white contaminate significant snr db fig observe spin perfectly signal merely guarantee spin manifold incoherent informally incoherence template sufficiently spin spin manifold entry choose incoherent dimension b spin intrinsic matrix rip spin construction rip combine lemma serve cover manifold low probability measurement cardinality upper geometry correspondingly specify bound measurement preserve em spin tractable computation example manifold input return location match fourier transform operation tractable approximate projection spin weak noise shot acquisition fig spike undesirable sparse
reach result centralize u I readily assertion centralize denote iterate adapt process evolve adapt auxiliary eq hence hence conclude hypothesis exist let note readily w tt introduce adapt evolve reduce also fact complete proof note adapt lemma observe hence adapt action conclude construct satisfie process bound event exist functional contraction application yield conclude contradict complete lemma immediate consequence characterization achieve simulate behavior example agent cardinality control pair stage one stage sample state action uniform simulate near communication link place circle simulate trajectory trajectory trajectory sampling probability past illustrate path centralize centralized centralized randomly solid factor among action consensus
great correspond number pearson coefficient value value zero space pair one orthogonal diagonal engineering coefficient monotonic sampling rank multi orthogonality measure variable particular orthogonal exhaustive point hypercube possibility prescribe apply uncertainty divide range value centre randomly couple result discrete variable quite simplification optimisation mainly discrete domain simplify space significantly restriction sa restriction different value needed level equal homogeneity result appear alternatively author mixed programming generate design evaluation criterion way sa intensive reason develop recent work topic design far genetic optimisation bound design establish
infer prescribe structure grey description random direct graph favor prescribe structure translate convenient replace extract perfect compression necessary perfect always amount threshold sample show partition overlap correct different plant block observe threshold lie region
work simulated statistical employ sec remark two issue propose testing behavior l evy stable evy simulate stable pdf stable mention variance evy irrelevant straight correspond asymptotic evy pdf two empirical pdfs visible fourth moment simulate random fourth converge infinite fourth evy stand simple present indicate former constant behave
lm lm verify misclassification n lx induce nn next derive classifier classifier misclassification sure kernel b converge sure pi rf pi rf dominant pi rf belong deal derive misclassification unsupervise classification fix assumption misclassification plug tight rf pi pi verify upper
channel expression ss pixel nonlinear simply pixel difference pixel neighbor hyperspectral organize sample angle denote matrix spectral channel diagonal element vary band easy product unnormalize pixel yield could unnormalize pixel characterize unnormalized likelihood diagonal adapt transform computed series rotation final computing efficient pca adaptation artificial treat equilibrium map energy attractive energy distance constant function pairwise shape result attractive force formation force adaptive potential act field thereby nature point nature distance negligible map I map range force map behavioral distance appeal intuitive property differentiable differentiable pair map e tangent inverse approximate decay unnormalized generate grow map unnormalize bound spherical field shape harmonic planning center origin force field continuou unbounded invariant similarity unbounded field map coordinate pair equilibrium strong force field search configuration unnormalized yield multidimensional artificial field elastic relate motion illustration force force mostly effective short propose entail multidimensional
break imputation observe social friend reconstruct information explanatory social ht ci ci yes yes yes yes yes ht ci ci yes l c yes yes yes cm de sa e sa author email
poisson binomial dispersion poisson tail alternative allow dispersion normal allow dispersion allow binomial less law unable poisson
find cone dual cone cone follow cone cardinality regularize least square literature example thresholding recently zhang propose penalty solve well finding unconstrained least successfully cone particular propose constrain programming sequence generate minimizer method finding method iteration local propose penalty dynamically accumulation local minimizer outline paper
use select fan li choice scad mainly scad simulation come fan last two size simulation examine smoothing ps ps simulation random subroutine software choose wavelet spline code code local matlab knot select initial knot factor fourth stable tuning present scad fits cm ps ps eps spline slightly cubic spline also observe penalize spline
attain subset observation membership sub application model select bic classification membership ccc est run comprise know run e label select even bic group bic sub cluster among three group suggest cluster introduce parsimonious model cluster latent structure explanatory parsimonious version combine constraint constraint factor likelihood estimation algorithm sensitive maxima dimensional step hierarchical utilize likelihood word artificial regard latent
subgraph valid define reach realize abuse notation allow universal constant essentially bound xy lemma condition along chernoff relaxation remain include detailed suffice achieve appendix favorable give dependent direct value preserve special satisfied possible define universal constant conjunction localization similar employ remain conclusion remain valid bind excess prove concentration via analogous base aside possible arise specification entropy preserve validity substitution derive constant essentially appendix preserve validity brevity detail omit validity theorem arrive except replace preserve validity theorem omit brevity computable note precede imply reasoning argue find remain replace definition essentially brevity running algorithm main maintain showing sufficiently reduce guarantee state request component follow application algorithm value upon obtain complete round denote h g eq total mh furthermore event I sm j eq imply next event j u j value base inductive event trivially imply hypothesis imply eq plug particular always establish j complete imply particular keep least er suffice theorem statement event includes state furthermore event note bernoulli least prove er trivially otherwise classic
part equation lebesgue constant depend theorem follow easy prove minimize achieve adaptively slightly finite reason higher high begin result extend manifold depend eq positive form case embed hilbert space orthogonal complement describe tangent bend map mt tangent orthogonal complement crucially case dimension complement k high greatly well analysis slow order performance datum key region manifold point flat guarantee vanishing infinity approximated tracking tangent space fundamental region vanish diameter going infinity unless take root surface curvature arbitrary k domain type link tight manifold bad method easy quantity less restrictive region curvature cause result present care take throughout constant constant
rational quadratic additive combination considerably suitable scale grid otherwise grid divide six performance simulate real affect finally simulate credible single simulated choose unimodal tail challenging separate figure model narrow gamma mode gaussian middle la importance la integration density median line statistically difference first mode plot practical interactive figure show correspond credible follow set visually la la mcmc leave log cv performance across importance laplace galaxy datum improvement perform similar slightly density galaxy tail observation look mcmc
degree gs similarity regression preferable tune competitive one bind difficult bind benchmark specific allele testing outperform affinity machine yield training build predictor represent represent bind bind function output bind affinity real training method represent output give lose energy true bind regression fundamental draw accord predictor small access assume tell predictor risk suitably call predictor predictor measure accuracy complexity elaborate tell subspace span q change efficiently computable feature induce extremely large dimensionality running point kernel correspond space normally value normally mostly denote product product space protein kk fs minimum coincide gradient vanish kernel ridge semi inverse exist
inconsistent domain transfer target impact help conduct fine grain analysis different target domain show first rating tail really grain cf without fine grain usually preference user method respect long case instance well rating user historical rating benefit avoid none apply extremely compose record result table utilize domain type domain observe compare book part information interest movie preference wikipedia record graph directly relate still netflix source observe wikipedia record domain transfer method transfer compare source domain
insufficient autocorrelation allow principled outcome binary instead combine sign undesirable one deal use allow outcome maximization chain statistic develop alternate change e degenerate method preferred software validate posterior quantile estimate move network autocorrelation present parameter suggestion complete paper process influence channel member message widely model assume social interact realistic large connect people distance influence refinement approach outcome individual explicitly method simultaneous residual eq outcome
base use recursive formula adaptation possibility iteration mh pdf choice well section seem correct ergodicity property future degeneracy first iteration poor collect produce mh assign among mean update
let consider node collect information centralize optimal write operation centralize across impractical centralized architecture highly rely fusion fusion filter distribute algorithm distribute algorithm base classical neighbor estimate take diffusion denote node exchange weight I cardinality update adaptive I ta represent filter combination mainly combination rule laplacian ik ik first disadvantage sensitive statistic mean I algorithm focus rule nevertheless individual evolutionary predefine besides reveal distribute sequel graphical exist give distribute evolutionary biology differ property equilibrium widely process communication streaming spectrum approach dynamic equilibrium period strategy ess ess stand player player equilibrium ne stability moreover
trading summation appropriate us rate strongly regret lipschitz mirror without function bregman strongly norm regret regret stability require strong continuity lipschitz strongly square norm continuous diameter bregman generating function incur algorithm bound r use optimality strong use side forward formally previous note strong condition convexity need select r r q optimizing prove online objective
important arise approach ordering partially order significantly optimization final network demonstrate network census weather accurately dependencie domain graphical acyclic dag extensively wide variety instance gene medical machine vision behavior probability random
generally call exhibit practical important collection text sound broadly use develop step conditional variable derive stochastic empirical follow organize document simplex denote topic topic infinite collection associate distribution simplex simplex entry index draw topic proportion assignment relevant expectation dirichlet w v topic assume document exhibit proportion topic topic proportion draw model lda exhibit assume dirichlet exchangeable improve prior analyze amount capture describe degree exhibit assign explore large document central researcher develop method include markov monte propagation inference develop variational collection illustrate topic lda collection document corpus compare algorithms lda collection inference subset document collection variational well derive two represent categorical like assignment document assign th likewise word document abuse review distribution simplex one factorial dirichlet parameter derivative log gamma derive put dirichlet variable set document local variable proportion assignment hide topic complete distribution variational conditional conditional topic assignment approximation variational topic conditional th dirichlet assign conditional dirichlet variational dirichlet dirichlet document different proportion conditional context global th dirichlet topic hyperparameter topic global variable complete conditional
fit break unnecessary parent output depend causality fit choose additive linear aic order build spline gaussian automatically optimization residual series combination one correlation thm correlation insufficient fail run determine bandwidth median distance input heuristic fitting choice unobserved model one try discover causal whenever version repeat able remain exclude useful investigate principle remain mostly require fine
replacement might randomize careful accumulate increment select iteration many increment specialized case toy scalar apply initialize incremental stepsize replacement use sample square toy illustrate scalar consider stepsize perform replacement replacement inequality always close expectation sort toy example gaussian replacement subtract side multiply index multiply risk namely demonstrate without replacement respect noise sequence simplify thing take expectation
course incidence incidence diabetes relate adopting may continue year consideration simulate simulate carlo applicability analysis strictly separate pde mention input exploit predict accuracy structure inherent affine interpolation incidence work affine incidence diabetes age year
experiment change transformation purpose drop network present make network interesting order motivate natural change influence content network follow coherence network content begin examine coherence edge drop add involve change structure analysis present coherent connectivity view measure enhance overall graph affect connectivity protein belong use protein class network average number scatter measure original statistically significant connectivity plot class performance prediction explanation help prediction protein connectivity original connectivity original improvement auc prediction transformation functional connectivity class accuracy average particularly help improve connectivity prediction original show important transformation enhance connect early utilize score identify
nonlinear switching process generate weighted variable construct experiment successfully discriminate dynamic human extremely process occur person subtle pattern observe movement gender person remarkable learn generative dynamic environment brain infer propose generative continuous dynamic online accurately discriminate dynamic implement implement winner many switch active associate variable parametric equation movement primitive observation standard
interval assumption population four step apply value formula simulation various fr distinct three compare square summarize via significance er von ks pearson summarize table illustrate new compute coverage probability summarize illustrated sample shape
big outside determination confidence interval event allow confidence note bayesian determination frequentist
inference infeasible feasible numerical community individual similar overall diverse context protein human method know interact distinct community across community structure representation diverse analytical candidate partition depend display yet understand consist reveal reflect case procedure belief propagation perform classification node consist partition block belong block
pose low solution np current impose make amenable via exist alternate small represent capital letter letter column transpose default norm denote spectral trace represent pca sign prediction etc reason applicability memory flexible modeling c amenable parallelization despite largely heuristic approach problem b complete low provide show recovery minimization attractive additive
outside simplex ice write covariance control tradeoff good simplex simplex choice version worth note geometric either volume regularizer enforce abundance datum cloud regularizer fractional update introduce cca boundary cone concept cca start eigenvector large eigenvector basis belong cone convex minus imagine hyperspectral sensor type tree spectra almost spectra associated spectra pixel consist perhaps represent true nature vertex region interior design find hull unless fail happen rely design devise identify hyperspectral class convex piecewise approach design latter next rely fuzzy cluster assign one cluster allow every fuzzy assignment membership two point objective attempt spectra membership ice abundance analytic update formula membership multiplier fractional one still solve minima update formula version pure algorithm simulate data set illustrate pixel snr db follow pure simplex pure simplex distribute bottom mixed set produce good pure nmf produce degradation bottom leave still significant degradation pixel close remove finally produce pixel vertex set reach mixture mixed yield simplex statistical powerful high computational represent spectral formulate statistical case hyperspectral blind tool separation tool hyperspectral abundance hyperspectral datum abundance imply applicability hyperspectral signature ica
let vector exponential sample exp construct write functional q equivalently mode weight denote distribution anneal van level canonical thompson walk set energy energy evolutionary liu liu slice sampler uniformity contour exhibit global local example derivative difficulty additive slice sampler boltzmann chain example
exploration asynchronous helpful convergence case careful perhaps idea adapt tree store message inefficient many convex cover problematic message pass algorithm dominant arbitrary message cover finite cover lift minimize domain contain even objective use trick theorem force computation twice differentiable second partial derivative semidefinite demonstrate computation convex guarantee independent twice continuously condition convergence min scale extend subject future without diagonal break irreducible argument eigenvalue existence definition dominant cover scale positive symmetric cover definite loss value irreducible frobenius scalar radius nz ic consider c iy ty iy combine see imply word walk root let potential bound depth diagonal eigenvalue ii contain need ij far multiply potential multiplied possibility leaf
statistical work take structure evident successful conversely goal may successful configuration variable right column disadvantage pattern lose level relation branch view kernel mark node anchor west induction anchor anchor west anchor west rewrite node anchor fill white l l fill proof tree oppose sequence branch devote generic index lemma goal dissimilarity goal ml pg analyse case rewrite mul n big rewrite exp big seq odd move h move big odd odd square goal rewrite big fact big add next consideration pg extract pg goal discovery area machine irrespective feature extraction pattern tool limited exception rule proof extraction major respect allow goal pg choose property shape shape goal feature uniformly rich may see library number list library trivial rewrite trivial inside double five consecutive extraction pg name link step hypothesis inductive lemma track extraction goal shape still infer goal user treat goal step ml want might advantage apply library adaptation change library
evaluate horizon learn generalize outperform worth well report associate optimistic shortest find totally policy mean index score bad arm arm times decrease ucb reward play preferred suggest optimistic paradigm multi ucb horizon table give train horizon percentage count observe learn percentage real percentage better numerical core perform whole take hour hour symbolic case take minute bit less hour symbolic explain careful way obtain optimal symbolic algorithm cpu able rapidly reject strategy armed formulation
piece automatically extract result learn behavioral twitter occur network piece feature automatically extract behavioral role learn time collective twitter united conference importance role visualization clearly analysis twitter day role role find regularity pattern arise thus spike presence anomaly small spike observe type role step locate step pattern importantly towards relatively trend whereas interestingly pattern twitter involve conference drastically change conference come un conference form connect dynamical completely manner systematically behavioral world ip role membership individual role past behavior role measure formally role interpret membership measurement component coefficient degree contribution time contribution average evolve behavioral trace
assumption recover signal arrive sparse sa na induction conclusion trivially least imply u ss argument proof combine arrive l ls imply ss induction assumption ss ss note arrive step
orthogonal subsequently anomalous greedy anomalous link anomaly correlation link anomaly operator filter traffic anomaly filter choice wavelet fast fouri e accommodate change see detail comprehensive test estimator leverage sparsity cs plus decomposition fit incomplete cf l minimize pseudo unfortunately albeit natural typically nuclear singular adopt since one controlling appeal develop anomaly offer traffic subsequently task exploit spatio anomaly procedure ht roc curve leverage jointly via anomaly show blue move implementation generality miss sparse encountered pursuit also refer pca available aforementioned however superposition challenge cs paradigm recovery arguably dimensional adopt missing entry implement link traffic measurement use aggregation operational paradigm adopt limitation architecture information raw measurement communication translate miss raise concern central isolated failure reason motivate embed anomaly per computational task locally rely link link message anomaly attain centralized counterpart
baseline helpful preference map show classifier weighted vote diversity indexing claim appear appropriate prediction modality show competitive fusion incorporation order preserve constraint sometimes beyond pac theoretically indexing ranking etc corollary cm
go go vice small vice error small suggest connect threshold impose additional b whereas relationship show approximation easily equal cauchy conclude complete whereas nonlinear equal right similarly step representation analogously improve negligible specifically addition third condition type jt w depend derive nonlinear follow q relationship property select every proof mi e attain weighted e
gibbs partition drive proceed conference complex combinatorial nj formulae gibbs exchangeable appear analysis exchangeable gibbs triangles science approach estimation discover cm gibbs exchangeable exchangeable th statistic game volume cm brownian motion characterize
recommend quickly refer back let denote radius smoothed version small form hausdorff reason commonly strict distance analogous familiar subsection self intersection reach weak find reference gradient whenever define let ax contain aa bend circle slightly create straight reach corner reach set homotopy continuous map case smooth write nearly homotopy k extensive let ax jk f stacking length conversely vector length stacking think anti hadamard calculus jacobian q fx ff
approach result use method I I average along sub learn rule em extensive derive correlation benefit correlation modify deal application sparsity zero row mutually parameterize multivariate gaussian nonnegative hyperparameter control definite matrix correlation structure partition reader convenience quite intra inter inter intra inter correlation
create subset document represent difference market similar level look problem model technique represent document collection text represent vocabulary uninformative contrast description ir key naturally incorporate example create guide semantic meaningful despite suffer many bayesian incorporate perspective parallelization scalability remain aspect sparse concept incorporation side semantic feature improve ir capture derivation however discard employ sampler birth hasting sample gibbs gibbs j generality concept come well ibp exchangeability write factor dm call simplified word document word document explain supplementary material metropolis hasting flip value since exist let concept
sample form table du matrix singular pick fail choice discuss choice topic conditional information document satisfie unit return algorithm permutation illustrative appendix remark boost confidence convergence polynomially failure repeat sufficiently interval scale factor sample depend bound factor eigenvalue approach much broad view w k conditionally column specify else conditional hybrid depend style cm fill black sep hide name name style minimum inner style hide name name observe name h follow degeneracy general remark view dimension notational stick case mean develop estimation topic
rgb rgb blue fill blue shape blue shape draw red thick thick nlp view reduction purpose view supervise unlabeled datum rare article york time recent year easy classified finance need human feature supervise unsupervised view occur
time news influence account micro platform view message account follow novel message collection account major select abc france news multiple account retrieve account tweet focused tweet contain chemical intervention pick show hour snapshot source two account tweet relevant black line tweet short user content long period first tweet keyword tweet period tweet contain relevant black line tweet user diagram user tweet depend past past solid tweet box denote period star tweet establish ground news influence user tweet manually formal performance message arrival network user activity although continuously might twitter see figure without one delay minute period merge news source active binary tweet contain tweet contain send describe inactive period precede model past entropy logistic fit denote computed manner overfitte description penalty parametric entropy quantify conditioning within could tweet tweet keyword period example likely due fitting news user activity correspond negative remove criterion infer positive tp tn evaluate
identify cluster fourth lr sake completeness add column basis lr intermediate table group correspond trait describe ordinal multidimensional concern ask health scale develop compose equally item category minimum low loading aggregate suitable matrix accommodate item treat dim section optimal latent aim avoid employ comparison differ rely increase bic start est est est multi maximum number parameter bic lk bic lk bic basis adopt correspondence small fall repeat vary start possible high log estimate bic est concern good logit link logit index free difficulty completely multidimensional est multi est np lk bic logit second
vector proper get zero full proper joint spline write present simulation carlo nest approximation structure conditional straightforward conditional metropolis sampling technique adaptive sde unfortunately sde care next key hasting adaptive sde write possible conditional b c na b I order approximation approximation proposal conditional n sample package integrate laplace handle model compute estimate mcmc technique hierarchical observe likelihood marginal construct without
learned stage dictionary learn task task character among character induce since interested task tune validation task simulate overlap target value rr go code optical character miss employ capital character digit regard space interval level divide possible digit random pixel value image transfer vs vs atom dictionary learn tune show trace ridge worse sc apply pixel analyze
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characteristic apply person simultaneously pose part assign orientation detector body location identify pose detector precise many variation pose people model pose encourage diverse predict single person approximately pixel possible select frame people camera serious person annotate head location reference orientation angle value part pose dpp instead build factorize treat pose root head arm body part singleton pairwise quality derive model part tree expect pose control hyperparameter hold quality score train image every location accord encourage example arm near detail overlap pose diversity location reference space evenly ignore orientation reference pose reference align width pose hold compare baseline pose normalize pose choose pose incorporate select pose maximum encouraging diversity pose normalize independent pose overlap pose pose overlap cover pose pose max long pose achieve might perform poorly people overlap stand generate spatially diverse strong visual datum randomly test training score well baseline irrelevant baseline pose total precision measure predict part expert head leave correspondingly fraction predict encourage diversity expense score mean metric tight radius acceptance perhaps resolution diversity obtain significantly baseline curve tend model improve pose give acceptance radius part match interval recall circle illustration cloud color probability marginal pose main loop synthetic entirely pose similar marginal pose successive left extreme combinatorial present dual normalization marginalization even memory limit mainly reason move become big natural analog vector occurrence dimensionality propagation reduce diversity cardinality dpp dpp cardinality denote structure note sampling mainly eigenvalue require eq structured must label part empirically toy diverse diversity optimize plan problem structure city factor singleton stop base google city popular consecutive stop preferred order path assignment arrival assign diversity feature singleton city result appear path country path tend along east increase emphasize show draw leave correspond city distance project sample dimension relatively projection approximation dramatically projection theoretically empirically justify projection black variational project complete graph diverse salient depend citation paper thus period article experiment possibility discover trend social medium image video overview collection mean insight object instance people image noisy manual tool query relationship continue automate key efficiently effectively weight edge diverse salient tracking program lead like link detection text collection however address probabilistic tractable work tracking division slice generative available evolve sort document distinction compare retrieval extract document collection find cluster entity take input stream article topic seek extract sentence occur cluster document contrast extract entity sentence focus propose news set intersect map event content specify query likewise build individually assume collection perhaps input make document collection diverse cover aspect length setting finally efficient memory projection allow simultaneously goal direct graph node edge define weight control dynamic likewise feature case might convenient feature node application context diverse diversity independently identically dual dpp sampling citation computer paper news comprise large approximately paper citation node remove
suggest report bic polynomial polynomial degree bold highlight highlight accord bic estimate round scatter fit component c mm brevity sake consideration reference mixture place collection nine area order quality life place health care united seem due prominent overlap group aspect evaluate clear group regression ht std nominal suppose forget directly consider display bic correspondence good consideration summarize
datum regard tolerance treat tolerance x positivity failure respect calculate impose scenario inner evolution solver termination condition inner loop parameter set constraint loop constraint sum inner loop terminate cx produce constraint impose solution object cx maximize failure exercise protocol numerically replicate denote optimizer outer quickly search instead force search value optimizer report zero nn iteration equal tolerance subsection report implement california institute technology small impact brief form smoothness ball angle impact impact pass fully say topology opposite situation merely refer shorthand complete common use impact complicate experimental scope find relevant mathematic select scalar post impact cross area measure optical convention mean distinct triplet response protocol tolerance observe independent failure fall area sub agreement grey criterion plot markov bind feature agreement markov maximizer
manuscript training svms linear svms radial try map solve possible must solve conduct robustness heuristic parameter kernel inferior polynomial parameter linear rigorous alternative classifying predict simply essentially distance vector margin equal probability age old patient maximum classify negative sophisticated smoothing polynomial contiguous point smoothed consequently directly polynomial smoothed point unnecessary result average filter tend information mean derivative examining assume aid discrimination example fluctuation something nature employ throughout three ignore apart use replicate replicate investigate construction detect disease protein aggregation regular longitudinal construct evidence aggregation exist solely threshold aggregation decide upon optimally address svm besides issue stop process possibility detect incorporation additional patient disease duration construction disease machine rt current test detection marker reduce specificity associate term induced exploit brain patient induce shape aggregate reaction aggregated emission rt create longitudinal interpret curve occurrence national
avoid membership apply child reach leaf leaf reliable vs score assign report sample predict tr fr fr tr tr propose drawback label problem sample tr fr boost widely categorization evaluate big flat
solve mle run polytope algorithm maxima run objective ordering occurrence experiment begin work problem problem whether non negative among mle mle boundary none lie lie model description determine polynomial irreducible component closure component intersect ml far algebraic algebraic could ask exact description seek boolean inequality find resolve range van illustrate nest factorization q irreducible component algebraic boundary ideal height determinant define study look topology affine variety mle regard ambient simplex parameter product parameter parametrization identifiable semi algebraic dimension fact topology maximization method find exposition emphasize operate entirely
conclude image figure display residual reconstruct comparison basis basis comparison image construct patch image rather eigenvector patch eigenvector clearly explain eigenvector patch white every patch image patch lastly patch clean image pixel denote similarly nm coordinate real eq patch coordinate nonlinear thresholding coefficient denoise clean patch patch pixel ball radius overlap center patch correspond pixel word patch exponential weight patch center neighboring patch variance connection translation invariant original shift reconstruction translation denoise eq method denoise problem process define patch wrong explain heat kernel solution diffusion notice expansion reconstruct base similarity expansion understand encode amplitude exponentially fast geometry
pd symmetric vector submatrix satisfy collection conditional conditional fact factorization statement either implicitly joint variable capability mixed random variable ideal undirected graphical ideal method implement
angle correspond around big small reasoning apply pick lie plane angle function metric ranking angular angular equivalent distance preserve grateful
correlation subject multidimensional scaling generate embedding common canonical analysis optimize regard fidelity multidimensional scale fidelity canonical enforce canonical correlation formulate generalize space calculate subject different respective canonical space embedding lack datum lie investigate relation datum experiment generalization around million article pointing explain language article
reference come disease axis co disease duration represent axis subject line word line time parallel triple duration respectively disease life change henceforth illustrate subject show birth disease life disease life
model cell map feature lp pool activate lp pooling method pool c traffic sign house classification multi stage ms branching figure
indicate min inferior almost result lack present max performance affect average proportion effective covariate fix size parameter set value roc true less joint separate method benefit penalization former complicate compare max two comparable sensitive note point represent average sensitivity specificity replication select maximize conditional log dimension proportion covariate strength non vector position sign magnitude separate level eventually almost sensitivity specificity point separate extra uninformative covariate size
sum involve positive exist existence eq argument strictly convention minimizer exist unique since equal constraint attain proof detail regard present ridge equivalence integrable rkh continuous operator derive multiplier banach space suitable lagrange multiplier existence lagrangian context lagrangian fr lagrange multiplier n use f hx hx problem become
subset subset usually superior however well fitting likely superior include suppose sure well regression estimate develop desirable use improve least scad fan li adaptive mcp zhang ridge subset sure screening guarantee fortunately appeal monotonicity fitting improve circle contour square leave orthogonal solution right example arbitrary fix mx difference proposition orthogonal penalize square cx ci iteratively
important finance decision central financial market simple value period available variable distribution cdf monotone specification cdf iid specification lag l pl lead dynamic probit root represented lag extension reaction correction var discrete unobserved typical forecasting resource recursive typically behave existence consistency estimate estimate inconsistent diagnostic goodness
krige express covariance correct krige orthogonal interpretation calculate conditional knowing note prove eqs plug eq difference
discuss contraction moment interest quasi link identity canonical response canonical link lead acknowledge robust sense influence canonical convex binary response function quasi hazard decrease hazard quasi loss clearly constant compatibility constant normalize design denote column canonical correlation span combination combination geometric compatibility never eigenvalue compatibility eigenvalue
political facebook approach ignore known user gender compute consider alternatively construct capture user proportion live relational though decrease accuracy even political link derive relational compute user view label initially typically refine infer cc cc inference many gibbs propagation ica survey concrete facebook special emphasis cc transformation task set relationship classification anomalous resolution figure relational find powerful elegant link decompose interpretation former predict interpretation part construct node link task task primary graph illustration summarize organize large link new intuitively facebook user value feature share political link people increase improve accuracy collective similarity neighbor infinite random walk second type graph facebook link thus figure result weight strong identify probable link rather weight link link may assign kind discrete label link related event link label influence feature add kind link count occur number link labeling perhaps feature construction section technique link special emphasis link model user friend technique section additional link figure discover discover facebook might friend people relevant characteristic node associate link could way though away short length original new link propagate algorithm cc apply identification group apply separately finally link influential weight assign friend eigenvector g represent graph give estimate estimate relational via representation inference arbitrary value add instance facebook friend friend count friend would autocorrelation feature might friend conservative political user friend feature political identifying feature essential performance prominent prediction interpretation turn ij refer vector node contain summarize notation relational transform link label node remove seek instance objective completely specify improve transformation four prediction present link link task link partially transformation task simultaneously create produce new node feature represent label introduce notion relational node construct link without sometimes article intrinsic aid reader summarize often focus interested create modify motivated way incomplete interested goal discover represent predict whose common subsequent learn seek evolve connection might interested object spatially pair individual successful summarize prediction summary predict weight great original link step link step link often yield uniform show compute link section describe non topology base technique exploit link predictor exploit relational define create exceed pair high similarity object domain cosine similarity represent node use cosine link show link link compare leverage
optimality condition inactive bad optimality problem within comprehensive description assume g essentially identical viewpoint smooth suggest assessment iteration current assume set current complete zero hoc fail otherwise optimal follow compute algorithm option computation optimality gap duality follow gap figure elastic net short penalization duality gap bound rather fairly coarse unless solution reach gap assess rough good computationally tight optimality gap provided guess optimal step current complete complement minimize magnitude finally low solve duality
sf bind expression rhs rr r sf last j inequality chernoff hoeffding chernoff hoeffding fact inequality use combine sf e j f combine lemma dr dr I dr define history play martingale define first never happen martingale notation us time play l bind
discuss detect fouri paradigm correspondingly empirical cdf representation think work given find unit execute outlier remain portion incorporate unity create exponential family correspond smooth fourier spline overall proper pt circle general edge furthermore discrete density feature device spline try regard see parametric circular spline empirical preserve metric borel topological empirical histogram haar number assume exploratory descriptor absolutely major derivative haar topological etc pose look mapping topological exist haar situation integral integral indirect fashion major reason tool fast transformation fourier transform let function support finitely definite induce turn space shift representative unitary
pi begin generic namely independent algorithm specify although useful purpose mind analyse provide useful later vary admit target interest investigate desire multi level smc mix strong often kind simplify state generate proof find generalised always hold show time get estimator short variance hand long long proposition stop summary tradeoff convenient trying level particle tradeoff serve flexible vary use smc within multi smc within density density consider stop process sample k accept accept ki pi p n addition proposal irreducible modify notation algorithm pi generic form enhance add update backward
replace matrix process call employ cause generic solve load loading vector repeat describe formulation step burden produce load monotonically use fully albeit use purpose besides conceptual formulation theoretical certain iterate load depend state giving framework previously literature provide order variance algebra involve open implement parallelization strategy formulation architecture ii serial computer core serial benchmark measure speedup core gpu codes section parallelism purpose compute
middle summary summarie unchanged rating overall randomize amazon hypothesis topic summary currently summary estimate superior lda estimate interpretable plausible stable approach modeling explore obtain quantify regularization within poorly rare word usage topic expectation summary dominate regardless content across topic panel rate lda panel panel refer fit topic show assign occurrence score dominate order logit panels lda assign total occurrence contrast frequent pattern across result big work quantify well within fashion dimension content word alone stop space quantity estimate base occurrence bias difference strategy lda normalization usage dirichlet topic regularize word usage analysis interpretability output stop word presence summary versus bottom arguably one propose stop word topic become negligible word design induce sensible prior vary contextual corpus stop topic emphasis stop lead appearance issue propose stop post amazon coherent proxy evidence support summary interpretable
give equation comes obtain small sum arm obtain state relax mab number show repeatedly arm fractional bind recall mab problem indicate small upper summing apply lemma fractional highlight previous notation previously introduce variable fractional time arm within fractional note show equation equation equation produce behaviour budget mab within particular arm cost mab accord limited mab mab limit fractional within mab domain fractional practice counterpart cl cl cl cl
arbitrary part sequential tree location new location grow enable pool reduce size discard crucially analytic enable large retain leaf yield discard preserve property discard subsequent stay arrive demand would available new tree local nature subtree nearby loss tree affect single leaf take subscript l leave enter usual leaf fine way wish consider recursive equation like represent stand inverse assume xy keep memory statistic crucially dimension additional multinomial discard may indicator vector count obvious manner sensible detail unfold preserve distribution non posterior unchanged never discard update demand loss argue limited
strongly correlate redundant rate preferable f l l l spam lasso c c l l r spam c l l sub ar table ar lasso method dataset e p select red select redundant propose width must choose carry choose propose comparison lasso delta clearly outperform input genome since value
stable point show requirement translate show highlight fact hypothesis allow generalization bound hypothesis space capture double useful choose probability guarantee get application union yet q note confidence tell choose yield proceed via predictor lipschitz first variable satisfie invoke ss n respect value es gs n fact lipschitz function f
variable express denote two observable define likelihood latent marginal prior include express reveal estimation plug measure distribution kullback construction bayes form error learning estimation difficult parameter good performance successfully word assignment situation posterior distribution bayes go zero theorem decrease variable
predictor spline express term basis plus panel consequently lie near span spectra pure near informative cosine problem methodology encourage use decomposition penalty strongly various many two step example component regression project predictor basis modification spline towards functional extend longitudinal publish longitudinal functional predictor proceed truncate represent spline longitudinal mixed model incorporate random predictor decompose visit accordingly longitudinal principal longitudinal outcome coefficient operator inform estimate unbiased predictor precision longitudinal penalize bias obtain weak compare propose evaluate effect explore confidence band probability evaluate partial real section implement observe
te tc tu tu u together tm tm tm ty tc tc tc tc tc predictor vector predict learner e regret successfully contextual bandit payoff learner predictor dimensional rank accord consider contextual problem underlie predictor good aim contextual armed bandit e thompson ts heuristic around old numerous reinforcement family matching algorithm ts contextual follow consist reward pp function ts simple way achieve play arm design generalization thompson fit use mab irrespective derive thus ts ucb interpret
obtained unfortunately preprocesse difference neither word advantage induce word gram try hidden word hide activation probability last sentence hide activation validation superior embed cm crf meaningful primarily syntactic representation short window enforce local semantic syntactic group word maintain small distance week digit year year visualization describe design word representation sentiment probabilistic document semi supervise sentiment capture mostly semantic word encode combine feature
principal cart split biased categorical allow split ordinal split variable unique ordinal multivariate cart include guide avoid select set approach make guide fit article guide longitudinal variable briefly guide univariate longitudinal fix concrete section compare prediction deal occur predictor analyze extend longitudinal school compare accuracy idea two longitudinal stress child conclude remark univariate response fit idea sign fit variable residual versus exhibit sign depend piecewise residual cluster end center obvious contingency chi test group indicate dash form form table count chi value small chi split serve split example test lack least exhaustive search reduction side practically important search well recursively large
propose rank correct low completion overcome nuclear confirm improvement correction presence play nuclear least foundation structured correction sampling great extend also interesting acknowledgement efficiently density function say elsewhere eq n fu operator eq otherwise reduce detail spectral operator reader thesis constrain continuously lk give feasible constraint tangent linearity lin equivalently l constraint difficult verify problem b previous condition reduce accord condition reduce characterization optimality decomposition orthogonal projection directional derivative nuclear like find rademacher bernoulli probability less estimate event hoeffde far imply proof theorem let notice imply direct calculation yield b ms inequality choose constant arrive intermediate result derive estimation deviation sum random norm version control operator norm characterize tail matrix extension find exponential z f exponential meanwhile direct calculation q calculation remain
throughout frequently relation every existence vector hence trivially obtain case hence q may distinguished possibly atomic agreement reliability model either q part prove complete determine carefully meet reliability estimate precede even hold put precede value could spurious would modification long precede add spurious interval bad distribution category
square motivated achieve property presentation change unify arbitrary parametric another parametric high sparse additive segment bin acknowledgement gm partially support grant fa collect well text reader use vector denote index support set extension notational denote denote ij model input exist literature ignore
row illustrate movement bottom previous connect tp green distinguished node movement static support node movement reflect static table due lack centroid temporal cost compare although static temporal require static four time present significant addition base support website cost notice table mean centroid cost evolutionary spectral achieve temporal static encourage mit reality mining social mit access equip phone medium access nearby device five proximity construct week mit event participant income student university business rest due number display clutter encourage reader website idea network first week class group incoming student separate work building incoming student proximity remain expect student affect separation group clear subsequent separation switch see color correspond create discuss notice group separate group another group time separate participant evident
add latent belong add variational single rx horizontal expansion wish break conditionally variational due applied scenario arise output believe example event version form hierarchy recall come set automatic relevance ard describe extend present output gp assign ard vector index hope encode assign relevance idea level obtain network obvious ard neuron learn ht general architecture cascade depict simplification demonstrate mapping leave intermediate include leave graphical variant variational covariance prior couple
aware machine tu group institute mathematics fu institute tu factorization component demonstrate real world potentially gold avoid spurious canonical cp multidimensional singular decomposition science many name discover discover
report experimental verify demand recent problem demand fact fast availability raw speed efficient problem smooth determine respectively predictor function fit sense cost decomposable sa bfgs respectively fast currently available incorporate area curve rating speak perfect sequence datum batch acquire could day alternatively large batch sequentially notation subscript write th practice know effective algorithm suffer mention regard variety proceed scale understand
select criterion filter optimum subset search good subset black embed use learner recursive elimination framework score g perform node single tree feature selection however limited ensemble tree accurate extract ensemble recently ensemble feature select forest redundant concept iteration use embed framework building acceptable desirable selection method reduce type
ergodicity impossible contour geometrically ergodic decay exponentially regular contour stronger complicated require theorem whose mutually exclusive behaviour tail number central limit coincide application variance bound geometrically ergodic hasting exhibit behaviour inference large like argument proof although loose geometric chain geometrically ergodic sensitive change understand qualitative result prefer variety addition justify reversible desire lee centre thank research grateful helpful proof markov bound reversible markov kernel early variance invariant theorem minor state space omit suffice
observation assume covariance design write effect specify overview resp equation term mixed simultaneously good normality pdf random ml q alternatively positive preferred numerical bad condition variance matrix inverse solution matrix unbiased blue uk k l u u g special conventional normally distribute sr
marginal I fw generalise iw effectively technique another eq sampling introduce dependency fortunately generalise latent use metropolis hasting hamiltonian monte carlo rejection use logarithm please number cluster advance turn update update update update give w mh denote accept move w jk jk k flexible obvious counterpart lead construct finite dimensional base item atomic existence rate atomic atom locate mass representation simply subsequent previously pick none partial bias measure exchangeable may exist specify section process measure gamma normalise show auxiliary list sampler derive correspond sampler extend general measure extension heterogeneous capture heterogeneity preference mixture degenerate item ever mixture component structured section mixture college show student interpretable preference conclude proposal work college degree handle application college office study
segmentation potentially conjunction denote element wise division derive pearson pearson testing histogram estimate histogram symmetric take bin virtue harmonic lie remove use harmonic original goodness statistic freedom regularize gamma cumulative cdf unlikely happen one analogy base harmonic although enjoy analogy since compare intuitively distribution empirically kernel degree work similarly degree refer paper form sum represent give approximation repeatedly plug side eq geometric straightforwardly see rather trick approximation full rate slow
powerful signal chance detect maximum window compare roc scale average roc curve scale window scale raw scan fdr control aforementione slice fmri slice color color color study inspection datum appear right slice leave central slice middle top slice slice shape seem false fdr false rate region previous paragraph many likely false positive region cluster spatial scan cluster paragraph spatial scan reveal shape fmri panel slice shape
decrease value way effective implementation slight original allowed additionally attribute importance present benchmark achieve forest internal assessment several derivation classification attribute equally disjoint relaxed bayes formula strict practically combination large object impossible simple assume independence
b b n position say random ii iii rescaled parameter density probability proof theorem proof step first step iv integral bound argument nm v restrict integral n n n n convexity inequality expression zero turn previous b b b dnn n n eq q ii satisfied mean eq last valid quasi n conclude theorem assertion reduce lemma iii c normality proof satisfied exist conclusion former sequence truncation argument concentration give w u w n w u first z use right go zero w w inequality see take sufficiently value n j dx ng j n ns
correlation among contrast separately pattern bayesian reason different latent relation type capture interaction latent factor object relation binary matrix consider latent factor observe enhanced hyperparameter gibbs space initialization stability result miss link quality clearly relational confirm relation prediction conduct impact variation improve overfitte result tradeoff model efficiency factorization dataset trend choose number object connect interaction within distinct relation france
coefficient matlab expand toeplitz intra reflect empirically calculate average ensure calculate denote fm real intra tend positive far constrain intra correlation average reconstruct fm rule due constructive people
explicitly page importance nothing change evolve al approximated stream outside node equation utilize transition probabilistic introduce walk popularity capture external popularity implicit human dynamic namely human rapidly change topic human large update receive considerable tensor
regularity sde assume typically stationary g unique unique identifiability sde unique minimum remark restrict process result concern framework derivative sde time let mh mh w h w h identity q write define account er h g jt eq obtain conditional imply obtain lemma distribute zero estimator state approximate reduce reduce restriction discretization comment end converge construct order estimator approximation reason formula high first moment variant multiplicative adequate weak convergence performance conventional linearization scheme bias ordinary differential equation equation various
correct p small report conclusion bind site breast cancer repeat validation split fold optimize selection repeat note extra training set separately receiver curve auc auc repeat fold ideal would choose histogram resemble si please represent histogram signature exist
continuity combine computational three subspace change sufficiently perform accomplished grow prune tree split merge branch bound adaptive nonparametric time branch increase high merging reduce make consideration residual penalty splitting algorithm split odd bi form figure data node represent parent means perform minor use prescribed residual partitioning leaf virtual child similar possible initialize root virtual easier virtual thereby recursive square expect small historical evolve control residual tolerance vary underlie control usually detect change varie track produce sequence allow merge splitting track increase track residual track residual close gaussian adapt generalized ratio change generalized procedure detect residual assume normal mean formulate point post quasi survey expect
agnostic construction bound frobenius apply agnostic theorem constrain seek r b cr define positive present algorithm select construct eqn compute leave considerably improve ratio deterministic svd first step run algebraic spectral rescale opt opt opt conclude need I multiply employ greedy column convenient matrix set vector u symmetric dominate index eq achieve search need
highly non parameter unlikely perfectly describe set mlp unit layer patch refer filter connect unit mlp represent patch datum mlp interest attempt input activation conversely cause activation layer unit output last hide generator mlp architecture purpose mlp detector generator detector detector generator bottom separately detector generator detector similar generator detector classify detector small pixel dot feature detector scale shift dictionary denoise explain patch htbp detector detector sort feature detector low noisy detector normalization display detector mean dc component auto train tie force learn detector feature generator intuition reasonable might generator allow mlp dictionary output coding suggest hide ask activation hide b activation mlp mlp evaluate berkeley activation center random mlp activation mlp either function mostly activation later measure neuron entropy distribution plot entropy bin repeat mlp entropy absolute reverse true unit absolute mlp high mlp absolute also behavior
monte general introduction smc shall smc smc sampler particle close iid seem close iid particles mention little exact posterior iteratively simulate particle parametric generate modal smc sampler idea mode simulate bold determinant transpose trace preliminary scale may facilitate assign read one work likelihood likelihood function difficulty next either likelihood include involve symmetric second determinant evaluate fouri integral third describe section regard first cholesky low triangle obtain compute quickly substitution cholesky operation completeness mention solve conjugate gradient involve unfortunately alternative approach evaluation satisfactory use approach importantly front decomposition scientific give magnitude take author introduction importance sample evaluate integral quadrature poorly numerical sect yet
intuitive rough bit suggest subsequent favorable non differentially learner subgradient generally score asymptotically gradient secondly estimation efficiency estimation subgradient centralized argument thus also apply derive first view conditional saddle theoretic optimization result upper application efficient mirror technical defer complete outline technique remainder precise notion devote information observe unique section constructive trading notation definition set assume give distribution point denote element q distribution support lastly symbol denote formal protocol information notion focus statistical formally access subgradient intuitively communication follow three vector subgradient variable ball cover variety online optimization approximation allow perturbation live radius regular kullback leibler view adversary controlling say publicly available provide mutual saddle conditional meaning saddle direction formalize notion collection regular q saddle calculation supremum infimum formulate notion differentially private setting define privacy infimum regular private take limited consist suitable set sensible choose minimize mutual definition specify receive belong guarantee mechanism study effect method sample assess term risk describe obtain uniformly gradient gradient excess risk distribution randomness perturbation gradient mask regular sub set denote loss belong random minimax loss giving
axis next imply entail eq j f easy kk jx explicit expression depend coefficient call adaptive lasso allow lar recently penalty huber criterion enjoy oracle pairwise tend group exist regression dominate combine ridge penalty elastic net en en ridge penalty propose version elastic net en property elastic oracle propose version huber criterion call huber relatively small error contribute grow hybrid shrink coefficient provide optimize objective nothing show ordinary huber account heavy tailed enjoy huber addition encourage spirit create penalize group avoid individually en
rr optimization thus applicable could could note proportional subtle rr constraint employ rr coefficient sufficiently small ridge close optimization employ rr algorithm interesting examine approach density detailed derivation conditional model model solve model employ lasso lasso employ implement employ software http www negative posterior follow likelihood employ eq eq conditional follow algorithm negative bayesian gamma normal normal consider general support
nonnegative learn matrix position observation clean allow sparse solid theoretical justification solution knowledge work datum additive without position rest organize section propose algorithm iterative optimization method justification method column represent nmf product minimize multiplicative objective
distance measurement sum population eq q pick decrease element entire empirical covariance close continuity learn subspace large th show norm canonical angle learn interpretable robot result model nominal detail robot control evolution govern equation generalize denote denote observation noise translation rotation robot nice expect observation dynamic easily derive motion robot pose nonlinear inaccurate optima hypothesis multiple make ideal common difficulty lack guarantee formulate range relate state several need weak motion hope track optima formulation learn desirable optima bias general expand dimensionality identification focus localization
number hyperspectral core similar step project complement extract strictly identify vertex use function maximize randomly reason solve point b enough extract isolate make rather algorithm recursively try volume hull volume induce extract approximated algorithm assume locate therefore noiseless ill condition confirm synthetic set allow highlight separable follow matrix randomly function svd tw hence might way set zero different position column remain one average geometrically hull column column outside contain dirichlet choose dirichlet generate boundary column hyperspectral mean construct separable four table describe total ht exp exp exp compute percentage curve figure generate experiment exp
none directly instance equation resort perform rigorous contrast inherent hardness motivate applicability section space branch extend approximate examine extend distribute kernel pair correspond product induce whenever reduce method metric efficient reduction strict subsequently desirable applicability near candidate every kernel previous object cosine candidate still query notion characterize hardness sp dp dc sp sp expansion effectively surface hyper sphere expensive scan linear runtime
sufficiently joint successful methodology length burn establish burn iterate retain autocorrelation within outli occurrence cut detect recall ga variability belief time setup time informative variability process
apply derivative pricing infer triplet neutral observable datum triplet completely distributional option quite differently find evidence calibration day section sd explicit estimator self section interval assess simulation apply discuss conclude sample variance defer basic evy stochastically stationary increment evy decomposition write brownian drift denote drift jump jump throughout absolutely lebesgue jump component variation therefore evy evy determine call triplet reduce one well dimensional characteristic depend pricing throughout neutral martingale exist strong finite activity self decomposable increase decrease additionally sd satisfied k
generic matrix mind inferential interest potential uncertainty evaluate measure approach miss maximization adopt ml em complete likelihood function particular expectation maximization carry equation approach report remain solving form derivative adapt conclude paragraph worth potential spatial discuss issue address estimation bias covariate part variability even sample covariate algorithm list namely provide likelihood asymptotically multivariate case normally distribute operator replace confidence problem estimate run pattern simulate enough
framework brevity setup iid poisson logistic couple power penalty spectrum nearly penalty shown save detect sparse occurrence along path predictor perfectly predict response therefore largely combination except variability second logistic convex penalty short time penalty separation early figure display select glm define negative false negative select positive number rough glm convexity tend lead appear significantly improve although admit predictor overall non convex penalty square error mse parameter estimate empirical comparable penalty selector mind result simulation vary numerous signal etc hope tool facilitate comparative study numerical package material run penalize glm generic combination convex mild path gps connection ode approach track smoothly avoid choose bayesian empirical quick besides extend
two treatment group result informally index assign fix treatment subject treatment outcome row treatment affect outcome help make asymptotic outcome treatment assignment model population thus treatment estimate effect ol coefficient ol regression paper analogy observational give analysis regression indicator chapter surface leaf consume weight quickly leave small simple random mean unbiased estimator expect motivate adjustment way leaf weight adjustment measure covariate measure population help estimator vector b ib connection adjustment remain mention despite agnostic much adjustment observational literature simple
classification generation set optimization analyse separable show margin lead good oppose order analysis control unbounde early adaboost sensitive choice condition goal manuscript minimize element risk convex empirical I risk perhaps available example relevant risk place theorem input statistical however briefly losse exponential sample suppose attain infimum subgradient descent employ cone descent adaboost iteration suffice suboptimal mostly descent lastly manuscript theory times b goal allow odd task bounding characteristic map countable perfect albeit I mb exist cm black position anchor west fill fill node north fill text simple appear example determine fail agree convex statement fairly additional regularity
tx difference arise moment maximum dependence key final next estimate belong modification apply bandit unlike chapter convention arm interval realization random similarly chapter rewrite denote strategy unimodal necessarily monotone decrease unimodal general unimodal choose return perturb bound repeatedly however exist belong irrespective regret incur ready version section chapter chapter operation financial engineering usa edu di di armed bandit unit allocate payoff slot armed american allocation multi armed must repeatedly bandit basic instance sequential fundamental trade exploitation sequential player must action past give high payoff clinical trial must decide treatment next create play domain service target benefit adapt service sequence request concrete domain page website website deal design font image payoff desire behavior ad pool ad bandit change time ad choose path send payoff path computer playing many move bandit version bandit huge tree game focus promising play survey cover many variant basic arm correspond forecaster select arm receive arm focus regret pseudo sequence plain regret fluctuation hope contrary control use formula let armed exploitation intrinsic seem explore actually see exploration exploitation principle uncertainty many sequential decision uncertain environment forecaster accumulate decide act plausible concentration inequality favorable identify base favorable plausible armed armed bandit weighting bandit function example attack armed uncertainty construct choose arm need bandit obtains call ucb assume reward hoeffding turn ucb first three indeed imply bind use obtain straightforward conclude denote leibl
even construct representation combine much design elaborate feature recently encode demonstrate much aforementioned provide broad theoretical offer explanation success result call feature encoding cross achieve popular benchmark however feature especially appeal produce encoding multiply triangle threshold slight idea modify term square take constrain triangle eq version threshold validation soft threshold similarity triangle feature computer vision approach encode assignment van version idea quantization rather hard assignment quantization scheme success threshold feature setting encode applying unsupervise triangle compression version level chart threshold detection recognition respectively
preserve integer stand span sample datum eigen denote eigenvector assume dimensional supervised reduction joint sample independently ij j main randomize rather input particularly reflect biology finance intrinsic appeal explicit control computational efficiency rapid convergence achieved investigate statistical highlight input focus practical sample population subspace information population algorithmic subspace generalization describe power iteration estimate span increase factorization problem fewer large deviation effect randomization argue computational combine implement computationally efficient shrinkage capture large estimation randomize study literature simplify result science discover approximation algorithm propose extend randomization capture scenario capture project basis set random bi validation project compute onto tu u
choice cutoff theorem give discussion proof useful tool limitation supplementary throughout notation integer random scalar distinguish constant ratio bound cardinality scalar generality mild condition surely borel also subset small assume write linear functional k l u interval natural quantile regression function formulate quantile function go generate admit quantile restriction location obey quantile obeys quantile restriction design suppose eq denote restriction estimation principal x dt consist expansion principal score
b u b piece formula corollary statistic north university nc li modern produce structure digital imaging flow frequently measurement underlie address scientific arise take matrix form crucial hinge coefficient signal approximated structure low lasso highly scalable algorithm develop facilitate selection demonstrate real nesterov nuclear modern produce unit include digital imaging contain color consist intensity cell channel motivate example eeg two normal subject stimulus channel subject scientific association pattern glm predictor include covariate gender classical glm deal pose na turn large eeg sample inherently e channel correlate dimensionality complex regularization covariate dimensionality preserve variety year
happen region combine example explain list pos contribute alone n contain redundant execution eventually replace small easy search tree depth depth backtrack partition object take never twice backtracking step backtrack computation involve split operation take complexity significantly relate indicate far backtrack equation bad hence adopt design algorithm seem heuristic may currently subset select e unfortunately happen frequently test alone region given generally subset less prove give reverse sensitive
group quantization know quantization representative I indicate distortion two need highlight quantization application consuming mean converge stop small discuss real seem big quantization could code quantization result denote center lsh try guide matrix adjacent group pick random natural pick adjacent median adjacent hyperplane easily associate plane previous projection generate projection usual projection select
bad forecaster share hoeffding jensen convex hoeffding substitution step sketch multiply examine difference inequality update third sum fact I recall convention conclude case forecaster hand hand extend generalized sharing technique way loss adaptive tuning paradigm choose learner expert classification special case online design difference cumulative element criterion
bundle maximize subject constraint bundle type linearly value concave explain agent behavior help predict future choice agent pair budget example function p b p every distribution example every set observation say complexity bound polynomial complexity learn algorithm learn mechanism bundle equal bundle agent select section learn predict require bundle exactly receive rich learn observation probability produce eq additive typically normalize lie function particular provide upper function low immediate start characterize linear utility intuition price respectively denote
chinese restaurant process crp customer chinese restaurant infinite number stick break resort constructive way construction stick break
pa rr r h e x x put together since henceforth pilot kernel pilot symmetric variate density square integrable come optimal pilot pn show everywhere asymptotic minimizer relative lemma analyze begin taylor use expand eq make expansion third section selector substituting yield lemma h h pn remark multivariate region contain derivative estimation receive excellent practical generalization derivative due bandwidth multivariate derivative achieve advance analytic tractable derivative behaviour explore application drive technique combine shift along argue exist bandwidth parameter exist derivative introduce validation method optimality plug root domain recently smoothed selector automatic small univariate surprising suffer equally literature bandwidth recent
p check x last finally conclude proximity require fortunately proximity subproblem q solve implie ball extremely well study hard fortunately solve root unlike proximity I generate approximate mixed proximity ultimately motivated author build upon q matrix formula lemma norm old matrix know invoke conjugate triangle old latter
also entrie nonnegative preserve normalization nonetheless class infinitely see infinitely concatenation analyze na need definite infinitely infinitely infinitely definite infinitely link space q moreover notice definite way suggest infinitely formalize semi exist contain hilbert metric infinitely notice normalization matrix theorem hilbert infinitely negative depict gram employ entropy path normalize alternative negative definite establish function quantity interpret functional far attention entropy gram certain operator act converge entropy population entropy bilinear fx fx x xx normalization eq orthonormal schmidt compact case extensively
sdca non hinge plot horizontal divide data sdca sgd work non smooth hinge sdca hinge small test error sdca sgd smoothed hinge axis correspond sdca gap ex descent solve svm theoretical guarantee dual ascent software package good paper present dual sdca sgd analysis sdca application associate regularize minimization let scalar tell number convergence iteration large specify examine proof small close close dependency occur deal sub optimality objective solution primal solution primal
approach candidate criterion costly remain refer extra choosing infer include square hyperparameter tb distribution estimate gp minima local minimizer multi modal minimizer
follow extension nonnegative sparse pca sparse conventional pc especially environmental science biology follow element recursive strategy separate series small respect need convenience rewrite virtue nonnegative model differ sparse evaluate matlab pc ghz cpu gb memory os list well synthetic evaluate recover truth sparse extraction utilize benchmark generated create observable intrinsic utilizing contain apply extract pc perform truth effectiveness algorithm adopt toy pc covariance apply schmidt randomly first two set
likelihood estimate kalman many specifically display phenomenon variance approach new forward accordance degeneracy ess stay ess drop proposal consistently forward run backward filter smc estimate superior show smc explore smc calculate utilize rely rejection
suggest direct may competitive penalization inference come validity relationship imply graphical rely assumption conditional course
freedom input pass sigma delta follow diag specify list r know column coordinate column indicate similar omit fit implement algorithm call eps scalar specify contain specifie start parameter specify iteration specify termination user argument expect hence attempt search well termination criterion control loop terminate criterion reach current log log default argument option likelihood display fit termination set detail argument
output draw white normal independent reconstruction accuracy select bottom bottom functional alone toy marginal coordinate label investigate enforce rescale determine size degree fix optimal validation positive irrelevant root error select versus value respect variation smoothness lead term smoothness h present set experiment vary unchanged account training space space kernel corrupt regression depend white sample rescale determine ratio fix value relevant leave unchanged ratio always hypothese selection versus decrease visualize plot minimum necessary clear visual inspection high want evaluate hypothesis unchanged set exception vary step signal noise display versus different hypothesis decrease hypothesis chance error number want evaluate effect well parameter theory unchanged polynomial fix relevant regression randomly combination degree involve display feature indeed present aim method model multiple
form exactly pairwise independent obtain upper moment inequality nr symmetry brevity term use bound show bottom divide numerator rhs obtain pe n u n z part large distribution distribution side compute note show derivative attain equal use pe imply interval arbitrarily long rate choose theorem encoding decode refined codebook tail
embed minimize introduce pairwise measure widely move row node give equip node eq sensible measure arrival site start match precision coordinate large scale eigenvalue eigenvector trivial include eigenvector eigenvalue dominate scale structural approximate aim estimate become maximally likely trace node specifically sum possible assignment free functional procedure maximization maximize maximize ideally gaussians easily lie ahead field field start addition introduce energy
generalize enable rooted tree dynamically multivariate generalize exist modeling flexibility conduct hasting build around overlap group technique em model model stock multivariate generalize induce period volatility market fluctuation stock build conservative portfolio recent scenario large irrelevant appropriately literature extremely popular seminal lasso zhang priors
respectively incremental dct corresponding coefficient coefficient correspond compact dct reconstruct representation follow last fig car define final likelihood criterion sigmoid ability model confidence compute score candidate bound visualization normalize location fig map obvious modal peak observation discriminative filter particle update observation tp frame state normalize state solve maximum evaluate video compose image video factor lead appearance illumination rotation motion pose etc sequence experiment main goal verify propose situation evaluate adaptive capability appearance change matlab intel core tracking normalize computational state particle module particle patch scale image pixel factor set buffer setting remain sequence achieve track state compete tracking decomposition online adaboost incremental l state utilize strategy slide inference code l utilize adaboost classifier select
sample controller increment measure use velocity start prior generate gp track horizon r gp gp gp gp gp filter color explain latent even outperform commonly use ratio robust unstable analytic gps se function implementation gp linearization density mapping nonlinear incoherent smoothing gps uncertainty
model term solve age specific publish calculation age burden dividing derive random stochastically problematic composite increment choose small accordance assume increment distribute
dependence compound kernel hilbert schmidt intuitively test cross dependence group separate cross exist group norm satisfy relation sufficient nuclear appendix theorem furthermore mc know nuclear norm topology treat quantify dependence obviously relation indistinguishable question still relation zero perturbation keep h apply correct incorrect eq switch incorrect correct relation compare hidden perturbation correspondingly strong close indistinguishable lemma correct relation
label label optimal partitioning may partition cut could tool optimize mixed combinatorial optimization poorly unary potential permutation true uninformative force exception case label graph efficient perfect dual statistical computing ise explore map mrfs partitioning provide partition cc partitioning label weight consider merge follow cost q twice small cost approach relate provide approximate likely segmentation poorly real image segment come colored subroutine difficult problem tractable
parameter finally discuss exposition writing sequel relevant quantity know support covariate exactly completely set well know closeness sub design define sub gaussian high partially solve noise miss estimate discuss covariate assume either know true assumption seem plausible instrumental whose row know rigorous asymptotic covariate independence operate design section iterative although currently analytical sub accord outline follow simple strong cauchy schwarz generic noise depend hence result subsequently estimator study finite bound simple corresponding instrumental variable economic performance indeed central paper able obtain
thank value ml mb define eq upper different successively cauchy schwarz every plug yield let xy py conclude proposition lemma use addition get absolute hold true imply q ps mx proposition union remain bind bernstein variance sup assumption eq absolute obtain separately constant absolute mx xu mx u hence deterministic appear follow every q nf mu mf remark follow u supplementary organize variance penalization detailed exact computation analyse fold criterion useful concentration supplement every proposition variance variance increment computation appear definition eq appear pm q oracle inequality hold correct fold penalization fold fold cross like suggest increase overall explain common
series h p soon representation trivial p h multiply limit desire follow observe consider reversible function positive measure satisfy whenever convergent hold lemma full proof give sake b metropolis hasting define check I f ap p pf qx qx exist x ap contradiction shall shorthand notation nx g zero variation p place acceptance remarkable fact see product metropolis hasting qx marginally implement marginal assumption may see allow interest aim algorithm equilibrium respective abstract pseudo illustrate one simple assume simplicity borel consist lebesgue consider density computed suggest approximate importance probability unbiased estimator pseudo constant seminal various application particle filter
boolean request facebook many score well request I rating often request statistically sound concept request demonstrate request user application facebook platform party application market resource average review description help phone hardware call history via restrict application application camera need application order study examine small map dimensionality visualize application analysis category relate finding request minimization low generative probabilistic request identify et correlation weak correlation average availability website application publish expand analysis far date focused technique identify apply static string et al build
smoother think right figure comparison include reconstruction depict mention believe induce change correlation figure case convolution noise results vary signal reconstruct original scheme matrix aforementione major come figure right see observation smooth worse change describe much rich specify move geometry identity observation follow complex reconstruction prominent reconstruction capture layer reconstruction reconstruction parameter well close model course add
follow derive assume therefore concentration event embed space moreover interpolation give supremum arbitrary satisfy satisfy universal substitute eqs probability moreover eqs far q side take assumption density choice probability less function estimator rate gaussian learning estimator rely technique develop involve stem expectation investigate ensure paper exactly need
bind jensen inequality help jensen difficult concave logarithm inequality conclusion complete tight maximize fix maximize guess maximize whole kl eventually current calculate later call devise
cloud future precisely cloud cloud entity show naive parallelization scheme propose provide parallelization vector order resource intuitive parallelization sequential therefore improve
learn perform depict learn thus indicate axis gray scale learn indicate
predictive order suppose give analytically web want predict consider dynamic stream thus dependency however gaussian relationship variable follow denote distribution analytically tractable condition
measurable p parameterize easy difference expand q jensen denote coordinate apply existence jk ensures hence write update sf c since recursion see ode eq quasi ode bound p easily satisfy theorem ode also g sf martingale adapt fx bound almost asymptotically stable ode fx fx prove convergent easily measurable martingale martingale jensen inequality jensen inequality fact argument convergence error term term n case summation odd show third claim follow ode see stationary lie follow show iteration track ode sequence obtain sf converge neighborhood stable immediately update
reconstruct provable introduction unlabeled piece leverage liu grouping merge piece introduce estimate rv nu ig represent recursive latent rv empirical I along ni constructed initialize take union span latent iteratively neighborhood tree run employ latent propose divide latent building fidelity optimize bayesian information search iteration limit quick implement correctness flexibility cycle contain many cycle tune latent page leverage cycle cycle node two cycle sufficiently neighborhood depict denote close span local neighborhood observe surrogate relationship child merge select merge discover latent distance select nearby compute provide asymptotic ise succeed potential ise bound potential depend edge ise potential potential consider counterpart potential expectation class
methodology system parameter local mode subject go population measurement straight forward example effect mcmc something department centre college bt bayesian computationally demand applicability application markov jump valid model convergence rate allow fast mcmc chemical formal physical applicability model chemical interact describe population network terminology notation chemical methodology model reaction interact ordinary differential fluctuation analytic form simplify hasting perturbation use local refer unknown paper nature metropolis hasting strong behaviour auto tuning good mixing adjust langevin mala hamiltonian hmc also metropolis hasting ess rate several however mala mechanism see chemical
may scale span use theoretic l matrix order decrease expand expansion basis leading mode analogous q completeness x correlation function kernel type significantly capability rare event remain step need give word theoretic generally manifold application establish overfitte address behavior shape curve context frobenius evaluate via lead considerable choice monitoring change propose via spectral measure measure energy usual order set energy spectrum energy information theory show mode
relate probabilistic estimation devote determination conclude remark auto property function q orthogonality verify property extend pp give quite focus pg gr suffice notice give semi auto highly set center diagonal semi auto model arbitrary dd
replace low summation efficiently use variational calculate expect hmm hmms sequence expect transition state state hmm maximize appendix update tuple include covariance base derive average base derive h cluster hmms relate hmms hmm center compactly center hmms leverage cluster positively break small set learn portion standard intermediate value directly direct time store second evaluate short implementation effectively parallel require allow model updating assume model requires learning follow intermediate computationally intensive considerable computational hierarchical implicitly virtual compatible generalize lastly virtual intermediate make virtual relatively positively run achieve hmms product hmms hmms virtual distribution hmms affinity compute hmms produce novel center suboptimal hmm example implement spectral mean point embed fail capture novel center especially high hierarchy leverage one obtain hmm cluster hmm center limited hmms I optimally center cluster involve hybrid learn hmm obtain h summarize hmms sc cluster hmm h accurate learn likelihood second drawback matrix hmms see music distribution embed costly exact carry I py alternative sum probability py dy affinity approximate distribution real respectively work spectral similarity show superior hmms
adaptive mdp dynamic give observe agent act optimally belief statistic lead exploitation uncertain unfortunately mdp dynamic action action readily derive execute real constitute agent respect prior occur planning find decision equation bayes adaptive monte planning forward tailored monte carlo search effort branch state base estimate clarity adaptive ba apply base ba simple search necessary simulation base tree ensure root originally partially observable search current compose represent
question answer general point short operate regime limit function reveal way answer density special unweighted short unweighted limit geometry implication short path unweighted accept pass high illustration opposite achieve application unweighted lead huge estimate manifold see illustration area compute unlabeled crucial exploit region distance
p pf pp p derive sensitivity subspace fit shape shape total subspace fit fitting note result tuple instance coordinate total sensitivity function observe project technical shape shape subset distance shape set abuse denote fit shape shape focus fitting seek integer shape tuple affine underlie euclidean projective cluster square tuple projective shape projective specific projective polynomially
simply treat player belief field kind towards point start clear mean offline equilibrium payoff map good equilibrium field speedup acceleration present observe time start point point fix speedup htb speedup summarize acceleration base sequence acceleration small start repeat acceleration seem great acceleration speedup satisfactory remarkable result challenge interactive system aim payoff satisfactory interactive user good good type seek solution nash criterion satisfactory offer alternative closely user make decision strategy make meet response strategy specifically option satisfactory action space methodology basic see solution player satisfactory pose existence satisfactory feasibility vector action necessary satisfactory na w w jj jj jj w jj n player necessarily target user satisfactory satisfied situation examine course
class verification may function imply filter sequence separable hilbert lag terminology frequency filter course converge technical sequence possess satisfy analogue theorems proposition establish previous immediate proof filter hx ty dimensional require belong process possess spectral operator consequently readily verify density minimize minimize employ infer operator consequently proof turn convergence integral tend theorem fix subsequent notational dependence limit assumption l jensen triangular lebesgue continuity b imply suppose exist clearly side v contradiction remain let turn first lag consistency result define assume proof tend lemma term jointly consider space x product operator possess yx x hilbert schmidt orthonormal hilbert schmidt show orthonormal hilbert schmidt compact class hilbert schmidt operator inner product h hx h define trace hilbert schmidt corollary remark
distribution mean necessarily interaction whose agent utility policy let oracle policy stage payoff stage policy consider total utility stage reward oracle reward special however payoff mapping payoff stage game stage payoff agent stage complete policy terminate end terminal termination payoff satisfy agent goal termination horizon discount reward reinforcement opponent select payoff maintain
section defer theorem rely theorem real target column construct svd projection spectral frobenius zero residual nc frobenius construct compute adaptive sampling base idea proportion arbitrary algorithm residual decrease theorem adaptive nr mr nr expectation randomize solve present follow selection algorithm near optimal
conduct topology large centralized localize community link although algorithm suffer requirement like algorithm practice use neighbor discovery recent research team community detection burden output detect community briefly discuss knowledge centrality heuristic view preserve locality location community community network lot attention detection detection community differ structural usually stronger sensitive tp fp detect community share exclude correspond person follow twitter friend undirected relationship undirected social vertex shorthand denote iteratively observe notation local call observer information depend observer facilitate discussion edge illustration observer node direct friend observer level node edge link visible observer thin observer within
cart tree cart cart learn classifier space h train cpu empirical il restrict test cost test learner extraction use weak use free auxiliary let formulate cost norm scalar assign weak cost extract learner difficult non non mixed sum recover commonly encourage plug cost relaxation norm sensitive single linear expand balance path show tree derive
mapping factor iff denote set exist semantic simplify semantic substitute equip edge model circle black scale
indicator accord positive entry remove separately reason remove remove statistic datum preprocesse attribute affinity use randomly unconstrained fill correct relation label quality rand guarantee compare algorithm unconstraine sl affinity directly supervise perform new try encode grouping projection trick sl report propose sl constraint encode laplacian use uci dataset largely sl adjust rand range quality mean maximum ari fig report ratio constrain observation algorithm utilize spectral hand amount consistently outperform margin ari able satisfy constraint algorithm significantly random fig quickly consistently practice fig ef suffer free unconstrained introduce small modify sl identify cut easily identify minimize outperform sl instance document write translate list document
calculate base algorithm treat calculate newly duration case locate life point direction duration might diagram person diagram relevant simultaneously depend age subject duration disease
difficult focus carefully neighbor confirm well fulfil robustness adapt triplet generalize metric xu come increase instance iid denote example lf lf loss lf generalize generalize almost imply notion weak pair sequence example learn almost recall sample belong notion average testing sense weakly fix weakly training example robust testing obtain
relatively fail boost predictor one fdr value argue opinion tool gene matter gene boost gene gene gene boost gene gene gene gene fdr fdr fdr gene gene fdr gene gene gene gene pls gene accuracy pls gene readily see false discovery interact gene expect cancer fdr pls gene identify non essential size start picture method fdr pls pool consistently gene increase sample ten gene pick gene boost gene boost know false identify gene false boost seem boost weak show majority exception rank rank ten ten gene determine gene chart allow gene cut ht model polynomial show describe taylor boost couple method success disease x show fdr pls three disease ht equal fdr boost accuracy x accuracy boost x x x gene pls x accuracy gene pls fdr continue relevant classify interaction cancer certain
posterior turn regard inferential analytic sde analytic available value discretization euler high taylor comprehensive observational fine stepsize value jj tt nj abc computational make turn previously significantly simulate sde soon trivial verify switching calculation propose proposal density simulate sde acceleration speedup kernel return sim relative produce considerable computational acceleration follow uniform denominator probability practice acceptance nature check verify simulate sde always regardless value fail accept proposal reject code initialization choose simulate sim generate sim sim sim sim sim sim sim sim increment modification considerable benefit particularly algorithm moderately accept obtain simulate sde benefit idea early rejection knowledge early generate proposal
dyadic discrete similar research et al base alternative approach relational apply problem essentially yu al variate gaussian generalize interaction relational incorporate capture globally property discover structure show benefit base optimization transfer algorithm complexity dataset evaluate relational still automatically factor cluster would marginalization prove form auxiliary easy il trivial form observe china fr relational application recommender bioinformatic characteristic thus source relational many world ask efficiently
corpus train lda hyperparameter place document proportion word symmetric symmetric improve lda compare widely hyperparameter except place look less design explicitly discover topic pattern capture word topic correctly identify non lda find quite identify topic topic interesting capture topic word topic lda actually adjust lda informative token ng change corpora
wide sde drive hilbert difficult common involve inherently term derivation advance hmc avoid strong assumption justification project justification analytically far beyond scope paper application verify mix sampler operator measure measure discuss sequel suggest coincide theoretical development construct definition hilbert space greatly extend develop carry sense separable formally state introduction argument think energy eq define hamiltonian dynamic hamiltonian preserve proof solution volume motivation direction velocity target space force end become show solution preserve hard carefully equation solve accept reject correct invariance standard work give rise volume preserve property look hamiltonian synthesis hmc table due mention easy acceptance invariant h hmc give calculate go ii nx standard sde give increasingly would vanish keep fix employ brownian path wrong case mala advanced hmc avoid degeneracy gaussian hamiltonian hamiltonian equation equation analytically analytically define eq small define define step proposal operator absolute law absolutely
decrease solve keep decrease iteration recover channel fista tv operator tv avoid list conduct mr multi channel validate conduct ghz intel cpu matlab matrix measurement mix white deviation evaluation reconstruct power original multi aid clinical diagnosis mr water water appear water suit imaging intensity weight mr correlate mr forest suppose fourier formulate conventional forest multi mr htbp extract domain frequency mr image mask compare fista fista fista fista demonstrate comparison among could model
zero multivariate gaussian n kn ni function j v work use cumulative control identical assumption hyperparameter characterize assumption expression posterior factorize incremental training approximation f nm ip ii site active basis note site site differ basis selection site optimization give initialize initialize basis selection site briefly add basis set incremental calculation carry
exceed paper powerful unconditional situation unconditional power underlie constant consistent conclusion preferable unconditional compare statistic note property simulate var unconditional variance test bivariate var generate write variance exist causality variance angular exist causality almost modify detect note positive definite instance check property lag length know autoregressive commonly ol light study table c c asymptotic nominal sample c nominal seem reasonably compare nevertheless outcome generalize recall size confirm almost nan hypothesis size
three cost examine dataset disease dataset consider minimum bm bp datasets bm tie l bm svm heart eight namely examine breast diagnostic breast spam cs exhibit tp auc tn bp svm breast diabetes spam dataset svm bp svm cs cancer diabetes imbalance ratio evaluate svm could exhibit tp tn auc compare tie experiment bm breast svm breast survival sensitive dataset past non mail performance table ad hoc consist subsequent algorithm rule prune sensitive ed bp algorithms ed suggest two sensitive box convert insensitive resampling accord kernel ed bp p ed bp black bb ci bb example also sensitive generalization cs svm cost cost result avoid superior dataset
subsampling object multiscale context large decision parameter multiscale across allow capture extra train representation edge abstraction pixel provide hierarchy observation level object capability assess automatically produce spatially naturally precision base extension neural network transform share weight justify combination region region enforce convolutional shift impose pixel wise label dense system complex typical pixel million naive convolutional view context multiscale convolutional extend scale multiscale pyramid construct multiscale pyramid laplacian pyramid process neighborhood zero convolutional multiscale per scale output scale normalize
unify os optimization os reject refine mode mode proposal dominate represent optimize find line refer far rejection usual mark stop far trial return accept second refinement history accept value trial history last provide detail decide accept mode accept enough maximum mode accept base trial line trial reject refinement hmm word observe noisy depend state gram language define x optimization huge need explicitly context
neighborhood lem I either technical require go topology auxiliary set nest assume subset requirement loss generality satisfy extra statement subset valid main ss measurable algebra decrease arbitrary claim validity efficient nested theorem since validity I nest xu valid combine previous display eq text automatically confirm discrete keep algebra text closure affect function continuous particularly light either paragraph predictive new measurable finally follow suppose iff iff iff put iff claim k j u interpretation measurable similar important ii random validity enough I shall produce optimal intuitively value want want typical mind present goal goal frequentist observe datum parameter uncertainty statistical unclear uncertainty come interpret level measure depend produce probabilistic unknown effort inference specification fisher inference variant inference calibrate recent focused incorporate frequentist
prove proposition least ensure prove force mathematical aware outside classical field acknowledgment point source solve difficulty challenge tend empty three algorithm solve anneal method combinatorial result backtracking solve solution however backtrack student mathematical science simulated anneal share common enjoy applicability projection feasibility problem convex programming optimization optimization combinatorial much backtracking annealing alternate np
example obtain send classical formulate counting nonzero encourage hard one pursuit ds constitutes match mp pursuit hybrid stage algorithm least hybrid accuracy high
proxy define note relaxed som positive kernel indeed condition impose nevertheless range string heat som act compare via walk day become several simple mixing description snapshot change adapt artificial strategy tackle stationarity risk structure dissimilarity flexibility artificial paradigm artificial originally traditional set difficulty evolve minimize quasi newton leverage
estimate give rise case bayesian without use cell single point bar one target responsible high expect contribution amount come decrease perform low sample negligible formulate posterior mean weighted posterior estimate concrete gaussian generality behave generally hamiltonian
strategy train predict take vote classifier receive party baseline favorable point margin support continue reach separable communication equal size run round ref stop reach underlie report communication cost accuracy applicable communication report number average run standard report send another incur word word describe assuming send classifier incur cost base find within small one value dataset test incur assume round send node reach cost round first account player observe small round
expect ei ei point max algorithm batch ei dynamically condition select explain algorithm suppose algorithm query start gradually ei pick second unobserved formulate q q ei close light sample close pick sequential ei line second know outcome whereas select ei method know parameter around curvature zero due closeness corollary square mean estimation show optimize summarize hybrid optimization batch
subsequent r phase eq therefore conclude fix pick unit particular follow turn later bind express coordinate depict I ii first inequality term first term various bound therefore thm orthonormal eq note suppose call subsequent call eigenvalue return permutation correspond permutation actually base case ignore assertion observe therefore assertion fix assume inductive least permutation inductive precede least call therefore quantity must triangle side separate another first assertion inductive inductive j assertion inductive thus induction principle exist hold complete variant stop key difference explicitly iteration converge within high condition tensor sphere iteration update eigenvector denote frobenius order ai ai symmetric stop eigenvector show stop condition condition hold I eq claim claim I k intuitive mean rise population approach prohibitive complementary take newton start often asymptotically latent evident well match observe intractable system multivariate fortunately class rich structure moment amenable descent include certain
general differ limited acceptance event characteristic integration variable experimental usually integrate symmetric although problem solve appropriate information incorporate act adjust smoothness unfold pdfs unity convenience indicator often commonly b spline eq substituting fluctuation account histogram observe experimentally value covariance bin column response
begin state measure limitation expansion wide parameterize parametrized parameter expansion taylor etc intuition statistical cast dynamical state special observable system output formalize follow make use necessity apply system light filter measure filter complex available approach polynomial spline smoothing filter implementation
index furthermore proposition issue see consideration develop poisson mass q function unity introduce variable poisson sampling use connection rewrite generic inversion assertion side sense one side use decrease integer set nest I I notation assertion alternate calculation show c summarize side via give belief plausibility I plausibility fisher true describe I frequentist observe plausibility show type ignore randomization issue correspond singleton assertion like tie
since core benefit cardinality suffer additional would boolean clause cm ti add pseudo boolean concluding equation describe result clause clause solver previously encode frequent constraint enumeration find frequent note encode exponential growth enumeration strategy opt adopt expressive enumeration prune part transaction approach rely heavily frequency threshold center frequent enumeration compact anti direction rely enumeration strategy solver output finer relevant valid strategy concept frequent frequent maximal either frequent follow upper option example may redundant transaction add must exclude base subset frequent frequent largely reduce impractical growth clause clause significantly high clause encode clause previously q clause directly find dual problem medium threshold fashion item frequent choice adopt
expand applicability general international dedicated research learn output space object difficulty encounter structured usual case reproduce elegant overcome kernel structural information relate kernel kde kernel kde reformulate et project structure rkh scalar regression input feature structure output pre kde beginning reduce component base measure kde kernels space approach permit exhaustive pre encountered kde kde kde minimizing capture closeness output
corollary proposition markov hasting mh inference challenge rejection mh scheme density proposal reject crucial drawback adaptive never proposal happen lead never converge overcome limitation adaptive target distribution strategy simplify sequence proposal algorithm outperform mcmc hasting mh rejection metropolis arm bayesian implementation smc know filter active area carlo powerful tool numerical stochastic simulation rejection mh single mode despite pdf problem adaptive never add region even procedure construct proposal g region density depend good imply indeed adaptation structural solve support mh increase markov partially discuss arm arm bound approach incorporate chain direct modification inside region mh learn past except current furthermore construction pdfs reduce effort require almost everywhere guarantee improve whole point introduce
spectrum conventional curve event estimate trade true discovery significantly hand speed essentially unchanged comparison experiment duration unchanged evident show present extend event spread probability repeat spread time recall represent arrival single scan expect number arrive bin single scan henceforth assume arrive give cause observation span generalize observe event cause one span poisson variable impact observe distinct event event drop impact event term vanish subtle impact weight observation observation weight include probability event exist identify event one form log weight event calculate set weight union time zero weight write event
core statistic us compute cn explain argument bound follow large uniformly hold prove proceed argument lemma n rough denominator since probability lemma fisher compare bind appendix lower desire speak several complex see operator et calculus orient circle center radius open domain whose calculus operator formalism prove contour complex plane lemma sequence define upper bind applying obtain refer counterpart calculus look definition contour change event lie circle circle almost work come nj obtain turn sketch eq decomposition eq proof get last us acknowledgement research partly b like
binary code output way replace definition sub policy binary learn complexity carlo training example state label training provide sub set combined final policy alg estimate ht kt algorithm identical slight distinction keep usual multiclass line map onto binary learn replace original action correspond code global define accord note ensure stability
forecast horizon ls stage forecast forecast approach good forecast spurious coefficient var contain spurious presence forecast less reliable column l l l ss l ss test week week forecast horizon air application concentration four air year california air observation previously component different var correlation domain ar partial correlation air specify range exclude pair stage coincide result bic var partial graph concern spectrum ar graph never result var contain zero spurious limitation substantially fit limitation partial since usually execute exhaustive pattern spectrum reach feasible numerical google trends ar selection dash apply display modulus I well see imply recover pattern estimate figure display finding agree series mainly large reflect major role intensity co relatively discovery finding underlying interaction air reader model
match letter individual index file file indicator ia unobserved verification record linkage match assumption latent simplify mixture field extension ci reduce need accomplish one restriction probability match denominator experience record linkage informally select linkage suggest et experience datum record linkage site study census record whether match find census record linkage application characteristic population study varied area impact linkage across percentage record record match correspond across across site among however variability
conditioning level th quantile laplace ratio different monte estimate change model change reference ht ad ht less dependent particular occur highly dependent row compare stem specify change asymptotically possess strong dependence estimate occur variable conclude induce dependence htbp cc ad show
replace function right equation obtain equation report value parameter marginal first enumeration acyclic graph graphical graphical often estimate involved step depth explore detail structure structure graphical graphical stand among model topology represents otherwise naturally split call separate step step graph structure arc direction common element depend however mostly multinomial variable local univariate bayesian markov former associate wishart respectively bayesian derivation identify arc dag arc element correspondence graph probabilistic inferential derive
correctness assume depend interaction notation emphasize let kernel underlie brevity instead thank plan g restrict draw accord abuse I j k r r r notation linearity trace k r f optimize drawing algorithm write return consider parameterize prominent dimensionality parameterize covariance per x traditional algorithm interval g discretization space leave might perform discretization space number continuously parameterize introduce kernel result suffer minima programming address fast search however combine kernel continuously parameterize similar describe randomized continuously parameterize similar denote plan k k brevity general resort sample auxiliary denote kernel parameter auxiliary dirac
assumption pass change bind surely outline procedure computationally tractable risk aggregate preference advantage datum construct relevance score preference discuss selection web information allow rank overview structure attempt preference preference relevance vector fit rank aggregation show limit aggregation whether reflect population detail framework description alternate david aggregation construct relevance phase preference skew symmetric matrix entry encode item recommend derive observe preference hadamard natural select square objective yield problem unnormalized laplacian preference observe decomposition I matrix note continuity solution likewise lipschitz skew symmetric variety aggregate pairwise datum skew upon specifically comparison mean distribution structure skew matrix point aggregation preference let grant count method score count time particular rate give skew score skew result ranking suggest one budget pairwise preference win opponent la eigenvector begin form reciprocal pairwise encode multiplicative preference item idea ratio strength generate empirical
n cyclic problem proximity project warm inner initialize nature initialization empirically prove guarantee discuss minimize without expand variable variable work p write advantage rely regularization overlap unitary furthermore ht q replicate much proximity depend overlap numerical comparison present aim family apply replicate index consider group amount entry take exact solution compute cyclic dual project dual less tolerance mean computing repetition situation convenient projection tolerance outer tolerance outer convenient resort dual optimal always subsection propose proximity benchmark group
assess measurement valid unbiased estimator dispersion assertion question determination location parameter minimum unbiased estimator huge induce pdf location control normal parameter denote estimator example generally address aim measurement world world conditional pdf vertical bar
kind concern efficiency inference reach I proceed proceed specify association structural model describes implicitly drive write result association describe subset know good assumption support combine specifie dependent assertion concern subset belief evidence plausibility readily magnitude assertion clear definition plausibility frequentist procedure plausibility plausibility consequence region nominal assume
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duality whereas duality gap notice duality function result methodology duality gap provide notably lasso optimistic path homotopy require solution enjoy exploit piecewise precision initialization invertible approximate direction x jj j reduce cm great direction
approximate map net definition notion message infimum ki li il il ki ik li il il il ki ik ik il il ij achieve message ideal message construct transformation subsection laplacian width graph simple notion matrix pass tree laplacian subgraph induce understand edge share subgraph induce cluster split decomposition exist factorization pass go dominant respective sum dominant approximately satisfy inequality message element assume non ideal message lie range intuitively effective node electrical roughly speak node net subset transformation note row diagonal definition row sum obtain stay follow approximate give pp net next prove round message pass recall make follow addition take tell round q far straightforward going follow readily induction k integrate w go
approximated value hold component approximate ready infer analytic stationary fourier domain truncation introduce transfer matrix obtain fouri obtain eq previous derivation inverse fouri fractional integral psd fractional integral independent process point way aim bound integration introduce truncation evaluation partition integral correspond rectangular evaluation read eq accuracy formula increase accurate scheme field treatment mathematical applicable satisfactory important
generalize cluster study simulate cluster define compose explanatory response joint partition model convenient type joint py py section broad canonical specify particular second derivative link marginal hereafter logit inverse thus variable binomial distribution apply discrete reason mainly interesting cluster differently mass moreover mass estimate gaussian finite denote generic result fact marginal marginal group extension
neuron spike strictly positive neuron period spike output come neuron neuron environment neuron spike neuron neuron notation reverse procedure perform else include neuron neuron neuron let external environment arrival positive spike rate mean environment trivial nothing happen also otherwise inactive time
author pay training require svm multiple requirement place label low desirable notion cost previously recognize barrier scaling notion characterize new source add example increase linearly pairwise similarity function label prohibitive propose jointly motivate extensive scale match extensive synthetic er work zhang valuable comment usa serious almost entity er prohibitive cost similarity exist literature er challenge er uniqueness characteristic synthetic state reality must scale superior constitute contribution entity novel contribution previously motivated er statement elaborate run multiple thorough world direction define dimension
consecutive finish phase action similarly distribute plug call mdp reward horizon node two denote latter implie expect mdp accumulate start state accord policy action union pair yield correspond take policy induce exactly finish phase apply action go last horizon variable know hoeffding hoeffding q verify bind markov way maximize coefficient decrease set lemma exponentially fast sample turn induction induction thing correctness eqs verify eqs iteration exactly bound eqs substitution respective eqs stem hoeffde partial sum e ty recursion q leaf node constant decrease valid rely modify modification induction induction satisfied grow beneficial might exponent expense coefficient
frequency c consistent single assumption tumor evolution process tumor population vice versa neither c frequency frequency figure either rule frequency triplet tumor likely pair tumor error frequency read note rule frequency rule estimate frequency tumor describe read count read copy profile tumor read implement non generative attempt explain frequency model visualization alone sufficient implicitly incorporate assign read column column parent parent child order always true brief parameterization allow represent discussion represent evolutionary distinct ii input reconstruct take union frequency copy appear aa indicate copy copy copy population frequency
wise prior mass pa distribution duration parameter prior collection duration generating exposition dependency component omit focus transition kronecker delta otherwise duration length switch model choice duration illustrate
insight reason unknown use trivial evaluate assignment validity suggest need easy answer need discover approach incorporate cope incomplete representation axiom answer query axiom formula verify nevertheless suffice complete restrictive impose concept utilize true probability formula game concept fair game prove theorem limited decision invoke axiom essentially system simultaneously reject valid preserve proof limited running assignment formula time either evaluate note bit claim run correctness first valid interpretation sample consistent summary see sample formula least hoeffding formula axiom solve accept may notion somewhat hoc weak appendix computationally notion likely infeasible purpose possess close decision solve invoke integrate implicit reasoning semantic standard largely due resolution turn excellent system surprisingly effective remain attractive natural operate clause inference resolution clause contain one appear infer find derive clause know clause typically actually use empty derive resolution incorporate hypothesis formula syntactic intuitively restriction wish lemma derivation syntactic capturing proceed recall give clause clause acyclic corresponding clause corresponding clause derivation clause appear input therefore prove derivation dag dag dag root edge correspond syntactic intuitive early possess decision restriction first establish resolution restriction effect proof syntactic formula axiom correspond clause derive restriction
smooth question answer question show bound infinitely go follow find give residual entry prove fw complete wherein solution sub problem initialize compute condition rate need problem dual iteration round minimize n w exactly overcome modify feasible accord change guarantee objective f k strategy additional spend sentence time wherein number get infinitely close go adaptively call total improve iteratively warm accelerate sentence check stop objective go infinity go theorem decrease take th decrease objective f immediately follow get fw precision fw h fw round substituting infinity without fortunately converge iteration converge iteration assumption usually stop time spend sentence successively respectively since iteration precision cost th wherein iteration empirically summary wherein round form factor therefore method scalable nesterov
process classify dft ds base empirical classifying threshold analytical empirical sample accord band different receive filter specific besides exclude psd band filter detection class different process classifier result correct classification characterize probable existence differently correct
random nn excess sketch hull smoothed marginal know rademacher smoothed trace norm rademacher yield theorem give supplementary material weighted specifically choose excess term smoothed weight trace norm new max smoothed norm simplify suppose choose excess max excess max come cost error guarantee require rich rich test local trace smoothed noisy entry trial draw random sphere moderate
reduce hdp random geometric explore nb across count restrictive nb nb constructed explore share nb analytic posterior comparable conditional posterior process conditional infeasible represent address atom gibbs gibbs nb block nb brevity infinite problem address small small constant modify integral whole risk level avoid truncation slice dirichlet auxiliary slice sampling nb count mixture amenable impose slice leave study block sampler slice represent simulation sampler employ collapse inference atom collapse usually marginal usually collapse conjugacy prior word atom likelihood atom develop collapse nb collapse rule future variety crf hdp latent marginally distribute fix learn crf fair discrete atom iteration gamma initialization enough last uci toolbox restrict five document corpus select word jk v great work appropriately averaged testing partition
power modeling law kernel generalize laplacian distribution mechanic positive definite insight regard correspond hilbert rkhs significance economic field law different finance traffic definite kernel study literature information quantity define measure base interested turn laplacian cauchy machine us term inherent
combine variance decrease rapidly enable ascent order cardinality cardinality observe reduction pca easy pca reliably apply rigorous pre processing couple life typically exponentially decrease allow many feature eliminate introduce block
sample sgd form j suitably scalar definite assume sample around locally optima dimension problem separable ignore diagonal omit gradient rewrite sgd unit variance h derive rate minimize loss whenever direction learn overall usual close optimum anneal schedule automatic almost surely converge subsection trivially parameter separable appendix full useful component
condition reasonable assumption tie interested trivial example exist inductive ml appendix brief mathematical technical input output x x l definition convenience convenience compact eq set I probability often adaboost within boost condition rarely condition useful specific part regard relate adaboost example weight perform essential main weight zero state later implementation adaboost consistent implementation update stay away low weighted hypothesis hypothesis close function make mistake easy imply negative incorrectly classify weak think converse third perfect stop weight w implication condition presentation state list dataset input note part convenient associate representative adaboost margin set whole feature call extensively use error hypothesis mistake equivalently mistake say mistake mistake eliminate mistake removal sound dominate hypothesis select dominate weight adaboost call fact mistake mistake associate e turn convenient classifier adaboost similarly mistake equivalent state early replace matrix form row column construction vector mistake hypothesis differently encode incorrectly whenever learner hypothesis effectively reduce common scheme apply pruning procedure remove mistake contain repeat repeat unique representative say mistake mistake mistake repeat mistake select adaboost execution adaboost select adaboost weight remove mistake equivalently correspond call subscript state previous frame adaboost system fix element representative update sound dot w boost context say differently value weak learner want emphasize simply learn adaboost adaboost classifier optimal characteristic margin introduce make traditional initialization replace surely drive fundamental almost e denominator adaboost depend round numerator adaboost decide whenever error implementation mapping dataset selection implementation scheme introduce hypothesis matter presentation adaboost cycle learner behave example execution state early informally adaboost appendix case say implementation adaboost implicitly deterministic mistake mistake implicitly mistake mistake low exist proxy define adaboost adaboost implication preliminary definition useful mathematical example tx tx adaboost build classification correspond effective label converge infinity hold grow replace change another margin formally bound positive weight round expectation hypothesis normalize similarly tx tx converge state practically present training say something strong behave optimal adaboost classifier effectively converge input
naturally generalize covariate estimate historical sized bootstrap deterministic still variation tail binomial draw analytically complete calculation straightforward numerically give size draw tail see appendix bootstrap averaging observe sized guarantee step tend infinity interval f analytically typically describe bad issue partly effort alone evidence plausibility misspecification conclusion interval reflect inherent difficulty correct tail select compare comparison approach example fully description plausibility law fail likelihood statistically wish produce confidence aid decision averaging pose risk inherent tail inconsistent appropriate frequentist currently insight draw rare event law variance fluctuation tail event large generate synthetic particularly challenging great fluctuation fluctuation perform even sample less event decay small may deviation estimate percent approach historical observing event global database number highly
bfgs hessian fall optima quickly figure make quickly variance large large update huber sufficiently find result analytic gradient factorize px contrast analytical draw average analytically optimize hypercube solution proceed choice far rise specific quadratic go replace objective smooth optimize typically smooth version make smooth classical approach potentially constraint relaxed relaxation optimum combine relaxation variational
set proposition let rademacher uniformly collection rademacher average gaussian compose complexity conditional bound scale rademacher every continuous additionally conditional comparison function point relation due zero separable gaussian gaussian essentially say bound another first maximum analytically difficult easy order let iid accord overcomplete useful bound dictionary overcomplete construct difficulty cover encoder stability concern hold enough guarantee sample weak exhibit proof overcomplete learn depend learn brevity quantity lastly proposition empty rh whose stable code guarantee code strategy fact strategy al turn notion provide good visualization illustrate guarantee code sd code stability code margin
run draw reduction determine input generator hence generate obvious generate show generation randomize take input access uniform n c central proof follow random consider second describe draw permutation compute use walk start output reach end suffice first least second assume eigenvalue graph imply location reach conclude range study polynomial finding several remain outline direction goal wide class several class seem investigation intersection note generation counting intersection independent pac know integer tree improve standard tree fast time note assignment tree reduction rely crucially implicit distribution investigate free quasi branching program similar determined computational generation wide range threshold future goal result wide range understand intersection monotone note counting former latter result natural language quasi branch program gaussian intersection pac know another interesting match boolean satisfying approximate bipartite preliminary suggest range namely color approximate generation admit rise forward inverse generation markov mix rapidly cut partition component sparse cut small random space least example belong uniform since generation possibility give physics formal ise critical temperature temperature spin come reduction fix boundary bring mix existence cut chain know color cut state approximate uniform generation even chain generation mix interesting chain mix polynomial yet cut formally ise mix hand spin make acknowledgement helpful approximate suggest signature scheme helpful acknowledgement helpful theorem claim definition remark pt nsf award foundation grant california berkeley author berkeley fellowship university support nsf university generation focus assignment function inverse problem assignment belong formula goal output variation sort type uniform satisfy assignment inverse generation type call speak approximate uniform query inverse inverse signature certain negative literature formula intersection function formula forward generation hard certain type code hardness inverse combine construction plausible hardness easy generation computationally generation easy combinatorial research theoretical decade complexity know generation fundamental topic wide approximate generation perfect e linear threshold instance problem uniform describe briefly approximate generation generation combinatorial object object speak output must put every taking
equation finite one exist number cn c number initial k surely exist k c exponentially exist k l analogous summation fact random l decay coefficient cauchy real great integer stationary coefficient first elementary component eq mix proof impose immediately result think generalize original multidimensional justify expression eq stationarity covariance depend n k separately move representation summation index exceed cauchy schwarz decay c indice small thus sequence satisfied l normally normally zero fact notation proceed asymptotically normally asymptotic infinity lm step devote first stationary mix strong process condition device convergence multivariate l converge mean apply limit mixing obtain converge l absolutely note treat restrict one exponential reasoning show latter term observe cauchy schwarz step r limit chebyshev every proof
cluster outlier discard algorithm place partition traditional graph construction employ set statistic neighbor denote distance list number neighbor theoretically near lead randomly split datum calculate eq versa rank step time final low indicator smoothness pdf thm minimum average degree point cut typically rather successfully solve capture area cut ensure degree distant unbalanced cluster unbalanced
eq simple adaptation either ml em continuity argument use consistency adaptive describe condition set maximization addition addition continuous r conditional p hz py conditional true continuous require parameter log maximization function run dimension assumption parameter consider empirically order identifiable surely identifiable surely remarkably parameterize correct know non sufficiently set guarantee equation eventually enter set see space gaussian parameter context compressed consider lasso max many performance mmse main motivation gain compressed sense example
f fa f ff sa fa argue sa redundant finally follow supremum set bound f h combinatorial envelope redundant sa fa fa sa ia sa sa trivially picture rather combinatorial define whose face set symbolic sa f value combinatorial envelope positively decrease obviously restriction odd much relate actually norm different possible generalization group overlap connection exploit parameterize tuple precisely norm provide address immediately values propose abstract explicitly presentation norm support introduce view generalize w fa illustrated figure would induce pattern small support motivation call support fa view relaxation cover rigorous link rigorous envelope l w fa unit cube give combine fractional yield since low combinatorial envelope
weakly convergent attain limit pair x g v I integrable p g v integrable integrable apply lebesgue convergence h nk ng n k km l k k l h h n thin l v k hence mx h nx h mx weakly converge pick last hx assume measurable nf want b consider integrable n x apply theorem n nx integrable furthermore get dx k furthermore fine concentrated trivial bound think integral check integral lx restrict measure fulfil establish well state rigorous preliminary definition bc kb c mapping song define rather song eq song state theorem song smoothness hilbert schmidt boundedness although scope xx yx xx state exist n h schmidt vi use cauchy schwarz follow together lead exist integrable fail km z contradiction case justification derive
maximize response last third parameter expectation unique generalized maximization semi effect mean distribution parameter coefficient via book association et al one variation act response source effect definition multi effect know measure similarity effect structure unknown arbitrarily associate component th identifiability additional constraint restriction element covariance write array covariance rkh space
kkt condition work fix concavity value derivative decrease modify active method appear previous organize introduce path mcp penalty section detail regression poisson present lasso simultaneous selection estimation broadly section glm additional design matrix intercept coefficient find maximum exclude little abuse notation therefore minimize glm assume twice differentiable check kkt q easy condition q decrease poor predictive follow construct decrease logarithm kkt activate coordinate together formula calculate appendix logistic intercept penalize warm start keep mind outside additionally typically correction necessary get solution correction user
distribution heavily thus sensitive capture capture work fine synthetic multiscale dataset reflect explore effective affinity agglomerative u perform among robust affected link mnist c add image sc cost bold statistic show c c sc graph base cost fast sc link comparable image initial detect outlier color correspondence robust object vision integrate correspondence cluster range potential application recent agglomerative correspondence cluster acc graph gs respectively present acc gs outperform sm greatly
bar cd color mark option solid dash explicit mark error title xlabel ylabel cycle legend pos bar explicit mark option thick title ylabel black white legend pos north west blue mark thick bar explicit mark solid bar cd x cluster title xlabel edge ylabel list name black legend pos north west bar table mark mark option solid thick bar cd mark title xlabel ylabel list name black legend pos north west blue mark x thick cd mark color mark option thick bar cd mark expect two number ten complexity
concentrate early learner improvement mn l remark supplement x concentrate constraint need since usually seem complicated show learn provably subtle simply bottom cube integer quite join follow measure measure cube eq lebesgue measure notice component define
center let column entry specify dispersion parameter independent wishart determine stage diagonal linear ij ij g ij estimate generalize linear pooling represent variability determine parametrization predictor x subject covariate I equal specify old reflect response underlie correspond partially circle undirecte depend circle variable circle hyperparameter repetition corner specification linear mod example identity link linear mixed lead vb importance assess center parametrization center efficient gaussian partially response express offset approximate bernoulli response specification response quasi keep posterior beginning variational pass machine vb factorize conjugate density exponential without derive bernoulli assume belong pass
hard inequality hold problem special appear considerably hard concentration iff ph thereby iff ar clearly obtain estimate measure begin serve introduce quantization exist compute extend produce sec focus two measure prove sec mean intermediate optimal quantization broad exist
bayesian formulae formulae equivalence frequentist bayes eq find several formulae q jeffreys
detector practice might reverse initialization aspect heuristic appearance improvement yet almost par detector multiple scene suggest wide detection cc split split template spatially flexible template appearance characterize connect learning involve step box part disjoint possibly interpretable profile vision critical discriminative reflect correspond really experimentally might actually reverse aspect show
series multiple physical taylor expansion determine avoid physics device illustrate procedure calibration information present effort include apply g scale level reduce computational explore quantification compute research partly department nuclear security award de university support also thank prediction provide network method multi physics integrate adopt calibration uncertainty extension various scenario natural uncertainty due uncertainty calibration physics investigate calibration form interval determination parameter expression physics model multi experimental resource limited expansion calibration due structure know save taylor series expansion base discussion likelihood complex physics calibration aforementioned calibration modeling device interact dynamic computer model quantity actual quantity distinction represent specify deterministic contrast
explain vector discrete probability dirichlet index different base multinomial label w component unity zero nm r label multinomial classification straightforward observe class posterior classification derive na bayes address hide represent mi joint
forecast forecast high forecast greatly reduction sample mse mixed generalize often high want use eqs process fully stochastic researcher predictive simulate realization simulation condition eqs initialize pt go simulation simulate spatio temporal simulate realization fig couple column influence start soon similar right trace show solely fig pattern qualitative mixed reconstruct spatio
need use common function extra etc commonly method implement package primarily literature reasonable preferable bound g second namely ij conclusion term accuracy make focused attention sample simulation table ahead motivation study section simulation finite performance group least estimator select group sl penalize estimator apply select estimator call estimator minimizer penalty theoretically implement cubic spline evenly distribute knot penalty q certainly option choose cross aic intuitive implement pl estimate automatically
phenomena number call curse especially np hard present section latter use protein investigation probably important limit phenomenon molecular model exploratory former intensive select heavily available prior distribution size size arise model encode reality context informative prior posterior undesirable regard advantage flexible specify assumption less dominate non informative well understand posterior ease system biology monte
lee xu tree scan get traditional refine term alternatively extensively compare approach optimize fit reading structure ultrametric think read observation hierarchical base bin bin bin search operation endowed operation case constant time ultrametric dendrogram carry logarithmic ultrametric address big massive data
stack tensor tensor minimize eq form term second independent describe let j kb ti ib ib ib ib ib ib ib z ib ib ib ib ib ib order generalize cycle sequentially minimize index minimization cf consequence somewhat consider minimization yield yield r ax yu ib ib ib ib ib ib ib analogous expression hold complete provide user rate test attribute rate close return explore slightly rule introduce conclude rating
divergence enough chapter supremum sup give conditional guarantee band roughly speak require desire efficiency measure band minimax finite coverage efficiency standard regularity drive resembles confidence band regression inference trivial band exist hand band mild distinct band nonparametric band nonparametric interval bootstrappe constant include bootstrappe none furthermore produce fact aware provide free band optimality marginal validity predictor introduce notion efficiency section band selection conclude remark band extension region
rely assumption marker measurement longitudinal roc datum suppose marker patient suppose repeatedly measure marker marker marker marker u auc auc sensitivity define real function derivative statistic compare two longitudinal marker linear broad roc auc evaluating marker auc auc statistic allow patient marker marker time two marker point linear high marker positive bound
neighborhood svd pmf variation item average item nmf tend less variation two neighborhood cf perform considerably significantly user slope insensitive count pmf variation item count mae rating fix user display near mae bottom follow baseline item remarkably perform seem svd cf dependency density level dependencie density difference prediction item slope one pmf three nmf relatively slope pmf significantly baseline previously trend fix quantity arbitrary examine dependency prediction count density mae item fit multivariate dependency mae count figure contour mae axis panel cf panel
u du let choice resolution lemma sm sd du ax true eq odd eq coefficient main map prove u u du u u notation du b du coordinate expand du nu dt pn du du du dt k substitution du u pn du pn du du du pn eq schwarz lastly eq function complete contain sd u sm complete
around maxima maxima quite small advantage pdf well suit optimisation intrinsic multiplier sa difficulty find lie
establish hypothesis enable distance apply lipschitz boundedness evaluate hessian log hessian suggest approximation ij define parameterized lemma appendix vector write choose give eq conclude consider equation bound lemma recursively relate constant
grow least see also theoretical algorithm smc approximate pseudo likelihood incremental weight observation subsequent transition nu adopt drawback algorithm grows typically maintain must advanced strategy investigate rejection note differ smc particle shown every realize versus mention smc term smc article modification notation static markov carlo operate standard distribution interest marginal metropolis q concentrate presentation abc procedure step resample also proceed depend store p k n accept reject probability accept ki ni ki ni ki approximation estimate smc extend include summation extend
fix drift establish situation transition share point transition example communication although never situation lead whenever inequality hence condition require drift satisfy markov equation theorem set visit infinitely define family lyapunov w satisfied indeed obtain note loss c vx suggest possibility precisely characterize return form drift know cc v x convenient prove exist hold cm w satisfied c follow indeed consider deduce x consequently existence exist deduce c existence x upper vx vx role play control practical relevance possess desire control driven approximation recursion describe root
square form ordinary obtain term maxima random apply norm q event uniqueness concentration let projection subspace entry mean sub universal second absolutely continuous unique suppose hold linear strictly minimizer turn cone z solution concern second trivial conversely assertion third fact lebesgue probability use state uniqueness follow position recall conversely hence hold contradict show treat event less set state readily fact optimality problem yield result immediate reasoning indeed scheme appendix bind establish proof span let span negative sequel threshold note contain event span corresponding fix positive conditional controlling usual exceed assertion establish coefficient kkt optimality satisfie employ minimizer low handle back system maximal component cf fulfil indicate negative lasso suggest literature minimum coefficient obey probability shall build previous paragraph whose gram constant denominator convenience constant well compute order give active j trivially conversely bind second latter scale optimality imply bind side similarly active set unique slight modification proof back substitution apart modification place eigenvector eigenvalue turn step active kkt follow inequality eq back
multi goal regularizer edge kt regularization gaussian regularizer drop graphical lasso term originally minimize original smooth upper provide multi multi task lead problem multi penalty non uniqueness boundedness maximization regularizer smooth task learning dual norm dual virtue swap min furthermore note solution get kt n k find optimum primal dual zero k block descent problem contain regularizer method descent rarely converge point correspond zeros precision minima among primal graphical simplify
column column single row general goal understand fundamentally hard version competitive side refine classification dual price key ingredient together lp bound geometry classifier pack input priori either reveal coarse input probability full lead yield management inherent accelerate development arguably study program application include ad management indeed uniform constraint online understand reveal much develop packing unknown require value column offline scale generality packing non ignore satisfy property column randomly approximation allocation ad
single correlate protein uncorrelated simulation block protein simulate protein control jointly patient change covariance mutation patient usual discovery method see outperform group moderate induce suffer problematic panel protein patient plot effect procedure grow become increasingly difficult gene interaction interaction exhaustive exhaustive run section gene dataset patient patient patient microarray along gene use gene measure multiple find interaction cutoff find significant find entirely small roughly nan hypothese true small regression surprisingly drop however unfortunately marginal interaction
unbalanced naive cart boost forest method approximately response significantly unbalanced show estimation main present classifier build structure tree classifier unbalanced occurrence minimum support sup minimum four estimation performance ht sup specificity auc classifier area curve learn min sup covariate consist aggregation bagging error error correlate take area mine large database express potential valuable relationship viewpoint weak classifier derive already relevance real variable categorical nominal elementary notation pattern categorical variable hence event pattern probability occurrence coverage index
supervise reduction pca bring designing framework change calculate assume incomplete namely unlabele search set self look subspace give increase separability absence detail calculate sample word lb self equivalent c influence control way derive call include xy xx yy yy please section summarize discussion extension calculate design another dimensionality categorization audio index appropriate three appropriate correlation find class introduce cca
hand side side go associate shorthand associate contain associate shorthand associate association edge associate edge associate measurement edge edge go associate measurement flip associate either edge associate go associate convenient adopt shorthand measurement associate refer association index understand refer go prove hand side big edge associate suppose index happen repeat z time measurement additional I z I z z rt proof similarly associate one remain due association assume former similar prof conclude piece observe continuity strict
add package namely first author ep lemma department computer uk ac continue method binary prediction problem predictor calibrate independently property introduce simplified predictor chapter property form property complicate whereas version behind way validity call part introduce technique uncertainty probabilistic facilitate various loss interested measurable z n ni measurable equivalence output denominator predict predictor hull sometimes refer variable take perfect get predictor satisfy equality say calibrate small
enable readily interesting show correct paper discussion author sparse popularity apply readily
infinity ultimately strategy theorem ucb value optimal well future prevent grow ucb bandit one comment contrary armed horizon small able interesting elegant derive request good mass minimize request give answer request number request oracle close assumption ucb probability least informally lag oracle restrictive assumption expert variation next another clear good ucb ucb use recall ucb attain expert note get fact event use request expert uniquely see
average either average finding capture cf model scenario herein apply gaussian example example label use comprehensive recommended index ari use ari evaluate herein present datum chemical region physical chemical r fit setting well bic ignore bic bic classification associate respective respective classification four merging probability slight ari go average inferior ari good probability window discuss seven model ari three model component average merge good ari compare average window good model window bic good bank available six old bank bic use
prove quite cluster cluster important classification system classification major release computing figure google grow retrieve title body increase decade closely term figure cluster post detail decrease relate historical follow overview computer science school development lead google justification albeit cluster lead science replace entirely principle general automate service start year publish team carefully page york city company
undirected node edge recommendation user follow acknowledgement describe inspire make recommendation represent web link arrive movie movie movie movie direct node item information profile weight specify rating item tag profile category weight relationship mutual twitter graph collaborative
control evolution say know literature mdps treat transition matrix observe build assume identically gaussian inferential convenience process model interpret two quantify expression right controller error thus characterize mdp mix r rx rx ax tx separation state exist mdp observe maker entirely determined also implicitly infer govern criterion observer collect apply controller inefficient observer observer controller control transition unknown approach case however controller inference henceforth may policy behavior system person omit choice intractable integral henceforth invariant scalar eq assume selection scaling observe outcome alternative function utility interpret covariate exist posterior improper parameter identifiability example zero suitable remain zero sec probability
topic document nonparametric hierarchy together inferential topic model include collapse gibbs complementary relax exchangeability correlation idea technique work tag meta importantly multiple widely vision image hierarchy augment spatial spatially none generation multi previous factored sharing even compact factored diverse multidimensional cluster modeling topic build upon classic whose flexible user rate dirichlet number group little place
neighbor ising despite simplicity modern image reference parameter active research validate control boundary condition unlike mrf particular mrf compare different compute chain ghz intel matlab ease visual display logarithmic observe agreement
maximum variant instead dataset examine general depend rank advantage gp contrast neural specification hyperparameter handle principled automate determine hyperparameter probability reward observe logarithm plug optimize solver conjugate able evaluate partial incorporate regard indicator generalization capability measure equal flexible effective achieve fall quickly rank sr approximation sr possible penalize frequentist definite parameterize scalar parameter square exponential
address theoretical algorithm separability penalty unique coordinate part hence second directional exist close relate nonconvex model algorithm solution step satisfy limit stationary simple initialize criterion although graphical
tr tr network I tr k compare strategy profile description use kronecker always stochastic doubly cf eq conclude hermitian compatible dimension profile conclude relation list conclusion diffusion combination matrix construction leave doubly need construction couple select construction originally symbol refer likewise symbol refer second laplacian matrix selection another degree choice assign average combination rule whereby type l rule construction weight largely dependent degree connection influence weight neighbor appropriate weighting ignore across sufficient solely connectivity node design combination aware network devise strategy able adapt statistical adjusting combination assign relevance network formulate combination adaptive become adaptive perform reasonably illustrate consider generally block already view network appear conclude justify relate expression series scale property independent instead minimize alternative minimizing use express combination follow q node associate noise measure amount trace covariance shall noise solution refer intermediate neighbor meaningful large rule appear eq stochastic covariance across size variance across network simple averaging table table evaluate neighbor term know realization learn product particular happen neighborhood neighborhood procedure product recursive motivate recursion write eq side use steady consideration fact hand depend product fourth desire enable let denote one recursion one become product optimal provide adaptive construction diffusion rely neighborhood construction select combination motivate adaptive manner aware variation profile also perturbation exchange neighboring factor channel study degradation noisy straightforward proceed subsequently presence operation exchange account node indicate noisy node source flow mind receive neighbor vector noise scalar e send refer variable noise measurement ki exist locally model receive spatially zero noise process perturb adaptive perturb quantity ki original quantity quantity subject exchange interested examine whose recursion largely shall emphasize main deviation presence may diffusion case noise exchange account additional show general begin aggregate represent exchange combination expression aggregate exchange noise covariance neighborhood scale covariance block follow scalar eq absence exchange datum I aggregate noisy note perturb hold exchange datum reduce introduce exchange see version denote receive scalar noisy node mean contrast mean follow study general zero shall limit discussion exchange henceforth exchange estimate block diagonal block repeat lead arrive recursion repeat recursion link replace arise influence driving appear involve reflect exchange weight involve account exchange involve account involve account exchange exchange weight recursion simplify expectation accord recursion recursion encounter
layer pre compare na I classifier nb principal analysis word paragraph context speech conduct automatically resolve language lexical ambiguity extraction level learn addition typical engineering natural language coverage feature similar deep learning behavior belief conduct compare english lexical lee evaluate various claim vector machine svm classifier kernel analysis kernel feature result show na I
study auc consistency loss introduce calibration prove calibration necessary insufficient minimize finding distance consistent auc addition derive regret bound surrogate loss equivalence loss auc exponential find auc surrogate auc surrogate consistent surrogate loss proper adaboost infinite use medical exhibit well theoretically parametric semi non parametric assumption auc apply mining rank incoherent aggregated auc regard rank especially bipartite ranking prediction auc
inference incorrect make scalable area key acknowledgement national security engineering technology dr influence call confidence interact denote actor position position actor actor actor differ actor change specify roughly actor keep percent position percent original actor interaction appeal underlie parameter partition consensus eventually consensus consensus large common point partial consensus regime value exactly actor attract asymptotic actor partition location toward empty sense adaptation analytic replace take dependent yield focus apparent actor become latent second high dropping term q right side eq side summary
universal adapt play formulate online inefficient relaxation meta relaxation devote complexity localize particular convex localization play adaptive integral turn provide idea define penalization localize analysis play localization give shrinking need unlikely rise develop illustrate example emphasize localization idea place set set cumulative minimizer empirical minimizer henceforth notation time th block inductive bind game rather ix history fast idea form trick notice player history meta previous play game localize adaptive size successive ix localize ix might empirical define radius set enforce bound exhibit choose parameter initialize block local sequential complexity possibly rademacher meta rademacher property replace localize upper instance replace rademacher bound integral bound set instance latter loose pp p trick course advance idea appendix admissible notice grow linearly block quite objective close strong relaxation ball localize remark spirit proof function demonstrate localization case guarantee take without advance certain call imagine expert gain clearly winner minimizer game since else work
take robust consequence polynomially combine theorem robustness linearization linearization mistake logarithmic dependence make decrease tolerance achieve well corollary labeling small polynomially mistake unweighted star mistake irrespective specifically irrespective assign central star actual ready operates weight shorthand notation q span spanning weighted graph mistake satisfie run span ratio mistake relationship similar linear corollary quantifie similar corollary phase linearize discuss subsection span unweighted weight graph complex outside scope paper naive run memory tree describe phase linearize line initially terminal traversal make node leave construct add leaf usual maintain whose label reveal list figure constant implementation node adjacent bottom reveal grey doubly depict traversal predicting stop soon mark traversal begin keep go determined determine leave locate respect go traversal mark connect start go use mark find goes reach right via node within determined quantify key observation internal gets visit trial visit mark first second visit mark mark preprocesse operation show time trial linear trial instance first trial section comparison real weight different
euclidean distance kernel careful svms rbf show near neighbor predict amongst neighborhood
proportion vote entire crowd indicate consistently high baseline indicate effectively member crowd identify pick crowd baseline straight note separately unweighted performance value baseline demonstrate random yield vote make informed member crowd template initial subsequent added step scheme affect selection vote test effect q high latter additional select three remove capability whereas make limited rather tend third component seed random without quality combine voting weighted vote
indistinguishable quadratic base fail reject hypothesis performance integer cause occur detect sample right outperform well frequency reasonable practice yield test rkh distance distance coincide testing parameter energy rkhs machine researcher associate testing chi inference particular problem reproduce one e statement term directly term valid reproduce x product product start expand em xy x xx yy x xy xx yy yy yy xy xy xx yy expansion
euclidean space state apply respective member definite kernel readily definite ari light graph al positive definite structure review relevant highly forecast form probability future retain class measure hausdorff quantify probabilistic
nu nu break nu k false n alpha beta alpha crp factor lambda new I value home paper n net home paper true code home exp home paper home paper true home seq shape nu lambda lambda n sim net home paper sim home sim sim sim sim cn sim home sim home value sim home paper nf
estimator define distribution pool define argue x fy comparison density interval du density estimate distribution density choose distribution density start whose goodness test du du du yield gx interpret transformation du goodness
achievable rate parameter performance limit infinite asymptotically recognition error completely complete side exploit side difference achievable margin since improvement achievable margin model viterbi algorithm recognition limit baseline information error text simulation indicate recognition test label reach gain baseline efficacy incorporate access able provide gain
condition depend database dependency computation query considerable subspace framework recognition constraint point notation close generate rv repetition projection repeat candidate scan return main choose close nr q iid cauchy nonzero constant notational convenience write first minimizer comment several perhaps gap close subspace subspace suggest reduction proportional notice weak dimensionality reduction impossible state overall implication guarantee constant probability easily take failure drop exponentially suffice generate candidate subspace candidate nearest second resource reason especially small query desire amongst query subspace retain project basically query reciprocal nominal gap near negligible fix nr r great random provide gap clear enter current extremely bind issue lead cauchy iid cauchy iid
effectively independent empirical suggest approximate theoretical designing topic model statistical recovery parameter algorithm provably meet separability assumption separability require anchor probability topic contain anchor guarantee proceed step anchor anchor second moment word occurrence provable assume correlated choice evidence economic economic run sample empirical program anchor reduce anchor inversion unstable generate topic three contribution replace programming combinatorial anchor selection separability prove presence second present interpretation
back result ode proposition recently ordinary relate incidence equation incidence incidence ordinary ode describe relate incidence article incidence interpret inverse pose inverse
variance bins correspond group bar center minimum bin demonstrate actual observation prominent model algebraic characterize extensive necessary combinatorial observe compute transition explain explanation relate one technique aspect study aspect present identifiability largely independent matrix completion apply demonstrate argue exploit problem beneficial practical algebraic combinatorial statement practical ignore conversely algebra various interesting therefore involve field valuable rt new european european framework carry fellowship ex european research framework agreement author universit completion intrinsic approach apart introduce combinatorial rank whether complete complete interpret version extend exposition hoc way statement algorithmic implication quickly contain principled application read reconstruct matrix position occur practically collaborative link prediction netflix correspond rate movie reduce complete arbitrarily partial single adapt combinatorial observation reconstruction full validate algorithm identify low allow study via g include answer main question
value order quantity alternative formulae solve base recursive backward next explain backward probability hmms evidence evidence hmms formulae quantity convention backward forward quantity observation prove apply rule independence establish classical inference forward quantity hold backward straightforward consequence corollary sample recursively markov sake completeness present hmms evidence formulae obtain modify formulae backward hmms evidence formulae impose add formulae quantity convention si forward quantity sx rr precede constrain evidence b extend formulae
parameter blockmodel restriction asymptotic mle blockmodel blocks edge bernoulli edge partition membership specifie matrix observe undirected loop sbm evidence community small grow slowly two ensure edge necessary control otherwise dense prevent become grow tight connection slowly grow network allow probability necessary quickly quickly high sbm restriction equal let interval small high tractable large propose recall denote partition arbitrary
identify predictor improvement promise consideration critical office center center support office environmental office science random uncertainty replicate relevant estimate df df nr nr clarity outline model standard simple training group split effect avg avg lar lar yes lar lar selector selector yes rr rr yes regression ridge yes rr correspond ridge assignment avg describe lars ds average divide highlight significance previously section section paper lead basis pursuit
element chain maximum decay partition need relevant scale explore decay suggest attain require obtain sample dag dag partition reach short sample construction much chain recall standard take hybrid reasonably advantage chain possibility short example disadvantage potentially acceptance operation probability chance get acceptance move application dag example restrict simply produce graph meet requirement complicate restrict density speed uniformity remain graph enumeration graph markov restriction incorporate enumeration example impose dags pair record arc also reverse order arcs make check result slow dags fraction dag increase possibilitie grow modification remove enumeration apart possibly stage enumeration dag number incoming arc receive simple allow connection avoid dag arc change restriction incorporate dag easily reconstruct dags arc binomial arc separately limit previous dag limit arc one complicated arc newly add mean
predict acc classifiers tie evaluation perfectly match acc predict rd tied rd predict early see metric system validation bad blind table predict metric metric acc pre predict match predict identify fairly consistent depend interest investigation blind crowd aggregate pseudo gold classifier average crowd exploit rank developed crowdsource track measure measure vs alternative classifier rank accordingly rank finally compare correlation determine achieve great correlation evaluate learner expensive represent bottleneck retrieval ir expert foundation evaluating paradigm enable evaluation ir scenario collection ad hoc item arrive accurately show ir operational increasingly manually traditional least insufficient label shared ir particularly sample technique stable labeling bottleneck
subject figure nevertheless ml indeed denote substantial incorrect consistent score part plot well represent total albeit think assess sd quantile normal monotone ill pose variation case bandwidth score even represent indicate reach around score translate subject get correct kernel total commonly shape assumption strong negative skewness note hard moreover differ two subject form gender differential literature occur subject belong group different choose recent strategy illustrate datum come efficacy conduct center california study concern effectiveness useful part via bank ask agree agree agree statement indicate favorable preliminary directly
exhaustive explore adjust behavior table need efficient estimation procedure minimal determine order residual remain outli subset contingency table fr partially contingency table outlier poisson subset cell count minimal pattern likelihood estimate thereby expect criterion minimal position couple performance develop outli identification method keyword contingency robustness outlier outli omp cell outli minimal pattern result minimal pattern independence method discuss conclusion comment poisson may model structural column design vector give notion far
protein bioinformatics matching view return target node label relation role type information store learn edge bipartite ordinal label rank ranking nothing occur second relation represent namely differently relation kernel cover multi respectively ignore condition object know light explain ranking object define total conditioning fourth straightforwardly reciprocal meaningful domain main setting extend regularize square rank iterative exploit kronecker graph million label relation incorporate consider reciprocal relation via corresponding product namely study reciprocal kernel prove edge direction machine also reciprocal kronecker completely generalization show basic ranking task expressive function indicate likely generalize rank learning set scalability considerably large state solver fast organize formal theoretic review edge allow model two reciprocal base derive algorithm connection difference relate learning emphasis section present promise world term computational introduce label provide represent underlie certain relation take real straightforwardly appropriate vice versa notation kernel consider parameter datum cardinality input consider select hypothesis algorithm function parameter theorem minimizer admit dual associate rkhs feature triplet correctly aim minimize follow
rank collection reduce theoretic contribution yahoo rating well comparison fisher rank another schedule game cluster within division poor ranking conference scheduling experimental synthesis connectivity ranking variety alternative address inherent inconsistent consequence rank comparison old recent theory ranking pose alternative dataset comparison refer oppose comparison pairwise preference comparison express q data subscript rank satisfy collect expect rank tradeoff ranking follow question collect additional pair maximally propose algorithm rank design community lead choice include constraint specify collect rank complete vertex alternative arcs arc yield find weight desire spectral design second theoretic question primary contribution previous ranking amount briefly receive recent paper os assessment crowdsource experiment without ranking like collection preference propose datum construct example algebraic os
change exist analogous procedure likelihood every maximum lead q nan maximize nan following unbounded treat section whether change occur change stop detect receive repeat compute propose occur contain monitor discard observation become determine monitoring false alarm constitute change occur ideally would equal generally compute usual expensive procedure carry real corresponding store arise example involve conditional variable sort fashion small order threshold sequence
within copula model copula compute mean function basis appropriate optimize copula conditional effect procedure resort pt training model predictive density hoeffding mean elaborate volatility financial examine efficacy covariance model daily return index include exchange rate asset matlab application first scenario return daily price series business daily composite uk sp third business day daily effective sp month series benchmark assess model train contain daily return scenario training datum vector shift ahead series contain index remain
present present present large replace follow differ hence expense factor sample probability nice norm gaussian exponential tail column factor argue probability row factor follow optimize regression give construct minimize probability least entire dimension extend point subspace dimension note analog solve singular hyperplane pass take regression sometimes solve subspace multiple become optimal separate regression problem recover appendix reveal vector continue hold matrix continue hold imply desire shrink additional union input summarize fast construct solution run problem vector regression save factor regression approximation essentially overhead regression matrix separate regression regression regression svd separate regression regression problem take make norm problem reduce regression onto regression regression onto replace vector subspace replace
convex assume strong analogous exist imply guess know average stop know decide schedule maintain average time maintain returning iterate eq provably alternatively easily maintain iterate suboptimal average give parameterize average computed accept publication scheme slightly tailor strongly merely simplicity
none solution initialize meet constraint update result see return solution close knowledge percentage signal gap median convexity undesirable might positive portfolio study euclidean onto replace hyperplane motivation address quantum portfolio illustrate provable quadratic subject assume operator simplicity project
node state paper generalization operational wireless highly nontrivial special plot qualitatively change affect behavior small accuracy intermediate accuracy overfitting improve rate sample state accuracy operational accuracy parameterize fig balance classification communication mechanism behavior transfer broad classifier access protocol partly way complicated remain certain maximize may solution preliminary fig plot solution sensor signal know
conventional innovation period well know g drift estimator whole drift conventional reason innovation decrease value histogram limit nh z I innovation error value average decrease illustrate innovation state go order h u correspond one decrease e finding precisely summarize average exact innovation adaptive improve innovation large sampling innovation far illustrate l l l limit c ccc ccc average exact innovation conventional order ccc r r r r r difference average innovation conventional l c c iv exact innovation estimator conventional figure histogram limit innovation available conventional
flow financial economic market excess volatility recent attempt huge business ii extract trading activity remove identify hierarchy news differ dimension investigate news market one news aggregate manner account impact total news record period study release economic serious limitation significant could short news day temporal partition address simultaneous relevant financial trading activity raw text million news record thompson examine impact trading activity stock list stock period determine piece information explain activity regression news event volume explain news good correspondence evolution trading measure daily volume correspondence
approximate bad vx minimization modular correspondingly algorithm every get strongly minimization note bad complexity much practice h try cost strong amongst find choose capture relevance subset na bayes applicable meaning express submodular subsection regularize mutual observed laplace help without experiment find representative subset formulation entropy factorization formulation factor factored simple iteratively factor factored mutual information lastly na nb heuristic modular modular submodular submodular generally outperform procedure quite thus
form maximum greedy rank notion check greedy cast flow vertex towards capacity hypergraph maximal clique decomposable see converse naturally polytope constraint optimization form replace convex remain integral remain primal relaxed clique feasible integral relaxed convex relaxation primal constraint convex minimize polynomial constraint complex introduce lagrange multiplier maximize em cover constraint clique constraint therefore relate primal dual variable cost
large action short virtue exploration behaviour identify towards particularly demonstrate line fashion theoretic influence environment previously transition assume main relax calculation vector application exploration long assume precise transition interact environment address state advance computational point vector space action discretize however naive discretization soon become infeasible work partly agent artificial laboratory grant science foundation n research european part project www grow fp angle measure axis angular velocity nm describe q angular velocity mean physical symbol dynamic simulation keep scalar action discretize continuous domain angle angular artificial virtual physical environment fully various automatically example dynamic programming specifie explicitly goal want application perfectly reasonable sometimes subtle human sensible investigate coupling prefer resort task research idea organization goal drive agent universal rule optimize define reward kind behavior artificial inherently interesting step
randomness hold exchangeability family second among carry information training calibration proportion standard ideal proportion prediction calibration become confident variance cf
well remark admm scheme adopt quite different cc admm e e compare synthetic create proximal specialized code http mit edu code comparison sparsity denote percentage nonzero always small zero cpu times second report r dim cpu e e e produce comparable objective compare fast table ten sometimes four need hour minute
walk two group walk label start reach measure unlabeled show walk probability walk step define bag develop section suffer thus path entry bag path put term graph could framework tackle network et suggest avoid compute computing input small community obtain approach suffer hyperparameter community node compute reduce investigate work technique span tree laplacian compute enable address graph liu investigate relational latent dimension social twitter example approach promising label define last walk force outperform group centrality
ask probability note equally analysis fail vertex find least reality loop actual isotropic implie find sample vertex vertex input map simplex simplex isometry q sphere variation distance distribution simplex set let simplex orthogonal orthogonal decomposition approximate maximum eq q study maxima moment direction geometry maxima minima optimization like although isotropic n complete maxima third homogeneous symmetric direction conclude consider simplex identify via na v equivalent embed canonical hyperplane accord constant maxima condition equation say coordinate put order necessary eliminate list candidate scale projection vector maxima global continuous maxima lagrangian semidefinite restrict tangent square coordinate elsewhere
acceptance momentum flip tr relevant represent label transition net
beyond absolute position benefit increase possible vary region improvement tree recommend seem forest need point yet c balance diabetes breast scale diabetes incorporate position metric distance example extension consistency metric learn propagation generate specific unlabeled well ten uci measure cross constraint construct explicitly method inherently method efficiency specific approach time comparable position dependence allow
cluster extend gradually merge agglomerative dendrogram depend distance merging procedure linkage linkage first statistical noticed dealing cluster generally surely mean group seek center certain notion clustering assume parametric modeling g bayes partitioning regard motivation view see instance clearly component underlie mixture certainly way dense arbitrarily coincide several component need might well unimodal suggesting shape cluster motivate example helpful cluster separate density maxima mode modal understand provide precise goal explore next population cluster fully researcher lead parametric attempt modal level define
movie see long describe opinion influence opinion formation influence individual rating preference vector influence influence user attribute default also change member case normally assume individual necessarily decision social influence motivation apply group assume describe affect evolution describe certain social influence disagreement consideration member regime rating recommendation beginning may ap equation directly initially assume recommendation different factor
separate backward computationally exponential recalling seek induce input unfortunately find much hard output output method output long allow let approximate output extend let nan tw normalise h probable probable probable remove trivially return sort appear good shorter technique speech observe output iteratively store running prediction memory network store
find region ec reader circle cone black ec ec field surprising convex geometric argument come point ignore either expand essentially precede q ec density ec illustrate rejection likelihood threshold cut ec cone ec representation case easy coefficient cone center origin sphere geodesic probability content subset beta necessary requirement must intersect similar establish convex rejection region provide euclidean radius sphere radius spherical geodesic radius near contribute pointed ignore root
state enable model brief relevant inference extend hdp thorough parameter stick distribution parameterize notation hdp role transition dp draw link dp discrete typical state towards chinese restaurant assignment infinite hdp context hdp individually state reduce hdp purpose note reduce hdp transition hdp hdp hmm maintain mix correlation thus provide new alternative exist hdp hmm hmm towards transition provide encourage process elsewhere hdp allow duration duration distribution duration explicit duration remainder component complex factorial self transition emphasize allow transition transition simplify clear allow explicit duration allow transition duration challenge particular hdp model tool duration fit make bayesian semi treat parameter good bayesian later allow compare hdp case state duration index
infeasible differ support introduce use induce approach maximize provide clear set variational gps generalize group auxiliary induce task sense induce portion detail follow introduction group specify parameter n j hyper parameter likelihood observation low j j posterior variational variation completely free entirely additional derive simplified predictive ready integrate leave th jj g low achievable jensen equality variational
history coin infinitely identical coin minimize coin coin minimal monotonically increase coin log optimal description coin coin decrease coin coin constant coin give repeatedly coin log coin starts coin coin terminate likelihood coin log state get heavy coin respectively coin get get heavy coin get heavy coin coin eq heavy
sampling introduce estimate independence node rw strong adjustment applicable evaluate technique wide topology facebook art facebook guarantee available idea estimator sample node rather sample combine together challenge rw unchanged rw sampling se finally claim gray many network walk rw estimating property global property type node exist inefficient rw sampling address induce study participant field cost measurement concrete budget desire feasible rw rw use www networks offline derive sec estimate
necessary equivalence raw statistic equivalence justify identical correspond nan identically distribute bernoulli hypothesis I therefore infimum quantity nan statistic contain equivalence monotonically statistic contain statement lemma number decrease give intend microarray cutoff point equivalent observe equivalence great q equivalent cutoff equivalent cutoff positive recover gene hypothesis true hypothesis therefore q gene cutoff
condition modern area increasingly high dimensional challenge mean definite covariance arise rank throughout finance sciences matrix invertible inverse covariance powerful allow discovery multivariate encode conditional draw normal distribution covariance matrix surely dense high setting perform sparse penalize
tx fx kt satisfy tp gx k proximal many include available problem efficiently subproblem framework bootstrap previously solver loss suggest newton outer higher nd confirm empirically newton gradient expensive quadratic subproblem newton approximation bfgs bfgs sr statistic heavily generally set algorithm warm start find cost fitting solution kkt mixed opposed interest prediction feature build conditional full form choose pairwise pairwise independent potential r present experimental conditional sample true figure theoretical
arrival evy strictly transform arrival evy ie dd atom evy obtain gamma poisson stable often prior survival measure construct normalize refer g q great use application image directly instead exchangeable infinitely exchangeable sequence de infinitely sequence distribution sequence chinese restaurant de measure exchangeable exchangeable process likelihood integrate beta exchangeable ibp subset nonparametric version conjugacy distribution analytically chinese restaurant parameter represent analogy customer restaurant number table single table one customer customer restaurant table customer table choose measure define measure discrete day formal technical report criterion collection measure support distinct reasonably easy either computationally familiar converge posterior converge specification typically assume metric however hierarchical exchangeable hierarchical model collection discuss covariate index dependent nonparametric dirichlet frequently nonparametric also collection partition
extra content notice content penalty introduce operate completion admit explicit parameterization kind paradigm need content reader apparent simulation fact cf think widely accept model certainly simulate would favor approach understand interesting informative question suitable model consider content issue focus collaborative filter methodology impose alignment shrink attribute regression attribute produce novel insight recommendation mean thorough theoretical mention section bring opposed approach plausible generative rating outline idea practitioner easy manner support natural science engineering research project create associate constructive especially grateful em recommender recommendation either content matrix approach rate rate recommend
lead solve mode bottleneck factor matrix surprisingly show full column correctly efficiently use decomposition substitute rewrite ki exactly entry arrange order define unfold replace product way unfold unfold call unfold tensor tensor number unfold mode simplicity refer merged tensor transpose unfold arbitrarily product tensor mode reduce several rao product component immediately uniquely permutation respectively rao recover rao solve problem decompose apply component optimally rao set proper pre problem comprehensive simplicity subset linearly obviously useful appendix jj rao mild factor matrix mode likely full rank cause mode key
template optical flow template reference motion template event training sample sample ann knn coding feature feature location six sa knn achieve future extent additional image addition feature extraction like fast certainly achieve improvement feed grant plan science innovation lt city science technology unit theorem college computer information classify diverse video challenging ignore past realistic annotate six location classify consist four phase
annotation noisy search keyword unsupervise annotation raw tag feature purpose model text describe quantitative qualitative evaluation present mean raw use th feature eigenvector equivalent normalize unit cluster row nc hard obtain normalize way nc factorize nonnegative normalize assignment highest nc soft cluster document topic document use posterior lead hard map cluster posterior investigate kernel baseline motivate consideration large number image want annotation method tag truth paper unfortunately standard use namely tc view tag propose modal retrieval internet wikipedia labels annotation million imagenet choose collect publicly available query category cat tag dictionary average tag per image keyword tag retrieval collect ten imagenet result tag national manually annotate city etc important wide list tag ground truth annotation concept tag web category include concept mark relevant query dataset contain concept second view text web page tag appear stop list document tag dictionary also tag characteristic class whose annotation whose come image comparative implementation wide manually annotate come collect image inconsistent quality large vocabulary view visual
univariate multivariate margin unify however characterization describe term unit sphere statistically comprehensive max describe device propose generate family max stable distribution already vary multivariate concept generator development nonparametric parametric latter frequentist multivariate asymptotic independence accurately refine seminal behaviour maxima structure dependence related highlight aspect multivariate sample collect
procedure implement recommend complementary multiply carry measure data set section monte present p value section reporting make remark concern interpretation significance reasonably model homogeneity aside consideration multiple testing statistic homogeneity reject question
ibp consist plus learn latent ibp ibp ibp truncate allow iteration reconstruction incorporate reconstruction pt ibp ibp extensions data ibp absence word vocabulary ibp text statistic model corpora ibp ibp tailed corpora collection graphic hold replicate experiment perform original three parameter ibp negative distribution
chain operate representation space chain define advantage stem move abstract auto encoder good auto side variant variant mm embed kernel hessian embed visualization hidden representation well preserve neighborhood property local density neighborhood sample covariance empirical machine manifold window achieve test mean neighboring local basis eigenvalue predictor neighbor predict objective discover manifold
achieve propose algorithm conduct learn overlap employ whole pc comparable five common precision learn scale averaging framework reconstruction complete community serve meaningful protein gene well li department science china email engineering institute east china normal china email edu cn science china email cn motivation apply average currently model average super novel network principle divide comprise first perform robust community community merge contribution robust segment much small accord allocate depend denote closeness challenge pearson one introduce overcome robust small sub community current perform dependency intra community make intra unbiased credible analyze primary
simulation r r c random knot knot knot full parent spc spc spc spc parent spc stationary square interval time simulate random miss location simulation study two trend evident well spc knot knot knot even close parent wrong bad score region assume covariance parametric examine performance unlikely conduct study sample mat ern simulation study mat ern covariance simulate add spatially study knot knot fix knot spc spc spc sec parent spc covariance observe random
brownian motion bm epidemic decide adopt model bm basis fig sm bm interesting alternative criterion likelihood iterate filter subject particle filter estimate expect uncertainty credible interval month early stage sensitivity explore robustness concentrate
inter instead time average paper inter cell interference interference noise interference fractional beyond though traffic ix analogous active bs ix feasible bs eventually bss indicator exploit controller know bs switching strategy last knowledge detail section minimize overall bss shown consumption bs linearly coverage energy consumption bss consumption stay bs traffic vary bs traffic generalize summarize besides consumption bs consumption bs order avoid quality service delay formulate specifically queue flow system little active association consumption ensure balancing reflect consumption mdp tuple traffic controller mode otherwise bs correspondingly metric bs cell strength traffic traffic load transform determine volume bss meanwhile immediate cost environment feed bs controller discount state
constraint take programming problem example handle slack article particularly svm lee preserve incorporate address norm svms obtain classification svm mainly motivated adopt slightly exploit solve learn easily explain drawback depend image computation convenient derive wolfe multiplier lagrangian tucker plug q dual qp equal otherwise entirely example considerably small original introduce algorithm classifier point formulate computation radius image mapping dot dot wolfe solution formula square subset concept going explore immediately deep presence objective mild kernel ignore within I implement constant thus equivalence construction dense qp expensive matter kind one take account approximate solution within modify priori try specify definition tolerance ball algorithms scale bc iterations r kb exploit idea support reduction include expression moreover look k I evaluation lack reduce
north rectangle south east enforce effect center constant model gibbs covariate index possible knot observation region covariate posterior take informative penalty entire spline wider credible spline global responsible thin fix knot place quantile precision spline correctly gibbs sampler determinant posterior avoid thus penalty zero eq spline control smoothing shift lack smoothing smoothness gamma spline hasting inside sampler credible calculate quickly sample credible interval report coefficient spline credible effect interest credible interpretable forecast manner forecast precede take advantage day one hour come hour training week forecast hour advance value forecast forecast treat imputation week forecasting error iteration eq forecast sample asymptotically section chain check unimodal draw posterior even ci spline marginally normally interested contain credible informative look spline say zero posterior
effort call rbf encoder center mean step simplify basis center experiment value coordinate elastic give similarity parameter normalize mac alternate slow parallel mac qp achieve nearly processor reconstruction elastic run mac qp manifold loop mac c indicate marker speedup experiment approximately reach processor rbf autoencoder architecture decoder try search define weight include regularization selection know aic omit error training parameter decoder layer output dimension encoder equivalently output dimension center autoencoder lm rbf autoencoder network constrain point mac qp complex e previous optimize mac qp model net good potentially nest function plus indicate marker change annotated architecture move impose parameter follow minor architecture continuous final total incur large training function optimization result achieve approximately speedup mac drastically runtime nest jointly exist
present theoretic epidemic cascade prior super graph cascade cascade reliable outcome I epidemic bind variable find likely cascade node algorithm computation need infect learn tree node greedy explain cause large infected remove theoretic cascade approximate recovery notion sir corollary super specify factor cascade final sir epidemic cascade direct graph infection allow illustrate role compact emphasize thus merely thus system probability cascade regime corollaries union basis thus recover neighbor super neighborhood find neighborhood entire neighborhood neighborhood super graph epidemic cascade cascade
minimize difficulty non feature interaction redundancy combination optimization since completely redundancy interaction forward select search deal redundant detect presence start iteratively instead potential interact go problematic seem easier discrete consider feature interaction feature table however reveal dependency nonetheless unclear mi capable detect nonlinear cause property instead permit efficiently compute past exhaustive impractical univariate combine feature good strategy combine ccccc forward interaction dependency needed efficiency excellent forward backward exhaustive mi variation exist impractical optimization consist zero learn
order require hold therefore distribute unfortunately therefore remark knowledge still ht finding verify text categorization news article evenly topic perform topic label wish total number news article otherwise wish function step matrix pool satisfy communication subgradient consensus consensus consensus gradient proximal weight converge neighborhood achieve highlight
amount alignment somewhat alignment statistic reflect weakly correlate alignment statistic statistic influence influence quantity stability statistic relatively theoretical paper describe meet table fix assume page solution overview stability statistic theorem within away stability statistic adversarial structure outlier type note follow subspace angle subspace imagine advance pose oracle regression intractable l theorem even outlier compare standard formulation pca search minimize subspace minimize sum residual tends sensitive consequence contribute short section contain page argument prove type primal technical detail hard working dominate paper elaborate analysis idea replace nearby objective take exact result paragraph classic optimization see believe idea relaxation data function technical use target function perturb contain subspace perturb coincide technical result lemma show residual perturb minimizer close perturbation objective residual define perturb move feasible ascent lemma subtract second indeed feasible feasible apply lemma right reach finish argument eq second
graph n ns follow appear necessarily small q plug expression shall j asymptotically probability weak treatment valid immediately asymptotic proceed fact condition condition proceed end similarly numerical experiment dependent restrict bivariate autoregressive ar let bivariate set normal recursively bivariate version autoregressive previously procedure bivariate copula copula result since copula bivariate frank copula define dependent multipli sequence test move initially detail convolution choose median choice table monte
formula logical conjunction follow protein logic define hold redundant logic logic note every property describe static boolean asynchronous study functional aspect need guarantee loop describe logical name compatible satisfy property logical note enforce cause within logical fix look truth evaluate condition behind change steady compatible loop existence steady consider well work
coherence atom however still enforce atom r trace q coherence take partial derivative derivative minimizer objective run iteration successively I partial evaluate gradient computing dictionary ccc singular spectra self coherence coherent flat correspond svd lie sequentially coherence atom
reach change uncertainty stop fall point coarse variation explore adjust bellman update uncertainty keep track visit gp experiment rl algorithm fine show gp short thing learn optimal high sample always exception direct fit iteration method hoc domain examine explore gp perform initially explain generalization capability gps propagation car acceleration function transition iteration modify due dependent hyperparameter selection region transition car indeed position estimate gps
respective currently confirm subsection learn view secondary channel pattern vector sense channel slot channel attempt take interference message send discard receiver slot experience throughput interference sense throughput secondary slot configuration slot choose employ channel usage slot channel sum equal employ share slot channel selection complete selection describe round access throughput slot average expect throughput take prevent mechanism derive pattern channel thus interference cause show usage sharing channel selection error mechanism loss cause interference allocate channel scheme interference among high throughput high scenario throughput function number address spectrum secondary network formulate
deal time experiment effectiveness code publicly bayesian well hyperparameter fast human procedure almost set level significantly performance usually specify easily weight desirable another automate view tune black invoke expensive involve primary completion desirable making seek elegant approach function typically sample gaussian process posterior make pick hyperparameter next experiment optimize expect ei ei ucb
vary pair fix probability fix centrality variety description various algorithm shall score randomness outcome comparison comparison connect walk irreducible add centrality transition centrality regularize prior beta give regularization maximize solve wise logistic regression provide regularize centrality regularize mle regularize generalized count method si aggregate ranking widely break pairwise apply count rank count ranking generalize normalize centrality rank lead eigenvector matrix different base lead prominent top eigenvector rank mc mc mc
fold reduce additional challenge demonstrate generative rank optimal divide training feature remaining perform vary vs attain portion j accuracie bold maximum summarize poor svm polynomial rbf hand perform accuracy multiclass marginally generative discrimination text build way eliminate work multi interesting theorem exist paper large nature curse employ perform enforce density use jeffreys discriminate extend multi
convergence robust regression major minimize I response residual eigenvalue eigenvalue interestingly statistic california continuously differentiable onto separate set nontrivial project intersection newton interior medium modern application statistic potentially thousand instance optimization quasi newton illustrate several statistic formulate instance continuously convex closed include find iterate iterate generate drive strict unless c n pair lead penalization positive weight loss sum penalty differently useful constraint equally consequently uniform notational exposition distance require simultaneous point present ascent projection method exhibit ascent start c acceleration
characterization separation prove useful elementary omit brevity dag remove oriented form edge vertex calculus method interested detect strength parent instrumental absence let dag vertex separate pa share equal px px strongly correlate become compatible markov exist
line gaussian depend norm universal upper induce average sub reach much bound characterization learn rich term upper dependent lower bind hold rich feature adapt calculate distribution denote suffice distribution dimension every rich margin excess algorithm bind complexity establish margin regularize rigorously sample regularization discriminative sample bound decide label rule margin eigenvalue vector provide extend discuss sample complexity notation provide necessary margin dimension upper general must matrix implication proof prove set require type bind
update data enkf variant exist construct draw perturb equivalently kalman treat draw update distribution normal member update separately center member center datum actual ensemble combination distribution gray dot right frame see gray represent dot right ensemble draw produce give mean variance member enkf though first second two enkf representation appear right skewed enkf posterior enkf stage easily generalize stage enkf break information two twice twice enkf twice produce produce ensemble enkf twice gray right frame representation dot right frame see frame relationship cover somewhat evaluation
datum prior locally normalise would easy nevertheless least extent prior generalise qualitative shape normalise isolate early figure vertex figure contribution normalise isolate dotted learn dominate learn pass learn globally normalise although plot generalised random scope comparison student teacher leave case gp student globally normalise teacher normalise r enyi match figure show unity probabilistic similar case generalised figure locally learn teacher globally case two extend graph component qualitative globally normalise bayes expect uncertain vertex intermediate variance become receive kernel local exactly enyi law graph low peak normalise display variance dominant regard character package color conjunction terminal explanation graphic explanation terminal graphic ltb lt lt lt lt ltb lt lt lt lt lt lt ltb ltb ltb teacher normalise kernel power generalise exponent cutoff terminal option explanation load package package graphic explanation terminal macro ltb lt lt lt lt lt ltb lt lt lt lt ltb ltb ltb enyi teacher globally generalise exponent cutoff package conjunction
expectation come quadratic bilinear dt jt dt v jt dt eq w w dt proceed pp w w piece kp kp dt dt st dt w st first since turn whose adjoint project complement correspond st st dt st w st dt successively adjoint w st dt dt dt dt exploit norm put together upper dt p kt tp proof start introduce linearity trace diagonal reasoning follow hence give conclusion triangle definition sub hold z uniqueness assume j kt dt exploit dt j unique advantage exploit desire proceed dt p noiseless surely bound light lemma first follow apply desire proposition equivalent q exist
readily find q energy dx x p fr random clearly integer value fr event occurs see respectively imply odd infinitely often superior mean therefore inferior empty mean highlight fr mean set fail inner need exhibit property include true sequence realization subsequence infimum event occur follow approximated formulae clearly represent visit infinitely hence arithmetic expect every exist reach conclusion disagreement classical convergence
crp take unlike dp family result application exchangeable dd crp distribution represent dynamic evolution base user couple process dirichlet hdp introduce coupling use extend allocation hdp equivalently extension crp chinese restaurant crf crf couple coupling rich depend relationship evolution address parametric static time amenable evolution context recurrent crf scalability recurrent crf static problem medium focus around interest pattern medium use topic focus account social medium rich etc site recommendation assess interest activitie study user al model account popularity various relate deal preference attempt understand pattern twitter apart temporal dynamic study factor explore social model wide network factor social medium build chinese restaurant evolution application basic associate individual movie
function say enumeration close enumeration close equivalent valid relax proposition appear include term form condition continuity axiom general interpretation probability separate interpretation preserve proposition number individual axiom ns practice natural axiom type proposition definition property suggest strongly part valid principle advance sure idea inductive weak turn useful strongly development major alphabet sentence condition necessary sufficient discuss detail alphabet countable free enumeration hence universal condition fail statement read iff assume assumption get separate case separate say separate term separate prove prove set interpretation interpretation borel algebra topology need define alphabet sentence closed intersection topology borel form equivalently term countable algebra algebra interpretation finitely additive additive countable collection set hold ia always countable equivalent continuity algebra probability sentence interpretation finitely suppose finite interpretation useful countable sentence algebra finitely claim suppose contrary contradict thus prove decrease alphabet countable interpretations borel let suppose sentence valid finitely collection disjoint finitely proposition alphabet countable countable borel algebra borel interpretation intend intend member separate interpretation separate interpretation alphabet borel algebra need denote close finite intersection algebra consider
cca version modal discriminative pca lda handle supervise meet generalize class framework recover framework define propose generalize alignment author domain event detection corpora tv entity corpora image versus start significant attention computer covariate long new one survey adaptation theory thorough survey relate field mention introduction label fundamental devoted aspect fashion benefit
subsequent criterion deterministic criterion view field generalize training net optimize limited datum conditioning set work answer train time rather data descent
landscape evolve toward parallelism mcmc identify end extreme independent parallel nothing chain limitation non version extreme alternatively chain intermediate use parallelism share information mix na I grain execution nevertheless parallelism possible objective core low slice sampling elliptical elliptical generalize elliptical slice parallelism dynamically desire evaluate method x use wish sometimes objective carlo formulate classical example simulate operator produce sample use compute expectation typically unnormalized notation somewhat chain carlo adaptive rely coordinate accomplished slice insight conditional uniform slice potentially easier uniformly among typically define leave slice describe interval expand necessary acceptable procedure phase efficient first direction consider implementation later detail unclear
relevance exhibit mixture original gaussian contrary principal figure hyperparameter isotropic review classifier hyperparameter classification moderate dimension dimensional reader reference framework tune optimize margin svm penalization optimization hyperparameter leave radius margin optimal hyperparameter objective follow later gradient class classifier indeed problem must vs prefer require optimal similar since gradient enter gradient derivative conventional framework hyperparameter detail use methodology consecutive
show ad value thus satisfy ad achieve candidate efficiency iterative suppose ad ad subset correspond self calibrate prediction different ad raw candidate ad randomly bid efficiency maximize query mechanism offline maximize np hard even minimum feedback arc player rank minimize player minimize number time player player player generality show ad winner ad maximize raw exactly also observe np construction perfect efficiency calibrate illustrate follow four ad tuple c guarantee show also need already query
median univariate variable empirical curve point median versus curve minute geometric quantile generalization unlike median homogeneous coordinate coordinate appropriately compute argument favor g functional fact datum exactly median median central instant consumption record week week compute wise measurement consumption wise median notice coordinate median week situation median fact take account time point compute instant instant use fact fulfil point median note define nice direction towards unchanged obvious extend geometry index quantile way interpret solve zhang effort measurement work subset henceforth realization inclusion contain inclusion variety direct element design
conditional deviation give second moment unit respectively require limit follow continuous measurable mapping nh hx tb initial drift weakly distribution continuous step half length point x condition usual acceptance indicator denote embed discrete variable variable come mh acceptance original benchmark keep
result key good linkage record firstly j step e compose equal remain equation replace log estimate maximum frequency count usual stop assess measure consecutive start algorithm take great restriction linkage em clear greater refer individual agree information record tuple necessarily probability information high parameter easy determine constraint take determine start partition fine low criterion note procedure use direct partition branch partition identify constraint whenever among probability diagram link size small reasonable starting determine maximum element maximum reasonable starting determine minus notice rather merely guide start em maxima take hold marginal observe left side set
space ds exclude ds ds svm forecaster forecaster ds filter quantity slight improvement locate accurately precisely rmse filtering dominate incorrectly via forecaster capture forecaster significant gain historical set interesting feature forecast observation time forecasting performance excellent attempt state forecast find prediction svm forecaster additionally laplace fit likely variance able deviation gaussian actual observation influential result ds ds misspecification svm historical dominate filter stochastic firstly svm forecaster training set relative system significantly initially filter
margin optima peak normal middle shape symmetric well margin respectively well margin behave margin situation change remarkable kernel zero value top normal margin locate evolution proceed zero margin capture good exhibit introduction correlation seems find many reliably critical population two achieve population capable multivariate information fail optimize margin successful though say aspect marginal role follow aspect report combine copula apply truncation
rejection period local exist local consecutive update period cyclic loop matrix local period cyclic property transition ergodic condition ergodic transition state integer define ergodicity ergodicity markov condition considered choose way choose actual however choice short rate sequential small uniformly far ergodicity sequential even simple algorithm ergodicity ise practically check compare sequential assess lattice fast convergence short parameter conventional autocorrelation nearly
te coupling coefficient strength depend internal explain measure coupling define entropy second term first strength depend belong parent analytical numerical advantage mutual mi parent mi input depend coefficient process condition te though three depend interaction external formula mit solely depend couple coefficient autoregressive generally additive mit depends prove coupling strength theorem section variance parent separable mix entropy part entropy could gray couple rather thick nonlinear mit numerically constraint full coupling reach next formalize strength multivariate sect couple link follow derivation lag link define mean dependence source process parent I
choice covariance one x diagonal hyperparameter zero elsewhere apply ern kernel know choose produce already decide denote arbitrary property gps jointly gaussian x complement eq lattice simplex size set number optimization b bp assume variance construct combination optimize optimize sufficient include probability improvement thompson refer detail main function search densely shrink surrogate intuition function discard
yy yy yy yy throughout derivation therefore last proof follow tail bind gaussian covariance dimensional jj inequality probability least simplify probability complete convenience notation cone yy yy function convexity loss equivalently eq argument contradict proof figure support use f score norm present table glasso glasso norm glasso glasso glasso c c glasso glasso glasso glasso glasso theorem corollary remark pt section department statistic study partial convex establish competitive empirical exist various synthetic real copy unknown covariance
label assign instance find categorization organize problem categorization together brief explanation confirm second use alternative experiment server recently generalize claim extensive section manual multiple exhaustive think occur topic match abstract description category example medical deal mesh keyword work label admit belong music possible third label second occurrence label third song degree label phenomenon initially capture binary third look contribution label category combination binary account capture label crucial give rich instance every define assume binary content instance represent label
compute assignment business attribute compare user business user business user business role compute agreement assignment sign penalize alternative would pair set function equally sized set conceptually differ attribute see enable share standard new framework new role cost hybrid anneal convert term assignment eq terms business assign simplify auxiliary apparent assign contribution role assignment substitute assignment ik k eq directly compare business cost compute substituting procedure arise recursion turn tn ik gibbs multiple repeatedly find cost business information role attribute optimizer user role mining role group together irrelevant attribute infer without attribute theoretic relevance random assignment variable business generic job business quantity entropy mutual quantifie assign entropy business attribute mutual gain knowledge attribute compare entropy small reduce relevance bit knowledge care observation relevance business high imagine fair na differ considerably one compute attribute observation user highly relevant compute value sufficiently measure entropy different le access job unit attribute code index kind contract initially would user business task compute result depict histogram attribute relevance reduction much little role experiment time
proposition u proceed continue terminate inequality hence rewrite solution rule like relevant know square result regularization also cross validation motivated vector independently sparsity recall select vector ease elastic parameter varied necessarily
equivalent recursion start subgradient end adaptive may boundedness line convergence argument gap extend denote maximizer recursion convexity smoothness g note equality classical equation desire proposition consider optimize iteration extend search recursion proposition proof assume constant one v
rule year extent language order relationship partially rule another matter cm remark question university universe analysis
degree remove eventually interesting arise conduct ratio limit fundamentally consider hypothesis constrain penalize ny q endow hilbert reproduce trivial bivariate z g z basis limit distribution calculation fr g w g g g g g g ds w fr derivative additional rate verify assumption satisfied g main local present likelihood satisfy nz pn restriction local proposition explicitly reproduce uniquely rkh certain facilitate calculation example explicitly specify remark depend bias specify estimation fortunately smoothness true satisfied fourier coefficient satisfy phenomenon type since counterpart practice section counterpart simultaneous band originally density estimation method require relaxed translate
filter covariance subject wide range also datum subject trade optimization initialize use leave manner subject training use allow method cluster interpretation perform analysis whether subject kullback leibler kl kl matrix distance matrix user large difference may test small subject accuracy may degree stationarity performance significance description covariance test matrix subject train subject fig analyse user square angle subspace tu exist equality principal angle amount preserve project thus indicate subspace angle picture relation restrict angle tends extract type discriminative filter prominent direction measure similarity discriminative random
lc lc lc lc lc lc cx grey posterior mat ern field square krige see panel leave show ni lc lc lc lc lc lc marginal b lc lc lc lc lc lc lc lc lc lc right krige definition see carlo satisfying correct nothing distribution approximation ni handling cl bc bc tc ct ct ct ct ct ct cl cl bc bc tc ct ct ct ct green configuration configuration mcmc simulation use posterior ni twice estimate sample curves green ni ni method setting twice color show two carlo perform finitely posterior discrete indeed error full configuration ni cl cl bc bc tc tc ct ct ct ct ct ct cl bc tc tc ct ct blue green
multipli procedure extensive carry nine absolutely f multiplier routine moment likelihood imply classical regularity van appear q information equivalently respect unknown implementation gradient package detail subsection numerical multivariate degree freedom f dimension define statistic transform fit realization q size parameter continuous namely arise distribution g continue generation determine minimizer scale fix w shape parametrization use random size family multipli goodness carry von statistic kolmogorov rate column
bottom optical flow move backward mainly bottom identify due resolution sample optical flow one region prediction forward backward stop encode colour axis represent optical problem describe turn joint making prediction decide low state conditionally fairly greatly complexity memory incremental multivariate gmm arrive sensor learn incremental g around likelihood explain threshold component covariance optical
weather induce traffic speed conclude road traffic weather trend confirm quantitative conduct devote follow analyze draw speed forecast weather section road link road link classify well road class importance connectivity demand concern c attribute major road road secondary road road road importance road road minor road gps device position sensor sensor car traffic
flow term box constraint switch hour controller take one second pick iterative process low air moderate level process start repeat separate general denote interval usage describe effect relationship average factor xt nonparametric relationship scatter plot energy represents scatter energy hour xt allow probability simplicity quantitie
runtime total runtime prox well primal descent prox sdca derive conjugate sub whenever sign sign therefore q zero q sdca minimize accuracy prox sdca accord prox sdca mirror online therein closely average sdca
try mention theorem run version available page help look conference seek conference format format certain specify instance body copy live area center top cm leave
relaxed setting suggest learn time choose mnist fall completely top tune fall top training propagate interesting competitive energy pooling network stack joint layer capable distinct capture transform auto issue negativity form minima inference different training contribution context convolutional sparse learn manner simplicity integration pool amenable channel produce sparse feature minimize hyper map form give solution pseudo two operation produce large influence small neighborhood constrain make compute weight neighborhood map max except work treat paper neighborhood overlap overlap brevity operation
slightly careful counting index collection size n feature distribution satisfie n assume consistent ordered term uniform among old order derive ibp n final poisson number time draw exist formulation particularly provide infer partition partition specify block characterize survey vast clustering indicate height observe generative dominant specie th region indicate overall weight height region measure block guarantee uniqueness posterior rule carlo mcmc desire posterior prove true equilibrium gibbs sequentially sequence note exchangeable remain z nz nz rule full crp cover worth note exchangeable rule programming induce integer label parameter partition assignment separately lead similarly follow generative feature block let appearance belong parameter parameter review motivate example customer describe economics science describe book science book cardinality great customer book set book sale book picture train picture contain element directly might serve appearance remain ny sampler ibp encoding parameter every requirement argument satisfy consider symmetric argument argument proportion fraction stick
also mechanism template merge learn new template fusion pattern error take final rank system majority vote realize show interest fusion recognition visible color convolutional dynamic seem dynamic demonstrate dynamic rate capture sum powerful literature fusion powerful yet term rate architecture fusion interesting stop otherwise capture reach rejection acceptance score mechanism present verification modality necessary face acquisition kind architecture hierarchical multiple layer mathematical sum min
criterion coefficient index generative reformulate q lie element signal represent overcomplete joint signal simply overcomplete combine optimal atom index non row alternatively indicate solution actually overcomplete use benefit dictionary iterative complexity dimension large least guarantee boundedness origin r kp shortest distance kp kp let nan space I minimum singular value induce support index boundedness need unbounde x
throughout cover exclude special day validation set day keep exclude day correspond public day access forecast period omit unit consumption time hour explain operational constraint concern half operational forecasting lc day minute median graphical representation performance right symbol represent similarity definition several possibly benchmark procedure serve comparison benchmark compound expert size j category model get expert varied behavior expert expert nonlinear approach load effect process weather add forecast seven change gradient temperature short lead group generalize implement software idea adapt consumption parametric priori behavior expert account economic drastically expert univariate weather functional forecast day similarity expert depict bar represent sort value expert scatter frequency expert operational variant correction expert inactive prediction part consumption generate operational expert active period practitioner period length period always active report
gmm color patch gmm patch patch finite mixture al equivalently soft bregman soft exponential family mathematically hard clustered cluster interpret isotropic al mean et mathematical likelihood exponential mixture rate distortion expectation special von duality bregman divergence assign local update convergence monotonically discuss initialization strategy describe basic notion bregman divergence bregman divergence study base generic mle describe section discuss proximity assignment mle mle finally conclude paper discuss parametric distribution q density sufficient denote inner matrix q natural order family order
avoid eps b eps eps nearly method sample explicitly include calculate constraint I base reliable biased hand I large simulation imply I exclude physics mechanism assume former put desirable aim optimal useful information determine maximize bayesian method suitable model one situation previous generalize play robust estimation theory core entropy obtain introduce quantity quantity principle quantity adopt theory advantage prior bias exclude effectiveness
covariance cc correspond separate appendix ht bottom symbol configuration simulate size hmm spherical emission evaluate mse mse conditional evaluate criterion mse indicator classification rate posteriori rule hmm merging regard provide bad satisfying overlap explain merging suit ht well provide good estimation group far show
exponential family map neighbor neighbor two I kt writing form acknowledge advanced probabilistic traditionally exploit width graphical exact approximate possibility highly connect context relational probabilistic clear formally symmetry exploiting mean formulation typically preserve mathematical group concept graphical notion
propose combination mask reservoir store delay use later step nonlinear previous delay hardware use paragraph give brief system represent leave operate constant power nm device encode level optical act delay reach convert scheme experimental setup reservoir reservoir analog layer green part generator amp scope ni acquisition multiply mask level generator correspond add producing digital intensity state computer
logarithmic batch corrupt mini batch online lipschitz bound exchange message nesterov accelerate achieve unconstraine problem characterize present assume objective function lipschitz convex topology message introduce extend gradient present result experiment paper function sequentially receive loss datum subgradient unknown
des est des est centre de en des es dans la est la les la une la la de la est par la ne les les pour ce en
ph generated control hamiltonian reinforcement learn stability knowledge actor near drawback generate actor find balance actor physical setup due boundedness learn currently present freedom rgb university technology cd com g r reinforcement base system intuitive way achieve passive storage consideration calculate complex partial differential pde issue balance loop energy preserve pde matching specification hamiltonian control control law actor enable near policy close loop landscape near standard make perspective incorporate learning thank parameterization
ideal estimate vote house separate political color measure political fit house correctly vote correctly vote big predict hill r vote understand point cut side issue hill order top connect ideal row panel remain segment color still vote hill vote consistently within area education vote political spectrum consistently vote united position hill health care position put odd many particularly ideal hill position classic spectrum certain vote financial policy education take position traditional traditional model goal develop model usual political illustrate kind top panel point various political position bottom position model representative representative posterior give voting available ideal roll classical popularity point encode discuss ideal issue code topic model detail popularity ideal issue landscape represent issue value one describe change example leave left issue vote issue adjustment adjust ideal point vote use find inference find sparsity adjust issue issue probabilistic issue ideal point draw issue
f g hx quadratic compare alm optimization practice guarantee complex admm use default alm parameter need close easy fact five dataset alm admm slow alm mainly alm fy proper possibly non convex gradient method proximity operation main develop max possess advantage term gap aim narrow paper
justify along way carefully acquisition design feature train profile neighbor gaussian error suggest benefit aim future stroke feature computer affect identify resolve design decision use modality modality image front camera usage pattern research support intel science foundation center artificial university user interact phone behavioral raw demonstrate behavioral interact phone e learn user reject current achieve inter test week experimental mechanism long implement part modal user computer device typically user face challenge input dominate perspective interact device usage mobile device short activity email read simple weak increase device completely recent attack break entry successfully quite record eight reading text phone geometric discriminate apparent difference come stroke pressure cover extent mobile device computer implicit would passive user device ideally device complement monitoring accuracy substitute grow body
biological sample center scaling procedure simple normalization normalization independent sparse create penalty option preprocesse example multinomial sparse group sparse give model class cross subset approximately estimate expect generalization simply differently compare non estimator block fit one fit set order compare cancer datum estimate misclassification non right approximately expression cancer divide tumor cell select tumor sample unbalanced see lasso run evident group group however number zero
create loss unobserve datum low payment triangle jointly estimate make assumption feature illustrate mixture however specification framework explore augment assumption specification scale distribution incur loss comprise year variate p log incur joint loss denote year specifie low tail frank member source family give consider distribution payment development year mean prior distribution year portion correspond triangle payment loss incur loss independent precise particular auxiliary factor also augmentation predict payment remark incorporation payment incur loss copula independent model conditional evaluation analytic evaluate equivalent model analytically posterior distribution mcmc comprise gibbs update copula resort augmentation scheme able perform via mcmc hasting section mcmc static mass acceptance likely improve iteration progress autocorrelation estimator integral functional several adaptive distinguish feature algorithm standard utilize kernel past variant metropolis adapt history chain present ergodicity simple ergodic appropriate condition ensure ergodicity two strategy kernel stationary correspond static parameter start condition sampler produce draw distribution iterate tend infinity sufficient converge tv jj pd
estimate causal genetic account event restrict baseline project several attempt call analysis baseline imputation work however account clinical trial variable determine randomization present causal problem encounter clinical allocate follow graph affect analysis outcome explain intend include participant trial per participant include illustrate individual design control age control create structure probability individual depend graphical presentation draw individual case incoming index condition health status dependence take account estimate population
perfect loading possess perfect structure error datum n respectively rotation via loading eq six correctly example rotation yield dense maximum often enhance sparsity employ likelihood estimate rotation take loading give theorem express enhance modify control fitness solution penalty control amount maximum rotation technique near penalization dense model paper apply nonconvex achieve penalty scad nonconvex sparsity consideration
corner fill gaussian mixture nonlinear trade statistical linearization depend dynamically split quantify induce linearization gaussian linearize direction specific linearization performance estimation nonlinear statistical linearization filter bayesian estimation n measurement measurement actual measurement vector process white assume describe mixture assume framework namely step q step transition dynamic dirac delta determine system acquire k likelihood nonlinear exist density component substitute mixture give rise vector replace
obtain survey gradient estimate difference paper problem estimation base result describe related problem fact whole depict figure successively may possibility understand variable difficult obtain completion analytically apply normally monte notice simulation another arise reliability system engineering element keep fail lose work one working maximize expect relation element directly retain sample expect system illustrate depict important may solve activity network network
observe control chart play significant role though difficult uniformly depth range fit gamma distribution fit except lead lack due resort bootstrappe computationally save time fit point pt pt pt
orient thin corner high distant thick play player corner corner corner corner distant present compact reality large always shape move omit pattern entirely interpretation unless context observer course match pca correlate uncorrelated one significant pattern old corner approach pca low corner high modern pattern game perhaps play experiment sample corner immediately obvious visual significant player shape sequence obviously center orient thick play somewhat certainly correlation large correspondence pattern table coefficient pca limited reliability pattern feature carry c significant however extend normalization significant dimension elsewhere accurate contrast emphasis occur frequently require extended normalization notably specific
estimator consistency hold fix supplement consistent generality assume clarity sufficient stand x case impact handle limit distribution direction main corollary corollary usually typical random moment assumption x x bn c bn n n n n r n b n suppose tell theorem estimator enjoy meet magnitude incorrectly estimate effect penalize estimator limit one addition suboptimal estimator whose corresponding consistency normality n assumption n third condition two improve step challenge main region degree freedom asymptotic confidence region keep asymptotic confidence level property condition interest last tail supplement first
node mix graph analog jj jj ij laplacian nothing use w give auc give along hyperparameter paper principle probabilistic mixed propose highlight show usefulness author anonymous section usa provide tend connect label edge connect node ir relational neighbor graph deal mixed well world descriptor I pattern evaluation graph relational unlabele assumption violate depend underlying relational pair
usefulness model education exercise identify relevant covariate party exclude party origin exclude consider summarize htb gender g substantially opinion employ inferential statistic adequate drawback quantify base statistical law use support inferential attempt party outcome covariate model parametric limitation concern specific functional link outcome limitation aspect category covariate assume easy
share share connectivity number chemical perturbation cell line response enhance pathway interaction wide condition highlight cell biological activation introduce characterization response genome scale search modeling reveal gene utilize interaction limit guide signature functional unit network gene involve simultaneous activation signature vary precise whereas observe potentially alternative activate condition since response detection efficient proxy identify state task detect characterize measurement state index signature gene association observation underlie lead n triple w define shape fluctuation frequency gene signature feasibility assumption predefine difference expression predefine channel array use
theoretic rely concept approximate behavior mathematical avoid stable outcome steady behavior alternative diffusion social network see reference possible interested instance paper particular concentrate mention variety problem research computational relatively little graphical address success practical begin availability datum collect result complex individual people individual group system g trading internet individual customer device demand future considerable technical assume availability payoff availability arguably availability observation motivate theoretic formal player concentrate call e steady pure stable come behavioral lead potentially goal aim give graphical deal relative broad behavioral address arbitrary emphasis game introduce theoretic generative game behavioral seem capture validate argue exist considerable amount evidence capture learn illustration please constitute parametric address player help bring increase infer game behavioral player observe player application vote game refer reader far show learn logistic characteristic influence highlight influence member party member opposite party dash line unite tight time current vice display party bottom corner bottom leave opposite third new york arcs influence prominent explanation term allow focus influence business prominent member summarize pursuit framework behavioral game deal player hundred objective line sense steady joint making behavioral might end steady attempt effort behavioral state appendix far strictly behavioral action take require payoff agnostic player unit office company etc like un vote record behavioral entity single computationally provable
outperform standard probe level expression involve thousand rapidly collection reasonably annotate assess set collection type disease public microarray group class singleton annotation describe retrieve scalable preprocesse availability term sufficient available include origin batch approach set retrieve date file header tag laboratory day assess specifically array preprocesse addition probe set probe mapping probe use follow sd replicate change percentile gene change obtain slope slope concentration versus intensity medium intensity high intensity slope fold nominal log change nominal fold change nominal roc positive standardize intensity
loss bin large require reliably increase decision principle working power distribution world quantity set scientific domain include organization power merge branch specie interval volume ice power ten stay within year number plus bin spanning day stay omit wind united state accord enhance ef wind speed united states interval human census california measure amplitude area km per disease associate power fit indicate statistically plausible text denote next std ccccc arbitrary wind ef scale max wind knot knot population city logarithmic km disease gene logarithmic naturally bin raw analysis conclusion analyze primary law plot fit several case include bin boundary entire supplementary conduct claim shape inspection suggest claim summarize include statistical hypothesis law good fit power law behavior law still confirm heavy tailed law stay wind disease quantitie law law quantity fit multiplicative mechanism normal exponential
value addition notice side monotonic computed h equation self strictly sided investigate work regularity algebraic asymptotic relation conservative q strictly usual I characterize proportion reject distribution obtain hand evidence value hypothesis specify asymptotic dimension adopt threshold relation cut actual logical arise leave namely evidence value replace equivalent confidence state axiom seen satisfy axiom axiom nan propose analyze light abstract calculus abc propose abc symbolic know nonempty interval characteristic subset disjoint subset basically calculus property axiom
time es chains change variation geometrically study evolution strategy search es child add gaussian parent call good covariance adapt iteration investigate cumulative adapt adaptation evolution cumulative introduce make exponentially sphere dynamical problem optimum randomly
previous know netflix rating movie recovery impossible study iterative project svd incomplete surrogate lead parallel replacement completion since long minimization quadratic order norm matrix small recursive spectrum lasso enjoy
less investigate pressure genome association diabetes control phenotype think relate c pressure course body mass treatment individual various observe phenotype conceptual status understand genetic marker valuable suitable statistical analysis limit phenotype analyze perform snp phenotype analyze methodology determine associate conceptual truly phenotype variable phenotype longitudinal point suggestion phenotype treatment longitudinal significant little use treat remain replace person therefore methodology suitable snp associate estimate phenotype probability consistently suggest relate snp rs covering suggest minor allele snp latent part
recover normalize view rather suppose stress size svd algorithm subtle straightforward derive b invertible claim say canonical positive return subset canonical proof hmms eigenvector hmm reduction view reduce view find integer sum average document second third moment empirical moment singular vector matrix large singular reconstruct z remark exact succeed run primarily x lda follow sample permutation permutation column normalize accuracy alternative small element smallest relate obtaining currently accuracy probability sample
summation entropy joint product set n py variable obviously negativity information use theorem entropy identical empty conditional define follow mutual inequality distribution special transfer entropy n extension
arm auto intuition like behavior arise force trying already especially stack rbms deep belief net dimensional representation good obtained walk vector modify train reconstruction minus unnormalized small poor concerned try deal certain parameterization represent density trivially everywhere potential find look conceptually argue eq equivalent parameterization denoise yield tie obtain weight demonstrate energy tie lead integral parameterization denoise auto encoder flexibility well connection involve score denoise exist denoise auto particular family denoise yield stochastically case minimize square perform score matching x px width corruption reconstruction eq criterion say function per derivative denoise smooth I appear desirable undesirable since score
maximize optimization arise trajectory seed monotone modular successfully scenario submodular success library become submodular unless note effective people previous orientation clutter also position relative submodular crucially environment maintain property always algorithm contextual control library regressor I n figure diagram train environment row feature attribute environment denote loss slot example environment particular beneficial within classifier input environment ease understanding walk
figure three previously quantity test exact expectation fit progress cm procedure kernel cm together product expectation assign set fast apply exact expectation minima exhaustive search superiority min search slow seem boundary marginal polytope small convexity constant cm obtain mean match without candidate compute namely setup cm consider know
trace strictly well corruption bind corruption max ideal tight relaxation max suggest cluster spirit compress sense guarantee subject input class cluster objective clustering disagreement objective locality ratio cut try globally typically np hard ahead monotone principal component affinity transform advance algorithm might change dramatically incidence iff otherwise belong incidence e block write phrase correlation objective certainly cluster relax
candidate use undirected encode symmetric adjacency entry derivation avoid whose explain quantify community possibly overlap gd entry zero positive scale define encode community interest specify individual basis binary connection member community individual essentially two row conditionally treat individual variable restriction way addition imply edge clique clique information inferential computing involve estimation matrix basis element
seek follow absence light past I time previous play time spike neuron differ neuron base apply circuit spike motivated specific circuit probabilistic formulation employ link see approach pattern repeat count spike align model randomly ensure poisson bernoulli suggest however design setting behavior stimulus spike blue short vertical bar neuron fire stimulus nonlinear polynomial model review spline point view sparse sophisticated procedure overcome challenge theory simple add additional randomly process control consequence pattern pattern cover possible could point process create similar
phase notation arm phase occur one type slightly accept arm algorithm would reject event look arm whose least n type stage arm accept contradiction occur symmetric bad occur arm gap leave accepted arm active
family euler lagrange must form iterative parameter example take develop guarantee optimize updating euler lagrange equation make deviation algorithm parallel hardware factor runtime regime enable parallel update partially kl depend function implement conjugate gradient descent perform
probability surely conclude sum assumption pair standard note update expert ball dual simplify lemma estimate argument upper get inequality jensen simplify last corollary update expert unit dual scaling conclude conclude bandit pointwise also regret inequality arm pointwise maximal loss conclude strategy show rademacher give choice minimax expression compact triangle introduce ball let event coordinate leave far equal event q write attain prove lemma rademacher pick admissible rademacher pick strategy respectively rademacher pick tr admissible relaxation mention equation regret play statement provide vertex
two unsupervise external internal outside validity internal cluster cluster cluster separation tell separate combine since explicitly external include datum cluster external measure give cluster describe fall entropy set cluster compare set set compare paper consist wide web handle etc result matrix run value
life approximately month states disease disease present interpretation state absence backward require disease chain markov admit limit chain gibbs sampler cardinality longitudinal analyse presence specie therefore briefly section indicator presence absence choice correspondence analyse main counterpart analysis matrix correspond associate analogous correspond mle zero set nine target partition six sub scheme give walk metropolis rwm rwm heavily efficiency correctly curvature algorithm sub block current markov scale adaptation scale equilibrium conditional use forward algorithm acceptance conditional disease backward disease particular conjugacy dirichlet conditional posterior
aid economic represent direction comparison performance guarantee serve measure predictive event block iid data increase separation widely spaced block nearly expectation theorem prediction var linear collection sequence ar truncate error truncate need truncate notice analogously risk need row stationarity eq supremum class publicly economic necessary require create four description unit availability service consumption business hour population individually fit smoothing convention series show maximize filter surface rough information parameter constrain lie plausible normal strict estimate low upper corollary theorem definition assumption economic support grant r thank david valuable institute bound forecast many autoregressive move space compete high probability well make assumption generate motivate standard economic financial
proceed close suitable regularization type admit hilbert admit minimizer conversely contain uniquely minimize family admit first uniquely decompose eq minimizer observe setting belong
follow complete proof adjacency example remove direction match relation bind term equal contribute upper ji dependence recall show equality symmetry ij complete eq logical rhs average independent bounding key one example bind start independent noting lemma n bt lemma imply pick pseudo fast long history success statistic theoretical convergence theory multinomial value two area national security name detect physics literature review review partition cut modularity focus probabilistic overlap community perhaps study node define adjacency
trajectory generate simulation agent interact start state tuple action accord tuple proceed incremental solution initial proceed instead weight solve equation solution difference list converge deterministic state know condition violate nevertheless treat derivation implementation deterministic solve discussion regularization equivalent output one consider candidate function search possible candidate parameter weight quadratic solve consider mean regressor one choose approximate denote give submatrix tm point om operation relevant term distinguished unsupervise unsupervised incomplete cholesky
measure fourier interpret borel corresponding section estimate compute accord simply enforce unity level set support consider regularize energy summarize contribution estimator energy observable estimator reasonably balance reduction nonlinearity continue definition key concept energy system well measure support nonlinear theory space behave leverage connection control relationship previously certain forced system reverse drive invariant sde possible object
resort field variation inference recover variable context generative reverse inference variation factor visible observation latent variable think real variable element function unit interact interpret form visible bias respectively energy fully specify energy include factored representation yet highly dependent three binary spike adopt interaction group local define encoding block use binary effectively subspace characterize imagine dimension like edge detector
important develop theoretical develop achieve theoretical programming improve conservative practice develop offer empirical analyze program bilinear term empirical build bilinear programming optimize conservative norm compute solution main challenge mdp consume impractical return parameterized function address issue idea easy optimize small domain maximize minimize policy reason policy bind return base frequency represent fraction spend frequency
random eq generality oracle reliability oracle relate suppose variance comparative outcome distribution side shape realize motivate subject pair aid tune familiar query novel pairwise finite bandit comparison work work algorithm evaluation nearly match prove function empty intersection merely lipschitz bind noiseless bind relevant problem convergence match aim gap understand spirit simplify aid proof respect randomization every sufficiently infimum
set rich propose algorithm sampler vb super sparse interpolation bic bp camera house unless sr patch hyper standard uninformative e truncation image use house apply use interpolation color layer cb cr kind bp compare base interpolation gold sr near interpolation bilinear base representation also image fair change noise variance edge make image interpolation might boost remove beta construction covariate close factor assignment removal bp super improvement bp learning patch dictionary dictionary consist algorithm gibbs sampler train near c c bic bp
curve overlap empty bin become large curve overlap derive mat verify theoretical curve shape empty essentially h curve e mat overlap experimental theoretically mat mse confirm ii accurate hashing without replacement mat curves approximation confirm unbiased somewhat empirical validate hashing add hashing overlap validate expect bin occur goal hashing reduction expect nevertheless strategie empty bin integrated r dataset hashing procedure sample deal simply replace original code almost numerator drawback course strategy couple strategy review detail sec encode take strategy strategy essentially
structure tuning stability estimate adapt selection consider smoothing amount sufficient upper bound empty search problem range goodness fit replacement denote many frequency stable parameter link variable exchangeable factorial model choose take illustrate tuning consider scheme structure four represent column second g mainly fp tp negative precision matrix link element positively estimate tp tn false discovery discover whereas look estimate element covariance fact indicate scenario multivariate time presence absence tune efficient solver infer propose graphical tool analyze relationship universal
combine complete assume q q similar wherein satisfie eigenvector normalize sample scatter xt numerator normalization n recall numerator surely pattern exist showing generalization ability determine training reasonably conclusion size quadratic sample population covariance linear important pattern recognition
lemma b prove bernstein
f compare hence counterpart replace view large increase select also reduce convenient ready u pe pe tune lemma upper achieve optimal explain selection tuning toeplitz toeplitz diagonal exact toeplitz twice except gram first adjacent origin hand compare apply adjacent subscript long theorem b hamming diagram diagram visualize successful impossible density characterize require call interestingly pp approximately number signal rare large hamming procedure recover phase diagram rate hard numerically display diagram b fig toeplitz model view special gram convenient extreme ill pose hand exploit take proper work extreme small eigenvalue carefully detail retain different construct estimate I pe pe j k continue order node behind introduce component split small set separately patch performance cp stand change choose tuning parameter hamming satisfy p tuning exponent phase minimax hamming dominate hamming distinguish isolate hamming distance distinguish change triplet triplet change right hard boundary
experimental split evenly default univariate kernel compare test ol independently apply output outperform leave identity optimize output mass learn scalar ols gm tradeoff numerical class categorization literature target author object discrimination main experiment allow subproblem respectively note negligible subproblem update comparison effectively unconstraine psd cone hour insufficient progress make subproblem precision efficient implementation per iteration solver appropriate rapid progress accuracy fact three classification accuracy highly report
inequality first feasibility admm review comprehensive separately turn field datum machine intensive practical apply method linearize accelerate however classic assume value reality ignore
candidate form h multi map mutual enough set largely risk empirical risk different small hold regularization tr important normalization explain least average operator eq second task task allow fix limit dimensional subspace standard learning support already bring benefit sphere constrain
cone deal simple background gamble contain reduce condition bernstein discuss interesting elsewhere consider framework exchange possibility gamble preference fit impose open favorable gamble possibility gamble fit right discuss gamble gamble favorable gamble correspond strictly desirable gamble axiom look axiom formulate desirable gamble gamble factor recall consider trivially finally combination closure exchangeability reject repeat finite exchangeability illustrative trivial actually form intersection situation theoretic find towards model agent form accept reject one reject framework expression natural extension gamble background status assessment acceptable gamble regular regular side north east inner sep ex ex sep ap rp xshift rectangle xshift south west yshift ex south rectangle yshift ex north east xshift south west right xshift north east ex yshift east rp illustration nine work assessment draw reject gamble acceptable gamble acceptable gamble gamble reject status acceptable gamble cone accept cf actually framework brief overview axiom axiom framework acceptable background conditional constant gamble background ef desirable gamble contain gamble background closure axiom really gamble background necessarily finite closure imply elsewhere axiom say slightly look show correspondence axiom ignore call set gamble gamble factor homogeneity avoid partial sure scaling combination closure turn identical acceptable net use mathematical finance specifically see scale low risk closure requirement fall scope framework deal gamble way via expectation operator background argue restrict attention reference turn connect focus coherent gamble constant gamble identify notational convenience theory real fair first mind agent see exchange either second mind acceptable gamble agent higher translate cf real partition ph ph ff linearity convex segment
try behavioral pattern ip may action attack device business run email spam ip enter email service problem far important repeat rapid reaction email record length historical create informative record window multiple attribute aggregate predict behavior future prediction base property current aggregate transaction transaction estimate base transaction allow send unique property speed improve depict reference ip record g historical window length small historical integer exponentially grow historical benefit record size window window carefully choose smallest cover fs ip aggregate historical record contain feature set fs ip fs ip fs include email actual depend service email record take address error window change behavior spam vice versa play role ip ip change spam versa goal change stay long small value alternate back indeed realistic single behavior ip record train learn classifier window attribute relational
centralized order go observation kullback leibler kl every furthermore remainder describe decentralized detection literature classify systematically fusion second combine fusion center version communication alphabet integer threshold center additionally stationary independent increment detection rule fusion threshold choose alarm recursion multiply unit acquire rate stand rule study take sensor time easy optimize threshold threshold optimal sensor threshold way fusion detect support bit message policy fusion center sensor min shoot asymptotically alternative alarm suggest walk ki since determine decentralized contrary possible
importantly error contribute eqs acquire rate cardinality mind situation n c h eqs experimental model correct object context difference co occurrence context regardless target purely observational value select object select episode eq occurrence regardless refer reader ref mechanic mapping limitation allow analytical extremely bar word correspond storage capability make performance subject control experiment sophisticated capture human observational interestingly value law minimal report child map limitation impose frequency acknowledgment
observation measurement series random target target I evolves denote definition newly target detect target detected finally define measurement detect th mapping target measurement target observation target indicate detect target target detect time target name style name x name name name name x death target observation write mass lebesgue law main aim computing mle also static treat evaluate metropolis hasting move present online important iterative calculate follow step step repeat converge show calculate statistic arise omit notation sufficient statistic estimate tracking relation make explicit later expectation w solution depend matrix target velocity plane direction record outside interval detection reference pf
kernel task task kernel study latent model linear idea allow arbitrary task due similarity neither used perform refers water meta attribute task powerful present shoot theoretical investigation zero shot attribute label attribute internet source attribute paper kind linguistic source google yahoo evaluation approach attribute generalize task category equip category attribute help boost categorization heavily rely share part generalization maximum analysis find share latent modality attribute
eq basis vector quantitie overall partition use connect label unary probability necessary mapping invertible binary previously lead suppose vertex homogeneous unary
addition kde addition also continuity inequality find ne ff f countable k np approach suffer curse dimensionality tradeoff misspecification bias fall outside conversely parametric approximate moderately combine place mass family derive constraint match impose around example probability mass concentrate around result exponential contain added penalty encourage sparsity making within statistic vanish true family approach unknown
agglomerative compression minimize first group program encode membership underlie subspace infer cluster combination point c solution writing combination point expressive combination point representation fact number great rise infinitely nonzero refer combination direction ideally datum nonzero infinitely minimize objective n ij decrease infinity toward solution increase hard problem count number efficiently find consider prefer also rewrite program matrix ni ideally correspond next infer guarantee recover solve ideally infer segmentation build weight nn weight similarity ideal ideally correspond subspace point combination may necessarily particular make sure get post describe approach ideally row whose eigenvector symmetric graph step similarity normalize sparse norm point select euclidean value euclidean point nonzero put emphasis keep stronger sure ssc edge subspace weak ssc connection subspace subspace find lie program corrupt normalize spectral corrupt entry due hoc datum perfectly motion segmentation trajectory corrupt noise entry large
report reflect see sbm model perform one network however network structure homogeneou model show single improvement sbm membership perform single membership membership membership higher less accurate mixed membership show supervise times blockmodel network class mix membership
note momentum represent variational boltzmann weight eventually stability clear dynamical either discrete eq time several neural converge start corresponding point globally unit q specify lyapunov eq case follow
obvious prove number wide web internet tool life activity education virtual online handle numerous piece hardware software together internet evolution user interact evolve behaviour give privacy medium life social facebook boost web usage increase overall store expansion web current facebook million million massive stream digital text attract social communication rely access perhaps equally access although situation web weather long notice reference previous paragraph fairly start piece finding present weather point resource particular make weather uk location way automatic dealing amount interpretation become human feasible form lie notion text world define inference occur simplify amount simple conclusion apply artificial mining field improvement framework relevant context effort purpose project already sub constrain aim goal might seem address proper justification hypothesis course amount subset practice early stage ph project finding initial answer project dedicate develop derive conclusion inference form extend model retrieval processing massive amount aim reduce consider event pattern statistical perform case quantify fraction life measure reaction opinion supervise learn abstract discovery project aim discover specific applicable could carefully depend aforementioned question aim text inference extraction play important several work thesis attract health news medium research public shape perspective twitter mit ball new social medium detect daily release link paper review study forecasting article media observer pp impact research chapter thesis review ph three list provide infer chapter european conference machine principle knowledge database tool detect uk exploit twitter describe improvement publication chapter transaction methodology explore web two uk chapter social network social medium news www france associate score real life amongst evidence connect give normalise document raw term term normalise hold retrieve normalise document ij nm calculate document appear document retrieve inverse frequency use formula scheme nm ij text text basis variant merge something help reduce vocabulary index text approach use removal one make list research research researcher research remove keep character create distinction comparative since automate rule might addition desirable semantic mostly quite article proposition english text stop list generate frequent english manual word automatically maximum word spam decide set minimum frequency depend threshold frequency word behind remove stop word dimensionality increase behind appendix notion scientific artificial intelligence machine scenario input datum regressor show nonparametric bootstrap lasso likewise bag address incorporate whereas give consistent algebraic form entire removing help dimensionality feature statistical reduce mm two first project explain value method project collect impossible derive web website suppose update content resource article video type evolution complementary tool web twitter characteristic character post experimental derivation thesis collect use twitter create new account per twitter allow message see maximum character default private tweet publish account tweet interface type account public account people people distinction public one private sided tweet character user also reproduce tweet incorporate topic tweet start hash something topic twitter refer justify twitter content twitter social comprise sub dense one close friend latter influential great twitter identify vast twitter human short quantity connection direction united uniform population represent activity seek user connect interaction collaborative tweet work environment several aspect twitter encourage interaction student great applicability twitter news persistent nature enable break twitter communication political real life extremely limit twitter rich opinion mining sentiment track intelligence source easily regard political opinion tv start combine particular twitter temporal reaction political twitter preliminary approach due author tweet valuable medium connect twitter build tweet real life occurrence come conclusion degree importance twitter side relationship mechanism allow rapid spread conduct online make behaviour stage work work reason force collect store essential describe reference library really feed extensive language content manner recent development format support twitter use atom content retrieve atom match atom feed retrieve twitter actual tweet feed tag explanatory example date tweet language prefer orient framework establish interface useful already tweet collect study tweet tweet request
lead less problem show experiment achieve report strength joint near neighbor distance mutual reduce effect curse dimensionality effect orthogonality follow review detailed state art wide family section relate description cite principal project direction maximum whose usually difficult determine computed dataset arise threshold automatically
give weight mse assign situation truth ground selection fully gain compete
extend ei ed em hypothese enumeration infinitely often output make collection hypothesis restrict fail symmetric list hypothese output extension hypothesis must
heavily unbalanced supervise classifier curve supervise assign object roc positive roc curve gradually false negative specify classification proportion threshold mainly relative misclassification area roc widely curve lie strictly perform area admit several interpretation randomly member
regularizer realize neighbor classical tree group mini batch neighbor iteration apply parameter split rating similarly choose randomly optimization correction remain testing estimate rmse mae score provide neighborhood affect relation column align neighborhood increase rmse rmse result validation surface fig illustration mini size summarize follow neighborhood surface fig imply validation surface good indicator
eq anti differentiable square necessarily symmetric property directional derivative symmetric value separable possibly smooth non manuscript attention selection denoise rank matrix close
stability yield fix eq compute apply positively homogeneous jacobian first local belief propagation coherent usual bp convergence connectivity theorem impose decrease mutual explain sparse variable prop prop prop prop express field
subject censor considerable back censor also presence independent censor idea paper censor covariate replace rather strong assumption survival censor time iterative base principle mass survival subject censor exploit underlie martingale mechanism quantile simultaneously impact covariate work cope censor employ mass idea cite common rather involved rely heavily important far present paper sensible weak certain kind derive discuss remainder deal quantile yield substantial penalization identify predictor regression considerable context censor quantile develop penalty unconditional survival censor smoothing fix sparse context censor independence censor assumption many inefficient name concerned distribution example predictor
hold choice get maximizer relate phenomenon initial partition row update cluster lin keep make long locally row label profile likelihood iteratively perform converge optimum perform practice likelihood order conventional sort operation comparison algorithmic cost spectral computing indirect proportion misclassifie pl simulation poisson rate pl pl misclassification mean km sim profile likelihood partition separately initialization simulation block row column row column membership block q sparse present size deviation scenario profile criterion method sign although sim
prove existence jt minimize know np fortunately build reasonable suboptimal assign arbitrarily jt existence least possibility text depth ex var var thick hmm build jt cluster name instead result jt height ex var node var var label edge edge edge thick c edge thick edge fig jt jt cover also intersection less illustrative
dark background noise version intensity area reciprocal big pre contrast means variance image di perfect lee whole test summarize form show situation analyze
conjecture strength role determine minimax really around minima motivating convexity strength direct dimension reduce task stochastically dimensional sign exposition interest unique corrupted hence boundary switch point fx g probability linearly ie hence exponent provide obey like active classification query label excess symmetric difference modification proof probably show use see appendix x x originally prove provide nice
forest rapidly forest situation show band look image city area deal apply image manually four manually case degree polynomial employ statistical parameter detect usual shape etc detect four label heterogeneous homogeneous tend heterogeneous approach image spline contour vary degree specify area coarse calculate segment performance term image acceptable addition edge quantify estimator intensity component consistently nine model detail robust estimator contour
standard assume consider th induction least apply choice ks classic imply large project direction fall interval put k constant passive exist hypothesis find focus isotropic non isotropic pass whitening ultimately algorithm output consistent prove prove ks consistent classifier example include whose fall length project density bound constant need chernoff bind draw example fail member enough total number constant factor complete proof point
matlab n inf double os windows inf inf inf inf inf inf inf inf inf inf inf inf inf inf inf inf inf inf mac os inf inf inf inf inf inf inf matlab windows inf inf inf inf inf inf inf inf inf inf inf inf mac os inf inf inf inf inf inf inf windows inf inf inf
probit scale control bayes hold far py pp commonly make e normalize expand taylor binary linear mix binary trait scaling scale true correction back ht ht ht c keyword reference double mixture ridge mixture contain distribution normal distribution scale respectively scale respectively combine random capture sample structure normal distribution list table include refer different effect distribution keyword list fitting height tc obtain error height standard calculate iteration replicate intra family split phenotype computation perform core intel ghz comprehensive many suffice provide rough guide computational burden simulation snps causal snps cd trait height tc cd ghz cpu million deviation calculate replicate replicate intra split table ht error standard posterior ccccc cd il usa statistics il usa mail edu edu abstract linear mixed model widely recently genome study two situation assumption arise specification hyper monte phenotype explain phenotype value combine advantage sparse regression phenotype outperform large previously suggest implement available summary genetic variation characteristic quantitative trait human production improvement help relationship ultimately lead clinical practice present help tackle challenge mix
discard parent additional evaluation reflect trial r pt pt pt pt pt trial pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt
model experiment deeply crucial achieving introduce sharing share neighbor overlap within convolutional filter input filter set work much big image sampling set remove overlap comparison architecture neighboring offset diagonal considerably make set filter offset keep constant filter simple extend follow bottom straightforward stack top express eq discuss due factorial another high state aggregated aggregated posterior sampling consist top
begin ensure penalty focus coarse grid infer apply hierarchy argument satisfy number sake element large combining contradict satisfy penalty never minimizer oracle right equip restrict attention choose class select carefully reason present complete theorem necessarily smallest bind bound turn satisfie substitute assumption assumption verify remark establish sufficient look observe irrespective term due form minimize regularization fast rate computationally inferential modify motivating budget bound exponentially exponential rate use assumption fast hierarchy dimension loss square boost assumption integral consideration identical conjunction conclude budget expect argument heavily estimator process positively correlate constrained set difficult relate previous early perform
multiply obtain bayes discuss validation correspond conduct independently rule observe essentially probability observe individual factor multiply assumption logarithmic scale factor bayes threshold bayes cost decision reject hypothesis appropriate help reduce assume correct decision wrong probability true result bayes collect validation may evidence reject may reasonable alternative probability hypothesis exercise pdf pdfs model calculation helps correct bayes different metric hypothesis test start pdf quantity hypothesis denominator affect numerator standard variable negligible base eqs substitute statistic combine eqs obtain bayes cdf similarly cdf freedom choose letting threshold hypothesis
change likely detector observation likely drift control positive generate false positive roughly stream conclude approach concept pass overhead control although extended streaming predict object whether correct stream incorrect correspond detect drift beyond use error stream treat underlie black neural machine modular drift detection drift detector discriminant scheme detect propose however either change adapt weighted chart recently present general chart stream explain situation present describe chart drift name compare recently move originally detect increase variable observe change stream proceed estimator essentially form control old independent deviation occur away
condition study effort aim efficiently observation respect set backward hmms lead quadratic recursive formulae illustration apply time temperature interest influence height ex node var x var thick thick edge thick edge dash dash start hidden sequence heterogeneous
bootstrappe network enough bootstrappe quite slow fairly tight present classic nuisance present nonetheless derive limit sparse graph simulation confirm information criterion former derivation aic asymptotic tool theory suggest fold global network good way set miss positive inference present principled block ratio block degree network excellent correction point selection network divide naturally module community connect discover modeling range
nod fuzzy memory forecasting forecasting shift memory represent long exploit month value previous see fig fuzzy consistently low forecasting number increase shift continue improve fuzzy step fuzzy shift node numerically qualitative extract memory system successfully learn month information short correlation sufficient essential scale fig order represent fuzzy future begin use forecast distant predict prediction treat memory representation point end predict series begin shift show generate node continue cycle fuzzy memory represent sufficiently capture two shift signal describe overfitte regression extract coefficient require span forecasting perform row red shift ensure split extract coefficient fig compare plot red goal within shift although contain self evolve resource associate complete shift evolve moment subsample spaced extract corresponding series turn forecast sufficient red fig extend temporal memory outperform low come long scale rather pick help overfitte fuzzy information forecasting generalization environment stimulus
take rate regularize matrix affect seem method measure seem converge pairwise identify causal outperform ad hoc left study well world project recently devote causal connection among vector variable exploit present recent compare
adjoint sum may prohibitive particularly surrogate introduce section address expansion either admissible indicate pl expansion former logarithm obtain complete derivation input affine basis typically uncertain main perturbation analysis derive gradient incorporates show analytical estimate gradient experimental stochastic approximation deterministic physics base partial differential numerical able impact extract sample prototype iterative resemble appear rate necessarily well focus special discrete variable development estimator practical setting largely assessment conduct design practical suggest sa sensitivity step choice class comparable substantially sa design approach difference algorithm selection property introduce underlying present bfgs challenge evaluate expansion numerical bfgs sensor involve conclusion relative strength choose sequential allow experiment next paper continuously parameterize space infer indirect observation perform continuous random density evidence assume lead
metric distance similar keep apart ordinal rating label object likely employ label compute set implementation aim maximize average optimization experience local optimum ascent much consequently penalty detail use rating euclidean metric observe feature object object feature underlie metric
produce appear smooth trial trial produce recursive minimization objective indicate level give provide collection uniquely smoothness etc function give kernel smooth hyperparameter determine pt range e relatively large begin specification trajectory pt eq q range let define restricted partition I encourage smoothness partition unbalanced tree equal parent encouraging level less robustness child locally think level encode fine denote covariance notational simplicity inherent conjugacy one hierarchy gps marginally gp encodes define covariance two location contain distance level fall example determine band influential restrict evaluate specific
naive truth high model experiment section problem pick tb l international book com www com co store com com book books book usa model author recall estimate l round round voting compare result book author good accuracy reliability ranking compare accuracy truth value additional source source dependent assign tie reliability accuracy peak gradually
close strength hard optimize energy unary pair ij ji strength pair wise strong energy average energy negative energy poorly bind among used motivate focus swap sim scale character use partially multiscale result run anneal gibbs anneal cc show energy relative energy baseline multiscale combine consistently energy variance b experiment baseline percent strictly energie pair energy define energy perform chinese represent pattern represent energy efficiently test energy unified act primal energy dual energy addition visualization chinese character energy highlight submodular energy make multiscale expand result percent art fraction energy address frame video co
threshold start motivation understand md analysis separability separable case online sample erm erm batch understanding whether rather guarantee arise argument erm algorithm one online erm gap analysis sense mirror provide well possible desirable understand mirror contribution provide online improve regret use agnostic hinge even online guarantee ensure also erm focus choose minimizer rather match low logarithmic require
test homogeneity homogeneity testing homogeneity I dependent attempt contribution research reduce strongly mix mix homogeneity sequence say error immediate type pd n also growth choose result probability condition theorem let generate x least strict linkage target strict generate pair total number statement obvious homogeneity
arc galaxy arc big curvature curve linearity p make arbitrarily long short evident short curvature identical across straight cloud point point nan cloud fact value attempt cloud additional galaxy global projection distance curve notion physical origin scatter remain red galaxy highly property galaxy importantly fact shape depend correlated arc track separate early galaxy tell track material produce recent series galaxy green find formation galaxy separate star star form prominent color emission star formation galaxy population appear pca green density red color fig average keep arc increase except green stay decrease continue decrease monotonically formation emission equivalent drop move emission last pure form region branch separate pure star arc length basically nan activity agreement activity cause formation classify roughly lf come dark matter central galaxy galaxy galaxy power lf finite end galaxy hand shape function particular high mass central whose
far method nystr om transformation need number sample point force theoretically point double rapid bottleneck employ learn fast implementation number old take gain accuracy dataset set iteration take second average second medium test gaussian however tight instead seeks encourage overfitte gaussian solution repeat constraint repeat lie medium dataset median min randomly select scale cross dataset get exist work
within coefficient within consider turn symmetric process nine control chart lr chart apply chart chart chart depend list attain four depend reference bold comparison bad behave control chart sr chart lr chart small sr chart five scheme value lr chart behave good show assumption behave chart change generalize scheme chart chart behave reference chart chart lr sr chart choice magnitude advance chart ar chart structure chart
analysis however important group bi van zhang zhang zhang introduce concept superior standard lasso sparsity sparse recently conduct analysis lasso inequality restrict show bound class group sparse vector error enjoy excellent prediction general satisfy model zhang fan point tend coefficient one selection lin lasso minimize prediction similarly group select false false false construct base without penalty functions bridge frank dt fan li mcp dt zhang penalty estimator least assume bridge fan li penalty obtain bridge scad mcp penalty zhang problem mcp invariance multipli ensure amount regularization one mcp penalty scad adequate attention regression
inference provably even choice take variational bag comparable choose vb objective corpus use corpus trained lda test criterion convergence tp comparison depict vb much take less however vb often overall vb quickly significantly heavily vb suggest vb tp sparse infer proportion fraction infer latent depict inference always sparse topic surprisingly achieve topic reason multiplication solution vb provably theoretical experiment row goodness inference appear
constant give constant determine finite polynomial large simply request build strong classic exact sequence boolean instance small concept test definition shannon term check read boolean e must definition checking test target require test distinguish obtain individual lx generalized basis exist generally simply projection individual checking check test compare alternative read exist bit stand boolean permutation
divided subset site q keyword generate transaction begin mapping site send remain transaction site transaction request site candidate fail generally coincide site class transaction termination home transaction start next choice find user inactive tweet furthermore rarely user
road valuable flow planning road due cost dedicated trace equip match road real store later complex task address discover part road conduct understand movement well reason network trajectory briefly
especially extend neighborhood graph total span tree construction belong enyi entropy simulation number knn knn make neighbor value rp distribute individual group precision estimation accord estimation precision illustrate fig estimation notice estimation rp moderately estimator challenge deal dataset modality filter version
categorical dimensionality thereby share dimension new dimension use encode hyperparameter benefit categorical mathematical lose enable clean software engineering keep track space similar compute combine kernel straightforward use k iff pair input continuous include ten categorical heuristic value sort enable deal decay elimination clause discretize eight base continuous different exclude optimality gap preprocesse categorical categorical categorical simplex half categorical barrier parameter mostly configuration restrict cutoff second run default parameter instance benchmark instance amongst machine model prediction configuration run per node ghz intel bit processor ram cutoff second take requirement cpu year run take second year sound method large yield surprisingly prediction base datum section vary one distribution benchmark amongst one configuration ten machine fold previously unseen rr sp rt rf rr pp rt rf pp rf rr sp nn rt rf additional correlation log likelihood ccccc forest quantify solver runtime varied observe forest
infimum inequality union least discrepancy guarantee combination hypothesis combination weight favorable guarantee lead determining follow regularization standard qp available software practice require bound also hold close discrepancy spam filter union additionally case remain unchanged period cycle task spam political sentiment problem major
classification system produce range I lastly validate anomaly learn classifier transaction anomaly interesting trend classifier effective anomaly anomaly preferable rate anomaly index classifier found probably offer six apply variation distinct classification gm classifier across test rank auc rank rank result construct multivariate show dataset anomaly support anomaly base high anomaly curve reflect effort learn effectiveness additionally converge amount current contain transaction derive transaction divide subset transaction next partition sized construct finally result improve increase improved achieve monotonic dataset different gm reach classifier examine domain univariate approach anomaly outli outli anomaly unsupervised anomaly none instance apply one paradigm output algorithm calculate factor instance high local instance local suitable purpose outli acceptable widely repository type two prominent similar
process minimal maximal know degree structure process well series reveal periodic maximal randomness possess degree physical quantify statistical measure effective measure distribution detect
even summarize used algebra operation code segment efficient high additional execution high gpu processor cost use two cpu implementation matlab cpu call implementation precision cholesky hypercube scheme implementation power recent release relax allow section challenge effectively gp repeatedly solution lead suppose generate
nc edges superiority ht connectivity model see nc nc ad graph simultaneously multiple maximize log necessary condition block screening develop efficient multiple employ multiple scheme propose demonstrate efficiency screening explore use direction shrink method warm find initial improve possibility section theorem remark multiple graphical simultaneously fuse penalty encourage graph structure motivate brain network brain control brain patient cognitive ad nc identical brain formulation contribution condition decomposable key property screening present subgraph allow subgraphs cost demonstrate effectiveness efficiency undirected graphical explore relationship vision analysis
dataset trajectory evaluate importance drop vice versa compare segment analogy trajectory similarity graph similarity graph weight graph represent e e contain trajectory loose devise strict similarity edge e segment mentioned similarity graph discover road segment cluster analogy segment generator briefly direct along unique loose graph edge seven road across region
reduce twice ten cifar convolutional pooling follow layer neighborhood pooling pool layer layer unit softmax layer class convolutional cifar dropout strong regularization fourth pooling layer layer layer filter dropout softmax input weight imagenet dropout cnn horizontal augmentation help seven layer convolutional pooling layer layer pool convolutional layer pixel neighboring neuron bank filter input input pool response output group connect channel convolutional layer image filter pixel two max nonlinearity fourth convolutional layer one normalization third filter connect subset channel pool convolutional layer group unique channel produce layer globally neuron layer
negligible hour subsample large collection expense parallel convergence training invoke friend user single power scale facebook shall analyse sm multiple modality arguably single metric sm sm hold collection interest context capture user interest task easy latent like allocation lda visualization normally improve classification large amount train bad bag word stem dimensionality reduction necessity picture aspect link label small collection way use sm sm avoid dimensionality text come play statistically plain baseline setup ground truth interest sm predict vector user exploit support classifier input cross four collection take hour train normalize word document summarize
ep via factor approximate factorize distribution natural property exponential interpret factor exact link message repeat minimize I information incorporate update w kl projection datum clean kl minimization relaxed propagation factor describe relaxation iw I much relaxation relaxation unnormalized replacement adaptively handle outli accurately I
easy acyclic relaxation coincide polytope subgraph keep exposition cover g detail preserve idea involve decomposition straightforwardly subgraph contain grid graph energy energy uv represent way function q smooth subgradient I computable constitute subgradient one primal kind smooth g soft approximate acyclic vector marginal I q subgraph consider special positive tree reweighte free energy show e special acyclic depend probability gibbs maximization primal estimate iterate vanish entropy gibbs associate subgraph entropy combination free separable r
structural network combine acquisition strong similarity applicability recommender simple regularize practical beneficial knowledge challenge path various differ hand variety processing decade measurement suffer high cost low accuracy large infeasible two contribution rating network completion quantifying path investigate rating ordinal large advantageous metric rating carry streaming
edge hybrid feature combine neighbourhood explore great two language l exploit report implementation cl k n n set accuracy propose adopt approach row mae five obtain statistically test set method report result predict adopt distribution report probability rating number rating ccccc e error report mae adopting distribution unbalanced svd c c c
model follow l l preliminary positive risk multivariate student shrinkage estimator sign stein hypothesis classification however implication exist however pearson multivariate student multivariate cauchy distribution concern particular mathematical al et estimation parameter belong sub stein preliminary belonging describe
suit situation different consist random size let population assign sample perform detection aforementioned nonparametric rank empirical candidate straight maxima note identify section simulation assessment edge run computer intel operating programming language version graphic produce shall parametric situation rectangular window distribution window compose use span variety encounter practice range smoothed look situation edge algorithm differ texture situation window detect describe absolute difference locate store array
lin undirected dependence concentration fitting matrix empirical variant depend want element unobserved jointly effect source instance observe stock price price two term marginalization variable second proportional since trace
achieve rate source distortion design matrix varied trade distortion length space per distortion distortion decay sparse achieve distortion computationally encode square distortion compress sense compression shannon rate long goal information storage codebook complexity channel superposition essential terminal code build computationally theory develop computationally compression show function source small near length rapidly distortion encoding source alphabet code rate decode excess scheme slower survey paper technique quantization code attain
control remainder organize apply inclusion principle analytic poisson wide application address problem binomial variable rule possible effort eliminate confidence limit stop result appear conference throughout notation integer denote notation proceed propose analytic stop random specify confidence level estimate virtue sample number sample define absolute integer n sn rule nn n mean termination unnecessary rule purpose sequence n n eventually stop termination sampling procedure stage integer sampling sn appendix stop
suppose fire remove table partial fire let manifold equation small converge h ise model experiment compute seed identically distribute sample fig kl kl expand ise seed computation learn retrieve structure define follow stock
serve binary certainly gaussian clique ise deal interaction essentially uniquely table number node graph graph bernoulli ise multivariate member predictor eq covariate build model predictor valid log likelihood q generalize regression thus logistic coefficient vector square implement nevertheless hessian fisher pt negative multivariate logistic two coefficient hessian negative
cr award american google supported award fellowship nonnegative row one factorization permutation write offer strong coefficient row suppose great reach line I number tell representation row distance great I else introduce extra monotonically extend hold row list may well nonnegative assertion contradiction minimize construct satisfy representation c kk follow occur constraint conclude sum fall factor separable
particular point expression moment follow moment triangular set equation hold thus formulae direct gr basis elimination term generator basis generator parameter map substitute previous alternatively check substitution compute expression considerably simple probability gr basis execute coordinate factor define simple except denominator reflect meaning factor iff iid coin unlikely one iff hide subsequent model flip subsequent two model identically factor occur equilibrium define equilibrium hide restrict yield parametrization parametrization newly occur vanish lie meaningful matrix computation gain specification affine minute ideal take memory gb exhibit parametrization trace version formula generator integer marginalization
hellinger read parameter large useful actual parameter know appendix b paper subsequent repetition estimate though improve increase testing draw seem feasible probability distribution discrete focus testing assumption latter application display fit significance mind section simply consistency trying decide likely false try alternative handling nuisance always exactly repetition extreme substantial nevertheless amount associate seem experimental enforce repeat integer value value value value combination possibility know proper priori permutation specify bin entail sort frequency whether simulate many freedom permutation compute assess consistency simulation draw obtain root square run simulation conduct follow distribution experimental generate step calculate calculated empirical
assumption meet aware standard gibbs reversible reversible irreducible irreducible undirecte binary symmetric indicate fast sampler graphical part sampler state probability state standard gibbs directly gibbs sampler move standard fact analytically fast show symmetry fast several stochastic variation chain rather argument simplification method theorem value metric couple markov markov chain numerous application considerable amount devoted couple insight coupling graph
experiment fig confirm nonzero unit nonzero normalize specify averaged setup dynamic figure dynamic reason outline section experiment report magnitude three see thresholding almost performance comprehensive thresholding result enhance understanding approach linear response future extend fundamental bad coherence model orthonormal banach banach space value
number loss quantity positive table paper result density q formula joint variable evaluate visualize investigate copula degree copula precise loss continue marginal distribution display copula leave skewed density multimodal become distinct skewness readily write skewed mixture skewed mixture unimodal gamma modal influence figure display quantile numerical integration root solver setting dot line indicate claim number independence monotone association equal gauss frank copula grey solid indicate expected policy
brain prove brain region recover challenge recovery ill address univariate predictive domain fit use logistic lr multiclass problem quantity amplitude fmri linear hold estimation suffer particularly signal
use q depend complete pack proof quadratic previous bound number perturbation sphere complicated argument aim proof hausdorff hausdorff v follow relate hausdorff distance hausdorff convex every pair clearly gb gb fx gx
real number finitely arise finite latter exact arithmetic mathematic solve thus quickly approximate answer lead notion us truncation actually algorithm exact error tell run moreover forward forward bound backward expect condition much optimization scientific seen develop algorithm well pose unstable turn much partial history recursion calculus machine lead belief formally capture qualitative highlight role logic study turn rich many algorithm run lead notion np np hardness np completeness lead qualitative question turn problem intractable np sense sort provide possible several way prove algorithm relaxation relaxed program combinatorial convex duality implicit refer meaningful interested want learner sensitive quality interest provide meaningful ill pose solution meaningful noise one function specify geometric exactly parameter solution quality regularization manner lead natural trade optimize fitting though often hard
bound similarity learn norm term inequality advance secondly function specifically loss square frobenius bias zero minimizer xx equation z f combine together imply theorem c consequently imply yield consistency true function application aspect equally notion natural
correspond tend theorem clique clique disjoint contain node therefore recover plant clique size clique size clique clique plant clique model agree exist minimum plant clique provide match guarantee consequence exact bipartite subset subgraph induce bipartite disjoint subgraph set subgraph divide root equal disjoint disjoint density subgraph want characteristic sum disjoint clique pose nonconvex quadratic program let w relax semidefinite program block index nonconvex clique instance disjoint graph know weight edge complement vertex construct sample two plant define feasible objective describe plant semidefinite disjoint subgraph complete graph feasible sample plant scalar scalar maximum density disjoint tend exponentially tends plant comprise modification accommodate
component sufficiently incoherent compression restrict isometry hard subject henceforth admissible sparse draw certain random satisfy random collection restriction amplitude sufficiently formalize exposition henceforth accord element row q high practice specify take whose value recover coincide follow bound rate bernoulli addition incoherence constant generality inside special case hence expect incoherence subspace eliminate theorem section compression give rise fall column space admissible take account vice versa could condition deal benefit incoherent price e follow cauchy likewise obtains follow incoherent generate form uniformly collection partial independent incoherent cf clear incoherent suffice small close column unity attain single element aforementioned compression consist unitary entry subsample select ensure require
marginal specify numerically maximize routine conjugate gradient optimize type ii alternatively equation prediction computational phase hyperparameter marginal expensive involve input cholesky decomposition computation computational part depend hyperparameter evaluate ccccc storage variance complexity reduce g subset away induce plus nearby iterative speed give fully conditional recommend improve fast transform amount ignore complexity full nature paper gp gp apply size description possible alternative randomly cluster point complexity suitable
suitably scale convex hull norm carry kind induce matrix imagine take matrix hull general relaxation tight trace norm ball obtain trace bring paper condition raise obtain oracle poorly well effort algebraic nonetheless oracle estimator rank tractable viewpoint
ik definition approximately control theoretically improved note design must know prove test take perform compare control make design point mostly concentrate point stage stage empirically give trade design median fine scale include coefficient scale threshold plot access set family wavelet basis implement moment dot rough recover shape rough improve point level visually large take plotted run median design outperform consistent compare function level together present difficult design block significant still conclude spatially acknowledgement anonymous valuable
detection add anchor ii projection square obtain face cone current normal normalize lie hyperplane tb algorithm separable nmf separable ray point criterion cone residual identify extreme lemma column cone extreme cone form lagrangian tr lagrange project current extreme current kkt use lagrange hence ti right side ji I regard correctness add maximizer extreme ray select iteration identify extreme current diag strictly positive inverse j strictly positive I jj least remark maximum generate extreme add anchor point
step ise ise seem cutoff critical approximate question good sample output control approximation make close repeat median building calculate ise unbiased variation w
ise discrete mrfs slice image mrfs investigate slice field particularly suited segmentation generally intractable grow exponentially hyperparameter homogeneity image large contrary homogeneous region interesting know drawing achieve use introduce correlation may hide deep markov field gauss adapt memory resource introduce pixel number priori kf associate mainly depend application explain fix image noisy set produce bayesian propose assign coefficient work unnecessary priori theorem follow proportional unfortunately mmse unknown think receive alternative use sample
q sparse notion subgradient restriction minimal need acknowledge existence restrict imply convexity obviously subgradient determine invoke axiom subgradient drop context slight abuse terminology restrict linearization sparse function define align say linearization form strong rsc restrict smoothness condition quantify bregman divergence similar rsc convert bregman divergence rsc vice various rsc important difference bound bound globally unlike rsc require vector subtle difference invoke rsc neighborhood parameter require curvature main result furthermore suppose ii magnitude indicate estimate close magnitude analogous guarantee noisy accuracy obviously properly imply statistical datum target rather
iteration linearly sg iteration cost suit sg optimize iteration train uniformly among yield assumption suitably decrease step size sublinear take write sag iteration momentum sag recent momentum lead decrease momentum gradient sag gradient dual lead size rather author sg value asymptotic efficiency newton rate step size combine iterate display appeal epoch sg average objective remain sublinear various option accelerate smooth accelerated method conjugate hessian free rate deterministic discuss sg decrease convergence part sg iteration iteration basic sg size relationship work converge additional accelerate sg related advantage
furthermore point gram form pass alternative although reasonably sized prohibitive application realistic system however schmidt obtain advantageous pre computation e dynamic carlo dynamic integral evaluate analytically operator act state show integral term act operator efficient perspective path carlo aim estimation allow maintain
process intend enhance noise scene result wishart trace distance divergence neighbor present region come produce reject reject filter
stationary fundamental compression world assumption implicit rather modify stationary promise meta automatically generalize exist suited allow low begin terminology sequential source alphabet finite empty symbol string concatenation source define satisfy compatibility whenever define familiar rule j notation describe partition segment say overlap another exist segment notation piecewise stationary source partition source
mean know estimator regret raw moment median bind moment good trade estimator computational indeed constant update median linear question mean good moment requirement variant heavily rely unclear similar tail bandit focus attention concentration work decision nonparametric monte
trust significance reason automatic new remarkable cosine trend phase fact time perspective clear part cosine model properly course hoc remarkable high order small improvement clear quantitative qualitatively essentially decision could high statistic coupling deterministic trend synchronization relation model mean consequence highly make begin hypothesis hypothesis try find usually situation neither decision whether exclude almost daily cycle exclude long phase time project synchronization support university grant member mathematical image joint phase synchronization grateful extensive discussion synchronization view plot plot phase deterministic term phase process close straight call stochastically nonzero exist couple trend deterministic trend couple relation case trend equilibrium involve correction equilibrium reference international journal potential international journal co correction university synchronization circuit physical phase synchronization extend system synchronization report l dynamical ed mathematical series digital process david large synchronization international journal r detect synchronization autoregressive ratio statistic root make var review white forecast biological neuron k co representation synchronization regard brain region abstract sense equation system one include
model post processing rule validate example curve algorithm complex controlling different try obtain validate performance base benchmark median variable validate accuracy display rule couple htbp accuracy tree soft soft previous input soft rule input see use marker marker code
candidate set ei simulator output illustrate example candidate maximizer ei grow design summarize simulator simulator output select candidate ei calculate ei ei simulator condition meet go initial run aid assess space step step gp ei integration stop tie remainder evaluation simulator demonstrate dimensional approach present global average performance base stationary gp performance design shoot simulator output estimate increase result present section I design ei sequential ei ei ei instead ei ei ei outline operational setting section setting discuss ei follow
go able category bias go recommend method address case guess absence strong estimate directly use fdr assign biological category reliably estimate biological category local discovery identify promise maximum find mle perform order go ideal mle category component thus order representation application method recommend category discovery
offline e vast volume learn kalman filter model online model time step spectral model sophisticated incremental make unclear operate architecture demonstrate parallel robot scheme overhead paper learn share behaviour robot gradient td hundred policy unique develop online base bellman computational architecture correspondence traditional mean dramatically scope life immediate work learn learn control predictive employ e g feature robot learn useful al thousand accurately mobile great
look co two gene multinomial moderate large collect hyper parameter combination attractive might certain exploratory em mcmc model surface drastically capabilitie simultaneous protein per cell sort surface homogeneous cell characterize technology array single device enable cell hundred classic cell intrinsic nature result expression heterogeneity carry important expression population heterogeneity provide snapshot system multiple simultaneously heterogeneous whole monitoring cell intensity typically thresholde subset boolean combination positivity biological technology single cell thresholding cell subset boolean large
eigenvector define easy frobenius bind n specify nystr om om frobenius besides intrinsic low cause nystr om measure frobenius large spectrum examine nystr theory paper structure nystr
standard ht homogeneous h eq standard observation eq equivalently ij
consequently situation justify possibility b rule attempt next stage ir scan attempt detect identify case person stage unnecessary indeed similarly diagnosis disease stage sense begin early treatment stage cascade measurement stage complex region complex measurement classifier illustrate advantage scheme sense modality centralize reject utilize nd sensor achieve centralized htb approach call sensor sense modality derive reduce cost acquisition sequential reject measurement classifier cast dp stage stage scenario produce consequently unlike estimate adopt parametric utilize go discrete instead go function formulate utilize wise evaluate decomposition formulate erm stage decision transform reject
paper portion final find describe section graph throughout paper distinct edge sum edge unweighted roughly speak boundary number weight edge vertex add create remove create strictly boundary complicated remove strictly decrease satisfy outside every edge outside great edge outside outside illustrate graph half place cut
interaction trace specialized call web usage mining mining user behavior website place context extract file web usage trace become critical website management need build site help retain current cross sale track leave logical service traffic flow etc account entire usage trace
million sensitive visually variation patch image instead reduce color effective efficiency color ambiguity randomly raw effective image size virtual generative many object recognition raw element color retrieve
political structure even usually refer configuration agent network height network measurement agent small send parent agent message message throughout make information aggregate information convergence star criterion converge exponentially exponent star tree unbounde height binary leaf distance minimum leave large message unit likelihood total probability convergence tree combination gives learn social elaborate rate follow rule break odd rule optimal however majority achieve strategy show majority question close rule differ way fusion become next consider learn agent level generate message send parent intermediate receive message send place decision make information agent aggregate maker
reduce parameter vocabulary reduction sample singular matrix small become increasingly hard number dimension point write u likewise u range u product make substitution j noting substitution get mn proof three suppose take pr pr less trivial iid hmm lastly bind square assumption algebra mn minimum bound
weak equilibria obtain hardness corollary utilize tailor game outcome utilize weak hull nash equilibria equilibrium calibration calibration tx td nash follow weak return ne constitute randomize arrange lemma
convex beyond paper big multiclass become increasingly machine algorithm establish much understand multiclass collection binary base relaxation relax convex statistical quantify incur derive explicitly relate excess misclassification excess exhaustive presentation generalize class several indeed interpret relaxation linear offer speed work due function quantitative similar restriction notably classification might lead consistent restriction
minimize norm risk contour risk evenly risk happen end begin high take respect parameter risk risk sensible solution place emphasis task strategy particularly appropriate learner relative amount observation minimize risk subset compositional relaxation minimax case exhibit situation small fraction fundamentally hard remain task reduce level minimize maximum minimize soft intuitively formalize
last term note assume constant confidence almost happen space choose tend ball small close almost domain f e square weak heavy tailed bad optimal investigation algorithm paper pair form construct rank functional define measurable barrier verify shannon minimizer totally fx barrier difficulty analysis overcome full feature large first
heterogeneous various entity mass fusion aspect improve one integration quantity model gp objective gps gaussian process dependent gps extend handle multiple technique exploit spatial correlation perform simulated need equation represent regression uncertainty subject modification value individual model location east north depth datum may demonstrate stationary auto set covariance set gps consideration respectively convolution detail evaluate gps likewise concentration equation incorporate output gp noise hyperparameter auto heterogeneous output main cross auto function gps address cite kernel isotropic relationship covariance smoothing may develop model smoothing smoothing transform stationary nn kernel cross two correlation respective mathematical formalism mathematically white gp smooth white show stationary covariance covariance equation smoothing ix square e suggest covariance convolution two covariance
dimension probabilistic produce solution problem algorithm offer representation exploit minima result provide minima outline regression problem examine serious analyze understand
far apply inference need hierarchical bayesian undirected discuss trivial either unconstraine change set parameter resort procedure could solve method focus whole theory deal factor include likelihood function general trivial constrained see example present theoretical use concrete develop restrict linear operator banach space ap inference fp subsequent statement tool primal relationship optimization general space result duality banach space conjugate duality banach function bound satisfy either supremum dual banach duality apply density summarize constraint mapping product banach function observation kl relax kl regularity condition difficult check hold examine depend correspond duality posterior compute though representation operator feature convex optimum multiplying term prior distribution coefficient optimization problem remark put depend model extra contribute prior constrain feature otherwise regularization theorem prior widely use bayes implicit bayes rule define could integral implicit flexible model constraint systematically domain automatically piece knowledge allow incorporation knowledge knowledge especially expert normally low behave skewed worth although generic usually make assumption primal solve apply along development model inconsistent present leave systematic consistency future example appendix first classification x put use
classify active inactive divergence instability solution perturbation break replica influence instability discuss later first second former provide indicate success information extraction guide marker inverse classify depend infinity keep around plant locally otherwise along form
step step sa stable sa step path imply imply everything omit sa stable even omit stable doubly proposal iteration implement walk metropolis hasting v target pdf distribution panel diagonal equal artificial aim switch compatible adaptive illustrate behaviour multimodal two weight represent version particular distribution solution near coordinate recover marginal make mcmc discard burn assess look restriction efficiency autocorrelation sample histogram marginal quite well figure unimodal number enough explore original seed hand broad seed indicate autocorrelation reference function mcmc ht pdf panel depict thin line indicate appropriate correspond background
turn predictor occurrence report recent spike manner even signal potential could combine discrete train continuous scenario conceptual analogy spike basically sum exponentially spike spike train apart decay calculation local corner spike spike rather elementary regard complementary spike spike lag sensitive instantaneous instantaneous lag eliminate pt spike train spike directly address analyze dataset contain train spike bivariate average distance identical train dissimilarity profile profile come linear small instantaneous spike train equal rate reduction level detail instantaneous spike train result spike train movie material two possible spike train dissimilarity whole whereas average lead overall spike train interval application mind appropriate whole onto single might high train could intermediate useful high local averaging spike instantaneous exception spike spike global picture fact different resolution slide window spike dependence spike account spike individual spike among spike contrast measure pool histogram value regardless spike train qualitatively reliability difference ratio train computer memory concern profile successive time absence train instant depend
economic recent blockmodel connection respective variable blockmodel network form fitting blockmodel block connection probability histogram nonparametric array term co article co misspecification generative separate exchangeability significantly generalize blockmodel generate definition vertex parameter affinity connect network observation identifiable hence indistinguishable preserve many model recognize present constant co blockmodel specifies approximation blockmodel exchangeable follow sm tn ij give blockmodel parameter loss blockmodel latent vector class membership index blockmodel class separately exchangeable blockmodel task equivalent fix co task estimator categorical blockmodel contain triple whose subset contain measure construction cluster array partition likewise leibl estimator
perform analysis stage aic standard regression summary ridge may attribute robust handling adjustment illustrate use summary top panel bottom panel infinite visual clarity open figure square region almost increase rejection relative unlike term statistic adjustment indicate efficient may concern model production occurrence term block largely large inclusion extreme whereby intersect point inclusion distribution inclusion generalize shape extreme consist number total create exploit dimension technique consider spherical shaped focus priori subset six consist empirical quantile inclusion complete due value sensitive precise quantile quantile close maximum subset substantial alg none pt number pls regularization integration cross moderate regularization ridge weak solution method include square pls ridge ridge neural network introduction summary statistic adjustment column good general offer provide slight improvement substantially along statistic consider computationally exhaustive enumeration neural adjustment
convexity valid display association q display minimax theory q note measurable test direct upper bound explicit address match small compute symmetric diagonal adjacency undirecte find whether clique np semidefinite lambda semidefinite programming scalar program canonical use point relaxation nonconvex program prove major relaxation day though change yield z source relaxation semidefinite trace schwarz drop original optimization linear canonical aforementioned relaxation building early somewhat first inequality remain stay use dual duality direct functional property variant detection deviation st ij follow element bounding lemma taking probability event yield
expect good reward reward empirically minimize regret principle rational maintain find arm compare arm high mean effect single estimate
norm frobenius row index variate draw q unbiased I well know inverse sample either good introduction liu estimator constant decompose estimator obtain put column possess desirable liu rate drawback adaptive tuning need drawback minimization drive adaptive variability motivation p sample eq major upper hard hereafter entry denote step modify procedure take account individual note jj jj side relaxed estimate conditional estimate variability individual dependent
expect capable demonstrate extensive computational great deal many study extend exist study conclude remark component denote diagonal form index set index row index addition semidefinite operator low minimizer nonsmooth approximate threshold play crucial role section point derive nonzero stationary minimizer order similar general stationary statement hold hold yield multiply twice recently chen derive interesting nonzero minimizer special next also minimizer stationary twice
trust solution relaxation lemma therefore equality calculation omit f x denote x x r f write ty poorly area strategy favorable environment actually adversarial guarantee speak dispersion visit overcome problem generalize deterministic environment finite space work determine bad trajectory lipschitz trivial compatible function search expression worst possible return lower previous depend dispersion trajectory conservative investigate np stage np exactly preserve nature generalization lead guarantee drop problem polynomial trust lagrangian quadratic prove relaxation
quasi analysis necessary answer devote induce connect underlie theorem like admit well exponential type nr sx field n sx I dt concern every exist essential role random index large estimate estimator order convergence dimensional wiener theorem generalize n sx handle I quasi likelihood analysis type estimator moreover type latter practical application pursuit field question h discuss involve set let c n ij pp fact exist c kn u pn kn p u relation c write number cn j complete c k fulfil fx jx kx f admit expansion jx jx fx x x near compact admit support support one dimensional jx j onto easy fx
outperform recovery considerably sequence compressed comparison task mt robust cs computationally efficient affine cs admm robust cs good fista robust considerable huber despite fact slow need recover compressed term speed triangle inequality immediate mm mm assumption lemma compress cs cope corruption cs model combine robust cs recovery newly formulate cs solve rather inefficient cs limitation cs cs efficient advance non furthermore formulation norm though fista cs fista additionally solve cs formulation efficiently differ fista update extend number affine norm powerful cs cope extension cs include fista provably admm cs formulation
validation fit publication com component careful standard cross fail pca characteristic present pca idea predict auto many extract manner locally project som describe curve subspace replace manifold nonlinear mapping focus auto technique technique approach
regard final assumption smoothness constant z region cube sake brevity investigate kernel gaussian precise result satisfied nx long defer decay fx random quantity fx nx n fx nx band width ill hand grow system fix wise indirect price uniformity easily function band extend time define df ty imaging function correspond model statement add fourier consideration fourier transform transformation nx tt db g k h tn additional asymptotic require kernel assume mn mn mn mb k moment satisfy mb mh ta h
cluster apply become heuristic picture state desirable large depend geometry example room state seek partition directly worth coarse goal macro state fine coarse coarse average path extent capture fine sense markov determine hierarchy shortest particular coarse coarse generality rich transfer hierarchical framework overview current relate concerned discover specific possible transfer share constitute correspondence reward task instance option transfer abstraction identify carry kind transfer set common formalism specify mind immediately within also eigenfunction domain trajectory eigenfunction function define varied since coarse fine within mdps expand describe derive either transition result value may store library multiscale mdps paper contrast procedure support transfer coarse scale handle framework policy transfer challenging scale consideration refinement representation always mdps scale present knowledge transfer mdps multiscale center hierarchical coarse problem mdps local fine coarse scale argue multiscale efficiently solve transfer localize solution consider localize scale across would expect effort subsection generalization compression access pre model involve average need relaxed entirely multiscale compression include completely set bottleneck detection regime initially local heat evolve exploration starting need estimate coarse transition reward process detect bottleneck proceeding picture successively add compressed simulator could make reach state multiscale may approach discrete representative state discretization general dense prescribe geometry largely multiscale translation laplacian build trajectory coarse mdp handle continuous discrete continuous fine include policy factor slightly usual action reward discount adopt take collect transition mapping allow later policy thought satisfy action deterministic placing unit action explicitly track avoid matter constraint need assign policy discuss policy restrict feasible suitably markov action accord tensor hadamard finally denote take continuous space haar measure volume given define assign discount horizon take sequence action bring expectation ps action discount criterion choice commonly function compute dynamic ss stationary policy achieve policy whenever policy primarily mdp determination usual approach conditioning apply markov stochastic transition discount see one green analogy markov govern restriction subset unchanged restriction sub definition outside operation expectation respect restrict respect truncation action although efficient fact define quantity locally interest quantity restriction lastly vector scalar follow individually detail connect bottleneck whose certain policy compress mdps mdps fine mdps multiscale construction perfectly next fine compression enjoy scale scale fine scale high thought level abstraction original novel mdps yield significant roughly mdp compute across section establish mdps solution problem problem transfer devoted subsection subsection overview concern implement latter read big picture proof subsection appendix detection partitioning mdp c induce equivalence yield cluster equivalence plus bottleneck state denote associate markov policy consist reward additional reach say discussion make embedding cluster visualize reference graph
power allocation pa vectors simplex work action power amount level author point performance efficiency achieve substantially report contribution level proportion mean rely local nash ne organized sec wireless formalize sec present formal analytical property iteration ne well fraction ne validate analysis c sub
ascent may expensive attack optimal efficiently terminate predefined validation follow section method dimensional evaluate effectiveness mnist handwritten digit recognition first generation covariance assign figure otherwise consist class experiment blue serve start refine attack termination attack plot rbf background surface explicitly hinge loss
panel green line interval great show load terminal graphic ltb lt lt lt lt ltb lt lt lt lt bp highly noise quantity power panel package load package package graphic explanation terminal need graphic macro ltb lt lt ltb lt lt lt lt lt lt reconstruction quantity describe panel great log spectrum row logarithmic version dark reconstruction right bottom bottom uncertainty without white theory linearity simplicity effect observational mask uncorrelate one sec first discuss sphere periodic discretized case normalization power spectrum show fig highly variance panel reconstruction panel reconstruction apparent former homogeneity since impact higher demand region well represent uncertainty high linearity signal variance nonlinear high level reconstruction sec reconstruct usage partly prevent scale drop dominate still directly affect uncertainty lack prominent signal dominate reconstruct dominate signal around pixel linearity
compute similar u j implementation size cluster represent update concatenation pixel alternative consist whereby sparsity impose propose context specifically ensure fit pca provide coefficient coefficient eq soft investigate alternate method multiplier algorithm adapt poisson though practice strategy factorization extract fine consist avoid cluster improve partition approach result redundancy region perform patch similarly enforce inside patch size matrix factorize author cluster drive patch increase load improvement particularly noise paper compare similar adapt clustering
tail thompson close ucb thank bernoulli thompson sampling apply bandit therefore ucb complex setting computable efficiently use encouraging conjugate suggest pose challenge often heavily dependent sample give work support national project european community grant pt pt claim corollary remark proof question optimality thompson open positively bernoulli comparison
figure vs sample value cost ucb sensitive h cost minimum show exploration greedy well exploration sampling prominent h aware empirically greedy figure arm sampling occur monte carlo mdps improvement numerous application argue perspective cr scheme attempt regret inspire ideally ab well speed aware
dominate state fix let yield truncate appeal theorem provide kf fact already nevertheless arbitrarily large smoothness form manner formalize rise approach normal analogously discuss gradually achieve imply arrive immediately binomial converge converge marginally unity far q admit smooth serve slightly rapidly kn mixed distribution formally regime theorem attain coincide obtain shrink toward rate agree corollary result general poisson variate dispersion evaluate linearly relative poisson limit regime increasingly decay sampling limit degree realize govern characterize variation achievable complementary role play specification law network pairwise bernoulli trial achieve thorough understanding population parameterize smooth realistic summarize conclusion implication practitioner crucially highlight
backtrack line start repeatedly multiple rare multipli theorem use combine version well balancing demonstrate series generate improvement controller performance aforementione common approach control forecast future optimize horizon forecast realize first forecasting repeat develop regression focus attention dynamic evolve discrete action minimize cost quadratic gx illustrate balance cart cart move along axis linearization absence force
occurrence estimation respective pr innovation standard theoretical spectrum fit residual summary adapt deviation respectively computation autoregressive handle quickly moderate summary std pr summary furthermore versus deviation hessian estimation illustrate table hessian est est hessian show variance deviation integration implement integrate involved integral however limit ii iii
non mle somewhat ht interval observation solid df bandwidth ht precede estimator status mle smooth integrate w discrete plug seem attain distribution somewhat whereas natural dimensional respectively see bivariate bias take direction usual bandwidth lead play seem attain local bivariate order theorem numerator
ef mle p inverse profile exponential x reverse profile make help understanding adopt alternative alternative jx j know analytically extract distribution mean preferred score constant treat along case model intractable integral log latent mc e location family shape write model agent noise alternative focus eq density concave concave accord log
focus belong consider membership membership feature convenience intercept logistic parameter presence feature attribute graph latent affinity tendency assignment tendency link
greedy reach htbp leave one segment segment therefore good basis control optimization segment start segment basis quite spectrum mapping object offer characteristic
scatter deduce cost zero reduce zero question besides measurement anonymous review comment quality national foundation china project excellent team support liu grant leave side coherence side term denote transpose refer real value dimensional small compressive sense community measurement formulate reconstruct unknown interest
location object scene bin capture typical reduce normalize random run cross pair notably std fold accuracy scene dataset category difficult scene vary foreground category image extract dimensionality preserve incorporate spatial information location normalize show enyi accuracy attain previous method decrease though applicability tb science scale simulation phenomenon occur exploratory experiment data flow calculate contiguous sub phenomenon stationary parallel plane velocity center latter label negative kernel hellinger distance base outcome parameter achieve hellinger hellinger evaluate slice resolution slice result velocity classification slightly region little canonical somewhat instance explore large phenomenon distribution region fig pick
describe decrease guarantee play role aspect future empirically unique see figure relaxed grid side c impossible identifiability distribution uniform optimal node assume initially dag associate delay graph network access even necessarily dag regular network add let associate unit access
txt color solid triangle index header comma include dataset lambda l densely triangle mark option solid index header sep comma match lambda l header sep comma lambda color densely dot mark mark solid col sep comma include lambda l header col sep confidence txt header col sep comma txt densely mark mark index header col comma include lambda txt index header col comma lambda densely style thick index index header col comma lambda include match avg txt solid repeat header sep comma include match lambda header sep comma lambda gray header col sep comma lambda txt xlabel pass ylabel area log pos east col comma lambda txt solid thick square repeat header comma dataset lambda index header col sep comma lambda product confidence thick mark repeat header true col comma l densely dash style header comma match lambda txt index header col comma lambda txt color densely mark mark option solid header col sep comma match lambda product ls txt table header comma lambda txt color solid mark table header col sep comma dataset txt densely dash thick mark table header sep comma lambda txt col comma include lambda color densely style header true col comma lambda txt index sep comma lambda mark header col comma lambda avg txt header col sep match lambda confidence txt solid style mark table header col comma lambda txt xlabel effective pass ylabel style legend true col comma product solid mark header true col sep comma include datum lambda txt comma matching product txt none triangle mark repeat true col comma dataset matching lambda l txt densely dash thick mark triangle repeat option col comma lambda l opt header sep comma lambda product l densely dot mark mark option header col sep comma include match txt col sep comma lambda green solid thick repeat table sep comma data lambda txt densely mark table col sep comma lambda txt index col comma include lambda confidence color densely dot mark table comma lambda txt table index header true col sep comma lambda confidence txt color black solid repeat index header col comma include dataset lambda true comma lambda txt color gray style thick index comma datum dataset lambda txt see regularization amongst method figure cut give keep well see far scheme mit nlp initialize decrease pass multiply evaluation oracle count extra pass appear plot method initialize set dual option hamming sequence match task gold hamming error supplementary randomize classic frank wolfe separable despite show duality frank subgradient however unlike subgradient frank wolfe allow duality guarantee outperform solver amongst interest solver tailor applicability svms svms graph combinatorial object due difficulty deal constraint rate single subgradient structural svms practitioner sensitive terminate iteration solve structural frank wolfe recent signal svms key iterate use subgradient cutting thus frank wolfe wide applicability method apply bregman projection space svms also subgradient cut frank wolfe exist solver like cut plane technique see frank wolfe unfortunately subgradient prohibitive reduce randomize frank
unlabele audio six convert gray shift gradient extract image segment frame frame second audio sample reduce dimension result feature process window constitute audio location digit elsewhere audio mention design regression yield solely solely goal domain predictor recognition accuracy two domain present restrict boltzmann machine corpus digit appear within perform poorly row nevertheless well hard bad single domain success predictor train rbm audio audio analyze problem domain domain setting domain derive result use expression domain showed present synthesis neutral face context demonstrate despite method design though audio sentence corpus subspace orthogonality satisfy every consequently every q estimator fix respectively complete claim orthogonality address problem domain
non conjugate guarantee present fitting convergent furthermore new speed importance paper explore connection obvious approximation mc approximate directly available analytically wish understand tradeoff case bias variance first expansion spread everything recover make randomness variance mc terms taylor equal analytic suggest analytic derivation term first estimator term approximate numerator variance three analytic typically beneficial worth mse give contribution actually obvious family bias variance exactly case interest suggest family capable even possible exact sufficient vanish variance mean replace use distribution recover normalize regression really correspond exponential true predict mc benefit gain related h proposition david stanford edu approximate minimize kullback divergence intractable provide close exponential mixture distribution make precise several tractable quantity marginal monte approximation kullback approximate rely analytic conjugacy variable conditional analytically member exponential applicable equivalent sufficient unnormalized log
training training test set methodology li segmentation initial pool around segment segment extract two bag dense gray scale descriptor contour foreground descriptor descriptor pyramid locally texture vector score balance fair fourier feature gd want segment gd table set accuracy average class scales discuss able hyper train attribute minute cc
belong correct add generation treat generate block orient degree correct risk reader direct shorthand computing appear difficult except approximate assume eq however determine challenge say law cutoff essence treat impose hyperparameter open structure fit network growth power law cutoff specifically direct instance nod useful network various way exponent eq vertex network undirecte block block average degree degree power exponent bind degree poisson describe upper normalize know block force infer fully
formalism set equation numerically formalism set contribute link read read related lagrangian label node instead label satisfy two fraction link notice several link link b rescale version c dependency solution clarity scheme case close network working node
select fit set tend interior lead addition particular pattern derive slight proposition end verification lasso assumption
pg tc tc tc tc tc multiplication bc kx bc generated figure illustrate solve homotopy four axis cumulative proximal homotopy method vertical segment indicate homotopy reflect objective gap solve regularization parameter demonstrate slow sublinear rate last slow convergence pg dense several fast end contrast maintain iterate stage homotopy iteration gap clearly monotonically pg however sublinear maintain plot pg explain pg stay sublinear become automatically exploit convexity pg behind discussion homotopy strategy improves compare still slow perform homotopy homotopy method inner precision take reach precision number optimality take inner stage lack strong multiplication one evaluate gradient cost proximal pg need require three multiplication method cost eight multiplication confirm
signal guarantee exist measure ratio rule observe mass become posteriori map locally minimize strategy test underlie wrong go interested failure decision learn underlying presence channel recall channel equal symbol occurrence channel recall node observe immediate decision two exist p j error immediate private use strategy complement law total generalize symbol q j away size node exist decision measurement equivalent assumption observe immediate previous unbounded infinity probability result exist prove network let note satisfy guarantee consider
simplify make assign sequence change affect examine asymptotic delay since kl functional notation regard furthermore consequence lower proof nonnegative furthermore polynomial moment theorem polynomial notion notation sequence exist nonnegative side constant could bind algebra statement statement b nb na na nc na n n nc na n log inequality statement inequality n condition n n event similarly probability union p polynomial assertion accord omit implicitly state allow equivalence statement regard prior constant introduce occur density note use convention
draw visualize huge graph transform preprocesse mention price recent variable absence edge node
outlier final error make examine functional base gain make real synthetic datum water equivalent ten near three directly classify conditional discrimination discrimination estimate quickly fast estimate less accurate estimate absence classification direct knn offset speed like feature extraction surface full non yet variable range neural inverse possibility become resolution acknowledgement thank manuscript education research project
elaborate procedure type boltzmann deep boltzmann boltzmann machine eq unnormalized involve sum rewrite partition follow eq notice compute analytical sequence transition one draw draw mm evolve denote transition alternate ratio partition anneal result datum eq time produce estimate demonstrate center deep boltzmann machine section deep boltzmann discriminative
show small well suffice section sparsity eigenvector moderately able select significant produce use eigenvector determination difficult issue section describe regularity assume ba hold maximize impose hyperparameter observe convenient trivial increasingly estimate describe risk estimator implication c estimate c replace compare also establish preliminary eigenvector c become establish conduct eigenvector thresholde additional technical certain circumstance slightly risk property seem well describe behavior somewhat significance notice space imply ba condition satisfy important emphasize asymptotic sense whose total prescribe hyperparameter apart supremum usual inspection ba one weak condition eigenvector require look geometry subproblem final
office research twice continuously ii lipschitz continuous sequence descent theorem apply proximal newton exact newton newton focus minimize proximal newton tailor newton include newton analysis inexact method e subproblem newton newton inexact proximal rate adaptive stopping exactly demonstrate benefit rich literature monotone newton unify close continuous necessarily nonsmooth assume attain proximal mapping projection convex entry proximal nonsmooth minimize iterate first include accelerate method simple composite gradient composite neither subgradient subgradient
general ica entropy yield decrease excellent relation frequency property importance temporal introduce technique ica temporal search minimize entropy iterative analytic solution optimum lag signal detect nonparametric macro pca signal classification stationary hold ty unit variance last plug proposition axiom criterion theorem exercise theorem presentation remark
process context topic document global possible relate variational stochastic variational inference factorize searching leibl parameter conjugate conjugate exponential simple follow stochastic inference sample vector conjugate conjugacy two motivate maximize gradient conjugate exponential immediately statistic calculate form iteration optimize direction index subset step among observation select optimize expectation subset equally probable w proceed optimize update convergence similar exception sum though batch exponential ignore select matrix inverse simplify old index subsample document subsample subtree optimize distribution subtree word allocation eqs collect eqs inference stochastic entail optimize local parameter document follow variational table case select latent word wish consider lda hdp allocation consume additionally mean translate subtree entire subtree reduce activate
dynamical include present use basic interact feasible political spectra influence international involved opinion opinion formation model collective locate endow spin could total state network name existence feedback opinion meaning evolve time link opinion change modify due topological structure evolution numerical continuous formation communication agent topology division feature opinion system model close reality additional apart opinion birth death division opinion social
quantitie piece compare reduction th data reduction median approach th I mean predictor high approach correspondence sharp vary find begin end smooth also end regard band problematic lead interval true plot covariance mean smoothly even order maintain separation suggest property quantile quantile I quantile th spline compute true standardize I overall superior standardized residual choose minimize lack closeness summary confirm even highlight investigate well improvement locally consequence show select vary propose clearly induce mean proposal accommodate tail characteristic financial algorithm new generate subsequent maintain continuity ji black mean mean green black simulate accord subsection smoother drive posterior show mixing assess
set micro day second parameter memory usage seven collect computed main concept micro summarize anomalous interested discover trend change day behavior final informative curve positively skewed differ mostly curve interpret representative give curve spread median characterize value day case detect observe box opposite depict box vary trend show
vary measure root true state estimate meaningful number compute figure show bar state plot pf serial implementation la la divergence la significant well scatter multiple illustrate accuracy latter attain much pf variant la parallel follow transition execution figure respectively perform pf la similar draw computation yield table run monte approximation respective perform show
shorthand vector hope minimize provable pruning loss goal pruning assign answer score become decompose transition form hmm model pruning segment prune
use frequently rest coordinate comment step setup initially block nesterov weighted structure subspace permutation identity enable coordinate notation work block decomposition pick one criterion natural write uniqueness view write definition separability respect block simplicity sometimes block block block space denote let far iw inner weight l uniformly fx standard block group block convex strongly q study regularize monotonicity variation choose block point randomly block comment begin set set set value precisely brevity refer set property get chance update doubly give descent composite loose introduction develop map comment nx introduction besides separability lead block stay ready technical expectation involve name develop complexity motivate concept section devote property choose subset subset assume choose degree separability sampling situation analyze monotonicity guarantee directly inclusion arise encode block
value strategy assume uniquely condition compare represent gaussian deviation kl divergence distribution finally obtain applie deal showing imply oppose error distributional armed bandit imply derivative bandit randomized strategy coincide show average general convex rely insight specifically thm lead thm turn function provide non stochastic section low look easy derivative fix randomized function euclidean attempt later intuition case optima pick close optimum order therefore far get easy lead
eq notice sufficiently constant dynamically monotone generally outperform counterpart propose lipschitz constant proceed variant exact parameter go step go step simultaneously b c violate em outer outer inner terminate outer convex modulus follow q
feasible since sum column solve another program see optimal entry separable state characteristic spread simplex x say nonnegative whose column column column matrix perturb perturb belong word order robustness permutation construct satisfy focus prove robustness fact attention admit separable identify permutation deal margin isolate topic context impractical near easy measuring say relate whose equivalent differ multiplicative
index nan old without step coordinate basis th row old column outcome multiply vector remain event basis augment sum complete set column column matrix index cc denote block dependent row top half row denote independent among differ row row dependent bottom require suppose zero result remove least eigenvalue symmetric contain linearly long introduce extending require
desire convexity machine example compute gradient pool suitably formalize even generalize well sparsity corollary result shorthand state condition rsc row parameter vector setup definition modify logistic finite pool suppose satisfied sparse universal constant q purpose et objective term strongly section regression previously ease least cost need assumption brevity shorthand characterize identically state use shorthand epoch universal number dominant stochastic draw iteration match sparse corollary approximately corollary early corollaries lipschitz follow cardinality weak universal observe corollary analogous obtain involve replace rsc reader implement manner achieve length unless nesterov address propose epoch also additionally strong constant fix epoch set budget version length bad past work function technical ease presentation state fix epoch parameter recall cardinality logarithmic optimally concern prove case least square turn characterize
depend explicitly facilitate presentation notation belong due paper constant assumption ensure eigenvector depending ensure spectral close concentrate effectively dimensional relatively space simplify q interested alternatively bound pca every eq
correspond potential maximize potential obtain collective prediction individual atom independently set exploit meta predicate signature effectively expressive relational stack graphical instance instance set stack present yield work margin offer dimensional appropriate subgraph act interpretation procedure section mode expand predict biological help drug kernel test perspective consist interpretation binary g molecular genetic half start molecular molecular radius ccccc frame atom element element h atom atom type atom type link specification domain predicate atom chemical functional group functional group via signature serve purpose simplify atom replace list atom signature twice atom play chemical tuple directional first syntactic extra domain learn employ list permutation atom tuple unnecessary list term whole kernel regularization report composition surprise similar expand molecular specification trivially background atom summarize table language bad atom presence group unfortunately unable refine hypothesis atom almost coarse grain optimum ahead expensive bagging boost bias bag atom bagging x code repeating time fold root rmse absolute fold outperform cart rmse atom extensively match publish fold run second core ghz disadvantage powerful thank kernel additionally due powerful feature set radius respectively signature analogy empirical recall signature dramatically start complement create predictor stack obtain binary fed stage predict
minimize discrimination bias consistent reasoning formally appropriate normalize normalize lagrange multiplier enforce restriction result nonlinear mean marginal rewrite compact quantity two way multiply
easy optima optimum present variational essentially locally identifiable possibly noisy signal identifiability identifiability global context learn require come closely tie tv neighboring component rank operator nan modify projection span suboptimal solution project subgradient analysis algorithm program example unfortunately change difficult face synthesis v technique seek local approximation keep repeat operator subproblem line line solve although convex solve matrix noiseless project subgradient operator subproblem subgradient value subgradient subgradient long needs project unfortunately find attempt differential projection projection row norm norm row row normalise random due normalised sphere onto tight frame calculate linear method practically work needs project
separately subject european pass evidence procedure snps additionally exclude snps allele miss snp allele miss snp snp mapping procedure pathway provide gene map functional database classify number cell current relate reaction start list gene least pathway assign within gene pathway illustrate ad human map base pair question snps map gene snp half snps pass within pathway use canonical pathway gene molecular signature map many gene around know pathway snps pathway map pathway exclude large pathway map snps highly redundant pathway subset pathway pathway pathway snps overlap pathway snp range pathway pathway distribution snps snp distribution snps pathways gene pathway pathway pathway include gene highlight review pathway snps map gene pathway number snps study list gene pathway snp hand one
adaptive seem scheme reason error statistical number trial suitable use good next large change p three invariant projective match scheme around small two scheme routine figure adaptive I sphere match adaptive bad scheme htbp dependence average green sequences calculation carlo integration radius two scheme appearance peak column right column run three right column lower behaviour cluster state great direction cluster around function update hilbert schmidt former mention true state measurement
balance space repeat follow namely complementary j py px line ensemble evolve procedure outcome code computationally inner loop performance autocorrelation serial move generic extremely draw complementary q px huge associate method autocorrelation autocorrelation especially affine acceptance fraction agreement acceptance propose step chain independent target conversely perform walk regard
follow strong compare dependent find sample almost thm indicate take classic subgradient stepsize strong stepsize sgd testing suggest argue proper yield rate stepsize compare nesterov behavior theoretically accord prop nonsmooth part nonsmooth apply nonsmooth inferior due run plot deviation bar first subgradient result assume take value vs
encode iff least sized partition terminate consistent terminate sized need able equivalence class state disjoint minimum grouping obtain one initially explain way splitting probability match matching figure split split exactly one think grouping split probability put put stochastically split put match split imply support consider relate concerned therefore want relate distribution grouping way grouping motivation partition tuple non state iff iff stochastic mention grouping give group parent say tree go start parent argument notational point say every partition exist relation partition analogous sketch partition define problem find consistent finding sized
equation since generality magnitude among constant w bm integer magnitude every entry c next require except syntactic sake clarity give complete hypothesis prove lie hyperplane prove dx n dx x create vector round coordinate near multiple rounding lie moreover take must value n w recalling dx relate community view represent th voting social choice correspond design final researcher efficiently uniform attention david give learn al show accurate theoretically suffice specify within theoretic theory generalization recent threshold predicate hardness approximation considerable interest community provably fairly recently give parameter fx sufficiently precise output integer fx efficient polynomial dependence magnitude see doubly exponential implication agnostic uniform pac run quasi polynomial elaborate integer weight magnitude bound approximate representation weight integer must get away small approximated subsequently improve tool useful context hardness construction object approach
periodic infinite generator use implement acceptable variant variant elsewhere random generator necessary resort device mathematical string principle long random sense string generate express web page www computational simulation sample population interested build section resource mathematics generator binary string test concept notion number concept randomness randomness section devote present objective preliminary specification accept complete notion bit introduce understanding present instance generate digit digit could ideal
leave range anneal accept replace previous marginal ip p sufficiently converge specifie topology example topology symmetric difference edge sp optimal framework consensus alignment intractable proposal summarize numerical illustrate calculation substitution aa computing equation compute f f v artificial site gap f character equation v use paper compute macro percent percent proposition corollary department division university california berkeley usa address inference molecular process generally marginal homology extensive integrate derivation difficulty probability play role accuracy overall
depict backward allow simple execute step number htp case ii path zero coupling meet hyperparameter perfect carlo randomly perfect estimate paper forward metropolis dependent collect burn burn period seem arbitrarily draw mean histogram show htp histogram draw histogram backward coupling exception make much
geodesic multinomial support high fisher arbitrarily singular therefore nice multinomial width proportional fisher information show inspection see positive spectrum come consequence denote except whose example chapter comprise distinct spectrum comprise satisfy typically replicate central importance exponentially fisher matrix vanish bin symmetry eigenvalue resemble mode dominant omit asymmetric potentially natural look behave issue typical simplex call face bins positive count face face span concave face strictly decrease normal unobserved face zero count flat log lack strict concavity immediate constant fixing b affine let v mix decompose direct appendix closure play computational geometry concept furthermore example boundary simplex
verify q q q apply prove suppose minimax identity leave hand saddle side asymptotically tight connection chernoff dr dp dp saddle never big converse intermediate three possibility proceed impossible give follow contradict empty think letter channel space either infinite whenever implicitly assume topological space borel algebra close set generalization finite channel capacity finite uniform channel geometrically interpret respect enyi divergence order ar extend channel capacity extend plausible extend minimax argument convex subset topological compact convex continuous quasi quasi verify convexity minimax letting never converse space sample space redundancy always continuous redundancy achieve supremum continuous compact attain convexity simplex semi achieve q r may center inequality must channel capacity redundancy particular channel shannon zero
already act noise noise fall class estimation problem posteriori map development measurement well proceed statistical bring ignore develop formulation nonlinear regression variance new exploit composite special subproblem solve implement series preserve smooth numerical smooth capability approach conclusion kalman know definite state
achieve notion consistency function question incomplete framework whether constructed show policy cost randomized randomization consistent must feasible able mean third rarely affect average idea adapt feasibility start definition I kn denote mean refer determined vector policie q population equivalence period coincide history force purpose force population
representation look eq normalization positivity think g normalization write eq inequality concrete complete transformation dot putting rhs ensure negative second large program polynomial feasibility equality obviously concrete upper factor determine lp representation thousand variable provide formulation mixture probability avoid write equation hold transition transition say exchangeable clearly occur extend joint string impose like
band bandwidth overhead gs genome wide small evaluation vector residual term hamming criterion optimality optimality mathematically demand yet paradigm gs hamming especially signal rare weak provide subset optimality advantage gs gs sufficiently adequate tie signal strength tune challenging subsampling scheme gs insensitive mis attribute important screen property screening pick threshold retain fraction hamming negligible mean except subgraph exceed original conclude generality throughout gs gs computation threshold proof lemma similar second cost step goal retain signal try minimize controlling threshold tradeoff high may signal screen computational burden characterize gs sure setting theorem gs formally depend setting screen r convenient negligible many component moderate reduce regression define notation obvious regression involve letting g j contain negligible negligible dimensional finally gs gs magnitude procedure low result inference constraint optimal optimal another tune different step discussion notational end retain gs subgraph p component summation subgraph probability event claim connect subgraph support event possibility connect subset hold I gs gs I least op op w w proof need probability control claim definition em numerical screening subset comparison hard experiment tie spirit refine iterate screening refinement behave poorly find
profile economic given see second density bandwidth useful estimator privacy utility beyond merely underlie goal release differentially private draw release differentially basis analysis original datum exploratory example previous however histogram suboptimal converge assumption true smooth preferred statistic lead statistical example regression task develop outline privacy section give value theory broad space discuss recall input database row database say write whenever differ element exist database adjacent whenever element may characterize private
forecaster player choose possibly action player simultaneously player adversary incur goal player internal total delay objective represent start problem optimization dimension paper consider website set ad select bipartite choose gain click ad easily example span communication well understand paper feedback adversarial arm regret online short path problem derive suboptimal term dependency derive discuss detail still survey overview bandit
unable evolution hide feature deep architecture address stack rbms deep stack rbms side train temporal autoencoder simple dynamic allow stimulus interaction hide layer different energy autoregressive rbm train rbm sample hide connection structure fashion visible layer activation delay corrupt visible
di author term exclude word question offer elegant dominate parametric author analogously
nontrivial problem tb problem multiclass laplacian neighbor first eigenvector laplacian clustering display follow fidelity determine select point fidelity point procedure fidelity fidelity evolve develop fidelity seed region configuration form nearly decrease run display average error per technique ghz ratio relaxed involve prior balance even calculation
cox ii meet n slightly self contain despite integer case bi bc l equation ad b b ni triangular q triangular support part
compatible matrix component e individual component present seek matrix extensively decade simply ordinary large create stack detailed sub unique since arbitrary proper qr decomposition hereafter generality purpose implicitly assume common solve cause additional hence implicitly indeed restriction worth ordinary equivalently matrix stacking matrix partition distinguished involve part restriction part unknown compute equation compute truncate estimating play role pseudo inverse need qr decomposition svd respectively fix norm threshold find otherwise
vector onto eq et rbf function detail category prediction actual function detail general least classic advantage predict example attribute probability eq attribute lr omit role accordance probability concentration fix relatively give totally among five dataset h ccc record notation interval give
worth vary typical give rise question discuss concentration adjoint adjoint refer adjoint jensen integrable value cm risk class put contribution rich scalar criterion thank concentration upper bayes pac derive elegant risk majority derive achieve art multiclass formulation framework majority
density verify calculation carry complete calculate numerator denominator denominator numerator divide denominator since correspond corresponding term evy process locate w iw gamma ease update update give tw w poisson appropriately rescale sampling q mh moves irreducible go probability use recursion backward forward recursion eq
package include illustrative population believe parametric diagnosis examine data package discrete discrete package discrete variant bandwidth fix select di mail age adopt form table life table raw extensively situation age counterpart present change agree characterize opinion form closely neighbor relationship progress smoothly
exist see nuisance rely simplify composite hypothesis alternative nuisance sophisticated eliminate interference cause formal justification theory see power limitation refer reader therein issue simple problem versus test base proof v statistical generalized reduce tie via reader notice connection cut program first cut within cut know np tree past decade notably approach semi relaxation cut metric minimum suggest second relaxation
emphasis paper self barrier perturbation rate careful slightly sense elementary point hand result online bandit feedback clear playing instead incur convex thus thank eq mirror efficient strategy specific precisely former improve result latter linear optimization choose randomize compact player adversary incur forecaster
predictor lag enable superiority lag variable lag lag package regard number coefficient predictor error snr diagonal diagonal set snr snr evaluate algorithm train testing pass training sparsity fold set inverse standardize quite summarize dense lag stay inverse choose cross lag large cause lag discover l lag snr error testing training error error testing testing section argue possible explanation uniformity adopt choice lag
n correction th subsequently correct obtain eq literature estimator overview estimator extend multivariate arise nonparametric result exponent satisfy inequality place incorporate estimation estimate exponent incorporate inequality independence constrain constrain denote turn
drb drb drb drb drb set triple bind drb ic score allele name sequence item appear allele pair affinity keep handle algorithm insufficient denominator tuple contain allele pair bind normalize suggest select show compare allele auc drb drb drb drb drb drb drb drb drb drb drb drb drb drb drb drb drb suggest substitution describe drb identify analyze detail classify drb drb substitution bind ray use linkage proximity order element proximity define choose weighting build leave linkage instead linkage allele treat cluster successively union merge leave separate family assign cluster cluster describe cluster two close identical st st st st st ten drb drb st drb extensive medical call oriented disease gene human ray sufficiently assign allele allele drb gene drb drb drb drb gene drb human assign allele g drb digits allele protein family first type
graphical prior posterior expect nonetheless extreme observation prior almost curve difference impact exception bs estimation mcmc computation option little least range extreme sensitivity posterior simplicity work multivariate take fit distance otherwise fit ab unit reason exclude individual daily class belong nonetheless posterior predictive one answer nine minor problem negligible people minor daily task unit properly summary value fraction observation appear problematic look find unit gender individual use site journal conservative replicate analyze chain spline thank thin dramatically effect connect spline characterize trace
see alternative regularity f satisfy asymptotic calculation show asymptotic schwarz obtain hand iff alternative small formula explain third perfect agreement finding hypothesis
image technology compatibility sentence edge compatibility define demonstrate assignment bipartite become cover sentence assignment constrain node aspect bipartite use goal sentence explain aspect rating aspect figure constraint review sentence constraint add additional unique aspect dataset separable discard discard aspect difficult constraint absolutely semi supervise unsupervised condition initialize datum choose option statistic labeling compare level agreement corpus monotonic minimize loss maximize loss diversity technique diversity introduce require bipartite objective match image framework minimize upper rating rating overall rating review rate review miss aspect rating understand user user rating na I learn aspect rating fully rate review review user mixed aspect appear review make apart appeal
minimal column vector bold capital letter denote indicate denote weight unknown space occur naturally audio video code mt state focus discount mdps learning choose receive environment change transition stochastic action discount reward start act define bellman develop value policy reward iteratively apply guess optimal bellman operator estimate transition square difference derivation state extensive computational space induce random gradient proof offline
backward backward pass conceptually explicit pass notably truncation backward approximation less require intermediate backward use inference believe respectively h sl university california berkeley automatic control novel particle backward considerably improve instead achieve forward framework challenge markovian state principle backward trace trajectory seems promise markovian flexibility must kernel path degeneracy sampler add yield considerably particle unfortunately simulation
eq iteration second denote j last il therefore next give know identity therefore distribution j g j concrete complete reformulate z include convenience decomposed view belong group recall j multivariate distribution fact formulate develop close remain note I expectation expectation z g use update decompose depend eq separately two subsection optimize optimize find q interval j j denote matrix ii fine accordingly optimization since immediately find j j optimal series shift shift shown inverse fourier transform wise conjugate equation respect reformulate derivative eq obviously decrease optimization form rbf amount find therefore
decision employ cost depend precision bipartite give round precision player cost fix denote cost log growth decentralize decision monotone precision bipartite matching nm unknown choose arbitrarily slowly arbitrarily close result affect limitation optimal play j total play I jt jt jt mm bipartite account skip similar I least follow chernoff false get put regret theorem l bind linear time monotonically illustrative establish follow computation imagine decentralize base could unfortunately player decentralize grow number use different growth decentralized arm reward arm
inequality error express dft pmf monotonically certainly work eventually process meet know eq rewritten imply monotonicity condition method move broad monotonicity integer positive kn pt type might first exist fit proof next calculate treatment unit store holding take
find computationally remain promise principal subspace analogously principal instead one manifold low upper row highlight existence condition statement complicate multiple pt subspace minor nontrivial similarly violate entire necessarily occurrence selection refer subspace search active bind interpretation sparsity regime generally second pca problem low sparsity mild become involve ball row sparse generalize combination outside sparse dense regime remain upper analyzing case state simple upper row row maximizer certain radius principal nontrivial serve prototype involve main upper specific state involve involve process mild dimensional since extent qualitatively analogous condition enough
equation kullback leibler neighborhood prior marginal hausdorff metric special dimensional distribution resp kp n divergence divergence specification p g kp g p c particularly usually violate conclusion worth depend remove result weak hausdorff wasserstein wasserstein distance subset pt tend degenerate leibl nx alternatively view kullback coupling pt kp kp follow kp kp kp n nk summation take pt c inequality due kp kp element nh nh combine display obtain kk p w assume nc appear essential exchangeable kp g gp j distribution definition kullback prove main assumption support bind say density kp pp j k q note similar
total additive column reconstruction additive noise ht reconstruction projection measurement reconstruction measurement technique compressive measurement via tv sense interest exceed threshold identification level
variant regard extension extend reduce hull minimum shall illustrate sample decompose uncertainty hull hold loss subset index define function uncertainty set present problem q solve optimization eq z need first function decision appropriate term rate hard separate step simplify analysis study figure derive conjugate dual late duality property primal notation equal infimum subscript drop clear clearly regularization may depend size introduce assumption metric set continuous respect deterministic training function condition satisfy negativity e subdifferential e inequality condition hold q satisfied strictly inequality shall sufficient existence assumption gap empty
snr level reconstruction rapidly tie l competitive signal model lag examine snr lasso roughly tie l limit large quantization small figure examine snr fidelity well snr first confidence deterministic reconstruction degradation optimum sparsity confidence high l performance confidence evident snr try evident performance measurement quantization advantage lower specify varied lasso performance mark highly
conduct maximization bfgs variational shall evaluate like david discussion way nonparametric introduce infinite random location break dependent measure hyperparameter beta stick author propose stick impose probable vector associate explicitly position capable regard
different ol ol criterion internal validation computational setting implement strategy exploit randomly iteratively vertex connect connected sparsity exploit arc iteratively exploit sparsity pattern path sort code encourage entry square one penalty penalty pair link arc encourage pattern number use implementation algorithm use approximately penalty group six left vertex link arc legend present top art often image patch natural approach introduce consist extract predefine induce regularization clean patch patch average reconstruct full set good choose orthogonal cosine transform dct organize order frequency dag neighbor go low high frequency large million difficult since exactly solution admits close soft tool proximal image step exploit arc large arc approach equivalently quality image optimize keep denoise table standard image answer observed computation factor include patch second cpu core I penalty require solve quadratic minimum allowing patch three fast process move significant instance indeed penalty superiority penalty denoise scheme base overlap evaluate quality individual opposite draw section thesis error without table penalty possibly penalty helpful denoise patch art address question choose successively gaussian bm course
expectation outcome product coincide experiment advance direction advance particular send particle coin choose measurement direction observe correlation run fair coin imagine actually outcome outcome four I avoid hoeffding inequality already imagine copy different locality cs cm angle radius cs cm prediction quantum certain selection vector lie plane angle choice difference equal positively correlated degree correspond anti correlation fourth ab ab absolute mechanics notable paris later coin aspect direction signal light consider start experiment available online ab statistical delta calculation check binomial count roughly locality chance ab ab b describe system spin truth concern picture repeat time advance leave many pair actual individual time outcome detector still particle fail could measure neither detect advance definite measurement outcome setting rapidly value occur close belong event event return whether picture particle separately really point quantum mechanic principle seem reason could equal equal experimentally recover situation first
mechanism martingale exchangeability construct sequence p small design simple mixture strategy apply stream martingale suggest use well aim martingale exchangeability present construction exchangeability previously experimental result test life exchangeability plug definition take joint infinite random exchangeable tool martingale exchangeability keep conditional
behaviour see fig right polynomial assume vanish improper prior approximate piecewise polynomial near assume piecewise coincide ordinary interpolation e near ordinary interpolation position grid interpolation four interpolation cell grid corner grid analogous high grid spline nd value position away posterior away sufficiently smooth grow distribution prior conjecture argument interpolation scheme expansion adapt terminology scheme estimator choose whose parameter et motivation rather approximation constraint make usage scheme assume uncorrelated easily adapt q obvious improper prior wang et approach constrain obtain unconstraine take limit wang et
c r ic ik subtract z al z km predictive liu computer validation lin h tu involve dynamic multi model new york discussion imputation york variance trial g carlo maximization family space spatial lee h lee h experiment high dimensional west university computer branch krige gibbs krige hypercube design et computer help response response help affect curve turn residual influence life distortion consider set residual different profile tool
evaluation bilinear text estimate error replicate times square square bilinear advantage square important difference bilinear save function correction support dms index definition introduce generalize make efficient interest sum bilinear allow reduced evaluation alternative bilinear paper extend correction sum subset sensitivity introduction index systematic quantity property systematic new reduce early work strategy mean due appear review avoid index value interpretability bilinear include index
training run dictionary long increase significantly far conduct experiment activity video densely smooth max pooling video representation approach comprehensive indicate higher report cite ssc code code subsequent combine max ratio variance discrimination realize histogram code fisher code improve code sift descriptor patch code video avoid processing extraction previous recognition video densely extract sift c fuse sc ssc lasso
decade work introduce setup property simply ms aggregation space dictionary value goal estimate observation literature available ease q dictionary certain boundedness function except strong design rule suboptimal paper propose star remarkably parameter rule design sample idea subsequently similar study enjoy star solution greedy enjoy optimal deviation extend various direction aggregate aggregate projection first put weight prior weight appear derive convex optimization approximately newly propose formulation produce deviation model
path denote vertex graph slot choose reveal slot point regret selection extra incur compare select minimum end end policy path cost policy present path tail heavy cost tail cost tail moment heavy tail tailed denote chernoff
chain estimator depend estimator think lebesgue variance eq term rewrite df hx df hx indicate ideal take select upper term valid efficiency formulate term large infinity embed index setup formalize random non negative integrable markov probability section rigorously formulate select eq mcmc sampler find gibbs rare criterion formulate one estimator bias could infinite estimator introduce walk exceed high identically distribute distribution probability convenient
value minimize substitute multiply minimization respect minimization minimize eq increase function negative tn tn
award research innovation shift distribution exploit relationship general approach mathematically elegant straightforward novel demonstrate real illustrate classification mixture shift asymmetric laplace perform mark model well finite mixture naturally cluster arise finite write density proportion density density usually type often gaussian density mixture popularity mathematical flexibility capture density herein
observation exist either linearization particle moment matching incorporation introduce additional tracking lead track refinement ep distribution improve decision expectation widely deterministic inference ep factor specify pass algorithm approximate q minimize ep provably message invariant invertible compete approximate tb dynamical three type message respectively compute I factor ti derivative update alg node factor three message denote
c inner furthermore rational alternatively run emphasize answer purely need encode nonnegative system immediate implication nearly exponential show exponential somewhat surprising nonnegative reasoning polynomial inequality plausible property matrix submatrix play instance nonnegative rank nonnegative infeasible geometry subset inequality hope infeasible infeasible subset yet case nonnegative al crucial nonnegative row map point map intermediate intermediate simplex triangle case instance construct explicit restrict point modification property simplex write triangle triangle triangle
product b column product unbiased matrix replace ab
reach matrix could ever reach transition long calculate equation eq know look operation zero diagonal diagonal calculate rest process hazard basic hazard hazard function first quantity next certain viewpoint provide visit consider reliability probable cumulative version probable exclusive event add cumulative presenting time intuitive interpretation minus cdf state cdf must reach reach time quantity useful
track lc dl propose recognition suppose face x cn cc treat overall query identify code minimization scalar achieve recognition even act open code sample dictionary discrimination adaptive overcomplete pool linearly signal conventional dictionary learning learn
entry kk kk k inverse symmetry either alone storage initialize yield organize practice enough path follow solve ode successive ode notation f p jacobian ode differential ode calculation ode jacobian hessian df store light whole matrix incur cost direction usually constrain far application especially briefly proof proposition change equality program hessian strict convexity alternatively constraint represent p quadratic path segment satisfy ode non non cost derivative small expensive compute path qr either one leave specific function convex
high posterior meaning dirac mass final trace mode condition start proposal acceptance probability approximately effect empirical proposal acceptance adaptive accordingly small state explore big jump reject often small figure choice possibility express rather advantage possibility small static chain show rise ergodic accept initial distribution skewed proposal choice roughly comparable multimodal posterior advantageous mix area proposal purpose nontrivial application function potentially use volume explore severe meaning wave pde mean explain restrictive prior recover log pressure standard structure dimensional random covariance draw gaussian gibbs reader correspond posterior comparison algorithm forward mod mesh bandwidth solver operation page draw keep allow ms draw ei value trace prior acceptance around clear generate algorithm converge demonstrate importance assumption solution subsection image idea observational conjugate parameter setup correspond closed noisy datum
dense focus type mrf integrate extensively direct graphical model normalization apply low robustness related spike mrfs modelling organize hierarchical bayesian mrf hyper parameter experiment set mrf widely use mrfs log parametrization potential associate potential clique graphical equivalently remove learning allow consider subject infer connectivity two commonly correspond penalty optimization base sparse special approximate like one sparse mrfs spike point mass spike dirac uninformative component mild shrinkage
randomly kronecker delta use top storage algorithm bottom retrieval explain introduction explain try overcome world rare mathematically translate capability recognize short pattern person white
invertible hence cast observation form namely matrix induce question multinomial factorization independence translate nonetheless might aspect context possibility research piece come novel approach combine
note taylor add subtract inverse f multiply side expansion write simply roughly expansion first hope decrease sufficiently fast formalize assumption convert order control expansion term decrease quickly recall event goal move lemma prove constant respectively immediate occur recall union imply constant let assumption matrix define norm begin jensen inequality assumption cauchy inequality thereby obtain immediately indeed event cauchy schwarz lemma obtain complete inequality event occur know eq shorthand bit q triangle q recalling remainder high leave application couple series jensen schwarz proof guarantee c lemma inequality note without take decomposition begin inequality theorem definition average vector minimizer minimizer empirical eq parallel condition conjunction lemma couple indeed apply lemma statement remainder establish lemma provide auxiliary exist throughout notational shorthand big also shorthand optimization error recall lemma constant prove moment expansion lead
conditional distribution determine add result figure empirical predictor approximation w optimality poor actual bad affect concern performance discard layer comparison bind dataset layer deep optimistic thus final intuition upper architecture layer rbms upper generative use rbm result confirm likelihood rbm dataset alignment thus compare final vs validation likelihood training vs discuss early regularization could optimistic validation training upper training upper mention figure log difficult training almost optimal picture likelihood attribute increase upper still hyper training validation predictor notably lack optimistic maximize inference parameter display really ok replace well color really ok idea future validation still auto validation network compare upper reasonable well compare high might validation upper perform selection validation vs log experimental set small summing however exact layer situation mode distribution mode obtain validation since display sample dataset display true nice compare figure show bind optimistic successful layer
omp add backward statistically work difference backward object add significant extra gain ensure correctness specify loss target building modify kind singleton element entire row remove backward step whose removal kind compare forward correctness immediate removal understand reward step inclusion appropriate
resolution refine approximation static give accurate annotated boolean formulae membership equivalence respectively main loop choose approximation use query query teacher give query abstract presence answer teacher answer another practice nevertheless loop infer empirical evidence example dramatically technique predicate demand abstraction parsimonious invariant algorithm terminate loop invariant predicate example atomic predicate drawback loop invariant predicate essential loop atomic predicate otherwise formulae teacher never answer address predicate framework essential inconsistent free formula expressive predicate interpolation atomic predicate refine abstraction initial atomic predicate loop atomic predicate refine abstraction answer abstraction refine abstraction answer throughout sequence happen
mobile phone macro mobile phone activity section homogeneous performing extract though previous extensive city make attractive forest approach describe test lead forest useful ability time make time number efficiently forest moreover introduce frequently feature exploit later large share location denote blue bar forest vote count vote forest obtain input series implement vote vote denote vote count vote vote prediction input series calculation matlab implementation forest feature location hour mobile phone hour average feature location activity day phone activity test validation forest fraction
corpus corpus partition give fold call test testing performance mention validate fully nest fold partition fold block repeat produce fold validate train give validate fold outer validate transform fold rank gain perform accord lda suggest estimate matrix fact diagonal lda task run major report similar
combine natural fix gradient number convert unique natural suitable parameter write sensible metric treat likelihood x imagine suggest uninformative input gaussian distribution well generate metric
signal compress measurement matrix gaussian sparsity vary datum every pareto iteration reflect new extensive simulation approach compress
approach formulate finally important limitation automatically tune project foundation grant st decomposable efficiency guide optimum building sampling candidate incorporate run solve similar reliability much area automate technique previous basic idea problem strength statistic statistic bias instance strongly solve study metric practically decomposable class key technique
expression interpret small exploratory view structure possible vast automate human closely transformation automate interpretability transformation combine way view weight visual interpretability scheme investigation automate visual semantic exploratory present visualize axis city new york usa york usa pose practical challenge visual analysis increase dimension irrelevant redundant costly collect store exploratory prevent user explore visually extraction seek suitable help challenge create feature amongst extraction visual various exploratory pursuit multidimensional scaling evolutionary constructive reduction design intuitive measure study match human consider
audio separation law device utilize device device load device collect correspond periodic device background superposition component show segment simulation periodic two exponentially decay circuit circuit state prototype subject offset device speech overall raw audio combination noise depict ideal
ci result propose ci correct accelerate skewness particularly converge ci ci hypothesis reject significance ci dissimilarity reject similarity spaced range perturbation time hypothesis reject percentage time reject average repetition use base bound show fine perturbation need estimate bound reject similarity ability similarity distributional fr near neighbor knn mmd measure curve query observation number query knn mmd rbf choose video unique video decrease decrease drop add noise frame lastly rgb
relative profile carlo sensitive fix comparison different unity fix plausibility acceptable table interval classical approximation third order accurate plausibility marginal plausibility shorter confidence mean mean shape source lambda pl sim lambda cccc method first marginal plausibility interval plausibility describe allow construction exact frequentist parametric simulate variation carefully lee method compare many give answer correlation example example outperform model distribute portion come sample nuisance hierarchical hierarchical independent write likelihood free suffice carlo step readily plausibility parametric tend difficulty plausibility near sided bootstrap true get close increase particular shall coverage probability
show work take instead issue project project constraint conduct fix jointly happen similar dimension degenerate case positive iff condition convexity region term suppose trivial prove minimizer henceforth take place domain slack variable thank schwarz basic cauchy schwarz hand complete optimization nonsmooth nuclear norm spirit literature parameterized differentiable envelope distance norm differentiable multivariate center set datum nonlinearity make contribution laplacian term highlight phenomenon book history major predict cross sale volume aggregated test book aim predict sale book predict sale co product graph one market binary item graph co graph product co
eigenvector energy choose sample privacy privacy pca capture whereas gap become quite ht un pca subspace scale utility perform however comparable projection indicate point show optimal differentially private investigation future differentially additional assumption verify truly effect ideal privacy total variation distance differential yet analytical sampler empirically account bring specifically result related capture approximate pca low possible packing utility bound perform space classical dimensionality component rank vast amount collect user behavior survey subject contain sensitive typically therefore discover take study guarantee differential privacy gain significant community privacy privacy risk output database differential sensitivity data proportional sensitive change change smoothed set sub query
order accuracy bit bit hashing reduction online platform online platform accuracy batch package face training exceed common repeatedly model coefficient bottleneck load many disk os hash high another loading feature time follow notational online sgd training svm q replace stochastic popular time point update sgd code initially careful every observe epoch dataset sensitive increase h th epoch hashing present accuracy regularization achieve original perhaps epoch reach accuracy hashing hashing reflect load data loading dominate sgd converted format oppose batch report loading time helpful communication
db db db db ccccc bm db db db db db db db db db db db db db db db db db db db hill db house db db db db db db db db db ccccc db db db db db db db c db db db db hill db db house db db db db db db db db db db db htbp ccccc bm mlp db db db db db c db db db couple db db db db hill db db db house db db db db db db db db db db db db db db db compare bm outperform ten image bm image achieve result bm house method outperform significantly well bm well outperform table suggest prior method perform method bm high db htbp compare mlp bm bm average image improvement whereas improvement middle compare achieve mlp bm dataset berkeley db bm compare achieve mlp bm bm
indicate discover underlie sparse thresholding entry might expect constraint stop early constraint compatibility provide justification series accuracy dual value layer enumeration link bottom center variable cycle greedy start first fundamental cycle cycle stop degree cycle dual factor reduce contribute estimation level contribution primal coupling result quality cycle satisfactory beyond percent size temperature value cm ef error bipartite layer fundamental number dual plot value plot primal cycle basis propagation independent cycle propagation bp bethe bp bethe propagation reflect principle absence improvement temperature direct bethe probably possibly insufficient away future width ab curves modal bp ising turn original motivation calibrate model inference complete historical plot ise mrf associate obtain map cumulative inference perform index ise study variety real alternative mrf mrf happen high wish ensure algorithm propagation real inference concern mainly follow purely approach rely constraint analytical coupling picture correlate modal possibility relevant bethe possibly correction modal kind pseudo moment explain modal like relevant mentioned surprisingly perform determine sparse estimate inverse direct amenable
separate precisely estimate modify gauss explicitly definite ignore see form subproblem constant proxy natural obtain parameter datum conduct place reflect surface naturally cast square framework transform record discretized receiver location denote column approach discretize locate hz typically ratio situation bfgs method modify
conversely syntactic emphasize language dl distinguish framework table language dl dl cl clause clause dl dl safe tight dl safe head dl dl dl role dl dl dl reasoning calculus boolean yes implementation yes semantic dl part cl separate connect minimal interface kind coupling illustrate derive previous external evaluation implement part couple achieve failure picture dl hybrid dl cl read comprehensive effect precisely dl answer answer occur rule reduce answer ip argument recursive answer combination logic role must non dl propose reasoning dl complete answer dl occur
series third parametric assumption topic lda overview tensor decomposition lda topic mixture hide approach impose constraint degeneracy paper furthermore parametric topic sub idea paper topic employ matrix recall establish topic anchor uniquely anchor degree requirement presence word matrix variable mix idea dictionary learn ask pair et study full square noiseless knowing enjoy sparsity state clearly recover explain I idea leverage counterpart assume intuitive arrive parametric consider case framework sparse work refer structural collection associate progress identifiability identifiable moreover gaussian combination identifiable et al gaussian normally work deal latent contrast identifiability linear learn object intensive investigation past community vast biology economic learn extensively year tree tractable likelihood approach test score direct undirected relaxation variable challenge latent undirected latent model
exclude characteristic cluster exact achievable spectral bring consider ht cccc cluster vary child vary vary vary vary none total analysis look e child interact child interaction thus point consideration work data effectiveness teacher run category simplicity ground use answer cluster measure spectral second although visually inaccurate secondly figure take potentially less well display odd run entirely unstable set spectral offer important business address separate cluster similar become datum mining range bioinformatic dependent medical imaging aid vision utilize surveillance represent population attribute leave company
perhaps reconstruct note follow claim still see express ignore incorrectly express uncertainty irrelevant play express divide characteristic variance resp write contrary situation real background equal eq equal sc x p se mp se mp se mp n p sn put part yield analogue odd roughly evidence population highly expert working see interpret rewrite odd eq sp sp sp I retrieve likelihood depend prior quantity note however quantity prior quantity take obtain likelihood course thing prefer care likelihood useful rather complicated weight evidence evidence turn prefer without frequency effect weight population contain evidence role frequency weight suppose belong long
idea collect verification return candidate extension fraction begin candidate solve trial verification oracle order exist good polynomial shall address condition extension rank easy height rational k counting determine height polynomially linear counting problem exist fully randomized approximation problem count approximation number two get algorithm incorporate extra eq approximate compute k ji k good whole complete verification return depend position fraction remains preference sort information give order preference compute matching algorithm make verification u stable return notice verification totally iii time stable matching verification oracle algorithm current discussion preference list analysis cost trial considerable room improvement unnecessary way almost even unbounded computational power create initially create verification direct path verification note gets make query uniformly match strictly positive probability instance consistent far verification oracle sufficient go could prefer previous could happen preference preference preference randomly preferred averaging prefer put together probability strictly need large comment stable preference bind refer trial formula clause unknown u formula solution verification return false first computation satisfy assignment exist oracle target also search standard clause n terminate oracle assignment ask verification confirm terminate program return idea elimination extensively pac formula proceed propose clause return
first use optimal parameterization within hamiltonian learn uncertainty clear experimental hamiltonian illustrate advantage tractable involve presence result show error experiment irrespective rao frequency characterize unknown perhaps importantly unknown end show region accord perform term experimental conversely fail cost time vary advanced optimization search advance resample technique substantial reduce experiment extension control system united acknowledge question apply infer quantum system parameter trade resource avoid storage post processing learn change process rao quantum processor first require accurate characterization dynamic device code tool implement quantum carry quantum computer unable simulate call analog quantum validity depend simulation dynamical simulator present output expect characterization generation quantum discuss really know nothing highly unlikely typically insight form hamiltonian process additional knowledge process
successfully recover wavelet full wavelet note problem nonconvex q eq give easily modify objective basically generalization solve stationary
coarse energy energy procedure construct pyramid principled algebraic look matrix coarse act coarse label yield end equation one key pyramid straightforward coarse energy multiscale heavily aggregate fine assignment
put emphasis plot different effective iteration divide size performance comparison trend worst prove strategy still pass iterate worst average second typically among
ff precision error statistical adjust regularization proof adopt thm fp eq triangle cauchy schwarz q imply thus union n argument rate remain iteration front change error constant w eq covariance conjecture present thorough convergence graphical kronecker implement penalty enforce lin covariances kronecker generalize flip factor iterates converge local maximum ff establish fast ff glasso analysis root target al dimensional slight glasso diagonal matrix penalty glasso et report analysis significantly rate infinity kronecker show outperform ff glasso rate ff glasso achieve hold
bring much enable hierarchical approach even configuration significantly reach hierarchical database database dataset show satisfactory world number analyst work exploit cm cm cm universit le du france
exploration random discuss detail regret unlike benefit look close see exploration budget exploration lead exploration similarly start explore imagine dark room room little little whole exactly speak hence point far previous I choose select search exploitation close optimizer baseline like unknown costly classic apply frequently
gaussian signal purely model evolve accord random distribution g walk generative calculate datum identify fit uncertainty infer goodness fit extend measure time standard true event event event estimate uncertainty event partition set signal logarithm base reflect gaussian strictly quantity e gamma shape appendix em gamma rarely sufficient specify procedure canonical prior mean determined inspection curve shape scale fig
jk calculate held word word compare actual rate mixture nb count appropriate hdp base marked performance follow nb crf beta outperform nb percentage somewhat intuitive showing share dispersion document nb hold various lda nb crf hdp mark nb active topic word model document learn algorithm framework distinct mechanism document weakly couple indicate topic non topic sharp either close zero control n jk count usually indicate rarely observe use smooth gradually control jk modeled allow confirm
cost fitness evolution capability learn fitness code sequence give agent code fitness landscape bit distance ham landscape fitness fitness landscape fitness bit zero environment evaluation fitness exercise evaluation fitness function grow fitness landscape capability grow subset evolutionary slow evolutionary hill growth growth
merely hence trace simulate produce trace see figure part trace property make important making parameter zero iteration reduce significance unseen zero divide parameter alternative fraction strategy unconstraine progress converge use course motivate second checking platform human genome project vast biological pathway biological chemical abstract demonstrate model check handle space biological model efficacy dynamical comprise five reaction decay mass
skeleton direct orientation equivalent skeleton keep edge complete introduce essential one complete let least isolate undirected subgraph isolate observational direction undirecte perform intervention measure complexity chain graph study equivalence complete play role construct complete transition modify completed carry graphical six edge edge direct direct edge complete modify edge operator represent undirecte complete except modify section supplementary pe na introduce several modify operator chain move happen complete stay order complete modify characterization introduce alternative exploiting complete might complete also valid valid valid direct acyclic dag accord belong unique operator complete valid define operator modify occur complete operator guarantee operator condition result complete condition imply valid complete skeleton use complete dag every undirecte terminate undirected alternative operator modify material include generate modify creates complete consistent describe supplementary material constructing complete operator complete construct subset complete valid operator operator complete apply markov markov start arbitrary complete repeat complete result complete stay transition
converge exact bound perturbation argument require plug polynomial method moment introduce speak estimator find mixture observe moment parameter system challenge early undesirable moment reliable year restriction distance polynomial complexity estimator base require resource exponential mixing dimension degenerate multi view contrast base separation condition degenerate degeneracy condition discuss degeneracy condition weak separation parameter close degenerate sample polynomially natural quantity measure closeness degeneracy condition
tensor q explicit proportional decrease carry choose form parameter full rotation matrix obtain step flow equation hold define final point vanish pair vanish result matrix degenerate possibly problematic tensor operator expansion
respectively illustration nature carry exclude mcp scad eliminate nonzero covariate model mcp make smooth scad penalty make transition g scad change mcp representative path mcp group group straightforward fit scad mcp need replace multivariate thresholding update update method list f f recall note expression possess every respect see iteration scad penalty furthermore lasso minimize convex minimum mcp scad convex nonconvex similar interestingly show even update sequence converge orthonormal update group initial produce multiple converge considerable lasso mcp group scad algorithm logistic recall simple possible iteratively reweighte typically equation diagonal mcp lack simple context depend hessian let positive logistic bounding consist
norm penalty element clear generic minimize sparse dictionary coordinate minimization follow many represent code apply every overlap patch neighboring patch pixel shift thus redundant introduce sparse formulation convolutional convolutional coding provide function predictor function convnet element predictor consider predictor final unsupervised energy input follow dictionary filter step fix inference carry sparse code fista propose iterative shrinkage thresholding improve size momentum domain adopt
review part hold get validation computationally demand though lot exploit parallelization practice usually employ speed heuristic may local minima test see quality local heuristic cross datum reflect large finish effective heuristic require error subset minimum converge process subset reveal way take candidate automate propose size substantial testing control roughly speak number dropping criterion speed result method less significant loss certain optimality yet availability vast guide region take learner section discuss section fast testing state synthetic real world set conclude skip self hoeffding test evaluate remain confidence lie outside perform drop similar approach pair false devise pac emphasis application domain kind suit costly train evaluation direct would one learn half half maximal model procedure utilize remain benefit necessary evaluation whether belong top extend upper procedure nearly infinite domain evolutionary configuration concept boost bernstein extend algorithm concept evaluate increase probable wide domain reinforcement relevance topic hyper model hierarchical previously observe performance candidate configuration historical apply deep belief potential second lead learn toolbox auto search like concept combine armed bandit seem way armed bandit identify large choose
pose necessarily hypothesis might machine insufficient limit number regard treat technique sensitive reliable allow establish sample single voxel frequency voxel radius frequency increase irrespective voxel optimistic manner data regularity voxel voxel member voxel voxel simultaneously contain voxel voxel member voxel voxel equation voxel member center member non voxel contain prove diameter voxel less equal central voxel spherical voxel great exist second voxel voxel contrary assume contain necessarily vice versa lemma voxel imply reasoning every every voxel voxel case contradict requirement numerically
penalty n x generalize freedom correction adjust covariate iy wang adopt idea evaluate perturbation derivative wang liu perturbation quantile function time computationally surface show path fix explore show construct respect piecewise linear bi optima computation time figure display convergence propose multiclass method measure convergence
would still appropriate importance sparse carry implicit instance weakly correlate spatially order variable apart time little assumption example variable association possibly reduce notion large clearly degenerate matrix small perhaps property reduce largely bound estimation jump finite eigenvector eigenvalue establish norm hold appropriately small continue pt pn r similar introduce rank detail arrange arrange henceforth number plot calculate section justify exist detect jump jump gap level definition estimation aic criterion special structure jump spectra jump decay treat offer jump counterpart less study exist eigenvalue via back asymptotic analysis grow mostly
x h ranking regret analyze classifier rank reduced subset show equal classification calibrate yy every proper therefore calibrate let r logistic therefore calibrate term associate balanced loss q balanced loss distribution analysis logistic loss f f combining suggest balanced regret optimize justification situation usual exponential minimize balanced loss rank
employ admm estimation problem derive optimal base effectiveness future formal sparse oracle convex application sparse namely arbitrary diagonal consist u imply matrix pair additive easily verify clutter denote context trace norm lemma derive matrix summarize lemma complete bind location set entry denote entry entry
increase focus acceptance rate obtain unnormalized rate proposal improve exactly cdf inside technique term e methodology target unbounded case transformation px ease exposition generality illustrate rv know unnormalized class previous limit necessary sufficient ensuring limit equal us bound unnormalized transform htb unbounde suitable increase unnormalized pdf vertical restriction combine two subsection obtain pdf unbounded unbounded pdf higher vertical denote section complementary analyze suitable unnormalized rv belong monotonic inside mean invertible interest domain transform rv unnormalized pdf assume without loss pdfs case pdfs support ensure monotonic decrease may difficult define sake monotonic pdfs pdfs unbounded pdfs case pdfs case imply assume normalization unnormalize unbounded support unnormalize obtain monotonic inside unnormalized sufficient condition sufficient equal complementary case unbounded monotonically decrease vertical unnormalized bound support rv monotonic unnormalized inverse rv unnormalized term pdf unbounded second term transform finite vertical decrease imply therefore high order unbounded subsection demonstrate rv inverse target pdf set summarize subsection transformation target pdf either c vertical transformation pdf pdf none monotonic monotonic fast monotonically decrease
signal separation literature like propose elegant approximate propose hessian directional characteristic hessian provide complete noisy ica sign case since extract square definite fourth point relate specifically refer tensor fourth tensor simultaneous view single scalar similar expansion fourth expand x algebraic share real value cross know r c ix
combination perform publish global ensemble analysis combine explicitly combine know correlation consistently make complex object one simplify analysis tool complex effort go task complement analysis satisfactory team later aim flexible well development scientific new physics significance observation often limit observation common determination estimate credible represent compatible hypothesis hypothesis fit quantify describe statistical procedure
order instability desirable replace version like identify like perturbation serve always satisfy zero although area cs terminology derive pt pt version non cauchy schwarz relation equality attain opposite continuous identically continuous sensible illustrate essence sort plot triangle axis profile see geometric way panel plot level sx align axis effective cross select number iteration level tuning solution erm differ soft sparsity angle cite method theoretical
category ai db ec ir net graph cut directional community explore direct define weakly reach regardless meanwhile direction call e satisfie se v te te propose type edge alternate backward call connectivity sequence e htb direct adjacency connectivity concept analyze centrality web page world connectivity regardless role develop hold alternate connectivity two terminal next source terminal set maximal call source part directional directional directional edge terminal direct belong directional tb asymmetric decomposition directional directional box membership terminal directional source terminal directional show figure way partition adjacency exhibit directional source terminal node role terminal require except terminal surprising work find directional achieve simple directional terminal appendix algorithm direct directional prior search one drawback search directional real network directional one expect absolutely community order community external edge cut directional community directional community adjacency matrix notice cut emphasize cut cut community
p w ard p jk iw linear covariance ard obtain go go general choose whether nearly gp model prior modify hyperparameter variable residual plus minus residual base distribution hyperparameter unfortunately latent overview classic metropolis easy implement dimensional distribution proposal efficient appropriate neither big generally unknown generally
aggregation mean differ differ advantage induce kernel poorly example kernel hand perturbation improvement compare kernel offer free improvement wider observe induce advantageous differ dimension kernel fine characteristic benchmark independent ica angle gaussian finally pass variable bandwidth able case indistinguishable sensitive quadratic base fail addition increase frequency make small right kernel bandwidth type independence occur high set heuristic practice test informally exponent play gaussian dependency small median marginal capture point dependency establish notion embedding definite interpret mmd respect type readily use product class investigate show family choice dimensional particularly learn statistical side testing
amount line orientation document character probabilistic capture part identify however removal level character character vs though pattern consist character represent e therefore least mechanism character discriminate reconstruction possible solely single page factor corruption character character character instance character character type discrimination character become especially strongly learn character character character performance contain instance character character consist alphabet representation feature absence individual pattern variational learn
fill height width corner em thick fill draw black thick fill black nash game algorithm measurement perturbation perturbation payoff perturbation perturbation modify vanish perturbation sure equilibrium extend nash non perturbation function perturbation nash previous general payoff write markovian section case I estimate stochastic perturbation helpful node try follow action propose perturbation game node perturbation method discrete state converge locally nash equilibrium vanish payoff concave find even provide discrete ode remainder provide stochastic algorithm ode example provide paper proof summarize notation h symbol space payoff decision
explore gaussian compute miss computationally expensive span tree observe fill value em imputation apply problem database contain miss value survey database simple deal discard however technique suit valuable value imputation note extensive imputation scope propose lin focus show value imputation prefer preserve machine piece take account instance indicate consideration mind address presence describe assess discuss gaussian
eq decomposition problem feasible least assumption pd method nice pd become numerous trivial pd reformulate eq ready pd penalty arbitrary apply find subproblem perform k go step set x k observe thus reasonable terminate method practical termination relative terminate iteration method enhance one execute outer find th outer start kx k repeat terminate outer iteration next b notational index iterate block choose solve x local minimizer vector observe affine together minimizer generate accumulation saddle exist affine statement eq accumulation exist subsequence q use continuity imply together saddle
sn example definition see fa di identity lx kx obtain eq generator see theorem align apply sign apply statement lemma appear tv obtain state proof observe sum rewrite side term sum belong partition apply complete department mathematics department mathematics university article volume title definition david obtain appear generator derivative copulas copula due copula inner
concern half however purpose relevant express function certain associate explicit relate property asymptotic decay rapidly explicit expression product converge slowly accelerate give show compute give computation comment generalise characteristic limit sequence concept merely applie character specify
reach challenging figure input match node learn trial slow base mutation slow macro initially grow reach around see average static receive achievable form combination action conduct select extended action exploration action modify purely running ga mutation report great fuzzy ga fuzzy element accommodate modification range way action update ga execute usual good compare criterion output illustrate trial fast none drawback exploration conduct high predict online furthermore weight include last discard parent evaluation rapidly converge traditionally encode environmental associate beyond number fuzzy artificial behaviour kind artificial biological paper dynamical within term genetic
huge try box quite robust huge scaling mh possibility design symmetry pdfs improve note study proposal find analytic usually within appear target good note prefer numerical computationally calculation weight indeed impossible proposal clearly choice help portion mh section pdfs proposal tackle distribution opinion promise use chain select function consideration adaptive independent pdfs fit already important proposal freedom specifically confirm work literature candidate generate sequentially whereas condition
fail case permutation field h h site order permutation recommend type field series ensure simulate c site make one explain simulate marginal h h field h advantageous herein simulate field site nearby mean space construct nearby site priori within known assume assign maintain desire include site connect site site portion sx x rhs recursively sx sx probability marginal use smoothed state undesirable replace use portion picture special case zero term h space I different covariance site connect notion neighbor markov accumulate neighbor state one assertion one match collection probability covariance correlate ease future sx sx reduce sx sx auxiliary sx x form
infinite obtain rate analyze infinite laplacian small given define tend infinity integral embed subtle manifold tangent one hence inverse intersection projection onto union tangent however still restrict piece tangent convert tangent follow let tangent jacobian point resp sufficiently analyze simple smooth give near neighbor boundary normal sufficiently boundary interior hence laplacian near interior point oppose difference plane
approach outline article marginal acyclic nonparametric little explore vertex edge edge exclude encode sum clique fully nonparametric specified possibility remain assumption discuss possibility conclude two exist joint jacobian function make identifiable demand variance jx jx independence encode respect obeys x center form thus let matrix rewrite covariance hence diagonal zero identical example density three different family transform concavity density much rich normal lead dimensional estimate enable distribution
change eq take argument size cause inactive enter small size become support sign update update step homotopy every come compute solve compute know construction cost application element homotopy step instead explicitly product add recursively addition rank often suffer issue become close number close stable cholesky factorization qr support cholesky involve homotopy support section present homotopy iterative change wish new weight incorporate change replace propose homotopy homotopy change phase old one homotopy take also homotopy computationally initialize optimality column small diagonal respectively move
laplacian often localize precisely identify graph spectral notion study trade focus extract static ability localize transform lie euclidean space aforementioned processing vertex classical major obstacle processing technique dependency fundamental significantly challenging translate analog change translate option vertex useful generalized translation fact graph shift translation graph signal multiply translation analogous trivial define intuitively create capture structural addition previously application order process localize operation process weight localize transform leverage year process research develop implementation localize transform extract high graph address algebraic concept harmonic literature spectral graph e therein focus opposed signal finally localize technique manifold pass operation enhance decomposition level continue signal processing overview next way encode graph classical frequency survey operator filter translation
relaxation verification acknowledgment research nf nf nsf view conclusion document represent either laboratory reproduce herein lemma remark property conjecture berkeley edu imaging md compressive accurately original class lagrangian minimization relaxation expression enforce exploit interpretability example group block different entry coefficient reconstruct since
mle mle mle logistic set vector goal noise release simplicity q estimator obtain solve adversary could logistic reduce use noisy overcome mle section require condition function hessian distortion low estimator linear represent nx account discrepancy predict outcome give known likelihood represent total design transform odd see concatenation require lipschitz lipschitz log bound xx ba matrix hence equation adversary mle construct write attack operate construct attack attack similar logistic operate logistic attribute private attack achieve attribute exist every logistic every private run
swap operator phenomenon give belief pair advantage score address causal improve identify orientation edge identify skeleton factor cause network causal cause belief inherently one believe path separately marginal infer wish joint give belief belief axiom compute belief uninformative incoherent coherent induce joint efficiently compute take advantage learn causal relation facilitate learn skeleton propose operator quality several regard causal variable absence belief prior network argue
symbol interference frequency channel probabilistic pdf pdf coefficients p account prior pmf bit conditionally conditionally independent capture derive iterative receiver inference fig receiver beliefs pilot symbol channel probabilistic variable next assume run bp obtain message symbol message pass nx nx
subsection consider anomaly anomaly analyze ad introduce product order anomaly type exist machine u anomaly type refer type denote expect tu u suboptimal kt show anomaly exponentially decrease exponential
km occur terminate km south gray cluster way measure add diagnostic tool highly finding consider extend include delay model high area also behavior identify classify help risk occurrence great acknowledgement thank comment suggestion connection useful suggestion help mm mm mm mm http project mm z via diffusion ann university verification weather forecast mm
trick apply allow work space give da evaluate toy inter svm iterative illustrate difficulty minimization source action
maximum p residual take eq hypothesis exist recovery omp omp namely omp omp stagewise omp ols literature often analyse ol remain rare reason ol consume omp therefore ol compressive ols atom dictionary atom close ol omp contrary correlate inverse significantly differ ol argument motivate omp although imply atom weak exhibit behavior omp ols general analysis omp specifically
center contain uv arise batch technical heterogeneity lead spurious association differentially expression could cause partially cluster new uv strong effect end predict sample begin end well patient accept combine study heterogeneous merging sample remove uv difficult consider uv batch essentially factor gene lead variance bayes use replicate estimate batch replicate center remove replicate batch factor factor also influence uv interact poor factor difficult estimate uv factor effect equally lead conclusion variation unobserve covariance gene matrix gene address interest intuitively good long uv factor variant recent propose uv microarray affect variation analysis state method recently effect variation interest factor jointly induce penalty add relax constraint jointly variation effect ice iterate factor maximization yield ice finally factor observe difficult indeed want study cancer batch cluster one use large set sample identify subtle uv know advance interest person start become uv use uv interest axis principal variance snp uv correlate observe variation project along batch factor
incorporate eeg exponential valuable information nmf addition capable devise group nonnegative update use competition apply frequency bin find kind
would acknowledge popular restrict boltzmann however scheme activation activation interpretation sort globally coherent inference translate employ success entirely fulfil
scale parameter also value interval perform ridge adjustment abc improve assume credible large coverage interval even credible central credible compare precise abc estimator true location abc care suggest suggest determination hoc practically intuitively sensible determine confident produce consistent cc cc cc width width cc width adjust coverage probability frequentist credible possess frequentist coverage procedure parameter correction asymptotically unbiased central correct
rkhs induce function rkh dense universal e universal every every induce linearity every analogous nonlinear treat feature map detail standard use define long entirely effort practical notion distribution work function derive function certain recently rbf despite success classifier therefore class subsection svms amount expect kernel I alternatively k empirical sample respectively may
original margin margin original hold margin accurately optimization use dual datum ratio random worth random technique extensively compress theorem knowledge soft deal curse rr project pairwise preserve perturb distance argue projection recently construct hadamard short hadamard matrix power matrix product matrix column construction factor project important prior preserve orthogonality transform take advantage recent projection generalized embed use run sparsity integer appropriately construction start let let proceed create I concatenation stacking row matrix I matrix finally diagonal construct stacking namely
iteration rejection sampler quantile quantile table run replication realize chain empirical truth interval ccccc draw ccccc sbm ccc sbm replication assess estimate quantile setting quantile subsampling simulation result substantially effort block estimate run markov chain iteration median three bm sbm take second time rs sense often substantial barrier approach alternatively bm sbm software implement let stationary probability space fs ts elementary k sn n mixing
bayesian integrate hyperparameter expression available usually estimate optimize mention early integral calculate analytically gp noise certain laplace approximation two insight laplace approximation comprehensive comparison exact monte binary cost prohibitive learn information linear technique ep algorithm apply classification find approximate distribution q diag tf nf ip ep site site cavity tractable cavity f ii identity mean variance cavity site site py tf z moment ep ep z expression conjugate technique give various loo cv optimization criterion give optimize distribution py th computation discuss average logarithm nlp loo generic probabilistic
bias behave differently behaviour variety arise merge apply dataset merge independent examine range unseen classifier estimate upper proceed bias experimental methodology conclusion clarity paper author assume instance predict possible instance cd create explicit induce misclassification predict formally misclassification classifier create likelihood course approach bootstrappe hold error low estimation present behaviour classifier follow rate decay bayes combination less dataset periodic fit predict point value variability significant deviation law extension significance permutation fit result fit bound pac begin iid division case empirically measure error eq vc dimension algorithm probability effectively make inherently classifier think arrange higher relate maximum loose development similar exploit
iteration reject inefficient move slow probable space method inefficient metropolis acc acc acc parameter acc acc acc evaluate monte acc acc acc acc burn acc upper tune instrumental end burn consequently transition fix ergodicity stationary conservative also describe direct parameter difficult nonlinearity slow evaluation candidate sampling calculate linearize convolution also correspondingly replace similar adopt deconvolution convolution spread partially deviation
multiple must independent structure shrink cluster variance mean tt contraction direction heterogeneous contraction pattern past mean intuitively appeal able unless successively instability diffusion dimensional establish dimension order instability examine use energy lyapunov widely dynamical equation energy discussion follow normal entropy lyapunov functional first order dt I lyapunov non decrease complete law thus shrink physical unstable density
variance square smooth often simple give mat ern modify length vary range slowly extremely treat essentially fit gps synthetic normally gp square exponential ml square variance lead surface estimate generative indicate
well behave choice section possible appear lemma calculate right hand numerically jacobian family find must prove composed step also solve newton problem continuous allow optimization define shape define therefore generalize fairly
message strategy base perfect knowledge interference channel formulate ability exchange infer bit inference bp field distribute channel precision decode pass message connect determine link complement write letter vector respectively hermitian hadamard covariance system information receiver
distribute identically document exchangeability basic document topic word distribution topic suggest give topic recursively topic formally mathematical lda vocabulary topic select select select join opposite document topic
sized accord table nd scale cauchy cauchy contaminate contaminate location sample exponential location asymmetric dd nine classifier discriminant neighbor mahalanobis ms mh dd dm correspondingly motivation package mass leave cross validation relatively calculation depth exactly calculation done circular avoid choose dd degree function dd never phase distribution evaluate pair classifier perform box figure depth arise dd assign throughout kind bad assignment rule advantage lda dm figure leave behave perform since cope dd perform like dd performance dd cauchy location alternative close attribute classifier picture contaminate leave dd dd slightly
dual dual transform kkt duality kkt corollary remark regression technique find sparse dimension feature sample extremely solve remain problem safe rule inactive predictor inactive reduce inactive inactive constraint dual polytope projection mainly uniqueness inactive group currently screen synthetic inactive lasso various scale daily life extract datum build interpretable desirable popular explanatory widely square regression appeal equivalent feature follow success wide dimension large solving memory aim optimization usage exist roughly divided safe screening heuristic guarantee discard discard sis association application correctness strong check kkt repeat screen safe screening discard safe screening safe base key search effective safe dual result discard inactive efficient screening rule screen rule safe discard indicate dual formulate dual onto polytope elegant accurately g
observation cut maximal observation cut recursively decrease easy warm yield efficient solve simplification find vertex maximal clearly observation conclude cut consequently cut cut instance observation cut define cut exist case empty matching matching observation leave area practically part vast address issue major practice view
form application analytical derive ig element estimate towards label unobserve factor variance matrix update use detail family stage stage convergence extensive detail derive penalize maximize weak number bic intuitively traditional detail criterion asymptotic identifiability general parameter identifiability mixture criterion sufficiently close cluster parameter
scalar recall pde pde powerful indeed ordinary du c eq introduce sort location say thing concrete write projection pde pde solution besides pde rarely pde issue decomposition pde depend difficult depend address computation pde stand alone follow pde statistic development respect integrable integral sign baseline density eq u ix e pde couple technique readily easy see somewhat calculation fisher ignore term therefore scale similar I case simple auxiliary variable association pde likewise example highlight observable dependent variable auxiliary variable shall case association solve approach seek map orthogonal column easy satisfie differential construct conditional I build important unnecessary dimensional quantity detail I characterize inference translate briefly construction many iid efficiency auxiliary though typically variable fully unobserved characteristic motivate demonstrate reduction development dimension reduction combine counterpart call validity establish property I familiar sufficient dimension I work say two classical problem work differential technique reduction beyond ordinary besides development framework notion idea statistic valid
proposition r control dependence immediately obtain formula simplify fix sequel iw v v dt ik jt vector omit w vx vx vx vx vx vx hx vx vx vx w vx obtain w h hx h assertion apply derive individual hold vx vx vx vx x x vx eq prove vx vx dt euclidean obtain vx vx vx x tx ty dt w therefore vx vx vx correspondingly vx vx fix estimate lemma euler proposal direct assertion proposal
summary statistic examine estimate indeed joint r package fit transform mixture component spherical perform bic care dimension tend bayes selection histogram posterior specification top panel marginal derive spherical quantile plot automatic statistic marginal estimate although perfect improve considerably summary regression regression work even high obtain flexible modelling place compare early
use expert guide constrain selection color image success width task instance fact computation inversion require much denominator fraction practice local recover equation add identity small value show simple work ignore classify function ignore possibly incorrect label assume suffer serious relax replace inconsistent namely allow partial expand fill location relax inference inconsistent penalty penalty convex unconstraine rewrite accordingly label inference optimization classify mathematically formulate let know equivalent consequently locate mml world multi setting empirical mml supervised ranking perform principled component pca reduction optimization harmonic multi similar assume function come fix rbf ik mml approach determine importance task measure mml evaluation micro average measure micro real belong synthesis description extract sequence natural scene image belong category fall text mining consist text
equality type parameterization gradient programming equality exponential parameterization necessary condition program strictly convex nf claim prove strict guarantee uniqueness continuity path turn imply uniqueness continuity original dy concrete check span generate hull constrain geometric program trajectory solid contour contour ode solver matlab evaluate derivative seven solid contour dash contour semidefinite linear minimize complicated involve nonconvex way enforce semidefinite convex simplify assume unique triangular equal ji kl express denote formula I kk kl lk term triangular path problematic surrogate add decrease possess minimum identity deduce accommodate notation
approach flow anomaly protocol collect anomaly noise common temporal pattern flow traffic difficulty flow utilize section comprise assess spike select entry traffic internet anomaly depict anomaly specification matrix take previous centralized centralized datum centralize scheme fig depict relative control trend convergent behavior small zero almost end enforce internet application however delay infeasible one end delay dependency delay low nearby end bottleneck link distribute processing motivate delay test internet reveal dominant demonstrate entry drop delay relative compressed completion principal pursuit capture traffic anomaly trace large spatially distribute cognitive cost stationary point solve identity q equivalence advantage follow alternative norm follow explore lagrangian constraint notational accordance structure matrix
formulation relate directly effect negative characterization demand regularization interpretable quantity hierarchy regularization freedom effect force hierarchy another sparsity aim develop focus problem interaction model scope address specifically tool problem previous interaction selection fall three specify hierarchy requirement relate paper proposal regularize implication impose framework real datum distinction draw hierarchy typically ny also include ridge discuss may involve notational pair produces guarantee motivation proposal hierarchy th symmetry add hierarchy undesirable relaxation replace solve add give establish relaxation david cox total interaction effect additional advantage relaxation restrictive good make hierarchy interaction relative main effect desirable fraction tuning modification net supplementary material write similarity group penalty
thm thm thm thm science china grant zhang china grant office development extract possible one motivated study point subspace result transform search keyword sampling reproduce describe phenomenon reconstruction long way rkh approximation wiener question consider point practical ask strategy pursuit extract number question next subspace span known extend operator transform significantly computational section lead search sampling commonly space
occur even though easily accomplish refer reader many factorization component analysis agglomerative recent subspace ssc provably norms fix minimum frobenius iff frobenius generalize norm remarkable svd moreover remain optimal invariant frobenius product block decomposable easy work frobenius norm much thin svd complement ap ap minimum frobenius solution unique iff large largely adapt frobenius decomposable question whether frobenius norm unfortunately put
total jump unobserved jump observe become normalise hx remainder evy e hx px reveal constant use substitute simplify substitute variable let simplify get term term second fourth equal reject subsampling superposition separate take define process evy random measure evy evy merging impact evy jump evy argument detail first divide overlap use nm proof adapt measure mean measure introduce dependency finally dependency composition normalize measure randomize model random kind hierarchical model dependency get correlate treat stochastic model many
even contrast rely model candidate solution spectra affect degeneracy forward accurate bias pdf degeneracy g degeneracy algorithm magnitude however residual combination poor four model separately rich degeneracy bias relate temperature nm spectrum star star deeply assume make otherwise star spectrum incorrectly star rich star poor star might expect systematic two presence residual reveal weak degeneracy probably either effective temperature explain colour versa degeneracy degeneracy method unlike degeneracy reduce bottom panel star mean absolute bottom row star svm open circle green closed circle vertical line spectral f change panel g star especially star star sensitive form star probably consequence increase spectra h degeneracy higher would degeneracy competitive particular star low star row show mean star symbol circle red circle vertical indicate green surface generally responsible svm although degradation considerable dominate well magnitude helps constrain turn infer star accurately star star
topic well however base suggest construct beta achieve model decade hold decade decade hold find decade maximize dynamic model static version train static predict decade hold choose baseline select decade table increase par substantial take force nuclear activation table word peak united topic peak prominent address address show able multi modal topic framework prior retain exist conjugacy dependent literature specific effectiveness exchangeable latent time vary model superior dependent poisson dependent topic model predictive
di intersection hyperplane independent set prove linear independence exist regular q proper affine proper hyperplane discuss follow statement equivalent cell cell whose closure agree prove first follow useful proved induction non entry exist affine since affine translate enough since exist hyperplane let prof let statement regular strongly accord independent solution q strongly particular therefore hc contradiction item corollary hyperplane statement h follow let hyperplane intersect hyperplane intersect easy cell namely euclidean hyperplane hyperplane dimension intersection remain hyperplane operator
clutter obstacle continuous traversal perform minimize length continuous placing maximize traversal know clutter call brevity clutter motivation variant information require sensor technology necessarily nonetheless still clutter instance research know clutter set adjacency integer unit start vertex target intersect disk obstacle perform either disk devise length traversal exploitation capability call discretization statistical reader history fall rd algorithm continuous high provably instance suboptimal computable adapt discretized function q indicator expression disk probability disk obstacle ard algorithm would short walk disk might course traversal final trajectory walk since enter disk disk remove disk center obstacle repeat reach clutter obstacle clutter side obstacle disk center clutter obstacle disk important consider know clutter clutter example locate place disk unit obstacle clutter show circle solid circle truly equip sensor respective disk obstacle obstacle gray scale reflect disk true obstacle sensor traversal color probability show lattice discretization traversal ard obstacle clutter pass avoid point two actual status obstacle clutter side circle middle obstacle mark disk color obstacle lattice discretization traversal experiment disk radius unit disk center ensure clutter mark sample mark particular setup possess characteristic call extensive realization mainly identify category inter interaction tend pattern point category section describe six process background clutter disk spatial find brief spatial excellent intensity region pattern plane intensity call process four two disjoint independent problem possible scenario clutter start toward vice versa line toward simulate modification intensity location two process intensity intensity plane independently cluster cluster context existence point specific probability give star galaxy clutter formation encounter poisson randomize case cox mat
major subroutine give order new scoring different propose order define eq equal maximal combination parent consistent mathematically score scoring subroutine part graph function score consistent order operation avoid consume logarithm score lead incorrect generate max globally well consistent globally need construct thus avoid reduce decrease compatible parent indicate bit slow experiment bit huge order precede therefore vector parent notice compare generate parent range see increase limit parent number speedup parent
tree routine individual briefly mixture guarantee outline l l c product mixture separate random matrix isolate b h w j l jk let b r c triplet computation k h u find pairwise tree subgraph coincide tree liu exact marginal routine pair compute span tree complete complexity remark also briefly roughly conditional estimate decomposition selecting separate suitable threshold statistic similar graphical obtain analysis vector empirical vector I bind concentration bound denote norm frobenius norm sample result success exact ia h r q computed operator assumption observable u particular root polynomial invertible result imply isolate
well geometrically ergodic irreducible product probability lebesgue measure integer exist eq obtain eq convolution q prove prove every borel proceed repeatedly ergodicity
theorem grow exist candidate candidate incorporate considerable redundant group kind however deviation applicable feature agree constant regularization hellinger mainly specify sub mean continue mention critical dc lasso two modification focus often accelerate commonly fast solve sparse convenience follow depend supplement introduce whose involve become lasso preliminary
plain indeed difference regard stability robustness still maintain possess pe experimentally demonstrate detection even plain experimentally investigate artificial human activity sense twitter sensitivity issue algorithm red asymmetric green divergence change pe detection show p divergence behave imply forward point illustrate propose implementation median sample combination grid validation experiment contain manually sample e change mean number origin change step matrix change significant early
mix choose accordingly mdp learn colored bandit eliminate mdps parameter mdp learn mdp maximal transition translation knowledge know translation function color color diameter bound analogously additional assigning aggregate structured diameter mdp upper mix mix mix arm close reach sample reach necessary reach trial recall prove mix mix mix prove support upper mix j recall horizon confidence achieve
root bootstrap counterpart denote p p n give quite kernel degree proceed useful arbitrary kernel kernel arbitrary whenever notation define kernel arise theorem restriction theorem establish conclusion pointwise asymptotically valid theorem appendix fx gx g iii gx gx g gx fx gx nominal uniformly behave uniformly size greater strong convergence asymptotic approximation coverage probability poor
activity nucleotide bind activity act movement bind activation bind dna bind bind stress bind protein bind activity nucleotide bind activity activity heart protein development pathway activity bind bind bind activity bind activity pathway activation protein positive cell cell development anti cell communication negative dna pathway cell protein bind act incorporation molecular incorporation specific dna bind bind bind bind activity gene gene protein bind activity bind bind bind rna bind nucleotide development dna bind bind bind activity rna ii activity rna bind nucleotide bind binding via bind
classify apart node label cluster part figure book political orientation book political grid neutral book subtle categorical data som world data relation common dissimilarity categorical graph aspect knowledge mining prototype recent overview method tackle neural particular extension map
add b k k hence b g substitute k b b k b k g last equal far expand g k k rip eq term lemma gm x x use x rearrange recurrence relation b small rearrange bind choose mechanism omit detail brevity observe merely condition signal recovery constant could signal describe applicable wide attract considerable discuss instance utilize instance indicate kind gain offer practice signal overcomplete union orthonormal spin solve orthonormal basis expansion basis form algorithm solve efficiently depth basis commonly refer
develop comment constant proof continue arbitrary exposition instant laplacian require satisfy emphasize greatly analytical n ti devote eventually obtain accurate update may rewrite let successive iterate ni section generic possibly follow value value adapt processes contraction quantify behavior integer consensus subspace admits adapt satisfie depend constant weight state adapt deterministic sequence constant self sequence conclude assertion complete property inequality adapt hypothesis constant make recall stopping time constant stop deterministic give uk fall proposition event follow zero proof involve requirement trivially laplacian dependent dynamic drop evolves process hypothesis last inequality use fact lemma immediately preserve readily operator
usage sa like criterion seem optimisation process criterion complex annealing criterion differ involve movement position latter choose one drive metropolis criterion stand reduce multiplicative iteration reach evaluation guarantee achieve nevertheless frequent also decide result deep search reliable criterion originally develop achieve result optimal bad property estimate mutual criterion comparison attain variable design reduce randomness optimisation optimisation design evaluate plot figure devoted optimize criterion rectangle result criterion repeat figure give easily ease criterion recall rectangle eight plot individual order ml cn plot correspond restriction apply design plot
color correspond production material minimize well establish monte carlo hasting na implementation membership find drastically improve local partially structure propose label choose neighbor subsequent case minimization strategy belong propagation bp large approach
describe matlab using combine ad test alpha evy evy propose tends check exceed maximum model l evy ad exceed alpha via approach exhibit test evy confidence follow evy parameter depict htb illustrate section last ad last contain c yes evy stability obtain constitute l evy evy stable distribution behave ad indicate
analyze classifier classifier adopt classifier learn employ previous measure focus classifier unlabele contrast unsupervise nearest nn classifier unlabele classification derive misclassification generalization nn focus average classification measure misclassification error comprise plug encourage boundary separation weight induce laplacian region plug
vector nature base graph pairwise similarity scale memory usage objective probable pairwise observation usage pixel square memory usage method decomposition instability capable difficulty usage complexity concept rank greedy lead eigenvalue kernel efficient local develop directional scheme computational indicate complexity prohibitive efficient solution hyperspectral challenge embed overlap patch patch coordinate pixel reference merge result hard pixel contain cover resolution sensor poor overlap signature cover incorporate characteristic model challenge appeal numerous maxima formation explain observe create local still need neighbor maxima show weak map close convergence maxima handle short distance hope increase magnitude local report design algorithmic insight new dimensionality introduce kernel capability capture hyperspectral disjoint encode transform field demonstrate general applicability problem idea motion motivate short range map force act pixel iterative gradient energy configuration propose acquire propose desirable preserve topology induce strong structure disjoint spatially motivated cluster formulation popular assumption work visualization trajectory demonstrate overcome
past odd ci participant year response datum firstly test participant run ii estimator naive estimator effect regression ignore observed evidence dependence still replacement bias issue issue sampling reach spread dependence response take
survey proportional replacement rgb rgb corollary section rgb pt title derive algorithm develop case population assume
hard constrain decade gain numerous area example compressed finding system project method regularize programming cone conclude nonempty ball present technical close smooth whose arbitrarily respect projection map onto show statement gx lx lx next project arbitrary set project use gradient strongly modulus convexity continuity imply let let monotonicity immediately exist
fan fan scad regression scad penalty spline pose minimize fan initial minimizer scad singular origin step quadratic q eq compute third numerically fan li drawback stay also burden non regression knot generally spline knot error region hence value value place knot binomial numerical
likelihood among selection analysis analysis mainly conduct actually adjust agreement original rand index rand comparison observation group pair ari agreement negative indicate present parsimonious write datum set consist matrix detail ht c c datum note remain bic model model large good component latent estimate parameter sort number select horizontal hereafter factor bic
vc subgraph mention hull interest explicit condition relate excess risk express term convenient simplify inequality implicit convenient state distance excess risk express introduce g condition existence often refer satisfying condition often formulate abstract result weak replace add condition use place surrogate loss function explore condition satisfy satisfied function particular state condition classification nonnegative thus inverse nonnegative continuous easily type extensively thesis wang linear specifically disagreement coefficient class b wang reader refer discussion coefficient lead pg measurable van van distribution minimal pseudo van triangle inequality universal satisfie satisfy h h h p take set denote cover q since universal satisfie need monotonicity satisfy definition q arise abstract set vc gx pseudo statement convention say subgraph van interested concern function monotone vc subgraph van vc vc q apply vc arrive relaxed h use pf r xy xy h xy risk vc class imply apply log condition calibrate let erm er em note special loss value simplify xy bf vc minimax passive turn theorem xy j vc bound arrive regard vc class universal calibrate letting
necessarily see positive definite relate final imply lemma separable second fundamental possibly ambient complement although countable patch function always minimum let geodesic prove follow patch appropriate geodesic triangle neighborhood geodesic small extension mr orthogonal patch convex curve smoothness clearly establish closeness geodesic neighborhood hold minimum smoothness restrictive geodesic neighborhood eqs use region adapt old dx x manifold aside control interest q choose balance paper eight page long exceed nine page review consider presentation conference automatic line generation file version specific paper please number file version since early year blind file author
carlo draw multivariate predictive need map hyperparameter step compare case quadratic well present rejection enforce tail zero improve additional importance multivariate exact posterior posterior scale negative separately adaptively skewness distribution split observe need speed first principal axis take second sampling compute computational demand experiment code importance number sample use minute method gps speed grid large form exploit kronecker covariance covariance evenly spaced point form covariance th product evaluate unfortunately kronecker multiplication exploit reduce correspond eigenvector construct eigenvector
subsection kernel positive definite therefore feature word hyper value transform gs fact gs semi provide say alphabet symmetric convolution b kernel string alphabet ai lx x x symmetric semi definite positive gs except specialized exponential rbf obtain convolution eq equation distribution one still convolution omit paper gs cope pre stage pre computation stage complexity gs tuple number pre complexity form see gs follow position string q computation involve operation complexity programming recurrence therefore operation consequently complexity gs close approximation gs post kernel require inverse guarantee stage indeed positive greatly speed prediction far similar kernel vanish exponentially rapidly distance approximate gs
show iteration build weak domain base variance mis predict decrease ensemble hypothesis rest section mean absolute mae define also mae facilitate optimization previous confident consistency scalar compose hypothesis learner consist additive weak learner build additive subsequently derive scalar derivation omit index represent equivalently loss transformation zero updating rule also adopt similar perform level selection likely task boost weighting proportion single notation instance music rating book rating movie rating netflix wikipedia log netflix record
decision adequate accommodate two equivalence explain diffusion person influence neighbor network equivalence model person influence neighbor considerable model real explain effect autocorrelation ambiguity network choice reflect regime attribute consideration actor opinion take present plausible autocorrelation logistic speed method quick conditionally biased direct directional distinguished counterpart analyze autocorrelation outcome accommodate network component coefficient sign must redundant significant network
state symmetric error estimate correlation contiguous adaptation namely mh hence mh average location proposal htb average proposal target configuration sake simplicity target pdfs specifically pdf gaussians different gaussian mean
focus ability gene group issue eventually formulate network strategy fitness interact update incorporate treat network pure strategy update neighbor regard updating summarize correspondence player player combine neighbor equilibrium convergence state player strategy update fitness player locally interaction player baseline fitness example fitness interpret quality utility determine predefined utility intensity relative fitness strong fitness equal death death birth db I proportional fitness birth replace death process strategy death strategy proportional birth process proportional fitness kind match kind update filter network degree include node strategy proportional fitness term adopt information calculate case rule update node I update
conceptually minor difference usage proximal nonsmooth sparsity notion forward stability convert stable forward exist prove regret simplify exist quantity algorithm online consecutive iterate far closely uniform define stable well uniformly stable interestingly stability forward incur access next go make attain q counterpart bound online concept besides right give bound relate third show stability
publicly capture check knowledge study include heuristic lack deal problem investigate decentralized clustering author thank discussion theorem theorem united ct usa university bayesian dag capture structure particular challenging bayesian arguably
want infinite forget far estimate try way maintain acknowledgment david nsf grateful comment calculate hidden show break exponential family depend second local proceeding imply focus write natural context yield noisy exactly context optimize ascent wang wang david wang scalable technique class allocation collection article stochastic handle small bayesian counterpart apply massive massive raw million book store want text book build datum website million description item want recommend item user continuously collect want classifier sequences million people make hypothesis trait illustrate challenge high structure observe finally never stream address challenge elegant visual language observation graphical explore organization book trait recommendation model powerful connect strategy quantity scalable address massive set store require computer specialize hardware topic collection document topic collection exhibit probabilistic model structure infer structure collection illustrate algorithm probabilistic vocabulary article analyze massive collection like traditional posterior article article behind study topic build variational transform traditionally iterate analyze inefficient stochastic noisy use variational inference show give subsample call variational graphical stochastic variational dirichlet
differ instantaneous ik x mu k tx instantaneous instantaneous instantaneous ii differ instantaneous instantaneous together introduce observation assumption bs inference infer causal functional call noise causality exploit additive instantaneous effect instantaneous propose algorithm insufficient datum mostly avoid answer code upon request finitely series whether causal influence review inference iid apply instantaneous describe causality instantaneous common
replacement provide elegant markovian incremental algorithm replacement probabilistic tool compare replacement scheme conclude conjecture arithmetic guarantee without replacement another spirit randomize algorithm exact solve projection medical device consist incremental nn replacement expectation average order tuple similarly replacement define inequality geometric inequality hold replacement implementation return pass iteration document arithmetic variety setting establish tool
reflect incidence widely know calculate solve cauchy euclidean figure incidence year year expect reveal factor work incidence stationary independence net applicability care health limitation need duration disease duration response disease disease duration play major role age disease diabete cause
binary pre unweighted addition numerator absence reward case similar set term avoid bias measure protein interaction majority finally generalization network conservative estimation weight much wherein original see common protein evaluate measure framework graph protein methodology evaluation first measure strength protein input operate higher interaction higher structured constitute thresholding performance next run function go simple neighborhood inspire score protein annotate protein relevant cross roc curve score separately section study subsequent explain trend interaction protein transform detail transform describe number connect ensure network bias variation protein primarily score edge weakly measure transform create network
preliminary flexible dynamic representation small along limit flexibility need could parametric discriminate potentially model smoothly combine kalman track multimodal impact switch delay attribute limitation discrimination perturbation actual change particular response cf even perturbation stable improved distribution increase unnecessary dynamic person active transition time mostly discrete principle dynamic kalman filter discrimination process illustrate simple motion human
cdf q parametric associate unbiased n distribution condition show pareto hence cover thus situation quantile establish quantile well hill tail denote statistic eq author standard stand convergence situation real necessarily tail
bayes flat coverage mean probably principle confidence big determination principle frequentist coincide whereas
definition usa paris france e microsoft centre france multiple type label stochastic label correspond focus scenario reconstruct correctly transition belief insensitive threshold know correlate absolutely fully graph underlying recover belief phenomenon implication lose detection sensitivity belief noise second locally resemble ability propagation retrieve structure
appropriately main idea e section th expression get irrespective precise error term let rip see finish theorem rank theorem v v present essentially decrease bit iterate iterate particular replace iterate remain lemma imply span exactly modify also iterate version state suppose span subsequent span
rely type linear scale light material occur fact resolution fine enough material completely mixed measuring depict material detector measure band spectrum weight spectra weight conversely physical interaction light material occur light reflect object hyperspectral illustrative derivation model borel show product sufficient bilinear material consist material mixing level average individual lead illustrate leave material layer devoted mechanism real scenario nonlinear back year find material scene infer signature pose rely scene hard spectra later aim kernel algorithm mixture design sufficiently nonlinearity degree radial polynomial expansion inspire successively e occur scene term differ rely firstly multiple vice secondly signature signature extraction achieve strategy neural reduce parameter fashion reference therein seem sufficiently cover wide however assume prior extraction attempt conduct introduce algorithmic n simplex volume compute geodesic distance approximate short path distance nearest even recently geodesic datum fully mixture cite one assume either present accurately mix parameter great evaluate usefulness beginning expand past decade discuss rest processing hyperspectral dimensionality classical determination inversion sparse determination inversion brief light correction intrinsic property stress span spectra band identify subspace
j slice markov b g gray robust optimization parallel slice discussion r rate metropolis sampler application van j wang b university reconstruct landscape carlo j theorem axiom exercise remark summary via slice sampling south u edu edu work school business
variance equal definition precisely gauss unfortunately neither algorithm asynchronous message sufficiently ij notice min converge demonstrate reweighte pass compare typical min min sum illustrate asynchronous reweighted message asynchronous message asynchronous always converge min converge behavior apparent asynchronous definite increase decrease reweighted mark legend entry legend pos north east xlabel ylabel style number cd x cp plot mark axis width entry style pos north east xlabel ylabel format cd plot plot asynchronous entire positive region asynchronous require graph sufficiently matrix observation many leverage end standard approach correctness duality ascent tree reweighte fail walk variance matrix show exist monotone convergence open future open performance reweighte algorithm definite conjecture large force min
pg branching encode globally convention first branch store column indicate branch highlight bold sum procedure explain pg obtain ml pg invoke level feature input pg number ml pg pg assign numerical lemma provide blind assign consecutive library cell concatenation element numerical sum odd collect ml pg proof design prove concept interaction several process separately explain ml pg detail pg pg user familiar pg useful pg machine choice make tree explain extraction demand user pg extraction disadvantage engine ml pg datum interactive extraction three proof library send data mining engine extraction development pg ready pg build modular complete environment format hash table name lemma second sensible engine concrete format necessary engine equally popular machine matlab file case file extend learn extraction proof ml pg flexible use invoke engine ml connect machine mechanism connect ml pg ml ml pg choice matlab ml pg advance
problem whereas policy prior kind thus simulate resp bandit episode resp compose armed bernoulli expectation draw bandit draw two bernoulli arm expectation reward distribution change interval standard rejection obtain policy policy introduce policy previously design bernoulli hyper tune policy refer detailed policy policy knowledge kind fair choice careful policy default use nothing enforce constraint automatically identify constraint symbolic maximal lead apply described formula among strategy result million candidate strategy formula equivalence distinct algorithm report
compact behavioral representation additional large explore structural formal structural dynamic report framework exploratory conclude remark direction graph automatically behavioral sequence flexible learn representative search instead choose implicitly component structural network time edge inactive node make edge unweighte instantaneous duration node temporal snapshot active order snapshot represent record ex ex set feature matrix ex ex ex tt discover role large discover representative degree unweighted weighted aggregate exist node sum mean aggregation retain extract formally evolve extract feature active feature snapshot matrix property network graph discover structural nmf minimum nonnegative positive intuitively
general case solve induction conclusion suppose l note eq continue assumption set conclusion follow subtract side side equal iterate side arrive use l relation use x x combine continue accord induction provide
volume carry common flow dl scheme attractive flexibility utilize dl motivate far suppose predicting dictionary supervise form count regularity link graph gram count flow flow use correspond snapshot incomplete count suitable l ls count regularizer norm give laplacian regularization explicitly write w b encourage count accord typically adopt regularization encourage lie approximate count relate cost solver well available readily apparent imputation depend dl historical describe canonical form seek dictionary apply learn use count structure abstraction similar operational b unbounde smoothness regularization validate although convex descent solver update fashion dl enhance sequentially train operational prediction phase summarize auto thick thick circle inner sep minimum mm fill draw blue minimum draw corner thin color height corner thin bottom color height cm l b c node red line width south start
histogram library rbf tuned fusion experiment svm linear fusion margin sum unweighted vote q series rbf layer represent kernel see compare stack tune result computational performance pairwise resolution base fold cv cm france
conclude probability finally multiple hypothesis substantially complex consider elsewhere presentation air office fa grant nf foundation grant dms university california department mathematics corollary proposition section pc almost optimal almost sequential test composite pt pt minus sequentially nan composite hypothesis density sequential likelihood ratio sequential test first weight minimize term appropriate specify minimize negligible minimax leibler divergence expansion operating characteristic lead
discover ann exchangeable development theory volume page institute mathematical statistic cm brownian bridge uniform partition science pages institute california stochastic et mathematics ann di able provide conjecture central central number local
onto interpret whose entry bx bx g ok g eq q begin ridge together verify part get small unique r ok homotopy explain part hence notice dominant dominant eigenvalue require theorem get follow denoise shift go great detail approximate path mesh move along toward maxima consider kernel collection trajectory trajectory gradient path conversely mesh trajectory shift replace advantage log constrain shift mesh repeat hx decompose b default bandwidth select mesh remove mesh mesh provide
impulse structure non lie get turn turn potentially asynchronous manner useful pass sense cs change vary several change exploit enhanced ability benefit limit signal condition algorithm capable understand role discuss limit category focus category combinatorial paper support support denote index non zero
describe figure place normal analytically inference flexibility form feature flexibility toy learn concept base image rely corpora explicit image semantic choose co occurrence language within english vocabulary three sec infer word concept word probability co occurrence create vocabulary expand differ translation see rank word rank direction english vice pair rather word dimensional model combine corpus concept semantic performance use incorporate several coherence concept primarily server white concept order perform monte sampling update primarily concept document end conjugacy exist throughout performance force long another infinite
team intersect people car email repository routine book file lose start person change fan book fourth graphic pass never need explicitly independence u eq b yield care handle exploit view go via eigenvector respective eigenvalue give though essential difference perturbation argument defer appendix event eq event union bind therefore ii p lemma distinct eigenvalue j jj define bound k j orthogonality make establish subspace perturbation orthonormal singular similarly x x claim assumption claim column orthonormal orthogonal complement fourth claim claim
hmm dirichlet allocation language multi cca guarantee datum conditioning model feature provide vector work well lot nlp throughout use cca hide non state cca despite still end estimator look example view generate similar view previous solve
information feedback quantify achievable rate source noiseless control relevance testing portfolio theory pair message content relationship message parametric direct information independently information complexity wu sample mutual plug empirical analogous graphical bayesian network node causality autoregressive independence causality identify instance conditionally network direct preliminary proved address uniqueness estimator graph although explicitly pairwise condition extend instantaneous direct network use wu al algorithm parent whether autoregressive topology investigate liu autoregressive lastly direct correctness denote alphabet alphabet collection process alphabet easily denote give similarity condition remove clear remark consider distribution x kullback kl condition follow p I mutual however quantify correlation sense quantify justify direct formulation causality note general conditioning conditioning channel although use condition entail relationship examine base deterministic characterized influence state evolve induce analogous system fully condition notation condition e mn causal positivity avoid degenerate case purely argue strict causality relevant strict causality essential factorization strictly generative strictly form variable causal
model configuration response configuration maximize respect em implement describe denote unobserved configuration complete denote free removing adopt decompose subject column element conditional maximize respect expect substitute x expected denote explicit maximum fisher illustrate follow add respect expression matrix canonical add quantity compute cf suggest initialize deterministic start strategy class multidimensional lc depend parameterization term link iv item aspect suitable criterion suggest mostly rely lr criterion lose lr prefer satisfie regularity large sample size criterion bic parsimonious class logit function difficulty parameterization basis lr lr test may trait collapse
penalize basis complicate derivative fit spline possess attractive stability boundary connection spline markovian flexibility author smoothing spline generalize spline adaptive spline spatially spline also spline smoothing impose spline ordinary counterpart p knot smoothing window regression mixture smoothing bayesian equip adaptive smoothing stochastic differential brief form notation nonzero conditional neighbor zero relate allow numerical sparse major inferential enhance link nest integrate fast accurate inference connection
different involve sparse g therein gradient bt ingredient proximity form proximity operator projection ball minimization function otherwise minimization require problem operator see base independently generic follow whose replacement equal prescribe task sphere new seven code task report learn sc discussion quantity generate new regularization parameter repeat time average plot method figure clearly indicate outperform experiment approach
statement
diverse diverse game reasonably time make kernel advantage though component kind frequently good field case allow factor computation possibility yet structure bad doubly exponential subset exponential assign pose challenge need technique structure dpp dpp share message pass min sum replace special compute quantity factor normalize structured variety new application diverse instance good translation high perhaps aid code rna possibility representative world pose pose tend overlap salient citation paper want number major corpus text article article cover development structural assumption make give rise message sometimes demonstrate technique projection dramatically maintain yield improve baseline remain modern computer number item roughly particle universe course type perform probability subset jump ground dpp dpp implicitly part instance position element might sentence give leverage doubly exponential possibility think part finite possibility player position player discretize position th challenge long define diversity decomposition present recall nonnegative diversity feature specify structure structure build factorization analogous clique field part part factor factorization score decompose argue quite tracking prefer high traffic allow enforce path part clique mrf define multiplicative graph thus mrfs binary mrfs structure model diverse diversity function could coarse position player enforce instead norm bias towards every bias seem practice diversity balance high diverse structured contribution especially significant imagine unlikely remain go roughly position unique dpp serious combinatorial introduction dramatically increase item subtle might factor quality likely structure probable mrf nearly identical sample likely contain truly diverse set describe technical develop intuition study result motion track task follow simplified example motion assume particle discretized trajectory model trajectory involve depend sensor simplicity determine quality trajectory position measure smoothness primary mode depict blue curve leave quality maximize particle move essence path start position smoothly time trajectory similar trajectory pass near trajectory nearby position feature diversity feature diversity effect quality visualization expect trajectory draw model row panel trajectory independently probabilitie evident due preference diverse tend still smooth start near fig particle trajectory top quality particle make usual intractable dpp small feature diversity could marginal linearity add contribution however sum number expensive substitute factorization turn pass order statistic standard pass special product eq decompose structure message graph pass graph factor graph bipartite graph type nod structure model associate similarly node distinct edge graph connect factor relationship auto f minimum cm f minimum circle minimum draw f edge dot dot tracking node circular factor depend appear factor allow assign configuration whenever graph merge go forward tree convert factor bound factor describe introduce weight goal belief propagation large structure structure assignment node factor every assignment nod single indicate assign recursive function pass along depend whether versa factor variable message come message local propagation pass phase call forward pass backward pass root pass edge prove inductive potential note pass message edge
precise mean classification herein ij ij reflect uncertainty component bic numerical mainly conduct cluster classification adjust rand rand rand pairwise divide observation plus group pair assume value pairwise true membership perfect interpret ari allow possibility correctly classify ari would relate continue artificial real datum algorithm repetition initialization polynomial artificial limit case application regression moreover posterior membership arise
feasible change even change redundant constraint bind near first redundant secondly suffice total short hold differ geometrically check vertex cube along adjacent form redundant redundant redundant observation constraint put way figure isolate relevant redundant furthermore every far away redundant point close feasible respect isolate length neighbourhood demonstrating give redundant say x euclidean hull compact face axis corner redundant redundant feasible contradict feasibility similarly contradiction complete closure rectangular close see heuristic redundant hold enough content redundancy concern feasible constraint set z say eq observation say bind extreme feasible say extreme redundant bind bind relevant converse implication extreme point redundant black dot show dot bind unchanged constraint non bind provide result maximizer q bind g maximizer observation q maximizer feasible claim note converse introduction infeasible still
difference profile diagnostic tool obtained plot age examine curve profile present pattern combination ht patient plot clear correlation measure enough confirm aid detect confirm separability specificity datum reliable test robustness reliable however well cross fit robustness derivative hour inferior measure maximum increase exploit patient age hour way effective high hour show classify remove reduce considerably change wish read simplicity threshold suggest svm recover find hour discriminate calibration secondary likely test reading overlap value confirm distinguish neither conjunction classifying limitation manuscript suggestion extension surveillance rt compare use marker protein confirm year sd year patient year diagnosis classify clinical rt every minute hour longitudinal patient f year lp date date duration death datum rt technique specificity rigorous suggest exploit reliably superior protein increase specificity potential issue address figure replicate obvious positive henceforth note tend remain constant notably
classifier root leaf reliability formula reject categorization information manually code three category topic topic code category organize hierarchy basis business economic political document category depend investigate assignment categorization assign category also split level region tc
verify solution verify select distribution generate verify critical computed homotopy instance instance user berkeley code analogue matrix interesting critical point considerably explain rational entry algebraic solution thus algebraic extension rank symmetric abstract group symmetric group case six write imply explicitly complex critical analogous identity hold namely likelihood variety topological ml version theorem true develop global negative practitioner restrict non comprise diagonal row algebraic relaxation closure see tensor format distribution random negative column transform form identify far inclusion maximum estimation use compute point function
bottom topology graph use clean operator manifold perturb weight access try reconstruct graph topology associate image filter patch unless study energy eigenvector first scale amount image denoise compute eigenvector associate add scale intermediate image denoise eigenvector image display eigenvector visually pc build red il function radial leave fig display radial frequency radial coincide supervise construct classifier improve practice pass denoise fig study number use two stage patch introduce notice visually eigenvector another patch patch patch becoming perturb eigenvector patch minimize patch eigenvector require texture aware limitation patch size patch ambient compare effect patch image pass noise mm number neighbor patch
graph check problem graph variable many standard interaction connectivity vertex factorization conditional independence statement factor consideration discrete precede paragraph algebraic geometry distribution independence subset
dissimilarity angle respectively angular distance scenario deduce ff consider extend explicit angle transform leave transform angle invariant triangular apply sign reverse
distance distance experimental datum experiment demonstrate utility relation learning new point label relation learn point run leave carlo calculate describe generate kind test choose space e english dissimilarity dissimilarity dissimilarity figure document base indexing frequency inverse figure relation axis get solid learn manifold matching
individual subject subject several thousand simulation time diagnosis function occur approximated integration ex minus ex address minus event death diagnosis disease death disease article move closely diagram history detail current
good cm pooling lp pool perform insight vary single max pooling yield validation htb stack convnet convnet ms small convnet convnet convnet
tp mutation status status gene rank p q q propose ise covariate covariate strength association smoothly covariate strength link motivate limitation covariate interpretation discover covariate focus extend categorical mix involve continuous another condition neighborhood regression versa frequently understand principle research support grant dms dms ar support nsf dms gm omit separate literature hold version population also proposition j satisfied assume moreover uniqueness
record hz measure movement acquisition delay appear movement measure signal compute sum band filter movement keep channel segment label movement compose movement amplitude movement signal task functional response learn label recognize recognize recognize movement movement operator polynomial hilbert multiplication operator computable operator inverse integral eigenfunction regularization
sis fs fs angle fan sis popularity screening number select control denote sis fs sis sis initial new dimensional well fs estimator repetition criteria eight coverage percentage repetition respectively table local well particular local around sis significantly improve note inconsistent fs small rule predictor improve cr ii sis fitting many basic fs fs iii
misspecification dynamic misspecification affect marginal alone test inconsistent continuous case know alternative part continuous f f z fy usual assumption automatically therefore proposition check satisfied equal r noting exist function k b f f cm theorem assumption diagnostic effect bootstrap dynamic feature dynamic entail method tailor
conditional variance would part covariance correct batch formulae adopt n z
depend moreover level therein condition small already play derivation least avoid case large improve bound actually indeed want require condition hold mean loss drop depend practice apply treat estimate absolute deviation condition strictly result assume element note theorem detect word proportion nonzero sufficiently absolute nonzero eq positive quasi similar imply positive
find relational count predict link link apply structure feature topic link traditionally use document infer mixture latent topic document inter similarity metric link however topic capable perform discussion neighborhood global predict link summarize metric simple metric approach local neighborhood devise measure topology similarity link show metric neighbor share strategy normalization nine common neighbor perform well nine metric new use normalization yield average two case assign highlight importance metric specific dataset investigation evaluate random coefficient type network base perform relationship hierarchical perform good link find propose walk large I walk variant show base sophisticated computation node count short path demonstrate even fairly metric effective anomalous prediction statistically initial measure metric require walk node label relational unlabeled accuracy add probability walk roughly heart engine metric propose interestingly metric base required walk start walk efficiently recursive set supervise combine meta discuss arbitrary technique decomposition global modify structure reduce approach spurious suggest score node unseen meta global metric future neighbor surprisingly learn combine instance metric feature classifier naive approach simple link metric supervise probabilistic propose method network report simple metric global perform combine metric promise discuss sensitive biology costly incomplete removal issue analyze propose different domain web analysis citation community many subsection perform attribute key kind feature mix relational use domain study web page find relational page similarly short predict link link must consistent rank wider select important training diverse new combine kind network network feature construct use relational feature yield combine new pairwise protein interaction tensor linear useful domain might user preference recommender system propose relate euclidean simple diversity relevant link systematically generate search learn link logistic link equality continue add long information score improve find search co citation search avoid challenge second link prediction supervise relational topology describe apply possible link flat link make prediction involved link influence nearby newly predict node use propagation compare intensive note get belief propagation significant simpler involve repeatedly predict node link label prediction logistic approach possibility walk directly target social link study likely recommend node behavior initial link probability walk arrive walk process argue reduce graph base show outperform final unsupervised yield new reveal project low existence kind adjacency
level solver compute elastic net grid qualitatively regard display top middle represent ratio dark region indicate time region fast run reach large gain active penalty reach net soft algorithm behave evaluation stand publicly package choose package implement substitute implementation resolution bad package publicly available web available benchmark package elastic net reference use since solve problem rely solution penalty value empty model sensible return large regularization due numerical instability encounter extreme square mostly rely spurious case restrict henceforth along objective
observe suboptimal play arm arm multiple help binomial thank chernoff hoeffding equal let chernoff hoeffding bind variable common na trial happen dx inequality define play outcome trial chernoff ix k inequality chernoff bound refer I let denote play sum j
scale logarithm cosine dct frequency dimension traditional dct alternatively analysis linear regression magnitude either balancing generate sparse coefficient usage refer encoding seem combination might represent music signal meaningful lasso interpretability generate alternate admm version interpret weight dedicate quantization bin representative codebook cluster distribution space vector codebook encode binary select neighbor create soft similar quantization ambiguity threshold use quantization add flexibility instead every code trivial representation distinguish actually code adjust unlike lasso adjusting depend code code binary mean histogram rich show encoding transform song representation sphere since transform product kernel power histogram encode whose communication reconstruction signal necessary bit encode retrieval alternative form dictionary use calculate dot since force norm cosine act pattern frame select frame cs serve normalization cs unnormalized magnitude dominate frame pool verify normalization performance frame low unnormalized dot easy interpret encoding linearity less maintain introduce leave strong nonlinear thresholde various feed dictionary learn training codebook dictionary current dictionary newly encode instance normalize examine various song application hope successful task linear machine method find song representation sophisticated logistic semantic tag positively label song tag song tag train tag relevant song song vector song tag likelihood know semantic song retrieval rank average ap per general tag ap song repository song retrieval repository song possible distance carry distance mahalanobi metric recommendation algorithm training matrix optimize rank author usage collaborative filter metric follow collaborative song song label learn learn metric apply scheme range song piece even though piece music call tag tag tag human k label song necessarily tag evaluation song filter tag k last fm collect chapter base song song relevance metric dictionary audio file music experimental label tag encoding without encode codebook leave evaluate gmm assume bag codebook gmm current fold baseline h c auc chance encode audio pooling mean max gmm measure practical recommendation system user look top result supplementary material bar represent deviation five fold query tag feature compare axis use axis encoding pooling represent encode diagonal emphasize tailed test array tag support comparison show statistically advantage drive projection dct main train method music high advantage encoding encoding encoding baseline add plot codebook performance measure material remainder low query tag encode sensitive sparsity case result much pool max similar measure pooling apparent small codebook encode result clear especially codebook effect transformation inconsistent trend auc encoding demonstrate adjust dramatically see significant advantage non linearity max performance pooling peak l encoding cosine select gmm baseline lead pair array fold tag pca title encode supplementary material encoding increase codebook unlike query tag advantage pooling get partial codebook decrease effect fix pca stable performance tag advantage pooling pooling pooling stay stable pooling decrease stay cs sensitive select system method select perhaps control cs adjust little informative power highest consistent supplementary material query parameter encoder present consistently lead encoding method search representation consideration prefer shrinkage cosine similarity total multiplication euclidean depend iteration feature procedure converge encoding encoding method add admm verify runtime computer pc cpu core fit super use performance tag price runtime give slightly tag performance advantage use feature agnostic music increase result improve encoding sparsity indirect achieve code suffer decrease adjust smooth control adjust density efficient achieve comparable consistently achieve tag parameter aspect representation easy music text codebook audio retrieval digital music become automate system essential music manual preference rely retrieval key component successful capture informative audio representation enable storage indexing fast search music easy compute enable user design audio traditional adding encode stage pool namely representation tag successful recommend lasso music audio representation quantization become music source music exploration huge generating enable recommendation query tag query tag system rank music word ultimately free search query describe semantic content specific item music song create ultimately interface enable user music music search annotation require meta music title annotation file add repository title track duration expert genome project music expert manually relevant tag service fm whereas meta intensive costly inconsistent past record preference either user recommendation new recommend filter usage recommendation rely never suggest new user user preference large music rely system recommendation meaning signal meaningful past decade dedicate construct retrieval annotation music song mostly utilize audio work retrieval audio spectra frame feature codebook stage code song informative local frame pool representation whole compact regardless tag work frequency audio popular audio transform describe summarize per representation capture sound harmonic music frame principal component whiten scale spectral heterogeneous acoustic feature amplitude audio typically integration perform take entire song sometimes segment classify majority vote segment integration system require compute another temporal integration compact representation song song structure song time generate generative gmm convenient former retrieval generative later process straight forward although way like generic way compute song generative create song describe mixture result representation low use calculated codebook audio music quantization sparse code variation apply audio either heavy computational lasso algorithms match also multi propose bag combine modal audio codebook combine supervision network combine supervise fine audio process layer encode compare audio music examine audio encoding sparse music annotation superiority sparsity ng usage training encoding explain successful sparse code encode nonlinearity thresholding examine density extract show extract feature k network classification performance system
turn determine consume especially scale problem propose suppose integer e eq main let easy column scan rule explicit suppose arbitrary sequential result strictly improve instance label take eq set svm rule corollary since follow rule svm integer eq briefly strictly rule supplement detail rewrite include easy tight instance identify estimation l bs b bs inside vi conclude strictly improve rule supplement enhance set problem corollary find e application q know rule measure screen rejection ratio instance sequence spaced compare rule exist notice sense discard evaluate experiment synthetic illustrate effectiveness original forest pick seven row present synthetic effective discard support vector largely evaluate synthetic toy toy toy red dot toy toy dot observe class c solver toy toy toy row present stack rejection convenience index member blue red region recall datum toy apart almost speedup please toy instance non speedup gain challenging mention include table svm small identify member solver solver solver total total solver total ratio far support instance identify indicate gain instance indicate table speedup gain forest gain fig identify solver solver total solver set ratio gamma speedup computer table time speedup effectiveness propose rule screening class study formulation inequality rule svm substantial extensive experiment real supervised prove ready proof lemma base lemma order show therefore condition finite convexity us support indicator sublinear close lemma due close imply close fact contradict hence conclude complete lagrangian multiplier simplicity actually result kkt condition second column result treat strictly technique use detail vi variational constraint similarly consider eq clearly half follow easy tight strictly follow optimal eq similarly convenience call rule enhance first consider eq feasible make use lagrangian multipli method notational convenience exclude possibility otherwise make contradict strong clearly feasible case u thus q point opposite direction plug statement complete none non follow argument lemma svm although effort devote solver challenge nice result rule support analyze inequality screen safe sense discard screening cost negligible solving solver detect vector knowledge currently screen outperform art screening rule effective discard speedup gain rule magnitude popular effort svm scale problem pose screening screen well known identify substantial memory deviation regression major concern provide plausible square regression paper unify regularize general inactive coefficient solution speedup safe screening support vector svm convenience discard inactive essential nontrivial exist screening exist safe safe non support associate feasible consume novel rule problem share advantage discard guarantee identify homogeneous special case sublinear statement requirement function self supplement rewrite w due strong duality sublinear lemma indicator close follow sublinear nonempty rewrite replace duality kkt notational convenience call vector call kkt condition component set z c x problem
unknown iteratively union detail mode subspace nonlinear crucial innovation though batch mode mode iteratively align mode collection subspace approximate nonlinear iteratively video operating linearize problem current fix multiplier rewrite constrain sparse augment lagrangian lagrange multipli current estimate subspace jacobian admm thresholde admm penalty enforce monotonically converge summarize solver section identify estimate along geodesic clarity exposition section core geodesic mode seek effective regard differentiable everywhere augment subspace admm first see alg verify geodesic gradient important derivation singular adding leave right finally equation geodesic section batch image jacobian geodesic initially image accurately learn transformation new subspace reach e summarize alignment may consider adopt step convergence initial orthogonal corresponding initial maximum iteration estimate align jacobian weight linearize transformation admm gradient w j solver locally tolerance converge correspond jacobian admm tolerance admm outlier linearize initialize cache break tackle difficult nonlinear iteratively union subspace update iteratively illustrate locally approximated roughly transformation solver locally take along geodesic discuss subspace update geodesic way approximate video summarize h orthonormal span transformation subspace transformation align iterative update jacobian q locally linearize transformation admm algorithm subspace e subspace u q align simply align appearance category object surveillance massive processing easy small image align update remain subspace algorithm however stream typical video less subspace stream accurately usage require storage matrix storage thin singular finally jacobian per need memory store jacobian memory store subspace size pixel use mid large alignment maximum image jacobian normalize image outlier linearize admm conduct variety verify efficiency superiority algorithm cope illumination digit image face deal video foreground test want align image illumination head take illumination add random fig align algorithm canonical frame pixel simplicity result realistic face remarkable pose aside illumination align canonical frame transformation rotation subspace align face alignment align image red experiment wish generality handwritten digit database align handwritten canonical frame fig align digits original digits significant variation variation appearance outlier capture variation desire could dimension apply apply algorithm different superiority regard speed requirement object unstable camera author virtual camera quickly track tracking camera stationary record frame align unstable camera record show task video unstable camera unstable camera translate original well axis plane experiment compare frame pixel camera rotation range perturbation compare give align canonical frame subspace algorithm align visual comparison run achieve run fast intel ghz gb ram align superior regard width std std std std htb frame select perturb align foreground separate recover foreground separate use perturb recover move illustrate subspace approach foreground align perturb separation foreground camera camera without alignment video capability video apply describe image detector cause inherent detector rectangle frame case cause pose rectangle affine align show pc intel ghz gb ram frame per fast detector transform face target pose target limited choose tight frame frame camera align frames subspace frames mode algorithm rest frame heavy ghz cpu gb ram divide frame total frame track subspace important asset come variation change illumination change dynamic background motion change illumination slowly change iteration slowly change cause align recover foreground separate simultaneously alignment rank match align image transform decompose extend incremental fast image vision though present image computationally art real present step truly remain validation another interest fix estimating conference notice alignment obtain approach video successfully align video view unstable camera identify rank align merge two toward track camera object scene quickly vary frame get piece background scene use low incorporate camera movement merge modern natural science like acknowledge ph show outli know find area vision image face recognition massive image database datum volume control way pose serious computational incremental three component foreground transformation rotation half memory requirement face well camera surveillance learn method multiplier video surveillance ease put database rate per million end database certainly surveillance city collection pose processing face recognition detect activity anomaly datum pose serious computational challenge video surveillance foreground background background represent appearance illumination move foreground object popular camera camera low problematic pca detecting move camera problem accurately background transformation frame video generate camera robust camera htb video augment multipli optimization alignment align use align subspace author
classification prefer preserve sense viewpoint however ignore projection enable possibility matrix satisfy certain propose desire feature theoretically analyze trend structure nonzero per provide conjecture confirm extensive synthetic follow introduce property random sparsity evaluated propose predict vary sparsity conjecture propose sparse matrix real data reduction microarray document paper briefly review evaluate reading indicate follow define pattern inner two bold proof minimal integer maximum integer decade corollary origin let projection roughly two typical allow high improve projection illustrate weak part involve feature selection class matrix level matrix imply bad formula lemma also two constraint formula prove first corresponding constraint integer formula increase would product distinct possibility characterize goal seem perfectly generality two number infer reasonable characterize limitation empirically redundant coherent rather absolutely accuracy detailed later difference redundant widely involve great real aforementioned assumption product row calculate respect vary related paper lemma include viewpoint maximize arbitrary usually euclidean mutual equivalent vector convenience high ideally contain element subsequently part coordinate element redundant drop intra redundant expect derive problem minimize intra part maximize desire formula practice characterize result relaxed value limit amplitude act sake regard approximately take binary distribution formulate sake comparison evaluate requirement sparse lemma result distribute random matrix selection obtain nonzero desire matrix understand relatively complicated nonzero probability please appendix one feature element sample vector element practice implement location section implement also odd increase quickly towards clarity vary vary ensure consecutive formula please allow vary bind upper allow sufficient element element please share explain imply outperform former interesting setting norm nonzero hard determined usually ratio f position simply hold formula nonzero element nonzero present detail clearly much exist projection random however practical compression projection much dense share comparable selection performance tend high probability equivalently large advantage propose dense propose matrix bad obvious advantage verify column maintain explain require relatively classification synthetic datum synthetic feature relation impact redundant area text binary vector distance random matrix gm sparse simulation repeat improve selection projection taking run simulation four test decrease high necessary contain classifier randomly select class htp gm sm gm gm sm datum experiment follow dimension redundant element element two generate class pointwise element redundant element introduce redundant relatively precisely converge zero decrease challenge conjecture type evenly clear outperform ratio preserve obvious allow interference redundant dimensional evenly projection briefly analyze note develop feature load low text reduce dimension value dimensional subset examine face take vary face face partially dataset face suffer illumination face largely vary database different evaluation evaluate face face dataset capture variation expression face serious pose face besides slightly varying expression slight pose ar gm sm gm gm gm sm gm sm microarray across sample express dataset hold document modify document category version dataset document reduce select c c gm sm gm gm sm c gm sm gm sm gm dna microarray document observe consistent state perform datum note threshold threshold performance worth vary around moreover note inferior smaller low regardless extensive theoretically conjecture hold projection enough achieve feature element number usually aforementione hard practically
selection ever engineer high curse suffer tackle head usually impose penalty add early solution family commonly refer ridge regularization determine extent respective penalty parameter step difference numerous least focus even much dimensional dimensional respective meet existence need offer little drive selection theoretical performance turn particularly traditional selection criterion criterion validity furthermore even assumption satisfied approach become feasible increasingly rapid parallel paradigm platform rely literature select estimator interest among variable practical performance often tie base fitting linear variable want minimize predictive minimize empirical scenario contain identically incur multiple result minimize consequently closeness sample stability opt commonly statistic recognize broader analysis dynamical consideration emphasis statistical instability much statistic property estimation vary considerably converse certainly vary certainly cross validation devise stability estimation norm reciprocal function statistic criterion choose worth computational similar suited platform big datum three approach respect several dependence datum selection particular excellent validate predictor often time provide comparable positive work three substantial develop measure relate well across select remain introduce tuning paper modify former recommend scheme bootstrap hundred fit contrary datum scheme generate early fit assessment infeasible extra get key exploit employ scheme multiple set case fold perturbation representation index path single care must correspond fit vary lot sample poor choice amount correlate lasso considerable spread h effect cross scenario path package lar solve include cross later penalty work comparable pseudo opt index start move h alignment fraction cross worst give computing error natural thing high especially combine primary therefore estimate study stability stability early possibility look pairwise estimate sample formulation v converge somewhat really regularization different move another overfitting criterion truth estimation automatically exclude trivially agree sample occur panel drop vary norm bring normalization hypothesis test statistic specific often value student regardless statistic away standard relative normalize variance metric old preserve right select variance panel introduce panel minimum statistically fitting theoretical criterion locally turn whose large minimum note unless solution suggest solution converge solution incorporate minima improve commonly meaningful behave like pick negligible additional getting validation computation lie path assess fit instead seem counter interested however performance make sense consistent small non parameter value correlate combination give feature statistically unstable assessment pick picking would suffer validation fold pseudo way bootstrap original third apply penalize perturbation penalty bootstrapping example add experimental affect datum costly quickly get expensive stability metric regularization cross computation perturb variety linear search choice potentially lasso often fact least pseudo instability want evaluate criterion spaced domain simulation strength signal strength design scenario compare regard model measure include perform explore bioinformatic plausible model choice datum split commonly usual drawn separation selection selection identify measure simulation times aggregate across scenario favor strength signal exhaustive range strength include problem entry diagonal constant dimensional note correlate well observe translate leave harmonic negative negative path pick gets drop relax positive precision pick pick variable cut much expense false se note performance correlate pair report htbp c c prediction selection error c effect ambient note feature change coefficient comparison predictor scenario term selection much improved case since tackle classical close respect fall continue wide constant correlation variable complex block block toeplitz among design different qualitative prior variation prediction always predictive quite gain correlation level deep measure positive false much false poorly regime university california berkeley randomly natural image fmri visual voxel voxel visual voxel transform replicate predictive informative look prediction voxel select predict score pick say reduction huge pick less number voxel understand individual
dataset present whether nan significance error type reject although statistical test prescribe global equal divide homogeneity testing kernel hilbert account map distribution onto mean element rkh mmd reproduce hilbert mmd embed distribution rkh besides calculate distance mmd statistically nan alternative mmd indicate coincide mmd quantile distribution statistically determine need hypothesis evaluate statistical hypothesis simultaneously overall type testing maintain prescribe independent comparison significance test tackle control hypothesis mmd compare two vs comparison distribution comparison compare main contribution novel mmd close prescribe show window entropy application however jensen divergence lead mutual vice notation need section distribution mean covariance operator kx f v counterparts population pool operator call homogeneity consistency form hz kx kx distribution generality proof hoeffding theorem normally z z hz hz h user optimal change maximal nan mmd correspond size mmd sum independent variable operator statistic mmd al fisher homogeneity mmd distribution discriminant define statistic p w behaviour hypothesis et one hold obtain similar mmd statistic c use regularizer mmd sensitive high operator applicable sample test analytical obtain high power divergence asymptotic mmd nan mmd hypothesis happen generalized mmd moreover fix require power mmd sample power regard mmd parametric mmd consistency accuracy guarantee comparison eigen matrix statistic time comparison assess test obtain mmd effective test efficiency since hard limit limit efficiency denote well author assess work article attention small size section n e kernel mmd mean give small report mmd experimentally generalize generalized kolmogorov traditional top compare mmd experimentally present world benchmark dataset method homogeneity group periodic decide draw perturb uniform density become hard discriminate nature tailor periodic type compare truth power homogeneity periodic expect selection replace previous kernel report compare covariance periodic investigation mmd report agree mmd justification mmd periodic kernel tune hyper parameter hyper periodic gaussian process median distance point use cross tune change simulated test similarity power large maximal report mmd suppose draw equal medium run interval replicate small mmd control statistic alternative small mmd fewer average periodic periodic periodic depict detailed mmd first sample area uniform discrimination become range call characteristic tailor periodic kx tune procedure mmd frequency type mmd large periodic base periodic periodic kernel base size middle right mmd benchmark handwritten library library instance mmd run family result high record eeg visual detail preprocessing contain signal difficult high assess combined gaussian hypothesis confirm mmd depict infer eeg image technique comparison belong high categorization assess discovery use mmd fdr fdr task mmd discrepancy kernel generalize mmd mmd consistency test especially case size mmd convergence fast estimate experiment
tw base nmf ii nmf initialization initial approximate compute object meet break cost regard sub outer rand real compare six quasi nonnegative least project least symmetric rank method initialize generate randomly generate another matrix approximate average htbp explain matrix less time reach sr set rank sr similar sr comparison figure learn initialization sr slight fast numerical quasi newton improve sr decrease competitive experiment six plot nmf originally motivate lee process many nmf process recognition application go image face size image gray take image original reconstruction sr nmf figure learn sr reconstruct procedure sr quality still htbp order analysis text nmf text dimensional datum number htbp htbp present comparison except hand sr several much nmf active relax fact undesirable constraint show relax avoid addition approximate symmetric experimentally well aspect approach symmetric hessian rank function iteration maintain decrease synthetic wide factorization necessary rank approximate stop help numerical lemma corollary factorization dimension widely text propose newton type direction approximate moreover decrease fast numerical experiment compare nonnegative confirm active symmetric technology approximate bregman frobenius due favorable property many base divergence numerous nmf classified descent alternate lee rule alternate least square solve obviously nmf focus learn method square square example subject satisfy kkt tell method finite least symmetric quasi newton synthetic data part forecast nmf negative square regard programming gradient rule use kkt iteration update update updating stop matrix curvature gradient matrix take sub avoid realization eliminate regard lin bfgs gradient object bfgs consume researcher experimentally observe symmetric rule quasi newton update rule sr sr let consider
drawback inefficient resource large gain machine per mini speedup roughly trend return overhead nearly far online counterpart seem decrease great reduce fix reduce machine reduce overhead work reduce free parameter exploit weight learn tend structure completely choice advance maxout motivate well observation tend local give feature pixel average advantage store feature intuition dedicate neural represent weight product parameterization I carefully construct performance key expect see natural impose structure choice good drive extremely drop distinction static update mini value expensive entire process distinction easy static parameter across synchronization overhead incur deep compose transformation reduce factor simply derivative I rank factored redundancy invertible redundancy fix question answer feature low unit function value coordinate pixel possible regression function view basis base dictionary represent dictionary construct way obvious train layer unsupervise use flexible knowledge feature appropriate dictionary fourier wavelet basis encode expectation build prior achieve via restrict location enable entire ridge apply patch deep layer image patch hide patch correspond image patch index multidimensional indicate colour select location represent select uniformly tie well performance carefully select predict full notice predict entire parallel kernel length control degree smoothness pixel correspond element dictionary motivated predict however interpret activation neural apply motivation line discuss relationship interpretation second ordinary visible describe restrictive detector want area remainder naturally case predict form block predict feature treat completely dictionary tb column highlight abuse place think inside layer output increase parameter dynamic increase fact dictionary hide hide unit divide abuse take layer mlp produce convolution filter give improvement weight weight correspond patch enforce topological structure feature drive unit average initialize layer autoencoder construct empirical train pre tune backpropagation parameter projection random connection column dictionary predict first two mlp mnist legend dictionary main detail mlp use mlp mnist number degree permutation coverage softmax prediction softmax never layer predict construct divide unit connect horizontal projection iid ridge ridge autoencoder architecture substantially alternative dynamic consistency pre except autoencoder experiment ms hamming rate speech frequency along temporal network phone perform viterbi certain ignore show convnet size convolutional apply filter layer convolutional layer transform output apply third connect layer softmax outputs softmax fully topological construct fully connect come ridge square exponential ica overcomplete ica model linear autoencoder network effectively predict classifier figure layer architecture square exponential task able substantial drop ordinary exponential length parameter predict twice cifar twice gain recommend protocol several explore literature remove aim rather approach parameter use locally parameterization group weight convolutional neural tie technique appear double method al similar manner approximate combination reduction weight feature feature location parameter require represent
forward bl bl bc bl bl bl bl bl bl auxiliary neuron distribution incoming neuron mlp neuron nonlinear neuron tangent mlp attempt backpropagation ordinary mlp neuron activation neuron sample computation however user neuron differ output take frequent computational prefer neuron distribution auxiliary use nonlinear activation approximate expectation auxiliary expect output dropout force activation hide layer mlp add neuron mlp dropout computation leave backpropagation consider mlp training bernoulli fix connect auxiliary neuron hidden neuron activation dropout mlp zero logistic type way fix connection neuron mlp loss neuron mild approximation computing expect auxiliary neuron linearly expectation drop activation neuron unnecessary neuron geometric exponentially neural share procedure nonlinear framework albeit informally neuron activation pooling notice already well wise activation maxout extend dropout drop hide neuron multiply dropping probability neuron drop realize neuron training scheme hash autoencoder mlp reconstruct clean obvious special section add white ordinary autoencoder additional hide neuron neuron neuron sample standard additive gaussian expect since copy learn autoencoder without layer add common white gaussian drop small intermediate drop like mlp perform even regularization add hashing extract deep autoencoder bottleneck detail train white encourage activation bottleneck close exactly add stochastic neuron hide neuron activation auxiliary multiply connection neuron neuron ordinary neuron one ignore add focus auxiliary stochastic able extension perspective extend usage drop white neuron result show separate dropping white hide handwritten dataset use drop two see choose drop drop hide layer separate mlp split stop prediction validation adapt learn automatically tune bl bl bl bl bl bl bl bl test drop probability layer interestingly drop probability extreme dropping hide drop drop drop hide turn drop hide hide already b second shown affect first accordance research adding improve figure noise achieve preliminary well carefully amount different dropping dropout noise deep architecture add neuron multi mlp make neuron mlp change backpropagation use mlp turn recently introduce dropout auxiliary connection strength instance denoise autoencoder furthermore trick make bottleneck autoencoder semantic hashing neuron bottleneck layer paper attempt explain achieve mlp dropout differ ordinary auxiliary neuron randomness cause
lemma provide exponentially condition ii mix condition ii satisfied follow application theorem apply lemma couple identically remainder triangle inequality q control apply independent namely satisfy e eq application proof kx k kx b kx orthogonal onto sup space increase proof sup norm theorem satisfied denote projection finitely let active finally use orthogonal go estimate body definition invertible b full b b k p b replace inverse inequality compatibility spectral additionally b b b argument ii orthogonal work similar virtue part iii vi complete assumption kx p part I argument proof control supremum control finitely collection cardinality polynomially set define law iterate result enough bernstein sequence complete claim corollary pt nonparametric regression regressor instrumental regression ill pose inverse sup convergence allow weakly sup spline square weakly sup norm coincide rate ill low sup sup norm spline sup application useful result nonparametric instrumental weak dependence spline wavelet social science mechanism cause simultaneously conditional iy discussion regressor nonparametric technique consistently estimate permit instrumental paper unknown subject practical importance economic role literature pose unknown operator economic unknown compact call model indirect consistent whether unknown nonparametric conditional kind pose method series estimator reference study low show attain bind derive minimax bind loss large could ill subsequently achieve yet publish sup uniform sup sup convergence band functional currently series focus asset field economic rate pose convergence allow regressor weakly provide error bias extend yield sup spline similar sup coincide ill pose slow optimal rate ill sup know h provide sup old ball show sup norm convergence estimator sum weakly obtain uniform rate spline datum tail precisely moment spline nonparametric l attain minimax sup loss nonparametric financial series fall within linear ill problem operator vast ill inverse reference deconvolution see eigenfunction singular paper linear existence publish literature pose inverse optimality loss recently establish sup norm deconvolution sup present allow ill pose derive square allow section norm brief review spline wavelet proof denote euclidean norm large space norm denote norm eigenvalue generalize two number finitely many element dimension space begin regressor conditioning also structural chen b identification two space sequence span dense close span generate consider nice property section stage mx solution stage solve regularity norm ill inverse denote regularization unknown respectively sup ls invertible transformation normalize space kx kx positive root define let orthogonal onto characterization first decompose sup calculate section subsequent regularity condition ls separately let f dy density respect lebesgue bound infinity respectively section follow convention stationary ii moment uniformly infinity old continuous see old smoothness kx precede trivially infinity attain sup rate l suffice attain sup norm rate rectangular support regressor widely wavelet cosine series operator smoothing operator thus upper weak dependence matrix b k restriction brief restriction merely well define perform truncation existence moment slowly ill ensure estimate vanishe low sufficient dependent particular estimator ill pose require bias term ill smoothness condition facilitate sup include smoothness old smoothness namely smoothness radius bp b rate sup ill pose begin place primitive expectation operator th l pose case state l ill pose lower bind old norm next correspond minimax sup denote infimum prove sup norm large risk sup sup loss operator sup low minimax sup loss assumption infimum special case low bind model wavelet ls estimator attain minimax allow heavy proceed estimator induce matrix reduce unity general present control worth weak regressor implicitly condition assumption provide achieve minimax provide one regressor regressor mix regressor mix sup norm spline wavelet l whenever mix restriction mix naturally towards condition weak big tailed sup l sup sup sup loss sup convergence conditional mean partitioning estimator sup dependent tail financial subsection bernstein independent adapt random absolutely size inequality rate weakly datum dimension theorem value mix mixing coefficient process say rate make lemma mixing value r readily linear equivalence several application identifiability orthogonal identifiability establish conditional condition identifiability cast mixing sequence imply sequence identifiable argument argument establish certain follow identifiability linear let b kx kx kx p condition space increase achievable regularity tensor product spline wavelet product serie polynomial argument achieve restriction virtue inequality sum I nk sufficient spline wavelet power polynomial provide condition integer kx basis product spline wavelet basis geometrically sufficient spline wavelet product polynomial outline wavelet space multivariate construct knot normalize define de unity degree mesh span simplicity wavelet vanish k j nk q say continuously b spline univariate spline wavelet univariate basis wavelet basis equivalent norm wavelet coefficient euclidean p value provide segment b n inequality spectral ii noting assumption fix event b nc final recall vanish virtue ii application triangle yield vanish asymptotically choose trivially note cauchy schwarz together
million cancer understanding associate molecular medical issue modern scale cancer project cancer international cancer project primary patient newly survival resource improve multi fusion simply sample huge set comparison increase fully must principled analyse set range individual expression multiple possibly type bayesian dirichlet possibility disagreement extremely mechanism determine gene might result biological cluster partition identify share structure identify fuse cancer model selection add identify mcmc method slow mix sampler regard merge step post genomic molecular cancer datum rest describe improvement gene expression variation clinical matching type patient case blind bar publicly datum platform read single miss zero test determine normal correction publicly level generate platform probe copy number significant decide practical significant analysis publicly platform use threshold leave publicly give k platform sum whether apply correction keeping give correspond clinical file txt patient bar code develop regard analyse simultaneously infer similarity structure type cluster partition note infer major consider type copy contain item vector number one multinomial allocation mixture allow share similarity pair graphical single type type maximum set enough cost allow example distribution categorical great wish variable component responsible specify mixture proportion parameter couple allocation function strength association component allocation item proportion large likely degree recover mixture model distribution correlation analysis use make assumption expect noise characteristic independent follow subject normal gamma mean therefore follow hyperparameter multinomial model subject multinomial form hence leave marginal dirichlet hyperparameter potentially include therefore represent uninformative hence item marginal computation conditional model gibbs matlab allow us gibbs computationally execute gibbs sampler chain slow mix notice implement merge separately addition step increase require merge minor step mix partition previously package simple similarity produce linkage cluster number cluster regard convenient full encourage user c type recurrence gene copy micro rna patient google site distinct consensus censor clinical outcome death recurrence show plot multiple hypothesis testing plot consider clinical note recurrence rank test status point diagnosis capability particularly characteristic comprise item show recurrence new relative contain age curve gene survival outcome hypothesis largely patient particularly poor survival see patient diagnosis survival otherwise treat certainly large fusion matrix consensus gene copy variation inspection cluster difference cluster signature gene expression copy excess micro rna subset feature poor recurrence
select probable latent place sigmoid sigmoid logistic probit real predictive long analytically several carlo laplace propagation ep alternative approximate approximate gp observation call regression version promise object extend several binomial occurrence krige gp extend modeling observation prior poisson form solution resemble cox hazard replace linear prior model likelihood c l la ta exact logit label laplace poisson poisson negative mcmc chain la laplace ep vb ta taylor bayesian toolbox consider cauchy summarize gp prior unified multivariate propose ordinal probit multiclass probit softmax sigmoid gp latent single extend multivariate krige use interested prior works exploit slow convergence inference derive form exploit family create plug play gp extra connection assume output new depth interpret place gp place typical hierarchical parameter focus factorial tailed factorial propose allow inference use non glm regression glm point prior evaluate propose version glm generalise machine iterate reweighte variational novel connection use furthermore comprehensive present compare wide introduce assume bernoulli exponential dispersion e logit beta dy generalize wide output likelihood generalize glm generic independent output popular regression three glm affect variance control dispersion q canonical maximum dispersion usually know linear weight lead tractable approximate functional gp process word covariance approximate turn latter framework input latent model gp family relate statistic specification hand gp adaptively completely g naturally kernel directly link first derivative latent gp g table except student c set dispersion common select canonical closely standard given training n predictive explain correspond jointly n conditioning novel distribution I eq likelihood posterior analytically discuss kernel dispersion marginal probable select probable latent use adopt gaussian propose consider framework discuss criterion table change selecting type output beta etc l c e yy se ny e e eq link occur event assume mean logistic naturally uncertainty fractional furthermore change link probit substitute probit arise probit logistic bernoulli function negative change well model counting output canonical poisson poisson com poisson c function ideal recover exactly rbf kernel capability extent rbf linearize represent purpose figure linearize standard poisson result datum linearize follow limitation model handle binomial variance reduce poisson satisfy hence loss negative binomial represent level poisson parameter dispersion com interpolation close numerically optimization model com include dispersion thus com count datum trend prediction lower com dispersion com glm thus com latent consider base whereas variance hyperparameter negative function gamma hyperparameter parameterize shape fixing scale hyperparameter gaussian negative real inverse link interval mean parameter enforce logistic function beta logit estimating logit gp output real logit logit gp previous link simplify calculation assumption map poisson modify linearize select input accommodate gp figure gamma likelihood due partition c c play link notable apply arbitrary multimodal latent preserve learn whereas assume link certainly principle link link link learn learnable parameterization exponential mcmc computationally intensive instead consider posterior closed base taylor show justify common heuristic pre process taylor bernoulli regression taylor regression logit transform beta finally algorithms previously generalize approximation finding find suitable gaussian substitute posterior variance inference q matrix vector rewrite note particular novel close approximation first derivative link function derivative simplify joint latent likelihood nd taylor evaluated define joint vector remove approximately q iterative interpret transformation iid different explore taylor speed target observation occur close distribution choice simplify see appendix approximation transformation appropriate noise give insight assumption output heuristic gps difference gps hand preserve approximate approximate integrate derivation q modify early additional term arise derivative conjugate derive gps transform taylor specific likelihood derivation binomial effective agnostic approximation binomial target taylor inference irrelevant classify predictive canonical point constant prevent take logarithm effective taylor taylor close poisson shape thus equivalent use observation output assume use taylor canonical eq log monotone beta st nd derivatives derivative agnostic function target logit taylor inference alternatively canonical expansion since noise dispersion target laplace laplace specific q maximum target laplace expansion initial parameter posterior mode positive canonical maximum convex finally iteration derivative approximate unnormalized site site approximation eq ep instead update site site cavity marginalization indicate since computed moment variance py require calculate py follow subtract cavity yield site q iterate observation iteratively ep guarantee although usually behave likelihood initialize analytically tractable fact convergence unnormalized true remain approximation posterior site site approximate true unnormalized hence suffice first unnormalized minimize kl argument nd moment match normalization moment subtract site update iterate site I observation maximize bind kl gp classification extend posterior gaussian maximize low expectation observation log q expectation respect term remaining maximum satisfy mean reformulate parameterization avoid invert likelihood also conjugate derivative derivative plug rewrite expectation gaussian expectation link canonical link expectation close need expression expectation alternatively expectation laplace laplace latent likelihood mode variational mean expectation similarly laplace effective nd derivative st around close general require approximate whereas kl expectation consequence simplify derivative expectation expectation numerical novel lc cc derivatives taylor py py py py py b py cc marginal py py py py b k py implement matlab extend toolbox exponential ty moment approximate numerical necessary integral quadrature hyperparameter dispersion parameter optimize code make available provide experimental comparison posterior efficacy method evaluation metric dataset ep approximations latent example gamma f leave shape monotonically derivative posterior order ep concave py derivative rhs monotonically function slope gaussian monotonically laplace approximation mode zero mark star taylor method geometrically tangent convex line taylor laplace monotonically decrease skewed ep match laplace point taylor zero tangent expansion ccc w multi latent posterior nonetheless average run experiment dispersion rbf dispersion randomly sample shape function point posterior calculate ep mean trial plot multivariate average I ta less dispersion increase dispersion derivative point tangent difference bandwidth increase ta la ep ep mean converge e prediction always large la difficult theoretically variance empirically toy gamma trend exponential inference ta ep exhibit see predictive mean order ep ep optimize unimodal minimize performance laplace middle bottom influence systematically mean pass middle ta error middle error ep end large middle taylor contrast ep cc shape scale mae nlp mae nlp nlp mae nlp ep difference predictive training method evaluate absolute error mae goodness evaluate nlp predicts entire region ep mae hand test datum middle ta accurate mae gamma mae result nlp method performance affect evaluation unbalanced skew unlikely dominate dataset maximize problem local optima log rbf four section four least two local optima produce similar surface common approach log optimize conjugate example lead illustrated ta optimum minimum hence initialization require search ensure optimum optimization initialization burden initialization taylor initializations initializations several optima likelihood unique initialization laplace laplace ep recover optima ep hyperparameter select show comparison random taylor auto two initialization mae r taylor initialization nlp mae within significant pair relative change nlp taylor significant initialize random initialize furthermore help taylor ep taylor initialization speedup hyperparameter maintain initialization cc cc cc mae nlp laplace mae ep nlp gamma gauss cc time mae nlp mae nlp ep ep gamma gauss laplace mae nlp speedup nlp speedup pair difference statistically variety consider experiment gamma inverse present range consider count taylor la hyperparameter select set performance affect likelihood evaluation metric testing show ep ep binomial taylor laplace binomial dataset record mm day occurrence binomial occurrence output binomial feature likelihood taylor yield minima correspond interpretations large likelihood use four use mid generally trend look depict remain minima bandwidth five optima taylor trends binomial extension replace cubic control tradeoff smoothness validation smoother use binomial optima attribute smoothness level fully approach spline automatic parameter place knot small increase value similar curve taylor binomial shape curve marginal smooth inference l ccc cc auto inf mae nlp mae exact ga ga ga ga ga ga laplace ga ga laplace ep l c ccc ccc auto inf mae mse nlp mae nlp exact ga taylor ga ga ga taylor laplace ga ga taylor ep gp log transform regression inverse use uci predict city cycle consumption predict rise gain table sample remain mean mae nlp non rank find hyperparameter initialization initialization inference transform present taylor however performance difference due likelihood former taylor marginal include extra dispersion four hence similar log output observation property auto dataset worse result nlp mae significance likelihood look nlp except one auto marginally small ranking mae difference likelihood likelihood g inference indicate function mae cc cc nlp ga ga ga ga ga ga cb mae ga ga ga ga ga auto cccc ep ep ga ga c ga ga ep ga ga cccc ep ga ga cccc ep ep mae ep ep ep ga ga cccc ga ga cccc c auto ep ep ep ga cccc ep ep ga ga datasets together ep nlp c nlp measurement ta la ta la ga auto ga ga cb mae ta la ep ta la ga ga ga next perform look rank nlp similar la ranking dataset rank suggest ep consistent ranking ta large ta look ranking mae ranking dataset ep almost ranking mae mae la statistically significant la dominate ta mae interestingly auto significant highly affect likelihood evaluation likelihood nlp usually dominate mae correct inference ep rankings nlp ep mae wrong highly affect spectra cast input sample particular spectra scale interval map via equivalent logit hyperparameter constrain beta spectra perform set hyperparameter learn lc train max std gp beta ccc cb small avg error std error exact beta taylor beta due actual available small drop gauss beta taylor using logit ta la ep perform show la ta difference follow ta marginally dataset ta ep ta std l cc training ta ta la ep ep avg std counting base crowd crowd count dataset dimensional extract people direction leave motion crowd per leave crowd poisson com mapping canonical linearize compound plus crowd right crowd poisson com perform due crowd linearize link well crowd indicate crowd crowd com well crowd difference com poisson flexibility control crowd l cc crowd crowd right crowd crowd mae nlp mae nlp mae nlp mae nlp taylor ep laplace ep linearize poisson lin com cl cc cc c crowd crowd exponential linearize mean exponential nlp mae nlp mae nlp mae nlp com poisson taylor com right crowd crowd linearize inf mae mae nlp mae mae nlp exact ta ta com com ep com cc
use equation nj compute g relate curve class curve mixture class maximize principle mixture particularly adapt compose handle curve curve curve polynomial variability class regime g number parameter number free training note practice selection store select computationally classical small minute piecewise regression notice regime adapt section dedicate simulated real diagnosis comparison alternative functional discriminant spline sr use alternative sr polynomial mixture misclassification curve intra performance approximated error k curve sub spline spline curve shape compose three homogeneous sub class curve regime compose complex shape compose regime shape observe automatically sub class underlie hidden regime flexibility regime accurately approximate regime regime regime sr class heterogeneous provide clearly sub change adapt color top bold top regime bottom c error pr table indeed intra propose discriminant polynomial mixture spline attribute flexibility logistic adapt regime observe approach attribute heterogeneous provide rough misclassification approach fact shape adapt class model modeling therefore accurate good describe propose em repeat attribute fact approximate homogeneous select case next approach curve datum interval constant sampling heterogeneous top curve show class approach sub identify notice set regard approximation relate class change polynomial database context monitor switch mechanism train one track accurately detect service electrical switch curve hz second e switch successive part mechanism start phase notice shape duration vary used compose real switch operation curve class minor critical rather class curve minor without accurate automatic henceforth number class sub estimate class curve change shape class see probability seem curve vary smoothness c real separately sub class regressor degree process similarly bold regressor logistic proportion consider approach approach c approach intra switch although attribute mixture regime compare however well summarize table pr sr effort polynomial spline analytic base learn fast around one alternative piecewise comparison mixture spline still fast algorithm require converge adapt change one build regime change consume complex mean software cpu ghz dataset regime expert switch curve minor polynomial well selection confirm present classification discrimination include regime discriminant shape present change dedicate algorithm datum alternative demonstrate effectiveness classification likelihood mainly interested rather maximize criterion approach model em incorporate knowledge complexity change time specific regime particularly handle complex regime homogeneous sub propose model explicitly heterogeneity change model homogeneous decompose several regime observe dedicated maximization comparison approach linear discriminant regression mixture spline simulated outperform discrimination significantly curve discriminant unsupervise involve finite domain system electrical engineering speech recognition study datum e paradigm functional datum individual curve finite goal visualization exploratory analysis cluster technique task unsupervise etc challenge build infinite supervised discrimination temporal present discrimination include unsupervised task automatically regime class segmentation indeed first application unobserved class way label handwritten digit write digit class digit diagnosis decide without label label happen mean minor provide tool gene gene involve biological profile unfortunately rough generative approach dedicate generating describe explicitly integrate dispersion regime generative functional spline also discrimination modeling process approach use site process direct focus parametric example extraction mixture regime take segmentation paper generative shaped class class particularly dedicate sub relate present mean functional discriminant hide functional derive functional discriminant discrimination discriminant learning achieve first discriminant analysis present curve classification parameter estimation dedicated criterion bic training class background approach use program homogeneous regime smooth change process approach however limitation shape extend discrimination linear discriminant single discriminant conditional density density process use discriminant analysis limitation shape class curve furthermore thank flexibility approximate able hidden regime discriminant analysis hide unsupervised em present sub class class govern unknown regime mixture curve homogeneous sub probability govern hidden regime switch point unobserved sub class th denote unobserved sub class switch regime regime encode manner index equal code four index group regime sub class regime logistic thus regime transition regime logistic control parameter smooth regime give th sub regression
case occur carlo degeneracy underlie move mix datum apply gradient ascent outperform long state increase distance value long filter robust preferred particle parameter agreement count united approximate approach account also event ascent compare smoother except implement lag estimate alg alg lag result ascent execute iteration short smooth method estimate wide estimate result understand times monte carlo variability smooth estimate et example range estimate lag equally two lag algorithm parameter increase fast increase cost compare estimate information increase particle technique kernel yield vector display achieve thus ascent scheme unbiased estimating apply recursively term parameter relatively insensitive score convergence computational improve root square initially long series model filter estimate state prove induction firstly expectation coefficient ts tc coefficient update gamma prior follow initial scheme proposition show score matrix suffer whose amount paper approach terms degeneracy rao crucially robust choice kernel estimation estimate within batch show improve estimate significantly reduce computational popular distribution sequentially add approximation distribution represent q estimate score iteration weight show calculate value estimate observe information mean rao replace like get inclusion important shrink iteration detail algorithm update mean x k update n w I I give give illustrate result quadratic monte show mixing produce estimate linearly density rao impact degeneracy score vector observe say algorithm possible implement store current help understand running filter function state function recursively value th solve recursion shrinkage parameter algorithm simplify additive functional property monte variance least linearly smc monte carlo contribution weak carlo monte additional estimating problematic monte score increase infinity condition hold define ty expectation respect give implement filter direct true assume condition respect take proportional px u x expectation estimate mean standard estimating equation depend also underlie note efficiency obtain fix smooth reduce increase lag proportional pt pt pt show establish variance increase increase note lag bias outcome give lag appear estimate range lag fix lag reduce mean variability associate rao scheme detail drawback section pt pt nn notice time reduce wish estimate exhibit linearly minimal introduce overall result interest reliable estimate estimate use ascent algorithm maximum estimate ascent offline datum set alternatively recursive estimate update new q get different value iteration sequential monte carlo ignore gradient first autoregressive model score lag smooth ascent estimate noise allow kalman calculate lag fix smooth filter newton ascent lag smoother fix lag smooth particle gradient figure comparable achieve computational
still refer optimum initialization initialization learn hessian activity nonlinear critical value activity constitutes convert department stanford department physics stanford stanford attempt bridge systematically analyze restrict deep linear show deep exhibit nonlinear phenomenon see long rapid greedy analytical phenomenon find exact dynamic theoretical analysis reveal surprising approach finite find scale initialization exhibit orthogonal condition weight time far propagation special edge realize application object natural difficulty presence minima curvature decay neural network nonlinear dynamic apparent stage quantitative rich determine scale speed depth greedy unsupervised statistical inherent answer expressive aspect practical network exhibit error network descent convex error exhibit subtle weight layer deep network important understand deep answer question differential dynamic find nonlinear dynamic intuition deep build statistical compare learn non analytical nonlinear phenomenon rapid indeed show previously nonlinear sufficient qualitatively capture see solution greedy layer deep recover approximately mnist finally exhibit gradient nonlinear even long gain neural exactly activity even regime act three output layer network wish accomplish square output gradient descent rule sufficiently small p tn n linearity descent constitutes couple cubic fundamental dynamic output correlation though focus orthogonal exactly preprocesse orthogonal representation decomposition reflect input whose contain independent mode whose order perform weight gain element neuron layer think matrix connect neuron element connect let intuitively hide come mode column go term arise descent display competitive associated mode strength magnitude connectivity term competition connectivity mode symmetric force distinct mode drive regime connectivity fix structure work connectivity nonzero fix unstable remarkably dynamic saddle global minima hence converge rank correlation matrix interest dynamical fix difficult initial competitive interaction input mode restrict condition mode supplementary dynamic three layer curve strength seven course trace analytical trace show full linear eqn trace network activation strength network evolve consist hierarchical describe five tree flip mode exclude clarity delay competitive half analytical time analytical half error bar initialization orthonormal differ scalar magnitude svd u arbitrary subsequent excellent trajectory straightforward even interact product condition mode evolve dynamic scalar obeys connectivity dramatically nonlinear error monotonically approach satisfies transformation simply plane point unstable fig show typical dynamic initial treat explicit random track obeys q ask within fix e weak cutoff key learning strength learn course invert obtain course describe temporal magnitude strength mode start display rise sigmoid sharp sigmoid linear start initial manifold exhibit competitive dynamic connectivity note though network behave analyze net act deep network make attempt simple gradient neural gradient descent write w ia weight special output input strength mixing mode overcomplete change variable evolve simplification network mode scalar whose dynamic obey energy analog dynamic set arise track obey generalization integrate complicated rapidly eqn study dynamic remarkably fix iteration require go continuous operate must stable estimate rate mnist empirically learn decay large see supplementary incorporate dependence depth surprisingly supplementary emphasize analysis speed base train deep mnist task calculate training correspond nearly complete optimize depth pick fast appendix full network initialize strength time empirically optimal slow regardless moreover incur depth strength association find mode strength large existence mode evolve learn arbitrarily deep network long procedure rapid deep start discovery regularizer solution excellent scale initialization though interestingly initialization exhibit supplementary analytically network condition supervise precise subsequently autoencoder module output subsequently simplicity correlation matrix svd matrix variance handle identity general end roughly balanced map arbitrary tuning interest input begin condition section possible submatrix raw mnist input submatrix start condition delay due condition deviation initial strength state underlie vector input task precise idea help task consistent map moreover evaluate right singular approximately properly initial association near argument straightforwardly deep consistency empirically hold mnist deep network learn start small time supplementary appendix experimental analysis completely network nonlinear approximately e initialization nonlinear regime solution expect mnist architecture scale preserve norm gradient pre top learning rate plane histogram element choose take singular histogram distribution complex visualization purpose contain remove dominate plot greedy choose appropriately preserve deep context norm preserve choosing scale forward backpropagation gradient initialization depth linear train leave blue growth make depth strength scale initialization scheme preserve find leave red curve prediction choose mode greedy composite initialization rapid pre show fig red scale yield greedy training red green indistinguishable scale despite spectra gaussian orthogonal spectra lie exactly unit circle complex plane right spectra disk complex plane distribution nontrivial represent propagation activity layer preserve matter layer spectra orthogonal layer yet singular different singular singular value remain preserve strongly closely backpropagation early act projection operator concentrate close origin depth discrepancy eigenvalue phenomenon occur eigenvector non random gaussian definition norm clear way error projection vector onto yield correspond early present random start appropriate associate act overall global subspace possible many possible closely notion isometry compress projection preserve necessary achieve isometry network initialization exact dynamical isometry value greedy pre achieve pre initialization application versus scale initialization application supplementary appendix high nonlinear recurrent thought feed tie promise approach objective partially isometry back linear isometry dynamical isometry feedforward denote orthogonal connectivity gain factor nonlinearity show supplementary exist critical gain decay strong nonlinearity activity propagate decay network nonlinearity odd approximately capture neural population cx g exhibit phase critical gain infinitely propagation critical terminology value population consist neuron interested final layer early decay quantify perturbation propagate singular jacobian extremely value backpropagation behave decay jacobian point activation iterate start activation jacobian symmetry expect depend variance jacobian analog variance edge layer combination nonlinear yield fraction value concentrate despite propagation isometry nice variance increase beyond isometry network pattern enter interestingly singular singular compare row panel fig numerical beyond good deep nonlinear input map dynamic reveal surprising rich structure nonlinear transition saddle point quantity independently evolve rapid importantly sensitive learn scale input unsupervise mathematical backpropagation isometry initialization achieve preserve nature random initialization thereby finally show dynamical good network beyond address phenomenon deep carefully initialization full treatment currently open one reasonably hope move understanding sense progress learn material treat strength reasonable initial though access dynamically invariant situation attract
response student know answer truth else response noise ground query instance amazon rating per query pair ask rate relevance possible relevance represent relevance ground house original query instance ground rating slightly break exclude map value affect ndcg affect mse dataset contain rating point query metric method denote rank corresponding query rather absolute truth ndcg evaluate list rank ideal sort ground relevance note ndcg mse ndcg well might optimum initialize initialization metric lower restrict implement ordinal configuration query treat query query category abuse difficulty well simple majority vote median result table highlight mixture perhaps surprisingly difficulty seem dataset ordinal instance difficulty performance query perform amongst discuss overall dataset model low suggest ordinal amongst multi label method indicate confusion beneficial l likelihood experiment spam ordinal mapping mixture presence htbp rating per left correlation right ndcg bottom main additional present ordinal variational inference model review state rating binary value truth experiment world contain relevance perform model mse correlation ii propose baseline joint generalizing account thank discussion would foundation label confusion em treat equation normalize additionally dirichlet extension experimentally evaluate extension denote denote similar except replace denote confusion account instance difficulty log odd logit bilinear imply guess model incorrect equally likely typically realistic case let follow co optimize gradient solver optimize impose prediction label correspond ordinal coin explore ordinal case truth ordinal confusion assume take lead combination value assign prior model confusion indicate indicate assume ordinal binary vs derive prediction mean college department approach wherein datum ground truth noisy label obtain multiple level annotation ordinal mostly extension categorical counterpart receive crowdsource account derive bayesian analyze ordinal ordinal collect amazon label without baseline median supervise task rank label unfortunately obtain label large expensive obtain label unknown level crowdsource amazon enable significant use crowdsource annotation domain language processing computer naturally handle evaluation ground combination although frequently patient detecting optimally form mean difficulty bayesian flexible complexity annotation incorporate modeling jointly label optimize parameter cf exist broadly criterion ground label categorical truth level combination idea model difference estimation bayesian ordinal labels movie rating query annotation ordinal mostly extension natural ordering label experimentally ordinal order value focus categorical even truth involve vary crowdsource quality look useful e instance account able weight combine use obvious image assumption variational baseline vote label systematically ordinal annotation real retrieval demonstrate outperform inference relationship model conclude introduce update image pair unobserve ordinal sparse index nm concrete example contain pair require assign query relevance low relevance annotation collect ground web query belong equally could query pair category might interpret category every separately instance treat correspond rating x nm nm rating describe truth instance precision spam across choice allow level noise annotation rating spam component probability value model draw simply large b specific threshold cause may ordinal dependence ground truth variance interpret exhibit low nm indicator gamma shape inverse scale bernoulli follow hyperparameter rating likelihood variational identifiability likelihood annotation intractable variational vb alternatively markov slow estimate vb eq expectation variational denote mixture continuous induce dependence since outside interval simply interval integrated ordinal treating variational update practice probit variational approximation naturally posterior gamma form truncate upper denote expectation moment involve define overview ordinal highlight previous
offer greedy local approximation apply column surface dimension slice surface exhibit truth still despite appear smoothly narrow range estimation via globally inspection reveal area row middle suggest cope benefit localize spatial leverage structure modeling choose sensible global design possibly smooth spatially step design initial distance input sensible default square distance calculation via newton like initialization smooth later stage illustrate require share design find local calculation newton step concern stem consideration surface multimodal guarantee rapid mode sensible priori constrain slowly however absolutely though aid scheme illustration illustrative computational effort snapshot stage histogram stage notice increase fidelity histogram take value problem variation base greedy initialize smoothed stage include novel place nn context variation reasonable inside towards find obtain also big nn accuracy speak infer greedy magnitude design although per expense store broad common computer response eight rectangular domain illustrative adopt estimator fast generate training call constraint try frequent page lead estimate suggest code take finish contrast greedy require outline design old krige neighborhood control accuracy really argue favor simple yet algorithm order large mean greedy spend budget indeed competitive computation design allocate uniformly potential reduce increase large decision calculation design say effort stage global one inference iterate additional area budget represent alternative amongst location option track variance stop certain response common spatially time relate local search practical consideration nn trade design flexibility design large exhibit spatial heterogeneity improve relative suggest beneficial pruning search smaller elaborate allow covariance structure strength grid globally greedy boundary highly simple rule along search close member option restrict search size candidate search time benefit parallelization computing core core gpu search yield efficiency modern gp applicability field site sensible build local design jointly provide package context challenge optimize box design grid acknowledgment author would discussion careful thank constructive support part number computer experiment engineering management edu approach approximate equation local sequential scheme dynamically predictor sequential need quantity build iteratively vast trivially modern provide modeling utilize thousand compactly covariance key design sequential update active compactly local neighborhood popular choice rarely attractive ability say computation generating mechanism appropriate usually often stationarity schmidt process convolution allow smoothly spatial several work burden kind search iteratively krige compactly support sparse contribution aspect association ad hoc krige neighborhood pp modern ideally computer recent stein chi active start prediction locally computer recognize usual covariance calculate great expense contribute consider illustrate sensible key heuristic recent computer lee localize nn yield accurate design result obtain prediction recognize modern nn attractive family na I one nonetheless compare globally suggest purely innovation global hybrid learning retain un analog retain stationarity exactly code counterpart highly proceed mean nearly outline gp prediction criterion design efficient argue simple heuristic literature treat conclude gaussian define yx yx assumption modeling literature convention base small pre determine surface computer deterministic depend isotropic simplifie exposition derivation separable family package interpretation matrix normal conditional bayes gain specify ig analytic newton leverage see likelihood surface modal predictive degree freedom student x fast goal use sub neighbor equation converge trivially involve computational marginally uncertainty usually nn fact nn hard involve compound expense simple comparison extra forward particular together comprise greedy decision search criterion must scheme remain comprise easy show take account prediction yx jx notation explicitly indicate arise calculate mle argument also approximation explain turn contribution place col come predictive respect expect fisher comprise plus expect jx j vx fold derivative b student match one current obtain require average analytically though context undesirable whose odd extra preference nearby sensible non design maximize determinant thus weight fisher balance aim predictive appropriate automatically uncertainty aspect localize quantity update quickly time partition inverse application column yield require fast key aspect time j variance may observe trivially scheme sequential decision newton like analytic local defer fisher new add computer simulation lee rather single whole space jx jx dx ignore special integral otherwise method burden choose maximize sensible heuristic variance repeat application design show subsequent author refer numerical integration require mean simple involve therefore sensible alternative terminology maximize important context novel since integration straightforward heuristic derive correlation consider surface study x xy figure show local input comprise agree site nine site qualitative box take modern trial os time invert knowledge primarily invert seven second aside would ht precision light solid line open circle nn illustrate
real atomic norm signal semidefinite affine atomic signal decomposition signal toeplitz toeplitz toeplitz give atomic norm high frequency propose semidefinite atomic frequency result applicable signal atomic use dimensional signal tensor inner primal minimization primal solution turn variate polynomial j l l follow elementary toeplitz one elsewhere diagonal square diagonal addition every certain hermitian matrix l dual strict strict inequality decide relaxation sum square relaxation psd dm optimal exist dm dm strict strict strict strict keep increase suffice decide check check primal atomic norm atomic atomic norm big optimal dual extend constraint solution equal immediately minimum restrict evaluate multi minimization semidefinite generate sample distribution frequency localization signal achieve minimize author university e mail discussion xu chi conference presentation introduce open xu helpful visit corollary compressed recover though particular atomic minimization recover sparse exist research effort equivalent recover dimensional grid propose programming minimization recover signal grid estimation matrix completion compress sense cs sampling digital algorithm inherent frequency sparse signal dimension j l represent signal value discrete fouri dft appropriate discrete representation quite frequency domain frequency center dft bins dft traditionally fine dft get frequency propose semidefinite atomic norm minimization computationally semidefinite guarantee norm unfortunately semidefinite characterization extended arise difficulty generalize beyond proof paper atomic decomposition block toeplitz toeplitz toeplitz atomic frequency precise programming grid literature
mn except sufficient underlie dense topology elimination mn require extreme conditioning test require large amount reliable figure confirm confirm cascade traditional independence sufficient competitive runtime bar light gray column second experiment mean ghz gb clearly expensive grow phase require test many variable source extreme case hill mn low elimination really effective large quickly last mn perform conditioning average conclude confirm empirically achieve complexity present hill increase ten repetition axis axis omit similar c measurement hill dataset ise mathematical mechanic decade domain vision ten ise ht bar light gray size ise ten repetition order measure runtime column respectively clearly learn hamming algorithm respect structural improve significantly cascade traditional run computer previous except mn well follow case mn runtime runtime similar cost positive mn benchmark obtain uci unknown distance utilize encode work density goal knowledge evaluate structural number independence query read structure vertex separation possible independence variable query match distribute conditioning conduct real dataset list one sort domain learn accuracy information attribute third dataset set algorithm well bold result tie mn mn hc train test balance car band algorithm contrast arbitrary application evaluation fitness well learn synthetic dataset hamming distance mi clearly outperform mi highlight efficiency allow estimate independence maximum posteriori avoid cascade error traditional trust central pose structure posteriori compute compare art method complexity test practical challenging result optimum markov confirm worth ib independence closure testing compare grant scientific university national university innovation special thank support appendix closure closure independence determine reproduce necessary markov satisfy positive q w property lemma relate proceed strong union remain need argue something counter proof counter apply dependence intersection take independence obtaining intersection long respective match therefore independence apply xx undirected determine structure separately absence two closure independence similarly closure assertion empirically landscape synthetic dataset surface ib assess hill maximize ib domain explore landscape ib look like sort show ib structure landscape indicate axis ib structure sort structure axis log equation ib low ib also indicate algorithm ham correct axis ib landscape observe landscape shape curve tendency leave right scale axis second landscape learn highest learn close experiment domain landscape ib score landscape randomly axis conclusion confirm maximize ib score landscape work ib auxiliary edu networks problem avoid exist algorithm proceed incorrect reduction quality probabilistic polynomial show algorithm quality complexity application discovery evolutionary use model population optimum present work posteriori robust network efficient hill together network compactly represent distribution list wide evolutionary graphical markov direct encode reason distribution importance structure hard enough attract attention datum draw however np super mainly know goodness structure recently develop algorithm proceed independence test outcome structure inconsistent present independence test mention independence distinct score suited density goal inference goal classification design robust correctness consider explicitly posterior test well posteriori long discard incorrect probability test may systematic experiment real structural representative art mn simple adaptation network note quality discovery test genetic variation replace mutation stage sampling generation optimization population distribution structure learn effective experiment markov structure learn present overview approach contribution synthetic dataset pose direction include end motivate capital bold domain potential set read say separate completely k potential parameterized representative format perfect independence discard structure outcome test outcome learn independence conditionally proportional involved independence use continuous task estimation reach complexity perform estimate another important assumption true unfortunately rarely contingency table exponentially statistic decrease exponentially statistic incorrect produce independence posteriori error completely outcome idea computing approach posterior combine remain section closure posterior next closure closure conditional determine definition replace closure joint exactly base assertion express avoid posterior term monotonic approach computable undirected structure present hill give specific computing ib score heuristic maximum ib neighboring present pseudo search line create ib starts loop neighbor heuristic neighbor avoid expensive ib score line stop improve score check termination variable assign variable ht score current ib candidate line define closure present next closure test number closure variable connect term allow non give closure formally determine union dependence variable union mutually neighbor assertion xy x independence assertion closure determine construct undirected set completely determine theorem closure contain number size domain decompose permit neighbor previous remain ib line algorithm reduce convenience section decompose ib pairwise ib function reduce neighbor ib neighbor differ ib one neighbor perform compute ib closure cost hill ib ascent result statistical total heuristic computation cost input current corresponding contain score decomposable construct pair return note one number see sum ib score per assumption make ib structure need w use ignore mild possible pairwise complementary structure posterior sum ib score make heuristic impact optimal course effectiveness would sub follow sub follow impact approximation avoid measurement landscape ib score synthetic summarize whole hill begin ib score initial markov closure cost incremental require termination measurement empirically grow test several dataset robustness systematic comparison independence structure learn independence adaptation structure compare algorithm test learn finding gs add gs two phase phase increase markov find currently phase markov true potentially false positive remove phase find end match correctness test discover parent child symmetry correction code page pc rank unconditional discarding utilize inclusion candidate inside remove elimination iterate examine inclusion find execute network
sample prior computation integral analytically reach I z formulation infinite use example output whole example serve set retained network table square rmse reflect component outperform ht mixture apply multivariate apply adapt approximate mixture learn value advantage multimodal cubic multivariate process dirichlet adopt allow automatically infer latent feasibility value provide principled pattern formally collection number joint process motivate need attract multiple relate different representative work process two important limited inherent distribution multimodal second computationally infeasible big since inference process number greatly jointly explain mixture mixture variate gaussian extension mixture yet fill propose gaussian note multivariate complicated variate process rest infinite hide perform report regression finally conclude remark multivariate process depict graphical infinitely belong component dimension rd include latent wishart parameterization gaussian prior process positive specifie similarity input component give set obeys mean whole difference use initialize step indicator update output gaussian three step constitute adequate stage subsection formulation let decomposition joint calculate term involve computation efficient addition expert prior expert directly conditional conjugate define output gp product belong component easy original gp reduce inversion use metropolis
fundamentally optical principle perfect resolve fine detail point connection utilize theoretical domain claim point receive community old hope cross branch toy example recover structure mapping invertible testing homogeneity wave section explain learn map make surprising statement super discuss relate wave symmetric arbitrary eq call coefficient vanish induce reproduce rkhs hilbert represent take call space point directly space kind optical imaging first taking whenever pd converse map identical kernel imply obvious whether characteristic kernel exist e kernel moment high order pd characteristic pd let distinct product reproduce property amount pd conclude mean among operation structure integral consider light map mean equal analog sign borel integral eq pd pd vice versa pd kernel characteristic brief consider difference sign borel reproduce mm kernel consider pd written borel write rkhs ix nonzero spectra distribution might differ I e distinguish distinguish translation pd corresponding empty characteristic obvious restrict obvious choice characteristic function agree interesting class measure wiener characteristic analytic imply know compact determine everywhere pd support probability simplification indicator interval correspond convolution support non govern partial foundation classical field situation couple field well wave medium differential another linearity major allow analyse study stimulus obtain fully impulse implicitly stationarity optical record intensity square integration long intensitie temporal take coherence simplify complex effect correlation scene negligible contribute plugging expression yield eq intensity impulse image assumption incoherent description typical imaging image inherently negative induce probability addition translation invariant incoherent imaging view derivative optical without optical object harmonic spatial frequency plane wave wave transform direction refer situation finite camera ideal state transform follow compute circular transform auto function circular box function square kind pd due proposition theorem fourier pd actual resolution determined size mirror incoherent insight section surprising support area least theoretically limit bound recover limited proposition induce translation invariant pd theorem note compute section present illustrate source optical diameter incoherent illumination superposition image impulse response optical system apart resolve place two dash record model additive toy noise leave object length object express fourier hermitian transform transfer fouri impulse object maximum I least show matlab z finding exactly shorter already optimisation unstable account employ non negativity non negativity good bottom negative least artificial double green camera camera consist truth mm exposure double angular separation double star percent star ms eight minus average frame reduce able double star similar truth panel c cc recover break di state accept limit prove limit gain super observe increase isolated author discuss resolution bound illumination frequency object summation technique continue cut fact fourier transform case impose limit criterion illumination convolution wave inversion operator division fourier reconstruction overcome retrieval positivity slowly beyond relevant bandwidth object positivity book discuss early limit conclude image several paper bound constraint
minimization technique form constraint introduce paper applicable graph cut fulfil obtain note apply noisy show non negative volume ga ig da aa balanced computer range parallel image segmentation cut criterion dc dc normalize popular cut balanced cut incorporate motivate optimize subject constraint discuss problem network cluster work goal vertex cut subsequent community runtime seed bind contain relaxation derive type biased seed around seed balance dc vertex balance influence get partition add complex diameter normalize volume constraint dc community sense highly separation search association divide association density size introduction thus denominator bias obtained assign prefer occur preferred bind team selection bioinformatic develop bind hard using equality experiment show specify co author dc analogously dc problem dc normalize cut constrain program continuous relaxation enable integration form constraint detection derive compete aware problem satisfies volume seed although problem relaxation ratio numerator extend set restriction concern either generalize without restriction handle particular negative thus write direct continuous achieve assume define frequently indicator otherwise derivation paper extend see hypercube rf c submodular b modular connection submodular rf minimization problem main extension see positively f observe unconstrained follow tight agree computed problem continuous process sf ratio thresholding replace practice may submodular second remain term replace exist decomposition difference extension non moreover sf optimal thresholding statement replace collect submodular homogeneous extend space submodular positively furthermore order increase note convex positively homogeneous write rf hold eq condition b always furthermore let satisfy def eq negativity division statement analogously ready b lemma imply equality statement regard negativity make program objective fractional make transform unconstraine term constrain fractional set program define zero constraint otherwise increase possible terminate otherwise positively homogeneous thus necessary unbounded norm play special case one convex programming provide improve stop feasible result let find proposition terminate decrease latter strict monotonicity assume infeasible analogously fractional fulfil relaxation discuss problem constrain balanced relaxation omit first volume hc c hc hc could seed inequality numerator however form relaxation lead lie treatment empty set empty optimal either consider either low equivalence result tight minimization moreover extension f sake brevity subdifferential prop index small lead f fs hc proceed observe denominator tight balanced cut follow inner q w positive replace inner give homogeneity ff mx mx result solve proximal explicit fista inner htb input v rs rs z z rs rs rs expensive part number subproblem fista v straightforward kkt leave one unconstraine tight relaxation globally equivalent every solve replace prop rewrite w minimizer rewrite u pseudo ls ls ls ls l amazon mm detection goal fractional program optimal relaxation globally loose improve obtain experiment report regard result case fulfil could deal noisy seed vertex random social stanford network collection runtime amazon amazon relaxation solve optimally continuous contrary guarantee yield set seed guarantee perform optimal consider threshold generalization compute locally bias relaxation compute explore perform graph concentrate seed cut guarantee result seed fair thresholde cut seed ensure volume constraint treat unconstrained experiment standard seed runtime initialize solution l compete margin initialization normalize cut competitive normalize cut normalize cut solution walk accord cut community accord extract community around seed co publication database publication l share paper avoid give co author find densely connect researcher connection restrict consider seed graph enforce densely seed know researcher member community area validate two l frank seeds community key community member group acknowledgement support main theorem minimization function frequently occur community typically use practice relaxation optimally often loose lead optimum
weight might contrast edge law function one build auxiliary intend small graph practical availability offer subproblem auxiliary graph body submodular function offer could position advance research aspect important open deterministic graph consistency probabilistic reference notably lot graph estimator consistent show arbitrary represent observation weight involve integral usual valuable estimator basic even small promising theoretical construct contribute actual expect work graph auxiliary raise optimization approximately desire simultaneously close span obtaining path cut etc suitable estimator linear wish derive obtain corollary area towards expect length shortest span etc insight help predict property motivate statistical subgraph combinatorial subgraphs specific problem encourage type nontrivial machine area obtain statistic example short tree help predict look scenario complete edge accord explicit weight subsequent refine graph suggest size graph statement bring certain structured expect minimum structure function current question investigation perhaps aim nontrivial statement encourage study move onto formal integer value assumed simplicity I assume add extra weight correspond vertex reduce collection subgraph ultimately vertice edge characterize often convenient depend span henceforth efficient additionally care highly nontrivial nonetheless justification weight structure subgraph member additionally include submodular express let continuously differentiable finite variance cost minimum consistency edge special class boundary let real say necessary question point blind lead establish norm behavior behave govern conventional illustration setup modify correction observe require consistently point construction inconsistent weight let satisfy per vertex sample g
correspondingly every computed cancer patient validate signature develop signature separately er patient patient predictor tumor er status tumor patient age publicly wang diagnosis distant clinical predictor day survival censor main figure optimal signature cox lead marker signature external median crucial external median optimistic interpretation estimate model yield large survival cox regression hazard survival tumor effect recurrence seven tumor er status survival index one boost approach result cox correlate boost estimation plot refer et van al base test dot correspond new breast cancer van collect publicly add use van validate signature free survival van et set five predictor tumor affect node er status tumor patient times survival time censor index cox proportional apply penalize less however boost lead ridge van ridge penalize approach new boost training assess estimate median clinical predictor status true patient hand tumor affect tumor result negative regard gene could correlation coefficient boost index penalize cox article time measure survival molecular become tool medical measuring quantify ranking value marker combination different prediction quantify agree numerically actual rule calibrate versa feature molecular signature contribution derivation marker combination consequently long traditional cox calibrate well rule conceptually article algorithms discrimination binary outcome curve auc fact measure correspondingly datum rely concept smoothed influential algorithm breast cancer apply stop candidate marker early issue automate selection boost survival marker numerical marker combination compute meaningful power signature acknowledgment author cancer www grant role study collection manuscript index boost boost relatively weak perform base accurate via aggregate generally concept boost drastically prediction single solution base later adapt stagewise final specifically boost outcome boost fitting boost flexible boost variety implement learner combine see optimize w version necessary newly develop indicator sigmoid weight compute implement family sigma sigmoid sigma sigma sigma sigmoid sigma true true w weight ng w event n return build object offset check stop family name briefly combination pre van consider publicly gene signature originally van ensure code carry patient fit carry package object evaluate prediction package implement al load package loading loading set response survival via datum predictor family sigma sigmoid approximate boost tuning convenience define mod boost trace nu datum stop change via indexing mod mod take standard department medical universit universit cm development molecular signature event bioinformatic although numerous approach derivation marker combination methodology might marker regard criterion interest unify prediction rule boost smoothed index simulation molecular set survival patient sound lead derivation signature development field number signature clinical survival signature survival molecular signature remain especially outcome survival process development signature comprise subset gene outcome derive marker signature evaluate subset gene address calculate association survival subsequent association signature fulfil form linear combination gene cox regression molecular direct cox consequently marker univariate cox regularize cox ridge task challenge survival mean long applicable censor several approach address article survival briefly survival vice versa measure ranking discriminate survival helpful aim patient poor medical index scale perfectly interestingly derivation gene signature survival cox hence evaluate index root receiver operate roc methodology practical marker optimize partial optimize therefore may suboptimal marker discussion propose framework survival analysis evaluation signature demonstrate index performance article development combination gene gene combination boost combination propose signature index performance basic article survival clinical predictor survival survival let censor time survival censor coefficient assume learn sample cox cumulative hazard expect predict survival combination base index general discrimination evaluation ordinal outcome times predict survival short vice versa moreover area decade gain popularity research result article marker marker perform well chance breast cancer flexible discrimination especially gene process gene compute gene yet various way rank influential one advantageous survival survival censor survival time ignore notable bias censor bias correlate censor overcome censor et modify unconditional survival censor estimator inverse censor numerical suggest censor estimator exist approach overview cox assumption violate free guarantee situation censor contrast boost core derivation combination base wise use gradient boost predictor aim algorithm minimize empirical risk marker survival result less regard tuning indicator decrease boost implement programming specification practice evaluation influential gene use boost challenge result since use criterion marker combination task advantageous perspective practical combination optimistic estimate accuracy observation evaluation marker signature two splitting optimize marker external serve elaborate subsample five fold split subsample datum next consideration precise smoothed version propose evaluate observation question unconditional survival datum combine principle simulate datum aim select marker variable check cox base smoothing smoothness sigmoid censor survival generate accelerated failure equation follow realization marker normal predictor marker actual survival four contribute scale cf problematic censor survival lead censor boost leave marker effect answer framework develop signature bias simulate separate set influential predictor later boost l I external evaluation prediction simulate separate first available marker predictor individual include perform marker suggest marker outcome marker gray white ccc set boost cox cox ridge index boost compete run result amount size refer censor marker apply propose boost result parameter boost able combination display essentially survival effect marker close range close true marker performance
root retrieval call prediction structure amongst example name entity language vision binary hamming balance majority single indicating define harmonic sake simplicity refer measure despite popularity experimental find moreover present exhibit desirable property statistical consistency theoretic I distribution expect measure trivial force infeasible would require checking sum researcher study problem label structured hamming immediate candidate surrogate surrogate statistically regret increase multi classification closely optimization apart optimize approach find maximizer last decade require independence problem like binary indeed like prediction arbitrary probability bad entire maximize paper section quadratic cubic time regardless underlie relevant application aggregation distinguish instance wise micro averaging carefully distinguish version optimize evaluation extensive illustrate usefulness finding label competition experimental winner surprising estimation superior exact length etc decision go look f measure orthogonal optimize f two recently asymptotic equivalence binary assumption interesting theoretical hold instance needs optimize f measure discuss nevertheless mention binary classification rely explicit output svms solve closely theoretical experimental approach measure inference context view already conference paper summarize framework theoretical formal give formal serve key element algorithm suboptimal regret optimize subsequently present experimental benchmark proof section f framework related encountered learn paper investigate call regret maximizer literature classical tool bayes however emphasize theoretical investigate training differentiable surrogate analysis bound surrogate loss analyzing algorithm measure paper losse internal notable important train estimate arbitrary maximize loss distribution reason technical maximizer minimizer bad case unique regret restrict probability otherwise become require comparison minimizer instead favorable case q favorable prefer exclude bind maximize uniqueness maximizer regret maximizer notion fisher sufficient exact solution derive analysis investigate prediction hamming loss interest simple regret optimize loss widely use output see general hamming hamming loss risk ham loss present vector bad maximizer indicate hamming prediction confirm function generalize label function appear many make mistake mistake label discriminate prediction lot exist framework multi optimize zero structure hinge surrogate slack rescale classifier chain logistic base maximization posteriori minimize subset risk prediction simply mode hamming look suffice minimize subset violate similar primarily connection summarize follow analysis prediction obtain bad supremum rapidly illustrate hamming confirm subset yield might valid alternative experimental originally set know similarity family similarity set index compute union q write gain kernel feature vector bioinformatics domain often utility multi remain loss contingency recently wise contingency without intractable index think characterize prediction utility f regret maximizer respect upper bind rather loose interesting upper maximizer predictive situation learn dataset relationship similarity thing practically mainly relatively reveal optimize loss surrogate performance type specialized assumption operate constrain sometimes justify theoretical secondly independence variable substantially simplify optimal solution high marginal examine independent parameter addition show independence determine exact independence probability solve solve transform h outer outer maximization check possibility effort solve need vector high probability remain computation exponentially count much distinct exponential result obtain cubic complexity cubic long efficient label greatest derive follow recurrence condition ta j programming expect complexity dynamic programming complexity additional share new version construct recurrence equation leave extend independence propose utilize recurrence equation technique contrast primarily focus contingency manner f maximize label none able guarantee optimality maximizer compute analyze method model maximizer distribution illustrate distribution probability measure configuration check respectively regret may produce let obtain independence regret supremum satisfy detail scenario regret lower tight going infinity summarize corollary prediction assume independence bad supremum take probability distribution world categorical entry ik categorical distribution reflect multi special case label applicable high zero expect marginal despite threshold scenario independence provide order accord probability threshold need problem polynomial maximizer label fact substantial sort supremum take find bad surprising light many find seek justified application thresholding yield prediction illustrate maximizer yet exhibit rather low easily complicate maximizer adopt maximization solve convenience introduce quantity label specific th correspond maximizer theorem solely constitute straight high result summarize input inner label take outer optimization solve top repeat overall straight forward require time light combine particular like multinomial reasonable consider probability q let element matrix dominate significance practically I huge clearly maximizer affect label word necessary quantity take occur comparison apart numerical might contrary maximizer independence violate disadvantage input number unclear perform weak nonetheless concentrate number exact approximate cubic enhanced present moreover cubic multiplication distinct mass give concern inference worst check assume sample loss seek joint refer joint remain programming denote fm exact algorithm count hamming subset run virtual core gb ram probability logistic experiment set test one figure right range good hamming mode mode optimal hamming however much since estimate frequency combination fm perform fm slightly need estimate th original multinomial bad case end fully training may additional statistically consistent notice phase get estimate need refer classifier label solve test apply plug four label set relate svms move effort virtual processor gb ram dataset test nearest different inference exactly datum mode fm assume near instance hamming inference marginal fm maximizer compute exact maximizer f bold loss mm fm fm fm fm previous tailor obtain loss small throughout loss tailor maximization fm fm latter near general sample case volume contain neighbor far away example explain table time include search inference present fm search fm fm fm fm next implementation tune base minimizing thereby produce fold parameter mechanism fm sampling estimating apply exact method table hamming concern tune fm sample clearly tailor obtain additionally fm sampling randomness method time figure fm substantial sampling expensive estimation much fast table loss subset index maximization proxy fm hamming inference fm fm fm fm cc picture present publish hamming use result summarize loss training time surprisingly well outperform measure regard hamming nevertheless inference comparable cubic dataset experiment contain small moderate number substantially train multinomial multinomial regression less include measure svms maximize phase method kind refer result previously publish basically train appropriately rescale maximized cutting algorithm optimization surprisingly get usually method dependency regularization fold cut plane report observation structure competitive get worst substantial base dataset weakly probably high interpret language time fully comparable table basically however cut generation costly perform independently base probabilistic neither costly training unfortunately measure among advantage compare result see parametric follow method costly train former enhance precede training kind time need efficient quite base efficient small demand measure time main advantage different measure ib fm ib fm ib fm ib fm pt ib fm ib fm pt fm ib use maximize mining competition medical article essence problem decide competition relevance finding regard maximization competition satisfactory competition paragraph briefly fm way range regularization neither competition column competition minor test remain constitute score competition generate table method competition rely prediction competition parameterization prediction voting test describe detail voting improve slightly third competition show inference interestingly well independence fm suggest problem marginal probability fm average already clear fm provide fm well fm fm vote experimental square auc investigate devote far paper predictive already complete picture present without analyze polynomial joint solution time perspective prefer marginal focus instance wise synthetic competitive optimize alternatively variant algorithm assumption concern multi adaptation svms maximize gets give critical mention easily tailor maximize easily use kind macro maximize macro measure optimally state author independence integrate burden high emphasize instance measure suboptimal micro f coincide specific constant test significant measure expect surprisingly report micro instance wise training set binary seems case probabilistic optimize thank proof work foundation co european support foundation prediction minimize ham take maximizer hamming minimizer practically contribution vanish constraint eq supremum distribution integer nonlinear program h integer h b adopt shorthand coefficient four remain convert minimization mix new necessarily cause keep key element allow program zero position position b differ tucker kkt program optimize kkt necessary sufficient define lagrange multiplier imply system solution always additionally dual feasibility restriction apart negativity lagrange b writing explicitly yield trivially result equation dual feasibility plug b soon c b c imply zero consequently condition solution dependent vanish c constant soon remark solution impose additional constraint exclude case worst bound case u supremum take optimization loss unique vanish choose arbitrarily zero recall construction coincide nonlinear contain variable follow arbitrarily minimizer solution act supremum reach technique oracle reformulate program form p cl verify kkt recall maximizer minimizer risk minimizer vanish primal define lagrange multiplier optimality satisfied complementary plug three q subsequently obey negativity negativity turn restrictive analyze function equivalence negativity satisfied observe regret bad showing obtain
observe force exponent hausdorff hausdorff hausdorff hausdorff locally bi broadly hausdorff measure example lebesgue hausdorff dimension make hausdorff dimension useful tool science certain development around hausdorff system although hausdorff important structure hausdorff dimension see hausdorff hausdorff dimension reason hausdorff close problem place cover employ set hausdorff difficult reason hausdorff dimension hausdorff dimension capacity box dimension along relate asymptotic number require cover length cube recover motivate definition length define box dimension define agree common value denote definition hausdorff infimum cover diameter correspond count hausdorff box frequently strict box dimension poor notion count illustration dense covering despite box estimation dimension formalize produce brief historical pointwise kind dynamical iterate cardinality intersection ball precise norm characteristic suggest dimension technique rigorous know dimension provide previously cite paper correlation sum prevent define limit difficulty purpose rigorous formulation clear try dimension fix limit equal borel q suggest entire correlation notation numerical reason convention derivation mainly dimension carry bit similarity point borel q reflect decay measure point reflect ball point support past read concept expand upon borel associated let highlight major correlation apart measure low correlation pose serious rely analysis address although hausdorff dimension set derive follow borel hausdorff borel borel correlation hausdorff lot almost support explore corresponding hausdorff dimension probability rough let suppose exist borel everywhere constant pointwise uniformity prove satisfying point limit pointwise dimension almost datum condition detail let locally bi lipschitz ergodic ergodic pointwise everywhere pointwise equivalence tell invariant mean basis ergodicity idea estimate gp estimator seem frequently apply present estimator finally drawback motivate pointwise dimension gp measure derive mapping interval linearly use dimensional denote begin calculate measure pointwise support calculation dimension interesting numerical suppose sample wish correlation correlation since infer limit sum scale contain correlation burden choose problematic potentially source bias experience difficulty htb sample measure horizontal reflect true sum total relatively precise seem informative scaling claim effect observable distinct interest dimension difficulty involve broadly difficulty arise important estimator mind correlation dimension desirable sensitive estimate pointwise dimension treat advantage beyond elimination dimension difficult reason depend behavior produce address pointwise fundamentally utilize neighbor dimension expand upon next carry difficulty estimate problem get pointwise expensive difficulty scheme pointwise dimension example estimator utilize cluster cost idea measure great detail difficult datum transform locally although develop purely quantity pointwise naturally transformation scaling certainly something keep mind paper dimension pointwise sensitive utilize pointwise dimension effective cope bi identify pointwise characteristic develop utilize point near process approximation effective class borel probability measure uniformity suppose every difficulty description far satisfying uniformity concerned pointwise borel uniformity estimate distance nearest special derive gamma see definition essentially uniformity uniform ball look uniform arise region true neighbor scale fine estimator mark show free give conventional function give vary try distinguish distribution denote sample distribution seek call neighbor pointwise measure uniformity condition certainly measure measure measure density enable build desire locally bi previous eq borel define eq nearest borel satisfy local uniformity condition pointwise dimension density disjoint support cluster represent example aggregate behavior individual behavior often necessary pointwise feature adaptively balance concern clarity big produce variational uniformity condition neighbor appropriate implicitly worth note marked impact assignment consequently weight parameter variable seek follow gamma distribution dirichlet distribution dependency htb data borel measure point use approximation pointwise hyper gamma density parameter dirichlet I number procedure list basic phase number treat separately considerable iteratively section connection two objective seek maximize hyper quality approximation well write find maximize divergence explicit objective lt I I z z z I z k z initial I I z I relate serious issue difficulty informative scale sensitivity transformation thing choose informative scale datum analyze propose problematic gp formally generate follow distribution sensitivity locally issue convex measure mixture pointwise dimension measure cause answer explicitly choose integer choose call mixture basically product coordinate set probability analysis generate five digits precision fold product c consists sample fold fold sample sample fold weight test locally generalize rather define procedure generate probability slope satisfy consist choose draw dirichlet choose summarize analysis set euclidean near make choose initialization really serious limitation present generate give true also agree decomposition demonstrate marked pair indicate estimate give cluster course pointwise separately present present exhibit trend extensive error fold product component error trend surprising fourth mistake point fold fold product various far near neighbor reach many path example set idea reliable regardless along line important consideration hope tool solution frequency avg dim pointwise dim summary dimension e set sensitivity estimator locally bi datum describe end summarize generate pointwise horizontal represent vary axis pointwise vertical bar reflect pointwise increase estimator far true expand invariance subsequent use naturally dimension majority lie apply development potential algorithm improvement suggestion improvement throughout text question section focus challenge estimator makes really know preliminary numerical present considerable apparent far concerned dimension truly blind pointwise dimension truly reflect dimension correlation observe finite dimension build amongst noise whether one clear observation class however seem serious extent regularity mixture measure exact pointwise general pointwise still art additionally measure dimension minimal provide description dimension utilize description hold measure uniformity whose produce valid measure plan free pointwise estimator probably lead generally seem seem term measure pointwise seem vary everywhere estimator formalism scale hausdorff raise fundamental really challenge devise pointwise approximate pointwise dimension natural extension hausdorff dynamical pose begin ask upon hypothesis already uniformity dimension dynamical measure one previous generate ergodic assume ergodic measure system systematic hypothesis simplify indicate possibility test ergodicity system locally reliable pointwise generic pointwise crucial shift ergodicity establish reliability pointwise ergodicity analyze map ergodicity h popular reader map h dynamic h establish ergodicity certain dimension estimate count estimate dimension clearly locally bi jacobian determinant ergodicity conclusion reliability normal observable long discard iterate subsequent set manner neighbor one pointwise pointwise dimension estimate individual seem apply highly unlikely htb datum pointwise worth two cluster set indicate contain proportion belong cluster also seem fairly uniform across set reason unclear datum two variation randomness serious iterate value would purpose ergodicity one conduct tool relate distinguish essential issue however reflect map remain embedding indicate stability commonly evidence behavior mark behavior embed dimension initial value reasoning take delay contribution arise embed rule matter call stability embed near produce instability generalize ergodic lebesgue exist every neighborhood current initial discuss pointwise valuable system application important dimension measure consideration motivate discussion broadly first distribute measure datum set equivalent convolution set wiener fix integer near neighbor nearest neighbor datum fix figure detect shape indicate estimator interval level neighbor substantially unchanged high near neighbor dominate large measure relatively large estimator detect cluster brownian analyze effect increase time delay embed dimension property motion estimate htb analysis limited circumstance numerically theoretical phenomenon normally embed remark growth dimension observe motion distinguish brownian devise systematic estimator pointwise also coordinate paper seem final datum develop survey exist idea estimation major dimensionality second limit pointwise object pointwise dimension
multiple predictor ensemble ensemble classifier confirm expect algorithm prediction ensemble real broad application domain provide prediction class imbalance b difficult classifier random sensitivity specificity random distribute training representative test labeling detail supplementary describe eigenvector probability entry balanced voting consider prediction meta learner majority voting apply initial guess relatively improvement balanced majority contrast mle modal local relatively balanced majority voting versus accuracy method decrease simulation initialize voting majority size predictor accuracy voting simulation show improvement start vote right panel belong balanced balanced accuracy run dataset biological financial application comprise available software split label unlabeled method mirror subset label fig meta classifier apparent balanced analysis assumption initialize initialize majority voting performance panel dataset fig verify almost slightly classifier interestingly dataset classifier similar capture two seem offer voting avoid poor panel dimensional instance cluster isolate remarkably case median accuracy consistent well start majority vote novel spectral statistical unsupervised combine reveal independence rank eigenvector balance accuracy computationally classical pose propose work principled raise inherent limitation finite infinitely perfect study effect classifier eigenvector crowdsource estimate compute joint observation classifier may work categorical quality exist algorithm third prediction consideration study hard difficulty difficult ranking modify difficulty insight effect ignore contribution due trading material mining visualization information acknowledgment thank feedback cancer dr american grants foundation foundation k support breast cancer project new york department ct usa science department mathematics center health bioinformatics york medical york ny usa equally mail edu unknown query set assess raise reliably meta classifier accurate assume classifier balance accuracy learner whose entry typically achieve classifier ensemble majority estimating likelihood robust group ensemble away unknown suggestion combine provide recommendation stock central survey genomic bind peak tumor disease diagnostic challenge panel discuss grant recommendation reject several human prediction answer query key challenge combine prediction reliability active decision science business scenario whereby potentially action several interesting maker reliable decision maker answer provide answer first performance pre absence classifier potentially possibly source yet list label either scenario different standard supervise setting machine datum rank accuracy assumption performance rank absence combine pre prediction science crowdsource address external historical assess available could panel forecast combination available applicable information whereby assign crowdsource converge furthermore guarantee question yield major insight standard classifier error test set entry correspond eigenvector balanced classifiers rank approach classifier likelihood yield ensemble learner represent unsupervised learner term learner unknown ground voting robust presence simulated motivate asymptotic unlabeled tend infinity classifier define accuracy entry equal imbalance insight v sign ambiguity hence accord sort neither estimate sample entry assumption moreover unlabele accurately eigenvector construct lead diagonal r linear observation upon practice look error approach follow perfectly symmetric stable small perturbation particular finally bound gap ensemble label instance estimator label error likelihood show weight label weight classifier classifier approach look classifier jointly sensitivity specificity typically increase iteration however limitation non em initial guess note study vote suboptimal desirable sometimes initialize eigenvector novel guess accurate majority voting coefficient second order expansion inside around balanced ambiguity easily replace novel learner accurate majority
trading hope engine asset fashion forecast keep merely let define ij com devise trading model gold couple indicator decode next probable markov produce couple trading hmm several decade hmms progress state transition depend history irrelevant look guess probable switch enable prediction hmms formulation hmms correlation couple markov shall show later derivation multiple interact sequence situation world phenomenon develop market asset market indicator asset hmm already hmm exchange analyze financial market extent financial market issue trading mean hold little action much future markov hmms solve state transition state progress analyze market feature notion could hope trading market enable maintain formulate movement devise accurately market hmm formulation I reproduce fully hmms refer hmm depend accommodate hmms couple together henceforth couple hmms transition current couple grey represent h us contribution constraint state affect prior formulation coupling hmms model lb output gap assign gap example probability interval give might probable path optimal maximize maximize extend implementation path need two optimal sequence quantity high hmm retrieve path need maximized array backtrack hmm devise em hmms make intractable optimization hmms allow formulae us calculation partial derivative unclear hmm formulae hmm present hmms new swap present power well two strongly uncorrelated fact trading choose time economic favor gold maintain intrinsic us gold national hold gold gold fact gold fall gold model word asset henceforth asset asset careful observation initially decide observation hard observation price upper limitation discrete price either past second degree change asset affect hmm must degree asset solve normalize value decide sequence index gold know percentage explain sequence enter trade bar record observation iterate increment falls threshold probable newly probable probable rule enter trade go level period entrie use bar strategy trading deduce probable base state hmm likely hmm hmm probability hmm state sum weight together target hmm high gold calculate way simply state predict evolve hmm contribution eq trade viterbi probable probability reach specifically quantity measure tell current viterbi quantity thing trade size multiply amount hmm switch produce period indicator range eight evenly space period stop period trade entry period entry move allow long position deal trade interval restrict shall interval frame use gold indicator frame frame select obey property predict capture implication lose would otherwise determine interval deal random pick trading argue strategy frame period outcome step choose back period look far super lastly range update min high take x state computational work system outperform strategy future indicator use loss fairly return h standard viterbi viterbi viterbi dynamic name viterbi viterbi viterbi viterbi allocation drop return drop fig expect probability next probable largely fig
closely member representative cluster well separate measure load centre cluster diagram q diagram sum load profile load profile cluster centre load profile calculate group together large denote hence useful little meaning used window operating system service cpu ghz gb explore nine algorithm see figure black load profile allocate calculate centroid allocate red cluster cluster cluster cluster allocate detailed table som stage self grid load profile cluster result allocate modified match cluster fairly exception cluster self create cluster application som produce map generate som load profile order nine final allocation profile intermediate som examine final cluster final allocate order match closely allocate detailed cluster lower denote profile profile generate allocate clustering profile allocate effectiveness future profile difficult som show allocate cluster technique produce cluster visually similar widely significantly technique number allocate see success generating cluster successfully cluster representative shape load significantly profile see allow usage lead representative demand side planning nine reflect input nine investigation cluster uk build effective distinguished cluster uk conclusion simple application som result measure two analysis concentrate slice may find cluster differently year member identifiable investigate quality cluster sensitive quality investigate exercise shape usage explore uk centre centre access college reference mm university school science bb uk load successfully apply preferred uk show nine identify usage profile visually break usage profile published collect around build direct demand management customer uk currently uk history design national reduce usage rapidly roll uk mix reduce reduce demand demand weather impact market offer customer benefit efficiency analysis customer usage price availability customer peak period chain typical electrical describe work form demand project usage profile kind customer difference individual profile usage improve electrical time usage peak time identify reduction daily variability usage period determine similarity day pattern e g usage peak day peak period determine applicability define cluster uk dataset define profile day week refer property profile customer customer plot half offer much hour pm expense increase day user take approach order pre load cluster cluster nine general investigation quality cluster try calculation index assess comparative cluster use area uk develop store original lose recently approach uk closely framework uk individual uk hour day recover collection decide contain reading approach replace normalise reading day focus total usage medium usage g normalised load profile read analysis usage split remain variability daily datum arbitrarily future concentrate investigation individual allocate change success day per future
unclear hold uniformly rate discrete show filter estimate open obtain agree theorem complexity linear scale also implication regime slightly state analogue result figure generate solve observation value record sign probability recovery average realization fix figure success instance matrix success plot sample empirical generate describe curve value analog theorem success value extension begin present analogous theorem basis prove let time learn differential time linear matrix non stochastic differential equation trajectory focus dynamic elegant concern collect dynamical specify consecutive ask condition regularize sign support meaning time recover sense precise let refer indeed control limit mathematical fundamental dependent strongly dependent model evolve sub sampling model latter appear technical uniformity reconstruction guarantee already mention whenever I accord portion row eq q apart additive limit cost coincide continuous theorem easily sparse equal condition modify lyapunov clear refer matrix discrete establish square high probability r satisfie stationary exist square sign take word dimension degree time last important enable derive confirm intuition continuous fine information reconstruct reconstruct sign support constant replace base reconstruct required property support use dense section section remainder great certain result expect draw ensemble clearly estimator perform without random subscript theorem assume alphabet finite example symbol suffice sign support special notation make matrix represent parametrize take variable unless specify law vector value identity conditional prove system state white gaussian feedback et al give initial general might write ix ix ix tx bind theorem complex easily computable process assume realization trajectory tx stationarity simplify namely understand regime sde dense matrix shall exhibit fundamentally regime let set bind sign model one change lack exponentially defer system take close estimator dependent similar one trajectory upper bind ij guarantee behaviour stochastic sde form p pf b sde solution ix recover follow apply learn follow stationary restrictive uniqueness sde energy dynamic hold world clear intuition assumption practice analogous concrete scenario upper great match illustrate dense linear dense h success generate sampling gaussian distribution lead complexity curve point slope influence think point structure try amplitude surface unit rest length system external model position mass let straightforward far write accord instance interested trajectory three evolve theorem figure reconstruct sec interval h achieve despite non converge enough mass size observation require reconstruction regular simulating euler top success versus length window size bottom reconstruction network exact sample uniformly reconstruction equal full neighborhood use require sample uniformly behavior agree even mass linear consecutive decay exponentially despite difficulty general stem know sde sufficiently nice success affect linear consist norm constrain look pathway environment pathway behavior synthetic support normalize consideration pathway cause cell come modify modification enable genetic move event chemical thought chemical difference outside forward backward reaction e stage pathway model use obtain euler sde form basis function consist translate summarize top value length specie interact interact positive interact interact positive increase false low curve interpretation recover pathway available true rate increase increase curve recovery rmse top run second good bottom evolution duration h dms grant fa fellowship part help document text proof case prove outline proposition combine prove theorem detail state regularize recover correct validity text sufficient sign support recover analogous configuration consecutive change although relate addition never observe omit regularize recover far provide consecutive checking condition regard concentration indeed expectation take prove xt generate length path walk move neighboring theorem clearly finally desire eq q lemma follow relation condition guarantee sign reconstruction kkt satisfies follow guarantee estimate contain guarantee element determine correct sign c easy turn sign concentration proposition lemma zero denote eigenvalue immediate let zero everywhere one represent one position kronecker mp p first prove ab copy block block block block mind version matrix compute calculation notice sum type matrix term trajectory instant sample function converge stationary depend vector I lemma bernstein denote given continue similar expectation start instant consecutive plus continuous system initial statement define let lemma apply bernstein x reasoning lead let take expectation recall vector write put expression ab prove compute statement two hold order condition assume combine impose fail bound use satisfied substitute expression must satisfy require impose corollary probability expression look actually hold restriction conclude need conclude proof theorem prove assume stationary immediate theorem sde bound proof useful sde base bind minimal upper expression satisfy lyapunov equation show support randomly simultaneously bind matrix prove random uniformly independently satisfies property calculation complexity spectrum probability notice unless purpose eigenvalue write eq apply jensen last close finally alone addition choose right side divide numerator denominator ignore recall adjacency regular sign support kp account entry know enough denominator finish numerator denominator limit bound enough notice hx covariance denote ix rescale last inequality finish give closely lower random make construct follow lemma define symmetric describe since g start appropriately replace low equally h already find bind subset support zero note evaluate var ix ix ix x expectation x ix p upper bound since corollary proposition drift coefficient drift parametrize high tend lower bind characterization mutual differential parametrize interact analyze regularize bind differential dynamical continuous process differential diffusion brownian motion dimension polynomially trajectory precisely scenario simultaneously goal condition recover support small allow prescribe interested achieve optimal problem give parametrize algorithm define respect act element wise apply mapping eq alone define role several science finance consequently parameter great brief understand recovery special class coefficient linearly provide stochastic chemical reaction ab x ax bx comprise tell specie effect concrete chemical model near trace fluctuation species proximity equilibrium chemical linearize interaction vector describe describe interact corrupted word see probabilistic parametrize low theoretic irrespective computational consideration put low complexity derive state key problem would select subsample interval spaced time challenge pose obtain careful limited information might conclusion take important sde contain covariance confirm way least graph adjacency outside diagonal vertex know describe dynamic connect complexity sign trajectory sde particular regularize logarithmic
datum learn perform function probability indicator joint learn optimize hill search see propose heuristic basic depict preliminary consuming find indicator classifier radial kernel svms classifier distinguish location relevance bayesian calculated avoid overfitte parameter add count summarize classifier svms obtain estimate variable protein use predict whose would preliminary location indicator output thresholde protein numerator factorize probability structure equation indicator location protein protein machine library learn localize protein dataset location use localize originally publish localize protein publish publicly extensive localization result publish experiment localize protein composition annotation detail localize extra localize protein location representative protein system system localize protein describe publish fold split total complete fold cross use use validate stability significance split run use originally hour run processor notably improve run fold cross validation adapt classification previously multi protein let protein obtain accuracy multi label precision evaluate well protein localize localize capture correctly location total multiple protein protein location location use predictor correctness denote protein denote protein protein score al deviation ptc width ptc ptc ptc ptc ptc svms location inter dependency location dependency deviation ptc width ptc ptc ptc svms svms use dependency deviation ptc width ptc width ptc ptc ptc pt mi svms svms table show accuracy comparison predictor et protein slightly low statistically significant thus top system capture location manner introduce new new score obtain location dependency correspond inter dependency localize protein dependency significantly high svms alone utilize inter dependency difference sample obtain protein nu mi protein table svms inter dependency decrease statistically svms incorporate inter dependency recall protein protein svms location dependency predict protein associate protein location protein classifier inter predict protein comparable location address dependency create multi utilize dependency system use inter snps simplifying heuristic contrast protein involve number small ideally suitable benefit dependency use location inter dependency learn improvement svm would location dataset use contain available collection localize protein location due protein similarly protein construct future plan develop location inter available acknowledgment readily test style latent fill sep rectangle black know cell understand biological drug predict single assume protein protein multiple multiple protein treat capture dependency treat location individual location present new method incorporate inter among collection classifier dataset multi localize inter classifier inter multi localize protein system restrict location training understand biological role drug target experimental green practice consume effective effort develop throughput wide location last decade focus drive available database protein assign single simplifying protein multiple identify instance store protein happen inter dependency help association predict location protein several predict location protein use I combine predict location protein protein contrast use associate protein notably method utilize location independently location account make dependency predict location protein inter dependency extensive protein prediction na I protein localize na I assign location transform exponential training I practical general localize training evaluate extensive multi paper present inter incorporate location bn relate location assign protein location regard location primarily correspond bn notably consider create combination support location prediction multi inter leverage inter develop protein responsible indicate localization inter estimate indicator location along calculate prediction estimate protein give far procedure provide bayesian biological paper use protein bayesian notation consist direct acyclic represent use recursive partitioning use development cm column sep gray scale gray gray fill gray gray scale fill scale gray gray latent latent scale black edge indicate dependency variable location capture dependency among node reflect feature simplify help inter dependency framework graph contain
exponentially law problem determine np hard relax transition dynamic weakly state weakly put weak behavioral f cost xx transition eq q uniqueness virtue moreover denote steady convenient regret analyze separately upper right analyze regret reverse unique invariant distribution satisfie reverse inequality think I naturally markov decision process provide along poisson distinguish chain follow throughout rest pf moreover mix consequence poisson comparison principle reverse appendix lemma tp tf principle tp tf tf tf f causality time bound term key set set find regret past eq new relaxation dynamic separate interaction common come actual dynamical relaxation bind position without dynamic associate separate within observe environment current past action know payoff relevant environment choose reason agent action view game game end relaxation online dynamic relaxation online x associate behavioral strategy tuple w causal relaxation state value action environment construct behavioral overall mdp time behavioral behavioral depend agent state computation function behavioral construct relaxation state main suppose hold family admissible relaxation behavioral derive mdps pass set satisfying associate separate online next relaxation turn theorem regret emphasize relaxation flexible reduce original particular relaxation construct construct relaxation expert agent combine recommendation individual strategy action reveal expert weight indicate expert previous prediction rwm expert rwm entropy term expert expert select step provide degree stability common rwm algorithm consider mdp arbitrarily main expert mdp mdp law approach computationally infeasible expert feedback depend aggregate action choice make sublinear algorithm expert environment feed end particular relaxation kind every admissible relaxation sequential I value sequence mapping rademacher binary depth future past already future binary obtain bounding follow bind tuned optimize result regret relaxation exactly propose relaxation admissible lead recursive weight recursively reverse poisson reinforcement randomize action start immediate action reverse equality dirac first inequality repeat establish x transition behavioral give consistent derive analyze relaxation hoc particular wherein divided phase apply strategy relaxation index contiguous phase phase phase initialize phase choose feedback use end alternative definition relaxation condition denote tuple tuple admissible also phase tree precede replace infimum bad future involve replace future binary branch per phase construct relaxation specify fix state feedback see behavioral enjoy majority rwm mdps decision maker knowledge transition cost adversary compute perturb sublinear regret computational algorithm policy computation program carefully tune technical theory linear contrast simple second regret regret choice guarantee sublinear horizon advance well optimal appendix unified mdps extension certain show phase relaxation similar spirit one agent loop behavioral tuple mapping tf f prove g proceed use fact behavioral strategy associate admissible relaxation expectation third invariant step invariant eq equality get note c arrive relaxation upper optimize jensen inequality second negativity exponential inside due hoeffding lemma w worst prove recursive subscript equality hoeffding lemma plug thus algorithm subscript easy choice entropy hoeffding substitute involve left q plug work behavioral strategy bind armed unique invariant b invariance repeatedly triangle easily fact let bind recursively eq complete arise subscript keep assumption relaxation relaxation markov phase admissible law mp write prove last lemma right get due behavioral admissible induction arrive form l attain equality since bound contraction eq sublinear sublinear course long phase straightforward algebraic calculation however advance ignore round fix minimum meet happen every assume large side ignore issue horizon well algorithm environment notion current action model common control framework decision process mdps environment year grow interest combine two framework consider mdp setting allow area develop arbitrarily environment development arbitrarily change cost construct advantage method minimax process mdps sequential decision environment mdp observe choose action system transition action state action advance reduce however typically paper space allow time mdp arbitrarily agent action minimize policy interest make uncertain costly also collective possibly agent minimization ensure online policy online fact dynamic distributional learn action control aspect influence state future solve past decade yu area new online mdp two method theoretical interpretation recover us toolbox deriving algorithm principled mdp deal arbitrarily feedback mdps function extension general approach online course decade treatment repeat adversary analyze minimax game derive sublinear constructive design separate relaxation recursive minimax know give general develop one short convert game extension mdps learn free nature involve mdps counterpart mdp problem minimax mdp relaxation recover exist new specifically inequality mdps evolve control chain possible relaxation spirit bound organize section brief formulation online major challenge contain recover derive research intermediate appendix start arbitrary markovian environment opponent repeat game law current state game agent state alone opponent choose side utility opponent assume move opponent environment common objective cost agent actually incur agent advance definition start suffer agent optimally apply drawing incur step adopt game theoretic terminology define close behavioral tuple behavioral tuple initial pair specify action p steady outer induce randomization agent environment inner r action interpret gap cost strategy achieve stationary knowledge arise induced chain introduce consider steady constant negligible long run restrict steady start shorthand operational value agent behavioral bad behavioral strategy game encode extensive minimax give immediately attain give empty tuple state recursive decomposition arise frequently decision view minimax control player control player promise derive value operator behavioral minimize supremum affine infimum worst intuitive tendency present risk cost infimum supremum involve strategy computationally minimize near develop compute mdp domain spirit admissible associate behavioral admissible eq behavioral strategy loop suffice restrict attention
vc number th conditional clarity row vector discriminant ij dimensional cholesky provide empirical limitation parallel cholesky use common indicate exclude document neighbor lda word count conditional augment gamma equality gamma efficient involve inaccurate infinite draw iteratively alg randomly markov final testing hinge margin loss hard hinge develop collapse augmentation formulation u augmentation derive include unnormalized augmentation rate mix integrate markov whose marginal collapse posterior conditional collapse notation u u q distribution ij conditional lda count link augment inverse distribution assignment inverse distribution transformation hinge augment link assignment collapse conditional assign topic exclude sampler likelihood drawing meet g relative unseen testing infer take replace collapse document pz nc exclude c c gibbs var c var var although hinge loss classifier strict field constraint sub em type efficiency solve svms effective loss gibbs new problem resort collapse sampling restrict idea augmentation max expect margin assignment u discriminative solve p u close hinge develop collapse augmentation loss posterior ij unnormalized pseudo problem write get normalization pseudo express indicate marginal collapse successfully use order improve space improve integrate chain equilibrium collapse bx whole corpus distribution collapse prior ij element u conditional generalized q iteratively network discriminative relational sensitivity various experiment three science total citation link link dictionary consist link unique dataset focus effect extension discriminative various special var logistic em link gibbs link likelihood fast approximation document gibbs sample hinge gibbs var setup deal effect fix draw unobserved example subsample normally regularization example tune testing calculate processor ram rank auc curve prediction hold document document auc document phase remove fold deviation model collapse unobserved effectively deal imbalance auc word perform topic gibbs example diagonal gibbs topic superior gibbs benefit regularization collapse without restrict ccc sample sample influence gibbs sample present effectiveness deal imbalance gibbs restrict gibbs make note pay long gibbs latent logistic feature link fortunately simple training efficiency link take almost deviation discriminative hinge log discriminative hinge e g auc score gibbs verify sampling superiority gibbs draw constant sampler inverse gibbs cost comparable omit spend loss fortunately develop greatly especially insight behavior discriminative various gibbs dataset gibbs gibbs see decrease sampling well observe auc much performance allow pairwise expressive diagonal large word slowly growth fitting fitness compare var test suggest advantage collapse gibbs sensitivity gibbs burn respectively rank auc converge optimum grow respect burn iteration observation burn sufficiently fig show performance datum total link subsample weak influence since lead train gibbs diagonal gibbs quite different topic gibbs competitive environment evolve solution genetic issue evolutionary strongly improve var solve algorithm approach job scheduling scheduling evolutionary module acquisition multi operator evolution strategy real time separation close transfer sequential neural environment asynchronous modify update structure construction planning act observable qualitative implication belief could suggest table suggest query environment evolve solution complex likelihood term grind truly link document top gibbs find var link task whole corpus also truly link query discriminative interaction introduce control imbalance incorporate hinge present algorithm restrict experiment network future architecture inference develop selection problem I automatically resolve finally focus static interesting extend dynamic challenging address acknowledgment support china grant cb cb innovation china grant grant chen receive bs china science university china learn department interest mine computer bs department computer science university currently associate project department university interest primarily develop scientific engineering member receive bs school software china currently ms institute school usa interest especially mine social zhang department automatic china currently computer university chinese china interest artificial intelligence network publish field present fold deviation discriminative loss two hinge gibbs predictive result gibbs especially greatly topic table gibbs ranked yield qualitatively category reinforcement theory category visualize discover discover topic topic represent document good reinforcement l genetic stage scheduling schedule code genetic redundancy evolve preliminary report rule inductive logic difficulty program cut logic recursive generalization great logical planning reinforcement learn difference robot develop agent plan act domain decision homogeneous induction active improve learn factorial recurrent net improvement inspire feature map resource thresholding mixture unknown chain spline metropolis net basic idea extension reasoning issue approach diagnostic system diagnostic technical domain reasoning zhang department technology university china mail mail edu cn com engineering network relational discover topic representation however exist limitation deal accuracy paper present allow interaction capture interaction applicable asymmetric bayesian common real latent variational strict present collapse sampling relational topic explore augmentation making restrict popular log network efficiency dramatically improve simple augmentation regularize collection vertex relationship entity name network network citation etc increase attract comprehensive survey task study attempt partially entity link could useful suggest friend user method propose work design unobserve use design intensive expand scope ease applicability machine fast interest spend learn along link parametric bayesian model network little text paper citation network web page one work account text allocation lda predict though powerful assumption could diagonal topic asymmetric perform deal imbalance network entity pair topic weak predict variant normally realistic present topic consist extension improve relax generalize relational inference imbalance present gibb explore classical generic sense discriminative representation focus margin inference monte mcmc introduce auxiliary loss gamma exact link max hinge inverse representation unnormalize pseudo collapse gibbs algorithm importantly make desire several extension relate work section hinge present loss present pac numerical extension mixture expert em selection base model plan act genetic evolutionary parallelism control processor neural feedforward monte gibbs evolve programming programming evolve report formal analysis genetic driven scheduling application super task processor topology circuit difficulty logic cut constructive inductive logic least generalization great clause trend logic logic program logical discovery engine reinforcement planning reinforcement theoretical algorithm use develop agent plan act observable domain exploration mobile robot neural quantization reinforcement self organization em theory learn genetic optimization learn maximization blind blind deconvolution backpropagation organize formation predict exchange back condition retrieval net reasoning approach diagnostic system base net diagnostic reasoning meta document structural theory adaptation proportion posterior p common issue estimation much
ad finally tree aggregate total child estimate hierarchy estimation dimension construct smoothing discuss adjust estimate user triplet triplet leave root ar issue bid method current bid idea attribute recommend infer greedy online bid ad request inside recommend group bid place request default bid price unseen site get activity decrease regular major bid budget ad request come daily might exceed allocate daily budget several monitoring process implement daily well slot exceed daily stop slot illustrate algorithmic flow chart ad request bid need million request process bid distribute computing cross center offline training utilizes generate ar well stream income ad request many evaluate bid price via detailed c c fc fc fc fc fc fc bid lead budget addition serve order improvement metric simulation environment verify propose bid eq request slot server generate rate win uniform line error daily fig ideal relative curve ideal curve daily budget c dc dc dc dc c dc dc dc dc dc dc dc dc dc dc proposal evaluate entire bid flat flat metric take click account metric improvement flat across seven baseline method week feedback threshold slot show lift would report improvement dynamic try approach baseline apply rate without adjustment bid type randomly select different lift individual select well lift present general bid due implementation handle million request real show improvement without integrate capability online user acknowledgment would song entire bid environment assumption daily public bid request restrict budget ad reach desire smoothly reach wide since occur rarely occurrence feedback delay goal present approach optimize try adjust bid price performance manner demonstrate recent amount public manner ad person context side bid bid request bid budget goal minimize click per ar constraint fraction avoid ad time reason firstly place bid evaluation bid need perform ad request typically million request hundred user short throughput requirement introduce extreme request g feedback delay specifically click delayed removal hand action seven day convert attribute click search metric paper apply feedback future constraint quality adjust bid prior detail online bid bid discuss formulate bid optimization linear bid bid follow request like represent indicator place bid ad request ad typically would budget time stop budget day fig budget able fluctuation consistency budget suitable yet widely budget fig across day main traffic traffic vary lot throughout half receive relevant traffic day uniform scheme budget able end force low quality budget traffic oppose resolve issue extent depict fig course day period day depict pick potentially cause traffic explain daily budget break slot allocate strategy assign request represent true request request correspond bid optimization budget formulate ad slot obviously offline formulation ad request incoming request receive online bid observe bid price please bid price incoming ad request determine price exchange clearly bid actually pay typically call online programming problem online matching packing resource comprehensive survey summarize couple follow constrain ad request respect budget approach ad request assumption impractical strict framework explore bandit framework environment collection information display propose online solver solution decide check true incoming request need constraint accurately high system online bid explain control ad request discuss select price optimize budget introduce try evenly uniform remain budget length slot I click look history slot function click assume per probabilistic time slot eq length simplify notice slot hence come prevent situation split budget two strategy always chance adaptively adjust bid maximize function subsection dynamic bid price calculation always bid price select request bid consider slot ad request click bid ar detail offline ar smooth require similarly go ad request income ad request click request fig online algorithm find threshold slot request ar fulfil formulate ar frequently introduce current ad request historical perfectly prevent situation first use adaptation second assuming state tt day statistic provide ar slot request system ar predict value request bid price request simply drop value ad request select meet ad request free change bid price dynamically incoming request enough construct historical slot represent statistic bad discuss subsection base bid bid properly meet subsection generality frequency bid public bid action price increase bid price adjust bid price safe region treat compare base bid ar predict request estimation next bid safe bid particular bid actually estimation quality big reason classifier high hence big price unless past bid action
literature depth improve investigate structural convenient represent volume control patient hundred volume disease activation distribution rbm factor conditional feature learning ability operate mode allow investigate go focus network benefit mode operation pre fine treating layer rbm train unsupervised way input tune treat feed forward schema max fine tuning operate brain volume fmri five fmri rate volume study volume dataset generative train learn useful fmri image look learn investigation learn purpose embed display way representative requirement usually differently relatively preserve nonlinear embed property diverse aim embed useful current outline constraint deep useful hard complicated processing preserve hard deep provide amount deep effect hard know packing molecular code constraint control attractive nonlinear embed dc dc treat later replica involved dc divide projection replica near location place idea combine location divide across dc define behavior projection keep near space effect learn leave general satisfy exactly dc guarantee stop find informative dynamic provide practically regardless point complexity neighborhood cm data research institute university al comprise patient site brain structural matter gray white template derive optimize applied gray matter view gray matter patient question answer classification rbm experiment rbm drive slightly choice connectivity model network unit pre via rbm fine tune model top top layer respectively softmax back accuracy fold model split subject balanced cccc raw train rbf logistic neighbor knn fine tune perform likewise perform cross raw summarize precision score trend depth depth significantly support general claim improvement even knn character manifold neighborhood need analyze representative significant potentially useful neighborhood embed display raw activation subject deep control apart display subject increase separation depth useful diverse facilitate conclusion mention property result map cm patient genetic disease neuron area identify brain person begin learn answer weighted site international site strength weight series voxel cccc raw template gray matter template control train fine three raw raw depth capacity ability depth bottleneck confirm observation depth effect yet predict evaluate ability table datum deep scientific although train discriminative fine patient color code medium embed raw future discovery disease application rbm already correlation group depth separation apparent differently exploratory reveal hidden relation find researcher baseline rbm ica fmri separately current rbm modularity apparent visually modularity average subject rbm great rbm also rbm highlight ht nm network nm deep advance representation toolbox success explain flexibility flexibility new area parameter feasible structural functional brain imaging describe dimensional parameter choice representation latent image natural science improve understanding amount measurement image come map towards drive seed base canonical analysis successful patient control diagnosis disease often merely correctness checking emphasis conclusion oracle deep break mining art accuracy decade however automatic contribute seem reveal distinguish feature deep learning acceptable currently dominate clear state multimodal either modality modal relation deep relation relation phenotype indirect deep conceptual level imaging brain volume static volume fmri subject comprise multiple volume experimental feasibility application building restrict rbm examine latent brain deeply visualize gain insight process flexible embed choose reflect learn gain ica pca average ica show well inferior tc likely negativity field ica rbm ground representative dataset thresholded contour visualization result rbm ica slight sm rbm tc ica connectivity comprise task inform sound series standard center institute head gradient standard head run volume tr hz dataset post process software package fmri image remove complete voxel
parallel result binomial mutual information relate goal generalize scalar poisson relevant numerous channel ray classification availability provide optimize scalar counterpart inspire derivative mutual scalar channel average construct divergence case bregman divergence gradient mutual counterpart associate conditional property bregman show classical divergence bregman interest bregman often channel derive key notion generalize bregman derive poisson channel light bregman divergence possible result poisson transformation represent channel output channel arbitrary generalization scalar scalar input output scalar factor dark scalar offer application notably ray document sequel information output channel e dark entry draw gradient counterpart sequel particular vector channel establish gradient gaussian obeys mmse respect dark current regularity sequel differential operator operator input channel respect dark current irrespective distribution multi theorems channel mutual input channel respect align ix respect dark irrespective hold mutual admit mutual channel term dimensional conditional appropriate interpretation precise construct generalize notion bregman originally numerous metric bregman continuously note induce distance kullback leibler mahalanobis widely bregman divergence generalization bregman include extension modular however domain range banach first negative convexity positive extension generalize bregman wide vision banach fr strictly bregman divergence associate function exhibit constant exhibit duality property divergence function choose bregman form easy mirror computationally problem idea bregman exhibit term relate mean bregman relate minimization strictly subset banach variable sub interpret sense fx visit poisson channel gaussian light poisson channel vector bregman divergence offer average channel appropriate choice bregman divergence mutual channel gaussian bregman associate recognize scalar poisson applicable channel applicable scalar corollary theorem respectively classical bregman one derivative scalar induce scalar scalar calculation deep connection bregman gradient bregman divergence possible dual idea essence mirror channel relate application briefly light involve classification vector count word vocabulary compressive rather conventional document define characterize count determination informative availability generalization estimation theoretic quantity reveal connection mutual key system counterpart generalize classical bregman establish link channel aim range bregman domain shown exhibit various classical negativity linearity convexity duality respect scale dark gradient address generalizations compressive projection design ray document edu electrical university college ac uk theoretic quantity channel mutual
independently count cluster conditional lead identity simplify ap al evident adjust adjust poisson size role standard parameter one serve make mixture process marginalization h h except replace mixture construct pmf influence whether regularize size clusters standard marginal auxiliary da fix inconsistent addition interpretation difference seem unlikely gamma auxiliary difference behavior discuss generalize show discount discount close favor posterior precise size addition f distribution behavior size count mixture well precisely behavior cluster opposite p h p connection treat tb cluster sample calculate base visualize difference figure pmf pmf evident behave expect pmf encouraging whereas increase region pmf towards encourage cluster behave discount parameter slowly rate mass influence behavior substitute pp h expect lead opposite behavior discount rapidly towards logarithmic law generalize mixture model show power size pmf hence always increase show pmf pmf tail decay increase different pmf pn ap p ap similar property normalize discount determine low ratio unit g respectively p h tb h clear asymptotically discount cluster encourage small fit decrease encourage cluster towards encourage ratio cluster encourage nb nb advantage crp probability analytic posterior analytic cluster crp usually augmentation cluster l dr use carry easy verify similarly analyze tb respectively cluster compare column similarly generalize different exchangeable sequentially assign inconsistent sampler partition iteration cluster construction inconsistent exchangeable generalize chinese restaurant investigation chain monte carlo scheme generalize parameter random measure version base hence pz z q posterior p posterior likelihood prior place f nb process count mixture place e aa become proportional nb count place gibbs discrete eq similarly point use al normalize gamma inference allow al normalize major count replace letting pa n pa count galaxy galaxy last collect infer fixing let record mcmc ratio unit tb ratio average number function discount infer generalize binomial mixture discount version position sign mean infer figure show tend discount ratio generally increase posterior increase decrease rapidly decrease histogram increase size unit cluster decrease large favor cluster large exhibit distinct behavior share similar trend ratio discount generalize count bottom posterior number cluster visualize density figure show tail large figure large notice posterior seven one usually cluster together process unit predictive density leave region encourage figure increase high density pmf towards encourage cluster figure discount density figure distinct behavior allow infer posterior support consist galaxy lowest consist galaxy cluster sample allow unique partition subject consist galaxy galaxie count mixture evy cluster structure partition function subset exchangeable partition inconsistent cluster binomial size exchangeable define fully factorize likelihood define chinese restaurant whose develop sampling scheme analyse negative control exhibit distinct behavior size cluster distinct construction binomial acknowledgement author grateful g helpful thm thm example department management usa school business exchangeable sample partition propose cluster prior poisson distribute truncate cluster control illustrate result p control cluster exchangeable partition generalize chinese gamma negative foundation probabilistic consistency uniformly replacement element remain practice achieve project return platform partition could drastically merge species four constraint careful success concept define probabilistic random partition element require subset thus regardless order exchangeable construct assign mn l mn inconsistent infinite move beyond various process specie partition propose review increment completely probability gamma marginalization lead formula advance completely employ produce exchangeable consistent usually calculate normalization scale mass parameter become redundant variable whose parameterized mass completely measurable space poisson normalize identifiable requires observe directly come element distinct permit inconsistent model number point parameterize compound distribution concept structure characterize model extend structure specify number independently modify consider binomial generalize nb marginally addition stochastic chinese restaurant crp develop discount discount control behavior model determine generalize mixture remainder organize material introduce cluster construct framework introduce discuss control behavior generalize present product infinitely way cluster count mixture base distribution express mass likelihood distinct sample shall without generality modeling treat variable mixture compound become evident k poisson dr n consider mixed function evy compound process evident compound compound count distribute positive pn dr appearance exchangeable cluster mixture factorize size exchangeable represent number I rule gibbs current remove tie would subset element replacement exchangeable cluster count construct mixture first introduce mechanism exchangeable fully amenable long remove redundancy fourth
second fact distribution fix mdp independent statement theorems average discount simulator run simulator also vc gives obtain relate pair q stand increase acknowledgment support fp project corollary claim feedback learner series series preserve series information shannon implication many situation solve significant starting stream sensor agent interact formal first assume moment interactive look ideal random independent series sense ideal maximize shannon stationary allow situation maximize original ideal quantity estimate show certain condition importantly estimation series maximize next allow action learner action enough one without representation easily work variant amenable vast mention ideal independence get hide unobserved hide state mention state deterministic thus necessarily hmms hmms hide deterministic function penalize finite infinite set representation call memory perspective distribute instead series preserve arrive bottleneck turn generalization dynamical information bottleneck formulate give consider representation problem relate mdp states state view state generalization aggregation presence absence treatment non find metric distance transition reward estimate markovian note conditional independence previously effectively use classification object ideal decompose problem worth quantity context series consist identically equals organization introduce case chain proof defer measurable space assume g euclidean simplicity infinite continuous well space infinite sequence sigma algebra stationary sigma algebra distribution whenever variable stationary equality understand concern time situation conditionally give define conditionally independent define help understand equality stationarity let conditionally conditionally independence define maximize sample indeed consistent estimator entropy example situation infinite possibly like action play gain make assumption stationary order process simplify considerably conditionally also moreover function maximize enough sake analogous independent chain case q notation successively property chain imply imply possibly maximize proceed since select arise requirement serie establish formalize estimator consistent guarantee sample fy fy entropy process mix sigma algebra process absolutely regular ergodicity tool use follow vc accord bound distribution let geometric stationary satisfy vc dimension gx stand inverse entropy defer section satisfie mix exponentially fast exponentially fast define need uniformly general mix statement hold action assumed observation provide unknown reward deal version goal preserve active
store information avoid small window embed make hard sentence token token exceed sentence batch prohibitive million development mini development dataset divide layer embedding fan divide set development follow white driving train thank child trade student disadvantage approach criterion training notice decrease development occur manually sign overfitte explain choose far increase understand examine subtle information information researcher syntactic syntactic section semantic list neighboring color use call name conceptual show thing group conjunction word usually english word form specific syntactic feature compositional semantic preserve learn related english moreover add seen character use characteristic word wikipedia corpora see perform bad category test train amount quality present vocabulary language embedding cover speech dataset dataset come domain wikipedia reflect speech initialization embedding language great training believe illustrate performance c valuable language resource language embedding solution reach near art performance nlp believe help researcher develop language express embedding believe improve embedding embedding conjunction nlp community release resource pair language future area include window domain investigate well strategy see handle performance real acknowledgment grant cs representation embedding competitive language nlp task work language speech english semantic embedding release embedding publicly help researcher nlp preprocessing representation serve feature stage complexity language develop nlp focus rich resource nlp tool rely heavily english test new language serious bottleneck approach requirement language typically carefully language complicated new language addition hard enhance recent unsupervised instead rely large plain lead art syntactic task embedding architecture well adaptation believe research learn word huge system mainly english generating embedding language art include release word embedding language language vocabulary contain word embedding publicly characteristic new language believe valuable resource amount example language study english speech pos conduct qualitative investigation syntactic semantic chance nlp consistent language cover embedding linguistic believe resource researcher comparative library researcher produce embedding setting wikipedia rest supervise representation describe section embedding section progress capture embedding show pos language body regard integrate class semantic improve transfer language parse nlp task corpus supervise learn induce jointly learner annotation l corpora mac derivation representation word slow amount computational several suggestion speedup embedding substitute nlp task offer parse compare system execution speed comparable nlp language acquire vocabulary probability embedding nlp task english generate architecture work differ follow english embedding language next linguistic normalization place linguistic preserve show pos release eliminate despite make combine approach propose address semantic sentiment embedding map index space contribute concept back choose automatically unlabele unsupervised language task start require distinguish phrase corrupted phrase score precisely corpus corrupted vocabulary phrase take compute map vocabulary index share represent retrieve size bias calculate combination activation therefore h nh generate wikipedia write language resource wikipedia article wikipedia resource free continue expand process wikipedia version engine rely probabilistic whenever default text algorithm token reduce normalization
constraint complete edge constraint set difference anchor south shift node anchor east edge anchor edge node anchor north shift mm tree anchor shift edge anchor node anchor south q edge conditioning distribution factorize r specify form factor detect independently calculation de jk advantage two reason significantly curse far construct parametric bivariate copula method parametric section introduce bivariate copula copulas parametric let copula density operation recursive define variance equation approximate inference select edge bivariate copula bivariate copula ideally would copula allow model assign level variable estimate consideration justify solve maximum span tree solve non propose could easily problem classification infer scalar express advantage task refine task illustrate estimate generate effectiveness propose framework matrix comparative result adaptation parameter validation copula learn common ignore dependence moreover copula amount increase significant improvement uci compare parametric gaussian copula extract perform average likelihood random summarize technique obtain exposure curse six uci description uci supplementary technique different baseline source target adaptation technique perform augmentation point twice task learn kernel propose minimize target matching map universal rkhs operate way besides label source task target task contain task summarize normalize square repetition method case unsupervise label outperform case finally bivariate copula execution require available point take minute regular cost copula reduce copula product marginal density estimate outperform alternative adaptation regression real circle height center copula semi problem present bivariate factor detect correct adapt model importantly efficacy approach technique human address often acquire people rely learn exploit similarity test phase collect label operation framework provide mechanism improve performance solve domain adaptation concern share task semi adaptation regression task problem object map available assume generative individual task block across different copula tool multivariate product copula copula successfully wide include finance modeling recently copula name gain statistic density bivariate copula function vary domain contribution fold parametric copula semi performance validate experiment art technique follow copula introduce parametric describe copula describe experiment approach world density equality nevertheless multiply describe possible dependence satisfie copula joint cdf random pattern depend dependency infinitely multivariate share underlie copula copula dependencie together cm pdfs map hyper cube transform estimate copula approximated estimation pdfs estimate parametric copulas frank student copulas datum often exhibit correctly copula lack copula illustrate alternative parametric copulas review dataset copula fit unable copula elaborate factorize sample observation use random joint place square
result inexact classic minimization define find iteration factorization formulation evaluate factorize corollary reconstruct local correspond factor optimal lasso minima solutions local minima rather stationary point nonetheless quickly reliably optimize use gradient easy rescale formula convex spectral singular adjoint operator large singular quickly power every assume instead random possibility try initialization operator eq sometimes see feature summary lr measuring versus initialization rank effect quality final cc c l initialization factorize rank perspective well stay impossible specified level class ahead guarantee however necessary adjusted framework framework compatible root factorization representation change lr per factorization lasso subproblem subproblem direction incur residual factorize formulation modify form line sample reduce every evaluation fraction small factorize approach actually approach store decision memory factorize dominate observe factorize projection frobenius art projection generality dominate still partial essentially factorize far inexact three modify projection code acceleration factorize version singular show factor equal norm subproblem local minimizer factorize subproblem minimizer lasso formulation formulation pareto differentiable residual claim derivative depend crucial allow regression maximum statistical relax allow contamination formulation convex correspond notion figure student freedom general context formulation constraint agent residual use evident figure cost penalty residual penalty square residual entire residual likely error residual result fit residual size worth large paragraph use framework capture interpolation heavily develop function eq discuss penalty non still standard factorization formulation smooth project width height marker axis line line middle width marker sample line line axis thick height marker axis line axis dash red solid solid density plot log tail influence every dimensional span column basis nonzero situation estimate subspace initial vector case apply nonzero estimate row minimization formulate zero projection span live complement characterized objective norm formulate optimize project nontrivial fortunately allow weight f multipli find effective norm nonnegative positively homogeneous include define reduce norm invertible linear nuclear weighting solve minimization netflix dataset formulation order well convex solver lr classic completion collect section test complete netflix anonymous movie unseen solved predict actual remove assess performance noise pre define rank rank last serve baseline riemannian unconstrained formulation regularizer get result without functional contrast rank formulation importance regularization especially problem conclusion first constrain tight source receiver miss trace subsampling mask remove slice fig scheme transform fourier underlie trace exploit formulation interpolation completion strategy minimization slowly decay singular value hold penalization formulation allow trace achieve receiver offset domain r offset mathematical transformation domain tight transformation transform hz hz source receiver fig offset slice fig frequency slice offset fig offset hz slice source receiver domain singular slice decay offset domain slow fig denote subsample fig fig receiver offset domain result offset solve recover interpolation frequency slice hz low slice adjust go frequency slice slice frequency slice hz figure show plot hz shot able trace figure evident acquisition synthetic dimension slice hz interpolation recover spaced content hz receiver coordinate show receiver receiver row source receiver source dimension row receiver singular select receiver coordinate solver interpolation show figure see high case miss nuclear solve thereby enforce estimate optimize rank reconstruction compare regularize formulation approach decay regularize contrast obtain well increase technique classic use classic lr write author formulation function nuclear trace slice extract slice remove interpolation offset h compare time factor classic lr lr lr see randomly remove entry synthetic show snr factor experiment lr formulation classic lr rank give significant threshold reality know rank advance fair available remove entry snr table case significantly fast complete snr factor frequency slice set missing entry l snr db c c db c db c snr db situation observe heavily contaminate observe apply mask remove contamination whose amplitude datum sub behaviour amplitude robust take explanation motivation example implement slice adjust slice compare play significant role relative comparable unable solution reason unable budget residual student penalty achieve good snr db slice interpolation interpolation trace sparsity take overlap slice analogously weight column frequency slice problem purpose basis hz slice figure slice snr know basis ahead proceed recover hz hz subspace adjacent frequency slice hz hz way db respectively alone next figure c residual versus frequency hz hz recovery shoot record e shot record shot reference shoot recovery use snr db combine pareto curve optimize formulation svd free netflix problem factorize formulation fast svd factorize also truly scale penalty show denoise adjacent frequency orthogonal minimum feasible neighborhood continuity map feasible feasible subsequence minimum result application robust system million column extremely scale application datum interpolation matrix typically innovation make practice leave practical challenge propose improve completion frequency aspect robust available lr collaborative netflix along weight reconstruction contamination significant impact many decade sparsity transform exploit denoise image analogously completion include system combinatorial interpolation denoise problem functional explicit fitting term impose eq may norm matrix predict take regularization know require procedure alternate use nuclear require acceptable domain especially practitioner formulation nuclear completion system interest svd volume create fortunately formulation costly computation formulation original factorize avoid spurious formulation factorize efficient partial computation incorporate factorization idea enable formulation extension general penalty e g see e measure completion formulation contamination completion problem use incorporate recovery snr subspace reweighte recover datum design target user solve recovery severe contamination robust subspace weight section briefly discuss formulation solve convex relaxation relationship minima factorize counterpart set approach result netflix formulation optimization together also variety reference model interpolation prior want return reasonable budget formulation requirement optimizing solve inexact bridge particular pareto quantity approximate result much broad indeed activity key active large big feasible inf differentiable differentiable give adjoint solve entry gradient take evaluate evaluate norm evaluate optimization key make useful precision proceed computation solve scale method typically project gradient ball necessary requirement tractable iteration subproblem inexact strategy problem proceed
value variety standardized counterpart dark region high rule figure red green rule exhibit opposite value zero previous indicate relate highly dependent value raw counterpart unit raw suggest independence several prop standardize unlike lift cosine show datum great raw standardized share standardize raw standardized measure essentially measure standardize nearly similar observe one h gold lift lift say investigate set transaction investigation affect measure prop independence reason expect pattern expect counterpart pattern random transaction item transaction default threshold plot standardized counterpart transaction measure counterpart scale shape measure three transformation nearly maximum small contrary positively correlate transaction tend particular order random transaction standardize rule surprisingly much ranking previous index rule monotonically appendix highlight transformation appear generally rule standardize show several measure alone standardize value value another standardize compare raw result depend measure rule maintain original order standardize index relatively standardize scoring datum positively rule lift cosine similarity standardize raw highlight example indicate close standardized quite indicate independence last observe standardize indicate relationship near raw standardized raw alone standardized aspect rule similar raw independent map raw counterpart surprising transaction effect specific two contain property measure possess thorough member family address aspect acknowledgement discovery grant engineering research early award innovation mm interpret range however individual restrict standardized account date lift provide compare raw version lift seminal analysis researcher compare quantify measure value range standardized value relative date standardize lift herein three association rule non empty association transaction transaction single wherein present transaction contain equivalently absence transaction also transaction contain item transaction contain every explicitly contain herein use notation measure association purpose outline important property every prop know independence lift cf prop large third property relate small unchanged unchanged five identify necessarily help distinguish symmetry prop property prop I multiply relate prop symmetry two measure imply prop odd prop prop prop transaction neither value measure measure association include include measure eight figure consider subset property one measure fashion prop prop prop prop prop lift cosine n lift also rule lift negative lift problematic lift lift exceed another lift support lift rule lift lift occur raw lift describe standardized lift else lift consider lift lift bound small difference lift rule difference narrow highlight issue lift standardized lift rule suitable upper minimum support threshold transaction herein cosine employ understand algorithm pair measure standardized measure note threshold exploratory may application threshold threshold typically positive rule analysis burden gamma difference proportion pair ignore similar ranking tie conservative description order herein order rule raw rule apparent relationship calculate report value plot available national institute road intersection feature road condition direction age criterion indicate identify make available threshold approximately length adequate analysis ten vertical bar base bar bar circle standardize sort transformation raw cosine cosine similarity horizontal indicate figure trend rule generally positively value though cosine lift independence trend lift identifiable identify positively correlate item dark reveal high near indicating rule low achieve value surprising majority rule value standardized rule indicate namely zero already value rule choose standardize include standardized figure figure raw vertical lift raw
page p assumption therefore stopping note actually derive tailed variable refer stop chernoff bound confidence obtain p p sp similar proof apply et lead et al reason stop rule introduce let value moreover stop interval general stop stop boundary shown affect follow span decrease boundary toward increase coverage estimate double associate double satisfy consequence min define max imply take expected stage substitute calculate stop rule determine substitute sn define stop rule parameterize pre p see flexible sample satisfy great less regardless want accomplish section obtain ensure pick stopping reason asymptotic close fully counterpart investigate sampling scheme integer sequential stage allow vary coverage usually follow desire proof coverage tuning technique establish guarantee x n I know central limit z optimality double virtue p result available prescribed margin coverage area rigorous hoeffding prescribed confidence substantial sample unknown close derive analytic cumulative scheme result sa see symmetry analytic expectation sample number evaluate coverage rigorous b second introduce checking adapt b explain advantageous complementary coverage probability chen procedure binomial proportion purpose prescribe guarantee infimum checking guarantee infimum p j easily exact hard p kn nonnegative great differ coverage infimum whole coverage infimum coverage check guarantee prescribe lack another issue induce substantial choosing parameter adapt check coverage bound coverage prescribe pruning branch provide exact solution wide b computing probability far reduce complexity check chen check shall description computational dependent subroutine whether small prescribe every interval coverage guarantee proportion binomial situation impossible evaluate interval extremely b purpose less require interval bc cb proceed st st binary intermediate variable left backward interval interval consecutive width try repeatedly cut width becomes prescribe relevant f though situation make extremely rare moreover negligible g backward minimum checking backward checking adapt seem involve evaluate coverage purpose working explain coverage since precision computing precision computing evaluating error involve complementary readily control american binomial prescribe approach chen strategy parameterize coverage adaptively rigorously check coverage virtue bound probability search coverage confidence coverage principle construct interval term margin error inclusion interval coverage stop limit readily see chen eliminate necessity checking coverage b technique chen response result final manuscript manuscript grid manuscript notational play tuning coverage infimum pp manuscript method coverage infimum coverage present check bounding technique check check coverage parameter impact computational let outer loop let coverage candidate recommend numerical double coverage stop rule respect various interval determining guarantee fully sequential display binomial proportion show indicate coverage substantially prescribe confidence consider apply asymptotic theory scheme close numerical asymptotic computation may adequate drawback coverage tend tend pre specify level asymptotic applicable tend reality introduce prescribed confidence fair exact guarantee example indicate nature coverage simulation evaluate double coverage confidence double c table wants give rule determine substitute design side function binomial side desirable table size concrete prescribe double scheme number stage range coefficient coverage prescribe double stage range c want ten stage sample obtain appropriate look consequently rule value stop boundary scheme display leave side conduct numerical investigate impact computational frequently margin double coefficient choose suffice tune respectively average sampling scheme side coefficient double sampling impact interval construct derive stop derive confidence appropriate perform derive pearson rule pearson interval interval suffice derive ensure take double purpose scheme plot compare stop interval stop interval middle chernoff bind interval inferior pearson double uniformly stop rule pearson interval situation sample p complementary determine coefficient coverage leave without complementary coverage application double sequential clinical expect size determine suffice less accordingly calculate path equivalently virtue continue clinical trial proportion confidence seven conduct record suppose patient p stop satisfied see green experiment patient add number response stage stop conduct patient group observe response get patient stage get frequency satisfied conduct stage fourth group response fourth fourth stage relative value conduct stage experiment observe patient response among stop rule terminate experiment believe difference report statistical involve substantial see check rigorous known insufficient prescribed size method double scheme sample exact percentage serious run count patient important group become scheme review sequential method principle rule suffice determine level family boundary sampling ensure prescribe tend establish analytic sample termination technique compare one gx z z definition interval l z l p z p z p p complete sampling scheme stop one virtue chernoff hoeffding making guarantee thus complete theorem show n n strong law number convergent sequence converge event sure sure event almost combine yield p establish central tend zero x n arbitrarily small show simplicity notation sequel n eq combine yield write n show gx x px complementary event side nm sufficiently small inclusion chernoff letting establish arbitrarily theorem denote l l make relationship l hoeffding follow n n p complete chen first exist family sequential proportion error uniform optimality family scheme theoretical result establish little sample expectation sample number derive address computational illustrative clinical trial proportion problem significance area engineer science reason concern fewer possible require estimation goal scheme observation specify satisfied evaluate accumulate take sample refer sequential estimation sequential particularly fix group increment stage group actually sequential method statistical unique science quantify uncertainty inferential statement error quantification uncertainty inferential method exact word computer exist estimating nature therein solution insight solution necessary asymptotic coverage specify confidence proportion relative sequential binomial proportion involve conservative bounding employ derivation en proportion rule en confidence test interval fix confidence decrease sample sample termination interval paper american binomial prescribe confidence manuscript general framework estimation chen provide chen specific proportion prescribe margin interest sequential proportion prescribe margin introduce exact established introduce inclusion construction concrete investigate connection new rule feasibility stop prove prescribed confidence binomial parametric rule tend minimum method accuracy numerical study stop rule discuss general principle rule feasibility expectation various evaluate probability parameter rigorous efficient present result various scheme method clinical denote normal case frequency variable make clear many scientific space binomial proportion binomial reduce formally pre margin respectively complementary clearly complete construction stage sample stage throughout stage stage stage termination termination experiment denote likelihood minimum unbiased sequential size integer group approximately take coverage tuning follow stage coverage proportion coverage bernoulli yield show p pp page equation chen recursive upper p p recursive counting exact proportion adapt quick determination whether check coverage checking bound complementary interval bound accomplish exactly coverage application complementary check complementary coverage exceed chen propose reduce b adapt interval I I complementary p adapt modification lb u empty nonempty interval splitting eliminate process elimination lb I initial sake adapt superior see accomplish adaptive maximum checking chen manuscript advantage working coverage check chen
subspace incorporate define flip retain informative face learn notice face pose visually enable across section formulate low transformation experimental public commonly evaluation conclude simplify n ic arrange arrange low subspace rank datum global encourage maximally subspace subspace introduce discuss adopt present first norm global transformation denote minimize encourage second encourage diverse desire variation subspace motivate use reason one matrix dimension concatenation disjoint intersect analysis consider disjoint concatenation matrix matrix objective maximal goal origin independent maximally intuition proceed nuclear norm subspace blue angle subspace subspace subspace associate true subspace transformation subspace individual subspace rank improve singular nuclear norm ball optimize often adopt rank literature rank see fundamentally affect angle replace nuclear prevent unless otherwise specify adopt normalization research throughout paper keep form excellent replacement consideration next transformation objective maximize different property lead classification present matrix concatenation dimension concatenation space orthogonal concatenation q minimum every pair orthogonal equivalently reach maximized equal improve thereby nuclear norm justify synthetic fig real adopt describe modern nuclear technique include transformation maximize separation deviation present angle subspace ht b separation variation indicate nuclear row class consist learn intra lda classes class non improve pairwise elaborate learn transformation synthetic compare closed transformation lda neither increase angle line transformation plot visualization transformation transformation introduce enable variation decrease nuclear value set subspace row fig maximize class subspace reduce subspace share methodology intra separation fig class small intra class variation angle tree usually reduce distinction learn size closed give intra show two separable lda learn cluster property disjoint consider intersect angle independent limited additional theoretical observation angle significantly repeat clean interpretation angle balanced orthogonality believe persistent replace concatenation q row dimension concatenation nuclear norm major advantage favorable popular norm nuclear norm help subspace optimize maximized transformation subspace distant proposition frobenius induce normalization big learn transformation mini mini stochastic subgradient sum obtain mini batch sample mini batch start mini transformation mini warm far square devise linear connect compressed sensing sense provide compressed paradigm result dimensionality transform mean partition underlie general procedure enhance subspace fast structure label know stage stage ssc use improved technique introduce current subspace repeat assignment stop change enhance apply beyond enforce purpose adopt cluster assignment optimize update point cluster update warm keep overall iteration subspace optimize subspace optimize minimization study present excellent subspace intuition intra deviation even assignment subspace ht assignment assign subspace ssc transformation return transform subspace robust r subspace recover encourage space intermediate learn desire incorporation decompose predefine affine affine combination represent linear r subgradient update perform pca random datum thereby computation please note transform reduce dimension projection usually perform enhance obvious correct many subspace reduction framework subspace digits ratio misclassifie point visualization purpose represent color truth ssc outperform low clustering recover structure accurate misclassification subspace digit adopt denote robust r illustrate method enhance r use significantly digits denote randomly digit ht online batch batch vary value remove discussion value batch mnist image digit batch various always subspace project subgradient first mini learn batch warm iteration subgradient fig online sec batch learn run ssc running time transform ht framework significantly subject class cluster subspace transform domain ht extend dataset subject accurately approximate conduct ssc descent transformation run subject ssc run learn misclassification ssc fig show explain distance small principal angle subspace nuclear clustering run ssc ssc reduce clean view show misclassification subspace number subject dataset significantly rank decomposition misclassification rate cluster subject extend apart transformation play ht misclassification transformation form transformation subspace transform data significantly ssc orders ssc call mean median mean ssc ssc consist type video traffic video video task segment video sequence move correspond motion dataset subspace datum digit evaluate project low pca compare ssc previous method comparable ssc ssc orders magnitude fast ssc adopt global rank report accuracy recognition increase original image good subject row test g row decompose omp classify error value fig reduce variation cause illumination third global perform base fact illumination globally subject art face ht lc nn omp nn omp side nn nn omp omp adopt classify subject pose profile pose transformation table pose flip transformation well g decompose low subspace omp error class transformation transformation dependent good performance setup unsupervise method test significantly outperform pose illumination variation illumination classify subject pose adopt comparison art face actual domain pose global face variation pose illumination enable reduction transformation learn accuracy small ambient fig exhaustive misclassification reduce subject present design e plan cluster connection investigate connection compressive rank transformation cluster rank criterion nuclear subspace art subspace provide result analysis case understand vs interesting beyond feature know achieve matrix matrix nuclear one concatenation contain orthogonal singular project subgradient describe proper keep mind development improve select already lead excellent detailed section significant art subgradient iteration iterate subgradient evaluate subdifferential subdifferential approach project convex project provide iterative initialize minimization sub term use use constant subgradient evaluate subgradient cost converge notice subgradient function discussion simplification find problem subgradient concavity term sum guarantee efficiency obtain fig prevent constraint dropping incorporate g multiplier normalize change function change gradient fig affect gradient acknowledgment work nsf transformation learn classification extensively dimensional intrinsic violate model address nuclear structure time force maximally datum variation within subspace propose learn underlying exploit combine
concentration addition previous admit form need induce measurable next vanishe although split diagram distance sample randomness splitting splitting way make low region x condition kernel assume support randomly ng diagram randomness sample splitting section take density diagram upper figure distance discuss b density convolution inference persistence diagram set estimator see explain interest intrinsic machine connect upper homology set set suppose smooth manifold let hausdorff identical thus diagram persistence diagram generators generator suppose circle radius connect one circle persistence diagram level seem suppose noise assume support unobserved estimator bandwidth explore indeed precise topological bandwidth retain topological like behave topological omit stable reason focus estimate stability band support contain kernel choose bandwidth density assume lipschitz use hoeffding use extra approximate persistence diagram diagram solve probability persistent homology piecewise finite form grid follow define linear interpolation persistence diagram persistence level use sense inference take albeit sample theory simple bootstrap follow carlo h bt ignore make topological want require follow correspond persistence circle life span component red triangle span estimator persistence diagram insensitive positive smooth rescaling consistent hence persistence diagram affect outlier formally let robust computational synthetic example persistence diagram section bandwidth serve construction persistence diagram diagram span component triangle life dimensional right diagram section uniform circle top band persistence ii satisfy method concentration one density bootstrap method topological feature method connect significant right persistence diagram circle span bottom kernel right density persistence describe top sample leave use different construction diagram case challenge loop top loop significant sample bootstrap fail circle satisfy provide band around persistence diagram top subsample persistence replicate outlier figure show diagram confidence computational discuss drastically persistence force subsample significant insensitive outlier bottom plot figure provide find recall cover manifold small euclidean radius require packing set may overlap prove na da dd eq constant manifold form euclidean center argument constant depend combine theorem show subsample toward end n remark nn I assumption contradiction thus subsample claim enumeration possible size b definition size n bt bt bc bt bt expectation claim center n inequality fact b mn assumption differentiable bound form n last hold assume show denote independence induce union bx r almost join distribution I bx r fact j x nx x nx contradiction n large display balance unconditional induced outcome splitting probability splitting q theorem random expectation measurable sample ft n eq devote n value constant accordingly identity follow ft ft n write last step constant event equally value j bc bc mb jk bc jk ph c hc jk b jk n v asymptotic statement notation strategy proof n conditionally splitting assume solve assumption bound away otherwise p h divide grid cube center p hx hx hc hc hc hc hc quantity zero hoeffding summing separate persistent homology like statistical advantage plan assess assess scale examine estimator kernel helpful inference investigate interval investigate topological quantify minimax persistent homology refined confidence topological parameter diagonal concept see method conservative fact conjecture adjust estimation topological thank anonymous suggestion feedback fr ed comment remark support nsf grant nsf dms air force dms persistent homology topological function birth death vary informally signal bring persistent homology topological class refer collection method datum protein image analysis middle sub death feature merge dimensional appear represent connect diagram black dot connect component leave triangle appear topology summary capturing feature homology homology etc persistent homology assign birth suppose homology support one center persistent homology homology topological interval persistent homology one persistent homology separate noise suggest persistence diagram sample n persistence metric persistence diagram bottleneck dependent confidence persistence diagram goal main goal introduce persistent homology key persistent homology topological visualization diagonal persistence diagram synthetic paper concept key homology find basis persistence diagram homology persistent homology example top later contain detailed medical procedure upper persistent homology rate persistence diagram follow challenging involve attention manifold embed compute persistence diagram homology formally model confidence interval present illustrate contain finally contain conclude remark exist dimensional close q q projection value measurable event appropriate fold random place finally brief persistent supplementary material detail topology coverage persistent homology persistent homology topology persistent homology plane death distance persistent homology topological change topological include homology homology homology merge component topological material homology level estimate confidence diagram persistence diagram produce diagram persistence diagram different way measure persistence diagram bottleneck persistence diagram supplementary bottleneck persistence diagram bottleneck stability finitely bottleneck persistence diagram bottleneck main work reader proof theorem wasserstein perfect rather supremum distance restrict wasserstein present extended confidence persistence diagram wasserstein strong close result let embed subset persistence diagram set bind particular problem infer homology infer hausdorff hausdorff play important role include estimate homology observe observe homology homology set nx bx bx quantity define density manifold definition reach zero infinity open neighborhood explicitly result
recommendation active user precision user test fraction present top recommendation normalize recommendation set much poisson quality recommendation recommend show plot relative consistent vary recommendation factorization relative recommendation compete vary next lda datum netflix function user user recall user percentile performance user activity light constitute majority heavy exploratory fit explore discover among confirm way illustrate discover scientific article new york illustration sort weight discover cut across york multiple business relate self appear business news unify e poisson mean generate movie library significantly outperform recommendation rating implicit hoc poisson massive differ traditional ability amongst item account amongst recommendation text inference analyze study popularity use model derive algorithm conditional fit kl conditionally model perform ascent iteratively hold latent conjugate conditionally variable complete item conjugacy weight exposure item similarly conditional user popularity final count sum complete see derive first complete conditional distribution gamma shape variational activity parameter popularity contain variational multinomial variational ascent hold conjugate equal function mean fact parameter expectation conditional count multinomial divide rate update update come gamma variable hierarchical poisson factorization recommendation user datum item feedback rating number view click develop variational massive performance rating movie read scientific reading article reveal factorization help otherwise direct item article product recommendation historical item rate pattern kind tend discover recommend algorithm recommendation easily outperform recommendation tailor interest realistic resource user c united fidelity nothing star episode iv star episode vi star episode back illustrate netflix netflix contain rating organize interest movie interest science movie star episode independent fidelity course movie pf infer infer interest new movie list movie include star episode ii poisson user preference item non assume draw exponential preference attribute user figure illustrate top specific plot middle estimate preference spike preference tend item attribute general variant find pf enjoy advantage wide variety item integer pf variant significantly include bias netflix movie fm music read paper article main poisson contribute first consumption finite view item user budget movie model carry weight item partially lack factorization systematically hypothesis zero practitioner complex factorization need modify advantage pf iterate item implicit factorization take natural analyze massive iterate implicit take advantage sized full netflix appeal amenable stochastic datum discuss poisson property scalable inference root poisson come nonnegative factorization objective factorize likelihood show maximization maximum estimation augmentation develop alternative allocation lda estimate preference prior multiplicative infer posteriori prior draw skew contribute good independently detect include variational approximation issue detail consider derivation auxiliary feedback merge technique neighborhood technique adjust informative negative example appropriately weight rating cause factorization special rating recommendation compare variety applicable poisson empirically item user rate rating give implicit datum otherwise user behavior click view factorize distribution represent preference parameterize preference variant factorization poisson replace basic generating prior attribute encourage towards representation user item furthermore place user specific parameter control hierarchical capture diversity tend capture user put user sample activity sample popularity attribute item pair hyperparameter call model poisson computational generative item activity mf poisson factorization capture star user equal penalty user star poisson prefer bring close movie score back likelihood matrix computation sparse observe poisson classical mf especially massive implicit feedback must iterate practitioner rating like user preference rating recommend content user discuss challenge posterior mean scalable hundred item single cpu would like posterior user preference item activity item computationally field complex variational family member close kullback make approach problem variational add additional facilitate derivation description user integer equal rate preserve thought contribution place mass item initialize parameter parameter offset activity popularity repeat convergence user update user activity update popularity item member posterior contribution represent count variable govern flexible govern variational gamma distribution multinomial vector stem bank conditional sum specify variational minimize field optimize computation coordinate ascent holding shape constraint sum need observation thank previous terminate convergence compute specifically item use expectation rating stop insensitive hyper exponentially shape hyperparameter evaluate factorization variety music movie user read scientific article read significantly recommendation compete recommendation exploratory attribute study feedback article article million observation cell presence article library music million million observation time user song york article observation observation view netflix contain movie rating star provide robust york netflix partially item user rating vary significantly datum netflix preference item user rating movie click count measure user give article indicator article fully rating number
independently accord q possible able appropriate gradient observe domain simplex rate vector apply proximal p proximal consequently theorem assumption inclusion proximal eq choice attain scale dependence stochastic gradient corollary suggest require mirror obtain apart attain turn continuity difficulty non calculation lipschitz continuity convergence single slightly achieve even objective fast convolution smoothed density respect lebesgue differentiable mild smoothed extra difference speak unlikely vector near sequence positive directional stochastic section mirror descent additional simplicity impose f distribution uniform ball radius assume analog compare gradient norm capture introduce logarithmic possible remove smoothing aside penalty gradient essentially gradient estimator perturbation exist constant sequence approximately function worst notably substantially dimension previously result theorems rate perturbation whether logarithmic corollary begin describe form sequence q additional randomness require statement simple choice class norm functional construction lipschitz assume equal ball combine bind ball ball corollary minimax proposition tight within parallel evaluation minimax ball second investigate rate stochastic problem lipschitz continuous gradient dimensional problem class demonstrate mirror logarithmic factor recall corollary proposition low bound match logarithmic low full access evaluation minimax low evaluation iteration impose function variance optimization function extension proposition optimization dimension indeed euclidean observation minimax low achieve accuracy minimax compare minimax rate scale multiple evaluation precede loss ball pair proposition phase transition return full information analogous rate applicable factor achievable corollary descent consequently analysis show order moment suffer optimal suffer unit single evaluation gradient preferable use full even gradient somewhat nontrivial provide result argument mirror iteration assumption yield two upper choice former summation quantity inequality jensen inequality lipschitz continuous error prove statement recall gradient imply vector equality average directional attained draw take eq jensen inequality guarantee simplify lemma bound moment uniformly radius appendix proper lemma theorem cf specifically control proof denote eq turn lemma give independent first become universal universal similarly may jensen term claim proof low bound information yu strategy hypothesis optimize one minimax bound binary hypercube minimize must le technique detail objective show possible optimize optimality estimate sign optimality problem proof inequality symmetry imply give probabilistic bind q denote next bind le coordinate conditional le cauchy inequality obtain remainder sharp shorthand bound inequality eq recall return final bind enforcing amount choosing guarantee eq lower rigorously give j j precede substituting claim choose immediate analogue denote place give except vector negative optimization final le derivation schwarz analogous inequality nearly immediate indeed pair analogy may inequality remain gradient next help vector classical vector consequence bound substitute bind holding completes reproduce appearance give complete proof proposition construct minima relatively separate hard distinguish element distant yet many support evident belong inspection separation analogous define available early f normally random absolute mean use convex completely parallel proof precede analyze oppose minimize improve numerical showing optimality smooth convex information also complexity though require carefully randomization attain sharp rate bandit feedback transition compute gradient use evaluation interesting understand grant nf fellowship facebook fellowship constructive suggestion collect use calculation immediate sphere clear lipschitz continuity independent gaussian vector high probability invariant let follow universal standard proposition leave expense eq complement use schwarz inequality obtain q suffice bias allow reduce moment convex unitary unitary lipschitz denote property convex value replace supremum take follow expectation precede supremum let increase return complete equality therefore last difference choice assumption claim distribution distributional identity inequality independent first term moment calculation numerical desire lemma lebesgue standard subdifferential consequence obtain remain stochastically dominate set let lebesgue compute otherwise dimensional surface density density convolution domain nonzero depend follow relation r rp rp give tp tp formula different base argue integrate contribute p us section proposition optimal vector elsewhere quantity notational force kl divergence chain define precede observational scheme function query observation normal thus q aside identical inequality paragraph equality definition rgb berkeley edu california berkeley work smooth conference several upper pair estimate suffer stochastic analysis dependence extend complement theoretic establish achievable scheme book overview explicit computationally infeasible impossible gradient minimize rather calculate problem machine bandit choose player suffer value problem additionally problem graphical structured function may despite procedure remain solve convex almost rate well focus location estimate randomize sphere paper recent work achieve however complicated objective well optimization available difficulty inherent single evaluate note independently multi black access value classification essential point f scheme scheme iteration sample estimator take procedure adopt perspective randomization receive stochastic mirror estimator point natural extension careful gradient analytic past obtain optimal dimension detail increase sequence observation
regressor rewrite model form target would g linear combination dictionary rate state sparse form choose technical regressor datum traditional series estimator h identify estimate via moment j orthogonality generalize huber framework cover true identify borel overlap value map open nuisance admit large solve analogue contain typically robust orthogonality score value nuisance parameter express symbol generally project complement tangent j continuously differentiable e iii local c score away mild assumption allow arise median impose pointwise function uniformly I nuisance estimator namely k ns w obeys c ms ps conditions literature primitive condition vi restriction sparsity smooth grow growth condition condition ii implicitly require formulate estimator via non ii relevant function iii main probability immediate corollary uniform lyapunov central array eq interval uniformly implication central uniformly corollary normal j condition lead band valid corollary immediately hypothese wise rate discussion covariance multiplier generate random orthogonality z benefit conditional replace unconditional consider px ir design perform repetition unless decay coefficient regressor zero post simulation prior theoretical argument instrumental work median false rejection rejection standard base confirm failure validity post procedure design separate happen sharp track nominal confirm figure compare post post standard post propose post perform square supplementary discuss extension implementation let measurable pointwise measurable index fw fw es kt theorem e prove differ appearance depend step iii nf envelope apply bound use jt last expand pick jj hz orthogonality ii w h bound uniformly finite e ct w j wish e hz envelope cover ct w c l nm condition iv right bound probability moreover corollary assumption comment section b choice provide mistake obey semi regular behavior coincide median penalization derive estimator dramatically otherwise assume robust furthermore testing coverage invert robustness respect moderate mistake allow uniformly paper array asymptotic asymptotic capture phenomena coefficient ensure robustness conclusion respect perturbation turn translate validity address parameter huber large instead exactly smooth limit theorems bootstrap result latter dimensional respectively denote normal dependent omit quantity confusion measurable let finitely f step outline post post penalize x run keep iii distribution I v I sf absolutely v exist I almost surely growth min impose distribution condition ii impose equation condition impose instrumental quantile require restriction regressor condition relax minor modification unity zero vi quite plausible
search minimizer problem assumption therefore line criterion complete show contradiction monotonically try satisfy search contradiction thus pt monotone search lemma let limit criterion decrease point subsequence decrease observe eq consider continuity boundedness necessary optimality convergence critical generate eq limit sum complete em extension monotone limit sense consider mild existence limit omit monotone minimize approximate sufficient function mm focus easily discuss commonly approach multi dc solve generate sub obviously optimization ms solve number outer especially problem class shrinkage name backward extensively problem general wide vs plot figure decrease fast speed adopt bb rule convergence monotone algorithm increase finally search accelerate comparable converge slow demonstrate superiority fast use monotonically classic k la la ng bb initialize line bb rule line sim news satisfy termination consecutive objective much large monotone thresholding class encountered step commonly bb rule monotone criterion greatly algorithm monotone search future work focus estimation propose acknowledgement partly cb grant lm nsf solution observe decompose entry k simplify notation via derivative element otherwise three scad proof sparsity penalty considerable recent superiority counterpart several sparse setting convex penalty big challenge use ms problem practical solve nonconvex large penalty iteratively penalty outer initialize bb size large application area non regularizer extensively apply successfully signal sparse formulation suboptimal loose address regularizer include smoothly deviation log sum concave penalty mcp regularizer penalty shrinkage thresholding penalty propose adopt bb rule initialize step size greatly line criterion extensive large set consider make continuously differentiable rewrite say differential differential rewrite function many machine assumption loss commonly satisfy regularizers table except assumption tb satisfy regularization w name scad dx dx w w w w w shrinkage generating follow q size problem regularizer close solution u k h ix k issue select
refer refinement rough laplace refined draw different typically rough example algorithm sampler reformulate gibbs follow univariate update refinement draw rough laplace updating refinement lead true gibbs general notation fully support satisfie variational though univariate refinement posterior ip f refinement dimensionality curse issue rough start refined find performance multiply bring close limit subset issue repeat several issue increase also gibbs sampler evolve initial towards might refinement argument refinement process refinement bandwidth attempt obtain q posterior inverse result posterior efficient refinement begin refinement proceed middle choose refinement use initialization propose alternative self rejection self adapt feature suffer drawback along solution formula average draw subset generate far rejection draw sample misspecification function give h variate condition draw sampler pm density accept draw rejection accord rejection repeat reason rejection conduct incorporate undesirable accept draw effectiveness rejection sampler variate subset rejection assume equal smoothing subset easy tackle bring pairwise draw obtain subset obtain set bring less density parameter equation distribution well therefore posterior obtain marginal posterior formula term parameter inspire combine sequential first combine draw plug subset combine formula importance weight scheme require weight estimate combine integration adequate provide accurate draw require htb calculate obtain execute example draw machine run sub new applicable smoothing dimensionality curse update structure step update brief interpretation effectiveness new error accumulate accommodate manner curse subset smoothing specifically chain monte carlo mcmc subset within inverse subset weight average iteration rejection specify rate I acceptance function consider evaluate mode model seven assess evaluation refinement claim logistic refinement bi real subset density accord portion adopt refinement refine density different illustrate approximation certain rate particular left bi modal adequate reasonably good refinement real logit fit set refinement parameter accord choose refinement quickly move truth distribution plot run regression broadly many categorical q correspond function predictor follow normal different correlate predictor subset zero gaussian moderately performance marginally carry way simulated burn chain laplacian adopt initial accord refinement refine draw refinement posterior select illustrated nonzero multiple total marginal posterior result evaluate joint approximation kullback leibler divergence two reference sample finally parameter demonstrate approximate performance rejection theorem rejection mode search sampler averaging might rejection identifiability multi gibbs normal point suffer exploration motivated merge section implement problematic switching move situation handling mode sampler mode analyze mcmc multidimensional rejection sampling sampler subset mode pick chain beta employ testing sampler rejection kernel parameter common scenario partition subset posterior density different value see fine informed smoothing appropriately rejection smoothing achieve level parallelization total mixture fix census extract survey census whether whether whole turn fitting via usual illustrate sensitivity rejection subset choose away approximation fig avoid potential advantage conduct step refinement initial draw iteration propose refined draw tune rare set prediction logistic positive great plot datum list laplacian marginal rare laplacian likely approximately predict contrary different sampler enjoy issue dimensionality curse attempt trading accuracy sampler exploration correctly identify original sampler work sampler aspect investigate justification strategy eliminate concern estimation rejection control approximation exploration thing small entail exploration ability provide potentially old differentiable derivative let satisfying transform kt taylor taylor article focus choose transform old smooth assume application omit theorem posterior satisfy variation distance normalize eq merge notation derivation divide w kf way product relative difference k f kf sum trick entail mathematical induction easy verify difference normalize eq error though asymptotic rough magnitude regularity asymptotic ensure mi ni consistent obtain follow hold asymptotically quantify elaborate definition q essentially statement note ergodicity coupling derive conclusion continuous bound f function decrease eq divide due continuity monotonic always set h guarantee able ad define lemma satisfie become equality notice qx define kernel I strictly tw dt eq straightforward lemma guarantee fact lemma list remark rapidly grow free article sampler mcmc draw subset enjoy tune provide sampler mcmc chain subset communication year modern ease result huge demand markov carlo face big due expense latent unit accelerate effort direction computation separately store machine fed processor approach langevin hamiltonian mini batch direction partition mini independently datum
perform curve segmentation include choose previous mainly curve keep difficult cluster computing vary regime deviation obtain misclassification versus present misclassification similar slight previous proportion mix proportion accord h mix proportion cluster colored misclassification situation misclassification gmm third fourth regime middle attribute constrain cluster curve figure top concern triplet present respectively like complete data constant approach free number constrain proportion observe percentage selection correspond approach regime cluster like number illustrate advantage like approach curve different compare study curve figure curve switch curve diagnosis switch speed train one control curve switch operation curve several successive involved switch figure switch diagnosis achieve curve however amount manual labeling concern propose homogeneous compose curve observation database correspond operating operate accordance degree polynomial regime curve neither classification preliminary corresponding switch operation htbp gmm take temporal clustering curve regard cluster especially obtain informative curve second shape therefore switch mechanism particular belong middle expert measurement default switch differently true class intra significant inter em em mean intra class confirm well cluster spectra spectra record feed work contain spectra spectrum curve six five regime segment segmentation one see retrieve cluster result close like surprising confirm upon amazon contain set htbp segment cluster use raw preprocesse author cluster use som som clustering segmentation hide help well understand datum hand profile result attribute provide include directly regression datum htbp piecewise datum segmentation regime curve alternative include mixture gmm term curve cluster comparison confirm general note current piecewise avoid slightly modify algorithm add interpolation piecewise dot overlap cluster however regime propose occur range regime overlap characteristic regime scale universit france universit la france fr simultaneous segmentation present regime propose polynomial piecewise within piecewise segment approach dedicated maximization em probability latter optimize likelihood criterion dedicated segmentation programming segmentation perform simultaneously approach simulate include background tool several maximization em mixture gmm equivalent identical matrix soft classification involve input belong therefore paradigm flexibility interpretation efficiency base etc grow adapt heterogeneous observe curve univariate available input domain include diagnosis bioinformatic electrical etc I n label simulate curve regime simulate compose regime cluster color according represent segmentation aim structured regime range see characteristic correspond change mean etc infer hidden method regime instead treat simple achieve regime change g change problem namely mcmc online approach concern concern single segmentation cubic spline knot priori consist model effect regression spline spline require spline knot spline cluster sample generative use em author base piecewise allow polynomial simultaneous clustering perform programming minimize carry paper well datum simultaneously segmentation homogeneous dedicated fuzzy partition segmentation maximize algorithm curve optimally proceed partition optimal dynamic programming briefly curve work curve spline mean like curve clustering introduce propose derive approach dedicate deal carry world curve compare mean like gmm context finite mixture mixture component suppose two observe via probability membership posteriori refer likelihood classification consist optimize likelihood classification version em e em membership hard way use curve introduce curve structure rely mixture lead however functional limitation cluster distinguish include effect spline curve mixture em em piecewise cluster like mixture spline mixture cluster either spline model one coefficient construction spline fully parameter model partition cluster maximize map principle mixture however address change within stationary behavior well handle regime change alternative regression polynomial basis range rather single spline predefine piecewise spline generally either place range knot regularity piecewise polynomial knot optimize programming piecewise optimal mean algorithm involve dynamic clustering spirit probabilistic address mixture generalize deterministic possible notice task segmentation govern enable another among homogeneous regression author propose optimal curve euclidean criterion multivariate curve cluster piecewise regime thank criterion segment cluster segmentation distance criterion segment regime segment belong segment cluster criterion iteratively minimize mean initial step piecewise prototype follow segmentation cluster regime additive segment additive specify present integrate regression model curve result piecewise curve assume piecewise among model distribution regression index cluster mix proportion polynomial coefficient transition suitable shape integrate piecewise framework thank simultaneous spline generalize optimal segmentation base datum dedicate however maximize log specific present second introduction likelihood perform form iteratively maximize standard particular cluster th indicator iff curve paragraph maximize curve piecewise em start g variation complete current iteration posterior belong computation compute lagrange multiplier give find piecewise segmentation correspond fuzzy posterior cluster weighted piecewise curve consist solve posterior cluster procedure update regression segment propose em fuzzy cluster fuzzy regime index curve assign maximize probability estimate summarize compute ml partition regime obtain include fuzzy cluster cluster maximize propose approach dynamic gene assign behavior formulation curves suppose mix temporal heterogeneous notice introduction propose another scheme include dedicated likelihood clustering estimation include log classification em adopt perform cluster simultaneously maximize likelihood parameter iterative model long relative first step label log step q equivalently integrate step dedicate three step compute posterior equation curve curve estimate cluster label vector likelihood respect complete optimize optimize mix lagrange update segmentation maximize present previous posterior cluster curve piecewise polynomial estimation perform propose mean optimize distance criterion optimize constraint impose identical piecewise curve polynomial hard curve cluster optimal constraint maximize take maximize minimize w label criterion optimize regime triplet use criteria information bic criterion etc bic penalize criterion maximize p
let regularization sgd enjoy package sdca accelerate proximal enjoy rate default work hinge hinge loss smooth satisfie hinge hinge smoothed hinge prox obtain procedure prox sdca hinge w w w runtime eq multiclass describe label goal score different class prediction maximal one th coordinate whose optimization multiclass svm smooth original hinge specify prox multiclass show calculate hinge n write x I rest also eq equivalence sort sort negative cumulative sort correspond w optimum sort j code sdca convenience maintain prox smooth hinge w z ig na ia ic ib grid major coordinate scale runtime sdca work regularizer strongly respect arbitrary norm acceleration extend acceleration regularizer acknowledgement author careful reading support grant institute intelligence zhang grant nsf dms nsf technique generality strongly regularizer run prox sdca option careful proof option option choose optimize ensure bad simplification iv employ simplification follow assume let expectation randomness choose element update write rearrange eq side strongly imply q expectation total obtain eq since dd require choose choose result need proof markov inequality optimality therefore probability repeat monotonically sub use prof round probability therefore choose choose apply claim claim problem proximal version stochastic coordinate ascent accelerate outer art learn multiclass svm follow minimization instance refer solve ridge find apply logistic term think runtime recent become significantly improve improve runtime accelerate smoothing technique solution runtime svm regularizer non regularize add regularization assume runtime square put runtime machine learning ridge regression previous lasso coordinate fista ridge exact sgd sdca idea proximal dual sdca ascent ascent pass work convention machine distinction two direction allow convex square euclidean consider general generalization useful multiclass nearly linear case iteratively objective stronger particular relatively make runtime dual ascent add contribution extension stochastic study consider method dual problem understand primal optimality dual sufficient approach later frank wolfe special multiclass hinge end ascent rate allow accelerate rate nesterov technique idea present attempt accelerate reference therein runtime polynomial oppose logarithmic polynomial dependence allow single pass consider set number simplify denote matrix norm strongly form define regard description sdca accelerate prox sdca throughout two smooth discuss proof acceleration rest proof coordinate procedure solve subsection smooth correspond example dual allow th keep dual coordinate ascent choose uniformly let lead dual write particular ascent objective maximize namely eq simplify respect objective maximize proximal bind may complex w show still pick dual objective decrease throughout decrease need ascent prox p rv follow option option follow option replace option ii definition j default w w tp theorem prox sdca sdca tt give prox sdca least guarantee tt tt theorem tell runtime runtime therefore amount time duality small yes would proof prox sdca nearly linear runtime improve acceleration far subsection regularizer euclidean euclidean generalization acceleration convex future acceleration procedure prox sdca iteration sdca tw large center around around plus momentum code parameter determined htbp accelerate prox sdca minimize condition prox tw I tw tw following specify experiment algorithm find prox sdca epoch check well prox sdca terminate runtime outer call prox sdca straightforward argument accelerate sdca guarantee assume consider might differentiable lipschitz technique let conjugate observation proper lipschitz dual conjugate claim note smooth stop stop guarantee minimizer stop accurate condition valid recall iteration accelerate objective minimize accurate derive eq vanish strong every standard algebraic every quadratic maintain quadratic function define every every inductive upper conclude immediately theorem define formula assume minimize rearrange get back every induction induction claim rewrite inductive lemma therefore rewrite specify eq choice guarantee rearrange term side negative convexity q prox sdca terminate average runtime similar argument fw tw eq q z combine every yield eq get average runtime prox sdca several popular derive several subsection lasso logistic multiclass conjugate conjugate hinge loss parameterized strongly addition hinge max function multiclass write unless hinge loss technique parameterize add max conjugate project onto project b b projection max hinge ba max hinge function soft max aa yield gradient strongly strongly convex simple regularization square regularization use plus vector conjugate popular regularization regularizer add slight formalize later maximizer q gradient easy solution sign side component conclusion q use accelerate ridge regression prox
function fast upper note requirement diagonal element emphasize practically regularization diagonal proof line search condition small combine eigenvalue global optimum lipschitz continuous enough line step decrease value gx gx td td gx gx td gx gx upper side integrate side q enough update set compute restrict fix brief entail step coordinate zero differentiable subgradient relate minimum solution must therefore norm definition fix arrive property free eq restrict occur loop newton modification number reduce sparse value huge computational gain essence coordinate update iterate set correspond descent gauss index various free partition suffice iterate pattern consequence iterate satisfy condition shrink mention experiment strategy thresholded covariance e ij diagonal differently diagonal decompose problem size following show updating detect free recall pattern exactly pattern thresholde show structure precisely block set fixed need check diagonal inverse preserve meaning belong decompose prior run cholesky htp variable fix set free ad ij cholesky factorization x cholesky behind show unique eigenvalue primal level iterate contain accord compact attain since strongly therefore unique optimal general newton direction mention framework gauss size denote choice satisfy global step towards proof convergent convergent subsequence accord prove statement infinite accord away generality attempt derivation follow satisfie q positive definite still proposition relate define equivalent optimality therefore algorithm converge optimum converge select subsequence necessary easy consider subsequence subset lemma optimum briefly nonempty subset constraint natural strictly convex minimizer theorem alternate linearization matlab another linearization matlab yield achieve glasso project subgradient source code coordinate inexact code project source first compare time covariance graph structure procedure covariance element nonzero correspond simulate set comparison dimensionality vary chain graph correct value discovery correct result five measure structure recover true define ground truth tb indicate false c tp fp chain gap require user request stop run exceed hour c pattern alm glasso chain chain chain well objective well false positive guess nonzero whether table initially million converge minute fail guess hour primarily graphical positive rate synthetic rate versus solve sparse dense absolute obtain efficiently recover ground biology art first reasonable relative second observe figure see super convergence overall time expect htp er dataset use small yield figure regularize mle want descent focus iteration fix plot free dataset drop fact ht converge small produce show thresholded diagonal problem glasso end diagonal thresholded block even explicitly covariance block diagonal moreover set slow replicate eight diagonal use covariance compare propose algorithm glasso glasso decompose thresholded matrix solve individually thresholded covariance block structure reduce trial glasso slightly increase glasso decomposable keep block speedup glasso sparse glasso cluster time decompose free able exploit glasso drastically acknowledgement nsf grant would provide alm accord combine yield gx gx gx therefore divide side take limit prove less therefore eigenvalue introduce diagonal term leave grow side depend eigenvalue get likelihood mle recover limited novel program largely information quadratic approximation modification sparse mle method convergent synthetic datum compare state art method increasingly parameter potentially important range biological brain connectivity interaction network inverse matrix also refer precision active line covariance propose minimize negative entry covariance encourage problem log convex arise high dimensional suffer sub since matrix entry determinant determinant function convergence art linear rate thousand million consider second part order setting implementation step expensive high secondly log objective act barrier lose positive unless regularize mle newton approximations descent computational rule sufficient stationary condition characterize optimal small manner preserve convergence second order descent describe free selection sparsity block preliminary appear conference conference version subsequent ii iii comparison curve fix thresholding detail setup section present descent summarize use synthetic instead exist vector letter space symmetric definite semidefinite respectively denote matrix real norm define diagonal variate independently sample regularize write regularize inverse encourage give nonnegative obtain p solve inverse require entry regularization detail refer part efficient hard solve objective constraint subproblem lasso problem nesterov propose row subproblem instead implement widely package glasso dual apply project subgradient propose accelerate descent method smooth lagrangian nonsmooth alm package greedy coordinate prohibitive handling large quadratic perform trick propose common characteristic iterative gradient increasingly ease little computation rate attract non regularization constrain method order project solve compare order solver solve primal use approximate newton subsequent generalization fista two strictly convex build composite second smooth entire iterative objective type method empirically hessian newton appear reason gx solve lasso coordinate shrink apply would hessian make impractical fortunately sparse follow special form exploit form coordinate full step key reason newton solve exist function smooth part descent compute gradient direct cost matrix regression simple x compute small dataset solve logistic regression sequel feasible problem coordinate exploit hessian next iterate characterize manner preserve convergence subsection htp sx fx partition free coordinate lasso order compute accordingly verify symmetric rewrite descent obvious way operation reduce notational omit derivation newton apply notation furthermore index coordinate update current newton variable preserve expand substitute contribution sd ij quadratic term rewrite use symmetry column compute let ij soft I ic computational evaluating term
actor share link complex contribution go beyond interference estimation exhibit interference treatment dependency analyst work causal effect level interest example explore unit level characteristic treatment experiment exposure mapping focus randomize distinguish treatment ii unit exposure assignment arbitrarily design treatment interference unit interference spread assignment interference depend plane formally index randomized perform assignment vector specify treatment receive select possibility attention define exposure unit onto function assignment trait exposure quantify trait separately unit specific feed exposure discuss uncertainty implication proceed interference heterogeneity may real distinct rise outcome interference heterogeneity amount assignment come treatment design population thing exposure assume exposure mapping mapping interference treatment unit unique exposure unique clear meaningful analyst interference fix arbitrary exposure mapping analysis interference heterogeneity allow unit treatment assignment exposure vary treatment assignment would possibility illustrative provide exposure unit exposure support generalized call generalize exposure exposure tell experiment design exactly exactly induce exposure discuss estimator unit define unit exposure whether possible assignment diagonal individual exposure joint exposure matrix joint exposure unit exactly nonetheless produce replicate exposure drawing randomization plan exposure probability unbiased estimating unit exposure average exposure outcome exposure analyst principle variety causal quantitie difference average indirect individual average focus causal design specific estimator natural current literature seek ny td td observe unbiased estimator weight estimator potential randomization plan estimator exposure thus construct unbiased k n exposure versus exposure difference variance identify exposure unbiased population one thus effect g nonetheless bias exact derive conservative necessarily estimator guarantee randomization unbiased thompson variance unbiased estimator bias guarantee small term center maintain option correction via inequality note estimator outcome exposure discuss consistent quantity compute biased case thompson imply refine line value line expression obtain conservative variance unit population growth tend infinity vary validity growth exposure estimator converge grow regularity condition boundedness exposure potential exposure bound restriction entail grow amount exposure exposure mapping scope bernoulli treatment receive straightforward condition closely unbiased variance substitute condition type confidence asymptotic growth consistency normality confidence straightforwardly growth involve design exposure independent partial interference condition follow exposure mapping size exposure across inspire scaling variance define causal var serve purpose boundedness var ensure result follow establish boundedness exist var un b average effect variance theorem less construct cover far unbiased conservative wish analogy sampling approximation thompson reduce term help covariance adjustment randomization exposure address invariance covariate observe predefine g datum value z sufficient condition great discussion unbiased substitution proceed estimate adjustment hand selection weight sensible representative typically form define exposure regression regression estimator total linearization linearize estimator compute refinement problem estimator design high often drive may unit shrink magnitude estimator value ratio eq unbiased estimator ratio estimator unbiased tend variability place asymptotic growth variance bias practically speak adjustment square proceed via linearization linearize simulation illustrate operate indirect effect link undirected american school longitudinal add health canonical relate simulate treatment individual school experiment resemble various study exposure mapping value adjacency modify subject exposure indirect indirect subject immediate fall four exposure cccc experiment include health student estimate population drop subject analyst mapping illustrate induced issue address connection underlie trait fall another exposure cluster exposure variance school activity potential control exposure network interference exhibit right standard scenario run simulated causal associate linearize variance adjust degree associate linearize variance estimator ol estimator variable exposure condition adjust covariate adjust hc simple exposure hc estimator thompson unbiase unstable consistent totally exposure outcome control correlation exposure aggregation heterogeneity effect interval thompson estimator informative estimator ol ignore exposure simulation thompson estimator unit potential outcome suffer result rate nominal level interval ols variability ci coverage ols thompson linearize exposure covariate adjustment difference mean covariate simulation approximation unit causal interference analytical inferential principled assignment situation broad range school illustrate interesting potential allocate monitoring forest allocation monitoring reduce cut unit forest exposure segment moderately monitor place orientation segment reason proximity multiple potential proximity randomly select potential receive monitoring segment moderately monitor segment location potential possibility exposure segment possibility dynamic time vary assignment exposure could unit history prominent period period treat subject future period subject want interference subject period current exposure mapping treatment period never three exposure inference believe effect period analyst exposure period exposure exposure exposure respectively experiment vary review medical political science reader concern rely exposure specification exposure mapping classical exposure one assume interference unit typical model nest exposure mapping allow interference mapping place interference permit testing exposure may method uncertainty uncertainty unless analyst less restrictive exposure additional interference enumeration analyst may difference average outcome associate nested rejection
design represent group patch functional learn characterized center centroid significantly unique minimizer cluster stability clustering mean minimizer cluster mean actual generalization error become generalization sparse complexity dictionary k characteristic beneficial atom level exhibit theoretic base section obtain preliminary report stability algorithm dictionary unique level hold true completely objective level instability multiple minimizer section furthermore prove demonstrate stability learn compressed recovery recover novel severe degradation projection interestingly greedy pursuit dictionary performance dictionary measurement perform construct propose approach subspace graph sparse conventional dictionary building discuss idea dictionary procedure square subspace note case cluster subspace constrain origin arbitrary corresponding centroid cluster stage cluster centroid training distortion centroid update stage decomposition j vector cluster centroid large singular cluster centroid centroid good algorithm valuable set extract determine pose erm evaluate possible configuration example erm distortion construct function unit length sigma subset probability realization space ideally cluster centroid distortion respect resort minimize distortion average uniform central cover polynomially dimension cover cover assume radius center therefore belong stability centroid training stability minimizer respect expectation stability clustering report geometry k stability characteristic centroid realize distance unique minimizer hold set fail cluster centroid distortion function clustering g dp outside depend stable admissible centroid arbitrarily close matrix number level goal lk l l l dictionary stability prove atom process scheme denote patch imply residual serve representation fix interpreted dictionary unit sparsity sub dictionary centroids cluster state level stop reach adopt notation criterion representation level representation error goal list notation element vector index stack serve level give combine equal residual lie ambient space residual atom lie dictionary atom possibly union generalize atom lie subspace hierarchy guarantee atom sufficient level second guarantee per level atom l level hierarchy optimally theoretic minimum description number principle represent remain residual residual total energy level level represent energy residual level assumption likelihood code location integer fourth atom practice order pick train patch subtract estimate number level maximum dictionary level theoretic dictionary progress geometric sparse code dictionary propose evaluate vector operation whereby compute operation order order atom orthogonal pursuit useful property procedure learning improve draw sample allow learn sub dictionary set dictionary training repeat extend pursuit level implement obtain viewpoint stability perturbation learn realize probability arbitrarily equivalent utilize stability level prove training belong actual generalize difference error draw stability level atom cluster k training distortion training sample prove closeness center clustering show stability sample dictionary set lie level subsequent argument training subspace lie assign zero distortion clustering define respect supremum grow polynomially minimizer objective distortion become l g g clustering sample center cluster pick term distortion indicate cluster let set clustering define formalize lie intuitive distortion disjoint g angle span center small l j j l illustration arc clustering I g distortion indicator function tb angle span respectively l cluster stability wise residual clustering belong sub dictionary clustering belong pair orthogonal complement l arbitrary l l respectively similarly l belong unique prove clustering belong level probability level wise show minimizer prove stability closeness imply stability clustering note residual clustering clustering identical give realization level probability dictionary space level stable multiple minimizer respect express erm generalize empirical sum error see expect datum close obtain also fact cluster expected validity inequality training probability coding dictionary atom maximum create atom draw level crucial effective dictionary experimentally study training compressed recovery dictionary since dictionary measurement minimization greedy pursuit measurement incur computational greedy pursuit benefit dictionary subspace dimensionality visualization unsupervise component locality preserve discriminant local unify wherein undirected describing provide subspace supervise unsupervised c simulation compress dictionary berkeley dataset patch vary convert image patch evaluate performance standard simulation forest belong procedure first setup experiment train dictionary change infer condition satisfy use second dictionary obtain replace sample vary set quantify frobenius respect change number dictionary increase train sample close guarantee image fix number level learn careful atom level improve benefit round round round complexity figure tb c c mse plot dictionary learn varied expect approximation error reduce increase scheme approximation patch compress recover random online online describe omp patch dictionary section dictionary train training atom recover underlie image omp average recovery obtain db result recovery pursuit omp perform presence tb obtain locality preserving approach forest preserve training code setup training affinity computationally denote laplacian degree embed setting
order order candidate set dynamic small bic value select bic order lag otherwise termination discuss estimate order list order kk optimum wise fit lag follow order series iid time consume series lag million obvious prohibitive var component series lag dependency series dr delay independently form delay variable occur avoid delayed response take cross weak modification fit perform pilot study explain relate residual eventually terminate residual attempt much fast exhaustive variable proximity something expect hold context dr reduce estimation order efficiency order dr time contain polynomial ahead fit ols ols singular thorough review technique ol solution fit determination order regression ol solution diagonal use eq eigenvalue orthonormal ridge pls rr shrink create pls shrink regard rr shrink percentage singular give example illustrate need ar identical perfectly predict select order identical singular perfectly strongly numerically multiply denote pls rr correct I pls rr optimal parameter criterion row consecutive give subsample consist segment error fit limited inspection perform rr complicated employ search new precede value edge consecutive less change four simulation variable benefit large order feedback small aforementioned split optimum combination method normalize mean ahead actual monte computation compute time give average failure indicate significance prediction investigation realization apply level record pair prediction second present mark accord parameter different system efficiency indicator test signal root select use measure var system decompose namely dr dr dr respectively present series dr dependence order exclude series order spread order even prediction give table regularization slightly somewhat whereas especially performance model consistent dr correlate exclude along give table method similar uncorrelated input higher indicate vector dr efficiency score bivariate time dr bivariate system pick realization since score table criterion well become rr improvement size max perform rr seem feedback assign dependency multi create way series series common multiply q create time post time sake clarity omit indicate substantial good almost marginally system max bad parameter mean involve good prediction regularization predictive strong size comparable without good perform well small three instead ar back max behave case give full explain previous delay prevent reach true almost order correlation actually perform eeg sec duration hour point hour hour third minute check predictive ability modeling period receiver operate characteristic roc auc statistic auc record window duration time prediction channel series use estimation auc compare early period channel ols estimation median record opt median heavily heavily skewed either rr pls decrease record rr first record two record possibly underlie value auc slightly estimation slight differ median channel change regard range pls record ols record channel pls improve performance rr bad moderately order var max bad regard discrimination phase average table contribute discrimination record pls rr pls capital international market develop north comprise daily index exclude year period calculate year period method value lag indicate figure show value market period ol par perform period method table rr ol completely respectively method var stability parameter rr near ol rr north american market market uk give ol european market zero lag lag predictive relevance response already select subset time sensible real world datum correlation decrease conservative redundant compare commonly practice method subset use linear regression forward stagewise genetic least shrinkage selection lasso elastic regularization model method specific nature initial stagewise bad autocorrelation build regularization lag internal project relevance study show regularization determine method show improve small sample loss scheme max information var monte enough case pose show number consistently prediction find exhaustive fail method gave generally eeg result except perform poorly marginally conclude regression turn consistently performance propose backward conjunction regularization estimation assess carlo simulation consistently prediction popular inferior human prediction financial market selection lag quantity measure different connect financial product index method use univariate autoregressive univariate straightforward entail regard condition delay depend far delay series
choose particle discuss reason importance particle filter particle filter variance particle filter report sir filter effective particle covariance matrix asymptotically infinite moderate logarithm play particle assume pdf reach steady find frobenius q optimal particle moderate filter even assimilation filter thus moderate satisfied else situation balance example I e already eq numerator already leave covariance noise center panel assimilation principle panel exception equally balance successful plot ratio light assimilation encounter accurate neither inaccurate improve vertical assimilation unnecessary trust much optimal particle problem occur accurate carlo applicable particle filtering induce particle weight particle see section accumulation past smoothing linear setting seem basis indicate independent step help particle filter fail particle smoothing low instance application particle filter combine connection approximation balance require q sufficient norm moderate condition sir properly normalize govern covariance steady equation frobenius eq sir sir sir fail simple condition region assimilation sir filter plot panel figure region assimilation optimal maximum sir function see sir useful limited find sir particle however argue somewhat unnecessary realistic sir particle become approximation e simplify implie matrix similarity particle filter sir confirm finding dramatically exponentially logarithm variance logarithm govern sir particle frobenius moderate norm sir particle interpretation implicit realistic smooth assimilation idea construction particle direct available datum assimilation mode pdf covariance numerical physical collect thin repeat condition norm assimilation distinguish strong consider e error assimilation variational assimilation var mode single applicable frobenius directly reason well realistic difficulty importance poor wish find successful assimilation formalism kalman formalism concentrate manifold linearity expect correlation amongst occur manifold realistic find describe manifold perhaps basis special error simultaneously combine state estimation situation far case consecutive assimilation particle smoothing problem perhaps specification concern filter examine one sir choice strongly function computational hard implement implicit particle optimal see broadly model multiplicative matrix case particle become whereas sir use element simultaneous state become sir existence confirm kalman root differ substantially nonlinearity severe ensemble kalman filter variational assimilation smoothing expect variational assimilation particle extend assimilation weight sequential formulation particle think variational assimilation weather observation various compete scale problem equation interested difficult theory covariance systematically various assimilation covariance assimilation consideration regardless assimilation quantify define effective assimilation problem dimension moderate else reliable conclusion assimilation induce moderate even analysis capture main feature discuss result study effect effective particle filter importance model particle depend particle effective dimension expect happen often choice essential else particle assimilation principle comparable particle circumstance assimilation particle smoothing weak moderate else accurate prediction variational assimilation particle smoothing particle filtering help reduce responsible filter linearity equation enough reality acknowledgement thank berkeley discussion make recognize limitation university helpful thank interesting office energy contract national foundation grant dms condition sequential assimilation sequential assimilation successful filtering optimal sir break sir work datum assimilation mathematics california berkeley berkeley national laboratory use assimilation effective dimension even variable huge datum filter well solve assimilation solve condition filter limitation science engineering prediction uncertain jointly conditional density pdf discrete estimate shorthand set state pdf kalman extend kalman particle interested situation datum assimilation feasible define small moderate moderate assimilation feasible possible distance outcome experiment dynamic moderate conclusion reach assimilation assimilation assimilation assimilation principle want variance remain large state model noise stream perturb study qualitatively feasible regard effective assimilation frobenius norm state posterior assimilation sense effective paper discuss particular assimilation solve review particle particle filter filter principle certain balance condition solve assimilation building fail meet filter performance smoothing variational assimilation well successful section paper dimension effective model stream assimilation effective dimension assimilation successful effective imply discrete iid initial may satisfy iid independent principle practice matrix recursively start ap na nh call formalism pair dynamic allow encounter steady state reach state solution steady kalman gain steady covariance limit short mean covariance matrix kalman steady covariance datum translate spread sample uncertainty frobenius distance let orthogonal whose diagonal mean taylor expansion extend formula exist inequality norm root eigenvalue determine calculation investigate spread posterior mean assimilation collect mean volume spread state various physical situation know satisfactory want compute uncertainty show assimilation steady data assimilation frobenius assimilation successful moderate precise want reach reliable conclusion effective interpret data assimilation problem approximation arise differential equation pde requirement connect expect moderate assimilation reflect wish numerical like experimental sample exhibit spread uncertainty spread experiment experiment exhibit expect assimilation fall ball center likely section assimilation study represent represent dimension connection particle independent assimilation characteristic however pdf assimilation kalman formalism give valid discuss limitation discover interpretation frobenius must bound moderate lead moderate effective assimilation later section put solve life would expect moderate else assimilation condition induce balance condition error represent balance numerator hand side stand acceptable balance condition generalization illustrate assimilation correspond level assimilation feasible fix error model represent vice assimilation vertical line inaccurate assimilation perfect general also assimilation finally norm unless moderate even expect condition assimilation uncertainty dimension frobenius admit closed form moderate norm effective dimension moderate assimilation successful assimilation semi assimilation formula frobenius upper requirement norm moderate effective evident play role assimilation unlikely difficulty away pair unstable treat nonlinear problem implicitly one construct accounting extent construct bound example choose steady state bind hope reach conclusion datum assimilation physical variable velocity flow field frobenius proportional underlie energy else information examine flow thus assume moderate argument frobenius actual measurement frobenius norm frobenius typically deal assimilation value come discretization neighboring know vice versa grow increase another perhaps smooth spectrum covariance decay quickly practical split subspace drive linear function variable state
bias crucial compete appear variable dynamic irrespective strength summarize setup nan accurately module make powerful used pay tail consider tail criterion goal fdr signal tail control strength determine em complete fig signal maintain second reasonable except approximate satisfactory encouraging attractive reliable example discuss variety relate mixing like simultaneous category tail focused density take intuition behind comparison justification connect fdr whether establish likely quantile take em dependence systematic theoretical still apart discrete potentially mid mid research basis concept inference investigation investigation could leave frequentist discovery rate large parametric author like several valuable author anonymous constructive pt corollary pa usa abstract new concept efficiently ratio density nan step consideration yet parsimonious fdr view density vast discovery one false concept density impact example application united comparison density false discovery smoothing modeling introduce smooth nonparametric rate fdr comparison pre smoothing separation statistic fdr elegant fdr nan ft ft indicator denote say clearly efficient local fdr amount research smooth normal mixture reach far immediately raise scope advance current state em paper call comparison discovery rate procedure build various alternative suitable wide class diverse application bioinformatics particle motivation consideration fdr create fdr impose tail question estimating might problem far expect robust dependent one fundamental crucial issue much separately take fdr directly rather ensure gain attempt tool challenge flexible yet simple modeling attempt address tool concept add interpretability easy implement motivation testing paper follow brief description fdr transformation section summary conclusion present purpose tool comparison discovery rate define conceptual tool comparison du g concept comparison angle naturally arise hypothesis goodness test convert new formulation testing problem testing du u notion help statistical act local fdr implication multiple remain secondly detect substantial uniformity false answer section introduce idea role alternative alternative way model fdr specification fdr transform fdr type class discovery introduce nothing step formalize ff u u u ff u ff u fdr require specification marginal quantile one main proposition note report interesting make also connect fdr strategy fdr gene panel application algorithmic step estimation main come sharp narrow peak boundary indicate presence em list comparison density suffer density spline polynomial heavily parametrize capture lead undesirable spurious highly problem propose suitable large heart lie concept htb density fit come parsimonious datum easy interpret beta elaborate develop ensure main step convert value technique beta capture rapidly estimator ability tail part already recognize allow key decompose part du f u f beta denote beta act interpret view fig beta p simple straightforward exercise fit beta generate smooth density orthonormal density smooth behave conventional series shift orthonormal smooth parametric density preference consideration ease clear estimate value proposition equality substitute virtue model spurious underlying select turn make easily density estimation write f pre non find definition fdr develop utilize nonparametric comparison begin state estimation criterion repeat output behind come nan comparison statistic quantify uniformity idea density tail right panel minimum close uniformity link straightforward functional play diagnostic say appropriately help tail learn modal combine efficiently handle algorithm estimate fdr get value I nan provide user em transform value p value u shift polynomial minimum criterion em fdr htb datum interpret consist tumor expression measurement express sample statistic convert z value smooth pre comparison density em consequently fdr representation em result estimate fdr separately estimate marginal density purpose estimate pool spline maximum likelihood central match normal implement group put proportion choice note two pattern fig fdr curve
manner finds iteratively result potentially instability initialize likelihood argument lead quadratic supplement achieve iteratively batch tn tn x scalable em operate load whole online em call maximizer correspond complete triplet online statistic combination x complete current log batch modification observation regressor arise equation evaluate fast batch setting accurate tuning schedule eq adapt datum fast ever converge usual ensure taking iteration large algorithm em pass stochastic pass em require technology form much probably else run nonetheless middle notable compare method may thousand form feasible first second sgd may argue compare actual constant involve simulated predictor normal loading subsequently ensure normal pass pass sgd pass decay processing choose similar em three live middle online entire conditional total datum process even error arise optimistic nothing still meaningful relative descent decay schedule bad especially couple tend merge sgd nonconvex tail easily version considerable even disadvantage guarantee sparsity convergence straightforwardly standard binomial connection bayes previously community local interpretation construction bit suggest interesting marginalization think strength thank property easily extend numerical instability iteratively especially initialize poorly approach miss quadratic numerical evidence supplement error iteratively least evidence parameter exist approach approximation manner find coordinate use solve fit exclude contribution summarizes value quadratic approximation augmentation trick handle prior normal conjugate log complete x x thus simply replace conditional moment gradient cg cg method cg force reach tolerance cg tolerance design j trick consider work combination versus augmentation coordinate method see sub bridge prior minimize bridge represent normal design cd cg axis solution augmentation axis simulate da cg cd penalty test bridge penalty set true alternate solution difference path data augmentation systematically provide performance observation use used predict remain observation calculate incorrect classification across overall compare four logistic multinomial logistic denote class predictor logit regression indicator response want function allow vary class time except outer loop approximation give yet improve separately choose median start quadratic current descent approach parallel fashion logit lead identifiability change odd category phrase maximize look item iterate cycle logistic section td k kt j conjugate gradient axiom criterion university negative binomial draw connection interesting previously easily establish latter mark summarize detail fix success model predictor suppose purpose iterate diagonal working develop distributional straightforward exact maximization role ascent em guarantee converge mode row solve td detail familiar reader particularly relate variational via pure appeal duality line give subtle bayes section introduce variant file numerical exploit include logit likelihood function involve response author fact logit representation algorithm key mixture define infinite transform eq arise cosine complex plane zero term variable construct via laplace omit lead gamma distribution q make concrete triplet respectively either indeed binomial case parameter q predictor fundamental distribution arise treat gamma appeal conditionally em algorithm calculation meanwhile maximize normal collecting completing maximize diagonal solve linear accounting also combine yield complete q alternatively start typically fast merely gamma value calculate laplace evaluate em sufficient previously converge em newton acceleration em complete arise latent remainder definite log gaussian inverse idea quasi acceleration iteratively approximate remainder hessian l newton like iterate next extra explanation quasi general experiment first fails converge especially initialize poor numerical instability evaluate hessian third em robust ascent equally fast time basic em connection side equation follow variational defining involve takes purely argument rescale miss different approach conditionally insight loop bayes answer former treat datum operate fundamentally inferential
coverage credible standard credible interval credible value claim fully property support choice coverage provide value estimate distribution repeatedly perform pseudo uniformity equivalent work determine correctness bayesian algorithm post abc analyse frequentist somewhat consistency statistical observe outcome despite aim idea frequentist develop article paper methodology detail improve idea describe discussion diagnostic section illustrate method justify determine reliable obtain discussion abc scalar credible appropriate present discussion consequence test version suitable setting throughout scalar parameter remainder drop notational multivariate value density present argument interval construct credible coverage formally univariate function define result lebesgue satisfie requirement useful firstly appendix avoid false positive hold coverage prior coverage respect appendix hold particularly abc abc rather see discussion avoid choice lie subset preserve hold coverage long convenient much strong hold note abc test distribution coverage respect iff require value marginal inference derivative mass approximate intuitive coverage face credible discrete simulate posterior credible credible like investigate coverage sense technical difficulty produce credible interval avoid require probability lie coverage property similar argument parameter coverage coverage mass mean natural definition weak eq almost property repeatedly whether choice however simulate abc expensive common make difference follow describe test diagnostic assessment rather notational simplicity make part integer yy adjustment record diagnostic approximation size pm cm test coverage tradeoff great concentrate risk prior increase final increase preliminary finding scalar occurrence record estimate indicator estimate probability post directly remove abc estimate adjust gm treat mild induce uniformity diagnostic calculation value two tail unchanged symmetry cause practice leave receive diagnostic alternative test tail sample rough diagnostic guide consume binary value produce coverage hypothesis diagnostic proportion eq central limit hypothesis poor value improve value seed value drawback highly unlikely log tail regardless coverage reject use log discrete random insensitive nature enough sequence dataset hard generally diagnostic purpose diagnostic motivate well specific diagnostic uniformity histogram plot base spaced partition binomial rate uniform diagnostic coverage credible illustrate appear describe statistic coverage independently equally split prior summary abc represent estimate triple half model diagnostic value panel right panel inference set known panel sample close coverage truncate statistic discuss also early coverage demand require coverage hold truncate occur value draw repeat thereby remove albeit expense qualitatively see suggest diagnostic coverage draw truncate panel parameter panel agreement coverage roughly value confirm shape panel centre panel illustrate truncate prior former detail analysis genetic choose model nan bottleneck simulation accept regression whether regression adjust post processing post support plot apart perhaps confirm deviation coverage panel diagnostic plot panel regression inference post processing greatly statistic investigate coverage small figure diagnostic plot panel approximately achieve except concern remain g post whether abc base assessment coverage plot human use post draw several previous diagnostic employ test r evidence diagnostic statistic fully diagnostic extend property methodology cover model aim whether good approximation coverage impact addition consist perform investigation statistic informative correct additionally parameter margin joint within abc package incorporate directly available http www ac appendix marginal distribution equal converse hold hold hold respect statement ai pi marginal coverage immediate I prove hold zero lie interval false side q lebesgue assumption
medium high sc opt regime sub high regime two regime opt opt opt also regime close fundamental characterization one htb sc agreement conclusion furthermore happen infeasible sc similarly restrict word opt show htb x see figure relate assume r opt refer norm scenario assume possibility wide vector figure option possibly choose different direction regime value addition choice precise vary opt opt opt opt completeness middle figure htb present relate present suggest could actually happen certain course suggest favorable certain application priori adapt beyond hard g similarly restrict namely result side prediction subsection b setup subsection choose possibility side figure htb agreement prediction couple numerical feasibility correspond subsection restrict regime early show side parameter depend consider first sc parallel run exact run hand early htb b change run mention theoretical break feasibility point instance majority show average instance part feasible instance change change satisfy run fraction refer range theoretical prediction breaking htb sc determine solution programming characterization refer precise error provide appear conclusion framework mechanism develop handle noisy example quantify nonzero component low handle precision paper school university mail system subject linear system system level sparsity cone guarantee noise alternative framework use work use precisely performance different obtain framework relate consider type recovery solid one get simulation compressed sense recent interest study linear solution application range image pixel camera design decode channel wireless communication streaming micro array g therein study seem substantial practical area aspect put emphasis amount rest vector writing assume regime regime proportional g htb linear throughout exhaustive much complexity I portion design see see polynomial recovery polynomial polynomial since bp fairly present paper slight modification adaptation possible moreover optimization suggest find sparse norm mention instrumental generating sparse establish sparse sense available type somewhat happen noise course special hard impossible top although throughout majority heuristic generalization algorithm scenario availability freedom design scenario find control scenario one noiseless pursuit maintain proportional norm benchmark currently way generalization omp algorithm paper sparse one assume either fairly highly bound known quantity cone programming analogue generalization one utilize say usual exponentially decay almost find vast reference briefly mention influential topic show one hold noiseless course language state sparsity recover determine also establish subspace establish course practically characterization bad paper analysis apply recovery mention generalization g recent establish nice could selector advantage one characterization well unknown program slow selector program algorithm much interesting important answering certainly scope briefly organization utilize relate show relate specialized sign basic performance show proceed major assumption clearly utilize majority present determined system I random bit approximately recover sparse nonzero due clearly irrelevant location sign component zero take point namely know solve proceed definition useful present done change similar convention presentation keep majority argument let paper heavily rely general characterization normal positive let follow solution arbitrarily constant fundamental similarly k k h sort tie sort break arbitrarily rewrite optimization way restrict amplitude problem instance long magnitude nonzero priori problem behave use instance rewrite analogously q w assume exposition skip difference proceed follow thought obviously know rewrite writing respect obtain give equation determine equation scenario equality term hand equation simplify recognize z combine combine one conceptually determine unknown appear mention rely thing resolve deal need standard normal easily constant show q also thing besides arbitrarily respectively essentially need inequality necessary consequently course setup instead systematic magnitude sc h follow three combination section presentation part portion look regime medium regime regime opt regime two base result regime r opt opt opt hold high pair sc agreement another make g sc determine point present restrict regime r large relate namely pair one opt hold choice offer assume leave sense favorable hand one choose norm present show similar get theorem choice choose vary opt opt completeness middle correspond b large happen reason similar present skip exercise certainly generic favorable performance adapt similarly theoretical attention htb large trivially skip exercise massive agreement predict present numerical similarly subsection several demonstrate scaling result exposition attention medium set figure course scenario sc difference run theoretical prediction sc conduct experiment regime low change consequently choose satisfy time prediction solid agreement obtain present course run result h sc r solid obtain experiment one consider scenario run show course htb also choose part everything else except vary part usual parallel run numerical htb b sc leave obtain happen regime could fix choose large choose numerical theoretical give behavior setup set possibility namely hand side obtain subsection r sc possibility namely show theoretical sc leave agreement develop specialized well standard random variable definition try sign easy sign element choose simplicity exposition differently solve one well unless otherwise assume feasibility well feasibility positivity replace visual present keep norm procedure previous possible skip presentation presentation definition turn helpful first section clearly write convention omit way provide various heavily characterization characterization sign let vector standard normal respectively vector one consider define let large feasible arbitrarily constant present pair lie sign facilitate exposition eq n sorted decrease possible tie sort course break arbitrarily restrict specific unknown amplitude component sign emphasize magnitude nonzero priori add would behave let analogously determine compute thought early point rewrite write follow somewhat similarly look derivative one algebraic give analogue combine eq early follow appear side far simplify recognize way combination enough easily remain appear substantially thing resolve concentrate also early follow establish thing need able besides inequality arbitrarily independent theorem way theorem sc sc w subsection present collection take look back discuss feasibility bit choice necessarily choice brief discussion part first couple infeasible critical sign q objective probability lie sign one look clearly unbounded must square
guarantee optimum priori characterization see relax constraint original cover constraint briefly admit equal represent satisfie priori practice suitably large impose prevent right good theory practice matrix interest relaxation rank utilize reveal cluster advantage norm carry become ideally find maximal happen structure cover underlie clear likely cover factorize indicate negative nmf cover assignment cover assignment node belong practical require evolution mapping still method overlap need version snapshot available cover solve snapshot rigorously solution expect practice online reasonably gradient descent complexity detail snapshot analysis matrix underlie persistent underlie long highlight benefit likely snapshot individually snapshot snapshot size capable detect predict specific temporal spatially literature present multi snapshot snapshot plant cluster remark represent node edge independent impose snapshot plant ij ij pn characterize snapshot plant proof space snapshot plant partition change overlap conjecture remove detect cluster experimental method evolution snapshot connect sharing community otherwise community belong scenario share hard generate vary validate demonstrate efficacy slice modularity four real international mit reality include international mit mining technical thing sdp synthetic consider detect generate community overlap community include node respectively theorem snapshot unable randomness network temporal community overlap overlap individually small temporal overlap experiment community explain show case recover completely allow clearly lose overlap degradation also four compute ground distance find overlap none representative cite community modularity inter slice strength since truth cluster change force small fail recover non overlap community recovery column post community find display trade identify american west observe community formation west year ex block structure american us interact west block significant year mainly associate finally network analyze internet topology obtained start belong large snapshot significant structure assign node clarity upper portion persistent right seem significant formation similar phenomenon look overlap consistently appear internet form device carry human influence social social human contact device contact graph possess underlie social aware strategy centrality decision utilize contact infer relation mobile however detection use limited community scheme limitation gain group friend co require connection protocol access delay overhead make community principle build content protocol much initial mobile network etc trace life relationship work social pattern construct refined feature existence persistent temporal detecting overlap theoretical capable structure detect utilize temporal focused unweighted graph study evolution complex network acknowledgement medium suitable semidefinite factorize upper snapshot always desirable lagrangian formulation multiplier equivalent solve sub operator euclidean onto ball guarantee geometrically step store adjacency factorization snapshot suggest dependence time compute take snapshot consider product sub operator take summation take thus complexity characterize rigorously simulation study treat summary least read write recall denote product equality hold prove prove uniqueness contradiction unique optimal contradict suffice optimal showing value matrix serve dual whose singular whose zero variance setup choice singular matrix guarantee subgradient w f discussion tw ij sum independent previously converge union piece complete university edu com present principled overlap temporal dynamic community snapshot network temporal constraint smoothness constraint relaxation result reveal community discover overlap temporal enhance complex network underlie stability group network attempt identify fundamental primitive network social network epidemic serve important understanding underlie often structure computer protein interaction content name concept community belong reveal multiple study comprehensive primarily communication network wireless social network community vary narrow gap provide efficient detect vary overlap elaborate aim grow useful network interest evolve could apply independently snapshot persistent noise static principled temporal incorporate community time detect subtle persistent community various notion world correctly community know evolve observe rapidly evolve much contact social day people daily activity family people persistent evolution network could design storage design wireless another devise real life protocol elaborate application describe key idea temporal community independently snapshot limitation well minor detect limitation past argue detect explicit smoothness constraint past partition small persistent propose detect convex structure maximize quality snapshot subject handle community generalization smoothness densely persistent formulation fairly smoothness metric naturally hoc unlike exist approach greedy heuristic result problem optimize partition cover allow tight relaxation trace result optimization technique enable recover highlight utilize information small community relation particular detect summarize detect critical piece quantify community snapshot ensure past overlap provide convex relaxation efficient work rely greedy heuristic provide theoretical guarantee synthetic efficacy discuss communication network detection approach present formulation modularity function static community matrix recovery typically trace surrogate relaxation cover static without deal survey refer rest focus temporal maximize constraint modify snapshot start propose framework evolutionary aim optimize combination snapshot temporal cost snapshot quality negative matrix cluster optimum formulation framework able use reasonable method typically modify predefine maintain modification cluster much longitudinal community detection update create community base overlap allow idea remove work heuristic hoc provide exist include function essentially snapshot quality community detect overlap detect time detection density high change rapidly might associate network mathematically let course know cover outlier subset convenience sequel make concrete solution cluster edge undesirable solution provide little produce another node reveal enforce overlap remainder precise development follow cover c cover unique matrix representation cluster assignment assignment cluster belong clearly correspond
control suffice function allow unknown control exploit employ regularization aim perform remain tractable penalization weighted approach post article example appendix feature method diag penalty loading function probit penalty loading good method implementation consideration need fact potentially regression post use device specifically regressor zero estimate post estimator establish estimator maintain theoretic variable proceed similarly estimator lasso estimator define near good theoretic form consider modeling part specifically expectation give function strategy exist nonzero approximation also grow series long proceed analogously outline estimator z define analogously link l orthogonal moment moment tie efficiency locally minimax parametric robustness post need use function perform reduce structural function efficient pp efficient influence moment trivially construct via name return produce stack estimator establish principal rich appendix approximately distribute usage uniformity rich generating theoretically stack give form asymptotically bridge allow include perfect apply possible mistake multiplier via copy distribute impose include correspond multipli w bootstrap satisfy plug relate computationally hold influence amount bootstrap asymptotically conditionally bootstrap structural parameter structural carry example index include plug rule estimator bootstrap delta uniformity delta via plug bootstrap consistently estimate simultaneous confidence band functional hypothesis speed positive vary term random space probability observe copy u space field depend nu obeys supremum totally metric collection map supremum finitely iii away namely trivial impact imply q denote brownian bound path uniformly assumption approximate strategy uniformly belong hold n gram form fx iii boundedness hold impose intermediate approximate well behavior norm primitive primitive address sparse extend generalize use boundedness make could remove cost strategy link belong p r p n fx cs induce gram form fx hold z obeys relations recall convergence appendix linearization namely asymptotically path provide large assumption bootstrap consistently next functional show multipli consistent rely derivation modify handle uniformity structural functional space map hadamard structural properly normalize also conditionally theorem mean tight moreover high nuisance set rich moment datum nuisance approximation standardize parameter validity validity method delta smooth functional hadamard function value parameter true moment condition borel borel map map dimensional assume nuisance approximately model modern selection obey analog I regular orthogonality simplest form additional continuity z u u structural general suppose convex say obey orthogonality respect condition derivative dominate q obey orthogonality orthogonality reduce general moment orthogonality property moment identify identify orthogonality project original onto nuisance slightly general primitive hold consider measurable convex obeys derivative vanishe definition dominate constant condition space law consist suitably transformation hold I obey p twice orthogonality set assumption iv hold p problem suitably smoothness n condition suitably entropy w w various impose nuisance condition function hz u un suitably entropy obey measurable complexity growth nuisance estimator framework index select framework behave thus framework sparsity allow crucial modern condition various reference deal nuisance obeys mention moment new deal dimensional nuisance one value class develop growth bias addition obtain multiplier suitable bootstrap drawing show multipli provide valid approximation law obey equation eq validity multipli bootstrap functional delta hadamard differentiable properly bootstrap conditionally consistent p moreover usual hadamard uniform notion theorem hadamard validity delta theorem independent lasso beyond functional transformation initial differ rest generic generic covariate facilitate logistic link response though link well regression though focus principle establish penalize post uniformly observation hold penalty estimator case binary response datum define possibly additional obeys growth go design discussion additional singleton error singleton strategy bc analog post singleton quantile bc level associate singleton level choice implement theoretically addition loading diag loading accord form define constant iteration initialize set xy j compute base diag l k xy fx k kk condition establish sequence consist generate hold equip sparsity cover p moment jx function boundedness p sp n r j jx sparse fx variety estimator link uniformly response link loading large post obeys bound exactly condition condition logistic take law determine copy suitably measurable logistic equip semi uniformly hold index growth p u dictionary follow regularity jx p jx n minimum fx characterize link rate response logistic loading algorithm estimator sparse snp link establish rate estimator conditional practical method estimation accumulate implementation interpret inference control therefore robust post agree standard moderately control well behave compare method selection comparable size break well behave confidence line validity method key accumulate heterogeneity couple generally preference high preference likely tend otherwise high accumulate unobserved conventional heterogeneity tend effect k argue plan take people decision would causal estimate focus treat k argue estimate accumulate cite dimensional adopt number term control related method develop resolution broad set relatively approach chosen consist observation net indicator plan raw size year education status benefit status home present five different set status status benefit status status home seven indicator specification definition k specification specification way indicator specification indicator specification indicator status benefit status home status fourth polynomial family size education orthogonal specification specification orthogonal polynomial set specification polynomial specification form control specification form interaction variable main interaction non polynomial dimension thus interaction polynomial interaction specification interaction specification specification interaction specification way interaction specification estimation effect stage reduce detailed section outcome post reduce outcome post report interaction plus specification observation report specification estimate construct loading detailed singleton detail variable report analytic multipli standard bootstrap bootstrap multipliers standard versus suggest effect polynomial specification relative everything low observe orthogonal orthogonal interaction specification polynomial compute reliably plus due empirical boundary favor produce polynomial plus orthogonal polynomial interaction specification nonlinearity specification concern nonlinearity specification error specification sensible predictive use specification add nonlinearity specification error seem reduce result overfitte indicator specification polynomial specification orthogonal polynomial leave give display figure display dependent result selection estimate net financial report interval look similar stable across specification estimate baseline polynomial specification flexible specification include make behave apparent meaningful draw overfitte variable roughly dimensional reduced find accumulate net quantile quantile look statistically reject effect hypothesis treatment treatment though band interestingly evidence impact low intermediate k couple evidence uniformly effect quantile substitution interesting richer rich procedure selection take convenient think shall multipli probability live need live track increase throughout paper mostly subscript sometimes subscript denote generic capital letter expectation respect index measurable shall often omit simply use fw v tw measurable equipped sigma p satisfied denote index dependency sometimes keep depend shall vary stochastic relation specifically process element uniformly bound constant shall uniformly pa np k pp n equivalent notion hold na p n p claim straightforwardly definition proof equivalence extensively proof sequence element take suitably equip measurable bound semi metric cover finitely shall denote uniformly stochastically uniformly every limit consider precede subsection multiplier whose empirical define assign z pf theorem theorem convergence take place namely follow conditional place namely hold datum follow notion subset space call derivative eq q convergent every map uniform small allow endow much difference dimensional inverse sense suitable obeys quantile differentiable uniformly require distinction explicitly impose example outside process parameter n pp nz convergence every z moreover stochastic delta method condition multipli bootstrap process pz nz pd nd nz pd nz pp denote law hold q bootstrap indicate previously r pr n part consequence split pointwise uniformly total pointwise gaussian apply subsequence follow process application borel inequality sum md mm sufficiently follow claim verify pz share continuity claim immediate fact measurable mapping pf share n pf also map envelope discrete qualitatively multiplication since note z hz hz hz p b h first assertion assertion b n n h theorem conclude iii dp rely lemma convergence uniform multipli central theorem put less convenient application multipli limit uniformly random index law expectation hold bl hx bl rely write purpose sequence function split sequence function respect subsequence law determine enter subscript distribution n z nz df z along extend mapping argument subsequence state form claim paragraph proof random element measurable totally process empirical measurable envelope allow dependent subsequence pointwise process covariance merely sequence consist prove depend verify provide claim nh n extend mapping theorem nh n nh extend mapping completely establish relative satisfie subsequence z z r nz w define continuous extend banach extension claim r subtracting pp nr fact h n p conclusion z n element law suitably measurable equipped envelope maximal setup measurable kt vc subgraph entropy vc class cover obey class map measurable mapping eq jx f eq long depend measurable conditional f fw fw finitely generalize probability supremum finitely probability f fw problem supremum finitely w q measure vary strategy strategy np argument asymptotic presentation might convenient reader place x solution g estimator variable v stack u linearization step u trivially z preliminary via derivative vx x v vx v z z h f z vx probability c n p z fx p z norm similarly assumption g evaluation compute norm condition iii z iii note iii z vx vx vx iii expectation orthogonality moment moreover uniformly property note b real envelope ii lemma cover trivially envelope monotone set vc subgraph vc bound note preserve eq similarly vc subgraph bound transformation preserve vc bounded therefore envelope calculation exploiting boundedness note I collect claim namely q n bound uniformly eq verify shall invoke suffice ff condition transform property iii denominator boundedness iii iii covering bound multiplicative cover uniformly union entropy say uniformly entropy hold second constant must equal u sequence inequality hold zero u u deduce conditional expectation argument p denote put point draw u estimator influence linearization linearization arise completely zero component linearization p establish z v z pn calculation obeys lemma multiplication namely envelope p n calculation z b second imply n theorem order suffice measure induce preliminary definition imply via probability exceed leave bound conclude u n p un w hz hz un hz expansion triangle denote connect vector denote th element un hz un last maximal class envelope n f n assumption linearization taylor inequality u ii bound follow step furthermore pp k old equip envelope constant envelope follow since element condition entropy envelope f u step define ii j u u z ii h z z iid old application lemma envelope n u z un c u monotonicity norm conclude paragraph w w assumption throughout proof index put mass n n linearization un w argument proof state obeys envelope change namely eq calculation u j un un iterate expectation continuous monotonicity triangle n b second n consequence order induced shall dependency invoke lemma lemma specific state appendix imply f ideal lk hold condition cover pn np iv proceed occurrence loading p yield turn c f jx jx l eigenvalue cn n cn lemma thus post estimator penalty loading jx nk sp loading th also state rate sparsity establish result result dependency sample similar imply I require occur e r x lk rescale n verify imply condition algorithm c np iv occurrence loading jx imply jx assumption probability jx f jx n bound zero choice q lemma since penalty jx loading imply primitive assumption random variable note cover throughout u suitably identically vary f jx pn jx f u u jx u fx following theory diagonal loading element j constant pn depending contain process lemma verify sequence nc p jx nc u vc l nn l fix going case non transformation follow denote cardinality let study property relation rely minimum sparse present technical generate proof lasso u lc property sparsity consider estimator lc c post post u finite associate estimator follow x nu outcome restrict coefficient restrictive provided differ counterpart analysis penalize relevant relation many primitive imply bound relevant refer bound convenience rescale fx r r fx I fx fx r derive sample rate let lc eq q nonlinear coefficient q eq result zero assume u lc c u fx support select penalize logistic provide provide u triangle z nz bind jx jx pn lemma control term away q last since u imply pn pn tail calculation ff nc apply envelope cover number envelope q bound lemma envelope envelope second jensen hold jx throughout u u u otherwise u bc integer kk apply step lasso estimator note u u c l follow note un pf projection nan empty identity lemma approximation well provide include old inequality relation bind system provide second derive similar structure maximum bc integer k kk follow let u fx relation uniformly relation fx fx proof fx u inequality calculation old fx fx u u imply inequality build idea bc quantile statement trivial fx q tf follow fx fx conclusion trivial fx n n consider satisfy ss st eq note verification pt comment section example modern rich many observation type nuisance lead treatment handle many inference relationship approximately outcome treatment variable identity unknown permit inference drive control post across wide validity selection allow model moment illustrate accumulated range process include selection theoretically orthogonal achieve multipli approximation uniformly delta smooth parameter establish validity multiplier uniformly estimator provide justified parameter analyse economic causal program outcome complicate economic policy randomly true approach estimate observational instrumental iv treatment assign service right control treatment therein economic researcher condition randomly must plausible typically economic suggest important enter identify effect situation treatment instrumental variable
sub sub element image represent line test possible learn x bf compute region line alg learn sub f k learn learn simulate behavior policy region many sample complexity depend classical slow model dataset evaluate database independently image split run baseline acquire whole art scene matching performance advance feature mid feature embed purpose validate obtain dataset figure measure learn exploration visit testing correspond acquire certainly relevant beginning tend start explore acquire attention show average half acquire frequently region carry hold spatial balanced visual test share similarity latent svm visit show visit right illustrate method adapt choice classify cc introduce strategy combine exploration subset image region location content one produce dataset strategy significant rgb adaptively spatial process region infer choice content capacity image allow image image highlight exploration mind representation bag level take recent human need image interpret human rapidly interpret manner sufficient classify simply select region certain detail resolution use less resource vary complexity orient goal forward importantly instance region process leverage wise classification policy rl speed region preserve acceptable rest highlight finally report challenging follow descriptor pooling computer vision bag formalism sift alternative code coding traditional pooling pool pyramid pool pyramid detector salient area region contrast target object method feature dense approach study human salient object recently approach dense powerful propose pyramid multiclass one binary method latent jointly image classification et al model region topic representation grid classify contain avoid whole image focus region computation reduce aspect short sequential sequentially choose visit previously locality manner represent visit region inspire reinforcement dedicated learn region simulation quite technique classification sparse sequential able consumption selection share idea classify use summarize contribution present model select region advantage method algorithm able irrelevant select classified multiclass region globally class exist technique experimental evaluation propose qualitative study discrete category exactly predict label region leave sift word representation experiment classifier sequentially visit select thus tailor specific p give fix budget region acquire trajectory aspect acquire acquire region able representation image decision acquire without region result describe classification give previously function aggregate visit length space x x tr eq concern image classification classify classification perform classification predict use sequentially exploration denote sub time sequence previously acquire multiclass classifier predict region region parametrize current acquire output index policy multiclass perceptron multiclass network could classifier classification sequentially new computing previously acquire region line policy central region category start begin sequentially begin able obtain aim e algorithm learn training uniform state policy technique supervision classification reinforcement learn adaptation
code enjoy simplicity neuron belief decode spatially lead inferior already allow suggest neuron term fire network like perform neuron update neighbor follow compute sum denote possibly pattern natural fire neuron noisy query pattern piece pattern result pattern separately parallel objective highly correlated minor specifically formalize similar code refer plane divide lie connect except boundary entry regard correspond cover neighboring field visual assume consider integer value resp resp eventually integer noise entry pattern able pattern set orthogonal pattern cluster plane correspond correspond correspond neural bipartite panel correspond constraint plane figure contain constraint neuron neuron neuron whereas capability keep architecture brain overall plane unweighted plane main retrieve noisy query fix look eliminate noise couple sake completeness briefly suggest subsequently method compose separate intra try within message rely constraint neuron meet sum constraint neuron receive feedback neighboring neuron voting basis summarize overall fairly could input pattern correct pattern easily get drawback et procedure round schedule similar cluster boost correction capability cluster together mention early need modify act cluster round move next plane whole repeat correct threshold reach summarize capable message neuron eq let plane e polynomial notation give plane place super constraint z get simplify admit couple approach evolution error plane l lemma definition h df scalar system average successful decoding achieve successful correction correction constrain system experimental result confirm system couple correction potential ep decrease induction result couple could theorem successful correction however require require depend accurate I coupling show scheme pattern length construct simplify notation number word construct plain pattern subspace generator compose block visualization building realization integer assign obviously denote let choose entry form number mainly spatial upon weighted presentation produce generate assign connection matrix fact orthogonal pattern l phase recall zero pattern noisy precisely within local plane rectangular window pattern eliminate assume constrain system neuron corner similar clustering divide overlap lie dot inferior couple side though difference architecture performance dot dash curve threshold suggest illustrate function behave minimum negative neural comprise framework analyze derive two threshold correct match threshold give main interest paper phase assess world setup thank mr dr discussion draw draw conjecture thm thm thm thm style mirror amplitude draw style draw thick design divide neuron similar architecture brain spatially code enable performance drastically exponentially work fail pattern retrieval storage present large truly rely sometimes brain capable memory high even limited inaccurate designing past core transmission communication goal pattern novel employ code extremely analyze model role gap code technique capable reliably achieve introduce redundancy among message later noisy contrast artificial capable back hold require pattern scale long namely belong subspace modular suitable modular couple interestingly look visual make recent development analysis spatially couple code analytical achieve previous arguably influential neuron binary patterns retrieval increase retrieval capacity introduce offline scheme binary result divide block improvement come price tolerance capability
another important system allow inner train although mistake much fully high correlation continue previously category label label per label wrong one expect accuracy system set much system bad return path leaf predict base still leaf wrong unlike measure nice set observe less rank operation pair set characteristic respective category behavior desirable c acc x acc acc h l computational scale reality contain circle remove run measure forced low depth seem similar problem avoid dataset complex differently interestingly ranking remain compare measure safe system table predict predict category natural system performance rank correlate observation completely per h c acc e h c c acc acc table result purpose remain affected compare table general path c acc c acc acc table main classify inner prediction per accuracy bad fp mention multi label behave differently predict small per document rank one system path similar omit acc acc acc c present system system present flat measure dependency treat give ranking hierarchical give absolute rank system issue study problem work key measure propose grouping measure order difference augment base measure generic salient common contribution proposal exist measure measure along one assess wikipedia characteristic behave differently especially multi counterpart support flat adequate evaluate categorization show rare pair hybrid hierarchical combine national center economic business france gr fr gr hierarchical address item hierarchical hierarchical among measure hierarchical analyze component propose alternative view novel one test large undesirable exist address classify past year emphasis relation class gradually particular partly service hierarchical yahoo equip hundred relate class scale hierarchical improve remain open evaluate classification complicate among error document severe evaluation hierarchical hc widely hc comparative hc publish early type hc focus single task object assign belong hierarchy hc focus evaluation one interesting insight evaluate exist evaluation within common address analyze hc specifically exist hc type provide generic overview exist hc hc art comparative empirical hc variety remainder organize present requirement hc measure framework exist hc use exist datum present finally summarize remain open present new hc measure characterize firstly requirement evaluation presentation framework denote style blue font yshift yshift yshift edge node node right auto draw font xshift yshift xshift mm yshift leave right left edge edge edge node edge edge class organize either case parent acyclic dags multiple cyclic hierarchy impose child belong usually relationship among relationship cycle cycle class parent classify hierarchy hc particular hierarchy classification predict class equally severe prediction measure issue present case node circle predict sub group take calculation error problem calculate third predict path distance use reasonable predict class alternative path multiple path thick minimum yshift yshift node right auto node cm thick node circle font minimum yshift edge auto distance style fill blue yshift leave edge style blue font leave distance thick style font left label must class node could score minimize could argue finally predict class match class true distant assign pair could pair class predict omit augment default pair vice versa distance predict exceed predefine threshold pair additionally predict n contain default calculate short connect hierarchy similar therefore severe elaborate measure assign weight hierarchy move class spirit pair return way formulate ij j ij states alignment class pair pair align solely collect class counterpart allow pair pair exactly one limit label class yield reason default class well bipartite respectively predict look match pair opt relate particular flow lower denote network flow edge leave associate total flow capacity flow flow problem constraint latter corresponding explain integer quantity flow value exist feasible flow cost arc flow guarantee bipartite represent source flow see source class default predict predict include default true class exist default require constraint capacity interval possible number network indicate calculation put differently flow flow show pair predict class affect constraint interval capacity flow true pair pair non predict predict align capacity mean align predict class interval predict capacity predict resp capacity interval lastly correspond compatible capacity impose ht bend auto xshift mm dt xshift label xshift anchor edge node pos pos anchor fill pos anchor base fill pos white pos pos base white anchor base fill pos fill node anchor fill pos anchor fill white pos anchor east east pt bend pos anchor fill xshift bend south majority pair measure deal single dag phenomenon simplest equivalently correspond ht bend pt distance cm blue font dp dt label source anchor white pos node base fill white pos fill white base fill white anchor fill edge anchor white bend center depict edge path hierarchy propose true calculation I pair predict label e distance induce text predict dag pair one multiple predict pair true class pair exactly default predict pair dag predict class take path class true positive figure present flow angle auto style font right white anchor edge node base node anchor base fill anchor fill white anchor white white anchor base anchor base fill white anchor base base edge anchor fill white edge white pos white edge anchor fill node anchor fill white edge anchor fill north bend pos base fill white bend north center matching fails predict one pair penalize pair distant bend auto font minimum xshift dp source anchor fill anchor fill anchor fill anchor edge anchor fill white anchor white node anchor white base white anchor fill anchor base fill fill anchor base fill edge anchor edge white edge white anchor base fill anchor base fill white north xshift bend north straightforward call label suitable present optimization easy class pair default category default set near predict next measure two case prefer take tp fp ignore prefer predict single fp gain find real undesirable two create subgraph two connect vice remove breaking graph h auto style blue font scale leave pt auto circle draw font right leave minimal dag connect set fluctuation remove graph ty py py ty py ty g py ty py ty ty ty py b py g ty paths py p ty p py maximization ii one subgraph limit use satisfy iv existence connect subgraph constraint subgraph subgraph way problem expensive present py ty py g ty ex py ty py constraint top redundancy removal bottom removal share procedure decompose return need ii connect pass remove already connect subgraphs arise predict belong extra address recall purely base actually bridge pair measure set predict node measure combine type section case order pair choose version precision base illustrate advantage limitation provide order implement tool source require situation present specific situation appear capture elementary case figure variant different symmetric recall version differ take give behavior undesirable measure affect version difference account negative match predict maximum ignore correct provide allow multiple ht yshift leave edge node node yshift edge node c undesirable predict misclassifie mistake figure bad figure class measure give hybrid use augment create take predict true ignore ht right e tp edge leave leave tp node edge node leave ht c thus augment tp measure tp tp augment become tp tp measure remain advantage path worth note simple case remain due shortest affect particular path behave low advantage measure right edge leave right study combination present affected method give path phenomenon affect hierarchical node low common share true show would h thick style fill font yshift edge node edge right edge show differ since connect leave connect two reason hierarchical version differ table version near node count auto thick main circle fill blue font minimum scale leave c predict often illustrate case edge length result comparison one increase reasonable change right edge right edge node node leave leave left edge edge right leave edge c c similar accord decrease pair method double counting severe extra penalization base severe common advantage handle true predict certain threshold assign threshold augment impose order least example measure reasonable reach hierarchy lead artificial order connect measure run distance discuss pair affect one decrease multiple counting undesirable discuss edge edge edge leave edge edge node node edge c max b predict true category leave present either predict category receive node leave edge edge c evaluation type one argue misclassification simple example figure severe category lead pair handle since produce measure handle modification propose count feature although time undesirable could behave base general behave least measure category set suggest instead discuss undesirable additionally serve benchmark newly alternative path long multiple count apply system hierarchical challenge system ranking affect first subsection final subsection behave among system provide page project human instance five hierarchy hierarchy small regard
sd tolerance query obtain guarantee active learning ask query sample query obtain learner obtaining relax requirement access sample framework include filter query tolerance tolerance tolerance passive filter active see q passive statistical tolerance need therefore algorithm operate statistical answer immediately transform simulate draw randomly bernoulli ask label ask label accord formally follow theorem algorithm give tolerance sample chernoff hoeffding bound multiplicative chernoff label give estimate tolerance dependence claim claim dependence standard technique least dependence claim case sample obtain multiplicative chernoff hoeffding direct unlabeled however complexity simulation share simulate query filter simulating scale query adaptively belong complexity reduce answer filter noise label show example noise sampling response active tolerance label decompose two tolerance sufficient affect therefore use independence label chance label obtain dependence algorithm simulate simulation easy place tolerance estimate p approximate address hypothesis suitably hard suffice strategy example active unclear deal substantially specific variant uncorrelate give example label intuitively uncorrelated query almost target distribution function say expectation coin query noise valid active query use sufficient x uncorrelated immediate theorem corrupt uncorrelated query clearly uncorrelated noise randomly choose distribution point expect enough logarithm size target concentration probability note appear view threshold interval express use active assume interval ask function tolerance response query reach tolerance tolerance unlabele threshold axis whose target namely case scaling target interval interval consider interval fully include query condition tolerance guarantee answer least query must interval search interval tolerance align active filter framework disagreement active classifier point region disagreement formally disagreement algorithm consideration statistically confident last essence round need error still disagreement number simulate query computation disagreement hypothesis constant hard present passive homogeneous proceed round round approximations correspond filter build current denote hyperplane orthogonal vx isotropic origin concave gaussian uniform density isotropic concave c h vx prove constant exist concave active exist learn distribution isotropic distribution homogeneous use easily appear isotropic log concave accuracy active tolerance learn homogeneous concave density constant indicator margin use query tolerance unit vx w w note active execute ask tolerance valid response query induction addition case true inductive hypothesis know arbitrary inductive obtain also lemma k combine establish inductive hypothesis immediately finish establish bind run query isotropic position general concave density passes generally learn remark obtain active require learn unlabeled rely costly general learn costly uniform unit sphere study remark unit log concave isotropic log algorithm uniform sphere careful particular uniform distribution isotropic dimension unless specify otherwise section expectation like mention function theorem give estimate since start outline non demonstrate idea well present simple efficient denote vector vector disagreement proxy see behave disagreement vx angle vx h vx v distance perturb basis unit v u w w claim approximate algorithm exist query statistical tolerance q imply w variant useful warm version active base measure perturbation current hypothesis direction combine measurement learn current example rate constant hypothesis whose denote area observe vx v w appendix relate distance easily tolerance unit filter tolerance tolerance together mean imply sufficient note monotone use binary value x vx exist learn tolerance run parameter vector distance normal iterative v tv tv w use claim step tolerance query lemma bind claim time immediate corollary theorem presence noise classification noise unlabele overcome procedure approximately find simulation noise target agreement rate agreement rate noiseless access value run expect randomly spherical known fact randomly imply u vx integral estimate draw random estimate vx denote hoeffding tolerance estimate least know every stop prove true since hence tolerance estimate I procedure require run lemma polynomial unlabele learn differentially learner access record participant every request request like notion valuable medical research goal create predictor person certain medical unlabeled patient discover medical produce reveal patient database modify single always element operate receive label upon request number request differential privacy make privacy entry preserve label al translate differentially private achieve privacy analogously function use tolerance query exist active database da first active satisfy laplace add preserve begin amount desire privacy guarantee affect query query modify change query modification modification answer privacy add quantity label need ensure correction noise specifically property privacy guarantee constant add query hoeffding unlabele label unlabele active total get sufficient sample bind bound complexity differentially private even simulation exactly privacy label unlabeled reason unlabele datum label sense public public address reflect privacy parameter denote sensitivity denote vector require differential privacy database differ privacy point private consider hard use sample independent immediate concave privacy preserve learn homogeneous use passive algorithm ignore consideration require example enough passive counterpart convert differentially passive statistical support claim alternatively aspect passive unconditional extension would useful differentially private thank support nsf grant grant microsoft research algorithm learn al require unlabeled basis obtain active give private need statistical base easy verify perceptron modify version care polynomially give homogeneous place around hyperplane query margin formally every large within tolerance answer tolerance mean combine observation use isotropic eq q substitute vector definition imply ready prove learn concave filter function thm therefore plug claim convenience lem symmetry assume half region satisfy point word point hyperplane pass origin surface integrate region eq conditional probability claim lem eq fact remark property post edu filter build powerful efficient active automatically convert random uncorrelated show commonly include threshold combine random exponential improvement passive counterpart addition show algorithm convert differentially private lead private exponential passive machine assumption human application massive area algorithm available minimize intervention technique present pool unlabeled pool drastically reduce labeling past decade development understand principle advantage classic passive learning currently well understand efficient super vc provably give improvement efficient polynomial restrict example label restricted addition possess number useful tolerance random randomly differentially private access get property target function learn algorithm nothing guarantee inverse correspond label correspond sample query classic model
v depend random finite tf fx remark notation straightforward computation require result estimator moment upper actual level confidence n level slowly quickly attain close level support national nr cm cm pt remark involve input sensitivity aim impact quantity use influence output estimation index encounter apply science involve important impact output sensitivity aim identify sensitivity reference therein belief turn output total variance hoeffding decomposition see induce computation open statistical sense hundred evaluation monte carlo quasi community include pick output model hold variable sampling pick replication carlo general allow sensitivity index random probability group behavior pick direction rank jointly well total index estimator marginal fully characterize rigorously account error second tool investigate allow section close estimator comparison dedicate asymptotic inequality theoretical example whole necessarily connect belong space measurable py index useful widely engineering science context variable simulation confidence region test next express covariance one close expression consider mean view able sum estimator show practically estimation observation consist precise moment second account define property index compares numerically performance delta empirical study q become u vector take invariant center next calculation sake simplicity proof I n ti apply call delta jacobian state define ki procedure similarly let theorem expression may thank k test know fact hypothesis powerful unbiased resp reject quantile random one toy power testing computation take z gaussian variable follow theoretical empirical function call negative naturally context carlo figure function spirit power figure h plot reject variable carlo plot theoretical power power ie test test figure estimate power function accordance variable exact index q fact give confidence c min mass empty basic exclude capacity uncertain speed ascent ratio consumption follow uncertainty htbp variable density parameter uniform ex density parameter take min uncertainty arrival minute plane thus previously reject inequality index dimension I respectively moment resp assume since assume center obviously yy u u u iy u therein n u come I
three dataset criterion however use worth database signal roc specificity suggest standard database develop potentially drug report wide practice report mm mm mm mm group school uk email ac division public health uk frequently report database aim efficiently report database report incorrect lag drug limitation occur report database implement reporting database health suggest database report system database supplementary incorporate provide recognize decade medical database database record database connection drug face incorrect reporting report report propagation neural item specifically association database type medical gp record uk gp contain depth prescribe patient practice database direct drug connection predict look drug take occur drug take drug age structure prevent effective method gp successfully identification incorrect record report database focus implement efficient detection gp exist mining gp investigate implement investigate drug drug fairly similar mining investigation outline explanation gp standard receiver operate summary exist database attain finding report report accord contingency ccc drug event event take drug event drug drug relative patient event analogous drug event drug standard se method natural logarithm calculation occur drug occur drug occur conditional drug interest drug error se calculate suggest deviation occur drug threshold interval great event identify event patient drug define window choose find drug two investigate medical expert predict occur people immediately increase period investigated identify rest event drug association standard receiver operate roc choose method gp rate false scenario event event non method successfully implement past roc plot method rate use describe side employ medical database work event typical example event level great event also see must know list event code relate denote code gp denote code mining technique method let know event code eq complement analogous experiment database database different database paper gp database patient uk practice record medical event prescribe patient patient contain useful date gender present death patient relate visit record patient span event database paper drug drug drug drug outcome drug record drug occur include often record mistake report period record range report number name contain patient age gender event gender age hypothesis segment comparison statistically difference receiver characteristic curve calculate gp database statistically receiver characteristic less database excess amount event false ten event investigate hypothesis test mapping database choose investigate positive true use threshold true false positive calculate roc auc test difference auc roc calculate use drug side possible great database four decade year commonly figure increase dataset method database give great return auc correspond significantly lead gave respectively significantly auc return value reject auc p statistical similarity obtain apply gp high drug figure could reject exist figure apply give method detect correctly contrary event increase overall
without text show activity send binary relational using discover obvious expect trait mostly also effect network centrality mix small less measure behavioral strong peak towards time centrality parameter vary varied week centrality combine value begin company remove edge score true email centrality short richer incorporate dependence layer account multiple jointly describe noisy perform latent variable clustering demonstrate layer circumstance life develop explore infer pareto thank xu suggestion utilize line eq line maximize line segment correspond let parallel exponent q go finally decrease mean close iii edu connectivity information relationship complement behavioral interest multi layer layer typically semantic application combination analysis technique multi flexible develop model mine noisy model pareto network naturally connectivity instance social knowledge link medium whether send user interest behavioral relationship information connect usage deal multiple cluster layer perform bayesian averaging conditionally write layer view consistent average back discuss objective variable pareto front function tune supervised optimization utilize layer combine stochastic capture phenomenon finally illustrate layer connection layer layer comprise vertex edge multi graph convenience depict case binary merely instance see measure content could specifically influence measure dependency describe one relationship user represent intrinsic adjacency compact description network distribution collapse infer decompose adjacency similarity underlie common connectivity model produce correspond view variable posterior etc simplify act conditionally independent likewise conditionally formally condition observed variable factor denominator perform side solution map map prior implicitly assign choosing affect isotropic proof isotropic solution use cluster weighted bayesian isotropic construct equal normal weight community variability corrupt edge specifically corrupt noise second corrupted form mix spectral find graph laplacian rand ari compute comparison time average compute network improve use expected cccc ari course effectively family map rank technique maximization solution pareto objective multi interpret alternative seek rank pareto pareto optimal possible single say dominate pareto front term pareto pareto front solution convexity condition pareto front show pareto front combination optima interior may research interested expect behave determine allow interesting community create intra exhibit strong group sbm expect fall known membership setup binary connectivity observe sbm probability edge occur call matrix form node symmetric letting membership sbm matrix sbm temporal sbm take advantage previous membership evolve smoothly recently introduce account employ kalman filter track membership sbm review noise bernoulli onto real transform invertible transform model kalman filter estimator sbm parameter innovation process jacobian complete map membership membership implement extension walk state identity dynamic email approximately half million email send make publicly sec investigation company constitute large publicly email represents examine addition include class multi layer discuss information behavioral extract recover week dataset separate send email form long send user oppose writing term tf score commonly q document term corpus active document similarity week create second dynamic framework thresholde great correlation create insight structural extended layer company membership know estimate bernoulli represent evolution figure represent behavioral week represent important line
measurement nominal vector tune image due model deviation kk span empirically number nominal residual use axis number vector detect eigenfunction explain portion region explain portion eq determine relative basis blind deconvolution linear image hierarchical white take positivity point side atom distribution dirac delta prior non gx image indicate pixel bernoulli rewrite conditionally upon q give conditional distribution q factorize simplify variational distribute interval interval image distribution conjugate inverse gamma prior impact quality deconvolution include hierarchy paradigm hyperparameter scale parameter fix unless assume iteratively update accordance fidelity reflect prior informative posterior derive exact posterior within include p q kullback leibler kl evaluate distribution approximate approximation maximize iteratively update factorize group subject factorization expression compact expectation notation except variable hide factorize induce fully section distribution approximate require detail iteratively due z z x I qx describe conditional give gx therefore bernoulli nc normal mean l use I qx k iterative require distributional assumption numerical fig respectively select density flexibility surrogate zero quantity output depict steady state fig show deviation deviation level estimate noise curve knowledge image fig reconstruct variance pixel simulate level vb semi blind vb non blind term vb method propose vb semi blind quickly blind algorithm mcmc convergence comparison art blind deconvolution previous nominal sparse assume error algorithm two basis kernel propose use prior suggest image prior exploit sharp natural high vb see algorithm motion produce reconstruction iterative true demonstrate computation reconstruction algorithms semi blind mc blind blind deconvolution semi blind black mc normalize level reconstruct low oracle black performance stage error plot semi sparse make raw voxel fig estimate deviation vb different iterative converging near vb comparable vb significantly blind issue scale ambiguity guarantee prove ambiguity notice nominal space basis approach effectively delta secondly solution initial reasonably trivial resolve ambiguity framework resolve scale divide vice blind deconvolution induce via variational approximation automatically produce necessary conclude vb mcmc require computation non algorithm vb blind benefit ba b function variable statistic cumulative variational framework eq q convolution tr tr exhibit pc uniform wide real line cover variance ignore effect variance add due apply constant l ir orthogonality kernel base weight statistic derive unnormalized normalize notation variational method image reconstruction point solve deconvolution prior reconstruction framework impose atomic importantly propose tuning clearly demonstrate blind deconvolution compare monte version significantly outperforms rely perfect force deconvolution perfectly either optical perfectly exist sensitive circumstance standard suffer deal unknown knowledge refer blind deconvolution deconvolution task problem parameter estimate regularization ill posterior extend work estimator monte trivial variational iterate scalable equivalent uncertainty model deviation priori apply deviation represent linear basis correspond natural induce act regularization logarithm estimate hyperspectral contribution mass continuous function empty pixel gamma mixture challenge strategy mmse estimator draw accomplished markov mcmc numerous imaging recently blind deconvolution two semi blind suggest disadvantage posterior exploit conduct properly design produce analytical mcmc variational avoid stochastic bayes intrinsic limit guarantee though mean maximum variational bayes difficult mixture locally estimator vb model intrinsic limit variational particularly
illustrate recent production band merge early release compact know criterion identification simultaneous paper focus mathematical prior likelihood follow future assign plausibility bold symbol abstract belief statement exclusive shall mutually exclusive give element take reliable seed target include index alternative j jk stand limit data nominal seed jj obviously association complete mutually exclusive compare quantity marginalization np exhaustive pose proper normalization treat exhaustive write understand belief decision classify write probability term term separately alternative explicitly integral combination exclude effective source within probability seed q seed contribute imply jk precision part prior previous classification non match consider contribution non observation goodness prediction odd classification define would candidate index analogously confidence association condition object contrary neither object quality action validation verification scheme would four rating rating select rating would potentially keep generation still need reliability split ex lot bayesian never state generation additional incoherent widely accept describe potential introduce part prior quality rating ensure affect routine potentially interesting drop case fundamental fit emphasize imply plausibility question model datum space p assume ni integrate decrease ni increase ni immediately obtain ni ni ni mass ni shall equivalent know logic flat penalization affect confidence introduction little constraint purpose implicitly assume moderately match set yield ni ni ni ni ni ni iteratively rate update determine group separate range find converge update conclusion ill exchange entirely class make set reliable result picture normally cope anomaly successively anomalie interpretation replace entirely set observe far reader notice belief unchanged application slowly modify science eventually anomalie counter development scheme counter identify whether research paradigm cause anomaly cope road enter
look environmental matching triplet belong local capture condition patch result descriptor part part image descriptor environmental case patch similar match part vary descriptor pair matching descriptor sift descriptor descriptor technique discriminant fisher descriptor learn simple handling apart wide environmental divide give descriptor irrelevant near near irrelevant near define pair control pair near important matching pair e proportion overlap near far near near match distinguished show far intra diagonal meaningful near true intra well undesirable obtain shape boundary lead focus cluster class separability distance local descriptor near far pair separate sift distinguished far pair lie contrast match especially superiority microsoft china microsoft com accord aim image environmental matching condition challenge category original relevant irrelevant vision descriptor descriptor represent characteristic transform sift extract descriptor compare aggregate descriptor close descriptor whose threshold descriptor descriptor close whereas local belong part descriptor apart pair still
ill far regularizer idea important remark would though valid iii obtain instead yield expression unlike valid distance needs handle bound discussion f j ij I j jx highlight though require intensive large still mle however computationally system intractable partition hand statistically understand rate convergence condition theorem rate hold ne provide behavior non application hilbert space proposition involve part nf detail expect assume irrespective turn since impose identifiability mat ern inverse condition impose lie contrast impose indirect assume provide rate least common naturally regularize ill observation improve smoothness issue discuss finite guarantee prove rate l rf hellinger require impose attain condition simple kernel contrast hellinger converge interesting unlike convergence proposition distance consequence distance interesting aspect various distance hellinger kl nice difficult obtain consistency address unbounded require modification discuss modification unbounded therefore handle estimator iv construction knowledge assume solve quadratic program ab ab system rate prove hold vs unbounded hellinger distance difficult practice appropriately unbounded situation theorem weak able adapt bound p due convergence convergence kl slow application kl hellinger satisfactory consistency whether minimax tie smoothness early non assumption see section orthonormal finite clear decay turn imply smooth interpretation space kernel necessary dr follow l mat ern insight let rkh section easy ern rkhs proposition r r derivative infimum proportional explain independence rate provide dimension provide optimality additional completely picture open improve choose appropriately characterization yield capture rate irrespective empirical fisher iv yield heuristic linear finite counterpart ensure inverse pose name note regularization help ill pose pose approximate appear estimator calculus since explain statistical idea square rkhs alternate appropriately provide follow g c gd determine smoothness use solve nice analog new hold n fully capture attain present reader example involve verify verify case specify easy check end consider obtain kernel extend easily idea specify convergence construct show case assume pp f nx see iv quite restrictive family attain exist employ fail attain result introduce consideration function endow define everywhere f fw p follow bilinear make hilbert space completion assumption f addition regard describe adjoint adjoint ks ks ki ks adjoint q depend almost surely integral define construct p ff proof consistency existence density hold hold rt rt result ki theorem situation coincide estimate dense open easy rt hellinger follow note dense f q reproduce h h f dx f thereby trace orthonormal countable class compact monotone convergence dx cf x integration cf result verify minimize iv since define h f ff f reproduce obtain form explicit h derivative h nh q ij ab idea prove h proof complete chebyshev ii proposition therefore iii use step I ix dd obtain f bernstein inequality eq exist constant theorem hx c I f nx dx ix f h proposition nf assumption nc rp cf equality proof theorem consistency n convergence hellinger p f f ii term exactly bind iii e n carry theorem decompose match decomposition verify similarly h bf reduce tucker condition dual feasibility completely form program nx p p q obtain simplify iv distance h follow iii interpolation proposition interpretation briefly space banach continuously topological hausdorff space interpolation space functional interpolation interpolation continuous respect measurable measurable function define convention suitable interpolation space inspection adjoint separable hilbert hold norm schmidt representation unit index b h I bilinear induce verify h I jt count obtain ig eq continuity transform use inequality l fourier transform prove convolution hausdorff inequality observation show f theorem self adjoint schmidt operator separable spectrum constant l l proof deal smoothness self adjoint spectra self mean define exist constant collect follow exactly bind monotone along f r f p equality dx adjoint easy show interesting minimize yield given plug notational density define f n h turn kf fw kf yield hilbert schmidt prove schmidt therefore hilbert compact straightforward slight abuse notation easy kf kf ii ii match counterpart iv word system b kf proceed proof restriction kf kf kf kf wherein term chebyshev kf kf kf h bn kf yield chebyshev first ks ki kf adjoint compact proposition infimum family affine increase function kf nn reduce bf rt follow kf iii define bound continuous however suppose r dd f kernel gaussian mat ern imply pick proof iii iv f p p f since f f simply well linear problem adjoint hilbert bound compact self adjoint schmidt theorem vi since f yield acknowledgement carry department mathematic valuable comment support grant aid scientific research area lemma corollary proposition definition definition infinite reproducing broad density kullback divergence element element technique estimation propose base minimize involve solve fisher smoothness pn propose advantage grow secondary kernel interpolation reproduce regularization score study e call infinite dimensional h hilbert reproduce introduction rkh generalization natural particularly rkh reproduce take statistic detail generate finite rkh generalization furthermore statistic first fr element operator operator nonparametric hypothesis interest estimation density infinite rkh contrast class broad proposition density e kde propositions corollary density lead solve elegant however ill address involve pseudo mle size leibler divergence drawback computational see discussion consistent rate side assumption handle carry therein density approximate expansion polynomial spline mm span integrable though interesting therefore suffer drawback discussion show treat parametric parametrization kde study easy implement poorly paper counter kde pseudo efficient mle minimize kl fisher information open kl de denote convolution diagonal proposition precise strong hellinger advantage density estimation generate distribution belongs give nice asymptotically know yet exactly open differentiable x main advantage independent simply minimize counterpart independent mle like highlight matching scaling require integration kde estimation estimate solving obtain mle infinite extension counterpart however require pose consistency theorem consistency rate interesting aspect estimator divergence kl hellinger total distance formally show nh x rkhs enjoy nh classical fractional mat ern see space mat ern example interesting observe unlike classical regression inverse attribute cover parametric regularizer aforementione address use complicated pn statement even specify obtain approximate family spline result abstract present result kde get kde advantage get notation proof define topological denote locally compact hausdorff say vanish denote function fm rx rx rf df df df e h aa respectively value call pd yx k x kx yx reproduce
simplicity density membership reasonable lower indicate information theoretic hamming hold log likelihood observation summarize I low entropy error iii indistinguishable hold log separate interpretable maintain estimate ccccc dash assignment hold poorly dpp density task velocity galaxy complexity density estimate use proxy assess visually separate log statistically finally classification specie four know separation two class model error place large result hold log three galaxy inverse learner trajectory execute try approximately reproduce set high perturb trajectory cover coverage focus motion angle activity aim reference pose diverse perturb pose build covariance activity center select pose new pose dpp scheme show example pose compute metric base pose dpp frame frame neighbor dpp within compare sampling gaussian despite diverse dpp dpp pose well average right assessment coverage visualization supplement require cover dpp pose dpp issue ccccc diverse pose sample top om dpp neighborhood dpp subset continuous approximated range low nystr om feature method utilize dpp approximate proposal correct gibbs utilize complement demonstrate continuous dpp sampling useful utilize gibbs sampling scheme pose demonstrate approximation computation believe grant bt support nsf program corollary conjecture figure figure table conjecture edu detail upon dpp sampler discrete list case dpp gibbs scheme detail approximation specification gibbs contrast additional additional figure cardinality provide representation denote note sampler loop exactly discrete pr subspace orthogonal n jk k v dpp kernel bb cn vc b identity formula conditioning inclusion write normalize integrate full dpp use random om characteristic nystr om clear choice dpp list mean exhaustive elaborate standard section approximations standard om laplacian nystr om nystr cauchy nystr om nystr om approximate dpp characteristic fourier vice versa approximate let straightforwardly likewise sample coordinate sampling nystr om dpp similarity r coordinate let example om approximate dpp exposition although straightforwardly assume dual gaussian similarity ij previous nystr om total dpp lx gaussians computable eigenvalue index estimate distance dpp absolute scheme gibbs sampler quality space value high low run thin cycle resample nystr visualization nystr om approximate dpp qualitatively location correlated cycle cycle high low nystr om generate htb approximate dpp proxy rate effective size movement chain mix movement correlate effective lag autocorrelation lag across effective expect show om see gibbs value benchmark lower slow high gibbs nystr nystr om low htb consider gaussian normal case component specify denote inverse consider univariate follow consider wishart inverse gamma modifying examine simply eq jointly decompose emission cluster indicator ny kk post output mixture emission set mixture weight emission parameter summarize gibbs write clear full case k e cardinality assign ny unfortunately dpp conditional dropping depend use equality cdf involve form nice density correct centroid center recover lead datum indicator center conditional put mass assign cover one exist cluster center cluster draw attractive formulation fact maintain sampler sampling draw normal wishart wishart hyperparameter location similarity take quality covariance provide visualization pose sample multivariate pose dpp cover broad dpp reason broad sample pose pose fig pose compare dpp draw nystr htb ccccc pose multivariate approximate dpp activity category form activity edu point focus diverse recently grow dpp sampler dpp scheme apply rank nystr fourier feature utility mixture pose span pt process set tend spread semidefinite give tendency capture volume nearly linearly less finite diverse preferred kernel recursively projection onto select process many consider occur phenomenon diversity tend grow hill space interest relate generative attractive appeal seem algorithm discrete extend formal operator key span except dpp continuous progress develop dpp continuous space scheme nystr fourier dpp technique prove useful place positive probability devise sampler derivation rely complement kernel broad subspace integral efficiently limit case characteristic similarity efficiently well particular review discrete sampling dpp sec empirical analysis synthesis sec discrete cardinality efficient dpp detailed supplement recursively eigenvector eigenvector subspace onto straightforward involve distribution eigenvector extend difficulty phase approximate orthonormal fourier able function either via approximations sampler proposal rejection make method inefficient dpp implement even rejection infeasible density normalization generic extremely summary approximately dpp translation invariant kernel propose approximately wide consider sec matrix low basic share nonzero eigenvector supplement algorithmic space order extend inclusion inclusion point know represent km j conditional simplify general difficult range nystr compute analytically cdf supplement full conditional make handle sampling dependent variable sampler mix slowly dependencie strength material dpp theory sample dpp apply inefficient birth death step pt evaluate nystr lx quality similarity kernel isotropic covariance enable focus supplement cccc similarity vary vary display distance nystr varying nystr om perform increase phenomenon eigenvalue light decay nystr om method perform matrix indicate phenomenon result behind
detection occur characteristic process continuously make observation process statistically change within area diverse economic sequential rule whereby stop adapt observe cyclic change detection address e assume throughout surveillance pre fully treat assume distant process change soon name multi false appearance detection false alarm control post change control setup emphasis place relate sr generalization sr introduce sr name sr sr sr I sr analogy terminology due reason sr prove cyclic optimal time detect brownian g late neither cumulative inspection move average chart possess numerical cyclic matter knowledge address particular analysis chart cyclic similar however question employ accuracy convergence ad hoc equally minimax cyclic optimality prove optimal special consequence cyclic optimality chart minimax quantify work efficient setup propose equation use standard e identity efficiency improve great false alarm stationary average detection confirm design gain insight aid need utilize response synthesis write detection contribute closely point detection practice process remainder structure devote aim assess experimentally propose method draw conclusion intend formally state distribution density respectively serial change give particularly assume always stop way risk run alarm detection denote alarm select expense many repeat exchange agree go repeatedly alarm show sequential stop alarm change false detection delay limit detection delay refer state steady delay measure statistical comment difference end problem description cyclic formulation instrumental detect place distant future false miss economic since define limit natural answer cyclic formulation bayesian formulation completely overview major formulation find e generalize formulation limit improper impose formulation relative add inside call equivalence cyclic formulation statement see obtain instead evaluate discuss show multi cyclic solve sr comparative demonstrate scheme chart outperform cyclic procedure introduce sr ever hypothesis give kk lr important role tt respectively lr derivative measure mutually absolutely rate formally original threshold false sr paper definition stress sr mean martingale stop one conclude show limit nonlinear accurate broad exactly formally optimality sr recently sr sr sr sr similar early turn sr put exceed sr false alarm sr become sr procedure reason sr terminology martingale limit approximation sr minimize direct formally r reduce sr remark steady often chart popular area finally plug precisely employ make much first multiply eq identity next obtain hand hand desire rewrite respectively form integral operator hand side completely know evaluation thus evaluate simultaneously extension combine complete characteristic main equation present precede follow equation equation form give know depend notational long equation observe obtain alarm suffice solve well analytical order subsection underlie interval norm behave equip thus deduce propose suitably substitution zero achieve choice algebraic equation appropriate independent acting project x point point evaluate iterate solution accuracy play critical role specifically sensible error latter interpolation particular basis eq applicable often particular kernel stationary tight fact exact state question reason polynomial nx j h h nz align h j cf functional xx xx h tailor strict bind x proportional threshold numerator simplicity roughly conclude roughly magnitude seem drastically offset denominator argued confirm experimentally reason linearity evident close compute accurately require reason piecewise linear error theorem unlike next substantially serve compute implement measure identity integral exactly subsection framework markov approach integral rule mu l interval see approach exhibit effect also end substantial compare chart original sr confirm scenario formally change density instantaneous lr therefore measure consequently implement also whether loss necessary establish subsection measure method assess alarm accomplish subsection confirm devise sensitivity method rough fine alarm low subsection interpolation dependent interpolation partition non overlap chebyshev root chebyshev polynomial small shift chebyshev measure j convenient form vary rough small fine consider scenario moderate alarm level extreme unlikely rate rely solution estimate actual u partition test matlab specific report one false column report report iterate indicate fail present method fact much broad false alarm change magnitude hence accurate also robust alarm lr nan nan nan nan nan nan rate lr lr lr lr lr point multi framework improve
correct blockmodel dc generalization sbm add additional control constant make identifiable constraint within represent twice link within block paper sbm th element belong dc sbm identifiable able partition regularizer rsc perfectly partition define population population regularize laplacian element way couple line lemma explicit matrix sbm diagonal define separate block notice describe eigen rsc matrix z useful fact direction different ij z j project onto u z j notice figure perfect left would heterogeneity star shape star shaped stem heterogeneity network htbp come dc sbm block row different correspond block origin leave block share panel share project mis regularize cluster sbm proceed close rsc close dc sbm laplacian build nc constant satisfied large proportion heterogeneous regularize fail e low reason rsc show choice degree eigenvector normalize frobenius adjacency sbm kx step rsc sufficiently material bind mis rsc mis cluster centroid define th centroid complicate subspace individual bad span estimation correctly cluster minimize orthogonal show define mis cluster mis theorem dc cluster assume mis cluster rsc bound quality equal node essence insufficient expect need b alternatively eq eigenvalue material summation mn simulation average correct adjust multiplicative sensitive reference therein score informative score row top relate leverage spectral recall score denominator score explicit small small arise leverage motivate corollary focus subset whose score exceed threshold corollary mis mis cluster let apply denominator minimum potentially make replace superior score large eigenvalue corollary thresholding thresholded rsc rsc rsc assign centroid k remark apply sc theorem sc sbm dc sbm improve upon previous blockmodel linkage block degree heterogeneity within already four improve result result spectral generate networks dc sbm power law networks sbm benefit political spectral rsc rsc rsc thresholding rsc sc perturb adjacency compare rsc score rsc set experiment heterogeneity affect performance dc law distribution indicate heterogeneity network contain define noise number block throughout degree rsc rsc rsc rsc network line assign improve rsc rsc outperform rsc heterogeneous leverage htbp sc rsc rsc heterogeneity experiment sample panel figure rate rsc average rsc rsc rsc demonstrate sbm without heterogeneity exception comprise political large network roughly assign rsc insensitive rsc leverage exclude small among leverage almost illustrate leverage try result regularize value perform simulation adjustment dramatically heterogeneous degree current moreover minimum degree study situation score choose degree compete objective comment support nsf dms grant nsf grant dms nc z eigenvector eigenvalue z regularize h separately apply concentration hermitian put apply concentration argument spectra let q part projection onto span rank eq diagonal rewrite column projection orthogonal matrix assumption least min min correspond population sufficient centroid close centroid sufficient mis cluster centroid eq triangle mail edu mail edu cm spectral recently variation node degree statistical extend remove minimum blockmodel plant degree characterize several spectral biological researcher deep mechanism mechanism generate community learner aim merely devise algorithm detection understand inference
capture structure deterministic e estimate define structure infer prior belief maintain profile structure representative along maximum entropy discrimination model give training extension extension multi along also nonparametric svms structures infinite svms classifier automatically resolve complexity number component feature margin max latent inference mean develop provide carlo restrict deriving algorithm augmentation margin infer latent max margin single focus infer augmentation refer observe iterative technique community seminal maximization likelihood mle miss augmentation physics wang idea find augment speed convergence augmentation speed phenomenon standard augmentation scheme work demonstrate augmentation construct markov carlo slice fast excellent broad augmentation select augmentation elegant formulation fully analytical present successful augmentation problem bayesian e review conference present brief overview lda hierarchical topic vocabulary cm multinomial denote select prior lda infer theoretical interpretation show cm infer intractable approximation carlo successfully various scenario deterministic objective could generally carlo classification set basically lda describe denote document classifier allocation hierarchical model topic topic word vocabulary document draw proportion z zero z dd proportion corpus respectively rule p pz rule solution kullback kl desired extend desire develop regularize inference computational model shall topic training choose space classifier weight classifier possible error discriminant define topic classifier classifier couple topic assignment possible distribution word prediction impose derive lda wrong slack objective classifier regularize bayesian constraint equivalently solve classifier constant solve directly conjugacy margin factorize monte q similar type iterative svm subproblem outline solve form solve lagrangian lagrange multiplier constraint commonly binary svm exist svm learner respect solve although derive dual objective due margin constraint lagrange multiplier towards collapse assumption margin binary classification strategy building classifier nice property infer assignment margin follow latent rule hinge hinge hinge training gibbs classifier could expectation posterior one hand describe margin function complete integrated expectation expect expect thus deal differentiable max fortunately collapse analytical base augmentation hinge unnormalized constraint want sample scale mixture unnormalized follow get marginal high positive complete posterior augmentation normalization constraint impose augment improper upper augmentation denote unnormalized augmentation unnormalize pseudo affect derivation infer although infer q rate latent effectively dirichlet markov propose collapse detailed augment formulation gibbs collapse assigned word count count document conditional sampling assume isotropic distribution k variable dimensional gaussian draw distribution cholesky procedure normally inversion efficiently large indicate exclude document discriminant count second supervise initialization randomly multinomial draw conditional augment variable factorize inverse normalization inverse iteratively draw assignment augment distribution root overall iteration total per common drawing dominate per finish outline sample gibbs target see however justify satisfie start intractable one require performance shall analysis theory infer assignment test compute content second equality hold topic apply approach training datum estimate collapse infer exclude start gibbs sampler stop burn latter stage prediction slightly section task learn discuss idea develop variable regression component lda lda present gibbs widely use insensitive margin insensitive assignment input resolve expect insensitive follow principle gibbs bayesian note put irrelevant variable upper insensitive prediction rule apply jensen unnormalized scale note unnormalized likelihood express gaussians augment collapse derivation classification posterior distribution outline isotropic prior easily inverse distribution classifier draw inverse distribution give first count supervise signal inverse similar jointly hope attract lot attention task latent representation application define task apply belong output prediction detail consider classifier binary task lda sharing assignment follow gibbs gibbs expect hinge loss iy gibbs good loss lead expect define binary hinge task separability hinge unnormalized classification model collapse gibbs algorithm unnormalized class augmentation task collapse collapse gibbs draw distribution draw topic assume isotropic gaussian cholesky inversion common derive lda observe factorize variable efficiently iteration application assignment large scale categorization challenge thousand draw dominate nice easily present gibbs classification wikipedia document multi analyze examine learn qualitatively set contain follow list binary deviation randomly initialize large contain gibbs use variational collapse unsupervised collapse baseline denote gibbs shall insensitive well different number expect posterior achieve accuracy restrict factorize magnitude svm fast variational several space low speed collapse save carry variational build website review predict global rating score manually part speech tag character review uniformly partition regression variational set fold dirichlet burn show full time two magnitude testing fast especially reason perform category category consist category document large respectively build exist multi class vs choose provide preliminary vs binary burn topic assignment classifier predict document belong category category e insensitive simplicity burn classification build horizontal axis classifier couple classifier denote clearly strategy build multi classifier give gibbs restrict another improve train parallel save promise since processor perform output prediction binary classification belong share topic given easily learn gibbs data multiple infer assignment compute discriminant document category show multi performance method vs horizontal axis denote classifier see task fewer score use implementation time processor core parallel vs fast time single task time processor parallel least fast expense processor processor core parallel multi excellent multi parallel vs present build challenge document build category million document perform topic discovery classifier jointly svm raw svm method lda discover document build separate svm document discover step insensitive class precision f distribute gibbs reason improvement vocabulary million dimensional document rise svm raw fitting wide failure category discover representation produce ability reduce overfitte discover margin discover discriminative step similar define task follow expect hinge define hinge dy iy dy separability loss apply task expect hinge variable unnormalize binary collapse multi classification careful various sensitivity effect burn penalty training different burn draw initialize testing stable burn burn linearly experiment burn use vs show naive burn quite especially linearly use competitive especially binary figure testing accuracy time accuracy fast step linearly burn automatically newton analyze show classification binary symmetric dirichlet topic number wide large e topic slightly mainly produce topic representation appropriately representation dirichlet classification different wide quite stable multi similar time classification test test number multi class finally also visualize discover learn common share multiple category indicate topic distinguish class document salient topic describe salient rank reflect mean category graphic category salient file graphic category observation c c pt team mr db windows cs article play file center mail si file price people probe game crowd mac people people medical people mb box association ground mb controller te reduce article patient ne current thing work cx disk ms graphics anonymous server os mail file master people anonymous file multi vote email display server image ad mit program service consistent file voting perspective car south people output return neutral tb length start
scenario learn result show cl il aggregate capacity close practical cl compare il paradigm coverage although challenge fully benefit challenge interference random operator add wireless centralized approach interference call management focus access bandwidth cognitive secondary user try control maintain handle interference reinforcement technique multi agent prior due typical wireless perform il perform allocation interference generate learn policy acquire exchange table action power system call power cl paradigm outperform il achieve capacity term learn performance q compare contribution follow propose power namely centralized learn power use controller gain system responsible power agent base global aggregate capacity capacity evaluate robustness scalability il cl two dynamic wireless environment namely random activity macro il idea present learn result discuss conclusion wireless macro receive macro base base coverage area macro inside macro enhance transmission transmission power analyze measure bit hz channel gain capacity achieve user channel gain associated gain cognitive formulate describe state action agent task probabilistic state joint agent determine feed back cognitive due environment thus assign task agent action define discount infinite action state process agent environment observe base select action randomly ts ts visit agent receive reward process repeat discount factor determine moment notice reward depend joint action describe one stationary however multi depend agent thus proof agent propose learn optimal policy e allocation interact paradigm agent agent I consider agent environment problem paradigm application paradigm share agent cl share agent range agent behind strategy explain overhead overhead I quadratic n k interference macro capacity performance assume power transmission scalar set transmission reward fed agent behind reward maintain capacity around bit hz reward capacity aggregate capacity explore depend value q reward positive state reward thus maximum q feed reward could could value action lead another action whose great decrease explore robustness scalability il cl network il reason share certain implicitly action independently il know behavior reach overcome make could decrease action agent il cl reward indicate interference measure macro user aggregate power reward aggregate capacity note vector put centralized control regard power centralized controller overhead multi reward controller controller power size form large power infeasible q reward qualitatively table il cl grow scalar exponentially reaction inefficient cl robust robust dynamic efficient infeasible cl scalable il medium convergence huge overhead cl large overhead il wireless macro serve locate area macro band compose transmission use receiver dominate exponent calculate follow assumption associate user associate core simulate maximum transmission maximum transmission level learn discount aggregate il il cl optimal exhaustive maximum aggregate hz begin infeasible action besides pair get value stop stop search due illustrate continuity robustness start every iteration reach add iteration figure cl paradigm il figure investigate already join initialized table use il cl cl maintain bit sec hz cl regardless
deal investigate prediction penalize observation incomplete world unclear minimization criterion regression elastic net response miss utilize necessarily modify regularization imputation especially numerous miss since simple mse via simulated extend balance generalize definite covariance imputation role balance coefficient parameter issue handle observation set validation incomplete test regression ordinary penalty useful applicability matrix vector j cyclic coordinate descent formula fit x handle penalize statistical analysis mean imputation base unbiased like contain contaminate drawback potentially meaningful imputation mean feature feature miss algorithm item systematic sometimes matrix singular appropriate amenable unbiased estimate coefficient error bound show estimate assumption estimator extra negative definite become attractive pattern random space miss ij ij ik standardized point rewrite ij unbiased non definite without condition replace rewrite equivalently remarkable thing result original minimize cyclic coordinate meaningful observation unclear apply problematic incomplete test incomplete point miss extra expectation inverse extra inverse negative definite imputation simulate instance multivariate distribution entry consider investigate missing type concentrate signal case denote average try efficacy repetition imputation miss result concentrated lead good manually c c miss signal high mi mi mi row multivariate proportional mean imputation beneficial new parameter balance basis definite replace covariance ij n z covariance definite estimate manner corresponding place combine l jk range small eigen method negative negative definite covariance use conditional multivariate feature combine become matrix bottom bottom uniform line dot represent minimum mse miss mi cm imputation scenario imputation method set yield
must majority vote various model expert simple scenario expert give fix agent expert make final together profile analogously mistake profile source randomness run formulate advance uniformly two regime confidence frequentist regime I I essentially taylor induce rule observe error I hence np order state term inherent restrictive condition indeed decision exploit highly expert moderate sample regime discuss begin finally easy analyze since hoeffde nontrivial tool inequality regime estimate formalize put weight induce raise immediate concern high I high probability value surprisingly asymptotically achieve multiplicative chernoff ip ip hence ip e ip union let range I I consequence let plug majority rule yield proof upper bound first hand replace formula drawback evaluate approach dependence event interpretation observe determine occur adaptive confidence approach hold predict exist upper player recover limit note expert large bind ensure case cause operator mas I I inequality property norm p substitute upper invoke refinement I I optimal improve fail independently profile produce together profile rule identical optimal define vote ab analogously display conditional rule w I probability correct iw almost surely frequentist interpretation although unable trivial coupling size small compare voting rule indeed marginally expert majority n sophisticated voting rule well provide natural ability exploit expert set experiment vs end maximize absolute n though f appear conjecture heuristic n ht f dominate essentially trial vector expert surprising require majority expect understand consistency majority vote continue challenge hope defer examine derivative f x denominator nonnegative numerator n verify calculus concave maximum assumption classical weighted expert examine sharp standard illustrate weight expert considerable theoretical rigorous study majority vote explore rule therein typical expert make simplify assumption independent assumption take truly decision throughout minimize hold rule appropriately give odd expert correct voting rule naive pearson raise question address precisely equivalence universal multiplicative constant issue handle rather solution frequentist bayesian frequentist admit empirical analyze adaptive vs additional weak confidence regime yield arbitrarily far cause denominator exponent make hand hoeffding inequality guarantee even instance heterogeneous sum phenomenon explore bernstein inequality suffer fortunately sharp see
specification program implement correctness methodology correctness program arithmetic development suppose first know library user pg manual rewrite lemma move rewrite show pg analogy important feature lemma box formulate step incomplete equivalence specification factorial already factorial factorial algorithm factorial iterative library auxiliary discover challenge ml pg team member stop naive proceed call pg suggestion pg proof iterative multiplication power natural number ml show trace correlate lemma notice strategy c ml pg suggest lemma lemma show helpful pg result user optimal top level similarity discover ml pg fact pi pi make pi proof rewrite rewrite loop pi sn stop ml pg like analogy follow consecutive determined apply loop pg suggestion team try reconstruct auxiliary lemma figure reconstruction proof second factorial step pg suggest implement power program loop notice concrete case lemma finish pg find correctness program pattern apply lemma obtain figure pg suggestion analogy e g obtaining correctness factorial find heterogeneous belong library obtain analogous lemma kind lemma could factorial total correctness sf sf pi sf sf n sf sf pi sf split h pg suggest particular suggest k algorithm analogy user interactive intuition kind development follow varie acquire experience expert experience problem similarity pg help use user pattern study user ml helpful domain reason lack library material domain concrete development solve project thompson library contain library big difficulty imagine want explore scenario theorem contrary pg snapshot library available technique domain advanced notion library advantage user scenario pg discovery tool already suggestion domain user something development pg library pattern positive library probably find attempt pg librarie user fact style therefore pattern arise clearly reason pg work plain encourage style library people big development thompson theorem concrete discovery pattern ml pg nature domain capability pg test cluster reliable result library library return user produce pg require user parameter obtain pg allow quick interactive pg equally library try irrespective subject library subject librarie study pg cluster homogeneous library heterogeneous library homogeneous clear analogous contrary relation among subtle kind way incorporate extension pg pattern mining co automatically discover pattern already first language language difficult acknowledgment like thank read suggestion us presentation theorem thm thm thm thm thm thm interactive library varied formal perhaps library challenge tool ml library proof user basis find library interactive name wide mathematical verification concerned computer number security efficient programming combination see g rich approach rely newly situation mathematical explain often one ml pg enable come wide domain development lead library vary formal mathematical framework thousand definition thompson library domain challenge expert non expert trace idea library framework develop e involve project hardware software style extremely helpful pattern library address challenge propose ml machine proof main goal concept pg package user work interactive option call pg base choose library significant exist lemma connect run number query thus post processing choose ml two way read theorem display separate window additionally pg form give overview pg ml pg substantially extend detailed description pg example useful automate proof different domain devise pg pg ability adapt domain scenario library come area range basic verification pg development library library library colour key formal thompson library contain theorem library independently library scenario pattern library begin proof development easier available library light library discover another proof pg library result might contain domain pg use completely different pg reality save library manually team verification effort translate correctness virtual prove differ scenario verification big number routine lemma task team automate often arise different notation lot see test ml pg team develop proof program power function relative team effort factorial relevant lemma around total factorial evaluation pg mainly interface contain pg statistically trivial library facilitate domain ml pg background thus request finally automate smooth curve user scenario allow interactive pattern work concern discovery proof statistical unsupervised rather machine tool various come experience neither theorem main instead user intuition ml pg interactive generator interesting non choose compare pg analyse user step recognition make pg proof community subject area illustration statement proof discuss case symbolic e pg introduce statistical family search unlike symbolic search go search template symbolic template theorem pg consideration search attention symbolic pg user ml pg knowledge form user prefer cf information pattern arise library irrespective current step cf user choose library user wish choice ml pg user interface pg extract low proof interface execution choice display user pattern collect significant feature area classify irrespective extraction ml pg extraction extraction interactive construction current library external library information proof within one relation library current arise statistic reveal lot strategy proof length also big may issue implement automatic proof patch allow ml pg property patch constitute detail extraction find focus detail ml pg concentrate patch five composite step pg learn pg modular extraction complete within format consist lemma relate include lemma drop element drop remove list take lemma drop cluster lemma proof auxiliary appear library map f x lemma count cluster proof theorem pg lemma kind lemma boolean find come library tt contain equivalence lemma cluster proof solve lemma example solve follow rewrite lemma analogy heterogeneous cluster lemma library pg big homogeneous size per contain library ml pg addition concatenation lemma move operation kind pg leave mul cluster rely ml pg proof lemma rule proof carefully base list type type lemma library list quite base case lemma apply inductive rewrite move cat induction rule inductive hypothesis finish proof discover lemma proof group case find pattern correlation cluster use analysis ml section whether strong correlation yes situation ml useful little pg help could modify cluster ultimately user previous library stage namely relation common library proof user know library e library pg lemma use knowledge abstract player internal play path root internal strategy obtain well overall nash equilibrium use ml pg analyse two library sequential equilibria game game general unlike benchmark file plain ml pg verification hope inspection library reveal library pg negative experience compare instead challenge pg second analyse ml pg file topological sort file mean pg pg find mean question interpret set way analyse relative cluster pg figure produce pg lemma cluster proof merge transition annotate feature proof patch cluster correlate box box state bi bi bi bi exist exist split bi group pg theorem outside ml pg annotation result nash library proof library library notice lemma pg one bi exist strategy game backward equilibrium player optimally node state reflect pg see show concrete pg first contain theorems theorems eq proof induction induction rewrite contradiction trivial give
generalization diffusion datum arise adjust fig learn stationary answer key question distribute algorithm optimal receive expect q assume obviously aggregate denote scalar utilize simulation assess algorithms use adopt measure follow optimizer reason excess excess ability observe randomness development stochastic gradient literature also call machine considerable research focus derive excess descent stand alone two connect topology mean connect arbitrary agent region denote optimizer drift associate network feature function consider risk describe optimizer optimizer reflect satisfied stationary list evaluate diffusion optimization manner neighbor include satisfy start true alg gradient scalar entry respectively requirement variation example step adapt step different q compare critical difference way beyond immediate diffusion case computation result learner learner diffusion see excess main introduce filter square environment stationary excess datum adopt introduce regard hessian time encounter time quadratic translate logistic oppose hessian quadratic assumption subsequent excess theorem weight excess justify square performance diffusion excess risk across stress mse weighting regard perturb perturb condition history variance optimum term improve power decrease second noise refer absolute noise estimate vector early presence ignore class optimize risk eq let datum satisfy feature equation assume time instantaneous denote therefore appear scenario optimizer change stationarity optimizer slowly walk datum function optimizer individual minimizer walk observe optimizer due component furthermore definition risk walk drift filter assumption autoregressive process behavior financial modeling search internet shift demonstrate excellent page sort capability relate filter analyze f introduce algorithms risk assumption receive node utilize receive focus strategy stationary condition first show achieve excess stationary environment step size q arbitrarily also make optimizer equation show eq use environment mean optimizer quantity step next mse generally approximate excess risk state sufficiently steady excess steady excess approximated symbol kronecker operation covariance notice excess due value approximately steady approximated utilize result risk steady free assume stack equality eq q invertible small step therefore conclude approximate steady state list metric choose appropriately square th diagonal element evaluate excess evaluation indicate th diagonal element w kn mn act individually special matrix doubly mean weighting satisfie steady execute state excess steady excess risk see combine adapt stochastic aggregated become small ahead next diffusion optimizer change excess risk excess alg reduction excess possible environment arrive track asymptotic er non stationary satisfie excess satisfy bound risk weight matrix write term verify cauchy step introduction decomposition negative symmetric definite bound recursion evaluate limit series additionally sufficiently denominator approximated approximate note mean square bound excess risk steady steady stationary environment arise decrease track right insight fact track stationary excess remain even optimizer walk remains bound illustrate context hyper plane separate logistic plane origin rotation optimal plane diffusion hyper remain within excess risk strongly use indicate diffusion alpha deal dataset split evenly choose steady state divide relatively large analysis steady expression match set size fig size attribute regularize excess risk algorithm addition centralize full gradient access iteration move average horizon estimate processor average require server estimate back consensus scheme every evaluate iteration average size utilize metropolis metropolis doubly utilize loss x list dataset learner simplify iw iw show excess outperform consensus constant size observe fig step size excess decrease fact analysis average close global strategy actual classifier output operate curve classifier computed bias alarm consensus tend close centralized separate simulate random illustrate simulate concept simulate instant presentation result since establish study decay highlight importance environment zero receive per metropolis weight combine amount label add function plot excess fig step cope stationarity predict constant track drift addition instantaneous drift purpose target concept target map see color attribute shape small amount also result regularization optimize carry library simulate constant simulate necessity step size environment excess track change fails know detector classifier target second fully batch metropolis excess error accord small formulation risk study explain environment excess performance proportional outperform generalize loss process optimizer walk increment model diffusion tracking process excess comprise increment walk term proportional term expect track optimizer relatively slow optimizer optimizer evolve increase diffusion diffusion show extensive simulation area roc curve consensus see constant change optimizer unlike describe constant excess environment expression excess advantage national grant achieve excess achieve conduct constant environment start rewrite perform get simplify
represent nature relationship protein protein interact fashion domain social offer sign tag user positively trust towards generally individual website rating sign design especially sign e g relationship among sign graph conceptual provide social formulate relationship classify amount g reference therein heuristic link network balance summarize sign edge social tend consistent partition set cluster connect node connect set heuristic strict bias practitioner context social good finally fairly viewpoint undirecte sign heuristic exploit sign amount storage even impractical large algorithm protocol query match graph wide theoretic active label unknown introduce model author span mistake factor theoretical easy second tree query large classification graph edge sufficient remain budget sufficient hence query optimality run preliminary medium sized synthetic real dataset theoretical finding inductive bias seem perform heuristic represent associated adjacency matrix sequel define introduction efficiency label undirecte connect assign consistency equal sign path connect equivalent say cycle edge constant way I stochastic assignment receive sign build undirected mistake consistently cluster assignment moreover randomly label mistake query receive query build label edge ever reveal learner reveal active label mistake set graph diameter denote within unique e sign root child subtree root sign sign learn link warm recall connection g useful important learner denote label set label predict edge contain active learner edge indicator quantity reduce every unweighted span span tree hold learner query upper constant time fairly complicated implementation asymptotic disadvantage force tree short visit choice visit empirically improve clearly prediction test take amount constant per whenever time see key aspect ability edge create short circuit path quantify te explicit edge whose predicting say predict circuit label stochastic label mistake edge uniformly inequality input factor whenever diameter light graph diameter algorithm query label label coincide distinct exist adjacent query arbitrarily previously predict obtain span label predict first visit visit visit leaf set h ti h ti ki ti long quantify mistake query scenario integer mistake operate adversarial model replace expectation occur graph integer degree random mistake well observe achieve training unlike mistake training factor unlike constant bad require running compare length prove mistake di il mistake di span little since mistake optimality one span consider span training simpler easily refine argument lead refined follow figure line initial span new subroutine select star result optimality bound combine set parameterized create repeat call procedure create exist correspond tree connect star star pair distinct star edge k edge truly edge moderately dense need span theorem within hard replace therein design hard scheme compare result factor order edge quick mistake optimality lower compare low span tree optimality get comparable amount offer optimality factor multiple requirement hold ensure v enough whenever performance edge span short span tree generate visit visit pick run list baseline heuristic among heuristic turn predict sign edge eigenvector value expand power path length one otherwise equal product combination prediction multiplicative time create digit dataset randomly label follow edge class real world sign assignment assignment label three synthetic delta choose world sign subtract user rating user cosine entry remove loop take node snapshot snapshot similar node reciprocal edge turn remain
spatial dynamic assume nonparametric aspect employ spatial disease identify relationship impose infer project result description study exhaustive consist discretization level discretization perform system datum disease sensible count normalize region series per desire level divide entirely space arise assumption level disease group spatial transition probability transition location different probability every represent entry whose assignment group often growth group uniquely growth simply series level govern probability entire give rewrite location group count entire observe eq rewrite observe maximize goal find transition probable grouping fraction transition level good maximize transition infer call assume build introduce transition identify grouping introduce prior assignment govern categorical concentration parameter weight gamma posterior govern concentration equal jeffreys another approach prior simply proportional strongly biased toward dirichlet categorical distribution member group share see categorical collapse yield nonempty region group group indistinguishable another concentration mass group nonempty depend possible region possible become greedy search fail desire necessary stochastic optimization grouping equation nonempty criterion bayesian aic independence bayesian robust different group many region number region way tractable computer exhaustive enumeration grouping use likelihood sum enumeration ensure successive single change count subtract entry group eliminate synchronization gray group always growth agglomerative combination terminate region increase one exhaustive enumeration intractable markov distribution state interact converge use value repeat calculate marginal keep probability likelihood technique couple markov chain avoid get
sr valid transform eq express u dirac delta distribution convolution insensitive distortion existence valid characteristic satisfy characteristic source condition rewrite right hand side convolution never mean sr distortion distortion use laplacian distortion informative distortion yield term specifically analytically differential differential bound x px dx sc general eqs upper sections general source parameter eq entropy reduce dd right side becomes arbitrarily transform mean ss dr sr specifically se simplify b follow dy r r distortion e turn subsection consider source mean source maximum dd upper r sp gaussian far depict source upper analytic bind gaussian entropy distortion bind suggest average distortion observe gaussian explicit see tight high trivial informative upper distortion insensitive distortion source focus laplacian gaussian strictly provide upper distortion prove distortion numerical accurate distortion region reasonable define insensitive address result insensitive variation insensitive also property work grant distortion strictly distortion insensitive shannon source differential focus distortion source great shannon upper distortion function distortion evaluation suggest shannon low distortion insensitive distortion source distortion reconstruct average rate explicitly source measure function difference distortion examine condition coincide limit alphabet source class source magnitude distortion annealing obtain insensitive loss introduce distortion dx dy respect minimum achievable measure minimization problem exist infimum density parameterize slope property insensitive loss order sparsity
objective initialized svm task svm able classify original task neuron algorithm virtual experience store acceptable reduce understanding characteristic affect activation function evidence old influence outcome consequently examine relationship find dropout modern feedforward neural net activation relationship place emphasis adapt old task maxout consistent dropout validate activation dropout find net net great consistent dropout decrease net subtle study receive much deep idea aspect modern net study one task finding pair kind similarity kind standard move beyond limitation net use task similarity dropout improve train stochastic training learn multiply mask mask cause mask mask sample independently time drop multiply dropout extremely many prediction resemble bag learn help effective one main reducing simply restrict dropout enable training hyperparameter experiment classify mnist dropout validation train without dropout increase net traditional net input activation learnable learnable provide input layer activation logistic winner take disjoint block tie maximum break tie use index easy maxout eight comparison deep obtain deep familiar practitioner select hyperparameter allow complicated dependence automate challenge search suffer curse hyperparameter space instead implement obtain art mnist sophisticated hyperparameter find use method sort study selector form examine train kind training try four activation sigmoid maxout eight detail case layer follow softmax include magnitude layer initialize layer hyper control decay hyperparameter reasonably know dropping drop visible around known able fail keep search hyperparameter sgd dropout slight search initialization scheme maxout initial bias make result filter bias sigmoid significantly negative encourage initialization significantly prevent ever non also positive initial bias help sigmoid thus use necessary roughly art experiment random activation activation maxout method may initialization method go maxout poorly initialize case old set improved epoch validation begin old epoch running condition possibility curve task old task drawing trace cloud old new pass set compute validation set care relative state art result possibility scenario value trace hull error low edge error highlight perform make convex convex naturally structure task deep language language exist person language neuron use agreement rather remove pre concept design classification permutation thus concept detector pixel net associate collection old connection pixel fig improve set net basically first weight net apparent conclude begin high layer change layer adapt happen case sentiment two category amazon review two text use classification present dropout validation model pair happen task test amazon mnist size example validation amazon validation amazon dataset amazon dataset input mnist give two dimensionality amazon improve pair dropout experiment dropout dropout performance old along tradeoff curve balance dropout explain train
effective dictionary iteratively matrix gram shrink enforce effective mutual acquisition shrink choose matrix shrink square root choose xu tight frame minimal mutual aim solve replace refer diagonal thus equal base note xu different large element towards every iteration adequate value dictionary normalize enforce projection acquisition optimize acquisition fix dictionary focus brevity seek acquisition minimize notation indicate restriction eigenvector svd acquisition tight singular row improve think diagonal follow enforce rank constraint xu maximum atom project minimize xu way iteratively shrink enforce minor difference shrinkage xu algorithm two corner intuitive analysis scenario consider acquisition acquire desire nothing case perfect xu identical large usual recognize acquisition already second scenario scenario toy example illustrate atom effective dictionary gram pair unnecessary decomposition atom ambiguity reconstruct irrespective atom think gram atom irrelevant reconstruct replace minimization column optimize pose optimization q propose modification xu algorithms make diagonal xu constrained near correlation explain note orthonormal propose modification original orthonormal become identical algorithm prefer original indeed svd imply lead acquisition difference normalize acquisition vector play essentially scale projection essential original compose unitary therefore inner product atom product ensure normalize coherent coherence atom correspond atom never reconstruct xu avoid perform atom normalization optimization less atom result basis svd factorization one optimal effective restriction atom thing happen atom atom norm zero dictionary concatenation dirac haar effective dictionary variation fraction shift wavelet atom effective dictionary norm less scenario section case algorithms optimization explain estimate matrix refer consideration propose acquisition reformulate optimization modification xu introduce constrain near family finance correlation possibly incomplete positivity require input matrix formulate interpretation atom dictionary possible nature correlation atom rigorous justify orthonormal dictionary orthonormal coherence guarantee rigorous improve name previous emphasize common propose contrary constraint near developed solving penalization summarize semidefinite enforce eigenvalue arbitrarily achieve technique simple convex minimize give paper replace acquisition first optimize projection summarize solution constrain penalty create minimize converge tolerance solved xu shrink instead xu give xu keep xu project shrink normalize gram shrink element decomposition extract I compute dictionary normalize gram enforce add projection previous acquisition optimize algorithm xu algorithms acquisition iii xu algorithm vi xu propose xu aggregate behaviour compose k svd train consist atom randomly select patch test section public database patch affected vector exhibit correlation similarity image depict dictionary gram histogram dictionary size room successful measurement data atom dictionary reconstruction follow orthogonal pursuit ii iii robust sl iv accelerate iterative thresholding pass square mse signal db low algorithm xu slightly behind match performance reconstruction bad behaviour persistent simulation create xu display poor small behind reader xu algorithms xu almost equally take account atom dictionary normalization effective atom effective dictionary norm greatly algorithm explain fig little improvement contrary essential structured dictionary illustrate atom normalization list challenge atom norm e projection actual dictionary capture concatenation dirac haar basis whereas non wavelet noiseless result obtain sl recovery good mse please sensitive largely behind emphasize perform dictionary particularly dictionary principal note xu dictionary orthogonal diagonal element absolute basis behavior dictionary atom effective dictionary benefit xu dictionary concentrate encounter seem nature recovery dictionary propose scenario dictionary focus acquisition compressive improvement three analyze perform optimally argue xu dictionary reduce coherence small modify instance single unified unit norm distance become problem norm problem xu algorithms iteratively gram structured dictionary near exist acquisition
present exploration property generalise pareto parametric usefulness nevertheless remarkable univariate draw approach concept match phrase definition elsewhere coverage coincide phrase predictor question exist phrase attempt loose probability exceed course loose frequentist analyst member select actual unknown construct confidence maker may notion may approach analyst combine prior parameter analyst belief simply member great construct analyst belief possible speak integral possibility candidate analyst belief extreme level tail integral equal desire see appeal without difficulty span datum influence parameter problematic moreover integration explore outside analyst confident numerically converge level case predictor base choose family ready member form location invariant invariant predictor specifically improper power invariant although matter obvious come usefulness complete priori somewhat expand family pareto generalise extreme theory set speak condition limit predictor wide application arise case question correspond take sampling bayesian predictor extreme tailed tail limit match remarkably outside match remarkably suppose sample generalise order indexing low statistic opposite adopt aim predictor return level data point exceed irrespective value unit normalise independent mapping onto zero exceed likewise focus normalised parameter may elementary sampling brevity heavy tailed write tail since location remain normalise exceed admissible heavy interest elementary reveal equation normalise match distribution limit heavy tail e eqn unity simplify statistic almost integrate domain require obvious equal eq choose might instead sample power predictor numerically simple value r predictor play draw rapidly nearby distribute predict actually nearby optimistic soon h predict maximum even datum adopt aim create predictor albeit example match namely datum prediction absence informative adopt construct predictor attempt match extreme propose simplex possible normalise match surface unity tend respective limit number interpolation brevity first pre condition extreme wide limit multiply wide matching limit predictor numerical reasonable respective fig candidate determine numerical h use combine via surface interpolation choose location invariant datum estimator predictor predictor analytical justification conditioning return level size thus number probability factor see match case return span low prediction I return beyond often engineering event historical dot level prediction likewise correspond aim actual plot bayes prediction prediction exceed considerable actual suggest underlie predictor chance inherent quantile dash prediction design possibility may variety outside family show datum point prediction illustrate draw normal sided variant beyond magnitude level size size draw normal sided sided cauchy two match increasingly general location pick affect good normal sided probability improve moderately two centre side require side highlight lie extreme much sided illustrate heavy tail little problem figure predictor heavy tail wherein good playing stock universal extreme match small also readily could simple contain population mostly obviously case propose sample match within might
thm prop thm corollary example conjecture remark electrical engineering science university california proof compute simplex slight abuse find j dominate sorting algorithm identify implement axis shift early solve generality solve alternate projection onto constraint nonsmooth exactly sort proof involve kkt respectively optimal follow condition obviously component zero small assume sort order positive give value optimal thresholding easy solution indeed satisfy follow essentially solution sequel stay guess guess kkt extend simplex aa ia matlab implement algorithm project dimension minus extension dataset nm cluster appear problem assign cluster alternate nk nk laplacian quite efficient accelerate projection take project onto
equation tend dominate therefore prove tend proof first maximizer show n large term dominate tend tend next maximizer first maximizer exist maximizer mean hence kind tend show notice dominate maximizer q tend one maximizer satisfy law number pn obvious tend dominate third tend example case use follow theorem wang et hence tend parameter hence mle p mle likelihood property estimate tend overfitte accord space mixture mle mle f follow sphere bic mle p via neural http www f statistics nsf conference statistics institute mathematical york scaled department university research department institute biological concern model new selection multivariate mix probability gaussian performance selection penalize modeling stem recognition vision machine important mixture approximate demonstrate chen convergence finite model slow rate yield poor interpretation component flexible underlie issue also significantly determine mixture aic bic property aic select number true component show parameterization penalize penalization sequence measure fit nonparametric distribution penalize chen kullback hellinger finite unknown ray suggest number burden heavy chen penalize fan location scad difference merge shrink difference similar true optimal location case study incorrectly component inference fully prior posteriori van put dirichlet favor mix associate unnecessary component toward determine propose like eliminate change weight consistency often likelihood mixture focus eliminate retain deal need type function would weight directly penalty zero change especially consistency propose rest propose penalize finite mixture study discussion gaussian gmm gaussian density gaussian density weight mix integer gmm via determine intuitively eliminate suitable retain however mix indicator observation expect complete log likelihood expect illustration change zero particular bivariate gmm two negative gradually depict likelihood respective log likelihood word derivative function approximately close dominate penalty htp cc mixing learn gmm minimize log know type likelihood regular statistical log suggest need penalty go component covariance matrix simplify penalty mixing model prior function sense mix exactly function continuous mix weight li pose bayesian literature study function function similar support area improper fan li penalty biased compare replace penalty scad function li henceforth iteratively two step introduce em step estimate eq likelihood update lagrange multipli equation straightforward update interpret irrelevant impose covariance avoid ill gaussian similar modify em component reduce specify extreme avoid mixture exist local maximizer tending condition maximizer maximizer number tend two penalize theorem unlike component penalize penalty selection li easier follow criterion always select tend model number tune lambda select p weight component together exhibit triangle initial estimate evolution modify em correct aic bic estimate correctly regardless initial depict evolution numerically initialization intermediate estimate propose c aic penalize function one identify mix c c htbp component true may chen generate weight mean parameter evolution modify em correctly bic robust htbp typical initialization e three intermediate f final estimate htbp typical htbp component mix c true true c component mix eigenvalue segmentation machine repository http uci create database seven window randomly draw instance attribute green matrix unknown weight figure suggest represent htbp b scatter green red light marginal extra green simulation run randomly em propose figure algorithm select summarize parameter mix htbp two histogram number component c mean ex green ex approach gaussian propose involve load attractive mild mixture scope practice gradually generate component necessary merge et certain final result em investigation classical newton hybrid
sf strongly deduce gaussian bind know provide implementation detail solve subproblem introduce auxiliary augment multipli penalty iterate minimization singular problem consider remark science york university usa computer university new usa mail hard pursuit htp iterative sparse numerical generalize compressive constrained iterate descent step thresholding enjoy term estimation pursuit past decade interest discovery drive force rapid development bioinformatics vision datum represent million must substantially imply impose structure capture impose sparsity constraint parameter efficient approximately generic constrain ii sparsity constrain regression measure error graphical sample fidelity cardinality even approximate solution particular square gain area sense include pursuit compressive pursuit pursuit successively position value via explore method develop sparse selection algorithm exhibit attractive square compressive processing function commonly graphical broad constrain learn select sequential fashion category date back frank wolfe objective greedy forward take forward backward selection compressive pursuit compressive problem type efficient sparse component success thresholde htp sense propose pursuit estimation model descent hard entry mild strong analogous htp logistic model htp greedy truncation th restriction e restriction modulus norm nonzero index modulus entry th row column index trace diagonal restriction wise proceed follow logistic conclude procedure approximately generate vector sparse typically th minimization minimize function continue iteration guarantee regard tuning minimize costly replace truncation operation lead regard project descent optimize convex iteration descent outline mention fast restricted descent square htp specifically descent ax reduce projection meanwhile reduce kx fx fx study two accuracy integer satisfy condition index set case condition compressive connection strong convexity greedy convexity restrict strongly connect restrict strong convexity smoothness differentiable two strongly indicate smooth convexity imply condition strong analyze convergence simple periodic lie soon establish appendix sequence moreover whenever strongly parameter target arbitrary tx proof provide make attempt optimize constant theorem loose discussion ignore constant determined reach geometric particularly unconstrained minimum negligible small unconstraine condition enjoy geometric rate rip measurement compressive setup htp rip compressive htp almost although make attempt htp condition fairly compressive general similar support pursuit type top entry vector large descent large cost support performance popular associate eq logistic learn high thus minimum conventional handle logistic loss avoid proposition modify version upper desire log determinant specifically utilize c unfortunately constraint addition constraint address problem solver resort solve subproblem efficiency detail defer modify formally describe theorem valid dominant slight dominant dominant thus union cone support set remain argument kf f devote logistic problem algorithm compressive sense well htp implement intel ram synthetic draw random generate bernoulli interested sample size cardinality art greedy forward select explore well type iteratively select atom dictionary objective combination geometric stopping estimation algorithm consider consider tend insensitive superior comparable overhead also compare sparse size test initial set stop criterion figure term rate superior term fast fast although summarize well efficiency curve employ precision entry generate equal probability number sample tuning time modify handle glasso graphical estimation measure frobenius well recovery magnitude h figure compare f cpu achieve show large glasso expect approximately greedy selection iteration convex glasso instability observe figure glasso computationally greedy inferior glasso visually algorithm identify visual greedy phenomenon subject response disease rd high survival long base estimate expression predict rd follow well reference therein sake reader briefly experimental training testing set division subject rd constitute remain gene normally assume rd lda score l l subject x hence test glasso graphical training use specificity sensitivity criterion tn stand positive negative rd fp stand false negative respectively large classification performance adjust htb comparison std cpu replication c specificity cpu sec glasso average standard criterion replication competitive lead average time list evolve determinant converge draw curve h determinant convergence htp main idea
right plot variation preserve rate plain stochastic step change point consider support recursion update dyadic perform cm cm benchmark dataset level regression outlier average divide equal observation correspond pass plot black dash line mark effective pass replication normalize one pass power size compare average averaged size decay sag dedicate rate average sgd decay except particular typically pass sign overfitte high sag sgd exhibit behavior theoretical size well significant lead objective sparse dataset pass newton technique stochastic still average sgd decaying square hard tune inferior sag tend behave well differ notably lack convergence novel consistently newton type worse later except hard like fix good quickly level reason explain inferior bit sensitive algorithm robust fine degree freedom quantify accurately quality non logistic regression avoid cm performance effective pass cm cm cm optimize good approximation assumption analysis fast large extend algorithm b extension implement potentially hard adaptive size acknowledgement partially european like thank schmidt discussion throughout x triangle expression sense also n fix one remainder term happen apply technique technique arbitrary explicit assumption indeed h e x schwarz inequality algebra measurable rely weak equivalent replace expectation see asymptotic field e technique expectation h h convexity measurable eq cost form satisfy recursion h lead classical martingale amount identity n k h h I inequality amount consider sum e replace h proof expansion assume noise process condition add I decomposition uniformly zero x expectation x r may strongly recursion type x h n cm r h show also covariance h h h order independent k lemma h triangle inequality norm r r imply tend moreover integrate sequence increase use notation provide initial expansion n turn adjoint nh almost surely p r n eq monotonicity valid constant proof recursion follow control expansion start measurable increase h h n lead desire result proof technique semidefinite h quantity r h inequality b h h r k n x p e p h n n p turn p close form h lead similar argument I h h j r p j h r r r impose x surely p r r infinity r lead use f lead thus decay power decay desire rely mostly approximate minimizer quadratic term newton section effect consider f f quadratic favorable assume favorable error show two initialization average step quadratic denote separate need start around f f step lead f n increase use may check expectation lead need behavior step stochastic bound fine deviation prop n r notation n n cm gradient proposition recursion q recursion lead q k r surely p p previous statement bound still recursion almost surely previous valid get e rely order derivative global non function behave weight equal norm weight hessian bound optimum value integrate note eq prove u leave exercise next weight semi z imply desire denote follow give approximation bound expansion lead taylor note follow excess expansion integrating leads result excess look prop weak order stronger grow f n h integrate traditionally specific notion see quantity point originally convergent slightly hessian newton improve newton newton f f optimize newton bind key bounding optimum h newton optimum optimum reasoning appear twice preferable get prop moreover prop prop simple eq e newton show bound h bind newton prop prop two assume one thus bound f reasoning zero name result section figure objective make sparse sag step converge cm theoretical step pass cm theoretical pass cm cm step pass cm cm cm effective mm paris france unbiased gradient know achieve rate supervise least construct quadratic function complexity provide asymptotic extensive standard machine benchmark show become engineering amount practitioner typically observation still remain approximation predictor pair see difficulty twice differentiable strictly low proper step size sgd achieve strongly achieve bound issue typical close paper convexity still play central context optimization smooth lead remain square rate assumption precisely regression desire rate construct successive approximation loss descent generalization experiment benchmark outperform approach approximation refer use average lead generality assume invertible minimal subspace surely arbitrarily global n unless stochastic square recursion define start consider average h n h denote adjoint operator e see note least problem square section note surely point share newton trivial adaptation equivalent surrogate n f thus approximately replace gradient twice complexity step aspect strategy em losse sgd decay logarithmic thus
condition nd th mnist computed spectrum online order stagewise order condition make gap develop top singular efficiency contrast sharp spectrum suggest magnitude proxy news error stagewise multiclass key work conceptual make show similarly develop distance prox nesterov free quite setting simplex amount minor modification believe old algorithm research monotonicity smoothness convexity upper consequence eq piece q mahalanobis also algorithm yield metric induced gradient descent part nesterov induce nesterov link glm need assume noise link seem realize base rich estimate convex grow reason make assumption inverse duality inverse function duality convexity smoothness describe use iterative maintain ki operation brevity denote similar prediction always guarantee establish boundedness condition tt proof little assumption duality lipschitz regard strong convexity conjugate strong specific brevity property linear weight upper error proceed bind denominator final step solve bind almost substitute combine inequality replace I require easy recursion simplify lastly consequence operator eq alone describe variance performance notably seem need substantially median trick point format use separate load implement matlab computation fit processor expect update information stagewise procedure value remove word appear predict whether news classify economic market belong role fold token already news perform preprocesse split rest stagewise pick projection order news batch batch show b though stagewise produce well condition microsoft york ny microsoft microsoft cloud services le song college microsoft work provide simple setting example essentially iterative update front effective use substantially order stagewise achieve package standard mnist pt develop robust feature quite minimization multiclass multiclass logistic loss solve easily deal single empirically find natural audio typically ill condition problem generalize henceforth glm slow dataset decay alternative scenario conditioning crucially extension learn glm simultaneously variant tackle difficulty encounter apply second problem develop coordinate style stagewise regression solve batch substantially fast several art glm value present hessian use free involve hessian problem theoretically immediate observation well assumption building idea global convergence despite convexity multiclass setting practically enable example current prediction svms somewhat quadratic representation update address idea stagewise residual demonstrate excellent mnist cifar cifar stagewise highly speed matlab software highly notably cifar procedure entirely development theory leave seek utilize class influence order variate govern importantly manner case hessian matrix couple issue hope fact label dimensional build rapid somewhat relatively high precision ill condition small newton idea natural style serial setting stagewise svms directly generalize generalized classification loss stagewise boost literature iteratively work square algorithm guarantee difficulty need contrast matrix throughout regression good probably reference think omp describe algorithms variant square henceforth glm address challenge multi set definition glm binary facilitate univariate argument let define convex calibrate loss give glm convex definition identity logit term weight suppose monotone expectation minimizer pointwise surely point since global function sx xx surely contradict intuition loss glm specify family optimal discuss choice context zero pointwise expectation model immediately definition elsewhere correspond link maximal monotonicity monotonicity furthermore glm immediately yield analogous calibrate multi calibrate loss observe multinomial logit loss fisher convex minimizer minimizer satisfie proof convexity restriction realize binary unfortunately rich bound lipschitz imply need class computational curse consider present intuition g g put dictionary glm overall let mean linear conditional generally conditional efficiently form accurate prediction weight improve answer quickly maintain ki prediction alternate fit step fit decrease time mention conditional base prediction polynomial noiseless issue issue handle satisfie lipschitz monotonicity constant calibrate update fact spirit learn yet prediction option fix estimation view might suggest handle noisy via cca generator iteration generate feature prediction scale face serious block coordinate descent fairly replacement fourier call repeat return stress think result block diagonal across class group boost project transform non linearity despite need theoretically variant clearly stagewise property subtle greatly example fitting frequent well fitting least frequent word challenge encounter analysis broadly mnist variety speed improve upon nearly art emphasize accuracy novel generation dataset performance stagewise less approach substantial text dataset strongly favor online cc time error vision nonlinear mnist mnist modern requirement nonlinear challenging instance k hundred memory modern machine representation train specifically feature various polynomial effective use calibration variant consist apply pixel variant seem similar consistently improve stagewise fouri block three alternative stage loop computation calibrate linear regression
treat patch reconstruct compute pseudo pseudo effectively dm per computation notably admm reconstruct dictionary method completeness train dictionary natural admm see direction purpose believe reconstruct separate processor converge hour training average nonzero reconstruction contain typical low edge scale reconstruction million patch imagenet reconstruct measure sparse measure patch current guess value calculate measure snr patch measure return h lar pursuit lar subspace pursuit admm lar map dm sense recovery compete noisy compete dm outperform recovery noisy correspond reconstruct natural accurately reconstruct experiment dm reconstruct tune natural equally robustness dm wide variety make dm dm combine combine projection simple almost demonstrate power combine recall put dm disadvantage dm require pseudo computation despite consistently support national ef university edu laboratory institute sense sparse overcomplete competitive noise high art reconstruction observe compressed sense nonzero case assume problem use sc overcomplete seek achieve cs sc pursuit hard thresholde multiplier admm relax convex address consider cs balance competing constraint present method cs sc dm wide intersection x ap bx monitoring vanish art performance nonconvex include protein paper sense compare compare measurement wish map minimum small minimum projection solve constrain costly motivation simplification minimum projection onto linearly qp lagrangian qp solve qp yield give finally plug motivation come non observation see inverse computed reconstruction patch significantly reduce pre solve stagewise orthogonal pursuit accelerate hard pursuit square admm final alternate resemble many projection appropriately give dm advantage achieve dm individual projection procedure particularly true dm map alternate convex intersection crucial nonconvex meaning note dm minima dm improve projection inside dm continue combine two projection alternate fashion dm dm intend consider paper dm perform core processor matlab implementations lar subspace author author cite implementation lar pursuit free necessary tune tune admm reconstruction dm use one tune choose recover average reconstruction imagenet dm grid search outside interval well another surprisingly appear equally choose well logarithmic admm power exponent power result image find parameter dm respective follow address modification quasi proximal case try approximately ten run performance dm reconstruct variety unit nonzero finally ask runtime require require pseudo include experiment attempt reconstruct ratio demonstrate middle signal increase undesirable minima continue close reconstruct vary result
exploit place autoregressive upon architecture yield gain statistical efficiency section describe review length autoencoder decoder start proceeding encoder inference leave encoder autoencoder encoder decoder predict encoder respect imply variational energy generative autoencoder figure three pick observation decoder representation decoder give letter variable shall later easily generalise decoder representation autoregressive vector hide condition upon eq weight decoder shall elaborate late concatenation vector weight bias autoregressive advantage later extra ask eq boost add alternate autoencoder convenience decoder distribution become complicated deterministic stochastic layer encoder perceptron hide weight scalar increase power simple amount increase capacity unit restrict connectivity adjacent large connectivity less share periodic weight share convolutional sampling start top sample ph successive sampling encoder opposite sampling successively without layer autoregressive visible sigmoid belief sampling present deterministic hidden fully layer omit furthermore hidden layer fast add deep autoencoder train principle yield autoencoder deep oppose expectation principle finding maximally shall first residual representation compress description source code theorem show description take hence representation would average denote back representation substituting recover pick yield code expect bit variational encoder posterior learning sometimes serves simultaneously learn often jointly train weight bias decoder upon calculate exactly performance eight uci repository model first deterministic activation layer connection rate validation step hide momentum train stop work well connect web evaluate mnist validation digit pair generate intensity denote number unit deterministic layer unit architecture unit unit decoder product condition upon momentum sample encoder ten confidence likelihood perform hidden obtain log mixture compare generative performance machine deep belief notably description column result perform unit result likelihood speed generation multiplication fold speedup deep layer stochastic unit layer deterministic use skip connection receive layer upper value train unit unit encoder pixel intensity probability log l rbm cd frame five play detector frame frame fully connect two remove layer stack layer locally connect follow autoregressive connect locally locally connect kernel autoregressive use order right object learn frame game deep frame generate representation different penalty rough outline car car except layer locally locally pair near likelihood game activation encoder hide activation row deep architecture capable capture high autoencoder comprise stochastic proceed joint variational free backpropagation sample train scalable convolutional objective sample approximation unbiased baseline inspire eq low unbiased baseline baseline taylor derivatives backpropagation requirement solve follow solution shape cubic high substitute implementation scaling capable layer equip connection enable exactly parameter minimum length feedforward implement demonstrate generative set uci data intractable autoencoder mapping representation back author probabilistic autoencoder generate iterative paper autoregressive network autoencoder independent decode
express term restriction long note limited beta binomial straightforward fashion multinomial binary feature unknown q hyperparameter number observe one integrate obtain g sum derive notation item normal colored partition posterior occurrence compute posterior overlap considerably provide dp posterior note general partition unlikely possibility full approximation correspondingly take enumeration purpose enable dimensionality feature choice hyperparameter cluster operator obviously method fairly instance item think exact serve gold evaluating characteristic large evaluate posterior posterior segment proceed towards inference acknowledgment mathematics box university short keyword instead statistic cluster yield co occurrence probability considerably exhaustive enumeration partition vast majority cluster dedicated high probability partition consider normalizing deduce partition probable perhaps clearly inference strategy furthermore optimal mass spread appear reasonable concerned choice rapidly make enumeration meaningful collection partition calculation base convolution probability kind proposition shall evaluate convolution actually partition latter proposition directly derive estimate partition consider use subset convolution variant partition model method find different dynamic efficiently posterior statistic goal search computationally involve knowledge subset convolution computing posterior pairwise occurrence remainder derive section numerical conclude discussion generalization wider present idea denote element item datum associate denote nonempty union order cardinality singleton item partition compute sum define distinction label switching item kind order intuitive singleton either unique unique assume unless note characterize particularly good adopt bayesian probability term evidence partition since alone force task carlo space accommodate arbitrary nonempty likelihood evidence show example standard however approach apply likelihood analytically approximation laplace single function empty input normalize note partition computational convenience partition widely prior convenient way strong used family express partition dp cluster equal order partition sum express function subset symmetric iterative word convolution summation partition write arrive cluster convolution convolution arithmetic repeat convolution operation call exact compute far use subset operation instead moderate extensive involve potentially lead large rounding error arithmetic rounding cause result fast avoid extra arithmetic integer software library subset involve goal belong lead partition merge numerous distinguish considerable meaningful conclusion partition summarize sensible average inference approach item evaluate shall co occurrence compute result occurrence detail
topological persistence diagram basis classifier machine persistence relation classify system periodic periodic phenomenon degree transition effectiveness detect conceptual follow time datum discussion direction fact differential system differential explicitly nc depends call flow equivalent flow onto direction parametrization vary equivalence flow happen occur topology variation refer value flow equilibrium local analyzing refer unstable unstable periodic connect equilibrium saddle global set appear merge split flow perturb external model equation sde detail mirror simple sde additive white intensity motion initial continuously initial old depend sde uniformly ode regarded brownian wiener force define differential define characterize follow coincide usual flow family hx x hx xx conjugate flow conjugacy periodic exist stochastic topological conjugacy entire crucially difficult reason number proxy temperature devise focus examine assess topological phase value parameter homology group instead path examine specific situation start system depend moreover topology homology topological system depend one indeed identical change cycle crucial homology due inherent homology persistent homology explain deterministic ode sde time condition evolve output increment sufficiently trajectory hence projection delay lag slide window give system consists shift ode small delay vector shift dynamic delay vary sufficiently intensity sufficiently sufficiently system static reconstruct dynamic copy reflect topological quasi static noisy interval describe topological tool measure robust delay coordinate slide want describe topology cloud cloud system cloud algebraic cloud constructs homology complex persistent homology homology generator vary diagram summarize birth death homology diagram persistence diagram necessary aid persistence homology identification rise combination geometric boundary boundary operator leave linearity chain space count count cloud vertex include grow gain diagram homology space diagram exist long correspond short correspond term birth death solely monotone construct overview algebraic homology book homology use recommend book overview persistent homology cloud persistence provide center move increase corner nest circle representation center feature indicate long imply dominant topological sequence step start along experience reflect want able significant indicator aim toolbox learn paper classifier study model decision region produce unsupervise supervise expect learn generalize unseen interpolation machine create example machine refer periodic noise tend trace trivial degree homology find trace curve phase system periodic force period trace surface trivial measurement system correlate presence type system capable detect presence highly homology feature produce persistent cycle periodic regime significant length persistent homology persistent persistent periodic quasi periodic value indicate recurrent b slowly unstable periodic vs highlight due plane window take practical limit terminate assign death stop choose assign avoid machine report herein choose persistence several persistent highly persistent bar infinite bar topological feature use persistence intuitively bar next bar window regime possible order pair length recurrent quasi regime persistence distinguish intermediate yield able around center qualitatively regime tag dynamical minimal intervention base collection linear unsupervise scheme train persistent bar persistence medium persistence low persistence value regime tend example call unsupervised effort tag supervision beyond save available work break significantly exceed effectiveness persistence automatic vary global lastly investigate temperature co record one classic plot regime periodic circle topological conjunction metric persistent homology bar code windows trajectory manuscript tool cluster datum window window regime confident cause decrease period cause issue highlight strength highlight region contribute uncertainty limit cycle exactly history model discover sensitivity initial find decade research real fix classic observe condition trajectory resemble classic classic topological could behavior condition dependence exhibit symmetry rotation axis symmetric case consist topological perspective well remove trajectory parameter regime unsupervise three take point compose time classifier tag window clean instance partition two class little partitioning near separate two central interpret occur certain resolution class ice core research ice core aspect year ice temperature proxy ratio temperature lag observable small lag poorly understand aspect record analyze window two regime record use window classification figure distinguish regime find marginally region regime possess definite trend fine enable short window aid sparsity analysis breast cancer distinct regime several regime internal homogeneity topological perspective cluster persistence vs temperature coordinate persistence protocol dynamical system dynamical topological powerful regime scheme majority separate
model blockmodel commonly evolution model central develop involve kalman tracking application augment email email statistical fashion temporal extension closely model temporal multiple independently membership sbm specify gibbs simulate anneal temporal mixed version sbm model node major paper probability inference static stochastic blockmodel individual adjacency matrix observe node edge member give time denote class stacking index entry index denote entry snapshot matrix form class adjacency node submatrix dependent setting membership assume simultaneously vector denote edge edge block parameter posteriori priori happen set posteriori include sampling switch use combinatorial membership large exhaustive extension blockmodel state time priori set equivalently rather entry identically distribute tn ab ty ta observation iid entry past view state generate model specify evolution walk commonly refer apply rectangular box unobserve quantity refer logistic function entry generating perform initial state mutually independent kalman kalman filter employ cluster procedure social million direct email week week send email role company g role place tp estimate probability vice notice week priori solid week edge probability suggest knowledge begin examine variation logit first apply variance examine temporal interesting trend increase week week confirm probability show week trend highlight confidence edge steady six role fall investigation oppose uncertainty static sbm interval next static link predictor time observation address static sbm equivalence node however combined predictor operate link move combination level receiver operate characteristic roc curves posteriori posteriori well priori assume roc link correctly rate fraction non predict alone accounting block level dynamic utilize dynamic network propose know blockmodel setting priori optimal extend augment email interesting trend examine steady situation investigation show email activity believe reveal dynamic many dynamic acknowledgment office nf xu partially edu iii significant effort development analyze focus model static snapshot modeling offer rich phenomenon propose dynamic network extend know dynamic modification kalman
adapt study learn dictionary problem direction name svd eeg eeg thus kernel case aspect temporal learning view conversely temporal shift dictionary base cope shift invariance temporal eeg paragraph already eeg temporal dictionary shift invariant decomposition try variability add degree improve multivariate account spatial flexibility consider context eeg mp omp multivariate totally different keep atom channel multivariate right atom form flexible atom multiply paragraph shift way formalism use section atom omp mp coefficient contrary multivariate omp omp describe previously current atom denote derivation maximal extension consideration concern index trial two omp multivariate step end follow article method shift improve eeg activity localize eeg additive eeg statistically stimulus drive propose learn kernel experiment interpretable low snr eeg highlight representative comparison competition iv task eeg signal hz trial ask perform four trial compose raw filter hz band dictionary compare atom sample frequency atom plot fig give kind atom choose channel content channel learn fit signal smoothly contrary atom bottom content give different dictionary subject learn dictionary omp dictionary atom one approximation compute reconstruction blue dot blue dotted green solid well green well phase atom finally adapt code take transmission generic power generalization test remain subject subject sparse compute plot look black star inter representation dictionary eeg moreover note denoise h experiment signal tolerance ms sample channel spatially filter enhanced pattern amplitude give similar whereas pattern temporal maximum pattern fig multivariate dictionary quite smooth exhibit supplementary behind difficult analogy pattern previously channel choose reference simulation signal create reference spatially filter eeg add ratio ga secondly estimation dataset shift correlation htb average plotted shift recovery ga ga pattern shift average overlap temporal deviation reference shift reference result convolution confirm thin shift extract due invariance shift integrate eeg processing often carry experiment rough temporal shift invariance flexibility improvement well say estimation eeg improve experiment activity interest apply negativity n seem prefer ga l estimation moreover observe channel influence influence linearly provide configuration ga parameter multivariate square b observe able opposite phase spatial plot source head extract fig eeg dictionary base characterize profile obviously eeg multivariate necessary represent fix eeg repeat robust user inter interestingly extract context property localize context eeg interpret distinct eeg signal entail huge signal temporal consequently good concentrate component outperform classical atom eeg simulate datum generate realistic eeg signal recurrent experimentally experiment diversity eeg consequently believe candidate realistic eeg generation secondly keep mind generic dictionary accurate activity wavelet various eeg represent dictionary flexibility directly particular interest potential potential conclude kernel informative eeg relate computer classical shift flexibility discriminative modify spatio parameter anonymous usage article address issue eeg way use analyze eeg drive adapt dictionary reach inter multivariate invariance learn kernel atom eeg measure flexibility moreover dictionary interpretable ability p learn pursuit multivariate invariance eeg potential eeg electrical activity produce potential old medical poor resolution eeg medical contexts brain computer eeg device relatively image fmri period low delay feature concern pt event potential potential electrical steady potential brain activity specialize brain example activation area know hz hz synchronization electrical device record activity wide area eeg indeed practitioner spatial good fourier wavelet dictionary allow spectral signal mathematical basis lack flexibility represent shape pattern attract interest temporal shift suffer flexibility represent eeg activity area consist complex eeg activity inter peak shape activity probably dictionary wavelet dictionary approach development focus drive algorithm take spatial eeg approach aspect bring generic remain eeg article eeg review invariance provide dictionary eeg compare interpretability learn processing eeg frequency review approximation dictionary dictionary paragraph eeg channels signal dictionary decomposition dictionary assume code residual redundant determine constant norm know mp select correlate scalar eeg dictionary widely eeg signal hereafter eeg
homogeneous homogeneous homogeneity preserve non continuous balanced constrain program undirecte two negative homogeneous negative balancing ratio function minimization generalize contain treat manner prox solve problem inner possible constraint moreover note homogeneous euler identity euler always minimizer prox ratio one ff choice define minimize homogeneous prox decomposition choose f u homogeneity kf k c kp restriction homogeneity intuition strength prox near successive iterate proof inner moreover prox ratio give prox choose prox sequence use produce prox show strictly minimizer special prox monotonically additionally nonnegative produce prox satisfy sequence terminate terminate rf kf kf divide converge convergence strictly prox convexity however contradict large condition sequence connect interested want produce prox f minimizer terminate iterate thus termination termination reverse implication hold termination allow restrict balanced cut though immediately problem collect prox purpose submodular f ss sc hold set nonempty extension maximal consider class symmetric property convexity lemma imply maximal balancing sf lemma always well graph balanced cut show theorem generalizing partition always spectral state prox terminate follow accumulation point directly relate optimal accumulation prox set get I accumulation theorem thus extension also boundary use reduce prox termination thus situation general prove prox thresholding prox p prox terminate finitely ff either strict terminate finitely terminate finitely equation primal dual hybrid terminate build vertex avg avg avg cut ratio cut ten prox initialization initialization interested often trend improve compare correspond confirm ratio balance bad performance subdifferential extension seven graph laplacian initialization graph initialization prox perform solution extension mean extension cut worse well cut minor balancing consistently cut prox method spectral balance loose tight
shannon entropy come generalized deal scalar direction firstly give general secondly moment bias fx h tx fx arbitrary one build transformation highlight score characteristic positive involve essential ingredient generalize obtain identity generalize propose derivation notion define jointly integrable respect absolutely integrable function set q ta tx x tx fx gx tx equality ta tx ta moment inverse side interest scalar rao equality unbiased respect rao play role characteristic distribution vanish without mean inequality equality tx slightly eq dual recover generalize fisher gaussian inequality measurable function vanish involved integral exist distribution rao inequality eq gx follow equality equality condition pair distribution direct rao minimum say prescribe gaussians entropy fisher information entropy entropy q shannon entropy continuously generalize I generalized minimize fisher entropy result let theory allow nice entropy minimizer inequality involve quantity generalize inequality generalize fisher rao corollary entropy version identity link entropy identity generalize derivative generalize fisher derive rao generalized fisher fisher rao generalized physics entropy moment minimize minimize distribution relate boltzmann moment generalize also know classical fisher estimation minimize distribution extend rao estimation useful boltzmann shannon information de entropy suitable generalized generalize distribution state notation wide classical identity heat laplace operator heat equation mechanic heat transfer mathematical biology reference medium diffusion medium equation include laplacian f equation doubly coherence typically lead euler self initial dirac doubly
almost give utility mix car forest decide noise randomness monte noise arise bootstrap effect noise monte experience error ij variance carlo careful ij estimator monte underlie develop version estimator ij need replicate rule evidence rule bias ij arithmetic estimating result forest analyze rule validate analysis focus bag bootstrap replicate form replicate studied directly bootstrappe usually require large base bag bootstrappe around apply considerable interest bag meaningful reduction bag fail bag analyze bagging produce however somewhat far present main computed bootstrap replicate predictor random analyze z ix iy spam spam train could mail quantity bag aim base datum version replacement sample respect bootstrap expectation form estimator goal learner make eliminate bootstrap sample contain bootstrap estimate arise call delta instead behavior distribution ever replicate natural experience estimate fortunately correct version correction application require replicate reduce replicate discuss average dot confidence band tree sampling variance signal reflect spike variance identify spike bootstrap dotted line ij compare adaptive limit highlight importance reasonable estimate unstable carlo noise fast less replicate monte performance sampling analogously accurate sampling namely well estimate bag somewhat bias arithmetic close mean forest widely suppose notation predictor forest extend individual auxiliary source idea encourage tree variance bag theoretically e variable split pool predictor split always random forest predictor choice forest particular allow class forest special variant bootstrap experimental apply forest reason predictor learner bootstrap replicate time auxiliary draw forest forest base learner check result hold extra correlation meanwhile correction valid random forest interval forest formula bag valuable use resource package mail spam part distinguish spam mail spam investigate dataset spam forest splitting variable highly forest accuracy gain deep get understanding ij error forest prediction plausible change drastically forest suffer quality could substantially conversely prediction appear remarkably error mostly constrain bias report certain mail spam predictor effectively classify mail mail probably converge vote spam appear forest forest prediction spam decision class spam forest appear mid confident prediction able panel display sample individual bootstrap forest bias variance forest trade forest california plot u variance across variance attain minimum meanwhile term mse choice trade variance minimization phenomenon idea back forest govern variance get substantial bring whole forest achieve fairly stable small correlation monte discuss monte carlo distribution bootstrap estimate appendix replicate control highlight estimator recommend risk ambiguity monte carlo bagging q monte practice treat computing remark notice bias linearly original limit get carlo treat ij variance interestingly carlo square computational difficulty ij bagging replicate exactly sample size extend drawn check still hold simplicity exposition restrict approximate term show appendix carlo bootstrap replicate primarily ij ratio monte carlo start error replicate matter preferable practice especially perform variance bias although depend modification estimate letter stand remove carlo bias correct bootstrap replicate monte mse make visit dataset develop ij participant receive measure well originally polynomial adaptively criterion study polynomial fit description experiment restrict decrease patient low plot derive error deviation compare predictor repeat realization immediately verify qualitative present rule monte stable ij u fix without introduce instability ij surprising estimator picture relative ij rule computed variance recall estimator begin simple develop bias draw expression suggest rest arise theory projection estimator projection insight behind effectively try estimate h immediately right expression independent connection projection originally practical recently appendix build case valid hold bar apply variance main suggest bootstrap suffer bias bias estimator decomposition decompose general point meanwhile suggest estimator term right second triple ij drop fact order situation exhibit take individually idea tree train sample ij however appear slight expect dataset fairly variance converge estimator figure discussion appear unbiased r cosine var mse var var var noisy var mse bias bias exception suggest interaction systematically overall idea mse emphasize heuristic plug argument use justify second high develop formal bias remain ij estimate demonstrate monte correct version appear well practice sampling view preferable method experiment random gain acknowledgment grateful suggestion three support stanford fellowship derive expression finite variance indicate appear biased bootstrap replicate eq degenerate thus meanwhile converge uniformly integrable verify conclude term variance variance namely estimate variance
compute latter synthetic breast cancer show competitive much sparsity sparsity several regularizer glasso fuse lasso elastic en see review glasso variant structure many two class suit problem often contrast en regression structure outperform en obtain validation pairwise simultaneously encourage may magnitude penalize pairwise prevent grouping magnitude overcome drawback propose cardinality penalty proximity regularizer allow proximal fista selection method lead algorithm k x stop satisfied algorithm acceptance enforce decrease report breast benchmark aim six absolute freedom correct classification pose less h mse show promising feature respectively repetition group regularizer term select feature degree en c en propose sparsity encourage zero magnitude accurately group shrinkage net shrinkage future lx lx
scenario intervals intervention rise spatio structure employ occurrence likelihood infer maintain employ track intensity treat spatial intensity allow evolve research focus model forecast value elegant present maintaining induce sample cluster hierarchical either value series poisson structure provide background poisson maintain margin capture alone accounting covariate predictor eq iid denote nonnegative variable eq identically distribute probability poisson initial finite independence yield stationary essence alternatively equation binomial produce binomial conditionally introduce restriction multivariate extend restriction produce margin independent poisson zero stationary binomial however poisson poisson draw produce dp construction variable write indicate member identify observe crp crp dp reinforcement property clustering expect crp dp dp thus series rate grouping series number cluster shrinkage cluster thereby yield prediction combine obtain generate choose distribution cluster reader alternatively membership one place weakly explore half reveal change dp concentration base rate high rate belief counterpart order autoregressive var compose process similar also possess characteristic match diagonal var parameter process show autoregressive determine autocorrelation coefficient especially var single variation note binomial binomial computation mcmc latent dp census var deterministic one expect model compare counterpart sampler advantage small sufficient derivation material posterior align observation know tractable count l ty assume c burden value count portion count use hasting poisson importantly count strategy exceed roughly induce restaurant crp sample specific identify cluster membership indicator crp last cluster assignment sum effect also conjugacy collapse indicator need rate auxiliary discard currently element eq occurrence month data notice parameter important vector discard burn thin remain result chain look mode reduction assignment hamming phenomenon census frequently spatially population dependent describe examine census census portion city central east compare census autocorrelation value calculate autocorrelation separately adjust value wide range slightly reason raw therefore standard noise raw raw small magnitude adjust multivariate leave adjusted describe sample mcmc ahead count mcmc mcmc average supplementary material past one week ahead week month week predict error indicate frequent value produce rmse high count statistically equivalent expect since well frequent rare summary average week ahead bias supplementary material produce small bias observe minimize square natural three reflect sensitivity department quantile ahead desire sampler value may provide prediction interval estimate distinguish require rise benefit intervention c rmse rmse frequency ahead last rmse previous show bayesian incorporate covariate forecast would associate interpretation cluster insight explore census explanatory census census population c incorporate population person dp prior yield straightforward adjust sampler describe population measure population size informative manner analyze count begin iteration indicate amount map person highlight feature person portion city person city center city city insight differ look emphasize count future decision important merely improve prediction week forecast week adjust accounting overall suggest model extensive add covariate benefit reveal grouping series consider measure paper forecasting correlate count series framework induce overall individual rate latter dirichlet encourage term grouping strength series shrink assignment remain evolve assignment create might examine finally broadly spatio datum claim across pt pt theorem significant predict happen aid spatial order temporal variation follow familiar smoothly instead spatially disjoint exhibit pattern region count serious motivate propose tool count value process discover within approach forecast standard providing prediction low count area across united city range ability interest occur long predict region dependence familiar instance neighboring experience smoothly region spread fine g track make devise smoothing methodology capital consistently united consist list serious report keep record make publicly website map boundary census census vary census status accord census due homogeneity census track surprising neighboring dynamic cc count census heterogeneity
multiplicative lipschitz generalize gradient also similarly derivative analytically summarize post prediction order test mean order hour discretization time interpolation gap ordinary split patient series objective ability predict value observation past absolute multiple specifically define follow true observation number various patient complete calculate pair observation assume predict time reading help formulate series randomly pick task times prediction see achieve vary improve optimal reach state exceed clearly overfitte additional sparsity help well fit htb series prediction traditional overfitte problem represent additional transition matrix result clinical health novel ordinary like include conduct addition regularization plan switch system research support grant lm gm content solely represent view like thank comment th aa ga ga department dynamical elegant modeling however difficult dimension small time overfitte method incorporate generalized descent map framework iteratively improve predictive compare ordinary series dataset support spectrum successfully purpose focus popular clinical aim develop method time assume behaviour capture transition corrupt value briefly time combination know priori real learning multivariate sequence matrix prevent issue present representation able depend hidden prevent build probabilistic maximum posteriori estimate series observation probability probability model make hidden capture linear emission matrix gaussian relation either maximization adjust prevent impose regularizers transition zero reduce actual state overfitte even hide pick laplacian laplacian element follow pa pa
stock al de de em r team available cm cm plus height em ex ex minus ex ex ex minus ex pt stock operation market play allow technology structure stock public time stock maintain education stock depend economic activity behavior stock price take trading stock propose stock pattern time suitably price forecasting strategy even automatic trading usually attribute consist traditional indicator compute volume trade stock feature technical indicator action cross three percentage percentage classifier exchange raw price maximum number asset trading concentrate preliminary focused stock integrate bm volume test outline subsection r environment team aggregate classifier tree voting system instance classify although majority assign threshold achieve vote forest tree induce draw original induce select split forest construction calibrate wiener default forest rate forest attribute indicator package move change roc day stochastic share stock follow occur price raise day occur strategy price discount strategy classify successful positive strategy reach strategy end net end return negative strategy adapt successful assign convenience notice optimal parameter stock strongly implement automate next usual one adequate observation apply leave except training occur form use observation denote observation reliable training process repeat forest time compare forest total return obtain confusion form represent denote classified ccccc indicator successful eq successful successful total operation return yield failure weight successful achieve datum class compute stock assume stock day operation setup final datum present stock great method stock return
explicitly computable however away greedy arm expect like arm combine exploitation xx use prescribe formalize regime hence small discuss regime arise would time horizon try exploration amount random keep hard satisfie reader p vector past p characterize reward satisfy away constant case demonstrate near optimal linear bandit assume achieve obtain bind result characterize sharp rich datum poor reward gap order behavior closely central notice scale upper large suboptimal namely exponential irrelevant tight dimensional poor regime number small nevertheless partial scale limited fluctuation limit explore noiseless degenerate cover case equivalent projection span suitable since component instantaneous reward cumulative wherein introduce base confidence bind cardinality upper improve throughout least arm appear good respect confidence dimensional regime even dimension regime estimate distinct phase approach regret suffer confidence ellipsoid confidence prove develop geometry regret incur arm horizon optimal matter geometry around regret dimensional short regime geometry quantify speaking require spread precise contain interesting closure hull denote ball direction concrete I refer cloud apply present exploitation wherein separate base exploitation reward incur policy well randomly present average realization negligible poor cumulative reward achieve theoretical right instead convenient risk correspond display scale extra multiplicative correspondence product arm netflix movie feature average user synthetic take simulate rating movie star rating feedback use simulation implement version estimate feature vector estimate free construct list whose ball uniformly list classical theory imply appear qualitatively incorrect regime reward behavior explain policy fairly robust uncertainty inherent qualitative approach would interactive realization recommendation unfortunately naive approach actual rating feedback user movie database rate movie bias useful notation algebra tt compute measurement posterior coincide reward unbiased eq jensen inequality simplification right side cumulative bound guarantee condition eq lemma proceed bound side cumulative q sub theorem rewrite inductive distribute independent zero eq hence bind numerator tx assumption take consider inequality last inequality give q sum sub note martingale cf conditionally gaussian give eq light condition employ define result lead desire horizon special adopt notation error follow lemma obtain eq recursion yield result linearity cauchy second norm chernoff eq incur mean split side bound integral work nsf nsf grant fa induce sphere fact support compute second equality linearity projection thus assumption fact closure ball assumption converse would support hyperplane hx h ball hx x high cauchy since pack disjoint volume yield invariance iid distribute pz p bind obtain employ union invariance obtain event distance indeed arm arm arm arm check correct mean straightforward chernoff norm obtain q combine obtain hold probability cloud decompose follow cloud choose fact assumption proposition claim service automate recommendation help collection new product video user history profile recommendation satisfy allow probe trade use linearly parametrize armed bandit propose policy low dimensional regime work bandit focus low figure provide simple establish netflix good prediction ever grow internet video scientific paper recommend history right compete allow albeit hard impact experience limit recommendation impact provide user choice practitioner rigorous mathematical largely tr tr
dataset occur population col population col show discrete laplace package ms windows worth repeat fisher evolution certain mutation single mutation laplace make independent rgb rgb rgb sciences simulate fisher single mutation discrete laplace laplace estimate frequency follow please visit discrete distribution discrete dispersion yx laplace see mass mass function h names allele sep mass outside plot rgb x mass mu mu g allele axis false type col col simple isolate modal allele around allele allele shorter long might follow surprisingly happen across laplace distribution probability laplace modal laplace normally central profile regard mass laplace define probability observe individual may evolution marginally observation discrete observe priori parameter jk mean effect estimate multivariate marginally distribution estimate central equation demonstrate multivariate multivariate visit package estimating multivariate marginally mutation model marginally discrete predict evolution mutation approximate population simulate fisher analyse discrete package package please visit simulate profile rgb mutation mutation population mu mutation trace sim note mutation rate range number frequent individual calculate frequency draw replace table alpha db type db db db fit compare mutation dispersion laplace mutation mutation dispersion dispersion exact equation dataset dataset population frequency b analyse dataset mutation change add rgb number sim false sim sim size number population rgb type n replace db type replace n
move mode mean component propose switching incorporate remove special conduct model likelihood model exchangeable bayes properly scale collection sophisticated compute modelling assume component evaluate empty implicitly chinese restaurant modelling monte carlo way dedicate harmonic importance laplace approximation substitution implementation study mixture invariance arbitrary mixture achieve valid intensive purpose paper partial answer specific estimator importance sampling importance approximate posterior mode estimate reach demonstrate method advantage symmetry reduce importance recalling section simulation galaxy use paper plug value close constitute converge regular accepted fact switch rao explore miss target quantity later generic correction permutation label hence perfect switching permutation permutation note rewrite notational permutation gain store evidence nest reversible jump ratio normalize suited normalise unnormalized posterior bridge portion bridge sampling improvement like shift bridge mixture split sampler vector bridge rao follow quickly approximation posterior symmetric multimodal rao drawback increase block type importance distribution usefulness rao inspire representation importance sample generate posterior maximum posteriori marginal map mcmc propose proposal equivalent computational produce tail narrow output estimator difficulty miss wide simulating simulating lead correspond everywhere positive rao produce alternative proposal unconstraine importance j conditional density representation albeit switch upon permutation conversely select symmetric mode simulation transform j artificial label necessary generating proposal hold q set computational viewpoint proposal term ignore importance underlie joint posterior design create label hyperparameter hyperparameter thus select gibbs correspond particle h n q alternative implement switch value compute sampling gain term thus ultimately acceptable difference dataset real use performance seven simulate x call benchmark dirichlet proportion calibration occur sequence simulation experiment remove simulation sensitivity conduct expect relative section seven large confirm cm construct derive negligible compare identify matlab study give table switch gibbs significantly contribute I md ig round ratio md ratio follow proposal label method density particle sample particle sample iteration impose scale effective ess ratio replicate modify provide equation stability proposal sum evaluation setup demand remain bs mixture regardless switch sequence sampling albeit see figure dual table always posterior label switch ever present reduce maximal factor demand bridge cpu ignore disagreement versus fail properly observation calibrate I occur gibbs prior correction effective mixture bottom effective bottom naturally occur gibbs outlier panel discard ht evaluate normal model summarize figure scheme agree separate improper inaccurate approximation observe increase exponentially distribution get accurate importance posterior galaxy tend long flat chance overlap magnitude provide efficiency datum c approximate galaxy ht bottom four fit top panel discard ratio six fit galaxy top leave discard consider evidence mixture challenge high mixture miss produce study exchangeable prior likelihood exploit pointwise perform poorly derive method second practically mixture scheme thank symmetry mode separate importance extend case gibbs close suffer consider extension parallel extend importance investigation model label switch marginal central drawing inference due chain phenomenon label
implement technique thresholde whose network integrate classification brain patient relationship bn cognitive recursive method topological adopt predictive recent deconvolution detect direct correlation direct indirect effect remove indirect length eigen publish method predictor competition procedure predictive interpretability occur time brain record sampling trial approximate frequency whose spectral coherence frequency row define notation representation deconvolution filter interest approximate split part balanced development spectral combination frequency follow linearly scale eigenvalue decompose svd rescale link indirect node distance procedure elastic split correlate variable introduce shrinkage follow phase follow step minimization cv tune cv b fr ij method competition benchmark novel sometimes bias estimate activity condition attention either trial successively rescale rescale filter balanced test trial report inside net selecting finally parameter split procedure analyze use network test method establish grow tree look geometric try classical I svm metric rf hamming net key matrix involve net activation discrete cosine transformation consistently finding alpha hz brain hz exhibit cv error classifier raw accuracy surprisingly rf probably embed I reach near chance imply topological cause symmetric nature network deconvolution except rf reach furthermore elastic svm hamming deconvolution dct elastic high despite recent increase paradigm classification fundamental characteristic topological devote network mathematical formalism notice might complex level like core validate network accuracy non hz baseline reach elastic net spatio activation dct basis I rf section via c com complex mark noisy dimensional structure experimental condition group distinguish arise convolution learn aim brain different individual sparse
counterpart difficulty infimum employ deviation varie appear amenable massive metric possess approach instead optimum moreover massive range univariate emphasize property univariate scenario probability convention follow close minimize support next since compact interval precede thus definition thus measure borel algebra notice show behavior check lastly dr choice univariate adopt notation notation borel circumstance take input univariate require rather avoid exhibit desire correspondence manually albeit keep mind define borel subset p x q recall correspond minimum optimality property measure element half convention indeed minimizer whereby supremum outside achieve mean construction p variable hoeffding henceforth discard optimum form distinct usual arbitrary result primal player define ignore integration ib satisfy option desire wolfe establish inductive grant upper derivation let mc tc la q simply note q provide recall simplification q bind whereby hold combine optimum least henceforth failure grant whereby grant discard failure least every h la turn meet la search candidate establish statement whereby desire follow state follow albeit line search first develop line q iterate focus summation whereby plug display collect result substitution end work could avoid literature assume whereby rather least arbitrary difference la yield consequently recall application neither rademacher voting apply early simplify plug grant q exist vc failure borel exposition please output suffice borel finite meet guarantee iterate basically establish mr grant la failure manuscript consistency variant replace one heart induce otherwise surrogate exhibit curvature boost combine effective theoretically popular reveal convex minimization problem due structure converge optimization minimizer convexity learner linearly mean singular fairly analysis must measure compatibility learner setting far arise new vc consequently hypothesis span generally infinite unstable topic great research establish adaboost adaboost exponential practical theoretical devoted logistic intuitive g consistency consistency current carry primary manuscript loss loss similar assumption practical regime coordinate crucially unconstraine early employ perform size lastly final iterate classification perhaps unnecessary separable along sketch use convert consistency guarantee case decay roughly proof outline defer logistic introduce extensive discussion particular manuscript essentially risk converge infimum margin come early adaboost work solution penalize former work particular excellent sample decaying condition tractable produce estimator namely merely take adaboost variety risk discuss fit well adaboost without arbitrary establish consistency decay risk derivative loss another way adaboost appear roughly clean separable relie upon idea relax margin develop manuscript concern relaxed margin behave share idea adaboost hard distinction weak find interestingly loss implicit boost constant present state fit modify constrain size establish iterate dual difficulty source appear unstable subsequent one help work must appear distinction prevent instead note classification develop collection learner crucial weight absolutely convergent formally consider hypothesis banach consider abstract additional placing weight elsewhere banach please vc dimension contain class finite denote denote conditional latter consider many case algebra always borel suppose define second notion contain every span dense topology collection bound measurable please f trees suffice borel indicator denote nonnegative problem source practice drop integration convenience simplification empirical function convex loss loss allow piecewise x core analogously denote denote definition value set margin algorithm coordinate relevant gradient scalar allow approximate learner step though state previously unconstraine lastly achieve return alg alg alg learner descent whereby la unconstraine separate surely margin weak probability iff control yield instance make fair mistake make norm moreover roughly axis allow unfortunately respectively constraint average margin separable quantity vary whereas study bad result bayes rate convention develop play analogous adaboost separable separable error tolerance stop least ignore choice om suffice achieve whereas om adjust class additionally empirical close grant correspondence pac empirical highlight problematic function threshold also suggest elsewhere incorrect suggest sequence property exist size om om something wrong seem quite easy indicate simply easy contrast carry confidence parameter prove prove cardinality hypothesis proof operate fairly topological adjoint first part dual property turn give infimum remove supremum good turn deviation empirical denote error convex risk quickly drop round dual dual correspondence precise rescale lipschitz rescale sum across result low quick selection go reasoning provide classification risk essential problem specify probability adjoint satisfie positive well along curvature provide make precise dual resemble include case logistic fairly concrete dirac entropy mt aa dual optimum suppose cb statement hold least exist r la inferior exponent guarantee force every help bad curvature reweighte positive margin random behave within removal display margin margin find curvature quickly course instead presence applicable weak role show development sampling h x attain rate well weight favorable risk consequence unfortunately look absolutely noise perfect nothing norm solution potentially predictor norm reweighte margin expression counterpart aforementioned curvature fact mean exactly study quantity la every adjust yield guarantee allow application modification ht rate meet thank mapping consider produce probability continuous another boundedness proof imply may also similar develop adjoint dual establish adjoint every recall element integral relationship counting measure cardinality manuscript identify crucial duality lastly adjoint adjoint operator space identify weak topology topology induce recall operator space elsewhere adjoint relation e pl infimum immediate copy direction within side display dominate dominate arbitrary eq might close conjugate last use early lastly property possible neither respective indeed banach closure threshold predict interval sequence sum indicator arbitrarily choice dense measurable density follow contain continuous lebesgue arbitrary closure contain measurable form proof lebesgue continuous merely slightly close assumption analog proof existence one condition verification satisfied whose hull indicator simplification suffice tractable decision condition turn proof adjust carry uniform distribution go exist unlike handle precede one consequently infimum lastly let satisfy avoid compactly support satisfie uniform continuity let partition cube let indicator arbitrary corresponding cube contain logistic gradient lipschitz twice check whereby line segment second gradient direct expansion inequality conjunction gradient identically time lastly convexity useful note reason increase dual interpretable suppose continuous whereby convex let origin everywhere continuous convex subgradient additionally next parameter whereby subgradient consequently proceed whereby drop consequently since arbitrary must hold close convex optimality separable somewhat justified suppose convex every everywhere take empirical measure arbitrary second recall property instance mistake order follow close topology topology I induce topology necessarily n whereby condition meet closure within space subsequence fail lastly additionally whereby eq nothing whereby firstly discuss topology remainder suffice I pl p provide lastly compact topology thus weak understand explicit metric prove totally particular infinite suffice totally reason infinite close consideration one arbitrary ball cover totally norm construction basic establish grant without duality established attain final p infimum weak subsequence weak weak limit infimum provide whereby attain minimizer weak convergent definition convergence infimum remain swap follow minimax topological lastly convex compact topological necessarily l structure z low also low iff measurable well whereby lower continuous since conjugate every lastly follow everywhere everywhere domain thus nonempty moreover simple pick subgradient along finitely finite continuous pz measurable
notation ar envelope proposal generic minimize tangent ar mail gamma know widely communication letter extremely reject variable gamma shape gamma sample attain currently probability pdf parameter rate gamma digital communication communication channel optical independent application random e e pdf generate gamma rv usually focus address several reject introduce show letter develop rejection draw sample pdf literature rate accept reject suitable integer draw proposal pdf provide generate arbitrary target alternative simple pdf rs work proposal key rs average dx c designing find sample thank depict envelope proposal direct generate accept discard otherwise first rs technique ensure rewrite alternatively present since finally take side two sequel contact function equal eq fulfil achieve obtain grow fast guarantee satisfied envelope concavity monotonically technique indeed independent gamma generality fair consider although describe method indicate eq proposal ar proposal variant ar h technique describe obtain technique
shift facilitate generalize gaussian q z k follow show multivariate asymmetric laplace asymmetric laplace skewness scale random w bayes factor analysis mixture modify analogous parsimonious analogous covariance matrix family modify component g g pg g g g g gp em iterative presence unobserve together iterate complete calculate maximize replace computationally conditional maximization step anneal surface enhance model base source component membership standard anneal auxiliary annealing em mixture attribute take make calculate ig z ig problem modify algorithm latent complete density multivariate density exponential modify anneal latent update expect define ig I moreover presence proportion skewness shift q need family model update g ig deterministic stop iterate user anneal deterministic algorithm deterministic accordingly classification component assign component membership component membership cluster eigen decomposition initial eigenvector initialize base inverting make determinant identity true scenario membership base cluster joint framework generality observation nk label q acceleration determine converge converge log asymptotic iteration acceleration information criterion schwarz popular selection bic maximize number free bic select factor model bic bic argue make suitable ig ig classification selection mixture herein bic fold family factor treat case investigate classification adjust rand assess rand use true predict unfortunately interpretation difficult ari chance value mm length mm component set cluster component pearson explain principal principal sized gender family group give respect good result classification contour notably classification choose merging sometimes identical merge non gaussian sometimes match follow consider bank compose six fit component component performance choose make merging component b development classification localization site uci repository variable region prediction alm distinguish I latent choose fitting table see scenario component f thus classification subset present challenge principal family htb result factor new covariance crucially deterministic anneal classification scenario superior family merge bring performance family develop mixture develop model factor herein acknowledgement support award innovation mm parsimonious shifted extend mixture impose constitute result lead parsimonious model criterion complete
method effort interaction datum research west south md usa calibration enable reject decision calibration require expensive resource component gmm demonstrate good baseline calibration unsupervise automatic trial part unknown part trial differ trial target trial negative trial differ environment like language transmission etc still discriminate environment hard decision resource environment expense supervise trial label calibration paper resource completely view straight recognition convenient model calibration follow target target score gaussian affine score target theoretical overlap decrease calculate reality label calibration score parameter straight forward logistic denote label treat calibration generalize component mixture model gmm let gmm class eq importance repeatedly remove irrelevant nuisance result find ph computational computation involve sum intractable conversely label shall numerator w simplify show use alternative help theoretically relationship expand plug numerator score denominator supervise hold posterior take q posterior give predictive remain peak peak rs score rs already score demonstrate dominant peak via peak suited low posterior peak peak assign form reasonable prior want assign likelihood eq computable zero hessian nd approximate call denote p multivariate gaussian summary mode hessian discard la parametrization obvious parametrization behaviour nd approximation maximum peak magnitude nd inaccurate wikipedia parametrization two corpus million pre corpus provide segment different different mixture diverse segment truncate million target million score gmm priori target enough choose extract infer calibration level since la finding likelihood optima know behave exhaustive experimental facilitate visual dimensional log single dominant peak exercise make plot part abc believe challenge proportion likelihood algorithm reveal la trick
partition different kind approximation marginal x minimize derivative analytically set couple equation derive descent fail structure analytically accord repeatedly furthermore x baseline induce unconditional unconditional program time encounter use pass instead ignore illustrate program variational argument default note logic parameterization stochastic outcome another dependency remain height height normal rand rand height extended automatically compute forward program call sample instance accord mean log accord reward gradient terminate trace accurate likelihood parameter mcmc everything regardless program parametrization emphasize three extension assume wish variational inference distinct ideally solve inference yield unfortunately help learn sampler implicitly instead fix instance know function find gradient similarly case structure variational field point complex still unconditional via program perhaps via rl gradient program online similar fashion inference separate latent allow represent document topic recall variational reward reward random approximately allow entire htb automate variational inference common benchmark network bipartite lda actor connect inference detail long gradient curvature far roll use descent optimizer stepsize stepsize scale experiment fast descent conjugate gradient optimizer fast performance solely additional baseline improve converge sample use also start conjugate mostly devoted case gaussian gradient program estimate highlight optimize suggest approach sequential refine require distribution automatically generate model inference generally conjugacy refine without process observation derivation conjugacy general hold arbitrary probabilistic program program variational efficient restriction probabilistic program particularly probabilistic program derive optimize perspective language simplify development resemble modern language mix element prior unconditional execution trace generate program execution trace output example program
real experiment criterion evaluation call demonstrate signal l coherence coherence measurement range uncertainty normalize range b cr outperform share performance compressive omp need computational iteration omp conclude paper representation cs multiplicative noise robust recovery signal greedy successful propose rl superior recover linear sampling exactly advance uncertainty distortion finite grid dictionary generalize sparse consider uncertainty signal optimization signal propose relaxation use relaxation relaxation greedy realize pre numerical datum life well performance sense sampling dictionary assume measurement advance however uncertainty affect assume space discrete grid reality physical dft matter space lead assume actual basis analogue circuit noise induce uncertainty signal recovery suffer model paper address relate pursuit perturbation performance dictionary basis process recovery signal allow operation similarly generalize pass amp matrix uncertainty several predefined structured perturbation non convex signal deal sparse estimation burden analyze uncertainty computational presence uncertainty generalize representation error representation error fitting norm solve obtain greedy convex sufficient simulated life signal generalize sparse solve conclusion instead sample formulate measurement many natural representative formulate representative entry significant vector sometimes nonzero significant sensing prove signal recover consider signal noise zero scenario analogue signal uncertainty result ideal sampling channel couple advance treat uncertainty representation result quantification dictionary dictionary advance error approximately treat variable take consideration error base additive generalized formulate representative vector vector encourage full np hard pursuit recover fitting match uncertainty degradation classical optimization lead incorrect sampling constraint I noise real fig tangent away one minimize original intersection contour minimize accurate multiplicative solution recover signal uncertainty semidefinite measurement without model obtained expect sparse q gaussian e eq new match yield sample representation uncertainty nonnegative balance tune validation optimization analysis example cause parameter quantization address error know p assume observe type replication derive consistent unbiased simple exist est address see estimate consistency error generalize newly find among satisfy formulate balance blue intersection diagonal simplified uncertainty variance simplify elastic net evaluate recovery kind solve convex relaxation achieve call robust cr equivalent formulation relaxation exist cr method convexity explain propose cr multiplicative simplified cr eq ellipsoid constraint robustness multiplicative noise constraint sparsity mixed tangent minimize ball line quite near one original coordinate axis additional quadratic induce slight term assume therefore convex relaxation cr rl optimization compatible vector rl cr achieve value n let see provide zero satisfied meet let sufficient bp cs agree multiplicative noise new measurement introduction enhance robustness gradient computational contrast omp choose one minimize residual e
subtle concrete g tensor tucker sometimes unbalanced tucker perfect condition random sharp square fraction correct certain failure black success nc standard matlab uniformly increment increment pair successful recover solve method augment alm accelerate linearize bregman include detailed plot pair white region produce outperform c sample non compare require compare complexity open relaxation speak recover object induce norm prove large lie result v n v inequality e n bin odd odd fix decrease k last similarly r k I valid argument claim u u last similarly c u split verify equivalent accelerate linearize nonsmooth linear nuclear minimization nonsmooth firstly objective e add perturbation nesterov verify unconstrained alm solve exactly objective easy regular singular unfold fold I ki k experiment recovery axiom claim conclusion conjecture corollary exercise notation assumption summary recover popular minimize sum norm substantially reliably length tucker relaxation partially succeed suggest also tensor completion nuclear norm perhaps surprisingly demonstrate minimize individual induce possibility reduce exploit naturally estimate multi index several continuous index high dimensional space interest much structure estimate variable classify audio processing mm ix pose progress exploit matrix tractable ill pose structured object high recover generic nuclear nearly contrast result tensor compute cp q general nuclear norm intractable formula work many study tucker tucker way entry q th tucker tucker automatically tensor cp recover seek originally perhaps substantially suboptimal regularizer provably ease state estimating element measurement I standard nonconvex strategy accurate unlikely unless substantial ideal nonconvex square require improve multiplicative theoretical motivation compress sense sharp rigorously efficacy scheme number qualitative realistic interest wide tensor literature structure surprisingly poor phenomenon discover et al recover object often structure induce nature geometric regularizers demonstrate reduce several jointly tensor recover near computationally baseline evaluating approach tensor tucker matrix unfold rank mx tensor pareto dominate say equivalent serve recovery fail occur significantly exceed number intrinsic freedom follow one probability theorem exponent cover satisfie q remain number perturbation tucker decomposition mode b u construct net net total net r r theorem follow convex algorithmic surprising provably recover underlie simplify suppose tucker succeed degree prove reliable unique imply tight negative efficacy nuclear norm although use direct much next recover behavior actually much discover structure sparse low convex relaxation relaxation individual combination significantly general nature cover regularizers bound correspond nuclear composite solution c g orient nontrivial intersection large precise inclusion geometry pt width unit x structure regularizer hull subdifferential likely minimizing single way circular angle angle various structure notice small angle dense contain small circular circular cone subdifferential regularizer circular large sparse r x achievable number measurement reliable recovery line square behavior affect make elegant integral problem phase transition around variant cone calculate property bind circular cone constant eq recovery proportional good structure whose lead phenomenon discover recover object tend powerful good norm follow subdifferential contain relatively circular central axis low mode tensor tucker lipschitz
since find grid residual interval example space h increase increase increase like pick minimize error cause pick small possible curve curve curve performance maximum curvature residual training another approach could perform calculation residual use batch validation point complete subspace batch batch simulation window choose residual start fit pair minimum table instrumental either subtract add index select randomly example calculation computation penalty correct ratio detect outlier outlier detect detect outlier outlier point perform dataset report rate correct suggest correct decrease dataset level outli outlier similar carlo simulation mean select point inexact correspond outli detect outlier identification propose output evaluate apply realize outli identification accurately trade aid suitable open space exhaustive consume way space robust pca feature alternate multipliers direction definition berkeley pose nuclear relaxation inspire robust framework take rank robust problematic suitable space need identification study within identification nice pose minimization interest nuclear give convex convenient square fitting minimize minimization compare dual partially scenario output author optimization fit study extensively learn principal analysis pca although identification seek additional subspace nuclear minimization outlier consider sensor agent solution formalize detect error iii minimization attack instant attack attack outlier order order impose trade simplicity attack nuclear robust pca exist point interest like include novel subspace identification regularization precisely optimization outside parameter suitable regularization derivation minor modification robust identification rest discuss detecting conclude section partition row equation frobenius estimate solve find matrix convex input formulate regularize norm q time instance measure output g equation detect output outlier introduce outlier specific outlier occur term term intend therefore detect formulation vector account occur norm norm enforce criterion version filter outlier allow estimate outlier appear valuable piece recover filter subspace identification section penalty force exist whenever whenever useful seek value space aic occur
level relationship still less retained share would demand model satisfy within experiment retain discussion importance involve preprocesse determine scientific broad preprocessing suppose vary broad work practically path preprocesse output feasible specify preprocesse purely computational setting conduct impractical generally maintain without retained proceed carefully exclude analogous include imputation realistic clear situation analyst select discard identifiability analyst know completely practical important establish fisher asymptotic estimator nuisance play compression lose compression question behave analyst procedure equation obtain use detail step asymptotic likelihood likelihood common support rank open contain second identity however crucial accumulation index constrain remainder term score identically requirement simply data suffice overview phenomenon reveal inference case unbounded actually eliminate nuisance see result profile likelihood likelihood also invoke asymptotically establish central missing determine procedure usual asymptotic mention performance loss hold analyst optimistic issue focus definition unlikely produce truth risk type regret risk community adaptive baseline raw risk truth baseline property asymptotically classical asymptotic inefficient long guarantee asymptotically uncorrelated precisely efficient yy rt rt view usual divide appeal preprocesse evaluating long restriction must carry idea idea cover idea miss provide preprocesse preprocesse even ideal play transformation change parameterization result sometimes linear preprocessing technique individually regime least reduce analyst semi dominate justification preprocessing appear analyst preprocesse feasible inference conclusion crucially setting accumulation individual nuisance describe growth calibration preprocessing effect efficiency nuisance inefficient inconsistent back marginalization nuisance typically even many problem minimal estimator careful preprocessing regime phenomenon stand imputation valid mechanism well principle inference base occur mechanism restrict compare traditional datum framework inference generally dominate relationship without behavior principled prevent well establish principle inference principle far nuisance parameter simplest eliminate bayes dominate base example would provide analyst limit feasible signal via g combination across could trade preprocesse remove nuisance improve robustness datum trade carefully design preprocesse utility original subtle many use microarray interpret unbounded processor analyst want consider nuisance ty inconsistent inefficient preprocesse dominate procedure presence nuisance asymptotically far belief technique rich primarily reference prior g invariance core preprocesse application setting distribution invariant absence censor cox require invariant transformation cox statistic excellent nuisance hazard preprocesse rank preprocesse rich investigation practical robustness realistic inference datum preprocesse preprocesse realistic normalization eliminate nuisance discretization greatly parametric appropriate estimator generally explore even appear regime tool robustness inference motivation extension setting case family procedure retain precise one dimension yy fx hx yy ty lead phase nice rare bayesian offer little integrating typically remove become approximately curvature analyst inference experiment individual approximation may hold mode thus remainder inference justify theory narrow model unfortunately justification many asymptotic concern implicit justification error away case apparent preserve statistic exist however approach possess favorable property consider analyst could design analyst analyst ahead input neither investigate burden final quality formally input actually second rule portion model construct input second analyst would admissible procedure stage result statistical fixed operate construction smooth strictly regularity admissible procedure bayes rule restrict conventional complete omit focus instead implication sketch proceed line argument geometry well behave satisfy strict finite real restrict face classical cover realistic show must broadly construction convex hull direct necessary admissible major practical difficult limited discuss inconsistent realistic regime however admissible dominate nuisance still behave way form input consideration world researcher database broad later input analysis inference well go rich area loose end simple necessity z x iy joint marginalization enforce x share information obtain strong result factor work conditionally independent yy yy iy hence conditional hold hence satisfy sufficient dependence preserve weak sufficient sufficient rather previously without sufficient minimize fisher estimator tight constraint restrict narrow estimate error utility theory little overall landscape correct core challenge theory require engineering insight deep motivate g preprocessing procedure nuisance outline direction look take moment look broad implication theory historical study inference role argue inference minimize rule play even decision fisher reject theoretic formulation interpretation fisher consider decision theory bring perspective fisher intermediate process theoretical pass previously generate final focus separation objective inference distinction build scientific reach interpret decision theoretic adopt phase certainly bring closure historical bridge think open systematic way potential preprocessing system practitioner subject severe degradation show form development call theory outline direction tool pass phase constitute minimal provide foundation technique sampler quite approximation believe purpose nuisance remain want robustness benefit preprocesse offer little focus partition principle underlie partial effect infinite dimensional nuisance rigorous provide robust invariance approximation technique little development make burden sophisticated significant effort computational obtain interest addition mind formal technique broad range realistic general upper pass analyst fix nonparametric provide alternative semi principled flexible incorporate subject trade share conceptual inference seek degradation inference agent attempt despite similarity nuisance core issue mi typical integrate analyst demonstrate nuisance practical role address nuisance parameter robust analysis challenge mi much draw history mi mi initially public mi separate deal make inference spread frequently tool deal problem inference imputation single analyst mi guide development valid theory trade equip develop exchangeable often inference true structure allow phase provide huge gain modern distribute example factor massive sophisticated yet parallelization necessary analyse chain carlo produce tool believe principle preprocesse theoretical potential become massive dataset acknowledgment would acknowledge award support nsf work win award david van stein valuable discussion feedback thorough enhanced wide analysis decision make constrain analysis collaborative involve party fall traditional phase particularly enter datum drive increasingly massive database become framework include imputation motivate foundation biology demonstrate inference efficiency robustness rich research principle tackle increasingly massive build solid sound principle investigation research treatment develop provide formal principle problem inference setting imputation extend situation information pass phase consist multiple imputation phase mi combine inference widely use technique within project biology science sampling increasingly build analysis project decide preprocesse analysis raw problematic decision preprocesse typically mechanism assumption constrain subsequent whole body preprocesse tool researcher deal provide throughput biology collect massive raw level rarely instead involve adjustment typically point become excellent illustration community project advanced obtain preprocesse calibration throughput biology face challenge share raw dataset size situation raw extensive genomic upon heavily preprocesse entire something generally underlie process separation translate effort involve preprocesse example analyst structure protocol complex well impose aim achievable efficiency practically currently represent serious preprocessing theoretically work fundamentally result represent great challenge build statistical statistical example analyse concern thousand parallel gene population upon rna gene use condition study change however raw consist intensity probe array group intensity typically model background normalization later scientific question address mechanism correction particularly crucial step typically move transform inference log provide microarray combination background sophisticated normally exponentially unfortunately available hoc correction search change phase power motivation quite scientific want become observation mechanism role preprocesse microarray extend beyond correction screen corruption precede technique affect analysis quantile array remove considerable systematic context indirect appear star form light star formation study investigate correlation temperature counter may correlation temperature may property underlying mechanism scientific continue hierarchical approach improper analysis issue improper lead incorrect estimate correlation temperature incorrect estimate appear narrow broad level demonstrate carry intuitive statistical inference wide range connection imputation miss datum formulate missing set mathematical conceptual need imputation address concept natural role connect comparison approximate back literature structure notion notion approximate relevant extension address combination excellent literature obtain combine multiple scenario development towards mechanic phase challenge bring contrast literature extensive design compression focus question yield relationship loss possible analyze class shall section formalize notion begin phase collection preprocesse second phase phase agent agent observe noisy experiment expression sequence intensity joint product factor model scientific mi analyst additional stochastic output theoretical description instead functional generalize carry procedure figure depict setup incorporate markovian process bayesian parameter nuisance involved analyst wants draw inference want forward inference yy sufficient development yy compare understand effect nuisance restriction underlie distinction fix impose process create scientific inference analyst observe upon analyst neither pattern selection scientific yy xx selection decision analyst design inference require tool design addition establish turn formally define subtle initially complete analyst restrict structure place far focus single analyst provide formal interface chain need indeed formally quantity phase direct practice build around property smoothness sparsity need output consist standard error carefully select cross capture inferential information influence capacity analyst adapt entry analyst unweighted analyst error adaptation formalize index entry select analyst capture analyst amount analyst foundation broad region estimate membership provide tool adapt procedure index correspond phase input drop definition flexibility issue long miss procedure use course consideration scientific amenable mathematical form basis focus statement meaningful naturally statistical include cluster often lack experiment involve underlie driving reaction measurement instance chemical affect assume create incoherence analysis create sufficient analyst analyst inference exclude obviously exclude possibility prior mean study current speak observation could completely separate process govern outcome replicate mean analyst sample replicate analysis biological replicate single biological correspond realization example quite aim open direction possibly theory find without trivial preprocesse tight monotonicity enable base clearly require bad news imputation rich arise imputation form narrow procedure mi miss consist draw observe markovian depict parameter second phase restrict repeatedly datum result spirit practically restriction analyst constraint intend analyst intend reflect capacity restrict analyst narrow principle special reasonably analyst family suitable analyst nuisance great theoretical interest statistical analyst analyst nuisance discuss would force nuisance analyst turn mechanic former consist rich realistic restrict onto analyst naturally investigation conditional broader former include restriction principle posterior interest believe take across variable interest accumulation preprocesse upon factored model explore furth nuisance role marginalization parameter largely extent preprocesse distribute explore requirement insight first language datum turn tool group researcher ease subsequent robustness measurement preprocesse maximally useful retain upon single phase optimal retain upon maximal reduction without information interest compression avoid impractical researcher first phase achieve preprocesse far complicated even theory preprocessing must statistic eq marginal sense useful analyst want imply require individual possess conditional eq definition baseline microarray probe intensity precede ensure obviously far say least scientific analyst relate construct scientific model correspond minimal statistic prior decide retain bayesian respect determine collection satisfy seek occur identify useful consider case practice scientific variable shall demonstrate long ideal preprocessing yet retain measure work xx exist measure assumption individual statistic e jointly additional form yy yy x factorization assertion easily establish analogous integrating expression emphasize researcher analyst require broad normality block structure allow analysis statistic sufficient nested greatest sufficient ensure validity perform check check specify although potential hierarchical still among independently class model across interest consensus suggest necessity communication practical even division permit hold formally factorize work necessary
reject detail augment omit section score measure rest xt xt xt nan score uniformly uniformly valid value collect reject hypothesis refer inductive method advantage first algorithm confidence part rank note function assume exchangeability least fall unless seem inefficient splitting greatly burden function augment result euclidean function z augment result modification augment case modify majority could define omit expensive splitting band band however score affect example mild random quantile band limited challenge extract information score optimality set even consider construct prediction third minimax vector choose score functional question computation visualization datum capture curve projection characterize prediction construct band band p functional principal summarize general give functional band eigen prediction band sample coverage guarantee function estimate used algorithm one abstract leave challenge characterization use prediction estimator unclear inductive simplify computation motivate mix probability mixture smooth decay quickly truncate fast decay correspond functional f emphasize band without projection mixture density function let proportion th denote density eq q ellipsoid close conservative idea mixture intuitively set make component overlap computed programming value show use search u b band implement consist behavioral center reach target environment curve show neuron record action potential detect specific array primary goal sort curve neuron neuron characteristic curve band three plot project curve introduce conservative mixture component well consist three plot analyze variant principal ease visualization summarize information assume observe exhibit non suggest model mixture band two component band wide heavy obtain density resemble account ellipsoid u kt n close use score however likely roughly prediction introduce pseudo distance measure look like density dominate nevertheless task curve estimator p space see approximated denote hx q ensure pseudo analogous pn describe explore first set typical see successfully shape group neuron high anomaly summary representative function maxima pseudo density figure signature fire sharp stay negative attention tree evolve change smoothly whose define level connect component vary index become leave single ordinary tree feature indexing density distributional density salient otherwise hard gaussian arise mixture occur subtle ultimately distinct leave within strongly component choose future dx plot belong extend variant inductive projection simultaneous functional band also view underlying region prediction reveal hierarchical salient investigate optimality tuning functional choice distance density show e somewhat coefficient cosine basis analytic analytic page sensitive datum partially grant support national science dms national foundation grant air grant fa remark explore university simultaneous band band tree prediction stochastic guarantee distributional density underlie ordinary computational cost functional real example research effort decade functional scalar vector perspective model longitudinal datum extend book exploratory visualization density nature visualization challenge functional curve notion ordinary component basis functional band project bivariate order band maximum sample quantile outlier
interval derive coverage interval consistently tune q interval interest asymptotically compare length choose computing coverage complicate close soft provide low soft coverage degree freedom either conservative consistent tuning fact coverage proposition degree freedom infinity base theorem variance symmetric n base infimum inside start interval nc interval hard thresholding knowledge close coverage instead bind allow confidence zero formula exact coverage sharp hand side display theorem correspond coverage know case kx k given conservative tuning fact case interval nc corresponding assume contrast hard thresholding derive form interval soft unknown derive variance immediate together sequence non real soft thresholding thresholde confidence interval finite fast tuning equivalence coverage eq difference right display theorem coverage know less apply length confidence finite proof theorem show q coverage square length thresholde finite asymptotically estimator length interval standard see tuned length learn like estimator tune consistent whether freedom independently fast interval enough eq limit thresholding limit thresholde soft bound thresholding provide consideration case unnecessary figure threshold compute length n minimal coverage list minimal coverage probability minimal coverage list numerically coverage length c ex ex parameter threshold bind gave compute roughly interval hard large line htb plot thresholde leave soft plot distributional linear potentially number unknown aside look valid confidence set coverage coverage sample error variance coverage straightforward manner display become move integral rule proposition derive ds independence relevant equation scaling factor replace cdf measure apply calculation eq term manner indicator indicator display apply proposition inside square dominate conclude expression limit distribution formula proceed square converge square prove expression measure limit respectively proceed inside square prove conclude expression limit distribution formula part lebesgue elementary calculation yield prove find proof eventually display converge lebesgue simplify prove part yield prove part distribution find proposition convergence give display converge since first limit fix imply work analogously elementary write converge confidence hard thresholde orthogonal regressor identify note result soft analogously make respectively mention proposition proposition respect concentrated proposition always imply claim hard variance discuss begin cf probability proposition dominate display kp k e since bind p n converge show step subsequence well bound imply bound identify cn large globally sa n finally prove claim p soft proposition equality dominate theorem b n soft proceed coverage dominate display prove n n proposition converge reasoning n also c proposition probability show step subsequence imply bound arbitrary fact globally lipschitz elementary inequality elementary calculation cn n cn cn proposition note proposition respect concentrated prove proposition concentrated imply conservative suppose suppose km converge cdf freedom eventually degree expression degree cm cm assumption technology confidence hard thresholding thresholding number regressor version estimator always interval interval provide coverage know carry asymptotically degree freedom enough set soft adaptive thresholding gaussian regressor thresholding early widely gain context selection country regression test whereas ridge generalized estimator thresholde g soft thresholding forecasting importance property thresholding square derive asymptotic act asymptotic adaptive consistent variable selection asymptotic smoothly absolute scad also asymptotic penalize tuned act consistent introduction exception point framework context detail sample various adopt move asymptotic paper finite estimator orthogonal regressor potentially imply lasso adaptive within linear error version estimator main contribution derive confidence interval short thresholding small adaptive small interval thresholding asymptotically arise tuned model essentially tune interval variance consider introduce length error minimal interval thresholding length thresholding base least asymptotically arrive conclusion estimate degree freedom relation thresholding estimator section result distribution thresholding interval summary overview non regressor may classical situation square estimator associate estimator latter define square definite asymptotically hard thresholding via tune thresholding number component estimator error soft infeasible counterpart cc soft thresholding counterpart define counterpart specific depend often independently clearly infeasible aside require regressor except fix satisfy enough really exclude asymptotic regime probability perform variable distinguish thresholding introduce consistent shall consistently case conservative selection tune proposition consistent notation extend cumulative pdf normal distribution convention similar convention square root chi degree freedom sample unknown finite scaling consider paper property confidence proposition base non mention proposition qualitative comparison hard hard plot dot mass part finite cdf h equivalently dot mass adaptive estimator thresholding cdf z plot mass proposition th absolutely lebesgue represent absolutely continuous absolutely proposition scale consist mean scale atomic effect absolutely unknown variance non paragraph remark distribution move parameter asymptotic yield introduction need distribution confidence interval base section scale choose rate conservative tuning vary sample asymptotically distinguish case estimation tends infinity finite eventually constant proof make tune completeness converge since directly relevant perform conservative imply tuning satisfie derive hard suppose give factor limit degree simplify argument limit look soft estimator distribution n conservative tuning large k correspond limit freedom shift freedom simplify argument give limit soft thresholding asymptotic adaptive tuning satisfy enough true n I cdf q limit proposition theorem proposition large perfectly capture sample fact limit functional finite finite possess learn theorem surprisingly coincide cf proposition limit distribution possess atomic turn case tune perform tune thresholding tuning large km converge imply cdf weakly r tune true km h weakly weakly weakly cdf q imply imply converge soft thresholding tune large scaling km h converge imply weakly
true positive negative false positive negative median model sp se sp ar graphical method stock yahoo finance available consist stock consistently stock standard health care material technology service denote price day median graphical correspond color different find stock member category generally connection latent stock imply stock particular individually close perturbation consider technology stock price clearly separate h concentration rate kp result find expression get kp eq precision relation give component truncation true positive open concentration ball purpose bounding thus test appear p hellinger triangle find test vs vs ball metric subset consider space number diagonal choose constant make q choose prior please large hence rate lemma establish p difference pd p I exceed upon term give pd taylor pd p identical regular tucker matrix element kkt follow prove posterior regular graphical lasso ij maximization respect maximizer give lm remainder taylor expansion remainder subtract absolute element hessian tend tend thus note writing involve note like place b tend get tend suitable tend prove laplace probability structure series expansion indicator bound simplify actual integral ratio go laplace eigenvalue hessian evaluate indicator principal minimum theorem estimate describe absence popular prior put mixture precision matrix model laplace lasso keyword lasso graphical laplace precision frequently encounter fmri array precision component discriminant lda covariance precision obtain invert handling method estimation precision recent method primary goal regularization base natural series far correlation correlation high situation arise ordering estimation inverse tool conditional capture undirected precision lasso invariant frequentist dimensional asymptotic normality unknown restrict dimensionality operator mass mixture loading model develop family wishart develop incomplete decomposable matrix term hyper wishart prior conjugacy include estimator posterior induce precision develop put exponential graphical gibbs graphical absence mass extremely difficult base reversible jump chain convergence frobenius graphical structure selection lasso diagonal entry include free terminology selection counterpart error organize notation require underlie parameter derive posterior obtain convergence posterior discuss non graphical laplace appropriate follow section proof undirecte comprise indexing define serve excellent sparsity notation canonical zero entry edge denote symmetric order cone nx px ns n os nr english letter non bold letter vector english letter stands put mass event underlie bayesian lasso underlie laplace precision maintain consider identically distribute satisfy first decay also specify individual prior satisfy metric entropy indicator give eq value prefer fewer induce graphical hence jump monte carlo jump posterior extremely prior dimensional mean posterior appendix exactly frequentist give tend estimate triangle give precision minimize graphical lasso give computation various use laplace expand coincide u correspond parameter clearly derivative vanish differentiable calculus find I laplace probability graphical derivative regular essentially purpose previous section differentiable least model mean index solution notational element lasso solution provide big means graphical refer regular graphical give notational convenience matrix regular model follow regular consider
smoothing depend correspond processor accelerate able compute linear value time reach architecture separability large show equivalence see assume note k z ki conclude remark propose coordinate coordinate simultaneously accelerate parallel propose processor converge iteration counter separability constant distance without perform full vector bottleneck accelerate depend average separability separability attribute new safe independent utilize exist algorithm technology digital device increase extremely size area include scientific decompose piece full popular break take utilize architecture result parallel huge block block regularizer e contribution design analyze descent aware publish proximal counter nesterov coordinate descent convex originally nesterov proximal parallel notable yes accelerated general et yes yes yes st parallel zhang st primal dual coordinate yes st al yes st inexact couple improvement yes nonsmooth lee yes st accelerate yes yes distribute liu asynchronous yes yes method algorithm accelerate last highlight single notable choose accelerate coordinate descent list research paper propose table setting method much accelerate variant propose separable let function complexity however value consideration acceleration identify inherent accelerated nesterov impractical bottleneck operate dimensional issue method accelerate instance nesterov justify focus accelerate coordinate lee avoid convex modification nesterov extra sequence iterate compute partial linear combination vector form extend lee case lipschitz vector describe new compare exist comment subsequently proof efficient finally comment numerical convenient operator act wish lift zero likewise project back formalize operation decomposition h nh ns write derivative coordinate belong norm conjugate analyze accelerate notion review framework method informally sample expect sampling ni n f separable simplicity family parameterize fix change capture clearly choose dependent simplify parallel systematically method require norm lipschitz norm norm enforce useful random span block belong big subspace turn many separability update step eventually fast block constant assume easy satisfy similar bound uniform let satisfy nice statement notice hence consequence q add formula individual amount inequality function straightforward fix jx jx adopt convention expectation happen let convention convexity equality equation functional vector block lipschitz block satisfie jx jx u j ta ta ax ax ax ji ax u I ji apply cauchy setup block list difference grow equality nontrivial drive assume list l ji solve possibly rewrite form practical random block z k proximal start may vector ignore way necessary ever assignment place proximal parallel block differ done result assumption hold iteration expectation exceed proof comment need gradient accelerate coordinate descent exist simplify q stepsize case hence improvement happen logistic regression lipschitz run x k mean suffice parallel update processor descent parallelization speedup separability speedup degree even definition explicit iterate sum recursively proceed induction turn inductive constant combination affine fy fy eq complexity norm fy eq produce produce vector respect keep everything else h z apply identity z k k statement last definition expectation side rearrange term obtain obtain fx last use fact operation define coordinate costly vector hence operation simple e accelerate successful idea lee mean note use set block k analyze obvious start iteration produce express form avoid operation use maintain residual evaluation univariate derivative arithmetic average gradient cost small simplicity easily show residual usually strongly convex single iteration favorable situation
recently power node macro promise wireless traffic heterogeneous considerable work interference management coverage improvement traffic load balance macro cell dedicate capacity exist cell expect connect exist limited ip consequently design management solution factor heterogeneity body small management work perfect near perfect rely air interference allocation interference division multiple macro author scheduling interference novel interference management technique connect macro inter cell interference order quality type wireless interference management focus party connection management air scheduling scheme traffic share propose potential however capacity heterogeneous nature pose question air band require spectrum resource guarantee delay party delay fundamental question link resource wireless resource allocation paradigm centralize self idea heterogeneous access focus load balance load balance distribute formation bss game control game far self self optimize tool rl central designing rl node self decentralize rely local little exchange author study throughput delay bs mechanism cognitive extended interference cognitive cognitive network interference interest paper develop self interference management user cell wireless novel interference management strategy wireless heterogeneous act decode heterogeneous split part I coarse neighboring cell throughput tradeoff accounting condition cast find equilibrium reinforcement self implicitly transmission fully optimize utility tradeoff delay throughput significantly relatively interference conclusion symbol represent scalar symbol upper cardinality respectively use use table mm classical air set allocate receiver transmission power aggregate interference delay act capacity fine action play player utility utility regret boltzmann temperature balance exploration learn transmission wireless locate center serve denote set macro small radius small sub gain receiver refer total th receiver hereafter achievable denote sub interference allocation generality queue generation transmission eq act user transmission rate mechanism also adequate reason capacity achievable throughput air user achieve instrumental cell require account consider wireless wireless type wireless band user band band allocate band convenient capacity always service availability contrary integrate exist service wireless air rate ns sub allocate wireless cell generation queue capacity capacity per mention free unlike air suffer enable macro neighboring splitting build message fig combine message capable decode coarse message fine coarse fine message neighbor reliably coarse fine mathematically express transmission coarse signal power capacity share limited compare capacity main motivation split df transmission slot signal slot coarse transmission allocate coarse refer air interference cause user fine message rate fine message term message wireless fine allocate message df direct jointly combine throughput eq account half operation sum consist three due delay coarse message link delay path fine message therefore delay coarse give respectively delay component delay transmission transmission strategy analogous wireless access scenario yield rate represent message subscript interference wireless subsequently fine allocate fine message calculation splitting delay need channel share limited capacity limitation suitable formulate transmission allocate fine notational simplicity hereafter action per cardinality action utility action set action player consider throughput delay delay use delay sensitivity adopt play action parameter throughput delay tradeoff simplicity drop depend influence air crucial performance node mind formulate perfect knowledge existence action play specific strategy offer strategy correlate equilibrium ml motivation introduce match mechanism existence author player game adaptive surely regret mechanism allow explore correlate nash equilibrium nash correlate nash equilibrium match action utility regret action mt payoff play negative regret play mixed strategy give mt perfect action observe action instant amount interest partial distribute rely possibly optimize utility action lack unable strategy relie feedback individual instant action receive feedback transmission receive subsequently play action carry instant available play receive additive mt accumulate history balance action higher explore action capture boltzmann mt exploitation maximize result exploit action consequently conversely equally play temperature certain explore rest term mt value solve utility mt mt instant govern estimation rate converge condition average ghz bandwidth hz generation transmission macro cell per model radius number transmission power specification noise channel variation four baseline p description learn approach information rs propose information availability capacity aid baseline network scenario macro entire full maximize split utility achieve compare propose per number availability rs rs f air implementation propose yield rs drop share suffer interference interference interference low average varie allocate allocate absence fig see affect increase rate scheme reach improvement decrease wireless perfect notable reach rate demonstrate upper rs cumulative delay split wireless rs represent wireless dynamically split technique hybrid hereafter hybrid baseline fig fig see good delay contrast use minimum level low delay good rate achieve average rs rs outperform delay propose achieve delay rs rs approach case achieve high low delay leverage nearby suitable rs compare delay propose achieve delay attempt utility hybrid rs gain assess axis display throughput approach channel away gain proportional decrease scheme throughput sub move toward rs leverage quality able rate message yield increase performance advantage propose scheme fig tradeoff wireless remain presence message wireless compare leverage neighbor limited capacity link achieve oppose rs experience wireless compare rs rs high hybrid select good wireless rate cell exploit air air depend allocation wireless allocate air air throughput configuration throughput figure wireless preferred available yield throughput scheme incomplete e attempt throughput throughput transmission fig throughput tradeoff wireless rs clearly see incomplete high due temperature speed mix achievable conversely large play begin due result inefficient fig playing action begin exploration
initial model wiener cubic spline admit integral second model brownian motion sample see k two kalman smoother rely insensitive measurement validation smoother implement exploit previous detail contain rapidly smooth average validation compute fig kalman penalty induce vector size panel respectively estimate heavily come smooth time prediction variance sparsity formulation statistic g contexts mathematical provide prior knowledge class way improve ill pose dynamic kalman measurement aim sparse formulate measurement sequence solve preserve formulate sparse smoothing two mathematic early case constrain formulation project structure approach consider smooth straight exclude part constrain interior problem slack variable lagrangian correspond dual kkt system diagonal modify preserve structure definite point complexity smooth impose norm penalty value propose feasible require exact solution system invert spectral project specifically repeatedly project onto spectral exploit point agnostic onto survey extension smooth definite subroutine kalman present nonlinear describe linear nonlinear section entire robust kalman nonlinearity handle gauss newton exploit discuss section extension release extension novel kalman sparse improve reader tool constrain method system viewpoint smoothing department department engineering kalman perspective formulate kalman least highlight special algorithm equivalent equivalence establish present extension smoothing system process outlier measurement change preserve computational part package kalman broad inference dynamical gold weather prediction national kalman book write address modification use robustness bad topic system amenable smoothing filter almost equation interval smooth kalman elegant projection chapter kalman filter kalman smoother broadly dynamical fitting graphical figure extension measurement inequality state constraint model process extension key designing viewpoint formulate extension though year linear starting discover implement extension kalman smoothing singular smoothing smoothing state kalman focus leave idea future example smooth online application present recursive really solve easy discuss extension special preserve par another classic equation theory smooth process incorporate highly robust error incorporate extension follow q mutually know positive matrix point classic case know q mutually section classic gain relax formulate smooth solve formulate posteriori use theorem q posterior capture entire state give definition definition map give immediately specific give special exploit kalman smooth structure agnostic reduce straightforward definite review essential viewpoint smooth r k define system c solve upper triangular two filter substitute kalman relationship see filter represent covariance priori estimate easy game quantity information less induction put algorithm smooth estimate filter smoothed smoother apply work kind focus variant subroutine become apparent extension preserve block subproblem measurement line signal line kalman circle focus signal range biological inherent technique process treat brownian motion time interest k differential smooth numerical smooth measurement figure measurement guide true nice file coin square intermediate turn nonlinear formulate posteriori later broad gauss newton broad convex composite model smooth search optimum approach instead practitioner favor converge never foundation exposition illustration publicly available code van ode develop entire initial formulate nonlinear section gauss present gauss general gauss newton use iteratively gauss subproblem stationary stop direction backtrack line pick satisfied fact essential implement gauss describe gauss however define efficiently linearization reader gauss newton minimum along make solid blue kalman measurement display circle van nonlinear compare kalman govern ode contrast generic euler discretization specific situation euler approximation q truth van ground state x used simulate ground specify direct first show despite noisy measurement good file state constraint approximately encode box constraint state physical acceleration biological formulate state incorporate information inaccurate affine affine smooth interior ip directly optimality theoretical ip linear constrain smooth kalman improve understand review constrain smooth constraint reference nonlinear smooth smoothing subproblem smooth subproblem immediately nonlinear smoothing impose intersection hyperplane box one tool impose formulate rewrite equality introduce slack tucker kkt lagrangian lagrangian argument smoothing find subproblem exactly block diagonal indicator note smoothing interior composite indicator smooth reader include nonlinear smooth simplified approach repeatedly subroutine location distance nonlinear measurement nonlinear k straight simulate velocity x position model model velocity measurement make location locate know x encode feasible plot smoother unconstraine smooth occur contaminate generally heavy tail jump kalman filter gaussian continue accommodate non density tail measurement approach asset sensor secondary source kind anomaly stochastic monte carlo filter method intensive section come laplace robust estimation gauss successful computational notation mean density change laplace display influence laplace density dynamic model laplace map drop map know equivalent write generalized gauss methodology section apply approximate new form subproblem use backtracking describe take form auxiliary negative program kkt central central path method solve system form vector elimination b diag diag b diag diag diag diag diag matrix block consequently exactly block diag diag diag diag solve system preserve discussion incorporate approximate programming function affine equal equivalent subproblem laplace apply study numerical interval use time measurement v contain generate denote fraction contamination model variance lack knowledge recover noisy outlier outlier standard deviation thick laplace thin line outli removal estimate dot realization keep truth fix method table kalman filter iterate smoother iterate correspond mse centralize iterate laplace nearly smooth nominal condition perform contamination filter outli removal inherent outlier assume present lead classify fitting plot estimation removal smooth difficulty illustrate smooth van numerical experiment take correspond equation ground euler ground truth vector give model k identical simulate ground ground c panel estimation bottom error thick smooth dotted panel show boundary k x v measurement h simulation simulate realization ground measurement realization procedure table gaussian laplace smooth visual demonstrate relatively top two panel panel go van sharp peak measurement trick really smoother examine variation kalman examine function mapping concave matrix penalty application nonempty possibly nb type
introduce measure compare list length top list intuitive list relate encoding give rigorous measure implicitly variability list include measurement extent position measure kolmogorov complexity top list approach assumption important theoretic adapt individual ranking paper organize interesting explain involve content two section present information outcome seminal communication top list scenario top list independent joint list take denote extent additional state bayes theorem amount list bound measure addition divergence list I give foundation list symmetric apply bayes property early deal content compression symmetry dependent scheme rather corollary dependent content respective list direct acyclic inequality three list east west east expand list rearrange get I constant independent follow know top conditional copy list shannon list message take second aim redundancy list take bit information theoretic similarity completely shorter overlap position permutation respect appear distinct handle encoding scheme partial consider ranking differential expression domain gene label element page engine index page internet remain handle two piece element domain bit assume sophisticated belief uniform mutually order since rank ordering treat part code book stage list mask bit indicate position transmission state intersection efficient would binomial mask encoding require maintain count bit mask symbol divide symbol counter increment counter state b digit factorial system know overlap appear permutation overlap efficiently mutually book code transmission mixed factorial base employ define symmetric group symbol rank order group label digit require position index digit property permutation range digits permutation digits range decrease code identify binary integer code define space probability total add give overlap element state communication conclude element sort know modification step remain exactly previously knowledge list label explicitly efficiently remainder appear union step compress give code permutation achieved refer list length list size thus dominate cc quantify permutation assess measure permutation group rule weighted distance sort information content permutation important note adjacent element sort account combine contribution permutation individual performance list consider comparison vs vary increment possible pair distance grow mainly respective overlap set grow contribution dominate growth cost sense increase two list go get measure distance monotonically increase cause element permutation overlap net decrease comparison previous value compare popular web yahoo ask select news report google trend yahoo text uk computing use experiment vary trend distance avg cost drastically information new compare list explore rank list address use address important consensus rank multiple source ask numerous suggestion rank setting list mathematical face introduce compare measure encode variability consideration arise overlap list overlap actual rank overlap handle search routine assessment variability ranking much decade cite
gaussians correlate gaussians correlate model full full dramatically leading commonly assumption tie gaussians add apply feature gaussian transform back cnn start estimate feature map transformation eq far transformation demonstrate speech matrix however task I transform back correlate cnn use transform transformation specific adaptation layer discrimination combine layer explore look combine fed train jointly dnn style input feature dnn cnn small dnn computer come increase gain achieve feature possible combine cnn type improvement notice apply log feature dd stage neural perform frame gradient sgd ce criterion ce dnn adjust level function speech task objective speech numerous study ce dnn gain sequence compare sgd optimization training unit relu dropout neural relu dropout provide entropy english dnn sigmoid non dropout propose effective cnn technique hessian hessian matrix characterize idea loss cg curvature version gauss conjugate run cg fall tolerance cg newton product technique network forward operation dropout unit prevent complex co unit unit depend activation equation input layer relu mask dropout factor drop correct layer eq try quadratic equation gauss subset cg change guarantee conjugate mask cg way mask different work large dropout mask infeasible seed save seed dropout mask rd th reasonable hide beneficial dropout compare dropout sigmoid dropout mask iteration achieve improvement compare dropout mask cg investigation mask slow cg experimental dropout mask fix cg linearity relu dropout relu dropout explore link fact ce necessary start stop anneal achieve compare ce converge ce unnecessary weight relatively jump match ce ce section analyze cnn relu english hybrid adapt context frame dnn softmax output target follow ce dnn system train apply dnn softmax dimensionality dnn apply discriminative fairly system dnn feature old dd sigmoid linearity cnn systems relu base hybrid compare dnn cnn offer hybrid improvement old cnn offer improvement old huge target ce dnn system transformation hybrid table system rt dnn old hybrid dnn propose english rt dnn use dnn target hybrid describe relu cnn compare old hybrid cnn improve cnn hybrid cnn dnn hybrid help hypothesis cnns propose relu cnns sigmoid linearity rt dnn dnn old cnn cnn incorporate sequence share popular dropout cnn com edu deep convolutional cnns able well variation confirm experimentally cnn word cnn conduct share weight state strategy third adaptation namely sequence particularly hour news cnn hour bn improvement good cnn acoustic vocabulary alternative signal cnns offer cnn architecture architecture goal justify cnn speech investigate layer share find beneficial convolutional locality benefit weight focus part layer numerous improvement cnns computer vision particularly pool generalization max improve furthermore scale cnns output neural vision explore cnns cnns must exhibit locality good cnns feature correlate transformation uncorrelated paper use log transform uncorrelated transform back relu dropout hessian free cnns relu dropout entropy ce employ speech recognition performance provide dnn ce mask guarantee conjugate dropout mask change keep dropout mask iteration english bn improvement task help speech third improve improvement mask dropout avoid gain ce dropout relative cnn addition bn task organize follow describe cnn modification experiment pool relu dropout present improvement bn conclude discuss basic architecture baseline layer convolutional pooling use connect hide double architecture cnn able across english task acoustic hour english news speech corpora note cnns trained entropy hybrid setup locally speech remove locality frequency speech locality explore transformation apply improve show cnns canonical offer improvement adapt input speech recognition task region share approach span small region discrimination require frequency filter layer share layer filter locality constraint preserve work point alternatively share convolutional layer community convolutional layer allow layer convolutional layer locality preserve fed compare strong dd oppose convolutional optimal help variation improvement similar simple location prefer unit per layer conv pool speech pool input speak compare pooling characteristic speech bn table pool essential pool bn bn pooling help input max pooling region activation pooling training pool pooling pooling pooling look activation pool max pooling pooling activation pool tradeoff pooling large improvement computer vision compare pool strategy issue max pooling normalize activation eq create probability sample pool activation
naive elastic elastic net component split give low choose table error appropriate predictor group example contain whether group keep component exclude signal signal partition indicator place quantify signal signal component lasso splitting predictor rate diagonal indicate l l l misclassification l l l signal optimal misclassification close setting predictor evaluate area tend molecular marker dna certain genome year abundance molecular marker predict trait regression gene influence trait individual trait genetic writing phenotype genetic genetic molecular marker study yield type environment consist observation analysis predictor marker environment center sized set determine compare elastic elastic fix predict environment l elastic net elastic net net net lasso naive net lasso low environment genomic marker environment sort connect heat covariance matrix genomic marker example component interpretability lasso relate signal variable condition say column nuisance noise fall recover signal within signal oppose exist group property return highly happen identical equal coefficient elastic property case author absolute component net component lasso elastic subproblem connect coefficient fit elastic net package naive mode computation scale divide connect without elastic net negative thus linkage apply cluster run server use repository scale grow slowly operation see membership inner example nonzero product feature coefficient plus need membership thus future component split multiple number component must estimate exploit solve accurate real datum support extension setting outcome update repeat least might could non negative constrain logistic analogous model lasso achieve component contain exhibit performance help contribution crucial theoretical acknowledgement author helpful suggestion support grant dms contract sparse method split subproblem vector negative least vector select elastic achieve recovery modular also parallel net component variable usual setup vector center column intercept estimator important modern prove successful limitation setting highly tend occur frequently real number predictor highly group practical overcome elastic net improve weighted norm estimate ridge penalty penalty elastic solve elastic distance coefficient correlate predictor net solution non lasso identity covariance connect group correlate predictor adapt situation approximately severe elastic net zero conditionally version covariance regression recently connect inverse correspond suggest use component penalize lasso connect estimate separate problem elastic summarize remainder eight correspond block block block equivalently variable path square split path plot tune relevant signal variable blue variable component illustrate block sample standard organize include simulate real present make paper possible extension penalize criterion subgradient equation inverse block block split subgradient subproblem individually elastic net combine block create iterate find computationally induce block zero see diagonal one definite outside estimate become large increase lasso component mse setting false negative rescale lasso ridge regression naive naive net elastic net rescale naive elastic net elastic response predict naive net consist parameter observation set component generate example correspond early elastic orthogonal lasso differ non elastic example simulate exactly use cm ol hybrid elastic net elastic ridge low lasso predictor within component advantageous rescaling naive elastic net example original diagonal seem component connect component signal component contain fourth
limited consider event prediction task use pair base happen predict intensity take place homogeneous baseline select pair active task baseline predict follow numerical closed approximate fairly ignore parameter involve might extensive long history window self negligible summation h truncation inference corollary lemma conjecture sciences university california california usa self describe interaction approach fully observable interaction participant infer develop participant event validate synthetic world participant event compares considerable interest traditionally longitudinal limit manual consume survey sense online service location code interaction deal challenge social incomplete ambiguity come limited event record information participant might scenario participant miss distribute interaction event pair govern trivial certain attribute infer label component event label partially observable infer observation base nature generate event poisson intensity identification event attribute process intensity describe interaction temporal pattern statistically exhibit trivial temporal self previously mixture constitute infer even moderately toward inference validate check location service baseline inference task variational efficient synthetic conclude considerable recent time study wolfe among cox hazard allow intensity ref assume time cascade conditional dependency research network cascade trace cascade rely utilize knowledge self originally suggest number diverse assess portfolio detect miss information point process variate describe entity al temporal setting know impractical world generate learn unlabeled learn ref event ref less ref generative form interaction would edge network pair consideration computation observe event hereafter tuple involve denote occur pair process intensity interact within window intensity function past follow separability spatio temporal evolution independently note temporal intensity spatial research influence preference activity regard stay time likelihood simplify description concrete intensity event suggest equation g summation rate event self increase observe future use family describe compare decay profile distribute accord pair mixture q c distribution component component ref bic score number weight specific appearance cluster weight use justified modality school movie interact role introduction actual participant event directly need infer together gaussian latter select consist event miss label unobserve closed expression instead technique describe em simpler minimize divergence posterior variable hide portion know describe set matrix event multinomial variational describe present correlation make calculation tractable follow set rewrite multinomial work iterate calculate variational maximize overall pseudo provide tb complete hyper initialize pair unknown hyper report experiment six duration use repeat follow ml ref also relax constrain new assign probability rank participant three method trial accuracy ref show overall comparison meaningful temporal throughout paper measure express event hide indicate significantly baseline also bad whereas b symbol center point examine temporal temporal datum multivariate distribution simple analyze normal varied importance limit location participant hand increase examine vary performance baseline consider method baseline temporal comparative study figure datum become noisy reflect real service rest organize describe dataset conduct identity square km division department locate city identify responsible attribute explicitly event include intend address occur well dynamic dataset gmm website share undirecte also enable co occurrence list friend two popular place remove rule active collect mention social information participant record report event infer participant naive discard base infer participant label however approach account describe location record participant understanding recover well discard latter validate remain involve pair pair portion participant participant reconstruct identity outline baseline learn fraction recover participant demonstrate see information fairly perform increase perform well suggest simple missing label elaborate label significant label information perform remain competitive line inference datum experiment interaction
unknown large neither analytical general situation expression eqs obvious r imply energy minimize definite four derive rhs zero minimize parameter rate procedure aim minimize recursion study section lead minimize get minimize conclude inequality lead tend molecular temperature produce correct situation preferable fold continuously get elaborate finally less drawback number four value minimization partially global effective recover allow recover parameter know contrast modify em amount new parameter effective use locally obviously one step deep practice recall generally class fast situation work exactly hmm belong generally conclude key check fact express probability among discrete analyze basic hmms symbol virtue generating characteristic hmm completely ml value hessian eigenvalue physics biology complete main ml degenerate finitely degenerate solution identifiable parameter might show whereas impose additional outcome exactly definite compatible type finally mechanic behind likelihood temperature certain physical make physical second ensure temperature equilibrium probability relate connection type acknowledgment part nf sciences present asymptotic viterbi hide ml work seek analytical formalism hmm continuously degenerate degeneracy thus automatic scenario compare correctly hidden markov model simplest hmm naturally recover sequence state give estimating model conditional method implementation viterbi ml estimation likelihood intractable practice maximization em alternative viterbi literature em hard etc seek maximize hard em adjust ml consistency well speech parse generally accurate although task circumstance prefer hmms establish ml asymptotically estimate observation large however limit make impose process free establish loose qualitatively approach asymptotic previously entropy non comparative shown correspond certain free furthermore obtain estimation parameter identifiable find objective degenerate namely recover identifiable ml degenerate inferior may partially correct stationary markov realization take assume unique ps long realization observation observe markov average instead g function fix maximize kronecker equivalent outcome outcome statistical mechanic respectively gibbs hamiltonian temperature viterbi respectively maximization neither case maximization locally via trial trial repeat continue calculating average indeed introduce instance viterbi indicator obtain behavior product govern number formulation generate linear space norm multiplicative imply via normally maximal singular term argument introduce generate take via indeed small calculate instead employ stress analyze gray indicate realization transition light circle realization process state besides produce probability
condition see g excellent overview seem learn alternate related context rank negative matrix perhaps shorthand set denote norm denote element magnitude row column element initial alternate estimating coefficient estimate sparsity thresholding thresholding dictionary dictionary instance replace sparse omp square may replace computational specify assumption brief sketch proof step start paper always satisfy rip spectral non assume zero matrix entry universal universal failure need satisfy estimate parameter regard rip establish analyze compressed subroutine eigenvalue literature proof continue assumption assumption note rip probability w natural non sparsity identifiability specify initial estimate require recent provide provable way please specify sparsity decrease alternate method local alternate convergence iterate sign ambiguity dictionary element since exchange guarantee recover algorithm recovery consequence recurrence decreasing imply result globally alternate minimization since also lasso qualitatively section recent assumption obtain initialization incoherent without assume assume pairwise incoherence j coefficient eq specify universal give hold result initialization combine give exact recovery appendix show assumption crucially admit assumption initialization minimization assumption assumption theorem one iteration update understand square expand error collection apart denote indicate serious p argument control term level control magnitude lemma require rip assumption order invoke result compress deterministic eigenvalue henceforth form approximately readily continue sa bound away condition compress efficient subroutine establish satisfying depend least consequence well crucial controlling error bind cn p along main argument specifically normalize p lemma defer purpose dot distance motivation follow due proof straightforward case rewrite plot iteration refer incur geometric shoot iterative reasonable alternate al alternate significantly initialization incur alternate minimization incur trivial give error inner concentrated sample alternate minimization assume enough success trial focus complexity alternate initialization procedure initialize figure success alternate regime provide popular alternate commonly problem combine overcomplete favorable assumption learn understanding design well design well coefficient jointly allow force across number element control addition number extend grow variety factorization alternate go back motivated arising indicate class style convex present alternate lemma convenience lemma along proof technical defer first lemma p lem error recovery estimator recovery far constant suffice rip hold recall singular appeal guarantee infinity second imply element infinity move lemma apply matrix irrespective surprising forward consequence concentration theory lem every w eq support maximum inner product note pattern lemma lemma diagonal element lem diagonal prove follow triangle four lemma note decomposition invoke complement complement expression bind plug probability cn p r distribution linearity rr I consequence extreme recalling spectrum covariance matrix numerical hereafter universal r particular bound greater well large small value root complete norm least follow control singular matrix proof probability rs sr rs complete statement combine lemma diagonal entry see lemma nr nr nr nr part straightforward bernstein I consequently least state first n nr p r without diagonal use least second condition equation r begin auxiliary rip incoherent matrix eq similarly formula section rgb bold title title title title title consist atom mix popular code keep typically dictionary fix estimate keep variant alternate establish optimum coefficient rip combined result dictionary provable dictionary incoherent alternate rip sparse term atom specifically give coefficient code code learn dictionary atom yield field neuron localize speech code overcomplete dictionary exceed argue overcomplete representation flexibility video employ overcomplete representation art sparse heuristic alternate vice minimization empirical setting carry theoretical alternate minimization procedure dictionary use estimate coefficient whenever satisfie characterize alternate succeed require represent satisfie number alternate overcomplete incoherent dictionary al al alternate procedure initialize output al requirement r et procedure et carry local al noiseless unknown optimum fairly mild condition turn establish presence ambiguity exponentially solution via optimum limited square overcomplete overcomplete true incoherent simplifie consider set analyze program optimum number difference work oppose establish alternate explicitly minimization complexity requirement weak guarantee
common attractive study utilize meta signal value highly meta discovery assess discovery obviously true assess examine follow report primary side meta meta sometimes reject rejected establish sound essential high throughput involve severe examine simultaneously choice wider intuitive establish widely either control control false fdr regard claim follow either fdr favor fdr desire introduce primary complement table result finding primary main finding rank indicate complicated discuss meta widely fraction assess design design follow recommend price powerful primary value adjust hypothesis follow adjust method power typical hypothesis demonstrate however obviously come experiment microarray fraction study snps conservative guess bind call typical genome conservative gain input base primary follow genome variation value ix tie fdr exist otherwise si proof application r adjustment value claim say fdr property detail value coincide least claim show example increase lead maintain fdr report hundred snps small examine primary population examine also primary differ scientific importance discover differ study follow snps follow meta study hypothesis stage primary follow follow chinese association snps china china control china china snps primary seven snps value snps associations snps seven clearly follow table support si values si respectively six five second example disease snps cd examine european follow case parent affect region primary consider si decide snps associations value snps association correspond small meta snp value column si mark highlight follow value point example type diabetes discover snps snps snps case separate fusion choose study previous additional measured control descent combine additional follow study study snps follow follow proposal decide association disease formal family examine primary divide sub hypothesis nan study study analysis denote claim claim fdr among satisfie condition primary except change include select hypothesis procedure study value procedure select hypothesis small advance finding fdr selection follow independent hypothesis primary primary step arbitrary within selection value follow compute replace unchanged solution see si implication fdr control fdr level claim dependency primary modification si realistic study fdr finding proposal procedure valid dependency view type conjecture procedure fdr nominal level dependency primary follow si furth maximum parameter fdr need argue consideration identical finding si claim finding make threshold discover finding extreme discover finding primary recommend unless extremely choice ratio adjust consideration simulation detail si maximize choice snps signal signal study power primary power signal threshold si selection increase si typical snps severe examine primary yet severe examine follow genome recommend fdr value low suggest control primary primary follow parameter great specifically control data input otherwise claim si control chinese table four follow chinese far claim association design study significance however weak study every study meta compute least meta generalize whether find study examine compare consider procedure suggest two fraction stochastically much procedure severe composite discovering across study conclude bayes bayes unknown problem hundred thousand snps dependency across relative discovery suggest list multiple study discover gene suggest summarize establish move away design potential quantify value value testing turn fdr accordance commonly fdr show association suggest primary study value rarely main proposal fdr primary value conservative control dependency empirical conjecture conservative modification unnecessary dependency cd conservative result investigate dependency primary study comprise follow primary follow primary way evidence scientific scientific plan quantify least address line power point follow fine linkage association follow study need combine primary study detect association penalty discover discover study give utilize grant research science discussion history science thank comment substantial manuscript cm
architecture dataset whereas consistency apply scatter abstraction principle reflect ml execution atomic consistency aspect perhaps one room design remain exploit introduce program accelerate bound guarantee correct outcome synchronization dynamic policy take dependency parameter parallelization synchronization exhibit require towards converge implement allow size able ml reasonably ml practitioner benchmark ml program topic lasso metric library forest code develop principled formulation convergent program datum view iterative drive iterative convergent fitness margin typical ml iteratively reach stop update improve computation output aggregate omit subscript loss data parallelism divide common big parallelism ml big mathematical implication parallelism parallel assign worker partition subset parallel aggregate via stochastic optimize sgd intermediate variational algorithm additive allow aggregated worker produce individually additive foundation minibatch asynchronous asynchronous key validity worker distribute ml program worker contribute equally sense datum partition worker parallelism take scheduling operate parameter omit brevity scheduling datum parallelism enjoy property correlate parallelism global carefully choose scheduling space dynamically change scheduling correctness e minimize parallelization offer substantial speed converge lasso weakly correlate converge parallel program update running process guarantee ideally platform offer access pass model scheduling avoid capability available hence consideration platform tailor goal practitioner server machine resemble preserve parallel convergence fine grained ordering essence application server convergent ml algorithms q represent represent computation aggregation detail show parallel ml exhibit several exploit distribute structure parameter parameter limit tolerance practitioner principle component worker server program write future parallelism allow scheduling pick every complex may criteria identity worker schedule actual server soon explain responsible update later discuss schedule determine exist implementation round loop fit another scheduling variable accelerate parallel type dependency parameter execution exploit convergence ml subset kkt schedule intensive computation schedule worker execution computation responsible need worker schedule parallel abstraction storage system worker read memory disk system algorithm batch point automatically server exchange share resemble single worker finish new future scheduling ps access model share advantage principle ps implement parallel consistency reduce imply discuss guarantee later frame scheduling execute ps server ps ps return frame schedule parallel execute worker ps server read perform write ps update ps server increment frame aggregation execute parameter server read aggregate ps server schedule central variable server ps ps functions ps read ps automatically additional programming note early rd party turn scale principle focus strategy broad ml easily without trivial ml program code high metric allow similar enforce book book art output capture aforementione proper many neighbor nn aware distribute try mahalanobis symmetric pair I learn mahalanobis problem try minimize mahalanobis separate parallel relax slack slack constraint eliminate yield unconstrained constrain treat iid pair via parallel descent pair iteration minibatch frame single parallel learning schedule nothing ps server minibatch sgd server frame show server system read ps automatically ensure throughput asynchronous consistency parallel worker copy worker review consistency parallel scheduling worker example schedule implement study denote determine non convex function standardize loss let shall cd lasso frame single beta choose l independent beta correlation return frame computation call ps z return schedule aggregate worker j cd lasso upon choose scheduling parameter affect cd conditionally parallelization occur converge interface suit implement low schedule worker run separate scheduling schedule core optimization easy experiment scheduling schedule accord kkt constrained dependency checking schedule schedule solution execution include open source library word topic table machine simultaneous parallel schedule cycle disjoint topic machine update matrix accuracy solve fix schedule model update purpose connect deep neural classification project parallel schedule worker use perform amenable dl iterative convergent parallelism key ml program dependency exploit theoretically sound program big previously factorization deep system often ml ml property abstraction programming permit influence robust minor calculation still execute synchronization delay worker old delay tolerance synchronization substantially implement parallel system machine parameter worker server iteration guarantee receive stop minimizer worker x convergence ensure possible et stability optimum naive parallelization parallelization cause inter worker update little dependency allow user scheduling function per program schedule subset parameter generic include follow feature assume square regularizer penalty th simply context coordinate without schedule lead divergence coordinate computable schedule propose small meet parallelism let bb r use convergence trajectory parallel schedule trajectory trajectory proof theorem find supplement coordinate block nearly independent model prohibitive fundamentally combination greedy impractical regression evaluate exploit program empirically sparsity becoming avoid frequent update zero exploit scheduling output change propose magnitude check convergent analyze context rewrite opposite sign j cm lasso parallel th sufficiently lasso computation schedule add support program big baseline fast e exploit implementation reach size ml server primarily allow ml practitioner parallel ml medium automatic focus accordance different lasso baseline fix number machine consistently time speedup demonstrate parameter least initialize dramatically advantage relative speedup large ml times platform writing version achieve speedup number cnn ml ml speedup implementation time fast difference mf twice run memory nearly speedup mostly scheduling dependency aware execution well parallel fast parallel panel vs fast platform could handle mf hardware support ml cluster versus machine leave compare support allow tail topic capture panel mf versus support mf baseline scalability factor parallelism server storage development two source line code basic efficiency time speedup machine communication propose parallelization sgd execution implementation converge fast machine evidence cluster vary specification demonstrate hardware core gb ram core gb ram machines intel gb ram wikipedia mf cluster netflix dataset feature subset imagenet imagenet regularize select coordinate parallel worker partial assigned simply update follow proximity coordinate dependency coordinate iff role analysis trivial denote pair pass dependency achieve rejection consequence step worker p tf f bind avoid double index optimality long decrease objective put limit parallel roughly radius rest quick idea tc theorem confirm coordinate fast demonstrate tradeoff among parallelization correctness course big less converge proportional greedy significant potentially small big word radius totally passive compare big denominator vs big big small taking submatrix nevertheless may need coordinate simply pick coordinate execution schedule propose parameter trajectory convexity expect actually parameter achieve difference update taylor around rd since
large family select family despite optimize former opt ability capture training classify flexible c aa aa aa aa iy iy expect hand characteristic datum summarize give tie iy aa iy iy x iy iy else iy iy qx iy iy iy right iy iy examine readily increase size match search algebra among family test obtained find reflect classifier include clear interpretation continuously varied result namely phenomenon single definition algebra five family classification task english able reflect characteristic new algebra algebra consist expression construct value follow albeit dataset spectral often consist english aa iy two sound sample approximately proportion recall element algebra classifier spectral prior criterion classification small tie tie analogous obvious resort evaluate classifier family
time optimum cost dimension achieve time prominent particular online adaptive high dimensional random matrix semi definite psd describe finally control optimally pair controller describe adaptive system early controller show optimal converge controller solve suboptimal estimate aforementioned controller controller parameter method control small asymptotic extend provide cumulative regret control propose cumulative regret factor provide regret armed low time regret poorly focus state reinforcement application general achieve estimator arbitrary inaccurate unknown particular system equation accurately general even loop sparse furthermore dynamic correlate gain notion result estimate equip sparse perspective even cost due regret cumulative bound optimal contrast regret system appear engineering particularly motivated field dynamical four decade survey partial describe translate sale problem temporal interest sale level etc rich literature devise scheme include spatial temporal temporal extended state dependence pde pde equivalent abstract system concern control either deal discrete noisy infinite state g dimensional modern customer complexity interact interaction change landscape internet information customer bundle variable interaction submatrix integer employ episode construct include episode control episode confidence use observation episode geometrically factor episode detail code controller choose episode control dynamic begin episode cost measure fidelity constraint bound construct episode choose least reinforcement precision identifiable controller eq estimate l controller expect interaction achieve implement principle choosing begin episode controller code summarize guarantee eq state proceed introduce define ph control system matrix condition let index sufficiently word trajectory think quantification high dependency consider influence influence state indirect influence weak influence exist vast applicability scenario necessary recovery impose signal system learn control consider equip l constant assume identifiable give identifiable bound logarithmic condition probability x n contain realization gradient hessian find condition regularize square exist regularize satisfie defer state infinity assume deviation mean assumption identifiability q merging condition conclude high theorem give gap separately write bellman programming xt side average occur consequently probability lemma upper event following hold proof stem fact event anonymous comment stanford fellowship let zero prove reader lemma proposition convenience stacking
refer vertex method schwarz application variational amongst recently include open domain prescribe material density find minimize also share motion mean algorithm input meaningful parameter carefully suggest reader idea however algorithm terminate quality note early relax monotonic partition partition locally analogue calculation preserve positivity frobenius side consist entirely positive term use must imply iterate entirely connect pass subsequence pointwise thus partition stationary choice prove undesirable consequence experimentally small indicate kronecker vertex heuristic pick approximately small laplacian hope precise future determine define subset cardinality approximately cardinality minimize yield approximately equal expect numerical intuition desirable incorporation supervise variant label reader may check proof remain point spread handwritten digit section partitioning eigenvector normalize realize matlab residual criterion option contour feature illustration mixture gaussian panel construct graph laplacian initialize converge global panel value right cluster eigenvector vertex good contour right illustrate namely wish assign assignment initialization low energy five initialization five similarity gaussian choose laplacian ten different random initialization low energy figure give scatter assign point eight initialization fig iteration comparison normalize partition low energy initialization website information dataset use initialization report desire require small objective value well iteration involve computation problem find ground demonstrate minimizer c avg iteration diabetes handwritten digit consist handwritten mnist website semi supervise remain initialization low partition converge approximately iteration performance initial energy quality image eigenvector eigenvector dataset unique confusion obtain truth label represent sum represent true image cluster c represent ground row assign column sum one true datum mnist initialization percentage report initialization small value obtained observe great report partition geometric typical already report ghz intel gb ram mnist label avg avg periodic neighbor laplacian precisely laplacian initialize generator locally partition energy minimize energy geodesic kernel construct laplacian use partition generator roughly iteration low partition paper non sum relaxed believe promise particular rely interior assignment representative extend establish case prove hausdorff partition relaxation analyze property immediate direction numerical eigenvalue could possibly improve nystr om chebyshev parallelization choice believe choice laplacian choose find choice acknowledgement von helpful support foundation nsf fellowship dms grant thank project section theorem theorem white convex optimality relaxed formulation identify relax apply construct handwritten manifold extension representative edge frequently goal identify one arise measure computable certain application dependent eigenvalue partition q benefit component fix partition priori partition size fix partitioning state introduce relaxation relaxed geometric interpretation novel strictly decrease number iteration local minimum demonstrate arbitrary moreover assignment consequently interpret representative variant informative mining image semi extension apply handwritten digit another geometrically manifold ability sphere open concern manifold geometric domain community derive geometry motivate formulation method interpretable cut pde processing diffusion map low via operator another approach powerful tool multi come material science successfully image flow believe model fit analogous discuss properly introduce eigenvalue introduce eigenvalue propose purely geometric discussion introduce partition consider introduce subset subset complement take parameter speak change vertex variation dirichlet partitioning laplacian expand arrive laplacian whenever unweighted describe various algebraic involve good novel method future might analyze geometric nmf objective find partitioning problem efficient problem relaxed energy satisfies minimizer v follow exact localize interpret domain ss eigenvector infimum reverse normalize dirichlet giving admissible set indicator vertex define graph partitioning problem monotonically bound zero compact exist able interpret collection attain partition continuous accomplished fix
partial derivative respect dimension way dependence write multiplier incorporate wish set across assignment write iteratively update batch variational present think take minibatch minibatch document summarize case become initialize b streaming vb apply asynchronous describe portion primitive parameter maintain master computation document worker copy value vb primitive propagation lda topic proportion consistency also distinction token refer instance refer let denote document document integrate lda collapse document pair length proposal serve approximate variational reverse kl joint minimization idea ep proceed iteratively minimize document process parameter replace occurrence document call distribution q iteration distribute minimization reduce solve equation newton exactly experiment suggest fast newton moment alg k unchanged lda main report try modify ep make token update iterate rather former modify well ep number see far modify slow bayes report always fail put ep put vb similarity work approximation fix think take minibatch apply algorithm streaming ep lda ep primitive batch primitive next asynchronous asynchronous portion ep primitive lda initialize copy master locally c primitive indeed besides present bayes streaming bayesian framework make streaming specify batch primitive usefulness primitive fitting allocation two inference pass streaming increasingly technology readily operation streaming past advance knowledge memory complex hierarchical practitioner mind collect progress make big remain inferential bayesian paradigm e hierarchical coherent treatment currently seem reach known modeling collection vb traditionally function stochastic notably conceptual topic although must advance undesirable streaming aim approximate inference scalable truly stream process collection update recursive application bring vb similar spirit density filter propagation step involve match computationally costly avoid explore vb approximation development asynchronous streaming streaming vb naturally distribute implementation point px b datum minibatch treat prior incoming save posterior streaming automatically old data model often infeasible calculate posterior must minibatch intermediate update calculation minibatch long desire computation increase throughput minibatch posterior perhaps parallel combine full give approximate update normalize inference exponential assumption update normalize readily shorthand family approximate primitive minibatch together quantity along prior family sequential streaming iterate old posterior approximation prior arrive stream conjugacy actually posterior necessary conjugacy intermediate computation algorithm gain computation computation processor know subproblem worker report master worker subproblem worker finish system asynchronous present asynchronous conceptual computation asynchronous worker collect minibatch local master master master receive worker family approximation master prefer asynchronous follow master worker continuously collect minibatch copy master locally posterior return master master receive worker prior introduce change master worker long posterior master exact return nonetheless find perform focus overall stand stream intend approximation parameter current vb primitive lda model document potentially share well occur document distribution vocabulary kl divergence exactly descent stream result advance number process nonetheless visit requirement vb local streaming would minibatch document arrive add minibatch essential basis instead iterate evaluate approximate metric aside hold aside test document word document predictive predictive approximation facilitate wikipedia corpora corpora document wikipedia expect word extremely broad available online document nature document present main wikipedia correctly size demonstrate sensitive minibatch superior steady nonetheless minibatch r log comparison four expect streaming capability loss much single pass utilize also report minibatch minibatch equal process round minibatch equal send per number asynchronous case analogously minibatch minibatch constant context see grow slight asynchronous indicate speedup indicate processing seem dominate master asynchronous essentially identical might prefer robust failure stream design datum full value performance particular value wikipedia typically advance imagine need start sensitivity top order tune parameter require multiple run suited streaming demonstrate affect interact minibatch early apply streaming mix poor require storage stream ep primitive hour wikipedia predictive hour nature around ep primitive combination useful bayes distribute computation bayesian stream primitive demonstrate usefulness primitive topic wikipedia nature
primary motivation development pseudo likelihood make distribution mn mn full regression advance massive introduce use avoid unlike maintain independent solution sub combine give operate problem problem combine solution contain create clique auxiliary mrf clique variable parametrize derive mrf reading mrf clique estimate relevant sufficient pre store fashion estimating mrf sub dense graph restrict boltzmann variables prohibitive mrfs cost perfectly acceptable effectiveness proper construction mrf contain clique requirement clear algorithm desirable clique marginalization additional clique exact difficult strategy construct mrfs distinguish induce clique exact structure original readily marginal parametrize clique lattice making requirement create order mrfs add many unary potential fail true estimate pseudo ise empirical likelihood performance demonstrate obvious thing main good exact pseudo likelihood different mrf likelihood maximum refer mrf pseudo maximum class grid finally uniformly fit likelihood plot maximum specifically sample run pseudo ise lattice bar several variance estimate produce variance parameter measurement plot also plot experiment basically indistinguishable number approximate sufficient mrfs use parameter section valid choose correctly connection probabilistic locally conditional function uniqueness impose potential rise concept gibbs normalize zero whenever section central role one relative maximum deviation subscript mrf since confusion increase mrf clique system clique interest clique parametrize parametrization respect potential representation tell vector distribution draw correspond provide class auxiliary mrfs clique maximum smoothness identifiability see certain mrf accord clique clique follow likelihood estimate prove estimate characterization maximum estimate estimate compute eq maximum mrf proposition define auxiliary maximum since log family moreover respect auxiliary domain parameter parameterization potential already show exist sp main section estimate parameter integrate distribution q first parameterization summing accord mrf unlikely structure learning mrfs clique estimate efficient clique neighborhood clique behave large sample size work relaxation technique technique future would derivation pac understand direction selection variable tie distribute implementation goal new markov class practical degree linear clique unlike parallel model require markov mrfs undirected model graphic computational network markov logic processing physics point application code decade impact model convex maximum term evaluate expectation evaluate exponentially moderately maximum stochastic drawing typically mcmc costly difficulty approximate factored area mrf conditional term maximum datum depend actual detail performance undirected product one maximal set clique clique exponential
lipschitz mini minimax much uncertainty regularity minimax observation observation centroid dimensional confidence bound uncertainty community confidence would evidence surrogate lipschitz u accuracy surrogate krige pursuit expansion bayesian common constructing find though intractable input possible computer difficult effectively lack box amenable close run attain grow exponentially dimension impractical impossible box ensure actually input sample small numerous krige surface neural surrogate second simulation design response surface domain response surface krige domain day krige circuit domain krige rough could would regularity could amenable observation impose computed observation function minimax set title mini refer regular mini agree optimistic regularity regularity agree domain uncertainty would uncertainty derive bind learn derive low actual box function extend mean restriction function constant value well possible subscript value letter real scalar lebesgue measure letter letter alphabet real w fw choose data good lipschitz f simplify notation agree guarantee uncertainty require strong subsequent bold dot dot line tangent attain slope line q figure panel optimal panel vertical blue curve twice point uncertainty observation approach observation uncertainty section agree lipschitz small additional give observation well reveal lipschitz guarantee within happen guarantee two element agree require attain everywhere varie far away intuition additional much constant illustrate constant fx
tx k type w min inequality bind tx lemma theorem conjecture microsoft com engineering university edu product domain several practical crowdsource boolean heavily exponential mixture distribution sample decomposition match crucial approach decomposition challenge correspond tensor instead need estimate rank distribution domain denote size alphabet coordinate coordinate component draw j type distribution special several domain crowdsource crowdsource application popular answer multiple ground truth answer independently goal quality worker learn interested follow efficiently efficiently however depend performance use address divergence distribution say constant distribute probably et analytical mixture hamming ball case exact address restrictive approach general require result weight time scaling exponentially practice beyond running problem behave condition propose behave polynomial run accurate satisfy efficient solving theory open pt iterative product output rao correctly recover cluster distribution characterize cluster provide provide number condition parameter sometimes spectral moment one certain whitening order tensor high tensor popular constitute quantity whiten high incomplete version incomplete moment pose low completion problem diagonal alternate minimization method technique alternate complete also solve diagonal norm expensive completion incomplete simple robust use moment completion analyze alternate moment exploit efficient square solution combine estimate estimating also grow problem application crowdsource community detection domain discuss side scope inspire base learn gaussians another topic model moment base difference product problem general mixture practical crowdsource recommendation however exist case small alphabet practical either inefficient provably method distribution general decomposition provide separation method hmms topic another ica proceed whiten whiten operator construct reveal reveal entry whiten matrix alternate method completion diagonal miss denote letter denote third tensor row denote order q spectral norm u singular decomposition recent differ crucial estimate appropriate moment approach robust estimating key moment spectral recover estimate g eigenvector orthogonal reduce estimate tensor efficiently provide entry value entry coordinate j j bi form computed algorithm recover estimate min correct completion p qr square r u u section describe approach finite crucial sample section alternate estimate block even whiten estimate entry fortunately avoid back one back block upper bound incoherence alternating provably bi linear u precise recovery minimization completion alternate I iteration satisfy incoherent estimating insufficient precisely let te p show recover exactly n recover still range tensor solve estimate directly tensor solve linear r pt show nearly efficiently block incoherence p u sample third follow x constant least f method decompose estimate detail theorem proof crowdsource paradigm large processing human computer video datum optical expert confusion give diagnosis medical greedy expectation change problem recently provable two spectral j p project singular empirical second moment decision sign get recently improved misclassification decay constant approach label spectral extend general classification black binary spectral even provide development completion recover provide estimator crowdsource provable corollary enough misclassification scale decay present mixture min separability noise easily number require learn boolean clear leave complexity another natural establish information theoretic open problem crowdsource application translate analysis general believe unnecessary weight ignore component trivial fundamentally matching method suffer tensor number necessarily believe second several involve block key e n hoeffding bind see apply standard get e hoeffding claim h abc abc jx ia moreover tensor furthermore hence recover e consider tm ir orthonormal orthogonal using
optimum finite horizon decrease slightly decay stepsize varying trick rely equip throughout hilbert gradient belong increase family function attain measurable f n among assumption notion convex derivative remove power arise notion local convexity zero assumption example I field also norm hilbert space note classification ny space recursion full generality potentially hilbert space recursion implement readily projection learn combination recursion write term overall evaluate approach load covariance compact eigenvalue tend strongly strongly assume minimax obtaining loss change help obtain constant loss losse section within projection particular step obtain context loose practical averaging analysis would go mirror derive high bound lack hard deal another difference decay vs constant horizon trick done strongly use proportional simplify strongly obtain step smoothness problem size proportional lead convergence convexity constant convergence asymptotically rao bound unless limit lack convexity situation complicate compact already well strong convexity review convexity constant large finally unless strongly hessian infinity descent average convexity low present compact diameter impose traditional analysis setting strong nesterov stochastic context average convexity average stochastic strongly strong convexity recall excess lead show martingale inequality moment last obtain result continuous strongly average iterate bind decay recursion lipschitz continuity n convexity n result expectation f consider note iterate constant equal well restrict predefined tail convex case fine convexity practice typically problem prove size integer eq appendix original base take power martingale inequality appendix derive case apply f make appendix alternative slightly suggest use iterate iterate necessarily note pp bound extra projection know available constant bound result self size equal affine tail distribution make assumption gradient deviation technique strongly convex nf strongly section likely strictly summarize convex time positive stochastic gradient last iterate eigenvalue unique f make bind compare interesting study constant b depend well strongly hessian logistic input invertible eigenvalue time however large practice actual constant eigenvalue assess lead limit times f rate cost odd nd invertible specification get improve mis quantity f readily improve average strong self involve iterate extend know horizon proportional seem decay logarithmic online b alternative trick concavity convexity plain online newton rate strong logistic regression though understand simple one preserve rate lemma relate bound traditional inequality constant markov bp bp eq use valid positive eq inequality bp well detail reference go gradient integrate convexity finally hessian exponential around upper taylor behave well minimal iterate may time differentiable derivative r integrating result maximize technique real variable like one f tt two proof take power inequality moment b short appendix later use suggest outline communication allow bind bad constant logarithmic factor refine derivation proof martingale appendix almost sure proof recursion convexity follow martingale increment martingale boundedness r sure triangle turn bound proof th treat recursion use appendix k km r p k r bound expansion p p p element element interval q r p equation index term expand constant less lead recursion proceed induction k n q p order p p p true imply expand write p bound k k p p k term ratio result induction modify recall notation inequality sequence slightly constant r pp quadratic n pn b pn statement clearly martingale inequality relate norm martingale increment field surely quadratic bad constant r r r n additional extension satisfied probability show iterate f n f e p proposition use fine derivation denote b take expansion b na na moment n e b n
receive wave code high throughput sequence score rna rna rna accuracy state far satisfactory increasingly method expectation break barrier structure inherently predict structure introduce notion capability energy iff rna date minimum derive dynamic feasible convex hull coincide energy parameter approach rna systematic inherent c energy satisfy necessary input sequence structure rna necessary condition suggest energy loop problem investigation rna rna rna prediction key cell biology level throughput rna engineering application rna structure prediction community increasingly complex method parameter model recently provide despite progress last decade measure rna energy novel reach intuition convenience intuition systematic assess energy surprisingly single parameter iff set rna date ray energy structure equivalently learnable iff test set identical towards sure inherently learnable well systematic successful algorithm need unseen structure well deal power work leave secondary model often scoring alphabet scoring yield predict brevity focus model secondary loop energy associate loop energy loop energy apply interact energy interaction rna group maximum probabilistic estimate margin passive generation utilize determine probability rna sequence training use boltzmann good temperature ensemble possible question ask hope ever accuracy answer reveal inherent provide verify polytope every newton polytope answer also quantify assume minimize replace hull hull set polytope newton polytope boundary follow assume polytope contrary suppose interior ball center feature therefore cx sufficient v existence minimize lie newton polytope experimentally repository relate energy involve derivative partition solve newton energy conclude define replace polytope hull power vector call newton subsequence nucleotide formulate divide turn programming polynomial product newton polytope convex hull sum multiplication union union invariant rna rna rna dynamic transform newton summation hull transform rna rna rna interaction interact polytope sake illustration explicitly rna separate pair trivially programming follow polytope newton polytope subsequence polytope representation representation half representation representation transform convenient hull upon determined vertex half equivalent transform boundary iff condition easily check checking membership plane vertex lattice calculation rational exactly rna rna rna v rna secondary particularly rna rna structure date rna select exclude structure pair case polytope line programming hull mention pairwise summation precisely necessary experimentally determine boundary newton polytope feature boundary polytope matlab correspond case lie distance interior newton polytope polytope quantify matlab matlab run parallel lack rna varied
validation determine optimal associated validate sparsity fold lasso dataset candidate cv associate bic bic much less cv ols bring performance measure test asymmetric report goodness cardinality j fa ht ccc ccc fa truly relevant predictor happen strength moderate high support theoretical finding cv acceptable bad behavior bic excellent recover much derivation use fact follow immediately beta conclude first observe j dt last term r bound value conclude x ab b ab x mapping show mapping scaling algebraic apply lemma series iterate monotonically together asymptotically close satisfied verify satisfie mean acknowledgement thank helpful comment square sparse regression model square root residual proportional group advantage procedure noise estimation prediction accuracy minimal square root group similar need lasso strategy exist scale support section remark high become area decade observe assume th observation dimensional corrupt additive assume correspond control predictor solve zero focus group naturally plausible subset zero general direct random whose measurement matrix predictor write treat group model perhaps method group lin al consist loss term proportional euclidean group let refer assign group partition group minimal group design index denote indice sequence lasso well van optimal correct possibility first theoretical estimate optimal original make context selection consideration square root approach scale zhang theoretically estimation procedure much appealing true moreover give wide applicability motivate group behind square root achievable pattern open guarantee scale scope mainly moderate finding collect show lose group square root essentially estimation prediction normalize diagonal cardinality notation generic denote supremum coordinate prediction square root discuss compatibility slight cone eigenvalue condition detailed compatibility meet compatibility constant compatibility design value clarity exposition assume additive analysis appropriate reveal define establish close variable obtain tune independent index event large notation multiplicative sparsity index analysis size summarize estimation lasso hold eq constant statement crucial tuning instance van de corollary eq could additionally note recovery estimation even correlate design scope directly group cf subset recovery group root lasso guarantee additional group additional say meet write compatibility instance restrictive essentially sufficient consistent support lasso refer precise component formulate min meet component sufficiently whole group component slightly condition event orthonormal bm hold proof root mutual condition et al recovery minimal strength noise quantify corollary bm derivation course state impose invoke mutual approach idea group root lasso independently particularly determine recall component small small depend result value generic consequence summarize event claim rate subset group lasso root benefit method free assume eq conclusion theorem follow lemma definition claim consider clarity exposition establish inequality belong gaussian noise make moderate case group establish group root lasso efficiently consider convenience without variant fix order cone et solve square root lasso matlab method package short accord experience slow inaccurate large slow perhaps descent lasso fast scaling update soft resort package result considerably every accumulation conclusion analysis fast specific form convexity exploit denote converge optimization root special case root lasso three computational particularly interested since compete exist group publish devoted variable uniformity toeplitz al zero scalability computation compute empirically potentially recommend matlab
develop coding analytically exist confirm compare learn reduce statistical highlight representation svm support image use dct wavelet take regression use image entropy code vector approximate increase compression expense big low apply formulation certain distortion scalar restriction error use profile consider frequency account domain dct wavelet contribute different propose couple report svm scheme rbf arbitrarily guarantee approximate axis svm code orient axis representation feature wise correct penalization restriction wise strategy actually component eventually transform regression scalar suitable standard regression diagonal dimension zero transform independence jacobian illustrate represent box determine independence among align axis highlight point necessarily imply lie inside region meaningful therefore suitable conventional consequently trivial leave suitable conventional review coefficient transform diagonal desirable preserve look desirable increase dimension work jacobian statistical case case result experimentally confirm svm report domain recently domain close structured follow review linear transform obtain independence jacobian suggest room svm report domain diagonal jacobian coefficient experimental code superiority conclusion final fact dimension simple statistical diagonal accurate description high mutual nature pdfs refer neighboring phenomenon formalize second nature ica spatially selective orientation basis despite name ica coefficient seem unlikely linear independent pattern empirically image decomposition function coefficient nearby spatial independence introduction beyond ica transform pca linear ica spatially localize correlated energy neighboring orientation width statistical domain ica process image global unitary remove spatial block pca dct ica notation linearity relation dependence understand summarize bank spatial bank neuron wavelet ica transform function account report section second normalization last image riemannian geometry linear domain independent illustrate presence relation linear frequency wavelet cross behavior periodic mechanism dimension visible frequency one right specific frequency induce reduction reduction sensitivity get frequency frequency band band frequency acceptable depend frequency neighboring cccc width width width stress biological organize sensor exploit process review stage process successfully derive redundancy argument code vision support non remove redundancy transform transform e normalization jacobian strictly approach break domain sub locally standard ica restrict local separate separate pdf jacobian stage feature around feature general current linear sigmoid transform weight normalize combination energy neighbor frequency eq top coefficient exponent neighborhood non energy interaction figure coefficient width width width coefficient general equation describe remain intrinsic transform jacobian diagonal nature summarize illustrate improve transform confirm domain gain domain block dct wavelet either statistically exploration formulation code direct linear consume ica large image need significant computation equally need linearity analytical analytical reasonable size efficient use large explore normalize domain dct describe competitive formulate state component nature relevance direct theoretically domain moreover build explain early argument divide energy statistically cf moreover ica convenient performance domain distortion curve discuss select support dct domain train learn low frequency big relevance dct accord criterion appropriately constant parameter behavior standard include experiment state penalization experiment response rbf ratio experiment figure describe multiplier code image different euclidean rmse meaningful eight sample however mse elsewhere meaningful quality rate already range clearly previously obtain high compression recommend quality bit rate visual strategy bit rate compression
decrease schedule use berkeley difference trend experiment main hide tend layer agrees observe train bl bl bl bl bl bl bl wiener bc bc bl bl bl bl bl bl bc bc tc tc tc tc tc tc tc denoise various researcher autoencoder model popular boltzmann machine certain case autoencoder type level effect depth reveal performance improve level high numerous dominant denoise whole image small patch extract clean possible wavelet component dictionary compute shrinkage element element magnitude patch recently overcomplete denoise code essence natural posterior noisy patch either compute patch reconstruct expectation posterior utilize probabilistic latent denoise deep multi perceptron learn patch corresponding clean art denoise stack autoencoder effective denoise deep conventional propose type denoise boltzmann image denoise denoise autoencoder extensively evaluate boltzmann autoencoder empirical type level describe boltzmann autoencoder increasingly originally structural boltzmann become increasingly since powerful deep train stacking rbms another variant bm outperform learning task describe bernoulli gaussian energy layer neuron vector boltzmann base learn exactly approximation markov chain find start initialize propose special compute posterior need approximate variational exactly cd denoise autoencoder special perceptron set tie weight try network optimally minimize encoding decode nonlinearity layer share decoder notational simplicity bias unlike ordinary autoencoder randomly add usual combine add gaussian randomly input zero training backpropagation objective programming deep initialize experiment initialize backpropagation way perform interested noisy denoise patch combine patch whole height channel element division denoise parameterize extract possible patch image construct obvious approach many patch call pixel consecutive possible opt patch computational describe natural essence model layer latent ica patch build code dictionary denoise two estimate unit patch subsection boltzmann machine denoise autoencoder bm visible image patch eq word visible visible unit corrupt expectation tractable q approximate patch noisy image patch bias unit neither computable analytical form propose utilize factor patch may feed five turn cifar dataset cifar dataset patch location collect try image format average channel make try depth setting boltzmann autoencoder hide hide layer size hide denote boltzmann machine four hidden layer denoise denote model structure train enhanced gradient persistent train backpropagation hidden detail procedure denoise paper knowledge level separate training train accordingly boltzmann denoise prior knowledge level two white simply add pixel black furthermore white standard image pixel wiener filter width pixel signal ratio clean bl bl bl bl bl bl bl bl wiener bc bl bl bl bl bl bl bc tc tc tc tc tc tc tc tc white train image obvious deep network deeply outperform power model noise regime deeply lag possible poor might dramatically outperform noticed image instance although layer deep neural show variance depend generalization deep intuition performance deep emphasize detailed tend capture additionally try use extract berkeley segmentation collect white noisy boltzmann denoise try support neural image suggest question clearly find outperform regime layer deep train separate property turn deep suggest denoise prior available layer noise boltzmann outperform outperform case hide twice well unit definite evident regardless counterpart future appealing possibility combine neural denoise prior pixel variance use stochastic descent minibatch equivalent cycle decrease
computing storage capacity include discover reliably high pass binary value plausible algorithm capable approximate message weight example discuss eigenvalue eigenvalue spectra display realization element play neural understand stability transition nonlinear high dimensional begin replica typical variety ensemble focus wishart nan model outcome apply dimensional many ensemble thought eigenvalue attractive potential obeys fluctuation statistic appearance projection manifold overall replica formalism play physics dimensional difficult difficulty section yield project datum distribution low alternate subspace ambient lose set ambient critical ambient preserve theory ability preserve end mechanic hyperplane connect random fluctuation low mechanic reader sense processing refer high dimensional discuss include image compressed array compress molecular resolution camera technology also diverse processing include semantic memory circuit sparse weight long brain communication replica theory remarkably unlike display increase sparsity formulate qualitatively dynamic crucial history minimization propose coding demonstrate sparse coding finally overview replica applicable perceptron learning compressed replica powerful statistical mechanic hope exposition replica cavity pass variety context help enable student researcher theoretical learn advance last decade physics spin context train define spin degree take connectivity hamiltonian q noise choose independent progress reveal picture temperature pattern equivalently concentrate landscape index characterize activity free activity energy start stay ergodicity broken time activity pattern activity maintain interested understand energy minimum pattern mean realization vanish limit geometry free activity activity turn unless self overlap indeed detailed provide variability mean across case despite distribution self average organization energy activity replica statistical useful energy suitable energy self free realization compute logarithm replica power outline replica basically activity fundamental gaussian variable applying overlap realization activity integrate introduce attractive framework present minimization energy spin configuration explore inner product replica parameter replica low temperature configuration differ realization replica realization replica break replica multiple describe two configuration typical inner product series scheme describe gibb nested figure describe possible scenario alignment prefer preferred preferred pattern fluctuation activity connectivity similarity hence nonzero yield activity maximization many competition energy nontrivial computing likely overlap via saddle yield self respect ps physical meaning saddle replica overlap overlap pair minima replica overlap geometry free hamiltonian symmetric permutation replica index row heterogeneity limit replica saddle derivation represent configuration temperature continuously phase transition correspond neuron activity plausible inconsistent physical replica detect show replica physical picture energy minima remarkably predict like minima cd ultrametric symmetric temperature hierarchical structure purely phenomenon replica replica turn correct replica analyze toy symmetric matrix break replica low stable fluctuation possibility useful processing several note fluctuation stable temperature connectivity induce low pattern either temperature processing would dynamic manner connectivity early proposal prescribe pattern eq reflect rule neuron neuron weight proportional correlation neuron impose impose upon induce equilibrium pattern ideally mass locate pattern activity network relax dynamic whose relaxation thus structure pattern store e subsequent dynamic energy landscape determine minima landscape correspond recall experience complete storage replica method store uncorrelated choose analyze energy storage fit classic freedom energy denote activity pattern completion free pattern state replica self replica spin free minima overlap replica pattern low landscape behave phase enough spurious corresponding mixture state characterize replica temperature increase mixture phenomenon illustrate tradeoff away increase decrease recall dominate landscape activity network operate device phase diagram energy analysis alternate physical saddle replica symmetric seem bit give alternate cavity provide physical intuition consistency general cavity indirect provide intuition derive direct replica involve neuron write local act neuron fluctuation full gaussian term positive effect correlate fluctuation due coupling idea behind cavity acting neuron instead cavity thereby leave cavity cavity absence write cavity writing term cavity cavity absence cavity know fluctuation cavity htbp remove replica field cavity neuron approximate activity course must uncorrelate unlike fluctuation presence motivated fluctuation cavity system full gaussian induce coupling simplification show fig term vanish importantly go make cavity system consequently full accurately single fluctuation validity cavity replica symmetry single landscape average detailed cavity extend scenario replica replica compute neuron full neuron term mean cavity demand cavity nothing cavity neuron yet randomness virtue cavity absence random mean computed averaging neuron realization limit denote mean heterogeneity cavity across neuron fluctuation cavity field heterogeneity activity across neuron reflect demand cavity heterogeneity mean neural activity model consistent two allow average quantity depend realization understand neural mathematically neuron distribution efficient message science compute marginal factorization property index could variable systematically index factor factorization bipartite either ax pass message edge graph nonzero connect correspond neuron utility iterative marginal flow message along b later justification single type variable message factor besides interaction think interaction message interaction unnormalize equation message denote fig alone factor account besides message factor induce interaction leave simply see fig involve iteratively exception connect initialize absence pass remain converge intuitive lead key intuition structure variable treat include approximate message well whenever interaction graph previously weakly couple ideally weak whenever loop case one remove path factor graph variable path independent make marginal chain spin chain spin tell compute iterate message position q special message initialize spin converge marginal demanding whereas spin configuration operation transfer bethe nevertheless loop yield whenever weakly removal factor context compress sense early pass loop variational pass bethe free energy gibbs variational review message graphical loop message nevertheless practice success approximate replica average cavity replica replica saddle point perspective pass message message take upon operate little let remain couple nonzero variable parameterize thought cavity field spin cavity system term parameterization pass dynamical cavity relation binary spin field strength reflect reaction complex reflect negligible directional update cavity simple sum effective presence cavity ready point cavity field realization pass empirical cavity field pair self mean self observe reproduce update precisely cavity I yield characterize cavity message pass arbitrary index generally cavity analysis distributional could approximation cavity distributional reduce side hand simplify right mean equation summary neural network spin replica cavity pass analyze concern energy landscape replica correlation provide application free correct replica cavity replica break free energy inference physics lead survey propagation find good free minima review design mechanic conceptual advance make perform mechanic example system playing explore viewpoint extensive perceptron vector sum incoming activity depend mathematically zero fire state geometrically separate input weight normalize train perceptron desire input output modifying find inequality eq solution remarkably rule main solution mechanic answering sphere alignment pattern positive positive one wide perceptron choice space solution otherwise count misclassified example gibbs temperature become htbp volume nonzero statistical mechanic temperature mechanic formulation vector expression gene across neuron hide approach span projection often determine upon e center center mass origin point maximal variance direction across beyond clustering maintain guess cluster centroid cluster set centroid centroid optimize centroid mass cluster assignment centroid center view alternate joint centroid cluster membership assignment case centroid write energy force close mechanic replica perceptron association association draw uniform radius natural significance simplifie fortunately analysis distribution volume low energy configuration realization essence sign desire input reduce jointly replica averaging overlap integral integral overlap volume overlap perceptron limit weight answer large volume saddle competition energy select saddle make saddle replica replica overlap independently choice suggest free expect analyze unsupervised convex zero temperature configuration degenerate intersection set symmetric approximation limit yield appear inside perceptron typical overlap weight pressure agree large small reflect large volume increase place energy entropy perceptron store association interestingly author replica perceptron learn make prediction analogy cell cell capable devote internal cell turn influence signal receive inferior input firing induce spike inferior fire input thought guide thus cell think supervise task cell cell prominent feature truncate percent implement mapping statistic input able derive however take elegant architecture perceptron optimally capacity operating capacity replica remarkably whenever perceptron implement maximal association reliability delta majority perceptron near rule cell either perceptron face nonnegative combine fraction cell pattern turn weight structure pattern perceptron nonnegative replica perceptron theory capacity output store cell function average turn focus hermitian analogy hermitian section computing eigenvalue hermitian matrix involve obtain eigenvalue replica wishart element unit space distribution identity dimensional spectrum average identity fluctuation spectrum realization converge thought integral exploit fact variable going perform integral consistent general introduce integral integrating overlap latter end become method saddle choice potential saddle right hand side field nonzero region eigenvalue proportional region mp high sample spread increase density eigenvalue appeal interpretation intuition dimensional statistic distribution e zero unique svd eigenvalue fortunately need jacobian angular integrate yield change obtain arise factor jacobian incur energy move govern logarithmic potential interaction eigenvalue spread typical range precisely consistent rescale mp section behave typical mp typical fluctuation behave maximal eigenvalue form also dimensionality fluctuation mean eigenvalue lie density fluctuation scale range typical fluctuation often deviation large eigenvalue right mp curve histogram blue maximal eigenvalue rescale red mark discrepancy edge mean maximal effect like fluctuation vanish could configuration mp eigenvalue preserve shape mp mp dominate exponentially entropy play compute maximal eigenvalue leave must much pair energy nice explanation large reader summarize implication formalism dimensional empirical maximal eigenvalue dimensional remain moreover probability deviation may careful look along project lead skip step responsible project onto choose direction dimensionality preserve remarkably collection reveal rp preserve structure generic along dimensional manifold embed dimensional consist fig cloud consists embed project appropriately projection distortion small make cloud low dimensional long similar lemma answer state pair point embed dimension course projection reconstruct original projection surprisingly mechanic base distribute object rotation scale another would fire brain region show preserve geometry number curvature manifold projection overall ambient dimension finite preserve pairwise give alternate simple geometry projection orthogonal pay price optimal course hyperplane geometry interesting low nonsmooth nonzero coordinate hyperplane preserve show preserve preserve rp might interested might point fig general manifold general computationally tractable achieve recovery signal exist computationally tractable provably geometry signal recovery geometry compress review high rp computation signal signal pairwise distance signal detection accomplish rp comparable perform reason remarkable preserve rp remarkable distortion rp sequence loose leave condition actually worst behave mechanic distortion manifold projection role projection play degree freedom observable maximal fix self average manifold ensemble manifold general class manifold goal section inequality discuss fix realization gaussian cloud consist projection operator whose gaussian cloud scaling distortion pair random dimensionality cloud distortion well approximated take typical universal variable tail vanish exponentially extreme slow growth realization conclusion distortion point obey value origin slow maximal distortion directly responsible remarkable distortion theory variable intuition independent ambient dimension ensemble hyperplane analysis let range first exploit invariance perform column axis parameterize dimensional vector projection column second linearity projection point plane suffice sphere denote distortion fractional euclidean constrain section typical exponentially argument distortion indeed correspond geometrically kernel lie course want high projection obey rp hyperplane fluctuation correlate variable use induce rp simple manifold hyperplane see projection preserve geometric dimensional manifold furthermore case signal fig recover high signal tractable statistical mechanic application thus top linearly relate bottom fig signal reconstruction linear response neuron pattern trial might recover signal point search yield signal seminal focused guarantee matrix nevertheless condition replica allow typical minimization also message pass residual mechanic gibbs eq enforce take temperature fluctuation free energy average fluctuation measurement independently randomly hold interesting free averaging far typical depend average replica replica average residual variable distribute I replica saddle equation overlap convexity reasonable replica saddle replica self perform effective hamiltonian substitution variable full give relationship gibbs replica temperature component signal replica mean theory solution class depend capture always capture cs vanishe suggest minimization exist occur regime due replica plane different method understand theoretic carry bit redundant perfectly signal times increment region perfect perfect perfect yield sharp error various bar reflect red curve plot blue signal fraction dot middle think arise fed soft threshold function reconstruction less requirement exceed inequality surprising reconstruct surprising minimization example rise transition fortunately continuously exponent rise depend zero distribution fig note depend zero understand make cs look component interesting temperature field hamiltonian hamiltonian limit soft thresholding otherwise understand intuitively measure scalar laplace choose datum exceed noise play corrupt interpretation within cavity add cavity must minimization reflect minimize whose quadratic cavity field effect go condition signal reflect zero cavity field component feed soft arise cavity demand across component cavity replica show formulate pass approximate formulation pass yield neural solve graphical measurement factor implement application message however straightforward complex component track message message receive contribution message invoke thus keep track message system message differ exclude effect factor assume argument suggest I message pass equation dynamical reduction main thresholding wise converge temporal reconstruction interact residual store neuron current representation store neuron second receive feedforward external layer feedforward transfer interestingly dynamic computational early would explore architecture implement type review basic storage machine algorithm model review rich surprisingly picture natural ask modify mechanic prominent reader discussion surface literature lie intersection mechanic simplify connectivity lack external happen connectivity become asymmetric dynamic many one theory understand activity interestingly asymmetric ergodic partially asymmetric retain reach point size lyapunov long temperature asymmetric dynamical possibility seminal show mean exhibit drive dynamically achieve balance individual lead state spike fluctuation nontrivial strength example external interesting monotonic dependence neuron biological neuron internal dynamic exhibit network diagram neuron characterize solely maintain consistently review lead dynamic possibility periodic spike train spike train possibility population fire structure vary strength neuron beyond entire trajectory analytically spike next allow lyapunov product associated class network extensive lyapunov lyapunov sensitive potential feedback effect associate rise potential heavily trajectory perturbation state large perturbation exponential trajectory trajectory network extremely subtle due lyapunov perturbation limit would yield negative constitute perturbation lead suggest spike review begin capacity input mapping certainly important past response generalize experience learn input never see formalize mechanic perceptron input generalization train perceptron teacher perceptron correct present learn mechanic decay example perceptron act theory architecture classification fall hyperplane mechanic approach analyze memory sophisticated mapping activity replica symmetry solution mapping multiple internal activity implement desire mapping mechanic success analysis architecture learn support architecture capable incoming spike spike mechanic carry interestingly solution describe replica break analogy replica symmetry break component component imply apart yield identical neuron reveal double connectivity double important implication incoming review mechanic memory section mechanic structure return pure would ideally performance pattern reliably mechanic datum play mechanic signal low space consist identity plus rank replica typical empirical covariance reveal ratio ambient dimensionality increase low work sharp threshold amount resolve interesting statistical mechanic setting separate input learning model mechanic approach yield practical state compute cluster nan distribution mechanic reduction connect maximally distortion correlate extreme hyperplane interaction eigenvalue exactly rigorous upper bound distortion prove tangent projection geometry distortion manifold tight loose typical distortion incur manifold example consist interested fluctuation maximal multiple plane random could relevant replica replica symmetry break suboptimal end mechanic inspire propagation simple inference despite survey propagation decade physics science lead lead computation field year thank foundation foundation foundation stanford message wish mechanic hamiltonian spin perceptron spectrum residual sense q condition integrating introduce interaction overlap useful possible configuration prescribe introduce q integral exponential understand yield power final write integral via saddle self saddle exponent effective hamiltonian variable simplify integral represent configuration prescribe overlap reduce saddle overlap configuration exponent connection replica two gibbs distribution realization overlap average appear numerator denominator identity end original degree sequence
concept sample dynamically stationary environment characterize stationary work exhibit unique challenge collect cycle failure line snapshot network severe external learning require exhibit physical meaning behavior external aggregate failure graphical provide theoretical failure location work consider spatial variable severe weather distribute energy index radial failure occur mesh status assume simplicity exhibit node mode cause external exhibit randomness fail fail process use characterize characterize probability fail increment change versa stay assume together statistically failure failure internal weather dependence scale failure cause external snapshot spatial insufficient specify complete temporal indicator event occur predefined node failure fail node certain region newly fail recover node failure failure failure occur occur failure recovery occur increment failure ft ft assume occur furthermore failure equal expect region city g example q characterize derive recovery reveal quantity model behavior failure failure failure failure epoch failure failure quantify intensity occurrence across location begin ft ft characterize per time epoch process vary function rate characterize stationary duration stationarity recovery characterize conditional duration failure threshold probability duration failure occur sufficiently rapid recovery failure rapid dominate dominate refer terminology analogous remain duration recovery characterize recovery characterize failure characterize entire life cycle time failure recovery birth death commonly birth death failure occurrence hold failure occur time last duration wind failure happen day day operation elaborate life failure process failure increment recovery theorem theorem failure duration number failure duration recovery aggregate failure occur result model moment function expectation life scale cause occur cause failure million center power area heavy pass collect failure failure fail circuit operational raw occurrence network contain occur time failure failure group entity occurrence duration preprocesse result fail entity failure fail entity refer node include natural customer across entire temporal process notational simplicity rate move ft hour rate failure figure hour failure vary rate occurrence failure occur hour hence per hour day operation increase failure hour next hour peak nearly characteristic temporal stationarity obtain network failure failure rate increase failure hour decrease failure stationarity time spatially function peak vary hour reach peak varie spatial temporal non stationarity order figure city reach failure city city characteristic different city reach peak appear consistent movement learn recovery temporal focus failure among size piecewise homogeneous equation vary occurrence stationarity non stationarity note small distribution duration failure occurrence accordingly percentage whereas recovery across stationarity recovery examine close exhibit recovery e g city city percentage hence recovery real another large cause united result million customer without day million customer lose utility company report failure new report scale minute report plot accurate power begin aggregated failure accordingly recall equation aggregate failure epoch determine raw datum aggregate failure sharp sharp sharp rate exceed recovery rate happen recovery exceed failure rate sharp indicate salient point lower increment region increment bind failure network estimate failure obtain equation failure bind cumulative aggregated recovery impossible vary aggregated detailed failure available consider sample step shape reconstruct recovery reconstruct finding failure failure stationary graphical region different failure gradually however failure exhibit failure occur group aggregate failure exhibit e rapidly decrease recovery learn failure non stationarity location constitute recovery rest network steady hour addition failure duration recovery lack recovery detailed aggregated failure accurately failure failure occur within amount failure rapidly minor pass dominate important response external characterize life exact individual failure duration infer reverse sufficient failure insufficient stationary duration deal seem temporal insufficient stationarity suggest enhance spatial radial failure increment minute configuration yet include power flow characteristic small sub accordingly temporal naturally large scale recovery particular location provide failure completely vary failure across region real process learn reveal failure failure exhibit rate component region utility network recovery failure finding subsequent distribute failure area dependency need combine detailed configuration far understanding enhance thank data cox discussion anonymous valuable associate national foundation stationary environment failure severe external learn behavior aspect cycle power model develop third two life operational failure infer real finding behave two network differently rapid slow stationarity contribute application grid failure learn recover weather understand power edge medium consist node external wide service occur year service million customer day rely primarily scale failure assess failure system failure example severe become failure wide furthermore failure understand difficult overall need characterize distribution external discover challenge quantify power distribution external external exhibit behavior result occur failure usually force wind gradually move hence randomness failure quantify stationary external appear large often external generate shoot external individual enable study failure failure external recent work combine algorithmic approach failure transmission challenge question answer large external drive determine problem effective drive make parameter physical formulation focus induce weather failure detail failure second failure structure occur small sub whereas beyond weather failure understand external temporal insufficient spatial group node area characterize life cycle scale spatial arrival failure process process immediately hence constitute completely specify behavior distribution characterization learn clear location failure obtain detailed failure life example cause failure south affect million devise scenario learn process failure aggregate location spatial aggregate parameter failure another cause power failure million people consist failure aggregate estimate failure rate stationary weather scale b parameter stationary rest paper scale failure formulation describe learn study stationary part datum discuss finding section conclude example scale stationarity recovery discuss weather failure consist circuit circuit transmission distribution system commonly radial component secondary power fail primary source secondary source fail source external hence failure failure component
experimental real data observation piecewise present point index np k thus coefficient degree associate segment dependent associate distribute variance segment estimation maximum segment suppose conditionally independent log characterize piecewise sum likelihood write k segment constant thus log likelihood criterion optimization minimize respect segment dynamic programming consider optimal segment union segment segment run segment accord cost detail expensive accelerate iterative alternating equation minimization separate regression k compute perform contrast k use current tm present generate discrete coefficient reformulate noise probability switching model propose logistic variable accord covariate nz ik transformation example component design probability exp temporal complex kk dimension class parametrization show fig control transition point switch polynomial polynomial coefficient vary model step process density parameter likelihood classic involve maximization perform step compute simply require computation step maximization follow maximization respect multinomial regression reweighte iteration nan identification newton consist vector q respectively author approximation hessian accelerate exact matrix perform notice consist gradient provide summarize propose algorithm threshold choose mi nk increment iteration c equation provide parametrization segment expectation parameter diagonal proportion q likelihood must set devoted purpose piecewise iterative version piecewise give first simulated second expectation compute ex ex piecewise regression denoise denoise run segment respectively order segmentation contiguous fix second period sample simulation situation second simulation table second table two situation htbp ab initialize segment iterative dynamic programming random initialization addition initialization correspond initialization stop increment criterion top simulate term fig denoise error approach signal piecewise approach slight relation h present signal switch situation phase switch homogeneous interval adapt signal middle variation closed involve estimate signal signal cc illustrate signal generate signal em original extraction time series switch mechanism incorporate use allow transition parametrization
direction firstly predictor arise subject value projection easier pre obtain sense broadly frequentist different randomly average perform forest nature could average computational advance possibility projection ideally add lee pareto mean shrinkage j handling g recovery incomplete inaccurate selector large come spirit proof check three represent ball density therefore u well df similar f h df n ratio n expression px plug z vb tf x u taylor checking proof q expression proposition obtain integrate n yield tf taylor remark em height high dimensional randomly predictor prior dramatically storage project response compress available mix strong approach show paradigm simulation application key compression dimensionality progress routine massive number predictor million setting response residual rich variety method et lasso et et pareto prediction tend recent show variable impose al literature encounter application compute intractable hope approximation markov mcmc sampling unless approximate computationally tractable variational bayes popular recently start notable disadvantage model lack justification accuracy dimensional compressed solve bottleneck scale trivial gamma compress parallel random projection different al employ predictive notably one extremely rapidly predictor question justification bayesian estimate predictive distribution involve massive dimensional near mention compute bayesian computable compressed regression expect excellent dimensionality reduction information bayesian compress inspire compressive representative article al construct data model approach achieve base lee al propose compressive facilitate ability high compressive rely fundamental involve instead want huge sample orthogonal unchanged instead compressed regression maintain privacy approach estimate oracle property size instead provide background predictor form scale projection coefficient predictor unlike project low estimate probability gram schmidt popular compress compressed regression gaussian replace eq regression interpretation normal long set conjugate particular normal gamma eq special x x analytically inference inference st heuristic motivation assign joint keep embed typical define avoid maintain superior experience justify section surface find x x rich strategy severe moderately elegant al cope instead free generating sensitivity like limit sensitivity specify randomly generate projection row respectively denote n predictive give posterior projection little observe expression obtain average density identity parallel expense random inversion quickly massive possible gain implement rapidly batch lose predictor pay huge gain address theoretical prediction paradigm study follow difference sparsity condition near posterior computable let response density distance df enough sequence converge assign shrink neighborhood rapidly seek establish basic notation let predictor assume non dimension discover predictor fit model continuously parametrization class probit regression standardize clarity many appeal sparsity covariate empirical put dominate measure study focus broad literature high convention addition let describe result n covariate standardize primarily impose restriction size dimension grow rapidly tell grow solely dependent growth probit linear regression linearly impose condition constraint prior condition quite restrict compression away matrix iii approach compressive avoid complexity assume iii prior define dr h theorem omit discussion follow evident grow good informative show predictor consideration rate proof routine linear probit regression n constant row theorem q consider probit independently prior outline satisfied enough q predictive average rr partial bl pareto idea conjugate average matrix compress predictor compress double pareto method shrinkage default reasonable choice satisfies step choice discard moderately case assess change sample standard rest rest coefficient focus sparsity justify predictor much dimensional subspace much last case motivate center standardized rr use package default choice suggest hyperparameter bl put six present average dataset calculate held generate computing error level level bl rr dense case compete shrinkage induce remarkably particularly sparse lasso bl performance level level bl compete six coverage case satisfactory coverage frequentist I distribution equal coverage lasso rr show severe decrease marginally coverage produce close rr short wider narrow predictive well frequentist pi maintain coverage competitive performance simulation case much second simulation study bl implementation increasingly prohibitive compressed double pareto bridge generating rest scenario average simulated hold subscript bootstrap bootstrap lasso rr lasso rr show sparsity perform poorly dense show excellent figure probability probability excellent coverage probability lasso rr plug sparsity coverage dense lasso rr suffer severe consideration computing compress reasonably ghz intel processor compute second advantage rapid instantaneous computation compression gram schmidt multiply low calculate large quick schmidt compression burden average choice processor parallelization obtains optimize code single ghz intel processor enjoy little advantage lasso become scale thousand million become increasingly comparison initial compression schmidt compress regression molecular gene chemical pathway cell individual represent mix united nucleotide discard variability sample cell allocate I analyze exposure agent iii minute respectively dna measure cell subject image processing software dna tail surrogate tail moment establish surrogate study et dna multiply center tail
derive separate diagram sometimes diagram standard site diagram short site diagram classify hyperplane clustering hyperplane property separate diagram separate diagram diagram informally cluster lie boundary cell boundary interpretation power diagram boundary hyperplane cluster separate power figure depict emphasize strength guarantee separate explicitly construct pairwise separability diagram provably minimal tie special point construction diagram separate power diagram correspond square assignment balanced square clustering balanced refer cluster respect clustering trivially cluster least size induction square term assignment diagram assignment diagram separate diagram least square assignment diagram minimal cluster maximize maximum separate diagram correspond square well separate diagram site give recall separate power diagram cluster satisfie separate power diagram guarantee separate condition constraint separate feasible recall arithmetic satisfy lie j tc tc tc tc lie contradiction invariant see diagram site tell diagram none construct diagram point cell contrast state multiclass support helpful strictly separate diagram normal hyperplane compute hyperplane denote program clustering allow separate power diagram feasible solution particularly interested clustering margin correspond minimal distance hyperplane define geometric variable constraint refer divide obtain point site site formal diagram satisfy constraint paper separate diagram diagram margin separate diagram informally ok separate hyperplane euclidean justification approach definition maximize margin convex site site diagram margin diagram cluster diagram correspond optimum fix site keep cluster margin separate power optimum linear separation margin number misclassifie point misclassifie point multiclass service intuitive next section section desire property binary investigate derivative lagrange helpful deriving site find power diagram diagram misclassifie point margin margin multiclass application soft diagram separate hyperplane diagram multiclass diagram multiclass multiclass error respect leave multiclass part point multiple analogously count number multiclass support relate fold support outli multiclass margin error care whether hyperplane margin margin power support vector close consider soft diagram hand margin separate red green multiclass margin scale dash circle pt pt blue circle blue blue blue circle green pt green circle circle circle red red circle circle dot fact red multiclass margin next soft diagram prescribe upper number multiclass margin point multiclass us variable among margin point multiple definition formally multiclass soft power diagram cluster diagram multiclass power refer good term function margin purpose version optimum diagram margin multiclass margin multiclass due feasibility close optima reasonable local optima define diagram correspond lagrange use lagrange n tx two rewrite multiclass margin vector hyperplane cell margin correspond case error cell imply tt arbitrarily small amount former vector error count property site derivative program far optimal yield power diagram maximal margin bind error multiclass similar vector one constraint multiclass diagram soft diagram diagram margin cluster let optimum soft margin point point analogously multiplier take ij tc saddle support vector thus tt argument count site yield soft diagram maximal bind margin exhibit immediate outlier square fundamentally diagram corollary feasibility optimize program application discussion local optima nonlinear multiclass dna training partition consist dimension partition training partition cluster program start cluster representative site cluster identify error corresponding consider pair cluster prescribe number diagram site margin margin point solution output diagram power diagram tt tx k cl outli detection soft power expert site mean representative mean report choose margin main program explain favorable run make time comparable ten linear program reveal number run margin sec sec sec ten close look tradeoff margin program obtain optimal value refer intuitive separability cluster diagram bind yield bind dropping yield diagram efficiently let one type minimal value diagram interval claim turn interest tp tt fact relate contradiction positively unbounded maximal error termination start minimal nest happen analogously nk multiclass power diagram point support except sum argument solution feasibility basis precede program solve stage number nice separability diagram large give site insight site representative diagram site soft diagram diagram diagram tt l cl return soft eq analogously represent multiclass margin theoretically maximal multiclass margin error besides measure balanced square information apply test especially sufficiently large us arithmetic site list misclassifie require c dna sec indicate dna put consequently fact percentile set conversely prove well perform similarly bad confirm soft diagram core application turn locally violate compute local arithmetic theoretically stop iterate precision report computation power outperform power diagram table balanced assignment design outlier observe site close site dna sec sec principle serve multiclass design mind margin state multiclass information obtain plan place euclidean careful choice partition task devise find identify kind site separate diagram cell aim efficient detection purpose devise computation power diagram classify outlier non free way aforementioned use site programming extract key decision making represent euclidean partition explain new outlier identification principle interested square assignment principle g devise efficient frequent special construct diagram lie cell call power application site diagram non sake completeness site special simple situation motivate way plane far customer less reason assume customer typical arise generally customer far balanced assignment customer cell cluster lie site call separate site assignment clustering extreme study circle circle blue circle circle blue circle pt green green pt green circle pt red circle circle pt circle circle circle circle green dot site least assignment diagram diagram classical structure diagram application multiclass machine literature kind piecewise linear separability induce decomposition diagram special hyperplane cell natural hyperplane customer assign customer lie context find assignment new customer exist intuitively margin small euclidean cell classifier depict example gray hyperplane present task power margin balanced assignment scale rectangle assess e diagram come exhibit brief outline separate diagram implement among cluster interpretation error multiclass soft diagram depict scale green circle circle circle blue red blue circle circle circle pt pt circle pt circle figure soft diagram six point use diagram general set transfer site linear margin pair hard solve essentially hope optima use prescribe parameter
reproduce six multiply normal kl divergence cs exploration interactive music wang david wang recommender act greedy high suboptimal preference potentially interesting successful system balance need present new recommendation exploitation reinforcement multi bandit user preference audio recommendation piecewise approximation variational benefit unified music study indicate sound signal synthesis research national foundation centre office wang department national sg mail edu sg wang science star sg mail star edu music preference recommender system user incorporate feedback recommendation serve objective user feedback future c recommender ignore high greedy recommendation consider example rating three recommender rating rating song external true rating expect rating song user recommender user recommender rating recommender successful filtering song give expect net towards greedy recommender rating clearly suboptimal recommender mean recommend feedback rating recommendation shift user preference good interactive recommender preference information exploration exploitation especially I song music repeat song unique music occur often domain article movie arrange order strong repeat cf audio content recommend song divide generation distinct cf next suitable generation interactive music learn exploration exploitation unify bandit systematically exploration exploitation study reinforcement recommender recommendation traditional approach rate user audio recommendation music rate new probabilistic music discussion section describe rating music present discuss direction conclude comprehensive music recommendation detail currently music recommender classify accord cf preferred user well summarize widely suffer start recommend preference recommend content audio prefer quality acoustic system user system become popular various user context e environment hybrid work combine music recommendation markov whose user coherence allocation capture latent cf internet recommendation generation differ aspect usage highly efficient update real life user web zhang try recommend generate accord adjusted manually system control user wise music preference need infer unlike consider prescribed training learn rl algorithm exploit learn armed slot machine arm namely player round round receive sample e learn predict exploitation player payoff thus balance multi armed principled solution simple armed predict choose uniformly elegant ucb payoff arm ucb history high exploitation select ucb arm call face bayes art counterpart ucb regard variable similar every ucb select interestingly form confidence bind difficult ucb quantile posterior bayes rl decision mdp generalize mdp expensive reinforcement recommend page book web feedback document profile term base temporal rank linear attribute duration price country weight mdp preference recommendation payoff web recommender history web page similarity payoff click news payoff news vector bandit model differ fundamentally music recommendation different recommendation factor rating make confidence ucb bayesian section offline rating dynamically human believe reinforcement improve music receive little attention liu mdp recommend heart maintain normal state heart payoff however parameter learn exploration exploration exploitation evaluation chi learning learn similar state category recent history exploitation tradeoff contribute much music recommendation mdp handle require recommend base differ tradeoff consider recommendation approach search conduct active optimize predictive exploitation optimize recommendation system reality exploration exploitation reinforcement bandit improve music content state user factor song highly relate audio content audio content feature without factor preference represent preference music user keep assume preference tradeoff apply cf distribution popular cf reason need posterior ucb use complicated study update new matrix fourth suffer song problem method cf capture causality explain song however science capture aspect causality music content repetition song circle piecewise approximation repeat frequency essence examine user collect box proportion repetition last fm side fm even song individual user history frequency rank plot scale type book make little music appropriately impact inspire assume particular song decay gradually recover last song recover indicate user rate explore learn process recommendation recommendation traditional static effect preference song content dynamically song rate rate song likely recover repeat library model behave accordance law product lead alternative model user recommend rate historical traditional suboptimal account balance exploration armed balance exploration interactive recommender payoff music recommendation transform bandit music recommender change rating cumulative rating realistic objective traditional music individual song adopt algorithm recommendation task recommendation eq develop posterior history recommendation sketch explain th recommendation accumulate recommendation lr ii bayes song finally exploration song cause lack music content obtain address rely instead explore exploit whole interactive present fundamentally tackle start yet conjunction multi arm sampling comparison problem usually comparable focus easy ucb inferior k lr develop dependency convention recommender gamma put put close approximate directly use simulation obtain every sample substitute histogram approximation easy slow recommendation develop fortunately simplicity u learn product prior definite graphical model conjugate objective np linear convention independent minimize first step moment e ix nr ix nr ix n nb normally compute integration two normally distribute trivially scalable content linear song preference function linear extend put prior modify derivation incorporate factor design music effectiveness study effectiveness greedy baseline pure always song rate traditional minimum bfgs rating baseline bandit assume rating ridge balance exploitation respectively contain ucb indicate cn ucb cn four three cn ucb greedy cn nonlinearity include discuss combine exist solve future ten video convert file rate song second feature size accept retrieval recommendation one feature add feature user expensive consuming conduct principal final feature thus performance music use music lack explicit deal implicit cross result significantly audio useful offline propose contextual assumption time independently distribute unfortunately song recommend keep therefore comprehensive study approach pass verification refine preliminary whenever necessary omit page verify recommend rate gap two recommendation prior uninformative study exact uninformative preliminary bayesian discretized minute decay characteristic people song define user minute easy month ensure compare recommendation recommendation rl recommendation rating recommend song en recommendation well different element sample range preliminary conduct regret algorithm pure base greedy cn cn nonlinearity balance exploitation cn ucb addition fast small use system bayes ucb improve recommendation performance good cn also piecewise comparison analyze conduct efficiency bayes cn bfgs addition include simple perform core intel cpu main memory programming r time variational inference linearly size fast mcmc significantly bfgs compare variational inference three find another approximate less finish update practical requirement implement efficient language time prediction bayesian greedy however cn
regime rotation approach converge achieve genome wide expression optimization matrix include independent ica many orthogonal recently mixture multi challenge optimize break typically costly operation present manifold compute operate orthonormal costly equivalent orthonormal relevant partial answer start show matrix single coordinate update apply local minimal prove gradient variant depend demonstrate analyze efficient gradient achieve number operation fast descent choose coordinate calculate directional cd operate matrix directional straight introduce apply riemannian gradient amount multiplying rotation assume differentiable matrix dimensional tangent manifold denote define u natural geodesic curve geodesic locally short acceleration point direction geodesic pass fortunately might hard orthogonal manifold parameterization curve skew parametrization euclidean riemannian derivative look definition along geodesic curve riemannian straight along geodesic step size amount use coordinate descent gradient possible directional update directional derivative show obey know dense operation angle compute cost multiplication rotation successively ordinary multiplication euclidean determinant decompose rotation rotation optimize matrix perform rotation cd specify scheduling coordinate follow recent coordinate paper minimization usually perform periodic minimize obtain minimize single bound interval random coordinate minimization technique differentiable fu final choice square riemannian converge number iteration proof auxiliary provide convergence riemannian riemannian fu directional fu tr fu fu u sequence optimum algorithm accumulation isolate asymptotically regard iterate periodic period compact differentiable convergence directional second sequence riemannian gradient descent function provide pca dimensionality reduce vector maximize z z drawback ordinary pca lack interpretability expression expression difficult common problem gene problem find constrain et optimal imply objective principal round initial jt full practice memory evaluate operation drawback require component sparse pca necessary develop stream treat give optimize sample previous incorporate give memory material pca attempt fraction explain multiplication arithmetic cancer expression consist level tumor compare method method optimize approach coordinate solution generalized generalize cancer zero cancer gene test streaming material expression collect human k gene measure spatial brain compare partition include test explain take account fact principal orthogonal version greedy use range converge relative tolerance early stop stop range stop stop tradeoff variance component datum dot perform well find blue left figure explain sparsity power find comparable max sparsity sparsity max blue square represent range choose explain dirichlet ultimately reconstruct naturally cast orthogonal task method tensor tensor general tensor decomposable recently tensor characterization recently aim focus low result polynomial rank manifold decomposable extension start preliminary index tv v finally use du u exist vector scalar eq symmetric decomposable show problem interest mixture model allocation rise moment decomposable infinite goal orthogonal decomposition decomposable tensor find scalar state consider orthogonal attain material adapt solve need collect identity q maximize function maxima random calculate intensive algorithm require tensor efficient online recently common task third tensor art method sample gmm rd moment k k mark red line mark tensor optimal number dimension component wishart covariance sample moment decompose reconstruct procedure outline cluster accord normalize learn compare optimal across coordinate minimization tensor intermediate mixture vary sample framework manifold orthogonal rotation parallel framework principal tensor orthogonal ica coordinate descent sometimes amenable parallelization develop distribute would theorem definition difference essentially technical difference indeed objective differentiable fu variant start optimum accumulation isolate regard
kb though triple ranking embed thereby entity score system successfully entitie exist system focus perform supervision kb side entity head side entity tail example kb rf term refer movie ie consist new kb task ie consider detect aim assign mention relation kb triplet rf direct say supervise distant automatically create detect text connect kb express york article automatically supervision common language especially annotation parse naturally use ie system require label numerous text introduce match train open ie supervision seed kb weak supervision also option al train relational wikipedia generalize recently text rely collaborative filtering directly connect kb share text kb protocol concern energy method low dimensional vector symbol entity learn one interaction entity model kb perform share embedding implement connect end either word feature vocabulary entity kb triplet vector denote letter character framework embedding score similarity mention relationship inspire adapt replace image label intuitively consist window easily score mention weakly convenient consist mention relationship embedding mention predict corresponding well suit interested building mention prediction system metric extraction curve concern across calibrate confident setup soft ranking optimize hinge enforce column sgd update step weakly kb connect relational entity relationship embedding score plausibility new entity triple work flexible training I relation kb learn relationship arcs kb translation plausibility rank high versus possibility h sgd score convert output choose otherwise entity test perform predict relationship na marker na add treat relationship composite prediction agree kb na score baseline relations york times corpus use entity name extract speech name aggregate keep relation mention kb around relation scalability reason relationship entity completely keep large entity importantly remove entity involve generalize relation version translate company place organization organization train triple learn sgd use validation calibration training take minute take day annotation display use combination slightly due relationship superior plot leverage basis improve relation relationship kb
reference radius network formation vertex prevent appearance multiple isolate represent unlabeled item present phase keep utilize formation previously slight label prevent become singleton vertex provide respective topological high level pattern formation instance occur high great conversely dramatically modify high class change quantify organization intuitive local hybrid classification framework orthogonal item follow decision tree physical similarity responsible semantic meaning vision pattern train mathematically membership respect towards low receive label maximize label produce combine test incorporate predict create instance belong maintain situation class still represent single various play term implement traditional technique variation neural little take inherently relationship detailed ability topological technique formation process order satisfy construct isolated component quantify environment cover aspect one component mind concept walk critical indicate length walk indicate item class responsible providing possesse order regard cycle proportion item pattern examine test responsible variation test membership explain appear quantify since walk member class cycle length th give arbitrary formation new cycle procedure class class share k undesirable configuration class problem post processing link share post way formation since denominator accord variation cycle length view membership value pattern formation result low membership classifier quantify variation length class range order walk possess far away start responsible capture local hand deep responsible capture component make mixture global unbalanced great sensitive term mathematically indicator argument view introduction mechanism effect unbalanced high instance phase represent nn length component one link calculate cycle decision may general framework introduce link classifier walk q domain upper limit use look infer walk couple hybrid length length weight length walk simulation assess propose hybrid fold example show mechanic hybrid end cycle small synthetic fig goal triangle shape item construct component purpose classify discard respect classifier utilize equip tucker rbf u level fine tuning parameter class red circular blue square proportion circular shaped carry geometrically lattice pattern produce svm item transformation distinguish topological worth create datum totally capture pattern c cn c n cn report exist different value pure weight decision svm classifier weight pure know kernel able classify item svm shape shape fail correctly classify straight line densely blue arrange empirically calculate red circular shape segment circular shape vertex rectangular shaped formation test cover vertex straight vertex fuzzy rbf employ classify depict item embed shape classified member circular shape shaped respect triangle shape choose item least straight square shaped decision decide shape simulation classifier construct solely use cycle iii high show cycle length clear act produce conduct phase give traditional rbf employ repeat mean advance simple class fig type cycle classifier construct cycle length outperform classifier almost wrong label start two class associate representative mixture spatial class classify condition classifier insufficient get classification slightly heavily impact network illustrate phenomenon region item region misclassifie pure traditional display level boost rate rate influence construction representative different relevance rate distinct classification class indistinguishable consequence formation representative possess e unique depict rate combination keep high level transition cycle explain exhibit traditional heavy decision responsible high classifier use combination memory question really necessary range feasible section end argument cycle display fig dynamical steady verify happen walk reach sense computation walk quickly relation interesting phenomenon two know balanced unbalanced fig depict fig display see length hand interesting divide three small proportional intermediate proportional iii steady cycle small restriction intermediate reach peak characterize vary capture formation topological chance window scenario length cycle explain fig walk already cover capture formation class scenario walk completely topological walk redundant classifier near steady enhanced fig behavior framework value three distinct level classifier irrelevant prediction reach steady length change term length satisfactory htb fold report low kernel framework set table detailed numerical attribute reciprocal euclidean employ example utilize processing combination optimization highest critical length two deal kind walk base one walk realize visit site long jump classification high level report result technique purpose evaluate different fold cross three indicate case obtain row low employ weighted cycle table walk environment visit site contain window walk conduct visit item memory phase exhibit classification term respectively sake clarity level achieve propose technique accuracy refine propose accuracy technique boost accuracy outperform version c handwritten compose thousand handwritten digit technique digits involve recognition compare shape conduct name provide use classifier implement use setup euclidean function eigenvalue specifically act together classifier environment goal reveal mixture rate neural reach increase responsible regard classifier accuracy propose even hard distinct level reach htb cycle digit information digit regard length digit variation red respect g digit carried draw mnist firstly digit box box classify represent red probably digit correctly digit pattern digit class form test digit generate digit variation measure form digit cycle variation component digit occur mean test component digits result test correctly digit cycle well fig digit classify propose novel combine high instance physical feature formation walk complex topological interesting technique term order increase useful classification worth walk simple still capture topological underlie network local memory occur cycle length hope provide mechanism representative rather go acknowledgment research foundation via refer nontrivial connection salient feature study offer dynamical network inherently formation vertex technique level equip statistical high pattern level pattern training utilize semantic specifically end complex fashion intuitive way work critical length large make cycle length interestingly able already optimize traditional classification technique recognition handwritten promise walk complex generating input construct instance supervise near neural essence train classify unlabele physical feature input technique isolate tend represent hand intuitively circular perform identify pattern mean computer formation refer space sub form class former perspective capture turn permit classifier reproduce class put form strongly training attempt various focus really make statistically share vision suppose datum uncorrelated classifier change content datum item look relationship literature stream several kind relational collective contextual classification neighboring assigning viewpoint mention approach extract avoid quantity understand force window
site human genome population analyse self snps spread genome estimate allele count logarithm follow constant influence allele frequency convert binomial algorithm relaxation quadratic couple quasi newton difficulty algorithm comparison assess measure root square rmse matrix nuisance estimation rmse compare square pearson matrix replace true run value regularization scale imputation initially resample miss using stop successive minimization tolerance validation imputation evaluate partitioned entry test build value output set residual compare quantity snps provide sum kullback divergence sample shannon entropy correspond gene indicate error cross cross approach generate simulated assess identify simulation project datum population choose project consider snps linkage population true matrix construct binomial individual simulate explore accuracy al run cluster basis european define european grouping united france individuals european populations european population grouping sample program group level population frequency procedure population distinct coefficient moderate strong individual simulate correspond snps addition simulate without create ratio miss model individual li genome association phenotype molecular use li factorization latent semantic indexing structure novel analysis na history european ray principal analysis population range sd autocorrelation population genetic sparse negative negativity constrain square microarray h nonnegative active method comparison na permutation genetic dense lee dd factorization li dm h gs infer genome variation ms j spatial genetic wang md squares price nj history price I genome study jk jk population rw population wang p individual integrate genetic variation material root rmse rmse dirichlet population express snps panel snps estimate top cross cross bottom estimate european cross simulation population european moderate strong hour hour project hour program option well employ sequentially test matrix missing predict use cross capability material indicate capability perform extensive regularization could moderate panel predictive cross entropy indicate ten great generally discard value around panel one wide imputation regardless lead project last accordance criterion project criterion project regularization obtain snps phase european population population panel entropy criterion equal lead equal figure separate east separate european distinct seed project phase entropy substantial level characterize level et entropy value obtain graphical display occur east separate group estimate project coefficient chinese american american employed simulation assess simulation project choose matrix create use binomial range square table level error regardless parameter indicate accurate output set size comparable square degree degree freedom cluster every simulated dirichlet simulate coefficient population population european population population program cross low produce presence estimate robust assumption robust equilibrium human datum well explain suited level european confirm previous conclusion provide estimate though approach nmf analyze obvious population show overall advantage flexible mainly computational fast genomic http fr provide proportion sparse least nucleotide snps matrix record derive number bit word encode factorial suppose population priori carry frequency deal factorial proportion sample focus individual allele least square ls estimate obtain denote problem singular loading proportion non entry equivalent perform non factorization l optimization norm equivalent perform nmf
remark department management mail economic discover markov endow pairwise fulfil markov model widely handle distribution discover markov central task field computational diagnosis field lattice undirected graphical discover algorithm evaluate feature top bottom tree way study find approximation distribution method solution parameter kind lead solve learn network remain hard discover markov concept short network relate tree content third part find network give summarize set adjacent vertex contain hypergraph neither fulfilled ic j j link concept terminology assign correspond cluster correspond triplet set probability value product lot assign associate pm conditionally independent give lm property conditionally gm separate independent mean occur kullback separate hx hx formula see leibler call fit direction concept introduce special q express index suppose realization endow edge complete vertex adjacent fact common two conditionally tree leibler write true write variable pairwise marginal calculate equal mm well equivalence realization positivity condition discover positivity fulfil positivity question gm lm pm regard assign tree sharing say distribution tree two vertex essential necessary gm lm endow pairwise markov graph property gm lm pm positivity recall sound example j kl global despite probability reader graphical hold lc information divergence value number possible probability mail example edge ii ht lc divergence figure endow pm miss kl see easy pairwise
decomposition small implicitly form descent moreover convert stochastic operation implement gpu interface reduce cpu transfer overhead big speed present fast method describe gpu extremely dataset million consist multiplication million gpu implementation efficient large community cpu sparse node gpu dataset propose testing discovery validate although notion standard especially use carefully tune output community mutual score use evaluate fact theoretic interpretation differ incorrect facebook table facebook consist around community membership time second consist review consist second much large collaborative consist million node two exclude minute file disk load memory compute entire comprise million read begin method node method possible architecture order dataset method consist format efficient setting sparse graph code undirected handle bipartite format bipartite setting assume homogeneous connect intra community connectivity suffer classification main strength implementation huge speed method model similarity document multiple community document edge generate tensor word new york corpus bag minute topic interpret occur topic thus present broadly setting al tensor current consider careful power base decomposition flexibility trade method dimensionality reduction preprocessing enable million learn overlap community focus factorization test recover underlie community interpretability statistical incorporate investigation issue explicitly form store subgraph node neighborhood work tensor decomposition store exact document bag topic document represent word condition document satisfy model u overlap among control density function mixed topic special exchangeability suffice count occurrence c order moment factorize moment lda topic weak employ somewhat complicated learn membership introduce hide decomposition vector sample distribution topic modeling set distribution specify overlap control membership vector mix membership special model unweighted facebook review observe moment membership dirichlet define expectation moment product column observe moment topic unified tensor second whitening simplicity empirical modeling mixed membership use topic membership co subgraph second tensor note graphic exceed gpu large gain memory running requirement approach third tensor form operation operation via dimensionality use stochastic spectrum post membership truth whitening utilize orthogonal moreover reduction tensor whiten tensor whiten bilinear third onto result multilinear get tensor whiten km respectively multilinear transformation triplet whitening denote kk n dimensionality speedup easily word vector community compute algebraic pseudo pairs pseudo inverse running storage first svd pair order product significant role speed multiplication product allow requirement figure eps reduction eps eps object object improve equivalent c explicitly calculate maintain dimension corresponding whitening denote index dimensionality reduction decomposition recover iterate loop serial use iterative eigenvector cardinality decomposition role since loss tt maximize additional flexibility tuning let point loss iterative learn substituting update eigenvector inner product multilinear figure ensure penalty orthogonality prevent obtain solution eps learn decomposition community role dirichlet I eigenvalue estimate community membership thresholding whiten bottleneck handle aim technique computation overview graph membership semi definite project get recall whiten whiten svd namely thin qr whiten thin matrix note similarly without give method whitening difference instead implement thin qr whitening obtain bottleneck storage gpu highly limited gpu computation support exceed gpu core operation whitening module solve multiplication resolve issue million although parallelization efficient advanced library gpu therefore format cpu random projection eigen library sparse via theoretically device eps intensive task carry storage sized problem tensor shown gain speed implement tensor convert implement efficiently tensor learn convert ii operation stack operation although update simultaneously parallel idea stack internal parallelism design n iii matrix qr operation module svd qr storage node community perform store gpu memory interface illustrate transfer involve gpu device interface code interface eigenvector gpu memory device eigenvector l compare device toolbox code implementation operation fast cpu among code notice eigen huge gpu device code gpu iteration run code device gpu overhead cpu overhead perform well code parallelization whitening convert htbp module processing execution module pre learn world membership clean tradeoff threshold tradeoff perfect present carefully handle bag york top recover expect related word spread belong topic word topic htbp loose entity anomaly article capable loo bad agree file htbp keyword top word member program real detail htbp statistics facebook gd ab v communities l variational pick significant dataset american bar american bar bar bar present method compare variational bipartite business use review star provide gpu community device implementation implementation cpu case community facebook read around second compare effectively exclude minute minute tradeoff business attribute business distribution side trade recovery business lower demonstrate recover business match community business high top rate category ten large restaurant category result star count dedicated review business american free make media fm st libraries st recover node multiple attribute type business hierarchical american recover bar recover method still open open category remove remain business category number category number business remaining remove category three notice community business receive star score review business receive count help top community open location count star select business category involve review star although gender gender gender name htbp r star score l city city user ground reveal employ visualization note user accuracy attribute available limited count infer gender count valuable information interest location useful study well user snapshot user attribute top sufficient least attribute main improve large recover overlap efficiently score remain l alpha l alpha ten community recover high house reasonably look relation facebook reasonable college student school student record various publish publication member model author author community key insight involve firstly approach systematic heuristic approach guarantee secondly moment seem implement implicit lead employ run reduce paper incorporate extension principle number application partition communication machine community support nsf uci fellowship nsf award last support microsoft award nsf award award w acknowledge discussion david david wang team thank david provide variational answer discussion obtain whiten operation eq online descent whiten set rate last eigenvector shift correctness pp j whiten center vector amenable parallelization processor consist algebraic operation enable decomposition tensor advantage datum hardware parallelization gpu implementation high transfer operation data gpu library implement algebraic massive parallelism core massive core parallelism run core arithmetic core act basic block gpu multiple unit core precision unit unit movement memory read cache unit cache mb mention gpu whose core gpu device cpu gpu interact express etc execute many software core partitioning size parallel core share access memory kernel cpu execute architecture gpu cpu kernel program wide variety party library compute algebra base surprising solver library another library mask enable development requirement design execution speed implementation library gpu implementation library dense singular decomposition eigen offer flexibility rapidly implement maintain performance gpu architecture rapid intensive gpu memory cpu carry via cpu gpu movement gpu buffer transaction useful gpu direct cpu intervention specification time gpu programming gpu interface care gpu synchronization come interface call gpu require subsequent unnecessary movement datum interface device interface responsible buffer gpu required data cpu gpu operate gpu program good processing lead interface iii pre computation svd carry svd projection interface truth normalize mutual popular overlap overlap truth truth categorical empirical estimate membership categorical community categorical binary estimate coincide column consider entry binary realization probability node hold q denote overlap community aspect error j recovery special pair extremely sparse dense dense since vice versa small sized figure recover sized suitable dense membership community community np employ normalize norm community grind truth estimate normalize statistically ground community therefore limit community truth community score soft validation union community score separability recovery score aim within goal recovery well propose objective performance contrast look perform therefore use correlation statistical evaluate dendrogram preserve distance section corollary detect hide overlap blockmodel implementation gpu implementation exploit parallelism dataset wherein gpu memory suffice transfer computation exploit descent multilinear optimization flexibility tradeoff validate facebook ground notion value membership compare execution report many execution learn wide also topic membership
extension computer speech biology form likely return return hard hard mrfs unary cost admit binary model broadly cut application energy marginal intractable propagation topology belief remarkably effective fail result propagation coincide bethe variational optimum bethe subsequently bp fix optima saddle vice versa demonstrate bethe free submodular mrfs bethe marginal quick medical reference model involve disease finding therein medical posterior presence disease medical treatment seek patient suffer condition could different arise maximum procedure marginal problem scheme marginal inference marginal np bethe energy bethe pairwise mrfs find discretized mesh cover possible optimum great knowledge reason singleton case bound stationary bethe energy bethe remarkably view prove multi submodular find cut application model variety heuristic marginal diagnostic method restrict graphical another approximate bethe free minimization bethe free belief prevent bethe important consider singleton however connection provide without recently location bethe energy stationary may even arbitrarily global binary mrf primarily degree restriction bethe global optimum recent uniqueness nevertheless aside rigorously global bethe work marginal incomplete derive key admit form location mrf reasonable handle assign sufficiently bethe bethe adjacent connect normalization constraint occur edge sign positive negative mrf minimize lead quadratic equation root notice relationship entropy collect term free energy derive recall sigmoid bethe low pairwise order q submodular discretize considering end consider parameter energy maintain constant new subset locally neighbor let else exactly end solve energy change affect pseudo location unchanged hence location energy singleton entropy matrix entry bethe bound bethe free marginals q right constrain bethe away edge stationary bethe free remark hold true consider q inequality flip inequality improve dependency increase since bound achieve global value consider lead rapid even densely add negligible time global alone edge probability draw adjust ij yield connectivity strength small individual potential make return bethe run algorithm term width ib ia run crucial later discretized optimum discretized edge write q begin extend strong notational add dimension value otherwise define express local consistency normalization use lagrangian q satisfy duality focus lc kp stop precisely note substitute simplify I sign observe strong interaction hence second strictly assume else symmetry check element mrf submodular mrf edge mrf fully submodular mrf fully submodular discretization let define gx fs ft fs fs result continuity set derivative exist derivative fully sum express term evaluate term dominate singleton derivative second derivative bethe bind discretize optimum bethe sometimes derivative zero mesh fine sure mesh within optimum distance use taylor expansion optimum remainder discretize optimum bethe optimum eigenvalue point bethe bethe mesh outside eigenvalue bethe box consider expansion stationary side discretized never true optimum facilitate neighborhood theorem diagonal strictly entry define ij term bethe box bethe expression bethe energy location eigenvalue large elementary bind relate let proportion non diagonal reasoning start ensure use bad bethe bound flow need cut sufficient cut flow hence approach dramatically performance depend specification w knowledge bethe free mrf range
support indicate since run consequently nearly vertical line plot figure notice small recover long recover intuitively low neighborhood minimization confidence recover high matrix reasonable sensitivity intensity affect scenario sparsity intensity fig intensity intensity discussion apply present deviation correspond snr row notice vertical line support choice lead probability correct pattern zero correctly pattern condition objective row sparsity pattern theorem sufficient replace synthetic accord define range spaced iteratively scan poisson count correspond iteration intensity calculate non run obtain row row pattern intensity l problem dash row recovery solid cover gap framework derive row theorem px ls hence follow eq l due definition neighborhood ls ls x pattern l without assume x k x ls x contradict contradict present auxiliary give hold q divergence express rhs notice leibler divergence iy auxiliary hold notation subtract start bind let obtain substitute back multiply bound bind j iy q eq lemma respectively hence row sparsity notice r iy therefore frobenius neighborhood x ml exact row row sparsity x ml row ml l x zero difference true x x l ml contradict proposition contradict lem lemma prop poisson noise formulate constrained optimization square ls framework perfect original problem relaxation sparsity maximum recover measurement single measurement problem compress naturally image multiple distribute compressive sense arrival solve notable forward svm alternate direction greedy method use thresholded restrict isometry rip recovery uniqueness noiseless assumption additive many imaging emission consider noise ml likely moreover ml solution fit datum lead beneficial balance classical ml function desire measure value optimization whose sum leibl divergence penalty solution poisson difficulty namely non intensity without result solution tend intensity moreover noise impose constraint extension make approach formulation framework ls framework follow reconstruction original row framework derive confidence optimization problem optimization letter vector column transpose element canonical norm vector p eq isometry constant kullback kullback leibler use formula norm sparsity pattern x number zero e column mean mix measurement x measurement vector assumption indicate number loss matrix quick row matrix interest observe poisson want row sparsity sparsity row develop recover review approach matrix matrix review motivate square well observe sum nonnegative interest poisson maximum aim maximize follow independent add subtract ij log function omit rewrite unconstraine great examine question may unconstraine formulation incorporate matrix unconstraine unconstrained force row way enforce sparsity solution l ml start norm q define importance norm control fit multiplier substitute regularization control fit sparsity sense produce similar regularize control trade fit row similar follow trade challenge formulation say trade result fit may non noise characteristic pattern matrix common reconstruction approach principle ls ml framework problem formulation issue ls framework observe discuss ml framework find fit observe solution framework section allow choose regularization parameter switch role use square form confidence fit restrict guarantee perfect pattern search l ml confidence l new formulation enforce fitting observation method guarantee exact statistical characteristic confidence pattern recovery present proposition low proof section matrix satisfy call set satisfy set upper true square confidence row matrix proposition suggest sparsity proposition l high confidence set row measurement confidence set free radius likelihood framework obtain let true matrix correspond confidence contain directly proof proposition sparsity inside ball change row want proposition framework optimization sparse frobenius theorems x row exact isometry measurement follow satisfie exact mixing satisfy isometry likelihood choose proposition problem theorem suggest ml sparse large solution row infeasible effective approach exchange original mix eq study discussion method original enforce sparsity enforce row enforce relaxation problem ml constrain section optimization find lagrange multipli describe lx parameter tucker kkt optimality give scenario however e lead trivial solution problem second enforce look simple outer lx gx suggest binary find constraint strictly convex unique minimizer optimal project subgradient approach first proportional subgradient current project nonnegative backtracking input lx r lx lx lx lx constant choose derivation entry obtain zero place correspond row e imply line method project condition lagrangian entry x notice
les com cs france scalable robust term robust degradation phase limited threshold robust oppose learn model markov parameterize use monotonically propertie wireless load balance wireless receive considerable year application landscape heterogeneous often need access connect low load advanced algorithm via selection example complexity heterogeneous shift resource management burden learn introduction mobile automatic release release balance optimization closely aim adapt traffic condition self operate operate correctly neighbor stability definition stability derive monotonic purpose self optimize association robust robustness direct since association cf learn obtain learn practical robustness property wireless level optimize end user capacity mean tractable association prove scalable practical operate organize wireless elastic traffic problem static version association tractable section develop scalable heuristic operate manner show effectively network propose considerably improve accuracy convergence describe system mac summarize consider traffic system user locate ergodic duration interference receiver user user bs scheduling receive equal throughput user locate scheduling scheduling allocate locate equal gain maximal scheduling process arrival mark file arrival file write area limit unstable ps processor sharing summarize load eq unstable scheduling stability optimize distinguish static dynamic association attribute region regardless system decision user compose location amount call configuration association find system rate due code discretized use rate rate convention write empty control rate allow enter partition discretize conservative discretize low imply instability borel integral user arrival rate dr ss q dr gray close rate away close central rate association determine proportion traffic make intractable require store vast exist high allocate load traffic allocate neighbor variable handle non scheduling proportion user three physical mechanism technology I user load problem correspond optimization encounter composition affine convex affine convex file law transfer active user arrival file furthermore convex tractable classical include transfer reader study extensively markovian briefly possible whenever compute simulation twice component line simulate observation approach discrete trace eq gradient converge association user possibility constant user impractical overhead class configuration I completely specify configuration system state spend transition user depend decision take arrival ordinary ordinary controller class subset relatively attractive controller implementation ergodicity file alternatively throughput otherwise policy file specify intensity transition shorthand arrival arrival user denote denote write user intensity proportion spend decision previously intensity note link transition intensity optimal numerically indeed grow exponentially policy choose parameterization peak load already decision load irrespective load number peak weighting coefficient peak assign positive justify choose good rule let well system nr policie practical parameterized policy furthermore value descent iteration already acceptable oppose random poor stage large policy balance consider simple propose central neighboring decision write take locally available know number base user cost network time explain previously aware active user fluctuation overcome cost sum cost per active user cost reward heuristic compute solely cost behind reduction heuristic random far affect emphasize merely ascent perform numerically improvement essential outer would simulation central cell couple either area rate mean file size mb previous file policy good peak connect peak policy load even traffic improvement account possibly admit already result queue bring improvement file transfer traffic policy peak yield good queue transfer impact gradient propose heuristic fix gradient gradient obtain step strictly estimate admissible ascent percentage estimate percentage gradient grow perform accuracy say step policy figure traffic parameter represent network able configuration reduce evolution daily traffic satisfactory operational traffic arrival rate region assume hour association wireless file association reinforcement
simplify interested parameter space eq latent kernel static distribution assume denote predictor tractable unbiased hasting mh use combination mh smc particle hasting intractable smc distribution make propose algorithm interesting economic idea use article build recently mh construct langevin mala add proposal gradient intuitively chain mala hessian curvature drawing optimisation ascent scale hessian simplify tuning remove costly pilot tune matrix walk mala e analogue algorithm mala two paper lag motivation smoother make weighted filter consequently computation method compare marginal benefit smoother demonstrate interesting short burn markov chain simplify length especially variable incorporate outline construct algorithms mh mcmc simulate markov chain algorithm change remain current acceptance notation distribution explicitly likelihood mh algorithm however difficulty assume marginal mh operating proposal unbiase marginal simulate extended target use particle smc carry posterior chain simulate mh target gradient hessian expression define correct acceptance ensure validity aforementione proposal posterior current hence q side complete hessian hessian discussion matter point quantity form estimate three different specific scaling costly trial curvature hessian use simplify length drift proposal make acceptance special explicit run run sequel single proposal illustrate advantage add length curvature curvature manner proposal posterior pilot run analysis property mh mala proposal somewhat strict know benefit auxiliary particle filter sequence smoothing make particle denote dirac denote importance weight sequentially repeat correspond generate index article make special particle filter adapt r x w tr tw note quantity step estimator particle proposition section unique markov particle however acceptance reasonable study address particle approximation method smooth problem log explicitly write fisher identity yield intractable smooth decay influence observation mean large lag decide approximation parameter assume log calculate analytically introduce estimator need bit brevity rewrite write form eq option lag conditionally previously alternatively approach particle degeneracy affect well hessian proposal pd satisfy far cope issue add diagonal shift eigenvalue common type shift eigenvalue type keep another replace hessian trace pd resemble adaptive walk use mh reach burn pd handle use handle burn phase particle lag e negative particle propagate inverse use final burn combine proposal pd estimate hessian end briefly discuss property hessian obtain smooth smoother biased effect invariance smooth former enjoy bias trade lag lag provide smooth smooth length run resample true estimate gradient respect use lag vary run previously lag long lag seem use systematic resample model gray contour datum define use systematic resample hessian pd result adjust pilot single simplify tuning course length length burn approach previously algorithm avoid potentially consume procedure proposal burn phase result chain reach walk previous algorithm initial component model cause high propose never accept result result slow direction exploration continue investigate chain stationarity autocorrelation denote autocorrelation lag burn discard indicate uncorrelated imply chain index autocorrelation satisfy original setting determined series pilot run comparison near median finally mix mcmc discard burn present median result standard version add extra information hessian improve isotropic see column analyse major year prior poisson repeat length discard burn systematic resample hessian pd choice important explore discuss method version calculate manner smc alg r median median pre acceptance rate large hybrid seem due length improve mixing
soon stems include fit penalty constrain regression looking association discuss keep interpretability fraction maximal pattern document predict value give screening candidate return topic rate result rule find perform well difference typically topic perform interesting contrast variable take intersection randomly efficiency exploit derive appear among observation class high sparsity making detect budget achieve almost typically force make min terminate branch tree interesting interaction example interaction detection interaction right use sure otherwise aim build linear build latter way average odd development line plan categorical continuous variable fix see section contain far branching firstly variable moreover entail relevance remark ensure probability bind contain observation union distribution expect pick q pick eq guarantee substitute complexity factor get bound possibility arbitrarily small leave dependence weak limit return evy continuity suppose last thm example find serious approach interaction potentially work start maximal include gradually choose retain exponent addition new idea min scheme reduce computational hashing sparse classification predictor class label predictor important classification frequently word suitable certain available converted format choose reporting exceed threshold discover without low interaction also informative precisely leaf find set eq throughout subscript indicate empirical satisfy consider interaction tree form may interaction target difficult explain pure force interaction size whether restrict interaction problem infeasible interaction build work partition whose absence good class computationally produce unstable poor success recover interaction order informative distinguish ensemble somewhat instability prominent variable examine variable importance quantify pairwise fail highlight split try thousand leave general though crucially nod basis base split search algorithm develop improve form market customer together subset involve advantage give certainly distinguish frequently compare class lack marginal relationship cause poorly force require look discover interaction rather directly common active informative present basic feasible interaction setting yield previous paragraph method modification reduce computational min scheme discussion technical collect search interaction intersection variable remove present often retain intersection intersection solution check tree type computationally edge form connect acyclic undirected loss direct acyclic graph node root root say root oppose general graph differ convention slightly path number root equivalently depth indexing indexing construct rooted tree child nod collection depth every visit root compute intersection index parent visit child compute reduce complexity interaction tree improve apply discuss black black early node thus show many interaction tree interaction tree association search improve single examine return interaction pattern frequently search many computation intersection tree consider computation compute intersection check member order binary compare tree efficiency gain sparse intersection offer root though improve nk low less centre rather general search tend restrict take permutation let active agree variable subscript observation min index second equal wise turn derivation permutation create matrix enjoy reduced variance compare would subsample respectively subscript infinity would subsample hash would roughly typically long building note need every parent min wise hash construct rooted tree level child total ns j stop node intersection intersection decrease gain price wise effort turn quality previously bound absence early avoid early permutation depth create root leaf early yet terminate leaf insight win game serve fail concern classification stem class application bottom forest middle force row add intersection area pattern variable show count five way misclassification win frequently forest depth tree misclassification rate contain win end player white state take black white trivially transform binary encode presence block encode dataset obvious upper corner weakly winner presence add noise variable work black class white early modification table take stop specify win let branch terminate collect effect vary noise variable win combination frequently choose pattern win state hundred million aggregation iteration tree compute class absence pattern eq observation odd calculate probability figure rate number noise neither tree depth validation give misclassification rate interestingly tree much worse deep winning factor read win misclassification add variable easy variable importance signal important determine slight
operating subspace quantify closeness capture notion principal subspace orthogonality cosine value eq affinity angle angle affinity hard whereas become decrease able handle affinity description affinity go sample two say course measure affinity subspace cosine offer flexibility angle example subspace intersection regardless intersection subspace expression problem obviously identify correctly sufficiently able operate subspace method introduce confirm follow normalized want normalize code assumption normalization normalization entry ease presentation model ssc scheme encoding similarity construct apply affinity matrix graphical concern noiseless idea reason lie convex sparsity denote th trivial solution collect outcome cluster subspace access corrupted version make conventional recovery problem corrupt response representation long expression rewrite perturbation sparse three noisy shall add step emphasize theoretical concern cluster ni ij ji technique obtain subspace clean measure structure interest technique tend cluster whenever false shall apply permutation contiguous case apply lasso natural whether unclear select fashion belong subspace like natural need assign lie subspace select exposition refer statement thus constraint false subspace sample nonzero lie find rule take prevent make range imagine dimension wish way noiseless goal select depend dimension come dependence parameter usually depend raise unknown reliable argument rigorous statement article simplify imagine lying span column minimizer nonzero equality make nonzero arbitrary dimensional subspace combination observation operate column scale refine argument high compute precise dimension adapt idea beyond scope investigate relationship vary dimension subspace well solve heuristic solution number divide see subspace stack yield true inspection near typically fraction unless course exponentially trade typically false subspace dimension independently subspace dimension equal exceed clean sampling level proceed dimension belong subspace subspace hence true discovery likewise parameter around heuristic work discovery normalize point different dimension vs curve mark red detection positive rate rate figure show around dimension belong cluster discovery take near noiseless situation operate roc curve mark red dot trade true two step return like estimate next theoretically dimension coefficient procedure algorithm exposition precise understand imagine noiseless roughly home look solve plot clearly volatility low value subspace next dimension value suitable shall see rule select guarantee concrete return run example ft effective indicate false dimension point subspace hundred dimension application segmentation computer subspace equal step theoretical concern step obeys obeys argue small affinity reason level affinity eq refer obeys affinity algorithm di di column indicate step example subspace remain equivalently angle affinity subspace allow grow almost linearly subspace intersection noiseless first subspace practically average cosine noiseless show sampling condition albeit slightly provably like possible working affinity problem challenge operate unlike explore property column corrupt study linear uncertainty modify identical correction solve ideal noiseless resemble property I word mean gaussian deviation hence consider reasonable constraint high would variance article parameter depend underlie shall resemble proposal heuristic around rate synthetic selector yield false subspace two conservative come dimension early result half noiseless draw conservative subspace smaller close exactly conservative effective fact selector need yield simple n follow work comment subspace detailed comparison three propose computationally operate near per large affinity subspace one broadly classify cluster purpose mathematically present tractable subspace heuristic concern essentially interesting programming novel still intractable subspace clear algorithm representative iterative method subspace formulate nonconvex well optimize iteration due nonconvex iterate minimum consequence furthermore noise iterative model seek likelihood mixture maximization em style understand share discusse approach term please tractable tensor entry computation limit understand ssc lrr tractable robustness key understand subspace nonconvex problem formulation tractable provably favorable super fashion decrease learn establish robustness dependence key super polynomial demand simulation seem indicate regression corrupt covariate key difference show change covariate modification selector I obey whereas column design matrix classification support whereas establish closeness solution sense short far hypothesis finally mine sometimes experiment main suggest segmentation capture sensor multiple time segment correspond vector subject activitie motion system use trial trial comprise activity activity activity activity hard trial activity arm show singular activity trial show activity ambient baseline correct evaluate knowledge subspace build ratio misclassifie total half frame side always desirable small split sample strategy cluster standard similarity connect connect similarity tt temperature neighbor pair neighboring distance temperature solve procedure routine publicly solve correct selector homotopy solver spirit selector normalize step around vary corrected selector vary around build similarity cluster explain error trial indicate location sensitive around robust version ssc baseline range value reason poorly cluster summary trial table ssc outperform subspace correct selector report conservative parameter correct selector need achieve attribute get affinity group value sum trial discussion open develop tractable algorithm provably fairly along per rather section cc trial ssc express ambient distribution offer leave suggest topic future close question find sample density near ssc accommodate order one establish fundamental relate deterministic orientation noiseless leave future publication trial work concern sparse technique full clean develop theoretical procedure step covariance joint simultaneous year progress regression open see learn parameter response covariate corrupt natural clustering provably operate deal corrupt column density heuristic justification ssc purpose explore connection direction highly run sequentially computation acknowledgement thank manuscript constructive hold brief present theorem definition department statistics stanford stanford california stanford fellowship support grant subspace cluster representation fit take paper introduce ssc cluster demonstrate correctness theory geometric show subspace requirement orientation per subspace demonstrate motivation engineering find dimensional achieve component pca make perfect long around express long model lie experiment cancer belong different tumor imagine distinct cancer apply pca unlabeled cancer find mixture assign datum algorithm
mean deviation sample close value process involve estimation claim triangle strict stationarity impose parameter parameter non empirical residual consistency I reasonable fit copula unknown parameter need copula goodness fit copula assume bivariate copula term replace counterpart canonical maximum correctness approach show consistency canonical copula residual get copula I ic ne nn function residual available goal estimation e quantile q mean claim prediction asymptotically justify unobserved claim predict error simulate residual one order process error sufficiently acquire cumulative claim residual function parametric copula simulation empirical mass b nb bn u n n bi nb b j j b illustrated see residual compare residual model htb year stand year capture parametric part consecutive residual n frank together student copula fourth moment model goodness regard cite power bootstrap enough nan extract choose accord copula left dependence transform margin seem value value appropriate pair figure copula residual behave like one transform density copula bottom right simulate residual copula benchmark overcome number yield consistency consistency prove realistic traditional numerically cx cm lr data thousand total slightly prediction estimate quantile outcome really suggest model different logarithmic function whereas nonparametric demonstrate suitable stochastic claim bring relaxation restrictive traditional method structure claim allow yield development require mention benefit contribute small extension consistency procedure approach consistency estimate show bring development usage observe begin observe claim procedure predict even generalize omit stationary acknowledgment research foundation financial support pt lemma conjecture remark note remark economic goal life claim development period lead precise consecutive square estimate copula example provide illustration potential benefit claim distribution dependency copula square c I I I claim aim serious quite aside serious independent error claim often observation independent assumption enable need mention generalize model equation handle possible among claim successive year extend glm panel longitudinal another time simply claim possess common e observation estimation furthermore depend difficulty generalize period currently method consequently distributional var suitable utilize model business contrary approach business account business claim business claim notation summarize section amount generalize claim triangle cover series copula claim benefit terminology call claim triangle year stand year development period correspond development triangle history consist right correspond amount cm year estimate claim important distributional quantile triangle comprise suppose chain denote j algebra historical period accounting period early notation despite claim consider generalization generalization generalization next model positive continuous n procedure hoc propose claim contain reasonable one trend remove dependent dependence take c fix absolutely continuous respect lebesgue consecutive density marginal density distribution play process copula stage parameter estimate free fashion distributional assumption claim concern estimation dependence center moment model eq estimate lie fact computationally feasible corollary provide computational estimate fix j define since martingale respect integrable allow number martingale array n j lipschitz continuous reach convergence provide kind ii boundedness eq conditional least array integrable depend parameter true unknown minimum n imply integrable law number array j nb condition value behind claim
previous experiment hyper prior consistent experiment cost hour hour sg reinforcement optimality achievable dynamic use dirichlet self interested model hold behavior use limitation generalization integrate practitioner produce grain outperform reinforcement algorithm reinforcement rl optimally possibly environment e act optimally exploration reinforcement notion policy select action agent respect belief belief unfortunately belief policy choice flat assume pair multinomial dirichlet benchmark perform elaborate limitation drive practical multi agent uncertainty cause behavior independence place static sensor environmental phenomenon measurement unobserve minimize phenomenon action phenomenon spatial chen make ill problem despite convenience generalization across limit space come interestingly parsimonious resolve bayes car successfully human drive behavior govern possible generalize behavior state contradict inferior state simple restrictive world behavior latent gap put interact interested behavior well belief solve select jointly maximize heuristic aggregation perfect utility exact mdps agent behavior restrict model simple light depend application fit model arguably come model cope allow domain design modeling consideration call integrate expert yield agent grain practitioner compactly across show bayes analytically propose empirically traffic world situation modeling opponent parameterize opponent behavior abstraction practitioners parameterization multinomial knowledge knowledge fine grain compact generalize opponent behavior across opponent particular update history interaction consist history opponent extension replace paradigm conjugate multinomial practice desirable posterior thus make tractable bayes despite convenience inform update tractable parametric belief still though necessarily sufficient derive see later sequence iv likelihood interact opponent finite perform hyper theoretically agent bayes policy parametric opponent section finally size formally opponent tuple r uv p action opponent immediate environment discount latent parametric maintain belief opponent stage interaction affect latter obtain discount ab belief straight infinite v eq put value approximate action behavior opponent continue optimally constructive constructive interested reader detail tu uk bb b k exhaustive enumeration correspond construct function parameterization analytical final result parametric generalize represent algorithm build family function represent parameter prove induction let p vc general represent parametric theorem constructive latter computed base sketch v summation closely impractical function grow doubly plan parameter represent crucially modification address mention issue generalize solver augment belief state yield modification result cell move position mention level strategy accordingly spend intersection discount evaluate generate step fully inform upper bind know take employ fig always well rational base particular inform rational informed informed always know step maintain slow confident collect cause initial parameter conjugate belief actually exploit generalize behavior state restrictive e multinomial likelihood inferior non several different goal self play equilibrium converge certain security criterion self work develop criterion self notably work provably optimality adaptive payoff security game contrast work convergence learner course interaction terminate concern learner converge practical appropriate reality interact limited disadvantage stability optimality exploratory consider huge e poor security aim learner security turn tight optimal reader detail compare performance paper allow act work bad agent polynomially approach worst performance beyond scope call integrate prior opponent impose practitioner flexibility encoding domain opponent show gap self multi setting mit provide ga fa ga since every step finite function give induction recursively inductive rewrite plug apply b rewrite verify flexibility general present form implement function interestingly practitioner effectively though exchange upon efficient situation look consume solution alternative rest opponent keep grow exponentially project function minimize alternatively unconstrained specifie projection project solve back cast minimize partial otherwise plugging express j surprisingly operation eq find project propose work game evaluate widely agent stage interaction forward go back coordinate respectively agent remain current reward payoff discount factor experiment art framework meta hyper context response empirical opponent accumulated fall game performance framework test whose independently randomly generate opponent run simulation performance outperform h
storage negligible minimal provide make intuitive sense try space close use knowledge correlation approximate mean matrix describe thus expression accuracy limit decision define typically exact sub first show goodness criterion selecting distance selection candidate optimization close letter instead find coincide interval quadrature concatenation I symbol class solid plot correspond classifier maxima minima optimization minima provide matrix classifier maxima minima optimization confirms space around necessarily confirm jointly flexibility vary accuracy snr show accuracy pair serve note ml corresponding pdf exceed accuracy adding also note classification accuracy vary snr fair snr approach classifier emphasize sample feature act conventional classifier letter discriminant sample gain exist approach close computational concept cdf edu letter optimal discriminant function distance use classification location improve feature predefine could address application software cognitive interference identification exist generally classifier minimum possible discriminant perfect approach sensitive offset hardware implementation address issue propose goodness k identify ks classifier new reduce find high removing idea improve classifier refer recognize distance utilize sample classifier sample distance systematic optimal maximize result minima consecutive consecutive accuracy vary weight choice balance complexity classification improvement address subsection vector want discriminant follow establish state finding scheme I individual completely cdf jointly multinomial pmf individual could
densely arise context fix posterior look densely sub channel linear type graph amp design empirically wide sequel review belief high dimensional marginalization low dimensional message possibly pdfs probably know operate rule pdf variable pass variable variable message pass pass node posterior bit iterate refer channel impulse message repeatedly message leave belief code flow finite message pass pass convergence backward alternating symbol mapping code node soft decoder sequel refer impulse execute schedule hardware platform four belief code bit pass symbol message I start bit message pass schedule pass exact intractable pass review establish later use require relate transform belief belief symbol belief noise k k previous k refer estimation table derivation k l k k k k p require estimation specify straightforward mmse h r approximations schedule similarly output schedule pass message impulse message pass insufficient due connection impulse underlie compressed forward backward treating imply eq frequency calculate mmse mmse pmf z noise mmse convergence ti td belief act mc decode inference mc sub suffice backward detail pass correspond pmf impulse next soft symbol belief subsequently decode channel impulse backward reduce symbol unchanged k symbol node code code bit belief view soft soft decoder treat decode principle decoding study extensively reader decoding terminate k pass symbol iteration decoder bit belief maximum receiver pilot factor easily modify might opt receiver use receiver see carefully objective receiver utilize execute step mmse attention since perform impulse manner bit way remove receiver expense see impulse model non suffice node execute impulse iteration separately iteration need mmse nonlinear mmse pilot output pilot nan soft impulse recover bit standard distinguish impulse pilot nan pilot stem primarily call whose grow per iteration dominate modulus nature dft step reduce like discuss likelihood term belief symbol receiver conventional receiver contrast impulse inversion require conventional pilot uniformly spaced mmse selective meanwhile spectrum receiver operate mmse frequency noise ignore impulse noise isometry transformation relate state measurement sufficiently impulse noise carlo performance slightly conventional dft receiver lower establish conduct numerical study investigate receiver impulse pilot otherwise pilot space nan place randomly generate one gm noise power db occur respectively govern state unless run iteration ratio refer receive power symbol corrupt I trace bit simplification dft receiver symbol pp trace mmse processing conventional good technique trace refer recently among pp whereas formulation channel treat trace match knowledge symbol subtract symbol symbol send column non carlo nan pilot gm receiver drastically conventional db receiver art receiver db attribute huge gain utilizes receive impulse pp impulse symbol mmse channel pilot nan presence mf within db snr demonstrate near simplification db simplification db bad db state art receiver success strategy model clarity trivial gain receiver pilot estimation reduce simplification absence pilot similar impulse noise nan manner symbol estimation metric k interpret follow background impulse plot gm trace imply uniformly gm behavior expect superiority gm trace db nan extract meaningful must accurately symbol easier explain gap trace versus receiver backward two label consistency compare gm db worse significantly lose versus noise I nan plot trivial channel gm significantly superior estimation fact symbol show par well medium channel b trace cause moreover degradation mc symbol corrupt heavily skew regardless art medium investigate receiver know since trace exhibit investigate pilot nan examine channel gm receiver simplification latter channel impulse pilot produce bad pilot location performance alone dramatically improvement pilot reduction coherence conjecture pilot gm several nan pilot corresponding bit use code pilot channel gm word investigate conventional dft simplification dft decode conventional receiver db db impulse cost pilot gm graph impulse noise corrupt joint impulse symbol approach extension large graph soft decode extensive receiver gain exist noise match receiver easily implementation recent impact nan pilot address variable z z intend approximate product node node jx p leave variable approximate algorithm detailed guarantee scope interested reader h derive mmse accord k eq derivation derive mmse belief l imply derivation pilot reduce meanwhile novel receiver division environment wireless system much receiver alphabet symbol bit near yet tractable belief merge generalized message pass soft decode receiver drastically db filter meanwhile complexity wireless medium additive slow characterized impulse duration one channel independent circular gaussian additive white noise circular extensive wireless wherein noise reach communication highly restrict employ division modern communication channel low consequence conventional time convert transform dft desirable dft receiver complexity thus symbol however long strategy work exploit noise stems receive use straightforwardly impulse domain via nonlinear mmse pass conventional dft receiver decode especially power noise power signal use loss explain attempt suggest iterate date show adaptation preprocesse ad hoc manner approach noise impulse received pass conventional dft technique compressive typical technique sparse impulse symbol detection impulse learn channel impulse symbol code ad hoc manner perform impractical hundred channel generally form pass sum implementation algorithm vector dft receive become hadamard matrix fact q across symbol time sparse channel cover wireless system additive emission event device model spatio temporal field extend field wireless temporal result interference follow gaussian gm depend network provide significantly collect receiver receiver inherently provide noise
time evenly square area occur probability duration event choose result game two replication utility whole action fast algorithm need sensor need moreover play always result action propose introduce variation kalman variation strategy one nash pure game path play classic algorithm game hoc sensor surveillance play well algorithm reach play agent work property opponent strategy opponent assume player maintain opponent moreover estimation choose predict opponent generality choose player opponent play tm ta tm multiplicative update distribution estimate opponent pure nash strategy interested game equilibria equilibria joint player player reach choose action opponent player opponent observe play strategy confident level want action enough respect opponent play belief strategy confident change want action probability hence eq gaussian white player action simultaneously optimisation play implicitly player variant predict kalman prediction pure nash equilibrium available action pure equilibrium sensor network surveillance game improve theoretical agent game optimisation kalman recent technology optimisation crucial multi control sensor traffic control scheduling optimisation use common high communication well optimisation cast equilibrium feasible iterative game theoretic kind reward maintain belief nash equilibrium slow implicitly assume player play particle cost application instead filter small particle nash nash equilibrium player empirically observe game algorithm classic play reward brief play kalman play kalman propose play game hill ad hoc surveillance conclusion introduce game briefly classic play extend kalman player play action reward space number use distribution mixed player action mixed player choose player choose expect utility gain resp resp decision rule game response maximize player strategy specific response good correspondence nash equilibrium equation nash player player pure nash equilibria equilibrium strict action multi optimisation task potential game structure particular order optimisation utility global payoff equality player one nash equilibrium hence player therefore feasible global utility act utility introduce formulate life global utility utility action player action player optimisation algorithm converge nash equilibrium converge joint player reward play player choose belief mixed initially belief update weight belief well formally begin game maintain arbitrary weight formula cl estimate formula belief action player use belief strategy treat stationary fix observation player represent play space choose decision player assumption autoregressive sigmoid opponent represent markov hmm predict state current common kalman state hence kalman filter model follow zero normal also enough relation player opponent action use form observation respectively non first taylor expansion rewrite jacobian transformation gaussian need estimate step step light variance update respectively element vector entry everywhere else single opponent play separate estimate nevertheless action simultaneously cycle converge equilibrium game start always change never reach nash equilibria game choose action matrix appendix break occur infer proposition player available nash equilibrium date play play strict nash equilibria play pure steady nash equilibrium belief choice strict form identically opponent player estimation therefore maintain increase action otherwise equilibrium player pure become concentrated hence nash eventually want play belief converge correspond nash belief converge nash player eventually belief player play converge nash equilibrium one nash error state game initial belief player joint nash know joint nash beliefs player nash include pure nash player iteration game dominant expect payoff regardless player opponent action include game pure nash equilibria case initial nash equilibrium opponent strategy know player simultaneously hence pure equilibrium extend represented graph join action connect iff payoff change nash game game initial belief player equilibrium know action equilibrium initial belief nash equilibrium proposition number player payoff choose new nash equilibrium player joint nash equilibrium begin affect aim track smoothly opponent game one examine play two scenario opponent smoothly game opponent strategy action last game rest parameter repeat combination estimate combined contour area error value distinct area wide dark area respectively narrow dark area narrow second
next adopt minor omit intermediate elsewhere unconstraine onto w w unitary let expand diagonal entry j v immediately leave side easy theorem characterization portion prove let z ambiguity value z enough phase shrinkage straightforward consequence corollary substitute expression c expression corollary characterize limit conjecture conjecture show shall utilize conjecture claim coefficient correspond bound repeat utilize expression corollary recall eq rewrite plus small q let modified optimization problem p utilize prove leave still almost side bound show know moreover inspection limit bilinear sure limit limit variance consequently almost limit theorem transform computation answer expression match theory eq involve require limit bilinear form leave begin note value progress utilize identity state hermitian identity ba bilinear repeat statement bilinear imply large lipschitz function random tail absolutely imply borel q apply prof repeat proof bilinear form imply almost sure limit sure consequently prove key aspect rigorously understand singular singular vector exhibit decay edge space apart recent detail wishart cm truncate svd optimal singular measurement characterize limit show compute large noise model I noise bring sharp thresholding associate shrinkage explain via thresholding always suboptimal gain measure rank extraction many many application low rank inferential signal form q transpose arise assume rank prominent role theorem frobenius solution constrain precisely vice versa normality signal additional toeplitz see excellent overview reference exploit place set ask estimator improve start investigation formulate finding say even though invoke denoise let entry paper variation denoise formulate approximation denote h I drive yield denoise denote vector absolute exist negative element I high solution optimality bi invariant proof reveal behave theorem apply learn set effective singular clearly edge less rank conjecture informative transition high almost sure principal rank u result principal estimate conjecture almost corollary relative estimator highlight reliably estimate absolute opt opt good indicate metric might matrix gap produce consequence transform validate prediction value trial result figure realize mse show validate gain corollary optimal outperform suboptimal oracle able compare normalize produce validate proportion miss meet predict regularize estimate nuclear optimization form solution thresholde soft operator solution respectively small moderate compare significantly suboptimal noise figure yield potential set understanding benefit singular author limit interval informative component bring importance accurately estimate equally important able informative remain open application exactly analyze extend approximation research rigorously establish
close theory adopt share hypothesis system observe sample unknown agent observe I accord solve agent set prediction predictor pac encode example observe cause good unobserved task hard domain solve make directly applicable ordinary bound since derive result generally parametric form showed learn available part predictor treat prior adjust pac require promise obtain formally new quantity compute counterpart difference quantity inequality sample datum task p pp harmonic size task bound empirical lf kf follow probability appendix bind training kf union kf nm nm bind well understanding rewrite complexity p understand role look limit agent access sufficiently come complexity observe task agent task environment task opposite converge still environment amount risk transfer risk important quality prior respect follow relate predictor vector minor capture previously theorem fix rule choose distribution mean identify regularizer center q equation thing first empirical gibbs classifier gauss error function would prefer classifier classifier provide since twice multiply left side obtain truncation necessary substitute gibbs predictor inequality elementary calculation truncate gibbs half define consist point image label clutter also collect task regression contain student encode binary also final dataset class feature term large amount overlap setup relaxation z use conjugate find divide score exceed optimize loss multiply experiment discuss truncate bound able optimize expression numerically approximate replace predictor result quadratic get transfer error aside evaluate section regularization use split task three part third jointly evaluate predictor strength baseline fold dataset report area roc auc big mean balance method transfer technique square mse finding overall pl pl comparable exist manually sufficiently able pl value figure many need convergence fast pl even well possibly choose figure pl show value variance play big pl strict hyperparameter conservative reason make pl study theoretical main pac bind allow principle subspace observable derive comparable manually design study unimodal related plan explore integrate realistic modal relax g difficulty thank part european union fp grant also leibl divergence q lf kf apply hoeffding factor expectation fix exchange expectation apply expectation obtain probability mm ex mm proposition theorem axiom lot learn community effective relatively theoretical perspective pac generalization unify transfer low dimensional derive principled algorithm yield result problem equally human acceptable human new task whereas solve motivate formalize identify one goal simply task unlabeled learner perform future must task task solve task generalization however progress understand machine many transfer find empirically work many exception well understand aim pac generalization relation average contrast bind algorithm task interpret quality measure task represent vector plus
recover centralize fusion pdf need copy communication requirement factorize perform incur leibler close inspection fr r interestingly show find hold information grid factorize joint optimization describe fusion limitation fusion gm fidelity recursive fusion pdfs gm pdfs let correspond estimate pdf give two pdfs unnormalized pdf simplify numerator naive fusion form single unnormalized gaussian concentrate around move small force grow gm approximate pdf ratio moment carlo exploit proposal sample normalize calculation easily select though necessarily nx qr kp k qr select upper well adaptive investigate mixture approximation fidelity match gm fusion kl divergence pdfs gm poor local fusion gm covariance intersection conservative fig e substantially low fusion whole like approximation automatically gm hoc gm g merging term match calculation z lin engineering ny email edu ny email com advance communication mobile intelligence expand network bayesian robust efficient agent challenge implement ad hoc topology mixture hybrid tackle develop pdfs conditional numerical motivated target thus great enhance reasoning network considerable environmental monitoring surveillance scientific exploration uncertainty rely root decision efficiently information task foundation decentralize sharing robot mathematically equivalent centralized bayesian fusion sensor send extraction robust failure recursive pass property successfully demonstrate environment dynamic semi difficult reason firstly network avoid old information tracking fusion track topology hoc fusion conservative method analytically whenever non pdfs g mixture nonlinear approximation inaccurate message exchange copy expensive requirement pdfs densely connect novel issue insight flexible factorize update simplify communication fusion pdfs fusion lead fusion result conventional net mrfs exploit develop novel couple enable decentralized mobile cope uncertainty efficiently agent recursive common sensor discrete brevity hereafter denote conditioning aware set previously send pt px z z z cx fact show exactly recover distribute variant send exchange compactly summarize receive maintain require explicit handle exact fusion algorithms monte carlo convert gm pdfs via maximization computationally concern agent copy either dimension complex nonlinear propose eqs exchange complex pdfs relevant efficiently accomplish term dependency pdfs easy arbitrary sub order grouping factorization together original pdf separate exchange various pdfs may certain exchange factorization dynamic hoc topology tracking rewrite cx jx think yet allow denominator common pdf imply pt pt possible common separately theoretic obtain configuration conditioning expensive intractable conditioning realization independence relationship state partition state pt augment useful lead factorize whenever posterior represent via modular hierarchical mrfs direct although full beyond scope illustration leverage probabilistic hybrid factorize figure physical mobile look open engineering target coordinate group together partitioned exclusive region region sensor hybrid bn robot update
lemma arbitrarily minimum value follow use clearly contradiction illustration discussion support claim noise namely see believe desire result wide regime show worst achieve regime error arbitrary variance rely begin denote function finally use denote c lasso optimization correspond critical large cone eq characterize suppose direction sufficiently f exist convexity imply use combine finite feasibility equality f q lemma lasso normalize term equivalently write similarity key constant pair corresponding lipschitz constant w already low form next match concentrate gaussian lipschitz concentrate show show complete proof concentration combine fact proof lemma statement prove result decrease recall lasso satisfy lemma probability complete proof restrict eq increase apply lm analysis begin deterministic statement notation denote let assume conclude restrict feasibility minimization q start f last inequality prove proof concentration lemma probability applicable find lemma function statement proof probability desire show lm q prove statement appear notational combine statement lower union bound probability c main predict lasso relate create mapping justification combine desire show normal generation satisfy simulation sparse recovery small fix investigate marker property formula theorem f formula observe flat right start increase get close lasso calibration expect f f discuss behave verify expectation setup vary robust everywhere achievable nuclear nuclear singular basically matrix map simulation averaging simulation analytical prediction nd n quite analytical prediction function penalization penalization parameter estimation use time bernoulli entry probability varied give predict formula small change robustness apparent relatively increase vertical dash mark expect transition possible extension promise explore justification behind formula arguably point explore lasso value upper empirical open issue even compute formulae regime discuss formulae subdifferential one may observe classic motivate include sum close measurement experience also setup provide sharp recovery precise analysis follow focus generic state term geometry provide low simulation hand example interested sparsity consider nuclear obtain grow specific little find lasso bind measurement subgaussian entry setup adversarial noise example widely entry behave norm interest consider formulae mse possibly require acknowledgment author point section rgb mm department electrical engineering edu edu edu problem estimate signal induce encourage aid variation provide sharp study fall generic zero variance lasso precise observation error enter formulae subdifferential subdifferential priori give prove achieve choice f formulae estimation structure translate abstract formulae sparsity gaussian process statistical duality fit order sense machine learn typical structured pick induce function nonnegative case sparse induce great powerful compress cs variation selector course recovery variation accordance structure consider generalize lasso view related attract lot community noiseless cs problem linear common concern recovery minimizer realization random proximal denoise try noisy n kk kk particular closely pose via vector merge noiseless proximal compressed pose common main topic penalization researcher criterion mention serve relevant yet aim close involve regard analysis form relate particular form version knowledge makes arguably lack distinguish penalty former establish argument estimate precise lasso relate precise performance cs denoise discussion short main relevant highlight contribution close spirit pass amp connection evaluate explicit asymptotic propose complex amp characterize sharp direction note next summarize work constrain prove bad derive sharp bound parameter sharp identify calculate problem regime fail go scenario reduce mean wishart particular nm parameter square hull subdifferential subdifferential summarize briefly commonly encounter setting signal rank exposition include find nonzero constrain choose norm dr frobenius block structure sum norm similar nature concept geometry statement discussion keep attention letter capital denote simplify recall introduce behind use linearization induce function around subdifferential convex throughout assume minimizer hence origin small substitute first approximation argument clear approximated symbol simple characterization subdifferential approximation translate obtain original precise characterization suffice provide sharp noise term require precise case interestingly denoise close characterize regime validity statement formulae perhaps technical ingredient work establish gaussian let independent function worth mesh minimum measurement require cs purpose require slight modification precise section observe almost directly applicable problem take discard essence statement lemma detail summarizes optimization probabilistic connection argue size independent greatly statement term attribute equivalently write appear reduce value close statement minimizer close quantity concentrate respectively lm lm implicit large lm key effort cost minimizer bring question formally lm remainder show lasso predict short lemma similar low predictive power power restrict applicability idea rely predictive motivate claim regard lasso idea behind claim recent context approach significantly analyze lasso generalizing function defer highlight technical find arbitrarily third gaussian subdifferential cone lasso replace surprising replace analysis consider second nonempty contain origin play noiseless cs mention prove terminology descent cone noiseless sense exhibit normalize mild formal elaborate state repeat version therein assume standard mf origin result measurement lasso lm cm exist independent lm q match snr upper snr fully consistent expect assume define conjecture lm f lm expression bound call lipschitz continuous f detailed discussion place cs quantify measurement grow lasso estimate grow imply suggest maximize statement valid guarantee lasso theorem characterize penalty elaborate behavior yet several include limit optimal minimize define propose inverse effectively lasso translate formula lasso formula provide partial explain behind simulation validity prove lasso error measurement statement proof discussion interpretation contain detail framework summarize discusse prove motivate fail present direction section technical defer elaborate interpretation implication able performance constrained solely prove furthermore value replace observe surprising via hence robustness illustration regime mse signal equal case equivalent normalizing formula multiply explanation know normalized maximize conclude proximal denoise interpret estimation characterization involve naturally play critical characterize operation identify regime recover measurement noiseless translate write regime describe minimize explain nm definition recognize distinct region illustration definition empirically estimate noiseless inverse prove reduction indeed sufficiently prove validity claim f interestingly simulation validate observe nonempty particular sufficiently empirical arbitrary formula empirically hard strictly set formulae mapping parameter lasso behave mapping region f mapping prove short mapping mapping translate formula lasso important simple computing f comment simplify even use lasso formulae three version appear formulae abstract presence implicitly involve calculation however regularizer calculate formula formula derive summarize literature second row result sufficiently obtain derivation also substitute discuss see nr tb dr kk f discussion establish analytic research analytic exist detail compute sum lasso effectively calculate analytically point scenario assume variance translate distinguish variance equivalently objective identical mapping identical formula reduce power penalty map basic specific particular later therein explicit respectively sphere solve solve simplify presentation convenient problem follow write q lasso approximate accordingly eq similarly approximate denote cost convention distinguish symbol important technical ingredient underlie corollary establishe process completeness center index slightly course original modified lemma closely purpose bind carry opposed key unnecessary repetition treat choose formulation accordingly lemma side namely require compact lemma large tight possible similar analysis restrict carefully purpose analysis high usual eq corollary simplification corresponding statement discard corollary well term conclude statement corollary devote detailed tractable recall approximate generic optimization p tp recall definition analyze q lasso maximization sphere affect validity derivation treat common respectively nonempty convenient distribution gaussian verify notation line define perform detailed summarize lemma setup statement eventually last probabilistic decide intuition proof lemma section q q exist l constant lm problem optimizer state else statement constant constant prove statement sequentially first establish conclude second conclude l accordance clearly combine third exist lemma ensure combine conclude statement contradiction satisfy approximate c prove e recall c argue generic generic lasso quantity direct closed statement cone conclude lm concept feasible direction cone closure tangent cone feasible proposition element tangent approximate tangent cone cone subdifferential minimizer
achieve considerably small problem nevertheless large overfitte cca car c car year decade quantify well hyper sensor reflect energy recent vertical resolution hyperspectral segmentation hyperspectral use site water work band narrow band induce area moderate scene benchmark validate hyperspectral unbalanced pixel available pixel power use simple classifier consist winner perform extract cca slightly complex maximum projection use cca need achieve rbf whose width fold conclusion space feature extract overall accuracy conclusion high accuracy spatially homogeneous cover cm temperature resolution despite advance sensor retrieval full contain require reduce whereas coincide song consist way voting song extract consider method type poor make evident evident subsampling enough relevant bad counterpart analyze machine song although improvement excess enhance large focus figure figure review extraction increasingly popular beyond linear projection analyze kernel extraction dependence recent make suitable life application supervise facilitate cut heart manifold method challenge exhibit complex manifold uci repository applicability moderate complete challenge life record signal present outline many science advance theory come author issue presentation manuscript work project gray lar ia abstract extraction field processing device ever resolution multimodal source extraction provide treatment several principal least pls pls extension reproduce space review statistical estimation deal problem applicability analyze pay special hyperspectral monitoring device source feature extraction become increasingly especially true deal acquire image situation heterogeneous feature stack constitute method scientific area goal algorithm find set feature learn pca square pls cca variable projection pls cca projection maximize principle prefer fourth optimality square formulate linear algebra standard eigenvalue iterative manner speed refined exhibit relation classify fundamentally different relation reformulate paper map linear state property call inner work explicit ar label european centre medium weather total schmidt independence canonical kernel fisher multivariate least mean square identity discriminant ls extract rbf radial reduce reproduce kernel hilbert mapping rmse j gram uci california u appeal however involve label incremental regularization manifold unlabeled sample review aim provide extraction discriminant review variant label powerful illustrate wide applicability consider available real scenario audio hyperspectral monitoring continue review kernel connection applicability illustrative evidence method discussion review connection dependence variable throughout center matrix respectively target matrix membership l covariance xy adjust variable least square input usually condition ls project input preserve information obtain transform set transformation projection project projection project adopt field method projection input maximally align target characterize objective maximize summary discuss solution apply maximize nd last pt pls xy xy u c ic ir vector principal widely use impose work variance information task alone extraction explicitly explain principle prefer preprocesse simplicity ability discard direction pls pls projection either iterative iterative transform contain number way define variant pls hereafter refer pls assume relation variable pls pls pls discussion history pls inversion deal justify acquire correlate maximize covariance cca output deal high variance pls project final pay pls know multilinear cca multilinear l state alternatively maximization involve projection output xy u method semi project input reward predict subspace learning contain approximate extract lda view datum simultaneously cca preferred pca pls cca reformulate eigenvalue package inversion extract cca common large generalized choose method remove covariance matrix already explain minimization problem version cca add sparsity extract toy scatter extract linear output first pls whereas mse input achieve kernel aim actually map feature map define suffer serious practical dimension large typically equation term availability sample n trick indicate denote pls line pls pls pls temporal implement map view consider actual well input deal original representation illustrative incorporate toy radial rbf width select distance feature datum see expect linear counterpart look may good regularization problem size respect maximization extraction become expensive large finally opposite extract may especially characterize far life scenario label tackle fisher cca classification coefficient problem multiclass appear high unbalanced heterogeneity unify besides year hilbert schmidt simple yet cross estimator seek minimize translate resolution eigen decomposition maximize dependence problem connection correlation via convolution window scaling range show certain reduce cf worth theoretic well could principle make impractical application extension deal situation critical moderate memory computation extraction feature depend generally evaluate thousand new acceptable dense severe require evaluation contrast induce variable solution sparsity broadly method aim reduce matrix nystr low lr rr indicate row originally later feature rise version among reduce set sparsity representation generative include variance covariance show depend use insensitive adaptation algorithm full impose representation argument rl phase simple subsampling avoid kernel additional advantage matrix rl rl rl sum requirement sparsity act regularizer ability alternatively name sparse maximal maximal alignment project impose constraint restriction exhaustive pattern significantly reduce ccccc storage pca none none none cca none none none none ml dependent kernel dependent smc summarize analysis help choose application firstly critical function overfitte secondly eigenvalue analyze memory reason extract extract approach unlabele kernel ss laplacian essentially standard laplacian nn nn nn nn unlabeled laplacian input domain sum correspond note obtain drawback several alternatively kernel method rely generative unlabeled building run cluster gmm initialization assignment
set update cm block define b block achieve load balance block worker update step converge contribute carry recent dynamic static correctness little couple interference divergence interference correctness balancing vary arise worker finish curse merging contain similar monitoring contribute depend ml progress example include residual fast change dependency scheme scheduling advantage cost scheduling bootstrap discovery block structure fast worker bottleneck descent popular counter coefficient discover take optimization loss logistic standardized loss generality operator lasso accord cm step intuitively change justification cm dependency update parallel cause interference covariate step parallel size turn choose size consider load quality decrease runtime collect update use mf use collaborative predict preference incomplete user item preference idea discover small use user formally program via parallel cd mf column cm mf column row minimal interference observe perform load balance grouping row function distribute architecture begin group block avoid schedule large balancing load block worker report block update importance distribution dependency constitute iteration machine describe implementation scheduling scheduling arbitrary ensure meet block idea behind responsible scheduling block proceed assignment remain execute four assign block accord merge block load balanced worker take return datum available every though easily store distribute value parameter assign boost library inter communication model interface interface shall object dependency k p access distribute carry benefit effective cluster memory need variable assign take serve worker round fold prevent big problem preserves load balance block scheduling specifie purpose must variable theoretical definition regression section worker iteration group job one cm th worker highlight since zero probability analysis opposite objective f approximately optimal update parallel positive maximize decrease update indexed scheduling proof firstly parallelization minimizing cause parallel update happen effort objective whereas superior scalability outperform model parallelism select parallelism parallel lasso mf detail parallel dataset real disease sample covariate nucleotide feature mf netflix yahoo dataset netflix movie entry yahoo dataset run compute specification ram interface mf multiple machine core mf single multi mf variable block core three scheduling block structure synthetic core objective block static static structure configuration static scheduling use strategy variable e correlation scheduling bring ad second fast point phenomenon sharp drop objective update objective rate converge automatic stop change scheduling scheduling core core count scheduling select highly static benefit core count static scheduling begin advantage scheduling scheduling load balance netflix yahoo vary processor core compare mf use load matrix column row column intend load balance netflix exhibit benefit core reason core row exhibit variance say much severe core block size drop bottleneck thus reduce yahoo music exhibit benefit load balance unlike netflix load balance actually core turn yahoo heavily bias strong power without load balance load balance parallelism high count schedule communication synchronization consistency scheme design graph limit due work scheduling consider runtime lasso mf extensively literature parallel parallel differ purpose dynamic
inverse exactly strictly positive boundary equality hold seem trivial probability nonnegative obvious parameter functional obtain apparent reason real hx particular assume lemma pick v aa tail give v third leave x p repeat outside give empty theorem fact positive retrieve simply appropriate probability also pick clearly apply use q v p original global follow binary distribution satisfy property smoothly parameterize eq open map multilinear infinitely jacobian jacobian differentiable product full smooth parameterization corollary binary general discrete notational simplicity extend summary incorporate edge undirected cm dash use graph remain parameterization undirected h clearly need prove contain fact result avoid vertex head since h partition maximal induction take suppose proposition h partial contain set comparable partition suitable maximal partition define weak partial set role iii h h let closed maximal define respectively tail separate let follow close without intersect edge form since closed bc bc repeat argument replace similarly next almost everywhere need form involve side define let topological order corollary element addition v vx b vx v elementary conditional separate b vx x vx x vx v w f h addition order property proceed thus suppose elementary law induction ax f f x suppose order property maximal h imply hence parameterization h h conclusion inductive apply follow subgraph far h px pt px show vertice h h affect suffice b px px equation yield demonstrate triple also w h argument maximal head inclusion section lem lem lem lem remark lem support grant national health ai acyclic contain direct may dag markovian criterion separation first characterize markovian generalize dag discrete parameterization characterize smooth markovian direct vertex pair say direct direct vertex dag dag recursive independence causal interpretation unfortunately dag unobserve dag model dag marginalization direct pair denote graphical understand visually acyclic mixed study dag acyclic via restriction read graphical criterion advantage marginalization mention dag order dag discrete understand asymptotic general constraint may challenging interpretation chain lead family parameterization apply discrete design discuss relationship markovian model marginal log linear study condition intervention vertex order parent head form head induce provide model f conditional multiply see generalize well dag parameterization enable discrete view undirected edge parameterization class difficulty remainder graphical ordering partition subset basis introduce contain brief discussion mix edge join adjacent vertex path empty consist first last vertex path empty edge oriented direction consist entirely edge vertex parent denote include set none nontrivial notation shorthand contain finite vx ax v write relationship govern property specify non vertex precede path say path separate special separation dag separation statement say global acyclic nonempty separate x cp separate separate hard global property x induce vertex vertex appear precede measure say one easily topological ordering imply far measure ii obeys markov obey order markov satisfy topological order dag equivalently state simple factorization joint something similar consider block partition arbitrary nonempty restriction exist word dominate subset pick maximal return collection set suitable immediate definition recursively define subset remove follow contain suitable either follow definition definition hypothesis check nonempty disjoint set suitable maximal thus application subset provide partitioning try induction include trivial w w induction reduce show repeat application induction lastly contain within piece piece partition partition partial repeat trivial strictly small maximal repeatedly give c respect finite dominate wu precisely equivalence obey property e almost everywhere characterize shall dag obeys x two respect head vertex head within head subgraph subset head head within upon tail singleton tail parent tail head tail head head head head way violate say define asymmetric exist distinct iv cycle requirement clearly path order partition suitable partial ordering expression upon tail example acyclic direct mixed probability obey almost formal result sketch global imply theorem us factorization expression may since factor give nevertheless x x f head sense coincide partition produce parameterization finite exposition henceforth discrete case special follow obey sufficient parameterization set induction quantity need intermediate write follow quantity g ab px h side look however expression vertex partition left conditioning bar
expert I introduce I use mixture dedicated class iterative square devote accord define hide process independently transformation vector covariate nz ik flexibility particularly goal segment quality time control middle ccc middle prove random parameter model involve since direct use expectation mm ik pz mx mx tm ik update maximization maximize multinomial reweighted maximize analytically square provide approximated expectation convergence devote algorithm simulated criterion simulation misclassification signal signal piecewise logistic hide fix interval signal assessment show number observe accurate denoise vary correspond h section accordance phase switch operation guarantee segmentation degree adapt different regime middle proportion original parametrization switch hidden logistic logistic denoise signal propose cm national institute
spectral notion ni kb bt ba mt b upon rao invariant clear denote u particularly linear thing everything carry field apply apply compactly interpret whenever notation generalize high tensor section become ft give eq outline reader convenience section analysis certain fundamental symmetric spectral product representation analogy decomposition tensor unfortunately provable guarantee characterize interesting computationally uniqueness eigenvalue repeat rotation degenerate lead expansion recover requirement necessary access addition linearly orthogonality modify share coefficient tensor recover answer provably ratio distinct additionally scalar quantitative eigenvalue make invertible even handle restrict image tensor call get application tailor everything everything compute singular corresponding output tensor matrix subroutine ideally like error introduce complex field phase valid give vector choose maximize tensor column maximum matrix example employ atomic etc sophisticated substantially involve normal particular issue robust recover decomposition tensor considerable care characteristic run previous tensor pick independent tensor tensor run tensor empirically expression tensor matrix factor decomposition ratio mix uniquely recover end express bound derivative low guarantee ratio part might generality already concept algorithm reweighte begin determine ica slight unitary unitary simply make place position rigorously omit detail clarity case algorithm compute eigenvector covariance reweighte fouri input fourier reweighte formally simple transform anti complex hermitian usual hermitian svd symmetric eigenvector examine separate use svd subspace svd care gap preserve give method determine reweighted covariance subroutine translate gap observable real nr subsequence eigenvalue least j eigenvalue accurately recover separate block must recover accurately perform fouri pca input choice size desire space apart component gap choose gap almost accuracy theorem model unitary I recover sign satisfy proceed transform fouri transform reason heavy mean control uniformly arise ica characteristic underlying variable reweighte tx leave diagonal degenerate general isotropic unitary carefully tu td nonetheless mix picking anti end series differ substantially anti term derivative notably term strong anti accord pair q column assumption isotropic complexity maintain gap correctness case gaussian quick calculation eigenvector degenerate resolve vector commonly fourth differ require moment different role exploit unitary chain derivative later sequentially q derivative necessarily numerator numerator thus denominator anti univariate appear similar weak require apply univariate anti polynomial let proof lebesgue lemma derive property chebyshev include proof completeness supremum value chebyshev know see affine construct transformation fact chebyshev polynomial minimizer translate polynomial must lemma use polynomial stay within band usual lebesgue measure since interval change derivative contradiction know lemma expand series remainder end follow lemma associate derivative proceed base ratio denominator coefficient function derivative induction assume fact examine writing observe part immediately expression inductive immediately return observe q claim absolute guarantee exist isotropic apply polynomial truncation error desire anti characteristic exist ki n distinct eq use none care omit theorem degree truncation brevity q likely I anti concentration although prove real see consider next truncation error probability eq use claim want true complexity first transform let vector let accord separately chernoff variance covariance bind sample draw last inequality use derivative final ica unitary chebyshev inequality since frobenius consider basis eq derive choice low distribution e generally tail estimation give ica orthonormal hence gaussian eigenvalue least upper use frobenius substitute corollary sample gaussian nice fourier modify pick gaussian vector compute noisy matrix modify output eigenvector complete robustness sampling error omit ica svd singular matrix tensor tensor first correct projection e e I omit routine integer give work error essentially read one note arrange meaning copy choose coordinate get slice normalize v e tensor essentially say recover unique tensor correctly ica row column sample unique parallel could replace small efficiency satisfy tensor tensor tensor fourth ica compute optima equivalent form symmetric fourth derivative hand verify decomposition fourth tensor derivative give case simply second technique case though matrix hard tensor property independent eq mix one component derivative term carefully perform generate complex difficulty moment thus moment would tail moreover real exponential quantity modulus space complexity tensor characteristic run derivative decomposition derivative tensor empirically simply expression derivative entry naive multiplication thus suffice reweighte simply derivative tensor entry derivative salient count rigorous analysis incur random finite characteristic td k dt induction return first anti concentration similarity fully determine case gaussian isotropic orthonormal matrix randomness work anti concentration diagonal vector independent component k whose unit independently event thus concentration sequel happen least later proof taylor remainder term truncation go likely u anti concentration satisfy q rhs recall satisfying truncation assume event follow condition tu tu tu conditioning event use straightforward corollary theorem ica identifiable matrix unit eq q combine interval accurately throughout independent tensor characteristic behave similarly claim form show approximation dt empirical follow immediately give tensor random vector component dm tensor derivative tensor light expression product good show good complex xx argument chebyshev remain unchanged value decomposition come want second use come union bind want union extend give ica x follow tensor sample v xu st td empirical u xu hold tv success k sign q run put derivative compute error eigenvalue eigenvalue matrix low reconstruction require theorem show concentrate follow proof alternatively improve increase particular bound f parameter tu tv play algorithm recover sign hypothese eq probability simplify bad union event happen computation estimate eigenvalue skip routine check version indicate extend thus prove noiseless apply essentially gaussian comment precisely ica q characteristic algorithm estimate th vanish higher error make work derivative account extra little extra thing change getting complete noise matrix ica sec omit detail precisely get gaussian th spherical rather use rather integrable without fouri mixture span eigenvalue orthogonal eigenvector original representation estimate complete obtain q require fx respect dominate analytic extension integral complex argument omit example expand orthogonal anti gaussian anti concentration complex exponential anti anti concentration complex exponent plane prove sufficiently span vector matrix project mixture spherical recent moment unit without generality exist variance high therefore assume center ii eigenvector entry correctness robust condition mixture polynomial condition reweighte gaussian shift difference contribution use collect generalize seem literature determine estimate singular singular follow notion canonical angle angle range denote angle similarly small pick remain technical give hermitian matrix describe whole speak lie matrix perturbation long spaced perturbation theorem homotopy typically version circle let weak whose spaced consider circle norm consider eigenvalue contain ball disjoint contradict exercise eigenvalue linearly due generalize let matrix let close eigenvalue also eigenvector associate necessary identifiability fairly standard linearly almost surely sketch remove v v formal determinant identically check nn w precisely lemma number power situation though slightly power rao keep redundant multilinear multilinear simplify thing formally nc ns keep parameterization straightforward rao property entry uniformly use column incoherence constant na isotropic polynomial http www degree sc kx prove also parameter bind q clear first bound technical claim vector let taylor components b rr inequality e g well md hence old conclude open ica full condition ica large inefficient bind gaussian bind subspace two distinct ica acknowledgement circle lemma conjecture method tensor tensor share application provably natural alternative mixture gaussian principal effectiveness explain rigorously consist form datum axis eigenvalue rotation work handle high moment pca provably wide special ica mixture topic ica classic namely linearly isotropic effectively rotation cube fourth axis cube isotropic axis provably dimensional component product differ one gaussian fourth general observation give polynomial transformation extension fourth know ica derivative tensor tensor pair decomposition technique derivative reweighte fourier give alternative mean component give benefit result fundamental diverse area range source understand influential comprehensive ica variable unknown distribution invertible possible approximation hope one direction consistent model ica differ fashion ica community vast comprehensive ica rigorously fourth away assume fourth work several al al noise invertible sophisticated none know see chapter exist identifiability fourth apply know condition strong elaborate mention statistic combination mean covariance say gaussian mixture uniquely identifiable go exponentially separable learnable moment spherical mean subroutine tensor note equation obtain equation tensor eigenvalue question particular decomposition knowledge subsequent fully determined ica literature ica employ moment require moment moment complexity high order moment moment mix invertible accord ica n sample probability simple roughly speak matrix reweighte fourier pick inspire finite reweighted logarithm use measurement system uniquely fix derivative fouri add phenomenon probability moment set source signal recall case dimensional ica technique iteration linear handle base define follow denote outer idea attempt np use structural place restriction share extract algorithm tensor tensor matrix column unit permutation run explicit basically eigenvector matrix note provable guarantee ica tensor ica tensor second characteristic
mixture recover set fundamental inference problem inference draw computer computer start recover parameter random dimensional separation condition order separation generalize require condition attempt separation component polynomial separation impose polynomial worth quite many small dimensional appropriate recently show learn polynomial configuration inherently never get dimensionality primarily space efficiently completely clear whether easy dimension suboptimal condition eliminate gap polynomially precisely gaussians polynomially long complex degeneracy satisfied sense generic polynomially prove degeneracy bad show sample polynomially identifiable generally dimension consistent contribution polynomial show sufficiently smoothed anti main technical ingredient recover product ica combine algorithm ica bound dimension certain main consequence reproduce hilbert moreover combine theory technique mixture thus establish exponential theoretic bound independent good knowledge formally rao matrix gmm suppose mi recover accuracy b directional deviation weight tensor structure base entry precisely entry iid absolute constant simultaneous work relate strong learn align know advance smoothed mixture gaussian complexity succeed least add success polynomially high degree point trade result reduction problem extensively somewhat disjoint gmm reverse hardness low generic gaussians disjoint combine reduction ica barrier noisy due one statistic ica ica act inherent ica typically determine recovery latent signal exceeds observe ica ica rigorous bound dimension presence ica establish ica subset noisy ica satisfy coordinate polynomially away gaussian covariance sketch hard hard complex hard constitute progress curse seem situation computation concentration mass case problem anti concentration enable applicability wide jx logarithm characteristic jx property jx jx gaussian let value mixture pick weight canonical write act selector gmm goal formulation observe random distribution ica extent possible hope recover flip generate ambiguity arise order coordinate permutation independent mixing permutation turn requirement sign ica gaussian necessarily spherical noisy ica operation natural roughly convert digit final product vector characteristic variate motivate rao rao power product arise coordinate ica characteristic appendix ica set noisy parameter signal access noisy unknown parameter confidence sign permutation give outline namely mean reduction part reduction norm sign combine preprocessing relie state appendix fix x p iy mutually gmm similar coordinate describe internal step gmm experiment time take component observable sum mutually discrete ica fail satisfy assumption noisy additive noise sample ica parameter probability draw rule rr create ica model restrict rejection sampling coordinate long independent interest noisy unable produce ica apply appendix recover sign hand produce sample make demonstrate appropriate able sign add gmm whose I original sample proceed define unit column role play normalize column ica basic reduction ica appropriately application sign column construction last tell consist coordinate last I n sign subroutine threshold poisson failure add tb covariance tensor spherical gaussian accuracy bind norm variable subroutine cd add noise subroutine w invoke access subroutine obtain whose permutation call completely divide obtain subroutine capture threshold large subroutine immediately chance failure go gmm noise kx kernel definite easy obtain reproducing see introduction bind reproduce kx nf seem already embed interpret function thus reproduce see h subset fill exist gaussian sum one case complement sphere radius lemma kf linear combination coefficient collect positive put subset sum let interval strictly interval easy kx first integral collect convenience affect cover cut cube basic see fill apply principle least integer coincide without complete proof model ica fail coordinate explicit dependence run polynomially precise remainder proceed noisy homogeneity property univariate invariance scale ii follow absolute well completeness polynomial recover column ica specialized ica sample fix recover denote introduce describe name define statement theorem must away negative negative value moment give absolute moment odd give upper suffice require absolute apply suffice sign column produce exist sign permutation pi allow sign really want mixture recall correspondence use mean column arrive recovery gaussian noisy ica column later give suffice estimate permutation pm map proceed replace dependency gmm full reduction dependency model try learn propagate recover recover claim alternative bound recover column close sample recover replace proof define unnormalized one strict inequality dependency sufficient desired return error apply note vector occur bind dependency give negative reflect mutually without total enough arbitrarily variation truly chernoff get result let iid every union bind give statistically come truly make discrete work reduction learn gmm instead require remain tensor spherical norm low fix threshold bi drawing sample subroutine mutually independent ideal column ideal case recover approximation true permutation need reduction still close total random density fy ig variation satisfie triangle draw high fail otherwise time terminate return dominate particular specify eq suffice choose enough satisfie capture essence situation let role play role satisfy suffice require e q iy mutually part binomial case property largely let index also scalar imply property exist independent cross follow integer r n derive somewhat number second depend count give remain upper poisson distribution positive generating generating gaussian odd allow gamma factorial variation word clearly bound sigma total variation denote density specifically density choose simply assign atom continuity empty instance ica theoretically low provide outline give exponentially point generate associate ideal information replace replace mean hypercube reformulate unit ball unit sphere recall poisson denote draw upper result ica model reduction exponentially variation conditioning variation exponentially condition fact total triangle variable distance total sample exponential ica ica without treat signal define able gaussian portion efficient small ica k I suppose nk return k denote follow multilinear eq weight property ica literature logarithm g order coefficient taylor formula ex tensor contain tensor case order ica early popular practical determined exceed ambient ica polynomial polynomial sample later ica provide bound presence
eigenvalue set b li I conjunction follow index index easy nice candidate scalar question answer present require trace eq whereas informative seem sensitivity covariance analytically pick evaluation pick study copy still independent copy component note q normality q since change eq simple calculus derive scalar central limit regular sequence proceed respectively resp asymptotically since delta u order self metric endow use bound obtain one euclidean l ki computation deduce endow r bind negative hypothesis ensure complete uniform take analytic calculus true sensitivity index simulation pick proposition coverage count proportion apply upper sum constant clearly sample evaluation analytical close expression interpretation c interpretation si first order independently possible input dot around curve hull difficult rapidly frequent respective evolve motivation generalized order vector confidence influence l situation useful index hilbert e random associate well imply trace functional decomposition trace orthonormal amount truncate onto trace orthogonal follow eq random variable distribute define spirit decompose totally statistic center follow mn p mt l q iid eq p simplify notation large hence n v kn c mn c mn mn compute bind series get large sufficiently consequence nk theorem fact prove upper mn fulfil depend moment u starting probability central variable value step limit decompose u n l u sum delta delta partially national grateful cl section section corollary section remark section let input output measurable hilbert either index belong nice isometry keyword sensitivity functional index inequality mathematic subject mathematical encounter poorly uncertainty output aspect assessment word influence output independent input turn decomposition scalar split variance input importance large index pick recently scheme transform problem apply give mathematical pick give decade generalization aim paper wish secondly index generalization vector functional generalization implicitly al start construction multidimensional hoeffding decomposition due restrict satisfy pi sampling organize next develop discuss example difficulty extend scalar generalize index property indice one trace operation well tailor invariant isometry scaling introduce index satisfy natural invariance depend group sign permutation drawback unlike scheme pick may interaction yx non generalization index straightforward follow soon measure respect analogously obvious sensitivity sensitivity identity notice sensitivity index follow invariant isometry nonzero requirement sensitivity requirement fulfil index see depend rank isometry deduce symmetric orthonormal diagonal assumption diagonal contradiction isometry finally check formulation consequence notice index fulfil soon natural influence scale support invariance
either match one report cifar haar reason set run ham green green curve construct neighborhood threshold unique neighborhood letter unique neighborhood possible feature give feature number baseline letter obtain curve gain set much set set run tuning hyperparameter stack simple exploit preliminary improve cifar another benchmark whether correlation rgb gray detector construction construct neighborhood feature neighborhood neighborhood edge subtract usefulness adaboost mh classification mnist essentially free cifar suboptimal compare well nevertheless outperform boost raw pixel boost haar filter construction use subset filter connect subtract neighborhood motivated filter biological artificial system abstract notion haar filter patch intensity high level inspire naive natural world next pick broken motivation come show pixel recover pixel order recover immediately algorithm explicitly use filter view go pixel without validate feature multi classifier combine act algorithm odd result multiclass uci hamming implementation suggest suboptimal implementation significant mnist image prior pixel order cifar suboptimal deep reproduce boost raw pixel boost haar filter try uci feature improve significantly describe formal input label denote raw step construct representation intend terminology procedure recursively stack autoencoder neighborhood filter neighborhood connect construct correlate neighborhood li nevertheless quite role whereas control edge response find result rather insensitive set rarely three manual feasible aside hamming tree construct description website boost instance weak k h iy k weight current boost easy turn weak classifier require less case decision important less imply tree second design binary value inner whether construct manner unless happen single class perfectly ham tree produce length carry cifar classification relatively large benchmark repository hamming tree leaves split validate way single since overfitte large really significant iteration report iteration hyperparameter tune experience small control full mnist grey digits mnist baseline raw pixel achieve run tree leave type haar setup among generate feature decision pick pixel white construct edge depict hamming leave achieve image prior pixel relatively proportion neighborhood ht white
drop prior histogram adjacent joint locate image center smoothed prior modeling find ccc convnet unary construct filter propagation set control joint unary distribution towards final filter unary product propagation incorporate filter face result poor convnet job noisy detector actually position location face image face lastly learn convolution convnet location location priori multiple maxima filter likely candidate person scene comprise still frame pose process amazon ground truth position pose pose unconstrained part often also mirror example manually box training bring annotation px et test state scale highest across final location training convnet symbolic functions cache mini batches gpu convnet keep gpu main execution gpu evaluating gpu processing mini batch convnet gpu ms per cpu per x spatial window window dramatically reduce perform propagation full test evaluate model et give threshold evaluate compare detector detector ccc equal detector joint enable detector however spatial actual decrease location accuracy threshold convnet spatial accuracy already never remove figure subsequently spatial model rgb database vision pose low detector combine outperform explore structural improve generic spatial mention intuitive domain speech researcher equal probability performance mainly drive emission pose investigate currently take context office research award google award york edu inf de taylor new york pose architecture level feature higher weak unconstraine computer vision improvement art meet case outperform traditional architecture discuss detector level spatial previously argue structure crucial purely spatial currently many researcher recognition ht database vision determine configuration human part due background common heuristic art system face side view simple body sometimes pixel background significantly pose include body detector body detector commonly consist stage extract level sift orientation patch pool spatial sometimes scale representation invariance aggregate vector machine engineering produce sensitive remain invariant various nuisance alternative feature good nuisance learn refer technique unsupervised extract layer representation purely several margin imagenet end system advance hardware imagenet algorithmic advance specifically prove recognition use pose make end pose necessity deep recognition system precise location information complex modal present body pose want stress I human pose estimation filtering whereby map convnet detectors inform hierarchy detect people investigate decade early technique slide window feature extraction apply refer complete new propose domain call bag feature neighbor architecture human parameter pose tracking field find technique extract contain information contain pose convnet input find convnet would unbounded alternatively hierarchy work poorly pool useful object precise spatial accurately pose pooling issue mapping pose even deep pose much dimensional capture seem space pose restrict net output class configuration convnet learn pose coefficient find body per feature result region absence convnet indicate body location maintain body part detector enforce pose way full body pose child relationship convnet overview network end feature connectivity share local learn learn input convnet patch contrast emphasize performance comprise normalization input process subsampling layer internal pooling layer help even small amount drastically b tolerance unfortunately application convnet offline body pose sufficient invariance learn stage total three stage convolution pool process deep
even design proposal simplify probability infer f remarkably functional approximation nested laplace algorithm approximation via grid search direction densitie probabilistic knn classification distinguish approach order framework integer ignore difference newton alternatively potential candidate style round real classification l therefore assumption th observation similar knn label new observation hide hidden variable similar likelihood explain give last infer equation first equation posterior e kp kp pz kp simply reconstruct estimate expectation I I gamma yield give observation testing new calculate unnormalize j j normalize jj nj calculate whether validate two subgraph visualize red circle figure algorithm large regard reference similarity reconstruct four different metric root error rmse leibler structural mcmc rmse decrease ccccc measure case knn estimate conventional approach c datum asymmetric knn optimal table demonstrate execution slow knn well eventually efficiently improve include knn knn use similar quasi newton laplace unimodal optima optima maximal use newton slow dataset modal compare conventional provide proper order contrast consuming since bayesian validation approximation quickly find generate quasi newton modal lastly p remark yield improvement approximation albeit expense acknowledgement knowledge technology national program gray probabilistic original knn knn uncertainty make bayesian indeed assess view issue density without rely consume monte avoid adopt yield real bayesian free knn amount assign algorithm validation drawback knn probabilistic example infer paper address indeed perspective date several different tackle approach include aic schwarz information criterion bic information aic bic number functional posterior approximate posterior demonstrate address find near knn order conduct fair point several benchmark algorithm addition improvement knn although conventional domain knn define inferential point correspond intractable constant physics almost always impossible likelihood research use improvement carlo technique target posterior distribution model generally demand exercise context approximation throughout effort aspect approximation consist section include k near knn integrate laplace approximation extend review knn section generic underlie search include apply generic adopt real dataset conclude section literature jump jump process explore reversible monte green model approach relationship issue prior density reversible similarity instance mixture gmm difficulty number estimation recognition k knn classify knn concept majority vote simple sensitivity generate problem estimate boundary order address boundary knn conventional knn introduce develop particular z knn denote represent suppose point fig network structure conventional subgraph phenomenon implicitly likelihood probabilistic knn propose
vs prediction investigate weight epoch mean bar prediction weight hinge log inconsistent average highly conservative compare illustrate conservative bar hinge conservative update log mistake accordingly frequency update hinge log extra propose predictor addition reduce update derive total mistake mistake regret method notion strength online admit similar forward test art sparse lemma definition employ dual subsequence error mistake generalization strength affect mistake performance regularization fairly internet online email spam email whether receive update fashion stochastic online performance pass induce desirable add regularization online induce loss function apply backward splitting extended nesterov generate significantly recently lee suitable manifold high sum hinge logistic surrogate often regularize bound mistake affect generalization performance share combination leave subsequence mistake testing phase deterministic leave majority weight predict key method update numerous scheme mistake capture online term small moreover strength also apply online backward splitting mainly example feature attain simplicity focus batch set regularization prevent overfitte sparsity precisely error difficult optimize often surrogate therefore bind online setting hypothesis previous online simplify subscript indicate often difference fix q fw ic k w ks mistake list predictor respectively description perceptron phase predictor make predictor counter count example process correctly test module voting unlabele use algorithm auxiliary classification mistake summation mistake counter survival time predictor predictor form employ thresholding storing vote costly replace majority single predictor weight average predictor majority going perceptron give regret next happen vector case q mistake perfect large perfect regularization hold trivially give mistake notation vector perceptron lemma without regularization note make mistake relative strength lead suppose generate vector strength mistake margin context loss replace order hypothesis derive svm special mistake analysis training possibly evaluate separate algorithm batch brief example prediction unlabele vote test output hence name let mistake example probability leave make mistake test I random sequence mistake occur epoch perceptron hinge loss test adapt natural nlp sentence nlp candidate mapping already otherwise wrong classification classification sentence classification binary q margin definition correspondingly error I experimental outline candidate accord baseline train predictor optimize include regularization learning predictor parse label score sample predictor hinge perceptron axis mistake perceptron summarize online implementation perceptron average perceptron train setting tune result
common sparse disease return disease status use guide discovery matrix group develop expectation result accuracy ordinal discover imaging disease ad meaningful ad significantly ad compete task treat separately bayesian heterogeneous prediction latent gaussian guide discovery projection disease develop principle approach high status discover disease identify single nucleotide snps high compete objective study trait status focus diagnosis supervise heterogeneous seek representation trait status diagnosis view detect association sensitivity specificity addition bayesian framework heterogeneous datum datum phenotype association make diagnosis disease meaningful association snps advance provide common source variation nucleotide genetic basis disease source molecular clinical phenotype reveal change association different reveal wide biology valuable disease stage patient record thus predict ordinal disease stage approach discover association study cca extension approach linear projection widely quantitative trait sparse cca relationship genetic association dna chen et cca pathway disease diagnosis dimensional lasso elastic net group automatic relevance determination phenotype status zero weight wide application follow factor study supervision status disease ad often correlate clinical trait status relationship clinical trait diagnosis subject ad classification logistic ignore relationship design snps ordinal genetic imaging popular simply datum nature address new approach sparse association disease diagnosis variation trait latent latent use process ordinal prior show learn sparse reveal critical interaction group relevant status multiple diagnosis may form pathway meanwhile via disease influence projection guide association source disease name heterogeneous learn develop maximization vb iteratively minimize kullback leibler divergence tractable bayesian provide estimate estimate enables automatically dimension principled accuracy recover association cca higher advanced cca elastic ad ad account age old ad cognitive ad attract association trait among compete furthermore meaningful heterogeneous ordinal snps note generalize subject discrete ordinal status vector study subject ad link assume sensible association snps estimate evidence maximization framework specify label give projection assign decide auxiliary falls projection ordinal ordinal cross choose rich link data feature predict link identify critical force I e sample reflect experiment similarly sample eq selection beta experiment specification joint give specify selection ordinal auxiliary generating framework exact posterior turn infeasible calculate posterior resort expectation maximization f factorize latent approximation minimize kl posterior approximate fix approximate refine projection transpose g ij r calculate ordinal update distribution selection probability specifically ij gaussian observe ordinal auxiliary I n region decide ordinal information incorporate quantity expectation calculate optimize irrelevant optimizing dimension bind involve save present easily equation use initialize approximate bfgs training candidate learn update ordinal label regard also second drawback loading adopt variational em update except set relate broad variable include probabilistic learn latent representation lead datum recent prior employ despite induce irrelevant practical type together although gaussian gamma prior flexibility suffer highly control parameter solution spike task factor beta selection indicator yet prior assign spike generally avoid issue element often mine classification correlation among medical diagnosis meet discovery trait diagnosis employ diagnosis guide association discovery diagnosis diagnosis ordinal regression model latent cca moreover heterogeneous treat simplification function datum synthetic ad accuracy view ordinal instance projection diagonal rest zero ensure row projection block first rest column ij iy art include cca find correlation cca prior cca output include sparse projection software cca base package accuracy ordinal multinomial elastic multinomial ordinal run lasso predict net run cca elastic project employ learn latent ordinal semi package lasso fold tune free run polynomial except stack ignore heterogeneous nature learn fair test experiment partition data subset average recall truth successfully link compete improvement spike remove irrelevant avoid laplace cca supervision probably difference association study ht accuracie significant improvement find reduce prediction rank last capability performance summary confirm power discover association heterogeneous predict
performance account response pursuit force nonzero pursuit relaxation huber regularizer huber loss formulation excellent guarantee next class optimization corrupted original problematic recover bad relaxation corruption model entry output corrupt something certainly various propose minimization non problem unclear scale robust arise fact covariate rejection might corruption outli broad approach form tune interpret residual regularizer recover duality z huber condition satisfy regularizer concatenation column standard convex h otherwise consistent fail model since illustrate let set index covariate vector correct disjoint true choose objective fy objective fy make proceed corruption strategy certainly specific entry entry sophisticated corruption illustrate concrete pursuit recover serve merely importantly consider new sharp contrast success complete break regression hardness look pick r column index solve output operational outlier algorithm end gaussian exist omp henceforth result set force condition necessary handle section outperform force discussion demonstrate statistic high structure identification crucial perform lie robust pursuit intuition select column inner residual meet stop mp successfully recover rely condition value indicate mp robust mp robust matching pursuit similar mp robust product iterative response top one match pursuit input inner sort select large inner sort select behind dimensional easier low induce outli way previous intuition simultaneous rejection discuss choose guarantee sub sub design entry additive parameter note general cover bernoulli bound distribute sub corruption model hold output satisfie pn correctly identify nonzero remark exact need upper definition adversary arbitrarily change even course validation wish essentially character note simple bind fraction median result corrupt every spirit knowledge continue support error replace validation also strong hold corrupted row section corruption conclusion still correctly support report zero correctly relative error comparison tradeoff interesting would analyze paper consider procedure aim correct corrupted entry fill find portion entry magnitude row corrupt procedure far set q figure metric outlier highlight difficulty detection cc proof technical proof defer appendix simplicity correct adversary value alternative technical objective n combine technical lemma bound random definition chernoff appendix probability second sub gaussian random follow probability absolute mean absolute write ny p inner corrupt may assume write ij x obeys apply obtain due sub lemma follow h outlier last combine piece pick index expression pick incorrect pick expression constant straightforward algebra show proof corruption adversarial corruption seem original moreover corruption challenge corruption knowledge difficult distribute corruption outperform pursuit well
column show space basic modify add modify beyond way differ suit address show include modification miss complement current carry next current subspace remove tw dt similar differ handle next recent change influence old similar choice eliminate rotation appropriately suppose scalar define orthonormal choice positive large explicitly eigenvalue matrix reference condition eigenvector obtain formula identical formula relationship normality datum good converge quickly algorithm step prescribe low miss average iteration algorithm algorithm run span generate initialize miss prescribe require singular performance metric axis detail equivalence svd perspective linear algebra update incremental identify subset component expect certain similar decomposition modify incremental svd miss algorithmic subspace tool several decade noise approximation identify consist application lose corruption bad communication recommender miss product yet care patient health status sample originally stream signal vary estimation online use approach describe development closely field rank reconstruct tractable formulation experience reveal incoherence appropriate algorithm maintain orthonormal update incremental explore incremental svd
interval odd need interval short pearson connect interval wide interval contain pearson however union upper base interval always short binomial proportion seem importance pearson refer implementation exact interval give commonly thorough review proportion th z present suffer property recommend inversion normal interval however obtain instead solution equation coverage nominal coverage use failure replace generally simple denote quantile tail credible prior give quantile upper make beta quantile similar pearson interval jeffreys jeffreys prior side frequentist correct recommend general close bind invert modify root find length asymptotic approximation pearson pearson expansion quite actual space pearson allow plan useful need achieve consider confidence sample size expect depend guess available give first beta calculate desire expected length ignore expansion approximation approximation require bias complicate give accurate enough yield procedure ease cubic yield simple give approach sample determination guess wrong measure sometimes conservative maximize conservative prior beta constitute flexible tractable prior b frequentist procedure prior low bias size determination example jeffreys prior put probability mass uniform put close decrease formula similar tolerance interval set interval example pearson boundary place interesting determine binomial proportion interpret compare interval describe score jeffreys recommend proportion author term expect expansion expression pearson interval pearson asymptotically wide interval describe jeffreys denote length q expand proof increase constant fix interesting somewhat expect length dependent wide size sample level expect jeffreys jeffreys compare increase require quite substantially exact plotted function desire increase substantial expect increase remarkably insensitive jeffreys fix give require expect side expectation limit prior require jeffreys sample jeffreys asymmetric show different value side interpretation easier cost bind bound modify root proposition jeffreys order modify note version interval omit pearson expansion exact case preserve reasonably let increase increase sided set small recommend serious respect pseudo often preferable argument make sense interpret coverage minimum coverage think reasoning line coverage part parameter space close boundary discuss g coverage statistical practice widely level coverage approximate variable binomial really guarantee credible either jeffreys mean coverage accept criterion interval admit frequentist minimum line serious confident think discuss risk approximate cost actual may drop coverage anomaly occur close parameter close may subset jeffreys either moderately jeffreys interval score interval somewhat coverage neither jeffreys score interval minimum coverage computer intensive coverage discuss decrease minimum thus compare size jeffreys pearson intensive jeffreys somewhat I require approximate interval adjust outperform pearson similarly pearson order adjust coverage result short detail adjust pearson interval outperform coverage coverage pearson choose aware coverage approximate study practitioner costly interval much sided jeffreys price pearson intervals insensitive stand affect either size substantially method exact consider pearson short sided interval often interval role study author would pearson approximation actual even close accurate place lower pearson quantile asymptotic asymptotic take expansion analogously proof cp length twice thus q collect obtain expansion score interval length analogue therefore rely expansion side find corollary confidence binomial short coverage coverage investigate cost confidence short interval first term desire length expansion determine size pearson interval investigation reveal mm keyword expect proportion binomial proportion clinical risk consequently side coverage focus interval conservative tend nevertheless proportion practitioner g far pearson interval risk actual fall reason require sure exact seem reasonable method interval construction wider require sample certain expect interval pay seek suitable binomial receive method computer intensive nature close formula pearson obtain intensive size give desire affect size contribution expression excess increase require come pearson expression derive asymptotic expansion asymptotic approximate confidence exact rest pearson give asymptotic expression pearson give size length discuss side pearson give expression expect approximate defer two sided inversion equal
fourier function consider fractional integral change inside vice trivial indeed see p brevity fractional operator rl fractional fractional transform reader find operator et fourier eqs apply inverse convolution evaluate fouri suitable transformation mean moment recently represent real goal filter white probabilistic psd sake transform extension give representation symmetric perform belong deep insight transform application derivative omit sake keep mind eqs understand taylor expansion fractional derivative reconstruct eqs fractional fourier entirely new physical fractional taylor integral eqs psd fractional meaning generalize taylor expansions integral perform axis virtue moreover previously outline belong fundamental full fractional target psd stationary assign spectral equation gaussian white noise correlation system characterize impulse response transfer indicate power respectively suppose differential psd arise physical phenomenon find correspondingly transfer impulse enforce causal violate remain stationary fractional represent sum fractional expression psd order represent follow virtue mean allow h fractional integral specify previously impulse transfer third dynamical integral discrete truncation calculate certain amplitude integral eqs approximate wide discussion truncation integral along bind thus obtain linearity operator inverse recall w introduce impulse representation stationary psd highlight transfer representation mean integral axis real interval integral inside process attract fact connect fractional brownian white plot exact approximated dot plot wide interval although impulse f wind engineering wind application want neighborhood influence present spectrum differential whose extensive composition firstly latter exploiting rule report q fractional differential white linear fourier psd write therefore characterize target stress contribute power return highlight plot show convergence sum psd process consider express time equally spaced amplitude instant read useful extremely gives interpret fractional equation mean substitute highlight series first term second filter causal single auto function compare worth denominator algebraic adapt fractional find analogy function easy recognize conceptually indicate characterize temporal transfer digital reveal develop composition read discrete transfer characterize model eq transfer correspondence stationary color show density fractional integral first easily spectra moreover integral external white process desire spectral also approximate differential fractional derivative gaussian representation taylor form express valid drop report sake fractional integral derivative fractional derivative euler fractional convenient vanish infinity reason fourier transform derivative reasoning valid moreover evaluate mean moment q fractional hold along axis part belong along eq function analytic inside inside fundamental transform commonly theory reader operator prove identity fractional integral derivative hilbert eqs find fractional operator simplify type particular composition work previous definition eq criteria formula di ed universit di build digital filtering filter density brownian fractional novel taylor show colored noise weight fractional weighting show density novel procedure stationary process differential white noise process filter return relate equation linearity depend coefficient filter equation psd might least deal characterization filter psd spread fact many phenomena engineer physical interest indeed psd field wind engineering spectra respectively output equation dynamical multi engineering reader filter average colored applicability wave motion analog spectrum spectral problem spectrum define psd
combination particular illustrate fisher fisher figure stationary curve continuous blue corresponding record take account rather outline time together outcome atom record generate follow underlie cavity see cavity define computing cavity state jump calculation restrict diagonal notation fisher involve complicated follow fisher information former minus derivative mle trajectory blue optimal expect latter remarkably measurement informative attain h partial count statistic red large consist summary ground atom iii number iv statistic density atom intuition brief knowledge property procedure employ type summary number consecutive divide experimental statistic essentially need use distance cumulative cdf independently cumulative distribution k slight trial store experimental corresponding let subset minimize trial build posterior result combination distance build individual reasoning successive atom regime cavity jump see average jump density therefore identify define cumulative explicit parameter quantify stationary general distinguish notable distance slightly cavity vanish atom count section abc summary ks distance measurement generate synthetic plotted shape ks perform curve h right interest ground atom depend may constitute estimation mean regime compute trajectory formalism detect detector click follow click compute sequence consecutive successive plot atom strong production ground state state atom experimental generate clicks type obtain consecutive define concentrated peak point h line panel account click count atom length expression local equation atom initial lack detect find probability ground moment atom detect read theoretical figure posterior histogram ground energy balance consideration cavity procedure atom count trial experimental versus likelihood line atom asymptotic fisher per statistic count vanish limited broad distribution practically remark vary consecutive versus abc together figure broad real likelihood estimate obtain dramatically zero atom use informative poor considerably posterior likelihood likelihood abc may become useful inference implement markovian dynamic simulation produce atom tractable physical interesting project would dimensional acknowledge university perform fellowship ep identification time system estimation compare hand estimator account term method bayesian distribution summary different compare choose exhibit markov building atom value angle measurement fisher correlation time identical abc select lie overlap typical identification estimate parameter dynamical designing monitor output arise quantum channel hamiltonian quantum open dynamical approximation identification play quantum distinct weakly measurement output quantum formalism quantum output measurement carry statistical inference formalism compute maximum output average measurement asymptotically explicit expression quantum fisher cavity interact subsequently produce continuous count fisher attain extend firstly full atom record record quantum trajectory formalism classical comparison total around motivated investigation aim informative statistic demand mle mind part introduce analyse type base abc measurement trajectory parameter sufficiently histogram abc statistic method produce mle capture number appropriately choose statistic free abc valuable relatively easy formalism process fisher section contain discuss scenario fisher atom count measurement atom number total number abc separately jointly two atom pass interact cavity cf incoming arrival atom cavity contact low temperature assume atom cavity grain master evolution cavity describe atom cavity measurement record detection outcome atom record infer strength cavity measurement use record sufficiently cavity reach steady measurement certain investigate measurement computation measurement although may purpose consider think experiment measurement cavity scenario investigate similar fashion atom basis statistic assume time atom decay field arrival cavity dynamic govern master four jump describe jump detection atom due emission cavity master dynamic satisfy basis give investigation firstly certain exhibit number change reflect period period atom secondly similar hard statistically state exhibit interesting curve represent cavity number master dynamic describe evolution cavity conditional equation drive measurement process master recover quantum mainly tool investigate scenario measurement feedback interaction cavity atom cavity subsequently detect ground assume ideal atom detect von set measurement start full environment scenario besides also detect situation atom see whenever state cavity quantum current new similarly ground atom emission cavity evolve pure feature system dynamic solely reduce case cavity initially jump classical correspond birth cavity atom need accord time atom equal arrival finally arrival ground state collect sequence encode label k cavity decide whether atom vertical vertical line time jump generate green dot intensity atom method inference employ begin inference classical basic problem distribute error er rao cr satisfies fisher sample measure importance cr lies exist law normal certain regularity explain popularity normality ergodic chain discrete trajectory process stationarity associate maximum normal information per unit process statistical extend notion consist conditionally conditional give mle discuss general later estimate fisher atom detection atom key notion relevant reading refer quantum quantum number copy unknown performing identical space find optimisation measurement call quantum equal parameter rough subsequently procedure asymptotically quantum information multidimensional quantum quadratic extend concept normality way original estimation transform estimate concern one dimensional consist identically output quantum markov process carry system ergodic general present illustrate pass interval cavity unitary parameter extreme identical repeat estimate extreme parameter asymptotically statistical fisher explicit continuous investigate solely measurement cf gap likelihood free use small statistic successive detail frequentist require conditional place belief prior prior combine derive bayesian viewpoint principle interested deriving function explicitly constant denominator task traditionally severe obstacle decade tool density moment etc perform costly practically infeasible perform provide approximate originally application human wide biology finance name therein intuitively method simulate various produce
average signal region uncertain exclude dataset collect competition record subject patient use unbalanced randomly dataset collect early infection brain image patient infection normal preprocesse connectivity avg avg cf avg exp mod pr ratio mod exp mod pr exp exp rate method c mod pr mod ratio pr exp pr exp mod exp cf avg mod ratio mod pr exp pr exp compare use discrimination first find subgraph within exact approximated frequent discrimination compare version expect feature top compute value feature mod mode compare mod ratio test discrimination criterion exp mod upon g compare method uncertain binary link include discrimination extremely default criterion experiment uncertain remain performance performance power score classification subgraph show performance value stand evaluation worth hard accord competition win prediction chance assign hard rate performance error rate improvement prediction rate chance subgraph mining setting subgraph mining subgraph upon uncertain frequent uncertain graph moreover outperform thresholding convert uncertain certain subgraph uncertain linkage uncertain different dataset different dataset ratio advantage pr ratio additional pr value generally pr good value without prune subgraph cpu exp improve pruning trend run mod dynamic force search uncertain graph scale linearly dynamic programming enumeration even eventually optimize computational l subgraph briefly discuss mining graph year research certain aim subgraph extract subgraph depend whether class mining roughly frequent subgraph mining depth map code subgraph many subgraph discriminative find discriminative classification recently data especially frequent subgraph uncertain mining subgraph uncertain graph approximately uncertain author uncertain work graph graph object consider subgraph inspire discrimination feature instead uncertain reliable subgraph near neighbor subgraph mining analyze brain discriminative uncertain classification general discriminative subgraph graph probability compute grant grant r grant yu wang ann edu attention construct classifier etc presence node world linkage inherently uncertain therefore measurement unable capture paper study subgraph uncertain conventional subgraph mining score feature uncertain challenge selection discriminative subgraph uncertain upon include median discrimination subgraph dynamic branch discriminative subgraph extensive perform gain structural naturally chemical feature represent node edge graph attract recent index research mining focus object presence subgraph graph subgraph application inherent linkage uncertainty directly transform uncertain graph human brain figure brain connection functional connectivity imaging step temporal correlation signal connection functional disease affect researcher complex structure human brain stage aid diagnosis disease intervention application mine uncertain dataset discriminative uncertain primitive uncertain graph object despite value discriminative subgraph uncertain discriminative mining structure major challenge subgraph mining need estimate discrimination feature subgraph discriminative conventional mining discrimination relationship also uncertainty graph within feature long discrimination subgraph uncertain example uncertain uncertain label subgraph frequent uncertain graph subgraph subgraph ignore uncertainty uncertainty rarely uncertain graph accordingly subgraph additional consider uncertain uncertain exponentially graph efficiently discrimination score subgraph imply evaluate discrimination pair subgraph subgraph mining discriminative feature uncertain framework effectively subgraph structure upon efficient base programming branch propose discriminative subgraph pruning space study fmri disease demonstrate alternative paper mining discrimination algorithm score dynamic result conclude symbol ii imply certain graph imply subgraph graph subgraph n g ig kk formally uncertain discriminative subgraph mining uncertain I deterministic graph edge discrimination subgraph g score subgraph indicate concept uncertain probabilistic discriminative uncertain graph discrimination accordingly long deterministic probability discrimination value iff discrimination uncertain measure dataset subgraph discrimination score discrimination expectation random usually frequent pattern mining worth discrimination score g score dominate probability extremely subgraph order extreme feature discrimination score discrimination score among eq median relatively robust extreme expectation median also quantile statistic probability likely subgraph discrimination define mode discrimination subgraph within subgraph discriminative score subgraph discrimination discrimination great robust example subgraph score introduce measure function calculate name g negative graph support g discrimination score table frequency write n number different subgraph feature base definition bound probability dynamic calculate pair denote uncertain subgraph contain subgraph rgb rgb kk calculate graph contain value calculate substitute use figure detail recursive calculate measure highly application graph could negative rl eq ht uncertain subgraph class label
distinguish face affinity affinity centroid face face dataset indeed diverse representative face tumor formulate complete vertex guarantee prove vertex distance connect let character refer counterpart element weight vertex note weight bipartite unit integer program q mean vertex serve weight program hard thus replace configuration point recover dissimilarity attention proof ball euclidean center ball least draw symmetric support ball dissimilarity agree assign ball satisfy center ball theorem dimension preserve factor euclidean space center separate ball recovery aware theorem guarantee beyond literature three relate contain align relaxation nonconvex partition clique notable find recovery correlation correlation agreement disagreement cluster cluster paragraph probabilistic guarantee block plant generalization partition cluster cluster edge include draw build union subspace lie overlap hyperplane origin program pairwise distance close probabilistic specifie objective essentially use derive mixture model hard parameter point whose contribution admit many close parameter towards reduce separation distance center distance gaussians intend guarantee rather complementary insight space recover mention configuration euclidean lp relaxation dissimilarity distance optima realize large known location allow constraint triangle obtain metric subsequent unless bound approximation criterion bound li drawing result metric approximation available algorithms area research relate guarantee round condition respect triangle next duality optimal program programming probabilistic exact recovery integer focus separate ball demonstrate efficacy approach recover analytical review fourth final discuss one proof space second close point write necessary unique coincide tucker condition introduction prove euclidean space assumption draw ball recovery regime necessarily close particular correspond ball obeys uniform center assume sequel denote preliminary ball square distance dissimilarity statement min min statement min vector min zero mean despite integrate spherical bernstein min min inequality hold moreover statement contradiction exceed min min unit center ball isotropic satisfie q distance dissimilarity assign valid obtain three statement ball eq q sufficient maximum boundary narrow requirement eq rhs boundary impose rhs exceed previous paragraph follow cluster lp unique assign denote complementary separately inequality span sentence hoeffding hold cluster occur consider obey statement eq ball rhs q provide sufficient condition eq easily inequality satisfy perform obtain contain proof hoeffde recorded separation ball distributional ball configuration record recover place cluster recovery simulation use optimizer barrier implementation table remarkably fail probability recovery realize kkt condition assumption difficult prove ball result note plot measure ball conclude exception towards ball make draw draw ball increase outlier prevent ball recovery apart consider considerably theorem toward fix ball probability recovery fix ball thus room improve center increase suggest repetition prove globally regime two point guarantee fall short success lp distinguish ball distance relaxation recover solution extreme presence different number thus interest us choice dissimilarity example distance recovery guarantee guarantee cluster ball acknowledgement li suggestion grateful point especially
extract within local image common template template feature match portion non element feature map activation pass pooling produce pool aggregation q motivation pool less location within map take pool increasingly object function use benefit function suit mechanism involve negativity response introduce pooling ensure specific location strong location choice max element drawback convolutional pooling many combine effect activation element bad strong activation pool response suffer drawback make generalize example pooling helps pool form activation region precisely compute activation region location within pool activation q illustrate region back back propagation pooling capture activation filter input additional activation region pass network stochastic ensure maximal utilize introduce performance activation weight eqn element weighting denominator pool conventional sum pooling weight pooling since test possible architecture pooling pool large averaging occur confirm weight compare one pass activation lead benchmark mini batch network label cost parameter learn extremely efficient gpu library rapid development network dropout convolutional layer per epoch aside train pooling pooling region along pool additionally pool normalization layer pool output neighboring feature map typically help extremely allow neighboring finally fully produce model cifar view house cifar dataset compose example approach subtract pixel compute cifar image convolutional softmax linearly original decay setting find cross validation experiment architecture model respectively train stochastic pooling unlike pooling compare augmentation dropout pooling require cm mm error conv conv net layer dropout avg pool max dropout behavior pooling compare cifar train size possibly noisy digit handwritten test benchmark pre processing drop inferior approach mnist stochastic augmentation method elastic use type augmentation performance conv conv elastic pooling mm cifar another image test cifar example per convolutional network perform believe art pooling house dataset set test goal task center color world digit visible practical classify house google database subtract pixel mean image see leave variation color utilize normalization rgb process proceed relatively despite significant amount convolutional train epoch feature map prevent despite art dataset convolutional pool train error conv net stage conv net pooling avg pool max pooling pooling mm far illustrate ability pooling reduce cifar half full pooling pooling approach test cifar stochastically slightly expect max weight valid location throughout probability model table computation train max average pooling poorly incorporate element maximal scale produce pooling see weighting fit pooling benefit utilize probability pooling test error stochastic pooling stochastic pooling stochastic pooling stochastic stochastic pooling max pooling stochastic avg weight weight pooling weight avg max pooling max pooling pooling avg avg avg avg probability insight mechanism pooling gain network novel visualization network component convolutional map back operation stochastically location pass deconvolution transpose feed forward filter tie encoder decoder weight input reach produce max average produce reconstruction example reconstruction throughout reconstruction max small local cm un feed lose stochastic output feedforward versus contrast feedforward fig
ct dx sign negative lemma integral section especially point unit point proof follow variation neighbourhood replace respect I I note expectation main specialized base calculus variation estimator partition q x statement probability let x polytope lie whose disjoint hyperplane without generality correspond global minimum x ic ij ic j dd nd equivalent r dx hand negative part lie half vary lie maximal claim prove contradiction ij ic ic I I face therefore ic ic ip ip ic iv zero contradiction set meet measure zero contradict claim contain set lie consist consideration projection polytope practice code step interior overlap leave hand compose continuously lie formula condition second e unit general code fisher matrix evaluate relation q vanish unit sign positive hence rearrange prove exponential family normal variable equal jeffreys therefore except boundary truncate jeffreys describe euclidean centre lattice writing lemma correspond plane construct try expression theorem reverse e partition possible partition parametrization become find find problem solve probably dimensional update random replace side respectively repeat change result therefore also consider mean origin estimator estimator dimensional family statistic data calculus code estimator global second use convex prove estimator family calculate variation formulae denote dot give q denote double dot eq calculate derivative evaluate derivative put component together derivative vanish dx dx dx r dx substitute dx r r dx g g r dx dx jacobian rearrange cited lemma give appendix begin general boundary field lie derivative reasoning dt dt eq lemma alternative volume apply become volume volume normal field exclude j dx dx dx dx complete lemma except perhaps minus definition remark conjecture mml criterion kolmogorov mml strict mml algorithm calculate estimator apply take sense continuous calculate difficult estimator statistical sufficient notation part code change lemma defer requirement describe use estimator partly define notation family estimator parameter density pdf dot
stop hand envelope optimal determine summarize fully envelope distinguish ii latter model former design priori shorthand x slight abuse respective regression design forward start x v henceforth simulation state simulation require introduce extra correlation approximation restriction common approximate global regression x approximate projection onto amount parametric basis estimator poor noise overcome include use regression radial spline regression particularly effective henceforth rely partitioning build partition splitting cell dimension shape contain distribution ss tx ps grids obtain create american theoretical state require whether insensitive impact estimate dramatically different propagate lemma schwarz lemma difference estimate set indeed stop incur double error payoff record due propagation usual strategy control estimate error bound soon payoff bound achievable basic estimate use produce quality design square convenient theoretical term class consideration speed original generalization remain open directly return minimize control approximation approximation law stop region naturally entire crucially dependent grid proposal grid location wish loss design numerical allow require review concept speak control approximation zero contour place grid boundary unknown priori adaptively fix design step replace guide grid refer detail stop induce exist optimally new induce hence take f n n designing since focus true example tt pointwise loss average empirical permit control associate optimistic variable propagation sample implement evaluate integrated evaluate understanding impact regression fit intractable rely local thus find minimizer costly task extra replace sequential adapt particular furthermore localize design add induce dynamic tree refine fit design towards provide posterior improvement rule sequential thus herein illustrative ultimately efficient classical put option design explain small collected fit location improvement grid increasingly concentrate specialized design substantially differ figure illustrate feature approach focus identify turn permit refinement estimating reveal benchmark despite eight small entirely approximation quality ccc begin cf bottom panel histogram intermediate design full methodology reader substitute sample via unbiased simulation engine consider surface posterior surface depend rely discussion regression convenient posterior assume follow empirical collection averaging posterior dirac delta th method competitive setting include forest particle intuitively space towards place maximize contour versus reduce boundary visit basic construct heuristic guide design active learning ei alternative generally ei heuristic ei ei score merge identify contour reduce contour via location respectively simple posterior sign active criterion tend either overall account combine ei score latter place result exploration possibility sequential eq define dual preference close contour reduce numerical seem need ucb appear combine probabilistic guarantee consistency must density grow address concern design analogous measurement main modification contour finding seek maxima hyperplane solution exist boundary implement sequential propose dynamic offer sequential non regression conditionally via response fit linear model flexible representation suit easy updating grow refine tree multidimensional space hyper nearby fall else rule bayesian input recursively partition comprise dynamic tree specify evolve particularly stream infer version available increase grow precise analysis across step future require update fit add overhead budget present full suit intensive practical concern note rough gold main ease rather single generate estimate dt trick path serve rough analytically case thousand simultaneously loop replacement match normalize anneal analogue issue pick initial score accord design simulate trajectory update tn tn tt tt x object could classify complement mc option pricing challenge yx skew payoff usually opposite actual american majority exercise preferable total simulation path affect assume put exist backward completely reduce pure carlo iteration result magnitude introduce feed regression contour step nevertheless computationally dramatically henceforth physical discretized operate normal correspond black asset payoff form option correspond geometric put payoff classical put black log normal therefore pricing contract payoff reduce pricing analytic payoff admit multi dimensional american payoff lc lc discretization volatility payoff highlight impact keep rest size cf significant optimize budget backward time american american put find unique value similarly put subset ts show location dt every design start run loop point batch grid dt comparable range ht dt fit bottom respective grid group dt design dt implementation unconditional contour require precision fit shape contour observe dt severe true extreme right practically irrelevant path phenomenon contour dt implementation site towards much rely respective connect cell regression severe contour smooth function comparable design total simulation cc indicate level contour highlight contour show grid group fit dt constant angle dimensional dt implementation use I local ii provide rf dt rf implementation use sense make term percentage total tx adaptive design comparable design rf could stop average poorly leaf generate random forest rf dimensional run finally boundary continue compare gain depend put simulation effort achieve precision summary statistic product stop e plus nonlinearity make projection simple include variable commonly employ put save simulation effort run dt overhead ht price computationally simulate costly ii set pricing correlation volatility explicit pair consequently discretization scheme must simulate path euler pricing put specify daily year exercise assume favorable ever go adaptive benchmark spectrum realistic make stop boundary resemble put horizon boundary put I everywhere challenge simulation hence sde put sde option experiment ii iii dt parameter candidate exponent initial partition dt batch leaf sequential compare paradigm view never computationally regression apply contour framework machine grid theoretical device facilitate proof piece combine regression localize moreover grid refined design already successful meta design optimize grid efficiency fourth focus full posterior unknown instead empirical estimate tree possible purpose apply stop switch impulse control dp back particularly extension single replace sequence usually small switching require extra sequential implementation enhance begin loss permit usage termination design back mention may focus boundary search methodology base vector sequential design active distinguished collection dependent problem index idea model eps remark nsf new approach programming envelope methodology generation path examine adaptively stop boundary implement refinement illustrate variety magnitude benchmark price pricing trading option algorithmic trading solver high vanish thus turn envelope representation reduce expectation q process moderate come dynamic programming dp process envelope expect recursively via tf ss seminal envelope substitute sample innovation proposal quantification gain american pricing financial mathematic setting implement system great flexible poorly become concern complex pilot simulation become constraint variety
hold estimating symbol nan nan randomization randomize symbol positive step randomize symbol sequence sequence reject tail cumulative randomized sequence therefore propose involve sequential implementation randomization stop termination criterion may reject termination require constitute illustrate propose randomly length bias derive three realization tend drop bias discriminate constitute accurate significance randomize turn accurate randomized sequence realization confirm test large finding establish know order prominent one comparative aic use kullback lr aic order information markov estimation aic order though know bic perform aic size frequency occurrence length expression estimator simple study ps observation lr term kullback lr divergence second standard lr term kullback divergence ks comparative denote sf chain monte realizations sequence realization setup markov setup dna sequences sec confirm illustrative realization increase lie bias approximation htb number significance bias display grey online randomization significance display dash half indicate significance establish detect show realization reject significance realization reject rejection improve towards general test conservative often significance limit simulation determine dependence symbol contribute therefore significant bias close broad right account sequence less broad right nan percentage one refer information criterion ps sf estimate simulation mm r r r c aic ps sf sf high sf decrease estimate correct almost success rate order fail completely improve increase maintain fall dna four dna sequence symbolic analysis large sequence contain code code together non character use sequence setup two sequence symbol order hard show respective simulation sequence dna gene mm ps generally e realization good realization sf bic none well sf increase fall high sf dna sequence r criterion aic sf fall criteria ps highest rapidly success criterion correct latter indicate limit dna criterion p large well ps much region well discrimination structure code non coding whereas sequence latter mixture region therefore expect region region consist chain order criterion computation gene estimate htb criterion tend large order gene region estimate order increase order three order aic ps pattern region respectively sf close order level sf range gene large sequence maintain chain analogous use autoregressive significance limit autocorrelation mild analytic limit work accurate markov build scheme iteratively randomization significance find criterion bic compare monte simulation vary randomization test conservative find testing correct small criterion higher follow show nontrivial structure dna along dna comprise gene converge solely correlation gene code short range make computation dna sequence give confirm ability confirm simulation work certainly randomization irrespective lack dependence correct surrogate one adjust sequence test e generate chain order test randomize straightforward computational disadvantage issue dna accuracy currently develop apart intermediate analytic develop randomization randomize symbol symbol significance applied reject carlo bayesian criteria maximal ratio divergence turn order availability view conditional information randomization tp generate markov symbol length chain criterion consistency aic however bic sequence length criterion bic determination wide possible transition setting instead find ps simple criterion like bic global relative entropy use divergence aic relatively finally ratio measure powerful chi ratio dna chain conditional markov increase allow estimation analytic analytic criterion order dna paper significance randomization method chain compare markov chain well transition matrix dna limitation order conclude remark subsequently entropies shannon define discrete value occur shannon stationary read possible symbol mutual information mi
measure subscript refer quantity gmm parameter group transition subsampling observation cluster cluster last time asymptotic brevity set old previous variance asymptotic limit deterministic follow deterministic least k update algorithm cluster set algorithm batch track dynamically evolve primary temporal cluster sequence batch select applie create create update assign order dependence construct old important monotonically decrease step monotonically dynamic guarantee converge k monotonically decrease comprise component currently penalty away time specify concrete mean determine derive capability bayesian algorithm relationship mean dependent dirichlet process dp algorithm extension sequential vary require identifiable across determine across track cluster similarity evolutionary cluster cluster past clustering present theoretically automatic adaptive form old cluster sequential tracking adapt suffer drawback typical particle batch evolve derive variance order magnitude probabilistic hard provide examine couple convergence algorithm use critical planning system nsf award grant cm liu university mit nc base dirichlet model number evolve cluster low algorithm guarantee mean empirical synthetic cluster real ad trajectory demonstrate order magnitude probabilistic provide examine dataset powerful tool cluster despite inherent order influence label assumption modeling spatially evolve phenomenon meaningful evolve monitoring evolution construction build development mixture approximate algorithm generalize birth death powerful capability suffer sample variational current scale size method analytic cluster priori ideal context quickly reliably volume streaming system classical dataset advance make flexibility yet model paper discuss dynamic spatio derive gibbs gaussian mixture scalability ease implementation along computationally tractable particle inference well cluster characteristic test comparison applicability spatio united dirichlet process general dp respectively direct thorough process dependent dp evolve poisson process transition govern remove may move become
phase learner learner phase exploitation context arrival stochastic train explore make contextual bandit exploitation phase context explore nonempty reward time learner need since collect arm learner explore classification explore learner high sample I reward lt r wrong ensure sample upper context hypercube arm eq learner parameter optimize suboptimal due near arm variable bound constraint complete run great since select learner learner learner sum achieve sublinear independent sample distribute facilitate different bound process generate I sample artificial process run denote outcome exploit suboptimal bind expect number arm choose suboptimal exploitation collect suboptimal th arm obviously always true arm suboptimal three inequality imply suboptimal chernoff hoeffding order bind regret sublinear regret due eq denote lt apply markov inequality event classification exploitation phase learner sum learner exploit therefore similar lemma suggest learner probability suboptimal imply optimal learner near outcome time arm run eq event suboptimal classification inequality exploration phase learner choose near arm choose total arm appendix exploitation suboptimal expect bound suboptimal lemma regret due exponentially minimize regret near combine control eq high order regret come respect lemma minimize regret sum although require make trick phase begin sublinear convergence reward e imply network security rate high distribute go infinity mean increase mean I know classification adaptively choose example security day accuracy depend regret even bind case arrival assume train learner contextual function train first introduce slot interval decrease hand otherwise learn able stream error step addition run presence online notion context capture utilize treat sublinear compare memory requirement keep reward mean reward keep number requirement limit set reasonable size high store keep bandit require classification sublinear regret bandit sublinear respect good classifier learn classifier learn necessary contextual regret even arrival heterogeneous slightly know context arrival process homogeneous among arrival suboptimal time suboptimal classifier eq q belong hoeffding inside since side see rate higher large expect regret follow provide concave maximize incorrect bad boundary balance regret bad difference data slice slice almost boundary regret regret tight prove arrive arrival soon process arrival completion capture formulation q delay slot delay classification index delay feedback delay support regret chernoff hoeffding deviation accuracy accuracy add classification delay delay additive sublinear delay delay requirement context infeasible adaptively partition section network adaboost adaboost call slide window security time attack run simulation context previous context learner cost set train consecutive segment security test simulation classifier error type give make situation appear classification function train old inaccurate numerical show improve accuracy function test reveal delay classification naive bayes rbf perceptron bayes rbf network dimensional security find theorem assume exploration learner table percentage percentage spend exploration adaboost slide window adaboost consecutive train way length adaboost learner prediction learner access learner learner limited may communication cost well adaboost prediction exploit information percentage context attack utilize observation newly activate hypercube adaboost window difference explore train learner phase learner train prediction time window make inefficient observation context good percentage error context shape phase note previous spike set fact exploration phase c horizon grow error paper develop decentralized classification sublinear application combine ensemble computation learner ensemble theorem lemma edu application require amount dimensional produce multiple correctly event phenomena characteristic learner certain incoming stream unknown priori distribute datum source process heterogeneous learner run stream locally classify importantly learner incur beneficial obtain well exceed incoming unknown dynamically time heterogeneous learner contextual develop online sublinear error without compare distribute online characterize finally illustrate propose online network security compare state solution mine big security surveillance health monitoring etc sensor network etc datum dynamically evolve paper dimensional process decentralized heterogeneous learner equip sharing cost make centralized mining learner wireless surveillance location learner resolution speed event happen frequently therefore send event heterogeneous characteristic incoming datum classification learner accuracy learner reveal slot goal reward classification cost term cost communication cost etc similarly represent file several streaming page classification learner cost learner classify maximize correspond system jointly optimize distribute design reward classification characteristic scheme use sublinear average reward optimal reward low rate average context stream learner bound memory necessary classification exact distribution topic rapidly result security result security several network day ip send activity capture classification accuracy know security application base available note traffic network security application action another context information describe highlight section decentralize distribute computational algorithm respect system statistic contextual partitioning contextual arrival several extension network security application result conclude remark work two mining arm aim act aim arise online technique decentralize combine learn observe learner access feature come mining learn develop learner distribute improve costly stream characteristic stream distribute within illustrative depict change relevant result operate arrival decentralize learner correct belief system dynamic converge characterize incur specific choose produce classification send iii reveal perhaps learner access invoke classify know function cost know cost know invoke another incur without loss application one delay framework classification possible learner order label cause processing whenever stream send another incur learner imply learner effect learn learner learner help learn classify send maximize utility arm alternative comment binary adapt consider restrictive since ensemble may header dimensional comment security security feature tradeoff accuracy improve decrease increase depend input stream accuracy although cost general increase generic represent delay etc assume context formalize term indicate similar lipschitz constant example require learner cite require cause appear exploit context cost model input observe arm accuracy translate bandit formally benchmark perfect accuracy regret solution minus learner tradeoff capture weight know evaluate minus cost context context element hard scheme datum arrival
pairwise orthogonal long mc eq prove assumption sc proof rely result build cover satisfied inter matrix show individually simultaneously equal p u pm eq lemma pair conclude hold take bound subspace rh rhs every v proof consist outli misclassifie bind outlier violate misclassifie bound union union outlier establish detect outli union I hold outli one basic technical bounding detection detect outli outli violate upper get term treat separately assumption outli outlier bind outli scheme outlier reverse triangle thus j verify eq choosing end finally resolve imply imply convenience tail frequently lipschitz l r inequality international true green conjecture problem false connection problem dimensional point union number subspace assume cluster adjacency adjacency succeed intersect exhibit robustness noise reveal explicit affinity algorithm succeed even miss log ambient propose simple provably synthetic datum major information relevant structure high illumination condition motivated find datum lie union subspace hybrid follow face vary illumination image area computer specifically motion corrupt formalize point q want assignment access noisy pca numerous method excellent survey introduction find excellent efficient implementation heart typical distance subspace algorithm tractable restrictive notable ssc adjacency datum minimization ssc provably succeed elegant reveal succeed intersect intersect analytical result theoretically sound noise important address robust ssc replace ssc penalize least step succeed subspace ssc noiseless respectively pose computationally apply adjacency correlation construct neighbor measure algorithm observe measure noisy incomplete intermediate misclassified directly node correspond performance measure lrr ssc pursuit ssc omp strong come term near employ theorem apply compare ensure connection massive noise subspace low application cluster entry factor ambient analytical handwritten digits mnist cluster construct correlation albeit analytical subspace affinity fit clustering build liu adjacency minimization lrr succeed subspace sum subspace intersect minimization lrr complexity ssc find less demand ssc al substitute ssc pursuit omp ssc omp zhang recover multiple subspace pose tractable though consist noisy consider performance cluster subspace term along analysis remainder organize section contain performance case observe describe corresponding result analytical result ssc far cluster ssc isolated exposition proof matrix stand entry column f ij mx l shorthand write indicate distribution path connect either near case base provide outlier already outli point step restrictive prior point subspace estimation discuss every cardinality let entry step construct z decrease metric j j x property metric vector lie formalize nearest adjacency remainder connect component correspond oracle segmentation exactly adjacency may cope noiseless establish ensure zero intermediate albeit sensible also formalize property connection ensure correspond two cluster parameter take false connection analytical ensure connection large within performance guarantee specific connection property noiseless x k orthogonal intersect zero eigenvalue laplacian sensible number subspace laplacian corresponding belong subspace possibly small possibly heuristic quality estimate establish noiseless automatically substitute approximation matrix index substitute ls z ls refer formal ls entry x element x x j follow statement practical spirit ssc ssc ls term ssc omp representation omp respectively performance noiseless noiseless outlier choose specifically j jj ensure spread avoid degenerate situation lie direction skewed towards sensible one assign direction separate assign express notion namely relation affinity notion angle ks angle recursively maximization carry ks affinity ssc ready first main obtain uniform n correct l n states segmentation subspace intersect intersect point intuitively confirm show asymptotically differently grow yield graph show component establish necessary subgraph connect exist depend connect interval possible oppose ensure false error ensure cluster correctly finally note appear indicated function apply intersect find analogous choose x c segmentation numerical rhs albeit slowly exponential weak connection dependency vanish virtue choose false connection least n cn cluster corrupt obtain point j ng false unlike noiseless point unnormalize unlike theorem ensure connection clustering succeed contain sufficiently intuition distinct subspace reveal massive noise ambient behind rigorous favorable subspace pair inner product e I theorem e j j j cf obviously application cluster often impact observation subspace make subspace specifically take previous section exposition throughout dimension point subspace vector work ambient product miss suppose u n set false connection miss analogous report constant replace condition concern observe seem ssc ssc represent point succeed outli misclassifie outli succeed condition choose drive datum drive propose essentially comparison generalize lrr comparison main equivalent ssc draw carry use metric ce subspace estimate assignment appear ce specific cluster indices ce assignment cast maximal via employ discriminate instance instance however turn choice problem parameter get almost consistently subspace connect adjacency equal connection section estimate heuristic reproduce available unless subspace r uniformly set demonstrate predict intersect facilitate ssc set orthonormal identify ensure intersection least u ce averaging intersection ce ssc well style vertical sep edge bottom left font mark solid solid index ce solid intersection incomplete miss point pair ce averaging instance result indicate connection subspace base integer orthogonal ds statistical datum summarize font cm edge width cm height file ce file cluster error subspace horizontal font style horizontal sep vertical sep edge bottom edge height file file file file file vertical horizontal axis vary additive model theorem depict style sep vertical sep edge leave width font meta min max file ce file ce file ce file ce file ce file ce cluster horizontal noise massive numerically choose datum confirm horizontal edge height font meta max file file fig ce huge file fig l metric vertical horizontal facilitate ssc parameter generate accord misclassification misclassifie outlier misclassifie point ssc apply handwritten mnist pixel handwritten digits subspace handwritten lie validate singular digit sort plot show singular ylabel value solid blue blue mark none blue solid mark none blue datum table blue mark blue index fig none solid blue datum none blue table column give digit ssc ssc use ce computed choose choose image digit summarize l outperform ssc image xlabel point digit ylabel cluster e e e e e e e e e standard ce handwritten digits face illumination condition stems take illumination condition would contain person face pixel image illumination condition ssc affinity ce bad lrr ssc latter lrr ssc different preprocesse result preprocessing respectively perform lrr raw bad ssc case preprocesse demand preprocessing acknowledgment would like helpful discussion result lemma would thank helpful appendix previously u adjacency high mention previously normalize perfectly yield prove false subgraph end exploit j j result proceed rhs ingredient datum choose point x unobserved depend u c numerical specifically u denominator upper graph insight formalize appendix pseudo distance n connection accomplish z exposition reflect bind violate final schwarz eq contain respectively u invariant unit violate accord last every taking make large proof idea graph plane near graph choose sphere equal neighboring region near neighbor combine contiguous spherical metric define spherical spherical metric whenever mean
engineering economic bioinformatics partition segment curve program expensive piecewise assume uniform alternative standard markov hmm mean markov activity approach noisy function acceleration far extend hmm acceleration hide basic observation denote acceleration markov regression univariate assume regression hide application represent activity acceleration activity control one polynomial activity another tp variable represent additive initial conditionally model regression process simultaneously component model form multiple regime class matrix multiple parameterize sub estimation likelihood maximization em thank attractive limit property considerable acquire make sample maximized pz maximize em context step log denote calculation probability sequence backward procedure calculation rule markov one hmm update consists perform weighted polynomial matrix posterior matrix multivariate polynomial ik ik carry validate main idea throughout segmentation classification acceleration series conduct perform acceleration leave segmentation consider achieve task ground truth activity thank different activity ask activity label switch supervise performance fold cross validation matrix annotate subject criterion result acceleration activity conduct qualitatively assess automatic human activity basis raw acceleration nine regression performance parameter figure segment lie sequence figure evolution confusion average highlight propose automatic activity transition activity indeed static easier recognize dynamic activity c c recall efficiency datum sensor percentage classify decrease bad sensor classify standard unsupervised gmm hmm standard gmm gmm mean well longitudinal correct mlp nn forest observe nn forest mlp give naive low show class encouraging perform unsupervise main locate transition length much short confusion human perfectly transition truth activity furthermore train explicit temporal model datum treat multidimensional consider dependency notice may lead significant computation activity unsupervise relatively performance activity upon acquire body sensor monitoring context come statistical explain interpret activity example estimate dedicate consider activity learn particularly exploratory cluster amount activity classification approach show competitive way work integrate context prior interestingly useful activity activity application promise perspective unsupervise undesirable physical stroke supervise activity body generally require label consume collect unsupervised paper activity recognition raw acceleration measure segmentation multidimensional markov framework expectation maximization activity label need account appearance acceleration segmentation activity unsupervise activity recognition gain economic impact people european union pose service great improve quality life independence activity home prefer stay home adapt becoming service human health monitoring security range promise application security monitoring amount decade quantify activity base sensor environmental object sensor sensor gain number monitor medical satisfactory activity laboratory clinical free environment micro systems greatly thank considerable energy consumption early recognition recent study regard activity make supervise unsupervised infer classification enhance activity recognition spatio algorithms nn multi networks ann including perceptron mlp radial basis nevertheless collection sufficient label rich try estimate density technical well unsupervised activity organize platform model parameter acceleration supervise study activity place right etc activity ascent descent sensor sensor human body sensor consist measure acceleration range sensor activity exceed large hz activity help acceleration collect perform activity unit unit subject pc master unit transmission wireless perform paris est cr six subject office environment rich subject
min nmf part follow novel method formulate objective function optimize nonnegative th sample sample class label nmf aim regard basic vector represent combination regard seek problem regard eq within pair belong I pair set I coefficient discriminate presentation minimize minimize pair meanwhile distance class minimize maximized pair minimize combine nmf q couple difficult solve represent coefficient distance lagrange multipli constrain optimal solve substitute difficult alternate slack lagrange multiplier update fix remove irrelevant reduce column sum solve derivative kkt wise get rule division term fix variable regard respect kkt follow equation rule fix remove term problem lp label ability representation consider pair class label pair class pair distance min extreme maximum within try class distance differently max min pick distance within try decompose coefficient traditional label supervise ability use discriminate nmf representation hope class pair minimize pair maximize regard basic matrix slack iterative nonnegative min attract engineering nonnegative try decompose regard basic combination contamination basic nonnegative metric allow additive combination thus could nmf wang fisher encode impose lee nonnegative incorporate nmf liu nmf explore constrain pair improve discriminate
channel source problem formulate energy problem dnn spectrum estimate think dnn check signal lie correspond manifold represent dnn dnn source handle training testing dnn another paper factorization initialization paper organize section briefly nmf method source result find mixed source two fourier domain represent frequency linearity source angle ignore sum spectra need nonnegative nmf relate nmf nmf find nonnegative matrix nmf dictionary optimize minimize cost good measurement nmf matrix one operation element multiplication operation usually random number iteratively equation frame multiplication nonnegative nonnegative source observe calculate use decompose signal gain update equation source wise wise initial signal use separation nmf idea source source nonnegative separation nonnegative source separation model lie cone nonnegative variability appropriate us nonlinear nonlinear separation neural success superior signal dnn energy objective separation dnn classify frame illustrate figure source namely output hide sigmoid skip clutter notation h dnn architecture network unsupervised parameter compare use boltzmann rbm initialization backpropagation fine tune criterion least input derivative separation dnn score source frame spectrum separate spectra carry source elaborate separation algorithm audio mixed calculate normalize spectrum gain formulate unknown energy different source satisfy dnn third correspond spectra source energy function quantify estimate model dnn basically come vice versa following quantify energy cause square mix negative solve energy energy choose experimentally solution energy instead parameter optimization rarely happen solve gradient dnn respect able solve gradient contain word row setup illustration dnn fitness energy measure dnn find energy setup dnn non initialize mixed signal initialize similar result use spectral estimate reconstruct estimate source final eq music signal simulate speech speech speech web site piece minute duration one piece calculate window length use first remain involve test file speech database speech music ratio audio level speech software initialization dictionary source nmf dictionary music minute dnn reason dnn nod hide nonlinearity node dnn rbm backpropagation epoch output schmidt matlab solver measurement interference sir sir distortion energy error reconstruct define original interference sir interference error due music signal sir dnn dnn take neighbor frame single energy music value db dnn snr usually around db sir high music reconstructed perform neighbor dnn dnn well frames db sir sir snr sir snr c db sir sir snr sir separation deep dnn dnn dnn framework improve dnn autoencoder neural believe near source separation dnn unlike study dnn classify frequency
model vector factor factor diag extension additional restriction well mixture appear year build common diagonal compare thereby far detailed method place matrix preferable application member upon component g large situation subsequently analogue refer common develop mixture skewness dimensional presence skew generalize squared mahalanobis distance limit skew arise value freedom yy gaussian feature k modify third develop skew analogue herein develop assume component set skewness degree place asymmetric discriminant analysis lin lin truly within wide mixture skew factor comprise eight constraint member respective level parsimonious elliptical approach discriminant careful consideration difference member form however necessarily bring three family parsimonious ccc parsimonious latter rarely impose scale load parsimonious flexible fix similar number grow plot figure parsimonious extent difference grow free grow especially compare feature analogous fashion step require computation value membership component follow conditional expectation framework membership update solve value I ig nz gb consist estimate loading factor diagonal update eq algorithm acceleration convergence acceleration log schwarz latent maximized maximum selection support factor analysis class therefore rand class group membership ari perfect agreement negative chance analysis herein apply agglomerative hierarchical start skewness initialization analogous also fit package reason illustrate model skewness beyond datum simulate latent standard set skewness observation comparison classification h colour shape true parameter fit arise skew start term bic essentially classification give ari value match breast match reduce component ari value classification breast herein analyze component start parameter fashion ari perform flexibility provide three expression class subsequently prior log scale eliminate less gene filtering carry latent factor ari estimate membership factor give factor membership membership skewness factor e freedom low may prefer impose moment exist practice confirm identical achieve model arise generalized representation elegant mathematically arise attractive feature skew elegant work focus skew mixture skew mixture account skewness
order crp without fix although seem complexity actual direct calculation bit bit look bit th th th take correspond use indicator kronecker kronecker lb kl customer share conditional xx satisfy condition customer share side state hand set assignment side show assignment customer customer vector diag diag diagonal basic expand system make hamiltonian non diagonal diagonal later quantum scheme add quantum physics work state problem solve decomposition intractable problem search draw eq indicate bit exclude th column summation summation quantum diagonal need another define eq mean zero formulation tractable expansion I expansion rewrite derive approximate p look state crp quantum part ms correspond term aim derive evaluate crp researcher observe specify network network mostly member communication member generally group outside connection outside candidate illustrate support kind citation network use use vertex citation dataset construct vote history wikipedia direct vertex vertex correspond vertex crp map solution combination schedule sa iteration schedule note see slowly schedule decrease know gradually increase decrease sa log sa width solid l sa outperform whenever higher dotted line horizontal axis crp run want compare search search width variational deal dirichlet hard function run crp core one random initialization try seed generate run outperform line find multiple effective example quantum depend table customer node need sa order parallel environment scalability second second sa core process customer almost sa therefore fast achieving induce optimum interaction approach sa value experimental provide I handle mixture ii heuristic easy implement crp apply relational promising technique rapidly regularize analyze schedule enable acknowledgement partially support foundation also program aid partly institute solid physics university usage support aid product finite general use product rewrite e particularly definition kk indicator e q show eq easy indicate customer table th therefore derive explain vertex class vertex particular connect link vertex indicate vertex accordance vertex accordance generation distribute accordance dp generation link represent aa iv iv z calculate assign sampler develop new anneal chinese crp extension anneal sa apply crp formulate fix mixture apply crp partition run sa chain monte maximum posteriori crp anneal process posteriori cluster topic fundamental difference learn probabilistic dirichlet enable decide chinese restaurant process restaurant probabilistic map search approximate markov carlo crp map use map extract assignment posterior distribution converge attractive mcmc crp anneal sa parameter control search sa schedule schedule slow practical sa affect novel stochastic search quantum anneal alternative science experimentally fast ise control explain mathematically framework quantum induce multiple interaction crp crp I model explain relationship denote formulation approach similarity formulate cluster mathematically derive crp whereas exist sampler derive introduce customer customer table customer th st th crp interaction tend customer compose element customer restaurant denote assignment crp assign customer customer
norm signal throughout appropriate recover exhibit complexity trace relatively low block sparse partition block th square tangent cone study optimization highlight quantity tt subdifferential penalize result various induce gaussian bind gaussian complexity sparse p ss p norm setting structure dim p k theoretical structure corrupted measurement convex programming theorem prove tangent structure tangent convex natural show approximately suffice general recovery either exceed noiseless entail conclusion closely recent penalty next sufficient problem success entail long penalize recovery corruption multiply nearly threshold signal model make term subdifferential upper tangent complexity fact cone norm practice find recover recovery specialize corruption practical demonstrate recovery recovery x v low bound nonzero perturbation either requirement either penalize recovery apply x norm section corruption exist analyze communication robust channel propose aim signal communication channel protocol message short corruption message increase corruption tolerance past recover corrupt densely message constrain advantage theorem reconstruct corruption entry cone sparse vector constrain recovery addition admit sharp compare binary recovery corruption prove become indeed noiseless unlike provide recovery set corruption signal exhibit analyze corruption recovery compress sense structured corruption corrupt identity block consider dense corruption regime proportion may need suffice signal recovery corruption exhibit fraction nonzero block entail important suffice exactly respectively probability dense corruption sparse corruption establish corruption admit explicit level corollary stable corruption parameter signal allow specify bind corruption level address adversarial matrix adapt ball tangent save extension corollary suited moderate frequent setting corruption reason distinction advantageous give block sparse corruption general could attain corruption suffice ensure corruption entry support solve q noise level recover due recovery corrupt provide recovery general corruption block sparse corruption vector block eq suffice detail recover corruption stable exact recovery sample orthonormal uniformly noise support incoherence analyze corruption vector gaussian derive constant generalize arbitrary structured structured corruption analyze nonconvex signal lie incoherent manifold isometry convex procedure minima polynomial give result structure noiseless either recovery discuss presence rather constrain corrupted presence absence goal parameter penalize recovery matlab specify program signal exceed new conservative theoretical recovery guarantee synthetic threshold align phase transition recovery corruption recovery wise corruption corruption communication protocol discuss recover binary noiseless corrupted fix message vary follow corruption normal solve success setting theory corruption four setting penalty depend corruption bind highly minimize expect square sufficiently sparse corruption precisely display sparsity vary reference theoretical remarkably nearly good set penalize offer nearly constrain program moreover sparsity probability provide neither corruption ps n recommend empirical achieve recover perturb dense pair normal entry entry corruption entry entrie record rescale recovery rise constant rescaled complexity rescale common value corrupt penalize case recovery stable recovery corruption add noise geometric gaussian distance develop interpretable theoretical sharp phase several setting perform side closely match penalize fully bind stable recovery presence bound sub question extent corrupt sense gaussian either incoherence section structure cone table refer vector establish upper bound gaussian square subdifferential hand specific subdifferential penalize relation provide subdifferential result bind structured norm admit lead distance via subdifferential tangent cone give square bind treatment index partition disjoint block natural encourage complexity set block estimate hold evident compare former approach size grow suffice subdifferential establish subdifferential distribute subdifferential result distance variable inequality chi since whenever binary norm choose q implies well typically establish subdifferential matrix tangent cone decomposition subdifferential value begin bound width norm orthonormal trace standard entry q finally proposition definition bind prop fix unit sphere step prop cone uniquely suppose last clearly set lipschitz function make lipschitz function therefore constrain recovery binary corollary cone apply binary recover prove penalize sufficient final form penalize extremely choose q since analogous gaussian apply fix corruption let eq distance suffice achieve hence check section tangent refer corruption tangent joint tangent version constrain optimization therefore rescale give obtain reasoning begin relate via lemma pn ng ph nn entry therefore proof next low expectation n penalize obtain low shorthand complexity combine let expectation normal use appendix either first definition square combine sign sign n bt take expectation prove lem establish integer general might fact q slack handle slight discrepancy omit lem prove leave great inequality w prove lem suppose lipschitz q cone convex eq lem vs contradiction since lem pn ng ph equality therefore scalar side integrate abc identity rescale whenever true q lem hold lem suggestion presentation support nsf grant dms assumption title title study corrupt sense corrupt recovered face recovery corruption quantify tangent gaussian subdifferential take signal penalize program constrain recovery signal recovery theoretical sharp phase addition sparse recovery corrupted sense sparsity block atomic minimization corrupt sense potentially corrupted see problem sparse corrupted corruption former face sensor network modeling broadly deconvolution arbitrary signal corruption ill pose one hope corruption structure corrupted aim recover measurement modern far ambient underlie face penalization rank trace penalization framework compress encode geometric property signal specifically vector recover noise level noiseless suffice cone proportional setting need recover arbitrarily exact soon eq close corruption wide corruption appeal geometric treatment vector sense comprise primary ability common vector exhibit complexity precise secondary target ability make vector literature give deconvolution bind corruption penalty make knowledge vector constrain corruption noiseless advance spherical geometry recovery program q treat recovery presence noise overcomplete notion form sketch problem literature simulate transition recovery observe conclude discussion direction
tuning prior exponential description covariate intercept specify correlation flat sigma ig ig report acceptance rate acceptance burn sample start sigma sample illustrate summarize accordingly thin report sample sample pass random surface plot create plot package surface match closely spatial random key objective previous metropolis algorithm triangular solver efficient precede section comparison minute scenario choose define call replace produce cast take rw iw rw generic let z low unstable upon long knot knot desired obtain improvement adjust difference full rank random effect w I denote specification remain process predictive process c key choice knot could fix aim criterion evenly across small knot subsequent therefore sensitivity inference different intensity range move counterpart simple modify logical construct generate parameter grid modify modify process grid call process process specifie place vector extent grid extend location knot r tuning tune modify sample sample n model covariate intercept knot mcmc flat sigma ig ig sampling report acceptance sample rate iii iv time candidate model remove predictive model run time overhead full minute also estimate comparable attractive range spatial knot covariance suffer obviously spatial array knot observation see compare surface surface translate parameter compare random location knot versus ci interval location provide interface predictor matrix purely temporal space spatio spatial predictor addition variability specification lead hierarchical hyper specification note inferential space matrix row accommodate common monitoring environmental full offer modify predictive process achieve process expectation variability full monitoring require rank representation via comprise environmental monitoring record outcome hour average wind knot illustrative predictor spatially residual miss gain monitoring location identify ht list symbolic represent time easily tune exploratory use helpful defining start include mat ern exponential specify describe precede accept value sampler argument posterior location n formula max list n p sigma sigma priors list diag sigma ig ig sigma diag starting get fit general miss observation intercept exponential spatial model beta sigma sigma ig shape sigma ig shape ig sigma ig shape ig sigma ig shape ig sampling sample mean acceptance acceptance plot often useful explore plot strongly maximum much small symbol circle predict median outside ci coverage last several assess comparison define version version offer function mcmc efficiency specification compare careful formulation focus avoid core accommodate encounter currently develop efficient accommodate spatially add covariance among outcome within addition hope multivariate predictive version develop ultimately specification spatio dynamic allow accommodate acknowledgment national science grant dms ef ef ef monitor grant univariate spatio spatio model point effort focus improve computational efficiency attention compute development sampler rate reduce decrease implement computational represent term beyond improvement model implement class spatio setting view scientific move access environment complexity broad collect monitoring resource management advance spatially storage system source diverse monitor locate sensor across scientific researcher challenge coupling system inference support economic environmental public implication correctly inferential uncertainty framework capable account various multiple source serve development book variety literature spatial association capture effectively dependencie advantageous source uncertainty use computational advance regard carlo method spatial exception widely apply point class conditionally autoregressive car become popular implement mcmc model suit sampler draw distribution fully specify popularity automate software offer interface perform identify direct conditional bayesian project automate expensive computation become large less gibbs paradigm convergence multivariate spatial dataset spatially multivariate involve start relatively package via comprehensive help point convergence spatial handling analyze spatio view convenient identify package package list task package bayesian term model non univariate spatial bayesian development hierarchical bayesian spatial fit substantial model matrix decomposition cubic spatial infeasible version little attention address challenge consume fit comprise substantial rewrite improve subsequent sampler decrease computation scalability implement represent spatial new spatio setting highlight outline package bayesian outcome gaussian version hierarchical possibly regressor family index set complete proper distribution bayesian inference proportional detail behind bayesian direct computation invert development avoid redundant numerical subsequent describe cholesky dense multiplication employ metropolis fast integrate construct update usually diagonal adopt walk multivariate normal transform entire l log dominate achieve analogue similar u evaluate number cubic strategy cholesky triangular feasible algebra substantial see employ multiplication avoid multiplication multiplication close solve triangular system however become address posterior convergence store b b b standard km map spatial effect regressor x b b numerically mat identity devise numerically w normal cholesky numerical prohibitive thousand recommend spatial model load cholesky positive definite cubic must execute mcmc example comprise iteration second cpu marginalization fewer require inferential cpu minute spatial demand specialized strategy specify model model model predictive integrate q sampler draw b b section involve utilize x eq parameter random walk metropolis w q respectively gibb converge posterior posterior sample achieve closely description available replace predictor construct ensure definite covariance proceed sampling predictive posterior computation involve retain update u k v dominate u avoid avoid redundant updating require cholesky factorization take say desire predictive posterior predictive drawing low precede section function leverage algebra library matrix implement sampler table correspond equation previously dense due careful formulation description routine cholesky routine solve equation b routine matrix equation triangular multiplication operation processor core intel kernel library exception intel library dramatically reduce sampler illustrative conduct mkl intel processor matrix near linear sampler use also package chain result easier symbolic
branch decision root leaf reach determine class sum leaf figure decision testing testing letter main decision bad testing cost approximation respect worst know admit logarithmic result hold general instance much achieve bad converse happen ask accurate medical depict cost equivalently consume identification consequence test bad minimization improve object cost construction decision consist input small recursively construct path look reveal simpler present avoid intuitive redundant step provide art name long describe excellent survey cost minimize testing cost minimization cost study cost approximation prove complexity worst investigate covering belong different know extensively investigate minimization bad admit uniform testing employ rely previous strategy identification al cost test uniform testing stochastic boolean assignment give evaluation common value provide term definition want decision minimum cost fit recent boolean threshold formula clause obtain monotone reduce read formula evaluation et consider also bad decision minimum possible bad context use involve small object let subset observation follow measure progress express object already perform concept object let object constitute pair formulae denote belong figure initially identify apply path decision tree object agree test object class must coincide object class unknown outcome special denote tie break arbitrarily set context object object object pair keep say either keep separate say cover concept define separation sequence fix set separate separation e cover cost cover bad instance branch associate us test want perform claim ct ts leaf path follow choose ct prove second decision achieve possible I e property minimum submodular non easy non test cover map object cover integer decrease modular adapt adapt greedy spend test spend spend spend bt fa k summarize theorem return algorithm fr fr e fr e concatenation total fr employ approximate submodular attain logarithmic approximation basis recursion tree root associate clearly expect testing line third fourth find return cover ht leaf return tree separates spend spend u maximize make else child u spend spend ct kt u spent spend ct kt spend c fourth line responsible fig call block loop construct part fig line induce set contain cover recursively contain subset build root child fourth loop construct fig selecting cover responsible building cover shall third algorithm block right allow corollary coverage recursive call subtree low right test obtained select loop loop execution next pair call instance decision instance prove expect expect testing first recursive way build decision algebraic theorem induction inequality hold argue pair call true pair cover bound bad bad inequality every instance pair cost bad simultaneously approximation minimization testing cost feature correspondence binary string test correspond string th moreover unitary instance solve tree minimum expect cost addition bad vice versa testing cost situation present strategy ps select number cover cover construct construction find sequence pair satisfy contradiction suppose total cover run greedy least pair however must hold ki proof follow defer appendix test bi budget decompose repeat loop fig instance cover pair cover concatenation greedy execute object cover submodular ps p repeat respectively set object ct bi lemma lemma complete logarithmic achievable standard unless expect expect unless reduction bad version cover class proceed value later purpose distinguish assignment equal argue tree put child two child leaf child leaf child leave expect upper hand decision tree let root easy path test problem provide algorithm transform decision tree solution solution analyze bad testing notice bad theorem admit new learning also name class determination build tree possible achievable close left minimum among show done test broader label accord task power perform shall leave end leave strategy
couple may fail identify connect tends identify create cluster contain couple mse lasso edge surprising almost perfectly accurate estimation lasso covariance number cluster choice repeat except diagonal c consider figure figure outperform violate increasingly violate graphical lasso improve figure reasonable htp block element zero block element equal set interpret gene true feature simplicity stock price yahoo finance available huge daily price stock index stock consistently period stock stock day element stock mean stock know tuning tuning parameter obtain edge present colored htp easily graphical stock red conditionally stock approximately price university base datum include computer analysis student student construct whose entropy standardize zero tuning perform graphical choose edge ease colored word computer email connect office phone mail within include school music lasso large fail phrase within gene sample contain pathway pathway correspond pathway correspond encode locate know pathway operate independently interaction expect pathway connect gene gene set estimate perform graphical choose grey node represent pathway pathway identify several interesting pathway mostly addition gene pathway connect suggest pathway addition edge among agreement graphical lasso identify connect lasso base graphical lasso improve graphical covariance lasso contain huge impose huge lasso tend connected suffer hierarchical lead consistent suggest identification detailed context leave equation indicate interpret problem coefficient could explore investigation graphical cluster correlation estimation leave investigation future connection provide university liu sharing use cut dendrogram height matrix element additional triangle ij ij follows proceed proof graphical lasso recall correct edge must correct set yield connect imply nk gaussian graphical maximize log introduce surprising connection hierarchical lasso step perform likelihood maximize determine estimate linkage linkage certain setting linkage cluster variable selection consistency demonstrate lasso simulation university graphical use social interaction correspond edge pair indicate give variable edge conditionally nod compactly complex distribution rest graphical observation covariance inverse pair th involve however dimensional invertible even zero fully connect information overcome maximize author diagonal serve equal undirected adjacency indicator equal theorem separate identify matrix graphical connected similarity element individual merge especially clearly suboptimal connection linkage therefore graphical single linkage cluster subset cutoff parameter lasso detection lead estimate also propose choose problem cluster component graph connection linkage cluster lasso modification involve discovery connected consistency procedure application lasso standardize let denote jj j jj hierarchical cut dendrogram denote connect establish lasso solution similarity refer dendrogram concept eq cut dendrogram far connect apply tune htp dendrogram cut dendrogram motivated identify undesirable alternative clustering let perform lasso denote graphical lasso result estimate matrix subset true lasso case cluster cut dendrogram height graphical lasso fold mention early perform graph one connect component effectively therefore graphical typically extremely operation often inspection problem estimate procedure amount penalize impose suggest edge denote percentile freedom empirical suggest fundamentally proposal guarantee break distinct component study sparse estimate unknown provide recovery well implication procedure block similarity establishe connect block select diagonal bound cut dendrogram consistent one consistently perform base obtain parameter parameter correct connect combine lasso selection consistency start introduce state need specifie undirected th connect kf j union diagonal element connect degree hessian form kronecker abuse submatrix ab k q diagonal element main lemma far f op eq clusters state improve indicate graphical within graphical improve rate choice tuning require conditional independence fully connect graph zero partial correlation note unique pair constant determinant precision improvement theorem suffice remark surprising sample suggest edge result empirical support finding simulation
exist though variable since might change frequently frequent computationally complexity nevertheless scale dictionary gb face recognition baseline coordinate fista accelerate homotopy method convergence speed predict four pursuit omp accelerate hard subspace sp orthogonal replacement include active record pg conduct cpu cores os conduct windows os fair c run run write matlab eight fista active set signal active fista speedup c set sparse record time c c active fista speedup speedup unless apply solution early stop th length sp need specify know ground keep compressive type nonzero sample produce additive recovery performance adopt recover recover comparison record recover report value w metric solution decode speedup fast conclusion fig converge fast sparse fista decrease slowly fista subproblem attain fista table explain significant speedup dictionary table small fista master expensive take need demonstrate htp htp htp conduct sensitivity record average however improvement active significantly decode non need converge technique default parameter average fig outperform baseline sparse decoding successfully include atom many non severe master optimize signal lastly recovery method conclusion htp htp experiment trial avoid z x rate feasible rip condition unfortunately scale lot value sparse omp bad greedy use however still bad mp design ultimately efficiency much scalability several baseline dictionary sr face form big first fista relate fista rmse respectively accord follow show rmse rmse sp signal fista coincide evident efficient sp second need second particular clear fista rmse fista solution reason fista secondly dictionary exchange cache memory master problem atom exchange memory cache memory much scalability sp exclude fista comparison sp average show sp compare synthetic compressive mode omp generate generate produce noise second decode signal well second second speedup second calculate signal second second negligible solve adopt besides well experimental image constrain htp c extend database ar database normalize pixel result image individual pixel image takes spend negligible htp database ar l l c c ar randomly person set remain testing accuracy table conduct winner indicate comparable well performance cause unstable pseudo ill l spend art need test world application contrary complete time fast lastly comparable l question recognition pixel experiment prediction accuracy table become ill l achieve condition c htp l spee sr signal comprehensive demonstrate recovery million second comparable mode fast would thank anonymous suggestion greatly research research future fellowship ft research grant de pursuit recovery sr present improve address sr sr signal consideration batch mode speed recovery many comprehensive numerical demonstrate superior fast batch compressive development sensing gain recently community vision mining machine sr seek recover r np complete researcher relaxation lasso last decade least lar gradient fast method proximal homotopy reader reference therein comprehensive sr problem simultaneously computation sr carry sr recognition compressive sr achieve test image lie subspace image dictionary denote face image core sr recognition sparse representation directly computational projection rate affect require face sr argue denote decomposition pseudo solution product compute develop compressive sense acquisition compressive signal allow captured recover original might expensive task image video sense real large sr compressive aim good training increasingly area learn represent lead large sr core pursuit computational issue dictionary first continue bottleneck summarize subspace exploratory pursuit take second signal million atom method present address large batch apply face task well database experiment namely master master might accordingly lead contrast take stop affect value present progress outer u loop pg obtain line pg accord choose atom large objective iteration e loop improvement bad relatively support address include atom atom exploratory matching efficiency adopt warm master give calculate choose stop sort return atom convenience hereafter atom achieve convergence sparse subspace search atom selection pruning atom keep atom monotonically lastly search sp properly select choose stop ground noise
panel improvement overlap confirm predict linearly tr complexity see regularization constant lie clearly linearly lr choice depend agree series series marker lie especially regularization predict correctly scale case perform approach mse mse success tr lr complexity tucker might think however rank tensor simultaneously convex connect broader rigorously decomposition minimization analyze variable q let optimal decomposition k present completely orthogonal unfolding whereas lie bound k closely follow tucker rank truth relate sum minimization triangular line second fact h old third line combine fourth schwarz inequality divide side inequality mode tail q union first rank note happen conversely kb k k k allow full rank k infinitely decomposition tensor decomposition tensor wide literature mathematically empirically theoretically true know small duality norm identifiability confirm theoretical predict behaviour mean error component channel ensemble face call multi alternate orthogonal application statistical open challenging tucker mode mode hyper tensor decomposition along matrix rank norm nuclear leave specify induce rank notice approach perform poorly give tensor specific see panel latent preferable tensor full dash candidate mark vertical mark dash note statistical latent minimum rank explain latent approach suffer less issue identifiability approach tensor mixture identifiable two namely latent norm generalize plain group overlap lasso predict weight variation tensor compare favorable situation well case empirical noisy preferable complementary mainly focus completion basic norm section present establish identifiability scaling conclude property need way tensor dot two tensor mode mode unfold obtain along vector mode unfold say tucker mode unfold regularizer group sense repeatedly relate mode norm derive norm key step involve norm assume constant minimization appendix square difference theorem triangular schwarz f bound decomposition rank latent minimal choosing singleton zero previous subsection relatively element variance addition constant minimization n depend constant present appendix choice latent truth practice explain figure use follow compare inequality depend tucker whereas grow tucker interestingly know mode rank ordinary q locally decomposition decomposition identifiable theorem partly explain difficulty assumption decomposition identifiable confirm theoretically experiment low generate tensor various tucker rank randomly core tensor normal
coefficient pearson finite scalar xy correlation relationship correlation equal variable consequently pearson association variable draw pearson well eq respective assess linear nonlinear two mainly mutual concept well extend also mutual calculation uncertainty turn mutual collect draw decompose overlap decompose coordinate rectangular th rectangular fall bin th function estimate fall interval function mutual take eq naive grid e integer mutual association also converge drawback note pearson coefficient exist formula maximization positive integer denote vector define know independent nonnegative modulus otherwise show coefficient coefficient concept arbitrary despite remarkably define sample distance observe power pearson summary general powerful pearson correlation measure carry largely galaxy dark matter evolution survey square region south wide camera european portion result galaxy star hundred galaxy acquisition spectral star galaxy well determination magnitude deep calibration list position magnitude object estimate error version website apply many include galaxy g evolution study star formation galaxy variable contain table list definition ex ex ex band central pa angle mc mc peak dl mc ex ex ex f filter filter run run f f filter run bf filter run rf rf list include galaxy complete galaxy incomplete omit study consequence galaxy exclude galaxy consist galaxy bb type open circle red triangle sign circle blue ex galaxy al base type template fig ex sa type type galaxy range galaxy color scheme range galaxy template elliptical galaxy four galaxy extend analyze illustrate galaxy galaxy ex type ex variable calculate galaxy pearson statistic package three measure effectiveness identify outlier provide shape examine association database digital figure four plot variable correlation relationship give distance coefficient suggest strong frame graph frame galaxy g galaxy effect pearson vs coefficient frame see pattern relationship concentrated graph influence shape shaped relationship pearson pattern display pearson vs galaxy range display graph shape pattern concentrated figure four list display galaxy pearson three range pearson vs correlation type contrast show shape pearson correlation especially sensitive sample easy consistency galaxy type measure shape pattern galaxy range shape confirm strong distance effective outlier great detail c coefficient dl potential outli bottom shown associate dl mc pt mc two dl associated mc detect underlie shaped figure high compressed phenomenon numerous variety satisfy method reduce multidimensional shape plot kernel space pearson shape pearson distance correlation define compress remarkable discovery special case coefficient represent shape investigation explain cluster great plot potential outlier subsequently identify correlation check variable coefficient pair association illustrate method select run obviously confirm association high distance coefficient apparent insufficient justify application measure association pearson relationship pearson coefficient close inspection panel reveal diagram vs exhibit curvature accordance universe middle frame seem vary horizontal phenomenon literature statistical standard applicable assume fourth pearson panel panel hence association position minor axis galaxy weakly minor major negligible coefficient distance database issue formula regardless n variable variable formula distance distance calculation consuming distance coefficient nonparametric underlie derive comprehensive description distance n population remain gaussian underlie trivial matter apply large case may shape figure represent rather shape implement inside database manner recommend energy package introduction statistical call numerical need discovery mechanism outlier remain point correlation compare analyze association equally application correlation galaxy pair distance relationship pearson relationship pearson regardless distance shape range influence galaxy hand display shape regardless size correlation distance identify outlier far examine confirm association high correlation pair weakly relationship pearson correlation superior variable database advantage ability detect association pearson cluster readily use outlier illustrate broad applicability database thank manuscript science grant edu st mx department edu university pa edu universit I cluster deep south thousand galaxy formation evolution detect
plug give minor part assess importance estimator depth analysis classical usually rely likelihood neighbourhood applicable normalizing whenever contain terminology regular noticed stability application effective inferential eq l empirical form find ratio test statistic chi limit two encounter monitor inferential state smooth md jacobian hypothesis value treat attain regular likelihood answer follow population theorem cover cover approximate test limit local satisfying assumption denote function distance degenerate test sure limit definite regard partition algebraic expression information kronecker jacobian without full md central chi freedom usage test distributional local test situation kk alternative thus significance level reject current local alternative calculation demonstrate adopt setting suppose level tool formulate help section motivated pool believe efficient inference individual sample strong improve already efficient classical quantile estimator support conjecture adopt satisfy yet population demonstrate loss regard composite specify rd rd population ease nan asymptotic power know sensible asymptotic fix sample hence carry assess power chi standard central chi stochastically dominate correspond non parameter powerful vice versa implement comparison adopt composite give limit information pooling power x r f n distribution matrix r r concern significance power approximately statistic nan model correctly local simulation set present size representative include r comprehensive r sample nan sample shape distribution approximation rate level gamma find chi permutation examine precision chi square n kk limit clear well chi square detect nominal way analysis statistic matrix nonparametric utilize datum partial likelihood limit distribution power scenario seven curve reduce sum comparable power inferior population much nominal test generate gamma pareto distribution parameter respectively pareto simulate setting ii rejection note gamma justify ratio partial contrast pareto family shape requirement test multiple flexible include misspecification high utilize nevertheless examine effect misspecification misspecification put call unknown nevertheless test hypothesis calculate ii notice test detect distributional particular approach much ht violate normal family population answer example conclusion simulation ask include repetition population hypothesis ii comparison seem powerful identify distribution comparison explore conduct sample generate extra population particularly helpful accurately estimate scenario along depict b scenario hypothesis power six gamma appendix clearly match intuition evident test ht additional specific basis expand expand issue consider simulation compare four distribution simulation table appendix deviation x nan follow case x x dimension power decrease function agree dimensional rate good issue balance ht forest product assess engineering goal note effective modulus unit product model interest change show seem choose include fitting addition histogram density fit agree ht population year cause comparison comparison significantly arrive conclusion account strictly interpret cccc class work development monitor north american need efficiency small lead pool gain efficiency flexible misspecification distribution confirm power many cox proportional intend reduce need achieve give inference carry task notation applicable k eq expression replace exceed sum entry h examine algebraic implie observe second third df u statistic approximated chi limit distribution key lemma w md md kf asymptotically multivariate normal hence center iid thus limit asymptotically covariance addition easy quadratic verify expansion pn pn integrable maximize expansion express nan equivalent md md unique function neighbourhood jacobian jacobian function invertible md parameter nan write nan expansion note information q combine q quadratic rhs u define expansion recall mean md equality use expression verify claim limit easily check quadratic form expansion limit solely respect find sketch distribution set satisfy k approximate rhs limit n accord le lemma mean structure core respectively mean k expand kx normalization ignore far simplify hence ignore kx expand rhs summing get jointly lemma equal half right entry covariance le satisfied local distribution rhs still consistent similarly consistent root n md quadratic u md local alternative normal matrix still limit md define equality verify claim chi limit central parameter md aa aa subsequent proof write population sample sample partition rd rd note equivalent rd local central central respectively moreover local jacobian j q upper upper find therefore suffice u u rd adopt semidefinite positive semidefinite space lemma jx jx positive semidefinite moreover imply semidefinite verify therefore claim true except last induction block suffice notice give algebraic algebraic recall x r first substituting expression expectation square semidefinite prove important property complement equality similar expression u rd rd second block get finally imply equivalent inequality standard positive definite respect thus rhs column rhs semidefinite matrix side
index density random scheme see follow scale parameter independent exponential gamma gamma prior non produce proper posterior q situation log quantile threshold order beyond example display propose consider threshold basically adequate analysis tail model simulation precision affect assignment prior hyperparameter density density modelling analysis think model density small smooth univariate simulate hyperparameter simulate adjust consider period small credible interval predictive density large figure density represent center accurately ht ht level prevent population exceed capacity flow cubic flow monitor minute burn period display parameter shape around simulation distribution distribution tail see density heavy tailed tail application real datum small process mixture control purpose bic component environmental trial finance posterior realization value number distinct probability indicate summary configuration q resample cluster membership close proportional make x interested follow sampling proposal improve value density standard gamma truncate acceptance eq cm cm gamma part pareto model flexible quantile finally real pareto dirichlet mixture extreme combination generalize pareto transition location estimation approach normal unimodal realistic unknown application propose probability reversible jump inference could powerful tool density accommodate wide variety fit expect
label factor fashion mixture clustering set software use herein available compare analogue fit mixture gmm mixture model via family facilitate comparison approach method several extension cluster among available rand ari rand corrected agreement ari class example initialize use algorithm start chemical region chemical bic give classification gmm select factor classification performance ari bic gmm label base gmm gene expression microarray compare contain bic model give performance misclassifie ari predict gmm gmm report angle pattern signal set take uci repository apply set applying lead gmm ari well although classification compare result report observation gmm report eight fraction nine area available factor perfect classification region nine give similar classification ari classification gmm data application apply consider gaussian mixture skew distribution mix mixture component location uniformly hypercube covariance generate add diagonal element interval skewness value gaussian skew poorly would expect well ari simulate skew skew normal skew skew classification select treat four give near ari gmm skew normal skew skew conduct however point carry empirical analysis specifically consider mixture skew mixture show bic report find fail detect component method bic model skew skew outline approach look consider analogue interesting analogue investigate parameter estimation finally approach stage matrix respect g mm factor development mixture draw upon criterion select factor well illustrate combination number therefore choice parametric mixture g g dominate gaussian mixture cluster semi supervise analogue gaussian gain popularity gaussian mixture satisfactory asymmetric long tail vast majority date place past year lin lin lee first analogue mixture framework recently skew skew normal mixture asymmetric outline limit parameter detect elliptical skewness addition flexibility skew distribution multivariate inverse gamma asymmetric laplace extension model section generalize unobserve miss generalize complete consist well factor allow datum step complete compute give convenience complete expectation eq stage update proportion skewness index parameter form conditional distinguish complete observe label stage complete log update require slow
proportion different way analogy discriminant conditional curve g polynomial model spline hide logistic time discriminant polynomial quadratic discriminant specifically g example spline spline represent associated design adopt estimate perform fit piecewise govern logistic present regime involve model homogeneous curve complex hypothesis description whole restrictive handle analogy adopt functional curve functional mixture spline mixture describe previous time course modeling use adapt spline regression model sub p proportion represent hidden parameter class present knot advance regime change transition relax spline lead knot optimize dynamic hand regression hide logistic homogeneous limitation complex shape approximated functional linear mixture discriminant regression limitation complex shape class curve via mixture furthermore approximate regime analysis regression compose homogeneous sub class discrete represent class functional class group hide regression hide sub govern process another regime belong ij logistic ph sub logistic term transition distribution k pz g component ij generative sub spline thus change within class unsupervise independent curve g j maximize iteratively dedicate complete log g give belong paragraph maximize start compute give observation estimation current g posterior pz qx class ph qx ij qx j update parameter separate proportion logistic proportion update mixture analytic posterior probability q variance x ij consist multinomial logistic single eq pseudo code propose pseudo propose regime g g increment nj update provide assign use rule particularly class compose class summarize approximate expectation k px ij ij r weight logistic probability sub information algorithm mix proportion represent associated class evaluation real diagnosis perform regression pr spline sr model single alternative functional criteria first misclassification error fold procedure concern equivalent intra regard approximate g intra I notice estimate sub class spline function mixture spline respectively curve piecewise three piecewise regime h compose sub regime model complex shape show propose automatically regime class regime allow smooth within see active another time one regime notice approximate curve fail heterogeneous spline significant improvement regime two mean intra class color accord partition separately curve sub probability bottom c intra sr see use complex shape approach regression attribute flexibility change datum introduce temporal heterogeneous class call class therefore description top class model class cluster separately bottom curve regard dispersion class change accurately spline diagnosis study label switch operation use class minor class one minor automatic modeling especially two compose model two class class class estimate corresponding logistic regime sub good regime homogeneous curve regime h curve estimate approach mean curve bold top sub proportion plot obtain competitive give table c rate intra sr approach term outperform attribute fact fit notice converge class present change unsupervised dedicated simulated benefit discriminant gene plan perform bayesian figure mm concern paradigm rather dimensional paper particularly model present propose discriminant hide handle complex shape class class regime explicitly heterogeneity via within
ball training start call argue complex problem objective maximize episode study simulator simulator operate simulator increase learn agent act player body current status every agent execute parameterize primitive angle simulator chen simulator agent begin action end gain start thereby rise episode thus environment simulator operate discrete step map onto time reinforcement speak reinforcement action environment discover solve learn way incorporate domain knowledge available simulator distinguish ball body position difficult adversary gain turn angle intercept range detail center field choose angle macro action team primitive primitive treat formally tuple countable action transition transition take macro simulator reward macro episode consist macro action na winner case receive reward always maximize episode winner macro macro macro action forward strongly weakly choose instance macro intercept use ball adversary global angle object relative angle respectively field object close line otherwise range adversary together see describe adversary note informative impractical due curse tendency pre part interact adversary macro maintain winner ball scheme near velocity adversary move move macro iterative description team combine reinforcement introduce final assign action return event action first action pair update equation traditional parameter discount reward meaning towards select high table one state pair illustration sake discrete state action pair require fill accurately state never also change action pair generalization arithmetic computer work partition state share field occur multiple space vector fall offset cut different quick generalization ability fine indexing different macro equal field traditional detail action index update error number span entire large amount memory issue randomly actually store task propose begin episode end routine routine iterate action line find macro sum field routine select macro follow simulator choose q routine whenever namely intermediate routine field value next policy routine temporal discount line routine step temporal return choose state field routine end initially episode add return terminal state lastly routine weight standard simulator version protocol simulator visual sensor player view location distance noiseless vision ensure reinforcement argue brief weight angle degree every layer offset implementation hashing retain information hash create episode realistic episode experiment time episode approximately hour episode process episode process episode greatly improve episode win less qualitatively learn rule adversary considerable keep ball adversary behind ball illustration second adversary front angle start advance ball forward rule see configuration macro weight obtain high experiment well approximately repeat experiment present compose field dimensional curse detail exponentially take show episode hour process episode fast average bad win test solution simulation episode thus approximately qualitatively learn rule opposite side adversary matter location consequently unlikely succeed conjecture reason reinforcement dependence individually may valuable ball much individually robot domain observational reinforcement refine ball propose reinforcement call agent fuzzy system apply intercept pass arguably two try latter try task closely learn task player opponent task half field scenario attempt team pose reinforcement propose learn deal reinforcement simplification agent begin attempt solution
rbf different train corpus annotated speech english although annotation multiclass predicting segment standard modeling technique overall multiclass problem subject preprocesse convert frequency plus derivative coefficient frame precede frame produce coefficient target training herein hyperparameter optimize development experiment protocol generalize extraction select class pair extract eigenvector thorough strategy heuristic yield uniform random ensure denominator numerator class hypercube multiclass error rate entropy actually subproblem create ensemble ensemble kl divergence match improves publish remarkable consider architecture stack layer transformation procedure ensemble success method considerable work extract little additional eigenvector complementary inclusion classifier especially modern size valuable extract moment high experience suggest moment conditional inform high class label scale possibly deal slice class prescribe sensible dependent pick arise matrix eigenvector multidimensional regression discretization solely easy domain exhibit label incorporate extremely important local localize spatial incorporate kind direction perform library statistically finite second inaccurate dimension practice second low implement well setting although aspect non generic non extremely subproblem subsequently ultimately hope convenient competitive scalable solving efficacy via statistical extraction numerical primitive matrix method invariant transformation exceed derive combination believe method introduce herein utility acknowledgment li experiment proposition enhance investigate induce focus multiclass classifier excellent attractive theoretical induce invariant linear build three obtain learn great understand employ crucially compatibility kind representation well text drug design speech interest raw signal art conceptually computationally create discriminative feature scale example even exploit simple easy sufficient advantage nevertheless empirically remarkably usual multiclass iid low notation refer encoding identify vertex multinomial possible statistic involve multiclass classification fisher lda application expect tensor moment way feature matrix maximize xx direction might similar class eq generalize eigenvector despite convexity robust solve part software package since objective assume class long direction able discriminate moreover associate eigenvector detector result discuss sample invariance invariant invertible invertible setting datum transform conditional ac ac ac u therefore original worth point linear lead greatly invariance provide robustness original feature class uncorrelated response orthogonality eigenvector problem connection eigenvector xx v v j eigenvector class generalize eigenvector maximally method expectation dependence become perturbation use expect q eigenvalue matrix eigenvector concern finite may rank estimation eigenvector unstable sample denominator equation trace divide empirically f w leave specify eigenvector define projection magnitude therefore composition equivalent classifier pre remark amenable distribute parallel scalable lda x xx version noise task resemble think approach procedure ratio symmetric matrix represent signal capture well cast component find model vanish vanish role variability class use noise orient framework type meaningful closely examine distinguished two capability whether discriminative pca discriminative lda sir could limited valuable fidelity use conditional oriented novel begin database handwritten digit visualize eigenvector provide intuition discriminative nature direction extract row class eigenvector sensitive circular typical remain insensitive overlap avoid pair avoid detector consist arrange detector pattern bottom attempt horizontal stroke typical would ten project onto eigenvector pair top image projection distinguish information know motivate expansion vs useful distinguish class completely feature mnist extract discrimination pick discriminative diverse topic mnist determine hyperparameter setting fraction determine
convex nature original strictly smooth hessian denote hessian zero unique convex good hope iterate monotonic tt unique minimizer convergence happen property particularly provide guarantee initial optimization bound imply iterate relaxation central ingredient property special remark fact guarantee claim keep infinity ensure iterate non use framework globally approximate precisely immediately follow suffice dominant minimizer substitute proof j substitute eq allow simplify negative obvious concluding sequence monotonic convergent get trivial eq since convergent arrive directly convergent convex regime something limit particular claim typical restricted radius probably possible stationary sequence suggest linear exponential convergence control establish combination later eq second optimality write eq eliminate plug minimizer semidefinite argument converge linearly estimate show behavior iterate note mathematical sophisticated elementary unconstraine moreover obvious extend try newton guarantee exhibit quadratic locally typically require iteration reach accuracy newton inversion significantly turn notice stable newton newton often minimum iterate never infinity almost converge output rare find letter entirely investigate improvement interesting proposition apply university nj usa mail edu call patch denoise result synthetic image consistently perform beyond moderate significantly iteratively reweighte iteration exhaustive convergent locally convergent fail rarely regime letter explain mean patch reweighte decade effective framework distance patch compare mean sophisticated bm reader comprehensive review patch index noisy affinity assign pixel non local pixel particular pixel weight motivate optimize iteratively reweighte square rule q heuristic extensive globally convergent locally fails rarely say former exist community letter adapt framework force
hard focus non linear depth factorization achieve product interpret grow deep layer top standard approach rbm create bipartite graph edge adjust create fire correlate create grow construct observe correlation bottom layer find cluster highly cluster natural layer add pruning operation scale column random entry vector refer sparse way polynomially precision principle product high observe condition multiply invoke high large least extend bound eigenvalue correct polynomial time construct bottom point property output go go layer correction go look round integer one solve exposition let diagonal round integer identity unfortunately recover multiplication show round characteristic two two less truncation polynomial simplify analysis study generate prove diagonal separate statement row vector follow hold n distribution row hold coordinate study bind characteristic normal nc polynomially take multiplied finally divide result characteristic also step random characteristic divide identically odd power present probability density bound random average high bounded value distribute analysis lemma technique independently know variable coefficient eq let lemma condition hoeffding probability I ease exposition sign combination induction number zero convenience add w non nd ic entry moment get simple c switch back right complete function z similar statement lemma layer vector high I maximum value adaptation st inductive uv dm dm round near integer dm multiply know bound hoeffding I simultaneously vector iv ie j lemma apply remain disjoint value lemma term among obtain round reconstruct hidden column row identify pair entry intersect recover connect correlate node common neighbor non construct hidden share exactly one simple pruning node fraction correlate node share one identical empty hide obtain weight sign sign fraction flip wrong sign correct correctness linear reconstruct deep correlated depth term interesting find another type besides level input x instead produce th top elsewhere produce produce ask ideal image image correspond turn circuit circuit kolmogorov kolmogorov binary string string circuit turn input circuit circuit kolmogorov circuit small restrict circuit edge consecutive layer node circuit circuit convert vice versa thresholded computed circuit circuit matrix bit require encode circuit edge need connect sense circuit capture circuit sign use circuit kolmogorov factorization multiplicative generator bit existence inverting argue weight layer network match underlie rbms linear weight produce go reverse direction give appropriately equal thus randomness long top initialize compute network fraction xx much condition dense matrix number polynomially bound however prove find upper value base base assume inductive hypothesis characteristic lemma eq know truncation eq st om st ease exposition disjoint share prove coefficient tw tw w influence represented precisely similarly prove position go maximum show om om om st om st n tw tw iw iw I nm nd om om st st low drop factor twice know induction statement since non dd recover edge join pair join node correlation identify let point j ij k x total share j negligible bernstein get want say let point follow statement know zero position share know prove neighbor remain discard layer share position hide hide identify non entry neighbor
ratio reduction gene among address select algorithm selection family suitable make use sa good maximize characteristic algorithm capability accept bad end powerful capability sa lack involve exclude alone microarray expression objective development efficiently computation joint categorical public microarray organize briefly review anneal technique review previous describe experimental vi interpretation end conclusion future anneal sa mechanic near solution sa assume part belong retain explore sa hill hill avoid equilibrium give probability state boltzmann ts kt act normalization metropolis worker stochastic simulate temperature neighboring make acceptance energy state accept situation higher depend enough wherein energy reach metropolis anneal schedule design prevent process get time initially enough equilibrium iterate consider schedule final reach near inherently slow mainly temperature review feature selection originally search sa combinatorial optimize maximize purpose sa take enhanced improve bad mechanism intend account objective cause resample pseudo code loop loop finish update loop reach min x x ht rand output I return inner compose forward iterate solution minima start procedure remove feature accept pseudo forward backward backward finish respective execution equilibrium reach another carry aim present endowed search look limited try equilibrium point additional bias add available contrary removal improve current another direct consequence considerable speed grow deterministic configuration algorithm feature select accept true rand e backward modify entropy information uncertainty entropy expectation another one mi explain multivariate another one conditional mi q mi success currently mutual entropy character theoretic relevance use elsewhere contribution microarray obey property property say entropy entropy entropy new feature consider current subset turn far advantageous full order value procedure develop sequel table right datum entropy incremental every involve throughout store form entropy binary initial entropy form indicate table split entropy value total current computation joint hx hx explain notation sort e joint use number calculate take order avoid observed stage reach possible run considerably nature theoretic discretize gene significantly contribute index tend smoothly I small increment consequence add discretization increment value merge significant truly reflect subset set keep considerable require hour process unfortunately report consumption scientific enable establish eight classifier mean resample design sized data set choose metric knn classifier linear quadratic support machine support radial kernel specifically interface run regular winner classifier another error sign accuracy c cv indicate obtained display eight classifier accuracy cv gene less value significant compare step especially resample technique comparison comparison present reference illustrative detailed gene tumor cancer nn nb w nb nb rs fw nn cv svm svm work use scheme subset number problem reference cv fold cv rs bootstrap difficulty reach comparable gene seem among reference gene front strategy problem able yield solution big gene expression level see visually one level gene gene gene tumor tumor tumor tumor h cancer breast cancer ab nm vi gene expression evidence examine relevant tumor member rich protein involve development decrease dna confirm cancer cell sequence minor encode cell cancer member positively identify progress role expression find cell essential human critical human cancer whose expression value gene gene scientific involve control communication highly protein essential cell activate disease cancer pair gene family play critical maintain genome cell dna record level loop encode member small bind box role cancer tumor strongly cancer express well cancer surface play cell growth cell highly act water protein integral protein case normal formation encode member protein involve drug synthesis gene report responsible degradation cancer cell protein encode degradation control induction synthesis free breast cancer encode member protein protein act growth cell gene encode bind growth breast
problem quadratic qp solve qp solver involve combinatorial rip assumption efficiently solve rewrite pursuit use angle lar new discard detail code denote define feature pooling response joint pooling determine sum pool uniformly pooling vision pyramid matching pool code temporal pyramid divide click signal design pyramid layer pyramid divide click click tp overlap layer end pooling stage global pool encode coding see offline jointly q lasso lar inversion storing method dictionary code harmonic response plan reach trajectory train minimize test part xx linearly dataset detect click different ht choose dictionary feature iteration dictionary rest root per represent truth global calculate h norm influence stage dictionary basis pyramid temporal click choice pooling pool varied temporal pyramid pool pool window specialize click ht ht use temporal pyramid pooling permit code rough propose work directly click signal signal issue couple linear particle accurate configuration plan feature spectral first architecture present bag framework vision global representation transformation thank acoustic truth feed anti delay base working detect click rough head orientation purpose record precise system position party typically vision successfully extract invariant click basically three part extract simply denote local equally sample associate local patch
maximization subsequent augmentation scheme practice algorithm generalization irrelevant deal measure sensor annotate entry remainder paper organize describe detail derive formulae variable discusse exist dimensionality reduction section describe validation compare dataset image conclude refer manuscript matlab illustrative team estimate proceed dimension way present onto intermediate prevent dimension reduction output account method category partial pls sir component base sir design specifically reduction determined method perform necessarily regard pls principal covariance input output eigen method propose semi parametric achieve difficulty low consider around model low dimensional play role corrupted parameterized regressor number estimate linear task response deal kernel onto dimensional achieve sir machine kernel variable originally view instance non principal analysis drawback require kernel ad hoc point mapping learn model gmm response linear unobserved supervised mixture expert formulate cluster scalar recently distribution interpretation factor view supervise variable observe concatenation denote namely vertical presence corrupt hand allow add motivate capture regression dimensional visual human angle involve motion train nevertheless contain responsible various aside properly quantify annotate account phenomenon image physical surface end transfer chemical physical spectrum simulate huge collection spectra perform hyperspectral require generate generally restrict small number main chemical incidence neither model tractable sound acoustic experimental ground input response realization lk plus use locally error capture reconstruction due approximation variable covariance transformation gaussian induce probable q analytically formulae density interestingly rely transformation assume model size vector become example huge nevertheless drastically isotropic implie fit poorly lead appendix totally unconstraine joint unconstrained mixture gmm low dimensional pt z edge auto w edge auto node z z supervise treat namely hybrid illustrate constrain classical affected unobserved observe must decompose independence write realization flexible local mle model parameterize hybrid mle experimentally advantageous hybrid response variable result regression instance hybrid either local covariance isotropic view variant show case mixture regressor view I instance hybrid isotropic covariance canonical correspond covariance component generative dirac hybrid generalization l diag diag diag link fig last fig diag matrix matrix block block unconstraine worth notice account partially notably mapping yield nature partially forward therefore crucial ingredient usefulness onto partially observe devise augmentation one facilitate subsequent maximum miss training augmentation scheme naturally refer integrate previous scheme amount information may em determine missing accelerate decrease simplicity extension alternate allow application close hybrid affect estimation namely latter estimation weight observation em general lead constraint number mle describe detail initialization marginal hybrid variant comprehensive toolbox illustrative team identifiability issue indeed change change affine transformation solved spread grid matrix dirac matrix respectively set one maximize parameter conditionally step follow sake model n nk nk conditionally show amount recover posterior replace namely detail supplementary material posterior virtue nk k correspond gaussian straightforwardly mapping detail material expression formula one imputation variance miss formula straightforwardly adapt unconstrained diag isotropic initialization maxima choosing choose proceed hybrid initial posterior complete affine em variant latent leave explain variant much initialize isotropic iteration em continue hybrid learn log likelihood denote complete isotropic denominator natural value minimize refer method bic require computationally demanding implementation could methodology evaluate hybrid synthetic retrieve pose information image recover hyperspectral mention mixture mle mle estimate em forward function estimate please repeat mle combine forward additional table mle regression gmm sir sir sir sir principal axis slice slice little influence polynomial improvement experiment sir dimensional slice cluster induce quantization slice replace carry dimensionality reduction inform preliminary may probabilistic kernel code preliminary determine optimal test hence evaluate ability method high consider situation function give image mle kernel use family dt dt generate function monotonicity generate choose piecewise affine assumption hybrid uniformly drawing draw uniformly draw test display std absolute obtain noise snr test point mle average return value spread interval consider experiment correspond repeatedly average avg deviation percentage error avg std avg std sir sir show consider variable method bic outperform practice function automatically select component function decrease significant improvement percentage extreme error interestingly bic synthetic marker observation although error select latent linear effect could choose error marker mean comparison illustrate range alternative explain upon mle increase beyond parameter become covariance correspond influence synthetic test distinct generally degenerate class reduce fig obtain manually set experiment showed always yield either cost also error decrease overfitte turn snr db increase finally method outperform overfitte number similarly extreme snr mle stanford face consist angle range absolutely integer keep stack task subset image train per mle covariance previously spline annotate firstly image pose image pose unobserved pair pose unobserve obtain obtain bic vary systematically train bic invariant face well yield invariant light yield similar observe upon latent overall achieve perform estimation face avg std extreme absolute light method dimension term light avg std avg std avg std sir sir sir sir mle run verify latent recover meaningful pose association recover visually parameterization set show image mle image reconstruct reconstruction hybrid encode way l input reconstruction mle cb cc cd face angle latent image hybrid reconstruct pose use mle visible imaging sense technique study record light reflect range location physical surface composition etc characterize transfer spectra allow simulate value goal scan sign material ice associate parameter spectra analytically investigate consider relationship spectra physical model potential hyperspectral image express database spectra parameter namely water ice co ice proportion water ice ice spectrum hybrid regression spectra ground fully hybrid ignore database choose water ice ice previous proportion mix parameter transfer tend estimation ice water ice exclude water ice co ice latent remain parameter hybrid sir sir validation scaling test mle minimize show validation
multiclass refined geometrically decrease course minimization factor specify accuracy run mean eigenvector reference p correct adaptive cluster multiclass display correspond accuracy multiclass adaptive structure multiclass energy evolution comparison simpler relax involve balance produce result evolution label multiclass multiclass system form fidelity seed incorrectly mostly iteration converge steady multiclass segmentation energy evolution fig energy contribution three green fidelity initial iteration energy decay place toward integer eventually minimization drive fidelity satisfy almost influence iteration picture typical energy evolution guide truth synthetic roll create randomly gaussian convert roll close neighbor fidelity table description multiclass spectral multiclass compose configuration manifold multiclass capable manifold achieve accuracy algorithm divide black red neighborhood channel fidelity multiclass fidelity fig white pixel black mistake vice versa image take different angle step degree pixel make benchmark summary red channel randomly select partition six leave image process rescale add set point scale close fidelity labeling point select exception supervise fidelity see great mnist compose image handwritten digits task image digit hence construct near fidelity term per corresponding multiclass average segmentation fidelity comparative normalize cut comparative supervise convolutional net deep net svm take digits digits fidelity competitive supervise preprocessing unlike method exclude however form multiclass interface method alternative binary multiclass local graph constitute adequate modify diffusion exploit affect order label accurate method fidelity representative long rely graph depend label assignment investigate interface conjecture converge variational nature functional acknowledgement research air office scientific grant grant generalizing interface model motivated involve minimize energy transition preserve symmetry among label fidelity term segmentation many task pattern rely similarity infer meaningful characteristic global partition category devote multiple binary develop alternative involve interface graph equation inspire phenomena expression functional minimization graph operator multiclass binary ii build partitioning consist successively reach considerable compute class partition contrast interface simultaneous multiclass build modify give binary interface multiclass close class characteristic multiclass incorporate method minimize kullback involve organize interface describe supervised discuss multiclass present conclusion interface base measure segmentation small arbitrary dimensionality represent write functional denoting double segmentation smoothing field adopt label jointly term towards value interface term transition deviation goal length interface weight lead interface approximate total variation tv formulation functional piecewise solution efficiency tv method interface tv energy energy approximate tv calculus graph introduce interface undirecte represent relationship technique segment separable neighborhood correspond constitute segmentation potential label class potential cluster label purpose use periodic li fractional large great periodic potential multiclass solution laplacian term modification multiple change multiclass framework contiguous vary accord phenomenon fig suppose goal class gray clear two vertical interface jump smooth high interface reason assignment interface multiclass symmetry undesirable symmetry class htb class half symmetry define difference correspond strictly energy difference periodic express functional generalization normalize write constitute normalize laplacian reason laplacian satisfy u j u expression construct reweighte laplacian normalize laplacian increase generalize empirically differ implement tree
empty circle plain follow graphical framework tool interpretation typical two recall regression adopt random vector z respective represent consist gene represent potential contrast gene co expression edge graphical correlation gene expression profile belong resp resp resp characterize neighborhood precision independent decompose formalize equality g translate conjunction combine neighborhood test equality testing reject level every assumption crucial correctly sense hypothesis reject reject reject model training breast dataset originally publish full microarray profile patient iii breast patient n residual rd develop response map distinct gene gaussian graphical patient conditional dependency medium dramatically another question tackle whether remain take uncertainty cc disease rd patients residual medium present figure associate neighborhood rely validation collect two clinical center pool patient homogeneity patient reject rd half neighborhood differ responsible respectively neighborhood c bb decision decision ex homogeneity among patient summary test bb ex decision ex b decision homogeneity test rd summary test correction empirical neighborhood surprisingly subset rd patient heterogeneity regardless neighborhood disease half significant heterogeneity rd patient rejection neighborhood summarize responsible nine nine four describe clinical literature drug lead functional biology suggest expression prove growth cancer include bb reject ex rejected reject rd multiple reject add validation patient lead neighborhood neighborhood lemma upper deviation call depend statement upper universal small eigenvalue two universal n eq upper universal exist universal deviation constant q l f fulfil proceed variable degree nu nu convexity nu nu probability sign two identity n prove first tail exhibit convexity apply observe derive turn solution u lead follow enough thus z n conclude control wishart symmetry recall need control probability enough statement gaussian vector get r low n freedom apply large constant freedom large l consider union support q hold enough condition derive observe prove result sequel main collection contain support f f recall regularization short analyze act control enough event slight event assume sequel tell belong compare take x prove apply give ensure fix kt since belong n last eq allow tell event recall enforce hold enough come estimator w bind condition consider l empty l apply small rewrite regression hold proposition enough derive pn line consequently pn define previously derive last line union large event lemma let get enough symmetry integer fix problem bring hypothesis consider power wishart matrix positive grateful associate suggestion st partly bs calibration cm cm cm comparison microarray infer graphical uncertainty adopt test regression rely testing selection two test illustrate microarray motivate linear regression particular lasso guarantee provide selection performance among effort turn construction quantify select area two unknown component design matrix gaussian unknown formal remain decomposition want include equal covariance motivate homogeneity graphical defer infer potential drug target differential vs vs however difficult network error real difference underlie suggest global infer formally global dependency characterize objective eq problem global statistically test entry differ adopt approach high regression sample testing solve high test mean introduce compare dimensional covariance analog problem high test perfectly basically objective consider purpose false discovery assess another derive clean half ordinary size dependency result sample variable control wise discovery nan hypothesis performance deal alternative nan nan hypothesis intensive nonetheless adaptive sparsity optimal covariate introduce design prove reach high competitive term write across local elegant split lead contrast global detailed defer stem fundamental support test successfully lead sample unknown detection logarithmic know three drive subset informative attempt parametric statistic define calibration procedure control use calibration upon permutation reach tuning testing increase empirical power control furthermore amenable small interestingly require half sample describe section devote well tool interpretation asymptotic power experiment compare performance procedure handle graphical breast cancer finally available notation positive definite matrix scalar vector besides make notation refer concatenation concatenation finish vary line dimensional design proportion adapt regression covariance structure linear deviation test hypothesis reduce eq coincide restriction collection low hypothesis hypothesis global fundamental observation motivate summarize calibrate consider testing collection subset prohibitive algorithmic since result relatively small hypothesis see well drive produce resort three parametric statistic sf pt good subset inclusion subset reasonable time procedure collection one deterministic collection ii satisfy though introduction could appear artificial collection mathematical practical among collection straightforward collection subset kind collection thereby deterministic rapidly size reduce search costly introduce j development datum drive collection lasso type collection proceed informally intuitively focus however also amount three kullback term evaluate conditional variance term comparison step various size convert convert proposition however inversion computationally prohibitive subsection multiple value type overfitte sake merely sequence global least calibration define conceptual allow derive reveal conservative collection bound option type outperform calibration nevertheless mathematical provide sharp use henceforth three testing idea parametric permutation define procedure capture capture discrepancy effect say consequently design prove well lasso compute along additional lar compute decrease jump l piecewise change lar collection follow intuition estimator drive tune lasso find whole find subset powerful sample tune estimator term path hope trade formalize v behind choice resp statistic measure opposite likelihood ratio estimator could symmetric obtain sharp statistic make intensive powerful consider separately value non reciprocal degree nan let nan familiar ready table quantity variance sf random freedom one simulation prohibitive collection subset give justify definition eigenvalue take finally approximate eq although notation mask whereas consequence recall choose q collection derive choice put variance comparison similarly replace equal drive function correction apply initial include replace constrain small use perform multiple simple drive eq perform although conceptually difficulty correction need provide correction reason type permutation order choose permutation get new parametric denote size respect quantile eq quantile summarize b bs drive calibrate permutation test function permutation simultaneously loss mention early restriction drive treat simultaneously test exactly would favor would value calibration assess significance contrast model accurate model collection decide keep equation responsible sensible reject variance coefficient part define reject small procedure easier illustrate calibration calibration sequel resp positive resp kullback leibler consider deterministic section devote analysis covariance collection intuitively leibler kullback discrepancy give kullback role discrepancy matrix large eigenvalue first calibration refer size support ii error zero specific power k control power expression far assess briefly accord powerful tell sparsity hand variance outperform play role fix power large simultaneously section together sparse imply knowledge exist adaptation simultaneously achieve introduction proof theorem power union statistic deterministic collection bias variance trade link cardinality need give distance note restriction resp power long q side therefore comparable powerful require advance inequality term form sake simplicity restrict subsection test burden prescribe statement depend eq assume sparsity analogous lasso dependency instead extension subsection large order compatibility closeness four large reject n tell behave nearly integer define exist positive integer reject restrict small front technical almost block control detail assumption necessary aforementioned collection simulate regression test rare parametrization adopt still restrict sample parametrize common coefficient sample detail test statistic competition experiment repeat summarize control test collection case level illustrate scenario signal none coefficient specific overlap specific global sparsity illustration pattern actually covariate beyond sparsity consider three generation investigate correlation decay pattern link covariate column independent pattern simulate package generate structure make intra extra connectivity generate structure calibrate covariate covariate correspond decay connectivity default option connectivity coefficient take five time ex ex six combine collection deterministic drive collection permutation calibration permutation put statistic equality freedom eq maximum detect difference really suit fisher except collection calibration st high base split restrict ratio increase stability must aggregate permutation single multi procedure table type reject base nan orthogonal correlation level estimate give interval
f semi definite fisher rao fisher denote substitute assume exchange order derivation definition rao multidimensional bias average h find enable norm compute rao inequality regularity condition conjugate second f result unbiased estimator rao power omit unfortunately explicit fx maximize side x precisely attain e know iff consider without generality zero unbiased take functional gx characterization generalize distribution involve fisher take density fisher let gaussians minimize generalize among moment similar al identity usually heat consider doubly recover exhibit uncertainty begin rao wave relation e yield multidimensional path home propose modify generalize fisher generalize rao involve fisher arbitrary gaussian generalize new communication introduce estimation fisher information reduce fisher show unfortunately form rao gaussians physics mathematic entropy subject moment generalize fisher de extend generalized fisher information derivative finally extend r due space omit proof possible deal measure source notice see gaussians gaussian physic maximum analytical physical sometimes distribution functional build unfortunately characterize fisher consistency notation fisher include partial respect increment arrive gradient fisher information involve
equation discuss td conclude discuss mdp terminal subsequent deterministic policy say positive terminal proper visit time terminal accumulate reward along trajectory mdp approximate discount horizon extend bellman equation point reader may present derive linearity point map onto td sequel follow denote respectively belong joint mapping eq may fix fix direct may popular subspace adjust transpose q whose estimate constructive projection onto weighted trajectory fix state probability probability visit weighted euclidean onto namely let fix stating regard contraction find next scalar result project contraction hold contraction respect norm let eq equality weight triangle second claim exist real norm finite norm finite plug plug correspond weight unique next moment base jointly matrix explicitly q project satisfie orthogonality write invertible guarantee projection simulation extension square trajectory mdp initial denote visit terminal trajectory denote empirical trajectory n fx next proof involve application well td mdp policy iteratively visit terminal weight td td update td converge size td section prove project equation weight nontrivial contraction proposition illustrate positive state exception state instead choose depict constrained approximation depict section successfully intuitive highlight domain domain continuous ball reach depicted control apply force cause velocity additive gaussian deviation coefficient shape elastic cause ball make domain rl benchmark domain use use velocity near radial state reach velocity trajectory uniformly second moment coding feature velocity estimate deviation leave naive total target place monotone furthermore turn stress domain go novel rl guarantee evidence issue investigation work bellman return rule albeit guarantee unclear naive policy improvement may perform usefulness problematic adjusted reward policy propose handle completeness exclude terminal sum reach end similarly uniqueness function proper well follow observe mdp reward rearrange give state claim denote indicator note hand reach last terminal state q trajectory expect eq let trajectory independent real observe triangular eigenvalue negative part eigenvalue real q e next satisfy differential ode globally equilibrium converge origin follow iterate remain almost hold convergence iterate ode part globally asymptotically formally policy criterion cumulative reward risk management finance control propose td reinforcement planning process mdps typical cumulative discounted denote application however maker
crucial understanding recent study well investigate nonlinear infinite derive bound least rkh operator note unlike scalar reproduce kernel value different task extend linear refer space require infinite infinite identity operator scalar value identity al satisfy hilbert schmidt note de complexity hypothesis issue study possibly notion generalization scalar subsequent randomized paper concern stability extend stability scheme multi algorithm infinite schmidt demonstrate result various vector provide multi task show introduce notation briefly recall correspond hilbert value rkhs require establish stability bound ridge satisfy give hilbert schmidt operator illustrate usefulness conclude paper possibly separable separable kernel mx z ic loss illustrate functional eq regularize eq definition kernel infinite hermitian kernel rkhs reproduce iv hermitian base xy I schmidt assumption start hypothesis kx op kx weak schmidt assumption dimensional long observe converse true hypothesis hilbert schmidt orthonormal orthogonal immediately lemma hermitian detail kernel determine introduce hypothesis training rest uniform hypothesis sufficient family multi need additional hypothesis couple hypothesis direct extension value task setting concern uniform stability worth point differ convenience reader present modification regularize convex summing eq summing lipschitz imply consistency stable tend focus attention infinite define q satisfie verify hold prove value algorithm task expand manner insensitive fx task lr algorithm associate least rr hilbert schmidt example multi output infinite algorithmic aware carry regularize square algorithm assume schmidt show section schmidt addition obtain hilbert assumption encounter infinite frequently encounter regression analysis entire functional regression take response infinite hilbert work predictor author hilbert value value kernel kernel linear operator kernel estimation kde define scalar structure output value rkhs infinite kde multi task kernel identity structure conditional distribution gr rkhs embedding conditional value kernel collaborative build range past author rank optimization cast spectral operator spectral attribute predict preference linear hilbert item operator want emphasize infinite dimension kernel weak hilbert schmidt hermitian result value structure embed positive define multiplication functional regression hermitian kernel even always verify eq orthonormal multiplication k follow value hermitian hypothesis schmidt fact hilbert schmidt orthonormal sum case hilbert schmidt one schmidt make kernel hilbert schmidt like basic operator positive scalar hermitian task second add schmidt schmidt
various component hold transform set hence fix otherwise inverse gamma model consider usually ensure sort regularity stationarity present introduce constraint affect general advanced tackle widely recognize label switch inherent recent study reader g accord cite impose inequality upon subscript henceforth stress strict actually component establish ensure coherence component constrain require allow inequality otherwise former could matter concern impose e remain coincide slight abuse proceed rewrite c equivalent restrict obtain coincide hence conclude mixture identifiability coherence restriction statistical enforce sort technical ensure series framework specification regularity constraint r reduce restriction k assume constrain n expression model constrain specify nm expression hold simultaneously design coherent need formulate restriction equivalent prior switch q ergodic transition adopt convention refer short allow markovian intercept autoregressive consideration generalize specification hereafter q introduce discrete group provide suffice q notation drop indexing instead notice exposition identifiability restriction stress consideration derive form coherence need equality restrict coincide ar process q absolute eigenvalue well ar within assume comprise follow gamma distribution row note result proposition one display formulae normal hold prior structure value within proceed allow follow formulae two ar represent markovian four intercept autoregressive specification common parameter naturally switch include three intermediate specification establish straightforwardly illustrative specification write regularity coherent coincide coherent specify research realize national performing proceed respectively parametrized analogously proceed analogously respectively pl mixture generalize specification component specify introduce structure coherent relevant coherent univariate primary derive specific three inverse gamma study consequence additional prior enforce regularity stationarity coherence coherent prior class switch ar bayesian compatibility switch exponential family constitute special former restriction collect parameter note include oppose transpose notational convention model analogously measure theoretic abuse symbol amount I regard bayesian former nest incorporated structure via conditioning reduce restriction say coherence prior hypothesis testing model review form compatibility structure specification solely whenever play derive former argue relate express ensure specification appear finite mixture markov switch specify prior desirable relevance incorporate issue compatibility far coherent switching counterpart within finite model framework focus exponential gamma derive explicit condition relate nested one enforce identifiability sort regularity order stationarity illustrate markov switch ar collect parameter remark comprise parameter subscript vector result introduce coincide component contain simplex switching form underlie transition initial ergodic introduce conditioning transition represent single normal conjugacy statistical apply lemma formulae relate throughout fix sake g establish coherence upon mixture univariate eq alternatively parametrize immediately corollary provide component k k accord nest constitute mixture q simple average equal eventually relationship dispersion amount translate correspond see reduce mixture see equivalently thereby grow number assumption proposition determine specification adopt additional simultaneously variance one hyperparameter nest specification employ mixture min min latter constitute consideration calculate sort compatibility ensure prior coherence specification individually derive coherence inverse gamma coherent gamma see b formulae special hyperparameter k kb increase scale fashion prior examine
h unit key idea combination jensen show capacity misclassification rate label cast allow n asymptotic except notational order negligible bind consider pose step set n give make prove prove converse first give inequality goodness prove guarantee learn standard proof shall shall shall heavy bound routine empirical note optimization optimizer incorporate bound clarity radius assume sake notational sample n write use difficult applying estimate hand nn j invoke powerful alternate representation decomposition powerful couple real x index r e n random contraction inequality state exploit prove contraction average constant dependent domain function linearity conclude rademacher complexity hypothesis rademacher complexity regularize close convex make rademacher bound result np p excess risk r radius oracle oracle risk since oracle inequality inferior goodness kernel desirable learn generalization guarantee sake brevity differ regularize empirical lie due perturbation expression nn theorem allow rademacher make expression k good dependence able show show radius choice combination go good combination vector training good inequality goodness ideally kernel good ensure result exist unlikely face prove goodness absence combination q instead jensen convert goodness combination function possible goodness goodness goodness goodness however absence convert goodness believe predictor look lemma prove predictor restrictive predictor kernel good predictor notion combination goodness seem prefer
x fact relation begin relation preserve stay I check straightforward show preserve notation r correspond part factor two inclusion r r r summarize monotonicity relation mention restriction property violate factor preserve r edge case directly construction r desire ready prove grow step relation since call lemma r show empty low red e energy compute forward pass message compute backward pass order segmentation curvature f protein side chain triplet protein notation take pass pass pass message half lower bind normalize initial compute namely fig comprehensive comparison replicate comparison perform make conclusion outperform similarly outperform technique comparable bad f code behaviour speed advantage believe family include case possibility pairwise depend strength natural modify individual would change smoothed graphical hope paper research area approximate thank experimental let need show message description procedure assume remove affect claim correspondingly minimizer xx expression term rhs proposition message edge store xx message accordingly move accumulation numerical error store xx xx update store second keep message two store singleton factor update procedure forward pass q update update send order check definition look reweighte new reweighte case fast sequential reweighted message original derivation generalization graphical result devote discrete variable represent sum map mrf generality many probably mrf inference prominent approach try solve lp call lp lot solver special sum diffusion short efficiency slow advanced technique message derivation tree namely monotonic chain generalize hard product node equivalent involve decomposition almost immediate generalized complicated introduce definition impose weak graph next use processing propose understand believe benefit pairwise conclusion prove scenario graphical nest factor marginalization constraint framework family algorithm block another message simple however indicate significantly slow discusse far ascent may property lot converge lp temperature parameter go augment gradient lp bundle mirror smoothed label subset function set restriction direct acyclic polytope implicit convention whenever denote independent state restriction sometimes emphasize writing case high j result lp relaxation tight function completely characterize tight relaxation tight extra relaxation pick edge latter edge would relaxation infeasible solve message eq thus mm obtain follow dual maximize ascent strategy message keep message show restrict maximization tree forest pairwise graphical case shape fix incoming incoming proper special formulate incoming simple message pairwise see generalize consider parent move call collection namely perform energy graph allow fast empty generalize child factor min marginal propagate child distribution easily case depend question via message implementation store vector factor income l pt set xx usual numerical stability additive xx affect behaviour repeat reverse swap update discrepancy justify edge change ii pass I update immediately update zero claim unnecessary proposition pass pass forward pass describe update I ii interpretation give update extract primal beginning mark procedure line assign node message xx currently label node backward pass produce similarly pass forward give extraction iteration pass keep track important question order totally use propose sort sort arbitrarily rule issue natural work process arc say consistent relation jj ascent
nuisance appear without property estimator risk measure via excess illustrate excess behaviour decomposition choice ensure excess condition lead exponential inspection highlight bias variance condition adaptive general thus bias condition rule follow constant proof control excess good minimize trade give noisy fall problem input deconvolution correspond proposition discriminant counterpart form otherwise deconvolution introduce unbiased deconvolution bandwidth excess variance minimax illustrate excess could extra noisy quantile interestingly rate quantity bandwidth usually use give adaptive state risk bound excess independent pointwise standard apply localize depend moreover measure thank propose drive localize localize risk likelihood coincide kullback divergence variance assumption relevant coincide consider z get statistical discuss drive apply context see involve proposition du elementary u last line old q tr point weighted precisely condition inequality claim sequel use simple inequality control margin margin q need version device last pz mention proof proposition indeed eq inequality give sequel proposition introduce element computation get r r remain definition union twice twice eq eq q exist universal depend allow take na eq measure denote function schwarz result thank satisfied argument use adapt need maximal variable subset real exist constant introduce deduce ingredient dominate simplicity minimum exposition thank yield maximal inequality rgb proposition assumption rgb adaptive investigate set design direct noisy mean deconvolution bandwidth deconvolution fast excess choice smoothness issue call empirical risk risk nuisance variable learn nuisance whose optimal obvious point optimal rate unknown index technique suffer deconvolution deconvolution sequence distribute deconvolution bandwidth estimator fan smoothness drive select minimax function vast estimator exact index suggest theoretical minimax instance view receive development intersection confidence reveal advantage traditional procedure validation application image reference therein deconvolution pointwise improvement principle deal observation reference density deconvolution mention complete validation different multivariate cluster thank noisy hausdorff thank noisy propose study binary work statistical unsupervised use necessary suggest deterministic get minimax fast rate unknown smoothness aim contribution procedure knowledge procedure cross aggregation principle excess contribution rule comparison risk nuisance allow adaptive result context could adaptive organize describe collection deconvolution deal noisy state adaptive reach bandwidth trade bias automatically extra seem conclude generalization rule binary conclude whereas dedicated group topic biology science real life sequel law lebesgue unit classical peak noisy contaminate problem purpose standard mean latter codebook excess risk define assume rest paper assumption regularity density lebesgue hessian investigate area popular procedure partitioning set center observation near cluster minimization study work rate consider suggest minimax assumption reason deconvolution step deconvolution denote transform positive kernel abuse notation vector suppose practice avoid repeat instance c minimizer deconvolution empirical risk convolution z restriction close ball compact choice adaptive bias variance error overview thank upper excess sequel p smoothness deconvolution term stochastic empirical noise empirical spirit proposition bandwidth bias trade classical completeness deconvolution deconvolution exist trivially fine sequel kernel multivariate construction kernel satisfy instance regularity express old strictly derivative sequel law old extension state deconvolution depend explicitly need constant lower behavior noise excess purpose moreover assumption transform pose deconvolution pose decrease characteristic see deconvolution finally fast type classification margin relate view exist propose euclidean allow use localization principle reach strongly relate involved study indeed satisfy continuous point condition limit theorem interpret regularity separate follow center cell let boundary cell continuous hessian boundary concentrate optimal related well excess introduce na satisfied fs exist universal n spirit control process assumption rhs see case rate reach pay quantity relate characteristic na derive margin propose use procedure minimax turn bandwidth excess similar value performance rise validation unfortunately unsupervise lack choose presence deconvolution cross validation bandwidth possible square estimation transform unknown minimize square risk eventually introduce model empirical penalization spline smoothing radius ellipsoid unfortunately affect empirical choice risk principle appear commonly tool build depend notation satisfy involve na sequel set bind
algebraic situation scheme generic intersection projective choose generic property algebraic scheme generic much algebraic natural phenomenon projective certain content intersection additional property scheme application one one occur variety open occur generic event ignore involve analysis computer relation scheme brain computer interface brain task irrelevant task seek separate brain activity activity environmental epoch epoch criterion epoch stationary scheme scheme generic intersection generic intersection intersection condition previous unless projective projective polynomial vector polynomial brief come geometry scheme replace variety author set knowledge algebraic geometry suggest connection grateful insight first scheme projective variety define write scheme choose suppose single also carry application remainder conditionally way assumption plane intersection generic upon plane involve equality I especially generic generic generic consist tuple incidence show dominant generic dense p next show remain incidence correspondence subspace situation incidence projective space dimension depend vanish incidence projective natural cell projective increase vector see intersection interior interior intersect interior use write incidence correspondence negative map projective space q subtract take except k kn k k complete study theorem yield distinct theorem characterization identifiability generic intersection common meaning begin deal homogeneous define tangent full coordinate patch coordinate vector homogeneous lie figure main ingredient depend ingredient coordinate give corner gram exclude never appear define equation tangent coordinate empty gram matrix vanish result equation enable write omit vanish valid submatrix index column index rational expression denominator equal minor vanish lie intersection finish scheme equivalently full choice equation patch open match symmetric irreducible q ccccc extend argument generic direct extension corollary precise determination identifiability assumption interesting eliminate extend variety tensor irreducible know apply central task
sub exponential get arrive union bind j b p e norm random min max q hand hence size exist take least base whose employ bind arrive eq q note less union minus pt pt pt definition corollary conjecture replica claim rgb regime small use penalize lasso search set considerable amount work devote characterize estimation paper question precisely address bound test early achieve special nearly design matrix approach build distribution lasso size distributional characterization distributional cope estimator covariance validate optimal sample distributional design replica heuristic derivation suggest strong gaussian random standard scalar let denote matrix parameter exceed small situation explanation topic design assumption arise consider sample sparse whose zero underlie row perform linear problem design design insight compress sense signal determine focus quantify nan hypothesis assign p equivalent state face two positive incorrectly negative reject arbitrarily arbitrarily aim optimize trivial establish arbitrarily make practice indistinguishable complement precisely interested establish significance testing coefficient vector intuition design n I significance power standard design several conclusion computationally efficient numerical significantly prove e estimation remark simple question non document propose zhang zhang b broad matrix satisfy ic word test factor particular answer paper answer positively approach base component paper online apart crucially assumption assume deterministic technique case gaussian namely consider regime vanish level contribution power power dimension suit design significance except replace universal build generalization covariance matrix distributional distributional limit strong derive replica physics discussion heuristic validate section simulation result broad defer develop design require issue beyond present appear limitation make form note paper require namely limit comparable discussion regression regularize establish typically see recovery p matrix eigenvalue compatibility develop hypothesis within focus asymptotic high absence characterize related recovery condition cc make statistical power triple stay regime observation infer optimally tune nontrivial leading resample method provide alternative assess implement idea superior present provide brief notation use integer bold resp denote letter entry column likewise restriction index identity integer constant define introduce subsection minimax need testing give family measurable reject design hereafter subscript whenever clear probability power type matrix reality type failure false adopting formally let vector false upper sparse upper bounded realization accept procedure note exchangeable useful use property indeed take supremum family take completely output test offer prescribe control error minimax check bind design minimax f pz scalar sn take gaussian immediate omit eq look corollary goal design appendix central present paper hypothesis ideal regime method know coordinate oracle appear hence loose tight least asymptotic mention bind different proof reduction coefficient pz versus probability per minimax function hypothesis characterize reduce orthogonal span subspace depend lemma theorem see ii tradeoff high reader provide context discuss numerical describe table hypothesis model regularization significance q large assign test construction step next section establish u estimate precise simple noise mean particular motivate I appeal notice necessarily bias eliminate direction illustration eliminate modify increase define sequence instance p np rp scale equivalently could two scale favor simplify propose level size power indicate following indeed p p converge gaussian design cf prove surely exchangeability column prove need would impossible arbitrarily achieve happens converge assume claim true surely side exchangeability achieve know os establish comment irrespective keep fairly insensitive achieve minimax tune standard scale subsection another choose distributional converge instance design let weakly furthermore motivates hypothesis motivated sample variance illustration set strength active performance package fit path rather correctly measurement provide use reasonable width table regression asymptotic establish theorem cccc avg std ridge na ridge na c base na ridge establish deviation testing determine minimax soft thresholding remark equation return value fairly value guess analysis try oracle correspond curve predict width alpha ridge table simulation report table conservative type error prescribe power procedure broad tailor design drawback expense method achieve positive realization power see design design matrix subsection justify gaussian limit appear extremely nevertheless replica physics show regime propose alternative implementation bound pt design p assign follow base design generalization u define design model tuple instance index dimension np say sure potentially depend hold letting dirac measure probability independent empirical weakly empty np states p standard distributional sequence challenge section discuss rigorous rigorous validity usefulness notion appropriate distributional np distributional defer strong almost result power assumption distributional p standard distributional assume prove surely analytical power procedure dominate distributional establish rigorously np ps max min weakly distributional take eq prove conference notice control allow use assumption instead bound away distributional limit le asymptotic role section copy base estimator sparse regression column available de also establish optimality center construct control bias variance meanwhile require asymptotically dominate contribution appear rigorous covariance neither assume however method standard distributional require contrast summary complementary provide characterization restrictive paper support subtle difference approach construction reduce mathematical ensure normality regime row generate give zero small use unbiased estimator amount ideal u consistent vanish fact asymptotically gaussian distributional limit histogram obtain gaussian behavior eps eps eps histogram p eps pdf histogram width eps pdf expect os distributional characterization simplify lasso correspondence normal analogous characterize analogy mathematical normality understanding point sample contrast limit start theory around high theorem remain normality approximation indeed normal asymptotic design width true test method uci dataset us attribute community dataset response predictive attribute quantitative attribute predict response perform preprocessing replace community eliminate ensemble linearly matrix pn p normalize design equal evaluate various know whole clearly power active inactive validate take community statistical summarize result small report whole nonzero ridge regression subsample community description plot histogram width community type c cm cm par relevant par feature base cm subsample community rate type testing arbitrarily exist equivalently inequality last inequality imply thesis generality p jj basis recall I hypothesis random compare threshold desire let aa matrix inversion clearly converge measurable fig jointly square degree p p distributional converge theorem addition continuity eq depend use eq let side respect law random change expectation linearity take p distributional limit empirical p u weakly hence argument follow expectation gaussian design column along distributional limit eq side inequality get definition solve claim remark independent enough ensure least surely surely large particular sufficient uniformly dominate eq p cauchy eq py almost identity last bound I enough use conclude surely surely eq last argument ii claim triangular vanish assumption next next fix large equality let second operator whose proof conference convenience borel almost define proof law assumption eq derivation acknowledgement partially grant fa regard explicit formula convenience explain soft define effective restrict existence uniqueness prove appendix discuss tune achieve state soft explicitly mean give q theorem eq instance equivalent briefly compare zhang zhang paper eigenvalue author projection assess probability eq immediately low design get necessary condition define paper q correct cf decompose bias regularization second correct hypothesis jj paper negligible probability keep far plug standard design vector jk use large singular hence outline lead claim set whereby separable namely replica set factor unique hessian diagonal separable formally check previous establish analogous introduction motivation cf np np np np change derivation let lagrange replica calculation aim moment eq per convex first temperature eventually replica limit growth expectation evaluate expectation assume obtain limit hold strongly get n per duality complete quantity weak triple cf calculation use rewrite q gaussian replica aim fractional replica compute order represent slight abuse r p trace take identity identity integral integration kn z saddle replica saddle invariant permutation unchanged partition cf yield fact expression separately term obtain introduce eq next careful saddle point parameter show limit limit denote expression variable read must saddle start get understand saddle derivative cf assumption derivative statement replica identity follow integration part limit pa ba
idea locally bias semi supervise compute eigenvector solution walk algorithmic nystr om eigenvector consider exploit solution appropriate reasonably low semi eigenvector compute efficiently must accommodate exploit inverse solution eigenvector leverage treat lagrangian solution lagrange kkt identity efficiency eqn calculate ff gd ff td gd exploit since present exploit procedure well nontrivial eigenvector spectral graph locally alternatively heuristic work component problem regularization semi supervise modification occur seed local neighborhood seed algorithm comparison system even though ranking adapt notation define usual eq verify lead generalize subsequent eigenvector accommodate subsequent solution eigenvector systematically obtain consecutive semi supervise eigenvector eqn approximate apart eigenvalue eqn explain fact interpret eigenvalue eigenvector initial already orthogonal failure happen start constitute component pose fortunately detect general turn experimental general already eigenvector challenge solution localize algorithm control mass seed threshold localize eigenvector characterize span choice applicability implementation define choose project seed inefficient seed one combinatorial substitute seed eqn follow plain expression scalable maintain process process influence show queue result large scale semi supervise biased machine learn illustrate usefulness example model parameterize low grid allow section consider roll call voting united base structure illustrate clean study area substantial heterogeneity locally common smoothing construction fmri fmri characterize high semi supervise eigenvector spatially biased incorporate improvement equation solver improvement implementation challenge web non concept semi supervise eigenvector laplacian machine application biased region interest nice global biased locally bias machine conceptually involve problem extension property illustrate due intuitive application eigenvector wide acknowledge centre university fmri clearly leave eigenvector leave side perturbation orthonormal seed unit correlation seed bound orthonormal span directly plain substitute eigenvalue shift eqn trivially rank point unclear semi allow semi derivation seed write lead laplacian combinatorial laplacian rewrite exploit eqn approximate extremely manner apply laplacian notational simplicity sample nystr om extension approximate result matrix node nystr om lead correspond goal risk nystr om large normalize eigenvector laplacian gray pt information provide want nearby region cluster partition image truth bias sort challenge eigenvector tool eigenvector limit paper eigenvector perform successively correlate input seed manner semi supervise quickly linear describe basic several demonstrating semi supervise locally recent global eigenvector want learn task nearby eigenvector popular tend serious reason laplacian inherently quantitie locally bias nontrivial slowly mode fairly computable perform call graph region perform classification etc bias pre specify analysis belong cluster edge refine set find nearby member along ground pixel segment background automate imaging stimulus analyze temporal neuron nearby connectivity topology model construct feature supervise sense specify relatively label interested nearby present considerable challenge receive wide application recent reduction kernel machine nystr om spectral partitioning reason eigenvector inherently thus limit one interested essentially globally cut poorly eigenvector thus supervise example cluster near base dimensionality local might kernel methodology biased well seed make nontrivial eigenvector graph useful ideally usual eigenvector depend application able machine make eigenvector laplacian useful locally bias formulate optimization variant include locality constraint orthogonality solve possibly seed informally would seed analogous nontrivial seed semi somewhat algorithm return define successively condition seed compute quickly equation extend basic describe several easier supervised scale extension involve nystr eigenvector iteratively successive walk strong supervise eigenvector detail one generate network generation roll vote basic graph widely digit consist fmri medical imaging method technical work closely develop original locally nontrivial laplacian empirically social partition locally perspective orthogonality bias cut somewhat locally walks start find graph internet application cluster structure wide graph spectral objective eigenvalue asymmetric unstable calculation span binary multipli similarity vision application work usual global reduction semi supervise setting neighborhood around optimize go understand supervise interpretation usefulness range application many local context diffusion figure diagonal degree without helpful think indicator target region graph unit orthogonal correlated input seed application semi compute already supervise semi dimensional semi supervise consist eigenvector quadratic equality point equality variable matrix span take identity projection respect become term second great non linear return specify would evenly across correlate input seed assumption relaxed formulate generally correlation eigenvector nontrivial along eqn binary constraint wants supervise form weight supervise x ff ff ff g ff ff ff sx tx g g xx present code supervise think indicator set compute locality implementation projection I span thus residual solution equivalent exploit conjugate gradient explicitly treat simply fourth small binary satisfying eigenvector able eigenvector compute nontrivial eigenvalue project onto solution run algorithm natural interpretation underlie recall linear equation orthogonal vector regularize serve precise well run optimize correspond rearrange gs gs formally powerful regularization achievable locality constraint locality correlation important practical precise manner seed capture input seed alternatively choose eigenvector seed seed number nearby give must large supervised formally need regularization choose via nontrivial
sign find characterization design define loss generality hold even case submatrix away zero require singular away comparable gauss selector covariance gauss deterministic allow design row covariance selector population q hence convex assume noiseless empirical exact lasso albeit design population share sign observation allow gauss lasso selector given begin prove property population estimator problem sign lasso estimator sign gauss selector recover sign property former possess lemma one exist happen sign standard characterization far motivated deterministic condition estimator n ny define true q require low ii v analogous indeed randomness matrix readily eigenvalue much large hold proof note since gauss selector sign return gauss selector defer level magnitude gauss selector recover generalize result q gauss precisely let selector treat rate community attribute community univariate attribute community response quantitative include population operate budget selection gauss selector step non attribute eliminate attribute attribute pn whole show negligible truly truly inactive active community normalize gauss selector model selection figure gauss form path truly black path truly inactive solution active decrease mark removal variable current therefore support lasso gauss selector least square restrict knot lasso knot figure lasso support truly false positive positive false hand positive positive gauss produce gap positive negative width gauss pdf statement lemma sign also eqs sequel lower begin w onto complement space since normal union u condition u sign true condition tail defer section eqs thesis recall hold thus variance q hence nc cs modulus return correspond last fact q second rest devote validity true inequality hold future version read eqs place employ verify feasibility modify generalize begin condition vector uncorrelated var total probability bound obtain true claim provided prove x event degree chi square fixed bounding employ tail prove I present apply prove section q per latter eqs level proceed along line acknowledgement support stanford fellowship award nsf dms fa fu rest definition stationarity eq substitute fu fu sign v sign finally read particular subgradient problem note u sign show subgradient combine equation subgradient prove subgradient read plug equation arrive write second prove direction eqs satisfie define moreover equation let x get far I ni p state prove lemma strong let gaussian vector let shall hereafter define concentrate last b denote distribution e ie tu uniformly matrix haar eq condition uniformly sphere last inequality hold expression ib c eq c appendix discuss section detail validity explain covariance check irrelevant show broad lasso size correct sign unless correct q prove gauss correctly recover support versa fail regime check p substituting check c condition check since satisfy minus pt minus corollary conjecture fact replica claim rgb smaller active estimate correctly identify active roughly orthogonal relevant quantify two solve least weak gauss correctly recover linear wish vector denote consider set denote support namely true model explain body development computationally development pose ask case rest gaussian design computationally largely square minimizer one arbitrarily omit clear relate selector interesting generalize understand constant unless formalize state I yu prove uniformly independently particular parameter column require allow general proving detect covariance necessary minimum random design provide model selection fundamentally unknown selection computationally aspect measurement orthogonal distinguish upper design formally degree size value hand reason succeed unclear necessary characterize restrict isometry rip relaxation paper prove strictly random variate write component response I correlate relevant orthogonal th covariate recover follow long soon include covariate probability gauss lasso large cover rather limitation gauss selector isometry restrict compatibility similar partial namely recover mean factor
decompose mse write py py f b tp equivalent unbiased conditionally unbiased surely unbiased identification unbiased estimate tb py unbiased imply unbiased let compute identification almost surely sufficient mse py p l definition systematically assess quality obtained summarize assess surely give unbiased mmse imply bias mmse collective development analysis identification obtain trivial discuss mse trivial integral respect admit analytical issue mc sake find completeness computation mc j perfect efficient estimate unbiased mmse similarly systematic quality generate I py tm module identification density sequences module module denote compare simulated assess outline brief popular bayesian identification dynamic static evolve realize review though amongst limitation summarize artificial tune quantify former propose rule automatically tune develop quality sequence identification markov interest assess identification use mc simulation module amongst parameter surprising decaying suggest mse mse module involve mse see less early estimation difficulty point smc figure unconditional module assume theorem smc fail unbiased except unbiased mmse wherein simulation figure smc unbiased mmse unbiased unbiased mmse unbiased identification tool obtain popular assess ca decade sequential chain mcmc non model er lower bind square mse use analyse bias mse efficiency far back despite use wherein computational density marginal associate development smc technology allow line identification stochastic introduce semi interior partial order induce denote denote laplacian process let open conditionally marginal density respect suitable dominate lebesgue represent unknown measurement noise notation include signal bayesian vector rely estimation calculate respectively recursive approach next mutually class parametrize finite open x mf w implicit recursive q eq pz associate estimate derive mse associate bound tm give
alternative avoid cv nest cv double variance perhaps surprisingly cv large prevent cv provably advantage distribution term estimator without distribution cv much motivate explicitly maximize internal reinforcement maximize inaccurate biased speed learn multiplier tune manually relevant choice representation typically hyper randomness want good instance optimistic cv various hyper parameter estimate nest cv result latter exceed nest cv remove set general bias cause rest notational estimator I section discuss setting section resample result biased estimate smoothed nest perform normal nest cv accurate approach I far widely estimate framework find expect rather collect paper discussion collect minimize bias necessarily mse estimator measurable denote xx even useful state necessary strictly bias set index index define estimator call whenever maximal necessarily estimator admissible identity normal finite pdf concrete world rather perform bias reason generality distribution prove arbitrary family essentially smoothly value sample whereas piece wise everything necessarily suppose well high outperform hyper sound manually meta parameter non avoid meta implication evaluation real problem estimator properly tune overfitte non performance datum perform available actually since predict mean often collect whether positive negative crucially discuss I cv bias estimator discuss similarity low bias perhaps surprisingly bias variant large appendix end I possibly bias conceptually implement I theorem instance possibility discuss necessity positive bias optimal I variance indicate variable worst set discuss I consistent consistent weighted follow v conjecture appendix average bias many large imply high since bias cv high variance small variant cv leave positive I unbiased trivially I conjecture slowly multi armed bandit website ad exploitation return quick accurate estimate important place ad may induce quickly mse average experiment bar fold fold fold cv fold fold one fold simplicity ad return click click rate model bernoulli ad ad equally unbiased mse variance click rate evenly ad plot unit depicted contribution deviation directly error cause indeed low cv ad I whereas though leave negligible interestingly correspondingly stay leave go go accurate variable relatively iid fold denote noisy input denote fit square noisy set inner cv loop error e ip ix p ix biased plot sharp previous good I cv choice far I cv guarantee positive accurate recommendation fold especially perform decrease bias illustrate set fold fairly biased note fold accurate try cv indicate likely often I cv estimate I furthermore recommend course possible counter I guarantee penalty compare parameter estimate select goal relate equivalent bayesian estimate seem reasonable small therefore cdf approach I positive even I uniform maximum skewed increase shape analyze bias expect cv preferable variant cv set extremely inaccurate
solution cut method available kernel p different kernel bandwidth practice cardinality dealing allow theoretical using prove distribution domain boundary usual density go polynomial compare available well base thresholded kernel importance sampling summarize provide inverse problem establish connection integral rkhs principled separate regularization simple kernel algorithm regularize support manifold bound rate ratio comment kernel potential extension artificial alternative completely unsupervise usefulness address unsupervised semi finally allow different area shift estimation integration connection hope estimation rich estimate density extensively old include estimation deconvolution year ratio transfer learning transfer covariate shift brief satisfy covariate easy covariate shift closely setting rewrite minimization problem write feature e recall equivalent eq identity hilbert type different rkh empirical sample experimental hard nice setting another square density choosing function distance density kullback unsupervised idea need body kernel inverse framework estimation density well estimation formulation geometry set literature integral equation regularization development function g kx rkhs key fx allow write combination discussion theory relate norm operator p approximate make precise type linear important keep notation refer solution computationally sample perturbation identity analyze use functional inverse problem apply learn appropriate reproduce hilbert write algorithms type combine evaluation sample q path every I generally eq algorithms formulation compute norm want benefit first derive sample still summation rkh regularization norm use formulation applicable function rkh problem formulation loss regularize arise unconstrained center sample rkh type integral obtain analytical type ii sample similarly kernel bandwidth type difference briefly kernel lead may certain advantage compact adjoint eigenfunction method spectral cutoff span eigenfunction subspace large going detail take eigenfunction diagonal matrix spectral require eigenvalue need regularization potentially eigen type appear restrictive important difference applicability ii integrate absence domain problem e involve possibly constant impossible unlike depend kernel essentially case norm available type result convergence regularization type rkhs gaussian modification basic whole case satisfy also require certain p pf set type regularization tx sec solution width require number eigen follow assume least space apply sub set compact ii point assume satisfy sufficiently confidence moreover dimensional sub manifold dt along adjoint complete h h triangle f give pf typical estimate immediately put two lemma q constant complexity procedure necessity choose significantly exceed due suited classification use splitting repeat obvious need regularization grid range width experiment bank fm news datum point apply resample scheme point set resample feature label information along subsample define follow sigmoid resample scheme pca aggregating class validation collection avoid denote function use digits set procedure usage result measure function number cross qx fold error fold fold performance x jx set kx jx space half space experiment compare method set experiment unweighted weighting scheme square estimate diagonal c c c half space linear half half half ol half half ol c c label c weighting method linear space linear ol classifier building weight ratio ratio completely also performance training change whole validation subsample classifier performance term prediction hand write class c space half linear c c linear half space half space deviation project principal class c label half space half space space label c weighting method half svm density datum experiment vary two method experiment difference estimate suppose known intuition behave method column estimation kde middle method right vary fix illustrate repetition different norm kernel close penalty outside interval uniform gaussian rkhs ht rkhs width know acknowledgement grateful wang valuable suggestion point frank journal unlabeled journal international conference page convergence journal american de inverse journal system page smoothed covariate matching shift machine page gr embedding regressor international conference page j review paper estimation american david nearest neighbor economic least direct estimation journal learn speech international conference page equation volume liu rejection estimate functional advance survey j machine optimization mit yu operator predictive covariate weighting journal machines laplacian adaptation weighted journal machine von neural information processing system taylor page practical distribution international conference early constructive yu ari covariate evaluate international conference page rkhs kernel define bind need fourier transform f isometry transform use transform sd similar definition consider manifold spectrum chapter laplace discrete condition denote volume also definition definition equivalence need eigenfunction independent proof implication thus cn give lemma rkhs unique te e mi mc nf proceed formula tc third enough q p suppose density satisfying give follow twice bound know tx txt need integral space projection space sufficiently still thus dy formula give let satisfy tx identity follow identity identity large simplicity dominant everything together proposition exercise ratio know average another closely transfer well method geometry say integral correspond reduce lead principled algorithm theoretically flexible theoretical analysis compact domain sub euclidean include covariate shift encourage experimental choose useful rich subject review parametric go back paper address estimate ratio another attempt integrate value typical equipped know robot perform robot
span thus see duality admit continuous compact multiplier u u plug function due derive difficulty fact u contradiction thus p know v maximize thus know strong duality theorem exist q show duality lemma problem minimization kkt problem plug eq divide p imply condition plug follow note du complementary view q lead contradiction du contradiction complementary condition expand hand imply view p imply lead consequently kkt hold slack assumption p plug e constant theorem divide otherwise note prove case clearly apply view complete prove theorem compute q observe therefore p therefore show b fact plug eq complete sparse simultaneous feature selection recent effort devote implementation pose significant effective logistic screening substantial optimization need negligible solving thus evaluate extensive screening solve logistic magnitude logistic lr widely mining bioinformatics medical compare reduce regularize lr challenge lr last grow due high dimensional datum lr equivalent regularize scale lr high accuracy challenge promise inactive substantial cost al propose accelerate lr safe elastic net lr rule rule lasso base safe special discard safe mention rule discard sphere test easy lr safe rule lr safe model call screen inactive upper inner feature lr accurate inactive detect accurate quite challenging insight safe heavily rely lr presence contribution upper bind admit close safe computational effort strong rule safe feature safe spaced tuning please rejection discard screening coefficient rejection effective discussion review regularize motivate rule via kkt fy yy yy notation form unique supplement kkt condition view imply optimization however general applicable assume result j word serve foundation rule screen rule discard need restrict show estimation via optimization admit derive novel framework dual I rigorously become easy belong tool theorem become strict see q inequality note kkt open j j k eq radius screen discard tight upper please thus restrict region feasible optimal empty orthogonality absence imply j feature discard solve rigorously feature substitute full hand input yahoo web page set et yahoo include set computer education science equal number sample statistic computer education science c safe strong regularize lr discard datum table test sequence spaced report run time run strong safe long rejection e discard measure screen discard rule data rule scale report implement matlab ghz processor experiment rejection fig rejection six identify inactive inactive contrast exhibit strong capability discard inactive inactive identify strong mention discard strong discard coefficient cm efficiency set include computational solver screen plot running time rule feature size optimization greatly fig solver identify inactive solver without inactive yahoo page efficiency roughly inactive improve efficiency effectively discard lr art formulation fuse regularize convex one like theorems corollary text sample associate label problem take whose consist slack formulate lagrangian find subproblem
cycle ml matrix edge liu briefly discuss idea refer node example give full cycle figure node graph context characterize graph small set marginal employ time prove complexity fully feedback edge every order node node feedback subgraph feedback long give minimal size author yield complexity factor explore learn sized suffice recover feedback either variable empirical matrix covariance slight set distribution node ml exact ml combinatorial span solve liu describe extension enforce intuition though tree feedback feedback tree liu also property simply whole complement liu obtain inverse structure exactly feedback node feedback feedback matrix among feedback proposition complexity proof f compute ml estimate computationally involve large unknown possible find hence select well arbitrarily algorithm extremely practice df feedback true learn structure distinguish thus without ml latent divergence node instance general onto observe latent whose maintain latent one distribution family clearly relate projection allow among feedback j project fit structure liu projection exhibit complementary projection information remain projection two interpret correction second expression intuitive iteration bottleneck invert carefully projection exploit power reduce per liu number per due accelerated version proof never accelerate rule span liu note jump chance get bad structure experiment section present experimental synthetic delay fractional motion latent brownian define span learn liu trees learn decay distance poorly learn exhibit span model learn node k achieve empirically proper sensitivity structure converge structure give divergence show nj latent visual clarity blue represent feedback red tree model examine observe span feedback information also generate identity generate draw run successfully obtain feedback size delay among delay come arrival arrival delay model delay first average day use note delay whether major traffic interesting delay correspond node learn figure liu average delay learn select air reason approximate span tree exclude break cycle span tree start select greedy begin order dc city several major influence delay well capture result demonstrate suggest lead provide specify incorporate direction work extend setting support state compute pass algorithm describe run correctness complexity give q singleton distribution marginal distribution p px I proceed first equal step f j f j prove correctness complete proof twice minimum complexity exactly quantitie p divergence distribution confusion omit slight abuse distribution conditional liu summarize follow topological order root keep ii jj find graph iii ji span know rp fix tree feedback node minimize span feedback f fix use arbitrary node verify equivalently j I expression l divergence gaussians verify calculus implication respect neighbor zero mean px px coefficient equal invertible find book fix span among span induce invariant optimal run liu input accord hence define among node weight span reduce next entry multiplication regular since q complexity extra easy summarize compute h j sparse prove b propose projection fit structure variable q q due necessary remain complete proceed liu f j f expression exactly liu proposition accelerate accelerate liu liu complexity main due accelerate liu complexity liu computing check multiplication accelerate complete proposition j repeat ty ty partial covariance proof node edge additional check feedback feedback edge node prove need ab b submodular department technology institute technology department institute technology institute technology graphical gauss field trade modeling
graphical describe explicitly relate undirected facilitate regard demonstrate reverse sufficient condition equivalence markov property reverse rule equivalent intersection composition view axiom accomplish result undirected notion duality undirected close closure since intersect intersect yield contradiction implie show whenever b ba sa complete singleton disjoint inductive sa sp yield contradiction without generality inductive b sa ba begin eq reverse composition begin close disjoint q complete claim induction also claim q prove closed assume prove claim section thm thm thm et analyse universit prove dimensional lack represent independence another encode dual except instance duality duality proceed extend previously prove important domain property weak intersection composition reverse concept duality duality relate familiar intersection composition understand implication statement graphical independence statement example undirected concentration conditionally setting use rule independent model equivalence global concept model independence statement statement separation statement occur say globally markov respect disjoint separate subset graphical encode reverse within encode specify pairwise tree graphical model dual formalize frequently obtain parallel undirected graph use proof could formalism develop result introduce investigate undirected graph rule use adapt graph result relationship composition formally generalize significant preliminary notation general take pairwise consider reverse rule detail investigate extension graphical language relation formalism preliminary closure rule relation independence relation however motivate example relation disjoint respectively say cp ambiguity variable bc bc bs bc cb random axiom b ab henceforth satisfy intersection bc sa bc admit intersection relate sequel conditional define triple nonempty subset vector index relation v disjoint henceforth closure rule translate closed rule proof compact relation statement definition parsimonious convenient proving concern variable condition triple triple specifically statement set triple technique weak rule closure axiom imply closure weak contraction union remain counter contraction relation axiom consider detail set vertex say say connect connect undirected disjoint intersection composition large undirected write un un bi bi construct eq think consistent notation undirected terminology pairwise encode context undirecte random variable undirected graph undirecte e un e un ab bi bi bi widely reference pairwise global respect ab language markov fix respect minimal satisfie undirecte closure intersection undirected markov true relation respect assumption composition concept duality result graph four rule rule reverse also b c reverse sc reverse version composition equivalence reverse composition sequel closure de ab c provide random notion facilitate introduce duality tool heavily sequel triple duality detail class graph note undirected dual vice property pairwise sense undirected graph guarantee closure closure allow closure pairwise property examine develop condition pairwise typical assumption equivalence rule intersection undirected composition graph rule lemma either weak intersection begin graph equivalence pairwise property graph treat differently undirected completeness markov relation pairwise undirected composition separately reverse logic graphical modelling relation choose pairwise begin equivalence originally consider result technical instead closure undirecte pairwise respect undirected global complete markov oppose demonstrate vertex undirecte close reverse rule place original composition rule proceed concept duality much technique way subtle proof dual sense reason provide contrast brevity duality reverse assume induction begin pairwise b theorem either generality inductive b v sa ab ab ab lp preserve outline see pairwise q undirected markov therefore statement duality global examine rule use relation undirected graph solely make reverse relation restrict closure rule reverse composition right one simple weak currently relation parsimonious closure intersection closure composition closure closure reverse define definition direction rule clarity implication letter definition equivalence satisfied denote fix disjoint eq finally induction disjoint claim inductive q complete claim equivalent intersection closure composition rule reverse intersection full close intersection apply composition reverse direction resp reverse intersection reverse weak rule intersection resp composition closure intersection closure however hold closure rule converse intersection imply rule converse close reverse intersection composition latter rule intersection composition along rule result global reverse inclusion say place literature inclusion say triple encode undirected highlight trade graphical independence undirecte weak markov relate tree relation
independently nx yield contain sampling analytic intuitively concept ambient coherence regime understand whether flat flat nh therefore coherence htbp close minimal coherence flat linear coherence show show contradiction use frobenius general tight frame extend coherence arbitrary manifold minimize figure b n real projective tangent flat q clear contain ambient system agree bound follow analytic concept recall coherence smooth flat equivalence statement suffice show least orthonormal consider variable q absolute eq open absolute proceeding statement manifold algebraic restriction going consider dependency algebraic make canonical transform analytic manifold piece together manifold proposition maximally incoherent behave restriction summation compute specific example flat prove unitary irreducible space infimum xy definition coherence symmetric matrix symmetric hermitian matrix hermitian embed variety span resp hermitian calculation statement particular maximally incoherent fact row span column span span exist low equality h low combine coherence variety relate notation embed call explicitly keep argument normalize x r contain symmetric diagonal subset map set irreducible algebraic respective range similar map nm r proof q hyperplane branch square n n coherence singular replace row span reconstruction calculation usual kronecker converge proposition function namely coherence interpret average whole set analytic yield kn theorem framework present broad investigation ask remove dependence keep mind result kind sparsity line fraction scenario ex give formulation problem analytic bound geometric measurement ambient derive low matrix compress recover acquisition process usually undirecte comes call come easy reconstruction question sense well known roughly frequency density least order reconstruction imply rate interpret average non sense much contribute compressed literature usually analyze theoretic threshold restrictive argue bound principle sense formulation two novel manifold random dimension sampling ambient coherence general sufficient show near analogy capture constraint appear independently probability term example compress consider contain always paper setup impose restriction signal fix limit represent dft matrix statement probability observation probability least definition simple showing rank mn obtain let q reconstruct make theorem apply low analogue hold symmetric show far either exhibit distance completion describe density need reconstruct incomplete describe theoretical asymptotic attract lot except point coherence nc rate constant reconstruct
widely use search subset include guarantee markov ci relation encode et require ci relation sensitive ci statement quite restrictive especially graph undirected cycle skeleton case cause relaxation respect dag satisfy et al triple clearly restrict significantly algorithm adjacency necessary sufficient skeleton pc adjacency neither necessarily skeleton correctly orientation infer attempt make modify adjust weak condition ultimately lead claim discovery scoring search high typical challenge search dag algorithm develop dag function mean score parent node scoring criterion decomposable attempt advantage constraint hybrid closely infer skeleton ci perform skeleton prefer sparse maximum restrictive method less error unclear weak condition propose score method equivalence search scoring denote edge ci infer observation sp weak condition connection effect sp constraint base ci relation sp testing sp pc sp equivalent cholesky sp noiseless penalize van oracle markov penalize sp hybrid ci data fisher z transform dag true skeleton frequently bottleneck permutation require confirm sp weak base guarantee weak dag rise partial permutation dag determine skeleton dag assumption satisfie mean sub vertex section satisfy lemma suggest select permutation yield small number parsimonious dag skeleton vertice permutation amongst permutation amongst sp dag dags presence failure sp note single ci test ci relations flip testing rather correct equivalence paragraph since sp decomposable search involve search set advantage heuristic search use sp paper guarantee exhaustive search sp choose restrict class already suggest weak underlying dag satisfie sp sp algorithm determine dag satisfie necessary sp sp satisfy assumption dag dag unique dag assumption fact score consistency absence use ci present result satisfy choose edge denote order output fail sp dag weak restrict condition imply algorithm find restrict follow restrict cycle ci x construct condition hand permutation permutation produce edge would permutation satisfy assumption disadvantage compare sp sp exploit remove edge cf far aware consistency strong much graph theoretical result sp pc remove sp consistent metric fail recover recover particular relate distinguished type error lead failure error dag dag zhang mean triangle make dags triangle skeleton failure triangle failure sp output dag markov equivalence one illustrate cycle ci ci relation x triangle zhang make hand algorithm would dag cycle sp output equivalence sp formalize path triple connect satisfied conjecture path assumption expect satisfie condition single path assumption every unique markov equivalence class include sure satisfie check assumption section comparison one refer encourage dag fewer weak assumption assumption sub dag respect weak dag satisfy prefer precisely two markovian separation separation dag entail strict super ci statement dag dag prove satisfie dag satisfy also dag dag dag satisfy assumption dag contradiction markovian dag first identical generality exist subset identical exist separate cycle ci relation x x dag belong dag satisfie converse dag satisfy encourage determine equivalence ci constraint explain infer ci sp apply pc x construct imply ci reject brevity present testing gaussian main apply ci test ci x z fisher build jk jk jk complement accord consequence ci sp rate hypothesis error ci ci relation estimate dag least algorithm recover skeleton recover infer ci recover make infer ci true sp recover skeleton I recover illustrate assume type miss ci relation infer ci sp recover algorithm illustrate type infer ci relations ci x extra relation analyze miss edge pc arise sp outperform uniform pc true assumption lead failure overcome zhang assumption ensure pc strong zhang line guarantee uniform sp replace mutual state assumption assumption define extend ci discuss assumption strong dag satisfie respect ci relation satisfy respect ci since strong ci strong strong uniform consistency sp exist sp provide denote partial correlation chebyshev distribution jk jk delta distributional large jk hypothesis ci relations ci assumption recover probability sp ensure consistent assumption weak sp vertex dag dd jk positive measure everywhere structural express I upper cholesky cholesky definite unique equivalence encode ci permutation equivalent cholesky decomposition every permutation k gaussian permutation cholesky p diagonal setting algorithm sp inverse cholesky cholesky np complete review establish cholesky matrix sp equivalent discuss n triangular entry min estimator correspond dag belongs generate equivalence approach reduce r sp oracle penalize estimation weak assumption suggest sp result small dag ci relation infer addition hybrid algorithm pc package package simulation study conduct realization dag neighborhood ensure draw ci relation fisher size empirically algorithm finding simulation figure display proportion skeleton neighborhood skeleton sp unique sp make comparison favorable sp proportion simulation skeleton algorithms skeleton miss figure sp recover true skeleton pc due pc support finding pc pc algorithm tends often edge sp algorithm dag increase fully dag tendency simulation result compare performance node consistency scalability thorough must sp search computational resource compare sp develop efficient search permutation remain algorithm distribution sp equivalent cholesky matrix penalize likelihood require check believe cholesky factorization feasible efficient sp like consistency parameter uniform algorithm weaker strong study strong compare min also condition geometric proportion involve also combinatorial markov ci hold x ss x x contraction axiom ci ii axiom page ci relation follow induction correspond q node non would intersection axiom follow induction contraction axiom ci relation contradiction edge assumption contradict markov separate order consistent dag sp sp complete
different kernel neural radial research et al kernels discrimination specialize conjunction al series model autoregressive dynamical input design kernel time method wind use experimental current identification cause measurement cycle ten independently sample estimate ten consecutive superior perform assign otherwise rejection use capture sensor dynamic array array drawback measure medical costly consume application discrimination refinement building qualitative metric temporal symbolic sequence sequence receive time sometimes monitor classifying memory process recent directly time stay sometimes associate view base classifier incomplete sequence et reasoning classify knn classifier classify various distance dynamic reasoning occur percent length recognition identify show treat early classify al challenge trade classification base partial address short prediction make result action early classify feature temporal symbolic sequence feature frequent select association rule build incoming match branch way accuracy achieve handle symbolic competitive length full disadvantage handle discretize online feature time learn clustering use guide nn without accuracy classifier achieve maintain nn length early although essential different identification entire early treat cause presence despite progress issue art take record process offline amount e label assign question trust far automatic system reliability present accurate classifier reject option classify take discriminant yield rule assign class posteriori difficult option allow third reject discriminant report report pattern correctly classify accepted cost wrong utilizing reject pass lack decision never discriminant minimize reject option view misclassification introduce real early signal come portion signal memory reliability first signal continue reliable make cost decision besides e propose novel threshold decision threshold dependency stability reliability something go wrong wrong entire address classifier agreement necessarily output knn furthermore advantage e voting threshold consensus play role reject outline build build diversity high diversity accept agreement decision accept circumstance stable make diversity among produce pool pick diverse svm rate diversity measure classifier pool least intuitive diversity classifier incorrect misclassifie measure know thus incorporation propose note time array represent focus wind subsection sensor protocol wind principle sensor reasonably say chemical spatio temporal ambient problem address discrimination identification accordingly utilize array endow record wind de surface surface induce measure operate empirical e computer sensor repeatedly candidate increase ii surface maximize iii optimum equal mid admissible chemical piece identity source observer localization individually process recognition chemical recently utilize sensor array module endow discriminate wind comprise utilize location call wind figure set induce ten chemical source regardless module collect wind platform wind field construct adopt protocol first artificial air flow wind fan us wind reflect response record chemical sensor environment air allocate wind module room perform air flow ambient quasi wind indicate figure hold minute start actual constitute preliminary utilize sensor chemical wind reflect sensor response measurement channel represent time sensor minute store repository remove wind open one subsequent measurement chemical pair cover hz main demonstrating real rare symmetry volume direction collection evaluate strictly symmetric early observation window series evaluate unseen also calculate series threshold wind svm classifier vary run classifier investigate strength drop series early particular second show x location wind encourage superior relate time order varied report average location wind comparative respect optimum std new classifier reject option ensemble accept candidate rely posterior propose discrimination focus two issue recognition classify grant pn ii organization system early environment challenge great importance signal process architecture reject capable decision without entire acceptable accuracy classifier use decide accept reject apply build experimental wind confirm device intend decade sense important development sense refer reproduce human array machine international instance risk exposure raw material
overcomplete prescribe basis estimate accordingly constitute tailor cf theorem kernel expansion iy lasso cf b iy iii effect capability select setup correspondingly drop yield define identifiable particular lemma design namely complementary multiple pursuit separable mkl spectrum illustrate utilize frequency version prescribe entail imputation entry entry available low popular relate imputation imputation achieve solve hadamard rank correspond vector singular convex motivated norm ball hull value norm transform place constraint term step base rely alternative bilinear implicit nuclear eq attain singular unitary formal equivalence factorized reformulate respectively entry prescribed estimate family fm n via kernel rkh correspondingly upon lemma equivalence generalize completion regularization term enable optimal scalar solve entry framework entry identically factorize q ci column priori r remove ambiguity index c solve estimator provide coincide completion across smoothing completely rely available reconstruct capability rate enable user preference item bayesian explicit completion summarize algorithm solve identify solve change variable algorithm randomly identity dimension b detailed derivation high solving guarantee minima transform convex imply global method alternative low constraint trick nuclear imputation basis pursuit overcomplete basis cope signal extensive need prescribe next basis learn plausible overcomplete unless exploit constant need represent collect ensure determine enable mn c via blind regularizer measurement attractive completion flexibility capability cope basis coefficient jointly lie require span top atom replace column eq interpretation bring close bernoulli account across time sample model generalization across amount although dictionary blind capability recover recent dictionary dictionary designed distribute ambient psd sharing obtain psd specify wireless propagation simulate accord depict distribution two show psd representative ht model adopt collaborative represent basis prescribe accordance measurement via combination consider candidate mkl intend capture resolution produce correspondingly decompose function psd fig precisely mkl reveal estimate row ground depict third row multi resolution depict fig two capture distribution affect usefulness spectrum sense basis serve frequency band psd map compare spline mkl adaptation resolution capability capture imputation test microarray gene point cell cycle extract expression level organize matrix depict loss discard entry actually extra microarray instead cell across alternatively form microarray gene aside place depict miss db produce capability recover recover present illustrate cross validate miss knn package discard miss recovery remain db db ht comprise utilize load aggregate collect column predict hour periodic day correlation fig training traffic depict reflect sharp notice e z f z z fig representative link record sample day base comprise link yield aggregate correlation pm benefit interval pm pm traffic away add valuable information outline cross sparsity signal processing learn beyond regression nonparametric counterpart possibility contribute effort include blind version viewpoint interpolation suggest impact large selection research impact illustrate diverse property fed ideal cutoff frequency hence apply nx fx nf design rewrite cost I I b discard column hadamard product use identity product apply yield gradient follow reduce derivation interpolation view point advance aware nonparametric pursuit leverage nuclear dictionary novel toolbox beyond counterpart possibility selection impact illustrate cognitive microarray imputation traffic reproduce estimate variational rkhs connection shannon involve alternatively spline rather present see viewpoint rkh estimator coincide gram field krige rkh interpolation finally gps define covariance yet increasingly popular processing completion datum organize due limitation build assertion amount sampling theory constraint interpolation incorporation priori recent advance signal recovery motivate sparse learn core present signal least lasso version compressive sampling norm regularizer induce regard additive modeling collaborative filter tool limitation contribute cognitive sense management user bioinformatics forecasting price load wind remainder organize review describe kernel trick present shannon deal mkl nonparametric basis capture general framework blind dictionary vi present test real simulated traffic conclusion technical defer review place scheme denominator reproduce nonparametric select specify q exhaustive hilbert space equip kx h sense nice simplicity compound large around addition term smoothness reduce substituting coefficient regularize n stand loss l cost serve error hinge serve non angle hand describe unknown base predict point krige mse z z view rkh appendix elaborate gram eigenfunction norm eigenfunction use trick show rkh establish eigenfunction unless reconstruct alternatively theorem fit possibly account mkl nonparametric lasso additive introduce henceforth generalization deal fidelity nearby proximity point curse demand hypercube hypercube motivate namely form depend entry problem affect curse additive amenable spam learn yield expansion ix ni spam expansion solve ik k weighted formulation linear block descent multiplier convexity non differentiable vector separately identically gain rewrite exceed focus minimization substitute minimize k univariate
share draw put label follow assumption traditional pac feature partition draw identically still edge independent draw drawing error matter assumption hold movie rating realistic assumption rating probably movie randomly participant member ask list movie movie way sample concept predict unseen define l local integrate idea find minimization measurable directly independent candidate approximate measurable find hypothesis subset banach erm evaluated decompose lf lf lf lf approximation concentrate challenge erm intuitively bring must small usually measure covering capacity erm sake decompose f f state notation cover cover compact hold error bernstein inequality estimate relate relationship training dependency training example adjacent satisfy exist author average equal learn weighting rely fractional example mix use usually example regularize restriction relax call condition regularize least establish present less bernstein inequality different distinct check satisfy certainly interested reference mix author represent bad cause exclude use correction bias hypothesis test seem plausible apply testing training due directly key large higher induce find hypergraph graph independence find dependency maximum independent equivalently match effective practice propose weighting allow hypergraph weight nonnegative denote denote hypergraph define linear call linear program form mention constraint interior weight hypergraph show equal size hypergraph match hypergraph weight weighting define new empirical weighted sample erm use prove bernstein independent train function say concave k concave hessian express calculate feasible k eq important estimating analogue bernstein sample satisfy necessary define weighting arbitrary inequality everywhere taylor choose prove mean satisfy almost weight erm associate discuss erm aim empirical risk f f take erm approach measure excess risk excess divide part error follow error vanish z end error lemma establish erm theorem bound q bound md u nf addition fact union bound complete everywhere hold statement detail banach unique erm sample another deal initially propose solve ill pose ill condition inversion obtain algorithms paper bind ignore analyze large weighting computable weight assess algorithm bernstein statistical use well exist independence occurrence vertex influence task author grant graph base author however share piece propose show well previous example bind label sample take I call assumption consider hold set interested predict ask movie see past new draw newly introduce movie past movie independent since
right bad input inequality coincide indeed lead uniform however minimum algorithm minimizer guarantee coincide distortion illustration motivate noisy tackle deconvolution organize present method standard numerically deal conclude problem thank indirect set deconvolution suggest deconvolution inverse deconvolution deconvolution notation plug deconvolution estimator deconvolution finally minimization performance study uniform excess distortion follow integer partial derivative satisfy ensure consistency depend smoothness deconvolution instance inference see assumption use base quantity variance process density allow propose choice choice bias open point want minimize empirical noisy indirect deconvolution distortion noisy hand indirect consideration corrupt data additive measurement order condition sequel denote center whereas cell result direct indeed necessary minimize distortion dirac propose dirac deconvolution remark integral equation condition nx deconvolution estimator first distortion directional along define denote cell dx bound function convergence exist nj spirit figure iterative noisy enable sample noisy algorithm deconvolution direct corrupt iterative step purpose estimate corrupt product consequently natural f I fourier deconvolution estimation build grid use repeat assign diagram close direct programming dimension fast fourier adapt computation multivariate deconvolution compute discrete dimensional discrete transform equation fourier th th stand density iterative evaluation algorithm choose highlight important phenomenon inverse phenomenon different usefulness deconvolution deal experiment discriminate separate corrupt increase illustrate section appear good section affect algorithm highlight simulation density tu concentrate noisy mean eq mainly cluster risk realization first show well lack error performance source explanation comparable level vertical I contrary interested fail problematic number failure big total run failure exceed illustration behaviour error run noisy outperform job mean explain last confidence mean risk highlight mean indeed ic separate law diagonal decrease experiment purpose run realization performance show detailed explanation evolution seem contrary situation study fail performance failure problematic mean failure big c precise illustration two run mean run run mean job seem fail explain convexity deal noisy seem clearly show deconvolution gaussian vertical noisy interesting spherical gaussian deal deconvolution mean design calculus deconvolution distortion deconvolution indirect counterpart deconvolution extensively two fast simulated various phenomenon separate presence noise noisy deconvolution moreover noisy suitable spherical gaussian algorithm convexity popular initialization affect due dependence paper tuning bandwidth practice available propose choice law need inverse deal repeat measurement omit progress easily available nice work highlight mean interesting dataset variable detailed experimental eventually argue could classification definition deal tool inverse machine community two deconvolution mean deconvolution transform cloud quantization decade life error occur social survey process medical chemical physical diagnostic nuisance
define equivalently robustness trajectory robustness def interpret algebra induce topology value apply robustness model obtain much semantic tell much degree number indicator intensity conditional average average equation goal extent descriptor distribution synthesis reaction constant semantic formulae behaviour biological implement applicability blue straight robustness average vertical four list specie constant parameter value table reaction ode stable steady state model system depend two equilibrium close boundary evident express state value unit formula linear secondary equilibrium statistically value confidence threshold trajectory cross behaviour robustness hence carry stress compare derive easier investigate behaviour degree order vary threshold correlate dependency follow sigmoid curve case evident varied threshold robustness degree estimate stochastic hybrid gene event production degradation bind maximum grid compute new add thus change termination happen improve experiment time use radial range robustness observation experimentally monitor deviation combination optimisation specification possible time obtain unit score range robustness evaluation run score learn behaviour formula heavily partially never author temporal delay attempt filter logical specification save duration indeed intrinsic parameter robustness score range show obtain flat robustness vary near expect parameter robustness number investigate extend formulae setting discussing probability alone enforce optimisation art optimisation reinforcement remarkably propose briefly formula goal find machine formulae goal like deal curse dimensionality ucb work use concept address problem relatively new line research exploit smc tool adopt plan multi optimisation objective interesting combine address possibility address design state partially nr thm thm thm thm thm thm reason ability inherent biological formal relevance modelling checking problem behaviour logic may occur capacity system perturbation change verification recently notion logic distance trajectory dynamical interest system discuss show robustness indicator combine address optimize order specification single inherently inside specie instantaneous model markovian discrete interval concentration least specie take model hybrid biological formal probability logic may stochastic verification answer question use operate despite importance difficulty formulae temporal quantitative measure yes answer logic true deal model notion determine perturbation nature issue arise consider address question deterministic verification several notion provide suitable definition trajectory property logic logic semantic allow capture whether satisfied robustness clearly yet paper provide robust approach check logic formula logic particular formula robust indicator example indicator probability goal optimize maximize indicator introduce material quantitative semantic temporal experimental result robustness formulae choose semantic system work discussion process object kind internal interact classical genetic process network social describe formalism reaction variable count entity specie specify description change reaction reaction give population transition specie derive generator see recall force simulate standard construct semantic term ordinary differential ode flow obtain know suitable dividing system size intuitively ode population stochastic situation case specie approximation give result genetic network explicitly machine strategy continuously keep reflect convert flow modify remain event model term piecewise process continuous dynamic consider mode identify transition evolve field variable happen exponentially time constant jump continuous update see population continuous logic specify formal logic logic characterize pattern extend semantic interval logic parametrize predicate role atomic proposition provide semantic return semantic recently et algorithm robustness analysis
least hand imply resp decrease zero class affine smoothness curvature square show inexact class able grow square develop speak measure e g interesting approach arise check arguably fundamental simple cauchy attractive describe property efficiency determine computational various gradient loss rather structured instance logistic regression correspond logistic exploit provably method see lemma claim algorithmic inefficient therefore much interest inexact gradient result convexity combine error linearly non cover necessarily fit much algorithm convex sample output formulate predict simple approach require hence nevertheless difficulty one strategy make full iteration incremental update formulae size reference choice descent guarantee incremental method form typically size sublinear descent e per incremental gradient convergence inexact minimize formula possibly fall framework discussion behind gain crucially vector many convergence condition rate et incremental aggregated linearly quadratic develop average converge strongly result require front linear constant step schmidt square error norm decrease convergence strongly work study well sublinear another work establish require strongly instance satisfy zero asymptotic convergence structure regression note error subsequently weak norm linearly yield away asymptotic sublinear convex include norm decrease result extend case objective function develop global powerful framework allow rate manuscript aware author error bound non feasible method mention early restrictive context minimization apply globally convex analysis convex apply wide loss convex satisfying assumption function continuously differentiable setup sample express logistic arrive go note strict necessarily imply convexity full fy ex concern imply finite invariant optimal arbitrary imply desire use inexact iterate goal possibility simplify exposition step equal first convergence behavior possibly monotonically error difference two successive iterate term sequence q immediate proposition random corollary since quantify would intuitive measure disadvantage namely consider gradient condition relationship towards problem strongly satisfying condition find exist convexity ex equality ex ex inequality b scenario strongly convex satisfie fall scenario proposition establish recurrence rate hold initial iterate deterministic sequence consequently first verify scenario sequence consequently realization depend realization e equality derivation mean q fx fx k fx x fx k rearrange immediately corollary show value decrease objective necessarily automatically translate set l fx e establish complete norm play role rate converge proof find consider sublinear sequence satisfie resp resp suppose sequence linearly inexact method schmidt schmidt way problem sequence norm k secondly analysis function sublinear
uniform rp environment though instead consequently environment experiment policy distribution problem feature scale lie lie reward state begin one face stop turn point outcome ever close state less terminal repeat domain initial distribution policy varied episode determine report run include direction west east north direction grid partition corner upper right elsewhere state state corner elsewhere obtain episode episode vary final experiment mark empty place mark row obtain location symmetry feature horizontal diagonal horizontal exactly triplet horizontal line exactly number direction belong player function result learn obtain game x player experiment report average run tb cm clarity figure compare entropy omit presentation perform show environment htb cm cycle number cycle policy induce improve contrast approach map compete approach less slightly also game opponent opponent show opponent converge opponent minimax expert explain try find make expert consider try expert outperform intuitively never fail policy curve opponent extremely robust environment outperform sophisticated illustration demonstrate fact cost support fully prior map approach inverse reinforcement inspire avoid bayesian inference dynamic markov observation interesting simplification policy optimality present reward algorithm analytically approach computational method alternative try nonlinear reward respect apply play play extend reward acknowledgement paper partially project problem act stochastic environment games agent task agent extend probabilistic learn simplify probabilistic utility posteriori prior result reinforcement act markov game underlie environment opponent type useful particular application library accumulate year reinforcement principled play expert strategy learn agent infer always trivial dynamic reinforcement inspire reinforcement know environment algorithm extend original unknown probabilistic policy dynamic programming maximum avoid estimate eliminate dynamic estimation scheme broad acting environment agent play use formally set discuss contribution conclude act unknown set consist k ts ts agent act accord reward function argument omit denote eq observe distribution reward function reinforcement reward act closely preference calculate reward calculate reward representative temporal expert take theoretic find discrete raw state posteriori global main dynamic investigate robustness testing method particularly domain dynamic inspire maintain differ specifie prior policy reward specify model rather direct observation define map environment dynamic difference action come expert respect overall posterior question concave efficiently value preference expert dirichlet induce preliminary allow prior briefly prior obtain reward make inference hard one integrate consider rather policy unique
spectrum performance guarantee fall sample admit recovery even contaminate effect super resolve spectrum allow accurate source sufficiently separate amplitude phase later multi however establish complex sign spike draw uniform robustness either yield multi guarantee perfect sparse provide spectrum advance completion aim low exact recovery possible soon exceed theoretic noise portion corrupt medical imaging apply fold complexity exceed signal work strong similar incoherence mc physical interpretation remove restrict frequency present matrix dimensional summarize section incoherence condition discuss low toeplitz completion present theorem short summary finding discussion improvement support model frequency throughout normalized frequency coefficient frequency spectral special dimensional parallel briefly uniform frequency sometimes assume define matrix vanish outside aim might perfect denote paradigm respect minimization naive allow perfect exceed freedom bad spike large motivate harmonic adopt enhanced respect every enhanced find replace define span row space traditional thus attempt enhance enhanced program semidefinite solve worth similar complexity atomic minimization frequency careful reader performance must square later complexity increase theory measurement contaminate noise make practically applicable noisy th k noise adapt perturbation say norm sample arbitrarily sample due acquisition failure attack desire portion entry assume formally random conditioning eq corruption location measurement accommodate outlier regularization show select enhanced sparsity via relaxation respective notation throughout singular orthogonal onto space norm operator nuclear basis contain fold verify contain enhanced basis matrix specifically throughout short notation encourage news incoherence enable recovery portion unless certain illustrate amplitude frequency q understand reveal measure incoherence spike irrespective signal incoherence occur incoherence among frequency spike closely locate separation line incoherence worth thereby applicable broad htp empirical eigenvalue various choice reader incoherence presentation dimensional model suppose uniformly pairwise bound indicate spike randomly spike close grow argument frequency grid magnitude class beyond theorem noiseless measurement contaminate bound proportion exact possible noise theorem location set measurement noiseless universal mild scale admit soon exceed factor refined fine measure inequality worth observation differ randomness guarantee solely associate phase spike draw manner report improve require copy say close ground truth end follow counterpart copy enhance enhance snr enhance subsample interested randomly select q yield entry due factor simple numerical usually well applicability illustrate spectral grid atomic approach provably portion datum positive exist robust constant however specifie depend otherwise however well via cross demonstrate possibility robust recovery mild incoherence robust recovery proportion corrupt theoretical separate sense frequency spectrum extend high difficulty frequency enhance define fold enhance verify enhance summarize noisy search fold extended kernel coherence analyse frequency closely small fold clear rank enhance spectral think recovery general sense numerous system language vision imaging concern address directly framework straightforwardly adapt matrix completion generality modify state follow rank continue incoherence small vector uncorrelate rank convert toeplitz counterpart toeplitz capture harmonic toeplitz evaluate examine application exploit practical conduct experiment phase exact small trial spike uniformly solver trial successful return carlo n number perfect horizontal axis reveal algorithm vertical rate reflect color plot approximately line case diagram justify applicability htp transition concern whereas correspond rate calculate stability grow stability respect compose compare pursuit atomic recover via linear demonstrate mode location namely mode circle unit closely locate grid mode circle case successfully recover mode atomic fail recover mode pursuit mode mode frequency recovery mode dft along mode locate dft grid mode except locate panel truth atomic assume grid assume signal fig spike randomly circle spike satisfy fig impose atomic meet give sharp phase separation phase atomic omit separation sensitive requirement without atomic phase impose separation atomic impose atomic separation estimation portion outlier conduct monte carlo trial phase ground truth location generate illustrate phase corrupt tradeoff spectral success entry plot tradeoff spectral outlier see region recovery guarantee outlier robust randomly phase plot success plot calculate trial work consider example ground fig measure low point resolution apply avoid estimation number mode greatly suggest promising resolution leave reconstruction resolution low reconstruction c conduct solver interior exceeds enhance tailor completion singular thresholding structure location set enhance tt enhance operator singular pair project onto fold consistent observe unfortunately illustrate superposition reveal uniformly give amplitude reconstruct ground truth normalize algorithm contain frequency observe noise amplitude ratio plot spirit multi present exact recovery require informative estimate analysis adopt rely incoherence unnecessary focus sparse slightly involve establish span complement way describe replacement concern element operator extend rewrite follow completion exact convex suffice dual state random sampling obeys section construct dual remain incoherent respect projection establishe immediately control reasonably condition provide employ location multi way replacement represent via j establish valid dual condition step step within secondly q next subsection introduce characterize relationship crucial lemma allow appendix exist appendix inequality derive show combine develop follow translate indicate far lemma inspire well seek plus impose duality rely construct entry location multi coming mention simplify sign entry prove sign obey incoherence sign nonzero succeed recover argument introduce pattern unnecessary theorem section focus analyze say set besides extend establish recovery guarantee suppose exist appendix reasonably tight develop chernoff sample matrix location multi set random set follow n j proceed procedure dual examine condition reasonably c derive plugging establishes rely eq last fact remain control remains put together allow last c conclude present efficient sparse pose low structured problem mild incoherence arise numerically constant super knowledge result logarithmic uniform considers directly subset sample take mixture cs translate rank whether isometry nonetheless technique extend similar super great numerical work grant university chi mr li fig rely bernstein presentation bernstein dimension valid perturbation establish lemma consider two bound plug minimizer still uniqueness minimizer would projection verify imply last arise indicate resp resp resp l n plug fact multi similarly last operator q apply I define obeys need bind tackle sequel observe last entry lie diagonal follow one well last bernstein eq high probability write variable q result n vector vector easily vector immediately suggest high numerical complete make union complete proof obey allocate enjoy satisfy satisfy ease triangular component block contain triangular triangular block triangular triangular contain triangular low block triangular leave triangular triangular fact demonstrate control similar analysis divide set subset allow arithmetic contain allocate must claim optimizer verify constraint indicate eq require far remain hand satisfy derive optimizer inequality indicate q exploit fact inequality eq n occur first instead complement useful specifically n k e case either amplitude phase circle
compositional handle unbounded language account logical quantification promise combine compositional distributional semantic type semantic idea represented representation mathematics semantic provide algebra natural map al recognize map example nd mean rd use syntactic distributional representation see abstract compositional applie generally form aim rely mathematic algebra major open semantic framework compositional combine representation leave space acquire occur word category require high tensor large number need take step tensor tensor word phrase sentence live different space advance simple sentence plausibility distribution logistic plausibility begin want representation representation work representation phrase sentence live allow space tie type syntactic come price recognize additional capture difficult see g recursive language present syntactic english notation mean left category category et atomic space hence mean noun noun phrase people noun meaning replace mean rd tensor syntactic case noun vector tensor contraction multi first noun cat syntactic black people c syntactic combine object combine subject multiplication sentence practice assign read type phrase th tensor leave investigation two plausibility sentence think extension theoretic plausibility sentence subject people automatically specific dependency head atomic syntactic distributional semantic build take plausibility plausibility tensor object vector noun additional processing tensor linear sigmoid plausible softmax plausibility concept create dataset example noun vector employ technique number learn size baseline adapt kronecker plausibility triple cosine similarity algorithm sigmoid output softmax probability subject triple negative describe generate make corpora google syntactic grams wikipedia wikipedia corpus content stanford nlp tool example extract precisely extract distinct syntactic root phrase filter obtain c justify corpus plausible example preference noun noun size noun wikipedia corpus present title role semantic encode word mean format counting occurrence window boundary window noun wikipedia time occur within noun noun word frequent corpus weighting time noun word I number follow train tractable rank testing subset evaluate range generally spread noun contain normalised occurrence place noun row noun occurrence singular decomposition svd dimension remove improve semantic similarity enable noun vector word conduct use repetition cross validation cv evaluate many pair frequent sigmoid transform value softmax vector bad baseline evaluate use measure plausible evaluate class hoc auc fair repeat table experiment half baseline justify justify baseline learn effectively noun score principle baseline mostly negative predict plausible particularly triple see baseline latent tensor produce positive negative negative noun decide plausible may noun treat experiment property preference semantic table frequency strong frequent frequent triple likely noun frequent see justify brief analysis noun argument rd semantics plausibility connection goal contribute preference framework al neural literature rd order trend compare competitive
pour des dans une pour ce ci un latent et pour la pour la la I alternative de dans de par dans la les la la des es instant une en segment les des la est un de partition en un est de de ce l en la fisher est de la dans ce est send est dans variance pour les dans les en du signal il est non par un dans ce en di ensemble coefficient un I I plus de dans ce la les les la dans ce est une en pour des tr co pour les de dans la un plus de un sp pour point une send de par un I le le se est la si des la le une est un la est par tr par une par la en les est les en la par fisher est un il la de se ram de par posteriori la par par les est par la convergence l une en es est des l op est le de est le par le ratio l fisher est en em de en une plus en se ci cm de la une dans les du les en par dans une de dans en un une les en une partition en un les en converge plus em estimation es si les la des un dans e en des es pour ce les estimation est et le est g les de transition un es des es h pour la situation et pour de segmentation les pour les et es dans le h fisher em situation situation le pour des de tr la le de fisher la ccc de fisher fisher dans article une en es une mod le r par pour la se par de cm en es noisy le universit article une en es se mod class classification en la par des alternatives l fisher r propose classify order cluster specific govern
default plausibility plausibility I elastic plausibility interval outline discussion something case follow continue logic correspond objective difficult credible less I employ sort elastic I I describe efficient brief notice I range interval shorter recommend I focus regular important e section strategy via set marginalization accomplish predictive focus regular association similar except nuisance hope ignore set introduce valid retain suggest z regular analogue result result I valid corresponding sampling imply calculation q validity side stochastically large function complete way construct construction plausibility kind consideration nuisance dependent nuisance bind degree I idea valid predictive new auxiliary stochastically introduce necessarily uniformly non regular achieve technique case understand formulate substitution parameter regular theory marginal drive uniformly base deviation turn process solution population respectively mean interest proportional derive I solution variety baseline association combine n f n make baseline problem side regular could apply technique present like auxiliary variable rewrite show stochastically let function version choose random use predictive plausibility validity proof conservative e bind large marginalization admit dimensional minimal statistic take auxiliary simplified writing define omit check gamma distribution base could estimator perhaps likelihood marginal rather moment solution similar likelihood moment straightforward distribution respectively correspond need find find stochastically adjustment trick projection define median adjust estimator stochastically half scale negative side theoretical claim available picture adjust stochastically picture bind tight small distribution mixture valid predictive default construct marginal illustration model I plausibility interval short coverage figure I comparable paper presence nuisance I framework improve classification introduce admit exact efficient marginalization accomplish using describe herein regular predictive maintain marginal I conservative efficient marginalization dimension reduce dimension sparsity way amount dimension consideration help problem associate anonymous suggestion dr helpful discussion work partially national foundation grant dms dms form assertion e assertion display event consequence auxiliary change scalar auxiliary variable require whereas association require follow I correspond plausibility interval match jeffreys assertion support nest nest collection consequently I formulate scalar fact default predictive random section notion marginalization formal I claim admit decomposition marginal inference simple make transform nontrivial constraint easily accomplish initial model fix assertion assertion assertion look v regularity look function clearly onto margin restrict choice margin predict characterize implementation say regular problem irrelevant marginalization focus marginalization might thing suppose baseline auxiliary hold towards equivalence auxiliary display regularity association assertion event drop importantly actually auxiliary dimension reduction come clearly marginal remain marginalization valid predictive association xu cx claim equality follow vector length direction unit sphere baseline association describe marginalization strategy flat I gain auxiliary marginalization orthonormal auxiliary describe express give I central degree c usual positive condition random valid find illustrate assign flat central credible interval show marginal plausibility scaled show evidence plausibility summary panel extreme credible plausibility gray mark central credible interval assess whether result scale summarize plausibility credible interval plausibility nominal plausibility bit credible interval coverage case stein acceptable coverage although frequentist concern bayesian even moderate choose frequentist prior goal I marginalization analysis start association marginalization obtain valid I proposition inferential I posterior probabilistic auxiliary connect auxiliary turn regular marginalization I validity exact several normal regular propose marginalization namely gamma nuisance validity interest slope partition inference nuisance modification call opt profile cox unknown point hypothesis desirable construct maximum likelihood style alternative account difficulty arise requirement something beyond frequentist develop default reference prior fisher argument model known continue regard posterior datum dependent generally applies mention provably finite fundamental behind I unknown equivalent view essential exact inference consequence need inferential meaningful property I coverage especially auxiliary constructing dimension auxiliary notice certain actually dimension conditioning propose consideration particularly implicit framework problem reduction sense I section discuss validity nuisance strategy separability interest set marginalization regular random set benchmark gamma problem notation observable encode joint vector measure I agree express choose expression general easily mechanism structural important model cover association sampling model association admit interest drive uncertainty next thing accurately I predictive set observe parameter predictive serve encode additional uncertainty predict definition belief definition combine candidate empty assertion summarize support eq plausibility plausibility assertion set conditioning alternative elastic amount summarize pair support plausibility plausibility plausibility plausibility probability choice I construct choice work detail reduce finding set auxiliary I validity relatively predictive support suppose without generality support valid use predictive need validity I validity plausibility let belief I I hold consequence plausibility region nominal coverage validity plausibility interpret note property depend validity specify efficient try conditioning example gain nuisance reduction without efficiency throughout distinguish concept reduction dimension reduce challenge construct random simple suffice marginal inference unknown unity I entire unnecessary also composite invariant depend u dimension obtain baseline association usual posterior association complete bit explanation set calculation need auxiliary bayes take singleton I come section suggest model regular sense sufficient marginalization case bayes I answer implicit section reveal regular valid help formalize discussion marginalization importantly help free marginalization must produce objective must actually marginalization though identify variable arguably suitable I valid predictive definition hold hold validity result cover theorem w w monotonicity give connect sense determine vary accord therefore inequality therefore baseline association random frequency property plausibility interval base achieve nominal validity ensure marginal meaningful regular let valid random probability dimension match apply baseline association prior marginalization association way marginalization appropriate obtain fill entire dimension paragraph efficient however big yield valid without efficiency quick review efficiency valid assertion stochastically intuition behind plausibility region keep singleton plausibility exceed make plausibility validity quick note since big assertion check efficiency lose question
policy probability lem discard policy episode time optimistic select episode cumulative kn reward receive begin episode optimistic average reward policy event inequality episode guarantee gap lem lem p step accord suboptimal suboptimal high move high expectation almost thm major difference policy accordingly suboptimal achieve dependency show whenever policy although thm algorithm discard suboptimal relaxed span require show trial episode policy computational policy optimistic bound lem overall computational space similar trial solve extended iteration optimistic complete linearly state preliminary propose mention previously generic enjoy approach policy regret scale fair natural take mdp actual like parameter optimistic among subset tight compare accelerate action succeed go unless cause action stay place direction construct action action mdp bad mdps corner mdp receive policy performance randomly place state notice compare rational state average regret size quickly effectively maximum demonstrate task change grid state demonstrate superior decrease fast slight periodic increase new trial start policy decrease accord often conservative scenario show trial number run whereas increase empirical demonstrating performance domain significant input policy interestingly significant improvement setting relate armed bandit literature optimize good bandit either bandit arm distinction may independent reward bandit evolve demonstrate significant rl decision constrain lie set objective maximize reward plan rl set proved learn act linearly size set focus leverage rl rl focus expert lie expert expert predict rl rather select follow policy promise guarantee interesting closely consider identify policy special leverage expert environment history whereas similarity though take average reward precise state careful comparison rigorous general set mdp algorithm correctly bound hold precise expression prefer allow rigorous nonetheless easily improve policy information particular action use update expect implication less compatible correspond number episode rate bad potentially point leverage policy still maintain optimal suffer regret suboptimal total always learn question nearly rl input prove scale sub regret computational complexity domain cs edu reinforcement policy learn experience algorithm reinforcement task regret empirical simulation offer domain policy reinforcement agent seek learn world world objective close rl user model policy discrete continuous action set sensor act market trading current rl prior consider member given extract reverse free related idea agent try policy task encourage performance work formal guarantee learn policy difference rigorous setting contribute uncertainty adaptively policy policy linearly algorithm suggest benefit large domain typically scale preliminary simulation impact tuple state transition reward action zero recurrent recurrent transition matrix single expectation transition induce markov corresponding span ss reward transition policy reinforcement mdp learning policy arise near mdp policy different policy space provide objective almost well input follow mild induce also optimal policy induce existence weak assumption assume mdp require policy input induce set due optimality span initialize initialize ft ic h nb v introduce alg seek use input yield average reward initially within series episode trial exploration promise one popular possible policy guess maximum unlike bandit run confidence interval however fail episode confidence bound fail condition specify mdp necessity eliminate policy trial episode terminate reward episode proceed average converge policy know advance trial lem hold lem together bind hold discard use slightly confidence need reward trial bind lem combine lem imply high episode total number select total violate policy discard trial number episode truncate episode number trial terminate discard trial trial trial length discard lem discard w trial terminate follow trial tn possible episode stop regret ready assumption bound episode optimistic total episode also episode time policy optimistic policy policy execute optimistic optimistic lem line total inequality lem horizon provide bind deal lem line prove total episode limitation omit detail final combining union possible space contrast prior policy good building informative knowledge along receive markovian nonetheless display action elimination mdps pac mdp belong space mdps set
bind convergence attain estimate gauss quadrature weight fit outline q issue many trait argue zero identifiability free possible nevertheless actual less occurrence mixture analytic slope identifiable rotation determine parameter identifiability estimate check maximize reach example discuss model free observation value preferable important proposal quadrature maximize purpose context equal across group mixture trait within implicitly estimate depend observation penalize parameter extent number involve parameter worth note adjustment check total number p frequency pattern common count pearson case frequency via frequency since datum likelihood gauss quadrature quadrature sensitive point heavily number initialize observation possible group initialize final initialize ten start select calculate worth note method linearly em stop iteration desire tolerance binary record age national care variable record subset daily instrumental activity first first activity care get around inside activity community house light house take response code fit datum correspond correspond record algorithm outline report gauss quadrature cc cc good bic trait bic class sensible group account necessarily c c contain contain pearson truncation well fit indicate truncation supplementary match see trait model class trait bic ht error select standardized median report aid interpretation model l c c c c c c c c c observe group consist activity survey unable heavy house group indicate adequate outcome bernoulli consist non slope activity slope parameter activity indicate considerable variability positively worth instrumental activity daily characterize people able take unable outside activity big response latent trait slope sign activity exhibit dependence activity activity activity heavy require explain outcome mainly people able activity unable house work activity quite variability within group trait median excess activity activity highly slope moderately positive activity mainly people get activity big group trait median activity activity daily instrumental activity people able unable activity great people activitie unable activity activity trait large involve exhibit positive characterize mainly people unable activity daily explain trait structure individual tend activity daily instrumental daily sign slope equal strong activity heavy house work activity activity variability trait parameter dependence unable house around inside activity activity quite variability explain within trait individual group activity negative activity require group daily individual include group allow individual voting u house rd recorded house issue house publication issue publicly available repository contain house issue vote vote know issue project aid anti aid mx education act south code response fit fit cc select bivariate trait c rr c party membership consist mainly small member interestingly correspond party nine suggest vote way worth vote voting reveal voting reveal probability issue always vote rate water act south voting versus voting versus observe individual mainly group opposite mainly group majority issue median individual concerned aid education group low additionally group concern anti test group low suggest latent trait group group show impact variation position vote influence baseline median voting characterize plot trait introduce significant slope large code person vote yes vote code voting yes case median outcome q q htp vote person vote yes give table supplementary material dependence show value negative within positive dependence variable latent mixture latent trait categorical special applicable investigate latent analysis trait interpretable em propose mechanism provide extra categorical continuous framework trait cluster criterion offer efficient excellent behavior national care survey voting set group intuitive interpretation acknowledgement national centre university sciences college datum application method categorical data class binary categorical interest trait extend assume categorical depend categorical trait continuous trait latent trait likelihood involve analytically variational trait strategy latent trait survey u intuitive cluster class latent trait alone offer coherent quantify use many successfully measurement non gaussian lin lin categorical include science trait categorical latent within independence categorical class suggest interpret trait continuous trait categorical variable multi mixture trait develop categorical identify trait accommodate variable particular trait trait restrictive parsimonious trait like model variable nominal categorical mixture connection detail introduce latent trait integral evaluated analytically propose purpose converge deal dimensional trait mixture trait set national care set voting class trait analysis sufficient two set however mixture trait presence group explain trait introduction provide introduction trait description gauss quadrature carlo variational trait model parsimonious parameter outline estimating adjustment pearson sum square pearson assess fitting outline application discuss section overview latent trait outline fit variable otherwise random variable observe dependence latent variable see component independent log describe estimation via correspond cluster otherwise assume variable behavior categorical observation assume function logistic give equal briefly review section gauss quadrature treat take n fix mix proportion explain detail require increase gauss write trait proportion parsimonious dimensional come arise rotation dd dd parameter parsimonious would group free latent mix choose individual characterize direct group group positive variable distribute heterogeneity group large probability response group account simultaneously outcome calculate correlation group standardize value dependence group group response compare two positive response gauss estimate variational lift evidence lift output posterior estimate wider scientific discrete parsimonious account correlation variable analogous loading mix proportion identical general variable variable use discrete framework widely statistic call ability difficulty trait intercept formulation treat univariate model usually univariate ability across
behaviour rate come corollary armed bandit achieve near bind bandit achieve probability whenever furthermore event strategy computation describe solve perform bound multi armed bandit allow armed arm node child partition child algebra sigma e product algebra arm receive reward goal wide general lie let dyadic dyadic node child dyadic instead allow point tree dyadic borel algebra recover consider armed special bandit achieve proceed fashion arm box reward reward box index arm uniformly box construct partition uniform bandit box product dyadic interval p armed description idea I box width space point descent precise n leaf terminate otherwise define check uniquely define box definition arm bandit easily note always generate typically algebra fine define allow onto index active box tb past reward x maximum reward add select box b time let fix note estimate choose tb show hold box db depth correspond constant radius maximum term concept expect tree armed act discuss implication detail problem conservative define empirical tb tb maximum active box box active box index upper reward box satisfy estimate principle function behave box always pair c use therefore reward width mb satisfy j satisfy mb b agree pair state x later tb w tb suppose remove replace box must terminate cm unbounded jump marker coordinate inf inf inf inf inf inf axis cs axis box active global break tie arbitrarily full relate select box play b infinite tree radius tb secondly optimisation include easy lastly tree box nod estimate box agree axis width ensure box split shape box shape require reward motivate bandit whenever arm allow argue directly compare begin need preliminary definition collection disjoint box say union box far say refinement b c box u I grids box fix ready state condition behave box let cover box db box b box satisfy constant c separated grid b quality firstly height cm width unbounded marker legend legend north east legend inf inf inf inf axis cs cs grid cs leave xshift axis cs axis cs right xshift g let x coordinate space dyadic note relate solve reward arm mean treat bandit primarily box consider bad fix metric formulation closely improve allow box construct single combination flexibility wide maxima armed bandit armed bandit result near condition control box assume construct similar tree certain depth quality improve upon hold collection box show allow fix secondly box trivial allow detect axis along adaptively combine tree work efficiently condition show near b box create axis active box condition box region box create later main armed bandit armed definition require leaf singleton tree tree armed bandit uniformly reward space equip tree whose reward class rate ft write ft ft ft ft armed I fix armed bandit match example paper begin show achieve rate state follow fix box extend multiple infinite detail within tree still log fix run hence box construct l must c box ensure desire thus cover cover certainly cover cover l cardinality first deduce neighbourhood continuous large pick must conclude enough hold continuous eq b contain box trivially g region g box continuity neighbourhood c take box g b satisfying conclude far choose b I agree except one axis must satisfy constant c union g grid cover refinement construction conclude low bound prove let armed k px prove similarly begin note assume eq u kullback u far inequality likewise bound fashion apply lemma v armed reward far reduce armed multi armed strategy cumulative regret denote denote ty b apply also r deduce desire give maximum simple argue low multi armed arm later reward bernoulli agree except maximum index possible distribution lie within maxima around location global need carefully therefore collection node follow pick child j many maxima four eq must show note since simple fix l reward cover follow box indeed partition check consider u cover x box constant constant let box separately u l box eq box result trivial trivial refinement subject set taking set k condition let event proceed similar box reward p satisfy condition regret make significant improve result event execution clean tb box c tb tc tb clean box enough show clean probability tb nb ib must box depth martingale kb kb tc dc q thus tb deduce n prove follow box ensure box activate activate activate time c clean execution b tb hold hold tb part prove width radius cover must activate case since c box tb tb desire otherwise first rearrange index must optimum part first box satisfy deduce second tb r tb statement induction statement p pt show contradiction activate activate note c c activate lie q lie c must activate since b inductive lie lie lie j box c axis form splitting I lead contradiction x within member j term j j fourth contradict conclude hence induction select tb time execution box clean execution activate time activate box b b eq activate q turn apply result activate cover therefore need box form tree child activate activate box similarly forest conclude apply activate part result part trivial otherwise q since result tb b db uniformly first third time b activate tc db dc proceed part activate occur box q box principle ready prove clean execution event execution clean know event set box third fourth additional simple similar q fourth prove part note execution clean divide queue cost queue otherwise maintain box therefore cost total part box activate internal box queue cost part let upper number box box computation newly store quantity width tc activate box box total activation part time remain store box b store update tb update remain bind b split axis op total box acknowledgement anonymous valuable comment suggestion support grant ep primary secondary global optimisation armed laboratory university noisy global bandit good regret bandit bandit also possible regret near wish value wide sequential gradient observe arm ylabel legend legend pos north east ensure expect well practical varie q control tr control bounding solution place region expect regret lipschitz suffer simultaneously rate armed bandit think optimally slot arm armed bandit long history comprise however recent also focus specific armed bandit x smoothness reward solution involve place think lie optimisation set area intelligence services yu unimodal level distance child child child node child child child paper noisy regret consequence discuss contribution arm nearly regret prove ucb achieve regret find strong show apply unimodal reward reward finitely many quadratic global space lipschitz regret describe maxima cover reward quadratic function try q power q bad rate try lipschitz wide upon bandit adapt reward bandit directly infinite space p explicit estimate maximum reliable proof adaptively tree adapt construct adaptively partition achieve
sa snr moderate high coarse could phase combined initialization inaccurate inaccurate mc consideration model channel distribute variable channel snr simulate channel snr diversity iterative continue relative stopping table sensor snr value initialization small assume phase perfectly serve figure propose similar approach superior surprising classifier lf account symbol result optimality multiple c c c stop proposition corollary minus depth li sensitive channel paper centralize hybrid ml adopt snr diversity multi framework robustness snr superiority approach respect classifier moment fusion algorithm mc deal determine noisy play cognitive communication thorough mc method single nuisance signal offset usually diversity technique wireless communication system effect argue e mc potential improve especially mid inspire reason collaborative mc propose distribute detection fuse fc two centralized approach linearly add combine signal perfectly receive moment base ignore coupling symbol fusion fuse fusion centralize mc complex base mc issue expectation em mc channel framework formulation centralize consider algorithm snr centralize mc superior moment increase sensor signal block symbol flat sensor locate apart experience independent perfectly filter sequence complex symbol gain respectively sensor denote symbol hybrid approach lf unknown maximize unknown nuisance let r represent condition give independent symbol symbol note symbol assume assume discard irrelevant page maximize final complex furthermore couple sensor sensor follow symbol follow estimator denote respectively hermitian complex form expression sensor simple symbol adopt treat unobserve em algorithm iterative ml problem ml intractable presence datum formally describe em let start iteration step symbol reduce ii tm deriving use te represent energy signal substitute maximization take first derivative information sensor
false recent availability time relate analyze turn insight able series run paper series track activity particular news twitter ask topic twitter news topic otherwise series news trend trend twitter importantly mathematical furthermore remark time beyond scope numerous tailor series simple approach term variety competitive various elaborate tree examine near neighbor classification boost apply transformation mostly justification near neighbor classifier expect well classify datum tend near twice consider near grow however examine go instead classified impose complexity structure present follow collect human trend twitter volume context source series latent posteriori series label series weight vote favor label series trend label drive entirely training estimate serve weight vote neighbor time training majority new time observe account observe apply online series classify stream topic offline two classifier suggest weighted majority observe classified mixture show require suggest latent goal lastly majority forecasting topic twitter trend predict topic trend twitter party identify trend twitter trend say detection twitter majority whether advance twitter hour minute twitter activity number think voting neighbor series model theoretical voting neighbor datum topic time convenience assume classify trend label positively label whether label similarity allow allow look outside scaling determine influence label vote tr rt st look train restrict minimize shift pre maximum allow shift vote window long need trade long neighbor correspond near neighbor train vote tr latent latent observe latent occur time uniformly follow add label series gaussian variety importantly evenly alternate different like estimate latent problematic example adjacent latent source could noise latent example mixture mixture use latent versus complexity mixture I posteriori map know noise make map exponent replace majority weight majority smoothed whereby consider shift exponent numerator main vote follow still minimize shift lastly trade positive generalizing result weighted voting thus majority voting think majority vote neighbor classify correctly account classify maximum far apart different series shift majority voting latent source training series majority immediate tolerance term access pool label pool subsample weighted majority voting time needs grow logarithmic latent majority source otherwise distinguish use classifier gap series classify classifier guarantee probability time classifier voting majority voting match nearest suggest weighted majority neighbor method exhibit agreement could neighbor overview twitter example trend news trend phrase appear month twitter choose phrase unclear trend category control pre tune weighted majority voting experiment majority classify shift topic pre process rate tweet news topic trend tend pattern show divide trend vote datum figure choice detect topic advance twitter early achieve rate false positive early prediction yield early balance part research award fellowship series wish classify v size exist henceforth elaborate happen source signal gaussian misclassification primarily identical inequality take use sub last line gap g repeat plug give step majority decompose depend term label near source see shift signal existence optimality condition hold q step piece together final result ensure least source latent occur source occur appear source strictly time bound gaussians wang gap measure true latent label translate guarantee term gap assumption variance near classification pool classify series high side ensure classify series high series respectively tr te te tr r q shall v v v r ta v r opposite label bind bound complete square scenario imply majority voting neighbor probability bad control happen yield final see twitter social topic twitter real popularity surface trend month sample filter trend list trend minute salient also trend gram appear tweet gram contain know unclear trend comparison trend simplicity size equal could tune weighted majority weight vote trend create series tweet topic approximate tweet place count raw summarize classification characterize spike spike city mostly soon emphasize signal baseline de emphasize define baseline signal observation tweet rate spike spike de spike spike normalize addition eliminate volume slide length spread topic person think branching grow exponent suggest series contain entire window keep hour activity topic hour transform correspond topic hour activity news trend tend fix divide trend trend test voting hour trend trend randomly trend detect early measure early trend hour series number initial observe width sized detect
select prior elaborate close remain et empirically method mean outperform point proportional contribution define distance competitive initial refined initial seed local notice distance metric point label metric grow locality sensitive lsh near goal dataset object image object return similar wide hierarchical try similar perspective lsh image lsh accommodate make possible preserve describe first circumstance sure belong dissimilarity similarity otherwise dissimilarity try distance definite point keep apart constrain newton objective part metric agglomerative well grow exponentially affect general agglomerative estimating centroid burden happen distance highly expensive cluster locality sensitive aim solve scale agglomerative cluster problem learn step second table agglomerative explicitly compute distance measure hamming locality hashing neighborhood substitute exact instance tb distance step cluster cluster proximity hash merge cluster merge row cluster retrieve hash input step kernel suppose dim vector bit create base mnist digit evaluate agglomerative via mean obtain handwritten mnist repository digit agglomerative cluster string intel processor ghz ram table trend observe agglomerative cluster fraction cause decrease metric performance agglomerative analyze effect precision relatively string length validity increase length hash string adjust efficiency effectiveness notice binary hash possible hashing split number cluster meanwhile agglomerative superior compare promise improvement speed true low linkage metric instance p dl dl pre pre bit grow calculation distance sensitive hashing preserve substitute exact agglomerative reduce sized hamming efficient cluster incorporation metric marginally department engineering operation research york ny large scale agglomerative
share desirable property markov dag equally consider underlie dag favor prefer simpler greatly search later regularizer task end cross choose local j top jx j j x p give scoring finding maximize score dag super infeasible number cover ordinary dag consequently form purpose utilize reversible mcmc method discuss jump dag optimize consider whole optimize termination fall deterministic basically onto bring explore dag generally model shown possess advantageous denote iteration reversible generate candidate accept accept otherwise remain unchanged chain irreducible main objective state simply visit satisfy mention globally adjacent reach acyclic differ structure score wise local structure idea exploit respect part add configuration add identification weak edge exist dag however assumption inferior dag hence sensible underlie dag get inferior severe structure motivate part perform whereas structure choose validation assess identify learn calculate vector train calculate outcome count reduce variability partition average candidate search two early modelling dag equivalent keep investigate predict execute scheme split part parallel chain empty initial simply identify high chain case compose six factor identify list increase high bold illustrate identify contain label label induce marginal dag synthetic generate systematically investigate identify posteriori estimator leibler kl denote divergence non measure equal divergence dependence dag well dag dag dag may generate label execute model evident indicate quality begin suffer overfitte reduced prevent pick evident overfitting effect restrictive pick black ideally curve choice model perform always pick candidate identify investigate traditional divergence distribution dag curve curve chosen cross pick prior converge outperform dag sized sample contain discover restrictive curve eventually underlie dag table table see curve coincide dag identify dag restriction restriction dag size dag curve eventually curve require add discovery dependence structure dag dag since allow flexibility properly idea independence graphical introduce label local entity investigate structure score combine global experimental agree sense incorporation model appropriate go beyond express interesting extensive search model outcome physical family mathematics mathematics department mathematics technology author acyclic proposal direct acyclic concept equivalence class learn factorization dirichlet develop novel appropriately reversible hill real synthetic acyclic graph specific directed acyclic gain popularity system despite advantageous dependency modular parsimonious present allow dependency node explicitly parent substantially reduce expressive author generalize independence term node manner go instead introduce configuration outcome model desirable concept efficient introduce learn dag enable analytical evaluation relatively fast structure reversible carlo hill computationally structure introduce property bayesian set conclude dag acyclic ensure direct node lead dag formalize network node direct edge correspondingly absence statement constraint impose alone circumstance natural role distribution behind focused topology examine asymmetric bring graph approach introduce graphical label dag store example worker note person attempt worker gender conditionally corresponding probability identical notice represent two specific gender hold probability imply gender person representation certain allow state formal provide notation contain direct variable term use letter denote outcome cardinality ordinary dag encode statement form disjoint denote independence follow direct property variable conditionally lead unique distribution low factor node relation notion context specific independence formalize let variable subset denote discover numerous capture statement dag node structure offer natural introduce topology visualize figure incorporation oppose formally represent dag dag x x contain except part edge naturally parent incoming contain label variable derive theory apply label figure illustrate edge strength approach generalize network utilize correspond way capture somewhat power decision unfortunately usually leave exploit next approach connect consider expressive leaving scope prove advantage represent table grow exponentially parent fail include directly similar complete rule distinct right column five naive approach require define configuration path distinct reach terminal right parent configuration rule rise mutually exclusive mutually exclusive path leaf variable part give encode rule read consider coincide specific mutually exclusive incomplete illustrate minimal bottom compactly tree context graphical merging figure merge tree situation arise correspond exclusive order reduce mutually exclusive recover upper mutually exclusive point configuration configuration rule rise therefore generally configuration label combine rule create method thus exclusive exhaustive x base representation induce outcome representation consistent graph go subsequently even readily recover class balance expressive interpretability sound interpretation naturally perspective particularly useful effort exploit refer query observe joint incorporate interpretation interpretation maximal part rule exist configuration parent context rule associate minimal reduce effect vanish label introduce maximal label condition maximal must add configuration thereby maximal independence condition ensure add configuration must I l restrict regular generality regular recovered markov local local local describe dependence dependence ultimately local accord must pg hold equality representation dependency derivation ordinary dag instead local verify concept separation sound independence separation concept separation introduce satisfied j context denote underlie dag denote subset separate denote describe separation dag separation may certain discover directly notice necessary perform reflect however separation reasoning eq eventually separation lack separation cut regularity occur throughout outcome combination label still dag separation discover easily lead conclusion q hold separation non independence easily discover situation special arise outcome split several early restrict substantially exist distinct encode dependence highlight distinct class ci class conclude class graph chain difference occur worth note edge essential dag base determine local dag equivalence correspondingly say form equivalence remainder belong underlie let equivalence underlie dag skeleton direct criterion tie concept markov dag regular equivalent assume assume far exist markov equivalent skeleton skeleton exist must l x exist allow conclude contradict equivalent must map markov equivalent without induce map indeed obvious check context affect satisfied check specific equal dag strict regular outcome dag consequence pose obvious vast flexibility reversible chain carlo method combined hill reasonable score use set prevent balance ability additional notation consist variable span space outcome denote outcome parent
random choose derive technique nesterov derive second nesterov form problem explicitly readily detailed generalization nesterov nesterov technique produce work convergence nesterov accelerate minimize give bad well rate nesterov special analyze type also target develop call randomized analyze nesterov establish give especially technical extending nesterov establish expect high complexity technique converge smooth minimization technical subsequently throughout assume solution nonempty partition eq assumption continuous follow whole g satisfy respect large convexity respectively convexity expect separable coordinate pick randomly n x n define develop regard introduction solve proximal optimality separability mapping establish base composite mapping pick fy dx convexity fx dx fy dx fx gx fy gx gx dx fy gx corollary uniformly q block wise take expectation trivial uniformly problem nesterov develop establish converge probability iteration imply randomly realization variable eq quantity measure set optimal follow block separable employ block gradient mapping develop method iterate eq furthermore denote eq side yield rearrange take apply monotonically far lead obtain eq due k fx side relation result present nevertheless straightforwardly relation respect side relation hand relation see sufficiently improvement show fx virtue run optimal high hold one convex let expectation obtain relation follow q fx fx fx definition definition see special eq tight method optimal run next output j together definition conclusion total obtaining eq implicitly establish optimal q iteration restrict respect accelerate randomized repeat claim define inequality hard well description come directly derivation convenient simplify follow symbol ccccc paper uniformly depend realization variable state convergence rate relie randomize deterministic nesterov establish case verify hence much tight nesterov accelerate extend subsequently randomized estimate optimal solution randomize estimate randomize sequence together imply conclusion em namely addition arbitrary depend pair hold know hold last hypothesis x fx v estimate sequence eq substitute let view l k k fy virtue conclude together quadratic drop arrive convexity two inequality recall inequality corollary eq finally suffice since
bad exponentially suppose disk principle prove proof bind sup therefore maximize geometrically picture relaxation whole disk disk furthermore disk picture easy disk directly contradiction satisfie perhaps circle circle transformation let center disk fact map q bound conclude want transform follow transform second polynomial contradiction qx c affine type relation maximum function thought polynomial population attribute probability variable polynomial term easy bound disk theorem proposition proposition restriction question theorem fs nsf innovation grant population goal string length coordinate preserve improve algorithm et fact show via corollary restriction access et al et describe statistical determine consist find skeleton suppose say specie choose string replace coordinate like string requirement string formulation program yet challenge show see approach later show give time algorithm algorithm alternate quasi polynomially framework sample need generalization introduce seminal hamming ball string flip exactly noisy algorithm phenomenon time exponential mixture decision tree problem exponential dependence recovery time recovery naturally investigation central learn learn restriction access interpolation box box restriction obtain fix al string recovery yield restriction quasi polynomial run time clause reduction immediately algorithm pac succeed main open question population goal close goal estimator would suffice match make particularly give efficient inverse review vector index row index estimate access choose know observe hope chernoff hoeffding say enough less ensure polynomially bound natural polynomial exponentially another remarkably work subsequently improve turn population recovery indicator rest observation know string whose least recovery string reduce everything unknown least first population recovery rough solve population candidate string crucial string keep observation et know keep thought sample stre recover string ignore zero mark sample recover one map one symbol question mark probability assign string ignore mark robust optimum follow crucial check exponentially discuss well outline early find local inverse minimize sensitivity bound crucial reason interpret value estimator abuse notation refer basis inverse form let final turn choose basis program four group dual index program make simplify observation minimum simplify equation polynomial lead translate absolute value maximum linear change establish uncertainty fourier transform g literature concern e establish circle say restrict I qx value interval polynomial large informally polynomial
well improve eventually obtain pass accurate variable converge domain select value domain impact variational accurate runtime domain message round fix domain long accurate assignment graph bipartite graph set factor assignment neighbor distribution pz inference compute marginal perform contain factor f kl approximate marginal f h h correspond saddle bp marginals consistent bp converge find optimum however produce neighbor domain far locally consistent correspond well describe property improve accuracy marginal marginal instead complete maintain pass objective marginal fix domain entire marginal point bp study message partially value associate domain pass marginal bp domain l ic computation message much whole optimizing remove obtain marginal message domain converge identify add crucial variational objective converge marginal locally saddle impact iv il iv perform sort update pass select value belief marginal solution enforce constraint ascent primal objective obtain v identifie add add respective update area affect modify domain amongst message message formally e reduction point use message part locally consistent primary scheduling message scheduling use sparsity initialize fraction select evaluate message pass dual unary pairwise tb grid run mm runtime approach bp inaccurate solution suggest domain desirable initially fast add significant crucial rate considerably time domain become utilize eventually examine residual residual consistent low throughout remain log near domain slow domain grid entity factor bp extraction avg ms entity entity domain relation neighbor assignment detail omit time average run small sentence help much bp iteration sentence contain entity significant speedup design maintain efficiently update reach point dynamic scheduling gradient improve eventually bp outline initialize use high queue remain message pass maintain message queue message
slide window paper discuss category application cover anomaly comparative analysis work new current traffic law nominal traffic traffic sequence markovian assumption base datum examine independently neighbor produce detector sequence flow base detector technique biology traffic cluster network flow depict flow capable flow flow group window identify anomalous anomaly detection method limit availability widely label dataset collect year change order software label generator generator evaluate simulate anomaly service attack describe traffic mathematical description anomaly depth anomaly present five conclude remark server element anomaly detection care user ip source ip address incoming ip format discuss start transmission vast traffic grouping series flow n size duration denote start flow translate relatively collection number frequently server surveillance infeasible statistical something enable network user individual notation address address define easily extend address ip address center final use flow user ip representation consecutive window appropriate size windows h x g ref flow use statistical section fall category supervise well mode supervise mode remove flow human inspection mode short window nominal ref g j x ib alphabet user flow flow flow surveillance gets map empirical flow state compare form normality alarm detector eq pearson sequence chain flow flow define markovian frequency indicator form markovian markovian sequence flow follow similar analog markovian appear model detector relative indicator I alarm detector pearson deterministic base boundary technique name separate majority z outlier qp generalize mapping input outlier format rather compact traffic remove user belong z reasoning measure user belong less distance besides categorical unstable practice radial r c reach anomalous indicator prescribe anomalous alarm annotate anomaly result package flow level validation dataset software package annotate flow record generator simulator use ns simulator simulation resource attack attack realistic way validate package format record test independently internal network topology connect internet internal consist server server generate level flow assume poisson arrival level anomaly network user unseen short user flow size try file server try sensitive file sensitive file anomalous dataset create use traffic use ns transmission traffic poisson process time exponentially parameter server internal infect investigate server request c server attack flow technique duration flow determine rate stage attack flow flow affect flow transmission normal traffic send short flow combination common show server window window overlap consecutive window cluster quantization flow duration graph simulation dash line alarm part red marker observe stable flow high identification resolution sense identify flow capability stochastic tune window adjust window size reasonably optimality rely large flow window observation complementary combine method get rough interval anomaly flow deterministic belong figure receiver operating combination roc combine two alarm alarm axis threshold simulation window figure normal algorithm observe individual flow flow interestingly work portion traffic effective rare attack total consecutive window alarm nominal assumption method start suit detect attack unsupervise traffic percentage bad traffic nominal affect large flow window five complementary common anomaly open source package software package level anomaly level attack analyze advantage false rate
moment offer maximum suffice pmf case fundamental method deviation determined pmf pmf mathematical probability respectively neighborhood hold exist z inequality x z z large z n vi continuous z provide provide proof v vi lr coincide chernoff moment main objective develop bound tail pmf column vector scalar denote pmf th column ie determined pdf pmf determined pmf great element pdf pmf moment generate function probabilistic suffice theorem z f discussion derive inequality remainder univariate pmf random pdf pmf exponential follow result pdf pmf sample let constant pmf z z vi iv theorem consequence random cumulative cdf normal constant shall apply lr important univariate belong x make fact iv bernoulli variable pz z z z constant moderate chernoff tail lr offer proof chernoff case q nz observation nz sharp classical argument describe unit draw unit sampling replacement unit find difficulty define actually chen lr developing say possess generalize show derive lr c say gamma refer generate ks show induction lyapunov fact k n derive multivariate partial vector yx say possess multivariate mild allow I x restrict integer negative taking obtain multivariate generalize let constraint p mass setting nonnegative provide ii proof constraint coefficient distribution accordingly k c c I number x application note binomial integer theory say possess restriction allow say possess multivariate distribution r define multivariate eq set integer hold iv see constraint nx nx I multivariate generalize multinomial c c define nonnegative positive integer possess probability x distribution define real number z z gamma matrix therein wishart distribution size function positive definite matrix yy yx n p n z tr z n p probabilistic inequality fundamental lr theory concept lr limitation lr method moment function apply wide spectrum eq combine indicator yield result let random positive I let provide lemma since suffice central z sequel restrict definition imply convex increase restrict positive assertion z z value lattice assertion n z z show complete notational pmf variable unit integer integer great chebyshev inequality z z n z lem provide pmf pdf notation sufficiently sequel small enough show q virtue x n z sufficiently chernoff complete assertion assertion z z x assumption combine yield x n complete assertion assertion iii assertion assertion assertion assertion I n x fx apply establish assertion assertion vi x n iv vi complete pmf define denote pmf eq nonempty consequence gx r bm assertion verify consequently gx ax n r gx assertion similar r fx bx fx make bx gx bx gx b complete follow n r r n rr ar rr r nr nr r n nr nr r nr n n rr hence lemma possess follow observe seek yield likelihood purpose define z n z check z z noting obtain substituting minimize z case poisson reduce exponential pmf moment generate x lyapunov iv x assume gx gx assertion z nonempty assumption gx first assertion z z ax need two case gx ax sx consequently meaningful ratio ax sx xx z xx case hold ax assertion assertion gx cb nonempty z bx bx need case ii cf sx define q x bx sx xx bx positive bx z gx lemma I z nn eq clearly ii follow lemma assertion iii chen assumption c e z ic c nn iv gm claim claim contradict assumption result iv thus quantitie pmf x fx pre fx gx z b fx gx nonempty assumption gx show assertion need case consequence k I define meaningful z sx z gx assertion cb nonempty assumption gx bx show assertion z bx need consider x sx x meaningful ratio c bx sx bx xx bx proof assertion z c c c iv argument x x use derive equation imply eq note fx z z tr follow manner complete lemma derive probabilistic inequality base bound powerful frequently derive discover inherently concept maximum also establish moment concentration inequality readily moment significance engineering obtain event tight vector event represent certain deterministic frequently variable bound monotonicity bind chebyshev bernstein hoeffding follow chebyshev let variable mean chebyshev negative x refer variable se variable real number ss e discussion seek bound convenient minimization derive probabilistic expectation I view crucial role bound I drawback mathematical random chernoff difficulty encounter value minimize w I method fully exploit information mathematical expectation summary issue probability probabilistic drawback I density pmf pdf pmf parameterize gx e central ratio deriving refer ratio lr demonstrate I technique lr idea I pmf multiplying pdf pmf e comparison see distribution directly involve indicate lr allow key lr bound tight amenable pmf e x g inequality respect
overfitte unit base literature relative interestingly latent estimator encoder time latent high estimate monte hmc sampler appendix convergence figure choose recognition see mnist inference straightforwardly optimize efficient auto encoding vb estimator advantage reflect variable direction hierarchical architecture g use ii iv supervise distribution p variational maximized contain kl analytically variational element auto neural encoder decoder output neural sigmoid activation mlp encoder decoder multivariate diagonal weight bias mlp encoder estimate long sample low less stage base new fitted monte carlo em gradient p procedure hmc automatically stepsize acceptance weight update step acquire update schedule marginal posterior oppose likelihood first low kl divergence equal posterior match compose rewrite rhs marginal expectation rhs obviously separate expectation component analytically mild posterior pg function q notational shorthand monte carlo therefore estimator sgd gradients latent center isotropic variational posterior eqs possible construct estimator model analytically result element group inference presence continuous intractable large variational scale mild intractable yield straightforwardly optimize stochastic continuous per inference especially fitting estimator reflected perform variational vb intractable unfortunately expectation variational simple stochastic almost latent gradient ascent technique case latent variable vb allow approximate inference allow expensive iterative scheme per learn arrive variety direct graphical per maximum posteriori map latent scenario variational dash z variational jointly dataset consist unobserved step pdfs everywhere unfortunately view unknown simplify assumption probability conversely even case likelihood marginal p intractable em intractable case moderately p batch optimization costly would make update even single e monte carlo involve loop relate efficient resemble variable value useful datum representation marginal kind require denoise super purpose recognition intractable approximate mean inference factorial jointly generative representation refer produce distribution probabilistic I rewrite term posterior divergence rhs write also I problematic na I monte type l exhibit high impractical section please technique case condition variational bayesian infer certain condition outline section strategy w yield estimator eq integrate exponential cauchy reciprocal analogous laplace student uniform express transformation normal normally gamma exponentially sum chi fail cdf exist require pdf see give network posterior center isotropic multivariate note multivariate bernoulli compute mlp fully single intractable take approximate posterior multivariate approximate I nonlinear I compute result decode mlp model knowledge literature applicable employ recognition approximate true posterior drawback advantage apply discrete computational receive increase interest variate reduce exponential family variate scheme reduce variational inference approximate auto class long ml case specifically case relevant autoencoder training autoencoder maximization negative reconstruction regularization make autoencoder learn useful representation sparse autoencoder variant objective nuisance hyperparameter decoder architecture psd recently auto employ boltzmann model I like boltzmann machines probabilistic model
concern concern subspace show result substantially high c cart cart mse std std std std mse std std mse std std sim cart mse std std mse std mse std std mse std time mse std cart lasso std std mse mse std mse std result pt htbp mm ii iii fy py ix define path want response future observation strategy elegant embed unlike expansion predictor allocate close center euclidean summarie center assign computing variance whenever recursively coarse split subset drop choose posterior regression classification cart rf
row permutation identification make need consider permutation minimum contradiction imply j provide restriction important case sample equivalent imply identification special identify full diagonal dependence source verify condition theorem interest identification algorithms order widely exploit type diversity derivation additional example paper parameter inverse see identifiability discussion section portion matrix formulation n k k mix potentially compact k source simplify q q k simplify notation random vector realization multivariate k multivariate extension follow extension cauchy schwarz arrive measure capture elliptical broad scalar quantity mean vector elliptical distribution frequently nonnegative make integrate elliptical elliptical gaussian model elliptical elliptical less source form directly eq clearly elliptical source separation hold second elliptical three elliptical covariance n successful size dataset performance low number versus simulate exact knowledge shape use except identity median approach size increase median sample behavior theorem away increase source dependency knowledge account move average k trial lag compare lag estimate matrix trial l lag varied lag use recent general variety algorithm essentially dependence way principal versus individually dataset increase set source identify align third maximize achievable separation source align bind separation clear gap diversity complex value improper assess sample dependency entry computation useful k assumption nonzero k matrix green nonzero along I k n appendix score scalar matrix elliptical let transformation utilize er pr r dr k definition factorization separation ica align unknown blind order uncorrelated bind minimization via statistic fourth laplace gradient descent optimize newton newton power exponential nc gaussian nc gradient nc nc newton nc nc imaging dependent interference power autoregressive component trick extension multiple term subject research also generalization correlation condition account sample result furthermore aim identification source dataset arbitrary ordering bound term bound well algorithm array application generalization term frequency bin concept achieve example formulation formulation term instantaneous assume within mutually possess identifiable independent source gaussian possess exploit source general present account dependency formulation iv review notation achieve likelihood practice term describe section vi use derive section identification bound generalization bind express compactly publish algorithm section future mention date pre achieve dependent analysis serve review derive principle review source type diversity utilize first source dependence across dataset independent extend beyond term cost second result solution linearly dependent use equivalently minimization readily generalize estimate possess high measure two extension summarize another dataset univariate maximize mutual vector propose transform kernel transform dependency similar extend permutation ambiguity non laplacian use source order source exploit sample lag find minimize correlation lag see domain indicate respectively quantity denote face bold face respectively vector mn nm transpose hadamard element division kronecker denote mn compactly stack row vector indicate diagonal diagonal entry representation row partition imply I notation variable expectation mutual use gamma generalization contain form th namely quantity independent write p np source dataset k possess specify logarithm likelihood block diagonal sequel n n matrix v recall normalizing minimize entropy rate regularization equally information responsible across information useful score function n k fisher information dimension compute purpose identifiability need around general complex depend complexity unnecessary depend source prove k diagonal compactly appendix block n kn form result complex
j j u j visit location also online reveal user activity divide day save hour feature extract share inner cosine tu euclidean tu tu tu two fact user tweet lot word usage make friend bag extract share use online inner wu wu wu wu wu feature link user totally different user utilize old user target sampling accommodate traditional inherent predict old mention address accommodate old user user method user totally different heterogeneous across non align homogeneous meet objective new accommodate information user old heterogeneous diversity great preserve user link sample heterogeneous target e user network sample old user heterogeneous denote old old network relevant heterogeneous network network user auxiliary eq setting auxiliary category many auxiliary aspect similarity user relevance old value vector besides relevance old preserved relationship average user q social link diversity social link probability old old sub indicator function originally link target old user sub old old sub ensure ensure preserve add regularization sampling maximize term I j user user preserve link decide sample social except social rate existence link decide user need combine diversity term old old maximize regularize old network importance diversity link prediction traditional link target train classifier classify potential social consist user old user information user old prediction old user usage theoretically could well target consider user possess amount would suffer long start cause new preference even deal information target possess link auxiliary old predict deal simultaneously section target suffer align source improve align recommend recommend align base intuition start term denote source pseudo target decide whether recommend align help align link align network recommend predict could mean start structure always utilize social linkage overcome mention supervised align align build social align account align simultaneously category link align merge expand together label build existence social work align target user preference leave old use conduct user addition mention method utilize denote existence utilize multiple old incorporate training assumption relationship access align intra transfer tweet follow challenge whether reality conduct align social twitter description dataset summarize twitter twitter heterogeneous tweet tweet location possess tweet network know well available link contain anchor link acquire twitter account aa acc cn aa acc effectiveness link new supervise parameter base auc accuracy comparative give description align source process compare another baseline information sampling name build target network old could old use network simultaneously besides method target compare build social baseline unsupervise cn aa use social information source align network link auxiliary preserve sample target regard old link group number link new user organize two fold fold use testing social link inside sample old inside intra transfer social relate old link heterogeneous negative link align network network feature merge expand twitter source use reverse evaluation method score use evaluate method evaluation use source old improve old could performance reveal result increase remain increase user become achieve work align use start could another align align start table twitter recommendation mining study heterogeneous et al author approach link however heterogeneous network develop framework classify tie bias propose anchor link network location base become recent year predict link network social link heterogeneous wang try social move user year al phase bootstrap deal auxiliary al similar available start paper use heterogeneous align old network extensive great success recent involve multiple link great link focus future upon snapshot day new link new difference user old old user new user network normally involve service time facebook twitter social active another long supervised align network align account accommodate intra account align solve heterogeneous outperform method consistently social become popular year many involve kind link among social link social connection among meanwhile network frequently potential link among user base upon snapshot network treat try link world network user service user active network leave impact decide active turn user away create good long old work link formation probability age node recommend link old link user lead study account period prediction new link prediction explicitly train real world social old may activity activitie social link auxiliary old usually activity figure old old totally twitter old upon user challenge user transfer intra inter intra inter significance social link social network challenging reason old user user link cause need source social link linkage yet cross align link recommendation prediction result difference user sampling accommodate solve social align source network inter simultaneously make improve paper supervise base heterogeneous network
permutation matrix see row basis exchangeable entry accord haar therefore hermitian haar measure semi circle scaled haar measure case geometrically write haar geometrically represent angle draw haar anti md md ii singular independent simple assumption row argument exchangeable arrive conclusion assumption exchangeability column proportional equality exchangeability generic corollaries elementary wishart next I entry except pick random opposite assumption hermitian block diagonal draw semi circle operator finer understand singular turn situation investigate behavior dramatically limiting aim however closely involve determinant involve permutation exchangeability actually apply moment recall eq provide moment satisfied appendix element matrix arise consider non refer reader subsequent interest view summary pure variable independent independent agreement block averaging behave setup haar measure mention qr implement matlab accord need numerically spectrum versus symmetric good approximation circle law r gaussian next block entry histogram spectrum versus semi circle law versus eigenvalue middle right th qr indicate leave black circle less versus plot resp resp note spectrum show eigenvalue spectral outlier careful sampling inside nr nd di ji flip sign ensure get spectra eigenvalue random histogram broadly subsection cauchy semi limit fall circle show course naturally imply nd matrix leave behave corollary kind average discretize simulate l please realization rotation grid minimizer choose q discretize equally spaced degree minimization rotation block spectra quantile figure h rotation bottom histogram right symmetric numerical might might eliminate possibility guarantee rotation sample serve uniform discretization rotation realize surrogate image invariant behavior noise define affinity resp eigenvalue pn pn ij versus deterministic quantification look use aspect proposition asymptotically approximate independent paper ensure entry matrix light appendix distribution explain histogram eigenvalue plot uv reasoning repeatedly suppose therefore perturbation since right side result exchangeable exchangeable exist clear exist cauchy inequality exchangeable assumption row exchangeable deterministic function deterministic formula definite great semi definite fan yield q clear q lead rank inside hand theorem conclude eq consider satisfy far imply approximate quantity take expectation conditioning care depend give describe value write svd invariance haar distribute induce dependence deal interest singular singular g ig follow eq content detail call call vector density density change th jacobian determinant call determinant circumstance jacobian density eq large decomposition wishart appear eq therefore independence moment particular independent random stochastically cauchy intuitively among entry dependence careful need carefully address average independent conditional action equality come distribution conclude depend previous similarly random variable result apply course argument eq manner start prove write conditionally argument similar give variable variable grant dms wu fa thank anonymous constructive lead substantial improvement definition thm analysis technique dimensional massive appear investigation develop important behavior theory cover block numerical agreement simulation connection laplacian new datum dimensional massive dataset use system localization conceptual generalization laplacian commonly apply learn analyze though think resolution picture high euclidean live dimensional embed understand generalization live space geometric property heat dm capable topological object datum analytic operating rotation subset cloud relationship direct take rotation account reduction rotation cloud parametrize manifold group heat laplacian bundle popular laplacian dm tool understand manifold practically give numerical introduction motivation address problem arise mathematics field noise important modern dimensionality existence design noise account natural seek impact noise broadly reader familiar give rise generalization kind sometimes surprising property motivate algorithm estimate discuss low dimensional curse space may interested extract gap indeed call density growth parameterize child growth ray x ray transformation eq parametrize r vary rv nuisance parameter describe patient general formulate metric equip call left group action satisfy operation nuisance act parameterize nature setup non literature viewpoint remove nuisance generally reduction nuisance underlie ray projection image parametrize sphere take trivial account dm commonly report therein benefit generally computationally importance situation might lead far improve dimensional reconstruct x ray direction see thus symmetry describe nuisance embed nontrivial alone non aspect topological tangent bundle laplacian approximate construct denoise class averaging summarize framework random denote build affinity affinity quantify nuisance among block entry block eq analyze eigen assumption influence I formulate group block q statistical property turn influence signal situation high important much fully answer question purely independent noise mention block block way motivate general additional circle particularly block dependence among light whose spectral naturally understand limit spectrum dataset basically pure limiting result furthermore deterministic first enough gaussian counterpart develop situation except row hermitian eq hypothesis state matrix analysis extra freedom value depend manner deterministic algorithm matrix develop aspect average matrix call diagonal entry give therefore diagonal eventually distribution except size turn block say matrix gaussian entry random consider matrix row row symmetric assume ni z check replace symmetric norm deterministic vector ni choice state satisfying gaussian ni ni gm ni ni gm method replace block diagonal condition thus apply need satisfied translate matrix easy reader block deal block block entry write th present sure appear introduce p block th I symmetric call th block row sufficient satisfied block assume symmetric assume th moment entry hold enough understand form type compose independent assume clear compose independent block length assumption independent ji j j appear appear block block far naturally eq uniformly bound covariance I use py pp hence conclude q row matrix covariance row bound moment proof block variable symmetric moment automatically satisfied matrix
united public school home half percent public company job drop big finally home percent office www game percent return word player york visit country start public hour lose company head pay percent com game school company right delta company play page percent home house big south book percent company play business lose job reason com school company american york lose country mind job abuse house home security york closely big topic geometry word significance novel topic highly data pattern novel projection method mild document project along direction complexity recovery art random qualitative document compose word choose distinct adopt classic bag generate unknown unknown probabilistic document vector mix weight iid correspond word realization topic frequency vector fundamental document topic matrix nmf provable satisfy separability topic condition suggest novel unique topic identify mean key insight word associate consist occurrence hull base project direction identify false multiple word topic issue belong linear frequency scheme complexity art average per contain world qualitative superiority extensive base several attempt joint suboptimal often approach model column estimate inherently approximation expectation propagation provable guarantee propose moment impose topic require prior prior topic agnostic empirical moment singular decomposition important provable separability word second topic matrix correspond extreme point scale get small increase empirical especially enough novel word independence extreme convex hull serious dataset co occurrence lie separability novel document rather co occurrence hull pattern associate technical approach appear mirror lda degenerate case low conceptual level appear word belong robust fluctuation occurrence approach projection point organization motivate statistical propose practical word distribution weight ai def word word geometric intuition ik ki generality word extreme topic matrix distinct calculate use linear ex proposition solve system specifically validate approach identify available however even collect enough document asymptotically precision section geometric mention proposition extract novel algorithm cluster novel topic topic suggest illustrate identify novel word convex body project body point choice projection simplify subset statistically document normalize threshold margin specify correctness define exist constant h input indice j j justification time proof sketch sparsity mn ic rp use extreme reduce require rp iid unit direction onto input index generate sphere max projection consistency rp word projection rp algorithm computationally efficient split sized bin maximum winner win j dp strictly help identify word outline input sized document jk w l contrast rp algorithm agnostic novel word significantly rp detailed justification rp rp find novel word rp extract copy scheme consistent let rd novel word great word topic graph correspond edge word cluster reduce procedure word representative point cluster novel could directly describe part modeling exploit consistency validate fact consistency j j k b describe step mild elaborate argument omit correlation matrix minimum topic must appear substantial novel distant imply two probability assumption rather supplementary section dirichlet traditional validity numerically logistic matrix randomize projection non minimum respectively novel asymptotically novel constant find outli sketch detailed justification provide supplementary statement high ij ic converge converge positive statement prove ex seems dominate basically proportion document j sufficiently remarkable similar bound noted complexity would decrease consistency true novel input row word correctly least proof supplementary positive word zero hence connect graph novel different topic ex novel hardness clustering using finally suppose give index distinct assume minimize compact uniquely function r r b continuity accord ex approach outline section dependent projection least require apply know construction maximally distant novel clustering could size spectral relative small typically add detail rp follow dataset image agnostic dataset rp htb validate synthetic word simplex iid novel realization iid iid settings topic grind good average rp nmf nmf practical provable nmf type topic depict bottom rp well comparable nmf second note rp outperform fairly meanwhile htb cm cm c cm cm cm cm cm cm htb cm cm cm cm cm nmf noisy dataset topic clean ground truth truth topic arm cm cm cm pos la la rl rl cm cm cm cm cm cm cm cm cm cm cm cm cm cm cm cm cm cm cm cm cm nmf clean close gibbs nmf ground truth rp recover look clean distinct position pixel interpret separability ground body background pixel value clean iid apply compare nmf fig discuss see nmf pure arm indeed compose nevertheless show error rp image fail clean datum extreme algorithm linearly possible linearly last row produce truth extract close ground htb dataset propose projection rp circuit analog circuit analog device gibbs circuit n rp spatial orientation cell visual activity orientation orientation cell visual rp learning error rp training recognition speech recognition network word hmm speech recognition acoustic speech position rp algorithm rp weather wind air rp character heart rp vote vote rp game play super algorithm different world corpus vocabulary average document another corpus new york times standard character english order prune experiment vocabulary size ny follow ny implementation detail successful depict extract frequent list two extract group fraction topic recognize rp similar observe rp panel extract weather meanwhile table define choice consistent find novel outli novel write converge detect novel define probability right consistency corollary fail novel constant cluster support row retrieve furthermore statement union center fail truly define distinct row b notation r r hand minimizer strict follow fact positive therefore q b relationship normalization verify involve convergence conclude sense convergence converge normalization factor column hence e column normalization constant assume simplify expression finally estimation algorithm topic group topic topic individually htb analog circuit figure visual speech mlp acoustic gibbs observe parameter similar decision structure figure extraction gibbs prediction nonlinear motion direction velocity head radial net architecture feedforward global gibbs gibbs activation time sound gibbs rule markov gradient object pixel gibbs fire gibbs patterns storage matching instance gibbs controller encoding element human gibbs filter character module gibbs cluster distance bayesian back gibbs random action action goal environment htb loss site fire teacher principal loop vector importance signal predictor concept greedy weight orientation tune implicit encoding selective switching occur anneal assignment correspondence role symbolic distribute spectrum code parameterize memory capacity layer probability risk history weight divergence mutual filter scene neural delay delay adjust hmm bit encode set neighbor split trajectory controller learn circuit analog recognition rotation letter processor list serial block orient competitive head formal subject structural dot character state reinforcement rp projection rp data music object object rp neurons spike rp video sensor rp rp template network input component rp model cart cell rp leave rp visual cell orientation rp neuron current fire rp margin verification signature rp eeg blind rp controller rp cell cell fire rp human chain profile song rp algorithms rp circuit analog rp state delay load neural network query dependency query rp sound localization head cell position rp bind structure rp teacher rp rp speech recognition performance hmm mlp schedule execution scheduling counter rp rp action action rp language spin rp contour texture rp color orientation rp prune elimination rp module unit share phrase rp character character processor processor htb asymptotic learning policy recognition training network cell operation model view string spike time neuron neuron recognition network maximum motion visual generalization output teacher length cell easily proportional dynamical low image local control model error probability winner unit black orientation visual cell mit memory neural neural weight network circuit vary set task cost feature visual figure cell activity neuron figure neural period ensemble backpropagation hide word time rp rp com american rp building house home room minute rp article separate american country american room rp plan rp pilot rp world rp team rp cat rp job office rp home shot minute rp team rp goal process rp human science call rp microsoft software window rp china chinese united states rp body head rp big business find rp weather wind air million rp shoot rp ask room rp school teacher program education college rp rp investigation evidence rp economic rp player games fan rp company million rp percent survey rp american history rp publish rp house vote rp aid mail rp vote rp rp claim rp find light image sound rp cell human rp rp rp california rp help rp letter mail read rp play production rp series rp game home rp rp york york city rp contract rp attack united rp media public rp black white american rp rp hour road car rp drug patient house office rp company market rp action rp security water tree rp com www mail online rp team game rp death penalty rp country party cut program rp al political rp united states company internet technology rp rp rp view matter rp work movie actor movie remain early despite
variable proof refer tolerance frobenius typically iteration depend easy calculate one option option use wishart replace prior forecast distribution time information diagnostic bayesian bayesian available absolute mean model capability base bayes criterion relevant nest incorporate prior tendency propose bridge simulation discuss criterion upon discuss bayes minimum compare discount factor e sequel choose hyperparameter depend exclude principle maximization include implicitly ia state positive jacobian ignore reason keep depend conditionally estimate former mode differ line optimum maximize discuss bayes factor west discuss basically odd competition odd application bf fy example west preference possibility differ discount factor monitor describe application threshold jeffreys p select volatility small sequential mean loading volatility sample prediction sequential portfolio aim find return minimize unconstrained strategy compute transaction realize visually weight portfolio allocation discuss west reference adopt criterion apply discount factor scenario carlo experiment assess model generate process estimation wrong scenario model generate variate repeat experiment matrix carlo volatility discuss sensitive rw bayes factor average portfolio risk rw portfolio discount range large posterior average portfolio risk model factor portfolio outperform rw basically illustrate estimating oppose ht rw risk firstly sampler burn stage sample draw burn monte carlo mode obtain onto portfolio exercise yield slightly disadvantage consume ij exercise average portfolio model rw good well although put absolute upon sequentially correlation forecast volatility show element indicate highlight increase volatility evident panel relevant initially figure increase constant center comment tolerance achieve methodology volatility volatility stochastic precision volatility wishart autoregressive unconditional autoregressive parameter methodology procedure volatility propose probabilistic finance financial automatic trading demand acknowledgement grateful anonymous helpful comment considerably version paper appendix multivariate beta provide detail multivariate demonstrate wishart aim financial aforementione propose multivariate modelling mechanism define wishart conjugacy distribution formally factor decomposition wishart integer wishart element singular beta attract considerable recent year refer x ij ij cx cx cx I put cx b new volatility estimation wishart consider volatility procedure adopt autoregressive step unconditional newton iterative suitable medium illustrate multivariate wishart financial last decade effort devote vary related literature recognize asset generalize autoregressive stochastic volatility suffer curse dimensionality review yu employ particle estimation yet reason consider specification issue secondly upon slow difficult researcher numerical differential equations carlo practitioner largely volatility bridge gap attractive practitioner work contribute suitable medium paper suitable become necessity trading section wishart autoregressive process precision develop variate develop ar wishart volatility process ar precision process identify arrive conjugate discuss diagnostic portfolio ar volatility limitation evolution et exchange five suggest volatility similar computational par recent consider al empirical find west et dynamic comment log return arithmetic return exchange price list value exchange rate return arithmetic setting conditionally denote historical follow evolution covariance I strictly definite assume wishart determine autoregressive decomposition practical application shown accommodate wishart ar multivariate next motivate random matrix property discuss discount control move specification follow beta great allow give dimension evolution similarity author smooth use claim walk expectation preserve expectation equals basically extend several discount factor discount slow paper autoregressive consider multiplicative autoregressive process conditionally equation wishart degree write comprise posterior ar discount require give mcmc approach gibbs hasting aim bridge gap adopt first distribution step posterior work support cause close application equation specification discount responsible magnitude introduce large west consider condition normal follow denote inductive conditionally discount consistent posterior calculate detail discuss expectation set guarantee model express consider west chapter shrinkage first preserve propose discount factor agreement claim expectation use difficulty discount estimate note responsible beta
google usa google com efficient constrain completeness brevity denote integer simplify label activation gradient eq box need sequel nonetheless positivity index drop solve lagrangian non multiplier optimality saddle lagrangian index solution need index solution zero thus close lagrangian clarity us monotonically piece wise monotonically decrease monotonically therefore slope value set set form admissible decrease namely decrease order definition next knot brevity maintain additional value twice step sort component slope newly encounter keep track knot need
relevance determination ard marginal complete fully prior new necessary unobserved namely appeal long condition covariance integrate crucially account covariance distribution integral directly alternative characterize propose employ integrate quantify introduce integrate applicability acceptable pattern recognition mcmc gp gp face covariance make attempt jointly effort satisfactory still miss comparative notice alternate p obviously likelihood marginal entail integration analytically deal elliptical slice ss define transition operator slice variable ss begin randomly choose drawing likelihood mean cosine latent slice start efficient gp remainder however variant hybrid monte hasting sample propose draw user proposal evaluate hasting accept reject previously pm remarkable state possible likelihood ratio mcmc marginal posterior mean expectation propagation get achieve draw sample approximate unbiased p adequate approximation grow exponentially limitation variance eventually lead acceptance likelihood severe slow convergence low aim methodology capable ht multiplication grey result first anneal approximation red remain second anneal procedure assume ss going derivation unnormalize density next intermediate unnormalized begin draw iterate finally q unbiased normalize immediately numerical safe note although anneal inherently serial computation analyze implement gp visually prior ard recommendation base implement spaced spaced transition involve highlight employ effectiveness deal problematic amount increase dimension balanced distribution non order variability estimator draw preliminary mcmc ideally perfect marginal yield degenerate variability helpful order concentrated span anneal confirm offer marginal annealing reveal estimate notice increase polynomially approximate drawing initial iterate ss importance require operation th cm cm cm c isotropic c breast cm cm ard pm set multi turn class window class label window repeat varying importance ard tune mh run preliminary initialize hasting la adapt acceptance useful avoid tuning mechanism marginal poor acceptance report acceptance switching obtain acceptance iteration discard result across datum general trend employ pm improve acceptance replace affect pm affect case consistently offer way acceptance present application importance gp importance construct variable methodology impractical demonstrate likelihood exponentially crucially polynomial importance unbiased correct real employ pseudo mcmc satisfactory general improves suggest promise research unbiased fashion acceptance overhead third indicate gp classification importance distribution investigation gp furthermore sparse inverse popular spatio use sparse attempt anneal sensible minimize focus covariance monte optimization systematically superior quantify marginal intractable unbiased discuss drawback sampling application importance marginal scale polynomially step development automate method pattern machine nonlinear modeling capability quantification uncertainty bayesian paper focus covariance gaussian carlo particularly gp kernel offer use variable integrate pseudo practical efficiently process exactly sample infer efficiency pm importance approximation latent gaussian poor thus large effect
negative exploit case consider definition kronecker product entry diagonal combine hand combine substitute apply logarithm small keep mutual coherence moderate use relation minimize dictionary minimize training perfect mutual know smooth manifold allow method learn concept refer interested reader riemannian euclidean let consider assign tangent pass element tangent riemannian gradient tangent direction ascent globally entire riemannian space smooth describe path intuitively interpret straight riemannian tangent iterate scheme search formula regard g consider manifold b orthogonal tangent read tangent tangent consequently accordance projection b bt endowed ingredient close geodesic implementation sphere great eq geodesic simply search due structure geodesic iterate employ offer acceptable direction equal iteration since space I I tt derive geodesic geometry tangent via hybrid show excellent counterpart phase extract image course patch zero mean random column initialization parameter atom separable dictionary noisy solve fista regularization compute solution pixel exist final clean patch among exist technique denoise dictionary use dimension dictionary table employ always employ separable dictionary employ predefine separability allow popular overcomplete cosine transform separable denoise corrupt level respective five fista right middle fista middle bottom c c besides along learn learn sparse representation image patch dictionary demonstrate capability domain separable face image face face database remain face five result ability dictionary conduct fista eq achieve sophisticated extract htb htb dictionary dictionary employ structure employ dictionary due separable dictionary learn dictionary task mutual coherence coherence propose exploit underlie numerical image denoise show ability experiment acknowledgment technical foundation de computer machine dictionary analytic structure learn dictionary often perform adapt consider signal dictionary patch capture approach drawback throughout process permit large reconstruction basic property mutual coherence explicitly separable reconstruction combination read transform coefficient exploit crucial assign dictionary popular dictionaries dictionary formally arrange column transform coefficient problem therein g predefine admissible probabilistic cluster comprehensive overview dictionary dimension dictionary inherently limited computational resource within vector multiplication computationally dictionary applicable crucial allow dictionary structure mean kronecker small dictionary top employ separable reduce cost computational burden reduce approach dictionary class inherently however straightforwardly employ kronecker fix notation rest dimensional sparse dictionary scheme product dictionary mutual coherence riemannian conjugate line dictionary patch yield denoise separable dictionary dictionary analytic counterpart overcomplete cosine one achieve performance show global contain learn face pixel face region dictionary costly unable deal dimensional signal dictionary review approach idea atom analytic propose atom coefficient impose restrictive enforce entire dictionary problem capable dimensional signal signature vary near translation invariance dictionary approach extend learn invariant atom hierarchical framework framework conjunction mention framework h overall sparsity impose regularization
obvious quantile htp htp partially explicitly I j j generalize kind selection partially conditional quantile bayesian component fit pre specification spike inference design partially collapse approach real quantile variable model case additive parsimonious dimensionality widely practice environmental apply intra load quantile approach partially identically dimensional predictor intercept univariate quantile valuable expand economic science complete description additive nonparametric bayesian propose additive model work focus penalization perspective number paper component inference express basis assign prior basis variable enable none work least square component quantile article ability separate nonlinear effect irrelevant quantile nonlinear adopt asymmetric distribution error selection introduce set indicator possible component linear remainder proceed also discuss algorithm collapse sampler regression laplace give introduce write omit expression impose identifiability model spline knot spaced knot basis kx nb kx separate basis identification nonlinear spline eq q follow component transform marginally apply distribution mean deviation distribution quantile quantile level generate fit replicate burn performance approximate posterior burn average deviation replicate function obviously indicator poorly nonlinear outperform indicator reduce regression however obvious present mean absolute regression quantile check datum predict burn quantile median similarly mean student replicate replicate error rmse student estimation probability linear table component component truly nonzero base dark grey area percentage nonlinear see nonlinear htp component demonstrate propose save present table display figure rmse ad rmse student normal student htp htp standardized transformation method sample datum economic dependent logarithmic year consider four variable covariate variable measuring consider variable quality drop category variable country area day month percentage country classify percentage country km ice km category category include country linguistic combine linguistic characteristic share language rate selection nonzero linear nonlinear level covariate effect quantile lower fit covariate diagnosis regression figure plot ten htp htp besides production et al effect development find role conclusion attention impact economic house include environmental concern transaction four physical lot house lot indicator population average locate country express
performance datum carry handwritten digital point performance ssc lrr fail achieve database within acceptable report time nystr nystr om subspace resolve issue problem ssc simultaneously reduce ssc lrr linearity problem preserve extensive effectiveness perfect perfect small fraction dependent zhang zhang substantial x l zhang cluster two ssc lrr scalability lrr recently construct similarity ssc lrr inefficient moreover ssc lrr cope low rank membership matrix ssc lrr overcome effective make ssc new scale code classifying specifically split two part cluster assign near minimal analysis show efficacy cluster cluster randomly fundamental topic recognition mining aim intra decade extensively linearly numerous kernel clustering belong low high dimensional datum could project space project space membership derive cluster datum similarity lie heart graph connection use generally metric build similarity computing value alternatively point regard robust outlier connect word low fix rank representation achieve scalability issue resolve framework ssc low lrr base lrr ssc lrr nearby without fix ssc involve point calculate graph matrix fast medium sized bring ssc moreover ssc ssc whole membership make ssc fast online lrr suffer cube effective lrr cluster base sparsity union span approximate span use without scalability believe subspace scalable propose scalability ssc lrr code classify part subspace span cluster perform ssc lrr near minimal highlight sample reconstruction membership fast online ssc lrr scalable reduce original cube linearity preserve extensive show reveal even though outlier word issue ssc lrr without loss provide review ssc lrr spectral spectral ssc lrr section carry dimensionality number cluster kk column transpose notation use paper researcher sparse task face work independent disjoint ssc problem set three equivalently data problem survey get datum ssc cluster ij eigenvalue get perform assignment computational ssc ssc homotopy optimizer homotopy optimizer one iteration optimizer consider task moderate explore challenge task diverse set rank extensively study q unknown know finite doubly exponential benefit development compressive could singular difference lrr low nonzero norm frobenius norm assume adopt specific outlier corruption gaussian liu adopt augment alm nuclear generally perform svd eigenvector matrix lrr alm lrr implementation lrr balanced desire get low solving eigenvalue k row cluster ssc lrr cope affinity come ssc lrr datum lrr fast online clustering devoted solve scalability one natural option cost eigen al propose nystr li et perform nystr efficient chen distribute original point tree chen firstly representative point randomly construct wang selective sampling technique locality preserve et spectral embed sec come perform subspace nearest classifier select represent popular efficiency focus one intrinsic characteristic develop scalable ssc moreover lin optimization quadratic lrr penalty learn low reduce liu time dimensionality zhang locality hashing truncate lrr sparse affinity representation linearly focus solve representation rather develop method sample problem scalability subspace cluster large verify apply ssc lrr correspond scalable subspace cluster scalable rank treat scalability ssc lrr sample classify small step low minimal union subspace space span data point small portion denote adopt numerous small assumption side coin sample scalability lrr achieve comparable ssc lrr complexity cube sample original subspace could span point get adopt random cluster ssc lrr subspace approximately span non sample euclidean space euclidean adjacency relationship among task subspace cluster low solve sample dictionary assign subspace optimal q recent show representation could competitive cluster linear sparse term call avoid zhang name show representation near residual assigning summarize ssc lrr scale datum parameter randomly denote ssc lrr get membership calculate residual subspace via solve subspace produce minimal subsection point fraction derive part lrr residual lrr succeed produce define outlier corruption linearly segment identify clean dictionary independent denote point lie new easy obtain subspace perfectly segment subspace affinity group desire show correctness theorem consist contain could contain randomly inter effectiveness theorem need homotopy minimization use eigenvector laplacian number homotopy optimizer mean need compute therefore put everything largely ssc ssc lrr number alm section conduct scalable subspace scalable low carry seven digital news consist set scale brief image lie manifold naturally satisfy three database database database vary illumination subject clean image subset face randomly subject capture simultaneous pose illumination experiment moreover computational ar perform retain feature uci unbalanced sample compare examine subject ccccc dim feature cope several scalable nystr om report nystr om denote om nystr om affinity nystr om nystr om column randomly select sec obtain run intel ghz processor gb ram codes nystr nystr perform center avoid pre partition datum produce cluster ground truth category match whereas totally permutation mapping cluster label entropy respectively influence influence parameter influence take level parameter prior distribution evaluation result assign value fail moreover range vary vary range follow experiment evaluate get adopt homotopy optimizer calculate sparse data optimizer
misclassification regression mixture variance intra level approach display series mean three polynomial illustrate contrast model subject change regime htbp cc cluster htbp allow train rate intra cluster intra cluster display cluster mixture misclassification intra mixture univariate change regime polynomial regime smoothly logistic likelihood clustering operate segment cluster regime fill stroke g universit de centre de bp cluster multidimensional popular implementation deal regime vary smooth regime estimate maximum method solve algorithm regard operate change regime provide time select segment solve efficient electrical consumption switching rise mechanism enable train track preliminary diagnostic identifying switch operation characteristic electrical consumption various switching kind refer context adopt successfully numerous domain maximization framework typical series regression random polynomial spline autoregressive series word autoregressive time study subject successive within deal vary discrete process extend paper via em illustrate performance form unobserved correspond unlike vector component series curve coefficient distribute kp estimate conditional expectation maximization partition series mixture model cluster change regime independently lie follow cluster th polynomial involve individual observation series cluster j logistic l k logistic way ensure regime give individual accord write lead segmentation cluster appendix segmentation contiguous illustrate latent log maximization complete specification maximize easily membership different regression model log initial em e log conditionally observe denote current quantity quantity maximize separately maximize maximize separately iteratively square newton respect analytically weight element diagonal three parsimonious rewrite compute unconstrained constrain regression cluster updating write regression formula estimate algorithm partition time apply posteriori map cluster approximate criterion information unlike model parameter cluster degree polynomial criterion estimate free coefficient point view em bic criterion high bic solution devote carry real algorithm clustering criterion intra ik k binary estimate compare em polynomial sum polynomial logistic coefficient initialize spaced segment regression th segment perform proportion initial cluster
adapt parallel setting centralize break distribute processor distribute cost ignore work providing show solution average twice convert approximation factor outli point normalization multiplicative factor applicable mean zhang et construct accumulation decrease communication large span height accumulation communication accumulation ccccc partition similarity partition ccccc ccccc random span span p ccccc span tree span span similarity weight base median mean provable classic clustering reduce small size method previous reduce communication topology scale set outperform distribute cluster classic clustering design centralized database video surveillance sensor inherently collect become crucial cluster effective setting algorithm distribute empirically summary cluster quality additionally site paper provable set entire center cost original datum center previously centralize recently imply propose median node construct portion lead share efficiently precisely base node compute node build dimension central site collect communication cost size cluster topology algorithm experimental result perform summarize node construct node sophisticated approach reduce another root communication height although span diameter diameter grid increase construct overhead need construct represent sample communication topology require quadratic merge ignore cost review weighted solution cost coordinate point dp dp kk minimize several readily algorithm distribute graph edge cost simplicity communication goal center keep preserve theoretical distribute avoid raw compute datum drastically reduce centralized concept formal set center set ask construct set combine would greatly centralize cluster construction mean extended objective distribute briefly entire proportional intuitively close represent center probability proportional directly adapt approximation solution entire global fashion entire sum approximation compute local proportional cost center solution I ip I dp b p distribute size describe namely subsection small integer bf sp pm fp fp following show implicit sample every fp get difference bound precisely centralize definition lemma directly suitable different p pf pm b add center specific union sample key show choose local dataset weighting verification discussion dp db dp k least want center type local center approximation local construct triangle accord inequality solution cost cost center weight know communication nk kb local point show error cost cost weight show median directly median dp b db ph dp change divide approximately inequality factor eq may w bound begin op detail lemma bound bound w expectation b optimum since combine op suitable arrange connected neighbor propose approach globally share collect share local root tree weight instance portion message information receive message graph respectively subroutine exist approximation solution distribute communication om algorithm style copy approximation solution communication communication communication construct construct significantly reduce also root involve operating approximation mean respectively subroutine distribute root communication median cost send send construction send every root total construct paper compare assume communication cost build union accumulation subroutine dependence median mean algorithm top dependence height root sensor algorithm synthetic choose center center world spam letter point centralized distribute site topology include independently spam letter random graph distribute site uniform global equal similarity partition site site similarity assign distribute site grid partition run ratio na I algorithm zhang zhang communication span pick root perform search sim run sim pre run sim tree sim pre focus topology partition theoretical get thus communication theoretical uniform combine surprising reduce
multiple view use cover corresponding estimating transformation frame horizontal propose extension extract inform depth quantitative concatenation frame stack row individual frame frame accordingly analogy previous section factor representation depth response x detect encoding weakly motion encode response motion depth employ represent camera rewrite since sum frame identity filter response frame value frame correlation thus case detect depth motion motion thereby contain fact exploit approach dataset encode motion represent model explicit across combine representation section use concatenation third alternative frame frame contain obtain channel unit allow temporal representation write thought contraction derive amount md contraction unstable due presence weight alternatively denoise contraction typically detector reduce number show recognition linear projection patch thresholding motivated norm homogeneous amount simply discard norm feature extract dot propose section conduct experiment depth benchmark training pair truth ground capture calibrate approximately resolution fall crucial patch patch size depth filter localize parallel learn horizontal shift learn logistic truth intensity patch ground classifier depth involve patch follow sample show depth estimate depth depth procedure boundary expand patch depth surface rich region shift region depth markov merely similar bag feature take detector depth map region region observe infer come similar information activity next implicit encoding recognition video category video spatio temporal pair fix ten filter shift another across view evaluation perform quantization multi rbf kernel feature pair densely video super result block sub block block reduce quantization evaluate encoding recognition primarily encodes employ md channel correlation separately representation average classification classification base primarily depth motion evaluate average precision classification table detector motion motion model outperform date observe past sift recognition spatio sift confirm interest consistent albeit motion interestingly heavily action ap highest ap ap due likely related depth analysis future work popular base decision utilize type challenge depth view extract different represent environment l ap per across method c none md none th md focus depth mrfs reason biology make use depth depth depth inference forward paper depth motion well publish deep approach task domain university joint image multiple frame combination motion well combination architecture type cell pixel across learn achieve hand motion margin rely establish multiple scene frame video difference typical across geometry variable rely find position another essentially learn try exploit practice allow develop maintain piece make information source camera video stream energy mechanism depth motion g elegant explanation brain progress motion energy among video however depth depth show depth entirely complex type activity analysis camera use depth response invariance efficient implicit encoding depth implicitly grind since application explore variety utilize implicit motion evaluate variation demonstrate hand energy computing weight
discuss basic interval define constant consider generate neighboring slice column slice slice define slice usage derive easily expression define estimator look summation motivation wu theoretically justify concentrated instead perform metric suggest simple determine empirical estimate randomly pick call updating process local optimal optimal still consistent estimate true numerically computational ht graph initialize choose assign assign block observation detail find supplementary concentrated expected neighborhood highlight important iterate ij bind bernstein conditionally critical propose explain otherwise equality sensitive block define approximated hand number contain vertex sufficient block require generate relationship number vertex guarantee within second different lemma derive lemma force evaluate show thresholding entry experiment evaluate estimate arbitrarily space block grow observation mae independent one propose require fair nod graph show grow error second experiment algorithm end generate fix repeat trial generate result attain mae ht miss link increase give miss wise average b average evident outperform evaluate report consider special depend study depend structure generate low property likely option cc new tool approximate vertex block build complete derive find effective blockmodel online partially award award nf research fellowship partially foundation post fellowship conjecture figure approach recently gain define often parametric pose observe propose network base stochastic blockmodel vanish size infinity structure heart recent service momentum informative tool study parametric community non array connect exchangeability limit object local call describe see first connect eq adjacency represent particular realization block
elementary lead sharp sense noiseless case constrain affine ensure noiseless recovery large result matrix show sufficient approximately minimization nuclear recovery signal active area machine application process medical recover constrain norm signal signal measurement determine method signal al base transformation rank measurement recover matrix analogous compressed sensing suppose measurement isometry define small recovery introduce literature different order rip measurement sense isometry x ij integer restrict isometry q integer define compressed rank constrain nuclear minimization include sufficient sharp sparse matrix higher high order significantly interest obtain sharp high elementary constrain establish sharp technical tool state polytope sparse vector sparse positive polytope hull lie polytope versa geometric tool analyze constrain compressed sensing norm non illustration htbp establish sharp rip low matrix noiseless approximately rank section minimizer recover matrix q norm minimizer guarantee recovery approximately sparse show respectively stable approximately compressed rest paper sparse focus low case proof technical theorem contain sense establish rip immediately theorem recovery model observation bound q commonly notational minimizer recovery minimizer ds follow minimizer sharp discussion signal statistic bound argument suppose signal study compare context oracle compressed minimizer stable noisy recover depend minimization closely sense compressed sensing minimization assume define nuclear sum singular norm equal role norm dual similarly sense recovery nuclear q noiseless see follow sharp establish noiseless exist noiseless exactly solution e constraint nuclear solution discussion sense affine analogous sharp rip signal compress sharp rip bind bind recover exactly odd coincide exact recovery minimizer proposition figure recovery noiseless guarantee natural question among strong special al concentration matrix show ensure among base possibly suppose vector generality unit one take large still also note w wu easy hull q lemma well property combination denote check eq ic right h contradiction integer shall widely know h divide since entry besides eq set express suppose denote c h leave side
intermediate task old training relate local learn solution chance learner configuration global optimization intermediate start inference experimental learning easy intermediate experiment expand train supervision example ar concept expect unsupervised discover happen train deep network difficult exploit extra modeling deep architecture case require trajectory hundred run solution output multiple local minima due different minima huge unsupervised reach substantially minima term chance experiment piece yield rather initialization minima numerous subset chance nonetheless experiment consider large regard biological agent function represent deep computation composition rely gradually learner effective local learn discover learner chance represent learn brain high act indirect supervision linguistic constitute evolutionary internal evidence beyond capability human tackle ai purpose human fail human rely local layer perceptron iterative argument favor hypothesis although fire pattern strength neural generally consistent sensitive minima train phase find example large look minimum simply ill call configuration effective local limitation highlight minima regularization unsupervise interestingly get deep minima number minima like actual local issue hard issue work brain discover chance mention human human concept nature discussion cognitive science sequence example learner simple example first learner smooth recent human subject indicate human use deep study paper level high level visual high area abstract concept consistent training neural network learn fact constitute main minima issue arises view inductive important ingredient obtain generalization explanation knowledge previously bias learner another focus reinforcement prior logical speed learning system individual computation individual computational efficiency limitation human processing manner substantial brain ml one volume claim volume brain might seem reasonably nonetheless almost impossible task learnable learner appropriate neural machine boost result box hyperparameter section learn vast whereas depend category focus rapid result recognition b one order generator recognize final detect location task become like operation detect type mirror f p mirror f mirror figure fairly texture foreground background notational convenience intermediate location final outcome body perform rotation image value three replacement accordingly multiple degree completion transformation uniform divided block transform block overlap block locate translation inside invariance locate simpler large mask initially validate fold example example descent learn svms kernel ordinary fully neighbor neural stack denoise auto supervise tuning configuration hyper perceptron neural architecture structure two p share connectivity identical typically patch unless absence patch p part fully layer concatenation window overlap nn decompose separate patch actually nn output p patch field weight nn biases activation linearity relu weight p overcomplete shape shape expect dimensional category rotation give patch train target nn nn overlapping patch whether shape activation patch gradient case nn representation patch nh nn p activation perform deviation minibatch activation use deviation unit deviation minibatch image minibatch vector activation prevent patch image standardized test separately nn nn feedforward mlp layer relu layer task nonlinear logical operation representation provide exploit presence semantics nn nn shape block type value shape figure human computer architecture divide intermediate sum training wise patch fully figure activation large neural transform output perceptron make htp output output positive spike arise location shape htbp unit boost tangent sigmoid function overcomplete patch coefficient bias nn epoch learn shape perfectly nn hide unit l penalty rate base p nn nn connect convolutional example perfectly htbp deep architecture target connectivity hide hyperparameter select log fully per patch nn training epoch train final binary experiment good nonlinearity activation mlp piecewise activation output softmax intermediate layer thing activation sigmoid use encourage competition local contrast normalization normalization layer competition spatial location enjoy observe experiment chance computational standard deviation hide unit order benefit activation specifically derivative loss output feedforward mlp maxout regularizer avoid focus object detector architecture maxout representation hope obtain set experiment experiment mlp layer type task come observed suggest become progress clearly something maxout maxout long stay maxout chance iteration test start object possible l l svm mlp maxout svm mlp l svm optimization otherwise error example increase study fix size minibatch parameter incur converge optimum cause near optima ground distribution therefore update zero optimization stream example without intermediate hyperparameter wise unit layer nonlinearity intermediate either nn illustrate architecture graph randomly generalize seem eventually get htp online minibatch intermediate adaptation stream end chance example online minibatch sgd p unit output per patch use adapt end number show table size seed overlap error bar hidden layer large epoch use activation mlp nonlinearity penalty importance decision generalize report hyper hence avoid regularization mlp large hide layer several thousand reach nearly achieve experiment shown evaluate add connect mlp mlp layer activation use see error error regularization mm initialization network big show effect rest experimental initialization training epoch epoch iteration test error epoch hyperparameter table result give htp extensive hyperparameter intermediate softmax nonlinearity test error dataset p without use adaptive nonlinearity architecture use softmax intermediate function likely nn learn patch architecture large provide presence patch library gradient compute batch mlp still chance layer generalization compare relu intermediate fail introduce optimization act bottleneck architecture seem algorithm perfectly encourage representation composition linear sub task linear operation similar neural compare exactly training intermediate capture essence deep furthermore train one architecture start fail generalize train sgd generate still get good minima initialization effective enough capacity initialization find local generalization hand architecture constrain represent mlp experiment difficult remain optimize effective sometimes domain dynamic tends yield poor capacity course expect discrepancy decrease still note preliminary sign yet figure example architecture bring support simple difficult learner often getting overcome intermediate remain without alternate extent core issue change architecture much easier clearly initialization issue minima initialization go intermediate fail limitation test try variant explore explanatory failure help answer kind learner discover combine partial solution discover solve strong could inspire potential mechanism collective human rbms visible patch likelihood weight annealing rate rbm initialize rbm usual architecture layer nn intermediate sigmoid nonlinearity intermediate train htp htp htp cart cart algorithm construct recursively belong category criterion validate depth algorithm parameter support classifier use svm svms hyperplane vector separate margin cross hyper search term weight svm control width rbf kernel polynomial control hyper two rbf seven result validation obtain test error rbf implementation layer perceptron library unit validate hyperparameter tree bag grid tree hyperparameter obtain learn implementation nn k nn instance select training example close assign test close parameter hyper either distance compute vote weight inverse result good validation implementation convolutional cnn convolutional pooling layer mlp validation domain filter uniformly used guarantee fit field manual validation dataset cnn epochs maxout select maxout linearity cross channel layer unit piece maxout unit decay rate start epoch hyperparameter evaluate hyperparameter convolutional stopping norm fan final softmax slightly well validation manually hide maxout x convolutional convolution layer piece maxout maxout unit scale incoming unit convolutional epoch htp denoise auto force prevent reconstruct corrupt stack result use corruption replace input learn tuning sometimes outperform encoder jacobian input serve mlp automatically tune wise patch feed jacobian penalty training batch supervise epoch penalty respectively e mlp stack non recommend nonlinearity linearity regularization keep reconstruction obtain robust feature share output auto feed mlp hide corruption binomial contraction penalty epoch fine epoch denoise auto pre motivation deep network difficult encoder greedy supervise op universit de op universit prior intermediate supervised network black
marker sample axis axis middle scale coordinate height marker axis line thick marker line line axis coordinate transformation penalty basic piecewise penalty example elastic penalty take obtain maximize take elastic net take explicit take contribution huber soft insensitive loss add together soft hinge bottom back estimator turn wide class nonsmooth optimization working equation characterize present convergence formulation constraint generalization cover important show straightforward interior moreover interior form order iteration common copy identity turn identification impulse depend structure specific formulate entry entry complexity ip number identification typically large impulse response monte matlab continuous discrete circle radius plane feedforward run every monte carlo get normal fraction outlier measurement whose standard quality dynamic denote impulse first impulse fit measure run rational transfer define polynomial estimator matlab identification toolbox equip option specifie adjust large absolute median divide purely quadratic criterion achievable select criterion purely particular range matlab function fed training minimize validation impose fed validation compute union cross validation spline hyperparameter via optimization impulse output define nonsmooth coincide loss hyperparameter stable spline matrix remain impulse quadratic hyperparameter two particular value varies space impulse hyperparameter union datum matlab quantile display error plot obtain keep mind tune contamination reason largely focus scheme equip outperform good spline introduce equip spline estimator equip loss stable estimator loss regularizer inequality impulse example ip finally significant stable spline lagrangian express condition detail inequality reformulate use slack equation kkt set condition solve equivalent vast kkt kalman filter smoothing interior kkt numerically stable newton drive proceed every q eq carry claim give interior use upper triangular right hand substitute operation dominate give translate impulse identification encode also consider explicit remark lemma lemma identification error suitably recently identification determination impulse information regularity stability estimate nonsmooth formulation stable identification context functional rich moreover constraint impulse method system identification iteration impulse coefficient usefulness system robust interior quadratic classical error identify approach recently cv lead identification see problem cast impulse encode knowledge subsequently interpretation least stable spline stability impulse model spline recently derive tc covariance dc kernel small optimize marginal procedure resemble theoretical support estimate impulse become available closed lead respect robustness quadratic circumstance may perform poorly carlo output obtain classical quadratic paper formulate briefly spline stable spline penalty generalize framework demonstrate ip efficiently impulse response estimate corrupt end discrete true input impulse propose impulse coefficient approach system capture rewrite vector suitable spline estimator regularize scalar
bi architecture topology recall f recursive c prop bin prop bin recurrent bi combine architecture detection respect overlap metric compare metric might explain short word agree explicit investigation around phrase time detect look phrase detection combine network well instance cause recursive level investigate opinion extraction task employ token recursive whereas neural supervision recursive network relative difficulty pre semi future word pre impact learn phrase representation learn window effective explore word architecture part fa contain herein express imply science university ny deep architecture neural inspire network summarize future opinion extraction token conduct investigate sequential architecture involve layer incorporate layer potentially neural application language process nlp nlp word represent dense dimensional space deep nlp sequence token input neural constitute naturally nlp recurrent neural apply understand recurrent incorporate past precede incorporate past token nlp token usually helpful current token recurrent precede capture dependency distant token investigation depend token token far would distance argument provide e phrase determined composition believe many nlp explicitly incorporate token operate structured input parse sentence detection give structural representation recursively token produce phrase eventually produce sentence alternatively phrase make positive sentiment recursive neural token associated token extend recursive neural generate phrase whole sentence toward leave structural applicable labeling opinion extraction acyclic g l l opinion aim detect intensity sentiment opinion opinion topic grain opinion analysis opinion answer opinion opinion consist explicit private sentiment etc table explicitly opinion usual previously opinion extraction labeling problem view sequence b opinion token inside opinion indicate token opinion relate field base crf recurrent network opinion architecture information decision token natural language processing represent token vector token dimensionality vocabulary entry distribute representation token small dimension generally manner wikipedia architecture generalization capability geometry word neural make spatio natural token type network hide layer nonlinear previous final output interpret make eq nonlinearity nonlinearity softmax weight bias connect output limiting include window another incorporate architecture counterpart output summary make decision perfect ignore capture term cause vanishing gradient whereas classical type backward output part separate recursive recursively structural set acyclic topological recursively representation previously neural recursive particular even acyclic recursive network tree initial representation compute internal child leave parent vector give distinguish internal lie tree combine output supervision initial incur towards leave extend aforementione recursive rest structure decision summary modify leaf add parent representation parent right child weight connect representation contain information subtree root summary decision supervision output layer backpropagation use update fine tuning word unfold goal architecture autoencoder however representation aim capture tree rather subtree investigation unfold autoencoder recursive neural employ sequential input vector view recurrent neural allow around error individual combine handle separately architecture cast separate opinion opinion outside begin inside class opinion opinion recurrent bi recurrent describe bi describe stanford sentence
select select learn whereas classifier entirely wrong assess wrong representative condition never accurate desire relevant application capture relevance evaluation predict equal actual predict alternative irrelevant match match match virtue irrelevant score relevance thereby propose metric particular seek mechanism quantifie predict outcome actual know metric individual accuracy test minor replace computation show base five predict outcome probability occurrence predict outcome occurrence dataset compute responsible g test general input base qualitatively different summarize table consequence explain thus quantify c outcome probability qualitative relevance moderately relevant relevant relevant case outcome give high probable outcome actual high outcome switch keep minimum case predict outcome outcome high probable outcome equal probable outcome actual outcome previous real equal away high probable outcome outcome consequence influence vary value plot whereas prediction respectively predict actual importance outcome importance fig vs close vary rs predict relevant scenario q upper bound rs prediction research prediction pattern factor context goal good inconsistent inconsistent teacher mind exist beyond consider relation instance expect low rs score hand distinction somewhat expect high vs eight choose almost output choice real output relationship output probability thus perform well rs commonly mention class rs metric select requirement high value compare critical suitable metric critical select drawback evaluation illustration similar application name machine suitable instance outcome prediction rs carefully acknowledge service project support thank providing application domain metric use reflect concern domain call evaluate metric analysis pilot relevance appropriate ml pattern pattern success many car utilize select location performance useful metric instance consumption combination metric directly real experience home office possible shorter fast ml desire two weather nonetheless alternatively pattern may impact depend relevant yet device identify relevant ml algorithm employ challenge user previous application sometimes evaluate prediction algorithm might gray environment service light setting contextual activity day collect environment two remark light depend even desire light context acceptable setting context representative case different assess characteristic problem broadly classification category input select class recognition diagnosis commonly multi problem instance example classification categorization evaluation label hamming car exist ml instance fall label acceptable observe context acceptable multi misclassified achieve devise supervise scenario problem
equation restrict boltzmann set layer tie conditional number hide parameter model descent log neural estimation value conditional share activation well range rbms hundred unit gradient descent tractable require density dimensional directly require method disadvantage compare extend layer trivial lack like add must yield complexity cubic make impractical look property task model input careful minimize objective parameter model refine later write order px conditional specify straightforward across model share attempt doe case inside operation autoregressive rewrite index dimension move expectation order part index th simplify practice value state therefore probable unbiased estimator training descent agnostic artificial real value rescale end update network avoid pass conditional order probability boltzmann network simple issue value dimension one unit know input feed order possible mask otherwise interpretation scheme strength share good statistical experimental see section agnostic produce factorial agree might source variability advantage order order inductive despite parameter construct multiple strong bagging stacking suggest straightforward generating input computational density linearly ensemble random importantly training remain ensemble bag moreover adequate choose adapt budget mention autoregressive datum logistic single ordering maintain variant architecture stochastically motivate propose sharing scheme rely generate size log conditioning allow architecture acceptable cost boltzmann generalise procedure subset predict give value similarity denoise autoencoder train however correspond input unlike autoencoder model train tractable connect validate tractable rbms visible trained ordering variable baseline take agnostic manner performance offer ensemble likelihood dataset pixel image handwritten unlike configuration detail manually run mnist rbms hide unit fix order minibatch stochastic sigmoid unit minibatch seem obtain slightly previously report marginally bad test train ordering still perform close estimate rbms agnostic also rbms estimate performance belief see also without input competitive lc rbm minibatch minibatch input hide order agnostic digit show field input mask contain region unknown zero mask mask possible input perform inference show marginalization imputation arbitrarily mnist rbm operation density fix able density approximate rbm mcmc method agnostic calculate time construct example mnist order hidden decrease field field input performance agnostic uci dataset heterogeneous drop dataset dimension log fold subtract set divide standard rate weight decay cross use grid value datum prevent stop observe likelihood high point run gaussian hide choose weight table order ensemble red white gaussian fix also patch train one order gaussian state art patch discard pixel partition preliminary manual pixel minibatch iteration initial table layer layer obtain extent knowledge performance sign validation started layer lc sample layer example order hide layer fit possible new train ensemble computational outperform variable nonetheless exact marginalization sampling unlike well ever patch dataset ensemble mild improve analyze thank neural autoregressive value competitive multidimensional across variety domain order begin order sharing ensemble model different immediately unlike original empirically ensemble collection considerable
analyze dimensionality many gene irrelevant redundant perform overfitting set preliminary group within approach approach top gene little model consider default devote good compare circumstance comparative performance subset constitute good htb r nn lda breast due htb lda svm tumor cancer breast r lda cancer cancer breast display table two table validate final select accumulate outperform version among problem find low sometimes substantial nn explanation backward version accumulate remove bad minima explain temporal character bad reduction use nn gene low good radial gene time radial gene average modification suitable iteratively quite different current evaluation select usefulness context context subset express accumulate decide namely feature high accumulate usefulness history make conditioning selection possible condition assign source result model effort include explore first discover feature interaction feature greatly task nonetheless entail feature appeal modification little help possibly solution unfortunately microarray mode research adaptive influence evaluation early last stage school edu effort cancer microarray method model characterize observation model select interaction constitute accuracy application growth feature web categorization internet thousand another cancer dna task limit medical diagnosis involve evaluation many return evaluation g select feature remove readily discard contain relevance belong subset depend given subset evaluate evaluate feature evaluation context ultimately inform estimation usefulness different along influence note contain idea evaluation conventional algorithm known search microarray standard search subset exhaustive search intractable efficiently must often disadvantage classifier consider py maximize find evaluation may task case usefulness use evaluation vary depend resample notation express suboptimal propose wide family iteratively locally objective arbitrary e set iteratively latter evaluation feature far normally costly feature initial contain feature evaluation define q definition compactly relevance way take present illustrative capture imagine influence team score matter player player team difference team conclude player price consider monte redundancy make cope redundancy subset generalize choice improvement achieve scenario many improve could less alternative integrate additional cost favor take notational simplicity lx remain eq q consider inclusion conversely go evaluate way lx removal individual feature consider inclusion current set forward backward reasoning evaluate current subset denote search call accumulate estimation q approximate trace evaluate algorithmic impact redundancy evaluation xx generalize conventional backward recover evaluate pure arithmetic importance context appear broad mask otherwise presence evaluation good illustrate approach give practical presentation simply accumulate accumulate reason algorithmic first discard algorithm number resp negligible overhead time accumulate counterpart yx kx nj lx x x lx x x l k nj lx assess modification accumulate statistical computing cv author perform repetition fold keep half feature evaluate final select fold loop example algorithm accumulate learner implementation radial svm default cv loop
equally weight standard distribution one center density separate natural chain length various square posterior calculate walk stochastic pseudo equal start iteration iteration respectively explore wang automatically decrease flat meet namely wang outperform approximation step rmse density deterministic version proposal target acceptance rmse various adaptive adaptation mechanism wang improvement might tailor acceptance across display demonstrate improve run long chain automatic bin propose advanced perform fix setting sensitive bring brief demonstrate adaptive parallelization automatic bin splitting within many exclude reader might citation sometimes stack improvement case parallelization proposal provide improvement wang simulated space herein poorly understand unclear scale temperature force idea instead efficient biased wang equal interestingly biased ensure partition visit equally ix additionally modify distribution coincide restriction multiplicative constant x desirable obvious calculate algorithm parallelization introduce partitioning wang simulated suggest examine improvement propose parallelization
covariance delta g yy x yy yy invariant generality expand variance exchangeability equal cauchy schwarz inequality term always negative asymptotic variance hold equality schwarz ie follow obvious consequence n xy yy finally decomposition define eq show efficient show third schwarz get conclude index estimator consider also robust decay normality numerical reliability cm proposition mathematical involve output tool model output limit estimator generalize output observable replace mod les font analyse de le impact une du mod le des pour influence de des es est de du pr et un pour de est de n du mod est une la est encountered science involve poorly assess impact assessment sensitivity aim account belief turn model output different hoeffding output variable uncertainty partial variable identify index practice sense hundred thousand evaluation model output carlo view hold interest pick sample replication produce monte method sensitivity general variable random study compare denote asymptotic regular replication generalization minimum unbiased see many one time interesting evaluation numerical run negligible typically generally replace original run true krige polynomial basis use index replacement original infinite population replacement original index double limit converge quantification early paper exact normality hold produce assess asymptotically efficient index asymptotic index computationally intensive practice paper review prove property benchmark independent integrable non deterministic index q quantify influence close influential multidimensional see separation input treat square integrable particular eq classical see view lemma identically distribute practically estimator account observation well base variance rewrite enable great ie round course numerically throughout sequence large normality eq less equal immediate exchangeable ie fx fx z pz z pz fx pz asymptotic efficiency extend rao enable cumulative distribution cdf exchangeable clear cdf asymptotically estimate introduction often costly numerically variable replace approximation perturbation random define assume non respect estimator subsection neither second give neither indeed almost surely second assumption consideration ie justification vanish case object n entail suppose variance give respectively remain subsection variance cn weak asymptotic cn asymptotic uniform resp analytically available interval estimator large obtain replicate consider know coverage converge go subsection estimation confidence subsection subsection well rkh krige subsection regression subsection illustrate normality coverage asymptotic confidence build use plot dotted line close level thereby assess reliability interval size multiply small interval conclusion agree perturbation standard lead index sufficient condition actually normal illustrate interval close perturbation normal suggest hilbert space krige computer analytical formulae necessity monte paper choose generally accord sampling design sample size potentially computationally demanding enhance quality interpolation rkh link smallest know constant assume design q pointwise error constant numerical illustration illustrate base sensitivity true point rkhs interpolation gaussian software plot exponential let carlo relation normal accord numerical even rigorously prove different upper empirical change clear illustration c coverage n identically motivate smoothed smoothing kernel euclidean integrate error regularity expectation
reconstruction vs sense budget depict propose line investigation examine deferred effort acknowledgment thank edu award consider task compressive compressive measurement noise interference clutter post specific perhaps limited interference available specific aim devise incorporate compressive information interpret design propose compressive design agnostic enhanced sensing notion compressive sense arise compressive cs measurement measurement interference clutter whose may describe measurement noise error indeed primary cs sense noisy say significant initial effort analyze clutter compressive suffice high effect clutter I case clutter model exception work compressive detection utilize ultimately approach clutter contribution whiten compressive aspect clutter scenario relate cs matrix iid mean facilitate accurate cs cite virtue suffice scenario equip e possess structured incorporated hand assumption underlie cs inherent measurement suggest manner process compressive remain assume collection support location nonzero value likewise prior identify structure prior demonstrate knowledge enhance design quantity design sense associate analytical simple simulation measurement experimental enhanced measurement design base notion formally main algorithmic enhanced compressive design section successive ii u consider setting knowledge aim coherence matrix extension idea aim design dictionary collection examine sparse dictionary examine knowledge enhance cs formulation knowledge work assume gaussian acquire minimization dictionary compose eigenvector sense design none statistical estimation theoretic time application qualitatively interference effort examine bayesian compressive effort along line examine design strategy application imaging effort utilize principle information vector observation criterion utilize sense design none aforementione explicitly clutter presence nuisance work compressive accurately setting sense prior information clutter random component assume mixture full inherently lie model form subset block group group worth draw could likewise assign prior clutter realization weight zero uncorrelated assume aim formally sense particular estimate via associate denote mse subscript expectation random quantity criterion choice denote possibly constrain class strategy class sense unconstraine scaling toward would negligible design choose rise compressive impose per theory task mmse give mmse second order statistic without unable close consider restrict signal covariance matrix invertible mmse easily denote transpose algebra denote express sense aim trace seem address investigation report approximation lead applicability qualitatively snr subsection setting various size cccc b vs compressive detail panel respectively snr value reconstruction dot line marker measurement strategy examine let matrix satisfy find decomposition u c u thin bit linear convex resort successive successively subproblem maximization main subsection solve linear algebra equivalent strictly strictly positive multipli water modification lagrangian symmetric lagrange multiplier maxima unique lagrange multipli respect satisfy take evaluate local maxima contain eigenvalue eigen entry diagonal entry convert eigen large eigenvalue subsection propose u evaluate dimension clutter model actual clutter
derivation argument centralize batch implementation assume centralize also albeit access obviously centralized complex procedure examine structure consider interested cost minimizer seek ki ki ki ki describe centralize transmission occur asynchronous manner agent cause agent coefficient status communication connect fusion center accommodate asynchronous arise scenario useful classical batch note centralized admit decentralize fully distribute ki j ki calculate intermediate iterate center center intermediate accord agent example agent require global adaptation step centralize whenever decentralized model describe rhs continue satisfy ii part appear transform facilitate sequel asynchronous centralize asynchronous diffusion network I second order value random moment mutually collect fusion ni mean moment require equation steady asynchronous asynchronous recursion asynchronous centralized mapping moreover eq merge find evolves recursion ki ki ki maintain notation shall centralize batch whenever quantity part ii replace subscript error centralize centralized solution stability mean stability asynchronous centralize recursive I steady state theorem apply directly result observe centralize asynchronous govern therefore stability strategy moment define part centralize investigate stability centralize agent part recursion size hermitian positive semi definite examine asynchronous ignore drive recursion square centralize distribute take error I easy hermitian jensen hold long asymptotically version denote deduce noise side hermitian n error covariance z equation converge stable steady get weight arbitrary positive recursion setting verify ii ii steady asynchronous centralize f rhs dominate centralized would form fusion coefficient batch view batch size coefficient covariance employ stability verify recursion parameter determine rate long determine distribute diffusion centralized batch combination primitive assumption asynchronous almost sufficiently size adaptation failure comparison centralized agent combination coefficient aggregate neighborhood random parameter assume section centralized batch agent distribute centralized strategy necessarily relate meaningful strategy connection role moment determine diffusion likewise section determine mean centralized connection moment step part coincide coincide similarly requirement connection moment reasonable random random part ii know correspond primitive left expression ii asynchronous part vector eigenvector element likewise c require centralize identical establish condition p eigenvector consist consist refer centralize requirement meaningful result answer follow primitive result symmetric positive interpretation valid explain positive difference definite ii asynchronous diffusion asynchronous pre n vector dirichlet dirichlet distribution logistic unfortunately simplex nevertheless inspire chain mcmc procedure construct meet combination asynchronous independently distribute I mean let I leave construct simplex whose mean covariance specification asynchronous solution enable asynchronous although unnecessary converse solution possible determine distribute solution satisfy primitive centralized distribute centralized level answer open challenge stem general systematic simplex pre specify method guarantee satisfactory eventually recursion part rate diffusion recursion centralize determine matching asynchronous centralize hold square diffusion part mean centralized batch match square asynchronous centralize almost f part ii proof part h batch steady distribute centralized solution verify diffusion network k asynchronous constant covariance replace asynchronous network assume easy recursion asynchronous similar mean latter establish part ii mean asynchronous part network correspondingly parameter part mean asynchronous diffusion strategy dominate ii f correspondingly identical part lemma obtain f get complete likewise ii steady state network since rhs dominate term asynchronous degradation asynchronous strategy part since diagonal imply frobenius eigenvector must positive asynchronous k p c definite know get convergence asynchronous uncertainty topology arrival however degradation asynchronous previous lipschitz mse cost quadratic stream ki regressor regressor spatially circular independent mean variable mutually aggregate square ki k satisfy asynchronous network illustration purpose part give substitute compare ki verified noise iw ki part k likewise clearly always great reduce ki ki batch solution n k ki continue procedure generate fusion coefficient specifically show regressor white e show step value probability part probability distribute strategy centralize solution trial trial fusion realization asynchronous plot match attain centralized operation asynchronous implementation steady error remarkably continue rate level suffer highlight failure asynchronous get asynchronous hermitian hermitian use inequality induce identity jensen respect fact function n ki obtain condition ki ki therefore furthermore part k hermitian hermitian denote matrix deduce symmetric jensen k get dominate c use ii verify conditional independence part ii therefore r ii step also ii k substitute r fact hermitian yield know I f h verify invertible inversion fact dominant simplify lyapunov equation square side lyapunov equation invertible lyapunov hermitian follow dominant part establish quadratic c p p p matrix mutually satisfy step mutually substitute confirm part introduce sub contain eigenvalue sub let mn hermitian obtain ii q get hermitian ii use hermitian get hermitian unitary matrix stable generally express eigenvalue upper triangular block similarity transformation know diagonal entry use eigenvalue result circle center frobenius enough circle isolate center great magnitude sufficiently circle use note verify h part
number cluster x dimensional four setting construct eight method standardize method ari mdp ari ari score list mdp show average ari setting difference mdp matrix well focus difference set cluster informative similar tendency ari set mdp mdp c ccc mdp means mean mean mdp table ari iii difference cluster variance difficult difference mean clustering overall distance vector matrix appear mdp ari condition consistency distance depend variance type balanced algorithm work well unbalanced case clearly acceptable behavior high iv iii informative iv show tendency ari set mean mdp c ccc iv mdp mdp approach competitive microarray gene dataset dataset cluster mdp type linkage cluster method product distance linkage mdp cluster algorithm straightforward show error table among compare real h table al mdp preprocessing study mdp split large mdp cluster split induce first eigenvector point important closeness clustering provide true infinity effectiveness illustrate structure variance distance cluster inner mdp usual clustering approach consider cluster acknowledgement express thank dr helpful discussion grant aid datum always reflect closeness contain cluster distance cluster al approach illustrate homogeneous significant many e et al liu operate selection g conversely focus measure fact representation fact closeness euclidean distance mean variance cluster classical method euclidean distance distance supervise unsupervised mdp distance mdp infinity addition mdp prove condition size cluster mdp focus moreover detect difference context possibility cluster closeness contain information cluster call clustering detect cost proposal give al tends follow difficulty usual sufficient mdp propose method cluster label describe effectiveness illustrate sample al also ks x ks ks l dominate ks et fact draw infinity base variance contain sufficient condition clustering focus difference cluster dimensional sample draw copy figure contrast become increase closeness cluster usual matrix center cluster consistency method partition object optimize mean proposition follow cluster label consistency inner product label converge label distance vector vector partition proposition contradict obtain label distance consider type product lp lp contradict
carlo use step hmc transform confirm empirically measurement efficiency dataset bayesian technique layer latent possible express parent lead rapid relative graphical direct graph dag supervise special bayesian paper inference g dynamic parametric include ep assumption resort sampling sampler hybrid monte property find mode efficient network transform continuous auxiliary continuous replace variable auxiliary variable original also integrate result pdf inference observation auxiliary large pdf computational confirm variable applicable design auxiliary include differentiable inverse cdf differentiable shorthand conditional mass treat treat typically variable bold write bold network random conditional dependency acyclic vertex empty parent compute bayesian network represent factor conditional j j pdf pmf appear value possibly nonlinear parent dirac pdf j data likewise pdf learn p intractable compute differentiable first joint maximum maximize approximate outlined finding map maximization em monte prior consideration hmc p mix approximate map p ascent optimize risk p pp hmc outline use pdf information currently network pdf factorize pdf dependent joint connected factor consequence gradient outline graph determine reach versa propose form base variable sake include variable parameter parent parent bayesian pdf cc z continuous parent parent j form parent auxiliary network auxiliary parent except see p j define eq word deterministic new show eq factor except variable joint everywhere interestingly important retrieve eq call eq pdfs parent parent factor function parent auxiliary variable argument conditionally deterministic reach spread information learn net variable pdf computed pdf straightforward topological value compute subsequently factor full gradient backpropagation manual automatic valid transform generate target inverse obvious p inverse available case cdf degree software package generating option pz j distribution univariate determine parent nonlinear valid e extension treat element conditionally analogous z e x illustrative represent z variable choose take perform pdf look eq new influence step
distinction black black pt pt f black black black u drift drift cross side bound exponential derivative zero precise suboptimal regret optimal drift analogue drift transition govern transition probability exponential apply ml algorithm specify parameter drift likelihood point close reduce adapt parameter drift es drift dependent data reference drift clear track algorithm useful criterion locally try well indicate generalise bayesian external loss expert aware three show probability loss block number predict well expert log incur infinite bad case guarantee mix past posterior share tracking expert unlike expert ever posterior expert occur usual interpretation slight combine control fluctuation difference fluctuation gain policy term large lot interval bad case low regret converse regret lower adaptive interval moreover journal bad adaptive generalise give guarantee share modelling switching dynamic express run length indicate expert example point discuss calculate forward compute sometimes state contribute state adapt write contribute problem produce word sequence hand share sophisticated switch run posterior observation later analyse hmm e project us weight grid hand expert obviously rich map time separately log loss expert task form prediction value use loss less hellinger obtain loss straightforwardly algorithm sequential expert transform condition result strategy possibly express unit case correspondence hold expert depend expert aspect motivate definition predict stock factor incur stock round stock capital stock use mix strategy expert proper may one cause formula substitution derive carry prediction cover portfolio finance provide cover seminal portfolio fit formalism reduction round prediction expert mix weight finance recover strategy learn optimal weight albeit different purpose depict learn expert augment expert space large expert process outcome cover interestingly method carry mixture easy advance make portfolio selection practical stock fall outside scope paper universal code reference performance individual distributional hand universal expert thank well apply setting distribution expert sequence es infinitely long amount expert time model good paper markov specify es prior reason hmms forward viterbi answer question datum graphical approach allow unified presentation many exist expert incur incur block use discrepancy current expert literature literature describe quickly assume expert order competitive gr suggestion improve thank supervise gr take research fellowship centre interest online minimax finance supervision gr institute laboratory university description learn lemma theorem extend appear author include use material request sequential strategy natural code describe language new efficient switching tend cluster scenario know jump typically relate drift contribution interpolation development analysis sophisticated code sequence expert fill water expert among ingredient code investigate two way ingredient assumption generalise somewhat ingredient baseline thing way split sequence block consecutive outcome encode good block core round sequential distribution outcome assume countable subsequently new outcome exist code accumulate logarithmic loss implement logarithmic loss forget round convenience connection code build code implementation terminology theory henceforth code emphasize box strategy set next outcome prediction prediction prediction item encode end game accumulate among strategy issue produce prediction outcome reveal combine literature code compression call switching combine expert use receive introduction lot universal expert case expert probabilistic discuss comprehensive introduction overview bayesian expert unify mention follow hide intuitive language obtain algorithms emphasize prior make theoretic diagram consistently allow design diagram understand ease moreover beyond paper contribution first useful provide early regret expert describe develop practical switching describe expert result strategy cast algorithm theoretical hmm show seminal starting point two drawback fix incur number probability need model differently explain describe switch isolate proceed describe affect regret describe associated share scenario change expert none tracking assumption relationship box expert prediction parametric various two expert seem nothing term loose end discuss exposition skip evaluation call expert well generalise prediction achieve minimal response prior random bold face note depend expert time strategy regret break arbitrarily eq substitution reveal hand nature process evolve period generate outcome bad day model prediction expert distribution expert good infinite call es strategy expert simple often desirable previous expert depict prior capture prediction expert future prove regret maximally prediction application involve expert general however expert include dependency x graphical language hmms display result diagram computational efficiency markov expert outcome start specify regular markov subset state expert state grain allow transition conceptually efficiently prediction joint convenience identify e q setup specify diagram figure draw dot black dot display circle expert assign prediction write forward past intuitively maintain subsequently condition transition proportional simple f show mass correspond expert vector shorthand specify probability reader likelihood coincide known prediction per trial efficiency f rd pt cm q weight expert predict q whose intuitively past strength lie assign reasonable forward compare loss incur expert prediction mix expert reference note regret respect hold block throughout paper bind expert sequence probability expert expert drop mixture reference application derive play role useful expert sequence bound r subsequently inequality maximum sharp job available say share transition intuitively make theorem cite concerned switching dynamic use model section set govern function state symbol map state intuitively transition large difference therefore transition throughout define tn transition count result ml something overhead incur reference application replace state sequence important distribution multinomial setting appear way exponential degree appear also inequalitie lemma expand tf differential zero term eq complete note number transition regret already available assign regret state expert regret outcome correspondence share regret combine prove loss switch construction relate interpolation first reading take three evolution interpolation interpolation illustration construction start label original consist bit indicate evolve next interpolation state bit start indicate bit state label show interpolation em c r dr pt black em pt black black pt pt black black f f black f f black pt black black pt f black f f black f black black f black pt define careful counting interpolation bad version remainder cost also switch let switching trick source code block share much per switching whereas switch except switch long define model q switching second actually switch tune increase solve overhead yield substantial bound substituting follow introduce appear suboptimal penalty convenient constant like well asymptotic purpose would priori fix share lower bind dominant quickly use get good asymptotic hand may substantial careful decide acceptable pay preferable bayes strategy model estimate one model eventually point bayesian expert uniform bind ultimately expert even happen assign event mixture apparent advantage nonparametric mixture overhead one previous must section somewhat probability occur describe simplification band achieve risk define model select true distribution datum decrease integer uniform set sequence block decrease follow desirable express last switch weak term optimal would sensible flip switch long guarantee convenient tailed come close publication describe except switch develop switch interpolation bind whereas jeffreys switching interpolation past regret switch thereby universal switching switching switch inequality via level bad jeffreys bound previous share switch eq switching independently derive also bound likelihood share regret f pt black r f pt black black f pt f black ingredient share switch probability moderately place switch reduce advance mean present algorithm quasi determine exactly advance run length code constitute distance part contain part switch regardless previous limitation whose switching normally decrease sample overhead complexity code subsequent refinement successive share recover geometric interpolation become reduce length block assume keep regret go length intuitively th interpolation f f black r pt black black q uniform assume abuse notation identify
exploit consist novel novel important application hyperspectral easier main consistently identify topic order proof contradiction word consistent remain j define two row vector b separable row strictly topic matrix observe topic key condition occurrence consistently topic sec claim imply topic diagonal reverse general proposition demonstrate dominant achieve performance claim make key idea high word convex hull find project direction select computational absolute summarize computational stay provable complexity term amenable distribute since aggregate row maximize necessity theoretic sufficient condition dirichlet image consistently necessary proposition strong condition imply play role polynomial provable consistency guarantee separable solely one condition paper independent theoretic consistent solely practical moment algorithm amenable implementation attractive scenario large database algorithm past decade seminal dirichlet allocation topic popular semantic formally iid vocabulary vocabulary mixture topic proportion assume sample iid manner dirichlet distribution reference whose topic whose topic proportion document approach find recent develop algorithms consistency column column normalize loss ex reference fix computational yes impractical practical practical rank yes yes practical ex additional exist either lack computationally impractical statistical strong ex separable statistically efficient ex theoretic consistently detect
wind wind steady wind turn near spread uniformly domain trajectory trajectory detailed description model datum side spectra numerical cause substantial apart perform semi parametric frequency time leave display fit grey vertical multiscale effect axis numerical spread narrow dark grey indicate variability light grey variation remain experience spectrum spectral substantially peak ensemble individual mean frequency light grey model dark grey spectra south portion highlight estimate stochastic day velocity vertical line apply model leave one portion day display right model motivate positive frequency time vary parametric procedure display image spectra window length focus include model occur white optimisation procedure unconstraine capture observe frequency peak vary spectra display model spectra white display parameter figure estimate six vary evolve time smoothly though also include calculate equation fisher hessian shift frequency particularly band close reveal band attribute time pass frequency peak higher another around energy frequency figure capture mat ern slope model correctly identify frequency shift local frequency trajectory discuss ratio select shifted frequency therefore frequency statistic display significance level nan consistent significant shift due time window reduce expense short window multiple trajectory spatial heterogeneity scale hessian allow parameter time display strong amplitude frequency largely uncorrelated amplitude spectral expect estimate decay quickly correlation suggest model separately combine estimation example near estimate ignore equation accommodate model process understand structure model mat ern process exist motion without serious drawback six demonstrate variability allow vary show numerically generate trajectory model process carefully able reconstruct variety encounter feature capture herein step full insight surface future work incorporate detect extract contaminate agreement example raise similar shift dataset energy model analysis surface trajectory obtain multivariate dataset large modelling estimation detail issue relate misspecification parametric demonstrate effectiveness world numerical complex value mat ern dataset include environmental imaging method type tracking spatial movement series encounter wireless observe vertical lagrangian obtain typically water contribute understanding treatment perhaps challenge temporal consideration make lagrangian typically highly evolve main surface follow water lagrangian deeply primarily surface characteristic surface arise wind prove display national www million point position hour right display north trajectory global clear trajectory north colour different location trajectory right white black grey day highlight primarily estimate velocity series period use data model respect multiscale superposition important temporal accommodate heterogeneity represent circular motion express time simple real series represent display velocity series corresponding display right anti spectral energy give spectra display indicate spectra multiscale structure peak near background capture frequency subsequently position period summarie blue velocity may consistently capture trajectory aggregate simple window spatial resolution quantify stochastic time furthermore exist implicitly markovian show generalise unified mat ern stochastic process whereas usage sample process largely real world require employ semi misspecification task idea modify herein accommodate series finally together procedure hypothesis wish due persistent act shift local rarely different detail parametric estimation accounting variability misspecification output conclusion direction future process model herein phenomenon many time frequency methodology motivate accurately superposition process aggregate datum I variance order desirable particularly window summary support already work acceleration nd order acceleration rd brownian power fractional instead markovian process ern background mat ern model case isotropic ern define spectral use denote mat ern overall slope control degree wavelet background name energy decay frequency ern brownian fractional brownian self grow power appeal become zero frequency mat ern drawback exhibit whereas equivalently behave ern regime value mat ern mat ern mid frequency observe short mat ern figure series mat ern process subtract asymmetric freedom frequency model density phenomenon distant part particularly problematic spectra dynamic mat ern cause considerable estimate subsequent problem expense nearby spectral affect back observable spectrum fourier away intermediate address explicitly account fitting spectrum account introduce denote term find parameter sum equation bias discuss form wish error nearby frequency domain covariance computationally expensive secondly option model fouri set summation deal misspecification estimation square drawback however account inefficient additional feature parameter estimate series q standard deviation decrease approximated estimate uncertainty shown later outline modified misspecification semi frequency idea discuss parametric misspecification various address misspecification variability statistically misspecification create frequency ignore equation account parsimonious may include include one introduce another semi omit surface originally regular many interpolation differ mostly frequency measurement error exclude cutoff model need sufficiently order resolve slope mat ern model display fit period north display misspecification frequency choose computation www ac uk statistics software period result parameter capture period local marked period shift frequency agreement appear peak panel six accurately band strong peak likely real trajectory see exist positive resolution scope access flow exclude axis physical within semi velocity vertical window dash fit extend vertical indicate peak low frequency accommodate visit frequency equation borel measurable replace spectrum
locality present bag exchangeable among observation text speech establish persistence obtain regime provide identifiability overcomplete model give observable certain topic persistence identifiability identifiability presence object gram bipartite finally criteria overcomplete regime introduce persistent topic persistence successive word successive similarly share word one topic topic successive non overcomplete regime identifiable persistent overcomplete become l l l n word overcomplete regime graph encode word topic overcomplete establish translate novel perfect topic bipartite topic addition size vocabulary topic persistence observe topic bipartite topic identifiability hold knowledge first characterize identifiability overcomplete structured regime graph dimensional vocabulary identifiable upper need limit overcomplete regime topic degree degree correspondingly small extent increase diversity support among topic edge distinguish another furthermore overcomplete moment certain persistent topic uniqueness cp tucker decomposition structure symmetry tucker structural topic follow constraint core persistence model constraint inverse persistent bag word general tucker tensor single tensor cp I persistence level symmetry crucial towards establish regime overview technique condition give specifically occurrence pair document vocabulary let word impose col thus identifiable constrain overcomplete order moment yield identifiability high moment moment equivalent whether imply show access integer overcomplete regime expand overcomplete identifiable hand impose constraint persistence identifiability persistent identifiability rao matrix product expand persistence central towards regime intuitively relate tuple imply col column topic word identifiable moment persistent refer impose expansion condition exploit tuple highly topic generate trivial derive moreover fail expand nd thus desirable allow diversity restrictive overcomplete column matrix possess tensor incorporate possess rank persistent access column combine possess rank combination column criterion agree easier relate incoherence j thus high incoherence column overcomplete subject desirable impose identifiability gram matching topic bipartite topic uniquely final deterministic overcomplete maintain enough word sparse sufficiently diverse support establish identifiability consist condition deterministic word bipartite long degree intuitively bind degree concentration word technical result presence gram match greedy recursive constructing gram matching overcomplete setting edge bipartite graph structure manner rao summarize recent identifiability work unsupervise overcomplete task huge speech computer however theoretical regard overcomplete overcomplete ica overcomplete regime overcomplete ica overcomplete uniqueness cp notion strict overcomplete strict cp uniqueness dimensionality much polynomially uniqueness cp decomposition identifiability model generally dirichlet lda recently et overcomplete ica eigen reweighte fourier independence source context tensor decomposition identifiability arbitrarily topic tucker decomposition cp decomposition identifiability fully factor generic overview decomposition identifiability learn limit singular observe et provide moment topic proportion topic dirichlet allocation addition hmm adjusted cp approach show recovery degenerate overcomplete albeit overcomplete cp decomposition however method approach topic class view available detailed description et learning mixture view view number view view limited mixture topic incorporate employ nmf topic uniquely topic anchor overcomplete work consider word work al closely less vocabulary dictionary atom propose expansion topic word incorporate overcomplete representation popular context dictionary code jointly atom observe frequentist dictionary performance guarantee consider dictionary bernoulli gaussian provide code task reconstruct permutation scaling order tuple support norm denote column denote kronecker section introduce persistence persistent reduce word model specify population structure word simplex distribution topic persistent hierarchy integer persistent within persistence exchangeable persistent value possibility persistent topic sequence e topic persistent successive view notational encode draw expectation assume encode topic word expectation eq collect matrix persistent moment select word average moment subscript drop persistent lemma e valid hidden one discrete persistent topic provide identifiability structure notion identifiability precise strict uniquely permutation scaling relaxed generic refer entry absolutely lebesgue pattern structure identifiable identifiable moment denote identifiability structure give th moment hide order full degeneracy distinguish node degeneracy assumption arbitrarily hope scale permutation canonical say fix population draw provide subsequently impose sparsity pattern structure disjoint edge bi bipartite neighbor generalize graph refer gram bipartite graph common bipartite graph set gram matching discuss match perfect perfect bipartite perfect gram matching matching enforce require overcomplete size matching node necessarily perfect gram gram match reverse bipartite matching perfect gram perfect bipartite graph perfect imply pattern appropriately variable identifiable intuitively mean distinguished matching identifiability overcomplete regime remark hide gram linearly bind sized column see connection identifiability fixed state follow persistence rely access observe generic persistent condition hide denote seen population structure identifiable least order hide persistence identifiability study th structure bag variable identifiable hold overcomplete match perfect overcomplete result model require condition degree require large word condition among topic bag topic recovery turn identifiability imply appendix exhaustive recovery recovery moment additional propose future investigation result random bipartite size bipartite graph establish match deterministic show achieve case constraint overcomplete tight bipartite degree bound c cp low bipartite ensure perfect gram intermediate regime identifiability following address access observe success identifiability constant follow notation tucker cp tucker denote column tensor core cp na u j tucker square persistent th moment moment follow persistent model moment moment define persistent characterize deferred integer core tucker representation decomposition tucker fully dense cp cp form comparison topic fair comparison variable varied th moment word persistent representation cp comparison previous identifiability overcomplete cp th persistent tucker core equation form persistent topic bag word involve tucker decomposition inverse word difference core bag core fully dense reduce persistence property establish overcomplete overcomplete core overcomplete persistent core theorems auxiliary perfect graph summarize hierarchy perfect gram condition conclude primary identifiability condition l degeneracy matrix mainly expansion identifiability briefly available persistent equation persistent degeneracy condition condition show expansion property identifiable expansion impose property bipartite nd redundant identical identifiability column rank relaxed comparison remark appendix gram generalize notion introduce establish relate perfect match perfect direction perfect match perfect gram connect index index condition argue satisfied appendix lead generic state identifiability result condition matrix deterministic combinatorial section satisfy matching size section condition require mention show follow bipartite gram matching random bipartite graph randomly constant condition condition perfect gram matching bipartite graph sufficient size union easy union gram degree scale scaling argument population structure overcomplete th union degree constant acknowledgement acknowledge discussion support microsoft fellowship nsf award nsf award award award nf award nf condition theorem identifiability base condition redundant row index tuple row index row redundant version gram restrict restrict p n remove explain gram full bipartite hide indexing indexing bipartite version specify expansion bag expansion modification appropriately graph need subset identifiability result state address remark identifiability equation identifiability deterministic persistent define moment equation nx condition property rank tensor belong b nb nb propose condition also vector equivalently th contain restriction support therefore furthermore jj accord definition remove ib cp high order argue improve constant eq expansion contradict useful gram matrix generic gram satisfy lebesgue ns easily generic perturb denote say fix expand I vector fully submatrix one j moreover independent assumption consequently ready claim every submatrix u h need contradiction entry submatrix nan submatrix contradiction case persistence moment general also characterize tucker simplify notation power integer encoding characterize moment word persistent topic view persistent among give encode write order result independence independence equation dm law expectation third equality persistent topic single persistence persistent expand relation p following tensor matching rank gram matching gram generic lebesgue property perfect matching bipartite perfect bipartite bi adjacency set correspond bi adjacency note row entry rank analysis support support almost surely completes prove submatrix subset equal former latter use necessity existence graph gram perfect immediately discussion perfect gram neighbor edge perfect immediately definition distinct arbitrary order tuple distinct e consider matrix zero entry column rank almost surely non determinant index keep decompose correspond variable early root lebesgue column also rank since generic surely proof sketch vertex furthermore uniformly partition set size partition small iteratively l partition manner partition set partition recursively partition process prove argument intermediate partition partitioning induction gram size gram original last induction perfect induction partition bipartite consider correspond figure induction bipartite perfect gram bipartite graph gram l ny iy x highlight denote gram subset cardinality take connect parent one addition impose set perfect gram consider perfect gram bipartite construct follow apply degree apply conclude perfect figure edge exist perfect step gram union matching yx proof analyze induction step perfect existence gram early gram existence similarly prove probability denote perfect perfect gram matching partition induction order perfect since exist inequality gram matching note concentration bind reduce rate explicitly perfect gram eq constant satisfying rows event subset inequality condition sum enough lemma conclude probability constant satisfy bipartite pick subset long distribution uniform union pair gram conclude bipartite gram size existence bipartite graph bipartite graph side connect condition exist perfect graph match bipartite similar concentration randomly subset uniformly new subset degree bipartite graph q q term success denote tail bind apply result follow first inequality sufficient relaxed generic bipartite relaxed term propose expansion
review sign theory global sign local triangle cycle start triangle explore localize sign hope propose imbalance exploit existence balance cycle section balanced sign network adjacency reduce sign section conclusion sign basic notion balance two task address adjacency relationship entity treat entry relationship might view exist sign entity entry thus partially eq sign heterogeneous network kind entity entity negative website kind entity video sign video attention kind sign reduce homogeneous instance network possibly video part sign explicitly unless specify sign sub behind exploitation researcher identify kind trivial particular one influential balance balanced formally triangle balanced contain belief configuration balance unbalanced right node right right node though balance specify balance cycle cycle balance balanced iff contain unbalanced base balance define notion c node node right right leave node node balanced iff balanced expect define balance perspective balanced incomplete possible assign adjacency far specify balance theory nice balance balanced iff divide two edge cluster actually verify balanced look cycle divide generality pass step stop pass ensure stop balanced network balance cycle balanced generality pick group group belong opposite cycle mark mark node mark group edge within conclude balance social particular argue degree imbalance edge fourth three allow negative strong weak balance network balanced iff weakly incomplete adopt perspective weakly weakly iff define weak balance reduce social analysis sign topology sign cluster entity incomplete network underlie network problem link evolve prediction consist network temporal another balance group mutually weakly balanced task within entity notice balance sign network network balance term triangle approach design sign proceed unbalanced triangle define number unbalanced triangle triangle appear van observe equivalence imbalance observe query augment define imbalance predict sign note quickly computable particularly test abuse shorthand somewhat surprisingly compute sign derive imbalance rely balance link sign balance theory imbalance feed sign describe fix configuration variant cycle term power matrix describe direct theory mostly concerned undirected imbalance undirected deal imbalance long define analogue contribution unbalanced simple cycle length decay like eq imbalance formalize fix unbalanced imbalance sign cycle definition allow rapidly cycle imbalance unbalanced unbalanced cycle direction decompose unbalanced cycle finitely cycle cycle unbalance true cycle negative classic use link consider sign viewpoint key augment set q include since eq j true use use eq cycle prediction use reduction stand theory interpretation use enough prediction prediction sign social balance connection base supervised friend network type like degree zero neighbor rely degree could possibly bias cycle generalize cycle whether look negative transpose possibility pair possibility guess feasible long supervise hoc quickly computationally infeasible soon beyond concern combinatorial raise intuitive interpretability say walk j retain undirected graph consider sum feature another way deal compute link logistic imbalance cycle length denote map logistic regression query cycle play sign make definition balanced balance weak balance complete weakly balanced define point local weakly weakly balanced whereas weakly balanced global obeys therefore sign sequel low formulate matrix begin complete weakly sign network adjacency weakly node divide group say group vector span consider exactly equal obvious eigenvalue linearly eigenvalue column linearly easy contradict minimum style font size pt mul style white style plus style fill white pt minus right east style east transpose right west east north south west east east west north east north south west west east north east south north north south west east east south south east weakly balanced express indicate adjacency adjacency rank recall try edge balance network completion specifically complete sign assigning balanced complete formulate minimizer look whether solve np recent surprising solve subsection solve sign possible trace norm relaxation solve sign surprising perfect incoherence singular value incoherent incoherence high singular addition incoherent entry sample optimizer underlie incoherence sign recover underlie sign high group start group sign network imbalance balance imbalance imbalance indicate presence group extreme weakly one group imbalance without individual large entry completion able recover imbalance determine possibility sign weakly imbalance incoherent absolute normalize identical sign unit identical u u I ia incoherent put subsection sign sign suppose set imbalance underlie perfectly recover weakly network yield might prohibitive solve singular projection attempt manner might balance enforce descent current project convex sign prediction kt complete approximately k tt addition efficiency suggest incoherent exactly recover weakly classical limitation ensure violate addition base sign matrix boost accuracy problem eq practical much well netflix fair amount million nevertheless sign force entry either care loss important resolve instead square order sign change sigmoid hinge apply sigmoid slightly improve square solve square become developed square solver solve subproblem hessian therefore various sgd sgd entry sgd iteration usually iteration require construction reduce ignore detail structure sign base weak theory sign group sign laplacian partially sign sign laplacian sign eigenvector mean get analogous laplacian replace algorithm guarantee sign laplacian recover sign overcome prove balanced obtain say eigenvector complete possess desirable weakly balanced satisfied iff probability eigenvector theorem eigenvector therefore perfect guarantee take perfect clustering iterate incoherent completion summarize surprising superior sign method link completion yet sign sign sign cluster local hoc yield long addition global matrix outperform accuracy cluster sign laplacian usefulness real life construct first balanced entry form partially control percentage specify sample two law partially life wikipedia users trust review discussion friend vote user vote sign sign network wikipedia see cycle sign network accord prediction cycle balanced life three real cycle likely unbalanced study cycle network cycle cycle cycle cycle table cycle sign calculate denote observed sign calculate number standard deviation value discussion table three cycle negative balanced unbalanced large expect cycle contain reader balance negative network value cycle balance balanced c property cycle cycle unbalanced cycle balanced hand rank theoretically network real rank great two network er generate network network except completion algorithm look otherwise element wise approximation choose low compare prediction imbalance learning cycle case become completion lr choose factorization hinge lr lr lr completion hoc balanced observe entry uniform sampling noise rest generate observe lr lr outperform base substitute balance whose cycle balanced balance perform poorly underlie hoc learns cycle make balanced drop hand lr show guarantee lr recover observation synthetic add observation see lr perfectly lr perfect recovery law distribution relaxation completion crucially assumption real entry law examine perform law generate arbitrary expect degree plot weakly unlike compare hoc hand law see balanced network method cycle hoc lr power law synthetic local factorization lr observe type successively remove method try predict accuracy show accuracy decrease triangle finally boost large end order cycle observe threshold higher go beyond significant method resort cross concrete create disjoint fold consist test logistic happen remove hoc accuracy rate accuracy average fold improvement hoc reveal interesting phenomenon indeed hoc significantly hoc show unlike hoc various threshold consider benefit well hoc point order cycle benefit furthermore network motivate network turn attention rank law relationship consider lr lr real observe outperform cycle hoc hoc less global method consistently hoc lr lr improve accuracy show hinge prediction cc hoc hoc lr wikipedia representative hoc lr show edge lr surprisingly prediction accuracy regardless method prediction accuracy compare require different method factorization large sign table construct synthetic weakly balanced totally million show approach easily hoc need sgd hinge hoc classifier hoc hoc lr lr hoc structure particular term scalability ccccc hoc hoc subsection sign cluster sign balanced note truth weakly balanced sampling uniform uniform performance calculate edge satisfy ground assignment outline uniform noise network apply sign laplacian evaluate sign recover sign cluster lr sign laplacian network mathematically structural global balanced generalize balance notion show weakly balanced group theoretical study decade network scale become study sign several justify sign network widely sign however counterpart link prediction network correspond prediction link prediction explore solve computing sign develop trust entity recently use kernel triangular edge reasonable sign closely completion substantial study approach completion mostly collaborative arise
serve highly shrinkage form property study answer infeasible general facilitate use seek modeling insight role answer kernel capability independent answer positive estimator attain bound almost sense impact arbitrarily specify generalization like smoothness organized provide review explain motivation research polynomial localize kernel xu property role generalization capability regularizer section conclude paper since certain solution memory illustrate draw noise totally test sample deduce regularization quality regularizer capability depend take generalization regularization generalization capability capability capability arbitrarily specify generalization computational relation capability heavily capabilitie regularization scheme capability heavily fig relation tune regularization may possess capability capability choice eps scale eps find kernel importance solely choose consideration emphasize course obtain capability regularizer several focus wu algorithms claim extra hypothesis essentially scheme hypothesis regularization strategy sample rate bound sample hypothesis respectively result adopt tackle regularizers method cope impose note wu generalization assumption regularizer zhang song reproduce banach least equivalent satisfie song regularizer concentration assumption limit within certainly capability depend generalization capability generalization rather deduce low estimation capability scheme serve generalization provide answer spherical orthonormal use function space ball sphere q n restriction homogeneous harmonic polynomial degree spherical polynomial dimension denote formula eq denote set spherical center angle exist constant integer cover satisfy orthonormal basis define reproduce kernel reproduce hilbert reproduce q concrete positive admissible construct mask admissible henceforth definite whose show useful lemma reveal possesse reproducing define admissible algebraic degree localization actually polynomial localize property arbitrary exist constant depend q possesse property arbitrary depend method approximation yield approximation depend conduct capability specify aim derive quick review regularization last remark main input relationship assume admit decomposition aim error purpose error minimize function since access example hilbert integrable know function norm sample main estimate formal borel enter competition establish hypothesis space h z accuracy loss implement kernel number tolerance draw z simulation label value small tolerance otherwise simulation fig upper right tolerance approximately tolerance low color tolerance area area color point dramatically theoretical comparison bind directly set sample generalization capability bad show generalization capability associate specifically show far concern optimal influence capability sense regularization almost thus application merely generalization criterion methodology adopt traditionally divide error aforementione style regard attribute characteristic learn hypothesis specific divide reveal negligible learning analyze yield almost generalization benefit may due approximation deduce near formula derive use construct inequality prominent cover error estimate subsection subsection formula second deduce probabilistic need lemma dimensional define spherical establishes formula unit ball whose locate exist follow lemma bernstein almost identically arbitrary number least brevity without loss arbitrary equality follow hence hold virtue lemma prove identically draw arbitrary equality least q introduce decomposition ef ef z ef ef ef e ef ef ef inequality exist hold z define ef ef ef z easy deduce thus lemma confidence confidence least depend also quantity element least net quantity cover belong vc set exist equal quantity pseudo function see relation entropy arbitrary vector apply follow everywhere proposition qx sample z nx least cover b f q hold yield q let similarly since almost everywhere everywhere q deduce hence together yield deduce proposition ef cn e ef bind follow constant proof theorem complete study regularization fundamental scheme capability methodology say kernel regularization attain asymptotically identical capability regularization generalization concern choice complicated kernel heavily answer completely whether affect kernel localize implement capability kernel possess current investigation hilbert space function orthonormal basis reproduce unique basis exist summation concern hand iy thus iy dd k lemma proposition institute system school mathematics china center department
pairwise tangent reformulate determinant avoid convex solve interior method expensive geometry riemannian manifold contribution aid introduce kernel riemannian solver rkhs solved tie stein lastly show coding obtain performance texture classification person state art feature riemannian locality preserve projection begin overview bregman divergence follow cover dictionary manifold propose finding divergence stein divergence lead stein kernel riemannian bregman define strictly bregman divergence asymmetric jensen shannon symmetric divergence negative curvature symmetric stein riemannian metric manifold thompson metric inequality unique geodesic p stein similar weak property establish stein divergence address riemannian stein geodesic curve riemannian geometry empty set riemannian kernel stein let iff reader follow convert discuss determinant compute cholesky decomposition stein riemannian give query manifold express idea rkh idea combination term expand consequently relaxation obtain code specify solve obtain code query directly aid euclidean atom dictionary label ie approach close code atom dirac code residual error alternatively label dictionary tie code query datum feed code essence problem riemannian set indirect sparse riemannian code various space method like receive euclidean propose mean iterate step code fix compute update dictionary atom update derivative far simplified term inverse computing solution exploit previous rearrange estimate rt avoid normalised second norm iteration dictionary stein kernel riemannian manifold stein iteration nk ng tr k f r j j f tr tr r figure angle compose ba face expression illumination bf riemannian descriptor face intensity position wavelet center orientation sr sparse sr case propose obtain high furthermore especially b face sr sr bf average sr euclidean sr tensor code texture follow generate nine test scenario texture texture manifold fx descriptor feature test class select testing datum sr obtain high slightly texture texture log euclidean sr indicate deviation example performance dataset respectively cumulative characteristic curve method compare histogram symmetry accumulation locality preserving use capture camera variation appearance people sequence pixel use testing descriptor position correspond colour colour several previously histogram accumulation feature riemannian locality preserve sr heavy load curve represent match propose low performance riemannian performance obtain riemannian obtain rkhs texture riemannian truth sample riemannian source create computing covariance source point riemannian select combine weight variance measure dictionary rkh interpret texture extract block train image block per sample process dictionary generation repeat average probe report fx create remain gradient sparse riemannian propose classified use classifier generate dictionary dictionaries k dictionary high dictionary considerable tb texture couple learn address riemannian manifold seek aid lead riemannian experiment task texture classification notable discrimination code riemannian locality accumulation stem riemannian geometry stein via tight sparse coding also considerably tensor tie stein error rkhs improve accuracy use stein solve margin classification problem manifold translate design stein extension communication centre program ie box school national vision consider sparse symmetric definite riemannian manifold relate sparse code manifold reproduce hilbert space convex kernel solve tie kernel texture identification sparse improvement discrimination comparison locality symmetry accumulation sr lead result image
descent another generalization mirror mirror descent geometry penalize proximity online q p descent update generalization proximity twice b machine information geometry mirror correspondence divergence family exponential family bregman divergence l mirror natural gradient riemannian introduce later combine bregman divergence riemannian manifold develop direction descent riemannian efficient family neither mirror algorithmic mirror descent order implement natural gradient mirror advantage section prove mirror key involve concept conjugate bregman divergence convex riemannian manifold concept convex imply strictly twice also motivation convex supremum dual attain represent system strictly differentiable eq co straightforward show bregman divergence table explain bregman pair riemannian manifold differentiable definite riemannian map riemannian manifold riemannian riemannian manifold parameter whereas natural p consistent mirror descent discuss consequence implication bregman natural along dual riemannian state mirror riemannian manifold recall mirror q finding term dual noting rule discuss direction manifold immediate induced conjugate far aware riemannian manifold mirror secondly algorithmic notice mirror method since prefer since computation hessian equivalence mirror descent potential descent exploit mirror descent efficiency mirror perspective statistical mirror see covariance consequence mirror descent estimation exponential family q strictly function term divergence bregman mention one correspondence exponential family descent minimized step natural minimize co directly mean natural natural mirror yield efficient er rao unbiased estimator base equivalence natural descent mirror fisher er rao illustrated bregman natural mirror natural gradient ambient direction gradient manifold constrain remain discuss relation descent step idea flow velocity usual state manifold lie along manifold begin along field take key gradient descent introduce guarantee lie manifold extremely differential equation consequently computable exponential first exponential map yield natural consequently mirror first manifold descent riemannian manifold equivalence mirror riemannian mirror descent er rao mirror connection firstly issue connection natural gradient performance mirror would determine precise descent norm explore acknowledgement gr dms mathematical institute sm grant biology gm nsf maximum objective desirable admissible statistical large regularize prefer statistical regularize belong differentiable prior map ridge regression early optimize original stop iteration choice manifold every bregman divergence b g j leibler cc mathematics institute genome sciences
flexible regression common categorical substantial interaction computational time requirement coverage ability answer question reference fall xx xx probability extreme environmental health density make characterize distribution allow interaction incorporate step challenge assessment tail risk complex primary x devote provide option develop problem involve ideally exist high dimensional black motivate inclusion interpretable impact feature finite general expert distinguish weight straightforward form method literature maximization overfitte literature seek avoid posteriori estimate inherent bayesian quantification technique dirichlet dp prior focus flexible mean regression feature subsequent dependent mixture weight small categorical development weight mixture develop map permutation employ stick allow probit breaking probit incorporate response feature circumstance estimation infinite notable logistic present method density prove approach challenge derive curse explanatory grow data fill associate value factorial minimal projection space develop bayesian dimensionality approach demonstrate parsimonious involve disadvantage bayesian tree focus mean common residual note question response address upon mix weight certain break across profile situation involve continuous categorical consider univariate categorical conditional possible exceed sparsity complete feature propose tucker factorization general density q row feature constrain regression density problem tucker decomposition kind tucker tensor characterize reduce focused development derive upon tucker drive characterization soft dimensional space associate across combination rely contribute information match govern map hard full factorial still enough kernel desire sparsity first information influence assumption quantify convenience employ augment likelihood soft examine stochastic search merge move proceed fashion inclusion require serial computation make simplify assumption numerical notably computational inclusion upon decrease inclusion impose cutoff inclusion cutoff ordering stage tendency clustering likelihood candidate pass consideration perform second proceeding second inclusion approximate marginal approach individual assessment feature inclusion cutoff use gibbs produce detailed predict around assess prediction vary ground produce simulate true three way feature combination ht equal produce base prediction underlie random forest implement package real compare performance rf pass first pass package fed treat outperform rf prediction show comparable coverage summarize show appropriate positive strong metric utility compare rf dna environmental exposure type remainder information nucleotide detail original instance cell chemical examine treatment time treatment nature cell record dna cell higher indicate generally exposure chemical exposure long appropriate derive quantile cell line exposure cell tail treatment time reflect learn logit indicate cell individual aspect copy major allele snps snp also two copy allele copy allele copy allele snps distinct leave analysis
segmentation prior svm show cast optimization extend generalize margin submodular mrfs correspond negativity cut search submodular submodular call clique course submodular impractical vision problem whereas exploit neighborhood structure pair regular reduce cut approach allow express prior cut involve pairwise clique almost submodular computer cast structured prediction spatial generic maximum network support vector svm among margin approach allow submodular submodular discriminant submodular discriminant discriminant information diversity engine abstract maximization excellent practice discriminant function mrfs traditionally parameter hand exploit regular mrfs relaxation program lp qp cut regular mrfs graph inference interesting graph coherent separate unary potential mrfs functional vision general label briefly summarize function minimized flow flow arc capacity slightly modify description cut arc additionally arc every arc associate residual arc familiar residual max unary arc decrease residual interior arc piece keep track clique unit flow arc residual arc always j cs ta tell optimize flow along arc search max method vision submodular path path residual find ensure short track distance node maintain unique short path short proceed alternate pass pass grow layer symmetric grow scan arc residual capacity distance find path via arc flow operation flow arcs arc parent arc tree augmentation perform parent potential arcs neighbor distance parent apply submodular arc flow increase flow arc reverse change arc flow arc either arc never proof supplementary shorter contain become normal create therefore maintain submodular flow current arc iterate parent perform flow arc maintain arc heuristic correctness runtime material complexity still perform capacity arc separate done search work step runtime fast review svm associate svm output prediction output draw loss function quantify associated image hamming segmentation predict labeling mechanism svms find discriminant pair derive throughout nx give always discriminant value incorrect discriminant value add slack constraint example intuitively slack example want add slack parameter slack moment label cover label case section learn sum learn discriminant single clique clique vector letting index define claim parameter enforce linear submodular asymptotic qp constraint discriminant margin submodular qp feasible optimal qp margin vector constraint qp submodular ensure clique potential maximum margin submodular clique qp constraint violate I slack max violate slack formulation slack qp intuitive view train slack qp compare slack slack qp solve slack qp replace slack constraint example include slack svm c b b tuple sized large qp precision cut plane keep constraint solve regard violate add long sum instance hamming loss arbitrary unary potential entire expansion reduce subproblem keep label multi function expansion take submodular label energy set write eq also submodular characterize energy function use otherwise let model generalization encode learn copy add final note expansion optimally expansion label expansion optima inference svm denoise interactive denoise generic cut provide use interactive natural use run arbitrary denoise improve hand tune denoise line draw original similar independent pixel hand tune cut pose mrf unary pixel equal clique square root clique unary image prior include unary root smoothness prior clique every cut possible define patch loss noisy tune pick value minimum pixel minute svm perform well training cut image vs sec cut visually look show svm input interactive segmentation image sparse background annotation set foreground pixel segmentation image comparison crf unary crf fit histogram foreground pairwise sensitive code
consistent cluster contain constant additive noise run probability consistent contain connect mutually appear receive dimensional concentrated clutter consistent enough note clutter recover intuitively much small clutter far clutter low case variance recover separate ambient possible via deconvolution approach neighbor albeit obtain estimator case usual kernel estimator consider unlike usual integrate dimensional full kernel follow satisfie metric measure vc characteristic density manifold quite mild relax include appropriate tail albeit complicated deal integral similarly n h preliminary showing estimate similar modification omit full density bound measure depend depend ball density measure depend eq follow modification careful various p h fx ix bx dx ball uniformly bind obtain first tree vc characteristic kernel notice unlike prove see distinguished establish consistency satisfy assumption level distinguish similarly give manifold manifold depend vc characteristic hold consistent eq approximate expand taylor simplify claim must reliably resolve around kind ball volume separation connectivity region width point noiseless inside manifold clutter intersect remove rv ignore contribution keep already inside triangle distance argument set exist pair geodesic contradiction condition omit satisfy condition arise clutter noise constraint upper ensure equivalent let use resolve sufficiently compare pick ix r first x r mx triangle inequality least finally mass least latent outside provide contains ball latent suppose separation successive geodesic since satisfied indicate modification detail radius finally observe state lemma precisely ensure accord section pick radius pick argue pick crucial ingredient analysis center n nr net convergence notice pick replace easy check lemma together estimator tree recovery presence particularly regard minimax hope address easy show achieve understand estimator help manifold simple modification use geometric currently extension theorem theorem theorem pt pt density near embed modify version near neighbor mild dimension albeit result density sketch spatially adaptive achieve rate cluster collection estimating refer paradigm attractive clear population quantity make typically number finally inherently object cluster summary consistent tree linkage recently consistent find connect show appropriately concerned motivation provable live generally due curse dimensionality dimensional spread manifold hypothesis adapt intrinsic summary contribution show consistent fast dimensional manifold require manifold identify salient sketch manifold framework study consistency concentrate near sample manifold unobserve efficiently bandwidth level simulation cluster back expand idea formalize notion consistency fractional consistency linkage consistent generalization wishart review effort nonparametric focus specialized fix imply cluster consistency hold level trivial therein consider pruning remove spurious determine low cluster asymptotic unknown assume manifold riemannian compact manifold main impose number normal bundle radius every number prevent close self euclidean denote mx bx z collection tree hierarchy c informally dendrogram result salient definition modification account separate nonempty along fx refer rely estimator cluster tree consistency resp say separate except restrict finite theorem establish theorem lebesgue suppose run output large particular resolve pair least reason respect grow edge adapt involve q clear notice addition dependent increase finally use run sample draw probability remark solve recover cluster level condition main typically dependence sketch dependence another aspect radius unknown connection radius mild satisfy identical theorem real whose depend lead establishes consistency recover entire schedule cluster distinguish enough formal mirror begin section show mutually disjoint main challenge curvature ability connection algorithm somewhat surprisingly consistent tree classification non uniform convergence ball mas vc inequality good ambient dimension inequality get obstacle insight ball ms suffice net center net ready section exist follow nb sp provide ball apparent curvature many argument intuitively state volume low upper volume q v point keep remove modification importantly still resolve identify eliminate throughout proof good ball hand mass r n mx removed prove geodesic apart geodesic connect connect geodesic pass least satisfied condition assume least connected connect lie entirely I sequence minimal exist density ball everywhere least condition satisfy guarantee ball least sample completely apart apart dimensional density show straightforwardly instance ignore upper suggest bind describe part top middle bottom respectively middle part denote sphere center part center portion describe corner intersection essence construction ball construction finally whose density total mass plane require instance discussion ignore inconsistent mass euclidean ball mass
potentially square version straightforward simply keep deviation interpret change prediction standard intuitively focus common estimating house house encode effect adversary powerful adversary adversarial set adversarial round adversary round learner predictor binary model exposition norm impose input lie inside second input scale weight vector constant input adversary w q unknown reveal regret bound output minimum containing always allow diagonal always volume predictor update way adaptive gradient descent necessary achieve bit bound update fit combine rule follow guess minimize guess adaptive descent fix ii derivative ii ti choice ti matter scale axis scale large thought however good assume differ descent drop diagonal determinant c ii particular choice ti regret rescale simple ii ti ti x apparent potentially order complicated let tt ti ti ti ti g last lemma lead perform compute optimal suggest potentially ideal choice q minimum potential sum demonstrate impact input come advance pass algorithm compute perform adaptive bad enable use weight vector utilize determinant choose ji fix know advance sum impact ellipsoid differ reflect bind first input encounter determinant regret ji degradation due adversary large zero dimension bad streaming scenario low sequence particular permutation percentile feature lead exchangeable diagonal let ti ti incur w bound rl realization ti ti quantile quantity relate make always logistic similarly loss adversary induce bad ti ti expect suggest much bad experiment size bank census ct bank census ct slice loss bank census ct compare gradient without projection step validation besides adjust either square task square utilize uci census uci uci location ct slice data song uci datum public normalization pre publicly relatively little pre normalization pre every evident normalize highly heterogeneous measurement ct exhibit ct slice raw single device range conversely dataset degree trend evident varie term burden search easy conduct likely height pdf different pre normalize selection normalization result pre normalization update indicate max norm much outlier evaluate unnormalized public provide adaptive apply normalize dataset simultaneously achieve capable algorithm adversary scale adversary thank discussion tw tw tw g imply w tw tw g w tw tw tw w tw tw x inequality fact root concave upper taylor solve use rewrite invertible simply choose eigenvalue c input quantity rewrite imply z ii bound c regret ti satisfy tw w g w tw project onto thus tell w w tw w w ti maximum could imply ti ji projection step tw w tw w onto definition imply must contain term x c ji ji ji ji inequality increase w w tw ti maximum absolute ti ti ti obtain ti ji ti yield lemma denote large denote percentile ti tp one sequence observe percentile equal percentile feature corollary bind remain rl tw ji ji ji tw ji rl ji ji ji use must ti prove like much adaptive w tw x ji ji ji ji ji ji g ji ji ji ji lemma tw tw w ti feature ever ji ji ji second monotonicity online second term scale adversarial present present w regret w tw x k follow concave choice choose ji ji ji ji g ji ji ji ji ji ji ji lemma ideal surface ellipsoid especially might pay increase norm motivate equation access sample input initially feature define increase factor well note never bad remain tw w incomplete thing might tw ti ji tw try slightly make satisfy analyze problem invertible eigenvalue x x trace x x diagonal x tx ti r ti ti choice g ti ti feature rescale ii g w w tw w tw tw tw tw w w tw rearrange term w w convexity w w w tw tw w tw w tw w tw tw convex x w w adaptive w w x g lr w w g perform compute add make show regret w tw x induction statement kn induction last use fact concave upper first g ji g ji ji ji g ji ji ji ji ji ji ji lemma factor surface ellipsoid feature case norm pay significant increase equation case access define intuitively estimate good also note guarantee bad factor good tw tw ti achieve ji tw might slightly previous adversary input ellipsoid adversary ellipsoid goal vector c w general set volume set constrain contain weight c qp regret recall previous diagonal minimize solve use rewrite assume define ellipsoid space direction eigenvector maximum eigenvalue c eigenvalue w define change optimization rewrite since ii z diagonal ii ii ti minimize simple coefficient thus choose ti ti ti regret coefficient let tt ti x ti ti ti ii ti g ti gradient ideal choice information far return minimum access come advance know gradient pass compute diagonal descent time show bad well tw directly g w w tw w w tw w ti first must tw tw g hold w c ta tw guarantee w w tw tw maximum ti ji prove equation ji factor worse know advance pass pass entire datum amount datum case might learn input learn particular pass proper future input able case know observe future would regret ji ji g ji x ji ji ji ji ji ji ji w w w tw w w ti ii ji ti use must guarantee w tw guarantee e ellipsoid input since ji ji xt w compete set weight whose must get guarantee w tw w ti ji tw ti ti I ti l ti ti ti ti thus w w tw ji ji I combine ti I w tw ji ji suppose high great percentile feature percentile equal percentile reason randomly guarantee top percentile thus fraction probability would percentile top percentile expect lemma microsoft usa york ny usa online feature prove regret useful normalize robust learning transform preferred applying transform standard deviation set batch unclear applicable input dynamically g demand primal enforce invariant normalization technique accept use inherently capable unnormalized operate define indicate case invariant monotone interest practical algorithm importance setting hyper parameter regard parameter normalize need hyper normalization normalize test particularly ram time
newton scheme multinomial repeatedly minimize guess probability second derivative derivative observation hessian within combine write center independence sum quadratic quadratic dominant toward li q ti original approximation algebraic framework convergence toward complete repeatedly update multinomial combine loop follow initialize p interested model value compute feature along proportion path near statistically behave long converge choose restrict cut require fitting similar variable unfortunately could practice essentially never tucker potentially back rarely improve elastic net row algebra thus regression step replace regression multinomial new implement describe package usual multinomial lasso grouping heavy write simulation intel ghz processor run vary class variable simulation true iid row entry group group group group group group value average group order fast multinomial take little group quickly sized problem descent efficiency package purpose penalize use quasi extend publicly implementation speed solve gene problem regression traditionally many fail case often toward find trading give generalize deal propose solve eq disjoint index vector group linear case might particular roughly important explanatory either suggestion among group row refer likewise future multinomial via generalize find multinomial newton reduce regression multinomial regression coordinate algorithm algorithm must multinomial problem coordinate descent multinomial incorporate solve regression descent refer partial minimize objective style gaussian include well initialize iterate eq term center minimization intercept multinomial
conditioning correspond consider parameterization augmentation sa give transform cholesky parameterization augmentation aa aa sample illustrate introduce noisy distribution individually parameterization sa slightly aa parameterization approach coupling variable issue comprise sample strategy marginal likelihood devise clarity metropolis proposal integrate already analytically result hasting estimate unbiased sampler correct result remarkable couple ht aa parameterization parameterization figure conditioning sa aa green pm effectively mcmc quantify prediction need could iterate gibbs sampler despite could still sampling verify unbiased expectation marginal ratio distribution mh hyper accurate small report expect small estimator assess motivation simulation likelihood relatively generally challenging guarantee estimator available estimate sample form intervention assessment convergence efficiency useful work gaussian propose yield correctness rearrange interpret hasting ratio importance regardless target sample approximate expression target analysis reveal interesting similarity propose propose however pm irrespective section pm characterize base sample sampler third aa sampling involve la ep square solid assessment length scale sampling approach approximation approximation consideration datum gp model use generate impose scale shape repetition importance importance ep lead suggest little increase variance ht approximation la number compute isotropic acc acc pm pm pm aa pm la pm ep pm ep ep aa pm la pm ep pm pm ep aa pm pm ep pm ep aa analysis importance gp combination covariate generate select choose prior hyper sake convenience introduce run parallel chain burn follow use initialize use also check meaningful estimator eventually acceptance move acceptance chain preliminary ep acceptance rate isotropic ard covariance namely idea pm likely variance indicate approximate employ importance already offer acceptable ess compare however rate small ht ard la pm aa pm la pm aa pm aa pm la pm report pm method aa schemes metropolis comparable burn phase show pm achieve fast aa comparable present wise achieve high covariance construction aa close facilitate trace factor plot panel evolution period chain breast ccc parameterization aa parameterization extremely pm aa remarkably base la key pm possibility ensure situation employ investigate likelihood base report five uci classes window non set vs increasingly comprise equal number across evaluate pm pm pm ep classifier optimize type ml ep ep code package front end library employ square exponential isotropic isotropic pm ep reliably quantify label predictive confident label decision threshold summarize ability reliably quantify auc area receiver operate characteristic roc curve classifier versus curve accuracy degree score capacity capacity auc capable classify test confidence probabilistic versus denote accord compute probability condition rest confident increment accuracy respect finally area curve capacity auc classifier might divide two capacity curve auc quantification uncertainty svms gp ep ml yield integrate trend pm achieve quantification classifier cc look obtain mcmc pm marginal generally enough distribution pm ep obtain employ ep mostly prediction situation derive exhibit convergence consideration aim unseen achieve quantification point make exact sense actually build upon deterministic consideration ep ep hyper small isotropic rbf grid ard instance employ derivative respect compute extra mcmc ep marginal ep iteration approach pm ep require operation ep cholesky gaussian need extra operation operation compute run approximation mcmc sample pm ep sample hyper distribution parameter similar argument ep ep scale paper methodology model probit working build marginal devise scheme hyper currently popular indicate propose methodology speed feature process possibility intervention efficiency drive hyper inefficient hyper investigation study avoid random hyper optimize candidate covariance function hyper integrate uncertainty classifier commonly community account extremely beneficial small scalability computational bottleneck computation factorization apply integrate latent argue run hyper computational overhead believe result cox extend gp model characterize spatio sparsity inverse yield mix efficient capable amount anonymous critical constructive suggestion establish fellowship award project grant ep dedicate challenge adopt make probit illustrative present base efficiently issue improvement exist simulate distribution superior quantification uncertainty prediction art confirm model chain carlo kernel method approximate represent model throughout paper working problem however base relevance gaussian modeling build tackle problem call hyper observe hyper parameter grid search nature usually latent optimize hyper ml approximation propagation ep bayes integrate nest laplace extensive scheme like ii integrate latent hyper posterior uncertainty particular date literature tackle limitation possible obtain inference close analytical integrate quadrature recently carry stochastic approximation monte leverage guarantee monte employ infer challenge still inefficient practice aim implement gap propose address difficulty apply discrete classification carry gp hyper posteriori characterize poor mix break integrate variable mcmc maintain posterior hyper ergodicity marginal pm sampling show pm lead remarkable hyper thus implement employ hyper achieve sound quantification prediction highlight challenging marginalization carlo building upon already direction treatment quantification hyper parameter furthermore version svm support integrate quantification uncertainty achieve organized review present variable assessment pm gp classifier classifier conclude briefly probit extensive gps covariate univariate response latent perspective gp base likelihood function gp evaluate parameterize adopt covariance parameter distribution isotropic latter relevance determination ard hyper parameter scale comprise length hierarchical condition keep report input briefly one difficulty encounter unlike prior integrate consequence directly predictive research attempt integrate predictive new follow yield marginal approximate distribution hyper notation give distribute make integration respect univariate integration follow probit likelihood briefly popular integrate latent propagation laplace la center curvature taylor logarithm latter requirement approximate equal hessian logarithm hyper logarithm perform iterative probit
objective utilize bind hessian guarantee smoothness e quadratic quadratic q imply linear long approximation hessian particular average capture follow proof appendix stochastic plug quadratic instantaneous loss fw z first concentration I nh w lemma set instantaneous number sub optimality dm obtain desire mild least linearly instantaneous objective convex generally objective objective hessian rank certainly non objective regularize instantaneous instantaneous objective nm behave distribute gradient descent aware require iteration generic newton theoretical objective believe generic generic objective assume strongly smooth assume w establish small converge sufficiently ensure set recover familiar weak account believe quadratic bridge variant enjoy local proof procedure replace step exist w small c mnist number term tune picking make close per increase leave future begin consider use dataset machine show behavior machine total example hence datum biased namely away optimum gaussian derivative monotonically thus moreover easy verify let symmetry turn analyze therefore monotonic gaussian q get hard calculate thus verify always strong convexity perform instance fw give one shot analysis fail modification simple averaging reduce bias specifically optimum subsample optimum combination unfortunately still correct still fail least sketch simplicity tail choice return determined distribution numerical verify scale get always eq iterate desire auxiliary q equal definite size use eq back q averaging assumption multiplying side result follow ready prove lemma bind get justify right optimal mean get justify pick average receive probability probability plug instantaneous strong convexity sufficient eq w consider denominator bind begin theorem conjugate w ready follow derivation first use smoothness third inequality second jensen follow recursively eq inequality come third inequality result corollary novel newton objective enjoy require reasonable evidence advantage shoot admm consider problem minimize population e machine I evenly among machine use approximate minimizer lie set optimization resource play processing machine focus algorithm alternate local average vector high develop straight forward machine optimize obtaining refer shoot latter correct optimum minimizer obtain although shoot much bad population compare minimizer seem address round communication descent iterate also accelerate gradient need attain polynomial dependence condition convexity might convexity overall size number round polynomially sample descent sophisticated utilize quasi bfgs still alternate direction multiplier alternate dual variable distribute manner augment respect local datum recent rate favorable communication condition algorithm mention orthogonal coordinate assume approximate newton geometry particular take local admm however newton immediately apparent rigorously prove gradient benefit objective quadratic objective round scale empirical minimizer roughly machine evidence objective say optimization empirical reasonable attain distribute communication gradient strongly carry population minimizer ask whether achieve stochastic optimization shoot average recently convex objective third respectively refer high lipschitz derivative shot define argue dominant particular scale shoot population single round rate communication moreover replace bias strong worse strong arise regularization sample size increase e regularize svm well convex decrease choose even small unfortunately substitute term dependence shoot estimator total sample sub shot benefit ignore datum distribution set perform gradient run shot average universal construction deviation output eliminate scheme appendix show distribute average iteration converge optimum distribute newton type rate regularizer g w solve w maintain machine gradient separate computed local iterate average iterate perform machine bregman q objective objective eq bregman divergence check w w affect iterate depend vary update function
carry payoff adjustment relation dual mix admit every interior perhaps integral payoff bound moreover boundary continuity write forward penalty discount player specific player tend mix performance score reasoning role discount rate game whereas affect drive game equilibrium depict dark red point dark payoff discount dynamic fail rest drop critical globally unstable phase strict aim theoretic solution conjunction concept tie payoff end response smoothly nash equilibrium nash smooth curve form terminology widely specification logit u kk begin level nash equilibria concern equilibria assertion interior rest give face forward restriction kkt level proposition discount play double one discount player assessment reflect importance player give level measure stationary player say capture dynamical rest analysis case game player payoff align sense game function increase along q lyapunov dynamic interior u algebra yield k kx construction assertion jensen satisfied lie restricted lemma support nash equilibrium contain converge game solution interior boundary proof integral dynamic indeed lie highlight score boundedness boundary game hand connection q k simply score score remain payoff reflect k z begin show give z k k x kx x kx invertible simplicity inspection also tie set evolution volume denote ordinary lebesgue however k k tu assertion solve proposition yield classical asymmetric admit characterize much dynamic context reflect player picture game say neighborhood finally pt nash lyapunov stable nash state nash decomposable player player choice property lyapunov stable strict enough also stable stable nash equilibrium stable nash break discount lyapunov clearly nothing contain interior lyapunov contain contradiction regard equilibria generality strict nash equilibrium consider treat q jacobian order z kt continuity rest stable kt kt k kt lyapunov stable trajectory u k imply lyapunov pick time relative dynamic readily substitute conclude stay choose negative lyapunov implication span show relatively open either provide interesting insight role attract hand attract one seek pure nash nonetheless seek player converge arbitrarily briefly player discount even sign different thing note also anti rational payoff equilibria nash equilibria repeat k kt restrict opposite expand respect volume continue hold pure I vertex point unstable discount rate rest nash dot drop non equilibrium asymptotically equilibria game nash vertex game equilibria rest point equilibria rational correspond equilibria sufficiently broadly tie negative discount attract vertex dynamic attract vertex asymptotically lemma set unit volume become euclidean near expand property interior time inversion claim proposition likewise claim note vertex usual integration integral absolute kt k nearby interior restriction property kt cf proof proposition lyapunov conversely pure lyapunov necessary interior case neighborhood contain grow open contain proof complete play repeatedly euler discretization recurrence track step scenario absence monitor discretization involve payoff cf summary drop assume possess unbiased payoff observe game perturb decentralized variant update cccc uncertainty payoff payoff game payoff begin size adapt split un take euler discretization conversely relate trajectory step martingale ensure converge admit strict lyapunov decrease take point assumption limit point string prop prop immediate lyapunov since multilinear ensures take converge score assumption first player unbiased play drop issue player game payoff player estimate payoff sequence play player select bound unbiased payoff kn note resource allocation payoff estimate concern examine player exploit initialization kx kk mixed strategy termination reach player th strategy get together rhs potential particular equilibrium error vanish proof follow show satisfied nothing furthermore kn kn c uniform iterate cn n n assumption conclude bound away boundary image interior action player difference player possess actually replace k scheme player game arbitrarily alg innovation ultimately show difficulty innovation instead try track focus follow implement directly payoff discrete algorithmic dynamic player payoff one starting require logit mix kx receive termination rule reinforcement equation step rh unlike evolve mixed support step vanish payoff remain away penalty remain whenever begin drop player payoff use constant fix hand algorithm choice never become small apply directly algorithm dynamically payoff iterate simplex account sequence nash equilibrium furthermore vanish thank verify immediately simply note innovation strategy lemma thus converge set algorithm iterate limit interior assertion importantly arbitrarily game equilibria scope player arbitrarily close nash take hope nash equilibria game probability globally suboptimal random strategy variant gray normalize point also convenience strict strategy converge equilibrium even record payoff player play payoff albeit relatively mild violate remain strategy occur period g propagation delay current examine player strategy since counter update carry replace computation allow payoff subject delay well perturbation player past observe stage perturbation easy check general decentralize variant discrete keep counter simplicity logit mind full support pt current realize payoff reach cf lemma homogeneous unique distribution become step aggregated treatment conclusion delay easy represent eq q player strategy denote rhs rate adjust dynamic rate dynamical process equal include dynamic rest also lyapunov unchanged discuss distribute need payoff player choose alternate player action update guarantee game nash equilibria convergence roughly discount rate simulation converge even small discount fig repeat gray rgb corollary conjecture theorem remark penalty dynamic penalty fr fr start heuristic learning scheme new penalty penalty keep exponentially aggregate payoff inherent duality variant evolutionary converge arbitrarily approximation nash equilibria potential traffic engineering discrete time payoff algorithm require remain perturbation synchronization player tucker ne nash equilibria equilibrium response equilibria ordinary differential equilibrium considerable decade procedure divide category evolve class include learn play variant infinitely iterate overview payoff literature game focus player stream instance converge set correlated equilibrium whereas error player pure nash equilibrium provide equilibrium reinforcement learning framework player base payoff play mechanism player game continuous extra keep discrete viewpoint player move score action converge call good correspondence oppose equilibrium point response map kind usually compare long term comprehensive introduction g guarantee counterpart usually derive possibly random cf contrary develop two process crucially look evolution player performance consist drift keep action discount payoff constitute strategy dynamic thank dynamic also variant stability crucially discount strategy case equilibrium equation factor equilibria paper concern implementation desirable payoff player subject perturbation date need decentralized protocol traffic pose significant nonetheless property player converge approximation strict nash admit thus characterization obtain form agent turn mapping assessment score mix end would high score good response carry happen e tie rule trajectory instance payoff commonly case theoretic process equilibria strongly act cost pure term irrespective origin game theoretic comprehensive account therein let simplex span boundary refer induce allow view hx negligible map derive simplicity presentation comment
increase rejection coverage positive point relevant consider statistic reveal element affect rigorous assess study sake simplicity brief introduction former situation pairwise would sample know nominal level depend nonparametric test approximation adopt approximation idea distribution interval suitable draw version f iterate possible computation simulation consume simulation area burden might attractive practically analytical discuss analytical possibility would element bootstrap nan hypothesis small bootstrap achieve accuracy hand theoretical bootstrapping asymptotically need fact resample cope aforementioned involve computation model obtain version require consideration odd resample avoid focus suitable converge eigenvalue outline inferential procedure speed bootstrap reconstruct nonetheless may nan overcome ensure reflect kullback leibler coincide formulation solve ps n ps detail derivation root primarily address reflect turn whereas bootstrap estimate estimate monte former replication whereas latter bootstrap replication computation equal strategy effort replace inner appeal ad hoc calculation consideration smooth reduce bootstrap replication replication ran deterministic full replication resemble generation nan e replace deterministic test confidence store contribution replacement version us ps ps sort replicate large bootstrap score obtain resample compute accord replacement index obtain ps ps us value contribution new resample inner iteration j computation bootstrap substitute outer desire bootstrap b benefit follow one yield sentence construct exploit magnitude minor modification argument ps therefore distribution may expand involve polynomial depend smoothly bootstrap counterpart population counterpart consider difference actual nominal magnitude provide notation minor limit n ps g bootstrap counterpart bootstrap counterpart show bootstrap asymptotic easily weighted exploit proposition actual nominal since make appealing obtain fairly inferential procedure burden follow feature address discuss theoretical bootstrap accuracy fact relevance asymptotic bootstrappe require composite unstable estimate accuracy bootstrap series practical rely test set bootstrap lack imply desirable invariance however point exact computational cost test matter bootstrappe bootstrap outer counterpart pairwise pairwise score contribution sampling avoid estimate constraint reliable lie consideration regard must embed confidence claim little expense result obtain yield statistic construct therefore order weight concern whose depend hull degenerate resample assign unit occurrence hull rather minor concern f statistic automatically second place bootstrap version outer counterpart avoid use computation value benefit must minimize sampling rather exploit integrable difference actual nominal level outline appendix note far result bootstrap iteration nevertheless computational estimate vector resample element functional form hull satisfy degenerate show occurrence convex minor concern aim account associate absolute pairwise counterpart aim numerical impact estimate pairwise likelihood quantile trial equal estimate test section example serve scope dimensional compound element pairwise ss b component counterpart level test exhibit nominal especially former ccc ccc reliability useful analyse behaviour probability parameter allow nan probability contour probability carlo figure display report cb empirical nominal level likelihood compute contour reveal shape decay although nan remain use shape nan quite distant nominal multivariate correlate practical one present suppose one store outcome think normally unknown variate dimensional evaluation integral ik ik respectively consider simulated accordingly draw observation set section counterpart rejection probability statistic test nominal one full provide also likelihood poorly mark insight assess non confidence probability spaced parameter coverage lead contour plot assign high see particular corresponding hand problem remarkable nan consider none plot compare neither uniformly statistic inferential composite offer specification computational heavily involve asymptotic variance composite problem overcome resample version regularity level third accurate keep confirm confidence pairwise benefit bootstrappe non bootstrap go beyond ratio however appeal view nonparametric benefit derive inferential economic business mathematic play analogue ratio test prominent depend base composite however actual may differ considerably region distant rather framework explore confidence suitable turn accurate
must highly excellent flat constant image filter usually high pixel window spatial equivalent look care structural information scene iii distortion contrast filter version variance intend value define scalar value range big observe intensity mean window report propose last image agreement require quantify loss possess regular coordinate dft wavelet etc require distinguish blind reference define interval small filter term refined lee intensity drive filter window pixel former six air force band produce figure look false color channel filter lee confidence noise filter effect color balance filter produce lee less filter refine well preserve edge leave region apply star shape blue refined filter refine lee evident mainly detector band fail light practice look figure although clutter edge filter image smooth neither fine detail process marginally figure detect filter refined lee respectively preserve detail shape object former variation quantitative assessment assessment intensity channel look l lee homogeneous respect lee regard look bad criterion increase lee datum regard three filter least propose simulate take account target among mixture simulated pixel image generate band acquisition sensor angle spatial resolution resp filter version figure resp figures resp resp drawback refine lee present assessment compute central homogeneous well highlight consistent observe sophisticated account smoothness level lie smooth r lr em lr c refined lee refined lee refined remainder false interpretable red channel channel representation filter domain visualization national laboratory evaluate band four look spatial resolution google filter employ patch window reduction clear introduce dark center area lee filter show edge eliminate area result filter selective noise reference image assess channel homogeneous result scene distortion channel r em refined lee decomposition aim property propose indicator plane divide nine observe figure entropy class area enhance discrimination good detail technique preserve area composition band sample area blue sample present mixed sample area filter plane band reduce refined filter similar filter reduce still low surface scatter make sample blue value medium medium span forest filter refine one classification cluster refine lee expense mixing manner smoothing lee discuss iterate preserve present original reference apply refine five column stem former original comparable refined lee filter spatial iterate five add sample quantitative good area follow refined lee filter tune index c r lr em lr em refined lee refined mean forest cross band present highlight deviation small c r r lr lr c refined lee refined lee refined lee preserve original reduce good associated loss number example use divergence tool lead statistic asymptotic distribution wishart hellinger filter patch obtain manner distance mean compare filter wishart law realistic observe simplify quantitative assessment verify look noise free appropriately image blind filter expense small refined lee competitive produce bad filter instance feature assess assessment entropy affect noticed filter enhance perform preserve complex area iteratively verify plane produce cluster yield separation treat filter window iii latter economic former also adequate good smoothing without target respect work proposal quality entropy acknowledgement grateful equivalent base hellinger core proposal complex fact hermitian drastically require calculate specialize r language filter mind time filter iteration intel core software excellent accuracy present de de sp de paper present reduce divergence main select ne distribution wishart describe extend weight filter test come stem compare refined lee real employ validate show preserve prominent coherent di system amplitude return comprised channel result vertical mode vertical mode phenomenon interpretation contain image often use latter pixel frequently lee require signature pose I filter neighbor homogeneous adaptively resolution requirement homogeneous area identify poor target lee reduction square mmse lee et al lee filter technique et al decision homogeneous intensity information matrix drive adaptive formation reconstruction allow incorporation sensor image property deal process et novel problem bregman distance variation tailor additive propose noise contamination contribution rigorous free improvement regularize optimization convex functional technique sense knowledge take nature pixel definite hermitian complex matrix use regularization employ scale complex wishart variation curvature li particle optimization technique either amplitude intensity present novel blind spread square scatter way gamma reduction impose fix look call whole et term similarity weight filter suit al kullback leibler distance mean unless last filter rely mask two patch contribution pixel several similarity gaussian assumption good square term extend chen data wishart author employ equality law look proposal goodness sample test use divergence turn distance soft reject present easily generalized use neighboring patch central patch neighboring pixel illustrate figure pixel observation wishart law way employ reject use binary reject mask scale setup generalize way local window present generality cost test central
three study current clinical biological diabetes failure heart disease history first public versus potentially relate statistical algorithm patient department body level subsection overall design patient notation notation denote set generality hereafter process map matter patient consequently label patient refer patient either patient name characterize age gender current data e diabetes heart failure name either vector patient patient see cell patient verify attribute take patient attribute mention set label store make patient unlabeled patient patient decision make engine numerical patient quantify proximity patient decision hard classification otherwise usual soft maker assign label assignment quantify label case approach decision patient decision need different patient nn technology assign label rely similar decision hard design patient reliability label notion discuss subsection ideally speak patient patient express respective equivalently treat patient sake measure follow patient unlabele distance quantify exclusive weight assign attribute similarity therefore rely step detailed simple label reference operate major patient sort patient accord patient numerical quantifie depend label base refer contain patient store respect analyze patient quantify make outcome make patient patient denote assign weight conclude discuss briefly set value step learn quantifie behave nn maximize deal nn hand rely new logistic lm useful absence outcome value fit response rely consider explanatory e lr assume exist lm decision characterize maximum outcome analyze coefficient reflect list far detailed significance regression define denote vector respective deviation statistic attribute label gap binary outcome predict equal rely definition pearson logistic residual refer refer distribution introduce notation clarity training estimate label different lm algorithm nn estimation lm must overlap focus methodology include list year start unfortunately patient cf fully deal decide worth population population lm keep set unlabeled methodology phase partition lm database build rely population square perform square database aim hand experimental criterion consider paper reasoning nn model enhance capture simulate five analyze combine two describe section regression nn also refer extensively tool weight patient start weight subsection attribute refer nn patient simulation result attribute patient material five scenario medical first analysis reliable second scenario uncertain tool database contain automate simulate scenario simulation summarize operate auc auc original monte index bootstrap compute computation involve study lm specifically implement matter summary lm select lr relying criterion design programming language interface ensure selection attribute estimation lm scenario summarize estimation selection attribute table relevant predictive factor past note past history factor value lm keep eight age heart disease show value notice medical mention subsection might clinical main discuss new meet decision one clinical show factor might study design automatically factor factor lm random estimation lm weight decrease protocol factor relate medical decrease introduction clinical stable decrease factor relate medical favor factor keep significant expect random create help assess lm nn algorithms figure lm method nn nn either use nn weighting attribute attribute context lm tend powerful nn lm lm nn relevant conduct notice performance change except nn matter decrease discard attribute later add lm could suffer difficulty optimally tune performance lm one interesting figure combination lm attribute patient matter scenario without attribute perform whereas test suffer performance efficient choose combine determine factor knowledge evaluate lm use decision directly high age factor lm method disease author solid diagnosis examine result support show present lm reliable solve lm methodology call describe lm posteriori lm differently matter knowledge lm latter breast diagnosis diagnosis lm use relevant compute attribute lm perform attribute introduce opinion pearson weight attribute weight description define lm patient knowledge lm residual reflect regard rely logistic perfectly lm lm appear join opinion believe believe lm solve process latter user reliable utility medical may medical coupling nn lr modeling methodology residual lr lr herein work automate retrieval optimize robustness especially knowledge database opinion integration introduce patient orient though essential medical medical reasoning suggest meet contribution base reasoning paradigm medical new solve usually solve provide every attribute extract help
rl worth point equation framework optimal actor order linearly practical compact approximate activation l neuron control actor vector actor replacement iterative form simplicity weight residual nn way residual force sense project residual onto set inner substitution notation I computation inner w u expensive thus especially competitive dimensional domain ix w w ix substitution note update design rl present mean column sampling attain enough nice necessity investigation choice rich real select else continue algorithm datum process offline policy method policy neighbourhood equation problem arise solve either system simulation present issue initialization vector experience investigation view drawback accumulate section algorithm use different control advantage develop independent I linear pde theorem corollary proven prove actor rl converge policy eq follow procedure policy theorem policy rl control note loop stable loop system loop develop design linear linear result algebraic control respectively q similar rl rewrite q learn kronecker term form stack column equation residual cost nn policy f apply benchmark q attack angle wind angle attack q iteration system solution equation activation vector l generate set interval weight converge verify addition weight develop rl insensitive benchmark widely control nonlinear pose translation couple develop rl nn function nn stop integral vector weight te close conduct figure give trajectory control curve converge rl develop time unknown internal system policy rl derive approximate rl lemma control nonlinear control transform generally solve approach approximately equation accurate costly obtain overcome difficulty reinforcement rl evaluate extremely promising purpose nn actor nn residual develop rl tested apply policy equation rl machine widely scope intelligence rl refer actor environment policy rl method rl rl obviously control rl unknown promise design past rl problem especially important report rl optimal suggest programming novel pi method discrete continuous present necessity know internal pi framework integral experience input neural nn decentralize worth think rl solve system existence reduce controller require rejection effective achieve gain controller past control solve equation pde impossible solve solution work policy iteration successively approximate bellman successively approximate linear solve develop constrain purpose point saddle consider extension wu computationally solution approximate taylor coefficient system model usually costly rl control find problem control motivation nonlinear develop respectively study conduct brief conclusion transpose x denote operator positive definite banach dt x w w p consist affine dynamical nu law loop stable gain prescribe call observable feedback less equal close stable equation iterative successively approximate linear control v indicate linear pde approach constrain system discrete obviously iteration loop policy inner loop update index outer iterative index loop activate convergent wu simultaneous control iterative loop word former latter instant policy eq solve worth note iterative theoretical converge go infinity establish obtain converge policy control view sum game problem control act player maximize game saddle equation internal system unknown solve online policy evaluate policy evaluate learn problem drawback real evaluate policy inaccurate learn learn policy error employ control generate impractical
identify vs adaptive sensing strategy non utilize accord measurement vector vector iid strategy obtain form correspond estimate base identify accord sub overlap enforce structure form support accord glasso glasso evaluate sense compressive sensing measurement adaptive overall different scenario evaluate amplitude nonzero facilitate comparison analyze apply variance assess correctly identify final empirical support completeness regard implementation measurement trial relie specification tuning regularization parameter evaluate range obtain identify due issue estimation lasso estimator reconstruct software fr sense procedure instance procedure fit unit measurement impose interpretation per se budget prescribe adjust effective per along one additional note may leave may rescale sense satisfied marker marker employ sense unchanged experimental trial logarithm amplitude sense curve cs marker note first expect sense four approach sense group structure finally exploit suggest utilize sense technique traditional improvement claim dimension increase corollary sufficient ensure recovery curse technique magnitude significant problem size utilize setup support occur tree condition state discussion section little constant instead result bound behavior evaluation scaling behavior namely achievable provide plot approach depict sense signal amplitude parameter choose implement generate sensing strategy corollary threshold record choice amplitude trial result successful support recovery sparsity amplitude measurement dark average trial probability result appear text fraction trial region word region regime trial fail white support grey trial support accurately give support comparison critical satisfied imply particular dashed line depict point result discussion experimental sufficient conservative additional behavior identify sufficient condition amplitude proportion result figure signal amplitude successful proportional look transition black white region comment tree implication date strategy idea behind effort compressive effort strategy sense sparse task compressive variant essential amount initially location focus set decrease tree fundamentally behind approximate signal subtle extremely constructive locally onto signal exist essentially start contrast binary strategy necessarily gradually onto become unlikely fundamental implication signal especially signal identify exceed notably verify sense compressive idea cluster structure require procedure guarantee small sufficient sparsity ambient imply recovery inherently signal noting block analyze rise distinction benefit localization information root regularity accurately component strength dimension equip dimension localization tree overall comprise structure necessary recovery inherent curse characterization structure exhibit favorable characteristic path future inference beneficial achievable probability accurate recovery effort quantify adaptive accord estimate signal measurement constraint frobenius establish noisy selector exist measurement ensemble sparse signal selector ds adaptive sense satisfy analogous context show mse logarithmic structured sparse signal mse accurate compressive estimation nonzero second collecting note apply describe establish follow sense signal nonzero exceed amplitude omit component equal estimate would constant produce recover class signal e small component strategy capable produce incur strategy thorough investigation signal effort grateful detailed thorough pointing initial potential achievable mse intermediate main tree root subtree complete define add yield tree root connect subtree proceed trivial contain underlie binary aim end child intend worth classical essentially imply child special result opt completeness highlight difference set intermediate result identify setting hold equality nearly signal follow directly kk kt full exception level node last constructive select manner integer subtree contain tree contradict thus indice correspond complete subtree nearly subtree subtree layer subtree thus level index immediately partially subtree describe least index contradict signal sensing terminate event occur event measurement hypothesis test word establish equal turn simple line symmetry disjoint utilize standard event place sign nonzero ultimately nonzero imply w employ straightforward computation fact lead step proof amount easy verify straightforward omit receive institute communication technology science electrical engineering usa towards ph degree department electrical engineering university research interest compressive mr award work research usa ph electrical engineering research associate department electrical engineering department engineering research interest generally include inference adaptive communication dr include company paper frank mathematics distinguish fellowship award complete technical communication section corollary lemma portion appear conference system computer shorter appear global signal material purpose request sense relatively small possess representation effort exploit location utilize measurement sense establish notion adaptive sensing examine establish tailor signal agnostic establish support tree setting adaptively strategy sense analyze fundamentally sparse signal sense compressive lower structure receive area share mean inherently simple structure infer perhaps compressive sensing collect project onto cs q describe error initial noise free reliably setting ensemble generate entry iid cs efforts cs effort cs measurement design original cs paradigm literature extension deterministic randomized measurement sensing strategy contrast independent past employ cs setting adaptive beneficial sparse enable improve non reference therein compressive free powerful canonical cs correspond exploitation additional location formalize notion sparse dimensional vector correspond subset cardinality describe signal support occur distinct generally speak incorporate either reduction article compressive sense first quantify strategy tailor work establish compressive vector identify weak signal sense agnostic effort benefit sense tailor task primary aim fundamental adaptively broad performance associate non adaptively ensemble notion structure sparsity phenomenon exhibit investigation index put index set rooted subtree tree dimensional signal straightforward underlie tree illustration sparse node root subtree motivate cs exploit align effort specialize exploit inherent representation various domain examine application employ dimension coefficient object coarse fine top work compare coarse fine coefficient sense bayesian design context image application top wavelet strategy compressive strategy free investigate scenario motivate assess performance noisy completeness acquire identity though extension tree orthonormal basis adaptively design unit different indexing observation instead location end measurement project nonzero location stack queue root nonempty remove project perform hypothesis child structure support hand unchanged fashion obtain hypothesis amplitude essentially location amplitude contain main result quantify performance signal setting corrupt noise provide scale behavior completeness procedure implicitly obtain regardless particular structure acquire acquire nonzero terminate measurement satisfie result ensure sufficiently identify probability support tree budget total measurement average prior formalize adaptive tree step sparsity parameter satisfy terminate collect produce provide follow condition repeat sense nonzero tree fundamentally weak state essential sense previous fundamental limit recovery observation design adaptively measurement iid traditional previous formalize let unique dimensional node technical assume underlie meaning level exception last partially focus tree great quantity formally tree define sequel simplify exposition shorthand leave tree implicit recovery directly recovery sense strategy motivated effort ensemble measurement limit iid expectation investigation allow explicitly note recovery fail probability least support comprise vector amplitude concern performance summarize employ outperform weak potential imply either improvement recover whose good hand analyze nonzero weak recover depict order time much dimensional setting along recent effort propose estimate tree compressive exploit fundamental among image compressive sense motivation effort strategy structure examine activation matrix measurement fundamental limit proof one examine recovery signal support comprise nearly level quite fact nearly scenario tree comprise one problematic distinguish cluster different structure rise threshold localization compressive weak imply notation localization impossible constant signal particular inherent analysis difficult sparse another contain rise threshold localization measurement tree examine examine sense tree support correspond demonstrate analyze support correspond far state sufficient sufficiently specifically recovery identify presence activation essentially weak reliably fundamental examine view support subset element measurement tree branching factor slight tree model contain support detection characterization type tree signal setting measurement yet open specifically sensing limit previous condition estimation capable whose specify quantity see identification condition recovery sparse signal adaptive strategy noisy recovery theorem support procedure inference non compressive sensing exploit cs structure predict fixed measurement budget discuss conclude section appendix concern scenario randomize compressive strategy effort concern vector sense strategy employ base support leverage adaptive proceed introduce proof root subtree augment rooted subtree tree formally define proceed theorem quantify limit randomize reduction limit matrix problem introduce describe signal valid bound instead minimax class separately nonzero close sense support pair cardinality
filter read importantly grow increasingly genome sequencing variation newly efficient hundred currently infeasible issue use cope yet room improvement increase design handle database light region cover rare read species completeness adopt read distribute generalize denote order adopt read symbol tail sequencing significantly choose mathematical x x x na particular identifiable ex measure fraction size p x max j r jj means index identifiable identifiable p lp x height ex depth la j x n height ex ex check ability identify database cluster gene refer assume available sequencing practice choice sequence consideration aspect identifiability distinct entire guarantee read gene plot figure uniquely specie short entire identifiable vast majority partially identifiable specie read length specie identify remain specie group close distinguished read far imply z reduce leave mahalanobis ie eqs need convert eigen orthogonal matrix diagonal matrix divide side square immediately depth box divide reconstruction sequence read species frequency block threshold partition th allow binary partitioning iteration specie randomly overlap restriction exactly collect linearly dependent identifiable block specie collect result block specie frequency keep solve minimization eq vector read simulation frequency read read varied perform frequency achieve reconstruction number practice indicate tight bound might reason bound particular frequency choose importantly small specie simulate may small proving solution challenge compressed sensing bound since incoherence poisson read fundamental analysis goal reconstruct comprise sequence parallel sequencing genomic formulate mathematically reconstruct identity datum read infinity metric assess quality aware divide enable specie numerical realistic term obtain accurate term specie micro community major biological clinical micro specie base dna either genome sequencing rna gene sequence highly specie database million may enable identification community possible identify specie clear analog sequence sequence throughput digital data picture read reconstruct identity quantity specie many short read ability identify mixture reliable recognition typically achieve coarse main read length current pose specie read specie align reference database sophisticated quantify specie develop shot sequence read ambiguity enable systematically mathematically community characterize reads sample know specie sequence string accord frequency probability sequence provide probabilistic read condition identifiability specie mixture read read divide handling scale hundred thousand specie community scenario hundred thousand specie million specie study realistic simulating mathematically consideration pair read publication spirit convex read describe informally goal identify present extract universal dna assume specie reconstruct mixture mark sequence specie length roughly nucleotide contain specie frequency specie sequence th define give dna produce million short sequencing read together provide specie database goal reconstruct specie frequency unique formulate capture sequencing read identically independently I specie specie dna x j length ease convert relation e I ir p simple construct sampling specie read appear sample assume read bias realistic sequencing bias error etc non still keep evaluate need compare reconstruct metric frequency may satisfied reconstruction group metric criterion metric account identity norm precision norm representative group metric deviation reconstruct criterion account specie reconstruct specie propose mahalanobis I j pair specie represent specie result mahalanobis true identity similarity specie correctly say l specie identifiability limit reconstruct frequency read problem principle species vector since vector identifiable recover frequency regardless resource available rise observe read question reconstruct long read identification rna seq different yet precise sequence specie read length diverse dna distinguish underlying specie informative enough region may specie read short formalize mathematically determine read see read increasingly easy specie assume compose distinct sampling read identifiable database sequence obtain read result obtain read specie identify correctly weaker successful identify correctly characterize ability correctly frequencie species specie may identifiable l proposition partially identifiable specie identifiability property real identifiability ensure specie read power finite read
associate particularly bound maximum transaction length bind scan bind dataset imply tight present frequency use fall compute true critical method behind let set proper must frequency resp thm thm thm I negative f da approximation maximal thesis order bound still modify bound least contain positive hold bound solve bound da da definition hence fraction chernoff bounds method present introduction another additional transaction transaction original fraction contain amount explain computed sect take dataset mining dataset frequency become stress realistic regard transaction additional information flexibility usefulness often spurious false tool learn develop experimental evaluation show positive extract huge direction find interesting definition significance pattern low vc collection mining believe generalize controlling probability corollary contact author frequent primitive fraction analysis underlie distribution transaction extract attempt call frequent inherently rough spurious design frequency empirical dimension identify almost experimentally mine standard chernoff binomial well guarantee keyword frequent vc dimension positive identification mining database reduce item appear transaction dataset market useful indexing instead infer scenario frequent facebook online survey facebook user take want association facebook population whole facebook online underlie answer question identify former natural dataset customer follow future general concept assume transaction sample define transaction build appear transaction sample true fraction transaction real mining frequency market observe customer customer want customer frequent contain appear among frequent whose frequency aim identify even view disjoint item contain pair frequency frequency least false negative include huge true false positive somewhat goal care achieve balance na I avoid involve binomial possible chernoff union frequency dataset tool frequent serious achieve transaction item take potential avoid consist portion sect clearly show refined achieve balance goal find contribution minimum develop analyze exist method assess frequent pattern specify limited characterize pattern incorporate analyse associate vc showing field application base assess simulated frequency positive report experiment perform also compute union outline sect contribution formally sect sect space proof lemma theorem report value frequency extreme capture spurious item discard spurious propose spurious frequent procedure significance transaction observe transaction infer partial include dataset frequent frequent false filter statistical since frequent represent frequent well co occurrence due chance frequency discovery assessment play threshold reflect significance statistical pattern rigorously high rigorous generative transaction completely wise false discovery rate fdr false among fdr however mining number preferable statistical difference kind g comment model clearly real give sufficiently minimum traditional return collection uninformative suggest non mutually exclusive information contain compressed concept work intuition traditional mining actually generate understanding lead interested statistical property reader survey identify filter actually survey remark minimum complementary measure significance rule interesting accord focus focus apply notion generate update mining process surprising independently work impose restriction testing procedure support dataset item threshold input user contain discovery suggest extraction setting false discovery rate statistical test involve noticed test hypothesis technique rate discovery association rule dataset resample find act swap keep transaction derive procedure generate dataset assume present adjustment actual test establish correction significance decrease available verify significance critical instead threshold threshold split experimental power try consideration inefficient platform adjustment test adjust value model consideration assume transaction conduct correction datum permutation problem employ direct correction depend traditional multiple entire accurate dataset computationally bound desire limit analysis single item one order I union end I guess rule extract range boolean express role level significance arise able rigorously vc something definition lemma tool use throughout work need later distribution transaction item bag transaction I analogously observe dataset fraction transaction traditionally extract respect set well true reflect find exact inclusion may vice versa try interested specified aim provide sense high success dimension subset outline basic refer work introduction vc survey let call bp b give approximate formally independent element sample belong sd construct point upper vc vc distribution collection accord use evaluate rejection identification phenomenon associate predefine accept otherwise reject priori hypothesis correspond nan type defining implicitly one evaluate extreme conditioning reject correctly nan define type I statistic particular transaction event transaction whose size frequency number control report hypothesis acceptance test statistic employ
individual equilibrium type instrumental establishing precision compute use case regard use appropriate nash minimizer potential kkt exist lagrange multiplier observe large optimal satisfie kkt condition feasible moreover n I ta n k statement nash kkt condition kkt condition I none negativity tight coordinate nash equilibrium derivative th mapping decrease two ti I statement monotonicity composition differentiable c monotonicity conclude na arbitrary linear unbiased scale unbiased estimator prove hereafter prove desire establish q first q pp q trace derivative multiplying sum conclude expression follow player equilibrium similarly ia n satisfie follow follow q distinguish subset ia ia ac ia ia ia ia ac ia ia q prove assumption france amount estimate gender data answer survey medical tool science g lead discovery disease individual may concern express trade comprise privacy incur release equilibria establish existence unique trivial determine concept stability extend markov conclusion presence statistical several science study area rely drug survey involve become aspect internet google amazon netflix database behavioral search query service turn privacy general public lie extreme individual collecting may wish movie political hand successful may individual collect evident medical study lead disease experiment service benefit consideration collection datum clinical complete service game focus formal analyst private medical test feature public gender etc q individual reveal private analyst company political movie add privacy attain accuracy linear model aggregate multiple individual balance utility private comprise analyst nash equilibria show privacy unique nash armed game price privacy class estimator square equilibrium extend statistic square minimal among optimality remainder organize present characterize equilibria conclusion technical mining history preserve datum early public release perturbation tailor mining task reconstruct association aware perturbation technique individual add framework differential study computation publicly privacy offer change perturb analyst perform individual contrast classic mining motivate perturbation analyst observe public subject determine focus meaningful notion determine price stability perspective study version subject close broad participant determine albeit study nash issue problem reference therein variance contribute individual benefit use public game involve discuss technical review key relate classic vector column vector capital letter usual semidefinite psd positive matrix write recall define order say f na da sum element denote vector gender express likelihood survey inherent noise mean random variance assume analyst infer sciences magnitude coordinate capture feature age disease capture aid feature scalar analyst estimation domain throughout two privacy twice convex twice decrease positive semidefinite convex monotonicity convexity standard increase I decrease high privacy decrease psd decrease relax technical simplicity fact composition decrease twice continuously particular context design eq satisfy r player characterize nash equilibria every potential equilibrium see set minima equilibria nash equilibrium invertible precision invertible constitute equilibrium nash equilibria cost avoid slight finite individual bind span enforce equilibria potential equilibrium game nash equilibria coincide minima proof continuous therefore privacy privacy derivative estimation write constant unbounded k assumption deduce conclude potential implication start equilibrium trivial equilibrium equilibrium uniqueness equilibrium attention cost strategy social I cost ratio bad nash equilibrium set nash equilibria equilibria determine price stability price coincide discuss equilibria game admit immediate consequence unique trivial minimize minimizer positivity improve obtain follow two proof technical begin stability technical report characterize estimation cost extend extension rely characterize showing equal nash privacy cost attain bad class extend privacy convexity roughly speak function grow fourth case characterize social optimum relate trivial nash equilibrium linear family game point analyst gauss review commonly blue give case reach analyst section e expectation take unbiased covariance l x generalize semidefinite variance identical strong argument square presence suppose depend ask analyst inferior analyst
loss game play suffer inequality strict easy check conjecture mathematical sciences school engineering sciences berkeley berkeley usa department college uk department berkeley berkeley ca usa department electrical engineering prediction expert analogy ask every expert round expert round game expert set large expert expensive stock get expensive small good expert analogy prediction expert expert space expert expert index expert produce player fix suffer expert expert play uniformly replacement get regret q price pay constant exponentially constant technique lemma follow exp index j simultaneously
equation te second project onto span prove p formula derive formula f b onto span b tb tf e tp tp tp tp frobenius trace simplify tf tf tr f theorem select simplify greedy generalize greedy numerator denominator denominator criterion I tb rt hadamard formula different make computational complexity calculate formulate greedy literature identify connection insight subspace datum atom sparse basic subset clearly instance generalize goal previous successfully greedy selection greedy generalize original select random performance feature selection use distribute basic encode span representation method work distribute greedy value approximate lead matrix formulation generalize calculate lead greedy represent singular call represent atom instance literature variable selection discrepancy projection atom sparse instance generalize orthogonal matching orthogonal greedy least define iteration column error square ta selection atom different sparse simultaneous sparse atom signal selection solve propose effectively use greedy subset select subset b ta ta p ta proof theorem matrix f bp bb bf tf ff pe g te te te ph similarly calculate represent hadamard formula express corollary subset good span fast greedy draw connection solve column column span formally column p project onto span
function dual multiplication objective convex hull widely optimality cutting concave cut plane include violate problem violate cutting training violate concave qp violate constraint violate constraint still relax nonlinear choose variance approximation bag solve former example take dimension output overall conv mkl need achieve ambiguity validation compute directly dual svm guarantee terminate threshold iteration svm terminate high recover computing take alternative sort visually toy bag bag c separate hyperplane specific dataset negative consequently dataset uci repository table dna treat amount class heart breast dna rna l heart svm conv conv conv conv dna svm conv svm bag fix conduct individual selecting splitting available bag predict truth proportion th bag tune conv kind kernel rbf tune small objective minimal c svm conv svm conv conv conv svm dna conv dna conv conv material bag size proportion challenge amount supervision hard case dna dataset rbf bag work rna compare experimental consistently bag svm outperform improvement supervision generate proportion bag supervise hand bag least fact reach stable solution equation pose super assumed regression result guess challenge vote bag conv bag table run long conv repeat svm pick solution objective machine core ghz vote kernel fold svm repeat anneal conv second show experimental many dataset conv svm marginally worse explain relaxation use conv svm initializations heuristic conv initialize preferred complexity conv svm improve solve loop warm start complexity propose introduce efficiently approach flexible framework due usage svm error handle overlap plan investigate preserve thank yu li wang anonymous suggestion group call svm explicitly latent proportion avoid lead integer efficiently one simple alternate relaxation size label proportion attention group bag bag individual proportion raise issue hand aggregated proportion across region feasibility learning raise proportion address instance make restrictive either parametric introduce optimize unknown label label efficiently relaxation method gain propose theoretically sound bag label exponential maximize log bag unfortunately hold behavior bag region datum highly dependent bag propose bag super assume label super poor represent property bag utilize margin framework figure highlight semi encourage prediction unlabele train hierarchical consistent inferior idea heuristic cluster proportion svm optimize bag bag disjoint th bag formulation modeling instance illustrate toy experiment detail note individual svm intuitive convex therefore find method method svm label classic fix become bag independent bag separately yield problem q step cm align reduction flip bag take pick small supplementary alternate solve guarantee due objective increase terminate
aspect svms relate characterize dimensional problem explanatory output pair minimize process determine expect unknown distribution simple yield neither minimize machine wide variety risk hilbert already kernel hilbert arbitrary avoid overfitte support vector base loss loss binary purpose purpose loss function huber smoothed analyse concern quantify incorporate uncertainty report want include individual recently mild interval interval asymmetric fix idea include include mean functional borel operator vector empirical distribution dirac distribution draw random evaluate replace estimate interest symbol propose use use monte carlo original bootstrap carry differentiable conditional law law bootstrapping consider dm bl approximate loss arbitrarily convex integrable converge probability tight borel measurable eq invertible two bootstrap measure stochastically expectation independence understand product projection coordinate product b factor empirical bootstrap symbol denote weak need sake completeness envelope n statement em n converge almost n outer almost jointly sequence nz nz precise outer range carry hadamard differentiable functional sense law law see delta hold outer probability list essential parameter tight measurable borel measurable invertible ff sf b empirical ng gx ig measurable step purpose theorem satisfied equivalence conclude use fact put part integrable loss q indicator bound eq span subset whose hence cover kind b notation svms parameter guarantee tight borel measurable theorem map necessary hadamard immediate converge measurable prove somewhat theorem term remain show obtain equal loss finite use notation x sum right finite g converge almost jointly e outer conclude almost sure outer probability know denote hence consider measurable variable stochastically independence
rx mention game concave concave converse concavity state matter nature payoff small consequence implicitly consequence corner optimal player reveal therein corner want mind note shot contain half extend corner general payoff intuitively average compatible payoff upper corner corner even bad payoff upper corner necessary force consideration corner comment payoff section characterization surrogate payoff payoff literature reference monitor eq characterization sufficient ar inclusion necessary hx reformulate context equivalent primal conclude game property wise continuous convex argument concavity concavity inclusion linearity corner partial order compact ball radius hausdorff mb rx lipschitz continuous corner hausdorff composition lipschitz game construction corner study therein implication contain contain sequence tend concavity entail function hx fan lemma denote maxima hx put thing prove h hx hx disjoint banach hyperplane space define shot monitoring suggest surrogate payoff gain mapping approach like one statement adaptation latter strategy payoff payoff know mixed payoff rx aim payoff shoot keep corner property rx hx hx trick play block convergence payoff lead strategy lemma careful take prove constructive conversely theorem satisfied set hx separate put thing prove yet sake strategy reduce form program however start computational size polytope intersection polytope transform polytope negative appendix distance rewrite lemma provide characterization together result view demonstrate counter polytope tt game pure denote dual thus mixed corollary set payoff take play dark shoot precisely parameterize correspond point p half space separate transformation subset precisely half shoot action negative parameterized contain hope game payoff sufficient characterization check corner range outside latter half characterization condition general many one direction unit shoot contain half space equivalent lebesgue lebesgue integrable state section payoff follow boundedness stem boundedness ready partial monitoring set compact intersection line exploit indicate appendix state state start direct implication dual primal concave lemma entail also volume induce lebesgue hausdorff translate euclidean hausdorff exist contain contradiction action conversely dual e convex theorem closure supremum former compact convex banach entail separate hyperplane half result generalization finitely many direction finitely hyperplane therein obtain dirac hyperplane partial set work every convex intersection space mapping lebesgue direction equally way generalizing rely play contain lead characterization depend acknowledgement science la grant appendix know property self completeness supremum induce lebesgue function constant cauchy schwarz inequality norm integration lipschitz cauchy schwarz supremum lipschitz lipschitz supremum converse implication two banach separate hyperplane form use last equality cm dual monitoring conference develop paris pass away theory paris sup paris monitoring type convex set space shoot dual payoff monitor characterization convex case polytope payoff function aim receive also arbitrary set theory seminal present player regardless opponent action monitor equivalent turn determine sign characterization state mixed opponent player shoot value game relate therein condition hold concrete strategy derive solve shoot repeat game incomplete partial monitoring use derive strategy incomplete value monitoring game partial monitoring case polytope primal light primal game monitor requirement every half shot show section monitoring recall monitoring outline objective provide space convex focus section primal hold upper technical paper favorable primal condition conceptual link analyze monitoring case strategy primal finally polytope generalization polytope inequalities polytope convex use support appendix recall basic full model notation value game maker player nature refer action denote round even obtain get possible player accord player say monitor I action dark refer denote major I refer notion mixed action nature intuitive end payoff compatible statistically put full monitoring reduce finally element denote player mapping short strategy refer ensure value payoff converge nature analogy conversely monitor need strong notion shoot shoot shot complement shot way convex set space modification statement round player stage consider define hyperplane mix shot illustrate h strategy expect suitable convergence payoff martingale convergence case von state formulate check strategy exist partial monitoring technical objective monitor property sequel call corner hold payoff characterization still characterization closed monitoring indeed monitor strategy calibrate primal section payoff payoff eq dual characterization dark corner r wise corner say corner component entail control whole bad payoff associate feasible payoff vector rx interested corner payoff corner monitoring course game monitoring identify identify set singleton corner norm proposition partial monitoring corner differently upper corner property monitoring direct implication apply ar interesting implication thus converse half ar original monitoring player entire probability distribution indicate restrictive payoff player already belong thus exist corner property r n r rx nr trivially satisfied already belong condition rx tr rx n martingale strategy nature illustrate compatible trick theorem corner divide increase length another converge play stage player constraint weight put positive mass least play informally payoff measure action measure payoff action matter indicate technical fact appear rate affect trick property behavior upper corner corner fail example corner strategy thus main play formal
admm ascent redundant ascent optimality least principle primal appropriate indeed confirm inexact variant inexact alternate method objective merely aforementioned example namely regression fuse involve proximal gradient function solution paper organize alternate problem use conclusion map apply subproblem update step lagrangian function indicate take projection add proximal pre specify semidefinite extra result direction method solve eq subproblem lagrangian proximal rank quadratic eigenvalue easy basically arise subproblem write form equivalently augment take fact originally saddle prove therein analyze iteration result hybrid proximal term within constrain optimal throughout paper solution solution ready equivalently follow generate subproblem let sum two result get eq completes prove let hold due equality know lipschitz add get z last give equivalently optimal solution algorithm ax hold definition equivalently subgradient point eq note fx k convexity imply inequality schwarz b obtain verify convexity hold therefore n fx define q combine dual smoothing lagrangian nesterov technique method smoothed accelerate solution technique use require smoothing require differentiable lagrangian barrier apply gradient smoothed feasible complexity show regression fuse regression interpretable solve interpretability ensures fuse et impose order fuse transform equivalently et programming significantly solve medium alternate direction one edu choose plot model solution none natural order show capability logistic scale create fuse plot mention mention report cpu sparsity fuse see solve fuse regression cpu understand compare applicable subsection unconstraine lasso splitting solve admm effort iteration two multiplication shrinkage operation matrix multiplication shrinkage operation lasso suitable subproblem among subproblem inexact admm still subproblem solve implement performance admm simplicity instance randomly implement create nonzero position run record admm inexact respectively run gradient get subproblem multiplication admm moreover solve subproblem admm admm cpu cpu cpu cpu c admm costly small well admm perform iteration two sense result first subproblem easily solve sensitive depend need properly subproblem gradient several subproblem however crucial emphasize admm subproblem easily provide something linear subproblem lasso coordinate subsection admm randomize among stochastic learning apply suitable fuse logistic problem proximal relatively table size may alternate apply mapping smooth find solution exist solver new namely logistic numerical preferable order fuse logistic consider block easy proximal multi block admm augment block currently future simplification step version whose find prove complexity acknowledgement fuse grateful anonymous constructive theorem lemma example remark zhang subject relatively mapping processing field structure proximal mapping smoothly direction direction return iteration method fuse test method statistic encourage indeed keyword alternate consider optimization arise later recent multiplier admm augment lagrange constraint admm splitting operator splitting splitting particular split split extensively variant pca semidefinite recently obtain survey et admm whether subproblem space identity mapping admm require proximal easy
result remove communication associate long positive semi mild technical monotonically individual global however elaborate make impact bring optimal add assumption need optimization eigenvector approximation ki derive minimize equivalent semi statement substitute fact pi ii assumption step simplify q complete innovation unity must note straight respectively intersection intersection observe innovation let steady lyapunov let lyapunov equation ap n step fact fashion error q definition q appeal plug complete definition pi always axiom conjecture exercise theorem proposition summary theorem university pa usa addresses online setting observe private underlie world dynamic evolve aim true small function update mechanism estimate tight individual bound characterize square function decomposition measure error learn attract wide variety economic represent product opinion vote sensor network observe signal period underlie stock parameter prediction learn relaxed stochastic social world vary motivation random associated social unity aim suffer small distribute converge regularize proximal fix decomposition give consensus update mechanism incorporate private neighborhood estimate eventually unbiased interestingly whole role outperform provide previous centralized circumstance dependence optimality highlight network ratio less unity constraint run loss alone concentration inequality asymptotic trade level mild learn hand communication optimal sense prove occur individual underlie world evolve innovation variance could potentially great unity assume independently period describe agent innovation update mechanism discuss stem hardness virtue reduce effective belief state world agent undirecte link assign let self symmetric doubly satisfie satisfy goal collaborative cast online global period tackle proximal end function update innovation refer private availability choice size persistent innovation perform scope present paper simplification study define stack one show aforementioned collective dynamical collective vector throughout denote large singular behavior estimator square establish unbiased matrix social agent change always steady network steady rate steady mean truth sense state signal weight govern incur due innovation due rich importance conjecture steady intuitive discuss corollary complete star cycle vertex respectively denote corresponding preserve note n substitute immediately communication ratio steady eq close highlight communication quality centralize steady kf equation simplify positive tight choosing preserve evaluate cycle agent predict
condition constant assume condition f c lemma list theorem special proposition prove proof acknowledgment associate thorough useful comment presentation remark section section support grant dms dms matrix significant application stable recovery low frobenius implement technique main projection consider possible high projection recovery rank application include face recommender identification reconstruct quantum low include ray reformulate plan discussion motivate several electrical engineering mathematic science low subset investigate et plan zhang study propose derive sharp inequality restrict isometry write q goal measurement dimension also approach recovery nuclear minimization bound noiseless feasible nuclear ensemble exploit entry ensure stable provide disadvantage design require storage ensure rank section another popular matrix completion position ii nj replacement respectively structural difficult matrix ensure completion easily completion unless nonzero row paper nuclear p measurement rank call easy storage ensemble identifiability condition noiseless high projection ensure require rate particular property accuracy norm optimal projection show consistently approximately estimator robust perturbation far include euclidean li covariance simplify symmetric know symmetric setting recent present symmetric rank discussion main particular covariance matrix covariance component I observe vector variation fan suppose observable projection surprising matrix aim recover rectangular section implemented study nuclear numerical rank confirm alternative procedure illustrate compression basic exact noiseless establish identifiability nuclear low obtain gaussian noise detail simulation brief proof lemma give supplementary recovery begin definition n ix x I iv na c b nb pp p important toward constrain low noiseless lead sufficient rip suited suboptimal discussion rip literature introduce boundedness exact recovery stable recovery condition satisfied boundedness boundedness constant observe constrain nuclear recover identifiability standard integer degree must lead failure nontrivial consequence rank matrix suppose constant estimator exactly least since degree freedom noiseless projection whenever early isometry rip linear rip perhaps p many rip al show ensemble rip probability oracle rip guarantee supplementary need rip since freedom rip framework bound ensure rank rip th much require rip rip nr guarantee matrix noiseless theorem depend ix nonzero noiseless ne aa rip meet early identifiability recovery rank rip ensure recovery al use property nan say nan z show nuclear noiseless case however hard projection gaussian minimization define p np ga follow theorem square frobenius z crp moreover cn crp expectation theorem imply recover matrix approximately low rank exist whenever continue perturbation small amplitude remain gaussian propose intersection nuclear norm minimization constraint norm minimization matrix selector various setting matrix selector selector combine retrieval chen gram matrix symmetric simplify symmetric eq wish recover noiseless standard constant whenever nuclear exactly p aa consider take standard normal nn crp satisfy low bound rank low z cn p pr f focus design distribution gaussian distribution sub lemma provide condition random variable define estimator symmetric I exist design probability constrain model extend sub gaussian c sub depend restrict rademacher exclude rademacher exception information contain section mention range application pca equivalently observable ie constraint norm rank px constant n focus moment covariance problem al fix setting projection vary technique solve projection oppose constrain efficiently implement carry numerical recovery begin bind nuclear recover minimum constant recovery specify interest purpose randomly generate compare ensemble range consider successful show successful recovery numerical distribution test analysis grow ensemble ensemble ensure storage far dimensional investigate perturbation end rv orthonormal column nuclear plot recover exceed b n range panel ratio frobenius gaussian noise base propose estimator constraint selector chen al except matrix low compare selector frobenius consistent remark recovery vary randomly one specify implement chen al see estimator choice knowledge variance may practical fold cross group size index group split apply group evaluate subsample group select parameter numerical fold cross validation figure tuning parameter approximation singular image pixel approximately rank consider mit associate mit rank projection reconstruct image recover constrained method rate projection apply possible accurately recover projection component projection work chen recovery paper finish noiseless rip condition rip symmetric chen give upper slight noise applicable noise typical constant right converge comparison
evenly image image image maintain aspect randomly split evenly remove word occur densely extract sift word dense sift extraction feature construct word divide grid position produce f annotation annotation work recall annotation ground annotation percentage correctly annotation annotation representation rbf learning emphasize use semantic datum observe extract annotation associate certain label topic unit average annotation average connection illustrate figure visualize word visual word extract association intuitive example annotation part paper supervise extension model bag extract meaningful model hide result interpretable advantage require approximate confirm annotation yu zhang department china mail topic modeling base allocation lda annotation demonstrate state model scene model extension increase hidden feature incorporate describe leverage annotation scene classification computer vision image tries globally label city annotation focus local whether car relate image car building water lot separately work problem annotation model allocation generative processing great success scene model lda multinomial multinomial distribution word meaningful computer vision extract visual word visual representation visual annotation retrieval lda supervise variant visual representation thus heart extract observation disadvantage become sophisticated trivial expensive topic sample actually assumption visual latent generative document autoregressive conditional neural expensive document representation feed generative document consider image visual annotation label successfully visual highlight confirm approach supervise variant lda modeling v share brevity decomposition leaf word annotation topic feature image extend topic tackle belong extend computer topic model increasingly use see review document propose softmax bag much lda scene classification lda neural aware consider classify hybrid neural model model belong predefined vocabulary image convert vocabulary sift descriptor densely train bag visual sift descriptor extract extract descriptor model conditional equation probability conditional specifically possibility wise number vocabulary respectively since address conditional logarithmic randomly assign leaf reach leaf transition probability use hide left right choice tree leaf internal node subtree otherwise contain bias inner model logistic output sigmoid balanced output assignment leave combine train latent representation fed classifier vision use class position dependent bag compute naive procedure layer exploit conditional computation hide computing sequentially regression thus efficiently inspire classify supervise incorporates learn hide describe exploit annotation feature unsupervise model lda perform bad visual appropriate pyramid kernel entire discriminative computer vision task issue literature supervise propose make image model neural network regular classification propose softmax connection layer differently visual word crucial neural namely hide used conditional average generative discriminative second encourage structure word practice solution generalize propose instead hyper average stochastic descent backpropagation derivative computation order order word bottom implication order word help overfitte well experiment need algorithm descent c unsupervise gradient play role understand car bottom successfully seminal extract distinct yield gain visual identity region word ik possible visual distinct implication word fortunately computation region annotation consist token describe content annotation people annotation
motion motivate draw rejection algorithm q clear impossible rejection integral accept reject without requirement use brief proceed make derivative respect give rewrite rather define correspond brownian motion motion specify integrable notice simulate rather adjust instead immediate well consequence path distribute remain occur suggest poisson occur thin poisson graph denote poisson find upper bind poisson follow exact simulate simulate simulate otherwise skeleton skeleton brownian skeleton find suitable way step require regularity successively complicated version restriction suffice omit simplify simulate brownian loss suppose former correspond brownian minimum distribution close simulate together attain algorithm intuitively relax like simulate reach specify member fall precisely represent b require simulation layer achieved see extremely discretization implement enable biased brownian reject target however disadvantage offer probability motion poorly target rejection finite value generator classified specify regular boundary process natural whose radial brownian nice property brownian include bridge construction density give I interest diffusion boundary generality boundary drift fix specify process replace exist u meet set choose select eliminate remark growth transition brownian brownian process simulate use simulate bridge time bridge paper enable exact biased follow poisson process rate lemma rejection practically occur check simulate I otherwise return skeleton skeleton far necessary apparent rejection drift rejection improve effect deal bridge later appropriate analogue assume choose skeleton law detail achieve radial brownian simulate transform brownian bridge fact simulate bridge coefficient sum easily invert cumulative ts apply repeatedly simulate I furthermore let law bridge ordering follow rich construct follow diffusion drift diffusion continuity remain calculation bound use condition brownian path condition simulate side minimum relate since fact dimension strategy expect generator brownian case performance successfully population diffusion satisfy case omit brevity conditioning yield routine diffusion drift diffusion boundary taylor expansion suitable diffusion assumption eq write inequality path diffusion comparison slight modification path order boundary condition skeleton two run generate total run skeleton coordinate require four suffer simulate require per accepted algorithm size rejection iii variate come distributional easy simulate average across accept path remain mark could complete r total cb various parameter accept path skeleton total r skeleton differ point candidate outside moderately attempt accept path unlikely sufficiently attempt accept run serious implication mb requirement quickly far newly e initially population cause drop vary less effect evolve boundary theory typically boundary example example diffusion currently want simulate candidate progress brownian motion assumption nonetheless still boundary occur partition converge boundary consequence bind actually recall must simulate layer read boundary boundary choose define generalize simulate law correspond bridge event event mixture intuition behind first equation simulate select bernoulli simulate brownian bridge within detail simulate brownian simulate bridge give close matter distribute accord absolutely bridge find simplify event compute exactly remain simulate indicator proceed simulate applicable boundary match cover application process motion path efficiency boundary I develop brownian perhaps greatest develop upper relax analogue restriction drift analogous simulate brownian bridge simulate remarkably attain rather exact work ensure algorithm meet beyond contribution brownian raise brownian motion exact useful
weight poorly features build learn due iterative example store straight neighbor set suitably summary find experiment nearest label guarantee far conduct music imagenet annotation image apply task nearest affinity embed neighbor affinity train similar embed dimension marginally small dimension unique believe useful large task imagenet fall neighbor embed give competitive learn supervise increase explore regard google york ny usa google usa linear embed annotation nature exist iteratively linear variant family give standard feature annotation document recommendation supervise supervise return scalar lead pair return reduce scalar g weight retrieval recommendation far
motivate universal scheme implement adapt sequential bring novel big grow add problem whole anneal attractive feature remove parameter drop worth ad hoc keep guarantee optimality might greedy inaccurate total need access training time training order magnitude boost algorithm include provide evidence comparable date penalization much loss optimization relevant differentiable respect form tune intuitive easier specify experiment choice large course discrete idea plan reduce gradually remove irrelevant facilitate prototype summarize start parameter update loss descent remove magnitude gradually reach step nonlinear increase difficulty involve elimination rigorous htb classifier vector extremely drop removal anneal schedule apparent dimension eliminate early stage save give removal classification ht n k anneal schedule slow estimation decay inverse schedule eq six difference choice ht proportional curve computation time mn mn mn n mn k parameter rate small iteration algorithm advantageous tuning fit computed wise gpu base could computation differentiable example I prior ht left interval selection loss differentiable huber introduce loss everywhere behave svm loss logistic misclassifie work practice extension deal rank ranking good mean r agree agreement help generalization investigate iy mean simplicity log likelihood clarity application possibly anneal schedule satisfying regression stand algorithm sense large value monotonically limit ii cf limit optimal solution fisher matrix supplementary smoothly penalize overlap regardless may adopt universal iteration long properly view minimum attain accuracy come current use inverse attain balance schedule mm algorithm induce scad mcp differentiable objective loss constrain loss function sparsity intuitive cardinality control contrast penalty penalize share similarity feature elimination classifier feature however significant remove converge necessarily svm offer decrease boost weak feature select boost already structure general variable selection boost gradually remove variable elimination schedule boost greedy section variable although numerous way removal design unique theoretical consistency another class stochastic descent optimize lag behind computation regression application normal version datum label incorrect six ram anneal number experiment conduct separable show curve auc middle curves run htb l c c l mcp lb lb l mcp lb lb mcp lb lb value yield contrast sensitivity greatly parameter tuning reduce ad compare prediction algorithm logistic anneal schedule loss anneal schedule interior www stanford number call routine time algorithm epoch coefficient feature svm implementation epoch choosing give algorithm iteration logistic regression mcp minimax concave implementation r package lb regressor regressor boost lb lb boost one classifier percent dr percentage variable average roc curve unseen c c lb lb mcp lb lb l mcp lb lb lb obtain auc algorithm reduce magnitude penalize need ten mcp scad probably sometimes descent size svm job prediction job observe learn almost mcp scad mcp scad c l mcp scad simulation sample normal obtain relevant average evaluate anneal schedule build lasso elastic build matlab ordinary give quantile convergence mcp mcp coordinate descent could consistently quite size scale capture structural conjunction type nonlinearity compatible response characterize response univariate function linear pl depend number bin bin learner bin b b min jx return piecewise write work nonlinear depend cubic spline obtain lasso soft thresholding group loss work impose instead algorithm work computationally jx j tx prior response aside shrinkage second smooth response show differentiable regularization huber instance nonlinearity linear learner nonlinearity e learner obtain piecewise learner rank intercept observation motion sparse motion trajectory number common method motion segmentation trajectory affinity motion dimension several video hard dimension project space accord segmentation separability tune many motion segmentation publish segment automatic segmentation problem segmentation formalize candidate result velocity vc briefly self contain frame velocity vector different dimension truncate svd range obtain segmentation angular trajectory separability set vc please refer vc propose select segmentation motion segmentation describe camera dimensional affine segmentation label lie linear trajectory space affine plane let use distance thresholded take otherwise inspire vc obtain sort change dimension third partition near neighbor knn distance space angular distance change different knn connect label total misclassification comparison segmentation ranking construct belong contain misclassification error two ground truth ranking feature generate ranking intercept vector coefficient variable use notation htb learn select segmentation sequence whose number build affinity segmentation segmentation result detection segmentation feature intend vision present center side corner bottom training face train contain annotate face visual inspection face annotate histogram orient haar rgb channels window center interest gaussian pyramid e power train pyramid pyramid point pixel annotation negative negative mining hard negative negative mining classifier iteration hard negative classifier classifier train weak pl classifier feature without detection away pyramid detect pyramid face evaluate slide window image pyramid computationally context example part face detect heart pyramid grid equally spaced predict use regressor output detect point regressor illustrate mm regressor regressor learner variable j tx learn logistic piecewise linear loss piecewise verification lb univariate piecewise learner select boost add pl svm piecewise variable show nine see outperform logistic piecewise time lb train boost slide show cnn detection method outperform detector method top method rely detect cnn train face detector detector without base nine detector pose obtain top prune within inter distance predict pose cnn descent motion segmentation evaluate extensive benchmark set trajectory video type figure frame video mm c rv ssc likelihood prior train motion median median motion average median motion average median use method compare rank truth rank subset vice search formalize make try rank wrong sum iteratively q score pool add weak use range threshold number bin video subset contain video divide subset video separate video motion subset motion happen would train motion subset cross validation select pick calculate time set table set randomize vote rv sparse spectral ssc outperform training boost use misclassification motion comparison misclassification test misclassification rate prior vc use moreover rate half sc cumulative comparable good vc feature selection identify irrelevant proceed anneal efficiently grow boost usually bring big datum
unsupervise train conditional label refer consist probable class eq class feature second th class vocabulary average type briefly discuss difference representation capture fine lead slow crf another fact depend token previously weakly never compare comprise coarse tag crf word feature hmm article sentence journal label crf ten year york corpus unlabele million token first data article test token domain importance come people unlabele never sentence sentence use crf come domain adaptation domain viterbi decode outperform capture information representation finally unlabele label sentence domain compare crf use amount source twitter add sentence improve differently twitter investigate weighting differently come crf accuracy obtain train crf word sentence domain logarithmic conduct part demonstrate label quantity domain domain still language process syntactic journal web robust representation part speech study representation way represent hmm supervise unfortunately suffer domain example section journal web drop syntactic parse entity drop lexical lot test test example token comparison token unobserve make labeling language expert transfer learn precisely adaptation unlabeled domain order train view vocabulary first reduce sparsity name adaptation speech syntactic parse representation domain adaptation semi paper mostly hmm training mostly viterbi decode show class
I du le des par la une pour de par les est dans de par lin en est par une pour est des variance est les est e la I par pr adapt ce type I est I reweighte pour par de em les es pour les est newton un et la le le la form de kronecker si de jensen il et du experts la le la par une un em de de en observer limitation une du par opt pour une la pour pour la I par pr de l en de les pour des les des pour des pour et pour les le et des de en le bic le de la de e est de il est criterion adapt de mod dans le les mod g une les propose les par dans le lin la est la q fisher et dans est r de est en des pour ce la une par l des par une de une est le est une ne de les mod le par la des estimation par des cart de ik pz se une est un des une non lin lin et dans le polynomial par des situations la par et de es et le les du mod les du par le bic de les et es un h les de pour mod le une des des la est co en cause de la situation dans une lin le dans lin la latent les de est li en du par phase le du mod pour phase I adapt des trait b estimation par par cart le signal ce est par r de dans le en les en cart pour la les par et la par hmm inf la signal de la situation pr les phase cccc situation par par hmm les pr send les proportion du es des des du le pr send dans la pour la tr la pour en le les par bic les dans une les et c cm r lin dans une des pour lin mod r grant et r les mod le de latent la se le la gr la mod pour le dans exp les mod les un mod par un les des des es issue une de les les pour en les en lin g national les des noisy fr universit bp fr un di une es es et es la lin est I dans de la pr le de est de lin une dans une est lin lin par de es lin se du mod dans le pour ce de I les mod par base de le perceptron dans de ram lin dans ce alternative mod lin par un mod grant un pr de mod une une les ce une mod des lin variable ce un expert une pour du une des mod le est est le une de du des lin dans la comment le mod de latent dans le des via la section es es est es du et lin sa une de le mod le des par la du des par dans le la la du optimisation convergent pour un de lin pour des une des le un le w des de la ensemble des la ce mod le de mod le les de des le du latent des du gr de dans pour par les la ne du du il une du gauss newton newton converge pour estimation la de du mod une mod le un mod est par si et les du es g es la I par du mod le un mod le une variance la une le de un une
recent character principled modularity maximization popular generative node divide block specifie generalize connection also arbitrary bipartite core structure context task detect module convert inference generative framework advantage approach bring capacity separate spurious community resolution block refined limit detection modular one lack popular heuristic modularity base inference partially modularity heuristic special care restrict purely structure greedy modular expense divide optimize mcmc capable reach equilibrium configuration technique agglomerative heuristic avoid high quality compare mcmc network sec networks ensemble divide empirical additionally specify module infer membership count equally posterior e partition network identical rs eq correct total entropy network expression entropy paper direct good model always incorporate minimize trivial useful task identify principled fashion separate overfitting way perform ref resolution hierarchy describe procedure detection modularity ref identical constraint impose need subproblem detail state aware describe obtain minimize partition feasible instead partition modify fashion entropy partition preserve move reversible sufficiently occur proportional eventually long situation may long good desire implement chain simple approach balance inefficient large vanish ref fashion move block block label neighbor recover attempt membership node block currently likely node see impose attempt find infer move proposal metropolis fashion belong e inverse temperature minima neighborhood belong belong block block move movement probability give implement efficiently simply membership select choice adjacent block opposite require operation node requirement incur additional memory decide move value require compute change modify node mcmc operation independent examine plant pp block c bb r bc control example mixing mcmc discard parameter move difference right autocorrelation two move curve average network consecutive independent realization show time one mix two order choose time relative optimize range varied fully network realization correlation move provide considerable block namely heavily start discover state also alone take hundred drop occur average scenario easier agglomerative start approximate evolution pp fully representative snapshot drop previously likely since fluctuation energy configuration one cb modularity explicitly determine step discuss merge together fig construct block count node representation node move attempt select obtain block face control bad allow apply agglomerative movement amount merge step e nb nb ne despite capable avoid come close agglomerative heuristic red line membership typical start possible pp typical outcome greedy agglomerative describe appropriate trade speed large range see choice value interesting merge adjacency bipartite preserve merging fully reflect well block turn phase merge markov heuristic getting discuss result present slowly anneal plant infer pp bottom circular modular plant agglomerative heuristic legend realization grey vertical line pp line assess quality heuristic compare bound pp plant emphasize applicability analyze circular ec strength modular periodic boundary assume agglomerative start see optimal heuristic large actual greedy fall behave range precise region selection minimum correct case rely alone fulfil discard since compact fig lie agglomerative discard example situation close desire heuristic able description empirical run algorithm agglomerative heuristic different mutual one run mc agglomerative mutual collect agglomerative heuristic different analyze assess realistic large human undirected political email large strong component direct actor berkeley stanford web direct appropriate describe minimize describe
ground systematically monte perform see success follow bp trial specifically formulate solution visually truth ground truth image figure coefficient matrix fourier image represent stack domain sdp solution namely show visually performance properly two recover respect sparsity perfectly noisy clearly much approximation outperform visually though coherent collection conduct intensity pattern make piece decide possible intensity intensity give give measurement setup figure result ground actually estimate solution however know within truth mark position dot recover ground estimate right compressive use compressive accurately improve exist compressive sense suffice present quadratic relation nonlinearity relate relaxation basis basis classical compressive also implementation acknowledge discussion want acknowledge sharing partially support european grant contract grant foundation fellowship grant ii fa thm compressive nonlinear traditional treatment nonlinearity via un dynamic accurately characterize nonlinear improve compressive suffice classical compressive sense second nonlinearity use quadratic recover exactly numerical recover signal order nonlinear considerably counterpart optimization equation combinatorial propose relax refer bp recover solution cs dedicate solve regard powerful tool detail refer several recently cs deal interested therein specifically ny sense possible apply principle cs taylor cs f derivation hermitian motivate ray see ray ray physical limitation measure lead nonlinear structural contain complex transpose mathematical traditional appropriate imaging ray quantum mechanic relaxation readily type I greedy desirable traditional exist nonlinear therefore give contribution solve compare achieve develop main present validate image sparse system nonlinear limited paper discuss greedy propose iterative simplex pursuit nonconvex solution concern local author generalization rip solve recent work generalization compressive cs propose refer term solve exist underlie problem decision particular semidefinite program converge optimum pr extension previous note different solution inspire relax nonconvex sdp guarantee exact stable recovery noisy nevertheless retrieval similarity technique convert present previous solve compressive phase stability practice present facilitate imaging use vector scalar transpose transpose th th rip let trivially theorem rip point rip rip rip detail property bind rip rip difficult operator rip high realization gaussian rip condition difficult hand mutual mutual matrix satisfy b ready state coherence solution b critical practitioner solver moderate sized solver cs nonsmooth gradient projection augment alm moderate accelerate projection alternatively nonsmooth alm augment primal expensive solve also family refer successively linear refine iteration add however type summary nonsmooth exceed capability technique paper nonsmooth sdp motivation scale fast motivate choice denote dimension rewrite equivalent multiplier equality constraint rule lead consensus step tractable orthogonal hermitian constraint real denote onto eigenvalue decomposition domain real magnitude sign act wise l iteratively compute admm iteration stop eq also value term admm loop bound respect comprehensive validate efficacy solve representative primarily
performance small region conclusion simple free mcmc sampler family proposal history local covariance state nonlinear distribution entire return accurate parametric exploring mcmc foundation thank anonymous comment proof differentiable kernel kx kx fr differentiable readily show kx df chain fr derivative product kx dy hx integral x covariance identity proposal detail synthetic contour sample ax periodic perturbation amplitude band around circle contour deviation quantile strongly average chain sampler bar strongly main evolution whole principal standard metropolis extract j use take walk principal choose scaling addition scale eigenvalue rkh principal eigenvalue j eigenvector rkhs norm appropriate mcmc scaling eigenvector draw similarly proposal integrate move j hx integrate I identity h claim sum kernel hasting purpose target nonlinear support chain reproduce rkh feature implement move integrate analytically distribution original structure require attractive marginal hasting compete highly arise real adapt sampler method often learn target adapt accordingly sampler study along sample base proposal center chain scale adaptive scaling strategy beneficial high e ensure proposal use direction low acceptance depend sampler support present sample map feature unlike early locally adaptive orient nearby simply towards near simply input inform evaluate unnormalized gradient evaluation applicable hamiltonian monte hmc metropolis adjust langevin mala brief adaptive metropolis covariance operator strategy rkhs main term hasting comparison sampler section pseudo bayesian classification synthetic shape background let denote additionally target chain term measure algorithm optimality heuristic adapt acceptance theorem positive definite reproduce map embed single extend mean embed many include gaussian since characteristic n nk c bc b pg pf pg learn kernel pca rkhs nk z covariance operator behave expect analogy linear proposal pca proposal alternatively rkh determine operator extracting generalize space rkh chain history construct rkhs empirical operator descent cost measure mean covariance chain history lebesgue measure albeit abuse f k finite support span canonical covariance measure think rkhs trajectory gp respect measure lie see detail see rkhs gaussian rkhs ideally norm optimization rather new lead computational make single gradient point exploration two gx x nh plot case minima vary density distribution white target subsample probability subsample x thm center matrix accept reject metropolis hasting acceptance ax proposal intuitively good dominate symmetric adaptation compute metropolis acceptance gaussian current covariance subsample chain distribution never symmetric proposal depend metropolis acceptance reflect langevin mala current chain construct computable easily complexity shift add center current state belong density modify drift density available unclear additional require possible mala proposal term mala example covariance kernel proposal gain proposal use scale empirical isotropic exploration kx consider encode first dominate close determine mat ern family kernel kind x uci sampler isotropic proposal adaptive metropolis bring stop adapt proposal burn experiment kernel bandwidth median heuristic target gp hyperparameter uci shape shape periodic perturbation distance mean benchmark leave maximum chain bar burn scale interval scale quantile confidence experiment illustrate usefulness context classification hyperparameter latent hyperparameter observe problematic extremely drastically amount chain hyperparameter possible integrate enable inference importance choose propagation lead uci window heterogeneous window boundary posterior projection truth hyperparameter initially burn keep chain performance chain algorithm evaluate four sampler large benchmark benchmark sample sampler output benchmark comparison mixed figure indicate benchmark compete high indicating explore scheme
concave concave generalize concave know problem result proceed concave decomposition existence piecewise algorithm optimal concave function domain establish piecewise linear decomposition concave existence piecewise decomposition prove concave piecewise aforementione fact good concave density imply polynomial recall basic fact log concave density suppose q arbitrary log density mass decrease unimodal exist portion density far nothing calculation length irrelevant follow elementary calculus strictly decrease domain increase log concave strictly construct exist address proceed establish couple eq inequality put interval suffice case sequence claim combine yield want inequality use easy flat decomposition increase super jj fx k j complete description construction proceed super super maximum conclude inequality least super constant big desire follow claim desire inequality complete establish super interval claim argue piecewise linear describe identically fy dy fy dy fx fourth use increase use approximate jj I henceforth define length super monotonicity length obtain carefully inequality q manner super super complete claim completes necessarily say non increase problem extensively past year reference therein significance point aforementioned paper analyze rate estimator mle metric yield monotone mixture complexity provably chapter conjecture similarly learn theoretically thing consequence learn decomposition exist fundamental sample algorithmic density monotone run sample output approximation relevant terminology ft absolutely every subset function differentiable condition exist piecewise degree easy lemma theorem setting since nonnegative increase monotone agree method also dl book conjecture easily computationally essentially gaussians proof learn parametric univariate gaussian piecewise easily agnostic distribution agnostic learning mixture give theorem actually piecewise mixture gaussians agnostic mixture guess mixture exactly piecewise distribution true right guess near say obtain class modal gaussians actually learn something know think distribution use output theoretically gaussian complexity discussion algorithm guarantee complexity result gaussians parameter algorithm agnostic succeed even far gaussians generality take pdf ci absolute distribution taylor expansion clearly piecewise degree gaussian contribute suffice pdf gaussian equal polynomial give convenience subsection td flat piecewise together provably work discrete define follow distribute piecewise degree close distribution opposite draw round integer pd relationship learn flat piecewise flat close piecewise construct mixture discrete use sample essentially strong technical give theoretically learn arbitrary flat logarithmic factor would mention recent motivated database learn flat efficient immediately problem modal say monotone monotone increase partition interval conditional unimodal building place learn modal modal distribution output hypothesis learn modal compare result algorithm modal give specifically quite poor essentially optimal setting hazard pi follow run draw show must sample say every log let essentially efficient sample result give yield like thank probability interval pi proof denote draw multiple denote large simply interval st end result cover argument region interval denote straightforwardly I consequence ph internal randomness least td many element easily concatenation construction prove prove low corresponding ease later take sample statement slightly tailor take follow bx x b polynomial pair satisfy unknown hand algorithm hypothesis rest subsection distribution detail whether mixing become roughly actual motivate polynomial fix let coordinate dx square dx k construction quality way function indicator elsewhere indicator univariate construction existence degree polynomial value absolute sufficiently construction employ accuracy interval universal suitable polynomial integer polynomial desire indicator polynomial bx jx bx kp alternate view recall follow fix polynomial jx jx k jx dx ready agree coordinate claim bx dx value bx x dx k x dx claim dx jj value bound value sum ensures entirely integrate remain fix conclude p interval polynomial refinement piecewise pdf equal elsewhere likewise distance return satisfy multiplicative chernoff bind universal multiplicative chernoff since probability finish analysis multiplicative let get sum cx pt pt pt claim fact observation quasi berkeley edu university ed uk cs edu highly semi learn approximated polynomial density interval variation polynomial specify sample run high output piecewise polynomial variation unknown degree must td combine program wide problem estimation continuous domain mixture modal mixture monotone hazard mixture distribution mixture gaussian monotone yield provably complexitie logarithmic parameter past decade computational theory address boolean art analyze extend study sample learn approach approximate structural piecewise efficacy show many well approximate type factor generic technique mathematical variant neighbor recent theoretical researcher estimation pac statistical framework access total discrete concerned obtain efficient discrete continuous translate translation straightforward notion make variation efficient piecewise accuracy arithmetic input complexity essentially nontrivial distribution piecewise certainly precisely tp close total univariate degree piecewise piecewise theorem statement piecewise give crucially rather degree obtain complexity degree easily piecewise phrase degree denote distribution sample prove logarithmic statement learn unknown use distribution exactly piecewise low apply define boundary evenly highly log concave modal mixture hazard rate mixture distribution mixture gaussian monotone density previous run polynomial problem list polynomial polynomial monotone distribution case complexity logarithmic factor description distribution number e prove learn describe subsection robust belong output continuous piecewise piecewise polynomial concave monotone bound monotone theorem distribution mixture gaussians corollary monotone hazard distribution concave poisson poisson distribution reference correspond optimal mean histogram partition note number bin technique naturally broad histogram instead believe generalization natural propose density generalization histogram seem likely applicability use computationally efficient learner wide concrete result high algorithm rather subtle dynamic discover degree roughly distribution challenge arise level intuition somewhat learn challenging pair target datum close able leverage carry careful vc inequality basic suffice program accurate additional challenge arise go interval course introduce carefully use box general efficient distribution sufficient piecewise approximation necessary existence modal result concave domain result density finally leverage recent sophisticated result obtain section describe simplicity distribution go domain define say individual assign atom hence piecewise behave value otherwise behave atomic well non density throughout ever probability probability probability assign function necessarily integrate empirical z piecewise fix infimum attained actually require generality argument always place interval partition e respectively say obvious contain need notation result approximation theory bernstein markov polynomial inequality vc pa ax ax aa say convergence family basic primitive decompose behave achieve sample behave procedure equal partition output pi main start theoretic inefficient algorithm small variation piecewise theoretic run intuitively intuitively interval put assumption intuitively variable absolute rhs reflect reflect pdf learn call uniform ii happen single lp quality solution behave ii lp lemma least probability mass behave multiplicative bind bind tells imply ii assume event go feasible cdf degree polynomial take py feasibility care easy cdf clear since pdfs mass constraints mi pi pi remain argue eq satisfied bound magnitude therefore likewise proof argue w r pi require lp feasible henceforth denote value bernstein markov imply prove moreover never large magnitude lp must sketch shall achieve see section f dd f lemma translate bind I mi mp h h ms claim use follow place across interval rewrite l mh iy value optimal mass equality apply follow observe term triangle write q rhs equal vc incur let pi f semi behave least partition call subroutine j show subroutine return way interval time programming combine different subroutine domain subroutine arbitrary transformation degree polynomial chebyshev rhs inequality preserve distance conclusion subroutine remain unchanged output approximately parameter except subroutine sub update store recover degree event subroutine succeed mass constant construct consecutive super entry return subroutine interval piece program estimate pi union event probability piecewise constraint degree degree polynomial denote together consecutive partition non interval correspond rescale constraint value mi ig I hold td tell similar reason lp subroutine partition correctly corollary partition satisfy proof close piecewise piecewise contain contain rh
simulation adjust choose relatively conservative procedure represent instability overall percentage rejection r rejection c instability test overall overall value zero close first previously conservative simulation scenario type I nominal goal assess improvement consideration truly simulate come display partition q display set value individual algorithm regression mean absolute simulation define true estimate individual random effect par mixed intercept time table specification instability test node split recall extract tree instance seven estimation considerably improvement coefficient attribute extract homogeneous contrary mixed assume parametric influence vs treatment reduce concentration result significant receive year long period lead per year among year duration treatment result reduction double vs treatment effective reduce interpretable along fit suggest heterogeneity traditional mixed entire longitudinal population influence several true observational clinical variable mixed effect characteristic interaction vary limitation section longitudinal regression longitudinal useful identify heterogeneity trajectory longitudinal firstly control take splitting reduce computation fit cut partitioning instability paper score mix response taylor series eq probability linear model second tp dimensional zero bridge expansion definition along mm diverse exist longitudinal influence longitudinal incorrect traditional linear mixed effect covariate applicable trajectory aim characterize homogeneous combination parsimonious way construct regression node determine splitting influence baseline instability split control asymptotic instability finite whole study longitudinal change among patient tree instability brownian bridge longitudinal study repeat outcome specific analyze consideration diverse several longitudinal may mixed longitudinal incorrect population common mask difference conclusion meaningful interpretable differential longitudinal differential heterogeneous technique partitioning take error applicable take subject population interaction vary covariate inherent drawback inclusion possible specify functional association nonlinear covariate determine parsimonious population profile strict information popular latent modeling alternative longitudinal characterize partitioning covariate homogeneous outcome longitudinal homogeneity covariance throughout article refer longitudinal display longitudinal tree longitudinal heterogeneous population three distinct longitudinal profile gender age gender longitudinal depend heterogeneous longitudinal longitudinal denote covariate respectively baseline attribute add coefficient reflect baseline x homogeneity population longitudinal change e influence non partitioning variable longitudinal longitudinal goodness criterion choose goodness split statistical test partitioning variable point total multiple regression call instability instability much partition categorical put well construct level control entire branch priori issue task split identify splitting choose optimum perform split additional assume good split step instability evidence heterogeneity parameter point evidence heterogeneity goodness fit cut adapt multiple via repeat instability partitioning control variable present selection instability idea evaluate parameter evaluate remain cut longitudinal utilize instability way first continuous partitioning score conjunction brownian motion brownian categorical partitioning variable parameter instability employ normality score asymptotic instability extensive instability tree longitudinal among base cart method probably extend model binary longitudinal datum implementation longitudinal datum regular structure propose multivariate spline longitudinal method spline longitudinal use longitudinal partitioning control type permutation testing permutation test intensive take model high sized merge difference step random second repeat two step estimate improvement exist aspect split control group difference subject reduce remainder organize longitudinal summarize test parameter instability categorical partition variable discuss separately measure improvement pruning discuss simulation instability whole application infect patient record continuous outcome covariate covariate baseline include baseline longitudinal association far strict attribute longitudinal interpretable longitudinal profile fit traditional individual homogeneity true simplify homogeneity common make rewrite intercept covariate I far entire entire homogeneous term extent ambiguity nature influence profile important decide remain next whether instability whether remain attribute partition true constant homogeneity instability instability h discuss instability separately whether categorical test instability number category distribute indicator reduction degree instability partitioning theorem er mean process score process estimate bridge outline appendix vector brownian process limit brownian bridge bridge weak functional supremum q converge rapidly suffice high instability calculate value raw partitioning partitioning case perform instability adjustment type value adjustment candidate variable significance please alternative contain multiplication limit non chi canonical provide appendix instability indicate towards instability intuitively split high instability propose order tree longitudinal step instability partitioning separately significance perform instability test partition significant choose partition partitioning improvement goodness criterion goodness step include observation subject subject longitudinal goodness ii step maximum splitting follow longitudinal advantage algorithms far control huge select goodness fit provide criterion aic tree terminal terminal obtain effect root node parametric covariate population measure nested tree test construct well evaluate significance tree tree come complexity terminal node
least distance loss although short appropriate time algorithm mean define euclidean density von periodic probability distribution second euclidean von share share eq datum obtain value compute put term final forest construct euclidean circular case use regression forest additional typically find splitting regression consider form cluster effectiveness euclidean target head pose point head pose represent manually box indicate head region image compute multiscale patch orientation histogram cell gradient compare htb htb test circular circle car dataset sequence various direction car specify car ground range multiscale experiment patch bound box remain circle pls rbf directly circle pls angle circle train regressor map target coordinate evaluate mae measure mae percentile percentile circle much circle notable reduction mae mae computed testing failure percentile mae percentile mae pls al htb htb sequence number direction direction failure due error trial splitting predefine forest head pose direction research grant office publication title paper change pose direction estimation grow pose split tree incorporate traditional binary splitting predefine trial find loss consider splitting rule determine splitting enjoy rule addition target circular space circular employ pose target car direction circular target state art successfully various pose output mapping predict target new space complex relationship regression task non forest effective various computer regression ensemble method regressor space leaf forest tree prediction average splitting splitting limitation limitation predefine maintain thresholde due limitation necessarily empirical overcome drawback scheme propose incorporate forest splitting cluster find predefine splitting preserve product procedure node child adaptively determine child binary splitting enjoy partition structure circular circular forest splitting determination test point head euclidean target car outperform point multi view car regression regression forest inherently head orientation assign increase pseudo class precise become conduct precision apply automatically target assign somewhat still suffer discretization difference pseudo approach pseudo joint output task experiment similar apply space locally tree limited categorical variable although formulation head pose use pose generative rbf network pose forest split minimize supervise rbf mapping learn square function reformulate function circular subsection explain presentation normally employ tree adaptively child present modification necessary task lastly recursively partition entire space splitting determine prediction partition throughout child create child split belong node essential regression splitting node train suffice split subsequent node set disjoint partition th mean square associated computed splitting partition tree formally represent output mention child split long belong tree index binary splitting rule correspond hyperplane axis predefine splitting minimize select major drawback splitting procedure splitting predefine rule rule necessarily overcome drawback split rely trial graphical splitting stage difference space directly take space partition task determine versus approach formally solve optimization cluster weight throughout training child node node splitting split hyperplane axis cluster find predefine splitting space child employ one necessarily consume step achieve comparative adaptively bic use different
framework exist table choice satisfactory yield extension worth graphical estimate well know selection size fitting gene direction explore whether ideally endow simplify meanwhile select well closed high dimensional showing generalize broad intractable integral adopt ii iii next choose large show plug proof respectively since k assumption show nc lemma uniformly nc n l nn inequality uniformly v np nn nr np nr p proposition chebyshev clearly tt n ts n ts ts n os tt j n technical still iii n rs n rs r rs os op n r n part iv part n ns n large prove sketch denominator np np bad proof large finish n identity pp n generality examine assumption imply w w nn iv uniformly low w imply iii approximate submatrix determinant matrix pp nd nd uniformly p argument n nd uniformly nd constant ni nd nd nd l p uniformly complete p step go need numerator denominator assumption nd constant iv observe prove q p c likewise desire conclusion proof c n n complete cm cm explore property bayesian setting consist place hyperparameter controlling theory specify probability model reveal reasonable assumed draw stochastically unified novel flexible display keyword phrase fully posterior consistency control generalize generalize gibbs credible response covariate link tn true nonzero zero ideally restrict model fully vast list representative I selection mean go link lasso ridge obtain unified regularize class penalty sure screen sis correlation bic multi consistency approach handle research bayesian one frequentist unlike treat priori approach probability achievable probability one nice search besides conduct select theoretically consistency procedure dimension sis ii step several drawback sis often one determine size even bayesian small cause motivated consideration reduction apply aspect selection prove place consistency theoretical numerical situation grow simultaneously conduct unified framework mcmc employ reduction include control mild condition provide consistency also selection performance well establish bayesian examine size control flexible propose type prior extend dimensional consistency avoid misspecification nontrivial extension study reveals computationally follow involve justify hyperparameter controlling model various include new type prior credible select present simulation describe clearly variable denote size normal covariate prior adopt mass zero assume section setup apply place prior prior extend situation candidate bernoulli beta assume covariate include beta assume include treat terminology small huge candidate model aggregate prior selection procedure novel assign weight control q value candidate model clearly powerful bernoulli beta large incorrect greater hierarchical distribution simplicity joint index eq high maximizing perform name screening ideally hope asymptotically great throughout properly choose lie face examine situation imply select significant select maximal square positive target useful n show upper select say insensitive eq st nk situation similar assumption call confirm strong assumption place suppose nonnegative furthermore simple grow word want emphasize low model heuristic situation theorem consistency hold word proper dimensional upper show uniformly rate posterior true next examine performance enhance flexibility simulation choose prior give true conservative commonly upper sparse dimensional want compare type fully useful pairwise consistency bayes evaluate thus grow consistency consistency setting two course sis formal additional thresholding believe adopt extend impossible select select variable one model n n constant tt validity adopt e validity still nice conjugacy violate specify therein induce consistent bayesian motivation hyper beta properly demonstrate hyper dimensional enough support lead calculation therefore g implementation choose mode satisfactory choice application initial important simultaneous suppose model goal credible argument assume hierarchical eq straightforward diagonal element credible coverage consider credible construct arbitrary nominal small mcmc draw bic priori additional mcmc fix uniformly practically still facilitate difficulty complicate model p fix gibbs sample draw conditional however full conditional involve intensive inversion extremely consume dimension need ease inversion computing avoid improve gibbs draw block nice property inversion specific control model nontrivial modification constrain implementation automatic control flexible joint ease suppose match sense j j conditional eq integrating eq sample marginal draw p follow size mention additional bayesian need verify ig denote easy conclude fashion programming sample j draw choice role implementation address chose ease number preference popular alternative assume prior introduce q form prior n c cn compare generalized generalize hyper prior popular specifically examine length sis scad scad median median report simulation nx situation represent relatively predictor benchmark consider n np n vector somewhat perform commonly high define choose choose examine setting first conduct take second compute gb chain see examine bic hyper prior v f examine also mention situation find r package edu reveal hyperparameter recommend correct great reason choose hyperparameter achieve high value v upon request satisfactory performance affect select accurately select worst somewhat even much sensitive htp c bic bic bic compute high use credible interval
begin consider name policy build stationary policy return g go algorithm alternative involve ii iii small ii designing trivially obtain two policy notation ready algorithm enjoy guarantee like one iteration expect long q rather simple deferred appendix show constitute induce policy dependency remark express bad guarantee property quickly like simplify non dynamic introduce difference consider loop infinitely focus remarkably turn almost old slightly different control guarantee identical distribution provide essentially underlying difference refer consider require match possible remark introduce close connection guarantee guarantee dependency highlight red nb alg usually matter constant sort hierarchy constant interesting implication beyond policy focus pi argue arbitrarily strong algorithm identify nice property observe well overall several pair proof since involve also algorithmic variant guarantee believe helpful quality complexity look move deterministic policy understand potentially hide natural side argument well prefer problem variability analyze big constitute future slightly optimal writing see put back obtain multiply side obtain back prove proof essentially iteration algorithm fact v v algorithm maximize side long ii exceed iii prove write use advantage stop put I k notation fact observe begin define multiply back v section discuss get behaviour slow state take stop far two variation identical except choose small step begin assess finite application totally abstract kind mdp encounter parameterized branching specify next state action uniformly cut randomly sample randomly component sample discount compute call greedy apply value implement noisy white project onto fourier basis value respect projection apply operator project greedy amount much want set state branching correspond mdps run mdp figure standard mdp since display overall figure display respectively give observation converge iteration much average tend standard big behavior difficult action space pdf std std std pdf pdf std pdf std std bottom infinite discount formalize process policy compute optimal pi via exist notably come exponential algorithm dynamic infinite horizon simplify version stationary period enjoy within infinite discount process mdp rich bound closeness compute error policy policy unfortunately practice implementation programming approximation like control norm provide express right side weighted come price measure mdp concentrate detailed discussion though effort employ constitute severe approximate conservative policy paper involve input frequency policy piece expert domain main motivation emphasize importance significance section program extend property ease comparison stepsize argue price exponentially motivate describe infinite horizon simplification algorithm particular enjoy similar like begin horizon discount markov possibly kernel discount max coefficient corollary iteration conservative describe stop stepsize policy return successive call greedy explain g k coefficient small soon like modify iteration moreover exist hold eq always trivial negative exist consider mdp dirac deterministic discount condition mild follow performance stepsize big state error may return line completeness immediate exception stop satisfy complementary highlight
distribution depend radius bounding request iteration meet iteration label eq label hold complexity emphasize active knowledge sub special almost sample unknown differ factor modification label instance lead order sample additional surrogate active learning reduce reveal noisy high exploit excess theorem smooth combine convex satisfie condition q obvious become subset q eq low noise disagreement define prediction verify epoch bind generalization sample vc bind iii q must sufficient address eq rademacher eq contraction combine notice probability minimize function excess use ensure require satisfy unbounded base arrive follow thus combine constant finite constant equality lipschitz continuity eq case q inequality thm corollary thm learning protocol learner allow able exponentially linearly separable propose loss reduce importantly empirical solution minimize introduction surrogate yield exponential well utilize certain priori label condition characterize decision imply active develop aim show achieve complexity emphasize assume available learner closely appropriate exponential still achieve binary risk know priori idea explore smoothness complexity get tight improve reduction active classified algorithm algorithm greedy active algorithm design select labeling space instead informative algorithm selective label long give selective focus agnostic perfectly adversarial noise noise study condition consider know convex algorithm develop several maintain hypothesis excess cost limitation address make maintain label differ extend theory important analysis active disagreement low capacity constant disagreement coefficient bound constant capacity passive active surrogate passive ease build upon develop loss stage learn loss paper convex surrogate improve prior know negative active small error generally make assumption binary assignment lipschitz linear surrogate bind excess excess special truncate quadratic affine e finally conditional loss independent long remain examine bind assumption I isotropic isotropic concave hold iii universal isotropic log exist c isotropic iii since sign scale invariant distribution find follow divide epoch hypothesis compute epoch specify sequentially scan pool request domain request class simplify direction collection label new denote half epoch half work previous improve optimization could expensive unit summarize random satisfy r commonly use learn complexity key result exponential reduction appendix active learning disagreement region hypothesis disagreement coefficient keep coefficient disagreement disagreement example differently
similar difference penalize select feature whole create meta feature average belong penalize apply meta advantage secondly procedure biological perspective choose meta large difference help overcome finding penalize refer illustration comparison select validation randomly choose position pair correlation attempt strong certain close three covariance dna study comparison easier normal second first range everywhere else configuration reflect choice influence especially influential summarize percent misclassifie feature algorithm correct select diagonal diagonal good among perform bad structure performance significant structure much misclassification demonstrate performance poor structure affect feature covariance dna dataset prediction fold matrix though feature truly drawback separate simulate around feature somewhat restriction sparsity classification nonconvex mean secondly counterpart misclassification penalized propose pre conjecture penalize nonconvex restriction implication estimator generalize research grant dms dms detail derivation follow maximize alternate ensure accumulation partial optima proceed iterate step u tu v j subject solve step useful else solution kkt condition solve additional vector close tb traditional respective grid minimize cross error exactly algorithm solution penalty produce several discriminant previously consider discriminant sequentially th vector replace standardized matrix element though orthogonality discriminant sequentially discriminant zero remain feature proof proposition proof hence b update correspond take proposition condition rewrite mean hand mean equivalent rewrite corollary know observation discriminant however implementation fail interpretation result goal provide account structure apply shrinkage result ascent sparsity highlight penalize constrain simulation alternative patient coordinate ascent discriminant gap penalization discriminant lda popular big variate group provide asymptotically classification naive alternative often several independence rule dependency appeal simplicity lda crucial understanding biological instead independence result rely inverse definite misclassification go preferable setting misclassification always equal misclassification relevant scale normal equally correlate correlation beneficial rewrite dimensional subset relevant consequence beneficial discrimination drawback discriminant change adjust crucial covariance however estimate propose limited lda version methodology way account dependency structure motivate patient profile dataset clinical consist dna patient process accord provide insight pattern dataset great formulation function penalize penalize estimator require additional computational optimization criterion misclassification rate structure group notice certain regardless choice proceed analyze formulation problem lasso method svm lasso motivate geometrically project solution onto subspace force certain component equivalent natural expect penalty show explanation phenomenon propose follow section review classical fisher setting solution compare compete method study application dna independent come mean overall within tn linear seek combination eq aim variability pg px hx discriminant discuss multiple discriminant moreover discriminant extension solution fisher problem positive penalty objective alternate derivation instead perform ascent method h give w v pl q k stop simplify lead feature advantage although review correlation ii advantage estimator preserve secondly two moment distributional assumption give estimator easily quickly regularize discriminant aside selection procedure identity shrinkage automatically definite allow desirable matrix limited methodology available achieve advance method computationally intensive datum analyzing result feature corollary existence always sparse overcome problem large component simulation result illustrated figure specifically least non zero behavior penalization demonstrate derive existence fisher objective ease relate incorporate modification usually perform reverse generally constrain problem therefore long sparsity level smoothly behavior consider positive eigenvector scenario eigenvector geometry visualize relationship constrain formulate point definition identify corresponding finding follow solution support hyperplane hyperplane construct hyperplane construct lie shape second implication set exist sparsity figure language theory define hyperplane support say duality gap function visualization eigenvector indicate line support hyperplane non simplify assumption quantify insight inferential lead feature derivation diagonal follow w tt lead find test connection near centroid
available approximation nice version fully parallel descent adaboost coordinate large gradient coordinate step greedy coordinate need compute directional actually medium dataset fast reach accuracy second allocate scale parallel coordinate fast benefit also unlike algorithm moderately processor factor nearly decrease processor show paper parallel framework partially separable suit give computable speedup demonstrate especially directional optimisation variable length control decrease slow increase processor descent combine fully parallel coordinate outperform norm know choice core receive weight weak correlate label shall row iterate stop grant theorem iterate eq optimisation attain feasible continuity schwarz jx jx jx jx q conclude modulus strong part remark proof difference difference follow proof main q fa one argument align rectangle q merge rt get state parallel descent definition adaboost descent use logarithm length convergence factor problem especially algorithm widely high hypothesis greedy descent method decrease classifier leave adaboost find large may gene gene original sequential version descent row relax divide row support machine interpret fully parallel method design adaboost section unchanged adaboost evidence iteration processor merge communication author processor adaboost work parallel coordinate parallel coordinate composite partially separable coordinate wise convex nonsmooth function together speedup coordinate coordinate accord suited problem descent nonsmooth call nesterov theorem logarithm parallel coordinate descent adaboost classical adaboost learning vector eq accept coordinate depend admit separable respect independent return tx coordinate partially separable box however assume problem adaboost row lipschitz nice j choose nice adaboost nice compute done read datum multiplication pair multiply multiplication get sum comparison iteration well start function update reduction due logarithm give give need parallel coordinate descent adaboost align f fa descent adaboost iteration adaboost
vary require plan detail specialized discover latent representation accuracy art merely offer expressive subsection zero friend social network work case demonstrate regardless decompose component manually component capture activity facebook post lack person friend correspond people post measure person day appear matlab parallelization parallel toolbox toolbox especially tensor experiment core processor disk gb ram core plot either dataset operate equation repetition cost repetition observe monotonically run algorithm decrease speedup maintain speedup increase behaviour additionally benefit parallel come repetition carry core probably speedup course law mind maintain relative repetition approximate systematic randomly pair speedup gain factor achieve simplification derive short see htb latent fig relative calculate add number entry portion execute experiment dense twice plot able maintain demonstrate experiment observe fair perform miss slow probably implementation issue encourage amount htp ill condition setting reasonable fidelity couple literature first issue compare et mention introduction compatible provide core solver apply aforementioned decomposition match dimension decompose spirit impose explicit norm penalty model couple analyze jointly consider coupling outline algorithm combine speed parallelization able tune core comprehensive study miss value plain decomposition parallel nature implementation mechanic behind toolbox powerful introduce handle decomposition substantial utilize fast art handle miss degradation moderate amount entry question behavior predict brain promise additional scalable operate conjunction however approach outline scale sparsity rely sparsity factor maintain good tensor addition design complex interesting acknowledgement grant nsf nsf nsf recommendation author necessarily reflect view science foundation like thank algorithm scalable cm activity human behavioral express latent variable activity behavioral response solve art along fold extend degradation voxel human property able predict brain friend anomaly knowledge map store brain specific jointly network social comprehensive piece rank work fast scalable contribution jointly show traditional portion take traditional time maintain accuracy traditional carefully derive perform scan couple semantic fmri brain activity feature combine variety mining apply evolve side interaction demonstrate discover tensor scalar rao hadamard sec norm dyadic people tensor generalization relationship record web project lead mode tensor three mode activity fmri mode activity measurement scalable focus expressive tensor tensor one early semantic couple mode tensor thick brain activity semantic couple work everything mode three function encodes idea behind couple seek analyze couple share dimension share subspace one call idea fix fix optimize converge objective small strategy update require square provide detailed similarly rest simple singular ht matrix size initialize text besides exist however choose decomposition easily constraint strength operate large provide one operate dataset regardless formula contribute fast intuitive relatively factor intuitive interpretation order get couple preferable sample previous sure intermediate three concept behind outline fit possibly reduce operate representative use three henceforth marginal sum tensor essence represent replacement bias preferable probable high retain representative set index mode essentially likely appear important couple mode randomly sure coupling say run sample sample index detail factor aforementione go due initially highly whose effort e tensor factor repetition size factor initialize zero index repetition rest mode similarly replacement obtain likewise factor unit merge likewise average likewise merge correctly matrix repetition list ideally inner conversely column considerably I addition contribution speed core simplification rao hold partition hold thing together substitute offer gain algorithm precisely would yield many practical corrupt brain activity sensor work majority sensor signal mining algorithm operate essentially factorization knowledge usually factorization handling assumption careful miss everywhere else optimization implication handle miss value suffice sense line scalar analytical therefore aforementioned essence carefully line entirely parallel generator across repetition repetition set line across repetition brain henceforth refer two brain human subject g house fmri level localize activity make pixel across participant record fmri contiguous stimulus stimulus acquisition preprocessing table million zero dimension extremely efficiently speedup strength propose simultaneously use word brain voxel human brain people scope present display region activate stimulus
feedforward give dropout train model consist contain distribution mask presentation follow different mask bag subset differ bag single make come average sub bag arithmetic obvious dropout fortunately family geometric mean predictive run divide softmax dropout deep architecture geometric characterize mathematically perform maxout feed perceptron convolutional maxout may hide maxout layer implement affine channel spatial training dropout perform mask multiplication case drop max maxout piecewise maxout hide unit also graphical work maxout traditional activation function design produce sparse see dropout training maxout measure never bound significant correspond function maxout maxout almost curvature seem surprising find train dropout excellent mlp universal maxout network universal provide maxout maxout hide unit diagram basic present piecewise consist group value difference domain real exist continuous approximated compact maxout maxout arbitrarily piecewise note match maxout maxout network ht error mlp dropout mp network manifold maxout dataset state mnist handwritten digits test train densely maxout softmax layer regularize model apart maxout hyperparameter validation record point minimal error set log obtain permutation invariant mnist consider mnist permutation convolutional pooling maxout follow densely connect hyperparameter extremely gpu develop set error rate new state mnist transformation method table cnn nn stochastic image split train whitening term continue cifar training match would thus match old maxout layer fully connect maxout layer fully set test additionally datum horizontal absolute epoch run augmentation time extensively validate hyperparameter cifar cifar entire set cifar test pooling number google view two identify reason maxout compatible dropout approximate average intuitive justification model weight give single softmax model average exact several inductive bias indicate deep architecture locally apply dropout dropout encourage unit learn regardless drop maxout dropout mask input clean input change mask change piece map maxout identity relatively rarely mask learn average technique activation everywhere model average dropout network incorporate function test maxout mnist dropout tangent network mnist divide weight evidence dropout average even accurate second maxout improve bag style training dropout argument motivate use maxout maxout maxout difference dropout validation mlp argue maxout easier optimize pooling verify maxout pool linear unit dropout carry capability large dataset training maxout try narrow maxout well increase pool ht proceed differently sgd smoothly dropout work rapidly explore one slowly promise direction empirically operate training sgd unit less initialize rarely constant block absence suffer flow maxout negative activation illustrate unit become inactive inactive dropout maxout always active negative activation zero maxout mnist filter pool group include constant max fail maximal pool value maxout hand filter maxout network maximal tune behave sgd require drop simplify hypothesis suffer respect dropout mnist output maxout combine result maxout deep maxout help bag lose sgd bottom activation maxout suited prove dropout attain average deep maxout exploit approximation maxout unit demonstrate differently dropout pure sgd designing avoid able deep show maxout dropout low ensure benefit bag five benchmark design explicitly combine average ed discussion theorem sketch proof remark minus height width em design leverage average dropout maxout input dropout facilitate dropout averaging
become matrix quadratic search direction problem assume definite require subproblem uniqueness later throughout satisfied without direction define call behave similarly vector composite eq reduce exactly depend different say matrix study globally continuously applicable theory direction proximal newton study metric newton define notational ease proximal newton quantitie variable strategy shorthand notation ff proximal generate iteration call show step iterative proximal decrease scheme generate moreover establish appendix assume unique solution scheme k also case e rate virtue proximal newton solve reach proximal newton iteration accuracy phase tolerance terminate quadratic convergence upper iteration also specify bottleneck phase phase solve subproblem convex one convergence rate g use iteration newton become complexity require exceed f k cc requirement fulfil k k modify estimate tight enhanced backtracking standard backtrack evaluation interestingly analytically access within quadratic hence switch alternatively enhance backtrack knowledge reduce backtrack side information without expensive pf interesting proximal solve quasi completeness newton need next bfgs q proximal newton scheme bfgs bfgs subsection newton method proximal quasi newton assumption unique quasi adopt impose unique strongly maintain statement condition sufficiently equation norm observe direction diagonal scheme appropriate size proximal follow lemma show scheme find suppose g k k obvious since relax actually simple study dimension apparent kf straight however lower bound practice meet e wise operator require multiplication concrete proximal incur separable ht input tolerance compute k cost prox procedure additional choose tracking procedure evaluation initialize g k g l note step need relatively application global theorem proof locally converge reality adaptively knowledge l l l k g last impose claim also empirical subsection nuclear direction twice sparsity prove assumption tool preserve prox oppose increase maintain property global require apply property generate ks statement omit k eventually modification check careful implementation evaluate algorithm quasi newton bfgs metric straightforward fashion selection notational convenience maintain nonsmooth compute formulate variant solution approach introduce early new derive formulation subproblem subproblem write min conjugate strongly apply project method rate parametric k recover primal newton surprisingly us cholesky course solve primal cholesky decomposition subproblem become u newton direction summarize I proximal pn cholesky attractive different computational parallel implementation dense majority entry size trace operation require evaluation objective achieve cholesky gradient gradient suffice projection op require iteration implement major proximal multiplication naturally gpu parallel computation refer reader important cholesky multiplication since subproblem become component define definite I summarize ht start multiplication cholesky decomposition inversion cholesky decomposition require omit easily satisfy transform satisfy subproblem express k poisson intensity tv regularizers method efficiently discussion note estimate base rule initialize k appropriate k terminate determine size k modify scheme focus unconstrained variance consider highlight salient follow multiplication direction quantity require multiplication product product omit optimization numerical variant encourage reader quasi newton matlab intel gb ram proximal newton search procedure impact proximal solve fista I four procedure whose detail analytic proximal newton standard backtracking backtrack search step value improve synthetic run report computational average cf terminate summarize iteration cholesky decomposition multiplication ccc ccc ccc ccc c procedure approach usually start bad therefore advantageous search procedure reach compare backtrack iteration line outperform regularization become note iteration cholesky decomposition advantageous diagonal subproblem tackle broad norm solve impact implement method fista speedup propose dual subproblem proximal newton proximal matlab proximal terminate exceed total execution hour report cholesky entry indicate exceed either time limit time sparse converge fast active variable small aside bottleneck high dimensional converge rather within medium manner achieve accuracy proximal practice regularizer algorithm consider configuration quantity last iteration restrict direction proximal method accuracy proximal norm proof theorem last perform test instance median show median condition gap actual decrease look condition case local figure final suffer contraction drop rapidly check ht regularizer improve describe art poisson intensity toolbox termination iteration illustrate parameter illustrate convergence count top iteration behavior inaccurate solution subproblem exhibit search decrease search certain practice tv operator order good visual reconstruction previously summary time acceleration obtain termination cpu ac superior cpu objective report use unknown illustrate lipschitz hold comparison art model nesterov proximal operation option backtrack due logarithmic converge linear term expensive prox backtrack operation bad illustration typical stop use obtain speed tp ccc cpu c c cpu gradient assumption highlight work correction match structure lead algorithmic test application composite minimization problem propagate hope effort direction acknowledgment european author grateful action thorough comment suggestion presentation consist technical adding last ff f ff lead ts ff know uniqueness strict increase convexity ff add contradiction subsection proof unify fashion key quantifying hold combine self property combine k together definition k reduce induction number k moreover q bound stationary follow ensure follow proximal f subproblem solution replace expression q formula fix optimality eq estimate require purpose definition note ki quantity apply follow note k proof triangle rearrange get assume easily converge proof part substitute similarly show rearrange k k substitute fact rearrange linearly g k k applying show imply converge super linearly g rearrange k k laboratory ed minimize sum smooth convex endowed computable operator framework rely highlight procedure concrete interesting numerically real gradient self formulation ever expand statistic minimization convex canonical assume smooth composite naturally maximum posteriori estimation model understand efficiently polynomial point transform programming semidefinite curse impractical scale prevent direct r c f l f f fortunately provably trade two among lipschitz gradient fig g diagonal fashion say full nearly analytical lf analytical st accelerate proximal proximal quasi unfortunately lipschitz solve easy albeit composite sequential quadratic subproblem method e line address solve gradient rigorous guarantee answer broad class global trade self self self barrier fm mm self sequel unless minimization reason application directly continuous composite enable constrain convex endowed barrier minimization f middle solve problem setting benefit scalable highlight keep mind list gaussian inverse dependency respect edge cf node covariance apply ise act problem none exploit cf sect bf consider processing detector wish reconstruct low noisy level non proximal fortunately composite time smooth leverage surprisingly retain original structure lead computational many contribution summarize convex nonsmooth rely subproblem first achieve monotonic size correction global strategy variable
respectively horizon mab approximation separate traversal decision obtain mid adaptive delay computational obtaining recently issue increase online present user induce visit web page update issue management cloud service delay delay feedback delay weakly couple broad delay balance exploitation decide due delay arm exceed play different reward simultaneously remain satisfy single double exploration algorithmic typically mab play arm reward arm play arm concave individual consider click user make influential hence click example prior historical status model armed play concave reward play concrete step maximum observe produce issue make answer yes indicate accounting answer repeat scenario define accounting arm reward affect decrease variant issue relate weak reward optimum correspondingly strong perspective design single arm small outcome index like outcomes mab application goal policy pure exploration exploitation optimize example maximize distribution state policy goal find maximize take initial outcome arise evolution path final choice extension correspond allocation arm utility bandit extensively introduction direction result one free task lose reward reward arm stochastic exchange move play arm play without loss exchange case moving create infeasible delay infeasible reward appear appeal adversarial help across time analyze arm delay encode future objective encode well herein maximize maximum policy use argument upon herein run bandit mab different mab arm reward fashion parametrize arm arm allow observe outcome arm play outcome prior play update correspond maximize reward step expectation play objective pure reward observe play make uniquely oppose uniquely set uniquely arm horizon state state start arm parameter draw reward therefore reward reward transition e denote number natural update part state goal decision playing subject expect step maximize satisfy rule standard martingale property reward except space satisfy martingale property choose update failure treatment click conjugate bernoulli mean family beta correspond observe outcome condition posterior happen correspond uniquely specified evolve state early case problem prior expectation observe ingredient budget trivial design policy arm play attempt arm budget yield word bind achievable arm count reward arm arm step budget involve play play observation section handle require expand state space nontrivial encode context delay even allow play arm slot receive play decide make slot reward play make regard feedback relevant case significant accounting arm play initial play time slot policy mapping involve play problem action richer subsequent overall involve policy include delay feedback policy contribution simplify policy depend polynomially bernoulli bandit policy favorable dynamic index policy execution action arm several action arm stop action state restrict arm arm remain horizon state reach horizon step let policy state system current play stop basic variant action describe let reward expectation take outcome play policy section arm description description analyze basic problem constraint contiguous previously provide horizon mab necessarily herein illustrative significantly give lp formulation scheduling compute solution lp lp compact base use throughout likewise lp bound lp lagrangian lp correspond decision execution play arm define globally feasible follow linearity play correspond observation constraint encode reach play time play capture play reach precisely expect play state hence relaxation bandit ignore clearly correspond since joint different optimum decision path consider develop technique solve solution preserve large interpretation representation feasible solution encode separate consider arm probability condition reach arm currently choose stop reach arm play satisfie single arm represent horizon arm reduce constant constant factor reward policy global proof martingale reward proceed stop argument statement inductive apply prove encounter reward recall truncation space suppose choice stop play system reward mean specify unknown reward play observation path reward play reward exactly play regardless observation expect accounting scheme since draw linearity path take distribution stop fraction play mean decision execution play yield integrate equivalence relaxation one arm arm subsequent possible work compact lp single arm rich action rich relaxation difference figure policy schedule policy combine policy arm policy remain order say scheduling reach overall stop account policy outline reward start play play remain horizon consequence arm indicate area mab scheduling policy linear arm optimum gap optimum policy lp situation type arm arm arm type otherwise lp arm play expect find continue rest arm play optimum prefer arbitrarily play arm keep arm rest horizon reward ap pa tt outline application duality recall take lagrangian obtain constraint policy separately c rp ii kt policy nothing kt duality single arm optimum r increase straightforward bottom dynamic dag horizon let condition therefore case child play correspond uv tp q nothing moreover show optimum weak instance combination value yield opt policy least constraint kt immediately base set remainder let root nan maintain number kt kt perform search property opt kt thereby satisfying observe r kt opt p kt opt opt initial use rescaling follow immediate mab accounting corollary additional single policy scheduling approximation obeys arm constraint mathematical need still corollary follow kt kt execute policy least identical lemma approximation prove discount bandit recall arm policy construct arm policy number reward value rp tp solve play make policy sum policy start policy work play discount version start solution computational contrast arm lp decision whether play execute decision impose arm break tie play traversal affect concrete traversal cost switch arm system start total ii switching decision path receive provable set switching cost metric setting early improve motivation develop bayesian problem take mab decide arm adversary start play visit switch cost economic traversal subproblem key benefit arbitrary adversarial order traversal traversal problem encode natural combinatorial snp difficulty policy policy switch relevant herein second cost transition mab switch analysis traversal approximated factor mab order scheduling correspond choose remain apply replace final create policy adversary order arm determine policy policy decision scheduling overall stop execution finite mab scheduling argument arrive expect satisfie ir exact statement slack weakly couple policy horizon balance slack optimally horizon switch cost simplicity play discuss end arm current arm arm currently ii arm switch iii stop obtain play arm policy play cost decision begin feasible cost well reward encode play even single say phase start consecutive play rest block full block arm idea delay every play policy convert policy without know know outcome play next begin execute play step step know previous step immediate play horizon exist policy I kt continue introduce delay free become policy free time early outcome subsequently truncation delay decision horizon subsequently delay policy mark solid block blue delay policy horizon accounting argue block horizon policy play decision end play let play make play play use know outcome know play outcome distribution outcome well stop execution make least show ignore switch delay mode must first block since switch within r policy I policie p kt consider scheduling ht scheduling arm I indicate policy probability active passive active ready ready nan find first play arm arm policy decide ready otherwise reach stop execution expect scheduling approximation exceed suitably actual play start finish structure feedback therefore contribution note ignore moreover reward exactly kt exact ratio discuss block assume play intuitively equivalent play policy policy policy play otherwise policy delay free play execution block block block prove block whose couple execution define outcome play excess know outcome type maintain block decision make play make suffer play outcome otherwise outcome outcome store maintain fashion play extra play simulate decision irrespective occur fix stochastically identical moreover block division arise play additive type block particular bad hold path say either policy within start manner proof use fashion boundary structure maintain play observe policy kt scheduling change step except horizon policy p kt scheduling least proof conclude rest rather delay horizon exist block horizon play arm quantitie omit consecutive beginning randomized policy state arm make lp encodes randomize structured make system single policy block block update return objective arm observe play choose arm evolve nonlinear arise immediately discuss choice handle scenario application scenario budget model natural scenario effort alternative handle total discard issue accounting scenario resolve arm reward policy deep run counter intuition accounting matter play separate application first slot arm play reward slot scenario refer play potential move slot arm posterior powerful issue relevant time slot stop reward slot observe time feedback set arm slot powerful feedback budget model observed provide provide feedback accounting policy single policy introduce scheduling different notion extend exist restrict account approximation optimum feedback feedback budget easily incorporate feedback arm take randomized fashion execution play arm play value obtain play e next policy expect reward recall expect play goal find randomized arm play play per policy consider execution optimal linearity expectation policy choice make value policy relaxation take lagrangian result policy arm obtain dynamic time arm separable arm always state define leaf child root argument I insight policy path path correspond reward check qx check x I suppose policy play reward expect negative reward execute step factor take expectation path prove claim apply important aspect lemma policy nan p first specify remain policy inactive slot policy observe outcome state decision specify repeat either make time slot horizon reach stop expect near identical horizon policy reward identical feedback arm play single obtain kt multiplier observe optimum solution choose solution collection policy policy randomize truncate horizon kt observe solution collection following observe compute choose scheduling ready initially current execute policy whenever arm remove arm suppose arm state policy decision schedule policy choose play maximum analysis observe place order mark marked completion path arm arm play sufficiently many exceed respective arm mark play indicator denote arm execute completion due marked count objective expect contribution least whenever play lemma combine play least important aspect captures summarize policy budget arm observe feedback problem extension simultaneous analogous section scenario state far arm program separately encode budget spend extend concrete lagrangian might budget subsequently observe consequence show optimal scheduling policy yield consider total choice define program goal find arm play observe decide value differ feasible presentation power assumption power maintain distinction update surprising policy choose reward power call consider half case well decision path contribution suppose contribute half contribution decision path choose modify choose clearly dominate contribution time budget modify policy value generate value contribute case contribute original budget must policy factor statement immediately find arm horizon b relax proof identical policy consequence two dynamic programming consider value ready subsection time measure policy conjunction account feedback observe scheduling execute scheduling problem arm pure optimize reward play give policy state path maximize reward execution version specify execute return extend approximation policy arm arm available ii obtain iii single arm arm arm final reward event arm section define couple reward randomize check linearity policy expect choice make expected play make objective precisely feasible consider lagrangian ip tp I tp duality compare optimum policy ip tp policy condition programming child final stop play arm outcome uv running inspection observe ip maintain subtree policy increase contribution policy respective maintain fact perform ii arm answer stop ip tp tr collect follow reward choose reward play account true policy define lagrangian arm policy arm arbitrarily execute policy move choose arm arm obtain algorithm accounting argument horizon execute execute horizon single policy policy consider visit arm play arm continue case case reward step policy accounting choose arm let remainder execute martingale choose contribute contribution therefore horizon yield reward infeasible corollary inspection base lagrangian would consequence immediately approximation variant cost switching follow possibly approximation also learn packing several finite formulate weakly use devise reward analytic relaxation guide result comparable standard question performance delay latter provide upper low strongly instance bandit problem weakly couple acknowledgment thank p al helpful proposition corollary edu nsf perform google p research fellowship award grant consider horizon cost delay feedback concave play explore optimal near run variant computationally exchange reward scheduling account critical fairly basic policy suboptimal context policy couple restricted arm ensure yield show relaxation solve fact final policy index policy conceptually policy satisfy exchange reward play per hold global restrict find policy global exchange property number technique already demonstrate applicability consider iterate resource effectiveness resource uncertain series allocation past outcome seminal contribution vast reference resource action uncertain take provide reward cause arm agent play horizon maximize paper equip state arm play description constraint input output maximize output specify policy ideally polynomial specify regard interest index policy arm index schedule easier implement conceptually design optimum exist setting see discussion mab increasingly action large comparison historical mab mostly alternative motivated medical measure derive recent mab arise content arm possibly machine generate vanish reward concavity recent force formulation computational complexity consideration bayesian bandit back finite armed form set arm reward parametrize distribution arm constraint arm outcome arm play update maximize expectation take play step update information arm encode encode distribution arm state yield reward observation play special case classic finite multi armed space update reward play state mab therefore canonical martingale bandit recent mab variant arise historical challenge arise define dag root apply rule observation play provide posterior martingale reward constraint application bandit due cost availability underlie action switching cost adversarial arise feasibility consumption sensor feedback delay pay time delay non play person multiple attribute influential arm reward take step exploration maximize step constraint outline fundamental issue property arm play exchange play application ensure played immediately play loss reward core mab provable scheduling decision without index exchangeability hold problem example constraint play arm may obviously reward arise derive arm function arm delay arm previous play similar occur switch another effective play index lead provably relaxation conceptual idea bring answer main couple decomposable lp
practical choose knowledge q classical maximum observed uniform simplify geometry namely move move whenever increase local efficient intuitively control order index prior start sort let sort indicator merely indice set local tracking definition imply iteration compare two compactly update tracking stop transition become empty operation attractive relaxed suppose hold validation see model equivalently description relaxation parameter recover efficiently characterize relaxation relaxation primal entire relaxation normalize change r validation inverse denote equivalently depend omit change term replace product path change point minimizer consecutive second convex wish minimize specifically change interval numerically path wish q interval dl search trade relaxed maximum entropy solve linear associate loss associate validation increase list loss decrease increase far need range validation obtain increase list l summary efficient admissible model list illustrate trade complexity loss benefit increase term improvement complexity path local example text note though observe sample numerous size algorithm sample distribution path path divide grow function check without keep corpus volume dataset article stop collection set collection support category plot log scale seem monotonic grow close size examine tendency index enter usage enforce gram like rather obtain concrete benchmark reader token alphabet token word dictionary character n gram use markov chain token root leaf determine modern gram first buffer count occurrence occurrence string buffer normalize figure distribution underlying face bias certain empirical short increase context length know smoothing second empirical level along secondary pruning context estimate inaccurate remove alphabet language storage retrieval burden model instance alternatively parameter might wish whose inclusion improvement lead entropy prune pruning procedure ideally budget advantageous buffer regularize parameter gram entropy receive cascade specific solution th formally convenient denote store tend normalization pruning gram use relaxed root procedure estimate probability token buffer context context task relaxation path relaxed entropy available efficient list lk lk implicit sub choose option list allocation rule let option choose allocation result specific prediction budget divide allocation proceed recursively receive budget factor namely allocate validation right previously estimate short n separate pruning execute naturally allocation allocation root sub context understand validation benefit let worth node allocation sub otherwise option namely allocation potential path prototype character symbol commonly character module predict combine content identity character depend alphabet thousand buffer character language buffer use validation gram size maximal depth allocate individual maximum compare hold method character tree control plot trade versus outperform art seem size maximum translation language quickly fast tracking discuss easily path perform focus track separable objective useful extension addition adaptation master solve efficiently namely euclidean distance distribution rational divide event multinomial accommodate instance outcome typically desirable formally accuracy tracking show place set simplex without note motivate requirement place positivity result problem straightforwardly accommodate perform tracking next examine choice change track objective examine clearly inverse must satisfy get dependency piece follow definition q answer repeat tracking procedure onto intersection ball hypercube entire relaxation describe acknowledgment liu valuable feedback thank suggestion final manuscript support stanford fellowship google research homotopy repeatedly intersection line regardless orientation intersection line section maintain homotopy tracking structure queue small global starting maintain intersection line place queue denote line intersect horizontal fig formally intersection define treat naturally additional concerned sequel use queue keep track intersection intersection already long queue arrange queue front queue keep retrieve intersection maintain variable queue process involve queue intersection examine case two line simply switch scan line pass swap swap position queue since encounter value line newly intersection ahead intersection add queue large updating queue current queue either since line homotopy slope intersection queue global homotopy tracking queue perform homotopy somewhat examine set start index front queue intersection queue become identify line intersection queue become queue current intersection perform update homotopy tracking continue go numerous intersection claim conjecture theorem example google com view usa stanford edu department statistics stanford stanford usa entropy concerned find satisfy relaxed constraint multinomial problem detail geometric description relaxation relaxation path admit realistic path validation admissible infinite discuss relaxed index relaxation know task cast relaxation tune solve choose alternative possible solve relaxation specific regularization homotopy support machine gave admit describe characterization relaxation path characterization generalize separable multinomial distribution relaxed entropy subject generalization relaxation also proceed solution equation sec maximum admits increase description provide validation entire relaxation give validation able infinite family sec illustrate experiment compact gram language model extension relaxation efficiently complicated computational vector bold shorthand respect simplex retrieve part radius call name incorporate repeat setting follow problem convenient general objective optimum unique disjoint depend bind goal devise reveal examine follow characterize optimum coordinate jj hold prove complementary optimality brevity assume associate lagrange lagrange multipli simplex assume strictly know lagrange multiplier positivity constraint zero min get index examine three case neither bind complementary saddle thus statement optimality statement analogously finally simplex partition tend gradually approach homotopy arrive notion main objective constrain implicitly completely term justified determine assumption inverse invoke equip definition rewrite follow thus determined depend lastly symmetry uniform number later stem section characterize follow toy characterization region segment return finally necessary case build geometric description tracking entropy next intersection line practical outline complicated algorithm performance maintain homotopy tracking defer implement straightforwardly suffice devise track function plane trace track search slope close intersection line line continue beyond piecewise rewrite q j index imply directly triplet readily write line slope initially slope decrease characterization line potential denominator infeasible intersection discard large last use next segment start calculate triplet prescribe
scientific field technology evolves collect huge consist observation might hand extract meaningful massive past decade important area achieve concentrate response inference receive much functional besides functional classification cluster heart ask subject heart read datum focus distinguish group intuitively able distinguish normal subject group stable heart reduce diagnosis stress attack stress particular ask heart group diagnosis equality group group let denote stochastic process assume often equal problem nan hypothesis often equivalently functional response purpose study derive statistic nan via involved addition test classical throughout subject nan reject nan reject point advantage degree realization hypothesis pre critical percentile limitation example pointwise significant significance difficulty pointwise pointwise test incorporate multiple comparison complicate correct pointwise intensive bootstrapping therefore desirable pointwise pointwise study show point pointwise supremum pointwise somewhat permutation critical bootstrapping dataset power functional highly moderately answer question arise clinical sample necessary first nan bootstrap approximate nan difficulty mean applicable overcome bootstrap method nan find skewed functional moderately estimate approximate reasonably secondly level e alternative hypothesis via outperform control moderately former slightly functional datum correlate tend information low power whereas less correlated summarize since moderately correlate therefore prefer explain heart significant solely signal heart aid stress clinical straightforward response paper organize follow power present discretization discretization power extensive study heart give section conclude remark proof appendix helpful approximate test notice pointwise variation singular pool let integrable exponent modulus covariance list population belong zero ambiguity subject satisfie tt require size tend total sample weakly pointwise pool give function write ts hypothesis w k discuss investigate test ts ts pool adopt significance percentile I number hence accordingly sample size approximate nan applicable sample regard subject condition nonparametric obtain base repeat bootstrapping calculate value random nan proposition thus asymptotically hold practice nan alternative may function hold sample statistic effort require gaussian repeatedly easy necessary skewed prefer implementation relatively study specify local kt kk kt tn alternative root alternative long say consistent good admit consistency consistent show also first alternative abuse notation proposition write proposition proposition asymptotic power upper proposition claim power show root proof shall use follow relationship statistic test follow equal small percentile guarantee high power test continuous may discretize discuss apply reconstruct discretized behavior therein smooth discretization estimator asymptotically rate section approach approximate discretization discretization alternative interval vector j test discretize statistic discretized vector discretize subject bootstrap repeat bootstrappe conduct nan mean mm choose tend mf converge hence tend asymptotic limit test asymptotic study statistic give alternative l proposition component l tend modulus asymptotic test tends provide condition simulation reasonably furthermore seem remarkable shape pdfs strongly affect correlation functional cause skewness display pdfs well nan shape nan pdfs test affect decay variance also sample minor kt deviation cc cc cc cc way purpose summarize power kn associate deviation size power control power power test except correlated advantage test study conclude work reasonably approximate control power functional moderately discover medical detect mi clinical evaluation typical surface stress exercise scan fundamental regard stress detect mi typical stress accurate limitation stress attack stress patient test patient test finding characteristic study mi shift range mi patient finding spectrum shift test study conduct assess procedure daily clinical practice consecutive visit publish patient complete write group patient directly mi positive exercise scan exercise scan comprise least one exclude criterion exercise diagnostic scan rate less end patient please detail acquire second visit signal hz bit digital db cutoff hz amplitude deviation etc record subject pointing peak gray curve lead construct subject reduce influence denote time heart index r peak detect extract signal l cubic eliminate onto r direction denote r direction onto call eliminate denote row adaptive th hypothesis typical signal confirm hypothesis power heart shift region study power spectrum length follow convention say entry dc positive frequency leave negative frequency length spectra way adaptive associate p test adaptive indicate fact lead effect phenomenon activity even transform lead lead place axis undesirable signal inspection furthermore inside vary proportional cycle decrease pattern lead spectrum cumulative confirm spectrum shift middle panel visually spectrum test conclusion test screen patient support lead viewpoint induce cost effective exist method emphasize introduction domain use advance completely adaptive recognize report near test screen stable p replicate lead remove lead spectrum cumulative spectrum functional intensive study nan control test power moderately comparative power test screen far large easy test functional functional zhang research grant division national office support science mathematics division center office wu fa v outline tt k rt uniformly since always f rt rt rt rt decomposition rt complete proof pool covariance hypothesis claim proposition kt similar proof interval proposition k tr pf ac ac ac proposition vector vector mn exchangeable define proof kronecker matrix via stack one effect first group lt f mr
consist wind speed datum span daily wind root offset velocity use year average velocity day wind form consecutive vector contain velocity overlap segment training construct ordinary square construct predictor wind p qp qp qp qp qp qp predictor predictor subsequently set ground nonzero kp order decay weather give insight dependency kp kronecker kp right temporal kronecker factor kp bottom leave kronecker factor bottom solution factor frobenius scale full range visual kp spectrum energy kp compact kronecker spectrum component height percentage mean testing period day nc nc regularize estimator shrinkage suggest eqn show track wind rmse performance blue green regularize compare ht wind estimator red day actual ground truth wind show offer track representative condition e national centers environmental available website daily average u east north south wind year wind compute take magnitude wind grid range number raw transformation specific effect result observation velocity ht wind data day year th fit root year datum period consist year since pseudo predictor test ground truth overlap full make estimate nonzero kp factor correlation weather spatial give dependency wind covariance top right kronecker kp middle kronecker kp temporal kronecker factor kp necessarily definite kp frobenius note visual kronecker spectrum kp component kp spectrum kronecker spectrum height energy fig day n nc range parameterize wind propose regularization nc optimize fig day unstable kronecker product tracking track wind wind rmse use estimator rmse average wind day offer tracking separation product penalization name outperform toeplitz decay convergence synthetic real kronecker estimator wind standard product standard predictor several propose kronecker unique specify amount choose stein unbiased prove kronecker sample preserve inverse extension low separation miss naturally rank research support nf recall version problem orthogonality k norm invariant permutation subject k n symmetric n nj nonempty rewrite q iff k nj ni properties nj ni nj k nj rewritten nj imply right weighted scalar result l objective rewrite orthogonality l l l simple equal contradiction follow sign achieve generality l sign assume conclude generalize thm exist subdifferential tr use symmetry projection uv u inequality trace duality arithmetic rhs assumption conclude absolute measure sense variate model permutation ti version define write statistic summation concentration note standard component f bernstein thm conclude net sphere schwarz far finish consider regime tail occur regime relaxed choose conclude regime complete regime define event choose r obtain min definition orthogonality sort eigenvalue eigenvector pair square eigenvector must j gram schmidt toeplitz projection k orthonormal gram choice orthonormal transformation lemma finish complete thm projection choose generalized covariance schmidt submatrix proof write basis f f use variational e f f similar algebra use separation generalize f conclude corollary proposition conjecture method estimating square kronecker rate number infinity separation fast convergence tradeoff provide scalable approximation covariance mse recently flip separation ensures present kronecker spatio linear square wind speed least square product decomposition square statistical analysis receive diverse time portfolio management asset pricing bioinformatics microarray leading greatly exceed observation search set much estimation kronecker product kronecker kp kronecker product kp channel model wireless communication face system collaborative main structured nonconvex optimization problem arise optimization adopt alternate al flip parameter kronecker assume kronecker product whose covariance kronecker independent analogous separable component component neither relevant channel wireless communication receive covariance system netflix eeg finally expansion kronecker bilinear decomposition optimization estimate kp derivation infinity call least rate provide rate certain word constant size form consistency fast covariance separation rate ff covariance kronecker kronecker product expansion generalize previously different kp establish kp sum achieve advantage simulate wind show order remarkably spatio temporal kronecker kronecker pca eigen standard predictor kronecker rmse predict day outline introduce covariance present dimensional present place appendix transformation operator j j qp operator fig ht original note permutation operator set semidefinite psd definite projection operator project sphere denote notation follow index notation statistic covariance covariance suffer approximation retain principal component heuristic suffer high specify penalize least square develop estimator interpretation constrain frobenius solution interpret e psd psd show converge corollary establishe dnn nc effective rank absolute n notation amenable optimization true analyze solution thresholde leave vector convert apply permutation numerically evaluate propose full empirically observe algorithm svd compute operation fast svd require computational scale desire next show consider symmetric probability believe appropriately simulation order small rate provide norm frobenius estimation n establish spectral matrix strong surely norm kronecker establish frobenius selection norm define appendix optimize interval deviation inequality characterize tail sphere carlo growth spectral pn curve fit curve great result provide tight mse truly sum kronecker approximation f np rate reflect extend fully kronecker thm rank thm dimensional naive expansion principal component full finally choose separation pn rewrite dimensional rank remain toeplitz covariance separation singular spectrum toeplitz operator toeplitz thm toeplitz arbitrary size onto iff decomposition f fundamental characterize estimating block toeplitz decay matrix arise random process block toeplitz submatrix process toeplitz nu rate holding least choose
result stable reliable emphasis cv encoding ridge al videos movie induce video voxel understand visual major scientific use cat roughly speak detector use solely field show image basis image appearance filter likely brain world location build bank encoding model neuron natural al fmri brain signal voxel fmri measure activity brain coverage cube leverage single neuron brain signal feature fmri movie experimental fmri dim bank model sparse boost prediction net machine easier interpret subject rigorous movie consist boost average replicate result fmri signal replicate complete use fmri replicate fmri fmri voxel subject performance observe encode movie reconstruction validate encode human pathway lie find voxel thing really easily different hard resource human fmri long also fmri collect hour call consequently conclusion candidate drive remove proportion perturbation without conservative scientific sake field field numerical dynamical system pde concept imply model necessity procedure child history statistic form huber contribution prefer actual form population least key development excellent review series situation propose unbalanced wu study series sampling start early mahalanobis framework confidence interval subsampling apply series subsampling process validation cv select along regularize modern machine analysis bootstrap series find book tu mathematical foundation perturbation central limit central theory proof available website perturbation argument trick find ode generalization law gaussian see concentration result stability relate explain mean say conclusion stable statistical stability define perturbation law agree perturbation subsample close bootstrap linear block subsample subsample control size detect difference conservative conclusion importance science acceptable even desirable scientific fmri voxel boost use function cv choose unstable compare among version unstable order unstable estimator estimator drive interpretation zhang multi fold prediction provide low predictor van modern predictor unstable subsampling perturbation correlate model parameter selector consistent low yu perturbation scalar lasso threshold path yu like seek specific yu term es perturbation scheme partition smoothing estimate meaningful line yu estimate expression apply function es aim cv aim prediction es statistic statistic es well yu combine es parameter cv smoothing es cv suit cv incur negligible compute cv yu indicate cv apply movie fmri characterize location discretize orientation filter discrete lag act compare cv size al apply es cv e l boost frequency lag voxel maintain prediction es concentrated cv voxel visual compose four sub performance build cv es fit voxel vector display scatter plot sparsity es cv es cv small cv sparsity es apparent overall minimum es cv cv performance es cv model cv relative book huber primarily concern analogy stability equation view stability generally fundamental huber break step huber dependent study distribution gaussian fmri mean tail tail robust statistic error whether stability fmri high fmri problem phenomenon seen seek analytical work interaction variability regime rotation al distribute q equation nonnegative limit mention leave trick trick prove derivation step prove normality prox form appear analytical et al fitting mse match view key phenomenon variability express express capture et word act double exponential normal dominant discover act gaussian ol loss l double find consequently contrast also unbiased achieve l concentration design work et address question obtain estimator ol penalize double phenomenon double phenomenon ols well fmri contain st vision work smoothing perturbation classical robust analytical tie together stability scheme include bootstrap subsampling stability drive classical make statistical consideration effectively reliable broad include different stability high variability error emphasis place stability conclusion scientific statistic paper current statistical stability area action instability scientific finding future future involve progress service field science technology road smooth road criterion abstract without acknowledgement author th le publication bernoulli author figure thank detailed helpful discussion partial support nsf grant dms nf nsf science science scientific often modern finding rely stability statistical reasonable perturbation method motivate necessity interpretable fmri signal secondly strong literature selector estimation stability es bring es utilize encoding interaction tail double predictor exponential ordinary ol estimator deviation estimator analysis technology really investigation technology year imagine view curse obvious reason curse obvious prominent always think science self information
pointwise prop word choose instance rkhs span st polynomial see define circumstance restriction framework specifically sufficient let concern kl start generalizations borel algebra support typically lebesgue exist real automatically assumption made rely well rely isometry cf prop expand kl g respect normal consider source equip topology question paths rkh respective boundedness z l come generalize assumption prop ii let duality field field field terminology random field path solution equation field divergence free free gaussian path kernel equation homogeneous correspond endow path differential b ode belong another ode infinite equation harmonic harmonic kernel satisfy ode example harmonic see input path sample path show absence minimum illustrate incorporate invariance within case available give convenient framework functional depend community either define square integrable intrinsic expectation process minimal rkh definite predictor describe uncertainty nk influence operator letter restriction rkh gaussian conditional distribution know invariant evaluation invariant direct distribution know simplify stand gaussian covariance follow section focus involve various zero insight kernel gaussian property consequence choosing allow illustrate distinct see recover reflect integrate square error mean give equivalent satisfying ode show evaluation one soon distinct behaviour reflect ode b incorporate prior figure show prediction observation harmonic imply locate b experiment learn harmonic term increase order index quantify variable response distribution sparsity zero invariance operator map popular literature sparsity account parameter beyond sensitivity index close allow effect significant hereafter less setting four parameterize variance kernel parametrized expand see variation furthermore distribute accuracy perform poorly explain tend come back prediction least sum sensitivity index associate main additive conversely suit problem include ccccc log rmse q knowledge sensitivity since norm literature focus example homogeneous may cast invariance combination composition operator conceptually class describe function recent give kernel field additive path invariant group perhaps surprisingly path field restrict turn gaussian various isometry hilbert field reproduce hilbert theoretic random field section involve kind drastically design appropriate approximate perfectly assume improve avoid curse thank regard proposition remark control second result broad several include path path path promise composition g g whether rely field work root stream terminology g spectrum design consume simulation engineer theoretical therein gaussian random incorporated kernel kernel depend say e case informative much say field path subsequent behaviour neighbourhood origin field regularity regularity could refer thesis exposition concern regularity property sample path link rotation say extensively spatial theory concern field path main focus algebraic geometric field action multivariate path covariance location composition field invariant action extend krige characterization class lead integrable random furthermore particular case field cover link operator path general result characterize composition process invariant demonstrate impose interest situation framework simplify need curse successful multidimensional nonparametric become simplify ix possess arbitrary positive additive modification path k give birth lead generalization invariance property class additive covariance kernel modification additive ia da hold ij j arbitrary I additive field ensure correspond path composition operator remarkable restriction particular apply field argument take covariance turning combination kernel operator kernel object operator less define operator correspond generalize approach lead prop enable characterize rely joint concern combination composition operator cover possible covariance equivalence invariance
fx overview robust bayesian discussion west university minimax normal prediction asymmetric poisson conjugate mm exponential family parameterization modelling journal american mm bayesian minimax mm pe logarithmic divergence family statistic pe e conjugate exponential journal american association communications york minimax statistic decision estimate definition theorem department mb r mathematics invariant help formulate unified underlie analysis optimal invariant smooth intrinsic conjugate one usual conjugate distribution conjugate prior prior convex belong could theoretical keyword loss bayes posterior sample density distribution standard specify however never without error usually reflect approximately prior belief robust acknowledge uncertainty single criterion selection robust maximal posterior minimax parameter one criterion theoretical practical science collective posterior unknown rx measure construction attention intrinsic intrinsic true intrinsic loss benchmark loss utility relate practitioner desire intrinsic function property intrinsic tool application function use unified estimation obvious estimator natural exponential leibl distance conjugate automate unified exponential entropy point transformation distribution inference invariant necessarily one invariant intrinsic result one transformation general prior case class case connect sufficient estimator respect independent prior underlie prior finally conclude exponential fx pdf value measure kullback loss log pearson affect via parameterization loss exponential family family calculation monotone density function bayes regret obtain note xx everywhere q case f h conclude result normally fx belong exponential estimator conjugate prior conjugate q refer invariant lead proper distribution property eq first exponential intrinsic smooth omit intrinsic smooth unknown intrinsic transformation definition give use invariance property therefore exp intrinsic entropy application record refer similarly intrinsic obtain continue prior obtain class view theorem intrinsic critical underlying loss observe see example intrinsic estimator one check mean parameter x stein distribution give j example show pmf pi e estimator result function modification resemble similar nature binomial sake completeness continuity similar distribution depend belong connect class belong belong decrease function suppose dm fx di I let continuity show class distribution extension proposition al nonetheless appendix sake completeness define fx dl let set prior bayes class distribution suppose bayes
relevance nonparametric linear combination orthonormal coefficient representation jx equivalent letting explore consequence lp let u probable quantile fy important formula define median mid slope introduce powerful exploratory tool informative quantile call identically definition continuous lp x tail distribution long medium medium lp tail recall lp small threshold choose extend discrete moment lp monotonic short tail I criterion eigenvalue lp test expectation use true distribution expansion du jx goodness probable usually mass density du du g ed estimate sided sample equal fair chi small outcome population outcome joint joint discrete joint rule bayes probability common comparison density yu univariate lp copula conditional practice copula drive build lp slice copula quantile simulate conditional comparison quick independence rx feature wants identify classify rank one start em pt thm example science college pa moment lp mid science bayes united statistical big like polynomial skew great previously apply enable reference science mix science comprehensive copula discuss elegant united statistical elsewhere application distribution estimate method stand extension moment quantile mid orthonormal function build mid rank modern theory sample mid distinct scatter diagram mid linearly plot scatter diagram bivariate display plot test distribution difference density ratio orthonormal score denote transform schmidt power orthonormal polynomial gram schmidt four u u function orthonormal yield model orthonormal numerically function datum science name agree utility aim utility hilbert regularization formula answer include approach apply less unified big
convolution use c c cpu cross splitting output c c cross meaningful answer algorithm case stability lose sampling cdf performance exist numerical conclude rare convolution relatively fail carlo slow reliability application inferior propose grateful anonymous constructive remark suggestion convolution output look denote definition density definition get obtain follow bound nu b b l end nu diag nu definition figure proposition theorem estimate convolution mathematics institute extensively execution carlo formulas available unstable instability failure probability happen capable handle compare keyword rare reliability appear life social computer network communication mc require cumulative cdf variable undirected vertex edge terminal set failure terminal call handle main edge edge represent evolution monte reliability calculation note sum rest concentrate examine unfortunately note suffer instability code observe code test exploit rely main idea system rare sequence event sample recursively avoid rare event technique especially design calculation rest note organize relative present finally conclude remark exponential use ratio algorithm nx ny ny ny ny let unbiased relative algorithm variable size conduct numerical early practical purpose fail model
collection second row world experiment conduct server equip six core ghz gb ram gb draw conclusion solve c vs iteration usage composite tv regularize interest finitely finitely support generate accord intensity convolution kernel experiment option end exactly explain z necessary keep exceed initialize various merely indistinguishable wide result penalty plot mark experiment mapping heavily tv regularization moderate relative relatively c tv experiment experiment axis penalty combine vs include c en wikipedia wikipedia camera www com help ref experiment c run max mean b platform intel cpu gb ram solver q support grant dms application two convex smooth want cone case allow bregman algorithm assume relatively minimize intersection ball motivate nuclear entire symmetric norm image discuss capable handle theoretical lipschitz euclidean parameter interest quantify discrepancy candidate notably parameter priori priori cone covariance recovery property sparsity order type popular tackle overview among nesterov smooth composite compressive sensing algorithm theoretical estimate nesterov penalize iteration lipschitz set efficiency proximal algorithm possess favorable geometry well domain efficiency depend domain gradient geometry meet outline variation grow slowly meet application proximal become case violate norm denote rapidly high satisfy include norm image large norm limitation rely favorable interest wolfe smooth constrained extensively study e therein easy auxiliary arise gradient collaborative filtering study formulation solution hand algorithm formulation issue study although aim efficiency guarantee follow along assumption environment algorithm efficiency present cone loose linearly tackle solve tolerance find pair super formulation co find arise enjoy special shall situation norm space gradient induce take fit many discrepancy specify get fit take eq absolute choice discrepancy logistic side quantify magnitude context use obvious substitute sided analogue assume represent routine return minimizer equal suffice automatically oracle minimizer one minimizer due minimizer segment remain section present overview property conditional highlight new since key algorithm oracle routine ball recurrence build iterate eq implementation generic quantity eq q course good value summarize property eq conditional attractive property presence convergence establish run search point simple answer meanwhile modify oracle carry algorithm current select arbitrary along iterate belong iterate clearly per nothing implementation set auxiliary convex induce integer cost machine ellipsoid arithmetic call eigenvalue eigenvector outline life rigorous maintain auxiliary achieve computation easy auxiliary cf case read access precede usually computational overhead product solve inner method approximate inexact yield difference work stagewise bind since case minimizer option nontrivial origin positive f sequel refer induce every minimizer form lemma utilize explain apply iterate approximate low policy terminate option pass neither option place terminate stage specify affine due nothing construction low q satisfie select first new stage origin iterate terminate solution ii admit termination q sequel assumption define minimal important induce priori see point k x origin conditional composite recurrence build generic simple implementation generic give recurrence denote recurrence recurrence admit memory iterate gradient iterate since procedure implementation note basic discussion option preserve state practical let focus nothing hull far precede add easy improve assume specify augment inequality moderate arise explicitly low nearly problem mind solve also approximate solution value good termination associate efficiency feasible instance description subset restriction nothing allow easy add assume advance cardinality eliminate projection feasible onto space variable stand integer gradient take penalty truncate conjugate nice significantly implement efficiently attractive large space nuclear matrix type completion recover aim get recovery relate semidefinite want semidefinite symmetric experimental restrict norm symmetric aim building rank trace proximal decomposition resp decomposition symmetric may become consume oracle eigenvector large much computing computing decomposition consideration algorithm remain practically essentially proximal attractive stem situation yield composite rank provide iterate stage interpret image real image subspace comprise vanish complement comprise paper extremely variation basic recover image problem role focus replace immediate replace complement span fix consequently reduce algorithms ball albeit convex auxiliary scale account proximal hundred stem treat new set discrete field image dimension contrast unit reduce solve utilize reference therein state known reader orient arc arcs arc remain arcs arc arcs arc arc treat vector external external external question read q incidence problem feasible say solver return optimal flow lagrange multiplier subtract entry since entry interpret zero image turn nothing minimizer nonzero maximizer estimate end fit easy tight convert upper need estimate bound follow select note proposition sharp case analogy grow x ff f place note inspection extend appropriately preliminary simulation completion requirement parametric completion problem specifically norm entry problem method variable exceed count memory version algorithm memory performance version conduct generate density vanish entry ij ij rd diag I sparse observation
reason monotonicity insight work logic string logic implementation logic rely reasoning monotonicity lexical particular imagine might encode entity entity certain dimension entity argument behavior learn kind similarity behavior much entity might encode lexical helpful ability semantic task infer infer relation past propose seven possible trivial relation might hold yx yx universe compare relation none insufficient interest nlp towards evaluation distributional exist present exist monotonicity inference label test due lexical ambiguity syntactic ambiguity resolution possibility strict task interpretation provide ambiguity explicit structure involve hard oppose generic contain element reference dramatically simplify simplify like omit key logical deep aside logic conjunction minimal model logical linguistic though substantially modular natural logic engine linguistic call signature signature show give substitution explicitly substitution lexical substitution substitution additional relation series sentence compare build engine recent project representation inference limit build word phrase crucially base distributional line standard deterministic engine evaluate lexical substitution derivation use representation learn publish date center composition construction semantic merged phrase representation phrase entire phrase sentence supervise impossible detect relation phrase slightly depict phrase build composition phrase feed layer feature phrase turn relation adapt sigmoid nonlinearity function different nonlinearity sigmoid substantial label phrase mirror provide comparison pass backpropagation wherein correct node pass gradient composition tree pool training rate use start l sgd term add way encourage hope sort vector initialize uniform distribution attempt initialize corpus tune dimensionality produce run additional softmax top network additionally phrase follow yield detail appendix softmax composition composition vector build vocabulary intend diverse variety phenomenon lexical manually label need vocabulary predicate design six logical unary operator take annotate relation label divide constrain pattern lexical item sharing describe four table mobile mobile european mobile cat mobile cat cat mobile european cat cat cat cat set predicate entail dataset lexical position alternate second argument predicate pair position position category opposite predicate side dataset complex phrase argument create involve readily manually sample extensive six predicate side every systematically simple randomly make sure dataset portion remain correctly generalize reasoning quickly test capable accurately capture experimental logical accurately unseen kind three substitution still dataset cccc european european dataset see indicate target evaluated hold hold test none additionally broad reasoning pattern substitution word second represent hold interaction pair predicate hold last source dataset result experiment entire exclude hold perfect target setting poor performance target learn novel performance room able perform basically support show unseen difference lexical training learn lexical relation pair sentence substitution serve confirm underlying rather reasoning provide ideal able logic unseen somewhat training logic derive strict datum weak less informative exactly consistently set one whose relation infer train something logic help include long construction construction kind training set formal thorough logic lead help powerful phrase acknowledgment every project helpful discussion pilot additionally run experiment potentially powerful separately parametrize universal basic include sigmoid nonlinearity choose one argument phrase phrase argument phrase appendix cat unable mobile live european seven type pair relation train word lexical monotonicity reasoning could avoid give evaluate sentence truth irrelevant sentence
fw swap use ht cm swap swap fw problem accuracy correctly test require non detailed frank respect measure fw second swap swap swap difference denote accuracy quantify testing swap swap conduct ghz gb running bit implement source available web categorization dataset paper minimal svms instance collection approximately train amenable scalability figure report accuracy wolfe method dataset collection illustrate theoretical advantage fw routine propose competitive fw fw seem increase monotonically fast swap clearly large swap basic frank wolfe step guarantee step swap swap swap swap significant advantage prove result algorithm swap take swap seem toward fw swap outperform three note accuracy fw frank wolfe swap find small figure actually swap small finally significantly percentage seem decrease series problem derive census predict purpose analyze scalability method pattern rate b b vector collection scale confirm frank tend number large swap swap reach wolfe speed fw significantly collection large dataset swap dataset swap fw median conclude one fast fw previous remark away step towards face examine performance fast fw result fw away swap swap fast slightly frank wolfe obtain conclude incur b result time speed description find presentation medium dataset include first subproblem include group put together independently problem already compare train svms examine testing confirm swap swap difference fw swap fast grow around among frank wolfe medium scale dataset achieve scale fw swap swap respectively advantage method family kernel propose effective order kernel square distance figure dataset use accuracy gaussian kernel thus incorporate frank wolfe method demonstrate capability impose right running time obtain dataset contribution fw introduce novel step fw practical demonstrate effective state svm learner expand fw learning problem variant swap swap provide thorough demonstrate converge globally swap swap additional fw variant demonstrate useful svms swap fast swap outperform dataset medium problem swap slow swap fast swap swap swap magnitude order technique basic swap run fast dataset found swap fw statistically significant critical around fw similar fw swap arise improve amount step speed fw collection swap method competitive significantly fw step times instance swap clearly addition competitive fast fw swap away boost fw also away step swap fw point swap appeal swap significantly away step technique seem useful swap choice since swap reliable experiment come expense expense accuracy time swap fw accurate report statement variant perturbation approximate aim feasible stationarity fulfil easily approximate concavity remark detail lemma basis modify frank wolfe demonstrate eqn lemma yield taylor since concave matrix semi non bind absolute obtain exploit analyze objective fw improvement swap algorithm derive fw lie fw bound guarantee improvement swap objective function swap mark add g g k lead step case add frank wolfe q objective follow swap guarantee swap drop improvement macro corollary inf cl frank wolfe fw successfully scale instance machine svms fw training allow important analyze fw way step accelerate convergence fw analysis maximization simplex form geometry namely demonstrate number away enjoy form classification method classic away work frank wolfe fw scale several svms svms binary regression note researcher solution quite present iterative task minimum ball volume ellipsoid zhang technique estimation method simplex trace back frank fw move linearize relate whole existence svms formulation fw speak move linearize move direction linearize suggest wolfe fw method modify frank hereafter convergent assumption find classic away fw conclusion approach improve linear challenge interior circumstance admit large interior prohibitive cope practitioner widely librarie descent sgd specialized ascent gain large non effective fw method due label problem j iy fit formulation eqn exploit develop easily fw address problem quadratic definite definite whose component I c g base algorithm geometry author number I regard sdca required solution size training exhibit linear remarkable context cost testing addition allow linear competitive software efficiency fw method fw demonstrate time minor acceptable variation learn application fw endowed overcome observe classic preserve introduction away theoretical formulate demonstrating use classic away converge optimal achieve focus classic rate recent side fw fw statistically fw method fast equal fw significantly step addition competitive fw classic away step robust alternative organize give overview fw new minor detail svms provide discuss svm conclude addition proof report denote index simplex term indicate index indicate vector denote compute solution problem iterate follow current iterate perform order ascent ks approximation rest initial fw I k fw fw discuss stop optimum globally convergent rather weak guarantee svm iterate derivative however procedure amount per large context continuously wolfe dual problem strong another frank iterate multiplicative dual gap primal metric analyze former value explicitly fw give find recently also fw stop guarantee close analysis use introduce solution say condition face primal gap far gap compute coordinate face start previous remark exist even svm svms profile store svms non iteration classification idea property expand contain problem scale cardinality subset span respective solving variable index satisfie discuss special polytope general instead generally job fw solve consider bc computational fw know exhibit tendency explain geometrically tendency nearly orthogonal face span coordinate improvement moving improve span work optimality ascent fw move point maximize linear move face move towards away must lie active whole paper fw fw fw k k contrast rate exhibit addition potential compute since fw method coordinate arguably property fw method formulation equivalence normalization function exploit adapt core enjoy remarkable theoretical complexity iteration termination size determine search direction current operation search overall complexity measure improve super time report empirically train per still prohibitive obstacle train overall complexity thereby since detail speed technique explore original fw svms algorithm become significantly depend external solver within predefine work iteration adopt significantly present theoretical guarantee namely technique overall time complexity closely adopt approach polytope author theoretically fw introduce use soft svm obtain attribute gray variant fw svms comparable sometimes accuracie state art similar technique recently allow variant introduce svms stream fw svms structure compete structural svm solver svm obtain remark method suggest terminate fast experimental report ellipsoid problem improvement enhance svms systematically sometimes similarly clear fw look way implement away keep feasibility satisfied vertex th go toward ascent increase mutually exclusive around feasible vice versa feasibility considerably modify face need far new away step preserve discuss variant fw q face span however explore sketch scheme implement away conceptual preserve away vertex correspond spurious remove move away move ascent iteration superposition standard fw k component leave rest unchanged call algorithm represent simplify search j find g k swap add swap k mark perform dash fill circle fill path circle anchor north east anchor north west circle current black current node pos black thick current pos fill scale swap black fw circle circle thick dash triangle swap thick dash triangle current sketch fw swap ascent vertex iterate vertex current iterate direction explore solid swap fw weight descent avoid swap update predict denote direction swap toward swap toward swap prefer use problem observe method search select require search iterate computation analytical furthermore computation involve search overhead modify fw introduce toward possibility objective twice differentiable taylor negative finding good g highly order frank wolfe sense note direction swap step need three hessian adopt ascent improvement worth line ascent indeed semi negative naturally restrict modify expression iterate simple search perform fw ascent swap already vertex improvement objective analytically fw step swap q already procedure swap kernel svm value relationship swap particular start demonstrate swap analyze convergence optimum present framework objective swap linearly swap enjoy number stop coincide proof convergence statement appendix develop continuously sufficient impose frank volume ellipsoid general classical isolated locally case maximization objective fulfil kkt lagrangian behave definite belong kkt specialized simplex problem assume I linearly key stationary lie analyze fw strongly concave difficult guarantee simplex eigenvalue modulus matrix hold machine strong sufficient b satisfy wolfe svm convergence method match specialized also demonstrate gram matrix involve definite remark constraint variant key ingredient worth also start iterate problem key fw swap satisfy swap fw hard eqn also rest proof prove convergence swap use mark swap fw algorithm iterate immediately swap fw eqn swap purpose sufficiently follow globally iterate always arbitrarily follow improvement function quantity predefine iterate swap add solution hold swap add fw iterate iterate swap fw step swap convergent modulus hessian simplex swap fw swap drop step improvement compute subtract right equivalently thus swap exceed fw swap swap drop swap add sometimes clearly step step initialization combine proposition subsequence subsequence drop swap drop step thank affect iteration need prove follow suppose eqn swap fw improvement loop eqn eqn result converse last l happen termination fundamentally look iteration perform condition eqn fulfil dual gap improvement improvement swap fw multiplying come fact two swap fw finite iteration first iterate primal gap therefore iterate independent
scale obtain likely training obtained attribute remove l norm different purpose detection attribute novel ii help classifier attribute single purpose image salient level output patch merge weighted entire image grid gaussian filter pyramid give around scene level level pyramid rather use entire keep classifier get single grid level concept dimensional pose good linear capable generalise category learn concept entire instance represent scene use manually label web task attribute recognition learn require depict capture salient coherent among outli salient image salient salient hold depict parameter fix validation select fold end programming result south height font texture xlabel ylabel avg style style font axis legend mark red smooth xlabel font label style font font style font font axis x line none blue coordinate red smooth green xlabel font style font legend style line line blue smooth include colour texture texture chance semantic attribute patch label label learn attribute attribute dataset first image return eliminate google colour learn colour last dataset annotate imagenet attribute scene mit scene use colour concept also densely ref complete norm non overlap patch normalization crowd testing grid compare bl return train single result show capture intra som som cluster elimination characteristic perform well imagenet entire image comparable method attribute compare mit scene dataset perform short google imagenet ylabel legend font anchor north font legend area legend anchor none bar legend line none fill coordinate overall image mit scene li scene dataset scene category scene collect state art without require implementation concept negative classification refer office store forest ylabel legend style font east legend style area north legend column draw none bar area legend none coordinate class web model observe chart map som cluster respect use supervise concept scale noisy outlier classifier sensitive low good scene directly concept capture localize video attack concept automatically web go beyond colour texture label learn concept idea discover able datum irrelevant map train concept outperform learn competitive concept capable supervision label continue limitation scale object recognition attribute helpful share category novel recognition visual attribute label eliminate human effort yet may alternatively attribute name web challenge illumination scale pose compression importantly collection well attribute important attribute ccccc beneficial attribute propose irrelevant remove intuition category define although list irrelevant visually coherent possibly correspond semantic sub cluster sub category attribute attribute image patch unit provide retain attribute category correctly irrelevant may outlier alternatively category patch inside salient cluster patch outli element remove sufficiently improve som elimination outli salient generic capture category irrelevant instance go beyond attribute scene aim scene one irrelevant characteristic complex material focus recognition object label category semantic attribute annotation cognitive science shot attribute independent category intersection discover necessarily trait discriminative hyperplane margin locality evolve get epoch alternative windows som literature q detection som small unit learn scalar dynamically stage definition salient total cause epoch neuron normalize unit high whole period high thus capture low outlier via threshold range cluster unit salient belong category expect compose capture suppose characteristic calculation score activation neighbourhood activation salient category neighbourhood term share category neighbourhood salient group outlier salient namely detect statistic weight group winner winner box distance portion cover instance whereas capable discard learn phase phase calculate runtime purpose define variation
stack map vector vector cover formula could proper establish consider q plug give strongly norm tu tu u tu tu due proposition lipschitz iii establishing follow consequence finally observe utilize nice iii give special let nesterov te nice number identity two identity q nonempty nice identity hold subset cardinality nc straightforward identity follow therefore combine fix last fact restrict scope fix estimate nice outside possible notice possible nice sampling one select possible choice possible remain nice nesterov separable q useful q brevity nice add dependency necessary assume block combine finally w j utilize lemma obtain substitute remain nm nice nice theorem value quickly whereas increase slowly compare section see translate parallelization speedup large processor processor comment possible draw link hold apparent situation section quantity eq generic link value establish nesterov separable uniform l x diameter prox strongly convex recall quadratic smoothed problem iterate smoothed descent setup nice nice nesterov separability tolerance iteration counter strongly convex additionally decrease function encounter strongly case may generic bound theorem formula w nonsmooth smooth nonsmooth composite smooth setup iteration counter choose argue ii need satisfy fx logarithmic counter x fx assumption yield identical first x fx need argue imply e briefly comment strong convexity satisfy irrespective strongly cover exception minor logarithm probability iteration ignore nonsmooth function method dependence solution processor excellent theoretical parallelization speedup clear processor get nesterov situation regularize separability parallelization speedup nice depend change constant prox dual smoothed coordinate special comment example share intel gb code asynchronous generate parallel mn norm utilize nice method simplex method easily make suffer choose fast test utilize method sublinear optimal advance otherwise slow simplex core subgradient smoothed theorem core dataset paper fast version need one coordinate residual complexity proportional core divide test experience numerical smooth involve potentially safe suitable update prevent adapt deal suppose already x reasonably iteration prox center ji smooth ab decrease decrease parallelization speedup monotonic present define choice non convex case replace diameter form appear assume perform numerical exponential collect sparse maximum label spam setup parallelization speedup processor observe monotonic minimize serial directional nesterov minimize parallel adaboost trivial decomposition optimization variable parallel version study generalize greedy number processor nice sampling big detailed apply adaboost processor depict processor processor fast nearly parallelization speedup demonstrate view look see parallelization processor additional effort processor increase little processor thm corollary proposition thm definition thm study parallel nonsmooth adaboost define sparse iteration fast notation quantity level need fewer average single per processor decrease need variable coordinate huge historical reason almost enough unable iteration inversion multiplication expensive instead attention iteration requirement parallelization scalability accuracy requirement moderate constraint constraint exist recent propose optimization convex simple coordinate assume partitioned update subset block formally mapping encode variant law paper characterize see uniform additional choosing improve mention assumption certain inequality smooth inequality take identity vector denote block belong otherwise say admit write give intuition current right quadratic block separable draw describe algorithmic move new point dimensional quadratic problem random subset iteration compactly interpret issue several composite admit computable simplify rise computation associate separable h x h w take utilize obtain turn satisfactory large hard try gradient precisely characterize directly translate speedup factor easy good amount finding size compute technique naive would unchanged fail sub since update describe decrease stepsize infer inequality method safe jensen must serial notational one coordinate q stepsize mean parallel serial counterpart separable subsection issue progress cast result composite initial iterate optimal keep value intuition bad dependence parallelization speedup occur big solve outline close decrease differentiable uniform nice bad means implement nice phenomenon relate time study converge case strongly prove nesterov regularize box nesterov nesterov analyze different constant capture coordinate simplify give coordinate descent method improvement et composite neither composite coordinate descent inexact coordinate proximal subproblem iteration mirror nonsmooth accelerate zhang develop coordinate frank method al nonconvex block ascent descent method know quadratic block nonsmooth part linearly section composite problem result show choose processor utilize primal develop et develop mini stochastic primal machine loss ascent method concave maximization naturally extend zhang give serial descent early parallel parallel review nesterov smoothing contribution nesterov separable span block inequality derive finally preliminary section setup ng product coordinate general qp know differentiable nesterov seminal reasoning solution minimize strongly minimizer write nesterov continuously eq maximizer continuous direct dual part devote replace easily computable interpretable depend small decrease extension side replace give computation inequality variant smooth parallel utilize tool lipschitz gradient primal dual space matrix define mention useful I q proposition analogously last chain inequality get fact aware complexity smooth utilize argue quantity parallelization compute importance datum discussion weight time formula block increase close surprisingly formula separable recall although function give formula term parallelization speedup lead summarize table c c problem thm complete say take least probability logarithmic reason easy parameter define convexity diameter iterate observe decrease grow speedup indeed decrease separable partial separability cost simplicity assume block parallel take operation k z maintain vector iteration cost loss entirely dedicate serial discover interpret randomize boost separability machine often suitable nonsmooth smooth composite method iii provable parallelization real sparse framework problem involve preliminary scalable theoretical parallelization hold formula constant lipschitz respect separability also interest smooth alternatively inequality lipschitz idea block iteration constant relevant possibly much well block constant nesterov case space subspace lipschitz subsequently dependent define generic form nesterov separability collection nh h nh ni fix scalar norm euclidean primal block refinement consist section primal norm respectively since one constant conclude gradient constant substitute identity view nonzero
location hypothesis role sequence may rejection sequence comment try limit testing subproblem e obviously latter problem intuitively appeal robust seem sort assumption regard rejection rejection coincide rejection probability concentrate whereas nan modify modification substantially affect rejection rejection continuous point differently rejection probability completely completely matter fact closeness especially continuous everywhere experiment converge odd effect rejection probability rejection limit however amount reproduce argument really cf location whereas severe arise next commonly autocorrelation nan ty square ty data satisfy toeplitz nonnegative subsection assumption weight nonnegative everywhere product nontrivial certainly allow truncation lag extension result dependent bandwidth satisfied g lag window lag coincide time rectangular discussion lag window typical sort test negligible surely positive definite event concern normalize also accordingly assign singular power absolutely fact definite circumstance every design let basis rx matrix depend assumption singular singular equivalently entire violate condition singular almost everywhere choice violate precede satisfied define singular property violate commonly break completely trivial way force design matrix autocorrelation test sense matrix lead thin matrix full take either consistency discuss robust apply autocorrelation infinity assume hold hence hold every suppose hold rejection every depend supremum way part nevertheless include rank inspection ar two sequence modify singular accumulation see restriction vector weight decided proposition generic one condition imply autocorrelation power part want e part precede theorem essentially respectively ii respectively put mass modify conclude put sense w claim work concentration discussion subsection exploit rejection probability constitute ingredient cf satisfy implication contain intercept assume correspond first restriction I nonzero apply whenever weight hold hence satisfied arise even hold applie show get certain part odd conclusion apply example lag window odd equal case nominal significance power guarantee simple often use try autocorrelation location play example slope able equal arise case precede column else satisfy test easy furthermore odd detail exhaustive formal value satisfy universe part impose regression intercept subsequent proposition important thus hold assumption define choice nan k kk zero e every satisfie trivial proposition matrix contain algebraic regressor draw absolutely w proposition theorem rejection almost next discuss covariance assume singular sense guide ty e satisfy rejection satisfy whenever hold statement away suitable supremum easily exploit theorem bound away power involve precede constitute column hypothesis regressor necessarily precede extend adjustment suppose apply furthermore necessarily satisfy relative nn satisfies restriction e nn define restriction w ty ty ty tc conclusion hold replace severe isolated case make version statistic amount statistic working add regressor regressor harmonic angular fact restriction coefficient regressor lie heart express elsewhere illustrate applicable typical satisfied adjusted whenever hold showing suffer severe power satisfy assumption adjustment apply theorem apply except apply adjustment procedure extend covariance accumulation behave see elsewhere subsection estimator weight spectral much focus partly consequence estimator inferior certain final estimator belong estimator modern class narrow class weight definite inspection remain stand replace nonnegative case nonnegative hold definite arise first singular statistic solely restriction verify ny inspection condition obvious change xy already definite zero latter tu replace regressor consistent regressor replace empirical moment variant omit detail test weight lag window like general subsection accommodate arise employ case call flat value allow cover result subsection test parametric reference therein fall extent admissible singular autocorrelation course theorems space subsequent discussion concentrate discussion carry discuss subsection remark negative size equal nuisance rejection really restriction strictly large reason motivation development autocorrelation misspecification correlation restrict interval positive emphasis unit process seem see discussion mention however close design provide precisely happen power problem moderate relatively inspection show respectively continue ar version theorem apply illustration involve intercept sense conclude note cover special intercept regression test intercept immediately satisfied require mention assumption require cf appendix ar ar impose covariance especially furthermore I size extreme still appropriate regressor back modification apply point iv regard precede discussion iii theorem large without away without desirable achieved discuss precede elsewhere distortion belong nan hypothesis harmonic subsequent remark sequence weakly harmonic choose maintain ar allow subsequent possibly random treat result test however assumption sense special meaning want could vary independently covariance size power large recent paper density irrelevant context autocorrelation location compare standardized help test frequency zero robust infeasible ill certainly fact statistic say unknown observe behave reason uniformly helpful sense principle close close ideal uniform closeness test wrong thing sense ill estimate irrelevant statement contrary discuss estimating parameter interest consequence ill paragraph question uniform closeness immediately model singular phenomenon theorem nonparametric satisfy arise case additional singular matrix condition equal equal furthermore appendix arise limit restrict theorem albeit ty exist particular one hence hold hold nuisance equal particular element matrix submatrix regressor generality assume angular correspond precede obviously absence way note finally e additional concentration space newly arise test encounter equally parametric even parametric employ describe structure test statistic ar well square ol use feasible ol shall square estimator away modulus domain value set ii estimator exhibit ls l behavior two guarantee cf mild condition require inversion singular k estimator q appendix furthermore q ols n appendix appear n matrix negative odd everywhere except everywhere event go finite fortunately subsection almost everywhere weaker satisfied formalize appendix region real biased point zero exist replace constant replace mean mean appear portion differ somewhat early tell give condition alternative far nan power difference respectively nan require resort precede subsection little ol high parametric expect autocorrelation autoregressive also severe precede reveal serious problem correct property correspond infeasible base standardize counterpart precede estimator restriction hence condition satisfied remark precede analogous design satisfied part subsequent satisfied regression contain quantity etc write follow suppose fix statistic proper apply hence precede hold view fact differ suppose e depend proper e nan precede negligible xt ols analogously apply well precede maintain e conclude set applicable statistic satisfied must critical theorem statistic see satisfied choice critical comment apply part proposition satisfy choice actually want subsection next result least model assume singular sense rejection test region power away eq hold whenever statement hold condition apply obtain application theorem obtain adjust property precede test suffer extreme adjustment mechanism regressor provide detail introduction considerable concerned e correction autocorrelation autocorrelation follow bad demonstrate limit test intercept restriction test involve intercept belong contain intercept belong observable ii contain intercept involve intercept belong span claim argument incorrect perhaps autocorrelation correction exhibit behavior mention test ratio direct quite method consideration geometric test autocorrelation perhaps analogue establish test tu adjustment analogous size test break much adjust moderate commonly robust test unknown thus element covariance choose equally replace g statistic later depend design typical choice denote diagonal projection matrix overview three convention hence irrelevant subsequent subsection satisfy complete matrix note singular equivalently violate precede analogous omit statistic ty hold equal biased nuisance particular difficult show choice design condition theorem negligible set omit formal nontrivial together exploit space e corollary already variance order uniformity hence concentration bound robust substantial nevertheless relative discussion insight result mention allow sufficiently briefly discuss test eq obvious theorem hold variant drop size unknown equal nominal line note apply e test suffer severe power square standard size give applie structure provide section invariance property play result next subsection relate condition highly concentration provide condition suffer subsection result subsection test correct autocorrelation literature derivation exploit group every belong invariance coincide invariance imply super furthermore satisfy ng ng ng may artificial context problem family cf lr regardless transformation borel satisfying n ng g equivalent invariance indicator invariance super group affine let arbitrary affine g also write see composition singleton invariant act make let transformation denote subset close composition trivial singleton group group statement convention denote least normalized square residual test definition consider imply consist probability clearly invariant useful continue replace definite invariance property subsection see proposition rejection sometimes symmetric assume borel every probability r nonnegative consequently part relation rejection invariant hold fix constant along translation pass choose invariant proposition note part rejection recognize maximal arbitrary establish function arbitrarily theorem converge concentrate interior measure satisfy rejection reasoning reasoning concentration remain part consequence invariance property weak convergence inclusion possibly nan alone allow mp find theorem effect see concentration reasoning course crucially expect mind way neither r n n hence expect condition subsequent satisfied assume concentration test form borel measurable statistic trivial case borel measurable suppose furthermore function invariance immediately statistic everywhere equal statistic assumption ii borel measurable test satisfy assumption part similar applie invariant sequence condition suffer extreme power apart element invariant see proposition subsequent theorem covariance typically always satisfied trace maintain singular application verification given define covariance subsequent remain valid almost almost everywhere almost subset sequence converge subsequence number complement size I mn depend away inferior every specify hold whenever last hold precede strictly power space constant space condition ii condition every eq inferior even without part one significance result statistic clearly nan thus significance level assumption theorem sequence almost equal irrelevant immediately satisfy condition boundedness theorem reduce e express appendix case sequence converge singular suppose accumulation converging limit nm covariance theorem satisfied precede derive result contain vast literature typically estimator useful estimator symmetric set estimator yy k n first invariance express empty otherwise least contain arbitrarily nan hence n keep strong note note empty test assigning note w lebesgue leading almost everywhere guarantee definite accommodate contain almost everywhere nevertheless say real satisfy follow shall much weak everywhere imply condition certainly everywhere seem rule allow lebesgue statistic sequel satisfied statistic hold element test invariant rejection ty ty ty guarantee empty rejection empty consequently rejection mn accumulation vector depend nonnegative almost everywhere assumption subset unit ball unit assumption expression positive hold assumption n test suppose invariance every rejection ty yy also ty corollary negative imply continuity property verify become simple relevant cf follow corollary assumption c hold simultaneously simultaneously everywhere biased trivial rejection I zero nonnegative hold almost everywhere every particular hold lemma hold apply rejection probability nan derive correlation apply precede empty intercept obtain column nonzero theorem typically hold almost everywhere remark statistic ty rejection sequence singular md column basis na suppose covariance e cf ar simplify subsequent tu class satisfy ty c rejection subset every converge singular subsequence number sequence rejection away almost everywhere every definite condition every equal remark analogous remark ii part suitable satisfied coincide exist satisfy satisfy show crucial fact p replace nan n invariance analogue satisfied square denote way first vector add obviously rise alternative feasible estimator estimator part precede choice maintain proposition specify concrete done case autocorrelation see enforce unchanged design particular tell conjunction auxiliary severe adding regressor implementation except imply proposition satisfy hold except suppose apply finitely element invariance r element equivalent remark note empty second consequence arrive part iii estimator construct originally regression apply alternative auxiliary define analogously state satisfied statistic obtain enforce apply versus recover already note result immediately extend impose assumption assumption weak ensure imply distribution less trivial extension unit sphere variable square freedom situation invariant group hold nan hypothesis obtain immediately concern rejection well e everywhere nb due immediately part remain rx I singular hold satisfie x singular hence next case rx rx ny suppose definite everywhere define group rejection x w w well satisfied lemma definite assumption satisfy claim proposition know view corollary space apply note translate zero lemma define x operation obviously however orthogonal equivalently write x x nb rx xx nb rx xx multivariate vanish nan e argument algebraic orthogonal cr example invariance everywhere lemma precede satisfy five hold suppose verification analogous part lemma polynomial algebraic thus analogous inclusion trivial establish establish description continuity obvious away modulus remark provide polynomial submatrix dimension iy j ny view estimator transformation estimator define n n nan set iv nan invariance upon observe define e inverse give diagonal next establish define n follow third fourth proper cf rx yx view complete value n yx view latter nan display upon define hence establish identically yx definite define establish remain nan note well n rewrite display polynomial upon nan trivial continuity property obvious prove union multivariate polynomial desire consequence satisfy similarly set last claim assumption assumption consider positive definite assumption satisfy n also definite coincide establish arbitrary polynomial appear zero rewrite multiply verify view conclude concentration establish corresponding size note satisfy require satisfied view assumption formula nk analogously remain claim hold satisfying multiplication side power include multivariate observe condition equation precede display polynomial equation power result equivalent obviously polynomial show set polynomial take fx e gx rx hand multiplying equation suitable equivalently equation nan mention suffice column linearly orthogonal complement span e rx e x equivalently eq algebraic provide claim similar contain algebraic hold find orthogonal assume equal e rx ols ol ol equivalently ols nan maintain proper ol ols ols ar ar tu definite lem extra first satisfy definite everywhere precisely concentration apply elementary give last side maximal invariant discuss formula theorem invariance ng equal show e establishes argument sign immediate dimension consideration rest r r full observe proposition similar calculation together multivariate gaussian measure definite converge may singular converging satisfie sequence converge nan w n property eq e w borel converge definite total case unbounde positive indeed lemma ii use almost sequence positive symmetric converging hold sequence real matrix md regular q ms result choice invariance addition invariance integrate r normal product observe lead combine display typically proposition dd converge scaling essentially automatic finite along suitable typically invariance property suffice order end converge subsequence de observe invariance reason infimum equal almost everywhere sequence converge first definite pass subsequence pass lemma claim part find expression differ converge subsequence subsequence assumption theorem sequence necessarily converge onto proposition limit inferior definite next leave differ sequence converge define hold limit inferior note use rejection cf appear exist eq suffice assume subsequence far necessarily subsequence eq monotonicity hold together remark subsequence k subsequence show k along former close closed nan invariant immediately immediate consequence invariance establish finitely continuity observation coincide open imply satisfy measurable invariance vanish unbounded consequently lebesgue hence first n ty n c ty ty invariance assumption ty n c ty x n coincide subsequence eq since p imply conversely subsequence eq ball assume argument ij assumption inequality part part sufficient condition analogous note statistic everywhere q part let converge eventually argument since definite continuous ny part theorem remark define md standard variable else well hence repeatedly shorthand b
stochastic tune achieve adversarial set assumption adversarial analyze minimax regret iid zero cumulative function exist parameterize subroutine batch v tt let assumption exist adversarial characterization minimax regret appendix bound average incur presence various change cost structure access range multiplicative range constant policy practical policy par fix heuristic chapter performance oracle sake nevertheless proof hold small scale regret sa refer stationary complexity coincide scale differ feedback summarize minimax budget c c strongly convex gradient growth scale highlight occur relative non stationarity effect regret go inaccurate variation budget knowledge budget prediction denote agent hold implies real estimate still naturally performance dominate cost sublinear guarantee long sublinear order rate epoch variation interesting extent design variation characterize characterize constant important open importance proof appendix theorem policy use subroutine proposition side conclude proof horizon batch fix batch first regret batch good batch analyze good adversarial batch decision batch epoch epoch epoch jx jt jt sum batch decomposition conclude restrict limit select beginning specify define horizon perhaps cost change batch sequence minimizer interior point hold maker observe tx leibler feedback structure constant proof appear follow distinguish f begin discrete throughout epoch history available begin clearly take expectation hold gx bc last conclude let subroutine performance relative action adversarial follow analysis obtain select proof subroutine see select next appear proof convex step accordingly analysis note batch except specify set minimizer interior inequality batch fix respectively accord discrete inequality theorem hold c c establish subroutine proposition selecting obtain part notation feedback step batch noisy sake consistency non propose first analyze structure noisy noiseless access feedback former action set eq iid denote euclidean take respect one inequality taylor convexity te te te te db expansion tf txt q hence estimate contraction take take expectation jensen summation hold epoch epoch h depend solely give exist feedback single xx substituting taking follow euclidean projection take expectation sum use establish low good adversarial rate optimal convex cost bind match establish careful convex quadratic select differently horizon nature draw uniform discrete apply throughout take expectation proof theorem algorithm bound kullback divergence conclude proof online theorem draw throughout horizon notation throughout deduce horizon proof one measure incur change cost different achieve subroutine without size policie sa practical chapter cost horizon begin draw lr decay pattern sequence independent variable standard deviation consider tx noisy last batch sequence step similarly consider sequence action epoch regret dynamic tx dynamic tf tx refer apply fix pattern action feedback calculate regret relative structure also include table feedback fit percentage policy average r step step fix step step average policy epoch one representative illustration epoch subroutine tx b decay b decay incur epoch subroutine incur feedback subroutine capture consistent bound range vary observation multiply constant value close dynamic grow surprisingly policy consistently policy consider right step variation pattern size may setting outperform various heuristic relative arbitrary variation policy well policy setting less percent gradient percent access outperform sa policy policy tune optimize least par policy consider pc stanford edu edu along term extent budget achievable average refined connection optimization traditional stochastic approximation paradigm set derive policy leverage quantify mathematically capture versus stationary stationary regret sequential select typically compact incur priori convex subsequent maker structure noisy realization cost assume reasonable expect incur terminal epoch constant work stochastic counterpart study focus sa abuse terminology area publication seminal paper diverse operation engineering science cf book survey sa almost note seek sequentially optimize bring fundamental question primarily temporal enough capture still mathematically tractable performance stationary epoch decision maker select action observe feedback particular paper canonical structure minimizer move natural measure stationary generate performance know function advance hence minimizer dynamic oracle become constrain temporal change introduce concept set budget eq speak one time next add horizon function minimizer variation allow horizon measure scale variation purpose analytical key insight far formalize notion dynamic role select sequence maximize dependence order characteristic policy eq characteristic run average incur period approach incur benchmark among require refine minimal signal temporal set achieve minimax multiplicative essentially good qualitative insight necessary sublinear show sublinear admissible conversely average notion temporal uncertainty support characterize order optimality non sa characterization deriving prove suitable policy essence minimax either strongly gradient specificity stationarity thing stationary environment latter former uncertainty regret c convex noisy minimax feedback signal stationary environment mark degradation general cost explain paper meta construct insight construction policy run rate optimal stream call adversarial framework select action maker constitute traditional pick priori hold nature subject typically relative coarse benchmark know single static pick observe nature choice typically policy action admit oracle establish former environment meta principle policy good adversarial adapt guarantee stochastic subject constraint regret adapt sublinear emphasize policy admit identify counterpart date stochastic include note say policy relatively traditional stream several include work cost minimax consider cost show minimax verify minimax order temporal sa chapter literature mostly machine community namely static idea origin development make cf literature largely focus either convex linearity policy variety function feedback provide gradient evaluate observe class see feedback access derive static dispersion nature restrict reveal maker action significant distinction concern benchmark formulation ex post static feasible benchmark minima change throughout time oracle single significant illustrative example policy static adversarial framework world policy environment change worst possible reaction argue operate establish stochastic framework propose herein notion budget correspond uncertainty action concept robust predicate see research optimization typical objective minimize square error estimating dynamic overview survey application characterize extent stationarity may sublinear dynamic benchmark particular whenever sublinear sublinear oracle variation sublinear literature kalman filter typically fall latter characterize formulation literature filter importantly consider constrain particular work literature consider concrete embed sa observation dynamic pricing approach application arise wireless communication area see overview study mention underlie setting may occur paper say consider pricing absence demand demand function accord know demand unknown current suggest broad sense current study stationary setting establish connect achievable policy adversarial linearity latter main strongly setting present conclude remark find appendix online already idea formulation purpose fill gap exposition need expect keep empty epoch let action tf tx access denote possess uniformly conventional tx variance counterpart tx vector admissible space k dt u f x feedback noisy mapping policy depend past history action allow dependence sequence nature mind restrict element sup q bound hx decrease normalization purpose refer variation primitive admissible budget epoch variation restriction evolution temporal rate pattern consider minimizer measure characterized variation assume select formulation nature bad sequence function say minimax regret guarantee independent unknown efficacy time benchmark good static action throughout notion admissible long optimal good regret expectation respect randomness distinguish distinction nature advance definition next benchmark target static action f tx hence oracle adversarial example suggest linear nonetheless online context operate non environment explore question constrain world formalize achieve constant exist admissible proposition state variation budget admissible must circumstance possible oracle mind variation budget sublinear sublinear achievable minimax sublinear set rich might significantly rarely sequence change infinitely policy achieve action consider formalize refine adapt generate epoch history tt batch repeat x analyze via achievable use feedback subroutine meta principle whenever sublinear horizon optimal achieve sublinear adversarial signal theorem sublinear single action achieve sublinear section surprisingly carry optimality connect environment good use subroutine batch describe argument lies analyze difference benchmark sa decision horizon batch possibly batch batch respect benchmark sum first side performance benchmark dynamic batch functional change locally budget intuitively subroutine sequence oracle balance tradeoff principle feedback noiseless access function natural question arise non variation rate develop problem achievable fundamental bind performance assumption feedback counterpart random cumulative impose feedback property available epoch structure impose part gradient eq random quantity kullback leibler establish consider way begin batch cost batch tune draw maintain enough batch yet sufficiently formally divergence admissible try achievable enable set set subroutine adaptation input decrease f tp value procedure mapping access yx achieve completeness appendix improve next consider subroutine constant recalling regret adversarial sa essentially direct al provide balance proper selection track good within dynamic action get bad horizon good action note initial initial take one last bind speak characterization budget
rx rx correlation discrete pearson mid transform pearson equivalent student mean equality parametric parametric model function simulate distribution quick quantile denote identical mean fit estimation probability simulate comparison diagnostic tool plot mid quantile always continuous true mid quantile define define median mid mid give symmetry medium short portfolio example see quantile estimate estimating apply corollary call quantile hazard quantile htb integral distribution theorem assumption true functional say norm goodness distance statistic limit theorem comparison comparison skew literature suggest argue orthonormal note orthonormal series estimator guarantee applicable gold du du estimator equation du aic bic coefficient fit u concept moment give moment j score definition interpret moment small conclude symmetric constant statistic diagnosis moment lp tail normality test x ratio deviation prefer conduct test use significance pre fx copula copula discrete copula density mass density copula mid copula indirect nonparametric derive orthonormal gram schmidt power figure cubic lp age one show strategy utilize compute display htb copula reject simulation figure extreme quantile biological classical go description quantile give age tackle non population define step mean big unconditional property mean conditional unconditional statistic sample hypothesis statistical mean pearson statistical solve sampling sampling distribution probability variance calculation assumption variance variance bayesian posterior mean confidence frequentist discuss population symbolic random inverting representation mean quantile mean n case think index sequentially divide call analysis n estimate normally distribute student degree variable observe sample formula unique value py py yy definition area curve sort successive variance verify adjust variance adjust define package variance simple formula application traditional combine simply observe variance want formula combine ny verify complete recursive combined mean consist ny verify represent ny normal conjugate prior formula state usually algebra interpret mean combine n mean omit pool write freedom observation interpretation straight scatter diagram xt mx equivalently xt xt mx xt important mx x value computation conditional student two mean population test population rank pool statistic equivalent may prefer distribution lp one score comparison type smooth classify function alternative model logistic provide start master practical implement high dimension approach markovian graphical remark ex em unite big college pa abstract big big united framework comprehensive traditional datum datum goal quantile age quantile mid mid quantile mid informative theorem linear function score dependence series comparison copula combine theorem quickly update formula extension traditional high mid mid copula lp orthonormal moment lp dependence classification em present statistic science interpret building advance old omit rise topic tool big idea include hilbert information regression nonparametric rkh especially modern job exploratory emphasize science understand scientific mechanic programming answer question apply broad traditional traditional science application idea model paper
log black generate log plot show feature b histogram learn random map plot obtain randomly mnist class component role linearly project random map propose accurate behind eigen kernel two sketch feature histogram exponentially function project linearly dimensional maximally capture eigen structure randomize fundamentally linearly project map tensor product generate improvement tensor sketch red green plot demonstrate recall hoeffding central inequality polynomial p r hoeffding therefore improve focus error significant improvement vector euclidean gaussian universal constant let mean decay turn determined verify equal moment moment feature map assume quantity inequality bind tail fix early assumption apply inequality valid reach high hold simultaneously trivial let preserve high pairwise product bind fix base dominant spend compactly oppose gain complexity straightforward would projection since gain improve random matrix way hadamard set basis random basis structure hadamard enable multiplication operation structure hadamard generation hadamard directly incorporate modification give function need zero close multiply entry implicitly hadamard row finally figure solve output example binary specify evaluated compute assigning projection representation regressor perform fold dimensional project pass error reconstruct plot obtain fold provide improvement degree consistently range polynomial error versus representation different consistently improve mnist substantially project feature explain amount mnist feature however use reflect substantial classification gain achieve sized highlight usefulness memory mobile phone show result feature consistently improve length cm c ts cm cm ts ts ts ts cm ts ts row h example vary set use map converge fast compare test scatter hessian map tensor heuristic record use core significant towards x project increase become dominant naturally encode compactly training approximate theoretically present map reduce gradient effective way large compactly capture structure mobile phone section theorem claim conjecture approximation randomize gain lot identify polynomial utilize project challenge accurately error demonstrate superior efficiently implicitly non explicit solution hyperplane classifier consider vector unbounded growth result increase mostly focus low distortion inner map vector sample randomize applicable approximate analyze well matrix straight modification structure approximate multiplication formulation concept geometry different recently
regression yield bias suggest effect individual threshold formula express use logistic threshold lastly variance equal transform simply divide population phenotype individual phenotype threshold value threshold plug equation supplementary lee block individual genetic variance distribution matrix within genetic perfectly individual case include individual accumulate process times accumulate lee result highly degenerate capture lee block genetic correlation lee simulation positive magnitude depend small typical environmental reality closely intuitively lee simulation individual full generative snp randomly effect phenotype phenotype automatically include study individual individual accumulate compute normalize genetic correlation phenotype run ten repetition combination note estimate observe less notably lee underlie h kp estimate lee dash see estimator correction early yield kp method figure completely study case yield unbiased simulation method unbiased similar lee al snps fix correlation batch lee realize correlation phenotype generation early simulation result estimate unbiased simulation determine realize genetic since around genetic might decrease initial simulation still unbiased snps small display h utilize genetic lee validate correctness estimate estimate bias latter easy simulation result underlie seem correct additional correct differ due strong therefore slope slope depend tuple box cox indicate h slope use relationship top correction behave seem apply correction reduce scenario unbiased term h correction wherein true underlie correct I observe correction derive apply top publish deviation heuristic correction study correction also apply correct estimate highly correlate see publish lee lee lee estimation web correct phenotype k correct ed ad par lee estimating lee lee web al genetic latter since causal often different q genetic bias correlation realistic yield normalize phenotype plug increase al detecting due snps miss control snps control group snps display difference exclude remove appearing list degree individual european reason remove individual rate step individual lee vector attempt due note thorough removed effect second decide principal association however assumes sample study structure overcome snps control variant component every snp expect agreement tag highly phenotype tag actual university supplementary repeat online population threshold genetic correlate environmental assume genetic variance person threshold phenotype include study phenotype proportion case greatly indicator study use assumption relaxed simplify study yield denote proportion control study eq individual phenotype depend long multiply probability health study involve derive individual denote phenotype obtain q conditional full select hence get numerator wish phenotype therefore remain possible phenotype multivariate gaussian eq determinant cc require derive last expression yield expression whose exponent denominator exponent derivative l q slope k satisfactory taylor already eq individual genetic correlation study use compute derivative phenotype might exposure environmental risk projection
payoff value state conjecture game asymptotic conjecture repeat inform player observe observe particular sum symmetric game receive public game particular inform player contribution repeat symmetric go hope existence asymptotic repeat indeed player influence payoff moreover player observe payoff addition hierarchy belief play player know belief second repeat information provide blind repeat game symmetric player blind player observe sum provide alternative zero game concern set play easy analyze explain neither converge section class repeat game element player signal resp resp proceed know player payoff stage player receive public continue repeat introduction measurable player possible j map player resp resp resp strategy induce kolmogorov uniquely respect payoff stage resp minimize game player payoff define resp np converge see pointwise game symmetric conditional player represent current state triplet role game player current receive belief k belief state game transition p strategy vice versa state map refinement payoff depend player control stationary player main describe observation discount discount equivalent game player discount equal control lastly reach transition pt node style font scale text center text width cm text text draw width circle draw text cm text text center draw center circle loop loop node node b edge b loop b adopted go td vice versa player expression transition game belief current play bayes belief resp play receive state auto thick font text text text center text center circle center text width center bend near bend right player action player resp bayes resp receive resp belief state resp transition describe state auto node distance thick style font cm circle width text width draw cm text center draw draw width bend right reach probability informally player want play immediately game variable denote resp independent success thus random order study combine get strategy let dominant optimal xy n dominant strategy computation maximization receive reach player choose player rr f reach unique strictly numerator equal sr sr h r go discount player outcome eq converge q contrary player opponent player dynamic make formally lemma converge state play reach small guarantee v argument show v false present game perfect construct belong neither idea completeness repeat game supremum inequality give take inequality bound able repeat action payoff action state r describe convention simplify transition transition control auto node thick main draw font node text circle draw width center center b text draw circle text bend b node player analog replace convention argue mr mr mr formal subsection moreover play transition see resp difference risk player risk game moreover induce proceed exactly subsection sketch proof enough deduce deduce adopt follow call derivative evaluate analogously proceed step mb c show asymptotic expansion since likewise computation omit dependence b f go numerator omit last large enough computation also pa let inequality enough computation prove r converge let nm nm r r nm payoff bound lemma theorem nm nm let go deduce q go go lemma inequality eq prove game might flexible without change provide asymptotic blind repeat game space transition c l c model player player q player moreover game induce discount inform state player player past action space set payoff c replace state state change follow strategy player play play state proceed asymptotically compact introduction relate state j control let pure strategy resp correspond play reach stage take payoff
model course normally specific mean parameter assume popularity clustering single grant national institute environmental health computational mixture eq dimensional mean inverse gamma prior mi sample allocate gamma kk kf mn I cluster run cluster source equally overall separately beta need posterior recall object belong eq generally overall specific clustering generally illustration assume skewed inclusion surprisingly inclusion genomic present I tumor origin first point estimate credible level draw credible mcmc draw appear converge quickly stationary converge approximately average I marginal overall probability mcmc draw converge ht article compare cluster four compare clustering table I respectively biological profile case partition association fisher association drive c ht ht c show respectively process column group apparent variable ht task source several object clustering consensus source motivated heterogeneous breast tumor cancer software available research heterogeneous mode measurement domain multi diverse source biology abundance activity spectra science text document cite article broadly integrate heterogeneous expand rapidly genomic collect genome collaborative collect genomic comprehensive cancer molecular biology section breast separate lack association extreme heterogeneity may capture exploratory alternative demand motivate statistic machine article exploratory tool hundred literature cluster source integration clustering furthermore object agree consensus see consensus multiple dataset attractive specific yet determine overall stage perform entirely clustering follow hoc phenomenon exploit expense recognize feature find clustering maximize likelihood source use gene dna cluster goal association source spirit framework source simultaneously explicitly dependence strength model specifically source rather elaborate distinction dataset give object goal multi dimensional allow probability may mean draw component component correspond assume parameterized put standard overview dirichlet accommodate available parametrize give random object represent overall object source clustering specific clustering serve assume control practice hence equally latter useful object belong application rule conditional define cluster represent generally source clustering represent intuitively allocate source p number source source integrate clustering give simplify equal dependent control association clustering appendix restriction surprising clustering bayesian estimation introduce conjugate prior choose default simplex practice markov chain conditional mn c mn c mn mn suitably modify realization clustering clustering facilitate interpretation cluster aggregate sampling clustering equation interest improve efficiency dramatically mcmc complete full distribution burden increase bottleneck method determine consensus differ consensus model clustering consensus cluster simultaneously rather stage permit source assignment r multivariate conjugate variance detail specify large realize structured use exploratory would identify measure overall find select knowledge directly motivate flexibility advantage substantial simulate draw normal realization realization uniform detail display true realization display display credible mcmc credible interval simulation section ht randomly generate simulation show credible distinguish weak substantial overlap separate finite dirichlet mixture determine joint dirichlet datum spirit detail article incorrect assignment cluster smooth display cluster cluster perfect relationship agreement hence serve bridge panel display generate separate joint cluster well underlie blue green curve genomic breast tumor tumor sample
learn manifold fairly return solver cost riemannian return write treat regression ode linearization thus uncertain evaluation belief compact introduction gaussian prior sufficiently iteratively refine derivation return value expressive uncertainty solver kind uncertainty initial essential building probabilistic thing probabilistic ode pde numerical demonstrate strength highlight open minor highlight connection conceptually apply notation though order ode experiment riemannian geodesic consistent value shift along geodesic uncertain version assign joint geodesic covariance input element matrix output semidefinite problem necessary general function minor function available regression derivation radial amount curve vary explain linear thus belief derivative necessary initial incorporate eqs mean algorithm move ode derivative belief bt one treat idea construct classic ode family metric uncertainty order caused construct uncertain move external uncertainty estimate block element j etc show conceptual crucially curve bound classic ode solver sec empirically albeit number iteration round change arise evaluation implicit important update lead fit recursive construction value expect assume limited euclidean cover datum connect small connect point thus bayesian fashion x ds covariance control regularity infer give rise rough regular straight logarithm final location depend prohibitive use pre grid grid boundary area grid alternative external solver thin mean simple lead confident classic uncertain black standard deviation computational sound usually small number modern ode solver rely considerable quite algorithm idea hoc probabilistic approach shorter one think proceed consist construct evaluation initial prediction lie construct ode ensure nontrivial cite historical mathematic arise straightforwardly currently hoc strategy probabilistic generative probabilistic example question framework probabilistic sec concept exponential logarithm map complicate geodesic linearly geodesic process logarithm use quadrature logarithm sampling method dominate solver mean optimisation gradient compute covariance probabilistic alternative covariance geodesic compute external external external solver experiment illustrative mnist handwritten body next page centre principal roughly dimensional local smoothly change use eq tensors state solver implement implicit resort riemannian statistic solver solvers datum centre close achieve solver particular long curve estimate probabilistic reflect length mean deviation plot matlab probabilistic color encodes length white dark set run considerably length color probabilistic slight decrease precision computational solver problem length hard dimensionality advantage grow computation experiment ccc scan subject smoothly change metric increase base probabilistic achieve minute fig show principal geodesic learn uncertainty six principal geodesic increase supplementary study solver differential equation boundary return theoretical currently bound structured estimate riemannian manifold include turn mean conceptual design statistic acknowledgements foundation education machine award arise mit gaussian select function ac cx cx x nx cx various combination radial function value function derivative retain kronecker perhaps widely way learn py py tt g tt logarithm derivative giving eq constructing scale rgb rgb rgb text sep draw minimum black text white thick draw minimum fill minimum method boundary initial statistic analytic ordinary equation permit lead principal geodesic mean enable art wide numerical calculation throughout differential nd essential tool mathematic prediction future states area riemannian manifold short calculate path riemannian ode solver sec trivial optimisation heavy biased mostly differential study historical overview modern ode solver seminal carefully fact ode solver numerical interpret intractable ode estimator subject error numerical bind end functional riemannian smoothly metric locally datum smoothly define norm dc euler lagrange satisfy boundary return
n euclidean cauchy schwarz euclidean envelope envelope vc l l functions envelope euclidean envelope envelope hx e j omit pn proof np hz np hz generality I p I hz hz hz I hz expectation write sufficiently control vc subgraph impose I hz u np np np u nx assumption may nx nx nx j nx p schwarz use nx establish nz proof conclusion pz uniformly imply h du nx nx establish proof sketch argument expectation equality variable taylor expansion map compact obtain hold f dimensional eq subgraph rw g lemma take fx fx follow decomposition np nz p establish second theorem efficient state process direct third figure table htbp l l l l l l l l l ex example section remark j participant bc joint conference development economic international support nsf grant study identification weight derivative functional quantile setting regressor interval bound characterize rely specification define without condition admits characterization outcome estimation regression interval censor weighted average derivative efficiency censor economic datum value record code interval analyze interval pose challenge weight interest loss give loss covariate interest observe identify informative sharp outcome regressor suppose value bound eq interval value similar development identification conduct identify restriction know confidence identify set prescribe coefficient contribute study estimation motivation common specify estimation weighted well form study variety parameter interpretation marginal feature interest average mean summarize slope structural presence derivative contribution identify compatible identify characterize hyperplane tangent function use economic identify convex prediction price may price example mean weighted price change impact median quantile demand price quantile price house house distance house another home air record code measurement location house price house characteristic put high relevant effect suppose value throughout identify set function absolutely nonempty interior convex measurable determine impose derivative e compact interior ii continuous differentiable us nonparametric impose identify characterize sphere main characterize identify pointwise w w unique pz additionally theorem suggest support inner product extreme pm pm lm come evaluate map subject maximum function good predictor z j th component regressor value covariate observe unobserved pair derivative regression assume pointwise fix identify make regression derivative hold eq assumption impose weak monotonicity generality assume ii regression independence inequality contain interior iii analog functional envelope condition condition may sharp average argument one establish I hold compact pointwise eq v v g iii absolutely measure denote denote borel p define usual continuously fr notational fr point tangent tangent span sphere parameter function smooth differentiable finite present notion estimator borel measurable element pp consider equal characterize bind usually toward notation regularity quantile continuously differentiable bound continuously differentiable bound continuously regularity trivially satisfied quantile neighborhood ii note exclude either discrete point satisfy efficiency support current censor efficient influence identify show q asymptotic establish efficiency estimation identify setting efficiency efficient slightly one set explanatory variable index g support depend therefore admit regular mass differentiable admit inference need song estimation approximate give smooth weighted average right proportional suppose choose pz bound constant cardinality smoothly maximum smooth average manner generally leave work section illustrate study focus interval parameter interest conditional practical parameter would otherwise value figure since scale may interpretation estimation derivative integrating thus rewrite estimator apply counterpart interpret instrumental iv replace kernel order loss hausdorff direct hausdorff hausdorff risk report simulation increase bandwidth consistent hausdorff large identify suggest set use still bias stay hausdorff seem bandwidth hausdorff report tendency identify relatively improve hausdorff manner tradeoff hausdorff risk hausdorff exist identification derivative interval censor either identify far support characterize estimate outcome censor practical purpose hausdorff risk vary choice bandwidth open type direction research interval censor appendix include proof result main use appendix let pointwise density finite usual norm supremum appendix theorem write eq theorem stochastically dominate similarly stochastically dominate iii unique ii maximization q pz p pm pz proof end support cauchy inequality I integrable p imply derivative note differentiable ensure strict sharp take convexity last iv ensure almost everywhere everywhere weight function ii I monotonicity expectation respect ii bind hence integration imply sharp proof theorem auxiliary establish result prove let continuously bound define eq pointwise straightforward show curve neighborhood far introduce nm p u expectation outline proceed tangent tangent assumption hold restriction neighborhood affect tangent theorem step characterize along curve define weak tangent let curve assumption neighborhood l u nz build continuously derivative map continuity l u theorem exist continuous argument neighborhood complete hold continuous l inclusion assumption assumption claim continuously continuous imply suppose neighborhood write thus differentiable derivative mx l bound continuously derivative continuously write derivative side neighborhood continuously differentiable show iii imply dx nn argument omit hold tangent space dense tangent parametric give suppose compact neighborhood contain support contain argument contain complete proof curve z z nz z lemma claim cauchy schwarz fr l ii neighborhood map eq last inequality equality monotone convergence ensure suppose suppose z suffice write continuity exist ensure z u z second apply
operator thereby conclude linearity dynamic check random kind drive example meet limit curve alternative stability every euler euler consequently condition control trajectory choose impose desire shape expect closed dynamic feedback explore drift bx apply joint joint angle acceleration encode point configuration normal latter assume discard information latter standard eq convert expect normal control controller exponentially controller occur every employ ode package matlab ode setting record energy comparison proportional provide maintain belief prediction uncertainty indicate dynamic reach iii controller belief I dynamic unlikely false belief example provide iv standard choose exp successfully exp c c p sp sp start normal endow rational automate ard hyper observational place ard reflect incorporate observational incorporate knowledge accurately identify stochastic pre round less less energy exp ax exp impact magnitude wrong endowed controller prior observational high length scales result depict expect controller belief consequently course could overcome actor ax exp repeat hyper maximize automate beneficial underlying finding sensible allow fast effort train either sp optimisation outperform hyper exp c ax drift control affine pair feedback control learn signal trajectory towards illustrate controller identification control illustration inherent encode dynamical controller belief exp simple selection burden approach achieve desire expect extend achieve within question investigation analysis impact keep prediction low gain cycle length finally assess thm thm thm remark new simultaneous control observable achieve conditioning process configuration identification leverage knowledge mechanic drift reduce uncertain loop trajectory dynamic normal regarded make decision belief deal uncertain change parametric uncertainty model yield consideration control contrast adopt bring nonparametric address exploitation belief manner probabilistic interpret classical control finite great grant flexibility inference rich encode lead problem analytic process gps year gps discrete dynamic system flip lead knowledge slow rate corpus collect offline extreme requirement cause combination applicability work incorporate structural priori lagrangian mechanic partial component identify reduce identification decide incorporation aside feedback outer complexity thereby reduce control inner controller controller e stability double expect loop dynamic decompose learning uncertain priori uncertainty model reflect uncertainty underlie dynamic become course belief dynamic assume conditioning controller control controller learn order end controller additional datum base approximated entropy occur second controller every limit enable physical numerical become necessary choose state observe assume decide one refer distinguish make whenever encode control two obtain derivative describe remove
ignore method conventional distinct g decade range survey paper book comprehensive technique publish provide oriented distinguish goal classify consider dimension motivate describe formalism conceptually distinguish employ varying decode distinction observation gap present derivation formulation network representation subsection network explicitly time paper goal broad identify similarity establish link exist concept abstract serve explore relevant acoustic researcher acoustic view subsection subsection subsection acoustic technique topology l observation wiener measure additional lead generality gaussian pdf assume variance base noise track residual analytical intractable separately observation become justify moment develop mapping clean derive fed applicable uncertainty network decode fundamental exploiting identify certain index speech adapt accord although numerator sake assume affine model domain nk n imply analytically practice comprise component b c many technique concept assume environmental consider previous speech relax independence conventional model read domain early room impulse response respectively part depicted figure rule viterbi due connection arrive decoder marginalization result analytically intractable integral maximum determination core estimate derivation decode routine l dash link derivation cross connection relax conditional concept example decode bayesian fix functional introduce speech vector exploit conditional independence property link drop vary state numerator turn head respective decompose link figure update turn simplified give approximation without modify model next technique due front noise call imputation either reliable vector estimate observation become major call marginalization component clean speech derive assume model algorithm depict np consider decoding arise approximate former sake robustness consider simplified omit general adaptation seem impossible analytic observation representation adaptation pdfs random draw depict direct dirac distribution map iterative map fulfil conventional decoder b l decode technique vary adaptation approach subsection however problem time figure b example besides map gaussian pdf since apply mention notational score g viterbi integral decoder integral become assumption identical relaxed case step l l l read static component normally description manner viterbi decoder network avoid clean speech vector clean statistic assume depend speech cf b mention approach employ deterministic distortion differently determined employ normal adapt accord decoder concept path pointed mixture variable subsection range analytical l approach subsection propose assume tail time domain distortion weight network interesting note analytically nonlinear vector jacobian w turn two model adaptation topology adapt assumption conventional inter model observation become autoregressive conditional observation figure c l subsection several acoustic employ present give paper derive cf subsection imputation subsection subsection subsection subsection al formulation explicitly state graphical consider neither arrive e provide language major connection depict show aim improve inter frame correlation robustness acoustic apply connection costly summarize important approximation allow subsection empirically approximation especially become obvious figure instantaneous arc figure depend deduce subsection bayesian paradigm bayesian clean corrupted employ topology description easily technique seem acoustic conventional robustness possible exploit robustness one review deep paradigm bottleneck feature possibility far research recognition error tucker square posteriori regression programming filter cosine transform model speech density model impulse mean piecewise combination mean minimum computing article network view approach decode automatic extend conventional speech turn motivated relate clean unify well new certain generic provide highlight similarity miss feature decode robust represent obstacle meet condition acoustic system acoustic term inter sub adaptation mostly parameter acoustic feature decode incorporate evaluation pdfs model exhibit distinct step relate feature g rule observation uncertainty technique give uncertainty decode technique topology formulation consider fill gap unified bold letter distinguish random pdf normally covariance q matrix depend mn n organize exist overview article decode conclusion perspective acoustic model
vb empirically find rigorous preliminary advantageous material see therefore blind deconvolution question powerful show vb image explicitly circumstance lead mechanism controlling balancing issue minima estimator image constant invariant take also suggest image jeffreys section perspective picture vb able operate many development formation vb view obvious fidelity easily place quadratic kernel difficulty uniform convolutional reflect thus propose vb primary work highly influential joint meaning delta assume reflect underlie assume flat map estimation ii terminology inference ideal equivalent herein turn inferior vb reasoning look provide begin help specialized gradient iid map estimation flat arbitrarily delta contribute reduce increase broadly image dominate function image actually sharp consequently map sep step eps eps width sep eps width sep impulse eps sep gaussian sep eps ground denote original delta denote optimize figure favor small refined concave world composition signal large sharp desire herein sort argue equivalent local minima increase minima moreover noiseless minima standard virtue noiseless unlikely sharp meaning vb poor well vb actually degenerate characterization cost analyze flat constraint set equip appropriate flat vb naturally delta norm vb introduce whereas heavily issue begin sufficiently assume exponent correspond reasonably point maximally unlikely nonetheless approximation sufficiently estimate accurately mathematically k pp p k result effect reduction penalty sensitive expect small image importantly generally generalize distribution estimate image statistic reproduce slice ideal spike slice enforce sharp relaxed varied simple fine require reflect delta small relaxed strongly range generalize image ensure success capture preferred visualize depict sharp preferred undesirable solution rate vb function indeed behave width exp eps eps exp eps width cm exp eps denote row vb per percentage pattern couple combinatorial numerous minima many try direct vb ultimately surrogate strong briefly vb overall conclusion picture essential map vb proceed emphasize none herein directly grow type inherent contradiction vb fundamentally justification latter directly highlight importance grow large integrate argument estimate alone insight must look elsewhere concavity minima invariance maximal sparsity etc herein vb blind deconvolution utilize section preferable automatically vb conceptual integrate unlike reduction vb mention obvious perhaps try mention learn without detail example source estimate much mention might detail analysis interestingly herein picture difficult suggest stem degenerate minimum explanation dimensionality vb jeffreys art local map xu estimation edge carefully couple avoid degenerate delta argue map optimal local solution distinguish test view address regularization minima sharp prior strategy regularizer pixel prior reveal xu produce adjust carefully parameter vb jeffreys important benchmark width eps cm bar comparison eps subsequent blind deconvolution practical improvement influential vb map limit thorough complementary investigation rigorous vb associate heuristic examine practical vb initially plausible achieve implement vb prove ideal setting assumption advantage vb principled image need image strongly sharp motivated sparse simultaneously discrimination desirable lead intrinsic coupling bad minima largely avoid completely viewpoint cause failure marginally contrast optimally selective sharp bad vb deconvolution fundamentally equip noise vb nearly free sparse demonstrate enhance performance function laplacian application deconvolution additionally conduct blind deconvolution dictionary observation utilize deconvolution relate herein possible vb nonetheless adopt property proof begin obtain remove value simplify concave derive guarantee leave unchanged update equally valid explanatory interpret vb attempt x hold kernel next update purpose omit use principle facilitate conjugate concave equality fact ji ki somewhat handle way close transformation evaluate ignore optimize vast majority amenable reason differentiable may numerically perhaps analytically leverage decrease motivated pose examine strict bound derive purpose account minimize plugging lead see high solve review originally minimize equivalently minimize rule mean cyclic guarantee unchanged standard interpretation update equivalent computing diagonal explicitly update somewhat special image scale formal maintain utilize specifically motivated k bound direction research omit pixel subscript likewise x ignore g px x first express result minimum hyperplane different form necessarily decrease use hz necessarily previously must non non concave locally minimize non decrease argument canonical separable concave regularizer conclude proof corollary proof direction argument theorem omit sake brevity regardless value large correspond converge g produce occur extra monotonically increase deal minimize small likewise non infer moving case increase proof irrelevant algebra unique irrelevant assume want x concavity assume theory z xx eq irrelevant definition easy function imply simplicity twice concavity avoid clutter objective affine purpose examine whereby virtue statement true direction negative ignore numerator quadratic dominate assumption conclude vb function fidelity unchanged rescaled value solution must constant exception irrelevant noise vb inclusion dependent form solution utilize use concave reflect merged algorithm theorem blind deconvolution involve signal fundamentally ill pose strong prior standard framework issue convexity problem vb strategy however value beyond inspire unclear exactly method difficult demonstrate kernel level penalty characteristic concavity scale allow rigorously explain vb exist provide perhaps counter reflect blind platform experimental conclusion blind deconvolution blind blind deconvolution blind convolution operator undesirable formation acquisition blind deconvolution aim sharp observation process additive observation commonly point spread framework herein blind deconvolution mostly filter invertible lose even blind ill pose however difficulty considerably image constrain candidate framework briefly class blind literature posteriori vb late detail include idea useful prior analytic study deconvolution notably seminal discusse address section conclude remark blind recently statistic e blind deconvolution specification find maximum ignore compute penalty desire image must invariance irrelevant poor prior lead degenerate frequently call guide algorithm carefully proper minima balancing level regularizer salient discuss section propose technique sometimes conventional blind deconvolution ii integrating accurately parameter marginalization vb marginalization brief vb way bind vb methodology apparent abuse reduce summation note via degree favor x variance equality fact descent maximization respect treat hidden assume however optimize intractable available closed show form structural assumption actually albeit factorize call field effectively utilize minimize two factorial course long ii minimize problematic marginalization solve iteration compute require operation impractical adopt equivalent vb enforce type approach bind rule appendix proof additionally numerous difference also statistic variance standard efficiently vb finite mixture update nonetheless equivalent present tb gradient level factor level xx stop satisfied qx ii h diag b convolution reduction motivated rely severe factorial may denote show gap explicitly k highly couple factorial begin energy involve integration argument nearly use vb schedule algorithm implement potentially rigorously exactly vb successful decide image operate substantially achieve marginalization statistic prior directly motivate provide investigate exact mechanism operate accounting approximation assumption involve concept direct motivate extension vb appropriate prior vb broad conclusion investigation bayesian advantageous blind deconvolution blind deconvolution vb closely reflect statistic significantly sharp image statistic regard explicit reason discuss marginalization latter cost formally ideal factorial assumption vb algorithm mechanism vb bad solution largely avoid even distinguish sharp completely perspective advantage vb vb reformulate gaussian extension incorporate describe nonetheless model success complex model principle draw completely vb well fashion procedure remove subsequent heuristic somewhat effectiveness vb exact vb represent estimation choose derivative domain performance order derivative sharp via derivative give convolution sharp derivative sharp derivative derivative kernel simplicity omit explicit follow manner rewrite convolution construct image indicate ji accounting boundary effect boundary prefer explicit notational keep subsequent analysis omit result carry quantity depend magnitude subsequent analysis concavity favor sparsity mean preference distinction nonzero concavity induce function intuitively concave functional extreme count meanwhile verify whenever previously implement representation root also maximization scale gaussian negative energy function treat generally determine ultimately role vb minimization latent defer fidelity combine penalty unlike incorporate standard unlike map image parameter kernel moreover mean remainder explore distinction typical vb possess via px I underlie vb counter vb deconvolution example vb ideal noiseless scenario solution vb trajectory minimize rigorous may affect vb briefly distinction simplify optimal zero cost exactly solution mention essentially deconvolution algorithm gradually towards minimal cost differently extensively vb lie deferred section adjustment curvature minima especially begin estimation bad solution largely map employ static penalty eventually chance conclusion simple later argue easily give exclude penalty vb base corollary vb effectively solve show term long determine globally reduce represent count element quantify vb merely instead vb gradually guarantee vb whenever become vb differently vb couple penalty map factor superiority synthetic signal compose spike random create observation vb blind deconvolution equal constant reduction test figure second readily apparent vb superior signal recover vb considerably subsequent theoretical say initialization perhaps couple vb suboptimal vb improve local width signal height eps eps illustrative aside exactly contribute success great analysis carry closed importantly property potentially affect sparsity result therefore structure concavity magnitude signature property straightforward highly concave heavily penalty former bad section vb unclear condition concave much map vb penalty concave concave non element zero many explicitly quantify vb vb sparse filter appropriate precisely vb produce fact associate decrease non concave concave move forward really understand vb deep origin examine concavity consider half z zero heavily regardless magnitude nearly equivalently magnitude penalize much heavily ideally relative allow simple theorem turn jeffreys informative magnitude maximization hyperparameter figure worth selection mechanism heuristic limit term ignore entropy becomes scale fix particular address publication increase formal concavity g penalty minima meaning reduced penalty actually concave respect way homotopy e introduce gradually introduce noise vb shape summarize view shape something properly norm conventional thus retain control modification augmentation exclusive vb address follow subsection partially couple address width couple eps cm eps f include plot couple practical vb blind deconvolution heavily dependent stagewise approach whereby repeatedly successively initialize resolution implement initially structure subsequent begin reflect correct coarse shape gradually detail sparse concave effective fine convex
category image category per category image aside describe literature one category task classifier image result description method dictionary pca extraction assign whose matrix normalize category extract several pc union support pc mi computing support pc pc pc interference high interference use describe cat interference pc pc turn superior outperform cca image neither image skew group sift discard note experiment implement prediction decide sift result look feature visual figure category show dictionary show row note well select purpose give spin horizontal axis belong category cat cat image cat cat belong cat plot interference pc image represent solid interference image pc blue interference cat neither blue peak category mean pc represent object cat true top spin discovery unlabele identify co occur visual signature demonstrate dataset top spin topic correctly assign framework category individually alm solve pca directly alm pcs pca parallel theorem remark centre grant mathematics vast digital support grant mathematics vast digital resource university technology propose topic unlabele image visual subsequently compute word occurrence histogram view row extract principal pc identify occur frequently belong topic part alternate maximization modify purpose extract multiple attack scalable automatic category encourage design method able topic ii collection belong instance database people know contain people people people clutter essential automatically occur together people wish database article article column equal pc percent game discover article business contribution image identification database visual descriptor sift represent visual word image identify interference topic start describe spin discovery provide numerical efficiency finally conclude brief contribution unsupervise visual object problem attempt dataset rely capture image content unsupervise categorization human remove categorization local database graph edge image unlabele allocation represent vocabulary sift descriptor fine description degree appearance spatial facilitate discovery visual propose unsupervise al modify certain modify several topic topic facilitate visual inference represent color histogram organization solve unsupervised categorization object recognition important discover category automatically object category framework recognition descriptor high invariance approach approach image bag descriptor far cluster dictionary visual occurrence category generate large hierarchical quantization use vocabulary tree leaf pca select informative word object apply pca select visual informative form refined categorization project histogram pc category recognition system pc co occur visual signature projection quantify extent co occurrence top topic via principal interference visual topic row vector length quantify extent pc define interference quantitative meaning control choose high separate collection adapt shall instance see image interference pc belong category simplify belong topic visual word green dot dark green dot light dot visual depict spin pc choose outside pc identify particular topic finally image I ls consist depicted spin inherent datum depict many diagonal naturally visual word consequence topic discover spin simplify please real principal analysis tool call point much row extract write measure employ especially datum contaminate pc pca pc topic desirable induce pc find pc incorporate enforce add open alternate maximization pca capable solve formulation include parallel implementation step maximization select sa obtain absolute lagrangian object recognition formulation work alm via sparsity run alm repeatedly tune suffer control alm random pc alm measure subsequently level finding terminate alm get plot suit beneficial second problem instance alm explain pc carefully efficacy top spin berkeley wireless dataset category berkeley category capture simultaneously frame proximity
pairwise give scale mcmc whitening tailor gp prior structure softmax derive could experimentally ess need hyperparameter sampling hyperparameter every ess well exploratory improve l base np ne category sentence crf whitening result generally fit hyperparameter fix tb crf recognition dataset datum consist video actor perform video label video frames frames video vary video crf video construct codebook frame histogram occurrence result kernel initially median distance average crf performances se kernel outperform crf crf comparable encourage preserve cholesky operation hyperparameter proposal simulate rate generate least attain sample save nine entirely exploratory high rate limited cf similarly save involve cholesky inversion decrease error valuable practice posterior rather flat require mcmc ess improvement sample hyperparameter versus full sample scheme effect scheme possess detail yield encourage performance crf task exceed crf video promise clearly surface possibility limitation promise weak learner subset underlie mrf prediction bayesian acknowledgment chen discussion knn rf department conceptually non design motivate exist conditional mn structure wide grid proof language video prediction accuracy comparable scalar category image dna sequence structure comprise simple dna rich relevant practical suppose label whether background foreground segmentation decide dna coding suggest perform consider structure statistical structure network mn support field crf structure problem cf figure l mrf random gp process gp crf crf crf mn margin network seq regression machine svm structure svm gp decade focus attention crf due incorporate prior prediction mn offer use drawback parametric model integration principled reason motivate maintain view crf contrast provide treatment crf order avoid overfitte necessity cross validation rich history crf implicit model parameter main bayesian gaussian modelling impose crf contribution conceptually gps concept cross drawback prediction describe experimental evaluation address prediction output context object exhibit sense consist output term structure reflect predict call influence model node node belong clique clique softmax likelihood multinomial distribution crf potential clique energy crf potential weight extraction rather clique potential crf assume function argument gp prior entire give softmax mrf modelling mrf factor mrf shape pixel less example order probabilistic parsing experiment micro label macro sequence micro experiment tackle text chain task micro labels task segmentation micro segment mrf per clique grouping learn grouping clique type chain clique pairwise clique unary distinguish unary clique pairwise clique resp unary non parametric unary alternatively edge position may pairwise denote need constant across choose parameterization unary dominate label want input therefore range belief propagation test derivation jk wish perform posterior elliptical slice ess couple experiment discard third due range micro case shape mrfs belief propagation yield refer ess full number step ess perform necessary averaging build body none probabilistic posterior point class appropriate commonly logistic py let classification model index latent latent desire class py f k multiclass normalise set label structure infeasible extension structure history successful subsequent section label define graphical output graphical crf crf log linear z energy energy effectively advance crf also parameterization forest allow present crf kernel define clique template crf generally difficult construct via cross validation adopt crf mn traditional crf training predict margin incorrect output one posteriori inference wise estimate crf point wise ml ascent instead approximate crf learn propagation procedure despite name like underlie mrf though sequence importantly unable hyperparameter map come benefit evaluation consist select associated optimisation second trick concerned run macro exponentially trick inside function technique come gp literature mention prediction output mrf kernel similarity process address output similar output model kernel consist kl divergence order scheme name entity
proposal discrepancy correction law correct require tractable likelihood code construct particle intractable hmm hmm sequence q ball uniform eq regarded reflect p tp bootstrap particle likelihood follow mle despite abc may evaluate clearly practical dimension extension convergent estimate converge choose abc mle remove real obey equivalence finally remark types abc accordingly noisy abc mle important mle technique work unit variance choice possible framework remark abc mle practical mle demonstrate batch version assume density u generate hide given smc hmm transition dominate observation lebesgue see differentiable differentiable henceforth define observe iterative ascent update sequence call convention z x whole ascent update receive subscript indicate requirement truly online batch online intractable discuss suitable alternative apparent smc implementation availability score discuss nothing substitution law particle filter unique particle call degeneracy show estimate sequence hold fix batch n estimate aim particle experimentally variance bound grow find smc implementation online finally mention score computational grows add lag gradient infinite variance transform observed identifiability issue variance report literature adopt specific perfectly let sequence assume follow transformation give aim noisy observation hmms rule u h importance important note I case become carlo numerical actually calculation assume illustrative dropping corrupted measurement p finite second instability transform process corrupt density subsequently transformation distribution represent shape skewness generate generating mapping desirable infinite use ascent gradient gradient ascent volatility estimate develop parameter online abc mle generate gradient recommend transform stability check numerically look monte transform confirm transformation ascent datum trace estimate stability indicate experiment mle noisy mle result run mle estimate shape skewness present noisy horizontal following quantile function standard sampling return bayesian distribution abc ascent applicable variance stable actual experiment notice behaviour whenever estimate heuristic preprocessing sample add ascent implement mle transform accuracy correspond carlo figure bias finite particle negligible suggest essentially horizontal line empirical next experiment abc ascent converge detailed mle numerical execute normalise noisy abc iteration figure show histogram bin mean value volatility return financial represent log volatility whereas observation process stochastic volatility heavy display model estimate therein scenario sequentially noisy mle solution actual function online converge around indicate horizontal daily rate contain residual abc ascent approximate add value separately run add particle estimate versus also bottom part create converge set ease converge figure trade estimate yield estimate less variance exactly infinitely infinitely many update result decrease large estimate conclusion variability mle different box abc vs box trace top indirect result sensible estimate possible instead unbiased likelihood negligible discussion ensure smc much choose shown abc result sense high model fit perform model check value abc mle ty plot uniform ny n unable calculation original experiment smc see three agreement likelihood solution indirect set location component horizontal axis black point indirect plot check take mean implementation ascent noisy datum implementation yield convergent ascent may parameter online smc method implement mle technique introduction essentially free transition observation cope modification density another use smc gradient iterate iterated method assume perturbation respect straightforwardly mle deal fully observation intractable however
experiment gb collaborative filter create netflix maintain weak proportional machine netflix netflix change scaling keep run set purpose note achieve update machine machine quickly rating reference implementation make matrix algebra transpose multiplication support return nonzero version matlab convenient implementation matlab code code comparison weak highly optimize matlab successfully netflix remain x pattern outperform run scale promise complete netflix service google data facebook intel microsoft oracle yahoo table subset join seq join table map row produce output seq table seq function key column argument execute return none exhaustive loading cm mat mat composition column mat mat sub reverse indexing seq index zero element mat seq none matrix scalar linear dot transpose svd support illustration eps scaling eps figure eps l eps class else extend final implicit super trait sensitive b rgb california berkeley university berkeley edu application challenge machine primary simplify high scalable relative interface variety minimal performance scalability recent ml increase demand nonetheless prefer language matlab language line resemble typically ad hoc robust implementation often require relatively heavy amount effort implement system cloud development initial restrict g scalable much ml ml naturally efficient fairly subsequent attempt high abstract communication parallelization inherent ml implementation excellent performance quite difficult practice heavily high level utilize identify low level need fast lead specialize system restrict algorithm issue system ml researcher yet widely researcher environment rely translate subtle algorithmic insight scalable unfortunately process error significantly furthermore many always pdf novel ml provide user end bridge ml development comparison pure nonetheless parallelization complex optimizer corner development par matlab low distribute make ml datum loading extraction implement ml algorithms comparable matlab cluster system distribute outperform match scale specialized system factor application technique range scalability review representative nothing inspire effort matlab combine tailor interactive execution interact project try context establish alternatively project predict tool intuitive ml distribute keep matlab limited multi core adopt ml system focus develop entire learning optimize effort highly efficient specialized directly ml method well distribute learn various library simplify state make iterative algorithm introduce low algebra algebra advanced optimization also suffer execute flow operation express system low contrast challenging provide operation hardware accelerate rely region code implementation common rapidly evolve learn introduce mild scalable independent local share memory implementation support interface build platform help load load format train subset break dataset wise partition operate locally high allow communication take common optimization encourage external remainder familiar dataset array collection particular boolean scalar importantly table interface familiar relational reduce operation semantic table datum item ml primarily interface mixed datum real world transform raw datum give transformation parallel support box decrease spend datum convenience type column convention treat vector text raw text extraction k result output recommendation core ml express algebra etc class regression ultimately dot vector vector multiplication mini batch vector vector provide data partition typically automatically determine require develop operation locally later share principle scalable globally distribute decide primarily perform algebra abstraction encourage aside semantic difference design matlab programming environment indexing slice matrix scalar algebra addition optimizer close closed increase iterate reader encourage model implement interface input produce object model would predict give new collaborative filtering might recommendation simple crucially one interface design evaluate well ml system binary implement choose platform suited intensive dataset characteristic many ml property due attractive automatic recovery failure necessity top implementation experiment three attractive length comparable matlab appendix argue wide variety setting fact diverse group regression elastic net variant therein add operator implementation scalable various example set implementation scalability distribute system small factor matlab eps ht l eps weak matlab dataset binary likelihood log likelihood sigmoid gd gradient setup amazon ec ram virtual region compare cluster matlab experiment via sgd implementation weak scaling train gb imagenet image proportional experiment far note represent approximately imagenet train system hour implement locally globally implement sgd implement gradient optimizer function
entity limit size deep language major advantage well disadvantage exploit rich syntactic semantic parsing dependency representation structure determine al present dependency order connect resource relationship extraction perform extract interaction though winner notice kernel dependency parsing tend base parse general classification relationship entity huge correspond actually reason would potentially lead issue candidate due exploit typically limit sentence entity entity sentence sentence aim interaction relationship identify protein interaction relationship entity interaction binary relationship candidate present tb possible reduce candidate role relationship company entity company heuristic typically involve tend drastically reduce candidate help involve candidate describe extraction divide main execution candidate solve quadratic optimization fashion additional label aim classify unlabeled document decision entity contain return kernel describe idea representation sentence walk modify case range clear feature engineering feature problem kernel method feature base classify keep original idea exploit object help technique classify example similarity acceptable property example input express task typically well structure g sequence parse graph interesting representation input candidate sentence graph word sentence pos tag generic pos tag word additional pos tag represent tag candidate represent represent short dark node shortest represent sentence entity n candidate edge correspond candidate always candidate able distinguish candidate heuristic entity relate detecting determine whether short formalize word connect syntactic carry information analogously exploit call graph return belong short entity like entity walk kernel pair walk count path formal objective compute expect graph expect label label label connect respectively existence three probability probability path end state compute number path match path vertex assume give compute I go pair demonstrate efficiently equation identity walk inner kernel generic recall sentence tag word pos labels type entity vertex short use kernel label label attribute guarantee entity short path contain short path version present label string indicate semantic equation edge add modification present finally probability problem knowledge distribution follow uniform parameterization produce three kernel short random whole sentence able capture specific short present idea short node short path mark thing structure generate entity actually interesting several empirically combine performance kernel report present present use kernel claim kernel individual perform combine protein interaction entity relationship text kind interaction perform protein interaction aim task extract protein aim interaction protein interact interact pair validation split aim dataset document candidate split pos pos focus measure relationship extraction correctly text text amount extract relationship disadvantage extract regardless amount text incorrectly extract sure ignore text relevant precision enter increase may versa balance precision represent precision interpret important use comparison kernel pair compare significance text book give split claim experiment allow different default svm control margin value module sentence stanford segmentation pos perform operation necessary aim introduce recall high shortest exploit accord test precision seem term recall precision test metric surprising distinguish generate reflect drop value experiment kernel report combination either combine kernel regard actually surprising good distinguish candidate sentence entity side analyze graph distinguish candidate sentence distinguish significance kernel combination indicate measure type try actually refine short candidate entity differently reason exploit redundant finally kernel understand combination kernel high concern observe difference combination concern gain regard outperform exception difference significant interesting compare linguistic sentence base kernel show precision evident conclusion obtain observe still outperform linguistic however metric test term though something recall significantly precision tend tend combine influence section result analyze obtain combination understand kernel gram entity entity entity easy dependency graph combination whether individual combination table concern difference combination indicate outperform know significance test significant obtain result significantly outperform outperform kernel case combination outperform propose label random walk generic exploit entity syntactic particularly carry regard relationship distinct solution comparable art gain interesting study different kernel distribution transition directly compose document variety type entity david extraction propose walk exploit previously candidate entity syntactic representation combine may interaction method art method storage indexing
mutation fitness algorithmic hand essential conclusion ji reach essential essential conclusion invariant genetic array interval let statement px f genetic algorithm population mutation rate treat fitness figure last set hypothesis last size mutation rate previous reject identically chance first give likewise identically chance frequency generation run see less reject nan hypothesis level conclusion correctly solve oracle conclusion approximately learnable query follow appendix genetic noisy bound marginally optimal bound one voting branching purpose find genetic straightforwardly query solve support claim claim sake completeness take observe recurrence relation inductive argument omit give last follow observation implication claim model mask sample likewise mutation mutation mask variable independent absence constitute bit back dynamic addition variable mutation crucially event allele dynamic conclusion flow readily symmetry ec argument crucial physic indeed accord know exact atomic nuclear deduce argument theory conclusion regardless fitness truncation fitness sigma etc symmetry argument detail symmetry cut formal course argument circumstance readily cost work evolutionary mathematic formal system insight evolutionary real world cm thm conclusion thm thm thm thm establish broad purpose noise implicit indeed show treat noisy membership fitness straightforwardly essential total reject significance relatively efficient evolutionary purpose non genetic computational broad carry implication turn purpose noise constrained query schema search binary string length partition subset call singular schema c partition partition schema partition stand schema simply define symbol schema template order schema partition low schema partition schema fitness uniform partition schema average fitness schema schema intuition schema schema remove monotonically average use separate partition schema schema partition grow sub exponentially still example schema partition point exercise search coarse schema negligible coarse schema partition numerous non negligible effect implicit evaluation respect small coarse effect amount capacity vast analysis interact implicit purpose heuristic implicit identify schema schema high limit amount bit index remain word pick effectively yield low importantly coarse schema search space use limit search schema fitness fitness coarse weak search secondly company survey propagation local heuristic state np close abstract heuristic case stand stay capable schema partition schema identify schema fitness unfortunately constitute formal formal evolutionary typically formally prove without make simplify fitness previously necessary appropriate response foundation rigorous prediction prediction find absence validate straightforwardly fitness able oracle make vice description schema theory genetic linkage genetic two phenomenon strictly speak implicit implicit parallelism kind genetic linkage linkage difference perspective reveal implicit schema coarse schema satisfy element adjacent fitness draw follow frequency great hand coarse schema negligible schema fitness go schema go derive parallelism absence schema adjacency schema contain unit integer set string string iff bit tuple obeys return bit attribute attribute say correct integer boolean return bit give argument hypothesis reject hypothesis adjust significance word
monitoring aim resource perform via framework take flow adapt observation link accord insight flow network monitoring sampling flow consider expect ny utilize monitoring flow link evolution valuable section idea flow suggested model rely scale approach practical compare tailor present find traffic view volume traffic flow carry capture measurement strategy horizon first flow problem stochastic framework optimization problem obtain rate traffic kalman figure significant exist tp bb pdf communication traffic flow traffic predefine path describe traffic respectively ignore delay scale round traffic fundamental spatial flow matrix determine discuss optimization detail tp scale vs naive time flow process process flow time purpose duration minute represent noise system evolution initial state calibration summarize primitive determined calibration phase autoregressive see chapter mention volume sampling say flow observation variable specifie flow give binomial give expand harmonic harmonic concave verify composition function measurement optimal measurement write density expect minimize horizon subject sampling represent available optimal strategy optimal sampling would history dynamic therefore exploiting estimation general specifically control quadratic primitive variable gaussian instantaneous equation eq relate density function primitive measurement rate optimal condition kalman gain calculate optimal calculate solution estimator decompose linear constraint dimension concavity us lie hull proposition suggest concave program certainly induce low concave program kalman gain get combine acquire kalman repeat tp namely I kalman filter still though node capture traffic flow follow I scheme evenly available capacity flow link square rmse jt jt average slot slot na I correspond error tp flow figure flow flow moreover outcome link traffic depict flow short advantageous correlation traffic rich something design additionally increase period address monitor resource constraint traffic exploiting flow network topology flow traffic flow design kalman traffic flow world internet advance compute lead growth array cloud video demand cloud ip time attack extremely view anomalous capacity service consideration everywhere
model give clearly family select yield ari likelihood furthermore yield note parameter wide imply model always right component fit right therefore parsimonious deal response statistic bic df range good agreement pick four ari parsimonious htbp bm bf om blue parsimonious model correlate family algorithm stable fitting computed tailed may employ parsimonious discovery science engineering mm mm mixture offer investigate heterogeneity dependency regression relationship extend impose error covariance decomposition parsimonious mixture regression family expectation parameter estimation simulate real become decade mixture exploit insight mixture regression package multivariate correlate response integrate correlate response illustrate large square result fit model decompose gain extend response eigen sec maximization describe sec simulate datum conclude remark vector response explanatory decompose normally logit baseline word model logistic may include covariate dd parameter variate undesirable eigen decompose give eq entry constraint geometrically orientation th table covariance model eigen structure htbp name orientation spherical align axis align axis g dd g g estimation distributional refer covariate regression incomplete em nz complete decompose calculate th complete note model good model parameter use perform quite extensively maxima random initialization issue initialize use lack progress acceleration value estimate iteration stop adjust rand ari classification ari rand ari cluster ari illustrate facilitate
merge jensen edge merge segment segment shannon kullback leibler full value compute jensen shannon negative jensen zero agglomerative agglomerative percentage parametrization target time segment representation structure optimize irrelevant conduct generate artificial evolve structure synthetic consist number vertex group cluster cluster split draw associate cluster uniformly interval graph cluster union vertex uniformly introduce generate weight dataset reliable number edge retrieve instance retrieve strong retrieve one amount retrieve number tend provide take evolve structure evolve randomly graph consider snapshot ht edge noise notice data retrieve cluster due graph vertex one avoid spurious record cycle cycle may st available website million model vertex day study cluster source segment expect quantify whether lack excess two neighbourhood toward east toward north majority make pair segment evolution time excess traffic daily traffic period lack compare traffic segment day surprising people office go among segment cycle major pm peak elsewhere colored segment color white segment france process hour depend interval stream pre treat continuous variable irrelevant stream aggregate pre temporal require expert critical look user tune necessarily short change structure paper evolve novel name grouping time co describe time simultaneously order image particularly assess experiment artificial dataset reliable underlie work extend co cluster add label temporal day week mi mi france universit paris paris email com paris introduce track structure evolve source feature whose segment approach lie segment evolution distribution require discretization conduct synthetic illustrate life propose exploratory mining interaction entity case student thesis researcher understand graph track quantitative actor edge correspond interested actor role lead introduction structural equivalence actor role interact actor actor vertex obtain simplify synthetic graph relax name actor interact graph exploit generally row column actor indicate whether actor call extract group convenient subject actor vertex actor belong original define characteristic case numerous build satisfactory favor homogeneous recent approach include indicator actor conditionally actor standard simple whose parameter early cluster automatically number boolean vertex static segment agglomerative grouping interval stochastic adaptation dedicate bipartite graph track evolve agglomerative temporal scheme use guarantee robustness relate exploit exploit technique noise graph bipartite consider simultaneously partition within avoid propose evolve build free vertex whose addition stationary synthetic representation optimize simultaneously co approach reliable time make globally addition asymptotically post technique exploratory tool finally real life dataset practical adjacency evolve edge enough graph direct graph bipartite graph undirecte two evolve unique suppose synthetic segment v tc te ni parametrization characterizing source cluster resp source resp vertice cluster vertice vertex e graph frequency segment deduce segment resp source resp specification third temporal discretization interval robust requirement choose exploit rank segment edge specify one rank co source vertex discretize infer well build posteriori image datum phenomenon overfitte overcome uninformative assumption enumeration constitutes distribute correspond vertex cluster cluster regular equivalence segment uniformly edge time segment stationary many fine grain source clusters resp clusters partition sum empty subset make vertex cluster cluster cluster product cluster segment resp image definition likely know parametrization formally hypothesis way image edge resp know every posterior similarly distribute time segment behavior illustrate image graph criterion theory negative
let random turn process measure solve parametrize partition simplify denote slight abuse apply construct concern integral eq q technical probability moreover exist follow converge bound q hence similarly bound keep sign lemma start write accord nh consequently theorem summarize close arrive complete simple wiener process nevertheless consistent namely omit random extra technical section quality help simulation value interval relatively coefficient c c ax bx x preferable intel processor convergence independent take bad ax x bx increase time coefficient zero seem seem certain coefficient estimator work fine rely denominator follow positive wrong swap keeping estimator ten although clearly improve conclude sign affect strong estimator brownian department statistic university mathematics department mathematics chapter drift drive motion discrete solution vanish rate fractional fractional brownian h equation subject active decade finance modeling parameter grow paper fractional noise surprising reference author estimator paper disadvantage whole discretization involve fractional brownian mention paper long book convenient chapter result global derivative increment stochastic differential drift prove strong section brownian integral understand fractional provide g gx define old strong generic change estimate fractional fractional
score label unlabeled loss disagreement view u p operate minimize enforce agreement view prediction function p p similar objective v u expansion partial derivative view clarity l u view definite invertible coefficient prediction view prediction pursuit pursuit view code rank author many circumstance simultaneously algorithm avoid improve circumstance imbalance give good regression method despite minimize appropriate motivated empirical rank pursuit algorithm slightly elegant employ generalization separate rank pursuit least error pursuit disagreement form section regression ranking case regression formulation recover complete matrix recover equal kernel pursuit algorithm art compare main speed popular lead depend point present instead minimize disagreement prediction function objective r I nonzero approximate kernel approximation subset regressor frequently suitable regressor efficient performance regressor obtain supervise pursuit algorithm demonstrate rank publicly preference rating assign setup user three rate randomly user rate test preference testing user g prefer lower group rating rate value different repeat experiment ten ten run experiment pursuit matching ridge proximal term match ss pursuit choose stop choose performance create set collaborative include outperform sign statistically statistically obtained supervise pursuit learn modification simulate half training view point rest setup previously observe notable improvement statistically test performance decrease supervised fact label pursuit pursuit pursuit rank pursuit pursuit approximately movie way comparable rank pursuit statistically evaluate term pursuit conduct experiment conduct setup mean squared mse performance obtain table rank pursuit sparse task ranking appropriately perform specialized difference ranking method accord sign rank summarize outperform algorithm rank achieve notably well ranking method preference learn ranking algorithm semi algorithm generalization match pursuit applicable circumstance obtain multiple optimal frequently biology language etc contribution paper combine supervise regression semi pursuit baseline combine objective lead performance future algorithm domain aggregation sparse institute sciences university propose utility square loss pair point generalization pursuit operate supervise near propose unlabeled solution recently order score class strict datum include retrieval collaborative filter web processing bioinformatic protein progress development preference far emphasis mainly interpretability solution work novel preference time necessity e compare counterpart rapidly develop area learn problem variable enhance phenomenon objective constitute crucial sparse development subset task sparse reduction algorithm application biology name tie preference relation object lead however explicit produce interpretable note frequently rank applicable expensive generalization matching approximate utility method explicit accordingly write vector label tuple define preference incorporate relevance particular point task j informally goal real relevance preference relation cost disagreement incorrectly pair denote pursuit preference consider training dictionary dictionary expansion indice tn k disagreement write ranking
allocation lda inference variational bayesian online variational inference monte collapse compare fast still big corpora people often search document library enable retrieval appropriate title keyword document keyword need therefore document manual option computer document machine probabilistic allocation describe topic posterior modern fall usually conceptual generate variational variational vb bayesian optimize kullback divergence mcmc vb subsequent section lda wikipedia fastest still use modeling assume document fashion topic dirichlet th document thick rectangle word word lda use analyse computing structure corpus inference subsection derive bayes integrate collapse interested denote collapse gibbs thus multiplication refer term observe token exclude derivation document guarantee maximization convergence hold see step document converge combination improve variational improvement prescribe reach corpus requirements corpus naturally suit would topic corpus topic modify desire equation note bring topic document divide individually maximize accord previous fix compute entire corpus eq number available document size firstly document update everything algorithm terminate document process compare corpora datum denote word th document run later hold vocabulary around topic sampling experiment converge criterion
thought price node belong rkhs kn valid collaborative function rkh alternatively upon price market eq regularization notational convention superposition usually transmission line reach rate power topology grid publicly available pricing whenever example typically peak wind demand transmission pricing similar characteristic hour week specification justify product relatively small parsimonious cf trace every alternatively favor low square cf efficient pose problem understand minimizer albeit analytic minimizer minimizer appropriate thus minimizer possibly restrict relatively small restrict feasible turn small optimality rest develop rank completion low rank solve prove reformulate regularization eq kernel tune cross kernel multi period depend consider space construct optimize predefine minimizing accomplish even find respectively rt lr l mr optimization decompose interestingly enough generalize result transform section observe kronecker delta function pn tn extensively interestingly analogue read completion recover replace f zero due rank impossible generic derive kronecker delta enable jointly goal increase minimizing evaluate involve expansion lr lr collect compactly write admit mr mr mr rt minimize enable coefficient mr regularization lr l lr mr mr mr operator side l functional compactly face challenge though entail secondly scale converge find price via approach unseen pair simply lr mt mr cf essence forecast predict store compactly forecast market pricing remove accommodate update participant leave addition imputation entry completion upon substitute justified sec influential kernel systematic prediction selection coordinate variable per iterate block block block involve maintain upon rearrange update convex yet differentiable exhibit canonical accord optimization minimizer provide valuable insight solve directly zero admit minimizer value tw solve back linear convex solve gradient iterate sufficiently iterate secondly concern rewrite alg cast initialize compute b l lm proceed carefully alg separability guarantee iterate become threshold derive multi test market ahead collect period day hour pool one laplacian vertex similarity connect proportional electrical balancing belong price neighboring price correlate adjacent base fig build connectivity store graph utilize information specifically name name market interface whose bandwidth median pairwise square independence kernel choose price estimate historical regard temporal publicly hour forecast generation capacity wind major city actual hour week hour hour market forecast pm forecast pm wind weather unit next forecast pm load demand pm pm aware weather save achieve pm secondly weather characterize uncertainty hour ahead predict quite accurately wave say yet start remain hour couple hence price design plug feature gaussian median euclidean kernel shift distance select step center diagonal stationary mean yet cope stationarity market price center subtract per hour predictor forecast though rather absolute price transaction forecast readily wide several publicly affect stationarity day ahead previous hour regularization market day allow typically instead day day day low predict express fix good tradeoff capability novel kernel whether alg indicate eliminate eliminate forecasting day identity eliminate hence couple across beneficial computed rich information select far note turn activate provide method ii ridge forecast predictor iii price derive sparsity leverage forecast attain almost low average inference mechanism price price hour matrix sparse facilitate meaningful apply market datum predictor publicly available predictor develop generic rank setup need across feature extension scenario interesting research direction focus application grid rank model demand wind period proposition function respectively belong f strict weakly hold r contradiction f r f r yield every admit converge accordingly choose feasible attain square root strictly minimizer feasible complete build reproduce l l family whose represent define allow define inner pp solve r lr cauchy schwarz utilize square analysis penalize proof problem equivalently express solution yield finally optimization singular q matrix diagonal complete remark yu zhang edu vision advanced technology enhance economic align end statistical forecasting uniquely exploit market spatio nuclear pricing hour systematically market wide forecasting beneficial learning coordinate solve convex problem utilize stationary computational approach alternative price coordinate trading strategy price moreover independent forecast solely publicly model service national transmission generic market setup generator reliability power demand price exhibit importantly transmission limitation across source heat loss lead spatially energy price price far series auto moving generalization linear artificial intelligence network hide markov
thm summary department mathematics sciences university de universit paris size ridge elastic correlate selection estimation combine strength adaptively group property achieve goal handle dimension enjoy property term coefficient study particular outperform keyword phrase regression response predictor vector transpose often parsimonious assume non selection improve interpretation coefficient sparsity predictor comparable sample frequently tumor processing fan li application like variable receive lot decade focus implement variable coefficient chen et scad li rapidly grow aspect estimator fu zhang yu extension modification ensure hand regression fast fan li scad enjoy estimator well fan scad dimensionality fan show lasso fan li yu establish overcome bias define penalization highly dimensionality highly correlate combine elastic net combine propose call fusion incorporate information redundancy variable study van lasso fuse lasso second penalty penalty four ridge cite van de et al classification efficiently via lar possess scad oracle zhang adaptive net combine establish dimension popularity notably van complement aim inclusion predictor take type alternative group similar develop account property spirit square loss adaptive penalty highlight select drop predictor together weak zhang dimension particular weak lasso performance comparison estimator property set estimator grouping correlation asymptotic oracle show achieve detailed simulation perform illustrate performance particular discussion technical proof brief account statistical summarize predictor two encourage group selection covariate lasso select net combine ridge identity despite popularity notably additional van author second former aim latter correlation base encourage highly positive good simulation highly correlate mention weighted fusion smooth van de former replace modify ji I ji modification fuse penalty term help tackle coefficient slowly surprisingly good coefficient unknown propose modification elastic finding problem problem augment square estimator comment compute modification construct estimator correlation covariate regularization consequently magnitude effort ols modification lar lar lasso version put apply lar lar select predictor situation limitation variable fashion irrelevant non zero lin condition generate elastic elastic ic define yu yu condition relationship elastic net weak ic van show component oracle incorporate adaptive equation estimate combine strength regression avoid bias tuning sparsity allow quadratic grouping estimator tendency establish grouping correlation lead grouping case grouping contribution quadratic capture grouping adaptive moreover univariate become net elaborate discussion adaptive elastic zhang zhang establish establish maximum definite depend denote reasonably matrix cf q latter risk cf fan scad zhang construction adaptive assumption n write c solution demonstrate moreover helpful generalized ridge oracle consistency selection j extract normality enjoy adaptive net normality special si term ol inequality cf respectively weight j probability restrict literature method dimension et van hand van latter van inequality quadratic section square main follow put describe compute important select appropriate order good prediction validation avoid validate pick small say lar solution choose give cv wang li show scad method cf fan li bic well selector implementation couple select bic finite different adaptive smooth respectively example example situation choice weight go table summarize accuracy make adaptive non estimation accuracy set small winner follow winner follow dominate mse setting however far well sample case behave substantially accuracy regardless coefficient term largely increase value especially performance method well percent behave way increase correct variable percent variable accuracy outperform different example zhang difference
onto prof recursive calculate projection project column span calculate project matrix onto span remain formula column use ta te tp p p equation prove ex rank base calculate low column express trace ta side express trace te te tr prove f novel optimize recursive formula section select column minimize follow find na I implementation error candidate column small computationally complex operation efficient formula column equivalent criterion decrease reconstruction aa f simplify tr te ie te te te te te te column subset column select complexity memory store residual residual select start iteration tt start recursively substitute prove product column residual direct p substitute derivation iteration pg ta ta formula column without calculate numerator represent base calculate substitute ex greedy score far column subset number column select ta ta complete describe big whose goal na distribute select column sub store machine send selection filter irrelevant redundant store approach optimize physical resulting select na approach selection sub column globally extreme truly representative require many irrelevant representation span big send machine representation projection column rest section datum approach employ curse let apply random preserve probability criterion measure big approximation error instead f store integer subset index index whose entry exploit efficient example matrix wise fashion th column rewrite provide one size minimize network carry process physical block summation also refer dimension generate j c present generalize perform machine concern subset select subset source matrix represent column reconstruction error base ta select f criterion theorem derive recursive formula rank give low approximation term matrix residual eq f tf tr tr f tf tr ex greedy generalized selection target column ta lp ta I recursive formula greedy iteration optimize simplify greedy column ta r denominator manner h g numerator denominator tb rt hadamard size outline base share generalized selection run input well represent block uniform distribute pass across approach representative briefly approach randomize original matrix carefully subset reconstruction select calculate derive additive reconstruction follow enhanced propose bound proportional norm sampling allow e relative singular computationally complex large propose adaptive update calculate achieve sampling sampling implement calculation calculate singular consuming computationally whole employ quantifie subset randomized area numerical algebra qr decomposition enhance stability column qr category datum triangular select theoretical select column column cluster select cluster representative calculation lead recently selection right singular data rank use select author theoretically rademacher present polynomial volume theoretical quite present deterministic deterministic complex time calculate lead singular distribute computationally qr moreover volume sampling infeasible third hybrid combine sample column employ stage hybrid subset lead singular phase employ sample suggest repeat provably guarantee algorithm lead right hybrid randomize efficient implement randomize relatively present qr greedy incomplete permutation embed upper greedy greedy propose first computationally representative representation representation random select lead singular make big whose however employ deterministic phase hybrid medium mnist face million conduct conduct sized effectiveness centralize state experiment eight mat format image six set centralized experiment collection process version subset handwritten digit process face besides distribute use chen data contain million image convert subset quantify set approximation good svd compare include available replacement decomposition column implement matlab qr decomposition algorithm matlab matlab qr decomposition implement implement matlab select singular calculation computationally use lead singular value datum experiment lead selection randomized phase lead singular vector probability phase number lead matlab use select phase achieve matlab comparable accuracy random similar measure base approximation set column increment randomness repeat show six qr qr compare report figure table comparable term scale well hand method comparable time much note comparable low state method straightforward design implement step algorithm uniform column bad perform variant hybrid algorithm randomized three column norm singular phase centralized column centralize sparse svd distribute lead singular extend work allow vector set matrix calculation singular approximate singular vector reduce time svd achieve use conduct amazon ec consist gb processor convert binary sequence key format store distribute show accuracy matrix term relative run relatively small achieve accuracy accuracy method note less dense approximation use accuracy sign select third time set lead accordingly accurate select highlight measure indicate bad uniform c propose novel select formula reconstruction greedy approximation matrix facilitate implementation novel propose selection addition carefully design big eps fill electrical engineering science department electrical engineering fast format big selection enable datum explore instance preprocesse task low present accurate greedy scale measure error centralize column novel error representative learn solve subset sub matrix reconstruction minimize demonstrate benchmark recent year rise advance hundred process store create discover useful hide represent format reduction summarize difficult interpret instance traditional thousand instance centroid hard instance cluster cluster concept feature concept thousand feature data analyst understand goal representative allow understanding summarize big datum select generally formulate algorithm select select analyst method going produce meaningful aforementione present fast reconstruction column paper recursive fast representative manner big matrix distribute column machine design execute massive amount store cluster ensure scalability tolerance trivial large scale currently implementation dimensional subspace orthonormal orthogonal whose matrix schmidt column svd qr decomposition q represent base column rank matrix calculate orthonormal column eq span whose represent subspace calculate embed leave
big topic communication big modeling task topic problem communication efficient order art lda algorithm besides combine architecture refer big task advantage lda objective speed memory usage experiment around achieve modeling state algorithm organize current introduce law compare several art parallel make conclusion document mini index document word label document topic dirichlet hyperparameter review simple cost bad lda label occurrence vocabulary denote topic nonzero index token topic soft ix label token k parameter topic hyperparameter smooth lda symmetric combine active descent document mini batch mini pz w sufficient online index multinomial parameter document topic normalizing batch iteration reach memory mini batch local memory mini disk load model single processor platform expectation disk memory mini upon complexity insensitive number number mini task subsection parallel extend processor document processor global share mini still entire vocabulary mini end processor next batch processor mini batch number iteration suppose use processor mini batch around meanwhile reduce processor add reduce model serious major previous lda algorithms batch mini batch parallel batch big may infinity huge parallel end mini parallel nontrivial reduce parallel achieve parallel lda optimum lda objective typical gs parallel paper communication solve big model communication complexity converge local lda within box word sort residual select power sort residual show dimension b choose residual become relatively element residual straight solution cost communication mini batch cost communication model mini batch memory solution mini reduce explain subset dynamically influence power law dynamic vocabulary power word topic ratio show sublinear complexity select topic criterion inspire residual belief propagation processor successive processor similar residual vocabulary word sort power blue box power residual dynamical scheduling mini keep remain residual power get message process eq topic one vocabulary topic select element residual process reach state dynamic scheduling nine element pass residual show element residual relatively element show iteration get chance pass message task subsection propose algorithm summarize processor random initialize normalize message line eqs line initial matrix message residual eqs line message use statistic line processor processor line power topic use sort find top large computation sort low quick sort complete sort speed vocabulary size subset word subset scheduling process residual threshold line terminate memory word terminate mini batch life topic normalize topic multinomial parameter processor batch bp processors joint achieve resemble eqs word x mini sufficient previous batch mini invariant eq sufficient converge lda sense lda log goal mini previous mini batch unchanged processor current mini almost rate inaccurate slow speed change reduce convergence speed offline ensure superiority offline algorithms cost compare complexity algorithm simplicity token value set processor overall processor simplify ratio computation scalability processor communication per processor bandwidth limitation processor increase bandwidth simplify computation processor use processor table mini require matrix mini cost dominate cost sort cost minimal analysis consistent processor scale linearly mini batch reduce bp minimum processor reach high local message document topic residual provide solution mini batch big topic processor mini processor speed enough memory processor mini batch relatively document processor memory processor suitable topic complexity big section token obviously minimum suitable big indeed experiment subsection insensitive contain topic parallel insensitive memory also make bad processor become tail refer major proportion residual appearance message iteration convergence see curve intuitively residual become message optimum minimize residual residual motivation residual law mini batch natural histogram axis plot log straight law sort residual axis rank residual fig small law small vocabulary word almost top account minimize convergence fig fig show confirm residual topic schedule c c yahoo variational algorithms source also lda precision format represent gs processors cpu gb processors gb bandwidth algorithms guarantee comparison set insensitive easily fit gb evenly processor imbalance publicly set wikipedia relatively big million remove fix vocabulary word rarely contribute vocabulary greatly word token reduce vocabulary reduce token fit topic gb processor set token number parallel fix word random iteration calculate predictive word count lower significant speedup achieving introduce word topic cost topic allocate topic show fix vocabulary exponential increase indicate result confirm contribute value predictive training change confirm play combine speedup g change scalability wikipedia processor converge fast times subsection always reach low yield predictive processor gs slightly high consistent observation partly overfitte gap set wikipedia besides set increase predictive world datum stream communication algorithm wikipedia processor see communication set word communication gs type precision format select communication subsection efficient base accord mini batch mini batch wikipedia communication suggest try minimize mini reach processor topic set around speed largely attribute three reason least communication show select word topic computation show speedup scalability processor processor speedup baseline fig processor although speedup early speedup phenomenon confirm processor speedup subsection scalability topic often limit processor processor memory lda memory usage processor batch lda may load document process hand topic memory dependent batch size provide accord processor usage use disk topic matrix strategy processor extract truncation propose multi lda
proof section segment sketch follow drop subscript write derive compact variable introduce obtain saddle formulation proximal bound negativity count enough trivial emphasis piecewise potential pass algorithm belief propagation guarantee address truncate potential envelope rewrite envelope hence compute envelope envelope instance filter resemble envelope quadratic cost potential affine label label expense affine w drop subscript function cut mrfs see main benefit construction summarize intuitive potential enable isotropic relevant application pairwise solve cut weighted respectively go ensure direct finally edge vice graph equivalently explicitly affine term note minimizer ignore jointly unary construction first expression without focus correspond side add edge asymmetric consequently write u equation convention bound rewrite q follow pairwise term explicitly simplex rewrite eq subject introduces identify discard immediately constraint state require less generic lp important potential completeness respective prior relevant main shape constant setting read subject transform infinity cut equivalently read plug claim compactly pairwise potential correspond linear necessarily linear program construction prior require primal without call potential follow program potential write elementary potential reformulate potential illustrate elementary compact substitute elementary provide overall htb derivation allow elementary potential min bilinear lemma bilinear linearize family convex trivial equality w w convex program analogously eq essentially f therefore optimal duality repeatedly min potential potential via marginalization respective variant important bilinear linear pointwise potential e eq pairwise potential form minimum whereas per edge equivalence two energy immediate consequence let local marginalization cost first pointwise minima sum rewrite trivial simplex apply substitute otherwise consequently potential potential compact elementary potential g respective convex labeling piecewise convex pairwise potential lemma role establish equivalence subject intuitive encoding I attain element branch respective obtain potential st st ki st ki arrive relevant main one directly read number per dual practical beneficial smoothness specific derive htb htb linear potentials htb eq potential exposition labeling task continuously formulation preferable computer vision difference discretization formulation closely relaxation continuously formulation smoothness euclidean count align literature use expect result discretization plane base grid calculus set vertical pixel horizontal edge edge vertical one thus horizontal vertical notational simplicity homogeneous symmetric potential simple isotropic vertical e euclidean consequently edge cut jointly horizontal direction imply smoothness correspond standard penalization hold translate also euclidean cost program approach isotropic htb q q subject reduce potential option convert formulation behave less present isotropic elementary potential subsequently apply construction focus variation potential employ node neighborhood show fig term convex result constraint potential select vertical differently prefer reduce deep minimizer convex leave proximal utilize behavior eliminate lagrange multiplier remain respective dual use sum complexity experiment stop frequently setup instance label piecewise linear smoothness contain label unary potential randomly potential solve instance use compare globally minimize stop early development gradient appeal smooth program iteration count set parameter carefully compact section memory energy smaller choose denoise depict unary corruption procedure contain five pixel consider intensity replace remain clean intensity fidelity utilize fig image image extract gb optimize gb graphic acceleration display primal dual may objective compact advantage htb compact describe address assign label fidelity thresholding minimizer functional smoothness regularizer adjacent pixel represent note constraint grid truth label fig fig discretize discretized fig display minimizer functional htb label compact relaxation modify return consumption also formulation bias address applicability piecewise potential beyond pairwise theorem assignment smoothness prior adjacent pixel exact lp relaxation number clique linear segment lp piecewise construction standard lp assignment solution discrete minimizer machine clique associate potential graphical prior exact generally research discrete attention solve program inefficient many specialized literature map estimation problem quadratic term quadratic belief propagation inspire pass schedule block approach iteratively increase objective maximizer dual validate stop rule
cope unimodal function condition existence approximate algorithm computation utilize paradigm partial section real defer concave could censor precisely exclude convenient serious left inspection normalize mass infinity maximum estimator restriction maximizer maximizer maximizer contain point censor classical censoring writing geometrically fast simple mle situation exist may equality point index otherwise search check mle one check yes exist describe mle suitable constraint start intuitive domain concave q x x x let lemma even remain via piecewise slope suppose j represented verify maximize replace function contain knot exclude situation index domain part lie exclude lemma least remove augment q moreover seem maximize augment useful fixed right finite otherwise observation right decrease search procedure augment log non vx xx vx tv iy denote borel write follow borel linearization eq q available motivation em maximizer modification theorem measure borel subset sm equation imply possibly iterate set latter requirement tuple suppose log either hull closure candidate even suffice function stop iterate plus change sub become easy lead numerical follow q unless index choose weight large may follow compute write work ta tb dt k analogously denote n np concave latter least nontrivial lower censor follow observation I ni px ni ia ni inspection one question theorem guarantee existence mle consistency obvious statement refer censor start traditional consistency assumption point censor number whenever special example weakly increase open ii restrict pointwise unless day advanced record ignore observation rest right censor patient survival estimator censor mm reduction step concentrate essential proof note concavity convexity exponential jensen leave equal right side x hand dt treat analogously important slight modification concave subsequence moreover data eq word index analogous case yield right tends maxima follow q say lemma replace subsequence necessary maximizer consider q q km km may assume note q maximizer either kt sequence limit would imply r exist function q equality easily lemma assertion index write end become maximal b b exclude existence observation equal exclude existence number continuous concave number satisfy concave upper number equation x satisfy e dx I lemma illustrate figure respectively strictly indicate line surrogate let yet specify real connect possible one two follow slope value become current verify uniqueness surrogate follow elementary consideration scenario exist imply inequality dx first dt dt ii one change slope interior slope n necessary ii describe iii imply iii prove iv special probability sm dm e x dx dx dx proof monotonicity asymptotic prove consequence know part part related reader elementary maximizer concavity note estimator supremum one imply analyze fix concavity point concavity since b nx concavity nx analogous yield claim since supremum converge zero remain additional b nb nb nx nx nb os nx right hand constructive associate remark theorem cm case censor censor allow possibility estimate existence mild theoretical aspect give
share gold standard merge gold region subset gold initial initially edge compute train merge repeat match call loop training epoch epoch fast generate edge classifier flat call primitive use learning expect agglomerative learning map probability edge orient boundary orientation calculate edge orientation segment boundary map calculate orientation segment orientation addition channel response mr filter bank bin em label hand contour software boundary manually segmentation category voxel divide voxel train label boundary sample result strong load adjacent segmentation separate calculate single create histogram quantile interpolation histogram bin include pixel additionally central jensen divergence mid feature orientation angle angle convex hull use ratio volume main paper since question evaluation active commonly boundary match segment match true positive pixel automate false fp negative closeness precision precision recall segmentation boundary particularly problematic segmentation segment branch boundary irrelevant topological therefore metric though boundary result segmentation literature rand evaluate gold agree difference boundary little whereas sensitive rescale useful segmentation variation vi entropies truth understand ground truth random voxel vi rand vi natural limited vi quality rand especially image rand index topological variation em unlike vi scale vi comparable volume pair vast region near vi interpretable vi value average neuron size segmentation vice finally vi distance space vi distances candidate vi definition break false merge term introduce axis axis tradeoff similar pr curves vi line weight vi find vi suit agglomerative towards plot false mostly area compare gold marker denote star mark split circle vi threshold break term vi distance vi vice versa supplementary segment contribute vi vi present recall past vi measure optimal image cover evaluation ap area pr segmentation agglomerative define mean boundary segment orient previous agglomerative result difference train section merge check true dataset true determine large dataset denote regardless change rand merge implementation feature map learn strategy agglomerative use train dataset volume imaging volume cell boundary dark modality serial block em serial isotropic circuit publish work volume volume xu dimension circuit brain involve gold initial segmentation alone manually software purpose use map validation one volume total protocol mean compare active agglomerative figure addition agglomerative training classifier reasonable expect vi near flat occur start agglomerative indeed figure agglomerative improve vi vi agglomerative stay critical agglomerative training vi threshold agglomerative epoch star vi vi function epoch vi agglomerative minor significant vi vi isotropic segment publicly dataset serial section adjust rand place rd group attempt generate plane run box place st adjust rand group name group demonstrate general enough linkage despite isotropic linkage berkeley segmentation natural improvement art agglomerative improve evaluation metric algorithm error metric reduction improvement agglomerative em datum believe segment natural nevertheless slight demonstrate scale well dynamically adjust interpret vi flat mean measure vi figure boundary curve case agglomerative majority vi show support segment difficult boundary far ht ht vi image color despite noisy map additional successfully middle although correct failure case texture merge though vi top agglomerative grind merge scale policy match behavior agglomerative flat learning immediately apparent similar nonetheless conceptual gold guide rand index segment early train segment successfully volume might datum time could possibility train epoch epoch improvement supplementary small advantage supplementary recent work machine start liu merge merge hierarchy machine previously segmentation hierarchy epoch potentially error liu potential dynamically branch hierarchy effort shot region base use conditional merge hierarchical scalability volume exceed hierarchical allow segmentation large volume progress decade accuracy segmentation order magnitude human operate manually cut merge nearby scale hierarchy crf add human merge everywhere expensive possibility focus segmentation serial volume liu maps multiple section crf simultaneous linkage within segmentation segmentation smoothness separation linkage section necessary extension aim em pixel boundary improvement error em thin feature sum segment segment furthermore standard aid present direct segmentation agglomerative method map gold might bottleneck move semi supervised require less similar segment probable scalability direction availability find acknowledgement thank critical xu generation mat generation help figure discussion medical usa usa email abstract improve perform agglomerative segmentation combine scale agglomerative image demonstrate improvement image segmentation addition vision object recognition become increasingly essential primary circuit connectivity distinguish nm neuron range scale huge volume automate essential automate challenge adjacent neuron boundary cell shape error boundary neuron introduce right segmentation mean resolution every goal
set marker location represent represent summary set marker hash edge result final hash guarantee edge ignore hash intersection test set original list marker marker several graph collection incur redundant operation intel ghz motivate run graph hour collection reduction easily preprocesse record unique number unique l em p individual speedup several graph dataset four realize conditionally marker reduced marker individual marker indexing million simulation descent individual speedup graph marker interval process software negligible ab base graph set figure descent population realization substantial magnitude realize require indicate take minute run always set operation eliminate surprisingly gain substantial graph individual little note show single genetic variation graph therefore independent descent significant practice object permit operation operation allow nest complex test speed improvement eliminate redundant wish thank contribution code base rigorously help source available strong though large list operation implement work introduce broken act key work validity validity whether key marker iterate latter implement accept marker respective font em true valid validity hash false otherwise key validity key mark validity hash take intersection two validity return hash hash set return low great marker region great style true key hash valid value false otherwise hash key set remove return font style return hash form marker return false return marker validity indicate return location font style hash validity union validity input validity intersection validity valid discard return validity hash empty drop style return set original validity drop snapshot marker return valid marker return valid marker return tm form validity nan validity intersection nan corresponding author student statistic thompson mail university statistic grant propose design complex genetic marker class identifiability motivate graph structure marker connect edge constraint easily handle framework range use operation prove effectiveness keyword identity genetic genetic marker genome genetic marker underlie trait less goal linkage analyse dna affect trait location genetic marker comprise dna require specification location dna trait potentially trait phenotype individual key unobserved prefer large multiple location exactly especially datum structure instead realize pattern gene individual define graph individual genome edge connect deterministic lead obtaining realization computation use immediate advantage slowly vary modern marker density change realize structure long trait set realize trait observe subset individual trait component generally location also feasible lead analysis individual individual member population inference create merged power resolution trait potentially available individual slowly vary realize remain distinct recognition equal range trait analyse software develop efficiently burden trait magnitude key property test representative much fast many case strong sense intersection mapping practice introduce provably allow collection maintain function design hash return representative hash collection object equivalent collection index marker refer genetic marker could time indexing building marker difficulty introduce away tb link set second change marker marker arbitrarily arbitrarily range marker link location force specific marker infeasible computational collection look graph graph marker value label table respectively uniquely test compute hash essentially node c marker marker validity location restrict representation appropriate collection set vary example collection structure briefly mostly involve formalize describe basic function section describe marker structure marker detail available illustrate hash see short message datum reflect sufficiently arguably hash cyclic redundancy everything transmission protocol store along read signature message impossible deduce extremely message fast create array store table weak hash significant bottleneck array determine bit actually hash indexing hash relatively strong hash usually often file existence simplify processing application cache file file hash need slow calculate bottleneck hash equality application notably hash assume heavily hash summary hash original structure operation ensure hash arbitrary hash integer hash map query object original requirement see mapping etc later indexing object hash strong hash hash around object hash theoretically nonzero probability denote specifically h hash function research develop function satisfy also prevent amount object hash specification hash appendix existence hash operation integer operation combine modify summarize reduce hash sensitive hash nest hash invariance present early composition describe operation later mark must hash purpose function pattern return edge change nest single however satisfying come require preserve multiple hash present function number lemma let let multiplication integer algebraic multiplication independent random one every index one ready tackle operation part reason sufficient equation let generality ref hash sequence make identical done otherwise eq hash sign eliminate however far simple rely mainly hash hash function distinguish transformation trivially fundamental building regard hash value varie marker key hash hash refer wish marker represent sub marker hash hash value key marker validity sort object interval elsewhere something component collection mark object representative permit information complicate processing task dynamic simple operation efficient marker extract valid marker fall modify hash operation determine identical marker operation intersection set operation operation representative four easily explain present operation list powerful return every suppose marker key appropriate function hash vary marker single exactly valid marker hash hash specific marker location reduce use validity dynamic collection give summarize collection accurately collection collection include reflect fall affect produce set marker hash implementation central building operation useful equality collection use design give marker validity likewise location collection hash store skip list operation validity track marker like equality testing augment skip list hold skip value order link list easy efficiently level increasingly sparse link list point point level skip list validity level node overall skip figure skip list marker correspond level start node less move repeat interval contain query marker valid location present comprise marker entire hash allow logarithmic time time validity produce marker interval algorithm formalize augment skip correspond marker key list augment leaf valid function marker leaf valid marker begin hash interval remove hash maintain value marker leave
theoretically justify performance svms extension supervise view learn co laplacian learning method regularization parallel capacity class play view explain three respectively view appropriate integration effective unlabeled counterpart role term later complexity analysis besides give report follow concern term theoretical insight cover experimental report finally input label adjacency closeness input input neighbor act vector entry arguably normalize view decompose component correspond view depend ignore component supervise commonly acceptable good learn learner view extent prediction example adopt inconsistent multi formulate multi view scenario nonnegative regularization rewrite respective formulation replace reformulate term representation augment theorem duality mean lagrange optimization suppose lagrange respect program readily theory domain rademacher rademacher q function random empirical lemma justify adopt average view fix independently dominate fail achieve applie lipschitz reach conclusion also important role adopt derive inspired predictor function substitute function must l unnormalized unnormalized q convert employ give summarize uk uk l uk k uk k u uk u lk uk tr supervise laplacian co svm counterpart employ comparison method combine separate view divide label choose prediction performance test unlabele ten performance synthetic similarly toy appear view point size respectively classification test accuracy deviation well solely integrate usefulness regularization concern among perform collect yahoo content constitute view image sized gray text remove stop apply word fewer ignore text feature label unlabeled linear kernel co svm take unlabele clearly co classify web page collect computer department web four university university web page course home page home page web whereas point web page view accord extraction section vector set unlabele give test unlabele new supervised svms integrate convexity duality optimization classifier moreover indicate role experimental effectiveness special formulate combine mention direction common parameter hold selection currently semi quantity label algorithm intend usual one rest class binary adopt class
x addition impose condition get I strictly fact integration part also absolutely volume case expression eq case notice finite dimensional case fact may curse detail space independent process absolutely wiener h contain depend operational practice quantity quantity mm slight condition true integrable asymptotic percentile chi square degree part ellipsoid let bi xt eigenfunction see jt resp operator one brownian simulate leave contain covariate aim ellipsoid parametric agree greatly bandwidth call bandwidth problem smoothing cross problem semi metric covariate curve semi compute principal ht plot minor axis decrease size median real median literature site http dataset piece correspond curve analytical processing ht chemical chemical multivariate obtain analytical chemical economic sample predict three em nf study bandwidth minimize square coordinate conditional covariate dimensional vector coordinate estimator give propose kernel covariate routine optimal choose median propose function cross propose smoothing resp use test dx I curve smooth distance derivative nf correlation protein predict rather compare prediction criterion give nf cccc cccc c nf mean conclude table predict predict separately conditional independence non important nonparametric curse take seem value sensitive make adapt predict multivariate response fact coordinate median inter response vector asymptotic consistency normality type independence well quantile tool detect outlier covariate low tail distribution aim quantile covariate notation establish technical whose hold x give convergence rate q lemma ix xx apply appendix h desire borel uniform hold satisfied recall xu u g xu g xu xu xu xu xu radius center divide bound g n rate get exponential use uniformly k h thus real go infinity come obtain choose borel n n conditioning may xu xu xu view satisfied whenever treat nonempty n u nonempty moreover whenever end h lemma statement follow uniqueness quantity number x xx xx xu xu borel see concern nx markov inequalities iii triangular ty ty j ty ty ty ty concern get h h term write nx n h observe h x ii lemma denote device finding limit jensen inequality obtain w since follow analytic condition make write making obtain use one see h q cumulative series resp h df hypothesis gx ds gx gx lemma follow denote part combine write n accord conclude converge probability treat axiom conjecture exercise lemma remark summary regression na ib centre behaviour read de universit paris france reading ac uk fr estimator multivariate covariate dimensional predict rather establish normality simulation conditional median regression carry compare regression marginal covariate sure ellipsoid ball explanatory study explanatory instance mode widely quantile outlier quantile explanatory lie within decade thank progress tool come field observe kind consider curve book description deal observation whereas parametric view mainly generalize multivariate space useful area biology appropriate longitudinal many case e lot paper estimation quantile one paper quantile covariate invert distribution establish complete convergence normality set mix framework quantile adapting return decade study parameter quantile quantile statistic estimation univariate total median historical review comparison multivariate geometry multivariate little far transpose except continuity extension hessian accord see accord conditional respect estimator eq sequence decrease zero tend infinity denominator view respect remark infinity respect uniqueness equip unless fall straight strictly uniqueness point neighbourhood ball easy derivative nonnegative tend zero tend x h j jt dt
approximate minimizer e avoid indicate denote integer user consistent discrimination follow establish property decompose multiclass result accurately choose universal distribution give family property although focus difficult proportion error cost introduce practical propose methodology compare variety adopt contain anomalous method outperform datum contain anomalous class head exist base compete anomalous compare competitive offer experimental thorough investigation scope introduce implement vc histogram tree exploration tell receiver roc arises view roc classifier alarm slope evaluate become implement universal conservative classification logistic roc choice simply convenience binary svms empirical right curve fit proportion extra model roc denote corresponding rate regression eq cdf control roc quality domain minimize binomial roc index along roc slope fit case case average c c c project joint em kl multiclass breast cancer diabetes dna n segment perform achieve good multiclass indicating algorithm perform consistently well well sign rank find set allow class fig average rise anomalous anomalous material method experimental work demonstrate experimentally unlike able anomalous test estimation multiclass anomaly rejection fundamental grant appendix consistency show vc vc establish vc theory imply multiclass vc tend follow decomposition establish error k rf sufficiently permutation select grid use subsequent bandwidth save parameter maximize roc fitting employ bootstrap method provide roc eqn confidence roc upper interval correspond sum percentage fall percentile two side valid greater tight example th percentile deviation class count std multiclass cancer size manner plus minus minus pt pt pc result electrical department ann usa work two adaptation wherein example class proportion estimate proportion testing class problem label class us address adaptation namely option assigning arise establish problem knowledge work domain problem distribution testing study multiclass label unlabeled testing set estimating unlabele approach assign arise adaptation benchmark set state sample addition mixture critical proportion unknown represent datum beyond proportion estimation design space class achieve generalization motivate adaptation problem anomaly fall category recognition object know class predict decision reject challenge summarize discrimination experimental comparison another review convert method introduce use multiclass back introduce univariate weight distribution unlabeled idea estimate easily match formulate unconstraine proportion require belong simplex quadratic program proportion condition address proportion maximize test give kullback leibler criterion none cite unobserved provide theoretical consider univariate multiclass anomaly rejection option anomalous rather allow labeling instance two minimize rate zero learning classify unobserve semantic supervise classifier capable anomaly unlabele establish pearson enable class review measurable proportion problem address later relate indeed alternate valid decide toward end distribution say irreducible exist form distribution unique decomposition hold irreducible two irreducible hard essential infimum identity example support contain support still two density distinct mean identifiable irreducible study distribution iid strongly consistent sized establish almost note statement grow r sure show proportion estimation estimation require identifiability mixture irreducible reasonable assumption word probability handwritten digit recognition although overlap support class proportion identifiable via p call adopt weak follow intuition case violate say estimation accordingly estimator unobserved proposition
pose difficulty simplex impose label belong enforce lie contain zero elsewhere model aim note ratio pose proximal splitting detail minimizer ratio function subject indeed give subdifferential splitting require operator view state define function easily computable formula subdifferential asymmetric subdifferential find convex q simply quasi indicator cluster encodes simplex constraint simplex constraint barrier iterate proximal splitting q previous yield belong stand energy indicator much energy decrease rather individual energy proof split subdifferential fortunately problem play role process recent produce computing consist variation acceleration rely proper minimization solution mean iterative criterion indicate approximate ideally terminate k descent hold inexact may however weak energy finite moreover weak still energy manner terminate inner adaptively increase implementation proximal always implement iteration remain decrease projection simplex include computation practice denote gradient kf bf bf bf v fail f p bf old demonstrate standard basis comparison set matrix mnist contain point compare algorithm compare previous variation rely recursive bi nmf default recursive type leverage equal point zero otherwise propagate unnormalized aid nmf add perform trial report discrete use bias favor due initialization initial trial iteration follow report c c alg mnist percentage ground trial label matlab code standard run terminate change fall outer report construct remarkably news recursive outperform recursive art set tend noisy fact costly algorithm plan improvement variation future lastly find overcome convexity many approach plan principled line framework therefore alternative due foundation tight subset q indeed computation thus bf bf st f f bf bf summing q belong convexity suffice subdifferential end n k f subdifferential subdifferential particular estimate energy eq stand subdifferential operator also subdifferential add kf kf kf kf expand inequality summation minimization denote block diagonal graph barrier convex simplex convex denote barrier subdifferential may saddle complete square matrix form claim corollary david ideas image processing literature motivated rely total partition recursive multiclass paper multiclass variation rely recursion previous algorithm compare nmf approach rely pose np hard natural resolution issue many factorization follow arise approach relaxed differ loose relaxation match np processing literature new algorithm tight relaxation spectral rely concept total variation exhibit region relaxation employ spectral nmf therefore promise cluster precisely variation algorithm excellent two class partition recursive bi handle class unfortunately recursive yet art multiclass variation rely optimize easily handle outperform approach name multiclass weighted denote entry encode vertex balanced cut disjoint energy simple motivate exhibit reflect small sized minimum occur generalize set number balanced control obtain multiclass rise relaxed solution mostly sharp since quasi essentially value tight f constraint p therefore develop problem role variation play formation prove version use total f lf l unnormalized graph nmf positivity exponent appear consequence relaxation example bi depict vertex show observe solution total cut whereas model smooth exactly total tv monotonic prefer sharp differ
pairwise output tie improve run distribution distribution return operation algorithm select close collection dense sparse latter whose phase hypothese second distribution priori use run element close unknown regardless strategy select otherwise claim pick sample operation close phase phase take produces possibly element leave pair execution involve run distribution sample execution operation output least algorithm execute execute distribution statement theorem operation fast final claim justify suppose distinguish fraction least analogously close describe operation execute let distribution execution execute execute either never hypothesis step case final claim justify correctness fail union bind fail correctness two output assume begin furthermore hypothesis would still run correctness must hypothesis discard hypothesis lose ever happen case probability iteration stochastically dominate geometric round claim fail expect round fail happen remove nn claim lemma round operation guarantee alternate equal union claim least worst multiply step claim consequence theorem theorem collection mixture mixture theorem select among candidate execution pdf challenge access candidate uniform variable determine whether decide common candidate contain grid candidate form define candidate form additive candidate let inequality statement hold give denote median distribution since x symmetry normal low q combine rescale cdf sample distance mixture west aside conclusion proposition harmonic applying conclude sketch k distribution k use guarantee provide candidate desire first inequality fourth inequality list candidate produce product candidate obtain collection w ii total variation finally draw sample among execute generation outline lemma want branch boost probability collection repetition collection l scenario fit kf generate tv distribution extract desire discrete allow structure set map mapping value perform search represent algebra interval store concerned element sort perform modification later learn candidate mixture gaussian component perform probability density monotonically shift negative probability preserve kolmogorov distance implement monotone deduce fx fx gx I gx gx efficiently subtract monotone function suppose partition cdf monotone partition flat interval reflect update keep track associated interval process leave degenerate interval overall know name justify must efficiently interval process examine statistic iid distance statistic fall proposition sampling proposition apply iid interval x ni nj x bad event sample allow one close cdf let indicator second principle linearity thus far bad union result kolmogorov preserve eq desire inequality second kolmogorov next draw close total respect original suppose f ni nj x repeat proposition window sample interval arrive desire examine initially sample consider gaussian iw x x c cdf proposition cdf show latter corresponding show ii ff wish first inequality pdf gaussian hand side taylor error mass w desire uncertainty parameter property lemma analyze near cdf proof n j give statement competition follow subset competition carry draw mx draw fall inside draw inside draw otherwise winner return draw utilize correctness suppose competition winner competition return winner claim finally draw chernoff imply simultaneously p go beyond stop hence stop winner competition distinguish stop proceed notice stop winner competition stop hence algorithm stop winner competition stop draw draw distribution never potentially tie failure propose argue winner hand never competition argue never close union competition contain close close follow lemma begin execution fix realization ask happen execute follow would winner probability simultaneously away condition least suffice argue close match distribution close least first happen conditioning close hence etc condition output close distribution indeed choice confidence operation phase n though regime algorithm slow still regardless go constant set pdf distribution parameter make draw simplicity h run proceeding analyze condition conditional number draw run asymptotic guarantee output exponent run improved define replace follow exponent get arbitrarily exponent immediately cost replace access collection access pdf h nh operation question proposition sketch pt pt mit mit provide properly mixture two separability mixture distance logarithmic prohibitive et al polynomially parameter select candidate namely hypothese close sample run wide imply immediate improvement statistic science recently considerable attention computer mixture version estimating parameter run separability speak indeed triplet minimal separability gaussian suffice recover mixture author certainly optimize mixture algorithm weak mixture mixture notion pac al efficient axis align mixture construct kl sample polynomially determine range mean gaussians dimension l particular pseudo dependence dependence yet weak close unknown output mixture output distribution close near dependence mixture single dimensional obtain single gaussian run heart learn understand fundamental mixture amenable technique moreover optimal trivially distribution need properly immediately carry intuitively care mix weight additive guess every distribution step among candidate produce unknown distribution precise access collection running performance al continuous exactly involved almost number section note paper mixture would gaussians closeness guess mixture intuitively small distance candidate truly correspond remove know give purpose observe empirical distribution generate kolmogorov generate choice hypothesis gaussians weak tool generate strong proper hypothesis metric metric description variation require execute outline order produce recent result quite accuracy mixture factor weak guarantee properly learn dependence learn variation kl divergence near independently provide gaussian mixture linear instead single slow factor mixture obtain complexity improve roughly create component mean univariate parameter gaussian gmm w mixing assume correspondence branch candidate exploit summarize next variance gaussian minimum statistic grid adequate piece end extract everything arbitrary gaussian generate collection mixture gaussian candidate candidate among mixture candidate mixture conclude section whose difficult candidate proposition mixture negligible negligible unknown mean irrelevant draw planning hope perform accurate separately candidate small generating assume repeat multiply candidate essential generality candidate proposition fix suppose w take many candidate collection however candidate triple previous follow suppose gmm unknown gmm lemma defer generate triple least simultaneously describe candidate continue whether assume use defer establish scenario exactly gmm k iw mean kf contain must triple close trivially formalize
last note q complete proof application prove complete apply rewrite decompose let equality thresholding q furthermore combine q iteration thresholding simple contradict main tune noise descent main far say application study elsewhere exist q provide coordinate discuss descent employ step size situation employ approach along please discuss occur value occur value essentially zero access calculate confirm converge value descent improve gradient initialize size l dl l l break gradient iteration amp description mention obtain reconstruct test unit point amp find improve three parameter measurement plot amp contain risk amp noiseless fact actual vanish estimate different experiment noise deviation effect size accuracy noiseless begin estimate accurate experiment parameter mean user simulation wide value final figure grow well much improvement overall performance standard deviation approximate amp amp oracle signal tune algorithm tune automatically user equally space pick run amp name amp name converge final mse close tuning amp finally amp tuning scheme versus tune thresholding see amp converge mse well threshold solid green curve set amp fast rate solution amp square achievable amp amp optimally theoretical practical employ estimate derivative employ approximate derivative obtain risk benefit idea amp iterative thresholding suit compressive sensing tuning crucially algorithm paper message pass amp set parameter tuning user attain reconstruction convergence unbiased sure amp find concern fast convergence concern noisy acquire denote cs apply problem acquisition acquisition modification technology phase challenge computationally demanding algorithm amp simplicity appeal initial amp employ thresholde wise call iteration residual respectively finally transpose detail iterative practice tuning free instance tune properly major improper choice first obtain reconstruction bind algorithm property rip rsc potentially provide tuning risk practical often available second base employ step employ value main drawback must least upper consider favorable involve idea statistic feature tune parameter tune propose framework consider expectation confirm take capital symbol variable like ambient finally denote big summarize intuitively statement amp clearly amp write distribution amp play lead simulation figure exhibit prove accurate calculation amp theoretically practically mse amp theoretically knowledge mse stein enable describe gaussian claim sparse amp soft question shall threshold risk soft thresholding give maximally define issue even know exhaustive due necessarily behave algorithms gradient newton converge deviation prove minima minima ideal gradient employ practice mse address employ follow know unbiased risk sure weakly differentiable provide simple risk thresholding eq property estimate dimensional employ calculated finding formalize organization paper tuning threshold thresholding connect tuning amp include proof summarize consider tuning denoise connect tuning amp noisy variance furthermore accord give forget value lemma simplify derivative finite three suppose imply gradient therefore expect gradient step derivative minimizer provide limitation computationally demanding analyze gradient gradient simultaneously remark highlight implication derivative remain small enable descent point difficulty place derivative region first around small region avoid local occur risk phenomenon happen specify prove convergence require known proceed figure derivative ideal risk minima gradient go region modify gradient descent provide way avoid backtrack notational avoid tracking employ final avoid propose ideal claim iteration amp denoise toward start formal definition adopt emphasize ambient dimension goal increase notation call sequence weakly second nn np e appeal feature equivalent column converge amp observable observable mse converge amp surely right algorithm concern assume amp model law turn pseudo noise inspire amp function thresholding expect one main intuitive implication threshold amp threshold consider parameter optimal violate include case notational skip threshold fast plan iteration well achievable claim seem amp optimally plan however amp amp threshold plan iteration formally amp soft optimal noise algorithm calculate know continue risk estimate strategy inspire sure soft approximate descent address converge establishe estimate
alternate find simulate bivariate marginal eight equation suppose equation unique solution probability lie satisfied equation low long exist solution strict definite impossible build marginal necessarily multivariate multivariate distribution ib ib p give way derive lemma bivariate asymmetric bernoulli cdf suppose want add equation p fr let condition consequence symmetric bernoulli distribution symmetric b ib b u x x output allow convexity vector upper minor new eight single side choose simulate use hoeffding marginal higher write indicate low upper shown convexity bad marginal bernoulli bad build multivariate matrix three four bernoulli characterize completely dimension subset three mm science foundation grant draw correlation marginal achievable uniquely convexity correlation convexity parameter case bernoulli variable parameter fair fair bernoulli problem simulate deviation equal must numerous field finance applicability generation receive community lebesgue correlation matrix instance marginal hard employ copula general marginal copula typically marginal beta dimension distribution important achievable marginal bernoulli easily give grow exponentially use vector back li develop simulate marginal reduce existence marginal necessary correlation build marginal dimension condition next notion convexity give arbitrary marginal make bernoulli marginal chance vector let generate fr bind theorem equation mean ability marginal convexity matrix suppose cdf rl multivariate convexity use method generate convexity deviation correlation eq maximally logic cdf
column subscript recall subscript nearest integer looking guarantee converge recall guarantee complex obtain guarantee inside root right cc use establish sum right jointly frobenius parameter side positive obtain mm diffusion define recall data I expand mn assume entry stand saddle g expand mn invertible therefore invertible relation chen apply reinforcement immediate environment mean agent predict update clear gain agent increase bias variance policy restrict portion network diffusion strategy distribute temporal square learn saddle network form connect graph share environment agent actual represent problem large dimension size scenario agent agent environment every different actual follow commonly refer environment perform prediction form useful compute agent scenario derive suitable advantage low guarantee policy set agent multi network consensus strategy apply drift largely condition agent adaptation enable tracking diffusion enhanced consensus network network combine neighborhood grow agent diffusion local external combined focus remainder scale indeed apply algorithm even agent become demand estimation influence across policy different steady propose form characterize network performance constant decaying solution able learn reveal policy able centralized hand behave step sufficiently benefit agent behave differently directly sample network experience rich manner exploit literature visit infinitely often visit agent achieve interesting capability solution set neighbor operate setup application control wireless device water water influence decision device water behave circumstance device share work issue albeit example propose name consensus approximation herein long focus long give policy perfect build allow q scheme enforce size enable adaptation analysis employ turn work approximate difference td td agent must prevent solution agent connect connect letter denote letter g environment denote abuse notation specific agent add subscript environment agent vector vector denote matrix denote stand kronecker spectrum th eigenvalue long vector give probability distribution index markov decision process mdp characterize finite size action reward want generic agent action response stationary result irreducible interest denote probability chain ss prediction reward cumulative reward window effective length control vs term plan regard draw transition lead denote bellman ss collect transition agent currently ps r collect challenge aim challenge game computationally arise subsection single reference guarantee save computation rely span dimensionality original feature length original parametric parameter vector equivalent approximation promise mainly solution moreover approximation good g stack bellman constitute set represent full however denote issue onto metric x definite therefore different equation minimize refer already verify r w verify b spectral radius exist invertible x proceed vector agent knowledge environment process arrive gradient albeit fundamentally primal enable fully continue agent relate saddle equivalent b equal lagrangian lagrangian x lagrange multiplier dual unless dual minimize remove weight transformation optimize second problem agent able individual experience problem employ lead mechanism w multipli denote dual dual dual original problem lagrangian alternate gradient ascent g since agent construction solution need convert approximation express appear expectation substitute proceed weight matrix induce behavior emphasize depend agent aim trajectory however match actually lk lk lk kk k combination condition mean agent adaptation time ensure primitive I eigenvalue inside circle eigenvector show determined constitute diffusion fully step target step time take action combine existence uniqueness problem argument provide expression reasonable begin quantity appear instantaneous use aggregate length adaptation coefficient k k lk visit state state visit least bound visit agent start independent one state transition segment tend approximate simplify tuple ki ki refer algorithm visit able every visit agent implementation stability aggregate dual equivalent saddle saddle lagrangian must l saddle establish full w illustrate prevent entire state agent may unable agent existence uniqueness solution still diffusion agent scalar set subsection analyze subtract side k I recursion lk I lk across I n individual lead recursion evolves expect take expectation side I mn g g g guarantee c mn stable datum appendix stable mean converge mn mn input reward weight size error converge still ensure fluctuation steady use semidefinite weighting I kronecker find form mn rewrite f recursion weighting couple mean error state characteristic know characteristic l l model l fu mean recursion rewrite compactly mn w stability power square stability mn theorem root condition stability ignore mn mn r mn depend algorithm side obtain state lead h derive mn weighting f node block block size block block matrix solution global examine difference symmetric bias primitive eigenvalue sufficiently small ensure hold see come follow policy mean minimizer toward global nevertheless adaptation towards cost would adaptation would solution fix saddle lagrangian global q therefore behavioral policy see figure group form vary e obtain equal member lk assume combination bound state self grid sense radial marker namely north south east move receive negative north corner world agent understand reward go agent visit agent time consumption know reach possible denote agent low thus may worth try learn allow evaluate policy several parallel stream case agent constrain exploration space attract sample respective
belong successfully topic precision represent discover dot discover circle middle topic show infer figure use also represent topic explain train train document infer explanation valid intrinsic beside correspond topic finance assign topic early day mention document explain volume news discuss effort country confusion topic assignment belong improvement positive achieve precision recall evolve rely order document evolve topic start rate drop slightly drop maintain evolve document separate last document evolve word reflect news topic distribution change change word time period corpus topic fail result sharp drop rate achieve document ignoring skip edge long factor neighbor worth backward belief become represent eq though give could system rich typical attribute entity rich require modeling dependency become solve instead know equation thesis understand context variable base euler thesis base logarithm alphabet note entity take bit measurement common express entropy random variable give probability contain amount remove alphabet maximize variable achieve joint give would entropy give mutual want alphabet measure expectation divergence entropy know care triangular distinct number width decay decay decay kernel dp dp generate document variance time duration identity matrix multinomial rgb gray continuous university sc department information college engineering cm discover collection document model collection subset resemble wave discover collection span period capable model vary discover vary topic rely evolve topic evolve topic infinite model combine online continuous set model vary topic infinite dynamic sc degree department sciences college state cm probabilistic discover collection help huge collection resemble wave develop discover big spanning realize invariant capable discover develop process vary structure time dynamic evolve number continuous advantage dirichlet probabilistic change topic structure continuous favorable continuous vary structure acknowledgment greatly I I work research lead like support five give I right problem I learn lot wish read thank I refine work thank group use corpus make good discussion past research thank special thank give even though day move move forward finally parent life accomplish thesis continuous dynamic topic temporal evolve dynamic topic model evolve discrete carlo feasible enough system dramatically evolve inference limitation use predefine evolve time develop per topic topic evident medium business news publish reader rich reading experience list news huge manual category boundary search effective dynamic continuous topic evolve news tune accordingly evolve topic topic build wiener topic time evolve fine fast evolve impractical dynamic topic apart parameter multiple topic merge split topic overall topic fix use news place related belong care topic dynamic variational become expensive receive stream state transition diagram whenever process non document relevant process start action origin decade rise project google books internet scan book rapidly whether medium news way search present digital material usefulness service end free evident ever document collection find interest collection manual annotation categorization set automate find old word collection identify topic time reality dirichlet topic model treat bag vector term sequence build idea tend together tend carry technique indexing sparse get singular represent concept distinguish together semantic treat word challenge document correlation expect overcome occurrence latent topic grow linearly issue document topic semantic hierarchical text discover word establish link latent show document make represent represent observe variable non represent dirichlet prior per document topic document topic markov topic document word distribution convenient simplex sufficient conjugate help develop even though news year web news past understand year old categorization text manually evolve project analyze view topic composition annotation successful text domain application software analysis measure understand improve protein protein lexical inform matrix understand audio latent acoustic word describe audio scene text semantic answer stock model music approximate bayesian develop allocation facilitate system visualization view understand event set essential topic infeasible life practical due exponential maximal clique highly analytically engineering find resort case favor reach exact justify spend reach graphical probability subset neighbor suffice practical fall broadly deterministic practice tie even though widely use expensive approximation simplify posterior factorize therefore resource state decompose eq possible minimize kullback minimizing left side probability arise put family rich yet use method quasi newton method restrict family want factored independence factorize accord approximation factor substitute expectation kl divergence side going take factorize distribution distribution prior distribution follow q use factorize variational optimum gaussian depend expectation evaluate iteratively cycle optimum transform complex variational try minimize tangent concave use per per document topic filter variational factorize topic index equation dirichlet per document topic observation kalman topic represent lead time get fine represent multinomial variational give forward backward variational kalman filter backward q dirac delta variational observation done change dynamic logistic normal poisson build present non word popularity link pick model describe extended model period drastically topic cover branch topic dynamic evolve term medical collection old medical classify medical write reflect recent distribution topic topic evolve wrong wrong inference error could evolve learn old word topic recent collection topic change improve model document give alternative generative dirichlet prior multinomial beta document multinomial sample dirichlet give collection adopt generative generative present instead beta sample word model though change author argue evolution topic happen occurrence co occurrence topic occur form co occurrence happen big advantage topic discretization come good pick large document significantly increase arise discretization evolve may evolve point capture evolve unnecessary evolve grain make increase hand coarse evolve may starts evolve use fix word distribution extreme covering prior limitation topic evolve topic topic lead merged assume topic great actual cause topic one news article cause extra class reader cover make article cover topic appear undesirable article interest exact done use integrate implement distribution concern training every naturally expect document appear document something would expect single title publication unit publication entity figure assign author model lot remain collection generative topic weight document jointly co occurrence divide cover document region science order time analyze topic infer topic analyze document consecutive analyze change several limitation topic hand hard even relatively number become naturally rich topic correct segment segment manual inspection document time sampling learn parameter region system assign used tracking make topic independent previous analyze dynamic social friend evolve markovian discretization evolve markovian create order fix would take news vary infinite unbounded evolve topic accord markovian analyze evolution community conference epoch conference fall epoch application production tweet topic news tweet duration resolution within epoch different topic topic inference expensive discrete model could continuity prevent discrete time streaming modeling extension time representation chinese notation model chinese restaurant tie measure chinese restaurant number parameter value mixture use disadvantage news mixture overcome use hdp allow evolve global integrate q popularity depend epoch epoch epoch sense require epoch pass length epoch epoch place dirichlet indicate early document modeling integrate get chinese process topic make trend evolve hyper form eq evolve like eq conjugacy l concentration parameter component decay decay word dp dp generating sample new suitable trend evolve evolve brownian motion dirichlet multinomial topic suitable model document clearly incur analyze become entire requirement traditional algorithm variational inference extra counterpart traditional variational suggest kullback leibler online stochastic top hdp dirichlet dirichlet dp dp level another put dp concentration dp dp utilize document atom dp level dp dp document share topic sample word non stick hdp draw top level dp corpus beta topic base dp dirac second document level dirichlet apply stick construction document atom weight simplify introduce indicator give corpus proportion stick distribution set variational document entropy distribution continuous model use brownian motion though use algorithm topic feed aggregation topic cover news feed topic topic pre topic representation figure let topic index multinomial wiener formally map multinomial natural document inference sparse symbol definition word word time dirichlet sample chapter contribution infinite combine dynamic topic hierarchical stochastic combine property dim style chinese one dim sum new change ingredient mixture restaurant serve satisfy customer change availability dim chinese restaurant customer restaurant way assign differ ingredient evolve dim process model application word document map topic customer generative sum process proceed dim restaurant first arrive new customer restaurant new currently restaurant document restaurant one customer order parameter global chinese restaurant main dim sum process global keep whereas evolve dim brownian motion dim sum use notation combine two high implicitly measure ensure time chinese restaurant recurrent chinese restaurant modification evolve motion one time resort inference conjugacy probability collapse gibbs sample upper dp diagram topic operate document infinite early dirichlet model generative dim proceed level dirichlet hdp sample dirichlet draw atom put level concentration measure dp level dp utilize make collection dp measure level dp subset topic dp topic document word document method stick break give breaking hdp draw dirichlet document draw parameter atom dp dirac delta level level stick break dp document topic weight introduce stick stick distribution index variational optimize log likelihood document ascent wiener make evolve process simplex simplex posterior resort inference chapter need topic model well compete make corpus corpus exist corpus need challenge create corpus publicly publicly corpus news potential news media attempt successively news need collection news news news news valid corpus contain identification publication title body relate news news york meet condition though make advantage make news rich news tend source contribute world source limitation page rich vocabulary source produce writing restriction make vocabulary uniformity uniformity affect news make learn keyword news fall get topic news certain period could well different belong news page news tend political reflect lack come political bias news web collect news dedicate news resource section remove publication limitation section collect add fill publication syntactic vocabulary news relate belong news web rich medium text link medium format could place different content like news publication limitation could page paragraph could news discuss link news call external news website word learn model relate news carry apply collect news regard coverage news region cover political syntactic vocabulary match news cut extend source merge collect cover usually website version exclusive content exclusive publish usually easy news former web page content page source unlike news heavily external interested favor web page internal external source usually page coverage strongly news belong sharing keyword news cluster word rich set diversity syntactic large test different relationship correctly learn translate news natural good contain big rich news create birth cycle news manually generate manual news manually tool look relevant keep recent five related create manually need testing manually create call represent try seek match news create algorithms job human standard upon address baseline match exceed source like news france every second publication time correctly place news extend create contain ten day week able create set news process long get chain end chain chain chain first want get create news corpus news like website diverse news seed cover wide variety release news create pick predefine corpora corpus currently widely collection categorization make vocabulary news article come document accurate e manually assign news take place g people mention corpus name mention stock mention corpus name microsoft title news news throughout week news news news website corpus cover year average unique vocabulary word news identification news http www publish date title death continue body news death continue http news propose go per news rest document log likelihood accept learn community goodness right topic reason likelihood mean log news feed news article build document able incoming document real word log value desirable word online hierarchical setting compete next I corpora code provide run build package also corpus parse vary value close topic time number topic word infer entire document range training try model reflect moderate news find mainly discuss finance corpus consider topic compare spectrum cover news variational iteration document I batch value value favor batch note small figure two batch size include trend decrease size long period collect date rl per likelihood well small size batch double effect affect case affect kalman evolve batch document arrival arrival time document evolve note evolve arrival filter new batch value try batch size keep maintain gain batch negligible arrive rate hour would hour collect document hour encounter corpus day would enough use model parameter equal show discover early big margin test especially batch word outperform know independent likelihood evaluate document mini batch rely offline obtained evaluate train explain cover evaluate learn fluctuation gap period publish higher per run news present page set half corpus figure show per log topic reach peak higher see perform peak performance topic corpus reach topics corpus explain property corpora news high vocabulary news news vocabulary news corpus unique word document length unique batch follow trend value favor small size experiment corpus consistently drop affect convergence evolve likelihood trend show corpus performance return reach minor gain period separate word log good discover outperform take reach number per use fix per number topic discovery
identify variable equal conversely large sub point divergence projection onto lie maximal divergence model distribution support projection code cardinality kk distribution situation mix support disjoint projection consist support model block ji mixture product support partition exponential exponential motivation partition analogue contain mutually disjoint edge cube large deep belief difficult describe maximal direct tight theorem discuss behave similarly maximal family partition mixture homogeneous I choice vanish iff complicate probably tight expect fixing fill evenly discuss answer maximal divergence low family narrow layer never universal regardless j fa mathematics mis road nm usa review maximal I restrict boltzmann belief network class illustrate divergence model start super new deep narrow unit infer selection assume justify distribution constrain complementary approximation quantify dp value analysis prior criterion relate idea discuss design identifiable data model control identifiability coefficient instead focus datum irrespective making unique close class complicate estimate exponential study machine power neural appropriately place review neural discusse via na I star internal hide q tight bound boltzmann bipartite graph unit unit visible size unit binary tight I show deep direct interaction subsequent visible layer space q vanish enough depend universal exponential identify union hierarchical independence divergence understand exponential family model whenever belong condition equal bound dimension
orthogonality reality relate type I solve practitioner analyze model gradient optimal orthogonality play accuracy close orthogonality experimental compare partition generate similarly concern particular generate split feature overlap datum two partition block another partition block update gap dual gap sub partition find optimal datum communication gain final next reveal convergence communication ease simplify iteration updating denote primal variable dual q relationship essentially maintain whose start begin iteration due machine orthogonal difficult orthogonality update without loss e w term result regard global square hinge conjugate solution problem dual therefore induction interested dependence bind geometrically coincide one perform sdca individual curve clearly illustrate convergent small justify square hinge datum show curve least square loss convergent slightly set hinge obtain definition trivial empirical aid plot curve fix show also q also result stage machine convergence increase via manuscript progress particular establish practical study able speed update superiority practical variant partially speed still exist research analytical asynchronous convergence lemma r lemma q corollary com ascent minimization performance observe refer compare naive serious convergence practical empirical iteration superior practical reveal million distribute machine utilize concern communication machine dual idea ascent variable stochastic dual optimize svms logistic regression mechanism perform communication machine motivation speed fast would empirically refer variant analyze however bad paper practical orthogonal interesting relate communication optimization general show could speed increase naive variant update naive update evenly feature inner convex denote respectively characterize strong cast introduce denote respectively cast problem q correspondence proceeding recall important sdca hinge hinge smooth smooth least regularizer square regularizer elastic regularizer convergence optimization load initialization update iw facilitate sequel simply slight careful reveal function conjugate algorithm machine work total call update sample variant scale u j variant solve nx I variant naive variant dual primal variant dual performance utilize large updating variant increase objective dual problem employ empirical comparison optimize demonstrate versus establish remain open analyze theoretical well empirical justify theorem I strongly number effective increase improve heavily increase term present convergence
yet address expert present however company individual remove drug lead action thousand attention france elsewhere focus match involve heart control odd appropriate covariate information report adjusted index diabetes odd ci heart disease direction compute control risk ratio effect odd face serious compute odd ratio logistic simultaneously covariate ratio outcome currently basis heart claim dr dr heart argue probabilistic derive claim evidence clinical need decide heart disease address something scientific question evidence would capture would odd statistical support scientific hypothesis relationship issue subtle extended discussion cause effect cause henceforth cause laplace event cause cause author recognize distinguish inference cause effect effect far cause chapter sometimes statistical distinct problem considerably subtle build inferential understanding clearly crucial observational evidence shall possibility inference simple ann within minute ann ann information analyst henceforth want interpret need answer query regard though inform relevant analogous suppose comparative clinical indicate resp denote exposure resp henceforth term population generate ann whether minute cause careful attention many improve comparative hill control causality address albeit observational major regard support inference particular regard ann trial minute thing case causal inference simple particular decision distribution exposure modification adjustment covariate remain purely knowledge properly question problematic indeed nontrivial long knowledge probabilistic popular contrast response proceed potential value resp regard determination model together previously variable might cast relationship describe situation ann state might ann take thus regard shall remainder aside exposure occur circumstance series properly inform public service act argue early encouraging reduce burden health service matter universe policy question conceptual difficulty discuss simple cause take additional account ann take minute address much problematic formulate question nontrivial approach purely know fact knowledge probabilistic leave statistical turn potential response variable resp resp regard exist determination together cast relationship ann ann cause ann take regard aside regard response exposure occur circumstance properly inform evidence increase health service act argue tend encourage would burden health service policy take universe science conceptual difficulty even cause effect question actually ann conversely take formulate causal contrast observe uncertainty I ever become counter logical difficulty ambiguity uncertainty knowledge ann denote background knowledge ann probability chapter book necessity pn conditioning see attribute evaluate evaluation involve matter observe problematic hope assess separate bound indeed reader bound hoeffde copula inequality causal exposure outcome exceed deduce must exceed causality important sure subtle less ar potential outcome take ar e ar e close simplicity seem problematic estimate quantity sufficiently ann decide seem way decision entirely assume denote conditional independence information replace ratio ann weak replace adequate find imagine circumstance accept strong requirement ann would fail example poorly take treatment I ann knowledge acceptable observer condition avoid possibility replace denominator potentially relate ann refer ann whereby relevant individual get start progress handle justify would valuable numerator ann ar base subset treat share ann bold exchangeable pre characteristic ann trial regard ann comparable subject denominator nature argue ar ar ar clinical chance assume pre ann trial subject I regard response characteristic first axiom thing thing trial ann characteristic arm potential outcome treat suitably observational possibility discount require ann comparable fundamental argument justify population counterpart observational q equation counterpart observational make hold special circumstance detail henceforth consideration particular accept use henceforth unless hold chapter book necessity pn take pn sufficient ann require availability individual ann one regard randomize another naturally issue suppose exposure causal fact exposure know exposure fact multiply exposure fact yield strong place ann probability fundamental henceforth apply discussion treat value obtain formulae additional uncertainty uncertain interval different novel inferential far clear express make add material relate strategy simply bound around end people impact I interested take understood variety different statistical inference partly subject treat perspective usually assumption datum terminology henceforth distribution would joint comprise assign four fully determine problematic never observable consistently sensitive alternatively parameter invertible parameter point mass exactly particular identifiable prefer alone well estimate insensitive prior logical consider inequality might group individual available regard objective chance focus regard quantify numerically attribute specific ann focus issue individual individual example refer ann light change example individual regard exchangeable ann interpret far condition ann negligible ann condition chance similar applie think light thing say end sound estimate causal give proxy study exception issue world focused cause effect typically complex frequency plug totally account describe conduct sophisticated multiple adjust odd try regard successful ever estimate odd desire rare doubly relevance evidence assess whether drug heart even interest multiple notable effort united examine long term health effect exposure pass act require comprehensive scientific medical regarding exposure national report study status early exposure aggregate level standard study odd analysis identification exposure take motivate life diagnosis abuse life event exposure distinguish three concern relationship exposure relationship trivial suffer suffer abuse condition background approach interpret uncertain chance individual focus specific child take focused assess abuse take data study analysis address abuse cause sign issue sign abuse abuse uncertain need issue take target modify make justify weak use search relevant support easy support satisfied nevertheless shall proceed use assumption take credible hope implement software find good conduct several alternative analysis include exclude predictive model chance overall population chance purpose involve chance chance place evidence take consideration evidence relevant quantity treat expect deviation regard substantial uncertainty attempt abuse unconditional abuse bind code well incorporate chain burn generate chain would reason sample suggest report autocorrelation burn iteration autocorrelation take bivariate whenever negative association exposure happen uncertainty uninformative bivariate alone interval
two method shown agree double jump run second nearly perfectly move attempt alternative choice appear acceptance level mix involved scheme indicate double reversible bayes could mind development construct properly couple proposal reversible relate make move critical extend wishart truly high participant mini hold sl pl sl pl wishart conjugate receive considerable posterior prove new wishart development reversible normalizing calculation compare graphical two investigate receive discussion et wang li difficulty instability also hierarchical time accept probability moderate dimensional development reliably sampler gibbs wishart variate iterative scaling application usefulness propose jump develop involve use unstable approximation normalize resolve propose combine concept behind exchange reversible reversible jump article review wishart sampler reversible example confirm collect definite likelihood product abuse whenever wishart conjugate decomposition graph purpose assume maximally give prove general overlap development clique use extend create sampler block sampler work construct requirement section direct sampler block sampler sample conditional relative full independently run target question property note wishart know change relevant retain sampler distribution along move requirement clique determine store np hard problem jj alternative determine solve location put replace possibility hierarchical thereby form average build develop cholesky decomposition jacobian eq specify neighbor large mcmc current attempt move complete several asymmetric move acceptance bayes factor reversible compare neighboring cholesky spend development yield improvement normalizing factor approximation rather fail propose double approximate ratio metropolis hasting though appear approach exchange tool intractable exchange aid wishart similar direct consider exchange normalize calculation probability existence exchange approach reversible additional jump proceed lm see double reversible alternative reversible jump accord normalizing constant double reversible jump direct identical observe block well direct sampler million million gibb take eq expectation sampler appear element quantile
combine whereas distributional patient variation function ik ba compare svms dataset define another exactly validation grid optimize lrr report standard repetition outperform pool distributional achieve accuracy measurement people early six month trial device monitoring predict scoring disease patient rr rr score adopt experimental two gp total output patient depict consistently statistically see patient variation accuracy variation patient distributional propose theoretically empirically previously unseen distributional closely well interestingly result distributional svm account inter variation outperform motivate smoothly across apply sense assumption remain unclear generalize scenario task differ therein domain deal primarily deal instance collect multiple observe table summarize main difference framework setup transfer distribution ix ng ng n nk obtain kx n theorem consequently covariance regressor term operator endow reproduce covariance denote state variance follow smooth virtue second inverse x nk rescale optimization shorthand expect empirical classifier f assume recall rewritten take preprocesse transform pass largely omit apply therefore recall inequality obtain invertible define coincide combine inequality leave one accuracy accuracy distributional outperform svm possibly learn high domain apply previously unseen domain analysis dissimilarity functional output theoretic show reduce motivate experimental synthetic world dataset learn consider arbitrary domain unseen domain use flow cell expert identify patient however manual consume construct classifier generalize dramatically directly basic come heterogeneous cell exhibit cell attribute vary technical variation domain stable cell chemical attribute considerable make transfer therein cell idea population domain minimize main approach repeat consume diagnosis valuable asset informative domain extract generalize new patient generalization change marginal vary smoothly marginal still suffer perfectly functional relationship approximate sensitive invariant analysis transformation domain preserve domain task task generalization ability subspace previously unseen classifier generalize domain show generalize closely dimension algorithm include component theoretically demonstrate acquire learning domain therein availability domain contrast focus generalization ability unseen domain domain incorporate consistency theoretical guarantee sample performance setting typical learning individually adopt domain adaptation subspace approach application previous fully ability nonempty output domain probability define observe xy xy xy xy associated brevity let space kernel loss generality operator part distribution reduce dissimilarity relationship formulate capturing variance dissimilarity across convenient rkh kx characteristic preserve also begin generate generate ng distributional ng ng ng n distributional variance ik n pn n gram distributional estimator consistent minimize distributional variance require functional simplify k kk distributional sample capture inverse x q choose span previously map nonlinear function eigenfunction drop explicitly exploit covariance inverse regressor mild operator almost covariance inverse estimate supplementary affinity space act interested formulate term find solve numerator basis central denominator force thereby generalization diagonal contain multiplier eigenvalue constant benefit suitable high structured impossible tree framework entirely type corresponding kernel define may subspace maximize estimate eigenvalue special summarize component generalize eigenvalue inverse unsupervise reduce recover closely adapt applie transform k map technical assumption hold expect quantifie transform distributional variance analogous term depend distortion tradeoff distributional size denominator preserve
evaluate plus represent calculate operation since algorithm extend may name fuzzy scenario absolutely standard datum point labeling point protein cluster applicable case mean generalization extension none address fuzzy cluster relational fuzzy mean mean view vast simplification suppose distance exact vector distance namely objective norm vector case length know make calculate thus possible form thing know practically modifications quadratic form expensive th let centroid ia distance therefore generalization centroid mean abstract object generalization mean abstract distance
problem thresholding reweighte yield sparse target far sparsity concern somewhat concerned regularizer turn slow regularizer thus deduce form capability theory ask generalization capabilitie answer learn capability depend heavily impossible aim answer question widely use huge therein strategy may appropriate q process derive independent without unclear currently strategy coefficient study kernel possess investigation show possesse gaussian negligible understanding almost appropriately tune arbitrarily merely organize regularizer associated sample independently identically unknown use correspondence natural purpose minimize due square integrable least problem sense finitely associate concerned bind follow generalization impossible obtain nontrivial impose restriction portion proceed compact adopt positive smooth n know function eq arbitrary hold subsection certain remark four kernel role regularization capability rate enter competition estimator establish rate notice optimal method smoothness highlight bad analysis concrete fast c achieve rather capability rkh learn rkhs monotonically call variance gaussian infinite kernel arise follow coincide gaussian demonstrate two identical phenomenon follow rkhs cover description find arbitrary gaussian hand deduce used highlight deduce good thus rather equal address learn error increase error perform error regularization term force role capacity regularize consensus coefficient regularization bring noticed assertion always criterion possible consequence estimator need criterion take consideration point may generalization bring generalization bring generalization capability therefore classical speak gaussian obviously demonstrate asymptotically rate turn mean assertion surprising know depict empirical regularization describe pointed order appropriate capacity regularization choose hypothesis sample identical fig scheme possess path shrink estimator regularizer ball subsection divide compare result certain exponent kernel strategy cover number rkh associate gaussian analysis capability therein classification vector hinge gaussian kernel least scheme remain role solution specifically infinite least impose certain upon regularization square choose simple structure infinite improve generalization capability introduction passive operation technique generalization term basically error via follow find approximation regularizer employ technique regularize least achieve focus knowledge claim lead essentially space author conduct divide approximation error work derive adopt regularizer generalization cope property regularizer spectrum assumption sophisticated method banach regularize square regularizer kernel similar eliminate characterize generalization essential upper desire deduce essentially learn capability regularizer concern reveal capability theorem essential rate improve paper point exponent support hinge find also square far concerned regularize rate therefore compare topic study definite formulate practice capability regularization finally ef q rkhs associate decomposition ef ef ef q e ef z I upon make short sample endowed modulus smoothness lemma r assertion exist deduce rf c j k h hence modulus smoothness let define depend kx dd r g rf dd yield short hand know satisfy confidence everywhere almost f imply dm subset cover bt covering normalize x f two nonempty exist deduce arbitrary depend q hence arbitrary lemma confidence exist form thus q eq proof subsection give know z z q e z z f z z eq z e cc cc proposition ef ef pm ef ef pa set
op nc op p nu nu normality nonzero part cover include mcp adaptive lasso thresholding bridge satisfied scad impose scad mcp bridge contrary estimator select boundedness ni nf condition imply p law together proof normality estimation consistency consistency order oracle hold proof notation nu g c lead get c normality lastly achieve consistency stability notation respectively concavity th continue represent q subsequence pl keep function take subgradient partial derivative logistic corollary example ny sciences china functions adaptive selection certain regression penalize stability consistency stability suitable coordinate propose real datum competitive kullback kl fit lead well criterion distribution nonzero penalty among problem complexity computationally prohibitive attempt burden non scad elastic mcp extent exist classified category nonconvex stable penalty scad mcp hand identity convexity penalty nonconvex scad mcp extra tuning concavity penalty interpretation penalty bayesian penalty cf issue demonstrate unstable provide process analyze bag decision propose stable combine subsampling selection function generalized balance sparsity stability generalize model often consideration construction connection introduce penalty function family use develop call adaptively stability cover situation rigorously regularity asymptotically rest section likelihood connection example encounter generalized type algorithm short proof linear type regression include indeed simulation fall exponential induce throughout iid covariate covariate smooth dispersion give note uniquely contain negative modify penalty interpretation tune overall penalty parameter concavity scad li zhang exponential penalty interpretation define let constant posterior exactly good must scale hyperparameter speed decay adjust separately adapt differ conjugate aspect take conjugate additional sample away absolute redundant dimension looking encounter generalize elastic poisson poisson gamma gamma penalty gaussian probit naturally parametrize probit case penalty distribution tuning vary example role poisson penalty gamma probit fairly differ commonly penalty plot leave mcp sigmoid scad sigmoid penalty sigmoid penalty keep concavity sigmoid poisson penalty lie mcp concavity graph derivative common generalize scad mcp feature stable consider one true need control penalty log globally logistic mcp grey sigmoid penalty grey represent correspond mcp mcp sigmoid derivative control order maintain convexity stability necessary concavity nonconvex solution observation performance local unique stability still global want minimize ti n attain precisely minimizer paper r clear characterize asymptotic whether local minimizer weak stability minimizer perturb minimizer stay strong guarantee uniqueness minimizer strong multiple shrink high stability hand entail must high weak property adjust aic never possess property constrain optimizer probability coincide situation remainder include form negative penalty nm lie interior regularity convex function local derivative around provide sufficient type condition satisfy weak asymptotic sigmoid stability satisfied hold consistency consistency asymptotic stability generalization qp may asymptotic denote q penalize consistency asymptotic stability satisfied probit consistency sigmoid vary function simplify carry logistic penalize maximum asymptotic aspect lar wise optimize target penalty type descent convex hybrid newton descent coordinate achieve stop log calculate j else go transformation take zero calculate join warm reader strategy sigmoid pp I approximation quadratic wise dm l method satisfy parameter numerical sensitive case may regression remark recommend validation perform diagnosis get curve state theorem approach differ use term p p fall introduce bic solution convexity locally lie region produce balance sparsity validation choose good probit scad mcp logistic sparsity stability properly balanced patient replication report tp proportion fit cf proportion proportion compare performance lasso mcp calculate outperform scad mcp level sigmoid fp somewhat surprisingly lasso competitive performance attribute selection penalty scad scad scad mcp mcp mcp mcp mcp mcp sigmoid scad tp fp tp fp tp fp logistic regression figure path mcp mcp sigmoid adaptive short cross validation validation tuning scad smooth sigmoid smoothness sigmoid outperform scad mcp term smoothness generate subsection repeat times standard introduce evaluate stability evaluate level mcp plot lasso one scad mcp right j probit model report ratio model ordinary full mcp probit three result tp fp cf mcp penalty tp fp cf l scad propose datum response cancer classify remain predictor mcp sigmoid lasso mcp fold repeat calculate cross validation scad mcp
approach quickly occur converge h explanation typical approximate change vertical investigate apply sequence define operator bring together exploit property adjacent clear letting moreover second easily verify except first simple case q completely precisely rr q write correspond inequality let eq value fix simplifie desire result move optimize first optimize fix duality substituting recall piece deferred appendix notation index indexing go rhs treat subset start common vector treat indexing let th subscript indexing normalize n maximum confusion exact posterior quantity approximate rewrite quantity recursion simplify subscript introduce marginalization subscript vector product tensor think eq sense u u simplify recall observe q jacobian rule jacobian composition product precisely th ki nonnegative equal turning word row column lipschitz eq multiplicative complete recall n n measure equation desire similarly proof joint precisely ne j ne aa cb nb na stage similarly rule probability multiply place express sequence third follow rule replace multiply leave eq q rule combine get compact notation extend replace appropriately ready probability q respectively rule sequence q obtain expand replace function replace derivation expression contain devoted estimate lipschitz jacobian mm jacobian partial th denote matrix open component fix duality rhs record distinct index example propose formalism iterate present theory approximate pass sequential formulate support central role procedure bayesian interested interest sequentially point sequential data model graphical online manner challenge graphical message pass present bayesian concept arise property formalism connection bayesian shall sequel turn view instance system theory analysis exact message pass arise context problem note grow inference graphical pass graphical belief propagation product good exist pass present viewpoint forward iterate seem new general interest main defer sequential setup aspect leave consideration produce value think measure let coordinate dx prior stage new division q n mt take prove iterate contraction survey technique pointwise write nx nothing sampling modify around quantity assumption help determine limit general dt main result constant theorem stop rule variable graphical model state classical classical observe distribute accord density respect take goal find stop rule thresholde pearson collect nn asymptotically rule false fast change occur provide sample somewhat independent normally stop rule cast let nz q pointwise multiplication alternatively multiply express n c n n xx algorithm hence note constant distribute f gaussian probability application formalism point distribute setup briefly sensor node associate observe change connect node share connect share change two change share condition minimum write encode node wants change minimum maintaining alarm inspire change occur eq difference classical setting satisfy rule linearly drawback linear practically infeasible rare detecting next drawback develop exact message time step derivation iterative variable compute recursively independent allow private loop brevity omit employ practice exact recursion little obtain recursion used notation rhs rhs play role approximate sum message similar compute equation constant invoke joint j get marginal rule two turn meaningful comparison sequence pz recall make formal symbol denote polynomial description operator along assumption algorithm recall prior analyze
various would would substantially hard interpret reduce clutter behaviour description experiment bad round incur regret one end learn weight every round increase however top overhead competitive algorithm tune relatively learn helpful experiment behave theory suggest perform quickly acceptable although relatively observe numerous experiment regret round regret linearly less linearity regime stay grow row gradually behind expert accumulate concentrate round bound weight played concentrate quickly cumulative vary depend keep learn expert happen slowly weight converge intermediate converge sufficiently quickly enough overhead learning rate safe overhead overhead bound may remain concentrate average assume capable compete currently whether already satisfactory risk safe well safe average safe deal long bound unbounded suggest extend setting currently would value infinite number equip set equip prior basic repeat infinitely many replace identity get define expect prior mass denote equality sequential e consider countable take mass rewrite analogously use occurrence prove similar huge exist feedback grant assume generality obtain logarithm expectation losse basic bayesian probability item term except good expert last item let complete fix unbounded regret expert suffer say happen zero suffer expert suffer decrease remove iterated removal regret loss sequence list de van gr gr strategy stochastic bad strategy guarantee worse maximally provably good new way trick yield case algorithm achieve constant bad guarantee need range loss advance unlike intuitive invariant rescaling loss case gain theoretic variant expert develop adversarial scenario also datum easy adversarial prediction make typically suffer even intuitive follow achieve low case discuss version give overview learner derive expert nature reveal loss expert strategy choose expert denote cumulative capital letter bold denote learner learner expert simple put expert small far singleton well circumstance stochastic loss distribute follow bound time another expert happen large provide mean expert small sophisticated incur opposite regret development provide guarantee seminal crucially interpret rate infinity optimize upper round property simple way double budget budget g present relate reduction observe attain expert perform well expert guarantee easy substantially trivial case guarantee combine recursively see approach achieve safe strategy satisfie combination zero close overhead dominate stage similar dominant tune past intuitive invariant rescaling surprisingly clean strategy present section guarantee precise provably benefit concept analysis call appear work seem fundamental importance current stochastic big relate practical mention notable weight good expert dominate demonstrate experiment bind forecasting allow compete article weight safe safe share present crucial safe safe interpret learning keep gap equivalently keep next present analyse strategy loss analysis loss compare artificial present analyse scale translation loss initially losse normalise unit treat strategy simple refine weight uniform one posterior expert weight incur obtain good convenient tool mix loss aggregate track crucial ingredient tend find incur tie break divide mass mix loss mix approximation bound see decompose thought mix approximate mix loss proof analyse contribution separately follow lemma basic mix mix less loss mix mix approximate expert l l mix obtain mix bad use tune rate horizon cost factor lead remainder strategy refine regret learning balance cumulative monotonically increase block uses half start cumulative mix approach early much definition new note multiplicative weight long learn varie rate confusion specify loss expert tm tv kt l tt letter cumulative alg ff high loss moreover simplify analysis essential lemma analyse rate analysis become slightly involved mix mix incur final cumulative mix contribution balanced mix mix decomposition yield zero delta w ht delta delta delta mn inf else exp w end implementation task start round bernstein round express bernstein choose accord concentrated expert bernstein inequality argument reverse equality subsequently version loss replace concern add circular inclusion regret admit regret considerably clearly assume bernstein follow rearrange taylor leave side around proof complete plug analogous give interpret loss strategy long clear expert say concentrate variance decrease concentrate good regret loss important stand alone concentration strategy incur successfully proceed datum cumulative suppose variance bound eq lemma jensen yield eq term corollary bind plug proof theorem arrive desired express bind maximize dominant result alternatively bind dominant term provide translate gain run expert constant gap mix regret necessary adversarial gap case expert scenario explain discussion therefore regret loss expert concentrate quickly potential loop suppose difficult early relatively expert uniform consequence trial learn lead unnecessary behave incur substantial phenomenon regret incur reduce able guarantee really two may yield regret bad high safe case tend follow combine guarantee time surprisingly imagine scenario substantial similarly regret combine recursively fix rate yield problem choose round fail round strategy ff alternate optimistic investigate benefit identify circumstance regret gap loss mix loss tc change whenever expert make scenario describe get feedback outperform whereas scenario loss general case use behave decrease accumulate round flip regime subset rate regime bar value gap mix loss regime separately regime regime weight expert determine also round rate may worse preserve remain flip switch start flip mean regime keep epoch flip regime regime flip since flip subsequent epoch round recall flip epoch vice versa start epoch complete strategy implementation l zero ht lt ht proceed like analogously regret fact use either increase factor develop proof much v flip thick left node dotted font font font u font font font thin font losse two bound simultaneously regret decompose mix mix auxiliary lemma mix denote flip begin round flip epoch begin flip regime definition always mix loss regime write avoid double flip flip change accurately flip add find mix use construction regime trial epoch flip regime behave start current epoch value know st next change loss value gap cumulative analogous furthermore regime analogue directly rate equivalent prove loss cumulative sum variance satisfie add subsequently bind find
platform division center ghz selective close hyperparameter way choose crucial algorithm maximization future mean least least closely filter approximate filtering allow systematically extension modify underlying space observation model ability simplicity filter environment change environment among square algorithm particular ability despite stationarity inspire recently kernel desirable extend algorithm nevertheless adaptive implement filter grows process moreover explicitly minimize observation naturally class summarize format tackle grow provide understand tracking filter statistical literature broadly recursive filter fig class e achieve despite formulation allow systematically seek relate recursive implementation process naturally lead introduce enable inspired framework classic derivation evolve diffusion filter derive achieve posterior retain new allow observation section binary value observation factor ability tracking signal exist naive online regularize extension square traditional signal vector filter account potentially reproduce definite kernel error output problem gaussian independence illustrate input pair likelihood law number square coincide minimum mmse descent convex online practice dropping yield next e surely proper scheduling capability zero algorithm inherently show tracking derive principle slowly change explicitly dynamic parameter illustrate fig show approximate assume variance remark kalman filter wish recursively posterior since assume k single result evolution p efficiently quadratic grow prohibitive posterior concentrated posteriori isotropic previous simplify normalize rule normalize derivation identical frequentist convergence guarantee vector time complexity weight iteration derivation weight uncertainty approximate diagonal roughly equally couple report estimate update asymptotic provide frequentist track true k perturbation difference bind side get kk steady stationary steady case fail tracking need finite suitable learning rate explore theoretical explain environment latent generalize section instead pure add origin past exponentially result discrete analogue process auto eq absence constant function gaussian center around origin learn learn use budget parameter maximum pruning accomplish drop square exponential use prediction scan report indicate mechanism note expand geometrically old kalman interpret extra however significant benefit maintain compact effect drop pre representation maintain budget reach number section replace example quantify spike code great canonical link function negative approximate q posterior omit maximize stationary imply posterior reduce dimensional analytical solution therefore easily find exist overhead nonlinear shift center slowly track
figure observe large primary large feature secondary cluster method meaningful secondary cluster conventional association primary secondary identify sparse mean table high primary secondary cluster primary cluster primary identify secondary preferable secondary cluster meaningful relevant primary identify complementary although cluster outcome interest neither secondary identify complementary hierarchical clustering identify associate cluster produce supervise much strongly consider method identify whereas two outcome variable job cluster independent hazard patient produce supervise cluster survival choose tuning limitation supervise sparse tool meaningful detect importantly disease ultimately lead treatment option together available request correspond email implement version r package material available online http acknowledgment allow grant fellowship grant de study de interest ccccc sparse complementary complementary hierarchical sparse neither neither hierarchical cluster cluster cluster principal component pca number misclassifie observation se case complementary control cccc sparse primary secondary complementary secondary sparse cluster semi supervise cluster control control pca sparse cluster semi supervise cluster pca email identification sparse homogeneous one identify outcome fail identify conventional interesting strongly secondary outcome method also microarray cancer cluster frequently homogeneous information biological survival cancer patient study wish case clinical characteristic however mean may apply type relevant genetic outcome possible outcome gene pathway pathway biological motivate consider artificial form form cluster apply cluster feature observation feature identify exist detailed situation however intensive prohibitive produce biological way secondary generally biological information similar outcome outcome variable study extensively situation outcome cluster genetic observed surrogate outcome assignment artificial situation outcome variable mean outcome observation cluster considerable overlap variable rate identify secondary outcome figure identify secondary cluster set associate outcome accurate compete simulated world briefly exist set method wish datum differ cluster solve brief dissimilarity measure throughout propose q tuning weight mean dissimilarity matrix fix description optimal discuss choose variety outcome guarantee outcome develop cluster number call complementary hierarchical wish cluster method traditional residual hierarchical give height take remove high secondary yield secondary complementary variant methodology describe applicable hierarchical currently identify situation observe outcome noisy underlie world relatively cluster call supervised feature association example outcome nan semi partition mixture supervise cluster score feature statistic outcome use score semi conventional successfully identify study supervise unlikely truly define exclude irrelevant call cluster sparse calculate associated value testing vary eq identify across sparse give version assign cluster motivation identify secondary cluster dissimilarity illustrate p give obvious choice cutoff note repeat time cluster interest require outcome outcome outcome variant incorporate call sparse cluster strength association outcome variable outcome outcome survival univariate cox algorithm outcome similar sparse nonzero weight experience tend wide therefore optimize tuning unnecessary default default manuscript unless otherwise note generate performance compare complementary cluster complementary generate normal illustration represent primary secondary scenario simulation scenario scenario vary varied three scenario final modify slightly reason sparse sparse record set supervise indicator uniform binary iid variables scenario illustrated assign biological related misclassifie study always objective opposed set three conventional simulate return cluster sparse data evaluation assessment identify risk subject initially individual complete report period participant course description baseline study free individual measure sensitivity analyze see description total primary status include predictor variable control since participant develop follow period outcome control outline conventional cluster weight version sparse second manner identify perform secondary evaluate calculate nan cluster status second primary association cox complementary complementary computational supervised cluster cluster control randomly case partition apply data lasso predict clusters clusters association predict evaluate odd chi predict cox supervise microarray gene survival survival day
single receive subset activation word project activation layer linearly sized feed private filter mlp backpropagation adapt norm usual parameter thank pool closely special reduce projection average pool pooling recover grow ultimately unit relu well non maxout notice well stop pool activation inspire cell investigate vision conjecture optimal another property conventional radial b order motivate q say subspace euclidean dependent independence subspace dimensionality subspace geometry potentially object value unit correspond form center projection euclidean space vary basis shape remain draw partition learn divide instance maxout hyperplane piece space receive signal visualize space unit conventional examine class either mlp inside correctly identify red degenerate class appropriately draw case unit separate curve single space unit unit nonlinear specifically shaped curvature highly train unit mlp two unit separate translate unit able classify class mlp separate construct combination translate rectangle non trivial curvature clear curvature change easy htp boundary green order dash visualization four rate nonlinear red blue maxout green dash curve dot dash fail attempt unit sigmoid maxout unit mnist potential unit design binary non class mark train either maxout sigmoid varied correspond sigmoid maxout ten initialize difficulty conjugate c outperform represent e ten low training even less unit importantly order none run least succeed one maxout unit boundary piecewise b mlp shape curve learn non boundary boundary segment boundary perfectly solve task mistake versus low dimensional demonstrate stationary recurrent prevent result dynamic rnn non activation maxout author recently conventional rnn proposal notice instability linearity mlp associate rnn show bias activation propose construct summary previous highly nonlinear benefit unit feedforward translate rnns effect empirically later deep rnn unit section utilize distinguish architecture neural densely hide experiment expect follow adopt unit mlp unit dataset shift dirac delta number benchmark dataset result task reject connect claim order unlikely order value maxout pooling inspection confirm validate claim claim achieve classification feedforward neural network recurrent benefit state dataset std feedforward mnist ds mnist representative benchmark relatively induce music dataset unit deep recurrent neural understand order layer error hyperparameter filter signal signal table order unit confirm fig clearly even lot confirm interestingly order consist mode plot order initialize order unit three confirm try mlp achieve mlp two follow output dropout mlp mlp maxout mlp experiment rate although neither current permutation version report unlabeled find use estimation average train five fold hyperparameter clear near without initialized mnist phenomenon order average order mlp able sample well result mlp mlp classify four hide unit mlp maxout c fold result dataset optimize scheduling generally search hyperparameter mlp stochastic gradient library dot rnn train transition unit maxout intermediate layer illustration optimize schedule also threshold train art much dot rnns sigmoid suggest superiority suit feedforward acknowledge investigation unit draw concrete unit neural network ht dot rnn dataset conventional rnn novel activation max pooling case naturally recently relate signal important pre claim estimation order important order show define whose scale combination boundary curve claim feedforward network recurrent network test feedforward benchmark face test recurrent task music reveal order indeed dirac delta confirm would cifar computational resource com op universit de cifar investigate unit receive projection normalize interesting interpretation propose pool operator root max pooling instance convolutional cnn maxout unit recognition secondly activation represent unit arbitrarily boundary combine insight empirically mlp consist achieve art evaluate propose recently deep recurrent rnn importance design
naive implementation pseudo construct implementation pixel share however impact small multiple pattern spatial neighborhood spatially iv recall spatially approximated spatial loss cutoff locate position th discard element generate exploit end pattern eq sampling rand procedure pattern denoise pattern uniform pattern spatially approximate implement support figure ghz ccc spatially db db db radius spatially db sec db spatially sampling pattern intensity sampling idea intensity sampling h effectively compute project span notational project bin bin boundary word q sort horizontal boundary bin quantization sort black quantization boundary histogram validity nr sequence determine piecewise equivalent draw implementation undesirable experiment pick stage compute pick image ii main value image patch article noise totally trial compare article lower noise strong refer bm value independent sampling bm bm e db db patch show experiment corrupt spatially sample fact chapter f edu article internal implementation intensity additional root easy verify thus section way determine unique root robustness newton recommend identify initialize tf f sign replace replace continue tolerance piece find cost evaluate multiplication multiplication minimum reduce histogram predefine bin histogram define upper bin let element th bin approximated partition approximate bin common value advantage build clarity pseudo code matlab language begin naive rand generating
order operator heat heat kernel accumulation spectrum field vanish vector orthonormal basis heat schmidt heat hand classical increase basis set basis dot product invariant base mapping consideration embed compact riemannian manifold dim riemannian manifold field diffusion map embed allow distance clearly define diffusion limit vector diffusion behave geodesic dim close manifold expansion spectral isometry isometry class eigenfunction recall result diameter q inequality prove lemma essential pre omit simplify proof notation positive depend ed dx proof except completion trace heat uniformly bound also simple eq ad dm dirac measure hand decay derivative give finish proof positivity hold bind universal depend conclude universal define diffusion orthonormal eigen euclidean clear distance side riemannian manifold orthonormal eigen field span x mx mu expand eigenfunction laplace uv ii mt finish fix isometry riemannian manifold triangular finish definition exist subsequence hausdorff resp x ny separate resp resp inverse distance continuity continuity hence follow denote form integrate indeed smooth plug integrate mt nn nf tp mf frame around show claim q extend thus curve fourth come claim another field since construction arbitrary know combine argument finish construct definition vector isometry pre finish isometry universal inside denote closure subset equip canonical inside subset consist eq lemma hausdorff closeness subset equip hausdorff distance relate hausdorff distance hausdorff nothing hausdorff q grant partially award fa helpful discussion introduce massive valuable section thm close prescribe geometric heat connection bundle lead pre consideration close riemannian manifold prescribe square integrable series heat laplace refer past work diffusion manifold recently high shape introduce brief mathematical algorithmic diffusion map heat kernel associate introduction low aim reconstruct dimensional organization well modify manifold manifold close manifold connection
second trick separately w result bound let property ranking loss iff sample draw q stability ranking eqs plus square svm stability hence enough kl kl admissible proportional dependent vanish context algorithm keep investigate trust region paradigm p close black thin reflect trust region classical proceed model refer trust assess measure surrogate trust expand kl view principled adaptively control trust trust trust define secondly trust assess posteriori trust assessment adjust kl trust kl surrogate experimentally kl divergence experiment online kl es become adjust tb es precision bfgs variant evaluation marker validation es es quasi newton bfgs code default es multiplied e comparative firstly es es es active es comparative quasi objective legend es variant es kl behave three es display benchmark group distinguish moderately ill multi modal weakly objective function last good correspond virtual reach portfolio portfolio dominate es evaluation es es moderate modal weakly structured modal es es meanwhile bfgs performance good art significant ill besides demonstrate bfgs bfgs condition rewrite albeit dominate bfgs desirable quadratic bfgs steady contribution system adjustment hyper frequently call meanwhile accommodate differ ml attempt toward building ml face move successive yield decision es implement learn art quasi comprehensive examine enhance component g ordinal newton vary direction linear rank version separable condition weakly cumulative e proportion line indicate portfolio aggregation kl covariance devoted box es global address surrogate model surrogate model present learn surrogate kullback distribution former surrogate gain comprehensive ill condition benchmark state include quasi evolutionary algorithm kullback divergence es ml depend whether usual enforce never cause enforce usage iteration optimum slow circumstance might whole thus ml hyper parameter automatic hyper case require evidence embed optimization moderate objective optimization quasi approximate optima reach shall remainder adaptation es black attribute es invariance prevent evaluation engineering surrogate survey address limitation box surrogate surrogate hyper update schedule integrate coupling learn schedule surrogate along fast move assumption kullback principled approach empirically show
merge overhead intersect merging merge overhead p lda switch roll level c warm rt break news bin live video novel aid tweet top transfer media transfer lda extension lda core machine significant knowledge domain topic work explore domain examine paper advanced research project multiple program agreement number nf view policy imply research project reproduce volume medium site twitter facebook create demand dirichlet allocation lda handle short fast change transfer document yahoo news modeling specifically develop incorporate informative implementation scale demonstrate effectiveness social media facebook novel real channel people share broad million update daily effective way allocation capture powerful corpus distinguish social media traditional corpus great lda tweet limit character introduce text broad topic content input high volume completely lda would naturally poor occur document actual semantic generative learn meaningful training explore apply address challenge though limitation lda make document change furthermore label continuously grow medium mention twitter application refer sale model lda generate topic assign leave capability semantic feed short tweet message lack interpretation study summarize build discover give sentence use hide crp type window decay logistic decay customer external distance aim text without effort develop speed applicable scale propose extract corpus word consistently mean across contexts corpus extract source corpus utilize yahoo news web page modify nest chinese infer latent hierarchical mainly capability semantic topic document addition human organize work section summarize work direction allocation gain popularity extracting corpus document probabilistic topic represent dimensional generate pick dirichlet use topic indicate hidden word label corpus extend lda corpora address unlike document give hyper parameter topic label lda propose long lda represent topic assign path root path nest chinese restaurant path chinese customer equation path experiment actually encode decade share share transfer share self higher unlabele improve classification task limit label example unsupervise lda possess advantage generative relationship document concrete un intuitively one utilize share much utilize domain robust propose lda lda generative label training domain help build motivation apply transfer use guide share source domain domain medium lot miss feature share semantic recovered helpful well media characteristic prevent text develop overcome barrier aggregate attribute examine entity manual guide topic generation motivation unsupervise analyze social medium fail supervise annotated amount document robust noise exchangeability dirichlet exchangeability crp equivalent dp customer exchangeable experiment noisy occurrence document source structure user document category might see category hierarchy domain could produce leverage domain twitter list collection news category target domain document assign prior assign document two together keep separate label prior hierarchy domain hierarchy measure source topic hierarchy way cosine simplicity knowledge document cosine source root hierarchy chinese restaurant scenario chinese city restaurant table restaurant restaurant restaurant connect branching think unless restaurant restaurant tree restaurant leaf share nest restaurant high inference modification path sampling scheme document corpus document path imply crp gibbs word document path emission assignment thing change parallel processor facilitate part implement processor sampling gibbs sampling help sampler share inference assignment word document topic p exclude processor assign crp merge topic lda iteration state processor path conditional document likelihood give need state crp number assign crp cosine two infinite tree chinese pick tree merge topic tree merge base parent topic assignment count pick tree merge merge topic I supervision restrict document k unlike set number topic overcome barrier noisy sparse domain annotation effort model produce without additionally similarity furthermore model provide detailed unsupervised way provide knowledge apply deep hierarchy source hierarchy experiment supervise use source target text hierarchical know collection ap volume twitter domain yahoo news yahoo news science business health compute tf word category top tf pick target retrieval conference ap ap contain news ap ap corpus vocabulary divide document hold document predictive likelihood manually provide include category work document vocabulary unique randomly divide hold experiment twitter tweet twitter user e initial tweet message message keyword overcome character limit analyze use information n remove initial tweet cc transform term randomly document lda collapse supervise lda unsupervised lda topic yahoo category observation multiple topic map topic technology topic discover discover weight topic hierarchy fig result depend topic hierarchy easily understand tweet small hierarchy cluster nd topic rd key word focus th rd column resource informative relationship nod parent meaningful belong level ideally long dense interpretable twitter ap hold modify fig show node
q q u traditional base algorithm iteration burn draw posterior assessment classification repeat trade datum calculate average set perform intel ghz processor ram default select fit traditional linear svm interface tune hold grid test first approach mcmc class via combination simulate dataset new calculated figure comparable default compare vb performance vb trade svm average second vb mcmc second similar fast balanced vb effectiveness methodology correlate datum dataset clinical trial effectiveness infection patient group per patient evaluate comprise include mild belong observation severe degree belong group consider classification wish patient belong variable visit patient account intercept consider power incorporate effect svms mcmc traditional mcmc vb vb take average vb take second illustrate use vb spam spam mail predict e mail spam spam variable capital capital letter capital letter capital mail predictor standardize summarize conditional scheme retain slow illustrate inclusion variable visualize black spam bar although generate extreme inclusion probability agreement mcmc certainly one vb except vb select font slightly low vb select email inclusion term speed vb favorable take minute compare spam lie disease diameter one consist university medical disease patient age transform approximately split part vb chapter present complete case retain compare approach default case complete level vb datum illustrate vb seem default take second miss competitive efficiency svm deal method acknowledgment research discovery appendix scheme conditional inference element row matrix conditional eq mcmc conditional stroke support use ability handle simulate real easily machine classification utilize cancer diagnosis language likely strength svms formulation elegant convex problem efficiently despite popularity svms handle handle insensitive monotonic computational scalability vi deal irrelevant within ii vi deal irrelevant variable handle identically notable include unified approach adapt multiple g representation within selection miss data methodology model fit carlo method unfortunately slow typically problem vb computationally handle classical facilitate various extension automatic penalty parameter group respectively offer several classical svm sampler notation b aa xx xx modify kind comprising introduce vb rise hyperplane minimal hyperplane reformulate margin cause wrong hyperplane chapter amount discuss serve fit reformulate quadratic use chapter refer loss likelihood formulation pseudo likelihood contribution true remainder distinction formulation normalize formulation ignore normalize formulation formulation lastly nonlinear introduction advantage handle unfortunately mcmc slow mining model rich allow intercept effect nest generalize structure parameter th wish size intercept coefficient would random intercept observation would choose experiment place perform inference penalty vb method restriction approximate density restriction q variational bayesian parameter inference q n classical svm mechanism induce sparsity fit remove section incorporation numerous option zero consider model wish fit induce n mu degenerate laplace scale use introduce hyperparameter desire factorization posterior product optimal density function column algorithm take simplified form probability decide select q
file number nearly acoustic state sake build library modularity organize library core file acoustic open speech moreover file associate several gender range aim experiment objective emphasize section explicitly influence classifier describe encode hmms decision tree improve classifier improve corpus experiment take file english track come contain meta information gender age corpus information consider make corpus environmental create accord train start phase classifier represent hmm belong english language hmm transition hide state probability section encode choose distribution couple initialized iteratively value correlate reason string row notice give acoustic tuple equally train speak classifier tree compose c precision report correctly classifier correctly recognize recall feature tree put representative training classifier recognize meta effective correctly improve language employ divergence kl inferior discriminate compute divergence build obtain acoustic set record state distribution high entry c meta show confusion improve whereas recall significantly machine svm introduce svms regression basic classified support concept hyperplane attribute define belong example point simplicity svm represent belong linearly namely map high space many kernel polynomial kx kx hyperplane data algorithm regardless function intuitively point easy hyperplane attribute highlight svm several different svm represent training obtain directly svms extend across limit mining svms initially solve recognition svms detection system svm diagnostic categorization recognition extensive commonly realize like k tree evaluate simple able module sequential protocol network stack particular support many traffic accounting planning service monitor new newly collect select ip header precisely record define sharing source protocol protocol index type associate flow duration also record record direction traffic aim distinguishing extract author standard noise privacy differential privacy experiment briefly recall concept popular object example model unlabele time traffic flow ip arrival training phase partition select I geometric cluster observation point element cluster run centroid centroid repeat step achieve train implement privacy preserve version privacy latter two distinct recognize google traffic privacy centroid traffic com google com position picture centroid classifier privacy adversary distinguish whether google com traffic area close issue address privacy statistical database worth even formalize deal problem preserve q answer perturb distribution setting server server middle et differential privacy idea amount security due exposure linear decision provide aim develop record et class mention preserve modification data record achieve heuristic select utility protocol aim reconstruct randomize employ spam similarly attack spam aim understanding since inherent never introduce machine meta investigate single record focus correlate classifier suffer information privacy successfully involve define speech internet infer specific used training issue algorithm competition trade able variety control performance retrieve particular use unit ann value combination perceptron represent always call threshold make neuron perceptron hyperplane surface instance perceptron discriminate separable overcome decide bit turn neuron internal hide unit function weight unit internal layer set training go forward actual contribute constitute regression deal prediction domain value predictor domain certain finite type problem phase produce leave branch label leave root leaf conjunction test tree tree discrete represent attribute pair tree characteristic decision suitable solution great variety context implementation extend greedy possible start decision question root ask attribute example relate specific leaf selection attribute attribute separate algorithm evaluate equation value di la di di di ed universit di di computer complex improve experience computer recognize decision dynamic machine effective base superior training training paper attention reveal show infer ml classifier meta classifier classifier meaningful exploit example machine context technology ml approach distinct mathematical phase fed relationship correlation inside classify able historical biological medical diagnosis network anomaly safe release whether hardware property prevent produce example principle train may effective produce available software product along well understood replicate set sense make essentially stock depend valuable fair ask release ml hardware concrete training effectiveness extract accomplish typical ml change devise meta detect classify product get nevertheless analyze ml product consider engine open product addition source software privacy discover make compete though implement composed show meta reveal majority training people mark g american etc software stay ahead competition type privacy datum mining database differential privacy provide privacy novel quite clearly inherently open show something relate set surprising meta formally information mechanism prevent release train valuable propose way strategy ml extract several attack exist ml successfully internet traffic software markov hmms strategy put novel learning technique prevent ml problem attack section successfully train classifier analyze behavior differential privacy relate remark internet market classifier receive please detail algorithm neural bit take eight eight neuron backpropagation eight sequence eventually learn examine hidden possible eight bit typical backpropagation unit eight discover input thus look traffic include internet flow similarly system devise attack train classifier discover simplify label classifier train ann arbitrarily modify classification fact include definition adversary classifier reasonable extract plain namely meta encode meta feature represent denote case list support classifier information adversary wants learn preserve context diagnosis could assign dataset train build first generate possibly balance train meta describe input create corresponding start empty line train create datum set get add line train meta classifier l adversary classifier class belong adversary preserve thank attack preserve sort attack remark attribute essentially statistical among dataset attack extracting suppose improve classification filter optimal meta filter leibl section attack perform introduce probe software attack speech realize late traffic meta detail tree namely meta algorithm ratio total furthermore namely instance datum confusion attack strongly set phase unlikely decide infer training product matter adversary know adversary access employ version engine write hmm gram within
function condition define large eqn eqn hold side satisfy eqn eqn detailed discussion mcp design invertible basis invertible name mcp approximate mcp mcp mcp method eqn regularizer approximated regularizer derivative basis non decrease sharp sharp concavity still hold intuitively sharp regularizer approximate much relaxed condition sufficient sharp regularizers sharp concave proper regularizer se figure fast decrease important mcp mcp eqn se identical regard note special regularizers weak bound se se constraint infinity convex exist eqn hold optimal regression se popular construct expression restrict consistency whether become r u convex relaxed additional avoid convex regularizer example satisfy r result although regularizer approximation norm relax definition se contain se weak minimization recovery include selector recovery succeed regularizer cause regularizers se still give regularizers dc solution condition section approximate also estimation stationary less directional f pd r gap solution invertible se hold integer r sr r g condition except slightly also suitable basis invertible sharp concavity rip sharp concavity gap theorem cd method gap se relaxed error slope degree approximate approximate global derivative know e g mcp accord stationary since theorem regularize note regularizers norm cd regularizers restrictive eqn scad conclusion weak norm experimentally sharp concave regularizers maintain cd parameter choose belong noise regularizer mcp cd initialize illustrate show maintain zero gap decrease trial show iteration middle cd trial cd show regularizers three recovery support weak estimation regularizer verify high parameter compare regularizers fista regularizers mcp regularizers three regularizers vary regularizer omp zero cd stop cd regularizer application pixel camera image fraction pixel image discrete cosine solve rd mask dct denote rewrite rd figure one norm fista establish estimation non regularizer suitable sharp concave regularizers proper regularizer give estimation se estimation condition weak give cd explain regularization regularization work serve design regularizer convex global optima nan condition far r since concave follow u u rt concavity n r r modify eqn theorem support case ts concavity r combine follow eqn j sr tu inequality hold r x suppose continuity concavity sharp concave eqn n r lemma notation global nan r approximate hence r eqn proof step eqn non decrease concave eq eq hence lemma r z p eqn f k I non sum directional summing hence convex regularizer practice fact sparse sharp regularizer include regularizers solution eigenvalue global descent finally cd give solution sharp regularizers estimation descent e norm zero I first formulate estimation assume exist noise noise assumption regularize use estimation true non regularizer study list regularizer decrease indicator varie except give cccc gap satisfy table decrease use right leave weak se need gap estimation sense magnitude regularize eqn gap say regression guarantee sharp concavity gap derive sharp concavity sharp sharp sharp concave concave decrease proportional interval concave concavity sharp concavity satisfy strongly sharp concavity weak concavity mcp sharp concave whereas strong concavity hold besides sharp concavity sharp concavity gp sharp concave th sharp concavity derive gap nan concave problem list gap sharp sharp concave
theory propose multiple graph novel cluster related work subspace theory field analysis multi decade embed consist eigenvector space usually transform graph meaningful principal span top eigenvector laplacian inspire decade subspace notably computer discover work interested subspace interest manifold provide overview theory author analysis manifold work author framework semidefinite representative representation manifold application work author subspace indexing however work community graph representation generally challenge view group category combination graph author laplacian propose correspond adjacency individual supervise learn average combine individual intuitive first category exist work find representation multiple method author combine view optimization unified integrating canonical analysis cca different unified low subspace linkage achieve third try another strategy integrate view directly purpose fourth regularization graph present framework multiple similarity entity framework combine representative include incorporate information author cluster individual propose first provide multi step intuitive easily yet type work manifold focus comes link explicit link help second hilbert schmidt able unified concept three namely kullback leibler help understand finally merge framework yet discuss cluster helpful algorithm inspire study partition vertex subset undirected without generality connect symmetric entry vertex sum weight degree vertex laplacian laplacian interest spectral among variant normalize laplacian define favorable graph section vertex subset solve spectral q denote transpose correspond small column vertex behavior theoretically perturbation omit normalization affect derivation later normalize matrix contain eigenvector normalize get mean assignment illustrative example spectral single vertex sake simplicity dimensional matrix orthonormal usually view laplacian preserve connectivity connect graph map row vertex spectral embed first eigenvector laplacian vertex vertex find vertex task adopt summarize subspace whose relationship layer effective subspace focus describe subspace graph discuss effectively combine merge manifold provide merging subspace subspace map unique manifold subspace represent layer permit tool namely merging subspace mathematically orthonormal span two point define angle angle geometrically use geodesic projection represent comparison choose angle angle assume prior distribution angle meaningful projection interpret mapping preserve yy square rewrite come equality use two subspace map preserve natural take subspace third representation go generic merge manifold intuitive subspace associate namely meaningful originally projection next section ready merge information layer layer graph represent spectral embed eigenvector target number recall merge multiple subspace way representative individual vertex connectivity paper indicate projection naturally specifically representative individual subspace square individual give distance measure preserve indicate propose optimization ignore constant rearrange second rewrite modify eigenvector modify information individual objective keep minimum notice average suboptimal impose small projection merge subspace fact infinitely choice multiple step final vertex algorithm propose graph target compute l u kl cluster direct layer ingredient merging propose summarize implement example illustrate merge learn multi analyze outline link performance scenario subspace realization random variable govern clustering utilize consider closely contain indicator column projection understand negative statistical maximization dependence individual eq see toy toy possibly affect intuitively represent representative subspace let toy illustrate sake color vertice individual multi cluster cluster quality subspace connectivity cluster away find subspace satisfactory perfect recovery toy fig information cluster far away informative quality layer consider representative low cc toy distance toy example imply well assumption subspace namely helpful recover clustering namely provide assumption assumption seem world preprocesse dataset reliability graph layer synthetic real explain comparison multi brief overview form cloud five mixture represent near neighbor graph cloud assign connect reciprocal us cloud letter recover color datum mobile phone region record period graph measure location activitie phone communication gps roughly time device detect device window aggregate year period lead represent modality phone communication assign call matrix eight user email dataset contain namely consider paper two measure abstract clearly title represent word scheme cosine similarity graph third reflect citation among paper assign weight cite cite graph correspond field paper research english n visualize c respectively global view matrix represent dot taking see clearly cluster reason dataset create briefly comparative adopt art interesting detail trial cluster choose world dataset later follow layer criterion stop objective baseline comparative spectral choose individual layer apply summation kernel represent cluster namely summarize b scenario highlight performance higher dataset latter clear generally cluster graph limit consider building average represent smooth improvement quality compare iterative update individual represent step update subspace star merge representative subspace although analysis manifold representation update sense similarity subspace merge slightly merge specifically contain optimize subspace projection manifold upon convergence final combine approach merge information individual minimized layer combination mainly alternate scheme focus subspace require sensible end local final informative layer directly jointly need alternate optimization initialization possible reason explain experiment point involve iterative iteration find representative modify base introduce consensus representation still alternate sensible keep iterative discuss compare performance parameter implementation achieve performance range dataset algorithm permit furthermore worth note reasonably analyze transformation contain graph manifold approach realistic cluster multi graph technique mention interesting inspire suggest modularity subspace contain interesting subspace available graph however study partly center project ed mail entity dataset nature social common interest layer similarity modality graph merge multiple modality end individual tool subspace representation diverse synthetic dataset competitive performance
special proposal multiple try metropolis adaptive trial less demand also adaptive proposal interact adaptation within multi build proposal current support chain worth proposal iteration set htp via provide procedure partition weight tx tx mx I jx strictly continuous proposal note various specification find sensitive experiment tx tx unnormalized importance select proposal accept reject acceptance include proposal strategy proposal improve sake add iteration extend mh proposal adaptation proposal include history mh pp algorithm mh proposal transition leave invariant independent past condition section assume algorithm algorithm many approximate belong unnormalized identify evaluate change choose highlight ability allow adaptation scheme easy new proposal mixture density one draw select adaptation proposal available metropolis adaptation scheme mixture density disjoint support addition change shape instance feature scheme case proposal bound disjoint addition say mixture th improve scheme simple three adaptation adaptation rely interpolation procedure target densitie log let pass piecewise eq interval therefore region illustrate construction point add interval modify compute intersection straight able piece quadratic log piece truncate gaussians pdf obtain point procedure section arise need construct simple proposal inside pass extend straight formally procedure describe look simple since maximization straight form piece domain moreover piecewise constant straight express simple pdfs tail proposal also procedure adaptive proposal arm pdf rather pdf instance idea straight line pass two piece tail unlike tail linear construct calculate depict draw pdfs density algorithm graph section proposal generate interpolation target consider target bound density dx sake normalize rest unnormalized interpolation unnormalize unnormalized normalize converge infinity normalize prove jointly tx tx procedure section desired mention necessary tail benefit tail approach dependence also tail similar previous construction modify distribution control evolution see target distribution metropolis point computational choice first value target update parameter support rule investigate alternative output perspective incorporation point growth difficult produce remark shall resemble new exactly incorporate support point update scheme proposal update split proposal proposal second control similar accept arm exactly proposal accept arm correspond update fix robust specification multiple proposal transition efficiency mh upon mix adaptation price pay cost iteration possible strategy try computational decrease number iteration maximum phase adaptation proposal discuss strategy implement target mcmc technique need sampler generate sequentially conditional direction focus sampler application sampling iteration gibbs x x l sampler follow step gibbs sampler conditional ideally able conditional sample rejection general convergence say chain last gibbs achieve regard gibbs iteration play role validity algorithm mcmc validity arm gibbs set within initial accord simulation report sensitive use alternative apply multidimensional describe sake target support l element correspond domain rectangular piece build fashion incorporate update propose literature could ability algorithm simulate denote normal distribution ordinary method fail visit one arm distribution give performance proposal two inclusion indicate construction see form piece piece straight fig construction uniform piece piece value iterate metropolis chain remove autocorrelation proposal mse metropolis mode give construction proposal high support panel fig stay adaptation graph allow exploration panel full acc fig substantially mse panel autocorrelation lag see point panel confirm table arm intersection interpolation line adaptation algorithm mse arm poor proposal within increase increase computational arm due reject accept cc p good proposal choice autocorrelation high see point iteration however number construction one target estimate acceptance panel acc proposal usually higher add adaptation proposal improve acceptance cost reduce acceleration mechanism time require construction proposal inclusion support proposal sake brevity report use four section compare test control level point autocorrelation parsimonious term support method test random effort support chart acc chart point acc th iteration alternatively horizontal give panel test may lead adaptation autocorrelation acc exponentially update support bring implementation construction less efficient number efficient acc order compare mixture separate heavy tail mixture heavy tail symmetric heavy normal asymmetric mix denote shape respectively function otherwise control tail determine flat control distribution symmetric laplacian finally value tail dirac normal successfully field see year mixture increase arm test remove burn compute accurate start performance set point draw independently comparison purpose also implement matlab toolbox result experiment reference performance acceleration reduce good affect deviation sd superiority slice toolbox point bad bad estimate mix ii support confirm inferior arm fig mix mix histogram inspection reveal explore tail case initial account algorithm average arm ccc concavity presence skewness tail density mathematic life analytical numerical integration even high tail issue technique death life age survival law future concave pricing life pricing see life benefit death life residual life assume set present value benefit respectively arm generate metropolis remove burn skewness quantile result independent run metropolis shall arm instead algorithm summarize panel compare integration panel difficulty arm interest skewness tail example exhibit chart generate tail occur mid left chart arm high lag confirm second set set arm confirm support point already prove reject employ volatility univariate leverage persistence volatility conditional arms mh inverse mh parameter construction eq inclusion point panel exhibit autocorrelation lag mh lag one equivalence efficiency shall stress simulation purpose intervention adaptation proposal sensitivity arm affect arm ability visit domain panel chain confirm raw bottom ccc line empirical histogram bottom arm proposal generate candidate mix chain arm adaptive purpose stochastic construction ergodicity extend efficiency arm find crucially choice support choice heavy tail target explore proposal construction uniform one domain investigate quite control proposal project research education grant european fp grant agreement grant innovation author iii preliminary proof history index joint history tx actual transition probability balance weight thesis accept x tu ta eq define pi tv variation follow bounded dx tx xt tx taylor one hence discrepancy bound remainder th inside taylor remainder replace let assume next iteration assume split x thank binomial hence always decrease incorporate ensure become arbitrarily zero guarantee tx dx monotonic decrease inside point might decrease ensure tx xt take tail heavy tailed furthermore become since support inside tail decrease though increase tail distance goes let x tx tx tx tx tx moreover use inequality expression reverse reject rs initially accept happen accept proposal procedure arm everywhere extreme arm reduce must proposal inside allow improve inside add pdf interval consequently add support point reduce eliminate proposal regardless become fundamental limitation
arise directly partition stage choose simplex simplex draw concentration mix dirichlet contain local dirichlet naturally restaurant interestingly stage chinese variant customer popularity multinomial choose restaurant accord popularity efficiency number per machine cross chinese restaurant sampler super update ratio concentration indicate number balanced point varied concentration room parallelization view gain initialization randomly node mcmc hyperparameter finally amongst worker leave inference transition use multiple chain cluster although allow notation count prior multinomial component cluster hierarchical base hyperparameter constrain perform specific ask expensive mcmc update mixture modify technique move way move along index straightforward assignment operator cluster ignore million achievable typical run machine core focus elastic cloud architecture appropriate dataset million marker color datum clarity inherent transition naturally map describe implement act latent auxiliary intuitively update assume hyperparameter update hyperparameter cluster intensive loop use use demand amazon cloud experiment core gain parallelism communication overhead initialization small inference analyze communication overhead also rely purely avoid approach dp admit principled scheme significantly typical resemble subset amongst variable term operation sampler dataset consist truth sampler convergence see bottom sampler eventually convert slow log run inferential seed show text explore prototype implementation parameterize set weight draw dimension bernoulli draw collapse coin update reduce gain convergence support reliable probability generate show predictive density joint quickly concentration slowly consistent dp auxiliary encourage would interesting characterize regime occur approximate variational auxiliary size increase reach slow separate tradeoff figure dataset use process vector quantization run randomize top binary progress converge mcmc include rapid quantization mm subset worker representation cluster parallelization mcmc dp despite traditional reduce large compute develop prototype implementation amazon elastic cloud core explore run mm row model conditional search induce may new perhaps compute direction contract fa award partially google analog device name mathematical tool widely estimation processing dp gold computationally form transition operator parallel core enable learn parallelization leave invariant reduce test configuration explore synthetic run dimension enable tractable finite projective often construct balance process g process common building wide domain activity nonparametric model mixture cox approach appeal maximally unfortunately latent structure approximate monte leave mcmc move computer set projection interest mind monte bring among scalable markovian technique necessarily computation paper conventional important nonparametric exploit way introduce atom
train adaptively balanced training seem connection improve regularizer still axis argue dropout depict right level study detailed study discriminative independently sign discriminative draw entry first signal penalization parameter optimal penalization wang stanford university stanford cs edu dropout overfitte linear adaptive show regularizer feature operate repeatedly dropout well adaptive regularizer performance dropout improve review iteration feature generic successful theoretical broad corruption effect equivalent take understanding regularizer focus training regularization transform transformation effectively curve objective different regression rare discuss descent regret stochastic sgd sgd repeatedly solve linearize regularize close advance linearize dropout regularize regularization supervise discriminative fitting apply idea several dropout review regularizer unlabele complicated multi logistic chain discuss connection regularization glm define response give quantity log summary maximum copy additive dropout two component draw word dropout else integrate gives take artificial empirical provide act regularizer effect jensen key feature reduce depend artificial way depend model form regularization ridge penalization exploit penalty another consequence relate prediction decision expression effect feature penalty clean formally justified feature quadratic negative log perturbation indicate quadratic b logistic regression feature horizontal quasi step dropout depend insight type taylor expansion regularizer variance train glm bfgs train surrogate without issue optima compare penalty generally accurate penalty tend confident explanation phenomenon appendix training highly confident prediction find fitting surrogate give dropout logistic regularizer turn likelihood regression section apply fact yield linear p penalty logistic write dropout eq additive unlike method allow provide correspond regularization I discriminative dropout effectively suffice confident active dropout empirically document result summarize penalization dropout additive introduce penalty depend potential artificial dropout training suggest logistic perform rare intuition group rare nuisance feature pick normalization dropout product way dropout dropout penalty big weight discriminative meanwhile penalization less simulation table confirm dropout outperform know vector due first term third linearize regularize sgd discriminative run form sgd learn rule et al use g rare seem goal logistic alternative method turn deep unlabele unlabeled way sgd regularizer give eq regularizer center except eq dropout descent procedure dropout equal fisher word consistent estimate fisher use dropout linearize algorithm course perfect particular rate dropout appear goal scaling feature curve circular learning attempt feature consider sensitivity confident prediction cm ccc dataset dropout rt label drop r unlabeled bi
crcr meta header nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan draw meta mesh sep crcr nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan mesh table row crcr nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan model respectively height unbounded jump view scale axis xlabel reverse ylabel name plot axis axis line axis mesh crcr meta header nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan flat meta mesh sep meta header nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan draw mesh sep crcr header nan nan nan nan nan nan nan nan nan nan nan nan nan nan flat meta row meta index header nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan point meta sep crcr index nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan flat draw black point meta explicit mesh crcr meta false nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan draw mesh crcr header false nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan flat mesh row crcr meta false nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan width jump view axis xlabel reverse south east leave south axis line axis flat draw mesh sep meta header nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan draw point meta mesh table sep crcr meta header nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan meta mesh row crcr meta false nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan flat meta mesh crcr meta header nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan point meta mesh sep crcr index header nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan flat mesh row crcr false nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan flat black explicit sep crcr nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan point meta crcr nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan ig ia fig use except discount satisfy theorem obtain symbol slow channel average snr adopt satisfied accord hold monotonic height unbounded jump axis xlabel reverse axis z flat meta mesh row sep crcr header nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan flat point sep crcr meta header nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan draw black point meta row row sep crcr meta header false nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan black point mesh crcr header nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan flat mesh row crcr header nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan flat meta explicit mesh sep crcr header nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan draw mesh sep crcr meta nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan meta explicit row table crcr meta header nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan height jump xlabel reverse ylabel right east leave south axis bottom axis flat explicit mesh row crcr header nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan explicit mesh crcr meta header nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan mesh table sep crcr nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan meta mesh sep crcr false nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan flat draw mesh row sep crcr meta index header false nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan flat meta explicit mesh row sep crcr header nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan black row sep crcr false nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan flat explicit row crcr meta nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan g threshold show monotonic hold figure stack cardinality chain pr b b th constraint solve restrict online reinforcement detail purpose clear usually dp function transition rate assume preserve immediate value factor contrast paper essentially term unit cost discount factor mdp cost discount optimal sufficient monotonic policy consider traffic rate transition modeling corollary feasibility mdp transmission control problem nc random traffic purpose existence monotonic transmission minimized symbol queue transmission monotonicity property varied way firstly prove mdp g unit cost discount nc queue states paper utilize study process dp control nc accord submodular prove f sequential tuple case prove g first ml mf since show first since ji f convexity monotonicity convexity convexity monotonicity q know therefore l convexity therefore convex convexity f convexity convexity ia ia ia b suffice equilibria know similar q ia I submodular q q order monotonicity last explain ig similarly therefore submodular I flat channel work happen adjacent school engineering college engineering national university act email edu paper consider transmission control code two nc symbol channel discount horizon mdp transmission policy delay buffer transmission power consumption concept structure dp transmission queue mdp result dp facilitate discount decision programming code nc maximize communication throughput gain lot rapid transmission improve nc code channel nc compare conventional transmission total transmission numerous nc design channel nc dash color green wireless via nc scenario relate wireless g decentralize stationary highlight exist traffic ignore randomness tradeoff user code hold symbol increase symbol delay studied solve nc transmission minimize delay long run nc include wireless delay decision loss throughput transmission minimize transmission delay queue channel expectation formulate discount channel model transmission dp show service evaluate delay consumption rate physical layer wireless environment channel dash dotted fill color blue blue queue channel wireless channel dp information queue make tuple decision tuple queue evident intractable tuple state curse qualitatively structure optimal policy optimal policy monotonic load shrink iteration optimal simultaneous perturbation stochastic approximation general often optimal certain monotonicity extensively mdp dp basic induction monotonicity preserve maximization minimization adopt high instead usual queue concept originally discrete operational research high work establish sufficient condition existence relate certain uniformly traffic flow property observe cost cost transmission channel etc application traffic rate channel queue cost rise transform dp cost tuple optimal monotonic queue state tuple queue state queue rest state nc model channel dp dp queue example nc user randomly equip queue buffer incoming symbol user respectively control keep make symbol decision code simply end control minimize symbol queue transmission utilize code error obviously concern symbol would symbol code hold delay symbol without code future code symbol channel low snr seek rule discrete process divide call incoming incoming symbol queue epoch symbol per decision great traffic decision epoch queue store incur immediate denote queue begin I queue newly symbol drop call symbol lose markovian modeling snr channel overlap k ig channel model transition channel channel make incoming traffic channel time formulate mdp follow drop epoch state tuple denote symbol queue terminology action transmission forward queue forward queue symbol queue decision statistic model queue transition begin queue indicator current concern symbol delay queue transmission hold hold queue associate queue eq hold unit queue make count hold queue symbol lose queue say account symbol law proportional symbol hold arrival obtain hold transmission cost since code code immediate result form power tradeoff always incoming symbol penalize hold denote scheme e ip since happen decide eq symbol queue queue delay tradeoff form pose consider transmission addition policy always symbol whenever code consider channel form immediate function quantify concern either give structured term unit cost epoch mdp infinitely long discount ensure series countable mdp deterministic dp denote iteration policy iteration threshold apply e transmission mdp say tuple variable mdp dimension tuple tuple action major load number dimension consequence intractable increment cpu iteration worse cope researcher interested monotonicity structure stochastic investigate existence monotonic concept omit lattice verify denote equal form curve characterize formulate tuple entry j submodular strict insight submodular submodular nn general f submodular f fx x x n convexity convexity convexity submodular strictly concept coordinate coordinate il triangular one represent commonly model flow control network monotonic policy optimal transmission queue step follow stochastic measure describe quantify across contribute transmission prove monotonic property convexity satisfy definition property similar follow dp policy monotonic property v action monotonic queue control cost define eq since prove convexity appendix monotonicity iy transmission game model fact convexity lemma integer convexity convexity integer denote dimensional c game game obviously game utility strictly pure equilibrium corollary theorems fig collect symbol fig cost theorem monotonic guarantee monotonic unbounded xlabel ylabel name axis axis black meta row header nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan flat mesh row sep crcr meta header nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan meta mesh row sep crcr meta header false nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan flat meta mesh meta header nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan black mesh crcr header nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan width unbounded jump xlabel ylabel south west bottom axis leave flat draw meta crcr header nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan flat explicit crcr index header false nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan flat meta mesh crcr header nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan crcr meta index header nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan flat black mesh row crcr header nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan width axis xlabel reverse ylabel plot flat meta mesh crcr meta nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan flat mesh sep header false nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan flat point meta mesh row crcr index header nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan flat draw meta mesh crcr header nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan height xlabel ylabel east anchor west axis explicit mesh sep crcr header nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan flat draw black meta mesh crcr false nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan flat draw meta mesh crcr meta header nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan flat point mesh crcr meta nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan flat meta table sep crcr meta header nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan width unbounded jump view xlabel ylabel plot bottom line leave line flat black meta mesh crcr header nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan explicit mesh crcr header nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan flat mesh row sep crcr meta index header nan nan nan nan nan nan nan nan nan nan nan nan nan flat meta explicit mesh crcr index header nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan black meta mesh row table crcr header nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan width unbounded jump view scale ylabel plot south east anchor leave west draw meta row sep crcr meta index header nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan flat black mesh row crcr nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan flat draw black meta mesh row sep crcr meta nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan meta mesh row crcr meta false nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan meta mesh crcr meta nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan monotonic related consider queue e limitation extend investigation monotonicity main summarize ig q I ia depend satisfy adopt snr slow width height unbounde xlabel reverse ylabel name plot bottom line axis line flat crcr meta index header false nan nan nan nan nan nan
precede calculation sample exponentially decay hand suffer probability increase play bad suffer large loss imagine count lattice value eliminate I loss argument paragraph play horizon move action eliminate increase beyond action bind time action count must z ta constraint ensure eliminate eliminate action play depend kl geometry assume derive save put mass rigorously condition suboptimal scale trivial armed bandit essentially optimum know scaling less treat probability horizon regret bind state clutter presentation term thompson mild action observation finitely finite support finitely particle furthermore emphasize assumption primarily thompson continuous fine parameter core drive thompson action enough even prior posterior fine track region nontrivial task work action hx hx aa thompson bandit path thompson thompson exist ta concerned use self inequality help path lie account give constant case appear technique tailor scaling regret corollary quantify improvement naive bandit function marginal kl divergence parameter standard armed bandit arm suppose subset subset arm receive arm identity finite point theorem follow hold thompson ta dependent supplementary several bandit bad scale mab n interestingly optimal bernoulli bandit specialized kl ucb recently thompson prior action individual arm challenge aggregated across arm observe uninformative value informative provide regret observe max useful guarantee kullback marginal la l trivial additive reduction kl divergence turn max reward give play arm max thompson algorithm probability least ta observe regret significantly usual nn provable regret use combinatorial relate vertex dimensional find supplementary apply thompson sampling exploitation problem novel optimization generic regret thompson sampling capture implement sequential particle forward kl divergence adversarial regret bound inspire complex suitable example repeat bid reward learn construct bid another reinforcement complex state markovian promise demand state mdps solve thompson parameterize mdps work theoretical pseudo like armed bandit space rigorous characterization sampling thm multi arm basic arm decision maker play feedback arm observe feedback nature reward prove frequentist thompson support prior improve capture derive bound subset nontrivial feedback bandit learning action select long abstraction fundamental explore face extensively reward severe limitation see ad display maximize ad problem decompose car ad job scheduling number resource machine receive basic arm duration unknown sum average flow source complex action edge total different path inter dependent flow simple mab methodology tackle problem crucial unlikely get reward action choose hope aggregate action complex stem advantageous view mab arm frequentist unknown work bandit problem complex action algorithmic bandit thompson start parameter gets play posterior action correlation basic implicitly pseudo thompson mab ucb information observe reward processor job scheduling thompson merely need ucb basic arm extend ucb framework linear reward flow unclear ucb like treat action besides optimize arguably hard optimize sample thompson routine sort thompson result general constrain surprising almost regret reduce classic mab bandit generalization thompson study thompson particle maintain scenario bandit job idea arm bandit date thompson elegant relatively work notable exception decision reinforcement work thompson bound thompson bandit model purely differ beyond action focus general feedback novel frequentist account complex bandit complex mab bandit setting typically generalization tailor kind feedback general identically space e armed bandit section parametrize space play maker stochastic observation reward observe playing assume clarity borel etc parameter e denote space play action
curvature monotone key retain modular equation submodular satisfy xx vx inequality use curvature definition second therefore eqn also inequality curve normalize recent everywhere ellipsoid satisfie compute correspondingly query fx tight curve modular infer within oracle x ff proved show tight factor compute within use construction curvature rx nh chernoff distinguish reliably polynomial function imply contradict simple modular submodular notion curvature strong notion curvature fact follow definition curvature q series eq x immediately slightly fx sentence seem I form eqn involve true correct approximation complementary modular whenever average extremely however approximation fact tight modular submodular submodular curvature modular x x x size etc replace immediately imply modular use ix jensen inequality q modular function since modular ai also give vector ix q concavity particular derivative last hence curvature obtain weak bind modular bind modular root modular probably unknown polynomially good formally must reduce problem binary curvature j include sample value know bind curvature still go though monotone know singleton every run exist learn normalize version final proof idea curve define within curvature hold curvature similar line show polynomial factor curvature exist function collection hardness proof end submodular curvature polynomial example polynomial x proving result lemma general contrast curvature know concave modular immediate concave modular fx k prove adapt curvature modular reduce handle I part divide complement generate slightly let repeatedly flip fair q observe inequality point sample approximation generate label imply imply training many next minimization submodular constraint apply optimize surrogate approximate propose widely surrogate theoretically throughout approximation minimizer fx minimize find set part follow lemma useful essential combinatorial theorem modular approximate I try modular upper bind simple approximation submodular modular mx f imply dependent approximation solution minimize show thank lemma part lemma simple iterative yield performance practice approximation practically bad approximation submodular employ bind mx function modular hard modular algorithm minimize nonnegative important modular optimize optimize translate lemma tight curvature along proof simple bind monotone submodular function impossible polynomial algorithm factor bad directly observe similar hardness close corollary preferable tight exist submodular curvature algorithm moreover factor n nf problem ground distinct find submodular bad similarly tight provide curvature give curvature achieve factor n curvature adjust version particular hx rx chernoff indistinguishable submodular removal curvature achieve ssc curvature curvature lower bind result construction factor number sparse specifically surrogate exact neighborhood flow maximize mf refer approximation bad approximation theoretically network construction group approximation flow polynomial moreover modular convert form xx flow family tree constraint show curvature span bad directly corollary bind corollary case optimal curvature polynomial factor rx x min x n graph construction indistinguishable high notice minimum respectively well able contradiction apply combinatorial tree perfect submodular function corollary factor tight give approximation span span tree perfect indistinguishable ratio provide vary dotted solid bind visible submodular cover submodular hard approximation guarantee bad matching hardness submodular well problem perfect match hardness empirically demonstrate define hardness throughout rx rr hard specific curvature approximation factor follow vary curvature figure illustrate much well approximation lower upper figure factor improve bound find illustrate curvature grow polynomially theoretical suggest affect study approximate learn prove bound almost match submodular submodular optimization effect curvature term question learn finally seem curvature dependent regret submodular acknowledgment point early anonymous support national science foundation grant google microsoft award center office contract amazon services google blue intel microsoft oracle yahoo wrong order bind call divide dense use modular intuition dense bound diameter correspondingly shorter tight component hybrid provide exact interestingly submodular berkeley observation section corollary false approximate submodular pac minimization curvature provide refine proof use picture economic game recent year finite subset submodular recent year polynomial ellipsoid detailed summary submodular admit approximation even convexity variety submodular interest submodular minimization admit stand sharp suggest many practically case importance quantify amenable limit show improve sub include additional towards address notion deviation submodular optimization quantification practically curvature bound submodular complete picture curvature submodular quantify influence curvature moreover rely allow easy submodular state ground normalize additive submodular value space multiplicative factor learn like family study approximate monotone building monotone submodular instance weight cluster nonzero eq submodular maximization submodular
filter locally spline knowledge supplementary document asymptotic locally spline idea filter spline spline filter trend argue spline would trend filter due discrete operator explain locally spline meanwhile slower concrete primal dual problem locally regression course nonparametric toolbox highly offer regard preferred simply contribution toolbox locally adaptive balancing strength spline spline manuscript mainly comparison filter widely wavelet spline spatially trend filter become purpose nonparametric tool evenly spaced input extension supplementary trend separate another distinguish feature fall framework locally spline smoothing spline fall synthesis synthesis build strategy operator undesirable behavior natural former vice versa discuss context define extension trend construct synthesis example trend filtering outline throughout proper spline locally spline give example trend filter filtering adaptive remarkably discuss regression spline filtering pose identical allow exactly continuous trend always piecewise polynomial spline bound total trend filter locally latter already rate grow variation nonparametric idea trend trend filtering versus defer supplementary smoothing spline unique smoothing estimate minimization dt odd cubic draw comparison spline write spline follow computational regression remarkably unique knot recall spline last spline precisely function solve expansion natural knot product nd collection function knot ridge generally smooth spline trend smoothing expression fit expression trend filtering replace penalty matrix spline derivative analog like nd splines input spline penalty second diagonal equal difference smoothing utilize square analogous usual comparison unless adaptively might exhibit fine spline filter smoothing spline optimally tune smoothness prove filter estimate smoothing spline achieve optimal rate class trend filter simulated trend filtering spline program fast spline speak spline trend homogeneous smoothness entirely entirely throughout display smoothness spatial call knot space right become noisy evenly spaced filter tune panel level true smoothing spline considerably flexibility estimate able small degree cause leave side show panel top display trend spline degree freedom trend local smoothness rapidly display average squared spline range complexity dot line minus standard deviation aside underlie mention trend filter smoothing spline consideration recall contain evaluation natural spline knot input integrate nd fast spline smoothing fit operation primal filtering iteratively sequence path interior spline filter path section trend path vary add computation solve overall knot fit polynomial algorithm knot trend knot take second smoothing bottom panel figure take period trend compute smoothing spline solver quick calculation path trend less take second comparison locally regression spline locally intensive spline property name would order odd require start define boundary define locally regression derivative denote q recall total reduce briefly difference spline dimensional problem trend lasso estimator locally adaptive reader work explain locally regression spline minimizer generally locally regression spline spline knot contain outside outside though exist knot locally adaptive main study difficult easy determine restrict fortunately show result apply estimate space apart particular evenly focus mention purely example come filter equal locally spline e match also solve degree knot jump combination expansion contain derivative knot recover locally adaptive arbitrary q lasso take truncate let set truncate spline lemma truncate otherwise worth investigate spline somewhat last trend write sum formulation helpful explicit evenly spaced point trend really important trend lasso form transform spline satisfy order cumulative st discrete document result locally adaptive spline evenly regression filtering solution trend filter spline common problem different supplement trend locally order practically minimax locally spline converge evenly spaced regression sub constant denote sequence locally belong minimax minimax rate function well hope fit spline smoothing let denote denote embed function modulus working norm eq rate spline trend filter rate argument know spline set filter estimate spline trend exactly instance really trend term would follow trend trend filtering locally spline fit q matrix fit intuitively plausible similar know exist result lasso e predictor lasso end supplement outcome asymptotic drive primarily apply work trend filter locally spline converge evenly spaced input integer differentiable th tune fix supplementary document gaussian imply moment locally total variation last recall supplement reduce limit formula control supplementary apply corollary supplement conclude divide triangle recall arrive assumption satisfies trend minimax spline elegant involve derivative degree knot line argument require place minimization complicated contain spline function prove trend filter locally spline rate minimax grow total constant grow rely locally spline trend adaptive draw error integer depend denote th order order trend invoke variation say eq reduce imply show document conclude corollary supplement divide spline parameter th trend satisfy q achieve adaptive regression spline function grow result worth point restriction filtering result trend spline exactly yu universe observe great information line display spectrum curve gray spaced log true dramatically side smooth variance see error around curve spectrum alpha forest estimate apply filtering spline wavelet spline extreme proximity filter use spline produce cubic smoothing spline cubic trend wavelet vanish moment method wavelet interval boundary symmetry boundary behavior example use algorithm transform generally wavelet point log run top leave remain panel plot estimate top panel capture feature pick spike side spline bottom fairly fit magnitude wavelet smoothing spike leave top right panel gray filter job peak around fine detect extreme peak far computing squared draw panel spline attribute capability theory square achieve minimax seem region trend filter spline vary domain fine level example recent proposal see fit complicated allow ideal scenario variable spline division domain true function degree freedom put see set smoothing spline freedom side freedom plot minimum advantage setup perform par trend extension filter newly spline adaptive former fast locally filter complexity primal slow rate locally adaptive spline broad class one trend filter conceptually locally regression matrix something factorial dense retain locally adaptive minimax although perspective helpful original beyond give finish discuss briefly discuss extension filtering input filter discrete operator multivariate analogous truly even second point univariate trend additive consider contribution marginally estimate often efficiency extension often fitting nonparametric regression comparable trend worth investigate synthesis synthesis concept processing act add together fundamental undesirable term concept scientific synthesis characteristic formulation synthesis atom block set penalty represent problem representation filtering fall st difference representation synthesis framework factorial apply former filtering across basis trend filter identically quadratic trend filter piecewise argue perspective perspective estimator reverse also though focus piecewise input additionally zero chosen help analysis easily eq q sparse could taken sequence curve response important construct synthesis unclear yield another possess order smoothness trend eq panel figure difficult order smoothness synthesis framework synthesis versus future comment lead thank helpful encourage minimax rate thank help thank taylor place underlie establish discuss extension corollary support nsf dms study tool al trend minimizer penalize derivative surprisingly trend th degree adaptively trend really discrete input mind produce particular spline sum square derivative across locally discover filter local much smoothing remarkable similarity finding notably prove trend variation filter adaptive already converge minimax core together fit value share predictor introduction assume function input evenly space relaxed regression diverse well include polynomial spline wavelet nonparametric regression trend th trend filter optimization tuning operator multiplying facilitate hence filter trend fuse lasso pure fusion penalty penalty filter nonzero easily first operator word say st times analogy st derivative polynomial trend filter piecewise evidence later trend filtering check sign explicit th trend adjacent operator bandwidth trend filtering define strictly convex call signal exact filter estimate sign freedom knot section provide justification
kernel provide concrete kernel three divide distribution divide average together estimate actually provide family estimator choice result corollary specific kernel deal behavior enforce strong boundedness x deviation bound variant infimum attain replace strong ready associated assigning theorem statement involve three quantity quantity serve familiar finally integer tail decay theorem involve moment may arbitrarily independent square average somewhat complicated concrete interpretable specialized argument intuition typical zero number remain dominant decrease increase choose corollary machine tradeoff compare bind setting guarantee upper bias square two relationship type familiar empirical point yield minimax convergence equation yield without radius involve quantity triplet q term oracle know distribution plus situation verify alternate suffice instance replace alternative eq essence sufficiently prediction replace provide somewhat set approximation inspection bind roughly particular choose balance familiar estimation error impose ignore logarithm grow grow remark ahead generalize divide fine note regularization choice regularization desirable leave turn derive consequence class hilbert space broad outline remark balance optimal number finite meaning eigenvalue satisfy example kernel generate consider assumption hold processor satisfy q numerical suitably large square kernel rate know universal range observe memory low corollary decaying include space smoothness order kernel lipschitz smoothness polynomial consider hold processor upper final strictly include rkh processor bound behave partition see decomposition algorithm interesting split attain convergence less turn corollary aspect term error bind strategy equation standard follow auxiliary bound definition lemma represent constant assumption contain constitute theorem straightforward theorem present inspection enough solving take th arrive moment take identical result program rate claim sum make eigenvalue bind negligible partition boundedness inspection statement relate previous technical state claim add subtract variable proof old equality suffice inequality formula empirical minimizer standard rest subsample lemma denote side inequality proof universal upper allowed condition proof suffice property expectation jensen q basis old term combine inequality complete simulated datum design theoretical prediction ccc bc bc error size varied convergence explore subsample simulate normally distribute lipschitz reproduce gram identity eigenvalue convergence radial normalize vector select typical show figure curve release actual year nystr om subsample regression approximate two partition execute deviation time enjoy performance nystr om nystr om note parallel computation accelerate approximation appear trivial task regime conference publish know establish result nystr om discussion detail fast runtime ignore loading square bad performance use final averaging necessary compare latter achieve error rate negligible establish decomposition ridge achieve decrease theorem notably optimal substantial benefit subsampling scheme cost scale nearly require implement estimator require matrix evaluation contrast nystr om base subsampling factor roughly nystr om subsampling rate comparable situation compactly linearly appear scale arise scaling fundamentally grow substantially hope continue paper point mistake version manuscript facebook fellowship support grant generating operator outer f v devoted lemma shorthand jensen remainder eq q fr consequence conditional design fix define operator consequently since truncation argument dimensional expand since consequence equality hilbert apply show expression desire shorthand matrix lemma equality event n complete know along elementary shorthand apply multiply turn component whose operator factor little column cauchy schwarz take respect old turn schwarz use expand initial dividing recall note inequality claim us matrix value rademacher argument return numerical complete remain lemma begin instead moment directly triangle thus right turn term sub apply take root expression outline triangle elementary lemma place proceed eq q turn bound optimality find eigenvalue find event still term turn term claim follow claim analogue lastly note eq inequality prove recall matrix taking yield schwarz notably obtain lemma inner expectation remainder proceed divide let design outline usual expansion basis bound proof reference vector analogue remain control rewrite within provide q section inequality apply complete consequently yield proof jensen previous apply old equivalently inequality global minimizer inner product precede old moment q establish claim rkh assumption imply order addition optimality imply arrive early equality truncation argument proof second remainder proof devote expansion lemma coordinate I proceed derivation follow recall shorthand expand expression right provide universal obtain universal apply schwarz find eq establish particular yield universal establish em ccc zhang electrical computer science berkeley establish decomposition ridge describe compute average global lead substantial reduction ridge speed retain set concrete guarantee processor nearly finite kernel gaussian polynomially space allow substantial music complement benefit form real response estimate use future frequently quality error book estimator data focus reproduce hilbert rkhs widely establish estimation error rkhs zhang refine extended computation challenge large dataset implementation must require cost prohibitive sample avoid expense exact minimizer kernel cholesky nystr reduce prediction establish maintain guarantee detail second stop iterative descent conjugate early provide stop aggregate complexity appeal randomly compute hilbert even respectively moreover naturally parallel speedup processor function processor study include perceptron algorithms bootstrap demonstrate divide infinite regression solve independently regularization parametric problem care consequence demonstrate sub base nonetheless regularize bias local variance
experiment multiple several many pair covariance contrast genetic negative achieve exact negative se mat ern synthesis genetic approach commonly covariance valid covariance genetic derivation optimizer concept possible find good genetic perform synthetic show composite perform default datum focus synthesis contribution synthesis process non descent strictly execute tune computationally hyper additionally usage derivation support analyze compare approach enumeration compose well function often beneficial approach especially process acknowledgment thank lead center evolve gaussian genetic programming critical model model square default however lead composition genetic programming sentence derive approach synthetic find well covariance trivial hand tune integrate knowledge hand experience frequently function machine learn regression relate input model functional dependency produce value describe covariance genetic describe prototype tree proof yet evaluate suited kind problem indicate low gaussian function another recent contribution flexible covariance gaussian process composition base calculate genetic use evolve mixed find svms tuned cross grid contrast dimension model prediction solely inference must marginal matrix inversion must optimize often accomplish ml fashion optimize quasi bfgs gradient hyper additional prominent application genetic symbolic synthesis without genetic make possible structure combination parameter composite genetic follow genetic sure programming rule composition function represent covariance mask periodic covariance operator add scale covariance mask terminal mask effectively reduce match dimension check evaluate symbol range currently parameter genetic hyper solution descent process function implementation experiment mainly set forecast compare default tuned function set present randomly identify two computationally expensive
validation powerful complicated manuscript tune variable tuning schwarz scad papers wang et wang zhang et generate tend generating model manuscript procedure new combine strength stability stability enhance lasso know root cutoff pre need recently select manuscript et aim avoid incorporate strength manuscript organize asymptotic regularize numerical variable response center without generality regression sparsity subscript emphasize manuscript exist procedure selection consistent fact specifically yu li adaptive five mutually exclusive case tend sign sign consistent tend consistent tend tend tend select select lead might two degenerate exclude incorporate cross avoid fit et avoid fitting criterion aforementioned parameter randomly agreement c fitting contain empty full degenerate pre exclude pass run base b ratio grid play role assumption fan scad exclude pass distinguish et component ols prediction I generate ii scenario dimension example lasso scad set incorrectly select summarize htp select cr scad lasso cr lasso scad lasso scad percentage prediction perform well small also scad perform well cv show happen time seem select correct correct zero datum generate ii adding examine three ii ii I set average cr c ii scad pass exclude outperform snp large exclude pass snp large exclude effect case pass big scenario scenario examine sparse prediction summarize htp lasso lasso scad pass true selection yield become intuition variable selection similar subset effect would apply would drawback strength show worth note meaningful carefully scientific variable practically pass treat tool word limitation partitioned limitation order dataset stability become popular
kn kk expert item complete weight update success recommendation user new update weight correspond predict update model expert article popularity news news topic see far decompose modelling property process assume user mainly look last news I approach multinomial read news item read news user news trend frequent popular read least last news click equal click item click receive popular news website important tune news item news website news news unique news website two update first news news popular news news noting news neither happen dirichlet probability bayesian update recommendation recommender apply domain news concern news recommender one system execute user online offline change estimate news item estimate candidate context leaf recommender e set n nk recommender one context tree recommender build sequence item expert predict ct probable topic news dirichlet allocation lda model title news word lda news probable topic context tree generate topic provide score eq article system new set news consider news item equal remove metric filter due popular define ratio recommend essential read something topic range topic bring popular candidate read average interval dataset approach strategy recommendation fact integrate popularity sensitive number notice prior prediction make popularity behaviour recommender system size pool however update performance getting look recommender system item ct current important sequence root expert performance z news easy relatively accurate recommender system high accuracy offline recommendation play role recommendation trade may mix formalize specify accuracy dataset popular probability expert popularity simulate small recommender find h give evaluation illustrate h utility dataset utility parameter dataset performance dataset purely hybrid well third approach perform much else thus ct large rapid topic recommender system accommodate dynamically news sequence topic define expert popularity news examine propose incremental model continuously suit evolve trend reader preference require context surprisingly context recommender methodology whereby tune parametrize accuracy implement ideal interested trace qualitatively recommender future perform news website pt ct title news recommendation context th conference recommender page organization online create need filter article product news challenge news subject trend want article content insufficient reader introduce recommendation context provide news recommendation recommendation flexible challenge news recommendation storage recommender design news book little news article identifiable regular website strong user except news website already recommend article need available reader individual article recommendation usually manually recommendation automatically recommender collaborative adapt several challenge news news evolve quickly old recommendation front base recommendation update fully ct define increasingly partition context article tree context becoming grain deep tree actual recommendation associate recommendation expert take news methodology real website article want know preference make recommendation content item example collaborative filtering recommendation aggregation google algorithms latent semantic build give combination news probably recommendation content liu et news click behaviour create profile propose bayesian recommend interest news group combine generate li al bandit recommend select news contextual information strategy click click recommender tree originally apply symbol context closely markov click suffer complexity exist maintain approach scalable apply recommender system tree big al consider finite maintain generate recommendation suggest use specific mixture follow et al factorization chain recommendation complementary limitation big tractable many read news item visit unfortunately special tree key idea behind ct create hierarchy arrange parent topic article add create correspond old soon pool key expert expert prediction user read particular read prediction associate context recommendation website read list article read article read time sequence topic build corresponding length sequence similar next wants read formal context edge context initially read leaf subset tree fig main article product continuously maintain pool change use dynamically
solution contain view scale ratio approach could call odd estimation world problem marginal estimating likelihood obtain replace odd therefore identify sufficient likelihood classification derivation show likelihood ratio odd existence solution mainly exist unique possibility care happen integrable dominate fp integrable continuously differentiable derivative strict strict convexity must look case imply combination dominate q part immediate begin measurable q implication g h g density regard iv follow observe everywhere imply theorem ii calculate follow regard inequality rgb cm remark theorem definition law estimate unconditional representative argue conceptually sound alternative law probability unconditional probability total odd coincide literature odd transform unconditional population odd unbiased building block probability elementary whole call book probability author name frequently calculation impossible occurrence event might changed incur unconditional probability already know unconditional forecast argue however unconditional produce see fundamentally complementary unconditional training solely ratio issue present population class unconditional test conditional also write alternatively unconditional odd observation analogy turn special paper classification shift covariate concept odd demonstrate detail show population become clear odd need advance algorithm provide sharp covariate shift estimate sub field absolutely respect expectation interpretation historical g field represent field additionally application specifically might sure meaningful hold case regard exclude discussion continuity requirement general note event I exist equation eq let generate conditional proof give hence total odd unconditional ratio theorem iv odd population ensure normalise theorem interpret circumstance associate special countable read basically infinite rating probability measure sense corollary uniquely marginal odd respect exercise estimate unconditional class produce mean poor argue hence odd unconditional special total appear unconditional define absolutely respect modification replace nonetheless intuition mind total unconditional class likelihood proposition corollary contrast
sum relation affect algebraic counterpart homogeneous whether satisfie equation equation come relevance degree ideal rest identification cell counterpart add probability cell sum column computation carry ti contingency represent theory contingency repeat ideal generator analytical translate cell table similar repeat column quick inspection generator ideal degree generator ideal p binomial merge two next valid independence model ccc column indicator row example independence row since move process iterate twice contingency column prescribe merge last corresponding object third row define matrix row lemma remove column particular column thus immediate iterate process column cell table row new add equation address happen question example answer question address natural converse model happen generator think associate ideal ia condition binomial ideal find perspective study contingency table focus attention agglomerative framework moreover name bi available range application molecular procedure find pattern underlie essentially independence support determine cost statistical find application acknowledgement partially university research thm proposition thm thm row yield add part model paper aim understanding technique contingency table contingency outcome categorical rectangular non integer contingency table datum table ask way simplify therefore column contingency respectively literature cluster contingency row moderately small exist fall reference correspondence derive chi wishart density model implicitly count count cite reader several example cluster definition square exploratory lack underlie agglomerative algebraic order counterpart statistic use algebraic paper availability algebra efficient package polynomial computation contingency table define suitable set add independence encode special square table statistical contingency table basic analysis contingency table algebraic statistic focus especially term column new table fall independence ideal operation general basic algebraic basic idea algebraic geometric row contingency table merging column briefly devote statistical motivation row contingency section notion image variety parametrization irreducible algebraic ideal vanishing via produce extend notation get special feature reflect geometric matrix hyperplane cone variety replicate geometric property algebraic ideal generate degree assume hyperplane cone vertex remark construct enough show vanish cone contingency table outcome categorical variable contingency count contingency probability model table contingency table linear interior log ordering cell freedom write nonnegative extend
cover cover eq suppose explicitly use cover cover iterate cover ball exceed construction use suppose order iterate cover exponent fix iterate obtain sense order ball finite banach low technique incoherent dictionary good cover good mostly banach close unit unit respectively open ball center drop follow proposition banach q proposition concentrate close ball cover word prove banach discuss hold inequality follow construction proposition construction build incoherent hilbert follow corollary section incoherent smooth banach modulus smoothness make technique incoherent dictionary work banach ball cover consider guarantee cover banach space norm indeed give functional define describe cover satisfy prove indeed eq proof place get banach modulus smoothness banach property banach space define eq parameter banach modulus functional functional eq imply case q case uniform imply case case assumption discuss construct euclidean frame condition q need imply satisfy go contradiction eq contradiction lemma account specify get question hadamard popular correction code matrix order matrix hadamard mutually orthogonality keep multiply row simple hadamard matrix hadamard build order hadamard order corollary recursion hadamard system absolutely frame hadamard divide absolutely frame mutually orthogonal element discuss application incoherent dictionary normalize equip theory spherical spherical word sphere absolute value product distinct large spherical code call dictionary treat eq bind corollary statement coherent banach space publish dictionary banach functional definition functional dual dictionary uniqueness equivalent smoothness particular normalize equipped denote simplicity form row vector coherence column coherence fundamental result absolute implie thus banach particular modulus smoothness actually space without eq continuous solution otherwise section demonstrate banach modulus power remark corollary get denote bind
use activation layer activation hide visible unit temporal visible delay delay convention delay machine generative model visible hide seek structure energy bias layer hide configuration activation give configuration rbm visible hide vice versa exact average exactly distribution sigmoid extend modify rbm lead normal variance constrain value take deal rbms train likelihood maximize visible write note compute average intractable h v h visible chain leibler run mcmc maximum far involve intractable average become purpose autoencoder weight represent mse reconstruction layer iv I reconstruct visible layer sample reconstruct k descent cost autoencoder often corrupt perform gradient approach note pc u rr rr machine whereby visible visible visible layer conduct manner rbm divergence ball box discuss contain temporal model energy configuration visible layer delay delay expectation cd cost rbm estimate past layer layer layer way layer receive visible present additionally present visible receive input visible free whereas visible energy rbm il l visible easily date dynamical video cd seek datum allow model reproduce capture bit essential frame causal frame future latent noise property represent bottleneck denoise autoencoder past delay propagate activation consider gradient reconstruct constrain represent model frame static initialize frame obtain write descent train divergence fine tune complete gradient backpropagation layer make automatic package implement temporal mlp stochastic descent use batch r ti kl lm w th il b apply motion dataset separate validation train model evaluate frame binary rbm assess human motion benchmark temporal connection temporal frame train present require sequence generation take layer time visible initialize gibbs trial squared repetition table significantly outperform cd counterpart improvement far improve take take cd argue improve simply deterministic performance layer mlp architecture outperform keep back ta significantly frame visible frame mse still error ta along reconstruction prediction middle sd delay ta unit delay ta hidden frame delay ta frame htb generate motion capture great generalised forecasting competition competition forecasting range augmentation generate future successive train generate show average kind performance competition compare usual unsupervised ta continue improvement cd performance ta increase frame task performance allow generative network show achieve improvement generative frame dataset consistently motion able mse temporal alone ta seek constrain reproduce datum surprising consider autoencoder task beneficial context rbms representation fashion widely unsupervise pre rbms modify rbms temporal dependency consider bias structure denoise autoencoder frame corrupt version frame reconstruction improvement frame across reduction take estimate capture forecast furthermore looking prediction across oppose reconstruct believe approach autoencoder towards boltzmann restrict autoencoder great also grow interact require efficient scalable stream think discriminative predictive datum receive boltzmann rbm easily cd way rbms temporal temporal notable success learn latent whole mark success generating source narrow rbms generation general causality seek maximize likelihood give dynamic learn explicitly enforce
formally variable make together respect ratio wise change way upper write successive respect change difficult follow define eq remark supremum exponentially anneal distribution end chain q recurrence iteratively apply stationary take conclude recurrence adaptively side available error recursively upper initial consequence situation particular satisfied proof accuracie immediate writing norm successive closeness closeness total accuracy total measure induction walk chain act function function choice map expand variation increase inductive step tv random self barrier hessian easily calculate empty interior constraint self barrier quadratic barrier barrier sphere parameter constant self nonempty close form hessian barrier define construct distribute accord sufficient mind situation conjugate apply technique calculate norm furthermore set priori would accuracy corollary chain eq dependence especially geometry sphere side number datum dependence exist normalization large far walk partition additional concave posterior turn convex history recognize truncate nice normal provably furthermore track illustration study normal generally convex compact mean drift depend ball lipschitz constant depends solely aim accuracy drift sufficient quantify quite walk track change r drift achieve perform per model successful classical parametrize classical method may random pick mixture incur outcome cost strategy collection sublinear regret let turn good minimization next forecaster context exponential stand leibl proceed distribution bound consistency continuous stage follow observation loss regret easy efficient produce desire mixed use concrete boundedness condition choice therefore regret write result hold assumption randomize minimization regret divergence follow yet deterministic work back deal instead translate require barrier implement whenever match spend invert finally idea inspire method quadratic convergence interior point make curvature investigation development curvature alternatively let view rearrange weak writing square root proof suppose end contain lie possess whose q non convex multiplicative riemannian metric projective relation notion hold self nesterov segment conclude explain fix unlikely analogously reach step easily check transition conclude imply proof closely write apply rearrange fact combine kl simple show lemma prove origin ellipsoid ball achieve generality far generality lie riemannian clearly c point imply thus q technical likely argument constant euclidean hessian origin identity determinant norm deviation dt complementary error function less suffice show proceed union first prove exist origin observe least measure concentration interior nesterov convenience define barrier converge call hold barrier first eq theorem remark part grant dms walk rapidly track time vary develop step calculate propose update arrive fashion track vary truncated sample change repeat method regret remarkably walk track compact subset empty interior measure suppose probability lebesgue markov chain distribution comes show variety thus body arise notably sequential aim time vary datum posterior change use sample resource exploit filter therein anneal situation mostly calculate heuristic body present heuristic coupling method yield geometric decrease distance stationary know context finite state discrete circle via constitute important unimodal recognize g started follow series improvement recent advance obtain provably small walk provide free interestingly idea change theory barrier see appendix self close availability barrier separation barrier handle geometry mix diverse domain posterior conjugate dimensionality constraint constitute location well extension streaming technique via anneal example concern concave exponential follow geometry induce self barrier key give log concave introduce section contain appropriate devoted finally section contain markov proposal covariance approximate geometry geometry play point geometry markov similarity point refer introduction interior method subject center barrier barrier barrier point recursively barrier induce riemannian metric assign product riemannian infimum whose set metric ingredient possesse concave theorem ensure two subset markov follow remark convexity bottleneck fail walk get part borel field give initial chain collection let choose specify contour scale chain introduce adapt parametrize step write explicit implicitly sample stay uniqueness stationary simple detailed respect whose
estimate display true suggest population compare abc choice tolerance bayes bayes may apply random social network adjacency actor dyadic relationship otherwise edge connect respectively b posteriori evidence propose end experiment carry suggest extend situation markov field large exponential acknowledgement foundation grant article include program replicate exponential please see file contain bayes factor field due devoted bayesian inference conduct doubly intractable extend issue bayes yield performance material available bayes algorithm field statistic exponential network popularity complicate base despite wide intractable overcome approximate joint much literature address often example auxiliary computation name gibbs random term doubly intractable intractable un normalised choice class exchange innovation describe exchange yield statistical factor choice gibbs field explain statistic statistical field important role area ise lattice frequently relational excellent introduction popularity discrete problematic inferential point view explain begin value random field trivially situation approximate distribution show exploit intractable present tackle variable exchange explore use allow amount sufficient include evaluate intractable typically yield algorithm bridge posterior typically evidence posterior similar manner describe apply likelihood equation assume likelihood doubly one term problem motivate base exchange outline allow intractable interest augment might detail swap exchange close ratio exchange assume metropolis algorithm target move ratio sampling point implement exact sample obvious stationary justification exchange evidence constructing augment one slowly framework far define temperature target exchange detail h ns otherwise chain interact simple center obviously elaborate proposal base differential evolutionary monte allow efficiently transition intermediate connect mix space exchange target distribution swap exchange move target un normalise un normalise unbiased draw denote population auxiliary draw right draw draw suffice store development exchange draw manner write approximate usually kernel make high example correspond methodology extend parameter within social construct mcmc framework factor pair outline compete average normalise distribution exposition correspond sufficient additional context mcmc exchange apply similar full bf fy exchange draw approximate bayesian distribution area excellent development abc examine summary compare datum arise present situation need choose summary present algorithm allow inference specifically make lead abc describe accept ise size variable sufficient statistic q mean lattice point henceforth index bottom neighbourhood along lattice addition count adjacent correction neighbourhood lattice thus express ise lattice experiment neighbourhood
dnn perform randomization rather frame randomization mini batch descent ce randomization randomization order well sgd compute drawback hour task machine great amount datum iteration gradient subspace purpose explore algorithmic spend reduce amount iteration instance symmetric definite simplicity subspace hessian solver algebra refer readily extensively type design strategy reduce computational burden phase offset benefit proper effort computationally intractable hessian approach construct low hessian quasi newton bfgs approach cg exploit structure structural complementary bfgs directly bfgs implicitly curvature l bfgs training unlike cg cg scheme ensure flexible failure counterpart calculation gradually gradient base operate popular regime approximation select often quickly albeit movement spectrum technique computation gradient estimate well training propose hybrid capture use calculation batch cg geometrically benefit select give ahead variance extend dnn english task allow cg iteration furthermore cg overall extend sampling provide loss algorithm briefly dnn respect characterize hessian central iteratively minimize conjugate curvature implicitly efficiently neural cg search upon improvement curvature gauss definite cg due curvature definite via initialize ni implementation code closely gradient gauss product compute cg minimize loss cg base backtracking master technique cg per dominate cg discuss cg iteration nd hessian search method gauss newton solve bfgs reason approximate curvature whereas capture salient thereby extremely search computationally great cg quasi implicit system sensible structural complementary bfgs detail bfgs technique hessian inverse hessian bfgs small number outline minimize initial cg typically transformation upon cg easy cg definite violate may fail l bfgs specifically use formula cg variant require take flexible another gradient amount cg computation hybrid technique similar stochastic gradually amount propose sample estimate computation gradient output dnn loss loss training subset denote ensure descent make must expensive estimate simplified size sample within dnn dnn gradient training frame pass gradient square since compute become per approximate need add notable computation provide error error rate computing statistic link directly expect geometrically cg iteration development benefit selection sample iteration calculation conduct english news bn report extract acoustic hybrid dnn train adapt frame hide unit target bn study intel ghz operations intel mkl machines reliable time cg explore bfgs pc pc spend cg addition look cg indicate cg iteration pc cg move iteration pc appear cg roughly min pc pc require tradeoff amount use number converge little geometric geometric factor cg find cg good tradeoff reduction correspond cg calculation roughly tune small percentage geometric iteration variance method little available get reliable variance method tradeoff similar cg time cg full approach tradeoff training early training overall reliable section speedup bn trade baseline overall gradient cg hour speedup total pc cg hour news explore speed improvement hour american english corpus rt separately feature dnn frame dnn speedup bfgs hour bn task training find small cg calculation allow statistic bn training calculate correlate baseline notice fraction gradient calculation loss pc
membership node coin describe previous pay close death membership group interaction group describe turn link influence membership death group old one birth death ambiguity group change birth death overlap birth death member leave member join resolve devise birth death group state inactive group never active coherence group ibp name rise customer enter restaurant infinitely node usually customer ibp customer account death group become hand group currently active group often old would practically step give black circle inactive activity group member never disjoint denote active also active inactive convenience define newly group birth death infinite mean determine belong group simultaneously model group dynamic cc activity b black circle active inactive become node group membership denote combination member non member link logistic assume latent group membership membership evolve dynamic transition membership typically membership connection membership link define determines membership build group affinity link affinity tendency member member member traditionally consider probability relax node binary entry tendency reflect membership link combine logistic reflect vary density infinite tractable resolve subtract finite distribution figure component dd ibp group belong link active group appear approach exponential model use study dynamic bayesian place low evolution model contrast use variable stochastically depend membership allocate include mixed membership relational critical difference membership group group drift model membership factorial hmm add social process link membership next birth death powerful ibp representative direction research extend ibp factorial hmm ibp factorial ibp ibp dd birth death group posterior beta bernoulli kb k n kn rs group membership determine link conjugacy appropriate metropolis hasting proposal avoid move hybrid hmc guide membership vector update maximize compute via gamma death distribution sample briefly comment complexity parameter estimation computation link pair single sample interaction group group run evaluate different forecasting show dynamic character movie connect people appear publication conference year year people computer core network entire proximity interaction student conference wireless detector remove inactive slice naive regard relationship node bernoulli give time baseline static link independently snapshot dynamic recent snapshot network finally comparison network drift base infinite factorial hmm predictive goodness infer report hold compute possible precision task either run chain link final link average data auc auc auc statistically run value particularly score compare naive network intuitively trajectory snapshot combine information predictive make temporal link auc roc missing hold strong goal next time experimental protocol run mcmc result sample provide c auc auc naive result average generally exhibit auc achieve good previous conjecture network likelihood naive f statistically ccc group investigate identify dynamic character movie base create dynamic social character epoch character result group people correspond interestingly group evolve form start two place look acknowledgment share nsf foundation intel fellowship microsoft fellowship recursion algorithm indicate birth death respectively backward whole together active death chain pass pass forward feature probability compute forward variable q p ik ik c roc vote auc baseline vote vote stanford stanford relational graph evolve fundamental entity change drive group membership birth death dynamic network capture membership factorial explain dynamic explicitly connectivity model capability dynamic latent network improve model prediction network become problem network dynamic entity rise decay accurate structure allow relationship recommend potential predict network arrival group develop dynamic identify birth leave group explain infinite multiple simultaneously membership factorial recent dynamic birth group member non member model structure model flexibility link forecast interpretability obtain provide parametrization discuss related procedure section experimental well social movie death tend cause member group make never relationship would member coherent life member group exist relationship clique imagine member change regard clique group previous member coherence death arise relationship component next describe time dynamic time unweighte
f cx cx px x e ep x p likelihood n result growth quantify mild moderate health normal child country year old child aggregate statistic level obtain probit stick break change strength within across country make uncertain address public health surveillance health monitoring incomplete source report especially tail complex survey datum normal stick break contribute substantially burden low middle substantially nonetheless indicator measure deviation age health organization severe restriction child flexible model address population survey whereas summary statistic severe combine individual level accounting summary make coherent mild severe exploratory analysis across normality appropriate dataset normality could potentially induce systematic seem undesirable estimation outcome without assume normality rich literature bayesian provide overview focus dp specification across finite smooth feasibility dataset effort build parsimonious unbalanced probit stick break weight keep distribution important certain shrinkage preference discuss detail specify smooth change paper height distribution old middle trend health risk receive year development international health evidence comparative efficacy broadly weight cutoff g cutoff design effect proportion child aggregate complex survey account nominal estimate adopt somewhat ad hoc ess source effect survey report rather overall design aggregated design study total child million acknowledge limitation million surveillance addition proportion variability importance specific likely combine source country country year year aggregated accounting country use membership describe mixture distribution vary constrain deviation across ensure choose constrain interpretability interpretation across mcmc chain probit stick use standard cumulative function transform specifically determine manner start stick break stick break thus correspond probit stick breaking allow place still aside stick breaking analysis break weight mixture component constrain tendency something densitie skewed fit alternative additive country specific country level intercept determine normal country country determine country hierarchical prior term let change country component mixture vary country covariate study effect extra year old describe capture variability overall constrain zero correlation desired integrate part hierarchical structure nest nested mixture prior country country country center around level tailed country allow give hierarchy n equal identifiable strength unit simplify unit simplify contrary believe penalty effect toward toward specification characteristic expect smoothly country mixture component country component nonlinearity country autoregressive precision truncate standard recommend prior equivalent constraint extra variability trend flat improper time slope proper vector large study measurement issue see china mixture mixture weight account study country year parameter mi representative study effect representative country heterogeneity aside country study share systematic one issue term estimate country fully range term capture variability across range country design within country across age term relate country survey ess initial strategy observation survey ess primary survey variability analysis approach survey weight survey ess survey observation likelihood adjust weight survey ess nominal contribution normalize individual vs survey generally provide valid estimate ess scale motivating aggregate describe accounting dependence summary well survey analysis include obtain size child severe account weight sample e g large central derive statistic base population course complex survey ess actual multivariate thereby likelihood treat design simple sample adjust sample reflect information survey conduct simulation study enough normality report distribution rough take density summary aggregated hold study summarize simulated report show study see even normal aggregated obtain likelihood inform extent covariate reflect country country region observe motivate population level country level fig global level calculate average country country allow reflect uncertainty health pre child although status improve absolute improvement large relative status mid million million child moderately south sub complex risk purpose effort report uncertainty country year alternative candidate actually mean share across summarize combined probit formulation main effect prior etc reporting mix health concern power covariate five fold overlap appearance hold mix rich poor density year calculate absolute hold commonly occur datum maintain metric study exclude remain main model know across respectively difference country year cc cc p column nan hypothese difference test conduct independence hold value hold covariate study country test give country accuracy country prediction primarily region level primarily metric rather acknowledge emphasize statistically believe hold main low predict hold indicator another covariate suggest p unobserved mean un metric country informative covariate allow strength validity check whether interval hold study test expect cross assess quantification sampling quality quantification country cross validation extent absolute difference median severe course rigorous choose fix model exclude omit analysis extension inclusion index correspond group study combine individual group study report vs index include country proportion component density w ny ny trivial allow use inference report country level add three country specific allow vary linearly nonlinear study si model capture country country country country difference magnitude population scenario difference include embed trend show large gap east child strong region heavily region growth unlike region mixture allow fashion simply study accord age instead age mixture study broad age band country expression component specific particular country trend discuss capture depict green blue vary scale component normal remain constant breaking depict grey movement distribution one nonlinear hierarchical specify prior effect describe sec j place improper constrain identifiability country constrain identifiability specification along interaction add flexibility describe residual case suffice shape analysis date main main effect worth investigation effort health outcome incomplete focus indicator tail specify population indicator impose parametric combine source aggregate way accounting covariate country mixture innovation strength priori generalize level context important consequence desirable characteristic model favor suggest belief context perform show shrinkage exchangeable assess shrinkage due aside prediction token country
perform term maximize fixing follow regression proof convergence available I transformation convex proof immediately clear furthermore clear yield optimum symmetric lasso simplicity symmetry directly restrictive gaussian establish regularity estimator provably convergent section space critical property computable research preserve attractive property time lead clear immediately develop formulation yield advantage chance convergence deep theoretical property natural hence position tool understand objective unless section introduce aim construct start introduce objective obvious may strictly wise large provide tend infinity serve preserve attractive crucial existence solution graphical weight q jointly middle jointly function jointly element term eq put partial covariance regard pseudo function strictly theoretical guarantee always remark derive pp goal sparsity mention positive definite obtain give minimize hold variable proceed evaluate define compute proof contribution design partial coordinate subsequently partial h iteration initial threshold converge update q check select residual type penalty minimize sum competitive differently depend first sum row clearly similarly operation update require operation coordinate calculation product involve variable dimension essential calculate inner product quantify residual define eq appear fix suppose change updating require require operation residual appear expression element update operation update require section operation need appropriately achieve result competitive formally connect likelihood namely count different formulation observation matrix precision pi identify pseudo likelihood formulation matrix ii correspond unified log function pseudo formulation section insight different likelihood remark gaussian log scale hence sparsity perturb conceptually regularize penalty specify penalty partial covariance approximate concentration specifically q section illustrate usefulness correction proceed also necessarily unique discuss certain iterate function guarantee iterate e sequence iterate cyclic minimization sequence converge specifically vector degenerate strictly theory numerically illustrate dataset proceed glasso weight definite convergence issue ran iteration show glasso fast parameter typically flat though glasso slow glasso number efficiency space glasso due detail large non stopping iteration glasso mention demand average second glasso parameter see around glasso conclusion result fast glasso especially http web packages glasso glasso c glasso reason consider end illustrate penalize outside purpose study mean glasso set characteristic curve vary parameter possible selection frequently compare roc roc curve perfect perfect table provide normalize range h cc cc cc median glasso table glasso high auc glasso every remark simulate size dataset value run demonstrate simulation infeasible alternative could run issue clear auc median facilitate cancer illustrate patient breast dataset contain extensive clinical follow subset reduction achieve utilize clinical information together microarray via survival closely breast scale outlier seem choose partial gene target evidence connect central identify highly gene gene correlation table summarize top space place indeed relevant overlap identify notable identify mutation cancer reduce due breast cancer protein remark number top discover identification important finding target gene useful literature competitive also provable guarantee efficacy financial portfolio optimization stable follow portfolio constitute collection hold overall portfolio hold period weight return asset objective portfolio maximize overall return vice versa portfolio theory take portfolio asset return central goal illustrate efficacy shall portfolio portfolio optimization require thus compare estimation method portfolio optimization context portfolio respective portfolio denote asset period turn divide price denote daily return portfolio asset long position position long short position risk portfolio simply period q analytic covariance matrix practice portfolio selection make deal stationarity series strategy particular period compute return horizon portfolio hold duration start refer problem stock composite stock remove list stock approximately table select coincide day trading day period vary shall sample covariance glasso various particular n keep constant period glasso purpose cross validation estimation horizon average stock criterion reader cross quantity return realize sr table realize ratio horizon stand passive track index clear across glasso ratio bold growth trading choice growth another demonstrate trading cost estimation also low choice capital stock reflect higher normalize large estimation oracle adapt suitable dimension setting consistency existence suitably diagonal follow accurate exist estimate constant hold large theory valid consistency statistically behave infinity stand matrix let n ii bound eigenvalue uniformly e j ni jt necessary provide example satisfied line yu jj n c minimizer p c propose retain strength place highly space specific formulation show convex penalize pseudo likelihood descent objective iterate coordinate descent rigorously thus ensure always guarantee tend least establish yield insight attractive natural arise move away penalize glasso rather use note attractive glasso reason secondly computational glasso glasso magnitude selection performance may associations p j zero place objective q jj jj jj jj jj eq ij ij note root retain give let matrix jj jk jk update follow part among update q operation follow update clearly residual hence identity weight residual formulation follow h formula analysis without cyclic rely special objective function convenience completeness version bottom stack top z z z pi every block diagonal eq view also r uniformly zero produce symbol gene record symbol sr aa american ba company bank cat company company company company international business intel co company company united c date date hold hold period trading day th coincide either horizon trading day period consider hold period trading belong algorithm portfolio strategy past w day next holding consider stock precede period stock glasso explicit dependence method fold risk pr stock fold fold determine fold strategy give average portfolio entire horizon q portfolio horizon realize portfolio rate entire horizon amount portfolio trading portfolio weight hold proportion portfolio short standard short side period accumulate portfolio trading initial budget transaction cost cost account portfolio trading transaction cost trade stock capital position stock day trading return cost transaction transaction begin period daily percentage glasso return risk highlight bold glasso give row highlight standard highlight bold glasso transaction rate transaction transaction cost bold fact assumption uniformly infinity uniformly except ii jj nc multiplicative term jj n trivial place immediately consistency result accurate estimator obtain inverse conditional diagonal external estimate function convex descent lemma objective give lemma positive definite obtain exclude convex third term matrix hence follow b function lemma satisfied sufficiently regard parallel satisfied design incoherence column element eigenvalue eigenvalue bound ij n n condition true satisfy diagonal equal matrix eq ni inverse use fact ij four applicable absolute iterate alternate sample vector standardize turn successive alternate thereby establish non weight matrix thereby yield two graph topic modern statistic approach penalty likelihood regularize latter none solve pseudo provable clear corresponding exist computable pseudo likelihood base current respective strength novel lead comprise objective optimize functional rigorous using
learn sample unify large complete rank like inductive multi label recently lot attention complete learn rank real life popular include completion rank sense affine np hard recent present solution recover relevant problem sense completion restricted generalize movie sense satisfy condition least current construction large high storage work issue measurement hence signal encode well efficiently moreover inductive completion provide good paper rank measurement version alternate generic alternate key specific would imply minimization subsequent section specific property globally minimization measurement divide tu qr h h u qr measurement trace paper mainly singular goal recover reformulate follow recover standard hence analysis local minima show converge global problem operator rip sample reveal exactly inspire key alternate kk measurement let give initialization b ib ig I alternate special later step normalize form perturbation observe property get ready multiplying get multiply left observation follow h measurement complexity sample study sense important acquisition application area control etc design true operator I I number exactly several recovery matrix sense already however like stress mention exist operator isometry rip rip matrix require mean bound fourth almost operator memory store storage cubic recovery cubic computational whether rip rip sense use answer question satisfy rip gauss claim idea two independent variate sample independent z rip satisfied apart low match three property hence result operator n proof spherical gaussian invariant therefore assume basis concentration unbounde ensure spectral random co md ij md ij I ij ij ij rv md observe select definition z rate minimization almost powerful rip one drawback rip operator need believe end trade high success bit matrix completion recommender contain true matrix ignore information system user movie usage generalization model user benchmark completion theoretical inductive rank completion incoherent would method several even e information complexity utilize problem provide completion incoherent definition matrix property provide definition incoherent incoherent dx xx tu n incoherent rank ij j tw tx tr w tu treat condition w prove condition present mention canonical uniformly random j quantitie inequality follow eq incoherent argument theorem end initialization w prove proof property ij obtain similar bound hence prove analogously miss n l variate large generally even problem measurement tw inductive completion inductive standard completion type assume inductive datum certain incoherence assumption optima incoherent label constant ignore log sample recover exactly improve completion standard completion learn completion divide part prove proof mention proof orthonormal condition definition x u j canonical sample previous two quantity bound k k quantity sample uniformly incoherence uniform bind quantity similar property cccc rip measurement operator low significantly fast inductive completion plot incur vary empirically operator end signal generate measurement rip figure compare log time provide accurate recovery run base order rip demonstrate regression select label generate c incur test tw fairly small z least generalize distance subspace subspace give subspace case simplicity proof present update iw setting th recall qr denote obtain multiply side get q property get singular mention measurement operator use observe follow use along lemma reproduce completeness inequality fact conjecture microsoft university cs edu movie recommendation rating information age movie unlike completion able new movie problem inductive rating low matrix rank generic minimization otherwise guarantee
allow detect benchmark detection network eigenvalue vertex approach straightforwardly sparse object definition extend graph top project span eigenvector believe regime fail wide throughout backtrack grateful grant support european agreement nsf dms paris institute road nm california berkeley france algorithms approach community suboptimal community belief class backtracking spectrum behave adjacency commonly maintain relevant community even optimal graph stochastic detecting backtrack detect community module network block eigenvector adjacency laplacian statistical sufficiently dense hard recently phase community detect transition network degree grow average constant method thus regime statistical inference succeed current artificial succeed adjacency walk direct backtrack give well walk past theory adjacency fast mix backtrack walk belief rigorously analyze propagation problem regular classify however detection appear novel show way transition come label community analytic result backtrack approach condition radius deviation law lose unable community group label edge sparse affinity matrix stay study size subsequent section goal infer degree block extent moreover prove impossible identify parameter identifiable adjacency spectral assign dimensional according normalizing weighting way spectrum discrete part degree eigenvector community come randomness enyi asymptotically community structure sparse case picture reason vertex high exceed uninformative eigenvector result threshold grow grow square large proportional enyi generate vertex walk modularity qualitatively difficulty simple simply remove vertex amount information eigenvalue eigenvalue disk radius second incoming label contribution use rather address start contribute backtrack walk force tree yield generate block eigenvalue point block read spectrum disk fig e third eigenvalue eigenvector correspond direct vertex eigenvector income edge vertex succeed hold standard argument claim sketch regard property start recall adjacency graph similar integer tree correlation diameter closely eigenvector sign eigenvector income community obey equation give tend zero recover approximate eigenvalue approximate ensemble group fractional label lead spectrum disk real lie assign technique nonzero generally eigenvalue transition detect algorithm impossible community complicated mark transition one none group hard regime nonetheless easy drawback specify group denote branching branching explore leave arrive edge dominate eigenvalue disk radius grow number eigenvalue group appear naturally linearization equation transition bp algorithm iteratively message direct message represent marginal accord receive depend parameter block expect community simple sized bp update fraction vertex neighbor prevent converge fix vertex equally likely either community point give follow rule community affinity vertice community around trivial define give tensor operator matrix linearization term v incoming keeping track rather backtrack closely specifically unstable structure avoid vertex instability group general actual approximate inference update parameter learn difficulty depend block bp occur c cc spectral backtrack way modularity walk transition chance compare backtrack adjacency modularity base achieve asymptotically break symmetry permutation normalize true expect strongly mean overlap third essentially uncorrelated traditional operator real illustrate advantage spectral backtrack practical application commonly benchmark community circle radius square block spectra qualitatively picture network eigenvector eigenvalue assignment
way reliable fellowship science engineering research presence correlate serve generalize hyperparameter determinant vary immediately clear guarantee invert total positive determinant correlate hyperparameter set element wise multiply property matrix prove inverse prove determinant eq dimension follow determinant eqs eqs stand eqs replace equation stand indeed term eq prove eqs follow eqs prove determinant eqs correlate set greatly computation hyperparameter always correlate calculate solve hadamard element matrix hadamard hadamard inverse require element hadamard hadamard hadamard hadamard inverse become determinant covariance eq define dimension show rectangular matrix semi positive definite determinant check eqs eq group big eq one proceed big eqs repeat term determinant find joint analysis correlate construct design advantage method give multiple define wise hyperparameter rigorously recover original hyperparameter set toy set hyperparameter method systematic error set ratio necessity include show construct joint correlate analysis background e galaxy survey large scale assume datum simply optimal weighted inverse set discuss appropriate observation back joint velocity systematic hoc instance systematic reliable exclude joint high limitation assign hereafter develop marginalization marginalization force carlo directly include monte algorithm hasting non mcmc nest budget produce phenomenon recover hyperparameter become tensor background galaxy cluster velocity field limit joint correlate angular power temperature south moment term draw velocity flow survey underlie matter principle take presentation level multivariate present method section leave salient proof joint correlate make main apply straight fit budget systematic data behaviour hyperparameter discuss improvement method last suppose bayes quantity selection let hypothesis perform entire parameter play specifically preference scale list use criterion assess hyperparameter strength support substantial let collection survey quantity try hypothesis element difference combine form survey vector vector use represent covariance hyperparameter unity vector statistic combine multivariate gaussian parameter function likelihood numerically combine survey combine distinguish properly unbiased systematic method give propose assume another block hyperparameter rescale individual datum th rescale equation total become act weight survey explore explore systematic error sec become hyperparameter result set effect introduce hyperparameter th conversely significance th hyperparameter different correlation term diagonal section hyperparameter negligible include experiment hyperparameter rescale drop assumption covariance symmetric definite asymmetric covariance matrix hyperparameter first expand multiplying keep hyperparameter kronecker product covariance likelihood hyperparameter indicate hyperparameter covariance fortunately covariance invertible rigorous proof greatly simplify eq hadamard inverse matrix inverse correlation without hyperparameter check reduce hyperparameter likelihood unit recover likelihood simple straight combine improve hyperparameter error bar systematic reproduce validation hyperparameter prefer marginally see bayes offer preference two solid line traditional line contour black dot indicate posterior unity weight hyperparameter anti correlate level draw correct internal covariance set fig hyperparameter method weak hyperparameter simple begin extend hyperparameter hyperparameter middle pr parameter estimation red line hyperparameter blue contribution fit hyperparameter outside value approach heavily hyperparameter note bar factor middle estimation dash blue solid distribution pr two set correlation matrix posterior reveal hyperparameter inconsistent value evidence deal correlate uncorrelated panel fig error report recover true broadly likely hyperparameter reduce relative ignore hyperparameter despite correlate evidence bayes weakly strongly hyperparameter mis report sec provide method middle posterior pr red hyperparameter right pr hyperparameter approach provide bar differ section introduce systematic observe draw together straight line systematic quite apparent reflect middle panel indicate recover contrast two recover model joint space outside level evidence ratio hyperparameter systematic dash contour contour hyperparameter pr hyperparameter hyperparameter approach situation hyperparameter reveal indicate systematic panel systematic error branch branch parameter take ordinary zero systematic ignore set middle pr dash line hyperparameter right matrix c systematic bayes bar uncorrelated set standard correlated calculation hyperparameter sample equal hyperparameter covariance evidence ignore correlation specific please sec hyperparameter multi correlate greatly limitation independent important justify design illustrative sample take parameter full datum set original hyperparameter ignore evidence bayesian evidence value
misclassification equivalent contamination exhibit weak illustrate contamination report good equivalent rate contamination figure equivalently h line line solid distribute line line version color trade especially setting visible plot tend somewhat algorithm improve call essence large subset weight perhaps simple weight member rhs make evaluate nonetheless compare sample essence sampling dataset divide line sde solid color depict percentile dotted algorithm normally experiment nonetheless maintain sample depend good cauchy illustrate engineering quantity water aggregate variety concrete measure contain observation period date largely overlap bivariate scatter observation variable jointly denote member member square nearly run sde default estimator concrete dataset panel dark blue depict member version mahalanobi display concrete dark blue depict assign member notably value assign member outli hard distance observation observation index dark member member member member lie square mahalanobis comparison sde index distinguish index derive fail outlier third outli hard increase form member outlier third index experiment dark sde see index two overlap continue clear distinction compose outlier distance outlier increase depict index member previous assign member qualitatively contamination rate separate outli method seem setting see outlier reliably article pc outli crucially correctly identify contribution characterize subset use multivariate cloud point feature insensitive configuration outlier simulation focus affine find consider different given know prefer carry inference simulation affect majority capable draw cauchy article pc investigation support conjecture fit property rgb rgb pcs procedure search minimize design insensitive outlier affine extensive study real engineering pattern outlier analysis outlier parameter inference want outlier aside difficult visually instead formally concern simple outli sample draw multivariate elliptical reliably treatment article pc procedure multivariate fast measure majority pc index mean compute select observation outli approach produce solution significantly well section motivate pc synthetic conduct offer index subscript indexing denote mahalanobis observation eq way sde procedure case sde direction hyperplane case upon find outlier observation small determinant volume adversary place contaminate always choose form sde sensitive outlier place denominator equation small along numerator outlier observation maximum repeat cause value four normal obtain well sde subset first case fit include center star locate model visually draw dark confirm bias three propose method select new derive qualitatively pc along projection pc spatially denote observation hyperplane orthogonal denote member remove consider direction hyperplane span subset subset essence solution sample spatially disjoint spatially form spatially tend panel disjoint group hand panel behavior index belong spatially contain hyperplane dark blue light dark blue dot show dot belong spatially overlap member decrease denominator numerator overall spatially index crucially characterize spatial hold belong call parallel processor computing enhance user experience ex distribute package numerically sde package except algorithm small briefly give contaminate asymptotic affine contain eigenvalue bias matter rate contamination outlier also fortunately affine bias know focus contamination denote contaminate misclassification thus yield contamination mean outlier misclassification separate generate contaminate bias outlier bias spatial configuration difficulty shift constrain maximize adversary intuitively mass configuration adversary place omit adversary contamination radial outlier extremely generate contaminate since affine quantify generate package default parameter depict contamination one outlier shift contamination contamination determine case get always percent display bias misclassification plot dimension expect outli problem monotonically hard lose grid parameter chart point contamination configuration distance separate hard clearly nearby outlier distant chart misclassification rate solid colored median dot percentile base cover case bind majority bias algorithm shift normal perform bias misclassification sde stand move sde reliably
proposition theorem remark example th accept generalization sensitivity paris e pour une en des de pour des pour paris partial induce partial parameter identify rank multivariate index vector index decomposition satisfy natural sensitivity index property natural study monte carlo set x hoeffding kf u covariance orthogonality covariance output ie scalar interpret factor univariate f back km fs isometry leave I positivity positivity iii requirement sensitivity invariance ensure intrinsic partial variance divide covariance fulfil iff sensitivity good converse definite matrix isometry without diagonal contradiction sufficient isometry two canonical scalar monte pick evaluation case copy
histogram histogram jeffreys centroid otherwise histogram positive jeffreys centroid argument centroid unique histogram various hierarchical distance investigate cluster color segmentation hand jeffreys extensively non arbitrarily jeffreys loop structure manifold histogram report geodesic nest loop one loop indeed belong exponential jeffreys frequency centroid equivalently bregman centroid computation jeffreys centroid et mean divergence centroid divergence two centroid jeffreys rely centroid cluster report close form jeffreys centroid histogram section jeffreys centroid jeffreys report avoid conclude empty jeffreys first jeffreys weighted histogram wise using coordinate wise geometric seek minimize expand jeffreys divergence additive coordinate coordinate dropping inverse seem elementary logarithm fourth implement iteration reach play information histogram get optimal positive jeffreys normalizing jeffreys jeffreys require dedicate consider approximation jeffreys approximate jeffreys arithmetic geometric I normalize arithmetic jeffreys centroid normalize jeffreys positive centroid bin jeffreys centroid section jeffreys centroid almost coincide jeffreys centroid minimizing minimize instead design loop lagrangian enforcing coordinate iw I e notice I e c cumulative equation iw perform deduce approximation jeffreys centroid jeffreys centroid available jeffreys guarantee normalize almost arithmetic perform average error optimal performing search yield scheme rather notice point jeffreys centroid initially l experimentally fast uniqueness study banach ccc c frequency intensity histogram approximate jeffreys histogram histogram centroid carry quantitative precision consist perform intensity histogram histogram inside jeffreys centroids jeffreys frequency centroid average arithmetic result jeffreys centroid trial fine normalize experimentally open analytically bad scheme r implementation report double digits experimentally contribution jeffreys admits close jeffreys centroid guarantee jeffreys centroid notice experimentally jeffreys almost coincide jeffreys jeffreys notice monotonically centroid update provably centroid jeffreys centroid end converge jeffreys implement jeffreys chernoff divergence include jensen jeffreys lemma include fix source computer bag modeling histogram ingredient modern histogram centroid deal symmetric distance letter divergence investigate jeffreys centroids jeffreys centroid express analytically approximation histogram document task document category income text categorization document count word histogram per histogram classify line histogram deduce neighbor category document histogram assign category jeffreys kullback leibler traditional text bag instrumental categorization require create quantization datum belong give initialization assign update center meet centroid mass visual vocabulary jeffreys divergence euclidean histogram gradient summarize jeffreys divergence dictionary assign category cumulative bin histogram histogram frequency histogram histogram bin
approximation number within hold curse dimensionality dimensional reduction grant intra european fellowship within european ga university support mt national institute theoretical comment manuscript frank helpful compute point set output amenable serve train whole know onto span q recover q select orthonormal gram schmidt equivalently form q completely onto span gs carry physical precise alg arbitrary tolerance expression product natural h input arbitrary greedy point scope parametrize problem polynomial basis nest differential equation span approximate dimension parametrization intrinsic object arise evaluate function predict meaning agree build basis comment large interpolation combination interest physical physical chebyshev interpolation nest node include within whenever depend characterize computable lebesgue ref slow scaling say one main wave search quadrature perform evaluation dominant algorithm widely estimation study accelerate illustrate search quadrature show time fast expect complex signal model significant compact computing correlation large space aspect extract wave detector advance advanced cost grow several analyse bayesian great ensure desire reasonable parameter markov evaluate parameter throughout likelihood hence mcmc prohibitive use rather optimistic signal scenario evaluate strategy directly neural learn case technical cycle novel technique calculation model fine tuned application mcmc aim reduce exploit compressed thereby reduce generalization dimension cycle readily handle typical grow physical template cycle could template coherent light need cost scale length standard numerical non smooth integral observation exponentially likelihood thereby reduce computation quadrature parametrize produce quadrature parametrize exploit sample case outperform quadrature generic key space numerical computing overlap correlation cost speed follow present overview model interpolation reduce finally mcmc show considerably speed computation address generate method parameter assume stream multi instrumental posterior function density normalization word gaussian weighted inner density detector physics sometimes deal mapping posterior expensive technique space well expensive algorithm propose specific scenario parameter quadrature rule employ variation ref construction layer construct advantage include relevant interpolation basis relevant dimension nearly construct stream construct accurate datum within section put piece eq simple roughly deal classical principal component orthogonal decomposition history low specifically design parametrize whose advantage deal fit memory become prohibitive projection base identify coefficient appendix amenable represent intrinsic even would frequency refer physical one approximation form arbitrary represent furthermore generate guarantee nearly define negligible grow small typically see quantify worst many choice chebyshev fourier basis require practice provide lead global construct rule directly applicable projection amplitude width arrival signal ft describe build handle build q spaced unless one unit always rate parameter also present noise snr product point build pick error element find dense marker indicate aid compute within far opposed frequency compute full sampling problem interested review classical discuss empirical interpolation specific function finish interpolation find agree function show unique lagrange polynomial rate approximation accuracy trading optimally pointwise chebyshev node like basis interpolation point describe optimal interpolation set application dramatically parameterize absence pose basis interpolation point additionally accurate crucially select interpolation choice seek find moment assume explain proceed parameter transpose continue nearly representation define lebesgue decay qualitative outline interpolation proceed follow maximize f basis eqs interpolation together complete represent quadrature model discuss error practice replace quadrature machine precision sec quantify family frequency turn arise computation vertical thorough sum integration point family signal default top figure bottom noise realization versus correspond red realization parameter pure noise characterize signal arrival orientation phase affect space position orientation affect amplitude space arrival exclude ft form exploit search define function inverse ft fourier transform search enable integral principle guarantee however detection stream detector normally event comparable couple cycle handle denote without loss generality arrival window arrival extra arise build without offline computation therefore alternatively build value increase coefficient ht build continue ahead evaluate evaluate likelihood last handle first gaussian sec close expression expression build rule additional offline computation notice coefficient carry count application expensive full comment offline speedup find I quadrature use trivially build identify point inversion particular ref utilize triangular alg respectively inner vector compare cost compressed overlap respectively evaluate perform multiplication expression speedup expression great rule ordinary equation speedup size threshold equally space thus ode oppose aim chain sum use random proposal metropolis eq random move accept reject depend therefore problem stream include white take hz proposal eq span range prior vs proposal full result likelihood use calculation snr fixing table parameter recover full recover likelihood realization four define difference digit arise accurate difference statistical indistinguishable ask full consistent likelihood kolmogorov posteriors likelihood posterior likelihood ks evaluation digit recover posterior likelihood cumulative distribution snr curve lie top full test confirm probability evaluation agree full posterior likelihood variety case statistic etc completely likelihood ks cumulative computed posterior apply build value alternative approach ht pdfs employ standard computation figure technique detail c snr full full table search amplitude use deviation
envelope cardinality euclidean vector whereas euclidean observation lead study first recall regularization explain general comprise tensor trace regularizer matrix namely invariant permutation trace implement regularizer pose difficulty element author admm base auxiliary tensor reformulate nn n augment lagrangian scalar tensor lagrange multiplier problem whereas explanatory notation complete property proximity operator know form right vector proximity well prox choose next describe compute proximity operator calculus prox prox conjugate scale proximity prox gx prox x wish employ subgradient g consist iteration advance second project feasible sufficient formula k w method update iteration r k k k terminate assess whether tensor completion real validation tune among approach far tensor unknown use entry estimator generate tensor procedure tucker decomposition tucker decomposition distribution truth variable std create remain repeat average pair performance obtain always synthetic left mean square tensor propose algorithm furthermore experiment running generate tensor procedure outline quite high low try time ratio outcome demand routine describe increase decomposition demand approach trace first try education range student set school categorical think completion problem categorical attribute school gender band instance instance validation average norm regularization check conduct case tensor norm treat height video video case treat one test set repeat procedure approach outcome strongly pair test run obtain relaxation context norm prove tight argue regularizer may advantageous indicate method consistently improve tensor trace operator regularizer utility tensor multilinear acknowledgement suggestion discussion international cm pt ex minus axiom claim conclusion condition corollary exercise theorem solution relaxation completion interactive centre college uk science college tensor prominent methodology norm extensively learn limitation relaxation ball describe technique build alternate direction multiplier improve significantly regularization year grow tensor reference therein tensor collaborative tensor encourage low arguably widely extension trace tensor key behind regularization tight relaxation ball unfortunately difficulty norm stem compute relaxation different study relaxation rank ball show tight describe regularization direction multiplier operator present life improve trace highlight trace tensor solve norm eq trace nuclear namely singular tensor coincide trace trace lower singular convex envelope conjugate conjugate von discussion idea spectral tensor equation composite nature computing envelope resort convex insight behind appendix tensor convex envelope exist tr tensor choose function
present regret horizon respect analysis factor match technique imply bind aggregation interactive ranking click click close retrieve spend need make simplify list describe set item output randomize nothing position third minimize known work play choose round sum position element feedback improve discrete regret discrete expect regret argue tight section section result well et follow et permutation careful analysis explain section technique problem commonly nonparametric enjoy connect version abstract assign maintain weight round weight noisy noisy sort lemma regret state simply fix pair marginal multiplicative show procedure version datum property book model ground think low position observe denote indicator incurred view fact pair sense incur place define q horizon compare algorithm aforementione say identical since maximal non denote always ground parameter sort sort horizon additionally proof defer present nk nk follow exist integer make weak property guarantee derive tight binomial rate randomize sort procedure time choose return v p initialize work type study al permutation incur loss offline shall offline additionally na I obvious prediction ranking action track multiplicative scheme guarantee single choice assign efficiently real way vertex distinct easy highly suggest solve optimization problem choice reader work number history output order iid distribution determine guarantee regret td ta quick mention introduction suboptimal analysis carefully elaborate consider assume apply optimization view submodular bind ambient hence bind embed loss unnormalize ranking corresponding ranking adversary output reveal total loss nothing identify exactly trivially rank difference hence online aggregation horizon n done compare return equal uv way resp recursive event recursive recursive element recursion right event random also wu wu wu wu wu require xu loop equal disjoint space proof complete wu wu lemma namely precise order distinct easily verify plug fact provide proof abstract recall loss identical choose step sequence rank satisfy nu fu mx fu element mx fu e fu fu nu easily choose split fu fu nn fu monotonically chernoff state global integer polynomial possibly increase central exist cdf notation purpose rough reason verify cdf integer trivially fu give increase conclude indeed loss exactly conclude cdf cat noisy section decomposable execute binary ordering replace update nothing function assign commonly ndcg measure retrieval constant output well rank rely instantaneous subset family grow bind major open obviously set bandit set step matrix al ambient ambient ranking fix ranking consecutive clearly u prove bad underlie closure permutation additionally underlie exponential clear efficiently draw perturb single choice general
diameter degree call connectivity greater normalize laplacian generalize spectral intuitively criterion modularity detection cut size propose subgraph connectivity iv modularity base graph thresholde principal eigenvector principle community subgraph desirable property application outline subgraph belong hypothesis network pd probability false alarm pearson compete illustrate ensure practical achieving assumption optimum test involve pearson unknown observation represent assume observation observation treat simple treat vertex presence give optimality pearson maximize alarm test involve lagrange subgraph decision hypothesis measurement involve partitioning pd hard analytically intractable hypothesis greatly test consider unknown observation k pd integral otherwise pearson maximize pd maximize property lr maximize next section numerator propagation connection denominator principle insufficient term detect subgraph yield pearson treat maximize computing propagation simple exceed principle track communication arrival time view correlation constraint temporal model concrete example propagation give compute infer across vertex test vertex stochastic indicate unity continuous jump stochastic differential rate define positive time vertex time propagation track connection probability linearize relevant vertex determined track pt transformation column discretize time discretized column correspond track give comparable kt nonzero correspond essential track linearize track vertex combine track independent valid yield extend multiple track degree discretized connect asymmetric laplacian propagation boundary value harmonic operator harmonic propagation laplacian bi bb vertex interior harmonic direct laplacian discuss harmonic analysis detection harmonic adjacency method practical graph thousand subgraph system optimum pearson detector vector normalize detection optimality harmonic network compare address pose physical fact cut size threshold eigenvalue maximize subgraph background threshold principal modularity alternatively propagation harmonic equation represent boundary eigenvector rely cost whose os enyi surely harmonic inversion iteration computation inversion two detailed behavior full real partial detail foreground demonstrate simulate partially upon foreground close prediction accomplish subgraph enyi graph realistic network realistic essential attempt attempt behavior stochastic detailed detection detection algorithm stochastic blockmodel dataset necessity world detection network exhibit property power law world capture trait enyi exhibit law law membership stochastic include temporal realistic base network depict fig aggregation comprise enyi low dominant membership blockmodel community interaction approximate enyi model create law broad network blockmodel create parameterized time graph mix membership blockmodel connectivity temporal let time node discretize assign whereas half therefore rate interaction enyi j blockmodel foreground subgraph red determine binomial random community sparsity community determine per law blockmodel rate interaction j community simulation community finally ti multinomial example community activity foreground choose spatial real world interaction individual leave parameterized community thereby activitie meet number community occur one perturb foreground base community detection empirical result blockmodel independently trial set foreground background size foreground activity metric receiver characteristic roc foreground versus percentage background vary perfect roc alarm chance equal alarm spectral community foreground specify trial use comprise span ten community detail toward different mix activity law membership background community represent network foreground uniformly background foreground association community foreground life foreground interaction nod interaction belong community perhaps os must finally clique os enyi network nominal foreground foreground propagation use monte trial show foreground average time improve temporal foreground detector decrease foreground detect use constant community sparsity foreground make violate spectral level foreground n high foreground provide none assumption spectral activity moderate spectral well chance activity foreground detection partitioning theory address partition small develop compare different network bayesian introduce partition space time pearson sense interpret harmonic approximation node new examine compete notion detection optimality finally blockmodel detection parameterized combine enyi sparsity power blockmodel use foreground activity level varied hope analytic form gray minus minus centering skip skip bernstein bernstein mit edu em detection capability datum subgraph background characterize big discovery area year drive internet security activity specific address partitioning membership algebraic analyze introduce time prove community divide subset community analyze receiver operating characteristic problem binary vertex fundamental figure subgraph comprise member definition membership np problem however semidefinite sdp relaxation gp many subgraph cast quadratic eigenvalue spectral simple one global community optimize connectivity present propagation optimize probability false alarm optimum assumption detail remarkably optimal insight converse research use detect related network subgraph network belong network pearson method analyze detection assess blockmodel foreground detection community interest optimistic unlikely description community represent goal plain adopt operational procedure remain loss carefully organization broken branch communication office distant cell description organization group attack computer balance tree organization example distant part tie may observation network vertex terminal vertex homology topology incidence recognize operator difference appear incidence orient arbitrary orientation scale generalize asymmetric transformation latter immediately recognize laplacian numerous asymmetric play theorem involve laplace motivate behind several incidence laplacian laplacian product immediately manifold yield arise matrix outer product laplacian mathematic connection matrix across
finally comment corner achieve decrease reasonably mse consider additional mse overall difference practical moving generate multiplier red former fast discuss far interest work seed describe partial difference width width bandwidth ar scenario direct investigation experiment benchmark space accurately previously latter carry marginally form center compute sample generate copula carry choice std std define approximated sample covariance approximate represent resp bottom row datum generating copula ar copula report procedure estimate much affect like thank constructive early version manuscript support part collaborative statistical nonlinear dynamic research ns c compute ns ns df u I n n generalize implie complete purpose trivial ns nh q nu f ns nu ns fact q immediately sufficiently hence write proceeding completes start u sufficiently use inequality imply eq fact large u u ns u nu dr df obviously differentiable mean valid verify r immediately zero term remain remain let case distinguished nu nu previous equality carry dominate eq let since nb jj ns section prop prop prop condition prop example two observation copula appropriate scheme exist I bootstrap frequently sample propose adapt dependent contribution resample propose resample sequential thereby transpose setting include nonparametric test detection fully automatic datum adaptive estimate parameter simulation investigate resample suggestion choose product multipli condition strong multiplier serial keyword lag observation dimensional continuous capture dependence origin copula model margin quantitative management environmental name dimensional f mid among nonparametric copula frequently compute copula goodness respectively asymptotic procedure follow empirical copula detection generalization central greatest rewrite empirical ns ns ns nc u n coincide initially rewrite copula weak serial smoothness copula early key ingredient procedure ingredient replicate resample literature range multinomial multipli technique investigate far compare bootstrap mixing adapt multipli bootstrap block appear independently statistic interest latter connect resample technique bootstrap multipli scheme side result paper parameter block multiplier automatic copula I I scheme test copula detail subject paper finally could procedure markovian copula apply confidence band develop goodness hypothesis product validity bootstrap process obtain decay multiplier multivariate index leave adapt strongly side sequential serial scenario mild organize extension asymptotic copula serial multipli carry bandwidth multipli bootstrap adapt process generate dependent multipli central resample partially report carlo aim various involve follow notation sequel convergence sense resp represent continuous equip metric multipli observation consequence multiplier empirical adopt investigate bootstrap empirical resemble multipli main b multiplier suitable multiplier stationary exist symmetric satisfying main marginally define paper notation quantity symbol copy multipli display block form multiplier present sake generate say mixing inspire could regard extension multipli mixing index give appendix assume satisfy weak copy regard unconditional interest scope c margin usual f consequence q proof proof supplementary multiplier draw continuous strong f validity bootstrap establish convergence law van necessary approximate law simulation resample typically omit corollary reference include deduce bootstrap situation supplementary approach particular unconditional paradigm usual transpose goodness test respectively comment corollary requirement prove regard multiplier strong mix corollary unconditional scheme remain observation latter carlo experiment suggest multiplier resample capture observation process asymptotically multivariate representation side sequential preliminary multipli consequence actually asymptotic establish serial dependence consequence sequential consider draw weakly tight center serial independence immediate meet latter candidate weak pointwise derivative whole vector finally need back end tie serial serial dependence continuity tie example lead tie result asymptotic copula asymptotic mix immediately previous theorem strictly sequence strong whose strong coefficient condition instance draw shall combine state proposition regard multiplier underlying theorem regard n mc start state spirit partial continue satisfy condition constant derivative define process appropriate copy adapt strongly supplementary coefficient side copy sensitive derive definition construction nan derive weak define process nan weakly jointly limit key establish classical statistic ns mapping unconditional result material test copula alternative financial find maximally er von multiplier dependent multipli sequence derivative third subsection address bandwidth involve multipli sequence bandwidth assumption role block present bootstrap aim multipli bootstrap multipli expectation c moment unknown shall lemma prove adapt argument proof strictly satisfy twice continuously u v strictly sequence u condition u v u square derivative obtain asymptotically sum observable done adapt current empirical let integer determine spirit quantity n kernel parametrize lag n u q computable grid plug choice proceeding lag automatically detail base matlab page aggregation median experiment partially report section state section generation multipli way construct dependent produce multiplier satisfy implicitly positive bound around w practical reason development immediately clearly verify asymptotically additionally notice write sequence denote q hand ensure numerical several popular x rescale shift minimize mean multipli width additionally either decomposition cholesky multipli sequence obtain assumption perspective center normalize equal rescale represent rescale truncated top ensure definite see normal could expect move covariance approach respectively give use partial derivative estimate partial consist another coincide n slightly definition consider performance multiplier several mostly von functional partial derivative estimator define section condition proposition aim quantile correspond target quantile n ns pm mp allow
low rank completion tensor completion recovery investigate tensor recovery advance robust completion alternative recovery discuss knowledge include noiseless one special tensor identical tensor th otherwise solve thresholde easy particularly convergent noisy case true decrease quickly additionally heuristic determine rank unknown advance knowledge iterative hard rank convergence report preliminary tensor essential find scalar letter e letter e letter denote k j j hilbert corresponding ix mode unfold result tensor element map nj l point rank tensor mode unfold product nu l un express tensor transformation diag diagonal thresholding fast thresholding widely iterative hard compressed sense analysis compress recovery problem hard property error short requirement require acceleration choose improve speed space property variant minimization particularly perform adjoint operating operator singular keep specifically via randomized cost recovery inspire iterative hard minimum surrogate f ff k k assumption optimize decrease original word iterative k k k ii r nr r thus exact approximation first rank e unfold n operator iterative hard thresholding k k I concentrate let tensor unfold great algorithm rate begin concept svd isometry constant rank tensor I definition see element basis give towards rank x orthonormal matrix let round generate iterate rank matrix exist denote subspace span span set span aforementioned notation triangle second follow index prove term kx expansion estimation I kx r base I I u x large value j inequality fact k inequality fact iterate inequality observe given ensure convergence enough guarantee recovery apply denote mean round n c ii difference add term first noisy add term estimation term I cauchy inequality n use substitute obtain obtain iterate f proof random tensor experiment create nu construction surely tensor completion noisy low tensor completion n distribute ratio percentage support uniformly estimate appropriately propose heuristic kk ir tolerance sometimes increase note experience singular computational cost use code especially relatively low monte carlo develop compute svd reduce return large value ty ct output approximation leave balance computational mode simplicity mode hand although completion determine completion fp principal tucker decomposition square ie run matlab intel ghz cpu low closeness solution completion noisy normalize root mean moderately optimal iteration additionally noiseless fp hc regularization keep constant parameter stop residual decrease noiseless ratio worth assumption ensure find choose broad figure obvious cost error test noiseless completion problem tensor relative versus see slow heuristic determine iteration several hold comparison noiseless table present noiseless rank completion recovery problem time stand second respectively cost less low easy always error cpu slow need determine efficiency sr save additionally also good problem ie poorly little high tensor long inexact seven ie noiseless tensor fix rank convenience trial create noiseless indicate relative algorithm completion present execution easily far remove pixels image subspace estimation detail tool r image well original ie obviously long ratio result five result especially high rank approximation rank original image recover image fp tensor hc large rank numerical algorithm consider appropriate solve give
item view core live architecture recommendation million replace early version system popularity interpret easily procedure live system item base simple signal power distribution item description compute plausible graph collaborative treat prohibitive algorithmic fully signal pair unobserved item pair user observation amount real observation individually line contribution drop rely careful except expensive exhaustive rank uncertainty fall already body arguably rank everything see unobserved rank design item popularity unobserve item effectively utilize solution competition approach ensemble solution edge discuss typical bipartite generative collaborative model hide item consider address combine gradient scale probabilistic movie million version netflix competition netflix interaction typically live observe bipartite graph user kind sample movie appear movie edge absence denote distribution view item eq hold satisfy exhibit mark exponential cut scientific netflix take rate item five star degree calculate sum degree replace relevant repeat user pair model appear item like even though signal say consider item rule user item solution everything strongly beliefs distribution power graph signal bilinear filter latent additionally add user odd model logistic drop likelihood last factor binary odd separate bend angle minimum thick draw mm draw sep cm cm corner post g par label h leave post g edge post post r post ab ab ab background draw fill rectangle cm probability choose p various gamma belief explicit could parameterize power cut approximately generate closure b mn notation give sigmoid later appear bind obtain follow know occurrence devote treat place wishart feature various beyond simulate alternatively substitute marginalization deterministic disk approximate factorize approximate conjugate one stochastically connection roughly specify every like coin reveal coin reveal alternatively half constitute place graph type draw benefit algorithmic simplification exact procedure stochastically draw specify item vertex show connect propose simple histogram degree view user mark user replacement draw tree way degree effectively item negative bar generally histogram obey histogram weight example half unobserved substitution give histogram adjust skewed head edge item odd discard component edge connect maximize q need positive root deep statistic sufficient present optimization proceed user vertex factorize give loop precision distinction give recover computation pt back twice obtain subscript indicate need dominate partial gradient repeat stochastic bias natural give simplicity notation eq pt satisfy convergence pt avoid maxima well iteration pt finally marginal approximation update stochastically update algorithmic outline mn item gradient mutual dependence vertex update graph social require lower distribute across might vertex optimization discuss early matrix locate gradient user datum block separate wise one thin precision iterate full update keep incoming loop optimize long present vertex block collaborative future presence online model separate odd depend infer gaussian follow movie netflix set present form core recommendation criterion recommender suggest popular tail evaluation highlight item popularity interested contribution bring far algorithm way art form base dimension balanced user give value netflix star movie dataset pg plot present rating netflix five star rating around scale yahoo music noise explicit slightly skewed less certain predictive misclassifie slice require truth class present mn evaluation user item recommendation item popularity possible popularity exploitation utility average group evaluation netflix draw rank ranking score font odd approximated odd namely mn pg mn prediction q would place hold head list rank range singular decomposition near use second track competition competition recommendation regardless popularity miss probability popularity therefore miss item capture recommender optimize ranking specifie directly optimize come optimize still aim rank observe missing aspect recommendation generative model capture aspect structure manner group user group prefer popularity bias optimize estimate order noisy per learn perform poorly item superior result recommendation head item come tail less behind rank user item tail report average figure show decrease user movie hard popularity ranking
sg average dual primal subgradient objective accelerate sg later sg sg sg sg inverse scale however omit never step aggregated gradient cyclic weight give power weight regularization negative likelihood although search sag avoid calculation small global initialize leave right middle right center view colour plot pass observe vs sg method allow sg always substantially sg little progress pass contrast steady progress typically pass sg vs sag sag seem achieve well substantially obtain performance method sag continue steady progress pass method sophisticated sag method pass sag differ minor detail point sag sag counterpart believe sag would cause iteration regularize take advantage problem optimization descent ascent comparison em randomize descent coordinate sample sampling randomize dual sg sg method discuss convergence pass iteration since sg sag effective pass multiply pass method expense incur updating bias numerically coordinate optimization method observe trend top little coordinate middle sampling accord neither dominate sag problem give poorly among set clearly extremely sag robust cm cm center result colour analyze consider discuss compare sag sag well choose plot use perform little make convergence poorly unless extremely middle often perform consistently case bad perform perform discuss remain set slightly one poorly line section tend various constant strategy choose sag well sag sag size cm right middle bottom center colour batch sag trade mini batch fast obtain batch possibility figure compare optimality example mini step conclusion though theorem mini conservative mini account essential large large mini mini batch mini batch size middle gradient mini batch size center size right size mini mini experiment explore sag follow sag lipschitz method constant constant sag l form sag ls track sample least select initialize approximately pass never unseen method unseen function sample prevent initially poor step normally k entire sag sag uniform behave sag vs sag estimate individual non strategy give solution magnitude examine set context sense performance dual primal often lipschitz eigenvalue denote maximum also depend primal strong convexity constant minimum use improvement determined rate dual fast depend determined efficiently rate primal neither independent dual achieve rate primal applying cost problem dual variable depend compare compare rate term hence primal duality gap iteration sag iteration sag q rate depend case sag term denominator sag sag limit improve grow improvement point sag choice tend sag achieve unconstrained surrogate performing tend slow sag denominator slow sag give function gradient allow attain sag perform integer select convenient compose diagonal diagonal addition convention concatenation block equal block information generate sag f n lyapunov dominate parameterize leave coefficient coefficient guide cone validity symbolic check positivity certain constant lyapunov lyapunov evolve sag recursion np pc p lyapunov lead nb nb nb n show appropriate surely algorithm lyapunov addition continuity expand give e n get ns nf f sf ny k k f h gx gx g k k x f x n x l e k k h gx x k b b appear may respect obtain b gx k gx b cm x gx f g f k f x x b x x b b b decrease lyapunov c k convexity c c gx gx c x dominate q give c h dl na feasibility check toolbox cone program represent candidate cone program c author symbolic verify verify computation symbolic matlab discard impact validity assume regular sag suffice show expression multiply rational positivity na dl na derive symbolic computation positivity computing root strictly see matlab author express use check positivity dependence positivity check positivity b check similarly monotonicity term replace bind check positivity explain check positivity univariate polynomial dl yy na yy yy negative derivative b b negative derivative satisfied check sag positivity matlab exactly convex convergence sum iteration yield jensen note initial lyapunov initialization eq obtain note eq observe lyapunov function l l pt minus pt plus minus le project sup paris france stochastic sag finite sg sag memory gradient sag convergence rate fast box deterministic evaluation indicate sg uniform arise compute minimizer least problem datum arise modern extremely often amount class take sg theory sg method apply optimize average addition property regularizer form square scalar control strength eq result extensive smooth regularizer also approximation see iteration minimizer error fix scale sg cost suit modern may optimize iteration iteration uniformly yield standard combine display property author iterate non smooth objective sg option accelerate accelerate approximation scale method gradient newton hessian show first converge sg convergence tolerance iteration convergence strongly strong rate accelerate sg despite name relate aforementioned advantage sg accelerate sg use default decrease successive estimate lead achieve stay variant problem seek iterate fast propose weight achieve unstable treat pass batch size sg iteration oppose sag sag sag cyclic choice distinction al show convergence derive treat pass extension simultaneously lyapunov work show sag allow much require suitably convergence rate change dramatically improve method method linear method rate require sg size obtain parameterize dual line experiment although method sag poorly dual property rate obtain coordinate regularizer whether method convex regularizer show sg convexity correction general unlike satisfy publish closely sag term specialize smooth q sag algorithm sag apply use achieve convergence rate obtain rate pass direction multiplier admm variant beneficial complicated structure interesting relate sg consider rate sg sag sag storing variable store previous correspond fairly weak differentiable eigenvalue optimum two initialization setting express expectation internal randomization variable datum deterministic consider constant meaning differentiable require parameter thus add regularization term strongly problem achieve optimal standard constant sag initialize q q proof involve converge rate state average change function iteration also iterate valid imply bad cost slightly optimality remove optimum sag advantageous sg sag worse use particular set sag rate imply experiment minor sag early appear difficult strong convexity problem automatically fast lead local strongly optimum globally strongly convex problem global adapt local convexity observe practice characterize size order sag ability large sag selection lead improve basic gradient step bind cycle datum cyclic method sag somewhat surprising ill problem appear indicate date appear reduce multiplicative sag order rate sag strategy obtain iteration sag evaluation rate incremental surrogate focus condition sag example focus condition latter fast sag lead method somewhat problematic sag constant method coordinate attempt obtain sag l n consider stochastic ascent coordinate parameterize next parameter strongly sag advance sag strong convexity reduce storage cost handle regularization size uniform incorporate author iy storage cost prohibitive often gradient cost take eq store storage sparsity corresponding dense advantage sag time sag particular store iteration efficiently change update sag million zero total point early see point many uninformative point see converge lead sag appear difficult find sag beginning outperform sag sg sag information collect sag hybrid sg sag algorithms cost compute thus exact gradient requirement gradient regularizer dense implement efficient scalar multiply though prevent become normalize set efficiently operation variant time code sag keep track whether visit sum need implement ia let related apply form rather solve might well warm performance use gradient collect sag algorithm initialize sag scenario may beneficial setting around standard always often perform suggest though perform lipschitz evaluating run basic double whenever depend test avoid instability cause test neighbourhood size take initialize small effective never perform find rather add account parallelism architecture sg mini batch batch mini sag parallelism additionally dramatically storage batch batch reduction
develop adaptation particle adaptation sample closely ability proposal density area adaptive sir sir line parameter noise tuning rule kernel artificial kl sample around sir miss efficacy illustrate invariant slowly stochastic non exhibit good system advantage particular characterization state measurement respectively markov partially variable model process represent possibly assumption parametrize e moment like artificial introduce evolve govern artificial cone definite careful tuning avoid degeneracy estimation notational simplicity distinction make equation markov density markov density measurement characterize marginal series sake clarity omit derivation aim wherein measure output compute wherein arrive representation condition py py z q pz ignore compact write follow recurrence mse pz norm mse risk mmse except system state capability parameter paper propose sir numerically approximate line review detail simply intrinsic limitation fundamental consequence generate weight target particle trivial alternate function pz z pz sampling generate particle integral need use available joint nz dirac delta locate sample smc approximation give approximation respectively yield substitute independent outside integral yield dirac proof marginal covariance pd respectively remark mmse finally generate smc approximation pdfs marginalization gaussian q law probability substitute refer implication smc remark smc substituting substitute yield algebraic unchanged p variance important ad sir sample yield pdf smc overcome issue dispersion shrinkage kernel width replace width become plausible smc v smoothing approximation corollary finally represent w pdfs particle distribution represent smc substitute weight mmse parameter outline smc approximation correct kernel unclear suggest optimize batch ad hoc establish incoming tuning rule line paper minimization optimizer tune sample adaptation sir sir assign insufficient particle standard sir sir see filter allow sir different pz py operator pz py z likelihood sir particle fall kl divergence kl q however smc algebraic optimization formulate base substituting yield proposition dispersion make ad sir value readily place provide compatible development importance degeneracy wherein skewed require contribution resample scheme particle replace particle systematic easy implementation draw new particle replacement realize equality resample step independent return particle due particle discuss correlate particle accuracy mmse remark mmse mean avoid degradation resample systematic resample measurement process become available time allow present long estimate address miss predict mmse mmse outline represented step mmse th optimally proposition miss problem project address smc approximation correct value choose remark ahead mmse law probability ahead miss smc substituting discuss apply I nh particle set particle nn available line assume measurement complete missing discuss select pdf parameter generate identically sample use outline replacement generate identically particle associate outline particle set replacement pz analytical solution mmse mmse convergence beyond issue em mmse ball predefine accurate inaccurate converge see serious issue severe estimate hybrid system discrete mechanism include consider consideration make select complexity time n system behind simply next line develop successful linear advantage adopt variety author noise e measurement algorithm term particle appear highlight issue em force perspective asymptotic efficiency however solve step dynamical hour run art parameter either efficacy method case remark computational cost quantification introduce artificial might point assess parameter assessment mse confirm use tuning intend involve ad sir situation efficacy illustrate formulate linear mainly estimation study measurement estimation comparison simulation condition maintain extent particle reduce error smc mc simulation cccc c prior mutually study four run measurement mmse estimate along standard four estimate neighbourhood compare attribute value highlight ad sir filter wherein approximate posterior load algorithm measurement kernel smooth smoothing tuning miss converge neighbourhood sampling percentage computation take second ghz intel window fast comment make remark base wherein propose trajectory reduce figure validate proposition achieve neighbourhood miss another example efficacy percentage miss pz py initial q algorithm particle three case choice experiment select mutually variate large variance ensures include section mc mc mmse uncertainty high value evident neighbourhood yield estimate present neighbourhood second second parameter cc ccc rule highlight smc approximate distribution clear tuning project interestingly particle h tv stationary dynamic tune study density wide understood particle close limit I depend arbitrarily demonstrate efficacy tune non sir extension handle miss measurement usual introduce artificial smoothing algorithm smoothing importance resample different noise avoid advantage traditional natural science corollary proposition remark empty b computer material engineering mail chemical engineering bc mail role control monitor stochastic involve integral amenable carlo smc pf exist recognize propose line state handle simultaneous sequential sir approach kullback kl allow sir combine parameter line bayesian measurement recent advance fidelity dynamic implement advanced monitoring behaviour time processing parameter optimal filter kalman kalman filter extension year simultaneous advance provide non complexity line consider simultaneous system bayesian line state briefly review exposition certainly follow form state simultaneous lack ergodicity filter employ approach degeneracy smc dirac delta accumulation successive mc term grow reduce degeneracy accumulation successive introduce diversity add artificial e walk practice artificial appear line computational complexity particle smc
se expressive sm gaussian performance sm se pe green sm dash black red respectively mat ern sm wide density sm closely recover stationary point one mat ern kernel mat ern far gaussian process function exponential attempt integrate sm kernel mat ern kernel sm normalise compare generate mat ern correlation autocorrelation correlation choice kernel autocorrelation particularly lag empirical autocorrelation function mat ern exponential mat ern even though sm process finitely gaussians densitie reconstruct quadratic pe rational scale periodic derive square exponential pe mat ern gaussian reconstruct sample sm result heavy tail model one large period point justify fourth sm effect complexity marginal likelihood square exponential learn se stationary covariance machine gaussian kernel capture essential pattern covariance sm learn covariance ar gp follow pattern systematically periodic range view slowly smoothly covariance function density peak feature tendency negative automatic determination component sm forecast unit ahead function complex pattern show perhaps difficult exercise identify feature miss complete symmetry origin peak interference side peak origin peak periodic c learn correlation mat ern density sm se almost perfectly train blue sm mass gps mat ern se periodic predict reasonably entirely figure sm normalise behaviour pattern pattern learn mat unable discover complex assign high correlation nearby gaussian sm peak use unit peak structure distance square exponential origin green figure record blue wish forecast next short long absence force one trend expense trend short variation expense see mat ern trend sensible almost quickly learn se magnitude treat patterns kernel se generalize trend well sm kernel band rational periodic red sm mat ern since essentially density square red sm sharp frequency peak peak beyond describe large extent trend peak peak new effect air traffic detailed property etc number forecast rational pe sm se pe mse mse expressive kernel use process range kernel drop popular kernel benefit procedure gaussian powerful smoothing discovery nonparametric rich nonparametric naturally example explore pattern future work integrate spectral recently develop recent toeplitz sm speedup prediction david discussion rich interpolation process discover enable model fourier broad stationary inference analytic discover long co trend also reconstruct covariance framework fundamentally discovery machine perceptron simple neuron hope agent like automatically discover hidden datum human learn way technique subsequent rather analytically non classification often propertie e etc determine give task activation neural sometimes expressive discovery code often square albeit infinite smoothing device replacement agent feature context sometimes specifically representation network via build automated reason decision suggest inductive reasoning concept generalization remarkably particular expressive reflect infinitely expressive kernel process representation expressive kernel develop combine gaussian structure design g dependent gps induce complicated interpret sophisticated approximate demanding simple analytic sophisticated together restriction typically specialized restriction complicated overfitte addition interpretable change identify difficult bias composition automatic structure intervention covariance stochastic assumption covariance flexible go composition simple form useful bias stationarity kernel automatically discover pattern kernel stationary lead analytic simplicity many drop benefit feature understand air heart brief process kernel section model fundamental propose kernel discover co dataset covariance process joint distribution gaussian covariance kernel value joint entry function etc kernel se process differentiable trend gps square device vary learn density interpret discover generalize henceforth refer kernel spectral sm sm discover patterns model covariance kernel smooth interpolation improve likelihood alternative learn examine prediction discover fundamental difference alternative gaussian marginalization unknown gradient marginal section assume gp automatic determination minimize likelihood penalty log eigenvalue towards increase improve fit moreover sm anneal make easy optimize undesirable optima fully alternatively integrate markov estimate wish sm kernel inference effort popular square exponential se mat ern rational quadratic periodic pe fair likelihood suit dataset base sm training test compare mkl intend mixture se kernel correspond scale density perform well multimodal record use blue year green process gps tool human recognize hard code covariance look
inference generating proving method inefficient method df inefficient df power et recent study amongst view finding propose step line finding lag implicitly approach df break new discusse arise step df step df retain conclude generate stochastic parameter addition stochastically bound difference stock inverse exist ar decay alternative order representation q kt calculate df df test combine employ df eq extend employ q feasibility expand may trend miss trend x x invariance regressor power incorrectly invariance spurious important df trend feasible misspecification note lag mistake df first contain although calculate df unit end obtain infeasible feasibility l jj p substitute always correctly specify ls say serial corresponding provide misspecification efficiency efficiency observation employ lag study zero place little allow employ row l residual inefficient efficient u tr efficient common note df df df recommend eq however recommend less efficient need increase recommendation observation respectively df correctly usual df df sign square exclude lr lm control give apply construct version lm exclude df lr property lm lm include similar notation star l lm know regressor lm test gets go finite root original misspecification employ retain calculation df l residual original regressor demonstrate df l structural step employ instead residual estimation minimal even minor robustness df df algorithm axiom conclusion conjecture example exercise lemma solution em home university estimation df firstly usual misspecification df test new df circumstance inefficient finally two employ root autoregressive deterministic unit root testing discover efficient turn unit quite
delay information prediction show regret important maximum gap forecaster receive show case increase fashion gap delay consider show bandit delay also bandit delay monitoring extend exist black algorithm delay feedback assumption underlie bound delay adversarial non adversarial tight bound non delay adversarial full reward show enjoy algorithm enjoy delay subsample minimax delay see construct imply turn later satisfied armed bandit contextual bandit monitor extend delay need feedback prediction say instance exist ready one result delay delay bold reduce h time instant pick feed pick time instant feedback bold depend many number create bold create begin instant instance feedback g forecaster instant feedback instant create delay algorithm bold delay bold enjoy expect assume delay forecaster prediction bold concavity denote time denote incur instant delay delay choose ax fact inequality concavity substituting expectation meaningful delay back q generalizing monitor bind tight partial forecaster extra reward assume delay I sequence consider variance similarly bernstein inequality least union variance eq delay eq note delay whenever forecaster generality omit extend finite value separately outcome independent forecaster finite partial armed mab previous section feedback delay result additive fashion delay multiplicative adversarial outcome potential I sequence predict reward prediction I forecaster become similarly adversarial build delay feedback receive feedback core delay feedback buffer instant predict update instant buffer delay algorithm come store separate outer prediction algorithm run instant real prediction real feedback delay run delay delay feedback predict regret let denote receive instant making time furthermore predict time step make relate make time take otherwise instant instant instant buffer must empty would instant feed extra give upper delay delay stochastic environment upper run without delay delay bound delay prediction unbounded run base combining give lemma reward h I work run simulated reward right delay conclude proof convert one handle delay feedback modification inside delay extend delayed enable requirement box delay consider armed extend set ucb extend delayed penalty delay stochastic mab feedback reward draw I use delay optimistic different different type bound time instant reward delay instant presence delay use instant reward prediction instant delay version guarantee delay depend delay ucb trial concentration suitable use ucb use form ts decision rule bind algorithm call delayed expect regret delay ucb last different delayed ucb effect delay partial cover delay one delay feedback delay adversarial increase qualitatively low important determine number miss reward interesting note server infinitely type chain immediately result technique area hence work partial monitoring end lemma sequence preserve result sequence sufficient eq permutation equation since permutation independent law probability general need result subsequence future e I sequence sort delay lemma sequence observation observe subsequence decision subsequence future independent turn framework ucb delay setting analyze suboptimal ucb g make least enough use inequality form confidence unlikely suboptimal sample suffice high thus bind expect example inequality hoeffde inequality ucb ucb precisely delay work number time bound use concentration delay reward include instant inequality non delay use delay reward non delay depend delay demonstrate ucb delay ucb section ucb summation event last hoeffde ucb introduce kl ucb delay confidence arrive delay kl together obtain additive delayed setting regret somewhat need capture summation bound constant substituting last prove recall paper bound independent randomization eq paper bounding eq let back combine get delay reward consideration delay setting therefore let corollary feedback receive recently web systematic topic delay somewhat surprisingly regret adversarial way give meta algorithm delay handle feedback loop modification delay feedback meta low prediction make delayed ad come engine delay fashion click ad click send module ad delay among delay prove machine delay delay setup delay work concern delay delay mostly constant delay delay systematic delay feedback cover setting extend improve particular meta black box delay handle
nmf define matrix independent norm nmf suffer drawback list nest drawback globally nonnegative cone rank rank tend nonnegative cone remove backward nest applicable nmf optimal factorization cone forward different rank nest nest nonnegative nonnegative extremely complicated phenomenon recommend idea nest constraint backward approximation follow nonnegative decrease singular index define conjecture constraint replace sequence nmf least square discuss generally definition impose natural nest core part form search optimal equation challenge reason onto optimization problem share let solution feasible define problem turn sense solution always change slightly set easy handle analytically major nonconvex nonconvex problem subsection approximate reformulate identifiability want straightforward view svd formulation nmf uniqueness develop central nest cone rank approximations original datum negativity svd use nonnegative svd approximate variability among see projection nonnegative cone svd base modify nonnegative base nonnegative nonnegative height pt svd k svd base generate distance vector dx square proposition show span svd provide subspace approximate minimizer subspace proposition presentation show approximation uniqueness identifiable require distinguish proposition unique uniqueness approximate space component degree nmf nest approximation show visualization simulate setting subsection number summary uniqueness datum among ht pca svd method projection nonnegative cone summary average pca approximations cone rank approximation suffer interpretability approximation nonnegative frobenius approximation observation nonnegative cone suffer interpretation challenge investigate nmf study principal angle principal define span subspace span angle minimum eigenvalue matrix obtain qr subspace simulate realization projection svd report approximation realization cccc rank number projection matrix rank angle approximate summarize generate small average projection nmf suggest nice property drawback investigate angle note angle iid normalize point lie sphere apply nmf angle approximation repeat simulation table ht show summary angle angle angle point angle angle increase pca approach nest improvement nmf cone interpretability svd approximation rank rank challenging improvement problem investigate potentially interpretation visual device scatter plot score loading structure exploratory visualization tool author thank statistical mathematical science kind massive program author national foundation grant dms third author foundation dms program analysis factorize column column approximate prove square subspace f uniquely sign generality assume uniquely sign uniquely note definite standard convexity bp motivated nest nonnegative cone approach drawback traditional nonnegative cause cone interpretable nonnegative nmf issue nmf suffer drawback unique span approximation drawback determine number rank interpretability propose nest illustrate drawback traditional usefulness constrain functional nested object orient principal science come along context population tree lie manifold concept euclidean space boundary recent convenient property linearity naturally lie bring one major lie object mathematical oriented name systematic complicated object paper goal object tool reveal major variation orthogonal principal sequentially provide pc learn component pca projection sometimes interpretation interpretation sensible contexts impact type center pca svd variation however even less negativity direction interior orthogonal direction outside interesting nmf svd e gain popularity nmf suffer severe rank span nmf nmf toy toy pca observation green dot nonnegative realization simulation blue projection dash approximate intersection graphical box bound highlight intersection set actually plane panel nmf approximate intersection thorough middle bottom panel three highlight project outside rank similarly outside highlight lie nmf sensible notion approximate middle panel bottom viewpoint highlight viewpoint middle row provide nmf nmf approximate show face highlight reveal end outside plane degree far method propose paper nest cone middle bottom panel nmf suitably analog nonnegative reveal actual idea affine space subspace large mode variation rank nest subspace usually analysis data orthogonality cause projection easily leave motivated nest large mode approximation lie lie cone cone sequence frobenius residual ideal
variance improve try bias shrinkage add predictor determine small exhibit enhance interpretability usually achieve propose pass penalize compare distribute require multiple improvement train cross optimization lasso ridge elastic standardized scale column standardize average fit formally calculate usually store system memory eq q validation train cross validation reduce statistic I pre version www com parallelism cross validation implement job notice collect observation million confident feature thm lemma propose pass intercept depend include class
ref likely boost use cluster bioinformatic use community structure bootstrap boost crowd create case perform powerful simple greatly boost combine however try explain group simple perform advanced classifier devoted question answer ref use neural network modularity maximization improvement overall probability classifier voting rule classification independent application chance ref label weight weighted predictor latter differ well explain advanced classifier error decrease increase include cluster cluster ensemble clustering aggregate clustering ensemble method ensemble clustering ref construct fully connect node frequency place clustering graph represent edge cluster clustered act similarity result partition linkage partitioning lin community subgraph community discuss linkage nature place connect community detection candidate cluster agglomerative cluster determine frequency place meta merge node node node member candidate occurrence candidate linkage lose cluster hierarchical dendrogram list merge hierarchical node belong especially lie equally belong community also tree sensitive uncertain neighbor membership candidate community node find quite community classify correctly sensitive merge hierarchical manner merge node get merge early belong another community belong early merge detail regard simulation experiment propose method combination label fusion community lp discuss preliminary generation comparative lp sp ga complexity demonstrate synthetic generate structure community synthetic network value generation family mix community exponent distribution benchmark varied fraction community intra community community community community accord adopt synthetic consist five version ref firstly law secondly power parameter community use configuration model assign randomly drawback triangle social network enable study correlate effectiveness uncertain simulate community structure synthetic external traditionally include commonly classification drawback label single detect detection external community previously overlap measure measure correlation later former information much external vice versa follow ref label community joint distribution node obtain community structure unity correlation matrix classification let row row external neighborhood unity match advantage linkage label need remove overlap detection question answer lp merge varied node vary merge varied indicate find conclusion need network respect statistical systematic error dominate aggregate lp ga algorithms use vary community figure differ run number mix lp ga appearance sp previously know network quickly critical worth tail behavior parameter drop shift lp continue particularly visible small nature lp ga sp correlation computational mixing parameter lp well perhaps ga lp densely ga lp algorithm sp continue previously world aspect triangle conclude suggest modularity community modularity modularity frequency weight modularity modularity modularity modification manner previous version aspect community scalability desirable community therefore sp ga previously ga low modification ref use implementation lp merge agglomerative linkage complexity method merge theoretical run lp ga number discuss ga computational complexity previously discuss ref interesting fast small ga ga mix detection visible propose ga implementation difference lot make comparison complexity implementation ga advantage run increase function network detect name fusion community merging scale evaluate simulation study network especially lee ki discussion comment project force several source method detection aggregate ensemble ensemble community different apply method use community community detection apply low approach community nature regular interaction reason research provide method physics physical concerned quantifying aspect centrality measure degree robustness apply large example energy grid network protein social consequence interact body topic node densely outside community network school neighbor effort devote algorithm detect partition accept structure community ensemble method work community algorithm fuse accurate community present community conceptual community ensemble cluster possible definition community effective find merge aggregate run addition latter community insight relation structure merging aggregation community bootstrap replicate network ensemble method recent drawing mechanic discrete mathematic computer statistic thorough current community ref provide cluster g bioinformatic useful merging devoted community merge community vote merge manner good ref develop continue community ensemble detection suggestion give offer propose method algorithm modularity maximize agglomerative spin summary remark concern consist represent computer protein represent connection discuss community use bootstrap appear network later paper offer robustness structure additional structure many different relation citation quality obtain usually network randomly fix modularity vector community belong calculate member adjacency ij mn modularity networks nan modularity structure modularity take well community nan comparative drawback difficulty modularity discuss resolution detection consume reasonable effort introduce network example heuristic np also modularity maxima make global discuss effort refer community early manual
least fix simplicity drop terminal rooted height interpret majority evaluate thereby positive form compare principle develop artificial intelligence reasoning stop terminal output continue recursively protocol stop return terminal g consistent pose fact maximal cell tend unit cube cell observation imply classical consistency proof rely lebesgue global argument value however partition depend impose absolutely minimax bayes side efficiently model clear conditional attack ahead road start figure denote partition represent underlie full cell soon cell measurable tree root level k k n take manuscript cause confusion easy fact cell k partition remark k proposition diameter sufficiently infinity introduce play split tree start section book monotone crucial split also diameter result page make sure run away nonnegative fact prove infinity term aim fix uniformly vanish triangle continuous prove first statement proposition false notation subsequence without generality hand monotonicity statement contradict region tree collection strictly eq e na na n na definition rule accord since proposition cover q follow term tend statement proposition aim cell therefore large possible applying conclude imply thus fact hand q tend show fact consequently section adopt leave represent probability assume nonnegative integer large enough empty root th cut first fall conditionally distribute importantly eq q odd odd send canonical way g repeat median splitting create child repeat scheme construct root leave length leave deterministic already conditionally leaf symbol therefore integer subtree root sequence integer similarly statement choose quick I manner restriction cell conditionally fix sequence integer leave subtree root cell contain enough nonnegative use subtree root similar use clearly thus collect obtain cell cell cell lead replace invoke corollary proposition define cell one fact n na na e k k consequently eq q right four term whereas lemma see combine result statement na q na aim observe root cell k n n n anonymous suggestion section section section universit sup france tree give majority one potentially seem paradigm cell know different classifier asymptotic deal mind make decision majority rule part partition cell split upon principle classifier short parallelism ask purpose discussion various tree follow construction classifier motivate challenge involve issue role procedure adapt execution processor share memory communication processor need advance prototype take value finitely deal classifier certainly associate decision classifier unknown fortunately collect identically distribute co assume attempt notation many popular classifier histogram rule rule tree cite comprehensive introduction review among procedure view tree simple voting also regard indicator case method conceptually simple follow restriction huge paradigm address geometry basic notion tree relate pattern year mention fu I region majority vote tie break convention favor c dependent thus many great end node information tree penalty prune cart tree commonly mining create recursively tree final induction tree phrase mining literature far strategy topic concern manner make cart axis perform isolate second process call pruning cart non share allow explore thereby child sake clarity proof technical thing distribution tree choose dimension classical repeat region leave one leaf framework leave two set never leave atomic choice randomize function cell denote q cell cell take occur attempt make singleton need region belong study respectively lemma thing precise define path root nk cell median least since combine suffice establish tree randomize classifier mean diameter cell point n prove cut diameter cell take care randomization randomization order coin bit selection random direction bit tree tend level carry magnitude randomness matter set fine detail present analysis
large observation posterior starting want propagation denote transition typically unless idea flexibility arbitrary convert marginalization choice particle rare event time interested carlo inefficient call split one introduce sequence observation conditional probability give resample step effort propagate fix satisfy rather propagate particle satisfie unbiased particle aim instead accept reject importance approximate filter state basic imbalance present engineering terminology extract partial observation usually assimilation year markovian evolution white noise development backward space essential arrive sequentially start mid develop filter continuous long root occur develop recursive carlo method particle monte ensure sample among thing outside static ensemble kalman delay idea become research book survey en brief scope limitation many reference would process discrete simplify initial time independent density reference lebesgue counting homogeneity simplify state discrete transition usually analytically able simulate deterministic like slight abuse notation stand argument indicate involve probability measure range finance stochastic engineering recognition biology genome sequence possible reference biology financial state call terminology formula marginalization expectation recursive formulae verify consist propagation correction want computed special lead conditional filter update kalman gain change particle adaptively new observation importance draw start weight drawback locate position main mass unbalanced avoid introduce resample propagation particle weight basic call sir importance resample work propagate advantage recursion induction unbiased irrelevant particle resample one reduce resample simple variable intersection balance little extra cost whenever weight effective define justification step draw dominate weight let make particle compatible auxiliary new resample goal possibly shrink keep transition occur whole straightforward propagation unbalanced track though later create diversity unbalanced easily explanation provide advanced overcome difficulty filter approximation particle kernel construct instance new old component remove tie typically computational want use particle filter whereas monte kalman estimate moment propagate draw kalman gain algorithm however gaussian update systematic neither spread change ensemble kalman extremely spread complicated propagation computational force turn attempt make combine kalman filter kalman forecast usually kalman filter begin combine pass limit bayes relation formula dual symbol integral particle filter disadvantage backward py nh nh nh integrable backward approximation combine particle complexity innovation building block depend sampler slow whole filter approximation correct go metropolis hasting place ratio unbiased surprising error approximation algorithm invariant propose sampler filter give provide particle filter modify law number central
sufficiently small function extreme relation get argument lead derivative hold whenever number follow complete unless see give strictly follow unless writing satisfied establish follow proceed lemma suppose point write contradict cardinality exist decrease x x check provide interpret q let shall entirely analogous equal constraint yield supremum cardinality clear analogous statement hold function extreme therefore recall calculate knowledge sense try determine range argue identity characterize joint divergence restrict divergence prove short proof theorem anonymous proof elaborate attempt prove via f f f f mp clearly lie hull point convex hull write point complete immediately weak speak deduce slightly strong number pair determine reduce inspection proof solely optimization quantity achieve conclusion space cardinality strictly shall explain numerically maximize word divergence equal strictly convex supremum divergence exist strict eq easy conclusion section crucially proof equip critical application closure thus closure quantity behave divergence non finite drop constraint divergence divergence lemma let divergence shall inequality mm result denote leibl variation kullback say equal hold arbitrary show divergence handle unable resolve finite neither lemma function see necessary completeness sufficient suppose theorem let measure explore question l equation together rise equation appropriate measure imply give geometric deal e deal open possibly function empty concavity linearity would imply average happen fix notational outside equal extreme non empty check piecewise two concavity must segment concavity contradict primitive divergence actually equal opposed problem primitive divergence obtain inequality divergence divergence receive much mention area well leibler improve far kullback problem solve implicit infimum side bind squared hellinger distance triangular jensen shannon divergence total variation pair moreover variation right equal define corollary easily check symmetric side fact require fact imply describe another quantity let strictly strictly map inverse convexity strictly increase imply write infimum divergence sharp chi order case primitive motivated obtaining low every demonstrate divergence search low know inequality exist inequality sharp primitive bound problem primitive primitive subject divergence arise statistical le popular technique obtain minimax define affinity another technique application le obtain variation square hellinger chi translate common room tight divergence oppose improved bound solve primitive divergence divergence exactly convex set dimensional equality constraint denote primitive optimal constraint set maximization right side convex invariant show restrict problem obvious complete consider hellinger consider upper hellinger plot convex correspond quantity analytically symmetric sharp hellinger attribute plot analytically line clear agree simple maximize kullback take dot dimensional green analytic sharp b discuss limit maximum divergence leibl divergence consider variation leibl exist analytical problem straightforward solve program matlab surface square hellinger kullback leibler expect total variation surface flat vice versa flat vary approximately surface ridge intersection two surface see individual exist leave side coordinate informative upper total square hellinger kullback leibler strict pointwise require primitive solve hellinger distance upper leibler clearly variable space solve dot inequality curve plot lie constraint active kk kullback blue curve improvement blue curve panel square hellinger prove respectively red evident restrict constraint measure hellinger plot agree give rise attribute pair measure straight line discrimination check divergence discrimination correspond divergence investigate solve plot red triangle blue dot plot sharp analytic formula conjecture equality numerically analytic h triangle black dot extremely discussion grateful anonymous indicate weak general divergence commonly mathematical theory divergence chi variation paper maximize divergence optimization comprehensive unified obtaining sharp exist divergence improve sharp kullback possible hellinger question include machine goal provide answer viewpoint pose hellinger subject leibler shall unchanged element hellinger subject kullback leibler restrict attention dimensional optimization make tractable result leibl distance divergence divergence divergence virtue convexity two measure see choice dominate divergence variation chi divergence ready divergence hand side compute quantity optimization finite quantity main reason study quantity yield monotonicity inequality sharp sense inequality divergence area example obtain limit divergence helpful prove machine describe inequality involve divergence paper divergence paper sharp divergence opposed work generality popular inequality case leibl divergence less deal inequality primitive divergence primitive divergence obtain sharp divergence main divergence problem outline many inequality divergence improve sharp paper structure recent representation problem think maximize satisfy number part restrict extreme third characterize extreme probability measure finite divergence joint range determine solve anonymous turn base theorem weak remark extension tight obtain low dimensional also describe denote space probability require finite divergence see remark explanation divergence well form equal optimal provide theorem tight comment assumption validity attempt range yield eq common measure non every divergence denote associate precise every standard simply write twice differentiable short primitive divergence simple part straightforward check divergence correspond divergence write primitive primitive divergence variation primitive testing space note intuitively maximally separate mutually achieve eq moreover function divergence
long goodness measure distance estimator estimator study orthonormal noisy measurement solve isometry moreover random invertible way rip bad particularly serious close orthonormal application rip sharp estimation treatment cover unified serve basis minimization deterministic error estimator goal definition every minima compact confusion omit dependency clear article rely reverse triangle q triangle depend outlier square compare precisely outlier orthogonal decomposition suppose q obtain mf f q hence obtain yield h characterization recovery relate rank conclude find q x satisfie combined theorem recover concept signal relation semidefinite programming sdp bound next sparse verify dense noise arbitrary keep robustness improve response particular outlier reduce efficacy deal drawback noisy gaussian main strength motivate define magnitude denote word define minimizer inf reason constraint bring problem existence solution continuity function advantageous property numerically reason hold ng equivalently hence equivalent lagrangian duality eq finally absence existence multiplier sparse solution coincide unique definition enough pg n ng inequality convexity threshold outlier consider noise outlier comparable estimator term estimator bound simulation play important reduce noise let unique every thus b e dual combine claim depend dependency point keep matter term forward note equivalently inner equivalently since lipschitz characterize proximal step ns eq last equality deduce I h follow theorem approximate follow describe experimental standard accord standard fraction contamination draw light heavy type adversarial adversarial contamination create component contamination size construct index generate independent entry set method estimator compute ccc axis xlabel percentage contamination file txt file l txt xlabel contamination file txt file txt ls txt percentage draw percentage quantify plot light tailed outlier outperform estimator sparse contamination contamination raise right plot level tail well low contamination percentage contamination outperform dramatically contamination focus phenomenon contamination sensitivity heavy one examine confirm estimator width xlabel contamination txt file txt file txt scale xlabel percentage contamination file short file txt contamination contamination deep reconstruction link permit quantitative qualitative unbounded character robust statistic necessary work approach noisy modification nice like concern robustness influence every lemma obtain replace remark act regression matrix contamination restrict perform fine introduce inf convolution concern least outlier present convergent property robust reconstruction point inf convolution eq row regression give variable independent residual unbiased sensitive deviation normality normality violate interest statistical different error eq light tail usually arbitrary suppose outlier quantity represent contamination measure asymptotic since fit exist robust highest possible subject robust efficiently optimality act influence convex observation residual opposite face beyond capability size framework function nonetheless satisfy equal equal minus analysis remarkably outlier new good minimization residual globally convergent numerical actually behavior face outlier absence outlier noise theory sparse convergent define article shall notation
cca bias useful cca minimal bias estimator recent approximate regression requirement view state semi learn substantially improve wide volume collect social increase become non relationship cubic point randomization recently comparable cost machine computation intractable among nystr property arise dataset manually label expensive supervise extract structure unlabele propose supervise nystr om essentially way nystr another nystr om almost fouri quite step canonical cca procedure view uncorrelated bias work show nystr om view expect version simple supervise introduce outperform number label exhibit dramatically typical chose unlabele introduce nystr om approximation runtime point increase outperform average suggest extend unsupervised algorithm build elegant surprisingly despite performance view cca widely proposal multi view occur equip view multi overcome construct view cca regression nystr om view semi reduce computational multi view assumption many introduce intermediate random design empirical wish control view cca coefficient idea uncorrelated contribute eigenvalue eigenvalue decay rapidly e low intrinsic attain significant reduction unlabele improved performance weak view widely use difficulty naturally equip view construct view view result first kind knowledge exploit cca result extremely increase make nystr om algorithm equip despite improve art baseline variability number despite require additional similarity connection method approximate intractable method decade view theoretical comparison behind sparse compress algorithm learn algebra despite prediction widely difficulty obtain equip view multi furthermore construct view splitting set could view multi study kind cca extremely factor far computational difficult nystr om perform empirically variability number require tuning conceptually random extensive algorithm test variance theoretical section respectively point potential reason satisfy randomization two satisfy view result extremely state art factor comparison consideration consistently outperform nystr come equip range method improve error baseline also reduce tune total unlabele label feature matrix cca eigenvalue explicit get coefficient cca coefficient equally view cca nystr om construct view accord assumption regressor view map assumption good view canonical set variable cca basis projection pair vector maximize basis delta th norm define view couple ridge find estimator canonical bias regressor exist uncorrelated reduce variance formally canonical regression construct shrinkage across term whereas think unlabele canonical rapidly large nystr om instead learn linear lead speed om subsampling gram nystr om approximation gram inverse construct diagonal alternative nystr om operator proposition nystr space span eigenfunction nystr om view solve semi generate view consist nystr om label htp label eq canonical view generate random next view heavily cca introduce penalty cca basis introduce large correlate due cca obtain nystr om require learn step however cca linear program recent lead nystr om fix nystr om define mse ridge refer construct short good estimator om smoothed estimator consistent thus generalization control alternative nystr step turn line experimental operator user generalization intensive difficult since first gram nystr label unlabeled tune inverse justification note nystr om regression solve ridge eigenfunction towards nystr om variety problem dataset position robot convention position l repository take take c take exhibit method label training nystr example randomly select feasible dataset second dataset exhibit intractable importantly overhead task square mse classification report test set misclassifie considerably r avg reduction std htp present performance bar standard always improves label go
ht c average rand consist sized cluster l error three equally change diagonal multivariate primary perform analysis input apart th change method proceed goodness fit statistic reason prefer though run output estimate simulation estimate currently time package quickly provide additional change interval demonstrate identify maximize execute twice segmentation allow observation input output create change segmentation store update distance test update segment input output point goodness equation merge goodness segment merge follow greatly disjoint size merged agglomerative outline segment segment distance goodness merge candidate update segmentation penalize thm thm nonparametric nj ny email edu web many way purely package agglomerative identify agglomerative location detect distributional within key word signal processing title r change detect distributional arise modeling apply identify associate disease analysis anomaly classification multiple package provide analysis distributional determination number simultaneously observation distributional identify package instance univariate series although consider method allow change drawback term variety point analysis method change detect distributional full package package series change index recent however package design change mean tool change within linear regression detect package allowing minimize residual statistic fundamental change section briefly include outline algorithm package limitation change package univariate series multiple change absolute identically respectively employ distance equation copy independent identically distribute far independent distribution additionally independent mutual measure dx integration scale divergence degenerate hypothesis perturb applicable distribution develop method hierarchical point location exist segment tree node copy segment create significance point general segment location location point associate description help file point series procedure recommend require adjust argument resample change fast perform change similar guarantee recommend complete method hierarchical agglomerative point segmentation segmentation reduce allow priori observation segment merge maximize maximize change within goodness fit adjacent segment segmentation fit maximum goodness fit give computationally intensive detailed explanation efficiently carry overfitte concern goodness accomplish maximize mu period alpha member alpha member opt opt default generate result identically change cause bivariate distribution bivariate degree set library mu r period mu diag period diag period alpha ht period identity matrix period student matrix spatio examine dataset consist associated time dataset interval spatial times intensity intensity period mixture weight initially segment termination point obtain follow library library library lambda arrival matrix diag time mix nan count lambda interval mixing interval interval x member e member alpha run statistic scheme respectively densitie ht real dataset micro record series consist micro expect almost micro correspond segment result e another nonparametric change variability could apply remove miss replace average neighboring leave individual dataset library r use e estimate e subsample individual sized series plot line point location location dimensional identify first procedure observe phenomenon look intuitively segmentation place strong limitation change segment time provide span include change release capital asset management initially segment segment pass american act time index identify financial co
numerous introduce randomness characterization effect bias precision mostly systematic true describe dispersion relate true combine concept accuracy mean square quadrature expression skewness unweighted assume uncertainty measurement dependence unbiased simulation employ size square uncertainty interpolation base describe improvement estimator end employ throughout follow biased section unweighted formulation noise unbiased noise biased counterpart ratio sec scheme sample size conclusion follow derivation unbiased moment expectation th sample unbiased estimate central standardized skewness define consistency unbiased denote systematic consider herein random error refer error uncertainty uncertainty brevity name derive function uncertainty unbiased data expectation aim obtain measurement uncertainty measurement value measurement uncertainty herein decompose replace property section unbiased weight unbiased uncertainty skewness assume independent measurement uncertainty detail satisfy unweighted form substituting moment unbiased uncertainty skewness respective noise sample sample describe unweighted achieve direct substitution weight reduce unweighted noise unbiased counterpart eq consider depend definition unbiased eqs eqs large great biased compare ratio simulate error law moment compute simulate fourth power skewness standardize estimate variance without evaluate ratio root measurement uncertainty phase measurement leave u latter term draw uncertainty vary dependence weight herein size employ negligible result simulate illustrate fig mean skewness simulated phase sort w expense mostly uncertainty precision level desirable dispersion bias weight inverse weighting expression derive herein biased bias justify precision extent bias could mixed scheme eqs herein skewness expect constitutes weight scheme tuning offer limit accuracy signal conclusion phase weighted counterpart especially large sample herein sample consider simulation size unweighted phase weighted error accuracy case unlike estimator correct evaluate deviation weight unweighted dependence variance present component figure confirm level find case whole estimator standardize much great obtain weight measurement precise precise interval lead ratio apart appear limit ratio provide skewness high fig accuracy affect moment fig less precise normalization square reduce exhibit trend counterpart fig normalization improve bias great precision variance non skewness standardize avoid circumstance sample size herein bias nan bias bias biased appear unbiased apart skewness noisy uncertainty phase satisfactory improvement tune fit estimator interest view expression skewness provide unweighted formulation independent uncertainty particularly characterize regime simulation skew periodic employ unbiased unweighted phase estimator precise ratio involve phase able level scheme
author near use sdp near eigenvalue separable multiply full replace find condition near nmf noise case reduction take orthonormal rather analyze apply matrix identify index extract index extract allow error give assumption identify index course unknown nmf approximately semidefinite programming combine process would improve section numerical explain previous need imply cholesky decomposition step minimum volume ellipsoid origin ellipsoid via axis ellipsoid eigenvector square root formulate volume ellipsoid full rank hull column dimensional assumption noiseless dual te whose optimal duality theorem noiseless provide separable nmf continuously perturbation quantify optimal near later satisfy h ia feasible eq km bind equation eq optimal assumption denote limit low show sufficiently satisfie upper prove condition nmf make corollary lemma imply hence introduction derive situation use hyperspectral value respect frobenius resp resp singular near onto space minimize sum residual huge projection pick subset perform subset therein dimensionality particular beneficial technique processing equivalent frobenius column space avoid solve give rank svd sdp heuristic unless say noiseless case belong hull advance try solve whether resolve keep order guess residual equal zero extract ht separable matrix number constraint precision truncate r r I middle sharp change post resp post resp note even large noise level column hull post table average time robustness post explore challenging point perturb hull experiment give svd different provide set since therefore performance q vertex centroid see zero noise level matrix report robustness running algorithm post interesting robustness hierarchy algorithm predict theoretical development eq process clearly advantageous sdp dominate heuristic variant image column spectral signature image spectral signature material present linear pure hyperspectral image pixel hyperspectral separable therein section hyperspectral image band compose eight ht variant list perfectly noise spread standard value compute angle image extract spectral signature define vector scaling translation match report post green top give exactly compute abundance map correspond visually moreover compute svd signature extract ht slightly whose level assess real outlier handle separately assess hyperspectral dimensionality technique applicable run several hyperspectral differently ht hyperspectral common extract c sdp necessary algorithm terminate explain cost solve comparable note relatively especially datum semidefinite allow near nmf particular popular provably robust synthetic show apply image long possible would sdp first structure ellipsoid see particularly hyperspectral practically comment improve side inequality violate remains check r hold integer rr right increase whose give eq mm corollary nonnegative separability provably solve presence show hyperspectral near nmf span subset nonnegative approximately contain paper base improve nmf illustrate popular successive provably active hyperspectral nonnegative semidefinite programming separability robustness nmf technique linear datum represent approximately reconstruct interpret datum image weight allow additive reconstruction basis lead representation ill pose people nonlinear optimization technique hence come guarantee successful world nmf nmf exist separable separability exist require span although strong separability hyperspectral material hyperspectral contain material refer pure pixel separability assumption pure see blind source separation separability e input perturb important design robust separable nmf problem reduce cone set hull set point see near nmf nm j conditioning one normalize near full sum column perturb matrix satisfy arbitrarily identify projection recursive column project www www rw ill guarantee extract r actually www presentation general small theorem bound arbitrary also closely pick projection column far whose far require entry less illumination pixel discussion
intra stack complement inter stack average stack average output classifier ideally prediction weight diversity effectiveness tie ensemble include decrease quickly become ensemble subset base pool diverse successful selection establish classifier individual iteratively well actually improve al begin add predictor maximize auc classifier performance candidate maximum reach evaluation candidate ensemble greedy include top time replacement early decrease predictor force diversity ensemble reduce high dominating completeness candidate prediction combine move performance validation produce nest validation ensemble diversity member determine good diversity statistic thresholde probability yield great otherwise predict contingency correct incorrect correct pairwise statistic value tend classify correlate evaluate additional diversity agreement focus simplicity adjust raw diversity diversity clarity summarize aggregated stack good cluster stacking perform cluster size cluster stacking perform pf inter stacking size set stack cluster stack aggregated stacking stack selection well ensemble selection achieve size pf respectively perform ensemble much performance base high perform prediction however combine make poor classifier weighting stack advantage trend meaningful critical determine statistically significant methodology difference multiple determine statistically significant pair post hoc combine post hoc transformation assumption violate machine prefer use cutoff ensemble brevity significant performance difference method across table sharing label indistinguishable ranking aggregate similar aggregated stack greedy share rank distant approach aggregate stack statistically aggregation motivate improve suitable trend present base pf except forest generalize whose red raw adjusted diversity note nest relative validation stack increase meta bag result reduce overfitte quality meta computation motivate fold emphasize stack aggregation prediction genetic heterogeneous perform forest gradient homogeneous ensemble demonstrate heterogeneous improve aim predictive base combination ensemble apply decade variety difficult genetic prediction problem imbalance miss heterogeneous inherent biological stack statistically previous moderate effective even verification include predict stack connection stack demonstrate base balance variation stack accounting difference diversity maximize suggest effect heterogeneous performance diversity diversity tradeoff stacking institute financial edu present comparative study namely efficacy useful meta heterogeneous find statistically respective domain demonstrate balance bioinformatics ensemble stack combine produce task attribute prediction many diverse ensemble consensus outperform base ensemble consensus unlikely classifier ensemble available pool well understanding utilize diversity address popular bag boost example classifier build however well unclear instead wide tree heterogeneous meta stacking selection stacking construct model classifier ensemble incremental predictor balance diversity performance due ability superior across phenomenon influence lack consensus regard specific class imbalance measurement ensemble ideally world genomic dataset analyze ensemble important area genetic throughput work heterogeneou package interface classifier bag train use split balanced majority step boundary majority imbalance addition fold nest cross perform split create set meta result pool combine validation calculate area receiver operate characteristic curve auc classifier average later
show complete q verify must theorems box ellipsoid algorithm oracle present detail implement strong oracle suppose optimum thing leave would imply ellipsoid start always fact run ellipsoid polynomially strong f f call oracle proportional calls ellipsoid polynomial ellipsoid give conclude ellipsoid prove count raise possibility cut run ellipsoid check whether f heart add ellipsoid output radius final ellipsoid ellipsoid proof inequality inequality fact proceed ellipsoid guess check guess ellipsoid guarantee maximally linearly count theorem ellipsoid contain ball ellipsoid return else use ellipsoid half ellipsoid count f else return guess number approximate oracle generalize simple calculation counting step iteration polynomial polynomial complete guess succeed find denote return guess succeed return answer f sake ellipsoid ellipsoid run guess ellipsoid guess return guess inequality inequality convexity thus separate constraint final ellipsoid ellipsoid must present theorem estimate optimization optimum interior program lemma concavity shannon entropy concave entropy guess either run bit represent start maximize centroid fact entropy distribution hence upper bounded uniform thus estimate ellipsoid program marginal polynomially pre duality representation exist lemma interior good hope indeed restriction theorem useful max entropy oracle issue whether look inverse prove lemma interior show existence need let c lk constraint less one therefore integral henceforth guarantee interior must hand concave negative ingredient check interior separation oracle maximal hyperplane interior separation deduce separate first full center regular simplex scale oracle simplex center exist ball p give separate hyperplane let interior restrict attention affine choose simplex vertex ellipsoid ellipsoid obtain interior g empty ellipsoid ellipsoid guess ideally pass latter continue return oracle one volume get time first interior separation hyperplane ellipsoid technical ellipsoid radius become guess imply ellipsoid hence interior let program primal convex prove p satisfied marginal q eq concavity negativity description subsequently complete proof maximally linearly guess e x b repeat contain ball give ellipsoid tp separate hyperplane return oracle stop ellipsoid ellipsoid half ellipsoid return else stop guess thus run polynomially ellipsoid hence bound correctness guess return positive answer complete eq run ellipsoid guess hyperplane iteration separate hyperplane cut hyperplane clearly inequality cut assumption ellipsoid return answer eq claim ellipsoid contradict algorithm q proof interior attain satisfy q program entropy prove program multiplier multipli constraint lagrangian lp pg take dual become find infimum minimize e become satisfy duality imply strict optimality recall satisfied vertex hence restrict f p p equal complete proof prove close distance particular proximity distribution p ps p leibler define non negative kullback leibler distance respectively respectively hence equal hence obtain outline generalize counting problem weight solve max entropy leibler kl raise solution max entropy solve entropy convex straightforward divergence product p observe divergence distribution rewrite input program assume program approximate mf oracle bits appropriate linearly oracle kl z oracle kl oracle input highlight account proof interior value ellipsoid issue lemma lemma interior generalize complete picture access counting program algorithm linearly generalize count oracle interior optimal dual max counting oracle polynomial counting straightforward rely program shift objective approximate oracle convex mp p fact constraint convex lipschitz negative om p calculation straightforward since q towards claim use claim follow mp number ratio hence complete complete claim e complete note claim theorem theorem corollary gray email microsoft com microsoft research com compute max marginal arise applicability physics economic biology theory difficulty max entropy polynomially size description description condition subsequently count translate algorithm entropy count establish discrete collection block suppose simple principle good maximize shannon p argument observable less access obtain sample inform guess surprising show area economic biology information design find maximize ellipsoid time number interesting exponentially sized universe implicitly could span tree perfect exponential computing describe exponential good news convert max program additionally condition duality thus max main access obtain close give second handle equivalence counting focus approximation see polynomially bit raise whether description vast amount distribution example survey previous structure theoretical rigorously max distribution derive randomized round trivial problem entropy approximation approximation graphical progress problem ability max trees privacy tree max distribution question max bipartite entropy graph count count dynamic programming perfect shift algorithm count combine maximize given generate approximate counting problem bipartite basis restrict count problem open prominent perfect hard core gibbs study nice core core exhibit significant distribution lead several involve hypergraph fractional asymptotically behave enough max distribution subsequently algorithm arbitrarily entropy access count count oracle approximate variety compute concrete exist counting obtain span graph bipartite root strategy mention reverse direction show approximately entropy establish computing set max problem count graph bit basic thought thus concern long hope program require exponentially optimal solution entropy radius interior oracle give result state count count interior output setting keep length input oracle algorithm polynomially generalize count run run polynomially generalize product marginal remark level framework work dual separation oracle subset count adapt optimization problem interior program interior bit obtain algorithm answer question converse count access convex interior make input counting remark section continue hold corollary use oracle match polytope graph compute polytope start dual program indicator interior duality primal infimum finite hence pair oracle use ellipsoid relatively straight call oracle counting run polynomially rather polynomially part span clear bound fact lie affine solution direction space one impose lie thing favor optimal roughly diameter span bridge edge bridge however combinatorial argument exchange property bad implication obtain box sketch interpret v p live space interior radius contain nothing gives desire ellipsoid use marginal kullback kl divergence marginal access approximate thing count oracle translate gradient happen approximate counting equally good max distribution generalize count e z bit count problem problem arbitrarily relax generalized count oracle follow z ignore statement count appropriately suffice self approximate requirement oracle interior restrict since notion polytope appear answer entry reasonable c look combinatorial convex program polytope care notion definition combination vertex indicator vector central interest paper way maximize entropy probability give p see entropy unique unique moreover observe soon record definition distribution marginal denote relie establish duality marginal appear interior unique dual function change shift capture dd thus restrict value independent refer dual h compute good entropy marginal exist maximal linearly generalize count interior return max program polynomial represent ellipsoid program seem ellipsoid depend enough since call theorem interior max need perfect bipartite counting hold well generalize approximate solution assume count oracle polynomial input running bit need represent eq close project gradient descent upper provide choose ellipsoid proof theorem ellipsoid prove entropy approximation notion oracle interior mf assume run bit need give maximal oracle approximate oracle return run max entropy input run polynomial bit analogously distribution match polytope approximately problem hardness ask count max natural generalized counting yes provide marginal program program entropy program ellipsoid use proof count proof theorem affine k everywhere everywhere interior dual set linearly satisfied fairly proof requirement strong oracle project abuse denote latter access ellipsoid algorithm call ellipsoid property volume
replace variance maximize operator minimize minimize sparsity realization energy much coefficient compute estimator replace generalize order relatively optimize summation unitary definite optimization nearly although expensive stochastic modulus tend minimize produce across realization family explain discriminative transform integrate explain refined unsupervised explain unlabeled optimize unitary preserve expect transform compute classify realization unknown estimate block average block unitary average average average unitary operator optimize tend unknown need mx depend error estimation e ex prior available constrain partly adjust local stationarity wavelet outperform classifier oppose apply iterate choice unsupervise deep weight upon flexible pooling impose unitary provide precise adjust contraction admit sdp convex averaged label operator unitary letting prove must go prove non diameter fix since constant positive coordinate x contradiction assume ex bound contradict unitary let go prove equality block coordinate unitary give ex ex since prove prove aggregated px transform concatenation I I prove conjecture introduce transform model iteratively apply unitary operator perform modulus distribution show contraction preserve network perform averaged estimation discriminative powerful unlabele classifier address two unlabele integrating supervise deep remarkable produce many include image cascade aggregate variable update together label criterion play deep architecture lack introduce analyze property supervise classification deep network deep iterate contraction operator modulus unitary idea deep define representation adaptively preserve volume unlabele explain deep explain estimate transform whose body mathematical modulus feed network otherwise pool rotation contraction sparsity prevent calculate modulus redundancy preserve eq preserve square eq bound increase slow transform wavelet transform operator deep audio complex wavelet signal modulus wavelet exponentially process asymptotically slow decay exponentially decay slowly slow regime
adequate scale volume figure function increase probability bottom scale volume probability cp cp scale substantially section tell substantially maintain cp great cp check monte carlo simulation present comparison cp achieve great new achieve new legend legend htbp cp remark choose piecewise choice cubic desirable interval computation stage computationally convenient formula sphere pdf scale volume specify volume degree scale volume specify positive obviously piecewise cubic interval knot knot knot possess second knot reason scale formula integral quadrature normal statistic invariant american association g set stein communication theory employ journal confidence procedure statistical utilize interval scad minimax point thesis department set mean multivariate confidence set normal utilize uncertain prior journal planning sample utilize uncertain p factorial utilize communication property uncertain information journal confidence journal confidence institute statistics stein statistical good la correspondence department mathematics la university mail sphere multivariate sphere part radius numerically minimize scale volume sphere convenient derive sphere sphere mean sphere pp p prescribe confidence volume sphere condition expect comparison sphere argue tailor uncertain prior tailor uncertain common exist loss generality dp work stein review shape review specific proposal stein estimator bayes compare confidence sphere stein however piecewise cubic function minimize scale volume constraint coverage stein bt bound radius slightly volume ratio show function feasibility specify form give find piecewise cubic value scale b coverage never fall implement computation choice coverage implement adequate task check numerically completion type context present paper compute great odd
change hull formulation final step find permutation analyze matrix undirecte unweighted graph self loop permutation reach let entry sum entry therefore equality complete proof magnitude affect property fundamental tend less coefficient would evident use norm valid come modality address alignment collaborative difference multimodal minimization convex non alternate direction multiplier constrain problem use relate multiplier update minimize update fix subproblem decomposable matrix subgradient subproblem convex set general project onto computationally nevertheless solve version keep convergence guarantee case spectral linearize multiplication publicly estimate covariance information dependence numerous regard graphical good non solve lot fmri g consist matrix inverse share joint four estimator limit correspondence adjacency align general overcome limitation permutation optimization problem q minimize descent approach simple variant iterative thresholding last one present several scenario synthetic graph fmri adjacency traditional matching without weight original matching three technique graph representative wide world degree generate geometric refer describe bottom vary graph average run state large hypothesis h present study neuron neuron type connection chemical electrical map match chemical electrical graph construct use weight add show suggest art electrical chemical outperform capability deal multimodal lasso actual match weight completely commonly deal multimodal network different modality underlie assume connection fmri example multimodal match weight distribution add spurious weighted graph finally four stage multimodal appropriate evaluation measuring compare permutation free follow distribution intuition multimodal path short red dotted black application collaborative fmri publicly consist almost minute period per cc use extract datum test potential data matrix whole handle ground truth collaborative already prove successful take truth result collaborative collaborative matrix gold empirical matrix align collaborative inference outperform h minute solve new graph matching problem correspond result match weighted graph previous art formulation multimodal datum addition formulation pre alignment free framework common network preliminary work support nsf nc university nc la nc video biological matching algorithm inspire sparsity formulation solve augment lagrangian unweighted multimodal naturally technique problem observe come different modality compare graph synthetic graph multimodal brain connectivity alignment free fmri publicly scientific determining whether preserve vertex graph view yet graph contain hard matching finding therefore recognition vision area address new technique method relaxed version technique match multimodal inspire modeling group
mse select estimate validate give unable present analogue input value asymptotic acknowledgement nsf grant material measurable event lemma lemma ni pm depend min real q bound similarly bound positive symmetric assumption except going keep may large lemma q co moreover let mean find covariate value response parametric assumption statistical via synthetic object become domain would object many handle possibly ad hoc look perform task specifically map output look covariate end operator domain real interested take range nearby future take output health explore type expect interested price price price well especially thousand voxel contain functional covariate far beneficial f sf figure py sparse present covariate shrinkage make nature method effectively follow value asymptotic subject orientation subject matter mention take case analysis relate functional covariate evident develop knowledge study multiple value estimator covariate sparse produce search differential operator roughly speak across input functional need provide analogous fashion select lastly worth note broad nature spam search broad parametric though spam model covariate via evaluation work unlike spam well value regression lasso although simplicity multidimensional typical value regression w work functional presented possible inner approach use vector adaptive smoothness furthermore grid moreover case function problem show I order pattern shall subgradient fix show optimal optimization elaborate j p na ji entry h nh ji j j j aforementione shall simplification simplify take optimally follow zero lemma whose proof supplementary material negative r k ni constant eq proposition j stationarity truncation ss q first ss te ss te ss te unless row lastly lead eq similarly stationarity wish kkt p functions follow odd typical similarly set j compute validation typical estimate choose configuration record recover correct small small produce hence total input
extraction function non negative auto association cluster mean cluster th pooling vector pool pool pooling generalize allow non explicit drive cost minimize pool representation function encourage pool representation reconstruct pool representation reconstruction prevent auto auto auto way auto reconstruction difference parameter pool second activation auto bias cost pooling control invariance cost discard descent pooling score measure score introduce feature two measure invariance raw measure invariance pool invariant activation obviously ideal invariant invariant activity therefore distance truly invariant show effectiveness goal spatial feature gray patch video try video object dataset extract video qualitatively image cifar contain object frame patch part small consecutive frame pair auto encoder feature patch pool auto soft visualize threshold represent cluster similar also vary depend feature cluster edge detector detail detector replace edge detector inside thing auto pooling rotation next pool surprising put importance pool consecutive frame small increase role reconstruction make pool value invariance invariance stop category channel auto way cifar patch dataset auto hide patch cifar location cc extraction auto training implement map large pooling show auto pooling multiple region continue distribution spatial carefully however location variance location region detector make pool invariant rotation location ap clusters ap ap clusters ap sp sp sp spatial classifier pool representation label auto pooling number pooling size pool well grid pool two per accuracy auto pooling outperform pool time training substantial main pooling hand fitting increase auto pooling method generalize spatial pooling spatial auto pooling make coherent continue information pool auto feature spatial plausible cell video auto cluster invariance pooling feature auto
cross union overlap cross display bottom omp subspace vary overlap spectra spectra remain spectra equal cluster sampling subspace model omp omp set overlap role principal angle subspace sparse recovery nn sparse lie strictly union maintain omp perfect third observe gap omp nn omp admit probability datum uniformly direction probability provide agreement discussion claim truly behavior subspace description rely spectra display omp vary ratio nn observe ratio gap omp omp significantly outperform suggest gap omp spectra decay bp nn illumination face fix camera face capture well nn omp illumination pixel example sort image face place contiguous affinity row omp affinity matrix representation dataset alg affinity middle solve bp homotopy parameter small coefficient stack final affinity compute affinity three cluster partition upon corresponding affinity selection method well instead percentage result subspace database along top display result full dataset illumination trial illumination select percentage incorrectly classify omp admit nn find recovery rate sampling agreement surprising omp dataset rate classification illumination pair subspace result sparse compressive sense open question paradigm signal basis overcomplete dictionary representation accord mathematical dictionary sparse respect learn applicability whether sparse recovery arise learn dictionary block signal deal past year especially compressive subspace model compressive collection admit structured pattern presence underlying subspace subspaces ensemble datum utilize sparse signal cluster research coherence uniqueness dictionary guarantee suggest examine sub rich geometric dictionary angle subset insight coherent compressive structured suggest difference sparse sublinear angle spectra two dictionary originally dictionary admit dictionary learn employ classification classification signal representation admit learn belong learn dictionary dictionary aim learn collection incoherent accomplish minimize dictionary utilize incoherent learn connection current insight role dictionary us ability representation spectra decay spectra powerful predictor subspace discriminative advantageous reduce cross frobenius must spectra one impose dictionary provide well perform necessarily performance result affinity obtain datum omp well direction future another recovery nn comparable sparse sample nn suggest strategy analyze interesting future extend deterministic analysis setting study omp corrupted prove occur every omp selection greedy omp alg subspace point maximally residual include normalize residual span current develop rhs inner first interest rhs residual still lie portion lie rhs fact plug rhs arrive follow simplification function ensure calculus provide obtain final take root ensure require belong neighbor via omp admit follow put rearrange arrive condition guarantee alg stay induction prove thm union accordance develop tight mutual residual coherence residual expand z tackle term write simply sum principal entry since assume imply corresponding value let singular value note make require bound last simplification come fact unitary norm informative acknowledgement thank helpful discussion comment dr helpful discussion anonymous comment suggestion ed distinguish fellowship partially grant fa w nf remark recover point ensemble live study minimization sufficient greedy pursuit omp feature subspace characterize particular particularly suggest omp reliably recover feature regime fail cluster hybrid structure nd ny sufficient application must union union affine mixed illumination point structure signal low affine hyperplane signal different electrical subspace provide extension extension provable challenge identification live subspace simultaneously sift point point lie state subspace form euclidean neighbor nn summarize consist select live denote index affinity matrix live produce method local estimate local locally approximation curvature fit main either affinity obtain structure case build upon one spectral estimate mode pose affinity thorough review separable form stable subspace ensemble quickly fail dimension part intersection increase poor belong seek rely solely subspace live set estimate propose upon form minimization main approach sparse point form point consist assumption distance subspace lead provable guarantee occur intersect refer recovery due representation collection signal representation result approximate disjoint intersect angle subspace sufficiently large select bp develop parallel development cluster bp matching pursuit highlight tradeoff mutual point different interpret large point lie open attain thm intersect distribute lie subspace leave gap cover radius sphere denote star cover interior map sphere attain cover radius mark convex hull deep coincide maximal gap hull live extend analysis live refer subspace thm particular incoherent trivial intersection ensemble subspace suggest subspace characterize examine correlation subspace provide analysis omp contribution gap neighbor nn sparse omp synthetic sample recovery advantage forming way reveal affinity amongst might exploit capable provide subspace estimate neighborhood affinity form face live illumination subspace affinity illumination display illumination omp leave illumination provide subspace omp use condition occur thm disjoint uniformly selection discuss implication dictionary future line work vector eq write contain index reciprocal zero place take transpose dimension sub sparse ssc discuss generate unit span iy matrix stack expand subspace submatrix exclude cluster ssc employ relaxation minimization ssc proceed pursuit bp dimensional feature vector place cluster laplacian admit exact point constrain bp denoise feature solve noise formulation author procedure extend study originally ssc behavior omp detail omp alg alg signal remain omp index employ consensus omp feature feature consensus real affinity stack set ny ii affinity ensemble spectral laplacian omp know suboptimal signal omp low complexity alternative minimization ssc obvious exhibit convex enable large collection despite well carefully tune omp choice empirical suggest omp offer ssc affinity omp collection contain index set contain index atom pursuit residual definition sufficient guarantee contain guarantee omp interested determine return belong occur natural study subspace due fact true subspace exact kp contain require cluster formal coherence provide mutual coherence subspace mutual cosine angle support recovery omp bp union disjoint result principal angle principal angle e consist condition thm depend dimension enough guarantee author contain formal dual bp coherence direction contain thm minus radius live union coherence cluster subspace equivalent coherence thm particular omp coherence minus diameter thm gap angle cover diameter empirical tuned provide high two gap method nonetheless find provide omp offer complexity feature intuition geometry geometry guarantee must ensure atom denote guarantee bp omp support exact omp bp geometric interpretation project atom outside set convex atom atom lie sub violate guarantee reason maximum atom incoherent require column require local incoherence require incorrect incoherent point cluster coherent cover radius connection angle minimum angle characterize distance pair ensemble principal dimension small principal principal angle first define angle vector principal way angle decrease angle insight angle underlie practice however angle recursive principal angle subspace angle large value cross spectra subspace say disjoint angle dimension intersection equivalently overlap subspace dim thm reveal relationship cover pair ensemble reveal angle apparent produce uniformly bound subspace support pair subspace unit norm denote subspace incoherent vector uniformly formally require incoherence entry wise property require inner principal direction distribute point reveal section select point include bound datum subspace subspace spectra occur diameter diameter bounding way bound constrain amount point subspace spectra pair support equally admit however assumption principal weak require incoherent correspond angle spectra concentrate principal direction angle intersection subspace exhibit trivial intersection intersection probability admit dramatically suggest spectra play determining hypothesis spectra theoretical reveal connection radius principal angle subspace conduct study explore vary cover overlap subspace role spectra intersection vary two contain atom index atom goal subspace overlap pair equal invariant dictionary localize toeplitz cross spectra invariant dictionary structure incoherent orthogonal atom invariant dictionary
ta algorithm give full bundle vertex update pairwise contribute assume fast variety overall intermediate toward number could employ include possibly bayes length bayes large approximate bayes uniform nest divergence vertex correspondingly expect compare recover goal demonstrate classic sbm edge may important explicitly infer thresholding include weighted behavior capture demonstrate find graph block bundle see variation unweighted vary weight fit accuracy vary vary setting structure vi vi correct threshold impact differently result sbm thresholde lead result contrast threshold utilize flexibility particularly poor increase edge classic sbm perform poorly analysis similarly focus intra stochastic generalize edge family technical challenge bayesian approach recover substantially simple demonstrate apply weight unweighted naturally potentially present could extend mixed membership case heterogeneity variational promising membership sbm technique extent utilize exist sbm acknowledgement thank acknowledge support fa air force office research advanced project generalize annotate weight introduce approximate weight outperform common first weight stochastic model structure broad range biological role automatic role identify block stochastic sbm solve problem fashion classic sbm vertex group undirecte depend membership thus block assignment vertex connect wide variety depend diagonal element element great density generate hierarchical core flexibility popular tool machine physics sbm generalizations heterogeneity mixed membership infinite relationship latent relevance weight typically modern include em classic variational sbm stochastic mix effort sbm unweighted poisson distribute fit valuable information sbm without sbm annotated family thus recover block cause handle bayes fit dense thresholde close brief interaction compose define define disjoint one pair model parameterized bundle distribution determine large structure classic although parameter across specific principle learn edge bundle likelihood restrict mathematic broad include produce classic belong write mapping j r ta bundle choice prevent direct value bundle weight zero create degeneracy calculation edge represent pair interact interaction yet observe bernoulli random statistic degeneracy assign appropriate exhibit smoothly graph dense dense separate paper belief bayes vb marginal functional low constant calculate approximation expect log likelihood weakly constrain close
variant retrieval recover complex arise infeasible magnitude retrieval several broadly signal positivity algorithm scheme multiple transform suggest method algorithm alternate iterate phase measurement theoretical different recover outer recent relaxation section go empirically alternate outperform analytically gaussian contribution likely establish correctness alternate resample design achieve complexity kn sdp base also broad sense signal algebra example constrain alternate empirical however good analytical guarantee relaxation norm correctness alternate implication analyze error optimization run retrieval accuracy sample contribute nevertheless away sample simply option empirically non flow use iteration show vector section detail rest retrieval procedure huge attract retrieval early computational receive lot applicability spurious optical difficulty problem various practical still paper focus uniqueness paper resolve algorithmic seminal many iterate first suggest magnitude resolve phase success success iterate projection onto involve convex subsequent al flow optimize descent recover small resample measurement code measurement match report minimization recent problem rank system equation approach constraint trace make max cut convex lead state measurement establishe near retrieval lot attention recently signal sparse though problem compressed phase make compressed correspond use sdp phase retrieval recover measurement tight relaxation develop retrieval magnitude still alternate minimization low pca negative sign etc success provable guarantee minimization various dictionary etc though completion heavy subsequent algorithm propose manifold pt bold capital letter matrix letter etc scalar complex vector hermitian canonical conjugate paper say z shorthand generality pt measurement recorded goal c phase course recover diagonal convex know alternating minimize hence problem tt intuitive might converge uniformly initial underlie non convex fail initialization establish address challenge singular completion show address second actually optimum linear would closed much computationally accuracy since matrix mean use conjugate geometric time take sdp initialization likelihood initialization ht cc successful initialization random factor similar figure however specifically practice feasible since many application sdp approach face issue section aspect contribution paper provable success problem end use exactly complexity n though complexity problem use alternate minimization break initialization use away theorem nc require hence use correctness partition tt result guarantee vector great invariant decay theorem ct view goal second break two c see lemma second magnitude calculate phase try magnitude phase towards effect state standard u correctness complex sparse ask study exact still algorithm idea behind first retrieval solve present complexity ok ok kn ok quasi enough recover correctly standard vector recover show algorithm pick element appendix corollary sparse recover present advantage initialization
observe hide activate sparse representation overlap depend percentage augment p non equally sized group norm train overlap group model need minimize perform descent regularizer expectation penalty member activation ensure activation close equation q detailed use gradient penalty regularizer percentage overlap different unit whereby group unit empirically hand fig show rbm overlap overlap activation hide unit training unit leave whereby proportion overlap group attribute choice mix towards high process group towards zero datum softmax layer posterior network conjugate empirically architecture size hide unit respectively perform core server core cpu cache propose penalty offer create architecture task digit size mix understand architecture figure depict architecture utilize mixed overlap group however architecture phenomenon constrain penalty expectation th mn h mn mn mn mn sparse digit recognition limit overlap could easily overlap group methodology induce constraint digit offer scene categorization de universit la france paris universit de france de universit belief task write digit recognition effort optimize maximize advance focus induce approach constraint overlap overlap classification accuracy digit mnist provide estimation usefulness parameter rbms extensively diverse mainly generative framework range image scene dimensionality important rbms serve rbm allow efficient computationally architecture deep architecture although task curse various also serve way plausible benefit sparse norm regularizer restrictive interaction unit rbms extensively regularizer rbms norm group regularizer comprise hide unit increase rbms normalizing factor rbm energy bias stochastic visible unit conditional sigmoid connection hide unit amongst visible intuitively model goal constraint salient activation cluster data rbms train learn performing observe specifically train evident phase try try assess intractable possible use cd allow sample use gibb adequate follow gradient q cd constraint mix induce
turn end goal exercise straight decision several distribution motivate prior notation play role gaussians condition wishart multivariate density gaussian kt iid indicate definite book appendix wishart value wishart wishart precision mean form matrix instead normal model density definite precision gaussian vector kronecker prior diagonal column triple statistic matrix pattern recall recall assign form close eq wishart compare extreme remain pattern prior predictive whole thank conjugacy find use th k add informative limit decide possibly supervise label need wishart prior recall interest infer factor fact improper multiply get normal q
average acquire collect every p k average converge target want know one notational simplicity dynamic w rl interval visit matching match special type appendix empty independent reduce run parallel rational matter provide formal empty graph independent distribution real number dynamic weight translation circular unit cube fully complete ratio burn period formalize fully connect coefficient gibbs constant ignore collect begin convergence exponent exponentially joint state obvious marginal rapid marginal strong distribution seem proceed obtain average chain initialize discrepancy gibbs geometric geometric dominate discrepancy round find limit generic mathematical construct seem converge marginal datum gibbs illustration outperform complete particular joint approximated decrease clearly exhibit variable provide deviation illustrate match remove illustrate behaviour standard lattice mrf denoise infer clean value corrupt value take advantage pixel mrf ise rectangular lattice pair wise ij indicate coupling strength node neighbor mrf combine ising potential encourage gaussians mean denote couple identical chain crf employ skip crf dramatically semantic labeling figure crf labeling want dependency chain viterbi stanford name entity application mining entity replace gibbs sampler posteriori high reader viterbi pre crf stanford chain crf pre iteration average anneal gibbs attain viterbi fast viterbi skip iteration second second achieve demonstrate bad yield computation provide gibbs viterbi produce annotation example iteration gibbs viterbi business hold scene roll bar classified person organization gibbs full conditional random gibbs conditional synthetic denoise recognition provide neighbor bad storage requirement advanced gibbs need easy task certainly key version fashion indeed gauss version however core available strongly asynchronous likely outperform gauss efficient densely conjunction rao would enable attack task dirichlet process connection algorithm explore connect build node direct connection exist reach gibbs analogue normalize frequency instead usual expect many apply connection art need clear construct greedy lemma chen de facebook sampler paper convergence fully denoise mrfs name entity convergence connect model year progress randomize therein deterministic still many attack theoretical mathematical important consequence bring generators monte classical von architecture computer recently monte rate compute ergodic average deterministic create great application currently still narrow importantly unnormalized heart artificial intelligence carlo engine popular package several boltzmann popularity stem simplicity implementation would design sampler toward achieve simulation know empirical match follow statistic want choose normalize make update naturally upon denote position involve find component I output estimate guarantee right match paper popular drawing generate conditional conditional new outperform domain mrfs prove deterministic converge fully ensure
validation considerably suffer performance presence compete risk survival risk dependency risk risk fix force risk figure hyperparameter risk gp allow risk apparent infer risk towards independent underlying event hyperparameter towards despite lack know illustrate potential straightforward ard outcome compete risk covariate infer ard hyperparameter indicate patient function alternative many parametric directly relationship flexible provide elegant achieve infer survival rate incorporate censor truncate combination gp hazard perform monotonic hyperparameter hazard conceptually straightforward easy interpret interpret would noise free event plausible could randomly actually occur delay event occur record could event alternative interpretation represent acceptable interesting alternative multiple incorporate compete risk working call serious infer observe failure censor identifiability may perhaps density dependency alone event claim time lack plausibility quantity reality argue conceptual make event relevant second happen risk quantity marginal survival straightforward independence event time want risk assumption identifiability density reality joint useful illustrate flexible survival impose parametric gp specification hazard involve achieve great efficiency acknowledgment european fp ec grant agreement support survival outline simulate monotonic single competing present finally laplace approximation purpose gp hazard gp assume traditional hazard piecewise hazard contribution hazard negative q text hyperparameter section predictive distribution expression limit value rough explanation occur event density place place away limit issue suggest numerical computation infeasible numerically take hazard accommodate determine hyperparameter laplace event provide survival hazard obtain cox hazard monotonic compete risk performance finally give example dimensional compete risk time transformation c gp choose manually covariate covariate times q generate event finally independent censor select generate random number record censor time covariate gp datum readily exist tool visually clear fit good gp leave observation cox proportional survival individual broadly survival hazard hazard apply hazard dataset compete two infer indicate risk dependent characteristic scale slowly see lie event censor event event place risk risk similar effect figure c risk regard censor compute show perform slightly worse particularly predict risk slightly event event help prediction risk way event mse tend gp year particularly risk capable deal squared kernel relevance determination ard hyperparameter determine great impact outcome indicate covariate issue arise implementation gp instability occur censor hazard hazard write hazard give quantity unstable tend solve complementary numerically stable hazard numerical issue compute derivative gp hazard write numerical occur second gradient trick term second method partial optimisation laplace clarity rewrite negative derivative partial derivative ii f pd hessian definite expect note negative derivative avoid difficulty negative derivative ie log partial derivative eq note problematic censor first q eq convert process relate output event covariate event flexible cox model hazard class accelerate failure relationship covariate time without assumption hazard combination censor truncate multiple potentially risk survival event extent time specification risk simulation study suggest assume accelerate write corrupted specify prior view event time covariate connect quantity event survival hazard cox arguably typically capture cox effect hazard indirect need covariate survival whereas event negative consist posteriori map numerically posterior control construct hyperparameter incorporate censor observation event would occur study monotonic covariate time compare model model cox gp hazard rate cox effect approach extend compete risk use output gp output output output covariate gp output compete risk firstly measure secondly output would occur despite difference risk risk risk hyperparameter conclude risk truly reality nevertheless within value dependence follow show risk prediction occur examine happen risk event convenient give quantity survival compute compete risk survival commonly semi cause hazard associate particular survival compete since survival incidence similar hazard model random relative survival way analyse risk contain differ model joint avoid assumption structure follow apply survival risk outline hyperparameter interval censor data compete risk result finish offer survival event occur whereas censor total individual addition measurement monotonically covariate density assumption exist special recently base likelihood baseline model hazard cox baseline hazard recover cox retrieve term generalise linear implementation gp seek accommodate increasingly complicate covariate flexible sophisticated covariate completeness accelerate desirable apparent since transform make difficult uncertainty short time transformation due apparent half real line explore transformation hyperparameter learn appropriate output model apply provide powerful probabilistic method relate assume write covariance excellent find construct assume gp prior infer mean single case risk generalise via corrupted risk dependency gaussian noise follow approach gps share covariance noiseless output j vanish translation finally noise simple six hyperparameter l necessary appropriate dataset hyperparameter easier contain advantage gp gp intuitive interpretation output return censor event type gaussian censor independent conditionally noise free independence leave convenient complicated business risk similarity gp leave section write bayes obtain negative log laplace marginal support r f variable predictive time hazard question survival risk risk risk risk pathway biological system infer risk operate must risk risk mean rr risk survival risk cause hazard follow incidence risk whether risk see expression give switch risk risk marginal risk risk survival survival risk independent marginal survival gp conditional noise free always interpret rs survival regardless whether present begin apply single risk right censor apply model gene patient risk generation gp generate sample gp gaussian component censor
probabilistic graphic support inference computer vision symbolic description build bottom scene element extract identity recognition individual scene remarkably successful recognition combine identify character accuracy recognize possibility result system level modify character recognition change frequently generative parsing explore appeal integrate take like vision considerable design learn incorporate combination remarkably powerful example global geometric bottom detector graphic programming generative probabilistic program template scene graphic software stochastic compare output latent fidelity tolerance image graphic program write variant language model likelihood template write generative parse invert probabilistic graphic program instead hasting operator model variant combination tune analogue anneal reliable framework interpretation formulation combine probabilistic programming approximate constitute contribution efficacy interpretation character infer road representative baseline graphic program component write code configuration scene software likelihood enable fourth describe formulate image perform graphic program metropolis transition proposal induce probabilistic graphic later application application indicator presence absence digits parameter per spatial kernel execution describe I priori scene complexity step index uniformly scene variable propose ps jx ji fs associate accept reject pi fs ps px probabilistic program programming inference provide default graphic bayesian abc approximate process rejection formulation accept match hard threshold abc threshold extension cutoff likelihood incorporate insight approximately fidelity variability unnecessary undesirable treat graphic read short consist digit letter scene contain bank potential spatial identity ps ps x ps ps ps letter formally consist global beta standard favor small decision pixel reconstruction uniform challenging degree letter source graphic way incorporate break include letter result lack publish depend character optical engine corpus character rate dynamically fidelity generative accurate analogue anneal reconstruction variant probabilistic graphic generative deterministic energy minima fidelity global improve substantially line inference adjust letter minima f convergence fidelity right overall log pixel disagreement minima course scene letter newly localize max lambda pos lambda lambda lambda rotation lambda lambda external server load surface stochastic image pos pos rotation pos rotation present enumeration develop generative graphic single drive scene uncertainty variability need ignore input probabilistic graphic program scene comprise height road offset corner road arbitrary camera road road visible segmentation scene separately follow program image extension rich road ground experiment appearance center histogram assign pixel cluster histogram multiplying denote smoothed gamma per normalize input appearance scene accuracy low rich appearance compatible primarily generic hasting inference text although develop particular build graphic program sample scene show representative likelihood classification solely frame appearance geometric reliable road find typical inference result generative graphic rgb single appearance four report accuracy road vision face exploit temporal report classification appearance appearance per sophisticated baseline system significant include camera rough infer scene approximate figure probabilistic line probabilistic frame road width road server load image surface road frame surface pos pos pos road road height road surface appearance
ideally would partition situation positively correlate true actually lack partition poorly detect clearly illustrate partition structure rs bb strength phase exist even simpler isolated threshold lie increase begin selection simplicity perfectly ignore irrelevant infer partition regular plant approximately unchanged uniformly merge regular hierarchy branching ratio capacity plant fig well quantity h eqs omit fail network accordance provide good criterion incomplete tend strict correspond region refine incomplete perhaps eventually agree value sufficiently large easy network model criterion show infer solid line criterion average realization plant dash mark impossible prominent block method module merge regardless modularity maximally modular network module merge phenomenon detection method statistical scale plant exceed merge know pp generate block equally description presence model deal pp instead principle nest limit require description length maximally modular dependency flat minimize description one obtain eq block grow almost node k dash boundary segment right modularity mark star maximum remain dotted line various e overfitte remain construction isolate merge nested understand remain resolution limit block merge together despite keep separate slightly modify fully isolate internal arbitrary decide merge entropy merge block c n merge fig become resolve remain network detect flat modularity block consider rest edge compatible situation significantly split branch contain remain network merge level rest remain unchanged c influence merging obtain nest capable level branch situation follow efficient infer block efficacy network individually hierarchical structure block block method anneal ref agglomerative unbiased respect also complexity block know depth start low number level pattern satisfactory branching start optimal branching guarantee optimum perform independently also application network newly previously modification must partition e belong merged level partition level size remove node group repeat move keep track whether start mark succeed mark exist proceed level mark proceed lowest length impose general hierarchy global minimum found find case succeed initial simply actual final move necessary operation completes hence spend reliable consider nest usual pp inspire construction rs b le rs normalize partition network realization nest pp text star symbol infer averaged realization gray mark red detect indicate panel correspond hierarchy infer use circular infer square color indicate dark light mark incorrectly classify partition ref procedure generate detect match pp exactly become correct high plant infer hierarchy plant fig identical require kronecker use failure original reach possible know tend become graph conservative bring actually resolution capability tend spurious also scale internet system nest prominent core top node show material actor represent low hierarchy hierarchy actor label classify accord prominent temporal material b width l yes yes yes yes yes pt author mat web graph political book th american www pt c amazon gene internet actor bipartite pt political gr com berkeley stanford web power web author network yahoo co email obtain list bottom dash nest modularity nest span different correct block instead al usa individual direct exist political applie nest division hierarchy match accept division impose nest reveal picture connection pattern instance level possess composed cite conversely possess tend cite cite interesting large fraction connection level group concentrate act group capable reveal internet often private company body correspond information link traffic block network prominent observe strong structure connection act seem spread fig extract information database cast member distinct cast member single connection recursively remove fig bipartite network separate separate one flat empirical wide domain version trend correspond resolution trend size seem serve lack resolution previously length e tendency network increase large organization rather intrinsic modularity existence densely quantity connection contrast specific assume match fig network modular strong modularity possess indicate building block topological organization community clear internet probably partition much value partition ensemble case discard present simplify dominate possess advantage principle detect network nest generalize hierarchical structure assumption either possible show major approach modularity model module nest replace logarithmic come principle integrate desirable capacity actual structure spurious scale network detail principle link serve refined detecting well determine salient topological summary topology careful reading manuscript useful comment university bayesian integrate consider lead purpose usual ensemble define probability inference respect eq one p b give provide mean model likelihood inclusion quantity compute maximize instead one overfitte large dominate data contribution maximization compatible want agnostic count description block subtle flat choice agnostic practical evaluate one flat parameter sample large degree magnitude small lie constrain something rs b observe appropriate modify implicitly parameter block likelihood rs tm remain fix choice prior therefore finally flat prior rs rs become dense likelihood fully compatible although dense penalty therefore seem arise comparison interpret nest prior prior match compare network method focus plant detect concentrate good benchmark namely modularity compression walk benchmark block law parametrization ref power law impose restriction exceed intrinsic degree correlation choice network generate possess parametrize control connect block configuration parameterize internal external mixing choice parametrization block size degree approximately choice configuration significant appropriate keep value even one non identical indicate overlap vi plant indicate stochastic sbm nest sbm modularity maximization method block mark plant value average network realization vi several realization block observe vi number nest structure value infer increase plant exhibit systematically spurious partition threshold largely plant separate find spurious analyze desirable property inference overfitte spurious module fully nonparametric modularity combination resolution lack base statistical known suffer problem spurious module although try walk actual block topological fluctuation walk gradually transition ref put fact analyze motivated mention entropy direct read binary block half case approximately correct variant half expression case amounts direct replace r unfortunately level entropy description length need analogously eq joint degree node belong generalization adapt universit discover network serious actual addition observe phenomenon popular modularity validation principled selection way scale avoid limitation beyond current approach capable separate thus identification spurious module generalize purely mix hierarchical structure tree tractable advanced community community scale become perhaps science salient feature system evident giving insight evolution seem straightforward group often mostly develop detect great compete clear outcome modularity maximize partition internal inside cluster many capable apply heuristic drawback measure statistical evidence deviation separate statistical high scoring partition characteristic share vast majority solve task lack modularity maximization increase edge limitation salient completely degenerate network find fail large spend generative modular structure approach offer advantage dominate principle incorporation formally manner general overcome limitation intrinsic model purpose connection away restriction purely structure bipartite well straightforward cluster amount principled length bayesian spurious eq
also unlike current strategy involve portion chain operate greatly burn density heavily combination scope empirical demonstrating speed four criterion posterior px nm parallel combine produce sample iterate typically way metropolis would carry implicitly density product consistent draw full error analyze draw follow yield estimate density product estimator bernstein von generate approximate produce asymptotically exact third combine beneficial posterior von limit important asymptotic unique exist posterior concentrate approximated parametric serve good facilitate fast correct sampling density product quickly online previous method full quick asymptotically biased especially non produce asymptotically consistent kernel estimator use product eq density pdf unnormalized mixture set component component sampling chain sample reject index acceptance slowly way application could pair leave alone odd form remain sample dm mc tc mi quick bias make slowly number implicitly density large mr number small converge write bandwidth nonparametric pdf gaussians unnormalize acceptance follow procedure exact parameter procedure nonparametric give subset produce nonparametric operation pair increase acceptance rate section require present completion master combine incoming reject algorithm perform parallel volume single ever communication scalar machine carry machine third procedure sample fully nonparametric correctness bind correct square tend zero estimator therefore apply let old time whose first next bind bandwidth mc th use fact square distribution dimensional logistic generally unimodal multimodal present nonparametric latent dirichlet allocation hope domain approximately use used decide accept proposal small set adaptively exact compute sequential directly sample several parallel mcmc algorithm design topic require correct general synchronization combination consensus carlo perhaps machine independently explicitly reweighte final relaxation evenly mixture component set baseline section demonstrate empirically mcmc procedure asymptotically synthetic strategy typical mcmc sample yield yield sample average sample union assess sampling strategy iteration remove sample method hold multimodal moment mean follow density sample generate second take method method conduct cluster batch disk worker via subset leave generate biased result typical synthetic element matrix ii automate sampling machine one advantage c u turn hmc provide result product true posterior right approach product via overlap true posterior average systematic averaging grow left datum take compare pooling gives require storage burn chain generate reasonable step decrease however sampler quickly plot time compare chain algorithm procedure ghz gb chain bias right compare fast though chain cover observation chain minute infeasible predictive left task parallel high investigate dimensionality estimator implicit combination show relative take vs dimension synthetic asymptotically biased asymptotically implicitly density restrict density perform parallel mcmc aim posterior multimodal combination procedure suffer label component ten multimodal hasting
throughput evaluate approximately set run category balanced image evaluate set run standard protocol examine set score performance ranking transfer correlate medium variation correlate contain high auc neural site allow seek affected figure auc entire population variation medium already believe estimate auc manuscript incorrectly compute v version requirement effective representation effective natural community view brain benchmark success contain propose benchmark multiple utilize produce effectiveness order area superior area indicate visual hierarchy three model performance recent difficulty represent performance enable tool neural benchmark matching correspondence machine primary goal data space problem object speech pursuit goal produce source suggest benchmark success work provide measure success relative community incorporate insight neural complete neural formulation david original formulation sift concept process also history computer vision suggest way investigate brain work suggest specific hypothesis principle hierarchical network serve concrete mechanism field measure efficacy quantitative evaluation progress must believe main boundary accuracy measurement complexity dependent strongly affect accuracy decision boundary advantageous comparison activity see simultaneous order achieve reason measure low particularly exhibit sample relate kernel validate accuracy count support utilize measurement algorithm neural model approach pursuit mechanism aspect relate seek neural choose world neural hope neural optimize represent major computational external choose task visual series effort efficacy work examine visual variation work influence mapping major discussion dissimilarity matrix number publish account dataset seek cross performance compare algorithm brain important choose type measurement stream human specie leverage extensive behavioral neural measurement numerous technique fmri review process human process stream span area preserve mind representation benchmark importantly present visual facilitate goal intelligence representation effective brain preliminary goal neural protocol measure measurement publish model need ultimately utilize compose seven broken level efficacy pca level recognition variation rotation pose generation seven category finally three systematically sample low variation present position pose medium present multi wide pose variation level h result object introduce difficult current artificial contextual currently texture measure efficacy seek measure learn provide brief measurement kernel determine much lead principal component variation due representation variability task contrast little variation intuitively randomly make require form subspace curve curve advantage projection small multiplicative error important therefore provide assess effectiveness favorable present representation input category feature utilize kernel define x eigenvalue let drop solve linear square way result dependence kernel dimensionality choose square evaluate simplicity strong distinguish representation mapping use image generalize categorization strategy average error minimization proceed case representation auc image dataset consist seven object class seven break three level medium variation high face measure statistical subsampling class analysis auc value researcher data seven seven instance class broken variation medium class prevent fitting algorithm estimation involve selection training consist class object object produce training object background common seven category new image independent unsupervised tool analysis feature tool collect multi site neural firing variance presentation within variation human include feed optimize compute collection thresholded wavelet span orientation model sophisticated layer net layer sequentially perform normalization perform throughput performance criterion top performing learn million collect cell feature input image pixel grid overlap evaluate locally sparse auto contrast normalization million internet tune imagenet layer pixel neural supervision imagenet release additional test pixel procedure fed predict label representation measure neural population medium respectively level bootstrap indicate auc measurement level position variation neural v high variation medium variation increase difficulty task maintain variation medium sharp indicate able class level object task simple boundary discuss evaluation machine representation evaluate along medium variation indicate test l representation variation medium image image pixel patch base present variation medium variation medium medium across highly match estimate ht medium high est est like top ht le et et relate behavioral paradigm neural subsample neural feature measurement visual presentation brief typical increase condition long passive question bias competition current reach neural spatially code behavioral intrinsic code evolve neural benchmark accordingly measurement impact visual experience representation interestingly study experience object benefit million
level even level polynomial approximation hard unfortunately several show match often efficient algorithm contain example factor dynamic program show sdp give improved building guarantee algorithm show purely combinatorial use hierarchy lp give time testing give e far convex relaxation original case hierarchy always know advantage round sophisticated relaxation satisfie triangle tool high level help round lack round know sdp describe duality equivalent reach round tool implement weak full power work propose new algorithm progress hierarchy either positive proof connection give round computational hierarchy particular relationship review proof introduction discussion underlie sdp equal question non realize hold ask polynomial square prove proof assertion show polynomial consider context always polynomial author eq linear equivalently condition denote correspondingly give check efficiently possible operator sdp grow polynomial degree thus involve would operator establish equivalence correspond low gap relation instance weak gap output insight b argument capture system follow work hold instance small sdp detailed overview follow round input certain value solution combine support generally lift combine capture imply achieve round polynomial coefficient unit find relation unique game conjecture algorithmic application conjecture level semidefinite level related question analytically work hierarchy nontrivial give might approximation associate variate represent every hierarchy dot consider matrix logarithmic fairly polynomial optimize hierarchy polynomial try generally number obtain crucially solution algorithm serve equivalent quantum theory complex adjoint enforce symmetry would restriction symmetry simplify quantum space find classical state drop negativity resolve open dual quantum want test area solve distinguish quantum separability round actually find separable solve greatly simplify involve contain completeness short specialized real vector state proof appendix separable algorithm take naturally cube motivation informally hierarchy graph relate mean small know natural hypercube short graph inside dimensional many machine learn reference therein name isometry vs vs vs hard hardness approximation know hardness vector sparse say vector p l example wang find ratio relaxation existence support coordinate ratio proxy proxy choice attention dimension restrict isometry question kernel handle perhaps mild round maximize amount maximization problem sphere plant vector subspace dimension round program gaussian output recover plant coordinate coordinate necessary nontrivial hierarchy dimensional subspace nonzero output completely absolute constant relaxed current find use force enumeration enumeration bottleneck improve algorithm take opinion inherently sparse find good corollary informally output expansion away walk eigenvalue derivation meaning result expansion oppose vertex paper use solve instance lift actual positive lp view expectation rounding reverse good randomized round round way summarize conceptual give distribution sdp treat moment real often consider solution moment moment solution treat gap sdp weak make crucially tool sum combine problem assignment satisfy assignment relation generalization problem round sdp operate bit consider combine yet problem recover set high result approximation analytical show analytical plant vector corollary expansion contain certain lemma operator actual expectation write notation hierarchy yield bound simple condition norm sake completeness proof norm small expansion raise answer discussion measure measure indexing counting notation letter indexing form product linearity positivity sometimes refer mean polynomial consistent enforce constraint enforce traditionally one e program put design map conceptually round analyze come initially solution relaxation version detail roughly combine easy relaxation round combine every yield possible combine round relaxation optimize main lift argument round explain application optimize nonnegative coefficient sphere boolean hypercube unit map domain suitable generalize function linear relaxation typically semidefinite relaxation might semidefinite element maximize back approximately approximately round yield combine objective value direction distribution optimize program round typically general combine sampling getting turn round sized turn combine turn round convex nontrivial combine short consider moment function show nontrivial transform nontrivial transform cauchy schwarz fall proof programming hierarchy level overview sake focus rough natural problem universe subspace span recover show recovered linear framework easy recover finding mentioned polynomial euclidean sphere optimum thus combine algorithm closely orthogonal orthogonal hard every q therefore fact must eigenvector correlation combine rounding result actually square must satisfy latter fourth polynomial essentially prove appeal possibly bad constant even precise extend nontrivial weak correction recover original outline idea oppose subspace much subspace involved skip ahead give optimize polynomial nonnegative yes subspace time every use norm product equal intersection indeed one average easy find desire chance rather roughly speak combination mass inside random matching combination alone combine inner try turn assume coordinate turn specifically symmetric rhs schwarz get satisfy eq inner product piece property currently nonnegative pick function indeed inequality exist imply mean match turn level hierarchy obstacle appropriate generalization generalize yes case space apart project subspace round operation contain task nonnegative sphere nonnegative matrix standard counting get hypergraph maximize hypergraph since beyond hypergraph nontrivial dense guarantee log distribution vector achieve solution turn sometimes specifically fail sense simple time negativity schwarz together bound equal inequality part another drop value thing mathematically define dx look combine value least otherwise moment hence whole carry linearity replace use obtain hold access moment distribution operator round algorithm degree denote denote scalar linear meaningful notation functional stem functional semidefinite optimization p problem computing degree mx c functional sphere polynomial endow product degree denote spectral norm homogeneous programming theorem algorithm prove take show matrix simple follow step try find one direct fail conditioning must actual hold level require nontrivial though namely relation jointly marginal denote distribution independently hx standard hellinger kullback leibler would lemma sufficient x I tm x eq symmetric bilinear unit correspond together mx mx verify carry use cauchy sufficient violate actual automatically regardless combine independent copy ix independent monotonicity entropy lemma imply fail condition mean ia means conclude theorem odd multiply equal sphere odd odd degree constraint satisfies universe uniform f constant exist output prove combine transform round algorithm solution specify relaxation actual relaxation go output choose modify proportional every choice gaussian first moment hold distribution might attention actual expectation show consequence contradiction analyze round basis consequence big ib ib hand therefore freedom variance implication technical product vector hold schwarz vector schwarz argue linearity actual product lemma equality hold independence simply consequence coordinate even term actual round function distribution hold ac ad fourth moment fact conditioning progress pf pf conditioning point satisfy actual use inequality bind rhs r appeal completely inside want skip read every g need easily square much need extend boolean moment random coordinate round fail function round failure projection bound lemma support reweighte round bound hand side get level issue operation polynomial polynomial auxiliary enforce conclusion lemma design relaxation round combine step replace statement consistent application show vector subspace et plant subspace fact arbitrary linear subspace span think high run absolute solve high stage somewhat linear prove plant consist substantially notation great generality linear eq polynomial degree square return show degree thought vector reasonably correlate suppose believe result useful elsewhere generality also relation take linear less classic spherical state let subspace gaussians absolute schwarz imply recover cauchy schwarz could better sufficient take small recover complete suffice norm requirement require broad context generalize subspace meet hope result use state facilitate ratio ingredient pseudo note moreover even obtain requirement maximize sample whose write every know q proof expectation constraint existence degree constant get expectation cauchy tendency towards minimal early minimize amount linear vector fairly
variability trial due activity arise brain condition efficiently detect response signal trial detect individual trial multiple stimulus stimulus analysis past activity neuron period know period improve history neuron connectivity within spike history ensemble activity important process study simultaneously stimulus spike history activity goal previously develop space model e vary history develop combine density parameter stimulus spike history test architecture test significance estimate study spike step divide sequence bin ms bin determine th bin contain bin spike binary bin denote x x operation entire observation discretized activity ensemble spike use random joint mass bin family dependent bin denote subscript normalization simultaneous activity f n compactly rate spike specify probability spike interesting maximize maximization spin feature denote bin constitute evolution ar effect stimulus spike matrix initial follow covariance external compute nominal particular expect posterior smoother value obtain recursive eqs maximize expectation e e filtering smoothing step maximize e optimize auto stimulus history effect smooth lag covariance eqs respectively simultaneous see spike interaction neuron neuron spike excess spike delay stimulus response simultaneous activity test simulate fit equation likelihood select predictive obtaining parameter surrogate simulate simplify therefore method g circuit study practical fewer appropriate bin obtain determine interpretation bin recommend bin size confirm specific study method disjoint advanced method near ensemble activity neuron local circuit response response stimulus ensemble activity present gr un construction thank dr dr reading manuscript vary posteriori probable canonical activity forward recursion smooth density density forward first covariance mean unique smooth filtering lag log provide detail integral approximated used give circuit spike estimate dynamic correlate ensemble simultaneous e g spin allow stimulus interaction repeat experimental condition stimulus exhibit variability trial include effect neuron ensemble develop spike activity trial neuron activity stimulus spike history achieve process stimulus spike analyze internal make network receive neuron make neuron electrical action potential spike neuron circuit activate manner relevant process simultaneous activity neuron dynamically stimulus report spike neuron activity
simplify notation observational framework convention row intervention produce variance denote purely observational independent note easily identity log sum parent I log decompose parameter likelihood verify circle ex parent show dag involve parent partial likelihood maximum likelihood ki fix circle entry notation read eq invertible surely plug immediately likelihood observational dag thereby markovian distribution dag observational dag denote proceed intervention dag edge intervention tuple densitie intervention target md j f ix di jx I di dag target respect make strict triangular entry leave side hand side must transpose since hold aa ab restrict consideration precision triangular cholesky decomposition unique calculate perform cholesky continuous function intervention also cholesky decomposition matrix inversion continuous prove claim db p aim tangent denote canonical start derivative direction circle row zero see consideration ib continue calculation directional direction less linearly independent embed remain manifold p b conservative imply parameter immediate family intervention target true equivalent true true observational density almost surely solution class remark theorem v base v z rich application observational randomized type direct acyclic thank reasonable per intervention analogue partial identifiability identifiability implication tight bound effect besides methodology derivation keyword equivalence causal rely diagram direct acyclic graph absence true dag research often equivalence infer observational datum observe book important observational rather dag markov equivalence gaussian many observational latter come randomized intervention often observational individual focus observational thereby assume observational markovian link observational intervention calculus operator dag observational intervention intervention maximum observational bic underlie incorporate learn causal develop early problem observational investigate issue equivalence identify technique stage observational datum cope ensemble observational develop observational likelihood consistency mix observational case real observational variate following specify observational p regard observational derivation easily write intercept formula package option restrict markovian factorization refer observational dag follow factorization joint density gaussian conditional density distribution observational intervention calculus model dag allow intervention calculus calculus describe intervention realize intervention intervention denote truncated factorization truncate deterministic intervention intervention density intervention conditioning variable intervention consider u necessarily densitie intervention independent observational read intervention variate intervention deterministic dag intervention intervention value I intervention observational fully specify read quantity denote know observational linear observational kb intervention intervention thus dag causal stochastic intervention value alternatively constrain rather much dag likelihood regard nuisance sequel observational intervention I direct denote intervention target depend notation short intervention dag imply certain space likelihood expression describe nuisance minimizer depend dag distribution example identify observational equivalence namely regard family intervention target subset family conservative simple observational arise observational data conservative class jointly observational mind really dag markov skeleton edge one I intervention identifiable equivalence class family observational v dd markov indistinguishable belong different equivalence definition identifiable f u assumption markov dag conservative target rigorously undirected structure markov equivalence penalty invariant penalty outline section algorithm justify consistency equivalently f dag intervention intervention see read strong requirement infer dag identically evident consistency intervention realization value already nuisance set artificial nuisance independent realization intervention assume might surprising view family careful need cope without dag observational distribution xx xt minimum unique minimum equal statement although observational mind intervention target consistent selection alone let I realistic target small rigorous intervention dag corresponding whereas observational drift realization intervention tend could observational alternatively detect term variance need realization distribution coincide intervention true dag refer empirical confirm intervention away highly main difficulty markov equivalence optimization likelihood constraint dag cause optimization computational challenge allow dynamic programming enjoy nice statistical lead dag surprisingly problem dynamic exhaustive greedy forward backward turn algorithm step space dag rigorously algorithmic competitive keep available throughout evaluate simulated analyze protein abundance record experimental different condition different purely experimental perturbation cope latent measurement perturbation ground define aforementione framework hold graphical set frequentist stability significance glasso accept ground truth result roc edge matrix skeleton estimation comparable four b glasso treat direction pc comparable frequentist easily comparison paper positive positive potential discretization improve ground observational randomly draw causal illustrate consistency dag skeleton degree respectively dag correspond observational meaning observational variance generate total single vertex intervention ensure point observational allowed verify conjecture theorem sample markov essential long intervention expectation intervention choose intervention normalization expectation indicate observational gaussian causal simulate set namely n underlying causal set describe mention adaptation approach exponential optimize bic class dag node comparable runtime distance adapt graph positive negative skeleton orient edge matrix intervention show plot intervention match theorem grow point
field paper ideal researcher serious negative feedback feedback movie one user prefer movie widely collaborative basically like mean one deal topic naive recommender every high similarity knn recommend cosine vector situation complicated still knn paper centroid thus iterate score candidate example centroid circle centroid calculate knn recommendation target user say calculate interested although paper still relate least user item give index paper publish score scale scale follow paper area ml db recommend mixed recommend paper paper ml researcher researcher kind paper thing note may example ml c db researcher aim target publish researcher paper topic content publish conduct field computer base area year sec chi paper list title participant paper scale relevant perfectly prevent voting middle evaluate ask relevant thing usefulness recommend ask take read recommend lastly ask system question indicate recommend relate ask recommend research give research gave work paper relate research topic recommend system recommend four student publish student recommend paper recommend good ask paper read read read fact user content base lastly recommendation indicate research valuable deviation subject researcher intensive go paper system interface require two paper wants recommend show topic identify recommendation mean paper contain interested discover cause topic user user work interested information whether user able extend apply publication limitation recommend suggest interested accurately group discover even topic much show paper former student researcher researcher recommend paper also subject paper give great reason motivation read perspective minute take time improve similarity allow counting frequency tf candidate implement also helpful word dimension may thank paper system make three contribution retrieve paper web measure third develop filtering evaluation usefulness edu come lot circumstance base ease recommendation system article interesting introduce retrieve web base text similarity recommender collaborative filter paper recommender use purpose user profile prefer amazon com use recommender book system suggest book previously recommendation apply outside paper come lot researcher field research article relate google might article user intensive article research reduce develop similarity researcher article increase accuracy article recommender general recommendation evaluation recommender recommender broadly classify category collaborative collaborative use rate unseen preference cf cf far factorization accurate netflix content recommend include category hybrid collaborative recommendation effectively rate user profile concentrate movie extend citation list library provide page publish regular unlike page paper conference journal list challenge develop representation name solve rule handle order name middle full co occurrence author appear document corpus indicate word occur position naive bayes position identical incorrect title key heuristic stop almost every english
model objective form one transformation design original simplify question quantitative need boundary experiment generalize level construct boundary optimal design allocation theorem arrange explicit condition easier justified design design restrict four p w need check verify depend aid express precede prove construct four define allocation main w w accord known numerical checking valid analytic solution show analytic comparison analytic factor newton lift list main logit link analytic one difference large extreme long lift affected life suffer cost record u secondly show analytic approach value numerically figure comparison lift highly want allocation design cm analytical precise need order theorem need true eq quasi newton find boundary critical calculate precisely illustrate combination minimization region failure please v need fp ip n p n inequality p nf v f pn attain uniqueness strictly admit v g v v ii nx l x c l nc lead problem support x iw diag diag transform transform equivalent optimal original transform design part support determinant x w mx remove support design pre support determinant factor boundary apply determinant pre specify satisfied maximize design correct problem theorem achieve corollary lemma conjecture analytic approach linear provide generalize include special effect lead solution factor aid solution condition design quantitative construct boundary factorial design factor analytic factorial linear tucker use come linear connect combination factor either qualitative quantitative effect represent factor clinical run unlike depend good review solve local optimality replace sequential design level restrict interval combination proportion assign example factor wu binary show locally construct two typically deal design design problem allocation assign locally response two analytic level highly lift search locally specify point tool bridge quantitative factor analytic computation complexity highly algorithm analytic design optimality maximization large easy deal follow aim develop analytic solution design organize utilize elimination system analytic allocation develop analytic three boundary point aid section interpret coefficient section analytic solution answer eq optimal maximize determinant family design allocation constant link concavity special eq solution common analytic case v I actually case whose lemma allocation generality get eq get substitute solution u u u go back formula provide go formula list iv want derive go polynomial change combination replace motivated quantitative g design consist boundary x p p case generalize x factor design pre design matrix would general locally determinant row commonly allocation analytic eliminate cox little algebraic geometry complicated impossible provide class design design pre specify distinct row assume rank pre simplify situation optimal allocation optimization p np factor main optimal allocation take w optimization none sum guarantee interior proof
implicit relation label enables project sparse similarity discover infer joint augmentation arrive panel show label alone right middle infer joint inference cognitive describe increase water visually label enhanced car change visually car region fit thresholded probability semantic become space scene concept resp b number region per region semantic object semantic topic semantic semantic visual label label visual cascade follow context dirichlet label learn probabilistic topic learn appearance group frequently bag structural likely discover da posterior likely feature alone pass current label visual location co occurrence three allocation image corpus generate hyperparameter sample asymmetric crucial dag super mix label refer formulate appearance bag label representation project region label find near correspond observe label sift bag norm induce label pool capture visual semantic topic mix visual topic give model annotate semantic learn visual collapse estimating topic space distribution proposal derive assignment exclude token topic time within value capture structural link count super multinomial semantic gibbs proposal sampling topic assignment semantic learn label sample semantic distribution label return imputation task dataset collection natural image object frequency category consider category consider image annotate test bounding detector three histogram texture filter codebook dense sift histogram visual learn learn radius empirically visual estimation run use step generate distribution final thresholded ap gain table ap object gain interesting object text category object contextual highlight mean average scene label infer super estimate divergence compare baseline correspondence lda understanding lexical assume correspondence object implement supervise divergence low model conceptually semantic label inference match look confident verify improve output filter tree context retrieve qualitative method sensitive compare detector relation highlight sort average trend method well rich prevent imbalance frequently many natural follow law detector positive vs picture book understanding system capture semantics scene visual single statistically well task lexical visual accurately share context future usa edu contain widely vary name inform cognitive lexical label share semantic image visual near latent dirichlet art human parse scene object mind complex visual lexical semantic precisely mechanism yet across lexical key scene bar people possibly incorrect context object become road evident first object iterate visual semantic base facilitate scene understand lexical object name lexical space vocabulary object visual visual name contextual lexical environment context visually appearance end semantic connect visual interpretation top represent coherent visually appearance specifically hierarchical first image determine semantic visual appearance observe bottom observe semantic visual image infer appearance contribution cognitive share algorithmic cognitive entity share update significant contribution able object category particularly bar et brain object call visually modal lexical context finding interactive integrate visual learning aim natural object relate encountered search keyword ambiguity task lie keyword usually keywords pick latent variable effective rich scene due modular separation mostly remove scene visual hierarchy understanding try text capture complementary exploit quality infer good knowledge give fit content connection
acceptance region contract contact accept bundle assume order preference order induce order gx satisfied common payoff several notational convenience bundle wireless plan contract wireless user certain video audio wireless user contract datum less demand contract demand price price service tradeoff relate weight loss payoff decrease payoff function accept contract bundle boundary secondary contract service excess secondary user primary bandwidth hold dynamically change provide way type bundle payoff relate tradeoff boundary recommendation system recommender make recommendation preference either recommendation type preferred accept recommendation rating order preference q although preference choose recommender obtain reward number contract framework assumption bundle problem bundle I maximizer bundle maximizer bundle behind design type consist exploitation vary exploration exploitation second throughout time horizon example recommendation recommendation wireless service different optimal bundle algorithm clear sublinear average section exploration exploitation exploration offer bundle search bundle search bundle decrease horizon good bundle due horizon parameter k x sequencing distribution space contract learn lie simply contract contract contract accept know contract estimate exploitation offer bundle constant let exploitation bundle optimization q one maximizer choose maximizer combinatorial provide literature computationally efficient special old exponent constant assume payoff step contribution nearly optimal bundle step bundle bad horizon bound optimize u q even sufficiently accurate q convenience gx gx term acceptance define event happen bundle chernoff substituting sublinear sublinear want q choose eq drawback step usually offer bundle wireless service add new current thus total significantly exploration simultaneously differ exploration spaced exploration estimate phase exploration phase follow similar type zero initially basically exploitation phase form value counter number complete phase time check exploration phase exploitation start exploration bundle contract accept exploration contract accept exploration contract exploration phase htb phase lk regret due proof horizon upper function sublinear independent run eq contract secondary market author common type channel exist step x cube contract online linear problem linear neither setting another consider topological strategy since estimate reward prove bi old old boundaries eq q remark online contract selection sequentially exist contract contract payoff high payoff choose contract contract maximize preference hold payoff regret distribution type service online offer bundle sequentially good bundle offer good bundle preference preference stochastically paper type depend step independently step obviously maximize offer simultaneously payoff preference type observe accept compute problem propose contract simultaneous offer offer similarity difference
representation shrinkage covariance nx since complete word variance precise allow low cf stein would estimator one would greater slowly kernel small bandwidth rbf estimator impose resort quadratic unfortunately approach unlikely upon standard post weight often recently attempt kernel estimation robust huber regularize version mmd adopt testing result resemble furthermore f view generally regression operator work treat entirely fundamentally shrinkage play role shrinkage automatic let estimate shrinkage quality quantify show state shrinkage leave validation score simplify nn n nn nn nn x leave score take evaluate na fortunately simplify score satisfy weight calculate shrinkage leave kernel write span sample side solve leave validation leave product shrinkage compute diagonal low rank adopt second compute calculation simplify computational validation operation na product negligible optimization toolbox shrinkage product shrinkage rkhs covariance operator write covariance ny xy xy yy shrinkage estimator plug kx yy shrinkage mean evaluate generate distribution weight estimator calculation gaussian lin generate gaussians wishart rbf root median figure estimator kernel shrinkage eigenvalue case lin appropriately discussion similarly depict propose score slightly substantial large perform ccc ccc ccc pca first density matching whereas density initialize initialization return repeat pair test significance via achieve negative outperform relatively case provide estimate effort require optimize different projection kernel scenario shrinkage center center perform generalize eigenvalue c c obtain kernel hadamard product test illustrate result consistently outperform improvement compare sense intuitively change considerably effect reconstruction positive kernel I I kernel accordingly shrinkage categorization anomaly detection rbf kernel hyper choose fold cross validation repeat several report table report roc different mean clearly shrinkage evidence standard dataset small competitive commonly estimator improve upon theoretical wide demonstrate estimator namely flexible empirical propose outperform small paradigm estimation application stein transform likelihood stein show improve gaussian square several stein estimator estimator dominate although stein entirely frequentist view show bayes stein later stein shrinkage estimator usual maximum arbitrary stein usual give detailed shrinkage shrinkage estimator firstly formulate problem loss simplify leave cross validation give obtain estimator outline write write n kx k consequently minimize weight denote shrinkage remain shrinkage minimizer approximate estimate quantify whereas simplify length representation leave score full leave validation efficiently leave target n nx k nx require result leave turn derive leave write throughout virtue residual denote rewrite since span nx k kx j x kx n n jx side respect consequently score sample score often assume center map mean feature center compute center center alone empirical formulation shrinkage write ij nx k j matrix write compact center kernel matrix similarly covariance rkhs foundation kernel discriminant operator see measurable space feature xx yy exist unique cross cross n ny operator rewrite functional span e iy yy definition example ac bs ac com bs reproduce hilbert range analysis improved well phenomenon call consideration reveal existence estimator empirical outperform reproduce measurable ensure expectation unfortunately directly easily compute empirical q primary investigate rkhs kernel rely heavily rkh algorithm performs recently hilbert represent preserve basic operation carry g intermediate homogeneity mmd dependency kernel kernel optimal estimation minimum variance unbiased support find show maximum mle multivariate
one applicable design wise hashing technique construct compressed dot row focus work however understanding apply able derive finite risk linear logistic dimension min wise hashing though primary understanding bit hashing suggest continuous call allow construction importance one assess wise hashing compress output approximate column motivate hash dimension reduction create original constant maximal active surprisingly despite reduction effect row normalise version datum signal modification procedure typically need reduce interaction model extension numerical study conclude discussion amount concern feasible include approximate software package dataset implement may min datum prominent dimension discuss approximation square low decomposition sparse aim reduction compress preserve min try light interaction manuscript extend methodology continuous variable min hashing section principal projection begin notation regard index submatrix consist denote source randomness consider may min hashing variable compress bit min hashing choose block create column form matrix three step random column row order column record index index variable index first non bit odd number map number map construction illustrate index whose variable appear bold index perform matrix slight abuse block value minor variation hash replace b choose replacement map bit amenable since avoid difficulty arise map representation version identical purpose step implement bit sign permutation hash create scope paper go detail improvement would row keep create parallel hashing speed random redundancy min hashing sum block yield convenient work follow bit min allow create permutation create column sign hash sign identical min component toy ht l appear bold hashing bit min hashing bit identical intercept add include hash hashing sign former lead help scheme circumstance popular drawback component computationally demand almost one hope motivated reduction computational random hashing map matrix random typically gaussian result wide context sign hash pca interaction show matrix must necessarily combine hash show compress sufficiently many expected context row response coefficient number interest constant yield logistic help situation vary length binary exactly scale scale entry construct x min hashing sign hashing assume far assume unlikely add place force equal modification row along scale generate min unbiased specifically n sufficiently regard concerned storage bit min wise hash roughly upper roughly nature optimal equal sparsity rl seem observation recommend sparse value use storing study wise hashing random allow matrix rather binary random min n bit hashing applies non bound aside around ridge row sparsity mean restrict attention min hashing result min result matrix taylor series expansion suppose exist tb unbiased average approximation bias family alternatively help simplify appear signal multiplicative term involve approximation maximal row sparsity situation probability requirement show typically section sign hash work equally min hashing row linear model noise vector structural satisfie expense small preferred demand denoise type fit coefficient bound require condition observation avoid assumption perhaps simple way number stem balance ol reduction optimal sign hash well implication signal entry variable rescale value rescale vanish attractive associate direction large variance important add required consistency increase many predictor become encode word next gram interesting consider much increase block increase add sparse effect require keep substantial would apply lasso similar computational ol improvement discussion bind hashing obtained subset remain one discard transform require hashing especially fitting interaction give bind theorem look interaction add adapt however able since compress fashion matrix bag aggregate averaging experience mark aggregation computational computation parallel matrix fit stage scale one nevertheless specific variable look importance well interpret fit produce hashing create hashing component zero structural error present prediction store il lx il z il il need store matrix interaction variability k x could beyond design fitting procedure pursuit pursuit would hold instead predictive setting interaction bin bin bin bin bin bin exp rf sis iterate sis bit bit bit rs rs rs bold describe text hashing fully expand fitting strength exp exp coefficient exp exp exp exp exp exp ridge rf sis iterate bit bit bit bit bit rs bit various gaussian exponential exp sign modification min hash rs helpful continuous entry random hashing screening iterate diverse datum uncorrelate variable control design probability non zero binary take matrix draw draw create replace independently interaction control via independently set consecutive uniformly rescale version ratio method fold cross tuning unless specify penalty ridge penalty forest default sure independence screen sis iterate sis fit bit min wise hashing zero min hashing min hash random sign bit computation sis iterate intensive large scale sis sis validation take substantially dataset variable compute lasso min roughly minute small bit min hashing predictive consider fitting time observation identical column design want original fitting random forest fitting bit min bit hash hashing preprocesse take large preprocessing permutation min find bit min wise choose column full runtime model compactly store storing representation make comparison keep permutation hash advantageous fact outperform hashing evident however fitting bit theoretical representative show non design design replace across good result bold I min min wise start translate accuracy ridge bit min therefore computationally small ridge regression present performance random reliably hash superior bit wise hashing keep permutation min hash advantageous wise min hashing essential retain rs allow fit rs seem min interaction original result bit wise hashing interaction effect corpus financial volatility underlie stock forecast focus accuracy underlie financial view forecast log volatility stock return compare variable scale predictor using generate linear draw coefficient non group average weight finally apply result six different scenario generate report log volatility underlie way transform data z z normally correlation actual actual curve sign hash blue random projection linear hash random sign advantage contain design matrix comparison use projection normal entry similarly linear former well show similar scenario f hash example sign hash identification panel ridge logistic whereas lasso validate row report classification near associated lexical name token binary variable collect course day remain active least issue change propose go distributional change sign hash logistic ridge dataset acceptable batch day different dataset day test five regression varied hashing base average produce class drop four datum occur ridge perform bad hash day regression mention zero non regression lead
environmental covariate stochastically covariate consider spline reversible jump markov frequentist continuous individual constant approach I penalize likelihood depend ii term spline smoothing balance goodness smoothness via score specify candidate smoothing functional technique covariate investigate relationship probability grey uk grey extensively important specie top second application challenging covariate individual body mass affect subject numerous study dynamic isolate ease individual mark introduce likelihood model detail inferential include quantification strategy choose conduct challenge vary real formulate covariate section review three provide penalize discuss use spline implement approach three observe death individual death assume individual standard mark recovery mark identify recover straightforward survival recovery discrete capture notational convenience individual covariate condition array constitute statistic contain live individual capture individual array give convention leave side probability recursion survival probability omit subscript likelihood multinomial discussion associate extend three covariate vary covariate vary could correspond different time covariate common subscript drop express corresponding dependence parameter specific covariate two covariate live individual recover mark see line recover individual recover time know conditional initially time individual vary stochastically survival correspond subscript covariate indicate age probability dependent stochastically miss arise scenario turn attention specific vary history state system survival state individual correspond strategy summarize process survival individual initial observe survival know within unknown time individual observe assume covariate conditional covariate initial write survival assumed general continuous analytically however discretization range expression approximate arbitrarily increase become value typical mark approximate use hidden covariate schwarz exact consider interest deterministic covariate methodology multiple covariate may mark recovery link parametric covariate analogously general flexibility coefficient numerically polynomial spline polynomial fuse smoothly boundary manuscript cubic spaced spline consider allow curvature predictor modify adjacent need sufficiently structure reach long penalty integrate squared curvature type consider log goodness increase lead emphasis discuss dominate sequence estimate straight give consider difference general interested multiple model regression smoothing use regression scenario section coefficient combination spline numerically penalize numerical maximization know maxima likelihood individual covariate covariate consider covariate detailed discussion choice quantification part bootstrap implement sampling use capture alternatively array environmental covariate bootstrap new confidence estimate specific covariate quantile replication obtain simultaneous band function band confidence band simultaneous band pointwise interval local statement cross smoothing drive deal environmental covariate array see usually validation successively validation form calibration scoring apply calibrate smoothing parameter likelihood calibrate average score score leave validation successively validation often infeasible generate random partition suitable constitute calibration remain validation sample partition sample grid e pattern must allow successfully setting less intensive approach select smoothing statistic degree fisher penalize effective freedom account effective reduction penalization initially assess since individual individual capture specify initial age age age probability covariate could correspond dependent recursion q initial capture covariate choose survival either survival highly fitting discretization b fold approach estimate integrate two functional estimate obtain cross report validation tw simulation estimate use gray line exclude last covariate compare simulate survival boundary range covariate would couple ii mean relative environmental note consider grey year keep record array website support live study consider year age age age age age interested historical central covariate year temperature year year within relationship survival age array model parametric predictor spline basis conduct order vector smoothing yield lead class base smoothing identical obtain pointwise confidence interval parametric gb ram take exercise logistic correspond display figure agreement finding htb would environmental condition increase increasingly environmental relationship environmental covariate value survival constant majority year suggest though obviously parametric influence influence slope year relationship similar though less year estimate vary covariate search capture period survival specific vary note primary cause age age age class class function different evolve model fit recovery spline representation covariate estimate visually indistinguishable obtain analysis cross validation survival four age computationally infeasible scenario separate age follow age class corresponding nuisance nuisance initially estimate fully cross validation coefficient calibration yielded refine parameter repeat type validation hold nuisance ultimately yield effectively model information aic fit hour core ghz gb ram substantial achieve calculation pointwise interval via nonparametric exercise difference result find previous analysis slightly survival alone sharp survival individual comparable irrespective age weight could load seem rate model minor individual unify inferential consider maximum penalize constitute powerful alternative approach build extend stochastically vary modelling widely alternative capture recovery remove individual remove initial capture close population covariate real datum grey demonstrate nonparametric give new insight specie population fit drive environmental
sdca effective solve problem extension set accelerate version sdca ascent vector logistic obtain follow conjugate coordinate ascent keep recently stochastic sdca optimize optimum derive smooth sdca sdca stochastic variant solving accelerate condition find perform sdca bind scales sdca randomly pick update vector use mini batch batch mini batch neural sgd always mini multiplication multiplication operation gpu mini mini size typical mini compute author mind study mini sdca svm naive mini optimize might actually describe safe mini take employ nesterov acceleration apply mini show acceleration sdca mini procedure accelerate scalar result analysis require analyze work sdca sdca demonstrate sdca relate work result square euclidean regularizer q example smoothness convex assume euclidean parameter optimality optimal solution primal px assume side dominating compare bind sdca ignore constant c sdca sdca iteration sdca scale cost study empirical mention meaningful environment parallel environment minibatch sdca c recent year lot implement architecture discuss sdca machine node fact dimensional sum apply summation example message bit corner neighbor whose word hamming node away nod iteration overall iteration bit bit node parallel iteration neighbor therefore take discussion form implementation runtime table c runtime communication sdca sdca channel value reflect adequate tradeoff node communication channel bit outperform sdca perform smooth variant hinge label regularization package ph dataset paper physics physics classification collection dataset ph run run sdca primal optimality algorithms sdca clear sdca much sdca discuss parallel negligible like sdca mini process column top primal optimality value hinge bottom denote expectation update r noting next variance difference expectation round simplify e ti vector replacement therefore positively upper bind duality derive upper previous additional additional algebraic round bind smoothness rearrange combine convexity sum round lemma recall definition eq definition smoothness therefore combine get q sufficient condition theorem combine yield expectation apply recursively conclude ascent mini batch algorithm accelerate batch apply small mini batch give mini sgd accelerate sgd mini batch sdca mini batch svms distribute property however strongly smooth achievable sgd sdca ignore option divide instead reason practical take account
complex plane fire coding indicate thus number neuron activation compute output magnitude fire input neuron phase see figure example neuron simulator hypothesis play functional information mechanism receive spike fire output activity fire phase input difficult fire state input activation gibbs boltzmann machine conditional output neuron brevity aspect replace value complex firing analogously correspond phase generally message add real neuron input long fire account neuron strong less input total complex activation magnitude phase total biological strength fire capability network neuron net input decrease input phase individual neuron interference phase moreover I connection change desirable property biological weight neuron cause instability dominant negative lead introduce issue modify output first refer magnitudes phase reduce presence neuron connection never lastly give classic thus control possibility say neuron deep deep later value net boltzmann net two bind activation real artificial feature neuron brain dynamically correspond coherent entity visual scene phase analogously importantly communication complex message naturally agree message message opposite encourage neuron realistic visual arise interaction interaction strong input classic group phase gradually affect message gate dynamically depend input current interaction area depend member neuron phase another neuron input dominate latter phase particular difference neuron account complex plane contribution second phase equal analogous phase particular represent cause image simple role bind aspect network deep boltzmann machines undirected layer internal visible unit definition connection layer within layer stochastically sigmoid see implement framework adapt autoencoder inference probabilistic multi recurrent joint activation demonstrate work value unit magnitude describe develop principled probabilistic boltzmann machine well network explore additional appendix show role throughout magnitude unit image infer hidden phase initialize randomly experiment layer boltzmann bar draw whether constitute bar employ choose version bar bar boltzmann machine bar convert magnitude image unit phase activate bar phase code unit active bar bind distribute single neuron bar phase visible show code weight input unit necessity individual fully bar presence bar learn bar field overlap unit image bar supplementary video visible complex plot visible hide visible bar bar bar visible peak phase bar make three indeed neuron dynamically supervision neuron target semi supervise successful visible bar several bar phase limit notably argue aspect capacity limit third relate nature bar whether neural examine correspond bar response property individual maximal unsupervised approach recent somewhat discover concept cat unsupervise work analyze bar neuron represent mechanism establish place deep network binary corner arrange square corner draw corner hide discover field corner arrange phase corner multiple phase phase b hide image corner arrange randomly separate arise done rest control demand usage capacity activity play causal role similarly process dynamically change coherent possible interpret select reading subset dynamically layer argue extend rich rich specifically functional show potential mechanism support grouping representation impose object unfortunately role difficult principled interpretable understand aspect could recognize complex value firing possible convert without classic qualitative classic term etc representative example dedicate find arbitrary net favorable concept learn community useful develop learn work aware employ neural interpretation separate model however directional boltzmann state bind phase contour aspect attention principle within value extension implication interpretation extend desire unit switch circle perform qualitatively similar limited architecture development boltzmann translate describe primary visual though perhaps limit conclude much bind sparse result formulation actually motivation aware develop complementary broad propose experiment analyse rao aspect object unnecessary lastly apply experiment remain currently explore backpropagation feed benefit could carry state may detect alternatively bind backpropagation appropriate input cost function iteration order hundred thus training could boltzmann train consist layer small bar bar visible various magnitude input phase detail acknowledgement david feedback nsf early award support fellowship service support brain science visualization experiment follow supplementary video synchronization layer bar several visible synchronization shape momentum decay training persistent use instead learn decay factor divide mini batch encourage hide varied experiment value mostly early detail architecture bar section height width corner unit image hide global synchronization step synchronization stable mnist choose lastly mention main real lead synchronization separately image result either visible across image work result balance connectivity affect incorporate synchronization desire summarize discussion comment expand cover important issue issue biological rather specific discrete represent interference explain capacity item peak accordingly cope limitation usual limit involve dynamically group depend detail object group example texture focus bar capacity rest image change phase input propose input task ill binary alone rao point make issue rao number pixel contain simple pattern example two face demonstrate believe extremely show bind dataset mnist insight nature representation object rao al issue point motivation classic lastly behavior synchronization contiguous vs whole decode hide representation particular rao et object single agree proper helpful present activation network generally exploratory train value net net backpropagation feed autoencoder recurrent biological interaction v mathematical quantitative essentially final complex value result network probabilistic suitably extend classic term refer comparison experiment outcome analogy inference let total input circular unit mode qualitatively eqs stable supplementary movie analyze detail also phase matter great imagenet network still rich circuit mechanism suitable aspect play information incorporate build rich variety framework attribute fire latter property spike qualitatively think related processing bind representation experiment flexible mechanism approach successful language representation stage non represent world approach thus relevant computational example organization visual closely boltzmann generative deep processing rapid feed human amount label current truly rich reasoning necessary learn capture utilize deep organization computation spike
never forget index subject information application good language probability expression b arbitrary could parameter condition index sum product marginalization bayes db p marginalization follow require logical density write theorem relate factor represent recover normalization density essential beyond manifold principle apply examine manifold coordinate system prior alternate jacobian play irrelevant description fit force alternative must relative evidence net evidence preference usually likelihood marginalization properly normalize density particular feature model account become parameter distribution relative requirement restrict work peak another preferred theory space compatible measurement comparable fit datum reduce net give derive consideration gamma let unit likelihood transformation da define constant integral unity evaluate da p p recognized measure possess transformation intrinsic identical da unit da da evaluate heuristic care whether respectively identification similarly arbitrary one evidence hold commonly identify observable unity joint write b change map jacobian intrinsic coordinate one write b b unity dx dy evaluation dy b b marginalization b normalize beta b b n b optimal find k empirical rhs recognize respect beta fx dx x easily solve b b maximum x amount mode mode function non evidence hessian sequence proceed fa evaluate new manifold value db fa f jacobian transformation volume preserve particular value appear reason use likelihood consider care exclude manifold infinite assigning beyond article infinite extent boundary exclude scale measurement say sensible suppose finite absolute sensible symmetry limit interpretation member type select chance boundary us turn collection engine define link page normalize page link link page summarize log link significant relation engine rank fit way rank treat rank expectation gm display boundary exclude comparison correspondence indicate apparent distinguished marginally distinguished third nothing gain distinguish confirm page share tend rise rank genetic allele distribution parameter measurement member x x account unit joint mean dominant allele likelihood member single p b negative la recall nontrivial estimate n consist count q whose allele dominant allele frequency unnormalize equilibrium dominant allele equilibrium solution parameter prior exclude normalize everywhere similarly equal zero five condition boundary determine principle nature approach clarity various whose projection mode boundary certain datum density unnormalized evidence equal else symmetry net support unit percent statistic freedom datum significance give reject however require observation yield preference equilibrium approximately evidence inf inf inf location display variation truncate net evidence logarithm p interpret describe equilibrium close display break discard whether single remainder question expect boundary evidence analysis index remainder index inf inf inf inf nan inf nan inf inf inf inf inf inf inf inf inf inf inf inf inf inf inf nan inf inf nan inf nan inf inf nan inf inf nan inf inf nan inf inf inf inf inf inf relative identify display significant minimum add minimum normalize population display well percent allele significant population determine significant net give thorough would region sample population physical contact possibility leave exercise specifically average player appearance integer successful appearance record successful average product kx success subsequent form hand impose sensible limit domain sensible limit perfect exclude benefit exclude player type allow evidence play success observation expand turn consideration classify whose assign axis uncertainty chance event independent p margin normalize independently obviously joint factored express location become require correspond appearance average p p nearby affect galaxy assign form stage chance sensible irrespective spatial event clearly logarithm write x whose limit evidence write evidence p beta disjoint region span partially define use evidence depend gaussian recover distribution definition panel respectively likelihood predictor figure estimate panel normalization expectation b approach give contrary event likelihood identical ignore panel us ab let density directly see encode estimate display evidence see evaluate without regard observer draw surely production fairly heuristic absolute describe say indicate prediction location measurement location channel channel return evaluate interpretation channel prediction treat location quantity location treat nuisance p ask turn
non seek locally consider rbms two change variable equivalently optimize x form ignore affect similarly fold auxiliary objective introduce behave unary proportional empirically dominate quality problem randomized round randomize summarize round corner describe volume lie proportion sphere formally hadamard mixed gibbs sampler tp ax sampling two different rbms case mnist train rbm parameter independently intend mrfs rbms bipartite rbms gibbs sampler rbm relaxation round variant visible fit mrf notation add variable experiment latter theoretical rbm benchmark configuration algorithm compare gibbs solver optimize compare technique across three rbm weight rbm mnist rbm independently entry select visible modify small gibbs sampler ever mnist sampler schedule solver relaxation width execution update dominate onto curve relaxed sample blue true sum exp configuration obtain sampling proposal mnist log partition function typically unknown result via exhaustive enumeration benchmark advantage perform summation variable instance fix sum three take low map sampler proposal right side rough approximation sparse expect sampling help whenever near map remain overall still comparable mrf technique rbm base configuration succeed local initialize alone applicable generally mrfs efficient rbm randomized relax pairwise markov experiment boltzmann underlie sampler partition sampling matrix potential diagonal entry encode unary potential simply column unless otherwise find discrete
supervised dimensionality reduction validate present algorithm vertical line figure dimensionality covariate mention give description uci census census census service home dimensionality also ml repository concrete strength include water coarse dimensionality reduce candidate site south except month observation miss wind speed site four site predict wind speed candidate collect http www repository task record california office entire reduce covariate signature breast patient gene dimensionality technique gamma voting record choice outside suggest show green reach unlike reasonably without dimensionality reduction part believe reasonable generalize multi apart save sir without dr lasso voting sir without compressive method save sir dr svm node wind speed sir dr svm van breast cancer method save sir node forest proof dimensionality simultaneously feature variable help attempt dimensional attempt regression conjunction iterative procedure attain rapidly dimensionality curse dimensionality hypercube volume hypercube rapidly value indicate number fix radius happen decrease phenomenon recently dimensional dimension develop method try low high reduce covariate maximization present assumption type covariate dependent supervise reduction linear approach aim learn notation response define present might response focus evaluation response similarity dimensionality inverse regression sir technique principle describe however covariate brief overview section definition statistic sample distance formulate investigate loss optimize propose require property technique supervise reduction technique sir save evaluate regression technique rf nh tree learn dimensionality validate performance reduction technique along empirical evaluating propose discussion pearson nonlinear dependency vector dimension weight function characteristic clear covariance distance sample random feature response variable trace note computed response express formulation solve discriminant maximization formulate trace orthogonality constraint optimize difference iterative solution orthogonality require eigen orthogonality key setting find euclidean embedding preserve relation product distance correlation empirically plot maximize utilize minimize section iterative also fix prediction represent function sum iteration individual function concave minimize monotone minimum saddle point eq formulate loss matrix diagonal dominant lead separate quadratic w support iterate form function iteration inequality occur amongst additive term framework hence relaxation sum concave get iterative e apply update iterative
fold r news fair penalty trade admm cross conduct report measure efficiency run matlab ghz cpu comparison trial draw firstly leave observe two admm effectiveness exploration adaptive accelerate stochastic achieve slightly objective informative three observe adaptive significantly terminate save achieve compare column significantly slow efficiency table summarize compare make similar algorithm objective error value test error w objective value full world iteration second popular technique accelerate admm replace adaptive traditional admm propose significantly promise thm admm year traditional admm expect function proportional reduce complexity expect function plus bregman proximal norm admm algorithm proximal encouraging dataset confirm effectiveness efficiency originally introduce multiplier augment achieves convergence admm admm enjoy convergence smooth admm sense video etc computational drawback need make solve mining challenge depend achieve interestingly easy practice address issue admm online loss online first bregman solution bregman divergence proximal half square bregman way enjoy solution similar rate address issue new family admm accelerate admm subgradient theoretically admm effectiveness confirm encourage evaluation world present propose experimental supplementary material family problem equality draw traditional formulation paper assume present notation euclidean definite bregman divergence split lagrangian derive replace upper get optimality updating fact combine conclude convergence provide feasibility follow b convenience fact step completes derive proximal function low regret choose definite optimize restrict high may desirable advance minimize shall proposition diag attain advance receive sequentially instead diag use nearly update adaptive admm summarize initialize diag rate convex algorithm b h diag example round probability case equals derive one helpful
subsample describe begin use complete aggregating via sharp aggregate aggregation improve redundant consider deal readily minimax setting inspection theorem thm provide expectation depend skip brevity natural question one happen inf f view situation case aggregation inf f pa particular specify recover result theorem reveal intermediate two adaptive error study problem nonparametric model note measure problem admit section upper bound exhibit vc constant prove bind rather expectation countable element eq denote cardinality easy vc vc random take q exhibit rate theorem improve estimator real vector slow minimax risk minimax regret give design erm net dominant place overview somewhat skeleton recall find net skeleton subsample skeleton subsample minimizer within cell aggregate erm step net subsample well different paper take risk bound skeleton aggregation comprise projection hellinger aggregation desire successful setting explain aggregate erm simply aggregate cell us skeleton aggregation yield correct similarly skeleton aggregate satisfying exist q sample conditionally subsample hence relation sup relation skeleton indeed inf f behave norm use analogously optimal tradeoff introduce square cell get rate global skeleton aggregation rate risk erm aggregation skeleton erm pn finite skeleton regime method theorem aggregate optimize combined excess erm e erm bound finite shown improve neither erm selector skeleton excess rate erm suboptimal massive polynomially erm rate skeleton suboptimal case aggregation optimal erm skeleton suboptimal skeleton finite erm enjoy extreme massive nonparametric also unless skeleton aggregation improve erm turn aggregation skeleton rate global erm suboptimal role establish also al erm net bias balance present global version weighting extend propose third short regret density rich set distribution small optimal ball ball limit radius method state lemma localization empirical covering number sup consequence use rademacher empirical follow indeed apply theorem diameter loss sup absolute minimizing appear together affect right hand ignore pass prof involve extra affect term throughout generic depend expression cn get assume w enough previous upper inequality expect excess fix estimator cell convex function event hold tn square desire bound expect excess inf lf generality sup p lead inf lemma q imply bind together bind hold rademacher estimator proof lemma result remainder purpose obtain evaluate rademacher complexity difficulty sd ss rademacher another metric ball pseudo metric respect set net respect probability choice fix let denote least q taking give throughout sample fx fx ix ix ix ix fs proper turn product observe lemma integration go yield component exist follow I kx x yy x py x kullback obtain expression display constant check display hold choose satisfied without generality discrete within interval fix binary sequence uniform put distribution define hamming distance iv nx kullback leibler set contain sequence eq f n integral compose indicator fix r write condition entirely j j nd absolute localization hold prove theorem event denote pg kk pg event define solve fact obtain difference inequality yield radius entropy integrate n bind whenever depend class covering along take proof consider apply get sub pg trivially sub root p ng last mm assumption random design regression model minimizer erm appropriately inequality excess attain rate minimax estimation specify model minimax equivalent problem statistical enjoy problem rate minimax slow minimax oracle inequality rate type usual convexity slightly modify excess risk convex aggregation improve pair consider call aim estimator excess expectation measurable straightforward expect excess generic sign learn characterize agnostic right oracle refer constant front infimum key great bound excess oracle extension condition high statement level boundedness minimax point object write minimax infimum estimator instead goal competitive equivalent write regression thus interpret minimax context aggregation aggregation cf span hull initial suppose independent deal aim aggregation construct aggregate lie important nonparametric belong infimum sample fix give minimax risk minimax regret quantity magnitude answer positive interest massive class prove sense minimax risk rate violate rate minimax aggregation duality reality place quantity represent development mostly connection object risk minimax regret risk consequence theorem set transition bound minimax minimax aggregation close skeleton global proof technical consider satisfy class emphasize dependence bound admits decrease write thus measurable set real function pseudo cover respect supremum theory process ensure erm estimator detail chapter page function great cardinality notation positive absolute unless explicitly along inequality corollary estimation comprise step risk minimizer cell partition use aggregation radius take method minimization erm suboptimal section enjoy aggregation cell assume pseudo subsample cardinality clearly include totally net respect without generality I I break way least estimator exist modification possible finally subsample aggregate type aggregate word oracle lead sharp ms function sharp realize mixture sharp aggregation aggregation estimator sharp step next aggregation let localization radius aggregation stage exist absolute remark erm readily partition view overcomplete geometry instance cf set choose element erm individual rate partition linear subspace aggregate localization inspection oracle inequality way replace localization determining belong entropy entropy theorem polynomial risk constant cover radius
obtain initial input collect prescribe form appropriately h drawback contain result happen equally new mathematical real alg nh alg I I new lk trade exploration exploitation generation essentially exploratory action exploration measure current approximation time new constant q theory ratio towards change requirement large force large concentration lyapunov lyapunov concern cost function therefore attempt burden gene exist switch fast burden simulation discount set control perform hard computationally constrain purely offline realistic output output imply computationally update ability efficiently figure trajectory update trajectory stochastic trajectory average trajectory protein population easy measure exploitation decide choose number time sample illustration benefit investigate uncertainty model stable stable steady validation goal compute law stable steady binary current measurement compute current light require switch control set experimental trajectory exactly deterministic robustness towards choice specify steady proximity five trajectory learn depict protein red protein concentration equal green input point curve steady dash curve close target light occur essentially light light trajectory dynamic modify burden system realistic variation system simulate trajectory run algorithm protein simulation update dash line model exploitation towards control policy exploration generate nothing exploration phase explore big exploration begin small constant monotonically always simulation concentration protein equal steady switch successfully initial training model approach stochastic control deterministic present deterministic additionally simulation twice apply panel figure curve online equal curve switch long light outcome simulation algorithm apply consequence potentially take account apply significantly constitute future present framework efficiently switch control validation fact quantitative reached acknowledge ep innovation award ep g network control office uk co com mathematics college ac institute problem optimal adapt reinforcement system control collect create algorithm system deal gene network synthetic biological typically via gene cell e impose burden cell high induce severe perturbation growth intend biology highly desirable behaviour simultaneously art allow interact quantitative estimate marker protein g heat feedback feasible control drive burden minimal protein trade control maintain network network address author infer process moreover intrinsic expression translation add gene flat end steady state gene goal mode one infer interaction system drawback indeed interaction reinforcement learn system generally address concern hybrid first initial use datum mathematical control reinforcement switch problem protein space policy infer control system triplet sample transition simulate update online measurement past exploitation control signal depend markovian abuse symbol system control sum discount policy specifie drive control transition l lf advance central fit q eq condition iterative procedure eq triplets iterative every l tree generalise control outline simply maximum significantly triplet nk q c estimate pair benchmark switch switch mutually generic switch therefore protein product assume protein marker assume control implement sensitive controlling
hyper free introduce broadly successful loss issue parallelization loss adaptive issue importance minibatch parallelization return combination drastically concern grow architecture employ absolute address implement adaptive setting sgd schemes literature aim concern complementary produce scheme adaptive decrease quadratic optimal schedule preserve guarantee separable analysis dimension rate analytically curvature respectively use number estimate quantity equation memory adapt take rate ratio obtain pure time gradient somewhat appropriately minibatch leave compare online time minibatch parallelization core perform hyperparameter extent hardware bandwidth equation determine rate minibatch factor turn diversity gain substantial leave minibatch size impractical however fix minibatch minibatch get automatic rate toward mini batch green additional long red obtain produce mini batch one zero curve near noise figure effect dot architecture g penalty lead gradient non minibatch asynchronous sparsity equivalent minibatch ignore basis minibatch minibatch gradient factor learn reflect small minibatch case learn reduce translate minibatch dimension long figure suboptimal minibatch equation relative outer envelope one reason boost gradient come mostly reliably expect individual loss zero non expression move gauss newton diagonal pass purpose determine good hessian point regime practice obtain difference typical move scheme draw gradient compute sample shift increase h nh I far increase robustness intuition motivated curvature estimate produce reduce likelihood become curvature normalization signal maintain move compute encounter discard move statistic keep adaptive learning rate simply old reduced adapt rate threshold deviation increase combination minibatch initialization one update box without must able number elementary test purpose elementary stochastic optimization gauss curvature draw vary give case visualize see test range minibatch size gain update curvature level scale identical row parameter namely finding contrast tune reliably adapt automatically level deal adjust minibatch speed case broad benchmark deep expect performance world perform across noise smooth value algorithm need hyper task well variant learning rate account size gradient drastically parallelization also algorithm free unlike broad elementary box investigate adjust element problem dimension rely rank block decompose covariance acknowledgment want zhang cl helpful perfectly open national rgb rgb rgb institute york york empirically rate sgd remove tuning reduce stationary appropriately non stationary direction minibatch parallelization
sub imply player classical exploration choose might reward however player collect fail play application arm online ii polytope degree algorithm worst depend exponentially invariant denote regret armed bandit variate reward clearly round exponentially slow curse dimensionality avoid see function sub high recently ambient bandit sparse adversarial variate function optimal author problem bayesian assume relevant dimensional strong guarantee author vary reward b rank model reward special arrive variate depend reference consider function query independently parallel bandit variate although rkhs hilbert scheme comment conclude remark towards end armed case regret old reward prove author derive convex continuously gradient behave show local old maxima exponent achieve author reward contribution contribution namely achieve ok continuous term nearly discussion regret increase avoid curse dimensionality budget idea span employ play subspace derive careful allocation budget phase organization paper formally intuition along formal analysis approach regret finally conclude player radius time choose upon round budget player unknown denote k dr unknown reward u consider gaussian mean time magnitude strong assume lipschitz need smooth reward necessary formulate tractable make technical allow determine explain class satisfy assume row svd k unitary orthonormal obtain round cumulative regret play goal minimize phase namely dimensional estimated intuitively imagine close original would playing budget recover bad regret carefully divide guarantee describe phase outline budget generate closeness norm total regret lead say play duration eq regret regret incur optimal offset optimal make lipschitz ii f precisely appendix result algorithm lp ok formally external define expense incur dominate span know q u estimate convex norm selector ds nuclear norm sum singular operator large singular making theorem deferred appendix ds solution let arbitrarily actual estimate row singular quantify noiseless hold lemma arbitrarily prominent term start dominate handle stochastic concern require condition guarantee sampling reward stochastic total estimate mf total note obtain subsequent average reward change replace obtain lastly consequence duration imply state scheme phase time step lie consider play optimize optimal write incur account play bound ucb finite armed run duration step retain lie multiply employ duration manner proof attain bound form follow proof satisfy bound respectively order overall carefully precisely assumption notation use far achieve f suitable ensure regret bind plug hold indeed guarantee bind upon term dominate sub case appear unclear way let choice dominate appear hence improve armed bandit dimension model reward function pm create random j ij x obtain solution comment measure condition specifically singular imply indicate natural dimension decay regret exponentially fast factor undesirable provably decay polynomially regret et function origin full dd additive model function regard discussion corollary theorem summarize stochastic combination variable know derive achieve regret combine recovery literature arm bandit note difference function rkh bayesian idea span perform careful allocation budget amongst phase possibly employ bandit reward inf routine consider technique ucb work horizon unknown regret involve recover unknown subspace span lastly reward arbitrarily adversary fact
proof restriction eq differential long proposition distribution digital unlabeled label infinitely way contour unlabeled plane analysis stable assign label point contour speed need around contour contour similarity nontrivial contour plane regard piecewise differentiable parameterize piecewise uniform contour regular similarity two contour shape similarity contour contour shape nonzero complex contour span yield contour pre henceforth work hilbert identify measurable integrable center function direct shape contour shape contour unique curve contour take center arc therefore open subset hilbert omit identify contour confusion hilbert manifold embed hilbert eigenvector eigenvalue large projective eigenvector large derive specify asymptotic zero operator eigenvalue delta remain simple eigenvector tangent eigenvector complex take orthonormal q eq entry position formulation finite projection q component operator distribution difference explicitly however directly problem manner asymptotic approximate tangent apply arrive must full properly rather complicated drastically hypothesis methodology type al positive section shape contour probability equation population hermitian covariance hermitian shape infinite hermitian respect eigenvector tangent follow hermitian explain compute base contour contour ideally contour perform approximate evaluating function select interpolation yield contour differentiable denote order express similarity shape self dense shape contour hold purpose test statistic contour approximate work stop contour consideration work approximation select uniform hand ultimately sufficiently number vertex represent contour maintain match regular contour contour stop maintain self accomplish sort stop choose point fairly contour ensuring represent computational performing utilize compute region shape choose though cost extremely stop provide adequate contour significantly fig include ht contour number select appropriate determined selecting compare value number relative error examine however digital imaging contour necessary replace contour calculate point converge contour select stopping must properly ensure stop permutation point probability stopping converge stop time state successive time interval cdf uniform interval since immediately center mass center contour great shape examine distance self adjoint matrix consider contour digital contour pixel digital contour contour repeat distance calculate contour quickly show variability overall approximate unclear approximated distance shape indicator determine low helpful relative contour length quickly relative desirable keep error ensure contour well contour evaluate correspondence ideal scenario contour select point evaluate utilize contour hand highlight illustrate correspondence contour require stop adequate select stop contour stop time contain utilize contour stop previous scenario meet shape work preferred approximate contour time separate approximation subsequent automate automate allow whenever conceptually must take enable computer object process usually projection dimensional convenience current situation dimensional operator rather successively projection representative constitutes establish sophisticated brownian determine suitable value objective eigenvalue brownian prescribed achieve projection know perturbation appropriate approximate infinite scope consider could variety involve determine historical hypothesis test determine shape treatment mean shape historical mean similarly could perform quality determine determined application reach standard example choice instead solve reach role neighborhood completely neighborhood restriction small present contour conduct environment consider differ corresponding distance shape approximately hypothesis neighborhood consist shape scope ray contour shape contour shape calculation large value roughly large nearly ad reject agree visual inspection contour contour shape shape ad suggest nan agree intuition last contour sample determine reject nan reject nan small nearly ad unlike unclear nonparametric resample available region confidence illustrate methodology visually display plot ht reveal region wide variability top front see fig shape contour reflect region confidence region substantially ht intuitive processing compute elastic framework cost elastic intrinsic use root elastic arc curve step use either consume step intrinsic obtaining region result methodology perform letter shape et windows intel core processor ghz computation require elastic contour display vector denote coordinate sample square root curve well produce far detailed also al describe address neighborhood hypothesis population manifold hilbert rich study potential advance manifold hilbert space theory direct shape contour could extended infinite euclidean include plane gray image properly edge one image match point correspond match additionally technique two sample procedure hypothesis nonparametric useful shape contour serve important towards adapt sample procedure note contour paper maintain camera analysis projective shape instead contour commonly object slight camera substantial digital slight shift contour shape projective adequate descriptor analyze care contour object absence additional care help meaningful analysis grateful regard thank also subject section problem university nonparametric level digital hilbert manifold perform contour lying shape hilbert manifold hilbert schmidt operator contour general utilize digital imaging provide another method analyze shape contour keyword nonparametric bootstrap contours digital automate selection pt methodology hilbert procedure theoretical estimation testing hypothesis neighborhood hypothesis practice expect equal prescribe dimension projective hilbert schmidt shape infinite lie implementation turn shape brief discussion present good space lack review theoretical manifold statistical lie hilbert functional useful extension asymptotic eigenvector even al text utilize hypothesis model follow mean properly far arise analysis contour shape direct shape configuration definition definition notion fr approach follow le initially suggest op embed give motivated span regarded projective denote henceforth may manifold regarded restrict scalar structure space standard hilbert
rnn dropout annotation target hmm label character symbol emission obtain transform rnns likelihood divided factor character prior include lexical finite rnn compatible rnn treat hmm pseudo optical balancing optical language apply train corpus appear lm frequent rate evaluation vocabulary gram annotation contain gram language annotation rate evaluation dropout lm rnn lm dropout tend dropout activation lstm activation wider since drop strong activation make hidden activation check activation keep activation rnn dependency look learn dropout big dropout recurrent network layer reduce improved dropout lstm layer showed always improve rnn report evidence dropout behave similarly hyper much easy weight handwritten rnns program partly innovation th er ia de la paris france recurrent rnns cell currently improve recently previous show give convolutional rnn preserve handwritten database architecture recurrent offline text language processing module extract image word fed character sequence context reader review system hide hybrid hmms handle dependency sequence step hmms select carry rnn limitation recurrent rnn principle store representation event rnn inherently deep many layer burden vanishing reason practical application rnns rnns short lstm lstm carefully recurrent give superior wide fact rnn enhance lstm several currently meanwhile deep movement deep feed rnns dropout fully connect use rnn fully choice dropout made affect recurrent reduce rnn due performance technique generalization dropout also idea dropout design architecture dedicate system fed lstm scan indicate separately feed filter bias subsampling function convolutional element twice fully connect convolutional activation feed softmax layer softmax process temporal filter input enable layer lstm carefully design store period forget lstm layers network possibility context sequence explicitly architecture win entry competition recognition improve optical describe next connection rnn recurrent full share connection recurrent connection randomly remove neural let vector drop dm retain activation weight dropout value random dropout dropout connection connection construction dropout combine learn recurrent dropout separate output identical except stage deep architecture provide design appeal dropout drop connection design make sense drop convolutional layer share weight sample drop number input weight layer dropout sample bi directional lstm cell rnn dropout neural dropout slow higher recurrent improvement attribute keep recurrent dropout dropout seem favor relu nonlinearity however relu performance lstm cell database size isolate scale architecture rr full isolated assess performance character word compute distance recognize character simply isolate word rnn optical model log strategy employ dropout dropout dropout cl rr rr bold indicate database configuration rr c top dropout layer sample great small dropout suggest dropout size suffer overfitte recurrent train dropout set dataset improve hidden dropout help depict curve architecture rnn suffer overfitte validation dataset increase dropout end especially since training converge dropout unit possibility
subset datum recall close parameter strictly value space frequently map note fisher parameterization standard mean natural exponential hx x model include value exponential value equal core three lemma characterization exchangeability concrete lead small np jeffreys proper conditioning finite initial jeffreys condition length call e ml boundary lemma central parameter maximal essence varie result jeffreys equivalently invariant whole inside show assume exchangeability robustness family regular space interval immediately converse exchangeability relate exchangeability consider family exchangeable interior may integral integral exponential family converge n exchangeability lem continuity boundary space maximal standard laplace give converge natural exponential characterize early I element distribution I variable denote poisson mean value tail tail light contribution hence finite family family full shape family check check take xy immediate exchangeability gamma form gamma exchangeable exchangeability necessary exchangeability satisfie family space expansion geodesic necessary q two ready need exponential distribution occurrence linear transformation exchangeability location replace family family gamma exponential property natural exchangeable family arbitrary full exponential order generality indicate exponential family smooth determine family namely map exchangeable otherwise former pareto family remark poisson families exchangeable exchangeability maximal space family exchangeable condition exchangeability term look separately part family quadratic family variance negative desire note distribution gamma variance equation translation correspond translation exponential assume scale exponential family exponential family uniquely family non interior space admit sufficient go model exchangeable exchangeable multidimensional way model dimension mean matrix gamma family seem dimensional family without conditioning horizon hold arbitrary version allow kind exchangeability conditioning acknowledgement plus acknowledge nsf fellowship proof university berkeley business college university california berkeley technology study learn regular parametric jeffreys normalize coincide exchangeable knowing time horizon advance family answer one family exchangeability happen namely gb business college institute university technology california berkeley computer university california berkeley exchangeability loss jeffreys loss reveal forecaster assign reveal forecaster incur accumulate loss good expert reference minimize possible datum family distribution poisson geometric horizon result literature strategy subset integral infinite act start strategy segment regret finite unfortunately whenever drawback horizon involve possible drawback motivate researcher short assumption show act ahead ask look ahead game horizon coincide believe answer fundamental importance know strategy require solution bellman backward positive backward induction strategy analyze predict become bayesian jeffreys bayesian strategy jeffreys minimax show happen exchangeability however exponential relative generating give call natural space family proper family family extend outcome take distribution np define x never treat exponent arbitrary family however general statistic always express relative define exactly mild also right ensure q expression conditional distribution part simplify conditioning usually go generalization costly amount marginalization round furthermore horizon make eventually see discuss problematic strategy avoid jeffreys normalize short provide reasonably approximation point strategy regret step game
mean total number give set bernoulli infer version assume link belong one easily generalize link document consider topic form link generation content either treat mixture dirichlet exact avoid impose dirichlet paper easy equation literature generation treat network study like treat oppose one generate links treat integrate dirichlet prior fit popularity draw play although correct content topic depend appear text use mixture link define inner topic mixture nonlinear function logistic normal spirit topic binomial distribution eq correction large distribution way give topic number briefly several approach author extend relational unified treat generate treat appearance absence exponential attribute link graphical link generation expectation find maximum take link sum linear corpus simplicity version algorithm correct time appear log eq ignore denominator parameter directly balance content vs particular length contribution degree tend much balance normalize contribution vary study network topology document content section close topic overlap community trick change sum writing appearance due topic link topic equality give see detail time denote document appear nonzero step e document belong future iteration maintain simplify note come multilinear function link update product practice run high mixture infer integrated dirichlet associate recent subsampling approximate inference carry network poisson prior include update essence add word link know topic appendix posteriori map like model leave work infer membership label label infer link discrete label document topic let link twice normalize instance lin heuristic roughly discrete run test improve lin heuristic label take fix heuristic initialize variational hyperparameter execute set parameter infer normalize mutual information entropy respectively wish minimize rl sec kl kl algorithms correct topic link time run iteration corpus fast lin heuristic kl run mark bold maximize labeling return kl high good search try degree correction label kl heuristic giving increase decrease show varie link document solely pay intermediate broad showing carefully without algorithm implement grow linearly corpus prediction measure em subset link poisson link exist pair threshold agnostic threshold cost equivalently baseline auc correspond fold validation original partition subset link fold link train document execute task mixture assign zero impossible link degree assign assign measure solely pay attention maximize accuracy contrast content achieve maximize outperform horizontal represent achieve datum among specify interestingly show content important curve precision achieve figure outperform correct achieve contrast figure achieve link heavily low auc value outperform model latent semantic membership mathematical parameter scalable link achieve future I e presence absence word document base link text grateful mark david helpful z grant fa identifiability impose take topic determine lagrange multipli give link document distribute impose correction remain correction generate cause topic q however multiply side sum apply give lagrange multipli multiplying summing topic impose dirichlet think correspond prior correct contribution infer pairwise relation scientific consist collection word recommendation useful generate idea develop advantage topic overlap scalable maximization performing unsupervise exist art analyze minute document overall link popularity document outperform several variant addition overall e overall popularity set several scalability popularity test unsupervised consist thousand scientific outperform infer minute performance scalability modern contain pairwise form document link content document meaningful past mining learn datum relation link label physic take content topological help understand community stochastic assign label pair efficiently belief belong goal distribution describe assume node link node say fit classic topic word mixture word link infer absence link far innovation membership physics membership stochastic model treat community infer treat generate dirichlet distribution situation generate bernoulli treat number pair node derivative particularly
integer moment integral hence j power lagrange remainder eq since derivative power one include follow constant independent cause would integral hand side whole restrict integral small enough write proving euclidean change represent euclidean development extend exist relate exist relate st block obtain satisfie ensure kernel thus easily ergodicity backward q line inequality ergodicity q q approximation hence ab ba ab ab q artificial square bind axiom corollary example exercise notation problem proof cm universit perform maximum estimation model markov methodology score base artificial building sequential monte carlo dominate assume twice calculate component estimation build exact evaluate monte carlo identity numerous arise apply markov access derivative identity develop obtain beyond statistical require improve estimator state dynamic initialize parameter contribution fold artificial associate provide compare term optimal square second specific matrix compute smoother enjoy quantitative implementation follow introduce stochastically perturb random density relate equation score matrix artificial present norm f f f rely density likelihood continuously eq ensures go enough suppose constant asymptotic bound term compact detail consequence version show rescale proportional score establish theorem whereas theoretically finite rescale verified multivariate note make context obtain theorem approximation theorem model measurable dominate p yy dx artificial triplet alternative consist finite difference monte carlo sake simplicity consider difference monte mild additional chapter appendix point model independent sequential filter numerator positively measurable initial homogeneous markov whereas assume r log q latent approximation monte unfortunately carlo markov monte square could alternative extension describe extend allow introduce extend likelihood write copy introduce give covariance expectation far continuously associate dt assumption equivalent tu brevity adapt obtain artificial bayesian consider stochastically perturb model compact estimate whereas solve obtain exist q observe require use provide approximation approximation successive step sufficiently monte smoothing smoothing procedure generalize filter smoothing recursion motivation address fact enjoy lag enough lag practically bootstrap lag approximation lag rely yy x dx dx complex proof suppose particle partly constant
without simplify assume topic relevant topic emission estimate label notice mean emission irrelevant topic vector transition matrix hmm specific em keep unknown find point specifically collect economic economic remain irrelevant ten classified library database economic word size separately large wikipedia multinomial economic obtain supervise entry economic finance two task predict presence topic economics annotate economic document explain dataset system classify document contain text economic document processed separately discuss posterior require segment increment relevant occur notice use certain value constraint compute forward pass hmm subsequently threshold positive rate roc viterbi path single document irrelevant segment relevant occur viterbi path value false flexible hmms building concerned segment within document belong setup refer pattern document large pattern top k segment achieve segment relevant relax constraint retrieve worth segment hmms retrieval involve constraint segment tackle retrieval optimal hide path text associate evaluate document ground segment randomly perturb document explain appendix measure make popular evaluation object detection precisely topic similar object category natural box object evaluation challenge overlap area adopt segment overlap define ground segment intersection ground segment clearly close poor one correct exceed threshold illustrative correct get normalize respect average whole confidence interval repeat time create bootstrappe standard randomization involve segment interesting construct viterbi map path viterbi path contain priori topic top output segment retrieval segment evaluate viterbi recorded difference mean minus standard mean difference together segment corresponding path converge viterbi map path explain becomes present flexible hmms new make change new field child school medical school role integral education change service employ communication understand addition network education new service market communication understand connection demand model education middle box replace piece text segment economic show color belong classify correct ground middle box topic pdf star show classifier standard hmms highly analysis reporting summary hmms viterbi map allow technique augmentation apply posteriori segment useful tool genomic sequence target event type get insight cancer type instance allow system demonstrate retrieval future input construct alternative research could different hmm transition surprising transition infer zero could state hmms expect increase toolbox hmms mt trust innovation challenge award ref college mt uk fellowship ref mr statistical reporting hmms largely presentation probable find viterbi probable backward expand distribution programming call segment probability simulate segment contiguous possibly highlight explore exist fit use hide hmm signal finance fundamentally mixture mixing assumption rarely correspond generative process instance remain large central viterbi state computation summarize read treatment completeness generic hmm toolbox approach ever interest generic toolbox hmm motivation mechanism posterior inference limit report markov probable probability use summarize interest modification allow probable allow likely lead report alternative lead decision scientific conclusion describe method hmm incorporate constraint sequence transition show segment provide intuitive sequence allow diverse contain transition exist model case time illustrative highlight insight gain type sequence hide path observe state draw path independently emission hmm circle font b b b b b circle hmms structure allow efficient dynamic instance algorithm backward recursion implement expectation viterbi posteriori map generalization probable ml seek take benefit rely posterior efficiency hmm hmms algorithm novel exploration hmm efficiently introduce motivate interested number time occur global summation occurrence us marginal method approximate simulation ff bs insufficient pc relate task throughout refer segment hmm involve path representative path characterize additional representative article distribution problem programming introduce space auxiliary use augmentation allow apply allow provide elegant solution viterbi ff bs remain describe exploration consider constraint segment discuss text retrieval discussion future segment start define term present illustration section fix instance transition segment segment contiguous segment sum account segment result segment exclusive event decompose subset path exactly q conditional event segment inference find hide segment computation segment draw sample p task p augmentation auxiliary segment counter increment transition variable increasingly non conditioning sample accord delta otherwise refer count chain concept segment count I interpret delta increment hide path segment augment direct model hmm hmm augment latter constraint count final event realize add type reformulate decode hide find accord pp pass f event modification additionally maximize proof correctness statement special equivalently reformulate segment proceed normally second stop state visit segment problem count pair viterbi f ff implementation complexity take account count sep width font minimum b b circle mm mm decode decoding segment obtain map require overall viterbi algorithm apply max n n implemented initialize q equal message recursively auxiliary backtracking recursive take value configuration configuration check maximize whole require complete auxiliary backtrack x indexing operation computation viterbi joint forward pair call initial equal message recursion advantage recursion final computation recursion message take computation require normalization constant call message message useful em segment wish conditional ff bs base index forward final px n recursively go sample value message require time wish sample single forward pass complexity furthermore modification deal wish path k initially sample segment problem event use global guarantee least informative exclusive whole pc max compute pass augment hmm optimal must standard viterbi path segment probability graphical decode x n nm simulate datum fit three hmm estimate summary viterbi state color viterbi display contain path latter segmentation sequence principle circular two segment single segment latter final label precisely path illustrate path segment ff bs augment remark segment entirely hmm fit common simple set involve apply rich diverse application generalization provide hmm summary max states color show blue piece manner incorporate density evidence reflect segment amount count em write auxiliary subsequently derive ps n k contain learnable eq parameter auxiliary unconstrained simplify px algorithm current brevity compute message message initialize unity I compute store marginal pair wise marginal involve sum quantity hmms step local far deriving use six include segment constraint six panel blue dash optimize parameter notice show piece emission piece give observe reconstruction legend indicate incorporate segment parameter final obtain likelihood model toy toy toy toy piece form emission reconstruction learn use briefly outline gibbs bayesian previous segment conditionally marginally resort iteratively path augment hmm step simulate conditional unconstrained gibbs accept extension inference solve segment transition extract highly non markovian event hmm consist state hmms may associated type transition like term occurrence segment auxiliary hmm irreducible transition transition early segment care subset transition modify way think generate transition inclusion transition define dimensional initial interest hide event subsequently associate problem introduce count counting generate conditional compatible segment segment special simplify new clearly count segment programming segment inference problem equal one count ff bs hmms segment clear segment reduce hmm algorithm provide hmms decode generalize segment suppose separate row path segment hmm hmms nature subset state remain one might move nan return back nan extract next solve hmm group I jj sub path start state state intermediate path aim phase sequentially draw normal occur state deal situation occur increment counter counting hmm augment layer auxiliary state hmm triple work type associate count specify evidence final viterbi since require third modify I transition solve cycle state programming remain would like several counting display solid generalize segment inference recognize decode count chain programming also marginalization instance subset path simulate sample segment ff bs auxiliary different type problem segment generalize chain duration hmms consist modification hmm single choose randomly thought constraint hmm result hide semi ed hmms similarity methodology however scope approach case e exploratory ed counting variable semi markov segment hmms computation optimal segment contrast linear sequence massive bioinformatic point segment single segmentation segment theoretic hide define viterbi minimize loss penalty suppose developed subject segment theoretic tool hide segment word identification cancer genomic modelling genome classification dna copy cancer cancer lose
deal drift environment propose sensitive bagging boost evaluate uci set comprehensive batch propose counterpart comparison ensemble imbalance scope remainder review imbalance cost sensitive ensemble discussion extend non environment paper conclusion deal roughly approach former sensitivity consideration design misclassification minimize classification towards modification sensitivity calibration prune positive create conventional insensitive meta convert insensitive sensitive without modify one category insensitive incorporate due technique imbalance cost technique addition traditional sample iteration stream algorithm incremental naive binary discriminant analysis discriminant sophisticated algorithm literature regression machine besides learn bag boost also review bag cost version motivate sensitive ensemble framework ensemble base generalization ability classifier motivate averaging tend diversity present among different diversity bag boost adaboost bag construct date replacement usually diversity among introduce independently subset original constructing ensemble prediction majority alg hx mx bag adaboost focus adaboost construct series way misclassifie current correctly example equal misclassifie classified adaboost avoid update crucial designing boost online boosting since online directly paper implement resample fold boost bag technique second counterpart replacement distribution initialize train use mx n mx hx online algorithm inspire binomial poisson bag bag bag boost track misclassifie bag boost describe alg alg online ensemble framework hx mx initialize let train base hx mx sensitive cost bias bag learn turn insensitive cost resample briefly extension next majority one bag ensemble majority approach resample also varied alg show code gradually class learner special investigate h mx sample positive generate learner resample create synthetic alg generate positive learner diverse ratio replicate synthetic example vary bag iteration k train hx mx h k synthetic balanced eliminate vary resample roc sensitive counterpart boost cost consideration update particular weight positive misclassifie negative classified correctly cost contrary cost sensitive formula put negative one c learner distribution nc nd mx nc n factor hx alg kind treat way adaboost misclassifie adaboost modify treat initialize base calculate mn mx nc hx mx pre adaboost towards distinguish boost cost algorithm positive particular represent learner base alg class fix generate weight class learner unweighted generating example use initialize modify generate learner n mx mn mx n hx mx pt fix majority randomly remove sum majority respectively pt class majority create train mx mn mx mx hx mx ratio create synthetic modify distribution create synthetic example sampling fix synthetic example create ensemble cost sensitivity batch resample online distribution extension derive note cost sensitive I main generalizing straightforward straightforward ensemble boost update standard adaboost involve base boost implement learner key boost normalization ensemble property online reformulate cost bag straightforward large reweighte implementation online code alg c n learner mx pt cn pseudo code alg synthetic sample standard online boost synthetic positive positive proportional weight except base yx nh mx n hx mx cn cn prove insensitive bag converge counterpart theoretically converge batch counterpart base proportional converge converge learner batch mode follow cost counterpart performance roc auc obtain cross batch different focus learn approach answer whether counterpart one achieve performance online affect online observe batch experiment fair batch counterpart require well proportion learner batch online change cause change learner computation step classifier discriminant discriminant bayes nb learner parameter store online batch come specified learner parameter neighbor ratio ratio repository ratio select summarize include number number percentage pos ratio cr pt pos cr led digit difference batch bag speak divide auc large auc htb algorithms box auc batch ensemble observe close batch bag furthermore base learner good boost observe demonstrate bad ensemble batch counterpart performance algorithm ensemble appear consistency mode counterpart lda achieve performance nb yet consistently perform learner therefore indicate learner well performance auc bag boost example use repeat standard consistency get counterpart especially percentage large consistency bag improper base approximate training process base performance correspond explain consistency addition essential boost guess update way base train modify large therefore requirement boost learner stop ensemble learner algorithm stop since base error train small make online boost stop requirement violate example misclassifie decrease respectively improper weight base batch consistency show performance experimental fig obvious achieve base requirement learner stage online base learner online mode performance summary sensitive ensemble algorithms mode ensemble bag algorithms consistency performance bad update requirement non stream attract besides difficulty non successive therefore draw result difficult track evolution decision stationarity leverage concept drift learner select example base generative near neighbor select current concept discard perform base build adapt change environment base learner note learner approach imbalance framework propose environment purpose demonstrate flexibility deal stationarity possible research method quite straightforward main past use correspondingly p artificial method stationarity drift relevant uniformly distribute stream lie classified drift datum select example length gradually drift old new stream old repeat highly stream deviation ns stationary version online auc base modification overall especially preliminary encouraging indicate framework exist deal stationarity nb novel cost algorithm learn sensitive propose framework online theoretically sensitive perform counterpart bag term auc comparable performance combine environment artificial demonstrate algorithm accommodate even work environment imbalance evaluate scale natural science engineering research grant program health research sensitive
detail reference forward predictor small enter predictor inaccurate several remark mse model compare sum square value place age pt implicitly stop statistic associate multiple false discovery leave example lasso given enter lasso since leave predictor large include accommodate lasso center intercept term center intercept center step create vector originally assume center imagine cause assume careful product column center argument equally asymptotic sequential manner estimate predictor variable along difficult significance discuss briefly alternate derive alternate form statistic shrinkage helpful lasso sign simply shrink lasso representation alternate sign upon entry ss change occur inside knot moment lasso return plug squared testing variable previously forward term adaptively current shrinkage come distribution fix insight aside form next th denote sign entry problem sign concatenation key reduce sign sign provide active path satisfy cone ever check testing enter active column contain sign square coefficient sign appear knot actually somewhat occur square recover coefficient wherein third degree freedom fitting procedure fit freedom word sum adaptive degree freedom evaluate significance via something degree confirm predictor something quite remarkable model nonzero degree fit expectation happen phenomenon discussion statistic adaptively predictor degree freedom square estimate decrease degree amount relate quite lot work relate propose variable base split resample false positive general estimate big work employ splitting derive residual relaxed residual construct coefficient dimensional starting bias asymptotically marginally individual simultaneously way inferential group contrast deal directly adaptively select manuscript express confusion regard nan consider active variable particular model nan beyond nan consider nan must precisely describe may look fundamentally difference aforementioned traditional unconditional goal work also seem theory design test valid goal new base set start need set ultimately inferential statement author orthogonal predictor examine argument orthogonal extreme counterpart knot covariance far order statistic detail orthogonal orthogonality rewrite constant depend form soft thresholde abuse next special enter active lasso lasso path step covariance predictor enter active first I next lemma reveal remarkably time large v limiting hand throughout standard cdf equality eq exponential variate accord b b writing inequality complete tell covariance first third asymptotically importance fit rejection test step hence view correction nearly ideal nan exclude current truly inactive suppose true event truly nan idea behind argument nonzero enough hard rely truly add tend test depend inactive small study precise coefficient large compare namely q hold conditionally independently eq q weak convergence remain sample event hence essentially result position statistic knot knot moreover reduce explicitly knot form helpful knot sign decrease tuning trajectory join implicitly assume take error orthogonal begin study covariance wherein expression greatly sake mostly simplicity presentation column conservative enter path leave next step constant condition q equal since unit first enter leave active q inside drop nan significance enter case extreme approach orthogonal treat expression covariance nan process important entire jointly surely study interested variable event concern concern representation possible simplify drop notational convenience survival cdf examine ratio survival function lemma vanish hand converge define right hand induce bind predictor alone arrive assume subset grow ensure grow grows require imply speak assumption state variance contain correlate conditional subset disjoint subset position linearly simplification test step path less calculation integer contain assume assume estimate least knot indicator characterize step event comment sign change interval eq analogous slope broken datum enter varied number correlation correlation population block correlation simulate setup reasonably approximation truly inactive discard simulation truly inactive occur reasonably throughout active enter ccc c nd enter se rd enter c predictor enter estimate former use power would see simulation typically know analogy theory estimate square residual yx full nan numerator denominator asymptotically note unchanged fit value function meanwhile normality true enter path roughly equal difference know case show estimate distribution covariance examine predictor equal testing path column simulation quantile nearly idea square hope nan analogy dimensional rigorous simulation brevity far observe compare argue use validation necessarily anti conservative example work important issue estimate context covariance covariance real mention previously serious significance step result uci predictor size outcome forward chi forward predictor cd c ph residual density leave split test panel show set decrease rd predictor somewhat panel value well minimize stop six nucleotide treat target mutation model log base location drug tc examine behavior divide format figure covariance two confirm significance proposal support offer direction elastic parameter actually predictor entry active fixing predictor elastic net predictor elastic sign figure evaluate first predictor enter correlate seem case generalize cox function predictor measurement form via simply produce implicit freedom penalize estimate statistic mark predictor active penalize unlike gaussian piecewise algorithm enter knot path however approximate analogy asymptotic though rigorously investigate conjecture seem nan covariance knot look likelihood iteratively square cox penalize analogously active general likelihood compute proportional look true entirely zero right covariance step set versus predictor enter active truly predictor contain current convergence along limit distribution account nature usual chi square example light lar standardized predictor forward proceed fully predictor inactive predictor subsequent similarly coefficient lasso appropriately step intuition confirm look freedom covariance distribution reveal assess project natural consider value significance nan predictor exact reasonably behave agree extend along beyond global manuscript derive penalty nuclear component completion recent study inverse covariance interesting covariance test surprisingly forward regression greedy work confidence coefficient elastic net cox clear activity student hope broadly researcher joint effort set inferential use identity precisely take note side use plug second term respectively ss r ss j second follow side former sign inequality redundant rule also l complete proof lemma fix large enough dm dm term multiplying arbitrarily proof show depend imply tend result consider u ss j j r inequality use generality imply present fact really statement second verify jointly second come state inductive q c q independence v q variance use notice k statement k k must j proof drop simplicity notational j pt k drop inequality imply k side replace dm dm dm k dm joint fact k dm dm pt pt dm dm dm dm dm arbitrarily notational show brevity k k k k k desire conclusion assume k marginal limit pc sum require conservative start k implication x definition rewrite right ks sx x two inequality write generality proof theorem helpful discuss fact ratio unconditional conditioning conditional jointly variable independent none variance apply inductive hypothesis c q variance q conditioning complete concatenation solve row plugging produce leave side complete taylor series show v assume enough multiply side yield equal formula th hence denominator coefficient q numerator essentially numerator acknowledgement thank helpful natural science engineering support grant dms grant nsf grant dms grant predictor model model path test statistic statistic truly active current lasso result enter significant nan step place technical allow dimensional achieve active course significance nest chi adaptively drop stochastically much account lasso adaptive play adaptively introduction usual regression matrix coefficient intercept center column detail column order condition dedicate comprehensive category purpose short summary theoretical speak ensure favorable generative lasso major gap estimation real inferential construct exist estimate grow dedicated progress certainly many method resample splitting focus significance employ resample splitting instead use full simple proposal simplify relatively argument extreme theory treat general proof rely discrete section give data example survival conclude significance classic operate squared compute drop residual square compare know place often fix greedy unfortunately example empty enter step choose drop residual sum forward choose fix maximum chi square predictor would
part train convolutional sigmoid activation fc fc fc train effect fc fc network fc different rather example distortion feed display part effect distortion average feed adversarial remain effectiveness show purely network across hyperparameter emphasize adversarial regularization consist lipschitz demonstrate deep counter adversarial negative ability performance indeed adversarial negative indistinguishable adversarial negative extremely observe test adversarial negative thus issue address dark blue blue expressive recently art speech recognition learn paper report property combination high level various suggest unit semantic information high network fairly extent nature perturbation random different input powerful learning excellent neural achieve arbitrary consist computation automatically discover backpropagation supervise difficult counter intuitive deep concerned meaning unit previous unit maximally inspection individual distinguish useful indistinguishable neural coordinate generally seem contain semantic strong conclusion reach direction rich encoding vector representation stable rotation contain property concern stability network neural object recognition network robust perturbation object find perturbation possible prediction optimize perturb adversarial precise configuration normal variability arise neural network varied datum net adversarial still statistically surprisingly train suggest deep learn backpropagation characteristic blind whose obvious notation examine blind hide fc autoencoder imagenet al architecture learnable parameter regularization moreover dataset disjoint dataset vision rely interpretable color coordinate feature link reasoning analyze vision activation hide unit meaningful feature image activation aforementione formally inspection hold direction similarly interpretable semantic basis put variation factor first use convolutional neural mnist figure maximize direction similarity repeat network row unit combination although generate invariance explain rest neighbourhood almost distribution far unit little beyond certain representation network inspection lead instance model weakly analysis understand mapping represent train network speak unit nonlinear entropy softmax represent conditional distribution training present far argue stack output neural network encode word non non region pixel share nonetheless original implicit generalization proximity satisfy kind smoothness vision perturbation normally underlie neural network many simple able adversarial example perturbation correctly adversarial input already vision employ increase robustness inefficient way modeling training mining spirit hard mining instead negative round perform constructive way similar negative image label denote box minimize denote informally close obviously constrain bfgs minimize non case correctly predict correct incorrectly image distortion refer http image involve binary layer feature recognized recognize evidence quantitative mnist visually distinguish adversarial cross relatively example misclassifie number initial large misclassifie network adversarial universal overfitting might yielded train test pool adversarial subset continuously newly adversarial time dropout regularize decay alone subtle essential detail adversarial output alternate fashion maintaining update pool adversarial example layer separately addition accord adversarial layer compare manner consideration representative consistent convolutional model yet may behave well train bfgs first pixel weight add without fc fc high adversarial extreme layer consist autoencoder activation softmax filter tune average image pixel range adversarial give network fed misclassifie instance
group list evaluate dimension use multiply important value case remarkable lebesgue naive carlo evaluation result worse still remarkably well perform evaluation possible dominant mode resample integral accurate dimensionality mcmc integral naive carlo result volume even mode chain tool particularly sample motivate possible would particular costly inaccurate motivated present compute integral current appropriately eliminate error subsample mode integral resample accurate value original mcmc integral left volume naive elaborate size require dimensionality evaluation side require ever increase fortunately improvement automate stop core reach number base covariance paper currently classify award nsf constructive key point choice poor conversely accuracy simulated distribution follow select volume resample follow carlo trivial improvement application mixture accurate keyword computation method example propose model odd model normalization describe odd ratio count odd explain proposal almost never quadrature lead approximation property unimodal well multidimensional describe amenable motivated sample essence sample harmonic approximation remove dominate harmonic weakly partition perform likelihood application suggest extension explore intuitive focus variety integral monte carlo equation right side quadrature measure unity mcmc algorithm provide sample contribute quadrature motivate subset preserve successively hull vary preserve accurately volume volume addition sample new curse dimensional volume compare million low axis point laplace comparison value try implementation geometric mass eliminate outside self set within boundary requires adjust retain fix number therefore limit variance evaluation spatial increase width characteristic cell approximation inaccurate fraction recover limit high low bias lebesgue intuitive probability cell exponentially skew toward cell lebesgue imply volume lebesgue variant entire volume restriction laplace check lebesgue appropriately unimodal provide imagine widely volume center shape spherical retain tail original suggest initially dm distance sort coordinate entire min may iterate converge volume point significantly error several percent estimate appear appear change step prevent cell sample possibility original algorithm construct variance center unit hypercube center distribution x widely separate dimension two mcmc center hypercube shape multiple maxima
rewrite matrix source contribution source accounting instrumental ij pp distribute solution minima yield non factorization text mining audio source instance spectra mixture negative form distance account article come convergent descent negativity constructive allow share bss hope svd np hard numerous minima recover impose negativity preliminary advanced accurately art nmf aim management hoc regularization purpose introduce bss generalize analysis tackle negative bss extension additive contamination care constrain negativity unstable introduce negative constrain proximal propose compare allow large source mix setting illustrate perform also synthetic spectra minimization convex much update converge nmf design descent keep negative pointwise product non negative element division square write convenience convergent solve nmf monotone project quasi newton another alternating solve unconstrained project constraint eq easy decrease yield however lin project subroutine solve problem later method nmf approach sensitive state negativity sufficient actual mix ica independence source noise short presenting prove bss source greatly help signal content non sparsity express wave negativity arise naturally ms help formulate nmf enforce source sample enforce source active constrain similar provide website similar local minima without exactly enforce author type go perfectly therefore source may penalization case fidelity one admit analytic fast solver ratio time large none sparse explicitly contamination bss explore author enforce bss effective separate diversity mean separable sparsity source significant disjoint extend enforce bss deal source norm count nan enforce alternate unconstrained negativity thresholding source keep thresholding operator crucial use threshold begin decrease final inspire decrease motivation behind estimate source amplitude likely sensitive contamination way absolute source choose iteration active refine maintain continuity usually range trade denoise indeed sparse contaminated noise bss final leave source naive naive k square proxy converge stable couple deal fidelity term way sub next alternative exactly solve sub stable tackle project provide stable optimal problem beyond aforementioned naive alternatively exactly fix source quadratic term characteristic admit unique formulate fortunately calculus efficient type may locally fidelity proper proximal process forward backward solution follow define f onto solve I proximal take term skewed position thresholding operator operator induce replace rigorously operator propose sum projection orthogonal contamination snr name distortion reconstruction correct separation denoise little criterion advantage scale invariant next propose technique invariance know stand source reference source pair one experiment follow activation emphasize behavior final identical update role reference iteration multiplicative influence source numerous difficult prof paragraph figure figure evolution activation refinement sparsity possibility enhance representative figure coefficient modify sparse column simplify separation c c simulation source evolution iteration apply negativity constraint indeed solve neither converge lead lie soft thresholding figure apply suffer remains shift one large bias hard soft figure costly behavior activation suffer bias tend positive offset offset ground truth source figure source thresholded coefficient soft thresholding amplitude coefficient separation ill source noiseless noiseless uniformly amount term algorithm sparse accelerate publicly available account way paragraph straightforward sparsity ratio optimally grind truth source automatic way sparse accelerated truth run iteration good comparison include solve sparsity mixture bss line comparison reconstruction benchmark reconstruction activation figure activation contamination display loss facilitate visualization case sensitive benchmark experience one rate remain sparse accelerate ground available previous figure benchmark reconstruction noisy db low vary measurement measurement reconstruction redundancy help result far algorithm measurement benchmark provide sparse activation rate vary number perform initialize perfect separation initialize noise optimally initialize extremely within bss achieve performance separate compute source diversity greatly get source keep relatively however source structure source project concentrate thresholde significantly obtain reconstruction neither soft introduce source solution interact outperform take reconstruction low well range figure noiseless perform reasonably enforce get help reduce correctly source noise contribution quite robust reasonably large cost figure additionally figure well ill lack figure correlate mix together negativity nuclear realistic real spectra peak find spectral database spectra acquisition real spectra normalize spectra spectrum measurement e denoise become noise less behavior suitable indeed condition poor perform db peak sparse accelerated find bias vary since conditioning greatly large accelerate well article http tackle bss mixtures bss extension handle sparse
know distribution compare fit evaluate covariance regression observe thus mis final generate wishart truth generate x fit separate inverse matrix pool separate pool prior group generate three follow covariance plot mean check deviation weak get group size generate separate pool covariance estimate especially two variance correlation one separate covariance mean strong surprisingly case mis scenario regression moderate reliable estimate study reasonable motivated study association among health article study quantitative association among major health categorical predictor estimate covariance evaluation aic conjunction goodness plot discover four health outcome higher develop article classify population continuous covariance covariance method drive try normally non possibility semi propose though motivation arise health outcome jointly diabetes play role quantitative association outcome propose apply ccccc gender f black american high school college ht ht ht ht serious decrease quality focus health association problem characteristic joint covariance statistically quantitative association health predictor methodology discuss aic conjunction predictive goodness fit identify sub risk develop health problem regression serious build body problem include disease decrease disease detect treat slow disease failure quantitative define less assume stable ratio high presence direct disease population exist first estimating find across old focus occurrence study diabetes disease show horizontal population depend association different population association ignore economic association disease examine patient fit diabetes economic status education generalize evaluate heterogeneity health relate heterogeneity statistical covariance beneficial proper confidence interval test predictive ignore estimate third involve health evaluate heterogeneity scientific interest indicate quantitative characteristic figure among health measurement also vary investigate association help estimate understand health lead efficient sub pressure bp within gender education dot green estimate four health pressure log education green line covariance mostly develop context longitudinal study generalize covariance temporal nature conditionally utilize outside longitudinal explanatory comprehensive review covariance recently directly explanatory focus continuous regression accommodate categorical health categorical association among across us relationship health national health survey methodology selection present misspecification national health national survey health child united participants home medical center conduct health disease contain participant economic physical well activity behavioral major disease population increment recently view status three diabetes risk literature pressure bp level health study characteristic gender education proxy economic status participant survey participant due skewness health bp analyze summarize mean transform health health group variance correlation health exploratory indicate risk heterogeneous allow possibility vary ht cccc white american non th high college aa college goal describe heterogeneity cross gender age education sample group define size two flexibility gain allow desire express covariance estimate mainly focus predictor one accommodate type categorical utilize simultaneously mean categorical variable allow set procedure selection longitudinal several selection bic selection component identifiability method aic rank interaction covariance simultaneous require implement complicated procedure try search propose selection procedure try parsimonious obvious fit detail separate due fact normality mis select simple predictor model simple fit lack add acceptable serious acceptable repeat high suggest outline constant aic aic gender well way interaction gender age gender gender education age fix explanatory represent fit sure behind population diagnostic heterogeneity two white enough cross matrix argument covariance way gender age education pool wishart sample possible population homogeneou pool sample estimate describe discrepancy represent heterogeneity sample p gender education compare lie extreme distribution represent goodness first mean predictor together six gender add gender goodness generally goodness rank heterogeneity cross view relatively fit ht present equation correlation interval classify colored dark representation covariance color plot group age old summarize select finding education level high american year old pressure american year moderate level education high american age old level year education great variability year old great variability black year older greatest high however group correlation consist
vector I notice procedure require I error equation good look standard present misspecification break use parametric collection logistic feedback odd plot axis feedback odd simulation old prediction pair prediction noise center distribute dataset perform sum spline degree knot spread evenly jump odd e different feedback intercept intercept average fit feedback set heavily plot spike end accurately example feedback bar discuss possible work feedback world system try fix feedback automatically context feedback I lf set integral extensive problem us solution focus detect raw prediction optimize deterministic spline proof fully artificial priori term follow immediately ordinary treat proof regression theoretical independent google com live production may feature prediction feedback loop occur predict predict feedback causal detect feedback real conduct pilot methodology system currently search engine live production loop concern usually tune influence follow engine want simple classifier predict mean search people read news historical click rate problem engine start query occur feedback occur search page directly run feed priori way feedback source propose detect feedback loop source live measure future prediction artificial understand enable feedback prediction feedback dependence fit slope reason work construction turn general jump feedback discover relationship detect distinction mean causal relationship prediction randomize fully question change prediction frame potential treatment formalism often causal causal live predictive internet order need precise notion contribution provide outcome section model jump present discuss section mathematic perturbation conduct pilot predictive part period example take understand reasoning outcome distinguish prediction publish prediction never make chance affect environment define actually make would environment difference outcome feedback fy fy feedback plus term function relationship influence trend fluctuation resemble randomized ideally hope integrate system turn concern system add artificial time prediction I independent everything else put randomized causal effect way influence prediction practice operation act continuous piece treatment begin analysis discuss suppose intercept purpose fix historical system start prediction concern underlie goal feedback write counterpart without relationship perturb scale simple q want fit artificial noise reality get conditioning hand I treat parametric equation approach constrain solution expansion transform ordinary term variance square diagonal obtain huber fitting easy estimate roughly reduce simplified form scale roughly noise noise convenience summarize time noise I live non ty learn square show library make prediction difference intercept identifiable rather average include intercept add noise depend shape cost cost large ever add idea draw motivation methodology pilot historical feedback system concern induce bad detect integration logistic historical rule odd feature feedback assumption increase odd next half simulation generate period feedback additive knot evenly spread log odd space fairly reason believe may jump priori result detect shape bar estimate feedback parametric bootstrap accurately detect feedback scale world know detect feedback engineering view practical feedback feedback classifier system plausible change provide way feedback affect change cause wide range paper randomization real system add noise system put experimental feedback causal artificial noise thus feedback without feedback propagate
know date obey direct pp ready concern priori unclear correctly behavior hand merely definite answer convenient residual make dependence random statement order furthermore give theorem concrete imagine carlo derive lower established ratio practically tight would lower bind increase sharp improved bound plug right know upper equal formalize conversely justification use sophisticated concept tool matrix use elementary instance hold integrate limitation tight ultimately identify reduction monotonicity roughly say residual great hence apply turn property bad residual form extent sub importantly admit low section show affect spectrum value nd kn equal variable eq take setup recover yield corollary improved examine performance residual perform sample error proxy bad behavior separately set line dash previous resp blue line resp line top ratio hold approximation error reveal tight multiplicative factor bind concentration measure remarkably rule practical purpose deterministic practical b sense variability triple decrease concentration measure find approximate svd step approximation desire compute since form order fairly minimal reference effective pass access please therein decomposition qr decomposition search computation follow classical follow course new sure classical decomposition reader rapidly decay spectrum quite decay proxy poor rapidly see provably compare theorem similar block desire independent unit column form decay decay benefit sort state therefore trick corollary say error bad interesting upper bound see input sense mainly result throughout identity write save hence nonnegative similarly semidefinite say monotone non monotonicity dimension two fold generality follow eq section suffice input object bound mixed material begin spherical symmetry hence u fashion induction column need property prove monotonicity special positive g proof proof nz z z eq inequality follow detail resp resp theorem lemma let dimensional q row kx kx allow orthogonal norm reach give I orthonormal column svd decomposition orthonormal obeys prove also conversely index therefore column projection onto plane draw say pg g establishes establish together chi expectation wishart identity expectation develop characterize study algebra turn minimize formalize minimize supremum choice fix increase expect onto orthonormal sample choose property another rank present generalize inferior since algorithm multiply next expectation pseudo low set wishart entry position
table mae recommendation number neighbor well mae recommendation besides use mae metric also exploit q belief evaluation metric near object boolean build ht another thing interesting examine impact scale suitable recommender method investigate behaviour recommender mae mae mae require profile nmf investigate filter recommendation case reveal algorithm usage like assessment recommender basic like thank remark implicit author supervision new concept data svd nmf mae rating quality mae scale recommender fact recommender technique mf among mention modification exist negative nmf boolean seem recommender analysis use recommendation aforementione especially mae useful keep drop insufficient experimentally formal context recommender reliable term average error mae precision review exist mf recommender approach recommender close user methodology mf recommender validation mae last describe decomposition rectangular system vector user factor value disadvantage number last interpret allow new svd movie rating c c user rating decomposition eq great confirm nmf product negative widely area vector discover molecular etc decomposition product negative branch mathematic concept mathematical algebraic lattice object attribute object attribute formal mapping order union concept formal extent evident extent formal set formal context boolean factorization define denote conjunction binary attribute assume user clearly formal decomposition prove every optimality binary matrix concept base computation work much find factor number boolean decomposition matrix boolean product learn compute recommendation evaluate factorize base find original user collaborative formula calculate formation recommendation recommendation system estimate mae recall try item item rate user assign movie order recommend movie collaborative recommender movie movie movie history recommendation base make prediction base previously rate unknown rating rating usually user rate item normalize select item rate similarity since usually dramatically recommendation rating calculate demand similarity apply recommender simply input matrix datum boolean know interpretation several variant compare mae else rate else recommendation factorization rating contain user rating
image fmri activity brain imaging change brain trait treatment difference response deep understand simultaneous activity begin component ica extensively try decompose basically way participant fmri ica dataset participant ica ica ica cluster spatial extract necessary different participant fmri research ica create component ica size produce number spatial map come participant small mostly surprisingly well inherently contain high take characteristic analyze perform cluster suitable dimensionality give angle provide visualization explore ica logarithm reduction normalization distance meaningful connection create feature retain dataset row represent voxel part characterize preserve metric diffusion map matrix element determined try sum affinity calculate create get preserve decomposition u n stay embed decay coordinate point express diffusion take result spectral reveal sense reduction simplify cluster leave actual dimension relative precision thus first eigenvector separate cut internal inside manner participant music map brain activity reduction compare music piece experiment expectation brain music stimulus music information retrieval preprocesse ica temporal long participant number voxel across participant tb analyze methodology dimensionality region highlight straight vertical line perform space cluster agglomerative result result even spatial map mark symbol divide line along horizontal leave detect dense mark contain spectral clustering compare diffusion map correct create effect metric htb dendrogram agglomerative dimensionality separation visible see figure show evident cluster htb map dark area highlight voxel slice correspond low figure activity basis cluster internal matrix individual among member figure distance htb theoretically sound cluster component fmri imaging diffusion reduce dimensionality enable brain advantage map see visualization compact two propose methodology separate spectral agglomerative separation make solve interpretation useful create brain activity cluster automated diffusion sample
million class maintain collect perfect accuracy analysis far expert analyse infeasible classifier maintain miss instance costly reduce positive key instance greatly comprise primarily various interference change change short data stream completely drift perform within reasonable throughput stream datum must involve signal survey tool candidate sophisticated via ultimately discovery version tool lead discovery perceptron automate capable recall apply candidate imbalance either assign different thereby balance resample resample way al improve na bayes et weight classify stream find stream supervise paradigm apply set costly include concept stream base semi classify stream amongst stream tree incremental permit data stream dynamically operate output also conventional hoeffding attribute access determine user intend look algorithm framework implementation stream static bayes reveal perform stream take leave current numerical stream four see resolution universe continuous attribute summary label human remainder constrain actual incorrectly label survey label include segmentation gamma repository distinct level performance scenario assess fold validation fold test streaming vs pre order imbalance label test stream test test ten balance multiple procedure take line testing configuration table exception different pre static classifier allow learner sample execute assume instance learner label count positive negative check label label file prediction statistic dt nb pre label versus static train validation train l pre label stream expense across test consistently imbalance increase side effect effect see even stream label toward drop rate test pre rate slowly label appear stream train recall though classifier return train stream difference table non build stream stream zero remain classifier initially outperform train particularly increase pre classification actually rate appear pre classifier increase pre delay class class imbalance summary stream learner test accuracy almost majority preference rate rarely predict line short rate effect short l balance cm l balance c classifying stream capability heavily stream imbalance skew leave toward class improve find taking initially imbalance make inherently short suggest maintain return could modify accommodate increase recall expand investigation stream hoeffding analyse describe remove redundant may bank centre scientific obtaining mm mm mm mm pc currently significant amongst pose preliminary specification suggest final design incorporate rate tb crucial information store processing survey signal feasibility heavily stream use currently framework result learner exhibit learner definite potential stream stream ever volume modern rapidly infeasible store stream year considerable effort towards lead stream learner however stream balance completely label partially balance heavily skew face effectiveness investigation present hoeffding bind increasingly stream motivated community seek draw attention scenario ever diverse capture interference nearly separate likely spurious become currently international begin decade survey operation tb magnitude previous survey hardware store financial restriction survey low
cn average surface air temperature take national environmental national research grid network series field well couple singular term relationship field appear cn couple cn detail refer consistency cross present covariance cross matrix use tb eigen represent square operation length series aim term component reduction combination explain th eigenvector th component fig sort associate projection bar north decompose principal th e pc need dimensionality set tb singular decomposition svd follow rank tb xy couple covariance large bar north analysis spatially couple field couple find equation decomposition orthonormal set orthonormal call negative covariance pair explain large account couple already explain field expand couple eq expansion project complex structure typical internet wide science road grid engineering biology popular recently several field science complex explicitly transform representation refer derive system obvious grid functional network biology functional brain form recurrence study field present introduce distinct collection weather association political novel toolbox dynamic multiple link loop consist spatially simulation measurement predefine link strong significant statistical two cycle put pairwise mutual transfer event synchronization cn give denote exclude loop globally pair pair individually prescribe significance use study measure statistical undirected measure event synchronization couple field respectively quantification restrict pearson lag measure association field associate q prescribe delta pearson correlation univariate study frequently measure indicate particularly correlate ni display htbp local percentage variance first yield note display panel path centrality theory reveal short closeness measure path short geodesic small number link pass cn bc shortest pair define path cc energy air dynamic study signature la ni depth interpretation cn yy internal information adapt option cn link multiple node construct air temperature pressure relative region variability contrast statistical dependency region field result adjacency analogously q layer subgraph set belong describe internal correlation matrix dependency couple study variability north via ni xy yx couple pattern correlation map north construct couple threshold link internal quantify interact neighbor analogously degree net region location versa region cross major area field north analogously net generalization measure network cross closeness distance short path bc relative node define eq analogous expression interpretation couple cn cn notable lead analogous leading couple couple cn relationship eigen illustrate univariate couple couple cn temporal discuss eigen analysis lag information correlation decompose explain fraction cn approximation degree compare fig panel follow contribution cn lead positively first couple cn cross closeness display network threshold accord axis line fig set detect intermediate connect link cn field maximum pattern correspondence result inform information linear statistical interest degree mostly small degree derive likewise expand denote relationship area area directly decompose couple cross degree approximation show positively field still load couple appear super degree pair couple fig pattern pattern cross approximated expression weight related area pattern principal describe evolution associate project analogously couple temporal spatial directly field evolve temporal window slide similar strategy could cn analysis eigen network reason fold analysis evolve spatial subset window second fig evolve cn location hence derive evolve explicitly dependent standard slide window mode relationship yield information complementary derive study single pattern air data potential cn similarity derive mathematically specifically active strong correlation super analysis loading pattern example fig spatial lead cn degree reveal pattern high cn prominent preserve super bivariate analysis north strongly loading couple cn cross fig network topology indicate driven degree variability certain reveal turn fig correlation dynamic one variability covariance structure determinant link frequently study theory order base path closeness fig argue give insight speed propagation prefer pathway perturbation within study cn conceptually bc obtain systematically degree field correlation cc bc leading pattern considerably pattern bc fields degree explain view small pressure temperature randomness construct network correlation centrality closeness arise word spatially incoherent rise notable induce correlation centrality eigenvalue separate lead see bc air temperature datum bar north rule surface air temperature another frequently study set complementary aspect explain separate field strong spatial discussion property reflect lead firstly degree resemble case due weak consistently display fig b field partly resemble degree fig lead two bc display distinct feature f appear coincide loading along west loading third surface air temperature link panel small wave field signature temperature anomaly strong surface structure east boundary west note bc north appear logarithmic change large bc pearson recent analytical vertical interaction bc vertical air wind surfaces bc south highlight american propagation event may couple covariance mode display long could methodology research suggestion move discuss cn property eqs couple cn plug ij particularly measure fig illustrate complex easily understand view consider hard interpret insight correlation structure beyond complement study statistical analysis valuable standard tuning employ advanced study dynamical deep insight physical well cn cn analysis probabilistic synchronization naturally study analysis synchronization auto indirect reconstruct sub turn enable reconstruct structure sub interaction conceptual arise summary main article recently cn standard usually similarity cross correlation orthogonal couple pattern frequently use cn degree
rate merely tight specific statistic conclude study method work rough fine change assume mean deviation theoretical three factor confirm experimentally method handle performance integral kind numerical procedure detection rapid live sequentially characteristic soon subject quality control economic g e rule effect observe throughout entire period surveillance distribute common serial regard notation htb appropriately choose delay correspond problem three quasi multi survey g focus change determine performance procedure sr special sr analogy performance cyclic fair sr sr comparative sr carry far recently operate large cumulative move question method agree aspect question ad previous develop robust method study pre change stop first e alarm principal linear certain exploit martingale statistic accuracy tight change certain martingale method rather even rough partition interval respect change contrast rest state case study sr chart assess procedure knowledge comment extend well detection procedure section conclusion let generic expectation respectively assume change observation length alarm risk introduce exhaustive false entire lr change change joint assume kk lr observation play tt distribution cdf respectively lr mutually absolutely continuous dp rely later martingale procedure identity likelihood alternatively probability e sr define aa alarm sr recursion remark detection unlikely detection result asymptotically never terminate detection follow martingale see aa precisely establish computed purpose describe sr procedure regard sr fix chart give sr practically put gain chart sr procedure martingale result establish limit comment seem slope constant proportional otherwise negative value definition illustrate htb rr ar rr consequence method though slope linearly proportional however close rr almost direct consequence mention procedure operate develop numerical run correspond distribution e pmf stopping function interest moment e alarm propose build begin instantaneous th simplicity absolutely effort non handle tt transition identity dp tt x first importantly connection robustness equation fix notational brevity markovian worth long infer low right procedure early x markovian establish recurrence sequence recurrence sufficiently inside interval discussion recurrence rewrite equivalently denote e operator derive q recall last justify strictly characteristic stop equation moment stop consider stop time hence need e projection project eq worth note accurate point residual iterate question apparent basis error see factor interpolation big generic upper often far consequently potential possible obtain desire prove fan extend either sufficiently strict suffice interpolation readily piecewise end interval nz j align justify tight magnitude e square drastically offset denominator experimentally linearity happen accurate require method compute corresponding integral integral however identity integral recall subsection introduce later zero exhibit rate quadratic interpolation constant constant substantial chart sr thousand reasonable procedure first actual change next see one obtain regardless quickly formula necessary express everything employ perform analysis broad contrast range alarm moderate alarm proportional false alarm accuracy interpolation form chebyshev root chebyshev polynomial kind possible specifically non joint shift chebyshev shift left node interval evaluate method accuracy even range false alarm pn report various alarm failure expect quadratic importantly almost false interpolation equation almost respect work confirm hand method see converge slow require thousand high lr lr nan nan nan nan nan nan nan nan nan nan nan nan lr lr rate lr lr rate lr lr lr lr lr lr lr r lr lr lr lr lr lr lr compute specifically consider stop r present geometric would another see change e function sr asymptotically reciprocal alarm cf close large figure limit generally alarm distribution substantial actual limiting yet close almost indistinguishable close geometric problem correspond almost considerably geometric convergence change alarm moderate close p kk propose include trick usual alarm alternative measure false detection alarm successive observation index define k assume properly pick denote mt evaluate sr show generalize generic markovian admit recursion sx broad member right integral suffice dp
assimilation datum vertical spatial aggregation calibration observation spatial avoid high resolution interesting lose aggregate calibration decrease uncertainty depend importance several uncertainty error calibrate aggregation lead considerable uncertainty model uncertainty projection base process flexible computer attractive calibration cf unfortunately likelihood model become prohibitive dimensional expensive dimensional make moderately large location complete knowledge calibration approach challenge thousand point incomplete incomplete calibration due analyze set computer manuscript thousand observation large freedom effect aggregation example calibration aggregate enable investigate interaction error infer example process computational remainder organize follow calibration challenge calibration principal representation provide section goal build vertical potential parameter control output scale convert feedback parameter spline run relate sampling exclude beyond grid location representation average observational grid nearby point see material observational relatively location adjust potential adjust pressure potential temperature field pressure output temporal location convert observational depth depth calibration stage I parameter calibration computer output computer model relate observational computer observational allow systematic stage combine stage inferential solely observational separate calibration provide model domain parameter multidimensional calibration three parameter location computationally expensive obtain denote computer output tn nn tn z location one objective parameter outline covariate spatial e depth parameter contain application mean see detail fit define computer output set therefore predictive computer quantification interpolation model observational identically observational model model n regard provide cf pose without information dimensional considerable challenge expensive computation describe dimensionality observational data prohibitive na implementation explain supplementary output operation numerical newton infeasible develop spatial increase representation trade crucial basis feasible considerable drawback high limitation roughly three category representation method convolution krige process reduce formula relatively approximation field krige sparsity thereby computation computer recent readily applicable remain prohibitive knot computational principal component consider knot great principal need second may wavelet transformation point grid miss value miss addition difficulty computationally specification issue require translate aforementioned represent field use separately basis reduce low construct component uncorrelated principal principal require evaluation size efficient automate manner principal transformation broad wavelet transformation calibration develop low output replication storing model output scale p principal decide choose number proportion variation p basis r orthogonal us construction wavelet random exponential function partial range parameter leave percent configuration square exponential alternative exponential covariance supplementary field precisely arbitrary zero small scientific inference informative vary around specification find transpose r metropolis mcmc evaluation aggregation level depth computed depth aggregation determined explain try explain variation space conduct percent precisely depth reproduce give run indicate indistinguishable cross include root standardized prediction supplementary material graphical show principal reasonably calibration stage make integrate field density unnecessary highly nonlinear thus decide integrate observational receive prior specification try determine stage range depth surface km depth result give affect location principal component ensure variability explain identical calibration run carefully check summary g density entire run reliable challenge evaluate function deal datum current address effort discuss accuracy j j matrix parameter process multiplication make prohibitive stage case illustrate routine run system intel ghz parallelization note optimistic probably take indicated cost time parallelization experimental computing supplementary aggregation conduct observational datum choose pattern truth default mode cf compute residual observational output location obtain realization brevity challenge work even pseudo residual truth observational aggregate pseudo temperature respect water volume result simulate case specification drastically simulate calibration base datum depth depth represent density solid solid line solid vertical line synthetic respectively depth pattern line dash hyperparameter respectively experiment effect spatial average particular dimension illustrative choose base point repeat calibration material indicate location introduce desirable pattern drastically specification observational uncertainty prior assume result mean indicate pattern much predictive aggregated aggregation deep reduce uncertainty specification leave solid black dash red interval system intermediate spatial calibration calibration spatial use utilize spatial uncertainty robust specification aggregate valuable reduce deep several real calibration similar carry calibration use pattern pattern demonstrate approach deal orthogonality principal keep cost location parameter extend multiple principal density pca material density challenge devise approach apply science consist million common principle apply management matrix svd area research efficient cf make uncertainty effect aggregation projection make input economic uncertainty model observational virtue policy worth context component variation lead carry study number component suggest problematic theoretical valuable simplify separability surface depth separability combine geodesic euclidean remain research future reliably scientific conclusion ignore compute fully effect discrepancy variability conclusion calibration projection acknowledgment grateful anonymous associate detailed comment suggestion greatly manuscript solely code calibration manuscript design study publish process observational pt e nsf agreement uncertain characterize reduce
select example hash hash easy subsampling subsection signal signal hash different label element construction element hash keep along shift detection support estimation check similar assign bipartite node correspond bin different zero label assign regime signal position select independent regular select independently explain spectral denote assume select number individually uniformly bin see bin completely position zero spectral association completely apply bp still edge polynomial bipartite decode martingale decoder formulate concept integer variable hash notice check correspond representation lattice label interpretation element move axis hash plane span axis lattice tuple lattice dimensional lattice similarly definition obviously whose set lattice lattice proposition dimensional consist induction actually vertex cube imply continue let fix positive integer hash decoder similar number take without along subset element axis restrict along consider generate variable select coordinate variable imply decoder number th coordinate construction q way hash explain cover construction reduce hash construction regime overlap binary hash portion clear construct start give value terminology pick shift hash bin hash non zero bin decode value performance decoder empirically evaluate variety programming success trial comparison algorithm hadamard hashing pick hash random trial size run time estimate success hashing scheme albeit guarantee observe practice give value repeat experiment instead observer least deterministic per bin hash appeal change show succeed go compare straightforward implementation perform identify conventional large respectively function runtime computation multiplication algorithm nothing circular bit shift implement size transpose overall nonetheless unchanged product look give product unchanged finally subsample signal product column decode compute hadamard domain domain correctly hadamard high go also evaluate hadamard find considerable speed obtain length assumption statement apparent algorithm machine problematic robust taking sum let identity therefore linearity product obtain operator hence permutation exist complete easy otherwise fall fulfil unlikely unless contain element solution column leave top row exhaustive matrix support nan sub removing contain identity one vector symmetric thus normalize vector eigen result eigen result remove case notice hadamard submatrix matrix full arrange except row opposite sign combination row bin proof size pick object random denote markov choose k kk moreover bernoulli equal function consider obtain chebyshev obviously converge result infinity average hash check node singleton neighbor proceed event denote fail select decoder proof b algorithm hadamard signal sub support non component signal tend base property carefully subsample suitable domain treat code spectral formulate propagation bp channel tool code theory algorithm regime hadamard sparse decoder hadamard process varied compression user transmission compress mathematics fourier fourier dft allow fast recent dft signal particular well extend dft dft recover zero element hadamard later play development show subsampling allow induce design transform transform component create interference idea sparse pattern analyze result zero zero fast hadamard transform hash explicitly iterative hash output transform recover total integer letter domain capital letter signal represent denote expansion least assign denote binary field space vector inner arithmetic call sparsity signal index subset domain computational decoding correctly probability distinguish less know work complexity might know sparsity index sufficient automatically sparse success every success low index signal eq terminology frequency subsection devote sake completeness provide shift let q property subtle partially spectrum possible permutation permutation hadamard permutation identity find permutation equivalently column permutation non permutation property dimension subsample th subsample element eq subsampling label last group vertex imply sum spectral component along visually replace binary instead spectral account sign pattern zero basic hash main spectral hash hashing index hadamard operator spectrum hadamard transform pick bin require operation number bin hash spectrum nice hash decoding recover estimate let obtain ratio non coefficient value whether one component bin result precisely depend inner spectral hash identify position column value unique bin node non zero spectral original colored index brief overview fast hadamard explain decoder recover spectral sub give domain terminology row hadamard vector imply sum interpretation code bipartite picture one spectral decode dense bipartite check domain constraint correspond implicitly know position explain code bipartite collection bipartite suitable subsampling time different hash operation picture hashing output complexity iterative spectral induce hashing check terminology use operation check namely satisfied edge create zero notice decrease singleton proceed singleton succeed great call identify spectral code input signal count array du bl lc du ki u u diagram algorithm last subject regime depend sparsity hash bb bin imply hash hash hash hash node random bipartite node uniformly another label denote similarly nod construction bipartite degree node degree might ball ball contain bin bin ball terminology theory walk node edge consecutive neighborhood subgraph consist walk say direct neighborhood sparse regime hash different hash operation bin label hash uniformly easy I bit moreover assume denote hash check node label connect check label denote position uniformly specific another variable neighbor variable imply bipartite variable belong bipartite spectral improve reduce decoder explain bipartite hash asymptotically decoder upper ensemble asymptotically keep decoder see decoder improve bin imply decoder strictly consider graph rs hash connection node compatible bin asymptotically decoder ball bin explain support random non hash bin check node ensemble degree converge poisson variable ensemble show check let check degree bipartite degree polynomial instead write bipartite graph infinity connect node symmetry hash construction hash check hash proposition check node almost surely dominate converge integer output decoder equation decoder notice increase complexity hash increase computational apply singleton check contain component decoder remove continue singleton variable remain fail completely analyze singleton essence system bipartite time limit uniformly well concentrated analyze decoder code briefly node polynomial channel code bit module summation summation message systematic channel message bit bit independently bit pass perfectly bit redundant information receive check induce remove correspond one bipartite easy decoder recover bit bit receive perfectly bit independently word
existence principal frobenius modulus unity unity exist matrix eq span principal construction straightforward computation involve immediate tv v cf augment non eqs satisfie equal know walk unobserved matrix therefore walk terminate vertex prove harmonic detection network application especially track involve arrival time vertex graph may time set converse irreducible space graph illustrate propagation subsection augment create edge particular vertex across time denote unity probability state model continuous equation poisson define observe probability stochastic call propagation temporal vertex full space application develop propagation vertex determine vertex adjacency analysis graph position vertex determine pt u matrix discretize correspond kt form q cf nonzero practical block vertex kernel irreducible spatial space time equation discretize priori spatial graph asymmetric space adjacency irreducible imply graph irreducible observation specific frobenius theorem pose detection spatial unity also network activity seek whose activity correlate accord priori probability poisson use interaction arrive unity stochastic activity propagation space unweighted degree generalize spatial priori replace priori model temporal kernel time involve irrelevant uncertain ignore delay protocol essentially site spatio temporal situation discretized connection arrive discretize temporal replace equal time reference document vertex vertex vice vertex equal space clique detection treat binary decide maximize pd false alarm ensure detection indicate log achieve algorithm see previous graph transition treat yield pearson optimality likelihood cover optimum laplacian comparison pearson several laplacian optimum definition pearson maximize detection hypothesis measurement connectivity temporal foreground interaction background blockmodel trial moderately foreground network show roc exploit foreground moderate connectivity temporal outperform poorly foreground improve foreground connectivity expectation clique likelihood possess propagation propagation necessarily show propagation blockmodel spectral roc see concave pd roc convex except near concavity cause blockmodel monte carlo binomial variance real enyi realistic explore foreground space display topological characteristic world structure community one trait blockmodel rough capture degree reality belong law mat capture capture law characteristic world parameterize hybrid membership blockmodel combines network depict diagram blockmodel aggregate sparsity law blockmodel structure interaction individual order community individual fraction individual distinct give product term indicator blockmodel second per expected degree mix membership blockmodel determine strength value indicate community interact mixed specify among il expect dirichlet determined draw belong vertex degree determine exponent fix poisson within community dependent integer edge foreground detection temporal objective realistic foreground operate realistic foreground foreground network vary activity use discovery member foreground member background foreground background level foreground entire foreground actor characterize distinct membership background intend represent business home parameter vary foreground design detection perform rely upon exist realistic bayesian detection detection walk walk propagation definition theoretical follow immediately exact pearson optimality appeal direct benefit superior network propagation interpret walk equivalently harmonic interior hybrid heterogeneous temporal compare well know method examine realistic embed new membership blockmodel membership detection graph vary activity foreground vary example acknowledge consistently constructive comment wolfe walk rgb rgb address minus minus center skip center skip mit mit edu bernstein mit edu http networks http www network detection derive walk observation activity observation link harmonic prove introduce utilize spatio specific space lead significant demonstrate hybrid blockmodel introduce detection likelihood community theory theory walk dynamic method harmonic centrality detection analytic mesh manifold community anomaly analyze paper detect small derive vertex walk optimum pearson maximize space graph diffusion space detector activity detector framework walk original unified detection stochastic blockmodel hybrid mixed blockmodel simulate algebraic framework base detection subgraph framework know graph analytic pose context sensor network signal sense work manifold class anomaly detection relationship community exhibit interest goal plain necessarily operational remain hidden part approach detection simulation realistic fundamental new theorem maximum propagation nonnegative optimality propagation establish vertex subset mu subgraph graph edge induce subgraph adjacency adjacency undirected graph necessarily diagonal matrix vector degree neighborhood nonzero element th orientation map terminal incidence terminal vertex otherwise mu mu incidence unnormalized laplacian graph matrix laplacian recognize I negative physical laplacian physical explain matrix across application solution laplace graph walk chain stochastic harmonic state irreducible strongly describe harmonic boundary vertex special vertex partitioning yield pd computationally analytically general method invoke relaxation np detection propagation take greatly multiple treat independent test relate exist maximize pd laplace network vector distinguish may variety way graph appear base establish connection algebraic laplacian partitioning problem optimize subgraph section optimize detect subgraph criterion address avoid trivial propagation avoid priori approach alternate upon cut subgraph necessary separate subgraph membership detection minimal small eigenvalue fail discriminate subgraph intuitively degenerate subgraph zero subgraph principle show small many offset indexing subgraph threshold call graph call eigenvector analogous theorem riemannian geometry relate topological manifold laplacian topological tie diameter provide min explain connectivity imply involve size subgraph alternate criterion modularity detection minimize cut propose maximize connectivity graph modularity partitioning involve eigenvector eigenvector principle bias subgraph outline spectral framework invoke alternate relaxation yield practical semidefinite partition proximity et bias towards specific develop local partition locality dual epidemic observation problem determination topology occurs adopt fundamentally propagation problem arise disease may spread infected neighbor logical detection focus discover likely associate random yield arithmetic spectral partitioning observation propagation underlie probabilistic throughout bayes assumes give simple hypothesis form graph single entity bipartite heterogeneous bipartite graph comprise vertex email message message entity within foreground value discrete otherwise background foreground foreground subgraph decide foreground formally detector element induce subgraph call foreground background denote logical complement detector determine detection measure detection pd alarm pd foreground vertex model sequel context spatial whose vertice measurement however observation b ideally foreground vertex foreground background foreground graph foreground background vertex observation positive mutual likely vertex foreground member model observation foreground graph determine observation vertex delta allow vertex section devote development spatial optimum sense alarm motivate foreground graph foreground know observation measurement vertice bayes rule compute graph observation connection imply throughout graph priori neighbor probability diffusion vertex random walk vertex observation comprise vertex walk multiply connected existence walk vertex every transition implication well weighted indicator determine walk non repeat vertex definition capture walk variable average walks eqs observation illustration vertex middle show comprise simple multiple diffusion v propagation propagation walk terminate vertex draw definition model distinct yet stochastic realization describe linear neighboring realization describe walk consider propagation neighboring simple q notation eqs probability observe vertex vertex method detection exploit laplacian address implication rely provide detection boundary apply uniform recognize equivalently walk trivially constant principle establish existence unique probability principle connected observe
bregman alg lipschitz set measure variation deviation scale regret sublinear change correspond exist tracking variation tracking term track fact generative observation get analog enforce equivalent fitting bind analogous static regret require restriction without prediction repeat unity bregman dependent dynamical regime system scenario self process state observation account implicitly I therefore scale series dynamic eq variation idea online combine tight work static variation deviation track know model use may assume adapt environment tracking bound different dynamical regret time segment bind algorithm dynamic give initialize receive alg forecaster combine could primary drawback share upper implementation common alg run come expert algorithm exist thorough method share expert candidate dynamical assign cumulative yet amongst amongst expert share weight allow expert therefore quickly weight well switch respect alg dynamic share tracking measure deviation dynamical depend sequence sublinear regret advance v ti however know switch finite consider parametric vary denote word like jointly consider concatenation generating sequence use capture prediction model would regret would track parameter manner space resolution cover second dynamic show appropriately track inherent finite collection alg collection consider model exponentially forecaster cover dynamical divergence l dynamic choose candidate conversely much grow due q use norm get tradeoff computationally vary could trick set time horizon expert account change generate dynamical dimension however approach computation dynamical prediction produce mirror certain dynamic quickly convert describe applicable family mirror explore denote statistic dx know partition bregman divergence analysis let refer image sequel dynamical db tc alg minimizer dynamical parameterize dynamical eq prediction candidate dynamical individually need expert simultaneously track basic mirror compute parameter decrease bound convex assumption bregman tracking associate dynamical constant track dynamic nearly sequence specific loss dynamical online analyze anomaly streaming compressive video firing sensor physical incoming inconsistent many simulate correspond dynamic spatio within model autoregressive scene element motivate instance detector theory tracking loss inaccurate auto simulate video texture water flow denote underlie system unique texture desire intensity encode drive toolbox pixel water played play x generate equation finally missing choose every parameter define dynamical accounting datum reflect despite play hold bregman regularization run trial mirror dash line interval dynamical sharp interval md standard md facilitate model incorporate visually look texture mirror look snapshot water recover scene autoregressive finally water start visually spike big like throughput wish recent structured illumination sense principle however measurement acquire live motion sense image accurate reconstruction accounting scene compress scene fast reconstruction dynamic create overfitte improve series estimation batch demonstrate simulate imaging frame store take value correspond measurement sense architecture loss direction motion zero motion finally bregman square use forecaster use set experiment loss clarity dynamic motion successfully track figure impact dynamical baseline knowledge dynamic top representation truth unclear picture pick finally self likely action act node could neural neuron could neural self network much likelihood know parameter track simultaneously apply model trial generate except distinct element norm stability run know alg md additive alg result advance md alg alg approach estimate conventional mirror characteristic standard mirror even know streaming processing velocity stream big sensor fraction carefully examine identify datum inconsistent novel method mirror descent incorporate dynamical yield method applicable variety noise underlie dynamical share adaptively dynamical switch texture video share divergence eliminate additionally incorporate dynamic employ sequence alg arbitrary similar eq complete convexity schwarz inequality combine theorem matter denote tv q term dynamical mt track relative dynamical regret relative well slight modification proof loss l incorporate natural describe interval yield dynamical candidate dynamical entire minimize loss definition decompose use forecaster adjust instead bound yield notice variation variation give loss generality assume apply must assume lie interior minimizer project use velocity stream pose across application rapid anomalous recent advance lead converge unable adapt environment world paper address challenge yield accurate efficient broad capable adapt underlie scene modern collect limit rich physical large number variable thereby provide generate majority hard away typical hundred hour datum generate datum daily square array twice information send around internet daily science engineering setting recover anomalous accurately efficiently rigorous analysis pose major curse dimensionality exponentially dimensionality even setting environment vary memory resource streaming range purpose streaming lack dynamical dynamic environment streaming least square algorithm filter kalman update readily dynamical performance applicability rely regard learn place heavy restriction nature address incorrectly within machine community base universal perspective provably generative programming sequential forecaster new compute stochastic principle decade technical understand lead rapidly converging framework loss forecaster prediction loss next characterize efficacy accumulate forecaster total accumulate particularly interested computationally batch algorithm oppose future online convex optimization sublinear broad yield relatively static incorporate admit regret common form
limited efficiency model make high hamiltonian see mix furthermore bottleneck hmc hmc follow simple part develop hmc gain efficiency block sparse outperform random hope one introduce eq penalty norm optimum plausible recent solving name refer name respect cone definite constant laplace wishart prior matrix hamiltonian augment mcmc approach rapid strong correlation energy physical constructs hamiltonian q mass matrix energy energy discard sample simulate eq approximate step manually typically accept correct inexact hamiltonian number rejection develop hasting proposal gain scheme update block gibbs outperform block simple modification efficiency novel hmc part sparse improvement block clique sample wishart clique iterate cover draw k performance sampler probably clique connect isolate clique give well cover clique choice first clique hard clique require trade little significant speed paper heuristic clique facilitate mix keep clique yet build grow clique cover set maximal clique hmc choice challenge adapt positive hmc positive may infinity proposal cone remain could even reflect simulated straightforward non trivial path linearly cholesky decomposition variable hmc gradient precede ordering must slow run hmc draw near draw intuition good mixing allow move simulate length chain still move slow preliminary hmc mass freedom great hmc may well find influence perform short block show hmc ess ess ess ess ess ess data ess ess sec ess ess sec ess ess sec compute set hmc experiment sampler mc extremely impractical dimensional also mc number hmc gibbs decrease hmc tends expect use mass accurate hmc gibbs prefer dense expect case mc level hmc compare case require preliminary laplace wishart ess ess clearly hmc inefficient approximation quite poorly ess follow conditioning similar however preliminary hmc joint need change impractical wishart mass column demonstrate advantage hmc sampler employ sparse frequentist consist price stock market index day leave subtract main distribute graphical parameter space perform evaluate bayesian p hyperparameter analogous little performance sampler depend hmc slow try complete take graphical spend change clique spend value adjust preliminary ess practice little choice similar use distribution comparison test ip evolve version sampler converge start graphical minute graphical hmc empirical precision offer performance difference index volatility seem rather map demonstrate efficiency sampler describe choose cover gibbs real datum well log likelihood appear investigate far extend graphical would hard would significantly joint hard market dependence acknowledgment zhang discussion long discussion hmc uk science find
sr make six three case loss z exponential z therefore change percent almost hinge loss citation take shift g segment shift decode give citation need maintain repository signal citation citation need need label database database contain annotation database hz apart sample hz information contain hz citation need preprocesse hz filter citation base visual inspection filter decide hz cut hz hz pass filter citation need record normalise record annotation either database length randomly randomly sub sample approximately amount reason balanced class understand class since highly quantify patient underlie cause use situation exposure device episode considerably skew towards secondly prior prevent number class classification space classification non second length spectra onto perform vs spectra spectra space show long segment citation boundary polynomial kernel spectra reduce version high representation preliminary citation need achieve sensitivity average exceed despite achieve good bias classifier space magnitude spectra spectra space correspond bar good representation magnitude spectra versus versus magnitude use kernel length individual improve length attain agreement preliminary need fact segment form ensemble act segment shift second spectra classifier second window different form give window bar spectra classifier value segment extract long window short segment purpose classification take window ensemble form long great ensemble segment shift extract window datum label database show label database classify opinion classify correctly algorithm database thus database well learnable considerable label systematic classification perform spectra approximation projection consider improve reach short citation consider classifier combination accuracy sensitivity segment ensemble classifier segment interval achieve classification fp discriminant discriminant discriminant vector mode external modify likelihood heart ratio de health st database database association space label normal length fourier spectra publish improve segment little discriminate short minimum ensemble act segment window ensemble health cause death high death citation drug may challenge development form evident differ fundamentally diagnosis report unable test record citation limitation response citation error diagnosis discriminate include need study use assess discrimination view able differently hypothesis average make citation citation author wavelet domain citation peak correlation window short immediately classify citation create dimensional representation base spectral citation count turn citation fast ten citation need author propose phase citation need citation discriminate segment improvement solve accurately main citation resort specifically citation often limit exist heuristic representation another systematic investigation involve information reduction dimension understand statistical use heuristic preliminary fourier spectra combine citation benefit linear boundary may insufficient study citation suitable discrimination try magnitude spectra dimensional representation obtain project spectra respective subspace order facilitate boundary dimension vector prior report demonstrate benefit dimensional paper reference windows framework experimental label representation citation need wave heart rate human window normal occur interval increase heart rate citation heart applicable high window sufficient study window window consider assess observation bias consider magnitude spectra consider shift hz sampling close sampling frequency cause apparent dimensional magnitude spectra consider statistical employ reduction spectra form top low three direction complete high exceed perform step report purpose vector correspond jointly maximize misclassification surface lagrange create k compute inner perform separable analogously decision x sign e commonly grid unlike select give example width misclassifie gaussian lead increase flexibility flexibility kernel flexibility decision flexibility decision boundary grid grid coarse
describe sequence people mistake incorrectly specify inconsistent predicate planning planning plan plan tool plan valid plan plan hasting calculate capability resource response team largely incorrectly plan specification constraint people evaluation demonstrate robustness specification gibbs generative collection sample unlike write analytic posterior calculate valid mh gibbs explain prior mh direct difficult typical mh define case achieve select accept reject simple proposal need plan order tuple predicate suggest set predicate current one exist set select order move plan use move define symmetry acceptance current describe evaluate tool remain calculate plan fortunately analytic eq note analytic expensive truly care plan final percent sequence seq predicate relation predicate plan sequence g early shift position predicate arithmetic condition perfect file goal one robot file room enter possible great average predicate relatively specification cm allocation seq avg noise illustrate plan robot two plan execute collaborative pr participant use web collaborative complete manually plan infer human robot execute predicate robot action room name location perform offline advance scene pr room planning room member enter video burden programming robot team combine logical plan team enable infer plan average investigate ease infer plan plan investigate automatic mechanism raw take input thank helpful discussion air force contract fa interpretations conclusion recommendation united mit burden programming system people domain field operation frequently human operator translate plan machine translation burden team combine generative logical plan validation possible overcome challenge team planning validate human team plan describe robot people plan execute collaborative pr logical planning generative assessment identify major bottleneck utilize system aim careful execution bottleneck plan team frequently team pressure robot execute plan team translation burden infer plan process team combine generative modeling structure prior possible hybrid enable inference space team plan work raw form research area challenge planning plan team discuss plan pressure planning consist communication reliably majority final plan validate able team plan people plan execute collaborative task robot knowledge work logical increasingly web planning tool plan hundred audio text map final plan formally describe formulation plan specify without specify plan constraint action execute medical robot hour formally plan description involve material experiment concept room assess person specify information assess person medical room must robot room medical resource constraint resource blue resource room fix temporal plan conduct describe technical readily generalize later produce large achieve goal team agreement plan strength team discuss leave agree flexible plan multiple decide among team plan possibility plan human team collect person discussion algorithm machine human capture action relation work raw language plan human show part short robot blue robot red bm blue room natural structured suggest robot cover u robot inspection e red go red medical apply member room predicate happen predicate action happen simultaneously room room follow medical team room noisy discuss predicate explicitly sequence constraint u b place order u action regard plan code predicate appear appearance final plan u specify two predicate predicates plan parallel human human infer absolute predicate understand plan algorithm infer plan tag predicate represent action happen indicate tag order carry represent time plan approach plan inference noisy uninformative frequently converge team constraint planning team discuss relate plan solver produce plan sequence probabilistic logic base plan highly structure prior encode plan deal challenge amount sampling hasting validation subject combine year describe learn uncertain logic joint probabilistic logic share logic logical explicitly solution planning inefficient exploit
I corollary support fellowship nsf grant support nsf dms n e graphical propose subsequence knot path statistic asymptotic nan connect approximation sample test tractable connection linkage inverse focus interpretable statistic graphical maximize wise exclude restrict case scale correspondingly glasso recovery appropriate certain inference similar lasso base fit original enter statistic carry orthogonal statistic simplify enter propose along graphical path hypothesis test significance construct simple also demonstrate simulation practically reasonable theory tie statistic able method case linkage able statistic overview behavior cell derive exponential demonstrate demonstrate behavior later conclude extension estimation regularization level inferential justified presence simple set statistic resemble situation arise base statistic lasso fit enter statistic nan select test setting mention provide demonstrate part know graphical solution broadly speak point pattern dense move small group become connect previously group connect change decompose component lead naturally nest subset knot paper connect edge enter correspond subsequence order large would already precede element basis k never become true question find group hypothesis underlie recent construct intuitively signal converge absence signal nan hypothese empirically distribution size proceed result statistic lasso estimate experimental design correlation really presence mix uninformative variable correlation graphical lasso augment true none noise variable test statistic enter variation statistic distribution nan step conservative first middle line slope show enter realization enter connect real step accurately nan center show distribution accept realization plot edge appear attribute center presence conservative trend vanish base special general broadly formulation graphical change nonzero pattern zero identical simplify q knot sequence behavior global extend idea structure test n enter global hypothesis important piece element absolute edge connect previously second distribution grow fast converge correlation spherical pn begin quantity ij I ij ij ij ij large approximate lemma share lemma ij ni event intersection obtaining ij note ij two apply lemma paper notation establish proof support lemma appendix refer exponential true support empirically nearly proof conjecture unable apply correlation exceed matrix conjecture follow let cdf base scale strong sufficient km k believe conjecture conjecture stronger grow element gaussians rest paper hold follow theorem find work matrix share index event define correlation conjecture fix pn identically form expand great index define none share index j kx rewrite event permutation previous ij expand intersection note eq disjoint index k vanish obtain lemma k j j p theorems statistic distribution k present dependence relax structure allow assume dependence nan occur statistic nan signal careful sure variable present follow condition occur test k dd theorem rely conjecture probability limit spherical absolute step take edge outside v dd either recovery variable large share event lemma k k p correlation v rest outside number finite differ address go distribution statistic simulation carry variable vector center repeat conservative demonstrating reasonably cdf quickly distribution first close realization block rest covariance diagonal structure pair step middle plot nan demonstrate reasonably five demonstrating nearly show histogram nan demonstrate reasonably outside demonstrate scenario respectively simulation panel behavior closely suggest rely hierarchical linkage theoretical apply linkage linkage cluster base absolute pairwise start variable merge graphical lasso level sequence well test correspond height tree merge level statistic asymptotic exponential distribution graphical lasso attractive approach integrate yet develop construct test prove corresponding hypothesis finite extension method linkage absolute correlation test variable knot internal estimate statement important behave empirical make intuitively large nonzero note behave special case correlation nonzero statistic rather see size illustrate correlation nan first non quite large appear research determine graphical lasso setting construct broadly across thank point lasso feedback contain support theorem along ij nx nx
zero preserve scoring drive regard specificity indirect indirect realization direct detect give specificity involve randomize replicate sensitivity specificity compare require intensive significance much slow intensive get eeg usefulness series measure directional coupling propose measure publish mutual embed bivariate response show outperform rely test observe accuracy computation importance simulation channel eeg recent year time gain attention advance network understand system network interest effect drive sub distinguish indirect refer direct causality causality measure last decade accounting indirect phase partial transfer possible unbalanced production might increase add calculation delay embed delay transfer entropy te measure future te totally observe become eventually fail g stock portfolio suggestion dimension address propose good recent work drawback computationally intensive resampling use surrogate datum non derive bivariate directional coupling criterion neighbor knn entropy mutual mi stable neighborhood multivariate direct coupling reconstruct subspace non embed good evolution mixed variable avoid effect information measure whereas absence explain effectiveness compare eeg conclude want conditioning generally vector step ahead setting dense system lag within lag eeg natural maximum lag scheme term embed cycle component correlate give knn mi second augment component additionally contain knn cycle additional select quantify termination vector interest whether quantify causal q numerator future part mixed form lag drive account vector similar delay delay vector whole embed drive effect mixed vector totally dominate drive expect close maximum lag variable termination criterion selection maximum lag lag map observation lag flow smoothly time cover period horizon widely use work argue appropriate case densely threshold find simulation positive absence coupling compare adjust threshold select component cycle hypothesis randomly obtain give bias threshold termination percentile proceed terminate adjusted criterion length illustration coupling strength show weak drive free couple determine presence observational deviation sd embed balance seem practice optimize hand adjust well noise time well specificity compare causality transfer entropy specificity measure consider significant surrogate shift surrogate detailed stochastic superiority sensitivity specificity h direct coupling measure couple small indirect coupling deviation slowly tend significant htb non bar sd realization bar visualization probability detect relative find significance shift surrogate relative htb p c map number couple indirect chain setup challenge coupling detect specificity decrease interact get large show variable regardless mix always component two drive presence causal comparison continuous add vector autoregressive var component zero direct causality match vary realization var mm direct large true estimate still estimate rejection rate randomization shift surrogate much ambiguity specificity direct though four true direct four var setup show var var include time nominal high positive rate overall sensitivity high rejection small rejection nominal stochastic measure nonlinear setup table htb attain couple specificity coupling rate almost direct large lag therefore sensitivity increase rejection drop expect regardless map h time series dependent respectively parameter map use ahead lag present difference whereas additional result early whole symmetry coupling figure map realization symbol direct power detect biased evident noise sd add white detecting level direct oppose possible panel couple organization panel couple panel direct monotonically tend decrease point monotonic inter affected interpretation identification spurious indirect noise still tend spurious ii couple length causality causality realization randomization determine symbol legend possibly causal neighbor magnitude significance couple always statistically indirect significant give spurious direct causal attain sensitivity specificity proper add series white exhibit specificity indirect realization level significance termination couple differential solve matlab time unit strength assess trajectory delay generate variable represent option large ahead free white add white safe comparison hypothesis coupling rejection rate weak couple true dropping rate stronger detect well specificity good indirect coupling small significance rejection get large get indicate strong indirect distinguish coupling rejection rejection variable almost exhibit effect algorithm mix systematically measure persistent tend detect false series seem suffer specificity specificity indicate adapt threshold termination e candidate couple show realization threshold large termination drive consequence indirect coupling figure adapt
factor get uniform result least square achieved leave challenge practice small however reasonable limit expense zero case covariance significant increase due follow monotonic decrease attain increase psd take symmetric return psd close frobenius eigenvalue diagonal external zero budget perform however large algorithm decomposition library standard computer assume add free regressor full allowed provably simulate collect crowd amazon share request researcher ct slice dataset axis simulate collapse histogram bin single possible real value per example predict height publicly available chart post chose crowd might predictive attribute group make labeling encourage amount person regardless worker pay hour set average collect height fashion attribute long adjust count four attribute show normalize inter select attribute demonstrate combination useful noisy attribute exist provide weight indicate respective list attribute height weight error algorithm full scoring plausible baseline denote average plot predictive average set add time square regressor use copy treat different individual attribute symmetric large indicate say similarly entry let th otherwise define sample simply analysis label average note psd correct use similarly result eq define statement theorem lemma label average use feature random vector lastly variance x x q average diagonal right side hand side thus consider diagonal addition j entry conclude unbiased definition q guarantee assumption bound external eq maximal assumption wish rademacher md eq rademacher complexity individually rademacher generate application square q w relate net sphere note verify boundedness far get minimizer minimizer sm hold minimum g positive definite inequality lemma union immediately draw positive hold simultaneously finally prove main start recall examine feature probability label bind divide finally desire convergence get simultaneous convergence lemma mirror define index iteration cause increase prove induction trivially hold brevity since select q get induction j l prove induction hypothesis therefore eq case conjecture feature multiple effectiveness regression predict people height attractive publish term throughout attribute sometimes account important part otherwise capture crowdsourcing suggest include people henceforth set one limit justify problem toy multiple eq fraction attribute shape number color bias five people regressor possible budget resource test great coefficient color valuable phenomenon repetition multi generalize subset choose hard np consider nonetheless successful generalize approach theoretically world publicly height www com height crowdsource estimating object multiple attribute object use crowdsource natural source demand object object truth draw crowd pool worker object crowd budget limit attribute regressor unseen approach collect candidate attribute decide regressor collect accomplished step decide square per attribute estimate error projection accurate greedy multi minimize operate simplify simple rule incorporate attribute correlation notion attribute greedy provably attribute correlate perform attempt optimize projection crowdsource motivation would applicable setting input place quantity crowd contribution theoretically justify show feature work relate crowdsource measurement science researcher evaluate see instance recent reference attribute coefficient less suitable noisy variety combine crowd quantity estimate recent crowdsource million employ crowdsource objective image take predict key quantity standard compare assess reliability attribute perhaps coefficient determine reliability attribute independent set object object limit total worker object real value one object notational phase feature collect analysis trivially generalize finally attribute expect across ease presentation track discuss describe g font g unit represent feature attribute time represent object eq th coordinate repeat vector yx vector phase receive label generate set receive budget allow object distribution label repeat predictor operate predictor predict label begin increment project exceed simplify define call calculation vector diagonal simple feature empirical label score uncorrelated optimal multi henceforth similar repetition element
ergodicity human computer completeness recall format chain iid laplace even available normal implementation straightforward less rate code behind take five minute elaborate simulation special design wide generality go gibbs name initial implementation dimensional hierarchical group endow proceed conditional order produce chain hierarchical build hierarchy depend neighbourhood induce I typical observation eq index ij sampler joint distribution use conditional replace simple improper conditional joint improper growth measurement apply model compare model effect include arise hierarchical pair child acyclic bayesian model choose conjugate avoid constitute proper default thank conjugacy conditional straightforward associate gibbs sampler series iteration possibly growth right occur day understand conditional graph variable index depend connect index correspond modelling expert bayes naturally graph dag fall although principle methodology straightforward instance algorithm move open evaluated stage implementation choose model usual straightforward around within simultaneously chain explore challenging difficulty define rather simplex green reversible solution markovian chain space connection green correspondence move generate compute reproduce important acceptance jacobian mistake implementation simple space add instance series illustrate series stand model process resolution unknown associate lag root root number non root mean complex root induce root prior conjugate root root reduce extra past parameter q involve new sum move space wide enough jump calibration number whose much experience separate parallel determine separate like model comparison frequency accept illustration distribution end probabilistic structure space illustrate cover specific call abc short statistical population decade cover far assess addition bayesian computational carry label although reader deep numerous calibration abc end population intractable unable handle success develop computing turn link community method per se recent survey algorithmic accept acceptance abc replace acceptance evaluate proximity pseudo stress stage rarely since moderate close simulated parameter statistic simulate keep abc quite rapidly tolerance quantile simulated marginal density observation suffer stop summary bring close whole thing increase dimension increase discuss thing consider extreme note raw time vector would simulate horizon bring word force tolerance point build consider limit effort tolerance produce positive study behave non bandwidth slowly decrease raw setting recommend raw curse dimension operate statistic approximation even connection correct ss statistic addition monte approximation effort produce abc simulation result effort abc proper early external net non reason neural net component net ran tool opinion accomplish development currently summary quasi theoretic obtaining bandwidth particular argue estimate notice abc acceptance accept carlo n expect error vanish derive bandwidth algorithm obviously impossible estimate use produce sample costly bias inherent conclusion literature topic perspective eliminate pool statistic issue statistic problem curse ultimately approximation far major development come bayesian complexity complexity comparison formally straightforward compute posterior raise confusion community illustrate face perspective specificity sort set simplicity abc mc generate generate replication tolerance mc improvement regression comparison normal exponential pick double simulating choice avoid difference median median median absolute median later bayes summary outcome choice summary insufficient need check abc snapshot particularly dynamical rely partly quickly technology advance order asymptotic similarly bayesian field expand bayesian quick solution material hierarchical word time age grateful material part chapter chapter status attempt statistic biology survey evolve co author chapter request join universit paris paris france fr partly de paris grant bs paris paris paris surveys advance purely viewpoint abc expand particle method briefly chapter novel entry se challenge handle standard density block development bayesian simulation paradigm ideally suit core mostly historical development bayesian challenge limit understood tool comprehensive introduction book mention start posterior side proportional dimension discuss dimensional reader close even give pair representation computation case section denominator issue density easily conduct without test central address inferential indeed ratio likelihood support mean problem case issue inferential instance dedicate importance develop briefly notion bridge apply factor ratio integral integral identical space severe two posterior
walk riemannian figure realization actual target appearance variation pose cause red evolve david toy toy toy david car website face test htbp cccc frame toy toy david david movement fast team background player poor contrast stable relatively david david relatively rotation car david measure paper track overall xt xt xt xt xt yx xt work manually annotate frame frame change red art incremental al toy fail frame onto background pose non similarly fail follow motion towards reflect drift frame dark short frame face sequence expression descriptor example car background whereas illumination change dynamic unable account smooth template car close template overall tracking performance sequence image duration nevertheless track frame track change appearance comparable video website http www com pixel wise alignment well illumination illumination template norm align target favor stable region target similar target pose track hand covariance gradient intensity template descriptor cause method precise gradient intensity change flexibility scenario fast target mis alignment template add eigen short consequently non representative tracking scenario lot well firstly descriptor alignment alignment secondly accommodate template riemannian manifold evolve track second covariance feature dark variation otherwise descriptor may ill consequently measurement template propagation mechanism inherently impose definite process allow template evolve naturally target appearance quantify uncertainty achieve visual descriptor representation model covariance riemannian riemannian manifold outperform art pose maintain include robust change future include address speed adjust diffusion deal illumination change generative track discriminative descriptor explore acknowledgment would david share chen school member national b electrical engineering stanford associate school computer centre network receive ba triple members st college subsequently college science research usa include technology product group research mit science dr member currently head national ph dr ph engineering national mathematics interest include vision medical equation align track video challenge target model descriptor descriptor robust pixel pixel pose illumination occur template template random descriptor lie log transform free impose inherently semidefinite template pose uncertainty relate states template principled show change illumination pose affine outperform incremental principal fast pose change maintain target track particle filter generative riemannian visual range recognition surveillance interaction cover many aspect decade challenge pose illumination long video vary often common tracking task target illumination pose common approach deal appearance robust feature invariant figure appearance time totally frame target pose employ possible variation require advance scalable template gradually evolve template strictly choice template use use histogram appearance patch covariance eqn estimate update target template challenge template template frame accumulation eventually template drift one approach space covariance use track template incremental subspace briefly template propose robust decide update template keep drift template first match template template method well impose alignment template inherently pose employ template distribution account template template target pose outlier pixel pixel gradually template stable target template online incremental principal capture template keep test great robustness template due pose change illumination update evolve paper template sample ability variation although target contrast image sequence short target fast pose illumination change paper inherently assume template change pose illumination unimodal distribution uncertainty contribute template covariance manifold simultaneous inference template generative empirically discussion finally conclude explain motivation descriptor riemannian manifold descriptor q target coordinate directional intensity magnitude gradient extract patch descriptor gain popularity many recognition descriptor template target pixel correlate robustness view pose lie riemannian explain operation riemannian htb space manifold differential tangent space define dot product covariance descriptor riemannian c ic inverse exponential map matrix map riemannian operation collection model target space linearly target dimensional often manifold zero manifold visual target manifold could simple popular distance head figure descriptor distance well separate background well target patch large patch kernel background target patch formulate template follow template pose observational dynamical subsection descriptor eqn model center orientation pixel intensity patch gradient order variable velocity interact pose template model plane motion describe template dynamical transform manifold
disagreement eigenvector differ green red clique quantify chart node colored mixture green find nmf nmf mix quantify community cccc lead eigenvector colored star adjacency sequence time node importance respectively rescale estimate identical rescaled star highlight node incoming connection vector incoming connectivity assignment important series adjacency produce low basis force community series visually node provide plot display facilitate detailed analysis difficulty become window parameter interpretation income edge ik v jk jk contribution community due unstable assignment alternative measure community similar enforce negativity lagrangian follow u kkt w u briefly matter factorization cluster examine reconstruction base theoretically preferable intuition behind subset slice slice different correspond due datum structure employ two cross refer test care hold datum slice set identical submatrix identify hold submatrix cell index row set choose minimize toy validation penalty window modern require typically emphasize interpretability keep large instance algorithm suffer neighboring locally incorporate risk miss change detect persistent fitting sensitive short term fluctuation due number ahead setting allow highlight real example force embed colored soft nmf utilize part vast challenge phone phone record phone phone duration call challenge movement challenge identify five key individual activitie communication participant node use first see reference therein direct network daily edge receiver visually network structure rank penalty highlight persistent fig color accord community first row cluster result raw interpretable network visually neighbor higher belong community information different raw filter color factorization threshold community nmf overlap apply flexibility poor reconstruction fit network snapshot static keep clique snapshot alternative like penalize nmf provide visualize answer challenge however analysis win entry treat conclusion ground truth directly node accord grow spaced network matlab toolbox model represent connect generating framework whose law graph commonly area internet citation analyst datum centrality believe sequence serve basis exploratory tool help smoothness time plot node nmf distinct indicate importance time reflect display create identify research penalization usefulness display plot node appropriate analyst visually citation part graph physics period organize particular paper direct convention aggregate citation month month paper edge ccc investigate single due large statistic extract infer dynamic year commonly attribute paper move paper decrease diameter tailed degree distribution visualize node factorization dimensional adjacency score trajectory smoothed dynamic highlight employ important paper mostly string propose physics include theory see table identify work paper focus citation pattern reflect bold trajectory display appear uniformly throughout show paper fast correspond utilize eigenvector modularity community citation methodology mixture extract paper citation profile show time colored community paper utilize group view panel profile grow slowly observational paper type reference via google degree google string dimension topological schwarz string manifold et duality title anti de string alternative geometry et hull analyze directional country flow order express cross flow year space one community dominate six community historical event instance align former trading relationship persistent rapid close trading membership idea behind discovery strength benefit scalability factorization experiment enough visualization tool compatible binary ccccc runtime trade exploratory visual tool community rank importance connectivity display complexity node optimal tuning benefit give visualize topology fig topology feature connectivity visualization comprehensive important would systematically penalize version nmf svd nmf display preferable term consistent factorization svd relate deep topology combination penalty useful visualization precise mean directional edge nonetheless roughly cost evidence penalize functional increasingly pose visual exploration pattern vary evolve graph home display underlie scalable node accommodate topology dynamic real technique display meaningful advance become increasingly factorization base discovery visual exploration discuss approach address community series network important contribution community review field statistic relatively within connectivity low variable group paper evolve citation dynamic popular space utilize smoothness coordinate utilize community membership constraint overlap vary covariate predict link anomaly non quantify time advantage community addition quantification strongly suffer drawback modularity optimization network use create node dynamic literature aim enhance static drawing move node facilitate reliability ability change discover note methodology adjacency potentially appeal since analysis arguably compatible nmf interpretability motivation
low collection feature solely switch finite place expect self analogously hdp imply finite eq hmms focus ar ar define constrain transition amongst dynamic regime membership bp conditionally j equation couple process hierarchy var state form regime importantly time series parameter amongst time individually term representation specification summarize note treat bp ar exercise jointly amongst six subject category place arm circle raise variant result joint display figure skeleton trajectory contiguous segment two box behavior label infer true behavior two raise bp allow various motion behavior appear plot skeleton display contiguous segment segment box segment color skeleton modification toolbox identification share versus model true bp ar draw beta indicate behavior truncate regime article grow nonparametric membership literature cover main membership vary topic give corpus sometimes span likely popularity word time example article scientific question address naturally evolve within terminology perhaps newly phenomenon go specifie walk document specific topic specific word arise via specify transform weight topic weight assume evenly corpus evenly potentially sample brownian motion simplify author evolution topic proportion process dirichlet topic motivation importantly membership take assign dirichlet popularity topic static dirichlet measure static dirichlet originally propose evolve examine general stick break stick break continuous autoregressive stick break weight modeling function process markov switch infinite factorial collection markov chain define binary perform blind separation separate audio overlap hierarchical hmm collection chain model chain hmm base state external covariate gamma process hdp hmm regime switch capture repeat base dirichlet partition dependency programming technique partition explore form autoregressive hdp bp ar hmm autoregressive follow discrete autoregressive autoregressive select model autoregressive process mixture aim series contiguous autoregressive process implicitly formulation inclusion vary formulation mixture probability autoregressive via potential fix autoregressive imply take variety interpretation mixed membership comprise differ associate assign membership dynamic regime partial membership structure contrast focus mixed collection entity g word focus attention parametric latter modeling process transition markov series specific switching mixed typically corpus goal allow entity membership amongst attribute efficiently datum source associate entity goal article explore examine nonparametric time analogous multinomial review model allow sparse association series regime regime membership perspective previously lead interesting proposition california berkeley mixed series abstract aim associate individual membership collection entity view dynamic attribute present dynamic regime refer path underlie goal rich possibility recent development consequence membership series abstract form membership aim associate individual collection person various interaction people comprise mixed associated member natural characterize switching regime collection realize path characterization series rich modeling possibility development classical focus mixed membership focus goal entity relate overlap perspective series collection time proceed sequence focus relationship arise position velocity sensor place go exercise routine series exercise person select library exercise type routine goal discover exercise behavior regime person discover routine useful routine essence would combinatorial shrinkage behavior another arise mixed refer marker wish nearby marker genome capture series one problem involve adaptation hide markov switching series directly essence global library accord sequence state traditional dirichlet multinomial mix bernoulli model global drawing sequence subset state agnostic parametric arise multiple provide classical space select number select separate choice remainder review membership membership analogy time relate canonical dirichlet allocation review lda outline focus mixed membership collection brief survey series introduction relevant mixed membership parametric autoregressive use ar assume observation observation invariant time invariant normally imply refer closely process underlie dynamical process conditionally invariant process covariance likewise ar process multiplication consider sequence discrete useful hybrid discuss important formulation collection state regime behavior set merely briefly overview allocation membership document assignment assume describe mixed seek characterize entity attribute typically entity impose lda document collection simplify order ignore distribution word first select topic distribution select formally expect proportion document unbounded topic membership review approach entity attribute model topic topic specify equation want countable set dp distribution measure space weight divide stick length proportion stick choose denote drawing indicator examine indicator concentration describe crp provide insight dp lda draw define vocabulary multiple document might unfortunately word document measure parameter describe share entity specific drawing topic arise fix specifie beta conditionally document document allow topic employ hdp likelihood refer document break equivalent dirichlet name generate atom appeal hierarchy process distribution expectation stick hdp model see entity infinitely finite document subset hdp lda model implicitly alternative capture inherent topic variant context membership mixed membership interpretation build lda recall document model analogy compactly document collection series arm etc membership time model time regime assume time order fundamental description regime I describe regime section jointly time question analogy yet regime describe process via markov dynamic regime individually model switch autoregressive var switch dynamical community prove useful target doubly model conditionally emission specification independently membership model regime key markovian structure observation regime assignment upon proportion lda mix var observation conditionally independent give insufficient instead hmm complex phenomena attention article switch var hmms broadly maintain ar hmm autoregressive latent note standard formulation dynamical regime question would like add hierarchical allow collection hdp transition set hmm emission hdp hmm define evolution dirichlet part hdp unbounded encourage tie via create hdp distribution hmm link hdp hmm different persistence real flexible nature place mass sequence state persistence hdp amount equation see expect global state weight hmm graphical whereas hdp hmm var equation hdp hmms dirichlet process hmms dirichlet computation hdp switch dynamical consider allow potentially consider far interested grow field focus inference base might collect eeg contiguous epoch multiple individual subset exercise behavior individual dynamic regime share benefit joint may relationship among basic transition regime proportion proportion time row possible next distribution couple transition extend collection particular analogously finite transition transition matrix allow variability couple hdp hmms variant weak see specific transition around random series
induce block composition positive convergent know concept subdifferential convex salient notion proximity convex proximity recall subdifferential subdifferential moreover subdifferential consist subdifferential essential algorithmic development proximity difficult optimisation rare circumstance obtain proximity relate moreover composition prox readily available challenge prox essential circumstance proximity optimisation shall positive publication provide provide evaluation indeed eq use affine next extension observation applies case convex operator practical tool numerically matrix conclude iterate converge discussion issue contraction iterate applicability solve denote extension yet analyze update current estimate proximity gradient around iterate iterative acceleration describe bx process case backward forward function conjugate use iterative forward backward far simplify obtain use verify forward backward forward method minimization view dual duality relate proximity equals appear within restriction lasso prescribe call group correspond partition overlap case sum proximity case interest fundamental work inspire grateful fundamental first primal problem presentation svms representation straightforward consider svms variant give hinge view proximity hinge separable fact q ensure act svm relation consequently suited dimensionality obtain relate hence generalize proximity x terminology furthermore iterative invertible frequently either involve solve scalability scheme apply invertible iterative converge linearly forward backward accelerate variant nesterov method fista matrix version clear nesterov update also modify solve problem smoothing though need computation iterative similar svms provide proximity square loss lead respectively particular case convex method conditioning nonsmooth optimization sum nonsmooth composition smooth nonsmooth term regularizer important deal regularizer show competitive idea solve machine believe practitioner investigate convergence valuable multiplier several apply mention therein regularizer multi task leave issue support grant ep international european framework agreement induce learn laboratory department mathematics science ny usa college place bt dedicated contribution measure encourage solution r sa whose class arise technique strongly square prescribe matrix regularizer variety depend study function induce mixed norm identity latter well overlap group prove problem
small split efficient derive inequality establish ms cx far use thus sum rely original special light control bounding denote entropy respect constant outer rate integer na ns sum hence sum fix pn sum concerned bx bx k u u u n get depend design translate c positive converge prescribe consistency grow budget choose density level problem sum depend envelope expectation note na ns probability ns identical manner hence get converge unique converge remark equivalent simplify notational equivalently kl z kl nx respectively notational theorem consequently weakly uniformly drive behavior behave separately exist proof define w kl l sufficiently field generalize kl entropy respect cover assume supremum belong sub gaussian approach derive theorem dependent average outline position k p ms ms second n h h n display suitably claim n first consequence claim n arrive translate h ij j ij lebesgue intersect set argue ij ns n h ns analogous center hausdorff ball radius see appendix hausdorff fact extend manner go slow rate rate response model observation replication replication compute replication put coincide response regression dimensional deviation minimize eq abuse notation equivalent vector near tailor solve sensitive relevant provide solve prefer surrogate assume knowledge regularity allocation set bandwidth unknown adaptive dyadic tree utilize open research establish strong hausdorff would require recall ns sx justify ms term converge sl ns ns sl sl sl display converge probability sl sl sl sl fs rd converge x nx nx dominate eq bound complete bin kl x cx dx cx arise replace bound x x contribution dx study contribution right right side bound term display last sum triangle l jk il j block kl sx sx kl kl second hoeffding sx n pick sx get sx condition enough convenience collection apply cauchy v kl exponential inequality show display proof v ns argument na far bound eq third zero sufficiently apply theorem concern bx bx n class g lipschitz order constant hence eq manner sx hence eventually ns p argument proposition na ns ms c hence ns ms ns ms ms ns ns ns ns lemma consequently ns ms ns ms v ms ns ms know shown go corollary lemma remark nsf dms e baseline baseline typically arise statistic relate convex via fitting test contour study level response per covariate baseline boundary role rate convergence total analogue converge explore consider form arise baseline beyond detect several fmri seek detect brain activity brain range experiment measure low would response put interact level detect recover image computer boundary exception address property assumption point baseline convex non interior measure restrict shape analytical illustrative analysis impose certain convexity particularly convergence difficult need estimate along direction estimation level unless jump situation pre note level literature estimating set context contour density star shape approach excess estimate later recover set transform level range transform regard explore develop piecewise problem covariate limited without response early setting replicate covariate find effect drug dimension behavior study interact smoothness boundary play critical formally converge analogue setting converge slow rate bias point support fast nature novel computationally consistency estimate aforementioned fall edge detection set proof heavy deduce hoeffde empirical inequality usage spatial approach question extension work dependent approximation primarily address beyond convexity arrange formally define setting response set unless otherwise list assumption justify setting section respectively explore baseline region develop value originally generate sample use approximate suitable discuss converge level sense minimize methodology statistic inside normalize alternatively consider normalize fundamental normalize tractable avoid routine require choose carefully give class subset convex belong computationally expensive optimal particular axis shift segment move hull exclude segment exclude triangle overlap eq q successive segment among successive note minimize include contribution segment find recursively vertex I construct minor modification reduce seek test
multiply base whitening equation cholesky filter therefore multiply inverse cholesky inverse cholesky triangular cholesky eq coefficient easily coefficient least matlab central file processor mac perform conventional inversion p operation practical efficient square find cholesky structural compare least square corollary definition structural model communication multi audio branch express follow efficient coefficient review use section propose inverse follow conclude remark square model
synthetic hyperspectral ground abundance without intra generate take dictionary shift however modify fidelity prevent reference group spectra example group magnitude plus magnitude real dataset random magnitude order respectively figure ccc follow initially inner set either square problem three could penaltie inter dynamically additional crucial result outer stop admm stop error primal less red figure solve regularize either subset compare method set reference represent solve three column group average inner outer algorithm reference acting algorithm nonzero inner outer compare closely datum background three reference spectra multiply reference background magnitude ground ground square coefficient perhaps include nonetheless result magnitude incorrect regularize produce correct correct method accuracy filter ahead regularize yield quickly direct solution result little minute find decrease energy require outer run minute outer take bad construct k assign magnitude rescale free datum apply group abundance purpose allow use sized compute three dictionary inter sparsity intra sparsity intra inter nonzero intra otherwise iterate ground truth difference matrix accurately pattern absolute average column j ts ts table solution job abundance magnitude encouraging error abundance average signature turn consider expand poorly expand expand dictionary directly enforce group sparsity advantage penalty turn effect compute reference signature material want one material penalty obtain structure reformulate constrained objective difference strongly quadratic program efficiently alternate multiplier practice numerical convergence iterate optical hyperspectral approach penalty different level abundance like focused find variational approach optimization explore penalty sparsity parameter sufficiently interested gradually iterate seem avoid involve expand multiple reference material candidate would incorporate provide guide numerical pointing additionally want ensure modulus enough require eigenvalue search reduce time rejected decrease sufficiently projection method solve fy impose decrease derive add lead iterate include since long necessary greater preferable numerically scale positive definite also iterate fx rx dc concave function connection classical convergence result iterate include clarity intuitive stationary local sequence iterate bound sequence iterate increase take eigenvalue q iterate mean limit bound point subsequence along require strongly method want poor work theoretically well alternate direction onto split bregman solve hyperspectral application admm problem application write admm problem lagrangian saddle respect update find saddle subproblem k denote orthogonal onto huge newton subproblem write n helpful constraint set add problem augment minimize respect admm subproblem stop iterate error primal combine projection onto projection implement still project analysis hyperspectral mixture measure thorough analysis intensity assume average concentration relate intensity characteristic spectra length density integrate concentration denote concentration path spectra law write additionally key smoothly pass speaking remove structure keep narrow know spectra smooth narrow narrow take intensity pass filter background reference combine reference assume version notational change represent assume smoothly effect convolution previous approximately smooth yield challenging basis allow approximately account alignment consistently remove fitting coefficient modify deconvolution specifically pre reference rewrite term existence spectrum select make np hard direct method intra intra choose enough absence inter incorporate add smoothness intra penalty eq moreover group set way present section estimate strategy discrete fouri cosine smooth unlikely periodic boundary odd periodic rapidly decay coefficient encourage frequency frequency define adjust strength penalty single weight resolution spectral identify material signature hyperspectral band mixed material pixel material non combination signature pure material represent matrix hyperspectral different material abundance abundance pixel possible material hyperspectral tool encourage spatial independently add nonconvex sparsity equal put general learn hyperspectral image nmf however interested condition group th reason want contain redundant reference likely exist enforce sparsity involve one material restrict attention abundance abundance intra penalty think important incorporate variability way piece represent combination hyperspectral represent use additionally assume pixel benefit
regularize log dependence matrix reference node neighborhood complement use denote first exist eigenvalue understand ensure become also condition fisher incoherence ts ss variable variable technical use heavily proof glm nonetheless regularize mild partition conditional partition distribution satisfie exist partition pair b px n affect technical enable analysis glm armed show glm suitably behave proposition follow glm distribution satisfy condition pm lc recover note union high derive cardinality statement sample proposition subsection specific instance apply glm graphical family constant constant recall mrfs conditional log partition thus write set ise eq partition write lastly exponential node recall node constraint derivation ise theorem regularization lc extend ise model index meaning besides regime entirely sublinear regime result ise poisson graphical need condition poisson partition integer parameter x pa graphical distribution specify set regularization lc c recovery sketch theorem note program subgradient penalize program follow adapt certain set iff primal use prove proof set set condition construction support finish provide hold strict feasibility rewrite score mean row recall notation filter dispersion goodness consist associate label detect restrict capture conditional pre process positively specifically cluster linkage positively group median centroid node graphical model meta perform sparsity determine stability genomic breast cancer tumor growth major tumor interestingly tumor play relationship act act tumor breast distribution protein use measurement skewed transform negative demonstrate applicability continuous skewed learn flow pre protein learn exponential graphical select sparsity right graphical weight inverse entail association relationship protein indicate protein connect bayesian estimate neighborhood consist protein dependency also lasso graphical arise exponential network node strong statistical ise wide distribution statistical extension g analysis subtle may interest analysis study distributional sometimes place specific family additionally focus family bernoulli exponential negative binomial broad leave room future follow development denote see algebra side give exponential family side generally reason eq th proposition suppose sufficient bounding technique simple calculation show eq technique claim notational q equality expansion derivative n condition chernoff eq union eq provide strictly radius algebra event small u p di result complete going claim point similarly condition event algebra theorem statistics college institute application ise clear non gaussian categorical datum consider graphical node wise conditional arise negative binomial contribution estimator graphical rigorous exactly genomic learn via class derive graphical model know extensively domain physics language distribution product compatibility popular instance ise area discrete modeling question pick compatibility alternatively sub ise mrf discrete value range similarly continuous markov impose instance characterize skew thin tail capture finance tail cause financial mrfs structure approximation correlation fit mrf could variable mrfs transform inferior alternatively specifically mostly contingency intractable even interestingly appropriate model call spend exponential distribution skewed gamma ise mrfs derive multivariate node neighborhood distribution wise strong statistical model rest univariate family theorem algebra derive clique obtain node compatibility mrfs mrfs class standard ise discrete mrfs principled mrfs exponential multivariate multivariate graphical spend model exponential model gamma chi website disease report could model key motivating count genomic sequencing technology read count rna rapidly univariate typically model traditionally understand microarray poisson binomial graphical next sequence new variation gaussian single nucleotide copy micro rna generation sequence lead genomic relationship exponential suggest fit constrain conditional main mrfs note arise glm pose subtle multinomial fairly variable generalize analysis analysis interest outside model graphical broad hope question full generality relate idea propose joint conditional construction context belong distribution pairwise general graphical model joint univariate exponential family tractable dimensional guarantee graphical graphical strictly factor fully subgraph wise graph sufficient clique consist set discuss question translate discuss binomial outline answer condition rest joint exponential graphical encouraging condition recover univariate whose statistic base normalization exponential commonly use bernoulli multinomial square beta thus include skewed count ask leverage ability datum multivariate dimensional undirected particular exponential construction univariate canonical family graph eq specify normalization elementary shown suppose undirected belong graphical thus answer graphical exponential distribution conditional follow specify remain whether general conditional tensor sufficient interestingly argument consider node distribution canonical normalization factor clique size conditional tensor factorize tell graphical factor accord
typically impose parameter respect denote surface triangle parameter invariant gradient maximization similar combine parametrization exploit shape add constrain remain surface force thus act shape prior fit successive local regularization lack image deriving call efficiently soft discrete surface surface cluster vertex active surface shift external force compute compute convenient require annotated example object vary lot approach become allow model single diffusion wavelet build different drawback require obtain relevant indeed local mode desire shape optimize new scale behavior also behavior spatially graph whose analysis relation cf behavior strongly behavior weight wavelet computing operator graph symmetric laplace dyadic power output decompose basis coefficient project example onto pca appearance subspace base extended segmentation pca transform projected subspace voxel voxel edge voxel voxel let also label per walk algorithm probability manual deterministic assignment prior obtain linear contrast voxel normalize alternate since result since quadratic computed compose diagonal due adjacent voxel million solve iterative specific structure existence implementation regular machine k measure formally segmentation minimize energy eq like parameter make bad see compatible segmentation z k replace overfitte allow iteratively dual satisfying initialize iteration fairly iterative converge globally original specify problem set voxel neighboring appear subset
show abc reinforcement however apart calculate decision difficulty class estimation matter prior frequently simulator take simulator reinforcement framework employ abc overview bayesian reinforcement correct draw abc posteriors environment learn include simple quality approach reinforcement select policy set agent act sequence action observation reward interaction depend complete history sequence neither agent necessarily example agent partially observable simplify agent shorthand environment simplification shall policy observation action reward goal utility discount instantaneous reward generality optimal utility ill expect trying perform exploration guess obtain trade adopt environment particular describe correct lie formulate find solve exploitation prior difficulty adopt approach making find hard class heuristic thompson trade exploration heuristic exist paper version thompson reason exact intractable interestingly suffer reason class frequently problem reasonable simulator method find good policy simulation simulation perform simulator abc model detail available useful analytical abc dynamical yet propose reinforcement widely amount reinforcement learning problem include observable environment state space stochastic game develop simulation approximation policy bayesian cover simulator relax close thus posterior bound kl new applicable although use introduce abc discuss abc algorithm continuous state follow bayesian computation calculate via policy something property e posterior fortunately remove calculate posterior policy employ consider set probability always generate idea sequence generate accept require give detail reinforcement difference approximation remark use complete policy history threshold statistic stop basically see generate question statistic need sufficient alg defer thing approximate necessarily divergence divergence need particular case pass notion differential error hoeffde tight parameter marginal difference illustration approximate fix trajectory cm draw actual dot histogram estimate sample dot dash show accept threshold many abc ideas policy draw environment distribution environment execute enough step simulator arbitrary simulator approximate program approximate reasonable good expense alg idea paper sample approximate optimisation algorithm exact policy optimisation largely class type handle base discover work markov mdps policy iteration square herein sample important expense computation depend wish achieve sample require sample accept intuition abc simulator perform draw large simulator domain illustrate rl know generalised car bring car hill parameter low horizontal car car velocity acceleration amount present horizontal horizontal reward every generalise boundary maintain switch action mass cart amount environment actual htb cycle cycle trajectory cycle trajectory cycle average run confidence offline domain observe environment abc policy trajectory grid plot discount trajectory increase lead sample model take abc additional sampling increase quickly prominent car attribute investigation notice reliably estimate location reach may firstly trajectory uniformly draw environment secondly perhaps policy simulator real environment approximate computation control dynamical system particularly domain specify computation reinforcement include abc distribution abc abc reinforcement involve
mdp calculate policy sample posterior bound bayes policy allow calculate special sample calculate model correspond generate partition first straightforward create responsible second covariance suggest plug use suggest plug result distribution mdp tree secondly design generate trajectory policy basis variant temporal difference slightly efficient optimal require fact simplicity reward measure q term trajectory accurate capacity practice representative state difference norm estimate indicator calculate utility effectively approximate analyse computational fact firstly relate cover exponentially dimensionality expect construct secondly mostly depend take total construction cover must update fortunately logarithmic calculation root inference equal length contain depth turn calculate node bayesian require invert every step action look depend take take tree thompson operation online examine policy demand inference completeness thompson leaf tree wishart generation dimensionality take need matrix inversion employ thus complexity calculate partially sample small dimensionality think matter gaussian gp substantially higher simple model independently resort iterative cost trajectory prohibitive gp two experiment analyse offline online offline online online linear finally gp state velocity discount plus cart extension move cart without episode end cart area episode proceeding cart environment average estimate bar percentile order inter compare discover policy domain need order find fail excellent vast majority present stable behaviour domain relative even become reach near optimal performance order episode remain unstable reach policy although small consistent attribute firstly offline car start everything else exploration thompson online perform quite fig thompson obtain good offline domain propose bayesian approach dynamic order cover work another disadvantage gp thompson thompson performance much cover gp model overall gp rl show main reason tree sample exploration thompson additional advantage suited exploration problem unfortunately thompson practice nevertheless online offline thompson disadvantage low problem high dimension tree bottleneck estimation representative update practice seed approach purpose idea method policy promise run tree future exploration policy bayes search finally continuous action efficiently cover space recent tree search metric bandit may bound perhaps upon mdps consider rather base tree suggest observable see also acknowledgement like thank anonymous comment improve thank project sequential making propose tree reinforcement employ generalised model update combine thompson dynamic environment continuous space demonstrate gaussian square reinforcement agent must learn act feedback delay learn planning environment general online probabilistic near optimal environment far mainly generalise scale process tree tree structure sequence initially near fine partition low suitable cover investigate bayesian multivariate well benchmark dynamic programming consistently outperform remainder introduce discuss section explain contribution model comparative conclude discussion reinforcement act mdp agent observe current environment receive state agent action define history reflect discount discount goal agent expect exist environment policy ill condition concrete complex history policy bayesian environment environment addition select encodes belief environment utility make optimisation deterministic policy effective thompson know name stochastic greedy show suffer small bayes policy mdps reinforcement large reason firstly lead secondly reinforcement unbounded perform focus prior use multivariate since close non parametric discuss work markov distribution switch propose prediction converge mainly focus discrete observable tree tree generalise define previously bayesian reinforcement perform relatively square online approach partition estimate generalise gp predictive employ utility estimation gps computationally demand contrast structure cite rl marginal dynamic heuristic implicitly account notable exception bayesian quadrature finally treat dependent gps introduce apply reinforcement thompson mdp carlo bound bound function employ partition avoid efficient generalised endow multivariate underlie environment model linear dynamic use policy experience action main advantage perform property reinforcement heavy policy action construct cover tree simultaneously infer estimate cover efficiency partition density overview alg online bayes optimal step episode start stationary draw number exploration policy cover trajectory optimal policy episode new cover necessary updating calculate observe state add leaf p updating contain first cover build describe approach dynamic sec overall complexity sec metric thing efficient node correspond tree explanation cover require point constant correspond node arrange node let tree child proximity unique interpret secondly directly rise search root parent logarithmic efficiently nod adaptation create build new tuple decrease stop property child state explain node problem update fortunately solution prediction context specific action calculate notational simplicity neighbourhood
analyze expert discover pathway protein etc frequent subgraph scalable algorithm literature feasible frequent serious issue still need attention since limitation serious protein fact huge frequent subgraph namely redundancy frequent cause semantic subgraph differ infer mean significance frequent pattern subgraph subgraph patterns size initial subgraph though main define mining subgraph incorporate knowledge ability substitution work protein structure availability substitution application quantify substitution label subgraph clique sequence discrimination orthogonality base unsupervised help task approach dedicate task remainder organize discuss work area subgraph background approach describe setting present experimental result discussion worth note rest paper term subgraph propose subgraph discovery maximal straightforwardly select orthogonal redundant representative subgraph try high discrimination objective pattern term subgraph create allow com subgraph occurrence occurrence order strong term subgraph use quality select subgraph design select cluster consider subgraph weak base learner well subgraph selection base structural statistical discrimination often help dedicated selection discover large method specificity approach consider protein substitution quantify form substitution use substitution information matrix subgraph substitution represent though substitution would quantify substitution subgraph function subgraph pair score equal threshold preserve overview illustrate figure follow substitution feature novel protein sequence subsequence substitute substitute believe substitution impossible obviously impossible protein protein structure consider positive score generate pattern ensure unlike fundamental formal dataset node label call set frequent subgraph give alphabet substitution appear substitution negative protein substitution matrix likely give magnitude say elementary mutation measure stay obviously certain substitution certain stay divide mutation mutation possibility q substitution correspondingly substitution measure worth note pattern substitution give score possibility substitution normalize substitution iff user iff given simply merge occur correspondingly occurrence substitution similarity pattern since p sort divide choice threshold frequent subgraph extraction feasible limitation subgraph selection rate subgraphs frequent subgraph formally classification perform cross protein classifier I bayes simplicity technique conduct examine efficiency representative subgraph effect change substitution substitution size approach dataset pattern among substitution fold obtain average ds subgraph report show considerably subgraph exceed subgraph reach ds ds substitution ignore ds ds ds classification result help select really representative significantly huge ds reach almost ds mention metric support reliability besides substitution describe evolutionary effect protocol substitution subgraph subgraph use subgraph report noticed substitution whole frequent subgraph clearly notice small relevant subset also rate achieve protein substitution appropriate ds ds subgraphs impact result threshold check select classifier classifier decision besides na I nb protocol setting experiment different substitution respectively nb dataset frequent subgraph red comparison blue gain reduce considerably especially threshold number exceed substitution reach accuracy show even reach confirm selection contribute believe nb perform global unlike attribute perform attribute protein enable select subgraph group structural since fast selection subgraph perform pattern size pattern concern original frequent subgraph subgraph substitution use tendency substitution subgraph towards cut peak substitution another region demonstrate big evaluate trend subgraph representative subgraph selection com report substitution threshold svm iteratively discover subgraphs subgraph train com threshold report build show pattern outperform promise dedicate unsupervised well indexing use frequent runtime substitution threshold substitution threshold problem due substitution high substitution runtime way run fast parallelization substitution group subgraph size order treat separately propose mining representative frequent subgraph transaction pattern exploit substitution matrix select representative pattern frequent subgraph reduce considerably subgraph enable worth limited help subgraph indexing inspection promise could
high procedure exist tensor lexical semantic tensor suggest lexical section learn rank choose mapping compose component subject analogous noun case input extract corpus corpus extract represent etc combine th vector supervise technique require extra tensor high linear application idea higher encode recursively function thereby address operation case general propose tensor naturally tensor multiply first sentence dimensional return require input combine output input first semi represent construction like subject object variety object pair vector correspond tensor subject sentence matrix previous example tuple argument corpus tuple set j multi learn tensor function tensor outcome previous step learn tuple token token corpus distributional semantic small increasingly version function take tuple tuple identical tuple exclude demonstrate linguistic modeling occurrence concatenation web corpus mid english wikipedia corpus pos collect frequent corpus subject contain use one subject list item collect occurrence frequent content save stop frequent extract occurrence ignore co occurrence count transform word raw frequency multiply score weighting pick occurrence dimensionality distributional affect quality lexical semantic work vector problematic learn matrix tensor cubic input dimensionality reduction svd distributional distributional semantic fundamental multiply kronecker composition multiplicative nature unlike produce allow fair comparison singular decomposition pick first column reduce representation factorization non negative matrix normalize matrix implementation project lin square measure sentence similarity cosine adopt multiplicative additive method respectively multiply vector normalise addition consistently performance implementation compositional semantic live sentence matrix multiplication implementation formalism sentence sentence approach multiplication multiply rr regularize deal reduction nearly quite rr non rr simple produce fast speed tune minimize intermediate tuning example find combine subject construction extract corresponding vector normalise regression routine regression result second estimation tensor consist pair sentence would sentence lexical overlap rate subject composition sentence rating produce achieve noun construction successfully model performance nmf multiply multiply confirm see attain add considerably lower construct sentence similarity rating sentence regression perform nmf improvement kronecker also neither even significant human multiply regression svd human nmf kronecker multiply regression svd svd kronecker define multiply show multiplicative kronecker predict sentence additionally multiplicative extreme simplicity take multiply identical human setup although involve two component multiplication point hoc initially tie nature evaluation object stay pair considerably involve kronecker recommend statistical routine could regression could training phrase contain frequent limit technique tried tensor clearly vector dual application tensor contraction produce vector sentence form kronecker former second thus counter able say natural remark reduce counterpart multiplication model limit although perform main compositional compositional mechanism specific case common rank induce semantic model evaluate exist provide show extended regression allow kronecker subtle argument order quantification logical operation focus automated training set sentence count construction order sentence subject object extend syntactic application include paragraph want categorical scenario allow support ep independent acceleration fellowship ep mm mm compositional relate formal semantic method tensor find analysis nature learn suitable solve subtle distributional model internet call text subtle sophisticated language orthogonal approach semantic complementary strength formal implement content logical define systematic syntactic composition expression form syntactic compositional whereby mean phrase however reduce adapt language application detection classification relevant task truth logical logical expression contrast distributional linguistic state mean mean examine context token co occur frame token unlike formal distributional composition provide syntactic development distributional compositional outline brief history semantic overview tensor base compositional traditional semantic section experimental evaluate approach distributional semantic follow future paper researcher derive since semantic challenge attract composition proposal sum phrase simple wise effective semantic operate mean calculus composition reflect syntactic formalism distributional et formalism test method component multiplication kronecker argument sentence tensor argument construct corpus kronecker outperformed method categorical kronecker method efficiently implement provide meaningful indirect distributional noun idea noun rule context formal np np vi l vi follow like typically lambda abstraction inversion remain translation language predicate theoretic multilinear map geometric modern semantic example serve illustrate function correspondence property algebra determine produce multiplication multilinear map correspondence tensor matrix refer tensor rank illustrate superposition describe likewise tensor see vector matrix element element underlie element freedom tensor superposition superposition weight basis
hypothesis multinomial contingency parameter multinomial contingency assume array observation equal z parameter constrain mode h write support support hypothesis hypothesis modern mathematical integration horizontal result section test conditionally b x conditionally value table calculate elementary h figure contingency table priori equivalent uniform find bin bin upper horizontal convolution show support find discretization use vertical support procedure second relevant use model h b respective elementary independence discretization b vertical w w horizontal w vertical discretization significance acyclic model result graphical learn structure mathematical value dag ci ci perform fast incorrect true reject independence de universit sp independence ci receive computational intelligence indicator include network especially task propose test test alternative frequentist approach diagram acyclic dag help model compose node represent relationship help understand problem good involve variable write conditional base relationship conditional involve sometimes learn structure test remove arc connect return dag minimum motivate accuracy learn recently hypothesis hypothesis elementary component hypothesis point hypothesis region space evidence support complex component eq arithmetic arithmetic joint multiplication respective cumulative two variable use marginal convolution bins return bin operation would large size bin without convolution show h calculation horizontal select assume cumulative represent bin attention bins bin necessary tail bin axis procedure bin distribute space second bin vertical algorithm bin uniformly vertical sum w f convolution variable denote cumulative normally eq
repeat importance score genome give weight experiment effect base accelerate gain long cycle begin operate gs marker favorable allele heavily avoid recommend approach aim rare favorable gain component region popular try part fashion approach study transfer successfully method incorporate model evaluate genome turn good importantly keep program useful genome partially kind select possible genomic specie family genetic award theorem study distinction make genetic effect additive individual since expect genetic argue advantage marker introduce genetic information additive effect use genomic design post genetic genome genetic mix marker additive additive genetic model successfully genome marker information always study empirical study bi population marker response express design follow normal kernel marker reproduce regression connection recognize feature refer trick say variety calculate variety common choice linear polynomial kernel function though available real kernel exp expansion reveal marker genetic use incorporate additive effect marker order marker additive implicitly additive complex marker show accuracy usually increase effect lose additive kernel argue advantage marker effect information local chance percentage produce narrow produce line narrow broad gaussian matter trait marker task local semi supervise obtain many local differ kernel calculate aim incorporate believe local local genome section way information genome wide utilize propose multiple single multiple genomic kernel model good kernel commonly use function calculate space calculate interaction perhaps component effect interaction vanish except additive additive f additive genetic linkage justify notation element multiplication weight combine case principle technique fisher square suitable propose heuristic weight pearson coefficient alone alignment estimate attribute j g jk jj g incorporate jk jj effect component group calculate g g case kernel estimate g e account effect weight maximize restrict reduce maximum estimating multiple model marginal model component one vector twice likelihood component calculate regularity nan degree freedom alternative show distribution equally weight degenerate study contribution well likelihood recommend practical identify region phenotype repeat detail level nevertheless develop hypothesis sequentially family discovery keep scheme hierarchy hierarchical control family wise adjusting level test continue significance node factor fine level provide improvement include local discuss utilize marker partition random correspond region marker final coefficient function value zero matrix rewrite value important use hierarchical form genome region level introduce square regression etc essence focus marker genomic snp marker million exhaustive reduce nested region genome calculate separate marker term linkage combine genome divide linkage merely proximity separate sequence grouping effect low level trait allele individual presence marker guide hierarchical incorporate membership probability marker group ht refer marker window consecutive marker let matrix marker partition respect cumulative marker kernel marker specific kernel form kernel position calculation involve calculation kernel select marker genome marker adjust smoothness locality single segment genome structure come build block schema predict use structure fitness trait genome favorable lose segregation marker region singular ill condition well shrinkage shrinkage
exist family choose random follow distinguish drawn success distinguish sake follow let associate condition satisfy distinguish give provide could impossible task distinguishing draw versus run sup cardinality formally disjoint size st pair indistinguishable sample recall closeness proceed least heavy part light note parameter save factor heavy roughly show heavy learn without know heavy infer element inherently incur extra heavy heavy relaxed use minimize total individually achieve support sense approach achieve subsection start heavy light truly versus truly threshold consider write shorthand want rhs expand result elementary possibly side inequality bind rhs q rhs bound corollary note distribution chebyshev sample probability except probability triangle inequality rhs dominate reverse role use easy pl corollary show theorem probability low frequency evenly b tb equal probability pick check heavy light version probability pt pt conjecture corollary observation berkeley edu university ed ac uk stanford com com closeness test two precisely two set versus far give factor factor establish sample independent basic setting distribution want far henceforth closeness run time sufficiently check closeness correspond natural naive size require match theoretic might closeness sense indeed give history computer science date dependence previous logarithmic fundamental time critical resolve complexity closeness factor also closeness problem previous contribution similarly closeness setting allow closeness correspondence closeness low bound require oppose robust may closeness proposition idea closeness useful g provide estimate statistic considerable cs community decade work recent survey closeness property property monotonicity paper explicitly question journal pose closeness uniformity distinguish uniform uniform connection expansion uniformity subsequently give bind resp correspond define resp vector poisson result closeness access run sample least versus theoretically see hand closeness testing uniformity value know exactly intersect closeness also theoretically case easier hard distinguish case require continue hold upper trivially distribution proposition analysis distance complexity theoretically note outline follow closeness stress yield closeness seem possibly achieve support closeness two element mass essentially part second generalization two step closeness filtering since improve via reduction suggest estimator see numerator otherwise fact term conclude define cauchy schwarz lead support define variance due I expression moment expression divide yield variance since fix note p divide except replace sum poisson thus sum expression th first complete proof proposition establish theorem view apply chebyshev compare square show consider expression dominate consider expectation chebyshev expectation multiplier ii I mp bound need compare least nm closeness norm sample replace easy distance versus occurrence element return characterize establish theorem observation x I involve I easily independence occurrence domain element I ix formula moment somewhat compute variable mean x n run take variable poisson thank combinatorial kn kind algebraic q polynomial equality chain k prove eq well prove use compute variance unbiased wish quantity equal w wish analyze bivariate map poisson one notation wise schwarz chebyshev return within probability
validate h eq recursively lemma replace proceed w replace display substitute back get incoherence complete exact joint incoherence extension namely approximation structure joint incoherence similar sparse hardness plant capture importance row interesting relevant chen comment support nsf feasible satisfie duality sub get proposition must op optimum op eq op q display lemma mean fact incoherence moreover q similar bernstein th column eq follow statement incoherence similar treat bernstein h large enough similar fashion p inequality union h p number p inequality corollary proof differ except incoherence statement subgradient op op op inequality rhs positive op proceed show nc similarly clearly f prove lemma eq suppose line p inequality line lemma incoherence prove complete theorem indicator variable operator adjoint eq apply bernstein inequality constant inequality assumption similar manner lemma th column matrix get fix zero random variable assumption w h ab describe plant clique adjacency ji plant subsample non matrix joint recover special decomposition polynomial find q mean recover plant simplicity integer suppose uniformly ij n th row word row easy rank standard incoherence follow equal otherwise leibler convexity kl abuse q divergence distribution parameter direct eq randomness last hold n low randomness lemma completion show restrictive study incoherence log recover applicability extension projection improvement plus decomposition plant intractable interestingly joint aspect observe entry recent demonstrate remarkable certain incoherence possible exactly reconstruct previous incoherence necessary requirement prevent incoherence seem interpretation condition semidefinite parameter require proportional instead incoherence artificial constraint eliminate standard incoherence semidefinite incoherence consequence semidefinite high nuclear improvement achieve base norm define maximum column differ obtain strong theoretical projection completion structure supervise problem improvement norm broadly follow play crucial incoherence relate ask matrix necessary clique decomposition require joint incoherence plant clique study widely imply separate rank inherently incoherence condition reflect aspect briefly survey work detail present first norm theoretical alternative completion consider require incoherence extension svd structure completion supervise inspire improve upon result problem subsequently prove incoherence result necessary statistical plant clique establish principal pca take submatrix organization incoherence need completion projection matrix turn matrix show aspect defer notation bold letter capital th universal dimension nuclear completion include easily translate factor set index arguably completion minimization sufficient optimal equal row avoid situation become assume incoherence svd say satisfy appropriate result ij ij dominant factor natural incoherence restrictive several setting require incoherence incoherence comment completion exist pair distinct matrix rank parameter determine know ahead incoherence condition uniquely determine incoherent computational completion recover alternative recovery q complexity proportional incoherence qualitatively necessary ensure column concentrated assumption matrix right explanation often affinity matrix cluster discuss discuss multi access space structural ambient denote column assume orthogonal unit modify standard incoherence parameter program recover since pn recover additional writing translate incoherence parameter avoid dependence discuss ideal whereas strictly discuss completion interesting application structure semi cluster object affinity svd observation column span extra improve affinity link structure consider distribute bernoulli satisfie incoherence incoherence due affinity unique rhs take fully thus possible exceed restriction undesirable unnecessary eliminate plug succeed last rhs multiplicative ignore small moreover completion problem decomposition completion incoherence incoherence follow convex least provide cf semi requirement incoherence naturally dual joint incoherence require polynomial connect connect node pick hence clique clique graph plant clique overview regime polynomial despite effort widely intractable polynomial hardness certain utilize hardness computational submatrix adopt computational plant clique plant graph polynomial probability plant conjecture success proof appendix follow statement decomposition solve encode finite bit modify plant intractable theorem holds plant assumption standard decomposition therefore unlikely intractable semidefinite note
principal respectively eigenfunction feature extraction principal analysis processing pc offer many disadvantage pca extract favorable sparsity actual sparsity interpretability different expression pc sparsity interest sparse network machine bioinformatic enforce constraint pca term pca problem literature modify nonconvex lasso tackle function present semidefinite sdp augment favorable consider technique solve thresholde relaxation iterative conjunction truncate present prove sufficient sum scale identity e update utilize inspire work auxiliary present scan polynomial candidate solution original lie candidate retrieve result unit problem fully hardness principle auxiliary technique condition moreover novel compute version complexity interested sparse constant positive semidefinite identity always rewrite decompose eq matrix write consist index nonzero element support inner submatrix maximization denote singular whose principal contain correspond left discussion turn hardness identification optimal exhaustive among support compare grow complexity exponential indicate hardness solve develop search hence sparse complexity value even grow rank principal time trivial optimization become element one index hence integer length index operate select index large conclude subsection note optimal support compute consider case hence generality nonzero element hence hence development utilize auxiliary generate span interestingly unit polynomial polynomially solution obtain auxiliary efficient technique translates rank critical rank value meet collect belong one element metric obtain build major work show cardinality bound develop build construct compare n constructive proof present subsection begin parameterize radius c equivalent metric index intuition behind vector actually manifold absolutely large give point continuity discrete expect retain around formation interval absolutely sort occur intersect sufficient determine construct candidate support lie principal retrieve optimal intersection combination element pair illustrate partition matrix interval sorting interval dash point intersection create interval exceed support sparsity partition region adjacent curve support far check exploit feature implementation goal construction sparse interval step determine intersection examine imply sparse principal rank even statement length constructive auxiliary angle hence metric support correspond support obtain complexity select absolutely intuition auxiliary notice element function element sort give point sort retain sort around cell lie contain intersection set exactly element refer illustrative cell create curve cell carry curve normal determine set cell collect identify cell return desire vertex ignore cell follow compute cell least vertex determine recall vertex intersection say u solve equation sign unit vector ambiguity affect set intersection space linearly simply index correspond neighboring ambiguity regard particular belong due combination combination one cardinality n b rr include build build intersection cardinality mention build fully component principal subsection induce intersection support indice individually present reduce exploit candidate neighbor order curve intersect conversely element order candidate adjacent interval neighbor formally lie vice versa order th curve precede set neighboring differ g neighboring candidate pairwise intersection curve sort intersection sort curve intersection successively consecutive intersection determined appropriately update illustrative step examine fig keep track highlight black vertical intersection change correspond region depict support set consecutive candidate one change intersection implement serial present counterpart candidate intersection point step construction finally successive operation serial parallel disadvantage ij candidate pairwise distinct candidate associate aim reduce intersection compute examine complexity remain modify serial execution set side denote index large curve differ element curve intersect member small sake curve continuity exist lie curve among th candidate
orthogonality imply identifiability version n dd jt cp decomposition lebesgue cp algebraic since irreducible cp decomposition algebraic hand side resp dimension count q explicit prove cp observe cp tensor question strictly algebraic proper tensor atomic decomposition signature briefly relate define th dx dx coefficient taylor moment linearly random transform real statement moment analogous q equality follow hold consider independent independent independent restrictive several exposition reformulate algebraic explicitly term imply atomic decomposition I signature atomic decomposition signature return imply strictly fine uniqueness correctness proposition uniqueness singular illustrative purpose introduce extensively study noisy reduce w r r orthogonal decomposition correctness uniqueness decomposition guarantee signature improve repeat possible signature average nine symmetry problem presented numerically cope introduce singular value computation stability respect govern pseudo inversion step also degree projection furthermore orthogonal reduced singular decomposition make decomposition identify rank model acknowledgment thank discussion join j tensor machine study product decomposition across relate decomposition non efficiently reliably decomposition practical appearing time discover name process extensive survey recently application orthogonality literature identification moment statistical estimation task survey orthogonality branch impose orthogonality obtain optimization tensor author obtain decomposition discuss decomposition language specific orthogonal tensor wise orthogonality constraint directly singular decomposition give decomposition existence reduce decomposition variable reduce series singular theory theoretical numerical tensor notation ease basic definition tensor useful transform application tensor regard index kn tt tt mathematically give arise fix partially tn take submatrix furthermore create tensor tensor n cb eq outer product tensor useful calculation follow similarly outer product compatible let cb ci c briefly notion different slightly generality b identification product check product identical scalar product compatibility n entry jj orthogonal prove cb I trace cyclic put ia kk decomposition exist compatible introduce compatibility orthogonal compatible signature atomic strictly compatible signature direct checking compatibility scalar orthogonality index exist unique compatibility exist unique rank ingredient uniqueness singular convenient cp decomposition orthonormal singular e change sign unitary span condition include let atomic signature atomic signature small
th represent connected distribution th entry entry similarly see consensus infer exploit label correlation analyze c meaning prediction mod label identity analyze property perform label infer rank r optimizes rank loss obtain consist node connection proportion instance result walk probability node node wish establish label look perspective th column th group eq construct node label g step person choose group interpret similar walk case reach person reach interpret person choose end induction another iff th label start node label sum td th group reach cd jk label probability base v compute connect label relevant irrelevant relationship irrelevant pair relevance prove minimize rank score define posterior accurately ranking loss optimize briefly review metric describe auc area metric one greatly number classification thousand tag couple tag irrelevant adopt formally matrix nn n entry correctly fundamental difference rank label two rank label matter necessarily optimize average propose combine perhaps simple multiple average prediction matrix simple error base formally adopt last equality nothing square attain take application simple combine dependency base fail dependency phase average solely prediction motivate label group loss account pair rank contribute rank irrelevant one within pair pair pair indicate consider indicate b indicate portion therefore enforce label across indicate generality example derive relevant irrelevant py certain extent enforce namely py py product py preference instance follow tackle relevance correlation need partial correlation label symmetric estimate objective goal minimize employ average partial correlation latter formulate inner two yy problem take take obtain produce problem reality mle order observe treat independently multivariate estimate covariance community effectiveness summarize dataset consume account correlation cccc medical certain error rank regardless label rank label relevant statement perform precision evaluate retrieve subscript ignore retrieve average precision method baseline evaluation metric compute base base denote bm baseline report voting sequel average comparison would able improve performance effectiveness base model correlation outperform follow base one prediction performance r average next baseline observation compare bm one boost even use simple combination improvement surprising method method sufficient correlation especially correlation improvement improvement superiority baseline prediction task choose different consider outperform baseline wide applicability cccc method rank avg cccc avg bm cccc avg bm cccc bm precision bm attempt address learn classification predict relationship treat handle multiclass prediction relevance paradigm inferior prediction category relationship label utilize label parent fully multiclass drawback label increase category ensemble excellent relevant simple method voting copy bag base boost explain theory ensemble method skew mining access decade probably present bayesian infer model matrix factorization similarity maximize prediction diversity label apply solve treat stand learn thing label drawback address challenge prediction correlation optimize prediction fail challenge former correlation optimize framework algorithms experimental demonstrate superiority algorithm theorem pt plus pt incomplete task source help effect robustness model storage consideration circumstance multiple model raw consensus effective situation focus label nonetheless usually combine algorithm capture correlation classification popular classification task effectiveness model source incomplete generalization ability world purpose focus privacy bandwidth storage test finance aggregate benefit however infeasible analysis bank individually prediction paradigm situation abundance hope accuracy exploit strength access test focus meanwhile categorization bioinformatic importance although handle focus build combine without training need gap art classification correlation help exploit use prediction base address various evaluation loss desirable measure point metric translate address propose correlation base fundamentally different exploit label optimize rank quality per relevant image engine treat combine g image describe ranking since might combine purpose formulate show optimize contribution paper problem combine prediction access optimize two far work address baseline percent percent increase call furthermore performance metric hand metric fundamentally align correlation prediction method optimize base correlation testing wish model wish algorithm infer correlation importantly metric loss explore previous work nonetheless design label individual preliminary pool task treat independently multiclass prediction seek prediction base loss generality label construct bipartite prediction base apply task instance bipartite annotated letter letter class th denote connection node classify
useful incorporate hand averaging change main generalize max unified map dual entropy satisfy b b kl kl show sign tight equal substituting complete proof form max maxima corresponding naturally marginalization provide traditional marginalization sub routine problem enable derive message marginalization avoid inner routine b chain rule view node regular inference free sum generalize inference reduce max empty together remove optimal marginal tend low configuration interpret marginal obtain distribution unfortunately subtract cause subtle mix intractable calculate tractable dependency optimization difficulty marginal marginalization sequel tree secondly conditional entropy hence mix concave create difficulty optimize optimality strongly convexity smoothed b small smoothing map primal define exploit negativity kl formula positive transform map obviously hardness new deriving novel either relax outer mix tractable tractable focus energy mean graph exploit section framework adopt advanced like clique bethe liu advance like start characterize marginal map node satisfy parent sequentially polytope equal polytope entropy dependency rule energy decompose singleton pairwise easy deal graph motivate bethe bethe bethe involve truncated tree justification give exact usually surprisingly regular bethe approximation nonconvex pass provably idea construct subgraph assign tree show concave ab edge appearance replace approximation outer free always bind knowledge convex use bp integral I globally solution sketch tree marginalization apply argument proof arbitrary suggest tradeoff concavity find optimum small enough optimization bethe em cause difficult apply likely large mutual point energy less likely bethe difficulty tradeoff concavity bethe approximation excellent derive pass bethe energy instead energie version anneal generic value pairwise bethe free energy via general unified inference correspond product objective max product correspond marginal sequel product bp role singleton determine bethe vs pairwise bethe appearance initialize correspondingly perform update singleton belief message pass solve lagrange multiplier assume stationary fix ix sketch kkt multiplier consistency give mostly bp singleton map run directly gradually decrease iteration initialize take message interesting bp hybrid update list follow weight calculate singleton obvious message follow temperature let fx x ix plug drop intuitive message correspond marginalization special serve product max max currently local solution summation single take marginalization maximization problem parallel marginalization maximization advantage variational naturally marginalization hybrid passing differ replace product message regular message message optimality product viewpoint move pseudo belief leave ensure fix proving optimality bp mix bp mix bp belief map typical mixed marginal explicitly follow simple algebraic transformation concentrate continue belief interpretation mix beliefs eq substitute three mixed constraint product constraint currently x constraint ingredient enable local require max node subgraph ij ignore entirely max c ij provably point maxima configuration satisfy differ sketch proof consistency fact summation analyze transform inference task similar max illustration marginal energy mix proximal iteratively smoothed distance force divergence nice converge relate take proximal point solve energy update I proximal inner truncate adjust opposite anneal anneal vanish interestingly duality pure marginalization transform entropy respectively provably interpret proximal mix valid proximal provably bethe effect solve belief although bethe form provably global proximal convergent inner loop marginalization convergent norm accelerate leave problem treat maximize see ascent start introduce marginal remain polytope arbitrary b restrict maximum value small I connect em distribution second optimize marginal happen go back primal rewrite hide connection em coordinate ascent variational objective various approximate field variational obtain bethe relaxation equivalent subset special fall discuss represent extreme theorem encourage solution likely become local optima restrict mainly clarity bp derive similarly undirected call assume case without generality cluster graph call approximate replace higher locally polytope marginal k consistent intersection clearly tight polytope entropy linear entropy respectively far overlap max max call correspondingly ccc abc marginal eq derivation mix give message decode b kx bp special bp maximum task detail work diagnostic challenge bethe include state art mix bethe regular sum product max product bp max bethe max algorithms converge initialize initialization solution product message report run proximal product bp bethe bethe algorithm bethe implement valid span tree span method proximal message inner loop additional art use maximum searching step trial initialization trial sequentially maximize predefined approximated bp initialization normal control strength result randomly max generate find span tree element draw non sum node show hide globally panel energy optimum fig respectively globally tractable relative define b test diagnostic construct select version mix product bethe proximal bethe relative error percentage vary mix bethe bethe bethe outperform circumstance three almost always optimal dependency max make difficult explore mix bethe degenerate coupling probably worse less accurate able phenomena bp message pass bad mix bethe proximal bad bp worst pure interestingly performance max bp opposite trend max bp bad bp gets couple bp subgraph bp sum bp view probability trial bethe bethe coupling cycle couple cc cycle cycle max part approximate relative obtain proximal strength structure bn b diagnostic bn framework marginal direction improve truncate optimize optimality component convergent learn science foundation ph fellowship lagrange multiplier lagrangian define directly plug sum globally map ii therefore conclude proof theorem belief consistency semi global optimum note maximize map apply max provably secondly maximize condition denote node parent eq interior old equality satisfy ib ix b ix bc b jx db j b ix show ad remainder liu liu lemma thm definition marginal maximum posteriori posterior subset problem uncertain unfortunately np hard marginalization map naturally marginalization easily extend variational pass sum transform standard marginalization globally optimal bound objective empirically algorithm significantly approach local pass method hide bayesian random field powerful reasoning biology construct answer probabilistic computing posteriori np hard case algorithmic advance include development variational propagation provide circumstance type involve task posteriori probable explanation mode joint include normalization evidence focus seek configuration remain marginal map problem marginal play role scenario uncertain example arise model prediction robust optimization variant observe treat framework task list difficulty np complete speaking task efficient reweighte dual book attractive pairwise max inference hard alone hard max elimination order marginalization reason less map marginalization problem serious problem problem partition propose novel hybrid product message convergent iteratively marginalization present variant clique also discuss highlight maximization theoretical subgraph numerical exist hybrid message pass local expectation straightforward node get sub art method elimination mini message mix map max operators style stochastic propagation optimality relatively complicated introduce propagation minimized knowledge provide variational bethe reweighte mixed analyze convergent discuss em section factorize index interpret exponential family factorization represent undirected correspond set clique connect purpose mainly restrict sum calculate marginal single generally straightforward summing variational typically set marginal principle unique unique satisfie form abuse denote distinguish clear key result variational rewrite global equal marginal original sum inference free negative transform marginalization continuous calculate
formulate loading square loading pm th loading maximum available result rotation display lasso yield observation orthogonal may produce mc correct lasso model aic mc lasso mc lasso lasso mse mse mse mse mc mse mse mse lasso mc mse mse mse illustrate represent factor loading mc orthogonal model select estimate loading relatively mc loading respectively ht exploratory fail disadvantage come loading analysis handle monte carlo simulations simulation propose mean square solution often loading loading future interesting construct penalization nonconvex complex structure observable variable lasso apply mathematical lasso topic theoretical criterion complete em regard penalize likelihood n n ni n tr old old old ne n n e old old ni old old old old n old old old old old old old old old il cm center quite penalize approximate penalize penalize loading factor say use loading closely loading orthogonal solution type researcher least descent remarkably fast algorithm nonconvex penalty coordinate entire coordinate descent explicit maximization explicit em algorithm utilize likelihood formula update equation correlation loading unique correlation expectation old old old old old old ii derivation log function function usually coordinate utilize dimensional update maximize ik jj equivalent follow close lasso penalty carry degree interpret estimated factor loading fit reasonable yield large adjust tr df experience fit turn improper improper solution make slow handle add respect basic tr occurrence improper selecting difficult
cluster unique intersection else add v ready learn invoke either sign incoherence high complement svd contain advantage quite whereas guess notation condition e let direction unit dictionary remove singular disk proof overlap yy ii recover note problem subsection succeed even direction direction maximum variational value q imply separation apply theorem even close high return imply empirical corollary recover need iteration bottleneck elaborate noise uncorrelated dictionary construct connection make inner roughly preserve overlap combinatorial make connection decomposition early presence exactly hope sample locally incoherent dictionary plausible compute refine svd denote submatrix index span column j inner product j key suitably suppose incoherent true probability universal succeed simplify constant whose precise establish claim correct sign recover support j besides imply column incoherent simplify sample matrix I disk vector l directly suitably convenient expression simplify compute second desire analyze denominator whose invoke bernstein first first xy set certainly intersect least yet claim ty ty conclude unit unit imply angle claim time computation involve repeat time neighbor w u extend triple common test succeed technical intersection need way intersection definition key analyze order analogue analogue analyze probability collection contain least notion g bounding concern suppose intersect prove number number point analyze many way collection set probability need intersection intersection set crucially part one point break tie fix intersection furthermore pair set remove lemma remove find algorithm run succeed analogue suppose corollary depend polynomially break event family another event let family former invoke invoke part lemma greater large probability asymptotically small probability set intersection new distinguish tuple bind show provably around know prevent recover vice versa currently run slow alternative overlap clustering truncate edge experiment recover enough yield hybrid succeed often thus algorithmic assumption seem empirically violate acknowledgement thank discussion various stage provably large nevertheless important variety besides clustering initialize much find dictionary believe algorithmic idea empirically agree strong wise support check number common three common intersection positive common intersection negative still unlikely intersection constant size triple triple wise ok ok tm om set shall u u u u old positive however positive connection filter argue chernoff either sample connect filter concentrated edge number vertex going pick bad set contain high claim neighbor large examine step invoke oppose distributional furthermore find overlap lemma hold location set ease exposition expense lastly average variable zero result recover enough still slight still sketch sketch modify invoke ok ax r try trade major take value away depend upon weak moment among nonzero instead anti every coordinate different connection overlap think community overlap find community pose condition constitute community outside member find community meet quasi node belong set polynomially think leave community stay purpose polynomially purpose apply polynomially albeit notion constitute natural find community provide whenever share common neighbor community correctness condition corollary fs dictionary notion processing machine application compression resolution learn draw polynomial overcomplete dictionary previously provable give rarely dictionary seminal work incoherent inner product unknown know moreover quickly true dictionary substantial incoherent g polynomially find sparse representation natural language combination choice dictionary include wavelet edge curve common hand basis redundant overcomplete building design well dictionary compression discovering refer sparse machine dictionary design identify dictionary often correspond provable guarantee nonnegative topic come guarantee design dictionary np hard combination sparse easy building uncertainty dictionary incoherent show incoherent incoherent vector refer incoherent incoherent give incoherent dictionary wavelet rich body devote incoherent dictionary basis pursuit recover subsequently give give incoherent dictionary pursuit solve weak also rank trivially focus incoherent overcomplete dictionary extend rip major provably learn incoherent depend hence assume dictionary requirement cost increase additive noise dictionary solve variant alternate minimization approach direction mod maintain step provable guarantee difficult initial basis converge incoherent heuristic elegant provably rank redundant full independently give provable overcomplete incoherent dictionary minimization converge dictionary special case generally paper incoherent dictionary initial version dependence work square sdp dictionary ica provable non rotation overcomplete provable rely generate overcomplete require support u v u depend rely able recover noiseless section first good assumption bad statement distribution interested coordinate provable polynomially however suitably average derive formula update analyze instead analyze rapidly almost norm denote use set throughout large part intersect idea product prove classic variable variance determine intersect negative large restrict non right think vector whose entry zero intersect minimum weak condition allow distributional implie intersect disjoint imply randomness sign connection pair necessarily meet condition graph positive connection consider mean coordinate connection identify combinatorial decide respectively intersection straightforward together focus claim lemma triple recover follow common let support need elementary claim suppose pr idea second moment bind claim establish lower expect neighbor triple intersection set probability remain element contain positively contain mm positively set
version weight x qx choose function yield discretization spatial valid continuous spatially position resolve discretization scheme sx construction sx mark refinement discretization feature new operation position integral accordance eqs scalar denote scalar scheme analogously apply discretization operator act ax representation discretize analogy regard operator address proper discretization operator normalization position integral implementation code library certain grid field follow introduce comment abstract class replace abstract resolution grid computer environment therefore need structural relevant initialization prevent write resolution code exploit transformation greatly harmonic laplace operator flat basis basis conjugate space class check six derive abstract geometrically simple possible grid think use default conjugate limit regular arbitrary periodic dimension length specify origin symmetry whether fourier basis conjugate space fast position yield fouri versa fast spherical basis angular quantum serve harmonic grid sphere root gauss define bin hierarchical sphere often product order list multiply subspace grid cast allow along axis rl name field apply return scalar apply return scalar scalar field dimensionality field draw space transformation power purpose discretize field instance specify target default conjugate use array class information field method space example scalar apply weight volume address sec two multiply see instance standard implementation combine exponential space spectrum statistically represent projection onto specify represent matrix form aa represent response domain perform field field generic concrete operator capable field transformation operator concrete specify target operator code check field match operator method concrete part explicitly computer routine derivative linear operator calculate problem individual denote multiplication possible random probable originally infinitely sample find acceptable accuracy computational implementation arbitrary scheme compute take default enable trace improve operator internal correlation suggest work parallelization share memory parallelization parallelization within turn parallelization library ccc scale scale scale image scale image h figure create use wiener shall serve covariance find expectation signal map calculate filter derivation posterior covariance show time range legend marker run bar variation marker solely package extend solve choose underlie illustrate implementation fig qualitative power apparent quantitative depend ccc gap show red dash line green solid gray contour panel fig reconstruct green solid line interval contour uncertainty reconstruction interval operator define eq involve explicitly major effort visualize emphasize uncertainty mask sigma wiener classic datum generation multiply mask despite wiener spectral reconstruct dimensional non background library programming resolution freedom achieve object orient comprise among abstract support preserves limit consideration concern offer formula thereby development cycle code include framework successful application wiener filter problem flexibility successfully whether regular grid moreover already thank discussion support medium support space op economics technology research make discretization necessity implementation confusion concern correspond discretization identity equal kronecker delta draw gaussian equal intuitive field inverse volume inverse aa implement user concern librarie transformation grid currently future version spherical harmonic respectively library support transformation library select library library sparse cg inverse def technique conjugate gradient dim self def inverse adjoint g grid get spectrum kk power signal mask assign noise variance diagonal diag rs adjoint reconstruct cast min max plot reconstruct min inference universit universit software package enable operate regardless underlie orient framework write library discretize act normalization field take care automatically concern derive field theory permit rapidly prototype code world operate set space harmonic counterpart product combination diversity demonstrate wiener modification try reconstruct experimental set arise numerous problem know modern information inference formulate applicable scenario resolution physical appropriate numerically analytically
transfer corollary source policy however policy return policy need tradeoff policy well illustrate divide group policy element mdp need cost exp derive third recall regret policy build mdps term function assume mdps correspondingly distance policy mdps mdp perform net previous previous mdp define mdps mdps execute keep simple policy mdps similarity mdps distance bound optimal mdp construction vice versa goal policy another cluster element minimize representative therefore arbitrary optimal correspond bad case directly achievable proof develop metric worst derive function mdps action follow kk monotonically triangle lipschitz type give preserve metric operate almost ensure two policy probability derive analogue optimal function justify derive immediately mdps transfer run set well perform mdp hence policy ignore function directly instead proxy cluster task diameter diameter define centroid belong cost cost justify derive previous e transfer take randomization exp good performing bound encode previous discrete motivate around around clustering way guarantee cluster sampling chain approach comprehensive introduction metropolis chain short use simulate anneal temperature change mean schedule convergence clustering letter letter realize chain state space chain integer chain homogeneous chain chain call periodic set irreducible pa yx kn important idea stationary enough eventually end hasting mh depth via probability index eq check target auxiliary temperature schedule connection detail particular cost normalization note repeatedly draw draw element small element solve problem course stationary irreducible discuss define transition parametrize irreducible appendix kernel distribution hasting auxiliary list auxiliary optimal initialize irreducible periodic establishe draw acceptable state x x x kt establish nx initial state derive appendix parameter start result simplify independent set path need maximize difficult specify flow process decrease neutral favor note respectively seem initially sufficiently increase ideally reflect ratio difficulty even heuristic affect anneal carefully objective specification uniformly exponential empty irreducible ensure mdps combine far full list phase solve exp source cluster satisfie run search clustering task input either exp cluster mdps unknown environment exp transfer efficacy table section trial various h c exp combination full kind follow threshold construct cluster mdp seed add mdps low cluster cluster problem c exp clustering different refer reinforcement learn illustrate effect choose high exp choose nonetheless weak highlight difference irrelevant two mdps policy reward difference distance mdps indeed theoretically south north cell front motion agent move strength wind location strength wind wind goal mdps learn mdps mdps use present domain domain cluster sense despite wind goal goal state show find h goal state run mdp speed axis color indicate mdps goal belong cluster surveillance green surveillance cluster surveillance v reward optimal similarity domain location obtain otherwise figure hill hill location automatically level hill location extend domain group location simplicity acceptable acceptable mdp consist trajectory mdp shows incur trajectory incur reward greedy cluster poorly surveillance refer discount episode section look effect exp mdps consider remain graph rest figure curve exp transfer curve transfer confirm parameter actually figure show exp run number parameter affect exp transfer experiment title figure show transfer lowest optimal intermediate remain figure low deviation framework represent mdps mdps online source element cluster extensive efficacy discuss paper domain translate apply need pure rl distance exp transfer cluster treat mdps policy box algorithm point particularly additionally unable one develop metric work development end point task derive multi accord equally implement scale type plan future proof exp number arm across part proof correspond theorem deal arm exp let tc I opposite direction randomization take randomness get reward put require remove remove hoeffding see exposition draw random sequel denominator exponent draw satisfy triangle inequality union transfer remove every arm eventually remove note coupling paragraph require mdps three satisfy triangle two proof li mdps action proof metric proof md md definition imply previous arm exp transfer knn equivalent complete proof irreducible particular path ny hence ny I fy fy put together eq irreducible periodic irreducible x kt begin diameter indeed transition transpose fact imply inductive equality inductive converge complete fix integer set value path must paths positive length finite number step cluster spread across respectively create hold optimize give definition define diameter relationship definition finding finding mdps clique clique clique cover partition minimum clique graph reduce clique cluster mdps define proof cover np immediately find complete theorem identify mdps trivial satisfie cluster invertible identify mdp way optimal definition mdp cluster diameter denote show l I ji ei I ji I edge diameter turn clique denote clique correspond iff I j clique cover clique need reward order take mdps identify mdp vertex way clique iff recall denote belong ji ei diameter clique collection clique clique clique let cluster show cost g clique show clique complete polynomial computing present increase ia ia q reverse point cluster pa j pa I pa pa us case pa detail policy show keep begin dominate showing compare figure title compare title detailed plot figure compare exp learn detailed plot present general observe show title graph summarize title describe summarize title describe figure transfer transfer title setup area deviation curve h figure transfer title transfer task experiment setup area markov process learn subset cost transfer form net mdps mdp regret transfer optimally give learn mdps framework consist measure mdps use exp iii convergent validate surveillance reinforcement transfer mdp discrete optimisation rl mdps framework modelling transfer rl mdps rl target gain previously comprehensive learn agent efficiently learn possibly task benefit resource spend learning transfer wrong task transfer accumulate need compactly achieve gain motivate surveillance large appear location surveillance goal agent surveillance expect rule former case take old surveillance determine learn scenario pattern compactly sample case possibly state differ reward distribution motivate surveillance pattern know policy mdps mdp episode accumulate episode mean try policy mdps mdp one use large policy call representative policy form analogue space mdps distance present policy mdps choose source mdps policy mdps become policy priori choose representative task purpose task particular transfer exp source policy hence measure size cluster define mdps perform hence mdp choose pairwise mdp low speak inter distance np markov chain monte extension metropolis auxiliary short thought simulate require know schedule thereby schedule summarize mdps us use mdps transfer algorithm make transfer interval mdps policy convergent optimization brief note exp transfer exp non multi armed bandit fact regret bandit cast reinforcement policy exp ensure transfer never reinforcement survey transfer reinforcement reinforcement explicitly context work aim robot learn initial situation every episode policy softmax accumulate reward extend exp source focus well stationary policy key ingredient policy base rl look represent complete task similarity cluster heuristic algorithm simple toy cost principled optimize exp algorithm greedy convergent recent action sequentially rl setting error task rl rather rl term transition derive task goal exploit derive define mdps algebra transfer mdps preserve transition triple mdps unfortunately pure absolute two mdps reward mention way issue another identical action accord modification main base method mdps ultimately determine note mdps introduce space function value innovation mdp learning function learn task function value issue difference value policy term necessarily relate measuring respectively present algorithm transfer algorithm use definition tuple reward transition take discount ps mdp optimal rs agent act state state state canonical reward agent loss generality call regret transfer mdps transfer mdp mdp policy mdp similarly denote mdp reward mdps fall define policy algorithm exp problem exp exp armed bandit exp set mdp payoff transfer policy case policy source transfer introduce pr policy reinforcement idea follow choose
use svd feature weight drop offer replace expect effect nmf I recommender compact application however yield offer lift nmf fast predictor com lar university display ad increasingly ms rate fast propose use predictor click relational reduction offer comparable conventional scheme achieve usage fast recommend exploit recommender bipartite one people ad platform great possibility innovation user request specific online start numerous participant compete serve participant bid gets reduce reduction predictor click focus large bipartite website singular decomposition relational sparsity impose click throughput importance database require operation benefit present trade speed calculation versus cardinality investigate nmf compression user website svd nmf zero store computation computation offer cluster nmf option wants use offer interesting alternative lift nmf predictor yield fast also usage run time advantage need nmf usage logistic low fast computation great key enable graphic gpu would day schedule demonstrate reduction performance area observation label click sparsity use collaborative constrain object similar interest ad website usage user website add click user website building type profile click resource factor click example construction believe many additional predictor prior etc help click task I historical predictor ad along action click click build probabilistic model click dimensionality induce click focus introduce reduction bipartite prediction namely logistic reduction decomposition v k factorization receive popularity comparable svd dimension decomposition component select approximate unconstrained achieve good document bioinformatic nmf model relevant model cast co cluster bipartite group benefit co statistical infer datum e dirichlet discrete dirichlet j beta mn nx nm z cluster vector limit model solution chinese restaurant crp gpu moreover crp truncation capable predict rate logistic sparsity instance model single per py learning become overfitte solution skew intercept correspond model newton little newton solver differentiable due work penalization logistic speed I predict scale element considerably side store weight memory consequence binary desirable prediction feature database transaction transaction display unique user web store transaction click period final training pre process transaction transaction binary mode unweighted undirected represent e graph repeat quite consume transaction inclusion unique user transaction bipartite denote present svd dense vector matlab eigenvectors supervise feature nmf decompose factor objective toolbox decide I e number negative investigate nmf order toolbox default tolerance gpu computation estimation cluster separately modality specify dimensionality use thank aforementioned gpu clusters cluster acceptable day self challenge specialize implementation capable decomposition click benefit dimensionality reduction click summarize ref feature request predictor vector zero user visit past encode specific vector specific svd decomposition vector decomposition full feature logistic dataset matrix number fall click click unbalanced also learn intercept advantage predictor predictor way select regularization regularization strength predictor regularize henceforth short bernoulli measure likelihood report respect outperform cl ccccc lift
logic via standard induced see friend friend social friend size take resp second reduce resp take resp colour refinement however take theory fractional partition linear program main partition colour partition comparison colour desirable algorithm fashion optimisation colour view wolfe convex optimisation also give algorithm colour colour happen program program hierarchy interesting open question colour certain implement efficiently section em em observation definition pt pt university cs tu tu university cs tu colour algorithmic routine subroutine vertex colour class way colour colour tight colour refinement fractional colour extend exist algorithm colour colour program lp transform potentially lp colour colour refinement colour refinement greatly program colour k naive vertex colour routine iterate colour colour colour unchanged result known strategy processing establish correspondence graph fractional colour outline soon fractional surprising theory equation graph colour refinement vertex ideally one like mapping colour refinement preprocesse programming transform effectiveness experimentally method potentially wide course effective problem graphical arise model inherent exploit approach e propagation fractional link prediction social boolean formulation second third colour refinement matrix entry irrelevant denote matrix iteratively column set partition define class put class I jj p direct colour refinement suppose partition obtain colour generally bipartite colour colour refinement directed ease presentation adopt colour total slightly terminology partition combinatorial partition equation satisfy colour refinement partition refine result enable colour refinement dimension program correspondence permutation column doubly satisfy fractional conversely partition connect graph underlie everything compare dimension matrix entry fractional robust equivalent come fractional fine equivalence equivalence relation matrix idea linear program sense feasible lp feasible programming partition call finally arrive lp lp colour refinement refinement translate space translate confirm evaluation benchmark spend lp reduce small method symmetry substantially consider put matrix indicate iterate core correspond multiply see check minimal related optimisation attract lot attention g focus integer linear search symmetric survey lp method barrier present build give rigorous colour connect symmetry fractional result theory fractional tie already introduction column call mapping equivalently transpose doubly matrix associate sometimes call write direct submatrix row column component bipartite square direct underlie undirected strongly sometimes doubly weight edge write recall partition column use express follow simple combination connect let convex let contradict sometimes consider combination element rational matrix bipartite edge nonzero represent let weight edge vertex note iterative refinement yield run refinement round round significant improvement go deterministic maintain partition keep stack still refinement initially refinement step colour stack colour least accord replace partition among structure carry colour refinement add come total involve stack vertex time colour colour cost refinement step unweighted w w doubly relate fractional every v w vx qx partition let implication converse implication partition balanced balanced prove fractional part partition undirecte connect intersection part nonempty intersection balanced joint fractional fractional let partition restriction ia contradiction strictly class equivalence relation immediately fractional symmetric corollary relation connect balanced joint usual let let joint thus bipartite bipartite prove sum imply show constant sum q doubly lead contradiction connected v satisfy balanced otherwise sum prove equation similarly prove backward direction matrix diagonal entry entry fractional entry diagonal diagonal entry equivalently even every style sep v node w xshift v node w node v doubly stochastic satisfy leave reader claim application relation fine partition equivalence one denote matrix partition conversely partition define stochastic doubly doubly v scale define dd aa multiplying place multiply note hence number solve determined matrix usual compute relation identity matrix dimension claim surprisingly relation obviously core closure observe imply small partition leave efficient partition equivalence column index partition indicate thus eq might question whether equivalence polynomial compute remain open fractional equivalence let program dual focus ease presentation v entry satisfie cx solution feasible program observe first v qx hold dd ac second solution assertion feasible nonnegative hold dd px feasible optimal solution c dy feasible thus x dd conversely feasible solution dd reduction vector simplicity follow program program matrix feasible furthermore j j qp q pe qx reduction satisfy general follow multiply intermediate clear entry make row index entry observe partition w v w v prove come sequence b j illustrate algorithmic know decide equivalence fortunately apply reduction may search iterate spend yet systematic describe want reduction computing partition directly partition partition colour refinement partition colour symmetry et argue intersection dimension program project symmetry matrix fractional least via project method colour refinement
learn domain quadratic iteration number important benefit strategy impossible apply domain category technique unable dataset transform base category handle source draw similar review relate adaptation transfer consider rather change adaptation often similar principle idea combine exist combination force svm show scale introduce world across domain additive adaptation datum class target source category start subspace geodesic source domain number subspace kernel contrast capture domain category learn additional datum paper investigate adaptation visual work learn technique near neighbor work asymmetric transformation number target consider transform show learn transformation parameter margin category quadratic feature large even datum learning apply scale example learn introduce transform training generalize denote linear source hyperplane scalar similarity slack variable directly transformation divergence scale impractical due number new optimization exploit optimization dual coordinate row let hyperplane soft margin exploit modify descent solve problem dual variable consider svm incorporate dramatically operation augment single eq explicitly maintain essential easily coordinate descent step step fulfilled iteration target augment impractical vision efficiently hyperplane inducing derive recall dyadic product hyperplane equal category seek rank source update first easily product eq cache correlation hyperplane task see combine dependency category transfer update formula translate update direct opt bregman opt pt I loop pg pg optimization without shrink briefly coordinate descent transform available shrink heuristic maintain dual likely summarize bregman depend time iteratively take account number either target adaptation approach unable run describe previous original formulation identity regularizer lack cache j fast update next account dual suggest value iterate normally value solver maintain convergence accurate iteration briefly domain whereas imagenet image category name search category object fact consistently show domain severe adaptation method target hierarchy imagenet domain name maintain dataset description description map challenge node bound box without context allow easy result use bag visual feature imagenet furthermore bag code imagenet dataset category contrast imagenet create internet image keyword total object give validation claim pt significantly fast sect scale achieve state geodesic method sect transformation large scale sect standard domain train class domain furthermore geodesic kernel integrate neighbor share compare medium scale always code max margin transform following scale min scale comprise category dataset technique state art domain split example recognition target recognition furthermore state arc recognition experiment category imagenet technique number sect outperform imagenet even example category world category without adaptation rate improve obtain scene provide target example test truth box image weak type scene image target exact number svm benefit imagenet domain learn outperform imagenet adaptation advantage use visual category provide learn transformation apply dimensionality applicability method set setup imagenet adaptation compare see difficult
entirely unsupervised feature extraction human activity sensor acceleration segmentation expectation activity central understanding human service fact service population gain decade economic facilitate daily dependent people home adapt become solution service health monitoring security etc example sensor reduce early health status main use quantify human activity sensor recognize human activity advance collection activity consequence technique base sensor gain attention activity include medical diagnosis sensor advance micro greatly considerable consumption sensor satisfactory human activity laboratory clinical environment static lie dynamic etc activity acceleration feature deviation etc however static exploit velocity method dynamic activity recognition transition detail approach find recent activity classification distinguish machine technique provide activity feature bayes machine base gmm markov hmm gmm emission activity segmentation measure activity give person component record time regime time regime activity recognition reformulate activity acceleration time approach dedicate raw acceleration specifically regression configuration correspond activity hide configuration vary acceleration activity label state formulate efficient latent result therefore kind particularly adapt perform unsupervised activity statistical aim activity recognition acceleration activity markov task observe dedicate know describe recognition acceleration activity approach perform mlp classification descent introduce prove hyperplane neighbor parametric efficiency near metric semantic attribute classification machine classification train besides exploit temporal dynamic speech recognition govern current one previously formulate series point series segment change segment piecewise model partition segment characterize require dynamic programming expensive assume noise variance segment detection type detection reject use set testing one hypothesis independently test approach well know assume arrange state activity batch modeling hmm extend multidimensional setting acceleration joint segmentation multidimensional limit require addition formulation activity criterion particular recognition hmms series segmentation use approach alternative hmms online multivariate hmm raw acceleration section observe series hide process state acceleration measurement represent propose univariate multivariate case include polynomial order rather polynomial offer acceleration datum univariate state value take variable control activity coefficient polynomial polynomial assume detailed reformulate polynomial state represent acceleration tp rewrite observation associate regime distribution movement logistic accord multinomial tu adapt capture change activitie logistic relevance flexibility observe normal parameter estimate likelihood classic assume independence time log complex nonlinear logarithm context maximization maximize give complete variable dedicate logistic consist expectation current iteration compute kk problem weight matrix probability regression estimate close maximization multinomial logistic reweighte square em square multiplication addition loop inversion hessian em code propose increment step compute equation kt series segmentation estimate regime generate model state guarantee contiguous approach acceleration expert current practice expert propose mining activity optimal bic acceleration measure performance alternative recognition evaluation segmentation segmentation truth subsection sensor acquisition unit include measure acceleration range represent body well place show near acceleration different activity limit frequency hz sufficient hz assess daily physical sensor fix sensor activity etc ascent etc specific combination transfer connect master raw collect activity transmission receiver carry wireless est cr subject age activity store file acceleration analyze activity transition illustrate ground lie lie lie ascent activity activity involve part recognize duration ask activity sequential note static activity sensor unit acceleration series record regime associate acceleration activity transition transition discuss apply acceleration activity whole describe consider consider recognize activity activity show hmm cc model acceleration human scenario transition estimate probability correspond acceleration acceleration recognize activity different transition beneficial activity homogeneous markov latent activity segmentation person instant go detail analyze activity study transition activity precise example activity add scenario four activity rather activity scenario transition transition phase analyze pseudo activity activity observe within phase previously transition stand time satisfactory apply hmm acceleration activity scenario four activity one raw acceleration illustrate use sensor hmm particular close show compare sensor sensor study three sensor segmentation cc propose whole time measure scenario consider second pseudo activity evaluate quantify section automatic segmentation carry acceleration previously six perform activity sequence activity know bayes svm supervise hmm temporal segmentation unsupervise hmm activity transition define separate state set fold classifier train ground acceleration directly truth classification acceleration unsupervise class obtain minimum classification notice unsupervised acquire preprocesse feature acceleration extraction implement additional additional feature perspective table correct classification mlp nn segmentation however nn significant distance acceleration approach computation unsupervise supervise
encounter processing generally sparse convolution address spectral spatial information reading summarize unobserved scene band reduce band mean prior scene fuse image infer observe target derive investigate maximize lead map instance propose focus moment propose relevant fusion require distribution normally covariance consequently datum fitting determinant acquire possibly heterogeneous assume conditionally unobserved scene noise covariance recover decompose whose band assign project assign project ill pose design correlation spectral note later choose prior kind successfully fusion multiple coupling prior function quite obtain synthetic well numerous conjugate inverse variance conjugate distribution expression inverse hyperparameter whereas hyperparameter estimate assume distribution hyperparameter define include fusion investigate adjust carefully fix hierarchical subspace covariance hyperparameter wishart distribution interesting signal processing work following reflect regard mean jeffreys prior indicator improper distribution justify provide full statistically fusion follow parametrization vector compose project scene noise compute hyperparameter conditionally highly obtain variance posterior complex mmse propose collection asymptotically distribute use precisely number burn estimate sampling accord metropolis within gibbs sampler easily implement relatively hybrid community property determine distribution interest sampler procedure involve detailed hybrid sample inverse wishart scene recover conjugate prior unknown lead conditional multivariate fusion consist conditionally fu move demand mix property rely hamiltonian monte carlo generate directly hmc simulate lattice technique sampler improve especially exploit momentum joint momentum pdf hamiltonian define negative logarithm distribution f function define logarithm explore scheme gibbs sampler procedure compose move gradient accept hybrid band resolution image contaminate ms depict right htb bottom leave middle propose pca use paragraph matrix subspace illustration htb experiment hyper choice informative prior integrate respect parameter hmc mainly govern stepsize stepsize adjust statistical acceptance window counting vector tn accept adaptive proceed adjust cross initial trajectory stepsize adjust stepsize trajectory potential choose quality investigate dd snr large fusion vice distortion estimate obtain belong distortion universal quality similarity band distortion reference image band image band band spectral distortion quality q size ms band distortion distortion dd dd target bayesian infer hierarchical make assume block band target introduce forward model compare fusion art algorithm fusion fusion result depict dd see hmc term propose method covariance bar dd method generate gibbs sampler mmse estimator estimation track power summarize require knowledge paragraph devote robustness zero course regard adjust image display performance propose bayesian uncertainty db map quite respect consider spectral resolution pixel acquire optical project european water band paragraph like snr band ground fusion obtained display good agreement wavelet bottom hamiltonian mcmc c dd time wavelet quite ms use paragraph image average band snr db propose report interesting bottom middle wavelet dd c method wavelet band forward concept encounter conduct pseudo ms misspecification involve forward fully unsupervised algorithm incorporation improve estimate resolution would acknowledgment dr zhang sharing codes centre spectral acknowledge valuable work handle visit notation france paper relate resolution characteristic fusion formulate framework consideration introduce scene posterior monte dimension hamiltonian fusion method evaluate several state art spatial resolution hyperspectral image fuse high resolution hyperspectral image super hyperspectral deconvolution monte carlo image low resolution fusion explore year image generally resolution address decade active topic recently hundred contiguous band target benefit problem explore decade experience merely adapt ms fusion conversely ms challenge high process fusion differs exploit always visible red spectra practical frequency spectral band band nm nm whereas nm ms generation observe compose ms conduct multi band substitution fusion challenge fuse objective demonstrate formulate intuitive interpretation fusion distribution problem ill methodology offer problem appropriate scene ms improve zhang wavelet zhang maximization maximize unknown image account artificial fusion incorporate distribution assign estimate propose instance model homogeneous markov estimate relate propose unobserved improve spectral explicitly exploit acquisition specification exploit properly recover assign resort image devote linear particular spatial resolution assume low dimensional suitable scene material bayesian generally mean square generally intractable conversely posteriori mainly mmse estimator data ms design posterior suffer presence guarantee paper mmse estimator carlo
genetic variant phenotype however potentially convex challenge become substantial association perform hundred thousand turn consequently exist computationally phenotype calibrate strongly significance novel software fit residual error build technique univariate extend multivariate univariate compute individual number phenotype overview consider phenotype phenotype marker effect phenotype error identity matrix environmental component matrix row column marker turn marker size phenotype zero alternative computationally compute statistic phenotype require obtain implement type like nr combine every iteration increase fast nr supplementary figure nr supplementary apply repeat snps computationally impractical moderate substantially burden repeat snp operation snp detailed univariate implement algorithm software comparison diversity four phenotype tc human trait high crp strong large among exist indeed substantially example minute compare compute dominate initial eigen min min min come exist might day aware current method per snp avoid snp estimate component supplementary univariate mis calibrate simulation univariate magnitude large calibrate demonstrate despite could local optima maxima minimal impact obtain calibrate require nr value systematic account alternative pair trait hour finish phenotype almost half finish trait six hour consistently significant univariate approximation use large individual marker effect min htb red blue gray area indicate wise order nan power red nominal simulation snp type indicate snp effect direction phenotype quantify phenotype opposite opposite trait pair trait datum phenotype method use phenotype association analysis subject considerable recent demonstrate gain vs multivariate indicate phenotype drive association association rather may powerful test association univariate multivariate likelihood implement however lie potential show compare four phenotype vs six phenotype analyse six test four phenotype consistently powerful phenotype analyse four phenotype truly two eight four phenotype powerful phenotype phenotype actually phenotype power phenotype correlate associate phenotype four phenotype phenotype univariate pass significance correction snps phenotype signal phenotype phenotype analysis compare phenotype univariate phenotype snps univariate phenotype phenotype analysis snp four phenotype phenotype phenotype consistent idea univariate pairwise powerful genetic occur simulation powerful prefer univariate complementary rather compete phenotype imputation phenotype typical study many individual phenotype remove address phenotype imputation supplementary miss phenotype apply nan individual fully phenotype phenotype datum phenotype imputation approach alternative drop individual miss phenotype simulation achieve phenotype drop similar phenotype present implementation genetic practical first software phenotype unlike value limitation fundamental univariate counter part addition algorithm phenotype could could remain g require barrier phenotype moderate assumption could rank sparse strategy topic computationally eigen step compute eigen amount physical become intractable matrix requirement kind phenotype hybrid birth phenotype tc four million fully miss phenotype exclude snps snps try retain option us individual snps mis center phenotype quantile transform instead log robustness trait snps phenotype high crp snps software specifically exclude individual phenotype four phenotype exclude snps minor frequency miss individual snps phenotype transform standard transform residual replace snp product transform single phenotype standard misspecification guarantee phenotype jointly practice phenotype datum snp turn rely software modify code estimate component produce calibrate real phenotype nan simulate phenotype component phenotype turn base partly impractical check genome use phenotype back original phenotype identify snps phenotype one phenotype phenotype test genomic evenly space snps snp specify effect trait explain explain trait variance trait snp either trait effect trait opposite simulate effect back phenotype phenotype phenotype snp phenotype value simulated phenotype phenotype positively phenotype phenotype present phenotype drop used phenotype phenotype phenotype pair correction power phenotype two phenotype simulation causal simulate phenotype effect phenotype causal affect phenotype causal snp two phenotype trait half affect phenotype trait opposite trait effect affect phenotype simulate phenotype scale factor uniform back phenotype form phenotype phenotype analysis snp phenotype calculate correction phenotype analysis minimal wise statistical significance adjust account test perform ms thank j making phenotype study institute institute institute manuscript reflect university institute phenotype phenotype covariate include marker phenotype residual identity together matrix eigen knn knn corresponding diag transform phenotype follow stack column kronecker phenotype transform follow interested parameter test marker phenotype test present potentially dimensional optimization along path implement often complexity make impractical reasonably phenotype instance phenotype base accelerate nr variant information ai stability type algorithm iteration fast px ai package include package px ai per fitting phenotype em ai exist inversion solve nr problematic perform address recently introduce trait cubic avoid repeatedly snp variance nan likelihood statistic approximated ratio true statistic variance parameter mle novel burden describe univariate simultaneous canonical px block nr effect specifically reduce per univariate snp reduce need ai implementation nr computation element log consecutive nr detail previous snp test nan moderate px considerably nr nr marker px em threshold thousand nr iteration often take px em couple nr snp notice threshold maximal list adjusted optimization stable iteration guarantee increase slow converge work px iteration maximize difference maximal value use fail alternative analysis use px em close nr nr algorithm good bad start approach algorithm start nr algorithm moderate px algorithm time nr every marker describe likelihood estimate mle em estimate two view joint conditional update conditional th derivation obtaining calculus view miss likelihood q conditional distribution value expectation introduce parameterization remain computationally evaluation quantity involve make cubic trait avoid trait uncorrelated addition transformation perform refer canonical transformation reference simultaneous perform eigen e individual transform phenotype rather subsequently l px describe often place newton unnecessary algorithm log likelihood restrict slight equal calculus list obtain derivative partial derivative likelihood derivative calculation derivative respect partial derivative respect basic block
lebesgue open write define real analytic analytic letting proposition b measure lebesgue taylor let outside precede although omit corollary well definite dx convenience follow definition standard denote sphere say mu follow notation central plausible intermediate open pl ax bx argument nx x nm desire proposition u n x uv note let hyperplane intersect consequently uv hz r v v fact u h hx u u open proposition cover finite exist close compact know closed
present discuss straightforward individual differ across simply modify column amount penalty column encourage group group pi figure detect perturbation toy example constraint element white zero red blue column non norm result entire row non entire many maximal challenging similar co contain element let denote formulate problem encourage diagonal simultaneously unfortunately encourage row column perturb support row column overlap induce matrix check indeed norm also k symmetric non uniquely decompose amount row column case little abuse encourage small encourage point figure estimate apply entry penalty derive group row additional property appendix discuss perturb encourage network share co task precision serve amount simply lasso encourage similarity among amount encourage q set perturb condition detect jointly solve optimization problem nonnegative encourage encourage support row column interpret encourage encourage additional graphical penalty call interior solver program examine use square group unfortunately algorithm formulation group lasso penalty overlap minimize square subject overlap objective involve project unfortunately involves discuss proximal overlap form outline propose outline admm admm easily operator rewrite augment forming minimize respect keep follow use dual use detail admm refer detailed derivation update rule complicated rewrite variable term jointly augment dual variable augment lagrangian symmetric q know proximal operator formulation solution follow introduce difficult jointly lagrangian corresponding h initialize dual I I I optimization inner k f admm per complexity compute svd intel ghz cpu interior minute code time thus admm run datum describe require terminate require terminate total b observe time never exceed several total termination indicate initialize network admm poor initialization initialization htbp iteration run second function involve convergence admm admm area mention extend admm group assumption future consensus algorithm involve lack convergence group convergence present previous work moderate problem appropriate tuning value block diagonal permutation share optimization iteration subproblem know block know partition block necessary solution block matrix demonstrate speed graphical involve necessary gap require know necessary get condition tune throughout index non entry say subject permutation cardinality complement display solution matrix support additionally simplify present sufficient diagonal minimize necessary theorem contrast theorem sufficient graphical share diagonal suppose must simplify block diagonal minimize formulation gap sufficient estimate block support general class optimization include case block block diagonal let denote sufficient matrix solve broad give condition theorem penalty theorem condition diagonal speed construct check row column row correspond conclude estimate optimization reveal block global create covariance range computational speed figure display formulation generate axis optimization surrogate display time take exploit true ratio number display positive sufficient lead computational improvement ccc equally sized display problem decompose surrogate edge edge demonstrate simulation study two perturb co provide create select node randomly serve co definite indicate eigenvalue matlab seed create step generate modification concentrate top community mm mm ij p I I I column ij equal event equal metric base perturb co perturbation metric relate apply insensitive appendix metric use wish successful estimation far performance counterpart well extent perturb identical simulation display correspond correspond colored value approach parameter ease interpretation range identify positive correctly identify ratio perturb perform smallest unlike exploit surprisingly among algorithms condition note outperform range outperform sample perform worst colored line beyond contrast sparsity situation occur counterpart necessarily section outperform metric color varied axis detail result network section colored value axis detail scale color fix varied detail l color correspond varied datum reconstruct gene disease specific include cycle cell cell growth play development try gene try gene gene play control expression apply publicly gene sample raw gene technology format genome website raw perform study log transform correct software large analysis set evaluate gene gene process focus gene contain gene suggest result expression pattern network indicate highly perturb two perturb tumor associate evaluate gene use identify annotate gene gene call set contain datum co co detect htbp gene contain pathway include gene frequently gene three column column htbp b g pathway identify gene correspond perform display four gene identify wide knowledge base project university process student goal analysis perform q log suggest achieve comparable fit appear well perturb contrast phrase b colored line varied edge validate display htbp student eight perturb label color square student two convex formulation node joint lasso real world application learn multiple context rely overlap matrix encourage efficient also project formulation permutation meet break small subproblem two approach graphical share possible direction focus either however believe pathway activate update address set guarantee investigation lead thousand future speed accelerate break problem independent subproblem tight lead greater exist stability apply formulation aim jointly performance context adaptive improve lasso option adjust iteratively algorithm yield improvement see solve linearization explore implement dual dual skew matrix equivalent dual bilinear set compact swap plugging subdifferential note solution matrix optimization problem hand suppose indicate subgradient one loss solve obtain p c c qp follow note optimality subgradient subgradient kp zero without generality satisfy symmetric subgradient support triangle get subgradient use yield k note condition prove must k simplicity general simple triangular figure note overlap contain matrix element blue norm p j rewrite matrix rewrite thus work penalty group elegant convenient derive update admm apply formulation augment hold fix update rule rule definition expand operator update derive fashion thresholding derive p scaling easy skip augmented rule rule primal derivation omit derivation rule couple note I I k f I figure illustrate metric describe identify furthermore perturb perturb column insensitive perform value lead perturb figure quite tuning index display remove size indicate
convex polynomially establishes number identify replica problem blind access many calibration e literature replica bayes mmse large define possible marginal independent hard transition carlo chains message pass intractable amp design moderate numerically grow prevent tractable dotted blind mmse sample remain transition dot qualitatively mmse rather sharp mmse bayes leave derivation mmse computing computation however mmse replica replica process large mmse potential zero unit single element eqs potential completion gaussian potential simplify mmse maximize expression allow blind hence independent count lower require regard transition potential maxima marks gibbs message ascent determine secondary show limit mmse reach local case limit pass compressed term lead like distribution simplify full elsewhere instead reader follow final blind learning mean marginal amp read marginal variance take expression learn uncertainty mu amp bilinear current value uncertainty run mu amp variance run amp repeat clear implement neither paragraph apply amenable asymptotic cavity physics state compress sense give approach eqs one ascent explain mean transition arise show approximate compress sense state rigorously arguably mmse theoretically correction test agreement region correction roughly excellent note minimization give even perfectly correction prevent able mmse prove compress improved reduce matrix complete aware analyze reach agreement ec grant triangle la consider blind calibration signal create dimension replica method appearance phase transition impossible match performance show tractable decomposition seek etc property theoretical limit decomposition tractable still poorly towards determine iid vector perform summarize element
b round join cycle anchor north west node join x pattern west circle pt black pt node overcome fitting way w w plug problem new detail ax ax x compute lasso second problem latter quick calculation correspond discuss way already solve author derive finite place selector survey instrumental knowledge explicit concern geometric interpretation exhaustive comparison well formalize invertible invertible rewrite statement form well moment deal later non p real eigenvalue positive half plane eigenvalue non laplacian probability high notation matrix zero thus reasoning imply possible em em observation square algorithm use difference way statistical via instrumental geometric view moreover bellman modification markov practical solve feed state description feature reward abstraction map discount reward show abstraction system may piece reward win piece individual game may human function express value motivation computationally intractable even completely prohibitive need behaviour aspect scope expert system behaviour stochastic decision act manually basis information directly applicable action reinforcement provide trace additional control updated influence previous clean cut line converge seminal van connection fix point iterative td formally prove td evaluation bellman td extensive bellman principled way policy albeit strong provide interpretation difference automatically give certain van discuss exist literature none differ find fix policy access linear obtain reward transition state row th element leave denote eigenvector eigenvalue stationary correspond eigenvector define function expectation consider standard ergodic application convenient general stationary long example derivation state denote feature matrix whose state trajectory repetition discount operator derivation begin instead original vector model reward model look approximately give optimization correspond case new function exactly e expect discount equality know von strong namely contraction know I proposition one see eigenvector eigenvalue hence zero entry continuity general beyond spectral radius expectation sample matrix definition q one elsewhere exist invertible implicitly enough sample solution applicable transition reward task separately learn bellman define value vs exploit seek briefly possibility thing would exist distance efficiently compute project sample need come equation bellman motivated relation relation work see approximation tell may mark describe various sample formula assume linearly column strong visit nonzero imply full appendix comment exist assumption rarely bound constant fundamental need approximation section compute trajectory state correspond trajectory infinity correspond implicitly realize method introduce consider x correspond assumption role transform side w eq instrumental formula plug begin bellman vs approximation bellman regime follow convention column write td error tw current expect column satisfy brief consequently principle external argument verify td look mechanic derivation accept observe expectation observe iterate q correlated term vanish two correlate ol requires uncorrelated input mean zero good lie fact multiply side fact detail intuitive interpretation instrumental method rewrite contain ol project column term enough derive interpretation subspace modify equation stress projection see equation component correspondence method one interpret instrumental observe projection amount residual space yield reward value along space column space correspond put recover project correspond smoothed reward way define project true produce project vector approximate notice projection vector another call formula projection complementary subspace way follow linearly invertible full exactly since two subspace common contradiction right space fulfil substitute interpretation outline full invertible line join round triangle cycle pt line join cycle pattern plot north west north west anchor north anchor north west equality p estimator equality formula equation term iterative correspond iterative expect td os w os os following eq update desire td resemble problem priori justify one treat reach td trace quadratic value approximate obtain invertible introduce substitute definition w v v minimizing would norm solution repeat reasoning define valid minimization way originally show subspace feature explicit span influence estimate derive early transform
margin fact connection neighbor multiclass perceptron algorithm motivate insight svm smoothed proxy make near neighbor definition let say might expect mp termination unitary make w x lemma mp twice restrict analyze mp execution induction initially restrict restricted update w w c w x p cnn essentially special algorithm say cnn suppose change one correspondence update begin line code mp line effect equal x yy replace I ip update without number update theorem feature feature good essence exactly accumulate cnn margin radius cnn representative accumulate algorithm multiclass perceptron algorithm multiclass fortunately multiclass perceptron input existence explain training function r r p x family correspond eq q know page vector behind boolean eq want summation dominate correspond gaussians mixture typical constraint mix coefficient drop mixture might nearby gaussians gaussian point entirely summation every zero limit subset act nsf temporal nsf science center lemma theorem neighbor cnn store keep accumulate bind multiclass nearest nn assign close every point arbitrary nn twice impractical huge training memory complexity nn reduce set preferable entire set dimensionality classifier entire tradeoff find training set rule subset minimum heuristic approach nearest cnn simple meet multiclass small though obviously train set much need cnn naturally drop understood stream overlap class bayes size grow linearly
link approach contribute social structure intra information take network detailed attribute law age people directional co work element take form e count unit unit lose information effort direct involve introduce role show normal integrate entity term link count observation geometric count monotonically incorporate rich datum sensible integrate correspond entity reflect hidden relational individually hide mixed stick breaking method community conjugate property effort special design various include choose rich embed thus performance illustrate section structure organize introduce necessary describe integrate entity propose link datum discuss assume fix whole community realize multinomial distribution binary link two entity determine dirichlet replace various capture amongst notable branch membership entity hence mixed class entity assume entity potentially work feature entity generate work variant subsequently extend nest chinese restaurant process build community give htbp number number discover role indicator membership significance role compatibility depict generative branch blockmodel base elaborate k r stick break mixed correspond generation detailed form attribute c entity entity equation impact likewise age attribute age make entity neutral mean stick integrate membership suffer conjugacy inefficient logistic gamma indicator stick breaking become q joint j k condition comparison place community author stick break generate insufficient accordingly incorporate normal function state way approach generalised model incorporate replace unified property efficiency confirm extension popular obtain make assume z ik e stick traditional stick generate feature seen hide specifically beta underlie beta conjugacy entity motivated stick breaking contain reflect use break state importance indicator opposite stick breaking beta individually single state introduction directional instead discuss derivation supplementary propose reflect compatibility community encourage put discover j ij k value unit link generation case discover model supplementary explicitly precede strategy membership community condition finite community mix membership promise membership entity counterpart replace dirichlet parameter contain element model community extension complexity information incorporation analyse three world mit infinite behaviour implement slight variation mcmc prior generation validate ten ten entity capability loss testing auc roc score derivation material perform learn successfully hyper initial hide indicator latent sample use c cccc testing test auc reality use denote model distribution locate contain relation work basic label exist provide status gender office age practice conduct inferior result attribute perform training capability include burn mix attribute geometric attribute state htbp small amongst indicate phenomenon attribute form office school gender mit reality mining describe entity towards proximity proximity subject correspondingly set proximity minute per day accord generation directional datum survey entity activity life reason link necessity ccccc ess reality htbp trace early status desirable detailed besides mcmc trace interesting observation mix stable chain active update variable mcmc measure autocorrelation ess indicator monte carlo target
give uncertain recall identity uncertain give uncertain location dependence enabling describe log form rr give laplace attain gaussian pr pr take inverse evaluate construct may cost section require perform technical aspect reader want skip normalization sum choose expression prior expression quasi hessian require laplace algebraic quadratic kernel equation analysis storage derivation particularly involved term use vector result stacking storage array multiplication take square use evaluate overall cost form inversion iterative conjugate method achievable linear feasible even active principle belief process belief active mean integrate dealing mean integrate ii delta ignore hyperparameter way compact notation denote seek approximation require marginalization hyperparameter construct tractable optimize ap reasonably determined alternative derivative prohibitive analogous turning firstly choice inconsistent constraint variation expand around give separate matched mean illustration change value zero prior generate scale away training point hyperparameter scale repetition additionally compressive strength respectively train remainder else exception ten burn marginalization negative displayed see provide superior likelihood posterior exception penalize predictive conservative variance lead likelihood consistently confident refine length scale share axis display length sample legend find uncertainty select variance simply objective variance whereas reward follow apply linear hyperparameter embed describe demonstrate dimensional uncertainty corner right concentrated embed dimension search search likely minimize u variance henceforth map signature corner corner match signature corner quadratic covariance similar therefore compare true difficult performance apply learn embedding sequentially laplace square function matching draw global embed weather model form dataset comprise community repository predict historical census survey discard record slice repository task slice scan miss zero location vary discard leave slice community machine unnormalized version embedding whose performance alternative negative dataset c synthetic temperature slice average dataset successively maximize objective fix input select uniformly across optimize slice consider process transform box min noise observation compare learn embedding predictive intend negative predictive accuracy average embed expect box embed extremely magnitude mode log previous table active prediction advantage task hyperparameter addition integrate hyperparameter result address need bayesian empirical efficacy synthetic real dimension much rgb circle black center width sep fill text black fill white draw double thick rectangle pt pt circle fill black size sep learn discover low dimensional task increasingly severe practical difficulty hyperparameter yield hyperparameter mis quadrature low embed domain learning task modeling process quadrature approach remain exception problem exploitation notational reality way iteratively select informative location bound corrupt py propose comprise belief laplace quantify section hyperparameter include embedding hyperparameter mis specification sub applicable marginalization finally previous select reduction uncertainty simple build estimator wide lasso selector method embed dataset learn function evaluation explore embed via dimensionality visualization blind solve factor latent dimensionality consider find dimensionality associate contain discover
column group unnecessary perform selection parameter remove remove therefore change remove derive analogously fa quadratic penalty great admit advantage adopt penalize linear convex suffer multiple I suitable estimate specify component mml selection criterion minimize zero probability component component gmm typical expect complete likelihood approximation involve consequently fortunately penalty produce objective component idea adaptive em minimize maximize gmm change th drop far desire lot recognition message drop denote specify integer factor factor highly large penalty weight free derivation reasonable j ik gmm converge merge certain em fa four bic quick ic quick aic cv generate bic aic cv quick bic initialize let trial quick cv take second quick appealing histogram find quick contrary quick bic always bic light quick quick give good htp aic quick cv bic aic quick cv quick minimize mml mml two datum trial trial generate initialization shrink fig bivariate first set quick second cpu quick probable component small process mixing indicate consequence actually significant remove message htp quick ic e quick ic quick ic mml determine factor local repeat trial initialize result dominate consistent noisy quick ic variational initializations quick ic take minute produce clearly overlap quick ic generate separate divide htp penalty resemble motivated traditional penalization greatly formulate quick approach finite sample quick ic criterion criterion suitable demonstrate computationally efficient result ic quick ic regular zhang mathematics university classical bic demand study hand penalty strength penaltie adaptive exploit penalization sample coincide particular case penalization information extension apply mixture traditionally adopt suitable criterion enable adaptive mixture aim choose candidate mathematical simplicity traditionally bayesian criterion bic aic mml principle optimization exhaustive intensive complex candidate testing cause high impractical effort adjust continuously regression apply shrink various lasso shrinkage mix weight however usually moreover study finite would one develop continuous penalize approximately coincide bic quick information quick ic selection mainly fold regular penalize likelihood approximate perform penalization would save especially need solution parameter one approximate quick ic quick ic complex gaussian selection traditionally difficult large make logarithm penalty datum illustrate consideration regularity consistency tend parameter penalize pl penalty produce estimate scenario select consistently stability magnitude condition penalization lar entire select cross latter sec ic however iterative algorithm correspond demand impractical path penalization simply convergence hand estimator speak parameter expect change approximately indicate suppose model I proposition penalization none maximize although elimination help ic quick ic finite parameter quick ic ic however remark firstly propose rough consistent reweighte use reweighte logarithm penalty locally secondly quick ic example resort logarithm corresponding penalty correspondingly replace auto var provide convenient causality analysis economics etc ic impractical quick ic investigate quick ic contain bic like quick quick bic first find lar evaluate bic quick noise bic orthogonal uncorrelated pairwise random trial iii quick bic bic agree quickly surprisingly chance however seem statistically bic find computational long quick bic htbp quick number bic
estimator abuse notation clear consistency result average excess r nf nx fx r ng ny exponentially z ce choose consistent capture estimator entropy regression couple estimation suppose assumption decrease tail assumption suppose nh nh rely error yield early introduction smooth ensure continuous rate nonetheless generally setting tend suppose n f nx denote assumption derivative bound n tail support outside sufficiently follow inference additive requirement various distributional focus inherent difficulty believe vector leave situation causal early extend primarily extend distributional tail condition trivial extension conditional entropy additional integration step carefully work investigation procedure explain early without distributional omit experiment selection method matter causal bound clear bound also independent sufficiently density note two lemma assume c since h old jensen inequality show claim fix randomness standard fx start pick interval pick let q function dense f h sufficiently b property assumption discussion statistical call work concentrate establish noise receive causal fundamental acyclic distinguish independence causal elegant causal causal every parent unobserve infer give two assume function noise independent plus noise term initial focused understanding distribution work linear independent identifiable nonlinear marginal absolutely support generalization term post properly work procedure mostly successfully validate mix artificial infer clear side unclear whether causality situation identifiable particular functional relation successful recent appear initial consistency log causal particular procedure algorithmic situation work focus difficulty achieve algorithmic section hence inherently typically meta fit residual decide decide reverse hold vary measure procedure employ usual detail section want detect sufficiently sample consistency rough sense face subtle four estimation well understood observe approximation sample detect estimator ensure usually e nx fx sense instead close independence depend independence influence error regression employ previously mention family test sum entropy derive entropy consider fact clear possible generally bayes yet structural arbitrarily theorem deriving quantity appear affect seem strong effect verify control simulation denote entropy abuse r denote analysis residual regression factor capacity regression variance causal couple b tune validation bandwidth estimation everything procedure tail validation become regression tune cross validation see seem causal version meta divide half half could either half couple entropy consistency estimation entropy estimation reduce estimation generalization algorithm erm functional rich couple remain regressor shift everything large converge entropy locally continuous difficulty follow estimator distribution residual thus problem entropy residual section employ regressor properly entropy sufficiently result consistent tail additive decrease difficult mild assumption polynomially tail interestingly analysis convergence causal likely decrease tail meta consist relate residual norm residual henceforth let polynomial bound derivative note regressor appropriately maintain technical general consistency suppose meta understand converge error proceed residual lemma entropy residual property residual verify consideration
seq laboratory stanford university http www artificial inspire segment signal intensity differentially gene figure average segment estimation simulate resample artificial segment line indicate right rand binomial recover segment rand prove segment segmentation however loss constant reflect rand choice implement perform development moreover term algorithm allow long signal genome cart rna seq laboratory publicly sequence http www annotation available genome sgd http www validate distribution poisson binomial select segment sgd loss tend select outlier segment contrary segment binomial gene sgd figure none exactly annotated boundary increase genome annotation rna seq validity read count root square scale choose segment choose segment gene proposition follow binomial term precisely expectation give sequence ks leibler hellinger x ks ks k l ks ks ks b ks h ks yield h proposition get use decomposition e j j j cauchy schwarz ks ks negative j finally cauchy l c control cauchy equation equation binomial cauchy proposition negative wish thank st helpful discussion statistical insight biological cm example remark paris france mail fr mail fr binomial poisson important penalize construct performance assess rna seq mathematics secondary keyword estimation change rna seq distribution introduction suppose draw distinguished might piece subject unknown change want segment follow observation different differ motivate example sequence rna seq experiment read genome genome stationarity area genome etc wish significant poisson rna seq new literature datum sequence include approach test segmentation numerous criterion hmm penalize segmentation segment minimize contrast convenient contrast segment choose crucial example penalty segmentation criterion instance version criterion show base consideration last extensive influence introduce model penalize procedure amongst good term consider various context least large exhaustive parameter theoretically practically adapt particular need list number penalize true inequality organize precisely penalty poisson binomial along exponential perform segmentation rna seq proof intermediate partition st p define collection partition segment segment length define amongst risk natural kullback ks respectively assume ks ks express accord appendix oracle construct exist ks procedure see see u ks ks complicated deal model decomposition control term separately chi characteristic deal classic facilitate direct chi square denote purpose effect expectation recall derive bind subsection establish framework variable z therefore z case te z te case introduce segment j f n double addition restrict first follow random distribution segment control noting bound accord distribution parameter j dx e dx u pp rna seq question annotation genome number start proportional genome however return criterion compare criterion
possess property compare chart sr exponentially throughout surveillance iid pre density indicator optimize compare chart sr cyclic sr compute performance change scenario question end extension propose range broad detection markovian clearly proceed employ numerical integral admit recursion sufficiently k integral x equation narrow chart introduce section integral present kind analytical solution however explicitly parameter supremum see compare sr procedure respect stationary delay sr considerable degradation optimize minimax sense confirm figure recall sr minimax sr sr sr significant drop optimize w fix sr sr sr performance optimize present optimize w r lastly case post lie certain affect benchmark sr stationary sense optimally sr mind misspecification acknowledgement work support air force scientific fa reduction project nf national foundation u research office grant nf nf california department university schmidt university constructive feedback version paper point work effort spend special mm g mathematical sciences york york usa department corresponding receive abstract chart examine sided optimality criterion conditional multi delay equation formulae formulae bivariate constraint optimize chart set cyclic conclusion chart fully optimize competitive indistinguishable procedure move chart point change concerned design procedure change observe random sequentially behavior change within subject area branch science economic see system name change define observe stop effect detection construct sensitive much use maximum lr sum theory sr sr focus exponentially weight move chart geometric move chart chart apply raw base motivated consideration chart change statistic observation refer autoregressive powerful detecting noise chart chart inspection hand ignore chart importance past observation memory chart turn base assign apparent chart chart main current rule value shift empirically detect change brownian motion optimal slightly conventional consider thorough chart carry present center employ sided chart turn chart accomplish also extend formula conditional detection delay add good first formulae formulae factor consider optimize smoothing problem formulae operating characteristic derive formulae apply obtain formulae simultaneously problem secondly chart compare multi cyclic setting serve benchmark cyclic focus work formulation formulation cyclic procedure sequel suppose refer draw pdf know serial therefore distribution decide effect alarm challenge decision soon limit true statistically sequentially occur moment never option accept occur continue first construct ratio let kk n next decide sequence turn detection detection statistic lr improper prior overview choose appropriate statistic first sr eq threshold sr statistic sr start hereafter sr derivative sr start design stop deterministic detection sr sr unlike sr chart raw certainly also build ratio usually observation stop side proceed criterion additional notation point scenario minimax formulation delay delay class alarm fall desire priori set level still sr sr sr stationary asymptotically chart order asymptotically
construct matrix exist remark construction order proposition fairly straightforward span singular thus subspace r orthogonality combination apply fact along average one please tensor processing step careful statement lemma product cover prove e lemma treat want satisfy intuitively identify co dimension sense formalize projection follow definition spurious cover vector dimension dim conclude many projection robust span dimension spurious singular let matrix comprise orthonormal contradiction leave combination column use small column orthonormal stage matrix satisfie ordered previously leave column rest subspace zero far subspace project extension large dimension j lemma formally pt initially orthonormal also pt height pt pt convenience without report fail fail hence fail nm ti impose orthogonal column orthogonality constraint pick th linear combination result learn pick decomposition overcomplete tensor solve precise subsection describe construct relate sample unlike multi view whose expansion nice term restrict distinct gaussian align variable thus precisely w I idea part tensor piece roughly order inverse polynomial recover mean perturb far scale differently find take zero another place perturbation useful analysis match dimensional wise anti concentration along coordinate perturb suppose perturb unlikely parallel imply perturb p perturb projection let partition suppose divide equal vector exposition computing denote vector restrict restrict concatenation scale agree claim entire weight repeat hence compute obtain error please establish recover weight know many end solve covariance entry equal procedure apply recover nearly equal consider portion similarly know I perturb condition rao product rw dimension suggest extend spherical axis align rgb claim proposition question conjecture theorem remark algorithm author nsf institute advanced support nsf grant dms innovation fellowship tensor powerful tool generative significant matrix tensor algorithmic unlikely hardness decomposition overcomplete rank exceed challenge develop tensor highly overcomplete polynomial application polynomial overcomplete setting main smoothed product perturb robust polynomial result mixture axis align model choose formalize perturbation believe since overcome usual decomposition central illustrate usefulness uniquely recover access unless require factor rank tensor general condition perhaps due review method commonly parameter generative contrast g hope recover rotation call issue tensor around decomposition hard approximation matrix generalize tensor subtract rank tensor approximate rank tensor algorithmic decomposition matrix column decomposition rank tensor met mixture gaussian however traditionally give robustness analysis give proof basic work tensor independent concrete get order tensor hence follow operation matrix new tensor factor column column fact tight bad operation allow dimension overcomplete case technical natural smoothed rank case tensor immediate learning mixture model study decomposition adversary choose ia assumption convenience perturbation rescale inspire smoothed analysis understand well realistic application intuition component algorithm various give algorithm spherical smoothed without dimension virtue tensor technical n rao add smoothed crucial application tensor error arise moment method achieve exponentially perturbation two polynomial analyze et al tensor smooth moreover additive error run succeed least discuss numerous traditionally hence case however get work component view view expressive et full like speech dimension small distribution perturb analogously perturb model succeed mixture align section covariance mixture al give pac mixtures axis gaussians time gaussians full turn smoothed mean mixture axis align gaussians axis align mean obtain complexity succeed believe algorithm overcomplete decomposition far framework study distribution easy observation et yield tensor overcomplete another control alternatively overcomplete assume exactly many sample good distribution decomposition main application ica condition hold failure showing depend polynomially failure smoothed perturbation prove main recall perturb leave negligible complement core orthogonal complement dimension subspace reason low straightforward onto space approach non meet suppose projection onto orthogonal complement exist technical constructing help definition intuition reveal column significant reveal add complete description rely basic argument review discover simultaneous column far decompose analysis span entry span contradict uniqueness pick write operation move factor run decompose state thus analyze equal factor column stable show intuitive condition algorithm inverse amount noise decompose appendix condition I u I tensor efficient return preprocesse slightly presence top suffice condition since requires see tensor handle overcomplete tensor n u additive corollary decomposition mode condition handle wise rao product become decompose vector follow robust analogue know tight bad handle vector much strong v u tr three appeal uniquely involved application exactly rank analysis rao product prove let column parallel hence surely overcomplete tensor want prove possibly end theory albeit rao perturb perturb vector formally let perturbation e suppose perturbation rao product hold omit rao allow repeatedly theorem say apply truly idea follow rao eq states analyze one leave one singular factor size span probability span prove perturb onto fix long dim projection one product rest perturb vector subspace dimension tensor nx square
bm minimize may thousand stage tractable objective kept firstly guess classifier learn classifier obviously discard step actual initialize step svm primal form parameter guess kernel rather blind information come global step guess objective individually become svm ignore interestingly consequence svm supplementary kernel introduce yield discard scale thus sign class show discard contribute final rule margin objective vs rest initialize parameter pt number sample test guess learn svm order svm x algebra square constrain might fulfil valid multi denote follow strategy svm sample yield find practice sample object standard discard contribute eq deduce optimize svm primal x number besides primal form bottleneck initial bm moreover also note step vs benchmark context task descriptor introduce detail intel library library summarize characteristic attribute normalize logistic lie procedure supplementary uci cost little testing split benchmark set sample split test testing separately descriptor cm attribute descriptor divide sift opponent color descriptor relative attribute author object category except texture color semantic attribute contain class overlap randomly maintain image evaluation average descriptor use codebook descriptor classification svm descriptor imagenet setup bag max descriptor perform max pooling accuracy vary amount initially randomly conduct test imagenet extract conclusion conduct baseline replace generate random achieve performance cm input discard different baseline vary decision increase fitting amount believe generate use properly regularize fix close svm lack fix justify discard together kernel efficient evaluate performance initial already note around descriptor informative region reduction feature length descriptor report vs pos sample split impact report set balance accuracy proportion insufficient observe descriptor kernel descriptor vs learning class decision also forest uci implementation iteration intersection kernel ik align left bin quantization observe quantization report evaluation code projection contrast learn locality original descriptor linear perform poorly criterion retrieval approximate adequate mkl predefine report comparable approximation cm observe perform approximation descriptor type descriptor attribute feature descriptor already descriptor predefine similarly actual feature length conclude outperform normally achieve outperform uci imagenet achieve compute whole report test relative time achieve level fast original map decision calculate projection opposite uci dataset poorly descriptor attribute thus optimizer could optimizer scope method predefine overhead score two difference compete perform perform accuracy combine amount simple randomize non efficiency svm descriptor achieve descriptor generalization capability exploit descriptor efficient svm demonstrate capability kernel common descriptor histogram attribute descriptor uci imagenet type achieve svm svms object stem well design right combination appropriate crucial depend descriptor familiar mkl base mkl might complex inefficient mkl avoid explicit kernel approximate map thus around svms coin learn binary decision forest equally well benchmark kernel svms select dataset benchmark descriptor histogram attribute quantization achieve comparable hand descriptor moreover fast select emphasis scalable svm aim apply trick lagrange multiplier classification score k vector inner strength svms max margin cost support vector latter quite expensive computing matrix dataset try vector create rank part final binary though base supplementary valid arrive classification two class well supplementary induce operation two constant evaluate mapping generalize feature advantage distance depend bm equal discard complexity aim particular collection define random achieve excellent descriptor
dag world original branching growth relation cm multivariate count within individual group count issue modelling field biology branching denote modelling identifying appear contrary mutually exclusive identify independence parsimonious relationship particularly goal probabilistic independence ensure kind direct partially direct acyclic identification frequency use mutual reference therein multivariate lasso et poisson log dag explore heuristic hill visit graph eventually review parametric literature graph address lee graph identification continuous restrictive subgraph chain component univariate poisson mixture singleton graph among multinomial family component parent family univariate multivariate parametric joint uniquely search dag hill greedy ascent improve account dag define operator addition direct operator specific add vertex one hand parent child np c n covariate discard n c comparison
directly result purpose project vector several regression j pseudo follow traditional ordinary least sum eq avoid least ols linear merely column perform gram schmidt one matrix vector normalize space
implement much conditional give overview conditional gradient new linearly convergent offline optimization analyse analyse online convex step complexity new nearly vector denote row lipschitz smooth fy fy optimality imply strongly convex condition twice fx mn p vector henceforth shorthand notation clear linear simple polytope iterate lie thus projection set observation analysis remain term keep shrink force decrease know enough consider intersection smaller observable linear show polytope dimension quantity call decision maker point maker incur emphasis loss arbitrarily even adversarial give maker maker full maker learn standard goal quantity tf tx tf tx cases maker make regret take randomness maker length game convex scale like attain convex rule function offline achieve strongly quadratic convex loss slight also take form minimize smooth rule algorithmic problem good date step differentiable everywhere suffice loss everywhere also point minimize convex direct query sample independently optimization strictly stochastic could convex reveal iteration tf tx tf tf tx tf expectation divide denote fx tt rate rate complexity offline conditional suitable either offline online polytope return smooth strongly iterative make rate nearly optimal online gradient online round function whose round randomize round art non optimization specify function q strongly norm distribution stochastic generalize hold convex optimization present analyse offline optimization polytope oracle call analyse convergence rate algorithm lemma smooth hold fx fx fx fx fx definition oracle fx convexity subtract fx decomposition additional call operation depend call number convex thus point operation consist scalar follow linear iterate vertex current theorem vertex need invoke iteration rely oracle compute find decomposition oracle invoke follow local iteration per convex suitable decision polytope present convex convex imply smooth optimization subsection bandit full set informally information plus receive tx loss theorems reduction describe corollary arbitrary linear function algorithm output eq theorems determine also observe smooth induction lemma hold convexity tx tx tx tx tx f plug give ready play xt play achieve overall zero tx convexity tx f triangle lemma convexity tx ht hold tx ty hx x g strong convexity tf tx tx tx tx bound tx tx ht ht ht ht tx ht ht tx ht ht f ht follow ready respect x txt tx definition ht tx tx tx tx tx ht tx ht ht value tx observation tx bandit basically technique scalar assume chosen adversary history randomization decision maker positive uniformly play closely analysis reduction tx tx conditions h tx apply theorem respect tx convexity tx lipschitz hold tf tx play definition get analysis tight type smooth converge sense solution combination tight dense n simply coordinate solution solution inherently produce iteration converge tight improve logarithmic strongly bind defining thus thus simplex approximate nearly tight aforementioned acknowledgment thank numerous early paper european project linear optimization matching problem algorithm whose counterpart hard admit algorithm motivate optimization offline give strongly single step enjoy rate rate answer open algorithm gradient project subgradient method theoretically inferior information infeasible computational descent onto projection euclidean hypercube simplex make impractical setting convex optimize prominent example phenomena polytope linear optimization polytope convex hull weight rotation bound psd linear amount whereas svd decomposition phenomenon motivate algorithm optimization contribution algorithm smooth enjoy improvement h offline offline offline strongly smooth online convex loss online strongly loss maker require point choose decision maker incur adversarial optimal benchmark offline fix round maker offline know new linearly converge online step guarantee term answer exist information algorithm also imply offline set result offline smooth convex smooth date frank wolfe convex whose convex domain recent work consider simplex
stock movement value lagrange multiplier introduce function address testing model reality stock period design market less stock market stock market forecast daily indice stock stock price index utilize performance stock stock index option stock stock broad provide effective risk individual stock daily utilize market process publicly internet exchange collect yahoo address precede stock amongst stock market forecasting index daily price individual daily exclude besides stock market exchange yahoo finance international respectively market market align market deal redundant daily miss daily empirical period unlike method window window short training year end year period period daily normal determine three day day choose three market index change delay market comprise element affect short period forecast ahead detailed indicator give table calculation daily movement direction stock exchange stock categorical movement price e eq value fig histogram respectively cumulative plot component illustrative stock period scale principal component origin greatest dash principal vector eigenvector axis uncorrelated helpful stock component reflect stock observe highly stock cluster g kt kt sub branch branch stock stock red decision kernel polynomial radial rbf kernel experiment cause data value therefore choice study rbf kernel svm examine effectiveness method accuracy also ann rw principal component ann table perform moderately positive svm ann original ann ratio forecast svm reflect drawback ann volatility hand iteration period moderately rw index pca svm ann ann rw pca ann ann rw average std forecasting direction carry pca ann rw sample bank table unlike market index individual high ratio ann summarize movement direction iteration pca svm svm ann ann rw std iteration ann rw std ann ann rw std pca svm svm pca ann ann rw std pca directions stock stock price identify internal financial forecasting experiment show movement ahead prediction window long period available stock american stock study theoretical study method stock investigation performance forecasting study feature selection stock classifier another acknowledgement author china fellowship market serve recommendation short system paper stock employ component predict stock framework identify stock market movement classifier economic propose stock price experiment year st predict direction notably ratio stock american stock principal analysis pca aware stock market simultaneously hope market financial regard crucial financial study market stock represent market prediction regard movement return portfolio early short drop artificial intelligence tackle demand mathematical frequently adopt support draw interest several decade specification model make frequently stock financial et stock et system gold drawback price price complex stock address several article efficiency fit stock information implement structural often overfitte stock experiment outperform predict stock yet prediction could price report remarkable ratio svm predict period however conduct set unlikely verify phenomenon show good fact connect stock market external stock market price factor daily datum obtain analyze reality generality stock access external factor daily account factor article secondly contribute stock aspect compare paper organize pca svm section detail description conclude paper discussion structure show column stock daily principal component cumulative rate
monte jump introduce basically carlo efficient proposal observation integrate nest laplace approximation close hessian direction log extend work address k order resolve cluster domain use eq step algorithm property ignore still round operator real gmm component x kp x rather difficult bad hessian quasi newton slow decompose variational indicator finally approximate k approach several real experimental investigate synthetic gmm wishart test number cluster demonstrate performance selection cccc explain selection interestingly aic bic fail whereas aic large build distribution apparent b mse concern stability approach mean mse mean five display figure find aic effective aic bic square mse although activity unite distant galaxy ht ccc galaxy graph histogram different figure reconstruct approximation algorithm time carlo propose scheme run letter use em variational represent component key cluster
focus show natural issue prevent convenience approximate limit compute may numerical notice obvious device camera device expect fortunately accuracy need device use iteration might reduce dense design work projection significantly estimate summary recovery recover coordinate nontrivial measurement practical store measurement costly recently projection idea compress sense use count cc maximally skewed nonnegative sparse scan theoretical sharp preliminary encouraging expect promise future research true statistics nj department statistics nj department nj compressed signal research topic observe nonnegative framework maximally skew originally computation scan demonstrate suffice precision coordinate number essentially nonzero focus nonnegative world nonnegative neither magnitude entry unknown stream compress recover magnitude framework differ maximally skewed generating sense typically gaussian skewed stable originally name focus leave stable random projection future compressed context design facilitate integrated hardware sensor sample like pursuit pursuit lp computationally expensive might desirable fast programming decoding require desirable maximally skewed stable sample design maximally skewed first maximal skewness characteristic I procedure mean replace heavy tailed distribution design maximally projection dynamic stream computation line work call compressed cc projection computation stream stream linear dynamic stream recover stream naturally handle stream measurement update eq pseudo entire stream mention streaming actually process histogram building view stream nlp natural language traffic important recover heavy compressed active reader mind recover nonnegative decode wise min I additive scan coordinate constant require coordinate literature know provide min useful precise l reasonable mf q sharp write precise proof z inequality check eq eq convenient bind sharp complexity min sketch comment follow bias eq beta plot bias merely theoretical effort decode package matlab use e look would computational author although l solver present fast could well use coordinate coordinate design generate I formula use l interesting experimental sample essentially choose option decode error q present recovery ratio panel confirm produce solid become l accuracy recovery estimate decode basically require scan package present l comparison efficient run program progress result normalize although experimentally h solid produce maximum around time make effort optimize matlab h
alignment fit simulated observation nearby two physical manifold collect plan material building file propagation simulator environment repeat step load coordinate need knowledge position source similar define extend coordinate vector element number original match actual modify arrange pair position source device localization user modify previous plan simulate map direct stationary reflect propagation plan coordinate emphasize relation source environment especially location environment perfectly two nearby usually distance similar far common plan physical common learn feasible collect environment plan material file plan position neighbor locate opposite point source weight load map remove simulation localization phase similar definition making alternatively inside coordinate modify set arrange pair source datum order run way device localization environment university france environment figure order compare nonetheless pre environment location grid two localization error sophisticate recent full calibration location static scheme set figure neighborhood spatial correlation plan manifold alignment platform france capability simulate wireless run ray resolution environment simply require environment different direct reading involve path calibration percentage load plan degradation localization calibration load degradation high load plan coordinate localization calibration plan coordinate localization simulate exploitation user localization drop calibration depict localization error localization observation curve error slightly nonetheless localization request localization collect database building depict figure variable performance propose user center calibration effort achieve stationary propose explain early localization localize read perform square algorithm obtain collection explain improvement calibrate estimate compare map setup depict percentage compare propose use ray plan figure depict improvement overall figure per load simulated figure plan coordinate natural plan localization accumulate accurate plan set change calibration load depict effect figure minor especially plan reduce increase moreover simulated greatly observation significantly joint localization construction limited load environment employ preserve number calibration localization perform manifold alignment localize calibration propose preserve namely plan move user plan also load localization also observation scheme improvement simulate load calibration phase work possibility effort simulate simulated coordinate add robustness complexity definition note wireless mail com france extension bottleneck implementation signal localization system effort map paper full localize simultaneously employ environment number location alignment source namely simulated environment correlation localization online plan simulate degradation load localization construction correlation manifold receive base localization system attract extensively promise relatively reference therein operation signal access environment wireless device system hardware appeal arrival angle arrival signal technique consist localization phase offline phase measurement environment receive mobile map user location building map consume expensive bottleneck towards extreme importance map evaluation optimization wireless coverage capacity try replace propagation nonetheless structure dynamic move change location consequently work model calibration exhaustive post acceptable map device despite great adapt complete accurate accumulation localization suffer requirement extensive finally future localization target simply limited calibration user full environment accumulation localization use exploit inherent spatial measurement amount localization without well neighboring position could reflect know collect transfer localization reduction spatial set simulated propagation reflect extent propagation decay map suffer neighborhood position simulate plan coordinate simple effort correlation environment plan coordinate reflect propagation datum effect achieve user correlation localization perform localization observation neighborhood localization outlier whether obvious less plan device enough correlation change correlation operate organize summarize description embed present propose localization solution plan coordinate localization paper offset variation environmental still successful densely learn map less effective require extensive period huge tree transfer device manifold require complete may plan map localization manifold base learn mapping characterize correspond use knowledge target assumption set correlation set possess common next section mechanism formulation manifold alignment manifold preserve reduction literature embed preserve neighborhood space embed capture dimensional small ni close solve close define neighbor tn chosen compute play significant scheme noisy outlier close distance far skew neighbor small percentage effect space localization mainly relate lot concept neighborhood dimensional size balance computing weight laplacian alignment source datum vector respectively pair minimize preserve neighborhood factor different component write define hard impose intersection index index element eigenvector small start follow point remain dimensional set embed consist eigenvalue yx base plan coordinate calibration transfer correlation calibration observation accumulate estimate plan reflect propagation perfect physical despite limitation propagation alignment nearby coordinate similar far physical plan thus make transfer manifold alignment feasible collect building file point locate computing plan number location collect result position position pair online localization server server receive localization request request alternatively vector match distance define follow pair point pair coordinate arrange paired concatenation offline calibration localization request input source complexity calculate eigenvector structured l source data row observation close smoothing group subsequent place enforce replace follow centroid immediate location simulate map wireless desire environment usually simulator coverage study network propagation shape material environment position wireless introduce map limit transfer spatial concatenation directly observation decay end quite suffer outlier manifold generally map nearby usually away common indeed make alignment collect material building file position simulator environment position weight calibration position position call position calibration localization localization server perform server receive request localization request
markov figure mention partial correlation believe include numerical experiment finally check coverage wrong edge rate well coverage infer graph assumption traditional idea aside way play relate reduce would show preserve perhaps extension beyond write nonparametric development restrict shrinkage taylor together end correlation correlation write hessian continuous invertible q variance partial except constant single correlation jk know note q jk take supremum give complete bound presence term avoid comment theorem pt undirected graph weak provide graph normality allow increase inference low sample increase inference bound accuracy assumption instead something less partial correlation undirecte glasso strong sparsity incoherence come confidence guarantee provide eliminate incoherence normality estimator guarantee increase bootstrap delta confidence interval partial style bootstrap indeed dimensional high moderate increase receive attention research moderate dimension much emphasis style guarantee denote notation edge false could use correlation mean yield accurate graph show principle interval conservative case inference whole weak assumption handle correlation contribution property depend incoherence coverage increase sample method optimization bootstrap improve partial correlation undirecte choose glasso provide assume paper namely condition method estimating make sparsity incoherence dimension method introduce bias correlation asymptotic dimension validity outline start method moderate delta increase dimension role section conclude allow assume denote element matrix let edge graph let stack quantity matrix product frobenius max variable sub large positive c ts sub incoherence condition b thus certain specialized serve seem incoherence especially eliminate correlation eigenvalue b large together may rule occur course price pay reduce construct identically never trivial detect equivalently confidence width must interval low partial correlation estimate regression usual eq intercept normality want interested assumption want want symmetric q set confidence q later number fix take fixed multiply kullback recall constant establish sharp bind q inequality quantile obtain interval correlation let confidence let correspond write conclude sparsity incoherence inference unless ccc pt circle circle controls controls control gray circle circle eq throughout incur rest term partial partial taylor hessian evaluate mainly inequality eq hence supremum practice e j complete assume simplify rectangle hence show theorem replace ib ny bs b sample modify bootstrap much usual bootstrap describe describe property immediately define confidence stress obtain confidence interval coverage accuracy sample uniform q approximate interval replication correction simultaneous mainly focus namely three note help get dense even favorable relaxed shrinkage motivate avoid inference correlation high inference estimator finally confidence exclude confidence guarantee case connect contribution inference relate block cluster node connection cluster restrict varie subset correlation partial correlation correlation correlation bootstrap valid inference require matrix construct partial correlation dimensional asymptotic coverage validity tradeoff investigate tradeoff elsewhere easy error shall simple construct use correlation connect user algorithm select compute bootstrap rectangle put let let correlation equation eq refine independent bivariate graph valid consider center describe undirected feature correlation graph graph half confidence repeat move feature either delta construct cluster graph improvement splitting datum introduce whether eliminate data open problem validity correlation half rectangle select alternative graph connection undirected bootstrap within
throughout entire set scatter plot nan hence concentrated level band correspond level e definition transform statistic upon make plot confidence construction band first statistic adapt regard standardized approximation inaccurate pose difficulty rarely attain index asymptotic test base converge provide slow hold ks statistic deviation rare asymptotic alternative alternative combine theorem follow result consistent statistic nan hypothesis fact corollary converge sufficiently test statistic alternative eq work whereas contaminate testing let first f recall concentrate strength n perfectly detect lead asymptotic plane point easily hypothese lr sharp sum error rate optimal precise knowledge importantly without hc mixture simplify describe let detection namely likelihood paper hc adaptively may considerably compare hc nan test perfectly separate sparse region side ks corresponding method particular ks recursion compute side sided algorithm supremum test include approximation side approach incomplete l nc package variable box permutation sort readily side evaluate right integral degree polynomial simple explicit straightforward th degree operation still implementation suffer accumulation nonetheless extended number accumulation error side actual day sided value supremum type sided difference hc statistic compute l type statistic recursive beyond detect lack standard significance right nan hypothesis shift change panel two side hc alternative significance level detect change k ad ad close power benchmark affect tail contrast poorly close poor ad hc stem etc end statistic give u hc specific hc false alarm implication hc test clearly large asymptotic error value tend demonstrate non extreme huge size deviation scientific inspection shall later variable beta next either slowly eq replace taylor similarly cubic combine simple prove study behavior hypothesis density beta respectively positive low inspection standardize play statistic follow magnitude location similarly standardize quantity standardize supremum attain left interval supremum implies rarely attain extreme statistic let union statistic ready start lemma algebraic q constant follow q therefore plug proof combine distribution fix standardized attain beta statement eq eq next consider statistic attain standardized attain location significant statistic hypothesis bind vanish n variance score lemma corollary give high respect first ready finish complementary fix every combine identical w nan n claim instead generality derive case bind number follow eq tend theorem end recall definition probability tend chebyshev conclude sketch next let observation union let totally increase sided direct integral shorthand value polynomial degree whose sum product list symbolic small clearly symbolic rapidly unfortunately simple close nonetheless iteratively polynomial store straightforward double suffer error propagate symbolic translate integration translate polynomial l l l l l accumulation error calculation degree basis translate integration yield define coefficient translate accumulation recursion slow empirically update calculation side recursion reason accumulation rather small double limitation overcome multiply final step size numerically precision summarize code procedure http www ac compare bit exponent bit help translate translate bit exponent polynomial extend exponent fix point thank discussion foundation section deviation kolmogorov weight distribution prove range mixture supremum side statistic simulation real goodness assess validity know continuous q fit one fundamental testing problem distribution broadly comprise distribution nx I x kolmogorov ks er von hc first variable orthonormal basis notable drive moment determine adaptive abundance ks nonetheless commonly desirable property good availability however suffer little detect deviation tail situation whereby contaminate generalization example high hypothesis popularity ks sensitivity several question ks hc way measure deviation statistic follow look weighted look significant independently method band turn proposal early author relatively often approximation simple computer long front nan consistency converge alternative supremum show adaptively detect broad mixture contribution devise section side test hc side exist ks test operation power ii rare weak test concrete introduce notation denote th sort u denote nx standard ks sided distance although supremum follow equivalent discrete whereby sided vary throughout small suggest weight deviation location
particular add cause greedy pre point reach drop pre medical application latter connect tolerance cost imply differ achieve sparsity via exclude point average individual learning algorithm extension via function denote norm specify tolerance instead mixed propose restrict extension apply application require low dimensional frame acquisition time position location build heavily kernel review result nystr om extension ridge ridge nystr om rkhs dy pf iw pf refers nystr approximate extension manifold manifold assign coordinate directly eigenvectors nystr om ridge regression dimensional embedding denote reduce let eq see element low embed nystr om algorithms ridge rich extension nystr om seek x p ensure formulate variable optimize simplify note n fx rewrite encourage achieve convex mixed encourage consist zeros zero ask norm e become lagrangian duality multipli iterative thresholding duality scalar variable fista achieve yield coefficient tolerance decrease produce projection surprising allow solution meanwhile define get similarity match nystr extension manifold lastly program incur offline result whose cost compute ridge correspond roll proportional vector relatively scenario method address run work hessian nystr extension construct laplacian nystr w specify near interest near radius om extension manifold application interest roll fig compute hessian near neighbor function probe ridge fig sample appear boundary fig predict point along boundary need tolerance broad influence increase ridge repeat roll vector grow ccc cc seq seq seq seq seq track patient cycle numerous imaging highly calculate acquire manifold incoming stream extension conduct acquire free vary frame capture hz image laplacian heat embed entire ridge sparse signal method frame remain frame influence reference versus error support also lead reference regularization lead result tradeoff operation first rest frame correlation support ridge repeat experiment ridge correlation number frame vs stay roughly vector imaging depend position impose restriction slice resolution acquire patient lie embed body head near neighbor slice position learn offline actual scan extension project acquire slice low dataset meet requirement kernel regression vector offer run whole body medical expert low pixel slice neighbor heat image nearest learn compare embedding kernel classification comparison interpolation reporting total regression clear small value lead classification speed maintain classification multivariate approximate act apply classification turning dimensionality generally algorithm expensive massive dataset ideally would find training
component b function use failure w final recursion initially turn equation op c storage run instance alternatively old keeping rank pca generate svd orthogonal diagonal element goal recover e view streaming orthogonal careful spectral quantity step entirely covariance vary block large angle two subspace respectively datum stream generate tb component extra energy lemma initialization least proof relax set let th step rank previous outside easily crucially distance rather initialization tight numerator require desire trial successfully recover model svd ny predict successful recovery inherent big explain average prescribe empirically tune random least lemma symmetric proposition independent sub whose random subgaussian bound multivariate mention lemma fix I ic bx ic bx ic appropriately assume enough manner block orthogonal iterate qr upper triangular u singular follow follow use get assume union p along p use furthermore hence conclude individually rhs use eq similarly hence follow probability q inductive decrease induction base trivially inductive assumption simplification concentrate respective consider use q second q third fourth get p k te iff follow u fact select bound note particular correspond global constant u h definition proposition lemma claim claim microsoft microsoft research consider streaming pass sequentially goal require memory storage meaningful context equal spike understand sample provably achieve meanwhile provable present mean storage compute sample kind successful much component dimensionality clustering procedure core singular half focus therein hence complexity recent dimensional dimensionality lead covariance largely influence draw low work also explore noise singular matrix succeed principal extreme bring focus quantity memory sequentially provable store pass sample must store empirical covariance availability massive application resolution length ram storage phone gb ram gb streaming sequentially store require namely covariance knowledge guarantee work detail perturb rank efficient operate recover maintain light point numerous body work statistical deal online streaming include minimization multiplicative approach goal regret improve natural perform batch pca however store multiplicative light variant typically guarantee order low save svd pool rapidly decay produce fundamentally approach appropriate come statistical covariance clear subsampling correspond fundamental column sketch gaussian vector straightforward recovery towards bad recently seek subspace every full rigorous come maximization incremental behavior know along quite popular go name term basic version principal via q top far perform rigorous guarantee analytical high make analysis constrain pca simultaneously guarantee competitive minimal provably stream receive store goal compute component probabilistic random sample vector mutually asymptotically consistent unitary scaling interesting major goal paper provide streaming match additional capital letter bold letter denote denote spectral norm line finite wise variant know potentially large primarily vanish snr reduction step one reduce variance illustrate rank case panel section describe stream
novel patch within field conventional layer use scan micro complex micro perceptron maps slide micro feed implement stack enhanced micro network able utilize average map interpret less traditional connect layer performance cifar reasonable mnist dataset convolutional networks cnns pooling layer convolution product underlie field follow activation portion output map generalize glm argue level abstraction glm feature concept replace glm enhance abstraction glm extent abstraction concept separable variant live separation glm cnn implicitly linearly concept live nonlinear generally highly input glm micro perceptron micro propagation result input perceptron mlp consist connect mlp field slide input manner feed overall stack multiple mlp element deep instead adopt cnn directly layer category via average pooling fed layer interpret category pass act contrast global interpretable use micro depend dropout convolutional neuron alternatively stack pooling layer map convolutional activation etc pixel feature stand patch centered channel feature convolution abstraction separable abstraction generally utilize cover concept learn concept however single impose burden variation cnn filter large region generate concept beneficial abstraction high level concept maxout affine direct activation maximization make convolutional separation well several maxout impose lie latent introduce novel network micro convolutional abstract feature patch slide micro structure perceptron patch design problem slide micro mlp convolutional sec detail concept desirable universal extraction capable approximate concept radial perceptron know reason perceptron compatible structure convolutional neural propagation perceptron deep consistent spirit new layer convolutional layer perceptron perceptron equivalent channel pooling convolution map go cross channel pool channel pool next layer channel complex learnable cross channel parametric pooling convolution structure maxout perform max pooling map maxout maxout piecewise patch maxout capability form hyperplane convex ball convex benchmark cifar cifar consist stack spatial max input regularizer dropout specifically average pooling apply al provide supplementary implement convnet develop splitting set et procedure initialization weight rate mini batch stop percent training rgb apply whiten maxout network feature maxout decay test dataset art h error pooling conv maxout dropout cnn augmentation maxout dropout augmentation datum network improve show dropout dropout add use cifar already previous regularizer maxout maxout dropout available method cifar dataset translation horizontal augmentation set cifar cifar cifar hyper layer current augmentation detail show table lc pooling conv maxout dropout compose color divide digit locate image follow per select extra validation remainder extra used lc pooling dropout conv dropout multi preprocessing consist layer follow pooling show test cifar adopt reduce simple cifar method dataset convolutional lc layer layer nn stochastic pooling dropout well mnist tune perform feature lie global transformation layer matrix subject back replace global remain dropout cifar pool without dropout worst connect add dropout fully reduce average conventional cnns conventional describe convolutional connection feed fully layer dropout comparison fair reduce network global dropout average performance cifar cnn dropout report replace fully layer percent cnn without effectiveness regularizer slightly regularizer argue average pooling demand linear layer require activation category confidence
beyond motion pattern expression recognize enable spatio interact great spatio length movie intensity time hour time exchange stock finance exhibit current expression series pattern point face corner space change reveal cognitive social effective human automate expression important I xt assume apart scale rotation translation sample linear subsection briefly describe place shape frame denote location denote consist global scaling angle rotation I denote follow vector opinion multiply stochastic marker position marker I reader kernel select low error svm within fashion gram expression feature date spatio temporal ica table experiment decide order early series performance length figure roc sufficient expression available find superior first frame collect notable exploit method one mixing may advantage robustness pose variation sensitive achieve promise early time roc whereas frame early enable response human early series small value sum kernel promise number potential novel optimization make time analysis research carry part national innovation
eigen value symmetric triplet cl x cl minimum composite q entry thresholded estimate usually tune state choice variant solving enjoy probability finally proximal sgd compute eigen vector dense usually involve complexity intermediate element eigen pair substantially assume run approximate sgd compare dd projection sgd extend sgd achieve strongly convex optimization factor optimal stochastic optimization projection gain order projection optimization neighbor analyze condition randomness round easy f f analysis gd eq take summation eq tf tf f corollary inequality bernstein martingale tm step follow bernstein next lemma summation substitute lemma plug first similar similarly extra care let usa state east mi motivate method aim computational bottleneck iteration enjoy make develop epoch projection projection less proximal speed regularize propose neighbor speed order magnitude see tool solve sgd computational subgradient sgd independent appealing scale psd ensure feasibility bottleneck sgd sgd constraint sgd claim perform final share convergence maintain projection namely parameter convexity parameter another research algorithm mostly frank favor linear frank wolfe exhibit rate problem present several algorithm convex online convex however polytope sgd resort reduction technique projection burden method extend sgd aspect develop projection advantageous smoothness projection conditional extension epoch projection sgd proximal epoch discuss utilize goal optimal radius consider sgd iterate return f psd could projection objective multipli analysis finally project cc ball compute standard sgd suffer average g similarly enjoy notable several recent achieve strongly optimization make sgd epoch make sgd enjoy key difference rely instead projection first projection proximal regularizer strongly strongly convexity f sgd sgd upon epoch intra epoch epoch k k intra sgd apply denote condition randomness convexity thus note total epoch satisfie q notice sgd sgd enjoy propose alternative second project domain center decay add additional burden sgd sgd g enjoy convergence q sgd comparison two real subsection proximal yield substantial improvement exploit previously gd variant utilize proximal proximal intermediate regularizer usually yield sgd proximal projection psd cone proximal enjoy close norm sgd provide sparse place interested square regularizer know elastic statistical yield application regularizer subsection verify eq product assumption update average subsection present one art neighbor psd goal near class notation psd define ax separate class margin belong belong share label end extract near
processor acoustic time overlap short frame characterize frame processor series continuous localize useful take account along feature represent sound shape signal hamming length ms choose feature sound sound functional see measure recognize give recognize functional present consistently classification major sound class put forward value feature perspective suit extend regularize context experiment sound improve sound far large interesting acknowledgment work education research region r project arc although receive multi little attention understand potential adopt operator kernel classification algorithm function per outperform classical multi complex method recently attract considerable machine turn suited output function multi context deal value kernel measure densely sample rather value constructed value extend kernel ridge dimension focus study operator value pay attention precisely value suitable scalar infinite adopt functional curve correspond consist finite motivate explore adopt valuable motivate practical great surveillance security classify incoming signal environmental predefine preprocessing characterize different parameter method feature functional drawback within employ arbitrarily equivalent index change inherent nature dependency work sound parameter consider thus behavior functional capturing characteristic contrary concatenation decade popular regularize nearly equivalent machine obstacle involve inversion hilbert space characterize operator perform remainder reproduce hilbert space discuss idea feature use functional regularize square classification case per sound present introduce multi kernel multiple task simultaneously multi value value example multi infinite well response input module methodology extend extension matrix replace function square domain infinite vector case value rkhs function value recall basic say hilbert continuous q therefore define follow consequently definite proposition proof design method base space example map input high input understand viewpoint basic atom understand focus space function space point project operator correspond higher possibly infinite high separation class project dimensional rather scalar square classification kernel function space functional learning value solve obtain come three grid solve however drawback exist sample way consist scalar case approximate minimization solve compute directional third approach domain obstacle inversion matrix invert matrix overcome study block value operator kernel adapt multi relation suggest identity et al operator identity able take account introduce multiplication kernel construct eq functional definite value label z eigenfunction operator equation eigenfunction experiment sound recognition task functional collect database break sound classify environmental speech music extremely business environment recognize surveillance security sound usually apply characterize classify sound recognition take library bit sample hz resolution frequency harmonic
decomposition sequel diagonal b term slice dimension generalize nuclear singular tucker fortunately offer tensor completion frobenius pm pm unclear whether relation capability tensor completion state nonconvex pseudo give column implicit solution rr r stress capability produce low transform multiplier eliminate sharing norm one ball responsible inducing reveal sparse c next property direct consequence establish p think regularizers adopt property hold need zero solution reference tuning parameter using relate atomic norm infimum name atomic complexity remarkable demonstrating induce atomic know column rank induce argue early fair comparison convergence future research design c turn enable provide global optimality ball still incorporate available recommender system attribute e similarity meaningful exploit preference description preference microarray dna imply degree among available prescribe estimate capable subject develop order integrate white gaussian respectively uncorrelated inherently present vector correspondingly estimator explore incorporate prior incorporate interpolation space reconstruct look p generalizing formally define family criterion adopt leverage admit representation correspondingly discard nonlinear approximation tensor connect three relative perspective capability completely slice share imputation rank point slice build expand original capability correlation entry dimension correlation slice give obtain aspect explore implement matrix priori alternatively estimate need procedure counterpart error useful visit probabilistic kernel similarity slice pp inner readily strategy preference age develop step cycle consider identify readily minimize term approach infeasible storing overcome successive choose simple optimize satisfie condition iii maximum eigenvalue iii place across product c b np np row standardize quadratic readily gradient accordingly load system solve parallel collect update c iii readily stationary gaussian deal poisson tensor data counting suppose entry mutually choice count divergence criterion couple bind feasibility miss nonnegative prior entry index b p understand entry wise aid interpret conclusion thereby prediction estimation carry imputation provably alternate optimize r hold sequel matrix np definition understand entry likelihood desire expression z aa mr mr b r available could principle resort extra iteration approach I r mr mr mr mr mr see mr mr highlights reason adopt propose coordinate descent separable admit solve carry virtue lemma readily iterate point cp se generalize cp focus prior term convergence stationary allow require cp without iterate synthetic dimension generate describe entry consist entry scale snr construct factor independent remove five percent recover miss recovery regularization depict averaged repetition vary repetition error fig successful entry db recover minimum induce effect confirm corollary describe describe section realization accord specify construct factor independent yield half implement show recovery exhibit db recover trend rank set internet brain repository tensor estimate scan brain percent together depict reconstruction db slice corrupted counterpart six contain covariance slice show miss db bottom priori parallel low recover usefulness incorporate array advanced continuous datum original recover position rna seq count reverse rna dt count rna genome biological replicate organized poisson count percent center depict bottom db factor induce vector argue numerically extra capability suggest parallelism rkhs obtain correlation among slice probabilistic criterion process minimize l respectively synthetic induce truth experiment image evaluate although way readily order immediately l part cost depend rewrite contradiction minimum expand aside matrix value mean expand definition b f put would du contradiction minima frobenius across outer product substitute tensor reduce focus minimization part cost thus varie mean scalar equality substitute show equivalent prof corollary invertible minimizer minimum rewrite vanish remove set q multiplicative property cauchy previous large enough q characterization characterization substitute substitute hand side hold accordance reduce remain condition recursive minimizer tm tn pp n bn r change p q simplify state combine kronecker hadamard convert product put respectively mr satisfy pair focus vanish establish iii consider expand logarithm see inequality concavity combination substitute iii evaluate ii readily select root usa array completion incorporate enhance capability approach accommodate optimum estimate sense gaussian kullback leibler truth synthetic complete imaging result recovery db imputation arise big diverse medical imaging bioinformatic well feasible low attribute capture regularity readily exploit organized rank
respect element model conditional expectation joint condition straightforward joint condition rbms state layer resort persistent divergence persistent gibbs sequentially conditional recover probably maximize rbms begin gibb phase suffer poor activation sampling induce poor estimate approximation somewhat increase diversity update way physical rbm mixing problem occur negative uncorrelated physical rbm wave implement sign state quadratic analogous boltzmann analogous set ising model boltzmann family map rbm state encode bias convert state draw wave result sample convert parameterization interface hardware use ise parameterization probability wave slightly difficult precisely wave change simulator wave twice time truly deterministic compare multiple physical explore face wave impose restriction physical element nearby element interact connectivity graphical model observe bipartite wave rbm rbm bias cause bias ht block pixel adjacent connect position pixel long explore wave hardware rbms primarily simulation differ wave visible hardware implement rbm partitioning allow visible visible approximation derivative autoencoder understand wave hardware h ht ht ht ht b pixel standard mnist connect letter physical computer simulator draw negative phase original pixel unless train sampling add monte although constraint wave rbm directly ise parametrization first expect sample train rbm mean digital case distribution physical exactly add bias noise could function dominate variance test thing bias less rbm draw sample training region poor phase sample either evolve increase parameter sampling rbm bias train reduce sampling able reduce estimator benefit noisy extend noise training qualitatively sampling increase major effect rbms add level rbm affect turn range train rbms magnitude stay whenever update bring threshold magnitude constraint little magnitude noise constraint interact explore around magnitude appear fact noise perform rbm may force rbm scaling generalize conclusion outside range evaluate rbms subset weight cope amount connection even force increase physical implementation connectivity instance wave unit unit connection remove result look rbm representative power decrease long digit fortunately likely kind structure connectivity get train visible unit much pixel unit try logical lead rbms fig digit well preserve case series feasibility rbm topology performance sampler phase time find limit rbm importantly restriction limitation structured wave system perform connection cause difficulty fully rbms suggest discuss noisy dominate bias constrain need verify suggest hardware concentrate effort reduce computer researcher effort designing cope topology op universit boltzmann rbms powerful kind digital computer implement expensive computation offer build whose drawing desire rbm avoid hardware implementation usually limited range rbm determine restriction simulation wave computer form physical computation suggest hardware computer effort impose topology restriction computer model rbms remain classify permutation invariant mnist rbms boltzmann dominant deep learn probabilistic miss input rbm intractable boltzmann machine intractable hardware difficulty possibly non boltzmann view implementation rbm approach physical share
google site fold validation hyper refer beyond examine learn pooling region dataset pool train batch suggest learn region decrease pool rich cifar good source target accuracy cifar cifar cifar table first widely rectangular pooling discover size cifar pooling strategy perform bad investigation pooling perform visualization smoothly show similar conservative use batch visual inspection localization approximation batch parameterization pool different regularizer train room classification spatial stage hand pooling strategy improvement margin observe baseline cifar state believe framework strategy progress pool publicly publication computer multimodal max maximize inspire pyramid pooling play pooling code degree translation preserve spatial information despite system progress fully adapt pooling scheme previously propose scheme particular investigate regularization show regularization parallel improve scheme cifar particular improve crucial role object detection system biology statistic computer vision method popular visual version object pool spatial pyramid unfortunately namely division independent amount boundary towards fully adaptive architecture choice constrain network intermediate representation line propose strategy shape discriminative interpretation pool popular yet freedom optimize jointly progress learn classification pyramid hand recognition framework solution achieve optimize superposition rectangular basis function pool discriminant individual classifier also large neighborhood information image class question restriction impose method strategy weak new pooling shape previously generality come memory requirement well mention possibility fitting therefore approximation therefore code code batch code optimize parallel hand pool regime dictionary capability explore spatial pooling strategy specific despite return improvement code also performance outperform oppose classification region arrive pool histogram refer code code encode patch patch encode spatial code pool feature histogram division largely arbitrary discretization occur spatially nearby code belong division make address operator multiplication standard division set zero instance divide recover respectively pool parameterized row configuration contain soft generalization architecture design aim access belong statistic code learn region pooling region adapt densely perceptron code every code connect th unit th unit relation pooling notation connect information choice dictionary similarly class term artificial employ logistic regression pooling label stack pool py option start large little gradually use batch batch weight train concatenation communication machine batch form combine batch boost accuracy call redundant sized perform redundant batch reduce greatly evaluate cifar cifar provide insight pool setup result cifar class sample million dataset work extract dense employ encode dimensional use regression region division pool furthermore division initialization learn bfgs subsection limit parameterize feed independently train pooling call pooling region reason behind transfer firstly approximation batch intractable transfer code learn dictionary lastly enable classifier one dataset classifier try logistic regression classifier svm benefit big difference pooling pooling refer select hyper respect dictionary accuracy big redundant cifar approximation set whereas consistent big dictionary baseline dictionary subsection possibility divide code batch use extract code train besides reduction benefit parameter iteration convergence performance baseline subsection batch compare batch dictionary observe drop improve dictionary attribute condition baseline interestingly add redundant batch perform use restrictive region employ feature r feature acc batch redundant dictionary size accuracy base
adaptation present interest global solution nonconvex problem rank use warm moving value descent converge monotonically efficiently exploit use trust region gradient descent minimizer nonconvex since minimizer rank saddle virtue kkt exist saddle algebraic direction exploit iterate solve stop reach propose important function saddle exclude gradient descent generally view fix versus trust cost reach trust algorithm approach problem generalization evaluate matlab arrange computing distance matrix remove distance random goal incremental display result trust stop relative version matrix accord distance stop cost drop plot configuration descent monotonic algorithm size generate accord distance algorithm run fix average run test intel gb ram millions million minute solve ht paper embed potentially handle manifold burden devise converge trust convergent encourage present research dynamical office scientific ac address completion matrix completion strategy embed result problem converge numerical illustrate good benchmark complete entry matrix fundamental recurrent problem engineering therein recently gain popularity thank netflix focus problem complete typical application visualization dimensionality behavioral sciences economics molecular name verify examine form cone geometry euclidean dissimilarity completion euclidean restrictive set unknown redundancy closely relate multidimensional scaling pairwise distance rely scalar product variant cost multidimensional completion involve consider multidimensional know relaxation tractable relaxation cast semidefinite technique formulation impose formulation appeal reduce optimization although practice heuristic good difficulty optimization intrinsic invariance due second normalizing representation penalization rotation computational low find new scale priori focus number contribution adopt geometric framework riemannian main result distance monotonically adopt riemannian manifold notation introduce section book qp q minima descent affect issue riemannian geometry reformulate unconstrained problem equivalence algorithm tangent space endow riemannian metric metric tangent give point complementary vertical direction class orthogonal direction restrict along vertical unchanged horizontal skew satisfy equation overall project onto require operation solve update tangent manifold formula give full rank exploit concept descent trust smooth function tangent apply adjoint descent algorithm gradient asymptotic memory demand memory handle potentially
way incorporate role play effect weight account work weight misclassification prior stress critical sensitive though implication choice hand concern outline want follow operate somewhat aim motivate far may counterpart extract motivate ht toy training extremely surrogate separate hyperplane near chance weighting allow one point assign find near toy show suggest predictive aforementioned extreme happen evolve wrong side boundary obtain probability weight reflect point outlier lead py conditional scenario bias hard check receive relatively behavior strictly monotonicity property minimize bayes classifier recover mainly serve introduce annotation human fix hypothesis learn practice overfitte simple second available certain error penalization assume hard parameter smooth version optimize state column optimality condition convex continuously strictly differentiable directly apply popular hinge unless see uniquely continuously ideally relation would differentiable differentiable hinge desirable latter figure unlike function differentiable like linearly fall convex region present tradeoff chance problem resolve tradeoff validation empirical implementation bfgs subset fix run aggregate mean compute difference split tune hyper coincide middle validation split ht verification finding handwritten digit discriminate pixel average svm svm svm bb impose kkt plot ask possible label translate score estimate observe additional expert help subset human human digit representation final experiment digit digit direction sample ranking replicate experimental ranking consistently par svm somewhat comparable remarkably weight give significant combine source additional translation interestingly possibility weight virtual score learn exactly attempt uniqueness closely well weighted constrain certain dependency incur sample svm encode training consider learning allow learn validation set extend experimental powerful kkt condition primal duality gap kkt kkt problem solution unique otherwise let expand yield employ proof concern solution follow expand full dual optimal proof kkt must kkt equality kkt condition sum equality lemma svm satisfy q I stack hence margin ci impose additional constraint compare soft thus follow may need uniquely unbounded low proposition construct provide problem imply multiply sum plug choose sign convenience lead proposition requirement necessity unique maximum since loss twice kernel define solution unique continuously uniqueness obvious tb optimality yield drop first computation note equation first function remain recall determinant component eq since due finally institute amount learning paradigm aim utilize framework relate weight training example replicate svm solution limited improvement form incorporation focus svm svms categorization review see mainly additional information supervise unlabeled certain information marginal distribution introduce reduce require bind loss give hence differently outlier explore give outlier encode via framework importance come weight cost appear naturally class unbalanced penalty encode knowledge weight try less point weight ultimately lead form encode feature happen framework uniqueness relation turn solution use realize serve purpose choose weight contribution work available offset reveal connection uniqueness choose go always equivalent hold reveal svm solution find value learn estimate risk procedure compute large learn svm highlight briefly introduction paradigm later body various task apply idea compute face however mind svm motivation entirely instance vast related online learning perhaps svm fuzzy fuzzy membership represent svm svm pre cluster instance weight svm also give necessary solution find svm matter discuss complement svm svm concern choose propose lastly present publicly conclude remark enhance base basic tucker kkt convenience latter provide study consider draw unknown hinge hypothesis label low information map endowed decision via correspond positive formulate context omit deal inner bold letter capital bold letter capital let shorthand stand correspondingly space orthogonal augment implement paradigm slack learn svm generalization instance risk f weight appear follow relation two solution surprising weighting allow relationship already let svm inequality eliminate lead hard optimization problem equivalent note maximum point svm explore uniqueness make respect equivalent uniqueness aforementione unique unlike offset latter svm constraint begin uniqueness solution essentially instance weight separate within range total value offset heuristic g allow range happen problem unique vector interesting show svm formulate give support vector classifier concern technical appendix condition source section give statement primal weight contain surprising concentrated suggest corollary become clear main possible solution discuss opposite reveal constraint sufficient discuss problem appropriately weight weight dual point good learn oracle sum whenever svm average great toy weight force outlier receive high weighted non opposite characterize svm solution highlight relation variable show equivalent h equality correspond svm dual compact theorem theorem suggest interpret effect impose emphasis loss non necessary b optimal choice feature svm eq optimal first rewrite average put higher change matter reasoning check one equivalent check non
robust uncertain environment novel dynamic high task action function capture unseen mrf framework demonstrate successful plan task interact robot user specify ask pour place sequence environment perform vary depend relative distance robot thin may pick pick success task ability way response development probabilistic semantic labeling enable manually sequence scalable variety arise environment environment attribute related give object similar human naturally suitable attribute representation similarly capable suitable task environment carry pour tight bring pour dynamic planning primitive primitive execute discrete primitive environment conceptually plan consist statement associate primitive controller action environment match primitive controller execute robot complete sequencing representation field train margin comprise controller environment execute correct argument five high task sequence task planning specified context plan plan category work manually end mix human task require complicated sequencing environment retrieve making assume state ar represent individual learn natural language learn language area computer vision modeling activity video plan symbolic entity formalize green planning plan planning generating plan recursively prove symbolic symbolic require encoding planning surface example suffer suffer real situation explicit label though substantial body semantic still challenge reliable represent describe object present datum robot object suitable attribute object study decision process policy give specify reinforcement learn ng extend learning try optimize reward frame learn max path planning prediction prediction much small state action sequence co planning learn sequence base free knowledge draw book robot accurate reliable simulation require reliable implementation note unclear occur environment become challenge take weight maximize expert policy sequence first program property atomic operation close primitive specific role primitive task accomplish illustration write program could format many statement however simple environment challenge release move commonly ai planning sequence start state reach current symbolic break environment next statement loop plan bring close state dynamic include physical object location primitive execute step designing represent correctness primitive parsimonious markov capture dependency argument environment node task top represents associate primitive past clique graph respective clique simply ta w step top task layer represent primitive bottom attribute train discriminant map cut cutting formalize slack structural optimization combine sequence select primitive argument discriminant cc cc cc cc release hold c consider seven depend task contain say pour sequence base attribute sequence interact empty see plan produce environment ideal size locate surface interact whenever algorithm learn robot object object c interact robot identify object pour directly variant able robot sequence present participant simulation mobile robot primitive simulator thesis scenario single sequence acceptable variation provide ensure base environment unique predict primitive individually predict program without intervention primitive multiclass cutting argument symbolic translate symbolic entity however pre condition action definition predicate symbolic planning symbolic planning instance reduce planning planning svms ground provide code still difficulty rule include fact need handle argument full sequence confusion seven quite correctly pair sequence table without argument able primitive last whether learn pick primitive would drastically correct select symbolic specify respectively make code symbolic rule handle encode capable encoding planning language carefully come evident correct unique randomly sequence detection attribute describe explicit program form make online test environment four manner respectively total able scenario tb percentage labeling error scenario robot observer help greatly
become inefficient dimensional problem inefficient live distribute uniformly live velocity boundary problem location know carlo proceed pick live point accept third point I n nest modal isolate analyse live set likelihood nf toy function define minima show assume take likelihood right panel panel leave plot axis problem fig function global minimum fig leave plot function run prior parameter take evaluation plot panel running show plot z thin degeneracy minimum challenging assume prior take sample plot characterize discrete nest pixel object spatial extent assume shape contribution assume additive emission instrumental contribution parameter template give list add independent pixel noise hand corresponding ideally infer technique nevertheless extremely computationally intensive value shape problem source dimensional numerous spurious object show nest single live project successive likelihood nest algorithm infer list object order panel thank nest address group cb maximum entropy perform difficult parameter modal calculation evidence integration nest monte calculation carry probability one carlo nest standard monte hill minima problem inherently modal distinguish spurious posterior evidence normalize dimensionality inference obtain unnormalized mcmc model compete compare respective ratio monotonically thus evaluate quadrature first volume unity value remove live draw prior contain contour define
critical nuclear device detect particle energy shape digital detect characteristic material detector material necessary classify gamma ray refer shape discrimination psd unfortunately laboratory ray previously event arrive source exist collect source background eliminate class take pure ray source proportion gamma intrinsic source change could psd describe propose noise draw common corrupt label clean loss necessarily random label generative noise impose distribution parameter use kernel distribution presence lack tolerance risk respectively suggest modification classifier develop hinge modification proportion priori proportion could broadly classification pac formulation support deterministic concept contaminate complexity change asymmetric basis co thorough challenge ii asymmetric label noise machine one unlabeled example estimation basic instance side certain sense proportion assume absolutely respect lebesgue respective assume every ratio preserved regardless take ratio proposition every contaminate threshold one range equivalently generate operate contaminated ii type circumstance concrete jointly priori probability contaminate regardless assume algebra easy equally probable priori contaminate pearson note roc classifier case contamination example point design respect yield optimal equation reveal satisfied asymmetric optimal respect contaminate case base set essential ii turn facilitate discrimination readily subsequent lemma problem unique plug identity hold apply motivate contaminate follow estimate note although possible converse exist unique assume equality plug identity hold hold solution easy check imply distinct distribution explicit contamination proportion initial contamination proportion alternate sense allow reduce far motivate mutual identifiable also question contaminate obviously trivial trivial maximally develop iid estimate address definition identifiable hold decomposition decide representation one therefore valid say irreducible mutually irreducible irreducible vice versa let unique decomposition irreducible respect q clearly irreducible respect h hx identity check contain support irreducible gaussian variance discussion mutual et consistent almost surely consistency us distribution corollary irreducible condition introduce identifiability mutually essentially equivalent irreducible irreducible role class solve produce deduce irreducible conversely I consistently via idea develop conclude contamination describe equation arbitrary mutually irreducible analogue proposition equation intersection half mutually equivalently maximizer maximizer distance explicit correspondence establish solution decomposition original representation total base relation obtain correspond unique mutually irreducible interpretation restrict contaminate ensure case identifiability condition characterize total separation irreducible denoise maximal separation geometry region proportion solution contaminate single unique iid define develop n sometimes converge classifier vc dimension consist converge vc al continuity development criterion classifier classifier rf number classifier avoid empirical family classifier universal histogram tree neural measurable give classifier set hold consistency discuss hold condition rule maximally version theorem consistency proceed approximation denote brevity f go sake argument realize probability go one provide embed reproduce convenience absolutely respect lebesgue density entirely take clearly make easy indeed mutually irreducible iff supremum ratio equal formula easy variance hard tend left solid dotted line third distribution roc slope roc equal point roc correspond hx less provide intuition estimator understand estimate slope roc proceed plug expression fx h fx hx slope tangent corner like kullback nonnegative equal likelihood like divergence whereas infimum divergence chain separation distance actually next estimation relate distribution density logistic dimension fit connect irreducible everywhere mutual estimate could link consistent supremum invert equation estimate estimation mixture proportion context sup distributional assumption label possible label conditional mutually irreducible false seem previous theoretical even asymmetric require mutual maximum denoise contaminate discrimination consistent sense performance tend performance maximally nsf grant european community u well nonnegative denominator positive feasible correspondence decomposition value satisfy proportion correspondence explicit constraint translate via value unique maximum attain subtract eq imply mutually deduce theorem sequence classifier set vc k coefficient mapping let let k km x z bf iid govern bf size since therefore q implication combine eq let brevity substituting consider et converge thus meet z z mf conclude define proof fx x yield fx manner existence
group phenotype phenotype key noise uncorrelated note global oppose shrinkage apply column shrinkage result gibbs update proceed update related reduce residual value rank regression minor modification regression update accordingly update bayesian infinite apply equivalent current save structure experiment hundred negligible account expect negligible metropolis hasting account straightforward proposal acceptance specifically bayesian rank infinite reduce decay exponentially clarity prediction response proposition term prove supplementary material eq conditionally write independent variable expectation let take follow summation independent identically equality result expectation easy substitution require material let consist column infinite exponentially detailed provide evaluate sharing weight shrinkage group share reduce rank sharing noise regression regularize multi l factor factor comprise along method snp predict gene individual remain processing cca learn snps correlation prediction unchanged snps reduce computational speed considerable require comparison encourage response model penalize rely statistical strength sharing model model effect implement information sharing effect sharing serve single task default cross develop genomic fair pre construct wise sharing otherwise simplified factor discard surely reach prediction achieve factor prediction truncation fold parameter control evaluate student pair variance gene could predict snps gene result baseline mse could predict accurately baseline least one use table table method share performance version cl ht c c c sharing group sparse statistically significant sharing model training surprising encourage phenotype accomplish completely principle performance towards training emphasize share treat validation hyperparameter plan whether improve even apparent approximately magnitude close term prediction ht problem predict effect receive rapidly find also application effect attain combine comprise thousand method conceptually share framework experiment aspect predict flexible effect benefit visible improvement performance integrate machine principle prior concept establish infinite shrinkage prior truncation bayesian treatment training hour realistic sized method outperform weak effect centre coin grant pm financial grant complex university project f fp dna control national public health institute thank ms ms author would like acknowledge science explain amount facilitate constrain structure sharing noise far effective shrinkage group context reduce flexible model genomic sharing lead accuracy weak example individual genetic nucleotide snps phenotype response mainly association predict effect currently appear consequently statistical weak effect impose impose call compare size effect similar challenge attain set ever application predict effect often advantage genomic phenotype simultaneous phenotype potentially valuable effect reduce quantify uncertainty inherent statistical analysis address predict bring principle satisfactory recent technique shrinkage prior group alone sufficient effective introduction conceptually share provide regularization input relationship intuitive reduce rank structural domain regard immediately predictor effect response variable response affect predictor affect predictor learn shrinkage matrix building noise problem enforce effect model target specific prior reduce validation parameter conceptually suit relate work regression many promise thousand sample require work recently phenotype field trait mapping association avoid prediction phenotype constrain problem approximation integrate hidden factor become equivalent figure display noise joint phenotype correlate circle node mm none fill thick rectangle draw gamma delta star nu gamma leave lambda connect delta star connect connect
update algorithm refer reader review therein discussion relation svm build algorithmic dual method new differ aspect namely currently shrink work decomposition extensive elaborate systematic elaborate frequency individual effect replace shrink heuristic considerable speed order remainder review modification extensive conclusion svm software multi regularizer basic hinge norm regularizer proceed describe training find primal problem machine drop offset drop box quadratic rooted mention usually standard program restrict variable problem solution sufficient sub algorithm perform coordinate ascent skeleton solve svms algorithmic improvement reduce currently trick track rewrite operation requirement inside make change flat inner loop provide detail g x w cg v define iteration elaborate heuristic fast perform systematic active sub proceed track dual operation perform sub problem update operation maximal tucker optimality dual problem express stop soon drop algorithm keep track gradient impossible check operation check time active thus variable check keep check equip shrink remove stay remove fail end run need detection mistake costly therefore conservative remove exceed active variable improvement modification contrast svm priori kkt progress execute shrink make hard shrink decision active problematic wrong remove share fine effort shrink amount predefine select inactive establish selection heuristic pick sense frequency possible still whether progress frequencie self heuristic find modern unable determine good current indicator utility turn preference outer schedule list index frequencie operation randomization variable crucial preference w gain preference change rule bound ensure convergence establish directly carry modify version preference take algorithm per dual modern work training resemble variant therefore average past move give minimum remove active checking thing compare time mistake discover much original shrink decision relative dual objective exist modification schedule g cg update preference false ii p adaptive frequency fair modification write aim speed problem counter code outer sake stop heuristic criterion section algorithms default default run rely medium binary website cover data accuracy runtime compare fair non regularization several perform computational exceed final time stable primary metric relate easy update step roughly step slightly costly original time number table comparison b news baseline update loop iteration font train range mark adaptive run finish one star remark actual big solver baseline news baseline number iteration font algorithm baseline frequency run mark finish case runtime want big c l l l second c mm mm red solid dashed behavior new lot often magnitude sometimes many become loop soft shrink effective original shrink heuristic shrink value close exactly target adjust shrink shrink wrong shrink really
operator present sufficient problem hilbert combinatorial optimization approach complex noisy exponentially generalize perform require frame author optimization program reconstruction absence separate algorithm transform algorithm operate semidefinite yield absence inverting compare performance rao present result incorporate frame bound redundancy reconstruction absence study several phase recovery problem approach different subgaussian additionally probability asymptotic behavior analyze reconstruction frame problem obtain upper upper frame give estimate computable singular somewhat lipschitz constant map robustness fix lipschitz bi induce nuclear operator bi constant elsewhere organization review exist establish theoretic rao robustness measure reconstruction present stochastic analysis reference real mh set concept completeness instance terminology property remain dimensional frame eq frame frame frame frequently frame large frame frame large eigenvalue first th dimensional say full frame reconstruct ask question relation constant distinguish analysis nonlinear map distinction distinction necessary problem reconstruct phase reconstruct analyze stability inversion give proceed review exist map generic z n nz polynomial frame partition phase phase expression fisher noisy white consider noiseless disjoint hilbert belong hence algebra fisher matrix bind rao low furthermore follow hence prove b go formulation constraint w without generality l j contradiction hence achieve satisfy clearly satisfy check eigenvector eigenvalue eq conclude proof mm remark seem typically depend thus exchange study study entry shall function nuclear interested investigate bound bind frame bind pick fix generality q whereas hence xu aa furthermore bind whenever eigenvector whenever respectively x proof already prove generality v w w w infimum set let normalize eigen q mm quantity subtle relationship remark estimate specifically turn motivation add study product xy xy nx xx tx p nn symmetric non note spectral easily useful equivalent yield prove obviously eigenvalue w prove parallel quantity x vx vx f xx yy tw lemma bind operator banach space endow frame fix generality vector hand prove lipschitz remark note bi point section establish robustness reconstruction bad scenario reasonable robustness dimension increase reconstruction reconstruction thus mostly question frame element bound element linearly preliminary f bound preliminary constant thus minimal impossible independent scale whether redundancy turn wang precisely let whose part result continuously mm estimate explicitly probability stable reconstruction redundancy thank discussion thm corollary
resample towards briefly approach expectation naive bias elegant smoothed histogram derivative bin use spline offset bin density recently generally comparison selection quite two bias let bound give simplifie raise frequentist framework distribution get equivalence answer yes mathematic manuscript try prior compound decision though none formal selection bootstrap flexible complex empirical joint parameter nuisance bias risk hold maximum illustrate categorical variable wide variety involve statistic complicated categorical simulate bayes bias drastically bootstrap order outperform handle categorical variable regression procedure simulate vary estimation procedure regime separate illustrate ht spline spline spline plot show empirical bottom empirical particularly red solid calculate strength estimate coefficient determination continuous categorical matrix response class least observation base entry specifically variance estimate use observation performance see perhaps extreme gain estimate shrinkage plot scenario insight rarely simplify simulation compare behavior bootstrap bayes approach cancer two method provide unfortunately gold hard beyond increase like array generation dna sequence typical attempt univariate disease control important rarely adjust result assessment become classical shrinkage stein unfortunately effect estimator empirical leverage estimate effect interesting formalism frequentist formalism connect remove selection resample lead accurate effect bayes scenario applicable apply begin quantile quantile second concentration combine corollary recent throughput researcher find thousand effect size generally extreme unfortunately effect account manuscript show estimator dominate likelihood resample show connect idea bayes stein modern interested effect size interested estimate paper stein show random shrink dominate average insight large competition small though effect size throughput experiment generally look extreme size attempt confirm effect effect statistically originally report selection bias explore compound among asymptotically minimax dimensional practical pursuit new computing popularity give elegant everything square th th order statistic stochastic index rank define extreme dominate eq bias manuscript bias read vector bias give q never estimate give bias estimate update ik parametric calculate bias vector mean accurate calculation improve
combine always rp list list c power public I list estimator subsample answer yes reject monotonicity public cnn treatment follow question test direct public speak question risk subject treat list public study figure estimate study difference experiment b suggest tight formal study study answer mean independence none statistically different joint fisher nuclear public speak cnn describe combine subject treatment b subject treatment answer question list answer two question treat point support report independence violate question list estimate se public speak cnn conduct subject study present first likely variability none difference significantly five significant treatment answer cnn behaviors question reveal reliable answer yes distinguish sensitive subject technique crucial identifying respect first monotonicity hold cut opposite direction different pressure support whereas conservative pressure second force concern recommend order list ask matter design effect reject finding argue direct second application direct independence recommend question perform simple straightforwardly setting post yu covariate adjust parametric residual question response appendix list bold question bold nuclear four energy people think united develop multiply let proof proposition substitute var eq proof assumption v apply I term ensure strictly joint test effect consider estimate appear yes direct must yes monotonicity effect meet subject answer yes variability sensitive lie treatment sensitive item power figure panel axis three proportion profile assumption design respectively control unit meet treatment control list treatment response list response generate plus except treatment list response equal sensitive variability response sensitive item four maximize affected conduct simulation reflect proportion nan power around simulation treatment sensitive list test test minimize ii test ii detect independence proportion panel yes treatment proportion control high power band around power red factorial subject study question study present direct question appear change conventional case support percentage ask whereas point ask se se se power public f study ten identical design start fail attention leave complete study table follow present question study none fail five four cnn fail second cnn interpret treatment answer causal percentage rp list list reduction speak nuclear cnn se h rp list variance public speaking estimate nuclear public speak cnn estimate public speak r se se se public due tendency popularity sensitive introduce item procedure drug et al vote al standard proceed partitioning receive sensitive receive comprise plus list treatment cover sensitive long whether aggregate question subject distinguish type nevertheless subject behavior question bias detail researcher question precision false subject former question latter list design experiment group question treatment direct direct variant subject direct estimator control receive question measurement question investigate ask list list experiment assignment fact report item design assumption require non list absence additional identify formalize control answer yes nan difference subset reject treatment assignment assess dependence direct question experimental demonstrate researcher previous goal decrease experiment model double item conduct subject thereby reduce covariate interaction heterogeneity construct correlate simultaneously variability avoid effect design non model estimate sensitive item propose least offer show ease exist technique survey amazon platform analysis experiment scenario ask question subject independently draw method direct goal report direct subject lie claim behavior lie monotonicity report direct question mixture report response direct reveal contrast subject receive number control subject states design outcome subject list treatment actual treatment assignment treatment ii assignment affect concern ensure mild regularity degenerate give degenerate represent hand represent behavior z v experiment section propose experiment indicator base result let non degenerate variance eq proposition consistent variance substitute sample confidence gain estimator upon asymptotically precise monotonicity effect treatment two test assess validity assumption jointly independence thus distributional distributional nan distributional equality eq monotonicity sided proposition calculation analogous answering yes proportion affect variance response probe independence answer statistically treatment e independence estimator hypothesis side follow calculation analogous proposition treatment assign accordingly assignment question assumption test property study amazon internet platform return behavior may relative internet survey favorable environment expect subject might face comprise experiment direct question experiment sensitive preference energy source neighborhood news experiment would expect anti huber
first tensor broadly line uniqueness key permutation interesting case tensor I third tensor broadly dimension core tensor matrix permutation look vector intuition permutation precisely prove sufficient must column non permutation scale diagonal state recall section robust bound far suppose satisfy satisfy nr prove permutation section c cx statement repeat conclusion verify hypothesis hold say span b b lemma robust lemma view three tensor slice weight slice cauchy slice th slice sum begin define vector lemma q error rhs obtain co rise slightly co b contradiction formalize consider let contradiction set inequality know definition prove sub matrix well j multiplication lower formally note apply fact row q define divide otherwise prove imply give contradiction define equal term span span column dimension error frobenius ii projection multiply obtain contradiction follow proof conclude good else v bb unit q since contradict case almost go whether argue number use c setting condition sufficiently also reasonably far say decomposition otherwise add rank contradiction show dimension submatrix submatrix us vector normalize r c since permutation follow exist permutation scalar quantitative ready complete prove permutation various dimension theorem satisfy scalar three multiply identity permutation contradiction notational correspond map similarly correspond argument along combination unit along dimension decomposition r use carefully negligible partition partitioning pick choice due true let since take give unit eq since vector along ab c easily theorem complete well value analog problem arbitrarily good approximation vector order rank I low decomposition low vector norm condition would recover conceptually identify suffice space force search use constant mode fix column outline top singular example carefully around point construct try possible claim claim top rank span mode choice argument inequality frobenius way rh tensor frobenius precisely j proof frobenius early observation claim iv cr cr nets size easy hybrid argument exactly involve length try candidate tensor argument tensor proof given find rank low decomposition many search length vector look consider orthonormal formally combination plus max getting algorithm search dimensional alternate last column matrix find force enumeration concern hence decomposition moment tensor close eq permutation get entry identity ensure eq correspond lemma r r lemma popular latent multi view exchangeable single topic apply model al third identify omit exchangeable bag word exchangeable view picking topic topic sample dictionary word variable word conditionally correspond document since view nc probability word identifiability constant document topic n r discrete markov bioinformatic etc state stationary represent time conditionally column eq q identifiability markov statement satisfy singular consecutive find far proof sketch cast hmm def nice three view show fig comprise view tuple view independent dimension conditional reverse matrix transition matrix rao fairly al occurrence property rao hypothesis stochastic rao continue precisely use instead matrix need spirit condition let add thus combine perform entire procedure rank obtain well corollary sample zero give explicit whenever condition advantage finite arise rank decomposition np regard uniqueness instance help particular rank decomposition polynomial e mutually orthogonal iterative least weak uniqueness idea algebraic prove zero tensor characterize note strong bind hope tensor result imply elegant even relate isometry rip uniqueness robust theorem involve perspective parameter satisfy robust order analogue extend general gaussians mixture thank valuable discussion helpful literature thank speech image primarily difficulty trivial include completeness suppose singular q vector suppose contrary column dimensional subspace contradiction vector denote subset span lemma first correspond orthogonal exactly ia z I v schwarz minimum submatrix satisfy unit somewhat argument vector spherical direction know zero anti hold rao otherwise k r multiply lemma apply establish lemma tight first orthonormal next suppose v l respectively imply eq hence v u lemma permutation know substitute last consider tensor rank necessary term necessary uniqueness decomposition b r show condition well condition error tensor look various variable every j bind difference consider entry tensor sample independent bernstein nc multivariate gaussians suppose generate satisfy rx r sample q element tensor since spherical align hence tail union conv var notation support nsf award university support fellowship uniqueness give approximately recover decomposition inverse error identifiability latent robust version immediately polynomially setting identifiability decomposition parameter variable efficiently long give barrier establish expect beyond explore central question unsupervised computation efficient learn observe polynomially sample moment pearson moment correlation moment lead meet considerable underlie degeneracy explain informally observation least position represent tensor dimensional matrix decomposition underlie degeneracy decomposition iterative develop setting violate small arise variable speech pixel class decomposition early statistic paper identifiability fundamental uniqueness tensor play crucial ensure correctly identify procedure assume infinite sample specify establish uniqueness moment tensor approximation size need uniqueness moment tensor decomposition decomposition contribution classic uniqueness identifiability identifiability assume polynomially distribution yield robust version establish well know literature expect robust application beyond setting robust uniqueness multidimensional e decomposition method since rank role algebraic decomposition vision analysis graph introduce canonical introduce refer one column require decomposition definition tensor analogous computing np hard good rank consist cp decomposition fact tensor well approximated tensor good overcome concept minimum tensor sufficient give arbitrarily small high minimum contrast decomposition svd distinct impose additional give cp decomposition tensor follow rank matrix form column uniqueness q several alternate fundamental tensor let decomposition uniqueness decomposition tensor end need natural analogue form bound finally measure closeness tensor frobenius please precise robust unique formal individually close tensor analogous rank robust component uniquely proof lemma component necessarily belong conclude work typically error thereby require exponentially reach polynomial exponentially tensor low purely possible avoid step rank practical algorithm hard tensor find approximate rank dimension tensor decomposition error view analog rank approximation property sum error variable identifiability contain prove identifiability broad class tensor uniqueness ignore tensor understand estimate within need tensor uniqueness know latent possibility mixture view discrete multiple observation view expressive markov hmms mixture tree latent exchangeable bag word exchangeable pick give topic accord topic I multi view identical hmms extensively recognition bioinformatic follow eq mention important e class imagenet speech typically coefficient incorporate build hmms speech recognition hmms much vector lie dimensional feature small unknown mixture spherical history overview necessarily focus mixture spherical much need certain separation center recently albeit run dependence necessary recent special decomposition independent additionally matrix low dimensional subspace tensor observation range hence associate tensor order establish apply moment view topic rank high sample compute gaussians provably good gaussians polynomial mixture gaussians gaussian require polynomial identifiability imply mixture gaussian variance identify polynomially technical contribution uniqueness tensor decomposition broadly proceed establish two permutation permutation three permutation theorem ingredient prove condition decomposition crucial analogous proof uniqueness besides technical permutation lemma establish column proof column intersection natural analogue would column subset column however inductive involve intersection particular statement step induction recover start small inductive whereby error recursion describe crucially rely condition observation proceed search naive exhaustive input first comprise vector dimensional space need correspond optimum obtain condition boundedness high tensor inside exhaustive search net time assume mode comprise vector find span decomposition bad never well decomposition work tensor algorithmic whether singular application particularly perspective al algorithms tensor decomposition orthogonal decomposition source tensor crucially identifiability degenerate range variable degenerate large tensor moment hide transform crucially need tensor decomposition go beyond degeneracy barrier identifiability interesting result successively hide tradeoff moment parameter seem advantage third whereas identifiability argue identifiability uniqueness large work pac statistically start notation tensor use throughout term tensor result tensor array tensor concept play crucial tensor tensor define rank decomposition rt tensor
family create scoring set simply include give high transfer select modular among use modularity parent graph parent biased toward encourage apply penalty geometric normalization sum exponential combination parent however form finite calculate parent value fix parent set parent parent etc maximum expansion outer sum binomial plug give edge occur therefore parent allow likely introduce map avoid deal typically hold bayesian similarity algorithm hold tune uninformative applying euler beta function q hold identity solution integral solution calculate combination value solver exist plug equation make order exist network normalization sum dags sample blue curve possible task discovery baseline baseline pool take leverage discovery pool merge apply sample network discovery picture likelihood affect true posterior posterior identify reverse direction true structure give network different vary experiment child edge repeat time produce network transfer training look closely effect size sample however posterior exhibit large bar curve show achieve network wide sense learn well separate size edge edge transfer raw estimate edge tend curve pool network pool modify edge effectively non posterior auc vs vs pool fp generative pool cross well estimate edge obtain learn feature edge tradeoff fp construct fp various roc curve roc positively difficulty positive greatest overall separation initially roc curve low positive pool fall false pool able give overall various small edge share true reverse blue green auc roc auc table trial problem trial bit compare trial look auc increase auc per trial give pair confidence winner pair test give large network mcmc try notably burn give ground auc versus vs pair fp tp aggregated particular alarm therefore order posterior roc figure positive curve well positive make auc dominate training dominate training size brain network fmri measure activity region interest slide discretized level mean functional network share discretized activity collect patient brain train discovery structure method learn identical much large many like subject large learn subject representative population parent calculate approximately score calculation posterior parameter burn interval tb complicate truth know trend estimate subset subject determined full edge pair various subset posterior able eliminate result network practice give present likely network software brain enough static read learn brain likelihood adjacency likelihood bootstrap experiment overall fairly amount however subject distinct difference visible visualization domain gain insight interaction among variable edge order modular currently bias term application sophisticated approximate bias could explore discovery propose incorporate biological calculation posterior attempt motivation order possible structural order term transfer show speed learn get knowledge experiment possible expensive impossible principled causal critical algorithmic present network structure algorithm leverage improve amount primary prior modular impose inductive structural prior intuitive advantage efficient closed evidence learn spurious mind network provide interesting helpful discussion grant dag child conditionally domain interpret valuable regard structural structural feature indicator otherwise super variable parent child parent dag tractable condition unconditional formulation dag order favor simple calculate sum approach involve summarize reasonable break modularity modularity px modular px ix ic modular common direct
orthogonality say orthogonal vanishing equation nothing else instead next series fourier coefficient take concept orthogonality basis elementary exercise page linearly gram schmidt normalise fourier q follow transformation convert chosen simplicity choose orthogonal regression discrete fourier relevant approximated integral latter task integral fourier normalise integration constraint case operation case orthogonal almost spaced function smooth maxima complicated shape difficult next taylor although polynomial initial outside lot derivative expansion small error avoid digit arithmetic transform interval coefficient perform transform data table square transform give table plot indistinguishable benchmark effort shall arithmetic digit accuracy solution divergence coefficient transform domain computing coefficient kx p typical program observe decrease
hinge eq differentiable linear point predict svms standard model svms multiclass call svms train class alternative multiclass denoting svm predict q softmax eq softmax multiclass objective maximize svms try margin use connect convolutional layer softmax notably paper semi deep embed net gradient activation input point backpropagation softmax representation organize compete initial period image expression winning win nd place team corrupt consist neural svm average capability preprocesse subtract value image pixel standardize implementation matlab file fast write google com face softmax svms model test split validation layer layer filter stage follow hide hide layer weight private vs latter maintain performance st convolutional h look filter conv mnist digit deep class classification problem example first pca dimension dimension follow softmax divide momentum softmax layer prevent overfitte lot add obtain error state softmax layer mainly effectiveness last svm layer softmax hide institute image color convolutional alternate pooling minibatch fairly relu unit second pool hide use relu difference convnet softmax convnet mainly svm constant decay constant learning rate select hyperparameter c convnet softmax convnet svm state around al normalization gain superiority ability model objective convnet convnet hinge square interesting entropy lead middle also convnet towards limited conclusion work softmax recent softmax multiclass svm understand much gain thank relu run experiment recently fully neural variety task speech bioinformatic employ softmax cross loss softmax base instead net svms result svms give gain dataset cifar face challenge learn softmax activation support combination net propose train use latent variable sample treat compare net top layer superior mnist cifar competition
toeplitz algorithm half datum thresholding toeplitz result swap repeat procedure penalize denote cca svd use perform cca select tuning permutation evaluate summarize median number deviation matrix sparse generate jointly methodology section precision tuning perform result swap procedure replicate summarize visualization replicate show l replication deviation ccccc svd outperform surprising assumption signal discussion define note ignore maximizer close explain I scenario assume matrix necessarily far consistent canonical dna essential tumor dna pattern frequently contribute gene survival characterize thus relationship gene expression patient breast genome breast consist dna gene breast cancer patient human site genome process level ratio intensity versus un du investigate expression clinical cca suggest select marginally status gene expression marginally disease status less select compare control gene separately operate biological pathway datum precision matrix cg cg cg cg lin cg cg cg cg cg cg cg cg cg cg cg cg cg cg cg cg cg cg cg cg cccc probe gene cg factor cg rna cg cell stress nucleotide cg region interpretation require gene apply eight table support detect genomic coordinate detect physical detect gene figure detect independently detect signature breast molecular van annotation detect pair canonical pair true direction canonical identify correlate expression provide oracle nature focus define signal define sequence set oracle probabilistic help thresholde oracle proof unless keep signal eq respectively signal q oracle sequence thresholding level parameter drive tuning depend choice oracle divide step first go bind oracle finite estimate number step outline singular corresponding quantity probability constant oracle actual identical assumption b induction induction generality since guarantee vector denote loss lemma desire therefore prove triangle least unconditional result vector exist sense give eq cs cs serve initialization procedure correctly pick one consistent summarize bi h h strong covariance strength among still guarantee consistency develop follow three state useful proposition prove capture much stronger specific strong coordinate constant result proposition probability prove result eq result valid second latent fact cp g second inequality eq far imply cn b cn combine cp cp b ci cn equation follow w h lem cp cp bound constant accord together proof proposition concentration ij I lead equation b ii applying inequality eq finish bound I ij ij equation union p ij cp last inequality finish p similarly j w b integer set large coordinate j k pick w q w imply constant weak weak signal line lem use follow b b two ns ns q proof singular intermediate see l h starting triangle last notice q qx finish lemma probabilistic lemma lem go matrix apply consequence helpful first step svd q almost vector notation write q step pair assume triangle side lem k therefore deduce desire oracle q therefore going prove obvious complete condition lemma lem conclusion lem conclusion lem cardinality lem first deterministic argument b nz ix ty b h ta nz nz h nz iy ny ny ty ty representation nz nz tb nx tb x tb nx h ty ty h row sum bound proof upper least union lemma kx k ty ty notice jointly union probability section h since k n formula proceed follow b us denote row tx tu q tu nu h tu nt row ty tv h keep notation proof q proposition ik e proof representation h nz nz iy nz h nx upper picking probability least nz h c q nz ix lem similarly lem cardinality together clear w equation tool multiple hypothesis test first need notation two probability dominate view give discussion need favorable favorable lower sharp kullback leibler divergence logarithm e equip integer unit covariance divide pick separate right generality switch pick favorable index canonical define I I accordingly determine later sparsity pick sufficient consequently coordinate q moreover ball therefore simplify rate bound together follow pick small op lemma rate prove inverse eigen cc plug unit fourth proposition cca receive attention relationship remarkably little foundation sparse active activity solution computationally procedure optimal breast cancer genome identify associate characterize signature decade amount development throughput group complex system popular range recently interface classical technique combination set center joint solve problem solve singular decomposition svd dimension fix difficulty impose structural direction cca canonical two vector effectively dimensionality interpretability many knowledge canonical indeed remarkably little theoretical study cca high setting despite recent development motivate necessary cca sparse decrease j sparse characterization explicit probabilistic characterization propose adjust thresholding simple good matrix transform adjust nuisance iterative transform implement sense estimating procedure establish cca nuisance dominating estimation canonical theoretically method cca propose drawback computational version propose heuristic would statistical literature account implicitly could valid guarantee consistency illustrate result account recover canonical precision canonical direction interpretation high cca structure simple refer correspond single comparison reveal estimation involve presence nuisance difficulty set absence nuisance contrast sparse adapt sparsity need method arise gene identify breast propose reasonable exploratory setting characterize sparse cca propose probabilistic second nuisance efficient attain section present investigate study technical lemma denote singular value spectral frobenius norm two notation along random covariance canonical direction solution nonzero maximization degenerate notice scalar identifiability reformulate write correlation correlation elementary svd transform reason omit routine sign jointly covariance inspire probabilistic canonical explicitly spike hand fundamentally estimate estimating spike study probabilistic latent specify set directly satisfie going consider precision second first practice apply swap datum half second final calculate average many bag stability sensible initialization specifically apply thresholding index think submatrix column index lead zero lead pair singular summarize pick ij singular c thresholding level specify constant allow adaptive location drive serve output pair provide span estimator estimator precision powerful tool among known recover undirected gaussian support among assume precision propose see detail propose go procedure apply literature bandwidth choose cross toeplitz entry class toeplitz naturally stochastic certain rate decay st average bandwidth pick bandwidth validation order toeplitz permutation estimator covariance row negligible thresholding give ij end present optimality range
survey factor context handling dimensionality var augmentation establish lasso precise oracle least square comment bound possible estimate consistently state big variable leave avoid bias relevant exclude sign pattern adaptive detect pattern tend square imply converge least covariate asymptotically efficient least inequality give dimensional eigenvalue special theory put forecast compare use result much often curse building var include attractive derive employ dependence assumption instead furthermore fix eigenvalue property asymptotically adaptive observation simply recent say forecast back informative procedure underlie lasso causality coefficient plan detail background main investigate validity conclude paper root assumption crucial root useful imply tail impose use moderate unable root equivalent denote large stack explanatory time covariate correspond dimensional parameter equation much say course traditional square bound whole might course let cardinality entry rx nj consist element denote large eigenvalue two let property study extensively see select asymptotically restrictive covariate device situation put differently remove irrelevant relevant minimizing plus extra denote let begin give assumption design let valid sequel yield turn derive whole gram inequality hold tend proof derive lemma sharp bound shall equation remarkable may zero non sparsity bind main lower bind inequality type g presence gram equivalently case square infeasible replace var var model restrict restricted condition satisfy cardinality minimum restrict large restrict trivially satisfied mean traditional eigenvalue eigenvalue eigenvalue like rank lasso appendix long close expectation condition close dependent k p fact asymptotically system explore hand probability careful put much emphasis finite oracle inequality least hold least variable exclude theorem non asymptotic vector smaller bind increase parameter ps almost fast discussion role type spirit connection context involve function dependent upper lasso classical discuss tend give reveal end denote pattern relevant equation expression consist multiply small oracle slowly reasonable call oracle lasso almost pattern estimate square beta min model beta min condition zero non basically coefficient bind lasso estimator follow whole constant least corollary whole counterpart estimation next grow one want consistency since yield e variable essential factor utilize result describe tend equation variable prediction suffice case sub exponential arrive polynomial sub exponentially sample dimensionality choose increase square root size still tend seem lag realistic sub suffice initial valid actually make sample make rely place iii condition avoid relevant necessary sense assumption rule beta min ensure exclude stage merely sufficient boundedness pp setting also min condition iv interest investigate estimator denote tend boundary unit deal estimation triangular stay distinct long slow unity firstly tend boundary secondly theorem case change subtle proof linearly conclude cover still give case vary alternatively apply easily bound assume reveal consistency lot rate sensible increase error vanish classical think possible similar fashion correspond univariate mention entire could arise var lag lag lasso equally truly non would idea regressor establish asymptotically set tend one pattern model give correct bound tend correct little denote minimizer th exclude lasso estimator second likely small consistency small penalty truly condition sign imply correct bind theorem tighter use furthermore convenient choice simpler consistent keep expression interpretation establish asymptotic consistency construct theorem precise small side hard sensible since increase dimension sign notice assumption reasonable expect detect correct calculate mc covariance setting sample experiment var truly behavior depend illustrate set redundant experiment generate var block block equal behavior often matrix maximal get distant lag conventional lag distant entry decrease oracle r mean root square ahead forecast table neither accordance clear hand detecting exactly correct much illustrate often adaptive share variable include still relatively find share percent initial exclude exclude encourage rarely variable many include put differently quite encouraging variable redundant use ridge often lasso albeit narrow margin estimator decrease error dimension reduction result reduction square lasso tend due second less oracle stem shrinkage forecast except setting consequence least contain monte replication equation parameter far leave bottom report include correct slightly biased square secondly due wrong top procedure still leave result experiment setting since possess circle ever procedure leave relevant relevant sample lasso irrespective estimator include lasso low covariate lasso estimator discard relevant variable adaptive oracle ols adaptive lasso always plain bias also possibility exclude lasso post oracle c lasso ridge c r c share relevant square root ahead adaptive ridge outperform lasso perform oracle ols provide ridge lasso improve exclude model experiment b lasso adaptive lasso least retain relevant increase relevant also always lasso discard turn select include lasso put carry rough adaptive always ols also previous forecast precise ol oracle counterpart mixed oracle ols ridge r k model include select c error forecast sensible procedure model irrespective explain parameter shrink parameter estimate precise forecast encouraging since type violate l lasso post lasso ols ols include root square forecast particular establish upper retained perform infeasible parameter grow exponentially next lower sign asymptotic consistency parameter allow exponentially least apply handle sample empirical matrix result useful curse dimensionality building increase adaptive applicable situation currently cover justified however work stationary start couple probability become norm useful maximal give norm maximal decrease result increase repeat application variance gaussian page tail fact assumption satisfy lemma expectation integrable integrable almost sigma independent define sequence equality ei next subscript minimize equivalently put add yield inequality notice next show verify restrict satisfy let semi b rearrange statement maximum write note value see first bind probability ne suffice far enough bind kp choose zero except zero except th position th elementary request obtain explain equal p multivariate covariance cauchy schwarz thing together yield let reveal display understand tm yield upon take subscript brevity confusion establish use assertion exactly argument p result equation omit subscript index operator give j follow slight state combination corollary valid theorem note suffice subscript brevity observe certain tends first note see part exclude argue proof equation omit subscript brevity tend side tend turn imply var tw ie f cc estimate finite maximum merged tend I yield suffice tend since remain similar satisfied constant norm lemma end notice tail lemma theorem shall equation omit subscript brevity set triangle bind first hold q side verify show c complete since continue theorem consistency verify asymptotically valid hold see establish tend verify q tend away term second tend suffice eq tend since away regard eq asymptotic measure result subscript brevity consistent tend zero eq probability tend right recognize suffice term side probability notice b bt sufficient tt corollary
kullback leibler divergence minimization simplify exploiting turn previously introduce unified analytically portion q low variational call substituting make one simplify result straightforward equation factor put bayes posterior normalize th correspond result distribution update reveal act latent model continue result maxima infinity assume follow maxima infinity small calculate analytically formulation report compare soon publish example university generalization bayesian examine relevance machine popular bayesian contaminate simple application model process provide rich similarity space kernel converge fast interest especially model output uncertain noise effect consist eq model similar q hyper e g respect consider one entail keep everything maximize determine
science reference therein minimize distortion distortion integrate measure paper adopt presence introduce deconvolution construct deconvolution deconvolution slight abuse notation I estimator plug deconvolution fix stochastic write devoted deconvolution high rate behaviour behaviour two assumption difficulty whereas regularity express difference residual side message highlight fast exact idea bias decomposition unsupervise exact organize regularity margin fast apply dimensional conclude discussion whereas proof result deconvolution erm originally discriminant construction deconvolution introduce dd fourier fourier deconvolution kernel empirical risk risk deconvolution restrict control restrict compact support great simplicity discuss depend moderately ill noise kind deconvolution pose deconvolution direction sake ill pose combination decrease recently deconvolution case follow standard arise proof relax fine sequel order kernel could approximation regularity old admit derivative large less old old regularity taylor know behaviour govern quantify space logarithm exist notion uniform rate really make small principle originally state inequality small excess exact get margin guarantee nice local theorem rate imply propose sufficient margin parameter wish non significantly assumption use class lead oracle inequalities time oracle oracle suppose deconvolution erm cn proof comparison order residual oracle risk residual pay standard rate deal intuition result insight quantization problem bandwidth deconvolution dimensional support could treat least favorable arise old fast equivalently optimize margin old challenging convergence also bandwidth bandwidth density deconvolution regularity ill asymptotic q propose rather bandwidth plug also theorem appear trivially satisfied fast rate context margin hold high cn phenomenon describe order appear give pay front infimum course rate tends see assumption theorem indirect loss variance precisely simplicity done restrict region control theorem noisy paper introduce integer possible center euclidean true sequel assumption performance minimizer nk means study excess recently propose fast improve ingredient localization spirit cluster deconvolution deconvolution deconvolution choose investigate regularity assumption regularity respect hessian matrix definite finite number margin lemma derive condition follow cell sup depend assumption related margin assumption assumption main cn proof remark dimensional lead term term principle consist iteratively concentration inequality finite pay avoid scope optimality way low bound supervise first attempt risk deconvolution risk deconvolution risk residual complexity hypothesis term behaviour previous consider introduce deconvolution deconvolution fast convergence another result learn noisy curve unknown erm design deconvolution principle new deal unsupervised step mean core principal tool countable measurable measurable every entropy refine consider core localization consist theorem small extend set apply excess class exact core general introduce eq equality slight transformation moreover need also interested discretization transformation sequel may exist constant u r pg g positive number event jt r g assertion rr rv r begin lower bind definition q write jt follow version r z g pg follow introduce jt restrict r r use definition r r rr event r equivalently obtain jt formulation allow exact oracle sophisticated dimension slightly algebra lead obtain g g notation u diameter thank assertion pl g assertion easy case independent dimension result check enough calculus thank introduce geometric g idea version concentration bound end check simplicity clear pg g g r g z z cg take infimum ingredient lemma lemma bound entropy line cg boundedness assertion follow assertion generic triplet give triplet get slightly inspection replace condition treat notation rise satisfie f depends bound generic
surely proposition evolution q collect bound standard limit surely ordinary admit surely assume ir iv consider ng ii g g r adapt surely evolution sequence differential lyapunov nash nash equilibria converge game spirit differential inclusion I payoff proposition surely connect equilibrium potential value definition response nash equilibria motivated project institute thank school author paris less sequence return discrete take satisfying set simplicity contain become argument extend space main positive change proof let notice therefore second identity pseudo since n martingale q surely eq surely eq almost surely successive recall hence iii go vector generality payoff vanishing exist without generality conclude let spectral gap direct sufficiently equation sufficiently surely positive line line estimation obtain read exist thus conclusion loss application mean value n ns surely sufficiently establish assume q go infinity part detail argument similar proof omit confirm suffice eq lemma hence go component vector surely let surely go clarity eq kronecker delta therefore easy fact conclusion go surely take eq assumption I cm cm rgb university economics de la fr section proposition remark normal repeat procedure realize payoff payoff restriction play converge equilibria game sum game one procedure theory repeat discrete issue play identify play converge equilibria body devote player zero games general degenerate prove player player large proportion approximation theory I al play analyze game I game payoff compact see agent game perturb game game define player response player know game know payoff inform action stage frequencie response assumption relax approach agent realize reinforcement proceed follow payoff receive suppose depend history play literature simple g game player equilibria homogeneous time decision mixed implement rule action markovian come player restriction action due physical player inform play explore action nash equilibrium assumption procedure player minimal information action behavior game action mean agent realize payoff procedure reinforcement addition choose decision mixed choose markovian meaningful difference reinforcement mix long choice variable mixed strategy good asymptotic move nash equilibria zero include payoff organize present introduce analyze markovian present result present extend appendix pure exclude good response equilibrium game value map game let play know game play payoff assume inform player reinforcement player time player observe consequence realize action update restriction play pure stage available action restriction action subset player last stochastic word player play switch reversible respect e irreducible let sigma algebra play end stage time payoff observe realize update priori far variable payoff suppose grow slowly quantity payoff matrix quantity turn example believe worth spirit suppose payoff along omit show every present iteration payoff matrix measure scale thick v v bend bend bend bend game select cycle game payoff assume matrix graph represent replace respectively show realization converge simulation towards display seem action interest player exploration matrix right c bend right bend even player restriction switch another action realization ne trajectory ne displayed realize converge payoff plot recent equilibrium aim introduce introduce discrete process space equip let assume compact asymptotic simplification observable value map nonempty adapt almost differential admit I e exist neighborhood every invariant connect define q evolution eq value map surely limit almost surely differential surely contain speak adapt see cauchy euler inclusion decrease guarantee vanish consequence dynamic chain inclusion admit stochastic approximation differential introduce adaptive call exploration suppose irreducible reversible payoff also player inform opponent action accordingly
cc fundamental role many crowdsource vote ask vote prediction forest bag give test error achieve majority vote limit exchangeable sequence vote write majority principle aggregate decision level vote resource time vote collect majority vote likely select trade basic small reliable general arise binary ensemble include bag boost connection voting arise space ensemble base aggregated majority vote bag forest test label eliminate tie ensemble classifier large nominal limit view target pay base must store evaluated cost carry variety different tuning select aim emphasis method bag despite close connection rate ensemble receive little bagging random regard rate aggregation rule boost algorithm iteratively focus voting rule comparable boost closely aware generate study bayesian analysis specify apart aware involve vote formula formula applicable majority exchangeable sequence theorem extend beyond relevant recommender online market choice voting vote weak exchangeability discussion exchangeable voting section test randomness play statement depend randomization bagging subset lastly write subscript omit arise correlation nevertheless case forest bag forest bag conditionally give seminal elsewhere abstract definition restrict reduce bernoulli proportion decompose sum draw clear odd formal distinction two generic sequence serve broadly vote scenario exist variable take exchangeable bernoulli twice continuously predict sequence constant due obtain directly continuously eq light term may sample define eq represent ensemble idea smooth transition classify difficult specifically test assign mass smoothness ensemble offer section turn technique second binomial although lead inferior binomial limit involve reduce let second expansion guarantee uniform boundary role uniformity let replace expansion express term contribution particular lemma section tend integral essential change integrating denote normal extra introduce give remainder term line upon every determining dominate write remainder lagrange continuity formula fix integrable consequently follow support base distribution
give equation recover uniqueness strong establish recently substantially decomposition continuous martingale increase follow diffusion interface alternative right quadratic variation interface interface approximation discretization skew brownian discrete markov probability time point indicate fourth moment proof line walk finite increment full remainder embed within general enumeration zero permutation skew walk embed motion skew rather exercise embed weak skew rescaled skew skew walk physical skew diffusion process convergence discretize process alternative suggest p self character conservative treatment formulation standard finite difference conservative interface presence term euler preserve euler conservative interface piecewise aid lemma conservative interface possibility interest simulation numerical deterministic apply interface available restrictive general impose outside interest find section et brownian special interface physical diffusion skew differential equation apply strong formulae give interface relation process check match operator characterization operator skew coincide markov regular interface primary article interface especially biological effect guide determination notion arise modification standard mathematical adapt unit continuity context process example modeling dispersion population consider different area mathematic application area involve distinct interface fit interface mathematical perspective already physical involve heat heterogeneous medium thin compose infinite capacity heat interface interface heat provide context biological environmental physical interface begin medium first theory heterogeneous taylor dispersion average dispersion physical science flow axis term flow profile separate geometry appear formulae effective rate coincide molecular align profile contribution effect asymptotically insight originally refine perturbation central brownian flow dispersion drift dispersion coefficient time particle heterogeneity currently interface average region boundary suppose axis matrix variable positive away average longitudinal initial involve scale eq ergodic boundary piecewise see accordance taylor formula uniform normal vector k x partitioning replace arithmetic second harmonic interface separate medium profile physical skew diffusion topic address observation result laboratory design empirically sharp example laboratory sophisticated different measurement interface curve require concentration interface retrieve interface dispersion origin conversely interface retrieve point unit leave investigate time brownian motion skew diffusion h recall respective skew stochastic time cite coupling rely specific conservative respective interface perspective concentration interface time skew literature recognition movement another reference therein point highlight involve specifically interface functional example fairly mathematic species united focus brownian general diffusion arbitrary network consider graph edge model stream reach edge associate strictly water velocity cross area diffusion coefficient correspond singleton root internal connecting leave spatio impose mass throughout denote restriction continuously differentiable consider join appropriate extension read water behavior boundary leave network channel flow occur remove mathematically require hand condition generator process continuous time brownian motion heterogeneity skew ne important drift whereby although stream average location population regard velocity population whole along channel individual persistence involve jump jump distribute represent water literature give view derive switch distinct recently however persistence regime net course channel interface simply motion drift brownian primary persistence define instability tree network wherein solve permit nontrivial event drift minimum individual persistence continuity exploit south activity narrow break south reach surface described see mathematical describe location simplify assumption balance equation surface bottom axis motion think diffusion play time feature bottom piecewise interface current lead correspond interface mathematically general theory interface indeed bt interface alternatively obtain article basic pathway presence process rich realistic diverse quantity arise science engineering short mathematical highlight effect attempt comprehensive consequence recognize specification result adopt default relate general dispersion partially coefficient may permit development along dispersion medium formulation probabilistic profile concentration express derivative case medium interface interface evolve differently fine interface fine coarse symmetry curve concentration coarse fine explicitly computable corresponding generality however expression skew brownian numerical piecewise piecewise problem largely biological conservative interface proper equation treat explanation theorem illustrate social situation change observable quantity refined process involve effect section problem network preserve branch comprehensive several extension formula occur across apply case occur example sphere applicability many occur case regard prove radial component process concentrate sphere second rank diffusion study diffusion skew problem yet progress heterogeneous water velocity use diffusion solution outside diffusion skewness limit analyze equation periodic outside finite region hyperplane limit coefficient determine turn interface side rich go perhaps skew diffusion namely medium define continuity skew diffusion characterize skewness several skew inside surprisingly continuous piecewise skew brownian skewness medium particle periodic infinite periodic diffusion dominate author grateful reading greatly improve exposition fellowship nsf mathematic application year support nsf grant dms section remark definition de scale scale variety span science term equation mass divergence spatial diffusion pathway special time theory achievable application coefficient consequence dispersion phenomena concentration follow differential particular coefficient matrix value describe scalar mass evolve assume take domain appropriate boundary condition development computation guide st equation inspire diversity development brownian stochastic calculus stochastic differential probability molecular tensor process probabilistic interest e physical biological perhaps framework problem still diverse way measure occur phenomenon admit determination molecular model obvious formulate phenomena trajectory financial biological experiment g trajectory variety first local toward express trajectory approximately shift motion locally particular fact constant solution generally directly form call science merely equivalent express suggest respective special integration adjoint discuss point present phenomenon piecewise dispersion term great perhaps application involve extensive functional early high brownian play albeit associate smooth process term dimensional dispersion across interface fundamental skew brownian motion mathematically comprehensive article skew brownian provide equivalent skew motion article build mathematically comprehensive skew motion quite recommend article indeed skew diffusion associate dispersion focus skew physical biological phenomenon differential differential equation computational scheme aspect relate accordingly amenable explanation prediction therein mathematics international structure reflect consequence locally result quantify large scale basic coefficient achievable model piecewise diverse range herein skew motion broad context dispersion physical subsequent section general arise naturally physical sciences free surface movement build complementary problem continuity medium separate xt one eq interface physical p check see nonetheless skewness around skew brownian since path complement countable disjoint union open interval brownian correspond consider difficult skew almost sure continuity brownian f markov brownian uniquely diffusion rescale skew brownian particular transmission may diffusion diffusion transition start self conservative interface diffusion although inspection transition skew sense skew consequence continuity skew follow martingale x similar albeit somewhat technical procedure develop constant drift skew detail theory naturally associate alternative path provide mathematical diverse interface begin perhaps mathematically develop numerical subsection certainly read skew walk follow address certain context outline lead refer reader comprehensive reference analytical solution follow variational bilinear hilbert consideration denote domain integrable u u v
indistinguishable n thing simple n n j put h obtain g expansion term approximate course expansion justify uniform fr see g van rigorous theory proof omit arithmetic random sample correlation coefficient n variance denote h show deviation htp simulate inside interval negligible htp perform gamma lead qualitatively author request reasonable scenario process asymptotically negligible hence chen dominate process involve goodness fit normality fact cm xt remark thm data transformation goodness right email cm department mathematics institute department university business mathematics north west south goodness fit censor strategy observation normality transform reveal property theoretical keyword empirical fit censor censor ii censor df interest goodness nan test test kolmogorov estimate conventional brain test test censor contain nice overview normality type ii censor already note censor method give statistic sample order sample uniformity uniformity lin uniformity naturally extend hypothesis use transformation uniformity quasi extra variability value estimation connection independent chen transformation normality combine normality uniformity censor test sample chen thesis chen uniformity conjunction chen transformation test paper section present indicate implement statistic deal type censor monte draw combination statistic transformation sampling property chen statistic put denote statistic seek type censor set appear n denote df lin set case eq r I transformation transformation aim stochastically nan hypothesis standardize complete normal latter wide also error chen efficiently base nz provide chen transformation justify situation appendix dominate part test normality depend parameter deviation illustrate testing censor calculate u denote transformation step normality test amongst df er von finite normality utilize smooth periodic statistic compare cf statistic competitive test fact complicated approximation distribution easily connection function choice become something cf weight also conclusion equivalence cf association weight hand impact power property cf bandwidth treatment statistic analytical quantitative suggestion require nevertheless recover already resp detect short long consideration extensive simulation correspond conclusion agree wang uniformity seem reasonable note chen justification kolmogorov would study know er von implement transformation employ censor depend interest df give simplified mle jx employ estimate suggest author size suggest whereby use linearization table extensive testing er von characteristic weight ms exponential estimation outline section alternative employ distribution three censor correspond size fact distinguish lin difference statistic statistic modification version apply censor von obvious last newly inferior power applicability critical censor proportion mention even standard censoring provide htp c ccc ccc ccc os c ccc ccc ccc os simulation gamma table conclusion draw os maintain ms base good power logarithmic gamma os transformation power gamma see power transformation however os stand distribution well distribution alternative alternative os alternative distinguish gamma observe transformation test mm ccc ccc ccc os ccc ccc ms os testing conclusion follow htp somewhat nominal level alternative ms os power lin ms os give power ms os detect reliably medium low censor preferable base ms os logarithmic censor gamma alternative suitable power slightly nominal nevertheless os high power base test transformation show add ds er von normal censor censor exponential although competitive htp c maximum quite bad omit mm ccc ccc ccc ccc cc ms os ccc ccc ccc cc os c qualitative message entry five uniformity transformation digit ms ms rank assessment three percentage rank alternative censor proportion test low give os look score clear low transformation ms os test perform combination test os htp os series transformation applicable distribution free conclusion recover power lin et seem good superiority compare transformation wish test advantage lie power censoring suggest essentially face test normality investigate reason validity chen
kullback series display stationarity row row row column stationary specific interest suggest boost model apply show remove stationary change making extraction remove frequent change difficult detect reliably propose number condition condition strength non irrelevant point apply condition interest discard nine augmentation pre extraction nine component area receiver auc column improvement change preprocesse enough reasonable many irrelevant background many change interest interest panel stationarity leave corner panel display row determine background change change case strength share device activity use device fmri imaging eeg eeg desire direction movement one eeg stationary deal level numerous machine learn question non stationarity stationarity activity result loose remain eeg activity highly stationarity study contribution activity particular stationarity contribution activity neural problematic nevertheless model present present apart stationarity achieve change eeg possibility display removal direction display pattern result stationary non remove pattern indicate generate however common remove difference estimate pattern background stationary source display hand smooth fact contain extend hand usefulness heuristic extent model choice subspace include background display condition predefine validity heuristic subject show propose paper study change eeg subject design usefulness principled far exposition innovation compare correction clear limitation condition background background subject condition weak figure observe present presence guarantee despite component strong component removal model type stationary highly stationary specific locate localize conjecture cause display non stationary visible display extra structure pattern heuristic conservative neural subject typically paradigm interpretation condition framework removal extracted interpret demonstrate eeg study pattern highly projection background overlap share lost requirement estimation independence estimation impossible independence origin realistic specific priori pattern origin hold stationarity publication allow non stationarity argue background non stationarity eeg moreover specific exhibit non stationarity background stationarity special direction however point important gain entire summary performance analyst discover change experimental specific multimodal co respective acknowledgement thank comment topic part training computation neural system focus technology adaptive stationary support foundation education technology stationary simultaneously statement k quantity bound line I sd apart row identity grow tend one density row dot product dimensional subspace entry dot thus term dot product span order root fourth moment decrease mean thus chebyshev conjecture corollary notation notation frank von series change induce light interest origin non stationarity interpretation wide ingredient application different temporal usefulness theory eeg experiment face brain activity highly variability attribute ica trend due due yield suboptimal paper neural propose enable extraction global behavioral change well specific far illustrate abstract concrete example leave use user alpha activity hz alpha activity like type change examine principle think change domain solved optimize subset propose task suppose experimental display condition highlight interest fortunately weak situation play important role datum far ignore sound issue framework aim contribute well understand experimental change focus model outline background theoretical quality simulation lastly cover generative underlying parameter provide generative multivariate superposition latent observable source dt stationarity moment mean stationarity covariance moreover eeg see ignore high filter matrix variate generate rectangular entry th contribute spatially column pattern eeg span note orthogonal recover however differ assume specific latter information introduce irrelevant inconsistent drawn series correctly capture condition separate record trial correspond single stimulus etc sequence although essential exhibit trial illustrate eeg brain record condition record single movement hand condition observe generate sum contribution q stay time refer stationary stated stationarity remaining imply random draw sample entity index series represent experiment sample separate background analyze whole mix mix background stationary allow stationary component background row subspace span basis stationary find epoch require decrease grow nevertheless well subspace eeg stationary condition system stationarity plausible system matrix may nevertheless plausible non control artificial ground investigate true analysis second detection situation non component analyse simulation accord background background transform orthogonal mixing choose system comprise stationary lag stationary stationary component lag choose model background consist correspond five draw spaced background parameter probability I switch another parametrize denote thus correspond occur short segment gaussian respective occur stationarity system first investigate influence two scenario high specific non error subspace stationary background condition specific stationarity blue line become condition system non direction stationary line contribution soon four identify specific panel degree make dominate change specific condition around stationarity system application realistic eeg head calculate basis place head international semi contribution et al simulated background edge
microarray datum define pearson point way define distance know single linkage element contain dissimilarity linkage dissimilarity linkage cluster cluster dissimilarity cluster dissimilarity detail note hierarchical root cluster terminal correspond singleton represent plot cluster merge briefly outline supervise known outcome namely label assignment seek outcome variable assign inefficient page one million classify intervention likely slow develop spam spam unlabele label inefficient vast majority classification rule unlabele discussion k mean constrain label implie denote proceed calculate cluster conventional assignment arbitrary mean label observation misclassifie constrain misclassifie misclassification iteration recommend exception assign near cluster even observation observation correct mean identical conventional exception simply use center partially dna microarray microarray normally one wish similar gene cluster pathway gene belong pathway perform experiment seek gene cluster method develop microarray specifically design microarray reference among know two place link place repeat collect subset experimental note generalization cluster one may cluster feature method constrain commonly numerous list various type assign follow repeat converge step identical conventional exception near violate assign cluster drawback violate situation wish violate particular incorrect propose solve identify must link constraint violate seek observation link constraint minimize must penalty detail minimize method modify exist constraint method contrast cluster modify euclidean distance distance metric two must constraint low distance possible far far specify collect situation select suppose appear manually examine document determine classify either link constraint suppose medical journal impose constraint article determination analyze small subset observation situation advantageous choose variant choose impose outperform generic algorithm semi supervise partly formulate differently semi utilize either e hierarchical hierarchy link hierarchy semi consider constraint cluster together hierarchy single return separate hierarchy constraint relate hierarchical propose hierarchical observation cluster consider order wherein certain order observation combine contain tolerance constraint constraint develop implement cluster semi supervise hierarchical little advantage development supervise hierarchical clustering remain research area noisy cancer characteristic may genetic survival risk survival risk considerable patient survival vice versa illustration patient survival time outcome instead identify outcome specialize outcome situation outcome outcome conventional center cluster great thus cluster despite problem identify identify secondary cluster identify outcome supervise testing association outcome variable continuous testing coefficient predict outcome censor may cox proportional apply cluster feature discard assignment approach relatively relevant set use identify patient survival datum effectively fewer choose cross validation apply cluster subset feature strongly outcome variable one could hierarchical clustering supervise recursively partition mixture method cluster supervise produce supervise semi detect see drawback fact discard screening step exclude problematic identify feature possible feature across weakly associate cluster outcome fail identify call cluster sparse unsupervised k mean dimensional produce inaccurate identify clustering mean objective maximize observation seek identify tuning choose clustering impose weight impose regression cause coefficient tune several reducing sparse choose feature initial word initial modification likely outcome supervise semi method feature sparse strongly associate identical supervised cluster clustering iterate procedure produce supervise situation weakly supervise procedure outcome seed remainder cluster without research activity semi supervise constrain decade numerous constrain develop microarray biological semi supervise constrained clustering appear extensive either method apply option genetic cluster dna genetic genome association rna seq generation dna sequencing semi cluster genetic associated outcome study future partially support grant thank anonymous suggestion seek homogeneous variety document modern cluster outcome datum many situation example certain cluster outcome review situation majority detail description semi provide
world demonstrate effectiveness show detector fundamental predefined object using gain label algorithm many art adopt detection receiver roc classifier threshold face area characterize vision highly asymmetric ever target million patch would thousand impractical researcher report false roc positive summarize practical detector primary metric directly optimize often ensemble classifier directly roc range ensemble optimize boost approach classification greedy weight however unlike traditional iteration place emphasis sample incorrect order auc yield area rate wide particular roc share conventional multivariate measure simple conventional visual transform modification code efficient cutting solver partial simple adaboost cost sensitive adaboost asymmetric unclear achieve principled optimize auc arbitrary false range especially effectiveness perform art despite fact detector two image vision object positive treat exploit cost classifier adaboost first weak asymmetric desire optimize descent address criterion carefully validate asymmetric parameter maximize false optimize propose bioinformatics algorithm heuristic develop optimize support outperform asymmetric principled fully optimize evaluation false positive knowledge principled ensemble optimize auc ensemble bold letter negative possible learner learner weak respectively predict matrix represent learner training instance row weak learner training weak learner score function performance rate briefly svm upon unless symbol area partial auc denote position cast instance order positive range consistent optimize score auc follow optimization projection project weak boost scoring optimize false set weak learner already projection function solve project output n difference new experiment dual eq follow kkt condition use ensemble generation kkt correspond condition weak yet appear current set weak learner optimal learner training respectively learner vice versa flip inequality linear close well weak minimize objective optimization exactly adaboost weight coefficient l indicator weight u w transform publicly line code supplementary classifier classifier node classifiers auc display approach weak show perform preserve decision boundary positive angle effectiveness baseline adaboost cost adaboost adaboost adaboost vertical horizontal partial auc train classifier illustrate boundary decision asymmetric place emphasis ensure part curve though choose adaboost worse since optimize weak classifier toy data observation explain adaboost though minimize classifier node often use often contain weak performance display display false approach place emphasis positive corner angle bt svm svm reported report finer mark report bioinformatics protein protein predict label interaction detail publicly internet contain protein interact non group validation repeat train maximum baseline svm outperform optimize either result face boost previously adaboost fisher cs adaboost also train weak classifier score experiment repeat time digit face extraction result demonstrate vision set ccc scene face adaboost cs adaboost detection approach datum generate window generate window node window orient gradient sketch self weight linear weak reduce time cross validate validate fine range could approach core pixel image pyramid window merge auc software false positive positive positive minimal tends advantage bt cccc range versus mean compare art algorithm compute perform comparable train detector scale close less slightly set observe bootstrappe cascade state train cascade detector train original detector detector combination previously cascade positive software sort propose adaboost reduce fig similar performance auc perform
conditional another formulae integrate configuration place write collection individually denote apply dimension denote sum kernel generate new expression repeatedly apply production rule kernel multiplication production kernel kernel locally periodic scale describe production algebraic notation good use apply second independent pf pf pf true lemma get series analyse load utility focus competition forecasting consist deviation overall temperature spike load related spike apply methodology temperature subsequent search explain short variation smooth periodic day since explain periodic component smooth periodic periodic multiply rational vary smoothly time increase detail explain periodic temperature expand sure go trace temperature temperature space understand interaction super
establish hmms adapt discriminative form theoretic principle requirement parametric form still discriminative max margin method model expense interpretation hmm concept direct density ratio estimation try quantify observation make likely possible observation observation inference algorithm ratio estimation carry base parameterization conventional monitoring anomaly detection rest conventional ratio nonparametric ratio supervise unsupervised likelihood section hmm monitoring improvement conventional inference available estimation markovian dynamic value independently emission backward frame use describe stage recursively calculate message step necessary message numerically become occur observation mean inference interested magnitude first calculate normalization carry iteration two final forward explicit every scaling prevent information normalization step formulation equation ratio express bayesian simply multiply likelihood forward equation correspond type ratio treat likelihood ratio eqs derive version forward expression term pairwise frame similar simply eq need st st complete backward intuition ratio reduce q backward specify px form figure show transition equal inference favor globally tw multiplication forward intuition map magnitude unity prevent inference hmm ever mathematically equivalent forward exactly formulation parameterization natural sequential discuss ratio carry learn density likelihood initial method efficient relate estimate individually discuss discuss square square exponential obtain empirically approximate average ignore regularizer criterion indicate square diagonal minimized estimator negative estimator practice optimum iterative would would relationship ratio unstable denominator estimation I px j similar principle scheme use likelihood hmms computationally square comparable kernel far less qx square training minimize ridge round zero qx though unlikely enough ratio sample two class quickly discuss density ratio hmm frequency remain hmm unsupervised carry iterate maximization px expectation give run procedure accommodate hard update experimental apply synthetic top improve incorporate right panel forward incorporate illustration inference simple switching dynamical regime parameter scalar follow left testing use slide window bottom result use likely find ir I equivalent individual sequence estimate estimate inference alternative nonparametric estimation validation conventional modeling observation minimize bic randomly wave example regime also hmm kde hmm gmm accuracy frame equal state deviation filter ratio high accuracy particularly ratio measurement intensive hour hour take sign environmental temperature annotate occurrence phenomena stop heart temperature probe period cover annotated period annotate baseline treat separate inference example training data period annotate factorial factorial switch patient monitor filter inference extensive knowledge cross area curve equal annotation calculate show table
behave correction estimation jensen maker strong well observer kullback answer underlie asymptotic kullback write particular distribution average old imagine observer trial tend trial draw estimate probability outcome kl average probability zero chance observer leibler resample impossible bootstrap estimate bar reflect truly deterministic signal underlie estimating jensen shannon behave jensen shannon divergence choose imagine observer old correct belief probabilitie uncertain room reduce jensen draw variable distribution choose draw jensen unity zero easy kullback jensen shannon always attempt correction take reference entropy particular subspace information grain appear distribution bias equation fashion mutual bootstrap correct bootstrap correct lead find place bound inside external measure maximally rational observer expect wrong infer fact bound oppose know maximally system outperform one ever observer old trial determine outcome observer know nothing bayes informally rational observer error decrease lin significantly world situation remarkable observer progress trial unfortunately observation formulation term observation one close multiple trial year period associate particularly close true value ease computed interest quickly become apparent system word possible category year period coarse close unity probability drop upon repeatedly overall obvious way fail fail text within remainder draw fail draw equally likely thing trial class second lead belief draw draw draw sub distinct derive q impossible simple version predict predict sample observation choose prediction equation expect predict trial course assumption wrong way signature trial ergodicity social observer sub measure extent decision ideal interpretation subtle useful kl interpret accumulate become infinite would expect world meanwhile divergence behave information bit distinguish outcome extend shannon divergence involve multiple true spin member cart result report appear eight event country member international security force extent day week draw remarkably open release lead effort characterize set study modal release amount roughly report record distinguish record event occur generate symbolic coarse code assign event record day code event record record rd spin one record device base fact assign code understand modern measure actor directly extent event extent play collaborative investigation minimal information share synchronization share time provide conceptual quantitative question human centrality study decentralize question neighbor kind pose measure two force marginal similarly label write mutual jensen naive correct coarse preserve observation happen produce precisely produce correlation difficulty many establish complete order prevent report relationship much method answer question starting describe characterize amenable case time mutual state code correct day event boost indicate pathway pathway involve common cause act signal include political national weather act usually daily make impossible know bit bit mutual influence flow consider mutual information day take day symbol date tie break mode essentially coarse day language rapid bottom consistent see reverse rise long month panel term constraint common mutual modal modal lead bit current much modal near modal day future potentially affect cause novel simple inequality transformation process grain coarse essential study underlie physic modification partitioning extend work compare bayesian bootstrap coarse mutual information include information diagram entropy joint hold recover nearly finite turn entropy formula analytically entropy entropy bias desire preserve equation directly sample characterize performance estimating observation divide show gain two table consider mutual fashion case compare sampling bit bit bootstrap estimator coarse bootstrap three sampling bit bit mutual information factor amount bootstrap consistency fast approximately poor bin equal grain coarse draw however largely estimator unchanged technical consistency analysis outcome old concern trial represent semantic none anomalous gain change meanwhile mutual analysis information flow highly codebook account parallel shift reflect underlying information axiom technical characterize reliability correction estimate uncertainty separate bias compare particular entropy jensen shannon informally estimate sample true quantity necessarily reduce bias simplicity q estimator prior discuss prior estimator performance predict want question lie deviation range reliability case probability relationship estimate noisy show sampling bit sampling bootstrap provide support bootstrap range error case coarse meanwhile comparable use example claim mutual range amount right edge panel band reliable present scientific collective phenomenon complex account role involve central axiom coarse consistency process tool choice axiom range theory ability outside environment theoretic inferential biological science domain measurement stage many relevant social understanding encode information many amount precisely task aspect environment transformation representation intervention design underlie reference engineering stability theoretic concept less obviously less central account phenomena physical fundamental notion proximity thing near coarse group finer grain systematic case social system principle dominate proximity necessary informally coarse domain group often need reliable rejection hypothesis social scientific reason study role lack knowledge theoretic underlying axiom axiom become important wants certainly preserve axiom exactly insight world thank david fellowship support thank science nsf fellowship acknowledge nsf grant ef project author appendix north sharp upon face live house look window north three drop north care I three cut send room open open take house five day north north meet together north say I would break home go I north old trial title text report spin send z fr site investigation cart report false red source source ref post gps management service additional multinomial subspace bootstrap estimator linear hard bootstrap aware approximate monte carlo correction assume equal particular naive monotonically bias strictly allow entropy rare note slightly small entropy sophisticated correction functional many correction correction equation consistency naive differently underlie bias entropy dash line tend dot tend course bias distribution lead entropy probability draw draw entropy bit change size shift entropy somewhat center range laplace prior hard range show sigma bar bin estimator tend entropy entropy overall method entropy source low bins skew towards entropy entropies institute road usa cognitive usa history university ab mail social encode theoretic remarkably axiom estimation trivial axiom spurious sensitive representation create address method estimate information quantity preserve axiom bayesian interest devote regime great one concerned bootstrap familiar utility axiom produce world guide information allow question technical paper use principled fashion well kullback leibler behave shannon quantify difference illustrative project information structure system consider extent behavior emphasize advantage mutual information reference process illustrative reference going rely well bias preserve axiom consequence reliable accurate guide ground account section theory deal outcome fundamental uncertainty draw shannon establish unique continuity additional entry monotonic coarse condition possibility outcome likely say coarse grain description weight outcome category central qualitative nature one description phenomena coarse language text vector dimension preserve ask question simultaneously description much lose go shannon discrete fashion axiom less demand require uncertainty monotonic go simple construction property slight abuse prove entropy bias naive biased indeed small reduce attempt correction
operator sufficient valid distinction ordinary derivative express notational consistency development choose spatially incorporation knowledge spatially vary correlation observational uncertainty uncertainty model gaussian represent express pdf use measure z technical condition measure space discussion approximation subspace element continuous lagrange inversion inversion inner must mass definite equip denote inner distinction make adjoint transpose also need endowed euclidean definition pde operator n dimensional familiar multivariate nm e finite measure likelihood give finite dimensional paper observable unfortunately govern expensive explore density extremely challenging require forward explore exploit impractical term role regularization problem pose deterministic mean pdf maximize posterior equivalently minimize cost function call point obviously degree nonlinearity condition map matrix parameter map mean thus pde inverse parameter forward many evaluation space discretization explore pdf hasting h employ point generate choose accept reject thereby chain sample present pseudo mcmc compute u accept every would proposal point appeal directly proposal know proposal proposal least reflect behavior sample increasingly dimension call proposal hessian log posterior adjoint solve column hessian make hessian free rank rapidly decay ill able hundred nonlinearity content newton dynamically sn sn prohibitive point adjoint component still linearize adjoint pde large highly informative compute hessian propose employ approximation hessian refer hessian employ locally gaussian proposal evaluate map point describe begin summary original mcmc newton mcmc employ target pdf rearrange dynamically change proper k discard summary sn mcmc iteration framework algorithm sn figure display black backward dotted indicate prohibitive hessian gaussian proposal avoid hessian find map hessian gradient note necessary proposal illustrate understand langevin mcmc substantial hessian adjoint pde compute pde solve gradient acceptance employ information change rapidly nonlinear observable less effect chain convergence numerical specific assess tradeoff modification mcmc map avoid motivated construct rank hessian take proposal hessian vanish map suggest previously literature sn assessment sn comparison proposal map evaluate current proposal bottom avoid sample also avoid computing map hessian pde solve however lead rate reject hessian acceptance newton sn proposal posterior minimizer k proposal random matrix iterating converge point term completeness independence sampler map note independence hessian mcmc method sn negative reveal hessian hessian vector root vector determinant determinant sn unfortunately linearize pde computation prohibitive hessian briefly previous work employ datum execute operation hessian pde space log posterior sum hessian consider prior jacobian observable properly adjoint form linearize pde ice begin describe newton pose inverse problem number decay rapidly often decay exploit enable seek hessian product crucially product effective hessian oppose note hessian form pair linearize adjoint pde solve ice outline rr correspond eigenvalue define form product amount adjoint linearize pde solve pde adjoint present employ pde pde forward adjoint solve construct form successively apply rr product cost negligible relative pde solve approximation hessian hessian see term rank approximation eigenvector small q hessian sample adjoint definition verify determinant efficiently rank hessian describe pde solve negligible solve forward adjoint pde need ill pose inverse ice operator rapidly decay eigenvalue spatially independence map fast operation hessian previous frequently rank per method forward adjoint pde independence amount single pde dynamically sn sample nonlinear adjoint pde computation linearize pde datum forward pde pde nonlinear stationary linearize solve permit permit pde per differently use number linearize pde discuss pressure incremental adjoint expression linearize solve dominant cost evaluation forward linearize one adjoint computation linearize solve namely mesh element pressure velocity component degree freedom velocity pressure forward incremental counterpart uncertain slide coefficient field compare solution system direct factorization adjoint incremental adjoint factorization triangular cg hessian sampling method start approximation sn adjoint newton starting guess slide newton method newton linearization require complete ice problem slide generate surface velocity I map decrease residual outer nonlinear solve inner newton residual average linearize addition outer require computation hessian sum outer vector map linearize hessian spectra chain point decay accurate eigenvalue sampling rank ensure accurate discard compare performance sn newton dynamically independence ice flow initial initial choose approximately result quasi ensure compare convergence statistic chain exclude multi diagnostic mcmc second scale diagnostic diagnostic averaged individual pool chain individual chain converge close individual chain variance decay however sample reduce integrated autocorrelation usual autocorrelation lag maximum summation scalar slide fourth column size ess obtain mcmc sample jump indicate result great mcmc computational seven linearize eight solve integrated time ess jump distance acceptance linearize solve performance hessian method sn problem slide coefficient sn sn large suggest suggest convergence use even large mean squared jump require solve sn sn surprisingly sn fact approximation increase forward acceptance delay adaptive far lack focus visualization interpretation highlight bayesian guide physical provide intuition particular classify observation qualitative insight two result visualize posterior marginal gray vertical independently neighbor spatial correlation structure visualization reason visualization useful marginal belief slide coefficient unchanged contrary available variance decrease slide region shift region infer width variation observation interpret insufficient observational belief accord subsequently gain insight poorly purpose influence influence therefore sort group naturally eigenvector hessian eq eigenvalue quantify square inform value correspond inform eigenvector qualitative hessian present selection eigenvector figure qualitative low upper half determine eigenvector primarily right study norm distinguish highlight inform information dominate eigenvectors recall say inform concentrate qualitatively nine see powerful confidence lie nine slide coefficient eigenvector eigenvector yet provide characterized ratio easy tail characterize although different eigenvector generally concentrate observable map insensitive part far away slide coefficient ice velocity observable insensitive slide figure eigenvector concentrate half see observable map occur even primarily field surface velocity one eigenvector group eigenvector prior influence affect perhaps optimistic medium fourier half refer eigenvector tail remain eigenvector certain variance assertion large infinite eigenvector similar counterpart therefore qualitatively limitation completely covariance covariance may reflect away insight return marginal color marginal already inform respect distribution eigenvector large group emphasize marginal plot mean due appear expect inform narrow influential gaussian occur mixed direction green observable map significantly influence depict marginal select eigenvector together gaussian marginal plot datum inform close mixed eigenvector mean marginal shaped contour map ridge mass pdf map marginal eigenvalue node cm axis line table bottom scale axis axis x bottom scale x rectangle color black mcmc address construct uncertain infinite govern newton extend way consistent infinite inverse investigate newton mcmc stochastic mcmc dynamically hessian ice problem govern performance comparison reveal newton hessian proposal lead term number sample pde also present interpretation high point marginal particular availability classified covariance extent versus classification informed hence inform nonlinearity observable dominate thank discussion rgb rgb em em paragraph em inverse linearize infinite linearization inverse use monte address sampling pdfs arise upon discretization bayesian inverse build newton take pdf give negative pdf construction component approximation compute low hessian mcmc compare independence conduct ice inverse mcmc rapidly original since avoid hessian hand expensive per however overall extensive interpretation informed bayesian quantification ice c inverse problem inference uncertainty map distribution encode assumption observational probability assign candidate true give rise sampling large high computationally inverse govern reference historical work recent survey replace forward observable map process response surface delay acceptance mcmc first stage employ accelerate langevin variant riemannian geometry accelerate create negative construct employ hessian proposal structure guide sampler region acceptance proposal highly contour typical ill pose problem parameter poorly challenge employ hessian explicit construction many large difficulty introduce compact hessian operator cost represent discretized field lead work employ hessian metropolis hasting position langevin mala insensitive mesh refinement maximize generate approximation use hessian information expensive propose sample require multiple forward adjoint pde linear adjoint pde solve beyond computational modify posteriori stochastic mcmc dynamically well use sampler attractive like hessian unlike
character word feature candidate candidate able detect almost character character character useful design prune world situation character safe remove child character versa child character preserve parent elimination operation recursively safe character preserve end character expensive character fortunately rather identify parent simply choose character tree namely eliminate accumulation variation competition character eliminate pruning section first introduce concept variation accumulation al connect whose pixel intensity increase level gray level new extract current level extract rooted tree variation branch root variation maximally region parent child informally maximally region whose unchanged intensity character parent child operation select low variation alone correspond necessarily variation common child correspond character parent tree character minimize deal parent low variation easily large aspect character low variation parent relationship aspect ratio aspect ratio character penalty base training dataset max htb htb b b colored variation tree color linear reduction present accumulation follow variation reduction algorithm whole recursively figure give root process tree segment work child child child apply compare child child reduction return show tree child htb child child accumulation child work accumulation child accumulation return else child return figure accumulation discard accumulation visit calculation construct character candidate single link particularly candidate link family cluster successively merge merged remain link close member distance merge cluster progress exceed threshold hierarchical forest termination algorithm character candidate text candidate feature q propose distance metric subsection introduce let corner rectangle width stroke color follow height bottom difference supervise label distance maximize pair specify cluster specify pair form merge hierarchical cluster termination distance distance top great member learn randomly initialize exclude specify single termination must equation weight minimize stable adopt objective threshold minimize assignment objective typical problem optimization initial design iterative algorithm involve call top optimize minimized algorithm begin first respect current assignment respect convergence label function minimize self algorithm guarantee decrease demonstrate generate iteration investigate subsection analysis distance competition dataset text text link link pair candidate learn threshold due whether converge impact stage correspond converge converge parameter validation set drop learn satisfactory detection text candidate competition show candidate text effective unbalanced dataset candidate character posterior probability candidate text remove probability character smoothness difference adjacent boundary stroke width variation height aspect character aspect ratio l character character candidate non character text p region reject tend correspond candidate text text candidate text candidate size text prior prior dataset measure task increase text candidate unlikely eliminate recall scene detection task preferred precision reach decrease occur explain decrease preserve eliminate propose scene text benchmark reading competition al reading competition read dataset scene detection contain note competition et complicated offer use notation set function leave room ambiguity evaluation quality use competition system et scoring method method competition produce precision dataset worth four win reading competition apart advantage list intel core cpu pc intel core tm ghz per per pc htb htb precision et publish method fully benefit candidate see absence candidate elimination major value degradation pass stage eliminate default extraction control calculate minimal remove measuring default set result major degradation explain detect low character tend region character second second speed include chinese english see figure initially image testing apparent chinese performance scene precision et al htb system present evaluation comparison character set character classifier scheme training character show chinese character implication character overall propose present scene method prune detect character even self weight candidate single posterior probability eliminate help powerful text integrate build robust superior performance art method research partly support basic program china cb national foundation china theorem minus em depth natural scene many accurate detecting text pruning algorithm maximally region character regularize candidate candidate cluster learn metric text probability eliminate text system robust reading measure state art experimental publicly available outperform increase percent scene text system scene maximally single cluster valuable exploit video mobile variation font orientation recognize retrieve scene slide slide method region slide text text tend candidate follow grouping character candidate additional remove false positive hybrid exploit detector connect character candidate character eliminate character group recently maximally stable character become project win benchmark competition report address character character candidate remove exist pruning hand room accuracy tend prune computable descriptor estimate pass character character complex character computational cause challenge generally hybrid
make rewrite continuously differentiable know lemma min minimization maximization derivative euclidean solve regularize euclidean v q ht improve function eq q kkt read eq kkt imply remove helpful discard end contain screen effective identify inactive estimation inequality section rule section discuss accurate via feasible solution projection onto feasible follow theorem admit ty interior imply next onto eq therefore optimization kkt parameter without generality variational write tell let see b b max hyperplane support support therefore know notational q show inner solution two hold statement trivial feasible point variational inequality lead see complete inside optimal q simplify equivalent inside notational convenience q screening identify inactive remove let entry kk distinct second inequality complete ready rule specific know view basic screening hold cross regularize grid challenge propose sequential section evaluate efficiency synthetic evaluate scale dpp state art problem norm problem obtain solution source code jointly randomly jointly nonzero response draw treat group solve regularize set try different setting distribution correspond warm observe perform plot comparable perform support recover distribution entry draw accurate indicate necessary distribution usually propose shall help multi default class handwritten letter letter specialized ball outperform coordinate project report attribute case second solve scale problem randomly divide letter set train set choose balanced report title validation small achieve evaluate screen group sequential inactive tuning screen ratio discard mention discard dpp develop magnitude specifically apply warm screening group solve time demonstrate run screen discard inactive problem generate I gaussian correlation column experiment evaluate performance effectiveness datum matrix figure present ratio dpp robust rejection inactive group discard reduce rule dpp dpp c dpp sr without b screen third report time screen report run screening time combined time screening improve indicate run twice rule need kkt denote dpp strong discard inactive please accurate size figure imply dpp rule discard inactive respect c sr dpp strong rule screen three total running time second table time average performance screen dpp screen discard inactive l dpp sr dpp different screening column report total second gain demonstrate accelerate optimization solve main technical include euclidean key ep two find ep novel mix accurate estimation safe solver extensive experiment powerful discard inactive result huge order magnitude problem develop algorithm study effectiveness world computer bioinformatics plan distribution theoretical value receive mathematic attractive application group problem challenge due inherent deal base gradient method solve regularize applicable value thus key ep ep significantly special finding problem efficient quickly inactive group may substantial reduction optimization appealing screening compare computational screen negligible sensitivity solution efficiency role many receive induce convexity theoretical great area apply mathematics grow interest commonly loss square e matrix vector belong composite absolute result regression non theory box slow focus constrain systematically regularization great develop smooth convex descent rule detail found prove sd sd converge sd rarely might desirable descent recent regularize least group group lasso cd differentiable separable cd note certain recently propose solve optimization p eq convex operator guarantee properly indicate descent extend optimize composite proven accelerate various truncate subgradient average average apply aforementioned online building solve address screening promise screen inactive feature inactive discard matrix substantial computational improve efficiency lasso support safe et al regularize problem logistic regularize problem although strong effective discard safe discard dpp safe sense group discard core optimal problem zero discard key idea dpp rule region I extend main efficient regularize via regularize accelerated composite favorable algorithm focus efficient solve global smooth method propose converge arbitrary multivariate unified consistency regularization establish screen regularization limit lipschitz show violate practice zero discard author regularize dpp rule rule region problem respect scalar denote letter bold dimensional dimensional th operator denote denote denote inner solving set propose accelerate due term stand online aforementione regularize key composite taylor put regularization gradient approximate point properly coefficient minimizer sparsity l algorithm accelerate key subroutine contribution thus study strictly unique summarize first directional give accord old eq therefore q sufficient condition main paper begin summarize problem q differentiable verify v tx easily obtain analysis iteration example contraction solve firstly vc q show solve find reveal auxiliary define auxiliary
factor design minimal allow let way example paper report rank fraction design contingency table whose otherwise denote binary contingency ccc cccc ccc c contingency design complete present fact algebraic contingency combinatorial property model algebraic notion algebraic notion contingency polynomial table negative ideal thank hilbert basis computation generator ideal compute reduce gr computation gr basis polynomial gr symbolic software gr fact model incomplete table finitely order finitely gr ideal union gr universal gr basis gr fortunately computer handle gr fast ti gr integer binomial primitive binomial primitive index irreducible circuit circuit circuit primitive later markov contingency circuit support closure see circuit binomial common simplify abuse call submatrix column note column identify design circuit submatrix support singular nan nonnegative integer combination positive part binomial belong ideal associated elimination gr elimination term order gr circuit onto replace weather interesting limited analyzing become useful study factorial note circuit analyze presentation nonnegative write ti software example ti circuit design circuit divide permutation circuit circuit circuit support circuit note cccc cccc cccc cell allow identify design feasible l contain function circuit condition vector length object associate matrix circuit gr among hold interesting coincide issue equal integer matrix totally zero totally totally matrix submatrix computationally pc basis circuit combination result basis zero running circuit circuit exclude come computation classical design circuit equal element class level cardinality less maximum circuit basis circuit element permutation circuit cardinality circuit check range circuit class support simple way interaction circuit divide basis contain support equal circuit element circuit permutation configuration without compute determinant fraction may check circuit pc line deeply look connection basis apply theory give projection combinatorial object need essentially define already fraction contingency table otherwise tool able margin projection theory merge theory select margin interestingly algebraic define check basic fact margin markov move connect table move standard involve contingency table matrix compute gr ideal term move gr basis cell special markov basis universal universal gr relevant table move start fraction matrix move move chain mn markov design margin classical metropolis hasting markov converge fraction support circuit discard remark gr coincide set circuit due limitation table move algorithm effect previous fraction universal gr coincide circuit move support circuit sample less execution desire projection theory extension explore extended design secondly connection study indeed circuit factor bipartite associated design statistical classical classical amount indicator level extension currently would interesting use circuit checking test array institute provide fr grant h factorial design combinatorial generate apply basis contingency circuit universal model consequence error nevertheless engineering become highly expensive impose point refer issue factorial circuit define whether avoid determinant design fall statistic application algebra originally present view use algebraic circuit ideal design polynomial originally contingency basis enumeration make describe account contingency experiment factorial fraction imply direction contingency investigation problem contingency
equality rank randomly result compare shannon entropy perfect observe shannon entropy happen rx f rx result follow change sum among entropy pdf unknown du figure maximum enyi follow shannon entropy measure science economics enyi enyi compare first concavity j dx dx concavity end relationship enyi would naturally appear analytical h enyi information case conjecture suppose enyi straightforward calculation px enyi calculate numerically h ranking error compare function value r enyi fix enyi information follow fx fx j j jj du information perfect great suppose respectively I jj j du u complete mm another interest end ix fx fx ix nf furthermore nf g f exponential pdfs ii nn consider kullback leibler entropy shannon establish analytically entropy show entropy behaviour case remain result example desirable context acknowledgement natural sciences definition j ir statistics mb university sampling sampling environmental etc inference content set perfect ranking shannon entropy prove rank well content effect investigate entropy rank sampling tool regard serious alternative design rank variant apply area environmental study combine simple random source auxiliary information help chance collect measurement span value underlie population rank drawing estimate variable ranking rank rank lowest carry unit quantify rank denote cdf variable pdf statistic denote wolfe chen therein fisher inference provide unknown calculate section shannon entropy rank section obtain devoted comparison counterpart show distribution finally provide conclude remark shannon q shannon extensively quantitative entropy shannon separate name theory shannon excellent contain indeed amount concern outcome shannon reader therein counterpart perfect without generality take fx fx dx ranking th explore property
call allow obtain rademacher complexity yield expression exploit expect ensemble penalty follow average shall hypothesis martingale sequence martingale sequence one inequality handle average lemma converse risk minimizer least n bn say ensemble online working bind n bn shall section complexity behave dc allow ex factor however significantly tight result mention enable rademacher complexity tight show consequently dependence formulation regularization provide confidence offer weak technique exclude fraction bound hand give enjoy give convergence function accommodate previous give online use follow example denote dual analysis pairwise require convergence empirical functional population fc rademacher modification closely batch loss function lipschitz guarantee n nn notation constant dependent explicitly start loss bind martingale bernstein formulation incur tt bind guarantee buffer buffer rule decide upon inclusion buffer stream stream reservoir rs henceforth policy allow buffer randomness randomness easy buffer set online hypothesis buffer capacity bound loss regret loss excess offer regret direct proof careful risk buffer use construct need buffer policie rs randomness naive unbounded buffer case generalization guarantee buffer bound require proof condition conditional conditioning randomness buffer conditioning stream stream subsequently analyze expectation randomness buffer part r penalty buffer online discussion empirical note able rs policy section scenario demonstrate input respective rademacher complexitie rademacher rademacher complexity h lipschitz frequently yy yy yy margin suppose yy contraction technique banach x classification classifier independence regularize complexity kernel learn yy w mix matrix class n x n classification learn use involve yy notion alignment two simplex rademacher complexity class function stream subsampling replacement sample uniformly replacement online present learning buffer buffer combine variant give rs randomness buffer buffer precede property claim expectation buffer buffer expectation take buffer property consequently hold relatively weak high reservoir sampling suit prove perform replacement overcome propose buffer buffer update variant regret ensemble drawback sublinear buffer open ex ex ex rs auc maximization buffer size propose stress enjoy perform practice propose lack adapt auc maximization reader split well small buffer size capability loss different sharp offer strongly function counterpart provide use memory online regret buffer else regret lower bind secondly idea buffer technique analysis lastly scalability work pose challenge comment presentation support microsoft microsoft microsoft ph fellowship hypothese z th bn excess manner hoeffde inequality analyze individually linearity nested performing coupling write et z inequality eq head inside sample add equation closed associated population fc rademacher begin theorem f risk mf mf rf mf mf f prove mf rf mf pf pf mf mf mf rr part loss complete work bound convex v nn nn c notation shall risk functional strongly apply theorem loss upon time bind give h ex type bind use use martingale martingale th martingale common prove bound lipschitz property th banach norm thus strong function point wise expectation specifically strong convexity population risk functional bernstein inequality fundamental martingale convergence due give difference uniformly write em denote notational simplicity useful simplify ad hoc ignore constant get use risk expression loss bound strongly convex prove sake clarity far motivate begin setup buffer online observe stream buffer buffer element online learning element incur state interested algorithm give I buffer randomize buffer reservoir step receive shall variable variable note stream completely index buffer buffer buffer buffer reservoir buffer result buffer tuple law time replace buffer incoming buffer establish copy buffer case auxiliary define variant rs decide binomial buffer incoming shall style ensemble work bound buffer proof execution order accommodate buffer construct shall follow whose rademacher average apply well show eq buffer regret give theorem decompose martingale application hoeffding analyze term simplify buffer keep buffer buffer copy precede satisfied reservoir without assumption q stream suppose buffer stream bind es would loose th st ig sg tt yield require trivial variable fortunately buffer induce buffer since buffer update index buffer figure step calculation traditional buffer construction e e least upon apply give add prove convergence algorithm offer bound train auxiliary right away task quantity proceed upon shall proof prove empirical risk index buffer step make simplify yet assumption buffer exact copy sum give write neither constant get portion expression involve step fourth step us eq r get z yy yy yy yy point wise lipschitz yy contraction en en en en en I fourth linearity expectation contraction actually prove constant subsequently expectation l iy every lipschitz contraction possible empirical average take derivation shall usual definition rademacher close equip f applicable rademacher expectation overcome cast modify behave linearly univariate f n rademacher complexity subset banach ball banach pr rr sake convenience regression x maximize roc hand translate situation hinge exponential apply rewrite hypothesis hypothesis use variety wish class use regularizer regularizer possible regularize guarantee auc lie rkhs b x require notion proximity metric wish learn metric yy wish learn variety aid yy alignment positive rewrite similarity p banach get average hypothesis sp p get p sp summarize amount effort case norm exploit strong convexity smoothness corollary rademacher learn additionally extended metric well alone generality combination yy hinge construction consequently popular induce regularization lie simplex l auc x rademacher average table note regularize bound worse compare learn kernel single deal classifier kernel essentially replacement precede stream prove argument lot generate style replacement analyze would offer formulate replacement situation propose give buffer section property x algorithm simply perform first involve buffer replacement see perform property stream buffer rs element buffer address shall first concentrate element simply buffer probability time step law law interpret buffer indeed identical step rs buffer claim prove ensure step buffer inductive claim obey law update buffer make replace element index p complete prove theorem regret buffer update policy step convergence type would penalty prove bound penalty proceed prove lemma use buffer incur buffer buffer buffer buffer stream exactly algorithm indicate buffer use loss buffer exactly rs buffer use buffer tt determine buffer turn eq auxiliary perturbation buffer perturbation variable application analyze expectation buffer element exploit rademacher average measure random update buffer union following suppose online incur buffer penalty buffer rs x generate ensemble probability random buffer h sum similarly hold confidence n complete suppose work sized buffer subset banach banach simply loss proof sized buffer generate probability rs clean suffer drawback inferior rs randomness usage use total bit step rs bernoulli variable buffer incoming usage consequence due increase step random variable drop moderate value variable slowly requirement generator become poor alternate b tb b remove alternate rs policy shall rs rs buffer uniform shall prove joint buffer
month informative gram dependent e music rare understand useful location page almost entirely gram location table provide snapshot e explore detail tweet hold training duration gap require test one gram test tweet success use show indicate add location field r lr description user lr tx tx ds ds ds tx ds tx tx tx ds ds ds summarize combination tweet text third row add consider improve km improve tweet text great success rate consider location field previous match find profile location field comparable description add tend redundant successful low bad tweet set yield one tweet gram randomly gram location tweet merge discussion independently wikipedia confirm google translate english discussion include good estimate bad nan n gram weight category top category city tx country rd tx south english word check letter ni word r j p http http http http http offer city name gram use tweet notably language basis provide signal offer insight base favor success rate consider add example location locate tweet location estimate metric scalable validate new tweet comprehensive implication well result implication suggest internet privacy mention country scale language privacy finally fraction test region q explicitly error probably specifically consist message database tweet gram combine mixture gram new message contain normal density fit maximization package component dirichlet investigate heuristic case work origin message gram gram eq gmm weight metric compute divide convex probable two section gram weight mathematically non specifically gram model place problem satisfied equality substitute plugging bring intuitive per datum drive tag gram gram pair parameter weight pass logistic gram n gram accomplish minimize compute mixture mixture weight function trivially gram error regularization regularizer encourage reduce overfitte experiment minimize descent denominator respect package gram compute metric find accord density equation convert n gram user field token consist character candidate category discard letter convert string min five candidate token candidate either separate word usage letter common pose difficulty leave future candidate assume text chinese usage twitter min create gram adjacent min min min explore detail potential slightly maintain boundary field treat gram tried sort gram frequency gram yield slightly slightly retain gram try consistent though display variation explore usa la macro pt cc social play increasingly critical public health management twitter tweet simple variant quantify propose novel accordingly million reliable calibrate intensive method roughly model tweet finally language location application health turning internet intervention content grow along california around contribution gram negligible optimistic play en e message point unique gram origin tweet previously gram example simple contain uncertainty quantify consider point argue estimate assess context answer four question internet accurately quantitative scalable gmm calibrate km competitive datum total daily twitter quality increase time include rare gram temporal find nearly bad gap valuable string tweet text weak offer tweet name place city remainder organize desirable location detail implication detail infer origin social internet content increasingly active area summarize primary line contrast simple location look user profile text location list find researcher service yahoo survey wikipedia text entity extract al report source internet crucial tweet match coordinate parse location service another tweet location accurate comprehensive match essentially actually statistical discrete treat membership token use classify city classifier state country language message city country city al combine classifier classify tweet city present fundamentally probabilistic evaluation additionally feature selection offer empirically fundamentally classify gmm region post specify technique topic et twitter work informative topic require infer et coherent considerable approximation global potential speedup approach focus solely effort cite restrict united english location limitation fundamental topic provide offer new insight strength recent work friend aid location complementary accordingly offer follow compare modal rigorously evaluate deal coordinate directly supplementary global language except chinese metric measure answer closely different question message message locate origin estimate uncertain argue answer quality near origin near origin within specify much distribution q focus distinct probability claim regardless uncertainty quantify within york city within useful even accurate precise goal discover optimize metric intuitive rigorous core select weighted track whole field profile well illustrate tight cluster inherently location generally poor modal good report estimator produce evaluate extend interval two dimension perhaps contiguous origin accordingly propose simply parameterized coverage origin tested claim upon estimate fall close specify fraction actually fall give observe coverage expect coverage exactly actual origin multiple expect section source preprocesse twitter streaming tweet origin tweet derive gps automate ignore tweet preliminary limited tweet find frequency remove location category boundary far cover low usage twitter separate chinese string min become min select option gram remove become implement tweet schedule example tweet train may schedule four length tweet tweet except tweet retain avoid frequent tweet test tweet day tweet testing avoid test length one day day test I due follow world relate family inference motivate summarize specific examine gram suggest suggest modal estimator probabilistic interpretation previously appear time fit gmm origin tweet gram gmm form tweet tweet weight gmm tweet gram carry high english poor several baseline tendency uninformative power informative gram assign information gram poor none simply assign signal fit measure try gmm fit try inverse product element matrix try number property carry forward discussion design specifically fit gram power good gram relatively weight error yield refer exponent latter give report seem optimize weight descent gram minimize accuracy maximize optimization three first feature far algorithm baseline gmm return operation gram experiment detail try useful evaluate algorithm thousand rt use processor ghz rr rr rr c lr gmm gmm gmm opt gmm opt gmm gmm gmm tweet day field experiment day yield gap gram good algorithm baseline directly rather simple even property optimization poor highlight poor former use gram modal nature precision far highlight well picture bad algorithm poor quite coverage short imply level may right result inconsistent metric highlight carefully simplicity superior calibration plausible complex well simple evidence
mixed function reproduce kernel hilbert virtue effect significant amount rather discrete vector densely densely sample various regression explanatory covariate perhaps response scalar al interest appear scalar response response al pay attention extend regression functional involve covariate methodology refer regression et et assume response functional covariate response space rkh kernel nonparametric responses present nonparametric functional aim handle mix functional helpful comprise remain categorical et al discuss functional response consumption event day improve organize multiple discuss multiple model functional present multiple model al model deal seek functional number covariate estimate center approximated combination basis coefficient penalize basis include basis suffer relationship specify nonparametric address nonparametric functional regression perform mapping space consider slightly precisely compose discrete function main efficient model reproduce space kernel see value definite converse operator rkh space functional theorem show minimization arrive scalar solve choose suitable operator difficulty adequate compose scalar identity space operator approach extension extend regression integral operator reproduce choose kernel choose kernel construct product construct extend value functional kernel multiple functional possible solve real value rkhs reproduce vector equation block matrix several functional explanatory predict rkhs theory discrete illustrate support contract de region j functional letter functional wang design functional task journal nonparametric computational nonparametric analysis w functional statistic asymptotic ed international science b nonlinear functional rkh artificial intelligence ai value space h functional reproduce journal journal generalize functional regression functional reproduce
traditional clear advantage illustrate median adaptive spline adaptive flat job tracking function probably equally spaced jump paper difficulty characterize jump ss function ss ss splines spline bottom smoothing spline ft spline performance traditional suffer bottom spline bottom smoothing plot penalty figure track traditional smoothing spline grey spline step section spline research frequency band hz supplementary material supplementary detailed green function outline full proof reader supplementary outline minimize lemma minimize far since unless show everywhere outline similar derivative time continuously combine show jt jt j kt rt jt jt green apply jt jt k kt h f rt I rt jt establish smoothing spline smoothing spline detail supplementary negligible asymptotic complete spline consider spatially smooth spline homogeneous arise evaluation accommodate show smoothing kernel result kernel traditional spline aid green interior asymptotic integrated square illustrate adaptive smoothing spline play central problem mean function point true traditional smoothing formulate eq control trade goodness solid theoretical widely smoothing major use global smoothness difficult efficiently homogeneous replace penalty since region curvature smoothing spline refine design datum determine optimal location spline develop bandwidth kernel smooth adaptive local variable penalize spline regression nevertheless easy covariate smooth spline analysis spline let l square integrable function endow product smoothing spline denote later generalize traditional boundary smoothing aid approximately minimize spline green make contrast adaptive smoothing spline yield systematic yet obtain expression asymptotic bandwidth function define subsequent equally identically distribute regressor law iterate logarithm empirical let necessary everywhere piecewise show piecewise exact additional jump well traditional spline spline spatially spline interior boundary green spline approximated theorem solve value explicitly aid function green solve derivation discussion differential eq boundary condition stochastically small equation kt ms ds mt remainder crucial spline time continuously strictly continuously differentiable smoothing parameter fourth quickly equally spaced identically regressor former subsequently assumption let smoothing first generality smoothing smoothing point close expression equivalent spatially spline estimator green show supplementary material increase possesse asymptotically equivalent kernel vary shape bandwidth point bandwidth smoothing theorem rt arbitrarily admissible give supplementary material mean smoothing spline convergence give square penalty minimize assumption arbitrarily impose technical mt td tt assume development functional become technical essentially exist constant establish minimum assumption solution strictly constraint solution bind ensure possibility avoid impose additional existence remain approximate interior smoothing knot take jump point unfortunately th derivative estimate rigorously speak valid however seem yield good modify sufficiently neighborhood one replace connect result piecewise view version implementation knot smoothness example come small probability first r package select consume knot weight smoothing yield replace estimate package ideally optimal intuitively make bit parameter theoretically prefer due tend suggest traditional smoothing spline smoothing generalized maximum estimate piecewise
value coincide along discretize bin largely exceed bin large none review instead family base hypothesis away contiguous whenever disjoint contiguous reference proof intuitive regression slope regressor uncertain furthermore vector unit furthermore regressor th improve need represent distribution inverse thus matrix assertion conjecture effective continuous usually use classifier overfitte ba provide theoretically overfitte efficiently evaluate maximum likelihood standard uci concluding ba assess conditional log domain item bioinformatics diagnosis cancer task application quality assess confirm instance numerous among graphical effective know multidimensional follow multidimensional gaussian network description classification propose estimate directly matrix maximum overfitte bioinformatic small instead conclude paper formally assess principle formal appear averaging strategy diagnosis spectra future line contribution classifier possibility improve use algorithmic benefit paper bayesian approximation variable des type continuous discrete index disjoint discrete index takes represent assignment furthermore variable index resp resp bayesian direct dag edge encode eq q vertex assumption alternative consider variable discretize alternatively directly conditional belong assign input value example picture classify construct posterior recognition successfully use construct structure paragraph restrict structure tree tree bayes however simultaneously classifier fix ml bayesian among selective nb strategy propose bayesian conditional fit datum view theoretical theoretical treatment bayesian equivalent quality probability measure correct flexible network gaussian learn let parent consequence resp discrete random parent continuous parent distribution multinomial parameterization parent pa multinomial vector pa parameterized parameterization py pd parameter pd pd pd pc continuous parent iii learning result provide answer parameter introduce index assess independently composition transformation formula ml assess satisfy detailed acceptable say acceptable cell pa pa pa pd pd pd pc pd definite intuitively acceptable cell acceptable summarize ml acceptable provide acceptable sample likelihood discrete index pd pd pd pc pc complete assess value attribute alternative perform bayesian learning assume prior eq bayesian assessment prediction accomplished family conjugate start observe show easy predictive probability follow assume independent assume note pa pa pa variable parent pd linear pd pd pd pd summarize pa pa pd pd pd ss pd pd pd pd pd pd pa result regression provide show determine whose give assess fact distribution linear regression provide hyperparameter pc pd pc pd pd pc pc pc pd pd variable use ba start suggest use proposition assess finally use dirichlet inverse wishart tool exact average decomposable present decomposable ii reason particular classifier learn thorough classifier heuristic join augment rest strategy restrict structure heuristic procedure na make bn bn introduce structure partition group assumption group assume group bn algorithm structure evaluate selecting maximize fold cross differ candidate step augment na start candidate already create attribute dataset variable far remove search follow repository class observation repetition assess number total classify conditional log logarithm give instance correctly adequate three ml classifier final learn ml interested time set acc acc win denote pair ba dataset
hope direction automatic scalable exact intervention accurately miss thank detail runtime kronecker also kernel experiment image consecutive movie assume input multidimensional kronecker kronecker product covariance store decompose p first kronecker let whose operator kronecker change matrix repeat eqs dimension note notation transpose require point training wish observation predictive eq q hand inversion exact observation give directly remain unchanged similarly gain insight spectral concentrate spectral around origin higher able ht supplementary material use example predict movie ht ht ht interpolation small enable multidimensional extend expressive human intervention feature discovery outperform popular alternative scalable discover suggest expressive multidimensional big writing scientific american effort develop notable allow basis oppose well adaptive automatically discover procedure lack framework upon network infinitely basis kernel often interpretable network use rich etc control interpretable kernel process success research typically account accordingly nonparametric nonparametric natural fit automatically calibrate specification principled probabilistic framework hyperparameter like unable scalability simplify expansion induce input simplify already particularly instance popular process expressive rich ask whether kernel architecture affect flexible tool manually structure popular mat ern smoothing discover likewise kernel hand specialized application model datum expressive pattern discovery multidimensional brief introduce expressive interpretable kernel structure multidimensional input inference exploit exist structure technique relate recent relax computation computation storage cholesky expressive form emphasize interpolation develop variety discover structure across intervention sophisticated exposure reconstruct region scene remove discover movie large training instance example alternative speed stress discover method suggest representation pattern pair input joint covariance condition yx yx kx obtain likelihood condition eq calibrate fit optimize kernel integrate selection process exactly function characteristic smoothness interpret kernel smoothing device input heart inductive bias function expressive discovery learn expressive scalable inference introduce section gaussian mat ern pair approximate stationary example fourier mat kernel origin provide additive limited expressive equivalent mixture gaussians scale mixture approximate component small highly flexible location gaussian transform mixture sm kernel multidimensional popular input hyperparameter stationarity restriction help higher dimensional component small shorthand total hyperparameter hyperparameter exploit kernel achieve cholesky gaussian decomposition computation size kernel imposes ignore cholesky decomposition example separate across eq hyperparameter storage standard storage operation multidimensional meaning relax array multiplicative grid kronecker computation p kronecker eqs eigenvalue perform complete number component run bit pc ram intel processor express data optimize bfgs draw frequency scale truncate proportional weight robust test separately texture instance test instance input output pixel intensity pattern texture subtle diagonal pattern reconstruct miss dimension show plausible automatic sophisticated though across spatial separability represent soft reconstruction produce stationary function frequency expressive component difficulty regard function unable reconstruction basis improve likewise input capture necessary hour take minute gps se kernel derive expansion see completely reasonably act possible expressive fast pattern se mat I rational model eq proxy per complexity significantly contribute shrink weight shrinking help indicate whether scale pattern training help whether help component stress alternative square standardized variance smaller well use pseudo basis component kernel slope curve indicate scale experiment close cubic expect slope input instance basis scale gap fix magnitude practically asymptotic second se input bit gb ram ghz intel processor input section small pattern horizontal represent miss pattern compare gps stress b miss gps fast inference conversely extract perform exploit gps consistently low standardize standardized loss note sophisticated contain periodic periodic kernel periodic example rational combine pattern train split pattern r mat train train mail test large missing runtime second shown recover recover truth movie kernel
enable numerous laplacian specify voxel matrix rw formulate restrict functional like minimize empirical use loss function incorrectly formally segmentation loss voxel cardinality input segmentation would minimize functional follow latent risk slack add hyperparameter sample far away reason risk encourage lie estimate inaccurate set empirical effect function saddle iteratively improve start initial compatible soft know annotation inference subsection use initialize give segmentation soft constraint label compatible probabilistic valid ensure constraint rw set compatibility constraint solve decomposition define subset subproblem optimization package globally subproblem subproblem agree refer paper soft segmentation efficiently cut method start specify training finds violate update increase violate predict segmentation efficiently segment correspond randomly reduce divide volume volume furthermore use appearance total main hypothesis soft segmentation baseline replace baseline parameter solve soft find hard base systematically decrease fig hand structure transform iii svm see fig well transform hyperparameter latent case case provide empirical tight value provide hyperparameter incorrectly structure close segmentation rw segmentation segmentation compatible segmentation allow formulate problem demonstrate efficacy baseline replace variable svm hard scale rgb dark medium blue pt paris fr universit paris fr fr paris laboratory paris fr paris paris walk rw easy segmentation combine contrast provide automate drawback rw tune propose discriminative use dataset challenge face provide segmentation segmentation challenge treat hard segmentation employ formulation challenge real clinical volume walk rw popular segmentation medical interactive year automate incorporate appearance accuracy rw heavily relative henceforth rw present obtain easily hard segmentation compatibility ground truth specify great voxel us svm local optimum concave procedure solve propose benefit baseline structure svm real volume voxel set segmentation voxel human annotation
implication phenomenon outline detect irrespective index correspond difficulty entry context compressive necessarily orthogonal criterion include diagonal characterize boundary matrix boundary signal irrespective strength threshold strength successful detection regime parallel theory boundary boundary sparse regime strength boundary component sparsity knowledge optimally characterize detection binary illustrate exist balanced gaussian boundary pr match boundary normal drastically function design regime irrespective signal strength situation detection boundary rate detection boundary phenomenon test irrespective strong strength alternative pt accounting construct low obtain boundary noting cast homogeneity binomial contamination roughly binary equal regime op represent index component detection specific detection boundary idea design weak column sequence characterize version high continue regime respectively design sharp regime transition detection boundary certain behave paper formally discuss strategy matrix boundary weakly design generalize design use subsequent sharp boundary regime analyze design boundary sharp detection regime weakly design present material design dimensional coefficient henceforth arbitrary distribution logistic logistic let alternative consider absolute belong throughout strength equation recall familiar bayes positive study regime say test asymptotically specify understand strength determine asymptotically powerful call upper risk pa hull prior suffice bad case appropriate easier worth note set test ratio expectation assess fix paper study carefully matching knowledge intensive construct test knowledge ideally one seek favorable risk inspire number say p pa rademacher take give realization prior direction strength extra call sided realization alternative express single rademacher support vector support stochastically signal irrespective quite sparse verify instance integer say mutually close exactly mutually member suppose eq paragraph intuitive explanation intersect observation row draw fail intersect support quantify equation test irrespective effect theorem quantify irrespective signal hold appropriate partition consist dimension test asymptotically irrespective suitable common nonzero color white mutation color black sequence rare heat map suitable subject common variant structure partition top orthogonal bottom matrix condition location tight assume fact imply alternative negligible far intuitively quantify ask design sparsity sufficient condition possibly answer binary permutation row exist design partition width specify condition asymptotically condition irrespective complement subsequent devoted analyze complexity section association derive binary introduce set informative covariate element row partition equation weakly parameter condition binary call design design comment suggest easily condition impose finally without exactly orthogonal condition deviation orthogonality essence design regression similar low design impose definition correlate structure compare part denominator structure allow row orthogonal essentially ignore use design condition allow bb ba still behave much rich class white heart provide definition reasonable sequencing calculate heart motivate dominant rare whenever subject mutation column size support boundary weakly analysis insight design correlate divide study main dense regime next essential boundary analyze later separate come separately definition design attain detection dense statistic eq op tight eq decide note test use information asymptotically sufficient power finite performance desirable follow datum incorporate part reject quantity pz moment calculation correct correction asymptotic continue hold generic denote survival j definition let high denote rejection correlate test I converge rejection form maintain asymptotic rejection see interesting test important regime rejection region shall asymptotic sample information test let high h quantity z g exposition step define test exactly argument combine correction concern design note design cast homogeneity population testing equivalent link distribution random detection test homogeneity proportion sequence rademacher induce deduce detection detection boundary design boundary proceed design part correspond identity side alternative irrespective sparsity regime strength arise sided consider alternative irrespective strength regime attain detection provide problem evaluate function number alternative test irrespective strength alternative dense asymptotically asymptotically random bernoulli distinguish early heuristic expect nontrivial dense regime effect symmetry requirement complexity regime treatment identity completely upper denominator test asymptotically strength dense regime quantify regime argue sake completeness test asymptotically additional smoothness rest two part separately introduce sharp testing provide detection boundary logistic region correspond detection boundary multiply appearance single pt j nan pt signal follow pt binomial proportion test exact design well binomial problem test let mention surprisingly nontrivial seem simply natural expand taylor around thereby reduce analysis turn complicated nontrivial application introduce previous divide subsection study regime familiar max attain sharp introduce next optimality soon exceed testing testing test powerful simplification design generic function rw pt original high value token ideally define test statistic cut region work discretized value binomial valid testing observe equality worth compare orthogonal ideal normal value asymptotic supremum attain marginally stochastically unable bind gap essential purpose attempt procedure test test reach boundary minimum binary order value value test binary attain max continue regime let suppose test powerful asymptotically max high fails attain perform note relaxed situation necessary prove study role vector correlate design sake brevity drop confusion recall concentrate design regime sharp motivated dense regime correlate design directly test proportion henceforth combination essentially treat orthogonal upper regime weakly test powerful note exactly theorem play design column correlate unlike heavily quantifie regime theorem sake completeness weakly correlate pr asymptotically support nonzero structure orthogonality suggest condition asymptotically irrespective provide defined ensure surprisingly attain since derive exist let define test asymptotically matrix weak theorem go satisfied expect condition asymptotically irrespective state detection statement upper complement early since design depend covariate replicate covariate study high max follow empirical achievable average though compute yield discretized q similar test base
n nn mutation mechanism reasonable likelihood even among well compatible yield proposal expectation respect expectation I sequentially analogue ratio function instead distribution population evolve term range mutation omit evolve particle unit h eq exchangeability every total multiplying situation require decomposition exchangeability setting argument decompose consist allele affect exchangeable family context lambda respect stationarity wise order recursion define process diffusion inside substitute rearrange vanishing implie give recursion simple full recursion final recursion approximation need evaluate derive proposal song use definition degenerate start evolve hence reach forest infinitely interpret pairwise interpretation however force note motivated approximation accord transition upon proposal q chain form equation simultaneous form linearity efficient quadrature approximation modification generalise proposal introduce approximate gauss quadrature four simulate possible cc run core resample resample size reach generic regard spread evenly among specify type mutation every particle run evenly spaced grid mutation span evenly spaced surface distribution fast converge wide confidence lack similar surface proposal truth tight beta figure joint heat surface limit true surface star expect repeat mutation remove figure infer infer substantially run yield look surface good match derive particular whenever reject hence base upon expect hundred form magnitude fast mutation slower accurate toy slow genome large size pac principled restriction correctness base substitute approximate frequency would pac fast enough remain gauss quadrature use approximate family count condition work method address issue permutation use pac model base show come pose surface column true surface column ht calculation seem would remain feasible pac magnitude pac figure joint figure much surface pac surface surfaces pac remain substantially large algorithm thousand thousand datum set trial careful verification necessary influence develop thorough pac advanced motivate confirm pac useful principled tool derive extend describe ease poisson associate function j n kk particle instrumental establish recursion site analogue frequency convention denote class list member member section distribution type stationary mutation occur describe vector copy relation interested frequency recursion type frequency multiple statement available expect generator immediately counterpart de derive l parent upon solve chain event encounter simultaneous remark k infeasible small random require compute considerably place mass simplex amount restrict number compare size algorithm simultaneous population retain rigorous inference unbiased flexibility comprise mutation considerably reduce independent simulation use drive bridge algorithm use limit restrictive tackle broad pac method work sophisticated approximation similar direction research process generator vary additive member centre university engineering physical sciences grant ep theorem mathematics institute cv department statistics uk span department al uk full conditional apply rely principled approximation modelling fit
stop coarse version use original estimation necessity good evaluation property scheme optimally produce original control variate developed monte paper notation paper estimator wish study asymptotic kind notation sensitivity context input integer quantify input highly influential eq copy follow sample resp rest op proposition minimal exchangeable estimator size evaluation evaluate usage concrete motivate study resp introduction combine evaluation would evaluation consistently estimate quasi also target constraint force function satisfy cost require evaluation make cost beneficial estimation hand unknown approximately estimate quantity sample rise cost financial mathematic asset neutral european option semi formula asset option methodology realistic q euler gaussian price asset uncertain l volatility correlation volatility coarse increment keep hierarchical purpose interval compare use proportional confidence estimate efficiency empirical estimation interesting computational reduction risk partially national program nr mathematical involve aim identify impact tool quantify estimated evaluation availability costly introduction many mathematical model encounter science involve poorly impact aspect assessment aim identify sensitive influence reference variable variable belief uncertainty turn variance hoeffding variance measure parameter index practice hundred output quasi approach sciences pick scheme
less dense always even matrix gs algorithm transformation matrix indicate incorporate similarity value save block partially block calculation similar dense matrix heavily theoretical proof generic theoretical type gs consist objective mapping map consequently gs apply gs gs affinity transformation iteration however limit done expand compact banach contraction nonconvex mathematical general g g compact n ia ia n know special tx q mean double second equality contain set trajectory indicate eq need consequently map n cx theorem graph shift gs focus discover dense subgraph noisy proving generic three key gs simplex sequence monotonic gs transform generated terminate subsequence expand newly mining area vision tracking feasible attract especially datum case speak guarantee set discovery constrain give call dominant dense exist gs add iteratively add neighborhood gs claim finite number none exist theoretical issue behavior procedure criterion closely thing convergence convergent subsequence certainly gs utilize long topic decade fuzzy instance strict proved axis variant banach contraction equivalence direct derivation besides intuitive gs operate implementation hardness capture gs characteristic objective condition map perfectly match requirement gs gs given provide property importantly systematic analyze function illustrate gs confirm prove gs terminate local value contain subsequence make organized principle gs property discuss gs gs verify perspective mining gs search closeness procedure recursively largely subgraph subgraph shift towards graph non zero denote subgraph node subgraph extent algorithm operate subgraph internal accordingly solver dense subgraph identification discussion formal predefined related facilitate gs define sequence generate solver specify analyze dynamic procedure break th subgraph mode reach result actually procedure eq gs iterative loop dense whole diagonal usually mode go neighborhood mode cluster algorithm set tucker kkt lagrange subgraph state implement evolve recursively procedure neighborhood satisfy kkt vertex prove applicability mapping definition introduce mapping effort depict correspondence proposition gs detailed proof proposition gs stable set compact set proposition monotonicity mapping gs along f strictly expansion strictly map proposition validate mapping close closed point terminate subsequence converge proposition continuous strict continuity proposition theory hold three satisfied give gs local objective implementation gs convergence gs algorithm gs share goal gs find subgraph implementation mostly similar whether select expansion gs due ignore focus key component hold generate lie strictly increase continuous display htbp ccccc gs provide first discuss discrete involve refer diagonal equation either subsequence converge g theorem strict continuity proposition define accord theory many gs objective function convergence break mapping part regard apply convergent ht verification usually monotonic gs experiment conduct mb cache gb ram operate gs similarity similarity interval also fully partially experiment averaged verify propose gs transformation dynamic running htb algorithm
expect motivated alarm shorter expect delay alarm method page rule theoretically alarm motivated alarm htbp circle light geometrically plot case change occur investigation contain force consume demand distinction likely cut expect delay predictive calibrate alarm modern investigation tool problem reasonable remove file quick knowledge merely possibly sequence block may force technique block pick block proceed property quality evenly character likelihood apart quantify character code character likely evenly distribute text file program etc measure character consider character chi cox character completely distinguish previous cluster depend may likely occur character code evenly distribute character nothing chi square statistic cox hypothesis may force may physical possibly practically impossible evaluate indicator likely accordingly file monitoring facilitate consecutive formally describe one assume code character character code occur quickly really accurately character character kind occur count occurrence character number occurrence occurrence pearson chi special central monitoring base indicator
rbm hide result rbms hide universal approximation binary rbms rbm hide work direct star leaf leaf write identify sharing z attain unique maximum note p x z j direct direct recall conditional dimensional applying show disjoint describe choice turn power direct ny share layer latter map far onto denote ss repeat integer last coordinate free equal q integer sl j top rbm divergence proxy discrepancy focus generate artificial visible simplex experiment dirichlet density distribution practice preferred generate visible test maximum ml maximum estimate leibler dp rather generalization property frequent line solid line marker divergence dash divergence solid distribution note unless tends infinity good maxima maximizer arrange initialization sometimes poor especially contribute whereby bound maximal divergence combination rbms rbms visible visible explain limited maximizer approximate network hard effort actually theoretical although principle able target accurately accord difficult remain accurately author institute mathematics sciences lemma theorem remark universal narrow belief discrete university pa usa keywords belief restrict boltzmann machine power divergence abstract recent theoretical work layer narrow approximate relax unit interaction layer direct top restrict machine ability application decade whereby universal receive attention narrow belief probability arbitrarily exponentially narrow one deep deep improve universal depth depth narrow represent arbitrarily visible unit way instead universal tolerance treat universal tolerance universal binary number allow incur low serve may channel color additionally discrete rich I formal definition proceed bound error bind sketch entail step power rbms study feedforward study layer feedforward transformation expectation present validation numerically layer receive corresponding definition proceed model dp universal maximal universal number imagine layer arrange stack visible layer l lx finite undirected direct connection consist joint parametrize l layer row block I interaction space bias parameter q factor unit joint sufficient concrete vector ix function state space probability intersection distribution possible top layer input map low sequence class capability partition geometrically simplex indicator unlike kullback families universal exponential family give refinement mn approximate element kullback leibler figure behaviour upper logarithm leibl unit scale remark dimension dimensional counting universal depth minimal approximation depth think tight factor consideration dimension class binary power small na I
even significance estimator consider bandwidth minimax mode dimension quadratic study mode construct multiscale applicable clustering mode cluster persistent homology compare current outline mode crucial standard persistent homology remark denote hessian obtain stack follow define paper assumption compact hessian degenerate finitely assume symmetric probability bound second main clustering let hessian stationary degenerate unique ascent eventually class whose lead mode formally path intersect ascent curve mode define define mode mode integral curve mode define see mode let define mode random fluctuation bandwidth one use diagonal simplicity introduction find path mode input point iterate q assume size follow use mode remark purpose validity could valid maximum splitting focus simple step hessian construct bootstrap ensure eq test reject confidence lie mode alternative construct instead replace versus q testing hence tend asymptotically reject mode hessian nan hessian region visualization hypothesis goal exploratory intend definite mode far hessian describe need region use bootstrappe pose continuously differentiable bootstrapping produce valid eigenvalue eigenvalue elementary polynomial obtain valid eigenvalue elementary conversely root q note write w step bootstrap eigenvalue repeat set j mode get minus mode explanation point result apply hessian need hessian calculate efficient bootstrap split data shift find mode use elementary polynomial confidence valuable illustrated completely mode persistent homology homology salient present consider smooth density persistent homology measure varies decrease mode homology high homology refer etc imagine gradually decrease cluster death birth note birth death time start density persistence plot plane mode short threshold bootstrap band around diagonal advantage persistence approach splitting form topological provide visualization dimension advantage provide interval hessian bootstrap never persistence expensive advantage useful method similar consider object contrast order persistent homology correspond aim different thing visualization illustrate row use bandwidth numerous significant evident plot random diagnostic mixture normal normal mode gaussians locate mode show eigenvalue see label two significant interesting mode mode spherical informative show example mode assumption violate infinitely separate mode spherical mode shape analyze datum three find figure bandwidth consistent persistence analysis mode locate show interval since selection challenge first select purpose briefly idea many mode identifie mode fluctuation decrease mode finding suggest way choose significant mode mode find test example significant mode top mixture normal normal maximize singular nonetheless three indeed mode tie hope mode encourage thorough investigation recommend theoretical rigorous open problem examine section main bind width discover paper find bound hessian degenerate finitely mode let density first derivative second kernel density hessian mean denote hessian enough property lemma assume finitely interior let fact prove bound theorem kx dx mode jx maximizer maximizer write eq tend maximizer interior furthermore tend interior eigenvalue mode suppose mode recall eq tend liu maximizer eq property transformation asymptotic eigenvalue root eigenvalue depend modification odd depend similarly continuously expansion bad polynomial perturbation perturbation confidence interval lebesgue outline write asymptotic show size follow hx show mode gradient show tend test include conditional bandwidth get spurious hypothesis behavior clear numerically prevent choose significant mode make get asymptotic uniform asymptotic might scope leave future significance mode idea hope deal provide thorough combine strength specific asymptotic indicate possible mode derive significant population towards
certain configuration supplementary finding retain follow relevant change transform domain preferable conduct typical lead accelerate versa variation ia associate universal discretize transformation smooth rescale variance correlation row duration joint pattern phenomenon residual amplitude go simulation see also see material duration shape mostly linguistic duration associate slope slope influence length tend normal effect accelerate word relation duration yield triplet look component indicate mid acceleration change duration easily face change duration phenomena duration sentence relate appear due linguistic previously think correlation imply higher tend last long previous finding obviously interestingly low effect something need careful flat upper middle combine nevertheless value model data estimate less duration influence presence adjacent every additionally also appear duration break amplitude curve c play major appear type regard duration curve type prominent effect curve high low shorter short variability duration long short amplitude component significantly type trajectory exhibit establish drift need associate dynamic irrespective type lead type also influence dynamic examine phase covariate confirm duration likely specifically edge cause ai give acceleration individually examine illustrate consider joint comprehensive covariate area methodology zhang result insight linguistic establishe try need care phase covariance linguistic linguistic need pattern joint linguistic despite spline amplitude variation ignore mostly linguistic rather reflect duration sentence relate analysis incorporate phase duration amplitude major statistical issue interpretation result inherent identifiability extra amplitude simply identifiability amplitude well contrast distinct structure identifiability usually need enforce pairwise amplitude quantify outline rise meaningful variation amplitude interpretability link linguistic important amplitude basis detect would amplitude capture correlation help regard joint lead linguistic interpretation supplementary addition issue identifiability mix focus discretization fundamental importance principal question residual optimality question come application aside ica become prominent could inherently complex lack certain orthogonality choice framework result choice rely theoretical assume belief human mix computationally hybrid simplex supplementary material information research regard procedure structure insight five prominent appear influential despite recognize include beneficial available inclusion interest nevertheless inclusion linear cubic gender effect break component substantial potential misspecification effect conclusion comprehensive modeling framework information due domain via compositional distortion effect major language acknowledgement support engineering sciences ep research nsf dms dms research science variation due component transform inverse gray around covariance orthogonality arise component residual covariance structure actual random require maximize log cholesky computationally expensive advantage structured formulation optimize measurement error magnitude result number random mle starting model dimensionality dimensionality decompose totally random sentence translate notation significantly structure multiply candidate boolean dimension assume zeros hadamard product express additionally express relative wide variance reflect hypothesis diagonal ratio formulate effect minimization penalize augment zero submatrix lead final analogously define cholesky thus work particular notion follow non dimension dimension solve triangular eq finally break mean break character equivalent regression break lexical break boundary mark break complete speech paragraph na generality core analysis area framework zhang result confirm assertion choice insight application amplitude respectively insight auc score insight amplitude auc time grey rd th functional amplitude functional principal compute auc grey st nd th principal component calculate rescale unit variance row duration proposition definition centre research methodology university university laboratory department pure mathematical statistic university california correspondence laboratory department pure university email uk chinese carry speech sample individual amplitude attempt provide description joint datum analysis model component analysis connect compositional relationship variation linguistic linguistic comprehensive diverse contour reveal jointly carry phase chinese million people chinese language sound lexical statistical language nature contour contour individual contain variation response semantic context synthesis linguistic linguistic effect variation traditionally analyses linearly normalize remove normalized curve subsequently analyze interesting discard treat amplitude phenomenon propose single amplitude duration give focus identify fold compositional component ratio time principal score amplitude multivariate compositional representation function turn histogram take advantage chinese consist wide linguistic linguistic consideration implementation computational chinese serve flexible linguistic joint modeling amplitude outline compositional amplitude contain role linguistic covariate synthesis also allow last future supplementary usually quantify measure investigation brief segment span throughout trajectory model linguistic linguistic linguistic motivation amplitude usage markov model synthesis unlike maintain linear explanatory modeling material covariate usual template universal continue variational speech analysis principal use model phase variation comprehensive corpus chinese corpora collect corpora attention corpus speech design specifically frequently lexical word corpus acoustic interest specifically fully raw curve length aside curve covariate b segment adjacent covariate exception break count categorical form count initialize begin subsequently every break represent count break fold qualitative description cm mark previous short break sentence position break sentence effect sentence trajectory material linguistic wide spread motion speech production amplitude model framework sample phase amplitude linguistic feature dense naturally framework nevertheless amplitude variation size trajectory limitation consider amplitude utilize formulation introduce curve give amplitude monotonically time domain inverse transformation duration curve realization amplitude transform universal characteristic individual directly length normalize subsequently linear effect application covariate incorporate adopt common approach across covariate amplitude set function differ pca common eigenfunction variation phase reflect score idea likely shape indeed carry oppose otherwise strong feature location term basis th integrable expansion hilbert linear common integrate therefore model analogously transformation different metric root velocity curve normalization metric setting see square useful stress make overcome significantly different order suitable function adopt step arbitrarily adjacent step rise histogram discretize function compositional compositional ratio transform discretized use reverse ensure requirement fulfil compositional sum discretized function instance compositional employ compositional geometric mean log compositional alternative log choice particular sum definition distortion acceleration relative summation impose certain transform compositional transformation mention amplitude variation decompose one amplitude eigenfunction duration linear covariate error structure particular measurement error random allow pattern supplementary uncorrelated phase amplitude effect believe compound symmetric duration compound easier investigate curve one linguistic linguistic effect suggest eq multivariate allow fix coefficient covariate coefficient sample diagonal hold effect correlation error kronecker full effect error term requirement I function monotonic average follow curve restriction minimize g dy I gd negative chosen li normalize length curve global easily inversion note distinct shape align curve separately curve essential dimensional time compositional make mix observe usual likelihood ml utilize restrict maximum likelihood accounting formula covariance take diagonal fix estimate mixed effect software restriction require enough complexity write evaluation ml exact supplementary material computational aspect smooth ie possess derivative line locally smooth interval present smoothing employ smoothing spline use cross smooth common occur
value estimate approximated likelihood approximation experiment simulate combination summarize monte mean standard estimate value normal notation bias considerably nominal certainly adequate bias approximation adequate normal bias decrease increase number l l search frequently attribute correctly limit number compute bootstrap generate weight bootstrap display fig top line table classical interval event improve coverage outcome limit central interval equal interval modify poisson improve coverage shift parameter skewness hand eps leave poisson describe compound property review event physics approximate scale poisson contrary moment formalism demonstrate various weight distribution estimate confidence negative immediately become include important application physic frequently weight relevance certain reaction describe compound review effect approximation use permit derive weighted poisson least fit frequently weight event limit acceptance detector correct weighting use frequently assign attribute associate limit sum goodness theoretical prediction computation histogram detector monte carlo simulation perform simulation assume require p event estimate simulated correspondingly varied weight identical distribute realize majority apply physics claim car correspond situation poisson property useful underlying poisson confidence limit bootstrap evaluate convenient use moment skewness relation especially identical thus homogeneity weight sum follow relation generalize far treat compound poisson variable application individually event variable poisson multinomial weight multinomial matter independent poisson distribution multinomial valid formula remain valid weight unity comprise differ combine eps simulation histogram display line easier indicate approximation show line histogram composite poisson partially effect truncate cut leave distribute frequency good model distribution agree reasonably globally jumps cause skewness excess take always row weight last equal define event weight normal l type nominal narrow case small skewness relatively correspondingly normal small observe bootstrap sample poisson bootstrap outcomes size reject attractive case situation bootstrap technique infer number permit quantile mean number simulate
steady state corollary corollary expression set detail topology sufficiently small steady cluster within asynchronous behavior quantity relate combination guide agent solution ensure desirable level get take side yield appear part original q term jensen k ki expect side q jensen inequality yield rhs large yield part matrix block another th kronecker nn k block useful property ease block yield eq rhs arrive recursion know property block hermitian denote th size th block block hermitian spectral verify see use diagonal hermitian condition relate block hermitian I diagonal block diagonal hermitian eq identify k matrix hermitian hermitian diagonal hermitian readily eigenvalue denote ki eigenvalue hermitian get quadratic k km deduce complete lemma part know desire show primitive primitive primitive strictly positive entry primitive primitive primitive introduce matrix nonnegative equivalently call matrix nonnegative kronecker entry matrix sum exist fact take value I primitive positive lemma primitive matrix form division c dominate since hermitian unitary eigenvalue upper region depend block strictly triangular either apply know order eigenvalue eigenvalue eigenvalue locate circle center circle center f assumption theorem enough j circle center precisely satisfying circle radius disjoint use note leave big circle circle right big circle segment horizontal blue eigenvalue dot horizontal eigenvalue first rhs dominate hermitian term rhs dominant apply side steady covariance dominant matrix j substitute sufficient l h b matrix verify hermitian j f property following verify mathematical hermitian type hermitian semi hessian assume hermitian semi definite hessian verify since hermitian easy hadamard matrix hermitian positive semi definite hermitian semi definite hypothesis kronecker hermitian semi definite hermitian semi must positive hermitian semi kronecker product hermitian definite must hermitian definite hermitian definite verify l l step theorem obtain k express term immediately inversion use eq first rhs lyapunov apply equation invertible lyapunov rhs dominate term substitute introduce fairly event topology failure arrival time turn analysis notable fact able converge desire fast iterate get demand asynchronous carry detailed mean asynchronous adaptation analytical expression convergence steady parameter asynchronous influence conclusion asynchronous agent near desire small adaptation asynchronous topology link part fairly adaptation allow agent within communication link turn topology vary randomly select neighbor share part explicit size ensure show asymptotic interestingly conclusion hold irrespective randomness network mse question affect occurrence agent still sort steady randomness asynchronous comparable failure establish small agent agreement desire steady state illustrate randomness agent asynchronous able close continue present proof letter plain letter letter inversion eigenvalue norm besides block product agent examine square stability diffusion strategy eq form global optimum denote satisfy part necessary describe gradient noise condition assume topology moments eq uncorrelated circular I conditional matrix iw satisfy I asynchronous evolve dt dependency iterate lipschitz mn perturbation factor denote th moment k ki see appendix sufficiently assumption part examine asynchronous network recursion express follow ignore note recursion determine individual constant argument argument establish result recursion rely original long see auxiliary gradient assumption asynchronous conditional I k block kronecker covariance give kronecker operation verify appendix block block appear relate moment rewrite side recursion stability derive block hermitian denote radius asymptotic mean guarantee coincide part conclude asymptotically recursion examine long let respectively appendix second obtain follow vector recursion evolve recursion mse evaluate I guarantee convergence proceed comment operation traditional illustration operation preserve therefore relate use block relate conventional operation abc c compatible hold size pair operation locality block whereas preserve locality block network covariance vector z dimension find interpretation recover covariance matrix individual matrix evolution extract asynchronous diffusion jensen jensen concave theorem part ahead conclude z steady steady steady give th likewise q I I I substitute follow diffusion expression relate closely reveal asynchronous adaptation expression highlight asynchronous network steady behavior early subsequent rely factorization dominant expression proceed primitive namely integer primitive primitive guarantee realization kronecker therefore realization connect self random diffusion therefore verify converse primitive connect primitive unique pair entry satisfy leave primitive positive eigenvalue inside
balance operating cost work load balance mod traffic variant input perfectly prior knowledge offline datum densely demand fluctuation perturbation frequent sale weather service order strategy fine grained varying time address little algorithmic development challenge use mobile demand count road centralize sense suited incur huge decentralize sense grain demand mod pt modeling pattern rich gp achieve equivalent sophisticated centralized process computation approximate mod exploit demand analytically exhibit simultaneously explore demand region pick achieve service empirically evaluate predictive scalability demand business service city service edge iff least road road start end context measurement quantify vary demand spatial latter world impractical sense resource determine actual demand practice count elaborate mod contribute count keep track protocol hoc wireless consequently region access hoc service city measurement extreme much demand pattern put rich nonparametric gp demand gp positive skewness easily practice reconstruct measurement undesirable resolve practice take log remove skewness demand pattern unobserve gp component feature hyperparameter noise variance scale delta measurement unobserved region mean column mean component transpose predict gp must back utilize widely variant demand unobserved log predictor uncertainty quantify gaussian joint exploit posterior demand mod area predict demand straightforward perform gp call scale due cubic alternatively decentralize scalable idea summarize summary received exploit though structure coverage novel decentralized fusion gp prediction close gp preserve efficiency specifically global summary column demand tuple define global local gp globally predictive measurement set unobserved variance obtain demand unobserve far augment local observe local ss ss eq local summary local posterior predict demand unobserved algorithm inconsistent demand pattern often globally demand prediction unobserved region assign predict decentralized variance define notational assignment globally gp globally centralized approximation transpose block let proof locally request request respective b centralize gp among efficiency demand equivalence light gp assume impose experimental demand demand pattern mod service demand sense select informative demand sample q ease walk observe store derive entropy joint centralized issue rely demand b walk decentralize thus load among sense replace sense strategy group walk partition large form contain within later fully assume walk consequence walk walk highly correlate potentially conditional explore predictive maximize determinant mod picking besides predict demand pattern able service communication propose gp construct measurement region compute execute demand construct local global unobserved first region walk length gaussian entropy derive u incur h u couple increase computational couple distribute among sized construct summary assignment request size entropy walk couple message comprise local datum failure couple sized algorithm trajectory central business service area road segment access region measurement count company slot trajectory demand cc demand service location draw demand similarly initialize pick randomly remove new draw user location mod operate fusion algorithm couple gp couple strategy conduct intel cpu test performance rmse mod algorithm compare mod control test demand low notational simplicity algorithm mod comprise three test service area instance predict fig demand mod gp use well indicate exploit predict nearby unobserved pattern show analysis indicate balance mod fig gp service bad prediction demand fig imbalance demand increase pick remove introduce distant demand imbalance demand observe balance demand average fig shorter large fig collect demand predict pattern achieve improve since chance pick demand sample walk walk service area average random three mod less walk total walk less plan informative demand mod gp incur computational load decentralize incur gp f fig balance demand improve short average trajectory short mod collect informative region ccc indicate mod well predict demand effect describe decentralized real fine sense mod analytically empirically demonstrate well balance time
sec sec iii function translation temperature patch patch standard relate gaussian estimation exponential difference intuitively patch difference weight problematic probable v around perfectly weight probable fail probable close weight slope far nonzero quickly weight improve unitary center exponential weight pixel weight p patch difference difference clean patch match perfectly true noise later unnecessary f patch clean patch perfectly match I patch case fortunately know approximated show straightforward q compare measure repeat expand pixel j happen two overlap let give different imply calculate consider criterion stage threshold search region patch theoretical realization map location six value pixel larger imply small peak estimate sample since clear combination region attain away average goodness test four correlate approximate theoretical p reliably similarity g estimate blue ccccc search classic stein shrinkage median temperature realization method table ii clear outperform probabilistic term noise propose weight superiority framework denoise show promise whose reflect similarity connect denoise type meaningful correspondingly denoise addition easily replace patch difference provide patch similarity way early termination critical also reject accept choice provide bm threshold pt pt propose probabilistic employ formulate patch choose computation simulation outperform classic many peak encouraging find test variant model introduced prove denoise classic image spatial contaminate pixel pixel estimate
equilibrium correspond model determine entropy distribution generative entropy predictive observe lyapunov function require predictive distribution zero term negative policy game player tail tail whereas cc payoff second payoff player good pa posterior beta tail play pair nash equilibrium posterior good response figure nash equilibrium superposition characterize assume causal structure predict consequence observational however model intervention setup causal imagine give light green red positively device control green analogously light explanatory power compete hypothesis cause red deal causal represent graph probability induction discover causal represent causal challenge control causal representation c meta use alone operate meta graphical causal structure investigate inference represent graphical tree model encode realization hypothesis meta agent tree depict interpret path root correspond sequential mechanism logic underlie structure correspond sure happen branch causal reveal might reveal even though never observe b c c causal observational completely tree probability statistically extract causal cause cause repeat light resolve place subsequently causal intervention intervention introduce thompson naturally execute thompson decision intervention reveal probability account repeat thompson tb adaptive action implement environment time belief propose heuristic equation contribution thompson show uncertainty agent try optimal unable example computational optimum treatment uncertainty pure estimator bias action trade thompson probabilistic express bayesian investigate adaptive thompson converge maximization operator explain however pick bias b decision maker pick belief coin bias account e inside another example belief role incomplete hierarchy meta incomplete type choose optimally reason player maintain optimally uncertainty environment formalize thompson uncertainty uncertainty unable refined operational policy important consequence maker variable policy computation policy dynamically implicitly exploit popular reinforcement base belief static leibl divergence though maximization belief outer kullback divergence think initial statement leibler divergence formulate describe extra generation try suitable environment one agent couple theory allow away learn equilibrium contrast evolutionary game focus equilibria one evolutionary theory equation represent function denote fitness determine fitness achieve compare interestingly formal q prior hypothesis likelihood fitness landscape fitness achieve evidence evolutionary game theory extensively show nash equilibrium equilibria stable strategy share evolutionary immediate similar argument interact adaptive agent previously dynamic generalize study process evolutionary theory treat raise distinction derive solution analyze hold replace dependent speak operation conditioning coincide random important devise optimal environment mdp action reward environment like first figure belief environment stay agent belief time choose agent optimally optimal agent exponentially belief environment instance mdp converge converge environment restrictive form ergodicity applicable clear ergodicity require learn act optimally environment stable statistical open argue treat calculus thompson thompson straightforwardly game causal induction derive simply probability theory heuristic study theorem proposition open action sometimes solve control possible superposition optimal posterior update calculus thompson consequence policy thompson study agent theoretic thompson sampling infer relationship interact fashion merely principled address sequential thompson patient people inferior drug testing treatment suggest adjust subject cut treatment inferior fluctuation exposure potentially inferior drug optimal sometimes call extensively human make environment rather consistently likely outcome subject tend probability suboptimal strategy know nevertheless thompson sampling suboptimal thompson think optimally reward thompson thompson apply general control optimal possible environment policy infer predictive policy thompson thompson regard consequence uncertainty thompson sequential naturally address problem analyze uncertainty unable study interaction adaptive agent employ thompson sampling determine investigate discover causal thompson principled make exposition case discrete stochastic simplify string environment formalize interaction uniquely probabilitie mutually influence produce action history predict provide stream role output sequence stream interaction know equip perfectly environment ta produce interaction sequence formalize economic maker preference construction value rise utility quantify interaction use programming choice utility uncertain uncertainty introduce index class model environment index perfectly predictor environment discrete simplicity stay utility reduce environment environment thus obtain choosing maximize case procedure effectively mixture environment law coincide law uncertainty environment unknown environment environment create find environment statement environment stay calculus treat variable determine tell act depend past past probabilistic action calculus equivalent act generalize thompson execute effectively place calculus play past observation action past environment detail calculus deal random importantly result basic causal calculus expect appeal adaptive strict mainly computational bellman prohibitive scale exponentially complexity assumption policy interaction environment policy construct lot resource evidence adequate practically often specification approximate indexing policy eq solve maximum policy parameter policy predictive translate hull hull span policy obvious greedy estimate policy refine experience deal exploration exploitation trade agent act estimate action policy produce optimistic let time find pre model essence trade agent respect uncertainty policy act treat overfitte likewise agent point reveal trade naturally trade bayes thompson concept bias point instead exploitation introduce see policy index dynamical dynamical causal uncertainty observation environment agent distinction change information conditioning action follow intervention unique
scalar determinant symmetric matrix singular frobenius grid represent node branch incidence otherwise connect real dc power flow power semidefinite grid eigenvalue one eigenvector price economic simple sufficiently representative latter determine ahead generator negative accordance limit low generator positive load offer elastic load elastic fix generality also power flow exceed capacity impose optimization phase ambiguity role maintain positive definite upon rewrite pricing optimal lagrange multipli lagrange multiplier relatively transmission approximated wide energy marginal component focus even mention capture approximately period interval triplet offer load min denote derive min slight abuse comprise component remove hold capture physical variation collect price lagrange difference period laplacian find price recovery yet complementary th reach low since typically transmission line period property column express node nonzero definition laplacian entry grid entry primal variable fashion update step multiplier gradient iteration admm step detailed next complete ignore turn provide close entail minimizer become close soft thresholding three find simplify minimizer minimizer whose closed soft value lagrange multiplier via type generator gr l mean bid topology recovery ex day min solver generator reference rest generation bound list table order transmission deviation deviation hour day independent st price construct among interval interval comprise choose entry yield price minimizer correspond repeat square entry run admm intel processor gb ram solver fig encourage scheme solely price collect price offer enhance em solve matrix value grid topology recognize price admit interesting regularizer reveal solve algorithm update scheme yield encourage result market market direction f eigen anti anti symmetric suffice complete lemma potential topology solely publicly explore market price dc marginal correspond multiplier involve observation vary exhibit laplacian rank leverage structure maximum optimization formulate include rank regularizer price encourage nuclear norm compressed alternate multiplier economic grid load wind forecast physical attack detection grid mining topology one currently underlie transmission grid update processor constitute foundation monitor conventional grid attack grid know inform market addition inter among pricing adopt technique reveal influential albeit extensive attack topology datum measurement recognize strength attack impact market outcome yet attack assume know point detect study overcomplete employ reveal distant scenario line
bound second symmetric subspace bad region region q q gaussian remain www jensen contain nothing spherical upper uniform sphere substitute obtain combine deduce finally discover consider normally label lie create hessian mapping profile linear anti symmetric th diagonal fail signal call non affine therefore help transform label carry information corollary first bernstein inequality sub gaussian random proof uniformly isotropic q gaussian fix eq substitute matrix obtain lie region bind obtain exist expand middle substitute expression deduce lie lie interior hull
tree simplify base triangular incremental extreme path figure partitioning figure th row always even expect partitioning element difference illustrative crp projective expect cumulative independent satisfie index allow index satisfy crp distribute equilibrium determine distribution exchangeable almost arbitrary posterior infinite develop south east west per x image south north west segmentation interpretation cover element quantity minimum segment segment minimum quantify information range block segment q possible segment segment segment partition reach near shannon entropy write statistic assume entropy zero entropy partitioning tree examine previous arrange accord extended figure node serve grid relate coefficient cumulative segmentation act score count act quantify much divide incremental projection subset partitioning induce subset keep entropy subset permutation entropy sequence involve begin b dendrogram automatically cover uncertain distribution gene expression condition gene letter label rp protein gene subtree grouping circle plot distinguish inner tail gene pt membership year block tuple appear nz east present cumulative permutation systematic element sequence develop sequence knowledge conceptually primarily aim cumulative definition develop respect theory concern various type engineering introduce helpful discussion h synthetic data image north pca south north west hx e picture reference nonparametric dirichlet university dirichlet distribution mathematic constructive definition markov r dyadic factor process z probability variation bayesian profile microarray liu specific mixture gene expression microarray shannon c communication countable entropy conceptual shannon information theory display genome measurement genomic systematically j international http www engineering rgb infinite commonly hard interpret statistic represent quantify segmentation summarize visualize infinite posterior statistic aim group similarity belong gene belong base nonparametric priors dirichlet poisson dirichlet constructions chinese restaurant crp stick enable inference mixture inspire several include make solution summarize infinite mixture bioinformatics profile linkage pairwise probability methodology determine prior entropy quantify section quantify segmentation finally generate summarize set posterior synthetic interpret allocation begin basic definition partition motivation cluster mixture component draw crp discount conjugate integrate component iteration assign new crp hyperparameter integral capture relation among aim constitute exposition obtain obtain mixture block mixture average useful three commonly appear first theoretically mixture bioinformatics count contain express information regard relation among another partitioning exactly sample difficult interpret rectangle rectangle rectangle rectangle rectangle south east image south north west south north west image east north west let formulation interaction gene empty intersection onto induce partition project onto say closely subtle allow informed develop block systematic analyze count rewrite sum block size arrange diagram result weighted average always take average partitioning repeat
entry consider matrix parameter high sparsity sparse small allow probably sake sure entry coordinate corrupt loose nothing functional equal term shrinkage index speak reliable entry distortion paper level characteristic like ssc algorithm ssc value bring numerical algorithm special mechanism perform opinion completely idea lie foundation follow reasoning error identify contain mark working intuition basis principle tell error trick sure mark make reliable error error implie necessity claim true value independent typical requirement entry still information list information idea move list benefit location amount entry large restriction greedy increase capability ssc algorithm greedy especially redundant probable contribution repeat entry consider previously select entry approximation greedy product span select many representation exist combination reweighte cs entry fail greedy pick entry consider reliable weight compete iteration pick recent completion highly multiplier simple give boost capability set keep coordinate vanish error entry dynamically remove false k ssc greedy version ssc serious drawback ssc may undesirable accurate error bring benefit present elimination external loop acceleration tuning bring capability initialization median eq c c give comment update get realistic fill approximate large accumulate iteration datum make entry magnitude coordinate fine adjustment density ssc algorithms devote face face ssc size mention efficiency middle ssc execution ssc final selection ambient however absolutely mean subspace dimension average dimension dimension ambient obvious exceed quadratic table formula find empirically cccc present two low ambient indirect perfect interpret low dimension introduce project datum coordinate great index lead systematic drawback randomness guarantee non rigorous reduction ambient give fig support reasoning actually guess cluster random location close representation ambient associate justification original ssc study mention case incomplete see mean ambient applicability sparse difficult function normal intensity distant way combination subspace face cluster acquire pose vary light condition sort subject obvious object try benchmark reach therefore try image resolution subsampling image represent individual face acquire different inside algorithm triplet conduct section trial serious processing table ssc subject median median subject mean median median column misclassification whereas th process group provide ssc misclassification misclassification tell principle reliable perfectly loss occur individual result group give triplet group triplet cluster successfully triplet misclassification triplet thorough evident ssc face option show lot way improvement give emphasize algorithm aware theory recognition utilize principle greedy ssc corruption image time misclassification recognition ssc ssc justification desirable development address among crucial topic lot self discover subspace soft possible subspace subspace intersection suggestion original ssc functional strong datum corruption speak error correction capability believe correction capability cluster quality locate direction solve correction would computational complexity one improvement adaptation bring algorithm capability one correction compress sense recently usa fast greedy subspace method affine difference ability know corrupt usage reliability bring feature previous iteration consume discuss efficiency greedy capability fast ssc algorithm recent algorithm model extend dataset misclassification turn ssc corrupt efficiently exceed dimension ambient sparse law rank completion compress greedy clustering belong linear low ambient subspace history difficult closeness presence bring hardness combine many problem subspace cluster information especially sort database like face character symbol segmentation subspace develop space intersection situation hope sophisticated algorithm whose accordance restriction input expect provide space restriction require cluster split cluster satisfy may assign subspace outcome model generation subspace reflect subspace generator proper affine ideal outcome space include point problem generator create represent zero sure assign put linear solve mean whose edge decomposition space linear decomposition remain excellent hamming find compressive cs thorough emphasize requirement setting absolutely irrelevant mention uniqueness course important minimization hamming difficult unnecessary indeed request direct hamming weight helpful unnecessary decomposition wrong column column allow perfectly subspace precision perfect decomposition solve practical intensive follow spectral graph laplacian obstacle elegant obstacle convex ideal uniquely solve convex clean magnitude compress corrupt assume corrupt entry significant constitute corrupted value location index corruption call importance think datum second entry magnitude much magnitude measuring requirement mention correction wish index different removal correction procedure become subspace split search cluster greedy rely principle ssc capability due slow ssc capability sometimes ssc algorithm improvement take specifically comparison subspace datum goal sometimes netflix solve within low rank matrix exist mention good inverse correct one coarse great section formal ssc greedy external construction ssc showing approach world algorithm modification extend reasoning relate ssc early ssc create subspace adapt datum allow even significant fraction provide similar ssc idea work ssc foundation clean standard cs presence reformulate finding equation identity problem correction solve efficiently solution unfortunately strategy straightforwardly measurement corrupt measure low exist subspace solve simultaneous consider admit rate particular accept solver error magnitude noise relatively low sparse noise replace
verify penalty program illustrate bridge penalty chen et derive affine second necessary minimizer bridge mcp smoothing newton meanwhile convergent regularize newton main global problem existence partially associate operator necessary condition coordinate numerically coordinate introduce dual rewrite term third relation active active explicitly iteration subproblem often small couple strategy five nonconvex penalty introduce strategy organize describe nonconvex penalty condition whose coordinate minimizer minimizer introduce dual rewrite optimality active variable finally accuracy minimizer case penalty study scad mcp early nonconvex recover signal table overview penalty operator cccc cb scad mcp combinatorial tractable penalty table drawback g lack challenge bridge penalty quasi statistical property scad requirement origin vanish ensure specifically expression selection scad view variant regular coefficient scad mcp mcp concavity vary penalty ts cf penalty ex bridge ex ex ex v turn minimizer existence practice full column technical function map let orthonormal let exist subsequence scalar subsequence divide every let w ti k minimizing prove end submatrix consist row bound singular hence show map w eq q give many proof minimizer five penalty exist discuss separately bridge scad penalty zero calculus decompose kp upon convergent subsequence continuity existence shall coordinate minimizer derive thresholding table thresholding early see e manner unified end omit simplicity attain approach hence bound accumulation accumulation give nonconvex penalty thresholde useful characterization minimizer next observe sign assertion imply minimizer unique assertion expression thresholding give elementary thresholding five mcp table except penalty derive operator end coordinate wise minimizer coordinate next coordinate q coordinate minimizer element minimizer thresholding bridge value otherwise identity solve precisely wise sense coordinate necessarily minimizer sufficient wise minimizer denote inactive respectively wise entry column whose list small singular submatrix condition summarize deferred converge coordinate inactive statement minimizer bridge mcp hold always bridge scad mcp coordinate minimizer minimizer active large relate hold verify enable minimizer penalty list table apply convergence couple slightly set augment lagrangian dual active zero inactive update construct primal ii active optimality e eq operators scad mcp x together variable formula appendix k penalty ex I x wise respective characterize uniquely empty scad mcp bridge priori unify two equivalent expression define eq expression ii algorithm inactive primal active suitable dual active inactive finally variable explicit expression cf example natural bridge however choose amount strategy mcp complete approximation guess find active inactive table criterion comment e second choice cf share identical nonconvex avoid may add term lead might important step choice end adopt guess performance nonconvex penalty simulate core ram matlab generation parameter choice additive independent standard deviation follow normalize set coefficient normalize dr signal fidelity sparsity meaningful experiment couple algorithm strategy small take choice much equal scale let take guess unless bridge mcp experiment well linear couple remark unnecessary algorithm illustrate nonconvex dynamic proximal forward splitting recovery observe decrease exceed vanishe exceed nonconvex size penalty hence sparsity compare multi general iterative shrinkage due al couple terminate note contain nonzero coefficient range case involve thus easier numerically evaluate realization result algorithm ten attribute examine indicate error work nonconvex yield satisfactory accuracy gaussian scad mcp e correlate random gaussian scad mcp scad mcp e e examine strategy local couple refer exact active problem monotonically initial generally penalty attribute ht cccc cb scad ce cc active bridge scad penalties control concavity parameter cpu absolute setup robust concavity vary robustness cb time e e scad cd mcp auto uci repository length engine stroke compression city et clinical receive example lasso matlab build nonconvex width weight compression close gold standard good nonconvex exact fail observation feature mcp width weight engine stroke city take nonconvex model intercept agree penalty feature age develop primal dual nonconvex signal dimensional include smoothly concave establish derive associated optimality wise minimizer provide minimizer meanwhile optimality reformulate primal primal active solving couple confirm second ill matrix involve hard might motivate far implicit solver extension sparsity analogue interest acknowledgement grateful anonymous constructive led quality paper nsf grant science foundation china five separately verify minimizer mcp scad treat formula see bridge clearly root increase lemma close minimizer root maximizer computation thresholding operator iv scad u g v thresholding v mcp let q iv also establish small small perturbation set u uv far inequality four note nonnegative small thereby rest identity q iii yield thus deduce hold cx I combine q inequality eq consequently mcp iv small hold iv
row design arise first generating observation adopt integral regression adapt randomly linearly generate model step proceed binomial training context prior use procedure row previously step addition integral sensible robust answer stress necessarily recurrent training space everywhere markov irreducible integral prior require posterior step sample subset posterior submatrix jeffreys proper furthermore probit log log select step posterior function close form simulation accept posterior submatrix simulate k repeat jeffreys size work among increase keep simplicity simulate posterior bernoulli integer augmentation scheme independence contingency intrinsic exceed advantage time row discrete binomial avoid simulation end much need describe happen row instead work recall rank row order choose rank row submatrix design simulate take beta submatrix row simulate simulate beta markov chain begin transition one obtain markov odd full importance normal maximum covariance chain kernel simulation importance deviation estimation association zero row prior level although probability full include intercept explain comparison odd ratio object medical center identify birth record illustrate association birth five reduce variable odd nine simulation importance birth nine coefficient except prior nan deviation prior show coefficient concentrate nan hypothesis chain length parallel simulation estimation posterior model deviation distribution successfully selection methodology two consideration apply intrinsic cancer example calculate full intrinsic probit similar answer variability example birth deviation integral stable intrinsic conclusion prior conservative medical try associate exposure property nan way since intrinsic around intercept intrinsic nevertheless function several link nest computation es support foundation scientific project partly project calibration ia methodology integral binomial often association risk exposure develop purpose effect exposure formulate objective factor construct coefficient methodology nearly automatic reference integral jeffreys markov bayes factor regression analytical exposure adjust use logistic log function preferable possible estimate associate estimation perspective quantify true purpose link automatic parameter indeed formulate base hypothesis compete sample alternative priori distribution hypothesis eq specification literature like jeffreys recommend reason see g page property literature behave problem normal link yet prior probit prior probit probit link logit complementary log one extension prior generalise
study surrogate essential system detail section computational ode representation describe express spline piecewise j among large variation solution ode fix obtain generalization estimate log exist define need obtain ode separable help number equation df tr trace criterion select force solution ode estimate l available gradient toy dynamical consider differential generate equally spaced interval noise maximum average deviation penalize close original negative mle true improve mle close realization vs auto chemical reaction penalize sample space noisy interval together simulation penalize spline fix perform parameter randomly good comparative rkh time ode approach estimate parameter run penalize rkh explicit ode explain work similarly notice ode rkhs rkhs rkhs rkhs gene encode tf activate bind dna technology protein level activity factor tf point importance behaviour partially responsible production expression level gene change degradation activity time model degradation gene gene additive account level nuisance micro array reconstruct level unobserve profile describe section gene use spline equally space element basis fit reconstruct profile fit reconstruct figure replicate profile different protein unit level estimate show profile agree gene identify ode system baseline available estimate differential proposal ode green main single step problem differential competitive scenario simulation biology illustrate method scenario hide extension proposition denote j rewrite first f j second rewrite detailed expand substitute simplify aim definition system differential attract interest field like biology ability dynamical despite importance branch science systematic statistical work measure noise methodology penalize likelihood differential reproduce hilbert space test unobserved factor gene ordinary kernel gene network tool science wide spread difficulty block last six year serious progress within differential measure importantly estimate differential maximum likelihood version bayesian similar whereby differential ode link differential fully inferential develop kernel approach explicitly maximum whereby differential equation introduce reproduce rkh constrain unconstrained maximization idea model focus implementation conclude discussion practical reproduce kernel hilbert decade use ode space function functional bound virtue rkhs definite name kernel reproduce force state external force k dt parameter depend refer system equal sample make measurement ode satisfie zero mean though trivial let indicate available value time indeed system differential initial likelihood generally ode solver intractable differential key convex add ode representation ode burden ode solver function jx solution fix rkh detailed proceed penalize rather differential unless force
much total test turn scan modularity amount turn modularity test speak framework large moderately combination degree calibrate test calibrate way former relaxation scan eigenvalue formulation discuss performance scan relaxed scan test time table detect n n scan test unknown scan relax scan test address clique achieve clique calibrate large clique test detect densely connect subgraph analyze degree scan realistic handle test technical unless leave assume situation degree basically bound far latter imply non vanish chance community powerful positive integer convolution sum describe function result start set clique meaning consider necessarily increase test asymptotically intuitive imply high least clique classical graph fine argument study clique second moment show suffice consideration aside detect presence clique clique powerful entirely alternative proof omit conclude see detect subgraph start regardless requirement scan test test combine simple power leave purely challenge simplicity though partially extend usual assume favorable test optimal derive follow test asymptotically say difference deviation larger moderate involve deviation sharp next scan asymptotically simple total risk non tend tend name modularity particularly simple degree scan distribution bind lead powerful study position together mean bound come yield powerful limit inferior scan asymptotically powerful proposition scan knowledge unknown scan size correction essentially powerful test proof enough room scan subgraph impractical know difficulty strictly difficult regime way situation hypothesis composite option detail test total expectation otherwise risk problem say powerful fix resp compute detection exhibit interestingly require asymptotically eq shall say requirement either total degree degree variance node maximum modify independent estimator mean asymptotically powerful expect boundary require parametric bootstrap scan priori nan likelihood implement estimate subset calibration compare scan simulated generating option modularity first option scan calibrate calibrate achieve require adaptation achieve describe combination scan calibrate detection boundary without even statistic scan nature know compute size graph compute large determine level answer convex relaxation detect principal assume sparse entry submatrix statistic np resort relaxation denote submatrix q maximum semidefinite n scan know simulation effectively fraction sdp large unknown scan statistic carlo hold scan relative scan scan scan asymptotically eq compare version degree establish call plant become one computational affect advance compress show tractable contrast plant polynomial detect detect proposition provably hard circuit thorough want characterize powerful test situation limit computable first come mind degree degree asymptotically maximal asymptotically compare proposition maximum either powerful scan degree designing test reader reference additional density subgraph maximize polynomial h strong scan test seem behave statistic improve subgraphs size compute np polynomial variant statistic power may approximate powerful positive depend fundamental statistical theoretic difficulty dense subgraph os graph poisson small detail quasi normal follow moderately sparse known detection polynomial discrepancy plant surprising interest study computational science rich scope binomial distribution chernoff chernoff bernstein bernstein entropy binomial eq result contain stochastically ball pick probability standard start reduce composite alternative simple prior subset clique average arguably base cauchy schwarz expectation respect edge exist go imply q eventually stochastically use tend hence eventually show theorem reduce size ratio expectation function still leave let applicable bound regime deviation binomial play superior bound instead fortunately fine suggest bound moment truncate likelihood follow cauchy schwarz precisely prove accumulation adopt us converge eq notice imply lemma force argument technical lemma divide depend behaviour moderate dominate regime dominate normal deviation lead completely regime regime require treatment notation exact moment mind fact depend union chernoff conclude rh different variational track expectation respect recall decrease previously know tend look second fact precisely consequently conclude divide tell p k rr lemma k np imply continuous prove different depend q suffice tell eq imply p little hard show bind couple force definition definition increase lemma show lemma follow k lemma proof asymptotic entropy hold give convex use derive consequence away bound entropy q kp third away start two get expression entropy k turning imply h conclude couple lemma uniform q use thresholded version subsequence accumulation converge moment rely remain square modify follow already expectation inside thresholded term q exist define need show four expectation expectation hold current need carefully use sequel expectation cauchy convexity get bn second straightforward focus entropy p p nn condition nn op consider tend zero tend zero let turn argument min previous definition proof useful result prove asymptotically powerful versus powerful chebyshev zero recall definition powerful chernoff ne go describe choose tend powerful constant imply show remain nan rewrite variance n turn alternative hypothesis tend infinity show n computation np p met argue second imply suffice scan powerful concentrated chebyshev conclude hence rr remain satisfy contradict index q chebyshev sdp say symmetric ij j bernstein happen tend high prove go proceed particular binomial concentrate bind go stochastically assume
decide trivial allow predefine know precisely exploration need confident algorithm exploration along amount exploration region value stop reach optimization regret exponentially generic synthetic confirm commonly heuristic david discussion anonymous st international conference lemma analyze generic global upper bound cumulative algorithm exponential like novel gaussian improve confirm efficiency real improvement publication article find main observe regret function page mistake consequence observation variable noise claim even center condition wrong martingale measurable respect lemma gaussian experiment remarkably discover mistake able therefore incur optimization design finance science select tuning input optimize reward previously measure query reward receive minimize balance exploration exploitation focus predict reward becomes challenge evaluation noisy expensive efficient tackle optimization procedure gp close value smoothness main prove sharp upper build suggest alternative policy cumulative mutual organize setup cumulative provide detail performance real heuristic practice set optimize find successive iteration observation location noise mean address via cumulative instantaneous gap sample aim cumulative high control underlie gaussian formalize variation gp normalize previously compute current covariance kx stand identity height gray area represent deviation mutual exploitation ability drive govern amount therefore show robust confirm tx controlling exploration bind formally name x refer read variance mainly compute x ty measurable theoretical algorithm regret bound generic incur independent bound cumulative section provide proof approach analysis define martingale regret gap gp assumption know self satisfy martingale obtain remark inequality combine proving conclude previous concentration generic cumulative define xt property bind cumulative definition last modify algorithm unchanged eq manner optimize exploration state equation concavity combine lemma cumulative regret gp bind q consider incur simplify inequality prove compare improvement task five synthetic initialize inference pick half subset way experimental task several order magnitude incur discuss confidence interval briefly assessment gps mat ern dimension bandwidth dimension mat ern deviation synthetic gaussian smooth variation isotropic mat ern peak thin sequential search function another function slightly addition ability trade represent function present benchmark use evaluate price global local optima synthetic like recent vertical extent wave suppose investigate employ
inductive learning learn unlabele discriminate code partially label use code procedure sc label al construct graph sc sc manifold label codebook simultaneously unify objective sc manifold term optimize try predict class discriminative since ability predict class moreover code manifold fold propose discriminative supervise discriminative label inductive codebook classify supervise datum report objective firstly construct denote n denote class sample sample label element l try codebook combination call sparse organize codebook code encourage sparsity complexity sample please note relax present sample denote organize c linear parameter square th moreover introduce complexity label hope learn label sample near way coefficient label coefficient problem formulate combine problem eq please label code directly assign class label code label classifier sparse code learn difficult closed solution alternate optimize iteration optimize discuss together codebook remove irrelevant fix xy e lagrange dual recover row code similarly sign label problem class column unlabele separate contain code label contain code label moreover convenience contain remain rewrite substitute follow simply large element performance code modern major interaction drug drug compound drug evaluate prediction compound balanced compound signature signature circular candidate conduct fold validation ten fold remain nine fold leave unlabele signature codebook classifier unlabele test one codebook compound evaluate metric specificity spc accuracy acc score metric set correctly tn correctly fp number false measure q please range spc acc value code coding unsupervise sparse coding propose code sc classifier propose manifold please code four five give clear unsupervise spc acc able discriminative code method supervise sparse code sample label unsupervise surprising wireless diagnosis network collect wireless circuit include make conduct set fold fold test set nine fold diagnosis unlabele codebook code unlabele codebook represent test sample acc class acc acc fold cross give see classification outperform method wireless sensor diagnosis task utilize unlabele much well unsupervised sc combine codebook class label directly train achieve attempt learn propose state method outperform supervise code unlabeled algorithm improve appear direction machine learning pattern usage mm code linear code new span label sample code assume code codebook code class label code two world pattern recognition demonstrate propose method code set sc bioinformatics vision try codebook linear sc codebook represent coefficient coefficient zero leave refer sparse sparse one usually minimize regard
dynamical model conclude strength limitation abc likelihood first irrelevant keep ignore account density full involve abc repeat generate draw abc apply sample sample either imply abc substitute piecewise reduce replace one statistic abc typically much small question approach use henceforth denote regime observation covariance abc use product though approximation consistent covariance necessarily gaussian asymptotically optimality normalise normalise substantially kernel estimate follow shape kernel proportional dependent desirable notation regularity bias I integrate case density density true determine bandwidth expansion analogous tune ten time manually appear gaussians covariance section describe calculate estimate multiply suppose instead section detail imply imply discrete integral interpret discrete imply replace approximate imply imply approximation observation denote remove conditioning simulate conditioning abc convolution respect imply reasonable side shape abc average small around much interest equal draw give since kernel involve interest suitable direct numerical fine avoid try evaluate check resolution insensitive choice circumstance may approximate draw principle normalise sampling select sampler approximately abc toy involve infer probability second cox differential investigate integer time available albeit enable model modelling chemical therefore dataset draw use density approximate close density numerical integration approximation integration cox stochastic describe volatility brownian chi close situation unnecessary include abc choice comparison equally spaced interval treat parameter achieve acceptance average abc bottom plot abc posterior together approximation agree skewness integration conjunction arise context model number patient follow autoregressive identically random variable operator denote binomial consider observe generate plot transform estimate abc smc length able acceptance mcmc abc good agreement somewhat poor agreement cause density different appear reasonably approximation integration posterior density approximation contour likelihood enable marginal likelihood abc easily generalise likelihood resort exact datum mcmc involve treat miss example discrete space time process dynamic think chemical subject death inference simple type reaction observe unknown inference reversible jump develop substantial user implement provide likelihood monte carlo smc mcmc stochastic chemical intensive reliably r package contain design chemical use package result abc example birth show bivariate posterior gaussian approximation kernel contour contain generate size draw posterior assume take approximately minute computationally demand sampling able draw core error iid deviation equal display figure bivariate marginal abc agree abc abc tolerance involve amenable abc construct take match sample prior make calculation gaussian adequate skewness least strong true posterior unfortunately abc normalise sense possibility test wide normality recent therein promise direction far devise property base preferable converge computationally demand however probably aside robust ask difficult offer depend dimension large whether might say gaussians mixture gaussian scalar multiplication perhaps covariance say property hand involve extra freedom flexibility multimodal take small feasible result enable calculation approach factor block observation relevant extent acceptance lead need use tolerance abc bring expect central factor point perform make single per abc summary statistic uniform ball noisy abc mass recently learn interesting term propagation ep share similarity abc abc expectation develop use proceed draw sampling update component data pseudo abc lead fast disadvantage sufficient ep abc approximation potentially ill suited likelihood approximation promising direction investigate adapt density estimate acknowledgement medical research u gap acknowledge valuable helpful anonymous hand side abc draw accept acceptance q likelihood pz imply abc xy abc ac uk ac uk complicated stochastic model hence conventional cost choice potentially use abc
penalization local combine maximization algorithm show author call nonconvex interpretation laplace recently extend mixture distribution method bayesian parallel gamma binomial integer yield selection beta bernoulli process tool additionally variance mixture evy sparse prior dimensional mixture infinite construction induce stable connection refer transform evy surely decrease nonconvex penalization model shrinkage family compound compound poisson gamma discrete compound base random variable evy gamma limit compound laplace exponent nonconvex nonconvex exp additionally define exp compound mixture process formulate give nonconvex reduce devise expectation adaptively adjust sparse solution simultaneously review evy process family compound process evy devise conduct evaluation conclude work bernstein monotone completely bernstein speak evy decrease mainly laplace laplace exponent evy laplace exponent bernstein correspond transform completely monotone moreover expression datum eq error vector estimate vector nonconvex follow shrinkage furthermore regard define laplace exponent define nonconvex induce regard form bernstein furthermore see connection latent parameter parameter formulation point proper density scale mixture tt lb pseudo process exclude strictly bernstein nonconvex exp exp bernstein evy measure pseudo mixture exp evy eq improper take compound nonconvex sequence let independent compound poisson denote compound compound process nonnegative point nonnegative nonnegative discrete variable compound poisson gamma limit compound poisson compound write evy gamma evy gamma bernstein condition ss b origin imply sparsity penalty eqn tt du du u algebraic case yield gamma shape exp list special table fractional give say dirac delta randomize bernstein evy priors improper define mass poisson intensity nb mass give denote family bernstein function derivative evy proof evy generalize induce penalty ss notational du except log remain function case pseudo prior ignore normalize improper omit derivative w approximation share help direct latent parameter implement challenge algorithm learning purpose assign full conditional gamma recall compute marginal pseudo bernstein prior rely long analytically figure depict penalize respectively step update conduct j proper update convergence monotonicity moreover pseudo step b p j gamma specification proper improper subspace span proof bernstein necessarily bernstein estimate well map global latent aim increment w stochastic treatment also ht conduct procedure figure I independent parameter latent share marginal pseudo illustrate result also ht bayesian step w k step follow j ht j k bayesian step estimate k w k refer figure respectively nonconvex exp penalty expression accord asymptotic approach thus alg alg empirically validate effective shrinkage tuning via hyperparameter simply log simulated standardized evaluate model achievable alg alg alg alg alg exp alg alg alg alg alg alg alg analysis datum consider medium pn five multivariate model employ standardized ability achievable accuracy correctly zero zeros snr tune choose report proportion nonzero predict c ability set nonconvex competitive outperform instability know prior improper proper result improper inherent alg permutation figure depict change see take show powerful definition lead construct compound compound gamma establish family prove family compound density solve nonconvex conduct state show dimensional estimation framework framework mcmc
represent quick plot reveal consecutive would reject return daily return confidence level consecutive daily frequency synthetic serve also include assess goodness residual entropy random pairwise date density entropy rate use entropy use appropriate lag model evidence particular analysis auto research refine study stable underlie structure institute institute wide application science finance note redundancy dependence information construct dependence extra randomness process compression allow literature restrict pairwise serial dependence stock period frequency synthetic recover datum compression apply time economic phenomena asset pricing try capture salient process theory follow relevant behaviour economic stock return number order order return bring salient generating price change return argue mechanism help important price reflect asset merely summary acting incur economic agent asset price phenomena making extensively test prove test agent cost act secondly term pricing model contradiction pricing asset price martingale propose walk stock interested return increment walk inconsistent return effect return daily higher pattern stock stock return return stock price adjust belief pattern return return unconditional belief stock return volatility though implication proper specification describe redundancy stock statistical model return apart substantial redundancy stock various stock exploit good knowledge able dependence apply discover relationship apply discover dependence return category real include serial method discover dependence include audio measure value correlation dependence occur unable correlation sufficient many mutual empirical characteristic closely non degree joint density computing entropy entropy provide evidence random variable allow attractive test dependence monotonic lag series joint dependence lag contain ignore lag specify estimation preserve generality testing introduce author serial rely propose relie speak additionally transformation compression ratio ratio certain quantile theory compression describe stock return offer structure single series stock stochastic process conclude acknowledgment author discussion contain everything representation usually bit compression representation exactly small also occur compression order value finitely computer line observable precision practice finer wider use call resolution discrete finite original encode precise compression shortest encode sequence x one average length encode length entropy possibly quantity theory know entropy enjoy remarkable one random variable remain stochastic lead whenever process statistic word stationary decompose discount propose contribution entropy marginal behaviour independence restrictive exposition consist iid variable n coincide marginal transition stationary p p stationary word entropy random particular coincide distribution typical grow sequence typical length give upper compression message ergodic stochastic furthermore length symbol give principle practice situation generate process generating might ask universal regular day day computer shall term basic class algorithm year speak encode object content last occur typical occur sequence remarkable prove ergodic denote implementation develop algorithm format fundamental noiseless channel stationary ergodic computer process precisely compression ratio compression ratio hand optimal ratio formula take sequel compression entropy reader convert convergence algorithm overcome difficulty able perform reader fast experiment serious overhead mass generate readily next generate perfectly represent shannon tell large represent sample bit sufficiently compression ratio task write file file compression table cs cr examine generate leave extra compression take cost inherent actually overhead cost arise reason compress file header need code file remove generate identically uniformly save file compress file cost compression experiment ht compress overhead overhead decrease negligible bias overhead cost converge sequence sequel small indeed estimator compression ratio overhead cost estimate overhead cost account bias stock exchange period sample datum day day daily day std skew detect describe bootstrappe compression insight finding serial dependence function measure consider big collection series section statistical synthetic goodness daily want inference examine order order longitudinal empirical apply transformation order rank rank span integer prevent sized bin usual reduce decrease sample sequence length integer evolution enable effectively space area could terminology refer indexing plot serial dependency value process scatter resolution return frequency figure plot interpret markov lag every form pattern daily return half near clustered bottom longitudinal return cluster return presence half period quantile pattern detection discover human combination commonly pattern set amount measure redundancy return amount allocate apply previous also would high validity statistical would propose know likely know return represent effectively possibility furthermore center concentrated compression transform combination bit equally choose resolution various compression pattern series pattern mean standard deviation transformation bit eliminate temporal contribution return return entropy perform random vary entropy block achievable return series frequency day ratio process compute length range generating process height axis compression indicate redundancy frequency return statistical dimension compression machine overhead file point panel identically uniform size block perform interval return seem iid redundancy interpret evidence return identically distribute compare estimate compression return possible conduct panel zero sense compression series lie block
capture learn representation suppose induce basis feature space map data basis function excellent vision norm remove whiten ica condition close relationship tx x frobenius ica constraint representation ica fail please rewrite contain bfgs cg problem alternatively optimize derivative solve seek approximate exact solution point part utilize mean follow respect become linear straightforwardly connection attempt nonlinear feature feature sparse major difference utilize encoding code optimize alternatively simple force pooling group meanwhile optimize give label learn basis label well discrimination namely mathematically sample utilize class belong rather regard utilize learn reconstruct constraint representation coefficient belong fail optimal basis consequently learn discrimination homogeneous cost minimize jointly mathematically homogeneous specifically select coefficient discrimination concentrate subset please maximize homogeneous basis belong class thereby power incorporate discrimination scalar control term sample representation meanwhile easily solve point extraction image public cifar patch term image could channel follow set reduce dimensionality reduce pool pool utilize svm follow common experiment implement algorithm table lc incorporate constraint lc cifar dataset include color image category etc addition randomly image field follow approach etc c improve code belief layer auto rbm mean g pixel color addition fold class unlabele manner mean maximize representation representation learn representation introduce unsupervised perform algorithm imply discriminative additionally utilize penalty feature information seven accuracy performance model raw x verify cross validation important facilitate dataset show easy outperform similarly experimentally cifar control discrimination supervise set achieve well minimize homogeneous represent nonlinear imply power similarly cifar utilize image classification similarly experimentally cifar polynomial histogram demonstrate classification different study c polynomial inverse intersection kernel show image sparse representation correspond respectively representation similarity measure euclidean since matrix wise take discriminative linear achieve reconstruction nonlinear bring unsupervised discrimination lead learn correspond represent belong discriminative standardize demonstrate set identity reconstruction constraint q without loss generality derivation pt depend hessian meanwhile z get j j k z z hessian positive definite acknowledgment discussion acknowledge le code analysis statistically representation also complete infeasible essentially utilize kernel ica bring supervised introduce basis different belong representation thereby learn discriminative experimental validate effectiveness characterize signal play role sense factorization sparse auto rbms component ica transform multidimensional sparse independent specifically e meanwhile sparsity dominant natural maximization basically ica drawback ica ica whitening preprocesse standard ica difficult exactly datum eigen ica learn complete basis dimensionality autoencoder rbms performance put ica disadvantage drawback mainly due mathematically utilize prevent basis row satisfy standard addition complete expensive error issue le ica complete technique infeasible discover unsupervise sufficient task fail association recall nonlinear project high represent bring information maximize largely two computer image reconstruct utilize similar face accurately category coding often corrupt may sensor poor illumination communication select reconstruct image meanwhile deal allow denoise etc ica show mathematically encode ball mention study seek input structure phase principal plus ica use reproduce nonlinear fail utilize additionally extension equivalent
obtain third term decompose event decomposition easy hoeffding maximal devoted maximal n dx conclude proof distribution sub satisfy mean know assume derivation thompson sampling decompose ti follow rest proof ip follow n bound dedicate task clearly straightforward integrate use hoeffding dx x together financial engineering edu stochastic armed bandit problem reward spirit stochastic bandit building sampling sense exist thompson thompson nice property let random variable identically take arm arm potentially external source randomness measure external fix integrate context drop dependency merely view latter agent must low regret formulation history extensive bayesian major relatively mean process independent interested later know dependent product prior perspective bayesian thompson multi armed bandit strategy history ti interest incorporate arm thompson binary reward paper run thompson sampling different prior spirit thompson prove apply integrated section idea thompson attain furthermore arm call policy satisfie attain standard ucb natural assumption ask thompson precisely thompson uniform prior similarly thompson optimally furthermore remove present step towards bound thompson generalize challenging remark detail decompose measurable identically integration deviation elementary thus b start integrate next step inequality reward obtain u du u du put piece conclude proof armed reward thompson assume word one recall thompson word thompson draw random probability
view value small bid bid enforce bid somewhat complicate example view simply require total practice search certain google search focus million numerical potential usefulness propose utilize use project approximation mathematically capture solution suitably construct propose carefully examine prove achieve performance certain state section make geometric lift point exploitation trade objective nonlinearity present non since successfully analysis extension organize achieve near input algorithm insight design algorithm strong algorithm validate paper rewrite offline duality hold feasible bind describe solution tie break arbitrarily arrival allocation intuition approximate project input entire rule explain nominal bid contribution objective e possess instead concavity require prior follow assume input hold position singleton say break tie point always add variable arbitrarily assumption proposition define notation offline solution xu equal fix u first u probability two term j j hoeffding bernstein see sample different uniquely follow many allocation profile exceed therefore distinct proposition allocation j b ij I I follow inequality concavity due concavity I lastly thus condition essentially result undesirable bs hold b ij fm fm I proposition ij ij fm competitive permutation definition possibility receive nearly receive none always true possible reward particular choose fm choose condition nice practice resort validate investigate like pay display base problem category keyword ik random category probability simplex choose although seem reflect major bid category keyword company interested represent multiplied reflect level also way generate report end performance rl key give condition expect rl theoretical asymptotic good average run deviation improve performance small decision maker refine policy subtle force choose hand bid may enough poor allocation period choose small follow next allocation show perform gradually approach insensitive still mean theorem slow compute first one need fix base follow generate follow beta beta deviation case input c c three rl decrease size policy also vary overall resemble robust toward approach dynamic concave return primal objective nontrivial well problem anonymous research support research grant hoeffding theorem sample replacement real number lagrangian strong continuously two must hand always achieve optimality versa case exist feasible solution objective obtain ij jx tie arrival small distinct fix second therefore bound distinct argue u show follow similar one learn therefore I probability less I mm k k therefore know I I l ki I lemma prove lemma first due show give argue exist allocate allocation allocate allocation prove exist however optimality inequality definition allocate allocation allocation I kb kb ic condition contradict solution prove conjecture chen edu wang edu match concave return online vast input learn datum dynamically carefully allocation problem decision belong dual input assume optimal reveal begin management problem customer know regime gain past decade applicability effort toward understanding reader match online theory review reader weight vertex whenever set weight reveal decision maker maker gain match maximize vertex mathematically make concave differentiable early application
ai wise loss conclusion rf unnormalized empirical divided form convenience conclusion lf kf lf conclusion mathematical induction variable trivially conclusion second equal natural prop exchange bipartite problem sort ideal ranking position remain position generality position exchange first relative explanation increment increment position increment instance increment prove rank fig fig show conclusion still multi error error equal divide rank bipartite unnormalized kf kf rf fig conclusion loss rank htp htp ndcg quality assessment prove ndcg web challenge ndcg rating discount ndcg normalization
increase aggregated datum partitioning fan node balanced tree commonly interface next develop optimizer decide job fan optimizer answer question job task fan aggregation phase answer optimizer objective program job public amazon ec finding question fan aggregation tree job design partition record processing machine record disk hold influence process iterative consider follow operator follow spend spend need spend operator assume behave linearly invoke often assume transmission behave assumption violate real represent load assumption allow notation express cluster job job cl record per cpu record load lastly comprise time tn f mn f mn already state depend fan machine solely parallelism aggregation theoretically choice fan fan aggregation take aggregation number parallel spend height leaf node f aggregation input easy large number tree independence similarly spend aggregation balance aggregation fan fan decrease operation fan machine tree machine hence fast also establish neither depend fan respective refine tn mn cn physical stay iteration fit aggregate main must distinct possibility disk phase perfectly tn minimized minimized derivative processing take disk incur md minimizer optimizer runtime plan essentially unbounded assume machine machine need record job ever secondary disk well case good md md mae md solution intuitively spend hence facilitate completely task iteration machine two hold optimal aggregation iterative know minimizer e cost fast aggregation tree minimize optimizer evaluate choose plan present experiment evaluate optimizer approach goal optimizer fan aggregation optimizer predict present scale formalize tuple empirical divergence amenable optimization gradient step dominant per tuple amenable world dataset feature vector contain representation l r meaning record map per task load record conduct yahoo intel gb network interface run machine connect switch pair task optimizer optimize plan job disk format aggregation cpu effectively aggregate operate million mb dimension time iteration perform use per aggregation result hence current state optimizer suggests interestingly also predict minimize configuration cpu remarkably optimizer fan suggest fan aggregation node object evaluate claim vary fan aggregate report fan fan vast case thus fan optimum effect node fan mb mb mb mb mb mb mb mb job create scenario record amount fit cluster report pick minimize time cost fan determined minimized optimizer furthermore job north west optimizer competitive current art neither resource cache like read disk experimental finding finding optimizer pick plan increasingly cloud environment argue extend allow program execution iterative illustrate automatic class readily express task develop optimizer execution plan local loop aware scheduling cost partitioning resource reduce optimizer program namely aggregation present competitive specialized implementation optimizer take kind failure comprehensive carry establish specialized implementation encourage drive class especially tune cloud change resource availability thank across along database query cloud big become potential derive insight wide business recognize rapid elastic scalability ever lead operate paradigm recognize support limitation lead inefficient inherently cloud either aim ad hoc class construct propose extension programming optimizer iterative machine step empirically task competitive specialized solution recognize potential drive every aspect scientific range increasingly everything theory scale key insight ever valuable grow analyze quickly identify size paradigm large datum many algorithm cast term fails recognize due computation execution message interface algorithm recognize first programming abstraction force make decide cache main approach ill cluster draw database abstraction consideration drive paradigm runtime support iterative program follow contribution formalize describe big runtime new runtime optimizer optimizer pick runtime plan structure aggregation since logic theoretical foundation optimizer optimizer demonstrate art programming traditional aggregation step specify responsible transform input record specify process group produce g scalar group programming model use notable extension correlate top close intend target build level abstraction many machine iterative procedure give body solely sum query naturally compute express backpropagation logistic svms rely sum function build interface extension programming paradigm iteration fundamental program operator produce operator main composition computation key operator input information input aggregate sense typically functional look machine code ensure extension programming operator body condition operator chain operator loop output input external job job feed job lastly training benefit programming interface support extension loop master loop body meet add cache aware task explicit avoid scheduling pass interface loading feed worker connection system outperform speedup line early sgd failure long independently store machine scalable machine runtime include aware optimize iteration cache aware format speed scheduling avoid iteration communication direct connection magnitude abstraction call collection provide language consist relational algebra project join support runtime publish stock stock cache partition aggregation cast exploit capture aspect develop parameter aggregation fan optimizer good plan capture iteration next given describe optimizer section physical execute machine plan template realize plan runtime consist process operator execute runtime split execute operate plan explore
approximate sequence look interested minute trade day return minute typical day note obtain deviation divide trial chart show payoff value good value baseline payoff height algorithm algorithm auto idea provable predict time library forecast stock trading return stock day stock bid best york stock exchange stock show minimization provable outperform bit stre concatenation string draw length sequence length least enough regret sequence absolute sequence length random hand payoff string length independent payoff expect least feasible disjoint align hold theorem since notational let align repeat possibility union size size union case semi adversary sign choose sequence bit sign number multiply value denote desire number show sufficient condition payoff instead sign magnitude real randomly randomly sign give payoff prediction payoff string seen let bit equivalence show proof loss tradeoff useful obtain tradeoff different expert round payoff expert arm expert payoff tr ask average regret tradeoff relate two regret loss prediction bad sided lie upper feasible max bad feasible expert feasible side regret expert expert look sequence produce payoff gets translate map map conversely convert two payoff payoff arm payoff expert side prediction translate expert problem gets translate payoff translate regret arm thus feasible conversely instance side convert two armed original respectively expert translate consider classical bit standard regret payoff predict regret majority guarantee regret length randomize experimentally efficacy predict bit predict bit past wrong prediction bit bit payoff per b thus shift value equivalent one stock stock algorithm surprisingly positive payoff sequence one hope give guarantee certain payoff expert correspond one expert optimal expert regret worse opposed expert paper classical result via majority formally height chart payoff focus trend sequence string consider sequence high many interval may partition sense state bit consider armed problem round expert payoff expert payoff payoff function denote concatenation payoff define partition sequence payoff main theorem absolute payoff expert theoretically optimal empirically find stress length partition guarantee individually net fact impossible achieve sense give trade value use programming feasible determine well observation cover feasible achieve fs e u fs running suggest payoff far random sequence replace difference two case theorem expectation different naive bit corollary main result require fix advance achieve special case interval numerous style financial formal appear want minimum first payoff divide string length definition break align interval align theorem recursively power stochastically shift length deal mid payoff second payoff stochastically shift separately uniformly say align align interval break part instance always discuss interval align shift shift denote accord payoff check satisfied show f remain equation whenever appropriate p hx variable distribute show term write equation line otherwise side equation integral thus substitute value hand around turn equation last substitute maximize term set need
predictor discuss resolve first fact since imply target imagine priori define model however inference want frequency pose exist coverage form select model confidence ever coefficient across sense event control predictor involve instead control control proportion simply zero remarkably coverage iterate expectation argue post allow inference linear begin section characterize union precisely specify sign select polytope sign turn univariate derive statistic result intend newly add lar path framework question select coefficient address post coefficient select unless correctness approximation truth selection lasso usual square penalty penalty non seek define characterize begin note sign sufficient satisfy kkt implicitly fact zero predictor however every set predictor turn easy union sign candidate projection rewrite rewrite kkt convention kkt condition necessary solution two definition give lemma remarkable say affine affine inactive encode form substitute constraint rewrite inactive let bm ib simply union bm figure partition select sign correspond union sign cycle cycle sign previous union condition sign condition polytope bm inference fine conditional interval conditional divide subsection condition allow look extend obtain conditioning general I test price observe interest understand term definition seem residual rewrite function function decompose polytope category depend affect encode since like since change condition truncate integral statistic make variable truncate interval define apply eliminate let z integral q conditional polytope sign union e scale cycle cycle truncate union disjoint gaussian truncate set union possible immediately truncate interval link theorem conditioning apply theorem confidence formalize satisfying claim interval j truncate ratio appendix detail condition polytope eq inverting interval coverage efficient wide efficiency notice computing less hundred conditioning sign mean statistical strong interval sign see actual expect post notice truncate gaussian basically recover nominal ol adjust truncation obtain nominal truncation generally prefer shorter short among coverage cover I e interval short similar tail rejection short confidence coverage conditional e family exponential interest nuisance represent dimensional theorem say uniformly powerful versus conditioning let z unbiased minimize yield condition invert construction detail selection diabetes datum choose accord post nominal fitting ol variable ignore conditional valid post depict use produce nominal ol interval strong datum wider wide ol produce short among method selective ol long accounting splitting adjust significant demonstrate h interval
unstable largely overlap test indicate big derive parameter ensure practice whether develop way definite matrix distribute subject algorithm component definite lead principal minor parameter diagonal enter bind work calculating associate find c ik ii truncate deviation upper enter twice determine limit minor k coefficient find principal minor quadratic parameter updating parameter describe element refer doubly mean truncation deviation k ij simulate boundary draw uniform invert cdf distribution rejection sampler draw sample target rejection constant side truncate translate coincide truncation match tail value acceptance compare efficiency exponential doubly upon truncate well algorithm along test still component randomly notation section histogram uninformative red conceptual cubic couple think represent weather fluctuation slow variable inside well perturb act effectively equation display sde cubic system parameter gibbs sampler section posterior start predictive estimate empirical estimate show reduce reproduce noise model observe approximated ability derive accurate order model plot correspond infer apply couple fast reduction strategy separation system stable insensitive wide choose display leave small parameter scale fast convenience inference show moderately though amount lead estimate inference total simulate fig simulate full reduce calculate estimate posterior give reproduce reduce fit autocorrelation collapse onto rescale interval simulation use strategy system systematic inference parameter enforce order constraint sde definite develop improve parameter conceptual apply useful procedure anonymous whose version manuscript study physical sciences rgb systematic inference sde applicable system globally stable relate cubic nonlinear definite datum conceptual global stochastic differential constraint reduce dynamic stochastic run resolution dynamical prohibitive mainly interested exact scale typically attractive molecular engineering partial observation time low curse reduce principle full dynamic valid form physical constraint constrain systematic physics govern law energy model principle normal form provide observation fundamental eq denote external quadratic nonlinear model prediction predict reduced mode reduction systematic derive closure take mode order split systematically reduce denote cubic wiener process diffusion estimation chain mcmc knowledge mode pose necessarily real perform cause stable meaningful become lot parameter infinity finite negative devise novel sampling strategy computationally lead reduce experience finite derive cubic stability develop bayesian physical develop definite mcmc inefficient conceptual summarize result structural convenience inclusion cubic quadratic cubic cubic term global stability cubic nonlinear stability global write cubic ultimately global normal model unstable unstable associate weather linearly unstable mode certain amplitude nonlinear ensure form definite follow reduce lead I I I I I I miss observation independence next section proposal absolutely function miss data block accept mh inter interval become one small block accept reject mh euler transition gaussian I I I j algorithms combine mh proposal availability observation one walk gibbs repeatedly produce increase diffusion observation pair diffusion proposal process bridge parameter equation purpose drift q linearization bridge j contrast bridge sampler give drift enter linearly construct gaussian greatly mix consider index example q write instantaneous covariance zero mean dp dp pe j two diffusion choose observation period trace plot text set
rigorous include observation formally extend model present notation necessary sufficient sample recovery application include theorem lemma letter realization scalar index indexing provide reference transpose symbol basis random th th sub outcome iid variable subscript indexing determine index set index total ks j cs j index set iid index assumption latent correspond zero linear impulse response framework p outcome emphasize give realization outcome independent outcome estimate randomness set depend consider define formally pe g pe pe salient error assumption utilize recover salient element salient set outcome conditionally variable index e formulation recovery assume except py py ix assumption valid problem analyze within I restrictive usually incorporate averaged remark support recovery exhibit structure recovery instead effect density consider mostly compressive sense estimate sufficient notation discrete replace sum integral generalize notation continuous case distribution observation relevant depend discussion derive required likelihood decoder true among reason conditioning throughout decoder choose likely error occur decoder probable decoder knowledge observation decoder early version carefully obtain standard dominate testing consider undesirable sense herein whereas analysis also report infeasible similar code block error event set upper throughout replace sum state generalize mutual sufficient symmetry assumption ensure identical partition sufficient sufficient average zero zero upper possibly avoid model consider bad vanishing upper error exponent describe lemma variable probability define exactly select decoder average largely variable error exponent bad letter perform taylor ml decoder methodology proof conceptual technical difference include arbitrary expression model signal represent significant consider highlight herein channel code difficulty separate item every fix correctly every candidate bound theorem indice conditional note instead salient define require bound difference explicit conditioning difference appendix bad tight order wise numerator approximately require denominator subset represent number need control account necessity denominator term dominate recover support support hard importance recover necessity necessary sufficient support change maximization recovery index determine depend precisely relate complexity scaling scale snr note necessity scale generalize addition additional exponent letter section letter theorem least condition letter true respectively letter condition let number sufficient average asymptotically exponent multi letter expression theorem simplification may certain partition ix fix letter conditional exponent p sequence ix letter useful check easy notation observation sum integral scale slowly provide exponent single letter taylor series bad condition second control necessity subsection linear subsection regression testing finally proof appendix necessary sparse iid normalize measurement element mean observation noise iid processing framework relate general generate linear combination non observation account snr necessity level dot satisfied fix linearly sparse iid b regression mutual give identical know gaussian sublinear snr necessary recovery necessity sparsity provide recovery easy obtain recovery another interesting aspect analysis bound recovery finite triplet practical recovery optimal gap theoretically performance reader detail regression relation task compressive application multi task vector rx n independence model section iid direct mutual identical regression model show sparse per increase fold inherent expect number look compressive sensing practical importance noiseless follow element support gaussian element bit output input regime probit gaussian measurement support linear probit iid gaussian tp therein present group e binary define item outcome boolean sum item identify arbitrarily bound respectively lead false lead refer reader information lead consider fully observe observe entry change bind six miss setup describe lower miss upper theorem miss number miss datum highlight flexibility characterization flexibility enable new variant framework noisy code recovery approach non combinatorial corresponding difficulty algorithmic conjunction tractable algorithm useful gap exist fundamental limit understanding aim give since fix remove consider bad case integral prove later equal exist lagrange derivative evaluate zero discrete variable sum sum notational convenience b si ix equal expression expansion imply write separate second noting trivially enough dominate specifically note choose go proof necessity variable match condition explicit ensure clarity entropy proof hand continuous expression differential conditioning include depend independent except distribute variable equality independence necessary great x expression expression ix derive ik choose write therefore aim note omit imply condition derivative second inequality prove note bound proof variable observation replace sum appropriate integral define nonnegative first equality denominator write use jensen qx potentially easy note trivially exponent follow multiply divide inside sum obtain note iid condition scaling extension continuous main proof iii yet generalize latent bottom end error exponent exponent furthermore missing term utilize strong argument appendix proof modify bound replace result idea make generalization simplify exposition extension reduce ix sx density cumulative function observation respectively let cumulative ml decoder continuous error denote ml decoder input error indexing utilize holds note ix mutual continuous quantization level discrete quantization take minimum upper thereby prove quantization convenience boundary space quantization calculus increase fine small quantization furthermore quantization boundary space write py let quantization bound function assume continuous convergence probability measure fact f eq follow complete mutual ss omit conditioning equality recovery snr readily also note jensen q necessary since k prove otherwise scale two sufficient normalize equivalent bad minimum sufficient assume simplify exposition analyze theorem mutual ix see reduce case information analyze information follow chain note expand define limit integral note evaluate replace q q inequality inside define limit integral even constant look follow simplify conditioning entropy explicitly conditional entropy expression simplify term expand conditioning assume two follow early follow rearrange mutual inequality follow non negativity mutual expand information expression theorem simplicity exposition conditioning simply z z expand expression fourth noting inside condition jensen note concave expectation also write expression weak mutual mutual derive limit include characterized outcome identify set outcome characterize noisy channel analysis provide successfully recover salient expression aforementione mutual expression demonstrate signal sense video genomic process conventional method dimensionality exhibit structure
x third embedding select column lead eigenvector nk nk x x nk matrix view pair nk nk nk nk plug nk nk triple nk nk nk multiply whitening apply decompose n nk nk nk nk nk previous section hilbert schmidt bound constant vector sphere q power yield eigen permutation perturbation bind eigen proof appendix ok fix depend observe method complexity latent nonparametric gaussians guarantee global spectral spherical center approximate algorithm recover discretized density well estimation histogram suffer alternative make figure separately validation various dimension setting gaussian density variance gaussian gamma conditional shifted shape furthermore gamma component relatively accord fisher cccc gaussian varied unbalanced become data set measure performance plot converge rapidly increment mixture gaussian set algorithm bandwidth fold cross validation gaussian gmm covariance sort gmm spectral dataset multi view heavily violate dataset subject future plan number acknowledgement song nsf gm support microsoft fellowship nsf nf eigen detail eigenvector initialization replace neighborhood vector initialization lead initialization method update successive v number eigenvalue compute update establishe orthonormal initialization vector initialization correspond choose sequel establish concentration translate use covariance whiten rd embed whiten employ covariance restriction pair svd exchangeability whiten procedure operator whiten perturbation line whiten operator since kk sample separately eq lemma note residual need parametric perturbation trial substitute require constant additionally concentration pair hilbert schmidt similar deal symmetric easy result hilbert similarly bind rd triple let schmidt deal symmetric operator see hoeffding hilbert space greatly advanced latent variable sequence efficient algorithm strong guarantee current largely restrict mixture view allow mixture multi hilbert recover tensor sample relevant thus enjoy pt latent variable range document maximization traditionally guarantee largely distribution mixture theoretical nonparametric exploit key tensor three covariance efficient distribution delta spectral rbf sense provide framework previous spectral complexity low order thus computational nonparametric variable model none explicitly recover invertible transformation focus predictive marginal making model property kernel algorithm previous correct algorithm term opposite incorrect margin domain refer character joint methodology case domain hilbert element mean view point view kernel include laplace dynamical system structured embedding mapping rkh embed element rkh map dimensional implicit embed rkh embedding product map joint two tensor space reproducing characteristic map distinct commonly property embedding exploit state embedding equivalence product latter product feature analogy clear context tensor generic introduction tensor notation please shorthand fix application tensor product mean argument hilbert schmidt define operator joint nd operation singular order manner orthonormal small rarely embed finite converge draw similarly x I virtue subsequent kernel gram matrix value determined sample much small infinite enable nonparametric sample embed expensive low approximation factorization effectively maintain multi view variable give multi view figure complicated graphical show reduce symmetric view circle sep fill hidden minimum inner mm draw black hide name name observed size draw fill style inner hide name x value rkhs conceptually potentially infinite value retrieve inner distribution factorize hide discrete map embedding factorization embed alternatively mild identifiable multi view latent model independent joint kernel kronecker delta kernel dimensional distinct identifiable scalar however universal work independent non incorporate component exceed extended independence tractable latent extended version clarity presentation extend
local maximum corresponding primal condition point always critical derivative condition statement derivative dual use condition rewrite equation report relation primal dual point statement ht corresponding primal substitute primal critical critical eq happen configuration make root correspond point primal plug always refer feasible space explain critical primal minima local maxima relation theorem first critical primal domain theorem critical minimum dual problem order zero primal understand minimum order substitution obtain obtain negative dual always always condition primal way minimum x order correspond h h critical figure change lowest double well correspond critical minimizer primal near visible certain minimum case critical boundary critical primal point objective primal critical boundary get want boundary big high value solution preferable reduce critical point local domain basically near case three critical critical correspond primal value value possible make minimum primal problem critical problem case quadratic function study hyper domain present canonical dual canonical reformulate duality one particular exponential also important may application canonical duality radial kind rbf analyze quadratic expand multidimensional rbf theory canonical radial university national university radial basis widely drawback supervision highly problem fundamentally difficulty generalize duality theory challenge transformation nonconvex reformulate canonical problem radial function result even well canonical tool network radial tool introduce interpolation last decade apply mean give radial unit neuron weight center two main optimization strategy regularization strategy center radial basis follow unconstraine neural high use decomposition global minima consider radial issue cross validation order validation try find upper potentially powerful biology science communication study canonical duality radial neural arrange demonstrate nonconvex dual canonical showing obtain dual original analyze gaussian radial address comprise radial formulation mathematical basis eq belong convex radial primal solve geometrically clearly map radial definition say canonical relation depend radial couple reformulate total relation invertible u notice connect generally certain dual relation mean primal replace rewrite canonical ss canonical term point notice third primal
game choose loss specify player choice round feedback set loss extend drift upper adversary feedback bandit feedback logarithmic bound adversary feedback bound follow principle feedback switching set tx switching guarantee regret without cost expect use arbitrary round choosing attain expect proof technique straightforward standard provide simply equal subsection setting since fix horizon focus control time handle fact bound define define modify player adversary exp bandit bounded drift regret bound guarantee cf bandit adversary bandit assume horizon slightly player define loss consecutive epoch epoch begin action use round epoch round interval give exploration round epoch detail appear appendix round exploitation epoch exploitation play exploitation first play round action round action let player latter unbiased epoch end epoch fed prove regret defer drift attain regret study expert type range adversary prove adversary bandit match adaptive memory information cost show feedback feedback action setting predict cost adversary slightly assumption introduce low question bound sophisticated notion swap adversary interesting adversary feedback briefly introduction exist case adapt player play armed value initially round change zero know set prediction adversary switch cost assume action choose expectation random randomization expectation focus subgaussian subgaussian z b subgaussian player proceed stage player maintain action total number play horizon loss action action define use union least prove hold claim base q prove claim use sx sx closely base bound order desire randomized adversary standard two set player player loss previous randomized player randomization randomize adversary introduce action e action negative hold technical lemma whether action condition focus term random since player strategy player either time entropy switch action switch towards hence relative entropy gaussian shift namely overall upper use bound conditioning replace q gives expect player adversary event bad action action pick least also player player time let action time lemma quadratic attain value picking tell randomized adversary deterministic player adversarial player loss drift case govern unbounded variable adversary pick deterministic adversarial plan adversary event sum realization loss prove let independent possibly player randomization already know approach convert lemma bind weak show strategy adversary eq horizon reach get adversary coin therefore adversary regret strategy adversary strategy last plus pick bad drift factor adversarial tell exist adversarial thm deterministic adversary strategy regret prove actually show adversary randomize player randomize memory every possibly player player adversary thus switch happen right side observable bandit set get expect game least randomize strategy possibly randomize adversary strategy expected bound specifically completely analogous least negativity adversary proof thm adversary strategy regret drift loss equal tx interval size namely rearrange term sum thm recall imply always exp hold exp hold choose exp q side rewrite thm drift note consecutive epoch final mini draw epoch end epoch action epoch feed regret q apply upper loss action particular begin epoch randomization separate consecutive play action play action suppose estimate moreover depend action loss apply last thing marginal configuration appear configuration case point point enforce completely exclude exploration point final theorem claim universit di microsoft microsoft type feedback player notion policy regret adversary behavior loss characterize nearly cost switch bandit feedback bad rate switch full novel expert bandit switch rely adversary generate player fix begin action draw player game version game round adversary assignment player observe loss choose adversary round player action adversary assumption imply specify entire advance player formally player shorthand adversary input entire round player feedback player observe round whereas observe far notion compare round player round randomize one differ eq literature measure player adversary adversary quantity interpretable sublinear imply force focus reasonably determine base adversary past adversary satisfy focus imagine stock day stock suffer player stock adversary amount measurable action example relate feedback version game stock end trading day adversary switch adversary define two step choose stock keep position stock trading incur fix generally situation costly discuss special fixed switching cost adversary adversary adversary constrain switching cost adversary depend arbitrary adversary allow action predefine formally memory adversary bandit strong building regret cost bandit defer step understand upper essentially consequence switch versus learn feedback hard learn full action dependence aware number space e recall regret bind adversary switching cost demonstrate must play control switching cost adversary prove dependency feedback switch adversary whereas somewhat originally prevent seem key memory contradict intuition control easier noted standard technical loss two allow drift round formally bound slowly relaxed result must continue logarithmic
learn stochastic gradient descent classification dissimilarity generalize optimize relevance dimension weighting indicate profile provide information dimension vanish relevance relevance profile light negligible drop high regularization relevance base east lasso descent lasso distinguish quantization vector collect approximate denote prototype matching prototype become point classify correctly transformation monotonically increase sigmoid identity dissimilarity replace distance parametrize bilinear element profile relevance obvious generalization scheme avoid write variant optimize depend follow norm accord lasso q solve r absolute x relation depend q term add recursion apply calculation yield differentiable approximation htp relevance classify hyperspectral wave hyperspectral appropriate analysis acquire spectra prove assess composition utilize camera spectra nm nm per proper image calibration image segmentation obtain norm band ignore nm type full profile yield start solution factor regularization enforce drop lack htp however decrease differ keep long accuracy htp heavy relevance regularization hyperspectral play band
quantile ergodic inequality martingale field exist nonnegative constant follow technical h j h k kx hypothese nx nh making lemma x ft nh follow sure similar proof consistency proposition condition ft nh similar decomposition proposition asymptotic go sequence n subsequently together statement condition statement decomposition ft study ft nx converge surely term asymptotic end nx ft nx nx almost provide condition expansion lie theorem x nh q kf get function follow remark proof lemma lemma use l l nx x x nx ft x ft x nx ft similarly nx nx nx martingale upper relative nx nx exponential respect observe eq condition px p l n jensen one mh mx mx ii constant x x c f almost almost argument surely q constant lemma nx ix n nh nh cn nh c therefore obtain borel iy n n iterate logarithm censor consequence intermediate normality define array establish statement n condition iii ft use inequality lemma examine equality h x dft I zero get dft v assumption integrate ft ft ft ft get get h almost surely ft ft x part n l old l double conditioning lemma get nh x algorithm axiom conclusion definition exercise theorem remark quantile censor ergodic centre behaviour mathematic read universit france email read ac uk paper estimation covariate value whenever datum introduce consistency estimator normality induce peak demand censor carry keyword asymptotic censor quantile ergodic process functional data interval martingale peak load analysis study development last kind appear soon one phenomenon reason economic since work development around functional kind possible statistical parametric nonparametric model strong consistency explanatory response curve normality alpha mode side quantile dependence response function rely tendency function analyst robustness error useful conditional quantile scalar nonparametric smoothed local censor regression conditional quantile identically strong uniform interested estimation scalar conditional cumulative sample almost forecast ni prediction asymptotic normality estimator framework provide dependent investigate property response covariate censor consider model censor infinite character use strong probabilistic calculation moreover mix property well indeed mix induce therefore ergodic regard ergodic datum ergodic know censor theory statistical introduce estimator application peak censor proof main preliminary censoring observe continuous distribution censor assume sequence censor df follow study indicate plausible censor independent patient value metric abstract suppose sequence framework among condition establish confidence interval quantile assumption field depend h u ft censor satisfied denote ft x f limiting depend quantile form theorem use practice estimator fix replace km nm nx nx band nx x nh daily peak demand peak demand important operator daily influence load load influence increase energy wind increase heat grid technology advanced peak demand thus optimize network evaluate localize peak forecasting need peak forecasting widely issue short peak density forecast arrival automate read energy point hour minute know nothing peak peak demand send customer peak day wireless technology receive source instance wireless receive datum peak sample censor peak demand demand temperature demand temperature easily sensitivity consumption customer weather figure curve demand day day contain day
simplify notation fact easy eq monotonically proposition distance since q notation assume assume define fact constant document component candidate define loss incur correspond symmetric arrange spectral norm let minimize respect set along set eq proof note q hoeffding probability least express empirical empirical q setting easy interest use hoeffding tail final proposition lemma eq result x tv proposition proposition handle eq di mean q hoeffding tail let q q course remainder q eq ignore strict elsewhere elsewhere eq argument small trivial upper clustering hence bind condition
initialization converge log value could appendix inspection table specifically rarely unconstraine em em far degenerate counterpart compare constrained estimating within ari table ari associate model component ari ari introduce eight parsimonious family please discuss herein performance constrain suggest extensive real study approach converge degeneracy eigenvalue require regularization placing involve study estimating eigenvalue outperform improved eigenvalue parameter eigen far include fact include prevent mm mm treat incomplete mixture likelihood surface maxima within base parameterized eigenvalue illustrate real expectation maximization incomplete treat attribute previously employ involve two attain complete maximization refer miss concerned model surface mixture model local go far like spurious tend illustrative fit behaviour solution study tackle small component covariance paper consider eigenvalue eigenvalue large mixture general impose maintain parsimonious cf eigenvalue application impose constraint iteration em initialization maintain converge degeneracy change converge solution log initialization application one range base improvement model must choose discuss section real observe model base become popular mix detail mixture covariance matrix give algorithm eigen decomposition ten give software package employ mixture software free member within whereas last include within mod shape free spherical variable spherical axis align equal align variable align equal variable pp g effect writing include tie volume allow orientation importance utilize agglomerative hierarchical start select starting mean default clustering arise expect probability map converge value e step map classification depend convergence bayesian select regularity condition development bic mixture herein bic free maximized estimate bic alternate group maximize repeat eigenvalue maintain monotonicity herein constraint large eigenvalue range ig constrain g member describe iteration constraint however use scale I use family classification rand index summarize ari rand ari indicate scale degree freedom heavy tailed initialization eigenvalue matrix perfect g relatively accommodate tail merge component appendix well respectively l classification mixture show simulate htp lie eigenvalue appendix
label hypothesis hx agnostic labeling distribution fx learning least typically learn complexity far label setting oracle randomize set algorithm one label consecutive example respect requirement set receive chain mix learn simulate power let time reversible discrete eigenvalue assume eigenvector eigenvalue less f gx f throughout consider product norm distribution briefly boolean boolean cube eigenvector chain orthonormal unfortunately case vector standard show may compute dimensional eigenvector power apply markov satisfie eigenvalue sharp drop drop eigenvalue even transition ise value temperature small temperature ise temperature spectrum trivial gap block end depend block expect order extract top eigenvector condition require eigenvalue matrix eigenvector respect matrix basis useful discrete j nn chain function basis extract eigenvector size contribution low e expect eigenvector transition spectrum extract contribution block separate block require among index spectrum basis exist eigenvector express b somewhat provide notice mild polynomial except small somewhat case devise simple stationary markov chain represent treat feature perform figure access mx input label forward solve let xx let eigenvalue correspond boolean function basis learn markov nk n arbitrary run polynomial give proof use fig markov whenever albeit consider arbitrary ingredient approximate time basis ask class approximate mrfs answer markov generalize product mrf boolean different take start take gibbs mc boolean noise respect mc alternative form noise denote sensitivity idea noise sensitivity e easily generalize mrf appendix mrfs example ise mc dynamic sensitivity respect family mrfs parameter every graph degree hold tf proposition lemma mc learnable desire propose polynomial field product check ise majority require fairly paper beyond large computing look three enyi model polynomial approximations eigenvector majority except approximation eigen eigen eigen tf ix w k tn nk pn low stationary distribution temperature respect mrfs sensitivity temperature ise family spin sensitivity stay paper place fact eigenvector office target ce thresholding approximation ise ii ce ising iii iv ce regression regression degree product localize small rapid condition sampler auxiliary make likely eigenvector localize pair close work one use work importantly require bar chart rmse approximate correspond degree variable gold blue degree green regression model bar four correspond random degree blue degree ise grid bar chart show rmse function group regression gold variable green graphical bar chart show function degree green grid interesting large eigenvector mrf learner body devote estimate mrfs use conjunction graphical model oracle suppose co receive section ok boolean cube track cause subset give variable abuse describe condition ise access identify fx fx jx I jj j reversible rapidly mix mc learn n x xx identify assignment abuse assignment event assumption least probability walk variable union unknown observe assignment zero unlike example rapidly variable cause happen show graph let let distinct variable configuration immediately rate rate color ensure rapid subset color valid define map modify condition let map element non block define notational letter eigenvalue representation begin remain verified checking j observe nk schwarz v since representation element position g deal base first thus minimize rhs rhs evaluate separately b next expression rhs recurrence observe rhs non set target function f exist function denote indicator position value rhs thus hoeffde bind g mx treat randomize arrange take chain treat essentially suitable form sufficient sum standard running theorem statement proof type max term depend sketch jensen suffice claim show system let subset moreover sufficiently decay exponentially argument moment imply probability proof integer graph hold prove strong know ise graph model subset corollary node node cauchy analogously calculation since complete sensitivity eq system subset identically distribute let hold imply event thus conclude fact positively identically condition occur lemma remark california berkeley learn rich connection boolean area theory distribution rarely encounter family distribution cube markov tool investigation central uniform connection mrfs mrf learn fouri area rely uniform boolean cube paper random field seminal distribution major principal tool simple fourier cube rich class several function connection application sophisticated boolean theoretic property invariance complexity relate theoretical include randomization hardness elegant impose sample rarely ever distribution independence thus ask question question represent field set field area popularity field couple extensive study sampling question study problem state real ask mrf algorithms begin answer theory learn certain distribution sound surprising mrf imagine expand mrf mrf mrf seem I eigenvector gibbs markov mrf gibbs mc reversible eigenvector orthogonal respect algorithm straightforward potential study rate sample sampler rapidly mc perhaps surprisingly despite power view vector power transition power distribution time power approximate mc power collection whenever focus easy access part think correspond unstable evolution eigenvalue short evolution reason classical frequency part fouri signal obtain probability give elegant characterization intersection threshold noise function mrf problem receive mc
load three easily devise describe simple index mode make multiple repeat strategy contain roughly word nonzero critical index mode index array slice weight array slice slice weight nonzero slice case slice node weight naturally weight slice whole array index mode grid hyper cube repeat whole generate algorithm dependent sampling equally size containing utilize trivial nonzero entry depend algorithms size load devise strategy strategy uniformly location size mode without method location weight nonzero entry array help structure appear component ensure mode grid grid k th latent add posterior posterior use repeat several final prediction batch procedure procedure entry infeasible array predict entry needs infer prohibitive bagging aggregate collection fast prediction bag tree find element finally aggregate whole array view prediction I k u factor mean weighted prediction application weight average batch implement essentially bag classifier weak array monte parametric multiple ensemble closely function prior massive multidimensional treatment local information sharing divide strategy gp actually strategy easily readily conduct either tucker decomposition currently hand exploit array fmri suit array sparse exploit sparsity inference fast prediction u series sub tensor sub variate prior type array regard world multidimensional tensor decomposition examine answer third cpu gb ram disk streaming grid social network extract news website describe interaction way contain non extract email way relationship receiver time contain tensor decomposition pa tucker choose latent datum approach area fold nonzero fold fold randomly choose repeat training set use hyperparameter laplace generate run online kernel mat ern tune hyperparameter mat ern bagging prediction version well furthermore alternative scalability predictive world knowledge basis contain project triple triplet acc access source version company log resource file action action record triple action resource htbp ccccc j b acc ht regard machine dataset examine scalability machine latent result machine linearly ht acc dataset cluster intel ghz gb ram disk implementation adopt ern c cccc I cccc factor mode acc acc choose nonzero entry remain nonzero entry sample bag size also examine size keep note less thus versa running auc depend affect figure train small fast gp communication cost training decrease figure different strategy comparable benefit array ensure nonzero outperform consistently explore accurate prediction propose bayesian tensor random prior ease art fast descent train classical tucker computing edu edu research com tucker decomposition nonparametric model array powerful multilinear include tucker decomposition partly capability nonlinear array despite sound theoretical handle massive large massive enable training train model array read security achieve dataset aspect multidimensional array file drug array person predicate knowledge basis array three mode subject tensor value embed g drug g response random elegant multidimensional array array exchangeable array array justify theoretically de array superior predictive performance model tucker decomposition cp lead bottleneck operate main computer fast scalability easily even infeasible computational cost employ massive parallelism cluster graphic unit limit distribute explore avoid datum suffer limitation multilinear relationship miss datum principle way although limited scalability overcome keep scalable multidimensional array datum approach hierarchical enable gradient scalability array element impossible enjoy linear scalability number testing basis web project company potential high computational describe multidimensional array task present array tensor stack make possible multidimensional array multiple unit per gp factor factor k variance efficiency theoretical sgd naturally enable array increase break need introduce na update parallel inference inconsistent different representation show figure give observation element whole replace multidimensional factor factor additive variate latent factor assume variate correspond ht probit presentation handle array datum augmentation decompose probit py n augment factor variational sgd variational factorize inference kullback
output component parent rate branch obtain stationary standard description use formalism express description handle tp give removal marginalization pointwise symbol next fortunately unary factor generalization last emission fortunately symbol thorough rest get denote unary assign elementary index eigenvalue invertible assign string normalization infinite sum unary define eigenvalue unary eq path length sum describe marginalization product implement marginalization operation marginalization operation take binary factor unary operation potential eliminate graph operation allow factor unary define matrix negative apply move pointwise unary unary factor eq tp clearly equal first consider unary potential claim transition satisfy conclusion product pointwise multiply fortunately pointwise multiplication kind implement expression operation elimination string value model time demonstrate gain efficiency several practice time alignment potential leave explain detail short path term achieve run exponent multiplication note intermediate factor make quickly also cubic removal use follow note preserve sparsity address pointwise product easier implement matrix product preserve precisely simple entry form take store linearly unary factor operation binary factor intermediate pointwise multiplication size store section make additional part topology perfect binary leave unary potential leave binary potential equal upper analyze discussion figure pointwise kind factor marginalization size matrix involve invert even priori sparse run rest considerably tensor assumption call state triangular potential triangular enough main star shape triangular unary potential standard invertible next factor triangular free create moreover outcome guarantee whenever triangular marginalization pointwise result proposition removal proposition fact inverse triangular triangular l proposition q lemma equation triangular sparse matrix entry triangular column back algorithm get argument use factor execute inductive exploiting expression method several arguably building well algebra asymptotic gain build top package access optimize library gpu parallelization library last least tool inference close useful type problem handle building acceptance ratio problem subtree string internal except node resample amount compute subtree hasting ratio serve carlo particle update also macro percent string develop treat informative investigate dynamic programming practical view different model prove method consequence keyword alignment string graphical evolution go beyond turning assume pre alignment appeal leverage pure substitution bias induce conditioning single alignment require score large trees monte mcmc bayesian search frequentist sequence tree computation sequence derivation notion fix observe priori infinite create challenge situation sharp contrast alignment branch idea composite develop previous new algebra obtain expression marginalization worst improve marginalization popular triangular develop new inference star tree perfect coincide extend rely assumption detail composite generalize tree related work include see outline sequence describe exponential approximation via mcmc two mix move matrix transition reconstruction prove previous string differently root branch shape string alphabet evolution conditional marginal parametric set arbitrary sequence bottleneck base maximum unary generality stop let triplet emission call forward step define triplet sum path string list character modify pair symbol path p q ns concatenation character remove similarly character emission string ab path set normalization review require formalize weighted organize shape previous show address section problem graph unary factor elimination break complex graphical elimination apply find value factor take space string perspective topology normalization potential normalization whole
survival accomplish survival volume extend contain contour recall integration survival function evident inverse integral evaluating likelihood deterministic sequence fig quadrature rule unknown must mc volume contour evidence nest perform live draw unity low live replace draw volume variable draw uniformly live prior iteration thus terminate contribution live tolerance contribution estimate amongst current live ns estimate posterior use live discard ns low assign summary normalize course similarity ns path reader importance refer overview branch modern science statistical main challenge computational ns constrain address rejection scheme prior suitable live possibly overlap replacement point live volume constraint accept ns run chance ns necessarily decomposition substantial flexibility geometry shape mode break relatively smooth broken relatively maintain efficiency simple identification mode subset identify distinct mode uniformly union particular ellipsoid check accept new account possibility empty intersection reject summation choice higher ellipsoid lie outer true marked drop maintain dimensional operate mode volume link volume require union efficiency keep every live ellipsoid despite chance fit experience accurate purpose ns value upon expectation ns evidence summation estimation highlight mode contour live particle sampling use ultimately pool ns likelihood nevertheless evaluate cost alternative context use rejection regardless constraint scheme importance sampling point iteration union indicator return decomposition extent dependent technique reverse logistic regression bias pseudo e iteration decomposition consist one ellipsoid volume ellipsoid however live point set possibly analytical calculate volume available mc whenever ellipsoid probability volume uniformly choose ellipsoid calculate volume note procedure evaluation thus computationally demand reference pseudo importance sampling collect posterior probability importance nest sampling rely posterior summary ensure draw feature difficulty constant efficiency I store describe decomposition centroid eigen eigen vector cholesky eigen store latter volume volume bounding assume iteration collect iteration current contribution come memory requirement need store draw account rather discard reject would contribution importance come iteration lie subsequent lie inside follow summation associate govern cf sec run repeat achieve provide I set significant reliability evidence describe oppose importance sampling evidence would represent construct distribute pseudo uncertainty applicable subsequent dependent live default mode maximum successive confident sampling ever dominant indeed reason detail efficiency mode recommend check repeat simulation evidence variation algorithm bayesian accurately occur particle dot low successive iteration mode function q dimension separate dimension vast volume fraction detection narrow uniform show efficiency mode sec live total default volume therefore order true estimate list analytical sec dimension constant seen obtain default efficiency mode consistent exception efficiency mode attribute potential uncertainty b start become inaccurate mode inaccurate cause live cover constraint consistent true mode efficiency indicate sec sampling distribution vanish region parameter space vanish approximation volume encourage challenge live notice evaluation constant efficiency mode start become default denote low successive multimodal resemble un plot mode challenge calculate evidence accurately numerical integration fine space efficiently calculate evidence dimensional mode agree default discuss live obtain fig dot show successive evaluation case seed sample seed deviation away due calculate region discuss sec absolute low particularly evidence analytical panel contour credible region next problem orient center uniformly hypercube gaussian accord dirichlet parameter analytical fig analytical regardless live dimension mode obtain default show default mode constant consist mode hypercube even dimensionality variance asymmetric shape unity impose uniform prior analytical show analytical distribution contour credible consistent test live ns mode fig see heavy evidence substantially true accurate differ unit evaluation availability vast increasingly play physics recently variety area ability challenge calculate discuss scheme change accurate evidence estimate particularly efficiency enable dimensional evidence live target achieve accuracy speed recommend distribution drawback eqs requirement computer acknowledgement uv work utilize university service education ff fellowship ec arc grant arc fellowship amongst ns algorithm limit yet marginal reverse state marginal draw likelihood use infer relative miss recursive deriving consistency proof separability develop estimator hence variance estimator achievable importance expense introduce recently efficient density refine runtime manner dependent superior adaptive scheme cross entropy proposal ultimately discard opinion perhaps build density via ns pathway sec aim iterate within specify density represent sequence importance early proposal increasingly later achieve towards inherent summation entirely approach elegant detailed successive grow significantly odd heavy proposal numerous assumption proposal historical particular follow variance strategy rough assumption behaviour impossible practice worth ns section ns rejection bound evolve live ns understand particular derive nested sampling ready ns experience text content draw ultimately likelihood reliability moreover subsequently various relate towards marginal begin pseudo certain responsible background last outline heuristic break term origin investigate convergence behaviour description text take ig ic proxy meet flexible cf assume live volume say point replacement ns live fact pool draw pseudo sampling proceed reverse logistic bias support live point live advance uncertainty mc simulation call arbitrarily improve negligible label marginal pathway density carlo sampling normalise encounter worse alone contain single optimally ensemble focus demonstrate order separately importance mixture simplify counter single point independently label density identical result set three draw independent inherently appropriately also unbiased estimator k specify e satisfy triangular decide independent sequence knowledge sensible option guess unweighted variance triangular univariate hand independent draw might follow come alternative n f h density univariate importance relative derive replacement proportion latter surprisingly recall course observe order label really understand analysis strict explain strategy combine modify sampling separately would convergence former much consistency nature decomposition limit stop construct enough limit every rational imply various sure em mean past intractable dependence obvious clearly complexity perhaps availability convergence volume convex sampling regular hand constrain decomposition ignore assumption ns suggest insufficient likely statistical perhaps matter entirely necessity limit ensure requirement exist proposal ensure acknowledge decomposition draw realization treat indeed draw represent ip asymptotically draw constrained sampling ultimately say towards supposed confirm law number draw convergence determine class simply three distinct live break variance inherent might thorough meaning significant binomial ordinary unbiased draw parent distribution additional apply likewise unbiased suppose
triangle two display error show isometry frobenius though currently rip relate norm high proving rip use remainder combine last assumption desire conceptual flow similar particular feasibility setup regularize parameter several concentration arbitrary live cone condition stochastic demand rip frobenius eq invoke rip cone bound step eq convert theoretic hypothesis test particular definition regressor pair pack q rhs minimax error suffice rhs packing separation cardinality information exist pack condition remain upper calculation obtain eq conclude two efficient match demonstrating rate several immediate currently optimality relaxed extension component optimization complexity tailor bring compare generic third challenge carefully future thank chen topic acknowledge nsf grant algorithm imply output recall section key perturbation theorem subsequent require perturbation inequality elementary eq perturbation guarantee version end triangle q follow stochastic thus noise stochastic stochastic recover let inequality combine q bound last rhs outline arbitrary technical step combining imply imply prove appendix coordinate w q quantity spectral norm ns write matrix bound support h sub boundedness nz suffice prove statement triangle infimum prove throughout define hamming nc pi q mutual follow establish minimax regressor reduce variable note I need distribution j rhs suppose lk I universal q schwarz inequality lemma sufficiently use estimation problem test obtain part vector lemma f k pr similar rr increase generality function sufficiently surely q distribute note distribution moreover norm equivalently p e variance suppose program bb event program conclusion w original coordinate h last coordinate p separately e prove high bernstein q boundedness bernstein inequality p inequality bernstein boundedness matrix bernstein conclude lemma need packing exist set also satisfy contradict index sequentially suffice density expand rh distinguish case sign moreover follow obtain order expansion origin know odd b combine display q gaussian universal cauchy prove algorithm claim chen edu edu university edu adversarial convex solution arbitrary noise bound assumption theoretically tractable tight mixed regression easily without significantly challenge mixed fall difficult one algorithm near effort recent widely particular specify solution regime balance half come match optimal arbitrary noise observation regressor produce estimator satisfie imply noiseless stochastic noise balance necessary ignore stochastic bound show convex optimization information theoretically particularly bit subtle fact phase thus qualitatively different behavior parametric broadly use array popular broadly problem mixed maximization domain still beyond exception consider mix noiseless minimization initialize grid recover noiseless extension focus noise cf initialization notable adapt sparse regressor likelihood em achieve optimum tensor author efficiently mixture yet approach order magnitude tensor intrinsic set component many interesting mutation gender child example theoretically minimax unknown component finally regressor identify future response example molecular response pair mixture obtain regressor precise basic vector satisfy recover regressor interested bounding regressor noiseless set label key insight lead basic concentration tensor allow recover indeed first eigenvalue similar approach give p fact stable close output formulation main noise quadratic arbitrary turn consider noiseless result immediately exact remove adversarial stochastic theoretically derive bold letter capital th matrix norm nuclear sum frobenius define repeatedly say independent ease parse constant entry covariate np sub assume constant vector assumption noise possibly consider intuition noiseless substitute desire program encourage result precise theorem summarize noise close provide quality produce exist shown hold recovery see rademacher possibility value distinguish possibility bound main namely substantially want ie say violate handle trivially suppose ordinal show blind optimal really restrictive interesting finally asymptotically observe moreover recover optimality number improved consider covariate balanced obtain noise analytically particular lagrangian square thus possible term first close constant satisfie square objective formulation balanced program hold proportional b nn ignore three theoretic phase sub come complexity show subsection minimax estimation stochastic setting show satisfie eq regressor first lie n follow noise follow least upper given establish noise hide label hold absolute constant c three theorem factor prove minimax eq notice phase ratio snr medium snr transition slow illustration power problem attention set noiseless focused add many measurement phase lose correspond model oppose
shared repeat numerous vary simultaneously solve plot computational relative simultaneous rapid rise speedup return interestingly without exploit share across amount simultaneous apply eeg example instance solve bootstrappe sequentially derive trial trial bootstrap regularization parameter varied included converge exceed bootstrappe eeg along regularization template element iterative converge note step show simplify correspond may radius maximize definite denominator guarantee psd admm expand k w v tw thus upon compare expand update update repeat solution solution equation imply column span et department engineering university york ny usa many bootstrappe cross nonparametric permutation improvement problem sequentially conventional generalize world intensive bootstrapping permutation analysis statistically classification http edu generalize linear learn reality application involve highly relate parallel cluster improve common throughout refer model may show simultaneously array primarily linear elastic arise regression elastic net regularization parsimonious problem scenario build alternate admm optimization splitting divide simple procedure minimize univariate soft across key minimization solver across problem template hessian inversion simultaneous newton step algebraic access screening derive amount memory linearly example overhead simultaneously regression thousand background pseudo algorithmic step usefulness algorithm application may categorical attempt importance measure prediction trial improve interpretability paper derive elastic remainder section brief introduction elastic seek solve loss regularizer assume datum specify list information emphasize dependence weight often dependent predictor tune convex however goal problem generally log weight clarity subscript emphasize variability trial vector adapt bootstrappe significance fall bootstrap validation log exclude validation permutation fit dataset seek objective bootstrappe permutation utilize characterize arise bootstrappe column contain weight illustrated weighting matrix major overhead extension regularizer simplify remove differentiable portion also highlight simultaneous major complete share efficiency objective commonly optimize quadratic around respectively evaluate hadamard entry equation invert iterative typically prohibitive assumed point take share problem lie range column qr factorization orthonormal column express convert system project span scenario exploit share algebraic expression briefly highly invert template incur template variability low numerous exist space voxel trial orthonormal trial weighting lagrange multiplier discuss regularizers move previously admm flexible accommodate regularizer elastic possibly follow lagrangian minimization differ proximity operator useful elastic net result thresholde proximity group allow one cluster distinct encourage exclude voxel define would encourage voxel contribute grouping denote feature belong penalty q univariate thresholding block coordinate thresholded implementation available online specify lasso whose proximity operator computable close operator form differentiable specify require see replace quadratic entry analogously greatly exceed problem identify pattern brain marker cognitive state relate present result briefly experimental dataset functional eeg fmri subject trial present hz sound stimulus experiment fmri collect preprocesse find decode stimulus category trial fmri since category logistic acquire fmri voxel eeg experiment force discrimination present image face car house phase coherence present corrupt
overlap take sentence datum hz hz band limited quality speech expansion set speech nmf decomposition metric aspect reconstruct speech sound originally measure human audio base poor fair good excellent short clean reconstruct predict sound correlation band theoretically range practice average report improve band limited speech fairly predict sound quality multi eight sentence remain give frame frames datum sentence report simple median smooth equally table capture accuracy alone complementary relatively c frame filter use estimation demonstrate improvement bandwidth serve use building probabilistic nmf variational em fit speech leverage recent development much speech section variational lower keep part take bin reflect new york ny generative audio spectra filter spectral classic filtering replace decomposition build signal processing operation learn statistical inference model derive mean free potential audio bandwidth task successful audio processing number simple widely use decompose audio broadly decompose analyze independently broad successful sound rest etc linearly come traditional typically decomposition basic transform cosine transform drive approach inference traditional filtering approach audio spectra filter spectral select convenient induce decomposition observe filter filter audio spectrum result compact interpretable proceed first rigorous review variational potential audio processing bandwidth task negative factorization nmf frequency coefficient audio collection magnitude spectra audio signal audio signal frequency window magnitude fast within assume roughly stationary speech audio model fold filter element spectra another attractive symmetry impulse filter convolution mathematically could implement fold spectra approximately linearly pool filter impose activation encode expressive filter partition filter active frame two main classic filter statistical position inherently filter might classic model determine local factorial model determine filter factorial formally filter noise activation filter reduce control smaller draw multiplicative graphical ht alpha kind example string body system model interpret try non nmf nmf fully nmf approximately decompose product nmf audio largely induce also meaningful interpretation nmf component likely activation play nmf energy come mixed make tool address source separation although decompose audio nmf fundamentally frame combination correspond part hand log filter convolution suit compressive great deal factorization solve optimization add online handle model hyperparameter control activation explain validation select hyperparameter impractical problem arise use p enable fit unseen training want tackle problem derivation find p intractable inference chain variational tractable leibler kl posterior minimize factorize family lt tune marginal spectrum expectation generate optimize low ascent variational use memory bfgs l bfgs optimize lower bind parameter posterior inference independent break problem audio spectra carry maximum free formally solve variational maximize low free inference optimize variational follow accomplished finding close therefore respect optimize consuming row value etc show conduct assess evaluate infer miss unsupervised identification speech corpus contain speech hz eight american english reading sentence overlap bin magnitude except try order bind large variational low slow train parameter optima run variational em variational bind demonstrate learn small value preference filter less frequently since place filter rarely harmonic periodic filter figure tend smoothly suggest periodic tend rarely fact normally filter fold filter
sparse randomly create equal element possible variable generate storage requirement simulation continuous multivariate contain four curve tuning count well sense real change fp degree true degree true specify show even graph increase roc stay nearly roc vary average replication weight penalize roc curve denote simple regression variable include interaction consider ignore pattern setting graph subgraph subgraph easily happen overlap zero unique vanish fix subgraph discard one simple perform outperform interaction term twice allow achieve regular fail capture method underlie include interaction regular dense subgraph specifically completely connect node connect give approximately zero signal roc curves adjust subgraph compatible regression ii interaction term regular expect main effect present music comes design capture consecutive short group texture window base texture window mean deviation standard texture analysis keep coefficient overall amplitude audio coefficient readily interpretable standardized continuous result observation exploratory datum analysis usage stability sample keep edge least continuous label variable legend graph intuitive audio densely interesting continuous feature label music circle seem reasonable find short circle group label infer circle circle highly circle optimistic edge connect music circle circle circle circle drive circle reading circle sense modify previous correspondence generalize regression overlap upper specifically criterion separately particularly suitable general conditional go little application substantially general primarily exploratory reasonable ability practitioner range recently development manuscript restrict assume view restrictive likelihood approach discrete discrete regular forest stability regression rest specify couple subgraphs subgraph outperform theorem corollary graphical discrete extensively little graphical continuous variable scientific novel flexible represent fitting group structure penalty demonstrate extensive apply music annotation obtain categorical variable usage focus word group mixed variable annotation conditional undirected sometimes network lot attention node represent ise classical high deal kind continuous ise ise rarely practice complex varied frequently continuous variable characterize association fit likelihood arise modern graphical simplify special proposal simplify version parameter yet impose assumption structure penalty since mixed group parameter fast penalty selection setting start introduction conditional dimensional z canonical connect moment relate canonical serve expansion depend cg markovian respect z subgraph complete rest organize introduce enough structure model discuss propose music annotation audio section simplify mixed penalize overlapping separate regression lasso quite expensive appropriately rescale x binary part consider density via canonical immediately sum full gaussian way order higher binary instead allow interaction full simple cg among vary covariance discrete thus empty everything mean conditional depend add I I one log regression much graphical specify distribution z q describe since odd log via logistic predictor response variable give z original estimate depends denote encourage dimensional choice penalty determine correspond denote n z ij optimize tune tune regression simplify tune prohibitive another simplify overlap group regression proportional estimate although optimization regression jointly solve overlap similarly create difficulty selection overlap overlap intensive optimize overlap group penalty easy penalty provide example surrogate overlap twice make intuitive incorrectly parameter zero wrong incorrectly estimate parameter edge approximation problem feasible figure feasible region b since function lie feasible four exactly point regardless hold subset enough identify practice surrogate problem determine problem separate regression solve
figure limit converge power agree theorem ndcg hand figure demonstrate ndcg discount experiment seem limit easy whose ndcg limit ranking see well behavior ndcg smooth decay distinguish even ndcg depict ndcg proportion describe let discount ndcg function ndcg always converge wang thank fan wang long helpful lemma prove complete rely key weak ranking underlie say exist positive draw surely ndcg old almost everywhere prove q part theorem I expectation lemma give eq assume f say rank pseudo relatively expectation two moreover consistently ndcg close function satisfie cut ndcg surely notational straightforward pair bind fix event ndcg function step ny df almost surely next detail discount function well theorem similar idea minor rely rank hold sufficiently calculation omit detail proof difference old continuous hold claim two monotone merely rather old thus almost discount decay ndcg sufficiently prove definition rank clear sufficiently rank label least clearly rank label follow discount e ndcg discount discount much easy discount former pseudo first nf assume similar next discount continuous n p continuous follow old continuous fs almost institute information sciences china wang edu cn university china li lee institute china di com school engineering university china liu microsoft microsoft china chen microsoft com microsoft china corollary conjecture central design widely discount ndcg ndcg function ndcg discount converge ndcg success ndcg application deep propose notion refer capture ranking pair function ranking decide ndcg logarithmic discount rank discount ndcg concept show ndcg consistent discount decay ndcg ndcg discount agree application engine recommendation name situation want measure performance regression measure evaluate difficult induce possible ranking point application focus discount ndcg popular ndcg advantage ndcg document relevance allow binary relevance ndcg ndcg discount weight position care importance ndcg measure evaluation currently area rank evidence optimize rank measure ndcg promise ranking computationally inspire optimize rapidly grow study loss motivate theory machine consistency complicate ranking respect surrogate define consistency ranking show pairwise pd furth average precision reciprocal contrast ndcg surrogate use notion ndcg ndcg bregman sense ndcg good rank ndcg normalization discount cumulative gain formal please weight rank weight decrease rank introduce discount view decrease ndcg ndcg measure speak ndcg family measure discount application ndcg discount ndcg discount function appear retrieval cut top discount rank ndcg ndcg popularity ndcg mainly field empirical perspective benchmark insight ndcg list arise point sound justification discount smooth ndcg ndcg discount combination discount decay study ndcg address question study well ndcg hope ndcg ndcg number object asymptotic include statistic especially rank area auc p view relevance linear rely represent conditional work ndcg generate ndcg change object start ndcg logarithmic discovery ndcg go seem mean standard ndcg good bad system may serious common web deep ndcg study property good measure believe need describe motivate two well rank hope rank top commonly randomly million crucial assumption evaluation dataset accord rank dataset well consistent intuition rank one almost formal broad ranking however thing complicate ndcg consistent ndcg always converge ndcg ndcg discount characterize discount ndcg discount decay slow measure power consistent converge rank infinity cut ndcg give theoretical explanation popular ndcg k discount discount decay cut ndcg view choice definition contain set correspond large represent sequence object rank score list fx ny ny ns literature draw definition ndcg give tailor discount let discount cumulative gain discount define assume ds nd discount logarithm decay logarithm ndcg constant scale important ranking preserve ranking imply vice versa ndcg function class preserve call version define property prove addition finally point discount integer treat variable view real paper ndcg discount discount analyze cut ndcg complete ndcg limit number every ndcg converge ranking result ndcg limit ndcg rank consider ndcg deep power ndcg different formal rank instance distribution say simultaneously like rank measure rank standard ndcg desired extend general finite pair old unless ndcg theoretical justification ndcg ndcg strong ignore scale previous ndcg application ndcg evidence logarithmic discount ndcg measure think feasible ndcg clarity subsection complete ndcg utilize discount decay discount decay consider limit ndcg converge actually good ranking positive ranking rank ndcg discount power ndcg ndcg follow old old ndcg discount condition satisfy high converge limit deal limit ndcg discount ndcg ndcg limit ndcg condition discount describe limit rank continuous proof limit ndcg depend lower affect logical analyze ndcg however rank measure discount function decay discount ndcg tend importantly xy x xy dr bb probability function consistently ndcg feasible ndcg accord logarithmic discount function ndcg ndcg converge limit ignore scale discount discount strong discount decay fast ndcg ndcg ndcg ndcg motivation ndcg pay attention rank logarithmic discount ndcg ndcg state natural ndcg combination discount seem ndcg issue cut set appropriate partial discount infinity investigate ndcg
percentage list high often entity likely correctly estimate hyperparameter model development fold recall score achieve drop similarity hadamard ask triplet relation vocabulary entity otherwise order create example switch entity triplet triplet correctly tensor entity initialize hadamard achieve model represent average leave future work new predict relationship basis entity via much performance answer thousand unseen database external resource benefit corpus even ref pt ref pt align cm ref pt align cm align leave chen ng computer stanford university stanford usa stanford base systematic relational suffer lack relation large corpus complete predict additional generalization given introduce add database unsupervise fashion entity present exist classify unseen relationship accuracy resource resolution answer generally extend corpus accurately fact use entity database fact relate entity powerful distributional word corpora capture syntactic database manually parse resource relate unseen entity type entity triplet rank vast basis external corpus many contrast extension al implement model benefit initialization machine parameterize full tensor begin describe entity continue entity pre word vector model free wikipedia text learn window co result word capture syntactic embedding see entity multiple word replace bilinear relate entity vector compute plausible entry advantage model input nonlinearity bilinear truth minimize q triplet triplet entity replace entity triplet corrupt respect l bfgs setting entity triplet compare al scoring embed b
forget table conv maxout mark black dash forget conv output maxout new height major gray legend entry cat legend pos north west legend cell leave font white ylabel xlabel vertical align none table px none table none green table px mark none mark forget plot table px maxout mark red forget plot maxout mark none dash forget px maxout none black dash forget px major gray legend entry cat legend pos south legend align legend style none font axis background ylabel xlabel rotation angle align outside px conv single new conv mark px conv single px mark none dash forget conv maxout mark forget plot table maxout mark none green dash forget plot px conv output maxout none dash forget table conv maxout width grid gray cat legend pos south east cell align legend none axis white xlabel rotation outside mark blue conv none px conv mark px conv output none x conv conv maxout mark red forget px conv green forget forget conv maxout width height major style gray legend conv conv fully layer legend pos north west legend align style fill font axis background style xlabel vertical align outside mark px mean x conv output mark none px none forget plot conv maxout dash forget conv maxout black px output maxout style legend conv fully legend pos south east align left legend style background style ylabel xlabel align conv mean px conv none mark dash forget x conv maxout none forget maxout new dash forget table x maxout rgb cs variant deep network attribute favorable dropout maxout operation partially change ask maxout unit successfully achieve balance claim benchmark cifar stochastic recently tool supervise performance range large behind activation network probability extreme bagging among train testing efficient possibility million propagation stochastic connection drop dropout decrease convolutional pooling operation change regularizer function dropout maxout generalization unit suit dropout maxout partly attribute maxout inactive cause perform piecewise sigmoid contain maxout unit easy maxout activation mapping pooling maxout partially natural question arise beneficial replace operation maxout pooling pool subspace pooling operation unsupervise g give rise interesting generalize maxout replacing comes discard desirable maxout unit piecewise linearity restrict unit regime maxout unit preserve improve propose maxout unit evenly among flow maxout feature mapping maxout unit utilize subspace consist unit match art maxout briefly neural feed class compute dimensional vector desire h l activation n n b n probability label introduce formalize activation previous activation maxout first mapping sub unit maxout maxout unit k mapping formalize clear contrast conventional activation maxout interpret maxout input map compute pooling operation maxout similar pooling ica within input observation maxout preserve desirable improve invariance property generalize unit since maxout activation maxout replace probabilistic sampling boltzmann mapping activation define probability k refer control activation mapping unit preserve selection maxout behave maxout dominate activation differs select almost probability active chance sampling therefore flow unit hence argue utilize dimensional subspace practice combine unit directly q consequently dropout explore spatial pooling author probabilistic order distribution unit activation probability location unit form location difference forward calculate binary unit closely sample pool unit activation activation note embed autoencoder back neuron generative operation discriminative need account stochastic nature network unit sample one forward network possible test clearly infeasible dropout deal amount dropout modify perform average perform relu maxout remove dropout dropout softmax layer evaluation average axis background fill white ylabel xlabel align bar cd curve likewise regard curve b cifar dataset change average label increase subspace unit close together invariant effect maxout achieve sample maximal activation verify learn unit learn consist maxout fig see belong seem transform learn maxout encode invariance however support extract maxout translate image validation normalize euclidean vector extract unchanged image randomly image experiment fig introduce moderate positive rotation describe depict moderate reach conversely conjunction maxout maxout deal try replace mechanism replacing result decrease reach try weight even achieve cifar benchmark protocol image contrast whiten train example proceed stop decrease epoch take reach method conv maxout conv net conv net maxout conv experiment version compare convolutional layer respectively pool top show slightly statistically tie maxout also additional image well add randomly translate cifar augmentation augmentation maxout contain cifar image million cifar content cifar cifar image group super per class less class cifar super similar setup cifar preprocesse procedure epoch section ensure evaluation sample cifar maxout carry mention substantially run experiment result cifar super could achieve prior view house digit google pixel task classify digit task considerably contain digit digits task conv conv conv maxout conv test contain less additional validation extra training large image consist layer pool
adaptive markov jump process auxiliary estimation filtering framework asset allocation typically shall asset asset optimally portfolio asset regime regime discover asset market derive continuous portfolio choice approach hmm hmm strategy portfolio decision asset trading herein find portfolio growth stock asset lead market contain handle issue financial peak asset shall return financial time belong separate within hmm flexibility however handle care reflect return describe asset allocation stability filter filter measure robust filter section conclusion review approach model stock index discretize geometric brownian stock hide chain directly govern switch regime chain associate canonical j I k p v theorem markov return price observation state space constitute k yy theorem martingale use go filter technique change chain dynamic k observation outline filter relate back world f density normal filtering subsection fast update partly keep keep filter jump processes filter determine adapt stochastic reference filter adapt h h k h general adapt adapt ff give column derivative state three jump time special first jump chain j sr k get spend finally auxiliary get estimate calculate adapt filter chain assume estimate deal obtain maximum adapt whenever information available update update denote process estimate em run time update batch comprise filter ml presence first event occur observation recurrent additional ideal distributional closely otherwise closeness distributional sense g fit distance kolmogorov ideally closeness compatible usual limit topology closeness ball distribution conceptually random eq variable distribute accord distribution switch setup situation distinguish outli innovation outlier observation consistency literature term stand layer distortion correspondingly sense general outlier propagate simple generalization variable law eq sequel instead distribution model suffice also entail error symbol interpret drop due something change fast implement index market benchmark stock batch ten self value mean ten point figure original step ahead time considerable include time filter em outlier see find affect outlier third severe plant severe filter set severe asset certainly outlier might due wrong price short period outlier overcome effect like need filter topic concept robust need optimally robust continuity closeness fact serve stability quite already context filter functional parameter simply distribution prediction range functional rather argument case topology compatible translate continuity context continuity call call reflect expression moment bias denote euclidean also robust close capture massive deviation call maximal cope usually estimator mle circumstance robust globally hand pay certain stability ratio covariance need neighborhood maximal mse ideal estimation respective ideal bind respective different want robust idea add argument observation estimator weight deviation consistency estimation gaussian c empirically e j possible achieve outlier need outlier place weight asymptotic recursive neighborhood consider reconstruct ideal realistic contaminate neighborhood minimize maximal mse neighborhood I measurable reconstruction minimax appealing interpretation unchanged observe e something sense hence unconditional much tend take accord keep unchanged modify measure avoid non use observation tune eventually estimation factor denominator irrelevant subsequent aspect want situation last initialization part filter filter replace suitably though preferable crucial individual influence filter kt filter value give store building observation time within grow triangle would increase memory put onto need classical neighborhood variance usually bias appropriately grow dominate grow avoid shrink shrink indicate increase rate optimistic one rather shrink defer respective determine optimally robust achieve arise normality differentiable score estimator q influence sequel fix notation usually step r n asymptotic unbounded hence detailed come equally contribute pass sum weight scale become location scale package eq illustrate panel location positivity maintain essentially first compare length nothing outli course I justify specify therein give weight sum optimal filter likelihood log hence I ik analogue way k x l irrelevant term minimum therefore estimate batch achieve replace square value scale weight weight construction start absolute eq name scale weighted cdf batch square integrable latter two boundedness square would develop triangular state influential filter run ten determine calculate determine save batch numerically cost general batch recursive burden store offer diagnostic purpose em respective parameter go capture individual information respective much estimating addition observation mle identify outlier fitting implement plan release build package thorough detail quantitative qualitative cope figure robust parameter differ behave estimate remain desire essentially outli done understand avoid aside already contrary financial time financial database peak market conventional handle contribution analyse em hmms highlight occur extreme initial hmm reference normal second observation build robust iid nature attribute observation situation lead directly reason complete algorithm keep store filter additional burden though term diagnostic purpose markov use keep characteristic arise include forward loop forecast asset estimate decision asset forecast handle switching regime occur market obvious generalize mixture idea respective weighted weighted future translate asset portfolio optimisation
base structural help solution correctly solve simply add variable box derive finally derive recovery polynomial relaxation condition mutual mutual q mutual coherence hold nonzero equivalently restrictive assumption consider submatrix n e hold correspond sect whenever mutual implicitly case state result characterize simple method solution solution solution conservative exact tight appendix proposition appendix proxy solution solution e assume contradict unless constraint j add precisely unique rewrite meet define index correspond nonzero due introduce rearrange rewrite remain unique note addition feasible include problem share unique polynomial exist either contradict imply replace j unique imply condition finally describe towards fail yield common practice improve repeat weighting improve sect sect sequence group sparse refine assume force towards sm zero though due recover solution scheme fail absence tune sm solve nonzero polynomial basis pursuit sect major solve intend combination base remains design limitation apply implement polynomial feasible otherwise initialize ki build submatrix repeat otherwise index least l combination soon solution tn n complexity complete returning soon could uniqueness could similar explore branch combination implementation retain initialize ki define minimize error update start add retain add retain guarantee equal return variant ls total bounding bound complexity find sparse sparsity rather promising sect occur purely nonlinear part e order index correspond specific case hold unconstrained sect however core purely polynomial solve sect obtain sect j actually involve small odd determined case solution analyze sect usually literature dedicate purely many equation relaxed interpret term threshold relaxation sect derive denoise lead lead still solver apply sect regard greedy sect noisy solve form unique focus stability recovery stability minimization hold hold result diag let matrix part path adapt norm must satisfy obvious rewrite satisfy zero due entry norm yield inequality upper group multiple derive upper box q let constraint positivity evaluate accuracy time noiseless sect sect equation implementation convex program except nonlinear iteratively counterpart sect selective sm sect approximate sect sect quadratic equation sm sm success rate noiseless follow experiment accuracy define recover system meaning rest draw accord quadratic aim recover component almost trial failure long method enforce recover truly solution polynomial show set easy method obtain reweighte bp achieve rate fast sect nonzero typical fast trial long handle optimization offer rate sparsity difficulty obtain gradient sm success sm time purely discuss purely quadratic table modify belong solution mean high estimate cube success sect satisfactory result greedy retrieval reformulate trial mean unit show work note many propose limited generic phase moderate set low greedy issue relaxation despite name solve polynomial system satisfied guarantee general show recovery success grow become whereas successful polynomial recover sparsity system equation information expect success increase constant sm method still recover solution sm comparable yield method effective obtain recovery sufficiently sm nn right sm focus favorable left plot range similar benefit much fast time highly clearly fig compute sm reach equal iteration exact suffer much contrary variant obtain server equip accuracy performance zero recovery correct precise estimate sect form db polynomial reweighte bp group lead almost trial despite presence sm error success sm method let db plot error bind even interestingly rate estimation provide evidence structural knowledge satisfactory curve influence perform method except greedy error fail perfect much large bp denoise sufficiently polynomial generic convex relaxation greedy relaxation sufficient noisy propose numerical relationship success sparsity addition indicate accurately case sufficient restrictive remain greedy approximation towards problem constraint another restrictive convex relaxation future nonlinear taylor expansion acknowledgement nsf project system center european research grant contract grant foundation fellowship corollary universit de france electrical engineering computer sciences california berkeley usa electrical university deal system polynomial equation possibly particular recovered group approach result cone programming formulation polynomial noiseless stable noisy second approach algorithm greedy short accurate analyze relationship ability solve system regularization recover make popular minimum however become minimal equation numerous e review entail great importance process name sense write minimization constraint nonsmooth np two distinguished problem basis pursuit rely relaxation greedy add nonzero result introduce equation bp develop solution nonlinear q deal taylor expansion entail
view function precisely good minimize allow abuse wish scalar th require inexact monotonic fx v complexity residual eq block inexact update calculate impossible example computationally update successfully outer iterative show via cd mechanism break piece total moreover subproblem huge iterative scale huge problem excellent scale update subproblem update overall keep allow update insight role progress stress couple technical assign assumption sufficiently guarantee discussion magnitude allow error level show error allow inexact block tolerance block moreover block inexact sensible inexact level multiplicative allow multiplicative fix algorithm block iteration ii block long assumption correspond inclusion update obey iii block satisfied multiplicative may become dominate tend show relate criterion update give numerical inexact specific instance order need incorporate error optimal every subsequently multiplicative incorporated subproblem example subproblem solve inexact terminate duality verify accept dual termination criterion iterative inner update bind stop tolerance loop stop frequently follow result play key role let function far hold k notice thresholded sequence trick hold case uk simple shift detail let monotonicity apply surprisingly iii decrease arrive q notice make small force take assume argument lead c take comment theorem finish process generate choice apply complexity may recover recover last also ignore result inexact guarantee confidence achievable bad bound arbitrarily arbitrarily see small restrictive ii analyzing form analyze c analyze yield central relate update vector inexact n ix ix ix ix hx fx hx n current f fx ix use remainder letting fix convexity assume hx fy x fx fx minimize decrease objective initial iterate apply parameter tolerance hold fx result apply ii together apply theorem f hx fy fx last final objective strongly error tolerance fx follow simplify result smooth furthermore provide expression ix ib substitute decrease iteration apply confidence tolerance fx substituting remain use minimize rearrange strongly strongly convexity iterate target fx method fx notice second employ function smooth eq set determine positive moreover assume zero make solve gradient cg improve appropriate cg definite propose th rank identity fast algorithm justification appendix expect cost see block technique cg inexact exact form cholesky two simulated column value stop tolerance ax update block I moreover block first experiment block decomposition drop tolerance experiment table result average experiment ic incomplete perturbed find tolerance table average briefly terminology time represent cpu second update divide epoch product token cg update block cg cg cd cpu block approximately time exact cd size notice cg demonstrate result c cd cg update cg cm cd instance cd memory token cg cg preferable return memory token require form cholesky store cholesky dense expensive cg extremely test quadratic angular world particular block angular transpose case scale matrix compare exact cd cg original c numerical show determine block cg case cg cd fraction exact cg exact c cd cg iteration problem fit ta nb il probability algorithm regularization update mean inexact find inexact update terminate section solve terminate duality gap conduct convex purpose experiment level make fair test order advance begin store index uniform correspond block case use plot plot perform simulated particular problem description termination clear without show inexact iterative practical advantage theoretical justification speed iterative purpose matrix eigenvalue section study matrix setup apply matrix prefer work semidefinite nonnegative rank say block investigate block break part part consider case td full rank contain I td strictly thin qr factorization orthonormal triangular ic tc iy I ir ty iy iy iy z full rank orthogonal suppose angular structure rectangular full diagonal strictly sum row basis subspace n ic iw iw j iw I ta define see rank state full row define eigenvalue tend I I v tv j let partitioning basis expand jj j I w demonstrate importance value small arbitrarily hence trade bound equal tw w tw ii definite great proposition corollary exercise theorem support centre software ep exist assume relax allow subproblem solve descent incorporate good update guarantee consideration acceleration inexact conjugate encounter become popular underlying application arise successful compressive matrix choice result particular purpose study randomize block form smooth nonsmooth convex lipschitz concept precisely namely inexact support produce random iterate inexact high guarantee explain condition detail show example subproblem subproblem encourage gradient expensive method gradient system equip iteration bound surprisingly algorithm gain main serial descent traditionally scheme study coordinate select useful approximate inexact update consider inexact gradient consider inexact smooth work block employ potential reduce running benefit inexact update inexact exact inexact method mm
period second term randomness transition mdp expect ks ks regret bellman bellman bellman introduce empirical km pointing try optimize careful decompose show also go compare optimistic use well randomly mdp provide state every set dotted represent mdp arrange show begin time successful agent receive reach policy attempt receive exploration require mdps environment accord prior prior term respectively environment simulation optimize account appropriate outperform table show regret carlo extreme problem interaction episode discount factor appeal feature optimal policy fix period episode episode horizon show figure remain establish provably motivated reinforcement algorithm irrespective conceptually incorporate feasible optimistic believe efficient statistically perform domain strong wide upon acknowledgment support stanford award family q regret accord impossible asymptotic frequentist regret theorem ks say ks ks happen ks ks ks ks ks ts ts n since absolutely lemma van stanford stanford stanford stanford provably reinforcement poorly understand action encourage duration update sample optimal simple agent natural state reinforcement algorithm regret interact try reward accumulate environment model process mdp uncertain mdp environment observe learn fundamental exploring attain na variable suboptimal exploitation offset provably encourage model high statistically plausible optimistic exploration since poorly state high effect optimality strong optimistic reinforcement algorithm guide exploration provide agent choose episode reinforcement learn policy sample environment episode select accord optimal variance sample oppose successfully multi armed bandit refer despite history largely multi armed bandit empirical variety theoretical great potential reinforcement dynamic appear know theoretical guarantee approach guarantee introduce complicated mdps combine exploration visit show always complicate originally satisfie solve optimistic simultaneous across computationally attempt explicit allow structure crucial exhaustive separate optimistic influence past facilitate theoretical toy problem analysis performance addition naturally optimistic possible state horizon episode posterior condition history compute episode obeys demonstrate multi believe offer inherent advantage optimistic construction confidence base complicated optimistic policy intractable resort bound allow mis simultaneously action pair rise set far conservative select policy probability optimal policy quantify approximated believe implement optimistic regret distribution mdps generality result prior sometimes bayes literature bad case link notion markov show appendix bound regret similar satisfied algorithm rl give ts far tractable interested learning task constant improve episode produce bound dependence observation episode identically relate depend mdp mdp fully history reader theory think know variable measurable contain posterior measurable optimal reinforcement clean way relate policy
imbalance training commonly imbalance inverse rule cognitive bias prefer different mainly project remain database database task fmri dataset object task arithmetic database account subject activation individual type avoid bias procedure parametric multiclass computer also protocol cognitive process cognitive paradigm aim cognitive fmri experimental characteristic explicit response specify experimental stimulus stimulus list category stimulus visual explicit shape digit track discriminate response none occurrence rise regressor visual stimulus modality mostly amount exclude forward capture negative effect comprise remain wise top clutter term primary visual stream map difficult specificity inference several phenomenon hard separate anti correlate inherent factor identically experiment interaction occur occurrence protocol orientation attention predictive principled go activity define cognitive process classification careful intend highlight map study share experimental effect use label use study previously unseen leave cross validation training highly general label half classifier naive logistic regression standard retrieval score representation inference specific cognitive concept solid derive come conclusion specialized accumulation overcome small cognitive assess study practice challenge engine curse indeed correspondence study go design provide rely label inexact bring benefit description enable simple progress recognition previous work multiple leave one factor across study state every state predict predict easier albeit little explore model study share cognitive subject bad subject partially mistake task drop illustrate necessarily subject place task certain degree include study common least study leave predict activation limit study give rise imbalance interestingly database study broad cognitive imbalance look close cite low database inconsistent term inference map work unlike regard hope paper show prediction paradigm description image acquire different cognitive domain prediction se pose foundation integrate many study accumulation give region support principle reverse map promise probably benefit significant region hope progress term cognitive mapping cognitive come database study bring concept grant link via base response incomplete causal come conclusion imply activation region necessary exploration various brain inversion introduce observed brain rely corpus study engine without contribute task cognitive corpus completely study brain imaging fmri systematic date mention accumulation cognitive literature module specialize dedicated face manually lack co challenge quantify alone incomplete cognitive region conclusion study demonstrate cognitive measure exceed single lack specificity comprehensive scale brain mostly coordinate base meta activation pool across activation maxima lie cover manual comprise text mining comprise paper occurrence cognitive behavioral inversion forward inference study image thus demonstrate principled study challenge face trend cognitive concept corpus well specificity sampling inherent coordinate meta bring spatial purpose outline strategy knowledge functional image provide reasoning brain co cognitive concept tackle challenge risk protocol choose describe study cognitive cognitive enable span across share cognitive challenge ensure functional specificity bias comprise different experimental brain fold result outline second cognitive paradigm study paper methodology establish reverse corpus corresponding paradigm section empirically predict description unseen study mapping discuss wide analysis fmri study result per response single subject serve stimulus explicit reading challenging capture cognitive study fairly unique language language general study engineer across affect condition effort cognitive concept formal taking object describe standard ask use glm voxel subject term observe voxel glm test voxel model response combination glm formulation effect thus specificity co corpus regressor term involve experiment activation map build description inversion go
review approach policy explicitly unknown environment explicitly learn generate advantageous scenario budget determine advance need batch collect period optimize schedule advance without strong thus schedule degradation suffer problem without gradient reduce baseline estimate statistically gradient reduction bias use expense increase scenario draw want sample independent sample separately use estimation baseline environment model approach fully accurately estimate amount challenging although transition deterministic environment range propose base incorporate exponential policy gradient policy overcome limitation approach propose practical base method transition state art superior directly input solution analytically system promise formulate review experimentally usefulness section rl review policy decision consist state density action immediate function pa take action action action pa determine follow transition discount history rl maximize classic ascent express expression give roll sample approximate empirical average empirical reduce gaussian gradient search policy expect return gradient update estimate proportional critical limitation history long limitation call recently exploration drawing policy hyper thank formulation drastically formulation expect represent hyper trajectory formulation roll policy pair linear parameter consist gaussian derivative rl method previously iw extension collect current collect policy hyper hyper current hyper collect parameter collect current policy useful technique define baseline iw perform base counterpart want transition cost transition estimate gp propose scenario mp learning rate consider approximate review review transition noise transition gram denote together noise determined maximization gp method conditional parametric conditional estimator conditional restrictive overcome linear parameter reduce use necessary follow minimize ac element include add regularizer avoid objective solution identity solution conditional modify dimensional zero phase input analytically compute true mini method outperform asymptotically matlab experiment illustration purpose walk figure receive episode discount rate use linear parameter policy dynamic deviation transition give q sign randomly three estimate weighting budget transition initial uniform next reward obtain immediate repeat process trajectory gaussian profile learn artificial another baseline estimation learn policy sample figure reasonably poorly illustrate iw method iw method schedule collect scenario illustrate choice affect iw schedule value iw policy well update figure schedule figure schedule use optimal schedule iw return iw step iw policy begin begin iw may improved scenario return throughout keep without cost illustrate ps mm ps evaluate performance practical simulated body simulator figure lead simulator base roll roll roll controller receive real value angle angular dimensional angle degree action vector mean initial straight reward hand sum multiply policy episode length discount rate e allow time send simplify iw distribution th vector distribution artificial sample control policy iw method preliminary schedule time yield high show outperform iw reach motion step noisy mm observe batch confirm perform horizontal method move compare learn iw include outperformed setup essentially budget complex dimensional robot reach right hand gaussian iw set return iw plot show art iw motion iteration policy distant successfully policy observe iw horizontal overall
channel operate ghz frequency technology output multiple receiver consider virtual virtual receiver base magnitude receive b magnitude link stream one figure virtual link presence adjacent virtual still model mixture lead reduce take treat profile link figure image prove efficiency capture location section give detail system architecture block represent module operation person receiver virtual extract feature system unknown implement magnitude send virtual value construct filter offline area phase associate entity location space area wireless send physical receiver pair stream traditional human stand location record mp profile virtual link receiver allow mp discuss discrimination location capture adjacent recognition virtual link use reduce discrimination adjacent profile rectangular mask sub height range magnitude figure size uniformly range location number fall filter haar feature space well associate represent trade reduce select good classifier phase extract train location adaboost classifier weak classifier misclassifie classifier focus weight adaboost select discriminant far computationally weak decision take implement check extremely select feature discriminate large joint logarithmic location avoid fit df wireless channel purpose module actual process confidence location classifier classifier approach want show effect median figure combination lead different noisy wireless channel different receive news determine rest increase accuracy tradeoff need reasonable draw deviation increase sub increase tune median distance error number filter filter reduce median overhead number boost linearly boost take value compare error system probabilistic traditional df stream median state different show advantage c median ms ms ms ms device track camera system system area camera base location sensor system sensor interest apply technique track special hardware df scalability area aim experiment control several entity probabilistic minimization combine spatial rely mac stream provide accuracy system information approach since profile location lead adjacent adopt allow adjacent location three leverage rich information large variation allow stream achieve accuracy increase increase employ enhance trade haar comparison adopt boost select overhead reflect run training boost whereas entity extension straight forward entity localization aspect practical handle dynamic require area capture dynamically store e technique tool df g device passive localization layer base minimal profile adjacent df localization receiver evaluation error location highlight time df currently expand direction integrate entity entity corollary device df localization technology entity device localization process df number stream df localization accurate df stream df context side art achieve accuracy least median error less highlight usage computer communication year gps require entity carry device device df localization wireless track entity device localization fact motion localization medical care monitor server receive current df localization computer image hardware df operate network therefore service signal mac layer introduce hardware device localization stream ap reduce
e jt j u combine posterior datum devise simple proportional long recognize functional covariate influence inference reference play covariate topic receive primarily joint linear risk model mean longitudinal random primary produce accurate expect longitudinal process influence functional longitudinal predictive risk decrease summary whole longitudinal trajectory formulation longitudinal event approximate association event parameter even appeal parameterization patient marker level marker trajectory decrease capture depend current trajectory slope survival parameterization association slope longitudinal common longitudinal trajectory dependent cox author depend elaborate history vary extend cumulative predictor trajectory time survival particular point area longitudinal longitudinal take marker assign longitudinal reasonable close placing multiply appropriately choose place logistic degree freedom student marker old formulation joint random longitudinal association hazard parameterization assume longitudinal slope set parameterization outcome increase longitudinal trajectory slope respect share computational close baseline computation numerically disadvantage spline subject nonetheless model choice longitudinal event common base information criterion aic uncertainty scenario force addition respect prediction several almost equally produce accurate profile prediction subject whose profile account combine structure concern interpretation structure focus prediction survival longitudinal structure calculate measurement baseline let jt jt average survival denote derive denote compete observe classic even time posterior well subject risk probable association longitudinal trajectory calculation analogously equal come longitudinal closed integrate material priori probable return utilize record level survival section h h material sensitive assume association structure great difference patient patient joint particular weight interesting make year contribute prediction three practically observe little dominate though similar five variability behavior would sample specific prediction simulation motivate dataset year longitudinal nine follow longitudinal cubic spline knot place boundary knot place follow b b b spline treatment group I survival correspond h longitudinal survival effect scale baseline supplementary censor uniform censor scenario scenario regard longitudinal assume focused outcome specifically simulate exclude censor meaningful calculate individual since remain patient longitudinal one survival current value weight cumulative density scenario study investigate association assume baseline simulate hazard ten subject originally point available longitudinal measurement end scenario include model survival probability compare gold calculate j jt true subject simulate square three scenario observe correspond great difference scenario iv prediction considerably outperform careful produce consider seem equally average promising misspecification optimally bayesian novel prediction record thus subject account single adequate quantify simulation study perform include list average derive future patient five simple outcome often several outcome record follow record investigate analysis composite operation death treat estimate two event recent joint marker multiple elaborate challenge combination longitudinal survival address model concern discrimination question accurately predict survival discriminate lot reference therein joint relatively calibration discrimination specificity roc curve challenge supplementary material available setting cm mm cm rgb blue nan nan nan dynamic joint longitudinal averaging department medical health policy medical school division school public university department model research receive lot year attractive longitudinal advantageous dynamically extra longitudinal subject interest risk assessment record fold first association structure event response greatly second prediction suitably joint different averaging feature subject imply prediction subject risk prediction time covariate recent form medical care increase development disease example numerous cancer disease infect patient patient available risk repeatedly limitation valuable discard offer insight dynamic disease characteristic medical patient dynamically relevant disease prediction event e capability valuable medical understand allow make informed motivating patient detail human intervention accurate adjust medical death aim provide flexible utilize patient explicitly longitudinal framework longitudinal attractive use model advantageous prediction extra longitudinal record subject longitudinal contribution subject consideration exhibit longitudinal trajectory subject predict prediction affect longitudinal beyond longitudinal whole consideration compete process raise ignore collection simultaneously bayesian average organized background describe motivate research introduce present estimation derive joint section introduce several longitudinal survival section illustrate result research center widely disease report either date year advantage excellent characteristic substitute low therefore disadvantage lead risk patient complex substantial si patient replacement standardized take follow average per patient measurement require discussion aim exist construct accurate risk prediction future operation record death patient correspond intervention group figure difference sub end depict subject profile intervention skewness systematic difference denote event censor time corresponding observe otherwise longitudinal longitudinal longitudinal outcome process longitudinal specific longitudinal trajectory longitudinal design effect b term effect survival marker history longitudinal baseline covariate regression quantify association marker hazard need hazard typically subject survival smoother
euclidean whereas e translation sequel shall example investigation result metric line metric completely transform kind metric complete complete characterization translation generalization general expand line main translation q characterization translation historical recall general compact originally integer due recognize result play thing start arrive metric translation invariant sense equivalent trivially discussion proof require importantly kernel metric convenience normalize shall later relax condition aim theorem translation invariant positive definite integral bounded borel suppose real recall invariant translation apply arrive segregation characterize give metric upon segregation integral rearrange k dt dt variable rescale leave theorem give constant arrive example dt right extension borel dt satisfy establish line extend separable euclidean space write ik dimension characterization separable translation fourier borel k ik borel eq q k e kernel de I normalization specifically kernel ability real first proceed define might cause begin theorem prove eigenvalue distribution without allow real propose compact allow ce ce fx dirac claim ac lem lem lem claim lem lem lem look invariant metric alternate invariant definite definition discussion case letter small letter arbitrary domain define r discussion
minimize set autoregressive strategy regularize strategy efficient evaluate loss regret various regret notion compound direction static sense decision compound outcome arguably difficult perform period observe outcome action treat expert multiplicative expert popularity box must play role count notion closeness develop algorithm incur infinite try invoke expert show present compete idea expert author distinguish static non static expert static show rademacher average expert characterization odd notion theory empirical tool optimization directly expert view mapping grow space non constructive constructive technique analyze example admit surface imagine far develop interesting parametrize choice parametrize prior note competitive imply competitive ratio introduction build imagine source well tell may view outcome lift sequence generate goal perform indeed model small rely strategy prior assume autoregressive usual attain derive computationally section compete model compete parametrize example question regularize index shift parameter linear follow regularize parametrized schedule online round observe outcome function history outcome strategy regret loss cumulative loss strategy probability correspondingly constructive sequential show relaxation paper real prediction round observe strategy tree mapping throughout rademacher tt simplicity sequential supremum use name state strategy bound statement visually far satisfy rademacher hand non static lipschitz rademacher potentially dependence full history contraction extend furth sup sup warm constant strategy rademacher expert outcome associate direct rademacher case markov outcome determine move imply understand strategy rademach serve start admissible relaxation admissible sequential use deriving covering tree notion tree value strategy small cover employ cover number closeness away restriction rest yield proof rate guide development come provide fail fix consistency outcome zero recover extend neither literature suppose unit eq strategy descent gd z inf obvious history regret question prove gd easy index sequential rearrange equal h old conjugate observe martingale order consider bound prove theorem constrain mirror descent rate mirror see mirror observe mirror give rate look parametrize regime unit constrain guarantee mirror remove section complexity otherwise follow attain convexity generalization perturb randomize see consume randomized bit calculate sum replace variable brownian end analogue rademach complexity calculate brownian theorem admissible furthermore give replace keep round time variable brownian motion prediction round game time regret setting strategy case bayesian prior regret robustness statement investigation sufficient past take set z stochastic estimate natural collect set may statistic smooth need sequential rademacher value correspond intuition dependence ever grow refer bernoulli datum posteriori round consider mistake expert algorithm discretization attain conclude regret minimization attain analyze new twice supremum make supremum achieve tree speak note expert discretization depend avoid discretization obtain end take bind admissible relaxation admissible attain q write realization sign supremum fractional deal sign either idea draw admissible relaxation obtain leave problem consider regularize least parametrize solve pair usual datum generality strategy end w simply square minimax parametrized online vanish regret set strategy one problem loss follow compete index schedule specifically strategy write close specify rademacher rule r pick unit every rich possibly efficient general main allow play adversary adversary round unconstrained adversary come restriction adversary mapping pz proof via allow far show property change sequence value path prediction sequential range path arbitrary value tree statement body ease application operator sup inf b fa value respect strategy
correspond polynomial evolve iteration current higher expand result recursively find search cycle cycle step model solution principal component add product principal recursively cycle find algebra non purpose conditional independent integral existence integral equation distribution integral equation depend optimal maximizing integral due contribution parameter find maximize integral equation use approximation maximize training easy see equation least dependent statistic integral equal sampling variant form less artificial corruption try single propose include regression unnecessary see feature find matrix vector principal select select correspond certain hyper parameter control constrain feature duality model original feature like feature step pca feature extend increase add quadratic previous repeat cycle solution component emphasize new iteration super polynomial original expansion add product recently simplify notation look like product super express super super algebra important property super algebra super limited could satisfy algebra simple complex trivial successfully
neuron onto hybrid spike trace addition field challenge spike sort spike dataset channel probe principal component take evaluate positive spike discovery feature artificial decide particular point cluster whether algorithm virtual assignment close approach splitting study apply distribution analytically compute analysis classification rgb rgb cluster face two high dimension curse overfitte poor application subset cluster membership informative feature subset datum introduce gaussian case close sort channel popular gaussian achieve maximization face curse poor particularly uninformative second impractical must computation daily principle approach suggest dimensional modify generative gaussians fit enforce form approximate observation reduce may provide substantial offer discard limitation different set feature assign hierarchical algorithm spike sort newly develop count spike sort identify fire time neuron brain signature record involve million neural channel dimension sort quite approximated traditional derive optimally although software million spike channel volume thousand development neuron detect total channel neuron spike simultaneous firing cluster independently simply regard missing volume produce capable million reasonably run em stage encode weight data domain topological potential zero assign start algorithm second stage datum replace virtual noise virtual mixture split arbitrary require virtual analytically cluster multivariate iy ik ki ik mask indicate specifically outcome vector classify indicate intermediate major advantage gaussian curse number channel proportional datum allow way typically simple spike sort use advantage spike must across channel software critical provide mask mask compute noise whenever analogously noise consist step modify replace point virtual ensemble point mask associate spike intuitively threshold virtual model univariate simplification however implementation shall suffice expectation log virtual act pass replace possibility curse value replace virtual thus contribute step ensemble require simplicity henceforth assign soft derive set index assign index straightforward expect virtual correction carry decompose variance value virtual act mahalanobis em plus diagonal automatically determine gaussian model parameter commonly penalization aic number free statistical estimate fit number feature free matrix mean single weight must sum subtle replace fix freedom define number feature let e estimate effective aic implement previously open gaussian term million high run hard heuristic majority split merge increase distribute component first efficacy gaussians dimension separate cluster function toeplitz matrix exponentially figure format raw confusion bic confusion em aic em vi various penalty indicate bic penalty
run additional set ccc spc spc spc vary spc well scale spc spc conditioning digit digit plot spc run fast spc involve solve lot conditioning digit spc spc even several spc spc demonstrate spc spc run much method notice slight spc spc come qr factorization cache minor running affect lot census census generate compute percent bootstrapping area spc trial least square absolute plot plot confidence even trial percent bootstrappe resample matrix replacement addition solution area spc digit section detail spc perform empirical stacking medium although lead redundant favor size although memory ram rare parallel source widely practice several straightforward skewed stack implement sc spc spc spc relative error skew six conditioning preserve apply medium spc spc perform sufficiently base objective however detail record measure six sampling conditioning digit sc spc spc spc vary scale dimension spc condition show accuracy increase size point plot sampling size subproblem size point miss capability conditioning qr ellipsoid round determined scale ram perform factorization round hence prevent census datum stack construct realistic roughly quantile provide digit interesting quantile education strong total high quantile age affect total high c ccccc intercept age age age education education evaluation size easily digits accuracy sampling competitive environment capability ram since algorithm medium subsampling construct condition main run nearly derive calculate norm recently propose conditioning introduce spc approximation main applying conditioning meet conditioning method spc digit accuracy scalable compete large million work heavily well conditioning acknowledgment office quantile response permit accurate relationship least appropriate interior find moderately deal quantile distortion empirical competitive medium sized environment size quantile quantile express analogous conditional regression covariate appropriate setting reason quantile area economic regression quantile formulate simplex efficient problem moderate large reliably need computational relative construct distortion embed form recent randomize datum nearly specify vector problem paper augment quantile equivalently problem single tx notational presentation distortion subspace general technical result importance probability element wise basis depend every one slow algorithm theory additional previous algorithm representation algorithm approximate norm prove depend construct dimension interest low evaluation state completeness third number slightly condition however setting show conditioning conditioning superior state problem plus subproblem main characterize high dimension empirical term objective function also solution quality exact subproblem quantile regression sampling conditioning permit dimension condition moderately ram algorithm apply compute sized moderately use preprocesse predict element original compute solution come guarantee sampling complexity depend require overview randomize approximate solution square recent construct ellipsoid round preserve approximate problem use cauchy transform embed construct low distortion sparsity use conditioning improvement method construct well condition basis sense solve use th loss linearity follow ax prove dimension general presentation brief review embed definition low embed polynomial strong preserve method paper distortion subspace preserve nonzero row distortion preserving follow introduce precisely basis well ax sparse originally distortion embed ns column uniformly choose probability time embed replace cauchy improve fast preserve condition ns basis round ellipsoid round prove net inequality subset know subspace bernstein bernstein present theory conditioning role conditioning conditioning method condition basis conditioning property run obvious determine row select indirect effect discuss qr fc ellipsoid round er fast ellipsoid round er spc qr spc qr qr run apply factorization ellipsoid condition qr ellipsoid obtain distortion embed low polynomial cauchy obtain calculate see matrix qr factorization vary vary trade among type conditioning name transformation sc stand cauchy fc stand fast cauchy spc sparse cauchy scheme qr alternatively well ellipsoid round round ax rx rx ax rx transformation matrix time ellipsoid rounding propose condition derive er type one condition qr er like construct distortion preserve pay obtaining since dimension much bottleneck ellipsoid round qr one possibility round big round acceptable satisfying preserve ellipsoid round still eqn eqn replace condition especially preserve run reduce lot round second possibility eqn expect condition basis factorization spc remainder condition evaluation start qr spc result omit construct distortion n basis via factorization full nr spc call spc spc appear lemma omit lemma rank distortion da ellipsoid round nr full construct low embed da via factorization full take nr satisfy eqn step well condition require obtain distortion sr da time complete run construct additional lead input distortion row three step high satisfie step least distortion sum concentrate claim come compute ar condition multiplying remark text since zero zero row follow size parameter subsection state main computing relative solution suffice distortion conditioning algorithm approximate main quality full approximated distortion via algorithm original problem constant distortion solving subproblem solution eqn third inequality come fact subproblem return claim actual overall claim state lead well running time bad trade situation bound vector return empirical next empirical condition medium sized order increase add appear skewed call skewed generate row canonical length laplacian block associate coordinate may expect produce acceptable real census relate people work work hour week section six subsection five skewed census detail performance term quality respectively varied quick summary quality use main algorithm among conditioning spc spc accuracy achieve digit row moderately approximate demonstrate fix conditioning always achieve regardless show accuracy monotonically reliable range moreover spc spc scalability digit amount several method sc spc spc four sc spc spc spc stand condition identity uniform completeness norm instead estimate permit evaluation observe approximate tolerance size plot trial axis plot range test skewed plot look digit accuracy spc need spc spc condition property perform surprisingly reliable size conditioning close spc estimate actual solution behave well spc reliable yield accurate expectation point description likely generate design huge note digit worth spc spc spc run three generate follow spc spc spc spc spc running spc fast follow spc spc spc present discussion say opposed vector norm see norm error exact surprisingly reliable bad even relative error change substantially discussion dependence subsection qualitative trend measure different thus save figure spc spc vary dimension change summarize use six varied datum set relative since omit fix preserve high vary relative parameter except change take wider spc letting vary tolerance sampling fix relative figure constant sampling number see use spc remain roughly magnitude summarize figure plot previous change set conditioning spc
proposition trivially count iterate viewpoint define index effect map element distinct r ig cover star worth note construction substantially factor begin along secondly systematic construction iterate f r h follow check via ic l non far construction arbitrary unknown often exhaustive consequently star efficiently pair discuss illustrate algorithm section discuss balanced star balance space factorial balanced cover star c ab cd bc ab ac ac ad bc abc ex cd ad bc ab ac ac bc abc e ce ce table check problem check ab follow step constitute spread ab bc ac algorithm exist pair pair map map ac bc cd ad common lie star factor design viewpoint plot star enough cover star design randomization factor common seven assessment factorial use four star rank complete balanced balanced trivial every trivial spread trivial balanced exhaustive cf pg c ef bf cd bc ab ce ac abc theoretic balanced convenience notation star balanced star follow balanced star develop restrict cyclic next star construct start brief algebraic proof mostly wish start write element cycle h spread primitive spread obtain primitive method primitive primitive result establish spread primitive flat nonzero root equal first flat correspond set nonzero multiplicative trivially cyclic part note construct cyclic root distinct lemma mod mod mod consequently primitive polynomial respectively construct field easier let primitive root construct primitive primitive polynomial degree root root root define set root basis define task need preserve primitive root enough fix root basis polynomial however since field thus root field claim note map root root indicate field multiplicative flat unique cyclic construction widely possible spread star design investigation star equivalence star generalize mixed unbalanced cover star thesis additionally paper checking equivalence ray flat string sort characterize design rank optimality criterion wu chen acknowledgment support grant discovery grant science engineering research design design randomization restriction design complete trial projective geometry subspace theoretical design check explore completely classify cyclic geometric star design factorial design check factorial factorial design randomization restriction split design trial stage factorial randomization restriction several unified factorial finite geometry characterize define stage randomization assess significance factor factor combination factorial design half plot effect assessment effect recommend enough normal plot establish cover many overlap avoid case geometric share overlap enough factorial assess spread star say star generalization spread focus cover star design require star effect star check star propose develop cover star correspondence cover star number check star check reduce underlie dimensional array establish achieve via completely cover star achieve spread star correspondence show cyclic remainder paper define balanced star spread correspondence define equivalence simple star show cyclic establish equivalence star total simple distinct way factorial ray way follow factorial effect star mapping gets map gets balanced cover balanced map already select notion via formally star two balanced star say exist cover star establish balanced covering check numerous intensive star total comparison equivalence sort array box sort array factor sort two array expensive reduce burden representation factorial balance sort remark check sorting array ii reverse value underlie establish
time getting expect coin chance head flip equal tight illustrate corollary prove great trial colored line colored dot dash horizontal positive integer solid colored never dash horizontal bind nearly meet value proof corollary normal cumulative hold distribute p write sum positive restrict open differential least strictly increase whenever inequality equivalent conclude suffice bind integer irrelevant assume hold irrelevant immediately give bm introduce improve useful occur plug definition yield simplify rewrite differential eq holds increase maximum reach strictly bind achieve give write conclude large threshold shape maximize end requirement yield follow furthermore kk g address lemma let suffice increase write proof combine lemma corollary last probability binomial equal instead q eq write eq theorem long present rigorous justification relative deviation despite discussion topic tight probability exceed role analysis unbounded function expect around trivially number trial sufficiently de tell substantial figure machine relative deviation deviation useful standard bound approximation bound unbounde original deviation publication give literature effort instead
indicate solve lagrange multiplier approach infer node role train class infer contain optima distinct role relate introduce treat classifier inference optimisation correspond blockmodel part margin role help task treat conditionally assume assignment network use expectation em expectation infer role algorithm converge update infer role vb vb easy vb variational lower bind restrict posterior mean integrate categorical role distribution family general exact computationally expensive update blockmodel due margin decision encourage role role optimisation optimal role represent specifically node receiver position converge role classifier coefficient perform belong colour assignment visual infer role block available acquisition expert incur informative involve part good greedy stage single add classifier incorporate maximum margin support machine involve relation employ simple represent example uncertain multiclass represent class give second query select small two four max bm link node web lead node comprise occur adjacent appear text web resource direct classification task dataset attribute attribute citation machine paper direct link indicate class one subject class add validation however role fix exact setting reason link therefore role twice selection label stage cost examine role discover dataset interaction node understand well network role understand attribute largely web show previous discover attribute discover label quantify accord rest blockmodel blockmodel information blockmodel entropy blockmodel addition selection univariate collective active strategy implementation weighted vote relational high network link regression classifier method node sum classifier erm select heuristic subset evaluate erm figure see perfect see approach perform particularly show collective classifier prediction centre stage proportion label quickly accurate blockmodel slightly explore half network achieve english tend degree find mutual tend node almost equal uncertain analysis quickly discover main role role node role choose node role choose order degree find majority degree high uncertain show follow degree task centre figure classification classification last primary homogeneous distinguish great variation pattern classification network node tend suggest diversity previous suggest well diversity diversity heterogeneity e role predict accurately learn scale size predict node unlike explore terminate time half network citation comment benefit gain allow class discovery except role step task adapt acquisition label consider discover interpretable flexible model assess performance discover role accurately predict across build connect network allow heterogeneity class still accuracy structure collective classifier link method discover pattern comment discussion uk centre virtual environment image way connect sometimes relationship case attribute node label relate wish attribute discover call blockmodel model predict mapping role maximum subset node margin base active strategy integrate role optimisation adapt role network classification explore english word occur decompose network way set often attribute tend political network addition type link network opposite web tend specie adjacency english tend link relation therein task understand link attribute analyse group node within understand label description aforementione adjacent indicate noun label word noun label tell something word link word class label something nod role link noun role come noun usually come noun case display heterogeneity role heterogeneous link class show colour link box leave illustrate homogeneous show class homogeneous role class role role pattern class scenario network link label label network etc subset understand predict approach type node link way blockmodel probability role blockmodel us structure two blockmodel allow mixed role membership role role example node direct link represent label distinct role label would network role alignment role role blockmodel incorporate e class label role class efficiently discover role employ role discover process reduction principal iid role achieve goal independence nature dependent role label work connect problem relationship node label network unlike work explicitly try identify relationship link classify network refer collective lot year class node collective relevant applicable attribute collective assumption either implicitly necessarily hold network markov allow hard break many easier relate locally iterative propagate around conditionally discover role group link pattern
pca section construct explain tractable unique perturbation solve pca large entry eigenvector set try inner vector sort keep amplitude absolute co sparse candidate support candidate rank nontrivial detail q simple maximization dot similarly prove helpful efficiently cauchy inner x variational characterization rewrite generate fact fix collect support would need optimal issue infinitely could prove simplify q transformation space transformation perform spherical enable visualize span rank phase loss unit angle complement opposite pose issue solution fig example value element span curve respective sort absolute change set therefore intersection curve exactly might region top support set c intersection point axis compute distinct locally determine intersection v need normalize curve sign curve sort intersection one methodology candidate sort equation total n operates simply sort vector compute sparse sparsity subroutine constant tune performance compare path omit show synthetic seek eigenvector sample vector continue experiment comprise million twitter manner large eigenvalue correspond eigenvalue sparse non overlap eigenvector pick first experiment repeat rank penalization penalization eigenvector report correctly support support eigenvector optimally apart nd rd approximation generating experiment decay rank decay perfect theoretical model relevant literature evaluate two gene expression come explain equal eigenvalue also output explicitly st microsoft microsoft microsoft microsoft google acquisition acquisition google google pc microsoft country country google rd rank receive twitter twitter great great great year sg sg sg g census experimental comprise million tweet come tweet list character tweet tweet tag list simple normalize contextual etc hoc also discard word less character corpus represent tweet vector consist appear appendix law decay cutoff observe law model decay spectrum twitter law decay law good guarantee empirically guarantee initialize covariance test computer sophisticated one word benefit force help fair fast test performance explain maximum pc measure lx come contain month capture compare table tweet pc generate test computation window approximation second rank minute approximation matlab times row term speed observe slow test marked pc acquisition microsoft european music census carry interesting general principal appear involve word algorithm interpretable pc parallel interesting future may may pcs tx intersections curve c equivalent relative sort surface point case equation dimensional solution element intersection locally optimal support set sort intersection interest become member locally support change happen coordinate vector coordinate support previous check curve intersect generate manner intersect check sign equation vector check tuple di dp cl dl l visit sort candidate intersection rewrite multiply vector compute absolute need tuple potentially set neighboring rank instance sort eventually element entry factor main idea entry need explain fact pose optimization q eq solution support top element sort fact vector order put least amplitude opposite component proportional drive arbitrarily sparsity constraint implication intersection account sort due fact sign I different recover support amplitude intersection point discard green obtain curve apart discard intersection blue elimination reduce problem reduction run large elimination combinatorial sequentially check norm step elimination elimination locally mention element candidate surface observation surface surface surface surface critical intersection surface could intersection curve boundary become curve curve point discard point surface check one high amplitude description build elimination norm amplitude accord surface norm amplitude point surface surface intersection curve move st surface high amplitude amplitude st surface eliminate row surface amplitude interest intersection obtain surface curve need may continue process norm code elimination elimination input di discard comprise prove quantity respectively establish lemma bound first use optimizer achieve least psd inside sum hence n technical nonzero calculate well element multiply elsewhere da low basic important ratio relate spectrum decompose part feasible vector obtain come sum value due divide bound eigenvalue q straightforward identity position get eq examine absolute metric v development matrix make break intersect dimensional matrix issue intersection equation requirement equation avoid show perturbation rewrite requirement depend subspace interact matrix bound instead work perturb g union bound mean hence obtain intersect metric obtain easy avoid original random sufficiently slight incur objective give twitter h specification unique million entry month million day k hour tweet character tweet evidence test exhibit decay concept guarantee set parameter hour day length subset follow find compatible rough well approximated spectrum scale impractical moderately compute examine lie eigen obtain provable well eigenvalue power law algorithmic elimination provably safe elimination consist million minute scheme match previous dimensionality project span significance pca partially first consist entry efficiently singular decomposition pca tool drawback vector interpretability interpretability document trend g use reason desirable eigenvector intractable novel pca provable approximation provably diagonal related equation bound large depend subsequently rely constant regime partially theorem exhibit desire accuracy time necessary follow substantial drop
extract label collection extract feature discriminant mixture discrete represent class mixture proportion satisfy gaussian bic likelihood give estimate signal new design vector maximize signal phase mixture estimate bic table htbp modeling rate piecewise classification rate signal approach htbp class component criterion attribute wide propose mechanism incorporate hide logistic function transition series parametrization accurate term hide modeling discriminant company department availability university technology laboratory bp france include economic generally technique synthetic approach discrete logistic smoothly dedicate maximization iterative reweighted piecewise markov context monitoring particularly switch mechanism acquire switch series occur finance engineering economic bioinformatic represent change work relate diagnosis train track acquire switch classify predefine represent electrical power switch see fig diagnosis switch operate switch mechanism propose switch switch see non regime piecewise polynomial parametrization segmentation segment characterize model exactly programming algorithm optimize segment well programming computationally run time fisher iteratively piecewise assume variance segment model piecewise another alternative markov however regression adapt regime specific regression hide allow transition approach switch link I develop logistic I hide class learn label operate classified map good performance carry cover wide use base program maximization introduce propose describe deal term switch observe piecewise polynomial model series regime define polynomial segment index shall model follow satisfie segment dependent random noise denote polynomial coefficient variance model define segment parameter piecewise sum likelihood write maximize log segment programming procedure expensive equivalently I programming series ik approximate write diagonal diagonal element hide regression fact consist phase order constraint assume hidden model assume markov switching otherwise prove conditionally regression parameterize maximum likelihood likelihood log maximize em illustrate temporal fig transition particularly contiguous ccc variation proportion parametrization transition transition via htbp switching time unlike basic permit polynomial regression generative multinomial prove conditionally regression density mean variance estimate classic maximization follow expectation expression simply require step maximize expectation perform maximization perform maximize analytically maximize provide multinomial multi reweighted square verify propose perform em require internal parametrization time series approximated signal diagonal accord q contiguous segment penalize section two mean simulated curve signal criterion model regard denoise call second criterion use assess regard signal piecewise run logistic segment choose contiguous propose observe second three effect level transition situation situation three regime regime smoothness transition tune see situation show second varied observe noise segment situation htbp htbp smoothness transition divide divide hide initialize initialize several segment segmentation segment fit stop iteration show misclassification smoothness transition perform piecewise approach alternative situation denoise misclassification
role community identify cause assign user simple usually approach social community aggregate project decade aggregate likely aggregate never active clearly limit project organization overcome fact interaction precise aggregate window day allow focus short time project history report popularity rich social community assess centrality approach one centrality interpret either impact amount degree centrality actual node sum distance centrality term role total short path pass centrality centrality node recursively influence direct neighbor connect central centrality eigenvector centrality library capture belong degree remove degree within isolated connect eigenvector centrality verify large remain illustrate project study highlight variation organization although indicate differ largely term social organization represent network apply introduce section research scheme investigate four major adopt tracking system correspond one complete resolution addition fall status history community report fail include additional within period basic category first helpful report eventually result centrality centrality report eventually complementary centrality user decrease eventually hypothese relation helpful centrality reasonable centrality possibly contribute helpful handling centrality hypothesis centrality centrality handling process emphasize community compatible hypothesis centrality likely report furthermore central influence receive increase report take comparison eigenvector centrality five present distribution involve hypothesis respective accept size detail p h hypothesis report month precede follow eigenvector centrality category denote similarly extract month quantitative use centrality position classifier use comprehensive centrality membership order eigenvector stochastic shift execute either hypothesis side give threshold reject favor none alternative hypothesis eventually draw centrality reporting helpful reject accept significance distribution observe project fix report eventually h compare valid whenever month precede prediction list individually row aggregate fraction significantly high fraction classifier solely perform nan model randomly strong project ht svm svm add classifier eigenvector centrality classifier report report part respective centrality score community individually row indicate classification membership inclusion eigenvector centrality generally recall score relation centrality report project support machine svm report nine topological eliminate overfitte available sample nine row project report fraction project high reporting technical target user mainly obtain precision respectively majority two project result project project ht r r svm conclude validity describe record project approach collaborative software engineering mainly heterogeneity target rather general without particular focused end diverse project quantitative yield quality contribution nevertheless collect analyze insight organization generalize community project limit record time user evolve compute measure accurate automate categorization study software production diverse beyond present relation process unclear survey send project confirm exist indicate criterion fix community confirm unfortunately survey dyadic I user receive report user handling clearly interact must proxy organization comment reason consider direct communication furthermore study perspective social quantify away report accumulate newly community remain concern fact high particular window investigate whether performance finally machine come avoid limited fraction randomly facilitate implementation social available paper extent community perspective evolve report project study evolution use resolution day validate eigenvector centrality report report software opposite project decrease eigenvector centrality project validate centrality reporting report community report classification nine topological closeness centrality clustering whether vector automate achieve fact merely quantify position combination automate accuracy see scheme grant like collection preprocesse community share insight rgb procedure important successful collaborative engineering project become open project time paper refer software process away nine quantify applicability comprehensive major community ten project valid quality report find automate integrated community vector machine svm identify nine yield significantly obtain automated highlight potential organization collaborative software engineering social support engineering crucially quality success practical experience particularly project number report resource simple report community project report eventually refer rather software report basic reproduce magnitude project call automate high precision huge practitioner filter assign report improve incomplete queue effort automatically automatically neither incomplete software engineering automate report natural temporal handling unique full history consider extent automate technique evolve property centrality find user quantitative position automate four study extend work study automate eventually comprehensive evolve community base machine automate aspect collaborative software extract open research address project collective performance development project distribution member project validate effort small core report large community contribution community degree furthermore implication future subsequent relation individual well team work study investigate relation centrality individual software method quantify topology handle author handle user result handle project identification cause software task needs solve obtain project communication centrality comment centrality extract related failure similarly communication collective paper social build failure furthermore find positive team structure insight software engineering indicator important measure social network classification handling report million support automatic report report project comment add day predictive compare pure chance model introduce comment predict close apart use technique human machine I simultaneous classify relationship apply comment dirichlet prediction category report indicator consider report fraction eventually automate identification recently successful prediction get rank apply machine location software author work social individual automate report identify open question address evolve handling report measure position report valid classification way million report project one report connection nine topological position large static grain event predict report eventually identify valid address limit combine measure improve precision scheme drive base collect community evolve
py parameterize restrict prevent overfitte new outli indicator probability various threshold obtain shot shot alternative would outlier classifier unseen run cifar image method ng obtain vector experiment shoot class zero shoot close shot class seen span take shoot learn hand cat map transfer map thank performance outli cutoff negative point outli classify unseen image split image unseen test accuracie train unseen classify accuracy chance novel zero vector transfer representation outlier project semantic manifold shoot classification framework shot accuracy fully unsupervised assumption ref align cm ref pt align cm pt pt science department stanford stanford introduce recognize necessary text corpora distributional language span semantic shot unseen obtain first outli recognition require define image ability instance unseen zero shoot learn useful activity visual car frequently vast world available unseen attempt people identify unseen object read read possibly briefly look object shoot see unseen ever cat image cat training idea image map capture unsupervise corpus mapping visual us instance class incorporate determine otherwise category category integrate unlike zero shoot various unseen work knowledge manually visual attribute class language unsupervise corpus briefly work follow cifar outline difference five map manually classify able semantic word classify unseen class extend allow setup al zero canonical unsupervised corpus learn word among multimodal boltzmann learn multimodal work able description representation word capture semantic word represent occurrence effective natural extraction cognitive wikipedia learn occur context occurrence distributional semantic detail evaluation al raw pixel fashion learn semantic membership project test data without former unseen class capture distributional unseen image minimize matrix project implicitly word query color show visualization space image unseen unseen cat cluster correspond shoot cat mapping class cat map
freedom sparse present appendix section interest distinct frequency discussion devote frequency x l pair normalize write form uniformly location norm norm singular inner vanish outside original perfect partial correspond relaxation generic effect possible long motivate harmonic specifically effective enhance enhance algebraic enhanced shift invariance harmonic matrix r attempt recovery via enhance minimize enhance program solve semidefinite program tractable extend higher without frequency fold l iy I w enhance fold enhance thus apply enhanced summarize fold practice always corrupt amount noise practically follow noisy model denote noise condition enable recovery copy denote completion incoherence matrix convention dirichlet analysis incoherence define enhance brief interpretation incoherence among frequency pair spread generate fashion g perturbation skew diagonal proportional weak introduce ideal fold structure reason frequency one location condition rely incoherence mutually incoherence main guarantee location noiseless condition hold probability immediate incoherence recovery near incoherence like even time stable close counterpart set probability theorem basically recover enhance close enhanced signal entry usually yield applicability illustrate evaluate enhanced correspond small pair frequency spike entry uniformly estimate calculated fig illustrate carlo empirical rate reflect color grow linearly respect phase transition applicability phase diagram generate programming large handle datum exceed correspond enhance one e thresholding algorithm h set initialize enhanced shrinkage specifically give thresholding model fold consistent project consistent entry spike entry reconstruction instance illustrate superposition complex reveal total I gaussian give signal amplitude reconstruct ground fig stability model consider synthetic amplitude low resolution low frequency truth width resolution obtained apply transform avoid resolution fig suggest promise super htp ccc resolution super resolution low reconstruction small map matrix enhanced object problem matrix completion identification processing vision medical etc guarantee directly completion analysis framework straightforwardly adapt modify matrix incoherence small eq basis weak assumption convert toeplitz counterpart toeplitz form harmonic framework toeplitz problem nonparametric object pose compress low mild enable precision conventional completion outline structure matrix exist theoretical foundation analysis defer respect onto denote orthogonal onto subspace span span complement replacement z operator obey optimizer dual replacement taylor satisfy j incoherent exist c corollary study object sample object ambient complex frequency disk conventional compressed suffer impose fourier develop nonparametric enhance start fold structure mild incoherence perfect exceed show fold information theoretical robustness dimensional approximated superposition time involve estimation object resp imaging system acquisition often limit hardware constraint resolution resolution fortunately recover object object transform advance compressed surrogate require often nevertheless harmonic many processing localization etc domain identify
interval time step determine swap negligible factor take maximally take thus estimation take time time summarize quantum machine scale imply situation powerful classification becomes perform separate hyperplane high correspond surface space quantum computer polynomial kernel simply time polynomial kernel construct trick polynomial hyperplane time nonlinear quantum accuracy contrast inner product space important classifier machine quantum algorithmic logarithmic feature formulation phase estimation quantum inversion speed quantum maximize datum kernel principal argument absence knowledge suggest bind square aside benefit quantum quantum algorithm generate necessary product quantum implementation important machine privacy neural work nsf quantum artificial intelligence laboratory author acknowledge helpful discussion nan generate hamiltonian quantum norm datum j x investigate rank sub optimally matrix w yy rank training solution speed many unknown take frobenius hilbert give assumption new already optimize quantum computer example quantum big inversion matrix spectrum unsupervise classify new case big feature operate construct optimal hyperplane space svm solve feature accuracy quantum quantum stage quantum machine approximate quantum employ recently develop reveal overlap efficiently low approximation principal arise learn stage accuracy operate runtime classify j hyperplane hyperplane inside hyperplane class offset formulation hyperplane subject tucker hyperplane corresponding support vector introduce central k x kernel soft margin study dot programming dot training quantum efficiently state quantum ram hardware operation access inner component processing play square section quantum first quantum nm discard desire simplification slack replace lagrange contain determine partial derivative lagrange arise offset margin usually quadratic programming machine machine quantum state describe hyperplane inversion classify classifier success swap quantum inversion efficiently j cc factor expand storing generate ideal storing respective eigenvalue perform control set coefficient state construct addition eq swap test construct p quantum matrix inversion matrix consideration contain due offset parameter offset negligible reduce definite invertible dominate eigenvalue involve offset
satisfy everywhere probability gaussians mixture covariance next immediate thus whereby hoeffding give k analogous let moment respect q eq hoeffding let ball radius statement discard failure convenience whereby everywhere p display contradict show region outer follow analog quantity additionally outer hold draw outer additionally outer useful carry since every individual upper guarantee deviation p u correspondingly discard failure lower conservative compare whereas q integral hoeffding estimate attain control triangle definition correspondingly let throughout discard set henceforth discard discard start fraction numerator denominator whereby probability numerator denominator note eq numerator choice fractional term mean mixture satisfy separately cover control scale eq cover orthogonal follow within first cover measure difference cover cover since cover inequality max dominate construct since additive guarantee cover together cover cover meaning size redundancy contain cross correspondingly close max guarantee choice secondly rely spectral matrix triangle choice cover ball ex r whereby nc control way closely discard briefly element union together mean grid candidate lastly precision meaning whereby grant suffice exist cover component cover meet component relevant cover element whereby thank property mahalanobis combine q term probability cover element lastly cover various end cover name cover provide hoeffding corresponding failure firstly next must control together kp kp numerical particular may c cover discard simplification cover additionally mean corresponding source cost decay technical control set soft covariance refine provide center fit say denote interval offer information firstly consistency global minimizer deviation gaussian thus amenable consistency solution sample converge optimum task finite boundedness moment fix deviation availability suffice hold heuristic suppose method carry method simply equivalently cost cost moment consideration heavy technical deal single center deviation chebyshev deviation center cost grow successively irrelevant consider integrable dominate grant ball way outer deviation whole suitable course nice get sense local standard outer dominate upper provide apply survey establish term connect theory soft section means defer work mention handle mean similarly provide adapt community guarantee extensively study parameter component involve guarantee list make amongst similar outer namely standard boundedness tool vc theory handle mean tool measure consider boundedness work uniform unbounded constrain consideration condition center near mass rigorous mean mixture moreover mean argument heuristic optimal secondary constrain rate recent year previously choice empirical set logarithm need control pair approach mean fluctuation single origin always finite empirical serve integral typical use naturally arise chebyshev course availability dropping rate basic primarily chebyshev generalize slightly beyond differentiable b fx fy fy x divergence handle regularity place modulus satisfie dual sometimes bregman gradient respect thus compute bregman denote encode guarantee either previously lastly mixture spectrum bound c mean typically bayesian ignore beyond potentially violate whereby real least f c satisfie ignore seem inferior make next bregman divergence mass discard due resolution tradeoff reference norm cover ball bind k outer upper bound function similarly satisfy secondly function intend discard dominate deviation center far statement hold least u c also outer scale roughly pick kb least definition consequently proof mean easy dominate kb b precede point fx proceed appear reasoning integral condition suffice huge well size turn removal center set implement discard fall threshold bind two kind budget shrink behave another flat elsewhere elsewhere phenomenon precede produce arbitrarily condition soft analysis assertion nearby gaussians possibility away reasonable measure k drop polynomial depend quantity cluster distinction log contain expression proper class distinction gaussian influence limited proceed acknowledgment nsf result deal slight moment convenience implicit connect course working moment moment finite bind measure ball moment chebyshev follow basic tool control empirical via moment boundedness discuss univariate connect early map copy even give draw chebyshev recall vanish nonzero copy amongst distinct number time indexing plugging thank chebyshev prove control combinatorial scheme however sum moment individual variable material bound specific involve control return dominate radius bind suppose let ball conjugate lie lastly deviation inequality outer exponent integer moment radius sample eq consider provide exponent map map plug proceed bregman divergence differentiable instance part property precede instance differ first characterization least lemma naturally start control center additionally draw guarantee center q follow center satisfy proof p kp henceforth kp properly mass together together exponent lastly respect map thus triangle q establish statement henceforth discard failure event thus statement fix guarantee element outer establish direction add center decrease recall control deviation bound portion uniform cover subsection bregman divergence constant bregman similarly mean definition rearrange statement q yx useful radius
standard normal inverse cumulative construct feature tc ig mr worth gene I try test almost motivation paper contribution central prove specific within normally diversity entire mean category verify corpus show art macro micro ig unbalanced selection tc formulae ig mi expect entropy pearson corpus assumption hold occur frequency much high expert think term rather mi influence besides mi estimation score ig firstly attribute tree theory study message ig original requirement proportion requirement ig less ig consider occur document except frequently ignore information student use statistically calculate class variability explain average category whole corpus frequency let text consist term frequency th document subject distribution e multinomial multinomial document vocabulary event document position occurrence dominate multinomial sample term collection tf tf kt ic kn central theorem approximately denote variance besides pool definition formula deviation whether e statistically difference big large implie average frequency compare frequency occur consider category specific alternate way corpus benchmark collection document unlabeled document skew stop word convert use uniformly distribute category content character letter convert weight three establish classifier comparison support svms knn classic centroid implementation similarity use cosine effectiveness widely tc ways macro micro micro precision macro category micro weight dominate category real corpus table list category life corpus closely content category belong feature category ig mi select two corpus account ten feature space reach save feature accounting group two good macro micro respectively mi ig macro among method show fig perform micro five decrease high micro ig skew unbalanced ig inferior mi macro micro ig superior mi comparative al show feature selection macro micro results micro micro ig slightly four mi comparable corpus macro micro method performance ig mi meanwhile macro worth note mi well mi fall dramatically corpus fig micro point tendency however consistent micro good among micro trend curve similar ig performance mi centroid base macro fig observe well mi ig micro slightly ig corpus outperform mi
sd standard deviation cccc sd sd sd sd sd sd sd sd cccc sd sd sd sd sd sd sd sd sd sd sd high replication sd replication sd sd bold vs eq double definition expectation take square integrable statement immediate q consequently prove tend tend schwarz sense result amongst precisely ta let n assume mr mr r return approximate thus note tend denote interval real q inequality statement technical statement second finite number generality denote integer k r follow eq lebesgue denote mn yield develop ix jj introduce boundedness yield soon independently proceed proof author thank joint anonymous constructive lemma section corollary g sup universit e paris paris fr universit et fr national health usa mail instead convex collection basic collective indicator training model specifically collective sense package combine present substantial excellent velocity variety nonlinearity year grow procedure research available wide naturally efficient combined strategy know sense relatively valuable research tool regard aggregation literature various weight linear select notion optimal bound risk treat replace loss also mention single aggregation analytic flexibility aggregation procedure machine happen collection machine might sophisticated cite pooling machine aside machine dependent criterion seem weakly collective might nearby similarly search machine search method outcome good procedure combine collective thereby context preliminary estimator predict distant response precisely outcome select concept clear toy plot circle predict known machine triangle point new along dotted threshold black two stress central nonlinear basic determine original training well formalize aggregation operate paper release implement package statistical exponentially weight aggregation dt fast exposition proof throughout n value prototype equip euclidean goal regression abuse however cause compete candidate basic machine subsample neural network naive forest hoc suggest experimental parametric tuning ask alone allow machine throughout grow order random convention weight collective see local average outcome unweighte whose assess everywhere shall assumption meet whenever infimum integrable combine well measure estimator combine infimum integrable note integrable link aggregation aggregation performance regardless k remarkable firstly term predictive risk primitive machine sense distribution smoothness truly behave poorly otherwise lasso job crucial clearly discard conversely predictor predict also instead agreement implement keep observation proportion machine parameter calibration global agreement heterogeneous see homogeneity select indicator possibly conversely predictive machine predict value response adopt protocol simple splitting device large absolute pool logistic scale illustrate discussion package estimation convenience ridge neighbor li cart rip synthetic eight design wide regression toy appear somewhat classic set predict inspire deal form nonparametric whole x k x error deviation replication model apparent depict ability assess wide persistent next highlight perfectly able price sparsity
distance wasserstein useful smoothly increase importance invariant assess detection rate distance add ratio db keep without ease last indicate noiseless figure repetition affect noise robustness conclusion poorly recover half atom identify affected presence rate stay evaluation effect rotation original atom noise dataset parameter reconstruct atom coefficient explain dictionary reconstruct noiseless perturbation atom decrease aim metric immediate tool assess full application metric dictionary experiment brain interface spatio offer eeg variation cause recorded head activity measure location propagation physical head highly able capture temporal eeg multivariate dictionary dataset competition imagine four hand consist trial hz subject give redundant investigate technique part structure around competition even individual variability subject subject specific characterization largely inefficient competition variability know competition objective dictionary intended demonstrate ready apply immediate experiment propose investigate cluster performance well orient specific variability dictionary dictionary associate training competition matrix subject metric similarity affinity propagation find optimal similarity apply approach indicate dictionary right hierarchical clustering hausdorff cluster combine ensemble approach unique cluster high explanation whereas subject differ result evaluate subject time decrease demonstrate variation dictionary cause conclusion begin concern trial robustness dictionary last instead previous affinity apply class cauchy geodesic merged ensemble ensemble show localization area hand side head generate eeg head properly learn offer intrinsic multivariate dataset applicable application contribution advance algebraic geometry suit metric learn dictionary distance invariant multivariate dictionary metric eeg empirically also show synthetic show metric consider rotation suit ground distance possibility link underlie datum metric could nonetheless contribution operate application process loss metric multivariate dictionary bring framework new frame research try cover entire packing energy learning add constraint dictionary appendix manifold manifold field space class ii indexing chart open differentiable globally differentiable globally must chart coordinate chart formally compatible basis ie compatible manifold every vector usual sum scalar endow chart manifold hereafter matrix column vector orthogonal manifold orthonormal matrix positive integer unit sphere n span manifold plane dictionary pack multivariate remark theorem conjecture axiom france france universit paris france overcomplete keep grow address overcomplete despite recurrent assess overcomplete metric underlie yet henceforth overcomplete manifold distance distance reveal manifold wasserstein metric space study deep tailor eeg signal brain introduce embed competition besides principled packing compressed dictionary learning manifold pack sense question analyze elegant mild hypothesis know dictionary decade expert wavelet available learn atom state thank paper deal overcomplete representation topological space live thus qualitatively one evaluate benchmark meanwhile literature comparison assessment fall short recurrent cross univariate recall gram induce mapping allow equivalence partial order nonetheless approach harmonic overcomplete dictionary call theoretical numerous signal processing compression investigate packing packing problem good way surface sphere separate pack wireless communication frame pack problem exploit theoretical result bring frame inherently overcomplete metric norm cauchy distance reproduce kernel hilbert svm pack subspace process dataset series multimodal audio signal hyperspectral spatial eeg signal dictionary invariant intermediate particular define principled frame frame pack first recall definition manifold supplementary material formalize distance overcomplete construct wasserstein metric ground detailed assessment multivariate dictionary overcomplete metric estimate contribution real section overcomplete representation analysis interface competition variability produce consistency paper algebraic geometry metric associate pdf notice principal angle small singular span indexing principal similar review principal angle non function three axiom principal angle general metric space share vector principal angle separate first distance arc length geodesic nonetheless everywhere take let subspace small q column orthonormal formalize set thus use embed use norm fail thus pseudo metric distance frobenius also metric pl small define argue pl geodesic introduce subspace small allow pack principal angle equal nonetheless metric canonical subspace subspace cauchy eq introduced enhance stability canonical analysis cca base summarize table definition c metric de geodesic stand everywhere reader familiar frame introduce fact furth development please deep hilbert indexing set consider finite norm tight frame frame bound normalize tight frame function frame operator synthesis invertible frame frame mapping adjoint establish disjoint packing dictionary without generality characterize redundancy informative absolute q remove inner divide norm frame coherence dimension hold j frame approach frame minimize coherence frame vector frame norm frame hold equality call frame packing packing frame offer suited tool packing formulation goal packing pack show line pass angle angle absolute packing frame link code propose signal measurement k equivalently reasonably sparse programming noiseless exact sensing restrict isometry matrix isometry quantity hold constant independent sensing rip signal small could obtain bernoulli measurement order computationally intensive require determine value rely demand eigenvalue theorem coarse link rip k meet equality candidate produce sense dictionary aim capture overcomplete impose energy sparse dictionary update different deal direction energy learn empirically jointly illustrate part process put term whereas pack come sense dictionary drive interesting connect two recent possible packing pack dictionary coherence formalize equation packing coherence constrain energy metric definition set section family j nd pseudo underlie denote sequel manifold hausdorff separable fact borel algebra support close topological approach hausdorff subset hausdorff distance set turn hausdorff know limitation hausdorff reformulate allow wasserstein distance set r j rewrite q equivalently metric belong restrict dirac us collection w coupling wasserstein define eq wasserstein dirac wasserstein relate follow equation indicate metric allow compute subspace span frame collection subspace span another two frame definition formulate subspace frame distance act frame act span separability axiom relaxed identity axiom space pseudo metric pseudo metric desirable frame dictionary set frame major heuristic assess dictionary dictionary attempt similarity two achieve pseudo rely hausdorff wasserstein metric easy multivariate apply synthetic signal metric could box algorithm hereafter metric concrete consider dictionary learning signal handle sparsity atom possible coefficient compute formalize problem np convex tackle sequentially match pursuit review multivariate case omp signal learn formalize solve dictionary spatial trajectory eeg signal use dictionary audio image hyperspectral univariate atom multiply coefficient contrary atom multiply recall decompose independently q coefficient reformulate parallel signal sparse estimate associate active atom atom consider thus choose thank sequel rotation invariance decomposition q multivariate atom rotation core omp eq extend call study spatial trajectory attempt metric frame several immediate improve rely dramatically assessment reproduce art show commonly indicator section could computer qualitatively signal protocol hereafter dictionary atom synthetic call atom dictionary produce extract atom evaluate compare atom atom create white uniform atom dictionary generate sum atom randomly add conduct make set return training experimental slightly apply training equation atom detect recover correspond choose case equation percentage atom denote approach common community dictionary metric recovered atom atom naturally include correlation nd detection invariant consider algorithm include rotation assessment dataset rely equation thank equation dictionary ground describe select hausdorff wasserstein metric atom notation change sake clarity
easily look sphere satisfy show close example problem svd solution extend convert become substitute iterative unitary svd minimize well achieve element separately yield nuclear wish otherwise increase frobenius optimization problem semidefinite programming fan wish matrix solve know constraint contain singular necessarily iterative monotonically adjoint modified fx achieve norm self adjoint rule usually norm show global matrix optimal hold project involve axis case constraint reasonable achieve self adjoint investigate extensively completion differ entry constraint investigate rank np hard nuclear global solution matrix completion minimize nuclear apply address admissible admissible complete norm approximate robust variety corrupted size randomly difficult original nuclear nuclear nuclear norm reconstruct singular decay reconstruct technology theorem completion whose matrix approximation recently work theorem extend handle nuclear norm orthogonality algorithm convergence global discuss require parameter completion applicable mathematic electrical completion problem statistic biology signal vision important existence completion important netflix problem find satisfy np hard relaxation propose popular relaxation replace eq nuclear iterative thresholding completion base solve approximate find completion fan norm large value spectral norm
markov associate compose component view recovers strong edge uniform define thresholding suppose small probability define mainly norm fact argument use symmetric interpolation see equation dominate nature omit assumption result class ball show positive pick small semidefinite enjoy natural penalize study theoretical property researcher never include constraint latent weak let variate precision hide latent matrix complement sl naturally identifiable weakly q particularly make condition addition assume universal spectrum abuse precision inverse however ij pa scale constant addition level result omit limit hold ii assumption scale scale selection exist carry still issue level use large theorem address previous presentation require pn response mm c write consequence graphical bias random vector popular bind norm dual restrict eigenvalue compatibility cone condition design compatibility cone approach small penalty analysis majority allow associated compatibility compatibility cardinality want impose design level constant design deterministic let number negative quantile let right hand side thus easily moderately imply c imply represent true summarize hold let penalty lasso regularity condition deterministic row c level cause finite however section nk precision theoretical even theory suppose sparse treat write sufficiently asymptotic result precision matrix front bounding inference result parametric root equivalently proof somewhat case absolute possibly condition like make able long set compatibility eigenvalue population gram compatibility constant pn consequence compatibility extra condition require restrict automatically propose projection direction confidence discuss also set normality model two covariance scale lasso equation second score pick obtain final use scale approach however thing try partial compare enjoy form main understanding cover work mu loss observe asymptotic recovery generate precision block respectively block asymptotic estimating perform discuss remark multivariate scale k ignore theoretical glasso glasso third precision lasso selection table report entry report glasso glasso replication replication glasso penalize package design matrix object surprising entry figure theoretical super demonstrate match lead ij n empirical match level matrix great parameter glasso training penalty level proper computed rate overall summary substantial false glasso glasso possible false glasso tendency however hold low glasso consistently maintain glasso penalty roc various procedure glasso method poorly addition circle threshold plot glasso penalty cross consequence glasso glasso glasso glasso glasso block glasso glasso glasso le finite n independent denote eq dominating denote variation affinity version step loss always special case rest zero construction element equal later cardinality k p since imply b mb nonzero per row diagonal identity block identity therefore prove together pn c pn order pn pn partially explain bound derive favorable avoid methodology propose improve literature replace least favorable theorem proposition nsf grant support dms university popular among attract great recent year paper consider fundamental root entry condition equivalently show long possible achieve answer minimal sample test presence edge graphical adaptive entire matrix inference uniform strength precision hessian tensor matrix precision theoretical roc glasso class model tool investigate scientific central gaussian graphical recover dependence relationship vertex consist pair dependence structure graph define covariance edge consequently precision variate distribution without hereafter address two precision recover draw considerable penalize likelihood estimation bound hessian tensor norm concave early precision matrix run selector rest recovery depend procedure less practical require norm property analyze literature recovery largely alone achieve closely propose scale level condition size asymptotically estimator converge asymptotic minimum general efficiency linear coefficient however unclear condition fail detail paper understand gaussian way maximum relaxed convergence require sample propose adaptive individual asymptotic normality efficiency relax constant graphical proposing novel briefly task see length sub subset consider observation motivate regression response sense match maximum likelihood estimator space model sparse propose pc ii satisfie estimator rate eq furthermore jj le lemma novel implication matrix matrix support sparse literature immediate consequence estimation plug estimator formally estimate correspond scale lasso univariate q weighted length explicitly scale vector free simultaneous estimation general due oracle apply estimate residual b cost precision computation great run single rest I versus mm ia lasso p sp computation entire run edge model outline property prove prove scale estimator certain thresholded possess global spectrum relax complexity precision maximum measure relaxed ball p node spectrum oracle difference oracle estimator condition hold scale scale constant marginally variable immediately yield inference set n pc c efficient sn section estimate
transformation compute adapt gmm use transform gmm would bias task comparative technique exist digit ten acoustic build leave right hmms diagonal building acoustic training composite dimensional frame build time adaptation perform noise separately compute transformation clean efficient need viterbi adaptation especially case correspondence slight degradation attribute operate individual estimation advantageous short testing fast believe useful many noise recognition improve recognition unseen noise extension computationally standard would like efficiency uncertainty n speech recognition recognition noise especially unseen case framework non dataset clean noisy finally computationally efficient time framework improvement model run overall hmms robust speech build noisy environment acoustic performance environment former adapt match either clean feature characteristic piece wise learn train noisy model use gaussian calculate probability gmm decide assignment decade maximum mutual introduce see training also usage address issue recent eigen unseen noise frame extract adaptation clean space classify transformation whiten sub clean modification base noise work literature modify extend record minimal degradation computationally achieve well hmm adaptation viterbi decode contrast dimension rest modification dataset discussion versus section indicate two feature gaussian density clean dependent feature mmse tn separately matrix derive mixture mmse place eqs simplify degradation correlation clean perfect rewrite perfect make case per obtain hence eqs mmse whitening term bias see step gmm use give every alignment gmm counterpart clean compute eq I however necessarily counterpart clean frame hard cluster high indicator parse clean part clean noisy unity belong soft build mixture dark represent high clean clean gmm construct figure mixture correspond assumption estimate transform mixture clean maintain mixture exist structure clean linear transformation mixture ignore mixture extension absence clean exist mixture whitening transform clean mixture sort extension straight forward cross expect build gmm global clean transform covariance weight lose gmm clean clean gmm representation use note used pattern retain diagram estimate transformation non feature transformation three refine gmm
constraint compress sense completion max constrain recover also topic research range application filter netflix system sensor localization refer discussion structure problem stack low trajectory say move error range beyond camera field view miss real life therefore entrie suitable lowest satisfy hard sum view analogue effective empirically paper generic recover provide recover norm relaxation alternatively small integer view matrix factorization column usual trace define norm equivalently note space recently collaborative filtering problem max superior trace consequence risk quadratic loss second randomly numerical model example netflix equally movie movie likely rate point user active rate hence sampling uniform behave trace regularizer incorporate location max norm convex rate constrain estimator match minimax minimax upper together show provide robust respect distribution also extra avoid see discussion norm convex effectiveness also study first involve outperform programming sdp introduce basic definition max estimation minimax bound result method work implementation issue discussion proof key lemma collect trace repeatedly later integer integer complement cardinality denote norm p p p kl equivalently j u inequality analogous u max lee definition characterization matrix see trace point frobenius spirit norm let known rank one sign matrix technical tool rademacher introduction rademacher expectation distribute variable definition gaussian consider matrix ball max specifically rademacher complexity completion random sampling independently general sampling nk ns uniform consider general sampling motivated entry q write hereafter brevity clearly focus relaxation rank unknown quadratic nm require entry constant argue lee al bound constrain although minimization program optimization incorporate implement recovery use thm mc normal constant sequel frobenius least approximate without low assumption reflect bound direct consequence normality noise least noise exponential great give max constrain square theoretical completion ball minimax bind rate sampling satisfie norm indeed positive amount standard lower bound appear constant show constrain minimization use program norm choose q exact matrix recovery proof analyze estimation risk function develop uniform sub purely max recover parameter norm logarithmic minimax implementation present hold minimizer approximate parallel line study penalization nesterov lin al restrict less recommend use lee tailor large correspondingly solve interior method fairly scale alternative factorization begin know guarantee number value less rewrite original factored product reformulate problem minima lee et al method differentiable argument generate iterate recursion intermediate stepsize still next current exceed exactly norm unchanged completion allow entry fm decomposition take step opposite apply rescale respectively project back change iteration demonstrate computationally norm max matrix miss quantity advance directly miss estimation fortunately many real netflix rating motion camera feature trajectory percentage entry large alternative recommend rank recall motion frame feature regard miss miss motivate recover describe implementation constraint completion transform modulus stop iteration solve full go final norm fix bind max contrary trace norm minimization algorithm uniform solve interesting open whether accurate guarantee prove proof technical lemma lemma give exposition write ambiguity note problem thus combine model yield major challenge consist part bound side training direct consequence mean variance bound uniform realization sample noise case sub realization sub exponential yield constant union step j regarded exist instead result n probability follow radius satisfy n n follow show uniformly view set sequence event f c elementary sample combine finish one hand follow n apply yield c estimate imply bind exponential basic banach contraction version rademacher ball index mi f
ix k denote regularizer ir different discuss however computer one devise section iterate computer update free establish labeling start computer modify inherently allow analysis various computer computer coordinates practice asynchronous section block role stepsize understand speedup parallelization distribution easily interpretable upper bound construction one submatrix likewise number parameter shall fix proportional would wish safe practice hard bound much ml dataset hand remark cover ignore implied expense translate value sampling column e ie si sx zero expectation identity view nd bound term plug inequality study therein parallel study convex partially propose efficient protocol distribute algorithm recently design batch dual coordinate sdca mini descent mini lead acceleration long mini help theory recently nonsmooth loss accelerate counterpart nonsmooth descent minimize loss none distribute strongly norm convexity subgradient show assume substitute lemma together h rest follow strongly big setting parallelism relevant shall comment several capture comment influence various computer coordinate instance dependent depend instance stepsize notice enjoy linear focus lead sdca set quadratic provide avoid good need good special assume several need pay partition favorable circumstance even lead get effectively remove randomization eliminate certain extent choose partition minimize proxy ignore estimate update take iteration rd htp illustration purpose several well updating coordinate plot axis red line update updating coordinate solid comparable line pay node node communication big utilize aware coordinate closely parallel difference norm analyze strongly convex establish size independently number parallel regime loss look consider sampling moreover provide exception nonsmooth convex extend nonsmooth loss obtain rate batch ascent mini setup otherwise see result large regularizer importantly primal whereas sl j j store value piece store iteration protocol sl formula refer th row node store locally hence node add sum basic protocol obvious identify parallel serial ps procedure communication serial fix compare fp htp reduce significantly take e protocol message nearby nearby computer hence place asynchronous communication protocol message iteration htp neighboring send iteration already know send node rule need slowly affect analyze protocol take propagate node store evidence capable efficiently boost execute equip core gb ram square sl control modification well protocol organization avg ps fp ps fp ps discuss advantage protocol benchmark advanced protocol compare iteration approach run core datum double precision gain overhead hence case speedup fp ps communication remark bad generate matrix angular arise average compare send l fp magnitude minute tb coordinate replace partially except pair submatrix column clearly th let fix scalar define claim block lipschitz constant satisfied h x k h block separability imply remain argue coordinate p x k k x step argue simplification ij one verify first second remain substitute desire form parallel analyze hybrid problem initially feature assign pick coordinate independently computer compute apply give approximately numerical lasso tb randomize popular include boost scale regression randomly move loss type setting clear modern share computer partitioning block operate utilize library approach extend involve subset combine coordinate recent coordinate
entire appear recently go selection validation big take prohibitive budget day conference try test thompson high select randomly test via kernel propose outperform design regret strategy room automatic describe focus treatment demonstrate possible frequentist conjecture much strong prior address maximizer constraint evaluation know multi bandit empirically counterpart bayesian emphasis model arm perform number arm large allow practical automatic present thompson good identification relative address smooth function evaluation generally corrupt form importantly budget within query adaptively construct element automatic common option ad mobile application scenario company offer variation small product entire base crucial subset find good product boost forest vector solving give big technique validation function attack three function important greatly exceed recommendation make action optima finally handle explicitly word concerned optimum work approach design goal frequentist counterpart place emphasis detailed situation number arm much number frequentist counterpart paper comprehensive good examine relative different previously frequentist broad overview recently great attention black box technique hyperparameter type combine posterior turn use construct query acquisition function improvement pi ei confidence ucb key strength bayesian capture correlation many bandit online optimization optimize function prove rate regret variant expand evaluate expand expensive propose contrast optimization significant armed bandit often arm action arm often attack bandit discrete immediate arm characterize mean act arm arm distinction introduce sequential decision round maker select arm decision maker arm previous tuple arm regret select arm denote interested find arm exact learner problem incur instead sampling evaluation phase round wherein maker make arm maker write exceed wherein arm assume depend arm write perspective round effect choose arm give generalize reward posterior interested primarily write dy access induce marginal confidence trade exploration analytical next arm associate distribute reward condition marginally dependent unknown level vector place prior analytically model seem restrictive include detail construct follow row restrict discrete implement pose attack software discrete action vector begin round arm marginally kt chapter green depict example interval form discrete domain trade exploration thompson sampling set discrete arm query build offer principled incorporate correlation whereas early independent begin maker equip low arm set quantity rise diameter arm introduce q high among alternative ultimately quantity regret rather arm gap intuitively arm e information arm time choice subtle pseudo update coincide arm whereas optimality hardness essentially point identify arm good detailed speaking bounding bound hold regret attain decompose term outer note remove e replace final simplify decrease solve solve term statement proposition upper conclude proof frequentist analyze bandit implicitly regardless avoid key thing require hardness quantity quantity adaptively control much directly control width uncertainty encourage hardness bound conservative posterior deviation emphasize turn step adaptive modify pseudo hoeffding note assume arm bound reward roughly quantity primarily distinction prior fast simple reward adjust finally relationship arm remain subsection application traffic experiment speed sensor south california working entire month also different due specify restrictive treat historical policy detail gp matrix
polynomial drive tuning technique fold estimator precise definition cv cross contexts especially risk selection support cross know instance cross use cross validation address lasso dimension collection inequality relate use treat aggregate result dimensional numerous penalize theoretical validation fold cv penalize support non validation inconsistent tend select many recover cross attain lasso implicitly selection share stagewise regression raise concern fail consistent gap algorithmic literature stable procedure induce like stable risk nonetheless practitioner cv inconsistent obtain consistent position generally validation sound cv tuning recover predict tuning generate regularize algorithm setup concern freedom scale via useful provide far describe property predictor class allow predictor similarly dependence simplicity omit subscript little predict random choose form response lasso generalize group analogously cv remove estimator lastly risk set nonnegative analyst need choose analyst interval select procedure practical nontrivial quality eliminate thus treat upper bind eq observe would column equal inverse theoretically well suffer rank least potentially include estimator consider interval main demonstrate tuning define estimate cross decompose q study excess risk emphasize new meaningful assume expectation allow quantify notion define integer give norm represent converse dimensional random center product constant independent measure indexing natural indexing include common moment high dimensional abuse refer sequence exposition validation sequence intend unbalanced cv fold prediction define f f set usually oracle predictor correspond negative bounding case discuss corollary right go tend put less mass tail fast oracle fast growth set increase fast validation set consistency prediction inequality validation validation fast additionally rate comment high state must increase slowly potential analyst use potentially quantify requirement essentially n generalize constraint nm p validate tuning oracle performance alternative degree freedom validation condition design ratio measurement variance lastly form correlation diagonal element simulation define also random amount vary signal define eq noise degree equal qualitative differ greatly predictive cross validate consider regression degree freedom degree bind independent freedom therefore reduce bayesian bic operational practice estimation quickly grow fan variance true aic bic available represent axis lasso tuning accuracy plot plot along connect replication indicate simulation bit surprising perform optimally task likewise design true relatively poor aic cv exception sparse low correlation cv lastly cv except h snr pdf n pdf h pdf e simulation snr figure snr e figure e n pdf simulation figure snr figure snr pdf simulation snr snr figure snr snr snr figure snr pdf figures pdf figures pdf figures figure snr e pdf snr simulation theorem risk result preliminary lastly prove main corollary rewrite formula form eq lastly rewrite use general norm r bernstein several need entry wise constant depend q induce measure mean eq furthermore take combine decomposition section part address inequality likewise inequality minimize use give sufficient instead straight q completeness equation normalize write random large q take nb om nearly tune however minimizer analogous case ball datum estimate choose validation unfortunately choose tuning predictor provide impose achieve risk yet analyst interested choose good work reveal interesting open
use use iterative frequency case signal array emphasis search uniform array separately inter resolve ambiguity projection correctness guarantee introduce discuss detail denote transpose conjugate transpose refer delta stand operator matrix problem sensor noise q distance zero complex identity covariance write l respectively eigenvector noise become exist lead late part briefly originally toeplitz complex angle root polynomial asymptotically estimator mode guarantee minimize iterative detail inter share sensor array efficient course arbitrary slow structure array enjoy estimate way h inter element domain theorem angular domain root still problem period root distinguish root source coincide confident unlikely consequently become source impossible exact angular two plane color blue correspond angular remainder guarantee one result angular consist line segment onto angular line segment line intersection intersection fall result segment find calculate modular modular advantage optimal iteratively try modular equation iterative avoid gets suppose output knowing marked candidate likelihood one h short angle plot project point near line segment onto angular select large simple propose consider target output point plane close close distance project point back onto segment example project onto project modular solution angular equation state likelihood final performance method basically combine project combine use plane probably even fig error indicate move project line shorter outside expect propagate low fall incorrectly project segment final poor explain threshold point phase method go decrease four point lie area project segment error target two consequently intersection dimensional entail identify point particularly suitable source array overlap angular even domain estimate accurate simulation root music er root music fourier source array fix number db db music fourier series mean square mse music root error fouri improve reach experiment keep varied remain see mse show root music music experiment keep varied snr db method db order favor last experiment step root repeat simulation target figure array fig different snr observe fig fig method reach db db order favor propose root increase really db root music array fig snr array db well music mention fig mse rapidly snr db effect happen db justify summary music drawback music fourier satisfactory suggest number fourier approximately signal distance array root music achieve satisfactory root music expensive even root music fast free exploit estimate music method entire angular map segment array china ph department electrical computer engineering university research statistical degree electrical ph electrical department electrical computer associate processing transaction receive processing award distinguish signal processing receive european association processing technical award array research exploit redundancy span large suggest array consist search direction arrival array operate array processing advantage
suppose small high motivated investigate phenomenon simulation code simplify geometry surface control physical simulation proof rely uncertainty adapt exploitation fast regret example criterion update conjecture regret unbounded gps initialization specify greedy performance estimation parallel batch analyze benefit purely cumulative introduce upper pure exploration algorithm strategy pure exploration batch evaluation upper batch improvement version constant dimension empirically convex mean observation real application input expensive evaluation challenge global arise system determine heavy maximize output minimize cumulative optimum horizon selection deal exploitation successful different address bandit particular optimistic bandit batch typically sensor available iteration core explore potential strategy plausible novel base benefit pure batch exploration component helps support location maximum regret algorithm base need derive suffer curse dependence previous mention reduce benchmark remainder organize background formalize describe ucb pe concepts section relate algorithm series synthetic address maximum denote optimization opposite query horizon standard formulation incur regret define eq upper far regret case low want function sample gaussian formalize intuition extension multidimensional gaussian write eq form gaussian variable covariance gp finite condition formulae gp location location tx rgb width height style align sep xshift draw black forget color mark option forget solid realization grey common degree kx radial exponential kx lk kind dimension posterior observation grey area deviation distribution f illustrate envelope grey area high contain relevant contain dot discard figure refer process modify region also formally instead guarantee leave future x kx bind pure query batch tackle explore uncertain suppose location select region aim gain location formally reduction know far find integer due efficiently select one never maximize easily posterior greedy strategy depend location improve procedure two represent horizontal dotted green maximizer query inside cost prohibitive drastically mean build always far mention approximation challenge theoretical article need adjust confidence probability least derivative regret gain fix variance cumulative initialization multiply equal query initialization kernel kernel report ern rbf de theoretically bound regret pure refer bound sake batch via proportional knowing prove divide allow high least c ern c ex refer point select policy x principle location observe point sum divide definition get gain select maximum sequential express variance deviation equality variance see maximum gain lemma fact finite compact q
mi convex straightforward practice cope alternative measure introduce mi pe divergence thus common non independent advantageous review cope expand express suppose express kernel denote kernel element expectation posterior model maximizer c denote scalar class normalization furthermore negative take post processing issue account maximizer zeros max operation manner denote prediction assignment say notable maximization cluster parameter systematically optimize maximization useful compute select suitable semi set must mean link link way assume constraint link constraint incorporate link share class let perfect commonly link utilize encourage link far utilize belong let incorporate link opposite way link strongly encourage must link modify increase link decrease link link modify matrix modification demonstrate promise maximizer analytically lead eigenvector see biased propose indicate tuning experiment optimized tuning experimentally sl method semi spectral post thresholding neighbor add choose exclude dataset ari ari ari number link deviation ari dataset group baseline performance link reasonable link allow algorithm find reasonable sl heavily systematic sl important systematically sl tune particularly binary perform problem post poor alternative drop increase phenomenon observe sl overall show promising method maximization utilize link method name advantage conventional post step post mean cause degradation furthermore systematically determine width automatically optimally sl analytical supervise although analytical experiment previous possess negative role acknowledgement technology support choice kernel develop would determine maximize use favorable unsupervised stage already review q go mass parameter save kernel basis randomly minimize yy regularize notable analytically follow q ratio depend parameter tuning nm ratio obtain without hold denote summation summation pair procedure q minimize one give ratio obtain institute technology ac cluster process paper semi previous square link propose link usefulness demonstrate maximization mutual semi cluster base similarity classic rather produce limitation feature non manifold similarity embed lack model learn assignment maximize unsupervised maximization tuning measure systematically maximization principle among demonstrate give situation regard cluster side tied link
well quickly close optimum full simultaneously reduce per extension strategy incorporate order fast improvement e focus extension quadratic compute update product descriptor hessian implementation curvature inexact solver also mini gradient sublinear tight fast paper organize review technique discuss particular theory much rate regularize broadly technique batch quadratic optima less obtain broad convergence weak assumption carry convex log iid pair vector value scalar true density find subroutine find iterative minimization compute regularization initialize subroutine z iy f r summarize every strategy compute outer complexity gain factor nonetheless serious issue subsample algorithm stochastic grow proceed method development practically quadratic immediately nonetheless analyze much sample good overall intuition support effective competitive analysis first require carry objective likelihood mainly focus convergence rely iterate simple preliminary likely py note weak batch size grow etc apply infimum possible batch satisfy vector x taking give apply batch select assumption lemma suppose algorithm context k full regularize problem k second batch relate bound direction individual contribution hull trace variant method sample strategy particular mini batch mini size kind explore implementation lemma characterize strong lipschitz gradient l bind h empirical straightforward hessian q lipschitz rate eq incur batch growth schedule conclusion denote take briefly describe system structure return instead compute implement mini iterative conjugate update use interesting invertible make consider inexact update justify quadratic therefore k x kk kk z iteration range tell system particular simply small serve initially small fully experiment linear gradient increase specify gradient curvature term update follow efficiently small number choose set updating bind choose stochastic algorithm rate advantage update inversion establish minimal tuning know constant condition otherwise limited bfgs quasi bfgs competitive descent competitive pre constant implementation implementation size lipschitz constant implementation use implementation adaptively search method choose scheme sag fast bfgs dataset task regularization dataset testing dataset remain belong show report rely run bind comparable code publicly present together several theory stationarity weak hypothesis particular convexity logistic provide convergence large particular develop flexible setting include develop fully
high selection cost reflect consideration second true freedom compute residual repeat recommend case fit prefer prediction behave reasonably even good replication factorial combination sd superior inferior normal prefer selection model factor outcome safe superiority strategy sensible perform sd preferred scenario practice consequence prediction decomposition cost dominate comprise proportion contribution simulation finish response logistic build aic vary simulation replicate factorial preferable predictor really predictor full work nan safe valid binary response conclusion circumstance full strategy necessary try limited mind generality building implicitly substantially model strategy since cost scenario furthermore building completely tend strategy sd safe might well analyse datum difficult impossible circumstance preferable performance indicate limited situation loss recommendation response group hierarchical serial analyst decide prefer involve analyst bootstrappe preferable experiment split preferred analyst choice range numerical graphical safe prefer clear insufficient split find estimate analyst approach switch empirical evidence give default necessary recommendation type explanation certainly arise interpretation change split part reliable distribution score strategy splitting decompose parameter splitting investigate simulation introduce safe use model datum validation uncertainty base sometimes prediction compute outside expression variation generating change alternatively broad fail advance model frequently aware overfitte avoid balance complexity proceed make assessment reflect failure discussion select practitioner frequently action difficulty problem integrate example box cox method select might lasso succeed combine variable get realistic involve graphical numerical inference impractical assign unless reasonably specify idea resample method pre specify automate software possibility split necessarily avoid problem software analyst element advance select furthermore gain performance purpose discuss strategy model sequentially discuss split use effectiveness present simulation split variety purpose wish generate datum author obtain future prediction certainly obtain realistic quality naive estimate fit selection quality try splitting prediction surely make estimate prediction quality whether loss competition netflix competition model could internet business challenge develop predictive case requirement trust analyst naive validation cross future author purpose call ill purpose distinction statistical clinical validity shall restrict concern hand splitting validation include also hypothesis testing see predictive purpose interval record whether fall tend full hard split strategy datum number selection fail entirely choice could estimate replication simulation follow replication full always component split insufficient estimate full split outperform selection understand split cost one pay cost high fit cost strategy datum split sd select safe valid validation correct select use generate new safe see sd parameter safe avoid hand loss safe severe confidence safe valid motivated motivated datum may reveal standard type model little type tend affect improve one suggest splitting activity use would nice explore effect split mathematically unfortunately issue practically calculation make determination convergence arise post become impractical rich hence resort simulation cox select index setup box cox finite interpretable determination replication factorial generate score simulation predictive frequentist distribution plot safe superior
maintain beta prior undesirable feature contribute evidence ignore update distribution ibp variational likelihood feature dominate update mixed expectation propagation style inference order ibp evidence truncate latent infinity form gaussian linear accelerate conjugate model effectively factor ibp take mix accelerate sampler ibp currently accelerate ibp latent instance nonnegative heuristic bs ibp sequentially add assignment use probability evaluate collapse possible assignment iteration iteration ibp nb x log evaluate three runtime hold gaussian variational include iterated factorization mean sampling therefore truncate method center input mean infer initialize deviation multiplicative average iteration matlab implementation algorithm respective author ghz processor create supplementary randomly standard deviation random bs ibp synthetic hour bs ibp small dataset converge among ibp parametric fast ibp ibp eventually outperform small dataset outperform large method converge sampler eventually mix perform include marker plot convergence summarize bc dataset bc eight bottom five size dft tag test real average five indicate size marker marker unbounde converge ibp perform ibp iteration initialize optima hard step beneficial ibp sparse converge order inference though dataset slow sparse take converge converge long outcome fraction time indicate ibp bc likelihood converge test likelihood visible perform dark covering face collapse htb shift equivalence multiply column column algebraic eqs equivalence limit equation fourth harmonic shifted equivalence class nearly identical column assign equivalence prior column turn assignment limit main text assume hyperparameter estimate place gamma infer equivalent indicate subscript variational variational yield inference exactly variational hyperparameter shown submodular examine hold evidence naturally couple change factorial column depend conjunction text indicate reasonable eq straightforward q subscript indicate dependency explicit add inner inner similar term add become indicator state main text subscript remove maintain characterization bayesian nonparametric need specify slice sample nonparametric prior variational amenable nonparametric variational variational must specify heuristic limitation empirical global start perform type infer feature true dataset intensity value subset figure image yield four initialize initialization perform spaced option unchanged convergence experimental top histogram number row middle bottom top histogram ibp tend experiment simple medium factor latent occur split comparable infer instance difference unbounded prior operate axis xlabel ylabel count black forget draw forget plot coordinate draw black forget draw forget plot forget draw forget plot coordinate fill draw forget coordinate fill forget fill forget black forget coordinate rgb scale xlabel color pt option forget plot evenly spaced increment hyperparameter option inference model inherently grow input feature perform map inference model ibp submodular function maximize via scale linearly efficacy dataset currently ibp machine prior equivalence binary matrix row latent face latent model ibp factorization observation plus formally linearly combine latent noise place ibp unbounded ibp inspire version ibp model challenge ibp restaurant assign observation I ibp factorization term inference enable least comparable variational inference converge structured material presentation result arise ibp ibp place beta entry prior infinite limit particular take equivalence matrix leave show ordering row examine equivalence right zero maintain zero ibp shift hyperparameter harmonic supplementary material derivation well equivalence equivalence shift simplify mathematic algorithm general global local rv global infer operate kl divergence posterior original problem update commonly let assignment I instead maximize local compute I recover bayesian regularization global ability point structure scalable optimization tractable ground set incremental benefit desirable discrete globally minimize np enable determining estimate I scalable submodular maximization present linear ibp inference arise column dot element specify eq ibp nonnegative nonnegative submodular maximization optimize conjugate nonnegative g assume supplementary I maintain constraint kl maximize evidence kl posterior nonnegative ibp eq q bind simply ibp specify benefit ibp affect inactive see breaking cause lower inactive kl divergence kl inactive kl evidence nk k boolean plus indicator prove proposition quadratic boolean eq eq nz nk v yield submodular yield submodular unconstrained submodular np local ls optimal nonnegative
vary range complete list store store equal index meet element large sort one describe form hence term concatenation length follow could collection note shall us list explore time updating simply read appropriate explore serial th node explored group order boundary respective range available word define wish boundary node optimal exclude long store separate exclude update consist node exclude computing contain select group hence optimal case node include explore computing th choose first choose element sum ensure indicator eq case region overlap consider step need reason store separate boundary clean overlap currently select perform explore combine value large store unlike perform maintain optimal thus b I correctness correctness correctness rule correctness rule correctness correctness task group set node store derive well select exclude property know group obviously group possibility precisely obtain active lead optimization well trivial choose previous maximize optimal obtain correctness interpretation group exclude remove node algorithm determine independently much time determined explore thus table avoid store number equal number rh effectively one term term merely term order successive differ element operation rhs total value operation need perform update fix operation equal whether contain indicator preprocesse sort index canonical check one operation step equal remove explore entire boundary intersection exploration significant cost ignore b early encounter explore algorithm ignore compactly backtrack require e group choose backtrack storing amount rule g f shall simplify use backtrack start work select group large choose selection store th involve besides variable allow large tell optimal ignore rooted subtree subtree root subtree root connect subtree subgraph example store weight root subtree use x define store start move assess leave eventually root subtree root consider inductive induction child leave value follow rule optimal node subtree root allow equal weight must plus choose root subtree rooted subtree child pick subtree child pick remain subtree child least child less child child problem root connect subtree node root subtree explore store root subtree leaf weight finding root connect subtree subtree rooted subtree hence subtree store pick subtree subtree equal subtree child subtree child maximize subtree current structure dynamic child dynamic evaluate cardinality cardinality root evaluate operation leverage regular tree prove dynamic linear prop program regular level hence root j program select sub sub tree select require update fx fx operation break regular complexity prop dp dynamic tree number sub jj standard backtracking store maximum regular complexity induction without tree child connect level node connect high go connect child run dp would see regular tree regular tree respectively level cardinality subtree level see illustration hypothesis maximize correspond form theorem dynamic program regular acknowledgement thank anonymous constructive observation thank provide group sc physics ph machine science college ed include compressive sense currently laboratory information system interest mathematic electrical engineering minor science technology work research period electrical engineering university electrical computer engineering institute technology hold position ed university well international structure interest sc ed fellowship interest include convex machine analysis statistics b degree electrical engineering minor computer work interest compressive sense energy challenge nb mit mit edu edu ss com instrumental recover compressive interpretable group underlie know leverage group dynamic programming furthermore lead relaxation generalization pareto sparsity computation trade framework relaxation structure interpretability compressive many appropriate basis sense exploit compressive reduce accord cs theory signal sparsity bandwidth shannon sparse cs theoretical cs sophisticated structured structured number noiseless presence furthermore facilitate term structure understand either naturally expression bioinformatic computer might genetic constitute tumor allow certain incorrect speed cancer sparsity collection variable j dimensionality intersection problem signal sense budget call group short projection fundamental iterative thresholding algorithm problem impose group approximation group constitute call selection allow discover group instead precise image technique circumstance correctly signal combinatorial polynomial find certain affect measure exactly concerned problem support irrespective affect would computation compressive focused leverage group number recover signal overlap model difficulty feature well understand relaxation approximation select complement group instead overlap consider cast selection uniqueness prop infeasible intersect origin intersect equal recently relaxation group support consist condition care particular support numerically group support might incorrect instance obtain code structure constrain propose name code code scheme namely sum combinatorial homogeneous pp cover cover set set cover relaxation x take completely discrete rely relaxation contribution prior version proof due lack refine propose discrete support enable group sparse selection group problem maximum instance hope characterize find guarantee present tractable leveraging base exploit novel solve whose forest indeed relaxation group relax group term concept program solver graph induce forest relaxed sum algorithm discrete constraint individual group group program discuss interpret framework convex relax cardinality however decompose approximation norm atom relaxation produce pareto different section concept section model connect analyze section relaxation example relaxation present section detailed description program indice cardinality indicator vector dimensional identity x normally letter bold letter totally obtain efficient relaxation program totally tu singular main building group collection index name g bipartite connect node example adjacency bipartite encodes circle thick draw thick sep sep fill white minimum pt sep label g g g g g g example bipartite group group text intersection connect cycle group bipartite cycle thick circle thick pt sep auto distance blue label edge node edge induce group intersection class whose tree forest acyclic necessary group graph ground set e overlap note partition belong alternate consider cyclic note overlap acyclic circle draw black white pt sep auto label node node n acyclic add interpretability cover argument introduce reformulate set cover group binary group least active cover restrictive group group cover guarantee cover definition norm define minimal minimal group cover exist cover group sparse contain interpretation sparse constitute hardness finding interpretation lead tractable interpretation positive easily solution problem acyclic structure sparse solve solve problem group follow problem solution change achieve j specify variable make cover produce instance give np hardness solution select cover however structure structure acyclic dynamic solve included exclude strictly correctness set long intersection group maximum coverage hard overlap generalize devise pseudo polynomial problem acyclic cost integer polynomial keep track weight relaxation allow obtain approximate sometimes computationally relax form b n hard totally concatenation tu tu concatenation identity tu tu tu structure lead totally constraint intersection constraint tu tu transpose tu role column corollary totally opposite column represent intersection bipartite two common acyclic structure lead totally acyclic structure forest bipartite overlap group totally constraint result tu transpose tu prop column partition set condition totally partition group overlap furthermore entry sign belong opposite sign via primal great practice solver may still fast bad another energy maximization tree forest problem formulate find probability factor node potential single probable max message root lemma regularize coincide satisfie also direct prop find solution also pareto solution discrete intersection pareto convex pareto minimization eq vector cover possible structure cover require exist pareto value optimization achievable cover therefore infer known solution admit support convex hull analysis find generalization x design call group literature weight atomic author recovery find trade group sparsity recover constitute support weak group define structure capture hence minimization standard sensing base acyclic identification approximation via relaxation guarantee group open characterize signal admit identification relaxation example group support dynamical able recover correct cover minimal cover g inner auto label label graph correctly identify group tu group cover decomposition unitary unique group cover use cover minimal leave future obtained characterize secondly show generalization generalize totally solver wide association find usual sense generalize individually select within weighted index hard turn structure allow acyclic groups dynamic solve program describe appendix polynomial frequently encounter processing denoise wavelet select subtree root root node pt size distance node child child valid type represent node consist impose overall discard problem relax dynamic polynomial dynamic tree child manuscript dynamic tree follow regular tree computational show fast bad case complexity find appendix budget obtain binary program regularization control active select solve time totally due zero tu preserve totally totally result prove structure totally use binary totally column appear permutation leaf group depth consecutive regularize approximation address link solve problem relax smaller yielding root approximation pareto purpose simulation limitation relaxation greedy correctly wavelet see wavelet coefficient image regular multi orient group consist child element apart scale tree intersection lead matrix right fig pixel actually discard cover ground figure pareto approximation error approximation solution totally tu relax latent problem dynamic achievable tu relaxation group yield lead great error need group order notice greedy solution dynamic program select greedy simulation right allow wavelet select variable triangle stand active group main plot blue dynamic program yield pareto lie hull three signal wavelet decomposition blue group zero solve still find impose constraint constant signal haar sparse vector coefficient propose totally linear relaxation use norm pair call parent child enforce satisfied favor report hierarchical p gp group solve equivalent totally assign support block regularization parameter solution different approximation dp solution discrete point pareto achievable tu group tu relaxation parent sparsity price select parent pareto constraint able group dynamic propose thompson finding root length draw implement matlab constant haar bottom signal hierarchical parent constraint parent second budget keep group characterize group find polynomial relaxation simulation relaxation approach relaxation group cover pareto original group turn convex relaxation include spurious one summarize remain question answer circumstance relaxation yield solution secondly assume basis learn compressive represent onto overcomplete coding extent overcomplete characterization interpretation proof similar line start intuitive understanding description proof correctness space consist group maximize contain term generalization coverage fact structure build solution certain class community account idea behind programming subproblem look structure solve na hope global fail next sake g optimal involve select involve element optimal involve select group element describe g yet selection group graph long dp yet group decrease also happen
monte carlo integral observe rapid growth unity change imply globally transition concavity minimal becomes couple mark start saddle entropy distance fix search compatible compatible distance branch branch globally saddle point fixing observe branch boltzmann measure unstable space isolate solution separate close become evident extremely find tuning couple curve infinity sr thus concave entropy curve imply stable show solution seem continuously binary perceptron determine solution isolate explain heuristic enough increase solution thus become consistent computation ref density dynamical replica break scenario go separation say isolate instead many solution apparent typical distance landscape study landscape increase solution grow dominate typical typical trend confirm message suggest solution concentrate dominant landscape weight clear increase distance landscape large characterizing case replica symmetric agreement replica computation message picture landscape entropy distance landscape random configuration reference clearly constraint landscape deduce picture refer ground energy minima dominate landscape expect satisfy isolated solution grow simple search heuristic certain interesting around isolated responsible hardness address theory landscape analysis check code division access landscape performance grateful comment early version partially sr give derivation limit note tr derive dy dy limit q sr positive dominate line reduce limit reference department intelligence science technology department technology water china perceptron input set binary difficult pattern constraint suppose organization landscape configuration ham entropy replica confirm numerical instance pass solution constraint landscape deduce feed forward either mechanic algorithmic extensive pattern constraint capacity phase vanish implement classification input find limit replica ref show capacity accordance still maintain ham quite perceptron perceptron critical search increase local search organization order landscape solution distance rich throughout paper refer hamming landscape study graph map onto node pattern learn see b graphical efficient learning cavity representation solution problem replica trick limit confirm computed replica equation cavity context apply replica arise pure mechanism understand however cavity focus yield replica remainder organize sec derive self compute landscape landscape number solution configuration replica rs computation landscape derive message pass instance cavity distance solution rs message pass discussion conclusion sec cm cm pattern binary vector value variable mean connect binary perceptron classification random pattern figure binary correctly output coefficient define pattern serve empty nj pattern map incorrectly energy convention otherwise unity without pattern transformation landscape reference configuration entropy landscape reflect organization concentrate solution reference sum function overlap limit saddle point energy transform probability recover jensen energy however approximation alternatively take operation input pattern compute integral overlap dirac nj ij saddle read saddle energy quantity limit input pattern replica trick compute first although replica generally rigorous check simulation replica overlap associate counterpart replica point arrive formula free saddle equation self equation landscape reference configuration term define eq landscape replica equation apply cavity define cavity trend increase confirm use pass consistency replica shape similar growth distance exist illustrate typical constraint replica result entropy vanish typical typical accordance instance finite solution evaluate rs check dynamic replica breaking define typical intra contribution weight two confirm sufficient rs capacity capacity population probe solution distance landscape spin attractive coupling configuration ref rich structure landscape equivalently entropy value entropy set eq coupling nj predict coupling field multiple strategy section rs landscape eqs understand follow use sec ref maximization respect lead saddle saddle equation entropy energy landscape pair replica replica configuration computation complicate w replica carry tx ab ab ab r computation b transform replica symmetric free f dy x self derivation saddle saddle entropy negative respectively analogy definition weight component state constraint satisfy two cavity recursive normalization constant propagation bethe free partition cavity probability b simplify simplify j impose normally number characterize property bivariate b computationally demand require approximated order constant determine vanish contribution integral shift due variable adjacent obtain shift addition j correspondingly solve sec bottom connect symbol stay numerical
literature em mixture em mixture issue unknown component among automatically capable select number sensitive regard initial develop optimize message length mml penalize log penalization algorithm cluster proceed measure penalization include proportion reduce mixture address problem overcome drawback propose problem become serious detail datum concern reduce rely generative mixture next mixture base introduce generalize analysis reduce include effect I differ one I regression unconditional indeed proportion mixture regression expert proportion know model logistic segmentation curve cluster I temporal cluster associate set curve number approach assume proportion suppose model spline gaussian arise noisy polynomial matrix ij tp identity mixture adapt mixture depend regression curve conditional pz kf mixing noise vector log training algorithm ik polynomial iteration expectation log start complete observe simply compute curve pz k k curve step update give solution n compute notice sensitive initialization cluster procedure attempt curve algorithm regard initialization proceed regression mixture curve cluster multivariate indeed curve reduce spline spline fit start maximize derive estimating likelihood consist term account govern cluster choose curve log represent class hz pz pz assume whole I additive penalize propose lead penalize k maximize mixture equation control model optimize cluster large fit smooth smoothing closeness fit less get discuss maximized curve training curve iteratively curve em step penalize log rely log likelihood ik n initialization strategy step compute current proportion proportion r proportion constraint solve multiplier update update mixing update small penalization cluster competitive discard logarithm proportion tend cluster enhanced increase proportion therefore entropy coefficient competition one another decrease discard small cluster less proportion stand gaussian prevent following adapt k n per curve ik regression consist solution square ik ik ik proportion posterior probability initial cluster mix proportion variance fit polynomial curve avoid middle sort I stop estimate regression two iteration summarize model curve htbp input n discard proportion q compute k non simulate curve th follow ij respectively class linearly space deviation variable proportion top problem consist th mixture robust curve majority arbitrary curve polynomial second decrease rapidly entropy overcome determining regression mixture result demonstrate concern curve mm concern rely successful perform maximize observe initialization standard multivariate datum gaussian spline spline regression mixture number proceed fold simulation study confirm regard actual exploratory observe summarize miss domain label difficult explore etc divide group dissimilarity another belong cluster prototype cluster cluster aim hierarchy cluster agglomerative cluster successively merge move merge operate prototype partition criterion variant fuzzy map som unsupervised visualization generalize competitive allow winner minimize account aspect approach rely approach popular cluster analysis density component problem assume estimation maximize observe achieve expectation algorithm maximize locally log therefore may addition em choose etc focus use observation temporal gaussian develop include mixture regression mixture cluster regression spline spline regression cluster em proceed fold allow external
whenever px encodes x f figure gray example gray assignment encode occur connect context encode together nonetheless ib statistical test pearson inconsistent vary specific correct underlie px distribution learn ib due perform gain cost test involve perhaps know create correct test error conditioning variable ib efficient compute recent improve hc hill ib hill start add reach maxima correctness improve reduce cascade test present child structure encode generalize search space context context generate nest loop outer loop explore loop test accord generation generalization correspond fashion add exponential initial feature example match discover explore complete dataset context adjacent x order conditional independence independence conditioning encode straightforward adaptation independence independence test practice w namely encode specific encode px correspond subset x factorize new x w remove w encode x generalization consist satisfied figure feature remove remove encode f notice figure feature f f x x b q b x explanation put together start feature context explore pc subroutine subroutine consist feature subroutine receive end return accord one atomic trying conditioning consist subset test quality statistical test exponentially variable feature x w generalize generalization feature feature feature context allow experimental design understand control part compare two ib hc compare network direct edge orientation artificial example control demonstrate ib dense consider clique maximum clique test clique size structure connect node context remain way contextual encodes feature parameter odd dependency force ratio w parameter triplet force generate procedure pairwise force x feature already dataset rao sampler burn sampling use synthetic dataset synthetic version available fair pearson significance hc evaluate order sufficient quality use kullback leibler kl lose qx qx learn kl structure complete learn ib algorithms clique learn pseudo interested quality experiment kl figure structure impact encode incorrect fully impact obtain incorrect present feature describe axis difference parameter kl empty fully notice kl difference difference order difference order kl generation kl experiment pc low structure magnitude hc clearly actual report length log several horizontal feature always near number result increase trend grow pc reach structure similar surprising pc equal empty structure optimize search use large amount hc empty structure correct algorithm fix column near rest efficiency ht hc pc hc graph ib log proceed generalize initial present explore context generalize feature ib include adapt efficient ib state add lee tree execution operation receive attention focus purpose approach efficiently sufficient context assignment condition assignment encode model central combining provide benefit structure log show art ib underlie markov markov encode efficiency important problem learn sample ib proceed statistical test conditional undirected correct
converge center converge stationary sequence standard analogue proposition converge variable check limit positive b ab unable hard previous case satisfactory could try improve write u collecting get quite poor regression lead experiment suggest quadratic coefficient know simplifie noise amplitude long perturbation white know omit case estimator follow linear good determination clear none range quite acceptable aware bias note positive iv give regression coefficient quadratic error explain fact nature change behave smoothly contribution much small multiply use multiply visible outperform advantage use facilitate usage summarize estimator regression still estimate estimator construct even proposition result variation wiener fractional brownian motion estimator mix fractional motion frequently short long parameter center range range increment wiener self similarity restrict huge dependence depth regularity paper kind word process inherently combination concentrate put wiener process mix many paper identification secondary asymptotic variation pure extensive overview give variation literature devote asymptotic generally linear study variation equation paper concern address aim observation power variation remark pure fractional mixed model directly transform sequence study asymptotic behaviour involve increment wiener fix statistical sure paper section power variation quality wiener independent integer study mixed variation thank sequence n stationary study summarize limit theorem believe theorem desire special odd even odd eq wiener vanish obviously case order rewrite mixed q behaviour fractional motion conditionally brownian idea distribution far q dominate reasoning define see get case form define therefore get study sure behavior variation brevity phrase odd case need easy check represent multiple fact l question parametric mix primary goal measure almost consequently denote variation ergodic variation behave behaves wiener case individually pure case proposition h advantage indeed easy unless normality estimator analogy write get eq reason exactly central limit asymptotically careful show converge nevertheless asymptotically normal estimator end well introduce notation statistic omit expand chi deduce normality recommend practically measure induce give explanation fractional consequence estimate statistic observe strong concern follow write obviously due deduce consistency move estimate rather estimator estimator strongly consistent h ji perform despite
sec value overcome preserve total result discretization approach hold self bound general elsewhere eq lemma gives desire depend q also bind term apply every f f theorem define independently basis operator every special inequality lemma sum square influence let first transform sum thm part slightly simple thm self bounding function range particular I fouri suffice almost sense second optimal use influence rather influence prove even depend restriction still close x function mean x gx show necessary function prove boolean suppose fewer depend part variable equal event iff least happen probability nonzero hand constant similar appear absolute function choice gx gx fast pac pac access measure generalize notion disagreement boolean pac learn every function hypothesis hx evaluate input make model multiplicative condition hypothesis variable unfortunately efficiently give way small uniform example find satisfying run example variable degree fourier rely crucially spectral function approximate must argument purpose submodular lemma projection whose namely establish influence f j minimize close operator convexity sec partial away variable fourier real let lem obtain apply lem empty lem f I q imply obtain degree imply identity submodular either former lem ready lem f f j easy denote obtain estimate accuracy lem chernoff obtain desire estimate confidence together close submodular function find submodular algorithm time influential total influence find influential time problem however special monotone influential random influential influence monotone degree real corollary size linear combination standard least corollary thm least run easy algorithm return guarantee example output additional property learner always return submodular submodular running observe theorem obtain n least two submodular improve testing provide guarantee query submodular yes far hamming return strong test submodular building obtain reduce set project submodular satisfie function set mf run run function submodular find iy j function confirm function submodular case submodular fail fail complexity since essentially greedy estimate multilinear error query random time agnostic submodular reduce sample brief review agnostic pac label agnostic say every give draw least influence absolute error lp norm uniform w tp choice let influence run use uniform p gx fourier degree degree lp minimize subject x spectral therefore choice px xt thm n confidence solve give query make agnostic agnostic agnostic problem access least know every range approximate existence learn influence access oracle error slightly attribute efficient approximation gap remain monotone submodular monotone existence multiplicative learning question monotone understand whether time acknowledgement thank anonymous useful suggestion product illustrate statement beyond scope extension fouri tool set let op gx us product distribution si sg gs g fx I fx gx whenever gx proof iterate obtain f op procedure variable sufficiently follow reformulate boost give produce product x step fx step deal change achieve guarantee lemma variable fx similarly monotone event obtain fx lemma difference track select variable appear hence contribute value rest distribution choose ensure lipschitz yy variable approximate within op theorem investigate variable main result tight note necessary hold total influence necessary application distribution submodular demand multiplicative target run factor crucially agnostic class study hypercube primary analog play role recently submodular algorithmic machine learn several fact theory submodular application return algorithmic function etc contain broad bounding inequality structural approximation boolean sec know value know hold recently result special submodular take et submodular formally consider motivate submodular multiplicative ask wide recently attention submodular well submodular particular general approximated short range approximate bound technique influence capture submodular self bounding prove structural result describe submodular function submodular depend show submodular improvement formal approximate submodular hypercube add enough subset variable boost uniform choice exclude marginal high bind exclude exclude concentration submodular replace exclude allow reduce process repeat variable involve relaxation demand monotone monotone broad class prove generalization well boolean subset variable j g approximation influence true statement polynomial generalization discretize function prove bound discretization imply refinement component bounding function immediate implication alone use bound submodular general necessary picture approximation c structural uniform submodular main uniform example least uniform run example monotone submodular instead approximate alone approximate whenever approximate force return apply recursively optimal learn properly input function submodular doubly exponential give function pac give access example output far fairly monotone influential detect influential exploit find hypothesis use dependence doubly function organization detail discussion main thm multiplicative thm give submodular implication result agnostic multiplicative factor refer sketch sketch section learn come give factor result algorithm achieve number clause determine hardness submodular impossible nontrivial strong concentrated lipschitz nonzero multiplicative approximate small since give small theorem require grow submodular uniform consider motivated release submodular lipschitz submodular submodular running query et al stability degree work approximate fouri degree pac multiplicative imply error multiplicative guarantee query pac testing submodular ok pac query special pac example largely point decision style polynomial approximation case submodular boolean linear combination imply function well recent expressive completeness detailed class appendix nonnegative natural approximation shift invariant submodular way nonnegative equivalently include nonnegative monotone submodular non monotone discrete also share function could monotone form small view monotone submodular function combinatorial cut independent fact form whenever broad monotone submodular function iff real value rs ia b formulation closed broad class fact like constant polynomially exist submodular submodular generalize class bound generally restrict attention hypercube primarily self bound bound self include function monotone sufficient bounding play normalize function range bound concentration currently satisfy self bound example self relate application inequality lipschitz coordinate appear arise average submodular note swap machine switching mean distribution discrete equal th fx consider relative make scale additive scale submodular submodular within base depend submodular function bring another bring prove shall allowed contradiction pick statement statement j h contradiction lemma concentration boost know lipschitz submodular concentrate submodular function result state second easily self product bounding lipschitz submodular submodular scaling lipschitz submodular eq distribution q decrease submodular denote multilinear coordinate produce small coordinate sufficient long deal monotone monotone variable time assign boost estimate procedure number procedure would exceed suffice function final potentially monotone let include define iff contribution include definition ss ss si imply exceed rt variable replace exponentially tool go q know family monotone lemma decrease must prove monotone prove lemma submodular guarantee j jj restriction replace respective depend submodular estimate distance observe h fx fx hx var class bad fx condition marginal top value hence bad bad uniform coordinate estimate var f j desired examine submodular taking consider independent fraction therefore equal application exist submodular approximation multiplicative require r notion normalize multiplicative error together multiplicative notion sketch sketch polynomially many bit multiplicative point function good depend minimum value hypothesis restrict refine arbitrarily provide refinement approximation submodular uniform precisely monotone submodular every monotone submodular observe submodular strong multiplicative imply additive except idea rely resolve monotone submodular sure whether monotone case variable f desire significant procedure produce random subset element independently q include repeat return fixing obtain function sufficiently within multiplicative cardinality order select w
condition need assume payoff arm follow guarantee hold triangle inequality number mab relax take instance dimension application maximal payoff improve payoff plus chernoff bound prove call arm clean least difficulty chernoff phase activate follow phase fix payoff play conditionally consist reveal chernoff bound condition union phase integrate union connect good arm play round turn allow play phase clean arm arm first claim arm cover clean lipschitz x tx put inequality note definition play thus else play radius clean play plug radius clean activate letting contain arm diameter cover arm scale hand q clean round phase choose obtain summing phase match ingredient elaborate radius tx tx equation compute mab problem relax set multipli reward effectively reduce analyze confidence chernoff appear q plug chernoff b probability equal minus play say clean mean round arm chernoff claim phase clean lemma efficient assume tx omit problem reveal define yx target x lipschitz satisfy x necessarily satisfie arm apply use conjunction algorithm target multipli c know need minimal payoff intuitively subset small dimension multipli set diameter ir rr bc bx bc r fr reasonable finitely point problem payoff equation multipli proof extend multipli regret bound play arm precisely assume playing plus mean reveal interestingly bound slight abuse assumption start mean deviation multiply distribution inequality precisely analog claim radius instead chernoff omit analog radius estimator instance improve analysis easy detail improvement whenever tu tu tc regret mass snp measurable apply subset separately else reward least sharp identify symmetric sharp neighborhood use radius tx c lead arm reward receive arm break tie arbitrarily define arm play chernoff constant contain estimate lie concern good mab instance payoff theorem notation algorithm exponent lipschitz metric payoff exist metric regret dimension metric show arbitrarily result space moreover instance introduction subsection concern dimension rely existence min covering dimension kl defer subsection metric space arbitrarily tailor design analyze dimension arbitrarily close min bind applie use bandit evenly spaced sample precise prove na I version I describe away generic armed evenly spaced sample proceed phase begin space armed prove achievable I I identical payoff distribution slightly disjoint consist payoff disjoint subset arm infinitely precise root correspond space child disjoint ball ambiguity root tree require child radius parent child exist strength well hard good obviously hard mab tree strength payoff payoff absolute hold purpose suffice payoff hold rest prove lemma ball follow easy induce leaf random tree subsequent child mab arm let low armed bandit exposition usage considerably mainly mab function rely subset set feasible collection subset exist mutually coincide idea least round whether incur subset correspond child ball payoff subtree root consider feasible mab payoff bandit regret payoff author analyze case payoff preserve bandit strength loss generality finitely induce function induce leave function subtree root absolute child radius recall payoff borel finitely fix derive existence cover notion ensure arise ball tree continue new connect infimum open cover open contain ball positive bound mab metric min dimension explain supremum max lipschitz mab space packing suppose contain size ball maximal packing would pack pick construct ball strength extended root radius center extended define correspond center radius dimension pack child child radius gain max space dimension min covering let concrete example involve rooted tree degree node every node child say degree degree every form degree diameter contain subtree cover entire exist dimensional leaf subtree responsible subtree subtree generalize subtree cover cut depend choice cover outside covering cover formalize argument metric open neighborhood equivalently open empty open fact every open cover since denote open contain claim bandit correct warm modification close establish general optimal dimension specific lot technology poorly subset arm locate inside cover active locate fix cover exist suffice desire regret follow phase arm activate initially arm cover maximal index tie sequence trivial set easily generalize sequence satisfie di generalize eq complete increase oracle open pair cover output mab metric contain finite dimension instance outline simplicity unique arm desire regret analysis cover w long phase cover eventually large arm let neighborhood many hand p phase w stay lemma rule activation selective activate cover arm activate algorithm compatible round carry clean round execution activation cover arm call covered round algorithm consider mab metric one phase compatible duration clean cover eq arm cover q regret bind dimension multiplier cover clean immediate section compact compatible eventually cover constrain eventually cover part section mab fix subset suppose contain arm clean phase compatible duration suppose arm arm cover depend cover set packing pick empty pick exist empty compact supremum contain arm compact packing moreover pick packing point phase well cover round round cover activate ball cover cover arm active whether empty packing empty packing round therefore arm cover cover activation clean duration cover contain depend instance recall packing packing pick contain arm lie pick packing point metric compact packing point packing phase duration prove cover induction round cover design round activate confidence ball entire metric round arm active packing room cover activation cover activation activate cover cover neighborhood arm packing contain packing choice room cover activation cover arm case cover round cover pass length metric subset product metric metric admit decomposition appear decomposition instead design metric subset metric ordinal ordinal set decomposition countable infinity subset min decomposition exist induction consequently limit ordinal every definition min dimension contradiction completes construct ordinal cardinality thin thick note subset ordinal induction sequence satisfy definition great empty otherwise space neighborhood close cover c compact instance length turn vx subset attain metric maximal non subsection phase duration ordinal maintain activate phase arm never round radius idea long clean ordinal subsequent clean cover desire regret sufficiently ordinal thing phase ordinal change index payoff phase arm check begin net net exist jj follow accord radius set arm define large ordinal heuristic initially arm active cover pick play break tie arbitrarily tx algorithm ball union ball ordinal report arm represent depth output covering modify ordinal bring subsection constrain eventually satisfy clean ordinal cover sufficiently long clean open analyze internal avoid corollary analysis claim small node denote unique fix expert set follow sequence fix expert complete infinitely happen infinitely happen positive algorithmic compact tractable expert double feedback exposition expert version expert compact order payoff function correspond payoff structural strategy value compact attain non close compact finite element initial attain strategy eventually play idea contain payoff space metric subset oracle output cover input ball center return element ball follow subroutine input call receive call point oracle round sufficiently subroutine return notation consider x rt chernoff bound run clean union oracle lemma cover suffice rt sx kt oracle claim suffice loss generality proceed doubly round first subroutine subroutine description fix accumulate least sum exploitation exploitation return exponential subroutine point incur round view follow let generality diameter payoff neighborhood high payoff baseline x lipschitz lipschitz tractable depend sample outside sample formalize strategy round event prove kl technique complete theorem ir space intuitive space compact finite limit order finite consider fix converge denote set well trivially x xx yx yx yx remain arbitrary need isolated segment exist xx reveal subset x ii require oracle lipschitz mab I access collection problem tractable feedback compact x f I omit subroutine output oracle receive cover play strategy exactly sample let rx sx large dominate strategy winner winner else output arbitrary clearly take complete increase sufficiently subroutine return happen let assume clean introduce payoff exist compact rank strategy optimal strategy lie claim therefore definition prove pick claim phase winner dominate large claim dominate bandit completion formally sake expert either tractable tractable feedback occur metric space basic ball compact space introduction fix covering ball cover center mab look set consider cover tune correspond way difficulty set fast grow account fine tune accumulate differently corresponding covering cover call mab complete claim induction base claim prove metric infinitely ball ball ball partition two define ball randomly subtract payoff ball statistically indistinguishable fraction ordinary ball randomly happen never covered ball ball mutually number exist number define payoff sign define payoff function function one element independently intuitively discover payoff element eliminate identity algorithm jj tt payoff tn picks set eq relation imply equal select account j expect bound finitely k k k tractable concern expert bx metric call phase phase play pick size phase break tie arbitrarily description covering see sake completeness explain expert metric algorithm achieve payoff set sufficiently case feedback chernoff bounds eq note choose ensure incur event guess total accumulate claim expert lipschitz version obtain via analysis metric expert duration let set choice theorem sufficiently essential set choose guess establish remain step exactly require use chernoff need many point efficient use chernoff apply eq slack chernoff scale take advantage call metric structure root internal internal singleton child satisfy ii cover fan cover break rule say hold child sx clean I separately trivial lb j easy determine slack right place ii clean phase ignore incurred phase clean argument upper bind clean phase clean diameter pick child since break rule claim clean phase root turn log cover dimension notion characterize space refine max cover space expert tractable tractable suitably ball conjunction ensemble idea follow space metric plug technique complete upper proceed exactly except use feedback fix space many tractable proof bind use packing rely fact nonempty nonempty disjoint radius positive ball radius ball collection radius every packing recursively consist finitely disjoint equal let disjoint ball ball let I ib bx ib every verify construction ensure subset define payoff bias define low expert obtain notion define infinite specify expectation achieve finish proof ball bt half ensemble recall tractable proof theorem analysis expert let cardinality break arbitrarily lipschitz hold log rather let arbitrary ordinal ordinal existence suitable decomposition exactly metric ordinal finite open union oracle arbitrary ordinal exist either cover return cover net successive call union call usage scenario definition one phase round guess phase estimate depth show end break tie remain phase duration construct contain construct net large point depth chernoff lipschitz use cover oracle construct subset theorem phase clean clean estimate depth estimate clean reason let show contain strategy ta regret similar mab motivated address question first refinement use explore exploit learn side similarity arm potential adversarial pricing strong detail mab attractive mab mab covering may mathematical interest design bind analysis contribute existence would paper kullback several body self sum convention interpret give term convention absolutely detail chapter follow kl distribution sum kl divergence tuple also useful henceforth follow notational convention denote probability lemma quantitative term away ec rearrange satisfy supremum use expert whose draw divergence p event mutually exclusive consequently satisfy less satisfactory property contribute property q choice payoff function define paragraph equation select state role play payoff history indicate sequence strategy select payoff distribution mutually disjoint ensemble variable q select time determined event time choose strategy lipschitz mab tractable likewise metric abuse subset mab double proof tractable mab prevent space immediately consider mab consider conversely design slightly fix play play let expect two tractable behavior follow observe payoff high specifically query success latter round look omit let instance lipschitz expert feedback tractable completion remark direction desire use easier less elegant payoff tb element pick bound event set let equal k denote times select eq q k bound finitely r k r tractable sake convenience two countable perfect iii circular subspace ball tree tree root intersection intersection non pick arbitrary distinct contain ball leave ii ordinal define sequence recursion specify isolate isolate since perfect empty ordinal cardinality exceed order disjoint point order ball construct topological imply topological know open large definition least distinct point arbitrary metric space example uniform remain prove topological segment topological well segment open metric initial order metric topological initial metric topology must infinite contradiction metric dimension wasserstein page sake completeness k infimum subsection prove net note rational denominator cardinality bound remain ball radius true every contain close near metric hamming cube even cardinality map arbitrarily uniform prove assign radius center move sum good binary g constant implie obtain point contain conference publish full report min covering nsf microsoft nsf multi armed choose strategy bandit small well understand investigation motivate practical online solution strategy satisfie refer solution armed performance mab arbitrarily version round payoff arm reveal multi armed bandit also theoretical modeling inherent decade visible impact online game arm define strategy bandit finite problem set still topic active strategy payoff problem strategy trivial natural structured efficient broad induce specific space interval general treat natural motivate thousand ad ad display match ad infeasible inefficient since ad organize category measure website generalize ad make inference performance ad motivate management see digital product movie software arrive offer large product inefficient instead inference form strategy payoff satisfy form period receive independently think reveal abstract metric infimum quantity finite path x payoff satisfie refer order triple work mab space implicit work bandit bandit case contextual bandit set context rather put bandit recently logarithmic bandit run difference quantity payoff play mab every advance upper na I arm partition space hence real value minor modification extend lipschitz dimension generalize cover dimension cover notion summarize cover property cover context cover euclidean dimension cover value study mab metric space theorem metric odd achieve mesh refine mesh gain useful really close raise reason payoff distance scale multi usefulness proximity metric expect payoff lipschitz question class instance cope strong answer payoff function differentiable finitely maxima regret achieve modify na I interval instead play reveal phenomenon space outperform direction discussion metric implicitly bad covering take payoff structure useful regret would help relatively rich metric perhaps call apart admit metric problem logarithmic infinite regret tractable natural alternatively opposite end spectrum metric space infinite cover metric space intractable admit tractable admit feedback mab mab problem feedback payoff arm reveal arm extensively name cover result expert problem metric optimality handle metric infinite covering come resolve q satisfactory arbitrarily algorithmic use history region maxima mesh ingredient perform significantly call algorithm self tuning require maintain mesh arm unlike payoff upper confidence bind use early bandit refine payoff perform ingredient bandit corresponding algorithm require exactly mab cover space focus near arm example arm instance small follow hold every cover expect payoff fall dimension quantify significantly example payoff cover dimension thin subtree infinitely short infinite cover cover whereas metric space payoff smooth unique neighborhood cover dimension turn need condition satisfy relaxed case derive maximal playing plus payoff reveal reward play independent shape far tailor infinite meaningful interested bound theorem characterization optimal metric bound characterization define characterization table compact instance mab problem tractable space finitely instance essential tractable table read respectively interpret individual min try low min small subset open large min covering subset notion regret consider mab bandit much least suffice time upper achieve I early dimension metric highly homogeneous ball achieve I early deal strictly cover design suggest generality web describe web category topic hierarchy improve cutting reduce cover extend cut open reduce dimension region obstacle algorithm impose set cover arm impact contain eventually sense neighborhood limit combine metric space gradually cover region consist sequence finite sequence parameterize phase ordinal sequence active payoff next ordinal feature algorithm space connect connect certain space support bound three metric interest notion usage feedback setup setting bandit instance lipschitz mab tractable metric let whose constant depend payoff resolve mab tractable tractable former metric vs basic correspond countable also conjecture exist would finite vs bind upper bind mab countable possible bind show metric question lipschitz metric lipschitz fix metric online topology algorithmic bound technique vs result identify topological property ordering entail topological entail property theorem state theorem specify contain simple interpret detail interpret randomized borel function history observation arm play theorem algorithmic oracle access open output pose metric space require metric intuitive suffice hold wide set amount survey background metric dimensionality result define lipschitz expert section expert direction preserve deferred background leibler divergence reduce lipschitz bandit expert problem metric contain tie low appendix mab thorough reader book bayesian perspective survey distinction regret formulation mdp survey mention among formulation distinction payoff payoff payoff arm dependent distinction instance inherent achieve constant consideration algorithm close simple powerful idea arm call ucb payoff ucb confidence exploration balance several paper design ucb armed payoff achieve regret even ucb many setting g appear many paper assume lipschitz mab correspond reveal commonly linear payoff strong essentially strong away accordingly infinitely I arm gaussian bandit mab mab detail arm minimize algorithm identical item sequentially item price price appear four mab setting arm bandit mention bandit payoff click obvious technical connection mab topological uniformly payoff property payoff background apart mab consider several mab background payoff formulation mab mdp represent payoff formulation computer science interestingly bayesian formulation offline mdp nearly bayesian formulation fully stochastic payoff adversary seed mab minimize arm well consider subset payoff space domain include construct embedding problem location dimension notion dimension counting notion discuss beyond scope need cover number similar notion aware technical cover notion dimension however function characterize intrinsic internet delay round cover dimensionality space study popular notion e notion label location al considerable amount follow appear et al work respect extension mm conference publish extension briefly mention conference version full available publish mm appear author aware extension spirit mab contextual receive context pick strategy user express context bound contextual obtains improve set metric allow expect payoff apart
situation additional new confident increasingly concentrate illustrate toy orthogonal present incremental incremental getting solution begin converge iterate somewhat surprising behave l mnist consist handwritten provide interpretation update mirror descent relaxation lem sgd project feasible frobenius feasible choose way problem form strongly convex convex let kkt lagrange multipli complementary equation complementary feasibility feasibility must way complete title thm thm study theoretically empirically pca tool many information representation original cost control aid fix subspace capture reconstruct distance residual give svd optimization optimize population consider set goal inside well subspace rather furthermore measure angle capture one population justify favor far essentially base online stochastic approach erm sa variant many approximation study stochastic approximation heuristic incremental justification incremental distribution suboptimal see careful regret online algorithm convert stochastic approximation paper present novel stochastic unify mirror descent variant update incremental clean pca excellent algorithm consider find maximal loss generality fourth parametrization optimization optimization access study stochastic require overall runtime solution standard minimization erm covariance column eigenvalue approach require operation svd interested time approximation sa project obtaining rate instead parameterization pca constraint eq hull optimum attain vertex I boundary optimum optimum solve suboptimal find result feasible treat sample sgd onto constraint entails choose point iterate update analyze straightforward iteration start eq expectation analysis yield eq pca x km mm k xx u u define operation per maintain date code update tt tm x svd step eigen project project eigenvalue projection satisfy onto feasible amount importantly projection operate find handle instead list contain section optimization central motivating array simply search threshold mm mm mm mm j j n j iterate array eigenvalue line sort possible increment computational cost dominate constrain frobenius md mirror update choice choose geometry potential trace constraint suggest von update refer fact mirror constraint either suit trace dependence furthermore sgd depend come mirror optimistic error excess avoid analysis runtime clearly runtime depend iterate achieve runtime cubic hand runtime fortunately practice rank projection therefore update increase iterate potential perhaps difficult theoretically iterate evolve empirically iterate relatively rank detail experimentally difficult smoothly decay decay spectrum challenge sampling basis orthogonal orthogonal maintain nonzero iterate evolve extra leave iteration enter suggest lead significant next rank nevertheless reason add
goal coarse label consistency cv respect equality fig coarse separate solid line produce alternate measure well global entropy cluster coarse grain separate entropy motivation cluster next side look familiar alternate form hx coarse axiom cluster objective coarse theoretic axiom amount even small nonparametric supplementary associate discrete nn reflect uncertainty see long near neighbor lie fig use small resample limited keep random probability end correspond nearest yield want refer elementary summation entropy lead lowest reduce expression refer quantity weight sec detail look simple vary perfect completely random partition minimize clustering natural partition cv cv magnitude unlike mutual quantity limit procedure derive principled theoretic briefly mention practical concern partition consider semidefinite program supplementary theoretic require exploration comparison scenario partition develop intuition meaning solver optimize recall sec fail correctly ratio discard coarse bad clustering desire fig hard highlight theoretic fig unbalanced radius evaluate partitioning mutual inspire r result asymptotically expect similar mutual information estimator incorrect split also actually quickly robust mi imbalance prefer range partition sec heuristic optimizer able partition recover correct ht plot ht wikipedia frequently w wikipedia user article make user reject conversely consist look wikipedia page bayesian mean discover cluster partition ccc ccc rand dim quality calculate rand candidate cluster truth report mutual well low heuristic uci dataset database although optimize objective heuristic competitive compare approximately benefit balanced truth cluster ground ground truth neighbor nearly achieve list combine theoretic characterize similarity purely information ad similarity mutual entropy use entropy correct argue knn valid average estimator entropy nn estimator guarantee sample therein average estimator fail wang demonstrate semidefinite attempt invoke loss variant estimator construct span method ultimately limit unlike mi base bias attempt intuitive method sensitivity balance therein conceptually principle similarity require notion adjacency neighbor advantage theoretic unbiased find near neighbor coarse essential entropy formally incorporate basic estimate cluster preliminary feasibility optimize competitive theoretic attractive make maintain operational datum unknown construct theoretic foundation learn carefully refine development idea contribute acknowledgment helpful g fa grant foundation research fellowship foundation ef estimator term standard estimator discrete l py expand delta n j k n hx n j n value expansion apply entropy cluster define definition uncertainty maximal instance case full possible detail goal way partition point group evaluate cv calculate landscape method unlikely heuristic number cv tractable semidefinite candidate n minimize th neighbor lie close together close well matrix near need force gram semidefinite gram optimization solve technique find round first take cholesky recover group vector unit partition way calculate partition one combine partition low choose partition rand desire despite unbalanced size cluster intuition conceptual cause performance amount increase return coarse exploratory nature like flexible enough reveal modal change ideally goal explicitly notion wide mixture version clear model invariant require grow clustering assign cluster mutual maximize approach mutual invertible define especially attractive main information theoretic mutual fundamentally derive principle non simple mutual information fail dramatically succeed mutual naturally interpret measure alone cluster prefer ignore intrinsic cluster estimator method eventually converge yield sized fix construct objective principle preserve theory axiom sample motivate axiom form entropy preserve infinite datum shall robust partition coarse important structure size violate quantity alternate interpretation synthetic dataset previous recover heuristic show achieve cluster dataset organize idea theoretic status compression start develop idea report work conclusion give sample draw distribution interpret bit shannon necessarily draw discrete reflect course bin narrow bin measure although maximize criterion many paper understand help term equally sized cluster easy see data point non parametric sec near due separate hand amount see mutual decrease cluster prefer lead use unbiased limit contribution boundary information theoretic objective like uci sec equal split become prefer b clustered precision
otherwise obtain see I ad put otherwise note g get cluster big big big well ready kk k kn hypothesis note sample see cluster away expect probability least point conclude call cluster long call prove induction generate initially leaf point clearly big splitting big small call assumption find split whenever big cluster e gb gb single assumption big leaf contain big contain big therefore cluster leaf big tree cluster algorithm stanford stanford usa run microsoft microsoft usa observe mixture distribution assumption sample current art separation generalization extensively recover line cluster theory address preserve tm mixture underlie topic word compare present similarity word recover see difference exist tm vector tm co occurrence model previously tm density word mm occurrence close tm vector distribution hard ease generalize something tm mixture mixture measure measure measurable j embed distance work mixture disjoint support disjoint cluster sample discussion compare art example medical sub disease heart location different status genetic exposure environmental sub type appear patient acquire disease medical record another break size pixel take underlie contain part likely picture component multiple sample differ preserve theorem present different design run result association present assume find wang pca gaussians use able suggest correlation feature feature align spread across project span distribution distance make impossible preprocessing combine yield separation improvement span gaussians restrict restrict support disjoint seem strict look center gaussian k support present project low keep well nature affine span demonstrate mild project select dimension present maximal component span return component mean projection maintain center distribution nevertheless apparent find centroid nice cluster claim cluster find classifier tree w return tree leaf else hx hx return tree root subtree subtree proceed relation abuse mix weight therefore minimize distance maximize boundary green maximize separate region region separate split trick recursively keep sub space stop demonstrate idea mathematically lemma explicitly find separate classifier break long suffice q split measure zero leave book however large break separate otherwise every every mixture mixture vector q separate gap bound away mixture least return gaussians synthetic space add normally distribute experiment experiment skew time select three normalize sample example run gaussian measure point gaussian infer several mean low second variance projection project k algorithm matlab means matlab unit dimension maximal large outperform difficult regime outperform projection time maximal k projection variance respectively variance create previous algorithm variance p problem cluster datum multiple
reason exposition misspecification correct world usually neither desirable expand quadratic miss cubic misspecification easy goal computation sampling achieve year substantial classification survey effort method examine limit marginally fitting class positive example whose average example negative control particular several categorical typically easy screening available wide collect survey calibration method computationally intensive purpose commonly control define bias generate subsample obtaining assign specifically generate mutually pilot offset subsample thus application rule relate odd odd odd simply vertical subsample valid exploit criterion informally good predictor get get right approximate panel line logit scale dash poor black fit ignore small reasonable well small log odd produce approximate match logistic regression place fit imbalance matter medical click section modify control unlikely everywhere shift dash complicated importance estimate subtract successful predictor converge predictor distribution criterion large solve population differ two way integral second importantly misspecification case equally limit twice answer subsample inference disease example exposure person develop rare disease population family half binary odd pt pt else column attention imply odd would obtain equal reflect right panel amount imply odd log odd leave odd differ effect panel precision test sample include weighted control thompson asymptotically weight succeed remove bias consistency effective size weight obtain pilot another pilot immediately control improve efficiency benefit pilot local differ allow selection degree experience observe px xy generate fit obtain logistic subsample adjustment justify subsample pilot selection motivate fisher odd maximize fair one effect local case conditional subsample original marginally pilot discard discard keep pilot fit day pilot pilot recommend section pilot asymptotically unbiased consequently pilot second per pilot dependent pilot local algorithm simulation pilot control role pilot guide discard keep conditionally surprising pilot much pilot roughly sample per reasonably offer pilot pilot expect pilot fit improve upon sampling support intuition correct consistent pilot despite improve two size subsample multiply acceptance w point correction estimate imbalance acceptance become marginal acceptance px accept roughly alternatively desire datum uniformly local consistency pilot subsample calculate pilot independence finer asymptotically estimate correctly twice full clarity letter place avoid redundant hinge eq function x local control subsampling scheme dx integrable integrable condition logistic schwarz dominate take minimizer separate hyperplane population control subsample pilot somewhat replace minimize purpose notation pilot estimate expect sampling pilot fix predictor population uniform subsample consistency pilot unfortunately pilot consistent pilot perfectly local control pilot evaluate population explanation role original example contribute full score evaluate score score probability exactly essence subsampling stand fitting sample suggest good fact see reservoir pair accept pilot possibly everything else local pilot accept pilot detail technical actually last minimize pilot decision pilot nearby pilot estimate reject pointwise uniform tend cover finitely ball q come prove estimate consistent pilot ignore everything case compact turn interior diameter section regime pilot pilot subsample pilot fit early logistic correctly pilot asymptotic matrix covariance name see integral obtain population expectation logistic dominate derivative dominate convergence inside dominate dominate applie note begin relation standard fix light h combine pilot independently consistent argument characterize conditional variance specify assume logistic variance mle independently hence size logistic regression regardless characterization case case estimate scalar offer accept contribute full sample contribute stand discard keeping assign weight advantageous covariate happen imbalance efficiency imbalance local control exploit imbalance outperform dramatically imbalance picture somewhat unbiased datum get correlation pilot affect serious pilot local assigning variance weight log similarly pay relative full unweighted consider begin matrix logistic incorrectly specify let misspecification matrix nonzero quadratic simulation generate pilot next pilot c cc cc observation see pilot must pay pilot repeat three cc improve substantially bias bootstrapping enjoy cc dominate estimate conditional imbalance improvement bias come limit simulate two class odd since unbiased introduce substantial imbalance demonstrate reduction advantage local generate table bias variance three substantially bias pilot subsample correctly specify variance roughly twice word control subsample roughly unbiased enjoy small case suit imbalance application spam local prediction actual spam website originally web page spam page design content marginally imbalance considerable feature transform offset reduce feature datum use pilot retain assess estimator uniform subsample document procedure pilot variance close marginally balanced subsample times sample twice local since pilot readily pilot experiment reasonably experiment axis index coefficient axis relative coefficient case substantial standard method conditional imbalance control subsample bias make post hoc coefficient estimate subsample way exploit imbalance inconsistent minimizer generalize address subsampling allow pilot consistent consistent misspecification local control sampling full local control practice way gain translate computational enable prototype variety try often validation bagging bootstrappe intensive procedure statistical relatively clear help point procedure bootstrappe make scale previously help basic pilot second extension principle pilot model must linear pilot fx odds subsample regression pilot fit important variable response cover logistic consistency pilot consistent nothing population correctly neither intercept pilot high local thompson come cost weighted case control suggest extension describe diagnostic screening false one bernoulli implicitly place emphasis odd near curve appropriate boundary relevant well model care curve subsample population curve could depend click obtain alternatively fit odd pilot iteratively subsample pilot could think adaboost classifier classify thought odd local q influence outlier hard classify adaboost fit logistic regression logistic especially regime natural subsample give acceptance large carry subsample arise glm generalize scheme proportional mean generalizations logit survival uniqueness w g neighborhood argument increase sufficiently enough different repeat replace begin selection compare mutually datum n inequality average take law eventually lie mutually bound hoeffding apply unconditional tend since arbitrary convex note rearrange obtain factor tend desired define triangular
propose framework learn drastically quality nonlinear favor triplet much easy dimensional poor approach subject outperform nonlinear learn small pair parameter convolutional neural weight stochastic design suffer optimality careful many order overfitte lead complexity face verification approach nonlinear stack restrict machine tune optimize use minimize suffer limitation digit recognition dataset outperform nn observe mahalanobis plug classification performance propose vector metric I respect distance ii mahalanobis author optimize thus psd constraint rank frobenius regularization overfitte approach improve svm propose gradient additive correspond building boost tree region translate translate seem robust overfitte achieve hamming distance learn value vector binary hamming binary code bit great work small neighbor sublinear optimize value dimensional binary sign learn transform network relative kb zero formalize loss objective author propose upper length optimize relatively sufficient code achieve classifier shorter maintain nonlinear heterogeneous capture one metric specific choose review see geodesic tensor crucial metric simultaneously make outperform come expense requirement address distance preprocessing partition class supervision generalization objective target neighbor measure standard especially expense local region generative local aim leverage model outperform discriminative sum asymptotic probability due locally mahalanobis semidefinite analytical regularize towards overfitte since independently scalable competitive poorly extend nn learn global discriminative bregman bregman divergence necessarily strictly twice bregman mahalanobi recover wu point mahalanobis hessian location infinite mahalanobi distance reproduce h allow classic hinge subgradient learn formulation learn kernel exhibit improve metric propose metric mahalanobis learn parameterized basis define ensure psd term weight iii vary smoothly assign basis positive weight fairly efficient procedure require eigen make intractable evaluate quite overfitte however euclidean anchor hyper default anchor metric use alternate metric map encode relative position mapping mahalanobis information eq decision tree implicitly adapt select split metric training tree drawback evaluation tree encode dataset histogram representation object natural vision bioinformatic bag bag specifically histogram prominent nonzero avoid division proper generalize introduce constraint map optimize subgradient optima experiment histogram histogram mahalanobis distance dimensionality optimize simple histogram bin histogram histogram location capacity respectively minimum effort move cost move bin bin distance amount flow correspond amount ground convex feasible flow represent triplet essentially sum experiment mahalanobis distance mahalanobi learn aim flexible set must satisfied q ii formulation bi solve metric solve amount ground flow stop change subject optima discrepancy difference distribution nonlinear mmd kernel trick use regularizer trade discrepancy face highlight effectiveness domain come structured opposed feature instance text dna secondary rna tree metric appeal use proxy without appropriate use vector metric structure string bag visual hand directly structured object capture metric combinatorial explain basically measure turning attract context structure define graph amenable parameterization string distance graph approach summarize yes string generative yes string string local yes global yes yes local tree discriminative local notation review metric learning alphabet alphabet finite nonempty symbol finite string nonnegative elementary operation substitution symbol operation programming similar include alignment substitution function instead cost operation reality task error alignment digit automatically task hand assign operation therefore update approach stochastic variant distance matrix operation probability output sum consider represent maximize pair via iterative procedure unlike rarely generative implementation pair estimate estimation expectation probabilistic string uv uv termination symbol cost penalty deal tendency estimator alphabet way level generative ii string iii probability bias highlight empirical conduct turn string operation done differ order field deal pair em drawback converge calculation costly alphabet string drawback avoid like homology possible time symbol align along deal gap procedure discrimination distant drawback objective nonconvex minima similarity require costly operation since depend cost nonlinear similarity psd learn essentially learn optimize distance seem straightforwardly adapt review approach approach string distance learn section string similarity discriminative rely em entire chapter parameter limitation unlike tree recover probability tree theoretical limitation author factorization incorrect correct consider instead derive em drawback approach overfitte parameter intractable node computation review trend briefly summarize promising feature reach indeed online significant role scalability setting task domain adaptation metric recent much structure datum structure indeed em algorithm dataset hard analyze optima successful formulation simplify highly flexibility probably good research light identify limitation learn thousand method infeasible therefore challenge recent rank matrix potential analyze learn classifier etc problem learn metric ask purely criterion metric noise spirit autoencoder direction relate problem characterize design show structure become network metric likely receive change transfer receive effort insufficient dealing drift different exist multimodal several instance perhaps versus similar interpret thing bring notion department california st fr st fr universit st france way measure recognition lead metric aim attract machine field past ten propose systematic metric pay mahalanobis metric powerful nonlinear learning extension survey address learn structured overview challenge year similarity mahalanobis distance notion use generic many learn near metric identify near prominent relevance base score clearly depend resp resp purpose metric cosine string often manual topic devote metric short really subject pairwise value mahalanobis learn semi triplet form link sometimes negative I ix triplets q basically find agree constraint incur metric survey formulation differ metric constraint regularizer use performance nn metric notion role diverse link reinforcement partitioning problem assess list large metric useful metric compare video bag word consist build word exist recognition visual annotation relevant query often document bioinformatic involve metric measure string alignment adapt mention outside survey metric learn mahalanobis one learn reader may unlike mkl predefine base regard oppose kernel mkl inductive interested reader mkl dimensionality aim reduction unlabele lie aim unfold capture preserve measurement low may survey point machine year reach considerable level practically early review recent advance survey surveys complement survey general core depth briefly review application hand survey comprehensive review literature cover drawback particular attention structured derivation present may metric particular introduce update knowledge strength practitioner interested metric information appropriate need code purpose applicable wide application address understand survey reader knowledge theory survey number dimensional cone psd real constraint link arbitrary psd nuclear slack alphabet rest paper organize lie describe body deal mahalanobis deal histogram cover metric survey discussion current limitation promise except metric algorithm essentially algorithm ability leverage scalability dimensionality generalization emphasis place metric access compose often set triplet costly approach building constraint metric provide meaningful implicit click engine link information semi supervise supervision typically unlabeled side overfitte label choice mahalanobis expressive limit easier global less overfitting often rise nonconvex subject optimality variation learn problem heterogeneous parameter large arise area machine scale reasonably however considerable refer ideally contrary formulation note early look fast yes weak none frobenius yes nn yes global none yes none weak linear weak linear frobenius weak linear mt global frobenius multi weak task yes multi global frobenius global yes yes none linear frobenius noisy yes nuclear none online yes frobenius global frobenius norm frobenius rectangular yes full yes gb full yes none local yes full local none generative mean bregman yes none local none histogram linear none frobenius histogram yes laplacian auxiliary yes probabilistic semi mmd deal metric attract lot nice interpretation present mahalanobis challenge originally refer incorporate distribution abuse terminology generalize distance cone psd otherwise express rank mahalanobis implicitly induce computation space explain distance attract major component challenge associate mahalanobi maintain efficient alternate step onto psd cone eigenvalue avoid iteration expensive scale challenge rank instead optimize subject regularization np positive comprehensive review discuss inspire online learning fit approach psd constraint seminal mahalanobis maximize keep distance project eigenvalue rely parameterization psd one optimize costly psd formulation triplet constraint slack allow constraint symbol denote slack trade solve drawback less mahalanobis learn furthermore neighbourhood component introduce leave stochastic near ii probability correctly classify q learn eq rectangular induce limitation nonconvex later metric maximally convex kl unlike matrix like onto margin neighbor al one subject reason popularity neighbor target keep instance euclidean target neighbor formally follow j j follow program control trade develop subgradient careful book keep alternative way practice although absence sensitive select neighbor highlight relation involve relevant component subset example closure instance efficient essentially within effort identify irrelevant mahalanobis I involve ii minimizing within limitation explain later handle metric introduce mahalanobis bregman divergence definite eq dimension remain aim key divergence therefore preserve semi dissimilarity close euclidean theoretic solve limitation pick influence hashing metric deal double space previous training time hypothesis online algorithm inferior useful tackle fail complexity bad hypothesis batch pseudo metric online mahalanobis learn well receive pair orthogonal projection efficiently basically distance resp intermediate possible solution compute project back psd mahalanobis develop feature tight regret spirit satisfied perform psd cone trade close solve computation several show mirror mahalanobis composite mirror online problem accommodate regularizer bind nuclear norm dm convex thus cover mahalanobis order task mahalanobis metric valid pseudo easily learn euclidean distance reduce formulation hand simply union case adjust information importance metric mt metric metric mahalanobis parameterize transformation low convexity optima oppose mt make apply result bit mt force low identify drawback strict mt preserve geometry ability propagate regularize extend tt triplet depend bregman metric identity von q matrix early survey von preserve automatic show encourage geometry problem solving solve efficiently task mt especially transfer source mahalanobis source learn amount label make relation task positively uncorrelate formulate term pair express task guarantee converge optimum fix solve fix second consistently metric outside category method boost noisy constraint finally metric aim matrix entire zero formulate efficiently entry level column practice matrix dominant sparse metric feature dd unfortunately regularized typically optimize reformulate min max fast achieve high unified metric learning distance boost call learner base psd combination one kind author popular boost adaboost quite since eigenvalue achieve require high dataset improve redundancy learner work investigate author cast optimization maximal solve large iteration eigen decomposition experiment competitive low clear learning method successfully constraint implicit optimization proportion training triplet incorrect triplet take word constraint hull definite program infinite descent psd cone incorrect triplet greatly metric rank relevant one irrelevant one set mahalanobis query sort instance pp represent ranking evaluate several roc curve average precision super training solve slack cut plane essentially optimize violate one subgradient descent however iteration dimensionality practice metric structural due nuclear presence
receive noisy challenge continuous interest year crowdsource annotation al method discrete draw crowdsource way example typical crowdsourcing crowdsource end often true rather whereas insight challenge work distinct play much role review conference heavily citation role seem preference present gain maintain quantity quantity assessment address future determining allocate good solution respective linguistic particularly issue study bias adequate student careful feedback build justify understanding rule perspective remain finally student student final set know inner statistical may hand satisfied reliable feedback open become ever critical address current student everywhere consequently free experience thank provide dataset nsf ci fellowship chen massive serve critical tool scale open hundred student despite promise expert develop algorithm course relate bias student student show assignment popularity massive access internet university allow video implement track progress remain feedback open assignment mathematical assessment benefit offer assignment thousand show recent computer interaction course demonstrate exhibit agreement despite room student critical lie reliably present date volume assessment create formulate probabilistic allow median rmse accurate scoring maintain say quantity influence correspond student affects amount detailed relationship small help may work collect consecutive calibrate assignment cover building require student correctly assess student student ground turn student evaluate detail refine additionally divide language english count english student student diverse student unite student around world hold experiment perform use dataset student assignment software accommodate experimental assignment student week truth super student network visible ideal system reliable assessment balanced student scalable hundred student broadly diverse collection formulate allow maintain principled assignment paper student assignment thus student student existence observe unobserved wish score assume unobserved bias tendency assessment percentage reliability close correct bias observe observable deviation residual visualization intensity residual box enough available compute present particularly put prior assume nonzero bias refer hyperparameter true score refer simple vary bias reliability pose relationship answer examine bias consecutive bias pearson correlation reliability hand model bias depend implicitly eq normalize assignment notice bias variance score propagate student assignment note dynamic assignment focus contribute towards unique student understand ability know student may cause place trust vice versa figure explore student somewhat section allow reliability depend note introduce student depend allow prevent overfitte student optimize variable gender well almost agree mechanism improve accuracy might coherence student temporal student pearson consecutive assignment coherence student scoring mechanism student clean allow score depend score desirable mean student consensus truth one interesting dataset mean remark fashion reasonable encourage trust student compare algorithm platform student receive specifically baseline bias histogram make baseline median scoring success dark bar number within pp figure show complete student indicate gain show reliability temporal provide accuracy bias responsible contribute small accurate allow would student increase student five rmse surprisingly model performance synthetic one four per student reliability variance notable impact student expressive reliability tractable reliability oppose student natural confident student allocation confident student ensure student get access feedback score fair allocation accurate like practice confident prediction learner pp actual well understand perform ground simulation confident bin prediction range pass report add pass prediction wrong figure demonstrate confidence claim confident model employ understand benefit could estimate confident assignment simulate ground truth run count confident pp student round confident means confident demonstrate student time spend predict student bias residual network accuracy belief distribution score reliability use influence ability future explore student residual score spend spend spend less surprising thousand spend significantly deviation standard deviation student spend normalize reflect less chance examine work understand residual also notable trend high assignment monotonically bias student reliable exception student get accurate student well bad deviation student student notably
matrix spirit reinforcement algorithm computation dynamic ease importantly specific convergence support reinforcement introduction reinforcement root mathematical application artificial intelligence control account chapter thing recall reinforcement reasonably something parameter work without learning upon minimizing call high quantum iterate reinforcement hand correlate typical supervised learn simple incremental correction though inexact signal iterate make framework economic recent scheme decision find thing equation stochastic iterative root control conditional appear hand iterative scheme simulate accord question incremental move slowly decrease latter averaging ensure limit situation wherein sum negative diagonal matrix stochastic cast iteration involve average amenable plain average reinforcement curse hazard become computation case wherein approximate vector instead article illustrate methodology context google eigenvector rank scheme frobenius eigenvector chain wherein page total node google constant latter irreducible excellent technique essentially base method along brief conclude define cp denote unique c row vector rank factor abuse terminology cp follow independently ni n nx px ode iteration cp affect define cp cp integrate cp z p dr cp dr equality iff similarly get lyapunov origin globally asymptotically stable asymptotic equilibrium asymptotic lyapunov let stationary generality necessary component accordance shall consider within rank output stop prescribed ranking shall iterate needed achieve aim flow differential cp z induce see markov whose row pick z pz ne suitable pair ni j n speed expense increase per simulate plot dot top index show varie conclusion highlight reinforcement already scheme transition completely drive reinforcement sampling desire accord since evolve transition
q l claim note principle estimate dimension calculate intensive sense sample perturbation perturbation ij kk kk k ib ib ib ip fact repeat analysis theorem result definition fy fy I fy f fy fy tv dy dy py dy px j px tv fy dy dy dy dy py py dy dy fy fy dy fy fy fy dy fy fy fy fy fy fy fy put fy fy fy fy conclude v invoke inequality chain ham metric together lipschitz concentration union dt analogou present hmm output parametric estimate accomplish fitting output stationary second step probability estimate finally support encouraging tool state know parameter serious drawback tend slowly recover parameter hmm provably hmms distribution notable hmm output include relation first calibrate laboratory speech hmms insight ergodic hmm stationary mixture approximate small lead hmms give hmm parametric mixture approximately convex program qp behind process markov operate empirical consecutive exact sequence gain mild output exact additionally perturbation practical light hmm hmms fit accurately recently hmm intractable mild asymptotically normally recent hmms also factorization method relate reduce convex consider stability setup learning appear detail defer shorthand similarly write finally positive discrete hmms alphabet discrete tuple p distribution order parametrized sequel hmm generate independently observation estimate transition distribution informally surely ergodic chain geometrically ergodic stationary vast relationship constant hmm py x problem hard computationally parameter follow structural hmm parametric output hmm reduce pass statistically estimate matrix hmm jointly estimate n fit solve computationally dimensionality chain hide constant b geometrically ergodic parameter distinct make impossible mixing learn use distribution pair j output impossible chain x already sample output imply observable realization output commonly em suffer maxima indeed viewpoint e trivial task general separation guarantee propose g polynomial exponential number imply hmms estimate separate perhaps hmms output stationary clarity completeness estimation warm case hmms size case replace state c mention analogy k single pass convex treat iid section asymptotically consistent iid draw treat iid vector bx nothing convex facilitate pseudo sufficiently bx minimizer nonetheless reasoning suggest bx k ensure take second taylor bx second vanish thus program convex ignore negativity lagrange multipli equality b ki eq enforce normalize note invoke qp away chain imply sufficiently large high small estimating require far observe large might attempt constrained guarantee normalization distortion consecutive definition py k k ib consecutive stationary pseudo ib kk kk kk program solve method kk problem negativity constraint essential hmms entry might true one construct qp bound ease perfectly probabilitie consistent satisfie assume namely recall argue positivity amount equation well question discrete small assumption length ensure entry strictly furthermore weighted program allow without qualitative k analyze quadratic quadratic program solution qp length suffice estimation observe accurate ht remark key value require many resolve via simplicity effect hold discrete end error error length theorem estimating length typically illustrate algorithm matlab help hide output cccc component consider qp none qp qp qp exactly emission qp emission estimate know emission guess obtain qp guess qp item guess guess vs realization figure number iteration qp em vs highlight hundred iteration inaccurate show lack accuracy qp comparable iii em surprisingly accelerate convergence accuracy iteration qp approach computationally detailed account theorem state stack thus frobenius definition recall prove geometrically respect fy l discrete bound function state geometrically hmm hmm start g examine ergodicity ensure rapid true discrete hmm follow stationary furthermore p note hamming directly account order var bind bind similarly take initial bind proof paradigm indeed kk lipschitz kk p put strong consistency state conclude go respectively minimizer surely sufficiently consistency minimizer surely program thus bound observation lemma change positivity entry goal ki satisfied ki lead correction vector non fact combine strictly positive loss analyze occur singular small coincide dominate unlikely I hold bind additional number sample negativity proceed v ba qp calculate
line line leverage increase respectively minimum line leverage combination leverage score equal value dot black grey variance tx observe tx tx tx ix tx x ix tx ii immediately partially grant national foundation yu sampling use empirical row column subproblem efficiency algorithm absolute deviation matrix approximation focus algorithmic issue bad running issue address effective statistical algorithmic leverage leverage unconditional bias leverage dominate well result algorithmic perspective superior algorithmic two leverage algorithm construct small leverage solve unweighted biased square base carry empirical practical improved leveraging lead conditionally observe unconditional dealing scale choose surrogate row thereby define random great deal develop analysis select row leverage datum absolute deviation rank approximately projection discussion algorithm leverage yield algorithmic benefit high hadamard code short linear widely library solve essentially implementation large amazon elastic cloud solution least quantile approximated problem base approximation dna snp matrix thousand individual hundred thousand snps none address aspect traditional provide analysis leverage paradigm context fit oppose classical theoretical phenomenon size hundred thousand million regime sampling method algorithmic leverage meet leverage perform analysis ordinary least subsampling sampling bias algorithmic leveraging uniform improve scale considerably analysis provide superior bad perspective neither leverage uniform dominate base leverage leverage algorithm term bias first increase leverage sample effect shrink leverage denote unweighted leverage involve solve biased subproblem achieve statistical empirical contribution detailed property leverage synthetic real set indicate good leveraging algorithm leverage improve conditional unconditional bias variance bias subproblem unconditional bias variance leverage leverage well box square sense I unconditional conditional review leverage estimator follow real set section brief broad context aspect leverage main review leverage computer start l exactly approximately exactly discuss approximately singular call matrix column matrix ol ty predictor via predict response call interest ix ti score extent observation outli result leave singular leverage express mse associate prediction true value subsampling interest relevant sampling computing l subsample e element typically algorithm approach meta call input return eqn e sampling l subproblem rescale row indicate trial observation set choose random trial equal thus describe process replacement apply subsample sample datum denote trial mean sampling construct dimensionality solve weight l subproblem solve ls previously sample uniformly eqn draw uniform subsampling replacement implement easy poorly parameter mass role score approximate interest solve sampling rescale estimator eqn probability eqn basic leverage algorithm propose construct subproblem motivate analysis leverage mean leverage rescale accord normalize sampling solve rescale sample solve unweighted ls solution rescale eigenvalue unbiased accord rescale lead use coefficient analyze statistical although run run depend solve former trivial depend recall score dominate computation score na I qr thin score take fast solve great compute main relative leverage score roughly eqn provide running depend randomized hadamard transform numerical provide environment demonstrate leverage algorithm qr decomposition evaluate variant environment resample resample desirable property method extensively whereas algorithmic leverage interested construct subproblem goal resample traditionally related include bias variance subsampling describe analyze challenging reason estimator randomness subsample ease analysis employ taylor subsampling combination sampling bias condition condition datum start bias variance ls probability l around value vector term bias variance well size size input leverage analyze leverage way construct l score leverage leverage construct unweighted biased l start eqn estimator rescale solve I probabilitie subsample interested vector denote entry perform multinomial q easily vector ol follow lemma weight ls yield q x remainder significance encode sampling depend series little although evaluate quality empirically currently characterization expression fail lose capture information may inverse invert hold increase regime large sample information score thing score designed preserve discussion last point available confirm two analogous expression establish follow expression unconditional two expression condition refer expectation variance datum expression condition traditional algorithmic leveraging subproblem solved given specifie distribution rescale unconditional leverage eqn state term negligible valid conditioning approximately relative l approximately unbiased true state leverage unconditional expectation result eqn leverage unconditional subsample middle score unconditional ls imply eqn large ls gauss lemma unconditional relative proof proof property leverage work focus provide leverage explicitly toward leverage provide algorithmic analysis reveal interested subtle lemma neither uniformly superior leverage rx tx unconditional expectation point expression number row second variance confirm empirically bias expectation expectation unconditional point worth variance sample e compare nearly equal leverage variance eqn weight several way strength would near want preserve help expansion avoid rescale leverage thereby avoid involve convex leverage normalize leverage e construct output compute denote form since involve eqn enjoy approximate present several assume extremely small could second oppose assume increase eqn still score score exact take fast algorithm reason promising scale different probability distribution previous bias leverage first modify unweighted ls yield tw pr remainder analogous different expansion perform term somewhat expand point taylor lemma condition find procedure w tw te xx tw x tw full unconditional unconditional approximately value parameter apply give time estimate ls instead square estimate roughly center around eqn random sampling one problem solve thus unweighted example conditional expression section part empirical bias subsample synthetic illustrate mse compare variance separately outline synthetic draw standard realistic fairly moderately synthetic summarize unconditional illustrate overcome problem variance method run generate lead rectangular moderately uniform score second uniform score leverage matrix multivariate refer ga degree freedom freedom table summary leverage score divide datum report confirm manner std ga ga ga e ga ga e ga e ga e e e k e e e e e e making tend leverage score intermediate deviation substantially less minimum leverage median qualitative trend base trend leverage score well small minimum fourth distribution ga leverage rectangular hold versus figure matrix similar fix generate multiple time reliable variance plot ga learn theoretical thing general square datum unbiased sense quantify ga somewhat quite differently indicate score uniform ga sampling subsample increase bias tend decrease roughly subsample agreement slow eqn eqn bias increase especially leverage score lead recall mse leverage deal consider uniform thereby leverage deal rescale subproblem except synthetic matrix ga data respectively worth ga panel panel panel difference consist uniform manner slightly range subsample theoretical axis panel subsample goal leverage substantially result leverage size range subsample size probability leverage around effect beneficial bias variance rectangular score result avoid grey consistently small variance emphasize bias variance bias suggest may primary goal unweighted bias variance bias various subsampling lemma direct perspective leverage consider figure present main conditional variance matrix ga observation ga exception conditional expectation approximately unbiased full ls estimate full unweighted ls variance subsample moderate large ga panel ga panel upper panel panel square black grey line line default leverage slightly unconditional bias variance consider conditional bias variance section provide additional outline result describe one sample constraint preserve leverage thing traditional unconditional finally real conditional uniform moderately leverage realistic describe versus lose course ls subsample construction subproblem describe algorithmic leverage guarantee happen follow roughly row sample importance approximate leverage score sense eqn high loose e leverage uniform loose well procedure somewhat sampling multivariate degree leverage keeping row leverage somewhat toy subsample size result summarize figure solid line panel rank panel rank scale trial worth first loose roughly less roughly lose sample subproblem singular particular many preserve phenomenon rank subproblem try subproblem try highlight lose obtain less severe fail capture outperform leverage quantity except axis low panel plot logarithm middle panel show figure worth bias comparable implicit violate variance effect minor bad variance increase gradually examine increase scale decrease bad perhaps fast randomized approximation score description return leverage appropriate choice hadamard approximation leverage score hadamard traditional deterministic algorithm matrix small several implement software environment qr deterministic pc processor ram window fast normalization I algorithm mean note hadamard sophisticated implementation parameter summary run leverage vary exact sensitive vary whereas contrast correlation approximated rapidly increase short run summarize plot varied see exact multiplication dominate running become environment fail run size permit even simple interested probably use hadamard projection sophisticated evaluate implementation qr bottleneck fast randomized predictor size panel cpu vary size connect line connect leverage dot line connect time leverage slightly variance leverage compute score lead algorithmic identical exact left panel panel estimate panel exact line illustration real set draw leverage illustrate make application previous pca snps illustrate leverage score rna seq set cell seq become analysis digital obtain million read rna seq summarize sequence short count find rna seq read map genome nucleotide th assume transform read ij b ib read ls consume seven subsample sample variance subsample subsample calculate estimate full plot histogram sampling quite suggest base uniform sampling panel empirical bias seven subsample subsample size method subsample size large panel right panel empirical dotted line illustrate datum moderately microarray present response remain patient select patient gene fit subsample nine subsample summary probability highly skewed quite large one leverage middle bias nine small interestingly approximately large bias subsample subsample comparable subsample size panel empirical grey dot algorithmic recently relate sampling empirical leverage paper perspective algorithmic leverage particular algorithmic analysis provide uniformly bad algorithmic reveal neither dominate leverage algorithm maintain usual leverage empirical demonstrate exist newly leverage
hand obtain eq q establish correctly anchor proposition element bound theorem put everything appropriate algorithm dictionary anchor every chernoff bound see disjoint bind q satisfy bound assumption dictionary rip let cx cx additionally fact q apply matrix diagonal prove claim section section theorem definition write bold pt consider overcomplete dictionary sample select sparse result strategy recover efficient algorithm cluster use scenario stage overcomplete dictionary incoherence observation coefficient unless impose subset element sparse argue sparse provide attention survey extensively study entail setting argue provide flexibility model great blind bss video dictionary overcomplete observation overcomplete present overcomplete dictionary decomposition cluster estimate form recover overcomplete post estimate dictionary condition advanced post develop subsequent consider non randomly uniformly element pairwise incoherent matrix certain sparsity knowledge first kind overcomplete dictionary special coefficient value solution dictionary constraint recover approximate dictionary provide recover generalization analyze alternate minimization subsequent outline tractable overcomplete recovery similar detailed relate work different community turn code establish succeed reconstruct certain code scaling efficient use condition dictionary al allow overcomplete heuristic dictionary practice theoretical iterative iterate estimation update dictionary optimization viewpoint alternate al optimality alternate minimization focus global combinatorial quality solution bound algorithmic stability sparse representation task predictive accuracy recover element closely independently work important require yield work et develop variant analyze subsequent work develop blind source bss mix blind imply dictionary extensively study source ica guarantee overcomplete source topic factorization various method guarantee assume topic word column expansion recovery al dictionary make work consider overcomplete expansion technique form involve find clique node clique find community detection context kind one free dictionary clique contrast work detection handle community overlap community overlap across different coefficient simple neighborhood state main approximate subsequently process give intuition underlying procedure step construction employ subset employ dictionary search clique dictionary element use core relationship coefficient coefficient dictionary bipartite encodes pattern coefficient map graph pairwise element result bipartite argue set fraction common diversity among expansion dictionary sample success subsequent broadly divided estimate accurately clique dictionary combined amount overlap coefficient argue neighborhood enough argue incoherence element accurately procedure estimate dictionary matrix sparse lasso recover estimate dictionary dictionary differ statistical assume exploit available deterministic combine thresholding procedure exact albeit value solve another c c I denote matrix denote column denote neighbor work recovery certain present correlation randomly edge intersection neighborhood routine dictionary ensure node neighbor correlation edge dictionary desire separation element initial u vector indicator unique return propose norm incoherent element incoherence element dictionary bound constant draw fix constant column complexity recovery theorem choose q correlation normalization incoherence rip constant lemma appendix bound coefficient make sparsity require recovery decay incoherence threshold intersect special guarantee establish something next establish clique clique unique dictionary element graph clique necessarily implication exploit dictionary element element two shorthand notation satisfy guarantee lemma clique triangle order anchor intersection choose among unique intersection indicate pairwise intersection formation anchor good element common dictionary element large neighbor amongst form correctness lemma crucial element intersection clique role provide unique intersection lemma establish sound high amongst event triangle rather anchor procedure need substantially small anchor indeed good anchor sample correctness naturally correctness correctly piece establish good anchor consider satisfied relatively key piece iteration great proceed hold pair ensure copy approximation clean dictionary assumption noise general setup present proof observation usual dictionary error mean even typical subsequently remainder fact assumption prove initialization approximately initialization rip provide guarantee equation obeys eq establish equation manner condition return verify obtain consider follow appendix satisfie immediately maximum singular show next guarantee model non zero since ready proof thing first initialization second pose solve assumption guarantee dr linear rs substitute bind mean linear pose proof novel base recover overcomplete sample analyze denoise sparse reconstruct exactly tie sophisticated room provide guarantee matrix randomly yet context overcomplete establish unsupervised possible hide observe dictionary suggest direction seem inherently important recovery another natural perform iteration follow estimation recover motivated processing machine provably correct procedure popular suggest microsoft fellowship nsf award nf thank helpful discussion thank suggest initial proof lemma deferred start first via contradiction l claim next establish lower bind upper bind eq exclude already pick establish correctness lemma follow common follow low probability numerator begin I event share arrange choose assign unique similarly remain logic algebra invoke rhs final low probability end observe py iy element l control different event bound probability complementary argument numerator event j iy
truth feature node activate network generative indistinguishable net truth exhibit generate net constitute autoencoder sense include decoder complete autoencoder encoder actually network run reverse appropriately threshold reverse stable noise net suggests level modern heuristic trick provably see analysis level hide include svms solve truth output denoise autoencoder code mention hide layer encode autoencoder generalization even seem possible adjacency matrix coherence restrict expansion autoencoder analog fact compress matrix compress contribution furthermore network practice neural net top bottom denote edge allow nothing much else paper complete proof layer assignment pick among become denote threshold else apply stand bipartite graph deep fig pdf net ground carry simple learner instead degree layer successive etc note network learn network true learn unable instead exploit autoencoder old adapt lot put globally problem reconstruct bipartite graph root np random speak leverage correlation together language suitable truth network optimum formulation seem useful think leverage local promise avoid usual nonlinear truth distinct level tend fairly disjoint set note neural net useful reservoir computing fact assume random bipartite carry choose denote wu least mean relate expect denote forward neighbor backward layer allow size degree simplicity recommend reader expectation happen bipartite neighboring neighboring adjacent version say show denoise mention net approximately go back representation denote hide layer logistic etc denoise autoencoder autoencoder decode encoding act coordinate autoencoder high draw shorthand use empirical autoencoder implicitly deep reconstruction actually generative encoder term net successive deep net adjacent denoise autoencoder autoencoder denoise autoencoder allowed flip every graph property respect neighbor property respect sketch decoder generative autoencoder graph assignment level prove majority neighbor edge edge lower network autoencoder layer efficient network search hide observe connection pick randomly put simply layer give node output node also layer autoencoder random graph allow flip output bit intuition observe hidden look layer property node neighboring layer neighbor connected positive node value least neighbor node understand hidden least autoencoder work margin stay stable flip denoise fraction convert fraction convert chernoff bound switch thus still noisy flip get need detail pick choose h uv du hence wu side unique thus case need notation pick fix flip outline layer learn layer focus sparsity support wise recovery h correlation graph use graph end observe generative assume observed layer layer via correlation step rule thing v uv output pair sketch vertex edge neighbor exclude parameter hence conversely cause respectively assignment union recover use edge detail encode satisfy output property neighbor neighbor unique edge edge find edge entirely unique contribution edge positive three sketch recover roughly one say relate triple triple roughly triple correctly consist triple bound state net hide layer run randomness detail still edge unique positive neighbor matter small turn classifier weight accuracy omit overall call use call decoder layer pairwise correlation observe triple share layer later show section show general give edge weight need generalize node unknown bipartite connect equivalent find recall hide need still later choose random among look happen happen simultaneously tu ts tu ts ts reduce lemma kind event every learn slight modification level layer higher correlation keep almost neighbor property roughly subsequent obtain define vertex least inductive argument crucially neighbor neighbor far occur pick edge depend layer intuitively pick parent result additional encode thus full repeat construct hypergraph let recover iff non typically unique neighbor operate activate look cause want warm bound replace bind use ignore I probability without continue bad maintain maintain invariant iteratively note bind expansion common first none know go back let b b u h I v thus u target graph set b sa intersection v reconstruction recover subgraph problem square pair node whose setting whereby bipartite whether bipartite graph whether share parent bipartite positive square hard set section solve vertex iff share parent find bipartite correlation give triple mutually detail resp neighbor give algorithm fu fu du uv fu fu say possibly neighbor second property basically say third property introduce property closely relate graph one common condition false property say statement property know connect fu fu fu fu learn successfully vertex cause algorithm successfully identify take take definition find v generate size neighborhood intersection belong refinement property graph graph randomly solve recovery randomness fu fu uv fu fu first property say cause except property basically say almost disjoint cause many say every introduce correlation sample graph fs fs md n ns md chernoff smaller easy vertex intersection fu fu fu fu fu fu consider edge know sample connect half outside satisfies property time statement say statement show unique know fu fu fu fu successfully find number cause successfully know vertex become become hard reveal correlation dependency correlation extend bipartite connect randomly hypergraph iff exist bipartite definition v create connect every mark behind rare correlation pair performance well correlation choose accord definition randomness wise expect use wise fu fu uv concentration bound hold graph definition satisfie satisfy pairwise neighbor statement false fu fu fu total fu ss fu relate finally notice algorithm vertex low threshold learning paradigm identify correlation recover weighted decoder form denoise autoencoder output weighted encoder vector sketch bit graph weight weight add pick time want kind lemma assignment sparsity state section probability weight choice common idea last notation layer distribution weight deep restrict sparse layer correlation compare difference vs h rigorous similar lemma randomness v h allow hypergraph hypergraph pair regression problem hence get satisfying probability weight learn correlation share weight variant algorithm learn weight even learn universal learn time give decode coherence go v probability expressive one output show threshold choice network simple observe parent happen ask height pdf rigorous beneficial rigorous analysis width spirit view net though reservoir concept autoencoder graph would randomness life
gs use message receiver immediately carry local library parallelization process know particle way parallelism pixel particle assign space balance tracking particle object load balance consecutive belong adaptive balancing size automatically adjust library implement like patch size circle corner balance eight involve computationally costly formation calculation impact pf base span visit load visit patch much small whole image compute pixel complexity calculation fashion one patch neighboring access patch share cache library sir piecewise piecewise capability library implement pf dynamic velocity instantaneous velocity etc dynamic appearance library motion motion range walk switch motion type occur use near velocity frequently use practice estimate position velocity estimate limited intensity impulse spread intensity yield length nm pixel nm nm acquisition ideal profile corrupt mix noisy point pixel location vector object tracking problem ratio snr image synthetic showing accord describe dynamic appearance pf track ground trajectory generate synthetic right quantify tracking library include dynamic precision six particle synthetic computer consist intel ghz v inter communication rna core rna maintain amount library implement rna exchange keep track actually successfully track arrange topology lose whenever object ensure exchange reduce runtime strong increase core core core variant per overhead simulation hybrid parallelism scheme rmse test pixel scheme use six six core scale particle scalability nevertheless less rna parallelism six library enable filter parallel library level hybrid parallelism combine show capability library biological imaging application library load balance balancing implement pf image library system easier programming simple pf presented distribute across core efficiency rna graphics gpu accelerate pf design collective library available web section team operational grant national foundation f grant organization scientific present filter software library distribute memory parallelization filter pf message pass level parallelism inter process load balance balance library difficulty program implementation pf demonstrate capability distribute pf library million particle core parallel tracking target processing despite inherently pf limit practical real address algorithmic improvement share implementation distribute library library platform develop pf contribution library optimize hybrid memory library orient architecture exploit hybrid parallelism pass combine intra process parallelism performance compute library load process balance process inter resample pf intra balance throughout operation exploit allow popular access programming propose facilitate demand pf smc manuscript generic pf resample briefly follow effective library application image pf library discuss pf consist sequential sis ii implementation part sequential resample sir unobserved ii model observation posterior square mmse eq particles amount particle weight observation sequential sis sis small successively overcome perform high fall sir sir sir trivially parallel sis importance ki sample k index resample include truly resample pf communication case local implement protocol label process communication overhead require gs match iterate particle receiver full move procedure gs j schedule sort sort identical algorithm sort gs perfectly cause overhead pair receiver thus find limit give sort library write interface inter parallel start interface currently basis also build library fig library actor iv tool module pf algorithms parallel distribution module module observation default library include sub application module particle method module sorting list etc link allow provide method process file I interface code implement pf application hence library parallelization easy library library library divide
course confirm prediction replica symmetry mechanic box storage capacity amount problem polynomial predict ultimately long belief present purely essentially powerful mechanism obtain bound storage happen happen combinatorial replica capacity storage also relate relate concept utilize result elsewhere relate version spherical perceptron neural mechanic uncorrelated spherical translate cover correlate pattern standard normal possible beyond setup utilize show would use central limit particularly elegant would exposition principle elegant little model create technique choose routine generalization primarily analytical quantifying capacity course interesting arise look view strength capacity alignment bind storage capacity constrain hard counterpart analytical consideration algorithmic mention algorithmic present discussion multiple throughout limit exposition avoid presentation main concept unnecessary discrete analytically vast easily routine direction elsewhere university mail edu long network analytical start initially mechanic approach prediction obtain follow later rigorously fact type discrete mechanic rely characterize discrete similar mathematically rigorous mechanic appear mathematically provable bound spherical capacity several lot analytical characterization appearance mechanic relate characterization simple tool analytical relate seminal developed know replica treat almost perceptron start course spherical perceptron accurate storage several different often spherical threshold correlate incorrectly either good quite somewhat successful result actually long appear special storage capacity know within pure mathematic real treatment appear confirm storage capacity spherical moreover confirm later prediction confirm rigorous bound hard perceptron type relate storage treat extension replica utilize make rigorous note initial call spherical long easy analytical believe treatment like mention treatment start design perceptron obtain certainly call already happen substantially result storage able match advanced version need provide prediction discrete limit study rigorously confirm obtain replica symmetry rigorous bound turn make presentation easier briefly sketch organize section mathematical perceptron operate class later section perceptron establish paper section type plan detail conclude remark easy perceptron need closely spin interaction strength site site call site follow without configuration strength know class unless one strength essentially make easy general scenario specialized amount spherical restriction mathematical restriction present powerful handle restriction avoid clarity purpose call perceptron often refer one convenient analogously perceptron set constraint perceptron operate strength perceptron constraint perceptron storage alternatively represent pattern pattern point bit govern dynamic couple restriction spherical perceptron alternatively would perceptron mathematical see mention variant neural possible purely various nice contain collection many mention choose adapt know try presentation somewhat contain case concrete elsewhere mention conceptually purely analytically create treat spherical proceeding study perceptron subsection look spherical know discrete brief presentation present presentation start recall replica characterization storage capacity present exactly characterization look assume length respectively large proportional obviously assume replica give decay away maximum rigorously neural pattern consider capacity spherical perceptron hold mathematical formalize large constant scalar constant mention early essentially storage capacity mention randomness however speak relate spherical spherical perceptron say network turn correspond standard conjecture mention bind storage spherical perceptron work briefly summarize fact htb far negative spherical perceptron section relate concern neural consequently emphasis relate spherical perceptron spherical thing perceptron technical presenting later already observe perceptron substantially first call perceptron equally important e place later view pattern uncorrelated assume bernoulli system q course large indeed match symmetry essentially think one column keep dimension extent loss generality I standard moreover mean characterize determine bound result extend thereby let constant arbitrarily constant scalar ignore infeasible eq feasible course essentially establish capacity rigorously confirm hand capacity probability high confirm conjecture question break capacity pair infeasible probability appear upper characterize normal independent scalar infeasible present figure illustration take theorem indicate visible capacity substantially e look essentially conceptually know discrete perceptron extensively throughout two namely bit technical detail present concrete know move spherical constraint feasible subsection presentation way presentation mechanic approach analytical characterization start although concern spherical perceptron observe handle mechanism also perceptron assume restriction convenient point pointed instability replica rigorously combinatorial argument open consideration storage capacity upper safe range parameter consideration replica treatment scenario great couple mechanic course already extension start break replica symmetry way critical storage give substantially mention perceptron perceptron far mention seem necessarily may among rigorous probably resolve proceeding presentation detail need recall probably basically storage capacity bound beginning concentrate purpose rely strategy ultimately follow follow axis last course last trivially completeness strategy lift mention probabilistic mention variant create low section substantially establish lemma respectively variable lemma structure introduce change follow lemma analogue also establish specify observation I scalar infeasible discussion combine could therefore storage capacity furthermore present eventually match one optimal well analytical transformation produce would surprising mention take numerical error believe emphasize completely mathematically rigorous bit numerical htb x mathematically rigorous result exposition section rely early hard storage capacity feasibility argue normal moreover continue work continue satisfied discuss could proceed follow exposition section maintain basically disjoint exercise keep exposition free unnecessary trivial skip concentrate affect well bind strategy bad mention absolutely necessary exposition exposition however view exposition way inside derivation follow back ultimately present feasibility interest determine feasibility early pose feasibility analogue spherical infeasible proceed namely everything fix really really find probabilistic problem ultimately relation infeasible satisfy relation mention essentially analogue fairly see consideration comment basically fairly g independent look interest obtain arbitrarily positive constant independent leave side assume summarize scalar independent let random let scalar q scalar infeasible probability present language ignore long eq infeasible exactly storage capacity perceptron basically establish replica mechanic rigorous assume replica operate course htb mention employ attempt find substantial present present present combinatorial bind sketch one obtain perceptron perceptron look likely satisfied previous accounting essentially row entropy union capacity base particular improve relate combinatorial bound differently study perceptron presented indicate correspond perceptron obtain substantially low example replica predict calculation type choose typical perceptron analyze vast perceptron version adapt handle mention exposition particular case concept leave presentation choose extra case goes limit digital basically digital constrain analysis reason constrain presentation easy try exposition section capacity constrain feasibility section ease exposition continue normal dimension moreover continue continue perceptron successfully proceed allow come basically would fairly skip concentrate mean bound basically feasibility fix previous exposition inside derivation go early infeasible ask analogue probabilistic ask infeasible probability negative usual randomness answer detail previous everything try small probabilistic problem ultimately first course conclude infeasible probability provide section analogue lemma course consequence fairly I comment basically g arbitrarily independent maximization obtain vector component zero linearity standard normal arbitrarily analogously dependent q need simple follow ultimately one assume strategy operational obtain look low enable bound directly look identity although mention equality last replace keep integral find
gaussian need acknowledgment author thank many theorem axiom conjecture example theorem exercise lemma remark summary optimization social I informative signal proximal kullback divergence online nesterov average purely identifiable scheme exponentially kl divergence highlight possibility consequence employ focus decade application range sensor economic scenario need represent decision global spread adequate recover neighbor sensor neighbor development lead advance decentralize generalize new principled distribute researcher ram work al dual average stepsize well social learning link two motivation recent author complexity involve agent receive private paper observation compute case agent explore building help problem mle product likelihood represent maximize know bayesian exact setup kullback prior add use proximal counterpart aggregate log agent step centralized belief aggregate identifiable connect specifically rate expect discrimination capture divergence aforementione state indeed stepsize stepsize use recover paper organize follow interact constrain dual iv stepsize conclude consist index denote finite belief interior simplex learn conditional govern signal ti private also independent state agent perspective identifiable log marginal bound triple number ti occur unique interaction agent capture undirected link pair belong communication wherein exponential th belief denote standard nonnegative entry definition entropy straightforward simplex update could view counterpart give rule divergence multiplication update since perform need leave positivity implicit lagrangian q perform prior however interested decentralize centralize study section distribute contrary slot agent communication begin slot agent accumulate observation gradient form slot contact gradient agent estimate let take stochastic equip kl define employ divergence belief opinion lemma bayes update stepsize stochastic evolve discrete closed let kronecker need complete argument form write close equation rule construct gibb state subsection play key role convergence time aggregate close index direction define converge sense sequence doubly product preserve hence show doubly magnitude entail distribute author point maximizer demonstrate consistent agent apply belief maximizer dirac lemma weakly accord negative fact trade setting stepsize must vanish consensus guarantee network stem grow direction influential gibb characterize convergence rate exponentially discrimination information log expect discrimination denote
hold frame present prove phase critical conjecture suppose frame vector trivial hold norm partition two empty set construct non frame instance times frame author support national foundation nsf corollary proposition conjecture property redundant frame hilbert perturbation token reduce critical cardinality prove non retrieval physics paper list network transmission hilbert endow scalar z equivalence ray form regardless whether two positive finite simply frame nonlinear phase magnitude coefficient global phase phase paper phase space phase frame necessary condition slightly different state aforementioned phase set phase frame generic frame clearly phase current art topology q critical author case frame review perturbation cardinality conjecture show stability frame fail note problem case equivalent least perturbation embed unitary space endow nu u outer denote symmetric rank note key complex denote eigenvalue large frame denote operator read magnitude appropriate frame lower span span h nonlinear consider eq start present lemma fix hilbert space mean u na nr n present I additionally ii vi ix add side theorem v real property invertible scalar frame invertible phase canonical k f h phase gx yx ii claims vi vi obvious obvious know interestingly answer example phenomenon belong matrix f matrix associate hence frame check linearly frame critical assume frame accord kf f nf f lemma notice thus finally estimate far
elegant accelerated solve constrain share effect capture apply procedure equal projection acceleration truncate svd fast difference rather evaluate effectiveness compare art experimental apply mse motion filtering server ghz intel processor ram effectiveness wherein original estimate h cccc compare cost robust size different rank wherein sparse build wherein entry draw wherein stop table recover low much cpu improvement significantly round ht background modeling correlation video frame low frames relate four respectively compose frame example video frame convert frame figure separate video sequence comparable minute sequence fourth around second therefore make scale light always image capture matrix show face rank real face sparse application pair video row row frames frame report part part generate assign close competitive transition regular behavior highly possible study ccc robust apply use decomposition frame sequence considerably perform flow surveillance video sequence translation well figure htb successfully geometric transformation detection tracking unify significantly attribute distinguish shift object flow segmentation crowd exist evaluate dataset include image scene image text medical text music sub yahoo obtain website website practical compare ml knn metric evaluate effectiveness second multi four precision fair evaluation consideration cpu knn mse table interface classic knn uncorrelated dimensionality dimension mse integer dataset choose competitive explore projection accelerate particular slowly asymptotic asymptotic convergence matrix sparse alternating produce make converge discuss far substitute side manifold represent wherein complement space substituting result entry eigenvalue respectively consider singular n fc fc vice versa normalize variable via general without normalization speed versa wherein convergence complete part asymptotic thus noise noise analyze low scheme modification diagonal singular part consider unitary spectral leave leave hence form deterministic singular tr nr bound singular decay deterministic small svd rank approximation modification approximate bound decay produce modification base analyze average approach power frame modification modification latter produce decrease increase deviation approximation ta ta concentration frame except see proposition eq partition orthogonal project deterministic proposition ta mi top proposition bottom block apply proposition spectral substituting obtain deterministic rather bound power modification proposition complete proposition draw hold invariant inequality calculate matrix variate r accord wishart proposition standard give note bind modification theorem complete proposition suppose standard gaussian deterministic study ta triangle lipschitz ta ta proposition event q event definition therefore imply ta ta obtain theoretical alternate analyze convergence behavior leverage reasonable able without selection objective follow q begin iteration equal optimize support optimize fix notation iteration iteration compute direction fast add optimize fix previously gain immediately obtain state sparse optimize hold decomposition suffer computation complicated structure decompose sum component incoherent structure firstly alternate rank part sparse scalable big form build right projection low greedy paradigm update mutually greedy manner significant improvement complexity propose nontrivial variant generalize derive strategy segmentation object sparse motion share multiple separable effectively score recommendation decompose low row study show real rank paradigm multi modeling segmentation generate structure provide semantic interpretation addition similarity thus robust feature play unsupervised recovered complete completion portion entry restrict sample reduction broadly explore cloud low rank researcher explore restrict expressive complex motivate robust summarize reveal global capture separate interesting decompose part incoherent complete dictionary versa two separable whose two building incoherent change incoherence view gaussian source lead identifiability fulfil class big complex firstly prohibitive extension invoke per iterate achieve incoherent dictionary encourage suffer time complexity achieve consume rarely improve scalability speedup pca subspace low projection column precision technique need lead costly determine low secondly sophisticated nonlinear geometry expressive wider explain current rich central interest application capture video include object behavior furthermore general largely rely transform fit evaluate volume build part incoherent big hand dense part stable verify noisy low part sub temporal identifiable usually role extension proper practical study decomposition rank overcome burden cause two projection update low recently wolfe paradigm update rank mutually generate considerably provable guarantee rigorous complicated mixture low sparse expressive variant shift track raw pixel far rich seem subspace ensemble extend rank novel address ensemble need learn fully subspace functional scoring item constrain row contain part effect item collaborative consume completion new item need strategy propose experimental problem effectiveness row mention recover sum exact exist incoherent obey select augment multipli accelerated method svd costly strong decomposition bernoulli noise approximated optimization aim highly version problem alternatively subproblem although subproblem stand singular projection subproblem update value updating need develop later alternatively assign cardinality hard thresholding updating via similarity player go part cardinality cardinality might introduce cardinality constraint replace support robustness appendix manifold firstly power modification time consume svd merely require multiplication invoke matrix unnecessary sketch adaptively determine stop error svd matrix wherein fast include inverse multiplication dense point operation use projection projection obtain new apply slowly poorly design modification decay fast share vector base approximation calculate qr e power modification five multiplication therefore qr decomposition matrix thorough bind scheme give cost qr per integer per htb la ty ty ts truncate although propose randomize free algorithm cardinality incorrectly pca randomized dominate slow scheme paradigm model start column optimize observation derive alternate mutually matrix column column specifically object decrease fast rank update greedy dimensional projection set possible biased select direction increment warm start high rank optimization furthermore mutually update simple yet svd implementation paradigm completion complexity rank noisy iii apply matrix paradigm particular formulate norm regularization soft update sort cardinality constraint element thresholding replace compute iterate subroutine greedy incremental paradigm iterate converge achieve fast decrease partial derivative row select decrease decomposition estimation rank decomposition capable tackle volume datum complicate exist develop combination incoherent beyond two strategy scalable several whose impose store move motion geometric share frame develop randomize extract sequel rank matrix update piece linear manner raw video frame separate flow recover transformation decompose separate move tracking stand step accomplish storing treat datum wherein stand sparse outlier segmentation sparse trajectory row shift reference due pose object flow reasonable inverse structured invertible transformation frame permutation permutation pixel geometric translation affine parameter affine wherein worth transformation beyond define aim background invoke eq obtained save facilitate tuning cast flow solve firstly solution aim equation albeit nonlinearity use piece wise transformation piece view loop include update jacobian transformation linear q iteratively difference leave emphasize select save frame affinity adjacent template another consider flow obtain belong background rule background complement approximation wherein accelerate base acceleration trick nearly area frame rank dense subproblem global via thresholde eq cx jj jj j cx j li update transformation accelerate leveraging position nonzero summarize predict method focus training mapping account correlation improve prediction grow increase sample insufficient prediction jointly inverse give sum matrix residual randomize part map correspond annotated row row explain
disadvantage performance dag margin bound generalization ability round support vector bound support vector number binary eliminate candidate remain possibly assign misclassification crucial point originally design top low result discard output ignore answer accord misclassification mention select provide class avoid indicate test require test low generalization performance unseen structure principle term vc randomly expect indicate sphere contain close set empirical vc framework class many svm hyperplane create suppose provide example margin pair model learn class obviously represent term margin ability b enhance carefully design utilize measure good believe use svms demonstrate classifier size training partition classification subset remain learn evaluate validate find example misclassification letter kernel trend actual technique classifier fig figure illustrate estimate cv sort generalization measure trend increase actual risk trend find method clear fig statistical fig bind low fold cross suitable apply research svms enhance approach base max order improvement elimination candidate filter enhance max increase utilize goodness framework times classifier apply however even wrong method weight select level minimum call acyclic initialization classifier minimum perfect class apply candidate classifier scheme chance wrong class nn small generalization discard binary svms weight perfect matching generalization class last remain node edge one output graph error subset minimum weight edge incidence convex hull give v mx minimum least large satisfie therefore candidate discard generalization sort element class discard output last output class calculate generalization class sort list binary sort classify classifier sort include discard candidate final classification employ binary test denote calculation define consider class accept less want filter candidate contain misclassification letter voting reach vote wrong observe study misclassification letter provide result score class target vote vote mis correct propose include equal vote example vote maximum represent percentage target class voting target eight example second around rank range varied varied fourth varied classify label random want correctly target class guarantee misclassifie value filter big misclassifie example first almost third fourth increase cover candidate class large create employ classifier hand may technique tune max divide part protocol discussion run uci repository tumor movement test add fold validation evaluate htp employ rbf kx polynomial apply page movement construction margin maximization software package create examine order order original enhance technique e se max art result table table pair among traditional represent bold face technique propose accuracy indicate confidence pair represent high baseline number symbol interval difference symbol htp htp combine need classification number classification wrong performance give incorrect answer due equal voting reach one select mistake include provide precise propose ability able classification technique measure margin method direct acyclic strong elimination classifier se vote enhance optimal class superior maintain testing next improve e sequence classifier select minimum eliminate assign se traditional enhanced call many generalization error ignore class ignore discard candidate employ give classification binary voting technique select competitive eliminate classification equal compare conduct use large concern high concern max term accuracy measure optimally mechanism fold learner propose discriminant analysis etc fold offline thus classification phase would dr valuable partially research school apply core acyclic acyclic previous attempt svms generalization via extract svms method build acyclic strong elimination se classifier candidate demonstrate high traditional one recognize art two time fast classification performance svm construct class approach solve subproblem difficulty increase number class construct train build train unseen combination classification process max one vote final lee nuisance vote vote vote fuse work label respectively adapt traditional rest correspond classifier rest difficulty calculate absolutely separate hyperplane denote string indicate unique bit string represent class bit care class classifier design classifier complicate obtain classifier decision acyclic select result eliminate candidate output ignore classification apply recursive apply last class misclassification wrong answer time produce direct acyclic triangular time possibly addition many measure select class construct multi concept li et preferred region investigate framework know max currently recognize art combine among classification reduce study characteristic lead wrong classification weak point opinion decision discard mistake last opinion output
blue represent sample represent entropy measurement great information circle characterize entropy white square region inside white elsewhere dark color play robot design take subsequent measurement shannon employ utility take next measurement bayesian circle replication metropolis hasting robot field record circle sample circle circle give light sensor collect consider fine location play location circle likelihood circle entropy measurement circle measurement great entropy set thus affect circle machine affect entropy efficacy sensor model quantify robot need parameter sensor indicate likelihood I sensor sensor region would white unity large case black gaussian distribution deviation likelihood na I light compactly measurement product likelihood experiment deviation white weighted light region field sensitivity complex peak far mixture gaussian black intensity completely surface completely surface surface unity minimum offset serve scale sensor sensor surface completely surface unity sum grid mixture gaussian parameterized sensor frame amplitude denote ensure unity find sensor vary six gaussian subscript estimate two gaussian assign student integrate write measurement define model generate discrete sensor response laboratory light property lead height measurement room avoid ambient cast sensor tb boundary illustrate use symbol indicate value top sensor package gray illustrate oriented look sensor surface direction bottom cm completely boundary white region process repeat sensor surface white sharp boundary surface sensor frame center black white region result process repeat four sensor pattern white boundary sufficient uniquely infer since line orient boundary four consist region white result measurement surface define relate illustrate tb describe method keep distinct inference circle inference prediction measure intensity light demonstrate improvement platform tb symbol figure use figure record dramatically sensor white sensor model obvious slope indicate record completely evidence gaussian algorithm mean list log model illustrate factor probable consist field center shift center due refer figure one wide axis along last demonstrate predict excellent white explicit sensor view inference dimensional incorporate machine system necessary circle first measurement location light sensor indicate indicate relative respect na I location indicate green square blue circle circle sample area white indicate inside circle informative make elsewhere figure later seven panel system accurate light sensor whether accomplish I sensor panel comprise left robot system view play use na I light sensor indicate square intensity na I green circle location white square probably inside measurement elsewhere plane stand equally shape known driving likelihood contrast panel comprise figure use sample circle partition enable sensor circle help accomplish I sensor quantify observe robot obtain precision light reveal take na I light inference sensor possesse demonstrate precisely almost employ simply noise quantify rise sensor naturally inference apparent careful design na I measurement circle interior circle circle location outside circle converge show simply search employ essentially detailed circle sensor answer performance estimate achieve consider I position radius precision bit measurement obtain bit measurement location method present inferential type sensor wide design likelihood incorporate plug fashion system force rely sensor sensor acknowledgement research grant would like thank preliminary effort arm sensor provide sensor extremely expensive prohibitive sensor sensor efficacy employ inferential employ arm field light spatially region sensitivity incorporate light improve inference present mind sensor quality sensor present study demonstrate employ sensor inferential improve quality inference make poor precisely inference bayesian inferential set identify probable inference function could rise consider represent noise inherently expect behave inference laboratory perform use sensor sensor spatially light I light sensor center black region center incorporate sensor demonstrate incorporate accurate sensor likelihood engine inference efficacy quantify precision characterization sensor incorporation sensor efficacy method light sensor experiment design machine employ discuss sensor sensor incorporation parameter select discussion na I sensor model conclusion incorporate sensor efficacy section begin discuss follow discussion technique use characterize light sensor employ vertical locate directly computer run robot matlab light sensor arm display insight right sensor play sensor height surface take measurement design maintain orientation aim surface tb sensor lead white red circle lead narrow ridge prevent lead ridge lead sensitivity sensor activate light reflect sensor micro intensity convert software micro controller scale spatially distribute field spatial sensor source view sensitivity surface integrate recorded light design sensor surface location play field mm mm experimental design hypothesis shape place playing field instead characterize utilize generalize couple engine inference record data inference select measurement provide information experiment radius arbitrarily place center radius jointly location center white black
weight partition polytope whose vertex clustering allow strongly feasible diagram correspondence polynomial view expect concept method mean point initial site ignore every assign close partition site arithmetic mean iteration exhibit favorable also subject analysis motivated discrepancy behaviour introduce balanced cluster cluster prescribe arise application well study combinatorial agreement adjacent total euclidean arrive cluster prescribe size often arrive analysis cluster world application allow identical repeat unweighted impose perform partial membership point consist three equal generalize computation square weight balanced cell decomposition power diagram diagram model multiclass informally determine exist exactly assign cluster lie place result classical extension run polynomial far smoothed complexity maximization favorable bad error key apply machine integrate outline diagram termination indicate bind without generality affine hull could affine hull c iy kn ny ic I tc tuple c cluster tuple shape center deal find optimal square cluster world like location cluster cost several classical least assignment mean assignment cluster assignment minimal balanced clustering degenerate distinguish size bound strict refer clustering treat combine diagram special kind generalize known diagram multiclass survey diagram cell define euclidean power diagram tuple weight set say special feasible strongly diagram support let cycle label coincide center power diagram degenerate associate diagram cluster separate single cell diagram diagram generalization cluster precisely size feasible assignment termination clear cell finitely visit algorithm fact state prove termination terminate cluster power diagram bind bad datum see also type clustering derive euler obtain fix dimension cluster computation run balanced give set site place heavy weight diagram study repeat state implication weight balance least square assignment variable whereas line assign weight polytope function write linear algorithm input solve return assignment compute programming constitute purpose characterization feasible diagram feasible diagram assignment unweighted square unweighted allow feasible diagram interpretation far reach extension characterization corollary let balanced least assignment diagram allow strongly I strongly algorithm cluster strongly diagram balanced polytope cluster diagram service mean site assign random site assignment accord low upper describe site apply current assignment site ij objective decrease go else return correctness termination standard site iteration readily square straight center fix least square minimal eq q sum distance strictly clustering finite point terminate correctness involve infinite clustering decrease suffice diagram additional tool prove termination terminate cluster center fix balanced square terminate diagram fix center I j j j tc n tc ij tx tc tc tx tc respectively site return linear note center decrease termination final return feasible diagram iteration diagram finitely twice termination iteration feasible diagram weight clustering different cell incidence possibly diagram precise diagram incidence stress diagram feasible involve weight number power diagram clustering upper pattern bind sign sign need polynomial real polynomial sign power power diagram cell site use eq cells diagram call define algebraic surface surface control incidence diagram define relation surface decomposition relatively connect cell inclusion surface cell power surface precisely class provide incidence surface vector correspondence apply eq hence upper assertion number corollary linear compute strongly diagram balanced polytope correspond strongly diagram give number balanced polytope share power vector cluster entry belong denote exactly encode restrict balanced intersection balanced partition hyperplane weight balanced also polytope polytope present contribute
information reduce perturb exponentially replace imply thompson similar receive shorthand vector respectively thompson prior distribute nd end observe aggregate good decision perturb perturb strategy involve sequence sequence expect satisfie expect let mx ms lemma give reward ms
array extension interpret gaussian give description define process underlie embed laplacian underlie u dot e product counterpart introduce tucker eq v covariance understand generalize infinite reproduce rkhs rkhs x exchangeable place restriction describe already parametrization random statement de gaps branch elementary notion field theorem within column behavior empirical exchangeable empirical vertex patch adjacency define entry example plot infinite counterpart converge defined development theory define metric graph since sequence converge turn weak define theory toolbox aspect define define lebesgue moment thought limit adjacency subset call distance distance figure limit modify let informally think illustrate simply reverse permutation function since measurable often refer clearly indeed limit converge weakly weakly equivalence partition new equivalence equivalence specific graph assign abstract actual precisely element exchangeable parametrization exchangeable collapse array take sample fact exchangeable analogy graph limit ask large reliable regularity graph weight summarize essential unfortunately valid possible form graph proceed probability choose vertex weight edge differ cut approximated weak lemma g result call restrictive mean weak make theorem applicable real around relevance network subgraph random structure compute array array begin define exchangeability due array dimensional array simply say k representation ingredient high jointly exchangeable array index collection cardinality e I index write u element index du collection array exchangeable measurable fu characterize exchangeable array indeed notational convenience fu ij fu u ik jk additionally notational array certain exchangeability array next generalization begin define array permutation indicator exchangeable uniform write element space map nonempty collection uniform array separately exchangeable measurable fu array fu fu j jk indeed assumption jointly array theory exchangeable fit power law network sparse raise integral decomposition exchangeability obtain exchangeable symmetry integral decomposition exchangeability group permutation acting array generate permutation mathematic consequence probabilistic exchangeability model define ergodic nice group act space distribution invariant call scale cm thick south north north west circle scale circle cs dash south north east pos north north west pos label dash north pos finite combination symmetry represent convex integral encounter term geometrically integral combination idea toy sequence invariance strong rotation regard sequence language rotation act mean gaussian deviation gaussian factorial distribution group permutation rotation ergodic factorial measure hypothesis de row yield symmetry symmetry statistic intuitively property identify sufficient statistic observation statistic exchangeable statistic sufficient compute distribution introduction reference refer every probabilistic symmetry exchangeability two ergodic representation sampling satisfied purpose two part procedure try randomness represent appealing notion symmetry sparse section invariance permutation question abstract exchangeability invariance network invariance mathematical structure exchangeable graph graph conditioning location informative break model simply vertex mark notion think stationarity root graph neighbor observer randomly walk move observer unchanged actual admit ergodic decomposition characterize measure abstract scheme graph describe seem believe property hold seem invariance constraint subset exchangeability study probabilistic statistical describe law intractable dependency law hence restrict sufficiently full characteristic exchangeable technical array exchangeable survey known exchangeable array depth theoretic reference exchangeable thorough exchangeable give exchangeability statistic probabilistic exchangeability substantial literature introduction reference exchangeability machine exchangeability markov limit build regularity lemma exchangeable purely analytic largely technical comprehensive account representation independent theory david hope design publish attribute version result considerably acknowledgment learn also useful discussion anonymous provide opinion greatly improve comment manuscript white model array exchangeable natural exchangeable sequence dirichlet nonparametric bayesian arise introduction structure generalize relevance bayesian modeling survey available application collaborative network sketch mathematical foundation method type beyond array exchangeable develop flexible toolbox process understand dependent address wide variety challenge arguably toolbox additional relational structure answer characterize bayesian give type parametrize probability partition array explain statistical theorem distribution application exchangeable de bayesian exchangeable increment evy process exchangeable array exchangeable chain infinite observation graph statistical property statistical inference problem relate course could edge identically distribute indeed perform within expressive compare problem familiar one datum initial case exchangeable tell eq generate x pool information make modeling assumption define frequentist way assume derive generalize regard sequence infinite segment tell break component conditionally independent turn permit statistical sequence substitute generating determine measure de exchangeable characterized distribution perhaps surprising characterized specific define distribution exchangeable equivalent recover nonparametric graph regard adjacency model exchangeable matrix exchangeable real value array space exchangeability refer object random structure applicable derivation review structure bayesian statistic introduce generalization theorem model survey close connection seem describe result refine parametrize explain array discuss sparse network exchangeability question arise far read fundamental exchangeable represent exchangeability property valid idea array simple shorthand suppose infinite exchangeability set notation informally mean particular sequence exchangeable space exchangeable eq right interpret sampling probability condition conditionally say recover imply number exchangeable exchangeable converge na fundamental implication represent exchangeable far condition random represent unknown every exchangeable rather statistical inference generate application look like measurement represent definition exchangeability infinite invoke de assumption generate source hence exchangeability assumption de sampling distribution specify abstract measure determine small mass take probability concentrate empirical measure converge specific interpret however generate procedure assume generate choose mass exchangeability answer de convergence result information quickly converge set complicated require assumption problem machine modern involve represent structure array partition etc notion exchangeability detail sketch exchangeable structure setup infinite general infinite representation infinitely finite model infinite exchangeability infinite exchangeability exchange column order exchangeability family permutation next invoke generalize theorem see ergodic ergodic ergodic sequence integral de ergodic random structure usually product retain key small sample ergodic distribution conditionally integral represent two distribution summary ergodic measure exchangeability characterize characterize representation generalize space interpret limit subspace illustrative exchangeable exchangeable object suppose encode belong solution partition index subset invariant permutation node u label label north south west north east north label north assign partition variable contain interval respective path segment interval manner style mirror style xshift exchangeable consist scalar satisfy ergodic distribution exchangeable scalar limit size de average recover consequence chinese example partition chinese crp crp time partition correspond generate crp stick break dirichlet difference stick break order scalar exchangeability odd stock poorly exchangeable certainly imply exchangeability assume exchangeable component process class value evy process piece left evy call process call increment whenever interval say independent increment evy disjoint increment I natural say due piece stochastic exchangeable continuous time l evy measure evy evy characteristic times walk exchangeability sequence countable initial trajectory transition process exchangeable mixture recurrence mean visit infinitely infinite number ergodic chain state process model markov exchangeable variable result dependency exchangeability construct walk reversible marginally describe covariate measurable exchangeable marginally exchangeable case whose process dirichlet another value process index although make specific apparent partition exchangeable applicable marginally partition fine process merge refer formulate rather cumulative see pos pos cdf invertible continuous u function scalar cdf special translate x fu fu less arbitrary exchangeable sequence uniform fu complicated structure include important exchangeable array array graph array considerably array usually statistical interpretation observe array network graph would induce random array exchangeability ask array invariant simultaneous call permutation separate permutation appropriate row column entity collaborative filtering may adjacency graph vertex would exchangeability analogue exchangeable array version jointly exchangeable array random array jointly exchangeable eq q sequence I array exchangeability separately exchangeable dash dot node pos xshift box heat map vertex sample order highly connect vertex plot particularly piece sequence hard two disjoint separate j exchangeable exchangeability treat row independently replace respectively distinct index replace index jointly separately exchangeable collaborative collaborative problem assign movie five star separate exchangeability movie representation involve state exchangeable substitute modify exchangeable array array analogously uniform distribution empirical make unit exposition vertex graph vertex random invariance informally think edge triangle five etc straightforward adjacency exchangeable let array vertex permutation row column precisely without loop let two argument u independent random symmetric q fu wu obtain exchangeable adjacency matrix random q independent vertex q wu indicate edge generation graph thus exchangeable graph represent integral decomposition exchangeable simple parametrize limit implication provide parametrize model case exchangeable simple characterize exchangeable measurable value ergodic exchangeable ergodic array exchangeable graph parameter space reduce ergodic knowledge formulate function propose regression formulate need recent work estimation condition beyond various type array covariate exchangeability hold marginally time apply exchangeability exchangeable exchangeability marginally reason exchangeable poor divide block apply hence distinct project different graph projection different xshift xshift xshift cycle xshift cycle parametrize unique distinct may perspective regard equivalence note weakly generally map weakly converse transform canonical unique every transform monotone proposition precisely representation argument uniqueness suggest yield canonical identical though random graph show model survey several category random build chinese restaurant process latter include range summarize restriction depict across value partition exchangeable exchangeable piece p gaussian continuous partition every homogeneous social describe application partition type kind movie partition group kind movie identify underlie user movie movie describe exchangeable partition chinese restaurant obtain relational parametric describe model array relational chinese restaurant choose proportional proportional parameter subsequently belong cluster cluster contain belong determine create new independent bernoulli represent arise restaurant process exchangeable invariant straightforward exchangeable addition straightforward generate array call infinite array process every partition array call simple literature g value conditionally array trivially merely identically similarly let partition every block put sigmoid obviously exchangeable base mixture family distribution mixture must place mass partition many definition exclude case straightforward relax piece constant nature function start function u u u parameter sample word family distribution generate base generate conditionally independently randomness randomization I take element independent randomization cluster bayesian array randomization cluster array membership determine infinite relational family bernoulli index e achieve conjugacy array simple array describe partition latter merely exchangeable generalization partition utilize stick break probability distribution sequence contiguous half exchangeable usually either copy partition partition define rectangular patch originally chinese restaurant process crp fraction cluster link partition process single stick feature array cluster base model partition cluster interaction row determine possess heart exist array ibp feature chinese restaurant fashion latent array special separately latent relational allocate feature early possess early allocate second allocate independently allocate poisson new allocate distinct set constant column possess row feature generate identically random bernoulli k kk respectively increase connection becomes decrease large exchangeability ibp define column ibp ibp exchangeable permutation distribution straightforward infinite cluster model block relax exchangeable class generalize partition detail terminology match term box partition generalize model special type exchangeable array exchangeable see
shift function depend decrease respect depend outline convex hull nest converge hull converge converge converge hull hull exclude converge argument converge repeat converge hull minimal let hull convex hull since nest convergence polytope therefore contain least exchange many exist finite equation impossible hull become new since follow hyperplane hyperplane hyperplane project straight line support hyperplane argument loss generality vertex tw tw w x fx tw update hereafter rest converge hull nest stage move outside current convex hull volume increase nest explicitly lemma lead receive attract say long receive influence point x generality converge become arbitrarily close hand zero side proof close close vertex product inner take side j tw v q q theorem guarantee condition cluster group since position purpose consistency general difficulty iterative update transpose updating eq denote consider corollary previous zero surely cdf empirical iteration converge define x dy dy positive norm sx fx sx n sx distribution x induction covariance factorize matrix assume equivalently denote gx therefore update iteration update outside point converge location present invariance empirical iteration decrease nice form normal consistent example proper converge iterative dimensional sample function integration original instead mean example experiment datum show process show deviation point drop nearly iteration process converge update deviation fig set summarize standard statistic experiment parameter time converge number order deviation deviation close value converge therefore converge seem statistic run order absolute set deviation standard converge process small suggest process produce estimate sample point outlier converge converge statistic result unbiased estimator converge process suggest produce rigorous shift prove go consistency proof point consistency prove study robustness outlier yield acknowledgement short discussion gamma shift shift version literature mean shift version mean shift remark mean shift vision kernel sample region analyze later apply well science community statistic community work recently year minimize shift shift update eq kernel weight study update rule mean weighted point shift influence nonzero datum single cluster switching meaning situation converge process
might situation predict alignment multiple although space parametrize pair product align represent predict predictive representative pairwise irrelevant predict eq problem change prediction predictive space create space word reduce gain pointwise gain definition reduce approximated type gain approximate pairwise alignment follow give type employ probabilistic toward alignment description moreover confirm consensus extend pointwise gain design maximum definition estimator delta gain ml centroid take ensemble hamming hamming evaluation problem centroid centroid centroid maximize generally cover number true negative tn positive negative measure function centroid principle note similar centroid problem formal definition prediction secondary display measure eqs introduce centroid estimator biological problem rna respectively estimator whose use rna respect model measure estimation probabilistic train parameter automatically measure f approximation f reader score difficult centroid estimator expect centroid efficiently compute programming bioinformatics alignment space maximize sum great moreover pointwise bioinformatics centroid consensus collect corollary programming centroid example seem programming maximize score multiplication division tn fp contain design secondary secondary secondary structure discuss binary probability space category estimator representative estimator prediction common alignment problem implement space sequence secondary general direct estimate secondary predict secondary rna homogeneous generalize exactly implement average homogeneous therefore probability accord alignment common prediction software representative alignment problem alignment biological sequence score correspond generalize discussion distribution formalize predictive space problem computational pointwise type sequence alignment approximated type secondary rna alignment probability centroid secondary exclusive et notion centroid centroid centroid regard work furthermore present several underlie theory bioinformatics plan present acknowledgment grateful also thank bioinformatics group national technology useful discussion bioinformatics alignment alignment sequence pairwise alignment alignment predictive hx h dp centroid approximate reference work rna rna secondary structure secondary structure section section space centroid software space centroid top include rna secondary alignment biological sequence summarize space often space parameter space every rna protein sequence I inclusion position mean sequence x space secondary rna eq inclusion secondary mean base pair pseudo rna branch index follow e topological q additional sx sx ix function pair transition either letter crf feature indicate reader speak hidden gap marginalization model secondary rna constant normalization generate transition emission performance q ss probability evaluation prediction correct sensitivity f true tn fp false write tn fp evaluation diagram estimator figure figure diagram definition bottom describe bioinformatic already publish explain reader summarize dna rna protein another bioinformatics estimator problem centroid accuracy centroid estimator suitable align therefore evaluation g ik align align backward alignment maximize align probability align basis alignment compute style dp calculating align optimal dp equal computational cost estimator predict pairwise without pairwise alignment centroid align genome alignment employ centroid false align compare estimator follow centroid centroid relation sufficiently al maximize predict estimator large et alignment accuracy estimator centroid estimator equation factor function function estimator gain centroid value depend align negative reference alignment align basis summary centroid basis suitable secondary rna sequence bioinformatics importance increase closely centroid introduce estimator centroid take centroid first follow relation estimator measure secondary pair predict base base secondary call outside whose rna sequence secondary base base style probability eq maximize base complexity recursion centroid software predict secondary collecting implement distribution secondary centroid program relation centroid expect gain symmetric triangular note specialized rna secondary relation gain centroid false e estimator possess use measure pair centroid superior estimator experiment author confirm centroid estimator well use structure experiment introduce space take easily relation centroid centroid estimator centroid topological formally efficiently secondary rna base algorithm use approximately theorem lead estimator maximize probability large centroid rna centroid alignment appear efficient dynamic computed corollary biological alignment rna sequence formulate alignment alignment biological predictive sequence alignment contain biological gap pairwise alignment biological common structure representative play obtain gain function estimator xx gap prove property relation process pairwise alignment alignment make tp tn fp respect align process alignment reference tn align basis maximize computed dp replace consensus identical la k la iterative exist randomly align identical score secondary alignment rna sequences rna code gene rna input structure input alignment secondary prediction alignment rna sequence secondary alignment definition give estimator problem obtain centroid secondary observe alignment extend prove measure predict secondary multiple alignment secondary tn fp evaluation common comparison secondary reference structure tn fp base much secondary employ figure secondary sum estimator compute replace predict estimator estimator predict base secondary consider systematically discuss see rna secondary target would alignment pairwise sequence alignment two align space biological besides align introduce denote space model triplet hmm obtain pairwise follow centroid however align base lot follow approximated setting gain length computation alignment alignment maximize sum eq calculate use enable compute type dynamic alignment estimator collect probability consistency sufficiently rna rna prediction centroid formulate rna secondary rna rna like make secondary sequence predictive secondary rna secondary assume common secondary naturally alignment basis also alignment rna projection obtain secondary sequence centroid estimator probability consider secondary structure calculation computational probability p approximated type centroid estimator equivalent approximate setting define eq gain estimator secondary compute maximize x h therefore secondary estimator eq secondary rna length rna employ reduce cost mention collect pseudo alignment structured output problem alignment alignment nucleotide mean structural alignment sequence alignment produce alignment secondary distribution section sequence obtain probability employ marginalization rna sequence space structural rna sequences space projection secondary precisely consider centroid estimator pairwise alignment two rna however huge computing matching definition centroid approximated estimator rna rna estimator setting gain function alignment compute alignment maximize probability define pairwise dynamic program note align check pairwise alignment collect align probability prove predictive index pointwise think satisfie consensus estimator estimator dd index theorem hold gain centroid p whenever ensure centroid equation representative finish proposition derive definition estimator http www original accept manuscript reading bioinformatics suitable discrepancy measure class estimator represent fundamental bioinformatic commonly sensitivity efficiently cover wide bioinformatics principle also shown interpret unified manner give framework design bioinformatic bioinformatics fundamental biological sequence prediction secondary structure rna tree classify estimate secondary rna minimum maximize correct drawback estimator propose centroid solution hamming conduct analysis present bioinformatic unify superior estimator bioinformatics estimator ml formalize specific gain rather principle successfully bioinformatics alignment secondary theoretical centroid estimator order centroid applicable problem abstract centroid define space centroid centroid extend theory estimator centroid advanced estimator multiple alignment biological protein alignment secondary rna rna secondary secondary structure denote align sequence basis point denote base align restrict see space bioinformatic formal follow space set predict predictive bioinformatic estimator introduce existence problem probability bayesian posterior estimator dominate bioinformatic year regard scoring substitution assumption alignment rna forward backward algorithm depend distribution bioinformatics centroid
maximize derivation obtain reweighte problem consider generate block continuous differently argue algorithm function cb arithmetic geometric also verify iteration leading suggest algorithm sublinear nonsmooth function transform simple additionally nonsmooth paper stands bind explicitly lipschitz derive problem application example throughout assumption problem assumption unique respect require lipschitz sublinear lipschitz continuity accelerate rule iteration variable unfortunately rate long applicable lipschitz subproblem strongly convex special assumption ba utilize block suggest analyze single variable singleton da argue cb eq mapping singleton apply second assumption bc cd verify satisfie assumption c continuous point clear generate argument block subproblem require convex without statement aa suppose assumption c condition converge accelerate improve main single utilize block must acceleration nesterov acceleration accelerate proximal interesting block problem acceleration choose r r develop heavily result unclear whether sublinear accelerate discussion without pick index subproblem make establish analysis framework herein difficult bind objective successive iterate overcome develop variant estimate c rule argue sublinear dependence impose rule establish continuous theorem next rule iterate good descent due convexity possible block distance change stay per iteration need nesterov proof must utilize suppose hold differ single block summing suppose update period c rule part simply new inequality square side proof combine utilize readily suppose sequence constant q rate bad far subsection special make besides smooth composite convex mapping specifically block modulus lipschitz constant strongly rank cover family regression problem compact coefficient total point lasso vector step presentation analysis rule sufficient follow inequality convexity g iw kx linearly version successive iterate oppose iterate composite describe follow composite express generate algorithm example lipschitz approximately imply bound time great sublinear channel matrix th subproblem reformulate eq say convex respect easy verify satisfied discussion rate form correspond easy block moreover scheme rule update eq obtain inequality iw iw subproblem follow descent cost gx w go express result hold let section mention sublinear composite present extension bound example q strongly together algorithm successively upper sublinear second directly require third composite f h kx set hold different type conclude remark analyze family nonsmooth form argument family type include converge sublinear classical sublinear even block convexity three establish example cm wang pt complexity general cover popular coordinate coordinate proximal update rule block successive upper bind nonsmooth sublinear index exactly sublinear rate without per block gauss nonsmooth smooth nonsmooth possibly partition variable feasible block descent whereby coordinate approximate version presence nonsmooth solve variant subproblem rule gauss gauss randomize cyclic involve solve subproblem effective exist require uniqueness subproblem framework block successive certain approximate block optimize flexibility stationary solution global regularity satisfy extensively function globally show classic rule linearly error around allow certain nonsmooth problem hold include setting accord literature iteration type algorithm minimization sublinear constrain sublinear term minimization block type g sometimes g multi block nonsmooth knowledge classic yet nonsmooth mention coordinate classic iteration provide unified type deterministic broad nonsmooth sublinear rate subproblem improve subproblem lipschitz continuity subproblem global without summarize constrain wise strongly gauss gauss essentially maximum improvement well introduce section method ns ns without ns valid ns ns k accelerate g ns c cm notation give matrix give contain use nonsmooth paper descent family fall block successive bind certain optimize block fix rest variable x r specify algorithm simplify auxiliary virtual optimize formally follow work iteration apply generally convexity reasonable ensure assume either hold true consider convergence flexible sublinear convergence optimality despite amount gap minimize gap update assumption rule constant rule part ba bb due last fact strongly sum similar yet minimize set far nonsmooth popular nonsmooth regularizer update follow kx kx optimality last subgradient schwarz u r r bc show define follow sequence inequality claim q moreover bc put three complete ready generality method well variant exist work cover nonsmooth close nonsmooth cover special update result suppose assumption g q rule hold let claim induction equivalently definition
respective follow contraction step bound kernel initialize every kernel eqn upper contraction distance us triangle combine eqn bound q plug eqn contraction towards outside ball chain move monotonically ball initial show stay inside p rt rp rt eqn rt rp prove upper hasting hasting dirac approximate acceptance look distance kernel apply rejection variation minimize keep tolerance ideally want contraction difficult choice acceptance since usage every iteration eqn loose bind sequential w compute correspond estimate collecting search across individual test sequential design problem straightforwardly grid conduct three detailed use move mcmc change birth move involve pick inactive pair death active setting discard probability pick value move give move move death move use experiment exact jump local minima initialize sequential paper value consider gibbs sampler every use compute otherwise set product speed sampling case sampler distribution gibbs upper proof obtain size kernel ix total variation reduce half discrete distribution contraction condition plug approximate field consideration densely variable argument potential table x x mini batch approximate approximate ideal impossible repeatedly distribution empirical chain empirical percentage assign tend probability end tb tb different amount variance reduce see towards exact gibbs approximate compute keep guarantee terminate control make test statistic respectively mini large enough eqn statistic multivariate show fit nj mini batch proof proposition mini taking account replacement trivial derive point mini expect test denote terminate
hyperplane separate separate origin due separability hull mean contain origin existence hyperplane generalization allow result kernel minimum sphere intuitively translation fortunately normalization mean equivalence ask spherical preserve hilbert normalization answer assume characteristic linearly normalization preserve normalization preserve exist distinct kx contradict spherical must gaussian rbf feature space figure depict normalization necessarily improve ensure section connection kde kde q include multivariate student well assumption correspond kde vector make correspondence kde class equip rbf isotropic analyze scenario conditions cf kernel large bandwidth similarly bandwidth different sample scenario cope uncertain treat covariance interpret kde representation characterize adaptive adapt less oppose individually scale kernel center asymptotic final smoothing exhibit estimator recover summary kde connection firstly fundamental difference anomaly detection compare anomaly namely knn np multinomial digital survey validation encourage fair try setting report performance serve algorithm employ gaussian experiment point treat mean synthetic usually practice knn knn anomaly synthetic datum blue biology corrupt figure estimate corruption tend true surprising uncertainty experiment deal uncertain might beneficial commonly fully ai community possible scenario digital www survey massive survey distant universe galaxy system contain image spectra galaxy identify studied replicate conduct dataset galaxy galaxy dimensional anomaly anomalous construct galaxy anomalous group galaxy contain usual galaxy anomalous group precision ap roc auc repetition show base average knn similar knn knn achieve auc anomaly fail anomaly auc sort anomaly energy physics fundamental particle particle describe massive energy physic discover know receive attention physics reference therein phenomena event background detector anomaly occur background background contaminate anomaly true detect contamination condition home se monte association decay topology represent momentum look different different signal rest group observable particle range anomalous knowledge depict knn anomaly tend traditional anomaly detection fail anomalous detect anomaly support represent behavior kde world achieve competitive firstly model secondly bottom group anomaly detection performance anomalous detect furthermore expensive suitable anomaly directly efficiently definition department max empirical institute system propose anomaly anomaly detection anomalous behavior estimator method solution particle physics benefit propose anomaly detection drive behavior characteristic expert understand anomaly interaction traditional anomaly anomaly refer pattern interesting behavior several principle anomalous easy anomalous group relatively normal group detect interested latter group anomaly scenario anomaly detection range digital survey produce detect star galaxy investigate universe large scale anomalous group galaxy galaxy reveal phenomena galaxy likewise phenomena high physics certain vast background physics detector investigate individually sufficient individual occurrence anomalous rare highly structured anomaly algorithm propose heterogeneous consuming spectra make uncertainty obtain uncertainty attempt apply anomaly detection require another possibility individually anomalous find rely anomalous thus anomalous perfectly group detector statistics family latent dirichlet allocation cope group anomaly distribution vary across marked anomaly scoring criterion define score group employ generative efficient discriminative detect group represent assume unknown base empirical map reproduce hilbert rkhs mean work higher incorporate empty algebra endow algebra assume accord work formulate anomalous training implicitly half space hyperplane work probability reproduce let promise represent mean element characteristic characteristic rbf laplace use characteristic map apply exist information intuitively group anomaly appropriate approach without rely heavily representation representation primal subsequently formulate analogous class follow slack outlier within fraction measure anomalous compare anomalous anomalous accept anomaly subtle need choose carefully effort introduce multiplier
objective overall structure schema comparison average early update discount employ approximation discount manner online action arbitrarily denote update q guarantee converge policy simultaneous perturbation hadamard employ approximation differentiable function like tune descent ai instant perturbation hadamard size previous implement scheduling average setting discount sake comparison approximate evolution attempt sensor thus bellman evolution note assumption sensor treat individually approach per assumption always evolve sensor eq represent future conditional incur object location thus find drawback dynamic programming dimensionality e complexity process cardinality incorporate ensure converge enough perform network grid sensor neighbor interior neighboring conduct component discount exploration co force evolve choose bn initial random easy ensure per metric algorithm sensor successful discount present average employ static slow analysis main fast convergence comprise effect recursion perform simulation analysis recursion parameter analyze recursion iterate asymptotically connect inclusion di present statement cc consider ode recursion constant action tuple instant distribution policy ergodicity process asymptotically minima within constraint I follow establish track govern compact subset recursion track e gs n gs sequence surely convergent martingale vanish asymptotically natural ode jt jt origin globally equilibrium ode unique asymptotically claim theorem pp let compact transition inclusion paper value map notation along diagonal denote row denote directional sigma converge close chain invariant project iterate use compact convex bn n bn recursion paper follow n n eq similar ensure ensure bound result see follow pt maximize network minimum markov decision state unlike discount objective criterion criterion employ approximation curse dimensionality underlie incorporate simulation value arise difference td employ manner scheduling policy comparison variant theoretical latter tracking low lot wireless network detection application time network keep tracking fill minimum height minimum draw white circle distance edge thin coordinate circle sensor centralize control set sensor simplicity sensor fully cover area either sense instant movement specify current location accuracy sense challenge balance objective sensor cost accuracy partially decision unlike discount average objective state behavior whereas discount study mdps framework schedule mdp learning comprise simulation good enough run refer comprehensive rl reinforcement specific emphasis sensor employ architecture handle curse dimensionality case scheduling rl scheduling involve state primarily nonlinearity simultaneous perturbation gradient discount simultaneous perturbation update policy direction well simultaneous simultaneous perturbation employ along td perform gradient approximation inclusion algorithm detailed criterion space schedule propose literature cost detail recursive numerical cost possess guarantee develop approximation analogue algorithm unlike possess scheme employ algorithm energy tracking scheduling discount adapt convergent variant algorithm approximation scheduling discount cost counterpart multi validate dimensional see consistent rest organize review scheduling well formulate run objective discount average present scheduling section objective extend discount conclude remark scheduling broadly resource wireless survey consider problem theoretic stochastic scheduling wireless mdp rl medium mac attempt maximize throughput whereas schedule object tracking scheduling target move author heuristic sensor cost tracking maintain track solve mdp studied propose scheduling scheduling object application propose programming like require optimize algorithm central operate rl scheduling mac full space except albeit mdp perfect e fully study steady primarily concern manner track word applicable observable mdp space aim long criterion tracking early scheduling employ scalable network curse dimensionality employ individual rl many scheduling management rl possess guarantee author derive bellman scheduling efficient curse space comparison close balance instant solution obtain performance objective enable steady behavior work scheduling rl update good enough policy discount enough balance long approximation handle curse provably instant vector residual residual sensor instant refer vector evolve value indicate second term assign configuration instant energy energy instant sensor long unlike special termination leave whereas termination action constitute mdp time location pass instant fall observation center special instant system action specify point statistic time note distribution object location evolve unit elsewhere idea evolution first know sensor termination terminology henceforth statistic observation policy instant admissible admissible suggest differential sum optimal bellman q denote factor expectation knowledge constitute system space far able effectively action space function learn continuously commonly class satisfy parameterize boltzmann q convex proceed far important continuous action practice shall aforementioned chain ease exposition learning use average employ representation curse dimensionality relative instant instant action let estimate instant tuple prescribe choose average discount mdps recursion see ii arise bellman mdps interested I converge differential function estimate stochastic state stochastic iterate optimal bellman cost optimal suffer space value look intractable action cardinality sensor get sense higher curse full q moderately architecture tuple dimensional value compact incremental stage direction keep stochastic update direction algorithm initialization next state policy grid major background style fill white ylabel xlabel align outside black near near style anchor node every anchor feature I keep sensor possible track prune select ensure cost tracking select sensor pruning action consider instant proportional action within action present scheduling subsequently convergent analogue learn instant fix state arbitrarily greedy greedy policy recommend cf several algorithm theoretically simple phenomenon unstable problem due arise min operation introduce minor cost separate recursion use place loop overcome technique good minimize approximate scheme gradient employ simulation perturbation difference td update joint td ensure loop duration consume slow overcome multi albeit outer loop loop run step loop small outer loop achieve nest loop ensure rapid draw rectangle cm coordinate thin node perturbation label slow recall policy perturb perturbation construct hadamard q show incremental recursive incorporate rhs perform proceed different update along descent use estimate update td like fashion choice certain certain compact subset projection operator size necessary policy update value slow slow recursion almost precise proof available appendix make markov policy irreducible ensure state visit rl ode convergence essence two fast analysis
key transaction agent agent user agent agent note transaction recommendation agent return back item recommend set recommend item agent request agent u u ii recommend action agent notational action correspond correspond set recommend item choose agent agent clearly ik define recommend assume agent form agent recommend item item acceptance item item identity g f context old recommendation group though old condition take large among get normalize item independent recommend along item unknown item exist agent item context fast compare since update every recommend dependent separately maximize reward priori learn recommendation agent optimal context context maximize one expect since agent priori contexts recommendation denote action jk rate item context recommend together recommend maximize definition agent assumption ix user past decision slot recommendation recommend reward agent sale get sale user item otherwise equal recommendation agent recommendation total agent get recommendation agent total assume therefore agent online maximize regret reward obtain agent therefore agent act agent cut price item chance recommend agent fully high user decrease case agent assume high rate maximize item high maximize recommend item avoid percentage sale price case percentage sale price recommend sublinear time e r scheme recommender recommender set item index agent set agent recommendation recommendation item maximize reward context maximize reward agent recommend recommend item high probability user agent item agent recommend item reward agent agent time recommend agent index action agent action item reward agent recommendation get recommendation reward agent get recommend user context set exploit user section distribute online recommendation agent slot slot recommendation another agent depend consist dimension agent rate partition independently context user agent similar optimal recommendation user locate recommendation estimation context probability call action item recommend grow polynomially fast subsection agent request recommendation context recommendation agent recommendation make agent phase slot choose train recommendation agent recommend set exploration reward exploitation select estimate maximize phase belong separate action action exploration phase agent base agent recommendation form need make sure recommend item high agent action might action htb dm tn I k l n l l l k lt I lt I lt user dx jt lk train kt kt kt l tt tt kt jt tt u jt n n htb explore ik k jk kt kt n htb n htb I kt kx n order separate phase agent agent count user exploitation phase agent keep decrease relate observation action specify regret mention agent keep reward sum price time recommendation exploration keep another f order sure reward sure user context collect action agent explore sufficiently agent exploit lt lt k lt reward action one e agent recommend agent item recommendation request explore agent action item maximize xx li agent let set suboptimal agent require optimize good maximum expect agent slot suboptimal write sum suboptimal lemma bind lemma limitation subsection online I tm small time contribution exploration agent select agent agent sum sublinear mean average classical finite armed bandit artificial good reward bad reward reward denote comment overlap artificial along old detail bound jk tm path exploit lt choose arm exploitation time since exploitation exploitation agent st w lt make notational q arm inequality condition hold chernoff hoeffding sum want sublinear small hence q hold obviously suboptimal lt j lt w lt lt lt equal phase exploration exploration agent exploit therefore lt therefore lt j ip bound suboptimal choose agent difference suboptimal regret result lemma choose suboptimal action choose think bind q space path denote select arm exploitation step use suboptimal arm suboptimal choose st v lt chernoff lt lt recommend set agent sum arm optimal bound regret arm choose recommend regret arm optimal arm recommend agent suboptimal suboptimal arm summing term suboptimal call agent agent want suboptimal agent run nu td order come optimize bound lemma indicate sublinear time however well dependence know agent nu exploration memory requirement contrast explore exploit depend set agent expect reward arm agent substantial regret another advantage keep rate action partition agent item item user agreement agent slot recommend item prefer agreement reward agent agent recommendation slot agent arrival together user arrival agent switch agree high high expect reward agent slot agent benefit obtain exceed product slot leave whether simple depend recommendation slot initially arrival agent agent hold uniformly partition independently give exploit control partitioning concentrated space great contribution come context densely locate context context close start context region arrive recommend give user develop item probability arrival much context densely locate region like due adaptively learn good loss agent expect reward agent maximize agent agent agent adjust comment adjust change little even item specific agent maximize round agent adjust rate recommend high constraint item agent reward comment learn online assume agent agent connect connect link happen agreement agreement assume trade even agent get agent payment agent call agent agent sublinear give recommendation involve policy agent modify network connect agent modified agent get agent get recommend agent scheme benefit rule scheme recommend assume agent link total item agent agent low agent exploration way agent recommendation regret give agent discuss similar n arm nu tm reward item et k similar theorem exploit increase make n impractical agent reach additional get recommend connect via result numerically refine version structure connect agent directly bind agent corollary agent run nu everything nu remain corollary fast indirect amazon network sale rank product frequently amazon website contain edge co choose amazon set product product set set item product denote product product type user present search search user thus item agent goal maximize item following product recommend co product product co product recommend context specific first item arrive agent arrival take get every since context group agent frequently item another agent user context instead frequently policy reward double train explore action separately perform go seem alternative total reward effect independent context agent recommend user great together learn recommendation agent increase agent illustrate change network connectivity price item agent exceed maximize total agent adaptively learn modify c reward subsection agent frequently context co c frequently co reward case reward reward gets average slow suboptimal recommend subsection topology use assume agent almost agreement agent agent comment request connect connect via item agent item agent reward get present novel algorithm decentralize effect structure sublinear regret user item type beneficial manner wish want decentralized manner achieve california receive sc engineering east university electrical engineering ph electrical ann interests bandit game theory university electrical fellowship zhang zhang candidate department economics degree double economic focus mechanism bandit formation design van van electrical engineering university california interest include economic game online communication processing stream distinguish communication transaction member topic receive nsf award transaction circuit system technology award cite award communication conference award circuit award contribution compression stream international activity definition zhang paper decentralize decision online recommender system recommend user include item gender centralized recommender centralize sale decentralized product user item another incoming item sale bandit recommendation realization well context item distribute amazon dependence item user collaborative recommender system contextual bandit regret powerful benefit social different form network share mutually beneficial group worker help search agent much individually agent operate slowly decentralize uncertain neighbor preference reveal produce class address allow decentralize incomplete information fully within broad agent network page time agent choose item offer user accept agent try agent try application likewise distinct mean agent uncertain acceptance able observe gender location etc offer allow let item neighbor incoming incoming make unlikely accept neighboring trading fashion accept recommend appropriately ensure side occur thus decentralize learn user preference recommend neighboring occur solely neighbor neighbor acceptance social agent directly another key unlike learn upon bandit learn specific difference reward agent learn knowledge probability sublinear operate regardless network connect item formulation involve decentralized regret sublinear regret develop demonstrating set connectivity agent contextual bandit consider centralize play focus time provide slot centralize agent framework differ contextual multiple feedback make selection combinatorial arm work rigorous agent arm agent contextual select arm slot paper select arm arm combinatorial bandit knowledge propose decentralized combinatorial us fundamental third party etc regret network contextual contextual ucb design author solve contextual perceptron sublinear apart bandit concern multi user arm provide work table contextual centralized bandit exploration phase exploitation use centralize bandit ii partition learner efficiently learner agent slot make agent exploit distribute multi rate agent necessary since rate em agent yes contextual yes yes arrival arbitrary regret sublinear sublinear sublinear yes action different agent recommender incorporate framework several armed bandit recommendation example bandit framework recommender preference user rating use linear bandit rating specific feature utilize recommendation consider update preference item time recommendation commonly recommendation recommendation predict preference high recommend
advantage sa incur analyse complexity classic problem sa construct irrespective establish scheme impact couple scheme canonical setting constant high empirically subroutine algorithm traffic combine sa bandit yahoo experiment step corollary demonstrate rapid sa scheme sa provide outline iterate extension next experiment traffic conclude know technique originally reader introduction sa accelerate sa independently incorporate rl td learn introduction popular td time extend replace iterate research rl improve cf computer dimension involve feature approximation effective sa meaningful sparse bound sa propose sa scheme descent sgd well high bound provide online sgd technique strongly regression highlight regression convexity size dependency much square problem propose stochastic temporal converge fast approximation sa irrespective value schema sa sa also sa employing iterate strong convexity height width white circle coordinate thin black cm block fill cm align green sa align discount instantaneous bellman cardinality popular approach linear architecture every td attempt onto transition underlie cf simulate mdp law number tend uniformly randomly pick uniform notice td assumption sa bound ex strong tt positive least work along along stochastic see decompose martingale concentration apply quantity analyze outline proof available appendix nh sampling martingale difference deviation dominant size fast sampling error assume specific specify claim deduce approximation constant sa choose c rewrite constant eq probability inverse approximate sa let denote true evaluate low conjunction first rhs least square sa theorem couple scheme low big sa know analysis variant sa necessary explicit search solution constant advance update rule variant sa except require employ scheme size approximate quantity iterate sa analogue iterate appendix cn iterate suggest error sampling average main sa sketch martingale analysis template appendix iterate derivation rate form bind use moreover dependence constant rewrite sigma lipschitz reward lipschitz constant dependence inverse eliminate reward let instant specific equality apply iteration constant invoke function follow appendix difference n x f ni k rest bounding martingale follow corollary specific rate c c comparison integral describe classic method least sample unknown notice unlike set minimizer empirical sa iterate sa uniformly size sa nevertheless derive approximation square sa definite eigenvalue analogue choose know mdps policy policy like briefly describe sample mdp action reward attempt approximate evident well behind study refer sa subroutine provide step initial use greedy adaptively choose configuration intersection road network order traffic road consider queue turn road mdp feasible sign configuration approximation employ handle control mdp road tuple correspond describe table denote queue network fashion c red green red green green green red feature selection queue length motivation threshold queue length precisely xlabel step sa ylabel height pos gray col sep xlabel ylabel legend pos south pos pos gray col point x sep pos north east legend code rectangle ylabel xlabel symbolic grid ex grid grid ex align bar style coordinate collect pick sa step sa significant sa variant motivated step road obtain fig sa throughput road reach sa runtime report road network sa runtime notice sa orders regular traffic observe sa throughput par establish place rate approximate scheme possess traditional make attractive data dimension control demonstrate low use sa like thank european fp system ep sa present approximation well denote rewrite sigma lemma constant crucial ingredient invoke f I constant return iterate instant give j j note vector fact since cauchy schwarz property final expectation invoke martingale lipschitz constant martingale nf recursive jensen inequality martingale inequality mention couple iterate give high iterate decompose martingale follow
extraction dictionary cifar fine grained extraction patch pooling encode formally stage extract patch encode patch activation value activation vector complete dimension patch activation classifier mainly focus encoding encode code mainly couple joint optimization simple learn later stage relatively simple dictionary value adopt mean aim patch code comparison complete highly redundant pooling obtain pool activation correspond usually activation show carry whole global image region extraction pooling pool output reasonably argue one immediate algorithm yield similar filter pool produce response figure filter response correlate pool response convolutional approach dictionary spatially invariant k especially hundred thousand code problem code color gray could model interested effective take consideration pooling stage pool effectiveness dictionary two first adopt algorithm size dictionary dictionary pool idea highly scale allow dictionary patch away redundancy pooling dictionary encode expensive extraction start dictionary pooling region obtain dimensional pool feature randomly pool way analyze post pool dimensional pool specifically similarity pool dimension code pool code output affinity propagation centroid centroid intuitively redundant pool translate version specifically find centroid candidate compute iteratively availability upon centroid candidate centroid visually show propagation apply approach cifar training pool feature centroid appear dominant factor contain translate like code column color vary center centroid solely pool favor reason simple finding dictionary size specification encode pool extraction image pool since extraction could view evaluate play encode patch dictionary patch budget dictionary dimensional oracle encode covariance explanation work pooling algorithm dictionary approximate interestingly think nystr nystr spectral enable explain mechanism field support recent observation vision already data subset dictionary limit pool though patch pool code reasonable patch subset dictionary pool output lead nystr om select start select centroid denote original covariance pool output select approximated implicit dimensionality code svd km wise orthonormal little terminology transform pre impose minimum overhead actual feature shape practice combine pool pca projection filter reduce yield zero coefficient dimension linearly encode however dictionary small explain learn pooling dictionary learning include performance systematically analyze grained classifying show gain cccc cifar extensively behavior cifar large testing amount unlabele image pool operation extract local patch whiten patch mean feature encode pooled follow code code capture pooling claim feature response filter response filter code pairwise centroid uncorrelated response correlate pool stage compare effectively affinity propagation take consideration subset response correlate preserve code figure eigenvalue approximate previous capture large original dropping cifar indicate axis include loose feature pca save extraction detailed dictionary table summarize final rather focus pool aware fix budget setting reduce dimensionality state serve consider pool statistic always well dictionary classification help dataset may infer selection local optimum cause codebook however codebook patch cifar cifar code pool correspondingly svms analyze approach incorporate weakly supervise selection extraction test algorithm fine grain file grain pose challenge classification localize manually design recent grain localization design al template grow performance whole communication grain whether unclear pre center provide box avoid introduce number training expand extract cifar pool cifar dictionary patch cluster center art baseline table feature learn provide boost localization fine grain appropriate descriptor local major factor subtle change improvement fine method sift baseline pose pooling pose pool svm
carry multiple algorithm carlo optimisation gradient optimisation available optimisation begin enable investigate investigate possibility gaussian alternative gradient hessian design estimate reproduce paper discussion suggestion mail engineer university novel maximum inference nonlinear iterate procedure iterate provide automatic exploration exploitation model good computational interested parameter inference state space latent define eq simplicity let denote likelihood estimate optimisation express eq density gaussian compute kalman filter intractable obvious address ml ml carry include method require computationally costly problem simultaneous perturbation algorithm ascent stochastic scheme gradient difference need another estimation moment reader static pf suit costly evaluate turn ml level discuss algorithm sequence iterate iteration step iterate compute objective tuple promise typically computationally costly estimate wish keep number evaluation likelihood pf process iterate discuss iterate acquisition rule predict automatically derivation brief see g discuss specific pf sequential particle locate approximation generate sequentially resample particle replacement particle put emphasis probable particle r step particle assign importance account pf sophisticated alternative pf pf write multiply divide draw carlo particle smc literature consistent unbiased central asymptotic proposition est particle particle propagate particle compute directly avoid estimate log pf introduce asymptotic carry estimate around true similar unknown validate numerical finite calculate estimate quantile blue draw naive create increase compute problematic smooth step serve surrogate possibly capture gps result obtain function could gps popular regression g distribute accord gp pf k kk log respectively follow iterate respectively kk save hyperparameter respect illustrate usefulness upper six sample log surrogate pass observe reasonable mean ci red lower propose evaluate iii choice consider curse dimensionality previously acquisition exploitation obtain gp recommendation ei exploration peak eq previous brevity obtain gp ei drop brevity cdf standard acquisition iteration improvement situation situation
compute would prefer cross procedure cross visually separate view instead course order refer impose therefore unbiased also collection average involve symmetric may involve observation contribute rate entire occurrence collection soon together cyclic permutation g entry come fold extreme case learn block entry test distinct learning consist index contiguous size observation ordinary validation may compute description invariant kn validation merely globally variance minimal like procedure cross validation call immediate incomplete balanced frequent fall away leave respectively produce among happen hold keep variance cross yet literature immediate statistic state sample reasonable violate irrelevant observation case maximal associate among estimator variance cross validation procedure leave variance statistic coincide validation repeat sampling leave validation computation arbitrary use full regular optimally also statistic depend therefore position outline formally covariance parameter optimally full identically generally two soon degree integral rewrite accomplished analogous formula minimal small kernel regular sharp attain variance coefficient present immediate general develop particular associated statistic sample variance degree split hyper geometric regular regular since parameter degree computation coincide involve careful already symmetric call quantity combination regular proposition achieve desire degree solely hyper mass whereas also prove show natural exactly analogy advantage prefer usual quantity hoeffde order variance statistic degenerate non degeneracy numerically check assumption degree parameter small kernel part need motivate assumption statistic kernel bias fit view reason failure estimator would severe power hoeffding setup fact would trivially zero hoeffding least properly one likewise global nan classical greatly notation variance short state alternatively explicit statistic split statistic vary degree particular define enjoy analogous particular reason split vary degree course need empirical analogue principle apply proposition keep rest write introduce special notation statement simplicity ordinary one statistic numerator denominator strongly consistent consistency statistic statement remains apply quantitative care statistic vary weak mean strong integrable hoeffding proof give surely almost surely consistent similarly tends fact estimate however property towards strong order tend u un decay behaviour show exist estimator distribute reason expression empirical biased estimator finally unbiased variance manuscript follow whether validity part theorem finitely multiply converge almost surely un side rejection positively bias conservative already relate bernstein kernel degree author practical incomplete treat incomplete statistic design approximate feasible necessarily symmetric order collection associate incomplete approximation entry draw independently aware concerned statistic order interest imply approximation soon specify precisely hoeffding theorem comma fix number repetition general time fit illustration tuning iteration remarkably statistic apart differently statistic apply one use testing explain kernel appear estimation precede section investigate datum stand tumor expression penalization parameter pre lead software internal difference great extent degree effort avoid numerical validity degeneracy digit side low reproduce page support science foundation support support de classification involve split statistic theorem unbiased asymptotically unbiased estimator minimal least enjoy property exact equality error algorithm deterministic algorithm tune lasso supervise statistical prediction value return rule learn typical patient outcome g tumor status response base marker e usually learn datum researcher want know rule set perspective unconditional focus unconditional algorithm classification unconditional pair section rarely thus estimate case resample estimation detailed overview vast literature validation go beyond scope reader learn variance estimator literature estimator allow g derive confidence interval true statistical algorithm latter crucial apply poorly split repeat show bias estimator leave validation estimator suggest critical simplify far cross concerned date estimation resample adequate answer view procedure available asymptotically exact usually algebra lebesgue measurable allow binary thought support pi investigation statistic product misclassification loss measurable marginal bound moment automatic typically residual score work unbounde interested error sample leave learn since contradiction
include generate un portion draw q slice efficient design sample adapt hdp use need infinite slice extend breaking ij ij tu label gibbs derivation material hyper place prior place place ratio concave auxiliary place likelihood thank conjugate property sampling slice feasible way previously distribution property community exchangeability make mix sampling dependent slice membership independently process membership help improve become large ht ccc ccc model community synthetic generate parameter equally partition truth group assess compatibility large value diagonal value diagonal diagonal compatibility four figure test run half markov chain iteration conduct chain implement package proportional I c also last comparison score chain stationarity diagnostic pass autocorrelation performance indicator monte approximation estimator autocorrelation lag point ht c cd whole burn manually integrate general value large help discover autocorrelation hand admit difference chain slice scheme compute iteration scale computing htbp c c case compatibility membership distance truth role table perform model time discover show sample total posterior directly need calculate value sample fast accordance tp c c select real world detailed type dataset htbp time friend like friend email contact friend truth mainly correspond log data c net interval versus classical one bold large perform well interaction period happen fail succeed interaction head tp bottom time bar tend dominate exploratory dataset linkage specification select close setting mark construct group right detailed simplex stay time compatibility comparison compatibility value blockmodel community mix paradigm realize adapt target analysis mcmc autocorrelation etc enhance verify effective construct role compatibility include systematic dim real interested adapting model sequence network binary persistence membership lastly extract correspond generative provide global share dp indicator community il li global represent exist compatibility marginal analytically hence need mixed membership concentration jointly stand node influence assume mixed distribution influence time activity current activity method hdp popular chinese restaurant crf explain crf analogy restaurant restaurant configuration model hdp place two scheme sample slice target due limit detail reader double
recently property novel combine clinical relate technique introduce simple paradigm electrical platform pattern deal concentrate platform construction issue eeg signal subject pattern particularly extraction spectral sr usually directly apply eeg achieve accuracy detect collaborative interpretation explanation mechanism paradigm modal clinical experiment subject change eeg pattern stroke patient potential mechanism stroke classification analytical study mechanism brain partly change channel study reasonable effective stroke eeg fix stable channel strategy verify cause adapt channel iteration record shift training consider band generally eeg conclude brain reveal post eeg significant phase adaptive boost deal band informative spectral band boost along discover expand tendency band spatial boost band combine complementary extract comparative eeg extraction channel selection reduce spatial pattern eeg eeg subject eeg different previous boost competitive eeg emphasize time connect change band component individual differently frequency subject eeg detect observe mutation reveal mechanism domain integrate combination improve stroke eeg modal propose experiment detect change recovery relate give month training novel rest part acquisition intermediate boost analysis finally brief conclusion seven stroke experiment group traditional clinical implement assess effectiveness diagnosis comparison supplementary material besides subject eeg collection contrast supplementary eight training subject training week day subject finish trial class trial imagine channel adopt raw signal record hz store converted mat file format paradigm eeg signal interactive paradigm paradigm reconstruct loop actual movement configuration paradigm material low ratio snr eeg common ground reference reference supplementary filtering necessary relate band hz correction issue often decide default eeg without band eeg signal configuration effect due spectral eeg case eeg set day experiment eeg material segment summary could universe possible aim find produce minimize convenience omit boost solve universe channel fc fc cp cp supplementary material appendix denote note base learner v kt kt kf kt establish base iteration function approach conclude pseudo stochastically partly good step learn naturally note fit gradient firstly incorporate stagewise performance original completely study heuristic adjust pool background iteration training verify simulation study weight mechanism classify supplement pool generate classifier split part x I I md mp p summary whole leave detailed determination iteration determine pick use base decrease randomness iteration increase ratio provide train local short copy adjust coefficient incorrect strong classifier much description ability square convenience future conduct determine loss eeg optimal base learner learner learner feed produce learner kf kx kf kx kf kf computation spectral spatial channel band band suppose universal possibility total lastly extract bad feature choose take consideration characteristic svm ignore fortunately filtering could process store offline analysis implement scheduling accelerate projection commonly use method subject eeg subject employ give st achieve see material appendix psd sr stroke subjects explanation psd spectrum sr closely change obvious stable eeg subject increment begin end exploit band spectral quantitative eeg sub band project onto illustrate day importance together obviously locate take cause channel present trend channel sign area leave stroke considerable start leave slightly initially take importance give patient eeg variance maintain constant lead eliminate explanation increment channel weight familiar normal band high band partly band exclusive eeg conduct dynamic band imply essentially reflect power change band change band pathway pick feed detect change band selection shift extract eeg complement ability phenomenon exclusive subject mechanism power appear
derive vector decompose posterior propose active wise form infinite latent derive covariance inverse gamma prior ibp list please variant simulated metropolis hasting serve move low rank space relate model inspire tool model soon available parametric factor diag plain anneal mh spike mh mh zero mh attempt evaluate since ibp towards random mean add computer vision arbitrarily adopt select iteratively alternatively unseen train choose amongst method accuracy impose projection likelihood prediction accuracy test speed show figure illustrate normalise complementary plot likelihood please align compatibility variant sa yield avoid time determine accurate include derive acceptable feature likelihood compare great prediction error cpu time iteration accordance isotropic sa fast variant classic way explain ibp iteration significantly speed high cause efficiency solution multivariate offer classic degree freedom overfitte integrate ibp factor synthetic control dimension parameter remarkable maintain da xu technology technology parametric domain input two b integrate prior factor experimental alternative remarkable growth variate high become volatility forecasting finance pose computer vision regression tool exist leave visible high elaborate model explore classic observation input large impose multiplication computationally numerically forward introduce factor rank factor improve parametric variant trial knowledge resolve parametric conditional variate optimal dimensional layer exploit ibp section explore follow section mention notion latent design factor latent share create factor turn dependency response low dimensional require propose regressor bayesian non model regressor high alternatively ibp exploit ibp enhance factor impose latent construct initially far extend integrate result harmonic whose binary equal analogy infinite customer choose either hadamard product noise covariance load comprise vector binary mask ibp illustrate graphical observation tend infer jointly posterior sampling metropolis particularly derive activate variable definition ibp infinite row efficiency tend active kn iteration due customer choice begin sample add posterior inactive thank active active linear result gaussian decompose former interest latter term maintain compatibility keep th element make equal zero inactive hadamard ratio ratio observation exist current observation ibp estimate ranging propose metropolis hasting step evaluate
ready full perform regression recover exploit strength compound parameter original form whiten base use factor robust eigenvector line focus observable operator recover moment work second regression expert restrict observation moment compound compound invert work appropriately transform projection compound work focus notably string weight finite string real develop idea regression explore recovery magnitude measurement construct power theoretical expert result polynomially draw regression far q dependence looks perform third moment square third show strength two compound convert actual robust basic identifiable identify moment optimality regression moment high parameter care dependent polynomial basis expansion might ordinary linear linearly mixture dimensional live bind compound build combination mix regression restrict convexity p suppose restrict forward strong convexity operator adjoint material constant bind adjoint compound factorization include whiten robust perturbation whiten operator find defer supplementary allow recovery lemma lemma control moment previous consistent alone attain high em expert end initialization simply strength solve optimizer algorithm em initialize plus final initialize em expert data follow unit actual identifiability criterion discuss normal different consider fit random initialize expert note converge spectral frobenius norm average instance one variance across instance minima spectral expert recover provide initialization study stability solution return expert recovery attempt typically enough almost always converge optima find true parameter little considerably varie get recovery improve suggest optima get find well highlight evaluate mis contiguous report error computationally statistically regression spectral power regularizer rank factorization actual empirically find expert excellent thank suggestion anonymous helpful comment problem zero observation p represent adjoint eq show bounded bound p f derive allow recall inequality tensor p show bind adjoint independent complete use x bias treat lemma take p robust eigen decomposition parameter orthogonal moment moment apply tensor eigenvalue whiten combine whiten let index simplify together completeness whitening show orthonormal whitening transform consequently orthogonal eigenvector eigenvalue inequality break differ element apply q construct eigenvector apply bind small eigenvalue whiten complete inversion relate apply q expression like follow require imply include completeness support express check difference inequality inequality jensen put everything together frobenius perturbation whiten matrix q lemma differently completeness operator exploit invariance note invariance eq whitening transform diag multinomial proposition condition discriminative learn optimization optima paper provably mixture linear recover power empirical strength relative em latent compact recognition human syntactic parse machine local broad goal develop provably propose moment develop variable include mixture latent parse idea method express tensor estimate structure permit regression moment reveal problem low regression power provide appropriate tensor retrieve simple prove consistent estimate model music hierarchical expert depend know
curve class dedicate address homogeneous change formulation lead mixture discriminant analysis entirely unsupervise hidden logistic supervise discrimination curve model give discriminant datum discriminant hide process label th curve observe time discriminant extend analysis functional assume functional rather label curve functional discriminant curve p g class define regression spline generative curve regime lead quadratic discriminant linear discriminant arise curve single spline spline parameter coefficient polynomial adopt e g fit govern logistic homogeneous present homogeneous class curve become restrictive handle datum adopt formulation polynomial discriminant model mixture class spline mix proportion hide discrete g g representation adapt knot regime point relax regularity constraint spline knot smooth regime logistic class compose homogeneous probability sub group govern regime model hidden assume govern hidden switching polynomial time result conditional j relevance flexibility detail key difference model spline capture observed training curve write dedicate em start compute log current sub probability qx ij k j qx qx ij maximize g mix logistic update q maximization regression separate analytic problem weighted perform functional discriminant analysis curve misclassification cross simulated curve piecewise noisy curve class class homogeneous class curve compose approach fact approximate functional process spline attribute flexibility regime shape class h discrimination spline discrimination new functional functional mixture hide process estimate dedicated benefit address shape compare alternative work concern sciences laboratory south france new specific flexible shaped present observe dedicated expectation comparison analyse
agreement division belong non cluster vice versa rand unlikely chance alone tie share spam year month rand rand create server connect temporal provide view gain useful insight finding believe previously unknown community community spam thank lee project grateful dr retrieval science grant xu support part sciences engineering mark chen j ann mi usa edu com date spam phase spam ignore mass acquisition address observe identity insight behavior email address social behavioral similarity use monitor main either coherent address reveal behavior discover study source likewise anti spam mostly base server ip little attention devote spam cycle spam address spam spam computer open identity identity indicate collect email address spam spam server spam social phase spam datum analyze project monitor activity use email address receive email project acquire email addition ip address spam server email make spam ip acquire email address project able happen appear email receive financial look group behavioral commonly identify behavioral group cluster choice behavioral email associate associate appear suggest resource spam member identify amount ip internet service indicate physical indeed identify project activity network web page email address embed web page human web automate scan page collect email automatically email address human investigate centralized project server generate email ip record send server email embed particular particular email receive one know address email address besides address spam million email address spam project locate receive email address month receive normalize growth project increase spam month notice number receive increase agree media report thus reader order social associate identity send email associate spam server use acquire email likely spam server assume collect email email address email summarize previously source email address sign account note email email email classify email content list build include name business spam receive spam spam ratio send number send exceed per receive label employ behavioral relation partitioning minimize natural choice detection refer undirected vertex represent edge indicate similarity adjacency refer similarity degree graph represent behave manner partitioning translate minimize denote group partition matrix let favorable association attempt association np note create argument rewrite formulate maximization optimal matrix near transform version eigenvalue follow discretization close matrix continuous detail commonly spectral particularly suit spectral cluster goal high eigenvalue relatively actor tie indicate relationship actor indirect relationship tie tie edge behavioral may evolve time need frame evolution cut choose frame point month month independently behavioral similarity spam server determine detect spam usage spam link spam spam denote send spam server send spam server number email address acquire variation send spam server send spam send total spam server send spam server send total normalization term account email address acquire spam address interpret exhibit indicate social another look send bin result indicate send hour send th normalize email address acquire vary different obtain unnormalized similarity normalize normalize similarity consist edge weight information connect together accord connect near opt near neighbor connect recommend choice improper case visualize result graph month note create direct year start month interval create spam connect show spam usage shape color indicate belong heuristic connect divide easier interpret component cluster ten modification make
suggest ill ill condition generate near repeat twice hence dirichlet distribution draw interval column identity entry follow distribution multiply column equal perturb perturb hull near matrix rank column cone meet rarely w language case rank although rank precise rank replace type display middle ht ht robustness projection constrain variable fast simple dirichlet residual compute zero large perform poorly robustness deal hull case middle segment identify experiment noise condition entry matlab replace nonnegative matrix step nonnegative matrix particular noise usually simplex hence normalization apply prefer near separable nmf successive although projection use broad nonnegative matrix hyperspectral near require among however expensive program solve recursive robust near separable motivate robustness projection discussion perform author thank chinese university discussion suggest noiseless separately author grateful comment fast objective word objective converge ht whose continuous initial x x strongly use although linear experiment simplex construct vector lagrangian multiplier eq hence sort bind b exist suffice take since obtain mb follow directly corollary definition near blind refer successive popular successive projection broad matrix synthetic world nonnegative noise hyperspectral nonnegative factorization nmf become tool two nmf np ill pose reference area document hyperspectral biology recently et introduce efficiently even presence near separable blind separation video hyperspectral index permutation separable identify reconstruct perfectly refer near separable recover application near separable blind presence pixel hyperspectral image pixel signature pixel light reflect signature pixel signature material pixel pure pixel satisfy equivalent separability signature pure algorithm separable nmf construction program successive closely paper successive simple fast solve separable nmf orthogonal complement later robust satisfying rank permutation randomly near column w www moreover replace algorithm lipschitz relate hyperspectral automatic generation successive volume maximization closely old field particular schmidt e advantage particular matrix enough column noiseless moreover condition fail see near nmf successive nonnegative drawback robustness show noise full second new show broad robust w outperform synthetic simplex drop denote context rw recursive separable see onto far difference perform near separable column extract strongly permutation j maximize pick function continuous gradient appendix implementation detail although require total operation closely canonical refer project point onto cone far main difference select e belong cone span two maximize column noiseless see page note actually exist criterion undesirable project onto column project hull column belong discussion perform hull allow however onto convex variant onto robust criterion direction analyze robustness derive normalized variant onto selection projection convex cone perform norm projection noise closely robustness definition need paper identify column among column noiseless explain behind derive rank apply generalize broad section choice near separable near put permutation normalization column presentation discard scaling divide column discard divide divide multiply unchanged since must also also make presentation proportional extreme ray cone span broad separable matrix datum hyperspectral image section experiment lipschitz vector yx induce metric point give hull follow eq fa fa fa introduce notation hull interesting notice able column column hull distinguished column even noiseless column imply noisy section behind working denote fa separable satisfying column identify belong hull column entry column extract always strict imply index extract note residual since strict unless identify extract need require fw exploit show full derive key column identify column column lemma lead combine lemma robustness assumption result easily lemma lemma imply satisfy fy q lemma satisfy satisfy lemma corollary fa useful lemma assumption attain k column column error close extract strong eq maximize let k extract perturb second inequality lemma since strict convexity attain f j j equation use prove robustness let strong extract permutation follow extract correspond induction satisfie let near w fact broad separable column replace robustness minimum residual onto convex hull distance residual triangle plane identify presence large last equal side distant matrix segment fact link denote eq robustness replace particular proof exactly separable let satisfy assumption lipschitz permutation use appendix therefore q equation derivation condition give let cubic form dependence order notice hence maximize seem challenge possible possibility index processing project hull identify column maximize update error separable make multiply improve noise w w along discard weight need reconstruct
arise form calibration consider non cox proportional discussion choice kullback quantify prior piece information cumulative q take form average serve future coherence case discrepancy kullback precise absolutely every kullback leibler virtue require case measure represent connect datum broad analyst distribution call complete used proxy call index complete proxy close analyst know family fully justify inference book comprehensive essence unknown belief via conditionally function bayes belief account mathematically via probability update work applicable case come density close bayes framework need amount analyst belief recover bayesian rule crucially maintain open many key rule equivalently open many perspective describe cross validation suitable serious suffer lack coherence ignore come obvious mean wrong inference extent prior construct know nothing possible infinite via find wish learn standardized cumulative suppose previously let consider belief model hence quantity specify issue irrelevant close true true since view speak open regardless view hold analyst wish express full motivate example analyst know wish belief greatest area loss highlight aspect traditional incorrect generalize whereby obtain approach classical function group link correlation entry parameter abundance equation equation equation connect obtain substitute essence propose type update mechanism implicitly limit minimizer hence ensure picture would make loss close setting issue apparent weight multiply arbitrary equivalently note weight loss combine different function well health economic cost loss idea discuss anneal gibbs label temperature power priors chen loss influential extreme extreme base validation appear gibbs posterior measure discuss value aid calibration additional term constant prior negative minimize regard give data piece make loss match whether take two choice piece loss calibrate ensure coincide connection find ensure eq eq value construction circumstance unit x way extend loss make since hyper allocation loss take eq perfectly reasonable assess seem accept operational quantile match frequentist base loss interval posterior reference posterior likelihood form probability stochastic procedure probability concern word need shall conditional base expert problematic information arrival reader paper piece assign piece answer define presence non put broad updating outcome value measurable belief assume variable say know assume update unconditional distribution version conditional essentially unique sure consequence nevertheless individual case exist regularity requirement conditional always space instance case measure continuous enable easily subset lebesgue distribution respect call distribution definition version joint available replace moreover even coincide know outcome also outcome establish one imagine model difficulty arise conditional process need circumstance bar crucial ingredient view agree certainly basis arise information want seem appropriate assess something instead variable recover case conditional density arise theoretic relate eq absolutely equal belong satisfy minimize self resort function form stochastic collect informative identify interested loss covariate baseline function possible failure failure individual time piece individual self logarithmic function take due motivation inferential survival time potential predictor quantile choice well traditional yet employ generate source genetic survival disease within contribution survival follow incidence cancer disease underlie trust centre cancer marker table row denote individual marker denote whether event censor association event employ proportional treat hazard nuisance parameter cox hazard ph widely cox log linear hazard covariate hazard seminal cox partial estimate event information coefficient loss conditional association standard practice genetic association genetic incorporation genome wide would detect marker linkage low resolution marker belief marker specify marker define belief straightforward incorporate genetic calculate marker zero marginal bayes marker quadrature carlo integral convenient laplace estimator hessian mode bayes factor considerable evidence marker accurate marker run sample indicate laplace appear surprising give association comparison cox ph partial factor agreement especially marker large size dispersion marker highlight region association colour likelihood estimate tendency marker information great highlighted standard log bayes relate reflect marker contain less return show marker fig marker association marker association arise effect single marker laplace monte marker variable multiple variation selection aic covariate model inference proceed forward cost penalize fan li partial constraint coefficient despite cox ph limit treat nuisance parameter hoc specification adopt et bic approximation model score important ultimately hoc enter aspect lose selection specification gamma baseline hazard formal inference conditional prior avoid q censor indicator covariate relevance vector quantify markov mcmc efficient updating mcmc propose one iteration current model approximate log accept run iteration parameter equivalent marker discard rate evidence single marker weak signal couple region reversible cox request illustration quantile notion loss update quantile coincide exploratory enhance uncertainty due sample start function learn distribution merely laplace fall within paradigm put come previously distinction learn obviously include constraint importance give q currently certainly update utility take use matlab file statistic toolbox plot record break country available matlab omit widely tool traditionally display use fact observation usa place median uncertainty infer fairly constraint use metropolis hasting show box credible interval dot line denoting interval comparison fig look credible addition overlap distribution usa general conventional france usa france observation marginal joint impose usa tighter include whereby update allow coherent fundamental concept recover precisely update rule bayes rule select self loss appropriate minimal act come within framework information log likelihood loss mechanism scope finding generalization implication ease loss implication need sample construct receive without need alternative receive coincide suggest really need generally loss robust equation appropriately rigorous approach think think mechanism perspective identically distribute maker minimizer know utility hence construct pick action minimize loss see leibl divergence conclusion acknowledge replace connect see section index close model believe fundamental parameter function restrictive narrow moreover support minimal construction loss restrictive reference journal association nature bar concern nonparametric behaviour model series york publication bayes possibly cox life matching prior asymptotic american robust parameter via university economics de b la population department fan j li selection cox ann censor survival w introduction mathematical nonparametric j american book new york r geometric journal statistics university bayesian posterior pseudo statistical h statistics ed nj j chen proportional journal statistic high classification mining ann l bayesian van misspecification bayesian statistics leibler longitudinal datum generalize nonparametric estimate transaction economic university likelihood application statistical model journal j york mathematical evidence j cox average proportional hazard assess stroke journal games economic university journal approach american zhang kl ann zhang theoretical statistical prove shorter distinct say otherwise degenerate thesis trivially satisfied prove h ji h pp point shall summing term obtain uniquely every substitute convex derivative go hold decrease impossible satisfying exist assumption every couple absolutely asymptotic general typical study understand happen proxy wrong precise idea posterior divergence true minimize eq direction de sure accumulation estimator maximum estimator satisfy sufficient close leibl restrict eq q result result support hellinger ball accumulation become sure find form stochastic piece information aid provide knowledge us return observe identically choose need whether contain scenario represent belief aspect also highlight prediction bayesian one know certainly come utility function move case covariate recover usual identically minimize infinite give equivalent idea case suitable sense mild regularity precisely stochastic minimizing yield hierarchical regression model would retain determine unobserve quite markov monte series set whereby autoregressive arise assume order know unknown correct take close truth bayesian update take fx repeat covariate assume update belief take close locate
importantly actual number topology traversal tree consume routine ab branch intersect multiple call number branch epoch amount spend probability supplementary pattern model substitution substitution extend heterogeneity character mainly heterogeneity intensity heterogeneity induce stationarity epoch model bayesian focus tree plausible evolutionary scenario heterogeneity different substitution epoch reflect process architecture although extremely substitution serial research limit restriction capable quantify prove useful detect change selective rapidly evolve dynamic software availability evolutionary sampling platform evolutionary analysis library code spread reconstruction available result receive european european grant trust national foundation dms national evolutionary ef acknowledgment thank critical insight end thm thm remark epoch substitution department institute institute evolutionary university united center national health md usa david school school health california ca department institute mail abstract molecular reconstruction time substitution process motivate computational convenience biological dynamic extend generalize evolutionary homogeneous assumption allow substitution time evolutionary bayesian framework offer great flexibility draw inference discrete markov trait impose parallel fine parallelization branch accommodate epoch graphic processing evolutionary evolutionary epoch nucleotide framework population epoch capture heterogeneity introduction molecular continuous operate branch tree call property current limited nucleotide frequently accommodate trait large substitution may process imply state depend event character infinitely substitution process homogeneous reversible stationarity realize evolution homogeneity throughout thereby treat induce upon frequency homogeneous instantaneous apply number parameter abstraction substitution ease computational recently relax process evolutionary assess accommodate example allow nucleotide composition vary include general composition large address nucleotide bayesian topology compositional heterogeneity oppose stationary compositional bias development compositional shift nod compositional drift across tree compound conjunction substitution pattern protein structure exist tackle stationarity substitution branch keep heterogeneity apply relatively rich possibly adaptation evolutionary simultaneously cut across consider substitution class transition prior belong requires measure molecular envelope sequence single patient evolve rate evolution sample author classify neutral upon approach several way similar evolutionary sampling tree package connect importantly epoch discrete accommodate recently incorporate average plausible evolutionary account epoch transition translate branch jointly epoch unknown history exploit trait evolutionary connect evolution trait marginal epoch advance sequence tree topology integrate evolutionary bad complex evolutionary reason substitution inference however parallelization graphic processing statistical complex evolutionary partitioning add burden substitution heterogeneity part platform evolutionary likelihood library effort accommodate scale parallelization computation mainly evolve population perspective demonstrate capture demonstrate examine life associate disease aim accommodate scalability implementation speed implementation offer markov substitution computational trait obtain state characterize transition stochastic overview numerically probabilitie discrete xt xt xt n probability state researcher often far process reversible homogeneity depend reversible satisfy balance return independent state constraint rate form define element trait branch set consider lie past imagine trait evolve independently scalar post integrate trait successive contribution assign either trait partially full root trait conditional serial order recursion equation homogeneous branch elaborate time strict molecular substitution tackle time homogeneity find usual homogeneity specify substitution characterize rate point change order marked biological branch change return say need keep greatly parallelization epoch epoch new transition eigen parallelization across branch branch epoch boundary break conditionally process integrate data r v rr along transition epoch homogeneous remain branch boundary augment boundary form substitution model action transition convolution reader integrate unobserved middle strict multiplication kolmogorov state outcome time arrive convolution burden branch boundary fortunately regular fine parallelization burden software package support state hardware unified achieve parallelization parallelism responsible opposed parallelism responsible consume coarse leverage coarse parallelism collect independent convolution across branch traversal simultaneously gpu include front routine keep track branch span epoch epoch asynchronous via programming interface calculate transition transition focus likelihood notion tree operate e index within post order compute substitution active branch branch entire single extra probability calculate eigen multiple form eigen queue queue parallel execution place queue track execution queue responsible updating buffer queue one buffer store matrix multiplication utilize abstraction empty complete return pool extra routine asynchronous amount resource available device allocate inference utilize exceed perform need store dependency process continue number extra hold transition store multiplication routine batch allocate routine update routine asynchronous queue transition create respective epoch alternate dot represent transition branch dot extra allocate consecutive panel routine update add queue routine queue fashion tree traversal queue execution use queue store buffer work replicate evolutionary history infer tree bayesian epidemic epoch illustrate two epoch substitution process govern matrix dark grey dotted line time put create epoch substitution dark area dot root year replicate nucleotide substitution test whether epoch identify homogeneous nucleotide substitution simulate alignment evolve model bias replicate date boundary substitution character chain start proper simulation inference coverage sign table quantify amount quantity reflect fall across credible interval coverage still heterogeneous substitution substitution recent govern boundary replicate arrive coverage nucleotide epoch epoch observe mse leave epoch inform less three mse high epoch epoch length latter major column along second major row first third model nucleotide simulation ultrametric model simulation c mse mse coverage lr lr lr lr lr restrict nucleotide also homogeneity epoch substitution check homogeneous nucleotide triplet coverage substitution scenario nucleotide simulation epoch three epoch sequence recent nucleotide recovered sequence datum throughout epoch end transform topology ultrametric root supplementary list row label sequence note ultrametric branch figure extensively infection patient previously drop consist collect patient material investigation patient stage infection lead hypothesis relaxation pressure need population state decrease target availability stage infection former distinguish ask infection patient epoch specification model two specification separate exclude patient sequence patient epoch estimating denote ratio rgb high density homogeneous decrease patient patient neutral even generally close patient patient patient factor despite credible formal test conduct bf odd odd individual average within compete odd bayes table suggest generally selective pressure notable joint evidence suggest favor bf accordance suggest decrease stage infection epoch diffusion infer spatio type discrete popularity year partly flexible implementation connect inference capture heterogeneity trait location epidemic seem continuous dimensionality parameter
numerically agree approximate hessian cost manifold description tangent hessian introduction low tensor manifold generic information iteration stopping criterion currently gradient implement descent free scheme toward riemannian bfgs nonsmooth scheme include weight cut relaxation dropping yield yield tight nonconvex rank turn space manifold riemannian l project yield advance redundant gradually global max cut sdp formal acknowledgment nb bm present science office support theorem p rapidly design numerical manifold suit orthogonality constraint machine sensor localization camera component toolbox piece dedicate simplify art particularly reach practitioner outside non optimization fast grow efficient problem search differentiable endow locally tangent inner smoothly number smooth arise n unconstraine equivalence constant define context manifold though riemannian solve point riemannian structure classical riemannian riemannian xx show manifold furthermore orthogonal yield relaxed search ultimately sdp relaxed formulation cut riemannian problem cut packing place sphere apart matrix compact orthonormal version dimensionality manifold nm thing prove one know span complete accord criterion rotation typically notably vision estimate pose admit structure propose exploit riemannian number address problem positive semidefinite space square formulate distance x without relaxed formulation sparse pca geometry cost move specify direction descent conjugate gradient etc emphasis lead numerical describe necessary algorithm come euclidean counterpart
investigate stepsize randomize sgd improved randomized sampling exhibit vanish change step size inside size immediately row problem describe expectation row recover tradeoff horizon convergence call sampling case well avoid allow also weight uniform enjoy iteration composite matrix w weighted suppose converge exponentially stepsize biased row selection convergence use randomized apply theorem weight gives randomize bias modify describe q I equation fully fully partially biased I I partially outline system residual explore role various five system attain case monotonic improvement pure weighted whereas case prefer paper make condition smooth convex discussion sgd randomize sgd iterate get complexity advance approach paper decay appropriate advance limited static update dynamically method gain relative although sometimes anonymous feedback manuscript thank white point corollary foundation nsf fellowship n partially support award rw support grant program award nsf award smooth lemma recall whose gradient page define observe minimizer inequality desire result theorem random employ jensen first expectation I lk yield result theorem section gradient descent objective linear necessary improve linear dependence dominate broadly show scenario randomize apply square rather original partially original sampling stochastic descent connect remarkably descent sgd unbiased body highlight linear convex objective access gradient unbiased gradient sgd attention especially objective strongly optimum optimum recently kind smooth almost zero exponentially polynomially reach require scale require iteration constant strongly term equation simplicity wide array range digital row proportional square use number number aim extend sample numerical algebra incorporation stochastic descent lipschitz gradient coordinate also play design name result euclidean row column completion sampling nystr om sampling framework translate orthonormal orthonormal assume detail inspire prove sgd variant weight sgd corollary distribution quadratic regime turn importance weight conditioning dependence average conditioning partially sampling dependence residual guarantee dominate good reweighted sgd regime show smooth objective sampling improve smoothness dependence smoothness objective eliminate lipschitz objective include objective suggest obtain explain improves know I row act exponential convergence exponential sharing sgd presents clear throughout manuscript unless explicitly specify otherwise index draw distribution residual function I I supremum infimum sgd estimate draw iterate unique distance study decaying size sgd sufficient ensure expectation long course need degradation regime scale expect condition quadratic dependence ensure though rather lipschitz l sgd iterate satisfy optimize iteration expectation l term simplify eq substitute rearrange need utilize rearrange yield note equation recursion co lemma factor second inside low derivative complete appendix replace uniform require iteration supremum conditioning hope follow recover less one uniformly take verify mean suffice method get correct sgd see conditioning source next average conditioning gradient assign index indicator discrete weighting probability mass multiplying construct sample accept reject continue accept accept use expectation property expectation component define q valid stochastic unbiased iterate sgd draw guarantee minimizer control weighted return investigate guarantee must involved constant scale supremum verify minimize weight apply must calculate apply sgd appropriate stepsize w exactly already desire dependence average strictly however contribute towards error might well relative fortunately original q constant stepsize weighted weight without introduce residual dominate substantially linear rather conditioning ask result improve sampling lipschitz q interestingly minimize I guarantee improve use sample biased sampling rely quadratic discuss magnitude necessarily lipschitz advance calculate sampling system solved repeatedly lipschitz computed pass acknowledge regime source imply option rejection simulate proportional additional weight accept accept much rejection gain rejection operating calculate actual accord dominate cost lipschitz mix bias partially biased initial tolerance constant estimate w plug lipschitz stepsize weight sgd partially k close add dependence multiply smooth convex objective particularly interested residual linear dominant also briefly survey relate interest necessarily sgd appropriately quantity rely dependence replace average lipschitz change weight quantity sub dependence partially biased suffice allow dependence turn lipschitz roughly speak
use chain energy detail arise technical distribute uniform lipschitz support transition exist heavily space notion dx dy wasserstein duality remark point curvature entire note interest sufficient prop general take tendency nearby dy dy dy kx kx fy kx bf fx fix markov nx concentration bf bf derive inequality time result tool markov drive transition measurable perturbation good mix property adaptive show late markov chain via argument curvature survey curvature sequence kernel markov drive burn satisfy tx b role difference inequality coarse eqn let respectively lipschitz bf bf choose reduce convenience appear absolute loss bind vice versa maximum coarse diffusion local kernel inequality satisfy difference chain far explain choose bind proof kernel stationary compact kernel kx drive start dx x subsequently inequalitie ty definition wasserstein couple markov start sx iterating time note dx k k since record continuity power curvature kernel w tx kx k w dx furthermore tf inequality trivial us k ty dx couple chain start drive kernel k couple dx dx induction I dx k dx trivially prove similarly omit k fy tx k dx line space stationary w tx kx couple point dx dx dx ex lemma fix fix dx kx obtain kx analogue moment bound curvature inequality kernel w f b dx g inequality simplify ty dx induction condition satisfy equation definition immediately bind inequality show w w w induction come variance condition lemma hold claim follow induction together x extension x b curvature apply see satisfie ba successively bf x b dx dx dx x dx k dx dx bf definition imply bf bf concept statistical mechanic move energy motivation autocorrelation mixture quite sample parallel target empirically plot certainly path rigorous asymptotic apply parallel algorithm rigorously auto covariance energy decay parallel thus empirical us density consider simple encode tailor general modal sampler inductive ergodicity convergence wasserstein begin recall define example follow let reversible fix density special step energy collection intend target distribution simple evolve metropolis hasting proposal burn make interval vx definition sampler energy kernel x couple accept make simulate vx independently behind energy proposal mode roughly sampler refer proposal q couple mechanism simulate give ix x vx constant vx v j j sequentially choose random satisfy associate limit energy kernel proposal energy band define ix vanish acceptance ratio difficult precisely empirical convergence agree example fit continuous potential imply vx curvature analysis simpler reason make impossible positive apply chain thus seem set conditioned markov x base implicitly sequence x j notation mean algorithm define merely suggest discussion entire write shorthand distribution chain switch begin chain energy case simulate couple accept mechanism vx swap move x parallel h node anchor north north node north anchor north anchor north anchor north node anchor east circle radius fill radius fill circle circle fill fill circle parallel comparison performance sampler unit circle embed circle target repeat diffusion kernel hasting proposal case definition sampler chain lebesgue free energy sampler energy lebesgue measure quantitative burn period intermediate sampler density b qx theorem conclude bias tend sampler underlie large comparison underlie chain metropolis chain satisfy bottleneck state minor prove claim need even fix parallel I maximal mix c discretized support document parallel prove interest energy realistic burn sampler least decay roughly former decay rate fast aspect theorem surprising familiar mcmc property turn detail similar decay tf proof proceed sufficiently measure stochastic associate metric either atomic measure choose check analysis relaxation c spectral proceed chain space see distance x slightly modify convergence support document inequality hold minor inequality equation let I fx b fx x combine inequality trivial x df extend bound measure empirical element distance translate chain state constant evolve finally event process h throughout argument hold proceed quickly probability showing event chain rarely far apart vx vx regardless vx vx inequality come remain computation concern uniform iterate next copy limit couple chain give every conditional accord computation dy inequality markovian coupling markovian coupling vx p lemma fix stationary stationarity independent restrict autocorrelation wasserstein fix function time generate fm fm fm fm lemma prove theorem lemma fix repeatedly condition hold coupling start stationarity fx fy coupling draw stationary stationarity couple use fx fy ac h measure energy covariance quantify efficiency sampler look infinite fx empirical scale chain stationary gap chain closely medium sampler limit variance mathematically finite often relationship hold energy sampler argue make well energy lebesgue measure proposal finally arbitrary x xt estimator bf fx sampler infinity algebra random energy hasting write abuse j sum albeit finite raise main study sampler infinite useful proxy one concern interest way asymptotic show energy sampler set begin kernel x limit p target limit hasting proposal metropolis proposal target metropolis hasting level general far tight let diameter section compact union ball densitie w ix iy bound rescale finite diameter largely argument hold compact regularity point much strong assumption distance assumption relaxed assumption change generally absolutely bound depend markov chain curvature see power find could replace hold section convergence limit fix assumption sequence tf e r lipschitz diameter event note condition allow chain induction conditioning argument curvature sf fx argument space lipschitz grow return proof sketch replace liu rely limit draw force close describe closeness strong pointed liu go latter kx closeness wasserstein metric much avoid simple dx measure variable since strict inequality couple dx proof allow construction measure metric b measurable b couple x dx b dx let v wasserstein approximation hold op ix p x inequality lemma recall describe rate volume rectangle function next fix assumption hold fix inequality event I start begin event measurable algebra b markov chain despite fact integer kernel ks theorem I item sx kx plug lipschitz w kx corollary r notation lipschitz f duality df combine r imply strong claim inductive fix constant fix k theorem part property close fix approximation failure g satisfy g equation definition statement induction fix inequality hold j induction mean complete complete note sequence prove fix integer constant assigning ib b guarantee upper fourth boundary define construction want choose modify bound theorem follow bind obtain inequality e choose I note limit yield proof poor burn converge strict metric many potential partial section chain strictly curvature example calculation multimodal circle potential anchor north anchor north north north anchor node north anchor east fill radius circle circle eqn vx define x hx x soon energy take condition energy move measurable check curvature couple quantile coupling mh move couple describe contraction move p
research research early award innovation mixture technique introduction year commonly family arise component decompose number impose component date effective maximization em parameter date family variational approximation information complexity associate minimize leibl divergence family literature bic base heuristic include hierarchical agglomerative widely account heterogeneity establish pearson lack computing probability take use model lee arise proportion parameter decompose decomposition probability decompose em algorithm mixture efficacy likelihood leave failure overcome rely heavily convergence family model em approach selection member criterion bayesian criterion remain chain difficulty encounter overhead model popularity deterministic approximate posterior algorithm sampling observe conditional minimize divergence eq factorize convenience approximate solve couple initialize component expect dominate component weighting consideration simultaneous arise finite distribution log eq number mixture quadratic deal parameter estimate easily exceed size covariance exploit mixture eigen decomposition covariance g diagonal proportional eigenvalue control cluster orientation interpretation rise parsimonious ht spherical spherical ax alg variable ax equal ax alg ax equal package implement model model mark consider assign maximum otherwise modify bic use implement variational approximation conjugate mix precision assign prior application assign matrix possible put I model put eigenvector von langevin von orthonormal orientation multivariate matrix method von von langevin von respectively von implement package model prior gamma gamma gamma diagonal diagonal k g al g v approximate density kl divergence proportion ig g gamma therefore value close g fisher integration datum expect convergence determine acceleration likelihood asymptotic find expectation chain posterior likelihood likelihood every iteration acceleration fail variational sampling increase random modify carlo iteration reach successive likelihood far extremely converge fails converge iteration despite benefit selection actually give algorithm compare widely r facilitate approach run variational simulate value model ari ari na na cluster hierarchical cluster initialization result table small bring back argument em package conjunction bic choose perfect give perfect comparison model also calculate select bic bic seem agreement run another simulation component ten consistently perfect classification ari max ari parameter table lrr carry tendency overall agreement bayes ari ari na na na publicly biological measurement specie width width body depth quite often among issue introduce initial step principal analysis principal transformation convert linearly uncorrelated htbp min na blue select rand relative species table membership cluster create spike thereby cluster species selection good structure originally nine region north south east west publicly r challenging set clustering take classification randomly ari range run range ari bic ari much use variational discriminant package analysis membership predict membership classification result range range na gaussian mixture outperform misclassification variational comparing build bayes close include despite simultaneously component model utilize select study conjunction bayes suitable start em note value play variational former gradually reduce start value explore bayesian widely model conjunction research approximation family proportion prior th assign von prior joint mean eq eq von g
speed km km road segment length speed limit comprise five relational road segment road topology size dynamic seven denote vector position standard noise kronecker delta size maximum select I randomly select machine remark differential criterion gp rank platform link system intel cpu ghz gb gps node root incur speedup incur sequential centralized first metric test f parallel improve comparable inherent assumption local remark definition fig gps order comparable among parallel gps efficient thus requirement time figs gps agree section cc g domain speedup parallel run fig performance fig f observe number machine drop assign machine datum scheme cluster machine well fig e order magnitude fast achieve explain reveal incur base gp machine increase speedup gp fig poor relative observe determine fail fig incur negative positivity problem observe fig order magnitude fast make magnitude capable real fig gp order magnitude fast structural number incur predictive drop gp drop smaller capable gp requirement parallel centralized counterpart improve speedup large describe load centralize achieve efficiency scalability analytical parallel gps achieve comparable result exploit parallel gps become capable perform server intel cpu ghz gb core memory slightly long plan http google com mit research r previously appendix simplify inversion lemma equality equality dy first equality interest predictive variance suffice covariance joint ji mi mi mi complete machine eq two expand definition expand last remain prove equality fourth respectively equality equality fourth equality primary mean predictive input ji mi mi ji u j mi first equality use trick due last gp equality third equality f fourth second equality equality inversion follow fourth equality equality definition time master local summary aggregate dm master machine u u parallel r compute master prediction master size local assimilation communication incur mf sized sized assimilation size predictive master store store unobserved sized demand increase u store last execute parallel I science university electrical engineering computer institute technology usa parametric widely perform cubic regression load scalability guarantee gps counterpart machine great analytically gps communication empirical real node parallel gps efficient centralize full comparable formal cubic approximation method suitable smoothly scale like method gps compactly capable scale prediction sparse approximate regression combine computationally impractical prediction time critical application system sense traffic system huge quantity internet traffic surveillance resolve consider exploit machine scalable scaling support machine attract local gps suffer different gps impose boundary restrict dimensional two low load machine efficiency scalability parallel boundary effect cubic cost full gps centralize gp section centralized among machine achieve gps counterpart rank trade efficiency parallel gps capability practical parallel gps use evaluate performance efficiency world represent dimensional observed gp gp specify covariance give realize input gp prediction output x uncertainty transpose predict use trace centralized well real time computational scalable approximate exploit adapt work decentralize fusion environmental phenomena mobile issue parallelization scalability datum present parallel overcome among machine common master master step send machine detail partition evenly machine tuple construct master support set machine tuple data sensitive expense communication complexity support u centralize independent transpose u proof result load distribute improve structural u impose less conditional independence u support remark output correlate comparable present gp machine scalable approximate gp incomplete cholesky factorization gp approximate semidefinite f cholesky step gp step machine step incomplete column parallel employ incur communication refer row importantly produce triangular incomplete cholesky submatrix store construct master cholesky summary master construct machine summary summary tuple master master give summary tuple u master perform master predictive predictive rank expense communication u gaussian distribution centralized u u proof appendix imply improve scalability approximate table approximate covariance matrix
choice weight formula guarantee follow path follow method long surrogate difficulty linearly replace length handle surrogate show compute asymptotic surrogate state degree freedom st close agreement indicate surrogate behave bottom data point disagreement efficacy quantify test probability incorrectly equal significance power failure datum nan could take express create normalize produce produce th transition trial power table label surrogate show surrogate datum entropy use entropy need computed trial suggest pt asymptotic estimate nd trial rd estimate asymptotic exact size symbol break separate versus exact method size significance quite need st nd test rd order asymptotic attain ideal high recommend statistic recommend computation transition entry improve even computer ability many thousand surrogate minute surrogate parallelization straightforward surrogate surrogate per entry recommend hypothesis markov chain heart test produce identical observe statistic make conceptually order freedom correction describe exact nan hypothesis alternate rely property build surrogate valid surrogate novel shot uniform useful probabilistic various chemical process sequence finance chain past series various narrow versus limit significance criterion bic ranking correction approach rely approximation efficacy valid rely distribution discover refer th especially contribution possible linearly sequence armed test generation formula compare sample asymptotic surrogate order take integer dna sequence bayes multiplied count index set count observe freedom vary shot shot hypothesis advantage know integrate literature order transition word nonzero block degree rely asymptotic limit finite possible sequence nan member
ignore accomplish iteratively compute equal density start high return improvement negligible uniquely op minimal explanation property db db db co op op db outli parameter integer represent size acceptable pair explanation consist main phase phase attribute set condition attribute pair mention pair accumulate acceptable explanation matter fact low support interpret notice value pair experimental effect basically basic notice early iteration interval overall rate slow newton rate em depend far step execute reduce explore practice aim effectiveness outli tuple bag briefly detect various run assign outli bagging outlier far simply basis tuple detect outli summarize combine tuple sort bag robustness base outli contrast manually g visualization justification outlier turn effectiveness outli explanation employ dataset uci repository dataset instance protein localization site cloud attributes threshold condition explanation table explanation property scoring c c report value column associate explanation figure report associate consider top explanation database assume object attribute report explanation dash line curve empty explanation significance account distance thing substantially object bottom property explanation attribute bottom report become table report execution sec cloud notice main identification degree tend overall affect increase parameter matter great support total computation tend value shrink become dataset ask whether computation kernel true advantage apply describe aim perform generate name consist outli distinguished rest equally distribute exception concentrate comparison attribute demonstrate property attribute carry latter attribute value report bin size varied highlight outcome strongly depends adopt measure number bin dramatically undesirable property determine bin challenge figure frequency histogram bin center case bin zero value frequency differently look histogram group frequent pointed scoring assign score unbalanced rapidly frequency spread large bin categorical value absolute get conclude meaningful detect interaction select subset overall population clearly exploit drawback method estimate paper numerical attribute respect sensible refined generalization anomalous respect population characterize object promise scenario scenario include characterize scoring like e exploit basic block future interested proper experimental property problem distinguish object advance take account case leave introduce compare relative algorithm latter characterize basis say differ rest approach normal worth focus concentrate provide problem intend widely investigate problem mining problem explanation notice explanation completely consider detection health feature body pressure mostly patient individual characteristic want input recognize anomalous advance virtue external focus structure accomplish detect subset intuitively object include attribute refer technique could outlier consider investigate identification outlier object homogeneous outlier give simultaneously share solution propose sub population individual attribute object overall presented specifically set numerical numerical applying attribute assign score unbalanced distribution contrast detect deal contribution work represent refined generalization able numerical categorical anomalous quantify degree measure curve occurrence probability worth let pdf cdf q height individual attribute associate report fig cdf probability assume result attribute define employ map integral cdf occurrence probable likely rare thus range one property mean exhibit associate horizontal cdf object difference two line detect frequency pdf wish measure pdf compare domain compare original domain attribute give justification anomalous characterize attribute behave normally attention
map order give characteristic extension hypergraph cut easy extension hypergraph maximum combination hypergraph submodular hypergraph reduce induce namely v induce group structure typically lead extension elastic net semi like functional enforce functional get functional laplacian basis method functional pp convexity clique star carries immediately write learning approach label solving aim label interval loss general recommend type cutting produce much hypergraph cut clique hypergraph give simple cut carry introduce application hypergraph resp incorporate balance graph hypergraph cut relaxation follow line research cut loose fact convex quite outperform globally loose margin balanced hypergraph cut hc symmetric balance hypergraph cut also nonlinear extension hc case part turn partition well balanced hypergraph extension write difference positively f moreover prop minimize ratio initialization r main q simplicity balance thus balance normalize cut inner semi supervise functional novel regularizer dual proximal main efficiently problem low recall conjugate similarly dual general refer htb kf k k optimization arise order smooth exploit g respective primal contrast proximal conjugate e e dual form give orthogonal projection linear proximal far algorithm n w ef k f k c ei subproblem functional separately solve hence primal write order conjugate q decomposition q e conjugate indicator thus exploit follow concern problem right proximal arithmetic operation computed arithmetic operation r sr prox term minimizer end increase arithmetic operation exist directional energy precisely hold thus decrease derivative stop vanish system pair satisfy index need partial choose eq r computation pair check side hand side increase every find hold simply set system linear yield note sorting algorithm proposition htb sort increase initialization accord output q prox f ef k subproblem restrictive functional seem standard uci dataset create numerical bin create l prop class use fully large weight gb hypergraph mb suggest regularizer laplacian arise clique expansion eq clique expansion due memory stop duality gap label choose via validation result standard deviation outperform interaction incorporate hypergraph much tv total reduce laplacian know recommend p cm p cm news deviation cut two reach approach clique expansion show majority vote well cut achieve hypergraph build categorical hypergraph problem similarity choose connect spectral hypergraph cc cc cc hypergraph c nn e hypergraph expansion cluster slightly cut small cut result expansion optimize run matrix hamming useful think room improvement hypergraph like acknowledge grant section corollary tool tensor paper new framework fully key establish standard allow relation recognize application vision bioinformatics retrieval relation help approach divide category use extension tensor mathematically appeal basic approximate hypergraph graph cluster semi supervise hypergraph clique expansion summarize state fully hypergraph prove hypergraph cut overcome limitation exist explicitly cut hypergraph hypergraph cut clique hypergraph extension variation enforce smooth regularization graph key supervise tight relaxation hypergraph cut derive lead cut hypergraph cut depend term partition attain split balanced hypergraph
rank singular large slice test three noiseless work reasonably add run loose increase initialized increment rank test dynamically parameter sr show slice slice consistently relative sr low sr sr extremely still perform fix work similarly dynamically update compare hyperspectral slice approximately good unfold large weight third dynamically one method set sr run table test noisy miss recover slice outperform dynamically work sr g sr two sr sr dynamically slice location entirely mode unfold entire impossible solver recover entire putting third could explain give slice recover look slice unfold recommend dynamically sr low entirely miss color video video treat video channel large unfold low difficult recover video large channel tensor entry practice fix dynamically update average report frame recover video hyperspectral low utilize significantly rank tensor nuclear base low mode unfold video produce solution among compare nuclear minimization quality converge paper demonstrate accelerate progress observe acceleration technique accelerate incorporate low rank table solver solve solve solve decrease solver solve solver fix solve strategy start decrease start h utilize number algorithm mode solve solve e rank strategy solver solve fix solve start start solver solve fix adjust mode ccc sr e e e noise e e noise ccc cc cc c fix sr ccc c sr e e c noise e e ccc h cc dynamic c sr e e e c acknowledgement author anonymous comment xu nsf grant support grant visit support china partially nsf grant dms dms grant fa lemma proposition completion naturally hyperspectral video recover perform underlie adjusting strategy phase plot algorithm variety rank sample recovery two art similar convex throughout model subsequence establish iterate kkt generalization call mode higher arise video hyperspectral search recovery tensor exactly miss entry introduce tensor regard low recover completion kind utilize mode result utilize mode norm svd tackle difficulty apply unfold factor alternatively much non specify convexity approach cyclic perform way adaptively adjust short mode cyclic adjustment tensor operation bold letter bold bold letter slice third mode index third horizontal slice second index respectively also example mode relate cp array nn aim recover zero mode unfold find ni nn nr adaptively nn consistency f contaminate frobenius knowledge noiseless rank must specify yet assume address dynamically adjust scheme scheme rank decrease mode singular start rank gradually slow solve update guarantee reliably recover variety limit iterate satisfy kkt condition regard completion completion partially entrie approximately tensor easy solve name synthetic world well nuclear minimization model nuclear define propose descent proximal direction method admm utilize low tensor demonstrate quality solve mode unfold propose square ni ni ni appear principal low tensor order increment increment matlab increment sr increment depict depict transition plot well apply look varied varie stop result give addition tensor rand second generate matlab rand show first dataset figure fix increase perform compare test difficult dataset matlab diag tensor decay kind appear dataset matlab diag decrease rank dataset alternate rank block pseudo inverse complement n product equality rank cause whereas underlying tensor provide dynamically scheme decrease e calculate eigenvalue eq gap n adjust work exactly rank tensor may exist tensor rank increase r nr nr r r slow mode estimate qr k r make tensor long decrease equip adjust section randomly start rank decrease small
regardless ne select ne agents ne game reward relationship agent analysis possible topology ne depict configuration isolate strategy pd isolate ne configuration isolated ne switch pd otherwise isolate exploit configuration configuration central agent play two continuously arbitrarily player additionally payoff equilibrium condition agent well avoid configuration weight play ne iv link player play cyclic great reward play respective game factorize equilibrium achieve discuss property main outcome stability allow exploration important game network agent play unstable play pure strategy configuration list fig stability analyze isolated correspond xy yx pd indicate neither would play g ne configuration multiple cyclic stable scenario examine co evolutionary agent system simulation fig connect furthermore grow small figure star player player recall block matrix play shown form diagonal nash action diagonal one start either depend agent coordinate conclude instability agent connectivity agent consider identically thus marginally one network dynamic vanish exploration examine ref nash equilibria exploration version strategy evolve eqs solid line configuration central configuration low red line critical temperature three dynamic configuration perturb star stable player connect fig outcome temperature sufficiently symmetric network versus region configuration plane critical temperature network configuration play strategy factor analyze detail ref particular games ne equation ne games equilibria critical rate one unstable whereas insight diagram choose versus temperature perturb pure ne equilibrium ne reward co network reinforcement learn agent strategy allow agent behavior fully characterize outcome agent game analytical player absence composed ne game dynamic allow uniform g play equilibria agent globally dynamic note strategy agent strategy profile irrespective circumstance certainly analytical furthermore extreme agent profile realistic cognitive load realistic profile interact present thank work part foundation study need game action jacobian block square respectively general simplify indeed consider jacobian eqs identically determinant factorize factorization stability mix need submatrix positive eigenvalue indeed eigenvalue mix nash configuration reasoning apply configuration agent mix prove player consider eq subtracting since monotonic repeat remain prove action jacobian consist block select configuration six eigenvalue determine analytically numerically configuration show formation player adapt tie reinforcement demonstrate co evolutionary dynamic system couple equation player nash equilibria game examine equilibria study analytical absence equilibrium consist block stable equilibria agent ne connect topology stable system generally introduce via approach link static endowed dynamic opinion formation treat topology separate individual dynamic fail indeed network network evolve co evolve attract interest behavioral economic community couple attribute network model network interact adaptive agent specifically network augment adapt behavior tie outcome produce selection exploration exploitation previously collective evolution context evolutionary theory equation use model collective interact action play agent ref examine outcome action simplest analytically characterize rest absence exploration absence exploration always allow ne correspond stability star instability configuration exploration indicate critical globally outcome dynamic paper characterize describe ne direct analog biology boltzmann tendency term equation reduce conventional follow q agent factorization agent game play make normalization equation collective evolution network game add payoff present agent incoming always reward zero reward isolated agent serve pose game tend try agent isolate receive round game payoff merely learn
child separate hyperplane separate leaf partitioning process complete define past regressor depth partition show construct leave region region indicator fall point otherwise generality region point label tree soft separate hyperplane regressor abuse notation combine rewrite xx soft separate paper regression leave node complete partition regressor instance third fig partition union regressor regressor hierarchical tree estimate q emphasize fashion piecewise regressor specific adaptively regressor update regressor size regressor region train approach separate node regressor simply calculate gradient plug piecewise regressor good recover process regressor region consider space partition use section partition regressor boundary dimensional regressor problem regressor doubly combine hard boundary train regressor merge output partition regressor e use rp tree generate combine weight w within depth asymptotically cumulative square combination algorithm tree value imply asymptotically priori different regressor region regressor converge widely separation assumption w separable follow mean algorithm chapter merge partition regressor combination piecewise regressor select direct correlation information combination label tree significantly suppose learn derive weight vector connect w w perform mild assumption sum get eq linear piecewise perform adaptation label tree empty string assume emphasize letter alphabet refer node string string empty string string calculated final final desire structure sum leaf compactly regressor string string denote define emphasize dependency estimate model simplify view leaf depth rewrite partition next combination piecewise regressor simplify manner denote node leaf recursively update recursive instead memory keep depth depth tree continue generality vector regressor q represent string weight hard used regressor state algorithm combine node weight reduce complexity total different leaf state location update introduce algorithm vector make algorithm complexity achieve result p p w vx tree find q e kp observe regressor sufficient estimate simplify final dp lp turn notice occurrence become emphasize achieve sequential regressor doubly regressor extend soft function train tree depth introduce achieve good computational regressor partition regressor complexity unable learn regressor since soft need cross change moreover node region boundary partition since cross correlation final get bound construct work feed reduce complexity omit label final estimate th find similarly give function approximation observe number leaf w pe tp pp p regressor regressor model adaptive region obtain way include accord eq update conclude outline construction emphasize rate consider requirement smoothly region experimentally acceptable multiplication significantly decrease therefore reasonable purpose put sufficiently either fall approximately one fall region region leaf observe solution reasonable use selection depth decision usually desire generate piecewise linear conventional use partition regressor guess order capture salient desire increase infinity partitioning regressor hard select minimize partitioning regressor locally optimal partition regressor sense select decision issue algorithm various model correspond partition partition performance depth scenario process map illustrate california dft algorithm regressor represent regressor represent lf represent cr represent represent fourier regressor construct regressor give vx subsection regressor cr correspond subsection regressor vector normalize aforementioned reason scale back fair consider order subsection california similarly depth experiment california dft regressor partition four leaf use cr knot rate equal computational dft cr regressor vector leaf node individual dft algorithm node rest I nod correlation node computational gaussian regressor achieve mass filter introduce nonlinearity power regressor result significantly convergence performance cr satisfactory knowledge regressor algorithm hand prior knowledge outperform trial subsection match signal variance function gaussian whereas regressor scenario dft cr ti partitioning generate demonstrate highly nonlinear piecewise cr salient characteristic regressor nonlinearity dft compare dft though partitioning perfectly match dft record text context introduce restriction fig sum force whereas present partition high whereas weight need datum operation desire piecewise specifically desire piecewise eq mean dft cr underlying point matrix accumulate well compare degradation dft importance regressor observe perform almost scenario dft regressor leaf regressor datum time accumulate boundary underlie shifted regressor partitioning perfectly update power introduce dft node unstable learn boundary learn b dft desire partitioning underlie salient characteristic generate piecewise model necessary depth perfectly process cr fig show scenario piecewise similarly fig scenario piecewise partition b illustrate estimate consider given behavior desired denote extended framework cr whereas normalize regression one whose salient characteristic since similarly use piecewise model equation known regressor try algorithm illustrate significantly achieve boundary whereas rely structure bank regression life california house california california provide normalized outperform rest aside california life set realistic link end medium f case action take realistic arm task angular acceleration robot arm bank simulator predict bank bank lf algorithm cr bank superior achieve much high superior regressor real life big regression deterministic tree regressor partition regressor regressor regressor linear combination doubly algorithm require upper avoid weight different regressor example one regressor incorporate method rp algorithm significantly improve bound regressor adapt complete mixture doubly partition avoid parameter directly minimize regressor partitioning nonlinear
context shannon maximizer show outperform strategy reliability policy rigorously however rely gp address uncertainty current pareto avoid criterion tune formulae avoid rely several provide method art finally advantage drawback response f p addition numerical estimate observation chapter chapter model line modification initial generate validate maximize update add meet perform difference minimum regard numerous solution global concept directly measure objective objective aggregate weight ei objective improvement increase new adaptation progress improvement measurement actual actually objective alternative sampling criterion aspect entropy minimizer interest performing location minimizer unfortunately expensive measure propose notion improvement difficult issue regard optimization current similarly shannon yet contrary volume little gain reduction paradigm minimize refer cumulative point volume without expectation suitable proposition subsection gp model observation depend value conditionally gp future conditionally observation depend restriction expectation threshold conditioning form gaussian bivariate lead proposition resemble conditionally proof back future hence expect term reduction improvement low prediction inefficient gain amplitude indeed volume current value region reduction high current gain volume toy gp build six improvement compute eq remain mostly unchanged indeed response would considerably volume dominate subset constitute subset separate dominate dominate objective cell couple consist consecutive illustration cell cell dominate say dominate dominate locate plane fit independence dominate part approach actual pareto volume tend define volume modify pareto front add dominate exist remove update remain dominate accounting modification pareto front complex value update I condition note leave aside know belong pairwise independence close notation additional th component th three arise dominate b ij k dominate new dominate ij ij observation dominate belong ij dominate first account potentially cell account dominate besides relatively computation ease report firstly carlo approximations integration integration quantity depend loop criterion rely must numerically efficient program task use non double intensive reduce grouping cell note finally grow filter retain type strategy applicable contribute critical substantially observation objective pareto dominate objective dominate cell cell observation quantity compute express factorize development case cell figure bi gps regular regularity parameter respectively randomly choose consider initial front describe section volume cell four belong dominate relatively obtain right represent objective actual pareto front front value bar observation pareto front approximate accurate resource art outperform gp limited budget performance criterion optimization series provide pareto pareto front objective realization six gps index regularity variance range take add iteratively result new base principle numerical strength gaussian share gp cope approximated stationarity efficiency greatly important model characteristic strategy gp wish emphasize propose computational loop limited choose
task family gene unlabeled instance extract gene multi class collective link relation combine multiple collective fusion relation meta validate collective heterogeneous collective collective exploit dependency path collective homogeneous collective ica use collective author link compare collective convert heterogeneous link type method link homogeneous ignore type collective collective collective implementation collective train collective relational collective iterative vote label aggregation perform collective fusion base collective infer link iterative claim instance achieve know instance meta path another dependency heterogeneous avoid overfitte claim illustrate performance without base classifier method number intel core ram l evaluate effectiveness collective heterogeneous collective report six dataset plot small large dimension feature base learner time significantly affected path method need neighbor collective slow meta path support motivation select meta classification collective heterogeneous type auto correlation among semantic compare version exploit path paper paper share similarly represent proceed compose citation complex citation link exploit I model collective performance path relevant collective paper topic conference proceeding likely similar publish conference conference year overall topic conference irrelevant research institute researcher area researcher operate system combination citation citation citation intuition meta expressive indirect collective collective heterogeneous conventional collective approach object link collective information structure object collective heterogeneous call meta able dependency meta study collective classification effectively boost heterogeneous important mining bioinformatics citation collective exploit link whose focus problem intuitively link path dependency indirect path among linkage dependency meta consist quality collective classification depend upon meta path accommodate large collective classification meta dependency path effectiveness propose collective classification heterogeneous meta collective exploit accuracy decade supervise classification identically distribute collective label inter connect label independently many example paper cite couple citation likely paper consider explicitly challenge collective collective classification object advance object multiple multi mode amount involve node conference five heterogeneous citation link collective node classify figure collective classification heterogeneous instance collective classification conventional collective method group among instance inter heterogeneous instance link citation paper citation proceeding paper proceeding proceeding conference proceeding conference author author institute collective heterogeneous major challenge summarize follow classify structure involve example figure link author conference link totally semantic conventional ignore type heterogeneous type path relational link relationship indicate author institute relationship paper publish network complex relationship treat structure mining classification study collective propose novel meta collective effectively one conventional collective propose object meta dependency type dependency boost network rest review collective heterogeneous preliminary concept conclude work collective relational network briefly discuss collective relational investigate class rather label instance independently sometimes collective approach exploit classification performance roughly upon strategy unlabeled attribute relational instance involve update relate iterative local classifier regression naive bayes dependency optimize entire relational attribute relational feature review please involve multiple network possess mining domain heterogeneous network many attract much similarity heterogeneous specialized problem heterogeneous network however directly collective convention classification exploiting object introduce concept notation formally collective heterogeneous network heterogeneous network direct object n heterogeneous network five link r symbol definition testing ji network heterogeneous conference multiple link type relation naturally range author networks inter indirect path path name similar represent path link table meta semantic study naturally path citation frequency number path meta unique sequence meta author represent direction paper focus collective heterogeneous network exist reason type nod quite type share label example patient instead node care assume classify suppose attribute variable indicate give know node set label assign collective heterogeneous testing attribute review require perform independently label closely conventional exploiting type author denote denote th meta dependency among instance link link collective effectively type heterogeneous meta dependency inter meta collective give meta meta mi instance meta path relate figure path meta dependency classify heterogeneous I approximate however reason particularly path collective classification heterogeneous develop mi extract propose collective heterogeneous collective meta heterogeneous small path type relation show conference classification paper dependence unique meta indicate however general grow exponentially path meta path capture linkage heterogeneous instance short meta really extract meta redundant meta path redundant path overfitte additional redundant path path meta construct path overfitte meta decompose short meta path meta path length great conference decomposed exclude meta meta propose network meta reach meta short meta extract heterogeneous show collective inference ica algorithm simple homogeneous network paper collective call collective l maximum meta path set training path first add short meta current path reconstructed construct label repeat convergence iy rl idea instance via path relate v j j probability instance extend build base probability treat I aggregation j collective link path feature employ instance instance relational meta appear related weighted paper list
large achieve low ii type amplitude setting signal duration audio extraction identical amplitude scaling instead base degeneracy tu product individual tu tu use whole sample degenerate mmd decrease cf size mark discussion dark medium france unit college united france com fr rgb family mmd test statistic subset test test combine favorable test block incorporate degenerate nan distribution transfer problem vs address similarity reproduce hilbert setting object discrepancy energy feature rkhs kernel density necessary determine similarity measure power nan unbiased take form infinite empirically merge unfortunately demand former cost constant mmd assignment pool small parametric instance pearson guarantee run independent also central limit give asymptotically statistic word lowest give much spectrum limit latter estimator look achieve datum data nan calculate test reasonable nan hence test test normality distribution cost dramatically much available propose two statistic degeneracy replacement statistic whereas expression suggest stage u degeneracy know mmd emphasize approach much broad variety easily fisher schmidt distance approach apply straightforwardly maximize code presentation brief overview mmd empirical provide discuss evaluate benchmark dataset advantage power provide ab experiment plot vary finite variance section normality green remain ks normality derive employ construction mmd compute block sufficient block furthermore analyze block rkhs reproducing define kf borel discrepancy borel borel square embedding eq independent copy kernel iff minimum analogy subsample notational index though present obtain variable although mmd compute gram matrix order central result accord hz hz block turn eigenvalue kx expect limit easy calculate test threshold beneficial well separate sufficient deal size recently develop goal interest bias estimate block threshold whereas concern bootstrap rt approximates generate process moment test comment central moreover converge ki ki remain convergence moment dominate distribution threshold however sum infinite sum skewness skewness skew positively threshold inaccurate account experiment bias cause low type require ess conservative I let every eq ensure fast underlie I point fast large decrease fulfil tradeoff sample nan provide error disadvantage test sensible heuristic emphasize assumption dataset comparison estimator kernel pt computation consistent pearson curve pt gamma approximation spectrum pearson curve pt gamma mmd median bar visible follow synthetic grid gaussian specify two covariance spherical prove parametric test rt employ kernel test gaussian match variance somewhat perform practice treat median likewise context learning optimize maximize power approach non testing set remain half rt approximate inform quadratic pearson spectrum curve spectrum quantile fix
uniquely either global around additional fisher great study inter proportion truly uniquely increase global three study large variability truly uniquely fisher around global differential meta rna seq arise study overlap list differentially express gene analysis account poorly performance variability glm effect gain proportion positive uniquely identify meta combination r call focus two condition multi comparison analysis consideration include meta difference objective population sequence laboratory effect concern rna genomic sequencing technique sequence seq dna sequence potentially order rely rna seq datum condition challenge jointly heterogeneous seq kind genomic straightforward ar analysis manuscript gm design study r package manuscript study author read final sequencing seq cm en france en et paris france universit france abstract high throughput sequencing rna seq biological power continue decrease likely conduct biological question microarray analysis differential rna seq technique binomial linear glm real glm well low study inter study large number study combination valuable tool meta rna seq appropriately account biological technical r keyword meta rna seq differential expression rely high throughput sequencing library read nucleotide rna seq yielding read arise continue seq perform biological replicate differential expression lack likely additional conduct biological question suggest need able among specific effect arise due difference library biological variability recent year analyze microarray arise meta advantage integrate subsequently detect within review outperform effect linear include area receiver analysis microarray directly applicable rna seq differential analysis microarray distribution hand grow body work seq binomial model heterogeneous recently method poisson well adapt rna seq dispersion biological replicate binomial datum rna seq arise model inverse combination meta rna seq binomial two extensive inter variability replicate finally arise condition consideration biological replicate differential analysis study condition biological replicate biological vary let integrated gene approach combination study analysis use differential within gene follow factor comparison different nan hypothesis per gene dispersion dispersion pooling strength raw subsequently condition exact raw obtain across assume gene correspond raw cumulative standard weight biological replicate study biological replicate attribute large quality nan subsequently control desire test define correspond nan hypothesis freedom combination classical desire implementation additional gene detail package glm rna seq study human cell three line hereafter refer cell read phenotype table study supplementary material characteristic supplementary datum tend library appear exhibit overall per gene supplementary appear figure per histogram state nan hypothesis second large discretization remove express gene peak filter raw study satisfy uniformity histogram raw real grey filter combine independent analysis binomial glm fix study differential analysis gene differentially express differential diagram present differential real meta analysis intersection package diagram compare differentially gene find immediately notice study fisher considerably large uniquely pathway www gene respect versa identify supplementary gene identify approach biological study study inter binomial mean relationship incorporate inter situation overall around study variability note effect fix realistic value differential fit human control identify gene overall empirical library difference gene gamma glm per dispersion fit value gamma glm overall gene weakly express gene dominate variation count dispersion nearly nearly inter observe considerable supplementary material four non differentially ccc replicate study rna seq experiment study analysis identify value variability per analysis combination assess sensitivity discovery fdr area receiver operate roc also assess add
deterministic omp bp threshold threshold quite alone various database omp bp algorithms corner incorporate coefficient sign image apparent image omp recover overcomplete linear constrain partially completely negative solution completely let denote correlation concerned recover negative bound minimum negative constrain small lemma expand substitute bs bi expand eqn triangle coherence rewrite eqn lower small element correspond theorem linear uniquely recover impose unique negative system exist theory sparse representation negative rest analyze recovery three support support negative general support unknown pursuit algorithm derive coefficient quantify order phase characteristic basis pursuit recover sparse unconstraine counterpart propose system orthogonal matching pursuit recover linear q vector negative negative unconstraine sub negative obtain non negative coefficient combine coefficient unique deterministic guarantee dictionary image recovery protein mass data portfolio name briefly mention patch represent predefine dictionary many image application resolution compress dictionary dictionary combine model propose section recover corrupted sign coefficient portfolio select capital risk recently constraint coefficient portfolio coefficient market combine effectively representation solution recover pursuit bp program express condition recovery derive negative property version orthogonal pursuit omp recover also major sparse coefficient vector consider solution uniquely investigation sparse vector threshold solution present coherence author improvement sparsity pattern sparsity threshold sparsity non corrupt additive model furthermore coefficient recovery derive recovery derive polytope span require row span threshold recover satisfied replace present negative omp combine omp omp omp algorithms factor improvement hold omp negativity threshold knowledge alone contribution result present detail combine piece complexity bp omp non coefficient random bp respectively bp recover furthermore characteristic utility letter denote indicate mean absolute column cardinality operator maximum argument size define drop coefficient coherence coherence side coherence span positive diag singleton main singleton singleton threshold geometric theory define entity cross successfully imply recovery program else solid simplex include span polytope non polytope consider polytope polytope correspondence singleton singleton denote convex vertex arbitrarily combination direction intersect vertex denote hyperplane separate hyperplane combine singleton investigate representation representation coefficient respectively representation negative coefficient coherence sparsity partially coefficient express define least square ls solution present sufficient define full zero derive knowledge q define become nn coherence satisfied singleton definition condition unknown recover convex k derive threshold solution loss bp since norm condition pursuit solution omp similar omp either norm denote correlation current atom update index choose consistent combine solution constrain compute ignore deriving improve c c omp solution proof inspire technique derive sub non give omp omp solution omp lead difference bp omp extend residual eq needs derive recovered threshold obtain column sufficient condition satisfy recovery depend combined threshold hold use slightly well already vice versa recover atom residual ready section omp clear make program discuss recovery use omp compare vary bp omp omp constraint drastically deterministic threshold omp omp perform gm ensemble realize case k gm random sign b sparsity c gm ensemble zero realize gm ensemble realize snr db snr c obtain gaussian ensemble non realize contour figure b quantify essential implementation omp lars solver source bp interface implement modify various factor parallelization signal representation step dominant computing correlation matrix computing coefficient step know omp complexity omp omp omp algorithm coefficient update perform operation subset whereas operation update constrain partially omp final omp omp remove constraint solution recovery consideration correctly omp lar solver fairly quantify lar coefficient support correlation gram matrix omp lar easily identify dominant dictionary omp lars modification quantify bp omp omp knowledge improvement experiment gaussian realize sign uniform proportion unconstraine coefficient combine representation test omp bp recovery performance algorithm recovery rigorously quantify compressed recovery phase diagram accurate various level image corrupt sparse atom zero realize distribution uniform case non negative distribution coefficient sign non varied hence total trial trial use four omp coefficient bp explicitly zero recover recover exactly coefficient realize increase omp omp respectively substantially big omp perform bp omp substantially improve recovery deterministic sparsity combination realization pair dictionary omp bp sparsity realize sign experimental approximate similar section consider measure counterpart omp
topic distribution cosine similarity result score take part main task system high score system examine target noun word compare gs class run class affect ability cluster comparison noun four gs create system cluster cluster sense mainly describe business third sense cluster either encourage finally lack number rich topic context induce annotated language cluster system gs early carry case gs induce though sense language pos create high performance cost language independent unsupervise space unlabele data train lda topic use test topic closeness induction v ambiguity try minimize vocabulary use portion vocabulary mean word context knowledge acquisition bottleneck corpus could sense effort area another technique cluster external resource rely word topic context document distribution topic closeness topic motivation behind observation help determine corpus infer sense induction lda discrete show level hierarchical consist speech pos annotate feature bag contain noun parallel lda model topic distribution cluster word sense space evaluation measure harmonic homogeneity consist belong gold cluster hand degree completeness see recall bad measure harmonic test sense induction case run
element low incoherent space provably iterative rank actually mass random coherent imagine adapt roughly likely incoherence completely dependent recovered element row score standard incoherence recover observe high nuclear sample single certain accord leverage column incoherent row space arbitrarily coherent immediately provably correct scheme assume leverage par complete coherent phase whereby draw score result able benefit nuclear unweighted minimization justification new bound involve appropriately row differ unweighted natural rank property capture expect generally algorithm vast body matrix big body review paper relate theoretical guarantee exact nuclear work incoherent subsequent provable incoherent nuclear svd follow minimization additive consider wise element magnitude later refined proportional magnitude argue preferable problem matrix column use approximation art involve randomize whereby select statistical leverage approximate fast need leverage extensively recently context name statistical sample wise spirit sense show sparse traditionally mutual extend incoherent sensing accord expansion quantification interpolation compressive imaging consider space allow adaptive require observe quadratic completion section two guarantee weight nuclear main paper arguably popular observe element optimum program singular value universal element reveal accord introduce score value whose svd normalize score leverage score row appropriate column column always mr mr n previous upper score leverage consider localize version incoherence ready observe refer strategy leverage rank element universal number reveal optimal minimization comment speak indicate large leverage score matrix discrepancy natural interpretation align leverage need score sample recovery know leverage degree regardless low uniform original subsequent incoherence leverage score uv art state unique corollary extra incoherence positive remove incoherence parameter improvement immediately matrix set column incoherent simplicity constant leverage leverage near number universal compute leverage e far unique nuclear whereas entire subsection show complete coherent restrict probability family say identical assumption mild typical scheme element matrix score bound eq score exist j conclusion infinitely condition least least similarly succeed shall universal nuclear recover completion one failure two failure cover restriction distribution score nevertheless result essential coherent highlight relate leverage score underlie recover arbitrary rank accordance application free priori leverage replacement leverage score generate sample replacement completed suppose give total budget underlie svd row score second remain sample two phase underlie incoherent recover coherent energy concentrate completion uniformly fraction uniform score axis vary small leverage wide low complexity opposed sampling decrease leverage matrix long coherent application like collaborative filter user distribution potential improve recover figure dimension phase occur sample compare assume noise axis plot demonstrate leverage score suggest well reverse way idea quantify reveal distribute row unweighted inefficient instead minimization diagonal element guarantee unweighte universal theorem draw nuclear leverage score scale rescale complete complete j observation roughly complexity unweighted quantify particular problem unweighted product form rp cc np cp recovery high unweighted latter nuclear approach succeed restrictive condition particular unweighted approach impose row heavily column precisely advantage empirically observe empirical nuclear minimization provide complementary advantageous unweighted serve relationship outline convex main establishing unique solution give rise condition weight norm maximum column lead completion simplify square matrix fashion proof simultaneously hold drop simply etc differ underlie several zero norm operator op f optimality follow unique follow uv optimality begin proposition condition construct dual satisfie p ij dual construction complete p proof appropriately norm norm maximum column norm need norm eq element magnitude concern norm crucial approach norm projection ij p suppose w lemma equip ready condition note ij w eq rewritten follow apply rhs w h rhs time similarly w uv ij uv uv uv uv uv large complete p g duality obtain q rhs tr op optimum tr op nr op display technical lemma bernstein frequent fact follow put hand q bernstein eq inequality follow bernstein bind te ie note ij ij te ie j ib nj bp ij apply rx union fix index norm proceed ab fashion ni bind sum ab quantity finally ij piece bernstein conclude union pick nn r leverage score row score also sample observe inequality actual observation corollary union part x number vary infinitely differ moreover differ n invariant complete
environmental covariance matrix strong marker gene interaction approach allow good performance assumption restrictive marker influence likely jk jk jk genetic trait correlation additional characterize care preserve positive complex topic ignore correlation coefficient prediction composite krige linear combination one environmental krige predicted phenotype compute phenotype individual training simple need prescribed method individual easily universal krige approach assume test individual trait curve trait correlation predict environmental matrix constraint environmental sampling variability sampling partition partition used validation auc trait calculate analysis clinical reporting imputation perform poorly snps outlier component individual baseline measurement imputation recommend default setting snps score allele change low phenotype subtract baseline baseline pathway pathway comprehensive pathway analyse pathway pathway pathway previously plausible pathway focus pathway reveal family member disease datum merge file recommend remove approximately prune addition identify pair include removal remove double snps snps double cross validation validation logical core search previously model area receiver curve apply genome validation baseline th multiply matrix component score respective thank manuscript describe manuscript obtain network statistical phenotype www v p acknowledge collection clinical trial full list www analogous krige use genome unobserved compute location close trait individual proximity correspond location genetic matrix genetic throughput datum trait ht versus true optimally weight gene expression optimally c weight solid line value line represent slope search contour validation roc curve auc permutation implementation disease genetic ok two optimally snps know ok double dramatically associate study fit comparison also permutation score genome wide principal implementation outperform baseline ht ok outperform baseline cd disease ht diabetes diabetes p cm cm r ci na genetic gene expression top ten cm cm cm p n n mean r ci na genetic al expression snps prediction gene snps ten cm p result double sd auc ci matrix na na auc ci na auc weight mean auc ci ci weight na na auc ci auc ci report al sd score ten auc receiver operating interval relationship genetic snps disease cd diabetes diabetes cm department usa department university il health studies il usa mail abstract prediction disease risk drug goal wide thousand broad trait trait propose novel approach genetic level translate krige call increasingly make comprehensive survey trait furthermore sometimes bayesian approach trust show comparable publish phenotype integrate expression substantially increase alone predict change level score summary advance development genomic clinical manuscript novel trait translate dna profile krige learning call wide datum comprehensive survey trait human trait evaluate growth alone show expression datum predict clinical response phenotype study seven trust intensive introduction trait trait contribute similarity association study significant trait prediction trait use height explain snps think appropriate trait approach phenotype analogy useful krige measurement location assume nearby site krige location along observe predict analogous individual application trait notion close tie genetic distance distance demonstrate human population genome naturally plot diagram analogy krige trait show figure method trait genome predict genetic phenotype marker individual marker estimate marker application disease risk snps threshold set set additional risk seven within divide implement trait marker association meet respective snp take phenotype score performance model assess associate include whole genome review de penalize method absolute shrinkage operator elastic net version penalty prior marker ordinary marker prevent squared cancer risk genome snps area receiver characteristic auc high include covariate idea back formalize linear use throughput several author compute genetic marker genomic unlike phenotype hundred thousand relationship marker measure affect phenotype normally marker combine allow marker large likely phenotype krige krige krige similarity base genome al use mat ern function commonly genetic measure extension krige integration furthermore integration genome giving subset weight genome genome krige tie additive genetic genomic cross validate sense closely reproduce rkh et al method integrate use connection krige less familiar usefulness encourage adopt analysis complex trait human trait growth response seven package implement similarity snps gene expression gene datum phenotype average phenotype snp comprise phenotype similarity test individually phenotype performance weight environmental produce prediction combine optimal genetic phenotype assign repeat method trait compute determination square phenotype trait curve auc pairwise report computed phenotype assess intrinsic growth commonly cell line phenotype associate differentially growth clinical european snps expression level level minor alone reduce r baseline model combine principal component negligible r combining gene expression clinical turn seven trust case snps approximately single successful guess seven area roc curve disease diabetes table determine validate partition generate perform disease result minimal improvement auc great improve variant generate snps kb disease national genome research institute phenotype predictive snp slightly type diabetes dramatically type diabetes figure diabetes add increase figure genome snps ten component phenotype double outperform diabetes double greatly outperform type diabetes great diabete baseline respectively system approach scale translate genomic similarity krige learning construct genomic environmental obtain individual provide krige interpret converse prediction component additive method component environmental component generalization modeling predictive manuscript grid approach reasonably similarity snps snps etc phenotype comparison report magnitude highly prediction associate intrinsic fdr surprising predict level stress importance consider phenotype population european analysis issue population implication estimation predictive genomic component environmental genomic component limitation expression study level likely limit snp r twice inclusion genetic alone demonstrate apply phenotype successfully predict clinical disease risk phenotype yield score computationally intensive time e processor minute whereas hour processor markov effect disease disease type diabetes snp obtain disease know relatively knowledge independent use select avoid improvement differentially marker trait information simulate zhang et recognize trait model reason linear consider approximation exceed logit origin gain efficiency add burden numerical unlike link incidence memory krige field standard krige motivate environmental validate performance rkh de structure ignore restrictive
review review define estimator since u van trees book information em posterior distribution element conclusion tell error evaluate diagonal compute equivalently rhs fisher q indicate convex u consider affect accuracy model performance measure ig vi receive review choose item likely review choose choose item proportion graph generate user choose beta prior pz pz nu ix ir label respectively item correct infer first size infer increase confirm graph connection inference curve approximately always accuracy constrain connection clearly experiment graph mode compare rooted rmse rmse approximately graph large low add rmse become bad constrained connection constrain connection inference study constrain cause system review service narrow interest find connection cause measurement review important customer decide service market reason online review field water review product service review review quality service always obtain g e service online ignore online review bipartite graph many explicitly implicitly narrow attention capacity connection jointly truth review truth rao low posteriori system vary different topology topology become follow review review consider always probability assume put different narrow capacity connect edge bipartite edge review
membership challenge measurement suppose come component model deal learn reconstruct graphical gaussian datum suppose come mixture dimensional setting regularize expectation maximization gmm way via likelihood however degradation high propose parameter technique eigenvalue cluster number totally among penalize technique encourage many provide sufficient closely model base gmm assume covariance approach matrix say result latent application penalize approach graphical cluster parameter cluster estimate view aim membership among additionally assess obtain glasso provide throughout follow introduce glasso proceed version em simulation introduce derive penalize statistical em prove consist variable denote essence eq mixture mixture component represent fall inverse density sample e parameter mixture namely parameter recover underlie multivariate consider moment ml write density goal maximize bound consistency ml investigate result mainly base result mle prove globally consistent compact high maximize complex penalize likelihood promise degeneracy keep space however make opt consistency prevent place entry function closure context estimate interpretable often penalize user define sparsity component assume complexity degenerate consist univariate likelihood tend infinity likelihood mixture whereby element tend ml want general cover consistent estimate set compact subset close mixing away zero consistent space eq second rhs continuously pointwise maximize log likelihood conditional expectation likelihood augment suppose penalize augment write indicator say follow compute consist see component tell actually step maximization turn yield q maximize thing formulate maximization component model cluster q consist get covariance covariance innovation formulate modelling modeling scheme consistency sample size well datum ht investigate em simulate proportion equal inverse covariance scheme q eq tp fp penalize ad tp fp graphical scheme size examine consistency base penalize different assign deviation proportion norm precision score false positive performance penalize em size increase ad norm precision indicate ad mixture proportion almost indicate distribution tp fp precision record component fix c performance different table observe ad suffer penalty higher record bt ht ad penalize penalize ad false fp precision graph property satisfactory penalty consider help subject mechanic algebra book mechanic vector closed book fit student indicate student subject fall group mechanic interaction consider cell contain protein cell collect stop min
improve network precision tradeoff similar graphical algorithm briefly assume perform population precision generalize variate datum graphical increase outlier dependency entry comparing bias control bias towards network independently approach towards gets learn recovered test synthetic network another create correctly identify precision edge distribution trial tb pdf depict figure curve differential bias obtain differential traditional high low precision curve training set yield differential precision independently compare increase little instance able differential many pay similarity bias different lead operate usual goal recover drop transfer bias weak task tradeoff control parameter control tradeoff recall due differential induce ten spurious one precision low true network true difference differential various get hard identify drop high highlight another usual transfer similar improve identify confidence procedure generate train learn repeat time calculate appear difference infer recall appear difference consider show transfer dominate bootstrappe reach precision regime bootstrappe bootstrapping bootstrapping learn bootstrappe increasingly bootstrappe differential difference occur one filter expert perform differential dependency cancer cancer quantitative usage ground estimate discovery fdr test estimate fdr first pool randomly synthetic population instance newly discovery splitting fdr perfect fdr synthetic fdr indeed real exploratory biology cluster complement biology body reaction system cancer response create tumor help response hyper state involve protein essential role cancer primary list protein associate appendix protein cancer instance well growth cancer show functional description name dependency protein e protein protein cancer tumor tumor process mention involve cell tumor body tb image activity brain region brain brain indicate appear question dependency accelerate fmri see interact subject ask virtual reality environment initially significantly identify fmri collect task reach perfect perform brain network perform cognitive task tb fdr varied low right identify difference brain estimate fdr rapidly close optimistic estimate confident versa expert confidence group region share pathway human location object pathway collection response separate network region identify object task pathway increase strength suggest result great flow identification network allow expert difference population generate many domain include biology traditional task compare discovery rate importance infer explore use transfer explicit tradeoff empirically achieve bootstrappe expert involved focused learn differential could conjunction cancer detect change cluster acknowledgement like acknowledge contribution university center health ed cancer h x protein protein complement protein protein protein cd cd sl responses h pdf nj co nm gain lot represent dependency variable look population specie population network compare discovery impose similar dependency network discovery acceptable difference transfer use provide natural smoothly adjust differential requirement conduct present study technique light learning algorithm enable visualize dependency identify difference various want understand region brain share person particular influential region accelerate analyze patient understand cancer biology learn dependency independently tend prevent draw reliable conclusion differential analysis find intuitive mechanism learn trade small spurious difference identify large novel tradeoff dramatically improve network jointly impose learn heavily difference learn network thesis eliminate spurious eliminate adjust filter reliability technique study identify insight biology find known processing pathway insight analysis learn post hoc permutation test difference perform far less expensive bootstrap discriminate discriminative interaction case extensively transfer analysis mention synthetic far explore dependency individual recover accurately interest provide trading improve individual dependency interesting orthogonal scope usually hold must confusion negative tn usually false positive fp false negative graphical trade adjust degree horizontal highlight well meanwhile indeed plot set various network change sparsity recall precision degree control tradeoff tb pdf pdf differential network identify
method give draw normal distribution seek covariance however likelihood inverse matrix goodness group model correctly show fraction varied group lasso group lasso correctly fraction correct small sample grow intuitively model axis rescale curve align say scale categorical person continuous gender categorical model auxiliary variable continuity regular exponential rearrange second lemma term active satisfie c cn statement use triangle thus exponential minimal thus strong extreme semi thing choice satisfy expression compatibility constant assumption apply check similarly assumption universal claim lemma right singular piece desire bound name theorem assumption theorem diverse area engineer dimensional encode low subspace estimate also fall subspace selection regularize special identifie regularize estimator refer consistent selection area engineer one learn motivate induce incomplete encode si fall notion correctness regularize convex twice continuously differentiable estimator refer consistent analyze decomposable consistency describe geometrically decomposable penalty discussion consequence converse necessity final devoted result multivariate regression mapping denote ball denote regularize section comprehensive work establish penalty q lasso nuclear decomposable develop result establish consistency estimator framework consistency include group induce possibly overlap group decomposable inequality generalize weakly process rich unify decomposable pe tp tt notion random measurement nuclear norm recently general derive recovery gaussian problem extensively commonly regression generalize estimation addition convexity upon notion extensive area framework establish estimator close eq function sublinear ball dual supremum hence subdifferential linear attain sum property sum express close penalty express set geometrically decomposable dual e contain neighborhood summarize decomposable penalty regularizer notation read contain span convex complement arise either overlap regularizer geometrically decomposable decomposable call decomposable section marginal seek capture model may low twice continuously differentiable cost convex problem geometrically decomposable regularize possess decomposable model coefficient complementary estimate decomposable b b h check possess develop say suitable strong beta shall way assess decay size grow second strong rsc loss rsc usually take restrict notion unified framework strong notion restrict convexity rsc p design rsc result subgaussian design dependency norm measure component imply sublinear simplify sufficient tx cx say inactive predictor ideally like predictor unfortunately orthogonality orthogonality require convergence rate require g usually allow parameter relax result appear ball resp resp compatibility rsc derive consistency must select norm solution establish ensures construct primal primal dual primal dual pair restrict restrict primal dual first order problem assume satisfy rsc optimal ensure restrict dual primal construct satisfie restrict primal pair zero condition taylor expand expand obtain substitute rearranging imply rsc substitute restrict also unique original seem verify sign slightly converse summarize geometrically decomposable assume rsc eq proceed solve plug desire solution say fall q deduce necessity claim generally show deduce necessity rsc violate ar violate origin origin lasso estimator family hold return describe information rsc smoothness condition subgaussian random subgaussian norm assumption strong rsc pn pn also sign dual thing simple similarly check satisfie also proposition union claim valid least pn aa regularizer decomposable prominent example regularization modification linear sparsity translate desirable property tb lasso possess analog lasso assume rsc generalize nontrivial let set argument turn attention recall family form assume organized group group q decomposable easy fisher rsc ii satisfy constant independent eq consistency regular kn define solution cn g min eq group correctly common regularizer motivate multivariate response assume estimating nuclear although rank norm alternative term convex sublinear weak geometric structure subdifferential pe geometrically decomposable directional geometrically decomposable penalty thing simple observation weakly decomposable term convex property optimal linearize expect close unfortunately strong convexity solution however shall linearize recognize possess pair consistent summarize rsc rsc solution change original inspection know q inequality obtain reach state return low multivariate describe
dynamic signal characterize produce look power two function mainly account harmonic ii stochastic represent music function white situation noise mix accommodate structure component inspire thorough million subsampling computation variance stochastic real produce simulated compression compression advantage paper make treat tool dr dr comparative expert music quickly medium type music business become technique show section remark value interval measurement expression define square tell determined average wave around level sound wave record store mean analog digital digital numerically analog compact code sequence integer proportional spaced sampling quantization round quantization power encode stream full reference wave wave wave paper commonly power figure piece song song start soft band start sound depend time horizon audio engineering community approach window signal form mobile sound ms music center vertical dash von discover produce periodic sum harmonic discover intensity harmonic varied length sum latter signal represent report sound change peak localize frequency spread peak characterize component record pass hz signal resemble empirical acting process stationary acoustic hz complex ensemble group average energy produce hz hz demonstrate example show observation essential motivate derivative signal decomposition wave acoustic iii harmonic wave white mainly hence spectrum moreover simply ensemble impose smoothness want periodic harmonic allow decay certainly mix feature discussion example restriction need technical reason theorem nevertheless moment strong variation something impossible analogy sampling nonparametric estimation sample million observation since procedure computation subsample subsample variance empirical subsampling procedure randomization impossible explore none perform iii block feasibility computing load generality equally adopt efficiency study additive exhibit correlation bandwidth define correction despite term increase small fix mix process h minimizer square bandwidth achieve regression e correction compute treat automatically achieve aim computationally subsample solid plot refer subsample plot subsample vary estimate ms discrete fourier transform measurement window log spread suggest vanish shape log shape spectrum noise music error rejection remarkable diverse stochastic structure within music take variance fact quantization introduce offset observable replace formula subsample la observation length would empirical agree grow subsample practical song series even principle achievable regular computer introduce variant subsampling describe namely instead separately subsample simple bandwidth step subsample length residual term statistic course symbol subsample explore however make variant selection block length draw subsample random usual easily bx gx b mean subsample variance justify consistency directly contain unknown quantity quantile subsample also interest weakly distribution quantile adopt show constant nature scale moreover consideration effectively music dynamic variation range ms protocol long monitor audio ms default start value therefore ms default next statistic efficient framework allow dynamic root notice subsampling preserve analogy dr subsampling sound wave instantaneous propose measure dr million implement computationally estimation computation suggestion quantity take sample index kn uniform replace current subsample k b I make q act onto express wave scale onto suffice add observe sound instantaneous hence value dr median rather toward want issue dr concept discrepancy distribution concept advantage procedure fit quantization certainly quantization operate digital compression quantization dr compression assess procedure know stochastic reference expectation bias write reproduce world music signal perturbation datum two record add dynamic assess highlight dynamic measure introduce dynamic achieve amount digital label specialized record guide source various track call track play sound huge play volume cause consider track roughly song differently song uniform path track impact record guide final track length dynamic test compression dynamic parameter compression signal reduce output power quality processing level song total compressed wave track involve though subsampling still require considerable ms dynamic ms obey theoretical change track impact result however load considerably seed allow subsampling induce seed result moreover boundary figure confidence reveal behaviour dr ratios ideal dr value dr behave level remarkable discrimination compression none band band compression wide interval case long interval path band I information dr transition go detect compression discriminate quality record gain price actually quality version promise claim name critical trend compression record dynamic compression think dr digital song especially figure mobile fidelity record produce music sound work company specialize
measure update optimization update use project coefficient span vector step correctness analyze mkl abuse kernel minimize empirical minimize use loss small imply approximated state result relationship quantity capture mb define dominate component measure zero element lead capture extend definition among matrix na stand span define quantity take non closely span denote correlation subspace span relationship minimum non prove kernel generalization regression kernel geometric let solution nf nf deferred appendix lemma geometric parameter bind define kernel equation satisfied unique previously rich mkl algorithms generalization convex dependency worth generalization differ worth differ respect sample recently rademacher tight mkl decay unify derive type solution minimize loss I fa generalization since could increase bind assume generalization concentration np nf nf pf take f ready prove obtain f q hence nf nf nf nf nf nf p r q result develop mkl coordinate gradient geometric certain generalization quantization nf nf nf nf nf nf nf nf f f nf j nf r h j nf j nf j since nf case obtain nf f loop theorem nf nf k nf update safe always change value change notation nf k second cauchy nf k n kk nf k j nf nf f nf nf nf proceed j j z j plugging nf rademacher property nf j j nf use uniformly large r g lr complete east mi usa usa computer engineering east usa mkl learn combination small large pool lead prediction error descent algorithm geometric convergence gradient previous error rademacher multiple greedy coordinate descent extensively thank empirical application kernel svms kernel crucial choose decade learn develop focus learn kernel example algorithm kernel bind mkl several kernel mkl effective kernel error encouraging mkl multiple mkl combination classifier predefine simple combination analysis generalization iterative greedy mkl pool gradient measure appropriate approach able geometric knowledge achieve several explore error involve mkl bounding weight directly relate pursuit apply exactly mkl I regularization mkl convergence propose except apply orthogonal pursuit greedy gradient geometric improvement contribution paper baseline greedy gradient application show stage fail two two choose two kernel kernel weight select copy copy since case unique expect argument two involved one totally irrelevant prediction task greedy coordinate algorithm select kernel search optimal kernel smooth due rate show lie choose coordinate respect mkl special convergence spirit share convergence rate mkl regularizer nf kernel optimization performance appendix middle otherwise
memory perform equip ghz core cpu gb aim verify lrr significantly synthetic focus segmentation synthetic segmentation first similar matrix kn average create eps b fix fraction outlier lrr lrr recover row outli criterion satisfy tolerance perform lrr column substantial lrr lrr lrr trial figure figure fix fraction outlier show minimal experiment lrr lrr successful trial comparable lrr lrr experimental image subject feature note model image person portion face corrupt collection segmentation dictionary matrix lrr lrr set result coefficient affinity top singular vector subject embed feature vote cluster lrr segmentation cluster label majority ground average respectively lrr lrr subproblem face require lrr roughly speedup accuracy lrr quite relationship vision construction calculate strategy contaminate vision utility laplacian regularizer encourage segmentation rely lrr variety large semi comparison scale benchmark video video category audio visual sift extract video video minute video positive video video event work keep positive nan videos video extract six feature video frame accumulate obtain representation visual sift sift semantic among available image associate provide tag concept tag version image image wavelet texture moment visual form single feature image eps three construction describe exclude scalability concern work already demonstrate inferior near lrr run lrr large dataset three graph section lead performance gain across vast feature demonstrate benefit enforce orient e infeasible thus employ scalable primary provably accurate subspace aim scalable subspace segmentation correctness lrr provably preserve theoretical lrr moreover divide comparable obtain computational segmentation semi lrr lrr derivative technique develop approximation formulation thm deterministic guarantee generalize probabilistic projection uniformly nearly coherent present proof thm norm make use condition liu et al corrupt behave dictionary well observation equal comparable submatrix submatrix coherent solution high coherent proportional projection incoherence column sample uniformly capture incoherence replacement constant partition subproblem coherent event column realize equal median index equal establish therefore remains show lrr hoeffding lem choice submatrix coherent hoeffding far cr support complement projection thm parallel thm begin introduce guarantee column equal equality indeed thm oracle constraint end derive orthogonal version matrix addition establishe consider select solve subproblem develop let lem eq lem proof lem next lem lem save next lem lem unchanged lem remainder proof lem save coherence lemma lem assume satisfie condition note unchanged except meet token fu fu berkeley stanford university berkeley stanford edu edu computer berkeley department engineering department berkeley range clustering supervise lrr convex small massive vision past aim rank factorization constraint novel subspace cope lrr decomposable constraint maintain lrr strong recovery implication scalability benchmark recognition novel segmentation scale concept art order close subspace might physical object comprise face illumination object occur production recover basis image motion graph one formulation segmentation lrr liu lrr segment strong strong problem due repeat burden stem nuclear encourage hope work distribute improve scalability lrr unfortunately technique tailor loss matrix requirement violate lrr constraint arise decomposable factorization develop provably divide account decomposable lrr approximation lrr subspace scalability divide lrr computationally tractable subproblem subproblem lrr principle divide combine decomposable cope decomposable lrr characterize new showing maintain lrr probability substantial significant lrr treat rich lrr subproblem lrr goal segmentation recovery correctness substantial see detail face segmentation lrr achieve lrr novel methodology lrr construct affinity graph attempt fail size leverage lrr propose construct task event image demonstrate magnitude speed exceed remainder approach lrr next lrr section highlight efficiency lrr computer task real datum present application lrr problem value orthogonal review lrr subspace subspace dimension corruption introduce task projection row term block lie multiple subspace segmentation recover lrr approach seek space lrr guarantee correctness meet sec detail lrr well suited affinity node draw subspace thus sparse affinity scalable call lrr segmentation lrr principle new decomposable lrr next lrr partition column simplicity evenly lrr solve subproblem lrr subproblem rank dimension step lrr approximation diag typical solve factor submatrix generate lrr project commonly randomize problem complexity truncate svd lrr reduce significantly subproblem relatively lrr solver return factor indeed complexity lrr maintain strong make technical require lrr
capture point relational integrate modelling node copula rest full use distinguish correlate recover htbp analyse three co mit reality mining ten fold datum datum train score material table show uv cccc mit proceed conference co activity year regardless co randomness manually eliminate dataset advance thus copula model mit reality subject proximity indicate proximity minute therefore asymmetric business school student student student portion encourage indicator obtain social study locate dataset basic network element label copula framework intra introduction copula pair membership indicator real copula incorporate effective predict miss analytical bivariate copula theorem blockmodel social relationship exploit social despite powerful indicator node know people membership incorporate individual jointly membership various copula marginal indicator interest detail number category show superior world community model topic include social medium discover interaction propose network group base wise directional membership community however social may well capture amongst role play facilitate phenomenon assume feature node use represent use difference interaction community community node membership distribute drawing draw final compatibility matrix row propose extend community restaurant build incorporate indicator pair limit indicator many social correlation towards category topic compare view towards intra introduce intra important time rest accordingly copula form copula subset pair maintain indicator distribution copula need copula impose copula update accordance analytical solution marginal indicator play core also new multiple addition varied relational blockmodel evolve incorporate infinite focus static blockmodel varied set rest article organize notation copula far provide base use real world social notation description supplementary htbp node discover directional interaction indicator receiver mix r ij l le k n parameter copula htbp dp ij ij mean phase phase generation common parent non choice criterion commonly uncertain break uniform membership indicator copula express appropriately membership indicator pair scenario copula large value membership indicator pair positive within encourage pair beta conjugacy become beta table discover community formal unfortunately bring mathematical condition variable collapse p ij ij explicit integrate q k rectangular outside note use marginal remain uv alternative collapse integrate independently leave classical ik ij ij ij obtain ij k similar page ba k u u ij e uniform ij v ij du copula independence recover high around develop variant copula concentrate intrinsic multiple dirichlet membership indicator therefore copula obviously dp estimate graphical classical model refer mixed membership computation vary copula extra sampling operation htbp slightly synthetic comparative general measure benchmark cccc p partial e type additionally approach comparison make follow
intersect desirable computationally demand product analyze provably succeed additive subspace subspace intersect distinct even massive subspace sufficiently simple outlier introduce succeed additional datum point result letter letter stand spectral ms r l l l x cluster briefly assume e outli scheme discuss point either comparable set choice find eq let entry j subspace heuristic adjacency normalize belong subspace guarantee correspond even hold strictly belong subspace correctly recall input analytical matter analytical guarantee impact relative subspace subspace j l l point obtain term subspace l k ks l lp l j nd cn c constant sketch thm separately succeed affinity impose restriction subspace success thm rhs thm states succeed sufficiently reflect distinct subspace assume favorable situation orthogonal inner first thm k lk j e j satisfy sufficiently ssc noiseless analogous namely comparison inner ssc employ find light interesting guarantee hand essentially identical bound replace note weight thm conceptually outli outlier lie outlier imply normalize trivially accomplish outlier scheme introduce noiseless q choose suppose choose correspond l n j le n l misclassifie due thm outli succeed exponentially I ambient noiseless remark outli even outli seem total equal orthonormal vary accord result succeed style cm vertical sep cm edge bottom bottom edge width height font min meta max ce file ce file ce file ce dimension subspace vertical axis em definition corollary proposition conjecture remark dimensional subspace outlier subspace unknown probabilistic subspace algorithm succeed case subspace intersect reveal tradeoff affinity subspace detection introduce succeed dimensional outli association dimension identify outlier assignment association straightforward extract approximation l subspace disease particular vision e motion vary illumination numerous intersect ls deterministic performance report lrr succeed provide subspace intersect
residual objective method obtain approximate objective detail important robust robust measurement trend sharp smooth incorporate aspect entire extend develop numerical model simulate conclude height marker axis axis line middle scale height marker middle height marker line line matrix n mu mean freedom definite laplacian student distribution tail discussion influence mean decrease eventually student student exactly kalman advantage tail scalar log statistical treat big cauchy proportional heavy tail fundamental advantage particularly application trend present kalman initial know constant measurement mutually consistent measurement interface briefly characteristic heavily student freedom track change state residual model mutually smooth find map general modeling residual discuss matrix notation make distribution sake across ability process measurement student assign penalty innovation residual residual gaussian student denote minimize degree approximation direction valid minimize solve qp subproblem matrix block computational result wide kalman index within density use residual explicitly approximation performance subproblem numerically stable know specialized robust student residual q hessian place density newton hessian general use information fisher student freedom approximation implement term present term random hessian fail hessian overcome drawback middle hessian rough approximation gauss incorporate size residual rest provide upon begin write composite f definite hessian reasonable take instead similarly index provide globally strategy outlier proceed shall exploit structure iterate solve eq semidefinite continuously smooth direction convergence stationary spirit term rely find subproblem optimality composite subdifferential fx subdifferential equivalence modify information yield semidefinite continuously define idea motivate gauss newton x nn sequence overall termination step counter gauss step dx terminate search set iterate return generalize term newton framework define sequence occur terminate finitely every subsequence none occur subsequence sequence know generality compact modulus rearrange since cluster generality nn must possible point bound terminate finitely satisfy f hold hold x bound sequence suppose unbounded subsequence tu limit contradiction smooth function twice differentiable immediately sequence subsequence subsequence subsequence argument arrive guarantee since twice differentiable boundedness establish boundedness satisfied denote necessarily contain sub matrix coordinate finish matrix g individual produce bind condition necessary smoother kalman smooth smooth ground simulated model integral white noise reconstruction bayesian cubic smoothing give freedom set generate nominal contamination present mse estimate smooth well optimal smooth always smooth contamination student simply decrease come case measurement contaminate uniform display notice spline interval c c outli mse mse student nominal reconstruction smooth laplace robust dot contaminate model uniformly xt solid symbol visible outside axis limit solid truth dot smoother dash line show dot plot bottom axis present van detail couple ode model give simulation ground euler n x k n realization gaussian smooth smooth nonlinear nonlinear laplace advantage extreme figure come nonlinear detail brief overview application laplace smoother qualitative robust smooth laplace smooth smooth outlier track target pilot place wave smooth sound four bottom location place track independently verify system pressure formula k east depth derivative time four measurement bottom pressure measurement measurement depth measurement deviation track gps verification thick smoother thin large gps verification thick line residual laplace smoother thin smoother thin smooth removal east depth give deviation process east north depth component conditional x zero smooth outli removal three peak east north removal three fit show laplace removal appear track use near gps use axis gps tracking time depth fine east gps tracking validate robust smooth fit robust presence smooth laplace track smooth outli removal east coordinate residual great deviation outlier limit removal outlier second fit peak enough influence result removal track laplace particularly depth reliable frequent robust smooth smooth previously behavior smooth proof two monte study sec root mse trend panel single run estimator experiment one jump wave panel reveal superior trend smooth panel estimate smooth line jump job smooth trend smooth solution rest nominal perturb bottom reconstruction trend thin application previous highlight trend filter track already build strong knowledge bad measurement aware sensor reliable subject contamination incorporate flexibility section gaussians innovation residual index gaussian different measurement transition process sensor sensor frequent subject contamination sensor rarely subject interface implement student outli direct measurement interface rather specify set contaminate contamination level measurement every result plot symbol ground show least smooth panel robust result track jump estimating jump panel obtain student double couple sharp double modeling residual plot frequent reliable represent appear axis range plot limit black show red least error outlier process effectively sharp double student follow smoothing residual error robust double framework efficiently state independently outlier robust smoothing track state contribution sparse advantage tail force non solve associate map challenge optimize objective even system contrast require iterative still within detail non laplace smooth optimizer outlier numerical initial experiment
n outer outer outer outer second loose loose phone phone total cut plane result instance set view two aggregated master generation bundle cpu remain number time computer pc core cpu memory machine run score hence multiply cpu moreover implement art conclusion outer table even generation bundle level instance almost twice justified instance overall level although latter total indicate solve flow apply context package network arc network arc hence generation flow describe cost present primal dual generation life set arcs demand must arc underlie incidence associate demand assign assign arc exceed arc addition depend linearly arc formulation typically life situation arc large solve outer never arc htbp act outer grid grid grid grid grid e grid grid relation available method master addition active show column percentage arc capacity active terminate act outer second cpu spend report approach scale cpu intel iv ram accord benchmark provide machine whereas machine score overall informative aggregate observe considerably although quickly become cpu instance htbp instance act grid grid grid grid grid computational mkl flow demonstrate wolfe generation convex include thorough involve address generation cut plane application namely aggregate master unbounded subproblem present extensive broad previously worth study involve extend problem define oracle acknowledgement study thank suggestion support foundation remark cl university dual general purpose generation master interior point allow suboptimal literature call oracle cpu typically column optimal small behaviour broad namely solve problem life context multiple problem flow publicly benchmark instance method result date suggest offer alternative specialize competitive large scale keyword cut plane method programming column generation iterative master price dual dual simplex solution variation typically generation active method degeneracy may affect generation drawback cut plane dual modify add purpose limit dual solve interior point column suboptimal dual optimality solve restrict master loose dynamically guarantee original encouraging relaxation stock window lot wolfe compact extend convex relatively master bottleneck hand address master oracle easy operate address implementation variable technique scale address contexts datum network contribution paper gain past describe solve publicly column generation cut literature address software available remainder generation outline cut prove deal describe mkl three section report computational compare art summarize outcome variable empty presence constraint specifically suitably partition partition describe later index extreme extreme wolfe consists rewrite ensure combination solution mp solve impossible moreover costly iterative aim extreme add start call iteration terminate guarantee solution mp mp extreme subset every outer add remove extreme represent mp component optimal mp use constraint dual feasibility mp mean cost mp oracle pricing subproblem possible unbounded extreme associate take ray rx eq obtain column feasibility solution mp solution use correspond extreme point dual feasible consecutive solution begin moreover optimal contribute close termination drawback similar observed cutting recall technique successfully master formulation ray separability reflect master master convexity situation aggregate master hence master master extreme extreme replacing master problem require exploit separability obtain decompose formulation use generate ray add subproblem one make master namely multiple problem aggregate master point principle general linearization mean form belong result observe approximation depend include define follow bound base finite hence convex linearization describe closely desire choose appropriately denote could closely master since typically column outer subproblem subproblem obtain master master apply idea variety cutting briefly prove effective address section describe variant cut interior rely price dual point analytic localization current localization dual space half space rely localization prevent unstable contribute deep nonlinear theoretical bundle cut proximity control bundle prox differ sequence iterate instance iterative rely piece convex give finally prox subject level branch problem find sub reduce observe primal mp call satisfie eq tolerance interior keep product center barrier dual restrict master tolerance loose column lb sp sp column mp outer valid provide reduce close monotonic upper denote tolerance describe two cut plane different compute similarity typically x similarity accurate cpu several multiple problem mkl thorough comparison context infinite development generation closely development keep primal mkl column generation component several art focus kernel margin problem structural hyperplane margin describe formulation single map dimension verify classify good discriminant associate misclassification vector classification aim distance discriminant function distance keep margin possible value importance first lagrangian optimality function relationship lagrangian notice dual due approach different kernel weight train svm benefit since consider discriminant describe kernel definite map mkl problem semi programming reformulate solve sequential optimization kernel kernel point instead interpret combination ii maximize discriminant misclassification associate write eq quadratic typically become scale nevertheless effectively linearization derive master development boundary convex interior function attractive interior development master direction minimize generation opposite observe master problem master formulation linear mkl pricing subproblem dual addition problem associate subproblem subproblem turn form single problem solver sp evaluate carry uci repository pricing toolbox experiment replicate kernel unit trace accuracy randomly generate instance stop duality drop stop criterion application optimality intel ghz cpu gb run infinite cutting plane simple subgradient descent method method norm extra constraint weight solve method look stop gd direction every time weight ex gd svm gd gd gd gd table name show cpu time second call svm call correct classification make discriminant solve average included present take result give ghz cpu use cpu indicate regard last call solver particularly compare gain use least high level accuracy kernel combine kernel svm solver due iteration solver translate gd instance purpose approach describe extended method art mkl belong family bundle consider gradient problem mkl data per ccccc kernel breast heart call observe database decrease small solve less average characteristic intel ghz gb ram keep appear obtain demonstrate art seem instance vanish effective additional experiment aside influence solver implementation toolbox subproblem variability time spend solve bottleneck solver unlike building cut plane master solve exceed tune svm solver implementation respect stochastic last decade currently stochastic programming formulate wide life refer program possible scenario cut plane deal pose optimize stage random hand side additionally represent define variable realization realization occurrence possible scenario
intersection hyperplane specify rewrite lagrangian multiplier define write side simplification precede p rewrite plug compute show procedure screen support vector list compute add compute precede subsection since accelerate easy theorem theorem assume sample primal vector problem enforce inactive multipli tucker reformulate problem l lb apply minimum minimum minimum minimum dual regularize l svm define reformulate primal formulation l relationship use close form l rewrite definition since q enter large necessary condition matrix regularize remove less inactive cost speedup construct set close differentiable close assume reason tight contain define know follow respectively substitute obtained define dimensional satisfy obviously region show indicate area space let respectively change generate theorems radius reach center project minimize also rewrite theorem hyperplane keep matter tt enable derive form solve precede since decompose problem therefore follow multiplier write correspond tucker kkt bound clear study list p plugging verify satisfied summarize case accelerate expensive computation accelerate utilize sparse figure arc red blue eq function write plug lead simply equation take side simplify obtain notice plug kkt specify eq
within engineering mm mm proposition analysis introduce paradigm analysis subspace consider group vary subspace combination eigenvalue project subspace give capable deal skewed increasingly demand across familiar biological application introduce herein technique illustrate performance reduction model cluster discriminant projection subspace introduce discriminant direction year mixture summarize combination original adequate herein address heavy tail generalize advantageous appropriate extreme clustering lie inverse sir datum consider identify sir covariance member combination importance via observation project capture cluster remainder outline present background outline dimension select combination contain subspace provide suggestion future work note herein carry use cluster base dimensional arise parametric f g component mixture half approach due dominate finite give past year non model asymmetric skew skew variance work herein feasible exhaustive suffice clustering become rich introduce wind generalize appear effectively extreme risk management application normal description reality multivariate extremely contain mixture hmm index parameter location g kind function density sometimes issue asymptotic expansion polynomial parametrization inverse eq relationship full parametrization section method unsupervise cluster supervise supervise discriminant relationship I estimation supervise learn label estimation label estimation cluster scenario none observation membership I cluster introduce membership observation generalize carry algorithm iterative estimate incomplete em complete miss ig iterate reach expectation maximization expect log extensive asymptotic namely q cluster criterion determine schwarz maximized likelihood represent observation membership posteriori model partial supervise analogue receive literature past year author demonstrate excellent real analogue similarly receive less classification membership label carry view arise discriminant observation first label observation form result membership discriminant approach class analysis herein restrict cf component know number consider flexibility make model latter investigate part reduction within mixture component gaussian find capture direction variation covariance pool carry package software ten parsimonious cluster constraint eigen setting cluster eigen analogue family general unconstraine degree define develop shift mixture g g third index previously cluster datum introduce herein dimension development analogue herein recently discriminant generalized mixture subspace covariance vary note covariance density mixture pool cluster cluster covariance direction span direction obtain eigen projection projection projection em eigen decomposition associate practical greatly direction linear offer selection feature follow subset bic good cluster value fit space usually feasible end employ greedy local search however backward main initial select difference well assume step amongst maximize difference iterate bic variable cluster framework consideration discriminant analysis modify function via hmm direction direction eigen compute project greedy discard return none discard herein step herein analyse true class classification class adjust rand correct chance agreement agreement take perfect agreement correction lead ari account ari correspond ari bad random first employ simulation two scenario analogue proportion sample ii adding discriminant result know vary appear ari discriminant analysis observation table generally ari cluster analysis slight performance noisy ht avg ari std ari avg feature avg ari std ari avg avg ari std ari avg ari std ari feature ari std ari avg avg ari std ari avg feature table demonstrate excellent discriminant helpful case run scenario scenario equivalent would run scenario compare dimensional generate three component distribution random normal multiply small iii difficult ht perform complete converge numerically ari ari avg avg comparison massive advantage draw structure repeat restricted course implement performance addition real eight method outline except mean model analysis dimensionality robust principal pair mixture eigen family principal loading projection pursuit computation well parsimonious package discriminative employ mean r function mixture procedure analogue mixture carry r package analogue scale agglomerative vary analogue decompose shift asymmetric laplace mixture eigen discriminant analysis use simulate membership otherwise sure move unknown varied utilize gaussian mixture discriminant analysis procedure agglomerative choose classification ari consist roughly number validation six leave top bank ari method method datum perfectly produce classification observation however perfect discriminant note feature ht ari component ari notable return component interesting component less surprising include mixture case another non discuss originally available fit perfect result htb height upper ari ari component da da record chemical physical package misclassifie reveal perfect classification ari discriminant paradigm ccc ccc ccc ari ccc ari ccc ari class da da da illustrate histogram direction clearly structure misclassifie give additional clarity ht breast fine
raise number facilitate descriptor pair scene comprise non previously benchmark constitute excellent self learn supervise transfer input structure patch around randomly scene allow self learning represent million patch scene individually see differ penalty setting compute gradient log method ascent minibatch ascent find train good minibatch gradient scale image subtract divide practice visual reasonable find preprocesse good identity originally mostly important want generative hide initialize bias minibatch epoch minute gpu patches unit drastically subtracting follow whitening stochastic train architecture start epoch architecture constrain denote architecture concern compact representation initialize take neighborhood factor gpu take hour architecture evaluation every denote label train scene roc term percent incorrect match find respective pair distance incorrect match nd brief nd ex sift ex nd ex nd nd denote descriptor line scene denote scene nd unsupervise method unsupervise restrict descriptor compact brief binary descriptor descriptor binary supervision number brief rate limitation place overall memory activation descriptor see descriptor form activation eq accordance manually descriptor rely e encode explicitly correspondence metric resort distance widely follow choose normalization patch vector conditionally bernoulli jensen shannon sift point detector descriptor performance kind vision serve evaluate sift descriptor sift descriptor report normalization sift perform descriptor certain difference peak order achieve optimize dataset evaluating descriptor descriptor sift half albeit cost descriptor representation descriptor comparable several art descriptor supervise see entry aspect considerably version second expect e consider problematic evaluating sparsity evaluate scene versa evaluate dataset observe architecture opposite scene much jensen shannon around report improve table entry scaling learn compact input multiple layer descriptor make compact suitable employ use histogram briefly comment filter column look computing scale resemble center location every build center projection figure qualitatively focus filter systematically arrange several filter filter place get systematically arrange around train nonlinear hide autoencoder autoencoder model stack rbms autoencoder start paper suggest feature find level descriptor image benchmark evaluation exist demonstrate real future deep convolutional feature moreover correspondence boltzmann continuous bipartite hide configuration r visible bias precision accomplish training cd approximation log bipartite nature aspect visible unit similarly unit compactly function rarely active encourage unit penalty represent strength feature hope model dependency triplet filter connect map hide element conditionally independent visible visible unit equation denote energy visible unit set learn estimation via sample involve inversion instead use hybrid hmc free de range modality computer unlabele utilize train algorithm never unsupervise image supervision unsupervise problem special restrict boltzmann machine rbms perform hand descriptor produce descriptor tackle computer viewpoint unsupervise benchmark evaluate subsequent supervise aggregate think assess direct
string enumeration incorrect enumeration enumeration string content output correct content since pe computable learnable construction family family omit simplicity concern fix pair partition convenience column remainder construction define list let computable enumeration triple number large label except say family depend stage indicate wish member record member set ensure ki cs ki ki ki label way infinite triple consideration recent two interpret label ki number choose member label contain set index yet assign suppose suggest currently kn cs kn h sp kn k kn far great kn pe stage label consequently set machine output label gm string content sg subsequent stage consequence member learn fail computable code construct reduction proceed describe arithmetic aware illustrate serve give computable complexity family learn specifically state enumeration family enumeration hypothesis hypothesis yet state characterize code satisfie formula learner hypothesis converge computable enumeration fail formula fail description completeness prove proof utilize learnable learnable description learnable hard code reduce preliminary section enumeration choose map enumeration column number nonempty enumeration versa identical finitely distinct identical subset consist subset consist collection finite set infinitely unary define new map reduction computable map rx let subsequence imply unbounded recall learnable learnable consist entirely set every contain less finite whole learnable learn finite let output content fed enumeration family either appear learner set may hard upper interest right arithmetic complexity learnable computable learner enumeration fail learn computable enumeration incorrect answer enumeration statement satisfy statement b shall computable enumeration enumeration produce enumeration construction fail yield shall use build enumeration complete enumeration counter length enumeration segment appear code distinct mf sx string k variable beginning type occur wrong wrong xx action stage pair increment equal stage hypothesis change finitely construction hypothesis include hypothesis infinitely since learn therefore use enumeration computable enumeration extension choose canonical manner enumeration construction eventually select two partial agree segment first disagreement extension stage perform enumeration computation enumeration learn partial g prove claim must enumeration apply complexity half result let learner set computable function let formula nmf correct later segment enumeration define formula segment possibility hypothesis nf two stage initial change sf sn f I f verify enumeration segment enumeration long hypothesis yet output learner output segment converge hypothesis content hypothesis later learn family low prove computable learnable learnable learnable stage stage step step learner complete fed explicit input initialize maintain column set marker complete member reflect next least find pick marker column stage string member b ab end construction possibility infinitely many subsequence compatible computable fail correct complete equal content last nonempty must code enumeration begin learn depend outcome construction machine infinitely complete code input string output succeed step define one content code otherwise code code derive greatest simulate stage tag exist complete enumeration suppose construction never satisfied satisfied update reflect addition construction always finite replace case tag ever never always infinitely step note enumeration learner enumeration often never satisfied eventually hypothesis set succeed wish make computable apply consideration computable learnable define conclude learner code contain learn thus learnable string enumeration fed eventually tag identify appropriate enumeration capable provide hard final lower bind description description description code learnable learnable initial computable enumeration family identify computable enumeration family computable learnable learnable uniformly learnable fix x e indicate complement identify complement computable pair call maintain pair search hypothesis equal string pass current stage currently four status two availability output hypothesis string enumeration neither yet pair content pick set least increase enumeration content next specifically marker inactive currently pair even number construction far great verify must verify statement learnable computable learn pair marker statement true prove second enumeration content member symmetric co odd infinite stage finally exhibit family family consideration consist either complement code f gd remain member mark marker succeed pair set remain unique marker segment enumeration either marker marker succeed identical exception justify prove completeness completeness numerous criterion arithmetic complexity question ask question candidate upper place observe decrease theorem theorem arithmetic learnable criterion notion anomalous completeness independent uniformly family learnable enumeration failure member amount class address mathematical endow consider model read computable enumeration take input output interpret describe read enumeration output
active ts main setup study empirically section present asynchronous start conceptually algorithm network equip operate go batch pick active current end select pool central server locally node select learner passive initialize q fp x active learning property generality rate operate coin determine ask formal use take unlabeled passive collection label example return update easy implement suffer drawback somewhat usual synchronization mean asynchronous offer drawback asynchronous version maintain node stream process example select active learner update arrive node delay produce appropriate instance training support machine model require train become consider decide example pass actual weight select example learn yet comparable since sift processing current cost online operation use execution volume passive active execution scale select example speedup sequential active machine benefit speedup parallelization neural computing dominate training speedup active update error delay example communication delay small delay delay negligible analyze performance selective use delayed update establish generalization substantial degradation long delay delay delay label learner formally describe delay weight example example probability everything constant initialize ts al change delay though apparent convenience delay take excess see match standard interestingly delay probability example delay batch size collect batch query use phase time expect dominate update rbf apply passive minimize fashion successfully successfully mnist datum albeit different modify active query probability obtain example correspond svms cause instability update change leave present distinguish example train approximately error report mnist test variant trade variant active parallel learning setup measure parallel show passive case substantial passive aim parallel enjoy delay update run simulation batch strategy cc visualize parallelization plot passive delay perform update get since large decrease obtain substantial go subsample parallelization ideal demonstrate network neural activation nod logistic input pixel scale vs gradient update stepsize rule subsample still mistake reach mistake expect reflect right gain beyond subsampling subsampling present design leveraging mathematic strategy effective remain effective rely particularly attractive effective experimental parallel sound similar gain analysis proof shorthand still bind start reasoning triple indicator history prefer choose xx analogue et pick analogue rejection p identical proceed induction trivial inductive inductive follow definition need inductive bind early combine yield statement lemma generalization statement need misclassifie rely base assume apply statement generalization bounding carry unchanged inspection proof lemma apart sequence appropriately statement address title microsoft york ny usa microsoft com number label generic search informative rely particularly attractive report preliminary last decade grow machine body successful optimization machine variant argue aim exist optimization exploit structure extent beyond I kind rely separate focus communication emphasis run complexity cover broad employ communication perhaps support set parallelism sift
adjust graph search ghz ccc label vertical four base number denote fig correspond row blue circle propagation clearly superiority nn graph algorithm number neighbor become large high imagenet divide tree division accuracy adopt neighborhood superiority become low second compare method achieve force sift approximate indeed present image rank good evaluate face face face require first nn dataset image dataset nn conduct neighborhood propagation nn label neighbor evaluate face label proportion face discount top various rank comparison observation neighbor hyperplane height side probability hyperplane also validate similar manner lsh slight tree discover discover true pass stable easily w ph I pd ij pt probability absolute large case use build relationship discover previous tree partition fail discover multiplication neighbor discover h I relationship discover simply multiply get neighborhood discover way discover accord lemma second discover partition discover ij tree neighborhood face organization distance face near clearly enhance object detect mm microsoft com nn play role increasingly popular drive task yet challenge scale nn emphasis accuracy build achieve base repeat several yield enhance accuracy efficiency deal large bioinformatics internet search drive organization object synthesis retrieval two near neighbor nn graph node two connect geometrically motivated hence suitable show especially construct construction pair denote denote slow application early effort complexity respect super linearly make impractical large scale research turn solution build indexing datum regard locality neighbor vice neighborhood methodology recursively divide merge subset merge neighborhood approach suffer overlap neighborhood graph paper approach nn emphasis justify theory scale vision divide randomly subset neighboring subset base neighborhood partition times chance neighbor connect partition differently number propagation local neighborhood wider achieve several repetition partition discover neighborhood propagate range division scheme superior term performance require neighborhood cover graph high divide worst constant propose exponentially quite high calculation calculation dimensional high case rich literature dimensional nn neighborhood nn graph near indexing structure usually tree rp locality hashing point suffer accuracy method unnecessary effort give query make efficient graph predefine overlap ratio subgraphs subset follow neighbor propagation refinement discriminate distant point contrast reasonable approach additionally subset together ratio balance randomly tree search perform expansion neighborhood fast use ordering method work build graph inefficient choose application division exploit boost index many tree differently propose technique build graph e formally nn graph l l present neighborhood propagate wide range achieve neighborhood group correspond divide form random nearby hyperplane divide set conduct division value manner adopt subgraph division overlap take subgraphs take division base serial isolate subgraph unable connect point lie exploit neighbor consider division interpret neighborhood identify division union neighboring write cover represent quality combine base graph division adjacent combination essence subgraph fig division yield isolate division yield isolate subset serve bridge connect isolated subgraph construct serve role connect subgraph implementation many sufficient make well subgraph number random large although neighborhood division progress toward slow shown rate point neighboring become suggest neighboring expand situation neighbor blue light propagation access identify gradually expand neighborhood queue near queue neighbor queue discover stop queue visit reach visit consider one process theoretic neighborhood proof hyperplane point one single discover discover tree discover lemma true newly increase theoretic justification neighboring point neighbor generalize intermediate neighborhood propagation kn discover true neighborhood discover neighbor discover neighbor stay grow discover keep neighborhood advantageous cover neighbor true discover least point discuss denote visited write name overlap overlap suggest choose direction diameter subset theoretically tend direction generate principal point compute direction principle
task individually summarize contain part converse sufficient necessary develop nontrivial specifically certain condition design satisfy problem union sharp capture support property also advantage individually task support differently k single lasso individually task task union recover individually case different union differently view response scenario task problem individually via need recovery via multi lasso offer benefit sense result task design matrix across task need per task nontrivial generalization consistent multi performing mention regularization threshold characterize lasso able characterize sharp jointly markov network sample base relationship node sample contain dimensional vector variance eq component represent node hence provide neighbor selection lasso provide neighbor justify adopt regression sufficient regression across task variance kk ks interested union adopt lasso linear k couple regularization characterize multi task problem introduce notation ex ex ex ex ex ex ex jk case operator kb ib support union denote index nonzero represent convenience use contain column index row convenience matrix notation contain give matrix min define th function matrix depend quantity c result regularize recover converse e union implication identical matrix ex ex max kk max parameter p define require sufficient recovery support regularize consider parameter union condition size recover regularize fail recover asymptotic regime condition recover proof theorem provide quantity threshold recover threshold support union behavior asymptotic regime provide recover union play role sample union analyze representative lasso regression jointly individually represent capture problem comparison lasso need recovery denote matrix entry study identical vector k correctly support union compare recover set individually see involve covariance feature share task recover per reduce factor grouping task regression view generalization task suggest task benefit support task fact arise design differently distribute reduce moreover set via task advantage appear task task vary support jk bound corollary version single disjoint q assumption need task lasso multi disjoint set advantage multi lasso vanish task support disjoint set corollary extreme share set respectively recovery union goes need single lasso extreme various level correspondingly support capture demonstrate behavior via simulation lasso union study regression x p b vector proportional dimension regularize support union ht ccc task need recovery consistent great compare lasso need task interest correct recovery next support equal pe x b plot exhibit although need task demonstrate level task affect correct union task task pe integer pe pe integer pe p pe pe pe pe pe pe pe integer pe I scale overlap extreme task share correct recovery ht ccc ccc experiment overlap equal across two task support let pe pe pe pe pe pe pe pe j precede keep plot support exhibit level across careful perturbation overlap entry require vary task influence result otherwise odd experiment fig size overlap overlap fig fig fig vary across proof framework base primal model develop express proof mostly bound across need tight next detail tucker kkt necessary solution follow provide suppose exist subdifferential kkt completeness appendix optimization eq subdifferential subdifferential jointly problem kkt obtain satisfy kkt follow guarantee uniqueness solution argument proof lemma proceed characterize j k ss tn cv jk cv kt following eq provide appendix j q equality thus appendix provide b later order high evaluation kk obtain central lemma q combine evaluation existence uniqueness optimal true guarantee eq suffice guarantee define q large step ex ex ex n distribute random precede ex apply set bind combine bound kb n derive eq probability multi development l already j use quantity assumption min proof hold suffice guarantee l max remaining take matrix side cs sm gaussian inequality standard find sufficiently inequality assumption c jj cs furthermore enough low converge v conclude characterize sufficient successful linear characterize union far social first follow derive n x jj substitute side furth substitute condition convexity partial column c c ss tx cs tx q ss k k jk k start large follow bind large derive follow tu conclude eq large summarize simplify quantity definition section useful bound spectral norm proof let distribution unit ex ex
online learner operator show regret gradient choose descent bipartite hypothesis hinge loss nature function function sub terminology receive update notice bind separable cumulative algorithm online constrain keep keep buffer exceed predefine history bipartite idea reservoir buffer maintain reservoir give buffer however cumulative grow chebyshev inequality use analyze risk exponential natural similar buffer learner buffer evaluate buffer buffer believe random buffer reservoir leave maximum example theorem similar completeness predefine sufficiently similarly extract hold choose sufficiently buffer fast finite buffer analog update weight update buffer use buffer buffer sequence eq describe algorithm th instance say pairwise condition buffer prove suppose buffer keep buffer round variation consequently technique past decade numerous seek agree far parameterize semi work metric propose analyze regret analyze batch space definite pairwise function hinge loss loss must loss descent empty work online learner projection bound give therefore subgradient psd th receive weight algorithm hypothesis notice apply learn general believe guarantee paper learner bipartite learn pairwise loss demonstrate applicable work perspective maximization easy seem want store build buffer explore buffer improve theoretical current bound achieve tight mistake buffer convergence investigate necessary buffer dependent sequence utilize partly nsf bound begin chebyshev n resort variable fx nc first vary concerned whenever great proof inequality take let eq variable combining chernoff random define next arbitrary online working sake hypothesis brevity write term therefore know tm equality hold define eq particular otherwise combine desire rewrite term follow martingale see nn n n n e chebyshev large variation change vary variation n stay buffer round example equal complete proof claim rhs follow thus q rhs already variation apply inequality get reasoning put department usa crucial build system receiver roc generalization bound provide dependent risk computable demonstrate bipartite natural use secondly online bound pairwise bound bipartite metric learning example independently draw measure find generalize small expect pair h ranking application problem ranking predict order call rank rule high vice versa pairwise indicator example wrong hypothesis ranking amount score example another come learner share close one far away extensively decade entire advance give generalization bound algorithm derive quantity idea classification bound algorithmic closely relate rank rank list author quantity empirical give empirical bernstein perspective learner receive instance predict accord reveal learner decade online algorithm study extensively learn set mild assumption learner refined ensemble possible realization generalization bind derivation h martingale difference thus inequality long course slightly online perceptron modify analysis area buffer size inferior retain algorithm round exist fail I question generalization online family loss round denote average sequence select online choose lipschitz loss definition section result online demonstrate analyze online perceptron loss separable compact space two separately part term martingale hoeffding term tm tm th j whenever side see rhs nn e chebyshev variance variable vary concerned see j bound complete proof rhs covering reduce deviation sequence rely resort bind I q bind start bind hypothesis next see lemma hold every replace discard hypothesis ensemble grow term put substitute tool good online small closely follow one deviation result chernoff argument convex appendix q minimize discard follow main hypothesis work pairwise theorem choose label example auc randomly batch analyze combine risk linear expectation use
notion regularity class dirichlet form fast quantity display tend regularity closely tie base complete picture geometrically support ng tend complete tool establish appeal general theorem establish number leibl result posterior perturb section strategy concentration early construction subset yield favorable show place cover wasserstein dirichlet measure new chart theorem lemma include bold upon build crucial sep sep em lemma thm n edge edge edge edge edge edge edge double double double double posterior question concentration number interestingly address variable grow hold method may problem non atomic borel narrow function space wasserstein wasserstein existence coupling achieve infimum theorem existence monotonic wasserstein every b highlight denotes value random g quantity element conditionally discrete measure admit abuse conditional g generative eqs integrating distribute repeat jensen simple establish leibler relate g c h r assumption suppose g g k note g property normalization j display similar log ratio n support state r na na c ia g entropy get gaussian multiplying depend kullback give result probability ball define wasserstein atomic packing dd gamma assumption exactly support proof defer asymptotic shall tend infinity long construct class density eq condition constant conclusion constant depend indeed consequence condition immediate conclusion allow discrete note boundedness support take small nc h h g third display simple satisfied study property boundary set measure typically variational purpose subsequent development robustness measure test give borel primary consideration condition iv nontrivial consider measure first suggest g case possible guarantee arbitrarily remain study regularity boundary base measure subsection extend geometrically c kb ib ii x g c display kp define proof sequel p b recall entail I q r eq soon short r display essentially display inequality r inequality display r conclude establishe argument technical defer q regular boundary regular base infinite amenable measure ordinary defer purpose bind establish quantity rate density worth easy opposite g proof existence suitable algebra desire existence basis establish estimator admit k sequence converge likelihood exist n assumption suppose cf eq display bound check prove tight part immediate consequence useful due dependence ready key suppose vanish sequence hold vary c ng g eq event observation q n q soon multiply proceed similar way empirical square deduce convergence convergence unfortunately kind posterior k rate df n basic calculation main lie fast rate take measurable cf ga first third assumption g random surely couple equality deduce g simplify define kullback g f q f kf g c b fy n fy g g fy fy dy fy fy inequality r jensen root respectively possible optimal coupling immediate measure f reduce hellinger wasserstein utilize c ec ec display bind part bind dirichlet process may distribute stick break p fy p fy f determine break display theorem addition q formally configuration set partition identical well know property stick distribute ease kernel nd n multiplying extend crucial lemma present end claim dy n dy dy hellinger compose net marginal density support part logarithm cardinality remain complete generality several step expansion let side inequality n may coordinate depend configuration distinct coordinate probability measure although match element entail nr distribution display square gaussian bernstein exponential constant construction measure toward nf hold dim nf nd nd nd display pick bound simply multiply invoke say density common space universal paragraph set display combine rd ball point write g argument side conclude g g proof side display long g lemma arrive dirichlet support admit derive value set consist whose equal wasserstein end construct radius ball q k partition subset small element one say q number ball need cover p k part refined cover metric element k ba q display consider cover uniquely remain index accord distinguished display kk kk show consider wasserstein q natural tc proceed scenario gb p p rt thus imply hold short c k gb kk g simplification radius disjoint also second display note part omit say density pp f f invoke thank law may iid stick break process v p p dirichlet stick describe g share point organize step let hellinger h scalar measure proof kp ordinary density parameter cd cd cd cd different consider construct sequence existence property note addition addition almost eqs follow lemma express verify tend remain verify construct sequence nm dm construct way eqs immediately tend sufficiently parametric rate two step final derive convex avoid upper cover number radius similar calculation carry mix stand yield omit derivation depend apply fm lemma section bayes anonymous helpful nsf dms key phrase process geometry hierarchical process base measure wasserstein technical report statistic version support nsf grant study concentration base dirichlet infinity endow dirichlet hierarchical establish wasserstein geometry support demonstrate benefit datum setting efficiency hierarchy improve nonparametric convergence include bayesian building block see probabilistic particularly structure also focus hierarchical become prior successfully problem group array dirichlet basic question convergence hereafter dirichlet associate dirichlet separable space equip borel sigma dirichlet dirichlet measurable partition dirichlet property dirichlet almost surely discrete measure directly instead mixture measure admit ix known kernel dominate take measure endow specification prior atomic construction collect distribution interest latent question infinity share hierarchy provide improve interest process concrete share support intuitive support quantify ask posterior concentration mixture distribution denote bayesian stand endow prior question hierarchy effect question show consistent sense make true fact atomic base measure estimation base measure somewhat dirichlet process make question answer technique atomic measure simple finite equivalent dim impossible finite directly fact dirichlet allow leave set atomic dirichlet hierarchical process appropriate dirichlet process hand practical estimation setting receive much decade hellinger behavior model remain primary concern adequate underlie account theory author demonstrate usefulness wasserstein analyze viewpoint canonical hierarchical distance tool g view concept shrinkage effect interest role latent effect manner hierarchical hide author address remain dirichlet behavior nonparametric explain geometry support dirichlet different theory wasserstein recall wasserstein measure whose distribution main summarize first vector integrate formulae suppose specified measure generate denote hold concentration fact achieve kernel quite parametric finite arbitrary mn main turn numerous application dirichlet process practitioner data se represent model responsible diverse topic text shall base measure true result work relative concentration establish deconvolution estimating mixture geometry gradually obtain g finite unknown geometrically sparse notion geometrically sparse characterize term respectively hausdorff packing arise geometry establish strength modeling suppose mild show hellinger c dirichlet process implicit specification share support discrete theorem section establish concentration decrease due finite grow sufficiently strength particular kernel obtain formal smoothness present particularly beneficial convergence variable far translate e establish benefit part proof lie dirichlet datum establish relate three quantity wasserstein distance measure notion distance variational obtained integrate latent dirichlet move lie cost move measure wasserstein one recursive provide arbitrary order distance one dirichlet set control vanish radius precise link geometrically dirichlet base require measure large show mass wasserstein ball result generalize tail dirichlet distinguish play asymmetric hierarchy optimal increase rate get presence numerator suboptimal presence density large fact exchangeable carry conclusion minimax theory sequence exhibit notable increase behind increase quality measure unfortunately get grow technique enough address asymptotic regime limitation root wasserstein dirichlet corresponding vector issue statement subsection distance boundary various wasserstein support wasserstein pack metric functional density employ leibl hellinger space mean either otherwise relationship sep width f drop iid accord generic density measure theorem fix contraction third concentration mix atomic consider admit geometrically k sc separate say support valid sufficiently cover ball bound clearly decrease say satisfie sc sg geometry hausdorff analogous packing main density minor sequel however additional deconvolution problem symmetric fourier transform simple multivariate increase list assumption throughout observe density laplace cauchy gamma almost first establish density subset infinity either fix tend infinity variance improve rate carry consequence parametric assume point much despite infinite set almost nothing assume except mild rate appear natural along effect concentration explanation phenomenon get degenerate calculation shall present entropy hellinger increase course optimality lack rate proper analysis scope turn basic gd fy gd fy hold posterior attain concentration rate obvious extend shall vanish mixture measure give mixture exact concern finite eqs marginal density parameter infinite ordinary eq admit parametric take obtain finite model continuous q mild identifiability one n categorical exhibit different geometry deconvolution account mix quantity geometric measure get slow appearance quantity play well estimate measure dirichlet capture previous increase integrate dirichlet vary bind kullback leibler g natural g tight quantity bias variance hand increase base give choice regime tend infinity return appearance grow define g numerous dimensionality term final base amount ease presentation hierarchical iid relationship among quantity diagram row em sep em edge edge show specification appropriately intuition simple size appropriately number benefit mix conditionally close dirichlet take
neighbor search hash application nonlinear substantially maximum likelihood extent recent bit code hashing application oppose binary sparse expect valuable variation include count sketch variant another future develop improve useful similarity popular vision nlp become algorithmic distance dataset transmission energy consumption influential scheme appear simple bin use bit develop bit code quantization svm efficacy evidence recommend code scheme high density cdf derivative note result eq complete dd w asymptotically q complete special eq q q know proof true lemma computer nj school department computer science university ny projection become popular large application project need bit significantly storage paper focus code suffice practice machine projection popular classification search focus similarity classifier influential code scheme multiply shorter consist pair input computing require scan data bi eq assume convenience brevity common proposal perhaps scheme simple quantization bin width large equal paper standard operation monotonically increase similarity make benefit project neither convenient transmission suit indexing normalize marginal decay cutoff represent bin width record bit optimum depend interestingly uniform quantization optimum code projection value mean feed b bit hashing course evaluation early day science use code locality sensitive hash lsh bin hash every dataset search similar lsh elaborate compare propose separate focus estimation offset write randomization offset compare bin accurate wide optimum mean bit quantization fewer influential prior use window offset probability quantization scheme always estimate estimation variance scheme conclude offset coding demonstrate largely performance quantization scheme certain comparison bit similarity basically confirm analysis present relate research conclude code h h practitioner perspective long similarity suitable matter expression denote express closed especially value quickly keeps increase undesirable code surprising well analyze theoretical precise scheme monotonically similarity one precision projection denote table demonstrate propose code projection recall attain popular scheme practical disadvantage specify advance might really optimum pair attain pair slightly well bit present small optimum attain similarly preferable optimum bit decay consider bit sign equivalently bit code analyze estimator variance var w var note width plot ratio demonstrate outperform horizontal axis scale similarity var accuracy code width visualize must specify quantization bin width advance bit var w var figure code outperform improvement drop region outperform unless believe around provide overall consistent var reasonably similarity var per bit require bit bit sense bit code estimation preferable bit follow cost bit scheme bit cost processing train expand observe mainly study restrict attention overall significant room improvement refine treat solve nonlinear maintain simplicity paper reversible bit recover information sign work code conduct experiment available uci original collect first dimension dimension code recall plus offset bin code project length projection vector fed solver recently bit hashing specify cutoff practically decay suffer cutoff report compare basically analysis variance small width scheme perform similarly use scheme suffer reduction classification accuracy dataset experiment confirm offset linear report normalize
see high derive formulae threshold function point answer corollary concern cardinality particular function iff real assume simultaneously separation define gx ne pp contain ordered line pass asymptotic remark asymptotic study author sometimes terminology lr panel separation dot set iff function call irreducible iff point call essential exist minimal unique fig number us line contain easy contain point zero uniquely adjacent point thus function point line pass line denote belong contain adjacent pair lemma formulae easily asymptotic coefficient define follow moreover form x tm assume line separate solution cell line edge partition possibly unbounded intersection cone plane cone vertex every unbounded represent cell parallel parallel former vertex latter cell generalize imply moreover asymptotic among since set apply lemma total line line infinite line irreducible give infinitely distant find euler plane line cell vertice c triangle triangle formulae imply q everywhere asymptotic triangular use characteristic acknowledgment support
usually low threshold number spurious produce hard thresholding hard rmse output spurious wavelet spurious peak rmse vanishing assumption linear sparsity function parameter logarithmic penalty application let question constrain ensure constrain strictly special name logarithmic penalty function sufficient ensure suppose definite semidefinite th diagonal entry strictly convex semidefinite rewrite strictly condition sum ensure convexity bounding full strict semidefinite vanish convexity penalty log view logarithmic strictly convex positive semidefinite illustrate contour function convexity apparent contour region contour induce contour norm apparent figure one point show star example yield maximally semidefinite low low example constant tight tight interest maintain convexity convexity semidefinite order tight maximize parameter semidefinite calculation inequality semidefinite eigenvalue constraint matrix satisfy proposition weight sum aspect standard semidefinite optimization function inequality solve matlab software often inverse signal processing eeg efficient store modify algorithm semidefinite row free likely time solve multiplication address nevertheless medium arise readily demand whereby iteratively convex satisfy use also aid parameter strictly vector optimality numerically illustrate scatter illustrate wherein plot make penalty ref follow approach ask zero write hence additive although sigma rule q white convolution impulse sparsity penalize assume input semidefinite semidefinite r logarithmic penalty penalty non norm penalize least square maximally maximally induce take recommend numerous sec primarily limitation nearly zero case practically example logarithmic function practically norm offer situation wherein sparse denoise deconvolution deconvolution system invertible frequency lower bind often overcomplete dictionary low applicability iteration previously active progress produce increasingly progress penalty become change element logarithmic penalty otherwise useful problem initialization record number check termination terminate sub semidefinite I penalty may subsequent terminate note less computation importantly induce therefore successively procedure reduce computationally signal length uniform amplitude signal input contaminate toeplitz matrix estimate signal compare denote support denote compute using namely accommodate zero se count false sparse se solution average trial trial noise se log deconvolution perform highly accelerate hard replacement algorithm case software respective algorithm non zero norm signal quasi method iterative reweighte without deconvolution seek convergence minimizer substantially effect demonstrate regularizer accordance logarithmic penalty l reduce logarithmic penalty negligible simplified wherein sdp also lead three run solution verify scatter optimality plot illustrate fig hence norm clearly relative bias solution together lie identity outperform se error though entirely notably attain false beyond solution illustrate parameter whether effective deconvolution structure comment take second spend size spend long norm time approach ill pose strongly utilize non convex approach introduce maximally maximally induce program sdp element involve convex intend minimization widely often desire solution find practitioner concern non one issue minima relate function cost surface vary minimum function jump phenomenon spurious spike wavelet hard reason favor formulation set entirely optimization produce quasi principled enhanced explore technique recognize deconvolution minimization reweighte reweighte convex wherein convex ensure cost solve rely individual lead except subdifferential thresholding threshold study straightforwardly first satisfy strictly require give q thank anonymous correction manuscript address processing deconvolution induce non regularizer ensure semidefinite sdp maximally maximally induce convex demonstrate solution substantially induce significance convex reliably many reconstruction numerous formulation estimation aim develop avoid pose regularizer arise denoise compressed etc motivate detection imaging explore reliably balance fidelity derivative propose ref denoise binary wherein parameter cost penalty utilize origin optimize maximally maximally induce maximally allow parameter convex hence describe iterative effectiveness extend rank ill dictionary deconvolution require suitable address parameterize penalty function propose suitable describe denoise soft soft thresholding threshold scad wherein function derive list proximity operator threshold functional estimation gaussian sparse essentially perspective design continuity threshold corresponding parameter threshold side derivative like bias soft solve threshold function iterative reweighted reweighted algorithm derive wherein penalty quadratic linear numerous formulation problem deconvolution convex penalty similar include direction multiplier admm minimization iteration several induce extension half non minimize convex originally penalty extend ill inverse penalty availability norm suitable norm reduce poor minima algorithmic dc programming operator wherein knowledge lead convex continuously except threshold threshold beneficial admit note algorithm beneficial sec approach let since subdifferential monotone convex threshold extreme case keep gap hence avoid value function note q equation note reflect relevant decay rapidly penalty illustrate fig illustrate increase derivative vary soft threshold therefore specify except negativity rapid identity lead turn explicit go increase also parameter identity derive go zero rapidly achieved obtain identity rapidly derivative specify increase rapidly identity
provide insight discriminant analysis laplacian jointly formulate mild trace optimisation probabilistic formulate c paper unified component mrfs component solve trace optimisation domain principal pca analysis lda analysis entail provide explicit feature useful generate merely product mrfs rest paper follow initially subsequently joint complete pdf co directional deterministic pca lda distribution latent determine use fully mrf connect derive derive choose mrf chain aforementione subsequently expectation em sec usefulness family technique give rise reduce neighbourhood nevertheless probabilistic pca lda entail formulation observation length dimensional deterministic find latent projection I optimization eigenvalue model relate variable sample assume isotropic motivation offer parsimonious arise observation projection basis minimize correspond exploit attempt formulate q loading regard probabilistic corresponding eigenvalue deterministic keep small one model reduce probabilistic closely lda define b observation drawback requirement formulation locality preserve projection aim latent preserve sample u ji diagonal map apart therefore minimization ensure near probabilistic space column aim representation usually first dynamic py py unified incorporate special produce probabilistic loading deterministic regard per class provide explain estimate per approach exist framework novel ml deterministic assume fully subsequently ml aforementione mrf latent latent mrfs node c x mrfs connected neighbourhood j potential sec dynamical motivation behind latent influential analysis make connection piece add dynamical use derivation ml method pt eq amongst lie neighbourhood fact vary translate potential essentially connectivity ml loop solve approximation dynamical treat autoregressive analysis theory define sec mrfs constant generality clarity notation sec without clarity e agreement replace latent mrf configuration specific connectivity easily htbp ccc simply obtain ca j infer moment latent posterior ia iy move adopt usual em adopt complete likelihood become separate logarithm subsequently choose update far clear pca well variable probabilistic case shift per order allow interpretation undirected mrf direct undirected mrf trivially fit em describe auto able bi directional direct resort straightforwardly solve enforcing recover neighbourhood similarly iteration unlike deterministic trace value propose probabilistic infer likely infer essentially log model adopt infer since mean exclude store average training provide I experimentally validate belong experimentally evaluate class synthetic dimensionality reduction toolbox detail correspond formulation lda mainly qualitatively match deterministic modelling lda clear col section rd col projection col recognition lda face extended database well ar database wide variability expression pose change use consist database image selection subject subject subsequently use relate lda lda show probabilistic method use pixel verify improve database gaussian although offer substantial deterministic perform well lda em lda outperform attribute model per propose lda face via typical embedding face intuitive understand structural face frame experiment perturb image gaussian unable cope add meaningful propose capture structure modelling infer face random gaussian neighbourhood connectivity novel probabilistic prior mrfs specific prior pca
chain property geometrically p mala combine show many mala geometrically mala geometric mala density preserve crucially acknowledgment thank anonymous valuable grateful green currently european fellowship development part support program ep fellowship section department mathematics university tw langevin simulate dimensional widely modern statistic analysis new langevin exploit concavity relate instead process exist mala geometrically density mala method apply density continuously increasingly process scope exist mala hmc algorithms method compute efficiently proximity mapping logarithm approximation proximal non resolution address exist mcmc markov proximal algorithm ever resource monte method fundamental modern markov monte algorithm apply diverse area range biology dimension gibbs purpose model arguably metropolis langevin algorithms mala hamiltonian carlo hmc mapping capture target explore advanced mala calculus efficiency structure lift riemannian isotropic calculus analysis log concave widely dimensional statistic thing compressive perform inference currently lot major high optimisation lead development call proximal concave function mapping maxima distribution high paper use proximal new langevin mcmc log possibly continuously useful processing address elastic net convex norm ball semidefinite cone remainder structure specifie class define analysis essential briefly langevin mala proximal concave distribution present mala demonstrate methodology challenge resolution extension admit usual lebesgue simulate satisfying unknown explicitly vector method proximity define q useful analyse term vanish opposite limit behave similarly move direction proximity mapping useful mapping originally decade attention convex capacity differentiable extensively machine mapping optimisation also great envelope envelope take eq several construct simulate approximation continuously ng subdifferential subdifferential continuously differentiable maximizer assume property extension envelope establish property generalise fact concave imply proximity decrease concave definition four laplace fourth polynomial decrease density property density tail blue solid useful mapping easy often conduct optimisation mapping example low recovery mapping mapping please frequently proximity paper langevin langevin briefly recall everywhere brownian langevin differential stability simulate unfortunately direct simulation solution diffusion euler control increment perturbation condition produce converge ergodic mala correct introduce rejection guarantee correct target sampling class mala geometrically ergodic ergodicity limit practically geometric space limitation mala method process mala capture concavity target well work modification mala suggest truncate retain langevin near add difficult implement practically recently variation mala implicit exponential manifold mala geometrically sufficiently small proximal metropolis langevin mala concave define mala geometrically mala converge geometrically mala approximate langevin diffusion differential replace regularity approximation wish simulate select consider euler equal interpretation discrete simulate bring convex optimisation viewpoint plus stochastic lead proximal point chain value set use gradient geometrically exist irreducible small condition establish geometrically ergodic proceed approximation gaussian tail apply property closely x xx illustrate ergodic sufficiently belong verify establish geometrically ergodic square hold continuous chain langevin converge strongly property approximation apply correct supplement metropolis accept reject mala give transition probability reject construction mala converge total fact chain irreducible though evaluate iteration condition geometric ergodicity ia I ergodic hold geometrically hold mala result simply check mala converge mala geometrically ergodic manifold geometrically sufficiently mala robust precisely regularity ergodic decay mostly continuous tail stability convergence study random walk drift hold ergodic geometric ergodicity note possible enforce drift equivalent smoothness assumption proposal mala geometrically yet poorly proposal mala target achieve rate approximately directly p mala similarity mala mala reasonable mala produce mala acceptance rate mala capacity efficiently proximity mapping significant operator optimisation model high signal statistical analytical example variation proximity list mapping please optimisation frequently art present framework mala use dimensional good acceptance g model g efficiently accurate h involve proximity signal processing formulate moreover h approximation simplify separable diagonal hessian lead parallel model admm exploit worth reduce mala mixing geometric ergodicity mala converge geometrically approximation bound drift application mala density mala gradient mala manifold mala geometrically ergodic target drift random construct mala hx ergodic converge particularly certain alternatively manifold mala figure chain mala implement adjusted pilot behave walk find sensitive poor around value tail figure mala exhibit good mix mala fail value slow lack ergodicity mala observe fail converge hmc mala l section first computation bayesian resolution present popular recover noisy image relate spread white ill condition admit sensitive bayesian deconvolution difficulty order use prior deconvolution improper variation compute horizontal encodes difference image linear describe posteriori use proximal optimisation region assess precisely mala pixel proximity scenario typically g g efficiently implementation present experiment deconvolution add achieve ratio figure optimisation technique sharp image region pixel measure mala use million burn iteration region uncertainty grey grey significantly concentrate contour boundary reveal presence sharp exact therefore particularly determine location size appear imaging image subsequently boundary decision mala mala partially differentiable use differentiable mala recently compare lag autocorrelation mala mala summary observe produce p mala significantly autocorrelation effective kk sample monotone ess mala almost expensive mala associate evaluate proximity mala mala mala mala normalise ess sample hour mala fisher lead mix differentiable prior l interval estimate mala mala predictive widely nuclear norm differentiable make apply represent observation contaminate white rank selection component object tracking rank limited seek convenient type define singular popularity stem nuclear lead log think matrix exponential accurately nuclear prior useful problem approximation viewpoint nuclear posterior predictive recommend technique check fit base graphical check visually application
function size truncation outperform course algorithm note exchange intensive become impossible perfectly lattice whereas still albeit considerable look less large methodology carry inference use exact smc implementation mean long perfectly critical exchange exchange exact ess ex correct exact chain iteration half use gold effective ess ise geometric truncation approximate exchange exchange chain partition calculate ise geometric truncation poisson approximate exchange exchange algorithm perfect ise geometric truncation truncation c exchange exchange partition calculate transfer lie surface radius form rotation axis identifiability represent hausdorff recent variable version mcmc inference carry draw exact apply geometric truncation sphere draw run technique also run monte carlo c ex ess mean agree method superior see possible sampling identifiability exponent unity large I importance idea statistic physics literature scheme doubly turn attention run method tackle determinant large matrix question decomposition determinant determinant infinite truncate mcmc theoretically upon inspection several difficulty associate unbiased determinant exposition purpose describe particular datum large type exact consider infeasible still run full detail suggest reason pseudo passive sensor model simplicity focus stage precision mat ern partial evaluate control fast product vertex allow spatial gaussian show end give need likelihood log pseudo unbiased truncate nn require overall gaussian construct unbiased challenge unit determinant estimate rational approximation require overhead need processor scheme emphasize estimate methodology reduce log shifted need system separate largely depend underlie arbitrarily affect small eigenvalue add shrink conjugate gradient solver convergence problem precision fundamental limitation attain system practically implement large amount estimator unbiased log shift absolute estimator slowly overall idea integer average unbiased multiply unbiased estimate exponential drastically model sufficiently exact posterior relatively suggest paper metropolis fail converge extremely variation determinant convergence solver trick unbiased nature concept approximate example efficiency full practitioner recently approximate kernel trade computing introduce sub big exchange induce recently review form give markov version limit exact kernel certain large number solver slow remain moment order quasi posterior scale asymptotically capability pseudo marginal intractable review ability availability estimate inverse wide date development return distribution truncation series intractable compose analytic proceed full restriction however form unbiased lose potential lack strict positivity adopt final correct expectation preserve monte carlo estimate work term achieve one future unbiased merely unbiased required computational parallelism require constant currently scale example abc expansion lie within monte np imply elegant remain come measure tackle methodology mcmc large statistical area simulation scheme acknowledgement grateful david motivate discussion uk physical sciences fellowship grant enable quantification uncertainty large scale inverse ep award sum let almost surely completeness set truncation compute state unbiased k kp p calculation truncation physics literature stop terminate variance infinite q nonnegative use trick jensen kronecker sequence np n bound series deduce sequence choose variance infinite computing analytically rule general sigma assume markov give markov surely geometrically ergodic hx hx hx px p law number markov surely bivariate device delta ic variance simplicity reversible roughly across hx hx p hx quick sum hx nx return negative hard accurately corollary proposition section result section problem conjecture example section section theorem replicate university contact author manuscript intractable model standard example include exponential scheme mcmc suggested review intractable develop yet doubly intractable take physics alternative intractable truncation exploit negative distribution preserve methodology review describe assessment strength methodology transition posterior intractable term illustrate constitute doubly intractable use inference lebesgue adopt p doubly intractable carlo estimate employ exact hasting design distribution analytic resource situation far modern day challenge methodology computational statistic currently hide publish physics deal marginal unbiased target might doubly current inference pseudo mcmc mcmc scheme section intractable likelihood maintaining contain experimental doubly intractable ise model fisher large describe complex dependency structure intractable literature formulate interact take introduce use disease green spatial integrate model field social analyse triangle massive gaussian amongst posterior likelihood term approximate compute pseudo form probability normally efficiently therefore scale bias long take case model hide model composite likelihood also massive joint spatially adjacent block separate computed computed parameter space expensive computation require inference carry however impact approximate preferable possible retain use unless justified abc likelihood model originally simulate neither likelihood doubly technique develop abc community simple propose propose generating set propose accept similar sample although attempt burden depend intractable p term datum remove although simple deal burden intractable significantly original parameter complicate impact computational eliminate several carlo approximations mcmc kernel wang estimate give posterior carlo approximations stochastic wang suffer curse estimate location need grow exponentially limit significant ensure achieved avoid approximation posterior complex area spatio disease approximation programming implementation inference drawback mcmc far ensure assumption apply cox metropolis langevin mala term develop approximate well doubly intractable posterior use sampling methodology intractable use reversible scheme get intractable univariate fy ny available bound introduce auxiliary integrating sum return propose get intractable term limitation firstly ensure positivity obviously generality methodology class however functional bound strictly argument bound tight difficult choice sampling g bound binary lattice ideally relax requirement interval long convergent generality far specific sampling doubly intractable extended proceed proposal intractable metropolis hasting ratio drawback choose thereby intractable extend joint propose swap method importantly methodology may return systematically effective follow section address issue use target develop positive importance monte smc non example likelihood long likelihood outline intractable nonlinear analytic reciprocal represent convergent term estimate unbiased stochastically bias compute produce unbiased strictly estimate mcmc scheme monte estimate respect desire distribution root several place physics physics unbiase unbiased unbiased estimate still unbiased iteration suggest estimate statistic likelihood write quantity generate method section expansion doubly come description intractable geometric bias unbiased correction give q finite summation essential geometric difficulty ensure convergent absence knowledge value guarantee establish construction loose convex ratio therefore implication series follow section practical run constant level pc alternative geometric series issue maintain auxiliary posterior write expansion define return sample converge quickly exponential expand introduction prevent series alternate sign exponent help return bind improve truncation unbiased series exponent scheme carry division log grow fast series generalise poisson estimator originally series almost surely problem situation geometric series unbiased truncation nonlinear rely availability unbiased desire estimator infinite sum introduce final unbiased von simple truncation define integer index g could case series variance fast subject moment unbiased truncation exhibit superior monte physics finite stop time sum implementation physics literature commonly n k p kronecker p refer relate design appendix scheme return unity fast example represent
unit dot mark density eq unstable project datum project locally neighborhood nonlinear pca inaccurate consistent projection ref project gradient minimize yield self approximate e ml due deviation self density error ml dominate give mae ml density force force molecular dynamic dft ks dimensional find functional energy force break exist come level although dimension efficacy acknowledge nsf kb rgb rgb rgb rgb rich systematically consistent molecular force possibility ab molecular simulation ks dft balance accuracy ks dft dft fraction total energy exchange correlation spin great dft theory bottleneck dft calculation ks equation formally interest free dft sufficiently energy produce greatly reduce focused accuracy requirement accurate ultimately density determined euler accurate proven task theory various generalize von attempt linear functional describe moreover euler functional due near chemical base difficulty bad local approximation energy incorrect limit tackle learn powerful successful paradigm density ml functional particle density separate self ml identity minimize absolute final functional nuclear ks local spin ref energy ref interact demonstrate ability live box variety analog center separation vary show potential united generate curve length place grid necessary converge dft low extract energy density energy need essentially energy far achieve accurate energy l e construct training evenly spaced show test expansion approximation minimize mae mae
consider condition convergent dependent choose initial get solution compare propose previous step involve converge easy convergent extensive generate form fisher distribute replication require allocation sec w optimal allocation update express repeat h user theoretical validity reliability extensive generate wide setting allocation case allocation deviation conjecture monotonic theoretically open r proof accumulation w subsequence dl I kx j eq empty subsequence q j hold theorem corollary one convergent accumulation proof w infinitely accumulation element convergent let convergent york york york equivalence journal mathematics kullback variation transaction theory exchange optimal experimental design development core language environment foundation http www design york multiplicative statistic algorithm monotonic alternate monotonic criterion simple convergence demonstrate reliability usefulness accelerate life regression modeling analyze variable one independent instance reliability often failure reliability characteristic use incorporate statistical depend random experimental one covariate intercept experiment item experimental parameter prediction plan choice x allocation usually design regression development box introduction development design idea comparison generation optimal modification however point convergence wu cyclic exchange design mainly continuous design conceptually prohibitive wu interior suffer recent develop numerical design yu monotonic convergence general reliable paper obtain optimal subject optimality convergence algorithm extensive reliability consider discuss relate concluding remark one approach parameter likelihood maximize subject express repeat measurement experimental condition mle information term aa xx wu li optimal allocation obtain w propose manuscript linear distribute plan analysis determinant fisher
ix base approximation augment lagrangian quadratic method bind well bind separability average constant quadratic appear lagrangian rise increasingly solve separable block link source stochastic block involve objective expectation block call encode requirement available model plan augment lagrangian constraints lagrangian introduce augment lagrangian simplicity arbitrary master lagrangian motivated development decomposition technique early work suggest augment lagrangian linear transformation cross aim lagrangian approximate original difference break attractive advance since solve parallelism acceleration development area technique variant schwarz recently scalability property decade recently coordinate nonsmooth nonconvex parallel analyze dual inexact include compressed system equation group lasso huge closed extend real constraint decision link difficult solve large million introduce challenge useful decision decision identity stack vector top moreover compactly gx ix form constraint drop instead gx ax vector penalty euclidean norm multiplier employ counter meet multipli optimization problem henceforth drop nonsmooth eq one old respectively main significance separability use case quadratic derivation enable generalization non quadratic finite difference lagrangian replace study generalization theorem happen smooth strongly situation enjoys merely study newly develop exist though approximate lagrangian much strongly show much vast least eq parallel valid maximum average constant factor large even form speedup factor come affect let comment dependence much theoretically albeit see show simple computing preliminary numerical advantage section provide separability utilize quadratic objective coincide complexity quantity understand separability quadratic row consecutive contain zero one degree separability separability analysis separability partially degree say convex two separability comparison result entry separable rest fix since exactly mean likewise row precisely build vi u third identity separable degree term separability section method paper lagrangian appear separable ignore cross refer quadratic multiplier solve amenable parallel processing notice observe compose product ignore lead separable approximation eq slightly less substitute replace multiplier solve determine intermediate eq comment step easy execute problem intermediate iterate intermediate iterate new far big would serious employ design generalization convex generalization simple establish identity generalize allow semidefinite form iterate possibly nonsmooth function separable quadratic exception convex quadratic coincide fx h set highlight main coincide block update highlight update able processor act optimize processor iteration interpret gauss gauss convex function apply equip derive convergence complexity augment quadratic strongly convex wider side employ explain accuracy need ensure need apparent minimize function value block clearly fail sided function iterate decrease function turn result step correction note strategy trust subproblem measure adjusted goodness linear find minimizer linearize taking value highlight difference present coincide partially lipschitz generalization separability separability equivalence method remark case cover theorem feasibility mean default translate slow convexity function case estimate assume convex convexity reduce convexity note may write convexity simple neither strongly strongly repeatedly e iteration big well complexity number apply long give rise let proper choose level target counter point expectation inequality specialized fully long vector oppose high partially separable assume separable degree convex fully fully sample apply theorem establish statement follow establish degree convex one generate q fx fx analyze parameter however stepsize notation argue much practice twice fast draw appear appear inequality loose view obtain iteration time bad constant compare convexity least make sure strong convexity hence convexity priori comparable sense explain analogue continue argue even theoretical advantage lipschitz parallel coordinate single processor available iterate update block parallel updating amount iteration complexity say constant high solve equal ask question partially separable block constant compute present numerical support finding parallel variant setting coincide recall compare stepsize stepsize processor primal angular structure appropriate size notice problem fully separable respect separability nonzero entry experiment sparse separability
affect regularize square power real exponent convergence ridge regression regularization exponent grow slow quadratic growth rkh extensively principle address know problem regularization ill pose work put regularize make regularization suggest different strategy among base space considerable amount gain square considerable success restrict rkh term exponent influence exponent machine main relate regularization exponent know rate exponent term grow slow growth rkh basis algorithmic consideration spirit focus algorithmic involve develop variable rkh exponent remainder present answering efficiently possibility contribution analytic exponent propose although experimentally compare throughout real reproduce kernel k n investigate combine power exponent classical recovered problem necessarily convex descent inversion generalize arbitrary exponent though equivalence theoretical devoted purpose define two problem recall become explicit spirit derive problem root notice direction minimum eq write recovered initial optimization regression exponent cm gram orthonormal basis n I employ newton reconstruct diagonal derivative verify q symmetric basis write equation follow possibly problematic calculate thus need root follow solution strictly root exactly show root solution analytically fast call variable exponent note power strictly apply objective minimize solution root find pt pt iterate equally formalize definition algorithm simplicity strictly convex function try minimization algorithm problem associate optimal important varie provide another strong equivalence algorithm theoretical optimality equivalent respective give nothing say consequence property involve stability generalization bring equivalent equivalent important optimization problem follow easy z depend may experimentally simulation remain unchanged vary rare degenerate illustrate generalization selection retrieve valid set stability derive extend cover previous property imply equivalent let let realization z kx hypothesis hold realization copy z reproduce reasoning stability begin original one generalize binomial define since get eq realization remain open future study explicitly conduct efficiency extract uci repository compressive instance concrete attribute attribute efficiency also generate scale root rmse experimentally fold cross strictly prediction use rmse fold validation likewise grid equally standard std similar amount capable achieve performance limited synthetic uci choose range step l cc dataset std std
thorough treatment provide foundation stochastic approach possibly parameter develop equip absolute loss mcmc recover recently determine use section review regularize rkh extend use connection numerical illustrate proof indicate obtain measure assumption often refer mean gaussian measurement pair noise absolute solid dash right zero dash laplace tail robustness laplacian standardize insensitive suitable random find case modeling variance independence turn posterior gaussian variance follow jointly assumption r obtain u one formalize minimum happen posterior density case could function follow show assumption let arbitrary negative estimate rkh consider map appendix right side belong subspace use mcmc real typically infer bayes often call hyperparameter optimize see estimate equation proposition measurement compute analytically hyperparameter conditional difficulty underlie estimation close possibility suitable proposal scheme model apply especially proposition eq thick bottom two panel measurement monte consist reconstruction measurement generate typical plot circle bottom panel simulate presence add random offset typical plot circle bottom panel experiment non e reconstruct equation measurement loss model reconstruction measurement model loss laplace scale estimate estimate method measurement loss scale rely statistic concept optimize grid freedom run fig different outlier panel reconstruction bottom reconstruction dramatically error significantly bottom deviation bayes remarkable confirm similar absence outlier minimum identical mcmc procedure third average decrease method around relative nominal top solid dotted nominal perturb bottom condition rkh fall generalization simple heuristic argument fact realization rkh whose minimum formal prescribe location location estimate belong training version link rkh estimation illustrate utility begin lemma instrumental proving suppose jointly proof property e density recall also h depend value q complete eq unique g g g complete definite proposition apply hypothesis g use agree thereby eq complete representation project recall give use eq hold scheme posterior posteriori component scale normal close specific gaussian use distribute classical computed position describe mcmc walk metropolis independent lead assess correlation function use quantile precision respectively recover minimum applying proposition follow realization
minus plus fusion school mail school statistic mail propose penalize matrix use discriminant penalty precision coordinate method quadratic discriminant semi base cluster inverse discriminant discriminant estimation copy pair let th class invert problematic impossible review estimate low condition exploit discriminant invert identity matrix pool covariance class trace operator penalty equivalence zero entry equivalence inverse zero regularization aim estimate gaussian another propose minimize ridge penalty entry wise similarity inverse matrix entry natural illustrate apply method cluster denote form evaluate p covariance tuning derive orthogonal iterative algorithm evaluate replace user specify negative multiplied multiply former entry although easily rescale objective strictly minimizer unique ridge problem infinity solution unstable parsimonious block leave rest respect zero divide tolerance compute descent iterate initialize initialize tune generalization divide evenly subscript depend though notation indicate minimize fusion fusion coefficient multiply pool estimate arithmetic mean ridge without fusion orthogonal diagonal estimate ridge fusion linearly toward fusion small eigenvalue eigenvalue classification clustering estimate inverse set random otherwise setup function treat unobserved algorithm covariance estimate penalize penalty introduce unlabeled datum penalize q graphical ridge fusion penalty supervise base analog find penalize denote iterate algorithm maximize complete likelihood subject next estimate iteration coordinate descent current em particular difference iteration iteration bad convergence proportion large iterate call convergence converge analog generalize em supervise set use unlabele randomly unlabele th likelihood subset derive say minimize allow parameter simulation fusion inverse simulation datum draw independent draw process tuning ridge fusion specific pilot test package implement cross estimator section package unstable validation element zero vector th investigate cross maximize validation likelihood outperform cross validation minimize lead tune study tune l ccc likelihood simulation section poorly ill table fusion fusion perform ill condition simulation entry section zero expect lack ill base replication outperform sparse condition entry th equal favor exploit even ill condition matrix l ridge th entry entry classification ridge perform well ridge fusion classification section much slow ridge fusion pattern second fusion calculate ridge fusion replication package simulation fusion ridge inverse covariance exploit inverse ridge fusion fast replication point ridge fusion ridge fusion entire grid second time ridge fusion replication
deviation bernoulli variable spectral cluster algorithm cluster cluster closeness ideal generalize approximate solution mean define k exist deterministic input matrix suggest similarity though completeness contain lem give theorem thm obtain theorem current apply lem k condition developed theorem apply work handle manner substitute bind condition lead schwarz fact normalize nonzero v loss generality v v v let correspond nonzero median lem discard event therefore must mis hand author anonymous helpful suggestion lead simplification grant grant fa nsf grant analyze cluster stochastic mild apply recover maximum apply popular polynomial spectral cluster extend correct spherical spectrum random conventional concerned describe modeling occurrence among actor simple dataset actor edge realize binary example network twitter etc email world inference stochastic henceforth expressive sbm partition accord realize occur probability community connectivity compare node real sbm certainly model network important inferential recover community membership solve recent researcher propose variety degree statistical particular modularity belief variant arguably widely speak adjacency laplacian infer typically means possibly form easy computationally demanding amount search fast see example addition spectral empirically recommend initial computer cluster standard solving plant sbm despite popularity simplicity sbm cover growth elsewhere derive moderately sparse block block node use result exist analysis spectral justification effectiveness procedure moderately network yield recovery node rely combinatorial demand provide analyze detailed contribution background degree impossible hope fraction prove simple consist mean form recover vanish extend result correct analyze spherical median among computationally assumption principal perturbation cluster sharp spectrum random thm condition particular allow bernstein inequality correct block organize give block result sbm section present modular analyze conclude remark matrix submatrix matrix community denote second large community count nonzero positive stochastic community parameterize symmetric label elsewhere matrix membership eq quantity cluster require estimator well community cluster simple heuristic relate eigen decomposition see distinct state basic eigen sbm full rank let eigen straightforward eigen follow clustering linearly operator constant eigen decomposition large absolute roughly distinct slightly perturb version apply algorithm mean np e value adjacency community consist absolute let input output edge generally speak community hard measure degree follow play hardness sbm edge node whether edge reflect connectivity community network average hardness community reconstruction imbalance plant sbm planted community easy see therein algorithm primary concern spectral allow recovery quantity notation community spectral assume small nonzero mean absolute subset lemma node correctness guarantee equation result include technical reason vanishes discuss long thm present change also different provide bind quantity involve bind reflect decrease quantity dependence community imbalance separation unclear next minimum least balanced community p improve need eigenvalue edge stay community spectral expect degree less vary cluster recover edge grows fast plant put regime therefore eigenvalue provide different reach barrier therein plant clique consistent community procedure plant recover membership sbm size recover simplification q bind imply correct extend introduce node parameterize triplet addition additional variability probability node edge independent formation inclusion raise issue identifiability view flexibility degree heterogeneity development spectral agree otherwise normalize otherwise cluster extended decomposition direction lemma analogue spectral structure lem eigen eigen p h cluster vector filter nuisance normalize difficulty affect entry identify community node overall fix condition quantity measure heterogeneity strong heterogeneity community homogeneity section general correct particular spherical row normalize time care possible row community lead order eigenvalue normalize output appendix b small least exist equal theorem probability constant equation immediately parameter minimum absolute solution median compare aspect condition manner likely sharp believe additional strategy consider community comparable heterogeneity overall relative worth heterogeneity corollary minimum effective speak fraction small keep heterogeneity additional spherical median argument analyze median moreover small stay sufficiently slowly comparison relatively recovery correct
sparfa collaborative predict unobserved dataset answer question explain signal course learner answer question dataset dataset label concept list course consist assignment resource randomly response fold fold use sparfa sparfa concept knowledge resource predict held response mean standard metric sparfa sparfa tp latent prior follow derivation substitute posterior distribution message pass procedure kl moment evaluate eq side yield similarly side symmetric thus last expression observation acknowledgment discussion work national grant air force office fa research visit website sparfa sparfa g z l v e r figure definition thm example n sparfa trace base education pass base sparfa trace knowledge ii learner induce interact resource videos iii organization intrinsic quantity solely correct incorrect response summary action learner g answer experimental dataset sparfa trace capable learner knowledge well organization resource association question sparfa achieve learner exist collaborative filter education kalman sparfa fit education largely learner efficiency unable feedback learner organization strength interest development achieve automatically mining interaction scalable education experience learner vision pls consist key knowledge dynamically trace time interact e g insight content organization resource content organization recently sparfa model la ca sparfa assume learner assessment govern knowledge concept particular q bernoulli slack probability answer incorrectly probit factor association vector characterize relate learner concept iii intrinsic sparfa jointly question knowledge iii question incorrect sparfa framework sparfa assume remain sparfa learner usual assign course sparfa framework induce interact recommend learner blind kalman la work approach illustrate la evolution state binary value incorrect response question available learner matrix perform learner intrinsic difficulty state instance learner resource concept knowledge resource organization association intrinsic perform sparfa statistical learner resource task estimate knowledge learner response question develop pass base filtering time wrong kalman algorithms maximization learner estimation crucial kalman approach case validate effectiveness sparfa dataset collect via sparfa learner estimate knowledge transition predict sparfa learner knowledge state time quality organization resource assessment recommendation learner relate sparfa kt evolution predict kt suffer follow drawback binary characterize learner explanatory knowledge la kt restriction narrow algebra generalize involve learn characterize learner knowledge state concept force state kt e analyze quality detailed comparison trace learner interact detail learner evaluate sparfa brief kt base extend sparfa framework trace characterize concept affine resource ii resource affect concept learner time concept ii question relate concept intrinsic learner assessment throughout course assessment vector number question define learner instance index activity shorthand question learner dimensional concept association intrinsic represent easy characterize value correct incorrect instance practice tt probit simplify write remainder sparfa framework impose assumption sparse basis question concept domain assessment interpret knowledge common orthonormal unitary unitary improve interpretability sparfa model learner concept knowledge throughout assessment explanatory analyze possibly long learner concept evolve time happen conduct experiment computer likely forget concept decrease knowledge sake treat reduce learner propose learner knowledge consecutive mapping index indice information activity shorthand study define mean learner resource time otherwise ready transition learner matrix learner state interact resource dimensional vector characterize concept interact eq multivariate distribution reduce identifiability parameter account world scenario knowledge state entry influence pre concept early represent towards course purely learner response low time concept resource cover concept cover contrast learner knowledge impose sparsity negativity property quality boost resource learner case poorly design learner model resource entry imply resource among different mainly learner concept knowledge response vary learn content organization pass approximate kalman learner concept simply kalman review smoothing introduce learner drop quantity shorthand quantity kalman solve dynamical consist state variable markovian derivation summarize latent variable observation markovian state dynamical system factor consist first pass kalman kalman backward message pass kalman smoothing phase interest via outline incoming message give node rule message message p derivation recursive message transition observation transition measurement matrix message stay pass recursion give assume detailed kalman utilize time instance tracking decision observation application also use word order backward node convention implicitly use markovian latent write forward backward follow although possible backward recursion common compute recursively backward recursion derivation unobserve message simply rest basic kalman gaussian observation forward latent learner sparfa value learner approximation make enable latent concept formula pass become denote covariance close update long perform pass within arrive tractable order approach covariance extend filter thus gaussian kalman filter use create sigma mean covariance gaussian mode hessian mode approximate approximated message employ propagation kullback probit tb close eq sparfa study inverse value logit link function logit due inverse logit prefer close expression focus probit sequel armed kalman filter backward kalman pass desire provide learner knowledge smoothing estimate question parameter however observed set convex technique latent learner concept kalman sparfa trace jointly trace learner concept resource question expectation filtering numerous practical sparfa perform iterative em consist phase I j state maximize observe latent e order improve estimate sparfa trace phase iteration reach change consecutive falls threshold estimation learner initial knowledge end likelihood determinant covariance since impose smoothing knowledge state resource learner indicate resource start induce sparsity impose take formulate augment denote matrix notational augment state multiply correspondingly notation solve efficiently particular iterative shrinkage fista fista start initialization iteratively maximum number reach perform first aim exclude parameter simplicity lipschitz denote gradient give eq backward assumption lower triangular triangular operate wise provide collection index learner log binary learner association impose norm function rr sparfa thank linearity probit simple utilize commonly kalman approximate statistic transformation know covariance generate accurate order variable sigma weight latent simplicity set latent solve result iterative iteration aim portion exclude ik k next fista convergence estimate concept exposition omit derivation additional concept efficacy sparfa trace synthetic world use demonstrate sparfa able concept knowledge learner parameter sparfa predict unobserved learner kt sparfa sparfa trace able learner knowledge resource content organization next repeat trial assess sparfa concept transition generate knowledge transition concept state instance question assign instance dataset consist learner evolve consecutive assignment interaction resource total experiment learner state learner concept transition question know run kalman part sparfa learner knowledge increasingly accurate time proceed trace decrease miss increase moreover sparfa still observe trace knowledge second simultaneously treat prior avoid issue arbitrarily learner vector arrive detailed fix vary number learner error sparfa metric learner
appendix respect maximize analytically solve estimate q provide q estimate curve appendix maximization multinomial problem solve time maximization piecewise propose illustrated graphic since exceed initialize first iteration em estimate provide setting reduce define eq em belong approximated parameter proportion th curve discrimination give label curve use supervise boundary curve describe g ig define simulate set regression two evaluation criterion mean ij jk criterion formula procedure evaluate first experiment observe transition level tune curve curve three simulated mean curve corrupt th table correspond increase value smoothness transition fig smoothness level smoothness experiment observe quality varie step curve size effect hide piecewise present segmentation provide h curve switch operation minor number misclassification rate test piecewise regression regression provide piecewise unlike piecewise involve regard involve switch mention obtain curve description limitation limitation show behaviour non class model parameter curve fig curve datum cf fig show simulate estimate curve h propose approach poor performance attribute homogeneous observe adapt class regression model govern discrete transition time derive maximum posteriori rule experimental acquire switch operation reveal performance term piecewise approach shown shape consist deal limitation shape class regression log compute complete th maximization write iteration perform respect analytically problem maximize update anonymous thank company especially availability technology laboratory functional consist hide logistic modeling curve parameter maximum dedicated maximization propose discrimination rule posteriori acquire switch piecewise term functional hide maximum curve increasingly available engineering economic present relate diagnosis enable train track switch electrical consider measurement acquire operation electrical cc diagnosis curve accurately summarize perform simplified class adapt switch curve present regime switch operation see find accurate consist curve set knot fit curve consist piecewise regression curve segmentation approach use segment segment characterize use fisher globally optimize additive regression present less include regime dynamic programming especially large generative present regression incorporate allow smooth work expert I function simplify posteriori discriminant cubic detail curve neural follow account introduce propose representation detail parameter dedicate propose deal experimental carry switch curve overview piecewise polynomial model context description partition segment regime generally piecewise use dynamic optimize term curve assume polynomial define interval whose index piecewise vector th segment dependent covariate associate classical accord represent polynomial noise point classical base within independence prove therefore give characterize log equivalent next algorithm minimize additive equivalently minimize n kp ij criterion dynamic consider segment index segment recursively accord replace initialization step consist matrix recursively optimal partition cost jk jk jk approximation discrimination piecewise polynomial curve curve value label set acquire class maximize belong proportion training ig base tailor curve present curve segmentation moreover dynamical programming computationally expensive present propose
construct ba nuclear norm constrain advance easy order facilitate local global factorization local multiple specifically assume estimate consist illustrate neighborhood matrix low region region closeness locally near connect show assume assumption map slowly old non symmetric unimodal parameterize bandwidth value wide spread narrow spread define ds j extend particular choice conceptually technique result require prohibitive local simply svd low global svd result c c global rank rank rank netflix dataset indicate thick dotted model color art recommendation netflix use whenever assume b anchor approximation local anchor publish increase law outperform graph anchor global outperform improve anchor increase college ga edu prediction recommendation system partially matrix analyze modeling improvement
pt center machines technology di characterize rely example capable seem propose visual recognition domain image transformation considerably learn prove invariant signature patch template store module complex cell estimate hierarchical architecture property capture compositional organization extend convolutional architecture speech recognition representation representation continuously appear nature technical mit tr institute materials architecture shorter isolate original module complex cell cell translation invariant detector underlie visual recognition expand visual possibly visual step conjecture patch invariant broad expression face body possible recognize object human prove module provide invariant maintain discriminative original module signature visual field inside signature invariant transformation architecture compute signature part prove locally globally affine module architecture focus characterization rest include interest result fully elsewhere intelligence supervise learn obvious ability conjecture recognition differ simple depth pose body face theoretical recognition task viewpoint illumination imply car well categorization distinguish easier image transformation equivalently special translation generalization face categorization evidence respect viewpoint illumination position scale accurately example sample seem propose stream try approximate oracle provide signature image patch transform like affine group would group simplicity cardinality slight abuse group element unitary action translation image say relation idea two point everywhere conversely none image metric obvious especially neuron irrespective group distribution thus discriminative unclear estimate neuron effectively implement high store template neural image classical si almost uniquely induced template induce set projection sufficient discriminate si say informally approximately estimate need discriminate induce thus positively template observation transform invariant signature recognize face different remarkable projection observation onto template distribution template transformation signature without group belong store template transformation input si dimensional pdfs si actual signature since identifie see si crucially mechanism capable compute affine learn maintain unsupervised storing update transform template element si way probability instead ni correspond condition moment characterize nonlinearity pooling simulation one seem si argument section begin theoretical pooling give insight something normally accord capture support database pool instead si achieve architecture inspire principle explicit incorporate include group transformation allow label face pose presence clutter build invariance translation scaling limit follow another variability transformation core rotation translation scale module observe part module invariance module scale dot see si appendix parameterized condition form dictionary group justification encoding localization si imply template requirement completeness motivation wavelet group property si notion incoherence autocorrelation module provide approximate invariance transformation rotation face expression si yield regime highly tune template generic template incoherence improve far module multi see architecture module allow recursive module multiple property response consider transformation property si k transformation whole unique signature si two architecture fig need compute object part affine locally pool module hierarchy imagine hierarchical reason include connectivity architecture unable hierarchical organization visual compose scene identity scene minor true object signature hierarchy must access enable categorization whole well image part invariance part uniqueness invariance stability different range bottom desire architecture hierarchical world retrieve theory hierarchical take invariant relate efficacy architecture deal difficult recognition clutter signature several size hierarchical feedforward clutter architecture need clutter aspect recurrent computation invariant signature know capability digital computer neuron connection neuron account neuron module window field si appendix object visual transformation experience simple patch one simple template movie patch transform powerful unconstraine transformation main pool key unsupervised rule together determine among cell correlation transformation continuity allow label temporal pool cell cell nx complex bin approximated distribution follow cell cdf moment dot product complex cell energy alternative fit template transform experience study formalize like tune template si appendix localization template specific transformation part module module body module localization template layer architecture tune template stage hierarchy theory fit storage template place via predict neuron face patch al visual theory cell cell complex cell traditional one example broad interpretation complex theory moment development possibility refine maintain development visual invariance virtual new learning object recognition cut computational unify achieve convolutional network feedforward neural architectures feedforward organization algorithmic level motivate vision speech include implementation characterization stream tuning area despite decade understanding stream visual prove paper stream object signature visual experience thereby allow machine theoretical beyond representation formulate invariant significantly thank brain version manuscript ng thanks carlo useful material work support nsf grant foundation nsf fa foundation invariance considerably classifier sample cover ball radius example cover dimension pixel translate meanwhile stand cover translate translate sample independent since image cardinality eigenvector dimension simple cardinality locally compact fourier window pixel usual invariant representation small definition let image image locally compact abuse action assumption justify biological normalize signal usually convenience dot consistent convention grey particular dot product different region isolated dot product image noise signature module histogram carry patch image maximum correspond spatial module hierarchy input module dimensional signature provide low layer module image contain module hierarchy notation response simplify notation suppose signature center dot template cell general pooling continue even two characterize module compute invariant signature patch important emphasize module always include study e localization linearization signature module invariant effectively parameter pooling g case continuous locally first transformation integral haar dot also eliminate explicit determinant jacobian affine transformation simplify divide affine uniqueness uniquely induced group observation signature compact dimensional projection projection template compact haar measure borel abuse haar equivalent iff ia ia ia ia integral haar implication intersect construct deal dimensional histogram give unit sphere unitary distribution two equal equal state probability choose metric eq metric measure easy main text histogram histogram finite number projection suffice discriminate template experiment mathematically characterize projection challenge partial question observe metric template approximation let c follow union fix random follow q probability union holding obtain note result projection template version distribution direction characterize finite albeit signature transformation signature equivalently histogram template unitary invariance template signature say signature stable continuous q ki kn signature map strong form choice component signature linearity product transformation unity sum component divide signature independent sum divide signature constant compute invariant signature template transform template pool result pool invariance uniqueness observable partially observable particular neuron part transformation window transformation correspond signature compact constant haar define latter distribution observable definition definition observable section signature uniqueness invariance common transform window observable signature repeat change observable invariance image observation partial invariance invariance plane uncorrelated pool range states localization invariance locally compact condition implication g b g template result raise weather invariance turn indeed case assume condition positive illustrate fig detail unitary translation operator template interval function strictly monotonic difference integral dot periodic reasoning shift equivalently shift localization translation transformation specific template template satisfy localization indicate pool window figure special consider case transformation instead affine assume image template contain range image translation scale fouri pooling range spatial localization template less template template localize corollary show localization template mean hierarchical processing translation group unitary translation discussion localization spatial domain invariance locally unitary spatial frequency necessary scale dot scale give interesting equivalent overlap compact support particular invariance self localization support transform rewrite dot product fourier zero template suppose zero big effect scale typically support non localization r invariance follow invariance localization localization note suppose generic image since xt xt xt possible xt b fouri depict fig big enough fact support therefore fourier support repeat reasoning translation statement connect localization template invariance template support fouri simple template translation range decrease note r invariance localization shift course localization support pool cell template localization range localization localization tune simple equivalent translation invariance localization outside length length fourier also sign uncertainty form principle lead optimal relax invariance ki suppose localization ki b ki difference localization decay gaussian optimal meaning theorem compute pooling pool window template remark spread translation scale q box wavelet plane frame localization template localization wavelet frame invariant prove show requirement maximal wavelet wavelet transform invariance cell analyze localization invariance case transformation localization computing transformation localization property hold smooth think parametrize theorem localization simplicity support center zero xt xt sufficiently rich incoherent note expect improve architecture property allow approximate arbitrary transformation clutter uncorrelated clutter interestingly condition type condition memory recognition beyond store template exact yield universal template hierarchy second regime invariance yield deal transformation top hierarchy large visual several transformation group pose invariance transformation localization transformation locally approximated combination rotation compact haar smooth twice image taylor around e identity operator jacobian approximately remainder range localize e twice differentiable parametrize pooling reasoning dr rl r ni last transformation induce plane approximation small key template correspond specific template template complex key template within rotation template signature rotation template affect implementation dot pooling key template complex thus signature component long key template correspond knot center leave aside argument classify term invariance compact plane complex cell pooling template restriction template template yield perfect range mild regularity complex template globally partially observable compact pooling locally shown partially shift fall pool partial invariance hold wavelet include translation group smooth smoothness imply operator around template dot template template transform plane depth number rotation template key template rotation correspond template signature rotation hypothesis exist w template remark invariance simultaneously scale image ideally preserve ideal transform call localization self localization invariance second apply module specialized specific group transformation invariance theorem image template template condition wavelet template enough noise template transform hold specific module nice object hold key template diagnostic also quasi orthogonal highly non transformation impossible signature general derive formal module outline proof local transformation architecture invariance uniqueness signature partially iterate first case associate window see dimensional signature signature signature signature assume module make normalize process layer module first uniquely signature locally consider see indicator signature signature window signature signature abuse construction call operator map image layer template operator nk correspond finite template available simplicity template template see patch complex similarly consider extension collect l sequence template complex cell eq indicator cell image remark iff intend practically take account value ignore always act fig reasoning apply order covariance module identical template module layer ig definition ig haar q remark covariance state expression intuitive hold transformation crucial define signature average last write transform signature group network theorem image contain predict stream appear meaning transformation early layer hierarchy locally invariant architecture invariance guarantee part invariance architecture sake template hierarchy locally suppose proof reasoning covariance formal invariance layer big big part layer subset template support big grow word transformation use formulate lipschitz lipschitz reasoning stability continuity recently relate lipschitz map author invariant form transformation invariant compact support hessian condition lipschitz transformation close identity transformation affine transformation fall reasoning parameter think value subsequent wavelet expansion point interpret feature image implicit visible assume fact observer assume massive imaging processing ball model assume equally eq exist sx ds tv tv sx ds r sx q
simplicity value zero point classic square utilize natural reverse lead systematic calculation explain increase generate hold whole order justify like see might quick fluctuation compare evolution trend definition integrable series integrable obvious integrable assume moreover lebesgue integrable possible exception limited value integrate emphasize product ex take satisfy assumption section relation take compare derive section compare return derive evolution b behavior two day figure take slide window figure show value return extension provide want resp accord determinant size slide window pick great join universit bp france f al alg pour estimation bp france com financial business management several associate beta capital asset pricing excellent remarkable paper regression time one reference unfortunately risk propose unified advantage mathematical foundation mild series decompose tool recent estimation identification theory processing utilize successfully therein derivation return comparison among vanish paper detail mathematical alone exploit advance direction would continuous say quickly integrable boundary integrable accord lebesgue integrable smoother integrable practice length correspond via classic iterate integrable loss limited uniquely always like replace arithmetic average original drop become eq replace say replace assume w
generation problem violate dual weak computational introduce highly learner generate useful weak learner necessary learner generate converge fast fast coordinate descent violate kkt construct violate pick violate sequential sequentially pick later randomly pick one simplicity iy equivalently binary learner dimension exclude variable step pick eq constraint solution simplify perform close solution weak kkt meaningful sufficient lagrangian w complementary simplicity column working violate working set randomly tolerance analogue violate tolerance work stop show value terminate kkt actually information inexact acceptable column generation iteration maximum algorithm unnecessary compute expensive able update avoid equally write update efficiently avoid else work stop maximum reach evaluate uci variety multi digit recognition scene traffic another popular boost tolerance experiment times time solver converge much achieve well adaboost slow uci dataset shown set second maximum dataset maximum uci handwritten handwritten dataset mnist test example randomly rest provide use label cifar cifar pixel set show scene scene scene scene histogram book window hierarchy manner histogram contain category rest testing sign dataset set evaluate work work iteration affect default fast converge parameter column set fast present dual implement generation multi train learner fast class achieve much fast arc grant ft publish conference van david present boost problem boost specify weak learner fast overhead fast boost conceptually experimental dataset method propose convergence rate coordinate combine learner boost extensively object despite inherently boost simple adaboost adaboost adaboost pairwise class direct formulation framework case work build learner sparse slow novel formulation separate learner learner much fast cost adaboost underlie multi boost lead fundamentally importantly scalable adaboost multi like boost propose descent optimization iteration efficiently apply al comprehensive scale class cd choice fast tailor optimization extremely easy sophisticated optimization toolbox boost specified learner share different class set generate weak class mechanism much convergence column boost derive boost class tucker violate apply boost stage newly add notation class weak nn weak learner associate weak classifier classifier data rule x clear framework analogue margin want possible loss also shorthand parameter control model encourage correct label label learn j xy x w classification weak learner compact training
denote pixel respectively range perfect spatial reference quantify loss natural regular presence another dft dct wavelet distinguish blind range assessment first proposal sigma reduction yu specifically sigma window side filter significance present situation describe homogeneous c situation replication looks highlight pt l lr improve sigma sigma improve sigma sigma filter lee measure filter design winner aforementione unless present filter band field intensity fig filter improve sigma filter fig row look real technique result regard significant look exhibit spatial variability lead l sigma assessment distance sigma filter filter protocol moreover index proposal sigma significance test along de com present window overlap pass goodness filter divergence sample sigma diffusion protocol simulation quantify employ equivalent look assess index pearson edge application also show illumination affect interference coherent noise b phenomenon interpretation accuracy analysis object single lee reduction square mmse criterion lee et propose pixel lee sigma solve sigma range present filter base employ incorporation prior sensor produce thus protocol et inverse distance tailor propose contamination main contribution convergence free recent al neighboring estimate pixel pixel white assumption central region distribution require change analyze inference gamma author approach coherence e et transformation noise employ select matching bm inspire bm estimate second filter compute statistical analysis essential literature provide comprehensive efficiently simulate plausible image noise constant truth present nonlinear introduce et term distance neighborhood pixel pass goodness whereas employ window solution propose nonlinear stochastic divergence employ neighborhood pixel nine treat area account likelihood filter patch goodness confidence user approach neighborhood pixel central patch pixel patch whose illustrate free observe law choose soft reject evidence rectangle text blue shape rectangle draw text width cm text fill black text cm fill black text shape draw black text width employ proposal distance consider parameter assume distribution function divergence sometimes requirement statistical likelihood nan reject test define detail hellinger triangular kullback leibler divergence eq check come comprise set reject
distribution multinomial integer z note multivariate chernoff distribution chen ratio possess x n number preliminary hold yield exceed arithmetic check partial derivative notation make use make eq prove make eq virtue z z theorem pt concentration generalize hoeffding inequality variable multinomial distribution concentration phenomenon formally x dimension fundamental investigate readily tractable desirable deterministic symbol yx throughout notation constraint random na transform random contain virtue operation vector remainder organize inequalitie dirichlet nk independent average define accordance establish number hoeffding
large several issue remain promise design merge closed center result merge cluster influence result strategy address issue propose approach power law propose structure threshold accordance cluster proper propose determine may isolate small far pair heuristic stop datum point cluster result mean spectral extend newly dp mean process mean address datum law overfitte order cluster spectral introduce related model towards law follow discusse include implementation center far find introduce prove follow section conclusion relation mixture sample dataset number mixture eq nk regard probability link gmm mean specifically gmm assume become consider j dominate denominator gmm mean assign nearest clustering mean cluster number take dirichlet generate prevent assign distance exceed fail take center specifically hyper take address write exclude allocate assign generate achieve corresponding cluster dirichlet know successfully local component treat component equally threshold contain however world usually lot systematic problem include complexity focus law approximate discount add discount classic dirichlet scheme py generative paradigm perspective object interest color ball contain ball ball certain put allocate probability eq pick number color exclude ball continue cluster joint unchanged preserve rich assigning color ball large thank discount color great py draw color py incorporate help tail behavior size subset come reality life wide include language intensity situation finding law would noisy interesting whole observation scenario denote distribution ordinary cluster trivial discovery determination slowly satisfy difficulty power law traditional size cluster simply treat noisy trivial whole method put mine good suffer hard classic cluster allocate paradigm straightforward quite allocation paradigm keep exchangeability property e probability size fix determine exchangeability benefit name induction quite mixture approximate assign allocation paradigm threshold accordance cluster compact implementation procedure check ball datum outside ball say get center depict htbp whole center partition share similarity datum employ determine later center corresponding procedure implementation cluster employ procedure update add tends term minimum trade accordance stage implementation point arbitrary would cluster assertion classic cluster determination center affect belong heuristic whose short cluster center datum recursively generate stop computational define center remaining become new center short distance remain corresponding come one center fix rule belong center formalize center cluster compute short get much close another one dense part special number adaptively cluster proposition evaluate one combine pre current reduce denote two center combine cost cluster penalty jump decrease follow simple prototype clearly cluster combine center satisfie employ cluster would remain detail implementation center run check cluster center prevent discussion include convergence analysis spectral local goal objective decrease iteration objective strictly local three cluster stage newly cluster center increase confirm cost penalty decrease reduce stage always decrease objective employ idea partial number number get partition maximum without inequality lead assumption get theorem local analyze step consider initialization center consist cluster center complexity size process sort bad assign cluster center cluster small computational feasible set threshold computational cost would heavy spectral first eq kernel determination eigenvector reach select eigenvector adjust integer measure e eigenvalue eigenvalue relaxed result group synthetic uci intel microsoft algorithms code matlab algorithm process accordingly pre cross determine label effectiveness denote dirac permutation category acc dataset manually contain procedure like accuracy relationship discover run generation property reflect assign datum vary case center employ cluster small center maximum identify experimentally set detail determination figure show synthetic run accuracy performance tp rate cluster default threshold discover reasonable also discovery relative cluster discover mean increase discovery decrease receive performance situation tp run test validate complexity comparable still variational running increase two validate uci uci type cluster equal sized law law dataset law limit manually dataset estimation law curve law size represent law tendency uci cccc cluster shape tune experimentally receive well value default
correspond four transition reward everywhere goal discount finally depicted agent blue car bottom move speed great move across central turn driving purpose test mdp five correspond driving car drive additionally previous discount tb trial environment performance observe scenario scenario relatively action negative observe also two around single attain world reward similarly correct hypothesis consider effectively overall performance observe sized domain scenario greatly contribute expert conclude applicability approach action reward tb one illustrate reward I imply x x j complete amount mass place fundamental non step keep contain repeat last computation expand recursion conclusion structure bayesian write denote correspond concentrated q write conclusion action simplicity computation yield since xx xx since strict prove bound separately neighbor state lemma set select randomly moreover say situation situation consider situation eq alternative immediately hypothesis equivalently j everything let inequality markov replicate theorem result acknowledgement project la team corollary pt address task enable expert informative lead theoretical illustrate applicability complexity discuss light exist applicability class motivate recent also discuss manner agent likely complex system artificial retrieve observe interact relevant agent lead reproduce part behavior numerous robot system simple consist target task unlike combine agent design batch typically acquisition process stage guide acquisition recent work lead adopt desired contribute analysis main agent interact accommodate different fact user unable demonstrate intend want user experience difficulty may explore additionally recent na I agent provide human obvious observe user tend agent reward reinforcement user accommodate feedback address discuss form expert policy learner recover efficiently target approach provide reinforcement error learning explore discussion remainder relate work learn particularly discuss exist paper sample provide discuss research report expert form human feedback change demonstrate provide user refer survey work reinforcement learn formalism seminal appealing aspect learner observe action learner perspective explore reinforcement learning replicate execution formalism task lead visit observe replicate infer behavior explore reasoning theoretic provable guarantee introduce cast compute identify target mcmc expensive reward avoid determine aforementioned likelihood derive gradient algorithm observe explore propose classification situation train work acquire take early stage guide acquisition aim reduce informative random predefine situation ask demonstrate behavior informative situation also enable human situation encounter adequate work incorporate confident allow stream stream sample uncertain informative query explore early uncertain unlike highly informative unlike query approach differ active accommodate relate select expert unfortunately determination gain extensive computationally costly address provable modify active generalize adapt cast multi sample provide provable extent conclude point learn feedback form expert explore learn explore agent see purpose observe observe may form reinforcement architecture improve introduce paradigm human feedback reinforcement extend suitable convergence provide complexity experimental form expert illustrate applicability scenario discuss applicability face result broad perspective trivial multiclass application background material inverse reinforcement formalism contribution mdp problem maximize reward formally tuple x process agent action x x expectation take induce mdp denote follow x ax x rx ax mdp describe maximize discount reward mdp deal reward give cast agent desire policy reward function impractical maintain accord reward kx q kx denote indicator function complement select action perturb version perturb uniform limit action update every case perturb policy develop determine mostly space define kx denote identify action assign action write hypothesis index clear prior induce equivalent normalize accommodate possibility inaccurate set word lie indistinguishable mean induce relation central alternative large alternative neighborhood x hypothesis neighbor case every word imply simplify particular noise x adequate simple define position introduce reinforcement mdp determine jx ia p accommodate situation rely fundamental lemma class c nh measure neighbor great neighboring state predict yet case informative coherence coherence parameter quantify query finite conduct hand establish measure noise history multi class fundamental fundamental require convergence soon apparent alternatively prove sub modularity ensure sample mass correct information concern convergence active theorem extend however due class bound obtain aforementioned work interestingly dimension corollary conclude specialize particular classification theorem generally multi applicability namely discuss feedback learner efficiently formulate mean manner hypothesis assess start neighbor action every connectivity graph induce discussion reinforcement reasonable work yield hypothesis sufficiently rich space hypothesis space already trivially sample generalization support option research focus negligible tb reward noisy highly query use rate version denote action presentation assumption informally correspond every reward implicitly work focus deterministic explicitly mdps action per property algorithm follow analysis multiple allow pose policy overcome difficulty update admit multiple optimal state equivalently likelihood observe x x consideration estimate get q conservative slow eliminate algorithm remain prior determine set ta unfortunately allow degenerate focus state least able retrieve complexity describe stop formulated correct hypothesis identification optimal following define admit devise recover strong theorem readily integrate feedback approach provided identify task formalism reward information pair indicate difference receive reward environment teacher another mdps action however include noise reward inaccurate give correspondence integrate bayesian set learn accommodate would time reward replace query query direct query purpose disjoint ar ax situation discrimination particularly evident couple informative contain dense reward illustrate complexity literature applicability approach use determine mdp perturb policy carlo provide correspond reward remaining build probability reward comparison purpose active select query criterion main accommodate notion bad notion query potentially require evaluate possible fundamentally expect method mdp pair small mdps first set sized mdps transition reward specifically mdps size mdps independent serve two purpose illustrate applicability method mdps enable quick comparative relevant tb curve accuracy early stage outperform accuracy clear view less clear idea
maximizer design generate adversarial completely arbitrary make flip increase omit version graph probability te te active learner argument number average span purpose v te result span prove let query span topology significantly work span sophisticated adversarial model address mistake bind budget edge edge definition subtree subtree root tree subtree sign sign circuit edge vertex edge belong circuit obtain edge circuit contain circuit input way circuit least solely circuit remain edge circuit induce adversary rise load one ideally circuit cover minimize circuit cover choose load sake presentation simple version version constrain circuit cover classifier circuit preliminary draw span tree query connect edge predict create circuit query test load edge test edge lie circuit along edge aspect tree load rely span draw preliminary subtree arbitrary belong rooted circuit prediction phase dash thick black circuit black line circuit belong remain label predict since presence prediction mistake circuit predict simplify constrained circuit build visit visit visit help visit visit te ki jk r incremental fashion visit visit solely backtrack current time backtrack edge node necessary proof visit occurrence terminal circuit select let circuit correspond correspond circuit circuit circuit part root load increase circuit phase increase figure label ratio ratio parameter edge label proceed label edge label last execution moreover execute satisfy initialize arbitrary component constrain circuit covering describe refined circuit cover classifier use reduce mistake step split query subset choose test consider subgraph connect component predict terminate mistake satisfy label adversary randomization beneficial two span tree accord span return uniformly adversarial labeling mistake show edge span constant optimality adjacent instance hard conclude requirement v prediction logarithmic case algorithm need loop parallel link sign graph social balance adopt index regularity upper lower mistake three active contribution notion circuit cover mistake working extension recursive decomposition social remark proposition universit di universit di di universit di motivated develop sign correlation index measure regularity batch algorithmic contribution introduce link circuit mistake sign social biological vast application include spam gene number started investigate negative relationship web interaction concrete network tag friend rating user develop trust another wikipedia vote cast favor attract towards sign determine relationship positive social link infer sentiment instance recommender early date conceptual theory understand network mutual classified network recently accord cycle correlation small node sign graph sign positive correlation sign deal sign study theoretic index characterize complexity mistake pool instance standard control observe efficient approximation practical link attention structural due many social context accurate sign variant hard design link simplify notion protocol correlation bound active sign learner classification guarantee sign active learner receive input predict implement edge henceforth label positively undirected partition edge red due cycle order bad cycle edge partition partition obvious quantification associate cluster partitioning regularity mistake cycle except contain edge node adjacent relationship individual link constrain bad clear fact relate give illustration bad moreover edge cycle removal bad cycle disjoint cycle proof give big give clique labeling restriction cluster two regularity minimum clearly least motivated balance long cycle multiplicative path connect easy relate standard graph laplacian degree moreover see e yet resemble eigenvalue eigenvector computation relaxation similar eq amount eigenvector heuristic build minimal sign classify matching resemble risk minimization guarantee heuristic odd edge prove new link setting express index active mistake protocol accord arbitrary learner receive predict reveal know mistake occur mistake bad within access dense edge edge mistake online exist label match version algorithm predict label edge follow predict assign either otherwise predict since mistake mistake expert lemma obtain majority expert mistake arbitrary sign v theorem prediction online state next theorem theoretical relevance computational hardness surprising compute implement unless use mistake unknown labeling training arbitrary indexing edge represent random training partition arbitrary predict put edge label give approximately bound mistake predict test mistake make predict edge training result minimize index sign z exist u cm give concrete find rewrite order moderate carry nontrivial reasonably tractable relaxation index restriction clustering partition make minimum cost cluster labeling clearly least social motivated theory sign rule easy laplacian matrix node hard hold batch
gb ht ordinal level hmc many assess focus reconstruct matrix copula particular outcome correlation imply look see htbp dim fa sample dim fa dim e order mix hmc elliptical slice necessary within copula behaviour insight another future explore method elliptical copula relate hmc acknowledgement largely anonymous ep department college learn gain modular parameterization distribution among copula combine flexible univariate marginal family likelihood hand g estimation idea observable ignore framework copula complicate point efficient advance hamiltonian implement way construct distribution contingency case sparsity idea combine building novel monte framework extend cumulative graphical modular parameterization say bivariate cdf rewrite unique define find statistic machine construct mix copula univariate flexible univariate copula far motivate use machine comprehensive discussion copula machine perspective provide overview idea back copula transform pmf intractable trick go hard check discrete cdf pmf continuous integral many copula domain reader familiar probit observable integer interpret copula gaussian copula entail issue general mcmc constrain field size experiment copula final fully parameter equivalent sample accord cdf obtain copula deterministic p distribution parameter underlie copula although marginal reverse constraint give fully method marginal ignore possible mild condition consistently application nuisance assumption dependence understand social base dimensionality second mcmc extend conceptually drop model remove necessary depend choice refine imply correlation decomposition latent variable explore sample data point condition univariate truncate mixing facilitate particular condition point boundary reduce space move improve mix sample column gibbs approach truncate gaussian though slow strong induce tight truncation deal n boundary later shall rank shall special exploit runtime hamiltonian monte hmc move bring sampler potentially place variable move hamiltonian context distribute hamiltonian physical sum energy hamiltonian compute evolution differential eq eq hmc allow initial repeat hmc chain position desire gaussians hmc plot truncate matrix general remain boundary velocity time particle simple ht hmc velocity particle sample find supplementary discussion position impact velocity bound hyperplane hmc initial hmc sample reach pick exist hyperplane velocity hmc candidate consideration bound large induce computation hand explore constraint problem constraint explicitly clearly gaussian task alternate wishart use oppose replace submatrix successive condition univariate truncate condition step hmc step gibbs via allocate cost explain sequel must satisfy extended likelihood discrete variable adjacent true constraint exist see variable deal boundary develop search efficiently practically time hmc particle decompose describe envelope space level dimension generalization level trajectory level blue assume red find blue curve find large level large root small green blue happen scenario repetition fixed level occur experiment thousand curve strategy take sensitive convex hmc summarize normal encode pt p pn jj jj
phrase scope decide phrase use coefficient prevent phrase form choose phrase pass allow long phrase representation reasoning involve phrase publicly york york news state company microsoft page google amazon reasoning example fourth phrase well achieve phrase several skip hyper dimensionality set already achieve phrase sampling hierarchical softmax subsample frequent token result summarize dimensionality subsample accuracy skip analogy train news achieve considerably surprisingly hierarchical softmax train frequent maximize phrase analogy increase softmax result reach reduce gain different representation learn manually model comparison seem representation softmax subsampling subsampling subsampling de master entity short phrase model representation gram reasoning skip gram kind illustrate capital chi city close token well skip gram explain objective input softmax nonlinearity train predict represent word relate appear frequently result previously work neural amongst author word et skip gram huge big skip gram model learn attribute order magnitude skip model fraction complexity c model planning skip microsoft token model gram phrase empty vocabulary several representation phrase skip reasoning introduce bag successfully train order previously thank computationally great learn representation entity frequent contribution negative extremely simple learn selection hyperparameter affect training work somewhat combine representation simply phrase token combination simple long piece minimal approach attempt phrase recursive operation phrase technique open source google chen google google skip gram representation precise syntactic semantic relationship improve train frequent speedup also word representation describe hierarchical softmax inherent limitation order phrase air obtain air motivated find million phrase word vector space natural word representation idea speech recognition nlp skip method high amount network architecture learning skip gram multiplication implementation train explicitly linguistic somewhat result france close paris skip gram show subsample frequent significant speedup noise skip model compare softmax use limited phrase phrase make skip gram considerably expressive recursive autoencoder benefit vector instead base treat individual token evaluate phrase analogy set correctly finally skip gram addition capital obvious skip nearby skip word predict formally give sequence word gram maximize center expense formulation word formulation impractical term approximation softmax softmax advantage instead need evaluate node use skip distinguish target data sample experiment small training large negative softmax property investigate rd outperformed task try corpus occur million information skip gram observe occurrence france paris benefit much less observe france applied representation train several million counter imbalance rare simple word discard formula choose frequency rank subsampling formula rare word section estimation subsampling reasoning task consist france vector france discard word specific paris task syntactic quick quickly slow semantic country capital city consist google discard
smooth next point determine viterbi alignment pass probability conditioning find present iterative error considerably consider likely hide thus simulation advantage iterative right viterbi correctly restrict coincide viterbi number induce present restrict behave correctly drop fact viterbi classified alignment viterbi either change expect error much pick know whether probability generally hmms transition negative question algorithm show corresponding probability probability lower also show presence transition classification finally occur shall stand emission discrete need clearly transition case initial probability depend stationary viterbi alignment depend backward time correspond let minimum corresponding reversible uniform unchanged general thus corollary follow remain obtained alignment tie break favor generality calculate case zero transition emission emission shall big strictly rest viterbi path state zero hence posterior imply viterbi alignment viterbi path optimality observation affect positive relax transition let emission call condition intersection emission disjoint cluster existence imply statement assumption cluster matrix meet element since primitive also condition general assumption vice easy modify suffice atom every primitive fix empty fully word time corollary constant word improve stop distribution obviously depend would however possible tail constant proof appendix distribution tail independent follow stationary process possibly variable exponentially identically follow stationarity proof classify accord chain corollary alignment since straightforward replace state possible original viterbi restrict viterbi alignment drawback substitute consecutive state adjust ensure alignment remain observation viterbi alignment calculate else state maximize nm calculate probability alignment first lowest find point maximum point classification strictly alignment constraint impose state viterbi else example iterative exclude believe phenomenon happen viterbi alignment high alignment pass prescribe viterbi range iteration low unconditional iterative preliminary point recall viterbi alignment decrease start classification error iteration algorithm viterbi likelihood restrict consecutive algorithm fix iterative available minus well certain need number ten iteration example decrease possible improvement half cccc log likelihood ccccc log account thus probability replacement big decrease correct method effect adjust alignment iteratively much additional iterative table restrict error state note issue iterative cccc consider probability emission sequence restrict viterbi viterbi state segmentation iteratively alignment unconditional probability restrict viterbi sequence ccccc behaviour restricted compare example average minimum restrict viterbi classification iterative average error demonstrate way threshold iterative take account log ccccc average whereas iterative sequence characteristic case recall much big respectively reveal imply example might due bad expectation little rest everything arise fix time increase hold q stand viterbi alignment viterbi alignment strictly big viterbi example happen hmm imply initial big later suppose positive begin state imply follow remain big back q viterbi alignment time restrict viterbi let find viterbi time secondly way viterbi viterbi viterbi stay stay obviously inequality recall right value difference increase accuracy usual backward backward calculate forward recursively recursion q beginning end choose big enough large big enough negative get arbitrarily hand grow away indeed q side imply number error recall state viterbi q arbitrary low together case state initial equation respectively eq thus use markov denote borel condition impose dimensional integer prove induction rx two start define markov sx cx possible chain transition determine transition transition element coincide transition sx ms c r eq irreducible take case conjecture criterion exercise section proposition remark mail probability alignment advantage approach iterative improve viterbi alignment reveal state alignment condition segmentation irreducible conditionally sometimes name sometimes regime observable emission usually borel assume density hmm stand regime since learn notation keep integer drop widely field language many hmms problem consist unobserved underlie markov mapping call alignment impossible alignment sense goodness alignment introduce give goodness alignment minimize risk risk introduce viterbi maximize I dynamic programming viterbi alignment unique despite popularity viterbi viterbi alignment pointwise maximum posteriori follow depend alignment pointwise viterbi classifier classifier purely disadvantage zero transition alignment zero alignment low even zero mention popular viterbi classifiers viterbi viterbi smoothing hence calculate big small number aim define new risk viterbi proceed differently alignment modify decrease alignment considerably introduce decrease trivially sure time hide even expect correctly classifier trivial viterbi classification viterbi arbitrarily data classifier together state viterbi must probability classification sum typically bind question answer matrix lower present showing
parametrize generalize know unknown robot experiment body typical robot fig complex reinforcement rl paradigm however rl thousand physical infeasible consume rl speed reduce interaction model rl q td use improvement interaction suffer resemble underlie model propagate policy inherently principled way accounting optimization prediction generalize learn concept situation key single robot game table generalize consider task policy capable solve task prescribe learn individual require generalize task learn robot task unseen test rich parametrization local subsequently achieve combine successfully rl use robot deal implicitly local policy successfully apply robot mapping source task new learn mapping meta across task independently elementary movement policie one task return generalize policy task controller learn task search unseen policy task framework learn flexible gaussian process gp forward model term achieve search successfully apply promising address policy dynamical x search deterministic step parametrize desire minimize task robot propose jointly generalize classical scenario assume share flexible aim obtain relate overfitte task hierarchical learn state task u generalize unseen computing solely change task line represent control red circle policy smoothly generalize across intuition generalization input parametrization five determine control circle control signal solely assume implicitly power represent multi high summarize initialize record update gp dynamic analytically policy parameter e bfgs robot record training initialize subsequently see underlying line consistently policy task augment relate two x state location task index approximate gaussian c state task serve controller although assume uncertainty reason define allow well compare induce uncertainty policy regularizer make overfitte approximate long averaging correspond task behind expect controller task controller controller necessarily good long x analytically joint p q analytically u pf iterate moment matching time horizon marginal long predictive tp c specific solve analytically choice gaussians sum deterministic analytic analytic fig gradient base bfgs analytic computation grow quickly policy repeat define x eq take derivative chain yield need compute derivative u control approximate experience time compute tb bar generalization bar hierarchical generalization boltzmann deterministic bar uncertain fig illustrate horizontal target position height bar trial per means bar experiment approximately cover controller neighbor cart incur hierarchical rw ic controller performance nn task combination failure nn ic could successfully combination nonlinear eventually decrease rw ic controller successfully perform balance task train fig controller successfully balance cover uncertain curve across cover average cost might cart offset cccc nn ic rw ic cost summarize cost task average nn ic rw ic reliably incur balanced wrong cart generalize unseen task lead generalization optimally policy blue combine rw ic task red circle green star rw ic generalize smoothly local difference rw ic network combination sense policy policy nonlinear policy performance tb stack camera visual sensor degree freedom base open close arm control configuration six duration camera robot robot stack block specified camera training stacking require multi configuration dynamic camera coordinate signal change learn robot use u degree freedom trial amount experience stack suppose stack test stack b tb horizontal axis control signal change block part controller block define teacher allow generalize trajectory single match robot observe distribution expert trajectory policy minimize kl learn small learning use light capable achieve system inspire context design modeling challenge robot control directly controller function comprise function share unlike vector correspond ball frame task expert single learn expert generalize demonstrate behavior task particular task ball region location blue blue box cover performance iteration alg ball center blue area successfully radius give task library cart location could controller target cart location cart location position
obtain finite dt eqn analysis fractional albeit exponential support define function solution due consideration eqn symmetric reference correspond reservoir furthermore cluster describe process newly create process eqn aforementione birth death process rapidly fractional ordinary differential variable know equilibrium cluster transition fractional poisson kolmogorov equation great markovian arise exponentially fractional function recover fractional time point fractional eq fractional twice usual eqn birth place nothing else complement readily recover normalization equation fractional kolmogorov fast grow branch shall fractional process eqn ode first variable leave hand poisson process infinitely consist time formulate eqn ode kolmogorov fractional discrete cluster cluster consist particle form one principle dynamic fractional poisson belong wide class wide self large time dynamical tool non phase far aim letter bring fractional poisson community deal equation seem present deal combinatorial inversion formula essential lemma define infinite poisson eqn follow triangular element eqn formal inversion eqn form recover probability eqn factorial moment give rigorous array inversion original recover infinite matrix desired originally converge prove eqn identity imply eqn infinite
process mutually subset definition weak disjoint multinomial complete poisson intensity show intensity exponential exponential family closely counting marginally independently family probability q respect family coefficient I e parametrization conjecture likelihood binomial regression mild discussion give satisfying case eq pareto refer logistic let uniquely call logistic property since proof give distribution show experimental binomial probit link satisfactory ht cc cc logit probit ht normalizing sequence c intensity likelihood estimator fail propose simplicity denote include hyperplane original problem space convex half compact penalize likelihood pseudo concave confirm concave estimator desirable property one set intensity regardless penalize likelihood become exist furthermore know admissible kullback leibl variant multinomial model multinomial logistic asymptotic theorem maximum asymptotic conjecture binomial regression maximizer q meaningful terminology maximizer additional converge binomial true describe robustness estimator serious support absolutely intensity fall priori full support become family hypercube assume compact support chance precede treating correctly approach consider maximum posteriori adopt admissible leibl prediction admissible density smooth helpful discussion exploratory stage denote x enough assumption converge since monotone uniform belong thus note conversely must contradict sufficient eq finally belong uniqueness follow concavity existence similarly hull open origin tend origin claim let fix boundary tend contain tt complete fix tt complete proposition definition remark logistic know paper binomial rely extreme logit exponential family poisson extreme family observable one function function distribution correspond complementary three highly diagnosis political g without covariate converge parameter converge theorem formally exponential family precise different setting asymptotically become unless converge indeed approximately process intensity measure paper various binomial logistic result become remarkable measure family family family geometry g definition theory
irrelevant quadratic apply domain sample code I codebook fixing come obtain code experiment domain method domain totally image class extract texture bag histogram conduct split split time semi test code representation accuracy notice poor around cross data email email dataset email spam spam due significant difference train unlabeled domain occurrence frequency email code target domain represent code classify time spam detection solid evidence cross outperform case one significant sparse code cross sparse code criterion domain utilize encourage develop national novel university state ny usa coding extend domain distribution impose mmd code encourage domain spam advantage representation usually real label propose share label help try representation use domain structural induce correspondence method feature et transfer component across domain via maximum mmd supervise recently code attract attention representation represent sample combination codebook number improvement n domain label target label codebook column codebook reconstruct ki ki formulated th code minimize class maximize semi supervise formulate supervise distance intra minimized pair source target adopt mmd distance code sparse
node speed cost operate bethe admm bethe admm add bethe subproblem solve lp operate clique type behave like alternate mirror descent bregman divergence quadratic alternate bregman divergence unit simplex divergence multiplicative depend term linearize show gradient w definition bregman divergence need establish g bregman divergence convex lagrangian e optimality satisfy optimality satisfied optimality optimality optimality showing converge bregman admm q bound sequence bregman define divergence euclidean divergence kl divergence assumption step sufficiently practice use establishe let sequence bregman assumption average report exceed admm plot residual figure admm plot runtime optimum software implement mac memory memory core problem runtime objective terminate optimize lp solver run several time terminate server memory increase especially scale consumption rapid similar situation observe server memory even parameter clearly illustrate memory mm c c bregman mirror generalize gradient inexact admm bethe fast program acknowledge nsf technical institute w university support yahoo require size sequence bregman rt rearrange rate objective residual ergodic c kkt assume sum yield divide side yield mm mirror generalize bregman divergence paper multiplier admm bregman unified framework admm inexact admm bethe admm convergence complexity case fast factor mass fast admm highly optimize gpu recent direction successfully broad range apply understanding refer reader comprehensive therein equality n machine cast minimize composite hinge regularizer nuclear mine split splitting solve augment lagrangian define dual penalty admm follow update computational trivial complexity admm lie amount penalty inexact generalize online bethe admm add additional bregman update penalty term far quadratic amount quadratic bregman type greatly boost use bregman e leibler kl quadratic dimensionality composite mirror bregman bregman term large amount replace quadratic divergence objective mirror descent bregman divergence outperform factor dimensionality function bregman point bregman bregman understand iteration dual accelerated rate like size admm point admm penalty bregman bregman divergence answer quadratic also introduce bregman divergence short bregman update variant replace admm bregman choose proper bregman divergence also inexact bethe consider special method global factor linear exploit lead parallelism even order magnitude fast software hundred server hundred rest establish consider illustrative application convex b commonly distance I ib replace penalty augment lagrangian bregman divergence bregman bregman divergence necessarily b l observation role standard admm augmentation add penalty update significantly goal update get use bregman require projection choose solve form kl alternate cast update feasible especially augmentation rather logistic function kl unit concern additional bregman update update allow dual update bregman generalize divergence share update update bregman bregman divergence sure divergence quadratic admm proper bregman close solution note argument problematic need linearize eq bregman linear convex need gradient strongly use gradient generalize bregman divergence rely specific idea update update respectively bregman base augmentation term sparse admm one iterative bfgs quadratic closed linearization quadratic eq mainly update linearize solve separable linearization simplex ball unit amount onto unit algorithm term kl divergence form solution
denote optimize number extract denote hold write identity row usual square identity partly unknown success probably possibility project signal uncorrelate projection procedure desire extract definition virtual predict certain order task intend planning purpose contrast aim approach build fashion slow extract behave highly consequence estimate certain prediction focus find globally low deal robust algorithm empirically criteria meet measure suitable default criterion motivation reinforcement setting place aim state scenario vision intend make vast incoming look information help plan need behave capable predict outcome action crucial control theory attempt put representation environment differential truly organize feature characteristic look pattern prove valuable field concern slowly vary sub signal classification many task much task self organization field whole spatial self organization cell drive force blind separation successfully perform basis signal use like select certain meet suitable notion like bottleneck focus concrete appropriate arise turn hard must optimal solution start tractable set related approach notion instance scenario agent retrieve invariant combine notion well code ica reduction component paradigm propose independent select well strength approach previous input extract inspire optimize linear mapping consist time point order avoid trivial constrain output component must uncorrelate repeat component avoid use training expand shift sum extraction rr equal transformation solve popular default regard approximated combination history identity denote unit ht ad like adopt repeat indeed expansion also dimension agnostic nevertheless mention strategy solve may extract massive advantage initially fit possible transformation extraction formalize formalize fitting briefly q sense prediction model sense q denote formalize formula fit write overview notation sometimes happen invertible regard practice away critical behind corresponding indicate eigenvalue threshold one multiplicative inverse proxy compact default however mainly inversion appear intractable every directly propose follow relaxation informally problem optimally input global write choose perform equivalence propose r global must reduce calculate relaxation gap depend manner zero signal usually overfitte offer overcome overfitte reduce overfitte propagation subsequent ground intuition one partly noisy prediction formalize idea thus iterate ce ef df globally tr optimally certain problem question know experiment improve quality time investigate formally subject basic would sense bind error involve overfitte plotted line overfitte dimension everything random orthogonal transformation add generate average prediction obviously indicate kind overfitte conclude algorithm noise make fit criterion projection projective simplify projective frobenius strong orthogonal prediction hold obvious criterion imply projective projective consistency criterion benefit get projective consistent r see thus since projective right analog relaxed like like projective agnostic model solve sort upper left generalization generally prediction projective hold line prove need mean ss rr ss formulate deal define bad q problem th extract would lemma global tr small eigenvalue upper every transform analog large preserve transformation perform lemma
rna seq read align position indicate coverage correspond decrease coverage read thus support site support pair span read pair specific connectivity genome site truth annotate gene convert assign atomic nucleotide see presence support rna seq read mask region alternative penalize sequence address model hmms recently hmms discriminative support training margin correct wrong sequence denote closure figure via discriminant satisfy property efficiently viterbi discriminant piece transformation real transition indicator function parametrization function constitute parametrization want enforce path wrong follow optimization complexity whose adjusted slack allow training cut plane grow subset work directly objective adapt function propose empirical j cut plane prox function minimization prox solve aggregation cut estimated optimization adopt elegant cutting dual aggregated cut increase estimate empirical loss remove cut detail subgradient hinge huber loss logistic outline bundle kb k k kk seq study high quality accuracy infer align rna seq aware alignment tool rna filter reduce number alignment annotate filtering criterion operation align segment support annotation filter site predict genome publish cut genomic nucleotide consensus apply recognize annotated site subsequently whole genome rna seq read coverage site derive gene annotation able assess alignment subsequent use subsequently repeat filter rna seq generate proceed genome read annotation genome annotate model predict cross setting filter rna seq bt svm algorithm utilize training quickly expression duality sufficiently iteration rna seq average ex evaluate rna seq species choose three whose extensively annotate quality rna seq annotation evaluation nonetheless assess reconstruct whole genome annotate validation detail criterion quality assessment infer nucleotide predict correctly criterion evaluate nucleotide annotation boundary evaluation predict criterion assess sensitivity predict define annotate infer latter proportion infer harmonic exploit quickly lead substantial training accuracy less assess accuracy effect confirm infer increase expression mean terminate dramatically subsequent third assess approach consist hour prediction converge benchmark reconstruction method evaluation adopt method comparative reveal always notably robust issue filtering decrease alignment drop fig infer appear diverse maintain assessment boundary predict nucleotide gene correctly whether read filter code see legend f ex yes ex c yes yes yes perform bold main precision alignment filtering train rapidly reconstruct show apply sequence depth clear modular genomic prediction make seq code assess extent find minor fig extension development additional prediction recognize site desirable label graph conceptually instead citation seq site error underlie read galaxy software grateful comment gr pm foundation ng grant mu fellowship also acknowledge project environment ib rgb laboratory max molecular biology machine group computational unit biology ny usa equally throughput sequence seq technology seq nucleotide resolution reconstruction technology may genome annotation mostly de structure primarily infer genome method reconstruction rna seq machine derive read utilize genomic site accurate alignment method code matlab predictor galaxy seq throughput sequencing rna seq
improve section considerably phase optimization generate phase run iteration optimization call iteration limit output describe property problem statement take stochastic oracle g firstly conclude call iteration sample size post call stochastic remain note definition q factor bind small term dominate improve statement define respectively set compute call stochastic provide part part denote lemma eq equivalent almost surely subsection deal situation order order oracle information see distribution smooth describe gradient denote q deal sp iteration reduce search apply nonconvex secondly stepsize policy would nonconvex respectively reduce establish call find solution eq complexity subsection improve complexity input point limit size mass satisfy output procedure post limit respectively compute call observation g f f x inequality similar inequality noting imply b hold call set n observe call bound small dominating one paper sa solve unconstrained nlp problem computation solving nearly phase complexity result specialize base addition complexity sp weak dependence nonsmooth sp approximation sa randomized gradient possibly problem programming show possesse consist post short generate specialized optimization stochastic simulation seminal stochastic sa solve sp descent possess class strongly sp implement stepsize policy especially sa together iterate sa exhibit asymptotically convergence refer sa year see progress sa sp hand sp necessarily development theory sa iteration et properly sa mirror smooth sp mirror sa exhibit solving method show competitive widely accept sample approximation outperform technique average convex unify explicitly convexity convexity play sa exist general sp whose nonconvex focus sa satisfy nlp assume throughout access set gradient call iteration stochastic observe worth set sp slightly study aforementioned sp briefly outline follow either nonconvex call sp sp problem function even respect represent nonconvex unbiased estimator g finally simulation explicitly black g moreover descent deterministic nesterov show al trust method applicable even x contribution aforementione nonconvex sa taking iterate mirror sa sp select solution satisfy substantially increase deterministic relation demonstrate solve convex sp discussion secondly deviation randomized post phase list run stochastic available long development method see reference therein type mostly directly motivated work nesterov nesterov first complexity result establish term apply smooth programming problem acceleration scheme respect random prove nesterov term nonsmooth incorporate gaussian technique solve finding possesse sp problem interesting weak dependence establish solve nonsmooth problem objective carefully smooth organize introduce sp method base problem brief conclude remark differentiable sa possibly nonconvex sp hold augment assumption assumption require convex one sequence sp problem allow incorporate randomization sa method randomize initial point support call oracle b eq first part history hence take respect side l note conclude imply part hold display k gx fx moreover l k fx inequality inequality rest detail possible sake simplicity let stepsize e assumption appropriately choose also assume mass note relation similarly replace remark firstly select rate use computational effort compute reliability effort secondly stepsize arbitrary easily solve nonconvex bound reduce often suboptimal upper exist deterministic nesterov accelerate method relax use line procedure enhance devise
invariant rotation signature element derive inspire set plane complex coefficient proof describe method suffice rotation write sum six signature signature product product keep respective rigorous stem bound identity algebraic part rotation order interested linear recognize geometric unclear interpretation proceed information already take I I theorem repeat consist handwritten digits input record consist device location connect stroke straight invariant vector rbf fold r number feature make theorem projection operator set coefficient leave unchanged rotation satisfy homogeneous show rotation invariant rotation variation linearly lemma rotation invariant rotation invariant curve rotation invariant correspond obviously let arbitrary full q hand use idea proposition x piecewise concatenation eq closed span axiom conjecture theorem remark remark ex introduce rotation base complete give six online first iterate object mapping map vision anomalous future hand naturally environment recover gps derive mobile connectivity see overview character input usually trajectory coordinate rotation character device fact extract task rotation long base center image connection problem modern subsequent method derive lie algebra inspire rotation series close curvature integral mention primal curvature sketch chain rotation iterate procedure iterate compute realization motion provide similar fouri highly signal fail impossible letter usually letter mapping reasoning euclidean curve connect start straight line write term integral define integration example chapter much consideration purely sensible integration theory exist
tb extension conduct relational consist united vice relational consist relational datum encourage future scalability practical independently tensor become approach various learn account adjacency gain approach benchmark extension significantly popular basis vary network recommendation datum tensor method field predict multilinear scalable easy linear approach dyadic result various relational task entity resolution dyadic entity size create q adjacency tensor factorize entity row hold latent entity encode latent interact unique distant instance propagate party party moreover simple matrix vector form factorization square interpretation variation distribution factorization bernoulli adjacency tensor benchmark relational interpret view entry regard seek e set log optimize nature function logistic bt font ai ai north west south parameter original correct error enable least basis sparsity instance million know fact computer
optimum solution restrict stop first optimality sequential restrict achieve scalar scalar estimation discrete sequential estimator achieve restrict paper unconditional time develop efficient decentralize sequential vector firstly case consumption scale scaling analytically justify secondly consumption prohibitive energy efficiency duration encode optimum organize section linear restrict regressor estimator different sequential estimation estimator optimum unknown decentralize alternative approach formulate condition observe value yield tractable tractable decentralize sampling scalar letter letter minimize observation regressor incorporate way deal diversity specifically dimension coincide estimator e write ml coincide l l estimator semidefinite bb semidefinite recursive recursive gain apply initialize pi represent find stop sequential estimator stop sequential covariance I monotonic order consistent positive definite mean mse frobenius norm handle restrict denote sample accumulate sample algebra algebra stop noise except noise case path attain adapt moreover discrete time observation attain sequential stop unbiased estimator unbiased estimator q satisfy unbiased estimator unbiased estimator true event measurable interested obtain unconstrained accuracy variance expectation respect h minimize cf optimum stop writing term account cost represent note conditionally markov optimal stop cost iterate eq specifically original problem divide subproblem subproblem sampling equation hold time subscript simplicity continue stop follow cost small average u find stop refer multi optimal prove scalar vector case intractable scalar scalar specifically time optimal u z cost function whereas decrease theorem scalar optimal time target illustrate optimal cf increase tend cost low lagrange multipli satisfy cf see scalar fisher tb select p c next multi intractable write expectation change coefficient hence dr nh z h indice weight multiplicative occur region I n dr line stop function compute cf neighbor grid appropriate line value shape separate region region continue move towards stop e surface uncorrelated separate become linear firstly stop region conversely region tb select z return decrease lagrange simulation follow satisfied use algorithm surface stop offline separate region quite increase find separate hand formulation optimum stopping solution regressor assess unconditional covariance motivate since realization satisfie rule stop I sample objective process realization thus minimizing minimize member noise stop among sequential recursively computation e stop positive matrix positive scalar specifically infinity give estimator hence scalar unconditional unconditional ls threshold problem threshold unconditional offline hence unconditional upper bind cc stop average characterize resp unconditional due case energy decentralize conditional sampling sensor fusion fc responsible determine due energy sensor fc main concern decentralize sensor observe regressor tw square give u general noise also straightforward estimator observation available fc process decentralized fc stop time report straightforward may decentralize setup distribute implementation cover form overcome entry diagonal diagonal define element vanish general might entry newly normalize entry jj last number rr fc e nj k process fc dd local hence propose report process achieve fc sensor approximation threshold simulation satisfy transmission decentralize decentralized level information accurate approximation performance fc conventional decentralize traditional uniform version fc employ sampling fc use need prohibitive decentralized achieve approximation overcome alternative sample encode single greatly decentralize non uniformly sample fc fc compute sequence I control dynamically determined whereas time deterministic period whenever fc indicate since last transmission sampling linearly encode fc index sensor mean fc channel sensor determine fc transmission e fc uniquely cf increment occur interval increment occur sample eq fc approximation receive sensor signal global dimension fc prevent divide sensor compute time time receive fortunately global regard th element know fc sake consider together sensor entry write sensor fc ensure fc receive message unit rule specifically decrease least fc whether b k linearly encode transmission delay transmission write slope encode fc km k instant k center assume fc fc channel bound ensure fc measure delay accordingly regard dimension sensor fc perform compute tb initialization md v j sensor stop level summarize sensor run procedure fc summarize sake sensor fc separate parallel channel decrease identical threshold respectively sensor employ fc estimator infinite system uncorrelated correlate stop propose two horizontal axis mse normalize euclidean estimate htb j uncorrelated scheme attain sufficiently stop decentralized performance centralize scheme obviously thank fig simplify close centralized scheme coefficient simplification section obtain htb observe exponential stopping time since observe cause q sufficiently large element r respectively assume centralized scheme know stop mse theory theoretical numerical due due multiply computing suffer similar centralized decentralize moreover decentralized match well decentralize scheme use decentralize still useful behind scheme centralize summarize sensor centralize correlation scaling increase observe combination thus grow stop decentralize increase uncorrelated decentralize close vector centralize decentralize formulation stop minimize treat optimal show moderate estimate formulation
amount assess know upper low bound question without information theorem fully asymptotic perfectly question problem one know together positive small randomized improve logarithmic growth would expect dependency attain logarithmic log paradigm effect term original paper regret bound bound tradeoff apparent interesting tradeoff vanish armed know arm permutation investigate policy sequential armed bandit know separate value sequential likelihood vs assume likelihood design subtle open limitation dependence regret achievable limitation regret seminal one theorem fully rescale know beyond multi bandit include example consequence deduce regret derive suboptimal dependency exploit show policy provably well concentration likelihood extent improve remove exploration one confident could explore option turn subtle argument bound theorem rescale throughout paper r variance I assumption inequality valid investigate toy agent know generality offer convenient build initialization regret armed generality second definition obtain pay regret trivial armed increase decompose first event use decrease simple conclude choice gives observe conclude regret simplicity phrase simple armed know good match theorem next imply case unlike compare ii iii know bound proof dirac kullback leibler divergence measure eq two absolutely continuous regret hereafter favor normally lead simple calculation family value c bernoulli long bind know access bandit latter obtain q quite surprising without logarithmic logarithmic appear moreover match upper first denote law reward computation obtain uninformative display yield rescale general rate rescale risk absolutely measurable one dc divergence read follow problem gap optimal previous reward lemma q policy generate reward respect I I em chain conditional eq respectively drop dependency yield last computation cauchy schwarz jensen inequality prove plug obtain imply one consequence observe reward well bound therefore exploration vanish acknowledgment reference enter yield q note quantity measure see schwarz three display yield observe scale conjecture axiom support grant dms dms center finance department operation usa universit paris du paris france department financial nj armed bandit know arm positive regret several
similarity fuzzy one fuzzy fuzzy fuzzy sentence ft sim fuzzy web availability storage collection separately matrix extension co work idea build cluster different view sim architecture sim deal describe prove architecture parallelization ft sim similarity propose parallel architecture basic node deal multiple thus distribute connection fuzzy document document site account relation occurrence ft sim sequential architecture paper organize highlight relate similarity view three architecture compute co conclude exist clustering matrix characteristic relation instance cluster deal refer approach extensively involve interact object set cluster occur interaction object view task challenge resolve limit method author cluster similarity along view permit cluster use supervise label modify close matrix perform extension sim multi object create similarity set split similarity match sentence rather regard proximity power break broken focus advantage model represent describe view say partition adjacent graph paradigm type explain matrix use relation functional represent way corpus describe document sentence fuzzy representation membership membership essentially triangular fuzzy sentence document word document define fuzzy bound assign membership membership slope opposite fuzzy follow directly sentence provide q large ft sim site deal matrix thus fuzzy trying express relation occurrence ft architecture merge local site site similarity sentence figure link compute datum similarity issue initialize document initialize identity matrix denote update similarity step sequential present execute execute create static dynamic relation seem propose similarity site perform merging simultaneously aggregation offer instance datum adopt site measure directly appear site document aggregation present collection let document appear produce similarity equal aggregation denote consensus similarity current consensus connect account create feedback loop spread merge process execute merge architecture parallel merging keep unchanged merging ignore architecture efficiently ft sim splitting show split base parallel architecture treat set split become process aim document behavior architecture split matrix sentence matrix form equal split want document divide gain lose solution similarity pair sentence loop architecture spread inter comparison core gain decrease need matrix decrease sim co task propose level fuzzy document share sentence sentence fuzzy proposition multi view cluster focus document spread instance similarity analyze three multi de bp le paris france fr iteratively similarity fuzzy fuzzy sim deal offer fuzzy development storage space provide site computation expensive parallel computing architecture treat multi source sequential splitting ft sim reduce complexity thank keyword co parallel internet approximately store mining one research
patch force learn see supervision expert transformation learn assume invariance turn image invariance change contrast change variation unsupervise network contrast via discriminative jointly train layer manner several transform method instead enforce implicitly force surrogate enable task invariant discriminative previously propagation predefine derivative parameter contrast dependent propagation combine manifold unlabeled supervise self entropy create algorithm feature unlabele come later randomly image sample region considerable avoid get colored patch apply transformation composition four transformation follow translate scale multiply color multiply onto principal principal contrast power within subtract pixel patch sample obtain transformation patch unlabeled patch leave corner procedure initially patch get transform assign discriminate surrogate label softmax layer network th network experiment layer follow connect layer neuron convolutional layer contrast layer gradually cifar mean whole cifar cifar test table compare first layer supervise cifar exceed accuracy since table comparable art cifar exceed distribution test close reach cifar reduce mean way video pursuit invariant vary training result surrogate per class show fig baseline filter sample bias zero bar deviation compute test dataset apparent trend class reach surrogate surrogate increase change surrogate overlap overlap difficult adapting succeed validity also surrogate rapidly grows support become difficult increase sample surrogate also train surrogate clear lead per problem unstable classification around per surrogate surrogate get sufficiently complicated training consistently well dependence surrogate validation dependence sample avoid clutter unsupervise learn augmentation art translate well probable viewpoint invariance inter invariance method level supervision number rich would merge surrogate future acknowledgement acknowledge start grant rgb cs deep recognition extra labeling cost help performance augmentation main component unsupervised architecture end separate extend trivial class transformation patch train neural discriminate learn network successful competitive cifar deep contain image thousand recent possible efficient average augmentation technique achieve state classification network train scene indicate supervise well know visual labeling require label gets currently appeal paradigm
concept pareto objective dominate globally pareto pareto front improve decrease respect pareto pareto front pareto front uniformity pareto front classic objective single importance current objective objective evolutionary parallel make modern computer widely optimization value return non dominate controller dominate solution determine good reality controller c control ax mx horizontal orientation robot camera robot robot movement govern periodic angular amplitude amplitude movement phase angular send ms thank cycle robot keep vertical control position controller fully describe numerous design simple performance setup nevertheless constraint inspire central pattern generator keep robot forward forward invert controller forward regardless function constrain feature optimize behavior remain center begin simulation trial performance straight long trajectory perform population straightforward generation parameter candidate diversity measure behavioral lead controller compute self training use library step function intuitively hard parameter dynamical simulate simulator predictor avoid simulation discrimination representation measure choose svms score input available contrary classic regression krige svms dependent simulation robot open engine flat ground controller use controller record depth hz camera candidate equally available solution assess ability cope failure leave terminal right half lose middle lose middle front lose representative appendix simplicity inspire fair experiment test real case minute robot individual key record see controller controller minute competitive therefore choose compare failure e preliminary controller preliminary change try robot choose algorithm controller random start replicate experiment real robot ghz core extension algorithm core record cx motion capture uk report internal algorithms measurement report controller test failure good robot cover robot failure robot turn case fall video behavior controller adaptation require robot distance easily c cc c self reference f median search self reference median test investigate report improvement median sum test line trajectory correspond value obtain video typical behavior available discover reference p value performance algorithm surprise mostly minute versus test surprisingly observe difference initialize data suffer lot iteration use long author mostly reality optimize robot go backward sign variant least fast time median second search run obtain statistically local search nevertheless replication algorithm execution compress time spend test working suffer difficulty optimize always reality self reality comment action result second simulation unlikely effect reality lose controller nevertheless controller stem fact robot ignore robot unstable function moreover performance predict population maintain pareto trade chance high mostly uninformative critical local search failure conduct informative find perfectly align mainly cover nothing trajectory seem intuitive fast necessarily straight fast achievable pointing instance find straight distinguished fast trajectory intuition fast investigate encourage trajectory actually straight direction mainly position begin control robot pattern sometimes step robot start along next mostly divide actual robot allocate hardware specification substantially computer experimental median proportion median duration experimental process median minute minute significantly policy power year minute sum action fast show new electrical less minute irreducible several demonstrate experiment many different validate combination principle robot time inside predict difference reality self optimize principle implement future identify achieve principle investigate search analysis show classic stem experiment local estimate robot internal estimate estimation cover greatly differ consequence avoid self match self especially sensor sensor redundant robot continue sensor inaccurate self instance stability robot behavior limit strong failure extreme find controller self update avoid reality gap robot also take record internal self combination concept also similarity movement understand cause cause know move human seem change reflect know people human behavior similar essence acknowledgement thank suggestion support complex costly require contingency plan ready discover behavior self behavior perform differently self reality implicitly search behavior evaluate robot adapt removal broken failure search modeling behavior robot assess thank robot minute consistently substantially inherently cope demand operate place house expert point system happen rather ask clearly handle move numerous greatly benefit broken example situation robot qualitatively behavior break tolerance classic topic intensive system design failure another reaction controller pre design behavior cope failure drop behavior robot discover new behavior situation numerous review constraint explicitly test situation adapt situation algorithm e g td behavior gradient algorithm reasonably fast well suited author typically hour lack cope truly situation evolutionary optimize reward space e automatic design literature hour evolutionary robot reality run improve strategy important al stage automatically build internal whole consequence action body intelligence self internal environment paper minimal horizontal controller make increasingly computer improve highlight mix robot model part irrelevant include second action behavior learn algorithm prevent controller stage accurate model adaptation instance adaptation still work modern computer situation optimize robot robot require robot internal perfect system observe measure purpose reality separate behavior optimize robot svm reality optimization objective performance maximize perform stochastic multi algorithm show concept design adaptation robot behavior rely part differently reality adaptation robot create approximated behavior work optimize robot behavior avoid behavior unable world besides recovery class self break failure robot assess sensor couple discover literature artificial intelligence interested evaluate possible reinforcement primarily discrete control learn predict outcome action experiment step action possible test orientation robot predict action robot orientation body camera stochastic accurately start action new stochastic optimization maximize controller controller initialize model robot require computer nonetheless require self identifying arguably require result author require consistently author orientation robot sensor measurement identification self robot thousand evaluation necessary test overall build self application robot computing optimize often robot experiment probably identify self perfectly world behavior policy discover original behavior initial limit result transfer many reality problem behavior learn real al consequence occurrence significantly self predict need contrary al new behavior update potentially solve perform fast come price post behavior robot obviously nonetheless reasonably robot body probably affected field probably evolutionary emphasis behavior within simulation simulator surprise simulate researcher evolutionary idea reality
many recognition problem character recognition handwritten face combination adaboost adaboost consist sequentially instance obtain bad classifier goodness fit improve make assign weight focus see boost complete become datum object one motivate training amount classifier adaboost short classification adaboost sense misclassifie classifier context label descriptor vector descriptor set weight time iterate adaboost n delta select classified give weight identify incorrectly exponentially weak er perform classifier work good forest forest efficient decision bagging belong appear machine literature building let sample bag bag less take decision possible forest create inside subset get suitable object feature procedure estimate classifier procedure performance one already new tree rf becomes extract car process summarize time autocorrelation variability autocorrelation cumulative sum start deviation period variability describe variance color continuous auto car car characteristic car stochastic relaxation describe amplitude scale noise car respect solution hasting extract feature million star sampling process solution less hardware resource restriction overcome simplify reduce parameter instead divide error getting solve regular one per multidimensional show car magnitude feature calculate eps convert pdf fit eps convert fit eps convert plot convert train able star long star non star rr star star match extract figure projection case separation star usually overlap fortunately variable separate b object predict variable star plot project many case predict regardless difference big big validation consist fold iterate iteration classifier fold test performance fold see return entire fold precision precision recall define fp positive positive negative forest regular forest use work car feature set cross validation model forest car outperform car improve svm rf ab rf rf car car car car candidate validate find match show regard extraction extraction reasonable database thousand field extraction run parallel compressed file extract file file within extraction run run thousand feature file calculate survey start event way several million star cloud smc build star period fr bin ii get star randomly remove feature band b band figure see group star star separate star separate star overlap examine projection cluster rr star star strong show comparative include adaboost tune rf rf ab rf ab rf car car car car car million candidate candidate get candidate find eps convert plot eps convert convert plot eps convert plot eps pdf plots eps convert plot eps convert eps convert convert eps convert pdf plots eps convert case model periodic star plot eps convert plus figure show star portion predict expand concentrated cluster combine b b list series version random forest candidate candidate candidate old strong candidate object car improve kind star car feature car overcome confusion periodic star false positive periodic star believe dedicated module periodic star candidate strong list candidate object utilize public project jointly department california national laboratory contract national science foundation university california agreement national group cat center institute science paris paris france school national present dataset feature continuous auto set know improve state million dataset candidate validate candidate list candidate get match main survey translate ratio general current deep survey survey new challenge manual capacity datum thus huge datum detect survey train apply train consist million select candidate actual improvement result machine decade categorical variable classification forest expert result focus search projection learn apply datum classify object feature series give classifier
factorization discuss construction clique integrate factorization maximal clique method clique accord already include clique appear reduction algorithm store storage point absolute interpolation transform n I size whole space n easy recall may think region correspond region marginal ideally difficult compute distribution n want store p dp root store evaluating interpolation interpolation baseline storage order store standardized value evaluation point specify method brief grid take difference notably use cubic polynomial interpolation interpolation nest evaluation point approximate full grid interpolation evaluate construct limit sum possibility grid question combine information grid representation univariate eq knot return multivariate interpolation give store evaluate approximate storage method chebyshev knot prefer cubic interpolation less store knot choose knot space large cover grid increase ensure knot somewhat ensure remain integrable stage bind choose large evaluation grid storage cost reduction approximate effect remove order remove removed stage large clique large clique take bad find reasonable observe covariate one tree simulate model observe match compete covariate independent standard laplace approximation dependence sparse storage cost increase dominate approximation point take laplace second second maximize laplace likelihood parameter laplace sampling log interested log pointwise likelihood maximum consider trace approximation difference plot length take approximation second extent hour converge sequential converge less sample converge conduct determine ability flat capture show part package ability binary find remove player minimize find upper either upper vary various approximation become hard approximation value excellent log approximation indistinguishable scale sequential laplace get high indistinguishable suggest penalize reduction may penalty model nest effect structure sequential special automatically demonstrate contain level model ig belong group show fit value find laplace k estimate estimate common approach rely likelihood information impact able sequential outline situation use sparse interpolation modification laplace approximation wide acknowledgement grateful david helpful discussion science linear mix integral sequential sparse mixed class dimension one replace likelihood however fail effect binary total integral intercept cluster write likelihood product integral situation simplification sequential method exploit simplify fast accurate approximation exist approximation method demonstrate structure generalize response knowledge linear predictor component link generalize rarely covariate may allow heterogeneity linear x u u nu normal element column effect linear element effect particularly problematic consist outcome player describe probit covariate lie generalize player player knowledge component row row unless dimensional integral may think focus information combination effect induce conditionally posterior value effect conditional involve edge posterior pairwise respect posterior give competition vertex rely observation transform normalize undirected dependence simplification vertex clique subgraph maximal contain within clique maximal clique write may condition case exist
eq actor efficient ascent reach traditionally framework optimize paper variance set classic actor drive natural learning evaluate include variance return begin state classic equivalently criterion without similar approach method next expression proof appendix together trajectory policy direction refer computation restrict adjust approximate formulae approximation return avoid interestingly form dimensional case compatible weight choose evolve action probability action visit vector norm projection compatible identify approach drive policy gradient product product write value gradient adjust case similar inner product describe rx appear define state action assumption add denote inner denote projection onto outline make use compatible locally consider sequence surely q sketch rely fast schedule slow quasi assume feature terminal replace markov switch expectation eq iterate square eq ordinary ode eq uniqueness return actor suggest assumption optimal let countable extension almost surely actor extend compatible rl first algorithm provably optima practical somewhat inefficient use incremental least purpose example another option td td modification require obtain weight procedure rl extension standard criteria global return hard avoid difficulty optimality drop sub reduce notational clutter proposition take gradient side policy therefore mx mx treat follow gradient recall state use term iterated expectation cx exist first property chain use state assumption theorem actor actor mdps adjust expect return extend set surely locally reinforcement planning process mdps typical objective maximize discount several know parameter need several framework model actor typical actor maintain estimate actor modify theory successfully domain finance control maker reward criterion account statistic total denote control penalty recently consider rl actor actor improve actor reduce motivate actor essential dealing space require real introduce algorithmic address actor penalize however actor estimate gradient drawback guarantee another drawback trajectory drawback gradient trajectory build upon policy go policy gradient theorem relate penalize propose actor suitable
normalizing factor express similarity normalize share term work well define normalize page contain frequency contain report equality see numerator rewrite gx gx gx fx gx large mean become unlikely use page count apparent everything get decrease increase everything apart great formula insensitive choice adjust choose parameter follow directly minus numerator google mention equal upper n fx gx gx gx gx apart compressed page page numerator pairwise state restriction every permutation violate cardinality nonempty satisfy namely yield world wide google find kolmogorov universal kolmogorov string symmetry logarithmic additive ignore term equal prove generalization normalize nonempty approximate approximate share semantics theoretic foundation fact kolmogorov universal great effective namely theorem logarithm universal equal term kolmogorov shannon equal length code negative logarithm great code kolmogorov close google universal well approximate formula former kolmogorov google kolmogorov know hence finite name object element theorem additive term universal among admissible distance member additive wide include remark setting follow priori distance would exclude degenerate finitely fast want admit go nonempty admissible I possibly computable limit adapt string google event constant quantification arithmetic case admissible incorporate admissible feature numerator logarithmic negligible denominator normalize similarity scale among approximate approximate require database occurrence occurrence challenge meaningful occurrence term world wide count occurrence issue page count issue day search process search application interface enough allow google cardinality require pairwise discuss portion distance distance among pair plus require query require calculation element describe eqn leave cross validation class reduce formulation require web exact google find answer especially absence knowledge possible internet use search engine subject daily limit internet count computing page page application google web interface interface different google perform google books corpus occurrence million book book ever publish achieve efficient manner gram gram file occurrence file occurrence file extract occurrence database enough extract useful count calculate gram result web page co frequency less google web page formulation google page count interface count google books corpus word determine assign achieve work three validation need compute formulation spectral unsupervised number arbitrary conjunction distance randomness model capture pick gap value intra distance distribute randomly datum spectral cluster distance cluster intra cluster calculate gap statistic intra distance intra distance randomly generate uniformly distribute dimension run set compute standard deviation adjust describe implementation gap gram despite scope use result use web htbp first question vs spectral find classified pairwise gram spectral google google web google interface correct group htbp question compare spectral find correctly formulation gram corpus web interface perform consider class classify distance classification google search statistic group local maxima group correctly candidate thompson candidate classify possibly spectral candidate classification popularity party thompson google google web interface engine perform poorly formulation google say quantifie way nonempty name page name page page count google use english dictionary search engine page count name object object name similarity semantic state phrase finite nonempty word call diameter develop show distance especially non world google example triangle name pair similarity equality google good name pairwise together color versus versus us world web google google google grams instance superiority equality performed view fact section proposition lemma conjecture national mathematics science university email google wikipedia engine aggregate search include phrase term relative semantic search application derivation kolmogorov normalize distance pattern kolmogorov kolmogorov object compression metric kolmogorov real free similarity many recognition google instead satisfy nonempty property certain version classify grow significantly well object file carry property without red file object represent view text name similarity background information google engine page discover word relative google denote page occurrence page contain page index google logarithm widely many reference google give together give page google estimate term page xy fx xy possible hence semantic google notion finite string different length interested element universal jx theorem additive string denote coincide come pairwise world kolmogorov turn determine object heterogeneous anomaly google search aggregate page count search engine new treat nonempty apply concern synthetic version handwritten character recognition significantly except translate semantics semantics phrase relative name phrase like google code non google probability close information latter similarity accord set color google set however reason paragraph use let search term page possible return cardinality writing estimate page index search low event every google define google event web contain return google return google page simultaneously see paragraph boolean finitely application google google google consist web page contain every sense direct occur constitute google semantics term course contextual page occur indirect ignore nonetheless indirect context web event background event singleton search hence page count divide
classification result classification train image input pixel reduce order finally massive training output significantly reproduce plot image provide illustration show validation colour digit shape alternative nn library library backpropagation technique hide contain prediction approximately overfitte large nn run consistently varied lastly create setup read save comparison seek file simple dark challenge simplify great star spread thus galaxy star keep description underlie galaxy assume elliptical major semi minor parameter set calculate competition however galaxy several regression galaxy star galaxy predict pair clearly network I galaxy ii galaxy image star galaxy provide consist whitening validation network different table rmse galaxy image full galaxy input even naive good software package produce method score hide beyond improve slowly improve number without content star yield hidden yield although improvement indicate complex accuracy result investigate rmse competition note produce box us nd reduce input remove star rmse absence infer spread sufficiently well underlie galaxy accurately note could profile order magnitude alternatively reduce variable publication long gamma ray almost core massive star intrinsic rate aspect detector pixel galaxy star pixel autoencoder output normalise pixel list correspond value collection comprise ccc rmse value another mutually autoencoder layer image variation slight improvement indeed account prediction large indistinguishable well consider construction image angle star spread amplitude need completely reflect ability autoencoder perform fit produce marginal decrease rmse fitting pixel comparison part galaxy even accurately number plot galaxy star correspond reconstruct figure feature construct autoencoder image galaxy structure feature negative train autoencoder investigate compressed value galaxy decrease less run many ccc layers rmse show disadvantage galaxy feature however accuracy demonstrate eliminate add unnecessary structure make galaxy star spectra deep feed include autoencoder apply supervise dimensionality reduction pre refinement network parameter variant incorporate derivative even compute store prevent overfitte estimating demonstrate capability classification reduction classic digit mnist measure dark matter classification ray reduction galaxy use produce task typically tailor future expand current work e pool convolutional speed perform learn thank stage thank utilize uv intel author thank early utilize service education discover pg fellowship ridge complete university ff fellowship newton pt mm mm training tool learn laboratory rd md usa laboratory jj cb road cb public generic robust tool neural include autoencoder range cluster empirically close follow far use adjust automatically derivative store complicate difficult backpropagation employ criterion naturally flexibility demonstrate number toy focus recovery identification ray galaxy software http www ac software increasingly complicated typically interpretation recognition many way task use account moreover artificial recently start category consist set training quantity output infer mapping know supervise discrete value whereas continuous observation property ia ib etc type energy classification obtain demonstrate measure learn beyond acceleration describe obtain parameter second expensive perform magnitude analysis g often item divide label lack causal begin end consider assume begin unsupervised infer latent discrete similarity explain dataset unsupervise pre sometimes perform accurately include observation wish determine instead regression mention supervise artificial neural network inspire consist group receive product weight constitute non represent input output structure perform feed direct input mapping output accurately approximate useful introduction feed nonetheless backpropagation many numerous hide layer deep model mapping numerous public release efficient robust tool feed forward recurrent network achieve training optimum optimisation variant newton package information improve store fast product implement standard language currently development accelerate generic completely automate tool see implementation degenerate reduce evaluation require order achieve gain likelihood replace specified sample train convergence ability predict specify tolerance fail sufficiently accurate future problem network evaluation much rapidly also obtain provide release also feed forward call autoencoder use perform dimensionality procedure train type task dimensionality reduction autoencoder classify handwritten task determine project galaxy gamma ray detection detect dimensionality autoencoder galaxy finally feed simple order perceptron pass perceptron kind map via perceptron nn input layer run call activation monotonic g essential expand nn layer connection connect bias determine huge universal three bound hence sigmoid well increase overfitte activation function wherein argue remove biological quadratic autoencoder feed layer input map approximate operation layer autoencoder layer input basic autoencoder arrange hidden layer node define autoencoder consider half network central two encoder decoder reduce central layer space central reasonably term autoencoder intuitive pca indeed pca noting also perform task autoencoder objective input nn layer complicated dependence train nn unable relationship highest overfitte training consideration optimal hide input find function node basic prevent obtain particular node compare error wish predict careful inaccurate approximately remainder retain information representative subset whiten easier start last output whiten subtracting divide whiten commonly wherein shift transform may whitening perform across standard maximum computed whitening whitening output since transform consist subtract offset multiplying scale perform inverse apply offset target bias posterior describe see encodes nn reproduce play role relative penalty form predict input network deviation use rather discrete interpret input belong achieve softmax unity scenario probabilitie true unity nn start determined network set modify make obtain bias autoencoder mind boltzmann machines rbms rbm generative model learn layer node map input adjust gradually reduce autoencoder rbms stack rbm hide next repeat central reach network transpose decode training begin fig procedure sampling indicate value sigmoid hide visible vector diagram pre use feed layer connect hidden autoencoder however randomly pre learn shown reduce autoencoder pre central hidden activation output feed pre continue default initial either assign optimisation initial set define algorithm bias define magnitude relate factor value optimal proceed adapt newton hessian free describe calculate another multiply similarly detail derivative log hessian identity prevent become region ideally seek optimisation iterate procedure maximum moreover behaviour linear rescale parameter poorly scale difficulty method semi term prior invertible second inversion size neural address issue replace gauss guarantee positive likelihood use newton second prohibitive expense solve gradient vector avoid calculate extra identity subject problem applicable additional method even network improve backpropagation note descent iteration use optimisation output output brevity relative predict practice correlation square output expect correlation evaluate validation eventually result quantity decrease evaluate divergence behaviour optimisation choose quantity error square include zero note error discuss compare architecture node distribute complexity increase reach gain reduce architecture peak performance equally suited practice simplest overfitte one predicted computationally suggest whereby one add train noisy prediction prediction noise prediction provide accuracy original train train additionally evaluate ensemble evaluation less converge residual output target nn start new time make original output estimate standard user add iii offset aid optimisation offset recommend add original method determine require network compare gain indeed computation oppose fast method describe incorporate public release order toy point range zero prevent exact hide contain full use determined increase square decrease hide beyond obtain per predict toy datum create variable less place condition percent per validation single train full three node probability sum unity determine compare correlation decrease reach node hide point correctly summary ccc mm similar classification classification principal pca eigenvector direction set combination certain large pca limit however projection lead construct orthogonality ica find onto independence non provide natural constitute intuitive reduction special layer linear quick comparison traditional consider example single gaussian assume check pca calculate result find first layer node whiten pre autoencoder fall maximum datum one varied limit encoding might expect eigenvalue covariance curve straight linearity activation layer note conversely linearly central rather result percent pca autoencoder network dimensionality original decoding vary encoding recover approximate small neither curve exactly straight line linearity curve intersect would principal correlation percent latter ability determine node autoencoder provide gaussian point form hyperplane manner autoencoder error correlation give hide node percent improve ccc square feature amount error increase analogous multivariate degeneracy accord originally present plot point noiseless fully method pca unable datum would along straight horizontal verify projection onto lie linearity autoencoder layer dimensionality architecture whitening optimal determined square different number hide show reach
process model outcome weight call obtain turn sampler efficiency able model practitioner dataset weight inverse sampling bayesian model simultaneously unit determine population treat model count population gp w j integrate quantity interest posterior population procedure unique individual map correct multiplying cell sample product appear additional construct indicator normalize observe individual unit equal belong cell expectation survey determine mind true cell survey outcome parameter value infer predict survey response q logarithm typically calibration response score model mean cell mean cell variance j alternatively vary denote component spline use nonparametric bring difficulty parametric gp gps constitute flexible explicitly gps theoretical conjugacy include special various regard spline default denote coefficient kernel hyperparameter control local smoothness sample smooth path value account cell special kernel hierarchical cell reflect distribution weakly cauchy center deviation assign cauchy set standard residual fits utilize strong information goal default goal inference generally informative half smoothness heavy tail size match population discrete population construct covariate predict eq unit posterior interested population sample total outcome variable cell predictive distribution population collect case bayesian estimator theory choice construct classical variance replacement statistical checking statistical use fit prior sampling perform bayesian correctly result contain true probability check draw hyperparameter population conduct choose illustrative population assign scenario ignore sampling realization case population select count cell survey posterior case warm binary draw chain take size computation sample convergence chain demonstrate chain draw calculate chain correspond summary change value algorithm check time examine rate compute coverage value tail simulation performance adjustment successively construct allow cell count weight see practice cross york city longitudinal population center city public american population weight adjustment gap education category education cell cell count normalize show illustrate model location motivation population simulation population sample proportional size bias classical classical classical calculate error table mean square error rmse coverage improve performance estimation apparent due estimation outcome classical repetition simulate extremely outli smoothing cell neighbor eliminate influence raise sensitivity highly informative fail possible treat outcome related estimation could discussion issue outperform estimator setting intermediate extra compare mean four support robust small size demonstrate case continuous outcome least balanced compare estimate mean subject large little cell empty cell occur observe represent estimation yield credible improvement goal large question ask classical large appropriately adjust setting involve pool family child nearly child address capabilitie child birth child nine design sequentially sample proportional selection main kind national city weight national randomly representative occur large city record city occur city calibration status education work city binary survey response public service year follow survey nonempty cell contain cell implement default chain mix rr sd baseline year use predictive distribution calculate side represent close cell follow recommendation check aspect posterior value l p significant evidence yield interval year error posterior child estimate service increase social child become involve red dot vertical credible dot sample cell top survey bottom summary hyperparameter large baseline illustrate variability weight cell structure cell cell proportion due survey correlation strong variable nonparametric population inverse observe unit represent proceed unit survey outcome process novel outcome environment expand alternative study bayesian perform well compare across cell bayesian rate survey start base inference fundamental many development fine direction framework census margin make census area model concern method inferential assume likely occur go construction partially pool inference empty cell information nonempty cell smoothing instability classical practical moderately possible stability factor effect interaction weight virtue survey alone subject future full nonparametric survey weight understanding include title probability model simultaneously gaussian study evaluate design find nonparametric smoothing weight sample construct calibration predictor probability inclusion sample give problem point associate discussion current weight interaction consider weight apply analyze file must weight inclusion sample survey information include adjustment account accounting survey serious method goal bayesian make inclusion probability fully nonparametric version survey get regression realistic survey key weighting set population model unit fully inference use survey limited work user publicly dataset generalize inferential uncertainty weight
happen easy leaf leaf set two case last variable aggregation yet aggregate child substitution assumption entry correspond achieve aggregate table join aggregation pick pick entry small set corresponding must block substitution choose incorrect claim argue extend aggregate take aggregate remain direct observation formalize reward focus new symbolic explicit relational mdps plan monotonic function algorithm reduction generalize diagram knowledge relational planning serve approximate symbolic solution perform preliminary experimental relational offer formalism probabilistic planning however work planning transition agent domain request ic motivate example domain company task maintain meet demand event service request arrival customer service request focus service request common problem air traffic service planning reinforcement algorithm number next branching request consider symbolic dynamic allow typical reinforcement sdp mdps adapt factored planning algebraic diagram add model require relational logic domain size event make order formulation complex work simplification demonstrate algorithmic symbolic simplification algorithm complete symbolic service transition model first take agent investigate relational approach symbolic domain assumption allow provide algorithmic support implement scheme analyze property representation practical develop inspire check sdp algorithm ic satisfy efficiently policy notion logic action immediate mdp discount policy vi iteratively bellman relational mdps mdps action free logical language logical logical predicate interpretation specify element predicate tuple appropriate match predicate apply appropriate tuple say ground argument constant domain atom ground ic notation variable clear relational sdp paper planning result policy generalize size transition model action parametrize tuple yield template concrete action simplify notation tuple action success failure user choose dependent action choice domain avoid instead plan mdps require symbolic representation compactly representation expression interpretation interpretation expression similar logic substitute expression logic object logic false aggregation quantification universal individually example aggregation quantification allow aggregation operator correspond expression enable treat aggregation formula portion separately use expression intuitively expression pick maximize graphical open close expression direct acyclic leaf internal first allow equality atom atom constant argument diagram diagram fix order label order capture paragraph one path leaf substitution aggregate aggregation function calculate object think aggregation block number path diagram think atom open formula portion representation aggregation illustrate diagram affected simply unchanged replacement effectively connect branch leaf false branch diagram replacement proof refer diagram operation expression diagram multiplication without aggregation correctness summation apart maximization maximize action obtain achievable ground head aggregation diagram step replace aggregation constant add object diagram operation advantage approach operation expression symbolic several idea handle aggregation safe use safe operation example describe main modeling automatically domain action act mutually apply equivalent representation capture ic domain ic ic level fraction average action replacement drive action empty fail variant customer ic facilitate figure assumption unary simplify presentation consider handle ic special specify appear agent reward function multiply add apart avg sdp complicate object expression count composition action event assume exponential simple directly sdp sequentially ground perform regression agent action generality provide sdp final expression variable ic template method first step denote sdp partially grind regression add variant exp straight plan analogy go completely fact argument template max expectation second apart outcome take correctly max step agent straight line start evaluation child due parent aggregate separately side aggregate return return value reach block evaluation force agree aggregate aggregated return simply expand diagram corresponding
property rnn rnn however nonlinear difficult train way address connection add shorter intermediate back connection rnn furthermore rnn output deep transition dot rnn dot propose parse recurrent consecutive state parse result deep transition rnn dt dot stack deep way stack layer top stack variant goal encourage notice dt rnn extend conventional different aspect separately easy limit incorporate case universal mlp layer layer multiple obvious dt rnn rnn dt stacking rnns stack dt formal recurrent dt rnn stack rnn transition simulate rnn implement element wise nonlinear matrix rnn rnn illustration build trivial highly consecutive similarly layer output deep recurrent rnn draw deep rnn dot deep intermediate stack top hide output formulation eq use state make depend well discuss early stack rnn briefly already deep recurrent network build approach predefine operator implement mlp vector return constrain dimensionality identical additionally another symbol many operator stick operator express rnn see plus operator operator think perform plus output illustration understand operator parameterize mlp linear operator rnn arise insight construct rnn may model plus paper algebraic operator train rnn benchmark dataset try task predict symbol sequence replace rnn predict rnn try different task music character model rnn character word language modeling corpus compare recurrent rnn connection transition intermediate white character bias model start initial time character music task music character level stack rnn dot rnn rnn wise feedforward neural ten time small either dot column deep rnns outperform rnn rnns rnns variant rnns benefit advance feedforward network activation build dot rnns maxout unit output weight training similarly train minimize obtain dot rnns maxout unit music significantly report recurrent rnn benefit feedforward detail acknowledge art rnn conditional model machine c dot character level free dynamic rnns rnn dots rnn outperform significantly task dot rnn outperform mapping modeling dot rnn memory lstm dynamic result report without evaluation art rnn architecture regularization technique conventional rnn advanced regularization rnn train hessian explore novel building neural rnn rnn reveal consecutive propose design rnn networks connection rnns design conventional rnn reveal transition dot rnn rnn stack rnn model achieve modeling music deep importantly rnn benefit deeply like feedforward winner task music suggest propose rnn distinct characteristic make suitable type future combine propose stack rnn construct combine rnn quick train construct able recurrent music possibility feedforward function recurrent experiment conventional easily dot rnn stack connection rnn become problematic depth cause advanced advanced promising acknowledgment thank support fellowship science coin et op universit e de extend rnn start concept rnn feedforward carefully rnn rnn deeply hide output propose two architecture attempt stack recurrent deep interpretation rnn framework operator propose rnn task music support rnns rnns become choice rnn use al thesis word embedding handwritten speech work explore basic rnn depth feedforward lead expressive believe recurrent feedforward rnn consider composition network deep rnns rnn multiple early propose rnn stack recurrent potentially operate nonetheless aspect consecutive state single separately implication base extend rnn alternative lead deep rnn empirically evaluate modeling follow briefly variant deep rnn empirically discuss advantage dynamical subscript dynamical respectively parameterize training minimize cross conventional sigmoid tangent illustration rnn gradient compute exponentially network compare empirical support maxout finding make argument apply recurrent network ht mm define feedforward layer trivially recurrent rnn temporal rnn output carry see fig nonlinearity hide intermediate transition transition deeply one intermediate two deeper previously plug multiple layer implication show network tend original datum easier temporal
px map conjunction x follow independence px dependency dependency px conditional factorize corollary requirement px map conditional expand q denominator numerator denominator whole present result feature px dependency factorization set factorization factorization model feature say reduce dependency regard requirement dependency context context specific factorization dependency definition require px w factorize factor conclusion factorization requirement equivalence compose match e dependency implication requirement construct sub assignment complement expert context sound complex context future explore alternative simplify present guarantee present context generalization believe worth implement structure mrfs structure complexity comparison conditional encode adjacent conditionally assertion dependency similar fashion px pairwise encode present focus finer grain conditioning set disjoint x denote eq interestingly assertion conjunction conditional independence context represent single undirected capture extended test validity assertion assertion contain px say px specific dependency input triplet x xx undirected represent completely connect graph k px parameterize follow distinction input log combination delta merging index dependency c capture context dependency px factorize potential function map exactly encode map assumption hold factorize potential clique axiom union axiom conditional independence markov clique last model px
hull fact hull form rescaled plane nontrivial nontrivial nontrivial solution clearly additional assumption count sort point nontrivial large intersection acquire hyperplane mixing least idea include additive preprocessing onto plane origin return keep nontrivial th initialize ig equation fitting plane product two obtain intersection form source recover fig noise three noiseless leave form intersection plane plane noiseless corrupt noise point norm tend get apply entry fall plane plane plane plot remain denoise project onto plane triangle right plot well may structure modify one belong yet vertex vary almost achieve actually correspond normally set smoothing box filter variation tv alone noise able preserve yield include spectra matrix source matrix last section assumption suppose work three source spectra chemical compound produce snr varying db db reliability plot considerable would remove removal simplest denoise slide filter except filter probably reduce image smooth reduction modification obtain neighbor depend neighbor select cloud disadvantage smooth away corner reduce denoise noise removal gradient signal reduce total signal remove edge removal shall use cloud illustrate total rescaling cloud include plot plot function restrict rectangular region cloud define intensity cloud noiseless proceed solve variation shall recent small minimizer cloud calculation follow threshold cloud depict show intersection cone total denoise preserve line away flat region idea total variation cloud dimension conduct experiment tv conduct comparison white db tv index separation paper bss blind source permutation validate noisy variation serve preprocesse blind determine bss invertible recent retrieve suitable condition thm assumption thm thm mixture blind attempt retrieve little separation linear mixture term cone possess stand alone peak spectrum fall unique property noisy denoise image principle blind bss source signal mixture knowledge mix bss bss chemical record differential optical spectral critical national security advance modern imaging technology pure spectral environmental challenge bss decomposition full unknown unknown spectra bss signal independence bss scale invertible research bss simplicity observation dominant location mix identification edge minimal hyperspectral image pixel material interest exist ground method term hyperspectral nuclear reformulate source non overlap acquisition simplification satisfy stand peak formation vertex cone let mixture singular nonzero stand peak column constant column estimation fact vertex datum point optimization column contaminate optimization solve column nonnegative hand high mean column high mixing though convex geometrically elegant violate primary scenario point scale column lie cone yet none locate vertex fig reconstruct instead component source possess precise study condition call identification cloud lie hyperplane intersection hyperplane expand identification flat manifold recovery extraction meaningful structure dual cone propose source signal source candidate process cone double method step estimate source large generate transpose author orthogonality source get sub computationally orthogonality signal
concentration private inference general additional step discrete bound principled guide examine privacy concern relate large privacy learning would learn leave apply adversary place expect simply space le equip pseudo use manner measure consequently metric construction distribution unknown extend set space suppose suppose also interpretation quantity q obtain specific family posterior proof compute thus parameter case sequence connectivity argument proof log item differ assumption become case product distribution parameter decompose part eq combine break part disjoint sized bind geometric theorem combine posterior assumption recall bind lemma inequality outcome deviation empirical eq substitute inequality proof processing adversary inequality bind term rearrange treat obtain robustness privacy inference prior without framework distance differentially private bound privacy adversarial give satisfy significant challenge statistical robustness training privacy study simultaneously examine want third party unbounded query appropriately satisfy generalize differential link mapping spam perturbation corpus smoothness outlier discover datum suggests simultaneously link mild security relationship adversarial privacy onto grow economic concern unify environment dataset outcome family regularity condition show change dataset change private framework bound distribution discuss work specify robustness discriminate give appendix supplement cast statistical assess select design particular adversarial minimax arbitrarily contamination neighbourhood criterion notion context minimax within contamination prior demonstrate robustness likelihood maximum principle whereby nature privacy database largely guarantee privacy differential respect query informally differential change mechanism differential laplace exponential private inferential modification private modelling interact arbitrary specific see complementary research paradigm probabilistic inference utility differentially private posterior measurement inference relate statistical laplace differential svm add differentially release infinite lie rkh make robust mechanism robust statistic operate global achieve perform differentially private release robustness privacy people suffer vote privacy user balance utility inherent privacy private query alphabet possible notion robustness firstly quantify amount query quantify true situation metric considerably privacy special hamming focus construct data parameter indexing algebra q derivative dominate perform select measure measurable response adopt mechanism distribution differential see measure distribution correspond posterior interpret privacy arbitrary necessarily space pseudo privacy differential differ standard x allow strong may come utility allow much broad pseudo sequel close party distinguish notion two smoothness metric let difficult see relax require lipschitz continuity constant require smoothness easy meet guarantee several family concrete ratio yield differential sample respect np differ item instead guarantee posterior posterior kl prior robust sense mechanism absolute posterior metric satisfying assumption constant kl show interpret privacy section introduce alternative assumption generalise privacy hold ratio private pseudo posterior differentially private pseudo achieve framework adversary mapping space know select require aware posteriori value sampling receive draw prove privacy response parameter example distribution component would answer answer would limitation effort require adversary neighbourhood dataset interactive adversary approximate sample attempt posterior empirical algebra arbitrary estimator describe might issue combine bind adversary kl measure fine distinction adversary draw posterior adversary distinguish likelihood
partition require instance third ignore subset however leaf select expansion select distinct candidate choose leaf search candidate dimension key forest candidate forest project point evaluate split search select restrict depicted candidate two create splitting structure cell structure indicator point choose candidate expansion stop make independently predict average prediction tree point contribute tree consistency train varie obtain classifier consistent structure average average possibly dependent consistency regression forest imply assertion straightforward average copy must triangle consistent greatly simplify partition structure point shape condition consistency conditionally distribution bound I boundedness apply dominate preliminary monotone marginals consistency base estimator sequence sufficient diameter fix conditioning trivial see support projection onto possible child split iterate argument expect cell contain hold prove describe two forest analyze originally adapt regression compare forest repeatedly expand leaf choose leaf dimension datum sorted choose expansion continue manner terminal refer support first expand terminal reach expand candidate replacement candidate gain split great gain feature fitting require part determine role partition estimation point main partition whereas randomly tree compare forest choose splitting happen factor effect forest cm feature diabetes ct empirically compare several insight impact use theoretical choose indicate selection variant datum split notable thing competitive standard forest split forest figure interest ct slice dataset closely image specify part hand prediction green evaluate location image body second label pixel implement depth body datum offset joint forest pixel offset pixel post shift final location avoid comparison model joint offset pixel location offset predict build resolution body along position joint body body image generate pose pose create replacement body without replacement pixel body pose evaluate mse ground truth vote arm pixel depth difference pixel specify variance produce depth figure candidate feature pixel offset offset experiment candidate standard joint pixel side uci simple forest turn narrow forest new forest prove extensive compare variant light choice theoretical performance focus consistency theoretical convergence forest problem beyond eq let cdf density claim tree cut infinitely distance leaf choose structure min max splitting mechanism active generality number child length repeat split child length bind derive assume cut dimension cut depth contain hypercube pick know depth side density contain positive set fix fall mean eq second rhs choose summary tree leaf branch contain arbitrarily large point terminate leaf expansion child leaf depth contain arbitrarily probability branch actually depth argument claim proposition definition definition practical forest understand contribute theoretically variant evaluation compare theoretically forest random insight importance forest ensemble base forest framework extremely successful purpose method theoretical forest variety forest appear algorithm even well state focus elaborate extension forest framework specific come guarantee paper focus paper forest tractable simplify also standard forest analyze community match theoretically tractable date empirical theoretical something appear insight importance algorithm tractable cart tree bag influence selection feature several idea early decision year random forest great field forest success work heuristic rigorously estimator surprisingly focus extreme random simplification theorem insight sophisticated forest classifier direct average rule view variant forest originally introduce original leaf open tractable forest survival forest version forest review forest comprehensive review refer reader forest prediction tree unlike combine sophisticated independently choice
characteristic represent forward reverse direction involve six cm denominator case become independently measurement accord identify vector slight measurement covariance matrix critical value right side solid cm curve principle supplement supplement guide mm carry accordance supplement guide supplement adopt like intend guide write would theoretically appropriate adopt take remove use goal remain supplement evaluation attribute vector supplement implication attribute describe way coverage say specify supplement elliptical comparison mean matrix attribute n mm attribute satisfied eq elliptical point satisfy indicate curve twice probability shift degree supplement shift scale degree need explanation involve complex quantity success supplement theory error rise estimate paper recall univariate allow component degree estimate bp degree freedom enable quantity attribute degree paper make estimate quantity variance independent estimate arise practice dependent extend multidimensional act suppose significant error measurement form multivariate mathematic helpful write form form imply appropriate linear simultaneous measurement result multivariate let applie form arise distribution covariance mean obey subscript appropriate summation bivariate example use approximate generally repeatedly make simultaneous parameter measure obtain vector indicate mean covariance calculate value h table mean measurement parent matrix parent find q give right correspond elliptical uncertainty centre mm comparison supplement quantity use involve simultaneous component quantity attribute recommend supplement guide supplement least principle attribute typical large reasonable monte carlo examine contain row scenario success observe rate method correspond might vast majority result show rate method figure drop indicate bold situation involve satisfied bold figure minimum success whenever scenario study rate observe accordance say practice uncertainty pay approximately region considerably reasonable conclusion individual think independently prefer success drop quantity region result far robustness correlation estimate success entirely adequate region correspond success situation notably result generalize simultaneously measure length approximated hyper vector satisfy summation iii q situation measurement separate summation simultaneous ii h iii simultaneous simultaneous q scalar scalar scalar scalar take freedom equation reduce eq summation run simply procedure guide alternatively obtain simultaneous measurement obtain simultaneous vector accordance applicable case limit I problem extend procedure type guide accommodate measurement quantity quantity present generalization freedom measure scalar explicit guide standard admit simplification expand give region supplement guide applicable guide classical see guide multidimensional important contract national evaluating incorporate full uncertainty propagation software entity call measurement equivalently uncertainty mathematical uncertain complicated series require propagate example handle j uncertain define uncertain software degree freedom display df complex uncertainty df degree uncertain propagation freedom display result gamma v freedom step j j j next four multi treat component associate sp label sp step use propagation degree freedom imply result g give degree figure cm measurement laboratory innovation guide degree calculation ii incur independently value dependent build analysis method multidimensional explicit two element guide practical procedure individual overall measurement variance combine freedom estimate applicable one quantity call scalar deal multidimensional common arise association wave phenomenon field electrical rf concerned propagate rf simultaneous form coefficient variance covariance calculate freedom uncertainty procedure guide form interval freedom approach mean guide scalar uncertainty estimate weighted sum scalar construct guide distribution w good freedom formula derive frequentist applicable scalar average measurement appropriate classical adopt type primary context guide methodology classical involve guide case case estimate letter bold bold symbol use say outcome variable denote capital letter purpose paper outcome tend think variable datum random writing possess distribution mean variance write draw write variable measurement outcome basic guide unknown approximated keep terminology estimate foundation methodology dimensional result covariance call well suppose multivariate covariance wishart freedom freedom degree numerator denominator denote outcome procedure choose calculate result subject begin statement basic multidimensional problem analogous description one guide multidimensional guide know unique generalizing formulation multidimensional sensible analogous notation quantity unique necessary quantity bring obtain write column column length sensitivity define quantity represent component differentiable matrix effort cauchy equation write uncertainty scalar guide generalization describe x cp law propagation
mahalanobis bootstrap cf jt jt jt jt ny I jt jt average guess arbitrary n stand n tm mean display estimation measure aforementioned real value monte define simulation table present integral step title derivative derivative study bandwidth carlo integral estimator base carlo two measurement error zero l c vector signal generate integral describe monte present estimator adapted report guess experiment initial guess space website carlo initial result variability guess program second execute ghz ghz estimator bootstrap drop calculate integral exclude local polynomial construct observation summary estimate estimator base integral approach surprising measurement iterative require initial guess otherwise time bad integral approach may theoretical improve execution estimator simulation generate merely understanding describe specie term consist take q represent experiment laplace table different simulation measurement unit unit repeat last line xt grow repeat reasonable small consist noisy interval last xt xt measurement unit setup repeat measure xt xt c c smoothing error initial run repeat display difference estimation substantial solid line correspond sign realization dash correspond solid correspond realization dash line ordinary widely life phenomenon spread population dynamic dynamic disease mention see reference address aspect identifiability specifically show uniqueness uniqueness seem notice linearity feature first model estimating need numerical solve accurate namely function approach numerical simulation estimator substantially well far true sense integral estimator complicated scenario accuracy furthermore see small execute matlab author website send request exist consequently w w w consequently obtain hence vanish thus contradict would vanish consequently th almost interestingly start concept complement side immediately clear tc tm equation plug xt b x gb equal square root square gx consistency consequently boundedness boundedness boundedness probability weak dominate q dominate consistency eq hence proof theorem inner sign define product diagonal schwarz w jj define hold indicator cauchy read continue another sequel side row schwarz show prove derivative entry taylor continuity partial continuity partial ns argument lead order boundedness g ns continuity need write bound schwarz formula term side second right respectively arrive lemma component vector term type side differentiable sum element replace deterministic repeatedly subsequently via continuity suppose let observation let variance measurable bound guarantee inequality mu mu hence sufficiently nb kb var fs ny jt w fs n boundedness imply guarantee cauchy schwarz jt var ft ny jt iw f w w n kb w equality vanish outside apply choice optimal rate obtain compare control precede satisfy x I boundedness prove probability q measurable acknowledgement support foundation scientific partly economic first author theorem remark research physics engineering characterize common sum product linear unknown parameter estimation computational avoid classic suffer smoothing consistency estimation biology physics differential system bottleneck dynamic literature particular technique rigorously ordinary equation take reduce substitution xt tt write h interpretability cox proportional system gene disease measurement case separability suggest separate treatment seem exploit practically attempt system estimation trajectory observe develop derive optimal convergence demonstrate via proof identifiable reference therein concerned structural identifiability affect e clearly particular identifiability exploit start equation give identifiability mean linearly dependent lebesgue formulation sign measure sigma borel measurable product integration respect introduce assume imply measurable function inner column identity matrix entry open eq symmetric sign conversely knowledge almost vanish consequently lebesgue almost interval nontrivial lebesgue almost conversely linear imply hence uniqueness require solution previously nonlinear theorem lipschitz consider positive initial value infinitely nevertheless identifiable tt analogy make proposition well inverse interest choose infimum attain minimizer polynomial xt xt u xt tb tb appropriate kernel local estimator tu kb b nb symmetric kx l mu nb nb iv need derive matrix automatically boundary easy implement estimator requirement px jt identifiable sign measure satisfy interval length second bound continuous define assume non singular condition variation sufficiently densitie lebesgue counting function provide boundedness measurement develop method base smoothing practice especially deal different make parameter situation measure hence repeat observation common
financial market asset return student df df understanding serial lp lag lag lp display order sum bic bic path make equal smooth lp auto capture lp considerably indicate original raw derive quantile use intuitive representation h kt allow correlation plug implement pearson estimate lp moment sum lp compute smooth lp comment lp operation term estimate order consequence sparse although flat smooth estimate due presence quadratic display smooth serial fine time copula expansion copula integrable basis v h allow copula density insight estimation structure copula know economic source lp generalize autocorrelation em graphical display lp lp stock panel absence autocorrelation finance prominent auto series nonlinearity say price price decay autocorrelation interpret display diagnostic lag square smooth bic select zero copula density uniformity two quantify h u asymptotically h h show panel contrast plot building discuss coefficient time treatment skewness lp transform computed investigate indicate presence slight stationarity behavior propose statistic also display correlation plot copula graphical visually get insight nature u leave right datum dot correlation dark quantile curve fit dependence coefficient u nonparametric quantile copula density plot dot correlation deviation help understand nature compute fit dark green albeit characterize linear nature correlation approximately simple illustration diagnostic tracking change quantify conditional dependence unconditional ratio elegant quantile unconditional reference application management historical scenario analysis create hybrid compare insight enable v eq say slice copula interpret serial copula define orthogonal coefficient density identity h trace asymmetric ask return volatility conditional change unconditional asymmetric volatility known leverage effect future stock past return stock tend stock price em display htb em utilize arrive nonparametric model simulate conditional accurately quantile quantile curve practical density panel unconditional marginal fy skew g reject histogram density shape long well conditional conditional proceed nonparametric quantiles simulate quantile quantile special significance sometimes conditional currently quantitative management base parametric theory hold return symmetric prominent asymmetric shape quantile auto movement know lag lp following interpret respectively u uv return dependence number display ar estimate f jt separately fit ar parametrization finally estimate function copula h serial spectral capture recommend measure approach nonlinearity capture interesting shape design time series spectra spectra rank transform look long methodology quite successfully univariate multiple gaussian covariance evolve maximize carry ar bic select description lp model volatility affect extreme many sign coefficient return leverage positively volatility provide comprehensive rigorous easy foundation united permit time create interpretable library purpose believe augment analysis student familiar central idea series modeling convert via orthonormal mid automatically heavy tailed daily return series modeling addition foundation show discover hide return heavy nonlinear asymmetric volatility market hypothesis manner unified light establish conclude reference article book tool algorithm prototype result thm nonlinear modeling learn pa usa college abstract comprehensive series base recently science novel specific mid polynomial like transformation series adapt concept daily algorithm systematically fact financial note researcher copula mid moment lp construction correlation plot height pt observe discrete time seek scientific understanding assumption obey law identical lag goal em
rbf bl denote weak derivative multi index even z inverse expect belong fact lack theoretical cubic rbf numerical experiment cubic rbf relative thin similar conclusion application work solely cubic rbf rbf norm fast however rapid interpolation irrelevant costly cubic converge respective algebraic conduct provide experiment leave nn quantify function define multivariate extensively graphic method compute optimal weight accord move least approximation relative scale via map eigenvector minimum reconstruction rbf rbf versus cubic cubic rbf extremely cubic h cubic digit red denote residual original rbf synthetic assess occur high digital handwritten digits handwritten digits rbf fig inspire novel interpretation nystr om nystr om extension eigenvector symmetric matrix use rbf interpolation apparent nystr om interpolation provide completely insight nystr om laplacian normalize kernel diagonal sum thresholded nontrivial define nystr om eigenvector x conclude nystr radial interpolation rescaling although new nystr important observation concern sensitivity parameter explain section scale attention involve reduction typical kernel keep entry nearest neighbor nystr thresholded nystr om nearest neighbor extension new unstable poorly om interpolation non rbf cubic could well nystr om author anonymous excellent comment support nsf dms partially award de reduction inverse map bi rely radial basis cubic kernel gaussian suffer ill require scale construction nystr eigenvectors laplacian base new interpretation nystr om map dimensionality radial basis om area research map extend example nystr om extension solve therein application nystr om bi nonlinear mapping rely via radial high contribution primarily small coordinate elegant stable contribution interpretation nystr om properly radial basis precise similarity nystr om suggestion consider mapping converge toward go construction map indeed need inverse nonlinear reduction dataset generate limit n seek everywhere address inverse toward go infinity terminology geometry coordinate map function given propose several use tensor product poor performance know node dimension exist interpolation radial function construct inverse mapping explore datum krige technique rbf application lack specialized rbf interpolation cubic representative radial scale experimental rbf computer graphic rbf reader drop radial exact combine linear correspond unknown system equation inverse obtain assess inverse interpolation interpolation true additional element radial basis system unique follow lead rapid ill increase discussion gaussian match commonly interpolation interpolation cubic point randomly right number point rapidly cubic function scale point randomly difficulty establish boundary discrete fill bad proxy distance distance neighbor relationship interpolation explore rapid conditioning fill conversely ill conditioning propagation rbf converge lagrange interpolation become ill g sophisticated intensive direct make undesirable inverse interpolation avoid rbf radial powers eq
require estimation base penalize suggest quantile longitudinal make inference random subject correlation asymmetric effect laplace propose regression allow varying effect subject efficiency inference quantile regression incorporate within subject correlation misspecification method inferential approximate estimate apply intensive overcome within quantile produce efficient paper quantile regression obtain objective newton iteration latter appropriately incorporate repeat longitudinal employ stationary specification use induced estimate work independence furthermore theoretical practical programming remainder regression parameter estimation asymptotic estimator intensive study conclude remark longitudinal setup response multidimensional repeat measure th covariate quantile estimating consistently efficiently possible measure obtain tn solve statistical software independence therefore satisfactory consideration longitudinal model diagonal th derive quasi solve estimate element treat slight estimation correlation specify propose estimating equation ia matrix ij iy ty ij ij calculated denote propose notice expression ia ic however estimate never may estimate become use iteration estimate large estimate use estimate solution find converge estimate adjusted method difficulty continuity high burden overcome difficulty induce smoothing extend quantile longitudinal counterpart update estimator algebraic calculation calculate ij r independence newton respectively evaluate mx compare newton much fast asymptotic original estimating smoothed estimator normal q mx mx regularity smooth smoothed equivalence estimator mx mx x I iteration defer report order conduct extensive simulation report balance variance medium correlation distribution error normal follow normal chi chi freedom quantile follow quantile quantile quantile estimator quantile analyze average efficiency different quantile work adjust regression deviation error estimator fall quantile sd sd sd se multivariate ar structure average bias relative quantile regression comparable assume small bias much become except quantile result standard adjusted normally simulation nominal normality inference similar compare report three chi expect outperform mean skewed tailed error propose median chi distribution particularly regression analyze report clinical trial total randomly line extreme pattern may course extent group height divided minute zero median group ci regression quantile assume se se propose regression report error confidence usual working standard error mean median except usual th parameter estimate give contradict significance amount use propose method respectively treatment help treatment grow quantile low quantile plot height figure longitudinal stationary auto reduce burden employ quantile parameter newton quantile regression inferential classical applicable study well work within furthermore corresponding mean analyze heavy tailed skewed real group reveal amount try within quantile effect different penalize allow add include study area parametric grant research condition outline proof measure absolutely continuous density away interior convex region definite mx tw ix let mh py py th I accord law number condition together mx write lemma
impact pass reasonable gd fast sag follow sag point gd evaluation full support gd amount pass provide gb extra c gd sag bfgs study gd sag row sag sag gradient begin practice sg sag sag optimum eventually gd regardless gd improve sag perform sag gd come choose stepsize sgd choose poorly gd gd three work tune new gradient descent gd complexity strongly measure unit strongly merely improve generalizing simplify gd gd exhibit superior method leave gd set inexact gradient appendix error lm q use get g iteration inner expectation respect condition expect optimal rearrange expectation algebra outer gd full summing multiply x h j f desire recurrence sure appendix elementary range completeness simple recurrence relation substitute section aspect performance regression write digits purpose vs artificial experiment perform label regularize regularization practical gd choice recall choice zhang demonstrate practice significantly well surprising possess descent compute enough reach implement choose numerically case drop residual computation need decrease stochastic computation include hundred essentially perhaps big variance theoretical insight comparison rate large regularize square regularization well work minimization run gd relationship practice previous confirm consistency gd gd sag zero size limitation run experiment sag gd sag gd gd sag analyze performance engineer big well first pass quickly inherent gd sag advantage gd sag linear convergence distance begin compute deviation theory performance gd find quickly continue gd constant evaluate gd epoch illustrate theorem theorem theorem school mathematics united minimize large gd stochastic run epoch geometric need pass equivalently unit run epoch zhang limited epoch stochastic gd need evaluation cast q feature incur big typically objective sake briefly review solve gd stepsize name prohibitive every gradient pick drastically unbiased full issue sgd consecutive stochastic gradient may lot direction performance balance sgd preferable gd acceptable sgd extremely fields sgd variance paper propose combine step gd le schmidt zhang zhang method reduction method smooth random coordinate descent composite work iteration iteration application generally sdca duality primal mini batch stochastic sag zhang gradient set epoch spirit achieve case sag classical stochastic paper gd gd analyze exhibit superior move contribution devote complexity gd hence choose optimally minimize result gd strongly encouraging gd gd paper convex lipschitz constant strongly necessarily gd stepsize limit stochastic single epoch convexity convexity stepsize low f jt outer method compute subsequently geometric computed stochastic gradient inner impractical big add ensure q stochastic implementation introduction implement gd superior include gd leave run run gd brief gd start run pass gd motivation sgd gd correspond gd sgd switch gd practice implementation sag gd heavy summarize main gd evaluation account output solution achieve run gd stepsize epoch gradient stochastic evaluation epoch theorem see match closely well constant convex result complexity nesterov sag sdca needed nesterov gd result expectation descent propose gd apply convexity recover high standard rate sgd derive formula approximately compute counting evaluation stochastic stepsize run epoch effectively limit evaluation gd compare epoch complexity gd gd gd case gd meaningful coarse behind box gd sdca sag unlike gd store size task problem gd gd perform optimize end fix epoch tolerance gd stepsize fx need expression choose verify case notice computation gradient slightly expression place eq claim solve form solution formula equivalent find constraint view search maximize quadratic stepsize dependence j strong may choose suboptimal stepsize instance one choose need small gd evaluation c c c c c c c c c notation convex affect eq gd perturbed problem necessarily perturb perturb lemma follow minor ready strongly stepsize sufficiently gd gd gradient theorem directly perturb conduct aspect practical gd practical substantially implement gd dataset gd several task repository theoretical datum condition particular l instance parameter zhang choose run gd gd convergence theoretical function convergence precision work gd speed implementation section formally structural composition loss point natural ask gd implement efficiently look sgd type let nonzero feature e univariate amount work method speed iteration sparse datum gd fully update costly operation irrespective level follow delay delay immediately know suppose epoch j finish j j notice never know coordinate perform coordinate appear correctly potentially update continue careful counter way forget fashion gd
process spectrum expect j theoretical spectrum way cause sample correctly consistency section usual fast fourier transform calculate dft operation likelihood complicate fourier xx truncation length effect multiply accounting sampling discrete due truncation finite form multiplication avoid summation frequency note spectrum numerically fouri triangle aside compute spectrum define complicated analytic contribution via fourier spectrum effect likelihood define extend likelihood fourier transform correspond recover fourier transform reduce estimation process spectrum exhibit nevertheless continuous type account equation kernel compute store perform numerical select effect likelihood advantage reduce effectively away frequency exhibit convolution range explore carlo simulation modify version likelihood value start complex component normally time spectra formulate spectra equation spectra transform definition bivariate fourier frequency approximate bivariate start define fourier implicitly simple see equation invert substitute rearrange equal simplify denote complex bivariate complex series similarly compute appropriately version retain series operation detail requirement remove frequency lose include equation include proper time frequency time standard consistent spectrum account bias correlation prevent exactly establish provide appendix proposition dft finally hessian converge real value series stationary twice differentiable consistent twice differentiable see somewhat slow weak assumption benefit reduction slow variance behave sum weight stationary twice density conclude theorem consistent equation analytic evaluate hessian remain substitute xx xx xx derivative separately deriving term establish invariance principle covariance dirichlet provide estimate normality supplementary positive gaussian much thus sum explore behave finite spectrum respectively control reduce take care perform smooth effectiveness range numerical section likelihood real note relate inference contrast aim infer memory fractional motion bias correction reason primarily frequencie certain movement surface exhibit interested infer one accordingly parametric bias uncertainty spectrum frequency practical omit estimation simple hypothesis spectra consequence domain testing frequency test set replicate parametric value vector scenario mat ern suitable likelihood differ coherence specify however prefer simple ratio check specifically statistic include equation ratio distribute versus appendix series yield statistic example study subtle care take spectrum range equation degree freedom special delay degree alternate conduct finding parametric practical many scenario number define model nest simpler choice aic use appropriate ern fractional brownian brownian motion inside mat ern slope mat ern reason value series aic domain done use size often recommend correction freedom length correction account degree replace number frequency likelihood fouri frequency attribute er example space apart spaced constant frequency smoothing must transform onto fine modelling well carlo simulated matlab use generate description simulation software section several likelihood use adjust maintain efficiency effectiveness likelihood mat ern parameter recall motivate introduction band appear colour scale strength lagrangian trajectory ern equally mat ern mat ern law behaviour mid contrast motion law mat ern low process self frequency addition model mat ern coherence q define begin coherence long mat ern detailed section simulation leave panel semi parametric exclude frequency mat ern coherence equation fit fit draw numerical panel top leave mat likelihood normalise spectrum model isotropic fit put ern experiment versus hypothesis isotropic ern procedure trajectory correctly reject sided confidence reject isotropic experimental control testing perform trial discovery simulate scale know scale transition behaviour begin convert division median day scale apparent solely lagrangian investigation leave likelihood isotropic first trajectory model capture quantify physical summary shall analyse series panel figure vertical radial california pair display component represent wave elliptical motion wave reference particularly capture delay wave model extend specifically capture series improper gaussian improper complex ar estimate autoregressive together spin magnitude value improper form complementary display capture complementary fit low noise exclude second improper elliptical particular wave leave radial trace record california usa radial time value trajectory blue improper complex ar ar right centre part complementary address implementation likelihood separate different frequency behaviour providing handle frequency characteristic example demonstrate primary process allow reduce computation efficient compare art yield parsimonious model physical fourier transform distribution fact transform approximately interest model complex remain series challenge extend value modelling effect spectra extract paper effect separate effect main challenge continue model framework work first find spectral increment component thus analytic careful frequency energy share shall separate sign write term component reverse specifie representation component complex relationship mat ern mat ern covariance simplify appear suggest specification bivariate mat ern ern equation mean restriction spectrum spectrum reasoning return component calculate last line frequency equation representation proposition analyse sample subscript discrete fourier transform dft choice dirichlet q fourier take rewrite split range start consider next term dirichlet behaviour q eq fourth resemble thus integral sum remain order spectral finite furthermore dft transform statement follow select eq sequence spectral domain likelihood choose frequency two note expectation proposition inequality yield ball radius invert obtain continuity see additionally outline ratio outline complex save description shall description special nan alternate hypothesis fall reproduce special therefore reproduce alternate theory start likelihood function frequency process analytic anti analytic part rewrite value process note simply alternate shall reality likelihood eq difference likelihood implicitly combine ratio simplify substitute regime covariance modification must illustrate implement acknowledgement introduce spectra value novel modification time introduce nontrivial bivariate value structure positive behaviour flexible property time series sample complex demonstrate improve value bivariate testing procedure transform mat ern derive interpretation value significant area optical neural blind refer advance application ability complex mainly place come parametric frequency domain contribution value address new way deal trajectory top trajectory north bottom leave work database global understand series complex top figure fourier complex value velocity velocity signal display negative feature present one behaviour direction spin significantly easy frequency domain key domain like surface velocity record development efficiently adapt consequence initially derive g subtle stationarity reasonably assume window stationarity motivate correct procedure accurate size likelihood novel maintain computational approximation advantage value addition relate procedure value semi parametric frequency relate construct test track nest seem theory method value special bivariate mat ern process choice model variability trajectory shall illustrate practical necessary introduce complex series likelihood account real complex series semi parametric modelling procedure flow conclude remark section reader development sequence angular statistically inconsistent absolute discrete fourier dft orthogonal increment process due truncation process continuous process effect I time convolution er square g effect expense correlation discuss trade detail result high observable frequency ignore estimate spectra effect section new likelihood incorporate sampling already area convenient bivariate series complex value real henceforth refer decomposition however contribution positively complex series year process sample property yy form equation alternatively complex fully specify denote bivariate apparent hermitian complementary transform spectrum complementary together fourier transform value fourier domain orthogonal increment transform relation spectrum everywhere otherwise motivation bivariate value counter spin analytic stationarity vanish three process derivation find relationship energy specific analytic negative frequency relationship nontrivial highlight work future relate relation sequence fourier transform relationship spectra process column value middle c complex yy xy decomposition differently analytic second anti frequency consider specify structure division seem artificial might represent complicated inclusion value reverse clear modelling many aspect specify mat ern real series develop capture define modelling frequency coherence quantify lead cycle coherence must hermitian symmetry modelling specify choice choice choose realistic would decay polynomial polynomial similarly odd simple across frequency note valid impose proper require strictly possibility fact rotation example generate shall subsequently condition turn imply refer specify coherence specify specify flexibility vary particularly datum unlikely correlate frequency expect aspect possibility decay complex coherence delay frequency model proper table require proper value zero frequency important aside additionally vanish model formulate later specification domain coherence employ random mat ern stationary gaussian process attention extension mat ern univariate mat ern three second hausdorff equal range variability mat ern limit behaviour fractional brownian motion whereas limit noise mat ern multiscale parsimonious yield modelling structure wish relatively construct valid mat ern mat ern covariance mat ern trivially specifie condition bivariate mat ern mat ern paper easy process second check valid time mat ern interpretable spectral relate define equation xx mat ern differ process equation valid possess invertible spectral cross spectrum mat ern spectra ern equation
panel curve error axis track iteration panel blue obtain composite gradient descent respect stationary early place optimum unknown setting predict fall panel provide tight cluster suggest lin lin scad lin scad covariate corrupt additive line predict show different initialization composite descent panel mcp regularizer respectively panel show note solution small set initialization optimization furthermore global optimum nonconvex program scad produce local optima whereas mcp yield optima note optima appear lie ball scad mcp logistic lasso logistic logistic mcp penalty c predict decrease plot plot trajectory initialization optima panel explore significantly violate take toeplitz choose result choose panel expect good regularization even convergence statistical cf composite descent small behavior see panel panel appear converge iteration compare curvature parameter plot panel slightly initial demonstrate could attribute h toeplitz toeplitz scad toeplitz scad toeplitz scad bad allow first establish nonconvex close truth imply solution variant composite optima direction generalize nonconvex regularizer cover bridge regularizer decompose coordinate addition would expand hinge nonsmooth nonsmooth penalty open near optima polynomial establish rsc beyond specialized property nonconvex cover specific regularizer give regularizers condition condition iv lipschitz bound magnitude suppose condition iii last come condition iv similar argument case verify inequality scalar trivial imply desire case assumption establish ft tt property last claim support triangle inequality remark thereby complete proof scad mcp regularizers trivial iii scad regularizer interval valid subgradient condition give mcp regularizer already may compute derivative mcp subgradient derivative corollary shorthand lemma establish rsc convex assume since lie hold arbitrary ft f obtain apply lemma rsc remain verify validity choice corollary applying value proof arithmetic establish variable ij ij c c last function underlie exponential series expansion boundedness condition claim follow calculation q standard plug lemma need assume apply proof support derive let unconstrained program g r iterate feasible duality force appear derive explicit regularizer program program write subgradient objective eq mcp parametrize mcp derivative agree expression show define subtract divide first trivially rsc imply rearrange along appendix imply eq furthermore bind imply imply rsc q contradiction imply combine implication also hence proceed base sake contradiction rsc condition convexity multiplying yield optimality rsc optimality summing combine whenever provide iteration belong inequality I convexity q combine shorthand subtract eq choose respectively iterate prove auxiliary rsc give sum eq rsc introduce shorthand value variable early taylor boundedness assumption regression rsc provide technical define suitably gaussian core prove take proof rsc arithmetic lemma whenever hold negative trivially hold pair rearrange choose introduce trivially unless bind truncation construction first tail gaussian assume consequently apply pair homogeneity lipschitz center apply p furthermore lemma c extend argument event restrict integer set relevant region definition auxiliary concern process lemma independent rademacher arbitrary useful next countable center sub equation suppose universal lemma suppose standard gaussians c condition bound ij variable nonnegative bind integrate constant desire show regularizers type complicate regularizer possess neither interest everywhere eq iii section cf separate upper rsc satisfy choose mention modification note minimum local minimum come replace consequently note remainder penalty regularizer assumption appropriate piecewise take outside subgradient define j q exceed ct substituting condition assumption lemma proposition theoretical optima allow loss penalty suitable penalty prove lie within underlie cover nonconvex lasso error nonconvex scad mcp dimensional graphical point composite within fast first provide high nonconvex year optimize function computationally nonconvex optima optima gradient terminate optima statistical optima theory global optima nonetheless optima various nonconvex arise behave insight occur construct practice confirm intuition lasso error function resemble strongly cone stationary paper couple regularizer interest statistical empirical possess multiple local essentially good new mcp regularizer previously nonconvex quadratic descent terminate optima behave optima terminate minima initialize satisfying produce nonconvex penalty show specific optima square stable initialize complete overview relate contrast appropriate within project converge point lie specific optima grow efficiency high nonconvex arise possess stationary strong statistical involve cite illustration theory panel gradient form regularizer red show mean optimum possesse final essentially statistical h scad mcp panel mcp describe nonconvex nonetheless qualitative statistical moreover predict geometric precisely modify composite descent solution generally propose review initial local optima whereas work establish successive iterate applicable regularizer smoothness entire axis applicability broad applicability organized provide nonconvex function main state corollary modification composite optima convergence result appendix algorithmic conference universal subset f norm subgradient occur tend loose take care write explicitly factor statement theorem develop estimator establish notation basic turning class regularizer nonconvex collection sample z parameter vector goal estimate regularize estimator serve enforce type carefully regularizer abuse notation eq q allow loss convex function satisfy low constraint global finally theory constraint contain state univariate tt line differentiable l except omit condition wide variety regularizer class exclude regularizers bridge derivative check local composite appear curvature control regularizer interest penalty nonetheless practice study past nonconvex regularization penalty mcp condition assumption weaker know restrict convexity rsc involve remainder expansion statistical rsc condition n strictly constant nonnegative understand rsc inequality high set rsc may still hence rsc inequality condition condition much family nonconvex quadratic see rsc range corollary section hold complicated neighborhood appendix however convex conclusion even weak rsc hold whenever rsc appear necessarily rsc strong condition past rsc enforce may nonconvex prefer rsc whereas rsc follow setup rsc hold subset consistency rsc uniform rsc prefer rsc hold fact establish rsc setup statement proof well consequence minimum eq lie constraint usual maxima term minima result interior local maxima main deterministic lie probabilistic section establish choice theorem require quantity rsc satisfy objective suppose necessary condition however guarantee discussion procedure obtain stationary agree familiar theory furthermore motivation indeed scad concern glm recall glm parameter family consistency optimization correspond give rise level since optimize glm optimize penalize observation glm give stationary nonconvex sub convex give rise optima optima provide theoretical result quality algorithm guarantee within proximity complicate explain proposition structure graphical lasso q possibly nonconvex penalty function statistical algorithmic graphical lasso value observation version imply even nonconvex regularizer point nonconvex inverse suggest graphical systematically corrupt modification corruption capture involve sparse whether draw graph variant diagonal entry statistical hold entry equally argument consider hold suppose base possibly q minimax frobenius sparse introduce shorthand bind feasible old inequality lemma bind rearrange hand claim yield assumption imply combine rearrange use old denote cone substituting eq eq cone inequality combine upper substitute version descent enjoy linear focus version function program objective nonsmooth produce sequence stepsize update may establish iterate large begin analogously define taylor require restrict n identical rsc taylor r condition past throughout coefficient ensure derive general rsc appropriately include exposition reasonable unless composite update state guarantee roughly square iteration guarantee statement square tolerance converge target perspective optimize beyond tolerance turn entirely deterministic empirical loss regularizer suppose global program n appear take successive iterate composite descent converge size satisfy probability conclude curvature penalty enter via denominator tight possess curvature close intuition theorem requirement certainly possible descent violate mild parameter must large stepsize behave stepsize iterative search unknown correct rsc
evaluate ability work model relate ensemble pair etc thesis way response explanatory guide adaptive rf drug discovery method random remainder organize set descriptor set assessment metric assess context describe final ensemble call ensemble present comparison comparison make rf forest drug discovery explanatory need group general explore diversity implication diverse conclusion four molecular library screen compound specific disease aid aid aid aid may four investigate proportion active proportion three tree run tend many terminal range drug application aid bind mechanism aid aid aid activity chemical correspondingly herein statistical distinct explanatory underlie drug drug compound relate molecular characterize chemical descriptor explanatory shape set descriptor atom ap burden number pair ph burden descriptor bit string bit certain see molecular capture descriptor summarize descriptor set atom pair ap number atom compute descriptor ap bn respectively table descriptor chemical remove give table hundred descriptor low continuous metric procedure rare chemical library metric misclassification unbalanced probability activity compound etc relate rank list resource allow measure functional versus proportion proportion show cutoff point superior value show curve apply bn descriptor early rank dominate ensemble uniformly clear winner curve numerical summarize outline summary rate rank want define average point rank tie tie rank order tie chapter rf clear winner criterion preferable evaluate ie proportion collection ie rescale precision measure conclusion give rank ie value large drawback ie therefore report ie ie rf winner train balanced fold group serve test remain nine group compound rank ie formation motivation group friend member depend upon individual strength analogy variable separate performance strength even number divide descriptor exhaustive infeasible perform look resemble variable cluster observation group original screening hierarchical merging screening candidate termination compound rf base ensemble formation rf drug discovery formation bag estimate assessment expense fit cross avoid formation reduce formation assessment measure final ensemble ie trivially generalize assume well dimensionality ap ph later hierarchical merge expensive consider quadratic demanding set give value initial classifier extremely initial rank ap fp ph descriptor name fp seven relate presence separate similarly group verify grouping provide grouping group bn descriptor weak computational burden group assessment criterion descriptor repeat empirical strong base competitive screening pass test consider estimated probability base use measure base fit single probability assessment assessment measure strong pass test alone group q improve remove group merge resemble iteration group ratio well merge union continue suggest merging illustrate group actually individual descriptor model variable merge group merge terminate original general candidate keep individually ensemble need exhaustive merging previous stage screen ensemble rf forest test ensemble averaged ranking show formation bad grouping involve fit fit pair fit fit formation already screen stage algorithm continue merge create new form fit new one need formation screen formation fit burden cause dimensionality descriptor greatly form moreover straightforward package descriptor construct three seed rf use package detailed aid aid aid aid five column ap total arrange screen merge run seed impact later ap bn fp descriptor base ap fp example initial drop descriptor bn use measure report rf description appear balanced validation split initially processor processing mean across binary set ap fp ph descriptor outperform bn rf however advantage great ap bn fp ph box plot run consistently outperform rf tree stage burden visualize gain descriptor figure descriptor balanced cross detect rank ap fp number show ie ie average replication balance consistently exhibit study three aid aid summary table rf descriptor ph descriptor rare suggest relevant rf cancer cancer drug whereas tree rf widely recognize discovery package setting aid augment inactive leave balanced active rf table overall rf sometimes bad still dominate rf c bn pt fp ph ap fp ph ap bn pt fp ph motivated assign majority class weight r increase example aid balance active time table little rf aid bn descriptor rf descriptor drug present formation descriptor name logical name initial form name descriptor index observation distance via cluster name ap descriptor aid pt l pt fp table base name aid aid descriptor small large performance rf neither aid descriptor ap aid overall argue strength ensemble depict diversity cross aid bn descriptor rank run forest relate order rank ideally would rank depict mid gray color indicate failure well px assign rank fair beneficial unlikely performance compound gray figure px report axis px bn descriptor rf constitute ensemble relatively aid highlight cell live multiple mechanism chemical structure form bn descriptor translate diversity active chemical sort absolute versus six favor structure particularly variety next stage drug adjust efficacy inactive identify show diversity inactive descriptor two distinct surprising include inspection structure show cc compound active determine versus helpful design datum little response explanatory information response drug discovery rare explanatory even use explanatory model
detection contrast temperature cavity transition find underlie message sufficiently sbm fraction even amount prior temperature rich improve line jump critical agreement cavity sufficiently one group node give depend whether group k e graph model joint encodes reconstruct toward give prior p unit temperature assignment function infer try state jointly maximize generalized hamiltonian term prior knowledge assignment since exact computationally intractable resort popular message pass belief propagation converge graph loop typical loop length sbm correct want find ground state marginal however temperature message label simplification belief propagation max least cavity preferred node calculate sum neighboring obtain cavity j pick belong membership depend due symmetry relevant cavity parameterize pass cavity seem resort consider pool dynamically rule specify essence message pass message pass scheme break one group write analyze temperature message message majority vote message receive plus type incorrect spread incorrect receive correct message neighbor poisson receive label achieve message incorrect message probability incorrect color regularize gamma indicate fix inspection phase transition single node q fact temperature intuition suitable locally correspond appearance second solution instability correspond message fig fit cavity albeit perturbation message curve randomly solid dashed gap two transition red plot threshold keep succeed whenever easy hard message inference community available one first every toward represent correct correct message scenario achieve modify fraction message automatically correct replace eq two move boundary hard regime jump accurate jump critical value amount information even small fix observation contrast ref detection jump point increase qualitatively cavity calculation zero message break tie reduce number give randomized message community threshold threshold reproduce qualitative transition predict cavity value move range jump cavity connectivity support part grant grant fa zhang sciences usa institute usa institute phase transition cavity rigorously analytic calculation cavity method since distribution transition inference furthermore whenever message break correct threshold reproduce qualitative predict whenever temperature cavity method analogous partially finally set correct fraction jump fundamental sbm network dense community recover circumstance sharp community cavity analyze propagation bp bp optimal community threshold hypothesis prove rigorously group correctly regime indistinguishable enyi
vector note independent lemma since function max q lemma sum moreover plug turn together overall mutual uniform induce follow bit summarize constitute individual expectation hold early build appendix begin thm distribution seek reference instance choose uniformly e biased coordinate denote computed protocol require equal choice presentation denote message quantity eq function round coordinate clearly every uniformly equal eq combine plug back bind justification inequality relative trial independent trial verify entropy definition overall describe follow assume independently depend note variable jensen inequality also eq reverse expression entropy upper term convex argument upper entropy equal mutual writing refer entry draw whereas number zero closed form entry pick independently therefore fact inequality upper require assume contradiction equal round commonly take bias easily see protocol take detection whereas scheme discuss previous least reach contradiction initial rather involved variant seek problem explicit protocol prove seek sparse parameterized detect correspond pick positive resp create coordinate thm seek seek broadly thm somewhat protocol since protocol allow protocol proof immediate instance feed memory online batch remain make memory protocol value replace tune attain turn union imply pick mean detecting since x j reduction define j I ji two zero choose belong non dd jx bernstein inequality since choose therefore satisfie observation sample pick assign unless pick probability seek definition ni deal protocol bm former simply instead latter replace thm replacement justify theorem construction use gap size support support detect factor already roughly lemma broadly top seek coordinate let message protocol low simplify drop message thus receive round instance zero value order instance uniquely indicate coordinate th eq since convex decompose lemma satisfied rewrite eq expression relative jensen relative argument entropy term expression equal mutual message coordinate see respect bind key bind draw message easy verify expression large bin randomly bin inside expectation make quantity briefly relevant value finite intuitively g conditional equality also I support size information intuitively variable carry reduction get define kullback two distribution relative jointly argument also satisfy rule easily bound variation context inequality many machine interact access g miss armed however understand fundamentally affect semantic memory perform bad constraint constraint availability learner currently constraint manner interact year among potentially speed ability cope flip machine require linear principal prohibitive high another gram effort develop analyze memory fast scalable online sequentially mirror seen often maintain see situation miss feature considerable online web multiclass learning perhaps well case bandit variant bandit example domain share constraint training notably bandit protocol g formalize price complexity guarantee lack information quantify constraint algorithm go one provably quantify get affect ability learn knowledge answer setting develop process characterize theoretic interact semantic partial information algorithm worse attain several specific problem new regret bind partial learning bit vector optimal matter coordinate bandit coordinate various semi limited linear feedback restrict partial monitoring etc interestingly learner allow choose retain bound quantify independent semantic pca estimation attain statistically optimal trade set interactive serial attain machine good knowledge formal communication budget large example come existence gradient descent statistically relate work much bound seminal also point directly translate wish simple exploit availability prox mapping moreover aware indicate work assume different identify case spirit per answer bit contrast low budget difference work focus include constrain strong also show apply need armed crucially information view loss semi bandit bandit projection stream communication constraint e unfortunately consider detect consider flip work memory algorithm guarantee provable trade work memory reference therein limitation require precision amount memory memory limit accuracy contrast consideration word th standard convenient constant log factor natural logarithm class constrain give access sequence vector function bit instance depend crucial constrain bit definition quite specific protocol stochastic algorithms size protocol round depend machine independent depend centralized message output style optimization previous send protocol receive extract bit armed bandit mini batch overall sequentially extract batch final process include provide bit unless know theoretic tool mutual prove contraction technical level divergence ar divergence perform time discuss eventually thm expert round learner know learner goal minimize partial information learner follow thm protocol constant model extract vector impossible regret interestingly examine choose lower partial allow view observation bind monitoring partially g low stochastic minimize recent statistical problem information constrain protocol provably pay price constrain toy realistic reason illustrate type information protocol involve realistic consider follow wish w concentration measure indice f db mb whose parameter protocol straightforward thm independently equal reduce detect early protocol happen gap protocol apply interactive problem know statistical sample direction large focus simple form zero case sparse pca coordinate maximally natural sparse assume covariate pair detect biased construction support intuition situation variant gap protocol estimator specific dimension unique pair empirical average dd dd exist theorem choose get protocol small even though arbitrarily probability sparse statistically size protocol regime algorithm early explicitly cost estimation interesting recently affect seek rely online protocol protocol protocol establish log information protocol interesting inferior gap constrain context learn partial explicitly depend bit extract believe first
minimum decrease try reasonably cf unique factor moderate also include figure substantial iteration follow conclusion estimate model factor normalize deviation extend case estimate computation quite rapidly situation type even though precisely predict expression analysis tumor normal arrays nd due wrong case reason rather reason force think old introduction exploratory factor primary aim allow factor explain uncorrelated vector loading mutually independent standardized uncorrelated diagonal normally score prefer start get also agree conventional square treat know motivation really formula agree provide distribution ml normality easily derive observation lead iteration appear precisely conjecture nice behaviour iteration likelihood equation model emphasis describe mean coefficient loading latent vector standardize covariance mutually uncorrelated covariance recent year increase appear comprehensive mention study deal propose assumption inconsistent sec artificial fitting problematic case basic component suitable purpose turn difficulty yield distribution free aim component pc sometimes pc regard represent define role vanish I determine matrix datum similarly zero know fa find pc estimate find scale invariant sec computationally replace fa utilize methodology property support equation scale invariant sense mention factor score basic naturally lead decomposition svd sec sec expression factor score recommend iteration successfully gene expression mention standardize concentrate population loading natural ml demand motivation classical require ideally get pca reduce rescaled rescaling moment iteratively observation cf know diagonal sufficient supplement space specify eigenvector eigenvector eigenvector loading tell trivially function give leave side multiplication formulae equation relate turn robustness argument strong argument also intuitive pca albeit dependent rescaling equation method take give sequel unless care equation yield impossible diagonal paragraph fa literature unweighted appear data yield identically estimate difference eigenvalue constraint case method different additional adequate estimating lead naturally iterative procedure calculate calculate calculation svd express history turn stop bad sometimes might good value wrong reason ml estimation maximization algorithm suggest iteration procedure yield advantage current recommend investigation difficult eigenvector update large discussion center correspondingly computation q orthonormal diagonal diagonal element singular square root singular form orthonormal fa svd affect whether form high thus express estimation combination follow express replacing yield start go wrong positive definite first satisfy tp high see contain diagonal supplement first cf harmonic factor proportion q subtract term relatively score row score regard know score score precisely achieve version diagonal scale cf good score well linear sec expect score precisely condition kept formula conclude matrix justified formula differ value component quite little precision score z residual standardize compare tell trace standardize normalize instead correspond freedom regard free method fit regard treat method quite uniqueness author impose contain satisfie fulfil conclude need singular constraint eigenvector gaussian global consistent consider true conclusion artificial constrain datum reasonable model light ml eigenvector score score constraint fit fit outside consistent illustration box consist
ns n n u n claim make rule proof lemma define n recall assumption sure strong sure almost show simplicity notation z imply n consequence either n use u define suffice purpose attempt notation consequence small consequence either show counterpart theorem n z z follow yield u mean v v n exist integer make rule position exist virtue n definition stop almost surely sure law number event sure event purpose let notation n v ny z e sure imply establish make proof suffice expect notation v shows imply counterpart exist combine q investigate property preliminary notation true moreover hand either must eq virtue monotonicity accordingly define happen iii iv early impossible virtue eq iv monotonicity respect accordingly lem shall show arbitrary exist yield constant n constant n n complete exist hold sampling scheme n I q virtue l I n virtue n l z hold precede q proof lem I I lem lem exist exist pp observing hold suffice hand definition stop thus complete lem exist rule since n l virtue I suffice use proof sequel positive lemma claim exist integer u u u follow z u u z large z z enough complete make position exist eq make weak number imply virtue n z n prove follow simplify notation z l assumption sure follow number stop sure event sure purpose attempt notation l consequence use must property establish sequel establish lemma lemma vi fact complete lemma j define z v u z z c u n c n virtue z exist define view q write c z must establish j make show statement claim exist statement proof prove sequel make counterpart group u integer z z exist exist integer use stop simplify z u z l u l e law number sure sure event suffice show eq consequence z make result virtue thus c u result virtue lemma thus property follow property property iv vi lemma j stop claim l l z u multiply inequality yield eq u z j j claim complete make counterpart exist number need prove claim integer claim make observation z u z u z z establish establish stop rule u l l z l z sure strong number sure event sure hence attempt write eq show follow must enough iii result iv develop sequel establish property lemma property lemma exist next z h inequality continuity assumption associate inequality c z establish claim use z statement lemma establish q j complete use lemma z follow exist similar manner complete position make large virtue n lemma notation define l n n l e l sure event law sure sure purpose simplicity notation z consequence imply either true thus iii lemma establish property prove iv iv vi lemma variable n q definition definition c j eq making result z statement exist lemma complete mean exist exist v w n w manner establish complete corollary engineering construct pre coverage propose statistical inference accumulate observational accomplish interval confidence technology statistical inferential become increasingly important purpose variety problem cast frequent familiar example recognition huge rule important issue determination sample overcome literature reference therein advance sampling accordance despite unified sequential requirement use inclusion wide problem cast level coverage interval refer controlling process confidence include coverage probability sequel rule inclusion usually possible lot formulate interval apply principle construct fully scheme inclusion principle construct scheme statistical methodology paper science science therein throughout shall integer denote integer denote variable subject concerned event probability always I take clear mention science engineering estimation random assume domain expectation interval estimator deterministic random reliability criterion pose view interval sequel construct random size n sample approximated satisfying depend issue desirable develop accomplish remainder propose general virtue concrete eq coverage inclusion principle include n termination sampling number respectively stop eliminate confidence confidence scheme property less virtue principle rule continue stop possesse moreover increase see non increase decrease interval respect show principle sec assumption define describe let virtue inclusion stop continue stopping possess arbitrary proof inclusion principle define confidence nx rule principle stop define let virtue inclusion stop continue less rule scheme possess property course derive principle sequence nu u construct approximation section consider note describe let number less result sampling possess appendix principle number less theorem interval stop first inclusion concrete construct sequence sequence decrease integer integer let coverage inclusion principle propose stop confidence include z stopping index termination refer sample provided eliminate computation confidence z actually follow ss respect attain iii consequence z ss c gs establish iii z g ss remove u since dm lemma z claim similar l uniformly neighborhood exist z z notation z gs rs g v claim satisfy requirement unique consequence continuity virtue q clearly continuous uniformly respect similarly argue therefore claim rs u finite independent ct v notation continuous follow preliminary n lemma prove claim n ns wu n n n v u f n enough establish claim lemma integer n virtue completes prove n event number sure stop almost sure event suffice notation continuity lemma true enough must either eq purpose result ii purpose notation u follow consequence either f observation v n enough imply position
intuition selective yet general practical impact converge sample choose stationary accurate decrease selective iid rates heuristic exhaustive also modal count heuristic perhaps remain nonetheless believe theoretical selective practical application author communication material david error course borel thm axiom function near constant know admissible asymptotically term neighbor character function linearity proceed intercept extremely require classic pattern appear identically bay bayes show near neighbor cover iid sample sense remove two near sample latter address technique accord rule sample misclassifie near simplify considerably conceptually diagram course take place wherein pool candidate odd reduce selective within broad selective term near neighbor al selective euclidean heuristic language abstract description heuristic domain pattern complex computationally selective setting assume measure heuristic pattern computationally complexity grow naive believe constitute advance establish set key relate practical indicate include throughout rest effort approximate classifier metric approximate set operate near choose cite much fundamentally algorithm develop break near arbitrarily critical recover area ball iff furthermore iff converge pointwise save briefly event occur probability probability phrase work term implication theory almost surely occur infinitely pointwise achieve random call determine precede elastic term countable dense immediate pointwise result understand arbitrary fail give monotonically design euclidean take newly newly inaccurate since become accurate intuitively reasonable method sample value modal count useful complicated setting connect connect inclusion iff rule near valid boundary burden boundary simply eq determine denote ie iff infinitely one candidate since occur lie entirely borel infinitely side q second borel infinitely often probability one denote contradict candidate small contradict eventually place less would force place candidate argument formally borel selective advantage iid indeed suggest iid ie superior demonstrate near value argument demonstrate mean select find cover cover metric measure immediately boundary either boundary outside lemma set tend geometrically unlike sensitive really want many elsewhere contiguous contiguous component also contain contiguous component contain claim eventually place almost lie borel infinitely claim isometry completion point way allow boundary iff boundary cross inconsistent certainly almost surely hence neighbor choose neighbor infinitely surely subsequence boundary candidate lie lie infinitely often candidate modal c cc remark obtain contain contiguous almost although derive contradiction place close contiguous component sampling contiguous component separable contiguous measure contiguous component modal borel place contiguous prefer place contiguous component measure place iid percent limit never measure force aspect avoid problem neighbor insight iff infimum completion boundary boundary define use boundary ball cover ball candidate lie exist close ball contain away infinity borel infinitely become certainly consequence fact think short yet contradict character except observe euclidean space rational desirable diagram difficult suggest involve diagram construct unlikely practical closeness set respect near define denote near contiguous component proceed proof identification neighbor per point least near predict infinitely infinitely ie side however fall back force eventually point consider however case remark intuition separable boundary union contiguous component measure near neighbor reasonable report exact derive value aware result although precede grow exponentially boundary second tell candidate appear contradiction boundary measure modal force force
latent reduction unified framework result baseline extraction conduct factor rank consistently meaningful baseline discriminant lda powerful dimensionality project low separability basic fisher matrix within class maximize simultaneously class discrimination deal subspace dimension multi effectiveness computational successfully face microarray dimensionality reduction label surprisingly prohibitive decade aid regression label svm supervise real bag labeling propose short fisher lda model mi multiple image video analysis require moreover variable great requirement surprisingly prohibitive good mi latent discriminative maximize fisher discriminant drive mixture one demonstrate capability extraction video event search dissimilarity minimize dissimilarity separability typically classification attract attention effectiveness al lda face project nan scatter maximize combines maximize scatter space space scatter separately get final transformation divide scatter lda easily trick analysis project point recently world consist pair construct predict output novel challenging decade semi extend supervise supervise datum aid semi supervise discriminant separability class unlabele intrinsic method preserve sample separating label semi geometric propagate example combine mean algorithm generate label lda utilize perform selection new latent fisher discriminant exist latent popular include latent discriminant mi svm model extend maximize joint combine unified fisher discriminant x x treat bag categorical assume finite set n j fisher discriminant subspace class search decide namely projection latent eq regularization scatter dependent categorical label know minimize need algorithm vice versa projection latent latent variable inference cluster mean inference learn extend incorporate instance class component gaussian gaussian parameter algorithm center class posterior discuss maximize one maximize maximize prior weight knn pointwise production hadamard fisher discriminant center update fisher update break compute neighbor discriminative weight return center update manner attribute assignment recall em approach use em variable infer follow jensen maximize em infer hard pz pz maximize hard thus special maximize pz variable embed step converge em latent model bayes decision jx jj graphical add lda strategy maximize maximize joint approximate argue video graphical perform experiment uniformly set description dimensional surface average bag category represent texture shape set positive example image draw reduce nn average ten validation summarize set mi svm outperform mi comparative mi table c set dim mi lda dataset event five consist human activity interact people place event use video human semantic loop explain setup extraction among baseline descriptor frame video bag model detect codebook benchmark mi use fast kind frame far away svm frames close refer svm randomly frame rand student old loop detailed representative discriminative mean event sure understand training subject extract comparison student trial require yes subject inform subject computer conduct image pair speed annotation interface video ask video video last annotation video video test video count comparison consider help frame fig well cm cm cm cm rand rand svm five voting treat yes image pairwise comparison yes cast leave else else
runtime requirement instead wishart big large number additionally design permit adapt encourage allow smooth change introduce hyperparameter relate autoregressive never access sample reason together infer inspection create time copy coefficient use gaussian copy strength base tune hold along hyperparameter approach differ hyperparameter autocorrelation encourage many encode matrix fully infer distribution obtain mode straightforward interpretability grouping feature clear similarity encourage coefficient go probabilistic lasso generalize prior improper jeffreys fuse difference clear autocorrelation coefficient resemble optimize encode belief allow modify base seek maximize inference mean variational derive log optimize variational factored variational distribution gamma b parameter seek give parameter coefficient employ bfgs newton variational maximize inequality give denote derivative respect eq interpret rewrite form hyperparameter analogous specify scalar role advance feature autocorrelation similarity encourage difference encourage smoothness sparsity compute determinant maximize calculate derivative omit bfgs reach value theoretically strongly weak principled gradient sparsity expect sparse incorporate term function analogous group consider overall keep simple two application text year compare baseline forecast regression train set past example non ridge year train tune replicate different insensitive coefficient available year slide size drift coefficient separately rather set hyperparameter prior volatility financial report mse set year development difference compete sign finance refer variation stock year volatility consider regression volatility transformation distribute interpret drawing apply make collection exchange publicly report period year report available text word feature keep community past return good volatility therefore stock response volatility publish tuning initialize coefficient training year training table summary set response outperform also outperform variant four major challenge choose relevant treatment strength autocorrelation trust demonstrate autocorrelation variational learn improvement variational feature feature previous volatility feature text time economic measurement word world average predict text write publicly affect economic sparse lexical perturb feature vocabulary background word corpus context feature easily coefficient effect correspond observe since text variable house economic frequency word world word multinomial multiclass logistic l apply assume prior connection might source k report report market us company head office two source primary body responsible policy market time discuss description economic national book activity produce discussion text book prior consist document report document book summary st dataset text produce unite text bank state serve htb various year development quantitative bank datum repository activity focus market market various characteristic addition compare baseline analogous variable compound model log dataset lasso year development six forecasting document early collapse ridge frequency background outperform ridge variant penalty improve predictive trend insight manually trend model figure word stre percentage rate correlation trend learn contain reading coefficient explanation present probabilistic model strength temporal coefficient prior task forecast stock report competing model word observe show achieve acknowledgment thank anonymous
sense fast order stick run code algorithm describe herein find compressed usage simplex solve simplex homotopy enjoy alternative tradeoff future sensing efficiently solve separately consider initial basic infeasible variant simplex arrive demonstrate sense stack allow compressed show kronecker sense perfect kronecker much benefit interior compressed recover sparse signal theoretical foundation compress sense work progress compress recover know system solve hardness pursuit replace condition discover submatrix index isometry constant convex exist solve bregman iteration several match variant combinatorial fourier develop paper optimization compressed name reduce competitive somewhat simplex appropriately take basic stack multiply compressed course multiplication sense problem fair multiplication need programming therefore compress sense although representation linear involve complexity complexity gain specifically section matrix see continuous block next describe simplex main behind compressed trivial solution large nonzero serve simplex rest sense compress kronecker sensing multidimensional try knowledge clear kronecker natural kronecker add stack put length sub generality right sense show kronecker compressed discuss sense column element rewrite kronecker product compress property kronecker sensing isometry isometry small equivalently distribution tx gaussian mean sufficient independent variance constant convex attain perfect careful measurement satisfy compressed stacking strictly multiply recover whenever program kronecker sense factored product completely product matrix introduce new rewrite constraint split convert see problem varied
reduce satisfied obtain adopt classical incomplete know label arise fact know observation bad source ig ig ig complete datum log iterate replace two arise g e th calculation ig ie ng ig rp ig cm nz ig ig contaminate gaussian aspect second calculation maximize numerical fix facilitate convergence way term eigen decomposition parsimonious eigen matrix classical q write computing environment give code differ package parameter package package preferable alternative solve start constitute consist select initialization position maximize complicated strategy select technique model correspond fix package initialize correspond operational view thank guarantee log always great log criterion correspond acceleration maximum log base whether log acceleration iteration converge good component I evaluate good eliminate bad outcome treat detection bad former proportion determine numerical search nz ig herein use analogous specify proportion outlier proportion advance pre specify realistic characterize far quantity fix nevertheless usual model adopt simulation distribution analysis adopt bayesian overall artificial set detect bad bivariate equal mixture add uniform uniform fall happen twice classify associate eigen decomposition lr component ex ex denote consider package good affected respectively bic obviously additional third affect bic compare recognize bic high represent view red popular consider measurement length cl ht perturb highlight perturbation cluster three compete report systematically perturb contrast mixture necessarily decrease extent perturbation recall th contain cf chemical region derive three clustering ignore ht difficult bad situation practically specify family eigen mixture decompose covariance package fit bic degree hold equal across recognize presence classify correctly classify correctly perfect bad view see bad capture surprising majority point bad generalization spurious refer importantly however family put gold although approach mixture comprise point bad point cluster separate used specification always impossible another advantage contaminate discriminant analysis option supervision could yet flexibility compete work superiority mixture gaussian contaminate give extent superiority contaminate gaussian mixture consider nature choice proportion sensible future facilitate contamination elliptical density paradigm acknowledgement work carry university visit engineering research theorem base contaminate spurious point noise bad herein contamination crucially contaminate introduce identifiability member family maximization outline issue artificial datum contaminate detection modelling purpose indirect application semi density see direct consider powerful device assume mixture theoretical contaminate refer bad herein component matrix bad insensitive presence summarize mean model weight mixture hull accommodate analyze drawback application bad lead assimilation bad consider uniform discriminant recognize lie outside fit model indirect mixture paradigm overcome contaminate contaminate probability align mm pp pp gp g mixture component membership ig ig ig notation base cluster classification note expect establish theory maximum investigate identifiability contaminate
could well lrr notable scheme limitation pairwise compare subspace linear select name representation incorporate distance linear representation select moreover analytic bs neighbor neighbor fig neighbor derive accepted assumption manifold linearly term accordingly linear fig toy illustrate get answer belong term approach I th n distance I produce inside high computing decrease discrimination subspace letter discriminate similarity consequently get discrimination construct step search graph treat assigning ji satisfactory simulation present verify effectiveness ar use near ar individual subject efficiency reduce speed neighbor firstly performance subspace preserve embed locality preserve ar train remain report evaluate see algorithm outperform high ar construct similarity obtain point database ac build lrr ssc graph normalize spectral cluster ac normalize ac distinct lrr ac ssc pairwise distance heat popular subspace limitation enhance intra intra enforcing reconstruct error extensive verify approach claim
ci find prediction accuracy study database predictive prediction recognize develop genomic signature batch correction infer effect power expression microarray depend set gene violate training biological class case research publicly microarray data removal develop package responsible promise genomic publish genomic finding correction develop design population genomic clinical diagnostic correction batch correction improve public genomic package basic research generation clinical tool make diagnostic group despite clinical signature successfully translate reason relatively variable genomic vary day responsible promise genomic major genomic finding biological batch effect recently demonstrate effect also recognize development genomic clinical remove genomic study remove batch two key correction correction level correction prediction biological time surrogate batch strength database challenge prediction standard clean use remove batch clean substantial biological outcome publicly available microarray prediction batch develop implement batch batch population refer paper protein abundance dna propose outcome relationship outcome indicator belong one form genomic accounting due factor modify factor factor call error previously demonstrate variance condition satisfied population biological must perform surrogate batch train pr estimation probability gene associate pr weighted let fit least square remove apply develop classifier genomic clean expression accomplish standard genomic application classifier new genomic batch strength remove augment weighted decomposition singular pre multiply result estimate sample consist first clean coefficient clean new exact calculate projection exact set grow approximation answer database estimate simulation benefit discretized weight varied weight subtle effect genomic ht parameter affect feature affect affected affected batch outcome affect affect affected outcome affect affect three improve equation specify additionally percentage affected outcome indicate simulation specify batch two varied batch database pearson correlation simulate batch uncorrelated database sample correction simulate alone database new sample simulated database commonly classify prediction build outcome repeat time robustness prediction display graph function figure test correction correction correction perform randomly choose simulation interestingly outperform however control perform additionally outperform correction outcome performance show minimal effect database outcome correction
say coincide regularization might remove sure conduct evaluate comparison note rule make error correct kkt check real synthetic datum simulate correlation gaussian distribution use face image gray face people illumination image pixel regression pick image datum handwritten digit dimension regression randomly set image image ensure correctness contrast need condition discard guarantee representation safe improve inactive paper propose safe variational three module derivation key propose usage variational propose screening rule upper sure removal identify module derive optimality via regularization upper discard th feature upper challenging plan quadratic solution also lasso path lar counter theorem counter com wang non zero safe technique efficiency safe screen attention formulation usually especially try inequality optimality lasso relax safe screening removal datum demonstrate effectiveness effective analyze area formulate loss regularization tradeoff let correspond I logistic unknown practical specify schwarz criterion lars path interior coordinate accelerate descent formulation series pre zero nature screening propose coefficient cost exclude inactive technique safe obtained eliminate discard variational safe strong rule monotone sure regularization feature empirical effectiveness screen extension logistic briefly discuss scalar letter bold norm infinity norm denote build dual dual eq component denote need say eliminate denote solution variable correspond respectively dual solution feature save remove small construction success loose upper discard feasible optimality condition small discussion extend elaborate building discussion derive dual equivalence verify follow relationship primal last dual problem formulate analytically start dual computed analytically computation upper construct close differentiable apply eq construct illustration please close tight singleton contain improve estimation set estimating solve discussion introduce prediction scale summation input figure line ball ed ec angle ec maximizer denote ex maximizer dash radius ed illustrate respectively angle supplement notation rewrite eq following indicate space say admit equal supplement theorem bind denote satisfie supplement c eliminate analysis establish firstly x th remove low feature remove wide monotone property propose differ feasible dependent inner term strong utilize correlation rule extreme dpp intersection ball center radius half ec pass safe ball radius point line segment dpp bc denote g safe make scaling safe formulation safe safe set follow relaxation utilize ball relaxation dpp add author motivate
cross negative example bootstrappe improvement scale per ratio detector technique confident window input pairwise simply calculate score cascade summarize evenly fisher cascade detector similar observe post post art detector author various combine discriminative boost conclusion feature object solely unlikely outperform feature train detector feature haar feature statistic pixel location intensity intensity orientation pixel map correlation coefficient feature feature encode histogram statistic texture project discriminant implementation ss ess good latter feature sophisticated compare fisher detector adaboost detector cascade detector train cascade detector set protocol outperform detector ss since original detector roc curve detection fig datum detector b inferior motion discriminative similarity part detector part art benchmark average number compare traditional cascade cascade classify per core consider scale feature negative chance apply positive distribute equality statistically point plausible diagonal detection j w w summary approach e object main reason first reason datum latter bootstrappe force visually ignore negative datum likely scale element impact c negative improvement coincide criterion lda detection validate always give surprising hypothesis arise really form would last try identity discuss numerical difficulty low well problem replace primal regularize qp clearly primal variable minimize margin maximize weighted margin may parameter experiment vary classifier improve experiment primal invertible use demonstrate regularization improve digit face overall explicitly take new object superiority label efficiently apply asymmetric computer future new exploit tu wise tune boost boost work work response fig hyper nm date nm hyper nm date nm hyper hyper nm date nm hyper date research fellowship ft detection cascade achieve detection moderate false rate requirement principled feature asymmetric node objective bias linear asymmetric optimize experimental verify detection real detection inherently large candidate process single image million window single imbalance detector reflect cascade classifier imbalance wu receive speed principled train boost cascade boost cascade significant subsequent attention li face wu wu cascade increasingly complex classifier false negative rate classifier whereby patch adjust achieve produce cascade make represent detection false point equation extremely moderate positive cascade node goal drawback adaboost boost cascade structure minimize false negative adaboost variant modify function negative adaboost still achieve wu lda adjust select wu fast node meet translate train strong classifier wu separate adaboost use strong adjust node conjecture improvement learn explicitly take account step propose implement idea verify version contribution simplify version minimax asymmetric importantly boost asymmetric basis fisher rather identify optimally knowledge method similarity sense ne originally propose lagrange generation qp special problem compare qp wu wu art list confirm conjecture effectiveness apply asymmetric analyze validity might apply rather cascade perform phenomenon show lda well detection explanation lda demonstrate detection differ boost minimize possibility designing purpose extend next real use cascade cascade target last decade seminal contribute time object cascade negative patch early maintain select informative train strong make haar cascade develop include cascade dynamic cascade cascade cascade recently embed cascade adopt efficiency cascade classifier patch reach th classifier weak cascade post processing enhance cascade cascade cascade improvement algorithm building node cascade wu use accelerate wu rare use online boost classifier redundant sensitive boost sensitive loss maximize kullback promise report logistic additive lin require location target haar object multi view face feature histogram orient gradient along integral detection descriptor variant promise human detection spend feature concatenation wang cascade wu tradeoff mixture expert briefly minimax version biased minimax machine asymmetric minimax section show design apply section conclude notation denote bold letter clear vector use project value finite element eliminate column weak output weak boost entirely directly interact largely write vector multiplying let represent w margin boost concept minimax machine n x minimax separation hyperplane express identify hyperplane accuracy datum problem efficiently formulation mis classification identical many application bias version modification class decision hyperplane classification biased well bias et show iteratively via fp technique computationally demand solve formulate simple program qp yu interested object robust wu algorithms theoretical yu general constraint form unimodal gaussian biased base distribution impose distribution show wu face output approximate constraint utilize priori consider simplify conservative biased special see bad symmetric symmetric unimodal distribution yu immediate consequence force put away bias formulate obtain q biased arrive simply enable close connection asymmetric classifier wu wu remove inequality lead eigen wu al wu linear start minimax symmetric brief overview post cascade framework wu solution know seek pair wu assume approximate relaxed last solve eigen solution feature output cascade wish detection maintain adaboost feature boost symmetric classifier verify gaussian cascade face detail theoretically adaboost follow weak verify result normality original mapping act I explicitly x straightforwardly maximize minimize express scatter project class reformulate otherwise exposition correspond training one correspond e rewrite constant convenience remove ill pose see optimization problem stage remain unclear infinitely classifier extremely program applicable lagrange derive need meaningful dual r give dual inverse actually eigenvalue zero simply diagonal strict strictly condition dual connection optimum must hard margin regularization cost duality solution coincide one violate add dual feasibility speed add violate follow q use produce well weak weak change note include offset final classifier find data search find cascade need tune offset guarantee cutting generation decrease objective globally defer appendix value dual would decrease accordingly primal zero duality therefore break master correspond new update increment x solve practice fast primal example exploit primal solve ne descent mirror exp qp object detector qp qp solver possible train detector amount majority bootstrappe optimum must exist summary primal problem variable world set large subtle difference place emphasis positive classified adaboost adaboost optimize overall consistent early test percentage asymmetric adaboost fisher ce wu sensitive adaboost rate adaboost baseline train classifier false time report parameter cross li choose asymmetric li da cost choose li da experiment enforce train enforce rate target barrier five set machine vision digit digits face face extract patch analysis total new car pixel apply capture experimental original pixel scene divide scene beyond code histogram wu represent hierarchy windows dimension classifier remove detection perform poor due cause overfitte section experiment eight asymmetric boost evaluate cascade adaboost alone wu fast alone detector adaboost cascade detector cascade li extension cascade training example order label follow acceptable node target node index current false increment node index detection acceptable yet classifier update weak linear coefficient adjust node classify classifier misclassifie adopt fisher post processing cascade node classifier cascade exhibit margin training margin distribution ce use node cascade choose multi effectiveness cascade lda ce conventional cascade post ce wu observe basic haar like image weak wu feature weak face consist validation large wu cascade ensure fair stage cascade consist index pre cross instead train cascade choose negative misclassifie cascade discard negative example background pool positive validation keep un ed face detector asymmetric boost cascade mit face false positive feature use implement cascade adaboost multi cascade curve original paper li compare rank legend rate perform intel gb ram hour adaboost take less complete
significant progress behaviour pattern subset datum pattern mining occur pattern several generalize pattern discover pattern sequence statistical machine community researcher try sequential property analytic behavior model model hmm dynamic represent deterministic lead modeling database linearly serious problem clear sequence simply sequence assume characterize deterministic individual sequence behavior sequential behavior news news news news database behavior preserve essential individual sequence avoid generative level effectively behavioral sequence paper probabilistic paper task datum paper discuss possible behavior behavior helpful form bold letter bold scalar database denote mn th behavior order behavior make item index type behavior web sequential behavior probabilistic relationship dirichlet reflect empirically initial prior dependent emission sequence govern database dirichlet whose row row detail multinomial hyper sequence index behavior multinomial hidden accordingly state level hyper generate database sample section deterministic hyper database maximize q latent optimize bind jensen distribution posterior latent variational approximate still difficult thus learn algorithm guarantee increase likelihood variational two optimization iterate respect line optimize respect hyper parameter converge important posterior product implementation hyper k k term inference variational inference fully factorize ff partially factorize pf ff assume mean pf inspire preserve inference ff please formula b step ff ii equation update pf iteration please refer appendix derivation updating formula posterior q n detail describe k v summarize updating formula omit complexity give pf ff form infer posterior form respectively hide maximum inference two pf ff form proportional ff form step likelihood hyper newton algorithm eq summarize specific begin line line fall reduce replace I I change step summarize procedure update exist behavior key hmms drop sequence sequence graphical contrast represent dynamic latent allocation probabilistic sequential behavior graphical simply dynamic relationship graphical firstly characterize secondly pf form posterior variational hyper treat individually individual characteristic comprehensive assumption hyper provide several mining task behavior model firstly web mining secondly adopt implement mb cache intel core node gb ram operating interaction restaurant recommendation e encode length code restaurant list restaurant similar visit category category news opinion air weather health business service news page behavior user hour behavior record user subset vary ff pf model hold compute hold model deterministic parameter first deterministic approximately infer hyper infer similar adjust log perform specifically fold testing fold process follow result hide chart pf ff slightly hmms ff pf well pf slightly pf may approximation generalization pf ff significant relate due simple form qualitatively speak pf fast form em cause early pf may fast ff need converge fast ff e characteristic individual sequence visualize plot diagram represent whose show sample diagram bottom diagram bottom individual slightly characteristic belong refer ei class sequence refer ie belong refer ei vs ei ie separate sequence class training class unseen picking eliminate roc report number bold surprisingly dominate possible model optimize thus model improvement competitive ei vs ie ei study year characterize sequential behavior sequence explicitly
requirement field theory shall element field link e might discretization resolution require sufficiently resolution rotation deal mathematical entropy properly normalize pdfs long continuous limit I divergence behaved difference entropy energy difference work concrete euclidean field knowledge field spectrum field give calculated theoretically field configuration determinant discretization interest nothing game via measurement response encode spread use transformation measure statistic field construct bayes q information hamiltonian translate theory technique drop irrelevant hamiltonian wiener reconstruction language information field prefer wiener role hand force language iterative algebra like computer wiener filter violate response covariance unknown datum contain couple lead interact many hamiltonian taylor fr expand part let value hamiltonian field expand expand expansion numerous diagram stress diagram line connect vertex numerically wiener equip tool diagram interact dx x compact diagram wiener filter diagram map interaction correction wiener wiener replace wiener map correction linearity diagram might also provide correction always wiener free perturbation lead well term prove perform infer reconstruction complex understand interact quantum helpful minimize gibbs energy transform basically logarithm partition calculate able mean reconstruction energy calculate gibbs convenient replace hamiltonian dispersion replacement turn definition result free give prove reproduce calculation develop novel e deal signal know name extended filter interesting minimal kullback leibler entropy information theory reformulate method develop vast mention list already map noisy galaxy dark matter space study wiener filter evolution condition particularly suit interesting characteristic signature epoch bayesian method trace calculate exhibit sufficient smoothness differential operator field act discrete simulation plausible continuous field datum produce field ensemble datum lead eventually student I give valuable lar anonymous helpful theory lin institute reconstruction problem tackle systematic way present base spatially field statistical theory permit signal recovery problem benefit technique quantum statistical diagram calculation potential physical air dark universe want accurately fortunately device
theoretical performance applicable modification performance like algorithm open learning estimation optimistic bellman demonstrate superiority bayesian bellman close significant learn theoretic reinforcement calculate expensive intractable solution approach gradient demonstrate act markovian state agent experience complete reward sequence decision mdp action time history denote discount instantaneous reward generality optimal expectation expectation reinforcement pose use guess reward exploitation trade bayesian viewpoint select measure quantity lie eq makes formally respect simple provide difficulty arise adopt require consider policy grow ng near focus optimal utility low upper monte carlo reinforcement belief upper bind attempt tight finding involve belief low algorithm relevant domain include gaussian suggest perform difference gps estimate direction solution gps transition appear suggest considerably process fundamental stem utility mdps calculate either utility mdp iterative procedure draw policy adjusting parameter method bellman function incremental mdp control dependent prior policy history write fortunately remove write mdps reinforcement briefly utility policy act act slight abuse belief policy value function mdps eq bound rl bayes try estimate utility implement either exploratory bound simple estimation select policy incremental version require expensive reason derive gradient bellman well computational effort idea stem must satisfy approach perform q bind trivial setting prior parameter initial ks aa make approximation take slowly almost alg step difficulty calculation sample mdp simply gradient td sample belief transition sample norm take gradient respect g basis follow twice derivative derivative bellman instead bellman working value write state state gradient q initial mdp sample act reward examine completeness hyperparameter principled experiment firstly possible hyperparameter perform choose high reward measure run unbiased method exploration confidence interval u hyper tune strategy decay initial value tune gradient algorithm require tuning policy switching employ standard exploration reinforcement transition normal action car domain grid employ discount ht car low xlabel ylabel label style fix scale style format legend column u legend name coordinate xlabel ylabel false format scale label format coordinate xlabel ylabel false label format coordinate coordinate ylabel format fix coordinate relatively simpler slowly
system streaming representation compact overlap presence linear bound assume set sequentially short possibly estimate block iteratively slide build interval estimate interval shift remove add active say form solve signal recovery way estimate spirit point lot system active indicate right active contribution homotopy quickly solve weighted minimization streaming recovery homotopy estimate instead new starting point warm homotopy formulation homotopy extend dynamic newly homotopy program sequential arbitrary representation weight homotopy close streaming transform previous warm homotopy sequentially remove measurement homotopy formulation scheme recursive kalman classical solve method solution admit representation homotopy solve update update spirit reduce update recursive kalman square signal available system stream algorithm greedy pursuit method support solve kalman original kalman use propagation update signal solve signal block error sparse jointly signal block homotopy algorithm restrictive nonzero formulation form slide interval update move new organized discuss basis overlap support homotopy demonstrate reconstruct orthogonal basis vector compact support basis depict support denote respective framework derivation lot orthogonal compact overlapping support orthogonality maintain opposite odd lot modify cosine function overlap cosine block transform rectangular window disjoint block artificial boundary block lot design sequence transition translate cosine iv multiply careful odd symmetry cosine iv basis respectively function orthonormal lot coefficient lot depict lot basis respective define orthogonal synthesis column lot orthogonal add overlap contain contribute top another contain correspond part overlap orthogonal overlap compact interval wavelet transform component overlap maintain orthogonality although filter bank assume basis shift wavelet overlap adjacent depict decomposition piece basis stream active slide minimization coefficient every streaming shift one system estimate portion leave active length active system linear short consecutive assume equivalent form streaming update accordingly measurement depict fig system use synthesis matrix example section discuss formulation recovery signal streaming basis active representation form diagonal important consideration decomposition overlap interval motivation coefficient length since fashion consideration overlap end align say lie outside relationship active depict interval overlap right lie outside active left align say however length want remove system could update overlap couple variable remove remove remove remove column modify accordingly divide q divided decomposition fig remove system modify follow remove modify part write error diagonal consist previous streaming overlap interval compute diagonal two tune speed warm start estimate streaming previous task predict location compute new location least square least system streaming equation suppose rest top consist diagonal combine error modify remove system remove modify location represent modify system q follow problem control dynamic select denote estimate streaming small portion predict signal warm homotopy warm start homotopy dynamically solution describe recovery streaming obey want follow minimization recover matrix contain solve knowledge regard assume homotopy give invertible quick final homotopy transform available solve original homotopy control varie end homotopy path build homotopy homotopy treat vector change vector elsewhere one piece homotopy path toward fact homotopy condition optimality derive subdifferential objective eq q subdifferential denote column optimality sign strict magnitude elsewhere incoming hold active sign opposite constraint satisfy sequence assume exist support element exist along path entire parameterized homotopy every homotopy step critical support direction maintain keep change violate add nonzero must small cause homotopy warm define minimum small inactive active index enter support small critical accordingly next immediately step homotopy compute update change equal homotopy homotopy come solve system equation size construction computing application homotopy one update matrix add matrix direction recursively addition updating suffer especially become close number stable cholesky qr change cost updating involve nearly homotopy cost application readily available update homotopy describe dynamically change measurement time vary arbitrary matrix appear recovery streaming homotopy warm start stream iteration use homotopy warm measurement eq small elsewhere warm start homotopy change system signal signal wavelet demonstrate signal performance homotopy solver demonstrate homotopy follow discrete toolbox basis sample frequency half summation generate zero lot lot streaming system measurement simulate compressive varying follow every accord snr becomes represent lot select overlap interval correspond stream iteration build consecutive update active old portion lot overlap unknown lot fig deviation measurement streaming initialize reweighte start homotopy solver warm description homotopy homotopy http edu homotopy homotopy multiplication computing use alternate solve weighted package sec use solve streaming select initialization weight code default mode termination criterion parameter modify code accommodate solver summarize homotopy solve homotopy use warm start initialization single streaming candidate experiment quantity error streaming product execution signal stream compressive measurement independent trial stream streaming average trial figure snapshot lot signal reconstruct signal presence different count multiplication matlab execution streaming compressed lot representation lot first lot reconstruct measurement approximate count vector multiplication second compress measurement lot representation figure present figure snapshot lot solver homotopy plot compare solver reconstruct identical lot transform base lot basis signal significant degradation compare representation middle plot fig algorithm multiplication reconstruction solver homotopy matlab execution compare homotopy reconstruction lot three plot compare solver homotopy significantly brief result reconstruct lot compare reconstruct homotopy simulate seed length signal circular shift interpolation define shift leave circular shift model toolbox rectangular smooth example shift build vary estimate coefficient stream select streaming compressive measurement accord procedure desire compression measurement noise expect snr block level divide consecutive coefficient circular convolution analysis filter bank every streaming build consecutive active interval old portion combine vector length thus measurement measurement unknown wavelet coefficient predict update warm accord use system deviation stream reweighted solve homotopy identical procedure homotopy stream compressive trial factor procedure number runtime average trial plot right
based far devise iterative almost extend learning adjacent remove insight structure shot relaxation approach carefully construct relaxation job classic graph cluster undirected unweighted disjoint edge higher graph arise prominent detection social identification search co document label pair object clustering encode identify cluster chen plant partition numerous different guarantee manner condition within succeed correct cluster break barrier identify extremely inherently setup size form clique polynomial require requirement still recover large extreme consist cluster certainly requirement previous main confirm intuition barrier arise chen really restriction shot technique use formulation initially recover cluster cluster cluster implication recover cluster intuition limited cluster vary significantly case equally easy aforementioned cluster identify node make cluster indeed main contribution focus plant cluster large precisely ignore small one notice threshold logarithmic arbitrarily turn disjoint size sure optimality solution easily identify perform provide converse precisely interval indeed free sense identify case imply exhaustive big necessarily rise recover prove regardless recover well provably cluster size extend case e small large hence adaptively free big contribution provide matrix datum numerous exploit possible even dimensionality combine envelope iteratively reduce use literature vast survey relate guarantee plant setup study model know block partition randomly pair whether belong random focus generally case minimal several work sublinear classified randomized methodology require cluster originally cluster minimize disagreement necessarily notion recovery usually study know hard prominent case decomposition motivate recently arbitrary ingredient surrogate corrupt paper author graph plant partition overcome cluster motivate algorithm setting instance learner cluster investigate obtain result erm running recover investigate differ throughout ground disjoint principal minor index generate model undirected choice denote determined ij exist p optimal km size defer previous previous treat allow matrix partial clustering pairwise otherwise tell fall range program cluster ground truth cluster fact need converse event input value solution look define black white represent black probability least follow bc partial induce truth cluster proof hoeffding simplicity bernstein use elaborate theorem long exist least cp make assume exist fall recover efficiently large exist gap cluster priori ensure cluster small sequence recover one algorithm elegant constant turn ensure recovery least require guarantee positive assume defer ensure recover number roughly next proposition tell recover cover vanish step proof imply size cluster step prove probability recover cover fraction bound recover cover number ng q rr v access formally mark define precisely node partition plant apply terminate detailed generate observation corollary cluster iteration exactly exactly table try version graph increase recover remove repeat terminate result node recover step cluster cluster say mid size behavior mid v recovered cluster mid characterize show gap theorem simple combinatorial true might search gap free cluster mid procedure algorithmic understand mid result say nothing neither big experiment confirm mid phenomenon real neither completely recover entirely ignore restrict obvious prove still mid study focus plant partition experiment theoretical finding generate accord provable guarantee particularly big merged interesting extend understanding barrier encounter resolution sparse recover formulation notice supplement refer distinguish operator subspace follow span matrix onto x denote set onto give complementary support rx adjustment present notation reader define project matrix contain feasible unique optimal solution satisfy c p solution know constraint cp higher satisfy f g f q p second rhs separately term p p block q combine bound p c strictly tc ci n I entrie second frequently check p deterministic entry r bound almost mean h r nt assumption b almost surely tt q p p I almost surely p mi sum variance cf similarly ci rhs p tn tn prove factor rhs prove b obvious property q program program cp cp contradiction cp c c assume k note allow q separate prove hoeffding tail assume possibility hoeffde properly union implication contain indeed would block difference negative trace norm function conclusion except set must conclude eq disjoint hoeffding inequality set indeed say tuple violate q notice possibility tuple assume bind size probability possibility contradiction strictly low conclude notice enough proving denote uniformly w h easily fix combination option bound size union possible prove uniformly probability tell strictly note rhs account final
original derivative therefore regard use http www populations aa dark water frame rate retain consist length classification observation spline basis basis confirm distinguish choose associate per proportion classification sample b gene bioinformatics li yu segmentation mathematic theoretical american mahalanobis vector european convergence pls nd vector classification discriminant lee gene wang ray curve wavelet paper functional classical precisely mahalanobis distance development operator space main mahalanobis functional mahalanobis functional use conjunction mahalanobis datum mahalanobis principal deal observation practice majority know multivariate treatment spline alternatively nonparametric adapt situation exposition approach usually advance introduction function hilbert endow datum little role distance book exception propose adapt principal square pls metric derivative distance frequently mahalanobis mahalanobis mahalanobis semi distance mahalanobis several analysis distance perspective classification decide belong use independent replication method provide classify variety paper observation principal method obtain component score posterior probability class method classify collection mean logistic regression sample song classify popular nearest et knn coefficient ba consistency knn particular additionally centroid assign close paper fisher discriminant functional class discriminant project classified bayes classification rule cubic spline multiply coefficient distribution pool mean mean expectation discriminant function alternatively al functional pls discriminant hilbert space method base probability function et al containing investigate svms functional al shape descriptor et classification approach hilbert contribution paper include knn use mahalanobis higher suggest mahalanobi organize mahalanobis mahalanobis carlo good conjunction functional mahalanobis semi conclusion present mahalanobis multivariate definite mahalanobis mean vector norm vector write mahalanobis euclidean mahalanobis account mahalanobis distance write eigenvector diagonal write component score way diagonal matrix eigenvalue mahalanobis variable term principal standardized component mahalanobis euclidean standardize principal component mention main goal section mahalanobis lead functional functional clear mahalanobis integrable closed covariance assumption exist orthonormal eigenfunction eigenfunction orthonormal score similar circumstance unbounded regularize operator threshold possible operator distance compact mahalanobi note express principal express functional component score state proposition standardize principal functional semi norm compute mahalanobis practice extend general functional variable functional mahalanobi prove functional mahalanobis q standardized component respectively functional mahalanobi write euclidean standardize functional semi distance integer follow fm fm k fm fm semi well variable functional mahalanobis distance process e v continuously interval functional semi assume observe follow obtain expression basis denote function basis functional close counterpart choice several possibility wavelet periodic nearly periodic basis adequate choice see simple effectively coefficient functional operator covariance eigenfunction principal curve mahalanobis ik k functional mahalanobi among possible mahalanobi introduce consider predefine split denote observation section procedure base known mahalanobis distance procedure neighbor knn one setting simple study et ba knn start distance new next find functional distance vote knn consistency knn space ba et al classification paper knn discretized membership conjunction mahalanobis distance functional mahalanobis semi classification case functional functional mean e estimate mahalanobis standardized functional k eigenfunction similarly write certain second operator functional mean estimate operator class function I functional mahalanobis functional g eigenfunction second consider functional score eigenfunction dataset probably fast functional centroid consists assign close functional distance operator square euclidean observation centroid compute common operator course distance class functional mahalanobi principal particular semi distance classification assign posterior come respectively pg different classify mahalanobis mahalanobi assume covariance operator particular bayes centroid assume multivariate different eq assign minimum respectively eigenfunction mahalanobis although mahalanobis semi apply principal multivariate et classification rule principal score expensive use rule monte scenario carlo different scenario scenario operator eigenfunction split training function respectively particular respectively second similar first third fourth one replace standardized operator convert observation perform four scenario purpose follow knn distances ba principal operator respectively mahalanobis fm denote centroid eight functional distance seven knn procedure linear bayes multivariate quadratic bayes spline basis denote simplification knn use respectively classification scenario precisely display mean classification carlo hand component compute semi comment case attain large correct classification proportion classification third scenario proportion scenario functional mahalanobi situation conjunction mahalanobi performance semi base fourth result point grid size scenario indeed generate appear assume operator rule coefficient bad large parameter dimension reduction may solution
get see aggregated section cast example weakly aggregated possible weakly problem variable estimate naive turn supervise assignment unstable naive approach simple situation meanwhile work often solution supervision distant supervision supervise extraction discover similar sentence variable prominent include dirichlet allocation idea latent allocation bayesian xu name include account vector propose method incorporate base fine grained side context search click side suppose click dimension blue click click click click good attempt standard clear generative use click solid nearly vertical line mixture would ignore simply know green observation could try replace discuss unable signal green blue dot want generative weakly possible subsequent propose click conditionally side membership user alone assumption graphical click quality affect click satisfied exhibit click assumption force information flow induce semi thick rectangle connect beta connect z affect click influence observe indicate build weakly top specify simple click bernoulli sigmoid exhibit vector possible complicated distributional assumption model gaussian cross independent purpose simple circle size draw thick alpha right alpha beta alpha edge connect connect contingency instead box quality noisy formalize graphical observe noisy outside human worker amazon result simple key parameter really click bayes generate frame procedure estimate instead define naturally section say somewhat complicated check one baseline suggest figure ignore latent let bernoulli create artificial word swap latent replace distribution set probability usual observation bin stability fit variation behavior group account fact click bad vice see lot power ignore I e coarse fine predictor approach happen click flat writing however perform regime guarantee problem often even estimator surprising discuss section show estimation full em em flexibility unimodal rather initialization consistently optimum review solve latent see worth consider individual step em scale handle million click spread ten information human stability effectively model infer meanwhile maximize e choice density put monotone equation aware work iterate evaluation way importance fairly misspecification stand evaluation bar group subsampling generate half without replacement choose subsampling parametric click bar sd instability begin weakly click large relative click click group moment perform naive estimate membership relatively per weak supervision human evaluation probability click look bad add estimate bin propose surprising meanwhile procedure click human em get solution without weak supervision apply distinguish click need name insight begin publicly available rely aggregated level fit appear statistical try want build access record individual vote census individual aggregated notation census vote model vote train separate exception nan rate cross validate state evaluate cccc direct latent error root vector record vote per membership member member union family member member white spread across factor frequencie union membership p display produce se obtain note closely public opinion research university remove entry vote original dataset row direct section latent par direct tune even odd surprising variable vote want interpret prediction variable close gold prediction low mean difference oracle average factor difference motivated internet company terminology run click level behavior spread thousand ask click click click skew dominate group click click weight click weight group undesirable consequence cause million click behavior facebook click divide separately click behavior bar evaluation tuning reason group click three bin fit fit curve phenomenon discover relationship click level confirm generally red click blue click circular click strong effect largely think click click circle provide signal discover insight click relationship alternative method evaluation location initialization important click less click relative importance clicks group contribution towards evaluation practice enough datum per start simulation find click highly understand design moment estimator click appear click close get underlie behavior number click per contaminated corrupted mean equivalent noiseless connection motivate like regularizer scheme noiseless row space span moment effectively act ridge way fix numerical ill conditioning treat variable discriminative condition
face outlier set model solver propose conduct ghz gb ram effort core matlab possible outlier via different randomly variable term chi side contamination sign randomly outlier lie line fit contamination case evenly side line sample around perturb offset drawn figure method centre radius th original fast propose fast initially outli run fast specifically second case time conventional second fix number observation vary execution table rapidly fast second art propose recently recognition contiguous solve tolerance extend matrix identity whose also evaluate collaborative representation base relax norm cast face simple coefficient class small experiment outli pixel leave list purpose statistic namely point specify number recognition coefficient recognize minimal residual ccccc ar subject two separate include neutral subject image pixel test carry subject typical ar outlier reconstruct show show various method exhibit superior perfect mm fail affect final face recognition removal method presence artificial noise region extend image condition pixel training subset image cover image face reconstruct copy table recognition run achieve dramatically method robust presence outlier performance sense variation mainly outlier example significantly et al face sub complete bottom initially etc removal vector central face initially etc image person picture pose illumination face image per use subset contain pose nearly front pose first subject first middle set bottom table area except estimator perfect drop black pixel technique perform achieve achieve accuracy result accuracy robust dramatically achieve demonstrate middle face situation face around form area around recognition cccc h identify outlier face computation face vary original original algorithm increase hour cost minute commonly use contact identify acquisition especially cause nd subject variety boundary package select segmentation area similar fashion boundary show circular region rectangular radial angular detect correspond pixel remove recognition clearly see dimension perform occur test image performance drop dramatically method obtain high table different efficient ccccc ccccc main drawback estimate outli removal outli percentage threshold huber nonconvex percentage take example vary table estimate percentage become image resolution mention visual consequently reject many specify outlier mm apply shown identify residual observation detect outlier standardize residual cutoff priori fitting iteratively remove outlier recognition main benefit removal bring highlight efficiency test many robust parameter outlier remove sensitive future outli acknowledgement work fellowship university correspondence address fig minimization regression school school science technology university science china view solve removal robust regression solve heart minimization slow large speedup high term robust previously break sub solve reduction number fitting square outli removal least square ls norm denote brevity throughout widely computer vision image recently face utilize however drastically approach robust commonly method huber minimize leverage leverage point space linear median square estimation high method object another class remove attempt require compute single tie solution motion estimation apply face recognition face image sim outli remove remove measurement generally optimization sim residual hence remove geometry consuming since second cone lp multi geometry lp approximate brevity observe small problem formulate quadratic relatively qp extremely efficiently particular solver allow removal problem vision often solve error minimal residual show effective outlier removal necessary improve recognition multi view geometry norm geometry minimum local minimum extend find example problem variable residual lead critical norm geometry truly necessary outli removal conduct first minimize norm residual minimax q f residual removal support problem outli outli consider residual I minimax strictly omit response outli percent measurement remove index solve minimax ti remove large residual continue eventually remove outlier removal process discard remove individual contain hundred remove small fraction good affect pixel incorrectly pixel outli removal practice support prove recall remove without present definition prove residual residual
e use deduce eq variable appropriate yield conclude formula analogously bind x nk analogously get triangle get assumption proximity run computation classical set algorithm implement use algebra library eigen empty solve new mix covariance old experiment problem ever artificial artificial datum different component parameter reasonable pairwise learn consist gaussians unbalanced world data mixture qualitative object condition histogram color real data difference covariance dominate different type range maximum majority speak close e influence step indeed sometimes solution solution differ log well worse artificial covariance world spread instance covariance couple parameter approximately behavior substantially difference still cf covariance expect estimate different initial set ten graph ten one set figure difference component fig type bound fig applicability surprising result forest datum observe couple round match quality page accept publication unlike analysis stochastic variant almost confirm model stochastic fast probabilistic central task machine solution use cf alg perform derive expectation decade lot improvement slow em algorithm saddle analyze em cf alg maximize second use assignment hide complete likelihood assignment randomness algorithm mixture exponential converge stationary likelihood stationary run retrieve mean small set contain gaussian author sequence produce neighbourhood compare reliable maximize efficiently mixture second sufficiently update yield present confirm theoretical successive simplified lead considerably run general suitable candidate gaussian consist k algorithm compute compute update maximum observation em fact equation prop sec hold would definite observation give assign difference computation algorithm state provide preliminary definition entry datum spread term translation respect difference z nk ignore normalization numerator difference contribution nk deviation analogously th q state fix proximity translation invariance thm measure thm thm deviation component proximity k w k would expect grow law state well proximity event proximity third two estimate covariance denote q speak weight thm ensure weight least dependence logarithmic
characterize support strictly divergence consider describe turn scale present maximum kullback checking come distribution produce datum comprise central reject filter filter pixel assessment performance evaluation monte assess simulate simulated filter filter look edge good preserve assess quality index correlation lee kl look image look lr em look lee lr em look lee lr kl filter respect equivalent look line gradient edge look universal lee present paper new filter noise lee filter behave outperform lee filter five six significance assessment distance pixel fit test filter technique homogeneous gamma lee protocol use quantify employ equivalent line edge assess filter pearson synthetic illumination affect interference coherent incorporate noise segmentation extraction object hard essential deal comprehensive procedure plausible see literature gamma noise constant characterize ground truth filter organize present assess result reference multiplicative outcome two characterize intensity intensity completely
average hamming goodness rbm train ml cd ip hamming drop dramatically grow interference serious limit rbms learnt cd gain increase section ip separate potential achieve optimal cd dimensionality preserve confident less confident derivation sbm rbm achieve space principle independent regularization neural rbm usage lead theoretical interpretation deep experiment bias outperform insufficient confident fail ip lead point rbm indicate robustness sampling bias far justification extensive handwritten digit justify ip ip deep neural completes calculate partition verify element hence er asymptotically tight unbiased fisher involve pair parameter share j g proposition follow one diagonal element next need complement bottom right block equal positive linearly easy ii w w ball surface center expect decomposition preserve rotation preserve assume tailor distance give trace proposition diagonal sub one trace among sub realize sbm coordinate projection give tailor sbm stationary sbm equation follow eq datum since flat preserve complete divergence respectively complete bm rbm projection equation determine know er exist belong parameter p qp also projection meaning mix uniqueness rbm flat find divergence iteratively direction fast treat rbm learning shown converge minimum choice hence mix rbm w bias I I subtract coordinate converge stationary preserve complete thus reduce maximal multivariate confident maximally preserve parameter confident estimate confidence parameter connection rao boltzmann bm single bm without rbm formalize essential density also layer specific iterative rbm cd ip series boltzmann geometry deep belief stack auto etc result various processing retrieval despite fundamental architecture capture generative underlie high require space overfitte usually empirically show failure big leverage understand recognize empirically act way region reach regularization procedure restrict desire region intrinsic unclear formally neural investigation lead explanation pre insight closely insufficient preferred satisfactory sample originally complex moreover become observed obstacle incorporate mainly main block ig perspective exist ideal parametric phenomena parametric derive low dimensional sub dataset insufficient perturb reduce maximally confident density ig concept assess unbiased rao indicate exclusive coordinate note ig capital letters index coordinate index regard subset stand indicate variable note coordinate another respect number coordinate q subscript indicate index position convention relation system formally transformation meet introduce distribution represent coordinate part consists denote th col fisher covariance amount carry parameter importance fisher give tight unbiased invariant another define information fisher influence uncorrelated technical orthogonality mle fisher j jk fisher information cardinality operator three information ij p etc use system dimensionality give target infeasible determine reasonable construct coordinate fisher confident distinguish low confident implement keep confident neutral equivalently usage strategy infeasible coordinate since coordinate proposition coordinate meet requirement mixed section proposition form fisher fisher matrix share l appendix diagonal upper proposition confident separate confident neutral indicate general parametric replace confident parameter neutral reconstruct tailor becomes tailor mixed fisher mixed ratio fisher tailor small ratio become coordinate respectively maximally preserve fisher neighborhood geometric perspective projection close kullback l divergence symmetric minimize direction tailor exist entail focus project bm show bm close actually case focus project manifold mix tailor determine maximally preserve rao previous binary use boltzmann bm kinds bm hide sbm bm rbm indeed algebraic neural fix bm belief approximately underlie parametric reduction specify choice next bm learn bm define stochastic neural visible unit hide stochastically depend input visible interaction connection self zero boltzmann normalization boltzmann coordinate bm sbm rbm bm visible parameter sbm connection parameter rbm set commonly bm likelihood calculate obtain gibbs stationary adjust sample denote respect phase section bm namely rbm theoretically derive principle help formalize essential part sbm manifold since impractical like part endow denote geometrically preserve confident part confident neutral tailor sbm space exactly next proposition explicit could maximally expect tailor coordinate sbm maximally preserve induce fisher rao sbm span maximally expect information sbm preserve gradient algorithm train sbm sbm learn tailor coordinate sbm learn uniquely sbm coordinate preserve investigate introduce discovery variable boltzmann rbm auto extraction learning learn one learning etc implicitly try question extraction guide learn redundancy hidden representation completely reconstruct manifold joint unit give current ml ip positive quality ml achieve accurate hand insufficient true bias rule update fitting new rbm update ml proper update ml reach projection update move distribution towards direction oracle step separate sampling mean sampling adjust direction immediately indicate bias bias boltzmann rbm order achieve abstraction unit capture dependency give architecture principle h greedy maximally preserve confident layer note confident preserve rbm guide architecture build abstract confident layer abstract describe flow transformation layer determine maximally confident process layer achieve tradeoff parameter find greedy pre initialize network application layer fall poor parameter empirically unsupervised training parameter well theoretically fractional coordinate regularize layer restriction highly confident density confident neutral value illustrate regularize fall search representation generate task boltzmann machine hence confident contain hidden alternatively investigate sbm rbm baseline cd ml adopt cd approximation ml bm avoid computing show run experiment kind dataset first randomly uniformly generate dataset collection partition evenly http com collection stop removal term frequency collection element occur document perspective ig cd corresponding coordinate get sbm confident approximate sense trajectory respect indicate confident contain preserve confident would produce good main cd confident confident carry sample assess fisher fire cd fire fisher information see consider practice fisher parameter realize phase guess approximate cd fire work cd performance cd size circle computation artificial investigate evaluate goodness sbm various divergence variable give offer qualitatively change contain tailor steady amount fisher constant underlie w family determine cd figure reliable parameter insufficient cd gain could result fisher reasonable sample cd tend reasonably gradually marginal affect cd cd well range cd baseline narrow investigate comparative initialization represent end cd true locate side converge note claim illustration may trajectory cd estimation term cd cd manually compute sample generate sbm use goodness fit train evaluate generate sample calculate position different cd cd cd significantly performance artificial insufficient rbm practically sbm compare cd distribution section compare ip theoretically section artificial rbm learning cd goodness rbm six cd properly phase bm hide scan times scan
phone social majority vertex observe appropriately divide heterogeneous framework email represent email correspondence send email account contain email account email outside email account spam email account group thereby define investigate see informally connect community detect community toy remain background link network independently probability run popular detection spectral ng spectral disjoint embed include vertex show figure identify community separate background community find background context community toy contain color normalize method separate community paper propose extraction community community core search identify statistically use tail probability derive configuration strength connection candidate idea discovery discovery search detect background handle practice output ease discussion undirecte allow edge denote degree sequence index vertex many detection seek score quality entire potential community extensive development give overview survey describe theoretic rely seek community minimize partition partition min community specify edge unfortunately min cut singleton community issue either cut community normalize cut np norm cut appeal spectral laplacian community detection seek network vertex seek class modularity seek edge maximize q parameter discover community datum drive maintain e also rely method community network parametric whose topological property integer fit describe recent review popular vertex symmetric entry stochastic network mixed membership significant development author powerful propagation survey level propagation block technique near least sublinear pseudo wherein consistent model dense limit histogram direction community detection technique extract search cover seek force place thereby flexible connect vertex utilize configuration currently statistical principle determined prescribe mapping open type method network remainder organize description propose extraction statistically reference discuss compete validate real four world practice arguably community capture benchmark assess specifically vertex benchmark background show network structure self loop multiple without contain link repetition vertex denote degree analysis derive vertex degree reflect assignment edge initially assign act half next uniformly connect procedure self loop even capture preserve strongly heterogeneous often encounter solely fitting configuration estimation graph beyond degree assess vertex edge define vertex asymptotic configuration recall total variation function sequence sequence configuration let degree sense graph fact contain vertex vertex indicate configuration statistic model value hypothesis interpretation role iterative note testing approximate core search community vertice seed successively update binomial reach unchanged final vertex identify procedure sequence vertex return collection repetition high search seed detect vertex adjacent latter detect detect background detect number detect range advance adaptively determine identification detect community community discover presence extent overlap update vertex informally community community fix identify regard map power fix rule start vertex success power set finite exhaustive selective seed explore space thereby seed set require implement lie community community structure enyi graph well stochastic block maximal degree neighborhood final collection uniquely use strength connection reference particular informally nan task identify amount hypothesis accomplish reject ensure number reject hypothesis divide control threshold adopt pseudo show tu ng u significance level maximal degree otherwise terminate modularity agglomerative search partition maximize stage reach sequentially modularity modularity stage treat pass community treat share configuration notably however nan optimally walk network minimize measure description walk employ greedy integer seek separate laplacian stack eigenvector mean apply row vertex assign mean advance real disk complex throughout manuscript characteristic informally extraction search theoretic extraction find maximize density vertex extract vertex disjoint find author correct detail inferential method compare give collection external edge collection vertex add cumulative distribution order fall vertex iteratively add away fashion procedure procedure run overlap spectral similarity compete instance specify candidate vertex belong use distribution connectivity community since configuration rely significance community whereas inferential network summary mention specification community rely false discovery summarize political facebook email detection widely identify satisfy assess spectral real extensive benchmark facebook visualize facebook color colored facebook author colored location individual email force software compare quantitative feature community include community extent extent ability capture specific feature describe precise setting gb ram dual facebook student california two friend addition college major year school table association display community closely individual accord compare detect summary community overlap find finding cd school pt average community vertice vertice proportion background vertex feature find detect repeat ccccc c mean mean prior specification discover seven detect broadly similar many small include community find well whose vertex determine vertex capable detecting total vertex suggests expect less vertex ability community interesting explore ability feature community community entire counting structure lee element predictor wish predict approach suppose method matrix represents give sample ignore treat adaboost tree construct evaluate ten equally sized aside classifier set treat calculate misclassification misclassification comparison table suggest community capture select past classification detect community network political network represent near undirected connect least pre classified author political political tend opposite force direct colored political match choose place community cd c compare community characteristic summarize take second large neither tendency note author divide assign percent vertex background set great within suggest presence background political cluster classification detailed report proportion misclassifie maintain suggest political capture keep strength connection political interestingly vertex still value weak facebook author friend facebook addition time period meet author file colored c available facebook typically capture activity individual facebook analyze specifically california institute facebook network view interact community approximately degree community average suggest little cd g background feature community figure b tend individual contain group author school final school f community location highly represent distinguished individual meet friend event author friend email email undirected edge connect address one email message send one address vertex run minute importantly well spam site outside spam email address network abundance background nearly vertice community nearly deviation vertex contain indicate moderate primary community community overlap plus structure assess network fact power employ heterogeneity present attention vertex benchmark sort flexible simulation benchmark extend compete mix external vertex exponent degree size degree vertex use overlap benchmark benchmark extend assess compete benchmark include simulated network density represent simulated exponent upper limit maintain community power set limit share outside vertex extent community mix become finally benchmark overlap overlap community overlap fall assess propose principled background embed community first simulate extent correctly identify lack none enyi model vertex link link prescribe community vertex block generative place vertex pair pair single embed generate modify modify network fix sensitivity method similar assess background combine block describe benchmark vertex block accord vertex disjoint denote probability vertex derive exhibit disjoint vertex connect vertex enyi range increment vertice configuration distribution probability node background within single ten background select generate embed range parameter network spectral community community use score detect community embed community result find size community increase eventually reach case slight finally simulation note vertex accord generate generate law exponent community accord exponent value increment realization pass input spectral normalize detect community background treat single community theoretic tool partition cover spectral tell thing method background point place interestingly peak around hinge vertex outside connectivity vertex low connectivity mixed favor give case importantly community background background average tool variety complex significance connection configuration identify statistically significant community number community discovery community extraction technique address identify role world identify connection significant feature community compete community variety specific complex instance identify individual political political importantly analyze potential simulation successfully capture overlap community well community former modern outperform method extend vary understanding theoretical research include
profile modification parametric seem engineering thank helpful early review develop remarkable pattern sequence event direction set refer put write unknown simple almost trivial early day summarie difficult parallel development inference readily great decade grow calculation concern standard method inference notation aspect emphasis composite suppose observable form unknown dominate whether typical potentially value dimensional usually considerable abstraction realistic setting work function likelihood mathematical ordering argument simply emphasize give provide plausibility various fisher range plausible likelihood I maximize theory statistical require consider ratio write fisher dependence notational convenience would value quantity could value quantity avoid somewhat distinction point inference inference density inference conceptually straightforward denominator associate marginal bayesian considerable laplace carlo simulation difficulty specification mean mathematical model way imply inference fact approach region test many separate analogous limiting among nested say model unconstrained shape parameter equal fit nuisance ratio log theory information criterion likelihood difference prefer develop leibl fit choose version suggest modification motivated quantity treat profile fact nuisance parameter variance normal dimension consistent finite practice way express equivalently often asymptotic useful generalizing profile accurate approximation perhaps importantly theory role prior three turn laplace numerator denominator integral fisher correspond nuisance parameter laplace satisfie similar normality posterior show adjustment nuisance adjustment profile isolate nuisance adjustment effect example independent prior profile modify profile likelihood improve frequentist nuisance suggest fisher make plausible model usual related function develop expansion change orthogonality orthogonality parameter need expression parameterization inferential statement profile accurate error approximation one motivation modify likelihood inference base marginal q outline fairly special base check little class relate development distributional scalar form indeed special transformation equivalent directional derivative determine nuisance approximate calculate account give implementation practical linear error specialize distinction whereas former likelihood suggest develop bound guarantee valid least high imply long research call matching prior inference straightforward complex light aspect include technique derive laplace integrate laplace though confirm likelihood computation inferential increasingly consequence emphasis probability observe response tool avoid construction information great goodness quality likelihood goal reasonably error use like outline difficult compute inference nuisance likelihood non longitudinal clustered datum predictor respectively effect distribution response require integrate effect suggest approximate integral penalize composite quasi specification variance specify density longitudinal work generalize likelihood discuss probit lead estimate likelihood generalize linear lee likelihood address penalize quasi dispersion dispersion theory likelihood somewhat advantage feasibility collection dependence relationship marginal multiplying version context study link distant treating assume sparsity covariance matrix effectively subset composite function composite sum property relatively composite likelihood composite marginal single conditional remainder repeat allow component event difficulty study composite generality composite likelihood result independent eq composite matrix quite convenient likelihood version context accurately practical setting full quite context subsequently process develop extreme record spatially correlate although dimensional know computable although
graph rao high lee price graph minor bound degree imply graph result slowly plant stable discuss discuss relation describe suppose small implie expect quantitative statement expect apart hence represent cut expect conversely function cycle represent cosine case straightforward would ff disjoint construct define localization interval continuously smooth support proof approximate step function main step approximation ok generalize partition almost interval sense smooth let finite undirected subset denote extend define vertex sake throughout denote volume subset threshold threshold define value abuse notation denote say function xt inequality inner product adjacency combinatorial laplacian unweighted drop operator eigenvalue furthermore non orthogonal hilbert space background normalize laplacian support choosing volume get upper bounding support variational support function q therefore wu k v iv wu f choose v ht wu energy energy clear context drop fact restrict disjoint interval volume eq q negative go threshold fu interval let parameter otherwise schwarz ei e wu vi fu wu vi fu f fa fa weak existence eq example energy distribution threshold threshold gap step threshold threshold let hand procedure succeed succeed construct support contradiction illustration otherwise argue know remain upper support contain say fu fu fu sum averaging argument contradiction next upper term call eq threshold threshold prove sample set set claim denominator numerator numerator fu fu fu fu triangle last x fu fu fu fu ready first schwarz put ready bound proof use prove prove strong version adaptation upper small function ff embed position cycle support dense separate region subsequent closed point region two f wu f wu v wu fu satisfy dense separate region define observe partition length heavy constant later say otherwise light heavy say balance denote balanced intuitively distribute inside interval find dense separate region construction heavy well region neighboring rest particular th note heavy separate dense summation interval balanced interval heavy proposition low bound rest ok let since inequality ok ok ok remove expansion vertex cut denominator ff iv ok threshold induce side inequality achievable linearity hand ft last equality normalization put together prove show analyze algorithm cut eigenvalue time graph cut cut modification ratio induce return cut finally iteratively minimum let cut ft ft notation motivated eigenfunction standard principle f zero support prove undirected graph hand side letting difference assume fact sign organize let step q follow fu fu say positive fu dx fu put together lemma follow simply adaptation support threshold small number say procedure succeed succeed imply follow argue already contain distinguish lipschitz sign summing thus induce cut remove fraction subgraph ft threshold technical cut throughout word edge inside maintain induce cut either induce cut remain l later induction remain subgraph eigenvalue furthermore assume cut weight kl l ft ft rl otherwise find threshold cut claim prove henceforth contradiction threshold cut remove sure keep ratio edge cut edge loop threshold fraction remove low edge suppose I threshold one increase newly edge th remove edge put complete partitioning cut eq therefore weak henceforth show side observe uniformly sign sign similar inequality inequality last follow approximated eigenvalue generalize manifold laplacian constant introduction plant semi model propose plant semidefinite programming relaxation generate adversary within sdp partition consider flexible sdp find cut sdp instance plant partitioning perform plant instance well plant instance two cut unweighted subgraph minimum degree subgraph partitioning apply call partition partition volume number maximum es b last inequality prove order therefore partition degree otherwise contain heavy connect algorithm edge induce imply cluster algebraic cut odd cycle weight even edge cut odd eigenvector eigenvector order large necessarily could cut relaxed near optimal cut cut problem small large cut stable instance exists cut tt ss volume stability theory eigenvector stable perturbation perturbation edge plus fact chi lee undirected prove guarantee spectral partitioning cut theoretical segmentation graph balance spectral unweighted weighted graph edge complement graph cut come include image designing approximation fundamental connection eigenvalue normalize quantitative extended graph influential application spectral counting improve eigenvalue laplacian every undirecte improve gap factor constructive find cut minimal minimum among threshold inequality show nearly could even unweighted cycle show strong though spectral recent profile sparse see generalization expansion improvement extend eigenvalue disjoint cut eq undirected graph I gap graph furthermore spectral partitioning implement efficiently guarantee problem explain phenomenon rigorously research towards analyze well random plant hide edge spectral approach partition partitioning proof detail stable cut
practically economic goal past improve decision rational environment online interaction social happen rather tool decide different action stick evolutionary mechanism mechanism keep experience viewpoint course ideally make well use gibbs measure quantity finite purpose state self contain give discrete everywhere variable tuple let transformation describe claim pair detailed balance gibbs everywhere detailed consider polynomial behavior interval exist recurrence relation fix exist end sign claim indeed moreover sign virtue strict monotonicity one give take q undirecte counting respect hamming lipschitz collective time social operate uncertain environment available environment sequence agent cost cost incur interaction incur agent stay environment must signal decentralize select action directly decision make neighbor regret realize well centralized entity evolution horizon polynomially neighbor agent social among vast cover wide variety capability allow able endow essentially computational power generalization signal noise large collection make past necessarily information decision emphasis shift decision share limit small group g decision friend environment agent receive stochastically iii agent select action aggregate private receive main question learn interest capability randomly evolve neighborhood agent underlie private model underlie static meaning coherent form space agent goal environment discrete make social network feature dynamic environment admit instead agent quantify neighbor environment take cost agent default mixed stay regardless distinction probabilistic bayesian environment distinction risk describe situation model variable available arise rational conceptual significance distinction effort economic formalize study interact environment interval aspect unable nature ideal minimizing really already assign belief agent exhibit tendency stick strategy sufficiently environment status collective decision presence assumption learning realize horizon well centralize entity induce composite function incorporate action agent default strategy give consider default mathematically agent interpret choose imagine fraction agent tend choose action default strategy expect cost random eq kullback leibler trade cost stick default analytically mean argument lagrange multipliers logit play prominent also well statistical measure section regime temperature support minimizer formulation agent know thus consequence agent bring rational come unable environment spirit must step cost take action however keep track environment instantaneous advance default action choose agent instantaneous incurred forecasting look agent choose rule instantaneous cost quantify gap step small cumulative instantaneous side per regret strategy agent quantify gap bad without round agent attain round instantaneous proxy typical environment mild allow use equivalent characterization tuple minimization dynamic account use instantaneous function correspond instantaneous decision deal seminal paper decision dynamic uncertain eventually act fix illustrative context discrete distinguished vertex action set path whether traffic vary agent pick path path traffic q higher consist edge small traffic moreover cost round computational round appeal round traffic agent round traffic good amount traffic edge agent path optimal eq stick effort need experience main decision network agent salient characteristic action entire number alternative agent decompose local affect agent neighbor social receive large cost agent neighborhood social network main contribution decentralize take regret time horizon agent decompose social network develop decentralized strategy statistical physics dynamic point literature economic evolutionary agent local interaction main parameter degree decentralized strategy exhibit favorable statistical physics recent chain decentralize combinatorial optimization decomposable correlation decay condition uniqueness statistical physics approximation scheme information problem assume instantaneous cost decompose cost decision know observe update agent interact agent environment generate environment advance summary process default local let uniformly random agent observe agent cf action environment specify logical immediately tuple x instantaneous incur instantaneous round stand interaction rule guarantee temperature network maximum agent social graph start centralized scheme shall approximate implementation function centralize recursive construction choose tendency instantaneous tendency stay work mirror descent online lagrange ensure work summarize property strategy q constant function give eq interval follow regret remark scale regret consequence presence respect variation show scale loss self part reflect instantaneous reveal strategy measure decay rapidly finally though differ rest literature emphasis proof centralize horizon recursive decaying past instantaneous time later ensure global rule bind uniform sum govern entropy drift scale centralize boundary discount cost q normalization profile conditional recent cost simplify write convention update instantaneous cost profile markov recognize accord step dynamic gibbs consequently balance give economic theory adaptive interaction see recent paper al os logit response include decentralized strategy attain regret interpretation regularity action profile counterpart essentially physic see typically rapid e decay condition drive spirit quantify agent change environment vary instantaneous cost regularity temperature maximum neighbor agent ignore lower centralized still polynomial regret oppose strategy induction eq showing expression fact hence show convenient gibbs span write hence q employ expression substitute precede exact upper definition write particular obtain inequality hold proceed proof briefly behind idea express decentralized strategy sum regret centralized provide main establish incur centralized action centralize counterpart distance small centralized invariant kernel condition conditional distribution ensure decentralized profile picture compact euclidean decentralize discrete draw riemannian walk contrast notion curvature idea sharp separate key curvature chain separable metric vary separable metric space equip borel transition mapping x wasserstein variational real trivial e wasserstein variation coupling achieve infimum e denote curvature contraction key tool kernel measure measure recursively inspection see relation recursive contraction thus use respectively tuple equip hamming q curvature coupling give indeed eq
nest generalize regression ridge eq symmetric solution easily verify wide sense direct prove include show smoothing monotonicity nest criterion stein satisfied imply ridge discuss low degree freedom unconstraine important note convexity dimensional axis euclidean respectively fit convexity requirement nest give devise key extent phenomenon interesting study regularization approach supplementary definition penalize formulation constrain modeling approach penalize non stein penalize lasso formulation constrain version variable stein estimate general monotone extent behavior exist adopt realistic observation characterization agree degree freedom regularization estimate negligible compare monotonicity see text detail panel function directly calculate stein two agree implication model clear tuning fact freedom constrain use stein right ridge ridge derivation admit counter example error generalize well mutually svd leave proof supplementary monotonicity hyper ellipsoid surprisingly get small project ellipsoid describe setup demonstrate let ellipsoid origin parallel axis determine setup top think ridge behind euclidean vertical component effect less observe panel covariance beyond illustrative setup scale hyper take accord normal form interval give start profile monotonic becomes examine unlike realistic ridge example require ridge guarantee monotonicity constrain admit counter freedom key concept context monotonic effective nest monotonicity preserve familiar scenario particular recursive situation less counterpart inferior predictive penalize regularization surprisingly every realize training appropriate value produce dependent approach reflect constrain freedom formulation notion guarantee community traditionally vc model give monotonically vc guarantee major give loose place error unlike fundamentally approach estimate loss besides concept expect negative change family generalization remain acknowledgement grateful e team science foundation fellowship bioinformatics university appendix eigenvalue eigenvalue unique orthonormal span via rotation matrix row respectively nest nest contour value euclidean onto fit close l modeling parametrization projection eigenvector origin radius around hence subsequently unity mutually uncorrelated eq jacobian component equal every mapping break project project onto orthogonal hermitian main jacobian stein theorem lemma aim improve performance modeling achieve bad error expect degree hold counter show simple situation approach degree thus inherently regularization freedom ridge accord parameter approach mapping error test criterion although examine selection problem model select typical modeling approach specify amount fitting fitting play role high problem degree constitute model choose undesirable gain training degree variance overfitte undesirable achieving produce typically bias training monotonically correspondingly decrease decrease remove potential less monotonicity great admit specifically typical degree model monotonicity implication counter intuitively inherently remainder organize formal definition review concept degree freedom effective monotonically familiar exhibit discussion regression subspace specifically former span explanatory span covariate geometrically nest formalize strict approach nest fit geometrically candidate dual ridge regression nest wide def relate self stein regularity normally extension variety expect jacobian think respective review material specifically form uncorrelated assume penalize singular design linear degree freedom linear regression add explanatory variable motivate freedom effective degree approach base concept stein degree amount nest ridge theorems monotonic regularization degree freedom regularization provide wide go deal practical relevance
lp principle lp pick anchor selection step small practice satisfie violate happen rarely never cone hence add set maximum combination identify use avoid possibility strictly empty interior continuous derivative bregman interest bregman bregman convex divergence generate satisfying assumption bregman divergence solve equivalent descent solve eq show result selection criterion combination maximizer unique step anchor anchor simulation much noise bregman generally projection criterion change variant meaningful interpretation condition addition separable applicable complete picture versus recovery color test perturb noise perturb noise dirichlet whose clear column laplace symmetric entry std varied plot robust huge highlight importance suitable next e I mention member unique corresponding bregman step recover anchor commonly use divergence correspond poisson report divergence since informative difference consider std increase signal noise anchor increase perfect recovery certain anchor divergence ij describe paragraph step report anchor index rate std ratio practically stay vary good color selection concern find summarize many representative variation illumination day foreground compose frame span video frame stack foreground model filter commonly frame stay half consider nmf inner constrain solution scale robust nmf restrictive equivalent video frame hope assumption allow video evaluation restaurant video restaurant move cast ground surface direction video video sequence capture office background foreground background roc robust widely methodology task vision addition local search approach search solve search use highlight importance robust refine dimension factorization method parameter robust report show plot three video restaurant foreground perform almost tie local bad almost similarly well robust dataset promise method foreground huge speed matlab time inexact multipli alm generalize algorithm separable bregman empirical foreground promise theoretical induce ij make satisfied divergence interest derivative positive nonzero strict converse consider anchor positive denote wise th index column follow regard anchor selection property anchor combination maximizer point select follow proof prove current anchor residual form lagrangian tr lagrange second operate vector condition I follow denote directly condition pt com ny usa family separability geometrically nmf extreme vector develop extension hull approximation bregman include foreground near pca selection express approximately non minimize divergence factorization small interpretable topic model mining hyper image source microarray exact np traditionally algorithmic nmf treat instance convex lead recently enable nmf say column hull word factorization assumption position equivalently right factor constitute column separability derive uniqueness separability hyper separability turn association translate every topic frobenius loss find expand cone empirically admit efficient selection iv preprocesse need use frobenius extend approximation family bregman motivate application video frame stationary background foreground dynamic frame pixel seek video impose residual foreground norm admit tractable state art nuclear convex relaxation impose separable nmf variability explain set anchor pixel restrictive frame separable nmf background separable separate foreground foreground algorithm text performance outperform work nmf loss frobenius proxy volume distance successive base nmf propose bregman gap minimize goal nmf cone combination pick anchor algorithm build pick iteration algorithm execute construct cone three identify extreme ray every extreme ray extreme ray next pick outside cone green project current point minimize term maximize identify anchor anchor add expand iteratively inspire lp general use projection cone separability expand recover end separability anchor selection demonstrate empirically noise member family precisely empirically superiority exist noise section pure proceed identify current expanding find column add anchor column project cone minimizing normalize hyperplane evaluate selection anchor use possibility residual choose projection projection solve negativity alternate direction problem negativity penalize proximity form thresholding operator least
hold sample exposure read gene assume poisson gene model shape subscript shape integer belong gene class setup rise read negative quantity think account dispersion mean offset depth specific write indicator across common digital expression replicate attempt class exercise share context information distinct gene gene infinite indicate corresponding conceptually summation go infinity appear class sample hyper follow procedure constructive resemble stick first weight correspond break stick break remain piece dirichlet base rapidly decrease rapidly single value value concentration slowly mass tend mass unbiased give prior straightforward share gene group accord share digital problem absence replicate pool together illustrate share distribution determine stochastic generation model identify measure binomial determine gene center mixture center categorical sample break observe indicate indicator list center weight monte large one markov chain construct repeat posterior explain infinite large set equal minimal truncation indistinguishable inactive active cluster gene inactive write update inactive matter active complicated involve respective posterior cluster metropolis proposal show likelihood negative binomial eqs indicator categorical k il il break generalised conjugacy update beta notice gene set order weight add mention center pa pa ac posterior ac ac employ metropolis furthermore advantage conjugacy normal initial k ac ac ac ac ac ac ac ac ac ac k ac first notice value initial b categorical single metropolis active see eqs discard equilibrium attain implement impossible publicly digital find http com supplementary gb four library neural cell cell derive different subject divide class gene cb cb cell neural implement library array module architecture modern processor iteration disk raw output chain specific indicator center constitute approximation posterior reach early stable explain respectively indicate inverse distribution center methodology present gene assumption value share sharing permit together pair robust replicate per class tag third constitute approximation small replicate sample chain derive although since sample generate distinct impose share different gene particular chain number course simulation discard burn use panel approximate observe peak around apart obvious library artificial never actual much truncation level decide infer advantageous respect simulation number two stage simulation iteration respectively exception super less cluster hundred class represent negative binomial concrete member illustrate histogram first sample table along cluster iteration gray observe whole index methodology sample expression modelling subset distribution figure log show algorithm generate huge volume relatively short fundamentally discrete analysis development method modify originally aim analysis research publish overview count mean poisson base negative binomial aspect dirichlet stick prior model negative force cluster elegant sharing gene analysis demonstrate actual biological publicly available gene cluster rest parameter currently implement computationally require complete core scale production due need chain distribution execution characteristic generally approximation completely operation vector modern computer towards gibbs sampler currently use avoid truncation datum thus great low software digital theory contribute development author would green discussion like produce public grant ep united sequencing generating expression rna library million differentially tag read constitute fundamentally expression low absence digital modify test seq analysis digital begin model gibbs sampling infer count algorithm biological together tag augment estimating decide along demonstrate fidelity public common truth knowledge molecular biology generation throughput sequencing tool aid genomic million molecular biological application study include offer wide prior typically category start sample length library throughput platform read start read genome map normalised differential gene review normalise read tag possible small locally together alternatively
intuitively place favorable circumstance zero search exactly case investigate exploitation exact connect physical confirm dynamic evolves encode describe choose site qualitative conclusion independent scale follow population human individual balance beneficial effect growth quality quantity resource etc slightly evolutionary site mutation function linear length site site interact random economic dynamic exchange project shift another material price portfolio asset describe gain rest portfolio regular lattice discretized heat upon j variety direct optimisation problem result concern concern fluctuation interested long velocity field discrete site example mention represent asymptotic growth quantity correspond population growth relevant material example call correlate random present language result reach exploration exploitation allow probe environment efficiently favorable hand benefit favorable last numerical problem system site periodic determine long much perturbation equation b large region approximation incorrect system predict particular existence blue triangle branching diffusion fix black large asymptotic perturbation dot reveal behaviour simplify exploration understand general lattice slightly evolution rule eq z jt mean correlation amount tree follow author generate tx tx evolution tx j particularly yield g tx tx operator tx j direct rise front velocity front fix tail term r r r impose without introduce harmonic eigenfunction ny equation write equation interpretation phenomenon standard propagate wave propagate velocity case therefore find unity arise temperature growth latter localization portfolio discussion determine numerically simulation see growth rate therefore optimum tradeoff exploration exist favorable neutral introduce expect figure particular analytically understand standard amplitude walk exploration different lead averaging reduce variance since correction trivially generic existence optimum exploration green square blue obtain triangle case solid curve grey asymptotic conclusion another
discussion anonymous thank make source available supplement appear list describe alternate density function allele arise theorem song department division university california berkeley berkeley california grant gm part research availability genetic experimental evolution study dna identify genomic selective pressure fitness challenge likelihood allele temporal dna integrate trajectory consecutive fine discretization allele numerically approximate fitness take selection apply dna genetic study exploration balance act natural evolutionary genomic pressure important genetic disease understand molecular devoted act stationary frequency utilize time genetic enhance allele frequency enable estimate example sequence sample evolution laboratory environment fast evolve advance human allele frequency analyze dna population unobserve treat noisy population allele frequency genetic variation integrate allele frequency evolutionary allele hmm use population size frequency variation neutral variation advantageous investigate hmm evolutionary consecutive rescale genetic obtain continuous diffusion accurately approximate allele allele pde depend mutation parameter pde incorporate aforementioned hmm infer series compute allele discretize reliably integral efficiency grid method depend appropriate discretization population another previous work restrict signature combine diffusion seem allele frequency close period approximate solution pde utilize song representation transition compute analytically spectral forward algorithm hmms population forward generator fisher representation representation leverage previously analyze conclusion consider advantage balance act call organize framework proof provide article simulate investigate aforementioned dna section model provide formal series consist denote population assume consideration identity derive allele use denote time notational tuple sequence allele evolve fisher population mutation allele derive reverse derive allele generality generation randomly copy allele proportional rescale approach constant limit population allele diffusion diffusion year per generation scale physical mutation correspond use scale diffusion frequency distribute density resp illustrate circle derive circle indicate trajectory population allele series framework allele unobserve hidden frequency distribute accord example allele mutation time describe transition parameter hmm allele binomial distribution introduce q joint density allele auxiliary observe forward allele fisher recurrence integrate allele recurrence finally observe integrate hidden frequency time intermediate describe analytic integral numerically frequency choice discretization discretization follow initial accord ny nb polynomial article allele mutation allow allele mutation drift arise article coefficient representation proposition dynamic representation appear vector choose detail finally observe genetic give section space analyze dna dna present observe copy allele maximize several simulate chose mutation probability series carry allele positively time course performance present four selective fitness arithmetic fitness dominant simulate strength dash true value effective maximum quantile box upper tend increase uncertainty decrease population easier higher reject dash line true advantage ten ten selection time maximum additional figure range estimate selection scenario various site extract allele determination encoding mc fluctuation allele time mc selection another recent selection sampling year sample pt mc investigate assume apply method mc mutation year allele frequency allele try surface allele frequency selective derive allele fitness selective perform set derive binomial likelihood set estimate marginal histogram maxima marginal fitness allele fitness thus individual advantageous minimal effect surface support good explain likelihood compute indicate histogram maximum grid proportion grid empirical quantile fitness indicate dash maxima indicate figure surface quantiles fitness allele fitness mc explain quantile fitness mc likelihood surface mc estimate bootstrap quantile fitness fitness maxima population time compute double refine chapter estimate eigenvalue repeat take power exploit come multiplication mc datum approximate eigenvector submatrix eigenfunction dimension empirically verify produce value grid figure adjust analysis report develop spectral series
ip set satisfie cost constraint assume otherwise necessarily verify constraint neighbor cover use j iv iv pn ip simply contain least recall constraint contain non cover hierarchy I part ip hierarchy ball center ball radius cover ball bind cover v j net ball cover center turn demonstrate trivially p pn I ip instead constraint lp solve solve lp complete lp hierarchy lp easily complementary count bind single always copy copy create yield additive error unless create suffice runtime adapt datum runtime optimistic linear generalization optimize classifier suggest analogue compute approximate question leave open reduce opposed runtime generalization bound notion analogous pca practical context constructive suggestion van whole whole whole em foundation yahoo award support grant science foundation adaptive metric contribution space nearly front analogue namely approximate ambient dual benefit optimistic central role rich assume either implicitly strength hilbert exploit functional place hyperplane foundation perhaps many naturally metric without distance banach general metric control efficiency metric cover algorithmic life learn tends lie close manifold quantify leverage dimension ambient one low via complexity bound present adapt complexity accuracy start simple x n tx p ambient separation margin distortion optimize hand corollary approach quantify algorithmic construct intrinsic pca cutoff properly train produce generalization significantly euclidean rich arise analogue observe von realize lipschitz obtain algorithmic runtime depend metric denote considerably restrictive generalize ambient generalization tradeoff intrinsic separate tradeoff lipschitz tradeoff address address optimal give formally devise approximation theorem distortion tradeoff within determine classifier optimize generalization bind al intrinsic dimension ambient dimension incur restrict theory vast survey scope survey aware provable mainly improve achieve reduction empirically speed instance combine rigorous dimension metric space heuristic may dimension attempt find distortion usefulness nearly general inherently distance furthermore highly aware metric remove set subset dimension spirit different require rather aside benefit gain merely address space body derive property kernel spectrum number recently geometric notion intrinsic providing low intrinsic seek quantify result statistical make work cite therein optimistic space nn dimensionality almost subset amenable sort tradeoff eventually enjoy proximity generalization quantify near compression bayes nn standard notion indicator positive function diameter denote small every cover radius subset strictly general denote metric ambient repeatedly learner example sometimes replace generalization term misclassifie rough admissible perform structural risk analogue bias take typically metric lipschitz contain countable every member pointwise limit supremum rademacher evaluate variable seminal rademacher complexity cover number integral essentially approach familiar algorithmic finding distortion solve pca hyperplane normalizing class x costly linear hyperplane dimension turn nh common assumption separability insight margin dimensional formally say denote whenever rademacher consequence expect lie nf exhibit seek large distortion tradeoff cutoff x x decompose rademach term proceed term function dimension lie ball bound absolute classic substitute second nf n second jensen n tx n prove sample probability hinge contraction hinge compute project tradeoff dimensionality distortion tradeoff pca singular runtime dimensional low project bind even without improved performance magnitude rigorous choose cutoff extend section metric receive training von prediction via extension near add dimensionality preprocessing motivate let von make powerful nn classifying make amenable rademacher particular nn efficiently formalize fix consist sf lf constant approximation cause degradation bound runtime stand benefit great deal formalize low rademacher nx elastic lipschitz function ambient dimension similarly euclidean rademacher estimate convert distortion statistical rademacher informally minimal distortion step apply classify algorithm near tradeoff dimensionality distortion achieving near low neighbor begin complexity lipschitz direct cover classic metric gx covering number hence diameter l proceed explicit function space nf f nf n ok claim distortion reduction diameter elastic let nf sf property compare rademacher case exponential dimension lipschitz latter tight fx fx nothing correctly regime lipschitz mapping diameter iid fx fx invoke margin il criterion distortion unlike well efficiently set approximately distortion since carry nearly predict value test thresholded finite point set define point set cost mapping dimension minimize solution time problem presentation step modify yet another extract require solution possess fulfil presentation integer ip ip vice versa ip relax round recover lp integral indeed solve thereby complete present though approximation factor technique yield create point point diameter hierarchy ss must possess hierarchy possess least modify yet solution possess imply ip fulfil significantly hierarchy cover property pt dimension cover cover extract set arbitrary distance construct include arbitrary within close hierarchy pack scheme imply must distance I finally must
estimate series thm lem context estimation impossible correct additional point show evaluation statistic domain bioinformatics detection overlap segment end point give point highly dependent assumption possible generate arbitrary assumption ergodic finite marginal assumption information change distinguish alternative impose strong change distribution generate provide scenario interpret imagine write speech video surveillance activity versus genomic application real interpretation theoretically experimentally framework make possible generate candidate induce partition simple assign near call distributional point remove redundant list output combination series establish link learn insight community change point problem general change usually restrict mix marginal change nontrivial setting address interpretation many world particular know however change incorrect unknown list estimator sort list present empirical estimate distributional turn tool study ergodic series definition section formalize section present finally conclusion borel distributions frequency distributional distance distributional distribution follow use partitioning decrease sum difference weight fine distance distributional calculation infinite fully b I indeed see formalize form concatenation unknown unknown ergodic partitioning disjoint every change consecutive segment distribution sequence completely finite different consistency distributional may distinguish point estimate assumption nature generate total unknown separation total asymptotically consistent list estimate change precisely list sequence I consecutive segment rr segment c estimating change denote distribution provide intuitive explanation work candidate candidate sort produce consecutive overlap partition consecutive segment identify redundant remove give intuitive list change whose change point portion segment generate point candidate apart thus generate segment segment generate candidate consistency true estimator estimate parameter list general may use stationary cluster center first center previously center center segment candidate obtain show require calculation segment computational calculation bring complexity proof rely lemma set segment specify every segment I generate large portion truth specify fix consistent generator apart candidate element pair consecutive complement true point right index equality occur change nb nb frequency fact hold contradiction length subset follow tn inequality
recover signal compress address recover determined equation efficiently recover simplicity precise establish geometry show measurement gaussian successfully successful property high recovery refer strong actual nonzero weak almost threshold simulation strong threshold introduce provably phase dense sense variation reweighte plain class empirically minimization sparse element modulus nonzero empirical identical result show limitation sparse signal modulus empirically observe fail empirically phase transition modulus nonzero element arguably signal modulus approximately possible signal variation iterative reweighte minimization transition plain two reweighted phase signal nonzero dramatically show stage iterative reweighte signal gaussian scale law angle iterative indeed exploit reweighte provably nonzero signal derivative certain reweighte analytically lift phase transition threshold plain minimization scale sparse sparse amplitude pdf derivative modulus approximate modulus element signal modulus minimization hard extract decode plain prior pass plain alternative information discussion work well compressed sensing though stage minimization boost transition whose nonzero amplitude th integer nonzero discuss whose strictly modulus taking sense constant modulus element idea matrix tailor non sensing scaling stability minimization compress sense show non bad provide sparse sense system summarize recovery stability sense new recovery algorithm section outline simulation give improve amplitude entry nonzero unknown signal define small certain threshold minimization maximum guarantee call sparse vector randomly recover type threshold restriction fix set kn cn state measurement ideally signals signal fix compressed isometry performance successfully propose compressed sensing consist I adopt traditional sensing multiplying randomly usual compressed let sensing signal iterative reweighted tailor matrix sparse eq replace reduce recovery appear modulus nonzero element amplitude reweighte modify reweighte minimization step standard vector remark possibility solve solution output sparsity beyond weak minimization capable recover identify element correspond finally chance section certain class recovery standard signal modulus nonzero element denote improvement successfully reader convenience step unknown signal overlap minimization carefully sparse algorithm support enough intersection good support vector amplitude namely ny proof recovery stability become recovery distribution iterative reweighted show significant scaling quantitative entry partition much threshold nonzero entry perfectly entry use measurement approximate possibility exponent failure main follow theorem detailed reader perfect infinity perfectly recover pdf origin present sparse signal modulus nonzero conventional fail performance sparse nonzero indeed distribution element follow element equal remark decoder constant modulus nonzero modulus modulus iterative reweighte minimization algorithm transition plain plain comparison algorithm one curve use mean minimization algorithm recover
example difference guarantee paper submodular optimization mm strong submodular subdifferential analogously subdifferential continuous case subgradient via greedy assign element notation permutation define yy entry surprisingly also submodular q define sub nonnegative define mm modular form current use maximization low bound tight optimize modular much optimize x txt constrain unconstraine setting upper maximize tight must improve iteration linear optimize hold monotonically problem analogously contrary subgradient produce solution thereby round certain addition relax instance rely define certain minimization unconstraine yield result iii respectively minimize step choose minimum author decompose represent restriction lattice great element minimizer lattice minimizer submodular denote iii iii return initialize arbitrary iii aa bx vx start element element remove element let I subsequent element remove similarly add lemma iii effectively contraction initial lattice henceforth start lattice warm enable start start yield new minimizer hold lx element generalize tighter small around proof build independent initialization select modular step word iteration ever remove ti analogous element ii never bb since add element remove least iteration terminate I must consider set exactly induction first analogously remove argue generate x local implication provide minimizer well mention guarantee submodular minimization instead lattice polynomial run minimum consider add remove local submodular strict pruning define modular refinement show ii converge every proceed proof local small cardinality induction contradict optimality hold equality result analogously must start ensure result stop switch initialize algorithms return remove terminate minimum minimum call analogously optima lemma apply element helps similarly look since optima within initialization particular ii terminate regardless lemma prune global contain possible algorithm initialize converge move lemma initialization straightforwardly generalize theorem general hand minimize nonnegative modular subroutine cardinality cut unconstraine almost constrain submodular minimization admit next factor achieve monotone fx return prove aa slightly cut notion curvature shorthand transfer curvature gx yield yield approximation constrain knowledge class remove minimize paper imply occur curvature modular function curvature fx x rank difficult practically relevant submodular replace know polynomial modular instance weight fx several bound far reduce curvature hold empirically find bad version proceed usually I run modular theorem terminate theoretical toolbox ground ii reduction lattice iii average vocabulary speech corpus vary case observe reduction mn accordingly speech dot represent respective take mn preprocessing show random weight order accurate estimate minimum min norm dot dotted bar cm mod case fix embed fig bind compute submodular many theoretically achieve approximation average instance result bad show cardinality complex algorithm realistic minimum submodular span bipartite four concave root modular clustered form good bs right speech curvature function sparse average consider square grid grid connect subgraph restrict bipartite dense outperform second despite perform sometimes experiment gain instance time preferable matlab differ versus c c unconstraine rp unconstraine unconstraine deterministic bi directional bi greedy member distinct subdifferential subgradient sufficient progress modular subject reach terminate within specific implication assume trivially monotone observation maximization algorithm specific subgradient subgradient pick permutation define stop rp iteration rp hold submodular permutation fs value sample symmetric subgradient permutation remain position approximate factor termination optimum satisfie must max showing set approximate analysis reveal complexity hard submodular necessary resort local completely variant permutation entirely order iteration schedule local implicitly thereby likewise directional distinct factor hold iteration order bi directional subgradient return directional chain inequality subgradient inequality belong satisfied continue directional show analysis show good counterpart bi directional greedy induce permutation order subgradient satisfie expectation randomness remainder deterministic proof monotone permutation subgradient number approximation already feasible general monotone exactly greedy result three analogous result hold constraint rely never bound follow monotone submodular would constrain variant monotone recent algorithm algorithm swap currently know phrase swap leave instance polynomial constrain ultimately instance subgradient generally polynomial submodular maximization achieve schedule submodular subgradient hardness imply fact submodular function polytope return set arbitrary subgradient fact optimality similar constrain constraint time approximation mild assumption constrain pose question would maximization large unfortunately impossible subgradient step hard return global optimizer subgradient subgradient solution solve np subdifferential express anti submodular involve sub express submodular subdifferential set anti subdifferential correspondingly equation submodular maximization test fx redundancy find synthetic instance similarity vary selection corpus string exact rs pick expectation repetition dominate though theoretical rp well rs bound importantly extremely fast minimize difference submodular function special divergence knowledge combinatorial multilinear maximization unconstraine minimization maximization however framework moreover detail discussion support foundation google microsoft award material part
simulation integrate equation evaluate examine energy distribution construct xy per bin regard generate regarded time obtain rotation couple dynamic respectively time fig decrease dynamic work track cause offer proper even integrate numerical naive integration examine two type dynamic design spin update continuously seem study difference understand capability dynamic mainly spin dynamic system couple spin behavior ref sampling exhibit reversible discuss topic dynamic modify reversible demonstrate spin observe ise align internal zero flip keep flip continue align moment second law natural reversible initial dependence dynamic impossible evolve contrary numerically examine finite size extend model xy dynamic various numerically infinitely examine future study dynamic promise present machine project r science science policy boltzmann spin implemented use numerically capability method physic appear monte second critical ise transition correctly take state successfully extension xy spin spin configuration normally number reason randomness sample randomly probabilistic boltzmann recently comparable conventional boltzmann machine investigate spin study utilize deterministic spin study firstly ise evolves introduce energy site update reproduce probabilistic behavior ise ise configuration ensemble system temperature although boltzmann similarity introduce generate canonical secondly discrete element lattice know rich spatio dynamical system degree freedom symbol partition regard deterministic spin ise dynamic spin boltzmann machine time dynamic note simulation ordinary computer sense pseudo truly since rely crucial principle costly monte pseudo actually randomness actually deterministic call algorithm large scale deterministic boltzmann machine regard mind capability dynamic spin model sample converge phase transition furthermore extend dynamic xy reversible good example discuss behavior ise although probabilistic deterministic place conventional monte briefly introduce spin dynamic ise spin model site hamiltonian spin configuration summation adjacent pair lattice distribution spin configuration give temperature difficult evaluate sample normally know gibbs spin choose spin update configuration stationary eventually obtain sample instead evolve change reach therefore internal dynamics eqs hybrid dynamical discrete continuous neighbor th move observe random instant consistent derive state node actually computer heat verify ise model error absolute error absolute empirical observed ise lattice size absolute total system error line gradient nearly almost monte carlo carlo decrease bias switch frequency numerically intuitively work absolute lattice decrease almost calculation perturbed multiply multiply perturbation sometimes system diagonal receive exactly position understand perturb confirm average gradient capability ref behavior ising know depend class belong belong dynamic yield ise numerically unit line lattice around intersection cubic provide estimate theoretical two lattice large use ise size periodic boundary intersect temperature figure cubic intersection fit lattice derivative temperature critical scaling follow figure fit log absolute temperature fig consistent straightforward extend general spin represent kronecker delta dynamic ising evolve positive point therefore jump state th determine greater accord eq state slowly duration regard couple interact essentially equivalent model internal reduce note ref implementation arrival system characterize site completely
trajectory trajectory prove free perturbation diverse trajectory optima use test random obstacle configuration seed employ learn predictor rate sort seed account relevance diversity employ reduction issue train low issue outperform build stochastic preference oracle recommend article minimize failure recommendation article user preference user membership perform five fold preference hold user test recommendation achieve significantly recommendation task character maximize coverage human annotate summary follow document understanding conference correspond cluster document contain document topic reference summary space performance reference summary predict therefore benefit reduction optimize approach consist capture sentence sentence compute square tf sentence absolute distance statistic observe dpp optimize test serve plot suggest superior acknowledgement project part valuable theoretical present begin prove monotone list fa fa fa let rl fa fa b fa fa prove randomized selection optimize submodular provide proof contextual policy class refer k rl lf kx f policy sample list f kx kx mx td policy depend form martingale hence lf kx mx x kx kx kx x x kx f x prove part additionally environment policy I sequence martingale take tm tm combine rl x I tm tm fact previous lf kx tm show must grow trick occur policy guarantee use lemma accumulate algorithm bx sample choose position list I benefit could accumulate fix construct list keep l k regret incur event event show generalized majority benefit z less root rl rl surrogate sensitive use domain recommendation prediction document submodular quality diversity provably near approach online regret learner prediction classifier agnostic validate problem include recommendation document range web recommendation identify successful trajectory predict list limit maximal utility ad ad high click pick trajectory extensive list item diverse diverse news chance like article redundant article little redundancy capture formally near guarantee work practice access list supervise maximize directly measure goal directly agnostic show produce learner list reduction lift class map relative hypothesis efficiency fully agnostic setting moreover exceed range prediction optimize submodular reward without contextual become machine diverse area broadly main function result second attempt simple identify parameterization match instance largely feature complexity combination model appeal solve potentially hard error attempt greedy set aim list utility full quite expressive agnostic generality come expense significantly assumption learn classifier position list enjoy benefit datum ensure agnostic online ad obey follow list concatenation intuitively add capture return shorter denote shorthand benefit item repeatedly arise take include predict option depend robot submodular good list unknown summarize contextual context current state quantify list expect submodular greedy find list statistically contextual observe regard lift hypothesis policy denote describe list quantify obey monotonicity greedy sequentially pick benefit list l list policy function call online internal list discount ts ts ts ts contextual subroutine regret exp contrast employ proceed state generate list via distribution item evaluate weighted benefit allow list beneficial theoretical perhaps aspect weighted denote first intuitively weight benefit position position adjust benefit position discount benefit later intuitively contributes equally omit discuss brevity ability measure expert annotation feedback setting value observe ad exp every issue full information case instance algorithm submodular online incur subroutine learner subroutine employ instance surprising stationary distribution greedy sequence state function define sequence incur internal construct list note sublinear fp l denote replacement fp list construct sample fp mf mf good list expect example fix use construct ratio close involve match especially contextual setting due contextual mention goal hypothesis item base within large policy list item select learn attempt generalize list construction position general policy perform length list construct pick sample user document feature sample new cost sensitive feature state list bs tw k ti ti contextual construct list policy free subroutine weight feature weight vector item incur cost sensitive example reduction transform task policy submodular pick list construct unlike work analysis lead several guarantee free feedback setting contextual exp cost sensitive task class leverage weighted majority maintain rate special policy tractable employ convex descent bind original briefly cost sensitive item cost convert weighted example loss transform ranking
linearization regularity quasi expansion chain approximation derivation approximation year l main goal proper relation estimation remain straightforwardly plug estimate e incremental claim furthermore imply unobserved amount diag n exchangeable correlation structure symmetry predefine accord covariance year diag diag diag diag correlation correlation structure need henceforth among amount e segment I n claim year year real data triangle year structure logarithmic function consider linear poisson gamma mean model fit compete model illustrate year residual glm might incremental gamma suitable six compete list half comparison estimate take brief description notice function exactly glm glm poisson sometimes call coincide difference mse partially cause covariance ht poisson year glm glm exch exch ar quadratic linear quadratic ar hundred comparison list linear quadratic favor work obtain prediction ar exchangeable structure reasonable ar table conclude year discrepancy mse prediction straightforwardly imply precision mse prediction distribution glm glm cl ar ar quadratic linear quadratic mse dependency year despite mention mse possible mse incorporate framework estimate prediction also estimate bootstrapping residual assume probable reason prediction resample cluster bootstrap yet remains resample triangle cf ar covariance structure similarly already criteria favor could g datum glm claim variance sake completeness six fit estimate total error list exch ar criterion ar correlation favor correlation structure variance could case independence correlation function however comparable ht variance independence exchangeable paper propose claim assumption development dependency work consistent specify dependence specific distributional compete directly often however fit simply number take account inspection part insight variance relationship also common criterion dependency within year year diagonal way claim triangle year notation need principle claim derive non traditional incorporate covariance bias shown estimation ignore bias estimate glm estimate glm variance whole cluster bootstrapping require possess triangle acknowledgment science foundation project economics g foundation p theorem theorem theorem section notation remark mathematics economic ta claim glm claim origin year violate classical may application generalize claim claim triangle amount year dependent allow dependency recommendation selection moreover discuss illustration benefit claim criterion I various glm tool classical claim year assumption point enable need mention paper suggest generalized claim successive year extend longitudinal another glm introduce method violate response nuisance correlation addition probability glm framework solely belong exponential together claim present correlate square claim moreover non way estimate prediction present result illustrate predict claim introduce classical claim terminology claim development triangle development year therefore accounting year year development period right year year j year development period claim run triangle comprise order correlate natural year hand year suppose chain cf create claim triangle type section explain principle use claim represent year development common logarithmic code j claim glm suitable claim purpose score vector poisson gamma integer multiply claim consider finally choose year decay time however situation strongly slow comparison covariance parameter satisfactory result particular glm nest structure test nest glm framework base information criterion aic bic see however since likelihood criterion analogy aic namely define likelihood datum equal aic independence independence easily provide software package criterion use
backpropagation neural mean square least selector estimation neuron least consider backpropagation neural way estimation technique linearly separable infinite linearly product space trick avoid curse primarily machine recently example recursive least least analog name usage life one adaptive area sometimes equation e ill pose regularization include shrinkage two lie regularization pdfs gaussian problem regularization introduce multiplication effect parent follow backpropagation selector via iii describe briefly principle base counterpart iv validate please rule adaptation labels adaptation spirit q ignore initial condition q trick product observation inner product follow write manner belong q free analytically cross dependent dependent spread q terminology abstract everything order another final layer without neural pass pass call back ease reader calculate output neuron output act activation function n popular regularization optimize equation target value respectively one elegant way interior follow comment follow subgradient hull gradient subgradient principle lasso selector subgradient similarly follow tn n n contraction ease reader evolve equilibrium consider equilibrium side define hence q large definition eqn q depend belong divide unit circle please inequality justify cause normalization valid bound may justify norm assume speed want eqn algebra would bind algorithm role minimize e weight subtract side eq please note adaptation factor positive normal get curve one report adjusted manner lasso nice aspect help point sense many figure pass channel noise modify pass pass linearity noise add coefficient linearity change linearity interference linearity modify term fast original train function maintain steady value epoch see epoch increase stay subgradient converge fast uniform regression contraction principle variant pass consequently white conventional variant worth instantaneous least newly contraction variant boost apply modification superiority mining algorithm
impose parameter informative gamma specify example identifiability issue without prior calibration spatial pattern temperature grid bilinear separability use specification except km identifiability density mcmc chain hour parallel core intel ghz mcmc adequate mcmc posterior estimate run namely composite enable computationally calibration normality adjustment adjustment common hasting composite use verification necessary attractive easy easy spatial observational helpful issue calibration make continue become evaluation computationally tractable computation become slow perhaps simple block analytical issue place computing process also regular output calculation inaccurate worth note discrepancy suffer identifiability issue impose discrepancy information scientific besides depend mix heat vertical keep uncertainty applied provide improve variability sensitive discrepancy ease eq contain j write k inference linear uncorrelated due negative l covariance model similarly n respect q th management nsf agreement center management adjust notably black credible interval bar wide computer uncertainty computer calibration infer physical output form spatial environmental sound statistical challenging composite dimensional composite likelihood computer pose several challenge calibration composite adjust composite uncertainty study computer often enable conduct virtual understanding physical phenomenon regard computer hence stem uncertainty value input parameter involve model compatible observed realization computer setting sound account uncertainty knowledge quantify uncertainty carefully rigorous quantification uncertainty projection output pose nontrivial inferential challenge limited enable calibration run face model form spatial increasingly modern develop recently resolve computer manuscript calibration use composite adapt model likelihood pairwise construct calibration method composite adopt composite likelihood rely component conditioning block composite allow reduction burden flexible covariance depend likelihood block bayesian outline basic calibration field calibration discuss relevant adjust posterior composite spatial change simulate finally discussion direction computer calibration stage calibration output well provide uncertainty take account interpolation observational notation henceforth field interested open setting computer grid since observation observational output eq covariate regression covariance mle interpolation provide location call flexibility model output process observational follow fit term discrepancy process location challenge calibration spatial exist approach proceed stem evaluate repeatedly like markov monte carlo become computationally prohibitive reduce cost overcome limitation dimension basis expansion exploit uncorrelated nature low basis reformulate rely block composite likelihood partition avoid practice block utilize block thereby effort adopt formulation assume conditional independence different covariance valid therefore define composite likelihood approach valid likelihood result divide block output note accord likelihood observational corresponding corresponding calibration stage choose proper standard hasting scale calibration calculation bottleneck computationally demand usually covariance covariance covariance block covariance spatial spatial discrepancy setting discrepancy respectively geodesic surface infer initially likelihood estimate potentially obtain improper objective identifiability issue explain far receive inverse also impose prior uniform fit receive note unimodal characteristic asymptotic therefore quite different discuss adjust inference justification normally vary reasonably smoothly model observational composite mean covariances ii collection consistency normality utilize result maximum cox zhang establish normality composite composite cl consistency maximum vector normality composite n composite likelihood regularity condition likelihood asymptotic absolutely theorem use ii ready state consistency variation mean n bn likelihood I follow several option adjust composite likelihood include post hoc adjustment composite curvature adjustment utilize moment inference necessary evaluate computation note posterior adjustment rely correct mode finite adjustment adjust open adjustment step mcmc mcmc another adjustment result mode suggest adjustment curvature demonstrate intermediate input air sensitivity important model diagnostic use projection economic uncertain spatial grid mean anomaly note model
application particular illustrate computer code analysis far natural way define impact denote yield index well go naive dissimilarity produce unnormalize absolutely lebesgue f divergence eq leibler hellinger distance variation pearson plug dissimilarity measure yield ar divergence sensitivity note invariant invertible transformation mutual divergence q information normalize study call squared mutual contingency actually indice ar divergence link highlight estimation index involve density importantly indice well mutual completely sensitivity come goal one density curse limit multivariate extension subsection besides need focus us idea estimator unconstraine importance popular eq measure measurable function ar rise different choice wasserstein total variation interesting intersect variation distance unfortunately easy ar divergence orient measure sensitivity unnormalized easy check aim quantify measure equal useful want design mutual criterion recently share deep reproduce hilbert independence criterion machine mutual shannon symmetric absolutely lebesgue measure mi zero mi able detect dependency unlike coefficient check jensen triangle simple vi variant mutual information dependence measure equality vi component analysis sensitivity mi arise index ar divergence dissimilarity input normalize product characteristic respectively characteristic moment eq invariance equal include interestingly euclidean concern denote ij b ij j write equation use function although biased specific correction cp cl space examine let particular distance retrieve universal z nx jk j center matrix like distance covariance propose sensitivity generalize kernel multivariate kernel operator operator see existence representation measure dependence interestingly equal kernel free select nevertheless since dimensionality relate estimation long pick readily kernel act need elegant input output one categorical selection include example perspective impact change variable contrast adapt fact possible dedicated datum kernel semi output variable act surrogate simplified code lie check semi practical principal learning feature selection detect irrelevant resemble screening assume whereas feature highly dependent precisely filter naive distinction make screen option independent technique rather approach hope new entail model additive selective overview generalization replace free identify technique pure target quantify selection involve jointly solve procedure feature time hand robust computation involve proxy likely limit effect add express minimal redundancy forward backward investigate mutual similarly backward mi replaced introduce purely author retain sure screening prove work sup sure mention base computation detect marginally uncorrelated jointly sup select input select sup feature measure sup repeat select cardinality reach maximum point another take iterative plan investigate full backward combination dependence center gram solve interestingly symmetric highlight strong correspondence standard dual augment discuss measure mention feature dependence dimensional screening nevertheless section remarkably well require reveal high preliminary assess sensitivity analytical virtual library experiment index sensitivity index fast version est total si normalize est divergence special correlation correlation extend functional pick generalization input variable index decrease eq compute sensitivity ar calculation pf analytical sis coherent top linear factor easily original conclusion small total sis almost equal sis small identify output fs pf size replication negligible interaction note index point total observe unlike detect computer code finally bring recall concern pick comment pf replicate result precisely categorical recover function pf replicate feature order test code control influential recall sis influential input replicate influential contrary completely fail detect exclude test notably perfectly discriminate factor replicate probability replicate eight influential method detail unable correctly impact accurately remain similarly select influential iterative h fs replicate size sis slightly influential identify influential fs replicate reservoir reduce parameter reservoir image basically solve inverse incorporate reservoir want example idea incorporate actually sensitivity uncertainty choose arbitrary dimension make reservoir test reservoir derive simplified seven medium uncertain multiplier residual assign prior consist ratio measurement give propagate flow simulator top day figure pick measurement begin production around obviously generalize observation thank
elimination subroutine elimination optimal subproblem quite elimination extremely invoke show good theoretical date ucb exhibit respect introduce ucb necessary gap arm problem fix reduction available great consider testing fix arm brief result studying term generalized limit threshold least imply fail great self simple appendix introduce procedure operate large terminate arm total theorem quantify arm sample st jt else stop ucb constant appear make make practice constant observe depend motivated theorem algorithm precise constant state logarithm induction k thank together chernoff integral hoeffding put k generality notation follow trivial step ii play define furthermore rewrite n observe hold sum obtain obtain conclude jt jt jt jt ji jt step play note hoeffding inequality thus treat arm exceed meet stop condition arm complete state method good arm behave practice confidence satisfy however ucb terminate ucb arm stop meet compare algorithm empirical would three require structure I change suffice maintain ordered list updating require contrast ucb procedure explicitly sufficient statistic per per step poor ucb ucb variety arm hardness case hard super size depth experiment realization stop size time observation perform uniform sampling confirm median make practically perform l l seem behave algorithm ucb ucb heuristic plot theoretical remark never fail terminate reality explore termination arm high arm measuring optimal increase decrease confidence three unlike empirical algorithm large gap successive elimination ucb ucb collect output twice tell still conservative motivate use appear across b plot number size right stopping time terminate ucb computational plot distinguish due nonetheless testing rely intuitive justified formally continue stop continuity intuitively low threshold upper proceed error focus threshold define stop x remove hold conclude corollary theorem department electrical engineering university department armed mab devise input regardless find must arm adjust well second well arbitrarily close fixed set total constant within fix arm history back decade provide successive find good arm within logarithmic design find arm succeed depend parameterization come elimination come logarithmic factor bind avoid bind loose classic answer imply procedure law iterate logarithm behind let arm equally random walk solve equation formalize specifically
sparsity coordinate loose lead coordinate strong satisfy horizontal dirichlet distribution sample trend finally convert establish via let avoid dependency turn control gamma cdf q copy event establish imply consequently imply choice coordinate q whereby define plug provide elementary dirichlet parameter coordinate denote dirichlet exponent coordinate threshold grow
framework predictive reward technique explain demonstrate approximate optimality criterion theoretic framework single reward reason prove article theoretic concept regret stand criterion fundamentally ill pose require utility feature behavior convex inverse ice polytope effective behavior criterion polytope criteria max family simple show require range market explore effect chain formalize motivate review many interested human contribution community describe contribution light combine market sale record census publicly estimate assume demand side preference price measuring like air determine effect production cost aspect preference market participant second equilibrium pricing assume technique value ultimately one derive market unfortunately generally available reasoning investigate market american price fixing measuring mid market competition guide behavior observe truth framework behavior explain people act oppose game modification player function notion latter integrating limitation recent surprising action utility absence surprising observation memory lead equilibrium response equilibrium concept player utility interested decision player know serve validate experimental hypothesis community human system work focus set known behavior summarize behavior utility number method introduce margin utilize optimal planning software entropy predictive guarantee utility weight community observation prior publication novel domain focus computationally efficient good observable quantity leverage game fine assumption regard preference present blind describe necessary background notation e game tool range illustrative game tuple nn game player player allow expand outcome contrast utility function outcome know allow real world scenario instance share specifie plan measurable distance speed intersection utility preference outcome exist independent traffic think portion joint situation play correlate play simple dynamic player correlate player joint unstable strategy condition ease notation two switch prescribe action internal external player recommend action randomize deviation player jointly write instantaneous instantaneous function class deviation conceptually benefit player typically instantaneous player portion write joint deviation quantify deviation call equilibrium think substitute utility optimality setting fortunately internal regret approximate regret polynomially sized proof equip tool multi assume notion player sequence observation accord player estimation player similarly structure initially derive true analyze appear assumption hope recover game game prefer joint joint assumption state necessarily prefer reason player lead approach unknown estimate utility spirit utility match employ inverse optimal relate prediction strongly rational prefer immediate another rational prediction rational deviation set equilibrium respect immediate conversely utility assume desirable rational rational prefer strong restrict attention rational bad agent act preference predictive behavior sufficient rational knowledge utility function translation requirement product vary fortunately equivalent convex equivalent equilibria ice ice polytope introduce reasonable interpretation assumption switch player measure strong standard ice polytope strongly deviation ice start generalize rational empty interior ice polytope polytope linear convex equivalence ice polytope strongly rational satisfy ice polytope provide appendix polytope directly reduce explicitly linear ice polytope polytope retain quality corollary formalize correlated equilibrium prediction ice polytope also correlate equilibrium definition equilibrium equilibrium polytope sum strong predictive requirement satisfy action distribution player game fix utility setting act independently external ice external nash constant nash marginal form standard ice equilibrium property property much utility assumption preference equilibrium modification capture demonstrate maintain preserve side constraint prescribe ice polytope utility preserve correspondence ice linear equality control utility preserving notable strong prediction thus matching preserve ice polytope converse match use match experiment use minimization behavior mechanism resolve accounting justify optimization eq principle subject know constraint choose salient characteristic affine convex field field within agent equilibria normal know priori select precisely ice polytope ensure predict ice convex feasible efficiently maximum entropy enjoy follow ice rational log choose proof derive program ice multiplier present program dual entropy ice derive dual empty duality entropy ice maximum entropy ice tb inherent advantage particularly trait primal primal still computing expectation advance game enable describe computation incorporate non stochastic vary game nature player player group game draw device class observation addition observation yet leverage achieve assume player next ultimately game reasoning regret need execute game different act game action similar semantic deviation game infinitely class decision regret utility quantify notion entail notion modify slight modification ice polytope ice polytope adjust entropy account entropy chance strategy familiar value conditional exponential family learn control result prevent unobserved game appropriate dual effectively primal hold approximately justification control behavior utility change interpretation feature remain behavior agnostic recommendation unseen interested situation reason model behavior sale pricing demand product sale utility production production line accurate unclear introduce approach behavior game enforce property true quality ice game predict feasible slack variable add primal program access would require may approximate observation cost associate observation inherent approximation sensitivity player accurately finite observe direction hoeffde provide observe r w logarithmic closely w r maximize home day office road segment upon arrival home total spend stop intersection utility outcome correlate equilibrium subgradient mainly compare accuracy ice measure log baseline vary equilibrium baseline distribution classifier parameterize individually predict maximum likelihood eventually since game ice constraint social optimize optimal outperform nature game behavior transfer display game add add game simulate city building add game keep change major share slack feasibility ice add add transfer approach apply general transfer compare logistic ice regression reference game strategy demonstrate behavior efficient size additionally ice beneficial setting assumption hold baseline market entry prediction competition player trial enter player simultaneous decide whether business enter receive stochastic payoff unknown player enter market receive round player reward receive human student play ten round student play proportional cumulative randomly fashion nash expectation would fashion subject performance experiment play predictive interested behavior game label multinomial leave feature game accord nash equilibrium learn baseline figure baseline baseline baseline sample multinomial slightly attain loss particularly nash feature alone ht baseline variety frequency strongly employ exponentially average interestingly summary gap ice size logistic good datum appear scenario behavior participant predict mid wish likely record total four none highlight aggregate population national measuring quantity restriction rate cost period vary aggregate parameterize simultaneous move player mid scale utility allocate utility action account observation one map respectively outcome quite action highly correlate multinomial despite four cross inverse train logistic match ice latter l quasi newton
rewrite mean unconstrained control base reference htbp I effectiveness simple unconstraine nonlinear control construct converse model converse associate equation control nn actor actor since actor nn k u system exploratory give signal complete control nn demonstrate representative actor wherein dash optimal actor loop policy cost demonstrate effectiveness datum nonlinear nonlinear pose couple give htbp iteration htbp iteration learn control policy algorithm algorithm select size eq initial actor conduct trajectory action control collect compute conduct loop input signal offline employ policy indicate figure nn six actor brevity give actor figure figure actor nn loop simulation control compute time reduce htbp consider constrained control unconstrained policy subsection action constraint system develop activation actor nn closed loop conduct collect actor converge demonstrate representative representative actor weight figure actor loop figure respectively real computed address propose datum prove base learn control system instead mathematical base contain offline actor simple nonlinear demonstrate control remark title thank address nonlinear technique bellman differential impossible bad mathematical overcome difficulty free policy control optimal policy require knowledge base thought actor actor neural policy respectively actor residual whole include collect information second offline policy iteration policy optimal problem method nonlinear demonstrate effectiveness control bellman control decade bottleneck bellman equation pde difficult solve analytic propose iterative convert linear lyapunov equation think successively lyapunov equation nn science chemical engineering engineering scale procedure prominent vast lack moreover accurate modelling impossible digital sensor availability direct design control practical decade technique rl computational intelligence machine artificial intelligence rl technique actor agent aim optimal response rl scheme approximate programming structure nonlinear dynamic programming forward avoid curse moreover rl control rl base rl suited rl rl design programming nonlinear si wang hdp linear system feedback iteration value measurement output fu adaptive system finite study introduce involve effect hdp apply feedback discrete time base iteration algorithm dynamic control nn considerably discrete estimating error rl suggest necessity internal along state lyapunov policy state trajectory online actor nonlinear employ design base rl require prior identification adaptive rl still general control problem nonlinear completely arrange problem preliminary present develop unconstraine finally test brief euclidean real transpose denote solution side rearrange yield function initial admissible policy transform solving require mathematical equation embed control thus lack model impact policy control learn suffer issue collect incorporate concentrated mean cost function policy advantage exploratory x v derivation satisfy contradiction start contradiction boundary imply real constant follow base equation actor actor approximate accurately infinite linearly independent usually approximate control vector neurons lx lx activation function sub actor u actor neuron output actor actor rewrite due actor yield residual notation rewrite form vector force project error zero substitution ix ix ix computationally thus monte integration compute ix dx u ix approximately substitution expression square scheme collect neighborhood trajectory exploratory select x xt accordingly note least e realize scheme persistent frequency similar issue community subsection develop present procedure constrain design
total gets visit setting set minimize mistake vertex certain mistake restriction open extension general active characterization general graph require tool attack reduce span span span know structure relevant passive correspond span simple clique clique say clique span graph star tree query star center span nature lead selection bad query gain currently span mistake baseline believe graph employ suggest combine active predict google google award ec grant publication reflect remark universit di universit di di universit di investigate learn assign factor minimize mistake query efficient mistake classifier modification query span tree active mistake arbitrary active abundance web network bioinformatic scalable graph prediction important topic area label learner receive must predict typically rely likely approach label induce g assignment extension reference version problem allow subset boost intuition set star star shape graph bridge big adversary graph assume center star strategy consistent far choose whole mistake query nod big star unseen show arbitrarily big devise place minimize mistake question investigate viewpoint et elegant mistake assignment unknown query unlabele instance example include system bind mistake note since query maximize must resort heuristic investigate active graph label actual extremely place query tree within trade learner modification trade constant tree fraction number mistake must construct query apparent obtain assign binary e j phase phase label query label remain ever reveal mistake number exist connect edge set obtain iff forest remove tree forest node tree adjacent hinge hinge node hinge tree phase exposition phase return query input label prediction ht subsequent connect component selection round generation gets store introduce reciprocal mention measure adversary viewpoint return end state maximize first component pick minimize construction maintain generate see step node desire reach also count query cause return label hinge describe predict connection node hinge label otherwise label node tie break query take constant factor prove subset operate node one set hinge node path belong choose label label reveal decide il mistake make given introduce denote prediction mistake make label deal mistake prediction force query mistake make though procedure competitive know optimality factor relate clearly interpretable regularity issue also query preliminary hold I size know adjacent arbitrary node depth visit node visit order extended share exactly visit node small leave abuse subtree node leave imply subtree select incremental recursive connect component leaf split construction split leaf add node child child merge belong tree cardinality leave let set ensure subtree leaf write subtree node split parent lemma subtree claim prove number node forest associate big forest distinct obtain n prove mistake budget tree quantity node tree adversarial make mistake budget large adversary create hinge tree one tree perform assign hinge label connection hinge tree adversary assign label mistake remain hinge tree agreement connection modify proof deal choose node depend previously assign query force method mistake expectation hinge tree hinge yield weak also easily rewrite order total tree step node tree node component mapping component map iff give equivalence sort selection moreover node set since union domain thereby proof node let distinction node capture capture extend node node capture reference leading contain turn would already let node turn plus plus bind clearly initial node select round sure lemma concern mistake query query set satisfie see l query optimality query mistake end minimize constant factor thus query number notation maximizer include maximizer sake contradiction ia prove immediate consequence query must hinge hinge since external concluding contain hence conclude put together set constant inequality apply lemma condition interpretable mistake bind mistake make satisfies label invoke made function yield need efficiently predict set cardinality compete set batch analyze total operate phase phase invoke assigned phase labeling describe predict tree even predict optimally tree call label consistent return label label ii labeling consider adjacent assigning assign otherwise minimize verify one labeling assigning unique build phase phase invoke assign assign phase task stop ask less made trade minimize query phase mistake prediction prediction algorithm query prediction mistake make similarly consideration selecting force factor minimize clearly section regular induce adversarial say elimination edge balance regularity balanced labeling imply get optimal modification optimize mean optimally mistake budget mistake know select query budget application modification efficiently search k ensure l kk operate build compute find stress mistake query put tree simple star always star always select center star pick hence provide result contain show prediction order mistake hold randomize labeling prediction query size randomize expect low exist mistake line graph choose give label mistake among label clearly impossible efficient show need query predict subtree algorithm maintain contain eliminate item resp take associated selection query node size item large create edge construction maintain find see refer choose first visit follow key sake simplicity root backtracking visit node contain get observe efficiently plus union child sum edge previous backtracking
throughout head result detector posterior position detector source nothing source prior separation appropriate assignment suit job concentrate position mix signal detector account physical write position source play assume source detector encode uncertainty idea square deviation principle say quantification noise merely something student assignment write q simplify predict potential assign implicit familiar chi function localization example trial trial subsequent accommodate detector signal already old datum accurate case advantageous begin source separation neural recognize early result formalism lead search algorithm theory optimize identify piece lead numerous parameter attempt reader present include search iterative markov monte ensemble variational bayesian technique utilize last aid well understanding recommend recommend seek paper array enable reader source separation call separation discuss advantage explicitly lead incorporate information enable researcher information physical world sensor make properly design comprise interesting design sensor application filter limit take lead separation arise superposition may infinite superposition may delay due medium focus effort methodology design advantage possess reach blind little assume detect work difficult information different prior source give methodology leave search bayes right degree specific describe could produce encodes part scientific call merely estimate parameter act term relevant indicate degree consider call theorem turn source look tell new prior next show bayesian prior lead ica demonstrate previously understand later derivation ica separation blind important blind assumption sense assume propagate distinct detector assume linearly linear furthermore model source record source mix detector physical keep blind see informed separation bayes represent entire represent signal give term assign probability solve search denominator calculation depend simplify writing construct process free assign delta states separation independent merely detector probability amplitude source signal super gaussian source without infinite perfectly good serve incorrect hold probability assign purpose assume q know nothing matrix encode assign ij long reasonable assign joint prior eq ready probability probable easy bayesian search logarithm separate log prior reduce number take logarithm mixing surely ideal write sign delta integral introduce jt na become delta substitute take implicit solve way ica respect ascent familiar gradient rule speak rule identical density mix interested probable probable ica certainly derive theoretic viewpoint derivation assumption algorithm nonlinearity merely amplitude density arise ica separate pure histogram severe implicitly modification density situation essentially smoothed density want include design ica detector analytically integration analytically allow analytic marginalization source elegant elegant break another arise yes additional understanding allow range applicability fix explicit demonstrate another modify piece prior follow inverse source detector detector know source follow propagation source detector detector element detector position detector detector angular coordinate eq rescale change specifically new probability respect rescale measure derive prior matrix source detector term delta inverse square assignment give rewrite integral detector wrong familiar improper go infinity concern infinity reader detector
term observe maximize approach time low classifier combination framework adaboost problem work performance classifier design design optimize novel numerical optimality robustness class imbalance like evaluation another behavioral science specificity sensitivity commonly science evaluation appropriate testing effectiveness rule imply misclassification misclassification find class decision bias get particularly important instance class one heavily loss diagnosis issue recognition mining difficulty skewed successfully neural svms poor context decision pruning remove branch relate class backpropagation neural gradient length dominate consequently think imbalance vector algorithm try one create svms svm demonstrate effort imbalance reference accuracy change distribution cost metric accuracy designing maximize choose measure specifically look classifier maximize f feature function problem propose minimum solve problem common misclassification appropriate severe suitable measure classifier good rest numerical represent possible tp positive correctly classify tn true correctly fp misclassifie definition tn tp tp fp precision measure imbalance indicate predictive meanwhile precision property since reasonable value true application adequate relation consist belong classify belong class train maximize measure must point propose estimate density density negative classifier maximize depend problem application boundary framework combine fp tn tp q training measure find region minimize extent define point datum find available quantity quantity sign boundary surface equation q smoothed task training find functional find write derivative smoothed dirac minimization solve euler specifically pde differential initialization steady pde reach equation detail equation numerically regular density level kind curve evolution distance keep classifier energy unchanged scheme usual corresponding minor affect result numerical scheme decision database next red respectively exhaustive level initialization component descent flow optimum database database multimodal negative distribution database select idea consider imbalance database variety shape distribution kernel estimation positive class traditional bayes subsection briefly choose consideration must account obtain see follow svm acc naive bayes typical classifier minimize problem class highly unbalanced unitary variance number positive class decision region choose threshold value one value recall measure dependency function tradeoff recall
importantly underlie aim cascade start hazard infection infection cumulative first infection di na occur time likelihood infect fact infect equivalently apply cascade represent infected term represent one window cascade likelihood cascade maximum cascade unique inference multiplicative hazard rate node hazard node undesirable property solution multiplicative eq dense infect bad pair node unbounde get common cascade rule infected cascade avoid unbounded infect infect infect disjoint cascade successfully yet even great include solve cascade pa infect one term laplacian log concave network additive multiplicative structure dataset cascade million month period multiplicative baseline observation window skip method model appropriate focus length window increase observe model jt ib media cascade cascade site spread select mention infer mention number site cascade several topic unfortunately truth inference cascade rare medium increase instantaneous infection increase decrease cascade record cascade disjoint additive cascade set build simulate cascade node cascade cascade test generate cascade multiplicative model none clear winner inverse cascade surprisingly generate cascade distribution infect predictive compare distribution generate cascade cascade duration additive multiplicative differ dramatically additive model linear get cascade size contribute towards propagation provide flexible additive influence include additive one consider nonparametric fitting use likelihood include dynamic goodness test principled theorem corollary lemma network skeleton spread disease survival network inference solve convexity generalize se ne multiplicative scenario increase decrease positive synthetic cascade model cascade ta place disease spread social network epidemic node network piece node infection similarly think spread propagation di vi observe infect observe customer product influence decision infection infer propose propagation hide theory generalize efficient validate experimentally encourage diffusion consider fixed node switch opposite represent count instantaneous infection e hazard infection explanatory discover take hazard node infection edge network additive function infection previously infect se general ever I previously infect node instantaneous infection relax risk rate node decrease getting infect observe similarly piece medium site relate news medium relate adopt efficiently parameter exploit inference approach infer every edge temporal consider temporal temporal li previously multiplicative node decrease multiple unobserved create cascade cascade infect infection generally node symbol infect observation window cascade generalize trivially propagation correspond piece infection node cascade infect cascade infect infection ti intensity arbitrary time decide infected intensity process ty ti define conditional note assumption word remain long infect infer record cascade discover network edge hazard infection tell incoming infection hazard hazard validate experimentally provide flexible process allow hazard argue necessity additive survival additive hazard hazard hazard infection infect cascade hazard infected force parameter negative hazard time covariate depend infected node model effect simplicity infection parent mathematically goal infer maximize likelihood cascade infer parameter discover network edge infection hazard rate likelihood infection cascade infection time tt infection di na infected fact infect informative add survival equivalently cascade end cascade likelihood cascade likelihood log cascade network define linearity convexity ga feature inference additive ensure infect parent otherwise log unbounded since weakly reward infect likelihood cascade positively norm heuristic encourage optimal additive generative di infection continue compute infection node gets infect infect
accumulate less user accumulate sum worker number update visible worker allow worker update worker though update within guarantee worker happen b loose absolute strong consistency restrict worker update see update worker worker combine provide strong ensure worker make difference version correspondingly certain level quality stochastic utilizes develop shall informally update accumulate server great propagation accumulate propagation operation p model noisy read worker imply pg propagation every read window worker counting update worker generalize parallel model consist worker interval difference argue early since convex f f tf suitable divide converge follow x tf tx say something employ cache minimize within share server cache asynchronous cache employ back model track library maintain entity represent server treat entity keep track asynchronous server achieve throughput message send might dependent default contribute implement unified modular consistency implemented perform different service accordingly consistency control logic semantic include response ps three type network communication server row server server update couple cache coherence implement prototype server unsupervise implement server weak conduct node equip core main memory restrict core gb machine machine relatively news strong scalability news show news token number worker assign result ps vs scalability conduct great potential th acknowledgment probe provide support distribute ml share node network overhead proper model correctness throughput exist consistency use either loose correctness ml fail distribute ml category randomly find distribute ml correctness consistency asynchronous correctness consistency model distribute server evaluate popular increase framework propose scale algorithm distribute amongst server architecture abstraction support broad server support read overhead proper guarantee desirable meet correctness theoretically power system implementation reason solution maintain classic provide guarantee meet parameter server correctness naive fail delay update keep consistency consistency require update parallelism application modern sequential guarantee distribute read write vertex schedule vertex carefully colored correctness utilize employ consistency make achieve lda theoretical correct unclear loose broad range algorithm recently variant consistency compose computation synchronization send synchronization go beyond share replica throughput sound descent asynchronous parallel improve cpu update synchronization barrier also system update benefit difficult maintain may ml sufficiently amount admit weak properly improvement paper throughput relax improve throughput rate propose tune consistency achieve concept consistency server unlike send phase whenever bandwidth guarantee bound update absolute combine provide asymptotic assess intuitively maintain amount less threshold accumulate incoming write make accumulated change user threshold compare provide fine grain guarantee elaborate combine hybrid present definition access worker parameter
extreme file rate substantially drop also observe rate tend prediction alternative range extreme logistic regression response explain extra vector contrast rely exercise model profile large portion handle information mcmc profile vector difference compare group respect common trajectory parameter vector extreme profile reading leave evolution shift old extreme refer discuss profile k extreme profile bar credible associated component salient relative k high dispersion generation explanation past old trend tend increasingly profile advanced conclude several meaningful easy interpret summary main summary way simplify longitudinal heterogeneity keep extreme profile allow trajectory extension group individual static characteristic time dependent dependent birth highlight show individual close extreme trajectory free life profile trajectory exhibit importance expect relatively old process consider birth estimating profile membership however membership importance monotonic old profile answer differently old appear furthermore differently previous show purely wave wave latent analysis analyze mostly wave uncorrelated longitudinal analysis transition state approach root survey characterize life trajectory across address informally choose profile section depend choose profile reveal really extreme profile therefore need report extreme range important approach index aic schwarz convenient counterpart difficulty impractical nonparametric favor sparse representation process mixed membership limitation effect potential variability follow way account categorical gender prior contingency covariate specification limitation model essence correspond importance pattern tie death one integrate profile characterize survival gibbs section sample augment equivalent shape obtain jk jk metropolis step pt probability distribution metropolis hasting step probability replace mcmc modify expression similar step pt investigation suggestion anonymous associate thorough ph thesis author treatment datum analysis graphic support nsf university analyze longitudinal population long care survey membership multiple birth order inter method trajectory tend later introduce estimation procedure longitudinal datum national long care longitudinal survey aim assess united researcher live duration age nature change answer question importance due increase private people relevant public potentially time change life individual likely age people question assume american people constitute population longitudinal capable account longitudinal frequently effort longitudinal method far researcher instead serie see attempt nature model longitudinal process heterogeneity mix describe ideal extreme partially pure longitudinal extreme profile time extension aim capture difference across allow individual mix article next present brief introduction description survey basic extension handle base fully section insight longitudinal panel design assess united year rough wave people wave serve purpose replace wave wave evaluate activity first comprise basic care activity involve activity within maintain determine functional status series indicate absence wave aim quickly operational individual present last individual subsequent assess individual community receive subsequent survey death subset six individual ability perform get six obtain link service preprocesse acquisition heterogeneity main trajectory idea behind trajectory broadly contain longitudinal measurement response contain dependent joint describe usually model age provide trajectory individual evolution technique trajectory curve represent present easy mechanism way heterogeneity handle perfectly attribute fluctuation formulation essentially say every construction actual individual belong conceptually homogeneity class mix small class simultaneously membership model develop pool combine mixed membership seek produce soft assume extreme assume correspond profile approach conceptually previous cross application profile ideal whereas ideal people way model specify evolve characteristic compose mixed idea profile mean case correspond exactly assume th unit way identify individual membership zero membership belong pass member extreme profile individual difficulty age evolution response index trajectory way differently one take birth dependence trajectory whole difference birth interpret use common enable inter individual index date birth covariate p ik jk specification replace specification dependent reasonably flexible let contiguous interval model handle population level vector p expand mcmc base basic rely extract datum individual receive survey death exclude year wave birth year prior clarity estimate relate offset c c c c define five partitioning range birth column individual salient feature turn span whole relevant birth year old year fit basic mcmc profile proportion last slight preference parametrization effect probability individual profile profile profile significant flexibility handle heterogeneity normal prior chain rapidly reach distribution slow mix case discard profile plot label switch although switch potential application modal due abundance datum distinct structural extreme profile extreme extreme instead age profile reach unable perform year subtract invariance profile extreme extreme profile high importance population close last jk jk sort posterior pt summary extreme mixed profile line unable increase aid result exact summary relatively small prior already strong priori relative dispersion drive surprising consider estimation model extreme profile population trajectory individual age remain extreme profile consider worth relationship trajectory describe note sort profile inspection closely
gx inherent limitation design necessary dynamic follow sequence reader convex minimal newton polytope provide translate copy newton polytope difference condition follow necessary minimize minimize since note v otherwise gx contradiction x gx yx gx gx proof constructive let support hyperplane convex say satisfied however sufficient would compatible estimate follow show selection compatible arbitrary reduction clause achieve polytope orthonormal complexity vertex lattice time moreover let hull polynomial b precisely every jx jx proof I compatibility show case contradiction similarly induce consistent make clause maximal compatible np randomized algorithm start seed input case keep member union require boundary however practice finally input static dynamic r b consist sequence secondary ccc rna know rna secondary structure particularly without rna sort one exclude pair single newton count energy polytope algorithm start polytope newton newton polytope subsequence dynamic yield result base newton polytope move turn take less return compatible origin hull protein rna complex rna bind ray resulting explain correctly predict energy approximately whether energy fall equation find solution formulate answer agreement energy structure develop notion energy characterization compatible rna compatible set give randomized energy g compatible energy open treat assess remain proof em em minus height em state department computer mi parameter rna secondary condition hull union translate characterize convex cone origin satisfy computing condition np hard rna database separate base counting energy include energy u u pair discovery key role rna cell recently rna rna determination prediction due consume accurate number date whole genome biology throughput overfitte recently give systematic method inherent capability parameter iff one rna structure date ray free equivalently learnable iff accuracy set previously condition
partition energy depict experiment top plot percentage calculate prediction preserve algorithm trivially minimum free applicable leave department mi rna incorrect due inherent energy quantity temperature equilibrium reliably ensemble reliable function complexity partition rna give rna rna sparse parameter turn throughput biology genome rna essentially rna code category structural cell comparable play sophisticated cell rna protein include role cell rna medium protein rather molecular biology rna absence experimental rna rna rna problems rna bioinformatics rna rna interaction receive attention develop bind site incorrect inherent equilibrium derive boltzmann rather likely reliably predict structure reliable structure obstacle ensemble complexity probability rna rna interaction recent progress sparse prediction free function idea calculate upper function recent interact instead compute rna interaction thereby provide rna interaction structure rna secondary monte density secondary advantage approximate partition extensive obtain partition firstly monte e g gibbs markov chain deterministic cf dimension size demand rarely problem technique efficiently estimate partition scale approximation approach come convexity feasible hardness kl variational hand replace surrogate hence provide approximate inference likely collection major see broadly show benefit denote index refer nucleotide subsequence nucleotide nucleotide call rna rna secondary base pair arcs collection feasible throughout free detailed partition energy rna rna interaction pair nucleotide however non straight existence free incorporate energy let family absence pair euler constant every rna rna rna perturbation include simplify incorporation term rna rna rna interaction rewrite rna rna identically constant inside minimization whose inside minimum free prediction albeit perturb base perturbation incorporation perturbation prediction exploit add calculation base additionally energy carefully apply algorithm add handle perturb energy triangle property hold triangle affect expectation average experimental result number rna length need expectation run memory energy prediction
event write correct include restrict actually use discard arm discard mean exceed quantity select optimistic optimistic discard arm optimistic model discard apply arm coincide optimistic actually discard grouping obtain finally apply lem gap obtain final optimistic discard reduce optimistic discard discard discard whenever exploit expect well testing testing confidence view strategy arm step coincide incur regret mild exponent logarithmic term improvement obtain perturbation bound orthonormal perturbation transform tensor thm bind prove step lem prove inequality estimate lem lem main transformation begin diagonal decomposable ease exposition perturbation bound prove bound norm coincide bound assumption rely guarantee positive eigenvector also positive definite whenever arm strategy ji ji j grouping together lem correspond lem last event coincide proof lem lem arm discard episode round j j introduction rarely alternative setting paper estimation wide particular deal progress regard main empirical order prohibitive order moment algebra recover knowledge form polynomial dependency excess correlation alg idea dirichlet distribution dirichlet variant recover multi task step label come different task improve contextual contextual step observe scenario resemble arrive sequence draw observe interact bandit piece introduce slide change optimally tune w bad expect transfer switch transfer average switch surprising transfer whenever collect bad result surprising know switch carefully bad task nonetheless bad ht arm arm avg table report fig france university experience improve build reinforcement learn task notably bandit experience improve intelligence task transfer receive rl encourage scenario task online fashion rarely consider material clinical adaptive paper understand formal learner act learn bandit interesting help education student online site ad expect act set knowledge task learner reward whole extension reduce sec variant robust tensor sequential tensor estimate long least guarantee pair efficient bandit avoid ideal advance preliminary finding problem observe distribute exist arm arm conditional e conditionally third p define multilinear norm norm max episode pseudo regret obtain formally episode episode model update regret mean definition notice compatibility discard select optimal coherent ucb always arm optimistic compatible current incur worse optimal I similar bound literature set optimistic incur eq display regret discard optimistic actual never discard significantly reduce arm minimum sec improve performance ucb whose episode approach arm sample run run model ji ji I tr episode estimate second third moment eigenvector whiten alg compute ucb sub bandit episode compute mean bandit compute compatible unlike available mean compute return optimistic model might arm terminate estimate ki task accurately estimate reward episode kt estimate whiten mild assumption obtain estimate w w eigenvector column transformation complexity accuracy error moment prove grow decrease need eigenvalue eigenvalue high mean every constant comparison improve previous move dependency dependency implicit dependency order small whereas polynomially illustrate relie estimate accuracy estimate computable practice usually introduce episode define dominate dominate arm optimistic j discard long episode among optimistic model discard want j sec supplementary report I arm discard optimistic model remove active e potentially discard optimistic ready episode regret set episode realization episode transfer knowledge bias often nonetheless bias wrong bad phenomenon aspect guarantee bad never suffer transfer even contain uncertain bias suboptimal exploit since episode potential improvement thm focus potentially arm gap big optimistic reduce I j optimistic arm instance I episode optimistic independently improve cumulative regret draw eq r randomization realization arm episode immediately thm show episode dependency know task could much episode nonetheless discuss never task exploit b preliminary finding mab report fig sect supplementary useful small making difficult distinguish potentially exactly mean discard might get arm show report immediately remain uniform episode episode discuss compare illustrate advantage per episode three episode discuss significantly correspond dependent model average three supplementary derive move episode transfer task boundary beginning select available task acquire increasingly confirm case complexity see reach gap episode accurate reach much see thm fig report cumulative show outperform tend approach paper transfer multi armed bandit bandit show fully bandit regret moment never tend approach complexity bound show preliminary question discard correspond large gap although discard observation guarantee tradeoff exploitation guaranteed perform well episode previous episode although strategy worse regret episode preferable early episode improve episode fast trade resemble exploitation single suggest number
et robot whereas robot planning assume expensive reward relate surrogate include original motivation build library controller later fashion intend video direct highly gaussian discovery mutual criterion optimization originally intend sensor plan bayesian introduce robot movement additional criterion trade optimization setup robot author attract reinforcement reward stability loop bayesian use gradient strategy reinforcement suboptimal performance minima instead rely test author environment contact compute explore use library volume contact predict recent algorithm bayesian nonlinear design bandit many application implement easy library compatible language art contribution application orient modification include library able small rgb false frame side bayesian field bayesian hand bandit surrogate standard literature thus toolbox allow contribution speed optimization good operating computer engineering science might multimodal bad outcome global evaluation costly optimization want value search pointwise optimization sequence able explain decision consider evaluation extend description relies express q interpret decision require belong family case q improve consider actual response point prior bayes represent rewrite fact expectation distribution base optimization relate also output recent show certain reason error instability make learn labeling label point analogy bias graphic interactive bayesian share piece review optimum case long analogy classifier design replace represent potentially parametric eq optimality c loss field reinforcement bandit reward low production main evaluation extra error reinforcement learn target bandit minimize thus rate setup reward cost evaluation result analogous stochastic greedy classical bandit optimization point make connection fact bandit independent optimization express abuse name parametrization dynamic etc reinforcement replace seminal bayesian numerical analysis solve complex interpolation simple put set go formalize previous single applicability numerical previously section application minimum function compact input associate kernel assume target optimization update model base follow learn hyperparameter cross maximize loo maximum maximize likelihood likelihood present estimate posterior likelihood base function posteriori modify distribution might restrict width restrict bounded optimizer include parameter general elegant applicability seem aim reduce effort toolbox c windows mac os matlab toolbox highly among execution time represent process cost purely matrix incremental inversion besides guarantee computation iteration element appear exist invariant whole query second carefully rely create criterion abstract thus newly fully library also algebra initial bias library isotropic automatic relevance etc etc surrogate student etc ei low etc design combine like gp criterion movement penalty loop criterion learn hyperparameter interface design highly many already test operate window mac visual library compatibility old provide matlab code language function even simple computer image compatible language program usage toolbox optimize optimizer summary library interface
approach totally amongst demonstrate drawback show solid line take draw line imputation imputation datum imputation literature near miss near neighbor predict variable unfortunately ask deal million missing predict feature fit train predictor response part train impractical iterate set usually case distribution imputation random forest indicator response naive represent dependency dependency relationship perform recent network automatic ray feature depend object miss complete miss value handle case depend able ad hoc classify word feature option predict propose learn different overcome miss outperform method apply generate fidelity section summarize bn infer build bn complete bn probabilistic special latent mass latent graphical local dependency random directional undirected random process explain etc star periodic star exhibit periodic source long reliably behavior regular reveal behavior visual inspection situation indicate conditional dependency encode intuition status depend periodic variation indicate periodic relationship periodic light periodic pass inspection interpretation originally bayesian cause let magnitude represent dataset product parent bn parent indicate parent advantage factorization example bn factorize corresponding respective parent unobserved bn probable sum sum variable elimination continuous combination parent node parent mean gaussian linear learning scope know behind node calculus eliminate bayesian eliminate leave idea arc order preserve adjust describe methodology adjust network adapt show bn know learn subsection grow exponentially number node possible exhaustive search work random empty parent complete allow parent attempt third keep parent node stay structure datum calculate apply impose use firstly parent set number datum parent figure parent combination value parent value take derivation bn side bn tell learn know work parent parent parent parent jointly gaussian calculated matrix learn linear pa learn set linear value respect zero f parent root model learn attention learn structure incomplete structure guess learn basic incomplete parent begin incomplete datum miss distribution include write likelihood express expectation optimize unobserved step create current generating unobserved expect miss value continue iterate change parent like case learn bayesian miss iterate improve show network structure miss md complete learn bn union perform arc ii arc union incomplete complete criterion fill independent strong subsequent fill value probabilistic structure section miss infer value train rf popular base tree bagging problem belong appear learn literature end explanation find description building rf tree forest number set training call bagging bag take split create separate element one bag create go classification become value star first imputation accuracy imputation dataset fact set method model fortunately computational occur learn take miss mass also car car snr description imputation square true equivalent guess compare imputation mixture gaussian ccccc less imputation provide attain deep insight show bn among among node magnitude red node miss datum detect dependency relationship could car model feature training object percentage use percentage miss value create seven star star rr table star rr deal take distribution importance accuracy precision precision tp fp negative cross show accuracy propose class indicate detect star rr periodic variable precision evaluate probe encode useful classify star classifier create extract band contribution train improve get list train without new list find candidate candidate match refined candidate improve get candidate match candidate previous list candidate matching candidate deal testing automatic one consider feature observe increase less per acknowledgment ia cat development understand arc think variance sec reverse arc explain add arc parent part new adjust value variance node remove
vote decide user friend interest topic evaluate user cut list vote user activity training category category user training method comparative precision cognitive factor communication infinite scientific article topic movie read book act bottleneck available human mechanism guide become ever factor guide frank com media finite limit incoming message friend pay attention recommendation incorporate uniformly divide attention diffusion spread able accurate item analyze vote news able vote alternative motivated drastically social medium twitter facebook video new message medium site attempt specific follow grow share contribute create media interest accurately stream e news twitter share item create cascade idea network analyze cascade item interested incoming describe important attention recommend attention friend interest cognitive select process attention online interaction act web email effort brain capacity effort popularity meaningful attention twitter display friend sort friend long continue chance item user divide attention friend divide friend influential receive attention make adopt pay depend diffusion divide medium user motivate introduce voting news alternative attention account motivate predict social medium recommend adopt user interest adopt interest addition friend item friend adopt recommendation share online user share share share recommend twitter voting notion social item whose social link interest social recommendation attention limited friend decide limited continue context recommendation limited friend recommend limit interest introduce lda salient element limited user recommendation consist interest friend interest attention item user friend interest graphical topic global item denote friend whose similarly item profile topic user friend interest interest capture finally process cc dirichlet generate generate dirichlet user choose multinomial pay attention profile interest specific inference equation collapse since summation denominator construct sample sample currently algorithm conditional probabilistic gibbs number item exclude assignment assignment exclude current assignment range index item topic assignment exclude current attention exclude assignment attention second allow learn interest account limited interest adopt global perspective set user user accord social network period user network special item source friend assign determines friend day attention parameter friend correlate friend follow intuitively item fraction budget proportional friend follow v share item cascade user budget check match interest cascade start seed adopt share share replacement allow friend item choose adopt share item vary synthetic able interest actual jensen indexing interest two one interest lda user item topic document run accordance generate varied interest interest adopt variety interest tendency distinguish interest interest lda lda cause pay friend subset friend low case topic value real find user follow user vote collect thousand front page evaluate dataset voting history user front page vote eight business vote user period vote collect replace business news business front page visible queue vote become examine vote front page mainly friend recommendation dataset k select result six topic implication learn consequence
variable intuitive independence equal infer appendix bivariate parametric copula family copula solution copula construction simply copula copula great flexibility bivariate copula belong literature regular factorization copula conditional copula form set undirected copula identify factorization tree sequentially edge associate index condition conditioning infer span infer share edge condition b style em anchor mm north edge anchor west edge node anchor south south anchor south cm factorization conditional density factor bivariate copula value cdf specify follow product bivariate conditional tree span highlighted bold left conditioning span tree graph edge pair share condition conditioning constraint build span condition conditioning constraint result factorization hierarchy bivariate copula span edge copula select tree directly relate weight amount measure span pairwise correspond copula deep conditioning complete construct full hierarchy may construct ignore copula density factorization copula constant pruning assume ignore copula equation conditional deep hierarchy tree copula marginal computation appear copula previous recursive jk de de de derivative refer still bivariate solution copula copula construct unconditional copula disadvantage simplifying construct bivariate copula dp xy ep frank closed pseudo address lack dependency copulas spirit one single handle consequently adjust linear v z neighborhood kernel intercept adjust leave one validation single make importantly cloud weather copula dependencie gps benchmark simplify dependency bivariate copulas ii would account pseudo process use hyper parameter tune ep likelihood kernel run leave validation search range simplify bivariate copula building extension among family bivariate copula straightforward family ep marginal scalar generative process first uniformly second bivariate good copula empirical cumulative level approximation generate well approximate cm generate run set dataset describe table copula superior often get dataset simplify stock table include outperform percent achieve interesting correlation concentration chemical cd cr cn chemical element co ti total water co cloud weather stock blue region weather pm return copula vary return major world stock index american chinese uci dataset repository copula specify copula conditional bivariate copula ignore construct world avoid world obtain well method state alternative acknowledgement equally helpful limited bivariate copula bivariate gaussian cdf bivariate marginal one quantile standard pdf derivative separately copula link ease dimensional copula bivariate copula simplify common bivariate copula relax dataset dependency copula copula becoming describe possibly complicated curse dimensionality copula simplify separate model easy univariate copula difficult require dependence exist copula family copula pair copula construction decompose hierarchy copula deeply dependency dependence parametric copula building block impact dependency likely specific thorough ignore dependency reasonably accurate indicate develop copula scalar technique arbitrary parametric bivariate copula
problem weakly hence surely remain problem feasible tail union bounding imply hence eventually surely obtain hence coincide far ny eq far proceed section next prove I fact obtain per theorem far bound last whereby replace tw tail supremum vanish let f assumption first hand last let mr rgb rgb rgb rgb rgb rgb pt plus minus proposition theorem consequence remark claim replica fit high procedure quantify estimate classical confidence propose confidence confidence interval hypothesis vanish de biased new improve special structure design throughput genomic publicly widely recognize problem require filtering successful develop problem suitably estimator necessity certain price impossible characterize exact large classical confidence however analogous procedure statistic develop high dimensional salient interval assumption respect area clarity presentation preliminary report also discuss generalization linear regularize estimation give pair standard letting denote classic method ordinary square yield ol ty ols directly ol interval resort successful reconstruction penalty arbitrarily omit clear namely ny lasso set ni pm everywhere else feasible come characterization tractable construct confidence property dimension discussion de de biased formula ty empirical covariance construct complete analogy instance e solve convex aim optimize theorem control control construct n x suggest constant covariance asymptotic uncorrelated design van covariance least technical bias already early need procedure instead term consequence choice apply structure contrast early compatibility high one presentation n ty coherence effective design generalize make bound bias cf theorem bias standard apply distributional hypothesis marginal approximately know standard low design compare cf namely bound constant away always central triangular simulation top open successful weak direction barrier arises rapidly grow sparse regression many idea limit investigate property consistency necessity instead depth quantify significance high far zhang zhang b testing procedure eigenvalue paper achieve significance show convenience suboptimal overall design achieve et irrelevant test address assume current response relate regressor vector main regressor nuisance parameter interest attain semi regressor sparsity closely relate sparse covariance regressor resample perturbation idea stochastically version call limit regularize latter perturb sample finally minimizer publication aware introduce definition notation let denote submatrix form column resp denote submatrix contain likewise restriction index shorthand maximum write entry position nonzero e cdf np normally whereby result sub random characterization subsection also estimator subsection estimator convenient begin broad notational shall argument require clarity quality course characterize pair min coherence terminology column maximum distinct column know context zhang sake simplicity coherence orthogonal emphasize classical coherence instance orthogonal slight emphasize deterministic design satisfies compatibility condition error zero maximum bound compatibility coherence former assume nearly establish hold subgaussian subgaussian ns max min min na proof crucially simplify somewhat restrict imply compatibility putting set interval omit explicit readily term establishe program prove residual scale consistent estimator sequence design matrix independent subgaussian subgaussian linear enough dependence last several make list reference consistency prove provide addition thus satisfy scale lemma straightforward appendix view straightforward construct valid interval namely significance let enough asymptotically valid namely number select index test construct measure significance error false type ii significance achieve achieve nontrivial indistinguishable minimax test index denote design define algorithm finally noise define integer monotone increase power randomly probability achieve scheme power level reader convenience design type error eqs power oracle oracle output computing reduce problem emphasize apply establishes negative upper test define eq sparsity follow true asymptotic efficiency apply increase assumption natural ratio test augment statistical interested perform simple stay omit far allow confidence lemma interval projection explicitly borel state valid inference aggregate attract considerable e design want achieve control trick let enough finally consistent estimator sense test defer interval value corollary broad write surely modify show validity suppose obtain significance hypothesis prove fix symmetric regard uniformly configuration measurement scale theorem realization confidence interval average average individual realization configuration report interval realization configuration sake interval width black confidence summarize false achieve configuration error achieve make error always propose contrast method positive splitting type conservative however come ideal testing allow control level positive rate test splitting ridge type show quantile versus quantile normal slope confirm regard entry plot cdf outside entry uniformly ridge type fp fp tp fp rate tp width pdf quantiles throughput publicly gene response logarithm production rate logarithm gene covariate production rate package similar previous scale equation intercept construct adjust find significant gene namely significance significance conservative produce large empirical value normal quantile linear function vector leave precisely instead last event complement theorem
day style align every node align grid style cm height coordinate coordinate popular gd use gd place sgd scheme two sgd scheme rate acceleration come square variant still drift discuss sub xlabel iteration gd ylabel pos pos style gray restrict blue col sep log ex xlabel ylabel grid grid gray index col sep xlabel sag ylabel pos pos style gray domain green x sep space various algorithm stepsize stepsize stepsize l sigma generate martingale difference z il interested iterate instant instant iterate jensen schwarz deduce expectation finally invoke concentration sum martingale inequality every lipschitz note lipschitz obtain proposition error drift variance f martingale nz extract martingale process ni kf inequality term simplify eq mean martingale put simplify high fixing rate choice n n stepsize bind error theorem theorem bound bound fourth term drift error q claim follow pt often study improve place efficiently presence drift couple attractive implementation strong need theoretical solution prove bandit bad convexity guarantee investigate adaptive sgd news news recommendation yahoo front page consistently track setting choose mean diagram illustrate ordinary square find computationally intensive part high least well evolve complexity approach computed algorithm give traditional descent gd purpose sample pose difficulty gd outline solution classic operate sequence choose give refer gd online analyse square instant sgd require track replace costly inversion efficient gd drift eigenvalue ordinary effect drift provide gd instant theorem essential guarantee gd subroutine matrix high cope situation eigenvalue algorithm propose henceforth refer gd online track operate similar except parameter update unlike gd theory demonstrate gd track solution gd classic solver gain subroutine bandit bandit reward parameter subset agent reward achieve appear improve short possible first bandit iterate ol exploration exploitation exploitation phase act exploration small bound regret gd subroutine improvement order regret white rectangle width em draw fill white cm coordinate thin auto node name fill controller green system controller output measurement node node pos near end white rectangle height cm thin auto fill blue right cm label align right sample gd right cm consider algorithm efficient begin replace gd state design situation agent optimistic ucb reward ucb gd devise gd procedure ucb arm result improvement observe gd iterate variant consistently track runtime gain sgd bound expectation sgd scheme machine learn propose regret convert convex sa batch refer none scheme track least grow batch scheme present gd outline early gd track estimate fig uniform pass randomly make boundedness initial small generally scheme square initially hence reasonably hold approximation gd probability expand initial error martingale sampling drift ni martingale detail initial error work sgd constants form rule step sequence drift eq square adapt ball third decompose g il exact update derive constant martingale concentration complete contain specify exponentially drift asymptotically sgd scheme dimension indirect convexity constant example bandit initial convexity dependence ensure knowledge bandit average arrive optimal independent choice strongly smooth bandit well set exploit estimate grow length strong guarantee propose replace least gd whereas phase incur incur input get require assumption function unit result gd b gd I get dm rest follow perhaps obvious offline size arbitrary amount adaptively gd attempt iterate lead
perhaps sensible onto pca close two batch experiment experiment accelerate point draw subgradient hour realization limit simple less admm assign connectivity parameter choose small connectivity assumption superiority least circumstance trace subgradient criterion note stop centroid iterate achieve primal loss less result second box root three scale admm three possess constant median subgradient admm fair subgradient admm overhead fast computation time minute hundred subgradient incur batch experiment retain except assignment weight step size result run second square root short run incorporation easier competitive even speed second minute admm subgradient introduce splitting splitting perspective encourage path permit centroid penalty invoke readily onto unit ball proximal operator quantify admm circle iteratively convex admm performance non block accomplished feature problem ensure proximal reason strong convexity computed edge without prior complicated accelerate nonetheless admm quickly admm strategy incur overhead event practical question al event identify case storage complexity cluster near edge cluster find neighbor conjecture would suffer neighbor computation approximately might fewer serve warm perspective centroid cluster accomplish add induce norm centroid raise except leave suggest principled assess quality cluster assignment resample investigation material proof supplement package implement available website chi helpful suggestion research support united public grant gm however sometimes drastically relaxation centroid another global convex alternate direction multiplier minimization formulation unified appear minor complexity significantly efficient alternate alternate multiplier mean relaxation fundamental yet price generalize effort cluster formulate point column center point difference penalty arbitrary fused use definition recover fused correspond connect separate component style right right bend cluster center begin cluster cluster possesse cluster certain value pass separate many centroid unless admit fast guarantee global minimizer contrast classic classical greedy suboptimal agglomerative problem entire agglomerative computationally demand suboptimal minima relaxation perform trace appeal globally tractable main new application regression little solve fact introduce dedicated use path note dedicated formulation distinct encounter norm design norm conjunction active convex frobenius polytope frank wolfe approach framework objective arbitrary solve problem alternate admm efficient introduce contribute way combine admm convex theoretically cluster give extra need also computational connectivity enable enable rigorously quantify efficiency proof intuitive path tie solely minimization regardless minimum c suggest choice enhance employ near requirement linearly cluster book review relaxation agglomerative classical come report fast agglomerative cluster assignment less probabilistic mixture assign linearly valuable demonstrate effectively merge although path cluster persistent need determining throughout scalar denote letter derivation easy matrix adopt letter upper paper solution path theoretically admm discuss acceleration cluster nice weight material continuously continuously weight employ homotopy find problem grid previous value warm rigorous example shown intuitively expect satisfy connect minimize column equal path guarantee agglomerative case uniform agglomerative example centroid frequently describe compute truly convex tackle minimization shrinkage convex criterion equivalent index centroid set variable centroid used attack alm alm problem include minimization alm solve equivalent impose penalty deviation feasible coincide quadratic term find minimizer constrain identify saddle alm multiplier strongly therefore ascent alm unfortunately jointly difficult admm adopt simplify subproblem alm update minimize slightly augmentation accomplish later see pay cluster overall block descent simplify update proximal map call map unique whenever norm explicit solution vector component group map require simplex explicit algorithm project unit projection make operation cccc simplex augment edge update l note condition consist bit l give system l set edge l l duality iterate mf optimally optimality short iterate terminate form make trivial evaluate feasible quantity l compute converge two admm converge broad guarantee provide convex ensure refer convex convex modulus feasible lagrange multipli constraint verify cluster dual material assumption sufficient condition ensure gradient proposition generate satisfy twice insight edge incidence coincide eigenvalue practice dense demonstrate admm algorithm prove assumption proper lagrangian saddle f mf primal reference note strictly next admm iterate meet lagrangian possess saddle global limit bound unbounded pass along subsequence contradict limit guarantee continuous function accord difference contradict unless f admit acceleration little computational effectively nesterov admm initialize l l sequel complexity specific sparsity problem wish duality single require vector cost cost operation finally duality gap iteration operation norm estimation nesterov variant consequently accelerate specifically asymptotic bound ascent per acceleration effort attain duality np limit node bad point near connectivity quadratic restriction neighbor count update outline argument update operation establish require suboptimal algorithm update require bad together situation improve consider alternative node correspond zero demand recall l definite course cache cholesky repeat admm update cholesky triangular since row amount per grow either storage regardless weight dramatically path versa factor gaussian distant uniform note positive act similarly sensitivity read centroid regularization determine assignment admm assignment store running terminate graph induce graph identify place connect graph synthetic choice quality solution limit al neighbor et
set presence fundamental limit heuristic signal middle eigen might suboptimal invert approximation yield large eigenvalue limit spectrum precisely multi limit noise compactly transform increase figure thus long principal eigen conversely eigen similarly eigen middle eigen gap eigen gap indistinguishable gap gap eigen depict weak eigen gap eigen gap informative eigen eigen gap equation eigenvector eigenvector concentrate interval successive large middle g imply vanish eigenvector associate eigen employ would eigenvalue depict interval notice eigen principal eigen extend whenever principal component uninformative employ middle eigenvector informative uninformative stay informative eigenvector uninformative middle eigenvector informative versa determine structure spectrum summarize gap iff asymptotically eigenvector associate principal exhibit eigen gap eigenvector uninformative eigen gap eigenvalue eigenvector whenever hermitian order eq surely compactly support q support small integer let hermitian symmetric invariant orthogonal unitary throughout sure hermitian lastly space perturbation give follow eigen phase behavior functional take prove transform value eigenvalue exhibit eigen gap identify eigenvector throughout nc prove accounting eigenvalue unit prove follow transition occur exponent decay eigen base detection eigen whenever principal eigen middle gap eigen informative let empirical eq measure compactly non support ip small real bi bi bi invariant distribution unitary right leave right unitary matrix leave left get vector singular phase singular exhibit prove accounting ingredient adopt outline take perturbation state furthermore asymptotic follow accounting denote prove accounting limit eigen gap fail whenever eigen suboptimal middle eigen gap reveal principal gap eigen gap theorem insight rank analog play determine singular noise datum show eigen justify eigen support interval principal informative snr large moderate snr middle component long informative eigen gap associate inference improve inference eigen spectrum exhibit identify eigen model eq matrix measurement wishart distribute model statistical signal application first inferential see eigen spectrum eigenvalue spectrum support interval form spectrum square phase figure value u clearly snr regime middle informative even phase transition occur theoretically desire singular informative great significant spatial variation beyond might conversely eigen range vanish singular eigen gap hypothesis establish informative associate eigen exhibit cm value plus inferential detecting embed signal great great variation often justify though plus noise good underlie take principle spectrum principal middle value informative support interval spectrum informative proper justification use consider involve heterogeneous mixture result suboptimal inference informative regime uninformative simulation signal relative infer estimate principal principal singular work great variation reflect content equip direction tackle latent rule assume simple modification signal approach theoretic problem employ lead component yield signal matrix give work plot model I variance detection work subject product n singular singular left plot informative employ informative correlated embed low rank manner facilitate describe correlated arise counter plot datum model noise matrix multivariate produce code list reader sigma diag reflect value separate middle principal capture greatest would fail signal product u I singular singular plot informative component employ informative precede informative middle sometimes spectrum norm signal statement underlie contradiction principal signal associate practitioner collaborative bioinformatic compute entire singular efficient big application researcher often invoke pca justification arguably use principal starting procedure really already component principal component may middle latter lead consideration lead road support standard derive involve section middle formalize produce exhibit look portion noise separate portion plot leave classification consider begin examine related eigenvalue signal eigen subscript notational brevity invariant unitary distribute eigenvalue utilize eigenvalue simple argument zero eigenvalue eigenvalue via eigenvector satisfie relationship z notice equation begin informative component picture far eigenvalue satisfy expression insight eigenvalue relate expression insight horizontal large eigenvalue denote function place non denote sure small converge support limit continuous spectrum successive eigenvalue zero reasoning picture say lead eigenvalue retain eigenvalue amount justify signal vector unit high inverting equation large recall noise compactly support connected tend long large edge
distribution covariance spatial smoothness fluctuation parametrization covariance orientation axis depend assumption harmonic n sphere harmonic basis onto harmonic subspace band harmonic parameter basis signal unknown another create require infer describe logarithmic incorporate power span element uniform spectral parameter respectively initially independent gamma pz z limit lead stable far variability spectrum spectra drastically mode shape causal might smoothness logarithmic specify deviation spectrum law slope per smoothness give consider flexible handle smoothness suppose distant source observer actual extent negligible phenomena contribution neighboring source approximation two close observational might huge spatial source negligible statistically contribution source spatial locality signal suppose single position discretization assumption depend spatially functional prior pixel source identifiable necessity complicated determination construction quantity exponential strongly favor entropy also bad likelihood regard log cf follow motivate say universe distribution would apparent reduce slope plain power furthermore galaxy light impose cut onto gamma prior derive inverse latter responsible vanish analogy universal demand add merged pixel prior still hierarchy suggest signal reconstruct nuisance need reconstruct contribution layer five scalar namely incorporation scalar dramatically discuss reasonable value scalar theoretical power spectra accordingly choice investigate style circle minimum thin text draw draw draw draw south u model describe yield well ideally uncertainty eqs point logarithmic however complexity posterior signal field allow infer involve huge space nevertheless elaborate low cost minimize spirit bayesian fidelity application sec coincide single suitable estimator use mode gaussian define eqs minimize hamiltonian derivative implicit equation hamiltonian see division vanish count cf eqs partly assign tendency derivative hamiltonian around approximate uncertainty form explicitly appendix filter spectrum derive hamiltonian formula accordance scale noise ratio drop become capture approximate posterior posterior investigate value field point field information signal field tuple separate represent effective additional formula account set eq convenient quantify appendix gibbs free equivalent kullback favor logarithmic energy plug hamiltonian evaluating thereby expand accord carry properly compare vanishe include power computationally feasible shall hereafter correspond consequence correlation equal read eq gibbs energy fitness formula take derivative respect formula comparison filter formula covariance either derivative derivative respect covariance close explicitly logarithmic suppose retrieve variational detailed discuss derivation discuss yield correction positive contribute logarithmic spectrum maximize calculation correction filter formula correction detail implication investigation perform reason choose expectation accordance respectively term symmetric ef complex many frame skewness whereby unity superiority covariance ordinary square root latter image scale l pixel show exposure mask bottom panel reconstruction different reconstruction figure exp exp gibbs ds exp ds exp reconstruct gibbs difference reconstruction scale gibbs ds ds top panel original reconstruction reconstruction panel reconstruction cc image dash line black dotted spectra line second corrections panel b ccc scale exp image u uncertainty top panel gray scale contour panel approach shoot gray contour bar sets filter second discard describe great reference future illustrate fig represent field resolution pixel include convolution like roughly mask virtual top fig gibb visible denoise field cf well noise instrumental remove present decompose gibb seem slightly well define euclidean error purpose normalize convolution approach incorporation correction slightly treatment gibbs outperform map solution figure illustrate reconstruction agree surprising one solution strong overfitte former intensity dominate spectrum give simulation spectrum suppose prior apparent spectra harmonic mode spectra correspond physical distance like virtual lack reconstruct indicate assign feature spatial component solely cause distinction basis spatial assigning assign like component reach boundary consideration noise map gibbs subtle give involve reconstruct spectrum correction formula gibbs consider roughly map tend high power seems cause noise signal influence influence reconstruction spectrum choice fluctuation order well track concrete point reconstruct field agreement locate less precise intensity although exceed expect shot overfitte low eqs deviation shoot fig reasonable high vice versa uncertainty poor curvature describe uncertainty source sufficiently landscape potential lead take possibility uncertainty simplification correction improvement argument reconstruction fig order reflect reconstruction carry accordingly parameter total error define stress change drastically partly magnitude moderate note tendency like capable denoise former perform map sign performance seem acceptable scheme combination denoise harmonic spectrum single shot algorithm reconstruct field capability foundation embed comprise assumption assume multivariate reconstruct spectra smoothness assume spatially inverse imply incorporation instrumental denoise exploit description five none driving discuss free estimator example yield equivalently excellent slightly solution consider l full regularization price prefer concrete computational algorithm carry regardless example energy analyze successfully decompose analysis yield reconstruction spectrum localization source determination intensity wide field concrete energy ray consider author regard release furthermore thank medium low publication package signal source inverse gamma still obey gamma law independent discretization continuous slope unchanged refine adapt resolution merge uncertainty associate mean accord hessian covariance field signal covariance introduce sec read covariance concrete correction involve couple inverse hessian describe curvature uncertainty speak valid potential approach derivative energy covariance read already lack order correction reason use inference problem handle fitness quantify mathematically theoretical minimal axiom demand locality coordinate invariance system free q difference derivation base temperature imply leibler divergence gibbs energy equivalent allow parametrize respect theoretical applie concept method enable field address mean variational onto within within inference demonstrate sect rather degree vanish gibbs energy yield eq define solution approximate posterior correct normalization I integrate marginalization integration might behaved comparison marginalization result solution since approximated solely style circle size thin em center thin draw xx center text width dash variational demonstrate stand describe parametrization signal posterior derive find solution logarithmic like favor clarity read h hamiltonian quadratic posterior covariance suffice e introduce approximation change causal depict posterior hamiltonian coupling trace describe yield agreement exact power denoise observation universit free ensure applicability fidelity realistic count show test decompose point signal respective emission raw image perfect suffer shoot instrumental elaborate denoise deconvolution subject severe cause difficulty discrimination noise challenge furthermore incomplete survey complex instrumental leave might exhibit gap spread vanish superposition commonly class source source smoothly correlation point contrary perfectly appear distinguish source background contribution intermediate sometimes classify extend arises cause ill pose without heuristic denoise deconvolution decomposition simple setting prominent identify source popularity fit source commonly deconvolution assume source emission optimally real use scale image b pdf c clean improve also angular spectra relation signal contribution probabilistic incorporation assumption often initial attempt reconstruct though sparse prove successful perform denoise task setting example simulate multi deconvolution background furthermore decompose simulated statistic regime deconvolution regularize square scheme tuning capable emission rely filter template filter exploit source position successfully technique mixture spline spline aim like equally propose framework field incorporate prior fundamentally point reflect prior correlation crucial signal count regime become low target simultaneous denoise task harmonic spectrum component infer contribute equally count information incorporate assumption fundamentally model field represent original signal appear physical infinitely degree position space computational need course signal except like source determination method detail regard implication performance reconstruction coordinate angle order represent permit topology prototype code sphere structure sec discuss I solve denoise
hence proposition let later r e eq definition remark thresholde clean conjecture proposition corollary reconstruct assumption component study regime influential fail thresholding confirm author rigorously prove succeed new regime consider vector estimate identically quantify interested limit rank simplify drop subscript inconsistent v phenomenon attract considerable motivated effort influential signal basis without basis propose follow ik k principal result matrix formalize require belong consider strict sparsity magnitude study within diagonal thresholding recover improvement denote pca diagonal thresholding achieve scale support identify soon theoretic year effort devote develop practical promising programming carry sdp satisfactory less sdp constant picture remarkable result result polynomial certain conjecture plant demonstrate consideration exhaustive support rigorous reconstruct thresholding succeed picture paper address provide positively algorithm proceed precise definition technical form empirical entry modulus suitably choose compute thresholded denote entry support briefly outside thresholding propose turn relate discuss rest organize follow full ease light drive simulation supplement section empirical p q ps convenience hereafter distribute accord number spike treat strength denote throughout entry assume detailed basic intuition subscript state splitting compute respective matrix along part consistent support first obtain estimate first dominate instance reduce must let w moment kk possible remove reason thresholde z denoising estimation fall short goal classical error wise interested result bounding operator norm soft thresholding affine expect decomposition approximately operator norm perturbation easy entry entry independent decay probability approximately norm consequently perturbation obtain intuition provide component use r discussion product attract probability study entry suffice rescale factor taylor instead non dependent concern limit empirical yield upper method bind give follow net develop bound maximum z continuous estimate flip estimator bring second exploit spike integer material support spike rank obviously standard union correctly individual support pose additional difficulty recover sign support technique support difficulty require spike roughly go avoid assumption question make sdp application help exposition challenging aspect indeed define show defer give v covariance eqs recover support rank covariance number conceptual improvement present strictly sparse table objective factor thresholding appear converge throughout region monotonically confirm knowledge succeed diagonal thresholding require plot compute appear probability curve thresholding appear decrease indicate thresholding become large success indicate sharp practical applicability parameter describe principled parameter purely variance principle proceed pn absolute argument snr reasonable ignore n rescale previous let constant appear well rely fact transform eigenvector gaussian vector q different share drive support size figure pca thresholding peak domain simulation employ datum version experiment box supplement experiment perform figure respectively covariance thresholding parameter thresholding curve dot notation preliminary vector low letter represent letter ne e ne ne respect statement thereby specific sphere definition net net number net every may cardinality net finite net dimension net symmetric net set various normal probability value lipschitz coincide f nf measure follow call I normal theorem hold bound hand follow q k v handle prove without treat way form z firstly cauchy schwarz nt nj estimate q use q theorem exactly define assume identical I I include union support g term component three term consequence preserve least enough least take let third proof proposition prop diag section result hold via union size bound choose proceed support hence thesis follow directly triangle complete lemma subsection support recall obtain outside rewrite eq g lemma large embed support net argument denote outside bind aa large denote favorable
concept sparfa capable generate purely drive manual labeling question enable pls automatically learner low knowledge level experiment indicate sparfa outperform sparfa vote bayesian inference descent computationally practice enable addition impose sparfa negativity sparsity framework topic method none however joint response start sparfa detail sparfa response question text generate association keyword sparfa consist learner answer involve column let question student relationship model correspond response learner incorrect respectively logit map incomplete response large value reliability response simplicity exposition address fundamental account scenario impose association assume concept typical education scenario negativity characterize particular concept answer question sparfa utilize post pre tag infer associate directly question correspond size vocabulary question entry model etc exclude vocabulary word occurrence model column characterize inspire topic question concept imply question concept rely fista detail subproblem fix subproblem subproblem fista gradient element correspond subproblem optimize separable fista analogous smooth represent element operation introduce optimize throughout sparfa sparfa cm concept energy water water percentage water water heat box represent question circle concept thick association concept arithmetic quadratic simplify expression equation concept simplify inequality algebra test thick line efficacy sparfa course high school algebra amazon crowdsource learner answer question question pair observe text word exclude common algebra user answer tag regularization together sparfa cross validation py slightly sparfa sparfa algebra albeit improvement reveal additional underlying question along characterize sparfa sparfa relate concept concept sparfa capable automatically interpretable summary concept sparfa top extend sparfa jointly associate purely datum manual assignment tag concept keyword extract question text e l z v c b l h corollary example n h sparfa com development scale learning recently sparfa knowledge latent content sparfa value question interpret latent concept sparfa post utilize tag keyword available tag question answer feedback approach interpretability generating post improve sparfa scale demonstrate efficacy real traditional education fit regardless learner recent advance enable provide feedback learner potential education building pls learner interaction material question feedback material question document sparfa component sparfa automatically iii solely graph extract sparfa pls learner reveal course original sparfa
rule swap experiment mcmc prediction iteration setup fix adapt iteration mcmc runtime order fast achieve record serial computation additional speedup drop increase misspecification align tree inference algorithm model lead sensitivity systematically vary qualitatively left column runtime proposal smc approximation gold fraction budget final competitive classic cart comparison could demonstrate yield similar process cart implementation two minimum high accuracy comparable low laplacian cart long highly cart plot cart bar log bayesian tree framework cart cart smc benefits fraction inference tree sequential monte classic resample guide tree growth especially state counterpart sophisticated lead smc expand way proposal intensive overall input balance exploration devise proposal getting explain contrast decision forest bag interpret explain tree significant important additive continue classic tree undesirable exchangeability incoherent stream alternative whose depend acknowledgment like david helpful discussion feedback international fellowship college bl acknowledge foundation I I densitie x I normalize iw w loop result bottom vs column vs circle dash represent proposal h accuracy marginal proposal particle increase marginal converge expect particle circle square runtime cart hyper consistently outperform result vary hyper change variant fix figure display column vs runtime circle proposal compare result text observe trend qualitatively text smc offer vs tradeoff well runtime predictive text filter bayesian decision prior show classic learning algorithm produce approximation modification markov mcmc monte smc behavior speed empirically fast tradeoff algorithm near art despite predictive typically specify hierarchical block input predict classical decision learn top cart learn combination decision forest method like recently cast problem inference place model common node indexing family gaussians conditionally bayesian interpret hand exact improve decision long local modification bias structure probability stand rapidly decision greedy success one article adaptation propose smc sampling classical pruning prevent cut growth tree focus attention tree datum produce tree posterior exist smc produce exact fast organize bayesian precisely smc detail test produce approximation conclude discussion exist probabilistic mapping axis block determine represent whole child two cut represent bottom red star circle represent block refer extent extent node dimension trivial variation rooted strictly tree finite root string internal exactly child child leave child node internal pp denote location intuition although choose dependence notational simplicity latent tree np np focus categorical take value correspond conditionally np dirichlet final piece prior comparison exist tree input effect exchangeability informally split grow child internal stop leave describe generative precisely capture tree trivial produce choose leaf leaf stop future leaf leave stop expansion rise I markov carlo online smc next grow cut choose stage identical node choose cut would depend train informally lead stage kernel filter proposal kernel call return section alternative proposal recall input uniformly sampling cut compute smc produce normalization latent joint deterministic probability property justify proposal proposal proposal p np unique denote number node smc p proposal per particle smc addition bayesian cart non uci repository recognition focus mainly cancer choose illustrate scalability predefine test contain approximately apply smc processing set hyperparameter configuration whose effective size ess stage never reach smc proposal proposal choose expansion single consider expansion per stage singleton nod depth expansion evaluate select marginal expand node low marginal likelihood perform first multinomial experiment systematic resample runtime average number initialization deviation show summary observe pt pt expansion resample potentially every decision immediately compare stop quite resample retain particle node stop expansion stop e require tree stop another expansion number resample suffer importance expansion strategy proposal proposal account resample rest plot
may margin present notation binary shrinkage may maximum wolfe effective demand price plot test perform wolfe search course example logistic show loss perform establish rate margin maximize margin mean without separability upon boost instance decomposition subset easy easy alone margin measure heavily appear guarantee numerous place reflect structure encode follow weight wise example margin negative margin hard exist aforementioned risk boost potentially parameter binary suffice instance attain minimizer improve state provide weighting margin minimizer improve margin consequently method margin neither shrinkage give size core close margin maximize boost discuss specific without iterate aforementione soft originally control try margin worth rate margin manuscript immediately raise number question perhaps margin efficacy margin certainly logistic tight analysis logistic question lastly show threshold right smaller reveal roughly dynamical system small acknowledgement helpful numerous insight suggest unconstraine support nsf grant proof concavity check stage taylor expansion eq e x e x e derive due satisfying shrinkage possibly unbounde nonempty nonempty interior since bound quadratic lie second exceed quadratic first proof statement guarantee wolfe line result instead directly prove later satisfie fix w mc la expression note finish follow choice statement size wolfe expression demonstrate wolfe see wolfe give statement time term margin low remainder subsection later material satisfy q concave bind whereby simplify term replacement consequently whereby size additionally replace term give proof note next q particular expression plug generic rest whereby grant desire arbitrarily close whereby extensively study adaboost margin suppose cf note whereby next simplify attention numerator finish imply combine h recall term usefulness capture iff element last q expression suppose binary adopting shorthand precede may vary whereby repeat finish convergence improve exactly line search proof adjust cover size theorem thing loss check margin q sketch proof specialized wolfe search constant carry compact cube iterate weak strict convexity grant modulus l la finish helpful setting size sketch la result old inequality e large margin split wolfe generalization quadratic size tt sketch quadratic line search give well due term however grant plug index achieve bad optimal rearrange finish give specialize margin exceed sketch let wolfe condition apply eq whereby remainder proceed cf handle grant existence large almost problem reduce consideration satisfy binary second follow concavity sketch proof type result rearrange manuscript adaboost immediate variant size shrinkage gradient variety intuition search hold loss similar loss notably logistic boost aggregate accurate efficacy boost seek margin generalization since attain margin make method carry guarantee equivalently optima scaling approximate separate deriving margin manuscript margin practice scale shrinkage scheme effective adopt manuscript guarantee introduction work function provide generally dominant study manuscript shrinkage risk margin subsection compare match demonstrate certain still margin separable manuscript proof supplementary line search boost first give albeit question rate appear literature come rate step maximize search without receive extensive survey literature amongst result concrete adaboost suboptimal margin margin primary exhibit maximization refer extensive summary manuscript match greedy distinction algorithmic minimization unchanged exist widely shrinkage manuscript concern convergence order rely heavily scheme risk curvature manuscript margin appear method bad unfortunately instance empirical weak learner assume specifically vector hx instance consequently regressor thresholde margin motivation problem advantage use form say gap primal primal separability subsection convergence method exhibit boost shrinkage iterate basic proof prove iterate factor indeed first quadratic implicitly give relative curvature analysis reason parameter dependence potentially mean quite bad choose perhaps convergence eventually stay refined picture constant eventually next wolfe analysis heavily rely denominator due extra wolfe specifically natural wolfe within statement treatment pattern fast correspondence unconstraine boost exist contrast convergence rate empirical risk condition make constant depend heavily upon unnecessary extreme
leibler bethe actually exact belief bp wish include constraint belief q convert real value study seek leibl minimization constraint constraint q stationary play enforce compatibility compatibility precede usual bp lead replace bp iterative fitting sum variant propagation classic belief factor compute send go cycle graph fix act mirror emphasize value successive drawback assume necessary bp behavior seem quite present node cut variable circle black circle shape fill draw circle rectangle fa node draw shape rectangle fa edge edge draw circle fill black black circle shape circle node shape rectangle shape draw draw shape circle fill draw shape black pp draw shape pp edge edge proposition converge two gray contain node however part still cut follow describe converge form leave understand describe paper consider increase rough road traffic repeat outcome make allow performance choice encode observe varie decode inverse copula case precision inverse r r ltb ltb ltb r ltb ltb variables edge style double double bend style thick package terminal load graphic terminal graphic macro ltb lt lt lt lt lt lt lt ltb lt lt lt lt r ltb ltb ltb ltb r predictor associate road rough description city network basically road road impact road correlation come road always specific loop road always begin precision encode cdf incomplete matrix choice cdf decode seem much loose efficiency encode difficult discard encode marginal ise e determining present optimistic road binary description remain unchanged particular decode em estimation message obtain unique derivative bound conclude prove fix would converge discard case trivial least one exist root root cross conclude converge recurrence apply case study fact much soon get node belief proposition trivial cutting conclude prove lemma well update lemma tree leave simply leave affect since send come back leave message integrate propose minimal variable incomplete cumulative application large partially observe variable objective scalability time encode system directly description rough observable traffic road rely pass demonstrate field propagation soft scale complex system different situation communication social evolve demand limited extent considerable see kalman particle exist limited scalability road reconstruction car real value time alternatively exploit spatial correlation multivariate restrictive encoding calibration scale road sensor adapt segment part operational explore possibility historical consume prediction available imply choice firstly prediction even message pass propagation inference stationary location rest network sensor sparse driven resort building avoid build real costly calibration prediction try run however traffic endow joint modal belief instead abstract descriptor parametric multimodal propagation ep stage complex require manually distribution procedure traditional particle filtering formalize follow state take never pairwise sample store go provide prediction stress latent noisy able infer observation pairwise random mrf ise physics font thick sep font si sl si sl sl si sl dash line plain ensure course distribution latent task assign actually less wish problem try compatible assumption question real construct latent perform prediction course build procedure resort approximate procedure rely belief propagation artificial intelligence community decode turn mrf algorithm value procedure nonparametric bp much bp bp tackle bp question find mapping shall definition encoding latent construct name mirror value address question random cumulative relate observation latent binary simple relate latent encoding directly conditionally simplicity limit monotonic choosing require latent random require order encoding follow encode binary variable associate since invertible decode map conditional cdf allow write increase choice function stochastically indeed probability follow prove leave conversely encoding consider cdf quantitie variable act random threshold order interpretation discrete space multiple copy lambda nature difficult criterion respectively information equivalently variable maximize word lead maximize mutual q turn proposition variable suboptimal maximize entropy equivalently limit get maximize entropy possibility latent maximize entropy ising see sense since parameter bernoulli outcomes admit pdf measure cumulative function maximal uniform cdf quickly encode position cdf correspond encode encoding joint turn decode simple obviously influence decode purpose assume predictor predictor simply median prediction definition predictor value decode base indeed estimate ml success increase encoding view see yield update cdf distribution define irrespective invertible may compute formula course invertible cdf understand commonly quantile eq choose loss linearity get without cdf equation eq decode conservative make span variable never predict rough proposition choice choice variable color either load package graphic explanation terminal need graphic macro ltb lt lt lt ltb lt lt r ltb ltb interest max entropy encode cdf measure relate question address dependency latent generally ise cdf pairwise notation distribution soon obviously parameter exactly follow mutual encode compatible binary bit shall quasi performance information strictly kullback empirical strictly soon use expand focus discuss come encode easily estimate carry estimation carry per admit joint pdf refer associate within explicitly maximize maximization building likelihood stationary obvious
act word layer markov chain dependent dynamical good top layer previous closely relate model seek cause update kalman filter propose neither model capable extract information sparse object propose extract locally building model boltzmann rbm predictive construct underlie principle like greedy wise construct encoder key encoding decode sparsity denoise sharing feed forward encoder forward avoid procedure encoder reciprocal bottom model robust structured image denoise image part network architecture layer state model usually interested abstract cause non relationship hide cause term stack act model fluctuation higher enter independently layer simplify notice low cause link layer layer linearly enter dependency influence latent specifically dynamical sparse state encode transition function infer stack several basic discuss red pool within stack visualization show patch overlap extract patch extract location tt extraction infer cause sparse track dynamic feature minimize involve consistent take spatial relationship neighborhood set contiguous patch add dimensional cause pool obtain invariant transformation like rotation learn capture dependency pool cause minimize energy constant state shape connect accumulate state frequently coefficient occur regularity activity representation separately combine devise unify infer update fix learn keep keep fix separately procedure proximal descent step hold alternate update relatively aside cause infer go back joint cause infer form sparsity parameter fix temporal although smooth optimization code solve code use scale like object overcome inspire structured use iterative fista key approach smoothness hence efficiently proximal method fista begin idea find linear smoothing approximation continuously differentiable smoothing write f solve fista infer generative model observe fista readily infer continuously lipschitz continue convergence fista minimize hold iteration important fista step maintain iteration guarantee optimal simulation reasonably good please supplementary influence feed prediction feedback cause time vary sequence transition parameter keep track update use gradient column avoid batch sophisticated conjugate gradient lead far build stage adopt strategy network patch network place large share consider input similarly input group together build layer emphasis layer wise parameter fix shift discuss layer arrange markov layer influence cause infer influence depend present reach equilibrium procedure instead top top n come previous arrival layer top likely cause predict cause top top induce coherence cause l wise inference perform minor namely cause similar elastic cause would like ability propose complex layer train network natural contrast contiguous pixel patch video dimensional cause pool patch separation imply pixel cause overlap pixel separation cause layer layer dimensional state cause represent primitive localize orientation position corner role shape classify frame pixel frame long shape long train patch neighbor overlap frame divide patch encode state vector contiguous patch pool cause cause input infer contiguous block video frame performance frame clean frame single structured poisson frame consecutive regard clean video perform inference bottom l scatter cause layer clearly cluster figure scatter video observe distinguish finally argue top help bottom unit true case largely able shape cause b predictive generative dynamical information adapt temporal dependency instability usually associate sparse help resolve layer improve believe convolutional high task etc video acknowledgment office suggestion form fista initialize thresholding parameter thresholding variable update variance element randomly poisson mean switching randomly
en est ensemble de les de la la patch dans les de un en mean patch dans dans en est send un de codebook dans l correspond occurrences un dans ce une adapt l et classification de exp exp des est des est figure est les j des challenge pour les pour les image une concept de un es des image la les les r pour pour I important dans les de est les ne es ne les la base pour ce I la validation en positive est par un dans la le de cr pour de les des un concept et les des n pour pour les de et cat positive et de dans cat la de pour une du par dans des des n un dans la les le de pr la performances la de la un des performance une une la des du des de es dans le challenge pour es dans les pr est la une des figure les tv la une la classification un il une pour la pour ce pr une pour es est une en la une adapt de annotation est pour les concept pr le plus dans concept en pour la concept e pour pour le des imagenet sa es paris des france e ce une pour es la annotation la la une de sources et dans l une est plus des des es pour construction de les les concept dans de image en et la de la annotation de paper propose methodology automatically build measure incorporate several conceptual contextual paper aim provide good represent semantic rule build encode hierarchical concept classification annotation result build semantic building semantics des il annotation de pour des dans un de des mis place une des ann annotation es pour le du la de sa une les de concept plus une du une la une un I de en concept de la nature des es une par la concept inter concept des annotation en de les des des en les se tr pour pour es les es ce dans de es la des les cat dans un n en est pour une pour annotation pour une ad le il dans des abstraction la source pour des es annotation image ce est dans pr dans un pour la exp pr send dans et pour di l dans pr dans introduction par extraction du ensemble des la de pour ensemble pour les la de de une es une de pour de les une pour la un les est dans pour une des concept une du une di g de pour les et les os est dans est pour la des les dans une en le en pour le mod allocation une pour dans une en dans pour une pour les les cat une des est adequate pour de les pour li pr send la et les pour une mod le une I est pour la pr information dans le fan al fan un la les pour la concept pour des une pour construction es est la est e dans concept dans de concepts est un pour annotation le se tr la mod la des en des dans une pour concept est pour le en les les dans ne la des concept dans par la distance le est un il est les svm des les es concept les du est la dans une le de concepts est la des et q les dans des la te de es dans la une une se un r est en des concept dans te la ce ne pour concepts est la concept dans le par une dans dans par se exploitation des occurrences du pr une un en ensemble pour des concept send un ce du des dans la est en la concept dans tr les des des concept dans concept ensemble des connect sa la est des des dans des dans position dans la de est pour identification du par les de la pour plus probable et la est par est il du la section des est tr dans un image en information concept dans des type image du de plus inf des plus par si une il est probable la ne send est la il de de pr plus pr dans dans mod par l est en occurrence des dans la total de dans base le le es par fr occurrence de le co par les pr par les concept si des et si l le concepts dans ce positive et les dans la dans par il la des dans la un sa la des concept dans la il par pour et es dans I la est la max en pr adapt la le est tr une sp le une un plus cr
state topological consist see theoretic structure discovery core single analyzing length single infer topology one transition value appear line mean explore source eqs generate process topology infer state statistical level subsequent example procedure joint kernel five table specify preference beyond inherent bias smaller estimate going topology result assign probability short series addition topology must take candidate sufficiently plot marginal specifically rather topology similar make color reflect black many band plane peak reflect consequence pattern stronger reflect quite create preference topology preference high modify employ sensitivity test reasonable sufficient make relatively explore fig illustrate state observe observe mean process previously source class alphabet consider structural proceed create symbol mean convergence machine consist topology monitoring give view structural agree begin state sequence allow leave despite calculate topology transition probability start estimate state topological color line mirror inference due infinite presentation minimal presentation countable process go symbol fig make topology suggests completely previously consider assume count topological reason typical accept topology calculate source series accurately unclear priori correct contain present posterior length single employ subsample black distribution topology probable right posterior peak come blue sharp peak nearly equally probable five topology five state denote topology five detail sequence length smoothly reflect provide supplementary plot fig parameter function mean first increase support complete versus topology previous inference complicated topology course perhaps surprising topology suffer select topology represent example demonstrate inference five binary alphabet include discover topology find sufficiently provide model structural mean estimate approach value effectively topologie broad hold reflect increase datum topology relevant consensus degeneracy posterior topology structure interest hide nonzero way aspect inference show topology five use topology topology follow topology topology come make small large never provide assign vanish topology accept topology accept accept build transition stationarity segment return data class source return return segment relatively structural point problem infer topology early return topology later return topology notably infer overlap switch structure sequel compare structural difference address expand candidate topological full explore inference topological library bayesian topological broad array single inference allow cloud computing topology comparison hour topology calculate candidate minute sample employ library candidate topology data automate method generate course keep mind inference model topology return engineering problem cite introduction motivated bayesian rely order broader require accurate estimate topology randomness bayesian structural inference application bioinformatic dynamical acknowledgment comment grant nf nf material em p material provide table unless note analysis single setting main guide three five subsample length initial segment long series allow view convergence posterior ci ci pair long reflect may large additional topology along subsample length difference complete candidate employ topology figure illustrate use draw one however topology subsample length source panel valid topology considerable finally sec topology meet criterion source notably many structure large topology accept e e e e e e b inference mean property e k e n n k e n n l posteriori gray solid h posteriori dash solid line panel even e e e e e e inference process e e e k maximum right panel line gray solid posterior mean posteriori dash indicate gray line property e e e n e e k n e e e e e e e lm dash indicate line lm panel black dash gray solid line posterior topology transition list output symbol state go transition state topology source plus pt end cm pre cm vector symbol pre ii pre ii iii pt pt false discovering rely topology exact restriction subset add benefit infer irrespective derivation expression estimate infer start compare posterior topology despite present effectiveness randomness reflect rate quantify estimation value finite process causal state introduce approach discover method topology enumeration remove topological restriction subset topology irrespective estimate infer compare topology despite internal reflect shannon compare former accurately reflect process well keyword machine discover quantify make understand signature model independent distribute iid pattern discovery testing iid violate discovery iid limited consequence discovery incorrectly randomness randomness discovery remove focus consist alphabet discover temporal occur state field science range bioinformatic dynamical assume field reflect molecular coarse grain often reflect word symbolic string make result interest topology quantify shannon entropy include sm produce inference candidate topology irrespective estimate transition probability cite single determine probability topology shorter prominent become light topology consequence familiar straightforwardly develop field range mechanic dynamical statistical element bridge overview concept sec sec algorithm topology model format structural clear inherent behavior system interest mean organization generate behavior incomplete nature precisely structure topology transition connection output symbol explicitly many topology accurately special model however topology provide unique representation survey topology process generate process behavior topology generate output return topology fig generate probability move move topology represent structured iid thus become iid fig topology behavior reflect properly example topology unique topology topology fig however six topology fig iid process exclude topology fig transition probability partly partly first introduction topological sequel adapt inference complementary effectively describe unknown start symbol start topology repeat topology specify list topology path short test fig topology one describe estimation symbol generate start obtain assume path eight topology cite count start way formalize primary sec analyze know topology topology length notably consider markov rich hide model hmm topology def symbol path state directly connect hmms many hide path observe analytic develop chain pass hmms include topology mixed state second computational aside alternative formalize candidate discuss correspond always hide series start include notation observe symbol path start I distinct topology obtain path assumed correspond us example fig bayesian datum sequence first infer transition topological must state requirement transition topology set estimate probability neither one I I I subset edge unknown set although make vanishe fig find neither path develop markov length choice case dirichlet simplex transition infer irrespective state transition transition numerator respectively call probability result prior subsequent application quantity determine start topology conjugate eqs x immediately posterior probability eq I prior topology notably probability completely specify since reflect high moment elsewhere posterior inference level probability typically variable necessary analytic average start state evidence eq evidence infer unknown instead transition q calculation state
create pick string consecutive create possibility pick consecutive gram include gram short bag also include build individual allow n gram construction partly meaningful semantic relationship noun place parse phrase content unfortunately approach insufficient deal occur automate appear group patient presence add challenge extract information challenge way area phrase contain previously illustrate bag corpus corpus result corpus word pair bar chart compare phrase observe bag phrase outperform bag show table pair varied threshold single cut gram occur cut gram less chart bar deviation plot create list create extract word train natural ask obtain significant simple selection significant bit good bit show comparable simple mutual perform select obtain feature selection second ensemble take hour std ensemble result corpus significant consist ensemble selection significant ensemble report static dynamic corpus pair vs figure corpus pair high mi discard replicate apply simple cut merely discard mi seem compare improve bag cut low mi somewhat effect certainly corpus corpus rank mutual dramatically horizontal two curve eliminate mutual std pair mi mi ensemble average cut bag single pair improve quality model discard score effect mi entirely clear ensemble dynamically select bag phrase show make employ cut cut effect mistake rare word bag counter pair mi table come cut rare word phrase mean pair pair mi static feature four rare pair different mi entry well namely cut besides word appear cut improve cut phrase answer reveal cut provide count count std phrase cut pair reject mi less high ensemble occur cut fewer static vs reduce feature score static table average accuracy compare usual threshold dynamic feature report feature select static choose initially long dataset essence system test anti correlation explore scoring table mark bold score highest dynamical indicate bold evaluation perform iterate trial find scatter vs model training poorly help improve training score voting replace vote usually explore fashion essence ensemble raw std voting ensemble consist size representation majority vote deviation seed machine increase score cut word discard consider word differ pair word use pair total training much high graph vote raw single pose train time parameter vary seed accuracy variation mid upper seed create get vote repeat different bar score bad hold good scoring bad scoring notice always pt demonstrate gaussians exactly comprise classify vote count patient green bar indicate vote correctly red bar far leave patient vote misclassifie voting bar mark green mark misclassifie classifier fold create different seed nearly identical gain examine individual patient give patient vote patient patient belong vote classify classify graph fraction vote versus ideally classifier vote always less vote group sometimes wrong apparent graph notable bar mirror bar confident patient bar patient classify patient assignment green bar bar essence classifier patient patient cope project appropriate optimize recall recall able identify positive expense positive rate reasonable harmonic group appear highly count aid word remarkable appear primarily difference group sample size appear often perhaps give word count promise future explore corpus inspire create bag create refined phrase phrase construct nearest incorporate part tag tag dependency semantic information lexical tag challenge construction evaluate significant phrase perhaps answer might applicable intend meaning conclusion training distinguish group patient sometimes achieve perfect set accuracy high task ensemble fold require train minute hour train individually adjust bring dataset sharp cut word mutual discard occur discard word contain acknowledgment c patient work carry describe create program primarily text format record health model construct genetic programming structure explore fidelity cross validation reasonably five fold ensemble average result contrast bag word contain medical record health center intend classifier aid patient patient red suggest build classifier extensive record patient patient record consist primarily free additional structured drug make improvement percent procedure description serious generate well entry three deep two aside quick validate deep review programming model model text identifie genetic seed primarily average use aside test remain score total answer focus distinguish several reason group word great clinical group impossible distinguish non medical care reference principle distinguish vocabulary distinguish well group model significant individually meaningful arise phrase phrase assessment carry assessment classify patient bag word find well overall besides bi gram gram train improvement see selection without pair improve fact easier typically depend distinguished score worse choose remarkable provide mode content obtain building consist stage bag count nothing particular medical ignore linguistic ignore depend checked well test order full set predict cut several way remove time perhaps indicator lose datum include count stem count singular noun patient grouping mi mi final step often limit machine irrelevant classify group difference irrelevant greater find limit prevent building pose learn candidate evolutionary h value occur case count mark indicate mean boolean moderate twice appear evidence increase appear twice patient classified belong five unique appear really representation many range depend representative cast determination follow vote classifier model part process leave use average five overall false rate maximize accuracy alternative seem particularly suited see appear patient population section low idea need examine document great detail three medical receive record group help issue span record generate upon care include correction visit bag white space phrase remove create ignore normalization nearly million word total distribute patient word patient patient word per record uniform character record mid mid difficulty record record per patient group record group record unique occur occur twice rough sketch medical fair name however word appear achievable achieve serve unclear remove building frequently make stem exclude obvious ability clear criterion base vocabulary cut count also remove word consideration purely word lexical occurrence occurring word word dataset word strong indicator know average word divide total count fraction word appear modify law quadratic fall quickly text normalize word sort indicate blue english language text book show incorporate pt explore predictive well ignore thus semantic unit big would adjacent middle pair discover equally meaningful word high mi word pair word typically word mi unit mean semantic phrase word contrast score pair occur around mi interact rx mi word appear next another due linguistic exclude mi pair consideration mi cut pair shape change similar show considerably quadratic fall build validation stage selection validation rather properly none accuracy various understand detail describe presentation stage perform count count simplify boost patient assign twice patient record see number bin determine average average vary patient provide occur patient one word two bin possibility bin patient time standard bin count filter datum general bin bin bin boolean value patient record feature three bin refer threshold bin specify threshold bin specify vary model threshold count specify create system value feature threshold dimensionality building bin assignment stage thousand time memory give record next affect number feature thousand maximize false positive maximize accuracy negative desirable mathematically five definition tp stand five equal group negative group group equal size desire quantity simultaneously ensemble concept way consist vote final make nature remain indirect insight individual representation combine make classification average behavior hold refer end want intend ensemble multiple vote accuracy model section summarize ensemble comprise ensemble formal mathematical patient patient either function course binary denote patient exclude classified patient let patient belong count positive merely connect concept nothing ensemble essence average value range inference use ensemble specifically accuracy test pos overview tune parameter exploration appear effect cut exclude gram precise number comprise case setting result fit essence train predict confusion matrix form training validation correct pos c correct correct fp pos c pt ten give representation extract keyword keyword reverse target distinguish keyword fair keyword error rare record presence immediate mechanism identify record unfortunately keyword predictive mis record unlikely future counter exclude keyword sometimes keyword role explore hc te ultimately video visualization alignment demonstrate notably split student teacher unchanged usefulness virtue visit volume ensemble model large pose use pseudo generator explore search result initial seed argument always way seed bar chart distribution distinguish accuracy bar bin dynamically initial fit gaussian typical clear gaussian fit distribution bar chart bag outperform much illustrate bar score seed show later model influence experiment total partition perform choose allow set deviation present model present detail section std maximize small remainder five consuming determine sensitive bin static feature static feature set matter sufficiently increase cause build ever attain appear score effect though dataset size less roughly parameter dataset large need focus unclear large size tuning model figure show typical c acc std acc std mean acc std effect dataset bag number dynamically select first count bin count threshold deviation count use histogram threshold dependence dynamical column three three bar word set pre select difference threshold threshold word threshold use threshold observe classification suffer threshold work well odd improve word occur appear predictive word less patient remarkable way raise variety way explore ensemble average arise little measure due boolean generate representation parameter initial ensemble share representation answer representation six half count depend much less show graph representation share high rank number rank greatest smooth result act another meaning rather word record perhaps patient record pick
c kn project rnn easy rnn oppose hessian character level generally lag behind need character rnns competitive find add connection first letter consecutive connection propagate long distance word character cross slightly par art estimate add first letter connection word phrase sentence clause investigate hierarchy connection improve character word sequence character well entropy propose completely give nonlinearity replace recurrent impulse response gradient pass softmax nonlinearity however token operation bilinear lm long language contribution soft diagonal use parametrization matrix fact like use advantage use implicit long weakly term entirely delay generalize learn delay think represent parametrization neural nonlinearity multiplication representation document assume document notice self recurrent encode decode fact entry encode associate decode every word due nonlinearity small recurrent connection parameter version dynamically adapt time direction new observation magnitude report corpus like gram dropout idea example never intuition adapt recurrent part add decoding formally define replace normalize tractable rnn simple jensen gradient exactly normalization cn far corpus cn consist incoming column cn use conjunction dropout author cn gradient step model norm strategy rate epoch decrease corpus project discover language indicate long cache length performance dataset use character wikipedia http dc corpora validation fold describe directly elsewhere learn model microsoft sentence completion dataset report hidden unit unless hide million large million commonly recurrent rnn model capture simple feedforward rnn language model training rnn mean rnn generalize low rnn regularizer allow learn spurious feedforward inductive bias feedforward perhaps model significantly rnn performance normalization regularize art publish usually enhance large kn copy independently highlight single gram low one cache kn feedforward bilinear lm rnn rnn cn cn cn report implementation interesting assess context effective contribution past division operation understand diagonal plot language nonetheless relatively hour gpu run larger know consist character word special token word turn estimation adapt neural read detail distinguish word word noise classifier base generative significantly gram lm low store also microsoft completion database rnn lm difference dropout long help almost gram usually cache gram case consist sentence choice around task consist project gram rnn advantage model efficient result project use agreement train lm perform report order highly surprising lm conjecture representation focus rnn range help verify drop code short correct task design lie sentence potentially context normalize alone state suggest analyze recurrent connection initialize unit diagonal block rnn keep possible enforce network modelling aim evaluate long context small good model regularization dataset capacity rnn recognize pattern language integrate context slightly use long scoring sentence result boost improve long context embed representation provide comment suggest point dynamic rnn really prediction computational college recurrent network successful linear rnns store pattern due explanation limit little data explanation art rnn nonetheless expressive without diagonal entry call impulse lm dropout keep past percent gap rnn sentence completion separate alone art internal optimization momentum main paradigm modelling rule learn model gram smoothing solve language modelling word appear fact model fundamentally recently model representation ability recurrent achieve parametrization function recurrent neural top score ensemble show individually slow time average dropout performance furthermore simple impulse lm regularize special unit rest decay rnn compose special lstm unit generalization strength learn word capture local recurrent token either character representation compute token zeros necessary world consider token representation use terminology rnn nonlinearity form nonlinearity derivative mean fully propagate aware run find nonlinearity unstable character behind art character free optimize adopt instead input smooth nonlinearity nonlinearity show character rnn back sgd rnns rnn
enough possible bag sample curve six show mind perform well overall two mind quite mind distinguish bag mind enforce select instance mind uninformative background distance belong difference mind always advantage mind mind disadvantage select instance difficult minimax approach reasonably mind performance except dd able training lot one bag explain examine bag successful em far follow mi optimization instance follow mind quadratic fast create take account minimax mind mind significantly select bag offer flexibility certain classifier interpretability certain offline easily instance might advantageous inherently neighbor combine paper propose dissimilarity dissimilarity bag convert supervised way dissimilarity bag bag attribute experiment different dissimilarity definition therefore dissimilarity average distance bag computational effort quite furthermore benefit potential end user impose restriction dissimilarity definition non dissimilarity counterpart property naturally approach question instance depend interesting investigate trade believe powerful combine make attractive group bag thank anonymous helpful concern bag individual supervise often learn bag generalize shift dissimilarity bag bag propose bag bag train treat dissimilarity show alternative bag definition experimental computationally yet competitive art pattern recognition complex part object various force reduction cause difference lose rather single instance multi segment vector potentially representation call bag belong bag bag standard bag instance bag label least bag instance classify unseen model concept bag point although instance typically fit application image wrong say concept use bag idea arbitrary several review concept contribute implicit bag distance bag bag instance bag instance kernel dissimilaritie attractive back unfortunately power lose dissimilarity indeed point suited concept still preserve class difference definition dissimilarity preferred type present paper discuss dissimilarity bag definition implicitly make definition type collect several restriction dissimilarity allow expert dissimilarity restriction user experience lastly suitable approach still provide art bag logistic classifier good implement decision dissimilarity mind example dissimilarity suitable method issue dissimilaritie bag multiple ib assumption bag positive assumption bag bag original axis parallel strategy point bag diverse target close need dd maximization guess accord maximize positive mi svm extension machine hide pose bag round label decide noisy instance reflect recognize assumption strict fraction notion concept well concept bag instance candidate concept similarity maximize bag discriminative similarity negative step assumption concept bag learn citation hausdorff bag bag dissimilarity bag kernel transform representation bag bag minimum last propagate instance supervise learner bag predict rule quite deal representation fig bag bag represent feature dd mi top bag representation citation bag dissimilarity approach classifier applicable bag need dissimilarity dissimilarity bag different bag dissimilarity distinguish follow treat bag set instance define distance attribute kernel straightforward every closeness hausdorff widely vision hausdorff bag maximum respective bag hausdorff maximum direct distance e symmetry point definition outlier hausdorff dissimilarity use euclidean computing dissimilarity bag diagram bag near diagram instance dissimilarity identity coincide instance satisfied minimum notice dissimilarity problematic bag neighbor metric problem view constrain step instance dissimilarity first dissimilarity direction measure measure way enforce obtain dimensional extend asymmetric dissimilarity identity distance bag kb ik radial type dissimilarity version alternatively instance space bag dissimilarity distribution distribution instance distribution density parameter intermediate modal consist dirac ht distance dissimilarity prototype bag near situation dissimilarity space demonstrate bag define either kb ik radial kernel statistic distance method restriction dissimilarity similarity incorporate pattern furthermore kernel bag necessary dissimilarity need dissimilarity bag choose expert subsequently greatly performance bag bag simultaneously maximize conceptually bag rather significantly furthermore bag jointly capture lose consider independently radial instance cluster distance sufficiently distance dissimilarity illustrate clearly artificial concept dataset bag bag background middle dissimilarity offer advantage informative bag discriminative overlap instance approach instance control explicitly might furthermore zero coefficient multi dataset several concept outside one need bag bag available concept bag concept instance bag turn use create feature reflect bag distance cm cm p bag bag dim avg web web graphic bag dimensionality average instance online bag shape responsible binding therefore describe surface property soon classify scene point water probably assumption sufficient bag imagine concept historical structure datum different object front orientation ideally concept segment part similar concept scene orientation condition object song consist whenever category expect possible specie classify specie typical bag contain negative bag topic nothing far apart feature concept sufficient reasonable web bag instance link recommend website bag link content preference concept need satisfied probably purpose propose bag dissimilarity characteristic dissimilarity dissimilarity receiver operate characteristic comparison dataset dataset behaviour denote stand dissimilarity bag validation fold bag bag bag compute bag prototype perform dissimilarity need default possible improve however cross value superior experiment classifier use svm unless otherwise dissimilarity hausdorff concept suitable strength bag dissimilarity success bag dissimilarity determine cause bag influence dissimilarity dissimilarity include table performance dissimilarity size bag life l r web c dissimilarity instance inside determine label dissimilarity multi instance although add benefit dissimilarity dissimilarity perform characteristic dissimilarity bag distance able well instance bag center cluster bag create dissimilarity however separate well perform bag object uninformative middle negative bag concept cause word contain away contain regular close select bag dissimilarity create performance dissimilarity bag somewhat
g likelihood unique maximum likelihood compute mle cell intensity call happen intensity intensity regular sum vs observe sum equal adjustment total adjustment c depend example theorem os cm os university paper iterative procedure likelihood estimation special sample space product cell proportional generalize iterative scaling bregman pt paper deal appropriate intensity appropriate model space categorical space specify cf sample arbitrary sample space subset appear deal comprise object possess feature approximate well maximum use special case technique use cf many case object possess context feature cf wu record reveal pattern association anomaly anomaly list affect birth nothing hypothesis perform within property review affected absence sequel model parameterization appear cell overall relational intensity regular family apply without family fundamentally review applicability relational iterative proportional fitting overall effect iterative sometimes thus relational effect generalization sum procedure generalize relational model construct projection traditional minimize kullback leibler minimize bregman fitting estimating parameter model find generalized output approximate specify variable cell intensitie distinction procedure maximum fundamentally parameterized component indicator subset multiplicative component present role include family form add therefore relational assume possibility representation basis matrix basis odd part degree numerator denominator homogeneous otherwise relational odd ratio dual odd affect overall assume relational property show regular parameterized q model exponential point eq sum adjustment property likelihood relational table mle newton scale mle use proportional fitting probability start contingency cell frequency adjust sum equal close enough value structure term ratio exist set block index sum compute relational actual like projection intensity mle iterative negative updating cycle marginal perform multiplier adjustment estimate proof relational presence overall equivalent equivalent non row hold loss matrix row sum ji ac complete feature selection apply matrix convert slack slack always add slack scaling generalized iterative require cell cell ai ji reduce procedure seem implicit whether row mention normalizing explicitly prove converge mle overall effect hold parameterization mle relational multiplicative denote probability combination let p ab bc abc feature variant exist odd ratio thus parameter produce converge normalize multiplicative imply isolated histogram sum equal constant mle relational parameterization model row sum fitting procedure overall variable multinomial relational matrix matrix consider empty compute cell parameter finish common u bregman divergence ip vector di di equation imply q relaxation sequence parameterize necessarily follow dd relaxation current cell belong desire transformation leave odd unchanged j continuity statement specify relational follow overall converge present know sequence maximum variable estimate cell point suitable
sense play success compress propose efficient compressed sense interested computing computing series polynomial bound smooth computational accuracy empirical algorithm output complexity exhaustive compress compressed sensing minimization restricted isometry guarantee recover signal sufficient space namely hold exactly recover minimization follow vector correspond complement satisfy q robustness recover property q index optimum programming objective difficulty obtain upper semidefinite transform semidefinite relaxation paper performance exact small polynomial time upper design greatly reduce achieve tradeoff result organize follow element pick algorithm coefficient stay section element exact result show improved algorithm method paper discuss future verify polynomial element subscript compute large ba ji value follow rewrite sum maximum th obtain pick chosen idea portion hx l indice index il element list compute find l suppose element appear right side k l maximum element actually pick additional increased become stay give sort pick obtain pick element coefficient problem provide pick element optimize pick optimize upper pick tight cardinality time define fact relaxation third come follow nothing pick optimize therefore pick upper bound element pick element optimize obtain greatly maintain compute execution bind bind meet reach bound upper bound lemma cardinality value follow family cardinality thus achieve problem objective value k newly add hx redundant always hx relax namely global bind programming base upper lk k sort th sort upper bind fix upper bound assign go sort assign big calculate go nan step upper step first element calculate subset cardinality upper every subset bound execution sort global never meanwhile stay unchanged come global small low meanwhile specify algorithm global among calculate upper calculated kk concave case value sign equivalent candidate apply major pick complexity bound sort subset sort bind time fix grow exponentially element rank big reasonably branch one rank shot discussion heavily compute low meet low execution bad case meet subset examine low meet use complexity exhaustive subset tend offer big turn sort subset tight upper sort quickly beginning meet simulation matlab intel dual cpu ghz gb ram os specify range randomly table pick pick element algorithm matrix run simulation value table show test different simulation complexity algorithm mostly pick element algorithm pick fast run pick reach exact cite result maximum result big step considerably search run reduce exhaustive method exhaustive think pick big list actual mostly matrix case pick find exhaustive take find exhaustive measure run spend subset subset exhaustive exhaustive actual operation table fouri bernoulli simulation various run fouri table table different choose total fourier simulation reach fourier example upper bound pick element element pick use case run time operation
single dimensional controller robot physics support answer question quantify low increasingly environmental influence behavioral dynamic depends view self direction explore dynamic internal means response current environment advance randomness replace effectiveness argue avoid curse dimensionality dynamic describe maximize maximize mutual parameter feed forward performed also fire dynamic act result feedback circuit neuron update lead behavior aspect principle behavioral level maximize loop nevertheless paradigm neuron specify average output activity close loop message work sensitive influence phenomena neuron feedback physical highly neuron produce phenomenon behavioral current limited dynamic essence feed parameter calculate complex measure joint sensor acceleration velocity conclude powerful tool express principle intuitive interpretation theoretic quantity behavioral level sensor explicit dynamical project support usa tool principle intuitive interpretation pi also call excess force exact update controller translate high system show decentralized robot behavioral physics environment dynamically decompose space curse dimensionality phenomenon key artificial human modify trait survival provide learn system improve cognitive capability exploration system extensively study area bayesian optimally conceptual focus thing way dynamical robot root deterministic internal generator chance exploit exploration body environment account building goal form behavior exploration exploration lead core system come interested quantify biology technical organization robot maximization paper study pi robot quantify experience define term predictive sensor high stream information shannon action lead consequence robot pi behavior become regime robot explore behavioral self sense simple strength threshold adapt modify pi process behavioral variability formulate feature pi principle make systematic pi maximization cause controller importantly pi go inherent mode encourage mode present theoretic optimize adequate robot process behavioral measure call application restrict restriction realistic without everything infer introduce phenomena self switch system high dimensional behavioral robot system intensive recent approach widely follow approach understand information flow brain measure quantify future action development problem domain drive pi large complex system recent book principle self organization mention self exploration effort robot motivation produce reinforcement task progress put play proposal challenge use intrinsic reinforcement fitness evolutionary decade see trend action exploit effect couple exploit behavioral mode entire dependent coupling brain briefly implication variability show surprising external far behavioral variability create idea pure randomness molecular pure variability process paper behavioral variation produce bring new free general information pi call time intend base window estimate special explicit controller dynamic batch derive sense shot gradient combine plus deterministic become part value define instant average joint random individual may pi exist entropy explicit pi usefulness pi development discuss early pi robot essential information behavior pi turn remarkable sensor already information pi see continue introduce specification pi process approximately start simplify pi information mi successive joint density realistic purely call pi drive exploration application pi adequate variety behavioral mode ideally certainly lead pi time window formally current instant length start distribution expression window probabilitie equality notation difference averaging would know sample update drive increase controller time ns neural standard concrete obtain dirac delta depend state treat property pi system turn pi already treat bring propose information quantity new basis propagation dynamic dynamical actual certain window define window capture occur time window principle consider shot realization figure illustrate interestingly linearization error derivation look entropy function agree variability term entropy aim derivation drive behavior toward comprise weight threshold dynamic execute omit assume essentially parsimonious realize low study get expression window learn dynamic eq self arbitrary application describe increase small instability elaborate call gradient ascent define however nothing learn dynamic never reach prefer notion dynamic parameter formula eqs gradient e replace little average valid limit different intrinsic self relate exploration sensor aim system behavior short window rough goal shot gradient favorable dynamic generating aspect maximization convergence state rich parametrization complex dynamic intensive dynamical keep finite landscape shape increase may landscape persistent actual exploitation use randomization decrease randomness acquire curse complexity randomness introduce successfully reinforcement approach policy system replace demonstrate relevant formally change window length give shot action demonstrate state rewrite z system show one cycle represent sphere decrease asymmetric saddle happen right system shift bring back initial shift diagram cycle see phenomenon effect stability see white noise vanish noise fully system window eqs rule ascent q speed agree principle detail briefly sketch salient dynamic keep converge towards maximum induced cc bias jump result feedback strength toward high interestingly restrict system observed fast spike rule remark time setting parameter depict typical depict qualitative window enough equilibrium basis short readily exceed physical magnitude characteristic exponentially barrier height process window maximal equal show decrease cycle towards reach convergence induce generic phenomenon capability already investigate provide information theoretic inherent let decentralized drive collective mode phenomenon chain mobile control define strength loop bias turn perturbation effective window infinite length entirely different rate allow effect chain well well exploration capability chain demonstrate consider freedom physics simulate simulator source neuron treat measure controller define angle position reality force limited angle substantially angle deviation exploration dynamic robot equip sensor angle robot controller eqs eq start mobile substantial mobile demand collective assess appropriately depend strongly center circular external influence trivial environmental force robot depict robot empty give view demonstrate exceed value mean deviation drastically dynamic include set stable external notable reaction influence robot velocity latter observe result control demonstrate fact stop soon put interval sec highlight trajectory start box move cyclic inverting row show situation second enable investigate controller demonstrate obtain recurrent neural roll give difference conceptually design necessary get performance away moreover mode sensitive invert velocity widely physical decentralize task degree freedom joint send controller angle sensor angle robot behavior explore develop behavioral depend physics environment dynamically embed cccc bar px robot normal environment ground robot robot bar robot happen video want quantity body controller characterize parameter configuration controller order fig use let robot min physical time without noise deterministic variation start pose straight slightly pose front simulation dendrogram plot base difference simulation environment pose front support role generation behavior physical reflect thus square value element qualitative behavior sign group situation seem plausible constraint drive robot behavior situation move bar controlling inspection behavior latter different video robot min control setting uniformly expect sensor produce dimension pass filter controller case
nu nu save save additionally tr rank could approximation mle posterior ml b r surprisingly much square nearly identical rank make leave mixing look plot give expectation average true thin black line include health behavior available diagonal actor actor use reduce probit model latent actor model unobserved latent convenient inference proceed scheme unknown respectively give depend prior density variance markov chain yx indicator drug period examine factor specify hyperparameter prior magnitude effect fast residual let brevity start diag seed start descent although naive adequate ready gibbs store simulated object compute dividing provide v lambda lambda lambda lambda lambda e lambda lambda fc full variable panel order strongly sometimes color character drug red drug plot circle drug user triangle plot student drug use circle student suggest behavior social network use variate example reduce mean example require variate von variate set call denote unchanged order order diagonal detail manifold therein case manifold surface density von fisher langevin proportional often von langevin sphere symmetry recognition vector matrix variate von conjugate via iteratively conditional mf use gamma correspond full j accord seed
use assess availability formula single permutation carry processor ii repeat ensure keep simulation frank bi parametrize univariate normal bi variate copula freedom ds kk copula dependence imply independence copula bi dependence greater lower lastly bi frank copula value similar frank copula dependence despite margin frank consider parameter distribution vary normal degree freedom vary dependence independence five point grid draw permutation proportion xlabel ylabel name title legend style south east none legend align solid mark option row crcr black solid option solid row crcr solid mark option solid crcr mark crcr black option crcr height scale xlabel ylabel north anchor south west south none none legend cell mark square crcr color black solid mark option crcr color mark triangle solid table crcr crcr black solid mark option row crcr width height axis xlabel ylabel right south west legend south east fill none align left color mark mark option sep crcr black mark option solid sep crcr mark option sep crcr color solid mark solid row crcr black solid mark mark solid sep crcr height xlabel ylabel plot south west anchor south east legend style fill draw leave solid mark square mark option solid table row crcr solid solid row crcr black mark solid crcr color solid mark triangle mark solid crcr black mark mark option sep crcr conclusion apparent firstly effect fold curve rank test secondly effect great tail construct frank copula gain power great set proportional lastly symmetric practically htbp cm cm frank extent extension rarely drive solution weight since favor great power difference existence context independence goodness copula minor yield weight goodness use adjust copula considerable practitioner copula example financial theorem test component er von functional empirical copula act tuning dependence test arbitrary integrable weighting formula relate conduct variety great deviation copula dependence distributional upon copula represent assume follow copula become many rank comprehensive copula implement consistent particularly monotone test characterize develop copy encode characterize copula functional marginal inspire test behaviour er von functional serial serial compare alternative statistic asymptotic efficiency er von base investigate generalize behaviour er von series application independence probe flexible parametric von adjustment test certain copula test exponential issue aim paper gap integrable weight independence enable switch conduct assess impact copula alternative great discussion computation weight rank section generalize er von state discuss computation rank section issue interesting simulation alternative section von copula form behavior statistic later eq copula empirical percentile appear asymptotic establish equip metric establishe copula refine partial continuous empirical converge weakly tight derivative th tie brownian bridge behavior independence independence process multivariate tie bridge eq multidimensional boundary detail example r von paper er von emphasis part test add make dependence power test goodness requirement place lie non integrable theorem characterize limit continuous set statistic brownian c www n degenerate neither joint asymptotic lastly imply q lastly substitute third expression integrable derive yield formula term directly integration substitute repeat derivation percentile rank q proposition general requirement impose weighting imply statistic choice raise existence issue optimality aim explore power wide selection future search optimal choice weight weight commonly formula asymptotic statistic addition weight statistic use weight add flexibility goodness fit empirical weight unit throughout statistic equivalent drive notational convenience let reciprocal refer independence set copula eq weight offset problem scope may motivated type example close importance assign tail copula frank family independence around median tail emphasis median large integrable meaning directly computational similar may result dm u copula et hoeffding copula rearrange meaning amount form independence copula examine decrease tail place emphasis great management tendency regardless extreme equal observe either extremely another make difficult detect tail interest reveal mean statistic concept upper corner measure coefficient upper dependence low dependence correspond
away apply may worth formally state every mean arm take arm expectation restriction theorem non allow procedure consider mean distribution arm imply set complexity must statement hand imply existence particular non require part one gap use fact parameterization adaptive failure indeed tight adaptive choose arm meet sample follow total sample convergent probability return control compare factor sample implication somewhat improvement useful event result inequality away apply hoeffding factor possible elimination bind observe never remain probability sum give phase first stop median evaluate collecting obtain require definition pac probably procedure require restrict mean pick arm arm arm estimator arm arm compare px independence x x mi straightforward exclude show arm I likelihood test standard gaussian inequality step constant gap I minimum maximize satisfy monotonically imply range maximize give mn gap consider imply complete imply drop except small corollary electrical engineering university department research arise broad mathematically multi bandit application interested identify situation find arm arm bandit multi armed adaptive arm previous complexity adaptive non polynomial bandit arm payoff large arm sampling arm realization random straightforward paper realization focus paper necessary find good application thousand cell problem search surveillance large social consuming costly minimize influential crucial quantifie new call succeed within show order great scenario grow positive arm sample complexity require motivation second particular interest arm unlikely arise smoothly decay depict left plot sparse mean gap complexity adaptive differ case gap shrink increase complexity sparse case show fig bind fix constant know fail without crucial show design biological application mention add burden notation follow convention throughout good appendix bind finding follow sec derivation least hardness parametric problem parameter hard gap shrink quickly grow gap great sample ignore condition parameterization good arm find conversely gap possibility sufficient well next arm outline multi phase elimination mention propose algorithm output median elimination
box traditional approach predefine box advantage object train bound box part training solve box excellent representation recently image classification detection agnostic experiment post classify less ten box obtained achieve art box predictor generalize unseen flexible problem vast agnostic idea address scalability recently achieve thank carefully rely template scale become challenge imagenet former evaluate potential address song basis share across detection good approach segmentation segment motivation use detection segmentation segment classification layer prove principle show deep lead superior result advance et box approach handle et mask aim agnostic scalable bounding represent neural dnn bounding box box contain formalize idea box encode leave box four normalize achieve invariance coordinate transformation confidence box contain encode produce layer sigmoid combine bound box location treat output sigmoid output bound experiment confidence box box suppose object classify achieve box dnn predict match ground box bound box number box ground location well assignment iff true distance bounding coordinate quantify dissimilarity box additionally optimize maximize iff match maximize minimized interpretation achieve interest combine contribution example solve box variant bipartite matching object less case optimize back example propagation compute r make significantly cluster cluster centroid use residual predict matching location find truth matching moreover also unchanged match prediction match usage matching prediction note agnostic apply predict box particular box box unfortunately grow linearly class example argue step recognize leverage multiple image classification mini batch train identical achieve mention previously use mean set balance might area coordinate map truncate final box similarity network million train million training set image ten million image ratio range equal cover box set explore generation select evaluate held portion random example mainly complex scene box diverse label box net classifier comprise million overlap object similarity label negative similarity box selection first round model size pass candidate overlap top high keep classified classifier pass detection box multiply score pass precision produce addition well scale max center select window size image budget show achieve plot image object boost c car cat al et al competitive train rest box produce way top box curve image obtain column precision class challenge consist location category calculate consist image addition localization imagenet serve recognition train achieve validation latter bring substantial post score box minimum time score sort score keep challenge criterion evaluate held portion metric classification allow produce valid criterion truth box classify box directly approach infer box class metric challenge apply represent window come box per class win entry localization classify window window top window competitive able approach box come appeal raw output scale need object never train similarity see explore train imagenet vice versa perform occur window class interestingly imagenet capture window versa imagenet much rich secondly box approach naturally instance except generalize understanding
discovery voxel alignment voxel beyond learn generalize aim discover group suggest set optimize separate individual raw align cross validation relative separately individual reduce relative glasso individual somewhat identify classifier voxel voxel voxel subject validation lasso tail show subject learn error suggest signal region obtain aggregated indicate positively blue voxel positively picture slice aggregate pattern connect sparsity pattern c proportion relevant significance thing glasso voxel demonstrate glasso suited fmri approach exist voxel voxel subject sparse correlate indicate voxel inferior predictor yield well involve proportion voxel identify identify glasso introduce recover hybrid pattern convex program least succeed multi task make inference plausible region glasso work penalty similarity functional fmri proof lemma defer prove result trivial show equality suppose inequality assume homogeneity respective decomposition inequality follow decomposition optimal none prof subset datum conceptual challenge much rough correspondence despite neither physical thus benefit handle fmri similarity dissimilarity involve stimulus align co alignment possible spatially large region fine multivariate relationship among voxel coarse description establish deal leverage information subject discover multivariate identical solution glasso voxel across challenge set allow unique task draw group discover subject slice subject red voxel sentence opposite positively picture sentence note highly pattern individual stem fact account since alignment perfect result aggregated error glasso tie single voxel group voxel location force voxel almost histogram select drawback glasso voxel tend group voxel spatially result voxel subject spatial voxel voxel glasso fact specify voxel account discover lasso chance chance respect voxel group voxel utilize subject force voxel perform activate succeed allow group reduce brain capture group use interest expert particular involve sentence behave different study reasonable voxel subject classify kind stimulus sparse spatially voxel correlate spatially voxel glasso glasso explicitly interest poorly expect recovering find voxel high cox electrical engineering corollary proposition remark depth learn task useful group select restrictive wherein organize accord necessarily suit call sparse overlapping select relate error error loss voxel advantage relationship especially useful suited feature task restrictive motivate subset notion necessarily suited suggest subset contain recover generalize glasso span procedure capable glasso use solution lasso encourage similar pattern apply disjoint limitation partition motivate example contribution sparse analyze synthetic demonstrate encourage similar identical accomplish task conceptually useful identify fmri application spatial point subject example arise application recover variable across application handwritten character recognition exclusive variable pattern glasso pattern pattern study involve fmri participant cognitive activity construct activity accurately predict expect brain vary vary vary guide across suggest neighborhood useful voxel voxel logistic elastic net penalty rest outline notation set regularizer property derive leverage outline experiment logistic yield glasso notation sequel bold subspace overlap group group subspace index decomposable support decomposition overlap compatibility given consider coordinate representation g upper compatibility respect contain give satisfie parameter lie group display within sparsity ignore term henceforth sequel eq onto rsc decomposable program satisfie general error lasso regularization rsc next consistent need index group follow g g dd degree chi look help square bind invert loss n upper regularizer formulation fact square make combine overlap group satisfy strong
summarize initialize parallel step non constant proximal convergence ergodic n nc analogy decrease converge kkt convergence sense iteration representation see logistic lrr relate parameterized subspace respectively popular order naive generalization straightforwardly discuss naive solve constraint suggestion fix parameter rest suggest fix sp sp run solution experiment run intel ghz windows numerically even thank penalty moreover naive relatively converge solution worse solve function penalty grow matter data setting algorithm efficiency project algorithm lrr database sp sp show see comparison second percentage run quantitie second cluster percentage c acc second evaluate nonnegative superiority nonnegative singular matrix actually follow nonnegative feasibility fa truth solution thus efficiency degree fa e pixel problem formulate singular image obtain show b image generate original gaussian besides problem stop see qualitative quantitative well nonnegative c corrupted db fa subsection sparse overlap respect prove variable cover group one two successive overlap remove generate support row statistically row informative row recover separable proximal include iteratively terminate gradient subproblem outer threshold terminate choice nuclear norm square fast relative iterate time truth comparison slow consume outer numerical accuracy inferior proximal pathway second fold cross validation pathway belong c c pathway pathway breast set breast follow contain gene gene balance replicate patient select tumor proximal adopt choice fast loop subproblem threshold outer loop use logistic predict select correlate pre processing contrast phase ten active ten time subproblem pathway linearize parallel adaptive linearly utilize proximal onto convex easily learn distribute computing advantage although inherently parallel proximal face algebraic computation learn interesting integrate exist technique incomplete cholesky factorization technique order address scalability issue lin liu support china foundation project program state key lin valuable discussion prove proposition tucker kkt n subgradient first feasibility duality kkt k generate prove check kkt point subgradient mapping generate kkt check kkt inequality supplementary material supplementary minus er inequality corner proving proposition let kkt problem divide side ready resemble k k first accumulation since rl j f j proposition boundedness rl I kkt sequence readily k kkt proposition proposition due thank proposition rewrite boundedness proposition accumulation due proposition accumulation proposition assume let I jj together feasibility j lagrange multipli proposition k theorem mapping zero assume hence n whose boundedness uncertain exist cauchy sequence initialize hold reduce feasibility q combine summing divide next frobenius technique k continue observe hand f divide j n summing k definition divide use increment divide side liu author electrical technology school university technology school software university convex program however traditional alternating obtain quadratic generalize multi extend multi propose parallel splitting penalty solve prove reveal ergodic extra refined devise fast generalize particularly suitable rank recovery low compute advantage increasingly range field e low kernel real e face recognition video denoise reformulate follow linearly constrain separable ii nx convex program f block use capital letter close proper onto subsection machine formulate rank representation lrr propose apply vision work sample high liu lrr norm sum lrr decompose salient mining collaborative etc formulate eq index select index frobenius norm recover low observe noisy see reformulate auxiliary matrix besides show form logistic overlap obtain linear classifier row entry zero sparse rewrite program fairly complete solve interior problem typical machine general lead efficient interior toolbox minimization nuclear computing order often prefer proximal gradient popular convergence unconstrained optimization constrain result method lot attention especially utilize structure objective function bregman influential method program convex subproblem proximal thresholding nuclear solution characteristic subproblem separable solution greatly unitary mapping e identity adjoint operator subproblem close solution iteratively optimization process issue quadratic subproblem variant linearize globally impose nonetheless exist number proof case generalization practice program occur robust see extra recently substitution iteration parallel dual step convenient first penalty practical program constraint difficult objective compare speed section section review case consist four update q multipli operator adaptively please refer detail extend separable program provide global contrary fundamentally block natural generalize straightforward unable naive er inequality back substitution iterate actually naive converge g problem analyze block provide mapping ensure fortunately modify solving update q rest theorem update call kkt condition specifically terminate condition feasibility derive kkt condition rule suggest along follow kkt bound general program bound actually even assume need specify upper imposing equal boundedness far global theorem necessity optimal function remove rate open quite bound encouraging general convex program rate subsection simple
become unlikely non threshold say identify start move value begin indicate level include visual low behaviour value test large measure would indicate observe datum reciprocal estimate around gradually become small empirical quantile focus reciprocal likelihood fourth row figure illustrate third test exclude double usage behaviour plot minor fluctuation small left slight trend bottom fluctuation highlight informative pareto approximate particularly pareto tail remain body clear plot figure replicate identify location obvious realistic notable measure dataset although apparent clear figure value statistic point highly point smoothly predictive determine plot examine fit may panel variability average replicate measure threshold true implement bayesian threshold near mixture er von fit test compare generalise pareto pareto life low diagnostic become model refer fit pareto threshold threshold unknown unknown identify exception threshold estimate credible interval know phenomenon threshold flexible specify pareto distribute true evident determine benefit approach see bivariate pseudo n angular pr pr w h mixing bivariate manner smoothly complete true threshold point dot grey line threshold generate model analyse mixture dark grey bar grey logistic illustrate logistic estimate posterior prior threshold obtain dot grey measure threshold top figure estimate observe use dot far line obvious compatible require suitably radial value dot appear value rapidly indicate test statistic observe dark grey bar grey black predictive model fit observe distribution far dot leave light grey bar predictive predictive produce value distribution remove although univariate angular perfectly generate mechanism observe left panel interpret true case mis specify mis specification observed occur mis specification distribution remain proper explanation examine quantile panel observation obviously small pareto tail apparent mis measure fy ty ty right illustrate consist wave km produce series observation preliminary unit margin assume independent inspection histogram histogram actually datum threshold low dot threshold around neither fit logistic continue regardless evidence suggest identify flexible whereby line become effectively great bottom measure visually histogram predictive model perhaps little sophisticated modelling plot become ty fy threshold estimate line grey black line indicate value choose panel densitie air ground matter city centre uk record matter cubic follow analysis analyse record analyse dirichlet assume ad hoc thereby figure illustrate predictive subset air dirichlet panel rapidly determine unable observe stay level unable identify threshold outcome measure suitable namely select produce sufficiently flexible actual general arise wrong biased inference figure exclude structure reasonably whereas illustrate roughly panel f environmental panel panel exhibit characteristic analysis confirm acceptable panel panel c behaviour character dimensional difficult suitable estimate produce air flexible behaviour adopt threshold dirichlet distribution hoc visually empirical difficulty increase dimension get large ty fy combinations indicate dot grey threshold suitable extreme threshold extreme potentially problematic modelling extreme radial threshold multivariate analyse pareto principled identification multivariate analysis demonstrate analysis radial approach stem predictive fit many possible produce posterior heterogeneity limit asymptotically underlie vice versa analyst compare several identify criterion comparison hoc able specific rapidly threshold construction avoid obtain difficulty balance extreme extend difficult diagnostic plot threshold identification measure base threshold correctly interpretation value ad hoc choice acknowledgement acknowledge discussion threshold identification support research mm fan theory use motivate describe process number valid exceed practice must analysis univariate method attractive threshold extreme propose quantify model without alternative approach univariate multivariate bayesian pareto spectral threshold theory often event area environmental science commonly mathematically generalise pareto threshold generalise suitable small observe identically fr margin process intensity co unit simplex represent asymptotically approximately region case poisson fit observation exceed suitable small approximate tail number approach primarily univariate offer comprehensive method give category approach fit order priori e plot hill general threshold develop residual life plot method propose include priori several model generalise pareto attractive remove make balance pareto dominate appear extend multivariate approach concern pareto multivariate intensity angular radial principle construct exhibit bivariate univariate histogram visually retain shape bayesian diagnostic threshold choice various reference alternative would extent compatible pareto small threshold allow comparison different amount datum require threshold almost exist univariate select use subsequent analysis threshold selection fully article section brief describe threshold approach compare determine datum consistent pareto section specify problematic come pareto distribution treat model able rely portion upper however semi obvious modelling bivariate high dimensional idea quantify huge literature concern classical statistic observe model framework integrate unknown parameter denote fy perform improper easy compute standard double usage issue involve usage posterior test manner full simulation ft posterior classical distribute weak uniformity close nan conversely lack argue compatibility nan purpose estimation threshold observation exceed pareto fr pseudo radial component poisson intensity function admit form family correspond g dependence bivariate w treat measure strength bivariate logistic bivariate overview however accept flexible accurate whereas two datum set q necessary algorithm describe simulation empty compatibility pareto compatibility specific minimum threshold reciprocal advantage easily model evidence pareto process threshold permit locate circumstance fidelity near statistic partial latter case dataset generalise pareto however obvious disadvantage univariate compatibility small pareto define threshold model threshold dependent versus determine examine threshold small strong similarity adopt sequential perform rather mcmc assimilation whereby sequence increasingly
infinity classic supervise characterize dimension indeed empirical theory show combinatorial broadly theory surprisingly erm recall concept characterize stability supervise conclude remark open basic supervised notion consistency algorithm measurable close interested minimize risk f endow algebra probability identically roughly speak solving square misclassification exhaustive notion rigorously concept hypothesis hypothesis space say algorithm eq measurable algebra arguably training give erm minimization add erm erm general define erm measurable misclassification exist possibly minimizer aside consideration say universal shift learnable definition uniform refer consistency hold bias due requirement sample follow universal let atom universal uniformly learn measure greater free soon minimizer solution universal learnable loss uniform either hypothesis meaningful approach necessary characterize hypothesis space stability suitable impose notion combinatorial follow complexity binary value dimension vc dimension form law number characterize classification misclassification uniformly learnable result theorem term central binary value show crucially square notion originally introduce function prove loss note prove result hypothesis erm relevant stability add historical remark refer quantification respect stability ill pose concept pose first quantitative connection symmetric notion seminal stability th replace uniform stability thorough investigation notion notion erm definition erm probably stability uniform replace leave set finally fraction increasingly rather clear function erm stable result essentially stability erm erm uniformly erm assume satisfie condition generalize theorem function characterization term dimension question stability answer section focus discuss extend let probability measurable lf identically erm supervise difference need function distinction measurable hypothesis intuitively consequence definition universal set analogue note uniform restrictive notion erm characterization extension characterize possibility l classic result equivalence contrary implication sec show hypothesis vc learnable erm vc space add vc learnable erm coincide probability erm imply finite vc consistency equivalence dimension strong notion erm characterize strictly
either category noun tag noun tag capital simply gram low noun side contextual extract compound noun close window token window window row gram nonzero contextual filter p phrase pattern md visit vb vb dt huge jj nn pt dt lot nn cc south lot md dt jj cc phrase five contextual step drop row equal value contextual five five order rank count row generate count pattern tie break part year way co life form another concept role syntactic relate occur narrow proximity determine distant generate complex pattern try capture syntactic connect nearby phrase generate contextual phrase step token token everything token right everything phrase simplify character tag vb tag contextual pattern phrase tag contextual pattern token tag token reduce tag specific token tag tag token tag token reduce tag pattern tag tag tag replace tag tag n compound truncate general pattern specific pattern pattern right splitting pattern point drop may pattern phrase one every pattern marker one truncate phrase pattern c general dt nn nn named x name dt specific phrase drop value yield show contextual last tie break column count contextual complex pattern great proximity determine determine domain example give near give indirect role case apply syntactic connect imply direct object correspond syntactic many row characterize syntactic run characterize different contextual note appear contextual value character appear row vector contextual space space union column column row frequency equal space row element three generate experiment four set involve effort ensure adequate handle experiment similarity cosine angle two unit cosine range opposite direction vector raw necessarily negative cosine weight negative weighting element truncate matrix semantic control weight factor less explore range use decrease high fuzzy neighbourhood sharp neighbourhood less domain long generate column small increment increment value researcher either measure vector space measure function feasible combination tractable middle alternate hold tuning hold improvement try tune tune try could tune advantage background task change need fine grain know optimize file file look domain exact look alternate function automatically form none alternate map zero vector similarity various way phrase want component component similarity balance geometric similarity geometric number cosine component similarity successful composition far relatively intuitive highly negative ensure negative element half element multiplication fair baseline apply solution row benefit apply multiplication vector multiply result row identity row wise express way multiplication nonnegative factorization yet nmf past space evaluate choice analogy evaluate question construct third apply dual phrase three capture intuitive concept space analogy college table question word choice example analogous person trust answer relational across inside measure simply constraint inside across pair main idea equation similarity indicate high domain low domain similarity differ considerably discover reasoning high function similarity person functional role person functional trust person capture sim trust knowledge source behind domain target internal similarity motivate constraint analogy convenience inspection relational understand without network similarity symmetry equation tell horizontal axis network swap holding would swap holding would change link word although would sim sim another way break inherently skew symmetry break natural domain similarity apply introduce inherently asymmetric equation desirable wish reasonable decide us value analogy might high though domain people certain abstract frequently become domain domain role belong abstract discussion mention together specialized manual construction two cause rise include question relational zero skip ten fold question fold question correctly question incorrectly answer student top ten past approach issue linguistic attain correct incorrect difference statistically accuracy know college interval calculate use majority unsupervise dual supervision binary training consist one positive example induce probability five choice choice tune see sensitive perform search one coarse narrow grid grid question narrow search value nine ten fold search value parameter setting search present minimum average search coarse search attain validation good accuracy fine grid evidence importance model sensitive variation stable nine fold select evidence function performance vary part answer manually part speech label label context trust noun noun split various none statistically confidence exact varied need speech space h wrong noun noun noun noun noun equation dropping constraint significant drop primarily test understanding verify reformulate question expand choice full analogy expand another assign test evaluate domain choices trust trust trust trust trust trust expand domain choice question question analogy except pair explicit new add choice select ten reformulate test attain confidence test insufficient far tune accuracy reformulate test domain table summarize base five question accurate column accuracy less space fisher test choice space space space five space yes space modify five yes five yes domain ten space space yes dual yes dual yes modify ten use modify dual yes domain choice question ten wrong modify perform art address issue linguistic reformulate word measure relational classification cosine measure near relation classification dual noun house class compositional multiplication neither class accurate class compositional class wrong answer compositional head function accuracy another whether brain classify testing head noun first noun look far accuracy general compositional significantly low difference h hyper multiplication different wise lack sensitive see noun include expand choice seven must assign similarity dual wise multiplication reformulate noun include row see illustration limit dual significantly accurate wise multiplication dual domain modify alone domain reformulate choice noun space perform alone table head noun drop alone accuracy drop space either modify reformulate noun question element wise seven choice address address linguistic subsection phrase phrase phrase noun noun noun object phrase rate subject low similarity degree h phrase phrase noun certain noun majority noun evidence noun noun environment noun noun noun noun city centre lift head object demand number phrase rating similarity vary high phrase represent phrase similarity q human similarity phrase pair rating figure domain similaritie model development set rating development evaluation phrase thus rate rating evaluation communication challenge divide phrase pair participant development pair rating phrase pair development phrase evaluation phrase phrase divide group evaluation people phrase value group people represent rating yield rating comparable number calculate vector rating paper describe believe rating people vector bias person rating consistently people score evaluation phrase type phrase phrase phrase phrase phrase phrase two vector input phrase score participant per group phrase type rating value compare dual multiplication addition similarity cosine multiplication represent phrase ad nn nn nn vb avg comment leave subject dual space multiplication multiplication multiplication correlation significantly multiplication addition difference space phrase participant calculate significance significance sensitivity pair test sensitivity adding human save h phrase group similarity noun certain noun noun great noun majority noun low noun evidence order add element wise dual address nn avg comment one space addition space manually pair automatically rate reasonable case noun noun pair interest object natural tendency assign rating rating would pair support order sensitivity addition order multiplication create dataset pair create cognitive experiment subject evidence brain seem similar music similar house associate word label associated domain measure degree associate associate high experiment three precede subsection three dual parameter measure evaluate three correspond relation noun measure look pair label would percentage desire three variation yield class top parameter top setting display sort list capture test similarity pair word sim sim supervise fold validation classify three summarize similarity degree associate cf sensitive setting complex thing suggest child section analogy dual relational similarity accuracy well significant reformulate question design sensitive merge drop choice noun noun compositional difference dual state element multiplication linguistic suggest gap limitation lack sensitivity order reformulate design statistically multiplication dual reformulate version show label similarity measure word similar support argue fundamental difference support linguistic capacity dual semantic relation semantic composition kind similar kind measure corpus approach similarity arguably relational similarity five version table well use significant level exact analogy water question traffic water choice recognize water high traffic share traffic water believe corpus phrase stand purpose composite phrase relation house house house way similarity phrase house phrase house third construction tie together thing depend nature comparison desire task stand phrase similarity connect phrase see seem phrase connect phrase dictionary kind perspective seem aspect together similarity similarity compositional order set word matrix row semantics vector row function simple normalize scale easily sublinear without contain element likewise similarity composition operate contrast work composition operation shift sentence sentence relational fit base compositional could analyse section similarity phrase equation phrase specific may limit growth quadratic acceptable option domain space another would construction composition word give option appear elegant section manually combine manual sentence reason construction automate solve mapping analogy atom mass list atomic atom automatically generate system atom mapping attain accuracy mapping similarity map search space mapping composite similarity effect mapping similarity figure believe automatically similarity dual example search composition subject constraint constraint sentence map align sentence experimentally evaluate work effectively similarity would semantic composition regard scalability dual vector size grow length phrase grow growth might impact long phrase tractable area experiment phrase parse search likely would performance noun promise structure another avoid use interface form singular noun certainly simplification sophisticated form issue treat case linguistic suggest arguably limitation perhaps capacity tune need phrase contextual research english generalize european language language challenge similarity use geometric composition explore composition dual space seem various phrase red ball truth bridge symbolic claim spatial question symbolic question yet join semantic relation address linguistic capacity achieve room research many kind notion overview word similarity measure multiple similarity composition alternative instead multiply multiply similarity sim sim way problem semantic helpful corpus space b available sharing question make interface datum intelligence research publish appropriate relation algorithm recognize relation analogous likewise house recognize house house house seem task relation space model match previous model relation share material house house house similarity way tend semantic capture distributional sentence house vector house treat way vocabulary phrase possible phrase people new phrase understand phrase phrase composition datum linguistic master able phrase treat gram treat ideal expression compositional gram vector representation house representations house house house house house yet composition phrase mean mean english text come issue semantic represent recognize semantic highly semantic linguistic composition recognize incorrectly suggest sensitivity relation idea proposal adaptive variety syntactic relation phrase example draw whereas draw composition variety syntactic properly give weak phrase amount dimensionality structure scalability unweighted treat kind composition flexibility mode composition adapt different weighted average tuned syntactic scalability means grow proportion eventually representation scalability map vector house house map multiple avoid hold bit grow eventually semantic noise significant radius number radius space theory message bit encode likewise capacity space limit suggest closely research unified handle issue linguistic scalability measure domain subject similarity function similarity analogy traffic water similarity relatively water relatively water role respective domain thing role thing carry thing recognize relation traffic analogous water semantic combine similarity noun noun phrase noun noun role noun noun brain degree similarity come clinical similarity brain briefly proposal measure apply various cosine similarity measure instead address information map vector vector space flexibility address capacity model compositional linguistic similarity recognize argue phrase measure phrase provide phrase hand past composition review section phrase representation component argument create stand alone phrase component believe hold progress semantic similarity vector represent similarity inherently two thing similarity connect phrase composition stand alone phrase stand alone representation stand phrase equally well composition issue depth survey semantic composition semantic relation separate create function question question similarity problem discuss question limitation examine semantic overview semantic semantic review survey examine relation introduction semantic linguistic semantic relation word sensitive affect capacity phrase kind syntactic relation word flexibility variety task similarity pair measure pair section scalability phrase scale neither phrase model relation increase phrase noun word composition cosine similarity take centroid vector vector relatively capacity syntactic relation unweighted addition propose variation sum word sensitivity scalability due different additive suggest element composition operation like addition element lack capacity nonetheless evaluation seven compositional two multiplication performance use tensor product composition product tensor scalability problem grow long discuss compact outer wise scalability circular convolution outer avoid circular poorly composition noun compositional square learn phrase linguistic problem avoid need plausible noun predict phrase noun noun generalize speech phrase semantic measuring phrase closely identify emphasis syntactic semantic vector task phrase decide phrase likely phrase similar degree without phrase exchange mean phrase similarity measure word functional role domain similarity model preference preference triple consist word preference preference phrase triple triple representation meaning phrase likewise triple triple transform typical consistent preference composition phrase likewise address linguistic information scalability measuring consider analogy traffic water water transformation recognize traffic water course model relational surprising however unified relation let pair relatively similarity analogy analogy suggest classify lexical map scheme something word similar algorithm order relatively close lexical hierarchy essence intuition also high material high seem analogy indeed high imply incorrectly classify relation hierarchical hierarchical domain similarity equation similarity past researcher
previous decomposition extension simple omit us union ccccc instance interval diagram six interval q cover diagram six interval formal concept contain I concept particular decomposition may one factor concept assumption since readily iii namely contain concept extend apply every boolean note formal non empty I equal issue theorem decomposition obtain even formal due exist contradiction ia contradiction deal strong search I hold compute easily fulfil whose contain column also part matrix establish important feature part first whose coverage factor focus second since small number particular build upon idea collection essential concept take concept optimal cover formal candidate concept utilize improvement factorization nevertheless exactly matter ia concept b f c c f c j u description justify lemma every I ie ie ff ie ff I ic I c ie ic ed e ff e ic ij ff verify manner call detail compute aforementione picking interval collection yet greedy manner start attribute concept try extend attribute loop restrict leave extension attribute extension accept concept concept interval mark remove cover remove greedy concept difference cover di cover consideration correct provide approximation evaluation comparison dataset paper provide describe algorithms minimum utilize modification cover large number still covered implement remark factorization exceed attribute vector row decrease exact decomposition attempt employ length principle cost analogously modification hence primary length small description length claim factor user frequent singleton find sort coverage yet author run size demand formal utilize improve cover avoid necessity formal result order magnitude implement considerably advance design decomposition stop provide solution noisy consist employ principle problem pattern pair cost pattern extend core core rectangle column sort list sorting row help randomization drawback fix sort boolean randomly prescribe density instance dataset set use table strategy number correspond table synthetic essential avg avg dna arguably decomposition literature numerous demonstrate meaningful term quantitative criterion recall account goal compute input reasonably way portion decomposition paragraph clearly prohibitive time concern analysis analysis particular future matlab critical part file paper column order compute asymptotically reason concept line search proceed extend similarly attribute follow recommendation choose individually require one choice frequent attribute sort randomization fast preprocessing utilize slow third slow fourth term however frequent frequent average slow selecting object attribute accord experience ordinary pc assess quality decomposition factor algorithm add desirable datum good factorization first factor portion synthetic average comprise algorithm represent coverage factor coverage representative decomposition display cover dataset coverage na na na na na na na na na dna na na na na na give guarantee stop find stop relatively indicate disadvantage large require couple well sparse couple dense dataset namely factor input slowly grow reveal coverage reason singleton singleton factor singleton first factor connection algorithm correspond behavior namely nevertheless coverage drawback similarly cf perform synthetic real factor strategy essential may draw contrary differently focus strategy justify boolean discuss design aim provide dataset comprise display dataset algorithm every level additive boolean flip contain flip entry flip entry curve compute algorithm curve average dataset noise curve coverage shift portion observe shift large shift context allow cover graph somewhat sensitivity sense cover believe sensitivity limited question question care explain result examine closure role suggest propose evaluation coverage factor small important present theoretical emphasize role part boolean boolean promising shall heuristic interval preliminary quickly note factor topic present beyond boolean general closure decomposition let mention three receive considerable attention present three boolean topic remark present new emphasize consider result experimental demonstrate coverage factor outperforms propose research topic boolean closure concept boolean becoming preprocesse heuristic technique involve provably however limited boolean heuristic examine closure theoretic lattice call lattice connection matrix viewpoint explicit essence relate notion examine section contain essentially equally entry concept lattice cover rectangle search base compute experimental synthetic real algorithm exist moreover closure order theoretic represent reasonable discuss throughout interpret primarily hence symbol indicating attribute find possibly product interpret exactly approximately explain read attribute decomposition rank recall norm minimize prescribe reflect factor need prescribed portion decomposition extensive overview viewpoint traditionally boolean one notion connect bipartite mostly complexity area boolean context lattice paper decomposition boolean design datum assessment dimensionality datum conclude among observation principal value interpretability tailor boolean among involve aware provable difficulty hardness basis problem interest mine primarily al complexity closely investigate paper show connection proved discuss author database problem involve relevant propose dataset employ use paper paper cx decomposition employ minimum problem matrix difference important topic present useful survey various rank detail regard reader refer paper addition work interesting recently role utilize reduce view decomposition matrix rectangle short permutation column say pair follow exist cover put th cover cover every accord set ic il consider rectangle l l b decomposition utilize formal concept interpret relevant aspect analysis viewpoint closure structure boolean matrix lattice lattice help decomposition issue play crucial role regard formal ii form subset call I interval q describe crucial role decomposition utilize later prove algorithm f iff iff iff iff iff iff b interestingly reformulate matrix diagram concept lattice diagram formal concept small formal concept reformulate graph diagram nod exception empty object attribute mark mark small exception path path geometric perspective illustrate
day plan control generate partitioning scan statistic situation count count hide examine alarm rectangular generate cell cell result simulation plan unknown flexibility partition strength across scan plan preferred circumstance plan offer keep thing plan stop plan child rule slight complicated scan plan converge therefore traditional cell control mean lowest report bold plan make easy trend note find scan plan influence early plan substantial plan close boundary shape change scan statistic less likely early scan plan scan pt rr rr method scan scan column simulation scan plan plan row column partition generation bad region case uk correspond centroid unit post address unit divide lattice marginal lattice plot span mean per cell give roughly cell cell figure cell cell count empty cell cell lattice day week factor explanatory fit count group aggregated group forecast proportional distribute model forecast start update daily day forecast present plan pruning figure day far north south east look day day day per day day week end reduce rhs shrink east south north persistent author request shape plan design detect plan effective robust detection relative seem lattice cell cell simulation report recursive partitioning scale age etc scan plan example find location indicate transmission public service location plan selection explanatory variable similarly number region correction correction aim bias improve plan future research topic reference e process car http discrete scan statistic letters b public health surveillance monitor health population health surveillance page university surveillance scheme scan statistics york adjust weighted journal american association comparison increase institute technology disease detection scan communication theory periodic surveillance use scan j early detection multivariate chart surveillance report unit cumulative approach journal pl surveillance disease transaction surveillance early high year shape scan statistic monitor health j scan health surveillance scan benchmark surveillance popularity simple time high disease detect scan statistic area shape plan detect generally offer scan statistic effort usual scan plan flexibility vary secondly series move average reduce exclusive rectangular away region significantly region prune scan surveillance weight move monitor spatio smoothing statistic disease intensive difficult scale high etc scan implement software variety permutation use scan time vary region spatio approach easy high scan statistic window scan plan literature size within cell plan advantage approach ease apply past count selection scan plan propose apply temporal count choose window scan section scenario use compare detection cover relate disease scan spatio plan regular spatio take paper memory plan paper compare roughly therefore marginal necessary detection column let daily disease cell row spatio window size cell distribute poisson whether adjust significance design first aggregate chen refine scan paper compare scan plan plan major way firstly cell smooth jt al multivariate datum homogeneous leave mostly cell trend similarly count past decay step smoothing let element spatial similarly smoothed smooth involve row forward step divide parent exclusive exhaustive sub smoothed count binary outline away expect recursively fail exceed significance otherwise determine partition boundary zero replace signal minus suffer small spatially smoothed roughly variance recursively longitudinal cell rectangular partition parent mutually exclusive count count expect however would adjust partition search later parent keep grow generation rule terminate generate parent stop prune prune good value use similarly stop apply two row parent space row partition parent pruning prune otherwise prune highlight detail pruning control control specify rule avoid effort stop split parent whenever parent grow prune count high expect rule partitioning partitioning terminate generation variable selection modelling area competition translate bias select scan plan forward plan bias subset scan competition explanatory scan statistic suffer window bias involve bias select minor suffer bias move scan scan illustrated paper bias rectangular region scan constant procee equal plan consider first generation consider partition partition partition rule scan plan need examine apply plan need scan plan fix work example use simple plan count lattice count additional add control count count cell count good row column convenient demonstrate count count expect figure let partition figure parent expect give clearly partition next partition respectively column value produce p generation top
common subgaussian tail wherein draw conditionally availability nearly asymptotically dominate power test take construct require establish unconditional respect method broad assumption arise naturally context relevant learning encode also compress model tradeoff prove hold covariance assumption general build hypothesis throughout refer p loss normalization test confidence index submatrix form submatrix contain maximum minimum write subscript nonzero entry represent nonzero standard normal z sub random eq estimator table characterize limiting follow readily model table design estimate precision estimator per define integer follow condition establish subset satisfie lasso minor instead let selector high give control formally rescaled get give tight bound population constant defer entry order fix define eq assign test follow related choosing estimator term false probability false characterize tradeoff attain tradeoff magnitude alternative condition high furthermore probability design plot versus monotone q give power zero row satisfy synthetic generate matrix everywhere else understand ensure vector subset else implement mean square approximate fold cross compare testing precision report mean test identically theoretical optimal power let ii quantile normal demonstrate distribution cccc avg avg std std c procedure setup significance realization os cs cc high turn note follow realization hoeffding f lemma apply bernstein center exponential side exist bound least last follow whose defer employ less subset union high plugging recall bound separately subgradient read imply ready triangle thesis begin row subgaussian constant column e union sub follow ij bounded let hold eq event probability e employ readily corollary ready moreover fix c per acknowledgement stanford fellowship nsf award dms grant fa plug tw slightly improved version prove stationarity condition function read ty equivalently sum identity generic position sub norm q cauchy pt pt plus minus conjecture consequence claim replica exceed successful lasso ordinary square confidence interval estimator address construct study improve art establishe size coefficient provable dominate particular precision require efficiently synthetic form design unknown interested parameter large small decade particularly reconstruction penalty select often context know perform well error address understand assess statistical interval p estimator necessity interested form
black average hypothesis index separate large particularly problem incremental nan generic two setting relevant practice verify pair medical meanwhile section boost incremental believe advance enable power guarantee finite testing mathematically motivated incremental one full rate belong support direct comparison separation large highest low fdr setting back occurrence early try setting signal include test statistic exhibit harmonic behaviour setting standard normal observation design non space varied simulation set excellent variable still plot power across three value least path vertical line four stop select genetic measure six rt subject drug mutation marker rule angle provide list relationship drug assessment validity select study main constrain selection begin drug use angle test select mutation location assess vary drug match l c drug point theory support suggest available procedure largely meaningful relationship method fdr high covariance harmonic design operate black proportion hypothesis index note behave nan part presence consider refer observation take spaced varied difficulty show realization superior take rapid decay control medium hard choice fdr effectively irrespective plot see power medium set almost value outperform hypothesis order require contiguous approach rate procedure testing denote control fdr specify level order procedure different nan meanwhile nan fdr error distributional setting develop guarantee many procedure nature hope way convert inferential guarantee important fdr except orthogonal procedure aware extend develop new sequential test g grateful helpful taylor constructive comment suggestion nsf fellowship b fellowship fellowship dms grant stanford department lemma rejection threshold rejection lead false define selection rule one hypothesis come continuous jump establish r enyi enyi order distribute uniform statistics martingale tell strictly let run sampling lebesgue monotone convergence surely meanwhile begin corollary proof helpful level term satisfy fdr close describe control sort statistic difference setup setup nan replace index nan hypothesis define r deterministic location nan analogous sub key accelerate rejection nan allow mix course compute number hypothesis trick corollary list infinitely list rejection control fdr enyi rule equality global nan r enyi thus draw immediately control equality present vary holding hypothesis figure vary term follow perfect moderate moderate separation moderate hypothesis signal simulation iteration unless hypothesis perform low low signal regime conservative strength surprising geometric nan p conjecture multiple order initial contiguous rule sequential stop none propose control false selection recent value setting nan hypothese control rejected order classical method test procedure false fdr order transform value fdr control statistic fdr control arise naturally implement selection path angle build add variable remove ask add idea path add question desirable model see nested coordinate order add add uninformative formalize spirit measure improve regression state fit comprise develop regression regression already write test study th support appearance character even aware path incremental context need seek parsimonious non useful subsequent hypothesis make easy full paper fdr procedure order hypothesis regardless angle nan test asset hypothesis topic angle lar illustrate simple linear predictor seek angle add need stop hypothesis typical exchangeable order etc produce control introduce fdr procedure successfully control fdr left panel number select level panel show axis grey order valid cutoff reject discovery rate fdr nan hypothese rejection scenario call whenever fdr transform value always regardless rule moderately robust misspecification nan index asymptotic second guarantee fdr family wise make single particularly decision reject uniform considerable seek large even isolate result gain extensive variant procedure fdr far adaptation formally fdr define constant power provide fdr among variety pseudo tailor directly rather provide sequential value section selection fdr note multiple integrate resample genome study might hypothesis across snp hypothesis contain correlate snp marginally significant snp carry response redundancy snp contain distinct important goal prediction select generally significance conditional value control fdr meaning list nan take get proposal value hypothesis trust fdr nan non one achieve fdr incremental value independently fdr section order sum backward test control reject procedure trust look last value specify enable last control mean strong fdr fdr vary simulation consist order hypothesis separation hypothesis varied determine scenario hypothesis show gray black proportion hypothesis think stop medium respectively hypothese hypothese nan simulation iteration index sampling replacement proportional small select easy separation medium easy setup strong separation setup non inter hard setup inter conservative fdr control similar curve performance precision reject hypothesis exceed guarantee restriction define reject get power reject hypothesis less one well aware emphasize stop value fail sized figure value performance note motivate order testing formalism result order add give option add incremental review proposal order grow applicability fdr controlling procedure statistic
mode near panel histogram color code neighbor grind truth finally algorithm object vary mnist handwritten topic remove appear category keep rand normalize mutual information result initialization mean homotopy result mode different average distance commonly homotopy automatically improve even poor early b rand index mnist summary misspecification shift centroid away create mode mode attract choice determine kde per matter whether kde centroids low density large true cluster multiple centroid yet centroid mode centroid move inside pattern large centroid look redundancy detect centroid cluster describe early mode centroid pattern feature common coincide bandwidth value neither average weighted single bandwidth centroid crucially set number cluster role smooth intuitively neighbor exploratory tool homotopy take say present smoothing cost comparable optimum place may centroid representative allow user bandwidth shift cluster find centroid lie high density representative neighborhood yet far misspecification mean shift mode bandwidth non fast algorithm formulation centroid interpretable beyond mode find acknowledgment award cm thm proposition prop conjecture wang science california false false estimate centroid mean exist return meaningful mode small able centroid cluster even appear outlier misspecification centroid representative shift try binary cluster cluster mean shift bandwidth start local mode centroid cluster mode user parameter implicitly mean shift popular application centroid outside create singleton mean computationally respectively particularly dataset shift active research prior available c concern validity representative continuously digit image nonconvex cluster high mean average digit representative digit lie manifold mode arise mean shift valid nonconvex manifold shape protein third centroid algorithm centroid centroid regard centroid centroid remove mean remarkable mode cluster seem obvious pick mode pick require pick uniform density mode idea assignment small bandwidth centroid valid computationally slow mean objective assignment proportional centroid mode separate naturally combine assignment objective case become mean centroid become mean use constraint maximize become centroid drive towards datum link intermediate minimize np hard iterative locally centroid assignment vice versa first assignment constrain separate point close distance distance centroid separate unconstrained maximization centroid proportional kde mean shift centroid tolerance meet iteration algorithm mode area fig dataset nonconvex separate since however move centroid kde would also kde multiple mode mean return pattern replace work case fig even wrong mode distribution edge gaussian long tail take random subgraph real world web page dataset degree vertex two character skewed power outlier outside plot obtain wrong cluster far right outlier law determine head step shift achieve kde mode correctly separate kde mode imply mode partly panel kde kde color mode vertical bar cluster kde dataset axis mode handwritten digit dataset run mode decrease fig show centroid digits identity etc neighbor interpret valid input centroid decrease histogram near neighbor show histogram mode bin mode centroid onto like digit
solve chapter finite convex section section improve shape iteratively bellman project cone operator ideal compute projection explain numerical approximately ideal iteration measurable next fix point combination existence contraction converge arbitrary underlie chain essential sequence sup context distribution cone contraction rest rest cone evaluate expectation challenge suffice rather two c moreover projection cone convex function obtain every also chain calculate step copy auto distance style circle fill font edge node leave edge appropriate condition discuss monte convexity fix show solve show convex unbounde finite sample example iterative discuss constraint bind help restrict cone constant straightforward therefore convexity linear instead throughout onto cone square finite optimization obtain initialize generate path generate two v k n nh k solve find thorough copy dimensional program stage reach next sufficiently size compact chain specification estimator converge grow random q show onto cone exposition vector example noise copy chain lemma optimizer estimator projection every radius ball subset state moreover follow ergodic property chain instead continue allow mis disk space show close subset projection converge infinity every sufficiently right hand correspond sample disk next disk converge cauchy similarly get obtain every eq second similar show truncate setting estimator theorem ready iterative projection value point sequence convex exist converge motivation contraction projection shrink fix point due conclude also observe suppose semi path two follow infinity see semi since contraction eq triangle converge exploiting lipschitz extension employ convex difference replace projection onto onto convex step value know estimate particular convex hilbert projection onto eq length variable similar set solve q construct belong extend length project close qp extension decrease variety problem queue service function monotone adjust close convex space cone qp therefore estimator decrease scheduling cost one fundamental problem encounter market pricing contract power exposure control party third responsible otherwise involve pricing result limited flexibility dynamically operate price fix scheduling note iteratively schedule specific pricing finance focus simplify parametric increase region run operator natural spread output time need hour spread price energy empirical suggest spread stationary driving process namely dimensional wiener poisson independent exponential degenerate volatility mode operation represent immediately start time slot moreover costly overhead cost mode switching let current operation power beginning slot numerically compute convexity policy numerical policy architecture span switch three approximation architecture fair sample cut plane report compute replication replication carlo compute last path truncate fix std std parameter dt compare point clear improvement onto subsequence project converge show fx fc ng nm increase bound ng show conclude convexity everywhere straightforward right converging since product argument fail difficulty possible bounded asymptotically property convex net large use also eq every ergodic initialize moreover conclude converge ergodic borel lemma straightforward surely hand surely hence assume positive recurrent construct ergodic recurrent exist markov clear almost surely recurrent dx number cover ergodic member assumption lipschitz recent every neighborhood correspondingly assumption hand recurrent ensure zero arbitrarily bind zero project respect sup n large enough therefore minimization large web www stanford edu web www stanford fully expect horizon value play important convexity function provably tends infinity implement agreement market concern estimation simulation infinite horizon discount play space huge even value fully incorporate shape dynamic programming e select essential cause effort propose correct estimating variety exist partially american generalize black property literature formulate process stochastically monotone discussion monotonicity sufficient condition provide probability stochastic exploit property value policy estimate function along know value function convex cone measurable process noisy noisy cone estimator require reinforcement fix convergence bellman go infinity extend value lipschitz precisely introduce
kind regularizer undirected graph sake comparison mention regularizer corresponding curve plot curve ridge regularizer grouping encourage equality magnitude pair elastic net strict like encourage difference successive guide correspond r regularizer net regularizer exhibit sparsity know point grouping grouping group magnitude order always consequently costly adopt solve complicate costly propose fista fast sort exact term element wise term art proximal review proximal splitting augment solve induce adapt active strategy arguably iterative forward splitting tend slow poorly research obtaining fast algorithm fista iterate theoretically experimentally considerably fast another variant reconstruction separable augment shrinkage address split transform unconstraine constrain alternate multiplier primal admm bregman bregman sbm image inverse sbm admm context experiment contribution regularizer sort make solve optimization algorithm fista sbm admm organization ii limitation iii iv bold letter p vector finally argument understand component new briefly let everywhere operator nonempty define f f identity fidelity pairwise encourage magnitude nonnegative controlling become behave category four induce induce depict fidelity example correlation contour induce grouping whereas contour induce compare glasso pre specification compare order net capability convenient regularizer fundamental proximity directly thus sort e permutation tie break notice equivalently perform within grouping averaging denote q satisfied ready term n exact proximity regularizer approximate proximity operator illustrate computing term w h view thresholding computation proximity operator obtain simple fast comparison difference let magnitude plot insight difference sort sort compare result show small shrinkage operation cpu randomly fast increase horizontal vertical solving smooth possibly nonsmooth special f minimizer six sbm worth recall inexact apply experiment fista fista thresholding acceleration fista u k x k satisfied fista fista term fista fast lead k k criterion typical acceptance decrease term term admm problem know algorithm choose k k k stopping criterion conjugate term term proximity algorithm convergence mathematically clear leave open practically behave parameter small numerical report show difference aforementioned window pc intel processor employ define iteration term x reflect possess sense sample aforementioned fista sbm admm stop kk recovered mae mae mse respectively iteration mse fista admm sbm sbm fast fast fista accurate study influence keep axis represent figure problem conclusion propose efficiently regularizer outperform version proximity regularizer difference analyze mathematically naturally proximity state fista proximity accurate fast mathematical operating proximity thank code corollary regularizer responsible encourage group regularizer sort proximity approximate art grouping operation costly storage reason guaranteed exact behave regularization wise appropriately group fast alternate multiplier proximity bregman introduction decade linear attract lot wide signal compressive sense name forward signal assume interest make ill absence address form regularization x fidelity regularizer certain solution type compressive zero ideal regularizer encourage solution zero combinatorial hard arguably encouraging regularizer convex approximation condition object cs regularizers propose norm reweighte norm
g converge denote differentiable lipschitz proposition conclude cauchy schwarz inequality convexity proposition proposition n f otherwise fact converge begin proposition rate adapt proposition let minimizer pick eq compute n f us e proceed simplify term concave pointwise concave pointwise infimum function jensen inequality r n r n n always lr calculation lr derivation proceed similarly proof yield desire surrogate surely converge expect convergence separately two proposition remark combine expectation sum n simplify go growth choose lr n r therefore quantity iteration block block classical surrogate frank old algorithm direction line cm smooth gradient lipschitz algorithm frank method surrogate convergence rate cm provide convexity f define use exploit nh ng therefore rate extension easily design present instance randomize frank wolfe popular proximal gradient surrogate algorithm surrogate exactly fista surrogate point show next cm convexity initialization surrogate na assume sequence f follow call estimate precisely surrogate heavily expand sequel keep induction prove existence value recursively scalar go recursively quantity imply remark g simply last true term appropriate value n f v hypothesis come combine n bb n lower obviously also three remain last rewrite n relation recursive depend describe computation show equation f na na na na induction devote method smooth probably stochastic sgd variant consider admit order surrogate recently linear sag algorithm smooth unconstrained gradient estimate iteration dual ascent call sdca perform incremental primal unlike sag sdca storing context incremental surrogate log update present proposition iteration surrogate randomly pick choose surrogate near surrogate conclusion surrogate index choose obtain inequality definition monotonically imply positive converge non g converging sum evy exchange sum sign front surely g argument strongly convergence prove several accord relation incremental exploit strong study first relation f come see proposition la f sum second e f n induction convergence interestingly rate scheme iterate randomly pick sag sdca even sag sdca smooth unconstrained sag instance lipschitz surrogate section experiment implementation cm regression intercept optimization regularizer name storage dense challenge website test software sag toolbox code run intel cpu gb ram double loading note issue component surrogate e tn surrogate rewrite p surrogate pair quantity amount z tn z upper significantly proposition notice indeed rate surrogate simply f tn motivate start decrease cm inequality satisfied c software publicly fista sag grouping run sag include heuristic spirit introduce stop consider regime present regime provide memory minibatch clear winner preference regularization consistently sag one fista already outperform sag sgd option proceed yield result provide rest material dataset require quickly minibatch strategy cm surrogate function problem convergence design incremental property solver plan incremental follow sparse algorithm particularly important dataset store past surrogate acknowledgment thank schmidt bin discussion program science agreement present contain various frank wolfe proposition give contain mathematical directional q directional direction write point admit directional everywhere feasible say stationary differentiable interior reduce reduce subdifferential differentiable f converse f say function lipschitz convex strongly note lead f proposition continuous imply gradient surrogate one differentiable lipschitz proof prove point hessian twice twice absolute role plug yield conclude f dt l third prove smoothness make continuous exploit nonsmooth function exploit differentiable everywhere twice differentiable differentiable everywhere call begin f come almost measure almost l general argument let function f function suppose f desired define function look directional stationary minimizer parameterize let differentiable strongly lipschitz second growth lemma sum gradient lipschitz sufficient moreover concave concave variant prove notation definition lipschitz q inequality thus admit taylor show continuity concavity lipschitz point affine regularity function continuous gradient convex continuous continuous strongly inside tangent simply trivial f sum inequality differentiable lipschitz basic us elementary technique combine surrogate surrogate linear surrogate l l l f last accord justify surrogate differentiable surrogate apply lemma study obtain surrogate convex use differentiable follow surrogate follow paragraph convex differentiable present lipschitz strongly show apply ensure continuous accord paragraph supremum convex nn separable surrogate pick search update estimate rate assume converge surely n lr n proof f r r jensen n relation give figure benchmark regression logistic regression figure cm supplementary section corollary definition conjecture axiom consist iteratively surrogate propose provide viewpoint wolfe incremental match solver large optimization iteratively objective optimum interpret view instance dc programming signal optimization generalize discover algorithm draw connection study smooth convergence condition convex convergence successively randomize zhang analyze family simple guarantee except frank wolfe rates framework incremental sag sdca scheme rule sag sdca analyze optimization scheme conclude focus scheme match outperform cut solver regularize subset present
regret curve appear super logarithmic short cumulative linearly uninformative credible obeys algorithm quality prior reward correspond true large intuitively prior fairly confidence show argument regret fairly inaccurate confidence bad super cumulative dominate logarithmic uninformative front notice prior human operator human optimization mechanism annealing attempt break optima near currently optimum decrease boltzmann temperature decrease gradually deterministic anneal choice schedule equivalent exploit context anneal temperature exploration exploitation exploration anneal optimum similar explicit implementation boltzmann eq case select boltzmann maximum temperature might stochastic make arbitrarily deterministic anneal choose schedule n heuristic value arm two infinite uninformative define regret formalize multi uncorrelated uninformative time suboptimal arm eq cumulative precede section prior may among arm wish diagonal fact perform experience across structure uncorrelated prior arm respectively estimate arm definite procedure generalizes correlate otherwise belief credible base univariate marginal distribution belief state uncorrelated reward procedure uncorrelated environment correct performance stochastic guarantee denote summarize arm statement formula q detail start ti express therefore diagonal use case include many hold correctness example belief perfectly correlate e arm reward tend quickly analogous human run spatially armed web participant amazon web platform select task website participant university website inform protocol inform participant play collect collect part game grid decision move element allow allow fast fast slow second automatically reward visible reward time immediately report dynamic experimentally dynamic condition task option time game option beyond scope paper participant block choice game dynamic block balanced design task combination dynamic second condition alternative landscape particular approximately participant assign assign belief second fast slow negligible task landscape landscape landscape dimension block landscape choose option reward choose uniformly range landscape peak point center choice cumulative participant reward multiple task participant participant block participant perform bandit compute subtract maximum reward cumulative reward use receive study human case performance repeatedly option classify form task classified behavior participant bandit classify logarithmic observe correlation landscape first across fit exponent participant nontrivial performance short horizon category task statistically two phenotype participant phenotype indistinguishable sufficiently may fundamental depend surface smooth participant distinguish surface rough identify good hard I value participant band represent horizon logarithmic law phenotype human law phenotype minimal encode prior belief minimal four scalar belief participant uniform thus encode participant spatially assume arm spatially reward element parameter spatial smoothness reward rough smoother interpret represent absolute lack choose schedule softmax action achieve regret participant schedule interest work schedule human compute bayes participant softmax selection temperature deterministic prior uninformative prior correspond close adjust replicate capture landscape reward case fairly uninformative surface agent moderate decision incorporate encourage option employ uninformative expect short horizon addition decision tend make agent example logarithmic appropriate significantly agent encourage confident less agent quickly reject area prior linear solid green fit simulation identical human landscape agent reward uncorrelated decision incorporate softmax reward armed section maker arm incur choose previous instant accordingly structure arm distribute region variation multi armed bandit extend sequence repeatedly block incur begin design provably efficient work arm armed cost allocation behind strategy maker maximize total maximize reward grow transition grow regret dominate ensure cumulative intuitively number minimize select maximum credible limit strong remove frame end option block frame length remain constitute length paragraph length block small characterize tuple identifie frame identify block select credible frame time instant divide block frame select block credible allocation round credible choose maker transition mean gaussian upper credible let block logarithmic formalize bandit transition uncorrelated uninformative expect time suboptimal arm suboptimal arm transition cost appendix respectively cost algorithm bandit regret compute run reward surface landscape noise uncorrelated option distance surface relatively value transition loose loose computed run variance minimal variance uncorrelated cost transition arm multi armed bandit maker let visit bandit I short node contain cardinality armed bandit describe arm credible credible reach arm limit undesirable arm allocation transition cost classify two goal upper credible situation path credible accordingly arm consecutive depict show arbitrary goal block select strategy short block compute frame arm frame frame frame allocation start block goal determine credible short pick frame block goal block short behind allocation logarithmic horizon context logarithmic transition logarithmic arm short credible expect graphical formalize gaussian uncorrelated uninformative time cumulative appendix expect regret topology line move regret use block bandit profile axis uncorrelated loose block switch graph simulated graphical block mean profile topology could uncorrelated multi consider three transition algorithm arm bandit armed problem arm uninformative uniformly expect transition among greatly enhance propose decision make armed bandit show capture five multi armed bandit namely ii iii horizon v environmental human decision make embed armed bandit demonstrate efficiently future human phenotype assess real experimental human spatial search allow use uninformative mean human overall present schedule thorough human subject correctness functional form develop algorithm human rich acknowledgement wish thank anonymous comment greatly author grateful corollary discussion addition behavioral protocol exploit bandit choose multiple uncertain address multi bandit armed transition cost armed bandit focus value decision maker mean reward credible limit armed bandit logarithmic cumulative uninformative good correlation greatly enhance short extend make behavior human stochastic armed cost graphical expect arm illustrate performance multi decision control imagine follow scenario order familiar ultimately familiar look interesting include day restaurant little everything decision outcome restaurant city unlikely close home may difficulty interact horizon interact environment engineering reinforcement maximize immediate reward often formulate decision process mdps agent programming find solution size problem often grow curse make difficult general intractable engineering mdps simplify learn analyze derive heuristic reinforcement provable task particular horizon dependent option far restaurant future discover although horizon intractable human restaurant efficient quickly inherently sophisticated heuristic understand cognitive may development mdps paper seek behavioral play model tractable armed bandit constitute mdps plausible heuristic mathematically rigorous context infinite horizon horizon solution horizon finite horizon performance establish armed bandit maker resource sequentially among compete stationary bandit maker instant choose draw select maximize refer standard multi bandit add arm example bandit clinical medical decision maker option unknown patient arrive information gain outcome multi armed bandit fundamental exploitation tradeoff indeed make scenario uncertain rigorous human arm task kind quickly relevant human armed bandit may facilitate specific likewise human operator human armed bandit seminal application diverse area operational armed bandit behavior environment show policy bandit well use heuristic achieve heuristic armed bandit allocation index select allocation idea suffer two drawback hard compute ii nature much recent bandit focus term decision difference reward play role expect minimize definition regret aware play quantity relevance analytical characterize ground break number thereby show cumulative work possible armed estimate asymptotically phrase I bound computation computation confidence logarithmic bound multi multi armed bandit sample develop achieve extensive survey take various related analyze allocation armed bandit study ucb algorithm use kullback leibler armed bandit accord cite frequentist perspective mdps mdps thompson uniform upper confidence optimality bandit uniform logarithmic prior armed bandit well armed bandit transition cost multi armed certain arm bandit switching cost index qualitative optimal armed switching sufficient base armed bandit armed transition cost uninformative scheme incur cumulative cost hold bandit selection expert performance arm spatially embed multi armed performance category category arm describe section vi armed propose set term analogy slot term option bandit refer among make reward maker expect equivalently I pick minimize suffice armed bandit exploitation refer pick pick successful exploration explore arm information pick arm armed bandit suboptimal arm least ir kullback leibler imply cumulative expect regret grow bandit gaussian assume e know process suboptimal easier conversely make reward difficult arm suboptimal one bandit variant logarithmic heuristic option reward reward arm ucb pick depict logic confidence represent uncertainty true option choose confidence example formulate act favorable c represent optimistic reward example option show appropriate term ucb logarithmic albeit policy term multiply close chernoff hoeffding bound probability armed bandit gaussian reward mean sample constructed term normal achieve chernoff hoeffding tail numerically improve result construct ucb provably rely new tight bound tail state frequentist horizon allow integration belief enable capture belief inform perhaps experience problem perspective function heuristic variable fx cumulative cdf give fx conversely provide cdf option unlikely mean fx fx tx sure function term result bayes bernoulli bayes uninformative prior choice speak choose suboptimal yield logarithmic discuss decision subject numerous study cognitive salient wish
expensive encode factor taylor complicated prop education google award encoding frames autoencoder frame compute give derivation inherent image element nonlinearity make sense relate transformation wish detect shall way autoencoder image multiplicative interaction encoding interpret define decoder perform input one sign information tie input two reconstruction filter fourier separately shall show absence pooling allow add bias term definition reconstruction useful representation contraction amount add frobenius jacobian feature square derivative respect linearity add hyperparameter multi layer bi contraction autoencoder make application contraction validation replace motion equation account explain term equation consider task video end detection motion wish local frame performance wide hand spatio temporal perform classic motion turn sum filter allow detect energy spatio temporal frequency turn encode independently move view hand like filter year technique video motion feature tend across task feature design video feature autoencoder cluster known yield structure seem autoencoder work notable ica visually see reference energy compute activity recognition task video linear permit feature encode invariance two perspective encoding view presence multiplicative allow conventional hardware method two frame video classic solve task energy computing sum quadrature multiple behind image content two independent energy may alternative originally cross encodes angle subspace angle thereby cross energy operation model compute achieve end encode transformation restrict space include combination orthogonality transformation implicitly transformation use filter yield relax exact approximate presence equation orthogonality transformation eq detect may look filter transform inductive reasoning step filter pair filter shift shift locally shift shift exactly identical phase detect filter video extended sequence detect transformation relate adjacent q necessary across deep layer summation nonlinearity plus nonlinearity shall discuss attain seem thresholde detect thresholded need two variance input optimistic half distinguish match zero become sum module deep way detect allow multiplicative response response regardless ability presence transform detection operation logical odd sum interaction entirely illustrate neuron interaction consist show efficient highly competitive performing equivalent local competitive winner take assignment center online k mind multiplicative interaction multiplication equivalent k winner prototype allow rule term competition among project come filter image patch filter learn video frames column row six filter nonlinearity global reason state sum product stimulus unit apply shift filter response implicitly pair figure tie contain video sequence tie equal weight enable motion multiple frame proceed concatenation frame stack row wise compose frame replace frame sequence update rule assignment account sigmoid experiment winner task sequence filter generate patch column row block similar model cluster sequence show six center center orientation frequency angle nearby angle alone sufficient pooling motion understand fair describe learn patch block give sub block spatio super densely video overlap spatio classification evaluate activity recognition dataset six train directly total video increase video class leave original video version activity belong svm ap dynamic scene category video version model mean svm table competitive simple evaluate wise autoencoder contraction precision auto mapping cell
noise content one graph adjacency thus original social contain tag dataset contain payoff payoff payoff user fm retrieval investigate usage heterogeneous recommendation system payoff strongly fm popular figure difference allow recommendation since remove neither frequent influence report number make difference recommender may market whose product significantly market share vs market rise law item logarithmic fm item payoff tag select payoff item payoff tag distinct tag describe item perturb large value term clique noise payoff increase lin clearly robust payoff lin sensitive high level noise e number perturb edge break tag word create tag three compound tag contain user may split unique decrease fm tag occur less ten operation fm already extremely test tag tf context independent item dataset retain generate clique scenario keep assign clique payoff payoff noise uniformly bound fm pick among nonzero payoff user comparison probability payoff equal context lin variant lin instance share moderately popular competitive fm popular lin normalize cut option lin weighted weighted inter original cluster together lin macro test different plot perform node lin recover lin macro report lin node provide cluster act regularizer influence figure select vector tune across appropriate figure payoff context lin robust outperform lin payoff noise grow lin world notice lin rely lin macro fm user give positive payoff item lin macro lin outperform lin effect macro moderately item lin lin expect former latter summary exploit moreover experiment contextual hashing technique contain say prove confidence function therein occur lin cb define eq follow u u derive therein eq uniformly cumulative r yield university arm great attention formalize exploitation arise generally strong social component want serve advantage underlie specifically strategy share payoff different experimentally variant art experiment synthetic prediction website play increasingly crucial appearance ever change nature popularity modern recommendation user interest content context content raise explore user create formalize multi contextual bandit become recommender system case recommender social provide recommendation interest user improve friend algorithmic provably similarity run allow interact share user properly reflect allow place node run contextual algorithm reproduce hilbert previously problem rely share network implementation guarantee principled describe drawback feedback sharing mechanism cause small social fully reliable behavior network network collect sensitivity two modification aim first pair scale strategy treat cluster able simultaneously achieve world dataset extract social service music platform last fm benefit social improve recommendation recognize fact recommender system model content introduce information contextual work work throughout motivate empirically section signal contain dm dt ta vc update maintain prototype vector bias vector tc linear bandit tc reward achieve estimation base suggest g actual rank adjustment precisely see lin bandits lin operate subscript replace ti say payoff receive assume prototype keep lin first kronecker matrix dimension compound vector description lin present confidence laplacian replace suitable explain k km ta accord bandit per node represent gets spread block contextual information available reward lin rely lin mainly inversion perform
root take guess possibility make density e gamma density estimator previously dependence recommend permutation remain permutation end pr profile pr residual yield generally rough optimization expensive justify simple iid location easily profile draw student center plot mode smooth pr bayesian dirichlet process residual yield evaluation expensive computationally practical automated repeatedly carlo assign proper prior technique carlo carlo likelihood strategy produce routine dimensional work direct opt section existence pr likelihood difficult answer recursive structure pr conjecture concave concavity unique maximizer moreover concavity could use establish pr include conjecture available important sense mixture normal heavy tail accommodate robust maximize solve weight demonstrate assign tend get support extreme structure along line hybrid scale parameter trivial hold respect proposal integrate respect precision inverse propose hybrid algorithm point iteratively solve least square repeat pr residual discuss em particular identify outlier case influential pr estimation density justify pr argue I I integral kullback leibler equal zero heuristic inequality nf divergence specify converge leibler small term reach increase give reasonable expect increase pr numerical hybrid pr method ordinary lm point fit show fit except weight density display pr model assign weight outlier standard gaussian less water construct nuclear see screening variable consider predictor denote guarantee marginally significant significance sensitive choice quantile sort usual way inverse maximizer nominal confidence interval coefficient indistinguishable confidence roughly long reasonable promising interval seem normality conservative argue provide list variety student mixture latter tail example exhaustive hybrid model compare variable square estimator l ml high intercept term table exp explore take scale normal mix maximize pr base pr em hybrid outli detection justify robustness estimator pr make open question regard asymptotic particular concavity various existence estimator pr limited iid explain section need work case paper motivate study setup grateful associate approach nominal construct technique nominal maximizer act denote percentile confidence since normality currently pr reasonably public economic characteristic public take variable percentage growth library public display outlier display little weight plot observation highlight point far see plot mix around mark help explain fit sim pr result sim l ml l pr pr proposition remark definition important nonparametric recursion simple computationally method recursion construct estimation maximize function hybrid predictive em method performance analyse em marginal normal scale regression response predictor row vector regression assume square solution lose sensitivity consider robust sensitive outlier fall huber huber sum square residual surveys technique include outli prefer distribution scale see expectation maximization normal mixture student specify mix write inverse square completely identifiable model maximum profile conditional likelihood produce nonparametric mix wang profile local marginal introduce reasonable joint west ba expensive several computationally alternative pr design fast pr methodology parameter nonparametric discuss recommend latent easy pr produce detection conclude remark pr alternative monte summarize mixture py density finite measure pr density step compute estimate ease implementation produce prescribe dominate pr mixture pr mix consistent rate
interpret worth unweighted draw density effective tuning probability numerator let auxiliary sampler auxiliary attain ideal use various euler set sampler hence shrinkage coefficient estimate performance auxiliary importance sampler sampler auxiliary importance sampler maximize penalize log improved regularize choose importance sampler illustration student maximum criterion stop prediction improvement stop ht maximizer penalize compute prediction back stop go go back otherwise stop explore look robust first sufficient sampler sampler use dataset euler sde simulated compute regularize regularize derivative free intel ghz times compute optimization successive default default evaluation allow sde equilibrium interpret volatility generate initial maximum root rmse estimator estimator path bias sample r table sampler regularize reduce bias rmse small introduce penalty regularize sampler red away maximum typically happen approximated sampler poor approximation small could poor approximate select performance indicate robust make easy path sde see section obtain sampler take second section implement second second computational grow note equal choice consider three wiener dataset initial condition commonly value state maximum estimate regularize evident regularize rmse regularize sampler perform drift generate show increase regularize sampler especially fix control path actual however sampler generating proposal trajectory second second specific dataset similar dataset transmission st ct month require density bias rmse improvement sampler unobserve context promise data sampler sampler r rmse regularize population disease cause important transmission mechanism deterministic propose sde propose display hold division epidemic occur epidemic loss natural record laboratory hence infect transmission sde infect assumption epidemic transmission transmission sde add population via wiener explain sde let sufficiently infection death furthermore covariance small hence square root quantity one euler sde result dynamical system sde respectively although provide interval approach goodness trajectorie death epidemic epidemic parameter remain unchanged sde remarkably job fit complex extend become complexity direct course entire biology people situation infection population infection monotonically epidemic equal infected population close case basic would expect spread infected parameter observe dynamic describe differential transition model balance complex importance order number simulate path sampler improvement keep measurement unknown carlo jump offer alternative particularly markov sde modeling diffusion jump dependence structure among simulation observation start formal tune find challenge simulated penalize simulate transition mle base convergence distribution future acknowledgement material upon grant ef lee support research office nf national security research utilize nsf grant thompson introduce division like thank either transition observation propose auxiliary parameter density auxiliary likelihood simulation illustrate disease euler penalize disease disease population transmission epidemic realistic transmission disease epidemic challenge sde extension simple chain several interact population moreover application biology economic bioinformatics inferential may especially multivariate sde importance theorem diffusion coefficient sde dependent diffusion development mainly slow dimension penalize computationally sde firstly unobserved integration importance sampler estimate arbitrarily true sampler improve brownian bridge sampler good recursive optimization apply zhang estimate sde introduce bridge sampler sampler area improve area inferential viewpoint practitioner multivariate unobserved b observe impossible costly interval consecutive even observe long regularize sampler choice cite determined importance transition sampler penalize likelihood select importance unbiased regardless choice importance attain mt I sampler approximate probability mt compute maximum sde simulate simulate performance three describe sampler construct simulating path euler simulate trajectory euler intensive multivariate sde sampler bridge propose euler sampler method draw procedure multivariate density draw multivariate
link view focus observe network evolve method consist unsupervised approach node assign similarity imply link attribute solely structural similarity structural neighbor ensemble path comprehensive review another supervise node binary whether predictor attribute pairwise compare supervise probabilistic model incomplete link prediction hierarchical relational supervised category similaritie link treat example particularly negative biological certain protein edge mean interaction indicate interaction detect interaction spurious throughput experiment protein new prediction negative formally observe edge rate kind fact estimate ranking estimate ranking sufficient many without highly ranking utilize topology organize link network optimize discuss propose link prediction node otherwise prediction either asymmetric direct version assume observe probability edge edge matrix easy check increase function imply increase crucial ranking positive recommender friend correspond investigate infer throughput case general criterion matrix describe similarity network later network h close node similar similar method network people tend friend etc valid feed web contrast plausible motivated assumption propose estimate tune first term connect key use loss negative likelihood quadratic true experiment conduct subset protein infer available modify otherwise criterion positive discuss partial sum refer rest h intuition eps cm undirected close similar pair combine multiple option find well range reason block network link bernoulli variable number hope base indicator estimate direct optimize criterion equation write form matrix iteratively define plug compute product applicable approximate use fact serve substitute undirected partial term updating direct truncate solve fast method dense descent performance consist hand subtracting logit link overall sparse network report figure asymmetric give network correspond undirected indicator bernoulli title edge true miss probability define similarity criterion prediction roc estimate false pair top without pair define show false range undirected curve roc curve benchmark sim eps cm sim eps sim eps sim eps sim eps sim eps sim eps eps full criteria little difference undirected partial sum always performance comparable gap unsupervise semi negative proportion large sparse roc dense intuitive explain large gap link counterpart confirm number network link challenge contain protein contain highly protein node construct similarity gene profile hybrid link eps eps sum criterion coordinate depend value roc random well criterion supervise outperform except small false positive positive sensitive rely heavily network rate substantially network topology similarity protein sample school eps school eps school eps school national longitudinal health detailed network contain high student connect friend average around latent variable network due covariate construct similarity minimize topology protein construct protein article link network ranking parametric rely pair range explore combine achieve robustness investigate ranking develop extension allow would allow example correctly ultimately
continuously nonlinear joint density denote respectively solution additive fix system ill pose operator equation unconditional given say use operator advantageous integrate g dy model g operator derivative local identifiability nonlinear necessarily imply identifiability condition refer marginal independence alternatively roughly speak dependence regressor vary fact real independent take value regressor q easily see unconditional conditional independent obviously interestingly guarantee rule independence local recall eigenvalue conditional expectation operator normally development random marginally elementary symmetry expectation operator map self adjoint permit th polynomial iv iv eigenfunction eigenvalue keep gaussian operator polynomial operator z turn surprisingly two sufficient ensure another additional copula continuous function equivalent f independence family density complete almost surely invertible differ conditional copula argue condition neighborhood may apply heuristic generalize discrete continue next binary explanatory r counting measure identity linearly rewrite operator return material condition result primarily source motivation present first hilbert discuss relevance condition nonparametric instrumental special source us relationship smoothness kx lx dy smooth polynomially exponentially analytic usually formulate smoothing property operator hilbert conditions linear adjoint operator scalar product continuous monotonically calculus notation l decay integral operator smooth kernel choice refer information context instrumental integral operator probability compose derivative typical I hence application operator infinitely analytic singular exponentially condition restrictive since decay super h source difference exponentially entail infinitely analytic guess polynomial fourier correspond smoothness desirable decay logarithmic fourier alternatively instrumental restrict banach operator map analysis regularization method hold h old source pp logarithmic discuss newton guess close depend appear introduction suffer minima frequently unlike theoretical always functional minima turn rigorous often hold lot problem regularization abstract application way kernel compose estimator kernel must fulfil strongly consistent enough strong consistency boundedness derivative joint density operator bregman consistency cone operator verification operator analogy operator model rate nonparametric instrumental determine density hence decay slow depend smoothness kernel operator analytic merely attain due estimator logarithmic continuous explanatory dependent nonlinear simulation solution approximate dimensionality regression simulation correlation term information instrumental variable although varie show achieve hold definition ever look condition hence formulation instrumental look necessity yield true discretized chose regularization stop principle guess exact density reduce error suggest identifiable solved method density estimate observe discretization discretization value figure exhibit exponential accord slow rate exact evaluate sample joint density density develop test approximate e reconstruction size explanatory exact guess produce become reliable enough quantile error median median mm let formulate deterministic respect approximation assume uniquely stop let assume exist notation k definition plug condition leave side k right side inequality inequality monotonicity e together stop rule bound prove induction arrange induction tx put necessary inequality side take thus convenient plugging side k tt monotonically k completes easily converge probability cone condition fulfil go convergence assertion theorem nonlinear de discuss solution nonlinear noisy instrumental convergence emphasis instrumental assumption replace independence demonstrate subject nonparametric iterative instrumental analyze estimating equation instrumental integral operator estimator available typically operator pose technique apply regularize newton numerical ill guess practice iteratively regularize suffer functional avoid difficulty local minima moreover newton eq newton problem regularization parameter hilbert guess start method suggest analyze old logarithmic condition reference therein general regularization incorporation penalty variation instrumental give basis basis constant main entropy norm incorporation structural close negativity convexity concavity term mathematical study quadratic number paper appear mention variational convergence close treat equation hence result instrumental ill pose operator interest application regularization nonparametric instrumental example nonparametric solution integral fr decay lee important integral operator analysis rate
bit concatenation bit interpret refer hash seed hash function map seed seed uniformly distribute trace function importantly technique rely hash integer ss seed resolve resolve next implement efficient hash construct seed modular division suitably since concatenation primitive generate iterate recurrence different hash practically calculation primitive datum platform equal addition integer write modulus multiply prevent recurrence arithmetic infer state step chernoff commonly specify incorrect lie specify guarantee accordingly give satisfy cumulative probability chernoff bind statistically estimate less express nm nm tractable h statistical checking platform experiment cumulative generate algorithm vertical blue mark grey multiple respective optimality wireless protocol aim device use point blue numerical model checking denote black indicate reveal demonstrate scalability meet intractable numerical true chernoff probability minute continuous mdps seem mdps although present respect budget chance limitation algorithm develop construct piece wise european union framework mdp process numerical often intractable present scalable verification memory facilitate scalable verification markov decision cost action real optimisation probabilistic transition state execute affect system mdp semantics action probabilistic may sequence every node fill node leave node node node leave edge edge style pt node fill black node bend bend bend node bend right p focus mdps check system logic quantify probabilistic classic mdps concern classic verification existence leave classic solve mdps check mdps programming action state sequence state intuitively choose sequence choose dependent operator intuitively true achieve transition achieve achievable optimal curse state exponentially interact phenomenon lead discounted mdps briefly check address checking smc probability proportion trace individually smc work construct trace decide property correspond return priori give statistical confidence chernoff test simulation trace hypothesis construct explicitly statistically divide computing architecture since probabilistic choice mdp whole smc see create facilitate verification storing essence possibly fully seed numerical verification algorithm mdps derivation statistical obvious encounter statistical demonstrate core smc practical implementation adopt budget learn discount construct action importantly respect reward probability check potentially infinite mdps explore bound however exponential action great current estimate successive maximum discount probabilistic explore error specify difference allow discount guarantee eventually terminate recent attempt spurious standard use approach limit affect scheduling make therefore attempt address mdp model author discount induce may smc store visit improve near optimality address standard checking mdps smc decide mdp threshold generate candidate improve trace limit action pair outer loop iterate optimum explore local maxima make reduce exploration probability optimum repeat outer eventually mdps
relate sublinear give algorithm path kernel support tool regularize widely tool along successful svms various literature largely attempt relate goal mind algorithmic respective sublinear alternative theoretical literature svms spirit trick insight implicit high simple variant equivalent kernel transfer svms svms equivalence exactly lasso input lasso identify inactive screening svms order eliminate potential thereby training study lasso regularization change translate path svm scale support problem column whose entry include commonly margin regularize offset allow hyperplane pass origin offset variant margin lasso variant square fix ball value constraint application interpretation hand usually column approximate single dictionary vector interpret input book recent popular focus equivalent svm solution simple preserving instance appear hard exist separate lasso margin svm show reduction require data svm formulation kind unseen goal lasso e turn explain significantly despite title address insensitive variant vector author equivalent become reduction work reduction choose unfortunately variant primal originally specialized area biology author already reduce idea ball lasso dictionary negative point formalize set unit simplex ball n absolute value vector euclidean notation ss nn na ba together binary illustrate partition plane site classifier write assume hyperplane precisely introduce important addition hyperplane want hyperplane separate define distance hyperplane among formalize onto optimization dual exactly think alternative problem linearization avoid formulation matter formulation feasible weight represent vector correspond variant property dual discuss subsection crucial optimize provide svm problem becomes attain hold mean margin objective useful attain quality take separate difference margin optimum gap useful stop criterion know separate solution successful soft margin concept importance soft variant formalize margin svm offset q introduce slack penalize regularization tradeoff attain parameter fix explicitly equivalence soft dual state margin data completeness lagrange svm problem hinge refer outlier penalize margin affect form practice svm variant lemma weakly svm one vector coordinate rescale length clearly attain margin svm bias variable pass separation trick dimensionality add fix value g offset nevertheless effect arbitrary scale feature value one also result svms popular anomaly investigate lasso problem two subsection warm consider negative dual non translate matrix obtain crucially domain ensure optimization preserve objective reduction direction reduce svm negative trivial relate translation explain subsection polytope vertex polytope represent particularly hull real writing write vector horizontal concatenation note several lasso n regression equivalent svm translation instance lasso preserve solution feasible svm vice svm instance instance reduction fact give proposition sign improve improve fact separate lasso proposition obtain sign improve negative strictly entry define vector respect value tc td ax b ax ax ax ax ax since proof scaling assume along strictly show negative tc td tc ax ax ax ax ax ax ax ax ax ax axis htb weakly separate angle weakly separate definition weakly separate unit product must w htb translate point vector claim pair translation ball particular contain lemma ii separate translation separate establishe extend definition also strictly desire implication relate respective return separate remarkable since size input matrix therefore precisely prove twice sublinear query necessary explicitly need every return sign lasso allow access entry pick hand open sublinear svm exist fraction traditional learn approximate linear exist discuss linear combination point give inner implicit analogous objective purely kernel product mirror translate crucial inner two kernel space kernel correspond lasso matrix space approach counter difficult use moment well add lasso kernel interesting relate lasso study application lasso svms early application instance translate sparsity svms motivation result classifier proportional classification vast literature lasso certain goal row sparse e translate sparsity svm characterize sparsity characterize svm support assume applicable type construct sufficient remain interest support direction svms example asymptotic hold lasso grow develop consist guarantee provable first translate svm discard started discard unchanged aware rule literature far subsection complicated direction reduction gain support svms simple direction main free svms good determine soft naive grid change develop svms popular particular investigation lasso simplex enable precisely problem maintain along path recently objective continuous parameter path lasso piecewise number piece I complexity inspire svm easier bad case parameter rescale relative occur change apply formulation vary formulation solution identical obviously monotone decrease grow large value lagrange penalize mapping monotone kind go appear pattern sparsity simplex parameterization unique rescale svm potentially occur bad case operation rescale practice preprocessing mean instance move
polynomial resolution aim dimensional application grow application order classical construct v call compose multidimensional degree try real evaluation least construction define sufficient evaluation condition problem priori adapt square bad ill regularize square denote j algebraic regularization regularization obtain another validation value decade extensively study scientific expand quantification particular tensor basis sample incoherence depend quantification strategy approximation precisely admit approximation random function approximation ideally combinatorial optimization certain consider optimization pursuit lagrange multiplier relate appear least contain solve lar namely solution non extract validation estimate rely formula modify lar briefly work lar modify lar zero zero j rely correction approximation tensor subset induce regularization presentation correction compute solution order tensor elementary n tw k square enable result rank difficulty mention square tractable relaxation tensor w propose subset approximation practice successive correction possible straightforward problem pg type tensor tensor tucker tucker comprehensive tensor parametrize r parameter basis coefficient solve square problem lagrange minimization successively fix denote function write classical induce modify lar leave cross validation approximation h evaluation max go solution approximation change construct replace square without square replace also one ridge regression optimal fold approximation variant successive sparse construction suboptimal rank however direct approximation format start proceed follow reformulate alternate successive leave one solve update step negligible approximation effective improve updating could follow correction space tucker tensor update yield improvement clearly dimension representation algorithm approximation vector evaluation evaluate use matrix sequence increase canonical rank procedure split approximately subsample evaluation test obtain sequence correspond mean fold k op approximation consist without induce closed minimization ill pose propose successive correction suboptimal advantage successive problem small iteration first highlight benefit greedy low approximation give sample need detecting approximation simple respect example estimate error rank carlo sparsity short total parameter tensor approximation benchmark uniform subset alternate aim estimate format evaluation upon prove optimal ordinary square evaluation scale degree support test function scale dimension polynomial robust scaling lead unstable low function construct ordinary isotropic space degree first space consider one follow plot repetition rule quadratic range find rule yield small value rule degree total modify size stable plot enable high rank approximation well feature conclusion sample rank rank evaluation enough certain smooth basis rich piecewise wavelet global rank one correction perform alternate algorithm lead relatively degree illustrate purpose polynomial allow rank approximation variable uniformly introduce polynomial polynomial degree partition orthonormal basis compose support element restriction rescale note interval admit piecewise correspond storing expect detect detect sparse approximation use illustrate sparse low rank ol alternative optimal without update type correction different denote dimension solution rich approximation piecewise ratio one yield find allow recover op various indicate element select cccc c error analytical conclusion algorithm able give accurate ol select effective smooth function basis appropriate simultaneous basis study function random uniformly space space polynomial wavelet resolution fold selection size find inaccurate increase function basis size show optimal sample ol important fully potential tensor evolution respect optimal figure different size able capture feature respect size analyze wavelet optimal fold cross direct wavelet approximation size approximation low representation low dimensionality interest use greedy representation reduce learn interest representation direct square regularization blue force compose boundary load boundary introduce compose element degree htb discrete coefficient mass unitary modulus parameter top right two structure subset polynomial number sample denote dimension constraint ols solid correction solid approximation giving line stable approximation polynomial good approximation ol constraint right ratio decrease polynomial degree partial ratio exploit especially coefficient quantity rank polynomial degree illustration enable increase high capture accurately tensor approximation stochastic greedy construction sparse
three similar choose provide goodness equilibrium identify drawback utilize frequency lie result least allele model due formulation estimate frequency fall natural first unnecessary modern snps intercept snps allele frequency approach significance snps population snps traditionally extreme allele although rank specifically attempt detect snps result form snps locate rank human rank study locate role distinguish phenotype snp rs snp human phenotype verify snps plot value subtle difference allele gene snps candidate tumor play show severe protein involve identify genetic bernstein breast tumor know type file call available allele frequency pca allele approach mean assumption capture allow individual allele specific population underlying factor propose computationally estimate build population term improve straightforwardly incorporate inference require well behave allele statistical inference equilibrium marker trait amenable complex population structure framework motivated well allele try allele frequency always allele estimate lie become genome fundamentally characterize care treatment text case pca example show population individual specific allele frequency avoid maintain relevant distribution logistic value color report bar side allele frequency form column column display pca across logit logit logit average snps error value scenario mark scenario take f cc scenario f fit psd psd pca scenario fit cccc cccc cccc fs fs bn psd e e e e e spatial e e e e e e fs e e e e web site individual identify second degree yield snp snps platform individual snps individual utilize simulate release consist european group china identify minor allele total determined spurious identify note plot apply form goodness snp ij kk apply calculate goodness goodness pool across data snps calculate snp form separate nan accord minor allele bin allow allele f summarize bn snp allele value snps matrix I reflect proportion row draw I I ip allele calculate psd analyze snp estimate marginal allele estimate estimate analysis al snps utilize row ip randomly among allele increasingly close assign individual equal spatial position individual simulate snps individual intercept place square beta scenario represent place individual corner create allele datum via snps pca take logit allele frequency estimate fs p software estimate evaluate allow difference allele allele frequency compare allele frequency specific allele frequency derivation population describe see arbitrary structure capture individual population allele frequency snp conditional snp allele individual good useful estimating plug replace modeling explain group accord example convention cc latent profile latent categorical population nonparametric estimation another convention could nonparametric inconsistent equation identify binomial linear particular begin work notably small genome wide substantially make model strong unnecessary wide datum quite calculate latent fit ref intensive dimensional iteration convergence twice burden make difficulty frequency several extension find algorithm genome poor pca datum directly aim factorization identify approximate translate interpretable computationally quite useful image human lee connect structure four simulate four scenario lose simulated column result generate value lose principal pca fit cccc cccc fs pca bn e psd e e e e e e e e e e e e e e e fs e e e variant rs rs rs variant p rs rs rs rs rs smc sp rs rs rs variant kb p rs rs rs rs smc rs variant rs rs reference rs variant rs rs rs rs variant variant rs variant rs rs variant rs rs reference rs rs variant rs rs kb variant rs rs rs rs rs rs rs rs cd variant rs variant rs rs variant kb rs rs rs rs rs rs rs rs rs variant rs rs rs references institute nj department molecular university nj equally present division institute national health md correspondence edu abstract abstract modern typically genome wide individual diverse probabilistic account population structure prominent focus modeling require interpretation formulate include well note drawback seek logit underlie latent capture advance human diversity make minimal modeling wide modern genome wide association study identify genetic throughout genome associate trait challenge analyze spurious association development comprehensive genome variation evolutionary rigorous understanding history expand ability signature important insight human genome diversity world genomic produce genome individual diverse systematically characterize genetic complex force drive fundamental provide presence population series influential method primary proportion p allele every marker individual instead allele flexible method include aforementioned bn psd maintain probabilistic estimate genetic pca genetic study pca allele structure pca produce allele frequency estimate unit approach latent observe include latent propose extend perspective towards interpretable allele frequency range summarie convenient bridge exploratory modeling exist allele superior accuracy allele snps population proximal snp proximal positive human snp mutation experimentally validate role phenotype snps human study disease cancer population structure frequency homogeneous throughout often frequency among european receive accord individual phenomenon recent explain difference allele snp th likewise latent variable datum detail make assumption detailed binomial datum minimal nature real directly obtain estimate essence form intercept constrain formulation psd let set decomposition svd note loading svd construct multiply f diagonal estimate f interval estimating variable subset adjust subset enough span basis parametrize perform snp allele logit important due apply recall form row mean w compose right svd
kx forward quantity activity rate go unit compute activity one modulus let let final choice parametrization output interpretation transfer backpropagation involve quantity kx jacobian activation k eq analogous define modulus index unit right block fisher costly compute output intrinsic simply output change k k k eq yield fisher obtain modulus construction intrinsic modulus modulus let input modulus metric immediate modulus modulus number computation backpropagation pass equation modulus relate approximation involve function could ill behave cross unit involve away term define intrinsic manifold unless additional affine riemannian critical prevent one approximation modulus fisher modulus relate try modulus fisher modulus point quadratic instead backpropagation involve keep involve modulus fisher metric op still matrix incoming parameter unit invariance intrinsic go simple define zero bias quick step classical backpropagation simplification somewhat composition range range make activity live back activity origin activity look invariance likewise parametrization basis value specify quantity decomposition change affine activity w ik ik separation formalize w change parametrization ik affine parametrization ik affine may ik parametrization activity intrinsic property ik vanish parametrization parametrization complex try parameter incoming w ik try simplify bias intrinsic intrinsic w parametrization scalar product ik k w scalar vanish affine incoming decomposition parametrization kk intrinsic new readily ia intrinsic metric fisher op assume compose affine quasi diagonal reduction inversion result diagonal quasi quasi entry gradient descent note cauchy intrinsic direction give dataset target average put subscript network intrinsic differential define direction give symmetric natural natural op op eq l la metric metric op op application activity sigmoid update intrinsic indeed even give intrinsic parameter intrinsic differ affine transformation amount ideal limit rate equation quasi diagonal invariance restrict choice layer ordinary neural network reproduce discuss define output distribution network choose fisher matrix average wise fisher explicitly train fisher output w ij fisher old contribution online draw output even section important variant layer definition towards actual upon intrinsic op contrary target carlo op variant quasi network unit incoming gradient op metric op op metric natural similar latter perform substantially modulus backpropagation rate latter course convenient summing outcome backpropagation rate general ordinary neural transfer modulus reproduce fisher term fisher incoming fisher unit matrix fisher matrix associate parameter correspond together classification interpretation perform fisher approximated batch use dataset one simplify backpropagation rate modulus immediately treat objective provide gradient gradient vanish degenerate invertible vanish interpretation smoothly activity rate rate usual depend gradient activity function write instance value trajectory course initialization behave transformation scale inversion unit stay behave backpropagation scale network scale evolve slowly conversely rescale unit go inversion way activity close evolve training feed stay backpropagation activity unit natural outer quasi w two network behave next simplify immediate intrinsic object trajectory op invariant affine activity obtain unit monte carlo op final output initialization thing backpropagation quasi newton approximation insensitive unit network traditionally normalize activity dataset unit unit activation fast highly quasi diagonal average weight give modulus modulus still invariance obviously take weight take natural op natural quasi invariance affine signal unit receive incoming unit evolve correlate invariance non unit incoming unit invertible activation q still parametrize dual transpose affine original step carlo gradient op non network network initialization place backpropagation quasi consequence interpretation proposition intrinsic construction note quasi tuple incoming activity incoming metric set activation fix natural gradient op quasi unit input activity income see set define metric singular singular intrinsic gradient non input weight least run incoming modulus modulus fisher modulus op modulus incoming activity see dataset prove implicit incoming unit unit natural update singular unit vanish e incoming unit activation systematically incoming viewpoint add definition apply thus ascent viewpoint vanish direction technique advantage produce thus formal article method depth symbolic sequence choose find fed input auto encoding ideally learn encode sample bit room output purely underlying method link hide middle scheme link layer unit layer string identical gradient parametrization non check invariance invariant algorithm implement activation backpropagation quasi op quasi sample metric quasi gradient equivalent keep diagonal divided sample small directly involve small affect contribute size would probably standard point sigmoid activation initialization initially response namely weight center gaussian deviation initialization incoming factor sigmoid sigmoid magnitude adaptive gradient metric implementation make divide improve loss initial value learn influence make sense advantage rate place backpropagation whole dataset run aside convert rough since implementation especially small value way network situation auto unit put natural disadvantage roughly lc natural metric quasi monte carlo op gradient newton fisher report end iteration interpret represent bit correctly sigmoid report performance run loss backpropagation gauss newton natural quasi natural monte carlo monte diagonal op well trajectory plot fig sigmoid implementation completeness run illustrative small elaborate competitive newton method implementation sigmoid implementation closely variation second inclusion regularization term invariance isolate initialization sigmoid differ double invariance bad layer directly instead perform competitive output task gradient op perform poorly set differ op gradient root invariance op natural well recurrent setting method layer gradient indeed reason network metric op contribute fisher contribute viewpoint op diagonal gauss newton different sigmoid interpretation quasi metric differ inclusion sigmoid improve gauss implementation setting internal unit input center diagonal metric outperform gauss even quasi arguably diagonal newton introduce diagonal gauss experiment one write invariant issue unit incoming unit cost invariance property mathematically activity neural treat manifold outer encounter task substantially outperform method use close gauss crucially differ inclusion diagonal gauss method substantially gauss newton anonymous reading suggestion fisher course output target parametrize activity fisher bernoulli interpretation bernoulli variable variance interpretation k layer kk layer softmax spherical fisher kk plug correspond kk ji ji kk yield proposition variation coordinate increment function v whose definite hyperplane neural direct acyclic unit activity belong activation kk manifold output induction kk output bilinear way fisher space parameter define riemannian map bilinear bilinear two semidefinite tangent space manifold differential map bilinear input likewise except differential network linear induction direct acyclic input unit us interpretation kt x kt define bilinear ii influence add contribution let metric since metric intrinsic object parametrization manifold intrinsic norm invariant invariance metric sc bx sc definition proposition lem lemma ex exercise four algorithm scalability principled invariant transformation representation obtain geometry either natural scale scalability keep mathematical train backpropagation backpropagation datum instance affect performance weight trivial restricted boltzmann machine ascent help center effect instance recommendation activation backpropagation fast reproduce input train backpropagation rate input pair insensitive activity trajectory backpropagation change sigmoid activation amount bias preserve gradient direction invariance design particular indicate transformation know include quasi change transform maintain prohibitive scalability keep limited memory property scalability develop invariant riemannian geometry output data backpropagation invariant require connect average unit fulfil task scale block income independently income back train neural network hide symbolic arbitrary distinct result adapt connectivity network newton approximations gauss describe intrinsic way stems follow reasonably small connectivity quasi remove dependency connectivity way sigmoid quasi natural output per whereas quasi backpropagation diagonal sometimes natural discussion invariant unique one proposition serve implementation principle choice riemannian geometry build network appendix together metric discuss approximate proof neural symbolic introduction invariant overview result build algorithm suitable backpropagation space rewrite distance backpropagation backpropagation minimal influence hand rather intrinsic natural invariant norm place network differential manifold riemannian gradient riemannian small enough learning improvement result algorithm invariant include affine unit invariance gradient diagonal receive give incoming unit output correlate input normalize gradient interpretation desire quasi distinction come separation bias intuitively receive ik ik might tune weighted average add greatly improve performance page arguably fa rate derivative effect gauss algorithm newton approximate gauss newton way article mainly quickly experimentally impact would perform build string feed ideally encode bit layer room backpropagation pass average sigmoid backpropagation reproduce bit auto comparison backpropagation computation bit sample natural impact diagonal quasi diagonal gauss newton invariance sigmoid diagonal gauss sigmoid implementation final somewhat close diagonal per input perfectly diagonal invariant diagonal gauss method invariant also exact gradient thank natural target fisher implement poorly auto encoding fisher numerical term prevent bad upon inversion invariance trajectory sigmoid initialize still overall affect though quasi sometimes reach metric maintain affine income highlight viewpoint network unit level unit point fire ji opposite activate common tangent refer mostly change unit dataset input layer arbitrary generative layer activation interpret goal probability define sum minimize output mean loss loss output activation activation interpretation must remark softmax backpropagation amount descent define layer activation propagation derivative indeed bias convenient backpropagation gradient descent follow fire network diagonal diagonal reduction preserve enough learn natural update eq take k ij ia turn follow compare course mini stand input fisher sample discount compute inverse unit use costly rest backpropagation algorithm inversion equation rule discount enough point evolve along set contribute small matrix close poor numerical inversion initialization great variation scalar product orthonormal basis coordinate gradient derivative quantity direction ascent step rewrite regular gradient yield clear scalar product influence indeed direction expensive account happen work partial derivative scalar orthonormal conversely norm direction depend guarantee decrease vector use gradient ascent parameter basis network ascent w space sigmoid activation eq bias activity sigmoid try numerical gradient value different update back ik follow sigmoid form apply ik apart obvious speedup difference backpropagation opposite small assume activity center around gets change need thing stay find solve high derivative ij f descent depend system decompose define change induce version fisher metric well present metric unit influence metric newton hessian neural manifolds riemannian metric intrinsic unit activity take assume typically without origin room multidimensional activation unit point always activate bias bias part unit decode belong implement ascent metric parametrization intrinsic trajectory object activity activitie manifold write intrinsic manifold
dominate provide go reversible f f f f distinguish eq establish q pt purpose f f pt f odd index pt finally even odd f inequality dominate summation establishe also combine extended average chain context turn mcmc mcmc try metropolis marginal algorithm provide theoretical context pseudo designing improve apply hence way successively cauchy inequality c statement fy u ty u yu u establish let complete iii generate pt jacobian value transformation continuously instrumental respectively respect dominate lebesgue mcmc ii algorithm expression u lebesgue eq auxiliary taking draw ty dominate measure regardless mcmc cover provide complete apply equal yu u function change equivalent fu fu fu fu fu u complete generate define case propose conditionally g iv conditionally give dominate nonnegative kernel dominate denote check u ty particular case proof variable reject obtain model proof presentation go remark abc context kk rr yy e support average evolve reversible variance soon pair order sense lag augmentation type metropolis refer hasting complexity within mcmc target normalize hasting markov reversible instrumental choose metropolis average reversible chain question order former dominate definition markov chain another general ordering propose homogeneous reversible kernel asymptotic markov evolve former integrable chain evolve reversible markov kernel work deal systematic comparison variance approach spectral state mcmc asymptotic augmentation propose pseudo marginal refer contrary pseudo marginal turn drive relate measurable lebesgue integral px induce two integral acting specifically fix distinguished denote f integrable measure notation brevity operator jensen recall kernel reversible adjoint belong state chain extend chain invariant denote order reversible kernel seminal establish state reversible chain reversible transition kernel reversible nevertheless space limitation say order implicitly page formalize order see concern idea homogeneous reversible reversible markov transition evolve mention practice chain evolve hold extend hand able dx straightforward satisfie condition consider even check establish upper th iterate develop function markov admit hold definition fact necessary sufficient development sufficient imply chain evolve geometrically ergodic evolve proof find reversible instrumental fundamental metropolis hasting reference therein situation sequel dominate kx derivative kernel kx theorem augmentation wish write u convenience analytic marginal computationally expensive letting component chain let instrumental define algorithm draw call draw ty family ty dominate nonnegative case typically dirac continuously dominate verify describe remark extension hasting draw move candidate construct draw concern fact special hasting algorithm replace acceptance obtain hasting small metropolis acceptance note probability mapping jensen ty computation metropolis acceptance algorithm proof rarely prevent metropolis explicitly approximate tool order theoretically construction carry detail sequence serve chain evolve product let essential kernel chain evolve evolve accord identity mapping evolve construction chain immediately u u implicitly associate check sequence generate marginal first evolve product reversible probability particular reversible reversible sub transition diagonal trivially complete simulate infeasible family dominate normalizing sample discuss sampling infeasible marginal contrary reversible due however unity accord alg term metropolis metropolis see resemble closely important difference store along marginal mc acceptance playing role hybrid turn algorithm iii draw auxiliary replace candidate previous acceptance turn metropolis interestingly comparison candidate accept noisy systematic unity translate component remain unchanged may embed properly propose abc discussion abc term abc yu desire assume
low bound difficult low candidate rather b problem therefore reduction bit polynomial need formulate point prove np nevertheless approximate realization make huge difference bad illustration graph complexity plant clique fix connect pair edge independently pick arbitrarily place connect vertex enyi plant plant clique clique pc clique find clique plant traditionally attribute plant size base distinguish polynomial plant focus prove algorithmic technique relaxation confidence difficulty researcher prove assume dependence approximate nash equilibria subgraph make plant clique level throughout randomize constant arbitrary polynomial randomness fact randomize powerful detection test lead polynomial sdp plant clique pc take time condition improve characterize fix instance plant clique potential extract choose right vertex among add new left vertex vertex every old vertex leave resp random resp result plant clique plant let rademacher variable random column put step bl polynomial logarithmic term level achievable positive fix exist exist bl k bl bl independent rademacher g x plant rademacher variable rademacher write correspond draw contain ball among type plant rest plant clique replacement counterpart follow joint ny coordinate iid close variation q together chernoff yield combine view hold distribution prove fix q support observe equality hold random hoeffding eq moreover inequality least display enough position integer check satisfied imply exist n tv n moreover theorem result fix randomized randomized polynomial test last pay use partially support foundation grant dms dms partially support wu fellowship tn tn tn sets unit support classical hold union desire result decompose diagonal formulation semidefinite eq diagonal ij get follow similarly bind diagonal yield mm corollary axiom partially wu fellowship grant dms dms financial nj usa operations research financial engineering nj component bring evidence towards computational signal strength detect computationally statistical plant clique class modern landscape past decade paradigm fairly turn interesting often lead computationally lead relaxation overcome come satisfactory purpose notion shift along want detect presence fall plus matrix plus towards complicated dependence principal pc method propose prove level develop efficient perturbation mdp recently develop semidefinite dimensional low former semidefinite unfortunately sdp algorithm plant problem hard focus testing suggest gap optimal detection achievable polynomial phenomenon focus exhibit price pay particular theoretic accept protocol hold synthetic problem tailor still accuracy aim general pc pc detection capture strength test independent copy direction around significant random bernstein inequality inequality sub example specify ad fluctuation around formulate unknown v center robust well procedure focus yet rely along unit hypothesis assumption recall family test bound need assume fix tolerance focus parameter integer define regime optimal I test ii notion optimality focus sequence variance along test
technique validate twitter relation political survey political peak micro services micro publication short share kind twitter tweet maximum character million every public sentiment trend news content tweet temporal text trend twitter research draw attention political great concentrate political science confidence political political political relate level political important implication political system make political study political despite popularity term political concept necessarily imply level political component general political belief distant knowledge never study twitter automatic approach measure political twitter aim measurement political political public opinion survey accordance political political particular party methodology political tweet universe tweet tweet tweet order validate operational tweet political indicator public opinion survey furthermore political news peak produce follow work summarize supervise tweet summarize public opinion survey validate employ extraction highlight conclusion great deal phenomena micro services recall work event twitter market employ micro work concern analysis political topic employ twitter author evolution attribute call profile suggest economic fluctuation discuss substitute traditional opinion author opinion tweet raw tweet sentiment opinion survey twitter sentiment find public opinion survey twitter tweet run present author count result perform predict present rely twitter information party candidate belong extend incorporate sentiment political author general compare traditional opinion survey similar nevertheless possibility author twitter chance analyse political political employ twitter quality highlight political opinion survey political concept political efficacy influential political political social change play efficacy crucial since political political efficacy create support political increasingly political making political reduce act vote political proxy believe supervise extract tweet employ political describe public survey political set twitter expert beginning collect tweet localize political end day regard political argument select political political meaningful political tweet political show keyword political result corpus record tweet collect political tweet tweet political political student need political neutral sentiment sentiment tweet label quite fuzzy reliable tweet assignment label student select tweet political nature voting tweet sentiment set decision label majority half vote account alpha unit sentiment compose label tweet knowledge represent dataset drawback limit retrieval lose accounting wide period drawback political tweet build article spectrum political view precisely feed history article extract categorization title news belong label political tweet political perform twitter community goal randomly extract tweet temporal range moreover active user user expand moreover profile thus prevent twitter could affect quality political extract tweet sentiment non perform ad create speech different step account section test efficacy crucially transform nevertheless identify feature problem task sentiment note political topic employ tweet feature extraction gram character separate gram string frequency fold validation gram option sentiment word employ count moreover process perform employ sentiment tweet perform extract wikipedia query way behind target sentiment date political political twitter able sentiment huge possibly update focus attention really really performance run ordinary classifier try hyperplane employ subspace task big comparable cardinality deal perceptron like setting widely result aforementione task classification negative sentiment sentiment sentiment online batch particular use speed extensive good cross validation sentiment c series th interval p series classifier measure time time adopt political sentiment obtain combination opt employ expert indicator employ relation opinion survey summarize identify political peak break news survey employ tweet subsequently survey date series employ three date day survey day date survey take pearson political tweet represent methodology highlight twitter able political opinion survey try name last la vote meta growth disadvantage half pd pi ci li opt il di il di project spread spread effort al il spread tu un la failed si di ai pm il il agreement risk many pearson political tweet result value connection model political next day take consideration twitter diffusion news result survey exhibit suggest twitter valid measurement political twitter political employ empirically cause political political daily medium identify peak series political tweet political peak employ peak improve quality neighbourhood instead qualitative associate peak news firstly create inverse frequency extract corpus news randomly political subset classifier political describe identify political tweet relevance recurrent news tweet obtain day two vector belong cosine select peak qualitatively news topic twitter day peak notice news effectively political daily trend however peak news tweet say happen whenever political concern fact
uniqueness focus cluster function scale invariance consistency work show apply function call formulate term distance paper focus community provide indicate self loop self loop notation grain see graph invariance consistency focus investigation quality notable modularity intuitively hold informally axiom cluster follow basic definition section discuss different six function axiom permutation invariance continuity locality modularity monotonicity locality motivate variant modularity lead new scale modularity tune resolution quality function cut unnormalized cut go zero quality similar mainly cluster style investigate decompose additive form influence algorithm investigate robustness study dissimilarity consistency introduce near consistency also base quality cut modularity optimize mainly resolution tendency quality particular modularity small size important quality phenomenon resolution therein family cluster show special case modularity formalize resolution quality axiom axiom node edge edge opposite convention loop cluster graph partition node otherwise write every clustering real convention quality indicate sometimes parameterize single quality family graph say weight potential axiom invariance would cluster quality equivalently clustering example clustering scale two style satisfy third converse axiom property prove unless specific axiom distance author reformulate axiom axiom adapt graph cluster identity invariance graph clustering j ef invariance quality edge intuitive axiom unit change invariance quality stay proportional previous definition relation cluster graph clustering quality function quality invariant constant clustering change weight graph cluster axiom edge formally call graph consistent improvement quality decrease improvement monotonic axiom consistency also natural node neighborhood discover protein human remain disjoint quality relate consequence total cluster graph agree preference quality write cluster cluster locality differ secondly locality clustering remove direction locality agreement give edge one locality resolution function resolution free call graph cluster compare locality require locality strong perhaps subgraph induce agree locality resolution limit quality limit locality locality replace agreement agreement imply strong property sensible quality quality leave another agree cluster solution use come scale graph perhaps intuitive write yield connect component rich consistent remove edge cluster within monotonic axiom reformulate graph seem however directly multiplication division division connect define monotonicity modularity weak relative monotonicity quality clustering consistent modularity monotonic clustering clustering edge increase improvement decrease g show modularity relatively monotonic modularity change variant quality scale hope modularity would monotonic suffer edge volume unfortunately fix modularity volume weight cluster fix volume negative decrease scale modularity monotonic normalization still edge scale family parameterize additionally modularity rich adaptive modularity monotonic adaptive satisfie six axiom axiom extend axiom cut add adaptive modularity invariant quality infinity model volume cluster modularity constant six axiom scale scale quality family invariant practice overcome resolution modularity scale modularity proportional q lot interest resolution limit modularity illustrate clique single edge correspond clique clique increase clique stem modularity adaptive modularity quality make modularity real situation graph clique problem build large simple two subgraph connect vary clique three cluster cluster clique apart desirable circumstance mm rectangle baseline node mm node b left node another heterogeneous clique subgraph clique subgraph random subgraph volume consider combination instance simple subgraph include clique subgraph entire connect relevant clique subgraph subgraphs internal edge volume total light blue separate red subgraph apart graph outcome column modularity modularity certain third split apart monotonicity merely control slope boundary outcome edge clearly see otherwise effect invariance invariance monotonicity locality modularity n normalize quality consist modularity property motivate modularity high modularity however adaptive modularity solution modularity axiom function propose exhaustive topic future survey property resolution vary well select significant necessary quality quality axiom modularity modularity close subgraph locality time modularity algorithm adaptive undirected extension direct overlap open axiom reason thank comment organization scientific modularity rich otherwise nodes cluster weight note modularity contain empty maximal modularity contradiction hand cluster cluster case cluster modularity contradict modularity half another modularity modularity cluster negative modularity hence rich modularity contain clique
create instance classifier regressor equal absolute datum expert provide absolute pair two approximation absolute expert train create replacement logistic classification table show give low except diabetes regressor thresholde c loss gamma diabetes diabetes loss diabetes error e e uci corresponding boost three ensemble classifier fisher discriminant two regressor regressor huber low approximation absolute e diabetes gamma diabetes hinge diabetes loss e gb weighted sum expert gradient three use classifier train set minimize except smooth add designing machine impact ensemble ambiguity decomposition explain expert ensemble arbitrary differentiable dependent diversity use regression accuracy function uci set encounter ensemble expert utility decomposition ensemble selection utilize diverse ensemble learn maximum approximation especially attractive diversity require develop unlabeled research understanding impact diversity introduce stage conventional supervise train set understand beneficial diversity introduce diversity classifier automatic recognition art finally characterize human expert quantify underlie world crowd involve extend definition california usa edu edu york usa us com expert ensemble widely achieve performance improvement however work characterize ensemble diversity link decomposition answer twice approximately average expert diversity ensemble explanation empirically diverse classifier dependent diversity present extension report accuracy present pattern set accuracy regression function theory empirically expert regressor single well project involve automatic processing program combination state art system processing application range parse text categorization compete netflix system movie million win system team compose independent theory ensemble diverse system task win yahoo ensemble bag boost forest lambda ranking overview vision multimodal classifier feature achieve compete age gender signal offer reason ensemble low state expert convex optima ensemble expert finally underlie problem complex expert ensemble list diversity act train say expert frequently individually explain tradeoff square loss expert quantifie diversity square weighted diversity equivalent network ensemble consist accurate diverse network relate say expect square regressor target term measure reduce ad form set mixture attempt diversity include ensemble decision tree vector machine conditional correlation meta another prominent create expert focus understanding impact diversity regressor ad present ambiguity applicable classification regression example case derive single different convex though link consider expert rely function present multiple prediction expert value class widely supervise close set l twice derivative taylor x remainder due second extreme value bound taylor desire inequality always represent limit domain twice reasonable ambiguity decomposition loss simplicity ambiguity ad expand arrive ambiguity ambiguity individual expert ensemble square error function prove ambiguity lemma ambiguity expert k let follow close small argument continuous derivative eq bounded include q side computation subsection derive regression reduce function square differentiable approximation use integral tangent approximate approximation set suitably positive derivative derivative maxima monotonically decrease maximum minimum smooth become well positive behavior maxima monotonic second minima type absolute compare technique replace expert model affine expert derivative increase reach monotonically q similarly adaboost loss decrease hence hinge loss machine svms differentiable approximation often second increase attain depend location expression various ensemble approximation understanding begin tradeoff analyze ensemble motivated approximation follow term right hand weighted sum term jensen understand perform common amenable prediction independent identically distribute unimodal give numerically unimodal ensure variance around varied pick loss depend distance extend carlo expert set median weighted expert loss figure plot diversity expect ensemble analyse ambiguity diversity diversity expert curve actual ensemble correspond move assume expert prediction comparison around diversity subtract three function approximation diversity unimodal peak expert diversity away boundary direction quantify spread prediction weight expert agree predict loss subsection deviation figure show expert ad rate use blue tight train utilize subsection understand loss approximation boost ensemble prediction add weight assume word assume weak contribute ensemble approximate taylor around write
infer technique avoid validation pruning add complexity adapt handle label problematic boost algorithm place misclassifie noisy instance boost instance support machine place bind wrong possibility instance expectation noisy noisy attention result frequently misclassifie ensemble filtering boost bag heuristic instance al remove surface correctly calculate network discard noisy correct instance noisy instance impact large discard instance belong examine weighting misclassifie rather instance clean correctly train attribute predict correct validation instead attribute value differ add examine filter inherent set examine filter large significance model assume class factorize modeling distribution instance misclassifie rather use require yield instance draw compose generally probable problem discriminative approach neural tree hypothesis instance posteriori hypothesis equation map hypothesis include problem optima representative true underlying possibility ignore thus label avoid probable label noise explicitly instance triplet maximize model probability show preprocesse approximated model give equally discriminative law quantity instance remove sum hypothesis multiply formulation infeasible though limitation probabilistic generative attractive kernel discriminant generative generally terminal option explanation either color graphic graphic macro ltb lt lt lt ltb lt lt bp r r r instance technique filter misclassifie instance diversity algorithm output diversity measure base score agglomerative algorithm default dendrogram figure height connect two cut choose cluster list perceptron decision locally neighbor neighbor I learner random reduction produce remove misclassifie first misclassifie instance filter ensemble filter misclassifie closely sum diverse represent estimate algorithm indicator class misclassifie ensemble examine misclassified percent ensemble percentage validation percentage highlight determine percentage would used iteratively add set candidate produce high classification train accuracy instance misclassifie idea learn greedy filter high use set tb use candidate initialize current empty return data cross time filtering algorithm uci repository non uci set filtering algorithm remove misclassified instance remove misclassified fold accord attribute type uci data bold attribute categorical contact breast w breast balance heart heart tumor car segment significance sign nature filtering examine diverse examine misclassification adaptive examine filter fold cross examine fold misclassifie adaptive cross result entire ensemble filter use filtering rather affects examine datum filtering effective suggest ensemble vote preferable show bold represent statistically test algorithm mlp ib rf algorithm bias significantly na I misclassifie mind reflect noise real world datum concern filtering filtering ensemble underlie diverse preferred filtering outline ensemble one perhaps adaptive filtering cross misclassifie oppose misclassifie rank statistical test compare order right l rf ib nb examine individually find robust filtering use sign rank significance great impact significant take random tree ib robust still filter ib near instance filter significant five mlp I inherent efficacy limit significantly investigation recent work neighbor data binary problem handle overlap calculate normalize overlapping total ratio attribute connect tree neighbor divide neighbor leave count class compare set hardness characterize misclassifie examine measure hardness rule classifier et significantly improve classification sign predict al hardness satisfactory add set improve add example determine technique neighbor neighbor share ds instance cover belong class total decision td decision cl belong feature class instance cb ratio instance belong equally ensemble filter voting show weighted voting ensemble accuracy voting provide generate filter algorithm ensemble common ensemble ensemble consider requirement vote ensemble appear beneficial ccccc mlp ib nb acc ensemble rf acc l mlp ib contact post op l filtering outperform vote examine percentage hardness instance hardness misclassifie set learning probability misclassifie accuracy datum set gain ensemble identify sign statistical ensemble voting training statistically train vote train ensemble filter filter include discover reduce complexity examining set classification noise factor need consider primary tumor datum yet voting ensemble hand great ensemble despite include discover examine use cccc accuracy l greater noisy greater less great equal accuracy value equal investigate training voting ensemble find majority vote noisy less investigate hard noisy instance high voting voting infer train diverse diversity often treat vote unsupervised train filter diverse significantly examine evaluation filtering add effective noise may impact filter clear also candidate filtering find add label datum increase examine hard set examine datum induce filtering result ensemble filtering outperform adaptive individually investigate voting find accuracy train majority voting filter ensemble robustness preferable filter filter adaptive bold refer time row cccc ensemble p less ensemble less greedy cccc value greater less greater great greedy p cccc greedy p greater greater great equal cccc accuracy equal greedy cccc ensemble greater biased p cccc bias greedy less ensemble value accuracy equal less biased great cccc ensemble bias equal greedy cccc greater less great less datum compare ensemble filter filter filter voting result filtering bold highlight filter vote filter vote bold column accuracy ensemble accuracy voting
surface handle non smooth surface see computational computational error say ss straightforward computation scale double bootstrap theorem double eq hold also confirm four geometric quantity well due accurate fourth error assume mm h therefore fourth order accurate case good deriving mention section coordinate thus section interpret geometric approximate h probability particular give give testing h rejection probability q third fourth accurate helpful discussion thesis also discussion denote kronecker delta index multivariate normal three expectation product correspond hand odd lemma direct coordinate bootstrap express calculate consider taylor u u u u l u h ex h l kl ex h kl u kl jk kl g h ij ij h ij jk ki ij jk ki look go right ij h h ij u k h ij mi brevity I u ij u substitute ij jk ij jk substitute mi ij mi mi h mi mi h mi h h h solve equation verify li mi eliminate ml actually ml mi h mm h mi h ml mi apply get rearrange get ij mi li mi prove mi mi replace h later quantity replace become theorem apply replace give compute replace give h h equation verify z h collecting term substitute solve mi ml h mi h grant hypothesis shape counting bootstrap widely evolutionary biology report bootstrap adjusting parameter attempt double bias bootstrappe another bootstrap employ attempt multiscale focus high order asymptotic geometry play role region find curvature bootstrap multiscale remove bias multiscale robust bootstrap replicate nan alternative unknown shape geometry shape geometric bc confidence interval geometric investigate asymptotic accuracy spherical independent dependency notation enough know degree centrality easy exact shape region parametric bootstrap spherical resampling replacement generate fall bp extensively since tree evolutionary biology name strength bp frequentist bias improve assume replicate bp eq respect denote improper prior specify rejection probability express eq geometry boundary surface cumulative prove rejection probability bias mostly mean flat curvature area false curvature little negative sign toward problem improve bootstrap confidence duality interval confidence parameter say spherical consideration bootstrap adjust bp calibrate bp geometric fact rejection coverage iterate bootstrap mean curvature surface plane sphere intuitively surface magnitude several bootstrap accuracy bp unbiased th asymptotically bp accurate attempt bp scale bp multiscale bootstrap bp bias correct geometric tool bias already refer newly propose bias correct double unbiased multiscale double bootstrapping smooth v confidence comparison evaluate chance example solid curve htb testing sign method bp ratio statistic pp two lr point side lie replace sign multivariate testing side power side lr sign lr bias adjustment correct could confidence eq reject intersection empty control error conservative like favorable configuration statistic v table statistic actually cone become away value large bootstrap method simulation assume asymptotically verify interest consider series summation convention j numerator denominator similarly flat axis tangent curvature eq curvature similarly origin expression section curvature curvature express q bootstrap assume alternative boundary surface tangent space sign express asymptotically represent expansion method express expansion term convenience define q accurate fourth comparing differ simplify argument assume sign curvature quantity hand small bp sign correct curvature verify section attempt adjust term consequence theorem rescale law introduce asymptotically eq h third complicated multiscale say fit h multiscale model compute fig geometric interestingly behave case seem work fine simulation vertical axis indicate multiscale z dot multiscale double dash bp bootstrap replicate distinction huge bootstrap calibrate mention later double asymptotically side third come vanish correct robustness immediately contour surface fitting model illustrate table shape mention computed carlo avoid look however similarly bias correct bias difference correction look confirm htb bp surface without argument derive bootstrap coordinate u ia q eq h ij ml k ij ml orthonormal correspond express definition follow basis coordinate element
instead merely generalize meaningful provide fusion feature space level rule flat region identically zero simple smoothed transform invert rule increase sensor increase function space relatively density diagonal allow calculation tractable also interest situation ultimately center long feasible solution apply decision close unconstraine optimal quantization function unique modify well example near actual optimal fusion allow usually cost quadratic solution fusion case numerically except density turn large cost whenever nonnegative p object detail section finitely support mh mi address cover behave function give something particularly detect compound inside mobile phone theorem say still way cover median statement continue hold bayes fusion long unique several property fusion sensor fusion fuse h k sensor risk output bayes optimal rule decay fast thing enable pointwise result theorem object cover perfect delta sense series problem simple situation object fusion even observe collection mean variance parameter optimal performance cv ec furthermore cost situation represent belief absence object pick sensor fusion combine soft sensor division combine establish corollary rule fusion skewed fusion mean satisfie sense h fusion directly entire opposed sensor use typical real sensor version fusion person fusion output see distribute application large sensor bandwidth communication sensor incur overall performance bayes fusion configuration internal center decision entire fusion bayes fusion example similar proposition type statistic exponentially distribute parameter performance hc I hc ec behavior case fusion rule entire performance extent overall increase identically function fusion fusion fusion sensor fusion follow fusion discrete generally performance numerically sensor soft decision fusion region decision bayes preserve object improve scenario context fusion center mapping pass bayes risk soft fusion turn unchanged information lose opposed h fusion sensor output decision feature fusion enter theorem cost otherwise word find explicitly positive mapping six countable number fusion fusion rate fusion non optimal space fusion room achieve probability pearson finally feature gaussian uniform h h fusion rule greater reflect performance h configuration scenario symbolic necessarily closed physics density kind put assume symbolic continuous point explicitly term probabilistic describe individual nuisance determine conventional property sensor density large come carlo collect maxima reflect accuracy tradeoff risk calculate risk preferable former pick leave large portion domain inaccurate accord distribution evenly risk graphical generate propagate conventional amount formula sensor depend graphical mcmc gibbs variant dimensional would impossible arise frequently jointly many type collect measurement signal look smooth represent influence cholesky delta never discretize pair produce histogram find directly hc limit dependent ergodicity let estimate confidence fusion cover q fusion performance compare concentrated correspondingly decay fast need fusion analyze optimality criterion diverse situation involve online know stream future theorems constraint make nonconvex simplify kf compactly write linearity additional constraint problem large space without minimize quadratic hilbert space solution solution unique pointwise dimensional lebesgue euler q finite check manner feasible optimal fusion long well define consider fouri transform sensor th imply h nonzero every stationary part g adjoint decay thus notation quadratic expand write third nonnegative fact unbounded bayes show minimize rule possibly continuity satisfie define compact entire order see definition taylor series around origin power euler compactly want show every say half plane order absolute another plane hadamard taylor expansion even power odd euler lagrange manner together imply weakly minimum upper convexity imply fusion give bound law number surely since surely feasible eeg optimal end appropriate cost enough act nonnegative eq hold constant linear since hold unit vector distinct find dense h e h v uv mu transform fusion rule I e e mu straightforward hc cv ec c c hc dc mu v calculation zero plane condition hold fusion input integral formula transform hc I md hc hc da da da c I risk evaluate ec mm c bp thm thm within sensor expectation fusion deterministic constraint property certain prove optimal wide satisfie asymptotic apply example determine fusion scale study multi modal acoustic sensor broad applicability keyword sensor optimal calculus variation probabilistic graphical classification many communication temperature surveillance track large gain achieve sensor sensor reach decision sensors wireless information centralize fusion fusion elementary fusion logic context due simplicity ease knowledge statistical sensor rule criterion achieve special statistical goal dimensional space sensor application sensor output incorporate randomness sense target common target several
parameter use value initialize hyperparameter study approximate update update hyperparameter observation eliminate component estimate eliminated repeat cluster map class analysis assess adjust rand ari predict classification adjust agreement classification simulate separate run bayes start give model excellent looking predict ten fit htbp simulate component well datum ten simulate fit capturing initialize hyperparameter flat effect run initialization initial initial hyperparameter classification initial identical density ten run identical component consist activity algorithm selection bayes model component clearly give modify third evaluate contribute shape relate via restriction ensure identifiability restriction arise conjugate exist f u u observe latent family normalization constant natural observation write ig g nz ig nz ig nz ig nz ig nz ig vector scalar assign mix proportion hyperparameter wishart ga g ga ig precision g truncate ig g gb approximate generalize ig misclassifie value cc htbp clearly fit contour capture estimate must consider present challenge dimensional perfect old available time duration old national upon visual inspection shorter frequent long start figure contain colour rw width body depth mm run variational bayes ari well ari family cluster b course variational model cf different measurement body seven different specie variable gram begin length end maximal maximal width length highly drop explore analysis run result inspection solution three table correspond exactly merge em seven selection four criterion htbp true bayes alternative past application furthermore symmetric variational base discriminant fashion paper variational mixture discriminant accordingly several approximation modelling situation add already burden start observation become remove parameter estimation estimation far computationally em possible efficacy cluster mixture possible research extend carry component analogy mixture mixture addition aim longitudinal contaminate within framework science early award innovation research computational mm normal inverse approximation em complexity uncertainty conclude approximation variational popular extensively date amongst gaussian limitation mixture model lin lee normal mixture within estimation maximization em em find estimate incomplete membership may also major drawback value problem minima dealing distribution furthermore use conjunction model criterion use bayes algorithm deterministic make decade simultaneous reduce computational overhead variational algorithm miss complex convenient kl negative kl maximize develop mixture mixture reduce associate modelling variational approximation mixture cluster model illustrate conclude suggestion future section normal ig result tail ig parameterization ig eq normalization continuous nu nu independent assign discuss mixture eq parameter mix complete write prior hyperparameter nz nz
bregman discrete order vector aggregation motivate relation new connection summarize generalized bregman divergence theoretical interest help score metric divergence ir normalize cumulative ndcg auc instance bregman unique property lb divergence notable amongst naturally capture notion high score lb problem connection structure norm connect lb aggregation past rank use rank closely introduce notion bregman lb permutation several similarity instance divergence also loss discount cumulative gain ndcg lb bregman metric notable amongst property naturally capture notion confidence exhibit rank application connect base web ranking instance closely divergence review define generalize bregman lb divergence main result extensive empty domain b yy series differentiable divergence strictly extend differentiable generalize divergence long derivative exist directional view divergence interior define subgradient yx differentiable notice relate variant x generalize duality bx divergence exactly divergence check belong substituting hence belong sub gradient strictly bregman divergence simple example generalize bregman divergence x subgradient different bregman theoretical property permutation permutation order call integral extension furthermore extension hypercube characteristic valid subdifferential let ny totally express define subdifferential nice submodular permutation hence totally subdifferential submodular vector define point notice parameterize related extreme hypercube permutation every belong lemma critical totally order belong consist subgradient entirely order define property lb insight permutation different permutation adjacent adjacent permutation share permutation key lb divergence position divergence generality lie volume totally subgradient fx n c x x divergence vector invoke order seem class submodular quite class class satisfy bregman score base divergence observe eqn subgradient one permutation see instance eqn shall permutation metric submodular satisfy divergence form totally resort subgradient natural subgradient map case submodular shall correspondingly assume case totally handle related generalize bregman extend bregman manner similar subgradient map simultaneously perspective bregman lb consider bregman list bregman lb divergence symmetric fx lb corresponding nice weight write jx j x closely relate metric concave induce class bregman I submodular large follow eqn observe function base visualization shall divergence ranking joint ranking case preference joint ranking totally order submodular permutation notice f combine follow element lb divergence improve cut yet nice lb ranking generator demonstrate permutation face say x definition submodular k small differ rank towards prominent cardinality recall belong start rank notion distance permutation often interested total list order lb list use eqn exactly overlap interested eqn another care ordering rank irrelevant document one irrelevant base bregman eqn define partial interested element rest eqn define lb extension partial ranking change model extend notion lb eq permutation oppose permutation lb divergence symmetric divergence invariant reason model permutation ranking lb divergence extend admit interpretation thereby normalize shall web rank x yy bregman regularizer want constraint bregman regularizer alternatively lb relate proximal investigate eqn apply problem connection divergence extend bregman divergence monotone subgradient ensures define interesting figure visualization choice preference bregman combination bregman list score ignore aggregation try permutation base also know application use unfortunately lb divergence close form exactly element bregman representative build bregman divergence somewhat however arithmetic score seem expect also notion retrieval community argument assume uninformative order I lb order instead ignore value permutation order uninformative representative see permutation though low bregman total variation arithmetic imply confident order variation visualization cut algorithm propose maximum therein loss instance lb document quality document particular indicator example title feature possible function particular order order interestingly relate eqn probability permutation exactly extend define eqn correspondingly eqn conditional aggregation context represent mixture ranking model homogeneous cluster objective obtain scalable practical amenable bregman divergence since bregman means go identically bregman demonstrate left demonstrate bregman dependence euclidean similarly provide top objective knowledge score preference base many paper interesting bregman divergence connection web rank integral extension special unlike divergence form integral finally world rank acknowledgment discussion anonymous material upon support national science foundation microsoft intel award electrical engineering usa electrical engineering usa extend recently bregman lb divergence distortion score providing aggregation cluster connection show lb divergence metric form lb commonly rank information ndcg auc traditional permutation base metric lb divergence provide representation involve aggregate bregman rank rank bregman numerous proximal many bregman divergence generalize divergence divergence function divergence parameterize via submodular refer ground submodular submodular attractive naturally arise operation etc see function grow nice structured induce norm concern yet application introduce context extend way connection problem aggregation rank social list aggregate fundamental machine ranking gain important assign score natural population representative fit interest combine system something translation treat output rank ranking overall boost ranking bit denote item assign convention permutation induce order vector decrease shall th permutation recently ranking distance denote permutation usual notion addition distance permutation invariance metric represent swap
choice microarray implementation property summarize setting cauchy prior seem optimal logistic moderately heavy tail treat cauchy hyperparameter reasonable meanwhile point difficulty highly bias subset difficulty use mode summarize drawback slow penalize still much room computational efficiency difficulty solution space pca low coefficient crucial solution devise method original method potential across effectively transition investigate pt discriminant gaussian estimate sample dimensional proportional construct transformation leave invariant minus ie minus set independently purpose independently interpret leave joint distribution qp call end discard transform qp hamiltonian along differential q keep unchanged preserve hamiltonian dynamic implementation hamiltonian dynamic discretize stepsize several alternative stepsize I independently series nearly transformation back ahead jacobian transformation unchanged qp accept metropolis hamiltonian current update follow draw transform step transformation qp connect along transformation decide accept qp result rejection last q discard hmc trajectory determine hamiltonian large transformation poor performance may move slowly rejection ad hoc choice reciprocal square nd account width adjust adjustment usually adjustment empirically close critical hamiltonian good thing choice hamiltonian trajectory nearly adjustment appropriate phase quickly look fast gibb fairly initial initial reject run random call instead sampler hmc move direction walk hmc langevin metropolis distant start transformation reverse direction back q along choose large move ie distance sample sum minus log reciprocal nd derivative sum dominate apply belief irrelevant coefficient close therefore trick computation even fairly computation time gibbs last number relate coefficient call trick markov computation trick essential important irrelevant coefficient ability hmc consequence hard large example choose prior df square scale gibbs length trajectory phase gibbs trajectory fix current adjustment nd posterior acknowledgement li natural research foundation pt macro dimensional logistic tail select useful great thousand area genomic certain gene different cancer base heavy moderately freedom microarray identify redundant gene leave validate high tailed hamiltonian monte fully throughput microarray easily genomic expression relevant cancer diabetes diagnosis disease researcher collect know typically large thousand look huge irrelevant strength test gene ignore expression datum see redundant gene include meanwhile gene feature fit model regression number maximize rather likelihood penalty penalty tail hyper penalty penalty dimensional article meanwhile method tail extensively tail prior investigate current develop fully dimensional classification heavy hamiltonian carlo article report result extensive heavy tailed advantage signal leave redundant among automatically tail hope benefit heavy investigate report cancer label heavy tailed prior collect feature label label integer model column indicator otherwise genomic rare df probit useful gene apply regression model marker median mcmc sample clear probably moderately simple hereafter small scale df express stand gamma gamma parameter standard exponential laplace parametrize interpretation exp gamma superior lasso problem article call describe generator description section primarily generate prior scale find look couple easily around list moderately heavy allow df tail coefficient tail good tail prior flat allow htp distinguishing divide redundant eps property moderately heavy tailed prior separate look constrain maximizer ie constrain find shrink contour toward origin line constrain df unconstraine path map contour path constrain axis laplace go close path base go look heavy correlate mode conceptual illustration simulate highly contour use explain label estimating coefficient correlate one different volume correlate problem contrast penalty constrain explain group correlate feature value helpful primary regression optimization algorithm mode arbitrarily become unstable sophisticated value sophisticated reality index display subscript column index value set subscript index denote collect index integer feature logistic introduce convenience hierarchy indicate th useful believe ig assign bivariate equivalent assigning explain propose recent shrink assign cauchy notation cauchy various induce assign exp close form q name go ig possibly without signal confirm ig seem rejection sampling substantially ig hour denote coefficient recommend reasonable magnitude wide range problem recommend insensitive heavy signal greater avoid adjust little cauchy distribution weakly markov chain stick region long however ie informative high hyperparameter logistic coefficient x therefore difference baseline class identifiability assign class variance implication may justify practical selection vary baseline common fix vary centralized thousand bottleneck challenge trick threshold update trick section recommend use formula importance pool different mode correspond markov importance estimate feature appear markov sample high correlate ability feature ability totally small recommend may useful frequency subset subset manually markov trace fitting divide list subset importance coefficient totally correct skewness compare various prior currently popular date fitting look predictive equal differential class nd non differential correlate first two nd rd st shape two value small validation pt ie ig setting chain small fast taking scale path coefficient package stand stable median skew large absolute page feature heavy tail good though problematic however fitting penalize method coefficient ht prior performance lasso rate error path lasso predictive really coefficient especially weak prediction conversely scale pick even nearly difficulty identify set label z generate draw call show common factor correlate run choice vary choose treat hyperparameter assign choice simulation n setting run lasso stand four group number logarithm scale horizontal indicate maximum feature nd highly correlate recognize another useful consistently believe prior almost contrast lasso well often hard identify mode correlate large entirely figure tail selection well tail allow go poor tail likelihood tailed prior summary select group small ie thresholding relative give substantially set small moderate less boundary chance weak signal therefore measure attribute pdfs priors logistic hand additional negligible base implementation markov previously take hour chain prior merely posterior standard ig possibly eliminate htp l microarray report analyze description order feature statistic run lasso leave compare fitting standardized setting
particular x rate f w fix q similarly accord eq concern point function absolute x actually second network representative reasonable neural physics work particle track receive delay particle htb receive input track half alternate layer solve right ambiguity still htb require event effect tail separate tail come repeat simulation result reproduce show gaussian tail leave right edge less ambiguity tail identification show separate region reliable answer k input example network fig architecture train exact leave track fluctuation train feed neuron layer hide supervised backpropagation neuron activation sigmoid present drift hide neuron slope track code meet reliability output increase accurate technique financial scoring quantitative method help likely due neural network impossible dependent typical try outcome illustration use publicly uci compose case find bad class attribute status etc code bad bad conversely put evidence numerically code neural numerical hide neuron train time learn evenly bad instance fig plot general network good estimation vice versa incorrect network easily bin end histogram reveal example show uncertainty previous obtain network network error predict three critical without look easy confirm refinement degree still occurrence estimation outside bar complexity research reliability predictive example high energy physics scoring perfect greatly predict fc com identify still hide system model ambiguity obtain degree widely importance create method reliability artificial neural responsible predict evaluate reliability error illustration tracking dealing rarely avoid social dealing uncertainty formalize uncertainty specification phenomena bar address several concerned specify output region variable answer score job asset situation cut develop region probable tell reliable network short
typically simplify explain process process coincide set input become multidimensional mean matrix ij f f input bayes rule marginalization conditioning first bayes function general likelihood iid function conjugate posterior multidimensional simplify computation finally marginalization equation marginalization latent marginalization input marginalization involves thereby condition marginalization property give I e solution covariance kernel bar natural sampling maximization svm hyper typically adjust different hyper gps learn maximization set illustrative toolbox fix j plot process area denote bar plot posterior thin blue right hand center lie view latent small bar leave effect point typical accurate bar train denote green plotted thin blue line thick bar become sequentially entire batch follow point readily predict furthermore inverse q correspond convenient predict easier prediction make scenario mean matrix obtain covariance recursively n proportional relation matrix store update formulation prefer recursive equivalent literature instance growth visible limitation formulation practical implementation limit growth matrix mean interval gp though replace spherical q gaussians tend infinity e away rewrite multiplied kernel matrix kernel matrix bayesian identity respectively connect estimation nonlinear explicitly indicate mapping covariance gps accurate play svms describe determine fast part everywhere else available hand already discuss kernel method bayesian rely computational covariance multidimensional simple covariance weight positive hyper together multiplication rely often signal application parameter radial different length input universal construct regressor term treat parameter covariance function fast transition mat ern prior proceed integrate section hyper parameter dataset eq integrate hyper parameter conjugate posterior hence although principled intensive example markov monte carlo reader alternatively maximize hyper purely bayesian sensitive nonconvex likelihood unimodal around ml solution ascent inverse covariance costly prohibitive ever scale database effort devoted decade allow inference scale linearly point approximate method gps full model usually use functional basis sparse initially entirely different modification prior plus block mention pseudo regard input flexibility typically flexible option active lie domain present worth increase basis algorithm yield general convergence gp active lead allow active unconstraine present reduce involve approximation accelerate multiplication compactly support covariance sparsity online signal active success suggest advanced pruning result gps flexible involve stock measure enhance gps include context process detail depend regard gp parameter maximize gp parametric monotonic traditionally whole learning might channel often account base non slide base gp describe necessary forget old become effectively sample store detail adaptive base hoc enable control exactly principled non sufficient covariance function temporal effectively account stationarity instant fairly augment model describe covariance augment spatio equivalent update online speed vary play r adaptive hoc use gp principled manner ml fig factor indicate mean toward indicate blue dash bar variance fig represent predictive represent filter channel fig instance communication rate scenario normalize spread slow fast difficult tracking time db additive tracking estimate new fig illustrate scenario filter implementation filter normalize square extend hyperparameter ml perform growth involve select pruning basis see mechanism quantization outperform margin scenario capable deal nonlinear show excellent convergence steady comparison tb simulate datum simulate db db simulate real use wireless digital realistic compose receive frequency front hardware generation acquisition front hardware platform correspond division frequency selective unlike simulated channel unknown measure depict indicate signal acquire signal receive use nonlinear variation simulate show simulate steady fig summarize set gps return change cumulative density probit respectively prior resort numerical solve marginalization numerical integration solve method ep performance nonlinear incorporate digital channel follow work optimally posterior like fed channel decoder assess result model fig probability b emphasize signal ratio db provide close hence channel decoder quantify gain clearly outperform backward illustrative perform tb cc b tb process mmse wiener amenable practitioner bar assume additionally parameter maximize hyper need tune minus impose strong accurate latent bar classification provide accurate posteriori probability gps considerably example significantly finally solve adapt environment method detail example reader finish gps also apply motion separation differential
award conference theorem proposition ex ex ex analysis university california berkeley berkeley problem constrain average value throughout pass message additive white channel decentralize conjunction spectral square required scale topology previous scaling laplacian good network constrain throughout message motivate sensor individual massive server memory store location typical physical e pressure extensive network consensus g paper work focus node noiseless study communication link paper reference focus constrain inter additive white various way simple question ask consistent average achieve consensus refined rate follow obvious reason refer near study perfect communication albeit suitable extension path noisy network analyze consensus topology much establish optimality scale high phase produce decay update vary within phase establish path simultaneously combine careful spectral stochastic compute diameter discuss detail factor substantially scale remainder algorithm state guarantee main convenience collect notation exist mean mean denote inner begin necessary identically distribute unknown sample aggregated central location paper consider network version model undirected sensor structure sensor exchange noisy communication directly node internal act pair send identically gaussian realistic noiseless pointed leave exploration algorithm message direction distance average per square node tolerance require mean characterize function number square question three type regular topology two regular degree topology figure form place node unit connect node euclidean radius random mm grid geometric theoretic short diameter path take see ensure connectivity explain later divide assume know aware center construction grid locate center p moreover square contain accordingly transmission connect adjacent desirable definition channel follow communication surely fix sufficiently grid comment worth reach necessarily mean point consensus since little arbitrary part b previous scale two analysis remove scale iteration value bad within diameter within set cycle graph average graph conclusion iteration also theoretic general consider diameter level phase outer phase produce iterate outer parameter total pass round round pass two per put everything establish phase average language round pass mse illustrate inner phase average choose average outer phase base average recursive update inner structure though give inner phase index inner inner outer phase break detail average single cycle right nothing decide follow copy I noise variance update node involve inner product time early argument structure vector meaning complement show depend noise mse final recalling average laplacian expectation take place randomness choice associate denote rescale previously update q doubly row correspond row consequently interpret transition reversible irreducible chain chance chain transition use study symmetric related large markov decomposition update show part intermediate section lemma core concern sequence evolve accord sequence q moreover bit algebra iteration part eigenvalue follow follow address protocol property cycle regular probability average standard rather consequence average phase ensure like laplacian iteration q guarantee mse remain technical observe fact recall piece clear gaussian nature I quantity recall cauchy inequality ii rescale establishe define write define algebra eigenvalue put piece first former identity uv subtract eigenvalue turn recall step cycle average path precisely case state reversible markov node coefficient edge form previous quantity equivalently markov form distinguished row xy belong row give involve varie row path piece conclude structure mm structured belong coefficient follow first introduce return different row path make since pick average square b substituting obtain fix square path node would give involve mb conclude case case sub b belong belong return therefore chapter variable absolutely integrable see book vector randomness update differential ode eq know ode recall cast framework I random v removed eigenvalue ode guarantee substitute consensus consensus propose neighbor size generate datum initial implement lead step square outer curve gap negligible phenomenon support precisely theorem predict gap n size outer instance
context probabilistic graphical motivate graphical global nx px normalization define configuration typically configuration many graphical model inference compute marginal parameter estimation assume access oracle posteriori solve weight evidence map np hard notice however np quadrature integral corresponding element volume compact access discretize restrict require bit universal hash familiar skip hold distribute think family function typical hash much bx h bx x hash space nice geometric interpretation translate arithmetic rewrite tb weight step value point rewritten implicitly divide either slice area sum efficient procedure create enough slice axis point way accuracy require slice grow x undesirable axis range geometrically size area horizontal slice difficult arbitrarily area must within sum slice slice area must n ib ib way strategy hash optimization approximate compute area carefully choose randomly configuration repeat hash configuration globally token fact weight weight configuration process hash keep allow parameterize instance prove constant probability hash working add outer loop th ia nb tm optimization fact approximately count np impose make hard instance set hard code code solve heuristic pass allow natural massive implication obtain suboptimal solution configuration natural partition induce fix special horizontal slice approximation partition query probability least approximation relie b n l map accuracy analysis write approximation know show add repeat lemma probability entire map scheme require construction new configuration construction grows polynomially increase variable runtime problem suboptimal solution output bind stop monotonically non eventually solver optimality competition augment cp filter elimination parallel compute core loop reach provide belief propagation guarantee implementation library compute cf clique uniformly far close strong range variable ground truth force truth around accurate visually overlap actual error tend method provable bar optimization parallel hour aware practical item next investigate quality theoretical although apply binary grid model truth partition variable attractive site uniformly report partition method drop coupling strength weight term instance roughly constraint however plot runtime width width cm p cm cm challenging arise number grid see figure entry row encode domain potential index contain subset distribution complete grid grid equal grid compute combination enumeration property symmetry follow simplify fix replace solver design unweighted support consistently add constraint success know reasoning powerful single propagation relaxed penalty normalization evaluate necessary model selection candidate early training start boltzmann cd digits rbm distribution unit visible unit learn cd sampling training epoch depict gibbs learn evaluate compute accord score mean rank visually collection digit note highly digit gibbs inference provide estimate rank reverse visually representative order introduce general set reduce intractable combinatorial constraint normalization small map query integral estimate stop early accurate acknowledgment nsf grant nsf research grant lemma hash let bx bx hash alternatively rewrite e uniformity independence configuration set subset value fundamental construction small reduce hash proof configuration give number satisfy choice sample hash uniformity linearity pairwise property add ensure provide follow chernoff realization result proof lemma observe b b j c j proof lemma number configuration satisfy chernoff inequality lb probability lemma conjecture cs edu center ny science ny curse dimensionality randomize general define exponentially rely discrete combinatorial use application demonstrate query function fundamental largely problem range biology physic
rational actor initially policy optimally expect utility transformation adapt formalize follow negative physics kullback kl divergence act factor utility variational influence transformation boundedness actor govern determine term kl perfectly actor maximize utility recover limit ignore whereas actor infinite gain principle optimally sum integral consider environment potentially state actor observe observation allow uncertainty adapt behavior correspondingly make straightforwardly describe rational agent receive solution eq notice utility depend indicate refer processing kl energy principle maximize take eq inner operator expand lead entropy interpret expect upper bind mutual equation distortion theory distortion flip see rate equation energy marginal relative entropy observation cost ignore perfectly rational solution rational assign since mutual see section high mutual term action actor observation mutual observation lead abstraction resource actor case rational actor lead good low alternative interpretation actor influence actor action observation low actor become uncertain value observation compatible solution variational compute iterate alternate fashion framework iteration unique maximum manner become computationally costly involve analytic exist distortion make task task fully value easily task define task assume task decision distribution utility computational constraint formalize self rational make information processing cost arise change behavior task behavior accordance decision make task trade mutual action action action utility task two component action q summarize informative utility suboptimal utility environment value temperature decision pick action task actor boundedness lead optimally put far fully limit perfect boundedness entropy conditional task actor pick mutual expect stay pick entropy imply equally probable drop bit decision fully abstract regardless utility analogy rate distortion expect maker action minimal utility analogously maximally achievable expect information importantly decision whereas suboptimal capability trade specify average bit process limit maker pick utility utility form entity translate form partition element group indistinguishable subset distortion partition share utility become essentially indistinguishable grid white colored colored pixel row column colored utility pattern color pixel score utility color pattern utility conditional action yield utility nonzero mostly ignore appear pattern would lead assign include additional pattern mutual reduce essentially simultaneously indistinguishable actor action temperature mutual information pick simulation section indistinguishable task expense distinct indistinguishable pattern reduce utility sample however potentially maximum share nonzero task yield utility task compare lead share abstract exactly framework decision cost distortion theory connection naturally capabilitie author distortion present straightforwardly carry treat belief inference limited information indistinguishable lead idea previously mathematically
rate liu nonsmooth optimization nesterov analytical solution globally alternate norm square multi successfully multi task graph fusion norm logistic regularization challenge norm minimization author directly literature widely many behavior selection semi multi also norm framework actually extensive study sparse solve disadvantage bring computational difficulty mix valid admit triangular unify involve base solve mixed convex follow lipschitz induce nonconvex lipschitz continuous neither problem unified find nonconvex fortunately also prove typical objective bioinformatic alternative construct pattern obviously notation matrix write letter th several useful norm neither satisfy triangular inequality strict axiom norm follow obviously reduce admit triangular valid convex lipschitz yet challenge framework area consider function solve obtain know square outlier use norm noise magnitude distant preferred choose choose intermediate hence reduce lipschitz objective minimization hence directly far know scheme solve mix unified problem denote write p pm ni reformulate except constraint q lagrangian multiplier stand kkt induce simple local minimization solving initialize tr happen update way let convergence decrease pt unique easily eq suppose proof let I p happen formula formula generate monotonically decrease respect minimization remark construction say formula lemma combine monotonically full offer provide support improve sparsity pattern nonconvex case algorithm easily solve b ty low theory enhance experiment public set brief gene obtain cg ng co sample available gene tumor available firstly preprocesse effect test classifier perform fold classification report htb top indicate base performance norm representation norm alternative especially empirically various situation validate
divide cube side box well discrepancy transformation influence priori clear mention explicit dimension consider discrepancy box often want distribution standard cdf cube resort example unnormalize cdf order would feasible resort acceptance rejection rejection sampler unnormalize generate distribute instance discrepancy acceptance rejection point project side side sample set test confirm set rejection discrepancy dimension explicit construction sequence establish chen discrepancy tend chen existence sequence number get sequence chain carlo chen certain completely distribute convergence therein discrepancy related discrepancy cube box additionally term completely box discrepancy invert denote eq combine principle cube respect general principle discrepancy construction point set discrepancy exist point direct carlo arc project help conjecture mu connection convex smooth numerical distribution acceptance algorithm discrepancy discrepancy spherical acceptance local discrepancy point empirical denote indicator lebesgue supremum star discrepancy test set depend one convergent well box know variation box place box like smooth boundary complicate therefore constructive box connection integration connection set relate sphere markov carlo measure cube motivation set cube cube quadrature rule lead one discrepancy explanation follow discrepancy use old modification inequality inequality due consideration obtain discrepancy modification approach partial derivative box let discrepancy discrepancy eq generalization variation discrepancy study bound schmidt explicit construction chen construction chen generalization measure relate generalization discrepancy respect discrepancy star discrepancy box q discrepancy criterion isotropic discrepancy respect set isotropic discrepancy define set lebesgue isotropic numerical case case box number discrepancy schmidt construction discrepancy case bind remain sphere way define account spherical spherical spherical surface case integration sphere argument optimal construction upper construction transformation preserve measure I lebesgue proceed map square spherical discrepancy result digital indicate dash show spherical discrepancy quadrature digital net map box differ sphere inverse shape change broken complement rectangle either north rectangle unbounded smooth boundary curve turn boundary part curvature briefly discrepancy cube discrepancy map sphere degree discrepancy smooth star discrepancy respect twice differentiable minimal curvature divide maximal chen generalization discrepancy discrepancy spherical
method obvious nuisance parameter none obviously write prior fit prior angle hypothesis determination thus determination outside insensitive compare hypothesis factor search particle frequentist way affect way determination summary difference logic deal interest physics analysis limit search phenomenon suggest term question old live like thank david van cm perspective mail l ac uk almost scientific field collect consist try extract determination determination mass energy hypothesis distant galaxy bayesian differ deal determination hypothesis testing example day physics approach back whose play crucial discussion fundamental physic try mass fundamentally measure physical sharp difference bayesian simple problem say simple gaussian plus experiment frequentist analysis approach mathematical largely base axiom sum something occur important intuition identical trial something two way attack definition large identical require unable probability single regard create would say replace frequentist question first frequentist physical also probabilitie dark constitute universe check suitable frequentist assignment assessment think something knowledge situation person coin ask coin tail I give quick tail estimate situation matter concern vary person person originally winner frank illustrate assessment person topic intensive go bar empty I dimension explain quantum might person go yes say attractive bar go ask go unlikely offer odd false assessment essential bayesian powerful involve frequentist imagine perform count fairly rare ray energy energy want statement conditional count give replace regarded observe surprisingly likelihood likelihood regard exclude uncertainty relate distribution poisson observation small rare event observe event really important remarkable look denote distinction easy interpret density example involve evaluate likelihood could select day day bayes say probability multiply depend reduce result probability probability example obtain even relate obviously theorem probability frequentist one begin parameter measurement bayesian posterior say bayesian fitness aim posterior want posterior allow strictly cause density multiply prior poisson result probably right ground break nothing obvious answer independent really imply likely flat parameter another mass bad prior theorem relate people laboratory incorrectly exist difference piece whether extract person database person chance another person similarly imagine coin tail give wrong consistent datum likelihood order extract multiply might assign posterior unlikely local delta even coin continue fall tail fair prior interval case insensitive multiply prior however choice little part material q want decay time likelihood multiply evidence derive motivation bayesian suppose plausible result option determining prefer limit eqn frequentist centre construction horizontal line repeat edge band observe value large value plausible scenario time band range acceptable shorter require distribution avoid ambiguity claim range probable different merely confidence fig experiment temperature fusion centre month detector detector construct experimental lie accept range repetition experiment differ fluctuation carlo analytically coverage technique use construct measurement coverage equal nominal g drop nominal value determine nominal determine accurate really method conservative particularly construction e likelihood poisson determine end repetition fraction result include frequentist coverage indeed occur frequentist bayesian mean nuclear negative branching ratio elementary must etc incorporate guarantee could physical year value limit frequentist unknown suitable experiment repeat range range mention end analysis want involve repetition unknown refer within cm fix apply range often try affect range experiment interact build nuclear rate early systematic answer nuisance treat nuisance manner measurement write multiply choose prior background integrate contrast start fully method analogy early region approximate simple construction tend frequentist profile simplifie profile set depend interest call nuisance nuisance full nuisance simplify inference integrate convert functional modify likelihood ref long deal way incorporate determination parameter determination decide two datum collect half new production addition fig decay could result quality unweighted event new property seem frequentist decide field discuss briefly task distinguish accumulate number consist likelihood assume hypothesis pearson lemma say suitably guarantee achieve incorrectly hypothese pearson nan fractional area great equal tail hypothesis yield value could hypothesis possibility unlikely incorrect test inaccurate etc acquire perhaps deviation become small decay follow decay allow insensitive amount might decay able motivate correction possibility statistically deviation mention cox extremely important correct probability negative comment value ease statement many wrong fraction unity two hypothesis adopt convention upper tail measure conventional low tail hypothesis intermediate decision strength ambiguity essentially probability successfully increase signal strength become strong become several possible search particle choose physic usually whereas perhaps physics reason incorrect physic regard weak car good look else illustrate define various allow straight line solid separation curve axis rejection ease four long axis rejection lie rejection dot straight contour ratio upper correspond lie origin mid separation likelihood c discovery correct exclude loss loss area beyond height ratio pearson hypothese likelihood ratio ratio hypothesis make composite hypothesis cause nuisance may base see simply calibration merely hand observe might think contain decide etc account surprising bit event day life fact likely decide begin day specific energy physics look search particle whose pre chance chance mass might observe physics elsewhere lee consideration field calculate chance effect time real specify exactly elsewhere
solve compare size lipschitz matrix ls coordinate backtrack search scheme regard monotone gradient operation structure function cost operation compute coordinate backtracking ls line gradient bc bl bl bl bl note version method estimate block size speed reason along block give use magnitude along curvature search direction l bc bc epochs bc bl bl bl l bl comprehensive figure method require reach different size range equal require pass standard bar epoch require large update size computation g measure however long computation large modern computer compute less intel intel library parallel suggest appropriate size heavily depend specific architecture cache computer small figure column operation increase fix three iteration bc bl bl bl l bl randomize dual dual smoothed dataset web site whose dimension dual two real dataset initialization randomize coordinate method pt theorem part discovery lin randomize method minimize separable nonsmooth method pick block prescribe associated subproblem usually certain contrast randomize partially curvature show stationary surely approximate sublinear norm gradient sublinear generate conduct preliminary regularize square substantially descent word nonconvex composite randomized algorithm coordinate namely type arise become greatly challenging expensive block method solve nesterov randomize promising optimization involve analyze nesterov convex iterate pick solve wise proximal lipschitz norm recently extend block choose practically slow spectral utilize stepsize shall applicable performance dramatically generate monotone counterpart nonconvex optimization problem motivate proximal form set nonempty associate nonconvex nonconvex nonsmooth function coordinate wise lipschitz constant randomize proximal method assumption pick prescribe necessarily subproblem replace spectral method progress contrast usual enjoy utilize curvature component uniform method run accumulation stationary almost stationary sublinear convergence gradient consideration show objective linear svm machine demonstrate outperform block descent method fix solve convex conduct experiment throughout paper domain close give definite euclidean follow nesterov assumption lipschitz constant respect uniform random hold uniformly exist satisfy relation k hold second immediately em ready first objective sequence uniform statement hence increase exist induction yield notice fact notice follow relation induction hypothesis continuity induction completed observe together continuity follow fx e combine eq notice eq conclude finally claim see k sufficiently approximate stationary point uniformly locally subdifferential accumulation stationary almost surely uniformly kk together example proposition use relation accumulation subsequence second yield eq surely iii follow view relation obtain follow yield know view eq statement sublinear convergence iteration define respectively observe statement solve structured assume convex subsequence k exist q together bound statement fact rest study version
phase retrieval equivalent treat matrix simpler shorter immediately extend broad gaussian discuss bind minimax risk zhang specifically nr possibly adversarial reveal suppose covariance well toeplitz constraint psd toeplitz psd toeplitz cone contain relaxation exact measurement state assume toeplitz constants highlight psd toeplitz admit specify toeplitz matrix soon large information logarithmic general toeplitz return randomly toeplitz state term norm setup worth first exponentially arise secondly able theoretical roughly speak distribution aspect sparse seek compatible intractable tractable relaxation support prove successful compressed sense measurement simultaneously provide list summarize exponentially high soon factor universal sense simultaneously estimate recovery approximately appear cs measurement estimation quadratic study model simple recover jointly recovery measurement measurement sparse low rank motivate adapt accommodate q surrogate rank structural state sampling satisfy c depend provide class signal recover signal recover performance establish gaussian sense sub sense vector somewhat surprisingly decay law fashion specifically power large c inexact exact perturbation bound recover highly corruption proportional measurement summarize psd symmetric covariance toeplitz sparse drop psd replace nuclear psd never toeplitz matrix scenario toeplitz isometry rank mixed isometry property dimensionality preserve strength acting way restrict isometry appropriately isometry lead rip occur metric strength al isometry call signal counterpart initially account analyze phase general rip consider nuclear rip leave rip rely dual mathematically isometry metric specifically norm measure frobenius trick treat carry slight modification rip rip rank matrix sparse rip plus class define rip small decompose component low treat superposition sparse measurement low unfortunately occur primarily measurement effect measurement operator exhibit rip presence minimal measurement let sample gaussian universal statement extend rank result asymmetric immediate proposition rip corollary argument omit reader constant universal rip obeys sampling rip obeys universal constant plus sampling universal rip obeys provide c recall subspace consequently stand via rip thus auxiliary rip constant establishe consider rr q numerical minimizer obeys constants universal establishes hold obey universal depend rip picking obtain corollary conclude rip concept allow kk rip constant satisfying turn toeplitz rip rip toeplitz toeplitz fortunately toeplitz rank detail next toeplitz measurement exhibit rank rip rip matrix near isotropic convert quadratic measurement isotropic measurement isotropic proceed toeplitz investigate near general matrix convenience presentation definition rip characterize rip rip suppose least natural consequence might refine due w obeys suppose universal bind operator type basis stable recovery basis soon theorem ambient toeplitz matrix motivate construct another focus subsection toeplitz low isotropic I calculation reveal isotropic facilitate matrix whose specify entry isotropic associated exact recovery exceed establishe equivalence argument detail isotropic combination toeplitz take q n one measurement matrix isotropic restrict define isotropic randomly select row submatrix set also conduct scenario solver figure freedom theoretic optimality htp gaussian comparison entry freedom psd cell line empirical psd b example concern recovery necessarily psd generate symmetric sparse level run standard sparsity noise bound measurement pair level set htp investigate signal device strategy energy popular e jointly toeplitz low explore retrieval indicate covariance recover draw soon complexity exceed fundamental universal phenomenon matrix structure highlight stability convex presence performance jointly notion mixed rip systematic analyze toeplitz isotropic operator future interest encode independence often measurement covariance whether rip rip fouri rip less measurement acknowledgment wu helpful suggestion chen discussion helpful discussion chen center science grant chi partially nsf fa google award derive low bernstein derive characterize concentration version repeat completeness similarly sub norm observe sub absolute indicate satisfy constant derive repeatedly furth moment shown derive rise ready characterize concentration sub exponential variant bernstein absolute exponential norm absolute yield constant gaussian norm constant establish compressed measurement focus isotropic rip bound arise supremum rank small constant process observe copy jensen expectation obey obtain characterize suppose random conditional conditioning last jensen soon banach space symmetric independent operator know theorem repeating conclude mathematical proceeding singular complement introduce rip write rr q divide singular singular feasibility constraint yield h derive bound allow put together let projection complement feasibility yield decompose collection yield recall universal claim proceed convenience characterize adopt notation entire sum subspace orthonormal orthogonal pointed compatible follow definition subgradient follow consequence feasibility constraint follow rank arise orthogonal satisfying property large singular exceed singular entry magnitude exceed magnitude rise rip argument scheme know consequence make goal combine reduce solve eq q q choosing satisfie toeplitz matrix q toeplitz entry correspond harmonic toeplitz matrix since norm exist absolute remain toeplitz conclude proof technical event toeplitz isotropic end spirit apply tail thus absolute recall eq absolute take yield constant supremum inequality quantity stand suffice cover yield condition chen distinction university electrical engineering university stanford engineering stanford stanford university research structure compressed chi receive ph electrical electrical china engineering department university author award award conference speech award award google award university stanford signal bioinformatics sm electrical stanford university previously electrical interest theory wireless communication field dr wireless wireless technology cloud communication high stanford receive award wireless communications award award award fellowship business journal award author wireless communication book communication cognitive publish university b engineering berkeley dr transactions journal trend communication communication wireless communication transaction wireless communication conference organization communication distinguish information communication stanford received currently serve budget force international inference accurate estimation rapidly change power storage acquisition device extract pass store explore quadratic impose minimal requirement preserve structure popular low toeplitz jointly respective quadratic potential streaming processing wireless phase retrieval coherent soon measurement exceed robustness novel notion mix restrict isometry property rip rip isotropic addition dense retrieval rip toeplitz rank stochastic ever dimensionality constitute extract signal acquisition device rapidly estimation stream storage low fortunately indeed possess dimensional ambient different structure structure list low account matrix monitor process metric covariance low toeplitz arise spike stationary matrix equivalent task wireless communication array g pairwise mutually exclusive sparse finance biology spectrum approximate attention recent development sparse pca closely recovery reconstruct unknown denote sense mm free number constrain acquisition could ambient wide admit bring computational storage comparison detail rest benefit quadratic model represent sequentially high acquisition device desirable input storing stream extract limited complexity stream force impose converge prior across consecutive independently exploit low pool small termed outline randomly nonnegative aggregate term stream rapidly first ambient cost datum affect aggregate compose instance distribute arise randomized sketch compressive snapshot order statistic measurement unable demonstrate allow theoretically acquisition motivating estimation place high regime obtain reliable communication operate extremely wireless system environment rely recover spectrum spectrum one observe average measurement measurement read observation match communication recognition encode cast subspace detector obtain optical imaging device measurement due frequency form naturally space appeal wave field experimentally low rank correlation physics form constraint optical rise recover refer convex e nonconvex enable exact retrieval recover magnitude formulation special apart precede aware rank naturally arise linear regression require rank aim near contribution convex optimization measurement variety structural assumption low toeplitz rank exploit tailor structure sub vector derive theorem aspect sense high noise adversarial multiple propose soon reconstruction secondly obtain isometry rip rip strength signal preserve respectively conventional point sensing rank structural assumption subtle simple approach complicated entropy combination covariance toeplitz matrix also operator universal broad operator include last scheme interest rank one small measurement universal covariance framework exist collection truth paper motivated success sense cs achieve sense parsimonious compressive robust assume approximately recent order consider et al estimating n estimation inspire recent development show succeed recovery type assume work show suffice result accommodate sparse framework put stream research decade inspire observation
obtain draw distribution rather explicit case plug functional computational interval individual logistic covariate platform core core processor regression processing second natural processor weight regression single equal require single accuracy particular style compute although motivated procedure bootstrap subsampling classical indeed fast subsampling bootstrap line sophisticated likely map useful way want kind exploit core tool providing continue recent notably graph collaborative notion canonical often map directly factorization procedure svd cubic motivation algebra devise hardware computation development determine meaningful particularly salient application matrix growth vast entry miss collaborative filter application rate book movie many analysis study divide factorization aim parallel hardware refer algorithmic partition column base method combine see central overall design retain guarantee base provide e model rank noise value follow matrix despite unobserved singular assumption assumption coherence coherent high sampling vanishing retain refer partitioning simplicity submatrix order address develop relationship point focus denoise important dimensional noise estimate shrinkage projection consist outer intuition exhibit statistical frame indeed widely problem science hierarchy relaxation tradeoff need connection move eq closed define eq refer hull combination set geometry relaxation tangent inside cone consequently pt establish gaussian statistical particular implication computationally less kind concrete formalism principal component entry entry associate pt detail conv super ball two hull tradeoff expensive procedure require polytope nuclear denoise set display relaxation sample order example seem method review line bring contact consideration orient massive mention interface explore set classification subset weight rule merge estimate able asymptotic equivalence also empirically significant divide idea family explore author model selection focus covariance reveal relate acknowledge reality massive dataset complex heterogeneous goal apply dataset massive full methodology massive massive risk discovery management achieve algorithmic character field acknowledgement acknowledge numerous perspective design increasingly couple inferential question certain time budget question identify consequence computational relaxation field mostly phenomenon big technology large increasingly inferential towards force seem meet challenge key inferential accuracy budget although field tool risk number different analysis predict datum force ad hoc science poorly equip inferential associate researcher rarely population inferential requirement comparative analysis rarely inferential goal notion save growth grow size perspective drive conceptual challenge address divide subproblem simple subproblem divide break subset challenge analysis wider divide correctly calibrate perspective whereby order back quickly result view quality algorithmic poor algorithmic quality overall increase impose budget challenge theoretically sound paper organize subsection divide algorithm section inferential evaluating point bootstrap material usual implementation intensive resampling apply notable virtue process independently cloud runtime require massive massive dataset processing may straightforward processor network appeal alternative instance approach subsample idea yield fluctuation challenge one fluctuation dataset subsampling bootstrap analytical confidence obtain sample subset procedure correction inconsistent broad motivation explore sample necessarily consistent procedure noise
data amazon dimension range column dimension ht precision recall limitation svms situation total geometry useful allow well moreover binary expect draw affine additive error true consideration confirm algorithm categorical table investigate structure set value dimensionality logistic six predictor use filter analytic commonly selection algorithm set feature high set feature single parallelization perform low dimensional original currently apply calculate study accept reject despite frame support inherent classify svms guide analytic oppose inherent geometry identify six suboptimal feature linear geometric achieve excellent feature feature solve introduction svms svms hyperplane separate new hyperplane category belong aspect determine commonly use heuristic due geometric tie geometric structure drive svms goal research solely inherent geometry first create optimal suboptimal difference identify analytic apart filter feature independence geometric make technique adopt domain understand learner insight guide field discover mathematical set precise manner know help maximize efficacy guide ensure property affine point generate particular include text sparse vector format format identify set whole binary word type achieve range within train average recall range range average set effort require address cpu believe impact apart throughout supervision provide marked hour deal carry optimal classification heuristic review et datum compute processing project increase dimension also correspond turn cause generalization expand idea selection capacity give preference feature low error emphasize classifier intersection sense idea small margin reduce affine discuss identify overall process seek identify key describe perform identify training consists manually label accuracy selection four problem every use subset svms inherently use svms problem vs consist way remove half symmetry leaving represent feature classification number choose kernel likely lead perform separation consideration implementation geometric affine sample us library application feasible hull small hull affine translate origin affine k affine hull dimension particular hull polytope affine hull write simple calculation easy computationally suppose hull somewhat affine dimension ratio affine hull lot project distance generalization observation lead geometric ratio difference introduce point unique value otherwise row organize belong refer consist consist point cloud cloud affine affine hull assess dimension choose exclude ambient dimension geometrically coordinate subspace cloud terminology define numerator ambient denominator affine l numerator ambient ambient ambient affine ambient affine dimension pp total purpose assess feature column certain amount remove column project select compare value feature train assess obtain also ht experiment plot ratio standardized case notice strong tp represent positive positive negative predict value manually generation clear relationship significant contain logistic feature selection whether suboptimal ultimately geometric property base algorithm include datum category store file represent consecutive previously column miss suit particular possible matrix ambient affine calculate intersection see detail possible subset lin linear regression lin write subset else identify generate combination symmetric remove binary classifier empty negative class program create create file store unique nest line subset training include feature example represent calculated predictor logistic finally logistic use forward inclusion receive predict standardized score appropriate file associate evaluation limitation particularly relationship text table strength use classify chemical gene movie sentence build language movie review corpus intend classify negative sentence six r bc sentence sentence movie table brief summary selection number raw set type feature ft list bc result list algorithm precision prediction make section generation accuracy tp fp fp tp positive negative negative evaluate algorithm optimally
likely nearby amount method well lose private cca cca one example face angle third random proximity train view predict view prediction relationship see narrow wide classical view extra zero proper predict good regularization provide problem especially term outperform term method practically suggest term important private always relation count bias bias bias recommender system early turn table useful type recommender roughly million entry minute comparable time report full validate collective factorization low representation collection limit relevant matrix technique avoid enforce factor structure share private notable advantage sample modify incorporate particular efficiently cca provide cca drawback relation entity problem illustrate recent multi relational datum acknowledge university foundation grant project supplementary embedding collective factorization variational notation denote expectation entry finally ij ik relaxed ik update term approximation automatic relevance determination approximation detail respect update eq shorthand mean x additionally variational prior bias approximation ard update update gaussian pseudo derivative update modification mostly drop modify without simultaneously representation base entity user item recommender embedding share enable individual share share alternate group support principled count observation focus approximate outer formulation recommender application many consider entity entity give whereas user least factor fundamentally equivalent rich case distinction factor analysis collection share entity name multi end bilinear idea easily three tensor example illustrate recommender setup matrix circular interesting circular additional view depict number patient ignore measure view kind common handling attract much likelihood use formulation meaningful simple view directly factor describe unlikely view generally subset matrix introduce constraint factor wise entity bayesian regularization automatic relevance determination ard control regularization free support collection type non setup interesting task pay augment multi view view space key ard regularization describe entity matrix entity row simple product e cm share low rank part share large symmetric private wise entity leave observe contribution patch represent matrix factorization concatenation drop notational simplicity belong matrix technique capable along diagonal unobserved crucial introduce explain extend low restriction solution factor correspond undesirable many practical individual structure since structure capture element wise basic entity th factor automatic put group create factorial similar private group standard private overlap sparsity implement group wise learn private factor specify likelihood automatically particular entity set especially column reasonable row inter analysis however support wide potential improvement closely relate early solution early provide maximum bayesian hierarchical factor wishart assume roughly equally hasting support limited case applicable factorization arbitrary collection worth describe typical illustrate practitioner might share column entity many relationship ignore throughput biology patient time language entity word column relationship identity relation proximity analysis modality even though representation fmri eeg jointly entity schema iii similarity feature reasonably model feature live along simple recommender relevance indicator recommender incorporate information help additional entity interest cc binary gaussian help private combine gain aspect value though bad large becomes furthermore tune need perform validation regularization use incorporate additional type alternative mean circular entity social column individual use high recommender actor actor provide indirect relationship start technical importance choose potential incorporate factor view setup goal conceptual importance solve task factor comparison show really compare secondary constant entity private setup special case two entity
covariance covariance estimate algorithm greater several currently available goal trajectory surface prediction miss must procedure possibility em chain mcmc automatically due computational overhead variational approximate initialize mcmc sampler refer rely density factor relatively posterior accuracy often improvement consider dependence among gain approximation loss provide reasonable agree author vb replacement consider tool approximate large situation vb value achieve fast interest close expression vb expression full mcmc alternatively laplace necessity penalty smoothness make vb speed minimal accuracy well remainder review analysis discuss parameterization unknown surface vb experiment forecast seven day website conclude trajectory goal regression involve eigenfunction penalize spline use e include dirichlet literature include one cite appear unclear analyze vary degree usual unknown noisy take ij ij ij xt gs xt x curve representation pc score principal component score choose integer unknown explain g vb though identical follow obtain pool fit cubic surface raw remove il il xt middle surface effect estimate matrix component vb notation parameter develop measurement update beyond experiment comparison mix mixed model formulation spline review bivariate nonetheless idea smoothing applicable spline spline order gaussian component specify grid integral trajectory surface follow spline domain kt kt evaluate integration ease specify matrix quadrature weight note row scale smooth scale lie unit smooth considerable univariate isotropic penalty penalize additive first make spline modeling usual frequentist spline equivalent place improper spline coefficient impose degree lead improper rank rank instability inversion appearance determinant employ spline simultaneously computation function e begin evaluation take decomposition penalty matrix diagonal eigenvalue dimension basis tensor spline combination form diagonal orthogonal column exposition x write ig gibbs exclude expression full distribution derivation quite omit update understand update provide update update subsequently apply draw posterior update density give trajectory scalar spline update penalize coefficient update response vb update attention rest besides derivation obtain full conditional overcome slice efficiently demonstrated slice hierarchical draw interval analogously especially recommend intuition second develop principal stem full proposal trajectory acceptance intractable specifically proposal part conditional study occur frequently proposal reject h probability section variational fitting quick closely reader parameter goal simplify closely kullback leibler derivation variational bayes rely parametric sometimes approximation bayes kl easy see e exclude algorithm one full update sequentially terminate change become sufficiently notice sampling conjugate helpful tool vb direct acyclic dags co dag calculate vb general update spline exclude calculation smoothing unknown plugging gauss quadrature quadrature implement grid quadrature approximation moderate strategy avoid drop take logarithm problem diagonal vb recall laplace variational explore denote vector routine formula give approximation dominant require outer taylor expansion derivation log monitoring convergence fit data functional covariate examine point equally response measurement examine regression surface case generate include measurement surface figure seven fit fully observe measurement section fit variational bayes curve vb except measurement order code package burn mcmc use burn simulate vb square report perform reason score difficulty predictor estimate singular singular cause zero principal score suffer vb perform trajectory scenario slightly make recovery make accurate trajectory sample file f sim pdf file pdf turn surface rise f dt inside hull trajectory plane performance poor mcmc mcmc vb mcmc optimal smoothing surface fit substantial vb mcmc root figure mcmc know entire trajectory estimate nearly scenario simulation scenario vb mcmc second vb file sim sim fit website forecast bid seven day digital take place standardized set period bid pay recover current item available previous amount table new enter least price plus give price six day logarithm final price try price hour hour price zero usefulness method trajectory ratio log six final day predictor forecast accuracy set rmse logarithm test training training comparison use ten surface display randomly plot trajectory additionally show frequency histogram part majority ratio early hour small high price surface flexible predictor coefficient function smoothing perform predictor desirable forecasting estimating reason fit rmse partition fitting vb follow simply assume poor especially bad fitting observe measurement metropolis within mcmc inference situation standard quite due estimate develop variational fitting obtain usefulness accurately approximate intractable input account trajectory bootstrap implement due concern perform slow vb mcmc future include investigate credible band bayes compare band bootstrappe promising extension binary response component spline begin
image truth body part optimization combine framework effectiveness forest support experimental diverse dataset concatenation compute nuclear one achieve minimum nuclear concatenation show frobenius show equality condition decomposition singular orthogonal write sum square one mm theorem axiom transformation forest learner splitting linear nuclear maximize separation thereby improve performance experimental variety ensemble weak binary left weak output weak play classification popular usually seek learner much dimension vision image handwritten digit trajectory object ambient space lie subspace wide approximated world often addition perfectly exhibit cast arrange approximately promising underlying structure realistic deviation subspace effort transformation decompose rank image multiple salient rank present discriminative split tree weak learner nuclear surrogate learning criterion recover angle class intuitively propose share lda significantly intra union lda art learner learn transformation help classification ensemble consist internal split evaluate leave child leaf point tree posterior improve capability learn free ic class respectively arrange denote concatenation nuclear adopt prevent effect research keep normalization prove lead excellent maximize angle reduce intra capability denote indicate assign subspace class b learn inter class affect weak tree separation class angle maximize angle different start present basic matrix dimension concatenation orthogonal result objective minimum space reach angle subspace maximize small subspace equal adopt norm rank norm induce frobenius nuclear reach disjoint maximally distant function nuclear norm good approximation reduce optimize category two transformation reduce class furthermore avoid descent matrix identity excellent adopt initialization public mnist handwritten digit face natural consist bit handwritten image extend face subject pose mnist extend image etc result learner context compare learner category discriminative transformation concentrate different fourth transform tree extend face dataset compare split half testing propose learner face subject third learner enforce two category class separate decision lda svm tree learner learner depth specify random forest rely termination tree depth termination prevent branch sample tree several art accuracy significantly learner try framework replace introduce learner weak learner forest accuracy increase increase accuracy linearly exhibit test significantly accuracy learner order thus fact order magnitude outperform standard forest framework randomly category discriminative clearly demonstrate concentrated class fourth first example test depth provide increase transformation learner model tree accuracy increase test hundred option tree trade far divide
goodness testing calibration satisfie rate test prior hold regular separation wide model still useful difficult point addition even improper hold case contrary bayesian apply detect versus testing consider frequentist instance treat ellipsoid smoothness sl threshold separation prior k n separation test test rely fast estimation lead rate design independent standard design computation piecewise counting study monotone function suit seem testing density exist section algebra condition condition define set speak constant study posterior metric define frequentist alternative set precisely smoothness compute procedure positivity discrepancy function piecewise k piece calibration posteriori positive possibly calibration lk lk monotonicity consider piecewise formulation way test lk lk neither alternative separation minimax separation term term I frequentist loose satisfied influence asymptotic great finite prior choice ease practical default variance inverse accelerate algebra j eq walk hyperparameter calibration choose moment datum flat variation great choice small size part difficult especially order median compute present property interestingly calibration test relaxed calibrate threshold tolerance available test author useful greatly grateful propose helpful discussion partially safe material proof use article available deduce end prove calibration decision rule want n np enough need define sl nf lemma material end come piecewise function piece kullback leibler useful study monotonicity constant n satisfy condition enough eq proof material either material turn q immediately enough satisfy theorem end apply first give consistency prove consistency assume df g increase piecewise gd nk nm nm give get end first sequence test exist define nf suitably chernoff net define moment central centrality nf eq get section concentrate throughout constant kullback divergence density define denote true density j jk f n k eq existence suit cover hellinger alternative construct apply either belong f p look eq l kx k together n c absolute choosing nd q proposition height width answer hypothesis well separate general shape test positivity monotonicity setting lead indicate rate far monotonicity positivity arise appear range survival shape theory back early subject notable enforce inference monotone density include grow grow frequentist point view propose critical test monotonicity look sensitive flat propose monotonicity make concavity primitive grow theoretical difficulty embed hypothesis separate particularly difficult approximated answer separate special particular shape last receive far good understand additional model mainly fact closure extremely detect note nonparametric many regression gaussian residual prior put f integrable sub thus prior illustrate choose spline log bayes compute favor way separate put
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process couple another observation parameterization slow typically recent upon allow reduce smooth real fast variety unlike binary likelihood general q parameter require parameter implicitly condition case learn several list constitute exponential likelihood spatial along country poisson gaussian cox process generalization model gaussian prior specify wish compute expectation observation obtain py maximize log automatic determination ard c l latent imply three column quantity subscript index observation posterior n variational parameter maximize variational log marginal multiply divide jensen parameter maximize eqs term list expand concave straight practice memory intensive primal jointly require result reduction affect suggest parameter parameterization moreover ascent limited require slow modern concave provably less eq approximation key interpretation one super potential propagation accurate optimization ep numerically unstable property beyond gaussian model show convex parameter experiment much iteration review decomposition form introduce equivalent introduce correspond lagrangian q constraint find minimize analytically close length involve statement unique maximizer eq q importantly collect involve get follow conjugate likelihood close example summarize effective likelihood denote plug ignore optimization strictly parameter appear might act barrier additive arrive form bernoulli logit volatility yes k bernoulli logit logit v k detail computational detail illustrative py conjugate conjugate take indicator constrain lie range likelihood give detailed derivation online case apply likelihood paper three plug eliminate substitution optimize quasi act barrier limit feasible avoid unnecessary function evaluation treat unconstrained gradient gradient descent direction direction goal keep restrict wolfe constraint arise implement next q strict novel world covariance parameterization optimize method naive much alternative multinomial logit classification setup outline logit rest paper uci repository category test exponential kernel th ij across hyperparameter hyperparameter marginal giving prediction error train train predictive carlo method star show reasonable fig trace primal iteration exist yet converge plot step evaluation expensive propose evaluation hyperparameter rate region poisson spatially eq hyperparameter intrinsic e form discuss region simplicity find hyperparameter log several occur trace much evaluation viewpoint variational problem variational logit super setup apply technology dual converge previous apply inference problem coordinate ascent coordinate interpret allow parallel maintain disadvantage variational convex remain open also aim covariance break barrier substitute derivative simplify eq equality additive constant eq add multipli term ms begin project qwertyu qwertyu qwertyu qwertyu qwertyu qwertyu qwertyu qwertyu qwertyu qwertyu qwertyu qwertyu qwertyu qwertyu qwertyu qwertyu qwertyu qwertyu qwertyu qwertyu qwertyu qwertyu 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generator majority nine health type sensible median total certainly game space randomly become content pose good present game lot control control proportion appear play extract player aim mm single playing labeling get varied cross beta facilitate public server survey people purpose five easy moderate hard believe something people choose category game select active learning game cc facebook total survey record game play nine five survey yes rate bad survey survey increase number also bad difficulty people bad survey label game cause disagreement amongst participant cause less disagreement indicate give beta select type category survey player engine produce play consist feature many play link question mention train cluster reduce game reasonable play label active svm randomly choose game active converge indicate acceptable game acceptable subspace consist difficulty category content decompose binary task active train five rf classifier adopt winner five classifier public survey survey game regressor regressor complex nonlinear em em epoch employ multiple rf classifier ensemble different combine form ensemble learner use sect simulation prototype illustrate rate sample define approximately effectiveness active scale xlabel ylabel plot mm legend plot cc cc active reach beyond lead fit specific easy moderate hard close confusion reveal misclassification adjacent misclassifie reliability cc performance hence use ip support target truth popularity beta player beta test reliability player ip fold cross play rf range decision exploit adjust boundary apparent negative error confidence threshold rejection depict rejection achieve approximately reject nature public survey play threshold xlabel ylabel legend mm scale ylabel style plot mm mm ideally ip play player video game expert player game ask play game number game beta present random game play ask yes base four play player game difficulty number feedback game hence preference player category game player score define content category game category metric model depict score basis player evident ip perform much ip player ip superior balanced inferior easy moderate suggest playing game public survey identify preference never fig public prototype carry proof propose suggest prototype mm framework relate novel framework exist framework neither term content generation control content create play play hence line generate style content exclude acceptable finally work certain variation content identical nature build lead evaluation measure content cc extract pre content game accord model encode drive drive create public lead novel content crowd public experience could evaluation despite line nature ip line new carry framework base style game whose playing game experience generate quality content framework test evaluation generator control enable technique provide technical mean though prototype usefulness algorithm cc proof lead satisfactory annotate apart exploration learn artificial facilitate annotation annotate game number direct content explore collect public simulation player type player experience play type content stage player quite preferred content present unknown target could affect factor feature pre define carefully study lead experience dominate development evolutionary computation contrast content next reduce carry differently employ function implicitly player behavior experience construct base player behavior content historical failure quality heuristic evaluation learn cc failure organization enable cc evaluation develop knowledge purpose develop novel integrate experience drive facilitate limit adopt experience enable appropriate already platform substitute enhance current enable technique present enable technique order challenge exist content incorporate player interactive via prototype overcome limitation explore relevant grateful anonymous public player feedback public survey player cs ac topics intelligence game among variety base dominate promising pose number range evaluation content problem technique exploit public test drive target player experience enable technique various develop prototype framework promise generating content generation evaluation public experience categorization game expand rapidly movie gradually point cut graphic game game many period generate character create manner content video game automatically content generate character basis upon provide stream player resource build game cost although often get create recent variety improve fall al give employ evolutionary find appeal player less manual traditional approach technique level like game game issue promise pose case need evaluation widely technique evolutionary nature regard evolutionary deal phenotype mapping direct game clear phenotype distinction preserve locality failure space generate content generation population content need fitness evaluation fitness ill pose fitness player term content generation aim encounter game unlike gain encode experience typical life cycle public beta interference end user experience rely generalization act controller tend ideal content result several considerable hard experience player experience main summarize novel framework drive exist enable framework person concept prototype sect background sect enable implementation sect proof issue relate section exist video game development structured incorporate production people go feature add reach beta participant behind game public opinion opinion bad public pass go end terminology expand parsimonious parameterize specifies generate generator level generator multidimensional health content generator generator likely large exhaustive hand content generator build target attain exist technique challenge concern undesirable map consist player problem describe optimize unclear extent purpose next content drive cognitive experience pose big experience accurately undesirable target experience much content avoid player experience behavior burden target player pre player trait difficulty infer define furthermore player feedback quite inaccurate difficulty player directly learn style type beta collection member often human build human play pool study challenge existence fashion collect public would difficult crowd deal one player drift adapt content target create super experiment successful half experimental protocol player report prefer dedicated purely properly online adapt attribute statement review game development exist development knowledge content space parameter avoid undesirable content limit space define player style implicit flexible categorization player crowd experience gain crowd player player style tackle player player experience adaptive aforementioned propose framework tackle systematically motivated process divide typical life stage involve concern public tailor target player attain goal behind encode public behavior control generator produce content player preference via depict fig tackle beta experience knowledge deal four player filter content poor quality manifold acceptable content manifold framework eventually content player content category initially ensure preference player ensure sufficient content subsequent generate game player detect tackle enable section systematic emphasize drive advantage encode sect enable support describe solution recognize description acceptable content lead acceptable randomly game train choose current record assign annotated update parameter general tackle limited resource want instead select game annotation apply partition cluster cluster spread entire space extremely play label apart assign collection aim search training train need representative game annotate constitute reduce acceptable content instead search purpose cc acceptable content pre content discrete ff fc f regression beyond pre learner cc th record assign find truth sect similar acceptable content annotate possible game proceed fashion sharing resource exploit result annotate motivation propose cc spread entire acceptable content area furthermore acceptable annotated cluster game game annotate annotate acceptable game ready predict acceptable game game ip use reject game acceptable high line content cc content reduce select difficulty role fold model public consensus confident acceptable identify reliable select g n span spectrum cc annotate game feedback variable feedback survey formulate play consensus experience assign beta player feedback play feedback multiple feedback reliability via positive typical seek infer adapt crowd name enable general extended categorical crowd em summarize algorithm substitute player actually play pf mm tn piece follows firstly assign factor beta secondly consensus experience regressor former utilize learn ensure experience exploit individual little back solution player goal target game preference provide player log attribute record play find formulate class public play correspond feedback record
model distinction indeed dynamic equivalent might online regularize suggest remain open follow definition non measure dynamical identity operator bind correspond tracking regret bound require contraction poor multiplication equal unity theorem novel sequence matter time trick whereby divide increasingly fix square horizon share additionally use dynamical model environment address challenge describe adapt environment establish class predictor segment q deviation good tracking sequence dynamical I mt tracking algorithm regret fs fs expert assign cumulative loss weight share amongst expert expert quickly describe term dynamical bind advance trick ti know rich history social analysis track roll http www com represent vector vote form log ise challenging associated correlation voting denote except agent loss c ab ab intuition two member strong become time dynamical trick power instead step allow value grow meaningful average per year window dynamical improve successfully high spike around drop tight form mid mid movement form sort align know political novel online mirror incorporate dynamical model fix adaptively select promising candidate track optimization develop useful regret develop simulated show behavior underlie optimality bind optimality condition follow subtract complete strong convexity bregman divergence cauchy schwarz theorem describe scale deviation dynamical family previous online accumulate track overall scenario variation dynamic mirror method dynamical form base scene social streaming range formation dynamical performance classical kalman readily dynamical track accurate dynamical rely generative propose model place restriction address impact universal provably instantaneous prediction method ensure good offline entire allow issue corrupt bound static piecewise reflect dynamic describe individual sequence incorporate novel online optimization prediction mirror evolve establish track another share scale evolve range setting estimate volume regime incorporation mind focus incorporate regularization dynamical understand gain ill pose setting experiment reconstruct motion compressive social network roll improve time sense batch poorly stream bridge sequentially construct prediction pose time forecaster forecaster define follow function measure accuracy similarly potentially loss forecaster minimize time efficacy respect yield goal online relative broad family incorporate dynamical admit contrast slowly time constrain static paper regret respect static static static regret static characterize perform say minimize algorithm access point however static dynamic study concept tracking regret compare output choose knowledge fair fit fitting frequently characterized variability sequence allow imagine series poorly conversely regret become static regret concept term tracking tend regret measure accumulate measure accumulate arbitrary static choose optimally globally moment intuitively track vice relationship rely general dynamical ultimately dynamical concept formalize intuitively actually generative get analog static regret enforce use
jensen inequality concave relationship rest derivation appear numerator discard iteration obviously jensen q difference zero maximum right equal maximize confidence become following come normalize good equal lattice lower lower expect whether best expect likelihood mle self solid bar confusion dash function affect graphical self include confusion rely expect count channel confusion mle middle expect well right c r model confusion self confusion confusion topic language take hour acoustic simulate condition interested resource challenge many set manual topic label create smoothed hour disjoint corpus blind channel proportion unsupervise b hour subset set experiment system use cluster gaussian tie model audio ml decoding perform three acoustic gram lm lm vocabulary reference corpus content frequently occur content information retrieval detection care content overall emphasis token frequency improvement compute content computation threshold manual corpus emphasize metric unchanged reflect correction accuracy sensible improve figure none confusion improve confusion significant gain frequency map gain mle supervise case content application finally adapt language language yield observe experiment confusion result content restrict appear time confirm improved language yield recognition focus incorporate across lattice self beyond improvement believe confusion plan work mm human technology speech mm cm channel mm cm cm support technology conclusion recommendation material http edu human model crucial translation particularly unless speech adaptation tune automatically audio consider likely instead good self training efficacy obtain reliable improved language self confusion modern automatic acoustic language million domain interest static make language change interest hour audio text setting must rely adaptation audio improve language self language output run audio training error error rate speech effort entire lattice self bias worse particularly rare content word content pose considerably adaptation adapt lie present topic proportion estimate probabilistic model improve well significant new existence rather train model present self introduce adaptation utilize channel estimate speech speech topic many domain manually need language model goal training base automatically audio compose speech represent confusion posterior speech confusion consist bin bin vocabulary vocabulary truly although lexical bin context bin confusion use model multinomial capture frequency text document equivalent eq posterior distribution compute maximize eq likelihood map place dirichlet dirichlet b introduce extra optimization prove tb fill thick inner style thick rectangle var fill sep circle draw fill thick gray sep thick rectangle var q var var var lambda draw thick sep fill gray sep thick var var w var var lambda alpha alpha lambda w confusion accord observe channel language channel recognition confusion likely word approximation phenomena speech variation lexical etc condition pt audio count probability top two implement task relaxed confusion confusion path confusion likely argue channel confusion discount
regret numerous include recommender learner together page choose arm offer user user reject accept item offer uncertain acceptance observe gender age user item must recommend agent learn particular unlikely agent get item decentralize item security system attack detect attack context characteristic traffic context valuable occurrence attack security dynamically dependent unknown traffic stream information concern know security contextual take request attack mis attack cognitive access decentralize user maximize secondary primary interference context secondary model contextual remainder difference learner reward regret space sublinear derive adaptively partition context type necessity training property conclude contextual bandit agent agent arm unknown reward slot reward balance exploration arm uncertain reward exploitation work sublinear regret sublinear bound prove similarity arm context partition non contextual bandit develop news ucb design payoff contextual bandit method develop mining perceptron sublinear regret choose adversary knowledge work first solution propose novel contextual problem adaptively partition sublinear design partition learn distribute user armed finite context converge logarithmic regret allocation consider logarithmic regret good static policy markov reward dynamic resource share logarithmic armed solve decentralized stochastic maximize cumulative cumulative control assignment subset agent message pass assignment reward action armed bandit propose bind bound confidence separately algorithm polynomially detailed comparison multi armed bandit contextual framework important bandit phase learner separation exploitation three iii context order balance learner consider phase learn structure train separate exploration place difference different consider work learner action global perform subgradient work share share slot sharing learner help addition work decentralize recommender relate address challenge recommender learner combinatorial probability decentralize sensor surveillance cognitive security recommender system etc arrival necessity yes yes yes yes contextual yes arrival arbitrary arbitrary process regret sublinear logarithmic sublinear learner learner receive arm let set learner index cardinality summary learner operate privacy learner maximize technology instance stream mining type use security want control network protocol learner learn satisfie privacy constraint optimal time sequentially slot arm call slot receive extend iii learner choose iv observe reward arm context learner learner choose reward context loss select generate context reward incur deterministic arm payment assume call another learner learner away learner know cost know know arm learner since cost net minus cost learner learner learner one learner arm context formalize old denote know contextual contextual learner learner reward context minus cost formally complete f f learner choice select context know learner know remain hard put scheme arrival learner learner accuracy eq receive learner belong complexity find optimal region exponentially try even hard know know markovian rely illustrate even information history reward observation history select time jt I random context distribute learner expect whose regret sublinear converge section propose sublinear algorithm form partition learner reward history distribute contextual uniform compose part help learner reward context determine partition partition observation hence reward first optimize balance aforementioned tradeoff form set dm k tn ip tt ip ia ix p jx b p j p j I x htb explore r htb exploit kt kx htb tx pt n j f jt jt slot three phase training phase learner call learner reward receive arm learner reward phase select recall learner learner learner reward without observe form reward need sure learner arm help learner reward arm learner form phase build estimate choice help learner maximize long keep learner except training phase time learner select context keep learner partition decide exploitation learner learner exploration exploitation phase learner update train learner learner time select exploitation phase train explore exploit phase make index time slot high training learner exploitation exploration learner learner learner set call learner balance accuracy learner select identify candidate learner estimate increase balance possible reward gain learner increase loss empty n pt identify empty learner randomly set without consider learner unnecessary especially adequate arm already learner learner hence learner explore learner learner control exploration select choice reward compute follow set collect learner exploration exploitation phase reward take let reward context union arm reward choice define k observation cost reward identity incur reward note need computed learner learner learner explore arm learner exploit learner I q logarithm base hypercube symmetry ip pp denote suboptimal choice hypercube bind learner hypercube j r e e regret regret suboptimal near lemma separately bound run learner small integer time exploration ii realize hence slot multiplying event context exploitation learner random learner select learner exploitation true otherwise learner z time w I bind learner select slot use suboptimal exploitation suboptimal choice realized expect suboptimal exploitation r w adopt notation event suboptimal arms b hoeffding since event sample take variation event suboptimal q inequality q get j sum guarantee start form estimate together I markov inequality order slice slice come boundary assume arrive continuous let arrival soon instance delay arrival completion true eq case delay time time slot random algorithm keep last label update whenever label result delay feedback delay feedback true label integer delay modify delay show deviation new label delay sublinear modify delay delay grow context infeasible large sample propose issue adaptively select learner carefully context previous beginning parameter adaptively arrive reward history go contexts cardinality systematic ensure variation reward inside balanced context lot cover edge length hypercube set denote level partition learner learner learner activate learner learner time slot keep keep activate learner activate learner partition describe define learner select learner context learner learner train exploitation phase exploration phase learner receive observe learner time contribute counter reward learner time counter pt exploitation hypercube control depend hypercube depend separate partition slot threshold learner learner division activate let de activate exploration exploitation enter determined compare choice select part time learner explore explore arm choose use mean define avoid except count set function form control activate stay learner learner neighborhood lot neighborhood low contain want learner choose learner learner may learner lot overlap keep learner arm use learner overlap partition overlap set k k il c ax I tn ct c n ct pl c htb initialize I c lp keep arm end slot learner mean arm even learner never train never analyze activate arrival hypercube high hypercube active hypercube hypercube identical arrival learner hypercube bind context whenever hypercube hypercube due suboptimal exploitation consider lemma omit lemma denote kt q lp hence bind learner lp mf f j lp f suboptimal slot l lp lp take combine level activate consider learner bound context arrival term arrival regret regret run hypercube context lemma hypercube define theorem hypercube balance incur exploitation context hypercube arrival bad arrival sublinear contextual phase I constant ii partition hypercube partition exploitation regret control ti learner algorithm similar instance cost consider learner context outside learner abuse draw later learner function construct linear since explore happen always instead accuracy incorrectly reward learner come learner regret exploitation slot learner exploit choose exploitation bound z reward sample learner reward accurate learner arm learner random bt bt bt bt bt bt q less give statement instance slot take arm chernoff choose exploitation slot hence regret linear mean hypercube hypercube inactive exceed active much activate time level create corollary create create hypercube trick make multiply f c arrival correlation arrival arrival om correlation hypercube activate least explore hypercube increase memory level split exceed call child keep active hypercube average hypercube arrival hypercube explore time activate child explore least activate requirement modification useful trend certain time concept drift recommender security income traffic pattern vary researcher two important speed drift amount change new cause drift concept category capture contextual mining capture speed drift drift speed concept drift focus incremental technique goal characterize advantage detection design drift drift drift change associate learner occur hoc provable guarantee framework regret concept drift extend jointly update accuracy weight although combinatorial contextual jointly optimize sublinear scope aim address future important recurrence drift recurrent old concept concept year recurrence framework recurrent know learner centralize contextual bandit call contextual create context slot ball lie exceed ball center author regret dimension account context payoff context occur payoff payoff dimension similarity arrival problem distribute center context arrival whereas position arrival active hypercube learner level contain hypercube simple need active region find belong different prove theorem reward achieve regret exploitation necessary armed bandit problem exploration exploitation arm slot term reward relative reward arm dominant hence arm high algorithm separate exploitation consider armed bandit show use decide explore contextual bandit equip want classify reward check obtain possible labeling human costly learner learner obtain classifier separation exploitation give another vary error tolerance come tolerance mis high ideally try shift instance course type future use deterministic learner arm learner explore learner explore compare could theorem open scope though reduce control training complicated reward come assume learner back slot delay delay note learner prediction learner learner affect policy request context dependent go arm arm instead even propose decentralized learner novel sublinear issue instance include mining surveillance contextual bandit raise centralized contextual control happen want context learner direction performance approach increase communication cost learner improvement large logarithmic complement learner learner set arm space reward well expect arm high learner
exposure parsimonious solution neighborhood exposure consider heterogeneous exposure fractional exposure exposure neighborhood exposure vice versa assignment core exposure contain exposure exposure exposure prove attention place absolute fractional population neighborhood exposure application exposure condition consider tie concept exposure estimate treatment experiment experiment range randomization determine exposure exposure exposure control probability know allocation randomization become thompson obtain clearly exposure exposure probability simple exposure exposure randomization independently treatment independently exposure treatment simply exposure highlight exposure chance exposure small intuitively exposure dramatically thompson necessity randomization thompson randomization create treatment happen number correlation exposure clarity discuss notation cluster neighbor n probability partitioning describe satisfy growth describe behavior arbitrary examine computed neighborhood exposure exposure become become exposure tractable challenge vertex treat neighboring randomization apply fractional connection bernoulli coin quantity obey question x make polynomial dynamic program runtime formalize neighboring cluster randomization w w neighboring randomization q exposure every consideration double exposure space figure like exactly exposure absolute fractional exposure unfortunately exposure unclear exposure formally nest corresponding exposure condition exposure probability formalize connection via generally concern exposure probability exposure core exposure pr direct exposure explore possibility estimator interference reduction scheme estimate exposure treatment exposure exposure variance estimator nothing fundamentally exposure aside intersection set make effect effect exposure namely control require bound vertex combination exposure write graph randomization outcome thompson fractional exposure randomization cluster sum double common neighborhood fractional exposure sum zero vertex connect vertex contribution give vertex vertex strength degree bound grow next find consistent result variance grow exponential maximum interference scale contiguous block vertex derive vertex connect mean n vertex non vertex vertex adjacent right joint center cluster z tell experimentally variance cluster contiguous block apply geometric begin develop randomization graph connect vertex restrict show cycle contiguous geometry crucial growth regular relaxed degree distribution weak still connect say singleton neighborhood hence growth condition radius growth space short metric space net mutually union accordingly call net net identify go perform pt vertex select vertex close vertex tie low index follow vertex independent neighborhood first indeed otherwise since close disjoint suppose way vice suppose distinct let distinct contain adjacent contain neighborhood disjoint argue bound growth inequality cluster result regular fact weak arbitrary restrict requirement weak observe growth bound whereby proposition apply effect response bound exposure reason regardless treatment scheme degenerate regular randomization randomization neighborhood randomization vertex lower bound joint least exposure make negative contribution term probability give become degree vertex turn linear graph ht net randomization restrict function graph variance give bounding n assignment neighbor associate hence assignment thus vertex contribution analogous trivially z upper degree desire derive graph weak obtain degree topic open focused b effect link social reasoning cluster randomization population lead emphasize graph technique graph arbitrary detection though variance guarantee suggest framework formulate objective thompson approach adversarial another minimize variance exposure know control clustering variance solution useful treatment dominate heterogeneous add another interesting direction exposure control ask response continuously extent exposure example neighbor properly take analyze framework lar facebook university edu lars cs edu exposure randomization online condition drawback poorly suited interference treatment individual individual work treatment interference begin graph cluster vertex exposure effect exposure focus first variance size randomization degree contrast growth neighborhood natural estimator randomization estimator effect interference service social inherently exhibit network value user inherently since user testing framework causal unit treatment individual response address formalism effect interaction trial color upon divide color scheme assume interference treat would observe universe b scheme inference behavior tractable change dramatically behavior effect case kind place universe place universe analysis contaminate behavior vice versa average network technique treatment service underlie population reaction service social otherwise numerical site universe service despite universe express formalism similarity formalism adapt interference mean group outcome treatment assignment quantity treatment two opposite ordinary ever truly key notion exposure treatment exposure control condition analogously exposure condition treat fraction exposure fundamentally introduce vector provide network exposure potential treatment universe actually place user treatment control universe randomization randomization base randomization high partition set randomization involve theoretic vertex determine exposure randomization effect explicitly compute motivate randomization cluster randomization first size remain illustrative dependence vertex degree randomization estimator bind degree raise algorithmic question provide carefully vertex graph class restrict graph expansion growth previously study near neighbor space design diameter let vertex vertex restrict growth say constant degree graph condition growth grow cluster intersect neighborhood cluster ball prevent ball pack closely cluster come density space space near control class restrict growth provide attractive type growth include graph vertex density edge within distance causal interference well recent develop exposure randomization suited graph show previous intractable meanwhile theoretic consideration improve randomization connect exposure graph randomization consider exposure necessity probability average treatment introduce restricted growth linearly bound conclude b whether intervention take explicit condition independently meanwhile exposure determine intervention intervention assumption exposure user arbitrary exposure experiment exposure consider outcome arbitrary completely different without control estimate multiple map enough vertex let outcome exposure view interest condition define set exposure set exposure unnecessary entirely exposure trying determine effect extreme exposure universe true condition require wrong happen average treatment way correspond introduce bias outcome even favorable experiment follow primarily exposure assignment indistinguishable immediate neighborhood fractional exposure exposure actual potential outcome introduce exposure belong pt pt full neighborhood
select use threshold manner look result motivate stop distribution well fusion sensor implement analytic decision engine bayesian network formulation stack analyze network discuss approach variant test distinct certain inactive author mr organization support dr operational valuable suggestion flexible sensor person enable tradeoff sensor network contain physics nonlinear acoustic along fusion fusion static combine dynamic component static time static component probabilistic physics model input feed fusion hypothesis test entire fuse sensor purpose performance fusion actual discuss fusion stack analyze efficiently material serve background rest section stage evidence accumulate stage show time compute dependent setup bank move target reach justify continue static fuse decision multiple configuration sensor operation remain unless call formulate bayesian acyclic random describe property conditionally independent immediate parent intuitive perform fusion sensor center sensor fuse one sensor sensor generally focus binary fusion vertex describe fusion fusion consist element vertex final decision child vertex formulation enable take different center sensor stack together ensure fusion sensor parent fusion center fusion center parent cycle fusion center parent fusion center intermediate ultimately figure bayes contain one fusion parent vertex use belief fusion parent child cover fusion child center depend center child probability static fusion fusion five elementary fusion majority pearson bayes five fusion bayes fusion sensor fuse choose rule minimize rule vote sensor uniformly distribute fused rule generally criteria conceptually vertex pearson fusion vertex subject computing ratio sensor decision combination partition bayes rule cost alarm detection mix combination individual decision find fuse minimize bayes risk wrong find simply look combination individually fuse decision find fusion rule discrete scenario network sensor target static network produce fuse decision sequential likelihood ratio make incoming decision reach lower sequential sensor static fuse output anomaly continue combine stage sensor fusion second stage stage include sensor add stage third acquire begin track clarity output respective center restrict fusion type low upper stop upper overall sensitivity specify write second stage decision fusion possibility k l empty define start static move outside stage force stop test run network stop affect
terminate option equal bernoulli factor trajectory form confidence interval pearson construct option use algorithm choose dimensional polynomial monotone initially display attribute non property representation policy use performance trajectory compare total high payoff proportional balance factor reproduce robust policy reward value pair marked novel solve robust markov decision knowledge capability previous focus exact suffer curse planning reinforcement reduce robust employ iterative approximation present usefulness approach robust mdps xu electrical department institute technology question consider robust paradigm show robust small sized curse mdps employ tackle planning develop fix robust mdps method succeed technical condition effectiveness pricing knowledge attempt paradigm mdps solve sequential making problem environment namely immediate reward reward bad thus surprising strategy significantly differ due true one mdp propose common uncertain member term uncertainty solution technical robust solve dynamic programming medium mdps paper mdps setup curse dimensionality practical mdps often large programming intractable many propose curse large mdps efficiently success broad mdps adapt robust develop handle high solve planning reinforcement rl mdp know still rl scale specific contribution mdps robust mdps mdp tuple r empty probability assume terminal policy represent expect discount action terminal term framework transition assume lie uncertainty set obtain mathematically tuple u transition implicitly mdp interested maximize define q robust px robust bellman sequel shall deterministic policy write improve choose action respect approach q iterative method terminal operator vx bellman operator see contraction sup find vector mdps programming intractable resort involve approximation bellman review regular mdp uncertainty bellman fix rx xx linear sum state let popular onto euclidean point assume form solve inversion iteratively eq correspond steady iterative converge explicitly approach yx yx yx x last transition matrix norm let bellman operator contraction weight contraction property approximation straightforward equation fix write inversion linearity exactly state resort base procedure popular literature problem apply robust mdp trajectory modify line use large together corollary convergence result omit hold counterpart uncertainty demand transition option pricing perform discount aware relaxation general emphasize assumption require transition assumption stop question whether exception terminal approximate cope hold unfortunately supplementary material fail single state iteratively note arise converge contraction case say approach type sup sup policy improvement state feature vector may equivalent policy evaluation approximation initialize arbitrary iterate n u w n x regular need may computationally addition need discuss robust mdps broadly option pricing stop stop problem terminate always transition terminal terminate satisfied satisfied supplementary material problem option american contract give asset time let asset
something aspect user internal hope approximate model consider causal mechanic every model coin add necessary represent state successfully various field structure spike train medium entity recognition natural complex use causal state network agent internal relationship simplify relationship output useful similarity utilize influence behavior decision mention internal move state computational mechanic model predictive capability state relax easier predict use state network begin literature use carry evaluate work research tweet reference generate user otherwise time tweet limitation resolution process tweet tweet sentiment mention behavior encode ahead past bin predict l problem use generate conditionally prediction simplify infer mechanic reservoir specifically infer dramatically implementation mechanic seek infer simple attempt combination desire mechanic space determine process two consider consider prediction outline mapping unique behavior call causal conditional know infer advantage computational mechanic way infer mapping implies infer begin model iid continue fine set result estimate distribution prediction give feed former easy lack suit represent system intensive less fix recurrent activity far simple output simplify process machine original output reservoir connect reservoir uniformly scale spectral reservoir asymptotically zero rather reservoir train diverse linear reservoir logistic function represent concatenation procedure present reservoir input reservoir matrix collect pseudo inverse weight twitter period seed user seed network expand user day fashion active seed etc tweet make est united stationarity window either day tweet otherwise limitation window ten create ten theory limit tell raw practical constraint datum computing tractable visualize period process trial horizontal vertical bar user visual inspection serve demonstrate original activity original time second together disjoint partition ten user filter active determine second pm tweet top tweet rate tweet testing partition change capture history use treat determined cross time spend tweet user tweet maximal history ensure joint stationary symbol practical reconstruct use length history length fold choice network create accuracy predictor loss causal baseline vote tweet vs tweet day baseline take identically random tweet predictor tweet twitter predictor causal state blue tweet tweet window contain tweet clearly prediction baseline user tweet tweet tweet rate great tweet rate estimate conditional density tweet group tweet accuracy blue accuracy either model state chain plus symbol completely causal causal top great occur tweet emission diagram resemble diagram additional property symbol diagram show circle state improvement state bottom tweet tweet tweet group emission observe causal causal bernoulli causal user respectively causal typical user memory behavior bias coin flip see label passive may stay transition user stay p transition correspond state exhibit rest tweet state user return transition state model typical twitter behave entirely beyond compare head improvement state network strongly user causal outperform user causal model outperform near behavior user model infer dynamic states characterize behavior capable causal state informally process optimally predict future period bit past complexity user predict average complexity state tend outperform near top user state baseline vs baseline state identity user outperform bottom show typical top outperform differ training rate entropy infinity thus approximated block entropy large block observe entropy block account range explain user overall state tend outperform see figure group absolute far follow flip day proportion become vice versa range increment training set differ desire systematically corruption approach two network degradation user indicate cutoff good causal correspond rate network beyond rate rate datum proportion bar indicate plus rate across improvement change many simple trend continue continue similar probability passive proportion instance short activity embed short period long observe corrupt sequence long fidelity well adapt perturbation maintain predictor proportion user predictive model naturally capture similarity behavior consider many prediction history user predict still account building state behavior action medium capturing model add description relationship diverse collection user derive medium indicate differently mechanic behavior structure change dramatically network seem give deep robust decay ultimately expect differ drastically paradigm provide latent behavioral able capture user hypothesis future restrict mechanic observe substantially present present user unit focus social twitter
position white mean identically I evy trial vary without effect kalman determine crucial definition science china mail cn usa e mail edu university california mail filter evy kalman evy modify evy work reasonable cost result present method kalman modify kalman gaussian evy state estimation assimilation kalman filter subject white tracking signal assimilation kalman either gaussian evy infinite desirable kalman evy contribution jump may greatly limit filter observation really kalman consist combine forecast error robust computational due practical practice must complete greatly fast cost kalman systems model state equation variable variable noise case non evy arrange kalman review provide kalman filter kalman review idea present filter section derivation filter find reference model give assume posterior expectation kalman assume correct minimize rewrite conventional filter noise approximated increment brownian per time increment evy evy l evy decompose jump evy approximated process approximately regard evy process jump evy decompose non evy white noise extremely filter l evy evy correspond practice wise operation threshold represent statistical replacing repeat kalman
consumption whereas great hand upper deal reward break give n payoff use valid summing obtain two observation first ensure mm describe primal set probability amount round resource generality resource whereas resource rescale fraction capacity resource horizon continue run least want create resource play arm denote resource resource play salient algorithm analysis rather dual normalized vector idea learn vector apply multiplicative round weight pack program online aspect multiplicative adjusting update typical instance resource consumption aspect know acceptable lp arm next tt run stop resource budget payoff belong imply self present appendix convenience value eq primal definition inequality together resource consumption stochastic must notation inequality lie modification simple time place latent confidence bound consumption resource consumption x similarly primal program lp let payoff would actual upper bind algorithm clean execution make need remain reward actually receive
sample learn estimate quality panel right response diagonal repeat procedure mean deviation root predictor report continuously less functional topic study air serious health air research topic year california air resources air year air locations california study city database leave display daily ground city hour panel give level day air data complete display highly nonlinear assess goodness additive residual vertical horizontal plot fit compare performance linear observation two functional establish minimax decay far additive issue could linearity line immediately establish find follow e f mm kp mp I leibler verify integer positive mf k result far see exist calculation order great imply choose complete proof linear subspace observe furth f belong proof exist eq ds ng g cg eigenfunction let recall yield proof omit detail constant stand whenever c fact conclude pn david functional flexible nonlinear well study paper attention identity linear rate functional reproduce determine reproduce kernel special linear jointly covariance predictor achieve optimal predictor approach rate analysis reproduce hilbert extensively response restrict interval cause copy assume slope I base component limitation inherent functional study call continuously case continuous bivariate nonlinear replace cause cdf contain case additive compare interpretation spline fit reproduce functional naturally measure risk possess expectation increase prediction closed prediction spectral admit eigenfunction minimax depend rate derive predictor attain paper organize minimax excess regularization show monte two end minimax lower bind excess reproduce space reproducing endow follow admit positive sequence excess prediction infimum take predictor regression bivariate restrict hilbert reproduce functional regression assume kt ts kkt kt xu k special coincide eq square closeness term control explicitly procedure nan space orthogonal complement jt may suppose qr decomposition row project onto follow eigenvalue score work study numerical numerical propose predictor exist rkh act reproduce thin semi regularize integer pair predict problem spline well approach spline tensor eigenvalue two closely spaced spaced spaced study response value contain thin estimator nearly identically
easy fix dynamic correspond impossible corollary enough corollary apply use behavior discover good alternative increase probability test alternative randomly choose three surely nice term successful picture succeed find argue choice probability thank useful discussion theorem axiom lemma notation theorem universit du france edu problem pairwise comparison reinforcement solution reinforcement win simple winner converge cycle datum alternative datum basic logical classical competition involve competition number comparison candidate prefer prefer latter relation good candidate majority rule define problem comparison attract attention field david et paper evolutionary perspective process period play alternative win future go add colored ball one alternative process discover unique alternative infinity alternative discover unless composition give optimal solution go even negative evolutionary instability mix evolutionary prove reinforcement original close equilibrium sample close equilibrium might interpret slow stability field process belong ingredient definition well martingale alternative fairly convergence alternative paper section introduce notion induce date game allow existence solution preliminary adaptive illustrate argument way toy treat deterministic result three alternative relation possibility occur set fix sometimes easy alternative alternative winner easily reduce singleton winner political let set probability random winner randomly term usually stationary characterize fact pp x pt symmetric game graph attract computer chen voting competition social formal remarkably precise result fisher use et follow cycle inclusion totally alternative pick pick compare thing color add alternative one cycle fast reinforcement thing implement describe fast reinforcement section rigorous argument justify focus trivial alternative long alternative alternative correspond limit ball type corresponding alternative reinforcement b c independently alternative reinforcement quantity calculus negative function difficult divergence rigorous converge optimal even alternative use discrete probabilistic three alternative reinforcement reinforcement realization almost surely give reinforcement reinforcement almost surely rely mainly integer denote number increase ball alternative martingale precisely q three reinforcement ball eq q sum pp begin alternative proposition martingale almost furthermore surely finite gp accumulation necessarily look hand side contradiction prove accumulation zero see reinforcement idea random sequence alternative positive realize x start ball reinforcement piece sure straightforward factor series bp inferior line big enough bp n happen still denote
simply bayesian cite therein completely identification solve deal bayesian minimization derive mc multi dimensional markov parametrization amplitude give cardinality symmetric definite matrix induce ss pz px pp py regression wherein formulate step ahead suggest square utility q mse p q test marginal admit compute method low next associate bound pz tp z assumption px u pz mse associate bound inequality proof low lemma finally problem identification formulate maximum minimum impose magnitude realization noise density choice estimate yield optimal bayesian u np challenge address dimensional state mutually sequence mean independence th px px px py x px u f tx u f tx reduce integral mc tx u compute mc yield optimal function almost sure perfect mc law choose initial mm u j py use design set covariance x min u max eventually smc give implement input transition purpose binary select give compute trace low quality input estimator figure low evident wherein simulation rigorous promise early smooth minima bayesian identification stochastic distinct propose input bayesian result use ca identification linear rao optimization parametrize propose decade great progress make statistic community make complicated model arise model detailed paper towards identification briefly unobserved process q independent give open density suitable dominate measure non series form
spectra dictionary atom close encouraging constrain representation atom quite scene make seem regard since spectra representation exhibit capability gain insight depict contextual atom activate fig fig activate observe active sometimes fall edge occur always contextual remarkable sparse resolution spectral band acquire motivate follow classification simulate level resolution assume linear combination spectral via adjacent set bin space roughly measurement obtain measurement cover spectrum coarse pre argument section sparse eq q respectively constrain initial train coarse sparse obtained obtain training obtain accuracy result decrease consider coherent may become coherent representation exhibit sparsity pattern serve proof statistically scene computation learn dictionary initialize dataset patch continue overall record machine intel bit processor code matlab accuracy dataset scene see consecutive learning affect ability accuracy increase discriminative code size matter early gap dictionary third speedup gain parallel share ram later short scale dictionary consider dictionary parallel dictionary refer fig resolution pixel band cover image fig train university consist band band training leave table classification notably center confirm early provide contextual cross attain patch contextual choose accuracy sample entire accuracy report result band band remove truth class image good significance moment usually small algorithm hyperspectral hyperspectral linear coefficient combination model exploit confirm effectiveness direction research linear little advantage representation testing supervise learn discriminative hyperspectral svms seem plausible regard classification spectral contextual would thank university provide datum dr svm receive sc sc hardware engineering work ph computer engineering technology technical service laboratory university current interest structure representation sm electrical usa electrical usa ph electrical engineering university west usa member technical work intel electrical university technology advance communication technology center advance digital medium laboratory mobile value service laboratory currently technology national international project international source plan numerous scientific international sc work sc computer university technology interest spatial dictionary hyperspectral image classification structured model hyperspectral incorporate contextual spectral hyperspectral idea hyperspectral image neighborhood call contextual combination dictionary contextual material combination constrain carry sparse pattern sparse contextual hyperspectral effectiveness hyperspectral experiment capable find hyperspectral dictionary joint sparse support signal model member dimensionality usually cause limitation artificial often real phenomenon effective research reconstruction task denoise recently art hyperspectral scene narrow band whose narrow band hyperspectral application many management pixel number apart cause illumination angle environmental determine present give device material pixel material mixture respective linear assume commonly unity essentially mainly reduce signal induce commonly formulation sparsity induce regularizer balance representation recently ability spectral encourage sparse signature code take contribute fractional sparsity sparse decrease number activate compose signature propose pixel similar total encourage smooth variation fractional abundance among adjacent aforementioned library pure priori dictionary e select manually eigenvector scene learn constrain experimental spectra differ dictionary fractional unity regularizer long survey hyperspectral potential accurately one hyperspectral develop tree unlabele inspire class spectral characteristic composite construct kernel contextual contextual successful classify image limit training graph incorporate contextual simultaneously recursive sample datum dictionary code hyperspectral pixel training pixel label represent pixel use fractional raw hyperspectral manner classification sparse fractional abundance regularizer variation code neighboring label discuss hyperspectral classification individual advantage pose remain challenge motivate focus simple learn hyperspectral incorporate contextual hyperspectral pure signature dictionary chen aim classification complex consist discuss employ window center pixel contextual contextual fact shall weight high spectral zero weight indistinguishable classifying dictionary al contextual regularization pixel fix fact dictionary contextual simultaneously yet optimization amenable dictionary hyperspectral aforementioned incorporate spectral idea pixel hyperspectral neighborhood contextual pixel inside often material belong sparse common sparsity induce regularizer view building upon introduce two sparse coefficient solution recent computer vision extract enough classified motivate finding classify extensive experiment hyperspectral model incorporate contextual set extensive hyperspectral infer enough classify art model infer resolution retrieve find use sample classification amenable section background introduce structured hyperspectral gain insight analyze view effectiveness extensive hyperspectral report section iv discuss brief short tailor hyperspectral describe parameter convex program sparse representation sequel capital letter letter pixel hyperspectral image dictionary atomic signal form atom application accuracy yet frobenius regularizer induce knowledge assume gaussian realization strategy update update sparse phase long independently sparse approximation compress formulation follow form reason treat value treat among et pose cone loop expensive problem alternate propose quite acceptable guarantee iteration function due linear employ objective update function use initialize point consecutive iteratively derive section hence convenience row introduce hyperspectral datum classify briefly hyperspectral spectral dictionary contextual alone partition predefine dictionary yield representation dictionary coefficient representation code classify linear correspond label classification like classification shall begin section treat unable capture contextual representation respective moment compute channel pixel class moment local manner view locally discuss particularly therefore concatenation inside width determine train spectral calculate contextual code representation train svm contextual contextual learn contextual center pixel contextual pixel computation power largely aforementione chen model spectral pixel pixel inside window spectral contextual dictionary explain drawback basic motivate dictionary take spectral hyperspectral patch hyperspectral partition way method vast literature aid yield simplicity future find representation svm provide validate propose structured classify hyperspectral namely spectral contextual support prove successful contextual rbf matching use dictionary coherence setting section collect distinguish employ second svm approach polynomial window five table variable window report iteratively atom dictionary residual window correlation explain unfortunately specific value report use complete use five atom number window dataset contextual initialize patch dataset high initialize patch svm multi class svms train classify choose collect use pixel spatial pixel consist across noisy correspond water truth class
value increase consider enyi graph decrease plant easier htb p blockmodel specifically degree plot similar rand closeness clustering rand adjust adjust rand grow average also demonstrate hold practice recursive p value cutoff ari expect community extract remainder change sequential recursive one manually collect nine induce node assignment circle hope circle examine work feature remove sometimes incomplete assign clustering ground extend truth compute subtree node cutoff divide subgraph fall note recursive comparison nine facebook network node nonzero ex ground truth ex visualize cluster structure row dark break adjacency plot plot row adjacency use political book book amazon grind common conjecture book political conservative author give core figure plot find blue part split blue separate node node split divide green blue row correspond edge simply put cluster subtree subgraph width width width subgraph adjacency provably graph blockmodel large suitably stochastic blockmodel significantly work theoretically establish limit statistic enyi correction together expensive parametric replicate nine dataset ground facebook nest structure vary discover political book book aim show find exist chen share grateful literature apply isotropic grant dms theorem theorem corollary university california berkeley usa university california berkeley usa self loop enyi length law law matrix entry define orthogonal ensemble suitable result law e limit converge nan enyi significance reject fall parameter estimate matrix scale orthogonal blockmodel blockmodel probability dominant lead edge consistency asymptotically hold like sufficient simplify proof let stochastic blockmodel whose constant independent propose structure htb network partition graph investigation eigenvalue adjacency small generate whereas case limit law enyi graphs enyi computing generate bootstrap step computationally expensive since recursion scale empirical limit compute shift limit law distribution bootstrap replicate limit distribution shift sample whereas shift use whereas generate c c plot bootstrap replicate empirical middle shift empirical mean draw respective well fit third good middle panel correction correction test scale variant estimate fit hypothesis formally denote law htb n tw conclude brief extract cut type tie test easily replace community extraction avoid expensive distribution provably correction finally difference author extraction complement community community recursively hierarchical nested community alternative test network I enyi converge blockmodel infinity wrong centering work laplacian behave large self probably limit behavior cycle use conclude enyi q use argument around approximated thus ta scale enyi work statistic c graph converge fast show convergence adjacency experimentally use develop year eigenvalue symmetric ensemble follow law eigenvalue density empirical population sufficiently use result upper eigenvalue constant limit eq triangular diagonal entry variable criterion hold rescale coincide weakly distribution h design match match center bernoulli factor scale eigenvalue mask prove isotropic eigenvector sequence random bound small power iff theorem eigenvector deterministic self loop half hold relax use isotropic law theorem consequence however theorem apply let eigenvalue inequality heavily ni nn eigenvalue eigenvector arrange p differ true average error q g result
choice dimensionality view paradigm propose select pca possible dynamic model raise question approach model dynamic space loading research longitudinal approach identify variable highlight evolve issue could employ evolve two fitting follow combine idea development vary longitudinal several many lie influential dimension reduction achieve autoregressive influential analyse molecular pathway apply rapid growth disease diagnosis nuclear ms spectrum peak peak abundance profile insight state system high spectra contain peak size easily contain one peak highly correlate structure repeat measurement observation appropriately principal analysis often exploration technique extremely field pca account apply point measurement repeatedly assume independent case pca look direction maximum variation variation act potential design study analyse longitudinal appropriately repeat complex employ pca local pca matrix analyse limitation associate difficult assess model longitudinal context longitudinal independent spectral peak important explicitly effect longitudinal statistical appropriately longitudinal increase dimensionality control dimension pca attractive probabilistic latent variable benefit reduction appropriately model longitudinal assume closely employ financial datum longitudinal study development study study drug purpose influential change appropriately longitudinal employ reduction employ change longitudinal article structure overview longitudinal present account measurement detailed accordingly specify distribution chain technique fit far defer conclusion year longitudinal human employ challenge powerful study subtle longitudinal determine long drug efficacy application describe assess effect treatment group help aim I model dimensional space appropriately aim ii output influential address analysis identify influential change illustrate briefly lead generalised period treat determine occur treatment spectra acquire sample spectra integrate exclude water final acquire consist treat bin time peak different shift value relate detail abundance associate illustrate single h probabilistic constrain span conventional pca underlie applicable extension develop introduction generative high function low isotropic bin gaussian covariance term underlie principle component pc multivariate distribution crucially span principal subspace loading conventional pca model output pca advantage uncertainty assessment pca model model time multivariate latent dimension depend assume multivariate vary score mutually parameter vary unconstraine analytic view constrain study explanation manner need motivate application practitioner time loading lead strongly understanding field maintain highly link variance equal modelling decision parsimonious assess fitting posterior model volatility finance time account repeat use volatility point variance assumption account potential dependence longitudinal datum motivation incorporation model longitudinal multivariate employ model model time assess volatility denote autoregressive process persistence principal dimension persistence covariance respectively innovation predict current volatility time component relationship time point parameter constrain lie previous fact facilitate dependence across component detailed correlation motivate error vector error volatility ar center persistence ar normally maintain reason constrain order motivating impose ordering peak unconstrained loading within loading mle loading result fitting identify loading practice prove identifiability longitudinal assess within treatment change longitudinal set issue reason clarity model km ig prior univariate metropolis first discard burn initialize datum volatility trace plot autocorrelation plot longitudinal trajectory insight visual insight trajectory model correlation across reduce time nature estimate latent loading material trajectory principal hence trajectory unified idea loading reference loading subsequent good match loading associate score illustration movement latent within insight principal cc individual subspace black solid dashed control solid line group dashed digit illustrate movement time visible separation position trajectory due actually result day stage also great control variability great insight change occur aim longitudinal treatment group fit task concentration change fit treatment persistence time figure persistence parameter relevant plot respectively mean positive significant base interval ci persistence respectively dependency spectra persistence horizontal illustrate establish third time achieve first translate loading large magnitude linear fit influential evolve accounting dependency spectra group underlie posterior loading pc rank five figure none five bin eight bin loading correspond bar credible mix eight influential spectral bin intercept cubic effect consider consider bin eight spectral concentration illustrate predict intensity spectral bin intensitie six influential bin evolve treatment identify evolve include represent bin concentration level initially study illustrate predict intensity profile bin treat e concentration decrease individual predict evolve spectral bin absence highlight time take fit spectra control persistence latent table show persistence suggest persistence estimate ci effect control lie evolve time pc rank select top influential spectral none evolve rank top eight fit profile identify evolve illustrate intensity eight evolve bin intensitie seven influential spectral evolve control spectral bin effect control decrease see spectral bin quadratic evolve predict profile seven evolve bin give material aim occur difference treatment six highlight evolve
soft reduce however threshold perhaps illustrate result thresholding software bt produces illustrate shrinkage noisy speech bin appropriate length thresholde experiment noise deviation db snr low block less likewise post result bt processing degree introduce computationally efficient group overlap algorithm translation without regard property though procedure conceptually simple admit use explicit part function procedure practice table shrinkage algorithm snr wiener post however include investigation function derivation proximal regularization speech denoise theorem thm signal amplitude separable tendency involve penalty function group overlap denoise translation avoid principle iterative minimization monotonically regularization specify describe speech wherein speech produce suffer recent year many signal deconvolution reconstruction utilize shrinkage thresholding function various soft nonnegative numerous derive mmse estimating property length deriving determined e wavelet fourier transform penalty know solution thresholde basis pursuit denoise scalar problem significantly shrinkage element viewpoint statistically signal also exhibit group property coefficient inter intra scale likewise group apparent amplitude invariant exploit group cluster signal act minimize group minimization mm extension successive substitution penalty implementation indexing avoid denoise conceptually parameter method select ensure variance reduce minimize square signal although intractable lack explicit functional form cost substitution minimization thresholding noise investigate devise section denoise include mixed describe overlap multiplier describe member use variable large base identification perform involve auxiliary call overlap group induce pattern employ several algorithmic mixture locally shrinkage multivariate thresholding variational wherein function minimize base coefficient coefficient coefficient overlap estimate invariant note interestingly design significant comprise penalty comprise group focus selection extension shown transform coefficient spatially multivariate simplest utilize obtain relate large member center block coefficient slide sub sample overlap second translation invariant may arise variational approach base fully algorithm translation function invariant sparsity cluster translation invariant denoise norm obtain convex cost strictly index index deal boundary fall outside overlap per slide window shift possible include ref size I show observe independent sign regard differentiable equal differentiable group equality follow equal zero moreover element constant depend method produce separable write equivalently th whenever care must take avoid involve quantity subset update group table clear initialize initialization noisy consider hence k initialize equal sect see go zero may finite precision remove overlap penalty solution gradually reduce toward iteration output note simply soft every multivariate step sum outer inner sum result multidimensional two multidimensional efficient computational group minimization mm cost minimizer differentiable problem relevant denoise problem contain zero differentiable strongly suggest equal note algorithm infinity particular zero go zero large expression behave sign small numerically reliable implement arithmetic indeed practice empirical compare prove small unnecessary regularizer form effective approach proximal reconstruction solve denoise proximal proximity therefore operator penalty proximal note implementation overlap overlap type yield noisy iterative soft emphasis largely significant problem common change sequence overlap utilize overlap block utilize block exploit measurement whereas latter sparsity extend norm attain slide window implementation regularization simple thresholding sec soft thresholding simple shrinkage extend notion eliminate large distortion effect zero investigate examine analytic soft pdf illustrate unity rule graph generalize deviation I standard percent threshold formula relate overlap coupling term group shrinkage analog available relationship array size illustrate apply standard numerically straight reduce deviation obtain value datum comprise depend graph store later produce signal length iteration accurate iteration use thresholde simulation intend one group denoise signal contaminate algorithm algorithm std group output iteration fig soft examine find group large negligible yet avoid unnecessary suitable sensitive similarly thresholde origin cardinality norm follow chi slope minor hand side illustrate em soft procedure shrinkage soft map nonlinearity contrast produce zero sufficiently pdf soft thresholding produce output origin mass tail translate origin element zero illustrate point mass reflect map pdf computed available sufficiently thresholde illustrate pdf exhibit computed histogram depend width thresholding shrinkage soft std normal complex datum formula zero unit normal chi value common transform image etc complex value reduce noise case example algorithm overcomplete white even speech denoise ignore table somewhat inaccurate penalty noise problem corrupt stationary colored frequency narrow algorithm apply problem process select block size vary speech denoise note beneficial high temporal high wavelet structured model interest illustrative denoise group signal fig adding indicate level gaussian deviation soft eliminate effectively eliminate amplitude apply thresholding simple intuitive certain straight regard note lead denoise denoise residual noise optimal low quality apply noisy five visible eliminate soft original signal rmse example rmse meaningful method value eliminate illustrate numerous ki compare namely dual software correctness usage note solve variety relate show exhibit monotone low admm iteration matlab file perform fast three matlab matlab fast indexing minimal auxiliary convolution implement matlab note admm parameter manually optimize call provide default
stack commonly drop goal hyperparameter way present attempt automate start use boost role motivate new approach scale affine image normalize cross spatial di histogram classifier architecture hyperparameter volume filter either projection train input projection input feature derive either sign di histogram partitioning span model remove space hyperparameter use hyperparameter condition inactive available notable backpropagation unsupervise rbms regularization maxout implement ensemble particularly suited hyperparameter see concatenation ensemble member piecewise necessary make ideal configuration setting generalization label hyperparameter indicator stand hyperparameter hyperparameter ensemble member notation set priori piece element piece joint complicated select configuration illustrate figure call normally method one loss margin decision nothing training avoid boost validation fit perfect validation chance interested validation round technique feature improve hinge reduce svm optimize partially boost suitable base boost fitting distinction vs provide validation performance strong compare learner generalize learner equally support face expression center task face seven neutral protocol training example partition datum perform hyperparameter regard website prevent score list challenge website round good ensemble proposal baseline strong experiment slow file typically configuration non degenerate trial take minute day ht rbm round accuracy create accuracy significantly rank competition among worth entirely design release choose excellent verification advanced state cifar release ability meta modeling generalize accuracy generalization accuracy individual set stay memory demonstrate boost ensemble member steady familiar boost loss operate representative hinge hyperparameter difficult hyperparameter challenge rank th ensemble mechanism raise rank ready date importance leverage conjecture wide feature initialization backpropagation maxout code regularization rbms token make claim add software experiment publicly representation acknowledgment project institute state national foundation national fellowship feature worth may say neural improve feature error improve extraction cifar data monotonic relationship extract accuracy classifier operating classifier reveal across pressure hyperparameter configuration give much response present acknowledge vector factor cost model ultimately response complementary automate even outperform efficiency reveal come participant competition subtle bias candidate impossible basis contribution thorough hyperparameter search ensemble say point neither alone still score exactly extend
show nan statistic approximated arbitrarily precision carlo extend variate several evaluate straightforward extend several encounter include miss diagonal entry structure extension datum discuss among correlation among positive definite derive statistic nan form statistic statistic statistic carlo simulation density zero normal py tr tr matrix minus hereafter paper mle draw achieve yy redundant value likelihood third full critical eq determinant mean derivative respect derivative point function let ia mle mle mle give imply mle nan unique scalar multiplication mle division scalar mle notational convenience diagonal point iteratively block able obtain ratio absolutely continuous nan likelihood time monotonically yy second stem mle satisfy third rely trace dimension identity since simulate distribution hypothesis invariant leave right transformation matrix since right multiplication diagonal prop base determinant multiplicative yy ty normal therefore monte simulation monte carlo q variate result previous class minor condition variate distribution mle variate variate say variate variable aa bb show mle mle normality result mle cone maximum normality mle subsection observation likelihood variate mle mle normality clearly identical test test reference zero power calculation section calculation type exchangeable covariance maximally stochastic blockmodel two model matrix quantile base perform quantile versus value exchangeable consider exchangeable correlation bind guarantee covariance grid interpolation panel power keep power vice versa panel keep high trace top right hand calculate hold value increase function latter increase increase bottom display maximally display blockmodel maximally kronecker structure demonstrate presence nonzero purpose kronecker covariance row compactly matrix compute range plot increase fix follow increase identify curve identity curve calculation always error popular relational stochastically stochastic blockmodel general blockmodel induce correlation blockmodel multiplicative specifically relationship represent row membership membership iid calculation consider node group w membership write bottom panel present power calculation property blockmodel since difference mean power extension equation reproduce nan distribution calculate statistic simulate article generalize situation two independent observation mp mp mp appendix variability along replication general normal matrix p likelihood appendix extend literature family due modify statistic test mh invariance transformation miss applicable unlikely treating nuisance likelihood alternative unbounded simultaneously distribute regressor dyadic identically vector regularity explanatory row motivate statistic identical explicit readily observe simulation residual appear asymptotically correct international residual position tool ratio test exp statistic diagonal reject binary protein far relational variate correlation node relationship adjacency protein interaction protein protein network connect node side meaningful protein represent normal protein protein asymmetric characterize latent receiver write consider variate immediate heterogeneity heterogeneity statistic rank capture fitting draw construct term provide description observe interaction rate approximation latent identically simple suggest high nan fuzzy long distribution skew capture little evidence rank ratio relational testing observation test separable versus test row correlation versus alternative correlation form leave transformation demonstrate maximally use assumption relax testing case demonstrate accommodate frequently feature diagonal application paper matrix ordinal variate among row mean unique specifically distinguish equal obtain analytic identifying likelihood author normal prior covariance second penalty code result section author likelihood unique strictly demonstrate solution rewrite equation explicitly state consider diagonal col tr c writing respect matrix familiar value partial refer derivative ki everywhere solution show definite solution verify covariance matrix hessian row likelihood function imply diagonal writing include mf always remain verify criterion uniqueness definite must point relative contradict scale likelihood pd try mp likelihood equations iy mp iy try hold strictly convex attain satisfy I mp try restrict domain positive large eigenvalue g meet positive boundary g approach point eigenvector eigenvector eigenvector match converge completely determinant thm conjecture record object formal parameterize term test row column dependence statistic row relational accommodate feature word actor frequently present entry direct relationship object scientific social health child market analyze business interaction interaction understand interest among relation similarity among long summarize iterate correlation row correlation among blockmodel small identify relationship group commonly suffer lack
originally model learn set topic neighborhood vector word view matrix word cast topic model figure view property well concentration bound appeal view independent condition multi specify view topic specialize importantly guarantee hope tensor power community modification adaptive initialization neighborhood initialization tensor eigenvector us noise star thereby modification connectivity consider need replace strong number sample connectivity satisfy improvement guarantee via algorithmic modification derive document within document concentration modify recovery tensor derive paper overlap approach recover community membership community draw mixed algebraic iteration tight separation number obtain sized weak drastically plant clique computational whitening step tensor ratio community make tight guarantee experimental million good magnitude variational work make make decomposition carry unlike serial moreover limited assume connect membership community impose membership suffer identifiable amenable impose membership across mixed membership answer question boundary tractable membership community acknowledgement action anonymous improve manuscript thank community aa aa microsoft fellowship nsf award award award perturbation discuss modification tensor modify modify obtain estimate adaptively depend current selecting initialize adjacency good vector involve bound procedure section establish tensor power procedure eigen norm tensor tensor good exist least one initialization eigenvector guarantee initialization requirement weak requirement employ random initialization subsequently satisfied see large perturbation ok ai independence bernstein inequality line various partition least event chebyshev however eigen direction theorem satisfy satisfie subset view condition chebyshev inequality q bind g norm q variance chebyshev perturbation theorem concentration tensor large norm matrix definition remain norm definition individually term property know frobenius bounded easy break sum one cauchy term bernstein inequality term dominate adjacency submatrix adjacency partition correspond concentration bernoulli bernstein inequality variable wise z bind independent bernstein f I concentration mean follow provide column definition note prove spectral norm variance dominate bernstein let probability follow distribution depend show initialization vector tensor I g jj orthonormal eigenvector vector c substitute regime require exact first observe r realization vector maintain unit ny py apply bernstein inequality event thus bi bi ji q improve bind dirichlet desire r r regime recall marginal regime agree spread sparse q sum z q q pdf pdf multiply similar acc dirichlet distribution ap semi eigenvalue small second ab f bernstein since real x dirichlet exist dirichlet suppose moment satisfy formula iv one bernstein almost vector inequality entry require chebyshev independent control singular singular theorem proof task observe interaction mostly restriction provide guarantee probabilistic overlap term membership community dirichlet unified tensor spectral base moment star simple algebraic e singular decomposition recovery membership model parameter analysis important result requirement stochastic community spectral tensor community interest music co formation start seminal tendency individual responsible non community attempt quantify property domain network exist vast literature community learn typically heuristic maximization practice poorly guarantee tend probabilistic term study learning belong setting paper mixed membership originally employ model provable paper mix set sufficient attractive property many convenient block edge assume give community individual different community stochastic enable work extent overlap among community affect performance learn community moment incorporate tensor spectral decomposition compare node approach mixed membership employ distribution membership regime extent control concentration mixed membership unify network size edge connectivity across community overview guarantee size community community intra probability membership community community overlap estimate matrix connectivity complete detail behind community indistinguishable scale become increase intuitive scaling show community grow quantify error lastly guarantee recovery membership identify presence present homogeneous identifiability access subgraph subgraph triplet leave community connectivity access implication learn stochastic special scaling requirement match separation previously optimization definite programming involve tensor detailed guarantee stochastic extent much extent membership interest university company network belong see practical requirement match degradation learn learn block unify limited note modification accurate network graph million approach guarantee propose order subgraph employ tensor count star star graph leave count occurrence star count relationship draw membership orthogonal rank method g compute eigen perturbation introduce adaptive subtract eigen pair neighborhood regime community additionally thresholde processing operation sparse community membership overlap community theoretically improvement comparison correctly community membership tensor inequality impose concentration bound work community discover mostly various subgraph typically quantitie star triangle etc instance subgraph count term identifiability degeneracy moment correspond star block membership dirichlet subgraph count tensor subgraph star label vertex consider aggregate scalar count allow subgraphs edge graph moment graph term exponential subgraph graph star triangle subgraph count normalization function suffer detailed contrast establish membership amenable algebraic power count stochastic block method cluster membership projection onto spectrum implement decomposition variant laplacian contrast work technique definite programming guarantee various try community without asymptotically graph define single community approach overlap graph requirement time run low polynomial serial computation assume form fraction outside fraction improve linear mostly moreover work limited within community approach limited assume homogeneity connectivity community make formation improve guarantee consider additional processing step recently provide considerably different setting generalization characterize stochastic instance general framework arise limit convergent term graph regularity propose block neighbor block overlap topic categorization topic corpus topic dirichlet lda perhaps assume document dirichlet mixed membership word direct document neighbor occur similarly link incoming contain establish model lda leverage development base establish learn algebraic base technique recovery guarantee moreover set analyze similarity difference model low clique constant query clique relate clique moment relate find clique correspond extend clique tensor general hide clique network albeit tractable reduction tensor scaling requirement iterative approach intra identifiability separation factor another learning optimize traditional em practice variational approach field efficient variational gain efficiency method lack optimize community maximum block extend provide consistency guarantee provide precise open estimator recent tensor fast recover community scalable million community membership draw introduce special let row denote support community stochastic model coordinate membership n k u iv diagonal high assign choose community assignment direct community learn assumption multiple community yet preserve membership give membership edge independently draw multiple edge community membership aspect mixed membership model community vector distribution serve widely statistic e dirichlet allocation conjugate make inference denote size entry learning parameter case mixed membership dirichlet fix single instance dirichlet thus serve concentrated along extent generate setting involve membership community much large individual interest university company network person mix moment membership describe learn star provide moment membership w v submatrix subsequently neighborhood tensor third tensor array symbol kronecker vector tensor refer subsequently tensor vector community first moment explicit edge count moment learn community membership interest star leave internal star denote structure part star belong necessary count star product leave head relate tensor community connectivity community moment partition node see map vector expectation decomposition learn parameter learn community straight moment exploit find another tensor consist star obtain community connectivity learn membership exact identifiability moment form level version modify tensor independence stochastic star say conditionally community membership draw partition ensure independence star g conditional expectation last collect obtain moment mix instead raw I edge moment obtain expression count measure extent overlap assume block third scale version stochastic block model tensor view center moment mixed moment membership normalize concentration community membership column relate membership star carefully decomposition matrix adjacency carefully choose eliminate correlation identity exploit algorithm quantify knowledge quantity tune via stochastic involve dirichlet moment membership involve linear expectation product collect diagonal th entry membership relationship enable learn outline section utilize base tensor use tensor tensor simple available section basic special orthogonal subsequently section modify previous refer third tensor symbol rank tensor cp decomposition tensor multilinear th representation tv multilinear linear dimension scalar linear use multi describe instance yield scalar tensor tensor cp form tensor tensor asymmetric tensor orthogonal another orthogonal decomposition tensor rank tensor decomposition fix norm orthogonal decomposition tensor iterate initialization vector fix map set set power see eigen subsequent vector obtain eigen orthogonal need suitably modify tensor perturb case discuss describe available section suitably modify perturbation moment employ tensor tensor employ obtain finally cp procedure obtain exact context modify cp describe convert symmetric transformation use modify adjacency svd modify adjacency moment matrix multilinear moment symmetric form tensor community count multilinear transformation moment whiten consist whiten matrix serve eigen henceforth refer tensor rank k c connectivity allow neighborhood rank argue degeneracy condition reduce moment tensor decomposition mix use moment network full empirical roughly sample I f overlap satisfie hold final take tensor place level mixed model moment apply describe enable membership I overlap exact membership star count star membership note membership mix use count describe modification handle moment tensor mixed observed method need adjacency community vector thresholding estimate community membership normalize vector g role tensor align common x g require star estimate membership thresholding along svd cr ac w ab ac k eigenvector reconstruction overlap whitening set compute star set leave star whiten except use value define q eigen symmetric iteration method sufficient get detailed modification ii adaptive detailed employ far perturbation third moment power comparison tensor vector moment perturb advantageous eigenvector advantageous reduce iteration approximately importantly make perturbation detailed initialization tensor guarantee recovery membership membership regime neighborhood good power behind concentrate regime consideration concentrate eigen direction serve iteration obtain eigen pair subtract current eigen pair eigenvector guarantee robustness power current power projection along direction eigen otherwise eigenvector estimate intuitively good current update direction already eigenvector note eigen oppose paper work l initialization vector eigenvalue iteration n previously moment stable eigen membership weak regime membership community p modification incorporate exact estimating straightforward membership p empirical available well guarantee row subsequently sufficient outline next discuss tendency factor formation many modify situation intra inter community community instance post procedure connectivity obtain averaging node belong community since edge use idea community exceed threshold evaluate community correctness procedure suitable provide correspond recovery procedure community membership strong presence adjacency community support membership support significant membership cf cf ci cx cx I role set note implement compute whitening top take multiply whiten tensor step tensor initial small dominate multiply recovery dominant match svd programming sdp connectivity solve via augment multiplier significant computational number small parallelization additionally rank operation method lead implementation run parallel result case community community location community community stochastic allow one membership defer dirichlet discussion sparse regime size bad latter state probability intra inter require standardized separation intra community connectivity note deviation condition compute thereby eigen tensor power assume number iteration power threshold community p decay fix estimate connectivity estimate algorithm event hold proof outline main ingredient establish tensor tensor f tensor moment guarantee perturbation eigenvector satisfie perturbation give refer norm scaling previously achieve match scaling block recover membership mix requirement grow increase assume post process recovery state guarantee assume membership small
work iterate plot provide material pass bring large signal loss loss regularizer way use unfortunately factorization argue amenable x compute concept consider size visualize dictionary element regularization encourage neighbor lead group variable pixel encourage together surrogate try computational dictionary effect show take perform batch flexible show necessarily convergence view initialization million practical value show outperform art solve open possibility believe crucial issue work project subdifferential calculus directional subdifferential use optimization probabilistic paper quadratic differentiable gradient find lipschitz subdifferential subdifferential differentiable relate directional derivative nesterov convex strongly strong tell appendix growth property strongly minimizer sequence probabilistic function classical quasi prove stochastic descent presentation proposition family e f x quasi martingale integrable lemma useful deterministic algorithm converge series inspire proposition otherwise proceed contradiction index enough n b j k b contradict lemma converging sequence surely series convergent non converge converge zero present convex analysis present minimizer quantity define recursively drive stochastic version expectation value word deal often literature auxiliary surrogate remark coincide nr n n definition strongly inequality schwarz surrogate show g nb n w w nc use induction lemma fact surrogate basic stochastic surrogate b sufficient g g g la f induction assume la e e g nf la n l difference useful relation strongly rest follow induction convergence involve key term entropy indeed provide uniform f I zero simply refer assumption ensure condition assume quantity empirical rigorously convergence apply eq uniform converge almost cost n prove simple triangle inequality fact e converge surely show call n nf n f w nf f n converge surely surrogate g n r fact minimizer inequality plot finally dictionary set iteration batch initialization figure reference ne corollary theorem axiom consist simplicity signal processing make deal scale possibly suitable assumption achieve convergence important several solver logistic algorithm effectiveness solve problem consist surrogate bound objective monotonically drive expectation build jensen interpret view dc programming dc stand difference convex bayes proximal algorithm suitable signal precisely address objective represent objective prove machine research consist surrogate turn obtain new em online factorization involve fashion another minimization solve store information past iterate scheme propose load possibly huge choose introduce smooth objective convergence horizon objective horizon analysis show problem almost aware gradient develop constrain long perform state solver dc batch sparse finally show factorization choose involve sequence minimize new convex propose alg technique improve reason iteration initialize nf x g iterate averaged option scheme remark practical parameterized variable minimize example section convergence analysis application provide numerical implementation perform core ghz intel gb ram differentiable proximal weight recursively eq truncate update write include gradient average past algorithm
consequently mr n r symmetry word unchanged symmetry set r n r space r mr equivalence matrix identify factorization separate matrix factorization scale separation scale behavior comparison later fix search bi r three generalize bi element element equivalence vx tangent individual manifold v u riemannian define product tangent manifold typical derive metric natural metric cost computationally costly simplify relate cost consider simplification contain full orthogonality structure play cost rr r block induce x x f metric r concrete conjugate systematically principle identify tangent along equivalence subspace r complementary characterization routine computation characterization rr symmetric uniquely operator ambient r rt tangent rt x extract normal metric un r accomplish v rr rr u z rr r subsequent onto horizontal operator couple couple solve efficiently perform mapping vector precise move horizontal direction nature combine orthogonal operation mapping tangent space operator horizontal lift q horizontal depend early notion equivalently cost follow direction iterate auxiliary n lift riemannian characterize lyapunov rr rr rr numerical computing riemannian nature matrix entry randomly os iteration similarly stop apparent rr note scale first simultaneous update motivated accelerate enable develop arbitrary unconstraine em efficient cm p cm p p p update moderate size similarly nice os differently converge pointing towards size impose exponential cn singular ratio become challenge instance ill require complete relatively ratio size decaying entry update fix combined instance simulation numerically surprising exploit square formulation like include rectangular matrix completion propose small matrix submatrix pick pick randomly simple os os os ratio full competitive difficulty computational note share truncate weight e matrix solving factor provide original submatrix pick consequently sign counterpart dataset million movie train validation partition train partition due uniqueness use stop mse row standard deviation ranks take fast rank row test rank give omit test score rank follow study ill conditioning exhaustive conclusion instance even valid accuracy riemannian completion stem geometry endow tailor riemannian set rank comparison conceptual riemannian cost viewpoint exploit van discussion theorem algorithm completion ac optimization manifold conjugate completion rank novel metric tailor least square numerical outperform art instance combine complete j formulate frobenius amount surprisingly variant vision name e much dimension rank among fundamental popular characterize invariance make many case differentiable manifold focus exploit scale datum abstract manifold non essence conceptually search smoothly vary metric role relate abstract notion concrete dependent limitation riemannian riemannian resolve issue propose riemannian tailor cost many matrix recent instance rank exploit constraint orthonormal first manifold mr
remove effectively rate reduce subproblem conditioning state require lemma loss band reduce clutter rest simply clean w pay zero hinge w tail appendix denominator b k cd ultimately relate weighted hinge clean outli removal subroutine figure polynomial replace qx qx x polynomial vector assign example fraction show vc polynomial appendix explain removal enable refined extent proxy formalize adversary large enough w w idea clean minimized fractional outli removal cauchy weight direction fractional soft outlier allow variance way effect clean clean summarize k k hinge loss loss vc yield w main distribution concave distribution admissible isotropic log distribution ball respectively satisfy unit vector u two du prove distribution admissible unit weight k request suffice contain special isotropic concave adversarial simple adversarial label except adversary removal adversarial noise except adversary outli removal learn adversarial label corollary marginal example clean noisy affect adversary clean classification change clean concern label prove fact rescale admissible admissible respectively idea traditionally get furthermore localization space base get time concept noise know model independently work mention regard provide noise classification selective label determine vector provide adversarial important difference noiseless word define easy sampling make show first balanced concave receive angle give marginal instance concave subroutine run choice admissible hypothesis call start state detail paper hold unit vector collect specification ensure hold theorem sketch inside band benefit localize removal induction implie define b w w r apply band recall imply induction suffice sample label taking subproblem analyze within body outli subroutine weight retain implement proceed lemma distinct obviously requirement solution infeasible find violate start chernoff member show feasible formalize length let unit ball put complete draw time two feasible prove separation first check check maximize iteration attention formula clean pay hence definition denominator numerator hence decompose work clean example output adversary round fall example unlabeled chernoff example fall noisy unlabeled complete variation distance probability difference event function example roughly keep normalize uniform make follow distance next relate hinge weight clean define absolute constant define generality fix arbitrary definition constant absolute since qx loss last inequality cauchy thus q probability w k w since w k algorithm k ready put everything proof element application turn outli removal iteration require true hypothesis part imply let v band q take complete induction suffice unlabele label describe admissible adversarial access show without generality noise efficient allow error oracle get label cut sequence precision hinge w v admissible constant polynomial imply cut k w request suffice case adversarial noise rest subsection figure define apply little first relatively soft removal subsampling outli removal may place large second analyze analyze noisy portion clean compare incorrect label marginal clean let include adversary round label loss absolute exploiting constant k k complete sphere obviously learn rescaling let refer subsection unit part unit projection onto isotropic concave apply concern span q isotropic concave fact log concave hold distribution rescale ball trivial contain isotropic concave x cx implicit isotropic concave want rewrite imply constant denominator satisfie numerator combine complete ready q imply absolute want integral use change get put get complete proof study agnostic setting describe imply approximation concept good opt x output hypothesis want translate via adversarial algorithm agnostic adversarial r guarantee x df cr along imply guarantee learn go vc tool say real threshold large use let md f know dimension bound useful next admissible distribution part union bind draw suffice prove lemma define lemma theorem theorem corollary cm designing computationally demonstrate tolerance provide time improve tolerance adversarial label noise unchanged label constrain algorithm learn uniform handle result isotropic polynomial adversarial noise also noise tolerance dependence dimension classification example whose exponentially passive polynomial active adversarial several localization rescale novel localize outli removal localization technique well deal active arguably popular correctly example understand like classic passive set active special learning theoretical seminal hardness connect formula design learn presence adversarial originally unknown see free year investigate show uniform isotropic continue adversary unbounded computation algorithm limit adversarial exploit active study paradigm receive passive active algorithm presence adversarial label open pose benefit passive challenge bring localization well outli removal theory localization refer practice focus increasingly possibilitie safe stability possibility relevant learning set modern scenario typical training model unlabeled goal produce also label hope active use request passive query keep unlabeled work output adversary learn algorithm draw adversary goal remain consider model active oracle oracle work random unlike receive separate adversary may x active label oracle theorem commonly passive sample addition want request depend quantify give design passive active passive li rate li show variant construction remain uniform sphere year closely describe uniform observation tend well limit adversary coordinate noisy limit removal use direction project training lead remove project algorithm uniform motivated modern machine massive amount unlabele significant interest designing utilize minimize decade substantial progress understand learn classic passive supervise noise free agnostic effort date provable guarantee result benefit passive polynomial unit suffice satisfy ball presence adversarial w agnostic inverse super apply necessary go origin isotropic adversarial upper imply satisfie w exploit active provide label algorithm unlabeled uniform label solve pose linear trivial exploit power localization designing time passive design active useful localization otherwise closely work isotropic concave adversarial na minimize act factor adversary opposite sure long insight iterative localization cause adversary despite half error rate boundary repeatedly example band key outside band furthermore band distribution progress margin active ignore consideration literature early passive idea exploit rescale hinge minimization small band adversary removal hinge rescale proxy step towards set pick hinge within band example notice noise within noise hinge loss vector adversary tolerance get tolerance adversarial agnostic example effectively noise tolerance case need deal uniform introduce removal stage next procedure indicate confidence hinge combine mention lead tolerance outli removal problem outli removal limit effect noisy detect outli removal figure spirit variance direction measure fact minimize band flat small limit adversary great removal weight hinge reflect infinitely many program technique community example particular hinge limit
covariance previous equation wireless channel vs decomposition approximate sum q accurate kronecker kronecker components matrix kronecker model element variance structure kronecker significantly predict video predictor organize introduce kronecker representation spatio covariance present comparative result video data paper application spatio overlap ml predictor appropriate kronecker video covariance strong affect uncorrelated replicate across kronecker highly determine predictor problem propose approximate diagonal matter turn element become problem low multiple intersection l notational multiply put divide vector kronecker alternate unweighted svd miss determine diagonal cutoff helps preserve apply rao bind optimal unbiased estimator iid give permutation certain coefficient yx portion q portion predictor possible reduction kronecker predictor predictor define assume row predictor thus asymptotic covariance result structural focus mse kronecker sum approximation focus mse appropriate slide window new one sample stationarity lose video divided image kronecker frame covariance portion likely due covariance show kronecker frame function htb computation slide window obtain new sample window near toeplitz stationarity video process point sec person arise situation video space datum frame ahead hereafter predictor ahead show approximation video rmse low sample video create somewhat covariance base mle prediction find average performance sample achieve perfect kronecker ls covariance frame video covariance kronecker difference kronecker prior kronecker match asymptotic performance htb rmse prediction consecutive frame predictor covariance regularization prediction use infinite occur htb kronecker kronecker kronecker learn covariance kronecker kronecker condition kronecker prediction unstable whereas correct kronecker use sample covariance use frame slide frame ahead monte correct kronecker sample covariance kronecker force train poor kronecker demonstrate multiple may arise forward available want incorporate forward pixel average pixel frame weight information since typically uncorrelated present kronecker tendency inter correlation covariance rather poor video average ahead correct kronecker lower covariance kronecker examine applicability spatio covariance video covariance propose result improved prediction video sample optimal performance assume kronecker use correct gave increase sufficiently
dimensionality entry denote often row quantity return work matrix define simplify ratio give superior bound imply c citation sample n hold stable readily need comparison key parameter reference value metric work wherein would sampling experimentally well opposite explanation phenomenon relative performance l sampling replace increasingly accuracy bring close give solution requirement r latter rp seek optimize readily next condition definition element hold change must jensen specifically jensen inequality meet e bind trivially b ij z e tb non entry possible thus triangle entry p ready third equality occur norm lemma yield rearrange third instead see couple minimize couple relate next term minimize insight logarithmic function minimize decompose within row e surprisingly closed form computable since set minimize fix leave minima consider lead call I ij nevertheless seek e equation I see unique every satisfying lemma pass pass ready proof minimizer also finish distribution deriving establish follow equation equality bound large equality definition characteristic summarize definition different cm cm synthetic image e e e e e email corpus subject line row word tf wikipedia matrix english tf represent single filtering row column user first vector dot correspond fact item popular entry item different distribution distribution refer early case l split threshold method ij ij technique analogous threshold although sensitive reason zero case top intuitively measure well singular capture compare singular give approximate hard significantly experiment method poor near perfect end small qualitatively indistinguishable htbp main never technique wikipedia superior insight rather impossible perform row seem highly option insight sampling discard small drastically however advance example matrix assume receive item stream reservoir classic problem stream instead stream stream could execute sampler point require randomize sketch impractical memory operation instead respectively sampler simulate item random replace appear variable zero storage disk processing generate sketch disk bound stream terminate process follow pair process operation bin uniformly item replace notice go execution thus stop soon bin irrelevant perform bit care sampler item track number whole update ball bin stream stack see thorough assign fall bin theorem fact yahoo yahoo consider sketch matrix attribute give form information allow non per entry mild provably competitive optimal offline matrix therefore might impossible streaming model desirable proxy mathematical preprocessing matrix spectral randomization overall typically cast devise big attribute zero memory original know basic priori actual non time streaming recommendation item give provably row row return mean matrix single simple generic consideration remarkably practical rare bit reason store bit store index zero measure result bit per usually less represent relative file list format combination budget result however grow tend refer appropriately consistently adapt phenomenon l analysis tendency align particular examine presence multiply trend measure minimize capture wise uninformative pick approximation long even far well measure minimize rotation variation vector idea isotropic packing much frobenius aim mean form entry demonstrate conclusion start work reverse optimal notion deviation inequality get fix form element let always estimator repeat matrix mind find mean yield conceptual sample theorem bring state matrix target sample practice element rest paper seek bound worst put perspective amount compete unbounded regard l l row compute yield pass moreover argument imply ratio well correspond attribute ratio ratio
take simplify yield preserve sum indicate little abuse eq discount know imply follow recursion sum analogously term plug simplify preserve sum seek definition simplify substitute simplifying hold kl preserve sum definition primary purpose provide also indicate following form low matrix e j
proportional perhaps even slightly rate ct parameterize threshold versus average run figure plot dot plant sparsity also increase show sdp away average dot dt circle dot lead eigenvector large sdp lead combination spike namely sphere dimension vector orthogonal use entry absolute suitable tend whenever turn tend proof orthonormal orthogonal gaussians coordinate task reduce estimating unit sphere tail proof chi record product high dimensional gaussian like independent realization independence lemma proposition matrix spike study extensively specifically fix chapter could either constant tend spike strength far probability tend trace consider range k kk plug example population exist tr solution sdp eigenvalues correspond orthonormal frobenius inner characterization level sec prove inequality tend tr suitable conclude pn prove tend bind sdp sdp vector rank feasible sdp q thus net single size net upper pn discretization fail end idea bind convenient argument another r ff f pi pn bb I complete q proving let high indeed occur similarly proposition f u plug get hence straightforward tending prove union arbitrary exist every length expand eq since independent schwarz contain ignore henceforth center net disjoint otherwise length volume grows plug complete spike become first obviously change since rotation would normally assume recall nu ni inequality follow wishart role apply tend tu tend plug bound tend least turn paragraph si row j jj nx suitably style theorem remark question conjecture support foundation foundation institute lead modern statistic many pca thresholding sophisticated theoretical recover spike asymptotic prove level reliably eigenvector sdp gap sdp usage recover level recovery wide science pc explicit rather give eigenvector suffer interpretation subsequent consistent inconsistent dimension comparable significantly large poor eigenvector population principal address drawback pca direction nonzero population eigenvectors efficiently symmetric pca combinatorial hard prove adjacency matrix nevertheless computationally regularization singular lagrangian diagonal semidefinite programming latter recover study concrete relaxation suggest two matrix denote ty ij x semidefinite rather et eigenvector first multivariate unit whose I furthermore eigenvector study nonzero prove coincide level whenever exhaustive subset presence prove question open outperform dt constant useful answer remain rank support coincide suggest pca problem computational limit coincide statistically sample page formally slightly exceed weakly indeed dt perform similarly spike arise outperform motivate greedy thresholding ct experimental suggesting consistent rigorously ct recover level eigenvector sdp diagonal thresholding define dimensional setting whereby nonzero follow nonzero signal infinity recover almost eigenvector large state either fix analyze alg sdp exist solution tend sdp denote norm tend contradiction spectral norm large bound large recall p arrive inequality constant appear necessarily reduce case nearly sdp sdp nonlinear input explicit sdp tend solution low tend regime conclusion one arrive next probability tend indeed rank prove similar significantly simple light weak signal strength yield rank solution provably conclusion computable reliably reliably plant otherwise computer literature task polynomial date clique wang clique hardness certain regime randomize unconditional main valid future development yield hide
present nearly optimal bound illustrate dimension weakly computed column sample probability small depend compute optimal algorithm eq hand tight publish sampling probability sparse similar achieve accuracy run theorems percent plot theorem plot quantity plot versus plot bound indistinguishable versus bar square triangle bind star axis label present relative randomization leverage leverage produce due randomization nearly probability compute leverage furthermore general small singular matrix optimal leverage score gram come present nearly uniform without replacement gram matrix coherence algorithm probability see although replacement replacement modify slightly tight application add replacement compute nearly probability replacement constant matrix orthonormal coherence sample nearly transformation independent random semi singular value bound probability check p write zero check j semidefinite apply determine linearity value value c ne ie bound remove change probability result submatrix replace give tight dimension apply tm nearly invariance norm unitary c start dimension real definite definite nearly optimal least check item define j version check assumption j f depend remove item substitute requirement requirement fulfilled divide relative error substitute thin view separate express uniform nearly sampling method present definite fx e cx j tc compute c conclude eq give decrease hold replacement derive similarly apply imply c x analogous remark example outer product computation arithmetic monte et outer product bound randomization bound dimension singular orthonormal singular rank unbiased answer probabilistic approximation follow overview result literature familiar review approximate specifically outer break outer specify integer column approximate weight sum outer weight expect well low intuition singular decomposition product suffice reproduce column singular monte chance contribution paper section monte algorithm establish connection linear algebra characterize specifically theorem depend non possible leverage however necessarily large relative numerical leverage sampling leverage obvious priori j monte carlo sample nearly theorem sample success probability dimension leverage probability theorem tight probability orthonormal compute surprisingly theorem always tight bound chernoff represent slight exist bind singular number correspond theorem error gram addition os element wolfe et al remove column importance minimize products monte carlo excellent survey review exist monte summarize table show column frobenius error f list uniform easily convert format sample sample aa aa opt aa aa specify probability without bound sample relative error reference bind bind apply orthonormal c opt u orthonormal row lower sample column specify strategy opt replacement replacement value list specific choice aware condition matrix row without replacement monte select select subset traditional decomposition motivated graph orthonormal rely form eigenvalue update determine view vector matrix apply barrier dominant stage et al perform deterministic sample second approach decomposition introduce leverage et leverage importance randomize decomposition square also approximation design bold ne thin tm kn unique definite positive eq matrix q inversion square large value event monte carlo connection exist linear since column submatrix express column question give thin tc distinct minimal opt solution matrix correspond sufficient tw nc c orthonormal q see special case leverage score special case column minimal coherence distinguish column equal illustrate index occur let representation matrix connection theorem diagonal opt minimal solution equal diagonal outer select minimal frobenius opt column weight orthonormal row leverage illustrate non column orthogonal column column submatrix orthogonal construct matrix opt opt opt review approximate compare two present special conceptual version present
borel sigma constructions nonparametric denote denote law dirichlet say henceforth partition particularly joint distribution predictive weight broad follow detailed present sized dirichlet draw dirichlet state find could n nx ny q iterate argument nr notational law slight abuse system component analogously rest law reduce market interact interpretation follow worth note array exchangeability identify row exchangeable exchangeable evolution interact market refer explicitly evolution share unit particle several market ease presentation consider single market unit market investigation section jx k unique current configuration implicitly share fraction market divide market restrictive approximated sufficiently remark discussion choose fraction configuration market possibly impose maker share useful context individual share position nc nc maker aspect mechanism upon share pick anomalous status occur share chain sampling unchanged either new markov also think sequentially update select special metropolis hasting monte chain reversible account market index market select random select previous new atomic mr becomes normalize constant allocation market new surely market possibility operate sample term empirical measure ignore unit case associate current market interpretation model market expansion vice cost expand different need operate certain market imply vice flexible reflect imply interpretation example location among high share weight function whole configuration interpretation arbitrarily relate parametrization unit threshold unlikely introduce micro however micro foundation view outline construction sec present generate convenient continuous make enable appropriate chain possess chain intensity time generator assumption recall generator process banach subspace limit carry determine state need introduce bound operator belong r rr value component generator market generator operator market last deal value measure provide aggregate time identify generator let element equal define q generator product product n atomic atom correspond measure value system proposition generator collection interact process brief induce flat close reader pass cost determine market concentration share eventually market interpret positive although rate mutation keep apart picture possibility qualitative market three actor structural initially maker correspondence line introduce new cost multiple distinguish parameter set equal equal concentration decrease competitive actor solid direction market dynamic graphic toward threshold represent political view structural change market market represent market entry comparison purpose entry still competition show opposite set competitive market particle mean hierarchical center measure market make establish inspection partially underlie imply place index effect one label fact opposite competitive market assume characteristic figure figure market space case cost entry market cost competitive regime case share dynamic point implement view behavior perspective covariate adapt approach recently present alternatively economic viewpoint modify add would section account desire behavioral status micro allow rich decision comparative dynamic issue economic sampler gibbs hasting chain procedure see space integral carlo integration turn monte carlo markov invariant discard approximate construction consider value x equilibrium component get respect pt constitute together process probability measure generalization evolution vector type identify q example compact genetic drift drift mutation selection specification yield classical type space endow weak convergence function f mf mf f measure onto bound borel argument interact interact extend collection model element denote index countable mr generator countable interact term r intensity represent rate mutation resample notation simplify f generator proof proposition multi market col process configuration market conditionally market eq removal probability becomes substitute respectively interpret nc r r nf nc k nc generator write nf r f mc ar r mc I f kf n let supremum measure denote rx normalize check tend let statement weak law topology acknowledgement grateful associate valuable lead also european research modeling share market set interact market stochastic contract share market phenomenon expand obtain bayesian collection mixture market partially exchangeable dynamical markov simulate mean study transition economic regime dimensional property appropriately rescale interact literature formulate later allow heterogeneity determine success accounting selection lead perform steady entry heterogeneity market share walk inter essentially equilibrium project construction dimension implicitly usually find agent steady state heterogeneity relevant difficulty reference dynamic market share share might share current overcome perspective micro tendency problem economics tool interact describe heterogeneity feature easy investigation aggregate economic keep tree interact particle financial limit empirical economic agent interact locally mechanism mutation bayesian nonparametric model interact particle unnecessary distributional involve cluster object current status share heterogeneity conditioning market share despite scope respect particle consider interaction construct emphasis flexibility necessarily imply degree dynamic adapt diverse framework population fact follow market motivation throughout paper micro economic besides present result aggregate process literature countable selection basic material finally worth nonparametric although natural seminal mention survival see powerful bayesian nonparametric
tool scale work generalize split introduce function handle problem sum monotone increase batch mode procedure probably memory start effectively deal appear audio video processing tool online day advance generalize average composite multipli method admm focus generalize online setting work train convex penalty regularizer optimization split convex thus iterative splitting hilbert convergence strong point process difficult inexact derive proximal nonsmooth pg forward linearization hx gives show hx hx gx hx together lipschitz constant subgradient hold continuous differentiable explicit q corollary assume continuous generate lipschitz let batch summarize z u u point hx x u goal achieve static receive seek regret learn point notation tx mind iteration q mean hx formula
suppose iteration b b n bm ensure stop rule theorem example finite readily available mcmc width rule remarkably nominal consider quantile methodology draw calculated coverage equal replication consider target I metropolis sampler exp proposal ergodic combination start run minimum meet additional checking replication summarize deviation termination result nominal suggesting mean equal ht sd sd e e e e e e replication nominal level summarize replication nominal estimate credible chain add interval specifically simultaneous confidence replication individual remarkably nominal coverage nominal due correlation ccccc length e e bivariate normal p walk proposal proposal uniform apply geometrically ergodic rule chain iteration check stop independent replication deviation iteration termination along proposal notice coverage setting independently summarize coverage coverage nominal suggest perform setting specify l ccc c variable c length sd intercept sd e c individual interval nominal dimensional simulation iteration add add meet independently coverage close nominal probability nominal except lack region encourage ht cccc replicate coverage nominal region absolute magnitude deviation width stop rule desire well variety setting specify small stop practitioner usually quantile single absolute magnitude rule would specify become instead standard deviation stop since easy applicable simply terminate approximately recommend excellent wide variety small appropriate simulation explore mixture bivariate normal sampler affect effort reasonable accuracy vary scheme behave usually challenge aspect therein bm overlap batch subsampling bootstrap acknowledgment grateful helpful paper theorem modification necessary stop q theorem assumption department california email markov carlo commonly employ challenge stop terminate width sufficiently magnitude stop develop validity terminate wide quantile simultaneous variety example provide recommendation practitioner estimation monte chain whose challenge practitioner determine terminate terminate iteration determine simulation alternatively determine estimate introduce motivate word simulation width quantity sequential stop note effort justify constrain simple width stop simulation ergodic say stop eliminate absolute value relative target width rule simultaneous specificity wish mcmc estimate entail construct homogeneous ergodic space popularity mcmc simulate finite outside matter long unknown monte directly approximate sampling central variance associate due correlation present interval assess practical sequential stop work first small distinct stop rule terminate magnitude terminate iii condition validity validity imply terminate coverage previously aware width quantile stop iii significant terminate confidence interval simulation stop posterior asymptotic limit carlo variance surely detail quantile rule sampler consider explore sampler toy illustrate utility stop final version logistic presence use terminate confidence nominal provide valid practically accurate determine practitioner implement multivariate priori excellent variety organize asymptotic estimating quantile example conclude discussion practitioner width limit satisfy strong central theorem weakly nominal coverage consider sequential confidence prescribe type rule absolute terminate insufficient terminate behave desire effort default absolute terminate yield validity sequential asymptotic weak stop expectation work well estimation section strongly estimator stop know precision stopping avoid absolute put confidence th magnitude rule large behave direct require simulation number research setting specifically poorly behave problematic another design end terminate confidence fraction suppose behave fraction unit would appropriate magnitude establishe suppose w p terminate additional strongly estimator expectation readily quantile discuss follow benefit second suffice estimate comparable yet informative criterion critical show simultaneous nominal coverage value enable follow normal asymptotic demonstrate quantile criterion applicable heuristic simulation primary interest estimation quantile first mix condition consistent estimation asymptotic interested direct algebra decrease hold establish constructive technique uniform ergodicity say concern ergodicity interested reader direct width consider natural appeal ergodic yield chain ensure fortunately frequently consistent estimation applicable
show euclidean space consistent sense demonstrate illustrate may refined sequence graph range blockmodel give embed via mean measure plot error embed multivariate plot misclassification misclassification give misclassification htbp plot indicate normal negligible clustering spectral method mean decrease previous accurate empirical cluster believe grow work regard analogous estimation hypothesis space spectral procedure allow computationally statistical wide though represent dot graph graph approximate exchangeable sufficiently exist original position link inner map position argue latent allow consistent consider strong eigenvalue conjecture setup dot latent outline construct appropriately position converge central adjacency os adjacency spectrum characterize walk eigenvector small laplacian min current investigate eigenvector adjacency dot eigenvector position loop grow contain limit position dot illustrate main corollary dot graph multi dot via simulation adjacency dot specific position associate latent presence absence graph independent link position briefly strong graph key ingredient fundamental graph construct subgraph work consequence subgraph spirit provide current receive much recent review result common position base link product dot dot dot graph true position influence classic os eigenvalue normal type regular adjacency prove mention eigenvector prove material difference result however entry nonzero eigenvector vector position condition whose normally difference remain eigenvalue relate logistic also belong class position interest information prove dot dot hold independent adjacency matrix condition slight modification graph important eigenvector normalize moment latent position control norm necessary dimensional exist hold bound large number event occur assume suitably sufficiently limit evaluate normal mixture consequence corollary os enyi os namely os hold need apply identically mean law proposition almost conclude imply convergence denote random probability variable converge remark ease exposition paper constant convention hide denote symbol change line proposition since q term normal condition last distributional position finite collection residual jointly uncorrelated word index mixture multivariate normal er ease notation general simplify follow zero leave covariance remainder follow bound dot graph position dot condition seek latent identifiable column decrease orthogonal remainder distinct strictly mild restriction impose technical motivated essentially embed decomposition large eigenvector similar case explicit small eigenvalue exist sufficiently event imply therefore denote operation inversion indexing hand sum independent two follow close probability yield diagonal adapt somewhat v q eigenvalue know q similar dividing establish scale position denote cumulative zero surely x j normal mixture dimensional give exist recall side hold let normal x n almost surely surely multivariate set ty factor time n derive lemma least reasoning conditional integrate display realization proof counterpart suppose condition setting suppose let index
approach interpret calculate likelihood response result sparfa summarize question learner cm concept geometry simplify simplify simplify expression slope association concept school algebra test amazon answer sparfa strength individually detect outli measure relative relative question incorrectly question involve concept concept heavily intrinsic incorrectly incorrectly question concept concept difficulty correctly question concept difficulty sparfa strength tag concept investigate detect outli enable pls learner identify insufficient knowledge concept answer predict provide answer tb question answer py sparfa value filter experiment learner response dataset consist answer response also rely cf parameter fold cross metric percentage unobserved response cm algebra accuracy sparfa enable estimate algebra superior learner little meaning phenomenon agree abstract visualize association graph emphasize sparfa provide interpretable estimate factor comparable slightly superior performance cf sparfa development automate generate interpretable learner purely model analyze rely predefined dependency primarily contrast sparfa dependency response multi question modeling association study characterize concept binary ignore strength concept sparfa relationship value factorization deterministic contrast sparfa statistically response likelihood exist relation include limited deal contrast sparfa multiple concept exploit concept sparfa profile feedback sparfa logit problem learner constant probit case brevity probit logit follow analogously probit operate simplify proof define multiplication response upper norm gradient eq triangle denote sign arrive conclude substitute note proof easily adapt case replace contain omit brevity sparfa descent subproblem correspond respectively plus probit analytic log continuity tf variable frobenius bound logit show probit analytic likelihood probit recall establish analytic analytic standard density real analytical consequently let kk x analytic negative probit probit link preserve armed prove begin show additional assumption finite convergence sparfa starting point start show every term meet problem regard strong subproblem quadratic parameter regard continuity subproblem show probit logit case analytic analytic consequence sub analytic satisfying sparfa specifically establish sparfa starting establish obvious bound meet final minimum consequence first mm ultimately active model equivalent whether noise recognize portion e since choose conjugate normal rx c normalization complete rewrite recall complete final probit adopt value acknowledgment thank xu helpful discussion insight amazon anonymous comment exposition foundation grant air force office scientific grant fa award program please website project sparfa h j thm definition example h b g coin learner concept concept model question factor involve difficulty response collection question underlie ill enable domain key concept leverage sparfa incorporate user tag question facilitate interpretability real world efficacy sparfa noisy probit latent assignment challenge bottleneck consume develop soon date expensive remain primarily say project provide feedback even importantly fit interest goal learner tailor integrate assignment pls continuously monitor resource progress example progress make past example hard code scenario specificity development progress make algorithm learner content overview article machine rapid enhance impact indeed base rather architecture pls algorithm simulation feedback learner moment outcome loop joint learner correct response term concept concept question graphical relate correctness question encode response answer correctly incorrectly mark due work relate abstract circle bipartite indicate question difficulty denote learner learner matrix intrinsic question answer correctness probit logit link function arm incomplete observation goal ill pose especially learner answer see factor overview first key pose domain involve key become relate concept relate learner knowledge concept abstract expert observation concept observation knowledge answer question interpretable learner strength leverage algorithm factor sparfa efficient produce sparfa b use factor factor abstract sparfa interpretation concept question abstract report range synthetic demonstrate provide collect learner platform learner answer science retrieve latent result bipartite estimate intrinsic link active entry question concept explain answer either low intrinsic difficulty nearly learner answer correctly incorrectly tag abstract concept evidence past solution change change concept formulation environmental change classify graph associated learner answer question response sparfa framework form arrive learner matrix measure learner abstract concept value concept graph automatically similar target estimate question difficulty property enable assign fashion problematic concept underlie conclude encode give correctly latent intrinsic difficulty question number latent abstract learner corresponding chance success question stack denote strong represent stack dimensional give definition represent correct incorrect denote inverse link success thus slack answer correctly hence case incomplete miss learner second learner attempt stack respectively rewrite form paper use link learn define define education setting interpret abstract large imply visualize connectivity tie question learner question question intrinsic ill unobserved inverse identifiability unique rotation enhance interpretability entry education typical level exploit learner question learner response parameter small broad concept detailed detail question associate concept domain course assessment word mostly learner concept answer provide entry indicate weak vice versa assumption likely violate transform help problem arise estimate assumption analysis sparfa complementary sparfa quantity interest contrast principal component approach sparfa quantity algorithm sparfa sparfa base probit quick intrinsic difficulty column augment coefficient constraint entrie convergence proof enforce negativity entry finally normalize matrix combinatorial interest problem relax move multiplier control sparsity level arbitrarily accordingly vice versa third include establish convergence sparfa detailed regularizer probit logit regularization negativity constraint importantly block sparfa proceed initialize iteratively optimize alternate fashion subproblem constant subproblem hold subproblem iterative iteration subproblem sparfa norm rr problem novel solve probit dimensional exist order probit explicit computation difficult thresholding accelerate iteratively give continuously particularly subproblem smooth norm regularizer negativity plus hence instead fista step simplicity exposition probit regression block regression transpose equal give fista step soft become common lipschitz lipschitz logit backtrack circumstance detail sparfa guarantee outer optimum difficult develop statement block multi sparfa starting point objective function well establish sparfa adapt subproblem tracking otherwise logit fista function establish reveal difference ready sparfa define sparfa negativity objective minimize result sparfa find sparfa point sparfa outer finite within close optimum p sparfa optimum bi sparfa increase chance global performance detail sparfa since sparfa outline toolbox improve provide sparfa improve regularizer facilitate however convergence fista sparfa derive lipschitz enable set backtrack complexity sparfa reduce take g inner outer due nature find sparfa picking solution objective excellent heuristic every iteration absolute gaussian initialize convergence proof sparfa include concept concept could select task predict strongly affect sparfa criterion bic validation detailed criterion result sec previously major difference sparfa framework negativity critical interpretation constraint make optimize likelihood outer sparfa optimize rr rr share similarity imputation outline however additional negativity sparfa utilize accelerate fista oppose straightforward efficient sparfa handling logit probit logit solve extend inverse probit link essential application e noisy compressive scale sparfa scale accelerate hessian solve bayesian chain carlo sparfa full posterior sparfa sparfa notable benefit context distribution enable quantity credible interest since explore sparfa hyperparameter intuitive regularization sparfa incorporate require enforce spike adapt loading q e xx ef hyperparameter latent conjugate enable inclusion use hyperparameter obtain mcmc gibbs sampler derive posterior learner must equip miss datum standard detail exception spike exponential next derive detail active rx represent sparfa carry posterior restriction draw k zero k f kb efficiently implement sparfa hyperparameter sparfa scheme way draw step perform second column parallel relevant factor time computation calculation constrain answer nature learner learner question learner user informative broad adequate space hyperparameter additional question difficulty question since sparfa substantial speed sparfa advantageous sparfa determining b discuss generation advantage sparfa visualization convenient sparfa method often sparfa generally mcmc nevertheless examine statistic make throughout factor sparsity component e indicate inactive approach use utilize choose negativity posterior closed form third tail improve away negativity constraint discuss non negativity dense loading furthermore negativity rather sparfa sparfa estimate equivalently partial encode initially rather matter principled post abstract concept estimate tag describe free association concept tag enable concept show extract tag learner efficacy tag real tag topic course subject matter expert learner broadly crowd general tag knowledge concept tag tag matrix column tag otherwise question association matrix sparfa factorize represent interpretability tag column ensure concept tag enable square negative variant basis pursuit denoise represent framework consist two gradient projection norm negative building tag zero knowledge associate concept estimate sparfa normalize tag concept correspond entry assign tag concept enable identify coarse mean tag learner tag characterize learner pls tag use learner tag identify tag entire enable course deal tag real example demonstrate efficacy validate sparfa sparfa synthetic validate underlying observation benchmark sparfa sparfa learner demonstrate efficacy sparfa logit variant estimate learner association learning sparfa collaborative algorithm unobserve learner characterize estimation sparfa sparfa test ground generate synthetic organize follow outline dl originally propose use sparfa sparfa vary concept simulate probit variant sparfa sparfa situation analyze measure trial sparfa aware exist novel setting probit logit negativity sparfa develop code svd negative pursuit omp outline enforce negativity constraint pick product absolute non square residual stage impose negativity provide svd zero favor sparfa practice oracle sparfa sparfa compare fidelity complicate sparfa output posterior permutation address concern post output sparfa use concern normalize column row ground measure analogously experiment sparfa vs concept question concept zero entry uniform sparfa sparfa synthetic sparfa show box four improve increase moreover superior sparfa sparfa svd sparfa b sparsity sparfa b sparfa complexity sparfa pc sparfa require summary sparfa suited confidence statistic key sparfa large immediate important sparfa sparfa impact observation probit version sparfa sparfa miss entry show sparfa svd sparfa comparable sparfa sparfa incomplete datum sparfa outperform probit version sparfa sparfa svd non zero scenario entry hyperparameter sparfa demonstrate sparfa algorithm suit application sparfa across metric sparfa algorithm k svd aware sparsity sparsity sparfa svd examine impact estimation match probit logit sparfa probit variant sparfa b k svd affect sparfa sparfa functional probit sparfa sparfa since probit logit sparfa box plot logit sparfa outperform svd sparfa since sparfa sparfa experiment sake often sparfa sparfa largely analyze consist learner answer course digital university fall estimate logit sparfa assume match course available tag association order interpret mean retrieve tag relative learner tag profile laplace transform impulse circuit association tag learner answer question association bipartite node circle indicate question indicate question ten question learner question correctly nothing b tag
specific influence increase fix constraint increase constraint fix product runtime increase product specific diffusion influence experiment average adaptive threshold achieve product product increase node become valuable degree highly gain node dynamic many possible figure budget per tends become investigate increase fix influence budget prevent make ad day boost popularity product node different perfect spread investigate performance subroutine structure estimate influence focus ccc vs accuracy relative core ghz accelerate report allocation clearly size evaluate scale million thresholding influence method depend figure allocation runtime large become short runtime allocation quality cost million increase speed w normalize adjustment pair pair user distinct specific pair group tuple budget constrain investigate four fix outperform increase monotonically group scenario community experiment balanced group estimate respect influence keep almost number group increase total influence product product limit fix product increase budget fixing user fix budget window limit fix budget allocation quality article medium million phrase publish cascade time medium cascade select group cascade product event etc diffusion network information cascade synthetic split training learn diffusion trivially learn meanwhile infer cascade infection optimize greedy focus hold testing cascade cascade induce contain cascade assign node allocation motivate select node result number b user window demonstrates find allocation induce percent improvement plot allocation figure qualitative respective media site com assign finance yahoo com etc invoke along media site com com com com product separate constraint constraint fixing fix window site study diffusion within recommend assigning cost novel intersection provable show perform synthetic real world normalize theorem specify still select infeasible formally denote differ multiplicative maximal note partition exchange therefore apply hand add time threshold consider stage add w claim combine marginal claim term lemma tt k estimate use evaluation return evaluation roughly influence product diffusion influence maximization output note current take runtime prove monotonic influence problem dp constraint active algorithm feasible last independent subset intersection plug fs ig g cg empty active threshold thus plug ig ig ii bind item setting exist expect edu edu algorithmic aim influential social shall influence user cascade platform face constraint budget unbounded reality recommendation user maximize within short extremely principled provable diffusion propose threshold traditional node mathematical improve extensive art significant margin play typically identify influential adopt user cascade extensively aspect online platform face unbounded reality requirement expect occur within certain window product requirement multiple product simultaneously entity diffusion channel speed spread like may group reach need pay limited amount maximization practical respect influence mostly diffusion asynchronous temporal become complicated argue improve recover also continuous diffusion influence prediction formulate requirement restrict influence submodular correspond constraint ground submodular maximization subject constraint product bipartite channel product unknown select user maximize user raise second constraint consider consider allocation problem perspective spread structure yet real scenario underlie diffusion specific network datum competition address constraint item focus mathematically rigorous formulation induce contribution formulation practical interest provable strong discrete use diffusion allow principled way influence aforementione submodular intersection submodular diffusion provide designing provable propose greedy algorithm number user large prove guarantee overall influence roughly optimization literature obtain optimization evaluate scalable million term allocation least alternative formalize modeling requirement provide requirement expect influence window different product different requirement continuous time influence time direct associate transmission function contrast infection begin infected source entail draw independently infect remain process thus neighbor continue pass infection cascade solid foundation asynchronous assume density diffusion function learn sufficiently flexible asynchronous pair classic intuitively window wide spread infection influential number infect infected infected infect take set submodular influence challenge graphical inference efficient set node entity diffusion channel may spread type product propagate diffusion diffusion time constraint set time denote product ai ij user mean constraint correspond matrix mean assign overall influence constraint set benefit product submodular independent cascade influence combination submodular submodular social subroutine maintain geometrically element feasible ratio marginal threshold gain come problem density lot much high small traditional heuristic keep speed element marginal gain add round threshold runtime guarantee expensive influence product randomize get fs tt l g z z intuitive non guarantee uniform analyze end maximization element become infeasible greedy poor furthermore influence introduce additional address whether adaptive turn element quality select select partition element element see second gain element large submodular inexact solution together approximation cost introduce tradeoff decrease fewer propagate ignore small logarithmic large suppose element infeasible select illustration fill rectangle fill ct fill thick thick mirror cm north ct anchor north yshift cm thick pos anchor greedy select algorithm solution partition still infeasible size two maximal subset arbitrarily among size otherwise exchange th apply since subset property analysis summary second marginal bound select threshold greedy select gain say approximately claim marginal marginal much gain maximization guarantee efficiently select marginal gain threshold user assignment stop almost constraint assignment product affect assignment product general possible case contradiction efficiently much hand ratio budget element element lead exist suitable threshold good
extract restriction path remark restriction indeed short path pointing accord decompose restriction maxima path restriction decompose finite path restriction strictly end apply sum sum length connect gx gx gd xx gx short interval path connect claim continuous connect contain define restriction union interval define connect contain edge dt xt geodesic inequality connect follow theorem lemma correspondence general metric restriction either time maximum parametrization geodesic either restriction geodesic geodesic contain short geodesic short geodesic contain moreover local part attain either connect graph set g gr figure tight diameter contain interior disjoint remove get continuous generality contain argument interval decompose piece consequence compact geodesic space fix edge diameter correspondence exist path consequence iy gd r obtain compact fix graph graph exist number edge length less short edge length compact let associated graph metric length metric short specify application e input apply illustrate first fix node root apply I I k k arbitrary think skeleton namely node node interval disjoint edge partially overlap interval identification perform split middle think interval lower identify upper interval edge dominate computation first base structure copy interval union find structure perform datum learn magnitude rectangular degree coordinate outlier randomly sample neighbor length graph euclidean equal use plane interval may distortion euclidean reflect gps trace move expect road neighbor connect component figure also graph pairwise distance gps trace big distance speed distance computation pair graph reduction ccc gps trace edge distortion distortion road directional different true addition gps thus road circumstance road directional gps directional stacking consecutive form dimensional high build apply road intersect synthetic simulate car drive driving contain stack along dimensional build neighboring length arbitrarily project st nd visualize reconstruct previous three recover road network b trace dataset pass record position trace trace trace reconstruct graph road ht propose metric hausdorff synthetic set get therefore distortion root root second current recover recently homology seem recover topological combine method remove improve acknowledgment acknowledge provide code acknowledge european project cg ec contract program china cb national laboratory technology foundation get number approximation reconstruction algorithm use persistent complex set diameter short homology induce map homology build homology build top homology homotopy correspondence persistence statement induce sequence composition map consequence h discrete space sample metric hausdorff variant approximation synthetic real set technology internet massive geometric area engineer business become available visualize euclidean dimensionality reduction geometric embed possibly dimensional concentrated manifold lie euclidean lie space instead space difficulty decade persistence rise visualize geometric without euclidean focus important geometric see branching precisely endowed assigned see collection trace road network concentrate around galaxy universe name capture approximate metric paper address structure call hausdorff unknown output prove close theoretical result geodesic metric function graph length edge span short edge large turn address geodesic discrete connect metric moreover nearby point know raw nearby equal short metric question metric neighborhood scope second graph usually achieve appeal neighborhood skeleton unfortunately triangle may overcome variant graph inspire recently complex easier dimensional address extraction cloud example multiscale recently approach corrupt embed inference focus geometric geometry sample curve euclidean space study embed self intersection metric whose metric real segment metric graph topological coefficient equivalently remove span compact geodesic let base I relation notice connect induce relation exist connect decrease partially compute approximate usually task issue variant share similar cover open closure dd closely construct graph change number direct connect finite turn acyclic complex present section finite degree base complement increase thus along edge assign orientation positive orientation pointing finally assign point graph
capture grid equitability without ordinary approximation computing versus tradeoff specify grain optimal roughly grid create column search contain grow find effect equitability compare runtime recommend set much significantly emphasize new remainder paper beyond regime maximal describe upper point n examine affect equitability search optimal modified long simply search axis row equitability suggest deviation equitability intrinsic rather compute find equitability perform noise use approximate list equitability carry modify intensive close equitability improve demonstrate gap score box set detect explore tradeoff equitability contrast elegant measure introduce belong method pose quantify association well done decrease decrease detect false rate hand across noise product moment well preferable thing statistic lack equitability ill pose appropriate measure independent colored differently legend six mutual equitability regardless al plot determination equitability color legend six mutual smoothing mutual variance realization result use estimator equitability noisy mutual less likewise outperform information model vertical alone scheme mutual score identical identical reach behavior bias al maximization step compute equitability element make mutual information measure mutual six noise suggest large gain significant decrease runtime equitability analysis equitability due current algorithm issue theoretical practical research equitability arguably want relationship without call functional instance statistic would suggestion nsf nsf foundation figure contain legend relationship sample legend analyse figure figure perform function rather add uniformly function legend analysis perform model name refer number size perform perform name refer department electrical engineering computer mit division sciences mit mit division sciences institute mit edu engineering edu department evolutionary biology broad institute mit equally type equitability identify within opposed non association sift thus information coefficient analyze explore equitability equitability exploration maximization improve range noise alternative mutual say noise equitability exploration type relationship sort one find important type equitability relationship important emphasize exploration focus determine maximal relationship relationship force increase available dimensionality become new coefficient simulate curve maximal correlation therefore identify association contain association sift clinical datum equitability introduce equitability aspect utility normalization time equitability use tradeoff equitability mutual mutual necessary mutual provide address expand perform comparison set size mutual perform mutual almost noise dependence receive similar score note equitability hard define rigorously relationship e uniformly uniformly distribute interval correspond coordinate noisy equitability interpretation assign set recall maximal pair grid denote cell grid satisfy speed much programming computing method state instead value b equitability functional evaluate equitability measure consider noise equitability differently four model range characterize equitability extend contrast exist offer insight perform well utilize six different find equally spaced along equally spaced axis noise coordinate add use interval trial provide add respectively contain function assess equitability noise periodic medium periodic xx xy xy xx random generator non x x examine explore feature equation normalization mutual grid consider grid set order pair row true partition eq upper mutual grid comparison grid xy distribution property lie see
metric quasi diagonal produce algorithmic invariance expression metric work update transition build first network feedforward recurrent feedforward train feedforward carry explicitly feedforward metric describe start past provide insight adjust likelihood predict q attribute network single straightforward cover mini batch gradient summing gradient metric step important start singular sum activate unit pass activation use evolution equation hessian information frequency plus quasi diagonal inverse formula look use sum constant term always go pass activation use initialize derivative modulus unit time variant induction unit index unit make weight start analogy update symbol read update orient always loop frequency transition unit reflect combination range linearization adjust though change time find place integrate apply activation learn enough step update try multiply transition primitive costly line number write write connectivity comparable ordinary backpropagation training activity hessian modulus identical computing contribute operation matrix contribute inverting matrix gradient step connectivity quasi describe term remove maintain point costly invariance property write activate unit thus transition become compute idea behind valid recurrent backpropagation simple ascent reason gradient trajectory orthogonal change viewpoint maximize value view namely equality approximation clearly represent different amount chosen depend define ascent suitable improve cost move direction natural gradient design depend whether instance feedforward share idea metric build metric network network metric distribution define fisher leibl divergence close recurrent network output define next symbol kullback leibler divergence probability actual network arguably compression ascent change stationarity ergodicity assumption x however actual training give easy summing monte fisher successive individual alphabet endow distribution network symbol te exponential property respect newton write discussion interested change activity iy metric iy hessian sequence whose quasi metric non invariance affine whole fisher costly network hessian method recurrent fisher invariance transformation parameter backpropagation invariant feedforward build unfold backpropagation recurrent network norm follow neural network unit ordinary feedforward unit unit original influence recurrent influence unit unit recurrent network coincide time decide recurrent network ready recurrent variation ordinary feedforward ordinary feedforward actually gradient update project projection orthogonal amount make ill training time metric feedforward feedforward sum metric associate metric describe detail incoming parameter incoming unit directly activity unit namely use recurrent network equivalently fisher sequence since network average training weight proportional length relevant arguably metric recurrent right network induce network recurrent induce decompose influence consider independently still account output explicit form describe symbol include formula product detail recall recurrent outer depend individual outer product provide metric outer op ascent direction parametrization counter thus adjust logarithmic output op approximation use name fisher discussion op increment op increment spread feedforward op large op income orthogonal op metric still invariant activity block invariance recurrent definition metric op parameter give outer network activity usual time derivative computed evolution define equation j recurrent unit component derivative parameter metric expression find metric remark unit orthogonal term orthogonal currently read computational burden rnns evolution product handling alphabet restrict algorithm third unit bias always activate besides derivative interesting square metric incoming unit activity choice recurrent recurrent metric right reason op recurrent sum recurrent general outer square sum recurrent recurrent set op decomposition rise rank recurrent op metric recurrent op sum time th computing derivative recurrent feedforward define metric result activity unit unit turn sum recurrent network set metric unit unit influence activity result induction readily compute change iy q proportional line activity metric translate initialize output express simple I equation easily compute equation recurrent network find modulus evolution network influence sum recurrent metric incoming incoming equation I recurrent metric income distinct distinct rnn parameter incoming unit incoming unit match backpropagation invariant statistical parametrization rnn replace inverse bring rnns close gradient ascent present incoming stem depend network sigmoid activation result identical trajectory practice invariance invariance trajectory step step activity sigmoid initialization explicit change parametrization obviously change parametrization procedure formally nice preserve traditional rnn example alphabet music distant example finally benchmark either recurrent outer metric reference traditional naive distant rnn poor unless directly compare baseline rnn softmax internal training via plain diagonal hessian learn frequency symbol gradient rare frequent symbol activate symbol equivalent unit divide root gradient report principle namely symbol section uniformly symbol decrease unit decay combination model rnn benchmark rnn lstm accordingly output random value size cpu since plain result slow rnn writing adjust rescaling input slow table appear setup obtain result likelihood near frequently specific reproduce validation symbol alphabet regularization construction rnn generate exact sequence baseline exact check online present concatenation bit use identical computation series parameter rnns span show attain hyper size network unit unit increment test network edge per alphabet latter contribution way rnn hyper parameter allow minute ghz use code example concatenation line separate symbol line write alphabet order letter order nine letter choose concatenation randomization rnn range minute likelihood come ten close rnns rnns bits difference bit training represent letter log line sequence alphabet letter confirm inspection train sequence rnns qualitative learn line difference letter sometimes letter rnns difference log inspection attribute factor omit letter letter arguably generalization letter block network increase extent rnn long symbol alphabet stop run variation validation log invertible surprisingly overfitte increase past rnn trajectory stay run rnn long time partially rnn hour bit rnns rnn considerably sometimes phenomenon music successive separate bar separate half dotted note hide three iv bar specific bar iv iv encounter successive set encounter namely independently bar validation law rnns variety describe minute log roughly close inspection output network see generative confirm harmonic bar display mistake bar long harmonic approximate reflect remain running hour bit visual inspection rnn reveal possible bar present still variation although train relatively may symbolic fix determine instance concatenation separate symbol make bit bit symbol take symbol bit result bit typical training goal correctly bit include bit notation rnn literature hessian free rate instance eight concatenation optimization pass computation alternate discuss recurrent recurrent extremely report score line score always likelihood express binary successfully independent reach pass rate algorithmic pass conjugate implicit hessian reference approach experiment intel cpu ghz straightforward parallelism training line separate symbol length take validation sequence rnn describe table bit come free whole bit however length infer reasonable law bit thus attain long twice log value obtain bit bit surprisingly bit cluster around unit log visual inspection train generative consecutive close sometimes kind reach unclear build counter take nature unit rnns design backpropagation riemannian gradient testing rnns recurrent recurrent rnn contrary different symbol alphabet orthogonal directly alphabet metric activity train various hyperparameter report hyperparameter bit alphabet rnn rnn lstm plain rnn respect difference log likelihood include reference text file incorporate concatenation alone third markov model hmm network interesting model rnn diagonal writing parameter symbol divide frequency adjust backpropagation correspond rnn pure backpropagation slow table first improve rnn train aspect necessary bring performance long alphabet dependency network quite train method conclusion capture algorithmic dependency symbolic sequence inspire geometric viewpoint metric bring gradient seem difficult need investigate effect training procedure need investigation influence initialization expert dynamical multiple regime riemannian principle seem promise scalable online state signal backpropagation exclude linearized regime effect suggest initializations ascent equation contribution activate feedback loop feedback reaction unit instance weight e activation linearize dynamic fix attractive small induction linearize activation level past exponentially rate insight lead start learn lead activation non behavior indeed yield control effective window integrate much order reasonably capture change set yield order seem independently enough stay regime assumption stay roughly yield shift training namely symbol take time empirically seem decrease viewpoint weight reasoning activation derivative probability respect backpropagation sequence give derivative respect parameter give backpropagation respect writing transition relation include activate partial derivative respect introduce unit unit evolve change sequence variation ascent order expansion rearrange yield jt induction write represent value parameter namely level relation p tv find norm metric writing weight effect find hessian value consequently express term sc bx prop thm lem corollary exercise exercise powerful dependency simple handle hard train riemannian produce design encoding ascent neural network riemannian variety context intersection type distant adjusting parameter graph initialization consider probabilistic observe symbol prediction compression hide hmms frequently set hmms simple sequential represent instance intersection recurrent neural rnn modelling limitation pick long dependency remain problematic short ascent use riemannian backpropagation computational rnn rather backpropagation gradient riemannian geometry adapt recurrent context provide improvement keep complexity identical backpropagation connect production depend produce next state current state currently symbol like arguably lstm connection control activation rather set activation level next effect model free internal hold else text devote riemannian believe proper major ingredient ascent parameter increase possible small mean number differ move become benefit self adaptation move affect gradient ascent replace leibl distribution come great free allow approximate yield expensive recurrent natural present
e concave employ solution reweighte reweighte statistically reweighte outperform exist affect remarkably affect htp w tested test randomly test different htp normal w find sparse normal randomly sparsity htp distribution randomly pc section algorithm attract lot construct example reweighte numerical reweighted solution cardinality numerical demonstrate reweighte look vector define system many state follow closely deal limited cardinality constraint wide range component generalized version minimization year idea solve problem continuous approach htp I see convex li figure successive solve main convex minimization method obtain sparse several minimization one mutual coherence isometry unconstraine investigate literature refer weighted introduce diagonal penalty component penalty weight avoid infinity parameter choose proper challenge improper cause small penalty big penalty recognize discuss numerical experiment later reweighte introduce refer unify reweighte cardinality linearization define li several reweighte reweighte matrix algorithm method algorithm discuss type reweighte discuss numerical function frequently field recently li follow approximate concave separable twice differentiable f x c I concave function concave linearization method conclude side iterative reweighted reweighte solve termination criterion stop add element challenge reweighte open successive linearization general terminate number creates prove proved converge also li range reweighte converge certain reweighte ahead verify check property see hessian concave reweighte iteration verify q diagonal every strictly base reweighte demonstrate test sparsity solution choice summarize success sparsity fact x ix follow element imply reweighte find exact enhance concavity without monotonicity reweighte algorithm begin one condition level solution min randomly generate vector test distribution distribute parameter gamma mention min differently cpu ghz ghz ram memory choice crucial test update figure affect vary algorithm cardinality value perform even low start cardinality cardinality solution set completely min fail cardinality figure successful finding perform generate choose big I start bad min concave approximation find outperform outperform choice
leave intuition serve interpretable measure complexity degree combine many fitting df exceed phenomenon restrict case present phenomenon contour effective consider df overfitte strongly strict df simulate noise unbiased estimator equation linearity last fit df estimate deviation simulation code subsection seed design pick process remain seed almost code subsection imagine false false degree freedom capacity overfitte bias contrary intuition poorly exhibit various monotonic exceed total freedom ambient arbitrarily show degree freedom convex method observe ny variety freedom precisely measure compare procedure freedom predictor parsimonious full bivariate univariate word size freedom fit intuitive wrong predict df severe value allow take strictly complex hard figure df plot expectation response close axis far exceed corner later confirm grow expect df full understand intuition review df original degree freedom dimension role classical ordinary say residual freedom error projection orthogonal residual n variate vector project onto subspace regression degree henceforth df df coincide redundant constitute overfitte exactly linearly independent df quantify compute unbiased small test contribute sort penalty model procedure free compare quite algorithm various author definition reference dimensional intuitively freedom entirely eliminate popular define df justify intuition df offer way quantify requirement df describe algorithm belong commonly nest lasso ridge union hard inclusion imagine df monotone ht monotonicity guarantee df break surprisingly break project ridge df exceed monotonicity df discover thorough arbitrarily degree regardless discrete less parameter predictor general df relationship ordinary analogously df identity df mean zero fitting technique fix identically equal zero summing fit example vector choose df estimate code motivating model noise equivalently generate one mean deviation seven generate substantially df monte ht carlo estimate df versus least popular ol df size case motivate example design scalar vector making fall univariate response times df error play reveal df unbounded figure value clarity variance black dot constraint iy ht ia toy indicator function set term
eqs give correspondence coefficient correspondence joint verify distribution eqs reduce appendix simulation demonstrate particle numerically system drift eqs eqs sde dirichlet sde cm sde eqs eq sde eqs wiener gaussian streams covariance w eqs advanced euler stochastic generalize dirichlet different initial motivation fold invariant new generalize sde invariant change demonstrate mathematically coefficient extra generalized dirichlet coefficient yield standard dirichlet third generalize dirichlet figure dirichlet stochastic nm use joint probability stochastic variable differential equation equation unit ensure similarly diffusion ensemble dirichlet physical model general covariance develop whose statistically generalize dirichlet subject variety biology stochastic wiener process diffusion dirichlet dirichlet covariance process general physical process may positive covariance unit requirement necessary scalar read denote scalar isotropic wiener increment statistically solution system provide restriction towards interior space together specification ensure diffusion diffusion drift develop potential solution introduce dirichlet diffusion vector wiener drift stationary converge sec specify put possible form constraint specification positive definite root decomposition equation eqs way specify drift arrive generalize functional may eqs generalize determine stochastic may yield choose generalize unique sde correspond converse uniquely determine sde dirichlet generalize distribution set univariate case yield distribution zero outside scalar dirichlet scalar positively sign sign sign dirichlet univariate distribution q process drift set invariant whose dirichlet process kronecker process eqs multivariate process univariate respectively beta belong family pearson start stochastic satisfie principle
treat bar one segment cycle compose cycle n finish direct particular manifold h diagram cycle bar code whereas bar code acknowledgement acknowledge european ec contract project equation proposition equation definition definition notation topology topological persistence persistent homology appear fundamental study topological persistence space persistent homology naturally persistence diagram interesting various illustrate last decade availability device tool lead even life possibly usually carry reflect embedded euclidean space come distance happen come know position thank may case give distance space abstract carry structure geometric embed possibly concentrate around manifold therein recent direction however lie manifold fail euclidean metric may highly space difficulty inference algebraic topology important toward give birth infer qualitative topological persistence homology tool homology topological space homology encoding number cycle formal introduction homology persistent homology encode evolution homology family e set ball multiscale represent diagram persistent homology include bioinformatic cluster usually filter available nested family persistence diagram topological signature exhibit topological persistence diagram endow bottleneck signature different relevance rely ensure respect hausdorff persistence diagram remain restrict exploratory persistent homology consider general persistence diagram realization persistent homology filter datum associate persistence diagram define persistence probability assume denote consider persistent filter bottleneck persistence diagram consequence obtain satisfie illustrate satisfy moreover isolated relie theory persistence prove persistence diagram possibly filter finite persistence persistence drawback persistence diagram build computed persistence diagram stability prove bottleneck hausdorff persistence support persistence diagram easily hausdorff metric persistence diagram metric use le corresponding rate persistent homology remains widely establish persistence promise homology persistent homology topological persistent homology von know meet statistical homology persistent homology recently context manifold manifold result spirit persistent homology compact space author persistence diagram tackle problem connection persistence well past framework propose method review symmetric reference topological set support hausdorff bx particularly convergence difference hausdorff support hausdorff study various additional another classical lebesgue measure plug persistence diagram diagram reach topology deterministic deconvolution attempt study statistical diagram point persistence diagram persistence diagram persistence allow variance diagram persistence pt notion filter persistent homology section convergence diagram space classical section give yx algebra preserve namely isometry metric correspond intuitively hausdorff embedding space hausdorff rd metric spaces metric set probability need low exist open ball center reduce check satisfie standard last build top space nest family depend topological persistence grow complete filtered top set triangle possibly face notice neighborhood graph serve vertex ji bx empty pairwise embed close ball simplex ball complement ball homotopy ball consequence provide evolution topology union grow notably family rest inclusion call extensive persistence persistence behind persistence increase connect appear connected cycle appear homology tool track identify instance feature component old intuitively relevant formalize bit homology homology get sequence vector many decompose interval filter complex interval represent segment segment represent n distance replace hausdorff abstract space embed metric persistence diagram probability low bound convergence persistence diagram upper corollary establish metric space moreover isolate soon obviously persistence diagram ab situation scope refer reader estimator situation context whose manifold framework complementary topological present fully drive procedure method adaptive control set shape smoothness persistence diagram inference framework intuitive natural estimation issue estimator support support set draw compact bx two main follow fast density boundary easier possible prevent boundary detail persistence interested whereas assumption almost measure accord rate minimax set adapt constant enough infimum take estimator level estimation dyadic side length histogram estimator concern knowledge prefer simple subsection embed manifold estimation recently several consider context diagram noiseless upper bound give hausdorff bound persistence diagram give assumption large quantity reach integer assumption include dimension reach volume eq bx r c convergence correct propose know optimal persistence diagram persistence diagram assume constant depend infimum take diagram practice illustrate persistence diagram endow probability convergence section metric hereafter metric euclidean measure interval unit restriction euclidean endow sphere parametric figure metric euclidean endow parametrization shape space gray figure gray project subset circular endowed uniform metric persistence diagram geometric persistence diagram homology homotopy ball diagram one bottleneck diagram metric sample embed practically compute persistence diagram homology discuss approximated homology diagram distance curve compute persistence hausdorff distance bottleneck distance persistence diagram obtain slope exactly notice homology homology persistence diagram randomly sample figure sphere know persistence build homology persistence diagram randomly diagram plot embed sample multidimensional carry cycle structure reflect persistence diagram point notice probably visible homology diagram top sample right number expectation point homology diagram sample axis axis diagram build sample homology diagram axis bottleneck build point persistent homology mainly consider persistence diagram exploratory topological data framework statistical homology give study convergence persistence diagram result open rigorous persistence consist recently persistence persistence diagram density sup develop persistence diagram direction persistence diagram space diagram call promise
code file test intel cpu tolerance default calculate dual objective eigen computationally embed calculate eigenvector pose iteration within eigen decomposition sparse structural efficient calculate multiplication fast descent become c turn small next accelerate considerably primal optimal discretized separately solve step user objective dual decomposition row descent bfgs spectral eigenvector interior point fast sdp furth eigen method long implement even application vision equality quadratic problem spectral method separate weight disjoint equal affinity degree classic c solution eigenvector original succeed density conventional sdp constant sdp discrete round method generate discrete obtain high show relaxation contain second maximum fail offer satisfactory loose achieve ccccc w problem result iteration become impact vertex sample half lead objective conventional bind sdp frobenius decrease far optimize objective value price slow convergence need condition graph graph range graph speedup graph score cccc times w berkeley toolbox affinity construct color similarity histogram spatial extract foreground group use marker foreground segmentation result compare time five sdp image perform simultaneously segmentation traditional recognize object co conduct criterion spatial separability foreground background sift denote sdp program find column formulation express pixel w n inter discriminative whose th affinity image lm sdp employ recover thresholding comparing image car front car image handle large standard minimize car back large score face co segmentation experiment illustrate l car car front co source match target match problem express match h source row formulate e avoid undesirable solution multiple formulation integer formulation toy firstly randomly translate stanford similar match toy time fast improvement previous reason formulation impact ccc toy toy produce bind conventional sdp spectral formulation toolbox vision demonstrate flexibility efficiency acknowledgement arc future fellowship ft correspondence formulate quadratic relaxation bind loose relaxation tighter present sdp desirable first sdp formulation conventional sdp efficient scalable spectral segmentation usefulness scale vision binary problem problem field mrfs semidefinite relaxation spectral convert eigen simplicity variety mrf loose poor case hard sdp tight method subgraph co mrfs disadvantage poor paper sdp achieve high sdp solve virtue solve quasi formulation similar sdp produce estimate relaxation apply formulation equality inequality constraint application area relate frobenius play simplify focus near neighbor interested arise vision sdp method find locally run fast interior co segmentation method achieve speed sdp trust accommodate problem simple globally notation bold capital letter case letter symmetric p wise inequality rank define nn nn x eigenvector onto cone sect efficiently simplify sdp spectral eigen decomposition relaxation often guarantee optimum poor relaxation verify author furthermore generalize method although equality additional hard semidefinite program p drop convex sdp relaxation prove value sdp advantage sdp constraint transform sdp scalability interior solve sdp impractical intersection extension one spherical
indicate penalize train bic lowest highlight bold regime penalize np sample due mm mm mm mm mm assess zero assess bic regime good train th mostly precision exception unless mean precision non analytic heuristic regime mixture box rand show dataset regime heuristic perform approach regime compare heuristic full rand heuristic behave heuristic remain agreement offer substantial gain heuristic exploratory analysis present base cluster penalization penalty method penalize model breast draw recommendation incorporate together bic select tuning exception tune train cross provide regime cv recommend penalty find sparse estimate cross slight gain dimensionality standard size result penalty analytically penaltie proportional proportion intuitively appeal size however cluster indicate sample mostly cluster cluster penalty lack convergence penalize incorrect clustering label relate em penalty offer although recommend perform far understand behave specific could pose difficulty discuss propose correlation inverse correlation focus propose selection cluster cross employ explore necessity gaussian model cluster high dimensionality indeed see benefit penalization already encourage sparsity lasso require alternative estimator estimator bias penalty non penalty precision penalty intensive dag especially biological meaningful hill extension idea cluster dag rather undirected extension graphical approach prior knowledge available joint model explicitly agreement current ep u cancer biology center grant scientific cm cm centre science ex department ex ex many sample may previously heterogeneity cancer biology differ molecular discovery specific challenge enable analysis forward whose graphical bring base clustering cluster penalization regime recommendation inference statistical decade increase motivation effort molecular biology variable together group together simultaneously variable together focus moderate high biological notably expression comparison various mean cluster popular root area attention year structural graphical comprise describe refer edge structural molecular gene network model review develop simultaneous structure question concern heterogeneity question arise differ implication heterogeneity understand partition base however practice molecular classification uncertain moreover latter interest disease differ couple network structure hierarchical model lead assignment underlie equally heterogeneity use graphical model set rich matrix graphical model propose setting provide review greedy backward entry recent regression perform sparse subsequently maximum estimation infer shrink encourage since precision correspond graphical suit molecular challenging large estimator behave add mixture formulation put root carry empirically formulation em penalize likelihood approach level show size particular offer focus variable improve investigate regime control sparsity precision result regime difficult choose priori result suggest recommendation remainder organize follow penalize graphical model base cluster propose regime tune selection finding area graphical conditional conditionally precision j identify location suppose ip f trace invert however estimate poor yield sparse place precision matrix follow tuning control penalize convex algorithm semi programming employ cholesky refer interested proportion unknown likelihood give likelihood expectation present distribute cluster represent order relation overfitte concern precision issue graphical parameter form term dependence proportion penalty cluster mix novel set analogous penalize mixture cv log datum define degree element cluster precision search bic prefer less value approximate rely cluster largely proceed first randomly produce parameter pseudo mean graphical use consider simultaneously third cv bic maximize minimize multiple value average consider scheme assignment second specific specific specific less concerned cluster structure consist precision base create zero everywhere randomly take create choose standardize result half share structure euclidean distance mean consider size reflect scenario cluster substantial difference display heterogeneity share across assess ability correct assignment simulate bic log independent test sample matching training regime table cluster graphical algorithm assignment function toolbox initialization matlab iii namely carry penalize likelihood penalty cluster analytically external package regime cm bic test hard assignment bic analytic cm mean km penalize dimension five cluster tuning value deviation parameter correspond large value regime grid increment bic high regime penalize approach np due sizes rand rand cluster take disagreement box simulate regime regime consistently cluster large size bic good clustering rand train low converse dimension mixture penalize well corresponding due tuning supplementary take uncertainty assignment interestingly penalty b
rx restrict estimator restrict substantially neighborhood set partition restrict formulate restrict dag r describe among additive structural additional identifiability set q function times derivative condition fx approximated hold structural lemma say one without study near linearity structural converge require note implicitly translate nonlinear close error identifiability statement model assumption quantity constant error variance approximation constant denote biased truth justification correct give second problem carry automatically assume reasonable nonlinearity additive believe also additive long nonlinearity kx n slowly identifiability due nonlinearity make hard exhibit less classical prediction establish potentially expansion truncate either appear mle cope dimensional consider allow notational often drop sub index target j j h h small require effect article follow maximum permutation define follow select tend ii screening assumption lasso penalty use basis condition compatibility beta min condition basis function compatibility identifiability exclude additive structural assume eigenvalue see finally typically weak require moment finite sum function likelihood j function appear k obtain theorem additional invoke bound require probability uniform convergence analogously learn dag observational implementation procedure feature discuss benefit preliminary neighborhood causal parent edge select considerable robustness misspecification structural intervention simulate randomly otherwise connection draw rbf deviation uniformly without standard repetition differently true leave unchanged provide author consider intervention see quantify correctness order infer count true dag dag permutation show eight connect step pruning consumption particular different address infer observational greedy equivalence conservative subsequent latter significance independence apply outperform method become dense number edge vary pc dimensional result function edge sigmoid close difficult process sigmoid identify assume expand dags class compare structural hamming true value disadvantage nonlinear discusse least focus identifiable assumption data additive experiment method case examine j noise simulate gaussian rbf bandwidth whereas parent figure expect edge dag algorithm truth algorithm become apply microarray concentrate observe dash indicate causal pathway network undirecte direct acyclic score interpret also record consider score suitable false positive gene pathway agree pathway finding prior good scoring edge additive causal additive causal underlie dag observational dags estimation substantially causal maximum misspecification develop computational variable empirical accurate structural dag identifiable observational closely additive structural adapt see permutation sparse autoregressive class dag hide structure allow unlike closed marginalization acknowledgment discussion issue subspace allow people european union fp grant agreement foundation pa dimensional additive structural key among acyclic encoding address sparse substantially problem search consistency allow misspecification class infer causal area thing size grow super exponentially major challenge generic tool sparse cf successively establish recent precisely acyclic causal hide causal diagram direct generalization hide unobserved formalize model concept equation equivalent true place restriction structural equivalence class dag causal model nice parameter structural equation general address variety procedure independence latter easily regard former linear model strong propose selection structural equation follow briefly potentially regression formula latter understand additive additive remain variable via mid penalize dimensional preliminary estimator entirely regularization generic within dag joint estimation structural level mainly high preliminary additive search restricted equation employ step search restrict skeleton section variable regression additive structural equation structural attractive derive testing propose develop maximum estimation structural gaussian fitting often practical presence additive present estimator know simple consistency equation selection consistency fast treatment high new denote set parent dag causal dag interpret absence issue allow unchanged general identifiability difficulty dimensionality although special difference respect nonlinear since stay nonlinear three dag exception identifiability fully case write active infinite variance true parameter correspond use statement slight abuse specify clear constant enforce whole function fx kk additive variable argument require function requirement depend occur function drop index cause later assume closed respect lemma analogue eigenvalue space distribution variable induce order sequel search thing define correspondence permutation connect dag variable parent node dag dag fully connect permutation typically autoregressive permutation identifiable low representation provide order gaussian even consistent sense true variable consistent class principle guarantee decide quantify closeness linearity important beyond scope work sequel helpful underlying nonlinear permutation consider projected form wrong project parameter obtain true permutation dag lead minimal lead would lead divergence project allow lebesgue generate nonlinear would describe misspecification wrong case gaussian situation require low restrictive approximation weak gap condition involve assume I I depending sometimes denote similarly g permutation minimize negative estimation function practice basis knot spline twice nonlinearity sufficient assumption ensure super dag dag j k estimation intervention dag efficiency estimate
semantic similarity term underlie formal identify underlie problem far scope section unlike thus semantic focus aim directly problem concept rigorous semantic currently beyond human remarkable capability understand partly ability precise understanding still intuitively semantic processing text heavily rely knowledge experience ai adopt know create computer program indeed prove see semantic lexical database specific exception procedure hand semantic background corpus interesting technique section motivation work realization consensus people relative many intelligence location generally strongly people may benchmark preference vs pair example annotate scale probably different double result semantic importantly ranking semantic semantic make impossible satisfy unsupervise hand moreover entirely meaningful outperform benchmark mention base wikipedia inferior semantic instance encode capable determining unobserve erm learn co occurrence background label result hyper corpus experiment show notable benchmark background corpora book literature assess semantic quite present world expert knowledge element semantic technique type lexical database project mid method corpus open project meaningful text categorization finally collection refer type elaborate benchmark greatly influence research collection currently quite representative annotate human list word pair common consist similar one r consist score consist semantic score evaluate similarity semantic relation past lexical corpus lexical maintain cognitive laboratory english lexical count manually serve lexical imply state link annotated lexical lexical relation certain score frequency include content ic set direct sr two contain attempt combine relation weight lexical parent information semantic two sum contain utilize measure calculate semantic short ii iii li three nonlinear definition measure calculate function occurrence raw vector node couple tag speech tag token stationary token contain ii token contain cosine measure newly kullback leibler propose measure relation weight consecutive edge semantic accord ps lexical coherent phrase mean term chain meaning refer study lexical six short wikipedia source article derive path hierarchy common propose result achieve term distributional wikipedia article tf article semantic basic example distributional currently frequently subroutine e utilize wikipedia two link cosine function tf measure utilize apply select represent article method call utilize wikipedia whose article edge accord dictionary cosine semantic add concept corpus representation document divide epoch day corpus compute temporal kolmogorov structure normalize occurrence google entire engine rough occurrence web occurrence singular svd compare meaning achieved quantify statistical co occurrence jensen shannon divergence estimate semantic free corpus appear tf feature cluster centroid semantic accord centroid combine centroid clustering know subsection publish whereby corpus technique human supervision utilize work follow methodology whereby instance generation machines svms regressor hybrid wikipedia addition google search learn comprise wikipedia base score google score employ genetic approach grid calculate four occurrence overlap coefficient syntactic template g derive count retrieve engine al whether take similarity function location boundary binary determining method feature use rank consider lexical semantic overall vector rank use classifier obtain correlation free achieve web document computation utilize core core attempt construct report consider short another co measure combination score correlation set extract term pair hyper precise report fold cross validation utilize available training summarize among work close formulation learn problem however consider sentence whole term phrase corpus new york ultimately goal automatically construct function correctly rank accordance semantic require induce complete reality two preference cycle perhaps make confusion impose hypothesis learn term pair related term pair otherwise along call restrict attention reason preference pair trivial denote binary context satisfy anti symmetry label classifier class whereby label unknown reduce quantify preference choice strongly answer answer quality extract score implicit scale mention justify accept semantic term occurrence document co occurrence major co require preference accomplish weight reasonable derive supervision fit rough refine preliminary examine occurrence index kl jensen divergence semantic base publish normalize implementation use wikipedia corpus effective appeal algorithmic complexity principle utilize semantic preference construct preference user allow assign impose constraint corpus constant q weight term manner namely encode coherent hard impossible common degree unlikely context e capture empirically question pair similarity relation ad hoc adaptive follow monotonically increase contexts weight algorithm utilize require property occurrence apply rely utilize erm learn appropriate consistent factor learn bad context resp follow weight accordance incur q decrease prevent semantic gradually observe decrease use iterate hypothesis exceed risk minimization minimizing initialize normalize require relevant occurrence classify bit gb wikipedia hash gb ram normalization context weight therefore due example bad scenario denote maximum semantic maximum exception iteration iteration necessary division certain normalization normalization min case case mainly number error assume computing computation classify st st st total training term negligible iteration effectiveness design lack benchmark attractive label human world involve even vocabulary experiment merge vocabulary resource train negligible preference annotate verify achieve leverage dataset want semantic check annotate dataset vocabulary write english text application size consist frequent corpus text project context call scoring project grow repository classic web book old project text try old merely purpose mention believe version prevent reliable generate score positively human together semantic score preference score definition text without modification despite human evaluate absolute see accomplish extremely training snapshot use old wikipedia available mention previous article filter article incoming corpus mention emphasize experiment wikipedia corpus wikipedia experiment ignore consider context whole preprocessing conduct preference preference preference choose fed output hypothesis consist weight include hypothesis calculate accuracy ground report time report calculated error mark realistic frequent wikipedia preference preference associate preference learn curve available mark corpus paragraph ii filter snapshot unsupervised score implementation successfully accomplish experiment take precisely rise high performance believe main large utilize preference line mark report supervise evaluated performance benchmark propose conduct label serve check score systematic supervision score obtain method svms horizontal line know supervise curve corpus mark level mark evident rapid wikipedia wikipedia enable internal panel experiment meaningful train partition size experimental available preference outperform well supervise wikipedia consume preference project horizontal mark supervise internal panel paragraph logarithmic depict internal sentence resp upper horizontal resp mark unsupervise resp resp paragraph context context consist perform poorly extent utilize still behind wikipedia show picture similar indistinguishable examine semantic pair score calculate wikipedia paragraph base corpus preferences wikipedia preferences wikipedia corpus wikipedia preference evident successfully train relation among prominent semantic strength accept via distributional typical term whenever vice versa contrast compute similarity score tend co occur question handle examine specialized semantic similarity similarity horizontal mark horizontal mark depict low horizontal mark line mark dataset depict result mark mark figure indicate achievable task distributional similarity occur co leverage co occurrence sufficiently increase encode insight weight arbitrarily optimize organize interpretable something human something interest try answer question study answer suggest semantic interpret utilize wikipedia extract wikipedia uniformly wikipedia corpus annotated preference article paragraph semantic preference uniformly preference semantic semantics rest wikipedia corpus paragraph level well examine player music music game release play replace heart player join band fan song run house score production mix sign topic consider music observe difference identify experiment target exhibit top term target semantic quite topic inherent generating wikipedia context belong wikipedia initial hypothesis learn semantic increase decrease aggregate labeling topic music aggregate increase music mathematical dramatically summarize merely organize semantic distribution topic reveal topic like classic consideration ensure sufficiently expressive avoid course result theory perfectly set training vc e conversely example necessary determine context whose hypothesis anti nothing gain essence pair vc appendix b permutation substantial improvement hypothesis huge resource preference small training vc dimension induce capacity place many permutation include semantic co
align well ordinary ols ridge l correlation transition setting performance design set experiment var medium var set family j p p correlate figure transition simulation structure var ridge var average replicate qualitatively overall accuracy ordinary least penalize fairly reflect ols ridge choice outperform l prominent ls prominent strongly correlate accuracy suggest accurate improve term fitting reflect ordinary ordinary signal ordinary block apply ridge ridge regression favor theoretical estimate generate stationary limiting include categorical predictor heavy concentration expectation entire direct implication control spectra potentially although order develop low process concentration non topic development v x ensure sufficiently start extend vector next contain closed follow j deviation combine final v high setting proposition v choice j v ensure v v derive number positive domain meaningful process decay condition analogous treatment frequency nice boundedness continuity etc spectra existence boundedness value transfer transfer excellent popular implication process stationary center follow form harmonic conjugate discussion strongly mix gaussian refer reader condition continuity jump violate impose restriction theoretical property stationary sparse refer result strong smoothness assumption spectrum functional dependence measure commonly process assume absolute satisfied decay verify rely stable automatically assumption temporal way transition assumption necessary assumption violate stable whenever derive finally b bound var panel var var symmetric var stable unit eigenvalue lie circle addition symmetric var whenever imply v c z ij eigenvalue diagonal entry ensure eigenvalue inside cone sparse cl conv close support support z u u u v union quadratic comma section thm thm proposition thm remark scientific economic involve dataset study high primarily distribute stable process investigate correlated matrix autoregressive var derive asymptotic bound estimation establish via regularization sparsity key technical stability establish dependent regularize technology increasingly structural forecast large gene course microarray volatility finance co activation human use analyze moderate point classical meaningful setting often without generate space high var notion often penalization variant estimate scale numerous year key assumption estimate series exhibit cross dependence incomplete challenge dependence affect measure stability framework regularize key predictor correlate log upper perform estimation mild stability rate affect introduce model highlight literature var high accord lasso theoretical property several regime form restrict eigenvalue rsc deviation sub mild regularity observation consider provide comparison assume establish validity var major establish validity class stationary result deep insight represent popular model apply finance simultaneous observe var capture series var instrumental system identification recently tool connectivity brain region formally vector var lag uncorrelated form possibly correlate innovation main var matrix transition insight amongst forecast var natural problem since grow parameter stationary rarely estimation carry multivariate dimensional matrix resort penalize stochastic important multivariate least square play important process make dimensional require violate least square base choice dimensional scaling stable interestingly latter lead fit measure theoretical behaviour regularize estimate capture dependence spectra phenomenon literature autocorrelation spectrum vice versa interpretability since allow expression core deviation dependent establish serve large help independent enough integrate theory mechanism result via hard thresholding penalty mcp structure nuclear minimization remainder organize demonstrate series stability stability bound subsequent analysis error lasso var examine square regularize var extension current framework problem estimate var proof supplement number cardinality denote denote unless spectral frobenius coordinate maximum absolute v conv write whereas analysis quantification dependence impact estimate condition process recent series mix show lasso capture several var assumption restrictive violate stable var importantly assumption beyond generate error come upper triangular diagonal diagonal band process predictor change error multiple asymptotic infinity however error capture via decay even exceed new cross absence cross exhibit regime show process behaviour moderate seem dependence significantly error function exist eigenvalue write underlie existence density density supremum satisfy invertible expression invertible value plane stable invertible appendix unstable representation gaussian quantify spectral density insight peak process function circle large stable measure stability satisfy consequently cross density define satisfie spectral stability study var also low circle capture dependence value crucial role high invertible function bound case essential reduce minimum continuity expression stationary follow accordingly particular radius absolute eigenvalue behave var quantity matrix eigenvector concentrate single assumption construct deriving estimation dimension underlie process covariance provide generalize univariate present similar toeplitz eq analyze regression estimation entry expectation sparse establish center series constant sparse process satisfy proposition employ technique theory matrix n ir note support since also n v establishes note separately deviation view process separately apply argument lead concentrate term density f w cauchy product apply stability stability x h h cross il f combine upper bound establish regularize around establish dependence capture recover datum however believe although exact asymptotic demonstrate dimensional effect tight sub tail interesting phenomenon regularize estimate way bernstein presence norm affect additional sup f coincide spectrum flat strong temporal spectrum ar come back behaviour tail low know chapter approach approximated q term process behave estimation strongly dependent term offset tail behaviour dependence prominent reflect presence correlate error deviation derive estimation result long assumption regime converge regime observation interestingly estimation dimensional commonly roughly appropriately vary proof lie cone whenever trivial restrict predictor constant n assumption fairly mild invertible replace evident sample require demonstrate design column independent case identically spectral spectral measure density second condition consistency coordinate concentrate around deviation uncorrelate technique large serial exist constant inequality concentrate proposition allow regression pc eigenvalue thresholde lasso dependence contribute additional fast thresholded enjoy rate shown assume beta min negative beta min initial aware next study assume var transition error error satisfied stable process consequently consistency assumption provide assume seem exhibit finally paper dependence mild moment condition functional predictive certain decay another representation hand boundedness satisfied process problem author square presence among improve loss incorporate square likelihood either condition verification discuss realization var model unit process spectral deal eigenvalue var uncorrelated simplify factorization model encode dependence separate ml direct correspond representation var construct var l practice ls estimate discussion estimate two penalize motivate general note var ordinary least version follow condition modify restrict eigenvalue stable long size symmetric condition curvature tolerance behave concentrate population mean expectation precisely penalize condition deviation bind thresholded variant mahalanobis govern set process size ii curvature tolerance bind rate temporal dependence affect internal detail quantity derive var bind var proposition assume realization probability realization accomplish proposition realization generate process constant probability insight mention early small clear constant
since lebesgue formula respect finite sensor work observable additive latter inverse differential choice two large dense contrary specify inverse solver pde operator operator smooth green operator three pde write follow control variation correlation prior posterior eq lagrange discretization parameter product denote inner definite element convenient notation endow hilbert operator sequel mapping n r endow euclidean inner correspond discretized discretization observable discretize discretized major component svd surrogate randomize opposed product aspect particularly large scale vector expensive pde see make dense usually decay cluster implicitly estimator provide trace see make large trace via carlo entrie possibility gaussian vector identically entry refer extension dimensional mathematical describe inverse finally inference accomplish trace present dimensional physical lebesgue denote expectation trace eigenvalue eigenvector function obtain optimal namely random field inversion parameter design well proper target sensor dimensional location associate discretization different classical formulation promise place inversion experiment inverse repeat representation physical phenomenon control thus prefer absence sensor solve combinatorial employ devise procedure introduce bayesian inverse since data weighted diagonal dependent collect candidate diagonal posterior covariance discretization particular discretize discretized measure posteriori minimization coincide call hessian independent assumption easily accommodate matrix formulate optimal previous trace posterior additionally penalization hence control cone fact penalty penalty compressive sensing adapt approximate design interpretation design namely minimize trace square mse mean average refer bayes mse concept frequentist unknown frequentist point completeness relation average mse derive hessian section trace gradient subsequently computation finally sparsity allocate sensor hessian play objective sum consider pde map stationary special sensor evolution discretize lagrange element instance integration pde observable space pde operator design sensor sensor weight write sn sn decompose different sensor location decomposition reveal identity estimator recall hessian symmetric linear trace section trace functional appropriately choose justification trace product consider use compute denote spatial summarize evaluation multiplication computation evaluation observable realization application numerical solve despite demand infeasible surrogate observable map efficiently exploit pde limit thus surrogate involve suffice smooth property prior usually bayesian fast decay pde speedup svd rank require convenient implement computation vector r equal r large us approximation due use instead need use low surrogate gradient adjoint pde solve compute compute j close remark concern computation involve pde require solver via inner product utilize application interval spectrum fast clustered eigenvalue cluster degree indirect practice sensor discuss see due whose interpretation sensor unclear practical solution vector vanish weight binary weight penalty define zero since penalization function origin namely order continuously differentiable penalty function value approximate height axis xlabel ylabel font nod legend pos east thick table thick color dot txt txt cope potential real number unchanged decrease topology structure optimal seek characterize absence optimization relax successively penalty outline rest design via numerical design model study measurement pde domain boundary outer face internal boundary map map condition spatial temporal diffusion equation u coefficient problem velocity show figure c problem side driving pressure right everywhere see e ccc rectangular field dot three gray block remove velocity field sensor location black physics initial evolution two leave middle correspond operator sensor infinite diffusion observation operator measurement evaluate discrete observable instance utilize additive mean characterize minimizer functional minimize deterministic inversion next adjoint observable adjoint adjoint triangular continuous space euler adjoint adjoint follow due large diffusion need factorization euler compute forward adjoint equation triangular solve problem build solver evaluation compute derivative bfgs summarize test scalability contain place sensor triangle parameter freedom inversion final interval discretize euler unless sensor take space specify compute counterpart lead rank approximation diagonal assign observable rapidly mapping accurately ht xlabel ylabel legend font legend south west txt width axis xlabel legend style font dr dr txt table dr dr txt ex depict spectrum sensor influence investigate prior right spectra lie counterpart sensor affect number sensor potential grid sensor grid correlation neighboring cm axis xlabel legend legend pos outer north txt txt txt txt discretization step sensor place correspond vanish numerical interior solve vanish ex dot place sensor vanishing surrogate observable repeat pde solve influence solve figure surrogate low surrogate even observable little influence design height xlabel rank ylabel dim rank interior mesh interior solve dimension fine require encounter affected sensor result table interior quasi iteration sensor attribute type mesh decrease regard candidate conclusion low surrogate number insensitive sensor design respect design nest weight binary figure weight monotone explanation decrease neighboring location merge weight width xlabel ylabel style font legend pos south east mark size mark txt green mark pt color mark table txt color mark cm square convergence obtain evolution decrease illustrate decrease design compare pointwise standard employ standard obtain ht ex manually sensor sensor correspond deviation design different design sensor arise design compute design strategy compare design choose differently report exact design report design additionally collection sensor configuration design consistently design design observation return sensor increase decrease function value exact trace trace estimator report reasonably random approximation compute sensor influence design design trace variation optimal computed trace reduce conclude impact width height xlabel sensors ylabel style font mark mark x tr mark mark tr tr design trace estimator visualize frequency candidate part optimal trace middle trace estimator sensor part decrease accuracy trace sensor trace empty dot sensor estimator sensor value width xlabel ylabel tr legend style right mark mark trace mark mark trace txt color mark x trace versus sensor red dot dot design dot dimensional applicability problem use show black dot collect equally diffusion discretized mesh degree freedom implicit euler step integration ht field random deviation blue find observable adequate large translate large use vector observe converge application iterative compare application interior iteration initialization follow auxiliary problem newton arrive drop quasi decrease interior point problem design note place illustrate effectiveness volume pointwise exploit optimal experimental design infinite govern numerical indicate measure forward pde solve candidate experimental consistently improve limitation linearity observable applicable approximated linearization distribution efficiency depend observable map rely admit indirect however combinatorial sensor computationally extension consideration experimental meaningful dimension observable observable map particularly linearization observable general parameter unique depend acknowledgment application root result randomize argument regard mapping also therefore eq equality follow thus freedom dimensional inner product dimensional inverse datum brevity variable linear
c hypothesis k z long true center composite center z z check equal neither two possibility factorization nonetheless surrogate rejection emphasize need center integral operator sum sum also appropriately useful hold na b n imply center k nk kl l symmetric nk l k l thm com uk definition proposition kernel nonparametric test three independence reproduce statistic straightforward powerful interaction alternative family kernel cause influence third effect strong influence especially suited model outperform compete nonparametric test detect structure widely much measure pairwise hilbert schmidt covariance canonical ask interaction become involve mutually consider question mutual independence implication I mutually triplet insufficient interaction occur two variable third whereas individually study three switching mechanism negative gene control third presence typically form variance false order broad modeling knowledge three independence embed appropriate sign reproduce third moreover test structure graphical interaction employ conditional pc structure detect pc structure markov absence original pc algorithm partial algorithm nonparametric independence test test test early variable individually begin presentation sign measure rkhs define may experimental benchmark matlab multidimensional value sign whenever trivial way coincide notion understand sign bivariate difference joint case x correct partition implication presence interaction possibility converse generally appendix important distinction absence interaction total absence interaction sign vanish hypothesis test total embedding sign measure rkh suit take value moreover remain valid euclidean valid positive detail testing test base characteristic kernel hilbert topological accord definite reproduce kernel hilbert rkhs denote banach measure extend kernel kf embed straightforward show sign inverse relate embed measure hilbert notion inner sign since np xy l np xy call schmidt independence criterion product alternative generalization energy article extend formulate test argument test test consistent throughout hadamard matrix follow row denote sum overview rkhs sign expand xy product np xy xy mean gram estimate p even second triple use dominant compute overall simple centering whether treat single product gram interaction similar address variable derive k possible express certain expectation gram summarize result derivation algebra exercise normalize inside three variable l lm ml lk mkl lm individual rkhs product derive various sign arise measure measure yy yx measure incomplete unlike center mean interaction k proof proposition appendix summarize hypothesis statistic demonstrate particularly x hypothesis independence vanish another interaction statistic moment interaction give rkh let n correspond independence compute time fix correspond lattice interaction correction moment capture notion construct embedding rkh norm statistic analogous avoid yield prohibitive analogy coincide high order neither moment general I ab ab discuss characteristic characteristic function invariant space similarly permutation dataset triplet random vector px p p dimension case pairwise dependent triplet increasingly difficult detect dimensionality independence factorization vs two triplet px pz use permutation test kernel median acceptance expect dataset pair detect dataset however appear significantly independence outperform apart dimension figure plot factorization z correction correction ii hypotheses structure nan structure appear interaction compete permutation nonparametric dataset pairwise case detect pairwise instance variable embedding sign multidimensional readily
theory correspond indicate counter theorem follow give generate increase follow tb j converge sequence also norm fix nevertheless nesterov present single newton per adopt composite barrier nonsmooth lie index homotopy proximal go solution selection describe procedure give perform newton pn k inexact pn k analysis first pn follow maintain assume inexact see start choose sequence converge next quadratic convergence region show parameter update eq increment consequently force preserve path theorem goal newton method proximal newton trial follow subproblem subsection give inexact proximal j newton j lemma inexact newton deduce deduce j fulfil choose c mm mm compute approximately subproblem indicator efficient optimization rule ii worst kt ft stopping criterion late section subsection analysis since bad give require iteration require f induction iteration phase induction lead round side bad analytical phase constrain self barrier relation follow proof find e feasible let eq consequently approximate suggest k k algorithm converge update apply solve stop give accuracy solving sequence ii analytical ii algorithm show hand kt kt case aspect next algorithm standard convex numerical example self ip solver application norm concrete optimization track approximate parameter fundamental issue warm main ingredient solve algorithm atomic subproblem structure observation exploit costly part exploit accelerate quadratic efficiency warm strategy solve distance suggest initialize warm replace acceleration k k derivation update quadratic broad nonempty nonempty closed endowed barrier equivalently convert concrete constrain follow program symmetric proper semidefinite cone endowed barrier possible inequality problem cast lagrange multipli equality write provide us newton direction coincide system standard consider retrieval upper approximate trace norm rank matrix g example test interior solver terminate generate size generate l test three report platform intel ghz ram sdp number rapidly consequently computational moreover slow transform problem standard clearly computational interior point solver enhance carefully implement strategy example solve problem unknown briefly vertex minimize explicitly combinatorial propose relaxation pose significant difficulty scale call max constrain correctness state art rule terminate respective solution return r pf pf self curse impossible execute large within reasonable scheme ht pf constraint thousand solution scheme parameter often accurate solution advantage vs handle avoid high hence cf number memory requirement fashion second solver fista obtain medium close warm scheme fm present state art non splitting code publicly code stop tolerance report realization complexity compare theoretical naturally self poisson learn satisfy conditional dependency hence covariance turn still easily formulation estimate tune obtain barrier subsection e g knowledge respect purpose homotopy multiplicative practice solution traditionally exploit guarantee continuous go though consistently adaptive regularization parameter pick range desire approach two size vs curve convex pareto pareto approximate trajectory apply newton five point relative table sparsity inexact framework minimize smooth gradient constraint admit barrier show slack modular subproblem tractable term proximal self maintain analytical loop remain interior subproblem scheme globally smooth involve nonsmooth constrain programming problem path self solver grateful anonymous thorough comment improve support european grant provide proof technical f tp gs estimate provide apply substitute result deduce substitute right check hand elementary calculation estimate provide substituting eq k q convexity last inequality since self last statement definition problem substitute rearrange j j deduce concave barrier function obvious use optimality property barrier optimal letting next exact moreover optimality g q schwarz k inequality together cauchy q follow combine provide sum ed mail nonsmooth minimization broad nonsmooth equip convex constraint solve without need dimension propose path bad subproblem show framework application interior objective tune inexact path tractable proximity follow include machine processing nonempty close set smooth n problem sufficiently mild mirror subgradient theoretically global counter size impractical mc nonsmooth optimization bundle potential candidate subgradient bundle scheme global nonsmooth approach sequential constitute usually require ensure algorithm interior exploit conventional feasible set parametric composite barrier decrease trace analytic central path solve hard self sequentially newton effort frequently application norm spectral atomic norm etc problem ip solver via nevertheless suffers curse concrete seek matrix rigorous formulate unfortunately add slack g nesterov semidefinite cone create memory sparsity dense kkt newton systems gradient proximity definition definition use proper e f ff twice function define clear cauchy convex strictly important self n self barrier degenerate contain self self equip self barrier nx f nf definition ready
comment lipschitz continuity boundedness function consequence verification continuity whenever satisfied quadratic follow denote suppose come iterated margin verify minimizer fix continuously around derivative remainder represent second partial derivative nf cx differently conditional margin encounter sequence q modify slightly probably analysis fit rely loss function bind lipschitz inequality hold probability covariate heavy state note loss function turn reveal satisfied quadratic margin even technique independently necessarily calculation modification note norm identically actually distribute argument satisfied sample identically one valid sake exposition shall situation l ef population covariate c n assumption theorem lemma lower seem particular covariate old terminology derivative old exponent great integer strictly old case consist degree polynomial possibility assume rather transform alternatively one case show ef state imply previously hence two validity existence properly shall compactly old choosing consist th polynomial hence furthermore ef bound restrictive either accordance since smooth nc n consideration choice ensure probability nn pn easy course desirable full rank reasonably eigenvalue part lipschitz impose boundedness trivially satisfied continuity working cover continue covariate compactly absolute mention assumption rather arbitrarily sub prevent heavy tail quite high subgaussian belong old support though sub assumption term one stress much develop covariate error enough apply precisely slight price pay increase generality tends exponentially point series estimation case series unknown function aware classical apart estimator another estimator e term slow increase exponentially precise require increase already though slightly considerable put differently location concrete usual inner hilbert every eq series f jx ss small put differently clearly considerably structural estimation combination upper part assumption p assume covariate support belong consist polynomial compact sub c remarkable finite error elastic mse depend approximate number large approximation restrict explain worth order base polynomial indicate necessarily combination topic appropriate plain series follow remark exactly accordance connection error distance increase almost boundedness may increase low lemma usefulness establish one tend restrictive exponentially covariate term moment degree polynomial corollary belong old polynomial meet corollary since condition satisfy quadratic loss resemble explain b verify loss derivative respect discuss begin I represent value variable f covariate valid positive except satisfied old since risk well model result difficult term hand away excess argument b tend long oracle penalize penalty inequality valid function stress see result deduce oracle use excess construct thresholded consistent truth linear give example setting fit quadratic allow addition precisely square elastic include propose justify extend quadratic interest practical proving elastic proof step possibility penalty multiply great equal oracle shall denote convexity loss simplify linearity rearrange eq side eq definition get one side get furthermore norm rearrange shall adaptive restricted condition case constitute step reverse adaptive rewrite get right hand bound inequality triangle convexity eq inequality eq rewrite inequality rearrange derivation valid yield write add get follow step argument repeat desire exist whose may throughout state strictly satisfy right negativity establish establish sequence non hence possess suffice converse reach assume satisfy convexity continuity corollary inequality also corollary follow precede probability valid remain side convex function quadratic already slightly involved state suffice suffice assumption argue inequality derive boundedness valid ni lipschitz take constant part estimate equality covariate proof first imply choose remark good linear predictor target course deduce tend q hand positive derivative show analysis shall choose general beginning section f respect suffice x get must suffice imply recall validity full rank argue turn suffice verify valid assumption theorem example penalize unknown use elastic penalty estimate generalize literature asymptotic estimator asymptotically thresholde variable loss cover contain increasingly many economic sample leave many financial nature high try control supposed find clearly linearity model handling set receive lot seminal introduce carry parameter study paper adaptive bridge recent review prove instrumental without impose mean moderate self greatly scope applicability datum inequality unify consider model focus setup penalize linear setup elastic furthermore focus non excess case target use penalize estimator elastic valid estimation parameter establish result risk order briefly thresholde version pattern provide abstract show contain finite square elastic series explain loss therefore main upper bound lasso shall procedure author loss elastic increasingly high set usefulness enhance paper organize put forward notation elastic net section discuss consistent thresholded elastic handle quadratic well stage setup exposition general precisely fact suitable suppose denote usual accordance denote intersection finally throughout shall assume convex meet cover set upon let consist cover cover see bound series vector logit model sensible negative return identically reduce plain note joint minimization instead population differently minimize f coordinate choice make transform convex example shall penalize linear exact discussion constant minimize elastic penalty plain regression estimate coefficient two correlated tendency include elastic net variable benefit formalize show elastic behave well plain next oracle inequalities fx quadratic condition bound hold margin shall margin conjugate development many find strictly lemma establish property extra sequel estimation remark result except restrict eigenvalue compatibility generalize oracle compatibility carry theorem impose adaptive restrict eigenvalue condition reduce adaptive eigenvalue net compatibility elastic compatibility front letting turn term depend hence follow condition provide equal imply yield shall worth pointing since even target member excess right side increase multiplication tradeoff restrict adaptive depend set minimize choose least size cardinality oracle satisfy eq pure reduce recall increase differently tradeoff bind follow give low example argument continuity contraction furthermore positive constant assume commonly use critical follow corollary valid technical exclude allow exponential size satisfy asymptotic tend cover theorem modification proof require bind lemma yield theorem conjugate basically large side reflect differently excess excess oracle reveal valid excess away order considerable
upper bind concave obtain role gaussian trivial arrive second weak relax relaxation trivial increase become dominate become accurate side become correspondingly right hand become negligible become inequality bound remain specify probability boundary quadratic subtract deviation distribution eq q square root appear drop side yield take root tangent tangent pf pf analogous argument pf pf pf simplify immediately apply result follow combine bind formula limit attention imply evaluate bind compare exceed trivial upon decay stem coefficient chernoff govern begin express expand approach gaussian tail lastly second combine deduce lemma substituting cross justify specify concave carry analytically substitute simplify r claim substitute equality expansion expand first substitute right low maximize eq note theorem corollary sense nonzero sparse provide gain resource allocation propose policy positive power policy non adaptive nonzero component quantify sense resource budget ratio budget rate fraction spend exploratory stage decrease vanish gain bind sense simulation adaptation tight adaptive sense adaptive allocation sensing refer control acquisition process ambient snr improve determine signal resource stage loss generalize allocation programming control stage increase policy lagrangian approach reduce sense analytical quantification gain obtain upper estimation low bound adaptation gain policy monotonicity sense performance notably signal one bound procedure sense support vanish discovery discovery sequential thresholding shown recover snr kullback leibler divergence sense gamma observation appropriate sense characterize primary budget sense compressive contrast compressive component herein recovery attention benefit estimation contrast develop resource policy capable gain empirical validation focus specifically estimation guarantee sense policy key guarantee herein quantitative opposed qualitative statement improve another intuition resource concentrate signal turn detect impact signal limit snr control sensing resource conditional observation error reduce chernoff coefficient use adaptive management detail stage bound th non sense sparsity nonzero either tend vanishing describe define unit power tight limit limit budget case confirm adaptive oracle gain allocation exploratory stage allocation notion sublinear arm case increase resource illustrate notably regime intermediate furthermore simplification stage policy minimize true nearly monte sampling increase adaptive first whereas tt effort policy assume sense horizon case minimize th familiar mean mse addition accounting amplitude estimating detecting component lowest achieve sense require support insufficient effort e nonzero resource signal two strategy uniform effort provide gain due interest fraction snr budget regard sparsity distinguish increase sublinear nonzero normalize snr summarize section intrinsic decrease budget prior regard fix improvement adaptation ability concentrate turn signal measurement theorem show unconditional weight shorthand refer chernoff coefficient exponent exponent choose characterize perfect overlap coefficient relate hellinger two distribution sparsity level validate numerical dimension set signal first determination budget allocate stage optimize mae proposition plot generally moderate independent prevent gap proposition agree suboptimal allocation accurate gain mae sense either mean approximate large gain respectively curve occur intermediate near unity mse db mae gain allocation suboptimal form policy gain gain stage bind optimal suggest mse improve low proposition
map choose provide w solve solve expansion multiplying regularity map solve equation determine system existence substitute big contraction mapping solution take enough done coordinate even odd eq embed choose enough finish embed embed future smooth manifold inside action definition thm thm spectral often linear reduction eigenfunction manifold limit many independently show connection laplacian tangent approximate bundle eigenvector converge infinitely datum manifold field graphic dim object dim brain fmri correspond source variability nuisance acquisition align graphic organization shape also nuisance shape transformation nuisance factor dm dimensionality reduction organization set nuisance eigenvector eigenvalue connection laplacian encode pointwise bundle limit contribution convergence connection pass manifold empty center le dm weight undirected vertex euclidean distance sensitive nuisance use invariant associate measure distance element map equivalence equivalence equivalence class metric invariant give isometry leave q isometry orthogonal unitary three reason guarantee hermitian isometry group minimizer invariant dm le transformation diffusion group exist usefulness manifold dim embed smoothness embed tangent bundle local basis embed basis embed tangent frame bundle point bundle tangent plane purpose manifold take laplacian bundle connection cloud embed euclidean cloud show approximate point manifold extend spectral constructing distance manifold prescribe point cloud dim since bundle encode nuisance total bundle space diffusion often medical imaging dim dim purpose classify similar basis right tangent plane direction action nuisance metric outperform mathematical bundle nuisance space combination space base bundle dm provide second addition show setup eigenvalue converge field connection result hold boundary spectral connection tangent tangent bundle estimate cloud list vector distance manifold cloud classify reconstruct prove nash point cloud bundle background le dm take spectral assume bundle effect handle section prove result bundle need estimate point affinity graph suppose assign scalar q diagonal un suggest interpretation follow characterize status endow dim status step absence define laplacian special coordinate status move become status influence rotation path group dramatically get transformation path close affinity contain path length square schmidt norm ti vector along path connect motivate ti affinity ti diffusion become hilbert define due negative unnormalized motivate proper degree associate please fact differential reader familiar bundle quick introduction denote dim empty canonical via geodesic bundle lie right act projection call bundle principal view act parametrization simplify confusion symbol p bundle qp compact connection metric determine mapping preserve interpret linear find coordinate frame bundle denote bundle bundle frame bundle bundle bundle understand relationship bundle tangent point point plane denote denote integrable iff x understand definition tangent follow back coordinate curvature curvature curvature fundamental open normal bundle radius automatically radius manifold notion f definition function define sigma algebra probability absolutely respect yx otherwise measure simplify hereafter point n identically independently bundle define interpret finitely discretization note bundle setup understand recover relate discretized tangent bundle setup measurable minimum cover eq q accord exist nf kernel decay enough characterize affinity estimation f identify practical adjusted reduce uniform approximate know result dm either topological nature concern aim fix affinity point riemannian ambient ingredient le dm laplace know laplace nan space embed study le dm bundle associate trivial bundle laplacian geometric topological core tangent bundle geodesic among nearby number synchronization translation laplacian notice similar dm bundle connection le dm dim smoothly embed induce close bundle denote symmetric satisfied detail bundle metric qr simplify principal vector principal bundle bundle tangent bundle construction discrete group take horizontal tangent vector bundle gx find trivial connection ex xx point bandwidth exponentially satisfy affinity take x function define affinity value call connection q symmetric entry matrix h h j recall quantity geometrically rewrite derivative horizontal lift appear reveal dm orientation principal bundle associate bundle e ie ix come orientation frame give eq manifold account recover double smooth double cover embed make appendix modify reconstruct cover dm trivial principal bundle bundle entry diagonal group dm bundle algebraic consideration mention bundle structure dm dimension spectral dm structure correspond assumption compact volume fundamental ignore pointwise normalize boundary empty find unify current base principal bundle bundle except bundle operator cx state laplacian assumption n h connection laplacian object suppose assumption take focus focus situation h situation h need f finite point appendix term theorem stochastic finite bias suppose assumption hold take h xx dx stochastic I stochastic I situation stochastic laplace unified theorem principal normalize empty setup argument du ix choose du assume relaxed except term become boundary dominate field connection homogeneous eq pointwise convergence come convergence enough algorithm theorem connection small tangent bundle vanish l le eigen field line normalize spectral theorem proof actually equivalent transformation denote th associate th heat kernel I finite integer mention existence ignore second spectral assumption choose basis embed plane denote view frame bundle parallel tangent schmidt eq coordinate denote kernel satisfy operator coordinate embed embed reconstruct bundle influence spectral study spectral answer question begin simply point cloud operator h ct spectral state h theorem assumption step step eigenvalue eigenvalue eigen field assume decrease exist I assumption b assumption assumption hold parallel step connection associate x analyst cloud ingredient proof come plane estimation embed knowledge embed bundle access embed tangent inside tangent plane basis tangent need embed tangent bundle cloud resource estimate embed tangent show approximation definition manifold euclidean depend second locally jacobian error theorem outline look reader bias finite solely indeed pca inequality choose note extent able v h behavior frame modification conclude event result frame enough laplacian conditional n step h gm award fa award number foundation wu fa nsf wu reading manuscript collect fact principal reader start discuss notion element onto satisfied leave word action jump induce name free totally neighborhood action equivalence exist space base space action canonical bundle definition manifold composition bundle setup principal action principal bundle lie bundle bundle smooth denote canonical satisfie note relation intuitively acts bundle trivial bundle choose diffusion bundle principal bundle dim purpose basis basis u xu v b x bundle orientation bundle frame orientation point way take disjoint call orientation form bundle base manifold denote equivalence relation canonical call bundle associate bundle induce confusion denote ff bundle principal bundle confusion take tangent bundle dd identity mean frame bundle tangent bundle point basis tangent basis notice view invertible map take mapping map bundle bundle always continuous associated bundle let map inner product focus bundle action introduce notion connection tangent vector denote bundle vertical refer bundle vertical bundle split choose splitting principal bundle connection value horizontal bundle denote determine word horizontal lift curve lift horizontal existence connection call along connection bundle matter projection vx tx horizontal lift connection smooth curve lift tangent horizontal existence life bundle parallel along curve interest derivative principal bundle curve parallel note provide explicit derive give appendix however notion connection define fact horizontal lift follow format q definition parallel equivalent definition bundle associate tangent bundle coordinate map way compare compare coordinate abstract satisfied resp bundle smoothly connection bundle mainly work preserve metric bundle product close preserve verification show structure metric connection bundle bundle derivative choose order simplify denote integrable dual bundle manifold possess connection tensor product riemannian manifold bundle connection compactly smooth direct divergence adjoint property heat principal explicitly order convergence tx tx suppose enough last decay exponentially note near boundary symmetric term nonlinearity understand care assume suppose divide slice ir u eq norm lead elaborate assumption take cut f sx f define trivial define regularity proof dependence error derivative regularity assumption next taylor expansion third symmetry finish analysis numerator matter numerator vanish symmetry property expansion integral price error x xx h taylor expansion numerator become symmetry numerator expansion depend finish ingredient analysis emphasize since laplacian term able rate provide assume assumption take point fx boundary situation hx fx hx hx k hx hx ig view un show clearly dominate uniformly bernstein hoeffde inequality take bernstein inequality take eq happen less clear satisfied h denominator argument indeed variance
improve estimate select maximize interesting current minimization algorithm support gm nsf nsf fellowship direct formally transmission infect infect invariant j exponential scenario information transmission heterogeneous time differ instance status inactive user day result transmission user inactive scenario transmission survival intensity transmission ji ji ji ji know ji ji ji advantage intensity enforce equal model define choice gaussian rbf kernel diffusion look model graphical collection infection contact loop diffusion cascade direct acyclic dag cascades dag collection parent dag parent true time infection graphical specify pairwise pairwise transmission infection likelihood parent precisely infect infection parent consistent contact replace node node pointing direct loop variance estimator union relation c ccc vs c vs label kronecker edge panel show time relative different every relative vary kronecker panel estimate window every label scalability synthetic evaluate scope kronecker kronecker dramatically produce small transmission high degree random output ccc core infect transmission ccc transmission ccc panel heterogeneous transmission function time greedy threshold lt cascade ic diffusion heuristic sp support pairwise transmission edge furthermore scalable average hour instead ic infection within window pairwise infection calculate ic edge compare infected become large infected window kronecker increase simulation running maximization network ghz core draw ns follow long hour dash qualitatively base time piece site spread million web month task requirement scalability address paper propose randomized estimate subroutine world datum scale network million node improve influence maximize motivated certain purpose accurately challenge cascade design network window time like million people month one sensitive requirement topology argue cascade diffusion discrete choice follow seem appropriate bin optimally discrete transmission hence restrict capture extensive show improvement recover diffusion cascade predict maximize challenge influence model transmission density markov exponentially size density extend transmission nontrivial approximation would unclear diffusion million especially naive inference round overall scalable diffusion heterogeneous edge transmission key idea view graphical reduce graph node node use logarithmic computation subroutine maximization maximization real estimation art allow art cascade model density discrete associate infection generate sample transmission contact begin adopt go neighbor edge entail assume transmission independent differently infect continue infect infected result contact cascade information induce dag heterogeneous formally transmission direct get nonnegative parametric function independent cascade later transmission times shortest correspond associate essentially infection time independence contact detail denote induce dag infection infection parent instead model infection transmission interestingly switch edge factorize path independent cascade variable direct source path node infect time path special infect relation involve dependent short window wider spread infection source adopt definition influence average previous source infect infect window infect transmission indicator estimation event summing direct degree user need integration continuous integral analytically heterogeneous resort numerical integration entry parent form without exponential entail algorithm randomized key influence ns draw run short average see appendix naive repeat hence million source summation rather short path fortunately neighborhood study science adapt randomize randomized location need edge transmission time infect source window node random label make also variable parameter equal variable small show estimator question compute efficiently source design least label query start first search reverse graph node find compare distance record distance small add algorithm label summarize appendix return node label pair element want small list collection expect label computational om randomize neighborhood neighborhood ti source sample need naive additional constant need achieve transmission averaging set q overall transmission j ji r importantly unbiased essentially loop transmission reduce practice experiment application arise source drastically label appendix time transmission draw indicate actual influence variance neighborhood number random monotonically decrease long match implication large large estimate influence infect maximize variable np monotonic return maximize add node source source achieve least number store nest loop storage randomize fortunately know follow confidence source opt estimate influence performance maximization synthetic significantly method synthetic generate kronecker parameter trace world network network typically physics hierarchical transmission ft often event survival uniformly order heterogeneous kronecker choose every edge knowledge analytical transmission draw ns near compare ns window times loop three estimation fit error increase figure c relative additional ccc influence vs core influence increase sample increase scalability naive ns run maximization outer loop two ghz processor compare increase select fix core edge window number source essentially influence source computation line finish estimate plot compare ns magnitude slightly additional range report core ghz ns scale ccc vs vs network runtime source nod density select source increase network fix continuous base discrete diffusion model
basis sensing theory often efficiently loss generality us video signal projection result vector know often show suitable unique minimization fact property match exactly original various concern perfect reconstruction signal signal dictionary incoherent atom fall category piecewise arise image little detail model piecewise difference recover concatenation directional difference tv tv use extensively image science minimization sense perfect apply frame nontrivial great proof successfully recover tv establish establish gradient signal haar wavelet modify isometry haar orthogonal wavelet offer recover signal remain partially decay wavelet establish answer open prove fidelity restrict isometry directly work space space condition mesh specify angle gradient tv minimization tv tv comparable minimization recover signal gradient draw gaussian proof mesh argument section recover sparse appear gradient matter support omit detail nan see gaussian build mesh mesh euclidean haar eq z l g q kn nk c n n n c pi pi q q pi tn ht c c q l x dx dy dl eq l dl dl number variation support result prove stability tv approximately sparse gradient answer fidelity total multidimensional signal sparsity tv angle framework establish vector current work ensemble deterministic bernoulli another dimensional conjecture operator number proportional support tv minimization work towards direction claim measurement tv recover fidelity linearly tv tv
wikipedia topic large encounter model document topic distribution represent variational represent intuitively vocabulary topic word phone topic possibility new associate take take represent need upon probabilistic commonly collection topic wikipedia article dirichlet sample topic word mention phrase multinomial sample mention identify wikipedia entity hyperparameter symmetric dirichlet interpret word zero allow residual probability assign training see content explicit entity name united seven wikipedia include text entity text mean evaluation use mention identify name entity entity simple lda generative corresponding outline sample couple document correspond learn mention co occurrence enable annotation approach apart ignore content observe inference lda ease exposition use lda unlabeled news article wikipedia article initialization article label english around article vocabulary vast potential parameter topic essential corpus require variational inference gibbs advantage efficiency online inference process brevity present content word lda distribution seek local integrate convergence bayes joint posterior computation maximize evidence log importantly correlation topic topic document follow variational fix topic distribution word dirichlet multinomial brevity henceforth variational perform sequentially topic pz key retain topic assignment element remain key insight enable local topic assignment inference full batch noisy average datum guarantee optima scheme update old one batch perform even update improve secondly discard mini save amount requirement store prohibitive gibbs hybrid beyond efficient document dimensional operation topic number operation compute brevity denote initialize zero k decompose sampling transform version variational document currently topic topic count word mass appropriate choice operation update summarize process algorithm sampling count corresponding normalizing line vocabulary begin pair overall count line current topic count count compute sample must normalize topic draw word topic change topic accordingly multinomial visit skewed component skewness govern act pseudo topic word discard burn k vs w si c kn w z z parameter value baseline minibatch work entity graph exploit wikipedia interpretability readily presence consistency line assignment coherence document score eqn coherence appropriately incorporate would impractical practice addition lda correlation prior correlation manner wikipedia provide solution extend scalability challenge alternatively sequentially document gibbs run inner loop interpolation change document update average aggregated model complete outer minibatch outer loop document minibatch present extension arbitrary simplicity interpolation develop automatic investigate optimal update schedule subject wikipedia md k statistical large parameter optima wikipedia significant vast initialization assignment entity system annotation wikipedia admissible topic page long character page article mention incoming link string amount roughly parameter initialization highly pair list token occur wikipedia discard occur article vocabulary word denote word v count word scoring accord notice represent initial model thus cross english corpus news consist document contain total token style name entity ignore entity boundary identify entity hold behave hold increase naive initialization sampler great even random mention typically tailed order primary topic extremely fine unlikely initialization heuristic derive weighted contribution document edge vote index candidate topic score select set topic closely pick entity english organization tag partition document development blind evaluation wikipedia entity predict annotation test document since optimize micro hyperparameter comparable achieve wide range act visit although topic work topic control exploration vocabulary denominator order robustness setting upon score like initialization around work obtain probably noisy initialization wikipedia alone along run alone yield optimal column gibbs wikipedia greatly annotation gibbs improvement base micro b micro maximize due skewed accuracy compare system extensively prove figure sim thank communication method blind report macro micro score system inspection error development partition clear mention gold annotation g appear annotated city sometimes country uk bag discriminate tend assignment per context could relatively straightforward weighting future goal simply framework achieve state art scalable different regime operate typical lda seek document represent topic model address topic attempt roughly time requirement art scalable framework document gibbs mb note directly topic pure sampling fast date report throughput document per corpora machine complex much certainly architecture principle plan investigate comparable week desirable area systematically wikipedia conceptually framework upon extended hybrid incorporate crucial wikipedia graph evaluation different usual exploratory discovery text different comparable lda topic parallelization date line investigation implement advanced local investigate interaction effect computational modeling explore alternative could refined acknowledgment thank valuable discussion google wikipedia content challenging topic vocabulary million representation gibb allow memory report public drive technique topic reveal collection inherent interpretation post hoc year phrase wikipedia map scalable understanding investigation notion gain advantage drive topic identifiable semantic person financial concept etc human insight interpretation basis discovery entity annotation typically phase entity identify assign alternatively upon possible already text name entity topic model address interpretability principled flexible wikipedia wikipedia article content inference challenge million ability stochastic upon hybrid inference combine gibbs sampling result online overhead online parallelization avoid architecture framework conceptually inference via join define inference purpose additionally document level link original modeling million topic hybrid exploit efficiency exploit incorporate report art scalability background section introduce inference scheme distribute conclusion focus wikipedia know two infer measure anchor text likely refer design depict news article connection wikipedia topics string mention topic priori character player wikipedia reveal topic densely connect candidate topic page way entity present lda entity entity link base identify wikipedia article topology drawback efficiency wikipedia entity report time large experiment document graph reporting times week gb goal wikipedia annotation reasonable focus broadly
element row column notation x diagonal entry notation term corpus number write v k multivariate trace eigenvalue matrix derivative determinant square parameterize probable say least say family bx probably convexity family decide perspective probable convexity proportion member family family member convex ax member convex convexity say family surely convex definition almost surely surely almost vice versa probable equally least say concave refer maximize concave history rich foundation often excellent consider call naturally broad many nonetheless neither convex concave intractable thorough find property concave indeed kk observation notation eigenvalue go notation probable concavity concavity concavity semidefinite derivative j x j k kk kk algebra say diag diag diag principle except associate z principle positive diag complete diag proof consider diag semidefinite consider concavity decide derivative suggest z diag constraint diag algebra eigenvalue kf use eigenvalue eigenvalue thompson inequality q show matrix nonnegative equal e k inequality derive use consider diagonal diag k diag z diag e substitute paragraph z diag equality minimize mixture approach condition probable concavity corpus compose th z fw parameterization proportion parameterization transformation vector fix transformation density function posterior document document give reformulate logistic make speak say posterior function worth maximize contain hence analyze family analyze member model kf corollary corollary p notation notation employ topic include general logistic normal prior therefore derive topic mainly concavity inference nature get optima close global one algorithm many inference reveal infer concave instance optima nonetheless good local optima nature select direction greedy nature optima answer three answer probable originally inference question support highlight concave benchmark investigation retrieve dataset check question take document default parameter avoid document learn explain efficiently concave able fast slowly auxiliary optimize document intensive reach convergence iteration observe rarely need reach reach see goodness interpretability observe assessment calculate topic probability document contain choose topic quality perform able comparable even objective behavior inferior tend interpretable topic increase seem topic fast learn significantly topic investigate topic depict advantage discovery interaction support able qualitative model pos dot ps correlation part variational employ next concave dataset previously totally instance investigation comparison take convex good try maximize variational try document hence quality subsection inference method criterion relative improvement well show find last three one perform find r average fail l intensive come require contrary iteration mostly derivative able individual problem fail solution return significantly domain always find significantly bad advantage good solution concave problem probable real family probable function member convex member practice feasible deal probabilistic convexity certain efficiently belief quality many significantly accelerate wolfe solve nonconvex topic behave algorithm successful suggest nonconvex well nonconvex hope highlight open connection nonconvex tu probabilistic distribution non pose attack introduce convexity contrary analysis qualitative highlight resolve efficiently might beneficial many context beyond probabilistic non stochastic estimation play conjugate likely efficient sampling prior estimation difficult topic popular approach cast optimization pose designing allow concept target reveal hard practice smoothly employ deal probable say family probably rarely meet
som apply update rate correction som sensitive control correction apply acknowledge limited become correction specify iteration construct som truly appropriately herein find large determine iterative change g tolerance figure demonstrate change nine cell encode evaluation cell however quickly begin middle stable map nearly iterative converge remain cell map within explore keep figure panel number map keep cell highlight cell keep map metric performance som bias green contain map size cell top panel maps panel som topology cell random performance metric add combination time map redundant bottom cell value mean right identical contour galaxy galaxy survey dot bar correspond bin width eventually mean number empty cell primarily subsequently score confirm present cell produce metric som simply single som technique full probability encode pdfs measurement table true recover use traditionally interpret reject hypothesis measure mode green red top bottom panel measure galaxy previously prediction concept analogy forest decision create subsequently aggregate produce pdf som score efficiently account overall scatter match different random forest explore final map update dynamically process call update updating produce hard limitation cumulative slightly weight som process topology rectangular rectangular show slightly likely boundary flat grid impose periodic condition topology cell naturally periodic condition provide galaxy hand som close unity explore cell construct give eventually map effectively configuration information map example ideal combination produce prediction previously cb employ forest decision tree learn galaxy use pdfs galaxy resemble pdf variety accurately error cut select galaxy pdfs create sample affect outlier present herein perform accuracy strength optimally different meta accurate identification present future strength different approach individual author thank careful reading acknowledge national foundation fellowship university support advanced fellowship acknowledge use resource science resource discovery environment national science number galaxy survey nsf grant grant som pdfs maps university usa paper explore applicability som pdfs attribute dimensional competitive multidimensional som correlation identify rectangular deep new efficiently incorporate compare result map well multidimensional provide accurate comparable art galaxy survey amount area considerable digital survey hundred million survey use considerably large quantity galaxy small albeit precision consuming hereafter survey increase development modern band imaging survey dark survey des survey volume galaxy galaxy decade review template ball technique estimation date technique provide error galaxy pdf show galaxy weak acoustic mass galaxy galaxy measure galaxy growth survey reliable understanding go also within two angular correlation oppose single measurement improve survey volume likewise discuss inclusion weak new tree near support galaxy concentration environmental include magnitude color reliable beyond aforementioned technique supervise algorithm magnitude color provide employ hereafter cb public use forest technique compute estimation tree determine input exact branch machine use g decision algorithm capable project dimensional e represent magnitude color attribute usually training preserve topology attribute multidimensional organize som single evidence technique advantage unsupervised nature thereby meta use pdfs another som ability structure similar map neighboring node source however still tool map configuration work previously originally herein subsequently som random technique use bootstrap training map aggregate accurate cb incorporate measured attribute galaxy uncertainty individual pdf desire associated explore topology rectangular grid grid spherical surface multidimensional organize complete detailed pdf describe methodology efficacy result approach analyze result capability summary discuss advantage limitation som introduction self map scientific description som artificial unsupervised layer mapping training characteristic som algorithm competitive process quantization map training neuron try closely train spatially cell make som tool closely purpose represent galaxy magnitude dimensional lattice cell neuron illustration som galaxy iterative galaxy individually specific neuron galaxy become galaxy repeat galaxy iteration separate basic som arise weight vector update present standard version som topology present detailed som input galaxy measure galaxy magnitude color galaxy actual consider weight neuron arrange dimensional lattice topology value vector uniform galaxy procedure produce self organization map galaxy component galaxy update weight entry within map direct simplified space galaxy batch weight cell update galaxy galaxy distance galaxy neuron weight map denote cell subscript close galaxy match neighboring region galaxy tend locate relation rate reduce monotonically factor quantify magnitude correction unity value node quantify near significantly affect iterative symmetric away matching node close pdf computation distance depend topology encode roughly width cell procedure apply result update last retain line training galaxy approach irrelevant accumulate summation galaxy matching identify fix vector technique map galaxy potential poor determined figure illustrate som highlight technique vector process manner common technique step batch respectively correspond spherical cell topology rectangular square grid surface include option periodic spherical topology cell rectangular training color encode galaxy cell iteration demonstrate som galaxy use end visualize estimation som galaxies supervision topology rectangular topology use extend periodic boundary six calculate distance euclidean center cell rectangular topology rectangular grid neighbor boundary grid last map directly surface dimensional area topology circle center cell respectively cell near cell learn forest generate subsequently combine meta forest demonstrate empirically train organized however collection cb aggregate similar manner magnitude randomly object available attribute alternatively subsample map reduce possible map cb generate pdfs training measure training attribute distribute introduce randomness map systematic manner newly construct describe bootstrap describe weight map galaxy process assign belong ensure represent subsample galaxy similar process galaxy map galaxy complete prediction total prediction contribute equally final online galaxy represent figure slight separation change final galaxy spatially self explore configuration demonstrate capability efficacy cb paper restrict analysis evolutionary probe survey multi low resolution phase ii imaging galaxy magnitude survey band france recently database release deep release source source survey france deep som sources galaxy bad filter response come two different survey treat galaxy target field end leave eight band set rest som implementation define metric accuracy bias characterize outli quantity represent kolmogorov whether underlie distance cumulative galaxy carry galaxy statistic ks decrease sigma absolute metric call som online meta first normalize range individual list table metric nine simplicity equal weight remainder paper simply nine configuration low score look lower accordingly way explore som conduct different test color deep three topology discuss rectangular spherical build four color use configuration addition rectangular topology use spherical additional eight determined good parameter som implementation bag datum similar cb example topology contain approximately cell galaxy average ten realization metric mean place column clarity highlight visually bias score symbol topology rectangular spherical online batch either attribute attribute inside separation test som highlight periodic rectangular table detail text ten galaxy symbol symbol symbol topology periodic boundary c topology np yes np np online yes yes p yes batch yes table albeit performance without subsample find remarkably cb superior trees attribute explanation attribute map forest explore attribute combination construct likely correlate introduce reduce contribute attribute large map construct map attribute color attribute determined run attribute som topology
analytically input choose informally cover crucially achieve induce small fully prior empirical eq convention whereby generate covariance identity oppose refer block diagonal choose induce selection appeal drawback interference hyper set crucial across attain start discuss gp briefly comment aspect costly evaluate track cholesky matrix trajectory possible cholesky obtain trajectory operation naive implementation reach intermediate operation scale avoid require multiplication tool use explicit obtain arbitrary integral follow expression standard treat gaussians predictive ambiguity factorize matrix pre overall chain use useful gaussian consist mixture gaussians prohibitive could opt learn cloud normally consider bx v difficulty multimodal since system normal run iteration particle capability baseline form green vs surface demonstrate apparent two e fact always case bi opposite sign plot smoothing circle correspond although function gp black accurately smoothness prior sampling fail capability dynamic generate step rmse obtain ground model table summarize report rmse smoothing trajectory ht cm ground truth mean propose parameter approach cart reinforcement consist cart force cart ordinary base corrupted although explore small dimensional produce predictive display crucially report space confident close nonparametric model make tailor characteristic approach transition unknown smoothness suffice quality smoothing describe dynamical reference engineer university uk automatic control link technology university economics dynamical system fully bayesian identification nonlinear place gaussian dynamic flexible able phenomenon enable joint tailor markov chain carlo state transition formulate analytically approach preserve nonparametric sparse greatly state constitute main impact external act dynamic relationship unobserved find employ nonparametric provide flexible particular functional possibly parameterize natural instance engineering sensor measure identifiability bayesian whereby entity namely approach gps tend infer trajectory tailor particle markov efficiently obtain smoothing learn form marginalization dynamic present refined gp likelihood model gps em base function wang learn find map hyper dimension vector state overfitte situation state smoothing describe gp notational clutter put hyper rely fully function insight system capture main property dynamic principle insufficient interestingly encode normally prior introduction seek I p refer sampling q particle trajectory accord density sensible may proposal formulation implicit correspond index particle particle normalize two despite resample step diversity far instant propose chain carlo generate trajectory employ gibbs trajectory conditional available tailor markov leave conditional ht mm draw conditionally kk trajectory particle suitable markovian rely standard pf particle particle sampling index unnormalize mutual independence form trajectory compute trajectory particle generate follow run result particle tp invariance affect autocorrelation drop moderate
line fact follow bm mp distance markov thus lemma proof event lemma notice last stem theorem corollary remark definition response covariate regression develop method distribution regression mean distributional assumption error measure excess rate value dimensional domain instead differ way importantly observe sample set mean observe rather predict illustrated let density law joint theorem measure risk polynomial distributional kernel estimator since way bind section intrinsic space numerical concluding remark functional new improve comprehensive review reference learn regression distribution propose parametric gaussians inner couple represent rkh use learn rkhs framework role generalize affinity dimensional hilbert nonparametric enyi divergence machine little even fundamental many let sample accordingly bandwidth definition specify precisely kernel eq appropriate borel call simplicity continuous belong old continuous functional lipschitz following lipschitz addition distribution mean concerned bound risk q covariate observation bounding call excess risk p l ph probability function first provide dimension provide kernel I random deterministic quantity depend provide eq lemma upper bounded introduce f k see similarly next proof supplementary piece together proof supplementary material specify old second term finally put everything risk without quantity quite slow huge concentrate optimistic small effective measure use every denote follow corollary depend side dominate risk yield notice meet reasonable distribution old give rate limited get grow noise reasonable establish demonstrate future serve proof concept demonstrate used triangle otherwise specify generate varied training contain learn skewness noiseless aware come skewness available appropriate uniformly choose bandwidth distance point uniformly test skewness true task
pn main difference come correction bayesian selection seem imply meaning apply consistency pay property agree view compare complex model model size tend procedure david sampling model close kullback distance bayesian model consistent example schwarz necessarily et b occur consistency also space inconsistent size commonly parameter procedure produce inconsistent respect prior al model provide difficult task repeatedly mention bring reality seem unable capture factor evaluate process suppose produce unless agree massive approximation obviously david avoid question still conducted landscape accept truly stage process prevent misspecification would worth discuss drive construct discussion de examine effective believe modal set answer final critical share likelihood already observable priori book reproduce point require p universit paris france write correction give rise considerable discussion focus procedure year later open model selection requirement imply avoid prior parameter continuous interest decision choose bayesian
assume perform value searching value design conduct demonstrate design efficient aim property change vary intuitively mean square error shape unfortunately lasso intuition detail number element increase size set decrease investigation describe answer rigorous formalize statement hold decrease quasi technique address cs problem researcher explore call amp iteration amp index iteration function apply wise parameter one interesting amp iteration consider three clear exist amp algorithm choice choice introduce threshold fix false alarm turn properly eventually nice false alarm policy straightforward however monotonic practical call approach fix detection thresholding policy element absolute similar employ sparsity parameter amp lasso sense unique amp conclusion monotonicity lasso formally regard amp policy organization contribution prove summarize conclusion capital matrix ambient transpose letter respectively denote cumulative respectively paper consider sparse goal analyze define main measurement iid subgaussian matrix simplicity ambient incorporate ambient notation sequence call converge distribution weakly measure moment impose purpose assume impose column solution converge sequence consider observable popular observable summarize far random extend general converge two exist restriction function almost sure limit scenario non random l name square mse alarm fa detection dr md converging draw x surely two soft satisfy follow first solution implication iid mild element calculate definition lead randomness measurement constant equation variance subsampling keep phenomenon sometimes cs main characterize surely interested observable use sure observable converge surely asymptotic normalize expression enable formalize question introduction mention introduction example inconsistent intuition description setting solution behaves expect converge parameter summarize speak increase limit active describe tune amp lasso next regularization mse short introduction figure exhibit mse description find shape shaped imaging find believe convexity mse certain lead algorithm amp refer mention iterative would like know discrepancy every vector different discrepancy amp converge amp provide sure converging drawn estimate amp surely hand claim long concern amp interested theorem establish converging sequence element surely right side variable discussion amp lasso detection point addition solution theorem thresholding mention quasi detailed quasi domain non increase scaling preserve convexity quasi hence quasi quasi quasi prove elsewhere extend extension random fix implication lemma unique fix quasi independent claim quasi proof quasi soft version manuscript modification mass take quasi convex prove differentiable function figure theorem sign change hence expect derivative even though converge risk instead main rest strictly q eq integral write r g simplify complete quasi far prove prove quasi shape yet shape sign change happen would like happen large enough therefore actually risk shape require summarize lemma lemma contradiction give contradiction lemma notation variable technique bind simplify sufficient q therefore would enough complete proposition quasi shape strictly write q prove enable break complete z plugging realize lead write q taking eq consider iii take lemma equality rewrite complete combine result quasi convexity function differentiable sign sign change sign soft respect parameter lemma prove sign change prove contradiction satisfy amp detection accord already contradiction contradiction slight lemma satisfy subgradient ax value c c play amp enough c amp converge simple reader explanation part remain start part ensure subgradient point amp surely prove construct subgradient z subgradient conclude q almost surely hence surely intuitively fact might flat step ensure happen algorithm general cover amp available reconstruction thresholding free e range measurement result let phase amp thresholding produce recovery correspond contain contain spaced consider spaced value use sample sample relative evaluate amp empirical j amp algorithm fix thresholde size measurement noise free amp exhibit blue refer successful curve phase color blue section diabetes year patient pressure different present mention previously problem decrease
resp know propose efficient square reference carry well provide take account mild centralized centralized diffusion match centralize classical distribute adapt strategy experiment subject type really correspond many problem recommendation desirable parallelization organize present strategy distribute dictionary experiment point claim vector capital letter physical realization covariance learn redundant carry characteristic property data coefficient associate dictionary distribute thank network neighbor include node node usually centralize setting matrix consequence associate column dictionary coordinate represent atom factorization ill pose potentially constrain observation little sparse impose redundancy sparsity complementary describe would maximally course dictionary generalization limit must offer fidelity datum ability choice priori share instance working patch dictionary l coefficient impose l penalization mild solution penalize ideally prefer solve penalize problem problem coordinate alternate possibility attractive linear translate choice pursuit backward pursuit denoise iterate estimate iterate tt soft step iterate dictionary knowing descent q stand large eigen pseudo inverse mod follow normalization propose k svd discuss sake root present solve detailed aim solve sensor assume consecutive record yield underlie eq column gradient sensor exploit neighbor eq average neighbor gradient respect update final intermediate sequel case observation simply identity choice variance diffusion strategy find square version would diffusion couple scalar vector factorization distribute mainly keep diffusion ensure every let setting sequentially assume available stand iteration sequentially node note know ni assume neighbor l sparse see various mm repeat update typically ii I mm dictionary random column node iteratively forward splitting iteration adjust node resp observation adapt step sparse compute usual converge accurate dictionary intuition numerical illustrate test form image compose activate true dictionary dictionary centralize average comparison situation note node learn dictionary dictionary appear centralized procedure even solve matrix common approximately identify principle rely communication conclusion algorithm solve dictionary thank permit neighbor exchange adaptation usual relevance improvement generalization consider
focus centre acoustic classifier frame sensitive little worth frame frames gap five acoustic lead improvement show absolute classifier frame benefit big case qualitative frame five solid condition generally show benefit db snr chance snr figure show train high adapt acoustic significantly close matched expect analogous method recognition h condition see show preliminary experiment acoustic gain benefit merging combination likelihood log likelihood place expect achieve desire improvement fit condition error error range sigmoid two range stream bold alone combine classifier acoustic snr upon low throughout compare latter adapt extend section recognition speech emission hmms recognition exception model remain suitable hmms importantly acoustic directly speech allow use obtain acoustic modification training system acoustic implement begin emission prototype emission global frame next emission state wise set state aim objective function emission model split emission double replace apart variance emission due acoustic objective maximum phone good adaptation provide performance match affect train speech adaptation additionally sentence exclude reduce ignore high representation representation pruning value indicate acoustic majority reject prune increase small reject fast pruning option need flat obtain train split stage apart acoustic zero passive method mean instance zero overcome zero parameter acoustic vector ultimately state avoid training subsection increase rate representation would representations previous acoustic classification value investigate suitable necessary duration acoustic previous investigation recognition hmms acoustic approach detailed section investigation component emission acoustic outcome contain significantly force zero stage give well consist final denote gmm legend emission method component reduce error range component gmm show frame concatenation model train acoustic initially vector cover every ms ms adjacent close standard effect frame acoustic five optimum representation consider recognition slight improvement acoustic ms beyond duration become acoustic baseline acoustic adapt noise advanced taylor conjunction representation respectively mean ms ms baseline commonly tune acoustic ms comparison acoustic duration baseline fix acoustic show result demonstrate feature representation achieve acoustic ms acoustic effect average acoustic establish model acoustic model fix acoustic hmms provide frame show rate db achieve combine rate robustness space acoustic speech separation acoustic domain using give remarkable result acoustic domain noise achieve gain acoustic nontrivial open consideration demonstrate acoustic information next compatibility gmm framework base co acoustic technology gain especially scale reason progress novel new signal front test hmms low level db feature loss towards significant improvement sophisticated modelling speech scope direction tie preliminary tie covariance extensive even possibility extensively average averaging frame tune uniform explore content discrimination explicitly class seem particularly model speech covariance approximately copy model use mixture constrain q symbol model amplitude gmm scale model eight although model rate us distribution acoustic use reduce broad confusion broad achieve acoustic success neural dnn acoustic robustness difference setting poor severe study et scope improve robustness gain achievable linear study valuable suggestion speech space acoustic improve robustness automatic speech recognition additive motivation acoustic usually process extract dimensional aid linear allow oppose develop linear classification result classification well snr likelihood individual across speech robustness major automatic system robustness substantial degradation environmental language would level speech effort speech recognition isolated investigate know human attain portion early process effect language optimally speech accurately chance already snr db snr human speech remain noise snr automatic able severe hide advance lack concern isolated unit study novel investigate lack factor front system step consecutive segment feature vector prediction initially remove lexical recognition resource result boost achieve massive amount computer order powerful speech robust message lose commonly speech good compress form like human reliably lose compression production decode speech compression coding problem fundamentally speech production manner unit additive distortion speech compression front end remove redundancy manner source code different speech apart considerably human speech lead severe degradation recognition performance correlation error speech introduce typical speech acoustic transform additional modelling noisy representation involve change challenge linear representation author filter acoustic avoid boundary therefore model short duration event illustrate power benchmark remain later model speech explicitly inspire switch dynamical digits model hmms even explore directly acoustic front end recognition commonly conjunction hmms practical assess represent transformation without potentially effect develop mixture length speech classification additive segment duration entire section gmm scenario start already snr recently normalise paper generative probability predict class great likelihood length acoustic segment probabilistic domain could speech possible dimensional manifold include linear many representation variant incorporate information distribution speech aim dimensional structure exploit build remain dimensional could gp find dimension estimate existence nonlinear structure impractical generic construct attempt approximation mixture mixture form mean impose constraint component weight reliable become already ms segment space density derive component gaussian number datum diagonal use model diagonal provide dct introduce diagonal retain diagonal arise consider sentence normalise sample sentence norm sample average noise away square norm sentence eq easily unit per rescale rescaling model precisely feature transform use feature window alternatively statistic sentence level require remove estimate consequence frame instead consider use feature long sentence rate hence exploratory match condition classifier impossible match nevertheless exploratory experiment datum come assume extract si sx database consist generate apply nine sentence level snr differ ten increment extraction combine accordance combination stable group increase addition shift version datum class segment extract shift sample identical effectively manner shift frame default inclusion increase representation frame close centre vector give feature acoustic dividing sequence overlap seven frame centre individually process dct use covariance dct impact investigate component testing consider noise low explain number average show acoustic also improve noise method adapt curve condition appropriate noise level compare acoustic giving db snr compare speech ms windows window form acoustic trend hold condition quickly point db snr representation db acoustic significantly well chance db snr dash figure
require inversion matrix value separable variable pre specify mean advance update evaluate complexity conduct evaluate algorithm counterpart five attribute attribute school datum synthetic instance describe copy covariance diag use different operator denote kernel identity square refer cumulative f I empirical algorithm split part obtain synthetic size increase batch separable dataset parameter use cpu figure mse misclassification report achieve standard table combination structure online require structure combination value report possible future derive definition proposition remark cm cm value setting reproduce hilbert operator describe take algorithm extend value setting hold output linear achieve good result low computational cost problem reproduce kernel problem receive community compare analogous scalar value decade attention value largely develop structured formulate value function paper value reproduce kernel detail important context operator rather encode value success prediction despite advance limitation kernel high expense value kernel associate reproduce output invert deal spirit ask online develop value base memory cost extensively refer review reference focus value little operator aim multi output situation make well use prediction recent exist online learning main extend provide guarantee address output sequentially value provide evaluation demonstrate section present throughout problem separable hilbert hermitian value rkhs call hermitian value kx I reproduce iv hermitian operator contrary batch sequentially construct f r x sequence iteratively value solution risk minimization write tf truncation keep costly however influence geometrically old control reflect cumulative error make make risk define admissible e regard fx z boundedness hypothesis loss assumption square note scalar hypothesis either tt truncation either exist truncation scalar several point group proposition due proposition refer reader theorem hypothesis hold consequence prove induction hypothese prove combine imply tx idea proposition prove hypothesis consequence consider dim calculation old truncation calculation prediction old sublinear truncation respectively truncation batch operator base algorithm cm jx k jx manner value structure
sample make define hyperparameter imply toy well da dimension different pac set classifier aim vote guarantee majority vote bayesian generalization classifier drawing risk pac follow mm good domain call related disagreement pair justify call c bind h formulation da recall learn vote counterpart disagreement margin vote regularization note justify pac cm ds e st ed st rewrite labeling bind become label function h first c see label da relevant labeling carry label want hyperparameter intuition pair label else obviously concern hyperparameter da make reverse circular validation however fold optimize justified minimize domain label tackle toy inter correspond one angle svm supervise da da base auto labeling come source use illustrate confirm da secondly label nn labeling appear focus density region match source word da angle cm cm cm cm tackle da algorithm disagreement da minimize perturb preliminary like method life perspective new transfer function adapt institute tackle pac bayesian adaptation da good vote set disagreement supervise setting involve disagreement da elegant divergence perturb justify secondly promise toy develop work tackle adaptation da different unlabele common spam one adapt divergence intuition divergence preserve source divergence
direct mutual inform representation space mutual demonstrate correspond thought configuration allocate sample way motivation need induce capacity expand ensure bottleneck even serve double purpose function contraction furthermore propagate mean optimization alone demand asymptotically regime train gpu dimension pre whiten normalize additionally cifar recursively layer deep take representation optimize objective perform descent minimize distinct mix back procedure free choose adaptively inexact exact computationally maintain gradient noise window example increment window minibatch new find momentum technique enforce result highly nonconvex challenging near solution distribution space difficult objective distribution converge empirical sequence maintain expectation stability accordance bias similarly rise initialize weight scaling match examine model mnist handwritten cifar color particular density emphasize estimate fully jacobian determinant mnist cifar test example te start variety unseen training example interpretable digit assign assign see calibration also investigate marginal final marginal extremely element round law k subsection combine real estimation density paper estimate differential geometry theory ensure learn interesting believe fully enable probabilistic draw class model class possible confident namely reject lastly rest space instead example demand little recognize manifold test idea raw bayesian however assign example provide calibrate confident achieve rate flexibility classifier normalize something typically energy deep leverage run algorithm dark medium ccc university university mit edu machine assumption insight new transformation factorize allow contraction across flexibility tackle variety task evaluate sample mcmc characterization many density discovery model construct practice undirected boltzmann compute latent unnormalized belief hand enable specify costly nonparametric estimation another costly procedure typically manifold characterize locally gaussian visualization notable seek unfortunately autoencoder probabilistic interpretation although approach window tackle combine directly difficulty curse gp characterize nonlinear probabilistic necessary space density latent integrate pre invertible alone density exploit invertible transformation rather limited project back gp approximate manifold discovery exploit deep rich flexible imply space factorize marginal ensures compute fully normalize density partition modeling direction generative present variety cifar mnist proof density possibility classifier calibrate class additionally permit exploit unlabele construct expectation provide fundamental concept model tool important connection dimension approximately factorize understanding enable inform selection interested structure manifold study directly present transformation factorize form assumption analytically compute rich map density flexible discover structure normalize density ensure often bottleneck density overcomplete span include mass problem whose map penalty univariate find fit tractable family hypercube representation match approximate beta peak distinguish closed penalty furthermore denote element total penalty pursuit sparsity activation representation undesirable activate unit evenly activate reconstruction example identically map force activation vanish force distribution contain small around another implication activity penalization attain induce directly achieve kl appropriately peak understand directly prescribe distance thus contour divergence advantage contour density standard autoencoder volume around example penalty invertible volume never allow computation determinant jacobian ensure must activation fix ensure maximum curse dimensionality become high dimension orthogonality find easily constrain ability representation space light eliminate latent
study lower maintain geometric property make extend challenge insufficient difficult neighborhood unseen accord behavior classical geometric carefully robust pick neighborhood simplify lp function decrease simple extend introduce geometric extension purpose add construct coordinate na I near neighbor embed last nn et wise require near radius nevertheless try refine much decide stop capability validation error cv cv build use single repeat sample drawback big though auto lp modification training phase diagonal shall significantly automatic greatly severe overfitte appear lp lp proposal apply solid lp iteration blue represent blue error attain prescribe lp require parametrization expert still achieve moreover add compare neighbor organize world end pyramid iterative processing encoding lp capture band carry gaussian follow quantization tight multi spirit method extend embed dimensional approximate construct slight abuse language notation result level distance function original choice kernel normalize general counterpart construct normalize row generate frequency fix construct scale residual capture stop residual small stop multi kernel new overall step iteration training denote identity limit I practice soon word stop iteration avoid fact decay work kernel denominator scheme relaxation function use decay fast algebraic overfitte cost independent validation start problematic sample dependence particular subset extreme I pattern arrive validation training start increase besides attractive held training validation case model estimate error ordinary matrix modification propose lp simply consist call auto pyramid accord previous formula validation eq working cost overall procedure algorithm algorithm k p iteration obvious evaluate lp iteration tell stop remove overfitte error effect spectral cluster several dm lie sc dm dependent similarity markov neighbor markov generator coincide manifold expect diffusion markov matrix fast use carry parameter besides strict dimension induce coordinate approximate dm project dm clustering usually distance step eq density q compute embed formulate elegant dms drawback rely difficult dm coordinate potentially approach nystr extension formula approximate formula I actual aggregated weather get learn community energy particularly total daily incoming energy energy company goal application numerical weather high input contain hour increment step dimension forecast yield testing illustrate dm coordinate normalize work gaussian whose percentile distance decide dimensionality keep yield visualize result dm dm apply decide difference perform entire datum extend dm extend dm sample diffusion colored target I prediction fourth ht ht ht first blue appear apart compare colored embedding value across band trend capture along dm every measure include fourth purpose test result dm entire dm figure dm color seem less coordinate new ideal jointly quality mean dm embed embed extend clustering see notice cluster reflect dm big embedding embed overall concrete assignment assignment embed look percentage assign match confusion total new accuracy c get back try predict dm dm build figure real dark win competition track approximation actual require choice expert address forecasting illustrate overfitte lp ht plot black dot advance one see decrease overfitte ht laplacian pyramid study apply processing rather cross modify lp training yield decide avoid overfitte illustrate show diffusion
distribution estimate candidate moment inversion fig estimate moment employ converge iteration slightly record number employ th iteration employ inversion correspond effort property moment website suggest exact generalize recorded estimate th slightly remarkably estimate moment th iteration exact derivative could expansion distinguish however convergence polynomial reconstruction slow stage unable advance plausible derivative list increase exception case estimate derivative scenario hilbert schmidt fisher derive bayes formula cf extensive analysis science fisher information concern primarily family translation probability schmidt moment plot employ hilbert schmidt moment schmidt moment parameterize moment show plot comparable identical family use remarkably moment intermediate moment moment parameterize family schmidt generalize states schmidt fisher obtain hilbert schmidt certainly generalize schmidt different like express institute physics computational support theorem plus width em em schmidt partial index system generic moment reconstruction suggest fisher study density despite yet rational http generate explanatory closely small derivative geometry death concern state additionally suggest conjecture fisher parameter cite pose state essence provide highly though still fundamental system endow schmidt flat hilbert quantum eq take dimensionality generic generalize probability accordance seminal result cumulative arbitrarily precision employ reconstruction nothing specifically know plot hilbert schmidt moment hilbert schmidt intend quantum long characterize develop formula possible able separability twice root pure preserve dimension classical low bound possibly additional intercept separability boundary study death begin integral reconstruction estimate least moment schmidt fig much slow balanced unbalanced moment function advance confident systematic computation great
extend random variable correspond two name random random variable posteriori classifier give mean px c py c definition dimension define gaussian covariance product try square distance respective center spread multiply equality kx x define determinant put kx kx maximize independence give fact appear fact use suppose accordance evidence conditional approximate normal product deal vector large original rule allow parallelization pt discovery eventually appear soon accuracy analytical show handwritten digit linear stand problem product rule analyze
amount mnist digit would mnist compare embed ii near neighbor embedding axis figure require time second experiment magnitude quality constructed measure negligible advantage increase embedding construct indicate indicate construct embedding embedding mnist handwritten minute mnist embed compare visually construct four hour comment tree differ consider instead twice whether interaction node summary correspond node interaction pair use center mass diameter cell speed embed mnist figure quality trade increase quality equal embed embedding roughly dual digit readily tree perform par set h implementation scatter plot similarity show substantial advantage possible million visualization essential analyst analyst visually explore datum key traditional histogram scatter visualization variable variable therefore datum dimensional nearby correspond distant correspond object parallel decade two embedding scatter base popular technique low high define gaussian preserving learn leibler divergence distribution original perform gradient kullback leibler body system point force limitation variant computational memory object practice limit set visualize large set implementation satisfactory require object approximate force embed algorithm commonly perform simulation reduce force need force away amount implement search partition use locality performance report opt input study speed spirit interaction like preliminary find tree par opt conceptually simple prior transform speed interaction force readily distribute embed divergence similarity object embed object compute object euclidean aim point sp herein bandwidth conditional predefine per binary tailed normalize student similarity volume space location minimize kullback distribution q normalization term point applicability limit datum approximate gradient similarity probability without negative embedding nearest similarity eq herein neighbor equal neighbor tree tree object radius leaf child ball store child whereas object ball construct present one whether current inside outside create leaf median inside construct necessarily search perform depth compute maintain current neighbor determine object node right lie inside current examine object inside odd locate inside right child target gradient start force force attractive force zero element ij force algorithm k j kf exploit construct rt tree height four small illustration leaf cell root node embed locate construct time leaf cell visit gradient cell far away therefore j depth assess locate estimate approximation obtained decide summary cell compare target preliminary experiment account rapid student tail find problem complex computation algorithm consider far use twice cell summary interaction inside perhaps surprisingly preliminary still need search storing list child tree construction costly present four set evaluate mnist contain handwritten digit pixel correspond ten cifar set annotate pixel ten class five image pixel delta extract
form bind value start low region define eq true value deduce standard cumulative term consider cumulative express term lie upper physical cumulative distribution determine contribute equation reach cumulative equation cumulative formulae formulae test statistic derive previously asymptotic author suggestion double side em em present general likelihood statistic incorporate test formulae case base test describe parameter boundary interest refer define convention letter bar e interest letter boundary side refer data increasingly test direction increasingly side convention side test one sided letter consider compatible statistic side convention letter statistic definition side statistic compatibility otherwise express test side test statistic distribution one statistic previously present asymptotic double sided due function present design double follow special bound side unbounded sided pdf define side statistic property therefore similarly find eq eq involve equation compatible couple term author nothing error example count make assume count model acceptance would constrain nuisance measurement poisson standard deviation suppose purpose seek nan confidence define compatibility define consider test global arbitrarily case value number event positive nuisance pseudo respective interest show dash line obtain equation curve approximation
nan index theoretically mean analog partition relationship I split analog cluster index insight us eigenvalue estimation quantity turn influence bias sample consistent denominator bias thus thresholding bias true thresholding proportional result anti conservative denominator numerator numerator conservative lead conservative set element accordingly numerator increase tend give hand conservative explain since determine spike toward conservative motivate value combine aspect soft approach end well hard thresholding generation single realization eigenvalue let minimum summarize eigenvalue affect study result name dimensional situation hypothesis control section datum e error incorrectly collection different signal summarize evaluate type nan multivariate diagonal consider summarize quantile value empirical true uniform whose big show quantile population strongly hard uniform anti quite conservative distribution understand soft anti spike less either conservative reason conservative material conclusion less anti low little thin situation case either alone method effectively type hypothesis dramatically increase method hypothesis show next recommend method basis comparison exist study come mostly quantile hard behavior generally hard one p report various single powerful hypothesis meanwhile combine relative conservative distribution hypothesis also gain curve toward corner bt conservative true overall subsection use direction direction combine figure result single hard strongly conservative nan powerful alternative nan combine bt hard anti sample conservative well combine method summary strongly hard anti conservative conservative hard hypothesis hard soft vary strongly conservative strongly anti conservative depend mainly quantity fortunately method frequently complementary e conservative result simulation section suggest combine appropriately cluster however cluster combine indicate cluster cancer current genome cancer three array array combine unified datum four genomic gene ratio standard contain gene every method cluster index cluster except cl method separate cl draw breast cancer include four hierarchical use filtering show value suggest finding suggest important division three hard significant find scatter clearly separate close h bt despite estimation remain soft examine newly propose soft thresholding thresholding framework soft extensive compare wide variety conservative hard would incorrectly reject latter occurrence complementary combine give well newly show error indicate datum come note reject version case report significant set disk diagnostic tool applicability appear typical application website project web package replication case simulate minute comprehensive genome university north hill hill north email com email email cluster lead bioinformatic determining represent challenge serious high subset gaussian implementation matrix lead suffer severe eigenvalue address soft eigenvalue improve improvement show extensive study usefulness keyword high cluster broadly include genetic identify exploratory cluster cluster learning yielded spurious motivate cluster evaluation significance fluctuation correct set cluster structure lie component resample assess hierarchical progress evaluate serious appear literature work monte design assess answer take distribution specific make cauchy cluster may strong otherwise situation usefulness bioinformatic see test datum ratio within total variation location nan mean statistic monte carlo procedure cluster computing assess article organize give brief exist eigenvalue carefully new base combine provide collect derivation supplementary material briefly review thresholding combined suppose observation hypothesis use index location rotation take parallel major parametrization diagonal essentially factor still relatively datum diagonal
tend correlation among liu although symmetric conditional nonzero accuracy coefficient study cancer genome et tumor collect consist primary goal coefficient relationship interpret dependency illustration remove miss less express et lee liu sample median gene top select training fit multivariate truth estimation predictive number select clearly small agree conclusion lee liu fact shrinkage positive among numerical report lee liu number gene display structure estimate precision lee liu graphical capture correlation among importantly lee liu weak correlation pairwise figure propose oppose joint formulate lead numerical asymptotic response worth point formulation distributional et al author thank lee liu hill code update cancer upper kk k verify simplicity lemma kp additionally exist p k n k eq k cardinality bound asymptotically min et al constant min p q op design r c j tucker must consider equation eq note similar yu al side brevity probability k k k min zero n asymptotically large bound min k k equivalently set k three component q assumption except tend k k set k et op normality immediately univariate paper propose multivariate propose multivariate covariate response allow simultaneous coefficient asymptotic also application cancer word selection small tool analyze dataset response decompose via model suboptimal correlate genetic common et appropriately incorporate multivariate structure multivariate assumption response connect entry precision nonzero utilize dependency among response fit regression fit develop challenge regression dimensional reduction assumption et li lee liu multivariate regression penalize formulation global tackle problem condition equipped facilitate multivariate augment regression package importantly consistency sample example also emphasis justification section contain appendix devote suppose I ip tn pn x nj nk response standard multivariate regression te I identically definite dropping also since fully response conditionally variable response rely proportion dependency response become little relationship penalize literature include et li lee penalize encouraging penalty jk u st optimize update separately party establish multivariate inverse accuracy sign quantify ss k jk aa matrix tune regression assume dm min jk jk j kk restrict eigenvalue al al ss yu fast dominate suppose set separate nd normality suppose condition theorem model normality effectiveness example cancer model adaptive li lee liu sep estimation accuracy accuracy b ij report active denote specificity score number positive negative positive negative nonzero refer penalize estimation numerical criterion schwarz tune shown penalize bic minimize equally space cross lee
plot figure true positive realization divide eventually stop replace inactive generally contain well accurately terminate light show accurate initial increase solution stability compute c update compute art inversion expense straightforward number method restrict goal paper generalization swap tractable na I complexity search among support differ greedy exhaustive search compete iteratively remove multi intuitively shall see maintain support performance depend support sparse condition understand guarantee define highlight advantage present regard main discusse extension seek column present theoretical search support find minimize norm understand important analyze minimize estimate state follow include non model suppose hold specifie number observation estimate proof outline appendix step mirror come since knowledge sparse correlation broad measurement computationally intractable desirable devise tractable offer performance collect minimum entry eigenvalue define define block clear correlation contain support active pairwise correlation vector project omp accurate similar highlight support clear accurate output loss statement precise play contrast compute support inactive reason appear swap inactive correlation characterize make inactive inactive us par exhaustive characterize performance summary identify sufficient condition recovery support equal motivate boost sparse theorem identify sufficient differ variable certain initial accurate support claim around boost sparse input support minimize loss variable condition guarantee support weak method superior omp correlation contrast kp support positive output true potentially improve recover initialize clearly support ensure condition enforce simplify theorem say output long state noiseless condition recovery reduce believe algorithm outline appendix rely impose swap decrease support support dependence union support recovery impose initialize subset support event kk ds loss support active achieve variable become weak drawback require support set possible support visit possible depend correlation additional similar chosen contain clear restrict eigenvalue e use sparse initialize thresholded select top cross use iterative greedy select choose large select follows thereby validate report trial respectively line correspond line solid predict able furthermore however algorithm selection correlation increase extremely difference algorithm support difference positive mean difference generally seem advantage use likely high long mean require increase primarily figure algorithm outperform simplicity legend reason expression reason performance select value select support contain one oppose plot versus sparsity elastic require regularization run two grid loss compare superior clear base well array dimensional computationally tractable measurement boost swap start estimate theoretically justified use regression quantify guarantee support use numerical real art sparse discuss structure sparse acknowledgement thank chi valuable feedback discussion institute mathematic fellowship grant w nf analyze exhaustive decoder difference rank lemma eq tail write q eigenvalue n standard k kp p recall choose easy support every sufficient stop inactive active define accurate support condition note expression third random see projection bound simplify e e upper definition choose substitute bind bind substitute union use result evaluate span fact variable find triangle algebra finally use fact use use expression interest simplicity capture depend easily suppose iteration want step intermediate support impose ensure eventually decrease ensure outcome outcome variable event outcome inactive active inactive analyze event establish upper bound condition ensure active variable active inactive support upper coefficient notational convenience know e similar c plugging kp see next upper define put everything together leave inequality let tt projection inequality analyze eq substitute desire denote eigenvalue block cauchy edu contaminate noise standard computationally tractable regression lasso matching pursuit omp extension highly iteratively prove relatively mild measurement boost several art selection learn give number know failure massive network representation graphical application simple observation unknown regression vector art key linearly dependent zero sparse regression situation processing signal admit significantly useful signal task compression image neighboring voxel inaccurate understanding connectivity expression gene correlate pairwise pixel inner low pixel intensity correlation clearly pixel intensity gene inaccurate x ix leave middle right figure expression round blue tumor cancer pixel correspond develop greedy sparse vector e zero behind iteratively way seek main reason able handle swap relatively mild initialize use could estimate sparse later output computational starting perform naturally play true output mild certain theorem unknown support use estimate support differ condition section highlight large potentially correlation see detail setup dash marker real art sparse initialize swap demonstrate swap true plot sparse solid marker solid line furthermore algorithm subset number iteration state computationally correlation measurement quantify use form eigenvalue review literature thresholded regression threshold non output accurate support condition typically computation computationally validation apply column highly correlate exact support support small example correlate actually true correlated however improve modification deal measurement solve prior empirically different version correlate understand superior performance main contribution develop guarantee thereby portion appear formulate relevant throughout refer known linear unless mention adopt throughout support outside support variable
technical left condition ok ok generalized thompson sampling analyze expert strong sampling exist quantify correct affect regret pac combine benefit frequentist approach proof online new rely loss thompson similarity regressor elimination difference require expensive balanced expert computationally much use weighted update prediction expert focus finitely expert motivate realistic continuous discrete may device cover direction work bandit importantly agnostic thompson guarantee reinforcement self boundedness boundedness logarithmic condition context therefore simplify predict predict reward bernoulli variable success r ratio show exist calculation log shift loss corollary heuristic solve demonstrate art lead interest heuristic paper effort motivate new thompson sampling expert loss adjust loss exist quite contextual bandit importantly thompson armed bandit unknown maintain arm thompson randomly decade art application like news online advantage robustness simplicity bound success finite thompson limit bandit nontrivial dependent arm bound bandit piece bandit use assume reward author factor contrast interesting connection ucb style thompson sampling property bayes risk fast confidence bound rely assume beta thompson nonlinear contextual none quantify prior knowledge prior accelerate address connection thompson generalize thompson contextual arm thompson sampling generalize thompson randomize expert thompson thompson sampling use general loss reward later certain novel application self boundedness function competitive loss prior come bad step thompson expert bandit formulate adversary tx setup adversary contextual reader reward make simple loss generality suggest receive convert pseudo reward remain expert predict reward expert thompson believe reasonable prior update logarithmic loss expert observe loss thompson function family regret analyse interpretation posterior yet show bayes observation thompson motivate incur generalized perform adjust select addition allow exponentially control posterior w tx ta update weight thompson thompson special loss another consider convenience shorthand x w triple ir condition x r x moment moment shift condition generalize thompson expect regret rt question bound shift follow boundedness interest shift thompson bound generalize thompson replace behave rest generalize thompson sampling weight weight sum change I I due expression inequality randomize finally hand imply last thompson bayes unknown rt iw
terminate number loop weight edge decrease important output satisfy property expand let I ks v I use conclusion remain inside lemma use notation see pre thick post thick color arc blue line width arc blue b p ks b case p third construct construct disjoint b b b kb b complete section prove try use actually terminate unweighted terminate iteration loop unweighted loop line line size weight latter happen terminate measure involve inside outside partitioning quality furthermore pruning end algorithm point partitioning outside none work study inside algorithm base relaxation result improve gap admit partition kk carry domain remain problem acknowledgement would anonymous helpful like thank read exclusive comment let undirected graph eigenvalue basic algebraic partition induce algorithmic inside high algorithmic eigenfunction large large use spectral partition subroutine disjoint subset partition gap recognition system point edge connect represent similarity vertex several quality diameter center etc fail unified property propose wu neighbors vertex small volume sep thick fill width dash sep thick style fill fill b circle width circle shift line arc blue dash arc blue dash arc color dash arc color blue dash arc color dash pt arc width blue dash arc construct cluster find quality lee al design way argue although inside induce subgraph large turn objective different cycle inside partitioning partitioning third inside cluster expand say one contribution contain partitioning guarantee disjoint subset prove furthermore find optimum cut subgraph laplacian furthermore simple linear spectral inequality nearly appear recently lee et inequality lee cover graph design partitioning cut recently assume exist partition unlike inequality cluster graph ask fact partition k algorithmic loss find ok unweighted polynomial lp partition subroutine expand induce partition significantly polynomial clustering partition first establishe inside existence drop exist disjoint importance arbitrarily require gap partitioning easy prove index index therefore partition enough gap partitioning provide partition gap show partitioning star partitioning ii partitioning edge clique clique set contain clique partition argument paragraph partition proper clique clique contain clique remark disjoint define motivate definition adjacent leave converse inside partition disjoint find constant fraction partitioning merge least inside need merge partition short developed partitioning inside lee universal support et q eigenvalue support let ji indice lemma lemma disjoint property disjoint
filter ml ml show filter unsupervised cm cm markovian prior general markovian environment label solution equation emission model label label second markov consider three different procedure viterbi mesh cut order isotropic iterate isotropic estimation computational gain accuracy coefficient allow robust synthetic come diagnostic interaction multimodal suboptimal non matlab code provide toolbox website paradigm model mixture agreement classification compression early causal mesh markov basically pixel markovian application often et review field common consist alternatively posteriori iterative gibb prior pseudo complete chain propose iterate mode segmentation smoothness et provide map approximation combine solution base prior direct neighborhood widely context particular binary segmentation problem object address successfully graph theoretical cut mrf framework cut extraction dimensional spatial interaction segment cut find functional np approximate move move algorithm deal label general functional date approach li al analytic hmm strictly near neighbor hmm decode share markovian probability path et al introduce homogeneity spatial bp mode comparable et al pseudo split causal framework decode approximation also make testing discuss al therein use markovian segmentation application consider isotropic neighborhood classical quasi isotropic neighborhood like segmentation neighborhood six pixel pixel neighborhood markovian labeling mesh introduce neighborhood probability pixel neighborhood isotropic three segmentation viterbi mesh proposal iterate propose consist stage initial labeling third causal training testing underlie markovian training simply however homogeneity coherence transition viterbi decoding produce discuss markov mesh implementation unify synthetic ground implementation toolbox website paradigm processing mrf domain labels priori mrf favor particular comparison another introduce image linearly order mrf sum unary potential pairwise field spatial pixel pixel image li ji ji site label labeling realization markov call configuration ps ij ps ij directly represent popular mrf posteriori minimize zero sake completeness assume intensity mixture emission probability contextual prior algorithm ml assign mode unsupervised segmentation maximization initializations mrfs generalization process local field gibbs define field member neighbor member clique normalize clique conditional corresponding pixel give markovian potential follow clique neighborhood inverse suppose pixel intensity posteriori map external field pixel iterate subsection proposal involve map calculate neighborhood depict nonlinear counting patch see iterated mode rapidly converge function close segmentation likelihood probable neighborhood suboptimal likelihood isotropic visit pixel give maximize iterate convergence first one ml term parameter coherence reduce maximum importance final several notation compatibility equation formulae second model incorporate one consider graph neighboring pixel additional add assign pixel edge separate subset separate cut capacity implementation potential function graph cut cut diagram show partition pixel ensure cut separate label ie configuration mrf move minimize repeatedly minimize energy flow min cut performs iterative cycle iterate run labeling increase ie li possibility propose pseudo expectation equation involve transformation constrain back state well decode firstly li et multiplying likelihood without dependency give likely likely consider change large discussion incidence viterbi decode original viterbi point diagonal pixel maximum emission pixel neighbor calculate sequence ready recursion save reach diagonal track probable path know report describe viterbi iterate mode apply unsupervised em likelihood real em histogram unimodal random ml spatial ml design matlab statistical toolbox em ml literature variance parameter window update delay site graph minimum code software version code widely multimodal initially likelihood goal quantify classification accuracy produce impose markovian relative improvement true pixel random allocation allow asymptotic compute reconstruct well classify pixel scheme good report ml first compare scenario ray make ray main smoothness histogram segmentation filter common area work digital ray abuse digital x ray often image excess analyze department capability ray typical background detect change density cast histogram b image illustrate unsupervised image histogram well fuse make synthetic segment unimodal mixture unimodal parameter visual assessment difficult moreover notably value mostly unimodal essential illustrate decrease extent classification turn noisy h quite observation final equation call approximation estimation execution al et decision possible segmentation reduce parameter sequence decode work supervise university density class make sequence decode compute accuracy allow probable relative probable minute intel show resource sequence interval panel relative second ray show set account material initialize automatic supervise automatic division make job separate histogram image mode histogram ray flat panel background work obtain processing study conduct confirm study x image used g three gray background old experiment region old expert hard experiment conclusion b high different value high value follow cut different maintain detect circle perfectly recover error introduce region contain merge confusion need work reality start performance class involve influence consider c c
marginally leave residual c compare substantially superior substantially high snr visible hard observe regularize also hard demonstrate effectiveness regularization group clearly preserve denoise second reduce noise standard sec second much low maximize snr low case however observe regularization thresholding evaluate speech sentence level speech fourier temporal overlap sample penalty set value sentence signal ref university website file add speech signal illustrate result size e eight sample effectively preserve figure single part recover indicate dot noisy compare observed estimate noise time second run second perform ghz intel core matlab find snr yield phenomenon noise particular reduce deviation optimize speech effective high snr speech specify group frequency temporal spectral maximize experiment perform denoise signal snr maximize group spectrum b size snr frequently however poor investigate size illustrate noisy file area exhibit temporal exhibit correlation fig fig group area area group size area snr group size inferior inter size negligible quality snr evaluation quality allow group size adaptively extension conduct rate determine group find snr snr rate snr large noise spectral ss mmse sub block thresholding bt persistent ps matlab software bt ps provide web page additionally evaluate process square snr obtain snr six speech db snr lr method bt ps average snr lr ss bt ps snr levels attain bt second high db quality ss bt ps slight sub however factorization preserve frequency similar effective snr fact inherently less bias snr db depend improve snr denoise still improve bt already quality quality maintain utilize improves fig outperform irrespective aspect ref convergence issue relationship proximal prove optimization arise reconstruction algorithm proximity appear fall effectiveness proximal deconvolution explore general signal convex overlap group real convex constrain speech open possess derivative note increase second satisfying also monotone increase g yu convert pt corollary optimization sparsity standard estimating take consistency therefore strongly without aspect unique improve recently develop overlap group favorable additive isolated rather value tend form group speech area isolate convex practice estimation formulation advantageous robust guarantee advantageous usually residual formulation generally due suboptimal minima issue solution e base sparsity norm g seek omp iterative function convex balance second second far relate formulation group convex penalty concave parametric identify parameter obtain reliably algorithm sparsity invariant due overlap per decrease cost algorithmic step lagrange demonstrate approach upon regularization sparsity numerous distinction two overlap overlap overlap overlap simplifie overlap couple define auxiliary variable splitting technique multiplier admm increase usage indexing overlap asymptotic algorithm sparsity balance penalty extend name satisfy condition strongly semidefinite program sdp incur balance minimize convex balance maximally sparsity seek primary capture group consider also computationally demand arise current develop finite signal denote bold write integer size boundary index fall outside f make twice differentiable concave maximally concave function parameterize scalar parameterize penalty assumption rational penalty fig side unit zero scale operator fundamental efficient proximity operator derive use penalty non still proximity operator closely smooth necessarily sec strictly wherein multivariate correspond penalty threshold function absolute call threshold soft identity fast soft threshold constant signal behavior systematic hence function asymptotically prefer derive penalty favorable ref derivative fig identity penalty sec positive constitute prior sparsity behavior trial principle avoid minima sensitivity etc straight overlap regularization component every albeit induced function reason question strictly strictly unique remark simplify couple wise minimizer soft address sparsity enhance illustrated ref strictly convex suppose sec strictly comment suggest function sec numerical result maximal method desire noise non offset large small secondly group fully overlap regularization minimizing replace minimization simple specifically mm base iteration q satisfy procedure sequence converge cost specify notation dependence qx satisfie illustrate substitution one verify use right algebraic component respect readily double shrinkage implementation mm obtain constitute use equal event occurrence divide occurrence occur zero initialization readily positive due propose exclude table justify optimal denominator strictly positive sufficient division toward never converge value divide subsequently occur implementation error update end precision base guarantee moreover extensive numerical investigation rapid regularize note penalty place observe penalty list sec reveal close list precede section straightforwardly two noisy multi size express become q respectively case become etc dimensional forward select use base directly seek preserve concept soft thresholding many zero exceed noise else use lose iteration reduce power output function empirically table record depend set regularization signal
capacity htb since case appear low bound powerful produce way characterize spherical let component scalar q infeasible present numerical take indicate improvement range value look mind essentially conceptually similar limited htb typical perceptron setup quantify perceptron slightly change spherical perceptron actually perceptron course work present sketch spherical original section recall practical need storage spherical previous normal keep linear inequality establish store hand view storage site however due switch course fraction fraction pattern incorrectly store mathematical description scenario replica approach namely give storage mention follow may follow prediction error back primarily value capacity pair curve memory spherical attempt respect allow essence basically characterize incorrectly store give number store think storage capacity fraction incorrectly store say exactly comment know storage know storage capacity negative perceptron fact indicate kind behavior approach storage allow may replica stability given produce incorrect incorrect good storage alternatively fraction incorrectly store say rigorous bound spherical rigorous storage spherical perceptron fraction incorrectly store allow mention end create prediction rigorous storage capacity start write analogue namely probably mention zero elsewhere logic previous feasibility infeasible pose probabilistic infeasible attempt answer present relation probability enough developed utilize study storage spherical probabilistic result process index let satisfy inequality random quantity concentrate around enough however part probabilistic treatment leave study presentation substantially exposition theorem easy similar consideration let respectively also let variable basically careful structure allow completeness sketch core argument part process combine use obtain arbitrarily look side follow q q trivial small along linearity normal arbitrarily analogously make operational need subsection briefly pg combination follow way derivation determine low prove mention present powerful precise I rewrite negative replace zero algebraic transformation operational storage fraction integral complicate bit helpful take easily q speaking holds replace quantity side equality establish probabilistic low long storage capacity incorrectly allow purpose subsection mention replace arbitrarily constant present early ignore infeasible exercise side think capacity error allow fact rigorous upper corresponding storage obtain predict replica present three namely incorrectly store go similar curve present primary interest storage consequence discussion elsewhere htb rigorous upper bind capacity spherical perceptron store store storage reasoning line show mechanic spherical perceptron actually continue predict mechanic come storage error error problem replica symmetry mechanic stop collection bound capacity thereby rigorously replica symmetry section true range range bound proceeding presentation upper bound technical previous section probably basically characterization sign storage capacity way mention concentrate rely ultimately follow fact slight choice axis q decrease proof slightly scenario completeness utilize lift mention low course since term bind variant probabilistic low version substantially look different component much proof mention main however introduce analogue constant q moreover establish stand able characterization eq consideration thing involve say since inequality say later mechanism reference therein show inequality skip show need really difficult opinion attention improve one enough section follow let scalar independent let arbitrarily small infeasible previous discussion note storage present spherical perceptron result illustration precision possible validity though either emphasize mathematically may bit computation need shown basically plot denote label theorem addition effect may indicate theorem ultimately optimization version storage fraction allow error three case zero assume dotted indicate storage possible certain storage obtain prediction mechanic replica symmetry basically storage error store pattern replica symmetry mechanic predict rigorous view improvement conceptually easily visible reason concrete storage optimize appear select one precision optimizing mention small line denote numerous result correspond argue optimize parameter f optimize storage spherical differently perfect storage consider case store mathematically characterize incorrectly essentially storage spherical various study focused mechanism statistical prediction obtain replica mechanic refinement actually certain eventually substantial replica capture many interest relate present utilize feature direction elsewhere present spherical within frame network statistical mechanic uncorrelated case study translate cover topic randomness strictly typical one easy mention paper great result present gaussian utilize particularly simple elegant adapting exposition framework relatively easy elegant however little routine focused behavior storage capacity analytical point focus quantify capacity question relate algorithmic strength alignment capacity problem error feasibility cast convex problem regime feasibility allow concern moreover design handle challenge mostly concerned certain analytical consideration algorithmic direction mention algorithmic spherical detailed direction theorem corollary definition spherical perceptron storage limit school west mail edu long spherical seminal work start analytical prediction mechanic rigorously important course storage first rigorous capacity call spherical later bit variant spherical storage original mechanic mathematically confirm bound storage capacity moreover bound present may range away perceptron storage spherical perceptron problem course spherical class range mechanic perceptron version probably easy mathematical use early type statistical mechanic typically replica theory quantify spherical typical know example storage many capacity typical interaction strength many exact prediction solid year rigorously e many solid g perceptron particular know storage attempt storage fraction error allow incorrectly often capacity storage error already nice observation mathematically confirm need follow organize mention formal perceptron operate perceptron storage recall rigorous use storage section discuss conclude feature spherical perceptron need closely work spin interaction site site agreement introduce typically go detail mention result dynamic course initial strength property general early negative spherical perceptron may purely neural network nevertheless problem bit storage capacity spherical seem capacity work confirm conjecture upper capacity mention randomness analyze feasibility recently various randomness present randomness feasibility formally normal regard bit detail
angle apart simulation entry subspace angle subspace prior estimator large std db mean std std mean std fr letter wish distance evaluate issue subspace joint namely distribution closeness square estimator subspace formulate simple posteriori present singular decomposition estimation signal belong arguably usually conduct successful noisy observe assume interest stand free range herein recover two subspace singular svd problem seek brings exploit subspace order framework distribution estimate angle assume column identically ix improper eq assume knowledge close prior towards write principal tractable chain mind integral sample contrary conditional belong indeed recognize lead draw summarize n h nh approximate however map appear iterative alternate maximize hold moreover q iterative may particularly h n x h kn decide consider estimation end observation maximize regularization play difference look angle frequentist instead von might ph von fisher however situation available distribution
minimum able majority coordinate two lp pursuit omp decode speed even portion assume interestingly merely regardless general gap sense recover measurement programming measurement heavy choose turn exact recovery recovery compressed area research naturally pixel camera stream many realize hardware foundation single camera proposal see site pixel camera illustrative image sparse reasonable example consecutive frame surveillance background detection often h line application stream conceptually rapidly vary know store stream video naturally database high total sale often detection recent compressed stream signal rapidly time model entry entry frequency geometric stable streaming may develop focus stream goal individual summary stable write standard cauchy sample compute q value storage theoretical small nontrivial illustrate stream fashion gap sort gap call gap set repeat observe iteration helpful assume coordinate measurement denote utilize define detect whether sufficient detecting zero significantly process estimate magnitude resort follow sort gap estimator time alg intuitive explain sec note directly utilize quantity reliably precise propose technical intuitive ratio use utilize define argument log close function approximation rigorously lx maximum statistic recover ij ij ij find ij panel tail jump cdf shift cdf vertical motivate outside away extremely tail extremely around panel tailed probability sample large approximate also provide approximate cdf stable clear sample either close iterative procedure simple without generality either reciprocal twice correctly cluster regardless bit long extremely narrow recover use practical procedure strictly specify must recover analyze estimator surrogate heavy tailed cdf jump near mean observation around detect whether observation estimate observation identical surrogate basically sort neighbor correspond gap lie narrow neighborhood iteration remove reliably residual chance coordinate understand minimum approximate normally enough almost certainly minimum gap absolute verify sec word explain rely estimate least estimator special lp orthogonal pursuit omp matrix probability use denote design lp recover know pursuit prove lp exact unknown computationally prohibitive lp optimization greedy proceed least maximally square end coordinate residual code find omp essentially modify omp experimental comparison omp basic baseline recognize compressed rapidly area promise recovery message plan validate alg comparison omp present coordinate two mechanism generate signal perform long reason figure simulation heavy tailed design comparison produce build solver reproduce figure sign panel gap identify magnitude panel omp lp omp costly method lp especially ultimately fail none nonzero figure accurately confirm reliable seen reconstruct h comparing algorithm omp obvious though give report coordinate figure confirm package find build time omp lp still omp expensive j ns I really practical iterative nontrivial well make major perfectly e effectively realistic example digit example try think calculation convenient truly basically simplify long interval recover estimator advance although heavy tailed show nearly decrease difference success observation exp notational convenience write follow binomial failure sure perfectly recover large coordinate iteration additional multiplicative remain q perfectly two residual make sure success least turn see first success iteration observation success notational intuition calculate rigorous conditioning coordinate coordinate simplicity multiplicative little impact encourage theoretical analysis surrogate iterative majority coordinate secondly even require estimator compress sense use compute lemma useful fix appendix figure es q figure plot simulate together lower sharp recall candidate entry min mind alg merely step filter false positive chance ij I number complexity match know false suffice affect example symmetry understand choice assume roughly eq usually ik nonzero coordinate minimum negative eq sign need equivalently signal small dominate simplicity sign signal minimum estimator long g nonzero first magnitude resort estimator gap intuitive analysis analyze q f always dash leave solid curve numerically evaluate stable close inequality suppose bm binomial start gap th neighborhood gap x z z I numerically k x respectively plot upper suffice use measurement sample computed mind merely analyze perfectly enough surrogate recovered basically gap ideal practical bind basically twice reveal perfect recovery sec good enough practical range perhaps fall extent practical close could away compressed sense common additive involve complicated calculation see sign gaussian measurement propose still lp omp add perfect omp understand insensitive utilize recover absolutely heavy tailed likely large likely large e intuition essentially consider assume measurement assume note false study projection slightly magnitude
allow graph partition epidemic since simply reweighte spectral assess quality reweighte use highlight one show cluster social community may boundary group clique minimize group assign centrality furthermore belong high centrality consequently reweighte clique node prefer cut cut quality cut give central epidemic thus dense accordingly great impact reduce epidemic quality cut epidemic create cut pick partition measure reweighte graph produce apply measure quality measure optimization cut produce restrict cut search entire subspace span benchmark minimize respective partition ground partitioning method recover underlie community display htb c cut normalize reweighted node connect opposite community difficult dominate low deviation operator identify traditionally laplacian describe epidemic normalize graph reweighte centrality measure globally node tend clique cluster split sort community partitioning macro community difficult normalize identify edge eigenvector centrality node influence node centrality limiting cut influential community grateful van suggestion air force office fa fa department office advance mathematic widely module laplacian closely walk new exploit epidemic epidemic simultaneously transition describe epidemic normalize reweighted reweighted eigenvector edge connect central partition compare traditional clique enable effectively community module spectral partitioning cluster exist partitioning associate laplacian imply walk take module module form basis partition graph normalize though inter module epidemic dynamic epidemic simultaneously neighbor neighbor often spread analog laplacian epidemic diffusion graph dynamic synchronization show couple epidemic diffusion couple partitioning epidemic equivalent laplacian reweighted edge old eigenvector centrality eigenvector large adjacency scheme equivalence laplacian reweighte spectral partitioning efficient order base select preserve likely cut partition synthetic know community spectral base traditional method especially many cluster epidemic diffusion probe property unweighted vertex link adjacency matrix construct degree eigenvalue property simple disjoint component small zero eigenvector assign respective eigenvalue eigenvector onto subspace simple spectral eigenvector matrix splitting divide base use median large produce ratio good another normalize symmetric rw name use complement consist disjoint want eq node degree several measure cut know cut cut np complete cluster assign cluster approximation simple relaxed given eigenvector graph laplacian ratio popular modularity maximization detection analogy base partition walk stochastic transition take place randomly node cluster walk stay cluster long presence partition normalize cut epidemic simultaneously current spread disease innovation social walk way first choose walk epidemic attempt walk replicate transmission law operator synchronization couple via epidemic know epidemic threshold system laplacian heat steady eigenvector associate also explain actor actor node variable node evolve motivate detection convergence
reconstruct extract descriptor shift compressive analysis requirement guarantee shift measurement focus probability paper particularly many result tailor set complementary represent scalar absolute value scalar cardinality matrix nonzero represent ii x ji nm cyclic paper unique cyclic shift extend noisy interested sensing provide clarity multi reject true unique true connection cross eq compressed give truth shift ss hence also hold accept wrong guarantee correct notice testing shift different recover true restrictive fouri check know see column need check conclusion formulate true make fouri denote make partial partial fourier ss denote th recover shift measure fourier remarkably scalar recover true scalar note fourier suffice multiplication require evaluate multiplication correlation conservative affect snr shift none shift compressed measurement h well shift fulfil recover corollary shift noise assume give shift great cyclic shift shift hardware compute therefore particular fourier shift perfect noise lemma projection shift column recover proof since shift relate shift identical contradiction shift recover objective writing shift lastly theorem theorem trivially next one conservative column rp clearly row last distinct element nonzero condition n n r n require integer column pr propose lemma maximize column circular step solve free proof corollary hence great recover shift measurement give equal equivalent recovery noisy trivially remark corollary retrieval problem signal form typically correlation signal compressive compressed show computation classical estimation coefficient mild suffice shift signal map active sound water take indicate acoustic shift sound wave reach indicate shift problem relate alignment traditionally retrieval problem maximize two use shift retrieval allow recover shift storage compressive scheme classical predict bandwidth majority guarantee reconstruction version signal typically signal however many application aforementione example need property signal example twice bandwidth answer scheme without reconstruct shift problem
buffer neighborhood e j denote finally note neighborhood one buffer buffer augmentation buffer separate define neighborhood decompose dash edge contour indicate set intersect clique form augment neighborhood corresponding lead appropriately argument dimension relax set augment augment buffer clique possible decomposition set difference semidefinite e pattern satisfie restrict submatrix two variance asymptotic large easily arbitrary neighborhood ingredient bind estimator local global precision neighborhood nature result dimensional sample ready prove lemma union guarantee hold local neighborhood frobenius error identity concatenation equality row proof difference study neighborhood shorthand sparsity define taylor kronecker product properly form next product norm eigenvalue due due property rhs bound sensor adaptive distribute matrix call concentration centralize computationally intensive approximate base pass unstable graphical framework distribute likelihood mml estimate maximize likelihood neighborhood mml message pass neighborhood thereby regime infinity show local high derive centralize extensive suffice centralize maximum structured graphical model principle compactly characterize among node structure use message belief make suited sensor social network study equally parameter distribution essentially reduce impose matrix know concentration precision matrix optimization network centralize impractical resource toward estimation leverage distribute marginal extension idea ml approximation iterative converge parameter consider surrogate problem process limited pass effort likelihood family paper propose general framework distribute likelihood within neighborhood mml formulate minimal message pass square fix increase infinity estimator asymptotically consistent improve dimensional estimator support extensive synthetic world distribute centralize propose upon computation parallelization physical near absence distance emphasize problem robustness small I centralize ml algorithm preliminary extend come attention field work clique generalize edge extend anonymous make aware early wang algorithm focus gaussian outline paper give centralize difficulty traditional inference technique propose implementation conclude letter denote set denote two represent submatrix index index letter index product symmetric ex ex begin background consider follow correspond edge connect satisfie pair conditionally variable vector correspond graphical matrix property sparsity gaussian reduce concentration define centralize regularize semidefinite program sdp apply iterative newton step fact ji obvious difficulty global inversion cubic structured expense inversion consider message propagation pass algorithm apply tree marginal condition walk etc sufficient converge unnormalized gaussian even learnable biased drawback inference tree reweighte bp motivate estimation compute form aggregation rule lattice two neighborhood relaxation right fill dash contour buffer node color blue local relaxation dash red due relaxation local neighborhood around contain marginal j sample mml marginalization relationship second reflect sparsity mml way example index mml arise marginalization constraint surrogate relaxation mml problem apply yield drop subscript simplify notation define neighbor complement q buffer set illustrate observation preserve local parameter one buffer submatrix general observation marginal concentration affect result concentration mml therefore relax feasible call mml surrogate extract specifically index zero row refer global guarantee form many neighborhood follow subsection absence e neighborhood consist node immediate bad neighbor buffer node fill affect submatrix leave definition edge include illustration solution relax mml neighborhood inverse local case relax mml show estimator literature graph know additional factor neighborhood slowly regime requirement partly small consider local parameter introduce result additional mention one framework violate practice investigate structure sample provide concentration mapping property minimal unique provide definite implicit consider perturb concentration perturbation enough perturb denote bias perturbation analysis define hessian relation inversion last identity condition also maximum bias follow high ex last model term perturbation hessian incoherence crucial incoherence similar analysis dependent incoherence relaxed neighborhood comparable conjecture formally prove positively discuss implementation note centralize ml well develop nature final low parallelization time algorithm sparse graph perform regression immediate major maintain parameter parallelization implementation distribute become increase slowly nn graph linearly increase slowly centralize dependence fast sdp requirement communication many centralized require centralized updating node mse lattice normalize small mse plot relaxed estimator improve centralized maximum evaluate propose distribute estimator literature code matlab page focus estimator coincide asymmetric version respectively mml alternate method multiplier consensus verify randomize sample gaussian initialize empirical square error bound worth note much approximate suggest efficient asymptotic neighborhood follow visually hence omit plot synthetic set consider three world application follow topology estimate sample report illustration corner plot near neighbor distance generate unit node finally positive lattice network regular spatial correlation field square ensure positive figure graph social network biological immediate generate mechanism particular neighborhood scale linearly edge loading ensure solver runtime exhibit scale runtime use implement matlab mse curve show theoretical prediction demonstrate superior neighborhood grid mml network two neighborhood relax mml graph hard distribute mml estimator margin estimator real dataset contain temperature sensor intel berkeley dataset fail sensor regular matrix sensor miss fail trend rectangular sample concentration matrix thresholded ground knowledge
manually experience completely vary allow condition formally go characteristic game yet characteristic modeling advantage create hard game computer far good human player promise player work area player create relevant tendency tendency risk represent agent probability win much capable completely high strong preference generate conservative represent player define specific game relevant variable behavior step model character human player definition easier need hard fact rule play virtual involve technique use genetic discuss representation player preference thesis weight represent virtual agent player capable represent capacity derive requirement applicable validity different vary model behavior infer observe task weight cycle specific generate capable specific modeling player chapter usefulness present game somewhat ai reason concern necessity difficulty modify avoid base present ai game discuss principle develop ai basic ai requirement coherence variety action try behavior feature reason level character ai combination relate paradigm develop limit game ai requirement show everything e enough modification representation see definition design thing meet simplify generation easy behavior variation fourth something vary applicable despite develop similarity final sentence usefulness everything ai however direct behave generic methodology preference fellowship chapter previous chapter final preference present execution chapter phase representation player define feature game select problem appropriate configuration generate virtual agent select preference chapter mechanic interface discuss us virtual game virtual human preference different thesis agent discuss create player chapter make clearly end generate play virtual remove human hundred impossible player collect indicator dataset main dataset six game agent shorter collect keep indicator game indicator aspect subset turn overall economic overall score gold need research gain per amount gain turn notice decision without consideration distinguish support decision fact feature mention basic name new virtual create call create chapter mean trend trend virtual agent preference preference preference high preference experiment mention discuss ml remove discuss virtual datum generate different player turn consist like virtual way play game gold chapter usefulness game characterize agent predefine attribute question control agent predefine preference result characterize preference base match play ai find describe game datum preference justify agent preference observe game analyze interest preference simplify suppose value preference indicator preference look indicator start adversarial impact player analyze preference phase relax correct independence apply analyze preference preference agent dataset preference value agent characterize preference compare agent evolution regression able generality main represent indicator calculate turn amount play use turn relevant distinguish preference select intuitively regression discuss characterization separate every understand game impact whether influence preference impact separate characterization preference preference use mention regression appendix confidence interval preference indicator feature characterization indicator gain per turn able perfectly preference select indicator indicator polynomial four derivative test regression decide regression simplify generality fourth high determination indicator respectively regression present figure great expected situation preference confirm distinguish preference suggest ml preference note preference indicator preference isolate exist game polynomial degree believe limit enough grow preference indicator amount water preference recurrent situation existence two initially phase world agent turn present able model mainly segment period indicator linear represent confidence determination segment successful modeling achieve coefficient equal regression indicator indicator able agent able great coefficient determination regression still agent preference large indicator characterize period coefficient equal confidence coefficient achieve able interval coefficient able regression confidence coefficient indicator discriminative allow period hard depend preference hard model believe discriminate building evolve player raise imply high generate peak indicator sum number water game agent determination applicable two indicator relate second phase equal coefficient believe high water hard lose reason conclusion preference successfully behavior preference distinguish agent different show useful able sometimes different preference give player state preference select indicator amount gold gold gain straight coefficient determination regression besides different confidence show regression gold linear agent preference receive gold evaluation able overlap figure gold model degree integral gold decrease gold amount store agent achieve indicator believe gold activity agent continue construct thus line segment go turn go observe variability segment evaluate indicator partially satisfied characterize indicator believe resource possibility evolution time preference analyze hard characterize model reason create city preference great new variation observe explain great fact unable distinguish gold essential resource whole characteristic essential reason resource way balance expect preference distinguish characterize evident explanation failure distinguish agent game variable insufficient gave distinguish player preference algorithm continue evaluate game discriminative motivate independently make answer different e impact present analysis regard preference validate previously extremely degree lose regression good great indicator agent obtain methodology divide indicator preference lose benefit understanding equal confidence coefficient match coefficient overlap lose present difference lose period characterize match variability indicator lose match agent coefficient determination probably preference evaluate well lose match lose match generate game match turn may analysis distinguish lose variability match determination already discuss growth confidence lose match despite subset useful division keep excellent characterization period unable distinguish agent decrease additional regard firstly regression indicator equal confidence characterization regression lose match regression present case situation observe match probably small gold variability indicator reason previously model line gold indicator lose decide keep preference variability previously probably match great match lose seem counter intuitive since happen preference influence amount gold generate expansion two lose get fit bad characterize preference division beneficial ml fusion performance natural automatically result accumulation match regard representation previous chapter able different behavior virtual observe behavior must discuss evaluate infer virtual compare weight predefine show behavior explain perform evaluation whether distinct behavior task would extremely complicate small change game decide topic game inferring allow easily agent file convert state provide file unit interface modify attribute define since preference game multiplier decide science different behavior coherent like preference without analyze game would theoretically affect preference expect high preference preference agent information available file extract respective file zero disjoint behavior useful show difference discuss evaluation regression score differently transformation expect model superior curve coherent preference perform regression straight coefficient determination indicator show indicator infer simplify assignment value agent possible discuss play operation select analysis generic approach model preference distinguish virtual preference observe manually feature ml indicator division match lose game chapter discuss ml chapter appropriate result regard player preference framework algorithm good parameter experiment compose virtual agent predict preference virtual agent phase evaluate virtual agent self player preference go fourth decide task preference additionally classify learn finding describe match preference player game indicator decide absence binary precise may preference two represent difference one modeling problem show algorithm chapter report radial choose far properly classifier input generate different may characteristic select generalize whole training model classify six different artificial player never predict human cross traditionally experiment fold set set nine report fold order chance accord fold last step propose chapter extremely parameter variation fold tool optimize chapter algorithm present grid tool look tool responsible testing pass correctly divide search require preference run ease evaluate match match word remove whole stress train dataset validation remain match originally sample set match match obtained describe belong test table map c preference agent gold growth preference know agent gold percentage match array sample experimental artificial divide predict preference fold generate design evaluate capability situation use allow chapter feature agent remove use presentation majority report name preference report table correspond frequent play agent classifying model class preference preference comparison baseline reproduce number validation approach baseline present baseline analyze result generate run preference good even demand considerably good accuracy chapter bad improvement occur preference explain gold player absence turn important turn perform analyze correctly classify available line review classify finally accuracie confidence unique preference preference preference accuracy preference similar decide vary lot preference another run costly statistically inferior accuracy fold cm bayes gold science divide cm cm preference class gold discussion interesting behavior conclusion well despite researcher able see preference additionally present low classification seem accuracy give rule rule preference virtual evaluate instance dataset compose agent generate recall preference explain second experiment multi unknown agent see process execute cm cm multi adaboost gold science th virtual agent accuracy preference adaboost gold growth observe despite state adequate player preference remarkable preference accuracy every instance frequent classify player preference slightly accuracy already discuss advantage verify ai remove start observe able preference incorrectly virtual behavior importance fact virtual preference list preference poor independence preference perform believe mainly removal turn chapter artificial preference turn present classify virtual preference discuss player evaluate player play require modify thesis discuss replace stress end experiment restriction regard play match play experience specifically game post test self preference play game write available player send satisfactory player long hour people say five experiment distribution game h preference growth game never suggest classified human learn dataset much short tackle ai ai train reason experiment train model classify player report result regard player observe result player classifier preference accuracy classify turn frequent preference discuss preference discriminative turn deal despite present bad different improvement since model game analyze decide classify wrong th cm cm preference majority naive bayes adaboost gold however point preference overfitte simple paradigm presented previously learn instance frequent half interestingly preference present study must specific always specific preference thing evaluate whether player self preference present ask list preference play match cluster group already play game lot confidence self labeling contain play well design great whether correctly player understand preference present algorithm able classify either close majority class obtain minus majority class choose wrong class consider matter present accuracy obtain accuracy majority naive adaboost growth class growth science division pattern classify differently example lead accuracy variation algorithm accuracy classifying equal accuracy occur separation self consistently classifier accuracy present accuracy obtain answer see classified execute experiment virtual player able problem overview generate agent virtual agent always classify human justify virtual agent generalize human effort validate virtual useful example classifier datum sufficient determine classifier poor classify preference player preference understand overfitte generalization tendency consistently wrong thing tendency irrespective signal overfitte need cause poor classifier structure assume justify since simple sophisticated properly stress assume impact topic surprisingly consider several generalize hard thesis classifier intuition common one think never feature perform present several feature select information chapter relate turn indicator analyze galaxy chapter present conclusion thesis contribution consideration list extensive main contribution organization generic player game distinguish evaluation classifier generalization capability apply virtual player state learn compare conclude choice player problem bad obtain high generate expensive preference player accuracy preference challenge field label preference discuss topic sometimes preference ask difficult another difficulty thesis play generate approach classify application ml discuss chapter test impact difference characteristic extensively assume independent sometimes match preference preference match believe result turn take add whether enough despite evaluation contradict say never benefit curse path future regard quantitative work develop trend answer chapter possible impact application development benefit could hierarchical level hierarchical organization higher higher promising indicator well preference additionally type player discuss main automatic preference appropriate interesting impact try indicator semantic discuss determinant classify preference useful concern automatically representative could unbalanced another also investigation applicability whole correctness label possible future consider intermediate player classify player understand classify turn responsible confidence player specific player evaluation several discuss thesis ml decision make much care still topic investigation present include range use power gold c agent interval game result determination interval indicator confidence general general general present ask play game ask post player ask player ask frequently know turn figure player answer finally game n j h n iii ex de com series playing fill post player preference label preference log bin n com prefer ci cc c c c em de ci date abstract abstract r say rapidly first basic trust believe many experience carry frank chapter discuss thesis objective overview concept general common graphic artificial intelligence ai responsible player responsible interested long effort graphic game ai new graphic non player character support behavior player discover opponent repeatedly game player challenge game great regard ai gap game ai performance new architecture allow ai ai researcher digital platform game platform serious cognitive platform ai game environment real use cost sensor attention thesis concern player action behavior goal style automatically goal work belief confirm claim discuss current ai state four key game ai research area currently ai presence ai challenge state challenge step create player despite much relevant topic several goal define extract game task important extract distinguish apply specific generic approach evaluation generic player identify generic huge field create ease contribution field extract aspect literature present organization phase generic across possibility show feature state representation game evaluate applicability indicator ml regression different preference observable several evaluate information indicator classify virtual preference generate classify ml agent player preference publish contribution th international conference computer game page united games digital g computer games digital games conference computational intelligence computer thesis seven platform secondly virtual agent work game design artificial intelligence present evident publish lack work present chapter propose generic goal preference game discuss generic chapter characterization virtual behavior useful feature use ml classify ml naive model agent preference chapter result chapter background thesis discuss game platform programming main methodology thesis use platform platform interface additionally ensure characteristic require topic deep possibility present player control reach player appropriate available bc develop encourage characteristic mention six strategy game main interact deal player agreement opponent six possibility game research platform behavior unlike behavior characterize preference preference assign six thesis automatically observe behavior virtual human player offer possibility resource interface give agent preference list file preference explicit reason select platform check interface generating replace preference indirect source retrieve indicator behavior preference indicator evaluate web dataset preference model available player example model player approach supervise supervised learning class preference together preference preference unsupervise learn datum example supervise learning label virtual game different technique apply technique paradigm main use thesis section deep classifier produce different domain model try separate subspace able use margin shift change class margin seek solve influence performance responsible evaluate misclassification example simple surface overfitte algorithm generate surface correctly classify training capability define support low lead support support whole support hyperplane probabilistic classifier make perform wide apart multinomial class assume independence calculate discover divide region rule statement time successively phase grow add positive example rule either false condition rule maximize calculate false stress interesting knowledge mathematical understand understand learner accuracy well complement arbitrary input learner boost present follow divide feed misclassifie take agree take differ boost properly disjoint successively give different weight instance misclassifie thesis main paradigm characteristic dataset concern summary assume represent feature distinguish set rule currently field organization name problem contribution thesis proposal goal work relate proposal field value help reader class present division author direct indirect direct focus thesis indirect approach research university california independently develop author divide category call generate describe intend describe intend distinguish classify unique already attempt automatically verify player regardless concern behaviour behaviour topic look since go far discuss aspect must discuss summarize researcher different contribution select common l l l online track action model recognition preference model implicit strategy position substitution knowledge player main player possibility model agent artificial want environment concrete game answer question know know position evolution evolution player guide action high modeling interpret relate abstract low concerned description present level moment low abstraction strategy identification high objective finish study unique important player approach ai improvement goal neutral relate collaborative player expectation expectation action player properly behave accordingly team game order attack make act player motivation know human fundamental player last agent neutral player interact interaction game world
focus filter datum discard simply closely incorrect model extension possibility model sparse allow high efficiently minimal change noisy manually automatically annotate biological annotation logistic show handle incorrect label deal annotation error center around filter filter train record eliminate svms presence long procedure clean example begin issue detect well know like mask influential phenomenon unsupervise avoid training cluster anomaly approach tuple eliminate unsupervised filtering perhaps filter justified example closely exception fit appear modelling assumption valuable lose perhaps handle identify extension regression author hide represent true nonconvex local optima difficult seek outli relate noisy datum technique function influence several unfortunately require nonconvex objective fail give advance introduce regularize robustness effectiveness adapt regression shift especially suited commonly nlp body inaccurate important come human contamination regression model simplicity vector feature propose value shift eq shift shift sigmoid perhaps annotate label negative analogously interpret odd shift probability method advantage individually well objective may concern application increase quite notice feature new otherwise modify train specifically I immediately logistic regularize wish pose show obtain usual normally cross development unlikely free find experiment shift may would restrict estimate situation regularization equation cross validate accurate error rate cross validate make overfitte costly informative adopt validate theoretically still good present experiment effectiveness range experiment natural dataset annotation systematic handling error please appendix logistic uniform intercept create label package train noise tune intercept development additional appendix yes baseline importantly bad learn parameter run almost expression tumor particular tumor normal examine relevant place select validation looking find shift correspond attempt false comparison note need time nlp word person organization name entity concentrate word person trivially find word people name dataset create take various article five amazon place produce train vote annotation negative bring around development test news corpus annotate detail pos extract feature stanford simplicity largely consist lexical current next character gram become choice regularization nlp wish verify work penalty besides recall robust wise optimize compare propose logistic mention label link latent label logistic label relate observe one two estimate true logistic classifier testing probability discard logistic offer improvement essentially substantial depth discussion automatically generate experiment development task sentence train system create recognize name entity gate purpose tool negative false fold tune ultimately pick attempt select gave nearly result therefore choice proportion nonzero sum realistic situation label reasonable robust experimental see robust regression offer improvement robust feature uniformly decrease explanation around virtue toward near decision concentrate shift robust note manually mistake certain incorrectly label word good together robust perform occur exactly way randomization fail outcome give nlp freedom essentially extra likely modify global major across grain correction presence model slack allow wrong separate hyperplane slack penalize reasonable approach add slack variable logistic svms slack approach significant vary take draw normal accuracy margin label random perform drop extend except shift could apply group consist parameter correctly label simple binary vote example preliminary benefit convexity preserve extension regression error maintain scalability train noisy noisy continue develop promising seem incorporate present model demonstrate annotation error acknowledgment grateful encouraging past insight suggestion stanford nlp student especially patient encourage careful fill label certain derivation largely author modify logistic contain relate probability per example graphical discard predict derivation method estimate variable log give likelihood true likelihood sigmoid log obtain maximize parameter derivative calculate position quite intuitive instance cast instance weighting copy number correspond copy probability insight perhaps sigmoid distance centroid grain appendix model design annotation particular near note otherwise experiment follow simulate logistic uniform create approach neighbor label discard train logistic classifier filter select set filter appendix logistic instance weighting simulation experiment negative baseline substantially match naturally evidence correctly learn achieve match near select lie somewhat less truly chance c c experiment draw intercept introduce set could
signal remain restrict attention sub straightforward signal independence block choice reasonable simplify dramatically capture essential regime assume source signal noise simplify vector concatenation concatenation total estimation equivalent square never expect recover root feature fast estimate matrix ix iii ix invert linear independence signal note block highly source recover underlie correlate depend signal estimation source useful property transformation quantity exceed quantity slightly specialized compactly eq thus variable linear transformation q give quantity range leave tight contextual supervision synthetic thousand energy function supervision outperform begin examine experiment separate independently uniformly correlate vanish compare bind somewhat loose recover motivate low component consumption come customer california estimation survey total dominant expect temperature strong provide supervision information pg e customer census parallel datum weather capture non radial basis energy day explicitly category multiply inherent usage window time intuition consumption evolve represent aggregation many consumption evolve smoothly present consumption week energy reasonably estimation air conditioning keep estimation temperature energy water bottom present energy majority usage hour day advance source requirement amount supervision reasonably source consumption low potential automate system demand develop achieve goal interesting future interesting explicit connection sophisticated supervise measurement amount enable benefit resolution load vast theorem new framework separation supervision input feature method correlation theoretically signal feature signal separation large amount explicit supervision motivating separation home signal contextual usage thousand previously effort synthetic outperform unobserve traditionally train blind drawback look use whole home difficult training setting algorithm propose framework whereby along contextual find often allow unsupervised correlation context air spike formulate directly likely separation theoretically uncorrelated different group recover correct high separation source usage thousand previously publish formulation show accurate separation application energy potential increase separation separate although set aggregate different component code factorial within category difference concern basis signal activation base signal pre probabilistic source base datum encourage hide markov basis basis typical whereas maximize learn basis conceptually base different basis activation effectively generate basis observe exploit signal exploit grow recently since naturally usage appealing type approach build require monitoring focus communication currently million limited usage low hour lead substantially relatively amount begin optimization contextual separation trace type air conditioning home signal formal algorithm unknown cast reconstruct term
e g iterative use involve multiplication fast recovery theorem time benefit much want recover partition block non sparse block kb ks qualitatively remark norm recovery bind per also base system e orthogonal since yield theorem statement instead sampling matrix suffice orthogonal weak suppose human angle perhaps handwritten input number imagine intrinsic lie manifold manifold example reduce parameter interest learn go example face face rotation rotation develop handle general class manifold dimensionality reduce map improve curve preserve manifold embed satisfy curve distance preserve concrete image tangent lipschitz geodesic length subgaussian geodesic construct preserve curve length also manifold satisfied geodesic distance tangent trivial see qualitatively know result implication numerical algebra compress base incoherent future work asymptotically end define overview apply proof idea subspace subspace vary proof idea ex x idea constrain square general arbitrary albeit factor mention discuss appendix benefit reader tool review helpful understanding constrain constrain square provide quantitative throughout paper hold ex ex ex ex ex ex ex closure set combination sequence variable mean unless identity operator norm frobenius remainder letter associate metric respect write euclidean diameter always appear remainder relevant bound gs sg semi metric functional distance infimum collection semi norm usually universal imply use denote ball point instead detail rademacher fix correlate exactly different give one choose partition independently pick non define follow supremum rademacher refinement u nu j x ij imply side isometry suffice isometry constant state denote overview go linear db arrive inequality standard gaussian concentration argument subspaces analog statement principle translate cover whether factor define relate ex tb cover space j I ex ex kb j u apply right side projection typically numerical literature refer incoherence gaussian function dimensional consequence least form multiply lipschitz cf q estimate right conclude eq cf estimate eq cauchy schwarz immediately prove assertion large constant recover small leverage may though linear constant subspace dimensional exist system rademacher incoherence suffice combine statement arguably strong norm semi abuse notation orthogonal subspace ex ex q ex cover duality cover number convenient work clearly see j q fix general q k precede prove dominate ex right side inequality eq sn cf respectively denote estimate kk q readily ex k ex combine eq arrive refer appendix give quantity tangent definition statement convenience reader proof essence slight taking give schwarz minimizer yield assertion clearly satisfy observation value subgaussian special subgaussian particular eq fix q hoeffding v subgaussian norm side find result minimization e score minimizer suppose least subspace immediately least subsequently q show least inequality equivalent clearly verify calculus may assume without generality sign proof bind new previous work consist block dual section study least program constraint encourage information case program reduce deduce old define expression sequence subsequently eq ni independent q display together solve minimizer eq probability q suppose sparse calculated example dense result holds substantially increase maintain embed alternatively interpret norm large large bad completeness fast idea small constant prove full replacement ex ex take element eq write set summarize net ex ex ta kk cauchy schwarz eq eq whenever bound j deduce apply bind obtain split three side random therefore ex ex intensity therefore value find combine mean square side q explicit interesting set q however ignore linear bound q similar calculation since condition qualitatively duration remark incoherence collection norm hence eq collect application become compare improve rely sense recovery show kb kn ks kn non number block one lead instead impose previous bad conjecture correct lemma special implication indeed finite estimate find therefore concentration final follow smooth map lipschitz distance want ensure sparse satisfie denote curve equivalent q point bundle estimate obtain assume otherwise apply result eq recall remark b imply manifold sketch demonstrate nf construction q map condition use satisfy section sparse preserve geodesic fix q respect fix take union give choice consequence subset sign eq least ensure satisfie rescale eq decomposition condition large coordinate hence since bi smooth specifically df operator I assume least bi lipschitz treat specify take linear apply satisfy choose care deal make discretization x w b w thus condition moreover satisfy appropriate ensure preserve geodesic reverse curve eq provide capture euclidean space qualitatively know application room quantitative improvement quantitative future logarithmic meanwhile work dependence reason result bound see one decay behavior quantity place lose logarithmic factor duality cover q conjecture year lose dual avoided avoid consider bipartite vertex degree ask communication leave point second visit second thank collect useful rademacher rademacher q th inequality frequently rademacher elementary independent combine rademacher claim tool cover set let close symmetric associate semi conversely convex q definition closed covering close finally lemma usual proof semi constant let write distribution fact close closure element cone denote cone cone states closure tangent eq clearly cone xt fy descent cone close convex subdifferential nonempty verify use tool cone consider let nonzero particular appendix program definition fourier transform n nx diagonal matrix low q radius associate special fix hoeffde subgaussian metric old q solving result functional definition let banach modulus convexity convex observation concave banach lattice e r r old readily associate estimate ex ex q I integral e q taking conclude take variation note element index every fix rademacher hoeffding subgaussian semi metric ex iv x md norm lemma corollary consequence r corollary theorem result qualitative discussion corollary eq worse well remark corollary lemma advanced nj transform subset preserve simultaneously geometry concerned qualitatively several lemma embedding transform algebra classical dimensionality array machine learn graph interior numerical manifold finding biology spam classifier email represent index dictionary words star vector light intensity point technique storage speedup analysis acquisition transmission computing dimensionality routine specific preserve inter angle preserve angle preserve euclidean norm achieve nearly cite space proof provide constant case could ask distortion infinite application exist practice move fairly question distortion distortion multiplicative arbitrarily employ stochastic gradient
experiment define section replace simpler known let positive deduce course vx v satisfy component transform white four matrix follow present readily extended assume briefly mention affine first resp detect useful illumination may achieve criterion remove intensity intensity patch feature invariant change illumination ad feature combine linear aforementioned descriptor location extract obtain rotation estimator rotation r correspond give rate exactly interesting concern equivalence completely match give criterion matching invariance concern identical situation look present admit counterpart von valid set justify von issue robustness presence situation robustness logarithmic function indeed thank slow matlab experiment simplify solver computation gradient time take six ht distinct indistinguishable well greedy indistinguishable greedy four trial plot fig methodology provide previous chose varie estimator trial estimator even pseudo confirm finding present motivated feature propose rigorous estimation minimax hypothesis theoretical appear however least theoretically section investigate statistical consider practice matching consider propose case matrix far collect theorem argument lemma identity denote confusion p permutation concavity logarithm readily inclusion well imply easily variable hold tail bind conjunction deviation chi square consequence pd bind hold estimator include hand side already concern general mutually absolutely define mapping separately integer assume generality construct large denote k lemma k get complete sort increase nm n monotonicity dc properly nk entail leibler eq view include choose readily suitably control probability impose right last part case part analyze follow mutually mapping differently vector packing permutation define infer consequence complete permutation view composition support distinct pair index leibler thank last inequality yield n r readily hereafter uniform define density first display follow inequality computation conjunction fact view union rademacher probability nn first auxiliary pair distinct index claim trivial always one permutation position notice one correspondence q follow bind integer integer nonempty continue denote integer choose permutation support odd much difference sum every index proof permutation permutation appear use alternate acknowledgment support grant thank valuable suggestion point mistake vision formalize view level dimension tight upper show setting demonstrate phase occur rate equal contrast consistency synthetic finally matching criterion minimax permutation hereafter refer contain coincide goal match tight possible accurately statistical allowing furthermore assume pair situation close measure quality procedure concept adopt spirit computer vision tracking carry example sift image matching object follow create simultaneous focus overlap resolution large sift naturally context assign estimator minimizer log modeling modeling noise level constant across noise noise noise unknown estimator normalize consider consistent similar condition condition aforementioned factor prove rate separation identifiability ensure proper bind affect quality estimation easy adapt presence carry small confirm outperform greedy square argue three estimator likelihood methodology computable measuring permutation true result establish equal permutation equal otherwise interest amount control feature offer evaluation moderately large constant rate regime dimensionality pack ball symmetric group quantity conjecture multiplicative factor us work conference estimation permutation analyze separation computational estimating procedure proof theorem lemma randomly generate model r task finding subset call belong ease mainly however carry correspond transformation turn outli eq admit counterpart level satisfy consider data generate estimate consider nuisance parameter follow write permutation measure quantify permutation hamming goal design estimator prescribe deduce hierarchy estimator bind multiplicative exist absolute bind infimum permutation state say expression confusion concern measure readily derive theorem know side adapt lower replace one tend minimax sense however analogue minimax switch discriminate procedure serious know level fulfil level estimator separation affect two natural arise start adapt possible depend limit considerably theoretical know substantially minimax equal permutation level estimator soon noise different particular permutation coincide event tail vector equal consistently permutation eq kind estimator competition determine minimax rate level cd n infimum permutation inspection show discuss question early concern distance separation procedure greedy similar superiority confirm simulation next sequel assume need order interesting compatible theorem strictly speak imply minimax latter exhaustive impossible soon polynomial coefficient von state von doubly every row combination permutation
problem subproblem feasible subproblem optimal subproblem value subproblem span subproblem solution invariant probably adjacent subproblem pool subproblem adjust maintain terminate pooling feasible subproblem find term bayes logit logit w ps solve determine namely proportion require w minimize rate hard proper ratio influence proportion weight monotonicity logit invert remarkably v objective subject simultaneously optimal log odd logit strictly monotonic solve determine may label un weight problem monotonic optimally proper scoring address concern reader may solve actually useful pattern recognition calibration utilize concern address non mapping reference cite another output flat invertible invertible invertible relevant contain score concern proper information logarithmic scoring equivalent shannon measure proper class monotonic transformation form monotonic infimum monotonic strictly monotonic rather calibration transformation role role form monotonic optimality employ thank calibration support follow define adapt family binary proper rule represent way therein representation let always interpretation integral reader notice proper rule member bayes decision redundancy normalization arbitrary scoring family member none interest pattern goodness calibration optimization calibration monotonicity regular score hold calibration independently prior pool proper calibration scoring rule much adjacent recognition contribution concern introduction define function calibration organize optimization problem solve calibration calibrate short discussion calibration pattern supervise monotonic parametric calibration proper ratio interested calibration pattern design discriminate classified classifier call make process minimum act target give transform calibrate first part calibration transformation context appropriate recognition case represent purpose mind decompose transformation ps non stage ps perform adapt application rule target therefore prior step calibration tool map quality effective goodness purpose quantify cost ability focus scope regular score convenient purpose appendix give link define q must proper scoring rule family parametrize almost everywhere later dirac delta misclassification threshold note moreover convex dirac proper scoring also score score salient strict minimum derive far find calibration via optimally firstly constrain preserve score secondly calibrate pattern produce map monotonicity sort input score refer one true denote every combination trial function minimize minimized problem probability denote p subject require feasible minimum already know special theorem detail state proper scoring publication already score scoring mention result without state monotonicity therefore transform let index go subsequence theorem subject tm observe let closed know load straight forward implementation section need subsequence start partial subsequence problem every subject monotonicity notational equivalent subproblem monotonicity meet feasible subproblem corollary every unique adjacent mean q step prove optimal subproblem partitioning total every subproblem minimum partial q concatenation subproblem solution necessarily whole recalling note j ji rhs partitioning whole minimum let whole total objective ti solution importance use short hand subproblem label subproblem solution exist subproblem important examine behaviour govern interval v subproblem q r proof drop let clearly prove property examine observation solely sign giving conclude strict property later subproblem solution subproblem index subproblem solution namely k constant ii similar adjacent must subproblem satisfy p j subproblem optimize every ii need index combine
partition cluster consider cluster partition note kt fp bounded tend sense eq reason k km put everything mention title link hausdorff distance detection propose hausdorff denote minimal hausdorff partition p partition moreover section note hausdorff inferior length segment partition l k mp give proof hausdorff statement pd attain thus p attain go back pt mit problem cluster segmentation video change focus implicitly euclidean base spectral cut goal mahalanobis lead availability dataset cast prediction regularizer iterative improve example bioinformatics segmentation image problem orient bioinformatics signal traditional mean linkage neighboring shift normalize cut segment algorithm change see specific generally crucial heavily metric trial recent learn metric directly supervision generative model reduce dimensionality guarantee lead follow consider partitioning sharing metric label label several already build often segmentation see bioinformatic see partition explicitly implicitly normalize cut detection partitioning cast rescale algorithm spectral relaxation dynamic programming label proper regularizer augment partially label dataset iterate extension univariate rather bioinformatic experiment video segmentation go beyond unsupervised learn base link consider label metric good unseen work link mahalanobis distance give include setting stack discriminative approach availability partition unstable stable margin supervision dimensionality cluster metric unable take label information relate label similar learn cut augment structured segmentation goal supervise unsupervised share conceptual like structured ranking section multi represent equivalent general additional assume contiguous increase vector one form model model overall goal distortion frobenius norm define assignment centroid matrix solve close form compute thus partitioning minimizing naturally parameterize matrix element cluster th otherwise cluster contain th order compose contiguous eq rescale trace km equal cm see point extra constraint add common situation estimate proportional cluster instance classical rescale following learn partitioning constraint problem constraint make polynomial programming general although get partition relaxation use detection contiguous change partition solve cluster extra polynomial programming e describe solve require namely area image left area initialize diag backtracking old matrix cluster optimize close minimum exact decoding polynomial readily remove constraint I relaxation orthogonal eigenvector positive orthogonal eigenvector eigenvalue thresholde less subset index intuitively suit rescaled naturally loss asymptotic loss hausdorff show structured output algorithm describe cast dot belong goal pair exactly margin structure matrix either section margin rescale standard may need regularizer solve norm rescale bottleneck situation exponential partition bx loss augment perform namely regularizer limit parameter input impose metric cm interpretability term symmetric definite present drawback speed prediction cutting method use project cut plane empirically outperform present extension balanced graph replace normalize similarity concatenation q semidefinite practical combination attractive convex parametrization spectral outline use eigenvector one corresponding eigenvalue threshold become w tangent optimize iterate process converge relie label dataset dataset correspond rescale equivalence situation start iterate dataset section label problem whole piecewise datum single detect change feature naive prevent consider th ix researcher link copy gene dna type change manually annotate error change identification metric improve performance conduct experiment improvement margin metric series change since identity hope weight segment remove give fraction information figure improve pca technique stack supervision directly adapt point detection performance method change set partially apply come old tv length series alternate video speak audio aggregated series thousand use show running time consider image stream audio case exist setting metric learn audio stream learn stream audio robustness three stand cm c c video pca cm induce ground
nonlinear fast implication appendix momentum thing regularize add hyperparameter weight validate draw neuron layer guess interval select run list notable transformation large size turn give iteration start decrease linearly connection multiplying layer examine big regular section dataset digits network hide neuron minibatch minibatch transformation update begin epoch update equation burn increase starting keep constant momentum decrease exponentially hyperparameter validate high accord variant seem high enable york university york ny usa recently neuron perceptron zero slope separate connection continue firstly introduce third transformation normalize analyze connection show transformation theory third speed performance converge output neuron close work return notice either sophisticated enough learn deep network perceptron mlp instance recommend even center connection center slope assume issue center nonlinear neuron zero slope include third explain usefulness transformation study fisher measure traditional transformation dimensional block approximated unit information matlab code mlp function nonlinearity gaussian additional supplement nonlinearity nonlinearity update evaluation help ensure motivate nonlinear activation similarly activation zero slope affect linear mapping linear dependency model many compete input g hide argue competition speed another fisher goal normalize signal normalize motivated observe matrix signal diagonal mean unity q element empty term center deviation operate overall transform similar equation second method natural decrease compare basic descent easily model heavy natural multiplied use fisher multiply rate transformation move fisher close zero element make behave gaussian depend related datum simple random rotation layer standard gradient weight decay network transformation multi network fix unity transformation hessian matrix show epoch roughly report along angle method compare eigenvalue transformation mlp eigenvalue angle second plot positive update close suggest epoch close unimodal suggest transformation vector evenly regular close tb conclude another whether addition back propagation clear transform compare standard back propagation network task sharing boost decay add layer line error classify set result back blue task network architecture affect result architecture three architecture result error especially dropout network regularization tb study auto encoder third pose neuron hide tend inactive beginning encoder despite distribution output neuron encoder see begin reach bottleneck nothing neuron experiment describe however seem speed transformation
imply rigorous degradation may label consider weak claim support unlabele include feature common reveal unlabeled selection label finding contribute text look degradation broad view believe ng discriminative vs generative naive bayes classify label document al text investigate stock micro sentiment forecast future movement lee opinion sentiment chen hierarchical training set principle incremental learning centre ai road china mail cn sentiment publicly active argue effective capable manual effort expensive consume effectiveness unlabele problem understand experiment bias broad know degradation cause unlabeled whole argue bias balanced unlabeled likely classification label besides application possess illustrative implication text sentiment sentiment opinion research lee sentiment start increase create various phenomenon largely attribute rapid web trend review website ever management sentiment become enable stock researcher examine financial report find future sentiment analysis hard sentiment formalize liu lee neutral machine believe research sentiment text classification address towards sentiment analysis important area gain consume moreover expert quality label label label infeasible usually paradigm utilize label unlabeled datum attractive many sentiment text classification task effectiveness unlabele al zhang li argue dimension namely label classification experiment systematic extra well degradation cause datum meanwhile conclude trade likely paper summarize gap research methodology dataset present discuss involve unlabeled section conclude widely accept unlabeled effectiveness diverse combine active algorithm able largely unlabele show amount unlabeled unlabeled classification naive em illustrate usefulness text show generally speak relatively optimistic issue indicate effectiveness unlabele optimistic point degradation cause unlabeled usefulness datum need mle theoretically mle assumption large likely unlabeled performance degenerate almost violate unlabeled degradation datum study may prevent understand degradation broad study effort devote research significant influence unlabele try understand effectiveness unlabele li focus mentioned sentiment analysis identify sentiment analysis well discuss underlie setting choose multinomial bayes carry reason preferable choice classification widely ng utilize unlabeled text observe seminal al study describe use utilize unlabele need portion hand follow form introduce variable tractable equation log aforementioned step step parameter optimum detail literature financial publicly financial experimental application sentiment become closely web network believe system continue besides sentiment become influential li detect business etc consideration financial firstly multinomial naive generate classification real label sentence financial unit label essentially low document almost always degradation cause observe simultaneous condition degradation secondly sentiment single sentiment opinion liu sentiment since study label would manner md k exchange dedicated company company generalization comparable sample collection randomly company reason avoid write company list financial report website md assume sentence neutral consistent sentiment lee engineering ax cs co de microsoft ms sentence year put pool pool ensure double check public label sentence unlabele pool label column amount company neutral sentence company regard approximate tend understand unlabele namely amount classifier amount unlabeled variance trade influence degradation increase amount presence unlabele et argue additional feature potential influence instance conduct every amount label variance break imbalance largely simplify seminal et al et perform varied reason devise consequence choice vary unlabeled influence classification fit label varied unlabele carry gradually able generate series confidence label division label varied amount unlabeled datum way fix include unlabele include unlabeled sample de ax ms unlabele decide purpose test varied firstly datum character one label test firstly add sufficient add finding include train several result generalization balance achieve satisfactory vocabulary diagram always add classifier
classifier high whereby optimality gap adaboost classifier achieve log metric algorithm weak whereby hold hence complexity gradient thereby extent adaboost satisfy condition subgradient regression give vector matrix coefficient high zero coefficient context coefficient incremental forward stagewise type boost arc updating present initialize r j property different stagewise algorithm fs greedy subgradient instance subgradient residual space residual convex r interpret objective gradient least loss solve initialize absolute subgradient computational factor instead shrinkage dynamically stagewise fs value priori square least hence subgradient mirror descent euclidean prox residual fr yield spirit imply interpret guarantee closeness coefficient thus know furthermore optimally interpretation property combine guarantee indeed advantage property ta define induce bregman furthermore section bind fa mit cat seed research nsf research fellowship mit cat de highly boost adaboost incremental stagewise mirror descent consequence obtain computational adaboost function widely supervise combine fashion overview adaboost develop application methodology stagewise establish two algorithm descent optimization well boost understand order new algorithm learn briefly development complexity adaboost appear related lot connect adaboost boost understand computational guarantee problem much work focus minimize et framework boost function adaboost problem minimize determined et show inherently link produce adaboost desirable maximize order develop show maximum weak herein exactly mirror descent dual pair iterate minimization set mirror correspondingly adaboost optimality case separable gap infer gap without margin rule seem herein learner always adaboost fail converge margin use originally prescribe search loss coordinate adaboost build cause fail maximum long instead exponential gap produce objective optimality gap limited line size suggest mirror different determine search separable apply incremental stagewise shrinkage correlate viewpoint ability tradeoff play important solution well forward stagewise well forward stagewise additional condition lead coefficient path accommodate simple question show also interpret square quantity choice shrinkage unit x ax subdifferential v pp eq smooth convex subgradient assume lipschitz primarily interested case define dual solving bound prox subgradient prox tx precisely subgradient mirror descent mirror euclidean useful adaboost tx x ne norm short format entropy follow state know mirror algorithm gap apply compact gap specify particularly mirror compact general bound sequence prox proposition specification fix rearrange bind subgradient sequence hold rearrange eq finally mirror size optimal hypothesis j good weak learner compute iy may adaboost base classifier perform base classifier algorithm initialize tw maintain classifier combination classifier speak base simplicity refer combination determine adaboost mirror descent primal distribution duality space variable represent vector coefficient determine exactly gradient utilize descent establish gap duality problem separable imply adaboost maximize computational sequence work drive function zero computational step denote th primal call eq format dual minimization whereby q whereby achieve margin positively normalize problem normalize also classifier adaboost log function equivalence arise primal mirror
performance dataset dimensionality factor dim red pt eliminate attribute normalize subtract divide normalize performance dataset fold choose fold hyperparameter train include stop reach avoid overfitte improve chosen number minibatch epoch comprise unit unit conditional grid learn decrease epoch autoregressive mixture validation method performance superior dataset conditional well unfortunately reproduce fold work mixture measure ability patch pixel patch ex patch output draw range lead narrow possible discrete order pixel divided take reduce value perfectly predict likelihood unfortunately likelihood previous dataset interpret preliminary dimensional datum always thin ability dependency dominate recent lead eigenvector measure comparison amongst discard train randomly subset validation patch subset stop measure log million compose image present subset use preliminary search minibatch minibatch epoch comprise start decrease reach momentum epoch improve early stop sign overfitte run sign overfitte well minibatch step towards parameter decrease reach stop mixture gaussians pixel website th column covariance good log average patch gaussian test sigmoid unit unit test sample pixel fourth sample figure constrain range sampling otherwise go away test much put probability scan order perhaps surprisingly pixel make difference pixel measure extract patch frame filter bank encode common visualization frequently speech could example denoise fit compare linear manually minibatch procedure gaussian log difference look like structure peak energy band core hide component horizontal display draw network skew multi modal parameter network grow target unless assume specify distribution sufficiently gaussians represent find problem representation marginal patch close different predict patch mixture predict context work explore scale restrictive unimodal aid size across application density red match patch green vertical line indicate conditional log conditional conditional average main drawback neural general decide hyperparameter adjust automatically efficiently grid several field applicable possibly show make relatively straight translate advance least inductive bias novel linearly task represent image patch excellent state outperform mixture acknowledgment thank thank pseudo density material http www com en research unit enforce gaussians activation calculate recursively scalar activation error calculate gradient z x tight slow high rate z function dd calculate conditional h model ascent log gradient automatic library find manual term cache store conditional th eq derivative layer calculate partial output derivative school ed uk universit conditionals mixture share learn tractable calculation density comparison heterogeneous machine involve model collection grow model generic probabilistic multivariate consistently mixture gaussian gaussian seem patch improve insufficient mixture rbms mixture explanation rbms form unit rbms number hide rbms approximate autoregressive direct feed neural introduce perhaps previously combine mixture flexible real point compute rbm flexible variance context linear component mixture attribute factorization network parent nothing perhaps conditional tractable compute gradient optimizer visible sigmoid network competitive approximate capable spike rbm future direction hide weight rbm model great flexibility parameter dimensional output mixture share parameter represent neural network discuss unit
dimensional combine inversion regular appropriate adopt dimensional inverse pose gaussian operator inverse differential problem still particular motivate operator inverse specify two either construction truncate kl inversion impractical require term prevent toward contrary differential enable build operator relate random allow interpretation gaussian field permit exploit sample employ solver discretize system distribution meaningful require pointwise instance continuous field q pde function green direct property green function one random operator along unbounded intuitively pde covariance green operator example bound green three dimension section provide allow straightforward discretization extract finally linearization parameter map define x mapping pde operator extract pde still measurement due lack reality discrepancy bayesian infinite problem bayes understand prior formula dimension use thus hold square inverse pde satisfy definite exist solution differential let hence densely operator assume require adjoint invertible eigenfunction form growth eigenfunction regularity distribution whose law almost sample continuous first explore statistical posteriori map measure dimensional setting define map maximize product argument show problem linearization posterior observable map reasonable scenario linearization noise parameter nearly nonlinear certain condition many lead approximation linearization initial metropolis relate employ derivative assume observable fr fr evaluate posterior map also find observable reasonable true around problem discretized high dimensional mesh converge way particularly choose mass euclidean covariance conventional sense self adjoint proper discretization counterpart inner ultimately development finite study field realization carefully mass due finite element discretization generation pointwise exploit rank lagrange correspond instead infer perform consequently infer simplicity symbol tm product mass matrix approximate infinite distinguish usual euclidean mass matrix product adjoint transpose ij ji imply q endow projection h l h implicitly lagrange function derive prior matrix entry adjoint sense position gaussian bayes since posterior natural measure likelihood lebesgue finite bayes omit express recall log counterpart inner wish store dimension large expensive reason jacobian map generally matrix typically pde intractable solve develop gaussian realization independent weighted sample gaussian generic gaussian recall hc nc method precede equation matrix ij consequently discretized covariance discretize read distribution build map first explore discrete use optimization problem amount scale numerical scalable reader detail particular wave inexact product linearize adjoint pde never explicitly inner cg gradient encounter cg backtrack solve pde scalability forward pde solve ingredient scalability increase outer inner cg independent mesh inverse wave propagation consequence hessian term term perturbation cg exhibit mesh discuss map posterior hessian hessian linearize pde linearize pde parameter seek explicitly compute covariance exploit approximation linearize parameter observable define result obtain gauss newton portion hessian newton adjoint instance adjoint wave adjoint expect observational large portion hessian hessian notation focus portion apply hessian might noisy datum rank translate exploit computation pointwise field pose gauss hessian gauss hessian evaluate like compact mode influence linearize present hessian observable perturbation show gauss hessian compact medium gauss finite mesh exploit compact construct scalable approximate inverse hessian convenient approximate hessian decay rapidly low rr r accurate approximation rank approximation approximate covariance give uncertainty uncertainty gain root filter acquire square root covariance linearize pointwise kk rv posterior detailed scalability low construction posterior dominant action linearize costly linearize forward pde solve however vector solve linearize pde term incremental regardless parameter solve linearize adjoint pde term adjoint expression incremental forward adjoint acoustic wave propagation solver incremental forward pde scalable vector eigenvalue dimension requirement number independent discretize continuous direct consequence prior gaussian counterpart compact wave mention acoustic pointwise observation operator show operator domain hessian smoothness smooth decay hessian number dominant eigenvector discretized field rank construct adjoint pde require posterior dominate discuss amount carry adjoint pde solve number state moreover since dominant cost adjoint problem scalability uncertainty quantification adjoint pde framework global heterogeneous acoustic wave speed surface rapid advance observational capability growth wave propagation govern acoustic successful global source take wave involve acoustic wave application discuss discretization adjoint wave numerical provide section km two representation current symmetric preliminary point determination solver real behave velocity model wave speed variation homogeneous reconstruct ground model synthetic quantify interest anomaly sensible specify surely discretized mesh mesh layer map cube mesh interface outer wave speed significant align weak mesh coincides wave mesh locally refine resolve mesh use determine map mesh forest library mean operator deviation acoustic mean therefore product determine next local wave wave speed source average material neighborhood effective wave mostly wave see deviation effective wave prior maximal deviation encode select near gradually weak observe obey similar precede choose km length radial direction smoothly sphere precision illustrate slice boundary partly construction mainly homogeneous boundary use construction square root reflect large close length direction ground figure comparable contour slice contain gray green function directly covariance note close ex order observable map acoustic wave density acoustic wave speed eq together boundary propagation choice velocity velocity elastic wave infer spatially wave speed receiver observable wave speed acoustic wave record velocity truncate synthetic prescribe uncertainty quantification computation derivative negative turn prior derivative adjoint clarity infinite denote wave wave observable wave velocity forward wave propagation adjoint approach adjoint adjoint satisfy adjoint terminal adjoint adjoint wave equation solve backward due wave equation computation act incremental forward wave value e incremental adjoint wave value c see incremental adjoint wave equation forward source gradient solve adjoint incremental adjoint wave equation respectively amount contain include incremental adjoint wave equation mesh discretized wave variant adjoint incremental incremental support discretization lagrange together gauss mass integration stage face replace face perform face project face consistent convergent scalable pointed hessian former limit mesh reason jump due introduce wave verify discretized gradient action approximation adjoint incremental incremental adjoint conversely compute wave solution restrict provide solver ensure hessian restriction operation adjoint discretization problem section require repeat pde efficiently algebraic ml wave employ poisson gradient require adjoint correspond computation store scale challenge incremental incremental adjoint incremental storage history use employ reduce storage expense increase wave core hour solve inverse section vast spend compute solution adjoint incremental wave equation map compute hessian wave propagation solve good wave propagation rapid scalability wave overall scalability inverse synthetic generate wave equation wave speed wave high order source locate km source narrow gaussian wave propagation mesh discretization velocity fine accurately resolve frequency hz fourier receiver location retain mode fouri vary receiver south wave time wave unknown material discretize mesh represent discretize rd million spatial wave fourth time along figure wave discretize wave rd nd wave propagation velocity wave speed rd velocity amount million wave discuss quantification uncertainty inverse numerically dominant former reason three simultaneous problem necessary sensible approximation constitute product solver adjoint wave order translate statistical inverse ii discretization wave essentially mesh resolve hessian consequently wave solution require depend dominant dominant eigenvalue observable associate become reduce scale observation pr pr pr pr pr p study expect reconstruct truth reflect assess knowledge gain data reduction variance compute posterior gain decrease relative uncertainty surface receiver informative core reflect surface confidence interface var var study see recover wave speed portion cover pointwise deviation observe uncertainty reduction wave
vote weak meet requirement bag forest weak binary classifier interest vote simplicity classifier slightly weak learner hold book weak achieve typical example seminal slightly turn capable achieve arbitrarily stand constructive sequentially boost learner certainly vote perhaps partially incomplete bagging indeed vote partially netflix netflix logic weak crowd majority important hope positive signature operate roc curve roc early illustration roc mechanism completely parameterize mechanism bootstrap bag average collection shall open ignore trivial pg pg say fairly classifier certainly individual f individual collection vote classifier apply theorem technical detail pay section estimate limit finite ref transition whether result theorem compute give determine whether origin phenomenon reader interested skip jump pg similarly limit space phase transition boundary particular jump go behavior l fix evident region whether small jump cause phenomenon universal regardless easily remainder focus special classification limit intuitively classifier indeed therefore base weak vote collection precede truly beneficial diagram show specifically need different side weak classifier word vote majority vote learner classify everything become classify everything whether green assume notice occur perhaps somewhat conclusion possible even figure bad vice versa point everything surely classify thing opposite conceptually mean may think weak learner average vote improve result obtain assume collection e word classifier collection arbitrarily weak conclusion total pf I obvious perhaps argument definition p similarly reality independence individual quantity suppose interpretation member large decay limit finite mention chain monte mcmc liu behavior typical percentage classifier achieve surprising clearly close asymptotic overall slow convergence although eventually achieve middle case consider section realistic essence collection realistic likewise long base become technical theorem namely merely indirect way correlation among classifier random paper prove thing even ideal majority
ease thresholding include thresholding dimensional index diagonal wu zhang tuning et integer estimator belong cholesky fan fan fan popular frobenius norm norm euclidean vector q attempt risk manuscript organize consideration conduct extensive fold base bootstrap technical cross especially validation dominant tuning cross first split un consideration validation datum argue asymptotically sample go reverse tuning regularize reverse select regularize cross validation reverse fold cross frobenius decompose apparent penalty bootstrap correspond frobenius eq tuning selection recover select bootstrap sample dimensional bootstrap well point bootstrap multi increase point also often quite remark formula unbiased sure estimation bootstrap difficult like derive rough approximation estimator hope accurate approximation let argument pl j eigenvectors eigenvalue eigen hadamard report well although derive frobenius simulation eight compare frobenius supplement mse exclude matrix structure figure four regularize perform fold cross unnecessary fold quite suggest fold compare subsection method summarize sure cross sure perform validation cross perform slightly cross sure figure fold cross validation estimator estimator sometimes fold much method slightly fold validation eight similarly accurate cross perform perform reverse cross estimator fold validation fold good addition significantly either fold good word big frobenius big operator validation summarize fold validation rough approximation reverse computationally recommend fold bootstrap cross group accuracy two norm operator base suggest cross operator reverse eigenvector delta l power side equation slight abuse note stand wishart argument suffice unbiased j manuscript use cm definition matrix fold simulation fold cross validation
information pair actor assign view actor receive indirect neighbor path consider whether previously contact especially classical algorithm spread requirement modification straightforward use length short path process ever take reach length direct course implementation chain information correspond temporal practice modification substantially lr avg pos avg avg pos avg vector keep outline coordinate link social line dash horizontal update view vector temporal track occur actor fourth pair node think well actor window illustration like much user typically day close may track absolute temporal view correspond describe rank absolute update indirect yield include might simply multiple parameter classifier feature reach reach create come twitter medium form communication explicitly public communication user tweet tweet user refer user twitter include communication e loop tweet user event tweet basic dyadic event stream tuple cover twitter include tweet introduce user dataset tweet news platform twitter list inform topic twitter account candidate office member us house total dataset date st collecting twitter mention tweet date range international social several online facebook dataset university static contain node event come student california unclear private public message dataset cover great majority mid mid mid week period message send period high predictor link baseline link predictor require one perform employ keep track directly many consequently statistic substantial geodesic decrease suggest useful local long exception trend boost stand accuracy predictor network imagine try direct since send message significance send message responsible benefit track run entire exclude result present present drop benefit combine stress new link form training exclude art operate set finer grain ignore case formation cascade approach co precise event largely irrelevant publication network twitter mention cascade drive formation event highly link event panel mechanism try b extremely piece send aggregate event static dyadic fine grain temporal information beyond relevant actor future track actor date essential modification reach indirect indirect update place communication practice restriction dramatically thus applicable network million actor event classifier exploit utilize wide panel well moreover already much learn apply real grow sequence supervise add remove intuitive performance advanced understanding mechanism behind formation specific e combine selection elaborate fair effective page u link survey longitudinal network university network u dyadic influence twitter neighbor j page message pass preserve order page f virtual page detection mutual distribute system pathway social concern f plausible size exploit locality maintain causality page efficient page world problem j collective dynamic network hill network human evidence importance twitter lee investigate online spatial interaction r link r prediction utilize panel social adaptation computing interaction formation yield date database management database application predict interaction prediction dynamic usually make panel former refer grain temporal hand finer grain relational event panel often survey typically outcome automate file mail phone twitter censor panel seminal define slice aggregate relational static expense employ tool static longitudinal crucially conduct co publication relevant problematic grain temporal may relevant justify pattern interaction exploit regard extremely piece send likely soon response highly relevant social aggregate communication link scheme demonstrate efficient operate directly appropriate dyadic advantage keep track date another flow confirm dyadic exploit fine grain temporal actor near outline expect fine grain data link link framework employ specify use predictor illustrative social contain fine grain dyadic communication event indirect e order order communication actor depict event mail friend plan subsequent message know clearly mind receive node subsequently send pm action key member indirect make aware maintain message make still indirect communication social several rarely central indirect communication third party essence approach diffusion update common exploit infer direct might first stream reason grain pattern micro come twitter find word twitter cascade tweet ordinary indirect detail node come contact final evaluation overview use combine dyadic receiver tuple predict node connect give previously event return edge example unsupervised predictor b simple unsupervised predictor interval possible link period link use roc curves evaluation argue extreme class imbalance curve precision recall natural primary multiple unsupervise joint brief supervised link work link evaluate reason depict bottom prediction feature period period relate previously link predictor set score accurately greatly link attain outline panel top distinct realization turn split train prediction distance geodesic put bin train bin quite common primary distance becoming treat gain strength follow separate link prediction grain exploit implicitly use e attribute introduction vector individual motivation track date person accord list basic idea grey keep actor plan ask detail direct indirect answering question keep track update receive grey node receive pm could received node let er tuple satisfy
resp general density score datum affine data xy datum score consider score concern statistical affect small quantify influence optimum briefly let parameter contamination influence contamination influence sensitivity parameter normal value preferable stable influence robust vice versa know density optimum old statistical score continuously differentiable let condition satisfy z ki mp p p addition optimum old defer old optimum score old score imply affine invariance applicability problem indeed characterization argument characterization property estimator introduce old affine represent mixture density score bregman old score old include h score applicable regression intersection bregman old old intersection property score prove old class show bregman score paper score expansion wide affine final goal composite reveal property identify score investigate estimator proper derivative proper proper score transformation statistical property old strictly proper author old score old hold inequality therefore linearly I monotone simple substitute f real give gx equation I substitute differential transformation convex lebesgue measure necessary condition zero disjoint measurable inequality assumption vector therefore equality hold constant non affine induce see order confirm theorem u bound take differentiable around equality h yield arbitrary exist satisfy number eq lemma theorem score sf gp score satisfy p pp kl lemma represent monotone sp p qx yx p finite derivation proof let prove implicit theorem subset general euler pde pde solution express lemma affine composite determine let g composite hold equality linearly contradict composite score suppose density mb mb number hence lead hold negativity positivity z say contradiction finally composite hold increase ensure composite k obtain hence property derivative integral differential vanishe hold affect influence via hence old mm em mathematic measure important task field score tackle proper statistically forecast proper score reveal proper bregman square composite name h old induce favorable property imply transform transform affine essentially system unit measurement characterize invariance affine old illustrate newly composite h old affine invariance prediction field probabilistic require appropriate tackle scientific finance prediction distribution hence formalize taking prediction conduct identically empirical optimizing procedure formalize statistical regard good score typical outcome proper attain mild score estimator estimation score element measure regard generalization square distance induce sort topological statistical structure riemannian affine consist relate statistical divergence since closely proper score major characterization proper produce bregman correspondence proper use bregman score probabilistic forecasting propose old score induce favorable property imply transformation datum invertible typical affine unit comparison affine estimator affine estimate depend characterization old score divergence composite divergence affine invariance among class score old affine transformation estimator derive old statistical problem point far receive variable specific property score associate divergence bregman separable variant composite way forecast old bregman affine show old induce affine invariant divergence conversely old affine invariance h old statistical include problem old divergence summarize notation denote interior number e sample measure denote f denote l denote negative provide function probability probabilistic measurable probabilistic forecast score forecast probability inequality expect widely term score score form density negative subset denote g gp sp pp almost composite density strictly proper definition proper score composite score probability density negative likewise score set simplify analysis composite composite composite divergence nonnegative bregman variant mild strictly bregman score bregman integral suitable composite name score investigate old score bregman empirical score condition produce consistent score propose composite score score also substitute empirical old show old score bregman score relation affine old old old integrable divergence kl score score score indeed bind sf score old score h old appendix old composite score name old come old non negativity divergence refer old affine h old function resp divergence divergence bregman need involve banach space differentiable preferable technique old continuous bregman old suppose separable potential old score kl differentiable bregman intersection separable bregman old bregman associate function present bregman score bregman old score score sign interpretation composite include score h old divergence often borel typical affine transformation matrix observe transform mean unit fair sense normalization often make induce transformation let statistical affine give transform unit satisfy mathematical change affine invariant composite equality necessity score affine affine invariant composite associate value affine transformation affine affine composite briefly provide affine optimum obtain q minimum dp affine invariant way confirm kl invariant score provide affine characterize begin
find global parameter instance gr base real via package weight match theoretical among seven real q close satisfying computation gr among also entry second list frobenius distance second intel ghz ed explain variety dual matrix imply assertion corollary informally set agree unit thus denote matrix intersection rank formula hold gap well conjecture irreducible intersection component rank correspond whose twice regular square volume fill table compute ed eq table confirm rational function table change topic space matrix low rank algebraic geometry formula generic ed degree variety matrix projective point binary form curve generic ed equal note function closure set homology show projection variety consider theoretic intersection curve hyperplane point linear define bundle bundle map bundle total class multiply prove ed define determinant format eq ed agree ed ed duality also ed degree weight respect expect instance medical problem find symbolic tensor rank write parametrization form parametrization every rank jump space specifie follow unconstraine argue rational ed library gr base avoid computation take critical hence return approximation polynomial symbolic critical real correspond minima euclidean take hour ghz cores numerical compute symbolic gr computation conduct return parametrization critical integer univariate polynomial formulation unconstraine problem minute ghz symbolic unconstraine formulation seem table unconstraine formulation study matrix row vanish kernel degree algebra aim nearby dependent precisely ed formula ed equal variety ed example ed multiply q polynomial embed compute ed degree similarly ed ed equal three matrix euclidean pattern rotation present ed degree leave generic know formula lead margin matrix c c weight ed thank project institute dms theorem example structure minimize frobenius matrix critical algebraic focus matrix low rank algebra real format frobenius entry close general complicated minima discuss example structured approximation typically good scenario happen property cf practitioner many ensure never accomplish square semidefinite cf optimality reliably optima aside arithmetic list identify point notably gr basis sort fact critical intrinsic invariant indicator run study semidefinite structure low close primary find ed degree always regard come write weighted variety number critical problem keep track highlight situation ed ed ensure isotropic theory organize distinguish either homogeneous equation affine affine case gr critical performance minima explicit arise affine algebraic include focus space require certain present let matrix goal solve unconstraine ed usual space section case exhibit r section ed degree approximation problem matrix take minima minima entry six algebraic root irreducible critical intersection six minima exceed highlight algebraic practitioner polynomial equation demonstrate solve gr emphasis lie subspace treat plus implicit critical u right singular condition jacobian model introduce lagrange variable equation verify gr basis c c c c linear c affine ed determinant affine remarkable correctness linear affine space affine solution jacobian variety prove generic variety solution belong hadamard scalar product live get contradiction whereas verify computationally gr package gr difficult already ed degree element due substantial growth rational number minima two efficiently distance determinant duality statement fix hadamard product entry critical conversely hadamard tangent variety point variety express writing denote hadamard inverse weight tangent variety statement proposition give usual geometry algebraic particular complement dense draw characterization define aforementioned ed discriminant degree proposition weight solve rank seek critical critical gr basis conclude matrix rectangular format affine complex matrix rank lm complement tangent space critical smooth introduce system belong normal span avoid costly finitely critical euclidean proposition computational experience gr compare approach weight rank integer pick entry table generic ed ed gr gr root c generic symbolic weight approximation three scenario difficulty operation gr solve unconstraine unconstrained formulation variety c symbolic affine section table report gr basis formulation ed degree face follow measure second gr symbol gr seven day computation running time value arithmetic need gr suggest ed degree accurate solving serve motivation ed carry shall arrive ed table ed degree algebraic start section build geometric theory rank derive formula ed degree affine projective ed ed degree cone fourth restrict since degree affine cone affine
median terminology immediately robust remarkable tail heavy present empirical qualitatively draw satisfie scale simple generalization median estimator metric mention first abstraction capture candidate large half independence random response statistically propose generate single small return robust similar rely knowledge b k scale circle circle circle circle illustrate euclidean half circle within determine k b random half contain ball problem computable quantity estimate may relatively access response weakly random indicate follow random statistically independent variant replace suppose return ki j b k approximation assumption k b union loss regression special suppose n strongly follow negative copy sample primary case interest l th I call well prove implementation ki union intersection let eq last rearrange probability lemma return easy smooth version proposition losse subsequent section dependence besides variant size correctly use I guarantee minimization implement note statistically implement empirical implement return scalar loss function loss population satisfie empirical l minimization therefore similar guarantee trivial objective necessarily analysis regression dd denote draw separately regressor generate covariance marginal regressor simple step expand implication ks I median random vector accord first ordinary guarantee mild moment condition product bound easily moment logarithmic factor assume parameter run q loss quantity low moment follow moment eq probability comparison propose however remove adaptation suboptimal unclear compute analysis square boundedness approximation include nb additive involve sure subgaussian logarithmic error unbounded propose subtle ball however evident tail derive simple algorithm suffice singular dr logarithmic remark boundedness subgaussian recent base convex loss upper bind must imply suffice interesting generally applicable assume loss strongly empirical especially around observable leave future variant least square loss l b logarithmic factor previous analysis analyze twice taylor control follow fix eq sake simplicity remove subgaussian consider dot see loss minimization follow simplicity q thus soon probability induce factor lead well reference sequel orthonormal dy l dl hilbert space algorithm algorithm assume similarly proof observable interested compare l b logarithmic indeed eq vb become minimax namely regularize subgaussian heavy dy j ds I minimization lasso subgaussian type heavy tailed use ss cd include dl define subgaussian ny fix return dc fix noise theorem follow fix design setting satisfie roughly empirical minimization implementation subgaussian analysis tail assumption class noise term lasso propose mild spectral norm frobenius trace norm return least minimize select select distance property might consider distinguished let w w w select minimize geometric banach minimize similar metric detailed factor procedure space banach hilbert provide low procedure compare guarantee account usually approximation approximation factor constant suitable procedure assume summarize w real product inner prove k bound real line point distance point minimize geometric median space banach banach factor entire intensive involve distance thus interest consider guarantee space several w banach metric general space let accommodate general case w set space addition w eq geometric prove base geometric banach metric banach space guarantee guarantee category type guarantee space necessarily show many bound match either distance bind banach distance base problem banach space I ia see scale cycle anchor west west anchor west west short path underlie undirected length multi problem choice therefore banach banach base procedure also approximation I ia h scale cycle anchor circle anchor west west anchor anchor west anchor north short path undirected line edge double line easy permutation index moreover least banach space simplex basis r easy thus p approximation hilbert depend hilbert achieve procedure achieve tight limit guarantee gap nx e regular simplex simplex ib n return anchor north anchor west node anchor circle circle anchor west ib ia see exist hilbert factor approximation factor answer exist banach factor space center vector p metric banach distance geometric median optimal large bound banach geometric median space base median different useful upper obtain upper take inequality hoeffding guarantee enough factor normalize factor normalize factor factor observe procedure geometric well banach space question distance hilbert implement median geometric estimation available computationally statistically core select candidate candidate candidate scalar without access candidate output bagging follow proposition l first assume generality approach require access unlabeled issue generate unlabeled aggregate predictor label find similar suggest several mean estimation particular rate distribution match max factor show heavy heavy tailed core estimate estimate work ridge class risk minimizer black box generate improve side get derivation brevity generalization concentration heavy tail distribution low sample loss without covariate application regression rank generalization tail minimax principle statistical bad expectation take examine empirical minimize class know specify family distribution square control examine deviation behavior heavy may pareto moment order commonly extreme event say weak may derive markov least remark statistical concerned guarantee subgaussian tail limit applicability even expect deviation concern heavy tailed boundedness apply control deviation low variance applicable loss derive specific square regression match without require noise covariate bound subgaussian concerned require finite achieve optimal improve logarithmic dependence number new generalization metric space least square yield split select one good fair chance good close behave marginal covariance detail bound
propagation w see back formula deviation calculate generalize kernel traditionally formula case formation receive weight minimize uncertainty value fit covariance indeed differ origin difference rely provide datum uncertainty conditional mean integration separation think principle unclear whether combine design take furthermore negative correctly knowledge uncertainty linear derivative calculated necessarily cover true fit function discover feature mean central limit central fact gaussian width maximum true mean fit tell conditional compare degree freedom sample describe well still statistically meaningful use overfitte eventually interpolation simply reach absolute uncertainty increase exploit though keep usually sampling sample p submatrix freedom full calculation variable parameter calculate function difference solely individual contribution type similar freedom make optimize arbitrarily characteristic optimize significant show fig patch random uncertainty pixel intensity boundary algorithm text principal axis model linear polynomial univariate sigmoid behaviour intensity accordingly instead method overcome polynomial large matrix space allow loss appear predefine split algorithm simply input demonstrate practically additional computation fitting split axis large cut likely produce degenerate place large derivative hard define multivariate appear nearby fit uncertainty model multivariate detect simple optimisation maintain numerical university describe input approximated sum uncertainty allow polynomial statistical significance combine phase splitting degree particle physics quantify surface simplify calibration detector response identify particle goal target converge cover radial generally degree freedom avoid pick statistical intensive fit global determine amplitude minimize form indicate sample respect power fix polynomial shift translation formalism original compress great reduction input generate
nn commonly reference indeed condition however make doubly portion rate among mild error benefit doubly robust modeling specifically sample complexity allow capture randomness approximately require estimator obey regularity estimator following need apply self sum verify assume expense strong form quality assumption slightly linear link meet group lasso directly verify logistic concern similar misspecification robustness consider hold theorem double behaved compare nearly identical result save main demonstrate lead giving variance potential plug asymptotic effect generate assumption additional estimate ref bt bt assumption true support randomness appear immediately uniform generate uniformly continuously function valid hence reliable crucial insight goal uniformity assumption approximation valid prove uniformity distinct treat formula define component simplification omit treatment generating process randomness support ref bt bt support randomness uniformly set condition theorem twice continuously gradient bound zero aim robust briefly discuss oracle put sound conceptual appropriate theorem theorem require treatment assume otherwise turn property conduct uniformly valid exactly true discover true support mechanism future way entirely instrumental inclusion general efficiency bias score perfect orthogonality bound intuitively correlate distinguish zero find bound particular coefficient vanish detail group selection precise discuss key state section select square solve discuss allow tt selection give sharp improvement lasso work thing may smaller cite formal discussion solve penalty penalty way jj weight invariant pilot equally perfect program theoretical choose norm probability set choice p x nr form heart rate concentration small appendix treatment decrease final stage doubly turn first option iterative validity x I eqn implementation option appeal form characterize underlying function minimize relevant formal validity moment sample empirical high multinomial restrict square cone small nonlinear respective contrast motivate define finally quantitie primitive condition often counterpart conceptually eigenvalue cite eigenvalue theorem event counterpart adjust instead zero notation multinomial linear corollary lasso lasso logistic theorem capture behavior hessian remark hessian ix control account population lead bias arbitrarily neighborhood asymptotically estimate shrink x close stem require sample analogous impact coefficient also capture maximal sparse latter able tighter weak crucial constant offset true dependent bound part post multinomial logistic condition save r capture lasso selection success compare restrict play logistic intuitive outcome regression perform bound group fit bind prediction imputation prediction multi supplement entire give selection follow regression procedure initial fit drop union theorem sufficient verify lasso assumption state result selection recommend reduce retain overlap ensure far various commonly asymptotic analysis display obtain asymptotic multinomial away bounded tx asymptotic suppose theorem iii ns p n straightforward common two robustness verify heart use concentration unlikely limitation mention group lasso improvement effect even could principle see regression practice may prefer treatment illustrate inference carlo exercise difficult average effect binary treatment use intercept remainder crucial aspect define scalar multiplier affect ratio small distinguish small control value panel coverage strong panel sparse roughly increase increase range coverage signal strength assumption retain robustness less sensitivity fold exhibit design see supplement limited contain mean group size cross xx illustrate role real numerous study rule discussion program hereafter reference briefly outcome include indicator year pre education status indicator consist seven highlight role inference interested specification keep doubly standard informally ex ex group ex interaction covariate specification model specification intercept education treatment large specification arm perform well accurate allow great flexibility estimate keep fail significance wide specification ci selection group subsample comparison specification standard exception partially intercept total group estimator regression doubly robust outcome begin treat unit score treat achieve effect covariate robustness error misspecification heterogeneous doubly show follow argue natural detail evidence show quite model work crucial plan choice becoming understand take section bound next understand sequence shorthand assumption define index prove additional randomness follow randomness turn require short argument linearization without additional eqn show eqn apply stage consistency use proof eqn eq parametric get x define equality follow apply moderate normalize sum lemma center third moment away assumption restriction give find assumption eqn inequality next proceed claim prior expand square use u tx w w use inequality inequality assumption consider variance define q old von mind decompose consistency von exist subsequence contradict unless note deal appendix eqn conditionally satisfy side apply follow jensen first tucker satisfy add subtract true triangle tx collecting sum side ex ex ex na schwarz cauchy schwarz inequality bias cauchy schwarz ex ex ex support follow collect least case suppose obeys eqn schwarz inequality restrict eigenvalue eqn note collect yield hand fail ex ex ex line third plug instead plug yield subtract deriving yield right ex ex ex ex ex ex therefore reflect dropping term bind eq side eqn class hessian fw verify involve bound hessian multinomial x I derivative multiply absolute observation v I inequality imply quadratic depend case schwarz inequality line lie r n equation restriction eqn nonlinearity coefficient dividing mean result q eqn cauchy schwarz set optimality ensure result rely rearrange many difference cone eigenvalue obey suitable eqn eqn second lemma latter collect hold collect low term eqn define minimum eqn q combine give odd score argument robust average treatment effect covariate h summary inference effect possibly complete keep file serve present section section shall understand symbol shorthand formal positive index schwarz consistency von x tx prove randomness short linearization verify apply first proof additional randomness tx proceed apply rewrite consistency consistency assumption old randomness apply add representation get I treatment deviation self moment moment restriction thus union assumption eqn assumption prior get eq term final apply proof condition expand eqn tx inequality inequality I define q old von assumption pair decompose von inequality theorem section etc eqn eqn residual conditionally side follow lemma follow jensen apply result eqn tucker norm subtract triangle inequality tx collect side sum yield follow leave cauchy probability schwarz inequality combine least imply convexity rearrange find divide collect consider case upper obeys eqn cauchy schwarz eqn note collect third yield constraint use first line eq back plug cone equation give rearrange subtract proceed derive upper combination eqn coefficient reflect drop final nonnegative second result turn eqn goal belong derivative fw w fw third bounding derivative hessian multiply absolute lemma logistic I give x aa I v two depending case inequality conclusion segment know I equation impossible restriction require contradict therefore hold nonlinearity impact equation divide union x triangle eqn follow note conclusion use bound use rely suppose rearrange give minimize argument quadratic bind restrict obeys eigenvalue inequality definition eigenvalue eqn place eq q step apply lead latter collect eq case hold apply collecting give plug yield solve minimum quadratic eqn imply bound log odd argument parallel otherwise note bound generic etc deal score bind eqn eqn residual definition follow jensen inequality old inequality assumption eqn obey tucker take triangle x side eq drop final nonnegative triangle realization side schwarz schwarz plug inequality eqn eq case depend nonnegative rearrange display return eqn rearrange discard support collect term hence obey begin cauchy schwarz definition eqn equation root appear dividing find eqn union q triangle fit obtain define theorem schwarz inequality eqn coefficient eqn bound find rely result q rearrange size analogue panel manually coverage fold sparse function current software choice multiplier xx xx e xx xx result web contain additional grateful feedback point work early stage discussion comment improve school business concern treatment follow model interval robust treatment class effect selection amongst possibly covariate attain efficiency appropriate precise selector combine drive give well treatment derive multinomial tight high sparse heterogeneous doubly place economic modern work researcher complementary economic researcher search simultaneously parsimonious many formal computationally infeasible response small specification matter inference never specification confidence interval mistake particularly estimate framework right correct post heterogeneity selection misspecification selection explicit require selector effect researcher theory drive valid third prove asymptotically impose evaluation inference sequence show rely uniformly speak validity interval idea theoretical practically imply great reliability application post break recent uniform change selection underlie fundamental shift inference attract make doubly robust name doubly reflect misspecification treatment combine imputation robustness extend selection enable error crucial heterogeneity average treatment treat differ present result third doubly stem us treatment propose focus motivate recover average binary treatment however develop consider heterogeneous influence allow offer enhance program lasso naturally already present pooling particularly group regression doubly average quite benefit require versa require selection remain popular economic covariate play crucial may unobserved plausibility exclude efficient set outcome necessarily assignment reasoning practitioner one formal attempt contradiction set covariate nonetheless capture capture control sparsity effectively specification unknown provide select yet traditional method empirical estimate multinomial regression couple group focus see nonlinear limited logistic error independent tool focus goal apply lasso condition hard logistic intuitive linear modeling mathematical put work offer numerical simulation coverage confidence interval sparsity uniform work accurate tight follow overview describe effect discuss show commonly use treatment present evidence proof supplement overview include section result group notation throughout treatment status scalar potential ex interesting wide fix idea effect binary section effect treat simplicity single selection regression away overlap treatment broadly unit treatment doubly combine imputation remain multinomial linearly select second include literature suffer bound rule offer discussion drawback give identification assumption must tie parameter inverse robustness q q specify depend plug condition identification assumption use proxy interest assumption hold group generic comparison maintain keep track special interpretation plug influence asymptotically formalize transformation allow transformation model may overlap vary multinomial log odd ratio outcome regression arise parametric require bias bias make q former object linearization odd great ts equation clear covariate generality remark practice know advance include example special case speak obtain uniformity
modify ise vary experiment factor ise show site observe uniform factor decrease factor possibility partition low wang need address future work relation investigate factor factor ising obtain analytical case graph compute partition one ise model partition temperature monte setup usually particle state represent stand configuration ising show consider site arrange lattice depict variable interact define configuration adjacent pair evaluate otherwise couple control strength negative spin configuration boltzmann normalization small low adjacent run adjacent interested k fx bc bc bc bc graph periodic condition box bc bc bc factor ising box box ise graph compute coincide physics ise constant coupling absence external value limit size ise arbitrary coupling markov chain boltzmann range interaction slow break analytical dimensional case much fast duality obtain ise ising value therefore periodic boundary end graph fourier dft dft factor graph factor duality finally ise factor end graph function bc bc box box contain symbol eq q bc bc bc unlabeled box represent contain symbol binary variable factor box label equality equality constraint replace dft per site ise sampling show path everything fig modify simplify construct modified graph experiment compute free ise original factor modify carlo cycles rather monte ising model model cf compare convergence modify energy spatially per vs ise everything
schmidt proposition offer unstable inversion replace numerical alternative regression multiple way define possibility important square express spectrum measurement define psd fourier transform least j well unbiased blue one estimator q advantage fact inverse covariance separate long infinity let introduce p ij ij product computing becomes process previous simplify spectrum psd psd psd psd suppose estimator show least technique filter technique aim maximize maximize fouri equation offset take method several computing matrix contrary least general minimization introduce discuss instability case estimator grateful centre proposition publish square propose well consideration discuss usual least square domain fourier generalize match optimal power aim derive data gauss mathematical discrepancy broadly understand propose definition square detailed characterization least square domain filter signal dimensional filter useful definition concern dimensional p assume g introduce kl ab let convolution notation l l result derive scalar derive exclude discussion transpose positively
change loss notice information online later appear model smoothness sublinear rarely change sequence model uniform exist constant space discuss assumption deterministic policy hold round suggest expert suffer full rarely well know mdp choose policy accord round change frequently shrink sd algorithm round expert initialize expert expert take expert adversary regret q follow game expert policy adversary expert learner draw expert learner guarantee adversary adversary choose ec lemma switch notice ex tv ex policy policy fact initial policy mix play adversary proof space cover assumption theorem l px constant argument proof eq get particular theorem thm conjecture thm berkeley ari university decision transition change grow root game provide sample designing regret open finite policy use action direct space x function space adversary learner choose adversary model simplify discussion choice assume version learner observe game learner suffer period round stationary mdp gap assume learner observe complete apply mix computationally sublinear regret obtain regret transition mdp aware polynomial mdps
subsection proof round let statement exponential compare rate setup follow assume specific regime range dependence confidence close interval neither optimally adapt remarkable phenomenon even description introduce type add aggregation problem I remainder aggregation smallest possible choice intersection summarize type aggregation analogy aggregation aggregation aggregation optimal em r md qr md estimators single solve aggregation partial form recently show exponential five aggregation knowledge argument show aggregate solve aggregation expectation oracle inequality simultaneously probability r qr rate optimal interesting indeed match low bound consider linear span bound focus make whenever modification prior allow deviation risk identity schwarz nonnegative particular orthogonal satisfy apply schwarz inequality observe eigenvalue yield follow chernoff expectation fix canonical basis q apply yield expression yield let inequality trivial follow v display bind follow definition canonical yield yield recall imply get side support one follow valid q c everything least hold eq replace display em qr qr treat observe particular observe aggregation linear aggregation aggregation ball contain sense obtain fix absolute value coefficient let inequality bound ball em I I non coordinate decompose disjoint support absolute since use empirical value copy notice realization return eq eq follow hold consider value b x b x em q put together get adapt omit gmm indeed regression natural estimator gmm main gmm make regression precisely know chapter local estimator smoothness lead adaptation unknown smoothness yield em see gaussian outlier sparse approximate unknown identifiability reason hope recover corrupt notational convenience kk gmm k r well approximated unknown family estimator propose affine represent sparsity pattern everywhere else stage sparsity example balancing theoretically nearly situation default theoretical analysis work experiment far finally detailed study case projection estimator es independent furthermore class aggregate fair compose vector understand suppose submatrix nine observe compose rademacher idea entry nine column aggregation coefficient normalize performance estimator unknown case tune cross conduct sensitive plug fold report correspondingly except produce cross validate true experiment next replication aggregation iteration term deviation implement bic validate choose ten package plus performance scad es aggregation obvious sparse es scad bic cv report prediction define like convex example discuss define aggregation particular weighting enjoy oracle dictionary design sparsity element example study use additional subspace dictionary indicate k ridge give filter diagonal call svd correspond resort form flat ps section exercise condition section section h nsf dms nj usa department operations financial university nj usa department nj department financial engineering university usa problem aggregate relevant commonly exponentially selection deviation aggregation may sharp inequality weak hold newly aggregation prove sharp oracle finally apply universal aggregation good bound aggregation include class argument gmm unknown know purpose explicitly stein starting point vast literature excellent manuscript independently variety dedicated introduce weight play survey estimator originally split various estimator aggregate advantage therefore aggregate mild seminal aggregation aggregation give goal euclidean result suboptimal understand various choice rely therein base original aggregate satisfie may choose attain may accurately describe risk especially limitation method aggregation recently enjoy yield aggregate aggregation affine pair estimator construct employ study light remarkable family affine mild condition previous sharp type bound indicate unlikely sharp inequality high succeed fail oracle yet regard rest give aggregation trace continue aggregate complete sparsity pattern universal aggregation aggregation sharp oracle prior result main aggregate remark oracle bit difference high one study free deviation estimator see bound sufficient sparsity pattern optimally inequality replace estimator weak oracle sharp aggregate sharp inequality inequality modify affine equip hard observe quadratic sharp obtain sharp illustrate weak tune hand side multiply oracle probably exponential sparsity aggregation assumption probability tailor order sharp argument carefully affine optimal filter aggregation prove computed sequel aggregation choice bound
series sort bold table fx daily respectively good method statistical difference rank test determine whether rank dataset performance pairwise comparison method test rank dataset segment label cd confidence confirm superior method superior static counterpart statistically plot tail heavy confirm show predictive analyze produce plot average approximate relative predictive likelihood method average daily fx early less however bad amount dynamic covariance financial overfitte problem financial observation financial asset price lower predictive major scalability sensitivity execution time minute daily fx order except dimension fail finish clearly filter trade sequential particle denote need currently operation dataset long sensitivity prediction particle dimensional substantial time desire amount aa another experiment wishart experiment previously author daily fx generate daily return index composite period index return step time series standardize protocol section experiment receive way instead approximated particle evaluate advantage method respect fx method highlight bold dataset good benchmark throughout overfitte parameterized performance outperform vary contrast outperform world predictive return introduce diffusion adapt market significant improvement recent wishart yield substantial enable scalable dimensional dataset prediction covariance however model suffer optima failure cost problem dynamic model covariance optima avoid change filter experiment financial univariate financial univariate capture extension conditional model topic receive machine development process return display dependency likewise capture multivariate extension model recent parametric wishart perform similarly wishart process model financial suffer fit maxima parameter value financial market naturally shift market maximum fit solve non address difficulty novel dependent matrix extend instead compute incorporate perturbation perturbation model adapt perform use regularize auxiliary allow change real assume finally document section introduce current covariance machine include experiment time series assume gaussian variance follow move process square flexible variety setting likelihood overfitte triangular restrict use overfitte constrained use comparison predictive version reference standard gaussian however financial heavy tail incorporate tail freedom mean ensure student graphical parameter market volatility past observation another volatility show conditionally latter generalization generalize wishart process dependency pattern evolution outperform propose perform filter online likelihood furthermore accommodate student particle sequential regularize auxiliary agree detailed paragraph introduce filter regularize hyper state explore take step avoid dispersion problem particle filter particle filter perform predict generate maintain less sensitive work predict representation particle shrinkage variance heavy tail parameter previous empirical inaccurate representative particle computational algorithm evaluate student variant computational maximum financial analyze daily exchange
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instance degradation effect phenomena process recover range popular advanced iterative algorithm take account ct rely ray enable reduction post process ct purpose ct focus produce process requirement access aim lead order version artificial ann also neural imaging particularly purpose ct overview aim intensity neighborhood take close possible mean find reference resolution trade image without organized ct scan reconstruction artificial familiar skip read core work dimensional piece sequel reconstruction filter back present section conclude discuss complexity summary implication process ct scan reconstruction plane plane coefficient material e measure perfect directly relate x transform collection straight line pass integral along scan coincide straight ray detector count refer acquire angle bin projection accord reconstruction measurement transform trace reconstruct count instance poisson reflect count detector random satisfie ex increase ideal compute measurement reconstruction image relatively contaminate appearance relate htbp object paper shall algorithm filter popular despite iterative reconstruction statistical expense describe filter adjoint back individual ram define prevent action ct need soft high pass noiseless case low effective propagate form diagnostic follow measure integral back partial region statistically ct approximate scan reality additive stem reconstruction computing map ct accurate approximation weight proportional variance square count refer clean expression choose convex huber bfgs matlab implementation mark schmidt ct medical ann use science layer ann cycles array nod neuron implement array output weight sum learn function produce come neuron node neuron second layer edge neuron th neuron neuron popular explicit definition regression layer ann training comprise collection vector weight sort e iterative backpropagation depict htbp propagation algorithm ann mid image powerful attack discrimination application area broad comprehensive overview medical imaging ann computer segmentation study ann reconstruction appear replace pixel et forward naive application ann limited input reconstruct ann imaging modality tackle ct large ref propose train maximum entropy energy electrical ann variety electrical ann method despite abundance innovation ann medical imaging ct problem rarely raw black box scheme se produce upon configuration section describe transformation noise recover measurement influence certain estimate responsible resolution tradeoff level image basic denoise signal noisy shift low pass convolution prescribe rotation spread parameterized kernel average noisy spatial resolution reconstruction recovery map filter back involve pass projection cut control reconstruct situation specific hoc consideration scalar apply prove switch rule select pass filter fusion filter switch smoothing balance assumption devise well pixel knowledge mathematical local reconstruction switch idea solution fusion consist know ann generalization learn location process input version small desire output predict ann employ matlab toolbox network classical sigmoid argument specialize method propose pass domain parameter control tradeoff influence iterative ct section approach simple denoise ct choice design piece wise beneficial signal easily purpose pass denoise length choose choose intensity step create convolution signal version htbp width setup train ann neighborhood ann sample process signal thus match ann signal extract describe test improve snr linear db db fit signal much interval center problem htbp variable shape width apply structure network question ct setup local ann boost reconstruction discuss error process ann minimize ann labels mse ct proper homogeneous region ct yet similar cavity small lose intensity image image training expense edge strong spirit general consist vector desire ann govern idea assign example zero air specifically maximal assign accumulate idea later strong air remain example accumulate patch ideal assessment ct study snr make computed region area active technique object specifically use ct large air diagnostic soft water chose good therefore consider fall pre process projecting similarity come standard appear know human reference therein compare image normalization intensity second numerical available spatial impulse function replace spread place take reconstruct spike width image response axis order number pixel intensity maximum divided refinement factor method apply back ct fusion low pass apply back pass filter multiplication fourier domain cut change tradeoff control window roll control reconstruction influence texture reconstruct increase cut frequency visually pass image spatial display value combination choose eight frequency last restriction ann filter fusion extract disk shape radius disk input ann stacking intensity neighborhood normalize produce value intensity disk shape image disk pixel cover disk patch average detail several list neural produce noisy reconstruction disk shape neighborhood produce pixel horizontal vertical put discard threshold region air patch observe improvement performance ann improve individual normalize matrix store neural constant intensity noisy ann small fusion image disk shape produce final produce ann outlier incorrect intensity regularization reduce ann test expect stable clinical ct body slice ct scan head visible project intensity level correspond section ann consist extract experience suffice fitting neuron pixel neighborhood radius come reconstruction image reconstruction mention would good reconstruction test away fusion ann visual fuse form close observed noise enjoy superior recall measure compute include plain snr high image plain increment db behave ann implicitly support fusion visual appearance since l image fusion snr amount local ann fusion reconstruction neighborhood fusion build neighborhood reconstruction cut learn relation neighborhood snr value radius compare fusion central pixel version necessity large neighborhood
distribution range mlp ready performance assessment independent hold follow continuous within range assessment scalar define action possible goal scoring use kolmogorov dissimilarity receiver operate characteristic roc curve ks function aspect perfect ks test metric indicate good separability distribution scoring curve metric classifier consist subproblem opponent point prevent opponent go outside fact trajectory initially desire ball point opponent goal reach cumulative ball go post ball short leave post line ball right distance post distance lie discard result network equation great agent ball threshold play implementation assessment goal score result c c neural lda goal goal goal loss lda goal produce end result goal present mining opponent scoring environment simulator knowledge mlp goal result ks roc curve large goal goal improvement assess alternate option help score promise particularly scoring mm center university email scoring goal depend simulator present direction scoring match dm methodology match knowledge embed score assessment approach previous artificial machine field use central research due mechanic aim idea use team compose player shall world area recent solution package inspire competition pose challenge offer varied abstraction simulator complex movement dimensional category agent act environment goal objective create team attempt action scoring influence opponent position ball randomness ball goal goal factor focus propose challenge cross dm methodology ball towards opponent availability play section characteristic follow dm show preprocessing perceptron mlp develop performance statistically measure roc ks curve describe implement describe compare know discriminant lda summarize research game player ball option could instance ball opponent ball opponent last decide give positive e increase game decide moment another name optimal subproblem determine enter goal pass situation interesting subproblem subproblem specific manner imply ball reach player besides path compose linear discriminant lda goal reformulate subproblem subproblem subproblem network reformulate capacity player decision second subproblem dm six major evaluation subsection explain analyze explain knowledge scoring match play contain result acquire agent towards year goal base previous knowledge take opponent infer speed opponent opponent player consider match exploit entire know exactly opponent look last player message send server message send server work discard without power support goal extract systematically generate fine median percentage treat miss inconsistent remove outlier systematically choose useful go angle right line go angle vision position position x ball position axis position angle angle ball go point player position vision add useful understanding problem semantic irrelevant acquisition class similar detect operate roc present positive classification produce along measure area
specifically due thompson prior particle select increasingly likely track dynamic reward simulate arm follow instantaneous high reward indicate arm arm hence track smc method low arm change around increase smc adapt show regret dynamic bandit induce diversity smc bandit one equivalent static vary stochastically arm reward come back play artificial dynamic gain bandit real online expensive smc offline precisely datum arm reward crucially bandit algorithm empty history initially bandit realization keep dynamically intuitive customer preference reward bandit cause decrease lin greedy another bandit dynamically recall force perform suited reward track distributional explore likely greedy slightly overall dataset confirm smc nature provide dynamic datum efficacy especially dynamic smc arm allocation promise applicability bandit situation million day click potentially conclusion scalable armed dynamic removal arm hierarchical specification addition structure significant monte increase bayesian inferential method model armed dynamic significantly flexibility large hierarchical monte flexible handle armed bandit hierarchical naturally bandit variant single inferential generality monte carlo inference developments monte show empirical exist cope additionally apply video recommendation art mab decision receive contribution fundamental address resource allocation field theory portfolio arise seek video country high suggest change mab origin imagine front slot time perform well continue arm explore potentially issue balance know simple naive mab full equally action regardless markovian pick past hence ignore relevant finding strategy randomness period subsequently suboptimal greedy choose probabilistic exist interval arm base feature thompson term select success failure reward arm bayes update failure arm promise corresponding strength situation covariate ad display user operate use approximate base ip reward function covariate covariate information commonly greedy remain strength issue reward acquisition enter additionally arise arm ability embed tune present flexible sequential able achieve flexibility maintain control reward impact covariate result observe trial thompson note thompson sampling beta conjugate setup elaborate posterior require carlo ignore binary logit probit connect success model structure induce hierarchical hard bayesian bandit contextual model parameter logistic knowledge create bandit hierarchy impose matching accord reward binary reward eq beyond approximation make sample arm arm thompson sample select collect extension fit add remove need expand cost include reduce reward recently node beta connect connect edge connect edge connect connect connect connect connect restrict conjugate close draw sample straightforward computational cost increase propose sequential monte intuitively quickly originally though recently flexibility adjacent particle reweighte ess eq particle ess drop threshold particle filter degeneracy resample straightforwardly issue smc method previously smc performance bandit problem area smc policy approach dynamic discuss later diversity naturally dynamic static situation mcmc similarly suffer mix algorithm smc probabilitie fy n smc literature advanced inefficient effective side amount randomly arm particle particle could monte bandit hereafter smc accommodate hierarchical smc alternative confidence bind naturally bandit additionally inherent bandit use model simulation cdf arm simulate bandit cover represent compare bandit link generation mechanism try ucb truth strategy vs cumulative bandit low cumulative performance sensitive bandit allows naturally extend additionally hierarchical bandit repeat
square relative north possible bind compare perform step recent estimate agent mean visit value optimize cycle return episode method run implementation place queue next update implementation ps computationally fine ps implementation planning reinforcement substantially efficient fine grain full allow tight constraint option domain theorem definition em em em planning play crucial role traditionally planning estimate proportional computation call reinforcement exhibit fine control planning empirically increase flexibility show improvement seek control maximize discount rl interaction estimate function effective planning planning technique vi perform action drawback vi infeasible many fortunately efficient large change introduce efficiency idea interested construct compute update alternatively receive significant want construct current subtract old kind computationally trade store plan estimate correspond serious restriction already memory storage advantage fine control plan effective quality vi empirically substantial severe showing per computation td perform performance introduce empirically td carefully step formalize mdps tuple immediate reward action define rl policy discount reward receive expect follow start state sa sa transition reward iteratively improve return greedy action action improve perform td size version prediction require storage make ss update state demonstrate three performing component base vs maintain note variable rely get long relation count currently relation vs relation still sketch change cause update full planning method time specify pair initialize initialize select action time efficient obtain store action implement time considerable reduction prediction theorem version prove relation sa u maintain update relation sa n sa sa hold update action finer grain small complexity small implementation state share disadvantage size state variance disadvantage affect outcome place restriction combine ps rl select queue maintain determine value main common call adjusting cycle per computation per occur cycle implementation queue state effect perform follow computation complexity update follow receive change cause compute yet queue queue full instead state triple triple queue queue much pseudo code ps raise question queue top element answer queue value occur last discrepancy state surprising form small action due algorithm memory simplify ps three indicate state transition initialize initialize take sa sa sp sp remove queue sp terminal td perform per small per state pair recent transition evaluation consist
last final weighted starting rule first formula capture conjunction clause weight weight derive fact contain alarm formula propagation define formula atom atom follow relevant ground program equivalent sense program result perform weighted mod formula formula weighted formula probabilistic fact correspond reader equivalence reader weight boolean infinite clause atom probabilistic weight alarm alarm example also contain follow six clause relevant program ground accord hard code probabilistic program query convert boolean formula reformulate weighted weighted formula exist state art algorithm counting model generalization weight task computing reduce weighted formula hold total evidence equality equivalence sum equal imply compute exactly formula logic program improve state leave study need efficient formula link concept background illustrate concerned logical family compute representation shift count circuit weight formula efficiently allow single circuit evaluate marginal circuit form root direct leaf internal label conjunction node hold two child share satisfy node represent inconsistent need every use exactly set smooth available circuit circuit language support tractable purely logical formula irrespective counting convert weighted internal multiplication replace involve subtree multiplication two child leaf alarm circuit alarm alarm evidence alarm node right correspond ignore leave indicator indicator multiply weight function circuit ready weighted count find node root probability evaluate arithmetic circuit alarm arithmetic circuit alarm example circuit obtain circuit indicator circuit root evidence explain variable allow add evidence top additional evaluate circuit circuit arithmetic circuit want arithmetic circuit circuit evaluation evidence provide atom circuit boolean strictly formula capture indicator circuit evidence propagation etc htb highlight et encode boolean formula formula arithmetic circuit main simplify boolean cf summary probability convert arithmetic circuit arithmetic circuit probabilistic programming community art namely approach usually connection property replace evidence directly circuit evaluate circuit merely count experimental result confirm superiority probabilistic atom compute reduce computing probability evidence conjunction atom previous compute arithmetic circuit evaluate circuit query atom separately circuit compute circuit require traversal literature simple optimize form retain involve atom set previous section circuit approach typically large resort markov sampling formula develop mcmc call solver mc ensure sample summarize currently inference unobserve atom evidence find unobserved ground program formula consider traversal traversal literature occur irrelevant w r simply truth associate probabilistic technique solve inference consider fact program set possibly interpretation learn far study probabilistic language terminology interpretation evidence shall term evidence partial let derive atom give observable interpretation coincide observable truth atom learn possibly partial interpretation formalize set partial interpretation probability example use alarm interpretation alarm p alarm alarm probability unknown interpretation truth true probability combine interpretation maximal consider observable count partial interpretation approach maximization calculate interpretation ground instance fact ground represent interpretation estimate number fact represent training partially receive phone know occur observable compute initialize unobserved equation maximization marginal expectation firstly fact dependency partial slow include n example third partial atom update partial learnable secondly observe parameter program interpretation algorithm likelihood em current complete datum estimating count p rand n mp I black work include one knowledge circuit hard circuit easy circuit furthermore evidence parameterization pass algorithm early work cyclic completion apply acyclic scale employ inference namely furthermore description clearly separate complexity exponential weight boolean knowledge theory exist heuristic decomposition probability atom evidence interpretation set spirit like markov interpretation fact predicate influence four program use rule cause learnable page learnable sure page direct neighbor learn big perform exact learn remove probability discuss six question algorithm end particular intractable curve end intractable q relevant rather complete question program formula derive idea behind pruning clause inactive query clause hence inactive clause body pruning average happen work complete boolean clause reduction formula loop cause remove ground break loop rule fast around complete beneficial come almost boolean rule question formula program formula implication scalability large intractable size runtime trend generate formula opposite fig question formula formula type measure quality marginal evaluate marginal let minute estimate size formula formula smaller drawn answer base preferable small large rule question perform question become intractable useful formula special query use nevertheless clearly c tractable intractable mostly inference formula proof inference feasible compact formula formula nearly impossible rule success implementation outperform proof work exact ask query vary large hard result scale well tractable e incur experiment time second runtime tractable repetition finish repetition finish repetition program answer program learn set program divergence independence probabilistic see mae l drop remain conclude capable recover figure domain question world obtain state run negative obtain four fold report result stand modify put prior prior default ccc outperform four four fold conclude suitable detail introduce relevant query evidence logic logic solver graphical cast weighted boolean formula expressive program advantageous employ optimize logic programming query atom evidence develop logic employ interpretation set third implementation unlike close answer set new one immediate point logic program ground markov logic allow mc contribute appendix remove program rule evidence ground logic program hold program impose certain program head rule hold remove inactive program rule show condition inactive every atom make evidence body atom make rule definition model fix plus rule inactive inactive rule execution evidence great respect definition say atom hence ground let mod mod evidence remove inactive rule logic program program mod mod strong mod mod l part prove e conditional fraction differ rule exactly fact range term conclude program semantic replace program preserve contain inactive respect irrelevant rule respect remove remove proof atom probability appear irrelevant rule irrelevant rule hence preserve relevant respect mod weight accord accord section yet present effect add evidence mod accord accord weight semantic underlie probability resp fact atom fall true denote denote definition weight weight atom hence eq prove weight distribution accord hard clause clause non imply equal exactly non accord soft clause expression I soft clause unit clause probabilistic fact clause atom clause equivalence weight program numerator evidence probability sum equal numerator briefly review markov logic logic formula ground part formula hard formula soft exponent formula likely marginal smooth atom child repeat node smoothness atom child transform figure substitute node add child create new link link smoothness use alarm restrict consider circuit smooth correct contrast smooth incorrect value htb divergence information theory information gain element one divergence cf truth restrict l program probabilistic except base interpretation subset atom q interpretation make evaluate define impossible probabilistic fact program fact possible k fact need multiply ground lp f simplify p l p suggestion thank discussion van pf contract fp first van department science probabilistic logic logic fact probabilistic logic interpretation address logic contribution inference task boolean allow reduce well count art know second contribution interpretation employ build art interpretation logic deal interest result field difference emphasis extension graphical model markov logical conversely logic result semantic task support common call likely evidence mostly without evidence furthermore learn set learning interpretation gap adapt perspective contribute interpretation algorithm relevant language logic key contribution step program convert equivalent boolean knowledge logic programming weight count result weighted second involve art new logic probabilistic inference weight much graphical logical approach answer often boolean second logic interpretation model use terminology lot inductive logic programming logic build paper separately inference use perform program later approach principled realize use al interpretation style integrate engine spirit employ weighted counting follow next task consider section two briefly implementation evaluate relational logic logic familiar lp skip form build atom universal logical formula implicitly call ground theory call call write form predicate theory interpretation also atom interpretation call formula quantify atom atom represent compactly free well least obtain implication set atom lp guarantee unique semantic logic consider everything implication interpret atom rule head atom difference semantic lp make model one intuitively make syntactic restriction believe expressive wrong semantic express motivate lp probabilistic logic programming base consist probabilistic fact logic rule probabilistic write annotated allow compactly specify fact fact statement conjunction call fact domain semantic ground probabilistic ground call atom head rule logic also model example alarm alarm alarm fact predicate predicate probabilistic statement rule define alarm person call possible program result base finite semantic ground fact obtain atomic formally atom total choice atomic choice see choice alarm alarm program choice total true second one htb l particular logic denote rule logic give exist semantic program choice program program case equal world logic program alarm alarm alarm hence world possible vocabulary language particular probabilistic language language logic language overlapping n definition language expressive respect allow particular require acyclic cyclic program type rule social exclusive restriction rule overlap alarm alarm happen system syntactic par system computing incorporate traditional query see hold system note case incorporate modelling system handle arbitrary base query carry system markov logic strictly speak language first logic programming nevertheless logic course language term logic drawback non ground inductive notion graph term edge plain knowledge inductive definition logic semantic logic definition closure express closure carry inductive definition language language markov logic markov logic network program convert first attention literature relational observation task known probable explanation query atom general give program mutual transform program however acyclic already task suffer addition base ground atom atom truth partial interpretation atom compute distribution atom singleton datum evidence atom program example know verify model highest verify choice hence approach convert program boolean result discuss step next take necessary atom compute take cf unnecessary relevant give capture distribution programming rule formula evidence define conjunction formula evidence correctness show describe three step take evidence program follow alarm alarm ground formula add evidence atom true section function boolean detail convert part make concept explain ground atom ground proof
mixture asymmetric generalized skew normal skew date form prefer restrict special word appear title indicate hereafter limit skew use modelling exploit maximization monte carlo note effort efficiency skew write integral form subsequently truncate package herein extend skew well application remainder section parameter component skew skewness dimensional use component mixture distribution model explain unobserve model vector loading matrix maximization find incomplete treat parameter step log update parameter expect estimate give formulate together source join denote q density skew skewness degree loading isotropic free constrain constrain constrain constrain unconstraine unconstrained constrain unconstraine unconstrained unconstraine unconstrained unconstrained skew write gamma follow require expectation computationally skew extensive detail skewness half write analysis write denote expect complete eq e employ integral believe offer advantage intractable stage incomplete skewness update update datum include latent factor load respectively impose eight parsimonious model matrix analogous note density mixture skew mixture parameter outline introduce efficacy skew skew mm corollary mixture skew offer choice sort skew herein skew
slowly explain ball interested might happen detect would wavelet lead suboptimal risk want simultaneous truncation simultaneously adaptive truncation let comment present phenomena truncation extend theory procedure finite truncation resolution gain work refined estimator discussion applicability mention truncation feasible coefficient calculate acknowledgment partially fellowship author helpful discussion remark k j j j confidence band jj diameter n band exist non carry contradiction complete integration r assertion write p induction find follow require exponential moment probability decay decay come exercise correct expression aforementione give constant slightly suboptimal probability nothing show assume inequality th moment q p use inequality k assertion suppose j k pn p contradiction must pn p l c j otherwise subset cardinality imply jensen take integer j j wavelet enough risk since eq term p bind ii consequently j p iii give iv ii j pr since adaptive loss risk thank embed remain show follow k j k j recall iv fix consequently p j p j jj nk arbitrary f f thus support pick subset uk exist triangle disjoint find theorem definition minimax either factor article construct without empirical coefficient essential simultaneous wavelet coefficient truncate completely wavelet thresholde crucial truncation although task drive truncation primary regression wavelet brownian motion gaussian nonparametric additive statistic extensively evaluate intrinsic white desirable interpretation minimax rate thresholding also sense consequently reconstruction make less appealing fine detail thresholding cf take coefficient keep tuning estimator smooth reconstruction spike behavior rather spike block tune respect pointwise spike suboptimal good find signal avoid large adaptive bandwidth choice thresholding ball construct achieve adaptive show adaptive wide thresholding optimal respect function exist generic optimal question practical naive approach estimator estimator differ bandwidth smoothness straightforward merge minimax regression construction achieve bandwidth kernel fix suggest simultaneous bandwidth simultaneous adaptation typically pointwise loss simultaneously cf derivative convergence minimax th achieve simultaneous say resolution negligible simultaneous minimax smoothness large level depend peak suboptimal converse keep coefficient resolution together coefficient far concerned level happen instead project consequently estimator coefficient work attain truncation level next idea section derive construction interval search boundary would simultaneously lead construction intersection build band impose exist finite tend proof derive contradiction use fact adaptive confidence version existence older possible h show target simplify whether confidence band band shrink slow exclude work smoothness index induce smoothness appropriate drive power view wavelet enough level wavelet wavelet truncation finite standard explain statement grow coincide truncation j impossible purely drive probability say truncation assertion truncate
express uncertainty covariance conjugate linear name square error mse filter use notation component summation filter reduce variance denominator conceptual inference reconstruction external calibration original eqs right calibration leave one sigma reconstruction use calibration reconstruction recover panel uncertainty show twice reconstruction location inspection reconstruction previous reconstruction iterate eqs use eqs sec gain reconstruction scheme estimate gray band panel inspection structure wiener wiener source equivalent formula wiener filtering argue optimal minimize remain uncertainty give wiener variance verify likelihood formula inversion ia corresponding estimator show exhibit noise position wiener filter regard realization reconstruction notational function wiener need formulate hyper spectra simultaneous spectra numerically method sample sec numerically reconstruction idea diagonal fourier denote jeffreys power spectra logarithmic formula variance wiener iterate accuracy spectrum smoothness combine non covariance covariance noise calibration show exist estimate covariance datum investigate sensitive signal identical covariance spectra wiener fourier filter mode filter precise unnecessary mode around unity spectra covariance degradation fidelity situation use measured order posteriori external determined calibration know measurement interest enough signal dominate sufficiently probe calibration uncertainty depend calibration consideration might inference non trivial minimize hamiltonian hamiltonian follow minimal r response calibration response calibration calibration enter eq hamiltonian calibration mean mean coincide calibration uncertainty external compare wiener eqs role correspond interesting n ss ab function calibration vanish vanish quadratic dependence later attempt calibration convenient regard align domain tx r tx xy xy xy tt calibration covariance space concrete respective short signal gain identical due gain redundancy gain degeneracy datum cause signal variation partly break report product response measurement essential break degeneracy degeneracy certain observation strong strength assume calibration equation introduce realization constrain datum new simulated gain specification reconstruction quantify signal reconstruction u statistic lr lr reconstruction wiener use gain uncertainty uncertainty calibration new known wiener line reliable gray region reconstruction gain thin gray fig use calibration fig provide calibration line correction solid fig despite result obviously reach improvement close sampling bottom panel uncertainty rely signal location happen vanish systematic let gain solution reconstruct careful fig concentrate simplified uncertainty gain qualitative insight circumstance calibration miss signal reconstruct reconstruct iterate convergence termination arise try joint posterior calibration demonstrate guarantee necessarily couple calibration indeed calibration work calibration signal posterior calibration result posteriori contain correction uncertainty canonical situation response know uncertainty correction uncertainty reason case source whereas source reflect calibration correction contain mutually nearly contrast correction uncertainty nature reduce thereby accurate illustrate numerical improvement regard incorporate computational calibration uncertainty uncertainty calibration asymptotically scheme correction direction also show far believe correction help refine many thank discussion generic calibration infer depend interested quantitative basis self linear calibration parameter practice external calibration solution reference internal calibration calibration find self measurement understand term maximize probability account scheme design accounting signal argue properly uncertainty calibration suffer sep furthermore argue noise filter calibration reconstruction common bin average improve calibration measurement device translation measurement impossible combine calibration accuracy gain physical unknown vary influence differ take change signal know precisely accurately recover response determination brevity kind uncertainty additive uncertainty multiplicative receiver kind multiplicative additive datum linear generic insight calibration derive apply paper classical interpret measure unknown situation strongly energy dimension impossible calibration domain possible exhibit auto correlation optimally suitably might indicate change external signal calibration self calibration scheme proceed first help reconstruction calibration criterion reasonable objective could incorrect signal calibration sensitivity reconstruct presence even hard external calibration essential joint maximization signal calibration signal stable combination calibration posteriori estimator signal coincide presence unknown turn problem skew maximum skewed systematically location indeed calibration bias prove inference problem ref however coarse approximation analytical formula follow calibration illustrative example frequentist bayesian perspective sep partly new correct different approach numerical illustrative summary main require theory close posteriori mathematical correction understand intuitively less formal rigorous sec response illustrative sec illustrative replace eq data noise gain independent stochastic separate signal reconstruction gain gain example sign positive denoting refer frequentist instance exist perform average towards average adopt essence calibration calibration infer average average realization give denote space moment path integral know calibration learn
comparison performance examine generate signal partition block determine randomly variable sum nonzero super position go beyond super nonzero independently column use square success successful trial consider successful great success noiseless employ additive recovery indicate early quantify dependency depict success different fig show original use respective respectively image superiority respective reconstruct wavelet randomly coefficient see close ex ex ex ex ex cccc ex develop new block pattern couple gaussian characterize coefficient dependency neighbor prior control coefficient coefficient hyperparameter encourage via achieve compare bayesian block superiority sparse wang recover nonzero occur arise scenario knowledge block develop bayesian sparse pattern pattern dependency control signal conventional framework individual hyperparameter associate paper involve hyperparameter also neighbor sparsity encourage solution hyperparameter expectation propose present uniform superiority exist method couple block compressive technique sparse extensively provide signal exploit enhance example audio block nonzero wavelet omp mixed behavior isometry coherence analysis suggest inherent albeit knowledge block prior exact structure address difficulty spike introduce encourage chain graphical boltzmann statistical dependency machine involve exhaustive overcome combinatorial greedy address recovery block partition expand block develop sparse pattern entirely bayesian framework model characterize gaussian sparse independently coefficient pattern couple neighbor encourage cluster isolated pattern em develop learn hyperparameter characterize propose demonstrate superiority signal organize hierarchical framework dependency expectation develop couple hierarchical propose inference method noise unknown iterative reweighted block conclude remark sparse zero sparse pattern block conventional encourage assign control signal zero place conventional hyperparameter independently assume independence potential encourage exploit statistical among coefficient bayesian hyperparameter prior indicate relevance coefficient proportional reduce conventional hyperparameter hyperparameter pattern neighbor sparse signal naturally tendency isolated coefficient encourage bayesian framework I q gamma choice switch keep determination bayesian make assign favorable set order pruning encourage solution large recovery ease variance discuss hierarchical compute q readily verify covariance element mean place mode em treat hide log expectation alternate observe e computing refer independent ignore entry denote replace current specify update independently function certainly solution albeit provide insight analytical update rule overcome drawback gradient alternative simple analytical solution analytical optimal examine optimality suppose optimality derivative individual note notational convenience subscript index notation mean simplify zero recall optimality side eq arrive rule rule resemble work conventional weight summation clarity summarize accord solution show performance away insensitive range simply follow admit analytical computationally provide insight contribute success conventional work appropriate tend small hyperparameter feedback mechanism keep decrease reach leave prominent nonzero datum meanwhile hyperparameter correspond impact tendency isolate encourage structure exposition assume extend convenience place already derive section covariance equivalent becomes estimate equivalently learn step observe value expectation form obtain know current estimate second analytical hyperparameter recall derivative replace current substitute back estimate similar form bayesian differently sparse learn mean matrix p estimate continue prescribe powerful regression firstly introduce regression address remove vector retain relevance determination mechanism overfitte superior series learn demonstrate signal present superiority simultaneous group assign multivariate prior group share hyperparameter control sparsity far improved accommodate correlate correlation see associate control hyperparameter certain hyperparameter know exact multiple hyperparameter encourage impose recover enable zhang block bayesian recovery problem work partition overlap identical address issue convert expand model remove add stack augment augment block conventional bayesian bayesian
dy dy determinant start concern second filter wiener pz admit tight variable terminal martingale paper interested converge variable originally eq fm application sufficient law adequate expansion pair another functional probability would property stable law imply expansion process apart various main mn need symbol play role call random symbol define deduce consequently compute order polynomial appear expansion symbol mix normality give let eq quantity denote associate ensure recall specify behaviour drop obtain martingale limit representation symbol subsection carry computation rigorously functional purpose p n coefficient polynomial symbol iv satisfy mm role truncation definition degeneracy dr dr obviously remark symbol admit eq calculus validate existence validate integrable hz z subsection result apply brownian one brownian weighted recall quadratic variation formula terminal martingale imply formulate assume bound localization prove theorem clearly functional convergence n result assumption satisfied computation require present convergence follow adaptive turn important derivation treat recall martingale truncation nu nu dominating turn simplicity nu v integral order two duality formula hold duality formula nu nu dt exponential martingale nu ex dx representation nu every together high derivative similarly symbol degenerate fact rank projection onto wiener z iv iv nz cx cx cx c cx step functional brownian motion depend equation later section expansion vanish consider variation process essential concrete identification symbol functional wiener refer diffusion dominate expansion second also satisfy denote resp resp drift manner brownian since polynomial expansion polynomial I rank mixed sum easy see cf symbol eq formula deduce thus naturally square deduce hold symbol define go dr sd rx positive admit expansion admit expansion recall variable identify stop stop domain integral e expand naturally truncation locally duality operation infinite validate exchange operator duality nu ds nu remark term associate wiener dominate cf shall nu n ds ds supremum product chain derivative treat section nu v nu ds thanks nu ds every derivative object limit c construction nu nu u nu solution du rd sx sd rx rx sx rd sx rx sx u ax du ax u ds ax du symbol generalize brownian motion hold fm hz nz trivially already subsection concentrate u tu tu take though work formula twice purpose see n iv u decomposition v n iv c subsection satisfied variable p immediately argument therein part introduce truncation part formula decomposition sufficiently small cf notice follow decomposition truncation obviously non degeneracy simply prove theorem recall ax deduce I rx r rx dm dc rx sd rx w I rx k f rx ia sd rx one check iii derivative aid ii suffice equation integral v ax h x algebra result p imply p p n p thus verify hereafter concentrate expansion type generalize variation power frequency framework differential aim expansion power variation type functional mathematical finance test power variation correspond function form finite recall limit continuously polynomial growth I ii formula sde derive central serve increment result interest expansion polynomial limit need ex g q later drift volatility type develop stochastic expansion appear may brownian perfectly correlated contain term next rather natural approximation central result derivation second expansion stable eq ex f expansion variation functional mathematical finance combine theorem see martingale weight require expansion expansion odd even computation symbol need function straightforward calculation give identity g ex pp p section quantity satisfied constant theorem note coincide random symbol immediately obtain symbol ds ds ex u du ds zero situation neighborhood expansion recall nf v nf rf px correspond symbol nf hz nz reason appear reduce estimation refine correspond expansion main expansion various financial estimator frequently area euler sde know euler mixed normal expansion potentially sde mention function dominate section complicated theorem wiener eq purpose use admit expansion wiener exposition begin expansion estimator expansion variation eq h h induction derivative z ex g dy dx dy dy identity enable compute polynomial ab b identity straightforward dy dy h dy separately since quantity deduce p cx dx dy obtain x cx dy ex yy dy expansion expansion cf frequency drift consistently time span applicability rely knowledge relate span expansion third quantity eq due similarity brownian except hence imply ds ds identity proof first type decompose nf expansion diffusion process taylor expansion deduce ex recall ds b du ds ex b b w ex remark treatment quantity obtain decomposition ds ex ds I ex I u odd deduce ds h convergence frequently measurable growth function let continuous result g measurable growth ga I n ds proof since ds ga ex complete k k hold n stable motion w mn obvious w f polynomial growth ex identity proof mark create national foundation support aid scientific exploratory research mathematic school university
join neighboring state higher see cluster west illustrate preference people stay stay cluster west part six state remarkable degree rest country new people rarely pair could partially attribute country state west west stay cluster next last rest add evidence proximity cluster dendrogram dendrogram fig apply define dendrogram pair resolution higher reciprocal hold reciprocal whereas become part resolution strictly merge fig pair york resolution seven reciprocal dendrogram attribute share span cf allow influence propagate cycle whereas former require formation detect cycle reciprocal people people move reciprocal reciprocal fig cycle rare united formation seven due share flow country reciprocal consequence highly reciprocal thus similarity reciprocal however reciprocal fig identical rest country dendrogram resolution whereas dendrogram resolution mechanic occur exchange direct confirm argue country apply similar cut cluster arise highlight red dendrogram green correspond west reciprocal dendrogram depict dendrogram correspond east method exception block resolution dendrogram cluster reciprocal case join north last join resolution single country reciprocal dendrogram resolution coincide order part vary resolution reciprocal resolution reciprocal state resolution exist flow infer resolution cycle compose apply qualitatively obtain coarse similar cluster axiom value transformation apply network dendrogram merge bound pair result reciprocal similar upon satisfy axiom value conclusion highlight color highlight resolution clear west define color dendrogram correspond map code show singleton resolution appear dendrogram resolution cluster flow consideration proximity determinant reciprocal california case california neighbor moreover immediate neighboring axiom transformation axiom satisfy reciprocal influence formation may flow reciprocal people rarely state accord way flow dendrogram reciprocal fig flow way merge dendrogram high country flow come reciprocal first direction six contain seven california york york indeed around proportional come four neighboring opposite merge state state direction thus merge dendrogram opposite east west division flow united merging dendrogram occur resolution cluster east west one flow two state chain interestingly west outcome direct linkage quasi fig quasi partition dendrogram new west state white singleton resolution quasi fig correspond dendrogram dendrogram merge region merge quasi direct single linkage capture formation also quasi resolution little interest singleton reveal asymmetric depict leave five influence would imply formation non singleton mechanic country reduce neighbor influence reach oppose influence formally capture similarly singleton influence arcs diagram hierarchy dendrogram cluster merging mark west california influential population partition singleton california influence west california merging map however california two state green cluster persistence edge appear california quasi partition give resolution dendrogram certain resolution dendrogram cluster allow west expect california force region permit resolution california rank state remain pair importance california order order resolution interesting form precede partial california nearby act force towards resolution whole one show decrease tendency become resolution original observe attention dendrogram dendrogram depict fig ignore dendrogram dendrogram quasi occur first new finally join coincide blue part dendrogram extend west california resolution join correspond dendrogram fig dendrogram decided extend west attention dendrogram quasi dendrogram dendrogram know dendrogram linkage department organize economic economic interact particular call input set north american input dissimilarity q decrease experiment interpret way combination input economic dissimilarity rely production say influence role function similarity corresponding dissimilarity algorithmic dendrogram cluster highlight green cluster appear fig resolution node resolution appear chain cost blue minimum service product ce management mc reciprocal direction g merge service scientific service mp mp correspond imply influence two service balanced service raw material secondary situation direction edge rl input opposite direction input come vice versa service location merge reciprocal dendrogram resolution fa big fa precise raw material product opposite fed generally occur consecutive production movement production fa however influence fa find resolution product te products ap merge production direction te basic influence represent correspond attribute te intermediate product example company movement ap back te resolution mi primary red form direction production mi pm pm mi moreover influence pm mi mining I respectively highlight cluster mainly service two paragraph occur mp rl mc represent management service relate activity fr merge balanced service fr sc come entity fr mp paragraph fr sc mp fr precisely mp fr input mp ce rl mc mc ce form ic mp fr sc sc ic come form ic resolution rl mc relation support service service cluster fig resolution red level material extraction secondary service mi resolution extend secondary merging occur mi vice versa primary pm mi resolution extraction mi come mainly distribution rt pm pm rt rt provide pm service product green compose processing aforementioned merging activity depend opposite direction product highly influence influence opposite direction cluster counter g reciprocal dendrogram merge ic merge product related situation outcome formula result dendrogram show ultrametric reciprocal ultrametric mine mi dendrogram merge dendrogram dendrogram qualitatively reciprocal dendrogram reciprocal dendrogram formation definite merge cluster resolution show grow merge singleton fp l bt cs cluster pair four dendrogram economic central singleton arise extraction construction co reciprocal merging occur resolution cf pc mp dissimilarities ii pc economic dissimilarity raw material dominant pc dissimilarity heavy input co project extraction grow simultaneous incorporation mp join loop mp loop involve new one mp mp economic come e mp come mp service co come mp architecture service sequential edge resolution cyclic influence pc diagram incorporation rl node rl loop one rl pc mp rl formation loop simultaneous go rl resolution come e g generator extraction depict loop one mp rl dissimilaritie resolution fr join chain sc fr rl pc rl loop exclude sc fr chain form appear fr rl resolution input come resolution loop mp mp fr sc fr dissimilaritie fa te ap cluster cyclic influence influence reciprocal pc merge ic resolution note paragraph precede section ic cyclic economic interaction formation look resolution influence node compose mp co discussion apparent allow cycle co cluster mp propagate permit degree reciprocal highlight correspond cyclic influence observe semi reciprocal output dendrogram highlight depict dissimilarity less resolution cluster generate draw reciprocal reciprocal fig reciprocal pair resolution high clustered reciprocal one co cluster construction product fm become cluster resolution reciprocal dendrogram resolution reciprocal dendrogram dendrogram inequality merge need fa product resolution reciprocal order merging ultrametric transformation reciprocal reciprocal allow cyclic insensitive influence precede subsection recognize extraction product direct whereas resolution cyclic structure represent loop merge service co form semi reciprocal feature reciprocal semi reciprocal dendrogram first merge resolution service mp merge precise merge resolution influence coincides merge reciprocal dendrogram increase service merge co mp blue rl secondary rl rl services service practice depict fr sc influence cycle sc fr depict economic sc come dissimilarity services sc fr whereas fr represent interpret fr act connect dendrogram join main loop reciprocal loop form fr sc I highlight blue green highlighted imply cycle formation apply cluster define formula dendrogram show cluster highlight blue red green highlight direct dissimilarity resolution cluster influence cluster put dendrogram reciprocal dendrogram merge dendrogram service reciprocal dendrogram resolution merge coincide cf must satisfy agnostic axiom influence merge single financial ft ft sc small dissimilarity influence sc ft comprise come sc merging increase extraction pc pc input mainly sequential pc water air come formation cluster around movement intermediate use mp maximum follow decrease chemical service support financial service activity important namely chemical five highlight fig every form blue either directly g form technical service input management motion picture sound ps services cs ac cluster mp mc follow sequential merging singleton cluster ps ac resolution form influential direct branch leave fig pl resolution pl need handle te chemical product form product join pl te treatment control ap main generate te te fu finally comprise product relate activity chemical outcome apply direct linkage network computed formula quasi dendrogram economic ten facilitate dendrogram dendrogram proposition dendrogram coincide output dendrogram network ten coincide ten dendrogram partition four merge dendrogram equivalently dendrogram fig dendrogram capture singleton resolution small merging influence reveal economic service mp service depict leave mp resolution would imply rl formation singleton pc sc mp diversity service economic engineering service production financial sc pattern turn influence qp partition influence service service fr totally sc influence fr since remain singleton influence preserve hierarchy quasi dendrogram co three influence resolution mp mp service join five singleton plus sc fr rl rl influence singleton keep resolution fr quasi resolution quasi influence cluster define partial every define relative state influence less resolution important resolution totally chain mp contain mp pc comprise fr contain resolution pc node cluster top red number quasi four cluster quasi partition dendrogram quasi year dissimilarity similarity outcome direct algorithmic dendrogram fig dendrogram dendrogram quasi partition dendrogram capture asymmetric influence resolution dendrogram singleton asymmetric relation quasi dendrogram finance combine precisely service concentrate furthermore qp business service every influence among exclude minimal influence form cf add influence must hierarchy edge quasi dendrogram influence reach trade service service health care social green influence green resolution compose influence appearance edge primary resolution main depict red seven span primary secondary influence singleton singleton join resolution induce block partial relative resolution min comparable green merge combine representation red sense quasi dendrogram provide apart hierarchical dendrogram dendrogram e dendrogram seem influence play merge resolution dendrogram main service except tendency decrease developed start generalize asymmetric dendrogram consist singleton small dissimilarity canonical node dissimilaritie two dissimilarity output dendrogram singleton dissimilaritie dendrogram singleton dissimilaritie node cluster two dendrogram minimum dissimilarity node map dissimilarity node equal dissimilarity node network node dendrogram singleton cost dissimilarity small output dendrogram consist dissimilarity network distance uniformly generalize hausdorff network identify cluster method subset several find dissimilarity encounter chain method comprise together cluster resolution exist chain direction contrast branch dendrogram branch branch dendrogram branch preserve branch merge resolution dendrogram network method construct symmetric dissimilarity cluster semi reciprocal chain secondary direction coincide differ link reciprocal chain cost reciprocal allow chain undirecte consecutive direct clustered together undirected reciprocal semi algorithmic reciprocal axiom extend axiom x x x axiom value x build perspective axiom perspective axiom property encode paper axiom axiom axiom source structure show respect give axiom show imply latter state axiom transformation requirement loop imply satisfy method satisfy axiom list satisfie satisfy regular prove imply agnostic axiom list properly quantification equivalent ultrametric turn notion output hierarchical original intuitive yield satisfied reciprocal reciprocal algorithmic combination read reciprocal satisfy desirable either subset reciprocal show output method method desirable property compatible axiom imply property hausdorff alternative axiom role reciprocal axiom axiom satisfy compatible value respect ultrametric yield uniformly maximal ultrametric family lie reciprocal axiom axiom agnostic axiom property influence reason fail constitute family intermediate admissible admissible preserve allow formation cyclic influence restrictive cluster control reciprocal clustering intermediate cluster generalization reciprocal method share reciprocal length single linkage linkage symmetric min algebra algebra regular power dissimilarity go node cost play major power algorithmic ultrametric compute first direct dissimilarity case opposite first power asymmetric dissimilarity opposite reciprocal throughout finite power construction insight preference internal addition apply interact economic rest appear dendrogram network reveal reciprocal dendrogram flow tight cluster east new proximity west state observe persistence reciprocal similar reciprocal axiom yield axiom application reveal reciprocal coarse east west separation fine state cluster california around around around latter ability capture one oppose method direct linkage quasi dominant california reciprocal reciprocal dendrogram cluster significant interaction service financial service reciprocal dendrogram separate cluster resolution start triplet service service group pattern indicate cycle interaction u mutual influence require reciprocal cluster cycle rather picture restrict reciprocal allow cyclic influence involve influence cycle yield around understand influence economic dominant financial service hierarchical asymmetric ultrametric summarize asymmetric define asymmetric associate asymmetric restrict show framework axiom show linkage admissible symmetric axiom axiom axiom axiom consider cluster asymmetric symmetric undesirable admit method quasi asymmetric notion axiom axiom framework linkage linkage prove quasi generalize ultrametric direct linkage compute power operation direct linkage understand understand influence cluster define permit observation direct linkage united network regular grouping california west grouping linkage california apply united linkage reveal prominent finance service admissible satisfy axiom fulfil stability desirable consider work include invariance requirement use dissimilarity attempt describe specification particular network thus give generative encode cluster network generate restrict impose property hierarchical output argue axiom network accord consider transform yx x x dissimilarity minimum exceed inequality arbitrary reverse complete proof ultrametric map ultrametric yx axiom xu xx xu xu ultrametric inequality inequality strong triangle write since situation satisfy axiom dissimilarity reduce split consideration whether ultrametric yx map xx yx coincide ultrametric satisfie immediate consequence xx x validity reciprocal ultrametric us fact prove proposition proposition strong divide recall combination triangle triangle since know substituting case strong xx xx yx x since conclude axiom q substitution prove valid ultrametric discuss paragraph precede linkage fulfil pick node network satisfy u qp qp must contain consecutive element axiom axiom dissimilarity inequality add cf dissimilarity reduce network reduce satisfied begin ultrametric xx x xx xt fix pick pair node let pair satisfy aforementioned chain chain minimize prove inequality network chain contain node q axiom show axiom network main achieve reciprocal eq tx xx image secondary tx c tx yx x xx I analogously dissimilaritie secondary chain semi reciprocal ultrametric compute chain bounding combination yx fact reciprocal definition multiplication represent minimum cost contain node reciprocal reciprocal backward symmetry fashion definition arbitrary minimize chain symmetry verify follow node show achieve cost secondary chain consecutive great secondary opposite moreover chain minimize secondary two self loop consecutive node pair back conclude claim notice secondary chain chain lx lx direction construct concatenation chain great node main link chain x construct main verify node chain equality rearrange consequently back pick complete claim contradiction symmetry order define ultrametric dendrogram quasi range valid quasi ultrametric attain well negative imply must identity satisfied triangle ensure x denote hierarchy x qp quasi substitute expression triangle ultrametric map converse result need quasi ultrametric relation identity property quasi ultrametric imply triangle furthermore respectively define guarantee need quasi partition nest domain resolution x know imply right condition continuity trivially satisfied dendrogram consequently valid dendrogram quasi ultrametric identity see true ultrametric network ultrametric quasi dendrogram belong class resolution either merge result quasi arbitrarily show identity exist axiom satisfy output least network qp axiom increase dissimilarity map dissimilarity map axiom substitute apply yield axiom thereby axiom analogous develop appearance definition reciprocal denote method axiom ultrametric satisfy cluster equivalence belong cluster proof theorem equivalence resolution x x z belong way fig nonetheless map class ultrametric calculate class belong since cluster method recall map axiom combine entail equivalence hence notation prove inequality ultrametric chain accord axiom equality successive dissimilarity justify define construction reduce map subsequent inequality combine reduce cluster axiom eq q equality invoke completed prove axiom immediate complete symmetry statement symmetry association correspondence prove statement statement identity assume element valid correspondence correspondence minimize must argue must converse imply correspondence nonempty imply know must inconsistent dissimilarity likewise value correspondence nonempty imply know construct bernstein apply guarantee exist force cardinality force must identity statement n yy correspondence exist pick correspondence correspondence conversely pick correspondence correspondence need minimize requirement subtract inequality absolute yield yield proof conclude statement secondary chain claim let correspondence imply exist correspondence analogous desire show case stability center anchor center draw black vertex fill blue draw fill green draw vertex sep draw white anchor height minimum single linkage axiom axiom space singleton small distance become cluster metric set dissimilarity represent determining term formulate cut different dissimilarity mean edge dissimilarity within alternative several laplacian matrix eigenvector nan nonzero community examine relationship perspective cut minimum aggregate asymmetric interpretation node far apart node apart close yet relatively possible network dissimilarity demand behavior proceed characterize method admissible construction surprisingly induce axiom network specify admissible method uniformly maximal admissible besides construction stability perturbation method network united u quasi generalize asymmetric influence follow section recall resolution various axiom cluster among axiom transformation axiom formally follow cluster together equal dissimilarity manner dissimilarity mapping level support axiom transformation node close arise axiom dissimilarity method axiom particular show outcome admissible hierarchical cluster existence indirect requirement direct axiom induce requirement indirect property instrumental axiom dissimilarity direction method allow cluster cycle dissimilarity direct dissimilarity resolution chain say resolution chain encounter begin clustered resolution direction whose cost resolution method rely minimax fact instrumental fundamental axiom reciprocal method axiom form form reciprocal vary specify restrict reciprocal yield output coincide linkage uniqueness axiom axiom datum derivation uniqueness true necessarily metric redundant imply two method lie reciprocal method reciprocal guarantee preserve give rise family method third admissible semi reciprocal formation cyclic influence sense cluster reciprocal cluster suffice proximity alternative admissible hierarchical axiom network dissimilarity alternative axiom node cluster dissimilarity framework section contrary admissible take agnostic position network axiom agnostic axiom reciprocal structure use output perhaps development asymmetric generalize concept dendrogram start observe cluster dendrogram partition partition equivalence hence derive symmetry equivalence construct asymmetric define quasi equivalence symmetric structure relation partial quasi partition quasi dendrogram nest quasi partition hierarchical map proceed study respect axiom cluster axiom asymmetric analysis quasi quasi equivalence linkage hierarchical axiom conclude strong parallelism network equivalence relation linkage asymmetric quasi relation linkage way case quasi former relate list relate correspond besides characterization axiom algorithm throughout determination power min max algebra operate field define operation scalar section th entry th dissimilarity determination similarly interpret minimax previously dissimilarity power g dissimilarity reciprocal chain adopt adapt hausdorff distance metric network compare equivalent ultrametric space case asymmetric network hausdorff quantify two method distance original dissimilarity every paper stable stability reciprocal reciprocal real year quasi propose cluster mix economic couple interaction example illustrate cluster axiom axiom condition compatible california reciprocal merge influence share area reciprocal cluster california merge influential state share area reciprocal outcome indicate satisfying axiom reveal intermediate information apply direct single quasi analysis quasi dominant california new reciprocal influential cycle reciprocal influence service cycle influence undesirable reciprocal motivate reciprocal cyclic close within reasonable direct single quasi reveal financial service rest dissimilarity dissimilarity dissimilarity asymmetric confusion denote network dissimilarity represent graph dissimilarity small nontrivial network dissimilarity depict define network partition set define contain equivalence always induce equivalence hierarchical index resolution previous resolution term require satisfy cf partition singleton sufficiently element q separate pair point condition together equivalence x dendrogram start form ever join stay stay keep increase dendrogram root cluster leave partition become fine leave root network leave separate node cluster part underlie denote hierarchical derive concept sequence node start chain say link connect chain end point coincide start concatenation operation cx cx cx cx cx l x entity intermediate connect consecutive chain chain minimum among connect cost instrumental dendrogram indeed resolution linkage cluster link dendrogram linkage dendrogram dendrogram partition build equal cost loop loop node dissimilarity quantity coincide xx loop dissimilarity metric dissimilarity symmetric triangle show satisfying axiom plus axiom state point asymmetric rich throughout axiom hierarchical asymmetric intuitive notion translate axiom dissimilarity node intuition form allow influence conversely latter nature dissimilarity singleton state map formalize requirement admissible dendrogram apply would resolution second restriction dissimilarity dendrogram cluster resolution capable form expect resolution node cluster formalize introduce map axiom formal axiom cluster axiom axiom element dissimilarity axiom state reduce dissimilarity cluster adaptation axiom mathematically representation identify ultrametric triangle formally ultrametric ultrametric triangle ultrametric stem prove construction ultrametric strong space network endow dendrogram define small resolution ultrametric finite furthermore minimum exist define ultrametric symmetry negativity strong negativity negativity symmetry equivalence identity property equivalence boundary triangle resolution small resolution x xx dendrogram follow substitute triangle ultrametric prove converse imply ultrametric ultrametric eq equivalence dendrogram partition satisfied identity property must bound partition imply finally technical may positive relation consequently dendrogram remain unchanged conclude identity ultrametric network ultrametric network dendrogram merge resolution merge result dendrogram since choose argument equivalence particular dissimilarity thus space endow ultrametric minimum cluster observe hierarchical cluster ultrametric correspond asymmetric observation consequence study stability say provide axiom axiom ultrametric satisfie reduce axiom admissible place axiom condition produce axiom dissimilarity ultrametric axiom dissimilarity network ultrametric somewhat interpretation virtue requirement impose axiom particular linkage dendrogram dendrogram equivalent ultrametric conclude single linkage ultrametric also write read linkage ultrametric chain axiom value cluster exercise influence node influence influence indirect chain influence introduce intuitive notion derive axiom besides intrinsic influence modality later e consider intuitive notion two part exercise influence cost link loop loop impossible mutual influence link link chain intuitive impossible observe translate pair node formally ultrametric application cluster ultrametric loop cost cf output imply cluster one form achieve formation axiom value edge go admissible together resolution arbitrary come define canonical asymmetric underlie dissimilarity depend whereas dissimilarity resolution requirement consistency axiom entail resolution permutation introduce extended axiom consider axiom axiom loop canonical n loop link link index minimum network two admissible axiom transformation regular axiom extend axiom compatible argue compatible follow axiom imply formulation axiom begin method admissible property imply axiom satisfie axiom extend axiom minimum chain cost network separate positive suppose cf block prove result contradiction point partition exist node compose since cost chain consider exist node chain small combine conclude exist chain therefore minimum must repeat dissimilarity repeat time partition node construction must contradiction partition incorrect method axiom permutation qp p p dissimilarity must substitute node definition observe least concatenation true loop already show guarantee two define sr ss r reduce validity exceed nonnegative r jk k lk il sr l kb dissimilarity combining imply respect immediate hence satisfy axiom equivalence respect axiom extend strong axiom together axiom transformation axiom impose admissible property derive axiom satisfy axiom instrumental theorem arbitrary canonical map must empty th dissimilarity satisfy xx must loop dissimilarity impossible contradict loop whose must e set repeat find dissimilarity consecutive loop time k x subset pick node otherwise x px arrive contradiction construction imply canonical compare great want consider minimum loop cost network cf constant separation x ultrametric axiom value dissimilarity consider axiom equality since map point fact imply claim axiom establish theorem also satisfy axiom influence satisfy latter axiom transformation reciprocal cluster ultrametric pair lead value argue intuitive lead influence argue natural extension cluster must influence indirect intermediate intuition formation form seem quite independent axiom require direct two mechanism indirect influence cluster mutual possibly indirect influence restriction indirect influence direct map reciprocal network satisfy axiom dissimilarity effectively linkage satisfy axiom analogous upon connection xu xx xx definition search chain every connect say maximum direction value reciprocal ultrametric possible chain recall dendrogram produce reciprocal latter reciprocal compare linkage reciprocal dissimilarity direct dissimilarity ultrametric linkage know axiom transformation ultrametric nevertheless indeed ultrametric xx verify chain exceed chain xx xx axiom reciprocal admissible ultrametric reciprocal formally cf ultrametric cf obtain xt network analyze see minimize dissimilaritie node twice possible xx chain chain go definition right side ultrametric tx fig dissimilarity great one depict intermediate u x x construct secondary consecutive node direction dissimilarity path minimize minimize replace secondary intuitively reciprocal whereas trust network propagate reciprocal situation influence propagate dissimilarity denote regard symmetric give transpose reciprocal search chain likewise cluster search direct minimum cost construct x operation algebra regular maximization henceforth product compatible size power dissimilarity ultrametric next ultrametric triangle ik jj ik represent ultrametric power play role construction indeed dissimilarity chain power cost algebra concept simplify quasi limit diagonal utility quasi dissimilarity quasi inverse direct minimum chain since already discuss section candidate reciprocal ultrametric operation denote algebra ultrametric compare compare inverse compare immediate complete operation q diagonal element consequently side quasi inverse finally prove reciprocal ultrametric dissimilarity maximization operation result ultrametric power dissimilarity transpose besides relationship reciprocal continue multiplication reciprocal th semi reciprocal ultrametric link terminology cost secondary chain maximization compute cost secondary look minimize compute cost ultrametric observe make recover comparison ultrametric reciprocal clustering reciprocal xx xx nu u emphasize reciprocal clustering sense perspective allow power ultrametric ultrametric interpret semi reciprocal ultrametric length secondary chain node intermediate reciprocal comparison respect intermediate admissible claim hierarchical cluster I ultrametric axiom compute simple combination correspond indicator position condition network ultrametric linkage ultrametric section output g reciprocal ultrametric note linkage algorithm involve power combination ultrametric admissible ultrametric computed operation regular power follow computed operation coincides take complexity use cubic relate method complexity reciprocal achieve leverage linkage span reduce set group node influence influence reduction element group favor dissimilarity group endow dissimilarity node asymmetric general induce search analogous asymmetric remove symmetry definition thus quasi equivalence hold point quasi relation term order term state quasi unweighted partition loop propertie pt edge block edge influence influence block influence respectively influence whereas notion one group dissimilarity dissimilarity least dissimilarity latter keep former addition dissimilarity whereas opposite dissimilarity need influence accordance qp dissimilarity block require qp qp quasi partition relation quasi relation q quasi conversely similarly theorem induce induce quasi partition quasi give datum edge regard quasi allow generalization asymmetric recall dendrogram nest quasi definition section quasi dendrogram boundary resolution influence equivalence x xx x requirement counterpart definition edge extreme empty influence loop requirement give resolution block merge dendrogram dendrogram set partition vary nest hence dendrogram case quasi represent quasi empty quasi empty edge space quasi space space preserve every vice versa study quasi cluster method suitably axiom quasi asymmetric equivalence quasi partition inside block quasi quasi equivalent qp cycle quasi partition cycle qp imply distinct cycle qp qp acyclic dag quasi partition quasi consistent construction partial set property ultrametric ultrametric ultrametric symmetry particular ultrametric ultrametric quasi provide preserve construction equivalence quasi map x equivalence map quasi ultrametric x theorem imply quasi dendrogram quasi ultrametric network set every quasi quasi dendrogram equivalence quasi quasi ultrametric quasi cf map ultrametric apart importance equivalence importance mathematically quasi easy regular preferable quasi dendrogram ultrametric block quasi ultrametric resolution set ultrametric class ultrametric ultrametric left figure quasi ultrametric dendrogram dendrogram dendrogram merge see ultrametric value cf resolution value appear resolution merge become belong equivalence vertex depict two appear value ultrametric fix resolution merge edge equivalence imply far x encode axiom criterion axiom direct version axiom value direct axiom transformation quasi axiom dendrogram x direct axiom network dissimilarity reduce allow axiom ultrametric mathematically handle simple axiom quasi axiom dissimilarity axiom axiom quasi axiom otherwise dissimilarity reduce direct axiom dissimilarity ultrametric justification output ultrametric sense symmetric axiom cluster method dendrogram node quasi cluster axiom development admissible axiom follow quasi output direct minimum admissible define axiom quasi ultrametric precede proposition show axiom pick node direction axiom ultrametric quasi dendrogram cf remark method next quasi dendrogram dendrogram apply conclude define equivalence define ultrametric denote linkage linkage method admissible hierarchical quasi satisfy axiom direct output quasi ultrametric eq show dissimilarity act quasi map dissimilarity reduce e axiom ultrametric triangle last imply arbitrary minimize validity equality definition inequality arbitrary pair
iw proportional baseline importance characteristic iw exist exist iw use reduce base assume corollary small iw bound intuitively contribute variance cause weighting bound iw baseline iw baseline usefulness hereafter experiment baseline gradient baseline part splitting work preliminary matlab iw illustrate toy stochastic noise controller represent immediate always discount adaptive plain plain optimal datum importance available iteration choose run initially agent choose agent transition environment trajectory repeat iteration gradient collect gradient update policy mean plain plain policy gradient update hyper deviation evaluate gradient influence estimate gradient gradient respect initial collect trajectory estimate investigate lm specifically seed correspond ph l lm experiment approximate plain sum agree mean error true obtain investigate bias iteration square iteration figure iw iw large agree upper large iw significantly importance measure importance figure importance illustrate estimate iw gap variance iw iw tend contribute reduce importance phenomenon iw constant tend significant iteration iw plain iw importance weight baseline variance plain plain agree introduction baseline bias previous inconsistent gradient bias iw unbiased plain iw bias compare mean learn hyper compare iw depict contour return return surface locate middle hyper large properly overcome increase sometimes iteration importance helpful sample converge rapidly iw update three contribute iw investigate properly iw always reach return middle iw extreme figure iw find reliable update estimate systematically calculate plain policy collect policy gradient angle gradient summarize figure red true gradient histogram true gradient figure inconsistent observe iw angle widely distribute illustrate iw angle iw concentrate highlight iw evaluate average trial approximate newly normal iw iteration converge gradient iw work first several beginning however tend large iteration iw iw next car landscape follow cb robot figure roll right roll roll controller receive angular velocity joint dimensional angle position position straight position object task target design q right cost result change reward car linear iw deviation usefulness later iteration plain policy discount rate reach degree body robot depict trial iteration trial newly draw use graph show iw plain slowly iw iteration randomly make iw improve observation mm also investigate initial th iw iw fast fast reach complete robot begin energy close object start adjust policy control freedom obtain step reach degree use right right roll joint robot distant achieve iw achieve improvement performance figure depict reach freedom iw distant robot right joint object show policy propose successfully distant object reach object experiment freedom reach increase grow exponentially decide allow difference reasonably thus iw truncate iw truncate iw truncation importance helpful method reach return higher large number number experiment dimensional tend iw promise although weight weight reinforcement number desirable high equipped systematically combine introduction usefulness truncation apply method low consider draw trajectory formulation reduce estimate full trajectory handle horizon another extension observable markov deterministic observable stationary stationary limitation extend trivial extend current formulation consider stochastic policy increase work weight scenario importance weight sampling keep hand optimal reduce use baseline comparison return baseline learn opposite introduce expectation e behind baseline subtracting reduce magnitude subtracting view carlo removal baseline primary improve compare gradient baseline current thus improve difficult often impractical baseline policy gradient lead value function feedback manuscript support support support due identically second follow variable upper could get bind elementary scalar q vector assume minimize iw immediately plug independent distribute know know give result mm theorem corollary example appear mm gradient exploration flexible powerful reinforcement method control give policy estimate previously datum variance maintain give estimate usefulness objective rl optimize policy among become search highly search gradient popular physical control policy change gradually obtain suffer recently novel policy produce randomness policy useful promise experimentally accurate bottleneck cost useful collect policy policy allow policy importance variance policy variance truncation suffer trade mean expense bias purpose systematically address policy basically weight technique first iw method consistent iw estimate achieve significant improvement artificial investigate combine reinforcement rl agent environment review adopt policy exploration agent observe select receive immediate result action characterize state next action density reward agent parameterized assume differentiable form denote denote note trajectory discount cumulative discount parameter policy policy follow parameter ascent standard estimate trajectory policy introduce cope problem deterministic prior distribution trajectory controller trajectory estimate deterministic q dimensional function denote transpose policy distribution return expectation hyper optimize maximize optimal derivative note logarithmic derivative expectation average q draw collect paper employ hyper allow deviation eq respect approximate gradient gradient update rule
report uncertainty region automate candidate consists build dynamical observe necessarily operate require explain capability identifying due vast presence loop consequence require accurately system present region naturally identification future fully external conditionally one characteristic often perfectly density condition nonlinear state represent deterministic stochastic difference identification condition observe difficult compute want e simple autoregressive model system observable output base amount state state drawback autoregressive generative generate new apparent randomness innovation innovation accommodate much identify instance signal significant tune characteristic automate simultaneously perform tune relate present brief introduction process identification integrate data dynamic conclude remark nonlinearity order efficient contaminate present incorporate carry manually portion low time combine see offer amount obtain quantify uncertainty uncertainty particularly useful typically agent identification probability belief successful field artificial heavily uncertainty flexibility make ideal formally collection random value location jointly normally parametrize degree constrain predefine shape relate contaminate exist incorporate regressor case covariance construct figure seven noisy infer exponential posterior equation selection refer hyper convenience attractive computational theoretical procedure maximization find integral likelihood specify error rather likelihood function side useful likelihood usually balance contrast marginal eq cubic limit great factor derivative hyper complexity hyper necessary compute posterior behind incorporates particularly name propose dataset location convention gp system identification nonlinear process describe marginal metric ability automatically model fit principled goal maximize likelihood respect hyper gp employ hill marginal logarithm simplicity pre process marginal become derivative straightforward process difficult regressor derivative pre hard pre process smooth processing per compute marginal dataset strategy whereby marginal employ gp datum process dynamic prediction show possible predictive efficiency induce overview gp initial guess hyper guide guess successful datum run step gp follow pre parameter subset magnitude consist perform subset gp optimize marginal subset datum final predictor equation pre obtain predictive aside autoregressive critical performance meet order order relevance determination ard covariance marginal irrelevant hand add hundred regressor cause experimental two nonlinear identification benchmark circuit wiener identification control engineering journal corrupt cope amount signal identification noise tailor benchmark comparison toolbox avoid particular benchmark underlie regressor report regard choose pass filter filter benchmark synthetic paper choose subset induce point filtering signal filter signal matlab toolbox pre take computation benchmark allow optimistic capability gp validate automate pre report times intel processor training gp fast provide freedom trade use point increase number risk overfitte present gp
notable symmetric word playing reveal play loss symmetric equivalent arcs cycle system direct instance adversarial inform direct direct sequence theoretic notion play important dominate two maximal size maximal independent denote associate simply view undirected ignore arc orientation dominate arc set dominate subset dominating dominating direct small dominate orientation undirecte explicitly cycle associate minimal direct graph graph direct turn minimal dominate connect orient dominate maximal arbitrary direct associate hard repeatedly dominate set lift acyclic subgraph give direct graph acyclic graph direct cycle undirected pair node arc cycle acyclic simple regret logarithmic factor hard adversarial direct though ifelse ti tp p p ti ti I exp mixing use divide probability probability exp view probability draw similar analysis bind shall two irrespective depend probability equivalent add exploration know prediction inform set simple result subsequent appendix adversarial exp satisfie system undirecte inform set irrespective immediately give adversarial exp constant unfortunately appendix theorem graph acyclic subgraph regret small lack something sophisticated upper refined way probability direct available show analysis fail fact call add exploration graph reason exp prior dominate turn require graph exist distribution number subgraph confirm hence find arc ready analyze inform direct exp index dominating direct induce algorithm quantity exp set let adaptively choose dominate exp use slight variant ifelse exploration vb dominate I tw r tp ti adversarial regret exp satisfie trick result number whenever see dominate observation exp dominate exp system regime bandit setting characterize prediction term adversarial fully improvement paper improvement inform setting refine theoretic provide solution rely analytical tool currently currently investigate apply system prevent direct unobserve many include see hinge express term suboptimal try adequate complexity corollary heavily rely generate term inform acknowledgment first author support advanced usa author support project grant author foundation united foundation theorem theorem claim universit di university consider armed introduce main direct model dominate independence number achieve exp operate basically symmetric informed inform dominate observation step graph need lp abstract study problem formulate player round assign loss fix set action randomization incur excess incur player round end observe bandit observe choose action action round possible bandit regret inf exp slightly elegant intermediate expert bandit intuitive action arc thus play action reveal loss empty reveal action regret independence undirected prove optimal variant exp graph ahead full edge program observe current number direct within factor case graph run dominate current computing dominating regret independence ignore quantity exp independence combinatorial interest graph set variant need current graph exp also exp set acyclic subgraph tight yet much simple less demanding variety correspond direct undirected connect select connect drive consumption addition sub abstract arise product know often preferred orientation person video game tv vice def indicate game system operate product case social interest however link likely preference person follow product probably person
avoid repeat optimization testing representation unique posterior maximize log proof simply fix covariance evaluate solve norm see parameter elastic net constrain elastic norm represent n net norm weight analogy elastic net elastic tradeoff norm net similar encourage smoothness encourage knowledge estimation give trace parameter elastic net parametrize elastic parametrization tradeoff trace recover hilbert elastic start employ warm start elastic net inference trace global explanation assume prior extension include short completeness represent I posterior observation shift bias parameter n find bipartite rank task item rank ahead negative bipartite optimize pair rank although applicability pair wise scale recently researcher pair list set item gain literature empirical performance inspire mr adapting predict relevance score induce correct suffice jointly transformation parameter estimation propose mr bregman divergence extension scope improve mr favorable good bipartite set sort set vector order inequality compatibility concept sort compatibility compatible sort j compatibility compatibility sort vector definition compatibility straightforward check binary keeping separately permutation sorted propose compatible vector let permutation next note sort order permutation strict ordering generate n eq dependence compare generative model auc correctly order follow nonzero maximize ex auto ex ex z r latent nr n rank equivalent variational bind outline restrict expectation problem parameter follow alternate optimize optima reach close require alternate alternate optimization achieve focus optimization term evaluate infinity constraint constrain task hence rank independence score arise cost invariant loss degeneracy constrain away score l vector equivalence form flexibility optimize permutation result dataset disease gene know gene contain unable insufficient storing kernel association gp kind know rank ability association randomly generalization know disease validation cross disease disease optimization trace maintain low optimizer employ improved performance learn bias training row testing hyperparameter space singular return warm select model allow dataset expense computation full computation motivation recall association follow sample disease association combine negative gene disease graph experiment graph gene adjacency normalize experiment observe identity graph metric association laboratory consume costly rank prediction practical metric rank remove gene average train total disease curve rank fraction retrieve retrieve gene position retrieve gene mean average precision result reflect performance gp experiment association difficulty reflect table fig gp suggest trace norm effective significant across metric disease training disease none remove disease interestingly seem unable experiment limitation fig new disease trace norm rank model performance list domain final fig especially interesting find model outperform trace outperform rank top elastic regularizer sparsity auc investigate metric sensitive explain good hilbert trace hilbert trace trace paper bipartite combine variate trace lead discuss elastic net arise constrained estimation disease gene significantly improve strong elastic analyze plan explore gp filtering acknowledgment acknowledge nsf helpful discussion thank student machine dr electrical engineering minor institute complete electrical university wireless communications lee student dr focus learn biology receive b computer electrical electrical engineering human development university institute technology ph california currently department electrical engineering publish book theorem lemma theorem edu bipartite generative wise inference impose low variate covariance close mean variate regression regularizer bipartite motivate candidate disease gene goal aid unobserve human gene gene gene disease illustrate find solution scalability baseline trace pt bipartite bipartite ordering rank propose bipartite extension wise matrix regression trace impose useful exploit inter relationship prediction learning domain variate process relate value kernel rkhs estimation gp variate possibly alternatively gp understand scalar across task gp link prediction application motivate disease gene determine identify human interact researcher thousand include cause standard discover genetic association conduct gene disease scientific interest disease receive response association association e problem learn task unlabele collaborative address recent rank ahead item ahead negative rank list produce reason gene pose bipartite task induce match observation profile assumption validate rank constraint several requirement na I applicability factor low structure rank factor solution non posterior without drawback factor propose jointly trace square cost weight relationship elastic net good knowledge application net bipartite ranking approach variate disease gene propose novel variational typically matrix bipartite knowledge first variational model combine maximum bipartite domain disease gene useful property product identity variate gp variable let row denote scalar mn kronecker product product kronecker assumption restriction improve computational enable covariance regularity impose separability improve reliability data iii wise special case joint product model analogous inference row covariance gp kronecker product extend subset complete finite arrange covariance r entry goal response column observe consist proceed follow z nz n may task eq index sample index training identity definition scalar gp appropriately complexity observe storing memory na I computation auto right ex edge variate process result hierarchical attempt see draw function fu f nn require expectation characterize laplace utilize large show element covariance overview relevant
random sum oppose must unbounded notion effective metric diameter context diameter spread give subgaussian class subgaussian diameter exceed essentially replace bad rest proof hoeffding method difference define v expand nx nx jensen observe lipschitz variant fy fy fy fy subgaussian diameter follow exponential optimize yield compare recall whereas inequality uninformative diameter put verify analogous x nx case easily subgaussian concentration albeit algorithmic I iid identical henceforth training literature assume invariant explicitly restriction empirical excess end notion propose stability general say z totally give n strong require metric fix z z jensen argument take prove lipschitz excess totally define excess r separately function lipschitz change combine lemma main totally lipschitz decay indeed albeit restrictive stability plan mix notion marginal denote achieve infimum one always coupling refine metric space equip define distance coupling dependent variable verify valid metric shorthand notation conditional I x x maximal quantity discuss conditioning set state main subgaussian subgaussian nh ij maximal subgaussian reduce definition respect special consider martingale couple infimum recall x yx substitute jensen argument third repeat martingale argument function convex vanish subgaussian converse consequence markov satisfy diameter distance extend straightforwardly finite sequence metric probability usual suppose n p concentration diameter show applicability stability unbounded loss give extension non process remain bad gap critical like recover subgaussian necessary subgaussian concentration lipschitz exhibit lipschitz match like compare satisfie metric nontrivial reverse question kernel totally acknowledgement I refinement thank ari correspondence manuscript extension unbounded notion subgaussian diameter method weakly nontrivial former generalization hold unbounded give concentration strongly concentration inequality speak sufficiently close datum quantify notion express relaxation use various strong mixing elegant powerful driving real value whenever lipschitz bound typical aside instrumental pac inequality stability result iid extension free nature attractive tool impose inherent limitation applicability limitation bound bound everywhere high constant still bad counter introduce inequality everywhere influential number recent entail analytical practical still concatenation lx space product borel extend naturally sequence associate define independent independent away random value say subgaussian small hold denote let centered subgaussian subgaussian diameter diameter diameter certainly hence
contiguous community one hand parameter powerful hand asymptotically satisfy recently focus equivalently theoretic degree base scan eq scan large present regime obviously vary regime remain scan wide scan broad scan powerful merge small powerful merge furthermore broad see two real exact scan degree powerful remainder regime broad scan asymptotically powerful merge asymptotically table visual regime fix broad scan powerful hypothesis fact triangle nontrivial merge completely test able large connect hypothesis merge asymptotically large connect table degree c large cc broad scan bound rely alternative anomalous deriving versus merge variant aa rely property consideration use separate model contiguous subgraph recent purpose os connect method seem situation moment limit remain paper follow notation concept probability statistic concept hypothesis study bound situation unknown derivation notation list change leave implicit unless specify limit hypothesis edge bound vanish chance throughout discuss situation dense consider regime unweighted subgraph equivalently notation variable positive part integer importance tail function asymptotically powerful risk test test prefer complicated insight practice indeed efficiently practitioner parametric besides reader obtain theoretical critical proof concentration chernoff integer binomial integer denote count contain stochastically consider establish recall degree recall total powerful asymptotically truly ensure naive regime scan play regime broad scan define preferable scan connect scan small fix know edge size roughly informally promise scan detail seem scan subset define definition exponent scan asymptotically powerful w kp broad scan powerful regard show positive broad powerful scan test asymptotically factor prove minimax boundary scan test scan scan scan shown bound away broad scan essentially result scan control nan chernoff lemma apply know entropy follow respect bind alternative w k powerful stochastically increase strategy connect cn cn w small suffice go consequently suffice bind positive due generalize stein define w jt first real q inequality large connected large component asymptotically powerful large connected component equivalent show technical hypothesis critical behavior converge slow power go keep exposition regime derive phase os enyi know q hence component recall condition stochastically denote lemma define apply max cn denominator denominator zero imply assumption shall connect component powerful rely moment branching ni fix sequence denote belong go proof observe use size probability tend conclude imply large collection component extract go node graph remove let component suffice observe conditionally comparison connected branching process finish prove go one chebyshev derive rely lemma upper node binomial branching statement leave prove process copy subset sx stochastically branching rely low r k x qp k k q line inequality stochastically branch random inequality nr nk argue decrease integer follow come turn proof last last graph term since conditionally continue hold sequel mean small may keep condition equivalent similarly keep condition true infinity large intersect dominate imply connected component asymptotically denote connect shorthand edge symmetry expect cluster branching branching need sx sx sx sx stochastically dominate binomial branching process set event outside turn small independent process get sx sx q q kk b ccc ki condition conclude component regime graph probability tend alternative completely test asymptotically max hereafter token n max max max p asymptotically prevent take analogous identity eq set prove rhs w asymptotically q theorem include subsequence converge hold also event since forest tree exactly size expect cycle go cycle forest conclude iii w iii stochastically hence chernoff min therefore q therefore suffice show min n p definition transform minimum kk min fact k p uniform start bind bounding consider forest component since connect follow label vertex formula suffice forest edge tree exactly way obtain tree fact forest forest small stochastically see imply fourth come use work let n w care analysis n n use remain one hand use leave k n b specify calculation consider separately ok kn ok np hence conclude imply cycle may cycle satisfies least two cycle cycle least cycle length denote configuration potential cycle possible possibility nod k sn sum control cycle occur cycle common cycle share edge observe configuration first possible configuration cycle less possibility n follow hence occur argument eventually moment proceed assume use together derive I ok ok k n kn ok ok w k ok k integer define k union chernoff binomial k integer eventually n sum bound motivate sufficient asymptotically powerful bounding moment come result dominate truncated ratio ratio versus risk denote expectation convention likelihood optimize cauchy schwarz focus bound case cover loss write fix powerful moment f consequently course cycle cycle moment rhs eq satisfied find detail total applicable replace difference degree scan calibrate way power show test degree test truly also calibrate asymptotic large also truly broad scan definition argue suffice meaning concentration inequality accommodate come see first case situation asymptotically optimal situation scan completely test soon bound versus provide although scan asymptotically inferior superior open plant clique broad scan sufficiently test close powerful tt q use conclusion us control take infinity n let go first result extension identity number label contain label label label forest satisfie double counting note tree two label outside order straightforwardly label vertex consider rooted size iterative add vertex root root construction order vertex label contain outside count modify put orientation first orient orient partially orient observe except subtree leave subtree simple claim partially orient undirected fact orient claim conversely orient child root tree orientation satisfy second part rely double forest label label straightforwardly alternatively choose vertex root iterative choose vertex root tree root result root final k ignore sequence tree vertex order k r k vertex outside unique orientation claim tree node obtain induction k build subset sequel sum configuration compatible event tree apply k configuration connect k q k p k k bind subset size low convention easily case n configuration identity lemma computation hence k q nk n q k maximize respect subset play consider sum dependency implicit edge k q apply configuration tree node give token tree c obtain let whenever two connect since contradict tree lemma forest q way tree forest component complete elementary k q k r apply last line decrease k e c r counting line use obtain observe sum acknowledgement thank discussion count research partly bs calibration support grant pt corollary
current fit model label label thereby label quadratic da eq classified jk eq unlike unsupervised assume notation section solely datum q carry empty happen case rule achieve success inclusion prove beneficial conclude reasonable argument classification incorrect inclusion rate construct negligible inclusion available label treat argument result label exponentially valuable mixture unknown mixing consider seem weight label contrary benefit misclassification roll phenomenon adjust adopt wherein control contribution population interest change estimator population exceed lead alternative dominate stein modify stein minimax estimator stein distribution inferential estimator traditional mle parametric likelihood subsequently author information population cf contrast assume population size version datum population identical pdfs I I iid incorporate population notice yield traditional likelihood herein positive weight relax maximum describe adopt weighted inferential view draw population traditionally primary pdf interest give z ij weight maximization l write expectation follow inequality side maximum maximum similar eq illustrate maximize cf section correspond cf empty correspond cf less restrictive bound update algorithm mm find maximize recall update maximize respect lead lead lack initialize primary group origin attain truth use adjust rand ari efficacy ari agreement partition account ari produce membership simulate ari correspond assign relevance maximize goal maximize weight q th observation example combination population calculate candidate low discrepancy come offer alternative aforementione simultaneously estimate differ keep discrepancy introduce therein mle adopt taylor drawing state aforementione solve share kullback kl entropy f quantification kl kl specifically density base f g practice provide mix sufficiently specification consider equally spaced increment datum simulate two dimensional component panel ease value plot figure kl ari right kl ari da attain da version rp pp relative drop separation cf conversely relevance weight less critical separate cf prove study ari instance choice relevance rp kl rp approximate relevance weight separation consequently specification improve stem motivate argument primary proportion require consequently accordance special case choice one case weight high average ari bias consider large datum plot see ari relevance respectively misclassification role degree separation along group compare physical property also measurement r package mass set label rp rp result ari compete calculate specie label black ari label classification da da da consistently fit result consistently classification put none find gain see ari comparison attain produce compete case attain average ari label apply label version plot ari six base perform three specie notably ari less half label model either specie label drop outperform da classification obtain ari ari yield ari whereas cluster classification da ari respectively correspond majority attain cluster datum figure outperform produce label gain instance take ari ari apart consistently set unlike one correspond specie optimal label label right herein flexible classification construct likelihood coincide result efficacy insight employ course efficacy weight illustrate aid label build theory variation value include special framework impose another say might calculated upon weight extension skew mixture thereby class choose herein static weight initialization throughout work consideration make wherein global l global local fashion ij lp z jj lp write leibler distribution section since kullback leibler always lp j p lp p local global maximize lp n p lp analogous department mathematics mathematics university traditionally supervise supervision sub supervise level supervision range
multi prediction approximation well often answer conditioning approximation quite ensure conditional learning define log use tight case recurrent net train justify mp learn single model mp train net run recurrent net inference justify mp approximately recurrent bag useful train useful choice variational underlie mp inference suit primary ensure persistent approximately evolve mp early burn accuracy mp easily break benefit mp easy mp contrast likelihood partition condition visit momentum schedule sparsity regularization hyperparameter hyperparameter constraint center require hyperparameter range minibatch layer minibatch fig mp consistently well less likely free momentum schedule change keep tune well center good mp add tuning train twice hide use generic follow entirely mlp evaluate classify detail amount center explore resolve demonstrate mp train trick training match still work probabilistic capable handling miss input answer training apply boltzmann require pass train layer perform task layer call train maximize network share novel trick term approximate input deep model consist layer latent visible unit represent form latent hide organize conditionally neighboring independence entire likewise point fast half proceed alternate update define normalizing energy intractable due summation fortunately estimate procedure whether simply rbm intractable likelihood interaction layer make procedure interesting connection layer entirely value approximate repeatedly two update posterior essentially expectation simultaneous component invariant handwritten digit conjunction test comprise cd rbm extra mlp top expectation mlp train gradient descent field unfortunately train deep boltzmann approximate fail naive cd rbm rbms slightly rule must predict well paper jointly train excellent classification specific extension approach different yield single answering query one model function outperform previous answer missing subset subset prefer single suboptimal influence deep layer attempt make optimistic deep layer boltzmann parameter set optimally nature unit share leave model connection factor multilinear difficult layer make probabilistic inference query classification miss implement stage develop software consideration kind procedure ability usually classify choose serve complement run black circle target green field graph indicate inference line run another mp iteration train possible complement multi mp sequence subset term subset remainder sgd simply sample factorial sense rich structure minimization net fix net description fig inference mnist apply trick receive iteration expensive run fix several order expense train long
imagenet fast produce internet represent significant resource inference train convolutional transform feature time convolutional pair set efficiently product network explore accelerate network fourier filter could maps overhead map modern factor lead speedup magnitude backpropagation standard compute convolutional three fix index feature convolutional forward corresponding gradient layer gradient respect feature eq operation consist circular letting fourier follow input convolution direct cn n pointwise require product represent index pairwise product though less convolution overhead precise input feature map pixel perform update complexity operation ns transform input fourier multiplication feature yield similar l convolution f ts fs c method come term show theoretical operation direct convolution size conceptually relate gpu needed section remainder convolution take input minibatch store store number assume memory follow ram ram use mb mb mb mb mb mb mb mb amount memory run series machine environment experiment gpu operation operation round acceptable compare speed size minibatch output measure see outperform nearly improvement likely convolution size image apply explore future run parameter configuration convolutional tuple indicate width image map feature input square size size highlight bold ccccc total configuration sometimes make especially inference obtain layer add fully connect account possible performance cccc total two implementation present fast implementation verify fourier domain domain remove accelerate power suboptimal must accept size fact speed large explore york york one employ vision leverage ability large
understand variance observe paradigm look statistic verification classification meaningful inherent instance provide evidence become wide individual population sequence neural subtle group question relate gene environment complicated input measure generation movie quality length generate application human acknowledgement thank david david discussion suggestion acknowledge medical institute program physics many formulate health gm gm national foundation find raw please edu background result image contain spurious image size result mask mask within boundary centre orientation frame vary obtain isolated template typical image manually step align achieve magnitude transform template translation transform alignment upon transform accordingly alignment achieve subsequently aim decomposition image make pixel however contain analyse tractable analyse subset pixel negligible dynamical accordingly subsample contain variance certain primary obvious truncation likely majority give frame compact transformation parameter embed embed seek kullback leibler transition space convex complexity perform start ensure however start return well calculate independently also make moreover entropy rarely accordingly allow original lastly median embed eqn embed bit find precisely construct onto von constant phase average alignment phase phase offset shift angle transform ht frequency channel wavelet pass hz cut hz entropy size near calculation ht width estimator embed standard von phase ht embed ht movie correspond align raw video display position within circle represent appropriate light blue behaviour text coarse movie real indicate subsequent portion movie indicate movie region table movie segment factor clarity movie movie movie movie movie movie movie movie movie factor instance preferred fig movie composite science activity describe term rely structure classify grind roughly discover state use result subtle difference nature influence decade possess ability vast way constrain thought action even potentially via limit control pressure robust search circuit begin action despite centrality existence largely lack mathematical quantifying dynamic map quantify lie coarse activity velocity barrier count frequency experimental turning leave throughput specie aspect behaviour often subtle effect apparent fine common approach quantification set category recently supervise technique approach throughput label human behaviour analysis assume class behaviour exist show action discrete manner ideally directly assumption consequence behaviour trajectory dynamic epoch trajectory near position represent trajectory stationary correspond action space move correspond distinct run head biological range ground largely part dynamic record individual sufficient resolve move body show thin side form clear diameter height flat prevent ability cover compound prevent surface find behaviour camera pixel keep move camera controlling position frame hour yield movie frame aspect interface isolate within imaging occur day place subsequently collection fig occur pm thus temperature framework background enforce invariance wavelet create spatio temporal representation dynamic neighbor lastly probability dimensional space peak confirm near peak state rescale align frame decompose series transform create plane lastly point peak wish representation dynamic start follow frame detail list method edge mask mask align cross template previously pixel number segmentation alignment body segment mobile degree compare image accordingly representation angle extract nearly project observe pixel euclidean apply frequently linearly span large eigenvalue rigorous mode find direction correlate variation mode see intuitive interpretation project image axis convert movie fig black highlight sign cumulative variation number projection mode instantaneous behaviour definition study series paradigm often problem temporal alignment relative component additionally certain moving time wavelet mode fourier possess multi complete occur scale particular periodic eliminate precise detail show example display fig present space hz hz comprised channel make correlation reduction trajectory embed local require long choose distant multi scaling service large scale possess embed aim much small preserving possible transition walk perform transition proportional kernel set restrict neighbor keep transition possible technical reason cauchy distance embedded embed initially position drawback incorporate importance set implementation detail lastly need accurately shape mode spectra overall multiplicative beginning wavelet simply euclidean two greatly amplitude however compose normalised mode hence reasonable leibler kl divergence b embed show embed nearby similar data three dimension reduction fig probability embed width location peak intra inter individual peak trajectory numerical move dynamic trace period quick space normal peak localize peak fig velocity embed comprise connect plane embed local density fig one peak last nearly region perform supplementary movie familiar classification extension segmentation category near region distinct movie vast point visit total less integrated probability within region region similar movement region perform visual movie periodic underlie dynamic produce fast algorithm similarity potential hypothesis periodic trajectory eqn region periodic clear hz spectral systematically investigate dynamic phase cyclic coordinate phase use hilbert phase combine maximum
cycle share root formula rather hybrid root choice consideration covariance square root covariance assimilation hand improve minimum eigenvalue ensemble assimilation calculate background counterpart process eigenvalue obtain previous calculate ensemble generate forward start assimilation inversion aim assimilation cycle lie residual satisfy convenience visualization background thick filter visualization plot scale thin solid norm calculate plot choose outside fig place result mean filter sufficient residual certain show eigenvalue cf circumstance g relevant matrix adopt highlight remain issue include nonlinearity operator former suitable observation constraint influence enkf therefore enkf perform aspect future like thank anonymous constructive suggestion author project realistic well financial background rmse mm norm f normal residual bound fig assimilation cycle rest assimilation international institute gate author distance aforementione certain condition indicate bound implication discuss behaviour kalman enkf covariances literature issue localization handle one increase hybrid way relaxation scheme modification back residual member name increase robustness enkf uncertainty assimilation improve residual respect dimensional opposite sign call innovation filtering enkf index enkf drop linearity discussion result present later might insight residual let notation q vector observation transpose weighted convert euclidean standard euclidean topological property euclidean g euclidean assimilation follow let state truth tr record observation realization assimilation da tr triangle residual norm hereafter satisfie da expect scalar practice though upper expectation observation upper evaluate matrix satisfie introduce correction residual analysis ensemble kalman see residual less accept introduce residual modify observation short state estimate show substantially improve extension examine enkf analysis ensemble kalman certain accordance scheme proportional background ensemble hybrid enkf examine norm one want work prevent circumstance instance fit multiplying presence extra may let move inside gain resemble kalman gain enkf use obtain dependent formulae residual firstly suitable one equal root secondly obtain inequality omit brevity scenario condition obtain variable case formula enkf therefore eqs residual aforementioned say bound respectively fitting multiplicative evaluate circumstance compute mean formula scalar positive definite suggest follow svd eigenvalue b determined kalman invariant eigenvalue account accordingly alternatively sufficient even lie interval may analysis residual norm satisfy focus l verify analytic intensive filter residual trajectory numerically integrate drive forward integration step discard step assimilation background background run
assign subsection spatially uncorrelated size uncorrelate coefficient ki uncorrelate across give correspondingly block expression whose equal remark covariance define hermitian subsection asynchronous arise model random failure model policy reduce consumption mode self save behavior model ki fix however social agent neighbor save communication interpret failure link agent drop bernoulli ki k coefficient view extension domain distribute take density pdf bx figure asynchronous assume take large ki ki bx shape assume combination coefficient k spatially range relevant introduce asynchronous asynchronous cover due neighborhood combination network influence summarize follow important present moment random size remarkable random combination randomness square strategy insensitive network asynchronous asynchronous achieve investigate stability denote error certain recursive bind factor unit bound step stable pdfs th moment decrease become establish small imply agent reach close desire steady interesting behavior failure asynchronous able desire actually subtract equation introduced j k conclude vector evolve mn agent within square stability recursion dynamic step asynchronous moment ensure specialized bernoulli beta admit size randomly stable network turn part derive explicit expression expression asynchronous establish useful fig close jacobian th respect easy z jacobian column w respect complex conjugate conjugate gradient hessian q hermitian identity play obtain ki ki ki entry entrie lemma realization step ki ki uncorrelated except th lemma ki ki side recursion get express value equation use ki ki asynchronous get condition ki property asynchronous sub conditioning ki ki ki hermitian definite large coincide eigenvalue condition yield divide side hold start I convergent upper condition hold bind upper k k hence get substitute jensen therefore size ki ki q condition moment parameter appear bind substitute expectation enough I use fact enough eq use write fourth moment govern quantity asymptotically guarantee sufficient guarantee hold hold straightforward verify hold k bound old k k substitute substitute k substitute arrive obtain therefore work support iii stability asynchronous distribute examine asynchronous uncertainty topology link failure random time agent turn stop update solution may stop agent order stable reveal influence network asynchronous centralized solution notable performance asynchronous degradation size largely solid justification remarkable face multiple level link distribute diffusion asynchronous topology link global distribute learn resource allocation decentralize consensus incremental strategy develop purpose range lead enhanced step necessary enable continuous enhanced explain exact actual size decay noise constant use unstable happen stability show insensitive topology randomly concentrate diffusion note extended consensus continue fairly general asynchronous already literature consensus presence asynchronous topology limited study strategy work early ability assume problematic purpose stream decaying eventually remove limitation also allow fairly uncertainty failure occur three asynchronous behavior vanish step affect sort agreement steady despite despite possibly steady agent much generate asynchronous stochastic solution derivation require due one systematic fairly asynchronous square arrive distribution behavior arrive steady centralized environment conclusion follow part asynchronous comparable case failure work analytically scenario component still performance remarkable intrinsic robustness material discuss body technical vector letter matrix plain letter also denote conjugate matrix inversion euclidean besides kronecker show objective aggregate allow value problem field communication fairly model wireless channel weight etc strategy w w replace complex extended way interpret value function argument base entry interpret w j analytic linearly ji ji follow analytic property individual cost k assume assume common minimizer function frequent especially need attain usual agent interact avoid common likewise wireless common survey interact track machine agent common still share minimizer agent subject condition information sharing agent may sufficient sharing agent ill conditioning enable desire strong ensure hessian away zero ill implementation base convexity serious limitation help strong though still hold require convex cost cost derivation argument demand opt main result work gradient vector representation argument hessian hessian matrix requirement study vector growth explanation hessian hessian assume continuous denote w continuity hessian globally lipschitz k j k traditional network study asynchronous useful aggregate form equation size I j k learning compute iterate intermediate adaptation share neighbor combination constraint denote agent collect condition matrix vector general agent sufficient gradient noise nature hermitian semi let covariance I link satisfy extend independent satisfy appeared distribute however employ explain several scenario cost logistic cost modify diffusion nonnegative coefficient random satisfy constraint compare ki ki ki collection time step collect asynchronous consist condition random matrix kronecker denote entry diagonal denote entry represent k ki I number process consist whose kronecker denote coefficient ki nm become random mutually independent topology neighborhood combination network general
instance polynomial attain value distinct point get algebra get jj imply polynomial attain express function expressive build deep network layer attain network tend deeply deeply gradually decrease principle control bias remain build attain e attain calculate process terminate polynomial polynomial linear function basis way construct orthogonal use schmidt orthogonal moreover differ augment center express generality fixing basis specify form training df use trick degree degree attain find span quickly run vector utilize architecture polynomial degree degree polynomial plus first layer network polynomial span eq scalar degree polynomial span attain layer every first value polynomial degree polynomial basis value degree polynomial subset linearly column construction algebra gram schmidt procedure column column specify nd layer column correspond correspond nd compute layer augment left repeat maintain attain new column polynomial degree newly layer attain polynomial maintain stability multiply factor scale otherwise iterate product value large specify span subspace stop architecture feedforward connection layer unlike many deep moreover although possible empty linearly f f ir ir n compute nz diagram computation top right svd column compute column call ir r ir ir basis schmidt find column together code tolerance machine attain degree train linear predictor attain polynomial early incremental build particular basic idea polynomial lot idea emphasize emphasis generator polynomial vanish goal nothing deep orthogonality end fourth describe far use make remark need advance instead one output exist satisfactory loss function deep constrain important nice backpropagation tailor easily architecture intermediate construct way construct advantage connection generalize still sufficiently expressive compute involve compactly complex principle choose layer product layer result geometry deep connection geometry finding area year aware basis polynomial polynomial polynomial algorithm propose generator vanish focus construct vanish representation get derivation turn property particular runtime polynomial set zero depth loop simplicity plus solve convex layer sum column span training vector prediction value assumption statement arbitrarily theorem easy linearly item whenever terminate definition bound memory explicitly entry implementation propose figure relevant item follow node layer exclude output layer output node node plus output finally final weighted output immediately derivation weight increase depth network exactly treat stop happen column span span training imply every degree polynomial polynomial span basis universal algorithm provably get trading size potential overfitting assume corresponding width target vector orthonormal qr orthonormal basis linearly ir ir ir ib w implement return pick indicator multiclass layer present provable limitation control node number instance drawback might huge computationally ignore huge modification width constrain column large span span small question choose unsupervise follow find practice layer transform data singular component layer standard least square seem relevant intuition value residual project simple batch vector iterate procedure implement algorithm precise code explicitly potentially large correlation correlate constrain learner guarantee case width adversarial terminate training happen linearly zero algorithm terminate happen indeed long position formalize analogous thm variant intuitively general linearly plausible entry reason happen formally distinct matrix theorem thm use general position assumption distinct point width construct depth number class memory plus require remark total arithmetic operation perform monotonically decrease return unconstrained except terminate drive iteration obtain mention perform svd e time approximate perform degradation svd construct proposition condition item way dd add gaussian arbitrarily variance surely memory requirement mild quite partial one procedure pick differ sophisticated perform even greedy work experimentally work remain actually good generalization prediction pick width depth binary take output label performance vc dimension know specify vc network operation immediately train note substantially improve case qualitatively speak tell reduce reduce overfitte intermediate connected network yet vc grow fast polynomial statistically possible prove generalization care g empirical square loss datum class combine theorem upper vc slightly additional output threshold class dimension popularity year principle form compute inner map via section interesting desired simply find coefficient network represent important runtime least expensive example contrast runtime thm moderately potentially require less memory contrast stop satisfactory correspond contrast combination polynomial thus use support layer deeply empirical deeply express complicated function architecture present preliminary experimental feasibility focus superiority illustrate approach couple parameter benchmark describe benchmark test deep highly predictor instance value dataset mnist digit recognition handwritten digit randomly patch real world pixel digits randomly patch image shape whether consist dataset refer training except involve algorithm stack single hide layer feed forward experiment machine rbf practical variant subsection publicly matlab avoid store experimental set hinge multiclass intermediate layer depth importantly check protocol architecture constrain width preprocesse project principal since narrow would indeed try worse misclassifie description test error layer report mnist correspond multiclass achieving less svm svm building competitive report minimal human intervention resource compare predictor memory generally order magnitude function illustrative play dataset tune train example example qualitatively similar investigate behave width regularization layer train choice generalize third expressive class quantity show expressive much overfitte behavior dataset depth predictor whose dramatically deeply correspond low basis universal monotonically trend tune important unimodal overfitte start perform
drawback reduce paper cite orient one numerical bind precise exist error explain sensitivity fourth begin context reduce basis parametrize partial differential parameter tuple solution invertible compute endow inner norm admit affine hypothesis require reduce fairly inversion aim query finite sequel reduce transpose proper many split two offline begin matrix second operation complexity independent dimension smoothness allow constructive ie computational approximated offline offline basis online q suitably sign infimum evaluate usually successive read computation complexity output lipschitz bind quantity compute offline section adjoint project basis modify bind give notation adjoint problem naturally q partition eq following begin expand clearly orthonormal reason minimize auto positive present offline online orthogonal phase begin estimate approximate large sample q take cm resp component resp basis mean nonzero orthonormal dominant deduce relation store compute dense n min max r min nk min compute offline simple one careful optimisation indeed discrete approximation quasi optimisation offline phase give dot approximate quantity offline phase n computable note cause error error possible correction reduce correct adjoint order application adjoint problem adjoint select adjoint offline roughly offline stability dual residual hereafter clear perform simply replace give computable estimate adjoint double est slight car pour optimisation pour beta pour pour du de dim des du pr si tend tend bound use e compare supplement correction reduce possible adjoint phase use successive require offline may optimization compute monte one decrease practice section probabilistic level classical residual cause work avoid argument winner well depend number affine decomposition budget failure one method sensitivity quantify cause replacement basis estimation completeness briefly refer accounting input index fraction index generally amenable analytic two computationally advantageous rather quantify error replace one estimator bind prove object corollary j parametrized equation pde typically element lagrange form pde onto condition pde usually encode justified write inner benchmark field variable denote steady profile state velocity velocity boundary well constant formulation find pose lemma use element subspace p bilinear leave piecewise mapping explain interest model bound output stability inf fair dual base involve online reduce multiply basis compute snapshot snapshot take ie take minimization different error correct accordingly correct output reduce size also dual new reduce reason superiority vs cauchy schwarz slope correct report bound allow choose risk competitive dependency correct size bootstrap replication confidence output risk combine result account estimation spread impact definition induce goal orient true value l index output benchmark pde eq choose endowed choose discretization step discretization introduce pde perform reduce step discretize q relation stability constant dual snapshot size retain propose comparison fair check dual error compare use offline problem actual correct size use sized primal conservative present new explicitly computable different lipschitz expense slight
inexact inexact subproblem optimization theory develop relevant global rate accelerate quasi newton fista match accelerate gradient impose hessian require two consecutive hessian fista requirement restrictive subproblem complex investigate accelerate version hessian randomize follow describe algorithmic method decrease inexact detail throughout paper v u quality prox play order assume minimize reduce solve notation accurate solution iterate optimize algorithm prox update else choose hessian choose fp kx fx kx acceptance inexact obtain algorithms rate smooth hessian sublinear convergence hessian see helpful hessian establish optimal solution proximal subdifferential q indicate hence q lemma serve bind minimizer closely definition subgradient order minimizer summing follow mm note subproblem accurately fu decrease iteration word objective amount achieved recall constant allow large turn take algorithm idea selection sufficient decrease iterate sequence moreover optimal proof include convergence rate inexact know definite long next ideally decrease bfgs approximation constant enforce bind sublinear iteration decrease reduce expense obtain possible proximal algorithm thus recover standard sublinear minimizer quadratic fact bind hold arbitrarily far let exact respectively side inequality apply replace eq hence present recursion eq convergence inexact inexact lemma establish global accurately left hold iteration set hand side move hence h fx fx ib hence follow inexact optimal sublinear convergence function correspond computation perform maintain subproblem subproblem duality duality achieve strongly proximal accelerated method termination iteration approach optimality classic inexact newton method proximal discuss point introduction descent lot less construct step take operation take cyclic gauss deterministic complexity particular guarantee randomize probabilistic hence terminate randomize next bring theory show randomize sufficient maintain randomize cyclic termination subproblem randomize randomized coordinate iteratively minimize choose randomize particular function dependent maximal model maximum uniformly convexity immediately subproblem auxiliary derive appear involve assume nonnegative independent whose lie independence jensen inequality square k establish jensen case lie hence note accounting establish key show develop section outer subproblem coordinate size bfgs work set th iteration th function bound analysis able prox apply note inexact subproblem apply know apply iterate step take side lemma subproblem lemma immediately recall application direct large value lead become believe balance describe approximate maintain special smooth step obtain coordinate form hessian estimate q definite define bfgs iteration matrix scale spend update prox parameter maintain backtrack backtrack smallest backtrack entry introduce heuristic efficiency comparable active subject future k fx entire subset element result subproblem stage algorithm coordinate piecewise special th th solve close iteration descent maintain take end accelerate step dependent need storing aim provide purpose extensive inexact particular hessian backtrack prox coordinate descent descent implement logistic solver describe package art category ensure implement search compare update prox parameter present notation backtrack prox algorithm search initial second plot optimality matlab interface modify code routine record run store array pass function add little cost also add call test except return automatically run publicly build intel core I ram mac later terminate choose subproblem terminate coordinate step work increase number pass receive much subproblem hessian fairly subproblem iterate move close optimality large almost bound large analyze guarantee linear work subproblem iteration figure far plot logarithmic scale gradient done follow framework large definite four real set expression show twice fast set method performance note decrease propose establish rate number denote hessian choose scalability yet four repository summarize uci classification determine person k census one artificial large often predict ct slice body finally handwritten recognition discriminate handwritten digits nine r cm zero census handwritten digit ct slice outperform third reach usage observe notable set rate inexact proximal quasi coordinate effectively sublinear expectation optimize subproblem strong sublinear conference rate hope accelerate relate study sublinear approximation lot cover algorithm large scale optimization however convergence rate replace prox update trust sufficient instead cyclic modify effective specialized remark assumption laboratory west nsf grant grant fa department university laboratory west usa author grant fa sparse careful composite quadratic optimize method coordinate bfgs proximal include method assume lipschitz fy lf exploit method requirement slightly clearly present
say early many end become fw result coordinate descent note active identify entry fw I understand union set working large dual q largely subproblem choose become nonzero fail ensure convergence enter leave purpose search adapt element positive otherwise convex bfgs hessian approximation coordinate iterate set q kb q jj kb gradient iterate piecewise obtain suppose hence dimensional problem form solution soft thresholding q accelerate use apply store diagonal th compute maintain vector instead multiply th little effort space store q n number number feature test scalability yet categorization corpus volume originally digit recognition discriminate runtime close training size bring aspect illustrate runtime plot scale fista fista begin fista near optimality work reach tolerance indicate lc fista specialized solver selection require definite ht interested implement matlab decide consist part one solve subproblem compute cholesky solving subproblem take first descent subproblem due iteration little reason state iteration denote inner apply subproblem actual hessian smooth part hessian different convergence descent discuss early enable accelerate coordinate practice define obvious alone coordinate add counter matlab report gene set similarly different precision consistently require order five instance inverse specialized solver exploitation greedy identifies solution general algorithm efficiently order information regularize achieve exploit low quasi large working complement allow size subproblem identify empirical art specialize twice allow sparsity pattern desire solution sake simplicity presentation machine desirable logistic regression inverse often common difficulty past decade effort aim development order accelerate gradient iteration size often alternative particular construct store hessian alone invert expensive regardless benefit nevertheless new sparse optimization problem optimality hence like approach subproblem subproblem note specialized implementation construct hessian enhance large exploitation hessian step training instance shrink propose focus minimization small subproblem later idea specialized strategy subspace behaves newton algorithm able line begin characterize phase minimization obtain descent enhance like idea subspace first take backtracking mention actual smooth along coordinate active another approach hessian require importantly help subspace fast size give return subspace exceed use control use large aforementioned one satisfy specialized heavily special improve idea strategy use set constraint similar decade svm subproblem estimate memory bfgs coordinate achieve maintain per method apply subproblem construct acceleration minimization step individual contribution thus adaptively maintain step good objective function nature help avoid update extend initial converge active hessian accelerate special implementation help hessian bring complexity limit hessian hessian exploit main expense every let th update use organized follow subproblem work selection descent subproblem instance selection demonstrate advantage inspire quadratic nonlinear optimization obtain smooth around positive choose taylor expansion maintain active fix change along coordinate
metropolis hasting proposal mix great equation evaluate metropolis complement likelihood f state sequence forward supplement sampler sequentially entry entry I coherent define enable programming auxiliary backward message explain detail feature eq q outline markov transition count equation belong family simulate posterior straightforward dirichlet explicitly normalize unnormalized transformation inform proportion influence posterior early emission conjugate across sequence let k k standard conjugacy sharing improve inference behavior description supplement sequence consider death reversible jump add new birth emission draw lead low acceptance high unlikely exist inform parameter recall ar hmm var scalar address proposal select window drive proposal birth death framework move sequence modify hmm parameter sampler avoid construct proposal away propose discrete assignment dimensionality observation show discrete alternative birth death move change combine move birth empty propose birth unique proposal efficiently backward forward programming emission auxiliary variable quantity solely overview variable discard stage allow efficient collapse proposal idea outline algorithmic presentation supplement birth proposal birth create sequence birth assign new however prior hand contiguous step choose give window auxiliary dynamic programming sampling state force new feature maintain death behavior require block sampling acceptance proposal via birth move note feature proposal statistic require additional compare birth death finally window birth death move choose current define transition reversible death balance present define scheme hmm sampling assignment efficient exploration change assignment simultaneous change sequence sampling additionally merge birth death improve annealing burn merge model dp conjugate likelihood conjugacy allow sampler operate partition emission use reversible split build assign originally random time gibbs update partition sophisticated proposal need emission even proposal often necessary datum however proposal alternative replace new remain either move create sequentially allocate merge model split well adapt feature lack merge inference sequence item split feature split merge consider equivalent cluster base possesse indicate therefore merge choose candidate propose drive birth merge feature sequence away sequential approach relatively collapse proposal prefer acceptance proposal distinct anchor item fix choice define split transition balance select anchor possess choice merge uniformly unlikely split move rare need bias selection split often merge select crucial select segment assign pool emission assign separate feature bias promise candidate lead acceptance f jk integrate multivariate denote determinant count sufficient equation process supplement especially f candidate propose whether split merge occur sequence possess either b pt k b p n f split iterate permutation item possess feature anchor enforce move reversible merge force dynamic proposal drive birth death hmm emission assign state initialize anchor assign conditioning stage merge thus auxiliary drawing final metropolis hasting acceptance give create merge tractable conjugate emission proposal emission require merge algorithmic supplement feature merge careful accounting correct reverse move bp ar merge death variable proposal accept form hasting ratio term ensure detailed balance condition convergence posterior effectiveness proposal enough cause merge merge require return original configuration anchor could return possibility unlikely even vast possible configuration toward merge recommend acceptance birth death modify start hasting ignore rapid improvement initial iteration decrease temperature iteration hasting fully reversible anneal several define dynamic switching approach building process hdp regime define switching space recent review analyze series I hdp prior switch transition extensive toy supplement unique behavior predictive develop multiple series via hdp coarse set topic dynamical assume topic extent alternatively hmm linear transition emission finite external covariate experiment series class broadly address receive perhaps difficulty treat parametric align univariate approach series use parametric dirichlet cluster hmms factorial define representation factorial widely infinite factorial ibp evolve accord markovian focus behavior dynamic instead aim align series motivated temporal anomaly hierarchical share series hmms nonlinear trace reference share series simple human synthesis visual tracking nonlinear dynamical collection binary latent effort dynamic behavior people rely manual way complex manually behavior sequence justify parametric exploratory examine record frame window component difference observation neighboring step supplement ibp hyperparameter every supplement merge drive discrete proposal method human circle recover depicted letter probable prior jump create ar alternative effectiveness several bp ar baseline implement reversible procedure hmm proposal prior jump merge move drive proposal death move detailed supplement merge birth death anneal hour least individual utilize parsimonious jump rarely meaningful initialize trace hmm configuration feature supplement detail evolution normalize hamming sample ground state sample hamming ground truth compute small alignment state log ham annealing run blue curve hamming hour proposal close ten substantial drive add new annealing improve indicate proposal local optima approach offer burn merge critical feature quality half circle annealing explain contrast segmentation assign multiple due jump proposal split merge could merge merge effective redundant unlikely move sampler find drive birth move rapid split merge move sm improvement nearly hamming error investigate initialize true retain truth label exercise behavior many iteration prefer consistently bend manual inspection reveal add segmentation local ar joint sampler conclude future concentrate capture hamming versus hmm raw observation hmm annotation bp ar mcmc hmm first difference present bp alternative assess gaussian probabilistic principal focus detection rather also consider gmm first behave observation parametric specified expectation produce maximum figure compare method estimate measure hamming sequence gmm result initializations hmm matlab toolbox bp ar hamming comes mcmc among anneal bp gmm bp ar hmm variability model ar gmm hmm assume behavior band bp due flexible activity early median length step comparison datum infeasible drive birth death move require special jump proposal initialization create merge behavior sampler sm anneal complete hour share move identify clustering series segment coherent produce lack manual improved inference explore enable scale promising box segment behavior bayesian behavior series prior dynamic additionally hmm merge move drive birth move efficiently explore ar demonstrate sequence switch var process markov switching process herein employ behavior emphasize however condition sparse collection process beta globally computationally rely area improve split proposal benefit sometimes recover root anneal address however maintain move acceptance configuration due parameter identical problematic grouping behavior might hierarchical behavior idea vary appear behavior case behavior motivate occur along rather portion e allow grouping become increasingly university california berkeley grant grant jointly relate dynamical behavior among segment region pattern develop monte mcmc predictive remove behavior novel drive avoid consider truncate promise segmentation motion focus potentially instead motivated produce motion exercise multivariate series type arm circle exercise describe motion individual global goal discover exercise type behavior occurrence individual discover sequence combinatorial involve manual annotation skeleton possible exercise produce manual annotation present manual observe series motion angle second hmm aim recover behavior use behavior yet describe assume describe individually markov switching switching field speech track human capture focus tractable class encode evolve discover behavior share multiple describe globally behavior individually exhibit among behavior global seek flexibility behavior encourage behavior motivate approach many potential var process bp ar also version refer bp ar emission replace conditionally chain procedure bp ar article furth bp hmm nature behavior critical challenge efficiently change jump add idea merge birth death nonparametric series domain birth death proposal assignment hmm parameter hasting ratio dramatically presentation introduce motion formal summarize truncation efficiently drive reversible jump proposal explore section split merge proposal sampler make improvement present experiment motion examine informative wish two angle angle time series collection serve sequence subject subject sequence unique behavior appear bend additionally human annotation exercise behavior time serve assess estimate analyze phenomenon discover behavior series infer stream relate share pool thereby improve describe dynamic share stream nonparametric prior address allow dynamic behavior across dynamic could hmm specific state conditionally insufficient human stream hmm probability bp conjugate realization atomic mass atom sample independently atom visualize result feature indicator encourage still variability seek transition distribution dimensional subset mass function define doubly delta switch pt denote define hadamard assign time indicate precede generative via finite contain nonzero entry reveal place expect mass self hdp imply jk abuse dirichlet infinitely useful index rather value unnormalize instead proper add remove working constraint specification conjugate wishart place specifically comprise inverse wishart degree freedom scale dynamic measure mass separately see provide
relate find high graph rely connected clique core etc clique graph clique maximal clique complete clique restrictive arc subgraph clique idea appear subgraph connect hard clique core instead specify clique present degree superior member core clique core algorithm compute core notion generalize cluster coefficient k base measure compute like shift mode area originally intend feature recently dense compute one intuitive density undirecte strength graph sum weight arc equal count divide connection forest density node method sum physics main physics immediate arc invert region graph forest boltzmann introduce index partition immediate derive formula apply area conclude possible weighted loop connect vertex arcs edge arc represent immediate adjacency indicate affinity compute relation reciprocal could well adjacency laplacian adjacency sum moreover arc exist direct two behind first set forest forest assign contribution contribute control smoothing low forest cost account physics formalism provide probability forest assign set forest define intuitively root forest subgraph node mark root forest forest deal rooted tree forest tree forest individual arc weight arcs forest arc contain individual weight low forest observe forest probability forest likely forest lowest isolated contribute illustration simple figure forest forest cost arc numerator numerator denominator forest tend probability low forest follow appear correspond statistical physics define delta indicate link present forest ten dataset belong dataset artificial gaussian center cluster lie three deviation give community community overlap artificial group shape separate finally graph originally list original database document three deviation small graph near compute euclidean pair transform affinity threshold arc investigate graph high relation give birth undirecte create graph adjacency matrix visualize dimension spatial coordinate node correspond reconstruct density try proceed density map embed visually spatially indicate reflect density extent area applicable visual checking firstly node exact density node assign dark present dark present high concern tuning give threshold nn density finally identify dense strength coefficient community display figure nn clearly cluster latter perform theoretically index arc explain clear result community correlation almost practically weight number index converge affinity index strength correlation threshold increase strength unweighted arc quite small handle distribute gaussian draw index much stable strength visual representative behavior measure show weighted community index visually highly dense identify well identify index graph even figure mainly dataset confirm recover identify correctly dense area community figure concern identical unweighted behind forest depend meta forest forest path physic form immediate cost arc efficiently invert matrix lead overall search area graph correspond center cluster density one strength regard construct instance like cluster investigate technique forest easily equation use diagonal acknowledgment project thank algorithm f school management learn universit de email institute email work introduce novel tree inspire boltzmann countable forest high cost occur forest high density around physics compute inversion experiment artificial real index perform dense mining dense forest concept particular social network biology world identify
coordinate seed parameterize correlation across correspondence correlate sbm parametrize respective membership transform row adopt index subscript grow next finitely perfectly cluster give sbm although result approximate alignment spectrum sbm parametrize assumption correlation respective adjacency generality block adopt assumption constant define q q embed adjacency spectral cluster align regardless matching vertex elsewhere recall cluster u lemma necessary result fw many finitely immediately clearly eq svd combine finitely term f nj q final equality contradict bi c follow finitely imply finitely proof implication scale fit align concentrate heavily one direction make sparse subspace remark analogue sbm explore effectiveness simulate algorithm measure latent fraction runtime achieve scalable achieve significantly exist matching procedure cpu virtual code need cluster full across cluster seek available accuracy allowed end run experiment sbm gm record running cluster match path cr exactly frank wolfe path path cccc cr path replicate used seed seed uniformly block seed match surprising achieve excellent matching experiment path convex good cr scale poorly significantly cr value time decrease effectiveness employ procedure would accurate cr yield fast less degradation increase achieve excellent matching cluster consistent matching cr par suggest seed important graph next explore effect decrease algorithm need seed seed unlike mis insensitive mis cluster step cluster performance correlate sbm mc divide seed possibly match j seed randomly mc replicate overlap heavily scalability issue path experiment gm oracle across maximum allow core average expect increase matching big graph cluster lead expense increase path cr well chance scale cr run significantly significantly good graph decrease performance matching henceforth focus cluster expect perform path cr achieve excellent performance though modification seed dash fraction correctly match bar solid bar graph match theoretically graph lose embed step cluster outperform sbm setting utilize across connectivity matching task cluster utilize position need seed draw dash curve plot various bar correctly bar seed perfectly across seed performance seed combination mc deviation seed need contrast match graph match sbm graph figure plot accuracy matching mc simulation seed vertex latent utilize cluster seed good graph match correctly seed reflect applicability seed robust also assume knowledge relatively low rank sbm algorithmic divide essentially match matching embed algorithm vertex simulate graph parallelization improvement degradation b first mean runtime explore run pair vary sbm block connection seed utilize core algorithmic intel e ghz processors costly high matching step relatively case roughly speedup utilize lastly graph second second second second divide core detail cccc runtime second core match calculate average runtime embed see detail match intensive aspect incremental match research gain implement parallelization strategy incremental effectiveness subject brain voxel voxel brain mask edge neural bundle voxel vertex connect component range detail reference contain therein prove sbm applicability heavy tail match heavy tailed rather flat across tail vertex correctly explore impact heavy tail collapse graph subject correctly match high percentage within subject pair result run match graph match comprise voxel voxel brain mask highlight pair note analogous example graph size match plot graph subject plot optimal embed dimension note cluster initially clear correctly proportion subject across run perform e ghz core display average runtime four c second subject seed match although run parallelization matching note algorithmic expect implement specialized hardware svd employ terminate emphasize even reasonably match cluster note entire datum seed pair however unable utilize seed selection algorithm pick match across chance explore utilize brain sphere cluster via mean presence means mean idea choose leverage datum mean accuracy cr estimate subject subject connectivity raw clean raw datum serve tool subject accuracy lastly matching post embed match brain carlo embedding scalability concern lastly cluster slow excess hour gb ram vertex rely able across graph infeasible fully condition simulate real effectiveness addition justify divide perfectly match flexibility choice match focused clustering procedure implement rest seed seed matching dynamically provide heuristic define extend towards national security fellowship university technology advanced project air force laboratory contract fa thank discussion suggestion proposition claim subsection em human language present seed graph combine embed exist state art procedure justify proving correlate correctly match seed divide simulated show minimal accuracy increasingly inferential graph broad include vision seek alignment graph inherently efficient problem determining allow assignment wide applicability exist paper excellent partial seed actor name allow alignment brain vertex across act seed information partial matching match across graph improvement gm incorporate even seed exist arise big demand scalable roughly divide match order set allow cut match excellent computational example operate adjacency match utilize solve practically resource match graph scalability requirement efficiently match often inexact algorithm small dimensional object prototype herein divide graph approach match proceed yield dimensional euclidean embed graph embed powerful theory embed asymptotically vertex cluster match fully depend property match impact scalability vertex parallelization accuracy degradation increase cluster hence core though employ example focus vertex apart herein vertex manuscript column drop subscript simplify submatrix index index concatenation symmetric matching step match output formulation though share two seek alignment preserve structure across set minimizing seek edge seek minimize permutation matching need cardinality see variety generalization match latent extend minimize accommodate subset partial estimate step divide plan future resolve adjust accordingly specifically vertex follow combine vertex size ideally cluster k sequentially work remove vertex assign desire non need implement graph within cluster match various matching denote solution could implement run need scalable computer specialize hardware software cluster procedure computationally minimal evidence modify exist automatically size refinement provide cluster original often vertex remark majority graph sensible within suggest graph treat remove match yield remark graph accommodate result modify excellent match subroutine parallel size
detail rgb rgb element norm precision covariance admm solve strategy formulation element appear decomposition next efficient admm solve suited optimization problem decompose admm alternatively augment give proximal admm algorithm equal statement true otherwise primal graphical
structure stochastic dropout view ensemble learn bagging unit input feature member combine ensemble would expensive admit ensemble crucial ingredient win several profile notably recognition molecular activity job competition also inspire work activation extension basic average regularization effect well hide unit recent empirical dropout feed neural generally employ recently activation expand geometric average compare enumeration approximation remarkably surprisingly accurate surrogate geometric importance geometric traditionally ensemble produce average prediction dropout provide difference immediately effect classification investigate replacement arithmetic approximate dropout training raise question dropout rule bagging follow bag member dropout unclear ensemble effect dropout encourage individual variety investigate replacement traditional bag ensemble ensemble share take place context implicit finally alternative criterion parameter dropout bag biased estimator gradient geometrically dropout ordinary descent feedforward architecture dropout train variable mask determine zero train sub gradient sample multiplication mask bagging bag ensemble prediction form voting manner tend generalize well prediction differ bag way parameter long train much bagging stop ensemble start guarantee train vast never explicitly average together arithmetic important come ensemble network predictive simply sigmoid special softmax mlp scheme architecture unit geometric characterize mathematically sigmoid activation network single apply six popular benchmark simplify fashion much architecture digit vs vs mnist validation occurrence test choose uci repository class vs first dataset nonetheless record task overfitte corner triangle moderately challenge additional mask enumeration tractable benefit simplify typically probability g average decrease mini early validation early validation scale investigate fidelity scale maxout concern enumeration due exact network randomly sample test task dropout geometrically average prediction geometric obtain scale hyperparameter yield network make visible relative different additionally fidelity nonparametric pair pair correction seven computation little geometric arithmetic impact generalization capability train prediction arithmetic figure seven proxy arithmetic discrepancy geometric mean arithmetic never investigation train remainder experiment capacity multiclass mnist employ layer dropout constraint income hyperparameter include initial range unit maximum weight perform bagging ensemble size share member ensemble mini see resample training take member effect apply investigate role single mask throughout perform hyperparameter configuration low obtain report initialize random seed train traditional bagging dropout mask hold throughout network evaluate test mask dropout suggest combine yield aside considerably small non hyperparameter train early stop remain unclear highlight high cost ensemble network autoencoder motivate robust slight transformation connection noise penalty question whether dropout whether perspective exactly effective error acceptable need otherwise train dropout objective dropout boost sub bag correct current dropout boost though reality tie dropout objective boost train share give initially train sub ensemble could maximize averaging obtain boost term member though optimize intractable appear weight scaling introduce estimator section small boost dropout use perform ingredient take view complex learner jointly optimize would employ similar bag
cause fitting audio file average hz half feature spectrum frame summarize scale log produce get cosine coefficient instantaneous derivative vector audio projection pre feature dictionary raw calculate feature either subtract dimension pool vector pca audio file segment process pool result fed codebook codebook codebook encoding encode various training regression tag false false validation optimize auc tag fold cross validation tag average five fold ranking split train slack trade test high auc set test average split pca song project low heuristic effective pc covariance notice reduce dimensionality dimensionality decrease keep covariance every dimension split song query retrieve rank query song evaluate auc query take audio codebook baseline chance
support detect framework via kkt technical dual optimal synthetic datum demonstrate identify moreover variational inequality strictly identify vector effective discard speedup gain rule rest unified extend rule derive rule svm real conclude entry index j j proper unified motivate screening rule kkt follow necessarily parameter notice correspond sublinear scalar sublinear sublinear positively
frame camera row background foreground background foreground recent extend transform face rotation tackle alignment pose linearize proven programming demand database tackle incremental fold alignment online memory make database rank throughout alignment removal video alignment arise illumination occur subject classic batch square severe sum rank robust low rank decomposition pose image alignment transform image decompose recover align image seek rank stack keeping relax convex surrogate minimize correspond relaxed highly domain per image component tackle linearize convex work batch despite illumination scalability alignment propose alm
odd monotonically therefore proof develop employ approximation precisely formula symmetric derive simplify subscript drop follow ease derive follow expect probability formula eq derive aforementione simple sequentially analyze respectively f ie deduce eq discuss sf sf finally typical projection maintain consider develop random viewpoint yield nonzero per well dense dimension considerable confirm classification experiment real selection lemma project dimensional bring attractive processing successfully
fig show four benchmark performance restrict linear often exception select close fewer rarely employ common estimation stability also via effective choosing dependent significantly performance much biology prediction subject believe well also generalize leave future c c c c c c c c c positive false false c toeplitz false bin yu california berkeley title validation dimensional however lasso lead unstable high dimension suit reliable free small optimal choice also enjoy
nan alternative assess eeg homogeneity important whether draw different application schema match identification like kolmogorov von capturing density mmd enable density base mmd string bioinformatic etc mmd two distribution multiple correction prescribe global significance greatly reduce test wants retain hypothesis mmd regularization term control power test statistic regularizer user mmd higher especially size preserve
box infinite lin project later programming quasi newton set symmetric use newton search newton method symmetric rank modify bfgs lin go idea inactive set place slightly say whenever denote current iteration partition projection follow
factor whereas factored rbms factor uniformly random option focus field interesting careful appropriate show encode optimization procedure possibility topology space dictionary dictionary induce induce pool structure network achieve dynamic idea advance dropout maxout create improve implementation deep bring united university universit de facebook usa ac demonstrate redundancy parameterization several model architecture
modify backpropagation designing method dropout another separate drop layer possible instead show dropping dropout hide extensive evaluation perceptron successful explicit hashing framework separate drop different neuron layer available drop probability empirically recently mlp extend conventional neuron need sample activation add neuron
dp q mild conditional relate early ill non decrease exactly related measure ill admit singular arrange orthonormal ill adjoint map relation ill closed appendix next j reverse weak reverse ill pose th h j expectation assumption ii pose ii use approximation error orthogonal property unable attain sup nonparametric series ls bias involve l inequality simplify presentation present pose case k j similar ill pose pose sense risk loss sup convergence norm ill pose power case subsection sup ill already show minimax include old ball ill infimum depend together estimator
mcmc series gene expression element make colour molecular set cancer datum contain type measure genomic combine capable capturing hypothesis multiple cancer analyse wide allow disagreement include disease apply genome consensus death event recurrence consensus recurrence log correction multiple test effect demonstrate integrate recurrence identify
nlp inference accurate nlp lead small mae different highly due systematic mean yield ranking difference statistically study approximate exist gp family approximate gp form addition observation link control appropriately trend positive definite computationally efficient justify heuristic gp label output logit outputs principled inference use laplace approximation kl divergence furthermore demonstrate laplace consequence since biased error heavily hyperparameter taylor initialization greatly ep conduct comprehensive experiment method large impact dominant inference rank significant future family multivariate gps classification appendix derivation taylor taylor likelihood substitute eq q use property joint rewrite remove marginal substitute look derivative hyperparameter derivative dispersion parameter remove subscript convenience symmetry yield appendix case effective poisson effective gamma expansion noise use agnostic q look st derivative acknowledgement ce city internal grant china claim b exist framework parameterize efficient gp form greatly simplify domain framework several gp taylor inference elaborate algorithm process gps classification popularity learn work promise classification gps desirable vision compare scale object recognition produce predictive e active gps allow local incorporation computer vision hyperparameter maximize expressive compound optimally combine advantage classification many vision recognition crowd anomaly vision interpolation pose space vision gp function predict crowd number real optical flow heuristic convert gp valid output prediction must generate proper obvious distribution developing require count beta
discriminant curve functional analysis principle g g provide discriminant maximum probability compute way analogy polynomial spline spline generative regime quadratic discriminant density polynomial spline adopt row represent consist approach fit generative model govern homogeneous present class single whole handle mixture analysis adopt discriminant next regression mixture discriminant motivate course gene model spline mixture class model sub function also conditional proportion discrete represent estimate maximization spline adapt curve knot regime capture regime point regularity spline smooth lead knot
solve however reduction monte investigate score investigate analytically calculate score kalman filter noise ar state linear noise proposal alg lag smooth datum autoregressive model algorithm computational code programming language run ghz approximately minute minute time lag work receive time provide fix lag degeneracy introduce storage compare store previous particle
free hence speed initial result convergence choose initialization optimize descent light initial condition carefully scale second discrepancy purely one coincide show advantage well initialization careful pair carefully momentum excellent try et convolutional combination unclear act improve generalization instance advantage appear descent random initialization deep three case orthogonal recall solution intuitively axis axis though variance differ recover treat well diagonal unit small draw greedy train layer tried predict index class elsewhere train eqn extent eqn analytical recursion relation variance layer dynamic orthogonality activity approximate neuron integral mean map fix point numerically blue via dynamic neuron per average computed left relation stable look intersection line unity unstable recurrence nonzero stable solution curve constitute infinity match depth blue dynamical
show ordinal outperform well real desirable rating experiment analyze rating different present dataset experiment query draw uniform number vary rating experiment figure spam baseline majority bad ordinal additional rating per mse top ndcg bottom propose ordinal exist baseline show median vote significantly ordinal discrete model ordinal ordinal whether ordinal use experimental instance difficulty instance qualitatively similar I variant spam component robust ii variant spam ordinal real value
evaluation raw measure deterministic computer report rmse sd monte experiment side table summarize carlo repetition set second stage greedy design library spam package utilize table parallelization slow local order average rmse illustrative randomness monte pair different average example second dominate lack former computational expense term local calculate parallelization contribution big amongst accurate observe cover pointwise coverage exception deviation however true deterministic mis cover contiguous input draw conservative seem solely denominator replace identical perhaps conservative coverage predictive uncertainty accuracy sized table increase similar local double approach limit due
matrix computationally infeasible art tackle basis dictionary associated kernel address atomic norm minimization frequency lie continuous provide spectrum nominal recover signal corruption minimization theoretical atomic minimization signal
evaluate application variation evolutionary replace population former stage population population current population associate distribution select good markov population test mn population independence variation gibbs avoid parameter mi advantage expert indicate neighbor variable test mi value sensitivity mi possible mi c critical mean fitness require conduct alternative structure reach fitness version test string reason fitness landscape evolutionary string road variable arrange goal fitness add third count clique use former fitness clearly c road critical repetition fitness optimum iterate several run time commonly benefit runtime ii fast fitness population denote measure experiment truncate size road road standard deviation road low table also always road fitness
integration indicator illustrative constitute circle graphical hyperparameter usually manually people control prior assign mixture mixture parameterization chinese restaurant inverse often flexible space far
rkhs lead boundedness ensure fully reconstruct image reconstruction immediate practical implication fouri nontrivial implication optical resolution hope exploit optical setup resolve limit institute systems university college establish fouri machine term kernel approximation identify square transform mean provide showing imaging principle generic super resolve imaging device collect incoming light finite impose light reach optical lead
special empty definition define formulate constrain apply note tight quantifie rational practice never rsc hand show unconstrained choice continuous theorem feasible problem second satisfy minimizer problem yield negativity contradict note non attain equivalence side continuous function also homogeneous write write ratio difference constructive form decomposition homogeneous modification require version order make self contain homogeneous l solve line inner sequence produce terminate l inner optimal l always attain boundary
minimal span infeasible suggest estimator suitably generic use function consistent generate suitably simulate variable estimate correspondingly well generic reduce fast processing theoretically characterize extent acceptable investigate main
consider survival datum essentially risk differentiable sigmoid control smoothness smoothed differentiable next iteratively fit learner typically individual learner tool specify estimate apply learner base component consequently base base refer marker component wise boost optimization smoothed marker offset example iteration counter marker via base learner good square marker component sl length learner current zero step base effect l l l go step estimate maximize smoothed behind descent span learner learner contain predictor final become marker tune boost sl minor boost length recommendation value boost usually determine validation complexity result avoid overfitte via shrink effect boost overfitte problematic relate
fm also maximize good model generate chain come mode perform good hamming subset importantly fm perform measure winner follow pay present run fm probability ht cc index predict x I test prediction optimize instance wise macro measure mainly like near neighbor f measure maximization method end maximization clear description correspond several see g none instance straight way correspondingly estimate f maximizer decision regression generalize output multivariate decision measure learn idea restrict discussion tree rectangular namely g frequent maximizer leaf use rectangular neighborhood query instance demand induction tree optimal split respect give example analogous cardinality computation notice searching estimate whole checking change ranking repeat recursively fall threshold course let bag easily apply bootstrap hypothesis return probabilistic approach idea repeatedly rule probability framework distribution marginal learn attribute pairwise search probable path always probable label algorithm method uniform cut mode mode method need observation get pick biased coin classifier sometimes sampling observation method neighbor example plug algorithm give optimize approach discuss parametric since proper multiply matrix weight problem class obtain multinomial regression matrix since
phase relevant devoted expression condition conjunction derive begin describe cluster state end expression density assume follow eq cluster indicator multinomial q drawing derive difference develop iteratively also density parameter cluster indicator portion relevant upon parameter always extensively rigorously prove basic weight follow posterior estimate posterior I I I denominator rule distribution variational reader detail update q numerically I dimension denominator integrate play role evaluation conclude fix number adaptively essence reader merge essentially attempt improvement department choose two merge concern make cluster whether case begin adjust split evenly among cluster propose merge merge add produce posterior previous unchanged new new cluster variational accept merge pseudo code correspond conditional basic initialize cluster fix large situation specifies minimum acceptable denote value objective implementation near neighbor call merge iteratively check function phase handle evaluate basic name posterior return normal I l
typically majority vote achieve consistently lead helpful survey group simplicity answer answer either standard terminology binary classifier answer term question vector classifier reliability ix set whose label classifiers ii instance class function yy classifier correctly predict positive fraction predict two assumption instance marginal conditionally classifier independent classifier label nearly may arise principle work well classifier I spectral base
indicator might filter complement improve far converge style suppose couple discrete distribution interval tackle work forecasting learn trivial task far three couple hmms triangular pair trading triple link triangular consider style leverage trading coupling relationship micro picture say min trading min link price min vice versa carefully start one list goal develop
reach conclusion combination map set measure cluster n location assign centre location centre various euclidean centre centre consider centre allocate centre centre change highly initial allocation paper mapping arrange load assign depend closeness position initially nod incorporate assign produce place together place order cope split split daily load within day
study drift diffusion theoretic low parametrize row estimate problem standard scalar shrink effectively select obtain diffusion normalize square sum rt square rt first term construct define derivative would require complexity coefficient diffusion linear penalize complexity simplicity state low cf analyze learn cf extension extension motivated proof technical provide appendix analogously slight abuse index usual norm support namely zero norm n denote eigenvalue symmetric throughout paper eq let plane diffusion measure lyapunov hence trajectory penalize square estimate stationary trajectory accord choose notion introduce depend stationary hold sign trajectory eq conclude reconstruct low say reconstruct complexity support diffusion low depend gain intuition drive lyapunov verify hence upper diffusion theorem characterize one subtle varied independently lyapunov dependency laplacian
protein assign localize development include comprehensive localize protein currently study inter dependency overall inter dependency comparable training classifier predict location restrict beyond protein multi inter comprise introduce present evaluation experimental summarize commonly context particular protein view characteristic abundance composition take et explain notation localization protein location l ps il feature value location protein vector j thus protein develop protein protein location represent localization describe
theory minimax minimax choosing cost future unfortunately general due number way derive mdp consider argue algorithm systematically relaxation quantity complexity plug upper give general term underlie state one modification term handle use markov use stationary exponentially forecaster lead bind derive use advantage horizon action go u u nonnegative entry index randomized feedback law distribution action transition throughout current action randomize eq interpret feedback draw joint transition recurrent induce invariant admit q supremum leibler u deal arise argument value online x agent perform control walk environment use strategy interaction observe mixed strategy simultaneously next end finite throughout environment loop evolution player observe previous environment
logic knowledge clause plan act exploration image tree induction carlo base event widely develop relationship document nice discover representative text link topic perform model pairwise block lda share aforementione usually formulate one impose imbalance large observe explore nice regularize posterior training another address inference conjugacy logistic variational augmentation marginal high scale strategy successfully explore sampler supervise explore technique collapse please method apply relational generalization conference consider binary ij consider structure multinomial draw word z non dimensional work define denote exponential function respect inner product topic document citation correspond show latent competition entry value understand citation would expect topic link diagonal link network make expressive asymmetric simple link use full citation allow interaction diagonal intuition citation likely citation link top scheduling reduce schedule requirement
objective one recognize different flat dynamic type bid bid price bid request goodness decision ar typically reach realize type try budget spend slot smooth schedule exceed constraint parameter detail show control daily budget time schedule try spread throughout day break slot budget spend slot slot approximately constant slot show assumption length slot choose assign incoming request spend incoming ad request finally total public slot request request number income ad request constant make progress sequential adjust next working slot recursive winning slot historical keep absolute future slot
insight investigate learning representation fmri would layer rbm family expectation method single quick small ica rbm graph artificial ica rbm ica spatial hide field dimension gibb visible normalization fmri voxel fmri define bias general normalize voxel zero unit fast choice affect quality interpretation fmri encourage regularization linearity setting facilitate temporal truncate cd rbm ht c section summarize comparison ica nmf map sm tc estimate ground truth rbm
extend bregman scalar multi notion bregman continuously convex empty pointed order interior strictly k banach norm fr differentiable fr fr correspond usual derivative calculus fr derivative dimensional offer definition banach fr differentiable bregman divergence fr generalize incorporate various previous extension proper cone
variable atom location write evy measure total beta widely generalize process discount parameter infinitely generalized borel suggest ap sx generalized become gamma l evy recover weight except specify intensity dr thus finite mixture point strict positivity introduce categorical latent observe categorical indicate membership factorize identifiability specific integral analytic analytic analytical expression calculate simplify conditioning variable factorize form mixture inconsistent become amenable posterior algorithm count inconsistent construct structure define exchangeable impose mechanism discuss completely resolve random mass positive point directly link poisson generate define scale construct
cost simulator set execute condition function policy optimal note chain entropy definition equality former former general thus get show conditionally stationarity simplification finally follow definition symmetry demonstrate conclude theorem shorthand generate mix process dimension vc
contain part traditional east two traditional word quality model quantitative embedding nlp task speech english capability learn pos annotate resource choose convergence change parameter use hide layer train tag specific word window size construct embedding occur vocabulary feed back embedding train universal tag one tag simplify language experiment train base label speed near competitive surprisingly
set normalization right relatively classic usually amount available task include representation marginal copula modify correct target identify correct factor marginal source marginal bivariate copula differ affected copula target simultaneous change copula limitation separately use address general identify marginal distribution conditional mmd unlikely draw mmd embedding rkh easily take
high computational become expensive due reconstruct correspond straightforward ahead illustrate advantage formulation user formulation satisfy db list right table table recovery fit factorize riemannian due factor factorize riemannian ht via find classic solver solver six explicit k computationally prohibitive netflix lr factorization result budget recover time cc snr db c lr snr db ht lr netflix show relative error cc netflix rmse db c c rmse sec order wide acquire process determine acquire constraint recent insight acquisition cost acquire subsampling receiver interpolation order perform additional removal improvement spatial key analysis minimization robust interpolation acquisition example transform cast organize source straight line source record shot record collection perform several source source duration hz restrict hz frequency fig hz hz slice receiver slice
similarity resemble lift cosine lie statistically attain value cosine derivation one odd ratio indicate occur indicate likely occur lift cosine symmetric standardized rule derivation context quantify eq derivation explore set obtain lift cosine measure clearly standardize making compare effect measure completely contrast standardize low occurrence standardize also closure item frequent subset frequent conversely item threshold alternative equivalence pass variation hash pruning
outline characteristic binomial proportion size stage sequential binomial proportion chen parameter solution special approach control refer coverage check virtue coverage recognize coverage evaluate coverage parameter rigorous computationally check coverage introduction publish proportion prescribe margin clarity presentation comparison chen remainder section develop chen original binomial prescribe error main coverage choose recursively computable bound complementary interval adapt branch tuning look prescribe guarantee bound coverage available branch coverage check rigorously probability associate confidence coverage level coverage since subroutine tuning check complementary coverage chen standard reduce improve adaptively many coverage coverage continue component sequel branch coverage binomial stop sequential control convenience q integer chen stop continue rule continue n stop rule stop remark principle bivariate take continue check rule virtue motivation introduce stop parameter form stop sample stage unnecessary stage take sampling stage sample guarantee stage
demand simplify local embed transform neighbor system suggest add matrix simplification denote sparse affinity cluster experimental art run fast ssc provide enhance use internal one transformation incorporate availability enable q balance perform transform neighbor search near low structure transform propose recover perform assign minimal omp predefine learn transform minimal present evaluation public dataset mnist handwritten digit extend mnist handwritten digit extended contain subject pose image classical motion contain sequence video video cluster ssc subspace performance adopt ssc similar public extended face otherwise ht visualization cluster color plot label indicate iteration ssc ssc ssc far improve cluster well view cluster digit denote illustration purpose conduct subset adopt digit randomly digit
feature death birth death plot result alternatively represent diagram topological summary datum persistence short consider topological large persistent homology summary goal paper summary homology sample material homology dense enough topological nice embed pt homology homology persistent homology nontrivial homology generator realize persistence length feature infeasible homology r complex homology material complex small pt maximal inclusion homology eq homology support birth death record persistence diagram formally persistence diagram subspace extend diagram topological diagram represent interval sum record birth death level set generally dimensional level let persistence diagram persistent homology dimensional persistence diagram persistence diagram record death persistence diagram appear axis essential one supplementary
recommendation form item discuss recommend like user item expect amount future highlight advantage correspond factorization sparse preference attribute user item tail user popularity user tend thousand question ask user assessment literature simulate set produce form user preference item attribute mf draw item factorization truncate plausible observation netflix datum illustrate activity observe red mf capture classical distribution measure distribution user item popularity advantage implicitly weight contribution interested contrast thus benefit classical mf square consequently feedback consumption factorization emphasis pair mf user
attain worse attain stochastic factor well bandit date smooth stochastic fast information statistical estimation establish achievable rate sharp factor factor organize multi rate provide smooth convergence scheme proof achievable technical indicate rv provide background class mirror solve strongly convex function define proximal bregman via mirror md method sequence iterate use iterate iterate initialize md receive eq throughout assumption standard mirror minimizer assumption concern strongly compact exist whenever lipschitz let denote subgradient functions subgradient mirror understand detail assignment mirror descent stepsize satisfie eq remainder explore difference obtain subgradient mirror similar guarantee instantaneous function gradient section first
instrumental base run iii instrumental median upon suitably penalty choice supplementary rely penalize estimator instead lasso iii rely instrumental median estimator penalize median another possibility post double selection union select alternative regularity validity behavior eigenvalue minimal sparse eigenvalue population gram impose vector well technical positive constant identically
convex call reweighte recently solve generate convex proximal minus locally adopt tr summarize sim class ten class transform multi set label first class negative execute cpu ghz gb vector terminate relative change consecutive iteration exceed matlab code available report objective
problem way briefly motivation theoretic justification describe extensive contain discussion idea parametric conditionally partition follow relationship distribution obtain posterior draw reformulate equation represent density subset partition relationship convolution approximate average adequate gaussian expect von non smoothing use kernel smoothing close multiply together implement suffer several drawback curse parameter sample maintain performance area subset posterior tail area distribution slight deviation tail posterior datum misspecification multimodal average component kernel smoothing method mode propose perspective good normality article typical transform pointwise approximate problem directly modify particular directly draw
segmentation dynamic representative constant represent assign step piecewise prototype see deterministic model later advantage sound advantageous generation fail structure advantage approach soft probabilistic incorporate distribution model unsupervise dedicate dedicated use optimal constitute segmentation describe previously standard piecewise regression segmentation partition curve regime polynomial programming thank follow polynomial describe parameter maximize regression generally piecewise regression assume curve incorporate regime index define segment piecewise represent maximize regime piecewise curve index belong maximize log programming procedure segmentation perform parameter estimate minimize additive criterion regression respectively segment th matrix nc additive segment optimize globally piecewise provide curve polynomial segment set curve benefit segmentation piecewise integrate
coordinate coordinate major coordinate coordinate coordinate coordinate scale grid coordinate grid major coordinate coordinate scale coordinate coordinate coordinate accelerate sdca sdca vary primal objective pass entire correspond stop meet pass prox sdca prox fista large dataset count dataset ph physics belong dataset follow table detail characteristic ph multiplied employ hinge loss behavior figure primal pass epoch fista prox sdca iteration prox sdca iteration pass prox accelerate prox sdca accuracy prox sdca often significantly behave slow fista sdca prox sdca much fista describe stochastic accelerate art case interest
iteration sign change establish unlike update iteratively select manner newton direction restrict optimum solution index optimality condition global optimum case satisfy subsequence assume equal dx continuity coordinate remain number iteration set large enough fixed set definition index never enough converge index follow constrain follow solve minimization mle satisfy update free establish lemma contain enough equivalent turn original minimum converge optimum asymptotic behavior empirically direction exactly subproblem solver iteration first er number iteration descent achieve begin convergence show get advantage observe begin eventually slowly observation stop descent step th use stop begin section synthetic art ram os alm
analyst exposure clear experiment exposure imply form ex probability root derive effect variance provide exposure indicator value nonetheless greatly exhibit thus adjustment provide prefer unstable assumption restriction outcome characterize design readily derive framework approach employ justification greatly extend randomization base causal causal effect unit estimate causal interference arbitrary assess empirical american discuss approach uncertainty interference experimental observational often interference researcher study interference researcher effect importance capital effect program carry unit effect indirect exposure randomization estimating interference interference represent scenario wherein treatment potential outcome control depend assignment latter refer treatment clearly exposure potential
utility dictionary application occur feature occur elementary feature refer dictionary sparse eq refer atom representative pattern usually count representation variety literature representation code successfully inverse name dictionary predefine union orthonormal structure overcomplete optimize tailor specific derive learn frobenius dictionary aid propose obtain addition scheme subspace level pursuit employ create dictionary residual define reach stem serve capable aspect viewpoint statistical good satisfie knowledge proven ensure dictionary draw probability guarantee possible reliably arbitrary word stability justification dictionary minimize proper minimum principle case approach propose model minimize optimize asymptotic robust variant ensemble dictionary
explore lag mostly lag type accounting closely close likely evaluate start compare determine give structure ordinary regularization technique pls ridge rr commonly aforementione problem combination estimation model scheme structure method well method carlo simulation discuss section application channel eeg capital conclusion series univariate former class whole latter class throughout series ahead var constant create lag regard prediction fit drawback simultaneously unstable component separately scalar remove restriction lag type prediction eq regard lag iid essence lag sake simplicity
condition compatible pdf straightforward calculation eq explain assimilation frobenius collect low likely assimilation moderate turn induce satisfied balance encounter sequential assimilation illustrate balance strong constraint consider use frobenius norm set lead small vice similar know important even assimilation strong var pdf ball sample collect frobenius applicable particle smooth gaussian pdf smooth zero realistic smoother large represent pdf sir particle produce logarithm q weight upper upper imply collect importance choose carefully put filter formula easy steady covariance stable steady accurately consequence note minimum smoothing always induce factorization choice section unlikely smoother blind one assimilation interested full trajectory pdf calculation argument successful assimilation frobenius balance model data datum
one popular mixture beta uniform assume quantile version write du du shape compare gaussian mis formulation mixture du popular polynomial review refer known density technique work propose entirely principle literature develop study article functional tool multiple hypothesis density connection local alternative ensures attain address discovery aspect fundamentally attempt model new technique pre smoothing allow rich drive tail add easily interpret angle density modeling heavy tail em primarily main reason raw fdr step
incorporate sparsity induce straightforwardly robust second online easily establish far establish circumstance outperform budget reader contain augmentation logit versus iteratively induce purpose complete sensible bar simulated logistic gold manuscript bayes mean normal compute central credible interval coefficient line dot posterior notably skew credible interval deviation figure show vb em essentially identical center posterior vb center mean either vb data line credible dot credible interval manuscript make follow exist method
truncate different value probability ht htp htp bayesian posterior intractable likelihood sample closeness typically produce diagnostic tool assess coverage credible interval coverage inference adapt abc analyse study history implement free unknown belief model update prior function pm usage bayesian decade build powerful monte make likelihood bayesian complicate wide challenging e common sampling base sample standard g simplicity l estimating avoid prior device see
regime obtain htb g decrease massive agreement predict present experiment early several run precision parallel simple show part medium regime center plot sc exact course b sc leave regime change consequently run theoretical obtain experiment x sc one run run figure sc right result obtain numerical correspond relate figure depend differently different possibility namely run parallel hand right hand theoretical obtain previous htb part subsection everything else remain except way parallel run numerical previous right respectively sc large chose show quite another figure speak may infeasible unbounded sense universal claim type happen unbounded everything work I turn feasible turned bound figure average bound one subsection relate relate well choose possibility
recover network incorrectly lose cm overlap temporal none ref detect overlap shrink community generate node include include phase community iii consist gradually leave begin unchanged throughout phase iv consist respectively community grow recover evolve show ignore aspect also parameter error fig observe method art slice modularity provide plot community capable overlap present real wireless base new scenario point primary three hour scenario several move six form physical team structure persistent basically instantaneous change overlap remain fair bottom snapshot densely connect snapshot community community snapshot contact reality mining mit medium student mit business trade create unweighted year place trade volume exceed volume feed
approximately sparse procedure build upon gauss specialized allow selection mistake rather moment gmm singleton moment insensitive nuisance q value nuisance moment perturbation derivative restrict estimator mistake moderately orthogonality condition history example setting nuisance parameter dr good gauss orthogonality question low apply specialized setup instrumental post lasso exploit develop post control extended effect penalization base upon orthogonality hold post asymptotically framework set moment couple forecast inferential effect formally broad perfect selection impossible feature main theoretical cover functional threshold consider allow interesting distributional single quantile uniformly valid moment moment validity consider via relevant quantile effect special immediately useful quantile partially identify limit theorem process functional central multipli validity uniformity build empirical bootstrap third delta multipli functional appropriately hadamard interest outside bc rate work extend grow result ff lasso bc bc accumulate similar range quantile method develop broad control previously accumulate quantile appear quantile suggest little impact accumulate save strong interesting allow rich control rest paper introduce structural policy relate describe estimate make functional parameter theory generalize form derive theory post use reduce form notation technical supplementary implementation application monte carlo consist outcome index give indicator treat typically view randomly instrumental randomly assign conditional observable conditioning notion employ clarity causal index useful datum outcome height growth health index tailor special simply singleton estimate treatment quantity pz vector line denote variable whose smooth approach treatment effect high couple orthogonal orthogonality deal estimator admit penalization motivated accommodate key functional elementary formation identify influence treat difference local become average treatment effect population thus cover special case impact encode offer program simply parameter special setting arise let indicator outcome describe treatment similarly transform examine identify difference quantile outcome treat treat treatment treat
following learn reasonable dataset obtain classify standard half outperform illustrate area ignore thus budget exploration accuracy policy w policy able particularly discover region acquire random acquire acquire sufficient allow acquire region percentage acquire decrease whereas able adequate detect whereas scene wider learn red slightly well let image final comparison sift computation illustrate
neuron plane value cluster variant error correction side neuron similar spatially code information try guess answer choice answer variant call neural correction side neural network unconstraine convolutional network unconstraine benchmark couple plane super plane message recursion node polynomial plane x correct prove I extend node across furthermore expression super plane noisy super plane know super
set network use close class pair default equal denominator take performance category operation oppose measure augmentation calculation augmentation capture relation augment predict adopt loss calculate follow flat positive negative q true positive write long rely class different differ predict class add eq latter trees tree penalty predict equation augment introduce true true class accord remove avoid tend favor system yshift mm yshift mm yshift edge right node version add predict undesirable happen precision low theory nod tree root font size yshift xshift leave yshift mm leave yshift right left node right edge edge fill font xshift yshift xshift leave yshift right edge edge edge right node edge edge dag definition necessarily dag path connect node define path path hierarchy figure worth single one extend predict low example class element element connect connect connect connect connect interested predict contain example predict min ex py path auto thick style circle draw font mm scale leave xshift yshift auto node style draw edge right py
requirement examine individual binary threshold active passive algorithm provide start present efficient convert form noise result demonstrate generality framework concept include threshold balanced concept class statistical counterpart primary homogeneous attract theory building insight margin active active statistical learning algorithm isotropic wide class play area proceed well function use certain approximation passive polynomial give simple substantially active perturbation current derive substantially tolerance filter tolerance active presence give provable isotropic complexity worse noiseless passive exponentially dependence issue generic issue give specialize differentially private active machine medical record sensitive formal address address define natural differentially private active learner unlabeled portion example record participant addition every element request goal request setup preserve differential database notion privacy informally speak add record remove record affect algorithm automatically translate differentially
center last yy us compute write taylor expansion neighborhood ny iy u follow obviously apply w obviously obvious denoting theorem triangular theoretical first minus interest precise interval accurate exist side resp take note true mean could
need exist mining multiple hierarchy code rather calculate beneficial separately record prevent event detect record detect reason code consider however calculate may could event reduce five event control detection drug two database reporting provide issue report hand database record patient complete patient find transform general form standard subsequently investigate receiver operating rejected look reject
combination layer appear hand behave behavioral centrality period behavioral behavioral account user relational direct reality communication third increasingly anomalous occur end period volume send directly come network infection study sbm simple clique interact appropriately dynamic classic sbm anneal membership fit mixed agent extended track result smoothed potentially grow interest basic level growth incorporate statistical
posterior decompose basis significant part outside basis reconstruct clearly see fig intrinsic ambiguity solid line solid line true curve achieve weight image initial fig achieve th reconstruct image pixel posterior simplify detect observed perturbation fig eigenfunction four eigenvalue reconstruct match variational deconvolution gaussian vb blind show blind location additional experiment image h h comparison sparse test level
odd country air odd class integral become small object fit scheme instrumental target could indicate candidate candidate principle arbitrary scheme overcome logical complement object need n object clear e human
near match descriptor versa relevant one near descriptor pair belong far seek maximize square space near respectively
space structure interpolation interpretation rate case highlight theorem therefore range versus convergence sufficiently smooth discuss section w estimator regularize fisher ignore completely fashion convergence kde simple estimator raise applicability though kde moderate kde estimator knowledge kde easy lie choose objective iv kde kernel cross cv constant method objective bar kde sizes gaussian score besides show advantage kde dimensionality propose dimensional consider dimensional generalization density reproduce approximate arbitrarily well kullback leibler draw propose empirical provide computationally alternative estimator suffer empirically kernel dimension rate smoothness address understood minimax optimality show improved regularizer minimax lower idea propose known construct provide leibl define denote covariance suppose show identity relate information integrate dt equality show norm kl distance follow g x q px cp p exist w
eigenfunction implicitly projection dpp value complex conjugate transpose two efficient sec low representation describe provide positive require loop supplement l bb dc vc must assume low need kernel distribution rank dpp approximation arise general rank kernel rank begin exact imagine approximate approximation yield product eigenfunction integrable approximation need approximate enable enable nystr om pt fourier approximate kernel independently jk apply factor e apply function characteristic rank cdf equation translation invariant characteristic function translation invariant supplement transform dpp nystr om sample matrix denote q translation invariant requirement similarity supplement example handle consider sampling supplement
g regime involve involve change numerical cyclic method time usual alarm r involve numerator differ hand equation offer rate build lr absolutely although arithmetic well td dd eq probability stationary process change dp dp tt early deduce around first importantly another compute eliminate subsection subsection brevity markovian introduce observe copy except replace markovian establish recursion cf sequence interval discussion note recursion equivalently rewrite use observe series convergence justify operator strictly sr seem first compute equation independently
dc stochastic blockmodel sbm sbm dc closely plant study regularization cluster demonstrate high leverage essential context network reference therein relate cluster provide analytic result suggest second star figure appear demonstrate project eigenvector unit sphere propose remove heterogeneous degree dc sbm throughout study unweighted vertex refer adjacency notation use denote frobenius two degree spectral account define subset induce subgraph spectral traditional
preliminary topic augmentation technique multiclass margin intuitively fix augmentation investigation collapse gibbs gibbs adjust likelihood class gibbs integrate dirichlet collapse collapse isotropic posterior cholesky procedure inversion conditional common give exclude topic first supervise initialization normal distribution distribution chain iteratively use condition root draw markov finish burn iteration chen zhang zhang predictor supervise integrated discriminative semantic unseen learn usually smoothness approach build supervise rely iterative solve latent subproblem desire distribution max max supervise gibbs representation gibbs model minimize loss augment variable conjugacy restrict svm subproblem algorithm analytical conditional experimental demonstrate improvement binary multi dirichlet machine availability develop tool discover reveal explanatory major tool topic vocabulary interpretability bayesian model substantially application various field categorization besides discover topic major make accurate classification task rating review get develop attract mle topic approach response margin discrimination lda
power may decrease range aggregate capacity capacity cl paradigm extra iteration cl aggregate capacity paradigm th figure order allocation il cl robustness scalability paradigm il achieve optimum robustness cl paradigm robust cl paradigm il paradigm maintain e reward function converge regardless old experience one finally cl maintain support wireless science department university box edu interference management share
trial mse definite imputation effective rate negative method miss high efficacy balance show incomplete test missing discuss observation convexity extend mean imputation incomplete investigation valuable setting pt investigate paper
asymptotic bayesian unfortunately open majority modern set considerably adversarial weighted rule guarantee weight assume learner expert still rule deep flexible dependency oppose recently expert adversarial noise directly majority expert analyze consistency suppose expert devote prove mistake f occur expert fails exceed since deviation probability
hash table convert format ml pg matlab result advantage software paper discussion ml offer ml pg unsupervise given expect user tag divide provide select ml pg change algorithm later various library similarity pg determine library user ml pg select value stand produce produce cluster number cluster worth certain library ml pg way one run may fact cluster happen cluster find run particular ml pg show come run experimentally important pg appear pg see assign value range example cluster ml pg proximity value reliable cluster enhance similar ml pg pg pg user manual detailed description tool user development library definition proof library concrete library context detect concrete study appear purpose terminology style library consist file number function manual inspection detect pg pattern analyse produce library experiment library cluster contain different library homogeneous similarity easily case distinguish also support extend
node get mean express term substitute get note assumption weighting eq minimum simplify notation combine network follow equal negative absolute scalar sum noting therefore analyze ability environment bound excess attain learner diffusion processing conduct extensive simulation illustrate risk excess gradient powerful iterative interest convex loss label binary predict label describe observation equivalently separate description goal incorrect accord generalization achieve label yet excess achievable classifier excess understand good classifier study excess derive relate procedure suffer drawback utilize environment relate regret size indirect directly size cope environment constant diffusion appropriately
topology whose often yet diameter think add new circuit cost similar circuit goal classification hard adversarial environment tradeoff main theoretical intermediate improve diameter mistake optimality test range depend fraction play crucial role span otherwise operate involve preliminary draw span tree subroutine vertex disjoint height small height visit tree visit internal visit backtrack tag height subtree root visit assignment root root remove along iterate
concentration vary approach grouping find maximize integrate tractable conjugate infer bayes grouping transition conjunction bayes concentration parameter fix observe transition count grouping likelihood describe search
regression difference distortion analytically distortion coincide take prove distortion source lie analytically distortion investigation limit demonstrate distortion bind distortion distortion function
sgd hard possibility activation function similar sgd bad eight method performance task origin possibility however hard circumstance worth include many function hyperparameter never probably limited activation maxout activation dropout appear plot three acknowledgment like resource google fellowship theorem definition remark minus em height depth machine train train forget widely serious investigate modern network activation
therefore account sparse unfortunately solution np way problem uniqueness basis pursuit bp convert well efficient pursuit case strict robust govern incoherence acquisition incoherent incoherence low projection basis incoherent basis overcomplete acquisition correlation atom dictionary atom optimize algorithm overcomplete dictionary propose modification projection improvement unify rest paper review perfect present improvement unified conclusion acquire signal denote acquisition effective dictionary mutual coherence inner coherence absolute provide recovery overcomplete coherence
desire heavy tailed diagram elsewhere predictor heavy shall datum ask predict predictor tail constraint tail limit mm construct predictor large appendix solution tail constraint define manifold possible satisfy progress scheme equal h result tail typically extreme appendix convergent line figure match possess limit individual readily know exponential construct uniform parameter see require
assignment centroid simplex I item soft assignment item training mapping appear case thm
em gmm summary modify follow start pre retain remain em iteration abuse notation component begin update em substitute update component numerical threshold though small time small threshold obtain final select tuning study fan li scad regression many select generalize validation fan li value generalize however difficulty true mixture normal gaussian model similar impose q q penalize derive avoid identifiability ill finite practically discussion mixture likelihood still way assign
penalty desirable high strength parameter hence unique cardinality sigmoid logistic loss iw assume logistic loss give go joint distribution term characterize behavior conditional algebra show expectation take variate assumption bound result proof find vanishe provide vanishing arise biology dimension comparable variate precision follow well know conditional equivalent conditional vertex element exclude conditionally field precision equivalent draw log constant write matrix sample matrix constrain determinant restriction diagonal cardinality integer hessian l kronecker show suppose product eq result motivated
x r r moreover gaussians gaussian mixture could notion freedom outside provide eigenvalue could still n extend expansion eq appendix square I assumption dependence term initial additional convexity fast traditional use size iterate chain technical iterate converge distribution invariant assume markov recurrent ergodic markov distribution slightly chain start imply explain solution pointwise extra bound say projection bound happen provide axis variable analyze technique likely robustness result expectation assume p moment need probability tail decay tail sgd constant quadratic f average sequence stationary typically converge pointwise condition satisfy loss optimize go around
imply minimizer minimizer utilize apply utilize hessian logistic regression direct matrix inversion expensive even provide repeatedly fit exploit indeed use diagonal bind hessian interaction class tractable property free ahead spirit view diagonal matrix utilize determine entirely datum case satisfie observe generalize empirical q state assumption natural characterize rate assume link amount want monotonicity monotone demonstrate copy result defer supplement depend smoothness strong ahead benefit well contrast qualitatively xx xx depend covariance accelerate discuss point observe also ahead benefit deterministic issue link unknown main difficulty restrict one weight prediction residual prediction simplex
affect hard reconstruct dictionary dm finish search start patch become table achieve stay fairly level patch achieve snr lar quality reconstruction fail reconstruct reconstruct enough error inspection reconstruction image neutral color perfect reconstruction black scale difference admm pursuit db db db advantage dm reconstruct dm consistently reconstruction
intractable proceed calculate gradient update autoencoder follow repeat require encoder decoder stochastic often practice empirical gradient stochastic gradient variance encoder autoregressive operation easily graphical autoregressive multiplication triangular train model binary uci set digits frame five game quantitative iterate exponentially representation stochastic estimate index repeat ten time per architecture description validation evaluate test
cardinality priori strong large probable occurrence show figure item order inspection exception dot also mode turn partition posterior apparent partition co take minute cpu time use convolution estimate full enumeration hour b item feature randomly inherent indicate reasonably probably belong together mode partition unnormalize could deduce
distinguish statistically addition cluster partition topological directly concern identify regime right acknowledgement thank dr ice data remark macro ef laboratory technology use study system change detail difficult impossible high sensible recognize transition qualitatively regime develop transition tag complex system particular stationary regime dynamical change behavior system arise phenomenon occur vast range temporal natural shift change shift population market indicate numerous rapid change european responsible year little ice provide example change instance rich reach water clarity greatly turn paper develop characterize detect
random th equation follow time scope membership priori along switch conditional current drop obtain simply substitute membership kalman filter linearize r posterior switch local search hill initialize membership step
performance adaptive kernel behavioral interpretation focus potential carry p paradigm competition visual stimulus target stimulus difficulty p toeplitz acquire channel filter hz rd additive amplitude retrieve use template pre prototype example potential introduce try spatial dependency channel relate learn eeg pattern consecutive variability ga signal q make flexibility iterative square restrict ms enhanced project second provide warm start length ms constant parameter point center ms stimulus update since shift carry
let implication either homogeneous minimizer pf minimizer converge generalize prox sf kf zero thus limit converge subsequence convergent hence proposition converge f lemma c eigenvector eigenvalue necessity point relaxation extension
mention possible second indeed case explicit generalize like one quadratic form ta ta ga z tb attain rao consequence obtain translation parameter rao density fx
figure data leaf bagging replicate report example follow methodology see dataset predictor mail spam spam spam spam train forest fit replicate california california dataset response house forest replicate point model package replicate implement base standard patient patient function random cosine treat q test label original produce point report set edu stanford edu stanford university stanford usa learners forest build bag predictor compute replicate work direct application bag bootstrap replicate version finally illustrate finding study bag technique bag variance learner compare learner study
regularizer formulate indicator encourage split key observation combine property operator idea obtain
consume dataset dataset group equally sized separation easy cluster rate separate medium less distinct determine autocorrelation individual process choose rmse percentage mcmc metric expect base simulation model forecasts nonparametric rmse outperform easy cluster higher help c thin thin easy hard rmse rmse rmse simulation indicate sampler find exist important cluster homogeneous cluster generate supplementary hope methodology report count census evaluate forecast run chain iteration different draw sensitivity significant
parameter prior model towards traditional kalman incorporate train training example combination coefficient introduce combination input make test laboratory predict paper follow review section outline extension time
move respective day strength operation strategy predefine market operation strategy parameterize follow strategy gain gain stop day operation condition price raise price price fall price day none occur day stock
address objective parametrize bandit item read etc feedback explicit five implicit feedback also play practice body literature develop predict item past concern see particularly rating give scalar product feature characterize item formulae capture approximated item construct explicitly derive feedback assume computed item recommend distinct hereafter user encode information advance user incorporate former information advance therefore linear recommender vector sequence wherein history
probability discrete add different dispersion location parameter independent laplace probability p db cluster location parameter fit dispersion parameter fit dispersion estimate one fisher primarily neutral population work
construct correspond conditional constitute cluster importance mean marginal follow symmetry integral respect evidence monte importance hold perfect symmetry importance posterior massive efficiency section permutation label proposal term loss efficiency term h motivate proposal negligible indicate high contribution appropriate h decrease approximated approximation approximation make size truncation obviously generate quality perfect symmetry permutation obviously detailed algorithm algorithm randomly
field subject fix regular visual ms competition four ms offset ms hz classify subject base restrict attention condition total balanced raw trial decompose hz interval hz bin use
discrete refinement every upper follow monotonicity property elsewhere choice make distribution consecutive outside interval incorrect interval incorrect leave right proof interval incorrectly form monotonicity integer grant p k computation provide lastly establish grant dual feasible duality thus constraint grant provide existence convexity search note plug single cf comes note la consider iteration cf positivity contradiction nonnegative therefore pick iterate remain class h coefficient follow vc simplify loss restriction iff choose finite mean map scalar mean iff follow primal may parameter value low error margin convex differentiable counterpart follow constant suppose c rule definition mean x hold mean turn proceed early la next margin cause iterate margin change binary counterpart give search candidate desire example h c c whereby assumption thus along la control size l mb la follow hold choice case nothing interval inductive candidate order consider taylor use binary expression
easily letter target pdf give proposal denote remain location tangent maximize eqs tangent straight line
mixture family parsimonious decompose component g gp entry proportional eigenvector package normal skew offer gaussian model significantly eight covariance alternative underlie latent mixture within random model latent analysis loading factor loading entry closely principal eight parsimonious setting gaussian impose valid constraint
whether notice far remove trial label target contain trial label calibration attribute calibration htb abc unsupervise abc c abc super c super super outcome hold unsupervised label contribute surprisingly little future
stochastic matlab easy sample hard know variable program sample environment simplify due inference powerful exact complex simple inference find close easy field approximation offer alternative especially intractable descent derivation computer
possible mean fig cr achieve performance especially obvious uncertainty mobile monitoring complexity omp utilize wavelet db compression divide reconstruction sensing generate standard signal omp compressive one omp coherence reconstruction measurement measurement fig show error signal measurement see omp improve much uncertainty omp htbp b htbp
model recovery demonstrate possibility exploitation simultaneous recover minimal hope researcher work structured recovery acknowledge university office award primarily adapt interest net rd zero tt result basic establish k c c u f inequality fact f u u idea tucker combine cardinality n u thus exist net cover kk nr c f derive proof basis k know moreover
equation uk model problem subspace identification set trace trace matrix correctly throughout paper block contribution introduce space use attain eliminate multiply side matrix follow generally order output sequence instrumental far include reformulate approximation svd realization recover
mutual first provide sensible theoretical must analyst obtain provide carry mi notion restrict miss aggregation summary monotonicity asymptotic scaling risk appropriate nontrivial procedure strong notion necessarily low bad sense analyst impose strong constraint example unweighted mean well error analyst know standard comparison experiment deterministic relation experiment informative loss relation imply stem broad objective partial deal allow procedure base attain estimator form preprocesse way procedure model traditional regime bayes transformation involve monotone regime procedure consist asymptotic regime principle monotonicity procedure analogous procedure neither mle procedure distinct correctly range analyst agree model risk inconsistent belief input nontrivial deterministic input uniquely set give analyst derive rule kt k bt tt kt bt kt bt unique comparable sense deterministic dependence overall biology illustration wide analyze throughput gene rank statistic another aggregate pathway replicate preprocesse rank statistic construct monotone thus bring construction unfortunately generate monotone construction monotone topic establish role illustrate framework utility demonstrate limit previously level intensity microarray expression observe intensity nuisance level additive magnitude markovian quite reasonable upon experimental protocol distribution correction log base scientific upon instance whether gene expression density spectral cloud typically response sensible base signal ground markovian preprocessing correspond temperature analyst miss essence setting degenerate function boundary little concrete failure suppose equally sized batch observe batch case preprocesse complete would simply select separation phase miss analyst chemical
visualize cover curve band approach sample originally tool online prediction major efficiency infeasible ordinary prediction characterize inductive implementation combine result band correct finite sample process many exist reflect functional slice salient datum classical closely level however functional space density dominate measure pseudo functional finite dimensional method mean mixture construct tree functional pseudo cluster free
consideration interval thresholde standard asymptotically conservative selection length fact large magnitude compare anonymous comment support grant quick proposition first chart except concern error chart depict tune namely converge chart variance asymptotic consistent chart behavior whereas unknown distinguish large infinity concern interval auxiliary one htp sample ex prop confidence ex prop c conservative ex c tune ex prop prop prop prop ex overview proposition h n h use formula cf regard factor replace apply perform second display integral display simply obtain integral respect integral proof proceed proof proposition derive ix use independence replace relevant formula cf fix regard apply perform yield display immediately
entry identity matrix symmetric r p r r rl rl stand root say nonnegative stand non letter set denote hellinger subset cover number size sample estimator minimize penalize definite penalty penalize consider follow positive order consider element obtain lasso estimator distribution theorem operator get follow context graphical distribution laplace independently impose graphical put put event correspond absence develop
long highlight compare second yet second dotted accelerate smoothed dash dataset consist medium wish non descent fast difficult compare remark variable involve way first orient method proximal descent accelerate descent solid blue line svm correspond matrix particularly suit coordinate descent sparse find stochastic dual coordinate ascent sdca use processor primal solution summarize sdca coordinate solid duality gap sdca summary achieve well parallel
paper international international wireless network conjunction nd heterogeneous conjunction nd international wireless conjunction degree computer communication communication engineering american university ph degree university currently electrical engineering prior university research interest wireless cognitive wireless security grid co book international conference journal author author papers award international mobile hoc wireless international conference monitor dr nsf award receive dr electrical research mobile centre wireless communication currently digital head communication engineering centre wireless communication interest mobile conference journal paper wireless communication vice communication plus pt minus plus pt plus plus minus notation centre wireless communication email electrical engineering department email interference management one key cell network whose capacity wireless mix
q kkt sufficient last dual interior point successively relaxation triplet kkt define implement drive kkt iteration attempt newton root finding root finding method solve solve effective kalman system obtain update note structure positive block matrix element diagonal solve back approach present simplify derivation improve see take iteration unchanged linear smoother present first impose box advantage state bound encode linear smooth modeling increase measurement situation figure constrain smooth avoid encounter unconstraine smooth middle end track smoother far bad track avoid aid file constraint exponentially bound signal linear trend use start remain include emphasize linearity mean fact box complicate smooth smooth measurement nonlinear smooth throughout simultaneously would like constrain convex constraint additional objective nonlinear
symbol generalize mask message mask encode mask end symbol update side achieve define another bit mask easy possible candidate already know assume candidate optimal emphasize permutation overlap handle step encode account overlap element rank list might compare list modify two track time bit remain column second track time probability r b b c b c
help model delta double delta dd capture far provide dd dd conclusion appear feature otherwise cnn convolutional aggregate discrimination cnn speech spectral allow explore convolutional layer show network network keep convolutional layer start furthermore cnns improvement feature convolutional vs fully conv dnn conv full conv cnns explore recognition sharing address limit share low high layer band share component difference convolutional something explore speech layer network unit require reduce unit connect keep constant hide increase slight improvement unit second vision task locality frequency region unit
vector square negativity seem since outcome component detail optimize optimize covariance predictor correlate split naturally identical lasso coefficient net elastic augment q predictor equation net component case involve split mse covariance maximize penalize kkt soft component correspond single linkage agglomerative cut linkage sometimes linkage agglomerative consistently component cluster linkage cut dendrogram produce elastic net fit remark partially optimize convex optimize optimize course
predict prediction dataset correctly predict ii top top top latent interaction efficient variational suggest general scenario deal process generalize event background rate cyclic patterns process factorize temporal might well support grant fa variational maximize variational satisfy enforce variational constrain lagrange multiplier multipli evaluating scenario begin possible similarly derive exclude logic numerator logarithm th combine convergence variational consist parameter look spatial spatial pattern single pair temporal involve allow solution
define read transfer matrix element call process stationarity probability observation via approach start likelihood mean trial due analogous true generate x ss ps unique reach learn employ map proceed maximize datum one try maximize probability sequence parameter observe maximize posteriori
substitute note naturally purpose matrix q ensure prove theorem induction definition fix follow lemma order algebra decompose diagonal follow triangle control control obtain directly cn put inequality yield recursion along complete purely incur entry avoids bind pt log initialization procedure alternate c requirement minimization ease minimization widely exist effectiveness world dataset three algorithm control advantage one shot linear alternate alternate gaussian matrix know satisfy incoherence spectral subset zero element choose distribution sparse recovery plot plot average step
disease examine select follow select test address goal goal detect address find science science discovery phenomenon laboratory order replicate experiment prove phenomenon laboratory lack behavioral behavior well comparison may opposite laboratory differently study e interaction laboratory hypothesis say non laboratory effect great concern half clinical fail success trial effect phase ii study patient obviously study iii ii genomic interest genetic study phenotype test population environment whether different measurement recognize genomic genome variant
width difference minimum possible index iteration strong constant edu com way efficiently wide spectrum ml program scale big big data parallelization employ fine grained scheduling processing paradigm specialized execution rely program system direction remain difficult program general purpose systematically challenge ml program admit convergent solution present unique synchronization scheduling program design modern program considerably size becoming extract volume big internet increase big model pressure beyond machine web web tb share possess highly fashion ml hand art cover semantic substantially improve high single slow rapid many new scalable remain wide mining nlp vision community especially build advanced suggest scalable execution art platform pc cluster cloud correct execution resource production platform abstraction considerable engineering limitation ml program generalize scale well programming interface yet grain communication show advantageous necessary
see classifier discrimination iy error classifier iy code rather iy iy iy iy iy iy
establish b proceed bind stack I vector take union finally episode episode eqs algorithm ready side proposition combine right side due episode length probability use choose union stanford stanford stanford stanford quadratic establish bind apart form
eigenvector write degree submatrix take discussion object partition exist finitely many partition outline generally compact attain quasi open set subsequently investigate regularity partition partition computational relaxation analogous relaxation spectrum fail eigenvalue laplace domain define use recursively partition cut local cut study partition share attribute analogue partition partitioning term curvature flow factorization find nonnegative application cluster proposition propose transpose asymmetric graph laplacian dirichlet partition collection dirichlet
document exchangeable belong accord condition fit global corpus posterior integrate local common complex normalize compute apply remain choose algorithm consider possibility vb ep dirichlet prior posterior exponential hence vb utilize lda subroutine lda k posterior vb make pass streaming represent shorthand give assume take vb approximate define collection posterior find difficult approximated coordinate vb appear instead ep like vb factorize pass store memory context next evaluation coordinate locally
pseudo distribute set share address propose pseudo combine averaging parameter mrfs likelihood class model parallel implement distribute replace certain condition true achieve satisfied theoretical exchange maintain empirical prove additional insight local whereas neighborhood clique rely author centralize arrive graphical clear beyond width exhibit
inequality hypothesis lemma right depend constant restricted range hypercube result input summing bound rescale eq acknowledgment xt thm corollary quantification approximate might arbitrarily large lipschitz
unknown block entry efficiently pose moment precisely diagonal fact observe rest third converge entry tensor cf least u tr whitening tensor pseudo remark third moment help ensure sample alternate moment discrete eigenvalue u see rank g robust g moment diag assume moment provide size pt min min n min necessarily valid might kullback min rl normalize distribution j drawing kl processing satisfy moreover
du du du bind du r du dt n e valid desire result event eq integral bound bound tail previous section equation n du r soon n term du dt put cm long em pt paper logistic use step low relie extend generalized function many modification average problem depend primarily potential strongly optimal stochastic proportional come form x convex covariate strongly restrict e correlation therefore
area normalization plot fig however clear case rna admit line lie introduce measure inherent capability give hull feature hull coincide transform revealed structure potentially energy violate suggest investigation interesting deep polytope application rna polytope symbolic also assess power remain department computer science
efficient cv square th specifie consume error th comparable accuracy appendix basic eq notice dominate implicit show desire derive step definition eq via term inequality convex compatibility consequently follow compatibility convex subgradient derive additive gradient first differentiable observe equation triangle inequality eq kkt moreover hold small additionally satisfie side schwarz term bound subtract equation cauchy schwarz observe claim claim invoke norm
medium high frequency low rate big necessary threshold big frequency svm work uniform rich ccc method domain vector row though give rise code paper develop diagonal jacobian regression wise restriction make sense diagonal domain trivial domain experimentally improve compression apply previously base relevance image tie ica approach non linear conventional restrict coefficient geometrically dimensional image suit scalar statistical interaction
computational encoder inference fully factorial layer hence take firstly fully factorial posterior eq patch reconstruct decoder decode nonlinearity recently try denoise sometimes favorable image denoise gaussian experiment boltzmann denoise autoencoder family boltzmann autoencoder depth gaussian additive capability completely apply three distinct database list present six term instance image pattern present coarse nearby road size vary quite color max size denoise separate image train patch
perceptron statistical mechanic unique minimum degenerate ground temperature limit fluctuation vanish interesting observable weight essentially unsupervise via replica method temperature cavity cavity cavity overlap mean unit upon total energy presence new consequently alignment show optimal minimization competition second optimize alignment first term prevent alignment change penalty old play weight condition within cavity identical function new finally repeat analysis yield apply dataset gaussian point cloud consist I multivariate whose distribution fig center datum alignment whose unit variance nonzero mass onto optimal similarly analysis find maximal variance alignment along c extra unity covariance width direction mean datum determine two centroid form energy reveal lack project along distribution however maximize absolute implie zero outside split along quite remarkably dimensional cloud actually replica however correction replica symmetric result replica exact summary slowly perceptron store also side hyperplane perceptron capacity association perceptron learn association capacity analog value discuss association perceptron iteratively initialize weight follow generality compute nothing iterate pattern pattern learn analog weight analog biological reliably code level analog np problem enumeration reveal perceptron capacity solution always find search exponential provably polynomial imply provably find time plausible pass joint weight desire factor example example number message pass system configuration correctly association would configuration pass feedback term pass obtain maintain analog allow discrete actual hide simplification form term follow convenience take avoid pattern nothing internal rule internal state iterate pattern quite similar modification rule concern situation single cause rule internal pointing right positively large absolute rule thus efficacy remarkably perceptron capability neuron pass per remove unable remarkable performance message whether signature pass via error error signal neuron close play role variety transition powerful eigenvalue classical ensemble replica formalism symmetric focus wishart high section whose element probability realization eigenvalue distribution realization average denote compute eigenvalue element perform dimension force decay think negative force field plane mathematic transform recover field relation potential turn potential oppose either derive integral potential replica appropriately care logarithm general hermitian
hour weather strong area prior time peak hour decrease rapidly failure per recovery characterize failure occurrence time aggregation remove duration failure occurrence sample height failure duration stationarity failure failure occur p day failure occur assumption failure occurrence duration combination weight mixture factor parameter failure family particularly appeal reliability theory characterize external shape scale respectively equation characterize short moderate recovery simplicity failure divide show vary failure stationarity failure stationarity duration dominate recovery day third duration failure scale weighting parameter dominate recovery failure around half failure occur day recovery within change non failure recovery comparison reconstruct actual sample failure closeness stationary approximate failure temporal stationarity variate failure variate failure
comparison experimental simulate enable train track represent electrical point final exploit switch operation see present adopt relationship adapt regression regression provide type optimize additive segment signal programming know improve run simultaneously regression various regression extraction
york filter compressive elementary compressive taylor bayesian median supervise bayesian parametric j bayesian normal gamma regression van multinomial probit van entropies density van w bayesian selection rate fit b map hilbert lee likelihood heterogeneity dna cells variational linear mixed h li l model j averaging distribution shrinkage bayesian relevance machine j regression subspace projection wang scale van l privacy yu define df sequence hellinger ball radius need cover define condition n show subsequently x x density pz pz consider result seek proposition spirit stacking
blue circle blue circle blue circle circle blue blue blue circle green circle green pt circle circle pt circle red circle hyperplanes diagram depict margin concept fix certain customer misclassifie identical figure separate power question arise prevent power question answer improve plan lot bad assignment customer one even assignment paper good completely plan answer paper allow construction despite misclassified multiclass pt blue circle red circle blue circle pt circle circle pt circle pt circle pt circle slightly clustering figure power diagram penalty term misclassifie separate hyperplane informally removal would change separate hyperplane separate hyperplane separate tradeoff margin exist interested separation complicated definition hyperplane recall multiclass diagram soft margin separation simple help sum pairwise size ability misclassification square actually
together bayes ucb cn cn reproduce law pattern cn effectively reproduce music recommendation study detail piecewise user learn ucb cn region deviation important multiply content factor scaling figure learn piecewise match analytic form accuracy explore preference recommendation system regardless balance important tradeoff news recommender effectiveness music exist interesting effectiveness type also cover rate product nonlinear linear inference model movie due consumption music user many repetition relatively rare movie follow law first diversity generate start interactive music recommendation exploitation recommendation rating music audio integrate music recommendation model enough update generalize diversity approximate variational accurate efficient approach start improve recommendation capture repetition recommendation variational low remain normal distribution symmetry easily p gibbs resort variational p q exponential family restrict family obtain expectation might q assume bit low multivariate
tensor scalar decomposition definition permutation tensor denote index ib ic form inner ia ib ic abc ib ic abc abc u fact du ia abc ia ib ic ia u ib last orthogonal form assume loss generality tensor inequality assumption attain diagonal decompose unlike decomposition tensor factor solution version start coordinate choose replace practice step principal jt min corollary center bar central orthogonal break orthogonality orthogonality
entity exist rely feature alone extraction ie aim basis kb compose triple hand entity head right entity example kb rf refer movie learn perform extraction supervision kb ie entity detect name entity aim relationship kb pair entity give triplet
web contextual devote spatial processing explicit deal desirable pattern mention classification present difficulty identify among item meaningful datum frequently local data relationship reason help formation consequently case topological walk visit past show work walk ability topological fashion walk apply discover pattern network totally propose classification paper upon implement classification exploit topological underlie exist combine serious problem network measure intuitive also inference novel high walk exclusive walk nontrivial advantage able fashion memory window dynamic far away global graph occur memory change cycle memory length reach happen walk network say walk dynamical property avoid assignment problem occur partial instant underlie combine vision show variance remainder overview computer world set manual site dimensional datum time site visit previous avoid deterministic
inclusion moderate fully criterion showing produce generate simulation structure european population moderate level apply program human loss estimate proportion run fast et wang efficient compute nmf alternate least active method nmf genetic provide trade implementation decide yield good predictive imputation computation cross value significance predict panel result program provide statistically show entropy discriminate simplified assumption linkage population factorial estimation avoid
paper regard paper overview importance call random mutual submodular go prove content emphasize distribution divergence transforming contain leibler equal entropy value q return kullback leibler divergence subtracting
various matrix operation complexity parallel model wherein number processor justify product serial computational complexity break serial would dimensionality parallelization preprocesse core qr svd result involve element product entry result lead overall core sparsity reduce requirement core core product perform perform eigenvector product core since core inner result independent core pre multiplication zero per effective reduce core product entry membership number matrix community unsupervise permutation may validate row statistically dependencie variable consideration ground adjust enough testing obtain via great define note student freedom true denote statistically independent test q set edge statistically dependence probability small nan hold perfect matching like match define eq indicator equal membership give performance give edge thresholding norm pair truth false estimate ground truth test former summation pair membership divide community discover test overlap discrete group people vertex centrality conjunction outlier bridge specification machine code hardware software cpu gb k gb release dense r perform stochastic membership around membership intuition theory well membership model case quite practical theoretically
q I ij powerful corollary bethe I hold marginal flip edge result yield I q q q substitute unless root tangent yield upper q I follow follow substitute q contradiction know derivative convexity j j pseudo whether finite prove flip pseudo marginal already iteratively sometimes local around node first eq stationary likely e probability raise establishes algorithm iteratively
sufficient condition ml improve theoretical confidence row true pattern poisson formulate tuning free fit statistical ml framework value row formula select provable free approach noisy identify condition minimization science foundation set x n j j side chebyshev get probabilistic although free term depend chebyshev inequality eq quadratic inequality solution account unknown recall property probabilitie f term sum independent type omit focus test series taylor due depict expectation return depend side chebyshev taking numerical obtain exist non row generality hence contradict x
reader action matrix cost ts ts sr distribution policy ergodic ergodic else hypothesis make queue else active grow infinity minimize uniqueness find reinforcement derive cf namely distribution simulation free note reduce continuous intensity cost know policy fix relation intractable curse family ergodicity optimal dimension note good optimal controller simulate computed averaging observe optima convex general require gradient th unit
transformation distinguish burn stationary phase empirically phase differ length prefer large stationarity opposite burn tend step rapid region probability appeal problem chain reversible time prevent accept work adaptive g phase interesting possibility relax requirement relate cause like optimisation transition work particle version manifold hamiltonian improvement mail united mail ts division system e mail hasting allow bayesian model monte particle filtering intractable formulation
expect contain contribution sparsity sparse close complexity force interesting would hope interesting sparsity computational complexity suitable choice expect order implication apparent take complexity hand function maximum computational guarantee search large insight define contain pattern situation independence occurrence independent computational force look scenario pattern provide drop rate size among predictor show preferred force computationally suggest computed depth set associate view exceed visit require pass subsample need fairly fast use min hashing apply describe leave aside end
properly snr discuss affinity affinity within minimal long constant noiseless true roughly noiseless word magnitude fundamentally great appropriately fashion albeit exponentially dimension become seem tight imagine regime accordance establish useful clustering negligible magnitude close splitting empirically figure histogram discovery value shape curve discovery coefficient numerous time concern explain fact similarity connect subgraph resemble os number subgraph regard cause expansion subgraph via ultimately succeed presence false please therein finally like comment fairly broad hence priori still conservative superior proof reveal proportional proportional statement plug value calculation merely mention variant give recommend proxy reason say subspace challenge point home consider noiseless close yield noiseless sufficiently many selector way perform step discuss selector sparse
independent markov true unknown error equal purpose name variety type regression variety conditional etc stochastic model copy time dependence suppose within row error moreover unconditional error insight mean link decay quite forecasting claim factor nuisance smoothed used decay least unlike independence link ratio parameter fitting relax arise allow finite development function time number thus
max k project projection combination belief write convenience basis onto space projection cast minimize respect belief analytically let c ia ji projection work parametric compact instead maintain represent belief maintain draw execution phase expect sampling b cost total significantly estimate offline step cast cost solve system interested approximated scenario inspire mark enter intersection road discretize grid cell level tuple ps either accelerate maintain speed speed parameterize acceleration range parameterization behavior show preliminary reader move current cell cell
compare complex symbol channel snr find generality unit receive symbol concatenation quadrature cdf denote obtain measure find define virtual give utilize variational form maxima
infeasible approximation merge message soft decoder optimal channel capacity channel decode bit code graph base tackle receiver optimally maintain receiver impulse symbol moreover observe exploit leverage message pass extension soft soft receiver propose receiver address receiver provide tradeoff suitable provide illustrative example refer channel impulse numerical denote sub construct column conjugate transpose probability rv subscript omit similarly mass pmf denote circular rv mean expectation rv font pilot index index code determine symbol encode bit integer symbol likewise scalar symbol note symbol code code bit allocate denote entire information th vector include pilot unitary time invariant impulse corrupt cyclic length symbol interference avoid discard cyclic
form independently opponent drop player opponent element use state describe play approximate discrete like response opponent variance observe opponent variance opponent evaluate strategy use estimation estimation opponent play action player opponent estimation opponent equation equation j use predict cm maintain estimation estimation use belief choose opponent action present game pure nash equilibrium player available example player opponent denote estimate action algorithm play algorithm opponent opponent play estimation opponent proposition appendix opponent eventually reward base
regularize problem integer non identically characterize high establish almost limiting assume show apply thresholding word vanish loss believe hold plus corollary exploit singular input use opt opt opt compute give q bias use signal define z hx estimate end matrix whenever describe theorem estimate
q dependent distribution optimally principled predictor subspace manifold want subspace equally free mode unit circle gaussian gaussian shift previous represent subspace function bind kb di di lm ib ib ii iy I promise future close weight minimize prediction prediction respect work transfer ridge equation algorithm structure constraint square report ordinary ridge experiment dataset
dynamic hoc topology suboptimal use guarantee fusion eq amount fuse various theoretic pdfs gaussians estimation always yield readily exchange pdfs decentralized team mobile control law figure gm along finite gm substitution pdfs find tractable approximated recursive eqs maintain end derive closed replace gm moment match gm covariance poor approximation
predictive justification mapping lemma condition minimizer f lasso order mapping approximation lemma substituting line support surprisingly rely minimizer proxy concentrate concentration assume fact remain around p may repeat e argue greater bind promising challenging topic show monotone mapping entire regime penalty useful discussion compact function define local nonnegative increase give f combine strict scenario function ensure continuity f increase statement continuous lemma hence guarantee consequently stable everywhere lasso show case independent standard lm probability however solely tangent basic lemma probability f hand expand eq normalize q rest lipschitz state understand behavior converse show approach infinity previously compress threshold noiseless slight modification entry f minimizer lead since positive integer proposition proof discuss assumption ff lipschitz constant minimizer argue minimizer away follow rigorous standard exist probability generation pick let f consider guarantee minimizer sufficiently ensure emphasize guarantee converse guarantee slightly f definition mostly nature proposition possible converse leave let normal exist minimizer
hyperspectral statistical difficulty transfer enable volume datum feature spatial km spatial noise twice measurement key specie whole develop pixel pressure width figure pdf challenge need fast retrieval approximately per day process european centre medium weather estimation spatial layer learn l width leave rmse different pressure pixel outperform probably presence area right panel reveal result value estimation inversion predict song audio recently subject much consist evenly country per test average rate music mp music hz process overlapping frame song use window ms ar adjust second song capture stack testing split subset dimensional partition kernel subsection performance linear sparse completeness scheme subsection adjust kernel ls scheme winner carry illustrate influence measure
algorithmic front aware key advantage new base slow characteristic base much quickly make ability adapt status precisely correctly convergence jointly structure load balance algorithmic progress basic behind architecture parallel lasso mf expect additional program subject structure runtime mf demonstrate load scheduling scheme experiment mf yield block approach block select column subscript bold letter bold also strength begin scheduling model
set close whose let www w important act upon within disjoint every partition manner resp order specify order order topological proper disjoint resp exist order construct topological k ordering let dd dd let may consistent note induction trivial close topological vertex lemma extend application c w h apply order consistent possibly order combine arbitrary order everything come everything order order topological add graph order lie degenerate contain edge pointing towards occur close direct path establish
laboratory bp france parametrization hide dedicated loop reweighted performance enable train switch operation signal
follow eigenvector sign full write eigenvector able similarly orthonormal complete span act orthonormal act particular product recover true version suitably perturbation closely exact version merely assumption magnitude let let suppose follow give r exist permutation phase scalar outline perturbation approximately lead construct leave matrix give singular space form orthonormal form basis span virtue construct yy yy phase permutation early ignore computation control omit detail brevity run two overall running claim statement formal broken step fact imply canonical subspace exist orthonormal leave orthonormal orthonormal last need similar hold computation valid discussion great achieve note inequality svd lb lb tb tb tb tb tb k tb closeness condition consequence b te f ty r yy applicable yy f l hence specifically give exist error permutation come column perturbation rhs term term use condition submatrix inequality lastly set thing length factor permutation fact projection show use put letting follow implie rank also k eigenvector claim applicable q resp us permutation linearly ica source fix normalization scale determine isotropic normalization fix
result model work follow input parameter gmm number various internal directional covariance internal need determined involve polynomial call give allow subroutine need subroutine return entire nothing failure normalize embed remove approximation gmm actual allow flexibility need close bound proof correctness brevity appendix first mean sample ideal use reduction appropriately high approximate noisy produce run ideal ica correctness draw ica time reject terminate base perturbation restrict work multilinear define base note subspace linearly independent line unit span span give polynomial span precise use eq q gaussian adjust difference pn therefore union c discussion restrict perturbation would informally say large close go beyond closeness cube point exist subset exponentially norm cube define object
estimator finish proper definition sensitivity nonlinear assume dimensional index respect f influence depend obviously seem output straightforwardly motivation introduce index increase easy collection scalar new generalize information denote output unknown positive integer positive definite denote input hoeffding f kf kl orthogonality denote variance
increase number term number much subsample essentially generate subsample reason run pixel output picks construct neighborhood highlight limitation feature neighborhood edge contain improvement spirit neighborhood neighborhood type important normalize deviation respectively proceed automatic hyperparameter way go especially
entire coarse tracking person general use structure focus come resolution extension image high image camera need video detector create body part adopt simple detector focus limited body body pose convolutional single simultaneously roll et people nearby body location pose near perhaps relevant work taylor al tracking video particle angle collect control laboratory dataset pose limit label datum million knowledge label exploit
c pz j pz calculate boltzmann modelling interaction show need weight vary strategy adopt conventional boltzmann modify transform undirected add likelihood name model benchmark six benchmark ccc name cross fair approach although bottom demonstrate reconstruct density subgraph density bottom density unimodal modal distribution select critical obtain number principle check
give subsequence example mistake th mistake respect method eq simplicity restrict choose number mistake use bound therefore combine inequality eq relax unweighted average separable subsequence margin define convenience since hinge subgradient regularization mistake perceptron extra factor two mistake tend large regularization tend small value
ad disease snps list indicate snps associate nuclear repeat study cancer last gene reveal relationship brain measure structural gene breast cancer anti one snps code gene ad association brain set snps opposite involve formation play formation term brain genetic status bayesian key datum I variation trait predict ordinal meaningful feature ordinal ordinal focus powerful extension cca wide labeling nsf award center edu university edu genetic trait important marker disease diagnosis ii association genetic association
outlier handle robustness support recovery requirement require example degradation robustness completely notion g seek generative obeys adversary provide model set fundamentally robust technique corruption elaborate point entry hope h dimensional corruption know recovery covariate isometry eigenvalue incoherence condition notably basis pursuit lasso solve square
iterate subscript matrix correspond eigenvector one eigenvalue eigenvector block yield vector yield additional equality definition seek root eigenvalue root quadratic
p thus give consequently n expression depend fact insensitive reveal frequentist pearson credible jeffreys seem sort continuity pearson connect frequentist previously argue jeffreys correct version pearson argument jeffreys mid pearson incorrect two another interval uniform essentially mention literature note central lower failure pearson similarly beta interval one success failure remove pearson uniform upper minus success one failure shrinkage pearson side interest distance side pearson figure expansion side interval distance nominal pearson upper simple guess proceed q good
fractional recall definition stress fractional integral pp differential process suit discretization integral dividing interval amplitude integral eq truncation arbitrarily discretized counterpart psd target obtain superposition integral white us band psd cut discretization axis theorem hold uniformly fractional integral directly mind eqs spectral well spectrum prove pm study white form pm pm approximated impulse transfer system eqs
death rate dynamic quantum trajectory leave panel exhibit period either low phase quick number atom active phase state long period phase distinct quick jump phase panel ground atom passive phase realistic detect one average cavity mix jump jump generator replace take account occur jump type atom detect un normalise trajectory satisfie describe evolution primarily behaviour dynamic memory cavity operator preserve compute restrict simulate birth death cavity count cavity state number atom draw birth death determine atom two consecutive jump time explain detail step describe evolution cavity monitoring time cavity jump cavity jump increment indicate jump either occur I atom
uncertain undirected assign existence edge denote probability graph independent though assume uncertain node human consist region uncertain graph linkage edge uncertain world correspond imply denote uncertain possible certain call uncertain dataset iff uncertain conventional subgraph subgraph embedding subgraph feature embed within subgraph iff graph say subgraph uncertain graph contain subgraph subgraph subgraph subgraph subgraph uncertain issue mine uncertain properly subgraph uncertainty compute subgraph score avoid exhaustive enumeration graph enumeration also infeasible fully uncertain introduce uncertain
convenient obtain tractable coincide multiplier vote two corollary unique ball unit take ball cluster ball eq condition solution role make vanish close solving use place ensure ball point close corollary solution coincide choose inequality impose extra upper extra cluster close break barrier impose extra permit strong large point note solution obtain use separate remark
network also select opposed dropout information art performance employ augmentation negligible overhead tune thus convolutional architecture computer institute new york university convolutional conventional deterministic multinomial activity pool hyper combine approach dropout augmentation image relative approach utilize fit prevent decay copy otherwise un regularization stochastically activation significant gain across reason efficacy fully dropout
inner parametrization family variable exponential family smooth differentiable e covariance hessian first r probability subset lebesgue measure make overlap lattice part depend lattice estimator minimize fact estimator partition assume global change lagrange constraint imply reduce change keeping q
stochastic consider identical leaf contain stochastically choose modification unchanged leaf particle independently stochastically average quantification study spread tree appropriately particle offer efficiency gain package implement forecaster leaf leaf suitable generally might study analyse practically especially estimate contour find spurious contour process predictive estimate response particle estimator student fit set large particle estimate mt xt x mt easily expect reduction particle degree etc ei measure guide grow design ei td put consequently optimize finite evaluate ei hypercube allow dense entire sufficient ensure ei setting anneal detail availability response priori provide online overall grid location empirical guarantee tt ignore recursive integrating combine avoid integration termination criterion ei score user specify tolerance pick consideration implement statistical propose grid across varying step incorporate pass
success criterion fail sf drop aic decrease rate bic ps attain rate ps improve datum demand symbol sf score high score ps failure occur mm aic ps sf bic fail completely success success lack datum sf score sf sf tendency simulation sf ps maintain completely setup transition ps score sf well small large aic bic symbol requirement perform situation ps stay rate
new cluster unweighted terminate order use free represent place cluster tune intuitively conceptual conceptual affect instead picking pick choice represent possibly time never represent maximum square distance
word regret technique bag boost stack initially set adapt formally concept distribution stream context treat hold drift without require drift require hence formalize mainly previously hoc distribute mining problem decentralize contextual bandit contextual bandit study single agent sequentially choose address decentralize bandit agent bandit name news theoretical result centralized user sublinear receive multi user bandit learn converge optimal allocation regret allocation select sharing also prove contextual bandit detailed work decentralize contextual important centralized contextual bandit framework difference exploitation standard centralized partition efficiently contextual learner learner essentially phase rate l c non none label residual improve online offline offline bayesian correlation yes horizontal vertical yes sublinear multi yes contextual arrival markovian regret sublinear sublinear system learner time happen sequentially slot
factor ambient dimension succeed high fully ensure subspace approximately pairwise orthogonal noiseless separately outli moreover case exhibit unit distribute outli employ maximum outli make rigorous classify product unlikely formalize unlikely misclassifie outli insight set choosing outlier hold provide misclassifie outli eq condition cf succeed outli outlier rewrite outlier succeed exponential outli rule spherical point near neighbor typically outli detection near appear though detect connectivity property x ik jk neighbor additive outli specifically conceptually outlier present trivially accomplish exploit l n probability misclassifie outli massive ssc algorithm non notable analytical performance spirit ssc finding ssc employ criterion adjacency point criterion make demand ssc surprisingly performance guarantee ssc come actual outperform ssc ssc see performance result factor weak cluster directly additionally establish already require entail range apply mention dependency noisy result
call density discover example particular interest exploratory obtain unsupervised may unsupervise gmm hmm emission gmm em early study thank advantage activity acceleration physical human person algorithm call hmm unsupervised propose hmm model use acceleration acquire sequence model sequence acceleration consist multidimensional joint multidimensional segment model learn acceleration acquire activity viterbi propose real world acceleration sensor place right additional
since nmf lee set effective pattern bioinformatic etc problem pattern recognition organize recognition perform assume ignore method discriminative ability supervise
source stage initialize dnn source experimental nmf channel separation deep separation challenge considerable recent training source quality separation source source deep many far solve single source model use gmm markov hmm hmm data assumption source fix computation nonnegative dictionary flexible limitation relate train source signal separation
theorem theorem loading allow dimension variant density follow cluster skewed skewed mark robust high excellent skew cluster set f mathematical tractable prove limitation difficulty skew accordingly lin lin lee suit large component alone model
sec run dependent among optimize individually find high optimize quantum superposition superposition state crp represent superposition formulate first sa regularizer among practical problem variational summarize crp crp mixture formulation crp key crp indicator vector state datum share crp indicate crp representation idea derive crp mathematically appear interaction derive number moreover processing represent
threshold gaussian give sparse corruption successful ignore small likely provide reflect theorem additive believe square binary signal corruption estimate minimize sparse consider measurement vary perform experiment signal vector vector normal constrain success complexity corruption recovery reflect transition curve ignore small constant gaussian estimate corruption theory vector partition index experimental zero constrain problem display derivation see theory accurately phase theoretical previous corruption optimization leverage curve ignore additive technique plus corruption corruption benefit leverage corruption recovery theorem display signal constrain penalize recovery noiseless type practice exactly practical recovery guarantee successful sample size correspondence penalize optimization unknown recovery yield penalize lie lead behavior knowledge corruption suggest simple strategy pick lead signal analogous
coordinate store thin iv r observe versus predictive percent regression know vector predictor outcome residual zero center multivariate model use mean cross assume endow obtain stack stack correspondingly hierarchical arise specification nm low involve predictive modify rank counterpart fitting setting use logit probit count regression linear assumption suitable refer spatial glm stage mcmc stage marginalization walk random walk metropolis low glm analogue poisson counterpart accommodate spatio temporal datum extensive statistical adopt apply series building include dynamic framework residual framework extend multivariate gain recent economic spatio offer dynamic location specification spatially uncorrelated introduce temporal vary regression parameter spatio generate capture transition
follow algorithm soon r sample remain level inf com application automatic diagnosis discrete function adaptively general read experiment next read design strategy accord build strategy decision attain logarithmic simultaneously bad spend motivate trade automatic perform market stock price volume volatility book among trade discrete whenever new scenario take scenario market take account scenario every action identify proceed associate might variable consume speed market classical call label task
graphical tuning setting challenge overview proposal leave perform roughly speak supervise g reasonable component sensitive corollary repeat contain draw randomly element j ik similarity base square identify select small value component underlie author lasso solution tend estimate zero established correctly assumption minimum entry identify component graph ii identify edge inverse connect specifically identify selection establish estimate highlight connect determine lead consistent connect component underlie network
comparable baseline briefly improve subspace method real transpose subgradient letter x x represent element wise lastly denote index introduce bring impose predefine satisfy programming solve bring transform introduce seek detail present basically add active bad master step detect iteration atom large record update optimization solve proximal pg resp conjugate adopt inner distinguish outer initialize stop large record index x c break matching relate stagewise omp stagewise add strategy atom thresholding however
denoise tucker root dependency empirically figure numerical precisely predict square unlike tucker easy scale latent therefore theoretically empirically latent approach analyze structure yet paper serve point basic strategy problem kk next write lagrangian k n equality diag p diag singular scale equality maximize obtain q
correlation resolve shape pattern range local universe galaxy elliptical probe deep universe understand galaxy formation distinguish galaxy need technique use association correlation galaxy property large database study cluster observation local universe deep south classify object band publicly include galaxy galaxy community association database generally correlation relationship outline pearson correlation correlation set construct pairwise confirm correlation suggest set nonlinear detect nonlinear database pearson generally detect nonlinear association introduce pearson correlation associations pearson coefficient coefficient
gamma pareto common shape ex c cc c ex setting axiom present testing population project monitoring build lead reduce chi nan limit distribution local alternative interest simulation misspecification phrase likelihood pool monitoring testing method create part program aim develop monitoring change interest affect way grow mix add come increase material forest vast include goal cost time condition propose exploit resource population year region ideally accordance american protocol suppose population single variable k use exponential normal distribution basis function x control empirical platform year study context quantile sample investigate test nan
purpose gamma component threshold eliminate et al propose kernel bandwidth extreme example extensive dirichlet flexible theoretically coherent year tool bayesian mixture density tail
variable third parameter skewness generalize x carry parameterization identifiability therefore parameterization parametrization distribution generalize scale quadratic free dimensional resolve uncorrelated latent great variability analysis I independently error diagonal analogous fashion factor arrive membership otherwise conditional observe together
spline mixture regard discrimination improve diagnosis system electrical engineering speech rather statistical functional concern paradigm entire finite goal visualization exploratory approach classification additional achieve unsupervised etc learn dimensional space discrimination temporal curve present focus help generate generative essentially include spline spline regression non parametric discrimination generative aim problem change model curve dedicate present extend discriminant present relate
state informative state extra variable increase solution dimensionality code modeling reinforcement however efficient investigate work like electrical engineering university agent end keep ball attempt use learn action meaningful variable macro describe strong adversary ability begin end ball determine action simulator increase asynchronous task combine fact line example see besides organize task onto reinforcement result
motivate nonlinear get magnitude nonlinearity sensitive practice smoothing flexible basis piece one briefly interaction beneficial address pair effective suitable order class distinguish force safe could class grow force computationally generalize statistically issue number guide pick subset generalize useful selection eigenvector class numerator discriminative feature extract top pick discriminative discard eigenvalue lead outline prove sufficiently herein merely use multiclass benefit raw eigenvector network architecture
denote restriction around window every demonstrate robustness mean incorporate idea fix consider use pixel formula wise generic space need patches pixel robust outlier patch improvement denoise intuitive understanding denoise
sign sign majority sign predict row small vector find magnitude neighbor recursively theorem claim ex proof height depth edu microsoft investigate matrix randomness simplification deep also unit assumption unit relation factorization factorization simplification non yx entry deep compression generalization network apply operation usually perceptron edge deep express circuit deep network correspond compute sx major machine deep include speech supervise learn train back propagation unsupervise pre variant
derive entropy change newly entropy effect consideration act function factor reward algorithm highly name entropy discretization accuracy improve also offer select reason design work supervise five microarray datum test validate gene goal distinction gene difference cancer might clinical behavior breast gene provide nature many genetic experimental test run execute subset reach function relevance overcome solution low mi less redundancy subset complete obtain mi preliminary ht cancer cancer breast indicate minute evolution temperature unstable soon
ii encoder train aggregate encoding patch recent year close example algorithm patch example pursuit algorithm click array c click dataset click click equal
regressor unconstraine model q prove formula multivariate identify lemma matrix non euclidean sum formulae correspondence formulae prove ii complete em hide already specify updating marginalization straightforwardly parameter noise add gmm weight nk k nk affine case option previous expectation algorithm replace algorithm cm provided connect indeed one notice observe nd key column eigenvector hybrid dimensionality reduction variant local dimensionality residual easier initialize posterior gmm step consume decomposition marginal image approximate di role regressor mixture mapping start parameter inverse infer forward interest partially response contaminate formulation augmentation devise procedure augmentation augmentation scheme detail inference view generalization reduction framework validate experimental method several
accordingly exist near neighbor local I global component relationship result vertex neighbor define whose diagonal represent degree vertex normalize limit size symmetric carry formulation fidelity enable kind setup semi characterize fidelity associate vertex semi understand norm fidelity non trivial consequence final metric functional term vertice fidelity term interface fidelity lead steady equation assign vertex obtain homogeneous region
sample responsible rejection tumor ht l rejected reject ex rd within correct reject work population covariance suit restriction difficult first amount consider vs set perform guess allow difficult impossible nan hypothesis toy I consequently imply rule equality regression test wide comparison graphical adopting avoid burden correction support joint resort elegant nest power strategy finally numerical approach sampling multi soon spirit involve consider rate regression opinion complementary derive sharp rate strong assumption kullback analysis test depend dependency rely likelihood invert inversion interpretation dataset identifying subset responsible network gaussian graphical high correlation encounter within miss unstable graphical estimation provide validate graph share comparable structural statistical obviously interestingly gene point validated promising target biology facilitate validation multiple pool analysis draw heterogeneity detect sample intend I dataset consider homogeneous hypothesis nan hypothesis degree freedom method admit notation resp span resp projection span consider resp eq hypothesis simplicity degree nan assertion covariance apply union integrating allow derive union exhibit nan
correspond norm inner basic idea work expectation another gx highlight version order involve ingredient context operational connection property
function single may equation eq difference work space independent approximate project equation admit equation write estimate expression trace give estimate similarity procedure omit td measure give estimating process constraint moment write operator explicit require negative state write demand matrix constrain constraint admit feasible state
infinite number exhibit uniform focus multi logistic satisfy cm ridge regularize square response individual operator value kernel rr link follow hypothesis hypothesis verify hypothesis c replace property respect line line uniform q hence equation stable map loss spirit insensitive fx fx p norm
specification constitute counterpart specification equivalent hand nevertheless consideration extreme reason two construction specification vector require paragraph prior individual independent consideration assume may improper posterior prior coherent condition note concern probability entirely general intermediate definition suitably group via fact j derive q correspond zero appearing collect correspond kx yield relationship hyperparameter density subsection attention family
lem lem lem lem lem present mkl kernel kernel ts target notion kernel good specifically domain kernel well kernel ik I notion first henceforth extension task
coordinate ascent pass branch mrf implement three try put effort appendix discuss appear factor virtual necessary sure process pass note intersection add general message pass operation implement certain cite require implement energy efficient incremental exploit message change energy energy pairwise energy take second prior unary scale curvature model unary protein relaxation energy add zero triplet generate implement triplet k protein protein try energy three
interested give localize use huber huber cf huber recently upper indeed huber hold look hope condition eventually first adaptive remain open core work adaptive highlight presence rate improve state localization see old deal characteristic adaptive rule risk upper bind pay theorem introduction clustering lead propose suffer challenge open investigate realistic precise calibration practical presence convergence old regularity isotropic deconvolution optimal difficult dimensional purpose challenge interest development bl test simulate calculation estimator noisy contribution heuristic
equation stationary signal connect equation polynomial vanish result assumption ambient intersection case epoch tend e reasonably omit scheme characterization epoch require uniquely identifiability uniquely identifiable result solve uniquely sharp covariance variable
rather challenging proof derive replica approach claim indeed np pa pa sequence distributional distributional obtain set replica class worth convergence satisfied prove hold block long replica method rigorous highly sophisticated probabilistic replica rigorous last year replica claim focus randomness cover theorem ten year replica result communication analysis standard rigorous group reason establish distributional require fact deviation converge distributional assume design supervise datum access vector accurately refer additional proceed equivalently let hypothesis attract several structural name focus sparse detailed description theoretical unlikely hence emphasize covariance procedure bound numerical return next appear would rigorous account covariance pt c c denote set experiment setup independently element symmetry choose conservative design result significantly error return ridge realization I average gaussian design table performance table plot histogram red white denote restriction active inactive plot exhibit asymptotically normal define width alpha pdf ridge design type I avg avg mean std c low ridge na ridge setup deviation realization compare related work order perspective method subsection discuss
spectral interpretation term rank base locally primitive interest notation discuss algorithm help certain semi supervise eigenvector section toy illustrate realistic analysis brief set vertex v g gd form imply eigenvalue small eigenvector let uniquely two iy technical statement semi eigenvector compute lead equivalently nontrivial eigenvector equivalently duality nontrivial global partitioning involve cut next solution augment constraint dimensional zero orthogonal matrix vector achieve optimal nontrivial usual toward seed locality constraint orthogonality constraint iteratively three interpret locally bias locality analysis require light discussion clear quadratic thus objective addition cut variant eigenvector perform graph formulate vector primitive
gauss construct ordinary onto gauss selector dimensional support n ny follow find modulus correct e condition weak design sparsity generalize convenient sign kkt least slack assumption ask early noise hold support argue literature broad never support summarize kkt justify broad specific show recovery formulate minimum coefficient condition side remark rest illustrate range discuss finally notation treat design random design control randomness result technical defer new cover discussion find
non set simulation numerical smc mcmc filter tractable design interest advance elaborate study propose assess quality result conventional er rao bind theoretical bind likelihood ml unbiased analogous commonly inequality provide next bound tp z
ba independent estimator course however affect cv split disjoint instance cv cv use q index imply value I source variance less sample single potentially select cv without sometimes claim attribute rather bias confusion seem cv bias unbiased prominent mean interestingly cv bad I bias cv
sub lemma assumption satisfy riemannian manifold approximate error notice bound functional calculus hard ff integral gaussian p last get q c concentration pp measure assumption consider least calculus sampling object f follow identity identity enough constant small compare z term put c empirical performance setting concentrate type use integral subsection selection throughout subsection describe procedure subsection expect compare kernel simulate set typically choose semi validate adequate proxy performance describe validate test measure measure various procedure however set linear poorly suited approximate well behave readily available follow function try coordinate thresholding experience coordinate rich adequate
theorem imply contain q imply small discard replace notation applicable discard coefficient form solution section lagrangian multipli pair via hold duality multipli give close straightforward general project onto orthogonal complement space immediately use multipli transform note strong problem admit multipli solve recover primal kkt derive case attain maximum p q notice although p discussion omit ready
regression thin gene thus lead learn study capacity efficiency efficient tree cycle size graph belief propagation liu capacity many tree thin efficiently algorithm np maximum modeling application network localization degree beyond thin family widely lasso regularization g grid efficient sparse guarantee study feedback vertex node author demonstrate accomplish use algorithm show full excellent main focus likelihood include two provide degree complexity
tuple triple relation close tuple intersection tuple arbitrary allow case close unary simply note triple closure rule finite relation would would example unary eq dual similarly global undirected global closure rule markov closure regard rule proceed provide specifie closure intersection rule reverse close decomposable clear direction direction condition show since ab b bs ab bc bc v union respectively bc reverse complete proof close intersection next complete pairwise let close next complete proof undirected sequel allow application
would initially fourier contain parameterize instead contain parameterization make make dependent word parameterization lie sampling process information manifold dependency coordinate key opposed sampling contribution compress sense generalization determination completion derive combinatorial sampling coherence achieve reconstruction irreducible projection onto coordinate least take hausdorff density statement rate recover
corollary claim statistical sciences institute technology direct acyclic observational base hybrid statistical restrictive skeleton base hybrid unclear whether weak assumption permutation causal order assumption also small dag compare base pc equivalence hybrid min hill set prove find permutation variable connection sp fundamental determine causal directional relationship system involve infer directional amongst useful simplification causal acyclic dag definition relate dag direct associate probability vertex node path connect triple triple form furthermore undirecte every non connect give ss v ci ci dag vertex satisfies respect dag x dag relation consist dag equivalence uniquely determine skeleton underlie causal degenerate throughout infer broadly classified approach score involve undirected skeleton identify skeleton infer complete
problem recognition literature major type approach deal signal apply play distance category classification mostly classify advance increase availability demand early precede occur air quality chemical occur numerous international event never health risk operate risk associate exposure besides issue deal large motivation investigation trivial time recognition however classification optimize propose base option early classifier decision incoming portion available next available assign confident pass continue manner manner worth compare
n r take form rx xx krige linearity unnecessary gps instance arbitrary jointly zero gps specify determine comprise instance gaussian mmse expectation condition compare one mmse coincide rkh decomposition establish integrable admit possibly kx ix kronecker use eigenfunction product two point kx xx form basis trick expansion matrix valuable provide ridge regression start ridge z give nd kx term product instrumental version entail eigenfunction solely term crucial importantly trick demonstrate context trick machine optimal minimize insensitive trick cf classification function integrable function e constitute span signal processing
method ignore large necessarily straightforward independent though inherently large use suboptimal novel nonnegative weight well sample example efficiently computable introduce statistical theory mainly method independent example network conclude summary contribution list notation minimizer global minimizer minimizer risk w square r square hypergraph vertex partition partition independence cover weight non
performance give partition copy distortion existence minimizer minimizer way thank distortion uniform number investigate seminal result regularity basic iterative seminal publish initialize repeat cluster adjust assign final cluster approximately appear spherical however limitation previous distortion prove iteration newton optimization precisely center visit algorithm natural distortion local principal appear practically
factor model exhibit albeit period give auto distance check presence period secondary one protein discussion robustness degree formula give amplitude stability period robustness say happen amplitude period plot degree quantity versus varied right application system tackle follow system satisfied tackle optimisation seek robustness encode control intensity evaluate checking optimisation gaussian upper optimisation varied fix ucb optimisation robustness checking point uniformly space ucb
first appendix k k suppose eq depend exist feasible assumption fx fx k min fx k q induction trivially inductive imply inductive step os fx f min k x fx note rest tool result cf probability finite algebra let converge bound yield satisfy sequence
horizon decision avoid policy iteration category reinforcement idea sub ascent use policy category inverse reinforcement take construct reward logical environment randomness instead policy evaluate handling add act environment unknown alternate opponent unlike either analytically model main technical policy programming simplification place function rather restrict jointly rather
attempt recovery mild perfect exceed bound complete rank multi fold toeplitz minimal resolution completion matrix toeplitz resolution class application superposition spike frequency acceleration medical imaging localization inverse imaging channel communication analog digital acquisition device hardware physical desire resolution reduce aim sample interest ambient distinct spectral extract signal collection domain harmonic matrix innovation etc exploit harmonic lie irrespective segment technique prior order frequency spike besides heavily noise sensitive noise sense cs recover ambient dimension provide enjoy surrogate popular noise furthermore nevertheless interest dictionary discretize conventional develop simultaneously invariance harmonic start enhance rank rank impose determined enhance partially small proportion magnitude solve minimization incoherence depend regardless respective coefficient incoherence characterize reciprocal gram around interest incoherence gram arise broad include restrict demonstrate incoherence condition recovery noise admit signal sample sample corrupt magnitude applicability super rank toeplitz matrix
softmax two divergence gold train plausible derivative tensor contraction matlab toolbox corpus generating product object positively label subset intuition average contextual extract interpretable plausibility object object noun pair cosine intuitively average cosine determine well intuitive corpus experimentally hence competitive cutoff break receiver characteristic roc testing example cutoff validation instance require
k est e le py une de le les pr des et par partition de le em pour la ne em pour em dans un de ik observation ik ik ik kt de le q es
full motivated take scalar amount ignore association eq I scalar singleton set predictive combine belief plausibility evaluate singleton assertion trivially easy plausibility classical observation interest I conditional define minimal statistic involve effectively hope auxiliary second property imply retain look lie exactly argue obtain marginal obtain curve dimensional reduction auxiliary auxiliary simplified section formalize normal interest set association datum auxiliary original auxiliary variable rewrite equivalently mind give remark normal leading direct general need assume regularity hold special regular clear eventually regular propose base regard like mean auxiliary thereby increase I combine eq assertion plausibility particular construct plausibility mention positive adjustment prefer free association reference minimal statistic characterize characterize
must span eliminate address successively amount trial define sec eventually bind span finally outer round keep execute guess span nonetheless compute bind span episode line differ average reward current episode whenever met episode discard minimization preliminary proof main begin prove average reward throughout average action suggest sample episode next step episode induce recurrent reward k sp h martingale sequence error side exactly trial use guess need discard high discard start trial corresponding sum
quadrature implement alternative monte method approximate component implementation variational trait manner easy likelihood trait analysis govern sigmoid involve variational allow approximate log likelihood nm x nm b nm nm outline algorithm trait approach equal likelihood advantageous likelihood use gauss quadrature advantage variational integral firstly quadrature carlo variational iterate form easy secondly converge considerably gauss quadrature carlo particularly large dimensionality variational estimate likelihood always true trait gauss drawback em converge instead function estimate gauss quadrature quadrature approach consist binary fully bring absolute always compare monte carlo trait analysis assume categorical necessarily conditionally membership group model trait come group come ie mixture trait form eq parameter addition assume
note horizon trick together result establish rate bandit discuss careful almost key idea store box queue tb b maximal efficiently maintain box indice box tb change ensure quantity update new preliminary memory box store point x b box store tc quantity point event newly store past run almost e uniformly finally second establish computational armed bandit continuity function elliptical maxima must continuity satisfied part large neighbourhood diameter deduce satisfy diameter deduce contain conclude f maxima p compact maxima continuous l l continuity condition finitely maxima let neighbourhood x diameter u eq finitely maxima uniformly lemma continuous establish grid condition partition compose dyadic satisfy j enough maxima l contain compact x apart also l c depend x grid box grid product dyadic length grid approximately place grid define choose box ib cover let compose box q grid cover
initialize initialization th eight base db perform search snr probability mid channel regime base centralized key performance mid db trade increase requirement synchronization overhead simulation significantly table average number iteration integer db observe
wrong failure voting label close contribute vote correct robustness majority voting make favorable prediction accurately place source achieve source correspond spherical gaussian model algorithm theoretical close source g step observe achieve least additive contrast depend separation even sublinear regime mt series detail wang model degeneracy substantially component massive expensive one instead subsample feed source still source source apply half source time test majority voting source figure outperform two performance map rate vary amount classifier also increase voting envelope roc curve tradeoff
computation topic scalable dataset semi hash underlie class whole separate discover split well discover handwritten digit digits agglomerative similarity label learn degree similarity comparison maximally able digit agglomerative
far finer grain turn speed conclude version give offer particular contexts network traditional except parent configuration contextual belief exclusive proceed beneficial discuss worth approach investigate numerous author existence accurately set expressive aspect interpretability facilitate undirected regularity local combination induce phenomenon arise context two label overlap label combination induce configuration induce type call exist configuration add induce additional add maximal must restriction form associate dag dependence represent dependence dag thereby say structure hold implicitly represent induce contradict condition implicitly imply induce contradict conclusion obtain reflect represent obtain configuration since without intuitive reach conclusion rule arise exclusive rule satisfy add label encode arise overlap achieve minimal mutually exclusive prove essential condition fail
accelerate nesterov expect randomized accelerate complexity block reduce keep common cyclic cyclic global cyclic randomized strategy smooth derive method minimize box propose minimize nesterov analyze separable complexity aspect composite smooth improve zhang randomize proximal ascent convex complexity pair solution
small refer sensitivity suffice within geometrically norm remove value ensure sensitivity small inverse sensitivity express directly program vector note contribution program program sensitivity dual prove dual translate show plane disk diameter maximum unit final result view kind proved tool informally al problem robust certain accord observation
queue message domain value total v f pass college university however facilitate provide accurate stop improve long converge propose message algorithm marginal scheduling select next providing obtain grid model processing belief propagation
u v described section window svm method window flow measure measure consist window svm detector help principal algorithm art anomaly input cluster use flow statistical g f ij control flow represent cluster every define initially empty flow ellipsoid contain euclidean cluster center create flow assign component center cluster adaptive update assignment flow become center flow flow c
bound function modify deterministic constructing parameterize x construct parameterized parameterized parameterized restrict appropriately choose deterministic monotonicity way use weight eliminate involve derivation extremely powerful I lr important obtain process method probabilistic fisher situation pearson obtain relatively counterpart establish lr wide distribution lr concentration family approach tight inequality inverse lr establish lr explore connection establish derive particular concentration inequality derive moment lr inequalities binomial lr concentration multivariate dirichlet gamma use denote generalize eq denote transpose matrix use pmf density probability mean notation proceed ratio derive probabilistic lr algebra true pmf pmf subscript desirable lr base result
analytically require encourage posterior variance estimator marginal minibatch find number minibatch stochastic approach compute connection auto look give kl divergence approximate act expect vector p equal auto minibatch draw dataset random distribution minibatch solve continuous conditional deterministic vector parameterize useful rewrite expectation z construct f differentiable valid q transformation auxiliary tractable
sampler x n summary quantile computed overview dissimilarity point
source special interest e case extensively replace source let score v v v unit q second gaussian dependence e versus elliptical distribution term shape possess score include case performance section dimension source shape pick experiment entry source standard theoretical achieve shape source total q sample local minima objective minima minima successfully permutation ambiguity trial trial across former source indicate permutation ambiguity trial
rare totally conventional upon single historical user transfer additional user source usually service join nearby meanwhile facebook friend involve twitter news account user account account facebook twitter account among align example two respectively account user reality account new social provide information user source crucial link little activity activity network network network align user account user information activity addition target create social anchor link account source exploit help improve link social totally problem pseudo start prediction feature detailed c heterogeneous heterogeneous heterogeneous yes user incomplete sampling handle pseudo start knowledge link attribute
theory characterize limit application get semi law researcher subset past moreover fair amount block dependent give e signal appendix show haar course tackle care appear hence carry computation instead organized block fair amount allow situation pure independent section go work result use sign denote equality matrix develop generally situation block infinity pattern generalize involved dependence call identity hold master suppose hermitian constant theorem transform approximation conceptual kk course martingale call involve hence appendix martingale theorem play hermitian random hermitian word theorem conclusion tailor hermitian theorem get simple consequence corollary follow hermitian compose hence sense correspond th transpose immediate need structure corollary kernel hermitian write block rise
panel combination supplementary programming find p computational complexity assumption remark separable knowledge rp impractical algorithm dirichlet solve impractical gibbs convergence general non propose rp term complexity list method time well recover decrease scale depend make comparison complexity appear rare minimum material separable throughout direct summarize proposition section run datum projection proof co occurrence add co occurrence running achieve vocabulary hash take document word summation matrix moreover word clearly ex recall b detail proposition rp operation projection index hash along component value rp winner bin rp bins rp ex proposition section please assumption section pdf definite eigenvalue compactly follow directly lemma show strictly non positive value proof result convergence numerator numerator denominator due convergence simplicity hoeffding f j g h obtain converge denote index row low j prove support analysis lead recall algorithm novel I ci j concentrate less vanish union vanish part less word vanish part rate j c ai k k lb sum ib j strictly zero eigenvalue
specification criterion distance power decrease reasonable model column performance scenario consideration need sensitivity modelling estimation medium volatility task present analysis five exchange rate us exchange daily exchange rate http year pre forecast adopt rw wishart refer follow specification correlation word specification combine varying model smoothed standardized residual element observe return specification element variance equation volatility carry information via specification specification square observe latent inference perform aim bayesian task compare
show simply perform q code calculation fig j j
intermediate namely employ transition intermediate unnormalized simply scale annealing report look see ensure transition computational likelihood pm approach posterior propose reduction ensure pm base scheme integrate latent construct anneal intermediate q k employ operator useful unnormalized intermediate model compare report employ pm apply laplace impose gamma
practice eq create appropriate scheme likelihood find previous previous iterate utilize combination geometric optimization c initialize dictionary separable summarize dictionary conjunction I negative white show achieve image different peak ratio ground truth quality structural similarity originally suggest zero reflect visual quality universal separable image content square accordance mostly literature dictionary separable dimension
design mcmc posterior study performance efficacy indicator determine effect encourage simulation asymmetric result perform base generating mechanism theoretically consistent finally alternative regression quantile level address choose asymmetric convenience joint algorithm integrate follow variable one sample j k conditional integrate j
time nystr om nystr om tune parameter algorithms nystr om nystr sec k margin nystr om run main lrr experiment group second take second ssc grouping offline assume lrr fail result case produce result sample chen nystr om database compare experiment investigate select remain sample accuracy time nystr om nystr om sec mean nystr nystr om sec nystr om nystr example subject database high nystr om experiment four possible reason attribute difference choice balance memory method number sample popular effectiveness complexity offline compute especially paper address successfully ssc lrr
change track diagnostic automatic group switch series electrical consumption acquire specificity analyze context involve accomplished parameter mixture hide model previously figure compose identify expert cluster minor series set accordance phase operation polynomial regime approach rate number show misclassification
support grant fa google research award thank le song proof follow state sampling space fp draw uniformly multiply side dimension point db dp dp dp db dp db similarly dp db sufficiently b datum topology column surprising use since local sample combine partition cost contribution cost outperform mean communication ratio grid result nearly combine similarity partition combine partition also merely topology span
two location way task nearby match similarity image position describe content around across common although require loop local patch amenable know phase energy correlation model quadrature shift view encode quadrature within fouri model correlation propose independently motion surprising consider motion transformation view frame encode motion similarity motion encoding date present view view spatio present hide
given accuracy lipschitz constant cross optimal histogram bin width potential risk validation input block optimal consistent estimate metric mae n detail supplementary key step present intuition goal follow bind
large integer eq consider besides minimization method fail recover fail develop zhang begin another rip roc small non sparse support base eq imply proof proposition zhang non immediately hand cauchy schwarz section compress sense minimization noiseless sharp
derivative hide unit basically remove row example hessian increase make training slow expect sparsity gradient sparsity become shared conditioning might location activation location activation bar chart show activation activation intermediate effect apparent figure intermediate unit hide activation nn unit sigmoid training decay sigmoid activation minima usually yield well compare architecture pre denoise auto encoder proceed unsupervise learner representation shape level unit nn pre use unsupervise relevant patch per patch layer encoder provide binary rbms unsupervise sigmoid nonlinearity hide weight auto training train auto unsupervise experiment tie weight combination none configuration test table complexity input design symbolic patch empty represent representation like feed mlp another one input experiment trial ideal nn nn job input patch represent rotation whole perfectly unsupervised job ten patch regardless product shape bit spread nevertheless easier read representation bit per patch differently object patch iff exactly transformation learn perfectly train experiment maxout linearity
quadratic outlier addition neither solution select impulse response great capability feature penalty motivate adopt penalty functional example elastic popular measure outlier norm insensitive cast unified model ip extend system identification propose impulse response stable spline arbitrary piecewise generalize work also generalization impulse response estimate ip output measurement procedure test
binary combine architecture loss binary bi precision complementary architecture long relative might contextual interpretation instance unless joint opinion table combine binary f measure opinion binary compare detect target proportional binary explanation information name entity fail link entity opinion therefore possible look name entity sentence opinion attempt token opinion opinion distance increase combine steady combine compare bi help well opinion
outcome probable outcome poor high case performance classifier influence metric simulator prediction consist data test set avoid bias low testing metric vs rs low condition observation low portion well calculate perform come successful e predict average find match find good prediction therefore preferable relevance outcome critical
rbm value dataset average log hold agnostic way order may ensemble clutter magnitude error due finite set standard error enough ranking expect partition baseline bottom half log bold configuration dataset train minibatch stochastic last purpose consistency unit rate fast momentum stop training likelihood validation iteration hide hyperparameter recursive start agnostic binary uci detail iteration consist weight validate
order number far finally select behavior step normal validate machine role every question situation result reasonably great seem svm case tend subset parameter optimize think room accuracy datum concern technique time set mean validation test observation create decide upon compete one hand subset interpretable clinical illustrative report report radial svms radial filter dataset datum
create estimate quantity space along reaction first sample distribution update version explore criterion di flat histogram threshold
surely thank let cdf closure let denote random follow compute influence minimal regular whose observation lie op theorem p eq influence efficient central achieve efficient estimator p variance asymptotically efficient estimating product deduce efficiency differentiable theorem delta follow proof deduce get ns let eq theorem justify boundedness eq ensure eq integrable u conclude delta proof proposition theorem yield conclusion delta
circle marker approach dot marker wiener filter green dotted marker clutter leave step red dotted triangle sense subsampling examine panel wiener approach perform kind sense former suggest incorporate sense rely investigation criterion note comparison maximize mutual observation specific criterion mutual specifically indeed discuss relate task clutter propose projection obtaining
adaptive various uncertainty topology failure arrival time agent turn uncertainty simultaneously mean size part namely ensure steady I denote minimizer replace part ii examine asynchronous expression relative agent reach agreement steady establish manner adapt question address asynchronous asynchronous centralized asynchronous surprisingly asynchronous uncertainty square rate centralize implementation steady suffer degradation network match centralize summarize various implementation result show centralized implementation asynchronous operation asynchronous asynchronous centralized implementation distribute centralized remark part study exist examine strategy asynchronous topology albeit decay explain asynchronous part cover broader include size occurrence source study address question pose argue asynchronous implementation occur solution aggregate
sufficient consistency random independent hierarchical hand combine label classical context closeness two reflect propose use maximal affine find split two et sufficient label consistency mdp case condition case g sufficient variance cluster next consider mean sufficiently discover consequently mdp cluster possibility
stepsize work training perform digits training experiment gaussian form auxiliary form layer transform converge interestingly upon parameter dependency ip generate I forward generate mnist experiment map magnitude auxiliary auxiliary form integrate sample efficiency especially variable applicable easy
weight first predict expert achieve splitting segment turn fix share share apply fix state arc fix use describe aggregate share mixture mixture em pt black f expert make expert history expert choose expert expert entropy take substituting complete clean way necessarily calculate switch time switching entropy keep binary asymptotic achieve guess switching observe address section share share mixture occur switch bayesian unchanged switch expert interpret mixture give interpolation follow natural switch bernoulli hmms switch really fashion lift predict useful tool build modular fashion interpret expert good bayesian process start share interpolation bayesian mixture become useful example coefficient state process possibility evolution behaviour step interpolation differ regular layer produce define input state I I determine dynamic use arbitrarily start switch dynamic state c rr rr interpolation separate concern switch behaviour reflect modular interpolation drop q mx correspondence expert fix interpolation definition sequence one
topic provable guarantee condition among separability condition separability separable unique recovery separability guarantee uniqueness recovery ccccc guarantee decomposition decomposition
identical amplitude decay base typical likelihood trajectory mat ern similarity across frequency generated series tendency cause energy ern generate mid indistinguishable h brownian motion top display real complex complex generate mat trajectory identically model process form aggregate statistical spectrum parameter correspond frequency amplitude smoothness provide useful summary structure physical process consider various five sufficient determine frequency shift occur mat ern determine parameter consist background remove complex variant test overall stochastic capture background parsimonious summary rich key demonstrate world outline series parametric frequency misspecification sample model lead ratio variant stochastic precede oppose reason modulus nn frequency exclude lose
p persistent least population identifiable prove deterministic see identifiable moment order moment moment size e order keep fix bound grow need flexibility ensure gram bipartite identifiability grow term difficulty hide theorem address identifiability bag comparable result exact use dictionary assume rank coefficient degree constraint degree identifiability identifiability overcomplete argue moment persistent moment relate identifiability uniqueness class enable discuss persistent persistent model bag persistent topic moment characterization gram kronecker copies rao prove characterize observe persistent model moment persistent moment characterization characterize defer integer q equation diagonal hide moment product gram moment moment moreover dense matrix persistence topic persistence model bag vary b satisfied note bag persistent comparing see structured kronecker product topic rao products overcomplete identifiable let moment rao dimensionality rao identify overcomplete overcomplete representation become determined expansion interesting rao product product operation dimensionality high overcomplete model example provide figure differently map word l l perfect l l l matching highlight simplicity connect show persistent reduce equation persistent desirable rao key establish model matrix equivalent tensor moment tensor allow compare topic tucker na simply write fix g stack operation tensor outer operator rank tensor cp tensor definition
network social propose family cycle imbalance order cycle cycle consideration suggest broad perspective social adjacency balanced network model discuss completion factorization via favorable via approach exist global viewpoint social possible explore heterogeneous sign entity heterogeneous question relationship exist network measure sign network work consider sign status balance natural ask theory acknowledge nf contribution occur remark conjecture axiom gray study social network research exist deal balance sign network certain characteristic exploit balance fundamental sign sign cluster social sign method measure imbalance supervise cycle sign triangle cycle relatively social imbalance balance theoretic sign modeling provide theoretical guarantee relaxation extensive experimental comparison adopt viewpoint sign highlight aspect sign multiple biology science economic mathematic form root major force online network internet web natural online increase science traditionally node entity entity respect trust representation fail encode network two opposite kind online review user either like review model sign great development theory algorithm network theory notion network break weight negative applicable applicable appear social theory balance relationship network
confidence possible correspond operator measurable eq eq transformation fourier define transformation derivative embed next let satisfying detail explain application problem ask product tolerance know success obvious estimator tolerance importantly reveal relation success error say tolerance relaxed success improve scheme theorem basically thing tolerance small scheme estimator tolerance thus indicate learn heavily bind thus generalization estimate almost short logarithmic factor regularization answer advantage measure phenomenon utilize accuracy change dramatically drop constant exponentially transition scheme conduct confidence without
set riemannian sr describe atom art task residual classification eqn conjunction dictionary three k dictionary first synthetic sparse euclidean sr data experiment half rest riemannian tangent manifold process recognition matlab execute ghz intel cpu term recognition sr generate datum space hence focus sr low high sr recognition sec approach represent hard b subset image image show
function manifold use riemannian manifold bregman divergence equivalence descent mirror mirror descent riemannian bregman apply estimation family er rao manifold correspond implement mirror online differentiable convex online iterate construct common update step size gradient descent euclidean family gradient ambient generalization gradient assume interest
vector k p result sampler instead develop search selection value give augment fix z h jj update appropriate mapping equal hard feature outline approximate likelihood marginal likelihood calibrate different clustering simplifying retain marginal within hard index eq q likelihood clustering move either split move merge decrease conclusion probability approach first
minimize cut impact address class submodular minimize via variant submodular flow express high prior expressive high order perform adopt formulate term cut art efficiently solve submodular comparison make interactive technique tune hundred value produce naturally two graph cut computer include interactive texture underlie behind cut submodular max however prior pair prior image field expert cut high prior efficiently expressive involve large parameter address question high prior
q distinguished cluster distinguish lemma result simulation salient sphere mixture mix sample density figure simulate low probability recover confirm size ambient complexity show effect success rapidly increase success grow full cluster prove claim bind straightforward approximation omit notice cover proof constant prove claim convergence point lemma family ball center fix nb union bind convergence subsample uniformly claim inequality put together net least n use follow see volume spherical sphere radius embed hold section prove easily check equal far incomplete beta
ii ti ti ji ji ti w additional projection step tw w tw analysis hold ellipsoid define step observe x xt weight vector bound must always contain project norm thus w g w tw ji ti ti ti ti x ti x ti tw ji ji ti result prove x ti g bound ji observe high percentile percentile sequence exchangeable value feature union observe percentile sequence exchangeable might improve dependency ratio percentile reason guarantee expectation percentile small fraction step would percentile imply later depend percentile might tw w tell w tw ii tw ti bound step tw ta tw w w thus tw w g tw projection guarantee maximum ti tw ji combine projection g fix gradient norm feature adapt x ji choice ti
covariate class parametrize might want penalize sparsity different discuss eq q refer differentiable advantage give coefficient model class interpretable criterion criterion convex moderate
experiment ess report need pm ep expensive la pm approach aa ess suggest obtain use pm aa try stop approximation report especially yield ht pm aa pm aa pm aa exploration advantage aa pm approach sampler time independent respect report reveal interesting feature characterize fast aa method repeat update time aa capable break correlation case ard covariance gibbs enough consecutive report uci capability propose approach effectively breast comprise class infer parameter set unit covariance choose prior shape b compare aa iterate pm la chain proposal achieve acceptance pm use la overcome arise get run initialize speed result uci set show convergence logarithm panel auto one period plot reach
similarity admm slow improve size time synthetic n ny randomly split dataset digit focus choose machine admm note machine rate examine machine admm also comparison use bias highlight importance achieve minimizer algorithm significantly suboptimal intel institute support foundation describe level construction speak shot average return training small predictor datum good averaging dominate convexity parameter convexity magnitude sample distribution dependent calculation specifically function return receive draw
range traffic wireless communication see reference therein begin scheme derive duality player strategie discount aggregate payoff obtain base dynamic dynamic devote discretization real span denote distinction write likewise family finally follow convention nash indice one mention start consist finite denote game k restrict reduce course player mix k case payoff player payoff mix explicitly player strategy notation kx kx context simplex polytope together tuple rely resolve prominent profile deviation formally restriction class finite game player payoff satisfy multilinear case game nash equilibria vertice goal section follow performance playing describe precise player action strategy payoff aggregation player strategy time moment model assign exponentially instance treat uniformly assign old readily represent tune regime discount rate favor observation favor aggregation limit indicator player choose time variable size see explore interpret past information move estimator dependent lead
model flexibility application likelihood concern accuracy inferential testing confidence straightforward asymptotic define composite require information accuracy composite rely fact reference inference asymptotic bootstrap general ratio intensive joint narrow motivate nonparametric framework formulation introduce refined standard highly prove test largely monte carlo composite difficulty time yield inferential review especially pairwise pairwise likelihood framework description section assess simulation brief discussion section follow denote suppose depend multidimensional log ratio fy fy composite likelihood negative likelihood
circular law chen wishart matrix compare used weight goodness divergence al principle wishart data complex wishart assessment paper imaging include phase distinct wave medium sense signal resolution characterize random locally model multivariate mean complex circular scene q determinant denote conjugate distribution besides definite characterize ideally look scene pixel looks prove scale wishart follow probability gamma give et term allow along hermitian likelihood due log likelihood let law ml n treat et neighboring patch pixel define pixel white euclidean central region neighboring patch window filter search mask proportional patch intensity temperature patch tend
utilize concrete refer tackle new specifically general solve consider case solve rely similar consists evaluate suggest case process part likely biology particularly design field health intensive advantageous reasoning exist case record first step cycle retrieve previously situation system link relevant case accordance traditional database contain consist basis solve overcome concept utility description new utility accordance analyse description system datum medical basis model utility description specific measure precisely consist illustration formal traditional retrieval logistic specify scope section implementation discuss information network end
rl learn cost exploratory nn weight compute accordingly rl offline thus generate real meanwhile term extra policy repeatedly thus efficiency view firstly control external rl external secondly control rl convergent policy control nn actor realization nn policy rl linear pde pde pde wherein analyze nn rl show actor reference omit theoretical linear contradiction contradict
adaptive strategy collection adopt interpretation denote sense adaptive non sense independent past vector non adaptive examine vector design employ measurable denote sequel shorthand knowledge sense strategy make estimate use sense ds n denote expectation distribution word essentially quantify accurately estimate observation obtain sense distribution infimum maximum risk element use risk bound regardless particular employ may undesirable make guarantee regard bad case scenario support exceed identify problematic parameterize amplitude necessarily task adaptive typically compressive theorem let sparse non sense minimax obeys implication recover support signal signal amplitude concern adaptive sparse dimensional signal minimax obey bind statement guarantee depict broad adaptive sense identify support risk necessarily list table reference effort line condition support strategy exceed yet establish compressive regime finally relate effort establish weak main bottom table correspond equation sample adaptive sense adaptive sensing see two salient point note summarize sufficient summarize corner adaptive describe technique accurately recover arbitrarily small measurement provide nonzero exceed sense support
matrix depend assumption accord dna dna accommodate arise bias limit library complexity frequency read species frequency together read zero small simple equal involve complexity dimensional database order collection string define estimator define estimator divergence maximize formulation scalable specie optimization real hundred thousand present hundred thousand therefore norm read limited fitting read norm problem replace convex distribution scale scalable divide thresholding enforce repeat truncation
frequency conclude completely performance example transaction know amazon unlikely impossible single item safe transaction online survey transaction question consist occur sense vc generate say produce impact performance evaluation compute collection positive time false false negative implement implementation evaluate dataset repository differ importantly original name repository c l work work negative table report false positive negative case positive mining frequency highlight like fact dataset false negative point compute negative end range repeat dataset contain extract false theoretical positive wants include assess fraction different true compare
product express linear feature apply span independent also inherent noise subsequently analyst aggregate report value turn access perturb correspond analyst give perturb feature analysis variable span differently information linearly blind analyst analyst degenerate solution square invertible invertible perturbation motivate concern grant private release analyst learn run medical disease beneficial consideration function action amount level perturbation notational convenience representation term individual choose cost comprise incur component take df
allow loss arm regret variance sampling game lagrangian
target word residual norm minimize residual ta closed solution select column greedy projection te project ta without loss column correspond td ta ta
denote ground use fx denote bag straightforwardly loss difference true simultaneously optimize permit add bag widely margin learning require restrictive label proportion cluster depend distribution naturally supervise multi calibration treat bag super instance assume bag k kk bag model albeit inverse though show alternative several limitation high distribution dependent bag
statistical assumption good property paper broad svms nonparametric usually general convergence informally state statistical arbitrarily true known normality svms goal kernel outer probability useful svms
forecast moreover infer characterization efficient polytope computational order full monitoring characterize wasserstein objective characterization close one sufficient note since notion coincide shot situation dual characterization form game satisfie series result section light repeat game discuss condition shoot payoff geometry close convex primal illustrate strategy result payoff entail linear partial focus half simple tie shoot half space half space structure well primal nature infer state differently proposition implication converse implication immediate dark eq b property von min exactly shoot complement neighborhood half respective structure devote state characterization key high dimension polytope primal characterization directly least without
negative super elsewhere see eq meaningful fuse logistic natural iterate subproblem easy solve note subproblem division ready operation x mb k k mb logistic intel ghz gb ram report fuse problem use regression preferable logistic simple create create coefficient draw create sign one solve fuse code propose liu www
jj j bind derive edge provide improvement bring view form common distribute world geometric neighbor view problem agent global state propose decomposition parameter state steady state square true key estimator indicate network bring decrease acknowledge program decentralize grant dms material
z z probability event feasible moreover n x z find satisfie probability satisfy get happen similar latter pz z p z p find depend constant depend constant constant constant value event regression x hence supplementary e n nn n nn np nn n np r rp rp rp rp rp rp rp chebyshev least hold know z f r indicator r r z rp rp z f rp r r least constant constant material basically material least n z z c z proof part
base solely implementation publicly report scene margin purely purely discriminative training I pure performing perform scene hybrid slightly vocabulary sift patch spatial run look trade discriminative exploit beneficial exception topic grid slightly grid annotation performance obtain available result margin margin report accuracy annotation report simultaneously classify image illustrate incorrect prediction
disadvantage two source monte carlo discretization bias grid negligible certain diffusion describe work discretization allow realization key idea use accept without simulate complete boundary goal diffusion certain one modification motion example brownian motion sufficiently relate replace brownian characterize motivating diffusion generator first introduce terminology unlike diffusion force whenever due asymptotic sample share include generator boundary substantially structure
successful many g approximately optimize recommendation contain experience affinity point
notational fisher matrix denote k begin stand e k verify nonconvex globally expansion iterate equal stay remark may adopt stepsize long regardless strong suppose inactive plain logistic label predictor space follow overlap either overlap feature neither guarantee overlap assumption orthogonal complement bit e u constrain schedule central tend unique bind limit term n matter enough e n tending stage let predictor tend stage give line overlap result save theorem medical imaging dataset million paper gradually remove schedule keep drop particularly datum loss variable extremely addition one piecewise account nonlinearity impose synthetic show art method regression rank efficient parsimonious computer problem classifier amount million observation big challenge selection contamination numerous big must obtain globally rather
unlabeled million online library perform second target twitter sentence evaluate token collect million twitter message domain label crf section three representation representation hmm
le des es send cm les pour la pour situation la pour de es la il de la les es plus pour I situation un lin pour des situation comment cart les de es les en la du pour la du pr send correspond une es h cc la par un les par les un le en des transition gr mod en observer et les es augmentation la du ne une augmentation est plus le mod
future optimize stochastic model possess mix propose membership agglomerative procedure reduce get inverse temperature turn agglomerative algorithmic complexity tend indistinguishable partition empirical network also infer suitable use conjunction technique heuristic agglomerative et variant restrict modularity agglomerative permit move correction done stage agglomerative modularity many block structure expense increase algorithmic complexity propagation approach possess situation quickly impractical application paper universit monte carlo fast mix much get state greedy agglomerative indistinguishable
denote construct namely count denote wise magnitude whereas spectral know scalar hermitian constraint rewritten find combinatorial impractical inspire recent completion relax sdp particular trace surrogate refer noisy result derive cs inspire derivation please rip rip q state contrary bind solution contradiction n value contradiction assume hold trivially
chain proposal denote later integrate rkhs rkhs explicit density long differentiable kx henceforth furthermore proposal pp analytically proposal proof proposition contour proposal evaluate point sample proposal proposal metropolis accept reject accepted rise unbiased without proposal encode target subsample contain subsample contour update subsample keeps adapt correct convergence subsample decrease theorem algorithm converge correct another set burn adapt rough sketch mcmc suffice subsample informative regularization case ill
td stage give successively challenge learning problem pdf precisely behave use semi agnostic degree pdfs behave degree mixture behave finally requirement inefficient draw vc target draw empirical consist since vc dimension least triangle td take degree polynomial interval inequality distance computationally time grow give time main minimizing involve infinitely achieve minimize achieve complement piecewise degree degree access draw internal randomness td theorem together carefully tailor construction probability meet proof defer appendix complexity well behave pdf learn run use definition uniform partition r subsection write write distance degree degree run degree call degree first find thought quasi quasi guarantee everywhere single polynomial need exploit full next subroutine distribution learn input approximately equal call polynomial subroutine z mi mi distribution generalize piecewise lp kind equally space across equally space across qx claim learn claim dominate lp single suffice correctness proof intuition program
heterogeneity slope assume population include interaction term absolute almost time compare tree include term longitudinal deviation obtain longitudinal tree include partitioning probably commonly offer provide improvement influence point apart estimation identify partitioning remain algorithm gray patient patient consider marker brain three region per suggest beneficial include education age cd count duration infection duration highly rna count effectiveness stage partitioning longitudinal tree discuss slope longitudinal fitting duration treatment significant year subject normally significance instability node terminal display regression tree terminal value
forest pls rbf use computer vision degree terminate number node combine set determine pls implementation author pls also determine fold cross significantly respectively computation single htb pls art work cross whole average mae htb c pls pls mae along mae select well good train regression sec cross validation training reference result
copy satisfy constant let nonnegative particular satisfy careful reader unlike place infinity actually proof appendix select true ask nan model happen assumption iv n p assumption concentrate state successfully outperform arbitrarily probability nan always prefer theorem theoretical simplicity hyper prior random depend consequently set represent set similar though consider selection rely derive section treat furthermore proper satisfie prove appendix result result treat proof give nest even able desire result hold assumption furthermore support happen choose satisfy iv f np motivated propose generalization motivate appear nontrivial demonstrate reveal shift range suitably choice exponentially grow say growing motivate prior actually
factor action stochastic mapping expect discount reward bound know unique bellman bellman optimality policy greedy function policy finally iff focus sense call sensitive case trajectory induce policy bound expect begin describe g coefficient relate measure parameter coefficient ci involve involve ii since generic term sum hand expect completeness significant second bind involve error
z linear weight linear represent represent risk measurable noise level follow assume obtain transfer target unitary transformation instance source domain classifier preserve lead computer vision acquire ii connect loss improve active iii reduce explore local minimize justify assumption ii optimal minimize measurable function minimize
misclassification tune penalize dataset describe analyze dataset available limit investigate dataset purpose apply dataset challenge possible idea penalize purpose dataset perform cluster start note big use meta feature relationship number meta appear meta non tuning datum analyzing select tuning investigate select seven one prediction discriminant consider sequentially summarize perform feature perform select correspond merge add score correspondingly correspond strong agreement fisher problem pose
coordinate value adaboost adaboost convenient consider lipschitz adaboost optimisation adaboost dual involve gradient parallel method x select convenient law nice processor subset coordinate probability good nice sampling processor follow uniform consequence method separable update
value use portion adequate repetition extract multiple retain go outline carry keep repetition mode apply rest sample independently repetition sample partial merge depending work redundancy index tensor behave uniqueness introduce redundancy especially parallel rank repetition different set set couple likewise obtain final ideally get normalize column repetition scale equation multiple correspondence output likely merge component correspond component might aforementioned normalization resolve correspondence establish correspondence sketch paragraph inner exactly match cauchy
format digit center additional digit may appear ignore digits digit test example extra build sample per remain digit local normalization follow mnist maxout hide densely follow densely layer set art provide method error pooling dropout translation rather maxout mnist preprocesse cifar preprocesse maxout fig find maxout improvement preprocessing dropout beyond filter model maxout benefit pooling times maxout ht ht maxout
model onto observation count formulation upon composite self sparsity norm positivity image formulation seem image barrier g selector quantum bias logarithmic composite well search strategy n basis pursuit denoise commonly image scan datum scan image poisson imaging barrier approximation statistical splitting attempt main proximal splitting function endow tractable easy calculate gradient proximal set splitting present monotone splitting technique backward inclusion gradient gradient fast proximal e g lipschitz continuity smooth unfortunately applicable approach augment lagrangian quite disadvantage tuning penalty augment lagrangian indicate alternate direction multiplier linearization view unclear lead rigorous direction composite rely introduce convex apply method assume positive prove trust intend exploit self trust region option composite numerically programming avoid slack introduce embed barrier smooth newton many lose pre conditioning newton scale nesterov solve composite dimension instead solve
relaxation couple priori switch fashion weak couple lagrangian problem remain horizon action play current number play observe constraint decompose quantity expectation path policy denote policy path problem since arm show feasible correspond need horizon expectation path lagrangian define v tp cl pp rp tp problem policy feasible claim play reward iterate policy distance cost path critical use inactive coupling never consider denote decision decision subsequent contiguous arm never simply execute distance path feasible repeat policy change make regardless policy arm transition regardless outcome play refer policy arm arm path make arm play switch arm outcome path visit subsequently outcome play directly play movement preserve stochastically identical cost policy arm subsequently repeat follow feasible regardless arm need preserve depend play lagrangian additive objective distance preserve decision preserve good policy regardless collection arm policy via metric start maximize follow graph correspond reward budget super play arm current super visit subject problem author show extend ready horizon armed bandit switching cost computable arm arm arm reward good scheduling policy traversal collection policy traversal observe powerful basic combinatorial triangle traversal solution mab play step encode respective state space switch traversal dependent handle feedback handle idea precede section idea policy horizon possibly confident separate eliminate introduce delay delay become irrelevant play early interestingly optimality bound increase play regime free approximation delay except ratio strategy conjunction delay factor proportional thus approximation regime scheduling circle connection rule delay feedback optimum regime relaxation structure preserve globally assumption compute policy approximation efficiently idea mab delay feedback delay constant feedback arm policy mapping ii step less previous change state without arm policy horizon step section collection arm goal play reward maximize relaxation current note
follow equation eq zero give since monotonically non solution monotonically thus set occur flat happen empty eq similarly solution attain path exist abstract solve feasible finding entirely relaxation path admit geometric sec discuss admit track relaxed objective eq prior examine q let dual moreover exponential multinomial coefficient add write tend sparse turn geometric path case introduce term equation amount proceeding would swap role distribution analogous characterization explore fix rewrite range partition therefore recall path respect unique piecewise linear linear sub attain study geometric plane combinatorial characterization interval definition segment attain bind artificial datum demonstrate sequel show remain divide straight line n hold add get induction recall inside extreme segment
study check approximate well functional way q parameter r odd help I consider specify integer smoothing latter time consume carry simulation n kn column top save space specify tuning specify properly list alternative fulfil unit correlate decay fast indicate simulated functional highly decay slowly uncorrelated moderate number specify moderate size generate functional nearly tail first way nominal kn pt cc cc c c configuration compute replicate test nominal significance nan process repeat size replication check pdf underlying pdf display pdf wide pdfs dash z I use usual pdfs statistic approximate simulated pdf nine panel show
wind wind speed rmse linear blue green regularize achieve average ht speed estimator blue red time day truth tracking compare consider correspond n raw outline fig result velocity wind day year fit square daily wind year implement pseudo inverse year truth non overlap kronecker estimate contain kp weather factor insight long dependencie speed sample covariance top right kp middle spatial kronecker factor kp middle kp kp factor positive kp wind kronecker leave right kp spectrum contain energy kp match component percentage energy mean rmse testing period estimator choose optimize rmse parameterized wind speed nc optimize period kronecker estimator
modern finding rely statistical scientific scientific statistical conclusion closely reliability community et scientific conclusion laboratory ideally mind however massive discover scientific fact involve code validation knowledge science mean people paper science enhance emphasis concern word know cause instability ol understanding help interpret reliably robust statistic scientific berkeley human pathway fmri brain movie et particular decode mind read computer thing really reliable interpretation perturbation briefly vast bootstrap subset sample review yu yu rise selector es cv fmri obtain reduction es
negative characterization random invariant finite act proposition case square integrable invariant corollary necessary rely combination composition us dt fields index let integrable covariance e invariant arbitrary note surely continuous indistinguishable sure almost path invariance prop prop tf iv g functions eq resp coordinate covariance brownian compose rotation angle ensure path field kernel invariance belong finite treat composition unit beyond concern sparsity path detail
family reduce essentially usual error existence distribution pdf unknown exponential intrinsic stein estimator stein loss estimator example pmf parameter pmf exponential eq give example distribution distribution rx intrinsic xx choice calculation prior section family intrinsic loss application rather nature one name intrinsic
lp orthonormal j yx remarkable correlation correlation equal pearson mid tie definition correlation avoid statistic express density parametric skew g g estimator form score representation entropy du du
introduce convolution matlab implementation method compare convolution estimation ce solve kullback cross method deal rare estimation ce sampling
value section precisely ray grid oracle tv problem surprisingly interior point solver dedicated try include solver become every pair arc arc arcs termination rely nonnegative decrease iterates terminate use true generate work pilot penalty explain work grid consecutive relative computable shift satisfy terminate experiment progress good goal solution yield solve work bad present experiment popular image summarize datum follow accordance memory value call much count present surprisingly sublinear convergence run quite interior solver take single get run solution instrumental factor reason see happen low working outline termination attractive single run penalty work experiment see work penalty ratio small
representation analyze lexical like valuable nlp real know relate another representation versa explore linguistic deep reasoning reason truth basis task comprehensive accurate language present make logical sentence five class crucially use meaning support accurate inferential monotonicity broad kind sentence understand useful way understand go wrong parse inaccurate section strict recursive tensor task experiment pattern reveal familiar generalize unseen largely brief proposal
fw sometimes fast swap speed particular outperform fw method tend competitive medium method evenly match slight swap swap outperform result comment example subproblem medium see frank wolfe fairly similar small obtain result fw run fast fw find svm tend well work swap swap swap fast fw swap run fast finally among build significance report adopt suggest conduct equally reject conduct binary performance non safe compare multiple indeed statistically accuracy algorithm low conduct test main swap significant predictive contrast fw observe practice although swap little test swap exhibit large fw predictive propose conduct tail cm cm value swap vs fw swap vs accuracy vs swap value vs value vs accuracy fw vs swap fw swap fw swap accuracy swap vs swap swap vs swap hypothesis adopt level highlight p test concern software sign rank test exact tie default suggest fw adopt significance running time swap fw favor alternative well swap fast swap insufficient swap fast fw swap accuracy fw favor swap accurate collection cc collection cc cc collection cc cc dataset time large solve svms
hash learn property attribute ultimately combine semantic appearance colour imagenet amazon collect web another colour name model represent colour name consider share basic unit appearance prefer patch level region attribute texture level noisy self maps som dynamics som inspire part organize visual simulate som neurons neuron sensitive type input location neuron som neuron weight call winner winner unit neighbourhood delta eq window som proportional winner gradually value large update algorithm
without interpret smooth another summary interest even transform second decompose nonzero matrix note clearly uniquely refer formalize notation q associate pair conjugate norm oppose coordinate novel setup still write block practical g often part treat differently block however introduction notational overhead pay let th coordinate use th key nesterov separability yu nesterov seminal applicable represent degree resp note block row I I convex nonsmooth optimization ii constrain ready coincide conceptual computation update compactly iterate parallel let remark actually encode entire serial parallel depend likewise flexibility choose coordinate descent descent repeatedly classical nesterov merely collect norm conjugate functional adjoint set nonsmooth describe nesterov smoothing approximate gradient technique rely introduction prox
estimator bandwidth asymptotic study typically albeit less early monte distribution value often expense attempt property autocorrelation test asymptotic cite result underlie something rejection limit test g size asymptotic literature feasible hold feasible subsequent finite robust restriction parameter finding apply equally well estimator allow autocorrelation maintain section mention require correlation structure stationary autoregressive wide appear literature autocorrelation cite certainly include stationary gaussian autoregressive feasible process restriction theory autocorrelation robust test cf discussion subsection mention design proposition strong correlation structure equally hold mention majority testing suffer negative robust discuss nontrivial possible robust via size equal translate equal discuss intercept regression autoregressive particular problem considerable body literature concern I construct correction cite error follow autoregressive process standard surprising typically autocorrelation subsection section autocorrelation despite correction autocorrelation exhibit bad lead case invariant include power strength correlation spirit argument unfortunately correction autocorrelation test robust test transformation concentration explain subsection negative prop theorem one autocorrelation coefficient overview autocorrelation lack derive regressor design dimension unknown matrix prescribe nonempty show notation entire depend uniquely determined element little induce collection denote possibly covariance respect definite well affine versus precise emphasize testing compound write stress property nuisance affine eq adopt definition testing model portion negative continue hold elliptical distribution paper one argue spirit unconditional counterpart randomize rejection supremum nan refer shall furthermore write capital bold lebesgue lebesgue view interpret interior closure take topology euclidean complement vector space define sign th denote dimension denote let symmetric positive sequel give rise set feasible matrix space require choice section investigate autocorrelation case study maintain vector consist density equal c density correspond contain autoregressive arbitrary hence shall section mild mention certainly case successive element autoregressive ar j get close constant instrumental establish bad property appropriate want stress restriction assumption I e autocorrelation test perhaps helpful gain phenomenon occur location location intuition picture want autocorrelation
select observe part jj inequality hold hence edu edu stanford university function whenever admissible limited fix assume denote functional functional epoch take fix algorithm except perhaps take establish conclude proof algorithm specify define batch perhaps single appendix equation hold conclude proof realize observation let epoch history follow fix repeat hence establish feedback structure cost assumption conclude yield batch accord assume batch odd assume access size suppose budget selecting batch size consider assume epoch access action generate consider part analyze sublinear generate descent policy linear horizon incur batch odd analyze next incur analyze decision throughout recall batch cost use hold c begin note batch therefore batch repeat eq incurred horizon q linear incur batch select cost batch analyze horizon decrease hold since incur recall throughout incur conclude section auxiliary adversarial literature often consider example include access cost noiseless evaluate section cost admissible minimize regret let lr mapping define admissible respect feedback exactly exactly action literature sequence adjust epoch advance
tail depend scatter diagram conditional htb measure kullback mutual distance provide integral diagnostic sum bold symbol age traditional chi square interpret smooth degree freedom sum drive discrete discrete start fundamental widely applicable especially traditional population population nonparametric statistic datum denote diagram combine population scatter diagram straight interpret mean index traditional approach expectation pool observation unconditional
projection particularly classifier single pass moreover learn classifier streaming code minimize true particularly powerful formalize maximally hardware use minimal communication framework well suit extend popular decomposition support useful well problem hundred datum kernel matrix low entry discard entire fast
detail order estimate generate accurate fix use early simulate individual relative risk rr threshold accordingly phenotype account effect change threshold phenotype phenotype value account account applicable breaking assumption introduction order magnitude considerably demand interval ci individual cover conservative case error estimate yield might correct exist estimate seminal scale threshold work lee note
outcome go probability often call probability game go reach player play basically stage apply game start play reach probability player hard stage go risk player risk fix pure stationary player every play play aim prove set player strategy strategy first stationary correspond strategy small hard play thus us payoff resp spend payoff sa tb quantity play payoff play term play game probability third resp
cluster occur fit pattern source allocate cluster giving condition substitution simulation effect distribution draw uniform estimate prior prior main surprisingly highly influence prior inaccurate moderately four procedure burn hard determine cluster allocate th mcmc sampling generate eq allocate cluster mn kf use gamma mixture cluster allocate cluster mn b implement cluster provide
form paper probabilistic problem initial problem ode uncertainty riemannian ode nuisance variable elsewhere beneficial replace distribution second remarkable solver present additional concrete riemannian fast standard solver implement matlab reason answer return probabilistic less smoothed cost dominate explicitly probabilistic sec offer evaluation matlab solver publish case believe example box role source design mind offer quantitative propagate numerical become increasingly constrain smoothly metric example shape pose
arise score interested model interpretable estimate suggest rescale theorem law iterate expectation support identify replace unknown expectation average bandwidth eq density implement derivative respect suppose function compute regularity support odd derivative exist z support nonempty interior boundary ii differentiable function derivative iv vi exist assumption impose condition remove additional assumption efficiency far q analog l use may follow denote hull consequence consistency asymptotic optimality let hausdorff due equality hausdorff supremum correspond pn associated support class risk hausdorff function conduct bootstrap mean omit brevity weakly conditional use side construct n may coordinate quantile performance monte experiment throughout eq independent independently control diameter
result rapid process state inversion adaptive stability capable drift vector individually yield fine grain identification benefit control moreover normal region configuration principle result euler mechanic generalize call proportional assume system always full acceleration immediate control angle incorporate kind lagrangian mechanic beneficial
concept uncertainty acoustic front b l l subsection map subsection detailed description text acoustic start distortion read account speech distortion underlie bayesian explain clean result adapt closed variety employ separate distortion network representation acoustic acoustic turn map viterbi frequently yield promise fundamental taylor series n network uncertainty firstly gaussian individually approximated extension refer uncertainty subsection deterministic reflect component observation transform technique promising field adaptation concept mean vector adaptation approach applicable
vb minimize increasingly solution zero actually figure dimensionality always feasible solution vb unless actually globally broad even slightly degeneracy vb fundamentally essentially small feasible expect vb examine previous fortunately offer undesirable simple factor specify justification inclusion proportional fact augment view effective report section numerous real world experiment reduction vb slight property appendix derivation free schedule offer natural coarse blind section conclusion tb noise initialize stop emphasize primary purpose formal analysis deconvolution se vb complementary nonetheless motivated herein briefly evaluate two simplify vb albeit theoretically sound perform publish vb considerably manual hope motivate usage blind jeffreys motivated automatically step adopt special refer vb jeffreys underlie prior improper jeffreys note blind final variant vb reproduce kernel metric quantify hard error true blind deconvolution hard estimate evaluation compare vb jeffreys describe vb dataset effective optimize respect consideration herein rigorously instead argue cumulative histogram figure bar bar ratio visually vb jeffreys significantly regardless vb exhibit especially benefit additional heuristic facilitate structure regularization boost discuss phenomenon vb vb experience reduce plausible relate algorithms prior roughly match sparse exactly estimate high vb extremely require produce error ratio feature vb closest jeffreys jeffreys may
ph bag maximally region local descriptor sift sift descriptor descriptor select visual descriptor close distance image instead image visual word appear frequency informative exclude associate visual form tf visual word occur many image versa might preferable pc formulation literature pc obtain solution optimization euclidean nonzero yet author source com scalable parallel implement described form version order alm propose grow alm define dot easy
turn pairwise tuple gaussian define subset unary notation true unary instance exponential take order gram length total designing mrf mainly design attractive follow appeal control design mrf practical number grow unary length input sequence support bar option hyperparameter cross define hyperparameter bayesian suffer hyperparameter carlo procedure predefine correspond low underlie mrf shape structure belief answer chain case obtain hamming micro wise micro probable impossible
computation sequential markov use field bioinformatic comprise latent transition density time variable dominate law subset hmms prohibitive calculate able despite hmm generate unknown intractable particle filter evaluate gradient respect subsequently review smc draw develop static algorithm gradient offline mle author finite ultimately advantage method log use intractable simulate using indicate particle indicate calculate exactly paper mle detail smc
x run hardware system drastically ease configuration cluster mode carefully dataset equally sized file number file streaming paradigm favor primitive incremental difficulty fairly easy reasonably netflix dataset user great care tune job correctly manner test difficult integrate store must build dependency manually copy software machine fit memory require extra cluster software program class generate build scalable distribute useful transformation local term ease computational implementation compare ml layer effort design simplify ml support part nsf award amazon web
assign organization determine standard format pre entity decade accuracy group manually specify program either produce extraction characteristic base rule heuristic extract language good result specific lot effort produce domain start receive exploit classifier responsible determine relationship major line work supervise approach try try computation develop able structure g though structure parse graph paper sentence representation contain rich entity deal walk able reduce computation equation proposal candidate entity connect syntactic representation particularly regard distinct make evaluate call aim
implicit power genetic mode argument q recursive voting proof induction inductive prove pass inductive independent eq note h r r ni correctly solve query approximately correctly solve time let oracle value line proof return corollary learnable claim claim claim approximately correctly solve make likewise claim complete constant claim take execute loop proof
volume flow express condition observation x u acquire find lagrangian optimality arrive eqs observable flow binomial relation state random measurement flow matrix measurement scheme covariance let arrange specify instantaneous estimation time volume flow instantaneous write
maximize measurement evenly specie originally introduce r width rw width body variable reflect width measurement cl predictor colour know group ari select component ari assume choose
minute maximal majority alone grained result overfitte huge come discretization nearly huge reason enable simplified exploratory obtain see simplification yield cluster reveal cluster general correlate within cluster case interesting source different illustrate characteristic locate triangle target city constitute see strong except front circle group distant assume business use exploratory illustrate specialize quantifie time leave aside traffic cycle day couple cluster negative probability
review mixed model wiener chapter drift model contrast deal fractional discrete observation fractional high frequency horizon infinity sure far maximum strong complicated
search near instead arise modeling include nearly g ground truth many find predictive exclude group protein property many domain retrieval document document rank important explore include optimal solution operate semi setting lead solution baseline want predict instance every query retrieve intersection retrieve associate different either empty empty interested rank retrieve task retrieve label
document collapse sum accord initialize one converge propose inference vb index factorize parameterized thick optimize evidence kullback leibler divergence coordinate ascent
forecast probable combination review exist train per basis market collaborative filter market forecasting cast several hour leverage market superposition component capture spatio base selection second contribution hour analytic result extend tool rank market pricing network balance connection meaningful laplacian base provide solve demand convex via leverage compressed sense exploit kronecker guarantee stationary forecasting novel letter stand kronecker stacking need outline forecasting block detailed forecasting market sec feature belong space basis nk rkh norm estimation pose regularization eq l whereas generalization balance solve fortunately characterize n kx
zhang jk jk arguments zhang inequality easy show jk zhang bn bn bn zhang use obtain yield pd bp page tend bn hence bn bn zhang obtain estimate augment j j equivalent latter obviously j j side previous j j obey last give since careful reading previous helpful author partially support mm mm section thm axiom thm conclusion
consider widely big consider aspect develop algorithm summarize select matrix manner algorithm minimize reconstruction present novel recursive reconstruction develop greedy propose learn select column matrix facilitate sub accurate matrix approximate column target pass communication overhead experiment organize describe notation background centralize greedy propose review section finally paper notation indicate letter small bold letter capital letter subscript indicate notation set cardinality row sub consist transpose norm column review column representative representative pre selection common much discrepancy approximate column criterion assess define projection matrix candidate matrix frobenius residual quantify derive residual present focus projection sub correspond note derive column approximate q find problem ta present analyst instance
besides law may deep first architecture big share serious dynamical scheduling non vocabulary map chi support science foundation education china team grant project city university topic modeling reduce space complexity lda parallel multi processor architecture complexity cost among processor lead serious processor topic architecture power communication topic architecture big extensive confirm big modeling speed compare recent multi topic model dirichlet processor propagation learning apply successful algorithm latent dirichlet allocation lda many biology attract intensive interest big become video big challenge reduce complexity traditional collapse gs big bp set contain store around tb consumption cost big fall batch lda fast observe small
dual practice experience verify whether reduce potential useful potential difference answer potential piecewise vision value label address illustrated unary one primal order piece primal meaning overall consider potential write pointwise correspondence minimizer one organize relevant notation briefly state constructive section pointwise pairwise potential technique material section enable isotropic behavior processing experimentally verify message passing stop early manuscript review domain set vision image application e neighborhood shorthand notation respectively convention indicate set density function
furthermore density number convergence fix eq eq density publicly give implementation simulate censor censor trick artificial pseudo survival function event homogeneous start standard exponentially increment present density survival censor sort second panel compare third survival function survival unconstraine likelihood clearly preferable survival simulated norm compare obtain quantity distribution simulate set poisson inspection supremum norm average error estimate benefit fact slight misspecification available survival
cluster effort devote generation pixel predict boundary object pixel meanwhile straightforward boundary contour separate adjacent merge merge long therefore reliable merge latter characteristic could texture merge merge predict merge find guarantee similarly get paradigm generate hierarchy segment scale determine want agglomerative segmentation merge obtain hierarchy part feature classifier likely manual combination complex set discuss compare gold segmentation sample independently similar yield explain collect agglomerative human generate gold correct merge scale outperform art segmentation variation idea learn arbitrary dimension first
formalize mean terminology transformation apply input done regard example valid transformation object special hash zero hash practice ref input specific hash produce restriction transformation formally invariance invariance distinguish input transformation word distinguish transformation atomic provable invariance choose represent hash use absence hash outside marker validity property operation single single prevent invariance property function composition key operation input invariant follow construction rd ref invariance nan hash hash hash invariance output hash hash eq hash label specifically invariance composition drop change expand reduction property allow add invariant output hash vary marker valid marker net hash ref invariance hash hash hash strength strong show node connect edge hash hash graph
one application make view support national foundation china project com machine svms semi view regularization usual extension svms learning view convert finite euclidean method give rademach complexity affect bind empirical complexity insight play world validate laplacian svms learning
quantile conditional compact subset extend unconditional tail bivariate conditional establish geometric normality purpose know obtain motivation study predict predict company estimation economic security maximum require price demand reason power demand predict separately hand variable curve maximum conditional problem organize notation concern include normality example median median median random vector value equip metric fix c given assume first however uniqueness straight line uniqueness sequel median
apply drawn training method I class anomalous estimate eqn result exist like previous anomalous extension vector proportion simplex estimate form roc estimate one eqn let roc th compare denote em distance kl divergence em method proportion set multiclass set performance permutation norm proportion proportion manually class range set positive proportion large original take proportion grow discard proportion permutation varied fig
balanced follow usual attempt enforce partition vertex easily incorporate label belong r matrix relaxed definition help mean function function nk edge correspond vertex entry entry vertex sense remainder asymmetric definition relaxation theorem appendix rely definition algebra precede form indicator simplex guarantee r form usual continuous following rely subsection detail total
draw gmm follow select select gaussian gaussians variation exposition infinite arithmetic draw gaussians computation pdf defer pdfs pdfs I kolmogorov distance metric kf x yx metric compare general valid state kf fortunately fairly learn kolmogorov inequality sample query cdf probability structure representation cost operation detail provide following suppose datum uniform sample kf x course access partition time suit easy partition representation modification metric respect kolmogorov metric f let deferred decompose collection contain candidate identify generation fx x suggest distance goal complexity size collection wish sequentially generate start candidate one accurate candidate candidate likely candidate close candidate component subtract thus single inaccurate start mix weight follow candidate value branch branch
optimizing lemma enable usually explain estimate way unbiased risk noise straightforward amp may employ derive exhaustive computationally hence useful estimate amp hence seek efficient purpose gradient one fix derivative prove section properly speak plug large intuition note introduce sensitive inspire suggestion parameter section approximate iteration issue quasi compare minima minima global minima address noiseless characterize unbiased characterize derivative descent section first concern eq close denote risk
lie achievable correlation dimension cdf cdf write cdf marginal hoeffding fr cdf cdf correlation achieve inverse minimum random method cdf achievable exist convexity simulate specify
dissimilarity importance reference show term moment behavior x w ps ps ps express though time observe time addition know want evaluate importance implementation know importance weight compute ai x I si ai w introduce use dual importantly section approach convenient meaningful network combine non agent operate mdps state node transition ki ki ki convenient agent agent network agent motivated agent global problem remove weighting aggregated effectively solve agent l k w x r aggregate lagrangian diffusion consist intermediate independently neighbor estimate single agent e follow dual minimize gradient lk maximize gradient n lk agent agent adapt combination weight
per topic distribution evolve long word reflect long topic closeness correctly successful labeling positive appendix introduce important thought represent coin success formally coin time multinomial categorical discrete time outcome multinomial science categorical distribution represent vector indicate outcome categorical distribution multinomial represent bernoulli effective respectively define variable q q dirichlet distribution multinomial dirichlet draw parametrize analogous integrate chinese q dirichlet process dp top show proportion share dirichlet tie draw construction know dirichlet integrate chinese topic word number share generative process chinese restaurant restaurant represent customer customer table chinese restaurant customer restaurant empty customer table new across rich add temporal dirichlet mixture model propose evolve decay time step time integrate topic evolve evolve non conjugacy graphical allow visualize complex tune event otherwise markov field direct independence dependence seven graph create algorithm example prior train three drug test example direct stock represent market affect speak represent set rest complexity become avoid network make bayesian variable analogously variable space clique normalize constant network identify markov query independence evidence find maximize assign underlying could yield first try search candidate trying require hard take entropy kullback kl crf hmms application name recognition speech pos relax independence make skip crf observe identical connect evidence long dependency would hard gram tend parameter skip depend model pos similarity use root connect identical adding would make approximate inference label give skip factor skip chain template arbitrary name chain hmm skip fall independence observation end skip word country skip china entity al rich pair useful entity neighbor state skip skip edge would
discuss exponential restrict boltzmann put visible take value see state nan state index distribution dp dp dp deep belief dark visible describe interact maximal code mixture
due bounding increase establish increase prove objective defer dual easy verify ix equality index proceed write strong summarize I nk randomness relationship increase duality therefore challenge come decompose norm account global primal primal define establishe let
sensitivity assess however case prior hence focus discussion cause effect view justify circumstance question specify make I relate ann could ann I outcome begin try purpose good experimental get good conditional exposure ideal inference uncertainty bound start explore understand triple uncertainty fall far short ideal situation much address alone observe might thank valuable ir cf largely law concerned effect quantitative cause understanding cause experimental statistical evidence reasoning reasoning outcome answer answer perfect uncertain lead kind still possible addition identify relevant contrast matter much science define towards illustrate keyword cause child fr one assign undesirable case arise study specialize evidence address unclear issue certain case complex scientific logical show good extensive cause accept condition remain irreducible uncertainty express uncertainty raise subtle issue
jump direct enable wishart dramatically usefulness develop normalize first mind slow spend move considerably fast dramatically rapid development form glasso would wishart structure decomposable graph critical bayesian double partially evaluation neighbor
adapt application formulate semi supervise extension basis lie preserve incorporate invariant optimize acknowledgment thank zhang anonymous comment suggestion significantly fundamental machine observation identically distribute observation come mutually assumption violate case decade many tackle scenario training shift well therein transfer aim knowledge target datum label transfer focus improve source although knowledge reference simultaneously especially example task simultaneously
coordinate spread may applicable gradually increase fuzzy yet
zhang grateful careful reading write cb national china remark definition regularization central say regularization capability implement scheme hypothesis attain learn asymptotically identical reveal choice might capability perspective specify merely criterion kernel scientific underlie normally speak system family property reflect reality trend rkhs rkhs hilbert pointwise effective available model evaluation rkhs regularize research activity decade
condition asymptotic minimizer n n thus strict suffice monotonicity equivalent q since u part suffice show exist maximizer np apply view nu nc right side cauchy fix term vanish fourth control regard term third term condition fourth conclusion hold first derivative zero take expansion term first third term positive condition order term sequence conclusion together imply maximizer give constant eq kkt lead necessary sparsity equation furthermore follow control
step interpretation two define produce observe nz n obtain produce follow lemma collection integer imply equal base amount evaluate value assign assign collection formula structure polynomial assume produce string bs q see linear complete proof note transpose follow mt row element jacobian independent row rest zero coincide outside entry sum sum furthermore sum desire hard composition recall use convention u
ht fourth last experiment third expert large respectively large well previous obviously visible achieve happen regret bound thus would adapt really main develop simplify moreover sophisticated weight loss learn rate obtain address develop datum briefly address place provide ask within constant minimize appear difficult would significant sophisticated version high rate resolve issue adapt sequentially hoc fashion universal adapt first ideal condition achieve use second implication remain implication continue hold broadly single scalar controlling batch setting diverse ridge nonparametric pac prediction sometimes cross guarantee extend treat determine fail consideration formally inference lasso ridge draw ideal adapting case currently encourage gap define employ technique ensure concentrated give complexity minimize predict sequential ex pt batch risk ex pt excess
receive track display track experiment similar map discard algorithm identical practical ability understand stochastic allow adjust non stationary alternate explanation model flexibility induce behavior model poisson filtering either gaussian iteratively recursive close sampling inference impractical high computational also multinomial logistic mean
cluster sparse cluster relevant associated survival merely result overfitte analysis frequently cluster offer biological use precisely phenotype hence treat disease conventional dominate despite relatively early complementary drawback hierarchical clustering mean furthermore complementary computationally apply gb section relevant problem associate semi semi supervised circumstance semi vary tune poor clustering fail identify association outcome noisy likewise drawback fail supervise produce singleton present supervise overcome cluster cluster vary tune experience question
well recognition pooling back adopt object dataset inspire understand feature pool recently nonlinear maxout output neuron act piecewise activation many art benchmark attempt generalize operator maxout understand output conventional however norm fix predefine value max understand whose separate boundary space mlp highly conventional activation boundary piece approximate separation expensive hide description perceptron mlp explanation pooling mlp propose mlp generalize sec analyze recurrent network unit object perceptron mlp feedforward
convolution challenge projection external inefficient projection present fast implement intensity approximated pattern motivation difficult easy already probability stage get two stage procedure valid
converge close riemannian denote basis denote denote vector basis identity n tm proof suppose ball hausdorff hausdorff manifold k tt distance vector justification dim mx fx fx mf n hence know smooth dim
bfgs see extensive instead digit arithmetic bfgs package compare noiseless benchmark characteristic unimodal modal without generate transformation run position optimum performance algorithm report fig initialize sample fourth surrogate improve range es es bfgs omit clarity behave ill
node topic level incorporate one document topic path generate topic original path nest depends assign assign high generating however incorporate supervision modify total document strength prior nest chinese process graphical specific relate customer customer customer customer table I simplicity sophisticated model generative table word level ii assignment th document collection crp restaurant topic th nest crp restaurant assign topic distribution vocabulary topic mixture first path nest crp refer label presence label transfer source hierarchy assignment document close document unseen probability already assign topic source unlabele equation nest
application bayes miss value k minimize kullback expectation likelihood formulate analogous let auxiliary augmentation approach q pseudo conjugate vb kullback th row order length later determine machine present address present note penalization constant section appropriate constant objective term unseen tuning include random approach increase effect enable natural svm penalty parameter handle intercept effect
traffic traffic worth set contain flow national google engine website phase experiment connection traffic great protocol highlight issue notice obtain thank web separate traffic margin traffic svm google attack several ad hoc well statistical property build flow traffic training traffic first train web direct engine direct google list namely dimensional sample support evaluate validation strategy mutually exclusive fold approximately fold c r classify google google experimental show privacy attack specifically adversary differential privacy sensitive information relate word accuracy statistical database minimize single privacy record privacy original actually adversary record information result differential privacy output adversary add subtle noise still must converge classify correctly see unclear fail adversary experiment
due np property solution eqn eqn rl conclusion corollary solution restrict use error regard method estimation tt omp relationship value decrease cd theorem descent algorithm solution ensure coordinate cd positive zero sharp concavity every cd optimize balance decrease step iterate th solve minimizer cd iterate concave property concavity requirement sharp concavity besides gap cd stop k give decrease property hope
meaningful layer trace quadratic graph vertex individual preserve representative layer justify information theoretic point al kullback leibler closely suggest condition stand mean ambient noise k subspace justification schmidt criterion random spectral row subspace distribution govern therefore gram reflect information subspace subspace intrinsic relationship imagine crucial importance treat subspace manifold permit find representative view reduction remark relative importance knowledge importance information graph adapt subspace informative introduce merging subspace lead representative capture intrinsic relationship involve graph analyze property see section success laplacian aim unified account information contain graph merging framework multi graph detail spectral spectral algorithm subspace graph merge representative contain
pass alg ei arm ei algorithm analyze proposal ht c alg sd slice c alg sd arm cccc cccc quantity di di panel carlo error autocorrelation lag three row mean alg ei mh iterations ei standard deviation estimate c arm david property introduce class purpose simulation target metropolis strategy use interpolation past metropolis control evolution distribution propose efficiency effectiveness structure keyword adaptive metropolis within monte see reference important tool field normalize standard produce crucial mcmc proposal heavily tail decade remarkable adaptive tuning procedure mcmc flexible rate adaptive e adaptive strategy strategy past rely auxiliary chain run interact metropolis generalization paper focus try reveal different mh move select proposal multiple setup target monte use contribute adaptive propose class metropolis metropolis adaptation strategy past adapt relie strategy extend rejection reject g interpolation adaptive mh algorithm arm
initial concentration initialize chinese restaurant magnitude estimation distribute room achievable would operation envelope suggestion memory hierarchy would machine third would tuning significant dataset gb roughly dataset want file advanced implementation beyond preliminary investigate auxiliary focus multi core implementation hierarchical hdp compatibility transition operator implementation enable
ridge correspondence interesting move logistic predict quadratic word encourage parsimonious encourage confident prediction away recall dropout correspond apply obtain dropout row attempt penalty fisher link surface identity surface spherical around normalizing surface spherical basis balanced graphical illustration dropout penalty connection previously glm additive
nan nan nan nan nan nan nan nan nan nan nan nan nan draw mesh row crcr meta index header nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan flat mesh row crcr false nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan flat meta mesh row sep crcr header nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan flat draw meta explicit crcr meta header nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan height jump axis reverse ylabel south east south x flat black meta row meta false nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan flat black meta mesh false nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan flat black table header false nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan meta mesh sep crcr meta index header nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan black mesh crcr meta nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan monotonic cost accord make hold monotonic queue message opposite control unit neither monotonic still height unbounded jump view scale ylabel axis black meta explicit mesh row sep crcr meta nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan flat black explicit mesh sep crcr meta header false nan nan nan
efficient numerical complicated analytical thompson complex setting impose support put mass trivial correlation divergence additive potentially total appear merely technique extract scale fact regret thompson sampling interestingly logarithmic characterize kl represent information bandit structure action simply separate complex bandit reflect result coupling regret thompson reward analysis bandit pose challenge overcome work crucially conjugacy normal bandit close form contrast develop novel technique posterior allow track distribution
sub well fourier class monotone submodular direct within problem submodular numerous economic pricing price object rate etc estimate machine sensor explicitly approximate learn example document score submodular machine size corpus result constraint impose cardinality cut span perfect submodular follow exclude alternate notion define form closely submodular finally notion notion tight curvature modular far conceptually curvature distinct submodular problem monotone submodular cardinality constraint tight show result hold tight solution yet address section picture affect maximization approximation bound maximal practically curvature explanation observe modular computer vision
computational requirement solid red dotted indistinguishable display cubic spline matching visually around slight nothing special phenomenon similarity hold sample theoretically tune asymptotically statement topic future consideration spline trend problem problem matrix trend filter analytical reason much dual interior path close compute path path require spline solve lasso power dense converted match would dense dense predictor moderately size deal hold solve lasso lar implement entire solve coefficient fair trend lar locally spline panel take trend filter issue scalability compute dual complete minute lar step hour size memory issue section show filtering express consider lemma input input evenly trend alternatively express filtering bounding moreover rise parameterization subspace function evenly space continuous mean weak filtering dimensional say trend seen point piecewise polynomial contain time necessarily though infinitely differentiable derivative visually quite function center utilize trend function plot show nonsmooth speak factorial truncate power basis fairly filter admit section filtering
square achieve partition disjoint subset gram total execute simulation plot black partition accuracy closely match size identical experiment choose local gap algorithm understand divide maintain degradation limited plot theory predict even partition grow polynomially grow constant show performance somewhat well thresholded partition polynomially optimality fail give number improve inversion sophisticated believe reasonable proxy exhibit machine gb processor take run deviation error rate magnitude error fail correspond inversion optimal accuracy yield computational prediction hold om approximation bar study year song base audio song song consist song track song vector information consist dimensional year song paper experiment gaussian
parameter modal g ard optimizer optimum typically execute well optimize map accomplish expensive covariance rational mat ern covariance complex function product genetic programming refer automatic program principle evolve structure frequently potential
heuristic chebyshev distribute lebesgue dominate lebesgue property validation manually lebesgue dominate ne c ix trivially pass r e r pr pass via pass cp validation cv aic package implement propose
high combine content hybrid build system expert prediction build probable topic later prediction probable instead item fix chain recommender google news click read recommender system measure recommendation news see obvious recommendation popular sensitivity ct hyperparameter second select measure methodology want implement recommender website daily range national international event span contain news anonymous visit create website lot mobile internet connection figure distribution visit go news anonymous news visit generate recommendation soon read
base would field contain possible extension probability extension behave might represent special field generate countable odd effort work assumption distribution really typically lebesgue count restrictive intuitive event impossible define conditional
ideal statistical counterpart quite consider diagram section derive interesting fact directly parametrization polynomial several belong let check possible configuration binomial configuration row th equal nice independence example behaved sum indicator column independence also sum column column come indicator equal q
hadamard always hadamard multiply hadamard matrix order remark hadamard area hadamard order still small hadamard matrix variety hadamard conference discuss illustration purpose hadamard order matrix hadamard hadamard need hadamard matrix kronecker hadamard let follow
dynamic essentially amount perform show idea behind procedure slightly let th usually train stimulus rbm visible ensure connection encode dynamic initialize autoencoder layer perceptron delay visible layer try predict current visible project layer essence delay delay constitute would hide current visible given network exact format sum complete visible hide train cd summary l constrain single frame constrain denoise autoencoder multi
become interesting arrival compute solution argument cluster drastically move argument base uniqueness empirical unlikely change thus easily component interact property give procedure measure term subgradient oracle scale art attain apply favorable annealing observe satisfied since hence whenever accuracy density chain permit calculate view optimization lipschitz schedule suboptimal question handle anneal learner place distributional initially low receive attention refer reader strategy cost exponential form amenable learner choose play study assignment predict probability outcome
dyadic undirected whereby whereby node collection vector respectively edge star configuration pair edge common also share practical sufficient develop methodology display sub represent actor model exchange outline implement chain overall generate draw draw temperature schedule implement take minute processor gb memory yield suggest strong draw
challenge characteristic determine work generally expensive obtain offset cg effective challenging circumstance challenge implicit focus explore element element limit mainly beneficial suffer issue pre require worker parallelization dnn offset far powerful improve merely scale parallelization bfgs cg residual store cg specify user bfgs cg bfgs adopt namely evenly distribute throughout different cg requirement need cg since
ibp add entities ibp ibp group remain original dd ibp consecutive either approach kernel modularity euclidean place space membership membership attempt include drift group membership drift factorial add social markov network link influence membership incorporate group develop markov approximate latent state transition fix represent pz sample backward recursion algorithm define deterministic pass one point time cache pass start collect need indicate death use add new remove life remove group exist remove
coherent posterior incorporate level likelihood fit programming computing introduce break mixture since include individual level value summary probability distribution heavily make move proportional full aggregated metropolis within gibbs use metropolis update sampling achieve sample jointly parameter conditional distribution require calculate start block allow adjust start hold allow reasonable effectively chain chain country country addition country carlo inherent likelihood obtain level country effectively across age group unimodal flexibility sensitivity result year band grey raw estimate density black line black credible grey statistic symbol uncertainty weight model fit example individual year model empirical density china tight normal non negligible weight relative china similar examine china poor nonetheless make
method performance instance substantial graphical certain assumption imply zero minimizer zero separate regression variable lasso fit neighborhood take I less neighborhood correlation estimation iterate partial correlation lasso separately indicate account symmetry consistent estimator regression approach graph example present seem really critical fail suggest coordinate approach correlation algorithm objective importantly comprise subsequent via strict guarantee wise descent unique guarantee quadratic rigorously demonstrate converge dimension estimator high identical attractive yield strength real datum present property table neighborhood ns space lasso property provide rigorous development pseudo graphical direct pseudo deep insight strength method symmetry guarantee remainder describe present fail motivate work propose section establishe illustrate procedure real comparison glasso apply recent breast establishe conclude remark k multivariate mean ji propose partial correlation covariance converge immediately establish property useful framework space deviation fix fix suggest choice partial correlation current update fix space minimize
recommendation vector movie user movie formulate study call user movie formulate scheme constitute bi rank problem sense miss use use recover find hard handle trace solve non convex successful sense svd matrix potentially square problem scalable minima recently assumption alternate global rip rank motivate theoretical success variant initialization analyze assume converge optima individually global optimum describe application level goal acquisition w p alternate sense operator
refer increase identify eigenvector income edge community membership exact well root hence leave find deal rather considerably reduce computational next disk entry via expectation expectation since hold fix conclude eigenvalue rigorously disk precise remark real conjugate circle radius straightforwardly density loop connect value singular eigenvalue singular value vertex degree eigenvalue spectral well modulus line represent regard spectral straightforwardly
start assumption underlie slope intercept value attempt notation hyperparameter set one evaluate geometrically impractical posterior carlo mcmc specifically setting framework metropolis way recover trace evidence integral function point harmonic spurious lot check ratio subsection hyperparameter hyperparameter original hyperparameter matrix bayes intercept interval prior hyperparameter unity weight hyperparameter normalize pr function confirm mcmc denote non hyperparameter hyperparameter likelihood eq correct correctly show pr hyperparameter hyperparameter hyperparameter unity play important hypothesis contain within confidence however ratio introduction
fourth attract include pc algorithm combine start throughout important characteristic exact situation exactly return index affine maximal lie away final rather improve use end subset index cm value increase default throughout suffice remarkable subset sde much projection reliably direction hyperplane draw entire set observation direction end choice point initial subset sde index hyperplane also hyperplane therefore exponentially depend experience slightly fast mean procedure impractical much four procedure
matrix project multiply trace suggest sensitivity sensitivity input generalize sensitivity eq measure
equal one frequency histogram jeffreys positive centroid state arithmetic geometric mean approximate jeffreys simplex jeffreys centroid jx follow definition jeffreys expand rhs yield jx w x w j I extend histogram jx theorem jeffreys centroid normalize jeffreys positive w expand j experimental describe
pointwise error e I albeit somewhat build basis relevant parameter space fig graphical large interpolation randomly parameter ht e point define max process continue hz four representative depict combination min evident error offline cost fast construct basis point stage dependent preserve interest number quadrature equal number function speak comparable choose exactly approximation sec standard interpolation spaced rule interpolation maximize quadrature comprise datum segment duration arrival sec sample discrete ft f f noisy know numerical discrete computational depend turn integral repeatedly evaluate template integral content template function spirit product study search set build noise generality coefficient compose product generation comprise
regularization experience summarize contribution discuss number define r denote tensor high tensor tensor letter mode fix index order analogue column obtain mode nj ready tensor nj j nj validation encourage structure freedom regularizer find relaxation recent work agree trace tensor
n c I ta tu give single bad respectively another upper assume bad bound recover mean study game single step allow compare algorithm online bandit setting optimize bandit embed choice note ranking mention online bandit al ambient polytope number embedding bad large function fact derive simple ranking share use derive ranking
use intelligence surveillance sensor motion attack device cell plan security paper model detection dynamic foreground membership blockmodel embed neutral business etc life depend community interact community control foreground networks blockmodel approach exhibit realistic connection observable active implie pattern organization behavior establish principle upon historical pattern make difficult meaningful indicate planning may individual subject ten result specific representative connect special confirm close use transition clique dense subgraph detection analysis investigation spectral pearson discuss define simple vertex mu mu mu adjacency adjacency degree application coordinate abuse notation direct mu mu incidence matrix orient
finite third concern follow page material convergence strictly dimensional spirit u ms eq necessarily strictly sequence strong satisfy proceed three observe construction expression q construction assertion notation q strictly stationary strong assume eq u therefore hand previous inequality uniformly term equivalently recall denote lebesgue whose start rate parameter imply abuse notation obtain denote previous application positive eq converge uniform asymptotic weak theorem asymptotically instance proof asymptotic construct continuously converge sequence converge index ns ns ns ni nf j u dc ingredient ns u ct n borel function consider bound extended h immediately g n complete remain q restrict supremum definition
graphic consider rank thresholding propose noiseless convergence rate propose determine generate real rank report show tensor measurement research netflix sensor solve minimization become minimize rank satisfy nuclear rank relax np bl recovery denote corrupt since minimum algorithm base transfer set atomic hard introduce among iterative thresholding easy implement researcher vision graphic tensor multidimensional low tensor pp author tensor unfold
resample leave item head sample analytically tractable bind deterministic bind create variational jensen depend double shorthand contain drop away evaluate case expectation formulate dependency purpose q know formulation cut sum finally gradient factor sequentially slow edge loop factorize factorize recover kullback precision functional derivative precision analytically suffer vertex burden sum full expectation run inside quantity plausible graph amenable remark pt refer reader require insight publish month periodic
change gradient need cm center dataset bottom give left dataset good colour use simple variety explore sag detail accelerate variant convergence convex rate convex accelerate sag basic sag size ultimately bad sag method proximal become assumption function could constraint use iteration form prox gradient accelerate explore replace sag apply experimentally proximal sag achieve convergence sag indeed scenario sag method closely work key advantage sg sag dependence wise offer sag whose sag newton sag approximation hessian expect order increase iteration multiplication choose distribute sag suited would explore sag iteration architecture preserve convergence serial sag suggest particularly delay failure non focus sag may converge strongly termination major sg slow terminate sag achieve sag advantageous term termination sag iteration termination criterion quantity optimality disadvantage constant large cause possible trust lead convergence behave optimum size theory sag iteration achieve sufficiently support european award mark schmidt also fellowship engineering research rate primal wise sag pass sg sag row write primal
sir filter degeneracy accumulation mc add artificial posterior also refer overcome density approximation dynamic approach together efficiently introduce sample non state space several transform add quantify fine tune er benchmark obtain artificial estimation priori sir filter ahead allow region superiority sir importantly poor filter couple cost compare sir filter often impractical move diversity monte avoid requirement kernel move introduce diversity target distribution base know illustrate restrict certain low maximum ml unlike state ml solving parameter recursive measurement respect refer state impossible resort suitable smc line point dimensional problem scale poorly term alternate ml method stable em exactly hmms smc class exponential distribution
gaussian analytically infer derive likelihood input likelihood estimate hyperparameter hyperparameter chain reference kernel discover pattern negative equation associate vary input value weakly stationary square continuous complex measure entirely determine substitute se se stationary correspond gaussian spectral origin gaussian achieve wide indeed mixture gaussians distribution dense kernel arbitrary motivate spectral location first substitute gaussians component pp tractable spectral arbitrary gaussian supplementary form inference expressive contain nevertheless provide specify contribution mean component period standard deviation length determine component
df specify df specification correctly specify recommend transform subtract side specify df inefficient df test calculate eq omit apparent increase misspecification induce asymptotically know intercept inconsistent although prefer structural
result regret multiplicative way additive meta algorithms ucb delay feedback organize follow delay adversarial analyze ucb result kl ucb delay appendix model monitor forecaster decision maker make action possibly reward delay reward side forecaster environment forecaster element pair indicate instant forecaster receive reward receive reward may singleton feedback end delay delay arrive definition forecaster prediction side side prediction reward feedback reveal agent observe together forecaster cumulative reward measure static select f forecaster performance consistent
theoretically analyze subspace scope rank factorize matrix exist meanwhile apply svd svd approximate square penalty usefulness toy nonnegative realization repeat simulated entry issue method simulation index nmf whose residual small frobenius subsection compare realization cell medium pca svd rank plus rank plus projection svd approximation nmf repeat nmf summarize small contain negative highlighted font svd contain e input nmf approximation rank apply lie principal angle
net require huge stochastic train choose instead user penalize
formally structure edge graph structure densely connect group essence resemble cluster group manner community think common interpretation nature social network think social group share tie important biology subject much devote develop fast detection verification community review field review field reader despite effort detection definition ref community community community problem question large mean versus another kind look related indicate lack raise community many scale one discuss modularity subsection measure existence include regard community underlie therefore exist dense social discuss identify community probably community network community existence
class rule find literature error receive attention wireless random mapping extra real flip coin classifier root complement leave allow tree leaf consistency tree split find median child cardinality stay send appropriate cell round dimension split random split consistent soon indeed specify median splitting split leaf clear decision fair create classifier consistent tree abuse marginal distribution median principle use randomization cell
bring close propagate understand proportional rx n estimate normalize auxiliary filter choice user ideally possible suitable choice particle therefore lead although variance substantial possible go auxiliary filter time generate particle difficulty filter unbalanced apply simple improvement coincide become transition occur contain state propagation add create
arbitrary divergence total variation divergence constraint primitive divergence explain give equivalent simple divergence divergence primitive equal constraint loss generality assume ratio primitive convex piecewise result interval two define linearity check divergence primitive determine divergence primitive divergence result give complicated equivalent write integral precisely characterize simple conceptually primitive least variation direct expression special appear let arbitrary primitive equal also problem q equality oppose lie lie last sign reduce finish show final term plug value replace need lie complete note lemma arbitrary total distance consequently inequality expression get much interest divergence satisfie symmetric include
despite remarkable place subject huber consider effect replace huber leave robustness tail sensing provide strength additionally uniqueness issue discussion process work contamination restrict small fraction contaminate completely hypothesis permit solve successful high random infinity ratio put deal outlier noiseless adapt sufficient normal thorough couple perform detailed take characterize
avg reduction fouri kernel fourier transform p equivalently interpret feature fourier construct approximate random feature nystr om major scheme random nystr om dependent unlabele generate fouri proceed label unlabeled data q compute cca view view contain probability theorem since consistently performance r std table average individually relative performance trend cca joint htp pt r performance nystr ever nystr om htp c david z nystr om
goodness favor large take signature agglomerative description argument package help file point identify change univariate normal seed library period r period output r alpha output r r permutation main col estimate red recommend lie general depict time along use code member alpha opt output merge marginal multivariate univariate package mean package r r period mu r period
automate correct correct bias simulation ratio employ measurement phenomena acquisition process contribute instrumental bias uncertainty statistical nature uncertainty propagate subsequent processing ratio weak fraction dramatically near away nearby paper characterize derive number skewness degree weight tail useful estimator sensitivity conventional formulation robust might
sdp solve perform factorization number compare successive projection algorithm post process propose see use remainder post et svd truncate svd extraction propose recursive variant refer hyperspectral code available intel core cpu ghz ghz take exactly zero row locate towards outside hull centroid column near matrix contain generate report fraction observe noise explain another post identify level confirm explain follow middle inside minimum volume ellipsoid algorithm perfectly sdp remain unchanged
genomic accurate predictive improvement predictive protein several critical family discovery genetic human l enable brain tumor human computational comparative ensemble diverse set ensemble address examine diversity tradeoff make simple enhanced meta insight complex drive wide diverse application ensemble begin experimental ensemble stack standard next examine connection stack impact heterogeneous receive conclude direction
finally project polynomial max entropy approximate solution omit polynomially appendix reverse count max entropy interior note ellipsoid algorithm highlight dependence entropy oracle raise issue count oracle irrespective allow interior work oracle separation separation either violate second interior interior recover apply ellipsoid target forward direction max cut ellipsoid overview couple remark unlike ellipsoid one approximate rest organize follow include convex program optimize max need solve program formally state max section combinatorial asymmetric feasible result paper program around origin ellipsoid show direction give solve max program proof omit body minimize respect distribution appendix introduce denote plain letter zero usage context reason hence also denote number set letter denote probability proportional denote emphasize additionally inner vector norm arise convex subset correspond polytope linearly describe former latter give separation require access separation satisfy separation say term oracle omit detail standard count weight weight
unitary modulus stable application audio classification impose shall cover convolutional initialize transform layer operator complex abuse unitary modulus layer provide encode feed unitary reduce complex modulus ideally group compute variable
never fall scale coverage sphere radius convenient formula coverage confidence odd contribution root scale sense sphere problematic contrast minimization coverage excellent compare scale expected volume figure coverage scale
potentially graph scenario affected vertex term match match adjacency matrix correspond find matrix nature graph tree diverse problem include relaxation review focus relaxation relax permutation hull doubly consist sum ij vector though doubly instead correspondence permutation linear graph combine relaxation relaxation
one back value instead feature associate ij q similarly functional may associate linear model coefficient also direct equivalently observe shall observe functional may n may function one projection decrease functional extension
instead pool architecture invariance complex rotation acknowledgment research modelling technology auto pooling appear image sequence make feature pool rotation video resource rich sequence available human birth plausible like believe pooling pool auto
subspace subspace small pair formally angle eq set cosine minimum angle measure cover subspace live cover projective distance cover radius distance relative cover radius normalize place span diameter angle neighbor geometry covering set form cover attain cover cover diameter geometric origin cover radius fig equip omp proof contain section maximal diameter def principal angle less cover first rhs coherence point attain cover near rhs cosine minimum angle pair diameter subspace intersect thm away intersection radius exactly subspace identifiable another alg residual close subspace cluster precise residual subspace cluster alg provide geometric interpretation lie convex hull span consider onto close disjoint subspace inside hull plane guarantee hull normalize incorrect guarantee angle length project geometric visualization disjoint show projection outside lie convex point along normalize point outside lie hull point subspace ensemble mutual simplification corollary diameter occur point section omp disjoint
mathematically principled theoretic metric infer assignment denote block recover vi differ splitting divide outperform alternative decrease fail perform well far infer correctly bayes thresholding sbm poorly test threshold structure substantial exhibit distribution find block thresholding noise sbm sbm find
experimental make fast number filter computationally perform retrieval noisy measurement vary noise recovery recover reduce error guarantee choose b show plot vs cc incur incur error entry complex decrease geometrically suggest like grant lemma conjecture section phase retrieval sign decade seminal solve alternate phase candidate despite wide practice guarantee resample approach geometrically
although constraint force activation zero deal distribution train digit comprise image publicly create letter comprise used mail work post achieve extensive task check ensure uniform figure rbms create pre manner computationally digit rbm test
definite integral use introduce supervise likelihood matrix wishart posterior parameter database
fx x fx configuration result store conditional configuration together conditional possible could develop maintain markov leave create vary show reconstruction error iteration gibbs field gibbs solve task bar deviation field comparison present significantly ij generate final show average gibbs share skip crf recognition name entity identification people text random crf conditional weight relationship
ij ard important covariate lot hyperparameter cox since capture hazard censor censor infer bayes pd likelihood truncation occur censor report time censor individual contribute occur determine posteriori optimisation base matlab laplace p derivative w take pd numerically respect negative hyperparameter log gp hyperparameter wish individual test q predictive additional noisy variable predictive event hazard may make occur numerically event give regard prediction exist interval censor survival parametric survival hence construct censor discussion failure family parametric handle covariate numerically example event gp readily accommodate censor interval censor define pt l st st take posterior ignore numerically inference purpose use assume hazard hazard cumulative base hazard
modern going burden improve important dimensionality probabilistic provably tractable technique preliminary unknown letter explore role probabilistic graphic graphic avoid inference represent graphic probabilistic enable resolution graphic many synthesis hope extension ultimately part analogous image acknowledgment grateful explore break idea computer history directly implement vision via possible short flexible generative automatically interpret world generative graphic program
limitation modularity ref modularity context modular infer occur develop refined block hierarchy serve information dramatically change replace characteristic enable much scale generalize hierarchical pattern addition allow arbitrary structure furthermore increase resolution attempt simplest overfitte spurious fully nonparametric sec proceed nested block twice parameter two respective edge average equally generative sufficiently fully stick variant convenient call exactly additionally specify graph correct network capable incorporate degree variability inside capable provide description arbitrary traditional count block block count node loop may generative generative level finally describe network resolution generative infer serve level elaborate tractable nonparametric flat impose preferred pattern restrict binary special hierarchy strictly modular argument variant ref describe model selection selection undirecte everything straightforwardly applicable direct network expression membership edge rs e edge realization maximize rs low entropy traditional
little communication arbitrarily gibbs give combine machine act communication final generate sample ability popular tool perform bayesian major asymptotically posterior grow point mcmc big store machine mcmc setting researcher primary chain parallel speed burn sample perform computation involve subset exchange sample greatly increase computation machine external tackle allow burn among machine
variation set achieve relative top perform across multi array system array record visually measurement separate pilot v stimulus presentation hz ms follow ms period movement sr good center indicate large experimental response image repeat result repetition experimental accordance national health institute technology raw neural raw firing measurement ms ms site fire fire gray response minimize deviation response divide repetition three complete deviation response calculate mean across repetition site post processing procedure guess possible neural decode influence appendix
distinguish square relaxation expansion therefore kind interesting construction code parameterize gap context eigenvalue eigenvalue gap graph gap sum work construction analytical vector small replace expansion implication solve regular expansion vertex expansion let c large square question relaxation display exponentially time aware problem low degree proof bind sum instance method instance relaxation instance conjecture concrete type tailor interesting plant unsupervise especially relaxation round algorithm analysis relaxation sufficiently flexibility universal round give round sufficiently mention progress analytically choice sparsity yield usefulness proxy plant conjecture optimize sparse unique game candidate improve question work suggest negative relaxation gap regardless round notion gap round used instance round study gap might question constraint mr section theorem conjecture observation construction question question theorem theorem chapter item fact reduction mm mm inf tr sdp sdp lp opt conv median I ph I f os p please lemma old david round relaxation square hierarchy connection relaxation square map possibly
bin external bin external event denote occur bin multiply matrix produce external time effect bin spike effect constitute density parameter hyper simplify column construct stack stimulus history row simplification tp namely select maximize maximization em iteratively marginal complete function
characterize observational markov markov equivalence object dag also observational assume equivalence identifiable distribution selection dataset complexity avoid potentially observational criterion grow know fx hx connect suppose exponential stand statistic tx row let family two intervention assumption realization x fx x also exponential straight forward calculation definition finish calculation express find immediately claim bic limit must different dag fit gaussian representative family natural write determined natural parameterized eq showing manifold satisfy conservative intervention furthermore manifold smooth say embed sense technical throughout assume l triangular matrix immediately clear composition smooth ib assumption
recommendation introduce scientific recommendation recommendation citation although recommender day still lack standard traditional system human computer evaluate especially experience flow system system apply bag word corpus method near knn preference recommend preferred paper get specify target neighbor finally section describe component h research research digital digital unique page year paper retrieve regular get list iterate
condition conclusion tucker tucker equivalent equivalent substitute get positive c solution provide software root simplification regard operation complex expression nevertheless would real number part able optimization follow calculate formula formula consideration design write consist th objective difference analytic allocation case case essentially case mathematically
co occurrence count joint proposal derive visual topic count entire intuitive count label random consistent outli appearance white match would image model image condition variable augment couple topic probability link semantic augmentation idea similar general iterative sampling guess posterior posterior successive substitution visual bag perform gibbs update topic complete eq region proportion image wide resp
payoff view combinatorial arm bundle choose interval hard armed arm number overcome spaced estimate ii payoff arm payoff sublinear linearly online contextual consider linear result combinatorial involve stochastic consider another armed space payoff distance many use contract framework instance wireless service recommendation recommender etc rating follow define contract preference
regardless stein interestingly estimator mle coordinate stein g reference therein I e estimate empirical average stein asymptotic rkhs regard rkhs although phenomenon dimensional true probability distribution specific well paper mean light estimator fundamentally consider leave cross shrinkage lastly fix mean empirical r algebra function exist estimator admissible estimator toward amount reduce estimator standard well risk q kx simplify rely important implication suggest exploit
lead setting ol careful view easier choose possible may choose give analogous label model structural omit simplicity orthogonality logistic eq usefulness hash min hashing sign hash explanation hashing form require hash interaction sign hash effect substantially need interaction model computational note predictor include start interaction large forest ensemble suffer similar fit p interaction note include model general tensor zero seem sparse let hash proceed exactly simplicity technique scale furthermore b w exist iii suited situation grow interaction tight interaction model case structural necessarily include high interpret main increase uninformative vanish require requirement ols apply would ols regression code interaction would complexity would norm allow large theorem suggest reason may excess risk replace
search type become case part procedure inference likelihood marginal conceptually solution investigation behaviour marginal likelihood laplace similar like collect national project would like discussion thank collect modeling model explain analysis mixed effect specify covariate penalize spline considerably choose drive contribution extend unified inferential population survival grey model environmental survival logistic regression interesting keyword array capture conduct understand population focus estimate survival researcher uniquely mark capture record previously identify assume individual recover individual survey recover population typically lie probability live capture capture extend additional absence covariate class model incorporate
supervised average non node hold fraction discuss deterministic bottleneck operation deterministic calculation amount hold calculate gradient vector us sdca sdca deterministic since cost sdca dividing however node enable indeed namely scalar implementation example node divide sdca latter
corner corner arrange usually synchronization hide individual corner visible due feedback layer connectivity first represent visible layer layer corner represent neuron part demonstrate need bar note without intermediate possible identify represent virtue layer globally visible activate space demonstrate content group train additional geometric toy triangle combine digit tendency network distinct shape shape triangle visible synchronization image mnist draw analogous run accord plane object advance k neuron assign read visible obtain especially noisy network segmentation bind apply principle arbitrarily abstract select neuron accord phase representation one layer image population simple decode main layer phase object representation end treat perform pass reconstruct though fashion somewhat apparent unit represent visible assume
generation ever complicated use think carefully employ consideration universe live parameter manifold improper fact appearance density fewer address evaluate summary relation distribution application reason beta frequency population classified long history continue encounter close analytically ia integral question want derivative derivative ia da ia da da vanish rhs evaluate analogue familiar investigation distribution relationship beta link know example also examine beta write lie justify theory reliability evidence come application model project study concern application methodology measurement variant follow
solve np fact cut encodes rbm hard optimize relaxed relaxation semidefinite optima qp value semidefinite sdp indeed relaxation project row round semidefinite round namely sdp
euclidean distance matrix form double double denote center follow matrix p kl q representation reduction formulation adjacency matrix yy ts yy se view laplacian construct weight adjacency adjacency row therefore express variance tr yy maximize
b pre admm require may empirical big later dynamic bregman previous approximation special minimize fix addition direction update b subsection firstly begin tf firstly term take bind
capacity generic parameter entropy aggregation hellinger van development work pattern back without assume output perform risk function place cast recurrent descent latter expectation call closely therein parametric covering number average expectation prove obtain fast minimal assertion page formulation lee show least attain instead class pseudo dimension additionally non estimator selector force class erm problem risk free statistical problem aggregation outline model derivation follow related aggregation recent development survey excess good obtain sharp excess risk correct rate remark convexity obtain sharp cf among mention erm selector suboptimal weight show suboptimal error involve find erm connect erm author part subset erm propose
follow aspect modelling problem control switch discuss detail generalise subject publication chemical reaction well specie volume specie interact specie molecular reaction occur reaction occur equal chemical depend principle reaction chemical model model stand volume deterministic volume describe deterministic biology preferable nevertheless useful volume hard input belong mdps belong entire state
gradient orthogonal gradient minibatch sample weight number time limit simplify align reweighte dense equation sample gradient reweighte maximally see illustration practice expense word backward pass minibatch thick value commonly unit normalization etc produce non smooth enough variability sample lead expect even though value loss
variate lipschitz reward nearly albeit slightly reward completely bandit dependence would polynomial prove describe thorough phase matrix estimate represent observe expansion approximate sampling construct consider collect far tr sampling x eq k n employ standard solver rank noisy linear furthermore note encode hope well row proceed demonstrate recovery discuss important isometry rip mean isometry measurement argument inequality rip
object digital consider shape dimensional shape curve et al gain researcher shape et et small shape manifold op intrinsic fr riemannian papers al riemannian preference riemannian gradient search close exist approach address make distinction population statistical paper organize follow data hilbert manifold object mean direct shape contour contour due dimensionality neighborhood remainder concern digital imaging address correspondence problem contour digital collected bootstrappe database end extension manifold analysis manifold capture digital image hilbert banach space hilbert hilbert hilbert space open transition map projective space model angle give map onto map open subset open use line hilbert
et et et et r dropout lm dropout presentation comparable remain rr dropout line classification plain resp validation resp training table rnn alone lexical rate lexical character system lexical dropout lexical constraint dropout relative vocabulary database open recognize vocabulary word model rnns network weight activation l weight lstm lstm weight lstm small dropout decay lstm
forward exchangeable likelihood answer parameter essentially jeffreys gamma g jeffreys minimax three one become equivalent ahead player observe organize mathematical gamma family strategy version sec detail short definition notation distinction short history define conditional density example lebesgue strategy define joint conversely define conditional come predict element reference set expert take family
monte carlo kl heuristic classification link run testing world citation top portion list corpora paper machine learn topic consist medical research paper diabetes three corpora ground corpus appear record occur document treat correct variant em implementation author try carlo well variety link sigmoid dirichlet perform quite e iteration implementation log observe try various observe vary document run iteration increase successive log number corpus large within corpus
variable interest end reasoning note expansion large ar reasoning conclude bayes substitution correspond particular case numerator come finally put together appear exist conclude assumption proof exist hadamard product look integral element equal theorem constant diagonal conclude straightforward bayesian assumption necessary
path obtain row generalize clarity segment interest black solid line row count clarity segment precisely exclude start point black solid inference fix hmms precisely analyze segment I posteriori use hmms fit inference tool ii fitting use segment training sequence arise justification suitable iii extended bayesian first question posteriori give system instance map ml action maximize expect utility u introduction allow utility regard question posteriori sensible hide interpretable class emission meaningful correlation class place hmm observe posteriori segment model sensible assume algorithm firstly rao sum forward recursion augment final counting compute pass unconstrained find unconstrained map maximize carlo viterbi augment wish draw b worth hmm data class correspond I observe em need phase conditional found label hmm approach hmm text retrieval ml might proximity could resolve issue assign state implicitly train unsupervised manner path unobserve second fit infer contrast add similarly suitable application minimize equivalently maximize segment represent fix incorporate either apply additional
poisson instance base bag alg c else train learner hx mx times base time alg online note online counterpart early stage example extremely therefore generate example online intrinsic problem distribution batch unless begin consistency batch randomly near instance instance cost introduce poisson begin boost key formulate weight formula observe alg online track poisson treat tp tn fp weight parameter tn unweighted code alg initialize mx mx mh hx mx normalization track unweighted error respectively alg initialize base mx p hx mx algorithm begin example first remain example get record negative batch boost derivation example class class
covariance eq consider concatenation achieve correspond note almost surely variable equivalently write k k k side helpful simplification notational write factor setup parallel major event introduce triplet triplet brevity event write intersection event decompose key triplet jointly collection variable replace tail second conservative limit e tt course relate calculation functional latter concerned admit simple bind define fix establishe enforce tail define assumption regression subset satisfies q tt trivially reduce therefore impose theorem former limit trivially latter nonzero conservative exponential precisely unconditional disjoint implicit argue remark likely literature magnitude tend carefully place assume one model path may sign zero pt path extend possibly truly rather strong rather precise nonzero still conservative covariance nan investigate general form path pt test lasso right respectively solid slope break predict predictor orthogonal truly predictor forward inactive average panel chi square apply provide approximation poor approximation simulate set standard se bottom corner se se enter conservative nominal actual error
additive perturbation sense produce perturbation last additive stability measure layer denote operator layer result convolutional let denote singular linearity hand jacobian coordinate expand correspond common operating regime conservative fully fully fully describe denote generic convolutional input input corresponding size conv conv conv conv conv fc fc fc imagenet soon blind explain generalize
approximately normal dimension tree round sometimes time recursion stop prevent axis ratio lebesgue cell contribution integral cell lebesgue accord monotone computation choose lebesgue q lebesgue construction choice result acceptable region volume volume outside offset decrease dramatically
continuous exception main exactly costly since fortunately greatly improve introduce nesterov accelerate care since converge degenerate column unity assign overall importance decrease applie acceleration contain one soft thresholding operate active strategy iteration threshold refine preserve continuity noise implement hard thresholding pseudo k k classical benchmark algorithms evaluation describe bernoulli activation control kind sparsity actual zero select case sparse signal become unless criterion criterion performance application interest source notice prior apply good contaminate noise source perfectly mix mix prefer less noisy criterion adequate measure separation al propose technique
way size much small people old college go fine median rd combination explanatory make impractical group want also pool component propose shrink eigenvector vary principal component mean carefully covariance model comprehensive review method alternative way parsimonious regression serve multivariate parsimonious explanatory simultaneous multivariate explanatory model definite matrix model special unit beyond additional variability restrict matrix rank essentially residual element flexibility effect representation uncorrelated model allow deviation baseline additionally independent allow flexibility allow without require
prediction feedback relationship suggest able result confirm suppose square apply give scale assume time period raw independent relax correlated error simple treat I use noise feedback extracting offset conditional moreover extend general feedback artificial noise reason natural reduction match state require prior feedback previous f tf feedback feedback prediction artificial term practice though observe I
ga partially support nsf grant foundation long feedback early manuscript european theorem definition conjecture rgb qr become area frequently discuss dimensionality specifically input low intuitive derive sharp estimate guarantee bound experiment complement think analysis algorithmic computing approximate qr decomposition development see probabilistic nature address great
quality recommendation consider else user rate else boolean covering calculate type ht besides ability near rating number boolean factorization try search near neighbor rating rating datum filter big mae recommendation construct rating filter method
united department centre research university united cognitive brain behavioral sciences mind laboratory imaging activity extract ica map voxel number spatial map spatial map usually similarity inherently explain variance develop reduction conjunction diffusion
domain extremely form massive star rare phenomenon happen unique perform numerous experiment scientific motivated search feed search periodic emission predefine investigation broadly speak separate processor frequency interference search signal interest crucially volume growth law growth reflect improvement survey specification survey fine frequency survey decrease thereby short period couple fine processing inverse criterion majority
advanced complexity novel insight dynamic complex interaction human global association project impact foundation project grant software discussion comment version fill stroke university department biology university united theory university couple cp covariance frequently detect temporal analysis cn usually cp formal conceptual difference network use air temperature allow cn complement eigen statistical supplement toolbox correlation understand large sir event statistical later measurement device rapid allow field air temperature pressure height index labeling aggregated time technique couple application range evaluating run numerous linear extension principal mapping classical last decade powerful extract volume quantitative rely reduction study series time
twice set moment analogous x ns resemble deduce distribution stop geometric though special mild recurrent monotone geometrically reciprocal first also compute desire computing practice turn solve precede form upper integration notational simplicity equation interest e g alarm suffice yield time solve equation allow analytical hence numerical interval usual uniform assume differentiable bound equip alternative see conclude banach ph thesis space approximate equation residual identically require proximity residual readily q remark
th principal finding principal component summarize use observational kernel representation determine specify kernel kernel th respectively choice unless geodesic measure km range reduce assume separability along depth result twice produce discrepancy imply isotropic discrepancy flexible capture use mean obvious advantage result correspond regularize simulation study automatically calibration discrepancy variance simply component therefore less examine imply write j eq gaussian lead good orthogonal basis since assume principal covariance assume separability positive derivation supplementary therefore restrictive covariance provide rich separable cross validation study adequate prior hasting allow specification observational straightforward inverse gamma
explain terminate iteration independent compute hash shift regime random careful hash check hash function random similarly bipartite decoder recover spectral component decode fortunately rich code bp decoder characterize perfect analysis step rigorously rigorously analyze bipartite treat variable fully random left ensemble variable distribution polynomial decoder variable node edge node around characterize exploit equation evolution show concentrated solution differential remain edge another argument show edge probability explain hash function component label return index terminology shift hash order far intuition hash subsampling ix b b output depend label multiplicative constant see bin hash obtain dimension successfully reason output bin hash output obtain sparse denote whose uniformly support size put component rs object replacement size sparsity rs infinity
partitioning yield pd np problem clearly analytically intractable definition allow general hypothesis greatly treat independent unknown across binary value observation b dr yield detection region pd application harmonic optimum pearson propagation prior function one observe vertex along event note ratio function equality threshold optimum become detection optimality harmonic propagation detection entire maximize yield pearson simplification treat multiple hypothesis detection maximize computing propagation yield propagation pearson pearson propagation propagation eqs optimum likelihood evaluation accomplish clique subgraph embed enyi theoretical world representative one performance specific set network therefore simple blockmodel insight characteristic evaluation time eq modularity detection standard receiver characteristic network foreground false blockmodel world structure order parameterize edge probability edge surely connect adjust r connectivity quantify detection foreground fraction illustrate roc performance foreground community embed matrix activity enyi foreground
individual incorporate dynamical effectively filter theory online establish tracking mirror descent characterize perform intractable algorithm operating prediction simplify adapt well either parametric class establish scale theory setting estimate regime incorporation dynamical key mind incorporate dynamical modeling role regularization regularization increasingly setting significant gain ill pose examine sparsity dynamic environment remainder formulate problem notation use throughout paper optimization dynamic mirror descent bound describe work sequence use dynamical family self point make conclude remark point forecaster generate forecaster reveal loss map space real loss convex function compare new possible dimensional assume incur reveal sequentially reveal
laplace assume draw wishart become spike typically something receive considerable one refer change reversible propose flip edge time transition hasting efficiency sampler hmc lasso allocate sampler compose run precision take gibbs posterior hmc wishart conditioning miss cover clique mc consist clique sampler identity run burn
author conduct thorough investigation numerical algorithm classify sample n n need take visit box finally number visit box box empirically threshold decide refer phase space n take heart representation important issue long contain beyond increase followed classification window fast reliable diagnosis length however systematic investigation issue citation visit box signal citation need citation need citation suggest short suffice citation need consider
infer team plan describe people plan execute collaborative design model use modification team two participant amazon great participant participant team scenario describe ask plan planning summarize final plan previously analyst review planning resolve member plan description necessary perform analyst review planning predicate final plan plan sampling plan gibbs sampling mh evaluate quality final produce accuracy allocation room infer percent plan predicate appear team plan rejection predicate team plan plan predicate true
note give step q obtain jk ij ss last bind bind op pe theorem convenience event outside block k provide structure still different work union replace sum necessary limit theorem depend chebyshev yield pp n op result bound logarithm bind large integral q conjecture op chen replace n therefore hold rely conjecture replace eq substitution integration part integral induction imply theorem px k n nx dx jk note therefore
couple difficult gets observe especially set situation remain whether causality note could lag good computation time demanding finally human ed rejection eeg signal hz sec ed duration sec compute period slide window sec connectivity brain estimate causality window channel robustness causality measure network channel fig subset connectivity strength see ed ed ed result e connection successfully causal intensive couple causality system component identify embed lag drive study detect true lag bivariate termination initially
selection repeat set copy baseline machine feature copy average crowd binary baseline present add set regular enhanced training label comparison random split baseline score bad baseline zero correlation assumption small less suffer reason great fitting probably baseline predictor number h test set collect access label feature label pool serve test arise value slice characterize future introduce average future aggregation majority future feature change crowd wish helpful discussion
ratio regret major challenge bayesian sample double happen factor close normal marginal double sort obvious illustrate normal double see bayes proper meaning correctly size value log double blue estimate credible confidence collection parameter achieve proper coverage advanced inference opposite inferential tool analyse datum nonetheless certainly address represent ratio integral computational get reason space series credible simulate impossible thus monte method von application law problem intractable large exist result lead compute simulate monte sample exhibit true interpret sense importance integral
informative figure frame representative track htp sequence wise reconstruct reconstruction row far template pixel pose inaccurate mis template drift target pose unlikely couple problem track target pose non however robustness life tracking solve state novel simultaneously quantify uncertainty pose match template automatically good template method estimate template part state space descriptor descriptor pixel pose illumination definite covariance model riemannian novel propagation free constraint impose inherently deal variation art incremental pca change target clearly organize introduction
predict dynamic large change interested detailed facilitate difficulty highlight advantage nmf static remainder article introduce static world article brief discussion singular connection detection give svd aim discover community eigenvector spirit rely low rank approximation search approximation relax namely decomposition compose entry refer advantageous visualization negativity flow relationship expression problem optimization descent algorithm nmf mine overlap community detection static correspond good interpretability always invariant multiply multiplicative nmf negativity interpret contribution mutual use instance assign assign community belong
endowed indicate contrast hdp appear document informally think coin probability outcome coin process result define property beta process encourage realization document interpret inclusion feature associate parameter collection homogeneous infinite infinite point see beta limit stick construction process q topic associate visualize discrete beta coin formally base blue cumulative draw beta blue flip realization correspond white characteristic beta trait infinite coin finite subset bernoulli show mass imply coin likewise likely draw though variability conjugate bernoulli process latent examine predictive bernoulli ibp marginalization multinomial chinese restaurant ibp develop portion ibp specific specification limit topic return focus bernoulli treat dirichlet solely indicate distribution set figure model building background idea membership model use stock volatility eeg dependent exercise switch describe
every concern fuse consider fall indeed intuition contiguous incidence graph composition choose symmetric rr form norm correspond proximity find uv respectively space obtain penalty choose nuclear transformation consider give turn attention machine svms
bin dimension since curse dimensionality real generally unknown extension assume average realize yield consistent obtain initial set iterative remain unchanged consistent sketch denote initial yield estimate iterative regression minimizer outline extension baseline situation baseline extend algorithm subset eq assume class response minimizer population criterion moreover assume exist fs condition fs lemma rule come establish level minimize minimize ns nx function context end consistent translate dependence transform show cost involve variable seek
equivalent last simplification q readily identify without optimization
local minima choose small sparsity satisfy direct gradually work hyperspectral column figure process remove reduce six hand appear pure unit penalty minimization pixel fraction sum error nonzero sum sparse solution trying move coefficient shrink magnitude sparsity direction ball job preserve magnitude enforce reflect fraction row abundance abundance hyperspectral expand consist consist hyperspectral california remove outlier extract give sense signature particular signature construct dictionary signature red signature synthetic construct truth abundance sparse noise abundance
sequel lemma respectively control term deviation remainder control taylor term statement strict feasibility guarantee strictly true suffice simulation follow poisson follow poisson correspond identical edge distribution repeat probability recovered sparsity estimation constant factor corollary corollary recover curve size align scale see poisson empirical probability successful recovery scale curve b example meta estimate exponential graphical gaussian high throughput genomic learn microarray throughput however even breast graphical iii cancer sequencing http short rna post measure throughput sequence highly skewed total count experimental process brief quantile correct adjust sequence bottom
fr universit paris des fr institute paris bm laboratory paris paris paris address surface neighbor allow thus shape translation term curvature cell enforce come impose
privacy whose policy statistic posterior distribution depend way firstly approach secondly smooth likelihood dependence around arbitrarily may statistic interested utility relate idea select feature statistic cumulative feature connection value mdp drawback yet hyper tune unlike bioinformatic interested find reinforcement simple paper trivially policy performance history trajectory utility mean order range hard accept two
rl thompson perform policy hence thompson limitation prohibitive computationally introduce methodology construct representative perform examine used one hyper feature subsequently combination algorithm domain pair evaluation gp uniformly draw iteration offline evaluation first draw start horizon collect fed environment calculate end last episode perform schedule preliminary compare online episode ht know domain car force vertical angle angular velocity three force receive negative episode successfully balanced episode perturb close specific discount basis suggestion action car top hill velocity randomly forward reverse receive reward reach begin
number sort pattern mutation pattern pattern remove merge remove though algorithm unnecessary pattern substitution subset contain divide group contain sort pattern check search bad complexity large group evaluate protein whereas negative randomly bank protein threshold formally protein summarize characteristic avg database maximal dataset ds protein ds domain generally approach aspect select pattern state subgraph frequent threshold among substitution substitution perform detect similarity biological
composite mean algebraic matrix composition matrix multiplication regression output theoretically single variant approach represent matrix involve linear two expression compose vector linear concatenation phrase reach task calculus rnn task specific label train rely manual nature composition rnn whereas present composition allowing finally follow formal semantic treat certain argument calculus semantic whereas treat matrix difference lead rich semantic leave study word meaning adapt distributional context distributional single adapt phrase semantic derive formal phrase represent semantic argument scope associate syntactic
q contingency table distribution high independence support hypothesis integration component discretization independent bin distribute empirical figure distribute respective find horizontal discretization conditional irrelevant numerical cumulative bin
spirit th individual favorable well use rare favorable preference among region term term correspondingly htbp selection index order ordinary phenotype good give guide individual region genome individual either parent region phenotype always locate allow occur block fast calculation proportional matter involve approach problem might marker collect genome linkage missing nest sequential view genome quantitative trait
optimal prove theoretic establish suboptimal closeness appendix technical section assume draw draw simplify trick begin closeness occurrence th output characterize establish algorithmic absolute distinguish suitable probability repeat exponential simulate distribution probability right distance next though normalization crucial though theorem result improve constant fair coin eq seen condition subject tail fair coin time variance alone coin value
minimum cluster motivation completion sparse signal incoherence support condition incoherence manuscript appear remove second incoherence much incoherence schwarz inequality important semidefinite psd translate previous quadratic require clearly regardless block semidefinite show simulation recover psd rank would possible incoherence minimum recover nuclear trial axis small recovery mention introduction crucially present structured completion svd projection completion unobserved step large row column svd return result concentration theory eq row corollary appendix small due extension problem application
element intersection solve occur intersection intersection change immediately immediately vice intersection determine curve none curve despite intersect intersect support intersection th th curve examine exist another intersection support ignore curve order affect one leave associate adjacent index intersection intersection point examine intersection intersection visit continuity identical opposite relative order positive fully intersection collect large interval collect support
w yield orthogonal make black box singular completeness decomposition tensor ds atomic proper computing decomposition factor ds atomic define note return rank th repeat tolerance value employ variant furthermore allow recursion result might sequence stability tensor ds atomic return suitable
address matrix instance number output frobenius label multiclass multiclass instance take account facebook employ unique challenge sparsity imbalance metric etc exploit improve accuracy simple pair wise specify occur hierarchical high relationship certain show connect evaluation metric relevance instance rank conventional classification algorithm label training method combine access need obtain final denote label specify simple vote favor certain classified vote receive vote posterior probability label average prediction correlation different label prediction properly information label lack
standard belief propagation assume fig optimality hamming coordinate semi subtree fig contain theorem ccccc c bp jointly subtree passing repeat message max product guarantee mixed product message decode mix bp belief satisfie crucially fix product similarity share interpretation demonstrate unclear maximize variational objective cause message ingredient detailed special optimizing function rearrange maximize message effectively asynchronous sequel illustrative toy mixed bp toy bethe bethe terminate forward backward pass track eq map rearrange sum elimination order easy property hold bethe terminate approximate function operator root cause max guarantee show maximize inexact addition different unclear joint single product hand belief propagation terminate tree optimize objective toy solution guarantee belief iteration maxima message eq objective toy effectively perform coordinate monotonic view parallel local theorem disadvantage guarantee undirected tree convergent optimize form transform
mc algorithm log tr explicit derive fix bfgs procedure orthogonal grid g solution select produce smooth surface compute use appropriate two introduce evaluation regularization degree freedom model selection aic
differ coordinate class condition moment coordinate hold random distribute differently requirement spirit spike encourage latent away avoid degeneracy proof sign say column permutation distance dictionary later dictionary order return column wise moment universal ny ax independent require run complexity relatively discussion section require event bound moment c follow incoherent cm succeed close run test whether whose sample correspond correspond large cluster cluster rough dictionary noiseless part heuristic mod cyclic dependence main knowing think look overlap since algorithm find overlap clustering combinatorial triplet recover correctness rely algorithm recover filter filtering much direction recovery average singular similar insight
inference energy count reconstruct two three medical motivation package present facilitate algorithm provide convenient numerical support package leave incorporate library purpose library user implement keep entropy free bayesian feasible carlo although original formulate language idea field freedom finite data internal lie diagram alternative formalism work without furthermore scale matter galaxy count reconstruct rotation improve stochastic calculation resolution easily differ dimensionality comprise commonly scheme basis orient preserve normalization operation multiplication position
location receive wrong instead result end episode please surveillance figure wish location subset visit location action direction reward wrong order instead agent block receive trajectory surveillance mdp sequence mdp solid target location line location block red take location red result mdps target complex measure previous mdps difficulty transfer v gain effect different clustering exp call exp comparison result presentation experimental two summary benefit cluster exp transfer present figure exp benefit fact trend curve previous lie expectation bandit algorithm number removal affect consistently gain curve use mdps measure gain discount reward measure final gain discount reward final episode target trial per appear task conjecture regard policy device arm turn policy reward exp cluster surveillance location mdps gain exp transfer negative policy mdps become exp transfer previous benefit reward exp location show dominate policy become exp transfer exp task section due domain cluster domain measure cumulative reward full exp transfer
effect see feature lift mode value train fast reduction fast reduction lift turn affected experiment feature combine lift outperform svd either combination fast total alone zero setting recall feature become classifier improve lift reduce weight prediction small reduction lift nmf svd run use full large bipartite rate demonstrate dimensionality reduction svd click
class count fp fp fp satisfy every strongly component equation combination strongly prove similarly review fractional graph undirecte adjacency doubly doubly square bipartite graph straightforward fractional pair w doubly stochastic represent fractional dimension perfectly similarity dimension doubly result notion fractional right connect close sum closure direct might expect something approach fractional starting graph vertex map vice direct sum every write fractional relation consequence fractional connect notion fractional five q matrix identical relate colour refinement colour refinement run disjoint disjoint nonempty nonempty joint use colour refinement joint balanced
world scene home message visual necessarily help scene understanding help learn transformation without overhead berkeley problem know domain internet database try scene understand consequence degradation classifier world show idea rank margin adaptation optimization easily category begin bridge internet object environment huge comprise million promise direction towards visual visual allow thousand category however parallel discover bias
activity true activity c classification note except direct test second sequence consume step quasi outperform statistical establish background encourage flexibility allow activity activity paper acceleration reformulate unsupervised learn latent joint use acceleration avoid flow preprocesse switching activity logistic adapt smooth maximization stable optimization tool apply activity multidimensional acceleration body encourage alternative activity current batch mode perspective train online expectation maximization also basis particular extension bayesian activity human recognition service human security etc grow accurate account limitation context activity recognition acceleration automatic process
monte sample tt update stepsize accept ensure simulated interest adjust gaussian might think solve resolution general technique rely include conversely hmc strategy propose ensure asymptotic hmc flexible contrary investigate pdf achieve draw sample know variance mcmc costly compare due hamiltonian monte project keep spatial acquire image image reduce band remove band reference spatial image band besides successively average adjacent band accord filtered depict obtain band model correspond perfectly signal frobenius band band ms snr bands composite green
simulation htb bold font snps phenotype highlight red snps highlight snps phenotype univariate perform highlight snp four trait crp rs crp rs rs rs rs rs rs rs rs rs rs rs rs rs rs rs rs rs rs rs rs rs rs rs rs rs rs phenotype analysis threshold font snps phenotype phenotype highlight snp phenotype snps four phenotype phenotype analysis test highlight snp trait crp crp rs crp rs rs rs rs rs rs rs rs rs rs rs rs rs rs rs rs rs rs rs multivariate linear mixed area attract considerable association fitting setting exist phenotype association novel algorithm fitting setting eigen complexity iteration optimizer cubic marker complexity
proposition large contradiction example grant dms contract contain exclude
perform perform rgb rgb rgb lee lee high dimensional condition task recover gene contain specie stage conditional network assume difference drive edge case intuitive interpretation network distinct similarity across penalty alternate set problem scale cancer gene gene multivariate grow finance biology computer graphical simply terminology feature conditionally dependent suppose corresponding covariance matrix th unfortunately estimate information conditionally similar structured motivating access cancer normal option basis fundamental normal network cancer substantial pathway effectively principle jointly way quite allow structure difference may scientific dependence relationship stock interested detect stock differential stock correspond field neuron past estimation zhang take approach similarity difference connectivity share see base powerful base fully exploit prior
compress mmse sharp transition mmse comparable region mmse decay three region phase diagram impossible perhaps tractable learning later tractable resort compress sense bp lead call minimization generalize il canonical simplification arise use central
considerable simplification optimization reduce linear program different keep unchanged instead suggest essentially suggests yet solve regularizer choice see characteristic would worth preserve invariant transformation tendency specialized generic paper never terminate define limit trajectory termination episode accumulate still give meaning expectation give rectangular denote termination assume eventually terminate construct distribution producing assume reach termination eigenvector matrix stay termination intuition behind describe meaningful episode definition termination element summation episode note fact termination matter state terminal formula
perceptron describe connection point accumulate cnn help first come minimize neither suppose cnn given select arbitrary without true terminate bad stop element termination cnn find
perform behavior request effective relation beyond currently link setting unify various kind model include effective advanced implicit link capability utilize extend integrate relational link categorical proper generation construction flexible way go way utilize proposition practical link embed hide rich attribute critical involve entity incorporation relational encourage entity secondly propose datum substantial towards model
obtain slice deviation less largely bottom display mutual maxima giving position next evaluation close average ten synthetic dim synthetic concrete select learn idea sensible utility entropy also mutual entropy insensitive entropy give particularly make computation computationally demand even trivial hand transform encourage hyperparameter cc approximate variance maximally sample variety
traditional minimization enable aic sample relate see penalize ic rigorously gives penalize ic seen suppose follow hold mle singular matrix maximize likelihood penalize become pl easy estimate ii free least change denote note therefore ic maximize penalize exactly quick ic continuous likelihood replace correspond maximize relationship quick ic suppose ic quick ic elimination penalization gradually increase ic ii quick vector ic
notation additive r hold v chain differential measure independence relatively depend residual conditional implicit procedure subsection estimator give decide paper ft dt procedure consistency let suppose nz hz procedure spline variety estimator estimator plot complexity estimation scenario sample uniform distribution open sample control control noise entropy repeat time repetition regressor geometrically
leibler ks risk knowledge minimize hope estimator risk possible close let collection e refinement every nonnegative penalty estimator penalty oracle asymptotic penalize partition absolute positive satisfying negative u hellinger onto ks remark treat kullback leibler possibly kullback construct model e choosing say constant integrate bind kullback leibler equation
choose throughout rest write also notation case z z looking solution eq form recall q kk know add attain example recall integral delay write formulae precede compute chart smoothing pair chart alarm level proper significantly htb first regardless detection delay measure minimize substantial varie detection delay minimize
require orthogonal check true z arbitrary rest unchanged subsequent stage unit give mixture view give view expressive like hmms mixture sometimes refer mixture view parameterize r assume notational convenience one pick dimension art multi view guarantee mirror gaussian show one al set singular non fail setting image dimension give smoothed complete magnitude vector give obtain allow result decomposition establish multi three apply normalize vector sketch estimate apply vector axis align dimension suppose perturb magnitude change mix align diagonal mixture polynomially length sample next
approximation dataset descriptor descriptor power kernel remarkably mkl aim base kernel learn combination year advance yet scalability dataset explicitly kernel matrix exploit fourier feature map relate scalable kernel learn come learn retrieval body distance next section formulation learn adapt descriptor learn speed distance method preserve locality space enforce image mkl jointly base kernel feature bm linear desire sequel possibility feature whether fall
application often skewed histogram cell nonparametric estimation flexible independence relationship cyclic dependency represent
generalize give although simulation
let ce induction ce ce ce ce require oracle procedure oracle polytope exposition case construct nx consider simplex optimally permutation pi pi pi pi observe linear optimization turn simplex assume point later maintain point v ir l cp v k ki b b combine conclude z I iv iw contradiction exist jx write c k hold exist establish condition condition since give inequality x l convex vertex give iterate produce iterate take denoting
range market behavior thus difficult assess period day ahead probably relevant evolve stock major svm input lie feature hundred thousand lot hundred stock considerable importance conduct dimensionality reduction extraction principal protein spectral reduction face reduction interestingly adaptation pca rarely stock common phenomenon stock return k market test stock market liu explicit verify stock stock among stock first
reconstruct quickly monte mixture cluster gmm parametric probability density sum gmm component k weight parameterize represent reconstruct
mean already scan coordinate perhaps present paper merely iteration speed utilize thus restrict ratio conditional joint ss sx maximize interestingly appendix use ratio motivate convenience plot prove result monotonically numerically f conclude ratio likelihood mle error complexity concern estimator
read direct reading ray propagation ray go list fidelity estimation obtaining plan use manifold localization depict collect device sure insensitive choose calibration offline set consist phase receive localization position error average grid figure weight calibration localization map plan significant determining level neighbor total grid strong weight computation neighbor practically far close computation outli total large small space occur neighbor loose neighborhood lot neighborhood degradation figure depict error observation propose map coordinate total localization plan coordinate explain plan coordinate preserve physical neighborhood noisy estimation represent determine result propagation represent apparent simulate detail low localization nonetheless plan coordinate avoid even depict percentage degradation localization
circle control shall completeness sample interval explore recall small result probability apparent asymptotic method call moderate derive confidence accuracy delta new result delta former case rather approach development dimensional allow simultaneous bootstrap throughout map write finite q define iv sub third inequality fourth argument need dimensional random gaussian anti concentration center immediate let inequality anti inequality union bound delta bootstrap bound
decay zero slow illustrate ad hc limit prevent near indeed significant nan detect hc remark hc problematic already note discuss several hc resolve hc contrast put scale detect roc rare compare result receiver optimal likelihood test unlike statistic require explicit lr may achieve question value four clear winner statistic shift extreme contaminate hc statistic second would characterize weak simulation test gaussian colored rate divide dark center gray dotted line detection boundary substitute leave right panel deviation random
national national grants rr mit edu variety medical imaging application projection extension interpolation manifold commonly om point propose interpolation phase training interpret sample approximate simple crucial medical application result function illustrate method clinical application fast mapping image dimensional application medical use segmentation computational little work
constant linear missing point batch perform svd dense matrix computer contrast streaming run plot empirical scaling theoretically model three xx consistent dimension algorithm svd result optimal figure tackle ny prohibitive batch latter million document vocabulary report extract dataset hour pass make statement call support let stream theorem let initial guess universal define noise
maxout replace universal capability various concept convolutional low layer feature network treat result feature classify traditional overfitte ability regularizer half activation connected training prevent another traditional feature category layer top take feed softmax layer convolution enforce category thus avoid average spatial global average pooling regularizer concept category map stack layer top objective sampling cnn maxout micro specific
wu et ga study two kernel performance annotate neutral peak transformation face gram ga leave maximally kernel parameter ga
scheme sgd sgd sgd intra favor bound ball project psd cone sgd epoch sgd epoch sgd nevertheless facilitate understand main state epochs eq sgd lower enjoy convergence sgd obtain bind constant psd cone choose bad factor compare projection independent conditional finally notable tradeoff number multipli
adjoint definite arbitrary block definite importantly converse true take kx present function rkh constant space square integrable whose multiplication property case provide elegant deal eq mapping virtue furthermore admit map correspond dot perspective geometry compare consider reproduce space rkhs rkhs reproducing reproduce respectively map see dot
decomposition include unfold way datum apply decomposition imputation build novel accounting nuclear two property carry sparsity tensor pass atomic incorporation smoothing capability rkhs explain acquire capability mean obtain information utilize fitting criterion completion implicitly assume adopt framework incorporation traffic genome sequence social medium maximum posteriori leibler l divergence remainder necessary decomposition definition establish induce incorporate rkh formulation lead finally present carry level image detail defer adopt bold capital letter tensor slice carry subscript see tensor frobenius norm symbol rao hadamard
log draw also intractable draw rbm digital computer exist monte carlo procedure cost number rbms region extremely rarely particularly problematic circle corresponding probability separate lower approximate require difficult slowly would gradient reliable thus whose natural behavior system rather explicitly computation simulate physical computation spirit analog states digital note different building rbm merely design kind digital rbm computation wave wave
classifier connect unit jointly iteration classifier practice however newton improve regularization pooling improve interpretability pool well cube well image weight whenever non spatial regularization hyper cross optimize demand cpu take image pooling unit propose make scalable big learn region little overhead standard approach test fine grain code location pool collect code reduce number code choose weight
follow present problem describe introduce devoted set dissimilarity matrix binary entry operator multiplication application optimal dimension dissimilarity potentially differ different cast problem set classical semidefinite embed q identify present nonconvex value reach solve constrain monotonic exploiting parametrization propose factorization
appear choose evaluate scheme section experiment come linear shape set tune perform weight validation average run experiment set validation split available reasonable splitting improvement error train split omit comparable set experiment evaluate three uci repository remove remain equal approximately summarize subset subset right setting digit translate datum weight learn par well svm split notably perform attribute surprising capacity rbf remarkable split subset performance outcome bring twice efficient handwritten digit evaluate scheme experiment weight rank label reasonable aggregation label human expert consider collect annotation rank expert human present
may complicated activity generally quite goal plan observe series approach base representation statement environment body statement conditional always move statement primitive statement inside loop single joint primitive ta remove program depend label attribute object allow infer explicit encoding plan discuss discrete mrf margin encodes relation primitive relatively sequence offer effective representation task see attribute follow specification task away single object object loop dynamically primitive
often set unity consideration see bayesian model likely large area highly automatically implement monte evidence transform integral accomplish integral contour write
observe class conditional distribution realization first mistake oppose label noise contamination apparent class contaminate contamination come apparent class proportion wish impose particular support review consider generality contribution element existence consistent noise mixture vice versa discrimination rule proportion proportion recover estimate proportion light condition contaminate role class solution complementary contamination leave label unchanged geometry argue unique moreover uniquely correspond contaminate separation condition maximally version establish everything particular emphasize restriction apparent view contamination interpret source realization probability superposition source
unless singleton group capable sparse variation whereas add component conceptually powerful effective correlate encourage weight correlate covariate assume correlated variable principle consist trait predict snp associate phenotype share implementation upon publication article integrate value output pair genomic trait trait pool regression elastic net regularization recently run mixture range group regression use phenotype genomic work among benefit require
feasible place directly varied scale choice proportional row replace linear information adapt frequency act soft shrink explicit shrink heuristic costly problem substantial constant universit ex un universit area year solver case implementation mark analog greatly come track method adaptation criterion inspire method replace shrink magnitude since
bind rao efficient error early computable follow denote define bad stability assumption total scaling property focus play role frame let denote subset note ss c fix confusion reference clearly measurement constraint eq eq stability frame frame eq consequently equal reciprocal proof follow impose choose
formulation adaptive shrink shape particularly idea extend frequentist affect shrinkage idea discuss common bayesian shrinkage frequentist simple procedure bayes situation empirical bayes intractable bootstrap simple variance scenario get unbiased select estimate corresponding statistic bias explore
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write j transform rank thank suggest key polynomial entry fact small algorithmic end run boundedness detail extremely uniqueness tensor polynomial identifiability hide size say find latent pick mean matrix capture practitioner typically expectation learn good model approach start pearson try moment drawback moment recent suffice call n proceed problem use iteration crucially rely degeneracy condition even case mean fail good bound feature fourier speech sift object feature could robust result decomposition go barrier suffice lie mixture aspect tradeoff get successively tensor intuition moment allow even achieve run well like hmms topic uniqueness decomposition imply two popular latent fit multi apply study omit multi view latent view conditionally expressive study model mixture variable domain view vector conditionally comprise I learn setting vector event al previous term moment eq usual representation decomposition say close singular identifiability r r give function note mixture weight large interesting space unknown also tensor proceed inverse polynomial exist decomposition know estimate max
order calculate score number parent allow small therefore function calculate local subset potential parent px subset take network simulation give pf space bayesian network toward learn network task among leverage task robust exist algorithm effective learn optimize joint encode toward task joint structure share task graph distance metric structural present challenge bias maintain calculate posterior like handle bias single specific formulation simplify
converge suitable present computing rule result primary polynomial polynomial regression formal truncate series expansion degree
fed svms fine paper model weight layer net convolutional net layer descent recursive svms every tuning top softmax beneficial optimize primal svm level essentially l l similar
propose canonical cca canonical pair methodology thresholding algorithm parameter optimal matrix review motivated eigenvector give symmetric lead eigenvector multiplication normalization lead eigenvector power generalize rectangular suppose rank diagonal vector step singular multiplication leave normalization goal canonical motivate suppose marginal structure consider target unfortunately covariance nuisance obtain structure dimensional literature covariance toeplitz structure later influence final direction split copy half second half splitting form eq identically explore result conditioning expectation instead accurate setting coordinate never span singular vector sample pca dimension simply cause variance dominate error optimal power iterative lead section without idea ordinary svd right multiplication keep coordinate thresholde thresholding level hard serve k tt scad w iw w tw multiply orthogonal rank one matrix value statistical level
pattern precision small otherwise estimator assertion theorem ensure lasso exclude adaptive sign adaptive lasso exist well ba b bt include number relevant answer course particular exist previously zero fast well know technique classify zero see series min condition shrink sufficient relax relaxed coefficient ii adaptive cause parameter asymptotically hence limit oracle reveal ols efficient discussion even dimensional adopt discuss adaptive remain procedure lemma one obtain rate satisfied notice adaptive lasso improve non square ols merely section explore adaptive practice benchmark whenever permit implement square oracle implement estimate lasso implement ridge first r fully provide bic bic df considerably equation measure report whole pattern discard irrelevant true procedure retain relevant measure since detect correct retain relevant leave relevant still fraction number procedure measure well rmse carlo ahead every parameter forecast denote root mean square forecast
evaluate model slow convergence gp training propagation follow impose gp gp approximation mn normal
investigate general exact empirical noisy rate depend asymptotic behaviour characteristic noisy purpose deconvolution minimization reach rate regularity quantization deconvolution quantization reconstruct processing cluster information follow law lebesgue thank risk analysis cluster assign however real noisy quantization compactly version well yet try deal paper exactly purpose dimensional measure point erm performance generality two possible fall quantization construction center n k quantization curve distortion collection parameterize wide range statistical problem
associate select every q q speak markovian sake later surprising natural put way candidate action ergodic invariant interpret payoff additionally payoff proportional play play move empirical frequency play call player game major assumption sequence nz I sequence converge potential payoff markovian enjoy sum surely nash converge game converge subset nash equilibria particular connect nash equilibria equilibria relationship limit response potential necessarily surely strict nash equilibrium connect modify equilibrium limit whole equilibrium player
involve algebraic simplification occur identically two theorem change integrate eq boundary vanish right converge pointwise polynomial integral negative absolute remainder line satisfy line assumption consequently suffice dominated formula grateful research discussion laboratory fellowship er thm
substantial long evidence skewness walk lead biased skew brownian motion determination interface reflect behavior way interface condition dispersion interface generalize interface conservative interface rise line obviously quite follow result exploit property brownian motion easily check intuitively reflect coin flip involve definition place quadratic borel modification mathematical area natural representation identify mathematical role context skew dispersion interface parameter natural equality therefore conservative interface condition spend contexts dispersion interface continuity continuous xt ax indicate exist q relation local use variation process skew motion see agree natural symmetric version extend continuity framework paper exist skew diffusion significance let skew diffusion thus skew determination view interface determined continuity interface dispersion network water water network central modern constitute population upon heterogeneity reach mathematically binary topological example topology skew shape natural skew
table percentage respective nominal level significance indicate omit likewise high censor omit remark environment r team table conclusion maintain nominal test ms ab distribution hypothesis os seem purpose transformation gamma alternative alternative ms os also behaviour censor alternative transform hypothesis iv power combine call decrease alternative lin occur transform call lin os transform follow unable detect alternative def interestingly os converse power
model condition separately equation result raw trial slide epoch epoch span several level mention epoch measure kullback leibler gaussians kl respectively generality function gradient originally develop stationary epoch epoch illustration background system formally lag epoch background condition th matrix epoch part stationarity become identify remove section defer appendix focus proposition form epoch stationarity number dimension remain effect equation covariance number epoch stationary stationarity th r nr frobenius specific subspace orthonormal column projection epoch projection sum epoch desirable derive measure proposition sense zero zero term
cluster determine patient cluster patient survival measure cluster supervise method unlabele review semi cluster type review primarily description hierarchical briefly namely mean hierarchical cluster popular datum quantitative square attempt represent number objective within sum square several clustering propose minimize strategy feature calculate mean cluster cluster step converge converge one gap statistic simple decrease simply minimize motivation value separate mean algorithm actually cluster identify decrease
constraint one cut plane basic cutting subset find approximate empty add violate qp solve svm continue violate quadratic program cut iteration bottleneck finding violate solve violate constraint trick instance break restrict instance new easy term space simplification reduce initialize partial auc optimize else extract cache classifier learner cutting cascade threshold adjust use validation ensemble classifier object detector
load year split otherwise temperature purpose together unit load trend
class anomaly monitoring sequence annotate probe consistent regime occur anomaly detection method hour annotate superior auc discriminative model particularly world covariate idea advance directly applicable modeling principle direction department science institute technology classification domain speech bioinformatic language discriminative drawback function hmms fast obtain performance
measurement rational observer four inequality two two mapping description random sum codebook consider structure eq system play interested mutual datum provide strict language deterministic one processing inequality coarse data stream extend relationship introduce collective behavior framework quantify conceptually noisy practice bootstrap conceptual code rapid evaluation publicly part characterize range hope apply amount distribution symbol elegant normalization common theoretic et suggest construct bayesian theoretic estimation arbitrary case give provide method entropy produce integrate estimate mutual wide range conversely fail whose lie prior wide applicability estimator unity example prior draw fail inference particularly problematic observer influence justify symmetry mixture make random entropy bin bin amount draw dirichlet random placing property draw partitioning rapidly contain drawn dirichlet case role observer coarse grain process grain cognitive semantic coarse introduce bin range grain level coarse
ice uncertain slide coefficient velocity move ice flow present exercise bayesian product ice equation discretization flow ice balance momentum state velocity employ ice relate stress tensor law flow ice ice boundary slide slide coefficient projection plane second tensor boundary accept coefficient velocity several phenomenon ice physical uncertain slide refer slide coefficient bayesian next ice efficiency interpret result ice ice together boundary top boundary drive g law exponent pa unit slide coefficient synthetic observation field slide field likelihood candidate slide coefficient operator flow top surface coordinate parameter map solve ice slide restrict operator add vertical flow numerical employ horizontal respectively hessian uncertainty reconstruction horizontal vertical problem horizontal vertical flow giving via discuss surface geometry laplacian call laplace projection plane eq unit surface length availability posterior derivation complicate observable ice computation gradient adjoint equation presentation pde constrain ice flow gradient log require variation velocity pressure adjoint velocity pressure vanish forward velocity adjoint stress identity slide velocity pressure adjoint
feature candidate consume pruning incorporate processing tree cluster scene hierarchical adopt simple design pruning character candidate reduce distance learn weight automatically character candidate candidate single character text correspond remove text elimination powerful text build robust scene text evaluate read competition first method chinese english dataset use competitive rest scene review describe read chinese english dataset remark present promise real project still limitation non insufficient candidate construction section focus problem refer advantage base traditional character candidate character resolution detect character correspond al present pruning stage segment tree function depth manner check
propose development regularize key algorithm ep ep two find special conduct demonstrate efficiency propose screen identify group group inactive problem scale appeal screen safe remove need run negligible compare regularize solver integrate exist solver exist case key include dual variational inequality bind kkt experimental efficiency order especially dimensional briefly regularization algorithm via coordinate consider group logistic develop boost lasso author share alternate spectral project ball smooth linear regularize task nesterov semantic via constrain via scale qualitatively ordinary reveal enough regularization boost boost penalty penalty family employ implement
totally totally totally couple less collect contain non zero entry totally obtain totally totally reader page e factor design totally arbitrary parametrization item nan argue totally totally interaction combine fact discussion prove factorial design totally simple design factorial page circuit coincide circuit factorial circuit circuit ti carry standard quickly heavily degree freedom interaction
parameter e chen ranking sample show rank may fisher rich content engineering shannon suitable measure perfect shannon enyi kullback leibler kl compare increasingly context order chen censor datum reliability life et al
buffer rs buffer replace incoming step rs eq alternate rs step pool without location item position thus complete present result benchmark compare rs buffer policy rs buffer continue observe trend slight buffer situation ex ex ex rs policy lem lem lem lem lem lem lem fig generalization supervise dependent one learning enable tight strongly memory algorithm learn subset order complement propose learn problem bound supervise metric bring keep differently function yy hinge pairwise h include preference rank auc practice algorithm pair standard analysis dominant idea convergence another algorithmic stability batch popular provide first apply loss higher admit order combine bound error however cover rademacher achieve bound cover analyze order extension complexity class reduce class suboptimal algorithm stage extend order setup penalty use expect regret update hypothesis generalization regret step proof hold buffer widely reservoir
mention system public disease surveillance provide enable include explore non perhaps multiple metric way combine location plan member thank anonymous twitter whose u gm accelerate program draw natural security department contract ac code gram list restaurant company ia en en optimistic play en ca extreme ga http tx date soon en news en tx en tx site se vc ir de http tx en tx tx tx home soon p en detail binary n gram token mean total vocabulary surface origin new origin valid density geodesic origin integral carlo procedure generate density q implementation implicit simply thus sample implement region region coverage use construct sort contain cluster cluster hull produce area projection
context hilbert space function construct basic rkh application space find et area output reproduce basic hilbert space evaluation continuity equivalent fx g yx eq operator value h kx reproduce
important characterize temporal left panel raw band calculate hz th power always noisy smoothed middle reconstruction spline also try program singular spline post insufficient change end keep meaningful predictor band drop pre band level even finding take minute latter need initial acknowledgement grateful associate constructive comment wang support nsf grant dms support nsf grant dms
see band band run experiment band ba ml vary repetition fold validation assess versus band accurate box ba band box ba deal ba ba class instance bioinformatic mass provide classifier generative discriminative line prior multivariate normal symmetric gamma inverse distribution inverse gamma follow v student student symmetric definite say variable value dirichlet follow domain multinomial observation experimental linear seminal
n f nd k conjugate gradient convergence ignore observation exploit kronecker take compute exact eq decompose compute incomplete approximate penalty eigenvalue emphasize determinant remain term exactly total runtime cost hyperparameter exact incomplete combine section embed dataset alternative particular spectrum method kernel many basis learn location implementation http www compare use fast implement popular exponential se rational mat respectively finitely differentiable gaussian combine use intractable structured moreover stress speed available
researcher quickly infeasible rw contrast attention year structure framework consist svm infer quality output would rw straightforward application svm output require human since tune structured svm desire handle propose novel discriminative rw pair medical
base orthogonal procedure base achieve construct statistic construct similar base strong information orthogonal part design substantially assumption sequence observe mutation usually linkage structure strong omit due limitation future need detection boundary matrix allow paper subject multiple nonzero research extend correlation covariate acknowledgement like dr li associate style graphic proposition theorem motivate genetic sequencing rare effect case detection matrix sparsity signal sparsity design asymptotically irrespective design sparsity context derive detection boundary regime show generalize statistical relationship introduce analysis use present day detection boundary gain popularity way boundary signal testing work context mixture sequence little generalize detection boundary testing context sequencing association lee interest sequence allow sequence massive rapidly study sequence
various derive optimal conditional method generator song provide include base permit merge event finite measure take shape see description specie equation section subsequent mutation permit generalise simultaneous introduce generality infinite simplex rate jump mass proposal distribution site principle approximation family heuristic principled generator consideration simulation algorithm site discussion
variate computed make evaluation costly computational gain interval build conservative use mean variance eq sketch follow q remain choose overall cost
denote say point integrate point assume still continuous continuous give follow close stop exist subsequence provide feasible verify especially one iterative widely convergence property amongst interested break gs characteristic verification set mapping nr function increase proposition close accordance proposition generate three property prescribe predefine set describe
aspect motivated calculation simulation aspect alarm simulate red line circle blue line observation quickly alarm second
simplex iteratively contain unit space distribution partition arbitrarily assume rbm approximate distribution arbitrarily block share partition block bound corollary obtain divergence prior divergence symmetric distribution bound euler consequence analytical leibl partition give emphasize experiment define neural hard replace
mode base bandwidth cluster theorem pt nonparametric mode university derive eigenvalue estimate information shape significance approach first valid eigenvalue symmetric polynomial lead set regardless suggest selection choose even density cross choose bandwidth method bootstrap mode persistence estimate mode true second simple answer question problem second negative right well separate reason example mode nonparametric difficulty mode raise form precisely px gx dx construct reverse location mode mode
yield choice rise basis amplitude eqs discuss generic basis basis note ensure identifiability regularity include flat amplitude reasonably identify importantly component reflect variation dominate usefulness interpretability analysis influence mode duration accordance pattern covariate duration expansion necessity curve finitely final joint formulate effect effect arise parameter assume matrix duration process summarize variation qualitatively amplitude utilize determine amplitude eigenfunction representation theorem amplitude eigenvalue determination examine later acoustic datum consider eigenfunction retain compute amplitude suitable integral practical examining mean correspond ie assume distortion conceptually approach utilize eigenfunction identify mode finite directly decomposition covariance eigenfunction analogous amplitude base base acoustic criterion correspond eq component external criterion purely statistical criterion fraction interpretable routine th distance select scale th specific restriction structure firstly secondly g subsample aside pairwise approach
analyze sum poisson demonstrate especially narrow approximation weight coincide mean value distribution quality compare ratio agree agree skewness excess distribution expect power distribution suggest accord central limit hold ratio expect
permutation equivalent establish uniqueness eigenvector note eq use establish semi vector p fact introduce mutually c mutually b use frobenius switch I imply eigenvector eigenvector establish induce hermitian expression rewrite eigenvalue two definite verify jensen get get eigenvalue therefore h denote hermitian k h hermitian corollary conclude f hermitian eigenvalue order large deduce f matrix I q get quantity express know size thus follow moreover complement invertible inverse complement inversion block equality hermitian get use
definition respectively expression four useful complete expand component expand fourth due transpose derive eq first second global summary fourth respectively equality third due trick last edu sg demand system promise paradigm sharing densely enhance mod grain demand present decentralized fusion grain demand mod system fusion algorithm fine balance theoretically equivalent sophisticated centralized gp mod demand though decentralized demand prediction achieve world demand algorithm achieve balance art become densely road limit private expand implement traffic delay
find critical significance adjust model variance reflect derivation indicate actual lead give raw noisy plausible eq weight probable matlab show superiority probabilistic map
nash function observed frequency opponent play belief thompson sampling agent useful modeling tool game generalize thompson design different causal generalize thompson causal induction base combine behavior hypothesis constraint causal statistical information unlike framework aim extract observational design agent interact environment discover causal far thompson mainly armed bandit problem parameter range also context thompson highly ucb sampling past mdps reward first solve issue avoid directly inference action pick high apply solve adaptive quadratic noise derivation show uncertainty calculus thompson sampling approach know maximization utility criterion highlight scenario depict predict payoff wrong guess rational maker place inside expect belief dot box tb two apply expectation
rest carry exploit introduce select diagonal square minimizer encourage low minimizer mean find extra structural express semidefinite dimensional market exclude efficient direction multiplier see admm reformulate differentiable cost constraint replace variable guarantee lagrange associate lagrangian predefine constant consist primal update
bind bound always bind look apply q bound involve derivation expand
subset yield minimize dendrogram entropy dendrogram entropy zero besides dendrogram summary sequence assess agglomerative standard summarize gene agglomerative informative remain criterion merging avoid drawback since entropy already subset test mixture experiment model infinite gaussian histogram pairwise dendrogram pairwise dendrogram arrange base clearly separate reflect qualitative first distinguish outer gray pairwise dataset specie obtain single
base reason discuss topic like compose accordance dimension complicated angle basis three e j generation unitary apply zero correspondingly zero priori probability add error correct magnitude normal obviously lot potentially process run trial ssc value space ssc capability ssc change linearly change image average trial describe range vector misclassifie much ssc observe absolutely handle free fig gray background ssc believe lie success belong line cluster point follow attract deal matrix e db processing input strong cluster still db ssc obviously cause work localize corruption error fig setting
scad divide necessary condition desire mcp iv expression optimality section traditionally penalize nonconvex recovery nonconvex primal establish existence class derive novel global thresholding system wise minimizer minor minimizer upon active dual relation active step primal active set develop primal dual active arise sparse attract considerable attention compressive represent utilize acquisition transmission storage statistic tool construct parsimonious model admit recover throughout matrix column denote meaningful approach look many novel issue sparsity basis lead nonsmooth optimization q denote gain popularity largely attribute fact admit problem design fast coordinate overview regularity isometry drawback restrictive signal bridge drawback smoothly deviation
chain theorem eq major difficulty likelihood mean approximation inefficient result variance importance numerically evaluate difficulty resort estimation chain present importance evaluate alternatively rely rough laplace type close rao simulation accurate stage year survival indicator cancer example logistic first classical analysis
combination equip adequate product see closed consider sample independent gaussian xt target restrict belong functional acting norm hilbert approach smoothing assume control hilbert space constructive green green linear operator dirac speak differential operator green dimensional span penalization maximizer calculus maximization give nn interact element external
clique plant quasi slowly keyword detect hypothesis testing plant clique dedicate recent detecting receive large attention important social biological science extract community fitting datum social represent sort mention concentrated goal inner connectivity inter connectivity detect sort set cluster limit I often insight extraction extraction turn procedure decide implicitly simple nan os enyi another detect clique plant clique emphasis tractable consideration community formalize undirected generality adjacency mean symmetric nan realization equivalently index everything else assume subgraph regime change community otherwise risk index say sequence resp practically speak test asymptotically substantially powerful asymptotic path hypothesis particular closely relate result
satisfy proof rely concentration process intermediate exploitation trade kernel cumulative satisfie general corollary equation lead generic rbf obtain remark assumption linear restrictive prior hence kernel use practice result cumulative regret incur gp high probability exploration describe algorithm incur rbf
derivative objective regard zero iterative code neighborhood iteration datum code perform code update codebook classifier fixing update code update matrix unlabele come nearest assume neighbor reconstruction coefficient compute code label codebook label learn optimization formulate th adopt alternate solve solve solve regard repeat procedure test class
costly inference simulate implicit computation abc enable inference method originally human species finance evolution name intuitively simulate let xt xt comprise point datum arise stochastic belief density proportional h dataset using accept repeat practical discrete else acceptance probability low take acceptance comprise summary statistic abc sx sx sufficient require careful acceptance abc balance involve trading decade extension original develop markov chain carlo implementations incorporation
u analogously second imply give log interesting behavior note exp eq composition exp composition still bernstein thus composition laplace exponent original two bernstein derivation accord mixture mixing accord mix additionally limit tt eq figure ht compound poisson bayesian rewrite joint assume lb tt give full clearly shrinkage w conjugate gamma experiment inverse additionally resort proper normalize q specifically always
test serial dependence early plot return much structure sample measure early sample minimum lag behaviour return acceptable sample log price brownian return grid independent walk show length close interval reliably point dependence stochastic positive ix fully pair autocorrelation lag base independence nevertheless one million result even detect compression drop lag half odd break identify pairwise also rate rate daily return maximum residual lag plot compression residual seem remove structure remove simulate lag estimator simulate exhibit theory applicability economic discover stock frequency period argue market research qualitatively dependence series rate proposing entropy
jointly learn basis subset represent belong sparse discriminative fundamentally replace extension usually improve classification capability basis enhance homogeneous representation simultaneously thus set discriminative basis force boost propose classification comparative brief reconstruction introduce incorporate discrimination supervise conclude review relate aspect code application code reconstruct relatively subset complete meanwhile code attract attention field meanwhile recognition etc
describe follow prior arm one q thompson arm play ti policy lebesgue variance ti inspire thompson show attain log term bind policy theorem generality let
favorable addition satisfy beyond illustrate problem optimality condition main arrival modify update allocation history matching period xu define following without generality I competitive permutation proceed define notation define allocate allocation period actual allocation entire bounding difference I relatively loose next obtain ik find allocation allocation concave last ij ij fm fm combine remark condition practice even
assume instance position remain conclude delta error equal exchange I
indicate loop terminate body many program express trivially facilitate construction allow ml visible express iterative ml special extensively importance domain support loop loop sequence probe situation depict flow indicate flow operator meet responsible model current aggregate step return control recognize consequence add fundamental construct demonstrate system program within specialized runtime general engine optimizer translate broad program ml program runtime execution plan environment drive valuable elastic whose change resource availability make manually program effectively efficient runtime exploit discover prior review open software job store job try task machine reduce job intermediate datum implementation operation support intermediate
reduce randomized payoff per randomized generalization real semi adversarial string number combine bad setting sign magnitude number majority integrate average predict consider high stock predict fashion mid price trading day regret outperform provable setup analysis prediction static achievable chapter extend significantly well expert ask also examine payoff payoff well mention every seek exploit
discussion particular geometric intuition figure lee fellowship stanford stanford genome partially support dms cumulative truncate define monotone distribution cdf exponential base monotone ratio appeal exponential verify preserve integrate side yield eq integrate side establish theorem assumption remark develop inference powerful core framework condition form valid variable include quantifie gene
impose constraint unstable summary globally definite cubic operator markov inference instance multidimensional furthermore gradient see update bridge constrain drift definite aspect sampling ensure stability dependency brownian euler point pair accomplish introduce purpose next observation transformation observation define ensure improve accord observation interval point interval give notation ease extend forward density pi I I I I I
exceed possible salient reliably salient processing analyze reconstruct vector transformation noise non also investigate channel quantization regression illustrate figure matrix correspond set easy see contrast hence model also problem miss miss entry take whether variable problem miss complexity observe sense boolean identify large example testing medical certain disease pool rather separate ideal positive research combinatorial pool test several group type salient test walk testing model depend determine include channel impulse channel correspond coefficient impulse index impulse correspond encode research particularly describe relate dominant research square assumption contribution recovery focus particular reconstruction relaxed integer bit quantization project gradient sparse miss form underlie conceptually unclear purely come herein complexity direct focus limit linear sense design alternatively estimate support sometimes
since correctly specify spherical gaussian perform restrict spectral algorithm automatically adapt actual gamma performance outperform dimension method favorable agree parameter hold dimensional view data mixture proportion exact experiment view evaluate performance cccc gaussian gamma mixture gaussian mixture shift alternative flow flow record light scatter emission hundred cell normal diagnosis disease group difficult distribution heavily skewed thousand cell separate cluster task view separately posteriori label manually therefore view
derivative primal omit similar effect hyper subset pseudo point local dual minimum primal critical least critical prove thesis critical low eq prove least critical global ht depend primal five case two critical canonical
satisfy literature type memory adversary sequence delay round depend adversary delay adversary merely strength understand player focus round bandit aspect assumption loss generalize mention adversary discuss full feedback bind analysis observe full applie switch full guarantee component player expect regret switch upper imply extend feedback bandit set e loss function fix study asymptotically logarithmic paper paper adaptive relevant extension adversary regret bind since regret competitive deal player past adversary unbounde sublinear impossible therefore competitive study weak performance competitive make work switching see match setting switching adversary
university paris prototype quantization relevance relevance hyperspectral spectral classification profile lasso latter upper variant natural quantization
conditional differentiable unique consider problem well value resp decrease appropriate conditional adapting emphasis censor pseudo estimator nx practice unknown estimator introduce field bx ix xu bx ix bx conditional almost surely similarly support set hereafter derivative satisfy nonnegative functional surely zero g ng j nf bound denote du em mx ergodic assumption nature impose regularity stand condition usual independence derive censor
simply replace affect bind projection orthogonal unit direction integrate direction
inherently model benefit constrain free parameter ei equal align vi align pp algorithm constraint slowly lift furthermore eigenvalue prevent degeneracy degeneracy member carry em maximum incomplete otherwise know complete involve mixture base likelihood step
say connect ignore end procedure eigenvector tx px non depend suitably provably efficiently rearrange x tx tx tx tx obtain algorithm oracle output introduce error give condition extract eigenvector error extract eigenvector respectively eigenvalue result eigenvector approximation use label example analogue distribution useful eigenvector key extract knowledge helpful candidate assignment boolean cube simply part eigenvector stable eigenvector often facilitate stable eigenvector markov oracle black box extract obtain fig access oracle output report elementary ising model x grid color function class decision arbitrary point ii eigenvector case degree bar figure gold iii blue iv choose algorithm error
map entry position r tucker decomposition tensor u kk tucker kronecker product tucker generalize tucker infinite feature variate variate location follow tensor variate specifically x u u encourage variate tensor probit map factor tucker decomposition core latent bottleneck massive entire store multidimensional array use sequential utilize parallelism distribute limitation global assume entry factor u expensive kronecker product avoid computation eigen couple u conduct online limitation couple bayesian local enable hierarchical allow sharing
form contribution metropolis hasting drive chain mcmc network develop transform encode countable work example drop requirement tuple string specific bipartite build let bipartite figure bipartite component depend element return factor use unary string value tp precise factor show type factor unary potential root string string edge unary factor dirac graph denote box interact section classical string respectively proceed extend define list variable language
entire mc integration ever nest mc calculation evidence also enable product allow simultaneous parameter estimation widely present ns technique modal posterior importance nest summation evidence magnitude n change explore accomplished treat pseudo include discard sampling apply ns keyword last decade arrival vast high quality datum facilitate physical process investigation divide distinct achieve sampling slice highly inefficient explore modal degenerate integration expense involve bayesian selection physics nest evidence provide carry simultaneous appropriate build ns framework especially posterior contain mode cost estimation model numerous discuss summation potential increase evidence computation magnitude ns www ac uk outline brief introduction nest algorithm sec apply sec summarize sec relationship mc detail account theoretic ns hypothesis bayes h posterior probability factor however take unnormalized instance compete compare respective posterior often unity
optimization use feasibility term check rewrite eq product first feasible respectively feasibility display eq conclude frobenius major numerical proof show get low choose accord triangle follow rhs recall inequalities conclude step reproduce large last display complete theorem outline give stochastic vector well shorthand support result high subsequent sequel step optimality define rewrite expand rearrange substantial result quite conclude step bind choose sum singular chain inequality inequality bn triangle need rip rip choose ready combine
present demonstrate speedup computational dataset family elastic bootstrapping validation extension regularizers function net lasso particularly real eeg genetic algorithmic future direction showed couple effectively handle scale connect another direction version example variant present adapt acknowledgment national health grant science foundation fellowship nf view interpret policy laboratory discussion suggestion eeg acquisition supplementary material method solver
large sound quality fair band nmf nmf potential way interpret attempt filter spectra believe capture effectiveness feature coefficient speech audio task include identification cosine dct spectra understand try variability spectra combination however dct basis tune therefore hope might identify finish one speak experiment demonstrate representation rather try system outside use learn dct obtain difference original treat classification problem frame
weight slight reason h job balance interaction double count subgraph appear music annotation audio retrieval classification set expert semantic music represent understand type publicly music popular music track include english language instrumental music compose year cover acoustic music labeling song supervise song include semantic tag vector partition short short slide ms window extraction procedure ms represent file music texture indicate amount coefficient
fs fs eq strictly ndcg lemma strictly give lemma theorem technical claim four claim eq discount let define r n n assume follow two rhs upper eq term rhs check q combine logarithmic complete next three claim large fix sufficiently prove lemma normalize definition expand expand due claim combine complete old constant eq key x fc order sequence expectation take calculation proof technical claim statistic yield chernoff bound df df df df yield integration part let separately first note polynomial cf r ds cn dr cn cn ndcg
learn cluster entity learn entity computation truth see case backpropagation network lastly instead space statistical share benefit modify unsupervise analyze embedding furthermore classification lastly factor
method cifar art dataset library mini start hyperparameter choice replace maxout unit layer begin cifar group maxout dataset make ideal start evaluate difference maxout conduct layer consist unit pool layer softmax convolutional pooling convolutional layer cifar start evaluation network validate temperature anneal lower good allow sampling layer performance select lower verify replace last maxout unit significantly height cm grid gray legend legend legend pos north east cell align legend style font white ylabel class xlabel align lambda bar cd lambda lambda cm grid gray legend maxout maxout baseline legend legend east align legend
precede negligible asymptotic process partially would lead separately whole loose robust contrary principle robust simply step find value cluster batch cluster batch use determine derivative calculate recursive skip obtain estimate likelihood sum recursive update update batch preserve l run next batch l next go next batch little initialization even basically prior amongst e informative hoc state add turn I approach could drastically outlier sophisticated classical interpret realization package individually moment independence component mixture determine assume capture non component retain frequency could situation one strategy small wrong work selection observation represent
quasi new efficient minimization however satisfy original handle complex quadratic bp quadratic lead introduction constraint polynomial relaxed method propose sect implement sparsity cone program efficiently sect benefit improve exact analyze sect occur minimization sect variant solve approximate subject priori thresholding projection simple interpret fix enforce necessary optimality constant globally lipschitz continuous polynomial coordinate method enjoy similar several dimensional polynomial become difficult greedy thank group sparse linearize formulation section section deal purely estimate sect sect efficiency result one find sparse sufficiently sparse fast relaxation benefit find system second relaxed sect determine polynomial
exact mention solve system coordinate exact cd inverse cholesky factor solve expensive medium dense iterative ix ix terminate accept inexact natural conjugate adopt system equation expect inexact subsequently apply fast convergence gradient find quadratic eq compare definite column adjust accordingly although equality nonsmooth smooth update nonsmooth subproblem concrete example widely arise statistic call lasso weight th structure ia ta I satisfy give k fx change variable minimize use preliminary inexact descent coordinate thorough investigation usefulness reproduce rather inexact update exact update scale cd ghz processor gb system angular exploiting
computational study optimistic mdps mdp r probability realize horizon initial state consider respect policy state indicate mdp associate reinforcement agent begin observe transition episode select action let make deterministic function distribution th episode action episode define incur reinforcement algorithm mdp internal transition reward assess performance expectation use reinforcement begin
occurrence likely insight want represent easier detect massive favor fairly general confirm outline distance alone appropriate discriminate underlie brain perform poorly recall score lack specificity logistic red horizontal bar interestingly switch laboratory confirm presentation prediction train regression term level bar evaluate spatial well logistic figure region category activation term despite report finding consistent forward comprise forward similarly map segment field unlike
base confirm gp estimation demonstrate acknowledgment support support first mm definition nature rx gradient exploration ac computational reinforcement policy reward rl directly policy collect hand learn estimate transition novel recently estimator demonstrate practical usefulness control future reward type policy value policy maximize represents expect learn iteratively accurately value machine technique employ well square robust value improve
location inversion location equally likely equation enhance window previous return even coincide spatial spatial probable localization describe performance art cover area approximately ft area ft cover ap mp intel provide construct location area test location location core cpu ghz gb ram detail result second cm parameter default mean receiver boost est spatial avg figure
compare prediction longitudinal survival flexible specification root cubic spline specifically mixed form b b b b natural cubic knot year knot year observe si replacement relative structure h ds hazard approximate normal chain mcmc discard burn computation version trace plot material credible longitudinal estimate relative variability association regression longitudinal continue calculation dynamic real life calculate prediction exclude fit patient old patient longitudinal trajectory patient similar profile gradient operation five next patient steady duration subject prediction I longitudinal survival calculate dynamic weighted operation survival prediction root figure supplementary material
g nc k say one square definite positive definite fix kernel k arrive metric satisfying arrive follow kernel invoke metric invoke
result arrive bind effectively root tree swap proof hold infimum equal argue introduce yield infimum argue equal expectation supremum splitting expression arrive upper due fact upper initial trivially infimum argue independent distribution put outside splitting result arrive uniformly pick predictor minimax expression consist cover therefore static strategy verify side conclude upper supremum write tree online notion regret
degree polynomial original infinite super feature algebra grateful mr search iterative improve become iteration consist convex
true fp tp spike supervised obtain machine truth evaluate validation false error receiver operate characteristic roc margin show verify indeed result run classical hybrid spike feature classical em theoretical well informative unlike global feature vary number scale feature find good sorting array allow
cost computation spc condition spc still construct summarize list spc spc derive method regression fc worse spc ellipsoid method describe condition transform spc condition spc spc hand spc fastest spc spc spc spc round long run decomposition embedding randomize quantile subsection technical quantile row non quantile preserve let condition condition matrix condition eqn prove eqn condition v I contribute inequality therefore linearity suffice u z z firstly state basis let diagonal probability least note change constraint easily show hold setting lemma sure z z z become involved failure hard subspace preserve sample induce see quantile eqn norm present norm nr well form expensive however adjust maintain except entire result nr r distortion ns nonzero distortion sampling least different
factorial factorial interaction exactly zero letter span randomization balanced spread condition factorial geometric star contain star cover star correspondence cover star balanced cover convenience star cover star spread corollary spread cover result develop focus prove factorial two design say transform rearrange factorial spirit star cover star equivalent rearrange factorial effect present formal let balanced covering cover e denote
eq e ds cumulative distribution q colored dot show horizontal line match upper need demonstrate blue line make binomial approximation
figure indicate belong represent logarithm role link node membership role row correspond network role half notice fourth column determine sufficient represent role term pattern represent interaction role word membership role approximately bipartite tendency english role majority role web resource interaction e role easier interpret right majority interaction diagonal indicating tell specie specie membership leave role compose primary role role blue specie respectively role level role specie distribute role distribution column row word role node row
solver compute constant subroutine examine vector principal admit provable able computation although moderately intractable value key algorithmic innovation provably safe elimination scalability million provably combinatorial evaluate algorithm synthetic million execute experiment collection specific word less minute million computer typically previously heuristic use rotation thresholding eigenvector modify lasso lasso produce pc nonconvex technique spectral argument motivate branch explore body semidefinite multiple pc arise obtain level sparsity desire fast develop establish optimality diagonal sdp author truncate isometry detection spike author spike significant understanding hardness np large clique challenge recover spike complexity spike recover clique barrier extensive provable approximation
algorithm hmms regression switch signal aim consist successive phase impose probability transition equation approximate use filtering denote pz compute filter series k ik algorithm diagonal take compute posteriori eq q pz ik backward introduce logistic define switch logistic vector generate independently accord covariate nz ik kx ik transformation flexibility
eigenvector centrality result provide draw project hypothesis whether report average reporting fall analysis project particularly centrality fix opposite centrality report significantly small analysis statistically relation centrality outcome handle particularly support centrality month semi report strategy quantitative highlight relation centrality quality question goal topological measure facilitate particularly aim whether report report status conversely semantic category provide task report base comprise different quantitative highlight gain inclusion simple large connect eigenvector centrality eventually make nine eigenvector closeness centrality illustrative evaluate term equally weight enable reader correctly interpret power fraction project report nine measure report l r fix consider
label visual class example feature shot classify category modal obtain et al first use design unseen distributional feature unsupervise corpora classify thousand image instance lot train different embedding relate sound et al use canonical
success cs rely actually specify object frequency cs operate grid disk impose discrete nature pose pre determine matter fine grid issue frequency grid along cs algorithms fine lead instability dictionary paper disk enhance solve nuclear minimization enhance condition ambient additionally guarantee great natural processing vision theoretic furthermore experiment super closely
partial tr n j tr see independent estimation inverse sparse sparse symmetric hermitian strategy adopt transformation state k operation quantum perform environment global swap discard environmental degree data range quantum big idea work employ machine develop programming obtain system central
pair compact center standard tool mathematical also provide divergence throughout cover please availability necessarily purpose propose property intend cluster probability center cost outer enough soon deviation value support bad function center ball respective center reasoning case center good ball give outer existence bound outer allow bound conservative suppose additionally center general guarantee note statement readily measure empirical center scale convexity suppose contain ball radius let ball norm suffice center q outer replace cost fix covering instead outer namely ignore turn place condition meet every fairly common arise
indicate offer comparable performance slightly well collection zhang liu edu cn work method base information gain reliable frequency count whether ignore frequency frequency within focus propose diversity entire corpus comparative two corpora new comparable ig macro micro classification tc language
average within basic otherwise understand refer closeness output point sense contribute outcome major suited pd context play retain prediction neighborhood r turn performance appropriate important devise little machine importantly notice combine basic different technique inspire addition let mention weight original opinion within relaxed impose agree sophisticated turn clarity mathematical decided accordingly discuss extra devote
maximize share thorough subject aggregate stable cluster ensemble color subject state art algorithm subject performance ensemble appear subject alone subject four figure visualization cluster notice instead apply linkage complete linkage hausdorff evaluate subject result dendrogram visible belong constitute metric learn competition iv seem share extreme conjunction metric bring qualitatively assess help community benchmark aim investigate day day variability train classifier evaluate acquire record competition iv variability heart classifier day nonetheless system day temporal problem propose help class dictionary dictionary wasserstein metric base geodesic cauchy explain dictionary subject give propagation cluster consensus ensemble cluster give time either say cluster say belong indicate either exclusive proportion ensemble metric yield global cluster ensemble stable ensemble propose phenomenon eigenvalue graph thorough embed laplacian near neighbor pick laplacian choose potential code link wasserstein cluster ensemble contain mainly clearly separate subject distant
computationally slightly global requirement give von indicate evaluate side sum observe contribute right thought length give since big fan
existence impact much broader derive fully thresholded estimator thresholded optimality without assume norm entire matrix inference detail matrix give regularization method study thresholding estimator establish estimate covariance precision new develop apply scale optimality spectrum statement sparsity issue necessity optimality support relate strong converted confidence unclear organize statistical spectral latent section support possible extension numerical proof summarize notation throughout write norm matrix ax smooth functional square submatrix asymptotic efficiency introduce extension functional precision estimate expect yield invert estimator dimensional th compose follow version coefficient sample mle course
database task test use hmms language decode clean gmm gmm seven range db experimental reference result pass viterbi decode level give whereas give improvement noise adaptation standard finally absolute degradation w r adaptation expensive negligible cost non
theoretic tool suitably pack integer exist distinct proof follow self rank noiseless require incoherence target consider approximately matrix subtle condition result comparable satisfy accordingly norm w observation trace approximately recovery satisfie properly absolute hold restrictive valid example netflix assumption movie constraint really practice unlikely accurate prove effective presence hence max robust approximate guarantee sampling actually reduce e matrix see follow estimator sample distribution uniform follow inequality hold probability
computer memory work parallelism cluster utilize computer parallel initially computer belong partition iterative coordinate locally describe partition belong store computer choose coordinate hence computer result overhead comment parallelization potential accelerate increase quantity completely processor increase speedup large speedup negligible may suggest whether second quantity characterize effect partition computable interpretable practitioner priori output solution ignore
set consist grid consider regularization vary example node minimum tolerance add third length svm square exponential unit length use compute construct test set size remain arm test specific validation learn task attribute experiment report recommend probability rmse proxy error interested recommendation almost thompson good ei ei pi minima randomization inherent thompson explore manner experiment bayesian budget correlation
substantial finally never sense attempt subtle acknowledgement thank preliminary manuscript helpful suggestion lasso relate sparsity induce substantial apply correspondingly choose tuning specify practice choose variant little high design wherein grow risk choose via cross necessarily generalize lasso performance persistence oracle regularization statistical become tool response matrix consider problem euclidean norm necessarily abuse notation form lagrangian lead contain scope picture highlight covariance author investigate lasso model
array processing area interest suggest acquire array allow estimate frequency analog digital temporal sensor spatial domain system analog front redundant sensor array element prescribe theory array integer integer great common presence spatially field array array point payoff array reduce array array array element share array represent array operate geometry array music however
maximum run directly behind influence author gaussian age physical uci repository match policy figure well without bad contrary polynomial dimension limitation rbf kernel phase consume visible decay iteration side physics moreover magnitude provide cumulative parallelization art gp strictly experimentally another approach gp guarantee confirm application
tt note systematic except sl value violate value penalty fall range parameter dataset extract uci consist spam spam word character extract uci dataset represent band dataset digit normalize pixel intensity digit face human face gray scale person link generate label couple
fast inexact subproblem efficacy demonstrate art optimize express partition quantity log linear central quadratic recently art focus extension recently system batch memory bfgs conjugate gradient quadratic need pass update quickly dataset grow size inefficient turn method computational stochastic descent sgd alternatively mini
interval fail predictive scoring propose use commonly field weather forecast scoring appear task analogous relate proper scoring intuitive difference represent predict interpretation response log likelihood discrete consider predictive ask future value set might might process specify become calculate z select construct future represent enable good decompose performance estimation datum use draw compute score draw n let interest effect weight put optimal bad
reconstruct unlike whereby local optima change characteristic assignment certain decrease objective ls consistently ibp furthermore submodular converge comparable maintain get optima assignment optima technique ibp model I approximate maximization insight exploit inherent evidence bind formulate quadratic boolean converge compete ibp various nonparametric dirichlet interesting research generalize work propose submodular obtain bad matlab implementation support united grants google microsoft thank anonymous comment material zero column place independent integrating let yield probability certain zero remains order whereby result shift propose equivalence
find optimal element boundary node step update optimal specifically concerned f rhs tell element boundary group maximize indicator problem entire optimal verify time unchanged backtrack rule order subtree graph graph boundary set say root specific root root root root tree negative exploration root root explore subtree fig procedure also node subtree bind rooted logarithmic node integer root induction case node inductive small minimum node root root tree graph root induction nod small tree node visit node encounter rooted node number really part adjacent boundary capture subtree additional connect subtree root boundary denote encounter pick let clearly enough induction base case suppose rooted encounter inductive spread form root subtree explore subtree induction explore subtree boundary encounter boundary explore root hypothesis bound encounter explore exceed pick pick exceed encounter explore addition root contribute boundary pick contribute node therefore total argument explore subtree maximum number encounter explore lemma maximum logarithmic establishe program run solely value run rule exploration run loop loop exploration ordering value single loop find among node child value recursively total time require represent hence subtree value take child e maximum problem constraint dynamic program regular furthermore complexity follow root subtree root assign integer root maximize exceed optimal suppose node
probability constraint satisfied learn consistent two normalization neighbor eqs belief equations binary perceptron propagation equation impossible bn I sum get eqs cavity log notice cavity constitute recursive equation compute energy bethe node normalization constant solve message landscape configuration accord densely affect consistent top horizontal dash line guide empty symbol system message random solid replica saddle equation landscape simplicity still behavior landscape entropy rs rs low maximal increase solution reach number distance intermediate set decrease
algorithm core approach desirable reliable indeed broadly applicable iterative likelihood model finite mixture model numerical simplicity reliable em reader indeed establish analysis som view som limitation variable good perform regression mixture main concerned model area curve rather analysis approach concern paradigm curve reduced goal analysis etc achieve statistical curve namely include mixture spline estimation mixture algorithm point
partial assignment function encode encode log model unfortunately feature parameter learn inference assignment conditioning set assignment encode dense graph result encoding improvement complexity present independence graph order design adaptation purpose generating set search use statistical generalize central theoretical guarantee benefit structure use learn distribution represent decomposable omit conduct generate contain ib distribution review representation ib
strongly moreover know moreover immediately prove view kb concern skip simulate parameter simulate trajectory obtain error I value possible procedure simulation intel processor computing table compare observe estimator bias visible surprising check use quite small table compare
f find influential fouri way thm find exist approximation give variable exist satisfy request lp require submodular hx specific uniform discretized use chernoff individual essentially fix example sure least suffice submodular satisfie gx return hypothesis reach assume step return fx fx I hx successful range express constraint value lp bound claim submodular multiplicative error fraction reduce execute recursively solve input equals least submodular see chernoff constant submodular integer randomly uniformly hold random chernoff tc f exist uniform time example outline level within multiplicative function f theorem confidence boost technique hypothesis indice execution example find suffice return estimation first observe markov inequality imply fraction return reach depth disjoint hypothesis satisfy multiplicative guarantee hypothesis guarantee point finish proof fraction obvious know satisfy inequality fraction prove inductive claim depend execute conclude failure run n simulate random require filtering mean note close start prove give
consider clean duration net play arm cover inequality imply need simple compact metric space compute net compact finitely diameter compute construct net point lie contradiction clean phase duration net tx need two thing happen intersect frequently play confidence radius constrain arm activate inequality prove therefore arm arm need intersect exist imply x contradiction attention full lipschitz mab contribution specify universe expect round pick strategy choose receive payoff query arbitrary upper restrict pick arm receive payoff abuse case payoff round essential metric expert regret tractable double feedback tractable former occur metric correspond strategy compact bind assume low feedback prove idea counterpart jointly investigate expert tractable mab upper whereas expert even feedback former occur expert describe whether cover regret question characterization matching expert problem set diameter size diameter cover definition root child leave uniform leave tree branching cover wasserstein k take wasserstein standard discrete retrieval dimension extend cover appendix detail metric covering sophisticated well lipschitz function dimension uniformly notion define cover hold metric lipschitz expert lipschitz tractable uniformly lipschitz part generalization bandit build metric lipschitz amount complete characterization metric analogous page characterization l completion compact countable cover compact regret concern theorem restriction mab tractable expert problem even double feedback tractable feedback countable bandit metric space classic point organize bandit expert auxiliary respectively topological entail topological entail topological contain isolate well exist perfect topology topological implicit proof part algorithmic metric space perfect compact countable perfect theorem provide appendix making exposition contain space space simple reduce space lipschitz completion space space mab metric mab state complicated space apply metric first remain apply therefore arbitrary lipschitz expert bind lower bind lipschitz expert perfect subspace problem hold desire perfect useful ball parent ball tree correspond confusion later necessarily radius perfect tree metric space subspace perfect us ball metric tree tree enough define path ball tree child child leaf child us ball payoff via sign uniformly payoff sign pattern follow uniformly independently sample child ball leaf suppose satisfie ball hold generalize applicable end implicitly function metric exposition considerably idea implicitly formulate payoff nearly indistinguishable lipschitz expert necessarily along borel measure feasible triple kx tuple pairwise subset borel exist mutually subset iii expert ensemble least problem bandit payoff feasible expert payoff function reason
dimensionality normalize vector norm expectation sgd except incremental experiment decrease size objective evaluate interested dimension pass runtime dominant complexity split incremental make progress per fast make runtime slightly incremental keep incremental incremental drop sgd worse careful good variant theoretically get suboptimal solution excellent recently incremental furthermore
difficulty theoretic way information measure bit capital probability clarity sum discrete various way standard ix hx hx hx hx estimate intuitive way still expression locally contain neighboring density volume th near norm factor match knn write alternate format ease derivation estimator neighboring estimator empirically amount datum choice discrete compression purely discrete separated gap purely bin arbitrary order information theoretic quantity depend without relationship arise subtle two intuitive split equally sized mutual mutual intend effect unbalanced scenario
present span space require projection discuss support example moment gaussians separation disjoint support mixture coefficient support cluster construct greedy first distribution sample classifier split another split significantly understand easier look know supremum split break cluster proceed associated leaf distinguish prevent overfitte one surely absolutely continuous enough
substantially increasingly grow cost may potentially gain enable experiment prototype change bagging bootstrappe intensive compressed discuss arise run efficiency imbalance attention form feature response imbalance email spam filtering train typically mistake neither occur give datum set imbalance focus second also common set example primarily rare computational hope subsampling way rare class implement care inference set control regression scan much contain roughly simple way reduce subsample else however inefficient importance control uniformly adjust promising control subsample imbalance costly fit subsample valid adjustment intercept however nothing exploit imbalance marginally imbalance look case discrimination purpose local attempt imbalance give pilot estimate logistic keep surprising specifically xx residual pilot model extreme imbalance generally quite subsample magnitude full logistic correctly specify
jointly guarantee conservative confirm improve histogram mahalanobis derivation learn aim learn claim illustrate may learn I bind unseen datum use prediction nn express metric classic label instance derive metric learning consist training triplet violate metric pair supervise mahalanobi iy work unknown quite strong constraint instead notion adapt bound limit regularization loose bound induce regularizer weak robustness necessary well lastly rademacher regularizer derive norm easily linear metric learning formulation deal regret hold univariate receive previously see point expensive practice buffer also rademacher essentially adapt metric open good knowledge linear link goodness learn al norm rademacher category deal make unlabeled constraint concern metric adaptation label accord source different leverage belong negative semi supervise follow review formulation incorporate unlabeled principle encode similarity et al construct laplacian intuitively preserve point experiment laplacian side drawback intractable dataset inspire improvement refine construct formally auxiliary weight regularizer laplacian et propose optimization converge metric tackle labeling give pair mahalanobis distance entropy unlabele regularization encourage low trace nonconvex iterative projection psd outperform supervise amount evaluate overfitte domain da different refer source situation real speech recognition spam sometimes unlabeled deal covariate shift label assumption minimize j covariate importance adapt compute reliably author situation covariate domain adaptation set case classic strategy bring source close maximum
interestingly use may depend score student put effort every student inference condition compute nontrivial correlate good bias allow well true order bias must apparent simple inference sample quantitie interest distribution uncertainty sample efficiency perform discuss rapid mixing chain discard burn maximization bias estimate parameter practice behave natural gibbs analogous score run minute em refer em pp pp std measure truth give bias well student pool residual ground rmse percentage simulation produce truth score hundred discrepancy consensus
historical also computation success fact old historical survey kind occur social search unclear assumption equally propose discuss differ essential view instance general simple linear scheme tune work cite outcome accumulate trace describe describe
hence complexity estimate extend seek relate enable solution reduce restrictive analogue entry definition empirical consequently solve qp analogy n unique asymptotic accuracy discuss paradigm imply chain nf hmm f j dy calculation since call effective discrete natural since consequently qp kk ij discrete obey f since ij advantage many cast ij additionally accurately estimate may require sample detail output next additional estimator know exactly estimator stable perturbation simplicity throughout essential qualitatively change defer
regression line increase intuitive phenomenon line flat slightly possibility reason score homogeneous greater great subsample size point probability illustrate bad parameter well leverage variance involve common intercept sample score building usually one mean column illustrate show score intercept include interestingly line element par common small leverage simply increase modify toy example simple behind define cc leverage leverage converge zero theoretically figure demonstrate example intercept increase illustrate variance get large triangular array one even pattern odd toy set start hadamard uniform leverage similarly matrix unless aspect extremely rectangular leverage particular consist identity orthogonal score large could remove zero seem think row point point score row point direction even bad example text trivial nice algorithmic perspective well problematic proof encode rescale point expand ordinary ls estimate part operator follow column block simplify simplify combine x rise e equal element q rewrite matrix algebra establish unconditional expectation expectation result rule double unweighted leverage detail employ taylor tw tw lemma follow take expectation yield tw te tw finally lemma rx tx x tx tx ix component analogously ii see rx tx tx ix tx tx
dictionary element recovery specify correlation require threshold dictionary present result recovery permutation note sign ambiguity exchange sign dictionary imply decay scale decay estimate arise error svd discrepancy element responsible even responsible analyze condition entry proceed coefficient sample dimension sparse problem suffice guarantee recovery linear procedure choice many concrete example work strong dictionary high output restriction allow condition zero element satisfy universal estimating place compare assumption suffice recovery algorithm procedure principle approximate recovery relate understand future present sketch yield move proof employ common connection concrete neighborhood dictionary
enough height satisfy property see choose thank totally four iff I value frequently density layer common neighbor hide thus support intersect pair node version provide need least warm prove accuracy clique vertex overlap clique analogous find potential hope forget strong share common neighbor graph satisfy strong appendix work edge except weight hold half layer density idea higher allow neighbor general satisfie proof common property know must early recover possesse neighbor say neighbor bipartite polynomially appendix unique less sure failure support strong property intersection neighbor depend neighbor randomness even layer proof depend encoder work determine edge denote back know case discover layer claim polynomially discover non edge complement find remove find suppose uniform follow proposition say probability choice hold
estimate master combine communication reduce independent introduce computationally incur global accuracy include rna l minimize inter resampling cause statistical generate algorithm l rna disjoint assign process particle step locally resample topology static redundant converge target track process equally exchange particle would become large scalability weight imbalance process particle particle overcome adaptive scheme particle number communication minimize reduce message optimize
speak improve reach prediction study variant general partition able replica symmetry consideration consider opposite mention upper capacity discrete certainly frame satisfied accounting inequality essentially union base improvement recall result paper mathematically perceptron setup introduce purpose need capacity perceptron feasibility argue mention normal early regime dynamic store briefly section feasibility logic pose question answer optimization relation discussion infeasible relation degree utilize capacity spherical probabilistic integral strategy develop choice version relate center interest concentrate average present part complicate complete study presentation simplify fact consideration I standard normal q set constant solving obtain zero equation arbitrarily constant need assume discussion ignore exercise show side prediction capacity perceptron theorem replica memory probability course simultaneously present rigorous mention consideration previous strict follow study scenario upper attempt upper subsection reveal phenomenon
row large connectivity assumption use alternatively cast mle equivalence identifiability eq side uniqueness hence unique aim recover closed tackle scheme period receive gradient perform centralized increase stepsize dual update gradient proximal function particularly proximal leibl know
endowed induce q endowed frame frame set perturbation equation phase set hypothesis hand phase connect firstly show connected segment let complement piecewise connecting prove obtain set
education knn reduction high mse yahoo acc c acc cpu sec ml knn education knn mse knn mse science knn mse business knn mse knn dataset score acc acc knn knn mse medical knn knn scene knn mse knn mse knn mse knn mse score accuracy outperform gap recall imbalance metric success metric cause imbalance mse provide dataset rw recover training justify entry item phase diagram cost seem item acceleration capable scoring base rating fast analyze monotonically project smooth manifold via cosine converge local asymptotic property analyze produce subproblem optimality error decrease converge proof firstly iteratively project onto projection manifold update definition angle sphere cosine angle convergence alternate manifolds eq refinement manifold close alternate intersection constant large intersection manifold iterate algorithm form manifold linear convergence cosine theorem determine
classifier eliminate output mis select answer discard remain possible propose enhance performance select binary propose elimination classifier se se remove ignore classification class ignore classifier helpful eliminate candidate propose elimination bad employ classifier round suppose active classifier remain class vs reliability opinion select always class answer give wrong answer voting reach large vote letter dataset section case equal voting vote classification popular opinion competitive high class technique voting candidate winner top voting score call output remain let maximum score score
present depend field predict sensor response keep mind characterize circle quantify estimate light quantify symbol aside robot inference collect fall sensor evidence find discussion inference circle number integrate parameter quantify support optimal use describe compute index collect result value refer additional assign uniform essentially calculation proportional assigning integrate five set relie measured likelihood take expect standard set record time employ explicitly gaussian four gaussians four algorithm iterate consecutive evidence typically compute simply perform consist strictly discretize estimate directly compute line section light laboratory
assignment ij go else return dimension number algorithm remark feasibility assignment stay subsequent iteration use warm start new primal simplex course many must force pair property way mean conceptual index assign preprocesse replace combination perform return set define third optimum site computation balanced error norm maximization far complexity compute problem open universit f universit algorithm subset spread cluster scientific business
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array common array invariant permutation act index collection user exchangeable sensible notion exchangeability dimension jointly partition class invariant act carefully exchangeable partition disjoint permutation class exchangeable k collection permutation cast separately exchangeable array recover exchangeability recover exchangeability characterize exchangeable array array let exchangeable collection complicate u collection uniform index cardinality agreement index jointly also encode empty generalize dimension jointly k jointly array write element nonempty index generalize array exchangeable require roughly twice even array observation introduction densely array overlap regardless pose exchangeable array array exchangeable fu special jointly fu separately exchangeable fu state general arrays array exchangeable exchangeable array absolutely exchangeable number represent finite vertex exhibit like law phenomenon occur though exchangeable graph inherently lack mathematical consequence random require exchangeable model development effort mathematic make symmetry raise challenge answer structure occur occur infinitely simple sequence conditionally infinitely bernoulli take fraction precisely constant exchangeable one zero single triangle five star subgraph never infinitely since infinitely connect quantify vertex random partial sequence sequence use typical array inherently social friend dense empty vertex undirecte sample accord edge x law dense limit dense closely exchangeability inherently grow define empty vanish one graph course graph simple generate multiply scheme exchangeable graph dense multiply specifically chen triangle obvious limitation equivalently independently graph
iterative digital gap condition present result convergence shift guarantee definition arbitrarily finite close enough convergence subject implication list originally mode result true view prove approach present consistency performance discuss necessary bring work mention step also consideration implication point reach point
free aid distribution ml markov context problem ml ml likely general expect estimator gain gain maximum expect q successfully bioinformatics estimation minimize facilitate maximize ml identical suboptimal extremely propose account centroid estimator pointwise gain gain centroid define iy iy indicator depend whether false centroid expect pointwise consensus estimator assumption gain eq space bioinformatics section consensus estimator pointwise function consensus pointwise index gain predictive bioinformatic positive tp negative tn positive fp false negative eqs superior gain score tn false prediction fp positive gain gain compatible score tp tn fp tp tn fp definition characterize gain function centroid define equivalent centroid maximize centroid special parameter positive gain centroid term depend theorem obtain follow centroid eq problem bioinformatics centroid centroid problem centroid consensus predictive pair position across base position pair structure
iteration q know update cyclic call cyclic recover rule gauss eq improvement contain denote minimum intersection nonempty uniformly make assumption regard approximation kx u kx lipschitz constant satisfy valid remark order indicate locally bind function classic application nonsmooth simple minimize function nonsmooth present lead call choice secondly strong requirement say block strongly fact satisfy interesting consider maximization wireless network base bs bs bs channel capacity optimization program water verify receive full imply type follow none simply remove problem
distribution linear however straightforwardly plug fig random walk observe mean infinity variance consistent datum make standardize size mini population variance even study apply individual sequential incorrectly whole incorrectly
dash red detect anomalous anomalous spatially rest anomalous possess box represent anomalous mix proportion look reasonably distinguish illustrate anomaly gaussian mean generate normal make far additional degree anomaly lastly generate outlier depict anomalous aggregate pattern proportion decide proportion proportion group individual perfectly normal group anomaly together look anomalous anomalous individual topic proportion different proportion anomalous figure uncertainty claim zero mean isotropic wise experiment encountered application
fig present combined model procedure fig th correspond cell contrast movement observe attain slightly high tracking additional sensor well track accuracy amongst accuracy number sensor discount set evident tradeoff exhibit behaviour significant outlier even instant solve bellman information carry learn schedule individual contextual information carry forward step regime sensor algorithm tracking system track propose evident possess theoretical guarantee see period worth note provide rate except know date even case show par illustrated optimize follow formulation similar long average discount objective steady sake approximation q setting empirically usefulness extend setting multiple objective sensor set individual sensor absence controller decentralize variant sensor algorithm pass sensor message alternative possibly group regard sensor time analyze paper essence
near bind arm exploitation call suboptimal lt f lt e lt probability upper exploration least exploit agent bind ik jk I tm optimal item recommend due near near number suboptimal exponentially minimize bound parameter jk tm f come near r z assume lemma bound partition concentrated region frequently context adaptively arrive achieve e type mention indicate achieve well know combinatorial final regret sublinear reward recommend item recommend request subscript recommendation f recommendation item jk ti item evident set map unique arm arm context nu since estimate reward separately estimate arm reward whenever arm partition context way unlike exploitation phase action exploration phase exploitation phase select phase keep function keeps basically count agent keep former time number arm exploit context agent choose item recommend remain select item htb tu dm I tn u u u l tt dx l lt l u nu recommendation recommend ir u u subsection user context th high agent let section optimal take part analysis regret exploit regret time exploit bind agent comment come comment regret much
satisfy boundedness I least context case variant describe z var rule difference bound quantity expansion high theorem proof write denote return iterate instant schwarz deduce put together update f recursive definite martingale show q noise eq result pt pt least temporal algorithm provide expectation also impact make setting analyse least regression combine signal solve equation policy reinforcement rl td difference computationally sa training analyse high well impact significant decrease cost appeal canonical
yield dictionary take view connection nystr om could easily theorem berkeley berkeley berkeley berkeley unsupervise art computer vision architecture redundant early vision offer dictionary invariance simple enjoy efficiency find well recent decade spatially pool feature extract either raw descriptor code encode complete simple linear pooling carry encode stack label single encoding set basis image patch encoding method dimensional locality patch like
behaviour particle toy obtain kalman grid line mark blue alg volatility suggest setting manually tune good evaluation converge evaluation reach neighbourhood iteration require algorithm ht indicate capture enable efficient acquisition comparison mind alternative important work acquisition
use statistic theory explicit state estimator sample use occur roughly context intuitive explicit arbitrary estimator minimal test asymptotically necessary equality rate definition cross incomplete statistic naturally difference soon satisfied exploit estimator derive asymptotically exact address computation approximation leave illustration penalty recall already familiar machine learn take however necessary need let identically vector average maximal design statistic cardinality call regular parameter number function represent kernel minimal unbiased etc cost deal many coefficient grouping together formalize true
allow adjusted restaurant restaurant infinite membership indicator generative global mixed membership distribution dp mixed membership ij multi community indicator ij community sample independently process base ij ji back crf analogy refer customer locate row likely restaurant restaurant
prevent must guarantee preferred feature channel select goal want learn model accuracy combine learn parameter channel channel enable interval hz expand band global band universe band splitting cover overlap min g min b min cc finite interval slide window strategy propose produce appendix c almost band band selection aim detect optimal band band produce optimal band study employ eeg band frequency tuple tuple spatial mainly contain step deal problem
customer draw prior customer select already try decide try exchangeability treat decision transform link latent reasonable generality consider covariance variance simple early challenge dimensionality yield add ibp infinite fit ibp mask determine whether weight entry ultimately respective dimensional binary
certain condition understand statistical spectral expert weak statistical efficiency serve integer use denote form take e dot product px symmetric row whose th norm sum singular frobenius max singular value x px dx ii ji j k though regression tensor simplify identifiability condition mixture regression mixture involve mixture noise distribution compactly proportion mixture observation parameter regression
limitation complex extend discrimination approach discriminant framework component homogeneous overcome limitation model shape flexibility sub automatically regime regime represent label regime sub logistic process accord
cluster spam server look group extremely group group extremely coherent ip indicate location green triangular ip internet service internet likely term send month average list correlation coefficient work together matter behavior highly correlate appear discover correlation email address agree cluster location c month reveal social discover community specifically cluster behavioral either divide community mostly non accord spam usage observe discover coherent temporal behavior ip address provide
conditioning average matrix magnitude value matrix two resp table give robustness c reason significantly dirichlet outperform time identify correctly ten resp times identify identify dirichlet middle correctly column three percent analyze road surface extract mention assumption hyperspectral take extract signature ht display abundance maps index algorithm extract six common ht well algorithm distinguish rd identify road surface st th particular percent eight reconstruct oppose focus extract extreme cone explain include extraction incorporation remove pure separable initialization sophisticated rely require normalize assumption matrix distortion
belief inference knowledge prior eq model statistic axiom show rational way vast might high take approach interested rational relax connect informally write discuss type shall report belief decision guide action outline probability function belief via assume piece information appropriate loss fidelity fidelity loss function analyst needs proceed remarkably kullback leibler von specific function surprisingly minimizer form complete usual loss maker need construction unknown state outcome maker concentrate task literature present since provide idea zhang zhang estimation name regard zhang zhang gibbs bayesian posterior gibbs posterior target well involve logit build zhang direction approach theoretical misspecification principled approach information zhang coherent kullback demonstrate incorporate stochastic cumulative presence non stochastic construct essentially idea processing rely present log compatible discussion derivation broad concern date back primitive belief express field bayesian assumption formal entail adjust model motivated misspecification pseudo bayesian update proxy conditional
avoid table primarily rapidly evolve occur potentially interact least capture substitution heterogeneity evolutionary history branch determine spatial focus explicitly hypothesis attribute pressure relaxation distinguish accumulation strong relaxation hypothesis disease within bayes factor hypothesis application discretization inference trait human trait previously infer epoch epoch appear central diffusion rate strong diffusion connection epidemic represent transmission dynamic specification alternate epoch fit homogeneous parametrization trait location readily additional epoch add limit incorporate reconstruction inherently inference approach available epoch allow among achieve specify hierarchical research approach sparse represent clear correspond times time epoch transition mcmc estimating may previous experience possible introduce quantity jointly specification add complexity evolutionary inference intensive evolutionary even specify evolutionary burden impose time involve library perform calculation
contain describe abstraction interface solver needs grow toolbox describe etc prevent redundant computation incorporate useful concept give weight weight max I sum
use normal create randomize various independent optimal size plot trial behavior row entry last entry residual vector residual residual vector entry mean entry mean variance residual vector entry case convergence randomized setting row far normalize varie trend monotonic normalize give test increment give purely still bad situation row cover medium regime slow theory hybrid sampling medium hybrid sampling outperform surprising hybrid sampling important norm decrease marker marker black marker star show need cut still
compact main obvious analogue fortunately enough lemma set combine bound possible bound situation work area way nice monotonicity energy sometimes bad time couple bx yx dx exist dx dx let complete acknowledgement thank division thank partially nsf dms definition thm example edu usa com mathematics adaptive markov important past markovian although establish ergodicity curvature establish concentration establishing give quantitative also provide proof finite property parallel metropolis hasting sampler mix time complex implementation require performance know guide tune mcmc algorithm adaptive originally build construct proposal energy move mode modal usually construct markovian many tool survey apply variation weak wasserstein weak convergence hold bound see paper ensure ergodicity see briefly two condition family transition recent show eventually stop adapt adaptation sufficient condition large central often information rate author analogue markov modal complementary bound comparable analogue time establish inequality markov contraction
adjust rand index use classification rand agreement ari rand agreement chance perfect would base excellent base know membership generality obtain model know membership unknown membership bayes know structure start component ten perfect classification start structure perfect classification cf ari ari model indicate start value utilize gibbs estimate cf spherical incorrect structure might let force spherical depict argue despite far
selection set select observe score large posterior include iteration step master construct machine summary step among partition evenly assign machine summary sm u eq support know machine predictive partially independent conditional sm u report appendix notation previously bayesian latter large cause realize output correlation combine intuition machine exploit improve correlated idea global summary know machine summary transpose predictive term improve predictive tuple experiment machine local choose center send subject sophisticated scheme communication improve
process analog measure device specify probability future present cast grouping past state iid say th suitable unlikely reject hypothesis observe imply refer equal reject nan calculate value nan
note rely recognize since former compare domain compare build property presence categorical automate property datum rest paper describe method determining discuss experimental attribute attribute object database attribute characterize fashion expression interval sometimes condition condition give w object satisfying definition introduce quantify intuition make attribute database attribute value pdf pdf typical pdf population low indicator whereas anomalous high analyze pdf key measuring
mapping large graph supervise explicitly section hypergraph analogy graph total hypergraph cut modeling graph involve weighted vertex correspond vertex e element letter incidence h impose restriction hypergraph vertex undirected weighted motivation correspondence graph total variation definition complement cut carry hypergraph hypergraph biased cut vertex handle exist develop focus work transform hypergraph suggest ce subgraph weight
recover tensor entry recently relaxation apply solve solution employ rank employ recently singular one adjust compare art method completion figure table numerical transition color algorithm range sample important recovery tensor kind video recover tensor along nuclear norm method tensor completion popularity availability superior solver completion efficient reliable admm code claim accurate reliable smoothing could solver appear theoretical completion tolerance loose relative test tensor generate vary
characterize dynamic accord stability provide conclude remark sec agent play game agent action action thus probability assume reinforcement reinforcement relative utility strategy round update accord reward value playing determine specify neighbor exploration mechanism probability temperature control maximum interact update agent term play strategy payoff asymmetric interested obtain eq collective learn agent repeat play pure increase goodness fitness selection population biology
yield piecewise regressor regressor partition union accommodate nonlinear regressor efficient tool several algorithmic preference affect life algorithmic tune carefully accommodate phenomena tree regressor heavily include region balance regressor uniform binary modeling regressor result extreme avoid direct particular partitioning average full fig hard boundary doubly regressor collection assign regressor piecewise linear regressor fix trade framework specific fix adapt theoretic consideration final highly successful operation algorithm highly provide application performance boundary minimize introduce doubly significantly outperform artificial parameter instead directly minimize rp tree framework rp tree learn tree weight progress introduce algorithm asymptotically good exponential regressor linear upper
important question framework constrain optimization let pdf conditional n eq simplify equation write bivariate cdf indeed standard bivariate gaussian zero hence apply normalize bayes definition hence definition expensive computer several objective new principle issue see reduction set high expect formulae avoid numerical showing provide krige design region output complex code objective typically
note schema graph schema computationally expensive exploit dependency aspect achieve classification accuracy classify collective exploiting network citation link link dataset improve performance exploit multiple type dependencie similar result collective meta collective consistently significantly meta exploit structure effectively claim heterogeneous path exploit complex dependency dependency extract boost classification furthermore able significantly dataset observe reach approximated path meta conference path claim dependency capture select representative meta path se time se show run collective meta path large meta path meta consider training time long method incorporate meta path slow path additional aggregate meta
summarize repetition figure type strategy conservative reach asymptotic different pt pt additional test gram achieve median multiple kernel broadly trend synthetic world audio result show test gram impractical statistically consistent computational experimental see conservative threshold figure table gain cf equation size influence observe drop distribution considerable overlap distribution indistinguishable behave vanish statistic
arise differential analysis however rna seq observe due discretization first difficulty package supplementary material filter appear filter peak value express together rna seq care suggest among identify differentially express remove expression final list differentially express global rna seq arise assume binomial dispersion interested testing gene dispersion obtain reduce binomial generalize glm gene study whether filter across vector filter procedure discovery rate additional
coefficient omp norm realize sign show bp omp well recovery unconstrained recovery rigorously diagram application image sparsity dictionary coefficient algorithm negativity unconstraine omp unique enter iteration residual condition hold lie lie equal therefore true sufficient consider atom pick denote value large bad case guarantee correlation define order derive low coefficient absolute success omp maximum observe series whenever q choose omp zero constraint integer minimum value strict generalize bad sparsity choose gram dominant number atom contain atom residual notational square
multinomial multinomial word word examine totally system expect degradation precision fine sensitivity word document topic topic
element leverage reason two sample phase significantly complete coherent matrix score incoherent successful roughly accordance surprisingly complexity proportional ccc recover coherent draw range uniform sampling note axis start uniformly leverage observe mark ccc ccc plot perturb matrix plot leverage without require knowledge suffer dramatically study performance phase law decay refer construction leverage score coherence incoherence maximal normalize plot successful recovery axis put plot well sampling oppose construct svd nuclear use generate serve theoretically justify augment lagrangian alm solve perform theoretically score score coherence low quickly
allele genome gene simply correlation method test similarity ability predict alone determine permutation alone correlation grid overlap confidence genome baseline top gene association snp determined fold univariate linear top first principal multivariate repeat unable vast baseline expression association clinical phenotype apply study contain dna either change treatment measure pass snp assess potential snps capture snps standard significant using involve separate heart et snps kb snps quite indicate significant snp single calculate kb snps grid use set snps fold validation approach well top snps respective half prediction expression measurement measure ref expression value gene improve choose gene pathway involve attempt improve combine gene pathway molecular signature database snps produce
online infer item probably obtain know review topology review jointly item constrain truth item pair connect bad good review truth truth
degenerate nature propose precision graphical precision regularity penalize simulation utility dimensional analysis genomic attractive genomic density normal analyze research currently undirected comprise multivariate whose precision satisfy conditional relationship node concentration interpretation find broad range world finance system structural rarely homogeneous
tool interactive exploration various control domain expert explore visualize difference operating depend even stage expert differential identify change protein analyze patient control use patient diagnosis technology tb pdf recall show tradeoff fdr cancer study learn network curve majority level differential much fdr setting reach acceptable difference remain difference real tb differential dependency cancer arc represent dependency present population differential dependency obtain ensure reveal relevant david ask extensive
mrf mrf discrete variable potential seek constraint trial show fraction sample experiment varied selection small grow penalize grow around example neighborhood estimator build notion admit consistency selection condition combine lasso knowledge asymptotic consistency nuclear acknowledgement anonymous comment lee fellowship fellowship stanford fellowship support grant gm nsf grant dms bounded sublinear sublinear solve since strongly substitute plug proposition q side substitute since ignore obtain combine eq proof taylor term q lipschitz first
spread power quality various measure sound latter nonparametric massive smoothing tailor assessed experiment compression music method show potential methodology music business keyword subsample regression music dynamic primary secondary music periodic dynamic level acoustic energy huge acoustic cause peak latter compression dr dr spread acoustic power dr translate loss audio dr field music dr measurement business death attract attention approach cause compression dynamic lose consensus practitioner audio dr various little dr section dr statistic signal detect compression computationally procedure statistical property key issue
coordinate sparse learn formally baseline introduce rate reproduce denote reproduce kernel denote instance convenience mkl combination kernel f loss kernel could combination predefine convenience nf f classifier note appropriately endow combine computed choose classifier equivalent choose weight learn keep classifier classifier choose large norm combination construct counter
neighbor connect via vertex distance space exponential empirically et encourage sparse vertex work implement solve optimization representation construct near tool lasso graph contain lrr lrr perform recover intrinsic treat affinity problem graph color sift sift ccc nn dataset treat video vertex create edge comparison nn tuning range graph semi efficient propagation enjoy art separate category propagation experiment wide sample propagation remain repeat random split perform use tradeoff lrr present require versus lrr subproblem lrr perform lrr gradually moreover lrr fast achieve issue lrr size accuracy
cc notational clarity counterpart case copula incorporation indicator brief copula supplementary material reader detail covering assume copula experiment flexible modelling achieve copula support refine intra context give membership equivalently express motivated network indicator correlate indicator interest likely belong higher low separate propose framework employ membership indicator accomplish copula
subspace algebraic spectral cluster sc introduction find central entry datum accomplish identify graph value two sc ssc introduce rely idea sparse construct low rank lrr ssc provably succeed noiseless elegant reveal ssc succeed subspace
state system assumption dynamic noise kalman filter use array medical may past improve estimation important kalman robustness dynamic filter interest since effort build huber penalty recent effort design track fast e g jump contribution laplace rather introduce evolution interpretable known process noise exploit nearly correspond outlier jump heavy tailed I necessarily non possible contribution convenient application student apply robust inference relate influence trend smoothing derive laplace function student less smoothed proportion trend smoother similarly student allow trend track practitioner measurement measurement fidelity different knowledge stability smoothing use student tracking differ include measurement overcome convexity student differ propose possible hessian gauss
optimization prohibitive solver special structure strategy focus arc capacity arc recognize cost depend additionally mutual individual literature wolfe decomposition description angular structure constraint subset extreme fully extreme rewrite master technique subproblem th extreme reduce path total demand source th subproblem length since subproblem subproblem generate new reduce sp sp master computational set instance grid path instance publicly initial degree set experiment core gb problem likely arc capacity inactive optimum identify problem know inactive constraint guess arc capacity classify successfully implementation inactive exclude total flow correspond arc capacity constraint become inactive flow arc experiment show name arc recall present percentage arc constraint optimum act number total second cpu total cpu solve subproblem instance second set spend formulation
proved intersection hyperplane equation intersect tt intersection hyperplane intersection half define inequality intersect big big color intersection color circle color
classification good approach htb cc normal ari c ari ari class class da da da da figure illustrate direction expect misclassifie interesting gene misclassifie although ari identify five clustering correspond effective technique mixture focused small capture inherent cluster simulate real perform exist case consistently outperform several dimension reduction sometimes encounter covariance one purely numerical update iteration em suitably cutoff future investigate alternative approach latter work fitting form encouraging outperform set discriminant supervise classification future author associate anonymous comment award grant natural science
middle dataset unsupervise beneficial computer task object one might object act proxy task instead choose determine matching descriptor computer vision descriptor approach optimize remarkable hand descriptor manner investigate unsupervised couple rely supervision signal sparse descriptor descriptor perform state art evaluate qualitatively mention test conclude future completeness spirit unsupervise supervised level compact autoencoder
content enumeration theory determine learnable code effective thereby establish decision notion certain natural place analog learnable consideration one decide learnable completeness reader note family compute code standard set string letter text treat string function feature letter initial logic enumeration appropriate switch order list say finite write machine denote effective enumeration computable machine fix denote learner symbol indicate learner four enumeration n mf mf mf identify member analogous everywhere replace first identify enumeration mf n analogous replace identify enumeration mf everywhere replace identify enumeration mf w turn fact subsequent
candidate batch easy setting delay process tight precise delay rather complexity result delay state need hypothesis space feasibility disagreement follow algorithm corollary delay delay bound delay probability carry example al apply elastic mnist henceforth mnist margin base classifier produce margin point select begin sift noise setting expect prediction risk uncertainty prediction simulate
hand powerful useful well distance measure hand construct discover collection challenge graph coarse fine discover consist sift represent word bag position word feature construction high spatial verification accurate yield matching graph affinity matching segment similar view group join segment grouping image segment reciprocal affinity propagation graph final performance object group contain recall f result see camera paper graph scale visual descriptor theoretical investigate neighborhood propagation appendix proof partition point neighbor neighbor discover discover
vector goal accurate regard interesting small construct I nonzero component convex review although force minimizer vast amount problem various assumption investigate noiseless guarantee sparsity sparsity whereas problem random design matrix work investigate aim variable individually vector regression group apply study generalize regularization I problem reference therein task column correspond regression correspond clear kb individual together block type regularization study sufficient necessary condition e union column characterize adopt investigate study empirical minimization lasso adopt analyze model study general differently model refer multi linear goal
also proof base mistake perceptron bipartite ranking secondly another online correspond generalization leave possibly online online online rest organize follow define state sketch describe bipartite buffer step depend buffer interestingly buffer hypothese q measure hypothesis hypothesis reliable discard reason simplify hypothesis term obtain hypothesis hypothesis generate online define exist radius remark e martingale concentration suppose cover fs fs restrictive take hinge thought bound define hinge inspire hoeffding
due g increase independent limitation theorem dirichlet base ideally would test measure obtain necessarily distance happen grow increase ready prove fact allow vary concrete concentration g plug q negligible compare term geometrically imply entail ordinary entail example nf base measure basic establish shall arise notion define wasserstein ball center useful notion hellinger integrate generic value relate hellinger distance density wasserstein distance role specify sufficient concentration dependence notation model accord pack metric condition prior kullback define process specifically appeal hold g nc nc derive nc previous display whenever constant sequel nc w nc en display satisfy condition verify turn rd dd cn last inequality nc check eqs follow sufficiently finite immediately display negligible compare condition posterior concentration geometrically support nf proof establish dirichlet process base perturbation measure theorem mn mn mn mn b mn quantity p tend complete ii proceeding lemma measure interest demonstrate gain perturb admit number cover ball wasserstein proof r dirichlet lemma atomic center measure present constrain ball center share support result polynomially rely intuition gr property hold r e entropy cf increase complement additional strength result linearly measure control r need hold
iw variance estimator lead plot select propose scheme exist vs small even find optimum value unless informative optimum right know conclude sign project optimum vary much around always bit project another practice know advance figure figure motivate uniform code scheme project region scheme scheme coding theoretically compute monotonically increase largely differ beneficial answer plot mean care highly
eq expression obtain asymptotic quantity asymptotically n substitution expression know zero minimal class pose complete u
approach utilize penalty utilize penalty induce strongly generalization minimum comprise non wherein gradually problem solution penalty simplification simplify inferior simplify propose would gain allow depend hence admit penalty solution quasi pursuit mp omp hard pursuit smoothed solve quasi highly method seek support index element calculate approach intend serve hard function family ill pose problem sensitivity threshold input threshold large spurious peak often appear result thresholde denoising reason threshold preferred phenomenon spurious peak quantify attain attain note soft however bias seek threshold sensitivity readily large decay rapidly increase proximity operator uniqueness induce
equivalent deterministic formulation produce adopt generative conditional describe obtain loading directional ambiguity deterministic pca firstly substitute likelihood limit map eq integral conditional constant q substituting separately reformulate simultaneously eq become ml solution lda optimize ml loading expressive local slow ordering solution lda choice natural let one case minor unified undirected em prior
continuously mala perform mala validate illustrative deconvolution denoise mala differentiable although applicable concave mala use simulate complex blind image split conditional similarly involve unknown mala new image bayesian bilinear optimisation investigate require appropriate framework preliminary mala tail algorithm mala p mala improve introduce geometry could position metric derive hessian log availability proximity euclidean function ta lot optimisation focus effort consider method robust mala mala make simulation proximity mapping differentiable many mala perform would applicable computationally finally acknowledge proximity mala use mala type proximity mapping otherwise significantly mala mala use shrink operator proposal atom generate consider problem relate processing similarly operator gradient thresholding
intractable mcmc major capability use consider applicability many perfect ise model relax perfectly sample instead auxiliary approximately doubly chain type due feasibility justification use literature approximate doubly exact perfectly possible perfect possible develop pseudo class method produce despite approximation metropolis hasting acceptance density density give estimate proposal doubly remarkable value chain joint simple sampling highlight development powerful wide applicability paper statistical cover efficiency acceptance quantum literature computational define configuration specific see hence unbiased estimate series unbiased series computationally truncation crucially goal physics literature use estimate series constant reality likelihood form proceed bound guarantee prevent plug pseudo return estimate turn well practical solution recent estimator exist unbiased value therefore need weighting monte carlo suppose wish u positive x absolute u likelihood accept q save sign importance
particular treat dft approximation accurately describe united atom limit complete accurate molecular force kernel prevent overfitte determined density training length term regularizer prevent eq q
obtain model provide alternate monotonic optimality demonstrate converge allocation optimality extensive study monotonic converge program upon appendix see kullback l without generality suppose j l global imply em
iterate introduction propose parallel instead devote develop allow update purpose formalize update iid two easy see necessarily pi refer name call satisfie analyze however family purpose paper concentrate doubly uniform nice candidate processor block compute fix scalar fix positive concept expect uniform design coordinate descent rise unlike replace affect verify separability hold eq iterate compute writing block set one first w separable assume section complexity cover definite otherwise norm norm nice resemble block
spaced root corresponding error good detail naive complexity precision iteration newton compute total complexity regularize regression crucial issue interpretation rescale regularization give indeed lagrangian unconstrained equivalence underlie necessarily notion equivalent algorithm definition equivalence match problem since set equivalent guarantee performance
signal location statistical foundation approach implement value contaminate reproduce network markov functional reproduce popular reconstruct noisy e regularization estimate q parameter location reproduce location correspond residual feature dimension infinite solution mild accord theorem section define specific define gram define quadratic admit covariance
tuning outperform ridge ridge label compare ridge fusion movement uci repository describe class type represent video movement hand correspond horizontal tuning use sample analysis ridge fusion observation classify incorrectly misclassifie methodology movement method smaller incorrectly unlabele misclassifie partially support national science foundation grant dms proposition
combine gives ensure correctly corollary setting give recall second large combine spectral laplacian lead empirical graph laplacian laplacian adjacency extra degree entry intuition close able simply find effect performance block explicitly track five exist develop plant clique imply find clique provide insight understanding guarantee compares summarize would interesting unified minimax compare technical proof standard inner orthogonal matrix sec pf lem hamming u second implie u u uk row row constraint hand member correspondence row fit argument
vary emphasize competitive sparfa capable concept resource organization pls provide learner strength recommend learn resource learner study sparfa sparfa prediction sparfa likelihood cm advantage sparfa kt visualization organization state instance learner gradually improve therefore concept seem verify learner incorrectly cover concept concept knowledge concept cover mean learner stage course remain advanced covered end improvement course sparfa trace pls provide feedback learner concept knowledge evolution course content organization learn resource visualize transition interact resource represent interact resource learner interact learner concept knowledge resource transformation dot unchanged solid characterized positive concept non knowledge level pre increase advanced concept decrease knowledge advanced concept resource course resource learner advance resource knowledge improve knowledge concept analyze organization resource learner sparfa pls automatically recommend resource strength resource course resource poorly resource resource easily
regime second parameter experiment simulate curve experiment consider curve observe transition assessment curve transition approach provide accurate piecewise third increase provide piecewise finally grow considerably curve curve evaluation switch elsewhere class h mean noise unit standard curve period sampling draw show rate obtain approach observe regression regression modeling test regression fig fig curve see
construct complete ill pose perfectly entry certain entire low low approximation express combination
check since signature invariant relevant image patch field module selective enough discard wrong signature match nontrivial constraint signature level match level signature match implication signature optimally incoherent interference module evidence brain responsible cell activate close temporal patch simulation remarkably assumption simulation suggest qualitatively consistent neuron experimentally complex nonlinear property template need test affine development need template could unsupervised assumption normal turn cause temporal assumption violate huge error theory find would test replace even yield performance translation li invariance experiment paradigm induce preference specific condition another rotation depth face theory theory recognition detail convolutional invariant piece completely accordance invariance invariance transformation module simple include perform class scene thousand convolutional competitive feature recognition art system behavioral account rapid scene categorization illustrative invariance stability uniqueness architecture signature bottom complex pool set cell center position language theory scaling include pool partial group usually lie cell cell template assume also backpropagation even number automatic hierarchy cell position position share construction result learn scaling everywhere convolutional perform object often invariance translation image principle recognition present incorporate translation must example transformation two list face exist already invariant model rotation depth face depth self invariance derive network architecture object face important specialized stream show store view template face recognize novel face example rotation face theory state face person layer build scale leave pool
respective behavior might therefore quite enhanced accord dynamic portfolio management pricing tracking relate beta preliminary q e small lebesgue series integrable infinitely rare integrable
stage totally consider newly add initialize work newly add iteration sequentially add working stage wise totally update implicitly randomly pick slow optimality pick cg initial fairly optimality coordinate cd serious generation boost kkt work strategy cd initialize newly learner iteration repeat work inner sequentially
whose good theory improvement distance improvement apply describe goodness stochastic assess filter multiplicative intensity outcome model describe specify distribution homogeneous unitary mean density gamma equivalent look heterogeneity allow look vary similar way impose call heterogeneity author use
shall convex jensen eq make q write complete lemma random
always challenge automatically determine approach promise incorporate bayesian clustering provide applicability avoid cluster address base process treat dynamically adaptively change guarantee generation merge adaptively complexity extension compare property validate experiment structure mean law facebook kind traditional occur recent phenomenon kind group collection automatically difficulty infer number various deal kind classic spectral shift user approach criterion lead research recent treat hyper learn give observation significance contribution parameter suffer design inference bridge gaussian model naturally determine create
consider depict corner expert add compare action pair state enable learner slowly improve indicate information action action target target learning may extract greatly action tb experiment reward information sparsity two depict fig function reward greatly exclusive sampling detect design reward propose reinforcement shape version depict extremely equivalent paper allow learn expert generalization expert state additionally consideration algorithm exist provable come guarantee summarize suitable condition dimension briefly modularity optimality fact shown use similar provable specialized particular way applicable class assumption simple consist action discuss expert combine reward bridge bring exist result reinforcement additionally general amenable integration example agent frequently visit mdp generally associate turn learn source less statement set equivalently suppose word hold assumption k ph k
correctly circuit right graph span root edge line subtree grey accord child edge connect grey white node explain return return subtree node subtree return visit input start associate record contain visit leaf node visit since internal visit backtrack depth first visit child exclude instance include subtree exclude three area create first visit ensure internal child already subroutine query predict ti form circuit moreover circuit least size circuit load parent immediately imply previously arbitrary query edge g query arbitrary edge
factor reach mode fast importantly concentrate hmc px poorly explore simulation count burn period observe behavior figure hmc count burn period px reach well mode fast much illustrate poorly explore true panel figure hmc evidence perform poorly px px might see seem use practitioner past work credible trivially offer principled check
softmax explicitly child random walk assign precisely word tree root let addition arbitrary child verify imply cost average unlike softmax gram softmax representation inner explore construct result assign code frequent observe grouping frequency simple speedup neural alternative hierarchical softmax language modeling loss ranking noise maximize softmax skip simplify
influence alignment meaning behaviour viterbi alignment also piece adjust adjust iterative considerably another piece alignment piece unconditional classification whether probability smooth random imply far conditioning influence find piece see iteratively costly way adjust viterbi advantage iterative tend adjust viterbi piece point small example protein chain six sequence come symbol emission alphabet emission iterative length shall compare behaviour algorithm characteristic table give minimum give posterior restrict alignment depend realization viterbi alignment substitute states low respective state viterbi restrict viterbi consecutive iterative would stop iteration algorithm
gradient polynomial augmentation variable computing mark product rule analytic policy update task cart learn stack ball robot gps learn cart cart cart cart cart parametrize controller basis share matrix weight initially balance target location cart penalize desire cart optimally require cart cart cause cost time step near learn desire learn apply controller close ic combine individual policy lead rw ic ic policy five I experience multi training
eqn constant argument gamma suitable asymptotic eqn concentration elementary pointing big find describe dynamic read notation cluster define
smoothing since exponential sufficient statistic smoothing mean family since belong hull see exist example pair let three define maximizer always likelihood estimator fail exist additive smoothing surely binomial
interesting investigate code novel domain end common codebook sample target domain utilize class regularization code adapt mmd implementation report dataset
satisfy condition kkt rt complete residual bregman kkt plug ignore yield right side term right side remove side give divide side jensen outperform admm unit divergence choose unit simplex satisfy kkt problem outperform mass cost assignment mass simplex admm constraint rewrite solve admm projection solution update operation admm mm admm linear solver randomly generate uniform run
square square top fit shape formalize follow denote euclidean else sometimes kronecker column state contradict orthogonal single component even initial write dimension coefficient orthogonal approach solve analytic straight find fix vice choose eigenvector eigenvalue write inversion choose turn stable reach predictor absence optimization replace orthogonal full rank quickly find procedure though
seq quantify structure resolution rna seq reconstruction become tool completion genome analysis differentially infer seq conceptually one rna seq read genome approach rna read genome actual basis applicable non arguably address difficult problem low reconstruct implement rna read considerable therefore rna across develop formalize apply namely way seq read characteristic genome site emphasis align rna seq read
solve point mass generator directly secondly smooth latter critical implementation theorem obtain refined convergence method expectation k nf side use conclusion simplify constant divide note g take inequality fact simplify fx fx rest similar detail stepsize policy mass choose eq relation inequality obtain n n fx remarks nonconvex sp convex satisfying weak dependence nesterov nonsmooth convex sp see
usually signature chen show almost iterate integral ordinary define iterate algebraic calculate rotation take care algebraic independence character iterate feature extraction curvature whereas iterate derivative
relational relation consist protein functional interaction evaluate party membership membership vice versa cross improvement multi
stop sample collect sample number level properly restrict stop history optimum sequential stop rule numerical scale decentralize regressor derive complexity constant consumption scale linearly decentralize achieve duration sequential wireless paper interested e coefficient stop regressor noise general deterministic commonly unknown another wireless access coefficient unknown channel pilot receive sensor network transmission source consumption transmission rate low decentralize decentralize fundamental minimize sequential oppose stop appropriate stop observation hence endowed energy unnecessary processing transmission decentralize mainly two topology fusion fc received call hoc fc sensor decentralize topology review parameter work decentralize assume sensor focus decentralized bit transmission decentralize
show two consider armed know arm arm threshold subtle previous round fashion pick arm arm potential r choice see round policy bound armed furthermore reduce yield bound guarantee small regret recover fact impossible scaling know general express
membership proceeding must initialize document sentence word fuzzy denote initialize zero membership according determine calculate fuzzy similarity document usually document account intersection sentence occur capture fuzzy come sentence directly fuzzy similarity provide work fuzzy form certainly accordance similarity intersection occur fuzzy fuzzy come sentence need fuzzy
decrease validation decrease repeat time significant error surrogate per error surrogate stop pre soon whole task consist image surrogate extract response softmax spatial pyramid train cifar training generate surrogate usual averaging fold report cifar
evolutionary approach directly controller measure robot reward external external already external h external search external external h gradient algorithm find local reward start controller user controller apply gradient reward control information iterate satisfy controller converge policy difference consider resp test increase decrease especially efficient mostly sophisticated successfully apply evaluation robot trial hour tractable freedom typically make robot computer fall hour evolutionary search optima gradient structure fuzzy exist many vast majority iterate iteration initialization controller population robot hundred robot hour dedicate evaluation hour aside author publish evaluation table regardless function device column researcher aim efficient adaptation measurement directly improve disagreement self action measure consequence robot update successfully discover loss loop action look
star series phase periodic series process star database system machine algorithm star indicator classifier adaboost classifier come tree explore well domain adaboost fit model object generate small classifier classifier simple nice property classification amount care area help many specialized car fit previous car detect car fit correlation car car give million estimation order considerable metropolis hasting suitable purpose gain optimize amplitude time multidimensional
fix laplace rely normal maximize write normal integrate normalize mix normal laplace error tend consider grow upon conclude reliable nearby tend approximation similar accurate individually compute moment nonetheless laplace approximation surface laplace fail sample normal distribution approximate guarantee converge convergence markov chain monte posterior computationally intensive difficult whether converge integrate nest efficient poorly likelihood method far
hold proposition projection actor early actor converge locally episode simulate trajectory mdp termination action let proceed maintain maintain actor show actor algorithm converge policy make objective bound furthermore local optima countable requirement iterate countable optima indeed satisfied
google gram base superior sometimes gram currently work google gram string string empty string large alphabet string member allow abuse notation abuse membership write nonempty nonempty member string increase equal mean element occurrence length increase kolmogorov conditional program complexity put straightforward law relevant page currently index occur million page vast page truly large google page divide page index google multiply number page frequency develop google search question google may corpora wikipedia english count wide web yahoo text semantic frequency linguistic random sufficient representative source wide diverse google phrase current singleton search singleton
observe hundred make difficult intrinsic indeed algorithm single threshold infer intrinsic investigate process simulate detect measured find approximation reality expensive detect cpu replace simulate determine give training pre blind total peak frame background energy angle detector output correspond unity detect architecture hide configuration well blind nn probability require statistic classification number completeness fraction detect correctly detect plot completeness lie top without know classification detect level predictor expect receiver curve detection reliable roc curve true equal rate contamination pearson connect classifier roc powerful figure actual roc curve roc expect completeness curve use positive minimal contamination sample alternatively curve example connect roc angle near wish relationship deriving show count nn agree detect full simulated essentially normalize exceed hour determination detect computation set often great redundant identify pixel measure able quantify distinct efficient finding autoencoder autoencoder galaxy challenge image galaxy contain pixel measurement star pixel order compression autoencoder simple require less train
treat inclusion indicator survey build predict inclusion probability correspond probability assume survey come proportional probability unique call cell since weight call calibration adjustment step construction unify sample cell national political survey cell age education category result cell present base datum cell treat cell characterize response let population let design modeling survey allocation depend population estimate express individual unit cell estimate unbiased realistic inverse calibration unit distinguish paper weight weight restrict sample long especially unify exist estimation allocation obtain size weight
th separately repeat detailed evaluate diagram interpretation aggregate block vary case generalize represent map convention definition operation safe aggregation operator simple safe formula apply represent use symbolic avoid enumeration implicit generalize iteration iteration get closed expression diagram variant template correspond open specifie execute predicate contrast planning axiom action variant illustrate action introduce new variable imply change aggregation implement every produce conceptually goal deterministic planning need potentially
mlp connection mlp stack rnn ht c rnn dots music word level minimize music model sigmoid nonlinearity character unit layer stochastic descent gradient training validation cost stop subsequence subsequence song hyperparameter schedule start decrease start increase tune correspond epoch pair hide layer sparse connection per unit weight rescale large connection layer well state gaussian deviation function rnn state layer sample white deviation bias initialize model white character weight output weight rnn
show let graph factorize z factorize contain represent assignment associate clear easily convenient representation log model represents represent potential denote feature make clear
function exponentially decay spectrum peak show show mix spectra computational rescale true computed method satisfy nn create mixture white snr db good achieve fig fig geometry insight threshold structure level parameter bss recover four visualize recover fig index normalize read nearly equivalent index compute compute four suggest equivalence compute one show source mixture white
require cdf absolute log inequality triangle use feasible prior similarity surprising lemma draw binomial beta prior proportion prior binomial purpose quantify give absolute ratio supremum seek however slope function minimum boundary attain supremum know complement symmetric concentrate outside max integral desire limit finally density exponential assumption variable q event
realistic task computer vision parameterize slightly specify size parameterize leaf size mean require scaling forest algorithm datum split ct plot tree axis bar compute dataset label use repository summary see diabetes instance third attribute mean five run five candidate insensitive parameter variant uci trend turn outperform quality set include forest variant splitting level appear however diabetes forest significantly improve difference performance splitting
success measure contain parameter relevant contain datum variance coefficient uncertainty correspond draw determined phase parameter parameter ratio value combination possible measurement situation set fold performance measurement phase mean relate different phase rather paper datum four correspond dependence complex quantity applicable situation involve form separate interval guide w ten form element diagonal g valid rectangular uncertainty region word meaning involve success area calculate method set regarded area apart magnitude depend reasonable describe present three part consider list
integral base appear already diagonal weight put mass observation dependent choice skip disadvantage version run performance chemical engineering literature idea integral method consistency rate estimator statistically turn square modification start result two xt px identifiable consistent matrix weakly estimator present away infinity asymptotically consistent consistency estimator parameter one need parameter require separate satisfied strong assume derivative follow r p differentiable condition continuous present condition fast rate mean next subsection base solution ode clearly depend illustrate estimator inconsistent boundary consequently local solution
em absolutely spectral domain relate white noise innovation autoregressive assume ar maximize nonlinear conditional expectation past expectation past expansion score schmidt power score score construct discrete approach nonlinear modeling univariate series linear series define summarize several inter step highlight require application daily return empirical begin model transform special analytic rule display normalize transform kt transform return
experiment unit consist reduction subsequent leave reconstruction conduct digit laplacian inverse fig cubic red low three representative digit scale table choose rbf outperform table like residual fig result poor choice reconstruct solely performance assess dataset digital video scale image normalize norm face test leave three reconstruction technique rbf method cubic
om om mx x h eq lie q strong law large know triangle g tm section classical examine scale response landscape paper propose longitudinal incorporate structure measure newton smooth regression consistent carry real self report two group quantile longitudinal quasi longitudinal datum study repeatedly collect subject longitudinal additional modeling effect still specify order
gd sag gd gd gd sgd gd sag gd turn gd gd epoch hold eq prove assumption gd choose let sufficiently expectation proceed recover zhang see latter major improvement former elaborate necessarily sum j inner take fact expect optimal rearrange take expectation algebra happen gd two computation side conclude high straightforward follow directly treat insight gd equivalently epoch inherent particular ideally optimization problem measure evaluation perform gd epoch indeed gradient evaluation epoch stochastic gradient view approximately fix find nearly fine obtain good depend plug optimize
representative value trajectory spectra display exclude averaged variability yield performance metric divide discrete zero cosine optimisation identical likelihood slope slope initial frequency marginally considerable computational expense optimisation slow burden million window complex time wish specify formulate significantly rmse accounting matter dynamic extremely close maximum dynamic case good process mat ern cost transform approach preferable estimating sample mat ern repeat mat ern length root rmse express true cosine lc r method eqn cosine cosine recent development propose trace embed determinant http software ern slope parameter slope parameter resolution slightly processing frequency numerically complicated code embed several code mat ern clear generalised ern perform fast version use trace oppose yield report bias deviation place poor standard ern appear poorly scenario table gain estimate ern mat ern square error rmse percentage value test cosine lc eqn c rmse standard cosine al normal version et fast version modelling bivariate spatial trajectory particle field quasi force isotropic simulation leave right velocity east
satisfy bound section square provide bind various case readily square design error cf measure measure bound q similarly quantity appear loss function penalty intuitively play role correspond result large curvature technique seem necessary behavior local alternate penalty limit version scad equivalently view state prove appendix reveal optima section scad optima nonetheless message significantly optima find degenerate finally convex argument regularizer decomposable regularizer norm generalization various nonconvex regularizer rsc choice consider systematically ordinary systematically corrupted version corruption mechanism include use population formulation correct additive choice replicate appear variable devise correct estimator readily assume gaussian contribution error covariance deviation bound subsequent dependence statement error additive whereas observe nonconvex program corrupt covariate define choose eq nonconvex least establish state whereas corollary strong holding early propose project stationary result usual corollary consequence corruption loss square much establish statistical consistency specialized algorithm local optima provably optima
rank probability rare essence subset goal rare chemical biological cell rare unbalanced spam email find relevant document drug discovery chemical library activity relate activity active drug discovery relate response variable biological five descriptor explanatory available descriptor chemical molecular thousand relatively uninformative thousand four aim explanatory cause imbalance successful drug partition large structure thousand million molecular make recursive combine widely drug forest attract particular rf method classify chemical study several machine drug discovery ensemble bag boost rf ensemble method comprehensive rf rank unbalanced two repeatedly rf shall rf competitive rf create consider build bagging ensemble classifier build algorithm exploit explanatory
message receive receive correct uninformative stochastic neighbor node generalize cavity message send q modify equation point cavity considerably around give taking whenever reasoning report tie break distribute link group well threshold rigorously true scale converge
aware distribute mini allow message vary even manner protocol protocol allow interact performance protocol constraint protocol seek ji pick pick coordinate goal note bias formalize immediate seek sample average seek detect theoretically constrain protocol protocol bias coordinate protocol coordinate protocol sample reliably polynomially exponential protocol thm online split coordinate sequentially go segment determine assume memory maintain coordinate allow analogous protocol dependence exponential coordinate bias return protocol coordinate detect biased instance require free protocol establish appendix high theoretic viewpoint analyze mutual unknown identity bias correlate coordinate instance noisy amount statistical consider constraint constrain since output contain moreover likely know provide therefore matter protocol
factor result ml sample covariance exploratory variable observation simultaneous robust exploratory simultaneous exploratory factor review address statistics se medical university red make eigen supplement orthogonal eigenvector update nothing else much appear problem let assume discuss rescaled covariance matrix matrix eigenvalue expect likely understand conclude exact statement trace chance trace correct simple state know sure
expect scheme property ii q iv property vi consider stage eq principle rule q stopping follow possess respect ii non increase number size stage definition bivariate virtue inclusion continue decision rule possess proof sequence stage inclusion express rule otherwise possess stage give virtue inclusion express general regard stop establish scheme possess sequential central idea inclusion principle process sample asymptotic inclusion possess efficiency prescribed implie tend infinity technique parameter pre concentration sampling inequality statement absolute use eq inequality fact x manner inequality combine yield completes exist constant assumption constant n u moreover n n v complete devoted scheme simplify assume ss dm z gs g c continuity gs g gs cs
prediction naturally motivate point precede paragraph examine sampling euclidean follow accord give modal diagram construct cell iid sample show construct selective method observe boundary elsewhere diagram latter achieve empty set frequency natural near natural much save say neighbor another ie neighbor yet whose observe iff definition suited analyze candidate predict differently preserve sufficiently neighborhood furthermore neighbor statement q let proceed iff boundary maximal contiguous denote neighbor
extend class consider treat joint discriminant formulate unified analyze consequently maximize class negative new x z ix I scale hence identity top z denote inverse calculated project subspace preserve space classify knn strategy project voting count fall knn subspace discriminative discriminative discussion accept new always representative frame nearest search initialize data
portion risk overfitte trend insensitive approach sparse effect capture normal whose restrict approximately covariance jeffreys smooth time fully usefulness gain economic forecast financial forecast text dependent economic notation distribution connection predictor parameterize task term link distribution group feature divide draw seek intuition ti view group explicitly independent give variance parameter autocorrelation draw multipli role
prefer decrease let formulation close parametric solve parametric simplex see successively solution solve turn split next replace substitution equal reformulate nonnegative decide either choice
good bad represent simple occurrence bad parsimonious variant impose eigen cluster framework outline maximization computational operational aspect discussion vector mixture th g assume adopt contaminate degree contamination interpret increase variability bad contaminate multi mixture component secondary free scale normalize element geometric determine shape parsimonious contaminate free volume orientation spherical mm equal align axis
similarity find transform dimensional step develop subspace cluster accord subspace similarity point graph spectral conduct
database new database surrogate clean clean clean sample clean sample computationally intensive application long period use cross able quickly clean propose greatly compute singular value behind computation decomposition save singular calculate sample matrix surrogate orthonormal project onto right
treat setting solver make package solver screening rule lasso equally spaced solver dpp solver screening solver combine result different present ratio feature observe safe screening rule safe able discard inactive rejection ratio comparable discard safe dpp term strong reason may discard component rule check correction necessary
intel cpu ghz processor detector achieve speed frame rank evaluate time spend extract extracting spend integral major bottleneck detector runtime core intel able average per believe significantly ccc avg multi adaboost cascade window frame detector pixel implementation vary cascade detection maximum adaboost subsequent node train roc detector improvement minor increase per window robustness weak cascade fisher conjecture perform follow later node cascade first node cascade achieve additional classifier guarantee conduct experiment two detector node etc detector apply two classifier previous previous detector criterion discard c avg fisher detection cascade discard significantly play percentage window detection wu et lda mit face wu fast criterion haar feature try calculate nonnegative easy setting mit work requirement perfectly especially sometimes perform even means consider wu et plausible explanation wu wu ccccc digits face average detection er regularization qp regularization robustness pe face
govern expand functional add lagrange multiplier maximize update variational inference pf form ff except express standard posterior calculate backward briefly review j forward posterior k p technique equation usa latent hide database behavior sequence chain control e transition emission matrix dirichlet expect database deterministic hyper sequence level latent learn deterministic approximate posterior iterative em step factorize form model sequence three world experimental
impossible alone impossible mean force belief mathematically theory context negative entropy need physical law unique impossible field guess field theoretically logic theory reasoning infinite concept develop mathematically function gauss evy process continuity physical nearby location similar location exploitation
previously upper bound action try boundedness easy addition sample easy bind switch mdps sample take switch policy switching interval directly alg via algorithms reinforcement correspond approximate low define belief euclidean derivative write mdp due linearity easy obtain
propose solution identical smoothing ls plot account filter exploit wavelet plot degradation signal result inferior kalman middle plot computational term total multiplication use compare matlab execution solver multiplication computation signal figure similar snapshot compare result homotopy almost identical homotopy significantly execution signal solve brief observe norm signal signal alone cost homotopy small homotopy time range result edu fix program streaming reconstruct sequentially small streaming framework reconstruction divide stream disjoint block independently infeasible inefficient homotopy quickly minimization block transform transform sparse recovery measurement slide problem coefficient add program homotopy warm homotopy homotopy homotopy setting numerical method reconstruct independent disjoint block propose homotopy term time incomplete arise see sparse nature basis encourage
remain hold grow regime fix constant b cp previously global g control try recover asymptotic exist constant partial clustering exist proof reasoning define follow cluster observation cp satisfy principle noting must disjoint least least end spectrum one enough omit summarize large least cluster recover guarantee fall small deep ng p ns sa cp induce cluster ground u v k u rr simplified version create difficulty increase iteration partial similarly support theoretical finding guide subject experiment report augment alm semi cp cp result
previous distance neighbor knn maximum eigenfunction neighbor centroid winner bayes standard principal number functional principal winner performance performance due functional mahalanobi regularize root allow write mahalanobis semi principal score several mahalanobis distance mahalanobi mention previously functional acknowledgement financial project thank helpful orthonormal eigenfunction proof proposition eq consequently function regularize mahalanobis distance functional principal score trivial functional mahalanobis semi standardized functional score supervise functional functional expansion york ba j neighbor infinite probability notion achieve shape descriptor datum curve nonparametric h pattern analytic j discriminant
latent work originally qualitatively rich insight baseline rely assignment matching far study information fully label grateful helpful discussion constructive manuscript stanford naive figure except line unstable show analyze propose variable research want whether people valuable experience conversely interact example search signal people go click give overview understand different conclusion address click single difficulty train satisfied click side click satisfied outside g human good click whereas binary click click grain want adopt weakly supervised guide
solve sequence binary fast sim removal measurement maximal residual away author remove mean iteration observe geometry propose lagrange pose boost outli removal add slack add slack solve draw regression removal image computational expense utilize illumination corruption discriminative robust detect review removal algorithm formulate sub removal technique conclusion norm examine norm close r column vector accord linear subspace probe represent combination efficacy application aim I minimization utilize datum easily influence seek formulation reformulate
likelihood compute compare two round theoretical sake compare would compute current high choose apply thm bind result fed created mean give artificial experiment combination lead discuss two artificial box dark gray mark gray
denote range good q pixel lee filter standard significance result summarize image six filter axis code
manifold visible completeness divergence manifold variable equation statistically hidden equation bm unit coordinate derive rbm since realize use simpler interpret rbm rbm target rbm leave unit realize sigmoid exp qx obtain explicit expression coordinate coordinate consist fractional valid relation relation coordinate appendix projection learn rbm mix fractional learn come projection fractional coordinate appendix back good rbm follow present rbm projection let put iteration projection illustrate figure guarantee ip rbm give alternative rbm invariance rbm iteration fractional mixed coordinate highly confident confident coordinate neutral mix fisher good system fractional mix jointly coordinate apply reduction confident neutral zero preserve remain tailor mix require neutral value zero sample underlie could implement generate stationary rbm ix h rbm rbm newly phase unit visible bm without hide traditional gradient
community size increase leave right agree disjoint setting study use matlab author http www implementation available http www software setting unweighte undirected default set seed generator http default setting undirecte attempt matlab matlab throughout text run matlab describe text run iteration random seed matlab implementation run resolution parameter range increment stable value choose facebook political network study available website modularity across repetition seed pt lc choose score optimize seed alpha level run analyze datum set political table identify text table statistic community c lc acknowledgment thank associate constructive suggestion sharing analyze claim grant dms dms dms dms st complex award grant divide community belong different investigate significant base strength reference handle overlap majority identify background parameter control use compare validation four datum carry simulation assess effectiveness various network overlap exploratory discovery community datum software available system individual unit vertex unit vertex network
general convert parametric associated b ki regularize tune parameter asymptotic review selection likelihood application model longitudinal likelihood nonparametric model type asymptotic quite discussion assume finite parameter construct probability particularly closely bayesian method adapt compute arise maximum carlo method investigate early complex approach wavelet thresholding depend general reliable recommender system area inference acknowledgement science
constant low light statement light observe disjoint ei ei k h I kf kf balanced unless prove get last equation support function union heavy restrict decrease prove partitioning include balance plant exist non function f furthermore support prove theorem generality exist negative disjoint assume proof part similar part exploit except remain part section example first ng eq side necessary n show iii polynomial ok repeat balanced subset union remove expand set volume remove remove vertex increase subgraph ok ft eq vertex induction initially
individual inform cost immediate allow communication immediate neighbor also connect give follow rest allow indirect path finally qualitative regularity state technique directly physics game graphical game game align view opponent minimize empirical e nash team common opponent bayesian economic presence provide basic concept development leibl divergence entropy eq reader detail ar simply reference book translation different large front relative entropy tight point concept g index context probability configuration state temperature gibbs useful later elementary completeness exchange denote maximum degree agent set action measure action measure default individual finally variable vertex initially single activate independently available observe instantaneous instantaneous interaction
strict produce optimization say strict sense cover family ridge cover level devise second nested sense approach wide sense nest approach sense depend hold value contain definition fit incorporate lagrangian conceptually regression form essentially equivalent constrained ball consequently ellipsoid span scale specifically constrain nest constrain regression large penalize criterion optimize strict duality penalize optimize denote
tweet etc denote selection translate select column act nmf work column combination compare different partition datum held representative datum express possibility anchor selection criterion report result frequent consist word class evenly document set varied step full show black family remove speed time fast remarkable give accuracy class give video axis highlight color problem foreground separation camera position assume video frame camera capture background foreground movement people assume stationary vary
abundance abundance data digital fundamentally quality intensity microarray next generation replace microarray count expression normality aim note fail also strong relationship count normality proper statistical gene adaptation exist aim begin process formally next generation sequence think thus describe poisson show gene large poisson account call poisson scale alternative negative binomial distribution poisson gamma incorporate model overview count despite decrease sequence digital biological replicate replicate although effort statistically lead attempt accurately
management paris france find exploitation exploration many scientific propose situation growth portfolio optimisation material find existence optimal confirm generic exploration exploitation tradeoff different field reference therein early management arm date switch potentially
fisher generator give generator reader diffusion develop method eigenvalue eigenfunction brief spectral operator define article mutation integrable generator x diagonal eigenvalue neutral coefficient recurrence relation explicit respectively b article generator eigenfunction eigenvector eigenvector see change basis basis eigenvalue eigenvector discussion vary submatrix size regime allele change frequency time transition devise recursively compute density represent basis eigenfunction ny k describe proof article
recover round variable pt intersect well variable half analyze round nest packing later need round number variable small sum step round variable round due round variable center inclusion n round ball round act level add possess pack add additional accounting point bound iii variable step hierarchy since z round nearby set nearby set nearby hierarchy fix ball ball point within I v w lp corresponding lp subsequently lp level round lp optimal utilize follow vector packing lp exist number program maximum may claim lemma
sort calculate change series framework note rate even provably impossible unlike traditional develop force rely convergence advantage framework applicable wide range situation algorithm datum condition conjecture factor well assumption work number possess consistency distributional distance proof red distances
threshold word minimization randomly signal recover perfectly whereas signal like randomly sense matrix signal well transition big weak recovery sparsity cardinality restrict isometry condition decode date restrict isometry weak angle characterize minimization show hold constant particular tradeoff term law compressive sense recovery first related element cardinality number suppose
big get general submodular minimization scale algorithm design solve instance simple algorithms e cut suffice submodular nature depend scalable aforementioned machine issue maximization framework rely state art technique provide class submodular also unconstrained nontrivial thereby reduce minimizer minimization preprocesse practical algorithm competitive framework offer perspective treat submodular maximization constrain rely discover submodular thus relate form important vision identical submodular rich set cut inference cover cover variant submodular submodular constraint alternatively reduce unconstrained submodular subgraph submodular cardinality hard cardinality seek connectivity maintain connectivity consumption span world
merely follow gibbs essential duration period require fact possibility q neighboring work employ implementation limited constant site multiply state continue neighbor internal simultaneously straight hypercube direction spin however show interaction system heat spin ise spin obtain system manner sequence statistic calculate manner unweighted sample every one randomly algorithm dynamic iterate induced flip perform algorithm ordinary
fa fa submodular function define list list denote uniform fa additive benefit fa fa fa fa fa fa j fa fa fa b rearrange b rearrange follow b b greedy construction strategy always list list value optimal surprising stochastically guarantee monotone uniform replacement list fa monotone fa fa fa fa list set fa j fa fa rl l fa j fa fa p fa rearranging imply expand recurrence b rearrange fact ratio sample lemma surprising also
year scalar development year amount link function fully specify need specify amount claim glm poisson identity distribution etc define consequently claim element come specify glm predictor outcome glm unknown belong third correlation form usually moment procedure empirically eq q refer consistent matrix biased method work close lead simple claim n j extreme extreme claim function stand effect
randomly etc indicate finally iterate q tn n autocorrelation accord contraction contraction banach recursion q every close point point assumption analyze hence residual tight inequality result tendency go proposal multiply term eqn refer instant q
spatial parameter limit cost run computer outcome spatial py j vector one location define degeneracy let vector consist define divide observation n separability spatial k parameter specific covariance computer captures block across setting eq component likelihood model variation normal ii var ik jk ij ik spatial location I term calibration calibration formulate observational conditional block observational discrepancy counterpart th block ij ik I observational block
base art schmidt independence potential rely alternative lot attention code day variance sensitivity physical statistical understand code dedicate surrogate however extremely popular measure limitation impact input summary distribution alternative derivative goal orient dedicated second index generalize unfortunately often consist scalar indice high preliminary screening modeling screening purpose screen propose concept dissimilarity introduce index comprise density rely density ratio access highlight dependence include mutual information motivate potential correlation hilbert schmidt criterion appeal measure multivariate variable replace standard
briefly arm sample accumulate apply jt jt naive size negligible implement fix run stop draw arm stop meet exponential proceed stage median elimination arm whose mean within arm empirical terminate successive elimination proceed exponential elimination stop condition ucb stop meet ucb heuristic run input
coordinate alternate let gamma I copy
significant effect regret portion ice act prediction multi domain demonstrate leverage efficiently display approach market demonstrate robustness consider future consider class game set player differ benefit current enable involve investigate statistical notion corresponding concept second game expressive game allow broad include explore dependency acknowledgement work national engineering acknowledge provide aggregate mid immediately lemma follow external player nash equilibrium marginal strategy respectively player equivalent statement unconstraine maximization set respect back bind give move constraint orthonormal unlike directly q science university pa computer science room il usa edu pa usa agents observation inverse technique behavior approximately solution decision use accurately observe similar agent unlike single maximize must game game notion regret principle generalize predict setting intelligence interaction party merely challenge
gradient operator matrix mean definite dimension u tx u u consider system u fx control vector dimension paper system generalize functional function present start admissible control admissible control result functional use monotonic odd control description control briefly rewrite successively provide prove e system unconstraine control controller successively worth choice cost solve unknown expression associate impossible
mistake label use together l label mistake satisfie hinge hinge least mistake hinge mistake square tag indicate node inclusion query forest subtree small proof reference explain black node capture node use mapping reader accord grey node latter depict node whole completely large forest include tree case component leave bit help node integer mapping connect mapping forest number slight abuse forest whenever singleton enable map node five would mapping auxiliary call connect let node select incremental query figure belong node tree contain mapping map imply actually mapping denote respectively combine choose split contain component contain
base parameter prior separation localization formalism separation localization many technique fouri methodology understanding go idea characterize typically record trial analysis datum trial understand implicit plus record index one index discrete rely assign likelihood time solving believe average source researcher aware simultaneous signal trial trial amplitude role variation neural lead signal neural trial describe backward amplitude change delay response differentially
previous suitable work naive completeness class design common adequate technique unbalanced post box section base realize run efficiently dimension run performance memory depend evaluation new framework present scheme technique addition scheme important property make interesting propose feasibility kernel
multiplicative additive first realistic use randomly draw record cascade cascade cascade infection variable piecewise likelihood parent infect need piecewise infection inference computing di mse edge quantify edge I ji quantify model cascade chi core multiplicative propagation compare infer accuracy multiplicative additive moreover value simply discover therefore cascade accurate estimate increase dynamic intuitively estimate additive di observation length additive cascade observation increase
dx dx dx dx complete return r dx l lp server ps ps ps ps server distribute library minimize network overhead process contain cache application mode server share parameter store ps triple ps support server transmission data ps consistency ps main function retrieve increment worker ps
generic ij ik jk measurement time group ik jk response times conditionally vector covariate effectively covariate component j finally unconditional gray observe white represent estimate admit z expression z jt show mixed membership algorithm inference extreme trajectory application logit specification specification notion monotonic parameter supplementary alternative specification discussion common vector property simplify computation adopt th extreme concentrated mean priori realistic modeling specification reason prefer datum specify interpretation consider entity also specify prior extreme normal priori specify variance basic advantage longitudinal attribute variation include aggregation individual distinct answer
rna rna decade increasingly complex early follow rich propose despite significant progress decade make measure rna rna satisfactory level computational role rna energy step rna structure inherently capable capable accurate highly experimental cause
ratio plot bottom depict distribution exhibit peak middle unimodal energy half less number middle bottom histogram iteration log majority sequence converge number need half length sequence less length energy practice iteratively upper algorithm fast
u u inequality rely denominator great combination eigenvector two apply bound matrix tensor lem likewise plug eq compute inequality tensor lead easily term martingale maximum pg main lem result lem w apply lem combined lem solve bind w union lead begin bound fact condition combine lem union collect never run thm model episode arm episode arm j never logarithmic order episode optimistic arm discard arm recommend process I I furthermore deduce together contradiction definition lem lem union restrict follow set
order minimum decision advantage process prior author modify gaussian different etc generalize student process call pointed reason observation example consider generalized linear form variance admit closed solution standard bayes g closed g hand assign parameter decompose sp option jeffreys limit jeffreys invariant precision however poorly bayesian optimization might
structure fit datum evaluate structure apply firstly take feature parent recall parent decide integrate use density appendix independence term substitute prior network structure equally likely expression fit datum well fit present probability de institute apply usa na probabilistic graphical dependency relationship estimate distribution dependency miss mass complete classification change compare traditional integrate percent keep cost statistic classify feature color magnitude descriptor become due exponential growth datum light curve use light
range information interest vote grow model improve improve active c c ask attention allocation vote lda reduce answer question present task user base influential effect heuristic multiply influential interest heuristic user influential user recommend vote datum user influence collaborative examine item recommendation discover recommend explanatory power recommendation extend introduce novel medium limit importance medium beyond medium
bivariate copula copula fully table show widely family expression interest v iv generate posterior f copula particular pf decide propagation ep approximate unnormalize whose update iteratively match refine quadrature ep conditional copula effect copula parametric ep dominate cost process approximated training covariance train input optimize
tail bind hold feasibility establishe design term variance broad consequence remark lasso compare statement constraint matrix objective quantitative sense biased estimator design well probability converge get define eq appendix converse lasso covariance sum term significantly first dominate hand justify want establish assumption remarkably design case defer quite kkt lasso read subgradient expectation decomposed describe formally particular model solve follow coefficient lasso cover control parameter role cover restrictive row covariance letting assume much assumption make case minimum singular value check quadratic different lasso derive
track error compare sgd instant fig file gd sag respectively evident report observe file correspond day evident scheme significant computational gain classic lemma sgd variant configuration regular ratio click multiply consider fact reward occur rarely converge ucb loss online scheme higher provide derive algorithm logarithmic strongly adaptively recommendation encourage theoretical adaptively sgd scheme support system ep european agreement throughout accordance combine proposition individual step detail extra incur
connect l n sum l algebraic equation update column l use choice lagrangian determine lagrange update l summarize update track progress stop material initialize l share update update ordinary namely l separate reveal h express proximal map proximal decomposition generalize familiar w function proximal indicator convex set set lead identity onto derivation identity l substitute alternative expression lead identity p tb tb simplify c l c l highlight long store look remarkably project gradient indeed actually perform convex derivation material least provide method rigorous proceeding dual often produce well clustering introduce prohibitive l
high exclude reason informative visual inspection need principal informative direction spectrum various summarize finding isolate separate continuous look component spectrum principal component informative continuous look interval component produce spectrum middle uninformative justify employ principal figure component informative step beyond scope detect extract informative middle conclude finding phenomenon interpret differently whose column measurement make array symbol period fluctuation fluctuation channel transfer constitute plot singular contain look portion spectrum finding component contribute play recognition leave understand extraction examine utilize analyze
formula derive sec hamiltonian energy phase configuration number need covariance inference freedom gets reflect landscape hamiltonian general parameter problem individually optimization sensitive convexity potential frame advance look primitive rather insensitive linearity problem field preliminary rough achieve reconstruct coarse field vary limit alternatively artificial feasible point locally approximate source visible datum disagreement dominate potential expect reconstruct efficient repeat like signal wolfe sophisticated scheme require hessian run might identify step gibbs elegant way rely random project covariance give symmetric minimum instability ts signal field analog potential compute term independent gradient useful convergence repeat step grow projection update gibbs logarithmic analog scheme cycle reach desire cycle loop increase gradually remark phase configuration huge therefore algorithm local matter result substantially start initialization point create start initial primitive thereby process rough estimate prominent optimization rise opposite avoid bias partially propose discard
p consequently concentrate proposition schwarz sum pc derivative x appropriately cauchy schwarz bind second sum reduce uniformly hard hold q eqs every suffice every chernoff random union pn acknowledgement grateful david partially support award nsf grant grant fa fa remark begin preliminary bound z event constant formula tx prove remark remark lipschitz already suffice hence q triangle z u remark remark symmetry follow remark lipschitz
learner assume exactly co occurrence association associate question negative make sparfa multi optimization probability term sparsity matrix norm simplify notation intrinsic add precision response sparfa correspond sparfa sparfa top e subproblem hold fix optimize coordinate descent sparfa optimize algorithm
child input along vary np parameter shape tree tree deep tree less likely cut location sample cut dimension choice dimension conditionally across model stage node cut non cut density decision p distribution deterministic smc excellent overview smc technique filtering description particle modify e let expansion smc leaf expand issue candidate expand stop state particle expand first manner process capture leave expansion stage already iy iy prior method must proposal approximation choice particular proposal
exponential operator z contain twice continuously crucially logistic loss x say unfortunately end behavior however unclear deal loss lastly relevant boost la define size weak descent direction whereby carefully follow suffice c every descent candidate la la la lc follow throughout perform minimize la minimize choice easy instance aid wolfe search nonlinear optimization binary precisely choice explicitly eq inequality wolfe wolfe require wolfe gradient unfortunately yield loss instrumental analyze presentation adaboost though true logistic relationship
matrix definite reduce correlation become outcome map cdf copula sometimes consider variable ta interest arise almost unimodal pt second bold repeat time experiment bias predictor weak correlation satisfactory cdf inverse mapping decoding prevent consider bias bias negligible bias prevent obtain connectivity get tree perform present sake predictor black compute predictor historical road complexity high bp deal probe rule seem connectivity situation non convergent rather strictly bp bp stable converge discard inferior package color explanation color graphic terminal graphic ltb lt lt lt ltb lt lt lt lt lt bp
group firstly fit function indicate orientation color strength unit indicate strong blue indicate randomly invariant unit unit basis spatial center similarly frequency invariant unit indicate color connection strength invariant unit plot orientation range indicate orientation department electrical deep impose generally capable adjust context address issue predictive empirically dynamic sensitive capture temporal dependency vary top dynamic extraction extend extraction pooling video high level demonstrate top connection robustness
dans la les et est de le concept des ne est dans l image est une dans de il est plus fr les une I un I un I une pour la l se la les concept dans une pour es annotation base les les source pr une les une un concepts dans de dans un est dans des relations parent les de une un annotation annotation un ensemble de cr er une adapt annotation plus des de la base et de les concept dans une une pour ce concept pour machine support un une est une pour est est
explore organization increasingly ii process finally effectiveness store randomness shannon interest behavior computational mechanic review collection system description bi capital letter exclusive originally define prediction call history eq history determine equivalence turn induce dynamic connect influence generator motivate synchronization generator formulation sometimes intuitive generator especially temporal hmm j hmm property property transition leave definition provide topology consider def finite state hmm topology restriction machine bioinformatics hmms topology restrict alphabet topology topology generator see sec example dependence motivate topological def guarantee transition probability equal probability procedure example topological exclude topology inference single state topology extend present sequel valid topological cr library full alphabet topological topological motivate application alphabet alphabet topology topological accounting eight eight topology however follow topology develop chain infer grain system demonstrate source addition nature extract
hundred selection suffice patient belong one take whenever expect likewise anti example mutual compute mi accept explore equally low mi select present depend tf strong root principle discriminate rare occur still group choice feature impact building model intensive optimize semantic search large accurately datum loop representation building start usage operator leave tree full well fix leave result well explore genetic evolutionary search combine hill time limit scoring program work add already tree current score dynamic program training effective dynamically highly five aside score work reduce train large datum equally patient patient prediction patient variable count tested set considerably know representation perform overcome variability representation ensemble get final input prediction vote ensemble version validity capture something essential text belief experience bag test fold validation order split subset subset patient assign selection training perform build build evaluate held repeat different blind validate explore later validation positive negative tn refer membership false positive incorrectly classify negative term confusion negative negative tn positive positive negative positive detect result capture way accuracy score address positive positive opposite positive minimize risk positive way score patient maximize
encoding autoencoder rbms static multiply probability define sequence rnn lm optimize successive activation intermediate reverse direction pass functional pass early future vanish understood transformation sequentially generally backward gradient effect nonlinearity sigmoid drastically much influence transformation rnn identity learn eigenvalue rnn character predict sequentially character rnns train require many pass rnn momentum study instead free demand take five eight value update effective advantage gradient find optimize comparable hessian optimization day gpu
instance create dissimilarity could total prototype would dissimilarity observation bag dissimilarity representation minimum distance compete try include display determined critical classifier bold significantly bad c dd mi graphic web result mind sensitive choice width perform across however obtain classifier rank rank mind perform difference mind classifier reasonably good minimax mind allow benefit dissimilarity matrix share distance respect problem dissimilarity approach dataset fail instance case whether similarity novel toy dissimilarity less dissimilarity underlie could measure string dissimilarity demonstrate dissimilarity bag bag bag measure preferable bag reliably dissimilaritie compute preferred bag specie generation dissimilarity representation principle bag complex bag perform logistic dissimilarity
opposite sign segment segment uniqueness early return variant obtain choose follow finish describe next compute maximum relational use j follow step go terminate st iteration multiplicative pointwise implement package adjustment alternatively present contain thesis author author suggestion use author grant national scientific h pt effect mm probability
ht algorithm obtain element sdp sdp programming maximum pt exhaustive matrix estimate total exhaustive operation pt pick pick pt maximum near base pick obtain element exhaustive fourier pt sdp sdp sdp low sdp program ht near near pt maximum pick sdp sdp lp sdp lp sdp sdp lemma xu
generator vary law accord drive entirely couple pattern currently exploration motion high system attribute incur behavior specify dynamical system writing function represent controller deterministic ms feed map process cs vc controller forward realize neural network concrete application use n parametrization adapt procedure minimize low contrast controller parameter ts cs potential derivative application window variable rule interpret term last anti structure output neuron th neuron interpret neuron moreover neuron factor signal layer neuron backpropagation algorithm layer simple study term dynamic pi exist latter avoid anti tendency term reaction strategy three consequence physical many freedom finally method robot various situation process average paper order consequence let derivation eqs rule consider neuron sensor value gaussian system maximization pi fix g behavior show
name mf distribution replace name mf idea refer density rectangular matrix variate nr selection estimation rank suitable bayesian
east leave west legend anchor south east draw color black mark mark option row crcr black mark option sep solid triangle option crcr triangle option solid row crcr solid option crcr height xlabel plot south anchor north west legend style anchor south east draw align left mark square mark option solid sep solid mark mark crcr color mark option crcr solid mark option sep crcr color black mark option solid sep crcr height scale axis xlabel ylabel test plot west anchor north
ccc complexitie sufficient arm increasingly gap strategy complexity range arm identification single distribution satisfy apply probably generalize open complete aware theoretical essentially upper sample complexity give tight limitation strategy drastically adaptive sampling complexity adaptive number surprising sparse
benchmark imagenet scale visual challenge win network box whole around multiple object number instance work agnostic bounding box single corresponding likelihood naturally instance use top predict location task computer vision paradigm address detector operate apply exhaustive across successfully train search scale pose computational hard grow approach train separate class vary detector cascade latter detector bounding box candidate
letter ib group set assume group group far least square logistic ease exposition p across overlap tradeoff support group vector achieve decomposition reduce overlap overlap vanish paper table insight kind prefer prefer group group within account group sparsity problem lasso method derive key regularizer independent loss basic norm decomposition imply decomposition detailed refer material dual far actual useful derivation norm apply develop
easily difficulty penalty fit issue prove strict operator convergence convexity notice lipschitz rank constraint guarantee suitable advantage splitting penalty subproblem low sparse storage computation unnecessary tune parallel splitting highly parallel strong simple convergence simple convex extra constraint devise practical version converge fast finally cope difficult proximal operation splitting however I mapping linearization unnecessary propose like linearization component moreover incorporate iterate focused parameter handle penalty linearization author prove tucker investigate rate ascent strictly convex low recovery norm work dual ascent slow
panel figure approximately observe estimate value mis specification estimate level impossible identify replicate replicate left right equal true around threshold threshold candidate behaviour rapidly behaviour display top panel value indicate variability value replicate dataset correspond replicate dataset candidate low candidate correspond bottom panel illustrate variability value replicate value correspond replicate candidate low top bottom threshold measure tool appropriate dimensionality bivariate wave air quality dataset adjust identification identify threshold list table residual threshold form somewhat centre panel show posterior value bottom leave panel posterior directional behaviour point focus evaluate life graphical panel pareto method less finally comment value pareto limit describe behaviour
every erm stable learn existence show part uniformly expectation quantification give simplify presentation review concern hypothesis extension therein category devise constructive suggest interesting study universal lemma remark n lf institute technology institute technology mit edu hypothesis survey
recognize house compare addition element multiplication automatically interface noun question house house multiple noun composition correct head noun noun noun choose choice randomly choice correspond least number noun also speech noun sense frequent noun house concatenation seven attract choice furthermore attempt compositional expression compositional impose distinguishing represent noun house answer question compositional follow illustrate noun compositional similarity noun house house head noun house head noun tend user dataset avoid plausible human reason house thing house house serve purpose language generally speech similarity model decide seven noun compositional skip set well correctly incorrectly answer yield accuracy approach composition consider examine paper multiplication approach include historical weight unweighted include element model baseline model vector latter try space dual multiplication multiplication function noun table represent space row gram house space house multiplication represent equation vector addition represent normalized length model good multiplication statistically test variation three space linguistic limitation multiple question row space serious compositional distinct distinct would reduce argue attempt frequent web include appear text compositional majority would row considerably gram build slow analogy question nine dual second static require corpus exponentially phrase rare corpora phrase large time predefine sufficient see language compositional include supplement alone dropping drop however greatly table take variation look drop h constraint difference domain argue suitable drop significantly due successful expression compositional compositional suggest gap entirely indicator compositional allow matter head noun house suggest house may compositional head character noun match five character character head neither neither head match head five character case classify character five character vary
row vector identify row furthermore compute see component ei even though look symmetric non component implicitly weight see already one decrease increase property observation derive provable hardness seem follow logic careful extreme requirement produce surprisingly however restriction outperform advantageous amenable term closure associate I assign set set closed hierarchy form complete form compound mapping concept lattice particularly line useful datum utilize fact contain building block efficient maximal maximal correspond concept maximal rectangle formal concept equivalent utilize decomposition bi kk th paragraph minimum exact decomposition consider every object find small database e approximation solution examine mention formal boolean reader rectangle element
process outline smoothed count measure square minus square root smoothed expected normally distribute control approximately constant let follow find either good partition decide row adjust amount search parent span lattice partition along row shift count parent ratio parent freedom movement variation parent influence movement count child along relate freedom search parent row partition one parent parent parent easy noise ratio parent count control vary therefore aspect next estimate conditional control temporal smoothed count find partition split otherwise partition value parent count column
without arbitrarily practically indistinguishable alternative trivial power significant provide sparse design minimax optimal average increase optimal power increase theorem proposition approximately variance suggest approach absolute value denote th corollary immediate consequence precision enough construct formally sub remove th define number non high sparse
predictor conduct parsimonious inferential key setting correlate criterion may broad scope discuss correlate paper begin present fdr justification rule procedure specialized take property control regression provide study hypothesis subset q control rate develop fdr show fdr sort let select key setup rejection hypothesis unless also hypothese motivation transform behaved list rejection transformation enyi exponential meaning sort list enyi us tool variable versa let global distribute enyi exponential like statistic hypothesis r enyi combined immediately rule fdr apply fdr state assume nan constraint inspire martingale usual
operate mean mode assignment centroid mode kde currently fact unchanged lower bound within besides finite assignment outer objective prevent assignment lead achieve use either user shift computational outer loop set shift identical proceed independently cluster process homotopy possibly several pick optimum gradually value b dataset fig mode put centroid
point value addition constraint equation fix project cone function cone square multi concerned computing fit derive estimator solve consistency sequence correlate satisfy show converge hilbert function behavior mis much strong result fully parametric adjust update generating path two type grow exponentially horizon lead see iv contrast markov chain approximate dynamic recent performance approximate value state finite detail rigorously shape constrain value dimensional continuous piecewise extend case explanatory sample along path correlate path mis specification convex
transpose vector element worth note element decrease regardless first lemma positive moreover minimization u accord unfortunately requirement proximity exact operator base completeness leveraging begin note consequently show obtain invariant permutation denote write sort theorem proposition q state writing element group group q eq group eq iii coherent split respect I except also
surrogate denote surrogate subset surrogate surrogate form block analysis property define g f lemma first exploit property present relatively limitation reason proof paper hand local instead condition directional direction condition building surrogate stationary last come second come limit accord surrogate summing necessarily accord surrogate lemma h fact inequality lemma notably sum inequality yield necessarily since accord directional direction n cauchy schwarz result problem case follow originally proximal algorithm strongly regardless n error proof technique lr f lr lr lr lr conclude strongly convex rate proximal growth lemma obtain f
spatial particular multi armed problem subset allocation regret uninformative decision armed problem exploitation human mechanism brain tradeoff na armed bandit subject conjugate limited memory look capture certain armed bandit subsequent show decision multi armed human different armed bandit parametrize human difference determine extent people optimal heuristic subsequent lee human subject explore zhang armed bandit bernoulli well capture trial trial performance human subject arm bandit decision ambiguity depend study human armed locate arm instance maker multi armed bandit armed cost regret maximum show efficiently contribution novel bind cumulative reward base provably cumulative expect bandit reward show e uniformly time slight logarithmic refer slow uninformative prior bad algorithm uninformative among reward performance prior capture inherent arm softmax selection uninformative achieve uniformly time obtain behavioral behavior fourth stationary time multi armed bandit locate neighbor armed uninformative logarithmic summary main contribution formal exploitation tradeoff bandit relation cognitive expect behavior thereby quantify underlie key behavior cognitive potential reproduce canonical large number paper armed bandit describe salient feature task analyze regret deterministic stochastic participant
detector unit allow energy rely non computation back prop also multiplicative interaction common ingredient motivation energy angle usefulness intra besides bilinear receive attention learn pi sigma response transform filter evolve column satisfy either column response involve multiple pooling detector classic example quadrature dependence position orientation entirely determine align filter pooling separately layer ht achieve present autoencoder q q
contextual bandit viewpoint payoff study contextual bandit run study reveal payoff statistical acquire spread rate communication limited action relationship undirected graph user link equivalently laplacian incoming equal otherwise fashion step receive context ti ta assumption make set arbitrarily past modeling assumption take approach parameter variable average choice choice goal give assumption user signal informative connect denote euclidean closeness lie
default student error distribution iteratively square lm package ml linear regression year range international phone mass name classic reveal presence line scatter influence vertical later display plot outlier nearly weight essentially line close around line near vertical outlier display pr pr em confirm consist predictor consider method observation residual versus demonstrate characteristic quantile plot least envelope b reveal assign
simulation study improve efficiency organize sde model regularize sampler describe present illustrate real begin multivariate sde unobserved let consider sde dimensional wiener g sde see state suppose jt kt unobserve observation derivation omit parameter brevity term except approach evaluate monte carlo integration distribution know application integration yield euler approximate solution sde eq size normal euler well two large enough euler property approximate approximated euler mt importance
use sum criterion specify similarity matrix situation believe external covariate natural though covariate node contain use decay covariate topology undirected network could cardinality particular measure turn real would informative neighbor except count separately correspond q criterion parameter fairly treat system could method algorithm would well system iterative block block descent partition block
power define purpose variational bregman bregman linearization differ point illustrate strictly bregman triangle situation distance metric meaningful interpretation banach e incorporation belong change formulate show hilbert imply variational close technical remark possibly case fast hence convex constraint improve capture oppose appear example assumption space possibly finite differentiable whenever main appendix concave monotonically stop small n condition must due subsequence limit fr know commonly
estimate regardless phenomenon confidence algorithm assign obtain coarse probability remain budget refine find good order subject prevent hypothesis evaluate region specify specify acceptable kind incorrectly define trace proceed simulation calculate whether multiple hypothesis cumulative error adopt accept accepted hypothesis find hypothesis h h give probability chernoff
possibly non weakly weakly vector obtain margin svm separable also margin variant weakly separate still easier final running solver perceptron contrast mention formally translate eq e result lasso objective original meaning discussion candidate contribute note determine svm classifier definition always hold geometrically small distance svm small ss da check linear preserve mirror copy entire polytope end lie separate solution lie simplex svm solution preserve feasible formalize translate know lasso svm contain know svm one correspondence feasible subset feasible construct value coincide feasible attain feasible lasso attain
g basis functional expansion exist factorial become dimensional interested dimensionality fairly discrete exploit structure space rank area space application quantification evaluate basis polynomial tensor multilinear constitute parametrization approximation encounter practical parametrization dimension thus approximation square tensor multivariate evaluation approximate term successively ideally although small effective combinatorial replace ideal use computed exploit low dimensional parametrization cross optimal construction reduce problem robustness propose allow number random limited model retain function robustness greedy limit exploit restrict
human population complex occur rather illustrate spectrum structural individual indicate membership three intend individual introduce unobserved observe snp take snp z e allele snp snp treat random allele important analysis p recently consider allele frequency snps disease may relationship testing frequency snp individual use develop test association song behave allele frequency individual population population population subject psd write allele allele proportion range psd focused interpretations aim estimate write individual snp allele frequency snp allele frequency combination snp define structure allele model model resemble response structure response essentially extension factor justification similar mathematical represent write log individual write immediate real entry form case number unobserve variable
natural gradient connectivity layer natural gradient connectivity quasi backpropagation general invariance implement use sigmoid output change interpretation output layer lie sigmoid random batch use involve take form propose train neural one symbolic fisher rely linearity backpropagation backpropagation transfer forward unit derivative k transfer parameter rate last hide readily compute w kk layer particular consider fisher backpropagation fisher modulus activity follow interpretation softmax interpretation incoming activate unit modulus definition matrix name learn rate value income activate update natural require distinct cost require invert take mini batch negligible batch formula per cost connectivity version connectivity cost backpropagation provide small compute w q cost inversion variable seem sum weighted unless activity case numerator interpretation follow activity center weight unit unit automatically average discuss introduction activity scale inverse fisher try compute fisher modulus backpropagation backpropagation cross term simple fisher modulus pass backpropagation associate intrinsic modulus fix activity propagation define depend network always activate unit sample method hessian gauss update define run incoming activate unit eq per backpropagation pass cost invert explain acceptable define rate unit remark gradient see unless change average activity per diagonal gauss incoming output newton restrict inverting issue add inversion add
remain yu ty yu reversible target although markov chain marginal note u ty u constant guarantee group metropolis hasting ed metropolis hasting situation intractable algorithm obtain perfectly cast role modification simulate transition simulate draw ty pt transition feasible consist let sequence systematically mix rigorously heuristic contrary auxiliary explicitly proposition pt reversible chain randomize mcmc terminology randomize mcmc reversible hasting sample actually cast create auxiliary deterministic
natural extension usual pc focus detection semidefinite relaxation extend mdp computationally section introduce practically plant problem hard well pc symmetric denote moreover norm furthermore unit finite write submatrix element th vector basis euclidean sphere vector identical draw replacement functional usual two number write center moment along direction variance say isotropic exist along
period tweet tweet opinion survey every week age education digit indicator political broad vote consequence percentage vote measure political political political table vote party therefore equal political alternatively political miss one code di identify political human tweet formal conceptually way opinion survey end label tweet match criterion tweet regard sentiment tweet message regard political party train classifier able time figure relevant tweet tweet previous relevant roughly speak give chain denote political able distinguish political political tweet result classifier
strong locality part influence existence degenerate require continuity quality every graph connect add connect continuous regard continuous six property axiom quality invariance scale locality continuity family invariance invariance axiom define quality family additionally satisfy axiom despite expression cg cg graph leave argument modularity invariant aspect within volume volume graph agree graph direct undirected twice purpose within weight take clustering q modularity change volume problematic modularity cluster decrease modularity modularity change monotonic monotonicity strong graph interested optimum small clustering
include invoke get natural ad reduce ignore spread say loss weighted expert ensemble bind involve tradeoff expert diversity spread prediction prediction ensemble point equally diversity perspective provide ensemble expert unsupervise diversity latter label make setting motivate q true approximation eq motivated theorem bind q target instance expert ensemble far accuracy expert prediction expert ensemble prediction loss diversity term fashion pairwise metric become inequality point metric loss get low diversity decomposition absolute decomposition diversity
rip per op vote wave acc ccccc max breast contact heart c heart heart page filter percentage filter percentage great filter majority vote filter voting column filter great accuracy ccccc primary vote remove prior popular handling instance remove misclassified filtering effect large limit run algorithm computationally individually ensemble select task ensemble increase classification also majority voting outperform unless amount filter voting voting keyword label voting infer generalize dependent world set often noisy attribute arise label account infer effect handle real inherently certain avoid handle noise create algorithm decision tree infer noisy handling examine limited
h h kk jk jk ki h h derivative ik numerator denominator scale h ij point coordinate coordinate u u f equation expression h u u u solve three factor l h u h ik u ik k u u j j u h u multiplying term respect u u ij h h ml h h ml permutation tangent treat ij x c ib gd jk expansion g ij ik ij ik h h ml
realistic fusion rule rule deterministic effectively nonconvex incorporate optimization framework constraint also calculation key modeling effort sensor configuration goal paper fusion bayes deterministic rule investigate mathematical property develop performance multi modal centralized fusion formulate abstract situation literature optimal quadratic result extension long scenario sub gaussian set fusion section defer illustrative examine type exhibit fusion practice integration sensor observe produce send fusion center false choice physical fusion minimize word world output typically random give valid sensor observe object h may close infinitely view former terminology avoid delta function object
g g g g ig g function ig ig ig ig ig u ig ig ig ig ig g ig ig ig ig ig proceed say randomly assign another initialize th initialize initialize expect initialize flat eliminate observation component eliminate repeat assign component give
look eqn extend consider restriction top divergence value order really value first value divergence zero irrespective bregman closely several commonly ndcg metric one widely web document ndcg function cutoff ndcg lb di di eqn form eqn choose hard decrease another commonly use area ndcg rely document set document order instance lb divergence cut bregman shall divergence show strong similarity enjoy combine lb divergence divergence lb submodular differ modular submodular modular modular function suffice consider negative monotone lb lb submodular lb distance leave invariant permutation bregman divergence show submodular little demanding
sampler stay difference centralize small value markov simply transform identifiable transform transfer transform distribute multivariate gaussian distribution distribution indicator equal therefore appear likelihood integrate obtain identity univariate normal variance maintain symmetry correlation cause difficulty evaluate variation variation predict feature normalize deviation term sampling alternatively fix around variable leave distribution reversible super invariant sampling page sample x fix py I p k last pdfs specify one full sampling ig alternate step give update jointly hamiltonian invariant q update sampling sampling straightforward sampling posterior differently replace concave induce transformation concave hamiltonian hmc hmc hmc greatly walk common major posterior tail prior many redundancy mode fairly joint conditional probably fairly hmc move contour mode
track angle degree track slope track test track error present gaussian tail ambiguity error second one actual indeed actual sense middle htb predict probable error outside effective track
conditioning establish tool widely process support machine opinion gps intuition classification second implementation computationally demand plain seem flexible enough signal expert stand practitioner gps mean square mmse wiener processing gps solve extend yet flexible nonlinearity sampling allow hyper last number description prediction divide summarize introduce mmse wiener filter recursively focus key aspect technique adjust third practitioner stationary relevant example communication paper way process find particularly wiener first gps natural processing estimating process mmse mean independently identically iid
leave choose horizontal vertical variable horizontal vertical remainder horizontal modification average assumption square horizontal pass token square token node head establish pass token special require within round receive token graph decide choose horizontal pass token square head head horizontal illustrated b three describe head select right adjacent pass quantity select adjacent square round n noisy version rescale quantity since average back head naive copy start time head forward receive noisy
suffer issue statistically reliable among sample practical number often practice performance crucially proposal domain specific expert approximately factor weight assignment graphical computational complete p complexity entire hard reduce dimensionality combinatorial behind approach great integer programming testing maximum posteriori hard approximate solve modern solver real runtime compute marginal need contribution call hash technique evenly dimensional space hash hardness technique solution general sum one
low upper available computational maker draw implement proposal computational explore system max intelligence allow separate theoretic processing costly establish make distortion theory distortion argue induced capacity scientific focus idea theoretic form reduce particular abstract many aspect size shape consider noise ability think intelligence cognitive behavior traditionally computationally
apply area computer vision pattern recognition biological etc extension norm jointly actually computational regularization matrix present mix pseudo generalization unify problem demonstrate computational choice vision
consider unnormalized set proposal digital net acceptance rejection would discrepancy point discrepancy intersection boundary respect set help discrepancy know point random chen markov chain chain consistently every chen update function ensure successive term
mass particle wider lower qualitatively resemble lee significance spectrum search extend hypothesis hypothesis bayes factor simple ratio composite integral require small draw fact calculate minimum extra smaller use cost hypothesis determine cost p accept correct cost reject discovery correct assign realistic posterior hypothese little usage particle physics search bic aim approximation bayes calculate usage experience use likelihood numerical agreement address whether physics thing nan regard account expect contour see ratio value gaussians measure regardless position identical increase peak keep separation despite cm frequentist method theorem yes yes short choice rule explicit obeys range statement range cover regard integrate extend
nonlinear start present method pick prescribed subproblem form achieve proximal j go step study notation exist matrix iteration generate realization relation observe inequality em inner loop method finitely clear divide conclusion follow know hand q conclusion follow square limit study continuity expectation
follow isotropic isotropic gaussian reveal take stochastic boundedness motivate truncate interestingly isotropic toeplitz isotropic constant version presentation demonstrate matrix toeplitz special establishes chernoff entail complement toeplitz toeplitz entail measurement form toeplitz component program tight I couple relaxation quadratic numerical trial pair repeat generate psd quadratic measurement sdp modeling return solver figure successful reflect color cell compare also red line limit turn close theoretic limit demonstrate cc time psd cell empirical red theoretic numerical vary vary experiment normalize square defined introduce b show average namely singleton projection set denote psd compare fig much trace validate repeat htp b justify toeplitz low matrices psd toeplitz spectrum psd toeplitz spectra underlie spectral spike pair e f unit disk value illustrate diagram trial successful carlo trial reflect cell degree freedom exhibit approximately confirm location generate plot situation rate averaging monte series carlo trial various dimension psd sparsity generate independent
combine project onto common column procedure onto projection average coherence matrix increase random low obtain result observe without replacement base solve choose divide nuclear succeed fraction divide representative numerical completion cc percentage reveal runtime error achieve nearly fraction cost way inferential map subsample choice submatrix work discuss goal achieve
summary datum precision identify suggest datum may well design amazon within repeatedly include include cpu time number increase first process dimension previous amazon author experiment raw several author rapid growth recall become slow cause though type label hold group algorithm ratio feature binary amazon data run machine classifier use separate example respective svms excellent use algorithm heuristic inherent help influence effort
user recommender choose hyper cross particularly heterogeneous arbitrary scale overfitte turn sufficiently private proceed explain avoid account another learn automatically factorize datum non summarize leave distribution update update update entry available element result slow update rule material schema likelihood spherical gaussian allow updating pseudo bound full derivative maximum give add section binary family special algorithm since view every share
conditional interest apply trajectory fit analysis website mixed measurement inference collect sample underlie branch concerned analyze rely observe frequently sample covariate regression involve commonly tx dt integrable compact intercept coefficient effect response recently additive remove modeling covariate ease interpretability tensor product two smoothing estimate multivariate number space subject little find longitudinal lda overview difference find regular directly appropriate take consider underlying paper examine notable nonparametric sparse complete trajectory functional datum analysis separately advantage across subject point surface across main success might think reasonable stage recover predictor fitting use main
distance maximize mention transformation modern consider search via inter subspace towards intra subspace angle two present divide category tree later transformation learner th split small available denote learner involve low computational testing transformation phase sampling randomize tree randomly discuss possible overfitte forest optimization divide node
constant posterior replication approximate behaviour finite hyperparameter nine plot monotonicity dataset parametrization datum draw sampler compound step marginal percentage rejection c cc ccccc ex ex ex less author choose last variance test perform although test hyperparameter level frequentist test arise fairly hypothesis deal theoretical testing relaxed version test prior tolerance
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task appear vector acceptable prefer implicit ideal playing acceptable game assign label train able correct order minimal number annotation cc come observation already feature utilize specify type yet model design acceptable content acceptable property group category content generic challenge categorical game depend feature cc feature address base cc content space cc need content annotate may know game explore representative game survey ideally establish select content category acceptable game select beta behind datum include facilitate result propose review sect public issue consensus select inaccurate assign confidence beta player feedback properly game public consensus worth game player probably extent model experience absence player back description ip enable beta contain rich player content play content feedback record play game correspondence able player take task game train predict preference player work yield target four target four player play cc categorical rapidly game preferred drift detect new categorical preference determine initial
switching dynamic complexity low regret respect accommodate large price mirror descent algorithm reconstruction scene video sequential sensing observation track dynamic social network test video image take matrix measurement coincide sense variance use tuning parameter shift frame dynamic bregman usual show rapidly dynamic prediction make motion representation ground truth wrong unclear picture pick picture dynamical
model way scenario topic beyond topic multinomial learn adapt remain topic select change relaxed mixture output begin self output maximum next modify confusion topic mixture well model expect base error log complete th th prior count next maximize q take equal posterior count posterior network bin note
hence call learner learner select positive learner select arm near select learner may near outcome bind number run contain learner l mf p near pt total due near optimal select suboptimal lemma time suboptimal arm hypercube bound slot lemma learner suboptimal arm need constraint z z constraint hold lemma trick phase length instance phase regret trick work theory next prove sublinear average dd regret go infinity dimension context e know arm one context parameter adaptively time security either classifier arrival need function whenever sufficiently classifier learner know test contextual allow usually often certain slot hand stream finite dimension presence classification learn notion context relevant utilize treat sublinear keep need keep provide arrival keep reward illustrate learner compare example need store standard keep mean memory high memory suitable arm arm converge distribute j accuracy accuracy standard contextual discussion arrival classifier corollary classifier classifier times call suboptimal q show datum belong receive hoeffding depend actually close enough expect conclude provide regret maximize incorrect computation scenario regret balance yield well case extreme fact
randomization satisfying admit cluster produce contrast show careful easy let vertex restrict vertex introduction derive exhibit growth growth assumption randomization restrict power restrict cycle vertex consider exposure average treat neighbor vertex treat vertex exposure exposure cycle admit structure contiguous vertex balanced randomization variance treatment basic exposure response asymptotic exactly vertex exposure control strictly rely exposure treatment variance double vertex independent come balanced e assignment whether share vertice calculation straight forward one possibility calculation handle separately omit brevity distance evaluate q z plot minimize cluster correspond degree extension cycle power cycle simulation th cycle sampling cluster agree precisely calculation cluster curve scale linearly able showing
illustrate static discussion fusion center assume constant speed need approximation parameter target move close sensor maintain first stage threshold target preliminary system variance base computation sample less especially target p decision fusion tradeoff cover decision sensor network close
mdp regular iteration set propose fix robust discuss state prove evaluation optimal policy positive let operator follow policy condition existence terminal give state transition probability evolve accord sequel mdp offline make number eq visit relate significance dd euclidean insight bellman contraction hold yx inequality conversely
md edu college md rand university college md edu media behavior despite interest social medium understand twitter seven week period mechanic attempt user past model process feedback user performance explore user model abstract social medium service view receive input way internal state output platform twitter stream event observe tweet observe insight behavioral user medium large amount observational key medium make behavioral available massive people fine resolution perform might user behavior possible view social great describe complicated system recently system capability point
index stability parameter conventional kalman modify modify filtering priori result show estimate position calculate averaging position filter simulation select
derive justify bound bound prove early clean execution factor find analyze algorithm maintain initialized guarantee
kernel reproduce joint minimax literature current model square paper note extend generalized leave extension future paper thin spline spline spline equivalent result hold simulation bivariate point reproducing totally discuss estimate consistently condition smoothness reproduce kernel perfectly align eigenfunction consistently
fisher go explain technical formula describe sampling alternative payoff let notice q formula appear exclusive event px gx xx probability observe game nice fix probability strategy strategy background material mind main previously win process positive integer ball formally associate q write pick pick random
challenge low well close high admit solution popular integral fu px u fu tu x hx hx tu integral approximate simulate u pz
px n I induce act representation vector pdf traditionally recover add constraint zero optimization discuss optimization solve briefly I laplacian raise sparse shape valid induce lasso minimize upper dependency pixel group model convex iterate step optimization regard objective independently eq know efficient solve among toolbox optimization quadratic equal atom unit onto ball straight approach calculate inverse propose employ descent dictionary suitable update atom solution atom th unit ball code objective separable overcome pixel neighborhood call contextual belong common long define I ix error vector column sparsity dictionary employ induce arrive parameter norm row viewpoint realization random member contextual normalization row row estimation map scope representation laplacian hierarchical lead
subtree correspond diagonal subtree subtree rise parent subtree help subgraph plot network log match image plot column figure hierarchical density h cc row adjacency well belong member split group disagreement group ground truth width width community pseudo likelihood p value cutoff cutoff clustering clustering extraction recursive cutoff cutoff chen find community put node blockmodel perfect cutoff compute dark value cutoff
ar model augment function p denote parameter I I model require specification realization gibbs specify loading assume loading model univariate element conjugate variance stationarity lie prior distribution sensitivity influence case persistence transform persistence employ persistence sensitive specification prior gaussian persistence realization loading innovation variance form gibbs conditional log sample hasting algorithm require full conditional variable detailed derivation metropolis hasting closely orientation taylor expansions proposal metropolis gibbs supplementary material analytic suffer identification loading give rise unless restriction impose attempt deal identifiability analytic detail loading low triangular element structure impose
bt wiener raise snr bt db db raise snr db snr wiener snr explain shrink wiener rescale slightly domain white signal therefore instead fidelity domain split admm take formulation definition forward experiment take take second run admm proximal value speech attain difference snr give preferable due rmse avoid noise increase use suitable speech intend illustrative example output algorithm without wiener post summarize table additional log subspace persistent ps matlab software ref software without bt highest bt ps ss clearly db clearly bt db minor ps db ps bt achieve snr vary sub due eigenvalue snr algorithms empirical wiener ss mmse alg bt persistent
participant release modification competition encode new effective reflect repeat work appeal intuition selection trial cross practice intuition typically informally may semi automate process grid belief constitute belief search improvement unfold international scale practice demonstrate belief play role difficulty recognize search
show extended accommodate non replication example illustrate international trade country country un un consider measure large measure interested relatively analyze measurement country country ij index year country difference country development available economic literature collect world bank correlation among figure moment estimate systematic panel evidence independent extend accommodate feature first eigenvector column proximity actor treat would allow treat whole draw matrix element use statistic replace zero diagonal transformation statistic replace square r diagonal replace square statistic transformation diagonal zero preserved normally distribute identical
moment initialization employ improve prove far power q q base ignore first observe second assertion fix inductive hypothesis take current estimate separate eq fact turn therefore iteration assertion inductive hypothesis permutation use per bind triangle assertion inductive conclude complete q constant line analysis improve property result error claim community perturbation bound matrix norm number e tensor vector eq q perturbation small result aspect perturbation claim satisfied exist initialization vector tensor equivalent therefore assumption denote subset node final threshold guarantee error z g perturbation q f lemma last entry close lemma guarantee q sub vector entry p chernoff row bernstein z j eq generative straightforward since e h norm proof subsequently go claim close normalization concentration bound low improve support estimate community outside sized pi ii suitably threshold stochastic claim threshold order mistake average vertex belong community accord intra error make concentrate edge average method require q q prove result average average entry order tensor b moment ab cr whitening correspond whiten matrix partition probability follow step control perturbation whiten perturbation concentration require whitening claim first term perturbation mean perturbation perturbation expectation close replace whiten dominant rank perturbation dominate together third dominate perturbation moment range whiten matrix perturbation q ok define probability g w previous rest note
possibility various introduce surrogate success scheme class introduce define surrogate gradient lipschitz mention admit surrogate f strongly amount gradient f gradient surrogate assume f convex amount dc surrogate differentiable surrogate order surrogate batch contribution stochastic scheme point usually interested finite form represent accord often define also assume bound draw point assume distribution point
allow search solve linearize guess r many priori numerical precisely decay instance rank one systematic ensure update way update move dominant one tu unit dominant right singular dominant compare state use conjugate gradient accelerate computation recent therein formulation nuclear present look exploit matlab ghz intel ram use rank
band theorem main prove hinge formula simply never clean denominator k adversary fall least number fall chernoff k complete loss generality case imply recall total variation assign indicator keep interpret normalize close uniform relate hinge uniform band project much body define adversary distinct fix lemma qx imply q noisy training notice yield imply finally assume b v hinge w since c notice suffice throughout clean remain though collect definition case b removal k removal subroutine figure polynomial retain identically proceed lemma loss feasible satisfy lemma violate chernoff imply member show vc tool show definition clean likely hold direction eq proof appendix polynomial easy whether pass check suppose maximize minimize subject polynomial iteration example next series hinge round clutter
minimum predictor error kronecker kronecker prediction activity develop spatio spatio pixel frame pixel nearby frame stream slide frame frame denote piecewise stationarity consecutive moderately large degree freedom sample covariance way handle sparsity reduce spatio confirm
spectral matrix go refinement os salient optimize mean keep paper sample special handling keep entry small pick handle depend magnitude progress overview sampling original matrix entry independent replacement remarkably handle discard entry discussion regard truncation completely avoid keep priori entry b ta quantity concentration around truncation amount keep discussion section measure iii leave determine good discuss matrix work propose
notational simplicity write
contribution come thus detail lower henceforth inequality plug bound get addition yield analyze use psd matrix b henceforth use combine low value drop sdp sdp big actually formally probability indeed inequality writing eigenvector suitable prove necessarily sdp zero elsewhere trace formally one tend feasible proceeding part condition covariance matrix whose population identity tend
probability number bind analogous constant logarithm orthonormal row nearly without replacement difficult two hand theorem matter tighter tight nearly thank discussion improve quality proof one approximation solution minimal frobenius proof imply opt opt tc sum outer product write multiply positive positive square root k tn f j te j cv special case weight te result bind singular semi probabilistic unitary
system concentrate generator q n interact generator sense section device scheme scenario regime statement hold particle framework section choose sample clear current market market reversible scan generate normalizing initialize within uniformly ii go update whole wants illustrate mention competitive share correct arbitrarily choose associate actual compose equivalent probability normalize nx straightforward mr nx average market note relaxed initialize select probability select c u ny yx ix scenario economic regime cause maker market interact market simplicity take discretized iteration retain figure describe market share time initial balanced competition among share
matlab depict convergence main linearization take precision believe conjecture logistic formulate proposition
appear proposal mix difference illustrate specify proposal mcmc sd cm conditional density easy summarize sampler close nominal significantly suggest sampler metropolis walk sampler show probability similar gibbs ht sd e provide package survey site short select six five categorical method five level iy denote absence choose prior interested credible coefficient run e estimate treat ht ccc e e estimating ii magnitude easy magnitude setting problematic parameter specify parameter choose two
right side observe term line put yield claim kind os also concentration somewhat norm bind occur desire possible realization proposition deduce exchangeability prop constant hence gaussian corollary special blockmodel condition residual converge position begin corollary correct blockmodel set support proposition exchangeability eq prove proof condition strictly
low technique collaborative filtering miss correctness task negativity relationship among question concept sparfa account negativity enable body theory question see overview main mainly context adaptive record management sparfa concept difficulty concept capability sparfa characterize scalar parameter consequently sparfa explanatory conventional variant propose algorithm lead interpretability estimate formulate content encode answer knowledge concept intrinsic question sparfa sparfa incomplete learner sparfa bi factor sparfa factor practice sparfa beneficial sparfa user tag facilitate interpretability estimate factor education efficacy sparfa quantity sparfa range pls identify level incorrectly discover relationship concept identify aid measure conceptual response detect enable pls feedback material efficiency sparfa framework variant ordinal binary utilize information probabilistic detailed sparfa interpretability response response sparfa b concept reliability sparfa connection probit logit statistically see I either equivalently possibly overcomplete dictionary variant capable handle miss negativity algorithm bit compressive signal sparfa algorithm wide education include analysis expression noisy bit fista sparfa deriving lipschitz probit logit link define probit logit omit probit logit follow first derive bound derivative individually bind factor use bound multiplying inequality decrease xx arrive fact let term function unique maximum derivative substituting result one lipschitz conclude probit case logit arithmetic consequently scalar conclude logit establishes lipschitz individual regularize probit logit detail transpose remain subproblem analogously
potential source group address combinatorial generalize independence allow provable user assign axiom empty exchange partition disjoint constraint formulate partition q describe formulation apply general address suppose user tree whose world policy assigning readily generalize subset family c pay challenge formulate formally budget assigning cost product correspond product notation element cost budget normalize define ground solution ground design involve challenge influence special maximization cost equivalently simplify uniform problem turn without form cost general submodular achieve time constraint submodular na I scale scenario time submodular polynomial problem whole function approximation factor quantify effectiveness assign particular specific user subroutine solution
topological intrinsic abstract space propose approximate quite restrictive compare come always output metric user approach relate call way visualize endowed implementation rely hausdorff reconstruct underlying point theoretical approximate metric tree construct span tree galaxy pt organize notion definition throughout graph endow prove section present recall space compact space preserve distance isometry endow hausdorff use notion metric space exist ii metric infimum curve xy xx follow geodesic minimize geodesic interval map finite finite vertex
dependence number relationship automatically equitability colored differently profile pearson correlation analysis mutual appear definition ask whether direct beyond size different information direct mutual exploration normalize score represent mutual infinity additional consideration finite al smoothing neighbor computation default mutual minimal noise size obtain setting examine mutual estimator perform plot capture increase amount determination equitability plot contain
numerical activity design change activation logistic tend thus agnostic meaning level rnn metric specific symbolic symbol management batch online batch stochastic present sparse network linear fewer find recent stack network correlation sparse initialization illustrate language language sequence subsequence structure impossible markovian model learning still problematic digits subsequence prevent capital ordinary backpropagation traditionally distant random symbol logical bit follow mark prevent detect one argument x bit line failure million music format intersection independent successive bar etc bar determine bar bar follow bar successive possibility commonly encounter result intersection represent representation constraint sequence exclude hmm block see temporal sequence exhibit dependency able minute set range thousand example rnns single choose sequence avoid marked cut music relevant cut stream computation respective riemannian text baseline code experiment symbolic sequence symbol alphabet depend set internal infinite sequence training value sequence step internal state probability alphabet symbol compute assign actual use internal compute distribution variant symbol might predict maximize symbol discuss recurrent rnns neural internal function network state time unit include always activate bias level activation rnn define symbol alphabet network
w exact test sparsity success test htp w exact solution randomly matrix different l gamma test htp minimization success sparse sparse matrix generate matrix test htp success find success randomly htp via exact normal test success exact generate
quick intuition carry proof heavily whose projection unique dot point distinct projection onto affine line connect contain hyperplane denote inequality translation rotation invariance positivity covariance fitting positivity df process projection onto least distance v subsections df estimate
equation diffusion verify specification constraint empty unity sum diffusion one eqs yield correspondence dirichlet equation base eqs stationary drift eqs specify drift generalized correspondence note
plane indicate birth represent persistence diagram diagram entirely since death occur birth even note technical reason part persistence persistent diagonal persistence diagram endow give persistence diagram infimum perfect match diagram mean partial point match bottleneck match interpretation persistence diagram prove compact one property measure draw measure compact hausdorff metric takes borel persistence propose fact product distance strategy introduction matrix distance hausdorff distance observation abstract metric consist case thank persistence diagram estimate support observation endow restriction estimator context discuss optimal hausdorff upper find optimality topological persistence diagram abstract hausdorff consider context line moreover exist
fast test usage bfgs b limited memory mb use mb segmentation represent encode pair cut two balanced encoding label pixel foreground pixel background marker pixel group together separate extension foreground pixel prior encode combination disadvantage incorporate furthermore explicit unlike incorporate partial formulate x f b vector foreground partial formulation particular solve equal round
graphical penalize likelihood maximize similar regime selection satisfy observation k e estimate posterior think soft cluster assignment seek maximize new proportion appear respect trivial update update improve local maxima method replace scale specific cluster likelihood converse initialize assign subject minimum assign use assignment step update iterate terminate increment stop reach cluster reach em algorithm local maxima giving assignment obtain assign cross cv criteria bic cv k penalize assess subset repeat choose via value learn datum case
likelihood selection subsection modular replace alternative subsection fit additive implement fitting r package pick iteration boost pick figure say parent reduce importantly true parent enough method alternatively search infeasible propose greedy start add correspond large gain current entry specify reduce allow regression spline ten function package score node avoid cycle remove fully dag correspond node perform thousand dimensional fitting include prune implement significance covariate function significance report equal independently dag dag estimate correct remove structural hamming reduce significantly hypothesis penalize selection step compare investigate
achievable sample complexity case preference agnostic process stay monotonic decrease handle question conjecture yes context co occur error another question discussion bad consideration consider preference term perhaps require million preference contrast semantic research solely rely sized annotated benchmark leave typical english speaking word desire two order magnitude costly believe semantic greatly introduce formal understanding still close able exhibit understanding mean resource process huge corpus example sufficient induce order q eq term consider x define mutually exclusive x xx x xx x h x x x xx summarize contradiction principle must set h dimension accordance node contain particular however classifier create cycle contain induce cycle therefore h forest edge desirable create h proposition theorem observation novel approach annotate example propose parameterized occurrence associate background corpus preferences method corpus collection text semantic user extensive scale indicate propose art year attention nlp semantic research greatly benefit among web categorization motivate semantic capability
thresholded j inequality reduce together assumption v bound derive false thresholded ls form random matrix invoke extend stable center covariance contain bound dp v later discretization ns ps dp part r proposition hold w w nh h j concentrate use yield argument term view proposition bound third proof take final strategy one important step estimation structural initial structure henceforth maximize respect regularization appropriately relevant structural problem l b b separate program problem regression way solve mention regression problem iy run dp ls end respect algorithm ml select l ij n single iteration amount lasso predictor ml one structural sparsity run graphical incorporate block wise maximum concentration term set observation use z norm together standard tail wise equation connection domain spectral
consider recall q xt em paragraph em em rgb pt design bayesian problem govern optimize location datum minimize parameter inverse uncertainty optimal design particularly challenge computationally pde exploit observable square root operator availability surrogate pde solve evaluate optimal trace derivative employ trace successively design characterize location spatio observation two optimal design pde solve sensor dimension problem solve interior insensitive sensor dimension numerically ill pose inverse rank trace svd recent advance enable infinite dimensional consideration place experimental inverse govern law problem challenge infinite discretized ill expensive bayesian merely repeatedly conventional essential I forward pde pde precise meaning constitute sensor collect concern lead criterion inverse average lead criterion reference text concern pose ill pose author regularize seek minimize design design build employ theoretic criterion surrogate nonlinear problem albeit moderate devise design govern dependent
highly imply present joint nan carefully interaction interaction vanish neither test test easy construct nan either sample compute testing e nontrivial factorization testing though deal single case reach sequentially namely hypothesis test sort reject hypothesis reject terminology kernel say induce domain x change kernel k whenever p z version independent integral two square compute little appendix quantifie statistic
convex solve exist unified barrier I end proximal follow method produce approximate accuracy inexact pn step adaptively regularization due smoothness newton strategy major research decade broadly optimization scalable different fast advanced free technique conjugate self follow box solve retain constraint nonsmooth slack updating attract cf homotopy guarantee minimization smooth rigorous updating regularizer weight easily adapt self extend notion handle form track consequence approximate control adaptively without manual tuning strategy bad case scheme vary direction bad analytical point function deal inexact newton fix framework inexact newton iteration bad section highlight strength
convex next let cardinality assume population matrix take subset even singular note adaptive restrict condition eigenvalue condition understand oracle cardinality collection define define case equal equivalently write turn convenient also approximation setting target know lasso quadratic beginning distribute sequel find exactly motivate approximation increase denote oracle minimize theorem shall actually concentration process process next supremum incremental q positive minimal margin assumption adaptive margin behavior excess equip often differently enable sup depend oracle concrete make proper choice particular follow remark rather give concrete example smooth precisely exhibit basis collection discuss full rank elastic use tie modify compatibility elastic regressor quadratic entirely inequality careful sense I assumption excess loss oracle error big comment
compute proceed relevant material signal observation two policy sense resource minimize develop two policy adaptive validate section conclude detailed find consider paper let basis gaussian I indicator proportion nonzero divide amplitude conditionally gaussian variance component allow depend homogeneous herein level turn considerable inaccurate knowledge arbitrary stage signal effort allocate stage effort resource depend tt satisfy overall sense budget zero comment
eigenvalue eigenvector connection eigen fix l bound uniform eq q take care series suppose assumption class denote class satisfy eq function notation algorithm compact metric preserve inner lemma tackle rewrite since uniformly choose contain hand converge law conclude lemma consist give prove put finish essentially empty uniform theorem eigen structure ni us field eigen eq come come eigen eigenvalue eigenvalue h finite compact compactly nx x pointwise h nx next calculation bind hold ny p n pointwise check condition compact problem vector finish direct boundedness continuity direct calculation last control ny z n property find h h proposition get convergence surely h reach show invertible binomial expansion therefore q put together enough binomial condition main notation embed argument much h conclusion finish th eigenvector step inside large increase q statement enough skip understand bundle bundle cloud possible obtain special bundle frame bundle algorithm cloud assumption frame bundle manifold principal top eigenvector x
may semantic infection induce infection consistent graphical joint infection cascade convenient address end alternative model infection collection mutually independent transmission transmission factorize contact switch collection specifically direct path contain direct assume infected obtain compute infection eq allow furthermore connected shortest weighted direct operation appendix suggest node node transmission scalable million compute result node maximization influence location multiplicative become quadratic network thousand million typical modern social naive additionally draw far impractical sort return list j ss sd sir h j b c output least assign search small label number transmission guarantee algorithm produce
signal vector gradient need frequently appear recover evaluate mse tv concerned performance possible support organize tv nan derivation guarantee tv mesh angle minimization multidimensional conclude discuss performance
effective distribute process worker document key map string sort encode key value worker grow new explicit parameter add store huge machine impractical store centralize communication propose conceptually store hold document content model value symbol inference symbol whose document symbol occur parameter pass issue represent full either g sampler I document wikipedia connect document thus document inference assignment compute full store model table interpolation old box continuous dash line denote complex procedure join operation fast time procedure intermediate output simplify execution care intermediate flow significantly replicate software asynchronous g become increasingly publicly machine take pathway store disk sum global stochastic use describe meta line transform eqn topic worker
consideration correlate interesting correspond denote k employ gaussian model prior separable diag discuss pose drawback technique variational approximate posterior heavily simplify maintain problematic relate simple exploit justification one em try impose joint prevent increase hand step inference frank wolfe theoretically optimal concave frank wolfe algorithm modify accordingly present infer maximize argument relate alternate solve compute tp f inference though good
panel match cb pdfs highlight power information general pdf sufficient add bin compute metric primarily merely characterize confirm systematic bias previous definition scatter pdf right panel galaxy panel galaxy share contour contour vary galaxy pdf use mean cb som pdf estimate tight generally median value except bin low pdf empirically quantify som use herein unsupervise illustrative forest technique outline cb som focus som spherical use cb generating pdfs som generate use galaxy color run color som estimation value surprising seem randomness inherent implementation improve full supervise som galaxy property subsequently use prediction table forest implementation superior implementation difference reasonable want combination explore improved technique discussion future combine template fitting technique estimation som som som galaxy magnitude color improvement pdf performance som unsupervised project attribute magnitude color attempt topology neuron cell map process mean target information building
function noise input approach attack learn address fundamentally challenge marginalization across markovian monte tackle section problematic strong target literature refer hence find expression marginal prior note equation condition prediction j look gaussian density emphasize depend typically turn inference start joint smoothing nature propose smc well markovian
let constant result proof use technical omit dependence estimator provide estimator eq constant square estimator constant jensen appropriate n thus notice imply q q event k find supplementary material
bayesian I many outside comparison therefore worth simplicity albeit preferred mechanism play familiar base accounting discuss consider ability david
theorem though amp algorithm major impact amp turn appropriately amp lasso alarm later converge let draw amp tn surely fast scenario since neither know model estimate far biased estimate discussion provide detection thresholding form alarm amp follow good amp motivate amp certain compare fix amp convert amp obtain amp correspond detection policy introduce thresholding amp iteration active amp two equation equation otherwise converge fix proof
believe far implementation potentially pt distribute dictionary noisy may useful context sensor diffusion scheme adaptive record observation distribute alternate beyond strategy present illustrate efficiency code network block variety amount high datum centralized carry another reduce extraction
representation clearly temporal correlation speech mean respectively systematically pca transform achieve dct acoustic show perform segment length classification ms diagonal dct dct make covariance may result short extract fundamental former orthonormal rotation preserve information whereas incur particular transform orthogonal magnitude discrete frequency circle value feature identity since already dct increase occur locate development single number component hence mixture consistent improvement mixture level subset give complex log alternatively mixture determined density model preliminary slight adopt weight paper primary train presence noise noise model valid good stationary estimate input snr white robustness noise model achieve noise corrupt acoustic specify combine exactly snr normalise energy per high energy reflect trace implicitly classification noise acoustic transform normalised white full covariance specify
example value section rkh coefficient linear variant combination kernel value require associate batch algorithm extremely inversion matrix limitation practical see sect key respect denote composition follow successively evaluation deduce deduce moreover easily
unlabeled auto tackle issue label distribution perturb variation perturb joint marginal perturb variation mm similar every instance consistent counterpart match maximum matching
energy must equal overcomplete space think parametric view learn produce invertible map distribution representation apply invertible nonlinearity perform entire composition function map optimize outline necessity expand subsection notational minimize current force representation choice ensure jointly encoder ensure term examine detail proceeding via representation choose advantage constraint rather overfitte explicitly advantage subtle live tail space hypercube
synthetic optimum exactly train automatic width gaussian require parameter procedure wide yield adequate width composition space distribute single three different simulation point large case odd set automatically refine figure show begin coarse iteration start capture reach relatively sense noise repeat experiment predict expect lp select conservative presence difficult capture essential level many dimension lie suitable adequate modeling assumption rise among
fast base attempt separability half seem rational one integral lack value rational integral reason concern correlation cf together upper base moment separability probability hilbert schmidt thus
well rule show rule classifier posteriori derive intuitive end respect
digit show like datum instance visualization orientation cluster one I construct approximate variant run rather substantially visualization set object scatter plot bound indeed future whether computation final limitation generalization exponentially say limitation visualization embed dimension relatively future develop implementation store addition adapt acknowledgment author support social advanced anonymous helpful
integral give equation cumulative finally eq come effect cumulative distribution line change give note simplified limit contribution small constraint last statistic form
significance sample symmetry compact unimodal version nan situation useful approximation room simulate test invariance reduce estimate diagonal nan therefore key estimation eigenvalue currently rank biological together background however recently many critical estimate covariance statistic proposal soft closely covariance single strong call anti conservative motivate soft context spike anti conservative give motivate advantage strength method wide setting simulation
solve graphical algorithm liu lee liu update conditional log develop jointly idea fact kk stay since joint regression precision formulate response response norm lasso penalty initial formulation propose q regression ignore minimum available adaptive procedure coordinate improve fit handle conditional graphical consequently refinement similar way set less conservative
complexity exhaustive p theorem support iteration swap variable swap total support visit note numerical inactive inactive accurate support allow inactive inactive rather constant inactive inactive satisfying control maximum inactive inactive highlight handle correlate motivate refer seek index support cause standard incorrect estimate calculation accurately perform definite also particular support however lasso lasso shall algorithms iteration need make theoretical know select appropriate exactly solution consistent sparse specify observation sufficient number iteration assume extension sample graphical explicitly take account simply upon setting exact feasible expect true may select conjunction illustrate popular present datum follow block choose negative
context respect ix ix naturally use predict difference set expert cover device cardinality expectation refer randomization exist thompson realization one expert predict without reward rt notation thompson require intuitively maximize expert expert ix tf r expert draw rt bayes
current partitioning move part search design b jj bp kb b j ip observe entire partitioning begin step straightforward induction I hand step inequality claim disjoint contradict terminate loop inequality loop partitioning algorithm terminate loop sake contradiction loop must put q remain
cycle find suboptimal em smoothness pixel assume causal markov state pixel understand past pixel transition probability transition dimension transition markov past instead pixel would one dimensional moving right thus whole observe labeling come prove relationship pixel depict figure imply operate neighbor suggest extension viterbi probable combination solve whole labeling field every mark l decode produce combination iterative algorithm finds give estimate parameter start prior guess contextual use segmentation choose viterbi decode path step estimation indicator decode et row determine horizontal vertical forward combine product vertical horizontal conditional viterbi
consider regularizer suppose strictly minimizer group k k n nn proof like contradict convexity simple function value strictly expand define strictly convex strictly convex strictly satisfy convexity symmetric corollary determine ensure strict convexity rational interval ensure one sec strictly scalar informative also threshold threshold ref deduce choose define absolute express soft consistency form neither concave log induced threshold magnitude identity function mild probability shape analogous ensure strictly result convex preserve strict convexity assumption suppose prop function
dynamic word unless strength general amount spherical follow constraint q question one alternatively question store moreover often impose strong threshold perceptron govern couple restriction threshold spherical perceptron perceptron perceptron purely neural nevertheless consequently result beyond purely find field nice contain collection mention namely limited fraction great detail recall spherical perceptron storage memory easy start replica storage show look consider scenario proportional denote obviously symmetric give decay characterization maximum positive fact storage relation hold course well rigorously either pure context neural spherical perceptron storage powerful enough storage capacity prediction formalize let let present form result spherical couple fact relate spherical clear automatically translate spherical perceptron
snr versus observation make bayesian subspace outperform small number snr snr increase svd accurately estimate unless snr large
failure undesirable e estimator significant method still perfectly reconstruct gap poorly robustness algorithm fraction failure course already figure omp poorly decode omp small decoding lp note display reconstruct h h figure display present gap min gap illustrate improve reconstruction h panel panel improve reconstruction many reconstruct error run time simulation repeat long correctly precision retrieval absolute lp nonzero coordinate zero negative ideally hope maximize minimize fp negative reality contain true figure mean positive always stage measure error indicate median signal soon omp bad produce perform keep fail accurate omp
eigenvector however precisely eigenvector efficient well cut dimensional advantage gain graph generate know node community micro create mixing parameter fraction connect macro community connect micro macro remain fraction macro benchmark systematically approach htb available graph take macro varied clustering often find world
carry trial integer uniform sure trial partial fourier shift successfully recover trial namely success quite remarkable recover multiplication maximize real part signal retrieval noise main recovery shift difference shift affect shift correspond snr conclusion sensitive use
symmetry away node node neighbor second unlike admit semidefinite row parameter estimate relax mml perfectly local agree come node local address ensure mle pass require pass pass converge variable respectively estimator characterize square relaxed mml mean square frobenius relaxed mml apply asymptotic arbitrary neighborhood include neighborhood neighborhood graphical wang point case completeness also comparable difference classical subsection number variable infinity high attract modern statistic estimator enjoy sharp rate true ml estimator maximum relaxed denote I local edge result standard convergence regime comparable slowly emphasize
combine classifier directly evaluate chapter ai publish loose structured paper lack precise terminology chapter classify focus start computer much limitation player another work year also study player decade area know iterate suggest player play start focus traditional game broad start goal recent deal application generation artificial game interactive majority classify sometimes branch ai implement tree search game type predict weight voting player create part game tree broad uncertainty game opponent branch relate able satisfactory ai several model good game modern strategy branching factor branching ai branch several order cluster opponent possibility movement activity make next player agent game player preference modeling focus thesis mainly focus task model strategy automatically character goal observe recognize order team ability obtain capable adapt formation opponent determine opponent multi predict player relate thesis model virtual game virtual try virtual present success model preference work thesis thesis player position movement common attempt present game rule resource recent movement position use inverse reinforcement player motion trace work player game fact work excellent topic position opponent possible player model although terminology aid activity base game dependency derive successful modeling player ai field application technique large generate environment possibility obtain level player possibility preference adapt game stay player generation tool extract game playing predict would stop play take find discriminant lda different game detect game hand mean simplex volume maximization player directly game playing test look correlation evaluate paper game use level drive experience obtain opponent challenge ct adjustment distinguish ct apply time change game generalize online optimization adapt employ tensor factorization discuss tackle adjustment adjust implement player type game scenario difficulty another interactive another interactive experience event play since implement player preference key factor management game use propose interactive influence weight characterize player interactive modeling automatically preferred model dynamically work player resemble rating additionally feature player take perform investigate trace preference year interactive learn select branch regard player applicability interactive many already state
learn identity poorly provide degree extra modify major neither appear help nearly robust method example choose discard succeed redundant incorrectly mark place corpus redundancy error systematic approach likely correctly appendix penalize equivalently q factor allow train robust penalty n datum label regularize adopt trick let train usual relative train lambda may practically master report stanford annotation grow amazon distant label paper extension possibility nearly efficiency name entity present almost annotation error commonly use processing quality common recent year
due synthetic generation signal energy visually begin smoothly repeat period signal two distribution mass sum window nonnegative generate specialized measure recover signal term decrease alone observe estimation model baseline nonnegative code code wave model report separation assign basis measure true bad loss turn
parameter choice include matrix rademacher bad typically well henceforth discuss write preserve vector amongst change roughly consider choose want wise achieve prove random long denoting describe isometry property extend give suffer dense mean dimensionality slow square multiply turn slow high hard involve multiplication transpose multiply small support finite multiply improve recently give expense matrix fourier come rip subspace analysis fast computed improvement achievable date maintain transform achieve worst good dense slow meanwhile bad good vector norm understanding general bring question address unit note seem orthogonal set make nice matrix base literature hope algorithmic nearly analog invariant reflect govern gaussian transformation stress sharp contrast solely transformation thus answer answer general theorem reduction qualitatively since theorem work
level parameter permutation quantify phenomenon minimal distance quantity suitable characterizing exist permutation even precise consideration call quantity skip e separation infimum section separation setting already four define follow x drawback result initial show avoid incremental lead refer fact turn cause serious latter noise I distinct vector easier also suggest available maximize nuisance noise proper choose minimax optimal case end distance bound minimax establish constant estimator eq q state result tell separation consideration observe regime dimension significantly separation absolute however
score non proper rule title suggest paper fitting data function square solve document case cost applicable purpose proof convexity note work positively combination necessary address implication place give although strict base proof applicable dedicated version state form logarithmic corollary theorem combine optimal meet requirement et every case score al every maximization p
matrix parameterize positive replace product cluster q replace divide use eq aspect formulate parameterize useful proper loss augment definite particular input simple way encourage go context find suited avoid compare rand sum pair pair pair belong frobenius distance matrix partition loss account concern optimize intra rescale dissimilarity partition hausdorff contiguous index index consider typically hausdorff widely variation association precisely contingency size contingency element moreover partition equivalent
fisher fisher matrix th th nonlinearity notice help fisher j information uncorrelated argument equation activation next scale affect invertible hessian zeros hessian still cause inversion therefore order direction basically combine get update method vice versa hessian demand experiment mnist handwritten
important utilize text attractive rapid growth community text sentiment limitation bias collection processing experiment bias characteristic finding document level examine research generalize another promising topic trade prevent performance degradation way generate well seem always variance therefore examine degradation cause consider promise degradation text classification implication text discuss support innovation ap suggestion f g et computer et c al financial li f li towards unlabeled opinion k employ
optimal weak primal course relevant since algorithm review herein treatment primal classical subgradient direction subgradient proceed determine euclidean depend ability subproblem indeed subgradient easy special subgradient problem formal subgradient k mirror subgradient strong prox prox use define one think generalization metric useful place mirror subgradient initialize mirror could ignore precise mirror involve
also prediction autoregressive neural conditional sharing restrict machine detail feed hide unit tie share dimension linear dimensionality share activation restrict boltzmann originally tie input give range extra datum extend generic autoregressive model continuous mention autoregressive mixture expert part sophisticated image gaussian use visible however network multimodal require flexible scalable derivation version mean field conditional approach isotropic
low hessian method forward pde solve quantification linearized adjoint respect problem dimension bayesian wave propagation observe order dimension appendix constructive find mean gaussian done begin rewrite structure root adjoint operator give eq n construction matrix identity self rewrite satisfy rgb rgb uncertainty inference describe posterior appropriately facilitate infinite discretized inverse prohibitive need framework build incorporate aim ensure infinite incorporate construct approximation inform explore scalability entire framework computational framework inverse wave propagation hundred thousand bayesian inverse quantification approximation wave propagation q inf dim quantification uncertainty seek uncertain broadly general input uncertainty initial geometry noisy map forward take govern pose unique stable perturbation input causal solution include couple nearby solution inverse great challenge pose causal set across uniqueness stem pde map obtain predict output also term use regularization parameter simultaneously interested consistent observable able present include prior dimensional proper discretization construct scalability seeks specify inverse
east region mirror letter section pg plot show weak imply majority vote trick describe phase weak meet requirement average correlation may need either acknowledgment support engineering would associate comment lemma example seminal never vote
g approximation replace unbiased derive unbiased estimator call apparent understand trade curse dimensionality dimensional adopt setting truncate eight tuning selection fold cv split reverse fold fold method
technique suit find bag ten random learn package classifier bag sample negative ratio positive another undirected I might tweet tweet would one geodesic link positively label train specify temporal window period snapshot therefore train window red bar day short duration shorter able realization period seven day leave day combine link predictor divide average perform precision leave sequence pre ci upper interested perform summarize outline treat baseline plot recall curve record predictor fraction combine record realization distribution histogram characterize much combined predictor average c lr lr lr combined lr lr lr lr lr combined combine sophisticated
finite bregman bregman associate bregman score satisfie ensure bregman satisfie theorem subgradient definition dual banach existence bregman score represent composite score bregman g bregman separable bregman strictly convex bregman potential separable bregman bregman separable kullback score subset suppose define separable bregman potential divergence density density score q confirm old power kl recover bregman sf sf limit recover composite conduct probability minimum call estimator special score may us score estimator composite score composite score equivalent exist hold strictly minimum composite probability equivalence bregman score equivalent
second ed polynomial degree explicit column set approximate great format square approximation row equal entry mini space symmetric rank st power form rational ed matrix display c c ed matrix chart formula indeed variety exhibit ed degree diagonal ed table verify gr basis computation run closely tie value ed gr basis compute fairly ed symbolic challenge growth size represent form sum power tensor format rational tensor tensor notably recent generalized algebra moment study image q regard indicate tensor
technique cast space generalize demonstrate low heavy also may metric guarantee aggregation appear simultaneous median space technique become median risk various generalize apply although retain overview use throughout estimator subsample high generate estimator fraction unknown even heavy parameter good single condition element fraction formally space distinguish v use metric formalize notion without metric access give application definition result discussion assign l banach norm probability average convex prove sample confidence return hold z result smooth provide theorem state apparent requirement imply convexity fairly tail concrete smooth linear space tail trivial requirement covariate operator
approximation input input matrix may freedom comparable I ik example thought vary substitution simplify approximate mean goodness ideally fitness like variable function treat derivative minimum
covariate efficiency search balance test commonly context drawback outcome assignment bias practitioner implement tool inference default implement code available request intend supplement variable flexible control misspecification overfitte example wherein define I tx establish uniform asymptotic inference challenge robustness flexible underlie efficiency condition distinct context discussion sparse stage place requirement doubly robust estimator yield product easy satisfy easier first depend level importantly lasso prove group selection selector lasso suit treatment group first pool substantial treatment may consist flexible series approximately build intuition suppose obey assumption dimensional nonparametric analogous familiar practice researcher hybrid cover complete apply separately multinomial select covariate select suggested work selector discard select linear vice versa uniformity select complex follow define bias assumption high tx meet find literature doubly smooth show sparsity step reflect theorem practical formal justification multinomial regression couple selection rest law variable constitute draw clarity adopt follow assume unit assignment ex tt j multinomial maintain bar either cardinality mix discuss change understand argument explicit usual tt logit treatment indicate general average treatment range fix idea keep treatment treatment mean many sufficient obey ref x arbitrary heterogeneity plausibility discuss omit selection three instead weak purpose gap excess moment case sufficient
factorization function represent factor graph half argument variable binary graph obtain replace factor factor general value domain alternative partition procedure partition bc bc bc unlabele
device measure make device result measurement read suppose suppose measurement filter pass vector depend linear go
choose learner suffer consider basic respect expert result expert switch lemma switch w ta tm tm tw policy rest
exhibit explicit integer denote trace rank cardinality real number indicator denote operator cone semidefinite flat simplex denote kullback convention gmm observe call deterministic family goal aggregate prove candidate affine illustrate purpose aggregate relevance body literature gmm assume regularity class class commonly include estimator description estimator know filter trivially specific result sharp aggregate affine base aggregation estimator note moreover infinite prefer case combination aggregate affine close true mean paper generalization static develop end
window historical window window stable window principled dynamic vary replace dependent dynamic add perturbation triangular specify process hyper drift prior drift move difference take constraint important heavy tailed tail arise distribution integrate tp heavy tail heavy tail variant place parameter correspond constitute
fusion figure order ann train reconstruction body head middle fusion present head snr table point correspond middle fusion fusion similar version measure somewhat well head snr plain snr plain image version neighborhood image visually versus produce ann slightly inferior image version htbp propose method reconstruction complexity fusion number produce activation ann ann output neuron also sigmoid require sigmoid ann local fusion ann method number produce reconstruction suffice computational hardware course regular reconstruction since depend iteration magnitude fusion marginally improvement general parametric fusion different setting realize fusion ann perform signal reconstruction fusion enable incorporate parameter improve remove tuning illustrate ct reconstruction resolution trade visual ct post cost fusion extent rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb
subproblem towards opponent neither get opponent go action perform inside environment characteristic factor lack opponent towards whenever chance score several time bad average simulator fact server movement target goal real environment wind movement step represent position acceleration ball random make movement non simulator match acceleration begin use extract useful match play
bandit bandit recommend arm select retain history otherwise select arm datum bandit contain arm sample retain retain bandit old user contextual user unite day week show formulate armed bandit click adopt probit reward smc variation probit static offline section evaluate reduce initialize particle ess similarly arm selector baseline bandit baseline ucb improvement baseline selector dynamic smc bandit static smc bandit
ps implementation computation dominant times cycle ps cycle approximated implementation bad domain explain full cycle probability lot cycle specifically increase position indicate circle black experience backward technique value datum whereas trace trajectory backward propagation broadly implementation implement one update cycle update
emphasis integrate consist relatively couple almost spirit programming discuss implementation rather relevant ground program program proof dependency rule boolean formula base section code evaluate arithmetic circuit logic mc code package heavily component essentially loop couple advantage art various logic whenever particular drawback drawback complex component programming language goal establish feasibility analyze focus inference aim work ground boolean base well use implementation parameter original system logic type fact p cancer program ground fact predicate relational dataset page need page rule program page different fact every fact look word word reason fact word involve class link predicate probabilistic fact dataset probabilistic fact learn previous use probabilistic graph edge label consist square grid horizontal vertical adjacent x model probabilistic represent find probability node way describe use three experiment constant people subset occur domain runtime consider describe report domain task program involve ground predicate interpretation predicate generator law random convert predicate interpretation fact evidence atom atom predicate neither query setup use atom atom query yet truth value well result logic complete predicate use truth vector atom query ground see probabilistic ask varied low right corner grid smaller long hard query time fact vary domain atom interpretation
label expectation different conditional cm step introduce family skew extension family comprise factor skew factor skew arise special parsimonious skew general
level theorem unknown detect prove property smoothness modification imply truncation organize truncation old refine introduce construct simultaneously denote write let compact continuously explicit basis et al expand consist index frequently fact give coefficient standard normal wavelet ball index coincide classical old ball function f norm directly compact inequality
calibration calibration vast classify whether deal multivariate calibration use review various find external external reconstruct device calibration interpretation challenge take variance properly account realize situation calibration obtain instant accurate subsequent change uncertainty internal calibration deal calibration inaccurate change comparable common change case interest calibration signal calibration reconstruction knowledge investigation reliable result fail practitioner certainly develop intuition build likelihood noise uncertainty analytically signal covariance result signal take determination tackle develop mathematical conceptual know mathematical physics example reconstruction unknown need calibration successfully effective treatment non
measurement signal run sparse denote bp result improvement exploit without know algorithm long success simplicity develop expand greedy pursuit bm mcmc reweighte propose examine develop knowledge block well properly high considerable notice reweighted pd acceptable bm omp mcmc measurement contaminate covariance unknown bm map omp sparsity plot white db yield subsection real certain wavelet cosine dct moreover representation usually significant evaluating effectiveness signal process measurement
diffusion limit theorem estimation variation normality functional various expansion limit asymptotic statistic diffusion study functional brownian motion sde power continuous sde expansion variation base recent martingale characteristic function associate sketch main concept term value martingale normal various behaviour functional quadratic functional would martingale normal limit diffusion volatility cf present functional process straightforward deal commonly frequency list involve order expansion ii asymptotic associate martingale external rely stable cf iii another expansion symbol symbol explicitly symbol show determine wiener iv check technical existence density element calculus section derivation expansion rely calculus theorems calculus martingale lot iv iii variation demonstrate devote
axiom ultrametric precede proposition axiom p reciprocal clustering contain consecutive definition axiom show axiom network dissimilarity u apply reciprocal achieve minimum yx dissimilarity observation ultrametric chain yx dissimilarity axiom reciprocal belong go back cluster relax cluster chain output x cost eq show fig chain backward go determine dissimilarity independently forward chain good respective dissimilarity across chain ultrametric properly ultrametric consequence properly triangle minimum concatenation permit conclude xx inequality axiom show admissible ultrametric satisfie axiom appendix denote dendrogram reciprocal reciprocal dendrogram nod connection maximum reciprocal cost dendrogram direct direct admissible axiom question arise whether construction reciprocal axiom method properly chain cf cf q exceed axiom state axiom network node reciprocal validity equivalence resolution belong far x allow dissimilarity compare dissimilarity member member pair accord construction dissimilarity resolution map axiom p cost either otherwise build chain small achieve likewise class analogous existence imply cluster assumption loop loop join join xx join small cost join maximum cost assumption opposite cost axiom combine accord map u xx xx inequality reciprocal minimum write observe accord dissimilarity axiom substitute true equivalently ultrametric inequality proof second minimal ultrametric clustering yield ultrametric clustering yield ultrametric value assign reciprocal method cluster resolution reciprocal symmetric space reciprocal reciprocal minimize reciprocal formally observe xx xx definition reciprocal direct minimum equal ultrametric use correspondingly ultrametric linkage reciprocal consider method network exist unique state define symmetric axiom ultrametric u connect invoke linkage unique symmetric admissible symmetric linkage axiom restricted equivalent statement hypothesis theorem consequence linkage reciprocal coincide thus reduce uniqueness claim corollary uniqueness difference loop loop achieve dissimilarity xx x x symmetric network symmetric version application ultrametric b correspond condition iii axiom redundant bind method satisfy axiom great identify build branch admissible form admissible reciprocal reciprocal rest propagate loop latter arise far admissible branch branch reciprocal dendrogram see precise constant compute cut reciprocal dendrogram branch branch cut branch branch tree provide piecewise ultrametric let reciprocal ultrametric keep reciprocal ultrametric xx x replace reciprocal ultrametric xx admissible define ultrametric axiom hierarchical clustering define ultrametric network axiom xx output ultrametric ultrametric represent fig reciprocal ultrametric merge join resolution reciprocal dendrogram xx piecewise reciprocal small outcome ultrametric u xx u u xx dendrogram cut branch branch use reciprocal ultrametric piecewise reciprocal ultrametric swap branch decision ultrametric method reciprocal cutting reciprocal dendrogram branch dendrogram entail cut dendrogram branch higher however function ultrametric b u triangle valid choice reciprocal triangle inequality alternative keep pair small reciprocal reciprocal small value alternative ultrametric axiom claim cluster ultrametric network axiom network edge dissimilarity reciprocal dendrogram cut resolution branch dendrogram combination propagation influence reciprocal propagation high interest want dissimilarity form loop dissimilarity link conversely reciprocal trust want tight form link trust trust loop completely admissible construct combination axiom indeed admissible respective outcome construct combination well define dissimilarity ultrametric recover ultrametric admissible obtain notice linkage admissible network combination network give equivalent ultrametric axiom two admissible cluster admissible ultrametric network axiom construction intermediate necessarily member member reciprocal axiom linkage apply ultrametric ultrametric large ultrametric think operation ensure ultrametric definition ix ik xx chain reciprocal influence propagate loop length clustering loop denote different chain cost incur use semi reciprocal xu xx cx consecutive build loop chain direction represent allow length secondary e recover reciprocal chain depict dissimilarity dissimilarity chain reciprocal axiom follow reciprocal admissible integer ultrametric axiom reciprocal family countable integer length secondary reciprocal reciprocal meaning
hill guide car compare importance natural importance weight importance weight baseline basically method optimal weighted plain without weight weighted continuous horizontal velocity non kernel dimensional correspond car car slope gaussian iw policy deviation employ car velocity car coefficient velocity iteration policy gradient discount investigate return trial newly experimental plot iw fast iw imply iw outperformed iw perhaps estimation iw perform fairly policy crucial plain car outperform outperform optimal baseline contribute iw iteration thank iw smooth policy among evaluate method nonlinear dynamic control figure simulator right arm object
autoregressive filter regressor regression gp approach e filter identification automate article pre sparse hyper perform simultaneously maximize probabilistic hyper autoregressive attempt consider input system identification pose infer finite technique dynamical face noise regressor make particularly normally process try frequency noise look avoid datum jointly vary parameter smoothness condition pre regressor leave
eq knowledge exp top instance prove equivalent prove importantly simple solving program knowledge begin turn treat hard adversarial direct graph self loop arc accord exp expectation occur adversarial adversarial regret exp strict generalization point concerned case low observation arc arc add arc bind applie fix exist low general quantity
variational inference maximum optimize simplex square solver task optimize na enumeration solve assignment sort selecting sort loss sort sort composition preserve compatibility convexity minimize estimation partition I maximum joint inference sketch square jointly argument addition alternate recover optimum model train score item ranking sort disease partition five fold model test hold fold train wise bipartite propose well human select disease genetic result disease gene graph extreme sparsity contain disease graph unable disease disease gene set disease branch extract identify derive gene disease similarity mesh result interaction gene gene gene graph disease connection
restrictive elaborate everywhere notion subgaussian latter metric diameter infinite bound contribution include subgaussian identify subgaussian diameter restrictive diameter bound algorithmic stability extend subgaussian diameter relate concentration give application algorithmic norm conclusion present borel algebra endow present discrete continue hold probability notation
max max hence tend assumption n n increase size partially broad scan large base regime show minimal essentially zero particular translate expectation kk n kk last line come formula chebyshev let let proof tree k p formula e fix imply k n long triangle test base count pattern simple least costly triangle topological life network triangle amount cluster fix converge triangle denote triangle result without explicitly triangle instead hold stochastically increase apply lemma ss holds consider stochastically chebyshev prove powerful suffice calculation carefully triplet count triplet number share rough square last much entail nan alternative merge asymptotically condition test powerful hypothesis assume nan os enyi composite parametrize subset comprise versus prior consideration imply equivalently express ratio versus decreasing decrease adjacency define cauchy schwarz inequality asymptotically suffice size
degree overlap euclidean criterion mle defined minimize equivalent kl divergence selection value attempt choose relevance weight kl suitable minimizing kl reasonable kl r plot label label observation label kl divergence order distribution affect r plot kl similar pattern middle proportion compare solution ari ari denote origin rp rp rp r rp label respectively rp simulation separation euclidean difference along plot red middle
expensive mp use base search possibility activation penalty training second weight unit see entire hyperparameter show result training without greedy final evaluate rao negative particle phase variant negative find reduce symmetry phase center validate new experiment three center phase center training three
gpu implementations useful computing transform task develop feature transform naturally decompose row column additional memory store fourier note keep pass backward however become expensive network memory necessary layer previous
standard whose deviation fig variation represent variation lastly mode make eigenvalue approach determine eigenvalue specifically assume noise collection error exist set upper bind meaningful column still remain correlation large resolution distinguish perform observed use representation mode series wavelet frequency wave pure frequency correct q wavelet function accordingly space hz scale via create euclidean embed cost descent allow system prevent individual create select point width space create group region select proportional integral perform training result wish compare individually set training embedding new vector accord cost enforce space transition divergence distribution
side obtain derive bound residual norm guarantee later q eigenvalue respectively recall norm positive lead eq automatically guarantee hold ensemble size space singular avoid multiplicative one least covariance rank reduce instance serial see enkf become insight constraint study consider family formulae coefficient differ kalman gain
recursion random multiply rely upper matrix mean stability diffusion ratio condition individual steady network k appendix asynchronous k ki sufficient insight affect square note size relatively behavior result step square robustness network sequel condition error asynchronous describe yield identical express likewise parameter mass gradually concentrate substitute yield possible large verify establishe moment bound moment noise extend appendix error recursion fourth sense fourth assume k k fourth appendix easy verify moment gradient noise I converse redundant verify imply appendix stability moment verify upper bind topology vanish
support algorithm architecture work provably computationally efficient require incorporate software package however recent algorithmic power lead effectiveness lead nevertheless heuristic decade cut network difficult bad manual build guarantee construct architecture rely order compactly provably polynomial amenable rely learner reach mild basic add layer bias process stop satisfactory obtain automatically variant specify advance rough already sufficient get train input predictor learn related function purpose good value attain instance layer network create build layer optimization rest heart present property architecture architecture preliminary bold face denote hadamard refer norm let column row column refer column dy j presentation easily relaxed error assume include square loss
computable solution reference offline probability risk denote center converge rate differentiable delta prove corollary section reduce reduce powerful reduction design computation numerical solution parametrize equation pde probabilistic reduce computable bind one application keyword model response model geometry external force enter pde many assimilation quantification differential equation large generally computation time computation use approximate pde ie way specify
size long often construct store invert use prohibitive regardless new careful new design hessian employ approximately solve subproblem none generally coordinate suited structure hessian exploit propose logistic regression descent unconstraine subproblem construct hessian number shrink scheme focus minimization subproblem favorable lasso subproblem another specialized graphical method refer follow quadratic working subproblem passes optimize approximately point function accept include theoretical analysis strategy analyze subject future follow replace line mild quasi newton approximation apply descent subproblem theoretically lack complexity guarantee sufficiently rapid expectation subproblem sublinear
handwritten nine first fista use originally logistic art matlab execute ghz machine ram os eight instance four range terminate objective precision set e quickly ht log scalability really benefit order iterate quasi objective curvature fista study trade fista terminate fista fall short almost always terminate fista
single switch mode switch among state switch autoregressive autoregressive hmms ar hmms ar evolve order var aggregate denote var refer might motion series behavior refer intend parameterized way may overlap behavior exhibit exercise actor variability people run share behavior series ii distribution limit transition set behavior describe transition switch constrain satisfie frequency switch behavior actor alternate behavior follow equally focus attention ar hmm specification procedure treat motivate visual represent series ar condition var form select share transition restrict series select behavior assignment advantage discover behavior share discover interpret relate improve behavior nonparametric globally share behavior explore specification feature challenge support framework feature maintain infinite vector informally think coin realization outcome coin sequence bp coin indicates select implicitly induce share coin finite coin need many share inherent conjugacy bp analytic vector distribution key bernoulli bp special sigma algebra idea generalize measure realization specifically assume collection completely discrete sigma improper result completely random draw construction interval feature
forest forest arcs node cost forest exactly immediate indeed characterize adjacency root forest extract weight root forest weight arcs new matrix logarithm thus root individual kk deduce forest next density thank equation define find kk basis everywhere thus node node column q apply simple vector contain index node experimental assess dense unlike classical graph suggest index
final constitute step describe embed subroutine input adjacency seed number symmetric embed orthonormal namely cluster let eigen decomposition tb b tu u ds initially align embed w embed adjacency transform embed vertex procedure see regime effectively large fast complexity excellent performance often achieve significantly extremely graph version procedure consider literature parallel step intensive gain gain achieve orthogonal computing decomposition v remark choose overcome automate profile procedure unfortunately require computation intensive example use partial procedure long detail insensitive provide consistently ensure subsequent cluster sized implement impossible ensure cluster work different cluster procedure setting practically procedure utilize indeed matlab ease many task consistent excellent unweighted approach direct graph would easily weight
specifically terminate high
add characteristic train sufficient dropout successfully train dropout benefit well gradient descent ensemble pair approximate remarkably average interpretation dropout dropout hide useful independent experiment share importance question direction able approximately share merely average acknowledge effort experiment also provide resource support recently introduce train neural subject attention simplicity remarkable effectiveness interpretation training network several relate
look audio representation tag fm encoding eventually recommend quantization performance audio representation compare include audio pooling tag query specify processing stage perform follow conclusion examine pooling scheme compact song piece comprise three stage song process extract encode code codebook song frame compact bag object patch image song pooling unify pool level frame vector monotonic short song song song represent reason code encodes absence codebook suppose roughly encode basis active frame sound comprised frequency band code pooling give histogram state frequency occurrence pattern pooling sound song appropriate zero encode cosine max function code commonly harmonic rather sound feature worth frequency calculate energy
solve imply applicable since unknown difficulty result relaxed serve propose rule subsequent include base start generate process refer outer one sequence point c I solve equivalent admit set please via variational parameter term inequality completeness inequality show z c statement add rule problem summarize hold statement analogously imply iy proof optimal parameter grid
basis geodesic important track vision communication survey estimation miss entry completion use incremental rank subspace identify outlier show even handle include transformation align separate foreground background align correlate orthonormal dimensional span low subspace call solve transformation image transformation jacobian th transformation standard subspace problem goal low represent k efficiently iteration via video video efficient discussion regard video constraint frame align subspace however transform possible subspace nonlinear form manifold approximately idea illustrate linearize fig high approximating image align
column outperform assume independent obviously real impact redundant avoid ensure practice applicability impact seem negligible validate feasibility extensive conduct various dna microarray document matrix present suggest currently attractive transform calculate sum q obtain odd decrease monotonically clearly numerous field concern categorization currently mainly two topic concern term distance sufficiently along specifically exploit clear attractive weak fact interest pursuit necessary mention ignore
voxel examine feature select subset drop add solution wavelet frequency orientation feature extra represent wavelet voxel htbp select clustered voxel area visual confirm extra location wavelet use make available resource bioinformatics laboratory pathway responsible pathway pathway directly pathway response pathway pathway thresholde seven seven regression problem thresholding pathway merge regression property demonstrate easily compare approach respect simulated cut term give biology less size subject plausible discuss parameter alignment come keyword
benchmark mnist eeg art partially human reward grant universit universit diagnostic universit molecular universit e sample come problem focus embed probability distribution characteristic reproduce embedding kernel hypothesis size power advantage mmd power enable look true multiple experimental evidence periodic gaussian power asymptotic relative efficiency elaborate mmd homogeneity comparison mmd discriminant additional justification
find large purpose searching rank one I current compute current inverse hessian current former go nmf square tw tw tv gradient gradient square directly inactive obtain scale symmetric newton search method object tw
remove connection layer state use randomly idea connection impose smoothness regularization na weight tend execution highlight importance introduce effectively state use particular relevance attractive way parallelism within operate completely incorporate future factored rbm parameter tensor tensor outer major factored fact predict good drop recent work network train sgd parameter machine synchronization drift
favorable mlp train scene autoencoder semantic important thing method equivalent mlp stochastic equivalent model mlp ordinary advantage propose auxiliary mlp connect neuron neuron learn consequently mlp backpropagation briefly auxiliary neuron mlp dropout drop mlp investigation drop stochastic connect fix
provide control control inequality k bound large virtue satisfy satisfied corollary yield theorem control suffice control bind yield triangle cauchy schwarz compatibility multiplication virtue schwarz inequality compatibility multiplication lemma virtue imply compact singular bp l together proof page case detail let basis regularity pair type wavelet require wavelet space function interior wavelet wavelet interior wavelet mc subsequently uniformly virtue disjoint equivalence eq q sufficiently satisfied result wavelet choose univariate tensor wavelet construct family interior univariate wavelet since univariate interior wavelet univariate variance immediate calculation k fact true norm virtue imply sup convergence regressor
signature biological mechanism drive level agreement type principal share pair underlying mechanism show sample chain first burn check adequate gene expression patient show survival highly signature informative multi datum may consideration modeling available com cancer complex drive genetic environmental responsible estimate
nlp com com l com com cb mae measurement l com com com right leave com com testing training count ranking different likelihood com poisson linearize model crowd versa crowd consistent observation experiment age net average per person appearance output leave one people dataset poisson low mae combination linearize combination mae dominate ep statistically approximation method significant test person method gauss laplace linearize poisson taylor laplace linearize poisson ep com taylor cm cc inference mae gp gp poisson taylor poisson linearize linearize poisson binomial taylor ep age l data ep ep c cc cc nb nb cc cc cb la ta la ta ep poisson poisson binomial choice approximate match domain link link dependent function looking method ep similar laplace comparable taylor sometimes yield derive simplify considerably generalize inspire regression bayesian regression method use exponential likelihood algorithm term create simply specific iteratively algorithm derive estimate exist gp framework model besides fold approximate algorithm approximation approximation generic exponential family framework output compare efficacy approximate paper discuss novel derive inference next compare approximate posterior hyperparameter estimation section demonstrate efficacy review relate regressor observation q gaussian prior place represent mean etc gp specify py f denominator marginal df f average eq ik noisy observation since gaussian gaussian closed form evidence ii average probable hyperparameter
logistic indeed assume curve cluster class polynomial govern process thus state hence curve notice hide spline capturing present unsupervise observed curve parameter density maximize follow log maximization dedicate definition complete datum propose g paragraph likelihood maximize em parameter expression require calculation simply posterior sub curve cluster index regime parameter maximize equation mix proportion proportion w sub sub class variance r logistic consist multinomial logistic weighted algorithm initial take code curve sub increment
lag lag smooth ascent newton gradient estimate lag lag difficulty sensitive decrease estimate good gradient value importantly converge slow choice converge slowly improve scale gradient pt ascent parameter compare use within interesting comparison perform prior update via ar set value state maintain initial gradient batch pt particle estimate long particularly
follow differential integrate yield lie therefore introduce cutoff mark obeys relation easier rough logarithmic cutoff generalize result output mode vector hide unit expect two equal vector solution solution network find initial satisfy dynamic arise go statement matrix time scale dynamic crucially extend derive start initial main text dynamic gradient start symmetric order one hessian mode strength optimum scale infinite time three learn eq delay l try contain correct train batch eqn mnist eqn propagate deep network computationally infeasible accelerate hardware overcomplete size improve power condition random matrix qr time nearly level limited optimize separately spaced pick yield minimum threshold condition suggest network train technique carefully scale initialization specialize long deep advantage variety paper
model map though generate ordinal mixture component key difference inference unlike ordinal labels label ordinal discrete produce rating et al real nm obtain I difficulty model nm nm n mixture prior use lead treat cm available extension account difficulty et suggest ordinal reduce ordinal nm k label ground restrict noise propose applicable scenario ordinal quantify warm none contain mixture good handle rating novel student
approximate distribution carlo setup hypercube k testing equation greedy share start represent dark dash outline dark solid average open circle axis size leave minus value bias typical gps predict nn standard consistently low confident variability method acknowledge inherent local design magnitude large design show benefit simple way analysis cover dense predictive location collect obtain obtain computation fast long even despite decompose build sequentially high may inefficient resource serial independence allow parallelization predict dense token loop nearly illustration cm cm posterior surface difference truth color higher low
propose formulation atomic grid organized formally programming minimize give simulation model atom unit signal atomic enforce analog lead semidefinite toeplitz tr operator complex toeplitz atom
variable exactly pc analytically confirm empirically get omit obtain markov mn subsection conduct generate systematic study provide problem structural learn false structure assess harmonic measure retrieval correct indicate algorithm correct total compute follow show complexity algorithms hamming result gray domain generate increase pair worth markov structure mn increase combination structure determine distribution clique quantify strongly encode forced parameter odd edge range measure gray bar light dataset synthetic size column row figure mean deviation ten distance measure plot order algorithm complex underlie see algorithm amount mn traditional independence sufficient low mn always learn distance case reduce domain many positive grow cascade see improvement strategy heuristic
algorithm proposal prior thus energy rd hybrid basic metropolis distinct posterior prior predict pz pz pz pz summation integration prediction simple z corresponding
overcome recover signal sparse effectively practically break limit use emission mention breaking opposed try increase resolution reader review novel machine square imaging leverage number diameter circular impulse disk wave way insight use infinity circle outside mean spatial likewise decompose component consideration say focus object detail limit formation system sophisticated hardware near emission topic want restriction
constrain relaxation unconstraine framework constrain network globally optimal guarantee loose spectral relaxation large way social biological similarity use base community detection fractional prominent problem subgraph bioinformatic turn optimization case show vertex recently locality seed integrate constraint maximum since mention combinatorial relaxation globally optimally due practical relaxation machine loose away optimal relaxation encode another exist normalize cut link tight combinatorial optimization agree result provide guarantee yield globally solution loose relaxation constrain
consistent q much auxiliary estimating consistently mean span tree infinity consistent much problem huge practically intractable first challenge theory back summary refer study different formulation wherein assume may
cox regression cox cox penalization boost compete scheme support view apply traditional cox criterion marker slightly table marker combination optimize sample distribution censor censor time lead censor rate result censor median censor effect censor training coefficient result get far confirm marker differ via investigate inside simulation marker empirical recommend default value ccc ccccc power result boost smoothing refer median value amount denote censor rate recommend cox cox cox implement predict distant analyze select marker distant simulation compare cox penalization additionally approach optimize cox estimate boost log boost cox use function package base learner boost carry sample split set order patient development distant
soon since impossible analyse property introduce h I h I shorthand relaxed standard p b set decrease independent fix subsequently result coefficient simplify contradiction recurrence equation introduce denote apart furthermore mapping shift one vice versa define unique correspondence objective optimum furthermore follow cl p contradiction global f respectively utility regret f maximizer bound possible double result let vector assume analyze define analyze family parameterize variable joint marginal probability capture dependence b maximizer correspond less straightforward introduce class series generality analyze fully permutation value mass consist solely one solely expect vector zeros q independence wrong obtain converge supremum take distribution observe limit last exceed equality worst low tight prediction put sort marginal take measure marginal positive zero consist apart return thresholding case candidate thresholding prediction consist zero apart mention et cs pl put pl university technology com amazon de university measure originally introduce binary structured optimize decision provide analysis bayes ham surrogate bad similar analysis show rely additional new computationally bayes regardless response analyze bayes prediction
dimension blind estimator discuss potential raise tool dynamical attempt understand question many know hausdorff difficult numerically associate theory measure roughly analogous classical theory bring notion dimension relationship estimate derivation establish analysis finally potential well question purpose tell borel borel space space exist locally pointwise source notion equivalence motivate pointwise dimension contain idea unit square measure natural metric pointwise apart borel may metric let pointwise q quantity agree call common pointwise dimension two pointwise bi lipschitz absolutely pointwise behave result middle denote uniform stage admissible measure support scale mean prove measure call class restrict borel measure reproduce iterate divide equal length iterate measure family limit measure uniquely integer scale everywhere many pointwise example define generate borel measure whereas pointwise dimension serious example say matter dimension reader quantity notion hausdorff hausdorff build hausdorff measure countable put infimum hausdorff exist
voting attempt achieve case one specificity case initial majority iteratively whereby fraction toward possibility raise question voting presence extent ignore substantially reduce effect know compose classifier satisfy previous attempt correctly predict balanced classifier population balance accuracy balanced accuracy classifier imbalance correspond rank two eigenvector entry eigenvector voting fraction hence robust vote example strength limitation
always capital drop conclusion trading help volatility thing quick analysis much probable viterbi viterbi performance differ slightly slightly viterbi viterbi could computational intensive actual value drastically gold trading coupling positively affect trade two implicit markov remove lag indicator generate trading common trading asset observation indicator compare hope beneficial make believe value strategy markov ability hmms market confident
exclude include investigation calculate mean value read valid calculate profile approach compare self som create som grid load diagram cluster uk uk som great datum rely start cluster seed run optimum seed sum square calculate total example cluster find obvious graph see uniform obvious possibly value analysis compare
compare assume regardless immediately behavior small proportional large vice versa get weakly behavior remain open stay limit system evolution family parameter ml refer normality ml recent challenge sample interval transition close exist nonparametric coefficient study little overlap simple ml dimensional interesting application biology variety chemical parametrize use regularize biological include functional brain focus dimensional fit development graphical high inverse use example lasso closely propose interference wireless network passive traffic concern sample interference correctly provide evolve discrete specialized emphasis effort dedicate completely different good beyond complexity deal sample raise new mathematical regularize variant consistency support upper develop paper two challenge elementary sufficient build present allow guarantee tend particular devote accurate capture correct scale paper without van autoregressive relate rate provide square hold never sde
value node parent denote consequence neighboring collection bayesian location classifier location indicator predict variable view except calculate classifier notably thus learn predict task value variable l indicator unknown svm classifier expectation location protein variable estimate know concerned classifier produce location learn describe conditional learn structure protein location protein value collection estimate responsible indicator location estimate svm learn directly objective explain vector train produce location
agent behavioral really mean irrelevant concerned happen simple regardless agent opponent agent trajectory insight challenge adaptive closed loop behavioral u conditional depend replace fully adaptive see steady choose relaxation ideally construct admissible upper conditional behavioral algorithm regret framework mdps difficult relaxation stem fact action reduce involved choose rademacher complexity algorithms static original derive develop showing mdps main overcome plan problem dynamic choose action plan several us markov cost fundamentally online mdp large large minimal kernel new differs type relaxation condition apply derive simple original setting inequality mdps let denote mdp I state feedback law belong future follow consequence law policy rather strong simultaneous restriction
less network deal imbalance offer handle imbalance sensitive specifically gibbs classifier links bayes write theoretical leibl topic build use likelihood prediction otherwise prediction optimize machine namely link notation expect loss minimal topic hinge error relational topic expect make hinge also regularization distribution without consider information use generic note sharing assignment possible learn distribution describe word understand formulation un discriminant pseudo un j ij normalize bayesian inference case play deal large dense negative link strategy log ad hoc pseudo link either conjugacy prior field variational factorization practice restrict assumption full exposition analytic accept reject present hinge approach latent relational random variable gamma augmentation follow pseudo indicate generalized bayesian relational e marginal include slow improve mix rate intermediate build chain whose collapsed collapse collapse ic
histogram percentile histogram bid improve safe region understand constraint tight ad request bid bid price public even bid frequently nothing boost bid reality exchange boost bid price great linear increase bid begin historical dynamically change thus base assumption good system ar section estimation without bid many firstly ar quality assessment ad decide ad secondly bid set ar multiplied affect literature feature triplet commonly node try behavior interest many regression ar level user improvement triplet leverage hierarchy page leaf hierarchy start continue category item
volume voxel cm rbm gaussian visible hyper show span field result unstable allow convergence sign spatially brain aid matter motion normalize fmri series feed mode series fmri ica remove pca ica rbm perform ica provide surprisingly localized recognize list section note negative regularize rbm start ica explain possible rbm orthogonality moreover material feature enforce affect cross run rbm rbm performance competitive art well proceed investigate deep learn
channel offer mutual scalar channel model generalize divergence also exhibit property draw considerable view conjunction compressive sense ray document recent various theory perhaps prominent reveal operational mean particular express channel mmse express mutual channel mmse scalar scalar poisson optical
require may introduce auxiliary characterized compound poisson process clearly specify every completely random evy count convenient calculate pm ap present ap ap ap j pt derive pmf ap compound ia p pmf logarithmic pmf becomes truncate nb positive express nb u ia pf n generalize kind calculate truncate nb sum logarithmic become sm first ne measure express ap compound factorize conjugate univariate infer recover formula crp construct gibbs exchangeable partition p z nb process assign table generalize customer generalize restaurant joint cluster equivalently ap aa customer
environment say independent refer homogeneous markov environment q policy stationary state x admissible policy unique stationary notation mdps mdps admissible mdp measurable conditionally every admissible weakly equivalently measurable mdps coincide get admissible connect conditionally obvious policy notation markov w dominate statement equality
rule replace digits symbol second token additional english character feature context requirement language require amount text restrict occur token ht chinese term coverage vocabulary frequent token special token five drop dramatically wikipedia article article coverage usage form usage high coverage achieve vocabulary chinese dictionary build section use choose window embed layer maximize embedding might force
transform cumulative quantile joint copula marginal wise support gaussian standard cdf function respectively generalize copula density variable method significantly affect curse tend address domain adaptation copula domain drawback address model decompose density non describe bivariate copula correspond plan transfer one different type literature undirected tree call condition
warm take singular system optimization repeatedly repeat computation feasible matrix extremely costly computation main decision rather give complexity dominate computing iteration practice operation much factorization outperform addition factorization burden aside advantage factorization identity projection trivial entirely avoid svd introduce minima correspondence factorize factorization nonconvex turn local un factorize appear context semidefinite sdp broad objective main require correspondence emphasize completeness optimization semidefinite change minimum seem psd interest sdp nuclear nuclear norm admit programming sdp nuclear precisely symmetric semidefinite formulation class characterize matrix continuous equivalent characterize clear feasible use write corner nuclear lr factorize formulation minimization formulation enjoy hand lot g formulation require identification instead formulation integrate solution factorize subproblem factorize subproblem relatively
relatively correspondence standardize measure mostly mostly table show value measure standardize generally standardized approximately occur item similarity lift ranking maintain standardized trend zero suggest high higher maintain evident upper rule sort lift lift raw raw distribution categorization available word reduce stem stop remove word stem least stem stem support threshold set high lift higher indicate similar rule spread throughout standardized concentration occur one plot
moreover chen need complementary large integer complementary virtue coverage great shall inherent connection demonstrate rule virtue inclusion chen binomial proportion parameterize estimator sn n confidence interval chen rule termination take termination stop rule control inclusion derive control propose inclusion principle extensively principle stop normal version simplicity stop chen effort eliminate limit context inclusion immediately stop eq consequently estimator paragraph consider immediately sequel stop virtue interval estimation p u sp p apply continue z z continue stop sa positive inspire construct p u general idea z x np chen page sequence yield continue confidence sp point sequence proportion since stop directly involve desirable stop purpose result assume confidence interval p consequence confidence stop method pearson confidence u sp u l decreasing increase p n consequently demonstrate pearson stop rule stop stop rule
stage rank framework extensive show propose outperform subspace classification transform applicable intrinsic area computer face handwritten trajectory high ambient lie subspace subspace problem cluster correspond mean applicable suggest subspace affinity analysis agglomerative compression linear cluster survey low dimensional intrinsic enable violate face accurately linear capture pose face exhibit cast realistic ideal arrange single approximately recent effort transformation transform component alignment application salient detection rank propose linear transformation via nuclear recover time force maximally separate improve reduce subspace accurate fig face low row face across visually enable pose follow subspace low introduce robust enhance method nuclear improve sparse ssc structure online subspace big extensive art clustering reduce nuclear separation
dependence implicit asymptotic hausdorff satisfy smooth away satisfie suffice mx p diagram denote persistence regard find confidence persistence diagram distribution may seem specifically complex outlier persistent homology set shall see find bottleneck hausdorff subset diagram q visualize center persistence diagram noise box diagonal alternatively visualize add band width around persistence diagram significantly band interpret represent topological diagram include diagram figure p diagonal put around outside confidence quantify uncertainty persistence diagram indeed diagonal imagine possibly purpose describe present persistence fourth take base illustrate subsampling usual subsampling e unfortunately nonetheless subsample confidence subsampling rather b nj nc
song reveal preference implicit user rate simple datum song click click consider version netflix original explicit rating implicit compete nmf attribute model space latent initialize multiplicative rule minimize leibler rating lda preference inference posterior mf netflix predictor comprise constant popularity fit squared rating regularization overfitte weight randomly initialize via use posteriori explicit zero feedback practice treat nmf rating negative employ user rate item fail probabilistic author report netflix minute day hour netflix alternative bayesian optimize algorithm negative vast unobserved prohibitive randomly rating hold comprise aside section recommendation score fraction top
directional gradient perform upon second mirror consider directional directional convex motivated choice nearly unbiased perturbation though remark optimize assume g interior impose property smoothing quantity appear statement hand explicitly norm finally previously smoothness moreover procedure gradient unbiased accordingly vector expect q term correction take make mirror perturbation build proof stochastic care truly receive gradient proof reasonably size multipli md specify multipli result independent follow continuity note similar next guarantee bound boundedness sharp inspection strategie concrete mirror compute imply characterize rate since assumption work corollary focus convex actually fast accurate instantaneous illustrate
iv equivalence gram matrix sparse general supplementary n estimator equally latter sharp choice former potentially material truncation lead preserve follow collection np p construct I define second hold possible ii explicitly region process uniformity region substantial exactly extend follow achieve impact work sparse regressor
criterion iteration tb counter l rt critical propose initialize size adopt bb hessian initialize outer monotonically accept monotone line satisfied variant criterion monotone possibly function previous inspire monotone line criterion lemma guarantee integer monotone line
posterior distribution run mcmc combine posterior small size suggest make subset density density adequate dimensionality product address density posterior transform posterior denote approximated variable derive density posterior component enable sampling conditional simply gaussian restriction generalization appendix density last distribution joint density sample sampler quantify dimensional derivation provide proceed identical smooth f differentiable exist constant normalize define covariance case lemma sufficiently approximate constant define vary even size increase infinity converge approximation effective refinement convenience straightforward
penalize log c incomplete likelihood convergence r kp em curve cluster segmentation compare piecewise cluster gmm approach ten em algorithm adapt cluster approach account synthetic real curve approximation criterion simulate partition partition intra cluster k em spline base corrupted noise simulate mixed proportion varied proportion class simulate shape simulation generate curve ij ij standard transition additive segmentation perform contiguous segmentation stop variation predefined compute datum optimize choose simulated piecewise model polynomial regime regression cubic spline uniformly knot intra extremely difficult retrieve misclassification however curve gmm smoothness propose approach segmentation attribute approach gmm em intra simulate different curve cluster gmm em correspond curve gmm curve simply continuous contrast adapt
sdca solution maximization maximization time prox ridge w sdca regression minimize constraint specify prox sdca option sdca logistic n bb b bt ip q q nx w ridge square loss solve problem positive regularization let yield problem fit eq accurate ridge sdca option update run obtain follow sdca z j j nx j z stop w runtime denote simplicity choose runtime eq become runtime sgd variant slow runtime fista shrinkage nesterov technique fista eq factor another coordinate primal show find runtime runtime well svm cast
direct maintain update variable ji modify compactly update newton matrix occur newton hessian reach examine solution eq time first direction newton call newton base value discuss newton direction subset direction eq objective adopt try definite cholesky cost need objective evaluation compute computation convention domain prove property govern establish enter show newton newton ensure converge current strong view distance decrease condition property property one per iteration first hold symmetric stand norm I e imply fix sequel define combine gx gx divide side positive ensure set level define positive begin show eigenvalue element imply combine
exposure mapping specification framework allow flexible exposure mapping come experiment suppose sure could formalize matrix demonstrate method complete micro finance field procedure random adjacency aggregate imputation combination formula paper propose analytical interference define assign exposure relate received choose make maximal design develop estimate randomize know specify three characterize assign ii exposure assign receive iii select experiment interest interference causal treatment randomization causal biased propose ratio adjust sketch alternative observational uncertainty interference extend trial assign select group interference operate provide randomization assignment strategy treatment extend outcome extension observational related interference hierarchical interference independence valid experiment carry
inter affinity matrix q label intra inter neighbor train neighbor sort base absolute correlation affinity embed perform varied size toolbox lagrangian accuracie dictionary subspace dictionaries subspace ensemble prove hierarchy superior penalty tb c tb c remark plus em height width depth em mail edu dictionary representation sparse representation use learn dictionary increasingly several availability provably dictionary asymptotically employ procedure dictionary code complexity pursuit demonstrate optimize framework suitably regularize therefore impose implicitly novel improve ill pose figure high patch comprise geometric compare geometric level level hierarchical quantization dictionary sparse subspace incorporate dictionary
create spurious correlation different spurious series result model criterion trading prediction average feedback criterion identify set bottom analogue forward regression centralize omit dr model former another way dr ordinary time world lag model approach world determination identification order lag scheme selection lag lag fit lag reach variable already parsimonious order var series maximum criterion bic ol detail determine appropriate follow go etc minimum bic repeat process component new value step intermediate delay etc arrive optimum model decompose begin
effective may assimilation pde e mechanic pde regularity assimilation practitioner correlation boundedness spectra covariance compact decay noise mass concentrate manifold must careful away decrease reasonable contain information dimensional indicator example concept energy assume component happen affect source component decrease small outcome new additional pdf specifically capital letter weight weighted particle target function sum converge almost expect pdf support particle apply idea recursive conditional factorization function lead recursion resample weight g set state resample upon sum equal one one near particle unlikely collapse logarithm argue rigorously logarithm lead critical follow choice importance generate step sequential resample sir filter sir filter measure induce target support event significant become rigorous sir filter particle sir logarithm choose
whether theoretical interestingly match theoretical albeit implement nan reflect sn proportion th sn exceed inference instead believe instability tackle problem ratio beta trick virtue reliable inference evident fdr separate focus entire curve section carefully quantify ht em discovery ex cm sn ex tail main criterion integrate comparison density estimate unconditional implement method fdr provide nan ask extra gain know demand empirically great em fig depict expectation fdr perform equally well apart
datum low second marginal tight given implicitly differ similarity identical upon interpretation yield treat evaluate treat parameter low argument complete datum conditional em bayes density conclude point bad likelihood lower translate report supplement mcmc mode true skew algorithm identify degree tool fraction introduce pg augmentation scheme penalty lead complete update step rarely still design proposal sparse regression see share thing treat device fisher iteratively current estimate form
statement ss throughout analysis bottom panel abc estimation influence first increase accept abc lost secondly approximate whose size extremely acceptance rate reduce datum vector summary statistic greatly inferential potential approximate approach approximately transform term adjustment inference transformation address open firstly validate abc accurate choose hoc manner suggest approach estimate extend abc posterior examining hold numerically evaluate coverage diagnostic determine likely accuracy circumstance coverage approximately alternatively
one look certain small constant respectively one eq unbounded mention feasibility scenario early constraint probability make exactly exercise fit mention scenario restrict describe nice connection present become unbounded infeasible unbounded expression positive thought obviously parameter optimization optimization problem call large positive feasibility breaking determine proceed easier compute derivative easy algebraic transformation fact give conceptually substantially basically thing resolve concentrate deal q constant thing besides expectation relate constant follow introduce follow definition necessary consequently precise systematic sc let solution discussion combination easy somewhat deal look recognize negative bound assume exposition result break feasibility subset prediction result split presentation result part theoretical look regime
unique entry row equals share mainly post process another cover assign overlap share many cluster consider contrast adjacency view mind think general assign smoothly node might great precise snapshot serve cover reflect assign observed prevent overlap measure cover consequently ensures evolve smoothly evolutionary life paper concave exist without modularity edge constant matrix ty ij detection difficulty lie quality handle snapshot weight ij paragraph function generalize correlation closely relate modularity assign node assign node share cluster assign positive observe connect case share prevent algorithm fitting generate many cluster overlap distance use change overlap intra inter vice combinatorial due cover exhaustive impossible exponentially many popular
randomly also control make treatment know accounting selecting control large raw outcome instrumental control power spline interaction allow causal heterogeneous treatment local quantile instrumental variable treatment take replace inference formal methodology effect informative without impose reduced form approximate impose reduce relationship linear identity sparse fundamental reduce use reduce estimating causal framework allow accommodate realistic exactly transformation broad valid trivial procedure single control unless assume perfect restrictive seem unlikely economic application typical selection enough distinguished near finite condition variable zero effect omit impact regard discussion paper inferential use theoretically structural identifies outcome treat state treatment interest treatment actually receive distinction substantially assign control average provide treat effect treat let interest family treatment quantile inverse conditional quantile treatment define standard briefly recall outcomes outcome treatment observe jointly indicate treatment outcome example think benefit instrumental variable offer generate decision respectively potential decision observe realize decision quantitie causal causal causal causal causal quantity section follow follow surely z p pz pd literature formulation simultaneous thorough discussion identification turn upon show causal structural q object parameter vary clear role strategy try model decompose parameter principled coherent normal easily indeed natural functional form affine product potential either approach form find difference two approach rest predictive high reduce form equation step estimation parameter effect elaborate strategy discuss modelling suggest large specifically approximation target take complement probit specification rich may size specifically condition dimensional regressor large technical control present could compose transformation spline polynomial various interaction chen chapter form depend dependence control constructive approximate impose approximation require small error formally exist size error approximate grow structural splitting high outline extend standard identity control depend specific
classical particular training idea automatically classify uniformly image region transform supervision label line end randomly set next region likely training region give drawback certainly predict classification contain exploration set word learn sub policy aim final remain consist simulate th acquire result policy properly e order
overlap individual inter cluster hope achieve correction new degree capacity neural click retrieval potentially randomly line lie retrieval modular sub come subspace author simple demonstrate recall phase paper think local cluster arrange neighboring enforce sparse cluster aim domain degree redundancy hand look neural module brain hand similar spatially couple spatially generalized code tool
call dataset instance exception comprise category lead maximize easy task hierarchy remove process map category format collection category stem hierarchy classification participant allow classify instance inner hierarchy purpose present basic three deep ratio instance comparable dataset respect category accounting term factor cat label cat depth report challenge flat well micro sign test hierarchical score instance system accord great scale compute ignore well sign account reason test subsection dataset challenge label measure system statistically significant correlation pair ranking rank flat flat hierarchical differently another ranking hierarchical measure handle present prediction per instance single label participant treat single section greatly label affect label measure handle label single label reveal lot reason highly treat something differ calculation perform augmented reason hierarchy labeling error c acc acc acc c h dataset characteristic affect measure acc consideration prediction system ranking compute fp without tp reason tend per way rest measure perform much calculation tn less acc e g acc acc b table system table something affect behavior
expectation pac pac machine differentially architecture know query framework strength private algorithm also differentially private class active algorithm include preserve privacy label request even classic passive model request preserve differential privacy theoretical active focus either complexity index factor computationally addition robustness selective reveal assumption bound query generate point converted distributional pac improvement label passive restrict pointed improvement addition define easy sampling contrast hard provably noise adversarial concept zhang amenable amenable disagreement technique existence class adversarial noise publication give adversarial first label adversary uncorrelated rate grow presence well consider contrast case deal noise sample run organization model illustrative balanced uniform give statement differentially function x boolean possibly corrupt respect statistical access active function point point
iy k iy apply get since u come priori denote replace bound theorem rigorous practical bound computable resp fully justify bad consider possible put choose conservative sharp desirable deviation section give theorem dimension let sequence center fr remark soon twice
great drug signal criterion however detect five event gp apply gp database database least case interesting gp database allow detect allow broad database offer able mention event occur drug event cause effect cause drug event even directly link produce positive frequently window potentially year window average drug false positive show database help identify new detect differ
correspond company stock stock company create demand system month accounting week month sec event baseline activity aware company event relational little connectivity high company group another activity end period suggest increase counterpart show vice relational nature tf could reason lot act cause vice interesting behavioral activity drop fact apparent relational happen send could
semi blind deconvolution structure distribution primarily minimize vb factorize despite limit widely development hierarchical prior student deconvolution include tv natural prior regularization mixture motion implement estimate kullback leibler synthetic deconvolution blind illustrate real force molecular imaging potential achieve atomic recently demonstrate reconstruction wiener iterative estimation drawback require known blind method deconvolution bayesian report real discuss finding conclude convolution equivalent noise vector gaussian nominal blind
exploration topic allow concept science anonymous mathematical procedure inference university institute predefine scheme refer single object refer match set set parametrize theorem combine way reliable handling discuss practical concept scientific step object
match three focus far close matching pair look irrelevant part however change viewpoint illumination resolution patch descriptor far
closure take rkhs norm kernel generate rkhs kx induce h modify third kullback hellinger total tv every induced lead generalization corollary define induce finite clear induced rkh role belong assume mat ern inverse compactly support non family index density therefore kl empty tv hellinger moreover choose continuous density hellinger see heavily notion presentation section e w fisher matching choose fisher show theorem dimensional linear mle practically due difficulty handle consistency section proceed assumption need da ix separable twice continuously density construct mat ern list clear include condition identifiability estimate fisher weighted square part motivates plug hold class ii iii give draw f alternate ii obtain estimator empirical turn principle practice however easy involve solve prove simple interesting obtain solve system turn system addition would estimator precisely inverse
approximation kernel wise fairly case biased linearity spread nystr cover experiment similarity nystr om dpp summary moderate generally om dimensional dimensional pt mixture use issue redundant reduce interpretability phenomenon especially prominent sample fix prior weight weight approach risk parameter location lead separated maintain rely define manner distance penalize computation fairly appeal however restrict fully probabilistic mixture dpp dpp gibbs exception gibbs parameter instead independently depend upon sec estimation dpp base synthetic discard burn thin chain supplement balanced switching process follow address gaussian separate poorly separate location dpp six see similar
measure identity improve false alarm expect confirm hope procedure acknowledgement thank anonymous mr department mathematical university york constructive quality author like thank dr institute mathematics technology national effort spend special institute technology scientific conference hold say institute acknowledge support conference edu detection multi cyclic setup choice time distant future method approach technique exploit change identity unique property robustness improve length alarm detection delay tight confirm propose vs vs design particular may gain great utilize sequential point detection concern
underlie spectral fast popular find various recently degree way regularization cluster statistical regularize correct blockmodel clustering rsc adjacency graph good eigenvector put unit onto treat point create th cluster graph laplacian rsc stochastic blockmodel rigorously help normalize b instead emphasize row introduce blockmodel stochastic blockmodel sbm node block membership definition block equal block node sbm node block degree
report act molecular health set engine good write pm ei enter automatic acquire gibbs max expect include categorization multi augmentation make restrict exist margin applicable build link public augmentation solve problem show improve efficiency interest develop scalable sampling deal set nice document architecture prediction model weight bring challenge minimize average discriminative discover text categorization solve exist solver strict normally subproblem procedure show interpretation successfully monte max margin field accurate need solve subproblem limit computationally demand svm analysis substantial effort parallel presents supervise topic develop inference margin adopt margin minimize latent rule prediction exist margin computationally margin develop successfully develop collapse without solve multiple latent subproblem close conditional draw algorithm augmentation development learn margin classifier learn generalize task collapse gibbs sampling augmentation significant improvement rest summarize review algorithm gibbs extension section discuss max structured decade research receive increase attention
share new one ignore thus overhead hand il bad reaction attain il dense cl scalable il il paradigm sharing begin term notice cl decrease fast notice cl early il paradigm cl take figure draw resolution notice cl il maintain capacity clearly dynamic cl figure scale notice decrease cl paradigm different centralize practical global optimum two il cl former
global minimum repetition se small represent bold mse cell denote reference se compare covariance ability test value section imputation cross equivalent design miss test every test miss convexity mean
dependent hoeffding hand side nearly straightforward prove elementary follow soft put substitute follow yield event expert shorthand convenient identity imply side mas contribute contribute claim also establish immediate define fx defer appendix iw lemma I iw segment chebyshev inequality atom follow imply goal expert
transition different goal quick within library ml pg interested appear attention library across case auxiliary list librarie correlation theorem library main nash library even present proof different reason development therefore lemma backward preference player acyclic suggest pg explain look figure user vary big homogeneous cluster little effect heterogeneous cluster ml pg pattern get absence library study theorem extraction ml pg automatically statistic team pg translate virtual stack abstract machine execute model program refer match end computing factorial program tail factorial scenario interactive prove involve load big proportion often notation lot pg multiplication power team factorial translate tool execute schedule indicate end schedule factorial number natural number one figure execute merely correctness factorial program team member ask follow correctness factorial factorial state contain stack associate program methodology proof virtual machine consist prove
write compute excess substitute q positive stochastic eigenvalue range combine conclude conclusion doubly stochastic like weight excess risk verify difference note assumption environment stationary environment seek leverage supervision entity access node share broadly privacy communication central node server track algorithm fully distribute choose aggregate regret similar estimation distribute rely consensus diffusion readily enhance stability robustness strategy consist combination average local adaptation incorporate result cost function size adaptation excess performance strongly convex oppose square study risk regularize logistic delta rule square distribute classifier optimizer tracking utilizes even show hold process
undirected r delta delta set real world never mistake delta mistake whenever one span tree exception smaller less preferable accuracy query range value since decide reduce use thus query average ten randomness draw randomness visit recent contain query provide currently possibly paper start investigation universit di universit di universit active algorithm motivated stochastic edge label obtain perturbation assignment node show factor mistake rapidly sign network sign network
multiple run configuration chain toward refined heat explore chain configuration identify dynamically series exhaustive greedy employ study reveal environmental pattern result complex process dynamic present challenge conceptually insight identify
induce rate distortion shannon refer shannon lower insensitive measure express probability density density eq differential parameter relate distortion rewrite put back thus insensitive distortion density
dropout pair dropout decrease size model try dropout additional help prevent task dropout reduce recent argue net stochastic activation less kind pair dependent logistic sigmoid bad always validate activation long general task dropout algorithm adapt old tradeoff task relationship activation suggest affect neural system train second task forget example
european reference acquisition overcomplete dictionaries dictionary atom nearest furthermore simulation improvement sparse recovery acquisition correlation possibility case basis overcomplete zero set projection arrange acquisition matrix element norm effective dictionary expensive approximate strict use desirable increase acquisition one coherence I diagonal element gram matrix state minimize large mutual effective large diagonal gram percentage fraction reason average value relaxed expense argued average well behavior
potentially desirable predictor whole range remarkable finally encourage current inherently optimistic put location exhibit extreme prediction span stationary general construct pareto provide extreme heavy tailed intermediate value tail predictor purpose derivation heavily particularly inspire prediction consider novel historical inference though phrase differ analyst limit historical ask regard event say experience whether
affinity respectively consist training avoid assignment dropping problem program average lemma
likelihood result simulation quite penalize function penalize consistency mild consistency model tuning need likelihood integrate reduction proof key idea define satisfy consequence proposition p p within envelope square integrable straightforward maximizer sufficient f ig ig ig expand n uniformly pn law large dominate ratio tend hence exist maximizer tend maximizer sufficient tend firstly term pn pn obvious third
force sparse via mild geometrically control target apply precision matrix estimation without accurate computational procedure research support nsf dms nsf li zhang nsf dms nsf triangle combine integration fy fy fy inequality prove desire prove consider restriction accord definition fx convergent periodic eventually constant deduce enough establish step proof consequence minimum restrict support consequence step q leave symmetric difference last obtain obtain fx x desire inequality b obviously strongly convex cardinality eq follow n
estimate around simply average point lead recursion section currently convergence change update may time e interestingly impose replace fast due linearity underlie see estimator newton novel situation convergence loss variable eq denote exist hessian optimum eq derivative derivative refer reference must loose x see hard I assume step average descent constant improve property sharp normally noise noise average replication average pointwise average excess decay indeed correspond average although decaying improve average consider output display plot constant excess reach indeed gap oppose curve
simple use batch demonstrate free quickly figure test algorithm importantly optimize c linear logistic calibration notably well runtime trade even regression regression small improvement calibrate cc comparison term runtime axis six algorithm take care attempt algorithm reflect loading feature consume generate feature appendix low little substantially learn feature produce classification error runtime linear calibrate consistently highly notable pixel test primarily instead dropout maxout algorithm increase size able linear without optimization image generate say along quickly thousand model obtain extremely b popular multiclass henceforth news four henceforth challenge vision dataset
less leave high snr get close recover true middle right higher able pre projection dm two elaborate result figure dm significant advantage art get minima reach good variety size ratio db dm size test dm contain exception compare
proceed total reconstruction amount entire decoder history autoencoder deep autoencoder provide justification autoencoder immediately encode accord metric yield compression making prediction also good sample free energy role unlike variational learning descent stack ever deep learn datum hide autoregressive precede precede autoregressive little computational generation mark contrast connection introduce undirected often
partition occurrence mode partition partition minimize expect define partition summing order assumption index permutation equal probability cluster pairwise probability equal q repeat compute co occurrence emphasis lie summary statistic partition slight convolution finding mode obtain among perform sum partition jensen programming
detect manner topological stable extremely regime extend stochastic version equation part approach computational topology prove world dynamical transition dynamical change stable cause stability regime particular realization variable current topological devise dynamical realization variable generate acquire realization challenge encounter world method detect critical homology prediction observational science dimensional slide window change system study autocorrelation measure window proximity aspect lack robustness analysis real phenomenon critical main development study persistent homology window subset distinct regime dynamical close yield topological appropriate paper break relevant dynamical persistent homology
analyze dynamic social physical biological phenomenon naturally represent effort dedicate network lead development many network research static snapshot phenomenon investigate aggregate view statistical long history among phenomena behavior researcher dynamic evolve dynamic represent combine model static evolution state
compose channel consider review link channel spatially fig leave mp spatial channel channel eeg event spread channel mp call select maximal mp select scalar product average atom contrary give channel mp coefficient phase channel phase mp select maximal channel deal base frame particular form shift also code kernel atom kernel kernel atom approximate adapt study describe dictionary eeg normalization shift multiscale dictionary shift shift factor dyadic drawback dictionary dictionary drive way update
laplacian relaxation general undirected cut write balance relaxation relation review far exist minimize relaxation case special new prox unify prox prox graph generally balance perspective relaxation power extension note characteristic extension positively
diffusion derive nice lead case extend de satisfying doubly differentiable continuously absolutely integrable integrable eq possible generalization recover integration actually nonlinear free energy recent exponent generalizing
learner forest assumption check application considerably projection presentation estimate variance functional average average want distribution place follow method independently become negligible transform bag equivalent approximation sum whenever behave say example fail match general interesting topic research auto auto uci discard entry divide test size random prediction bootstrap replicate forest moderate bag detailed description experiment forest train auto consumption bar figure give sample word forest new bar cross prediction equal random represent tell forest confident forest confident predict perfect bar surprising
propose sparsity regularizer modify version constraint pair encourage zero accurately shrink proximity
contribute spatially census count plot reveal interesting fortunately spatially disjoint smoothly across city count pattern become evident aggregate display count statistic accumulate particular versus rate census count across census namely low count uncertain methodology integer value induce time innovation factor specific cope adaptively adopt impose process rate flexible datum manner purely approach solely count within census account covariate population scheme simulate produce sample forecast model explanation outperform discover toward advantage ahead forecast
l joint sequence em follow statistic omit notational brevity differentiable differentiable two iteration need maximize pa eq norm every see decompose ga thresholding ga gx f f sub
per expressive stock consider trading day h indicator stock score present motivate extension inclusion incorporation type comparison grateful support de e reference cross procedure bm available forecasting tree
already notably develop recommendation objective policy focus limit go typical become rating policy pure exploitation exploration phase recommendation translate experience desirable property achievable let achievable opt aim address first upper low mild assumption arm extend achieve near regret prove formal type excellent formally relate arm netflix euclidean omit identity follow assume exist support meaning defer intuitively require spread ball draw gaussian
db db db db db analyse try analyse bic fit l cluster rgb good observation good cluster priori proportion proportion type type type dataset db dataset
switching permutation impose sampling scheme propose mle importance second importance evidence reduce demand produce estimate reduce mixture collection parameter mixture miss miss mixture observation associate simulation perform inference approximate via mcmc likelihood invariance identifiable phenomenon switching induce chain posterior explore sampler augmentation adapt miss ever symmetric mode chain explore multimodal output necessarily bias mcmc modify perspective switch inferential chain
problem predictive application attention trial extract demonstrate deconvolution match approach decode high pairwise brain sensor link drawback construction graph spurious association distinguished study pattern specific functional typically sparse subset
make nonempty whereby satisfy eq last dominate convergence theorem dominate since compact next measurable pz pz conjugate integral whereby ip dominate way away lastly choice like technical exercise structural topology convergent consequently return loss indicator occur verification first convex semi moreover mutually low continuous conjugate banach fr proof banach merely effective domain closed far occur closure adjust desire lie value discard always iff proceeding search search always involve finitely suffice form make never consider gradient sense fr firstly via singleton e equivalent copy indeed property adjoint course expression guarantee search wolfe manuscript may similarly expression minimum choice plug simplify quadratic lipschitz rest wolfe search direct wolfe side give derivation plug counterpart bind give thus discard main technique classifier sensitive refined contraction principle complexity define rademacher contraction per rademacher rademacher handle deviation uniformly loss albeit direction whereby establish sided deviation
newly highest ar extremely slightly provide result ever ar provide developed sampling scheme gamma random extremely ever project
analogue term e become popular denote parameter form density flexibility family modify refer parsimonious gaussian package paper mixture analogue paper material present methodology issue discuss illustrate real conclude suggestion mixture square mahalanobis asymmetric density addition
bar parameter predictive score million score test corpora eq alarm error ratio decision denominator small well decision report database unsupervised operating make also report supervise train high need calibration database
leverage unconditional mix belong atomic function valid program definition library probabilistic program sample probabilistic direct graphical since language recursion external library simple done adjust maximize divergence reward bind
cs component deterministic multiplicative correspond large coherence isometry constant successful omp match pursuit call robust identify index omp summarize stop standard deviation additive experiment datum sparse nonzero respect signal zero make signal uncertainty employ weight deviation advance get parameter straightforward iteration vary much l coherence
lie k recover explanation difficulty term come reconstruct right way balanced property overall requirement matrix shape square preserve let n j nothing select preserve structure tensor cp tucker low cp u r r balanced lead relaxation tucker recover sufficient tucker cr nuclear norm well improvement first obtain suboptimal nonconvex worth tensor length tensor capable generic measurement sum
zero subdifferential f therefore reformulate huber huber small subdifferential q take subdifferential nuclear q subdifferential separately structure furthermore calculate subdifferential rewrite subdifferential q find solve program calculation input residual error difference measure error encode position outlier
limit dependence dependence exist work joint sufficient statistic observation collection individual jointly sufficient individual sufficient condition nontrivial yy distribute preprocesse compression actual preserve scientific preserve preserving share although limit preprocesse mutually work independently obviously normality statistic must sufficient hold statistic latter long exponential family cause failure capture scientific control obtain necessary compression preprocesse substantially efficiency cover bayesian important note subtle general analyst inference belief analyst must analyst able analyst analysis affect belief require may carry analyst fortunately logical addressing analyst serious concern perspective theoretically trade coincide type dependency early possibility achieve collect distribute preprocessing redundancy dependency constrain work xx x unchanged researcher respective model seem situation force recover trick reflect yet researcher retain statistic part mean researcher sufficient mathematically correspond easy verify p xx technique apply work part demonstrate broad restrictive nature create redundancy observe actually need observe necessity observe copy retain use copy regardless dependency option open reduction retain preserve satisfie individual jointly yy x iy yy iy xx factorization safe preprocessing distribute sufficient preprocessing sufficient sufficient minimal properly however compression sufficient upper bind achieve compression scientific preprocesse scientific model scientific stochastic dependence among increase redundancy particularly I n observation preserve unknown retain sufficient statistic piece per force retain properly individual interested lead raw independent preprocessing correctly
detection mostly sample coverage index level region high knowledge first prediction visualization section band general apply efficiently construct simultaneous band good property cluster illustrate energy distribution construct future require may interested instead span visualization purpose prediction set xt tb finitely many exist prediction functional provable free general sequential observe random etc object test nan
imply weakly weakly proposition follow concentrated case combination worth thresholding fast tuning freedom limit scaling variance correspond concentrate absolutely estimator limit distribution always consistently tune severe bias dominant variability choice finite stochastic contain satisfie worth paper base thresholding interval illustration standard z unknown coverage estimator come coverage parameter section case negative finite interval aim prescribe coverage result every kp na na kp na coverage kp I worth coverage probability p c symmetric short though distribution error symmetric seem mirror least square estimator conservative immediately unique large picture arise stand n entail length confidence thresholding find construct simple
get last follow ratio result sense regular ignore approximate regular normalize consider exclude edge graphical model graphical tend laplace structure negligible laplace structure asymptotically tend depend additional result bind taylor series additional assess bayesian specify ar ar star connected circle generate dimension graphical appear graphical indicator assess median specificity replication graphical table tp tn fp denote
describe logistic memory ghz gb ram way approximate please extended cover assume work block size experiment regularize square separability degree nonzero element row initial appear processor size particular extreme sparse nonzero look improvement moderate processor finally see even processor lead speedup r r intermediate experiment special comment view ask ratio dense maximally lead instance speedup compare construct moderate wish nonsmooth technique descent study
obtain albeit novel distribute mechanism macro leverage cell act macro optimize suitable solution propose algorithm transmission delay benchmark aware resource management throughput thus lipschitz simplify signal interference without assume possible u xt xt xt henceforth satisfactory receive sc engineering university institute work dr degree communication engineering member centre wireless communication interest heterogeneous resource electrical institute work tv wireless communication university ph mobile pi fp currently pi european interest management heterogeneous heterogeneous small publish novel aware management enable user optimize neighbor formulate optimize delay heterogeneous property type performance gain term throughput heterogeneous wireless reinforcement demand speed boost improve generation wireless network
call extended extensively easily arise class huber huber obtain inside maximized take obtain maximize take huber take obtain take contribution huber take penalty panel integrable establish define necessarily calculus treat care important regard essential affine translate integrable measure function theorem function integrable lebesgue lebesgue density density nonlinear handle gauss model general kalman smoothing density suitable since decomposable analogous key proving set tucker kkt order necessary optimality solve newton relax decreased ip proceed iteration relax computational smooth smoothing come solve complexity key block smoothing ip practice motivate huber ml huber respective set measurement plot axis limit smoothed loss insensitive section kalman smoothing context application uniformly measurement normal denote fraction refer purpose simulate axis limit initial
topic primarily formally two group permutation rank correspond element ii th permutation order compare element list achieve fix contribution distance list define list give good grow decrease overlap significant metric insensitive absolute overlapping list compute distance overlap element rank ignore compare list attractive adjacent convert sort non metric penalty distance rank great penalty
pooling advantage overfitte pooling show huge success compare pooling stochastic seem improvement max gain slight speech recognition pooling offer pooling pooling explore compare prevent overfitte overlap overlap pooling speech thing overlap activation fair overlap match lot speech mechanism seem pooling pooling overlap speech frequency though cnns pooling frequency work deep time thing speech otherwise time another regularization table compare however large pooling time scheme pool overlap c pool max stochastic technique incorporate adapt cnns cnns cnns cnns bank coefficient popular technique reduce variability apply model
example belong remain elastic net perform pick correlate pairwise snr example simulate way main compare whether signal observation snr predictor diagonal give heat matrix example penalize example median mse false include lowest connect component significant indicate certain contain l rescale ol hybrid elastic elastic ridge rescale ols net lasso lasso ols hybrid rescale lasso hybrid naive elastic rescale
friend three city check location pair strictly detail analysis modal interact location different one experiment pair limited simulation baseline perform unlabele result baseline pair mainly dominate dominant distinguish comparing moreover active pair infer properly assign movement predict predict diffusion disease different task predict event th predict specific select option fairly large sum intensity across compare employ simply percentage perfect among pair prediction dataset poor able capture next poisson
hence analytical feature however detail case see generic transition transition read eq straightforward obvious zero simple employ identifiability model exactly non outline material eqs quantity still turn energy value depend hmm unobserved come jacobian one number effective
provide explanation method differ update dictionary instance lee al svd none previous work guarantee success recent et recovery rule overcomplete dictionary overcomplete involve dictionary element subset establish condition element et al alternate procedure distinction sample paper mutually incoherent local assume satisfie weak incoherence al alternating overall differ set algorithmic consideration parametric fit dictionary dictionary incoherent constrain produce incoherent problem sparse closely source reader extend survey procedure generally objective individually perhaps study problem carry statistic literature guarantee problem bi probability fairly
propose primary proposal primary search rare positive assign find low available becoming discover knowledge science amount potential verify often wherein promise throughput decide finding preliminary mind search rare vast proposal feature nucleotide snps association subtle bias affect association notably argue study similar sub population obviously split part answer concern arise area discussion reach prominent general new objective really replicate finding concrete easy rigorously versus meta analysis area
take expectation w iteration inherent randomness constant execution build scheduling program admit moreover limited unclear scheduling provide low programming yet offer scheduling partitioning simplify wide exist system support ml correctness medium medium medium medium convergent ml wide ml high paper building framework toward correctness program attain optimality space intermediate style resort iterative convergent computing database query correctly consistency operational objective tolerance absolutely however program argue grain tolerance consistency framework ml theoretic various operational explore early begin algorithm program stochastic descent determine variational graphical structure exist ml platform pattern schedule ml algorithm lie conditionally converge optimum yet statistically root parallelism ml tolerance strength parallelization non convergence step converge skewed core execute quickly ml fundamental environment framework perfect recovery support persistent memory
order speech discrimination range symmetric algebra value intensity invariant range spectral value range case family obtain
write independent eq inequality obtain desire ji closely proposition write symmetric chernoff get inequality proof lemma l controller conditioning psd term shown define martingale martingale since follow update way algorithm length logarithmic scale dimension present scheme achieve apart
fix concave compute closely probability simplex indicator minimum extreme achieve extreme index give exist concavity must perturbation optimality v well conclude v consider problem differentiable descent analogous illustrate iterate iterate finite algorithm converge normalize algorithm one algorithm strict compute eq construction hold eigenvector constant contradict terminate fix iteration fix locally eigenvector operator k admissible perturbation constant eq prove similarity laplacian indicate eigenfunction give indicator globally iteration
corpora advantage statistic issue approximation document acknowledgment thank discussion support fellowship fellowship berkeley part nsf award amazon services google blue data facebook intel microsoft yahoo upon work contract grant describe vb find posterior vb minimize kl approximate typically take find find descent form define wish describe describe topic document describe assignment document find kl divergence constant write hyperparameter follow equation coordinate ascent write functional derivative dimension say zero set q equivalently occurrence token write functional
potential parameter identifiable c derive value local sufficient estimate mrf observe index clique log clique interest quantity derivative likelihood feature expectation pseudo simple connectivity neighbor express th bit term expectation begin extremely evaluate gradient exponentially term intractable describe situation estimation datum compute however evaluate
q dx dx fx bx bx construction comprise segment roughly c aa dx fy fx dx fy w let lipschitz lipschitz consistent maximum possible call minimax uncertainty reality
algorithm u tu incoherent simplify assume permutation output type identity v show matrix span quite project span u first svd u tu claim w recall tr tr inequality triangular g imply put cr cr c rp et prove good estimate absolute normalization kl I side less w know n follow max desired bind kl analyze apply analysis identify condition sample tx
b b n cm expand n ba n b na bn b bn nb r proof control probability r moment moment valid gradient inequality zero rf n tail event regard r apply n express expectation split du strong add possibility strongly bound assumption namely theoretical namely low optimum term without require know local strong advance average method step convexity form moreover complicated notably possible size logistic globally linear show adapt convexity eigenvalue
minute run ghz gb ram rna calculate satisfied boundary secondary boundary case secondary type h scatter length secondary newton rna final newton rna boundary vertex big loop base consistent observe rna newton remain sequence inside newton fig demonstrate input dominant university mi rna
code require matlab termination output start convergence rely ridge estimate path comparison obtain inaccurate warm start obtain cost gain aforementioned experiment provide empirical advantage root parameter toeplitz p tuning denote th motivate choice recall define show need q independent variable ratio lemma control event deviation inequality show give correct involve f distribution choice error locate correction conduct fitting ol dimension boost accuracy
visual inspection confirm gain conclusion extract lead dct good yield avoid worth smoothed linear frequency ccc width width width width figure consider high compression intend limit recommend even reduce instance well distribution width width observe approach support support work dct linear compression region result approach variable accord rise natural imply follow take non rectangular allow couple conventional svm formulation condition
schedule encourage eq visible sigmoid noise force initialize layer single follow noise layer first stochastic backpropagation algorithm epoch choose denoise autoencoder regardless tie weight modify across visible minibatch size rate automatically fix respectively epoch decrease epoch persistent update gibbs step stage utilize already activation layer separate stage rbm inverse temperature intermediate base model least swap share unit however enhance use cast
similar saddle replica situation replica break permutation sum saddle sum summary replica equal fraction take problem replica summation convention make replica saddle determine minimize n give free need derive find minimum partition derive energy partition replica coupling occur variance obtain fact replica symmetric integration saddle example example weight distribution example numerator difficulty introduce simple replica play role numerator limit limit sequence free contour integral running axis simply tell contour cross contour take take xx z e xx first term cauchy reference recent advance challenge framework dynamical widely computational meaningful system review statistical science replica cavity physics way highly heterogeneous interact closely notion computer science distribute capable couple statistical physics computer science diversity context arise way within formalism replica review conceptual message model idea illustrate science provide function replica cavity high projection compress highly dynamical consist neuron interact scale scale order connectivity electrical neuron slow second minute beyond change induce experience stay learn extent powerful tool physic basic replica cavity interact network dynamical exist sake solve evolutionary fitness concept course biological biological priori complex nature turn idea distribute compute computer source network neuron distribute message whose single interact pass review pass related replica cavity statistical physics serve solve computational inference useful may way throughput consist classical machine large amount situation easily structure pattern throughput scenario neurons limit trial gene cell find statistically significant statistical system provide powerful understand formulate physics play role understand dimensional compressed tailor give summary fundamental cavity spin fix interested understanding activity find property term detailed realization connectivity matrix deterministic arise way heterogeneity understand replica cavity introduce provide perspective replica cavity many statistical physics joint equivalently computer involve message pass know propagation model may view pass essence version suitably message possibility understand significance exist derive hypothesis dynamic computational perspective idea well machine book length play role network learn play degree minimize learn structure depend realization training compute
radial topology failure second connect load fail failure tree line failure weather assume occur scale beyond time scale estimate rapidly force wind city city si diameter city speed wind hour wind pass city provide basis time scale weather induce c failure certain failure self fail recover secondary source build usually operate second failure due external manual field recovery environmental manual recovery minute hour day failure failure self network self failure scale dynamically cycle failure recovery scale failure occur show histogram duration operational bin hour duration hour failure last duration show failure occurrence failure region I b non obeys failure occur occurrence recovery occurrence identically distribute temporal stationarity stationary
denoise simulate compare devoted mm extraction paper incorporate discrete perform dedicated expectation em multi iterative reweighted square piecewise iterative connection I introduce transition I use expectation maximization algorithm organize piecewise model dynamic describe devote
quantile cell group agent quantile k k later dna individual dna quantify individual snps type linear response snps exposure predictor analysis conduct fold cross compete quantile implement fail modify due nature report select snps analyse performance compress quantile analysis response quantile explain evident high predict compressed bridge regression pls rr figure pi compete response high quantile rr evident coverage attribute pi suffer competitive response resp resp resp resp rr pi resp resp goal practical massive compression predictor computational gain pay price square prediction due part dense predictor
locally diagram margin error give least latter fraction margin error soft threshold balanced square assignment begin chapter basic terminology diagram classifier direct geometric approach margin cluster assignment efficient power site construct diagram count locally program also transfer fix obtain programming chapter point counting ls computation threshold program logarithmic theoretically error validate quality diagram discussion dimensional euclidean let site partition tuple cluster tuple shape throughout natural derive distinguish cluster clear simpler often cluster site associate geometric approach hyperplane interior important interior continuous hyperplane hyperplane separate separate strictly strictly separate hyperplane diagram natural diagram power diagram cell close call informally site formal definition diagram decomposition diagram provide helpful confirm hyperplane w j use weak
without since easy student chinese music per week evaluate sufficient stage every recommend song except familiar song rate scale subject minute ensure rating subject within study week subject spend hour total publication paper recommendation update immediately new rating interface evaluation evaluation regret simulation also popular rl standard th cn outperform cn since iteration cn outperform greedy cn bayes greedy cn share rating difference bayes cn exploration exploitation exploit improvement ucb cn greedy cn exploitation tradeoff effectiveness frequency interestingly stage ucb cn explore uncertainty cn exploit bayes ucb cn deviation song bayesian recommendation history uncertainty among iteration decrease expect uncertainty cn cn train bandit cn outperform factor rating recommendation addition ucb cn cn significantly suggest together well linearly repeat generation evaluate recommendation distribution algorithm
take coordinate gradient sample equal coordinate step least step compact algorithm manifold compact imply convergence riemannian convergence periodic lipschitz directional derivative derivative riemannian function g fu fu fu definition fu fu g fu fu corollary fu fu fu fu ti expectation side choice sum fu fu fu sum bind cost computation optimize rotation fast affect orthogonality show
triple well e entity france google york ny france paper extraction use operate embedding word knowledge empirically york article align provide datum predict task supervise detect associated label connect kb express employ mention approach known kb plausibility triple
trajectory intersect rule movement dynamic possess heavily site mean table order term I go near neighbor cycle period section begin end site lead visit however site visit without walk visit site would site visit visit enable sophisticated point walk visit one significant begin jump already visit time frame undesirable avoid visit site connect neighborhood probable depend configuration visit neighborhood scenario cycle nan walk approach new additionally pattern totally framework give quick overview level introduce algorithm environment mathematical notation discuss label train discrete entry feature descriptor goal commonly check prediction label item unbiased learn test review hybrid specifically phase formation formation extract use topological stage type utilize nn dense technique nn vertex vertex create predefine classify check within circular region center
gene expression compare encourage small computed negativity al program use programming individual spend memory algorithmic speed product evaluate time computer intel program processor develop run analysis world wide genomic population genome diversity project centre du extract analysis individual genome diversity panel analyze li snp filter remove snps include original file addition project project individual population whole genome project include
positivity condition give new early tree probability reformulate direction discover endowed markov believe may great impact field dependence structure complex expert insight use empirical
blockmodel formation community generally refer interest belief music friend actor community actor yet easily speed scalability million divide probabilistic contrast advantageous prediction link hide recommender system recommend product user rating hide community learn learn statistical allow approach tend base approach achieve mixed introduce scalable subgraph tensor decomposition moment work model membership hide markov maximization em variational practical carefully study scale method primarily document corpus dataset subgraph star observe network triplet decomposition tensor learn tensor proceed two whiten eigen first involve simple algebraic operation multiplication svd carry eigen decomposition iterative finally parameter operation processor multiplication since furthermore employ approximately svd computed dimensionality thin parallel qr community run parallel since million scalability two implementation exploit parallelism architecture cpu implementation gpu suffice involve implicit form large real statistical approach result ground truth available gpu parallel graphic contain thousand core storage cpu gpu minimize cpu gpu naive tensor huge poor running cost gpu never tensor carry
include useful bethe weakly sure restriction parameter potentially infinite correspond exactly reasonable entry drive cut strong belief careful scheduling link max technique propagation use approximate marginal addition may extend allow marginal submodular mrfs map mrfs theorem example inference mrfs np posteriori configuration mrfs efficiently cut inference class formulation bethe free location technique bethe apply pairwise discretized polynomial scheme optimization provide
one gradient matrix subgradient term take follow line present recovery problem l sparsity row solution relax exhibit behavior convex minimizing follow matlab intel core ghz cpu gb memory produce mix entry probability success equal matrix desire randomly I zero satisfie rip rip constant noise amount simply illustrate row sparsity describe problem suggest sparsity recovery least examine effect pattern setup value datum intensity section use recover evaluation counting number row pattern repeat time record deviation row deviation correspond correspond vertical line
association optimally distribute improve procedure dramatically advantage optimum prove mathematically improve monotonically evolution operational scalable linearly distribute perform effectively decrease wireless area reinforcement policy reinforcement division generation mobile service inter interference power control fractional enhanced code multiple access mobile base physical resource next generation self rd user resource scheduling plus division access max throughput fair min fair armed identically input physical interference file transfer protocol optimization file generation corollary technology se france du en
hybrid decrease overall computational poor pre condition algorithm capture trace use chain burn trace behave differently hybrid explore method parameter hybrid estimate estimate select discuss factor lag impact return discard burn parameter vary keep everything else robust choice certain discussion enough recommendation common newton add iii take posterior make proposal gaussian random poor inefficient number incorporate likelihood indeed lag particle smoother linearly ii stationary phase
exception interaction obvious black intersection circle classification show word combination intersection stem datum able quick topic seek stem simultaneous presence within evaluate document document train document stem document stem predictor topic contain document simplify predictor whether tf positive dataset let belong topic goal maintain away treat document belong chosen solution remove specific implementation consideration tree create hash min wise search cutoff pattern random forest fit classification tree randomness look restrict computational pure r speedup code currently work
develop cluster time write momentum use diverse disease recommender identification justification guarantee restrictive rarely meet although cluster technique come establish rigorous subspace literature computationally tractable whether less severe clustering rely sparsity sense please also long ssc since mainly toward noiseless situation exactly dimensional circumstance restrictive circumstance support free tractable cluster natural statistical represent lie subspace separate subspace performance explain interpretable subspace level term indicate sense near formulation point lie near subspace completely cardinality may partition subspace group belong signal snr superposition noiseless perturbation remove ambiguity noise assumption expression restrictive since say noise may move sphere arguably simple model good investigation noiseless introduce assume think component dimensional subspace fundamentally subspace subspace affinity subspace
mathematically speak j possess assume discrepancy reach parameter else interior estimate suppose theorem hold ex let ex implicit unique uniqueness nuisance vice versa joint partial consequently step repeat consecutive algorithm bring close modify require input htb cumulative triangle maximum n mi triangle year distribute variance structure marginal bivariate copula copula
inferior rely sub optimal gain risk forward go information hence case get lack information make gain forward insufficient besides strategy worse primarily focus criterion security disadvantage previously notably optimize utility take current future belief behave gain forward worth expensive hour exchange speed hour offline planning reasonable consider meet interaction player player shot play repeatedly move past mutually opponent conversely get opponent experiment opponent assume make bound opponent model
pmf condition multinomial pmf asymptotically pmf cumulative operation k proceed full multivariate pdfs require discriminant gaussian follow q refer bayesian approach store since dependent class contrast bayesian
gm pdf distinction decompose tp kk equivalently mixture hierarchical sample statistically capture bernoulli impulse noise equivalently gm state state transition pmf steady state illustrative generate bernoulli gm model trivial emission power db db background occur emission generate whereas exhibit statistic via practice slowly transmission efficient message pass bit decode see jointly code bit finite alphabet symbol channel exploit statistical pilot finite codebook posteriori decode know minimize symbol frame total write bit alphabet symbol code impulse bit factorization symbol confusion evaluation computationally due integral belief propagation bp alternative direct computation posterior loop exact forward generally np hard posterior inexact problem decode compressed sense large
ad hoc surveillance game table game equilibria hill reach nash equilibrium reward ccc u replication episode game simple equilibria pure equilibrium either hill converge play constrain sensor sense event sense mode schedule event cast area event utility sensor choose action range event sensor event mode sense formally express utility event q utility sum receive event communication event sense utility mode sensor sensor interval day energy mode interval unit away unit away moreover distribute
design benefit set involve rank structured develop minimax estimator noise lastly matrix reliably low signal break question answer would limit estimation question empirically trial plot realize compare oracle detector side compute use uninformative estimate realize comparison average proportion predict nature shrinkage portion shrinkage operator snr sub
resemble linear weight promising need space relate successfully assumption generalization use similar result reader familiar pac bayesian introduce relevant pac predictor set hypothesis predictor distribution denote expect associate e possible least leibler typical trade regularize uniform hold regardless choose refer distribution choose predictor choice loss regardless right side choice prefer prior learn
local consider likewise pdf pt sensor factor recursively update bayes pt pd dx robot copy sensor whole factorize fusion posterior cx dx fusion unnormalize joint fusion pdfs simple shown give gm move sensor time step decide perform robot show factorize need copy r robot manner hybrid calculation
purpose use f minimizer approximate c q recall last corollary regime translate result statement corollary begin implication independent use lemma triangle third c concluding assume invoke choose ensure combine obtain contradiction dependence predict divide region analysis formally distinct defer define related key importance important property serve reference nonempty compact contain origin convex attain unique point increase eq interpret require proved proof statement define three lemma subscript minimize formally prove explicitly formally definition make strictly nm formally n unique repeatedly make lemma function mm since strictly decrease ready define three distinct problem lasso definition f statement appear subdifferential nonempty compare proposition vi max recall minimizer approximate lasso denote corollary argue small regime translate suffice fix determine implication event satisfy lemma find satisfy subdifferential finally choose deterministic formula provide derivative consequently convex convex show regime observe reduce noiseless compressed provide case iff get dominant surprising
average dot product vector factor kernel c account probabilistic similarity scale number computationally demand gmm cluster still appealing view different capability extraction compare benchmark publicly repository california uci application process audio processing music impractical situation benefit benchmark set take uci repository orient important namely pattern kullback leibler test alternatively keep select classification information accuracy display l winner take compare linear method cca maximum next result extract cca perform feature poor cca covariance ill cca loading extract pca regularize cca completeness present result pls train extract discriminative aware next version half rbf fold rbf mapping even overfitte demonstrate superior discriminative capability completeness
boost correctness ml tailor parallel mf without modification development parallel mf yield performance future work accelerate principle analyze parallelism way interference minimize worker appendix program eq standardize eq equivalent optimize update small sample approximately low decrease update super iteration index coefficient update small j maximize disease explore aware parallelism build call aware parallelism update step
consist vertex already take yet start step reduce two single sum contain thus triple piece induction property also head remove stem able flip h easy prove repeat ac px inspection conversely c c px h x px hx kx equivalent order property simplex exponential prove probability theorem constitute induce smooth determine smooth conditional arise follow alternative amp graph sufficient however immediately necessary brevity abuse appropriate involve divide infinitely let value map
research de le france university technology electrical final objective incorporate process switch link
ica model independent independence statement result ica statement appear handle sec ica whose order bound estimate conditioning briefly unknown unit column da k adaptation median ica provable mild individually suppose high recover component high independent regard certain incoherence sense ica turn condition check satisfie randomly choose choose start power simple polynomial ica aspect use detect cause problem non gaussian great specific direct ica signal recovery application ica algorithmic primitive learn body come uniformly determine ica al solve regime previously ica noisy mixture unknown arbitrary reweighted give fouri spherical recover sample assume view fold spectra fourier showing eigenvalue gap significant sample remain integer simplicity instead slight loss oppose introduce along error difficulty tx unlike generate characteristic even branch variable characteristic jx resp ica distribution distance gaussians convenient measure moment exist characteristic admit vector index order thus value potentially detail review tensor index size invariant permutation tensor symmetric tensor essential result ica suffice tensor generalize tensor degree homogeneous tu I every form outer product q j always exist symmetric
ica reduction poisson recover remain recover relationship capture follow suppose ica component expand hold probability reduction let model keyword connection ica tensor emphasize specific identical smoothed exist elimination http www cs edu ica projection bad wang ica reduction content ica argue mean regime bs dimension requirement think ica clean reduction whereas really check spherical volume ml application gaussian http www smoothed justification smoothed discussion choice theorem move thm microsoft university computer science engineering microsoft university computer science science engineering learnable high precisely prove covariance polynomial fix degree polynomially learnable long certain degeneracy condition generic mixture projection barrier rely technique transform gaussians product component ica mixture combine hardness mixture ica establish exponential ica literature first phenomenon dimensionality aspect
different get index respect note good respect formulation spirit also good little reasonable symmetric coordinate secondly scalar isometry family row one element orthogonal isometry scale translation isometry integrate isometry first index nevertheless index admissible isometry sake restrict generic distinct group q isometry
pixel black none ht mark dot dot dot mnist image cifar color image object total experiment cifar baseline pixel achieve hamming trees haar figure run pick channel depict pixel hamming blue curve achieve none close
output stage compose multiplication learn follow point linearity relu output represent present convnet image aside tune hyper size rate momentum accelerate dropout present false positive convnet pixel area contextual position training limit therefore high spatial pose prior strong convnet expect truth within post pose inter human body throughout could right mirror convnet generate unary dense pixel respectively
asymmetric arise conventional modeling principal bring different rate value half yield inaccurate markov monte comparison via estimate classify test straightforwardly pz reconstruct mh unknown accord parameter estimate simply classify parameter ignore approach separate less jointly uncertainty classify approach reconstruct pz pz estimate unknown modify acceptance infer joint obtain uncertainty
perceptron achieve score truncate gradient choice especially sparsity ability course training regularization sparsity strong regularizer simpler model nonzero weight may online simple nonzero small many model reach illustrate trade sparsity adjust regularization hinge show overfitte extent achieve feature training observed error linearly secondly predict
conduct analysis diagnosis ad longitudinal observational mild cognitive ad ad care association measure subject ordinal state increase consist subject ad patient snps select brain brain surface volume software accuracy subject ad randomly time run fold latent low value large confirm demonstrate accuracy standard ordinal bad cca ht examine association discover prediction disease population ad rank measurement predictive particularly middle find power automatically region involve biological diagnosis analysis extension association study e separately task present association disease diagnosis
exponential force optimization generalize omp nk singular eq pe pz pz result thm pt distribute corruption allow corrupted include factor impose unbounded cardinality coefficient illustrate corruption obtain support matching algorithm omp notation moreover exist omp fail even might total square corruption bound bilinear tractable provable
plausible obtain identify dimension indicate specifically orthonormal span generate response newly reveal make convergence certain incoherence choice incremental compute add
level limit value eq give equality find reduce invert two consequently pearson closed expression interval package instance interval preferable formula numerical evaluation former easy next expansion approximation pearson upper bind accurate place upper pearson two side pearson invert pearson limit symmetry equivalent pearson interval binomial terminology interval interval admit form expression intensive reason pearson simply statistical package intensive pearson remain inversion tail two sided room least confidence exact short inverting interval still short
digital rely coefficient transfer psd lack author fractional calculus expansion describe psd easily find represent calculus density issue paper extensively latter differential equation base assume deal spectral psd fractional spectral show represent involve colored thought remarkable proper fractional calculus color preliminary fractional operator fouri fourier write let fractional integral derivative eqs verify reader mind
generate bayesian dataset parameter reject use distribution suggest simulate reject metric distance give rate comprise summary output abc abc careful large try abc posterior angle atom choice summary posterior full atom frequentist describe problem dynamical explicit unit extend investigation complementary correspond fisher atom infer mle first full monitoring description outline compute apply investigate asymptotic behaviour although record cavity markovian asymptotic normality scale simplicity call fisher asymptotic information associate count fisher quantum cavity benchmark estimator atom record emission process markovian observation atom certain functional unobserve cavity analyse two detection sub atom property characteristic locally properly count cavity subtracting limit fix limit local gaussian fisher
pr search feature subgraph discrimination bad feature replace prune sub root follow recursion root rl subgraph exhaustive enumeration derive subgraph pruning pruning prove anti prune q median measure simply perform subgraph pruning utilize branch pruning maintain subgraph bind subgraph update subtree root bound order uncertain test real image summarize performance approach uncertain test fmri brain dataset consist disease cognitive record fmri treat automated volume different brain region toolbox template spatially mm kernel data trend band hz spurious correction head
grateful song constructive suggestion support w fellowship nsf award award support song health rgb rgb prove draw ball separation hold nontrivial separation enough ball point thresholding fail exhibit evidence recovery form isotropic find point distinguish average np obvious recover local optima lp square dissimilarity use algorithm converge include partitioning difference face capture expression normal database expression background
distribution distribution feed forward encode additional video effective combine decay dropout augmentation etc fitting deep seem benefit layer vision layer fitting produce superior conventional pooling stochastic activation within pool copy local explicit elastic input image excellent mnist augmentation global transformation pool multi design architecture first novel pooling convolutional compose alternate convolution pooling
I lastly find minimize consider interesting place fairly mild restriction smooth description manifold value define function allow wide topology partition region u way perhaps close bound contain flow vector whose position see part varied
focus specifically recently beyond quantization calculus method mesh scheme stochastic optimal stop conditional zero set entirely drive high fidelity rather map formulation contour exploit design adaptively focus effort classify sign extract magnitude increase model overhead design ei grid approximate sequential localize moreover uncertainty universe tree meet recursive contour couple usual dp introduce extra also criterion paper organize rigorously framework review implementation section new methodology present treat option price brownian volatility benchmark discuss improvement stop basis arise discretized consider time discretization horizon use finite version stop horizon stop consist maximize x use value optimal go immediate stop level contour region henceforth induction classifier correspond payoff path algorithm tn contour define noise zero provide functional view regression thus sample iterate add approximate
chain apart entropy carry vice versa presence need account desire appropriate coincide mi frequency occurrence inherently suffer prove bias consequently mi two mi mi jointly bias two indicate respective mi approximate bias estimate symbol express entropy substituting expression entropy bias entropy derive see decrease order property chain determined precede thus lag time apart must order increase occur require far
time generative model condition df td old cover mathematic underlie development e subsampling number observe let measure asymptotic dirichlet serve foundation model gmm time step categorical class label
access partition take denote receive complexity increase exponentially importantly problem hard learner know know markovian optimization note illustrate hard put result subsection measure learner incur dynamic learner regret learner take rate total expect sublinear propose context learner call consider long horizon fine result entire hypercube secondly number space arrival partitioning algorithm classify classification determine large improve increase analyze form hypercube dimension hypercube consist learner p tn p tn k l kt kx r kt k htb explore kt keep time three phase train exploration learner update reward exploitation learner phase give arrival learner let learner learner forming need make sure need learner keep one datum learner
point nc w c nc total nc nc diameter partitioning describe application computer unit sphere region spherical diameter satisfy lc lc r sphere connectivity show point region spherical minor showing upper around contain neighboring illustration rectangle rectangle rectangle rectangle rectangle circle leave pt gray point ik cf paragraph w distance region least hold remains bound set statement fact nc note lc tail get depend accomplish bind p lc bind argument hold analogously bounding reduce theorem relevant establish provide establish accomplish upper vector x yield n hence bound q dependent lead upper argument set satisfied false connection impose step turn imply conclude remain step eq next use rhs follow eq step bind yield b u z note fy eq obtain take step notational convenience establish condition I accomplish follow lemma matrix approximately
age weight study list ground lie lie ground involve body recognize differ duration intensity transition subject ask activity duration vary sensor nine measure activity recognize human raw acceleration see time change activity time formulate multidimensional regime acceleration regime dedicate hide markov regression approach activity formulate segmentation multidimensional acceleration multidimensional regime regime segment activity present piecewise finance
maximize assign weight pair objective strategy min analysis pair distance coefficient minimize meanwhile vector maximize coefficient distance maximize objective impose minimization maximization
obtain check stage signal train dnn separation train aid source single source separation source spectra fit train mixed weighted spectra work model nonnegative like hmm limitation signal powerful properly another call speech bin classify belong field deep training network dnn input spectra frequency bins source soft use
analogue skewness common analogous development skew extensive eight parsimonious even parsimonious parsimonious bring advantage high factor fail big issue skew mind future coarse family develop skew acknowledgement support innovation grant natural engineering grateful provide access package proposition
indicate customer restaurant customer customer hyper crp customer select table value log kn explain derive crp key formulate crp formulation mechanic cost customer th restaurant denote customer restaurant customer th table restaurant customer customer denote customer restaurant quantum also sa eq count sharing show provide explanation inverse temperature customer share customer customer
research defer throughout independent standard expectation random entry analysis hereafter sphere evaluate generic norm ultimately norm corruption subdifferential vector corrupt notion convex cone cone feasible nearly coincide geometry closure set direction convexity cone tangent cone hull subdifferential entity complexity structure adopt set root root cone determine signal measurement gaussian width arise establish corrupt sense set interpretable describe back relate distance subdifferential normal bound tt typically gaussian weak assumption literature complementary relate together distance offer example show eq scaling unlike structured dense corruption ultimately distance set tight albeit closed bind approximate optimal square form dominate sx come square induced cone irrespective meanwhile norm achieve corruption analysis signal low highlight
instance always assume partial realization specify c w spatially spatially intercept spatial process capture micro collection location identically distribute follow local adjustment capture unobserved assume mean c site specify c include quantify offer mat ern spatial smoothness positive process usual hierarchical special mat ern precede conduct hence must specify metropolis variance metropolis propose first stage sample recover predictive fashion practice burn collect conduct illustrate intercept generate unit purpose equal surface spatial highlight manual specification symbolic statement require specify collect value adaptive option equal ig practice spatial assign uniform cover extent interval effective pass summarize package distribution credible ci sigma priors beta flat sigma ig
consider tree identify minimum theoretical result could framework coincide result label document hypothesis automatic diagnosis represent medical exact diagnosis rather diagnosis important e identify define subset identify class study tackle cast set object assign test object incur output number assume cost class object discrete object class correspond motivate choose sake uniformity work leaf associate belong leaf every root child decision tree set root identify path leaf
setting two block block block sparsity create block minimum diagonal matrix accord inverse sized block level generate standardized tuning algorithm present simulation present supplementary htp plot inverse average zero edge correctly total zero c small value component circle consistently identify solid circle large identify circle triangle edge mse compare graphical region interest also high fraction correctly propose triangle solid component choice identify signal ratio low cluster though due tendency produce
fit happen prevent stop denote stop fit omp implementation efficiency w atom product convenience partial correspondingly refer firstly fully exploit store atom atom retrieve c secondly cache memory away address explicitly store accordingly master warm master x I efficiency sr exist norm pg fista take per suppose contrary stop iteration large computational calculation reduce transform basis wavelet computational reduce transform short greatly actually x store calculate computing signal add time batch omp specifically take
matrix haar miss tb approach complexity observation author solver alternate multiplier algorithm range report representative measure solution prediction define bound tucker tr without ignore negligible equivalent tr tucker whereas run take sum rank singleton decomposition repeat figure result leave tr complexity middle show complexity
verify association important recently distance coefficient identify association variable set equivalent capable detect association classical correlation provide pair database maximal show pearson estimate accuracy coefficient maximal pearson coefficient zero wide wind generation electrical power database specifically correlation also another primary aim superior alternative discover association correlation database information describe distance maximal coefficient apply datum call conditional dependence among pearson since pearson measure maximal recent appear several approach measure include
semidefinite complement upper leave positive semidefinite ii ccc cc cc c limiting likelihood derive property test classical square limit distribution fairly power include distributional simulation powerful detect distributional classical test also find anonymous suggest semi analyze assume hazard clearly strong population comparison density limitation approach multiple easily include power partial ratio proportional test power superior survival size domain applicability multiple motivate overcome limit statistic hypothesis pool power test assess via give combination function l f l satisfied discussion numerical carry define base find profile write maximize multiplier
adequate mixture extensive criterion gamma part addition credible part section devoted model model pareto function cdf fx dp ig ib cdf
formula inversion initialized commonly initialization many package criterion addition number information use select number argument support bic select number analysis advantage drawback give analysis report herein acceleration acceleration asymptotic iteration l converge value model analogue
sub logistic flexible shaped present change summarize several regime maximize dedicated maximization comparison approach linear discriminant discriminant model curve mixture specifically use logistic functional derive result discrimination functional background functional functional discriminant functional analysis procedure dedicate maximization let label curve consist discriminant extend discriminant conditional parametric dimensional multidimensional functional discriminant principle assign posteriori
show period around define common propose task benchmark involve complex several direction extend practical scalability perform scheme partially information adversary review present future direction agent henceforth begin attempt gain subproblem domain main simplification player focus team pass nevertheless successful whenever face trial player within place position constraint gain go left bottom winner go field winner intercept ball winner episode player
train create algorithm nonlinearity subsequent level error understand art permutation method non optimization setting computationally robust suited test deep maxout multiclass task cover type art consequently extraction well feature approximate primal come last utilizing follow protocol hold portion summarize I self stand fouri randomized logistic relatively noted result instead apply induced computation g matlab second sequentially generalize second record task
exhibit convergence occur convex simple base must ask question regularize correspond argue converge minimizer tend word small close henceforth simplify say gradually tend speed complicated consider infinitely
therefore end prune high layer connect column zero column find sign first pick say call deep machine reversible output invert learn deep main rbms rbm encoding decode auto encoder beneficial behind able give random basically edge identical simple sign instead sigmoid compressed representation matrix image deep equivalent like simple different network hard translate encode view special writing encoding motivate network invert one
cancer selection anneal discrete introduce experimental gene five public promising offer framework relevance optimize analyze database size grow extensive use resample ever great de california new entropy present base relevance microarray gene expression context current relevance name implement simulate design experimental subset form meaningful microarray joint diagnosis tumor different tumor patient situation primarily diagnosis cancer expression systematic develop cancer dna microarray possibility diagnosis mining entail heavy consumption typically gene even thousand classify high
processing projection ht robust cc patch encode ordinary ol problem ols formulation extend include regularization term overfitte formulation analytic thank semi positivity solve decrease
database estimate squared quantify enable range close table show three cost high matlab implementation interestingly consider training identical physical recover hyperspectral synthetic proportion ice size water sir sir select database real spectra south ground physical evaluation well perform mle detail material appear paper partially augmentation starting regressor hybrid maximization procedure view span particularly map contain advantage simulate outperform road towards understand wide application complexity phenomenon merely slack latent probabilistic issue criterion bayesian impose costly far include investigation way high take investigate student finally assess behavior presence design regression test study situation high fully current art problematic propose difficulty role incorporate capture low role parameter high regression inverse derive characterize interest mixture tractable allow latent particularly regression regression framework formulation augmentation devise
functional shorthand time difference value hence modify strategy detect vertex change fractional gradient preserve result cause determine dt rand dt integer give fractional part result descent integer term energy summary set denote performance interface analogous classification circle generate two circle radius bottom half datum half circle embed component set problem normalize laplacian use near neighbor fidelity construct
correction alpha fisher statistic contrary use lead observed decrease actually test prevent overfitte statistic even know exact use adopt collection contrary permutation desire nominal statistic model collection splitting result collection c calibration calibration confidence conclusion behave split fisher combine calibration conservative desire nominal level magnitude upon calibration line permutation plain blue suggested red triangle stand deterministic empty point green circle multi splitting ht ht power vary magnitude common non calibration dot line plain blue represent test red triangle stand deterministic collection draw plain green plain splitting investigate power collection decay absence common reach magnitude compare base reach magnitude statistic prove efficient well subset variable activate suffice nan perform well large subset numerous subset collection limitation stem half limitation notice challenge ex ex ex power percentage magnitude parameter pattern uncorrelated result suggest combine blue triangle stand collection draw empty plain circle plain splitting splitting figure decay design correlate conclusion htbp setting decay ex correlate design observation collection square suggest triangle stand deterministic
enjoy monotonicity I coarse density transformation also process fisher inequality quadratic increment exist give j order score information f
risk mean work maximize ratio criterion understand utility exponent rl go rl obtain prominent area find space require sort art td estimating denote bellman td td jointly variant square td novel enforce approximate evaluation highlight usefulness importance understand organize rl setup fundamental
resp kx hilbert trace trace convergent schmidt family task algorithm include allow algorithm obtain learn property nonlinear reproduce hilbert task problem output like general schmidt precise schmidt functional regression design learn seminal bound literature algorithm binary function stem task output develop function become
enable derive turning transform prior hyperparameter guarantee positivity shape require handle end subsection positive provide one proceed issue shape k kk rise concern application prefer alternative gamma assume gamma parameter scale coherent prior eq subsection expression regard reveal evident monotonic dependency special wise equal increase concerned subsection inverse prior gamma work establish coherent
mean note goodness goodness hand perform see seem unlikely current goodness seem inherently suit accept goodness would exist modify modify accommodate goodness lem lem lem goodness distribution
coordinate ascent preserve remove affect claim pick uniform repeatedly choose via choose r lp duality vector correspond apply ascent vector optimal consistency natural criterion fig specifically I factor child except restriction keep beginning update follow proof r update increase devoted proof suppose operation step vector directly xx contradiction xx
sufficiently net ready introduce adaptive comparison noisy bandwidth choose rule fs l w adaptive coincide non adaptive pay pay see low pay estimation norm cluster choice concern plug conjecture generally highlight thank conjecture stochastic margin practice recommend could propagation standard principle major modification traditionally minimizer nuisance context empirical risk nuisance bandwidth risk minimizer excess via comparison divergence nuisance estimation introduce variable probability choose parameter bandwidth standard localization example bandwidth deconvolution estimator consider estimator
series come filter source recent jx central come superposition stationary signal identify stationary finding projection determine require uniquely connection geometry arise th first homogeneous fourier take value define property translate infinite homogeneous
substitute eqs simplify vanish derivative invert minimizer prove distributional coincide claim scale differentiable obtain limit condition proximal respect combine claim compare asymptotically discuss base covariance summarize table table table configuration contain mean deviation power across power avg std c ridge base na ridge na ridge na na na na na na asymptotic bind na base na na ridge base theorem setup significance design cccc type I avg std std ridge na na ridge c na ridge ridge na na regression na c c asymptotic na c ridge na ridge asymptotic cf setup significance gaussian cccc avg avg mean std std c na ridge ridge na ridge na regression na base na cf significance avg power avg std std lower na c c na c base na na theorem setup level gaussian need distributional definition establish hypothesis procedure since paper distribution somewhat support physics coincide motivate assumption distributional limit motivate distribution converge nan asymptotically stochastically sided construction argument recall lasso due simplicity normalize procedure entry n n sub gaussian step
semi eigenvector locally manner seed manner local structure b pdf locality semi dot blue curve illustrate highlight semi illustrate previous consider show seed node eigenvector black dot different locality parameter general monotonic curve monotonic semi eigenvector close eigenvalue supervise figure locality locality eigenvector derivation normalize solution convenient projection operator nan successive working laplacian normalize laplacian eigenvalue search employ monotonic seed kkt fail converge monotonic ff yy term psd increase state fail search compute transform great system new expanding algebraic ff td gd ff eigenvalue emphasize use projection operator explicitly present suited first construct efficient
require related selection propose give coefficient investigate hypothesis achieve level constant use design recent paper van analysis general design require mention alternative establishing assume suitable however also mutual incoherence much weak ideal scaling sign instance propose unfortunately approach low notation submatrix submatrix form likewise restriction vector denote operator nonzero throughout row otherwise section exist lasso contain motivated lasso sign selector correctly recover sign support recall x n eigenvalue empirical define defer happen
inversion depend integral depend dynamic characteristic property motivate bayesian identification mse associate theoretical low systematically analyse minimize optimal minimize refer minimum estimate derivation method error characterize estimate require clear decomposition let method surely
always cv implicit event eq stein w imply inequality third tight derive hypothesis value cross exist show knowledge investigate consistency cross significantly reduce bias bias cross wrong lead inaccurate variable arise stochastic evaluation formally independent mean sample clear quality sometimes unfortunately discuss exist construct average I sample average make policy learn
need average sensor uniform location distribution estimating relate chain integration hard directly possible deal literature te importance consider selection competitive detailed discussion view within approach classical case tool regularization hilbert kernel framework addition separate estimation thus certain setting operator apply analysis outline idea start importance note hand use idea e k tx tx dx fy another approximate integral assume know address reproduce kernel hilbert machine importantly integral approximate sum useful evaluation point hilbert formulation optimization combination point function evaluation perhaps interesting fact use norm linear coincide rkhs yield important insensitive approximate advantage result
easy infeasible together optimal feasibility trivial whenever rewrite thus feasibility solution feasibility give fy therefore clearly compact constraint set lagrangian multiplier eq e g convexity I strong claim strong q imply write q q complete j contradiction empty kkt write cone similar kkt set complementary see contradict nonempty feasibility e belong complement
efficiency family removal pass number propose node social influential node regardless unknown node variable matrix correction real delay mm field edge specifie call greedy unknown take latent variable decomposition algorithm iterate alternate correction though non convex carry thank far perform experiment delay capacity family balance work author tree variable one induce undirected set z reveal edge sample estimate gaussian distribution leibl divergence model whose
begin give alternate aside result strong demonstrate state bi bi bi un un bi addition vector undirecte un lemma simple gaussian markov hence ab bi un union undirecte un markov respect turn give un bi un follow bi use concern result state vector close intersection union tree global undirecte state un un trees decomposable assumption vector composition decomposable disjoint intersection ab bi un bi un bi complete bi reverse global recover gaussian decomposable close dual decomposable without generality decomposable close decomposable dual decomposable duality implication
theory state rate must improve extension imply argue want highlight conceptual projection counter intuitive light map guess na sample na guess projection coordinate sample span support easy reconstruct point require extreme need concept coordinate system intrinsic behavior lt european european framework grant agreement
equality permutation edge entail adjacency consequence triple case permutation give rise b ss bi j consistent dag sp first path assumption partition ci relations corresponding ci relation see ci omit sp ci hence go conditioning conditioning path consequence ci relations ss sg sp permutation dag paragraph ci two class ci relations ci correspond paragraph relation I sp every activation adjacency one ci gets activate ci adjacency part two path triangle sp satisfy minimal separate ci ci path sp maximal assume separate dag contain node separation relation x x hold acknowledgement thank valuable discussion comment gr national grant dms statistical sciences institute theorem exist
product contamination monitoring storage technology control identification air water recognition produce sensor pair order make portion available patch repeat confident label computational classification novel early series e system relevant importance time classifier reject option propose computational start hardware comparison contain conclusion recent wang convert directly send perform combination function distribute open compatible extension fu outer I inner amount parallel channel pca apply feature extraction reduction set class five green china experiment svms recognition
either u k effect reveal large enough optimal sub vector kx ni identically structure selection nonparametric setup datum reduction inaccurate add turning specify shape different candidate combination convex hull k define drive incorporate bring mkl sparse hull sum p lemma replace mkl resemble component via possibly mkl wireless communication section propagation effect cognitive sensing paradigm section basis depend scope space associate bilinear basis expansion constitute first sparse degree assumption prescribed model inaccurate knowledge also parametric address input overcomplete basis regressor certainly leverage accommodate practical pursuit however capability fit generally
several movie movie get guarantee naive weight input space maximum give formal hypergraph represent partition e cardinality e vertex movie would movie triple movie person give movie let iv may notation gender movie assign triple contain induce e I jj ie I jj thus example
measure accuracy consider operator consideration account task implicit error aim design contaminated initialization run lead figure panel show run performance see minimize noisy lebesgue convolution reason inverse empirical contaminate observable distortion study datum unfortunately seem codebook interpret cluster distortion deal coincide rise situation panel spherical right show grey
formulae domain range analog circuit also refine semantic paper semantic infer quantitative conjunction boolean interval atomic know secondary signal b semantic quantitative return quantify stress definition behaviour interest choice secondary formulae measure interpret behaviour precisely see consideration satisfying trajectory secondary space behaviour relate apply robustness definition predicate semantic extend measure behind distribution apply furthermore trajectory satisfy measurable refer trajectory trajectory satisfy
resp iterate norm sublinear hold realization bound consider expect sublinear ok formula approximation iteration pre square iteration sampling moreover hold tx b concavity tx b k since proposition together follow show resp make decrease suppose set sublinear decrease problem mention assumption satisfy rr directly
policy pair probability uniquely prior easy rp similar intuitively consider step equation start take logarithm solve see classifier domain game opponent obtain opponent set trajectory algorithm run amount trajectory examine amount mention uniform lead bfgs posteriori preliminary alternative resort local lead possibly experiment
auxiliary bound difference condition enhance apply yield provide next necessarily allow q high constant still bind deviation observe e appear chernoff bind g upper location n put together n consider introduce n imply constant high imply q n n high show proof eq indicate give appendix arise chen computer stanford ph department electrical stanford research interest include compressed science statistic chi ph electrical engineering electrical china since department electrical engineering processing award international conference receive award associate award hold position stanford university research interest dimensional datum signal application communication imaging bioinformatics algorithm chi explore aim random superposition multi conventional sensing issue issue develop structure prior knowledge start low enhance great natural processing provide guarantee completion theoretic limit central root estimation name prior order denoise exploit low rank matrix denoise
technique space incorporate syntactic wide type rd start support grant grant award thank computer laboratory represent semantic representation drive tensor semantic framework compositional semantic neural tensor obtain corpus argue extend beyond interesting challenging community computational concern compositional distributional representation combine distributional meaning traditional compositional formal property robustness ease ambiguity whereas
discrete successively perform via complete fisher algorithm reduce analyse est un importance dans tr le I ram ne de une les es le dans ce l est de de une partition en class es en une solution un de en
answer plausibility projection predictive random set regular predictive projection valid predictive say marginalization ignore efficiency point valid specification random predictive set iid interval valid lead marginalization construct probability dash rectangle predictive directly sample bivariate variance variance correlation sufficient express conditioning look auxiliary specify express fisher interval know posterior see interestingly posterior plausibility correspond classical sample inference simple version problem challenge consideration association marginal I trivial arise like I valid valid plausibility interval interval I alternative coverage association scalar inefficient fortunately marginalization row baseline give I change variable non central chi square freedom non usual predictive random measure violate I highlight difference sample write avoid enough empty plausibility base default plausibility equation empty
provide optimality determine much may seem surprising estimate reward consistent confidence interval let select optimistic correct lem long select reward discard increase policy appear constant order remarkable scale mdp space advantage prior policy performance build whole action prior knowledge markov markovian regret nonetheless display action elimination mdps pac assumption model actually dependency mdps policy belong dependency alternative dependency gap good introduce induce mdp gap average good policy bound stop unlike gap notice define
order allow multivariate ability parsimonious thus version mixture include fix varie accommodate latent another trait univariate trait use probit integration compute heterogeneous medical diagnosis additionally describe probit structure multivariate latent trait dimensionality discrete develop trait fit perhaps propose close connection response trait parsimonious quadrature mixture item estimate use latent gold use quadrature integration probably analyze trait integral analytically exactly latent trait em response proportion estimate return attain advantage drawback trait efficiently implement latent trait component density approximate obtain necessary double eq use ng ng ng density optimize variational z ng ng nm ng root increase ng ng ng ng ng estimate
wide motivation come bandit arm suitable contribution armed bandit arm also armed bandit wide interest optimisation bandit reward sufficiently behave finitely maxima call description function show obtain computation constant depend say another armed bandit adapt complementary armed bandit armed bandit give describe arm definition bandit measurable px x arm receive space measurable independently measurable conditionally event multi armed bandit additionally estimate x require measurable function strategy include define r regret consider arm space strategy remain maxima illustrate concept height cm width cs cs cs axis cs cs west operation bandit neither smoothness thus require new definition diameter axis q diameter compact continuous f finitely maxima f neighbourhood x small neighbourhood vary x set continuous essentially behaves maxima x one follow x elliptical maximum p elliptical maxima quadratic root hessian alternatively separable maxima allow continuity maxima well combination power motivate example
probability symbol fusion step sensor significantly couple ml monotonically initialization blind mc author mc respectively I regard format phase mc phase equivalent estimator special blind order moment propose perform coarse use simulate
list figure detect twitter early naturally depend high expense conservative expense tradeoff vary single roc curve describe tradeoff show envelope curve achievable fall conservative early fine experimental setup twitter news user list twitter tweet series datum tweet news news news find trend early majority voting still trend twitter theorem twitter mit series often competitive tree justification hypothesis trend twitter series relative access trend twitter massive twitter vote classification majority near neighbor account synthetic majority achieve rate neighbor forecast topic twitter become trend detect advance twitter hour
novel calculation replace distance drastically additionally certain metric learning precision recall major learn locality hash agglomerative paper background section present result conclusion limitation technique g expectation maximization seed address
finally use consider observe datum normalize ignore graph posteriori solution likelihood prior solve dag discuss structure respectively induce outcome space marginal express hyperparameter characterize must evaluate bayesian likely equivalent equally likelihood equivalence derive respect dag turn special prior belief choice investigate much play assume scoring likelihood alone quite ordinary dag prior likelihood performance generative however prior analyze score result tendency dense complex associate dag overfitting reflect rather may dependency thereby capture global drawback density basically concept vanish consequently construct size gradually vanish sample free underlie dag strongly induce support include uniform prior distinction term free implicitly underlie dag amount prior global adjust belief able express impose dependency
q addition note definition corollary author grant lin randomize block coordinate extend nesterov minimize separable type upon problem convex develop technique analyze inspire minimize block denote cardinality iterate pick uniformly block iterate respect precisely gradient block nesterov particular per type addition
relaxation bad must think domain get condition unfortunately origin polynomial satisfie interval origin polynomially small around move complex extend number close disk radius bounding observe disk transformation polynomial bad polynomial surprising polynomial whole disk reveal particular integrate polynomial stay disk origin particular polynomial
locally marginal continue marginal locally consistent bp algorithm variable factor domain message computation domain address concern return result try achieve intermediate unclear result error speed example involve track user point
flow well window method detect anomaly well anomalous flow tradeoff stability suggests yield edu present anomaly cover anomaly include vector consist nominal flow anomaly attack improve anomaly potentially traffic implication attack attack previously area concern network anomaly classified evaluate fitness traffic normal boundary support particularly class either flow raw directly aggregate flow cost group flow base
sequel parameterize natural parameterized pmf parameterized let negative take distribution range call eq eq bound demonstrate I lr discuss approach distribution weight tractable derive tight inequality problem another approach idea restrict pmf parameterize subset pdf pmf parameter choose function real yield restrict satisfying deterministic denote partial nonnegative ii eq note likelihood call monotone likelihood extensively paper derive connection lr method fisher follow exist function without see tight method moment offer close expectation pmf actually case pdf pmf determined pmf parameterized n relation xx x z appendix lr achieve situation computation
use binary variable auto trick work independently additional perspective considerably reach interestingly latent explain vertical variational bind per point take train intel effective train image mnist estimated encoder decoder decoder face decoder gaussian constrain output decoder update decay term equivalent likelihood compare recognition encoder generative initialize random stochastically use criterion stepsize base train recognition hide
pc choose aic matlab package table show concern mean amount table
diagonal pair element replace identification condition identification possible two source source factor expect generalization identification specifically source align original alignment dataset dependency identification dependence source allow indicate k gaussian multivariate state follow must semidefinite singular convenient statement lemma must hold lastly symmetry mention admit identification individually state identifiability require possess recalling determine source align across permutation commonly share dataset use
align heterogeneous formulation heterogeneous contain kind kind contain link link heterogeneous e word set heterogeneous link set friend location heterogeneous align heterogeneous network anchor anchor link anchor traditional prior want study new align heterogeneous social target link social set old user link want whether user feature heterogeneous difference user propose supervise predict potential social align heterogeneous network heterogeneous network feature social spatial feature feature relationship social social cn aa shared account lot strongly measure significance online location store visit extract location inner cosine location cn ratio visit
ni helpful cross th q anti symmetric proof fact row anti symmetric create good pick clearly anti block compose block cross covariance th replace concern row matrix make applicability study broad show apply turn know semi row anti symmetric following suppose hermitian block uniformly circle law variable important sequence block immediate guarantee asymptotically indeed modulus difference block zero result assumption assumption meet anti symmetric note row meet matrix us equivalent model simply scale matrix diagonal remove accord block simply operator bound row know clearly upper go semi circle case block haar apply check haar block haar circle moment haar take
fail rp collection rp job company rp project area rp heart rp united country trade rp team home give rp shot course rp car rp united rp program organization rp point lose rp decision kind rp bank rp song band rp abuse rp child look ask rp european france united al party attack music pass name south book big expect team business program correct service american home percent question kind program lose receive separate article independence line pay home join book kind public drop red matter home call place job version movie company com school million room york air word occur accounting percent plan site room open question home order analyst public return worker policy home house home security house understand department internet name pass financial company plan room learn list percent lose home red book home home important site company music human party team percent ray home analyst english lead business game mind united hour look com lose start sale home worker country moment change school htb times home super pay home half safe team game group percent problem word company person microsoft room child school
replace eigenvalue determinant remain n autoregressive convenience assume minor inversion ta beta multivariate ia derivation moment beta henceforth establish forecast freedom spread first order verify write furthermore identity eq verify thus assume wishart wishart ahead forecast inference unconditional transformation f fy fa forecast appear admit solution denote stack eq partial x gb g expression extend result logarithm matrix give right matrix e
suffice keep track newly slope cardinality nonetheless value
dimensionality exponentially dimension problematic present application low variance empirically reduce exponentially polynomially version pm employ pm mcmc automate pm present unbiased pm synthetic report conclusion observe binary response accord distribution function characterize jointly function sake gps latent parameterized specification nk view difficult parametric role kernel kernel assume radial automatic
dictionary unit euclidean norm moderate ambiguity tend zero global minimizer admit denote entry require mutual dictionary procedure dictionary motivate roughly coherence dictionary atom column character order relax average consider square element mutual prove experiment explicitly mutual coherence since furthermore within barrier dictionary implicitly influence denote validity since atom equation due cauchy schwarz thus
house locate convert house price explanatory variable age lot incorporate environmental display fit nonzero quantile identify component respectively price level big house house impact house price expectation price country price locate rate environmental indicator house effect covariate response heterogeneity age
need user specify find good specify choose configuration nystr employ specify kernel specify range neighborhood balance determine sec sample versus vary clustering perform subject report cluster sec considerable achieve image well om scalable except nystr om cope issue cluster new database ssc lrr whole lrr grouping ssc ssc increase ssc fail whereas come affinity nystr elegant balance cost although fast algorithm test whereas sparse et whereas adopt mean note result achieve half number tune nystr om nystr om ssc lrr ssc k mean sample ssc lrr directly whole image tune nystr om nystr sec ssc lrr examine
initialization relative performance regression favor datum generate reference series length cluster polynomial choose provide series htbp c tune follow triplet percentage pair percentage ccc order misclassification intra misclassification percentage cluster percentage intra average different misclassification differ
combine proportional combine communication lead balanced cost local site cost partition algorithm combine figure cost solution improve span prevent accumulation construct thus need communication cost achieve similar present appendix setting provide propose adapt clustering include send involve back communication propose summary summary distribute algorithm cluster summarie central carry consider topology aggregation scheme approximate sensitive preserve computation mean median cluster approximate set also clustering showed construct parallel merge
product filter image filter product response frame pair situation filter early motivated code motion estimation filter inefficient vast amount good filter recently motion multiplicative model learn autoencoder assume image let stacked row wise linearity sigmoid use multiplicative represent transformation autoencoder tie allow symmetric reconstruction extraction representation contraction jacobian sigmoid linearity contraction filter depth
piecewise wu jt tw random row column order sequence estimate high go infinity question estimator without wolfe consistency algorithms consistently besides simulation explore idea propose method blockmodel define property measure
get finally h basically except fourth term proof immediately imply real integer similarly qr sr omit see eq whenever since theorem plug get suppose therefore restrict isometry sparse rank rely polytope
art machine fail learn motivate work obstacle nature human supervision conduct favor mlp box chance supervise neural fail learner intermediate target form intermediate concept well explore variant point optimization difficulty task inspire effective ill minimum deep evolution interest science group individual learn way superior learning environment bit call share characteristic selective base counter issue difficulty difficult world help human experiment element agent become agent failure verify question relate broad human potentially machine artificial task binary present image contain shape figure machine could nevertheless provide presence location solve unsupervise pre algorithm variation architecture training one show pre independently different second learn architecture rather refine involve compose logical formula detection object network bring hypothesis form ai history computer science specifically ai create human paper investigate algorithm effective minimum either algorithm e serious ill abstract task likely yield local minima network hard general enhance
take adopt kernel hyperparameter typically tc deterministic estimator kernel contain obtain low bias impulse square underlie noise estimate impulse equal obtain integrate joint set eq introduce auxiliary rewrite relationship apparent impulse quadratic height marker axis axis line middle thick marker thick marker line height
boundary soft notion correct amount manually detection sentence run opinion target annotate separate pick standard updating sentence furthermore architecture architecture successfully token use embedding experiment word severe word experiment softmax hide layer linear activation experimentally activation belong space employ unit cause dense sigmoid activation layer degradation interpretation activation per choose number layer dimensionality layer pair
outcome prediction score base probabilistic metric organized commonly identify present relevance metric finally section drawback metric evaluating learning motivate dataset area light warm dim input external explore predict light success prediction select condition give observe model classification dataset time light user light four user
flexible variety inference nuisance task approximation carlo variational tractable preferable variant show art variety predict end order ignore conditioning variable factorial simultaneously possible cost share extract convenient task property actually ensemble explicitly even well procedure computational expense autoregressive probability variable product tuple permutation element first element model choose conditional autoregressive base regressor inspire conditional
informed usefulness use across subset avoid specification priori possibility algorithm simply merely influence stop well find outer gene commonly method fall category list rank constitute consider contribution discovery many preferable great take every evaluation normally costly validation describe five microarray expression set contain gene originally far merged gene distinguish available difference study primary breast
adaptive partitioning primary idea temperature proceed temperature pair explore pseudo prior ensure equal spend suit small
design fu recall quantity eq great reasonable exist eq yield signal regression carry np size critical normality estimation condition seem normality hold necessary evaluation comparable evaluation negligible nonparametric noisy interpolation preferable costly model hand conditionally invariant centering simplify set delta theorem delta justify g yy yy yy yy check thus follow delta
construct randomly trial experiment random four set measurement iid let sense heuristic form sense wiener estimate lr x lr lr v appropriately rescale sense constraint directly keep mind take estimation modify clutter describe separation entirely subsequent additive explicitly correlation describe whose signal model x model sense observation sub th compare
carry asynchronous adaptation topology failure random arrival turn compare asynchronous decentralize centralized gradient batch interesting stand result establish asynchronous mean centralized steady square suffer degradation adaptation adaptive match centralize conclusion highlight enhance comparison stand alone agent process remarkable various able batch asynchronous centralized batch solution network link failure asynchronous behavior centralize fashion conclusion justification benefit enhance stand reference therein remarkable uncertainty able batch solution distribute asynchronous distribute centralized asynchronous centralized mn r strategy I I I ki asynchronous strategy define manner ki ki ki nm ki mi ki nm np pp part respectively continue symbol part part present interpretation body
product constant label base label use linkage assumption jt sufficient mdp cluster approach variance alternatively inner matrix difference sufficient fact follow clustering mdp dimensional normal independently copy copy consistency mdp work propose via compare mean cluster type clustering
go log replace equal put trivial bring parent sake auxiliary observe reflect auxiliary g empirically evaluate relative efficiency auxiliary generative handwritten digits mnist form binary connect variable p sigmoid latent pdf dependencies similarity neural network variable
expert eq fix switch occurrence length block see length share distance geometric geometric binomial prior thick depict compare share parameter instead cluster show improve bad force switch carefully length subsequent per logarithmic later refined obtain switch occur equally switch relationship expert advantage arbitrary make intuitive expert together far apart consider expert clear increase gradually switch degree simplify switch expert apart practice expert drop expert outside expert specify suffice marginal sequence weight expert property current expert eq state sensible order state hmm expert implement analogue interpolation regret identify j carry fast fourier transform order expert provide time expressive kernel perform drift every switch event regret specify specify expert involve predictive expert expert indicate histogram arise view describe may propose essentially various value switch strategy therefore however seem reasonable switching drift combine fix expert loss occur parameter long interpolation shift expert turn use state allow convolution convolution drift sign
novel word behind result distinct novel induce question projection limitation appear elsewhere similar row convex condition second every row provable guarantee
vary sde give spectrum smoothly vary e process see sde vary similarly mat ern replace equation q model discuss brownian model mat ern complex process window observation series specifically window equation procedure capture vary semi locally impose similar spirit function yield window vary reduce drift around value appearance broad peak frequency empirically application formal error six parameter version select variant correspond use ratio methodology value hypothesis fit number extra nest choice procedure trajectorie numerical section world tool computation code www ac uk software include material series mat ern accordingly investigate generate wind force coordinate primitive equation similar
subset propose equivalent bipartite bipartite matching submatrix bipartite keep everywhere else support accord perfect almost exploit random perfect condition condition satisfied upper satisfied require complete provide detailed comparison uniqueness important uniqueness cp condition guarantee cp decomposition fully component cp uniqueness result adapt stack hand persistent write identifiability provide uniqueness size identifiability unique corresponding tensor persistent consideration identifiability identifiability matrix consideration identifiability regime uniqueness interesting gibbs latent overcomplete observable certain overcomplete greatly exceed overcomplete topic model identifiability persistence condition identifiability novel overcomplete existence perfect matching order identifiable overcomplete degenerate arbitrarily identifiability imply uniqueness decomposition tucker decomposition decomposition overcomplete representation decomposition machine representation representation extensively arguably vision overcomplete great flexibility overcomplete representation framework incorporate datum overcomplete probabilistic much overcomplete parameter uniquely recover crucial interest disease latent infer among observation identifiability predictive employ high classification identifiability presence isolate optima affect characterize identifiability incorporate presence latent document consist tuple word establish parameter use order third degeneracy non degeneracy imply exceed vocabulary remove restriction overcomplete topic model topic exceed vocabulary topic identifiable overcomplete regime refer topic capture
balance another effective cluster laplacian however sign weak theory researcher network applicable sign lin cluster assignment try move preferable propose agent basically sign sign network sign laplacian analogous ratio network iterative method solve sign modularity use sign prove problem np optimize disagreement versa correlation researcher observe sign learn active learning prediction study usefulness balance sign generalize act triangle interestingly balance global sign network connection balance section balance sign network consider problem relationship entity idea sign yield sign prediction cluster partition graph balance theory cluster mutually weak theory detail develop sign adopt social particularly global balance occur hope reader balance theoretically challenge design algorithm may adapt sign general signed however discover connection network sign network prediction prediction motivate straight balance albeit reason make sign synthetic world particular propose prediction social imbalance ii supervise iii use cycle exist use triangle fully implication structural balance measure balance immediately method sign network balanced sign reader version detailed research global perspective treatment organization global social
e college pa regularization attractive learning attempt improve generalization prediction capability coefficient choice application spirit investigation regularize hypothesis function attain almost logarithmic factor model impact generalization capability smoothness etc keyword learn g scientific frequently common explore number research predicate interested often train together empirical request view potential rule past decade generalization capability prevent shrinking attain value accord unknown dimensional coefficient regularization regularizer take regularization lead form regularizer ridge regressor smoothly toward coefficient
computer task expand sr euclidean attention fundamental building computer learn notable offer compact video covariance descriptor tensor image track cone curvature study riemannian negative curvature invariant riemannian riemannian transform widely accurately handle structure trivial computation riemannian incur burden perform coding manifold hilbert rkhs contrast directly approach riemannian geometry induce separately euclidean approach riemannian convert euclidean manifold tangent space benefit true distance
online algorithm mirror natural mirror generalization gradient descent interest lie euclidean manifold multiply standard euclidean solve mirror induce manifold select manifold let riemannian manifold family denote fisher riemannian provide manifold induce parametric detail thorough riemannian manifold riemannian manifold riemannian correspond prove name gradient
assess rf value selection chain iteration retain autocorrelation burn seed run rf show consistent feature summarize rf requirement nevertheless attractive gene rs rs rs compare snps rf though rf achieve low coverage metric coverage conditional level treatment mean interaction treatment copy major allele less exposure treatment profile conditional h
special solve optimization unary crf parameter describe near pixel distance label foreground pixel incorporate addition unary patch image cluster response patch submodular allow entire potential unary potential come simultaneously interactive segmentation dataset annotation truth sort validation regularization pick give training testing submodular flow mit perform average standard labeling preliminary experiment version program theorem property corollary exactly express goal training first interactive
mean ball reliably clean ball point us point radius resolve point necessary since remove happen q manifold input number us radius around access ball oracle fix achieve replace modify specify let begin figure draw parameter q separation choice prescribe theorem satisfy highlight exactly lemma remove necessity contradiction graph connect geodesic distance therefore geodesic net rest unchanged requirement automatically satisfied similarly satisfied far recover situation concentrate manifold argue manifold model set clear tree straightforward requirement underlie follow specify observe background clutter universal figure
center constrain testing performance adversarial adversary capable rescale unbounded common example project descent predictor instantaneous encounter suppose imagine factor predictor input conversely indicate experiment two vary first online despite adapt geometry variant gradient scaling make poor algorithm address grow existence motivate search computationally invariant rule second perceptron varie time free use divide normalize invariant scaling adversary online critical additional regret small algorithm dataset dataset little unnormalized rule advantage online update rule throughout label associate prediction weight observe w ng present ng add invariance making simplify standard vector update maintain ii feature make change exclude multiplying cause entirely weight impact scale attractive adaptive feature rate normalize version maintain sum square I somewhat scale imply decrease reduce update scale introduce automatic rescaling initially observe n justify well prediction input
one observation come eq parametrization get multinomial thus proceed note without never add row objective consider replace row entirely tell affect row mean row plug update
identity avoid invert factorization iteration ep base individual term likelihood approximate unnormalized approximate likelihood multivariate posterior ep characterize optimize loop approximate likelihood updating factor turn cavity leave closely cavity minimize follow kullback moment computed derive involve refer detail general classification convergence offer method fully integrate hyper analytically whereas tackle characterize estimate th converge joint proposal extremely unlikely latent hyper compatible observe therefore resort whereby turn briefly pm efficient achieve elliptical slice ss ss slice latent ss tuning minimum intervention factorize complexity variable hybrid carlo detail variant interpret hamiltonian carlo factorize simplicity employ technique classification hyper coupling induce slowly mix poorly illustrate
examine essentially euclidean update local centralized machine communication need iw substitute bregman divergence investigate align objective course place hope ideal get close bregman form close true difference approximate plus regularizer update distribute quadratic update rigorously quantify iteration round require newton quadratic hessian sort quadratic potential require provide carefully next objective also quadratic objective provide guarantee stepsize sufficiently bridge show quadratic objective objective without derive guarantee term set instantaneous
single immediate consequence boundary smoothness net trait boundary growth function requirement densely prominent entropy entropy ex strictly entropy approach logit response entropy hx game provide derivation subject fluctuation independent distribute random action maximize perturb variable logit good penalty perturb good stochastic perturbation approximate ordinary relative perturbation approach general context perturbation follow strictly smooth perturbation penalty initial towards condition bias rate kk induce turn penalty perspective difficulty always practical map agent strategy furthermore primal dual update rest update focus decomposable form maximization equality interior simplex little algebra penalty summing denote visual put everything obtain along dynamic name rate vanish asymmetric dynamic equivalence exist derive version learning field dynamic space dynamic context player appear perturb reinforcement dynamic appear differential payoff adjustment reflect past payoff highlight similarity correction mechanism set likewise appear absolute population score like
computationally inferential word composite deal canonical may theoretical instance spatial structure observable might specification straightforward evaluation cope difficulty specification composite great impact example process longitudinal involve component pl f likelihood conduct assess unbiased kullback leibler pairwise score ps ps pairwise function regularity condition hereafter counterpart score chi distribution independent main fact might desirable counterpart asymptotically depend obtained adjust factor obtain adjustment force freedom large likelihood place rate reference whose distant goodness
dominate mask reject close function negligible filter central dominate aforementioned effect goodness around central patch test central area par homogeneous maximum like li patch si tie ti densitie cone hermitian divergence every divergence divergence et measure scale prop er estimator detail test al obtain distribution test hellinger yield also et look instance square window base kullback leibler enyi produce almost hellinger expense load although know limit negligible filter statistic wishart preserve control filter implementation wang quality assessment filter hard assess filter inspection visual computed intensity channel wang al et reference look likelihood solve equation
provide reliable yet suggest nn database assign five possible lr lr weight base perform list deal relevant whereas suffer robustness suggest medical solving tool logistic model neighbor problem paradigm medical making system solely stage national list medical medical factor accordance medical really need automated make reason national confirm agreement access list influence medical study possible list medical decision support every patient start st patient patient status rely date first date description define datum availability
real exploration policy rl implementation complicate require policy online discard result efficiency drawback mention propose rl rl develop purpose pde derivative state linear pde side rearrange equation signal replace pde rl convergence equivalence pde linear pde derivation equation pde contradiction derive q q another thus complete solution equivalent policy rl equation internal dynamic rl design identification fact
define allow testing element support element since nonzero element signal overall minimax consider minimax ultimately assess performance testing hypothesis rely result low testing space leibl divergence satisfy minimax induce hypothesis obey need evaluate identify element observe assumption iid kl divergence mf respect distribution assumption mutually density factor signal log simplify exactly nonzero amplitude sign follow ni let equivalently element element lemma case follow ultimately correspond performance multiple testing hypothesis let measurable test map obey kl pair induce divergence derivation low minimum probability evaluating since lemma equivalently satisfy monotonicity quadratic simplify imply calculation see combine risk tree proof effort fundamental support recovery sparse dimensional context use distinguish describe signal one signal root tree also summarize sense directly formally recover strategy useful support estimator dimensional ti element kt I minimax risk one vector amplitude setting measurement adaptive sense employ claim provide validate theoretical result improvement tree sense analyze underlying increase scenario
general rigorous reconstruction read achieve practice simulation next frequency minimize aa mahalanobis satisfie bind rate depend matrix encode particular arbitrarily mahalanobis dimension achieve mahalanobis frequency vector assign specie highly solve hundred specie challenge even store trivial alone minimize develop scalable divide thresholding cope specie frequency block solution iterate reduce software package divide matlab package part hour mathematically obtain approach generic reconstruction region allow distinguish different example extend sequencing reference de currently database thus
simple correction suggest hypothesis choice correction hundred make correction impractical achieve collection empirical vc definition ground transaction requirement guarantee subset transaction appear set effect potentially low e transaction least transaction hence thesis I transaction transaction sure transaction typical transaction e negative define find appear unitary number transaction length contain define terminology transaction one label sort transaction computed scan associate use empirical transaction maximum happen sort sequence capacity item optimally know polynomial power vc specify currently available solver fast thousand
would use instead answer extend individual game moreover still convexity potential straightforward adaptation unique non uniqueness minimizer attain least unbiased unbiased trivial equilibrium unbiased square establishe amongst estimator w individual technique individual model addition game equilibrium gauss stability sub price stability attain cost include cover extend assumption go interesting gauss two way optimality impose semidefinite would case impose apply whose action arbitrary need occur individual analyst bring issue particular exist view induce report
substituting complete derive bandit allow reveal translate
index represent wise calculate projection matrix simplify projection project onto express projection p te ta ta ta td substitute ta sp te respectively
long decrease threshold alternate typically terminate iteration obvious possibility initialize c svm anneal loop gradually mainly sub throughout anneal supplementary consider absolute l k terminate optimizing depend practice pick transform convex hull conv solve svm conv svm initializations motivated change term objective remove center additional compatibility see loss
see e sure function equivalent measurable outer satisfied theorem pt pt mm support department svms unknown svms vector relate confidence interval tolerance mention svms correspond mild successful svms
connection game common player structure game payoff g restriction player nature action mix formally empty simultaneous game embed fit linearity hull latter write action nature give payoff define naturally construction nature denote corner interval right corner rx claim game nature maximize payoff arise answer need determine player average determine corner half shoot parameterize payoff rx incomplete introduced describe play simultaneously instead nature inform partial monitoring place nature payoff evaluate mapping must correspondence capture beyond scope full incomplete viewpoint describe simultaneous restriction strategy viewpoint mixed action nature think action notation define payoff product z rx recall maximize minimize
case linearize correspond proximal mapping function global linearize linearization step readily solve proximal mapping robust pursuit nuclear et note involve mapping graphical sparse log solution rewrite involve easy see subproblem regression sample condition intercept besides otherwise problem weighting denote impose want rewrite admm subproblem respect operation subproblem solve mapping problem admm proximal mapping example really proximal mapping exactly
early spectral drop notation analysis invoke bound quantity magnitude innovation observation let independent self adjoint adjoint let together boundedness noise regret least result time employ direction network contribute substantially section restrict estimator characterize intuitive idea steady let exploit edge network subtract
decay size increase decay chen chen phenomenon lie nuclear et constraint figure constraint improve technique positive definite general measurement matrix identify quadratic neither collect important lemma proof lemma involve material generate probability sub distribute essentially general cn c ready main easier noiseless recover give decompose polytope nan need satisfy q b eq n r r complete equal prove follow instead supplementary material exist constant least constant technical lemma property low low separate constraint z suppose
model let annotation annotation annotation image annotation bag word treat annotation deal visual specifically use leave annotation word training representation spatially annotation wish visual v tree decomposition probability leaf decrease select dataset dataset scene popular annotation supervise code available scene contain previous pixel wide select use construct tool pixel inside city open image randomly
acknowledgement helpful span theorem field simulate path problem idea extend class boundary innovation candidate usually path algorithm many field engineer great difficult problem rarely discretization intractable carlo apply euler discretization increment discretized overview develop boundary illustrate diffusion population growth section boundary sde follow interested diffusion apply drift unit diffusion coefficient dimensional note multidimensional may law denote absolutely brownian
learn train embed model use learn repeat build training
throughout time amenable large scale training classification extensive synthetic competitive date future selection receive bs mathematic computer project medical award author marginal interest computer vision image mathematic receive ph interest include statistic computing bioinformatic receive bs electrical engineering engineering china ph research supervision research interest machine computer bs degree mathematics degree ph statistic currently research supervision interest simple characterize tie algorithm characterize vector restrict include fit ad hoc design well ease resource suffice incorporate nonlinearity receive sparsity induce remove high consume dataset capture nonlinearity restrict run manner weak learner decrease certain design nature feature boost addition boost expensive hundred thousand ad hoc design feature selection specific class
well perform sequence sentence domain unlabele representation train great step markov chain word tuple expectation maximization million sequence message pass complexity pass train million learn
variance pour les mod le un mod le du mod le plus des en pour les trait es mod des quasi le mod de la mod le gr de du de la r du mod est du mod le ce le mod le une alternative pour I de lin si un adapt pour la section comment les du le par dans du la de un de pour les des ne em dans un la ce es l la
expense opt compare agglomerative heuristic value indistinguishable disease network wikipedia vote email political interestingly well result explain point average minimum use likely agglomerative never heat configuration even anneal serve agglomerative meaningful differ fig b partition across seem equally attribute minor agglomerative network indicate partition certain leave mcmc despite entirely unbiased towards pattern infer modular increase attention
mention solver cs pursuit several nonlinear system surprisingly experiment illustrate nonlinear like relate via eq possibly overcomplete quadratic give I noiseless equation perfectly recover ambient dense satisfie verify second enable employ basis pursuit well iterative also affect initialization finally advantage recover true equation measurement require recover solution
mala approach indicate target pseudo order impose indeed effective sampler subsample nonlinear distribution consider dx dimension example target compute measure similarly burn average acceptance target quantile quantile consider distribution moderately middle superior structure current illustrate mcmc scale quantile sampler superior empirical mean purely walk e tail target proposal though close true quantile resemble use example along fail dependence structure due isotropic two even highly thin joint
second expression cauchy interval well behave behave piecewise degree output subsection fact suffice key result piecewise degree degree let mixture piecewise proof theorem alternate remove precisely mixture piecewise degree run use approximately distribution find sample draw heavy heavy perform claim satisfy straightforward suffice extension require use sample output piecewise exactly piecewise work heavy learn I place draw repeatedly outside theorem trivial x assign complexity claim correctness recall infimum piecewise claim ba ap td behave piecewise succeed dd behave since dp td run succeed version range concrete study discrete cover select generality monotone density gaussians mixture monotone concave include give focus continuous adapt domain polynomial may class discrete paper approach efficient learning focus various kind restriction nonparametric study topic application area reliability see reference monotonicity concavity pdfs statistical application survey type shape concavity unify give aforementione restrict concave gx gx technique optimal generally concavity concave log concave density arbitrary draw least logarithmic g
adjust terminal exceed level complexity bring explore instability test performance whole continuous correct summarize various nominal critical percentile instability conservative approach nominal percentage rejection percentile r r longitudinal simulation uniform monte calculate nan th percentile distribution reject nan summarize table exceed nominal significance size test severe size test follow involve true base approach increase bias precise remain small however increase test trend nominal level error nominal reasonably report brownian bridge kolmogorov normality bridge limit conservative nominal significance small level observation follow instability similarly
formulation bic design model complexity result regardless isotropic gaussian unbiased density calculation likelihood node manually specify small throughout direction object angle represent riemannian target distance short naturally modify unchanged assignment training close centroid finding close centroid trivially short arc periodic arithmetic convert angle mean angular direction angle certain distance circle use find
model jointly obtain word credible construction proportion recognize select coefficient table summarize result control except show simultaneous interval reasonably credible become shorter reveal simultaneous credible appear much sensitive htp c two generate gaussian represent parameter situation median obtain sis scad report select median stable chain length markov chain appear mix median I scad scad prior prior satisfactory coefficient perform deviation prior stability htp ccccc sis scad generate strategy major lie explicitly generate describe chain summarize certain particular yield estimation produce htp ccccc sis scad examine prior role difference unlike use dimension fulfil stochastically procedure unify
defer appendix exponentially fast follow bind small learn conservative slow natural necessity description except since property turn straightforwardly specialized control nice value formalize summary well though opposite come increase time corollary iteration guarantee corollary slightly improve factor though performance well complexity algorithm towards like evolve horizon value empty infinite
least classifier space try need example ii excess excess non loss measurable function rf measurable induction label epoch sufficiently hold output total instance log concave achieve consistent subsection presentation bound relaxed condition section concentration bounding process let unknown
result concern variable dominant value generality indeed follow truncate solution far pa non exist eq choice simulation vary close feature obtained empirically come number non zero dot theoretical dash increase value initially parallel axis feature relationship explain decrease solve penalize subsection proposition lead propose feature cluster less specifically r j l r difference equivalent likely lead consistent intuition precisely close difference scenario versus proposition scenario vector interpretation proposition
equal give communication fast processor processor nice give choose grant eq grant need greedy gradient absolute computable section compare solve one available w medium example element number intel processor gb asynchronous parallel descent code
group word see along question highly group identify activation associate concept hold pick item activation pattern certain brain hold mostly refer cause daily grow ever pick house car cast soft offer low enable activity answer project expand brain voxel activity word entirely brain question randomized repetition let vector activity human left project brain predict brain leave two brain ability brain image correspond center although make encourage noun mean subject pair accuracy somewhat subject low brain activity
use maxout demonstrate dataset mnist cifar dropout simple large together deep task audio model average deep perform similar generally reliably yield model argue dropout slight designing dropout training dropout effective relatively regime significant subset training resemble bag sharing differ ideal regime steady progress another averaging design may thus enhance dropout call maxout beneficial optimization dropout four dataset dropout
within explicit exhibit pay attention subproblem algorithm aforementione real strength instance avoid graph poisson intensity contribution analytical away significant practical function show combine step backtrack search enhance characterization help adaptively switch gradient size correction step exist achieve decrease objective evaluation enhance path smooth convex fundamental present modification deal concrete impact conclude letter vector bold letter matrix denote positive definite semidefinite size proper convex ff convex subdifferential subgradient tool handle proximity whose notational convenience derivation sequel proper convex proximity operator nonsmooth assume value mapping due let indeed cauchy inequality lead self contrast global fig quadratic prevent rigorously accuracy transform dimension assume proximal subproblem inexact problem high higher nonsmooth proximity optimality done convex sufficient eq derive fix point principle sequence convergent
relaxation single policy execution policy efficient bandit contrast note naturally index require efficiently design albeit optimality execution arm define identify tractable arm construct approach fundamentally contiguous policy contiguous enable step essence study critical linear policy constraint across constraint interact infeasible many relaxation analyze produce produce relaxation schedule policy scheduling approximation gap feasibility step crucially weak indeed idea obvious play naturally execution restrict arm past play exponential length write similarly schedule surprising aspect lp receive max capture reward yet schedule step single play play good reward issue apply almost problem one linear fundamental regard program mab herein reward globally adapt gap accounting term gap scheduling bandit policy adapt observable policy gap factor observe version exchange play difficulty therein term within regardless nature term run take standard approximation designing policy per observe play np problem cost snp combinatorial term constraint factor programming alternate well involve incorporate way allow reward pose challenge formalize mab early arm objective maximize reward arm arm stop play introduce show argument improves result time edge define sparsity couple relaxation single policy arm although throughout level illustrative purpose discount highlight reasonable switch couple arm define adversarial market project uncertainty expect period change order place payoff epoch employ epoch change incur payoff similar worker priori fill switching constraint impose policy physical consideration move two adversarial policy horizon mab mab cost address agent flexibility order event provable reward ask policy horizon adversary order play return play arm policy maximize expect switch behavior adversary receive benefit metric cost variant locate space incur policy cost whenever arm already bandit absence assumption already encode significantly traversal guarantee section present constant become
homotopy correspond track formation newly find intersection intersection correspond move constraint vice versa track constant newly encounter equivalent update elaborate tracking section piecewise segment segment form straight line define variable per version identically compute however identification computation newly intersection recall leave slope current segment j illustration transition slope slope boundary consider index value homotopy sum characteristic take incremental tracking empty pseudo initialize j c j u c jj black exceed simple change complicated employ require operation track numerous substantial sparse tracking every induce plane equation path horizontal line intersection sort step search next line correspond next zero provide illustration line path plane toy horizontal theorem bad practical distribution total even become
nan decomposition l w q direct continuity gaussian kolmogorov say modification event old exponent old modulus modification exponent compact old achieve entry satisfy discretized hypothesis argument kt km nh k kt tt nf ml r l ml conditional modification old exponent old modulus version modification know old modulus old modulus u n q theorem definition thm apply university department mathematics university institute mathematics stanford usa study test functional asymptotic mild root alternative level pointwise moderately latter suggest proper clinical screening stress keyword test bootstrap pointwise get increasingly scope
constant thm collect datum result toeplitz bind discuss thm thus decay spectrum kronecker green obtain basis approximation optimal outline bind construct projection basis construct thm gs black construct dense matrix require eigenvector construct choose fig show covariance kronecker spectrum three full spectrum concentrated outperform across covariance kp leave middle panel kronecker eigenvalue spectrum concentrate normalize mse outperform solution achieve db show generate square simulate toeplitz show naive kronecker products toeplitz kronecker spectrum decay rapidly htp dense block toeplitz definite middle kronecker spectrum domain eigenvalue kronecker spectrum concentrate b outperform solution achieve mse db reduction kronecker spectrum much demonstrate real wind speed series
take analyze cause unknown mark record statistical likely involve enhanced probability often reliable suited interpretation stability consideration connect tail important case robust deviation mind work modern massive multi modal form advance van mathematical berkeley visual pathway author invoke brain white human visual pathway work pathway pathway computational vision task carry brain first encoding predict recover encode right issue decode fmri intensive extensive coverage medium read reconstruct mind mention one mind computer cover time movie encoding decoding employ former enough voxel separately
reproduce hilbert z construction extend prop broad current include integral subsequently serve basis application gaussian field define compact assume continuous measure say far act square integrable operator act hilbert space rkhs recall isometry completion hilbert respect topology integrable extend linearity isometry shown belong unique h pointwise linearity continuity isometry linear serve invariance prop state make prop introduce recall field kernel fulfil
theorem bayes intrinsic respect condition independent solution q x independent therefore loss distribution show intrinsic estimator stein usually estimation family intrinsic show one prior hyper belong obtain estimator bayes class example acknowledgement partial science engineering research author visit integrate eq integrating eq
scope exploratory outline modern mixed measure new mid mid inversion em parametric think quantile think criterion compression art compression quantile function probable small probability equal probable equal rank remarkable theorem
perform intel core gb ram sample g independent variance second multiply square table cpu cross entropy output unfortunately
entry thus vanish matrix matlab full memory tune output run iteration bound second equip intel ghz gb ram unable complete matrix regard summarize consistently memory prohibitive experiment efficiency c memory memory na na full usage usage mb multi classification belong exactly augment indicate goal capable vector classifier entry instance deal specify nuclear minimization th row report version solve sub appear consider factor sample entry index entry sample independently parameter similar imagenet dataset per train dataset convert dimensional art visual descriptor summarize regularization good cross standard optimality denote large value admit substitute actually termination memory present
know true unnecessary add difficulty potentially side know representation success array sophisticated nothing capture aspect linguistic reasoning evaluate train logical learn reasoning case promise capture logical reasoning nlp vector representation use nlp system question answer translation explore thing replace first predicate contain one reason work opposite inference least first monotone permit substitution specific argument permit substitution formally monotonicity position determine kind position
obtain multiplicative constant alone infer comparable often guarantee eqn result completeness hold fulfil particular svms reduce computational dedicate specialized svms minimum fw implementation fw coincide admit nothing else fw admit interpretation solve find svms propose swap step problem dot product space admit arrange solution unit simplex thus g hard point svms vice versa svm formulation interpretation margin svms admit margin svms author existence currently classic iterative algorithm adapt trick fw coincide note g fw k method additional write line recently train svms reduce key correspond choose geometric polytope discussion explore straightforward correspond swap iterate vertex polytope swap problem problem swap identical method fw swap iteration direction simplex exactly procedure conclude apply polytope swap equivalent method sense swap hybrid problem beyond svm problem swap fw method possibility step limit minor essentially converge linearly fw quadratic positive maximization simplex matrix definite hessian experiment performance public size class aside case versus beyond approach necessarily reflect address dataset binary binary report subproblem binary subproblem swap method solve point last svms rbf datasets svms correspond employ consist seek possible analyze behavior condition say reason avoid use
value principle far increase carefully discussion som similar outlier cluster thresholded distance activate outli algorithm learn formation activate neuron threshold kn perform win som kn win neuron algorithm eliminate ignore outli salient cluster cost detection iterate outlier suitable ability outli detect pass scale colour texture name label give entire image rather specific importantly irrelevant portion divide overlap patch sufficiently capture information
let indeed computable prove complexity interest need follow ji v separately e j I continue w old I substitute q simple useful twice form scalar say interesting informative natural capturing sparsity pattern subspace lipschitz constant lemma let comment theorem depend place theorem norm block sense constant although partially indeed define constant take partial separability pair hold also lipschitz continuity certain function enable smooth nesterov section technical involve large exhibit immediately apparent specialize expression allow parallelization speedup establish sample proper differentiable q fix combination establish equality observation diagonal one position connection uniform bind lipschitz use proper n substitute ii result associate block sampling object n ss
similarly event zero moment mn nu mm dm limit cf note kf else set lebesgue r f mc everywhere n n immediately precede display claim establish establish vx precede display except establish span column zero zero c n prof l hold establishe increase furthermore observe n argument dp result variable space span inequality definition constant invariance everywhere lemma part condition remark equal part prove monotonically satisfy statistic invariance z obtain imply satisfied r transformation respectively clearly satisfied hence hold remain show obtain p side belong z span equality side equation belong j side q hand part conclude concentration di j n clearly limit nonzero limit n expression regular equivalent show let diagonal everywhere diagonal everywhere everywhere obviously hence exist observe give establishing note eq diagonal element regularity equal respectively follow lemma ar integral extend covariance corresponding belong sufficiently arbitrarily neighborhood put mass close neighborhood show unit argument proof closure topology ar density extend high autoregressive multivariate autoregressive ac restriction often require f autocorrelation autocorrelation procedure develop positive concern large test correct test size nuisance generic design adjustment procedure artificial regressor adjustment adjust test suffer away zero classification keyword distortion fix autocorrelation test considerable two half decade test nonparametric account early nonparametric date consider estimator literature back latter discuss autocorrelation interval gr autocorrelation robust robust introduce literature autocorrelation statistic consider cite employ chi quantile square finite lead statistic nuisance arise asymptotic arithmetic least usual satisfy one proportional equal course choice rejection probability distribution converge writing observe e numerator weakly chi freedom denominator residual residual although correspond equal consider hypothesis converge numerator converge weakly weakly denominator numerator n positive rejection zero statistic since odd one depend n e autocorrelation also odd rejection close zero certain imply worth note odd hold second along certain invariance heavily exploit extent provide avoid appear
minimax noisy observation minimax spectrum variation horizon order linearly consistent proposition balance derive policy serve mainly tool general unify optimal may also take consider appeal practical rely subroutine surprisingly policy poorly environment see one fine achieve low eq tuning achieve optimal adjustment slow policy environment variation budget horizon step keep consider class adversarial modification achieve possible general adversarial regret achievable non apply procedure present aware algorithm space conjecture rate set stochastic adversarial setting access function feedback policy subroutine relative matching may next section support square matrix denote sake simplicity unified cost local play role optimization procedure adapt definition variation budget follow effectively sublinear optimality provide single completeness gradient convex class unbiased feedback stochastic gradient static feedback policy subroutine exist exist establish sa set policy subroutine slightly modify strongly gradient feedback adversarial single adapt non stationary benchmark part plug noisy follow adjust strongly cost strongly maker optimality carry adversarial rate operator k coordinate instead rate estimate tf tf define lr f cf chapter essentially
report ph report mix discrete massive massive utility demonstrate collaborative scientific modeling size child child age method polynomial conditional conditional quantile answer scientific level age dependence joint transforming avoid mid transform mid fx x mid mid non parametric orthonormal role mid derive fundamental fact show select look plot scatter correlation show measure correlation htb pearson rx x
gain operating embed product embed space dimensionality aforementione spirit effort problem several intersection random build provide compact approximate polynomial kernel demonstrate space challenge error hadamard positive polynomial list map p project polynomial kernel random efficient mean eigen structure kernel efficiency rank
component threshold eq value relevant vector formulation effect previous genetic environmental accounting adjustment repeating p p k standardized phenotype immediate yield derive need double integral threshold respectively integral integral dl dl dl dl w algebra check moreover individual fix effect plug relevant unknown control yield bias seminal
concave asymptotically player acknowledgment partially visit hausdorff mathematics economics warm grateful general thank suggestion theorem remark repeat game first person sum repeat second player inform long player able long provide repeat action discount game go aforementioned involve seven remarkably player payoff play notation rx fx rd gx gx person finite resp resp transition proceed game start draw probability resp receive player choose action simultaneously player stage signal player move goal resp resp payoff
include rna I datum reverse phase protein protein four different biological component represent genomic publicly available completely reproduce include four identify source detail draw overall weak association drive compare clustering cloud scatter overall appendix source degree seem justify blue symbol cluster cluster motivate flexible computationally scalable multi model overall cluster view form consensus traditional ability assume cluster sense furthermore specific know
form stack row row operation throughout solve geodesic start locally derivative corner statistical operation riemannian often compute statistic manifold geodesic logarithm inverse return geodesic normalise length geodesic affect utility mean manifold descent exponential logarithm solution practitioner rather run step rough principal manifold tangent space find direction perform unstable derivative numerical solution solver application original riemannian type datum live smoothly tensor riemannian imply smoothly metric
dominate term right equality hold combination lemma assumption dominate convergence integration obtain h therefore conclusion curve map lemma cauchy inequality conclusion let sequence z n z ep second follow theorem equality almost continuous assumption continuous give toward lemma fr dominate lemma follow equality continuously continuity q equality dominate finally dominate convergence lemma linearity continuity cauchy exploit ii continuous maximum combination final curve dense let theorem adjoint note measurable jointly measurable implie ensure p pz measurable differentiable therefore imply measurable assumption iv hence measurable pz jointly measurable z ensure linear theorem I hz iy theorem proceed ng auxiliary lemma uniformly step p proceeding piece nx
additional determine uncertainty e ax jx control time derivative describe let control see jx ax derivative add per capture set observational accordingly unless control aid want belief control desirable behaviour law loop law perform action law suppose derivative control globally differential leverage linearity exchange
observe feature acoustic depict go explicit dependency side discrimination capability acoustic word l l l latent adaptation decode modify conventional ensure mathematical modify bayesian clean latent b relation analytical incorporate distinguished front end fed model network pdf rule network would functional perspective take feature uncertainty decode adaptation dirac contrast feature decode approach pdf illustrate dependency approach share pdfs crucial reflect pdf arrive acoustic front clean clarity entirely article
detailed evidence however estimation iteration combinatorial moreover actually error contain unstable outline vb implicitly function initially proceed structure minima instability level error substantially kind resolution arguably superior base concavity keep regardless level coarse hierarchy seem across illustrate head head phenomena enhance parameter lead convexity function increase observe bound extreme without minima challenging broad image many minima situation penalty conservative sharp additionally begin structure penalty favor vb increasingly dominate small drop arbitrarily minimized extent great sparsity effect occur limit calibrate introduction map heuristic explicitly satisfactory performance add additional incorporate gradient thus step allow structure prune scale structure structure allow dominate structure generally speak map face properly deal minima regard smooth augment bad global interpret alternative penalty factor concave decrease domain analysis suggest simplify possess attractive concavity directly special closely examine concavity proxy vb estimation equivalently view assume decrease theorem correspondence versa examine relative concavity directly determine motivate assume theorem thus relative concavity directly nearly analog draw detailed section whenever affine importantly desirable special difficult dependency seem choose experimental conclusion increase gradually consideration justification choose constraint represent solution vb choice exact calibration fundamentally optimal solution rescale moreover omit carefully tune associate invariance additional interestingly
sift library extract descriptor hierarchical empirically take camera illustration purpose attention row figure interference category green bar pc image visual rd visual visual second third image top high interference interference top bottom image category visualize figure would project onto span corresponding topic clearly topic different axis portion top spin depict set identify contrast category pca moreover able categorization accuracy paper discovery category prediction demonstrate yield superior shot let odd
segmentation chain structure toolbox four processing task available noun identification noun phrase noun parse sentence identify word chinese entity type name entity person occur pre process sparse extract position per task except small overall split crf crf optimisation nest validation parameter crf cross validation power package matlab code use toolbox implement hour job accommodate crf nest mention segmentation fast get precise runtime comparison crf straightforward implementation language differ baseline grid possess hyperparameter kernel
degeneracy variance estimate mention fix variance growth hmms intractable address kernel density posterior smc potentially produce length method degeneracy practically remark recent non posterior unlike hmm set engineering sciences ep g support national ni x p static hide markov static gradient offline three intractable volatility return real hide numerical section cover three volatility hmms density intractable sequentially approximate tt particle
implement short interpretable aggregate primitive tree average together model tree contrast master iteration machine increase efficient perspective scale machine implement sgd data favorable machine figure cluster begin outperform moderate memory dataset minute specialize fast never time include spend production scale raw dataset machine exhibit close gold scale c matlab eps eps b ht c matlab technique recommender association let reveal goal bi solve fix fix know close solution step strong weak machine specification scale
linguistic increase digital source email communication report text short digital structured format extract represent format understand program system digital library ie scientific huge advance solution application semantic bioinformatic regardless ie activity composition atomic segment several biology interaction relationship art organize related try extraction experiment present conclusion paper kernel syntactic structure sentence concern parse error overcome propose bag kernel able parse kernel connect syntactic regard flexible candidate lead combination tag pos still point good performance linguistic exploit simple combine compare whole bag gram evaluate
rely rejection significance base conclusion chance false pt ptc pt pt ptc population population show grey comparison computational learn weak computation approximately integer boolean hypothesis argument comprise part rely significance query appendix recursive part argument genetic hypothesis testing let define generation generation divide generation v generation px x px invariant input element fx
expensive measurement monitor traditionally aggregate traffic volume observation level use simplify interest sample header extract record analysis aggregate high rate valuable resource consumption cycle greatly affect regardless monitoring flow I pair characteristic end end flow length flow arise location
sec package step component observation covariate support mean matrix normally distribute covariate regression coefficient matrix covariance correspond diagonal matrix generate value show statistic select
shannon prior length image interpretation length principle provide described criterion minimize heuristic start cluster per adjacent interval perform source target merge image improve greedy merge heuristic greedy heuristic issue implementation reduce optimize greedy heuristic fall local optimum meta mainly benefit algorithm different agglomerative exploratory propose consist successively way detailed cluster merge step segment induce increase infinity merging source vertex equal shannon merge
fix paper generalization random supremum pm ok ok supremum gaussian consider stochastic differential assumption lx ht fix eq eq possibly moreover divide
stage recursively p l equation minimize suboptimal fitting procedure validation subsection pursuit algorithm behind near ranking semi gain unlabele typically much independent different view process agree approach briefly search space label time prediction unlabele disagreement account regularization co regularization complexity view point description feature use similar l
change relative terminate converge bayesian fast vb vb algorithm vb reasonably experiment recommend acknowledge project cm artificial intelligence laboratory mail si
market solve build market forecasting market pricing ahead market hour market forecasting ahead explanatory relevant nature problem either hour candidate load interface another market natural nuclear wind location balance controlling comprise period pm st candidate pm load balance market weather forecast e wind generation hour day week month year capture peak market share weather forecast site capacity load relate whole predict price approach train prediction model determine transmission reliability limitation leverage market forecasting uniquely see energy market significantly transmission shift market period next
successive coefficient relatively advantage prediction accuracy clear coefficient sign well good extreme good moderate lasso propose selection many condition eigenvalue result consistency bind weak simulation generalize I hence subtract j jk hence j divide sn ki sn complete account bn bn proof must satisfy tucker tend definition jk j jk term zhang modification thm thm conjecture thm thm notation thm
column rank achieve term lead embed lead lead present simplify environment communication machine ensure analytical kernel job execute split physical view block run block group together phase process output job important incur execution typically round execute programming define scalability computational limit certain selection task facilitate interactive selection present minimization whose extend greedy unsupervised al recursive describe detail projection p project r e ta without ta ta formula b substitute equation tb nd tb tb b tb nd rd project span ta term calculate ta ta p ta tp ta tp substitute ta te ta ta ta substitute sp te te term project
combine stream compute word topic thus computation memory cost extensive confirm fast sampling yahoo lda bayes competitive high speed resource interpret framework basic em variable property checking concentrate run traditional gs topic active belief propagation sublinear despite datum anchor model topic speedup fast batch require memory store fast lda constant stream several mini batch batch memory mini optimum point variational propagation batch counterpart local rarely powerful architecture scale performance communication serious available currently architecture multi processor space multi share memory serious address parallel lda processor gs result gs parallel batch vb stream stream process load low far communication still communication big web
guaranteed implicitly requirement state space smooth e rate integral function introduce shorthand occur manuscript constraint extend iff convex conjugate l extension manuscript I redundant problem restrict state space assignment q nod unary pairwise potential mean mrf I find completely mrfs mrf proper mrfs label difficult hardness tractable obtain high soft obtain linear relaxation enable assign computer vision optical depend actual difference height pairwise potential costly potential potential potential w st piecewise segment show pairwise sketch
shape consistency computation foundation parameter censor qualitative censoring censor rather general assume concave concave borel independent constraint log rather enhanced require also review concavity value exactly one know right censor event viewpoint censor contain setting interval censor unit already give also contain interval censor leave open interval view censor mean censor inspection point assume censor benefit analyze interval censor unimodal accelerate rate compare estimator
blue volume densely thin path field small segment merge large merging determine likely merge merge segment anneal gold segmentation set scale map learn simultaneously compare highlight determine merge compare clarity feature indicate map region illustrate throughout pixel disjoint substantially cross true boundary agglomerative abstraction agglomerative long method choose hierarchical inherent merge region require definition adjacency merge initialize place totally hierarchical merging node merge edge merge policy node merge propose paradigm decompose v reduce step define gold standard assigning segment
adapt hash hash skip list node marker show store figure skip skip list hold negative hash marker leave currently leave marker list set leaf equivalently marker key hash include value property hash valid change hash marker marker value equal storing leave efficiently compute quickly full hash marker storing instead require linear valid store hash node skip hash leave hash node time algorithm nod single skip skip list hash marker th skip leaf marker node level marker great marker node level store reach pass equivalent formula node hash node hash hash object high skip marker skip marker I leaf prove validity involve detailed hash update move hash store motivate graph nod validity
incorporate exist idea svms concern hold probably improvement kind promise unlabele attract considerable collect unlabele collect expensive process semi supervise application video audio different view focus paper attempt inductive circumstance note view view still interested learn machine svms
ix h xu h ix u since comment condition mild estimation suppose strict positivity explanatory idea write adopt reformulate finite see furthermore many example determinant asymptotic property functional continuity order h suppose state result play almost h hold deal consistency whole give conditional median assume h h normality satisfied numerator say apply value jj define matrix n xu u u k rewrite h remark know nx k asymptotic normality
solve result follow continuity division consistently proportion eqn immediate proportion design discrimination rule iid random discrimination rule whose sample size previous address observe differ conditional q construct classic strategy erm grow family upon vc theory multiclass multiclass vc conventional vc dimension set erm risk write rf mf mf mf expression I mf fx mf mf mf mf discrimination rule multiclass vc tend
smooth function denominator sharp different exponent exponent f f tt thus prefer differ median total drive force formation function heuristic generalize multiclass simple quasi fact indicator total variation version indicator denominator sharp concentrate energy consequently indicator smooth problem know previous relaxation variation coincide np hard illustrate nmf relaxation contain handwritten figure algorithm four sharp smooth plot view
eliminate obtain component use candidate algorithm kolmogorov h cdf gmm apply obtain partition component gmm triple affect runtime triple subtract component rescale gaussian lemma miss defer appendix suppose theorem combine section appendix goal suppose select hypothesis behavior distance want improve run art interest make assumption whether distribution compare density definition mass discrete probability set selection summarize algorithm collection distribution parameter make nx nh number operation precede require modification algorithm provide admit though skeleton corollary al hypothesis work slow set require sample access hypothesis knowledge assign hypothesis large even description chapter perform comparable expect run algorithm specialize describe term hypothesis access pdf confidence return winner additionally winner winner property algorithm least hypothesis make
amp define approximate iteration amp rigorous accuracy approximate convergence gradient amp let denote amp suppose omit prove amp section include therefore accord lemma write yield minimize rhs amount decrease risk since speak seem sample hold variable hoeffding ensure value interested global behavior interested toward goal proof set first provide assume straightforward space step last triangle combine consider rewrite piecewise constant jump function monotonically decrease respect piecewise supremum achieve employ union employ opt e
rl multivariate nb marginal strictly great correlation applicable simulate multivariate symmetric bernoulli possible necessary create break unique necessary meet convexity exist unique verify satisfie
policy however evaluate except area position star compute grid square exact policy diffusion approximation exploration multi obtain follow exploratory move attractive reproduce feature surface show node show algorithm small neighbor diffusion benefit experience architecture fusion within centralize specific trade illustrate central would sample average maintain agent agent directly knowledge distribute communication neighborhood center multi communication diffusion size assume stationarity continuously mn r c mn mn mn e r ensure primitive stochastic frobenius c eigenvalue inside moreover associate c n c decompose r introduce shorthand perturbation introduce c form form except region
discuss new label get predefine topic topic reduce sampling topic evaluate evolve topic reflect discover far corpus reflect get topic learn document topic predefine improve word document compete log high evolve topic time rely document belong evolve accordingly reflect change word period document cover topic get change topic period plausible see real life closeness infer topic translate help word time period get big enough topic drastically match aware infer word real topic word log likelihood topic news base entire six hour news less finish finish document minute learner phase spend run test overall measure separately manual spend negligible spend news france news news monotonically increase document infer real learner newly document pass document point future train new news keep real could maintain limited history recently infer topic mixture negative system word topic outside limited document enough learner train live infer topic document retain affect run status receive system measure train I subset negligible result corpus batch size practically life almost exponential number kl grow linearly turn corpus corpus scalability two factor variational inference number document one note due brownian motion evolve per construction news near news discover topic challenge fall news importantly evolve rapidly short period set task document try merge vary always inferior approach evolve dynamically topic corpus infer associate event train use early experiment news corpus represent discover circle small dot represent document circle middle discover infer topic document describe early discuss month infer topic associate event steady office unable evolve distribution quickly enough keep track see date unable document overall unable correct non topic assignment news belong fail tag
q equality define family behave exponential family embed projection family give maximizer divergence among useful whose project lie via independence uniform maximize projection belong maximize exchangeable divergence
obtain well updating need primal factor algorithm exploitation orthogonality different machine practical ball machine smooth primal slight modification theorem benefit particular speed increase understand theoretical gram align machine diagonal eq dual regularizer split q orthogonality f update would speed machine although usually property
prior interval bivariate contour exclude case complete specify interpret prior contour random bind choose summary length uncertainty chance abuse distribution compute htbp choose bind prior display density minor density zero bind interval htbp interval posterior length quantity high exclude low interval upper display interval datum graph prior interval simple form accord valid expectation omit quantity interpretation follow serious drug market turn yet formally consider incidence effect affect relate distinction effect cause effect brief focus cause logic probability impossible bound however subject statistical uncertainty discuss illustrate conclude background year anti widely label publication popular book ir book health year hill focus heart high control examine million study heart medium drug hundred jointly les trial aspect focus company article issue cause heart
likewise return identical million million brief appear wishart fisher confirm consist length sl pl take double reversible scoring monte approximation set mc million run double reversible discard burn approximately ten ghz run recognize
functional subspace risk ij rigorous yet counterpart classification task collection point collection five distribution collection simulate form fig three unseen dataset substantial suggest cell recognize attack consist patient goal cell biological development cell population omit patient insufficient patient range cell dataset range development pooling pooling evaluate propose take patient subsample compare namely pool inter patient
whether generalize negative sum whose distance euclidean eliminate
r r qr version article make mistake bound statistic possess capability perspective estimator finitely infinitely space utilize obviously dimensional hypothesis bring difficulty phenomenon firstly observe wu suggest dependent call rkhs convert sample finitely many capability furthermore base regularization strategy greedy definite strategy regularizer property take consideration rkhs rkhs analytically smooth nonzero potentially take regularizer regularizer iteratively square minimum
evaluate scad mcp sigmoid quite mcp sigmoid error cancer classify tumor patient patient testing table although sigmoid penalty mcp sigmoid penalty attention propose argue balance result achieve consistency strong asymptotic stability usual normality issue choose standard cv criteria control concavity level computationally intensive cv desirable develop effective way select study approach develop parallel normality penalize log define let value notational follow regularity note c mild assumption covariate q partition intercept nonzero part ij l strictly th two hold maximizer maximizer op
precisely joint contract converge quantifie norm equivalently measure kl divergence accordance define produce absolute least geometrically converge geometrically probability give theorem I work sub either subscript q let vector belong denote eq know write monotone variable character complete present simulation verify effectiveness graph subgraph assume even simulation omit figure due illustrate path mp often might
lemma result loss factor guarantee outperform demonstrate factor substitution mean regret case valid relax yield select minimize front regret approximately minimize regret lead reproduce assume loss translate normalise assumption knowledge loss prefer normalise algorithm translate regret call regret go bound translate rescale loss rescale translate loss quantity denote omit prove assume towards follow also hypothesis round equal regime occur reveal remain change indeed note start flip fundamental corollary improvement loss let loss without optimally translate loss loss loss identity normalise four artificial learning keep deterministic consist generate bring close intuitively much well expert slowly property plot subsequently alg rate algorithm safe brevity note third horizon advance loss incur
estimate curve actual dark similarly bernoulli widely problem multi class certainly use bernoulli success write concave optimization efficiently track binary poisson bernoulli observation gaussian diffusion acquire wireless evaluate digital system environment platform receive node front end hardware signal generation front signal
average identify use patient conventional supervised cluster pca partition observation cluster semi supervise association gene survival cox centroid datum association predict survival cox sim produce scenario cluster first scenario cluster produce accurate compete two three produce simulation scenario mixed complementary identify secondary sparse complementary correctly identify whereas correctly complementary hierarchical two compete misclassifie error low method identify produce result conventional sparse principal score produce cluster pca singleton cluster simulation primary secondary
nonlinear activation deep apparent activation unbounded piecewise linear relu maxout unit find consist mlp weight incoming apply scalar eq define mlp hide q nonlinear input pooling operator widely convolutional cnn dimensionality convolutional use spatially neuron translation pooling operator pool max operator may view activation receive layer return scalar output max traditional element rather hide unit representative pooling maxout motivated nonlinear activation root discuss new unit signal indicate differ triangle satisfy center neuron illustration
random I naive random number repeat naive line avoid use perform replace predefine sampling pick return draw uniformly index noisy generate operation
bundle connection bundle result follow material diffusion map embed property define compare trace heat heat comparison spectral class riemannian lemma let empty close equip hausdorff map manifold close bundle satisfy connection geodesic laplacian bundle
gain comprehensive ill condition compare precision bfgs paper organize briefly kl es algorithm discuss experimental section summarize adaptation es discuss relate surrogate differ es schedule leibler generate refer working worth note denote space leibl define let model error k
domain knowledge reflect semantic relationship topic child death relate topic interestingly topic impose nd rd st child nd child recent furthermore topic strong association tweet use likelihood divide hold hold make fair apply apply hyper outer take sample apart use burn sample collect hold give outer harmonic mean illustrate twitter apply cardinality lda hierarchy well lda three case interestingly provide significantly twitter poor perform manual tweet assignment lda high topic node tweet assign topic lda twitter gibbs speedup overhead merge loading global system gibbs speedup process
k k q u methodology situation exist miss training completely similarly vb allow missing data methodology miss triple th vector otherwise datum mechanism respectively I p completely predictor general random case inference imputation inference mechanism adapt consider stored row imputation q inverse wishart I factorization restriction combine solution reduce multivariate u ti rl
software adversary extract instance technique scope attribute stochastic machine transition type process generate observable elegant hmms weather examine recorded ice sequence temperature hmms special g represent eq element emission markov depend state assumption transition emission observable figure show state emission probability respectively observable hmm solve type decode problem observe observable hmm decode well consist reconstruct sequence attack hmm sr process sound record acquisition sr audio etc technology tool exploit accord methodology detect confidence recognize speech two speech file word either probability contain predefine combination follow language model let typical unknown speech acquisition hardware provide audio converted term generate new contain move emission transition state right fashion loop make possible ease emission build
probabilistic consistency draw distribution however easy whether hold list g scad truncate regularizers gap regularizer strictly se eigenvalue integer minimum se restrict isometry satisfie avoid note n trial variance se regression gap parameter global suppose hold invertible decrease consistency se integer h order show se need much hope regression solution condition suppose theorem integer eqn rl
baseline generic extend numerous representation powerful mathematical tool entity traditionally graph relationship datum analyze light numerous form diverse similarity application multimodal relationship weighted share vertex weight layer multi unique rich single take expect combination improve relationship entity dataset I share potential unified layer layer graph common sake simplicity clearly capture combine adopt information layer method multiple subspace problem combine transform develop subspace overall representative subspace show justified dimensionality information contain multiple graph layer vertex solve relationship find unified illustrate utilize well address problem generic meaningful representative spectral cluster base representative real world advantage art beneficial organize work describe capture
cause add support point inside sequence critical point within add inside contiguous vary pass tangent respectively limitation arm multi dash line depict w xx tx contiguous interval figure support incorporate extreme adjacent tangent line straight tangent moreover stay first use proposal inclusion point different testing inclusion accept reject future transition property provide justification notice adaptive mcmc reach condition paper ergodicity metropolis condition proposal interpolation adaptive discuss issue update proposal distribution issue cost multivariate update rule conclusion suggestion research real normalize fix chain support propose metropolis htp metropolis build interpolation support point mh draw probability update eq iteration rely interpolation insight behind adaptation proposal accept reject metropolis hasting hence result algorithm strategy point see mh accept target proposal incorporate distribution zero cost keep bound iteration provide different choice quick target present
enable minimal communication approach require target parallelism world gain several tb per independent mixture mixture within gain dirichlet finite finite abstract dp discrete continuous component process form process allow possess cluster view exchangeable alone compute operator keep regardless allocation
send bi drop drop bi fail regularizer additional example original select discount work generative assumption relationship predictor marginal intuition confident unlabeled intuition entropy regularization semi describe unlabele improve single dataset see unlabele regularizer dropout unlabele train neutral dropout use unlabele benefit unlabele even amount accuracy analyze framework close
nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan row sep crcr header nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan draw black meta explicit mesh crcr false nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan flat mesh sep crcr meta false nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan mesh crcr header false nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan mesh crcr meta header false nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan flat draw explicit mesh table row crcr meta header nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan mesh crcr meta header nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan width height view xlabel reverse ylabel east anchor south bottom left line leave mesh crcr header false nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan meta row crcr meta header nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan black meta mesh crcr nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan draw meta mesh crcr header false nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan flat black meta mesh table sep crcr meta index header false nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan meta mesh
expect hx alternatively play suboptimal main basic ambient l input draw distribution hx thompson action state intuitive explanation sampling play setup action finitely action denote optimal index associate respective denote leibl distribution upon I play get simply marginal kl write count exponent loss play incur divergence upon inspection insight posterior sampling since ta mass show away
ratio close submodular modular analyze problem version call curve curvature decompose modular part curvature curvature indeed later since eqn submodular function evident definition monotonicity suffice monotone negative show negativity monotonicity fact part analyze hardness theoretic prove essentially indistinguishable depend random would distinguish enough ratio technique end curvature construction enable monotone refined bound approximate replace know submodular refine dependent bind table summarize dependent minimization refine new upper refine replace curvature make curve tight l modular ellipsoid lb n address approximate submodular submodular showing improve
repeat seem remarkable knot freedom knot rather filtering answer lie entry toward quantity equality estimate adaptively knot fit degree toward th spline knot give phenomenon fact great advantage solve primal solving reduce mainly filter argument well solve describe path solve piecewise path compute successive critical path operation matlab implementation filtering available repository summary know trend filter freedom much examine filter piecewise polynomial continuous order lower lead piecewise derivative knot nd rd multiplication resp derivative appear knot sequence filter piecewise polynomial st appear third represent evaluation rd section besides question trend filter trend achieve arguably adaptive trend comparable level spline adaptive spline propose suggest encourage solely would fix square error average spline perform locally spline question trend spline spline quadratic order trend filter spline spline derivative knot derivative filtering minimax adaptive spline trend locally adaptive practically indistinguishable similarity go theoretically regression spline lasso outcome trend
fold average reduce enough rate remainder regression discuss belong well provide several corollary exhibit consequence convergence rate kernel discuss rate minimax result aspect lastly far reasonably music begin background reproduce space brief reader book detail define hilbert strictly contain contain act mean norm denote eigenvalue orthonormal basis expand coefficient simple calculation rkh elliptical define non consist draw distribution goal minimize pair estimator combination square penalty square hilbert ordinary ridge parametric span function x computation dimensional matrix I work estimation description statement result namely theorem provide trace kernel apply belong space involve combination error trace class
necessary genetic quickly restrictive genetic programming experiment population length height initialization ptc gender proportional ml success pressure consistently accurate actually slightly program often drawback negative correlation forecast process covariance synthetic dimensional
wang guide regularize pt wang abstract cross demonstrate simulation hold consistency cross therefore sometimes procedure include scad fan performance crucially intend list aim select exist purpose tuning selection
evolve article branch create corresponding old article remove soon current pool particular recommend thank tree possible model iteratively small correspond hyper two hyperplane splitting half center one axis depth cycle possible axis analogy ct hyper rectangle center ct consider child distribution system possibly tree assign tree associate compute context tree identify fig expert recommendation active ti derive tree associate usefulness expert take account relative usefulness expert let stop th I probability
odd proof show sharp suggest classifier interestingly performance odd superior probability probability target unconditional standard solution describe give produce meaningful contain simple solving interpret ratio present approach often estimation imply iii implie
boundary short notation model define ideal model normalize especially column cell row cell column contain matrix label table study agglomerative hierarchical cluster contingency merge obtain choose since row merge cluster general agglomerative intermediate step merge step merge column use column constrain define independence
get let q remark q remark relation guarantee space smoothness q show smoothness dimensional banach polynomial example coherent take dictionary covering exist cover value radius cover time exceed space basis unique equation
dynamic learn network denoise autoencoder yield temporal towards quality simple way quantify fidelity certain generate generally suit certain fill absolute use quantify quality approximately train conventional across different modality take competition propose confirm improvement generative rbm furthermore performance increase limited scale memory encode autoencoder though greatly temporal propose autoencoder end denoise autoencoder
density lebesgue proportional indeed suffice therefore reversible desire stationary trivial ellipsoid give geometry procedure ball happen away precise upper continuous condition information valid choice say ellipsoid follow sum linear lipschitz affect condition radius ball notice condition calculation become universal change line omit sake order mix early bind general imply let reversible ergodic fy dy relate riemannian apart constant inequality let lebesgue satisfy size like outline early need poor markov step track distribution
explain posteriori experiment allow evidence calculate numerically integrate serve estimate evidence standard deviation exchange auxiliary generate likelihood importance chain towards important temperature important approach make transition choice joint result equivalent population exchange choose pseudo observe restrictive quantile draw performance ccc evidence tolerance quantile
cg use bfgs accelerate combine roughly speedup news attain informed statistic well algorithmic com become popular deep neural aim solver implicit l bfgs avoid bfgs iteration solver retain desire convergence counterpart geometrically utilize english news find speedup suggest grow extensively explore curvature network hessian free recognition addition technique recognition method great success dnn provide relative improvement
link exist latent assign latent interpret membership interest node belong simultaneously bernoulli membership model multinomial membership membership amount probability normalize membership belong time adaptively dynamic specifie dynamically join leave join leave markov node join membership evolve evolve entry correspond z link link occur account link member member join leave markov dynamic membership assumption hmm explain network link membership certain belong particular join evolve membership determine membership markov probability version carefully membership time comprise link
shrinkage especially extent compare country mean base great similarly dramatically country great dramatically country show cross validation absence direct country literature describe analysis context complex well country independent meaningful well spectrum shrinkage possibility prefer shrinkage effect thereby severe choose particular country posterior whose away toward specification ignore analysis child likely tail reflect nonetheless potential scope include additional covariate covariate lead assume constant investigate allow country autoregressive vary height age height z outcome rapidly time real poorly identify relative study survey conduct year even large survey differ observation fy ease notation distribution determine variance statistic turn covariance define sum central px px px px e p bit ny c truncate x
high guarantee approach symmetry furthermore novel place special keyword thresholding guarantee arise environmental sciences finance matrix high size comparable number often estimator life quantity inverse often negligible amount correlation zero reduce conditionally independent zero reasonable publish network identify focus many method definite however definite likelihood sparsity log setup glasso problem fast early year interesting fixing identify minimization example accumulation objective need contour finitely latter iterative start example converge quickly possible population standardize run choice iteration successive alternate non see provide edge zero summarize time weight converge implementation iteration weight exhibit simulation h c nc symmetric minimize negative pseudo likelihood denote objective penalty element replace wise efficient property yet establish lemma non symmetric bi parameterization jointly section check unless theoretical guarantee strictly clear descent symmetric especially estimate particular formulation penalize pseudo parameterize diagonal coefficient
addition inductive completion sense b label miss provably rank sensing improve rip sense problem motivate netflix address goal entry completion formulation world isometry rip see rip rip rip sample rip constant operator storage computational computational measurement drawback unlike operator need movie recommendation x user movie goal completion accurate user rate movie incoherent entry inductive require generic optimality sample e completion pt consider variate goal regression rank entry learn entry label
eigenvalue outside circle unlike decide outside particular circle indicator network generate suggest political book might group rather might group note clique count community backtrack important matlab reproduce large eigenvalue spectrum sign second eigenvector two community method community advance model g highly competitive efficiency sparse linear large belief close backtracking generate block spectral fact
original method strength well fit fit data correlation substantial indicate hyperparameter paper review multiple datum aspect scientific multiple review hyperparameter method combine inaccurate systematic set datum prefer non rigorously greatly hyperparameter weight hyperparameter recover inter set hyperparameter analysis correlate straight line mis report systematic difference hyperparameter illustrative bayesian matrix demonstrate bayes hyperparameter heavily hyperparameter context draw combine act necessary consider correlation temperature metric galaxy survey combine survey volume matter survey neutral survey combine hyperparameter summary hyperparameter unbiased
p normal line sde solid significantly eventually cauchy fail maintain misclassification line line line show mass contamination except reliably find look sde figure suggest sde line reliably misclassification reveal often n normal line sde line solid cauchy distribute show line solid show percent superior sde h sde consistently maintain bias misclassification insensitive low consistent robustness case practice place observation assume adapt member approximately direction sde equation effect outlier consider figure contamination maintain
resp resp resp proposition ie application definite exist converge center asymptotically sequence project
add one correspond bin histogram bin notational convention subscript generality non necessarily histogram I theoretic dissimilarity correspond histogram remainder operator divergence kl I I shannon entropy explain expect use hidden histogram jeffreys divergence divergence jeffreys hold histogram devote efficiently jeffreys centroid histogram
posterior calculation equivalence likelihood run begin mcmc general total offline chains simulation mcmc problem overall mcmc simulation describe negligible whenever fig show take full order fast ratio search light primarily evaluation factor implementation sample depend various aspect allow range parameter duration carefully sampling reduce observation compression effective speed typical elsewhere investigation speed currently red dotted computation time speed quadrature one figure mcmc ref stream modification design monte adapt stream expense model specific rule build file stream give set computation perform implement exist mcmc manner consider frequency wave increase fidelity ref basis represent stationary
absolute envelope every conjugate radius enough attain envelope strictly trace whose follow whereas fact obtain consider class tensor choose singular np remain show every use fourth identical discuss proceed ready tensor tensor tr tr lemma describe lemma tensor satisfy claim
regret take resp resp resp resp know hence call instantaneous equivalent easy recover optimization permutation case encode aside n denote define fix single unique ambient formulation cost ranking cost incur study general arrive regret argue update note regret thm thm system observe feedback adversarial st nd sum position
laplacian reason property harmonic harmonic unbounded role optimum key yield family detection furthermore zero equal number connect component though subspace strategy optimality pearson detection alarm refer alarm refer detector log detection develop detector ratio distinct mathematical algebra foundation detection foundation introduction theory laplacian problem partition connectivity analysis partitioning algorithm establish connection algebraic property spectrum laplacian subgraph connectivity subgraph edge necessary separate quantify subgraph laplacian application spectral network small laplacian fail discriminate subgraph intuitively solution make sense subgraph entire well offset indexing subgraph call spectral completely analogous comparison riemannian topological algebraic topological tie example
quantile simulation kolmogorov statistic dimension integral approximately grid space four consider bivariate recursively third autoregressive fourth last value recursively latter sp daily width width empirical mse versus bandwidth copula corner correspond follow copula consider copula value copulas resp resp copula tail tail chapter value consider supplementary estimator display order middle parameter segment corner correspond bandwidth ar look three copula use replace er von kolmogorov manner shape much affected analogue figure kolmogorov find resp red panel dependent multipli sequence consider move approach successively reason facilitate reading obtain plot scenario tend seem always dependent bootstrap covariance empirical looking accordance state low
advance convergence also present apply report theoretical numerical encouraging propose turn easy simple scheme analysis broadly practice iterative generate convergent rate noiseless numerical tensor application interesting investigate effectively iterative appropriate predict solve problem moreover worth investigate thresholde nonconvex model acknowledgement like tensor hc partially support national china grant pt school science china rank vision difficult replace tractable solve include fp
bar tight heavy user govern predict edge collaborative show incorporate extract largely popularity bipartite user model parallel pursuit employ rich similarly meta graphical partly negative alternatively know fix zero process draw adjust sake clarity management engineering rgb class collaborative generative form live architecture consider treat unobserved random connect demonstrate descent variational grain state art world mathematics paper highlights prediction association implicit live medium movies negative predict connection chance link introduce connect user miss odd secondly probability
mini batch batch minimum il l could eigenvalue small offset iteration mini batch sg uniform sampling asymptotically sag iteration regardless frequency sample bias beneficial change slowly quickly justify strategy lipschitz individual gradient place constant ill depend simple replace appear come sample extreme integer sum lipschitz across maximum improve function equivalent lipschitz simply original proportion lipschitz explore somewhat iteration depend however preprocesse use lipschitz size change lipschitz different lipschitz must sag sag compete evaluate mini strategy whether sg em pass sg pass sag iteration like bfgs whether practice evaluate sag sag regularize problem regularization range small would typically practice condition problem would focus benchmark binary website set website causality website website add mean variance measure pass datum time divide take advantage set examine time l variable reference synthetic experiment follow sg gradient regularize regression publicly method tune logistic since issue choose among power size strategy give decrease step size sg step power sg momentum
approximation interest computationally em space pseudo computationally unbiased practice smc block method artificial dynamic standard smc optimization involve tune degeneracy line efficient applicable per locally ml restrict model line pseudo need stationary degeneracy small loss per ml method estimate however numerical either bias tuning ml base compare applicability ml bayesian contrary allow free related ml method summary present next motivation line assume practice miss common available develop recognize literature unfortunately miss dynamic treat simultaneous estimation simply refer unless otherwise inherent base simultaneous state particle sir sir filter filter noise weight evaluate importance arbitrarily accurate increase particle several
blue green long decay away medium code however view human user use device flexibility recognition discover encode spectral mixture prediction essentially equivalent entirely datum sm kernel black discover data long square se rational periodic pe dash sm discover sm learn log seven automatic determination good spectral density peak correspond period relate month back little confident future peak prediction restrict rate reason finally peak month peak reflect red learn frequency identify sm se mixture origin poor combination
et course co allow ar ignore place fully iterate newton regressor co suitable df aware estimation call misspecification easily business recommend df misspecification business rely subsequent step deterministic
delay efficiently effect perhaps somewhat surprising fast scenario information delay reward monitor forecaster reward instant pick become interaction protocol delay bandit treat monitor adversarial model identically distribute may absence singleton delay forecaster depend delay may change observe delay assumption delay table ht full l side n side gap feedback adversarial formulation delay positive maximum step delay partial monitoring knowledge delay consider adversarial delay minimax run predictor form adversarial usual achieve consider set adversarial
nest span space introduce discuss section summary discussion element nonnegative nonnegative nmf define eq th column approximation svd component eigen mean vector q eigenvector covariance calculate value loading th large good addition svd identify fashion forward fashion forward pca backward svd decomposition sum non approach section triplet equation take note essentially unit presentation skip definition orthogonal negative even though lead nmf development nmf review nmf
lasso elastic statistic learn decade matrix py reason often
base field discrete mathematic physics statistic paper consider spin sp introduce ref greedy agglomerative method ga ref propagation label lp ref sp community hamiltonian kronecker delta determine detect coupling couple many inside scale community community minimize hamiltonian use anneal initialize quite intensive hamiltonian structure ga agglomerative hierarchical merge govern modularity determine modularity community label decide iterative step begin select random assign break tie neighbor repeat community densely community propagate spread secondly break tie responsible method lp essential next community share many area combine average method network
text material complexity find spirit forest could classifier prediction aggregate ensemble property true sophisticated randomization scheme start node tree figure cut decide manner median datum median contrary thus leaf region least equivalent consecutive split dimension rotation impossible force employ extension specify region remain specify aim stay child speak able datum fall detail calculation convention break tie cell estimate majority vote convention cut cell observation total combine
show particle self surely one grow monte carlo error method least relevant instead equivalent extensively g filtering consider observation precisely satisfy evolution multivariate field evolution pde interact associate evolve resample many state framework carlo complicate promising strategy consist recursively simple close distribution
divergence primitive divergence square correspond follow derivative operator every eq dominate convergence definition vector mid denote extreme important extreme topological space shall compact element concentrate functional finite continuous equipped see shall identity pointwise see observe pointwise indeed divergence compact subset easy pointwise define finite concentrate every monotone q assertion lemma similarly prove complete continuous compact equip sequentially compact sequence fix convergent subsequence existence subsequence pointwise rational denoted choose use fact lipschitz supremum infimum extreme equal measure conclusion also involve divergence finite theorem necessary equal space positive number I generality function respectively write
nonzero cardinality support often norm cardinality subset function proper term problem advance matrix follow solve indeed mild ff fy imply e nan lemma signal incomplete information unfortunately hard author condition eq index relation reconstruction theorem object unique necessary simple realistic dense exact
suggest point uninformative fully observe improvement nystr om feature table vs reduction avg reduction std discussion htp r l algorithm combine view nystr state depend label average order medium dataset gain develop cca correlation preprocesse nystr om since cca give criterion canonical informative index help implement nystr om c ridge report percentage average experimental width far importantly far outperform supervise l avg fast regression generate computationally
locate multivariate location estimate change multivariate significant detect argue recommend multivariate ht rand rand two rand evaluate examine segment rand index popular adjust rand point procedure segmentation univariate provide average report similar rand adjust rand package different follow r rand rand store adjust rand adjust index arrive c c rand standard simulation e consist sized distribution mean
time acknowledgment author thank many valuable comment manuscript cm l universit des centre correct uncertainty propagate lead bias basic statistic central moment fourth uncertainty unweighted skewness outli normality rare might help distinguish spurious employ central moment moment herein extend uncertainty whereas common series observational affect
extend add impose origin extract post robust index processing project matrix identify norm correspond matrix th update w process identifie improve bind proportional guarantee ill practice post synthetic bound affine read satisfy combinatorial output origin w bind analyze focus nmf separable nmf significantly th satisfy identify bind much process handle case active synthetic combine apply hyperspectral notice recent publish near nmf sdp might separable
tradeoff prediction classifier base become entry belong positive applying aggregate meta performance approach meta stack stack train classifier perform stack heterogeneous may meta classifier avoid overfitte typically intuitive weighted create nest split evaluate split prevent stacking use output single input stack instead variant stacking cluster separate first take level aggregation measure pearson correlation stack
ellipsoid ellipsoid reader detail ellipsoid half e fact ellipsoid affine continue follow combinatorial try understand interpret result v h existence separation polytope span theorem bipartite bipartite eq bipartite e inequality inequality hard follow cycle cover polytope direct direct disjoint cycle cycle cover easily perfect polytope bipartite correspondence generalize general odd separation polytope trivial characterization rao come polytope count problem distribution span successfully asymmetric algorithmic max cycle cover becomes computationally consider complete directed goal cycle cost formulate follow elimination h direct edge let solution make nx uv interior span span crucially polytope cycle cover see easy cycle cover algorithm et natural solution cycle max entropy marginal edge cc vertex return implement polytope sample distribution polynomial cycle technical condition bad case open polytope feasible box contain interior convex note supremum program hence duality eq sum complete satisfied center restrict interior radius
tail specify unique write lagrange associate normalization efficiently heuristic often unsupervise unitary contraction amount minimize decay coefficient discriminate supervise find operator maximize distance distance know class define optimize since
state page quadrature segment use computed quadrature segment present write coverage probability property determine empirical odd computationally coverage choose position knot knot remain coverage constraint implement computation
collaborative secondly fit alignment free collaborative collaborative exploit possibly multimodal application real fmri data matching present matching perfectly characterize sparse zero realistic tends doubly minimize spurious want metric non entry formulate lasso group group active remain active zero norm ideally match minimum matrix computationally relaxed
covariate support illustrate running group adaptive configuration increase grid furthermore compare support set matter represent water spherical represent water diffusion template matter voxel voxel covariate real white orientation white matter age coefficient voxel cosine
pool restrict rotation invariance auto detector possible spatial auto pooling pooling try meet invariance pool representation frame object property input pool representation could reconstruction pool representation sequence convenient frame image frames video frame
hold fix atom second sub shift atom atom shift invariant one construction subspace overlap pair atom case signal generate dictionary atom atom produce union ratio point span sub combine element subspace coefficient sort cross spectra monotonically let principal direction angle span draw distribution coefficient point unit sphere eq accord simply restrict total component ensemble experiment number subspace signal overlap probability dimension display logarithm row fig display coefficient vary overlap conjunction ratio equal per bottom fig sparse overlap energy subspace dimension phase display average predict impact subspace long occur cover obeys phase transition degradation intersection densely phase shift dimension intersect subspace sample densely cover ensure subspace point subspace overlap width common energy conjunction reduce bound union phase shift confirm intersection overlap another experiment reach boundary remain reduce transition nearest nn performance spectra ratio structure spectra top signal row show solid nn spectra
regularizer aforementioned likelihood term prior product conjugate normalizing observe give edge bundle pseudo prevent posterior bundle estimate categorical distribution parameter node belong flat form maximize q update independently bundle lagrange enforce set value initial guess tolerance z ta z
unit noiseless say give empirical ratio implement matlab respectively different measurement measurement understand illumination obtain illumination suggest choose discrete fourier transform multiplication measurement measurement measurement conduct experiment set vary signal generate magnitude empirically alternate perform similar problem robustness noise however analytically alternate bound establish represent albeit
rbm successful increase penalty focus mix theoretical practical overlap group penalty expectation probability consist mixed regularizer term define mixed expectation divide group group overlap individual overlap overlap issue norm sample behind application unit represent activation probability
observable live training recognize train although assume exercise generative distribution refer refer
connect graphical still variable method systematically replace conditional conditional match distribution auxiliary ease presentation assignment trivially employ scalar illustrate algorithm support ni ni ni ni ni keep value j nj nj initialization guarantee remain support scan step throughout employ fix traversal variable sample state make treatment graph distribution
simulated randomly number uniform record censor compete risk way consist censor individual primary end cutoff impose censor poorly suited assume monotonically decrease c underlie hyperparameter infer event occur note beyond end cutoff see great convert censor year interval non censor plot implement gp hazard rate support gp hazard infer relationship disadvantage hazard infer case describe relationship covariate hazard interpretation hyperparameter find uncertainty censor censor convert interval censor generate random infer capable non monotonic patient non effect level relationship examine gene infer individual see top function ignore definition square predict event report time event training superior unseen event
multiple write short graphic program computer graphic generative bayesian execution probabilistic graphic program implement generic transition library simple implement synthesis uncertainty introduction conceptual efficacy likely beyond variable currently proposal help train graphic appearance scene generator adjust fidelity graphic originally generative formulation combine probabilistic graphic purpose automatic describe read sequence character infer road camera
node binary useful expression necessary via discuss upper become arbitrarily block edge denote must upper nest ensemble encode sequence start select branch level one minimize infer eq appropriate possible lead trivial result one employ simple describe without principle well model hierarchy miss lb necessary partition equally compute assume case therefore block recover much partition choice consequence block degree correct low belong entropy implicitly uncorrelated block interpret description worth note poisson case correct need correct description easy hierarchy recover comparison ref description length nest flat e nest short predicate need method integrate ratio criterion criterion expect appendix integrate since stochastic nest block base capable block suffer resolution module formulate generate plant observe answer amount exhibit exist block weak partition model possible correct precision arguably case
move split sample average procedure bias capture quick low left draw generative split software package posterior quickly chain parallel mcmc provide experimental result partition onto primarily unconstraine theorem exercise communication cost synchronization requirement greatly paper markov monte subset sample four parallel mcmc sample provably full third package act post sample
experience passive question exposure protocol affect increase exposure testing expect leverage massive representation brain neural related efficacy stream determine investigation primary large visual well scaling measure performance provide fmri inference behavioral believe make advance recent intermediate evident supervise advance variation variation informative purely unsupervised vision methodology representation protocol measure implication discover necessary validate learn context domain may serve canonical artificial intelligence national science nsf advanced research project national thank le help evaluate appendix throughput class perform
correspond vector expect correlate unit basis share many coordinate handle broad decompose coordinate inequality term inside prove theorem therefore need combining imply desire vector linear search inner span vector aside happen side small optimum decompose coordinate coordinate prove concentration argument inside follow consequence measure stay inside set beyond closely hardness conjecture give reduction thus likely family graph dimension consider solve small instance hypercube short hierarchy hard instance candidate plausible span eigenfunction approximation algorithm family detect case polynomial coefficient appropriate relaxation nonnegative appropriate character let index eigenfunction eigenvalue degree additive approximation expansion graph relaxation degree distinguish set case constant weak algorithm yield improve natural three problem give time approximate variate spectral norm long theory lead find vector relaxation vs substantially plant subspace recover whenever recover plant nonzero improves require algorithmic computer science field especially systematic relaxation add researcher several hierarchy round enumeration
cell neuron know entropy spin however applicable non record analyze activity develop neuron ise method repeatedly record identical typically time fire stimulus trial fashion high occur apply recorded primary reveal neuron dynamically order triple wise depend behavioral decade neuron
number grow empirically figure support substantially mean visible observational notably measurement modification likelihood weight matrix global intervention determine parameter via calculus imply intervention reasonably since link intervention identical standard markov equivalence realistic degree implication tight infer besides derivation proof dag selection prove theorem exponential expectation write precision form circle circle pp canonical form exponential especially estimator causal hence calculate simplify use identity intervention formulae identity I claim check identity identity consequence identity formula fact every matrix dag actually r identity representation conclude prove finally single provide formula intervention
classify filter document slow manually dictionary result speed heuristic english slightly clearly role minor form dramatically remove ed document far ready main goal recommendation paper recommendation user henceforth observe characteristic describe key word abstract paper every widely system item situation netflix nonetheless accurately author conference proceeding
appendix even need j tucker necessary condition maximize np p equation depend verify v n get I ip ip two assume optimization unique solving solution satisfy numerical although answer provide totally numerically typical generality linear combination row l still valid base argument restrict l l h otherwise section pre point previous section
label model allocation hierarchy topic neighbor allocation encode label give scene multiple object context semantic single allocation commonly model encode encode direct dag topic level explicitly among enable fine grain occur together book occur room hierarchy get topic capture manifold capture spatially relation facilitate multinomial count avoid quantization keep multinomial represent near neighbor bag construct grouping geometrically location
contract offer bundle horizon bundle pay payoff none encode preference draw neither expect let bx contract accept represent wireless service gb contract month payment gb contract payment gb loss mb contract month mb course relate analyze either select contract reject contract accept know give bundle
shrinkage estimator upon regardless direction specify exist mean inspire propose systematically let kx functional functional g functional minimum one computation construct shrinkage modify monotonically increase shrinkage shrinkage span firstly zero correspond discuss call interesting functional particularly shrinkage eq differently notice fundamentally shrinkage well pose analytically theorem eigenvalue know n hence consequently ij n eigenvector kernel th eigenvector consequently
drop predictive seem simulation substantial distribution expect gaussian non scale data novel approach conventional methodology stage generate attempt produce large even one min wise hashing datum retain fail hard immediately derive min hashing hashing matrix computationally improve statistical reduction follow maximal sign min distinct complicate regression engine million computational ease standard wise hashing thm thm example section regression million become typically percent design zero feasible approach obtain compressed bit scheme despite encourage model vanishes ridge interaction modern powerful deal dataset may greatly number overview may thousand typically motivated shape size area define big increase arise text web million particularly imagine situation infeasible computational reason sparse majority signal dimensional indeed matrix contribute response yield demonstrate achieve sensible way task perform
datum couple suggest threshold model suggest drive individual function paper focus predictor specification effect beyond model include surface tensor spline interaction effect smoothing fitting however heterogeneity presence covariate express underlie explain temporal include example g gender resource availability condition specify generalise linear covariate probability covariate little investigation relative fit spline functional relationship frequentist analysis mark recovery analysis covariate none approach individual covariate evolve stochastically consider specific deterministic age corresponding proportion schwarz covariate substantial body fully parametric covariate approach approach derive covariate imputation approach integration inferential allow individual covariate unify focus
design example recently communication work repeat proof right side rearrange q combine next update equality gradient combine get sum eq round show conjugate function simplify definition lemma definition take proof eq condition lemma take apply inequality recursively claim remark algorithm dual
mechanism rich identify relevant implementation detail synchronization fire von reading spike neuron information neuron whole area relative spike carry invariance reading fire rate dynamically distribute content represent coherent visual demonstrate potential functional amenable deal realistic elaborate neuron mathematical naturally extend deep still biological value neuron fire across briefly describe complex network functional role employ success net train approach focus bind principle challenge overcome note value neural orthogonality nature attract attention benefit explore include potential fire read neural unit network relate input scalar output interpretation fire notion incorporate notion neuron receive train spike identical frequency plot b phase input rate average run represent neuron
coordinate appearance prior prevent support datum transformation count observation certain population infinite production uniform limit transformation approach specify measure course investigate prior b q shannon discard jeffreys b root likewise density similar give conjugate prior discount physical appearance beta axis neither jeffreys appropriate severe modification improper lead entire devoted appear paper brief analysis beta volume group engine whether rank site site genomic observable examine beta type conclude finding summary transformation discuss complicated strategy derive broken history physics energy momentum discuss
primarily optimization convexity come loose round still convexity ease sense sdp whereas qp model mnist dataset rbm modern hardware hence qp sdp relaxation denote qp
denote notation previous section reader strong point study scalar l iterative represent pair f dimensionality reduction save sir inverse inverse dimensionality reduction without dr giving low seven run vector utilize dimensionality technique dataset tune use cross cross validate root square
low us insight pseudo ai I I th adaptive admm plug trivial admm formulate I construct give estimation assign web list news website divide
proof conclusion rate exponent minimax sense exist assumption second fast rate I massive compare upper occur satisfy entropy variance comment consequence intersection dimensional roughly rate minimax risk empirical exhibit constant parametric class class theorem subset euclidean subgraphs type assume aggregation estimator three procedure enough depend excess risk type discussion rely either tight function estimator differ fix extra deal possible class study behavior geometry fix refer comprehensive design case intersection ball simplex radius type aggregation consist construct attain mc aggregation constructing attain ms sparse aggregation modify mi mf jj
activate concentration protein increase action reduce dynamic induction scale comparison gene hour reduce switch two protein coefficient degradation rate gene protein per light induction realistic delay less hard analyse state steady unit validation situation biological example population cell goal optimally switch couple trajectory amount extremely hard rough model fine exploitation trade outline
reweighted gradient improve function replace estimation procedure always available final hyper free generally form stochastic sgd among broadly learn robustness arbitrarily dataset many instead one local optima stationary environment change stationary even search local locate increasingly wide tool benefit hyper sgd
strategy smoothness simple obtaining simply use set bound ds remain hold care entry copy hold see ds f kk proof probability ds make imply perturbation state lastly assumption axiom conjecture exercise ball parameter mean smooth result efficient algorithm round budget assume know armed give subset receive accord reward assume round round goal performance measure expect sequence play
directly account nonlinearity manifold hilbert induce via hilbert schmidt via embed reformulate embed schmidt operator location hilbert manifold extend infinite embed hilbert space minimizer mean element mean I set extreme define unique point minimum distance sphere embed inclusion mean give respect call eigenvalue maximize vector minimum also asymptotic mean p jx final random mean decompose normal covariance develop space adapt test delta hilbert embed hilbert hypothesis procedure hilbert respectively fr hilbert normal tangent define testing et test type size hilbert space since prove test distribution give covariance
also layer lstm layer lstm try layer normally layer number lstm layer active layer concentrate lstm almost therefore lstm unit besides avoid dropout big report table since size layer rnn overfitte feature layer dropout decrease almost relative find generally helpful present database rnn sequence character recognition language greatly decrease lexical optical compare train
alternative jeffreys via fisher say effectively jeffreys see thing unlike conditional jeffreys use prediction lemma give jeffreys mx x exchangeability joint initial invariant permutation ask read three exponential define lemma theorem contain key idea reasoning result short appendix provide definition repeatedly unless state family mean geodesic space statement mean geodesic parameter
fit maximization another observe degree account mix popular slight present infer mixture impose update scalable sense iteration link word algorithm converge iteration make corpus variety include document prediction hard likely label prediction subset link ask document determine relative weight content set thousand organize generative compare conclude offer direction give variant string word link topic play content link topic distribution word document word generate associate link poisson
build derivative tool model estimator observe possible quantitative monte carlo extensive estimator acknowledgment like acknowledge denote center extensively odd use literature reference theorem base fix relie divide part expansion stem first us expansion simply remainder eq integrate multiply group q
cancer tumor activity gain gene identify cancer sequence microarray model hide markov intensity sequence state nucleotide microarray dataset bivariate snp location spread intensity measurement number genomic genome value normalize correspond dna copy one copy parent sometimes allele relative contribution allow lose gain sequence model denote expect signal measurement copy state copy copy super unlikely produce relatively super segment model embed chain observe real primary switching super state fully copy super ab bb aa bb full estimate map viterbi backward site wise posterior snp microarray log allele genomic non viterbi wise analyse exploratory information super obtain segment effect super exclude copy super mean represent probable super segment suppose state allow exploration retain site wise marginal probability apply segment retrieval document extract upon indexing latter keyword distribution mixture multinomial distribution latter aim etc carry manner several document document word nearby motivated construct semi supervise unknown content test scan retrieve topic order appearance text hmm assume topic relevant irrelevant topic topic one appearance etc
extensively task paper fusion ensemble art batch bag boost sound online bagging boost first unlike guarantee experimental evidence benchmark datum propose bag ensemble cost sensitive imbalance medical diagnosis spam automatically detect incoming positive majority algorithm work implicitly misclassification classify additionally often case much mis class imbalance within sensitive imbalance stream unfortunately require eeg rare brain activity high meanwhile clinical eeg must real though favorable adapt subject subject large memory positive rather static phase incremental framework example build deal imbalance imbalance effective stream incremental decade solve imbalance sensitive ensemble bag base version analyze counterpart show certain infinity ensemble converge long incremental technique convert insensitive straightforward modification propose
statistic enter versus nan cutoff would actual enter vertical quantile versus explicitly account nature appropriate adaptively example methodology anomaly general direct significance scheme split resample technique aside significance aim significance test predictor adaptively next statistic propose construct lar trace decrease therefore affine exclude affine span sign contain path column e assume continuous almost regardless define path knot active variable mark entry removal resp active index independent set knot path usually loose realization sign particular leave step condition restrictive e remove active therefore knot try consider active test statistic quantity predictor active predictor perfectly intuitively covariance respectively fit ask evaluate note restrict verify variable upon choice entry variable variable secondly ask statistic difference roughly think really empirical covariance small orthogonal last exactly hypothesis role seem statistic admit distribution section truly magnitude nonzero inclusion stochastically possibility figure fully convergence quantile match even expression freedom review detail cancer level cancer
column minimum distortion column error misclassifie reference show induced note show visualization adversarial network experiment conclusion example stay train hyperparameter adversarial error fc softmax softmax fc fc sigmoid autoencoder fc fc fc fc fc fc fc fc open hardness solely error distortion fc fc fc train fc fc fc fc fc fc fc fc study partition
parametric use four contour error volume separate summarize tree volume input model dimensionality second list sample lebesgue volume bias design fraction volume notably distribution case significantly cell result center sample table illustrate around relative posterior case table identify list error
negative essential solution carry art show contamination thank dedicate negligible tuning experiment setting algorithm competitive spectra article however sparsity work extend different fista national anonymous improve clarity help bss noisy datum blind bss research present efficiently retrieve sparsity enhance source produce paper introduce tackle blind separation negative show sparsity non negativity solution solution sub propose name proximal calculus constrain variety negligible tuning particular synthetic mixture spectra bss nmf diversity many mixture identify elementary order different mixture blind bss recover mix instantaneous assume linear coefficient
college education bp higher among old people highlight heterogeneity correlation plot present correlation see school white college age correlation largely show heterogeneity easy interpret interval parameter figure much interpretation coefficient compare group baseline interval similarly kk group baseline serve coefficient interpret finding figure coefficient significantly higher indicate summary age compare consistent finding group compare group ht ht general develop risk co pay risk common mis desire like model characteristic approach simulation basic
continuously interaction system extensively environment extreme example predict economic cause economic market concern metric financial actor financial market game metric feedback like finance education macro economic emphasis understanding interact feedback exception many artificial unable detect single author grateful suggestion interesting w stanford fellowship fitting outline forward standard expansion recall intercept construct construct however evaluate careful integration separately fast fourier onto computed q
freedom hence law proof put imply time subspace orthogonal challenge pose big study discuss dimensionality provide analysis performance concern desire measure either value root square singular value measure value svd dimension factorization complexity desire entry form approximation onto mean hope accuracy procedure residual subject well
rating rating test consist rate movie purpose dimensionality factor depend factorization cover rating cover factorization use q reduction result part original recommendation depend proportion consider svd calculate assessment show calculate clear factor improve mae ht c neighbor
go traditional offer fmri imaging analysis expanded validate understand music stimulus diverse fmri fail result author article original version appear fmri analysis learn process international work compact diffusion component analysis understand human eeg functional
process store case candidate absence real selection candidate possibility could delay possibly due support likely two stream stream stream candidate search stream summary specify label goal produce candidate n fx positively candidate recommend inspection stream candidate ever receive discard receive certainly substantial scenario stream feedback stream contain label crucially stream far yield
relation illustrate empirically field data air field high cn closeness argue valuable variability insight complex science meet volume grow like project quantify association linear pearson analogously trace flow surface air temperature detect community enable prediction prominent mode recently employ forecasting episode south derive early indicator upon recent variability cn contribution cn empirical couple cn surface air cn contain approximate flow influence insight tool science meet analysis increase volume observational like couple structured describe analyze eigen relationship observational lead cn conclude merge generation cause uncertainty study base www http www uniqueness retrieval originally degree anomaly resolution ice exclude raw set north method
early change trick procedure convergence procedure martingale exploit procedure moment sr sr equation exploit martingale procedure detection improvement carry complete accuracy provide show quadratic specific confirm accuracy wide moderate contrast large range remain rough de read valuable author grateful university comment improve manuscript lemma section mathematical sciences york york usa mathematics california california usa cm york york usa correspondence mathematical sciences ny usa mail powerful particularly technique asymptotic detection technique develop generalized procedure length integral equation identity martingale improve though
north play key role uncertainty observational aggregated computational effect unclear approach infeasible datum develop basis expansion uncertainty dimensional deep specification discrepancy projection efficient dimensional computer calibration complex physical process modern phenomenon uncertainty involve characterize observational refer compatible assign observational reduce uncertainty challenge expensive sound need uncertainty utilize calibration potentially discard information scientific motivating projection north dense north warm water persistent heat considerable cf publish perturb run university problem start vertical project mixing occur hence mix cf background depend uncertain calibrate instrumental observational onto grid interpolation parameter affect depth distribution informative
final node variable identify check problem particular proposition bipartite message bit fix independently message replace decoder ensemble explain decoder decoder probability non combinatorial namely check neighbor decoder ratio node modification prove ensemble recovery subgraph induce terminate successfully contain decoder remark intuition decoder regime decoder locally like similar decoder edge direct edge depth check case check allow proceed direct neighborhood direct pass leave head neighborhood simple edge ensemble progress direct depth around evolution decode show iterative asymptotically recover ratio stop easy point obtain apply ratio guarantee remain regime hash complexity zero case bipartite proceed graph singleton successful non perfect fails induce bipartite bit step analyze decoder hashing dimensional simplicity vector divide one
vertex boundary neighbor contradiction therefore neighbor value therefore minimum vertex eq contradiction appeal show correct realization vertex discuss prevent uninformative valid model weight propagation q propagate proportional propagation enyi euler first priori probability vertex I define operator connect asymmetric propagation boundary harmonic operator harmonic propagation generalized laplacian ii bi bb observation harmonic analogous fix propagation harmonic system provide practical thousand vertex time case ten million practice subgraph encounter discovery solver extremely stochastic realization interpretation propagation eq walk terminate diffusion model observe vertex augment unobserve represent transition priori diffusion vertex assign stochastic realization terminate vertex assign zero determine average walk ignore realization solution equation leave eigenvector right e irreducible matrix strongly simple invariant solution however frobenius apply chain strictly require frobenius state nonnegative graph require frobenius theorem form
descent subgradient bregman denote continuously divergence md generalization online gradient descent several consider composite mirror regularization function help md literature invariant refer static static static static static useful characterizing perform static static fail change time previously literature context particular output algorithm fit drift tracking need complexity complexity allow imagine series fit generalize conversely track equivalent static sublinear sequence vary slowly small tracking scale sequence model much broad propose receive dynamical dynamical dynamical key distinction analysis datum effectively knowledge tracking otherwise might prove lipschitz constant norm distortion tracking mirror descent
attention probabilistic modelling sparsity describe inference structure base modification cutting hamiltonian compare significant advantage simulated set life exploit fundamental structure relate learn discovery domain finance biology popular technique sparse considerable gaussian wherein zero field devote constrained handling offer slow optimisation recently budget address question method development graphical see
benefit reduce required prevent select citation need range kernel vary neither flexible flexible give cv cv cv eq specify select enable flexibility purpose allow degree starting allow optimisation around start ensure good scheme user select third split sized hold remainder classifier hold set mean test compute compare classifier multiclass discrimination binary combine predefined briefly code mn mn rule output capability distance able citation
widely logical plan probabilistic planning use team start variable must planning predicate order predicate distinguish model tuple predicate upon number plan necessarily valid plan validity plan predicate assign consecutive absolute plan work order parallel plan step occur plan step predicate step index plan sample predicate appear sample predicate discuss predicate specifie predicate follow order relative order vector index vector variable predicate order human refer absolute
px n pp correlation tail combine important proving na prove nx nx eq define x nx r combine fact nx nx furthermore strictly decrease interval ij series ij let divide expression behave mutually series remainder happen eventually plug back convenience ij chebyshev sum simplify pairwise correlation term correspond cycle index hard know apply chebyshev ij k density take advantage mutual piece lemma yield first show variable high correlation involve identically around origin event share
response enter mixed result effect threshold adapt inter dependence white add give high specificity threshold weak couple threshold couple identical delay system solve solver delay time delay denote couple consider delay stand evolution response exhibit ahead realization couple free noise triangular panel decrease latter coupling e give high rejection nan hypothesis coupling specificity high rejection causal effect specificity statistically remain specificity get difference become form couple x cx color black white delay variable leave couple free driving last
expand call give project maximize rhs sum internal attribute estimate correlation indicate great external large indicate quality estimate component sample per object unbiased external slight round issue seem arise external ratio scoring zero feature assumption practice derivation full similarly maximize base analogously psd implementation row column budget type score ji mx r dt bi assumption objective minimize objective alg objective train symmetric external covariance diagonal theorem stem additional
behave harmonic possibly f sample come respectively optimal optimum constant give difficulty difficulty bridge example need normal normal mixture normal track sum expression sort normal proportional histogram acceptable successive compare carlo histogram simulation double curve bridge solution valuable within equal model readily separate numerator reversible jump monte harmonic refer later paper impose impose distribution indeed simulation sampling poor empirical infinite reliable posterior density monte requirement target augment chapter volume rely fact positive recurrent markov straightforward equal know posterior remain
definite psd inherent positive normal psd define generalize accord assume magnitude process template spread manifold observation pose template covariance operator particle paper particle variable modal particle initialize variable include target I template extract comparison template particle propagation particle extract template particle target measurement posteriori mmse mmse resample htp descriptor good representation behind walk good target evolve gradually pose appearance separation visualize distribution target multidimensional scaling construct covariance construct visualization relative position red face notice together gradually evolution original us variation
encourage network approximation expect force remain algorithm similar multidimensional sequential observation keep mind encourage penalty factorize penalty encourage improve numerical therein nmf modify particular penalty include benchmark multiplicative implement converge slowly linear practice visually nmf derivation step present negativity lagrange multiplier descent tucker kkt condition kkt yield algebraic notable improve dense set undirected symmetry adjacency write diagonal underlie investigate satisfy underlie probabilistic nmf model additional counterpart influence versa reproduce task visualization community impose symmetry toy eigenvector modularity clique overlap toy
emission parameter collection center around hierarchy specification directly share occur hierarchy observation exist knowledge transition lda equivalent every formulation membership global collection dynamic series assign distinct hmms hmm comprise corresponding emission examine hmms allow fix assignment subset hmms membership attribute regime series formulation maintain variation mirror membership analogously hdp lda single assume membership regime infinite practice length comprise finite length set dynamic regime relate distinct regime exercise perhaps circle perform employ regime beta abstract flexible mixed share library infinitely regime regime formally specification endow dimensional f couple common measure coin draw determine regime result indicate select beta total encourage share similar space coin regime regime amongst series dynamic identical
cost comparable acceleration certain minimal proximal computation make proximal appeal computable describe proximity proximity solve efficient proximity accelerate efficient order organize review proximity fix convergent problem discuss section conclude approach nonsmooth problem move proximity regularizer euclidean inner
let take distribution xt px kl avoid behavior estimator boundary subset early algorithm random setting design provide platform treat later permit response fix uniform rate response precision eq response regression metric arise response empirical start summarize e density positive setting compactly lipschitz pp hold particular work whenever empty estimation side see strong note play density problem weak analogue useful adaptive generality sx act simplify independent distribution see minimize yield unique ms strong hausdorff require dimension function estimate efficient immediate proceed
least definite rewrite eq comma thereby indicate
incorporate encourage linear circumstance reasonable however vary good although course pixel nonlinear signature layer appearance material place hyperspectral represent material cover mix explicitly pixels reference material try contain intra assumption still material pixel full intra penalty penalty write intra assumption constrain pursuit matching use matlab bregman solve via solve increase iterate code decomposition initialize ta iy slowly magnitude l benefit initialization test discuss hyperspectral inter penalty acting penalty implementation intra inter acting problem
family important give univariate generalize define take exponential focus function constraint necessary countable continuous construction theorem univariate know exponential substitute turn derive theorem bernoulli member exponential substitute get ignore multinomial graphical ise previously ise impose since finitely configuration interesting member family form take get family entail word relationship exponential exponential statistic describe arrival event follow graphical imply exponential capture learn glm sample specifically assume graphical recovery recover individually structure problem structure recover neighborhood neighborhood mle rest related st ns n
division problem voxel modify voxel word multiply valid quadratic experiment solver first dark medium blue minus paris curvature averaging reduce surface
posterior policy abc inference extension free approximate inference main reasonable probabilistic set complex competitive abc rl appear even simple investigate abc monte close discount examine performance advantage would evident believe encourage methodology potential field induction replace proof follow drawing equal definition follow l second assumption z obtain final reinforcement prior model complex rl bayesian see extension planning experimentally potential
reduce euclidean space employ bayesian edge simplified context context parameter define set equal walk leaf contain towards eq form tree stop uniquely identify tree action need enter expression stop probability calculate recursively consequently path forward denominator whenever tree go update predictive backward distribution transformation current basis via linear model context pair marginal distribution prior covariance variable wishart extend classic calculate limit parameter integrate define posterior rgb rgb generalise context structure display sample find
integrate representative education author proposition look call characterize protein parse structure discovery approach frequent frequent subgraph explore far propose novel pattern large discover representative pattern incorporate evolutionary substitution subgraph effectiveness considerably decrease reveal protein sequence alone year various diverse descriptor profile spatial yet exponential growth database protein bank accurate help understand study protein evolution protein interpret study concept enable protein structure mine object graph trend aim discover subgraph
product tensor com contraction operation basis j two rank input tensor reflect elimination contraction multilinear tensor input tensor contraction argument correspondence map compositional first generative obtaining assign type represent contraction simple formal semantic type assign space noun sentence vector noun interpretation interpretation hence noun phrase np st tensor application sentence follow lexical syntactic tensor phrase represent direct arbitrary syntactic structure possible grant ability tensor encode leave open rank framework argument sentence leave mechanic one learn tensor noun describe aforementione learn
arithmetic also figure discretize axis figure convolution vertical b b b evidence nan independence discrete take test write hypothesis decompose elementary elementary contingency contingency
towards learn year penalize huge computational burden accuracy produce zero interestingly independence imply zero representation htbp article exist one marker toolbox web genome simulator genomic marker individual randomly marker residual trait marker additive individual choose accuracy score observe trait divide genome hierarchy experiment display local kernel finally display carry alpha record total year along marker marker training model calculate
least mx slightly kind obtain schwarz yield analogously provide large first thus approximate within number introduction combine immediately distinguish complexity match section remove factor might applicable give present closeness derive bound require follow moment distinguish establish uniformity need closeness disjoint subset notation constant kb ap ta expression dimension indistinguishable optimality associate value distribution p show optimality
prove joint finding plant part regardless argument interestingly joint incoherence statistical computational aspect prove theorem provide highlight innovation set construct require derive set n incoherence turn simplify prove exactly use derivative make arbitrarily sufficiently union occur respectively operator op subgradient optimality proof optimization follow hold op isometry requirement approximate isometry satisfied sufficiently construct scheme observe entry satisfy norm proof show quantity help norm tight previous solely norm constant prove lemma need sufficiently ready
small curve move th set curve curve proceed fashion support point I j j implement pairwise intersection step latter curve motivation behind th justify dominate intersection compute average intersection fraction curve intersection point reduction present sparsity number intersection principal computable semidefinite constructive proof efficient compute rank serial complexity present work implementation even possible equivalent find semidefinite polynomially utilize vector polynomially moreover technique
orthogonal atomic permutation transformation factor unitary compatible b special remove value compatible signature q statement atomic signature rank iii induce singular iii directly uniqueness proposition singular like stress different factor wise different rank combinatorial orthogonal enforcing
connect bipartite graph label classify connect connect class node top row achieve consensus among eq node group maximal value optimization consensus prediction lead improvement method combine prediction label label base correlation combination last jointly consensus exploit maximize consensus label abuse notation section encode entry predict th otherwise entry bipartite instance figure graph bipartite annotate letter instead node classifier represent expressive r connection fully instead break bipartite graph relationship nod detail r newly similarly reason definition explain next
log provide efficient inference marginal polytope pairwise singleton intersection q pseudo global energy intractable singleton pairwise entropy require know bethe convenience bethe polytope unfortunately bethe function reweighte bethe free edge span definition guarantee bind message pass bp reweighte restrict set tractable entropy precisely factor partition give unfortunately mean adopt find optima combinatorial problem find joint attain combination k problem remain inequality interpret locally programming differ lack marginal marginal sum let marginal seek pa call type mix variable elimination duality tb exponential time complexity marginalization marginal intractable elimination complexity although similar marginal map significantly classic example sum operator elimination sum eliminate max bad node may sum alone play practical scenario configuration b nuisance hide direct unobserved joint map case reasonable sometimes weather denote weather condition school cm weather b answer px wrong say person full
derive criterion penalization prevent improper implement produce package comprehensive simulation follow loading cccc dimensional zero diag model model n orthogonal model penalty select technique orthogonal
disjoint repeat pair lemma bound two pair sample support intersect constant location moment appendix compare triple expect common intersection triple intersection neighbor whose empty intersection sample support intersect take neighbor connection common decide support random remain identify pair exactly succeed probability np conclude conclude triple neighbor intersect intersection argument loop identify note identify triple intersection consider intersection identify identify necessarily build time finally conclude correctness show column learn intersect step cluster output algorithm either lemma complement respect uniquely intersect add remains show pair connect number common probability label pair node connect
try preserve inference limitation computer integral continuous domain field sec wiener problem confusion physics nuisance support inversion measurement process summarize noise deterministic noiseless measure linear position abstract context could image describe pose signal response set signal methodology implementation inference discretization extend domain grid grid conjugate list regular euclidean sphere grid representation field mathematically necessity calculus volume subset volume define characterize moreover discretization integral valid discretization choose
criterion observe policy episode choose go essence exploration path even switch phase another intuitive connection boltzmann distribution benefit policy connection issue take definition online learning choose episode perform formally episode stationary episode episode discount discount accumulate respect randomization goal policy policy performance solve frame arm multi armed payoff process learner need select payoff get goal arm minimize quantity meaningful sense expect armed adapt exp adaptation exp call essence exp arm policy policy episode discount k nr th mdp arm policy parameter discount return arm arm remove time discount run c tt tt jt z payoff learn exp exp payoff consideration soon determine input soon certain arm adapt known exp transfer randomization run exp policy expectation randomization tell exp transfer policy reinforcement play none policy essentially devote compute mdps proceeding take account bind normalize transfer randomization present cluster approach encode previous cluster clustering help purpose transfer worst worst empirically lead please turn mdps mdps policy exp transfer source mdp mdps exp
e nmf observation order idea zero somewhat svd become feature however confirm zero yet cl ccccc lift strength use penalization strength maximize classifier penalization currently address time take regression result elaborate reduction classifier carry bold value tune show symbol evaluate dimensionality train regularization table concentrate highlight bold see roughly perform another compare dimensionality achieve
much compute experiment describe last benefit program presence aim colour refinement already implement preprocesse code colour proceed colour compute dimension program solve program compression colour conduct machine ghz intel processor gb ram linear program evaluation relaxed integer encode combinatorial theory theory compute dominate hamming triple system fig clearly colour refinement reduce program expect look time reduction order overall program symmetry refinement take second high running reduction illustrate colour refinement reduce program function modelling make outcome partly lp reward receive state action mdp grid one reward consider goal whereas zero corner grid colour refinement partition colour finally consider
scenario domain completely extend large efficient dimensionality implicit use easy adapt internet database perform task domain small database collect internet inherent see imagenet database imagenet object sometimes scene truncation box transform adaptation learning adaptation allow adaptation especially scale recognition benefit learn category model domain transformation transform target introduce adaptation big formulation optimization although
mlp attribute temporal acceleration hmm observe outperform figure ground segmentation k ascent middle actual segment dot bottom probability segmentation ground lie lie estimate segment logistic scenario activitie true confusion observe probability attribute overlap hmm see probability show occur within transition activity instant confusion positive negative confusion especially successive activity basic activity easy detect like c c class confusion datum h activity fp rate combination sensor result obtain sensor confirm add sensor model acceleration govern switch another learn log dedicated algorithm apply real automatic assess alternative know classifier encourage approach
mcmc posterior result dimensional space hasting overcome difficulty monte differ standard metropolis evolution state acceptance ratio successive sample fusion exploit consideration derive obtain image hybrid sampler hamiltonian conclusion acquire optical imaging sensor measurement situation observe version degradation previously numerous work include spatial follow unobserved scene spatial stand observation measurement observe image either band band format convenience generality order version optimization however band strategy depend may another spectral degradation operation instance apply band e spatial give degradation frequently
slight abuse notation term evaluate key trick observe quantity multiplication consume calculation require iteration three score statistic g estimate vector transform compute z dd matrix nan z calibration strong statistic exactly calibrate correct follow critical test marker hessian diagonal element notice alternative experience correction score anti conservative likelihood phenotype although study fully phenotype individual may phenotype drop phenotype phenotype individual phenotype phenotype observe phenotype estimate value covariate covariate multivariate mn mn px nr em red pair trait phenotype genome six trait analysis thousand phenotype simulation I phenotype software package introduction multivariate gene assess genetic complex phenotype detect trait accounting sample counterpart grow potential association analysis detect genetic phenotype
convergent subsequence combination whenever mutually continuous b respect
red marker perturb detection similarity difference consider two structure node serve connect set second serve gene context due toy node represent effectively propose multiple matrix paper overcome h adjacency indicate differ material encourage difference pair inverse describe propose problem section introduce regularization break many subproblem substantial gene proof version formulation admm admm comprehensive set singular consequently author likelihood tune set positive definite serve correspond feature sparse formulation recently propose set goal condition certain allow structured way convex square formulation proposal independent distribute th matrix trace convex encourage among solve serve refer particular refer encourage network whereas encourage estimate share strength observation lead separately similarity arise powerful similarity arise pattern failure
rescale vector sign problem know least recover know refer calibration dictionary learn goal analyse inference provide mmse suggest test derive bilinear amp propose identifiable unknown hence exact several exact early rigorous
independent present interpretation rewrite minimization optimum equality thus bellman minus feature discuss bellman way nest two formulation equation equation construct describe section algorithm bellman residual correspond project state orthogonal un nest reason justification reasoning claim fact derive mean algorithm wrong show argument come chapter presence formula two w argument derive remain practice preferable sequence sample sequence estimate trajectory describe sampling sample expectation w iw iw row iw iw iw w b
cnn perceptron mp think class hilbert build iterative update w yy unit tr ty depend separable margin note update misclassifie construction decision
stable chain chain ess independent estimating autocorrelation ess calculate estimate half burn gain neural conference year co activity less randomness less manually eliminate symmetric count dataset link among entity detail algorithm generality effectiveness network structure topic social medium customer partition analysis partition protein propose problem linkage infinite partitioning directional inter
investigate additionally identify embed special active automatic determination numerous application notably work focus dimensional recently initialization phase objective good structure permit objective observe million separate uncorrelated effectively show insensitive input thus figure discover u u discussion restrict reality zero select space covariance induce isotropic quadratic square radial mahalanobis later relevance determination input
quick bic cm I cm iii especially complex one give quick ic give quick ic normality perform variable hold still penalization parameter practice parameter naturally select penalty speak criterion approximate parameter independent negative penalize enable particular parameter factor fa dimensional factor loading vector underlie error mutually normally covariance fit maximum expectation ml since ml suitable factor number give fa capacity avoid unconditional fa shrinking
absolutely suppose gradient consider function hold relate residual suggest tail term p sufficiently difference analysis state term appendix either differ q sufficiently small f previously note idea xy density notation notice therefore ready prove n nx hx ny satisfie condition provide learn e version regressor although box estimation sequence entropy
partition two segment construct obtain segmentation programming implement binomial package use tune slope negative estimator assess performance five rand index true segment belong estimate characteristic nb frequentist nb frequentist external frequentist external frequentist external bic frequentist internal cart bic external nb frequentist external exact nb external exact nb stand rna
optimal sr minimax sr chart benchmark chart assume alarm delay paradigm cycle alarm cyclic formulation specifically exchange maximal repeatedly detection rule alarm put change occur distant future false alarm consecutive false argue come surveillance stop alarm detection instant cyclic consist formulation propose sr procedure iid sr establish threshold sr note chart
projection anti conclude failure desirable since value meaningful guarantee hope dimension strong anti concentration attain vector wise constant follow theorem row span right proceeding trick vector non gaussians easily key essentially many say mass conjecture improve beyond illustrate idea perturbation high column orthogonality dot order motivate matrix refer consider span negligible moment instead variance result instance correlate orthogonal somewhat motivate definition orthogonality orthogonality property v say orthogonality property formally order orthogonal orthogonal satisfy th column projection span th order singular orthogonality perturbation proof section
literature select uniform threshold randomly contrast supervise svm combine may thousand level decision compare report descriptor affect feature yet generalization illustration compare experimental setup well conclusion distance produce change keep largely parametrize bm far objective high kernel learn smoothed learn represent indicate b image sort cluster decision differ substantially loss boosting optimize individually label add classifier parameter label kernel random
simulate gain nonparametric distribution strategy fast al et edge obtain modify dag construction comparison
vector find concern column
albeit slow rate iteration comprise optimize objective produce simplex zero entry semidefinite cone solution attract much community recent year smooth extension present regularize iteration frank wolfe optimal polytope away boundary point boundary interior much give weak condition optimum converge conditional assumption optimum stochastic optimization play point cumulative plus idea conditional update step online convergence full set online scale work recent minimize norm cone minimize gradient availability strong minimize intersection cone ball
principal internal utilize consider market stock factor besides market forecast internal stock market economic phenomena u stock exchange rate product trade stock market use study index stock market exchange symbol jj stock movement day direction carefully divide part financial recently article select order testing go year make besides compute ahead period divide explain utilize svm
paper build aic bic model straightforwardly variational expectation em optimal rather order gmm number maximum bic
minimum slightly min md resort compress verify improve bias correction improve curve present experimental study illustrate propose collect solver straight stand nonzero clearly solver robust robust essentially impact intuitive explanation formal future maximally skewed merely expect research arise issue
user define pair position pair map arrange paired concatenation calibration request output source order weight calculate eigenvector small nonzero remain observation localization set user subsequent smoothing subsequent place subsequent outli previous centroid immediate estimate location introduce device g server localization request device read device find quite mobile user localization mobile device reduce incorporate device sensor software indeed user movement position localization consequently device base localization begin time construct build send server determine store entry accumulate reach compute read point stop build calibration coordinate localization spatial neighboring calibration online measurement extent propagation decay neighboring position suffer assumption
graph possibility could estimate pair weak dependence define partial correlation something case validity interval version define basically however partial promise currently follow currently computationally feasible include believe future structural get intuitive capture stop namely graph good capture qualitative markov validity start miss obviously reconstruct qualitative detect fall permit qualitative preserve leave dense small entry correlation useful information course nevertheless correlation nan partial
statistic eq high view side sided transform u sort ks hc sided n uniform order expectation tend near center mask statistically poor sensitivity hc statistic deviation common variance index however beta converge variate monotone heavily skewed explain analytically normalization affect hc demonstrate ks hc uniformly deviation statistically significant transform statistic ks two side ks nan numerically evaluate straightforward package work recently paper however upon relatively define statistic approximation computer exact seem attention availability computer exact pose section derive several measured statistic
adjust accordingly sample thank medical imaging projection paper depend ridge regression interpolation optimize training interpolation guarantee square ridge parametric drive bind dimensionality medical classification training method new low om extension commonly manifold nystr om case om interpolation point low depend need computationally number use similar regression small reduce
sense initialization b using recover wise block integer eq iterate op storage requirement stream total tt th iterate obtain factor exposition already linearly component iterate principal component show dominate enough f b tight b point direction constant repeatedly individually stream define universal step universal prove provide initial note completeness hold probability least consider
understand accomplish visualize layer model correspond map category observe correspond pooling activation original structured object car feature category train category result fine grain demonstrate effectiveness global pooling learn base maps scene al network classification average replacement fully layer well pooling act regularizer prevent demonstrate cifar cifar visualization feature map demonstrate category motivate possibility detection chen computer national sg propose
perform associate grateful international accept spatio take advantage consider space derive similarity accuracy roc use pattern compress marker call
share class extracting belong optimize objective hinge balance minimize margin encode knowledge belong weighted summation intra serve purpose dimension relatively formulation capture solve exclude regularizer use purpose demonstrate previous formulate solver triplet second gd present project cone inverse descent appropriate distance metric psd cone write
sound surveillance item harmonic phone record sound break break phone temporal depict extract ms frame index frequency database similarity within sound class cm human phone child class select single run classify show value kernel scalar case choice scale number value eigenvalue eigenfunction associate adopted denote pre
error fit observe entry rank minimization np nuclear convex beyond enjoy practice miss entry also array generalization frequently encounter medical leverage nuclear tucker miss bind relate miss give specify matrix formulate matrix rank convex sampling norm norm equal lagrangian nuclear arbitrary attain bilinear building imply find solution however since stationary interestingly condition stationary part prove c minima coincide minima globally satisfy x globally play counterpart imputation since offer matrix aforementioned rank rewrite c rr bn r cp rank require tensor adopt define pp pm
complexity computer many limitation digital computer physical rbm face architecture aim rbm ultimately feasibility address question currently rbm wave physical suffer three limitation order limitation feasibility observe physical architecture restriction etc experiment benefit fast aim computation relative offer effort impose great practical rbm probabilistic unit rbm represent energy make rbm markov make mix slowly give rbm
architecture architecture investigate maxout take dropout investigation pooling mean standard selection spatial pool cifar still preserve visible regime big dictionary division approach eq two pooling region role cifar dictionary smoothness crucial good validation column select r cv acc acc smooth conduct cifar investigate cifar smoothness batch achieve good knowledge art achieve c acc site
gradient guarantee riemannian trust algorithms amount trust update trust trust region conversely iterate poor technical detail manifold generic implementation require riemannian riemannian direction connection notion directional riemannian vector assign directional apply formula cost iteration requirement bottleneck tuning trust enjoy convergence
check dual optimal iff maximize margin minimize svm gradually shift margin reduce weight may counter misclassifie I give find primal solution lemma one verify definition increase weight decrease would svm opposite I return return even reproduce line slope significantly svm characterization svm variable svm interested modification second intuitively require non objective convex minimizer multiplier similar reverse unbounded easy quadratic modify obtain complement modification degenerate exist computed framework reach observation classifier consider construct let admissible depend good weight feature yield classifier weight generation scheme interested
choose primitive argument primitive observer simply feedback option feedback choice improve whole top improve furthermore four task initial figure program execute pr environment section robot perform comprise program sequence e argument video human environment depend configuration environment dynamic planning represent design task environment weight obtain random field representation demonstrate dynamic acknowledgement grant microsoft fellowship nsf award cs edu human environments environment configuration object way plan
prior volume contain repeat entire nest prior deviation dominate iteration implement nest draw low live widely base iteration live
sufficiently significantly theorem k r probability follow use yield f least conclude k r result remark world label randomly corrupt work label noise proportion give weak identifiability class allow condition also discrimination correct mutually irreducible concept introduce limit problem argue pair distribution proportion estimate another binary instance assume mutual conceptually ambiguity source otherwise identifiable consider design discrimination rule presence classifier measurable assign classifier number depend infimum discrimination rule x discrimination rule difficulty extend performance frequentist pearson technique
gamma delta connect connect lambda connect red nu connect gamma group rank enforce facilitate bayesian assume select fixed avoid let nature shrink grow row row part global column flexibility finite facilitate similar column prior specify disjoint empty variable union includes place row matrix certain column row example formulation specify obtain normalizing conditioning equation element row group equation residual
surprising cover exception test seem odd time accuracy despite fact make huge weight distribute adaptation individual less meaningful drastically increase plot pose serious make support machine decomposition extremely software improvement situation observe representation advantageous primal directly differential hinge influential research direction cut gradient al possible problem svm linear counterpart allow
thm definition remark remark thm thm thm conjecture pt reconstruct dimensional hilbert magnitude redundant establish show redundancy reconstruct vector dimensional redundant reconstruct global phase magnitude f importance speech short time fourier transform appear notable reconstruct state mean nonnegative operator trace secondly hilbert schmidt inner symmetric language nonnegative symmetric
small intuitively sample mean spread true small order could approximate trick eq straightforward bias via monte though brevity reader manuscript second bias might de tradeoff correction useful see improvement past correction shrinkage stein stein great feature away effect contrast
potential subject answer question include subject continue subject fail quality exclude subject fail exclude analysis randomly study question list question question ask assign five display treatment condition bernoulli equal consistent randomization table diagonal subject group background age political education control well pattern consistent description survey list list prefer gender conservative high school college year college white american american prefer subject question pool average standard example percentage public list standard associate combine variability range subject question estimate generally agree provide experiment direct confidence
tensor central role tensor many mathematic jump dramatically go tensor two tensor potentially frobenius specifically frobenius square square since boundedness bound represent robust counterpart set linearly sub th little note somewhat spirit much weak restricted isometry rip compressed algebra onto vector singular abuse sometimes sometimes scale tensor length assume help simplify lemma statement involve track polynomial mention ready version uniqueness tensor rank aa k cc b theorem formalize really high analogue order tensor j tr j suffice find tensor practical ask dimension respectively approximation low rank tensor viewpoint note guess decomposition take well condition e guarantee naturally tensor version learn polynomial commonly view hold suppose sample multi def n q polynomial identifiability hmms hide learn polynomial mild condition please implication latent gaussian markov model consecutive identifiability hide definition observation distribution singular consecutive chain far r show identifiability additionally take algorithmic polynomial hmms require linear section straightforward
modularity hold space calculation parent variable unconditional obtain development loss transfer impose task restriction order toward evidence formulation impose transfer factor sum modularity additional modularity assumption u assumption number edge reasonable requirement four modularity graph property factor eq yet consume omit power gain factor change outline calculation score bias score remain unchanged reduce approximation network must score parent calculate
accuracy result know ol problem prove hilbert condition satisfactory arithmetic computational concept
face stochastic compare performance largely superior classification deep encode class specify discrete connect softmax layer total predict would machine formulate svms slack requirement unconstraine primal svm hinge l svm
space hard minimax obtain minimax model parameter achieve large coordinate sparsity level ball fundamental procedure simplicity signal vanish correlation bound require q natural angle calculate algorithm level sufficiently constant depend constant simplicity fully section along direction sec certain impose estimating weak ball imply rate assumption w similar corollary pick thresholding constant alg boundedness imply valid risk sense simple know covariance matrix simplicity otherwise establish minimax view algorithm alg optimal condition nuisance hard assume structure precise statement introduce eq covariance I op large corollary canonical direction w q method first scenario covariance comment scenario address scenario normalize elsewhere jointly methodology section step estimate precision first half covariance toeplitz estimate toeplitz move know denote estimator split parameter
equation omit subscript arbitrary prove corollary statistical high dimensional hold nsf comment discussion financial research analysis series create create author mail inequality error accuracy autoregressive inequality even sample exclude probability correct reveal correct establish estimate estimator estimate identical oracle reduction c last research statistic size non want able lot devote penalize estimator probably lasso prominent scad bridge bridge selector popular computationally review effort establish possess oracle oracle understand correctly detect pattern zero non zero parameter relevant include efficiently reveal true progress devote regression model sometimes independently distribute stationary investigate consider var shrinkage type paper slowly set concerned triangular row term sample augment vector constant omit keep notation var central forecast impulse response suffer variable leave satisfactory modeling observation least design singular construction regression run order information infeasible unstable seminal factor precise forecast macro inclusion leave evaluating
formulate introduce variable approximate lower bind henceforth maximize comprehensive
constant boundedness g z g use last suppose constant note omit use b fs sc generic use margin assumption dx dx last g z pg boundedness z third old space kernel follow proof g pg kernel constant sequel split decomposition subgaussian random u j c u cn c dx x argument assumption maximal prove z last argument arrive conclusion effect quantization recognition social
payoff player average move opponent irreducible invariant I player follow dynamic positive consequence surely equilibria converge nash equilibria sketch need introduce notion later finite irreducible matrix invariant spectral matrix play sketch verify map consequence nash equilibria argument pseudo inverse I control gap addition sufficiently general set admit two adapt considerably contrast invariant measure variable update overcome present present extended theorem corollary verify condition ii verify response adapt amount show q convenience proposition
binomial arise expansion include continuous general explicit formula order omit calculation term difference polynomial simplify fractional part polynomial lastly th bernoulli standardize rao define term know give formula variable z bernoulli relationship apart rely lemma remark
unique definite l part respective differentiable generator dirichlet conservative interface encode make adjoint probability via fy therefore evy brownian subsection outline refer reference arbitrary multiplicative measure denote green process speed piecewise let also f f equal skew speed give interface within must behave diffusion diffusion start interface interface make likely path evy follow canonical standard brownian motion additive functional function continuous martingale variation continuous finite x define adapt calculus local evident whenever trajectory brownian reveal effect jump interface quantify briefly background semi reader refer martingale local convention refer surely jump martingale bi basic denote surely nonsmooth formula use brownian motion x write follow representation filter motion skew differential equation uniqueness prove review relevance consistency place time sign right pt key relate differential term fact right satisfy yx
theoretically normal variance goodness merely functional process regularity covariance distribution note htp sample consider df strictly increase standardized term standardize quantile similarly mean simulated chen pp q difference chen claim simulated exponential approximated deviation deviation replication function nearly deviation rather appendix show function take whereas constant converge zero distribution compute stress variable instead seem converge value eventually dominate red green identical visually
simulate stationary visual simulate alpha subsequently scenario background brain thus give segment eeg principal component pca degenerate mix specific yield pca pca project obtain corresponding pattern display successfully alpha succeed pattern moreover non describe although general background reliably many still whether two circumstance reflect eeg describe simulate keep display show condition nonetheless final obtaining pattern correspond remove simply display third may cc removal stationary find pattern arise fact removal direction direction preserve method even reality datum condition refer detect change wide range segmentation use linkage
traditional unsupervised mean nothing however situation wish cluster mail spam suppose available spam one primarily spam primarily spam genetic wish genetic q reference common reference expect estimate deviation propose choose partition disjoint cluster structure form merge thus bottom level different briefly cluster method method description method agglomerative agglomerative cluster individual merge illustration agglomerative apply hierarchical describe hierarchical dissimilarity dissimilarity cluster start dissimilarity euclidean define dna
setting least summary well par state art simplicity unknown train applying improve overall detection performance c partial auc pixel partial pixel auc min auc score supplementary score extensive propose possibility apply detector improve fig corollary van sa operate false positive detector area roc outside label partial roc curve cascade classification moderate achieve define partial auc range propose use either form cascade experimental synthetic
trend might complicated else say take produce assume sum priori independent
either achieve increase probe ccccc hmm auc probe anomaly ability identify known observation method region high region significance use outli outli outlier zero else achieve minimize follow eq demonstrate experimentally train alternative estimate observation application hide hmm usual speech recognition bioinformatic language however accuracy
constrain distribution capture illustrate correct error range correction relate coarse grain equation coarse grain fine grain subspace equivalent term give identical expectation place constraint coarse bootstrap slight detail one bit average value study central synchronization old break trial report break result trial word go collaborative program nature naturally far trial trial likely happen roughly old break percentage observer would refine belief informally carry observer sensitive carry course great many participant social reliable event mechanism variable answer great deal make participant might purpose capacity answer measure hand trial outcome trial proceed identify likely focus lexical measure time drop order linguistic tool example noun word coarse grain build amount claim many example arrival carry outcome semantic outcome turn qualitative form amenable distinct answer formulate category aspect well second sufficiently
new gradient determine stochastic hessian newton method newton method variant metropolis langevin attempt exploit posterior make degree hessian reduce pdfs gaussian beyond variant framework linearize nonlinear posterior requiring extend dimensional setting care establish multiple efficiency ice boundary condition surface velocity involve equation ice flow ice inversion full explore three study various chain convergence adaptive newton hessian newton hessian yield fast sample progress study visually solution data posterior demonstrate make exploit knowledge many bayesian inverse problem expect visualization value overview discretization present base hessian ice gradient log provide insight ability conclude remark solve bayesian field present difficulty dimension must pose facilitate discretization dimensional posterior arise upon prohibitive high problem appropriately prior employ operator describe problem dimensional function definition gaussian evaluate intractable approximation hessian tractable mcmc inverse seek infer uncertain parameter pose uncertain domain observable forward problem follow operator solution seek make precise pose represent belief require belief quantify parameter differential lead spatial field
candidate text construction fall base character character fit bottom fully connect character filter run test edge character connect subgraph candidate chen character candidate cluster exploit straight fit character cluster character minimum algorithm candidate construct cut rule complicated post one energy incorporate improvement base scene lead scene text character use non character reduce minimize variation character algorithm present probability character non remove detail identify adaboost classifier train candidate text character text feature uniformity character candidate feature train measure propose competition candidate text partition classify character distance
follow unique interval clear accord intermediate lying conclude let corollary provide root second auxiliary follow define continuously ii iii invertible inverse continuously decrease continuously differentiable non inverse inverse function otherwise summarize key lemma continuously easy verify continuously show combine eq lead last follow find show root intermediate unique root unique root respectively accord however since decrease contradiction unique argument root optimal experiment unique root achieve current interval uncertainty need verify interval decrease see achieve cost find root find cost
table basic idea linear non exist transpose design basis design linear combination integer translate namely circuit basis circuit avoid useful need find generation circuit matrix factorial section notation provide algebraic absence circuit matrix report highlight relationship basis relevant section sample theory basis projection research direction let factorial factor represent term parameter matrix indicator
shannon entropy usual chen use shannon entropy order al stand sample shannon small counterpart sequel get note stand complete beta noting formula observed uncertainty assume size assume hx equivalently find
remain rs step apply rs rs identical ensure algorithm unable mechanism consequently regret guarantee continue randomness update binomial replace point pool location randomness binomial whenever randomness randomness usage rs match rs statistical rs analyze however section online work bound learn penalty regret claim however bind subtle mistake provide validation online auc generic online analyze constrained buffer online respect supervise concrete instance maximization sake restrict pairwise readily problem goal value sequential access nh incur expect aim specifically allow risk choose convex storing see large instead maintain buffer capacity penalty incur tt shall buffer policy reservoir sampling present regret give strongly w present novel present eq tighter well detailed note error exist martingale intersection aim utilize rademacher complexity face yet complexity turn head loss sample rademacher average yield bound risk perform barrier unlike mean introduce problem
worse great several metric different curve tail omit brevity difficult median report plausible poorly type tweet implication tweet locate column tweet al et et al compete median metric important limited tweet united create location stay location may appropriate message user message tweet user treat text notably improve yield tweet million datum interestingly vary median well higher et al examine light discrepancy exponent increase km km calibration suffer dramatically gram message exponent precision calibration impossible et calibration metric unique gmm demonstrate decrease datum test day roughly day evaluate description instance day tweet rapidly accordingly day appear gram figure show notably include improve explain tail gram frequency accuracy temporal gap test hold duration day instance summarize location linearly duration slowly
hilbert journal j tool b functional j functional york n data na na forecasting cm sequel project france universit de france france fr abstract functional functional et
information purpose penalize conventional information piecewise compute smoothing spline space unknown coefficient reproduce form piecewise plug select generalize adaptive compare different traditional cubic smoothing r spatially smooth equally implementation minimize site smoothing spline cubic smooth spline vary use replicate integrate visualize replicate estimate yield square error error replicate median replicate wise quantile outperform
record winner every significant ni counterpart loss dataset acc acc win ba acc acc win ba three classifier use ba provide ba prediction ba advantage ba reduce confirm ml ba cancer mass spectra scientific measure clinical purpose patient mass spectra present spectrum histogram observe mass z objective automatically mass spectra cancer patient individual high surface enhance ms contain control
kernel product operate input helpful class learn predict object removal mark show figure mask red green pixels channel channel run channel without pattern wish natural scene prominent mask without subtle access repository thousand similar objective sometimes cross purpose look g place water algorithm prediction test input process believe nonparametric large multidimensional extra structure inductive naturally kernel inductive bias exploit scalable play role model expressive scalable interpretable simple inference
label voxel binary annotation specify delta voxel training image dataset voxel rw probabilistic input model rw minimize quadratic reference probabilistic segmentation dependent appearance shape voxel wise adjacency voxel matrix block block rw equation reader rw
take test report fig however powerful test asymptotically employ correspond dense improve decrease dominate opposite continue different plot higher p risk triangle high star plot study testing alternative context regression depend matrix condition irrespective design certain bound regime develop binary attain construct test project substantial rare variant disease genome sequencing association study sequence vast across genome rare fu review association study see heart sequence individual consist goal risk total genetic variant genetic variant minor allele frequency vast expect variant associate testing histogram rare variant allele binary regression boundary explain paper strength alternative throughout design matrix show satisfy low presence sense provide contrast problem design explore
site pac substantial gain importance genetic introduce simulate identify et lead introduce approximation yield improvement inference lambda let mutation probability mutation allele denote canonical position zero elsewhere denote allele allele individual finite mutation sample associate copy low look copy jump denote configuration frequency along likelihood space leave compatible decompose mutation mutation
index view coefficient hold pick sample replication combine index computer differential thousand literature know krige polynomial expansion view see may simplify optimizer
scale representative candidate solution evolve axis fairly evolve dynamic vertex evolve behavior objective keep increase evolve accordance perfectly match start neighborhood expansion huge jump flat curve dynamic neighborhood expansion one candidate subgraph evolve denote e search subgraph scale affinity element etc little definition former later gs algorithm scale sometimes drop gs one scale scale
calculate fed extensively alarm argument rule make stop else stop stop alarm differently change stop alarm long
bayes universal universal factor bound maximum even hide independence mix completely close approximation case discrete ip ip jx proof mixture distribution closure strictly q qx normalization consider p p rbm k j uniform block mixture disjoint support block eq follow together pg eq whenever
high risk mode risk require make delta idea separate candidate mode test simplify hypothesis finite set mode mode eigenvalue polynomial note valid valid useful shape mode call formulate importance really approximate mode way possibility take eq small practice effect mode q depend bandwidth view mode mode course separate importance mode concern variability raise since hard one device relate mode estimator statistic
curve exclude analysis end delay duration within model yet smoothed mention contextual shape perform raise discussion curve amplitude issue restrictive fold contour limited likely contour account linguistic effect case covariate regard interaction effect specify covariate examine fix incorporate triplet interaction look break break duration phrase break cubic observed interact partially random intercept together word semantic examine incorporate sentence inclusion random justify factor age health mostly still incorporate sentence different question note nevertheless relevance utilize bootstrappe computationally size straightforward effect structure aic directly fitting entail finding group amplitude difference criterion third segment attribute amplitude five observe prominent shape great amplitude amplitude amplitude question apply retain component take calculate minimum exhibit difference eigenfunction reflect standard hz structure increase look analogy shape shape exhibit justify
extract reproduce exactly replace gram series explicitly term clear approximate suffer statistical fluctuation tie approximation addition poisson event experiment experimental prediction resolution parameter repeat complete select bin fit histogram bin expression poisson compound numerator
unit simplify presentation name unique sequel sub th whose sub vector element observe associate node union connect information symmetric p introduce hermitian c h h radius satisfied assumption q term dominate dominate term moment dependence eigenvector establish steady error matrix steady covariance rhs dominant appendix accord matrix approximate steady valid hermitian symmetric definite refer matrix random r mean complex matrix must complex approximate hessian lemma covariance steady vector matrix individual long dominant due matrix valid complex asynchronous steady state either steady
efficiency practical gp restrict demand environmental like monitor traffic also show though demand dual service relax communication acknowledgment mit technology equality inversion last equality severe hour system paris car sharing promise sharing specifically tackle cause private mod light city want simply walk drop capability general prototype densely city pick great user pick drop road service demand poor coverage mod cost mod system technical fine grain demand sense real
pixel weight pixel prescribe region rectangular eq include weak smoother simplify progress focus fast notice core scheme notice give reason behind fully letter rest
setup true probability parameter familiar notation intervention method term intervention tag random replace probability argument delta choose yield belief thus provide probability causal calculus policy agent policy third policy thompson agent investigate assume belief converge pa pa ergodicity requirement speak requirement mistake ensure make prediction couple behavior environment adaptive agent capture environment fundamentally behavior idea environment preference game agent simple shot action depend player equilibria classic repeat evolutionary game equilibrium appear view kind interesting equilibria thompson interact agent perfectly match agent posterior sufficiently formally define nash
communication blind sparsity laplacian linearly available field assumption network localize phase likewise magnitude constitute mrf covariance magnitude distinct consider available price multiplier constrain idea information collect successive matrix diagonal advance sparse low constitute second multiplier admm benchmark validity letter zero identity hadamard diagonal
hull wise expand second term focus obtain flip computed
project partitioning diagram imply v v v v v edge v v v edge edge rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle dotted path partition block bring depend previous whole suppose subset determine path result
simplification sparse noise admissible simplification speak find block option subspace incorporation still clear incorporation simplification frobenius add representation constrain transform equivalent diag last quadratic penalty function correspond lagrangian lagrangian lagrangian multiplier obviously remove cluster stationary multiplier part minimization constitute entire ssc brevity ssc good recommend ssc algorithm notation shrinkage accept entry index product stop update update consecutive value initialize due optimization computationally optimal fix explicit five unknown obviously diagonal affine subspace great ambient matrix inversion iteration unfortunately nevertheless straightforward explicit line update lagrangian update optimization move lagrangian alternate multipliers ssc give benchmark first miss clear
dual since necessary iii divide x x iv scad penalty mcp etc regularization control concavity significant numerical nonetheless much develop review nonconvex bridge scad penalty iterative convergence satisfie develop optimality augment functional diagonal study pursuit information adaptively square bridge popular iteratively reweighte origin smoothed alternative scheme also solve least expensive iterate limit optimality last li nonconvex penalty closely mcp zhang keep track solution statistical step involve reweighte include relaxation bridge several gradient enhance computational iterate generally unclear solver iterative shrinkage mcp illustrate coordinate step update component gauss fashion scad mcp smoothing nonconvex functional experiment
binomial model however extension integral methodology automatically prove formulate row ik equivalent model sample bayes factor rely prior integral objective minimal detail requirement take argument derive equation ideal yield equation derive unknown prior indeed chain consist invariant integral model associate
zero take equivalent main close green evaluation act operator j n difference different periodic hold j reproduce although originally differential likelihood interest infer might penalty reformulate j necessarily previous follow system homogeneous replace elaborate definition ode assume x collection trivial
condition arbitrarily apply go since consider bind call small achieve supremum turn supremum p p formula n ne get k independent eq go probability go negligible take chebyshev inequality subset bernstein lemma subset size derive large np except go one therefore binomial bernstein subset size derive go one compare np away thank discussion clique graph learn trend mathematical conference centre international de france office research partly bs rgb rgb subgraph dense os enyi graph probability composite connection detect subgraph term exhibit unknown testing aside detect bound problem direct denote much derive sharp detection boundary quasi tend apply display two natural arise
flat front aim maximum run real feedback compare easy optimization like strategy behave manner incur poorly challenge regret avoid get optimum like figure mention tt real synthetic algorithm heuristic probability toward present impact average regret value prohibitive build always mention restrictive challenging face
sparse seek sc many recognition dynamic texture human action speech recognition recognition annotation recognition assume sample thus ignore code recognition code purpose solve problem supervise sc code simultaneously optimize wang al sc discriminative code feature though label semi compare supervise explore effective unlabele
advantage side enable tolerance make less method density base abc use gaussian adequate hand kernel density consistently estimate piecewise abc abc tend infinity example automatic summary abc central determine balance control carlo error controlling bias summary piecewise abc aim fairly broad model markov correspond posterior advantage burden avoid abc practical abc factor full posterior estimating product density abc use toy illustrative infer experiment autoregressive
issue appendix evy measure consider alternative converge gamma shape lead mix let thus whenever chebyshev b directly ds establish x x x conventional otherwise consider expression inverse gamma definition penalization nonparametric processes compound family discrete compound gamma poisson binomial square laplace induce nonconvex devise effective great principled mode transform laplace express gaussian development nonconvex
series structure return compression achievable equivalently return try size return lie return series lag grow original give cutoff model lag tolerance block cutoff daily tolerance expect cutoff lag close word daily choose lag grow block plot compression adjacent size lag lag serial serial box plot compression ratio size otherwise approximation serial appropriate lag stock return lag model compression compression serial dependence compression measure lag block within away convenience cost redundancy test dependence consecutive return parameter hypothesis nan function transformation constitute notational convenience binary difference repeat variation empirical quantile nan therefore nan concentrate hypothesis quickly daily indeed
certain maximization sparsity ica maximization unlabele ica generally nonlinear identity traditionally prevent become meanwhile penalty point ica difficult ica sensitive whiten drawback restrict ica replacement unconstraine tradeoff penalty could sparse whiten invariant replace pooling encourage feature together scale invariance besides sparsity pooling network nonlinearity layer nonlinearity matrix fix prevent division nevertheless linearity complete reconstruction fail association insufficient develop representation discriminative class task motivate
sub summarize show attain I tt tp policy thompson regret try latter thompson assume arm various regime three step rest ratio bad evolve coefficient decompose
later fix horizon th bid page allocation allocation decision future engine concavity click allocate observed consideration situation objective reader thorough worth budget algorithm permutation assume permutation arbitrarily unknown decision maker intermediate path analysis draw identically restrictive online achieve performance stationary input model interest online permutation optimal th arrive match return ratio fix competitive bid keyword go
bipartite rank ndcg ndcg query rating rating error unnormalize rf
arbitrary resource iterative depict plan template flow internal representation g partition bottom entire include iteration responsible global driving across optimizer statistic computation balanced parameterize optimizer aggregate statistic pass sequential store plan template optimizer allocate computation fan plan template come optimizer take account mapping plan figure machine consideration establish parallel largely discuss execution resource aggregate choice must current across costly offer consideration available resource available cache performance significant performance improvement use file benefit apply inspire perform leave decision dependent cluster computation optimizer determine plan job optimizer number request job ignore job machine map
minimize satisfied parameter program table contain compute splitting readily height space subsequence recursively compute create quantity need appropriate dp payoff guarantee theorem achieve per space compute payoff draw advance pre enough make fast pre time similar check subsequence think store table step second manner align split observe align position setup part experimentally describe theorem require
ignore show interval blue problem select estimate unstable lasso elastic eq nearly lemma first rewrite q analogous notice rewrite selection elastic inference regard regression event characterize practitioner valid interval acknowledgement comment include linear model problem select tend sense regression setup response ask minimize coefficient describe
reproduce red observation interval red upon form eq set interval diffusion estimate posterior colored fall range rather expect around blue quantify quadratic posterior c c c c c range imputation interval broader c c c c c c c c c obtain bottom aim model problematic dealing solution infinity amount need cubic value record whether retain value marginal record look
acquire robust efficient paper aim responsible outcome index outcome salient generate salient sample analysis recover parameter limit identically iid dependent formula former perform code illustrate salient encode collection codebook code message channel message necessity paper analogous channel mention result interpret right side inequality reduction per term preliminary towards distinction estimation support discovery reliably estimate square variant conceptual capacity inference discrete object message give resort use assume sufficient well cover impose discrete contrast combinatorial dominating factor sampling rip herein always freedom sense isometry theoretic largely herein worth note author adopt albeit focused testing performance recovery include herein necessary error expression channel code framework literature sufficient literature former item approach consider wherein characterize precisely decoder use variable classical regression setup thorough
derive true counterpart step eigen eigen order manner factorization eq rank lead eigen eigen whiten orthogonal operator order kernel embed factorization tensor eigenvector completeness recover embed whitening design although infinite implicit matrix eq kernel map step nd order eigen decomposition involve dimensional matrix let cholesky result eigenvector cholesky decomposition eigen rd eigenvector embedding extend conditional view idea reduce multi identical give I variable respectively map furthermore
critical add substitute obtain collect duality rbf represent problem expression problem second duality conjugate linear total dual problem role p search bring exponential big make exponential go would never go infinity good sign exponential positive generality critical g
factor switch bandit begin bind adversary notation introduce denote sequence adversary switch cost rely feedback number exist cost drift appendix give begin construct adversarial equal shift illustration confirm range round xlabel pos east coordinate loss action walk random loss adversary player switch stay action gap loss round must total incur cost alternatively suffer plus switch achieve least show need little extra memory cost result action rely full round size range drift defer appendix feedback adversarial information loss action full difficult word equal bandit switch
classification relevance possibility differentiable illustrate method hyperspectral signature cm science mathematic propose prototype
day randomly sample daily load consumption correspond plot line censor censor variable dt receive therefore form temperature completely day censor day temperature curve peak load note quantile quantile choose kernel daily derivative finally optimal bandwidth near see provide peak peak triangle solid circle peak day author would vision project hereafter get follow lemma variable measurable almost surely need sum unbounded martingale derive result
complete probability subset q ss q inequality il I hence eq assume
see repeat family observation table four along three noisy classification algorithm please suggest parameter lead preferable estimation contaminate chemical physical contain five measurement separate gender superiority technique generate start em specifically circumstance constraint eigenvalue dynamic
corresponding remark mrfs mrf gibbs sampling mrf matrix propose walk none non improvement mrfs study characterize number node visit chain simulate markov chain discrete cube learn discrete cube pac threshold could let particular hamming distance hx ergodic transition I state chain mix graphical aid discussion nod energy assignment node configuration ise transition I node probability spin node otherwise unchanged maximal degree rapidly let graph graph every neighbor consider define transition valid nothing color choose color state step valid mixing let finite irreducible discrete mrf model serve stationary agnostic let possibly randomize label agnostic allow arbitrary
leibler coordinate variational update form eq index mode sub factor group latent eq optimize specific sgd factor investigate bfgs maximize well performance objective kk covariance sequentially similar omit detail sgd optimization implement zero simple update updating issue perform aggregate specific latent correspondingly procedure mode expensive entry node array computer obviously infeasible large store small store memory n carry variational update accord initialize aggregate make ensure
factor invariant standard elimination node must unary global normalization intermediate graph unchanged simplify graph call kind pointwise variable eliminate show kind figure tp factor apply one elimination hold illustrate integer state stop represent character index run entry encode index collection specify weight valid start product multiplication index order geometric root representation proportion symbol tp diagram index note index property well time eq keep basic weight
ordinary sampling consider consistency live strongly computational resource good confirm repeat simulation runtime expensive variance term mode bound rate decomposition towards reduce upon integration heart theoretic significance perhaps relevance ns e density brief measure mind begin existence normalise moreover common finite separable ns propose induce borel define transformation validity require e everywhere continuity integration valid absolutely reference ns compute sec ns absolutely agree valid monotone suppose validity lebesgue upon ns equality already without via theorem one trivial equal give interpretation distribution laboratory cb uk science inference main highly modal stationarity traditional markov chain monte second set compete strong reason key bayesian play role average likely area nest bayesian evidence transform evidence integral
consider phase retrieval generate independent ie ie unconditional applying noise retrieval sub noise bound use output program follow probability wise lose similarly handle blind z bind sub condition unconditional apply stochastic symmetric reduction phase retrieval extension scalability semidefinite noise linear constraint transformation set complexity regularize adapt mixed section outline perturbation theorem defer curvature direction demonstrate operator isometry turn convexity cone center direction generality correspondingly b rank equals define optimal solution either term appear constraint naturally operator feasible rewrite hold particular directly
result trial eeg spatio temporal measure hz stimulus treat pair feature result plus trial speed algorithm parameterize hold varied testing along permutation time figure plot speedup ratio voxel minimum verify converge never objective increase solve fmri along vary include computational speedup voxel include fast voxel never far tune run roughly converge exploit
run different produce filter case find filter demonstrate processing application band limited bandwidth bandwidth miss band limit tt inference band bandwidth spectra combine infer bandwidth full spectra human nmf use expansion bandwidth use decompose nmf limited part reconstruct spectra kl nmf leibler nmf nmf kl nmf order standard multiplicative nmf stop iteration decrease cost less spectra since power statistical randomly
maximum somewhat throughout paper type use identify parameter correctly consistency usual convergence estimate refer consistency consistency assumption rescale fit regression determine treat multiply divide column penalize consistency cite include evaluate compare roc fp false true respectively regression quantity set edge replace edge total number fix curve replication variable dimension discrete vary degree graph maintain total edge uniformity give case graph give chain maximum require attempt
clarity reason proof discount unbounded suppose limit ndcg exactly ndcg discount power ndcg consider assume cut otherwise consistent ndcg ndcg discount ndcg give ndcg feasible ranking theorem complete include ndcg suppose dr dr fx theorems top discount discount dr fx fx r dr notions conduct dataset web datum aim extent behavior contain click engine document task click labeling label click click click detail construct real totally document generate without set construction reality need search engine time train concrete follow choose separate training manner good rank bad ndcg ranking measure standard ndcg ndcg discount ndcg decay ndcg
comparison call two score correct triplet correct triplet max compute score result relation use apparent us issue overlap train interpretability sort entity relate triplet instance answer
subspace match without augmentation achieve maxout find current h l conv conv dropout conv conv conv net dropout conv net recently introduce maxout unit model challenge cifar maxout expensive scheme similar interesting possibility future approach work utility complex cell unit interesting paradigm explore believe towards activation incorporate invariance efficient average technique acknowledgment author discussion well provide b height grid style dash legend cat legend legend align legend style font ylabel distance xlabel blue x conv conv none green px conv output black table conv new dash forget maxout single red dash forget table px conv output maxout
shall examine implementation shall package thorough optimisation automatic retain desirable criterion like robustness p di red fm lagrangian lagrange minimization ss monotone minimize choice favorable dominate easy end determine p x evaluation situation obtain component saddle correspond multipli corresponding model helpful rf rf rf attain line asset price hide hmm hmm asset different regime algorithm utilize lead algorithm additive appear observe asset price peak asset return realistic financial market stock switch switch market participant result switch financial markov model lot switching follow amongst apply one viterbi hmms finance filtering develop
outside include nonzero index assume matrix allow constitute relaxation propose yield refine sect second solve convex diag easily formulate generic interior algorithm solve dedicated scale tell yield construction know sparse level appendix least vector imply need triplet q hand since guarantee factorization linearization solution check satisfactory define structural aim implication zero must optimization mapping mx mapping linear binary except sequence sparse lead order solve diag reason introduce weight sect relax case e reformulate software dedicated solver via even degree
result paper inexact exact update strong certain measure specific parameterized define constant precisely inexact require iteration certain inexact lead exact part focus inexact section assumption smooth minimization present section inexact provide step suggestion experiment section provide detailed matrix definition consideration model let decomposition uniquely vector give simplicity write symmetric standard dot throughout block lipschitz positive constant q separable decompose eq subsequent lipschitz reduce resp convexity resp follow eq strong optimal denote size iterate present description compute inexact iterate pick compute produce iterate iterate random depend iterate computed rest devote precise
action actually observe avoid issue difference mdp regret q expectation sum therefore regret reward poor mdp performance next tool bellman operator policy law give paradigm bellman mdp simply programming v ks ks ks ks I ks crucially depend bellman visit
glm naive bayes predictor voxel term rule voxel limitation hard scale predictor localize ill pose dimensional resort learn use regularize brain goal spatially voxel use spatially constrained clustering select feature voxel spatially quite parameter amount dimension help make tractable difficult highly class class simpler derive group category category apply train predict benefit suit highlight selective additional frequent mostly add
robot reach task scenario call fix appropriately design schedule free method quality yield furthermore change big function controlling approach maximizer action improvement maximize method update ascent tend result improvement variance policy gradient trajectory limitation rl long trajectory cope novel gradient policy irrelevant randomness draw prior policy prior thank prior base variance suffer instability call iw demonstrate rl method
stream detailed sub receiver detect human device technology receiver rich recognition treat capture effect interest computational server employ boost location single localization correspond distance localization environment rate brief physical human compare art localization relate direction briefly rely build core frequency call phase wireless superposition standard available market provide intel group
flexibility specification b spline baseline hazard express denote spline knot spline knot balance overfitte keep knot flexibility region greatest estimation bayesian paradigm inference markov mcmc derive effect account random longitudinal longitudinal response denote effect appropriate assume give history longitudinal censoring time longitudinal subject parameter form incorporate close numerical employ gauss particular longitudinal survival spline hazard prior variance prior bayesian specification derive either survival fit subject provide longitudinal baseline survival interested longitudinal response dynamic evident record obtain section correspond survival longitudinal independence rewrite note j jt jt jt jt
confirm map k identity precede correspondence versa precede domain invariance locally demonstrate translation invariant translation converse hold familiar inner definite metric
selector z z eq indeed verify simply switch replace large upper noting need last pass importantly pair depth define every check c along therefore bind expression twice expression suffer rewrite step pass choice derivative often omit understood range pass inside infimum inside minimax last q give yield upper repeat arbitrary appear repeatedly argument minimax assume infimum expression sign split minimax rate derive
transition iterative super polynomial feature super iteration feature polynomial exponentially previous parameter set paper propose improve continuous learning
vertical indicate bic confusion penalty indicate em merge aic penalty yield single number various factor penalty gave however systematically evaluate effectiveness measure variation vi value variety penalty figure f em achieve perfect penalty able produce merge spike sort neuron array firing yet hybrid dataset
different vary lot yield digit accuracy digit accuracy spc spc even relative unchanged figure monotonically yield investigate behave equally skewed plot demonstrate monotonically increase change extreme near spc spc spc spc yield low spc value describe behave start subproblem different spc spc spc generate naive additional conditioning spc spc spc fast become curve except spc understand accept sampling construct size indeed upon conditioning subproblem subproblem quite large conditioning base come spc least spc need fig eps compete compete primal preprocesse completeness design response manner skew replicate census datum obtain lead submatrix record result show spc spc spc similarly case skewed spc slow method namely conditioning subproblem spc ellipsoid round small spc rounding method notice skewed reason skew running size method except spc spc spc spc
context equivalence equality expect establish exploit geometric e filter star include balanced cover star let restricted check exploit spread let characterize factorial effect element specify disjoint exist constitute constitute span contain distinct constitute contain since I linearly since characterize burden consider say element j ordering e choose ideally find turn intensive working
reference unfortunately case bernoulli encounter proof general seem justification inequality full relative presentation ccc consideration theorem say coin
relationship node network probabilistic choice core blockmodel assume belong role blockmodel specify assume possible k link role pair draw draw receiver network blockmodel define link latent distribution role normalise role receiver intractable require sum latent jensen inequality distribution categorical conjugate understand labels blockmodel identify describe link find role map margin classifier support choose classifier strong range iid task additionally character recognition effective margin iid blockmodel representing relationship corresponding dimension property point machine optimisation constant delta represent slack separate misclassification multiclass
operate compute lead eigenvector input explain dimensional transformation constant solving maximization guarantee pca usually reason start description accuracy obtain decomposition eigenvector support matrix check maximum efficiently solve technical candidate optimal create support support submatrix row everywhere except principal submatrix index else large sparse pc table subroutine explain eigenvector lead eigenvector dx step elimination subroutine provably pc norm identify use elimination come large elimination rank intractable subroutine empirically observe around subroutine instead sparse give approximation upper bind model level solution time corollary plug bind family practical set like eigenvalue power
error accurate provide cc switch operation consider choose accordance operation contiguous degree appropriate signal switch fig top signal give fig probability close regression htbp k regression c devote classification signal divide base apply signal approach vector parametrization classify map signal index correspond operating switch consider minor shall
project category report methodology extract record via center report number field update user change user initially throughout report responsible list user receive subsequent update report million present throughout incomplete report total indicate report handle completed reach decision extract history change final fix behavior wrong usage rather already report resource whenever intermediate status report properly project status report report lack remain status easily classify extraction structure set contain associate report ten capture dyadic interaction arguably dyadic decide like user necessarily report add receive update special responsible focus update necessarily provide nevertheless structure project interaction indicate aware know interested
away car motivated find classify shot cover word zero shot image give overview describe predict conditional see unseen indicate set image new factor thresholding outli score training map space map class obtain px px
harmonic super conventional approach eigenvalue construct spaced accommodate infinite precision spike besides largely depend unstable presence noise differ algorithm super resolve object frequency end super resolution appropriately separate inspire al frequency assume randomness accommodate multi assume randomness inspire recover entry number theoretic limit algorithm nevertheless theoretical model direct degree
parameter eigenvalue arise possibility overlap eigenvalue small eigenvalue eigenvalue employ appropriately filter discard multiply outcome select analysis standardize kernel often effectively factor component admit keep principal important cutoff eigenvalue vector example filter discussion
admit cover ball ball radius covering let ball ex fr define set center estimate cardinality collection specify therein reverse inequality reason whereby cover state hoeffding least remainder discard failure cover discard additional definition bind meet simplify range maximize therefore choice consequently entail eq secondly last term combine piece defer rise throughout either direct consequently eq word combine follow scale radius lipschitz convexity respect uniform cover scale cardinality vice analogous minor whereby center cover provide probability remainder discard element lastly restrict throughout section kb terminology first radius outer construct use give
term corpora classifiers macro micro extensive predefine near svms generally often dimensionality moderate sized reach dimensionality cause curse reduce classifier statistical ig mi sound method tc task ig accuracy mi document df ig corpora classical whose term predefine ig separation
conduct replication uncorrelated design arise initial satisfactory numerous experimental involve adopt cut uniformly machine cut strategy respect test random seem uncorrelated average replication idea take mean risk since cut scheme generate practitioner compute vector compare super learner concrete supervise classification naturally research collective mean sd sd sd sd sd sd sd sd sd
indicate subscript ground distance wasserstein emphasize ground distance two ground choose invariance combination rotation principal widely accept manifold frobenius express classical f rate sign invariance transform distance ground distance hausdorff wasserstein metric uniform hausdorff rescale visually percentage evaluate combination rotation recover learn dictionary assess rate raise variation wasserstein well picture smooth dictionary initialize dataset original case limit frobenius early base hausdorff metrics hausdorff less sensitive consider set hausdorff change cause algorithm stay whereas wasserstein distance amplitude rotation invariance without signal ht rotation invariance apply rotation wasserstein hausdorff indicate slowly hausdorff evolution phase concern frobenius major metric wasserstein nan whole uninformative poor show component recover fail correct input signal coefficient atom rotation stay fluctuation selective around atom highlight extreme threshold frobenius distance affect rotation suit wasserstein hausdorff able capture learn dictionary explain wasserstein
problem approximate certain able approximate take review theorem full modify formally nm inside short sphere close point lagrange multipli force manner difficult
partial jj estimate assumption converge normality state proof fix estimator estimator residual c p precision ball n conclusion hold define theorem imply necessary carefully apply yield observe matrix covariance impossible provide weak ball parameter space need equal last g match bound norm support estimate graphical commonly require normal paper glasso support recovery glasso glasso correctly strength nc nm n n pn cm make support procedure impractical introduce oracle entry condition significantly remove correctly distinguish need define recovery ij ii jj sufficient weak thresholded precision jj set gauss
thus testing explain propose gmm condition framework manner conventional adaptation hmms transformation viterbi output recognition perform use adaptation gmm adjust datum gmm viterbi decode clean mean belong condition time p snr snr snr cm c restaurant belong noisy gmm
f ds rademacher inequality imply last bad process probability ds tn conclusion frobenius next bind testing distribute pack kullback leibler distribution km l eq norm kl distinct km r satisfie yield rd pf generality consist stack bottom symmetric kl succeed zero consequently pair independent bernoulli index imply I complete matrix trace theoretically numerically norm behave propose minimization max constrain minimax unified effectiveness method order program max
descent however none scalable size unable instance unable change assumption proceed method bound find computational section study nonsmooth regularizer definite typical encoding row sl loss sl logistic loss hinge rx choice model regularizer regularizer computer partition equal cardinality describe comment later assume comment computer pick independently computer formally iii cardinality summary feature belong update cardinality computer compute k kl ix
individual sensor set broad regularization bayesian enable truth observe next hardness use procedure outline benchmark replace gap exploration extension introduce wherein single monte arm select current pi select attack arm exploration technique compare purpose compare even objective understand empirically matter strategy horizon vary effect perform effort model arm well alternative address simultaneously learn practitioner technique predictive task understand toolbox intuitive mcmc preference
apply demand assumption particular leave one lasso versus degree freedom predictor consider correlated irrelevant analyze selector eigenvalue oracle oracle others possess risk consistency type say datum dependent persistence estimate lasso require om generality large contribution show persistent relative empirically model correctly pattern setting frequently technique freedom trace degree freedom adapt information plug taylor expansion risk selection piecewise
linear combine plane angular domain search simulation array array arrival array frequently wireless communication decade notion require step uniform array step replace music array equivalent efficient array separation domain music virtual satisfactory virtual array increase suffer converge rao increase special algorithm uniform array music process many degree freedom enhance receive
conclude proof lemma suffice performance batch task assessment come two initialize observation report bandwidth likelihood gps draw mat dimension mat ern variation gaussian mat ern kernel peak thin opposite challenge find peak exploration exploitation delay differential feedback highly make benchmark maximum vertical wave
similarity however unsupervise lack selection thus parameter need manually unsupervise supervised scenario propose call analytically systematic demonstrate usefulness cluster exist unsupervised cluster square assign label instance instance maximization try tuning regularization mi sensitive log
latent well induce university usa ny usa j research ny consider quadratic maximum base quadratic art method build convergence rate strong batch admit include sag average recently weight gradient method run parameter dual ascent optimize storage gradient minimize adaptation view update use
effect relatively effect fraction sd close particularly training case avoid estimator fall safe performance relative contribution describe estimate part fraction run cost account cost situation situation safe might make sd cox full strategy little take place choose amount estimation full cost significant full selection factorial difference full safe difference always sd although see safe well sd sd contribution allow split
submodular ls operate aa aa aa ls obtain great ls call step eq interpret ls n optimality guarantee greedy obtain ls perform proportional compare l force ten algorithm randomly skew favor near figure fraction fraction great bar ls mean comparison indicate optimization favor characteristic maintain auxiliary evaluate remove solution optimize operation arise add removal operation loose scale method three coordinate ascent technique ibp stick breaking ibp maintain couple beta allow unlike inference
future solution overlap exclude allow describe idea intersection group frequently group intersection store constitute store overlap group concern overlap belong also explore keep track may boundary although suppose store node turn explore allow polynomial key construct optimal subproblem indicate approach property group easily pick converse list remain cover could polynomial also cover problem difficult select imply element change independence give let boundary node know suppose group group boundary solution solve respectively ease explanation boundary boundary call need element group choose match maximum total choose contain equal group recover weight element formally disjoint argue component constitute small optimization global create two correspond construct select disjoint element element verify form valid valid assign ready prove correctness suppose contrary claim constitute selection namely element hence represent valid solution hence must comprise selection identical argument group element element node equals exactly prove correctness explore acyclic graph time store visit solution rule exploration rule take output explore encountered boundary update value explore node label manner define optimal maintain give solution store element count number select ib indicator vector selected exclude empty node argument mention practice serial track consist intersection certain set
simulation bottom pass algorithm average solid replica saddle point solve saddle fluctuation ten different initialization entropy concave landscape horizontal indicate vertical line mark concavity dot line mark line go part correspond min cm bar ten initialization entropy smoothly decrease distance behavior keep concavity entropy increase support simulation instance propose pass entropy landscape actually calculate figure recover typical take one curve leave however broad solution may responsible algorithmic hardness increase message pass iteration especially distance additionally expensive
specify dataset perform procedure generating approach essentially include spline spline organize give background base maximize penalize log likelihood regression formula brief background generally multidimensional finite observe represent multidimensional correspond unobserved miss class take cluster covariance mixture density negative mix gaussian mixture follow observe use em gaussian mixture model clustering properly initialization classification stochastic
gain approach graph value function markov network represent undirected numerical sub g c low px parameterize assignment x normalization model clique weight assignment domain assignment feature say iff associate indicator assignment j appear subset potential log obvious feature ht c finer grain specific conditioning follow random disjoint set assignment iff
worse limit come quadratic variation wiener different asymptotic simple expand classical similar lot estimator strong consistency begin impossible indeed mean behaviour essentially power wiener nevertheless behaviour detail contrast eventually clearly distinguish simulation classical collective behaviour end represent simply partition deduce center
order sis r hence every argue f ir work boost monotone submodular unless otherwise indicate return j fy fy use monotonicity lemma j finish nonnegative submodular e e proof monotone running subset every union derivative addition monotone x concentration submodular concentration bind submodular outside constant make monotone submodular also monotone submodular depend hx fx j eq finish monotone repeatedly reduce necessary eventually multiplicative use mean parameter variable invert approximation assume hold prove grow geometric much concerned work multiplicative remark constructive multiplicative randomized polynomial query succeed structural submodular section prove manner influence rely real function product refer fourier fourier expansion degree large fourier polynomial sx sx several value influence value f f influence satisfie equal prove self influence second self know fx self negative self bound polynomial integer q submodular approximate prove approximated degree self total show low close depend influence influence easily use fourier coefficient style also
available slowly payoff refinement far contextual pool click relevant may click minimize click combine bandit thing significantly click document similarity et achieve put forward arm metric implicitly implicit metric exploiting match implicit lipschitz payoff lipschitz payoff payoff regret constant generalize g payoff click click lipschitz payoff provide subroutine adaptively mab bad notion similarity particular accommodate outlier make elsewhere mind payoff ensure include mab bandit payoff function et possible lipschitz mab mab payoffs essential algorithmic dynamic pricing bandit various definition essentially self exception notion cover subject mab instance triple constant equation algorithm choose receives payoff throughout diameter set arm complicated triple neither reveal pick environment choose accord receive payoff also observe payoff query borel algebra topology fix choose supremum always specify list subscript denote similarly ball radius bx open ball contain finitely many cauchy sequence distance cauchy constant ball cover vice family subset contain closed intersection specific clear ball intersection topology call singleton topological image total non element classical concept extend natural infinity understanding notion standard von limit induction necessary material logic various notion metric space number cover dimension mab subset diameter minimal multipli infimum cover define diameter former robustness dimension often scale cover extra uninformative space contrary constant explicit allow numerically finite metric define base notion point least packing packing packing notion pack packing contain cover else cover pack remain pack use infinitely many set finite diameter dimension multiplier notion science g sake section give small infimum cover diameter dimension restrictive covering dimension concentration formulation chernoff space pick choose packing ball notion play specifically bandit arm phase round optimally duration turn analyze mab metric covering dimension parameterize sir phase consider algorithm round lipschitz metric net plugging phase phase incomplete accumulate refinement take advantage derive proceed time play phase correspond q expectation intuition available time confidence execution among activate arm active stay active phase remain specify thing arm choose active upper expect confidence note arm confidence ball maintain arm cover rule maintain invariant arm pick radius newly activate round implement use initially arm break tie provable near worst lower let arm arm cover problem multiplier satisfie moreover parameterize multipli problem specific regret except quantify lie multipli second example mild space example require example immediate mab triangle relax relaxed lipschitz
projection constraint reason matrix inactive optimum nonetheless e rank converge advantage recommend zero allow allow encourage implement implement proof act eigenvalue hence eigenvalue eigenvalue perform every check update incremental incremental maintain projection perform well incremental get suboptimal draw probability failure essentially orthogonality entirely iteration entirely fail algorithm
boundary hand give point test uncertainty even precise scheme mis boundary finite behavior link axiom theory make derive explicitly cluster shannon entropy begin property property desire property large grow refer coarse p p intuitively combine grain bin uncertainty within coarse grain bin original uncertainty al risk mean theoretic analogue dx py coarse differential standard notion coarse remain entropy quantity converge estimator unbiased consistent coarse estimator search coarse keep
underlying assignment algorithm sample low build cluster assume often pool improve limitation measurable contradict assumption forward expect operation rewrite measure chebyshev union recall j eq upon get several use chebyshev inequality union concentration tail bounds chebyshev condition analysis lemma gap away zero one prove least big aa aa ba b ga
ni array define tell n z assume occur convergence writing h h ni eventually schwarz proof n grateful suggest suggestion theorem nsf grant grant dms imbalance subsampling reduce cost subsample efficiently logistic adjust locally accept generalize pilot conditionally rare feature bias subsampling correct post hoc analytic require scan set inconsistent misspecification consistent pilot correct specification exactly select subsample comprise happen multiply acceptance great roughly experiment simulate method pilot control twice variance regression large two accept weight full mle subsample subsample imbalance case improve bias control general inconsistent contrast pilot asymptotically unbiased pilot present demonstrate yahoo consist predictor binary map linear less might expansion small set nevertheless real view population maximizer otherwise possibly matter misspecification
early mixed ignore situation far focus mahalanobi inspire work advance trend devote address deal method local histogram framework supervise focus mahalanobi attract often motivate due absence psd form psd symmetric depend cosine retrieval author idea optimize online perceptron satisfy criterion present follow perceptron form bound subsequent relationship learn generalized cosine imply use function projection onto matrix achieve pair back onto psd cone cosine normalization close base decomposition make costly regret converge iteration performance competitive base learn bilinear large bilinear retrieval require psd symmetric cosine normalization psd constraint unnormalize cosine bilinear similarity advantage efficiently mahalanobi instance query rectangular since psd manner simple belong passive triplet eq minimize clearly otherwise achieve medium scale unlike million instance regularizer derive similarity online learn matrix second matrix take angle focus metric metric good learns bilinear formulate efficient unconstraine reference randomly select point opposite margin learn classifier naturally bilinear focus set predict matching domain fix riemannian formulate square hinge carry manifold factorization manifold experiment conduct metric metric easy formulation global less unable see satisfactory explain metric address induce spirit svm metric interface datum get though analysis nonlinear pca short implicitly datum perform unchanged refer trick al trick theoretically sound unconstrained prove theorem another spirit sense space categorical result equivalence mahalanobis learn regularizer lastly choose
ask free comment understand performance comment comment figure comment sentiment sentiment word comment comment filter english observe suggest student write say mostly neutral suggest student make effort comment signal predict information well predict construct whether next assignment conversely student drop addition student reliability include bias reliability predictive area auc property student assignment aggregate noisy worker adapt answer key appear crowdsource combine hundred rich model constrain set perhaps use small work bias consider student crowdsourcing balance typically
usual involve respect transition work modify classical chain get th component albeit way article ex ex electrical engineering vs com learning gain popularity technique approximate dynamic know aspect distribute certain involve
give qp k true solution qp simply reality need solution affect error concerned qx solution bound equivalent bounding perturb give perturb constraint result definite minimizer different invoke bounding error k immediate counter suggest correspond variance estimate note finally stochastic behave output exactly qp precisely perturb aa c affect thus singular exercise lemma asymptotically ij kk mixed asymptotically fact matrix negligible k ib ib k ib ib solution qp want qp asymptotically repeat lemma get
demonstrate approximate score improve large exact solution straightforward obtaining due minor formulation statistic algebra gap provide perspective refined question algorithmic framework scale relate notion toy illustrate leverage section elsewhere reader skip familiar present relative sample size procedure meet give standard address spread combination combination somewhat detail combination efficiency efficiency follow observation ix tx assuming typically assume since definition analytical toy ask common restrictive leveraging asymptotically follow result compare term partly reason subsample state characterize relative efficiency lead asymptotic efficiency ignore efficiency thus omit lead relative ignore efficiency appendix asymptotic ignore tx asymptotic ec toy algorithmic leverage method toy seem artificial property realistic leverage behavior primarily extreme equal first term small score crucial case highlight algorithmic former problem perspective reason want empirical opposed ground deal leverage issue development leverage hand small score interested algorithmic leverage extreme small leverage eqn variance discussion toy illustrate thing set arbitrary case score algorithmic three let odd four four unconditional variance evenly illustrate panel easy variance lead term second variance expect similar calculation small lead component arbitrary evenly spaced toy space property matrix fill asymptotic score illustrate panel understand panel probability
j I yield q marginal denominator q easy q iy guarantee return ij yield q procedure return couple heavily sequel independent probability yield notation define set interest purpose helpful accordingly vector intuitively vector recover svd straightforward straightforward assumption next like amount show incoherence ensure vector bound norm vector probability lemma calculate note bind calculate far last q norm expect position size inequality contribution apart previous lemma fail random least furth two follow recall early expand eq show show norm ia use finally statement
example layer convolution backpropagation single provable learn rely support I control bound appear hide layer provable learn weak acknowledgment throughout stage fs thm thm thm claim conjecture provable deep net generative neural random learn polynomial cubic upon novel correlation infer analysis reveal random net task deep net fact np hard nonlinear hardness provable input hmms mixture provable guarantee net still seem neural net threshold depth net underlie modification viewpoint suggest rbm reversible layer autoencoder decoder methodology learn net learnt unsupervise generative follow training viewpoint reversible net generative hardness remain mathematical net autoencoder autoencoder hard linear provable manuscript must paper provably learn ground truth generative net sparse assignment upon successive bottom scheme autoencoder terminology equip decoder efficient encoder break assumption
module application system parallelization achieve parallel scientific parallelization enable job trend inter parallelism employ provide control avoid furthermore intra load balance scheme specifically design pf library choose illustrate panel process cpu per equal core compare allow core per hybrid keep memory coherent low particular application specific cache speed etc cost etc leave blue jt circle green rectangle jt library implement application choose
subsection follow matrix scalar constant let discussion essentially ignore follow small infeasible functional exercise omit really storage capacity bind storage capacity box constrain perceptron subsection mechanism upper maintain storage capacity looking fix discretize center design determine deal explicitly convenient slight far scale alternatively fashion make course fairly lemma let related well constant follow subsection first writing possibility contribute solve maximization moment assume e leave determine zero recall q ultimately eq recall combination leave pg imply feasible summarize subsection large scalar normal random let comment sign let q eq match replica mechanic predict replica symmetry operate correctly hand curve operate course follow probability htb feature focus property storage specifically eventually terminology capacity call happen match framework replica symmetry mechanic two perceptron refer box perceptron call digital binary storage
expect lemma enough follow line theorem proof decay chebyshev inequality least suggest fast agent state dependent kl study sequence informative dual update average recover connectivity belief true probability convergence exponential divergence observation true second salient contrary stepsize direction independence
refined frame set follow phase base theorem perturbation continuously rational go symmetric pass happen multiple finitely symmetry number thus conjecture phase comment hilbert
label feature denote nonzero nearly lie subspace upper cardinality constrain therefore decompose sample mse separate building constraint optimization minimization subproblem subproblem thresholde singular matrix hard thresholding particular build kl zero iteratively subproblem initialize subproblem fix acceleration preserve jointly since stage explore initialize kl j I kn py ty ny ty kl nk I ir group ensemble wherein define index group coefficient group nonzero concentrate label belong analysis set integer group sparse concentrate although via select nonzero choose properly set guarantee although low completion effective user rate rating explore model item user require whole low rating attribute miss recommendation helpful rate allow collaborative filtering supervise avoid consume completion new row predict rating whose column entry give feature rating user score rating study score user replace row effect anomaly user rating represent scoring update
correct output wrong misclassification vote large show incorrectly recovered base important performance properly base idea propose acyclic strong se elimination candidate max empirically traditional uci review framework properly classifier section explain conclude pair class voting class final output give maximum selection candidate maximum vote class ignore line model class three voting class large htbp acyclic dag acyclic graph represent node node arrange single final evaluate apply eliminate output remove select class classifier process class eliminate candidate final decision final
expectation inference section provide limited robot scan along shape determine location base know measure employ engine make inference circle parameter record intensity engine circle location relevant core inference parameter record location write compactly bayes allow side circle sensor ratio bayes since modify record result probability datum circle denominator act evidence write integral critical role term role compare I light sensor predict region return accurate light sensor sensor detail tb axis play coordinate measurement obtain light sensor datum black white na I sensor measurement indicate green square
line balanced polytope diagram vertex exceed basis point support feasible power diagram support weight algorithm vertex visit twice different diagram bound aid corollary strict balanced bound integrate higher replace set conceptual actual kernel argument high inner common conceptual change replace kernel u particularly site preprocessing xx solve program eq center output feasible center present handle new replace mean least square model
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box random sequence q special object may indicate possess remain possess exactly carefully iff represent object relate possess want relationship depend object assume array dependence capture exchangeability exchangeable index set consider array permutation informally exchangeable permutation underlie simple exchangeable define map probit straightforward exchangeable model array represent finite subset constant independent say array base array array either array base exchangeability nonparametric feature mass number base say exchangeable randomization simple exchangeable array compare represent feature randomization underlie allocation aspect call stick construction simplest break also variable concentration parameter relationship one ibp crp follow feature valid object box stick break ibp put every v ibp piece constant relationship feature assignment induce present illustrative formal treatment specifically partition comprise interval assume continuous markov embed process jump entirely cut cut proportional transition replace analogous transformation plain cut represent classical align possesse define process infinite probability one process nonparametric independent countable comprising along align slice agree dirichlet variable model bernoulli randomization detail piece wise mix blockmodel characterize structure briefly model gaussian function sequence convenience gaussian satisfy I semidefinite choose appropriately array exchangeable process however arguably real note mean process space put correspondence interval array randomization noisy probit variance family r many popular exchangeable array variable construct gaussian process nonparametric parametric introduce xu
final imply although still receive influence small induce accumulate influence converge induce example illustrate assign assign value odd absolute consequently system depend convergence iterative originally integrable weak suffice sup update influence point
calculate efficiently dynamic programming follow centroid estimator centroid maximize probability consensus eq corollary centroid program optimal secondary rna centroid secondary structure problem consensus centroid aid dynamic score alignment description centroid operational unit predict multi dimension unit cut every partitioning centroid estimation partitioning theorem consensus hamming topological centroid topological appear centroid despite difficulty generalize previous complex bioinformatic follow secondary structure rna rna length model rna predict secondary structure length probability implement rna secondary energy apply devise gain connect secondary rna rna order assumption problem different space generalize gain generalize gain secondary structure rna secondary prediction representative parameter represent predictive representative reflect data gain gain gain q homogeneous gain estimator representative representative homogeneous gain average centroid representative corollary problem representative centroid centroid previous representative section pairwise biological align sequence space possible
q fact immediately next argue inductive last inequality first desire part rule relation rest proof argument pt constant eigenvalue order bind smooth constrained unconstrained resp order cover nonsmooth e also equally applicable block carry mention mild relax block case reduce successive list assumption c suppose assumption must coefficient follow function second plug remark main result function upper cb true q gradient use lipschitz proposition cc use convexity cost go cauchy inequality inequality
together discretized usage maximum fig usage move away simulation dynamic line also make wrong accept test value smaller typical eqn error acceptance estimate eqn
kx program j qp solver see map k reproduce anomaly show rbf kernel kernel primarily sequel reproduce associate x group anomaly detection always I underlie apply gaussian analytically particularly incorporate observation learning section analysis discuss sequel analysis b sphere feature space hence probability mean embed convex hull form segment sphere three sphere rbf kernel mean lie sphere angle lie translation constant imply sphere hold show embedding cf uniqueness
assume slow recursion quasi static slow converge comprise using assume analyze slow asymptotically converge closed di cc establishes recursion track ode tuple instant chain denote equilibrium local govern recursion operator eq column parameter transition chain policy component transition inclusion denote diagonal along projection directional define define sequence sigma main govern converge theorem material describe section assume transition chain movement scheduling previous assume convergent scheduling approximation essence fast scheduling conduct slow thus converge transition instant ni projection ensure iterate fast recursion slow explain suppose known would elsewhere quantity elsewhere recursion logic extended know unknown true discount case section bellman sake implementation later discrete discount find bellman q average discount setting require incorporate discount
numerous similarity group recommendation base filter recommendation start keep mind centralized amazon netflix complete information moment issue pay separate recommendation share mechanism agent user recommendation system agent recommend another agent indirect agent important thing paper decentralize mechanism trust recommend assume decentralize user still recommendation source agent fundamentally l c item memory context ib mae pearson correlation pearson ib cosine mae pearson define c yes regret regret old decentralized learner agent denote set agent user slot agent website number ad place price disjoint agent agent agent agent topology item hold digital book movie video etc item know company item service video etc notion item thing google etc receive user formulation customer emphasize interpretation valid natural induced slot item user want list price age gender etc agent context space without generality take context asynchronous arrival rate keep index agent agent recommend item paper sale agent recommend beginning show basically agent obtain item privacy concern sale price agent restrict recommend agent may also preserve privacy item hence whose goal agent wants recommend item agent request create agent agent recommendation website format
z instant smoothness iterate constant establish bind recall main derivation find constant explicitly error average involve sequence step bounding lipschitz corollary perform second transform page display note page find I pass pick quantity size temporal td converge bellman operator overcome curse architecture every state feature vector onto space simulate process policy transition reward estimate iteratively employ complexity complexity trick inversion iterative theoretical approximation paper analyse
propose efficiently invariance learn feature redundancy code learn patch way feed nonlinearity art dataset explanation phenomenon frequency patch uncorrelated become correlated pool introduction dictionary limit code various reason computation purely feed encode speedup algorithm sized help true applie stage pool immediately layer beneficial compact address often convolutional usually find invariance mean operation instead simple inner patch account redundancy centroid method representation oracle explain recent nystr subsample
former neighbourhood peak area peak expression value ei mean ei desire behaviour acquisition previously iteration parameter obtain ready combine previous latter select also make likelihood e gp use mat ern mean kernel isotropic computation ei optimisation
data map z concerned algorithm random treat equally measurable algebra forward measurable learn visually separate test little reason necessary large size note happen view notational distinction extend expectation take realization boundedness also symmetric satisfying introduce involve word cyclic permutation permutation advantageous observation convenient ease case identically let recall nan alternative former usually unconditional nan nan suppose instance learn usually difficulty interpretation type unconditional plug ready make learnt sort error latter number
blind put anonymous interest auxiliary refer without generate portion gets think detail te sample bernoulli contain compatibility index membership indicator eq detail supplementary assume current index sample calculate restaurant analogy count un similar distribute kind table two indicator
high power hz day begin obviously second low become apparent importance channel configuration learner utilize construct improve similar boost eeg accuracy psd tracking spatial verify stroke tendency provide pattern current study stroke subject lack achieve change mechanism band spectral configuration eeg study heuristic boost collect clinical effectiveness stroke patient parameter model verify spatial band spectral knowledge communication human brain external device pathway accomplish base brain signal eeg
ratio approach elaborate introduce spike bias try ibp annealing encourage proceed ultimately fit comparison amongst q extend new annealing imply simplify gradually decrease jump ultimately probability add new feature denote please weight simplicity inference load follow graphical load posterior independently
dependence expert localize provide dependence model mixture regression mixture marginal typically maximization suffer optima next optima low regression tensor convex warm challenge step define sum mean produce know insufficient freedom power parameter treat another jointly provide contain fact definition consistent identify parameter identify rotation orthogonal moment zero knowledge whiten compute eigenvector w tensor return parameter
involve dimensional domain curve paradigm goal visualization exploratory discrimination present change curve multidimensional classifier machine regression spline polynomial piecewise understand process generate handle heterogeneity regime flexible new modeling shape
visualization calculate index later number suggest majority tie meanwhile small tie large community happen spam indicate work validate usage spam perhaps within either color entirely entirely particular appear concentrate strong tie enhance suggest cluster cluster otherwise label compute rand rand commonly rand measure label cluster number cluster different rand complete disagreement rand index rand index clustering problem rand index indicate rand index divide division produce table excellent agreement
focus application rather well original might beneficial use example noise well norm discussion would particularly depend differentiable apply successive nonnegative similar recursively project cone extract refer fold theoretical broad class nonnegative real hyperspectral image popular comparison test ghz ram separable nmf nmf zero contain another dense hence nmf decomposition therefore nmf separable separable separable good initialization strategy nmf initial reference condition close case result extract index near separable column illustrate apply broad class refer ill condition
bayesian inference view different parameter take close true idea decision maker discuss maker know would solve action maker proxy aid maximum estimate limit infinite ever decision maker distribute case learn later motivated close kullback leibler minimize consider follow whereby thought divergence hence prior assign assign problematic formal proceeding keyword generalize entropy environment become bayesian statistic major perhaps great requirement define datum objective statistic analyst model complete coherent general require connect mean ease shall terminology parameter interest interest lead conventional central complex increasingly force aspect traditional reliability analyst seek coherent proceed interest example median formal update refer suggest et validation serious back lack coherence another ignore come obvious wrong extent number al know none update nothing interpretation construct directly write whereby random denote estimate methodology make rapid development development complex complicated reasonable specification fidelity fidelity regularize et select approach paper
explore time detect temporal heterogeneity york specify alternate epoch interval matrix epoch epoch effectively produce parametrization factor h maximum branch color modal node grey represent across grey dark interval factor width reflect diffusion support interval search well support epoch spread bayes factor epoch suggest dynamic appear infer mostly within ps ss homogeneous homogeneous epoch discretized epoch alternate separate matrix largely fit marginal essential comparison improvement return epoch comparison material visual speed increase serial convolution analysis set equip intel ghz cpu gb gpu core run mb memory relative state dot transition matrix grey dot operation diffusion process epoch speed patient epoch figure gpu number turn queue number extra report relative serial execution update cpu device analyse double cpu set
globally toward many riemannian applicability track record build toolbox piece help researcher practitioner tool code toolbox manifold solver description need manifold pass solver
sgd depend complexity sgd replace linear ig directly subgradient I save exactly smooth objective consider composite partially linearize analysis return strongly avoid introduce dominant proportional minimizer weight dynamically beyond paper f proportional proportional optimize weighting consider smoothness thus rather also I l I yield suboptimal weighting proportional suboptimal factor two special l instead arbitrary estimate iteration first non rate lead provable convergence expectation extend yield convergence acceleration iterate weight reduction dependence conditioning
calculation potential apply repeat finite curvature efficiency prove theorem quickly small small concentration result argument around I tf f tf I top chain give something certainly rapid slow mixing first effort couple technique quantitative mixing open notion curvature analyze tool paper independence energy despite markov chain chain bound obtain adaptive chain main tool curvature markov general bound paper class markov chain focus energy contribution construct energy walk modal target show appropriate energy sampler decay parallel metropolis substantially small mixing underlie walk metropolis find burn substantially correspond parallel sampler mcmc parallel relative version analyze convergence broad sampler knowledge estimator sampler little bound mix complementary bound sampler period every sampler main show energy wasserstein happen mix energy sampler inferior associate chain nature technical strong argument e liu slightly convergence space strong curvature assumption ergodicity coupling system coupling spirit different setting notion curvature
fisher g g distribution gamma db ga ig g g ig ga ig db ig I ba ig I g g ga ig ig al ig ig g g ga g ig db ig ig ig g db ig k ik ig ig g ig ig ig k k g k c al g g g al g g g support natural engineering
u semidefinite uncertainty sufficiently ccc communication r compare propose gps analytically variance recall parallel base gp machine incur improve scalability centralized counterpart respectively base computational parallel machine centralize counterpart due grow unlike additional cluster keep machine base gp size raise gp slow gp memory parallel communication complexity depend entire old lack space greatly new stream gp advantage online machine contrast vary suffer matrix remark performance gps centralize dataset road road include etc peak hour
necessarily second count begin matrix compute th probability well therefore sequence natural logarithm p sequence method report generate impractical member path possibility frequently large repeat reach return
portion relevant upon resp db relative concentrate object attribute db detect form density kde numerical function denote respectively order compute formula equation bandwidth contribute manner pdf moreover otherwise trade sort domain greatly impact pdf window make measure proper block natural homogeneous represent accord separate density feasible adopting accord practice attribute purpose devise em draw location equation adapt procedure allow
cut towards favor vertex illustrate figure hypergraph cut balanced minimum cut cut unbalanced node clique expansion complexity clique expansion fully graph computation prohibitive minimum vs minimum expansion hypergraph perfectly cut unbalanced cut omit star expansion et exist graph cut hypergraph always hypergraph weighted uniform hypergraph matrix value correspond hypergraph cut take cut zero thus hypergraph cut variation technical element extension
scheme relatively adjust fast solution approximately rank preferred run discussion terminate output n rank n adjust simplicity uniformly experiment multipli lagrangian follow positive kkt proof mainly alternate non none give establish let prove let equality summing show hold space c reverse inclusion accord matrix size let ready kk summing observe x k k give section test solve real fix current iterate tensor mode overall fitting
yx yx game although value nash equilibria play potentially particular characterize bad outcome follow treat certain range examine impact focus play agent game parameter categorization proceed elaborate connection point ne ne equilibrium ne learn dynamic true one note first equilibria agent player factorize simply rest ne attain strategy furthermore game ne configuration ne play pure strategy two still trivial ne pd pd choose satisfy dominant
filter tree filter regression extensively investigate signal satisfactory linearity nonlinear suffer stability signal processing issue adaptive linear furthermore involve big require vector nonlinear usually avoid tree regressor elegant regression partitioning regressor space partition space regressor hyperplane complete nested regressor space nest regressor region regressor statistical assumption well linear truly upper combination restriction adapt extend final final ii iii merge minimize salient characteristic avoid bias particular algorithm regressor upper reduce learn structure boundary regressor final demonstrate concluding remark letter letter norm ordinary nonlinear give eq different regression partition hyperplane regressor however partition e hyperplane region boundary well regressor region continue
expression cdf make expensive improvement method objective nature advantage propose consider progress practitioner eventually interested possible uncertainty hyperparameter fashion address calculation objective likely practice accounting minimizer identify front dominate point computational model gp approximate output guide seminal efficient past input trade area extensively discuss efficient statistically alternatively reduce
author success collective similar show figure unlabele initial attribute current unlabeled relational prediction iterative meet value characteristic link validate collective l study extract digital library digital conference citation author extract contain conference first www schema summarize five relation link conference paper abstract rare word remove vocabulary network assign indicate category task base information involve another science include conference author title appear conference dataset bi extract computer science involve conference author link bag representation network assign label indicate classify local attribute detailed description please dataset heterogeneous integrated chemical gene disease effect pathway gene gene family belong gene
number gamma statistically selection set sample repetition block figure kernel choice strategy perform poorly reason distribution capture broad experiment aim latter table require rate additionally method fix five ccccc additional computation consistent pt pearson gamma gram spectrum evaluate heuristic equation figure error however conservative inaccurate selection coincide
versa positive uniquely compare different roc fisher method inter study variability inter study variability difference apparent pattern observe meta study variability auc global without low auc method yield considerably setting use lead low auc considerably particularly inter significant global moderate variability three resemble analysis effect meta analysis similar fdr supplementary seem calculate positive uniquely detect global effect vice versa set close variability positive find
omp signal ratio figure condition greedy algorithm small well reduce noise coefficient sign always improvement performance compress characteristic quantify recovery measurement realize gaussian concatenation general dictionary size zero obtain increment fix varied increment conduct trial choose uniformly combination recovery regime happen chance recovery contour provide careful analysis diagram coefficient transition improvement knowledge sign compare aim corrupted involve fraction pixel either case pixel recover divide overlap patch size dct atom assume negative matrix coefficient four dct recover arrange reconstruct orthonormal
affect performance increase stop sign experiment f score task since un annotated create topic noun test generate
projection onto relation row column remain leverage score orthogonal leverage score sequel construct element norm e eq projection form q incoherence assumption incoherence minimization bound programming extreme satisfie r nz prove combine eq n surely chen exact provable element restrictive incoherence uniformly choose leverage row perhaps sense perhaps intuitive way column sampling procedure first recovery trace un formulation completion coherent score nuclear weighted low study task collaborative correspondingly analytical subject development condition constant factor require subset
correlation investigation restrict assumption able limitation main krige similarity give subset snps sometimes heavy computational burden approach give analogy compare krige broad usefulness approach krige method krige framework different genetic marker gene encode weight component maximize krige partitioning variability correlation predict performance disease trait area operating curve assume probability follow lee matrix convenience genetic datum environmental identity package similarity genetic software compute component datum individual marker allele marker one marker marker correlation center trait effectively give trait different choice future work translate good trait use additive motivate choice phenotype individual q number marker gene additive marker noise iid identically convenience let denote effect convenience standardize covariance delta meet
review independently graph represent give item review item may observe review many time represent number time satisfy david posteriori variable factorize
profile effect quantitative measured component fall difference similarity two protein interaction occur among follow protein detail penalize graphical extra degenerate estimate penalize advantage development gaussian implement graphical lasso likelihood indicate mixture component unknown probably involve continue reconstruction heterogeneous use number population cluster
difference would manner tradeoff true true tradeoff help obtain e biological versus pair train true edge sparsity precision recall differential precision get tradeoff infer true fp mistake mistake getting increase four fold improve good individual learn perfect usually practice improve network figure differential tradeoff control impose similar strong weak differential strong recall strength control tradeoff tradeoff impose literature task inductive towards network dependency structure recall goal
weakly decomposable full define resp right singular decomposable consistency low multivariate model p rsc subgaussian zero subgaussian strong make final ingredient taylor nuclear say linearization piece low consistent rsc consistent tr unit proof unique primal dual linearize rv consistent invoke theorem linearize rv feasible singular perturbation rv r linearize problem rsc primal dual convexity nuclear linearize piece rp singular singular small ensure unit deduce nuclear generalize readily motivated bioinformatics application proximal type
dynamic overall else discuss internet upon measure track align subsample seed would thing report twice statistically distribution track answer subsampling testing go measurement apply location shift ad hoc nan dr distribution shift shift respect reject nan sensible confirm dynamic similar dr location shift result suggest reject confirm suggest limit interval version dr estimate smooth paper random advantage sharp level control statistic reconstruct real music pt remark di universit di dynamic nonparametric typical device energy
mkl algorithm select classifier remove impact mostly concerned j make kernel propose size counterpart f j b n define empirical classifier mf j incorporate part lead regularizer formulation suggest however unclear subsection geometric determine greedy coordinate mkl initialization f nf new algorithm choose gradient kernel algorithms
lrr liu rely key coherence lrr closely notion notation proof coherent coherence small information easy corruption liu lrr let lrr row whenever lrr recover column corrupt outlier allow lrr eps lrr lrr exactly recover column lrr broad fix failure depend whenever lrr subproblem establish lrr corrupt recover segmentation prohibitive lrr subproblems recovery lrr logarithmic allow little notably establish factorization column performance variety subspace segmentation complex design hold synthetic low incoherence corrupted column use inexact alm base lrr regularization report lrr subproblem column matlab run x ghz
variation mixed interaction binary synthetic exploration pilot associate copula accordance hence compatibility display matrix htbp cccc cccc interaction membership fold validation correspond supplementary material figure value sample one chain burn stage stand vice rectangular present rectangular smoothed distinguish enable rectangular shape color latent node successfully
algorithm note sound challenge address manuscript essentially replace step penalize step provably corrupt quite relative intersect construction require solution ssc lasso noisy construction require minimization nuclear significant demanding term sc adjacency term case succeed quite subspace
remain robust compare laplace heavily contaminate smooth track exhibit follow jump smooth convex deviation finally flexibility track important implementation treat method literature recently extension kalman acoustic laboratory laboratory tracking datum office code grateful rgb gray rgb gray lem lem trend kalman kalman smoothing heavy computational effort iteration grow contaminate modeling situation outlier track noise student smooth separately present analysis cover wide ingredient technique non mixed type algorithm newton student approximation guarantee definite convergent information size residual computing direction e loss conference proceed current discussion residual innovation use student expand experimental section simultaneously organize review advantage smoothing describe framework
stage every situation shape counterpart apply primal technique constraint primal problem interior show approach large due master formulation dual interested aggregated master scenario constraint slight substitution variable give empty mkl extreme extreme write extreme separability correspond extreme master problem problem similar one present dual subproblem subproblem take unbounded subproblem optimal subproblem aggregate subproblem unbounded obtain extreme ray aggregate use extreme reduce new give report stochastic widely format give regard number equivalent lead one vary last challenge simplex interior pt row base e e e loose e loose phone phone apply aggregated master subproblem scenario addition feasibility add cost equal entry generation degree experiment intel ghz cpu subproblems name scenario column cpu second outer involve last instance notice cpu outer level
derivative equation plug complementary contradiction ensure summarize note accelerate implement rule minimum boundary turn intersection could show form solution prove
breast sample status table r htb shape uniformity uniformity model give close particularly table discriminant performance run ht cc cc ari ht ccc ari ccc method ari class da da da direction obtain table plot inherent structure breast red side figure nice skew analyze expression microarray experiment array reduce package challenge compare microarray general informative gene dimensionality approach differentially express modified implement analysis compute statistic measure strength gene significantly cutoff significance user false permutation choose yield gene paradigm five misclassifie ari discriminant give ari give
subsequent supervise completely methodology classification orientation center capability concept descriptor excellent machine descriptor manually truth weak learner show excellent attempt image representation patch pixel patch patch sample detect orientation wide condition scale tp image surface matching multi easy unconstrained exact establish involve detail use patch result approach match sift descriptor raise evaluate patch similarity pixel
call represent success correct hypothesis limitation bound computational refer segment hypothesis learns three notion differ constitute successful enumeration eventually correct say identify enumeration stream learning hypothesis code section follow family arbitrary computable computable learn arbitrary machine immediate learn proof theorem computable learnable family existence distinguish learn learnable learnable column learnable input element element respectively great element interval correct code correct code learnable wish learnable learn specifically decide actually decide set desire learnable index begin presentation demonstrate arbitrary formula machine learner sense segment correct new base learn
active algorithm seek example need expensive easily selection advantage arbitrary building allow hypothesis class loss notably active delay condition delay et al gain relative cost obtaining latter typically point challenge arise synchronization overhead process delay study statistical substantially delay broad expect base objective experimentally demonstrate contain active passive protocol ensure arrive size
principal direction propagation pair point much avoid pair require pair hash overhead pairwise scheme highly near immediately sift sift collect extract maximally stable compute sift feature image sift conduct justify global descriptor describe texture localize cell descriptor high sift hence accuracy difficult nn regard neighborhood range well exact graph force cccc imagenet multiple divide follow recursive division cardinality neighborhood sec help hash easily comparison performance divide overlap respectively locality hashing overlap run knn
nu tu ex nu tu sx spectral ex ex n ex award nsf support gm fa r like dr university discussion fact edu design column regularize lasso characterize sharp threshold study regularize complexity special work sharp characterize condition namely support lasso correctly recover fail union sparsity statistical property task lasso regime practically recover study via individual sparsity study regularize seem exploration insight capture study discussion provide depth section vector noise vector across kb problem constraint advantageous oppose individual recovery support total need
hinge lipschitz therefore follow function function theorem minimize perceptron algorithm loss notice treat combine loss correspond buffer receive let unit cumulative loss suffer unit first notice follow combine yield risk follow ensemble hypothesis generate hypothesis confidence take calculate definition combine conclude proof perceptron algorithm recover hinge loss use direct let unit vector mistake eq tm upper solve start optimization select nature reveal
mass covering combination favorable become parametric dimensionality little control conclusion shall certain toward behavior translate behavior bound suppose assumption specification g c nm mc f nm iii nm condition motivate incorporation accord share base measure also closely conclusion theorem due conceptually unnecessary central statement part ii admit concentration mild compare claim r view overhead maintain latent hierarchy dirichlet claim demonstrate hierarchical suitably small rate mixture stand hierarchy exploit share translate favorable level conditional leibl neighborhood construction suitably theorem marginal density datum measure gain efficiency relatively small large concentrate effect strength arguably virtue hierarchical lie relationship wasserstein notion kullback leibler integrate mix link illustrate diagram sep em sep minimum em edge relationship aforementione distance measure geometry dirichlet measure test large section begin theory believe distance compare wasserstein wasserstein abuse generally replace admit suitable notion moreover dirichlet remarkable identity jensen bound kl divergence establish wasserstein tend tend geometrically result existence measurable respect line attack construction existence mix basis observe point admit bind exist vanish existence utilize suitable direct test capture rate need piece establish existence distinguish class measure wasserstein robustness incur previous paragraph formal test central
recommend experience accuracy verify figure report use quantization offset remove offset panel dash report wide outperform perform observation offset repetition dataset bit less h scheme report report regularization horizontal bit four code report figure panel value panel attain value observation code phenomenon quantization help boost attain product estimator useful allow highly linear
threshold rectangular formulae refine minimal also cardinality asymptotically corollaries special plane lattice line mapping zero take uniquely irreducible
approximately threshold therefore strictly strictly threshold solve multiply real root although therefore readily conjunction derivative refer illustrated threshold value strictly soft threshold function converge rapidly sensitive desire threshold identity logarithmic penalty threshold see threshold rapidly function logarithmic panel identity threshold go rapidly yet threshold function fast zero go like figure grow slowly induce large logarithmic penalty constant lead less bias logarithmic penalty logarithmic smoothly scad soft thresholding scad identity mm etc divide hence see function reader depth illustrate bias thresholding signal perform several thresholding c three fall threshold rmse
mean demonstrate pose work european framework agreement project ep agreement uk elegant principled create component construct popular study linear discriminant analysis locality preserve projection feature analysis literature firstly mrfs aforementione subsequently lda also produce joint select mrf observation algorithm exploit generalize product theoretical analysis framework provide material deeply easily build attempt seminal analysis fa mixture hide component unify unsupervised
could also compute evaluate application image show ratio standard snr depict thresholding stein unbiased rank denoise compare estimate image nuclear square error original many constant highly value singular perfectly six structural mention grey suggest capture use mala sample mala thresholding evaluate replace burn memory achieve mala problem figure trace plot autocorrelation time normalise experiment minute sample normalise per mala mala hmc practical reach state experience require perform mala normalise ess mala rate computing htbp l l predictive scalar langevin approximations proximity mapping efficiently log possibly continuously langevin construct approximate diffusion auxiliary langevin use euler mapping gradient modification
probability truncation q hasting interesting act require methodology various place literature bias consistent consistent describe available consistent reciprocal reduce employ extensively nuclear ray approximate doubly intractable strength inference form prototype image exchange access grid scale run inexact little practical impact ability importance sampling monte carlo smc index lattice notation summation nearest periodic boundary subsequent interaction q summation configuration infeasible moderately sized fact naive complexity lattice enable configuration standard metropolis sampling use acceptance normalise describe importance spaced temperature estimate infinite truncation exchange exchange iteration gibbs step transfer chain half normalise posterior comparison estimate agree well trace figure mix estimate estimate ess employ methodology large ensure negative truncation alternatively naive ise bind loose therefore impractical estimate upper chain rare trick small temperature level importance see subsequent autocorrelation
systematically report evaluate dft minimize ref fig state different functional show change know density potential parametrize effect occur potential evaluate density project onto black dash show atomic
wherein w minimization trace matrix variance parameter variability discuss easy check tucker tucker optimal l p obtain algorithm rule absolute stop monotonic multiplicative optimality establish yu alternate monotonic
vary control fully variant average run epoch far epoch iteration set demonstrate flexibility denote investigate plot refer parameter processor separability call processor one utilize early update block subsequently processor epoch need account processor investigate phenomenon numerically nonzero per stop three vary unit record h solid line far dotted line color figure dot run processor available require processor time unit importantly require demonstrate cpu curve visually indistinguishable vertical interested include dictionary g ix x
e concrete compressive dataset split two testing parameter cross choose grid range cross figure dataset situation exponent significantly slow lead least real propose good improve ridge support growth rkh future exponent develop online regularize step proposition rgb sufficiently smooth designing remain understand unclear exponent reproduce hilbert
lemma assumption hilbert observe rkhs quadratic interpretation measurement posterior gaussian covariance kernel rkhs provide loss huber specifically sampling location map white posterior variance review spline use insensitive loss support miss alternative argue view posteriori priori kind statement dimensional concept
negative fusion simulation perform fusion fusion section ridge fast fast grid fusion approximately second time replication fast result simulation fast comparable fast grid ridge fusion time fast ridge semi base compare generate unlabele ridge ability replication use rule form semi supervise plus
order extend correct equal community theorem paper least requirement implicitly impose spectral clustering correct present sbm degree correct block principal spectrum random bound ingredient difference eigenvector noisy assume small application presence statement care one probabilistic matrix spectral random adjacency node edge concentration inequality bernstein inequality combinatorial technique develop bound large eigenvalue os r edge probability major discretization reduce control bounding supremum pair grid decompose bound pair next bernstein union light pair heavy heavy combinatorial subgraph
trace box plot five learner estimation sparfa trace learner importantly sparfa trace relatively learner summary synthetic sparfa capable accurately learner concept sparfa kt learner record course value learner answer kt unable miss learner learner course digital logic concept full consist assignment learner concept assignment interaction since capable handle concept kt run without section inferior kt learn initialization trace concept initialize covariance validation randomly dataset fold consist learner question fold datum fold train kt sparfa resource unobserve learner use go observed response metric area receiver operation characteristic roc curve predict response average predict unobserve py learner area roc commonly binary area sparfa rather deviation trial sparfa outperform kt performance metric emphasize trace achieve quality content organization resource sparfa trace area roc curve collaborative filtering outperform kt unobserved learner ignore temporal dataset sparfa
logistic flexibility change model discrete represent polynomial matrix tm distribute gaussian mean identity process switch propose logistic assume generate logistic illustrate exp value particular regression switch regression within modeling transition regime generate multinomial nj mx ij conditionally parameter parameter estimate model assume mx notice control logistic dedicated start convergence likelihood cl iteration computation
recommendation matrix parametric smoothing primarily continuous space well know compress partially set nuclear norm
visible spatial observe estimate image translate immediately corollary motivation sufficiently differentiable transformation enough small allow signature transformation layer signature transformation g rotation range hierarchical show invariance follow invariance rotation impossible factorization transformation linearize piecewise perform high transformation global weak neuron complex cell pool cell entire image hierarchical architecture signature large large access match linguistic ability scene hierarchy approximate architecture specific advantage face translation invariant representation factorize together evolution mathematical invariant estimate term transformation simple unitary ki calculate template condition dot template partially observable translation transformation hilbert positive equivalent dot product template localize particular localization scale invariance localization translation simultaneous translation scale achieve template compute pool simultaneous e invariance template approximate template incoherent small transformation transform sufficiently smooth sparsity dictionary increase clutter improve encode uncorrelated respect transformation affine layer factorization range smooth layer transformation signature conjecture uniquely norm general remark theory regime sparsity consider fall first regime neuron predict tune complex feature perhaps hold strictly base assume various stage use sequence mathematics specific module localization old viewpoint invariant recognition general li transformation theory apply modality reduce simple cell template affine span position field window affect imply signature patch signature invariant affine window pooling patch invariance complex transformation transform theorems translation imply justification depend modulus special mechanism nice lipschitz property valid scheme property refine desire continuity characterize signature module invariant information image patch signature level hierarchy
formulae go different remark short compatible trend various achieve window linearity local short time linearity control many successful concrete value simplicity yield reverse would
termination initialize work learner learner weak learner variable kkt primal objective prescribe discriminant k boost need derive dual similar problem alternatively master consider learner generate weak learner correspond violate master write describe solve master obtain kkt condition dual solution problem generate add learner learner weak
set update filter assessment performance make carlo study assess see corrupted experiment simulate corrupt figure quality intensity estimate square big well pre width assess around compare contrast mean around former mean well small preserve relevant information assess filter correlation index
multinomial possess multinomial non integer number x x vector
table uci experiment tp l acc acc acc acc acc acc acc page acc acc method produce performance discount learning dataset validate conduct community economic census survey attribute vary aspect exclude attribute total continuous range manually interval label depict interval tp technique correlate parameter also true htbp ccccc acc acc acc acc receive novel modify propose address spectral address sophisticated issue discuss greatly reality institute technology fan edu view combine bayesian
ability policy complete evaluate policy depict confirm difference policy term target time method problem emphasize herein relatively concern optimization superior involve remain computation original medium sized mdps affected purpose evaluate medium sized particular term either mdp mdps trial tb performance remain curve execution depict significantly tendency environment action fact maximize ax since disagreement take toward identify corollary illustrate fig mdps structure reward set look domain problem literature introduce consist cell proportional range reach corner receive mdp action motion transition roll back cell move probability fail agent adjacent move two cell direction mdp world separate reach cell mark dark safe reach corner environment domain
operate resemble eigenvector computation reasonable similar use spectral cluster small eigenvalue write fact graph equivalent one eigenvalue eigenvalue heuristic edge compute sign bi connect sign classify heuristic much inductive analyze active phase construct receive predict protocol ever reveal reveal mistake make phase previous prediction correlation particular sign edge labeling graph active link label mistake prediction satisfie reveal bind drop indeed learn protocol passive active besides large span hence control span care tree give tree tree control depend tree unweighte low exclude
j j use see df correlation j v j experiment highlight hmc ask superior speed mix hmc computational provide application copula correlation modify expansion use hmc unlike hmc px fall reasonable bad namely loading conditioning factor hmc context binary probit extension remain involve step explore joint boundary perform row list sampling scheme intel core
occur datum result various gram analogy test table outperform reasoning slightly performance subsample frequent speed significantly argue linearity skip make suitable reasoning et learn neural improve significantly suggest non word time syntactic reasoning stand stand softmax frequency early meaning learn phrase contexts york token remain unchanged many phrase greatly increase gram previously
zero transition necessarily show viterbi alignment behave exist low stationary hmm variable independent useful prove asymptotic viterbi alignment rather small viterbi alignment incorrectly viterbi precisely proceed threshold viterbi alignment differ refer problem entail viterbi isolate time classification neighbourhood approach consecutive impossible transition alignment shall alignment differ viterbi alignment outside drop else section modification viterbi imagine possible figure realistic practice choose time follow shall meaningful classification wrong low misclassification find reveal viterbi thus turn start probability consideration viterbi
target location approach however learn automatically require controller lead nonlinear controller balance long controller sufficient learn block stacking combination individual generalize train present policie single policy generalize unknown optimization jointly optimization incorporate report promise rl standard benchmark apply solve library allow experience trial quality jointly representation parametrization rgb rgb college department tu department institute engineering university multiple challenge reinforcement impractical require principled transfer novel approach feedback generalize
differential whereby vanish eqn series differentiable domain divide eqn get convenient apply also summation eqn series eqn differentiable factorial know
extreme value example probit binomial exponential logit become uniform distribution family binary penalize admissible poisson problem exponential cox exist extreme long structure satisfy normal cauchy eq use sequence like indeed regression main observation
source ni formulation model eq iterative step optimize codebook fixing fix codebook reconstruction uv sparse code j rewrite efficiently
want augmentation note similar exercise augmentation term go bethe admm undirecte mrf joint node clique solve decomposable local polytope marginalization mn node edge serve lp inference decompose mn vector pseudo global sharing augment lagrangian iterate due mn lead due mn add bregman negative bethe update become admm propose dd admm lp binary pairwise mrf multi value mrfs dual need equality
imply solution stay criterion straight forward transform right q ready gap condition eq global implication theorem extend relationship still subject aspect future problem plan approach get mind would benefit keep car understand heuristic aim work grant foundation tp education national behavioral human overview raw signal might sequence consider normalize denote expansion consist set
microarray rna sequence thus able predict code expression reconstruct convert genome knowledge gene one start would atomic extend create dependent allow maintain set optimize certain read read coverage expression mark boundary boundary read ex ex color code model care former specific specific see label allow depend read support sequence derive rna seq site specifically position
note easily check set word factor exhibit trial adopt detail technical report note although case stepsize policy e g uniformly unified nonconvex sp problem recall smooth counterpart nesterov possess nearly term first applie method tight exhibit corollary increase stepsize speaking want decrease stop early well option expect policy deviation property run establish call constant dependence
brownian surely admit integral part yield integral space two algebraic usual whole space reason generalization characterize signature tree moment support give justification curve aim element
refer q q approach compute via partial gradient reduce currently require scalability alternating logit
regressor show parameter estimator formulation develop decentralized estimator constraint sample decentralize average stop optimum centralized estimator decrease concave z bound moreover concave bound z exist concave iterate concave similarly pointwise concave way concave non assume ccc satisfying except justify initial induction assume decrease similar showing decrease sequence monotonic g decrease finite limit monotonicity show decrease bound objective cm engineering department new york ny computer engineering wang sequential centralized decentralize sequential opposed level reach level counterpart conventional propose fc optimum average capability sophisticated observation sequential attain rao stop complete stop history
show particular regret know know investigate armed describe select finite time horizon successive arm yield accord optimal sequence indicating depend strictly eq observe optimal arm n hereafter reader survey variation investigate phenomenon knowledge uniformly horizon tend seminal regret describe know
document major retrieval community high similarity object text document exploitation relationship call build iteratively document build recent cluster three document sentence word represent datum dependency sentence another important aspect weight document sentence indicate presence word sentence encoding weighting scheme tf incorporate importance theory unable uncertainty proceed control
raw image video aim acquisition supervise set label may augment rotation color variation diverse contrast step create image surrogate consist patch convolutional neural classify invariant feature representation learn surrogate perform general unsupervise feature benchmark precisely assign
gap improve prevent rely part add noise algorithmic view controller model controller population translate idea improve section second idea correspond encourage robustness simulation could simulation perfect reality simulation among recently approach search reality lead optimize feed assumption search formulate reality robot rely model lift broken ground close find redundant less trajectory capture self reality map solution control descriptor reality machine reality robot internal exact loop cross reality propose performance un inaccurate robot internal measurement robot relate yet application concept rely main principle model robot periodic test three optimize simultaneously candidate simulation denote self model regression algorithm objective approximate compute test current population move update robot pick population selection reward technique use learn output numerous simultaneously objective
fold light star confirm another database substantially compare previous work database could classify extra candidate strong candidate find processing candidate day schema apply survey group confirm select provide critical galaxy black etc valuable study scale forest adaboost make diameter dedicate equip locate size along along simultaneous imaging non standard channel blue standard analogous transmission compare curve template software specifically learn combination combination classifier
factorization construct random review find marginal factorize maximal clique graph know typically gaussian combine marginal proportional approximate likelihood normalize start density product start dependence clique clique integrate maximal clique leave clique vertex clique give eq clique unchanged u form approximate choice make discusse choose minimize likelihood integrate factorization unique particular integrate approximate
penalize mdp know short discrete state terminal transition denote parameterize determine terminal action reward terminal reach throughout policy assumption gradient representation softmax let visit time terminal accumulate discount reward along expect go expectation slightly notation optimize adjust return
page exceed page prevent uniquely word overlap phenomenon singleton multiple search singleton overlap every number page singleton web page google page correspond search identical correct formalism enough distribution imagine instantaneous snapshot situation google associate equal unique code word google code term google singleton shannon carry meaning make object term count return red keep occurrence occurrence restrict kolmogorov approximate kolmogorov complexity file compressed version induce length google say google term use valid mention would value tend numerator tend denominator application purpose would difference difference information admissible share mild numerator admissible numerator divide normalize quantify normalize
panel curve black perform decode encode single layer autoencoder architecture clearly curve allow encounter green extend distance curve extend either encode feature monotonic dimensional draw distribution multivariate simulate degeneracy mode sum mode direction originally present obvious extent single dimensionality pca symmetry line degree pass dimensionality onto line thereby project contain broad peak however four single make non autoencoder autoencoder whiten compare number investigate perform margin top decoding vary limit encoding value autoencoder trace path value histogram encode data mode appropriate belong per classify raw accuracy mnist database handwritten digit technology consist handwritten digit digit normalise image publicly along analysis researcher set standard digit easy human train hide input transformation input correspond fraction incorrectly calculate set nonetheless website percent dimensionality mnist autoencoder layer dimensionality compression total error square error despite linear basis retain enough reproduce error demonstrate capable autoencoder network sometimes space illustrate obtain rather original train network handwritten number hide node autoencoder
survey value value inside implicitly probability thus estimator obtain estimate practitioner specification yield method reasonable weighting summary relevant fully population concern inferential uncertainty obtain estimate adjust number available give predictor regression predictor indicator cluster strength survey level situation enable consistent exception population common record size bayesian bootstrapping size bayesian sample design predict outcome even multiple keep capture outcome conceptually publish predict elsewhere weight cell weight weight conceptual jointly outcome incorporate use covariate make sense weight factor stage property computational
hold empty hold variable aggregate claim explicitly block join child substitution precisely include inductive explicitly aggregated mean previous extend value clear union together variable correct correctness detail edge substitution calculate join correspond table average edge variable edge win leaf empty set argue correctness return proposition identical return force concrete substitution reach sub root aggregation aggregation return rest return small contradiction hold lowest
illustrate work show word embedding neural relate neighbor reasoning approach input deeply replace extract order recently speech achieve employ deep hide making easy predict allow compact may able summarize let denote hidden output rnn feedforward hide state replace output layer generative boltzmann machine estimator feedforward intermediate effectively add input state rnn limited operation wise nonlinearity argue new nonlinear hidden adapt preserve past model generalize however nonlinear model hide
present dependency decompose certain dependency domain dependency px w dependency define pair disjoint truth triplet independence proposition relate dependency px distribution start map
opt problem cone condition source geometric develop cone noisy remove variation conclude future partially nsf dms consider behind problem linearly vector mixture e p example pass stand alone actually restrictive subset cone generating combination contain reader clearly follow span column vector vector locate method identify project column normalize origin convex
effort reduce posterior privacy mechanism algebraic choice place concentration level privacy research mechanism level differential family distribution assumption ty l amenable requirement scalar bernoulli requirement simplify find supremum leave side conjugate conjugate conjugate conjugate binomial binomial binomial metric consider network specifically probability outcome distance assign event private inference bayesian assumption private sense generalise differential construct merely
construct splitting randomness splitting require split choice use align split depend exceed threshold thresholded threshold optimize data leaf candidate split criterion evaluate choose candidate model approach optimize predictor explore leaf predictor beyond scope randomness dimension candidate randomize coefficient combination either threshold randomness sample forest unlike forest bootstrappe tree rectangular construction correspond leaf play role part allow internal effect make data
model zero covariance confidence straightforward percentile chi degree similarly representation result inexact parent consequently necessarily adequate address freedom case region expand uncertainty inequality obtain extend applicable error form dependency input quantity covariance measurement region freedom multivariate constant ip jk possibility effective freedom overall evaluation iii treat quantity dimension option denote applicability cp qp c also pn scalar distribution approximated proceed replace element give subsequently confidence complex simplify two example coefficient object common experiment fig reflect wave ideal transmission
make setup scheme construct accurate estimator choose estimator estimator replace equal integral run start run point order point following map identifiable observation let derivative component continuous moreover infinity well n ni step approach generalize step understanding accuracy repeat well error estimate study problem good example two simplification specifically model use across cell depend recovery compare integral aforementione study point ode system cccc integral experimental solve valid interval error variance estimator generate namely point remark selection local bandwidth
respectively mid covariance positive newly construct section describe lp kt organized informative quantile symmetry bi distribution interpret show b fit tail htb diagnostic number extend tail normal immediate fact detect lp moment datum lp cumulative square moment read tail use value order lp moment theoretical follow orthogonal quantile q lp moment belief p return seem tail
thin form family spline monotonically cubic intuitive cubic stems shift copy compose spline spline behavior away add polynomial slowly existence uniqueness zero polynomial easily form eigenvector rbf much conditioning consider might expect rbf converge contain combination span reproduce hilbert turn rbf whose transform decay gaussian cubic extremely space dimension odd
tell therefore tm ij ij tr ij ij ij dt ij dt dt condition cauchy positive take correlation avoid estimation make covariate generalization quasi approach solve estimate general method fact dynamic auto ar model covariate mean complete entire quantile regression outlier pattern quantile regression quantile longitudinal working loss efficiency firstly likelihood incorporate measure longitudinal
perform gd iteration overhead gd apply use regularize dataset label experiment regularization condition problem dataset edu tw benchmark experiment follow experiment decide practical publicly limited suitable broad class problem mark schmidt http www di fr also use stochastic stepsize choose gd stepsize get stepsize give sag store run function store scalar sag implicitly limitation although full plot various equivalent first remark make first confirm type reduce consistently bfgs gd differ explanation paper put assumption function associate author subsequently quantity opposed importance
specify ern comparison verify condition interesting mat ern family transformation amplitude temporal compression assume process introduce remain jointly stationary look take mat ern mat ern transformation original mat ern alternatively combine mat ern specification discuss slope parameter negative equal bivariate scale behaviour individually capture isotropic usefulness mat ern multiscale behaviour single subtle relationship decomposition specify mat force mat ern specify specify mat ern ern reverse spectra therefore understand process drive find decomposition estimate two drawback wish closely approximate log preserving indicate length process arrange univariate real value parametric model time transpose inverse xx tn bivariate domain vector theoretical denote eq parallel observe standard approximate domain seminal approximate fouri special toeplitz real likelihood subscript stand hermitian transpose continuous simplification scalar value frequency keep form frequency small frequency fit discussion domain I statement computational vast advance likelihood estimate problematic quantity inconsistent biased measure cause desirable rely behaviour prevent sample version model spectrum show
distribution covariance rsc constant bind probability readily computable apart obtain follow unconstraine correctness many regularizer convenient close scad program close mcp program take form taken take state involve despite fast restrict smoothness modification require rsc hold small radius together feasibility outline defer broad inspire modification nonconvex regularizers show long descent within optimum successive iterate lie ball initialize composite assume scaling recall early third combine establish decay suppose prove inequality divide epoch lemma recursion recursion q number obtain remain obtain reader rearrange perform algebra logistic function namely lasso mcp detail choose nr validation case corrupt additive mechanism additive generate becomes optimize composite compute use together update show scad mcp regularizers problem rescale theorem stack decrease scad mcp regularizers dotted choose penalty optimal base study mcp trial cc lin scale panel correct logistic p scad mcp result represent dot respectively corollary stack different number increase show optimization conclusion
chance contribute classification formation aim well relative rf descriptor adapting rf imbalance imbalance rf formation effective average usually great stand base drug discovery rf method combine various rf sense base rf improve potentially explanatory user base problem formation method acknowledgment suggestion aspect methodology thank help descriptor graphic pose interest little label explanatory moderately high drug discovery tumor rare descriptor variable inactive thousand characterize molecular explanatory information algorithm explanatory pass various role different contribute overall classification without compete another form working work different ultimately natural matter nucleotide snp suggest priori chemical platform
chance transition belief fix node belong cavity regime address analytically cavity require keep track entire message zero bethe message pass concentrated label equally label break external field order analytically emphasize message threshold algorithm nevertheless qualitative real regime many contribution message explore community
deep type gap protocol protocol question stochastic problem distribute inferior constraint current develop establish pay third interactive interactive rich interactive distribute within computer science clear statistical base fourth open estimation protocol tailor sufficiently behavior gaussian establish hardness natural generally remain information acknowledgment support intel ci institute science grant grant zhang helpful comment quantity theory make lemma subsection quantify constrain protocol message concavity variant observation suppose follow q proposition discrete remark qx ball ball hold chernoff imply particular let negativity bind integration get eq value state case build problem correspond coordinate message since chain side equal presentation drop message eq instance equal argue
aspect fit three depend zero fact height turn explain width explain three add random measurement zero factor variance really assume fitting study iteration microarray non sample gene available www early illustrative et al gene log gene none impose used aim biological conclusion try fit inferior extremely converged somewhat slowly even extremely require ordinary
v z follow argument need preliminary continuity z follow eq continuity b make observation v complete n notational yield hand observe position since follow clearly definition interval hence combine inequality virtue exist q n u integer n n stop complete eq q u e recall sure large sure sure almost sure suffice combine similarly small hence must small enough inequality combine lemma ii attempt notation virtue inequality n inequality u imply complete make continuity assumption z accord exist complete establish claim prove increase since respect respect virtue integer lemma claim exist integer claim note
furthermore always otherwise sample place ever satisfied every choose one clarity exposition near especially minimize tie break boundary second borel surely surely neighbor generalize neighbor two point force neighbor modal establish rest proceed use power second event occur infinitely meet seem simply intersection neighbor selective near near neighbor rule pointwise follow choice distance modal per neighbor finally alternative neighbor research seem valuable reason place set contiguous variable theorem convergence
nothing method rate video rank accord method outperform nature video scene people discriminant combine dimension unify extraction competitive knowledge video representation summary rather representative discriminative conduct thorough factor semantic consistently outperform cm minus height width discriminant reduction extend multi provide lda label training overcome limitation latent discriminant relax label level supervise extraction drive
helpful feedback early award author intelligence interior contract pc reproduce view conclusion herein interpret necessarily policy express section thm business usa institute pa probabilistic time construct novel sparsity variational financial text ignore scalable risk ensure fortunately definite sketch show strictly p scalability concern g wishart induce dependency adjacent grained base discuss
straightforward programming formulation instance parametric simplex method outline ht sensor formulate version kronecker sense element random sense publicly available figure point kronecker interior point simplex
convenient rewrite density identifiability state absence mixture identifiable due component contaminate elliptical elliptical avoid identifiability g represent good show identifiable straightforward joint contingency give way determine conditional arrange ht contingency table cccc without give solution p repeat g unconstraine mixture distribution covariance proportional latter give suitable g g st st st hence recall
pairwise helpful improve useful subspace orient become similarity similarity way euclidean alternatively assume denote intra similarity lot interest
situation database accuracy versus database database new require second encourage use particular set examine microarray study gene website except three standardized study equally database correct correct measure iterate interval result find table five study show confidence correction ci
viewpoint euclidean treat eq follow relaxation specify eq define ball monotone establish theorem identify feature removal small assume extend valid regularization follow supplement geometrically guarantee j main summarize x monotone monotonically regard regard monotonically illustration convenience axis supplement index convenience remove discard
cascade algorithm less hour qp qp iteration computational cascade training complexity primal qp need complexity haar patterns majority also experimentally solver qp primal second moreover cascade start previous warm start solver warm evaluate detection mit face shift treat otherwise window receiver fig use compare rate evenly positive factor cascade performance cascade bootstrappe superior asymmetric detection task find wu et observe processing either boost ed outperform boost boost note cascade cascade node carefully tune method guarantee cascade strategy actually boost incorporate adaboost fast cascade minor modification background contain annotate ed pixel additional pixel preserve human contour information window pixel human body block block pixel divide total pixel furthermore histogram fast block classifier fast fast classifier fisher detector cascade weak node feature legend detector sort perform compare cascade cascade weak describe wu post ce applied cascade rd select select ideally
sequence whole outperform art unseen competitive classify biological practical application customer transaction customer generalize scenario interesting investigate discriminative vector improve lda reduction extraction multinomial vector expand expression ff govern well term competitive classification structure abstract computation database paper characterize sequential behavior behavior internet show datum web server com row order discrete symbol behavior page page page behavior behavior decade effort characterize behavior
field value knowledge field ill field reasoning function pdfs properly possibility location carry pdfs functional probabilistic complex configuration physical field vector configuration integral signal discretized discretization yield path discretization fine quantity behave limit discretization resolution grid turn last
particularly unstable car see table total middle via perform account interval show bold tie perhaps thompson however use optimistic however remain despite simplicity paper introduce reinforcement bind computational approximate bind necessarily define bellman mdps competitive methodology whereby outperform alternative however simple
reconstruct reconstruct blue kalman filter smoothing break compressive db matrix multiplication matlab streaming compressed follow signal represent reconstruct comparison regularize solid line kalman smooth break line compressive db different multiplication matlab second signal present next colored bottom reconstruct along top represent bottom regularize kalman denote consist denote submatrix kalman filter compare homotopy solver convex identical advantage maintain fidelity however applicable streaming signal begin measure vector measurement measurement subscript indicate part streaming system overlap treat independent streaming stand alone represent cosine write sparse estimate adapt sparse measurement variety describe natural system representation system diagonal estimating happen overlap overlap measurement system depict overlap overlap constitute depict lot basis may constitute present two overlap streaming
step know second imply element may another element go cp small ij possibility give element possibility lower hoeffding special care need similar long give rise number bound union satisfy requirement conclude noise chernoff interval g one intersect interval let conclude mean h upper variance almost also ij e ij ij z ij bn q well variance bound absolute least pt assumption corollary thm thm thm chen presence correctly cluster sufficiently show really refined analysis prove
standard deviation link mahalanobis pattern number analysis confirm mahalanobi next li yu al dataset spectra http light consist channel spectrum nm water protein record classification separate content content content order derivative misclassification convert observation spline channel spectrum evaluate show mean standard deviation proportion classification via need maximum neighbor knn eigenfunction respectively knn winner proportion suggest perfect original second derivative respectively li yu classification rate second approach good rate account svms note take number functional functional mahalanobis slightly functional semi
term logistic class label potentially unobserved model result probabilistic xy noisy former natural setup motivate conditionally statement fit label grain application formalize weakly supervise fitting versus behavior group search able click divide distinction click pick cluster membership population click good whereas click result evenly call membership sub population surface understand click level coarse grain side information cast online want click ad customer visit store visit ad idea weakly supervised aggregated learn interpret click political want instead
differentiable satisfie say minimax sub sub cg add current compute residual determine violate residual add sub sub inactive measurement large violate divide set remain measurement solve recognition choose utilize speed software generate code qp algorithm convert problem solve generator problem restrict fix ever enable use iteration absolute residual sl max violate remain problem active problem move inactive measurement illustrate effectiveness classic geometric face recognition contiguous finally present several representative
involved update component use component thus factor mixture multiplication algorithm dominate dominate speedup confirm sec follow bound theorem nk n w difference covariance variable constant let q yy p discrete lemma obtain second
paper value distance divergence employ filter pixel neighborhood nine area central eight area possible within
cd number run ip adopt cd baseline illustrate average cd epoch scan ip train epoch average context order ip sequence divergence iteration trajectory figure cd divergence ml way converge behavior ip l divergence ip ip adopt converge select note performance ml converge e converge comparable ml significant large improve ip much short threshold local proper select ml sometimes ml follow negative updating produce much small gradient may converge minimum might ip introduce figure sufficient sample bias ip ml bias closely bias couple gradient estimation ip procedure separation sampling gradient conjecture ip potential cd c ml converge ip empirically investigate ip work rbm dataset rbm term hide unit rbm ml converge properly set ip similar
block fitting recently past year understand interesting require recently tool give network configuration nan find community equation one understand namely connection subset construct configuration vertex start sequentially half edge pair half first connect write replacement binomial understand without half connect uniformly outcome arise first find half vertex benchmark simulate two power exponent degree law range increment realization generate parameter community thereby provide advantage mutual result simulation mix community find value network community identify well network underlie community benchmark simulate range community big law exponent degree law exponent increment realization set detect cover notice biology protein interaction individual relationship organization application study naturally community informally group connect quantify connection illustrate disjoint fig divide community detection become increasingly useful study researcher field computer mathematic community review detection community studied assign another community allow overlap collection community speak type structure community detection successful wide numerous cite aforementioned review protein functional activity medium mobile
expression site type suggest time asymptotic investigate stable process composite selection adjustment devote composite recent summarize international nuisance example cox censor survival covariate influence model ignore failure event subsequent asymptotic validity view likelihood theory semi parametric seem view maximized nuisance function class semi profile likelihood parametric profile discussion insight extensive parametric likelihood guarantee accurate drawback
say stable even edge stable arguably meaningful instance stable plant easier stable spectral analyze cut informally show cut suggest algebraic small spectral motivate analyze performance gap partitioning perform partition small spectral bad case bind improve partition set nontrivial recursive find balanced guarantee find maximum edge sdp cut give ratio algorithm cut fraction generalize small find cut cut least fraction least fraction partitioning cut semidefinite relaxation rao sdp rao provide cut search subspace enumeration nonetheless bind hierarchy relaxation give cut apply partition line partitioning easy fast approximation guarantee spectral special another performance show
see step triangle inequality prove suppose parameter keep short wasserstein q finally recall couple draw coupling x tv measure cf therefore consequence consequence instantaneous loss substitute get relation decompose regret eq instantaneous expand first summation side q lipschitz obtain imply last next upper use shannon g need write decrease moreover online immediate vary tendency act lack evolution cost sum interaction term every agent inform cost cost immediate capture overall minimize achieve explicit achieve favorable scaling parameter use statistical physics development chain important conceptually quantify forecast even improve decision significant first rational basis
begin simple illustrative counter strict toy disk relax project line hand project disk since second formally model case nest supplementary panel phenomenon carlo normally distribute expectation still two persistence disk hyperplane supplementary section theorem address modeling concern big subspace contain modeling nest strict sense def mutually uncorrelated linear real nest proof provide follow thm show
cyclic coordinate residual admm procedure optimum since nmf r current cone suitable else remark remark current cone identify pure onto matrix else exist anchor select form tr lagrange multiplier smooth current cone satisfie proof ai complementary ji since kkt point correctness algorithm maximizer anchor separability let index identify hence j strictly one maximum maximum anchor remark correctness hold give find kkt ji ji ji ji ji ji ji j feasibility lp
restrictive gene share unique gene adopt improve dispersion parameter yet sharing assume large similarity proper digital take even biological replicate huge amount throughput sequence present digital utilize share analysis propose bayesian digital expression build dispersion dirichlet inverse stick prior construction impose cluster unique permit gene mean dispersion little algorithm dispersion form digital expression include rna seq tag seq actually apply herein arrange row gene correspond furthermore group experimental
tree theoretical useful mode coupling mapping interact explore external change strength numerous quantify benefit persistence understand economic growth ability tradeoff innovation g figure sup paris france capital paradigm economic strategy policy
allele parameter apply infer coefficient explain balance selection point derive allele subsequent confidence explore individual individual possible weak restricted selection certain model restrict get advantage ignore suggest maintain although focused could readily arbitrary population model suitably spectral representation multiple sample take light dna data green temporal inference gain adaptation human adequate necessary incorporate extend take link selective selection thank helpful comment
arbitrary cover ball define cover radius center point appear covered ball within property find possess minimum find point near distance cover small ball program near optimal modify relax denote dimension hierarchy cover distance possess subset equip hierarchy sub hierarchy modify therefore correspond ip impose constraint ip enforce convenient actually mean neighborhood let neighbor call define pack property formulate require possess property constraint ensure point recall program follow original guarantee cost great ip eventually remark follow objective variable state satisfie constraint dimension packing constraint cover tight turning
tn arbitrary generate definition eq center hand segment consecutive cluster thus generate simple class particular dimensional family commonly length x mr r long double produces value ergodic produce point every experiment calculate procedure throughout sequence change estimator sense
curve decode successful phase threshold plain threshold plain adopt nonzero I plain plain thank helpful discussion theorem theorem lemma compressed sensing minimization provable great algorithm perform well boost amplitude density th integer whose nonzero modulus value recovery algorithm plain minimization dense sensing modulus contrary bad sensing
office contract nsf award award amazon services google intel microsoft oracle yahoo powerful super algorithm repeatedly efficiently generalize old submodular take towards paradigm maximization treat analyze algorithm minimization maximization theoretical analysis support empirical minimization maximization discrete subset family set express span match trivially submodular case solve exactly subset traditionally combinatorial popularity rise shorter usage capture shortest cost cost span problem maximization arise extraction problem sensor document instance submodular maximization application motivation come near unconstrained date cost evaluate motivated possibly np problem submodular form submodular maximization np admit factor combinatorial mm notable
take equally dimensional calculate heat lattice phase site site heat line coincide implementation naturally extension value xy discuss possibility compose take hamiltonian xy th spin xy replace cosine xy basis internal evolve accord positive internal state reach jump eq numerical another natural limit internal spin xy evolve differential define although ordinary couple since continuously simulation system consider follow
item cost cost ranking item among obtain item high advantageous easier actual present contextual relate list sensitive loss collect pick regret eq policy mx constructing list position good list policy construct f prove guarantee pick r must achieve match outperform practice instead verify intuition surrogate convex extra gap convex sensitive instead convex q gap close imply good convex lead list reduction exist predictor near expect implicitly whenever accuracy apply plan freedom chance lead
choose working tend sensitive change structure correlation analogously use comparison work structure quantify mse claim within first directly j n neither glm link e logarithm sample exchangeability link conjecture unbiased mse claim reduce however prediction really incremental variance assumption incorrect simply bias simplification observation predictor need I vector unobserved amount datum bottom triangle future claim year prediction taylor vector
robustness motivate contraction multiply essence minimize correspond cost increase robustness number cost conclude algorithm give selector contraction simulation widely parent optimization ill conditioning machine learn contraction selector solve system linear base attractive batch counterpart actually pick widely
area consideration area randomly choose centroid spatial centroid geodesic distance centroid centroid assign densely perfect question composite similar density covariance composite spatial composite choose parameter truth leave run computer original effort consist covariance computer synthetic truth comparison density generate km conduct approximate credible reasonably density adjustment slightly cause one adjust posterior number examine credible adjusted composite dataset discrepancy expect become informative prior identifiability issue occur base observational
case concern water medium contamination mainly ten discretized screen simulation identify influential pick sample among take design number component author influential addition lead index soon variance functional introduce new base ar arise extended hilbert schmidt also rely interestingly new index input new screening dimensional expensive code suffer sensitivity index dependence compare yield index perform divergence well sensitivity index random sensitivity dissimilarity use ar link mutual estimation give review sensitivity example feature selection machine conduct
suggest follow develop problem algorithm ucb base finite figure explicitly account factor confidence stop ucb bound encounter epoch elimination arm term bind stop tight analysis arm gap large confidence arm require like elimination ucb main develop great complexity constant method sequential exponential elimination
fast linearly threshold whether summarize
substitute reduction training ice act outcome feature prevent overfitte feature build build bias varied utility give perform bias ice tie together model example player quantity instead regret sum dual treat implicitly incorrectly condition parameter ice additional loss compute fit hold perform logistic separate bias utility ice ice error parameterize offset quantity center show train space within solely acts perform without loss pair conclusion decision build performance act portion utility improvement individual optimal ice computational framework make agent perform framework ill predict framework cost goal typically consistently optimal rational influence type execute meanwhile statistical g regression equally match goal implicitly encode strategy learn recover compactly utility demonstrate leverage ill pose many reward rational everywhere reward rational far remove behavior operate theoretic bridge
actor u close simulation exploratory online k algorithm online state divide offline processing collect fact system embed thus offline control equation convergent control control simplify control system constrain derivative side rearrange actor nn replacement notation xt xt xt xt rewrite notation il xt expression square scheme base unconstrained design actor nn loop input z u unconstrained system algebraic expression
step store direct edge store record quantity simply large direct visit find let node compare smallest store three never e need eliminate item together finally undirected direct elimination perform depth visit completely proceed perform backtrack visit close assign direction efficiently find path connect th follow connect traversal mark visit reach mark connect query belong unique node node query incremental view query bind rely show overall spend depth spend node mark handling take operate phase depth visit start connection hinge twice add tag node visit started visit perform depth visit visit predict label store hinge tag
knowledge value lead separation result imagine detector accurate distance lead accurate probability mix element ica ascent mix act additive term rule suggest prior perspective regularizer viewpoint carefully design guess position detector take separate considered separation localization estimation eeg brain signal brain source current orientation mix source detector often much known flow understand model detail head propagate
optimal point well recall conversely low get recall database distribution database c design imbalance important database significant additional experiment algorithm vary weight precision definition present experiment uci red green blue face image age group middle old database database several class
cascade likelihood problem continuous cascade hazard parent cascade node infect hazard node trivially additive cascade model table transmission likelihood simple hazard set hazard map consider covariate multimodal enhance map covariate exist inference network hazard next increase hazard examine hazard covariate infection hazard vary infected infect instantaneous risk infection increase similarly risk edge expert simplicity consider simple try goal maximize cascade
update associate operation server update point usually proceed server may worker ensure read worker worker intuitively desirable worker consistency consistency order intuitively consistency ensure update order prevent worker biased update worker informally worker sufficient progress otherwise worker integer regular consistency guarantee threshold asynchronous server worker refer asynchronous
interpret activity inside salient infer slope extreme specification support summary profile profile avoid make priori restrict slope value implement slope trajectory profile trajectory supplementary profile trajectory account trajectory actual level figure random extreme trajectory profile respectively curve see lie trajectory product interaction particular individual form somewhat whole heterogeneity produce interpretable summary traditional individual explicitly outcome dependency posterior univariate multivariate sample predictive outcome individual refer extreme univariate predict profile pt predict whole response probability correctly wave univariate outcome fit six regression age additive assess handle end analogous ij use fold quantity validate
rna assessment generalization give characterize set parameter show select rna sequence np hard dimension greedy rna sufficient separate yield set experimentally energy give collection experimentally structure paper associate pair parameter ensemble ask nonzero ask slightly relaxed
rna rna n converge memory need usually dominate energy single rna rna rna complexity ph middle percentage bottom ensemble ph package sample need select totally rna sequence database compute upper partition ensemble energy
episode per episode time arm mean episode episode episode estimate episode uncertainty arm optimal moment arm episode arm optimistic give whiten whiten diagonal large eigenvalue eigenvalue eigenvalue large jj discard episode among optimistic non dominate arm episode probability never discard e implicitly assume logarithmic arm optimal optimistic discard optimistic event never discard end finally gap big gap lem arm never lem probability event event bound union last lem empirical observe confidence choose additional always select optimistic restrict optimal lem prove thm decompose sum lem
could interface language agree c double optimizer pass object must class double double query constraint virtual call pure virtual preferred nonlinear matlab analogous way initialize modify field detail language par par std par sigma student jeffreys combine expect improvement thompson par name par ml iteration optimization
arc variance edge adjust arc remove inference calculate order node n operator relationship mean leave ensure network make inference order reverse network f f pf j factorization calculate bn observe score relate applie classify multidimensional decade appearance automatic produce continue high fidelity advantage available available possible add band ray classify variability location time etc single thus miss traditional deal necessary feature member solve member dramatically reduce miss alternatively datum carlo
interest lag discard interest interest interest interest useful evaluating represent preference construct vote empirical serve evaluate interest learn similarity dataset dataset outperform lda interest close gold propose vote pre vote set vote positive friend user friend test min median friend many user end voting make extremely challenging vote model set user adopt vote lda
world account construct model copula specify term copula vector conditioning specify function scale bayesian learn gps previous equality hold difference multiply specific random variable distribution integral depend unique distribution share underlie advantage copula separate univariate multivariate dependence couple marginal easy learning range dependence pattern student copulas family describe
useful integer eigenvalue prove restricted sample precisely max min definition cf recall bernstein random statement estimate design let q ij ij obtain eq assume arrive follow shorthand appear q complement second since inequality nc state assumption n independent limit arbitrary vanish enough lemma therefore last since min n enough argument prove define prove feasible let optimization gaussian v u min inequality surely bind characterize interval variable sequence lemma appendix stanford partially award nsf dms grant fa fa notice consider component feasible plug optimize
ni upper go must examine drift must arrive control understood agent choose high ucb improve guarantee investigate experimentally gd place track gd derive update fix oppose variant give gain run would order computational article gd choose exploitation click yahoo front platform repeatedly news yahoo module anchor north legend column mm mm mm mm mm rectangle ylabel xlabel symbolic align outside
kernel gaussian near subgradient employ solve path invariance code make fair reach objective objective subgradient specific code sequence generate stop fall keep stop loss residual calculation primal burden monitoring create primal calculation compute primal require next solution drastically choice two near kernel evident number near many weight agglomerative intermediate half half near conversely neighbor panel lead path incorrectly hard three result path choice since visualize fit see sensible separation vote vote economic environmental concern different leave sensitive get party separation identify agree kind bottom top bottom top leave path choice leave superior
spectrum gap heuristic fail principal iff uniformly unit concentrate connect eigenvalue n eigenvector informative non vanishing justify set whereas component vanish component eigen principal component simple noise matrix interval depict eigen gap principal base interval thus measure random support sample separate eigen eigen enough smooth limit eigenvalue look successive eigenvalue go reason picture develop adapt middle eigenvalue exhibit deviation eigenvalue middle spectrum eigenvalue thereby look appearance gap gap heuristic justify informative middle eigenvalue
deconvolution conclude sec information infer fundamental produce infer continuous varie practical therefore capture freedom continuous signal might infinitely uncertainty signal signal combination serve characterize measure give field cover describe consider less restrictive mathematical partition calculus preserve mean denote distribution piece high consist count spatially pixel event pixel field would infer position observational within represent approximately plane patch point like component definition specify address point vary smoothly source two signal dimension area exponential naturally account positivity natural logarithm measurement process linear read comprise survey exposure area source coordinate severe information count per pixel assume statistically poisson ratio expect number shoot make detection source energy particularly challenge ray eqs logarithm favor clarity comprise field contribute equally define degenerate likelihood set intuitively introduce prior discuss contribution define like first place aid degeneracy prior primarily drive strictly intensity several I neighboring exhibit requirement normal obeys multivariate
union lemma term c close estimate treat z avoid ambiguity specifying imply far bind fix g c p kf z z hence analogously proceeding eqs simplify choose simplify net simplicity row ensure I e e thus consequently obtain bind prove appendix firstly ef lemma remark enough imply lemma imply possible choose explicitly entry exclude row abuse notation entirely analogous dependence
value question concept subject sparfa perform estimate tag limitation tag inaccurate insufficient inspire allocation text potentially reveal estimate concept tag domain incorporate information sparfa potentially paper propose extend sparfa learner question sparfa statistically occurrence question descent value response datum association profile keyword
suggest proposal however per lead increase overall runtime discussion computation tradeoff bottom plot runtime plot vs particle circle proposal solid dash proposal irrelevant performance proposal lie space dimension test validation dataset uci repository label test identical outperform particle budget illustrate bottom row display datum circle square average filter bias compare single regime improve suggest regime fix total operate entire unlike drop outperform independent across test vs mcmc employ hasting child leaf belong parent
stop return whereby margin nonnegative interpret simplify whereby bind simply term present suffice grant size cf discuss nearly wolfe setting whereby margin precede stop soon turn boundary choice constrain crucial wrong worth exist implicitly regularize prevent little size asymptotic margin margin value suffice e g suffice severe demonstrating benefit margin recall boost method separable iteration margin suffice suffice require margin comparison additive bind comparison require poor achieve margin multiplicative accurately present method brief check
maximize concave update quite estimate mean pairwise marginal ise estimate joint pairwise marginal compatible marginal cycle joint entirely pairwise marginal general graph cycle physic potentially use way approximate idea compatible wish bp marginal seem bp historical bethe closely candidate bethe usually rough correlation improve achieve theory instead modify single inverse role cycle calibrate phase transition bp maximal bp close section real convert binary marginal constraint propose modify bp similar notation interact node shorthand notation interpret bp probability belief site come neighboring update notation current normalization constant converge belief compatible belief algorithm namely minimize kullback
innovation state use sequence infer true term steady rmse plot versus run able converge true low code fail dependency consider code model show matrix leave correspond active time basis indicate arrange magnitude spatially localize dependent represent representation knowledge impose fix prior coherence etc purpose adjust main focus work emphasis prior bottom extract propose system extraction introduce strategy learn deep block greedy hierarchical build
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practice equally probable light large evidence explicitly average produce quantity compare topology forget infer structural prior level start structure model topology model infer start yet one topology conditional nothing calculate depend consider evidence term general provide model follow causal thereby remove important investigate topology alphabet topological machine infer inference topology distribution second topology short series uncertainty model course comparable estimate uncertainty topology expect want f use map sampling set reflect uncertainty primary goal inference infer dependence transition substitute interest provide desire analytic expression infer analytic expression mean numerical mean repeatedly obtain algorithm detail candidate interest summary statistic employ generate variety equal tail credible specify estimation visualize two topology quantify complexity asymptotic eigenvector normalize provide part start state infer use binary
word distinguish patient group would distinguish fairly rare word thousand time text select system attribute patient patient time negative patient present sparse way distinct evident total word highlight red bar word appear amongst patient text next green make distinguish patient dominate right partly ten red low twice fraction actually pt word exclude build create word wise appear patient record predictive word record location due predictive cutting dataset discard time case result std std show word row feature dynamically indicate parameter make figure figure table cut word cut cut rare word cut impact cut accuracy model completely gram issue arise learn sentiment negative keyword invert meaning sentence mean phrase look merely refined meaning mean something screen great predictive word phrase technique careful cut candidate feature much large chance may though pair mi score obvious discard word word discard word mi model construct interesting patient record central good view word gram bag gram refer corpus gram interesting distinction sort provide note figure result corpus gram consider gram contain gram construct adjacent reason properly word noun phrase compose actual solely adjacent
descent end gpu five speedup notice rnn perhaps neuron non smooth nonlinearity gradient momentum smooth sharp fall momentum without rate rnn suggest conjunction special initialization investigate parameter encoder relatively initialize rnn mb wikipedia gradient inspection associate rare character like mark preprocesse remove event rnn learn raw wikipedia gradient sgd image narrow quite wrong note solution maintain far detailed material character rnn rnn character c cache result l
distance match reliable information exception quite fig belong bag concept dataset instance informative dissimilarity involve perform reflect classifier dissimilarity evaluate property arrive consistent intuition dissimilarity behave distance still classification dissimilarity dissimilarity negative euclidean behavior euclidean nmf stand percentage triangle fraction web behaviour web datum behaviour highly vary bag use instance feature sample pick dissimilarity average employ favorable bag instance informative average bag bag bag dataset section dissimilarity bag outli cause object euclidean negative eigenvalue behaviour inequality identity dissimilarity matrix informative dissimilarity account dissimilarity aid understand dissimilarity paper detailed mind toolbox default case unless dd mi radial mi kernel svm significantly differently datum provide bag curve bag amount set
supplementary prove next initial st update di j I iteration iteration parameter update remain constant update ratio equals repeat right eq I e apply calculus argument imply use computing estimate probability intensity effect case finish step set finish lemma choice system second lemma convergence fix show cover compact multiplication
step upper satisfy program stop simulation table upper bernoulli run bernoulli run total test table obtain pick element algorithm pick pick element base come compare conduct median minute trial attempt ht show element offer tradeoff accuracy want terminate early versus obtain element gaussian provide nan design algorithm work interesting framework ht nan space
dynamic produce fix controller length define maximal get additional generate control dimensionality sensor less robot think master compound dynamic plus behavior long dynamical system visit globally observe cope characteristic method split principal discretization linearly component behavior observe increase length behavior latter point overlap behavior essentially change span overlap short behavior result explore behavior low avoid curse dimensionality completely drive information body interaction question raise immediately relevant robot ii find convenient ascent stream sensor give exact answer linear actual system lead predictive approximation still analytical way explicit exploration dynamic controller maximize yield landscape gradient calculate change dynamic landscape reason handle dynamic desire dimensional consequence dynamic slightly additionally self induce useful experiment control individually mutual individual effect interestingly formula exploration fully system basis allow control complex experiment control
v decrease generate orthonormal note r give row argument symbolic value think zero respective hyperparameter nu inverse gamma prior mle ed ed
depend copula detail low relevant application asymmetric tail financial well copula feature family interest emphasis observation tail statistic exponent copula family among outlier weight percentile rank obtain rest part making unweighted conceptually involve procedure estimate drive carry appropriate approximate limit draw htbp l l median tail upper tail disadvantage alternatively sample follow permutation sample copy uniformly distribute permutation independent significance permutation define consistency process nominal size test permutation statistic statistic lead permutation also equivalent
run arm identification problem term novel good arm minimum probability fig stop ignore quantity item elimination conservative low factor come factor note upper examine limitation time procedure biological motivate priori arm non adaptive require drastically
evidence support statement generally capture accurately single class agnostic propose novel predict convolutional extraction multiple box advantage predict challenging benchmark moreover location subsequent location additionally algorithm aim able localization recognition path shot feed forward pass state pass localization network follow categorization evaluation roughly cpu machine scale recognize competitive google com art box dnn bound
convexity nc substitute group stem sparse lasso intuition group activate uniform proportion coefficient magnitude retain assess squared regularization apply plot vary vary trial trial instance recover coefficient accurately glasso alpha lasso outperform outperform color overlap experiment star half second fmri retain stimulus yield process cross region prediction expert partition region subject voxel within subject assess whether leverage aid
tf leave side nonnegative note unconstraine criterion simple follow convergence ergodic mean solution unbounded theorem infinity consistent often constraint assume projection computable convex finally introduce convert reformulate vector partition lagrange projection choice magnitude well convergence sequence kkt summarize solve practical satisfied parallel note also thank analog algorithm ergodic practical solve tb set step update update multi subproblem function norm characteristic assumption always generalize interested decompose differentiable brevity solve since assume operation solving subproblem add proximal subproblem lagrange multiplier still terminate eq
amount sequential specify equally spaced alternative specify reach adaptive threshold construct observe importance decrease pre determine implement approach equally spaced univariate plot versus value evaluate compatibility multivariate versus produce however compatible intensity function sufficiently base threshold change manner construct additional component generate dependent result candidate examine utility simulate univariate univariate classical observe n density function pareto truncation observe dataset vary degree pareto exceed pareto choose close panel histogram dataset third row posterior predictive statistic versus bottom panel predictive statistic partial quantile panel panel variability panel replicate dataset compute various threshold burn posterior replication reciprocal statistic computed highlight base indicate rapid dataset around horizontal move indicate observe
class corollary condition erm strictly uniform sided sided correspond absolute definition idea extend definition relevant uniform erm uniformly stable definition condition minimizer subtle show characterize notion well establish stability hypothesis learnable speak hypothesis learnable I return
look domain show kind pair pattern word pattern relational suppose pair approach relational pattern author suffer linguistic model huge human daily linguistic within traffic perhaps road water channel however nine analogy question pair question second compositional semantic ten abstract relational category construct element context instance represent would order would contain segment cosine degree suffer scalability vector order ten sensitivity order natural word compositional phrase also word compositional sensitivity pair equation describe consist matrix context occur kind context context create hypothesis separate serve building difference space context contexts corpus give select phrase corpus window token token phrase context contextual contextual pattern candidate row pattern top contextual matrix drop match contextual final row term contextual count phrase contain number decomposition pt convert raw frequency positive mutual svd step page corpus contain plain facilitate phrase corpus word phrase select result simply select word phrase gram gram space gram gram character row select query word part speech tag phrase phrase step generate contextual speech phrase kind three kind step subsection phrase yield maximum phrase row several pattern phrase million filter fit ram ram sort file ram file file sort adjacent count occurrence counting list row step match step zero remove pattern contain calculate singular decomposition meet positive information frequency variation pointwise mutual column raw frequency estimate estimate definition independence thus product pure chance would word high uninformative orthonormal top sense minimize frobenius final form truncate svd behind space word syntactic phrase contextual contextual noun leave noun since word gram usually gram
boolean partial formal whose consist attribute middle three formal top represent one iff path go represent bold attribute object appear bold neither attribute observe true object node bold hence object bold one label diagram correspond concept empty mark clearly distinction e entry need obtain contain sufficient however yet coverage remain still prescribe say coverage coverage exception partial restrict essential matrix simply utilize demonstrate essential prove inclusion contain whenever essential concern mean identical row column removal simple useful remove redundant information easy decomposition readily hold matrix contradiction also cover lemma cover assumption arbitrary
region bottom respectively look respectively expect compare preferred correspond produce partition high row prefer partition column expect produce value partition count p generation total produce value row partition high see generation partition stop partition count terminate reduce potential generation pruning generally detect enough group advantage competition help bootstrap cell count poisson year separately generate fit bootstrap p p z exp fall partitioning scan rectangular plan fairly large cell plan search simulated count
assign infer computing confidence want recent develop de lasso define column subgradient worth note long regime denote present plug also covariance minimax optimality gaussian design broad physics de precision validity sample dominate program aim objective minimax asymptotically dominate require additional suggest natural partially answer identically
provide matrix statistic call along angle path remarkably surely angle noise nan test distribute control procedure though develop test remain also formalism complete use statistic power orthogonal covariance knot remove know knot guarantee recall harmonic asymptotic improve power procedure control however require regime may sample idea devise test harmonic statistic sequential procedure similar asymptotic context away statistic subsequent one control require control form give reject hypothesis choice anti conservative fact possibility control fdr nearly rule control fdr name start front back allow adapt harmonic know predictor emphasize guarantee test statistic appear realization angle gray
neighborhood digit mode digit digit style intermediate become right amount centroid valid digit dataset identity mode denoise denoise centroid redundant none centroid remove cluster centroid like apart mode kde lie redundancy easier detect mean mode remain centroid look like shape e note create mode particularly dimensional lie dimensional kde mode
present markov process consistency framework section exploit shape efficacy price discrete markov evolve general borel algebra transition chain homogeneous reward function receive discount space bellman sup q value suppose measurable onto cone respect projection cone denote suggest shape explicitly incorporate improve shape function satisfy section reward give estimation horizon discount sequence thus give sample point onto projection fitting assuming convex project dimensional minimization appear intractable minimization formulate program
group regularizer prior particular interest attract structured sparsity lasso glasso ig singleton reduce sparse glasso comparison variable advantage encourage appear elastic fuse lasso grouping gs elastic regularizer encourage grouping tv absolute encourage consecutive able smoothness regularizer gx z neither finally group sparsity regularizer variant x x j non graph r x er sign latter group regularizer
rely surrogate every cm surrogate well minimize surrogate surrogate sense sharp without make surrogate well obtain convex minimizer surrogate sum inequality come sufficient prove technique assumption twice eq separately prove similar proof follow mr mr last comes see mr mr mr sufficient part drop start use second n r otherwise rate cubic involve surrogate surrogate claim material introduce block procedure ng nf cm n final next algorithm surrogate minimizer conclusion proceed step adapt proposition surrogate g n g n ng ii e g g k n nf one
environment critical make multi translate reward arm ambiguity arm capture ucb relate ambiguity define size iii inherently possibly human signature general multi interpret pick arm horizon effect exploitation tradeoff sensitive horizon bandit sensible short take gain towards exploitation tradeoff ambiguity work environmental na important different arm utilize improve develop plausible human capture capture prior reward prior upper upper credible make capture decision comprise arm credible well credible estimate reward iii capture introduce deterministic choose horizon feature feature capture prior arm bandit spatially natural think covariance structure correlation encode spatial section capture feature make begin uninformative prior result extend decision make softmax exist feedback temperature uncorrelated prior gaussian arm reward condition reward gaussian variable multi decision select arm value limit select arm implementation f following know maximize total gain sequential option pick weight expected exploitation exploration pick standard utilize algorithm bound inverse variable appendix fan without numerically incorrect dash line algorithm logarithmic regret time formalize statement arm deterministic uncorrelated uninformative time arm satisfie satisfie guarantee algorithm incur logarithmic length small length logarithmic cumulative
low showing help report performance metric gets reduce fairly training respective training diverse activity compare mean cpu minutes gpu day training time cpu fast gpu also compute require video sigmoid average frame cpu gpu making practical possibly ghz ram gpu work motion video simplify evolution fraction exist motion show entirely rule product within compete may bi
use memory requirement project principal improve really practice significant limitation lin graph bre reasonable lin therein combination lin define cumulative probability bandit extra determinant result construction dense determinant independent see therein make thing situation nd dnn practice actual coarse upper relational approximate preliminary performance test dataset social web music streaming fm contain add pick create precisely
convergence pr practical average pr permutation datum sequence experience significantly increase computational nonparametric setup depend pr sort datum consider familiar natural demonstrate convergence pr methodology linear model nf tail inference base sensitive extreme basic put write pr density pr pr fast easy maximize equivalently marginal l section computation compact u error criterion sufficiently support restrict actually normal pr concentrate around
normal euler drawing condition sample poor sampler combination overcome limitation give regularize draw identity sampler dominate initially dominate depend study sampler similar small make propose choose successful implementation regularize consider auxiliary parameter importance likelihood penalize coefficient less level suppose auxiliary coefficient transition density let importance sampler adopt notational sample variation penalize reason former normalize affected datum choose practice show equivalent measure importance differ density
incorporate community theorem example remark fundamental notably partially create develop treat observe rate link node topology empirically include school network network gene observe link available see review recommendation friend member service facebook biological protein interaction usually consume comprehensive biological specific experiment problem commonly snapshot time predict second fully task miss evolve partially miss
besides analysis newton noisy second innovation instrumental independent scalar vector explanatory model kernel estimate differ instrumental study author g z w assumption typically question accuracy identifiability instrumental continuously distribute component explanatory variable binary continuously strictly comparison integral equation relate motivate instrumental variable source discuss smoothness integral old source condition form source newton apply simulation instrumental variable quantile explanatory unobserved assume assume joint exist lebesgue count z fy dy solve quantile study
store explicit mapping generator iterate generate subset sequence statistically explore possible implementation seed trace produce estimate specific resolve probabilistic choice multiple probabilistic trace seed separate unfortunately iterate choice iterate regardless state previous probabilistic neither dependent identify integer mdp assignment bit primitive concatenation
correspond movement change scale reduction translate lasso path still gain svm vector svm hope relate regularization future investigate svms instance obtain involve svms algorithms translate lot future hope understand correspondence dual loss soft binary equivalence case offset term hard svm formulation know lem svm instance matrix lagrangian soft negative multipli primal lagrange problem call dual svm introduction extend optimal use rgb title title proposition lemma inf tool learn equivalent margin svm vice exist algorithm svms instance equivalence know theoretical translate version kernel consequence lasso svm screen
detect sample propagation automatically find approximation improve performance paper propose low square approximation function noisy approximation possible rank approximations robust include approximation validation variable function evaluation propagation uncertainty square quantification crucial various branch science decade effort make development simulation suitable expansion function constitute e successive correction well construction yield suboptimal canonical decomposition extension low straightforward outline functional approach square approximate expansion section ability detect exploit suppose mutually structure hilbert tensor hilbert orthonormal approximation space space I iy
stack snp snp row maximize logit ik maximum fit logit pca singular two utilize value correspond singular need comprehensive study directly estimation allele frequency bn psd spatially population set allele frequency allele determine even evaluate pca row l calculate psd utilize able well computationally intensive heavy ref seem capture recover transform back compare able visually closely psd simulate substantially psd supplementary demonstrate psd estimate individual allele scenario base method simulation error root mean outperform confirm estimate frequency psd interpretation proportion propose also time fast notable implementation analyze propose individual snps bi top pc
agnostic physical interpretation activity activation frequency frequency neuron assume meaningful measure metric usually simple whole coincide parameter fisher norm induce scalable neural network incoming processing datum scalable break change incoming take fisher metric fisher scale output version output layer induce final result influence datum reduction produce simplify intrinsic reduce notation standard differential fully formal manifold activity activity represent tensor convention metric metric metric final network definite three eq income fisher call metric activity input output change correspond eq give change kk activity unit output layer change op gradient expectation distribution term readily backpropagation whereas metric target op metric product metric final output run measure desire yx metric derivative run incoming keep income define tensor confusion natural fisher op sense situation purely metric hessian loss function parametrization parametrization coincide unique provide gradient progress evenly op l follow property increment evenly spread appendix op variance network unit decode additional e softmax additional function metric induce descent fisher metric plain product op activation define define treatment norm strong invariance metric depend parametrization mix depend op op change define represent certain three op share form kx different influence cost metric outer ordinary backpropagation namely k k natural metric enough backpropagation q
k h integrate unity measure observe simulate comprise systematically however infeasible spirit algorithm abc instead perform denominator computable avoid abc respectively output reversible eq hold q proof hold transition check generate component homogeneous evolve product kernel establish metropolis include derivative reversible second dominate denote chain proposal reversible statement reversible reversible p apply ng ng ng markov evolve covariance tends denote two consist respectively f
kn proof follow exist follow rate detection size clique hardness yet specifically natural semidefinite let reformulate canonical precision interior elementary since together indicate however class computationally tight fast achieve randomized method instead partial answer question define discriminate low could clique evidence section sdp test upper polynomial potential efficiency phenomenon theoretical computer primary hardness align
community concern movement ask resource use reconstruct city region discussion peak medium work analyse political political political employ series extract twitter highlight political measure public opinion survey vote political efficacy political news high peak series possibility twitter political employ public opinion twitter automatically extract offer research notice political respect public opinion survey suggest political future achieve integrate improve proper label fashion furthermore include topology possibility htb di universit di di universit di di universit di di universit di analyse political political exploit machine
v otherwise imply cluster imply edge discrete happen cluster disjoint prove zero formalize clique graph adaptive modularity cluster non empty lemma yield graph cluster modularity rich modularity monotonic constant write volume contribution quality derivative non modularity monotonic plain fill inner assign axiom tailor graph axiom reformulate graph modularity six motivate derivation family quality scale axiom modularity flexibility kind clustering unnormalize cut general investigation indicate cover quality derive cluster modularity work foundation framework quality criterion clustering uniqueness theorem
taylor upper maximize central highlight loss perform sequential expert add incremental ensemble predict reduce utility situation expert example expert different use measure expert boost offer intuitive decomposition error boost decision poor boundary around true label note previous function complementary present world set reveal machine five repository list uci experiment encounter researcher diversity impact different logistic discriminant analysis lda matlab three regressor absolute huber analyse train regressor second consider situation expert potentially bag
ensemble train great dataset accuracy mlp ib nb rf rip ap ar balance cm contact heart heart heart set vote filter per refer percentage great gray ensemble mlp ib nb extra filter model use filter work investigate filter motivated part number store noise weight beneficial filtering misclassifie add artificial add improvement benefit filtering establish affect artificial generation may accurate voting diverse classifier noise datum confidence result ensemble filter set inherent instance ensemble high voting filter classification misclassifie present methodology conclusion direction future work inherently noisy design certain degree noise design somewhat infer
definition confirm normal toward distinction sign sufficiently uniquely surface approach q mi ij mi h ii jk ki go asymptotic expansion bootstrap develop call specify let express j coefficient four geometric also relate u u h formally side constant sign although fact contour lemma function denote ignore difference eq derive fourth section specify significance reject fourth accurate express meaning consider h q contour express expression replace equivalent h unbiased theorem inverse surface utilize fourier
small negative rule achieve treat easy develop theory space fusion describe cost want minimize bayes expect make density come operational density fusion information reasonable cost motivate increase proceed fusion characteristic clarity exposition classical cost fusion expectation establish proof foundation suppose interval set almost everywhere unique produce space classify typically think scalar discrete potentially camera sensor cover weighted delta deterministic generally constraint impose constraint assume least fusion fusion confusion matrix describe correct false classification decision diagonal likelihood fusion delta correspond involve result replace weak complicated stationary say
g nz nz ig nz ig approximate framework form ig upon conjugate appropriate approximate exponential hyperparameter assign mix assign prior inverse q ig ig ig ig ig ig u ig ig ig ig ig q u ig ig ig ig ig proceed initialize assign method initialize initialize sample initialize
operation convert permutation see order model exponential define rank amongst propose similarly model rank investigate mean paper motivate divergence notion permutation unlike permutation distance metric distortion function score cluster shall web oppose vector permutation say ranking informative available combine former concerned score latter take voting every candidate sometimes easy assign search often possibly feature provide application ranking call dx dx property dx immediately permutation become one order e permutation application cluster order divergence fit naturally permutation order purely permutation consider knowledge work notion permutation introduce property score lb divergence build bregman therein mainly connection
n markov take hour prior hour ps ps ps ps ps fold leave whole gene thresholde relative f statistic see rank small value indeed rank lasso narrow maximum small maximum gene rank end predictive performance table include probability prior difference lasso point htp prior lasso knn ps pt gene figure datum gene runs gene weakly useful correlated skewed value reason logistic coefficient fairly determine boundary slope log tail absolute hmc overcome ordinary second weakly chain separate gene none absolute useful generally tail multimodal compare look contain subset substantially well gene true subset good see useful separate lasso htp statistic pt er bayesian feature gibbs hmc
system one probable provide formalize application angle energy scoring signal take scalar value distribution index delay time track angle score set input feedforward connection strength strength however evolve consider correspond
thank university es ia fp people gaussian process employ nonlinear rarely gps regression nonlinear wiener establish important include recursive deal stationarity relevant digital communication gps art tool gps statistic community due become early interpret kernel additional statistical variable process latent produce multidimensional estimate draw yy long compute nonlinear directly either solution restrict come narrow suboptimal minimize overfitte square accord related distribute mean gp understand mmse additionally description gps regressor process label reveal latent gps characterize denote use zero prior
range stop roughly average additive noise diameter near optimal diameter channel wireless may interference memory channel behavior remain also worth type solve g study issue near network acknowledgement support nsf national foundation grant air force office sc technology department ph department engineering california berkeley include statistical modern code currently california berkeley statistics department electrical sciences mathematic university electrical engineering computer science mit interest code mathematical statistical foundation fellowship award mit sciences engineering research fellowship paper
exponentially dimensionality intractable know namely marginal function although continuous variational inspire physics sampling count reduce technique base arise inference recent suitable expectation combination perturb crucial aspect approximation showing expectation know efficacy partition ise model know truth reach belief propagation also problem write digit handle accuracy current set assign weight element wish approximately total follow compactly factor table factor access compactly I e
distortion limited capability arise perhaps constraint reaction size distinguish entity formation limitation induce capacity noise process situation resource extensively economic organization computer theoretic resource gain expect change arise mathematical discuss distortion theoretic compression compression separate decision maker economic rational distortion rational capability discuss present bound
unify solving reduction display step figure set step htb kind propose unified solve model validate unified propose meanwhile choice jointly structure proposition section wang recently biology etc increase attention
discrepancy explicit construction achieve discrepancy boundary one problem indicate discrepancy digital net arise respect chen want eq preserving follow inverse h quasi discrepancy respect box define discrepancy translate set sa box want box come sampling
search fig relevant possibility decay mode lee particle consider enough chance detect conclusion correct lee search particle evidence find frequentist exclude production physic exclude production new exclude ratio region exclude small frequentist price exclude regard frequentist line fig frequentist agree restriction remove claim sensitivity necessary different level typically impossible expect fig fall small origin possibility suit statement hypothesis light involve take nuisance parameter form ratio go hypothesis profile physics incorporate parameter apply bayesian model practice particularly occur alternative review physics nuisance ratio probability prior probability example hypothesis assign view prior prior nuisance usual integration parameter space likelihood narrow broad
replace establish sublinear rate divide case q need hold indeed follow hypothesis induction complete suppose follow lemma sublinear assume let know convexity precede k f define notice lemma result globally convergent give solution define lemma q follow take side give theorem illustrate regularize special suitably
stream regard variant presence however tracking stream finally zhang available manuscript rank via notion accommodate large toeplitz rank rest base base restrict property well rip bound section main defer provide conclude future proceeding brief notation throughout frobenius nuclear positive semidefinite psd trace euclidean inner product whenever denote operator onto toeplitz notation summarize summary parameter matrix toeplitz orthogonal measurement I recover pose effectively covariance preserve recovery restrict attention sense compose I copy express heuristic perform minimization encourage priori matrix since psd trace norm form completion retrieval relaxation stable approximately corrupted formally state eq simultaneously implication theorem list absence noise trace minimization program provide notice psd decompose say freedom psd recovery trace universal recovery rank recover absence noise highlight programming accurate soon exceed large gaussian beyond absence low reconstruction eq soon obtain intuitive understanding covariance th value obeys law exponent reveal return accurate decay broad rank reconstruction exceed reveal practically appeal
decrease improve give recent explore bag little target combine subsample take replacement form subsample plug proxy implement subsample bootstrap confidence subsample proceed subsample repeat overall bootstrap bootstrap conceptually would store size diagram moreover bootstrap see procedure subsample outer view map onto let subsample process processor virtue send processor bootstrapping conceptually create generally instead weight
hand r strength document sentence recall task recommend feature since pool ultimately exclude false table within number especially movie document feature return classify movie aforementioned feature close run almost linear full column increase case recognize column relevant exclude linear substantial amazon customer uci repository develop five author review entirely review even create decrease control inclusion feature number would review simply frequent labeling data
cca bias importance recommender method let ard unnecessary run variational measure square rmse demonstrating difference form size entity set second entity set finally first generate noise matrix factor ard predict incorrect noise map grid fold get contrast run vb validation worse vb hyperparameter necessarily scale separate scale infeasible validation view datum study patient breast two correspond throughput copy task predict third proximity gene location reasonable assumption
define spline th derive detailed score notation full conditional offset term g evaluation lead intercept complete nn rhs approximation notation derivative q equality derive derivative spline denote similarly denote laplace approximation compute expansion likelihood leibler divergence constant q third eq inequality jensen concavity matrix direction likelihood q term combine ten full vb page compute x nn b iteration reach l david alternative bayesian link functional covariate observe procedure covariate complicate handle update advantage approach estimate estimation trajectory covariance modeling although situation challenge
test train depth show classification increase notice tree accuracy already state dataset expect increase tree transformation forest transformation adopt provide depth part part background class body experiment dataset pose pose body pose point pixel use depth difference radius respectively form dim descriptor
boundary seem restrictive difficult come propose testing tackle test similar nan equivalent naturally depend give may priori reasonable could possible weight rule derivation small although rate widely frequentist bayesian indicate answer relaxed organize lead testing derive separation monotonicity positivity simulate parametric separation condition decision rule achieve even simple widely cover variety interesting selection
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play I noisy previously result explore information learn filter noisy training player threshold threshold challenging criterion reliability assign mean outli player tackle basic training train classifier datum measure weight construct ensemble summarize collect setting threshold classifier parameter record validation build individual play correspond discover ip monitor drift generalize target player ip detect failure formulate procedure content line stage use tackle various decision machine compact yet precise representation making establish ip design player ip via game user experience reliability game game group game content game rank play game log ip consistent enjoy ip state produce content game drift concept drift ip generalize show dedicated player quality game minimize game receive produce learn game public moreover ip determine ip would content enable generating content player emphasize enable merely require mm prototype first person game report refer name ideal platform purpose ability play software game line ordinal content
recover analogous sequence know accordingly express regret regret term assumption instead track regret bound approach mirror md consist problem subgradient bregman divergence continuously convex bregman divergence convexity md fit instance mirror regularization help md characterize max max q unlike track piecewise yield regret broad class dynamic dynamic mirror dynamical process
unlikely unlikely bin receive make truncate problematic solution lexical bin add suffer issue follow impose word exist allow give rise filter lexical objective define induce condition bin note although normalize divide condition denominator rely th denote bin come p I ir come l j r maximize appear denominator lower prove
context h old exponent hypercube slice adaptive contain learner train learner learner partition learner learner learner train candidate learner learner lemma remark van decentralize many learner instance characterize context context learner action provide request latter cost receive learn learner contextual seek reward trading learn action action account learn distribute provide analytic complete benchmark incur sublinear big mining surveillance sensor online recommendation system contextual user multiple decentralized learner assume information another processor process process total stream order sequence read limited storage capability stream mining wireless need information learner communication snr wireless communication transmission selection learner instance mining select transmission online multi bandit solution refer bandit learner equip learner agree give impose constraint learner know arm learner time different information arrive learner learner asynchronous time receive slot depend arrival arm unknown context learner maximize function context arm learner know arm learner learner benefit learner benefit solely arm ask processing learner beneficial contextual bandit learner significantly challenge observe learner expect arm moreover heterogeneous context learner lead learner distribute reward expect reward context rigorously quantify learn learner good arm scheme learner incur knowledge sublinear imply regret convergence reward context depend context concentrate
exposure exposure absolute exposure vertex exposure neighbor receive treatment fractional neighborhood exposure fractional exposure treatment receive absolute consider relaxation exposure treatment condition exposure correspond fractional neighborhood increase decrease reach exposure user exposure condition homogeneous much possible heterogeneity exposure vertex degree vertex neighborhood exposure fractional exposure adjustment exposure exposure universe beyond exposure also exposure turn may exposure characterize analyze core similarly graph maximal vertex degree heterogeneous fractional analog fractional fractional subgraph fraction connect thus equivalently degree fractional core fractional exposure version neighborhood exposure exposure treatment component receive condition absolute exposure vertex exposure belong core core vertex fractional receive component exposure perhaps exposure interference comprise include specifically note fractional core exposure exposure exposure case
start statistic alarm test income fuse sensor static I expand express manner generally grow compute iteratively likelihood element path compute store path still kp h kp k kp sum probability keep likelihood difference k stage h big practice decrease target
option therefore assume transition execution finite term fair price option infeasible approximate extend challenge option pricing truly historical trajectory stock option predict reflect price treat uncertainty well pricing stop apply algorithm pricing may horizon time set stand option option execute terminal otherwise transition state therefore write w I trajectorie never
friend friend friend impact behavior user tweet may become mechanic formalism simplify consider tweet occur tweet incorporate work symbol period longitudinal user implicitly behavior disjoint time see predictive exploring allow profile popular medium platform model future plan influence prediction interaction prediction context domain use send potentially piece go predict medium understand interaction medium acknowledge nsf pc understand mechanic social media david mathematics college md edu college seek dynamic future past behind construct observe predict behavior capture
modify kalman know follow derive identity rewrite combine kalman conventional kalman moderately operation else sentence outer product particle velocity perturbation velocity
explain term clean execution part partitioning index step I arm resource inequality sum jensen function value apply assumption justification
estimator ideal estimator spline ccc ccc space spaced estimate hold predictor iid data functional expect addition thin spline sample select regularization parameter benefit cccc weather weather temperature average temperature functional approach problematic eigenfunction smooth spline assume propose weather cc initial
sure remark toward variable expectation close respect prove core simplification prove different almost sure careful sure much worse know approximation euler tends imply furthermore restrict large take straightforward surely n last lemma converge behaviour x infinity notice never variance toward fix
priori set random ny mf tx u tx u dirac delta let unknown estimating formulate problem density pz input design identification
table accuracy aa correctly classify pixel accuracy confusion agreement random ten report train map depict fig observation spectral characteristic accuracy take account contextual importance contextual marginally kernel combine characteristic surprising report information comparable training datum inside dictionary near neighbor obtains cover narrow note chen al window pixel dominate pixel adjacent fourth since region fig contextual finally overall employ put disadvantage come narrow accuracy present far report test different accuracy ten avoid moment large dictionary classification decrease completeness dms use report accuracy dms dms window center pixel dms encourage smooth representation four spatially connect early group employ dms may accuracy also dictionary dms fair experiment overall learn dictionary fig depict
eigenvalue analysis case tell result invoke tend zero apply proportional area side length pn equation yield eq invoke obtain pn main step simple give equation establish yield dense degree element w large eigenvalue constant large section demonstrate hypothesis show plant blockmodel class much secondly show linkage grow linearly work graph setup densely connect essence blockmodel cluster increase
seven control set evolve spectral bin evolution relate quadratic intensity spectral bin clearly phenomenon spectral relate control modelling ensure volatility control evolution latent time check employ assess simulated predictive distribution observe treatment group compare examine deviation fit vast histogram supplementary group control compare view due reason parsimonious time course longitudinal problematic due currently limitation treatment variability longitudinal nature volatility two account high trajectory quantify evolve importantly highlight research naturally fit mostly costly several review area would mcmc point collect time imputation datum potentially principal
value use memory overlap summary minimal requirement favorable initial iteration explicit toward preliminary thresholding switch comparison alone overlap group serve perform experiment partially sparsity modify outer like systematically translate rmse numerically overlap fully overlap expect however overlap bad overlap inferior overlap case component group member component overlap group illustrate overlap group shrinkage fourier transform speech variation model arise may attribute isolated peak frequency improve due desirable isolate spurious spike avoid speech take account yu al thresholde note overlap like aim overall denoise result sparsity speech audio overlap denoise estimate speech dimensional illustration noisy speech illustrate noisy sample illustrate
include phenomenon optimization well point exactly search around modify feature ensemble intuition role interesting heuristic modification criterion mode establish present normalization focus component expression hold website paper combine ensemble accuracy valuable hope take hyperparameter challenge induce recognize technique improve bag boost
interpretability tie statistical inferential several existence group stochastic equivalence memberships blockmodel allow object belong relationship group membership object belong model relationship depend unobserve similarity widely formal object relation relational example multivariate ordinal normal probit link similarity row correlation multivariate kronecker specifically normal kronecker relation straightforward identity tie article evaluate develop variance represent among row maintain correlation row column rejection inference blockmodel normal fitting avoid spurious evaluate column correlation exist normal replication model applicable consist matrix versus give likelihood ratio
word membership node also community node intermediate value guarantee inference community intuitive decrease membership large block guarantee block community result guarantee section membership extend general homogeneous mixed condition involve network number community membership membership discussion set worst latter thus block matrix require maximum minimum community require perturbation tensor compute thereby provide eigen tensor separation intra inter connectivity special homogeneous mixed model special iteration procedure constant obtain membership vector stochastic regularization hold allow small pick community spread across community ready block say event probability satisfy proof ingredient result concentration tensor eigenvector moment eigenvector moment assumption recover eigenvector perturbation refer plant clique arguably simple problem size place edge clique connectivity plant sized applicable plant condition recover clique unfortunately need tensor whitening size drastically question improve community recover ratio summarize prove ingredient concentration edge membership adjacency around appendix hand establish quantity whiten computed moment recover span since main employ bernstein whiten symmetric establish around condition stage carefully perturbation bound detail various establish eigen tensor vector scale bound recovery special block step direct recovery community norm threshold draw carefully control contribution perturbation bind appendix simplify membership degree towards result concentrated wrong correctly identify membership extend membership appendix detail use tensor latent topic markov mixture include result several latent algorithmic improvement respect tensor variable among topic latent lda close
pair loss test weight begin zero consider describe table implementation publicly solver fista toolbox work well logistic incremental memory method epoch regularization regime yield figure provide rest material fista require epoch reasonable batch challenge previous classical way resort programming reweighte problem solution f function accord n p dc programming algorithm investigated note turn
nr propose motivate conjugate gradient motivate factorization full extend trivial riemannian metric riemannian geometry present report tune riemannian rank trade information simplicity building work factor exploit base although second order riemannian trust region focus offer superior example geometric notion list concrete illustrate implementation manifold set size orthonormal column non factorization decomposition relax full
band shape point hypothesis flat direction direction smaller find direction allow adversary extent possible unlike removal make weight noisy separation general technique optimization hold general admissible result concave distribution instance mixture w dl band w polynomial number passive set iterative operate round round fall boundary hypothesis use low first soft outli removal minimize appropriately description figure formal outli removal specific choice description preliminary specific removal htbp draw example work possibly apply localize outli work put reject end use polynomial unlabele imply passive set well ball case computationally allow example unlabeled sample sample cut removal draw example normalize hinge fx outli removal unit vector specify fraction r qx positive polynomial cut k pd km k ks w x noisy fall outside
worse allow small number nf cs nsf kronecker decomposition spatio decomposition impose reduce required tradeoff reduction reduce covariance approximation estimation product er rao sparsity exist markovian neighboring alone sufficient method spatio prohibitive structure model small follow normal spatio covariance
example correspond know global moreover ratio apply even rough ratio throughout th regard condition think attribute hand number expect definition hold enough regard easy unless unbounded variance grow merely hold condition acceptable replacement sample entry otherwise exposition discussion bottom efficiently function strictly conceptually simply weight sample replacement streaming achieve active though far matrix accomplish operation streaming matrix outline implication compare metric denote analog algebraic always
write way q trivially
clique second summarize figure picture work sophisticated sdp fail follow strength sparsity polynomial support tend follow give theorem motivate ct short intuition expect algorithm value deviation strength entry thresholding must entry fraction entry fact dt recover proved level simulation result sparsity setup single spike priori execution return divide fix average compare performance evident ct dt scale value prediction ct sparsity level
user compute estimator tn c ct allow select division zero matrix matlab probability probability corollary probability leverage minimize nearly view since score probability exact leverage qr computation probability identical optimal probability number probability c eeg wise present uci repository show error randomization average versus right show ratio leverage score leverage amount furthermore leverage tend differ plot run optimal versus vertical axis correspond plot ratio sort probability
yet extensively economic contribution date gs financial series pricing model exchangeable array term describe interact market market positive share enter expand contract competition stochastic market technology cost different construction achieve hierarchical derive interact scheme bayesian market mean gibbs interact market increment mechanism share determine competitive term relative strength market market deterministic percentage force away competitive great flexibility functional local information aggregate market rescale system dependent section model simulation perform dynamically choice investigate economic transition economic regime market impose policy maker interaction property appendix interact defer appendix collection dynamic finite complete endow
obtain solve problem show batch lasso scalar difference follow example randomly example certain add noise online
spectral variance consider bm easy software package bm output broken batch batch length estimate general bm estimator batch allow overall length simulation regularity condition geometrically ergodic require storing allow batch concern could batch reduce usage one establish bm sampling routine univariate associate bayesian estimation univariate cumulative little estimation outline natural distribution give eq gx q strong chain q order extremely broken plug well lee yu bm estimate
fix distribution cdf variate matrix illustrate block probability determine base membership blockmodel position locate sample adjacency denote row plot various colored membership vertex blockmodel show curve k htbp dash give curve distribution theorem estimate residual empirical covariance compute sample respectively estimate covariance tend limit covariance htbp block blockmodel theoretical covariance matrix also effect
tag estimate concept ccccc transform impulse transform learner tag tag tag entire computed average tag impulse could concept weak tag substantial sparfa sparfa datum concept sparfa reasonably broad hyperparameter rate correct learner enable adequate question estimate describe entry thresholded set ghz core pc sparfa converge sparfa require minute sparfa sparfa comparable concept association sparfa b tag association sparse question outli concept question intrinsic difficulty advantage sparfa sparfa ability reliability estimate decision action considerable learner ask reduce uncertainty assign sparfa box output iteration learner enable visualize question portion set level information learner compare learner learner question number conventional percentage answer similar concept due respective assigning question sparfa posterior learner find variance considerably learner sparfa easy difficulty remain question affinity roughly half learner affinity thus weak learner strong determine response sparfa pls quickly assess presentation present variance concept pls plan sparfa concept matrix enable validate extent across strong sparfa sparfa b answering estimation sparfa sparfa inclusion probability concept sparfa sparfa sparfa agree question sparfa link concept mcmc concept exclusive concept jointly sparfa find compete question resolve pls yes yes final capability sparfa school carry university amazon crowd learner answer cover geometry function manually set tag fully fully value logit sparfa
balance lead optimal large select quality ratio gain formally exist factor constraint uniform runtime intuition inexact evaluation first constraint active section apply inexact budget product try element gain include marginal fact select gain element nonempty add budget gain must active combining systematically evaluate scalability algorithm structural massive medium compare counterpart classic discrete degree gain allow type diffusion trace physics heterogeneous dynamic product different candidate target potential allocation million base heuristic degree treat natural large million twitter often considerable payment agree post ad sort diffusion pair solution simply continue search pair assign heuristic baseline assign target size network product
interval identify indeed consequence short go piece p p p pd gp p equal pd hausdorff bind hausdorff supremum set easily technical lemma end edge point orientation section geometrically connect set merging homology homology persistence connect path connect contain persistence persistence vertex persistence vertex create never simplex cl closure short connect xx short parametrization embed path restriction infimum pd p nx apply short path
less appealing square alpha variant neither normalization variant maximization show equitability three score plot point realization visually equitability couple receive might want noise change imply statistic strong equitability much effectively variant noisy relationship section plot determination relative noiseless type appendix figure variant plot range variant produce demonstrate combination equitability maximization grid cell good grid row never monotonic relationship
final algorithm theoretically step principle recurrent construction build riemannian framework network activity unit belong represent number prefer correspondence force weight voting represent intersection active put active unit put weight alphabet symbol intersection compute activity write specific symbol give backpropagation time time suitable metric take section section idea three neural article rnn step activation symbol activate unit bias ix exactly symbol backpropagation parametrization bias input input activate reading activation actually relate traditional procedure different procedure invariant trajectory idea recurrent activation output activation level rnns sum activate unit symbol amount rnn alphabet discrete sequence continuous linear activation hmm transition softmax I reduce hmm tx ga networks carry parameter letter alphabet symbol store costly factorization technique apply neural allow handle distant understand analogue produces activate occur stay activate feedback loop always dynamic f j x tf looking dynamic vanish call chart assume ia dynamic uniquely evolution income obvious follow include activate unit output q far integrate study linearize sensible initialization principle along build form cost backpropagation adapt appendix derivative turn metric weight diagonal reduction hessian information
different via cardinality examine sparsity success fix sparsity show solution figure jump back function reweighte either small improper weight failure fix sparsity set choice well tend big updating cardinality fix may find solution min updating outperform
come I pay convergence simulate monotonicity understand regression constrain see df monotonicity cause coefficient intuition df great intuition follow unique set process small project onto magnitude go variance remain tell df roughly formalize intuition ht minimize subject nf neighborhood exist
unity constraint potential system joint dirichlet dirichlet diffusion dirichlet stochastic matrix stationary positive covariance unit requirement times appendix eqs eqs correspondence generalize arbitrary q eqs drift eqs
view topological recall variation define moreover draw estimator line rest covering necessary packing ball cover packing packing ball pack measure get upper bind r upper bind corollary thank b bound inside na b b u bind dirac dirac clear denote nb p case treat bar code compose segment compose segment cycle b da bx bx close inclusion second f empty second accord bx fu find bx proof consequence adapt idea proof hausdorff logarithm alternative large belong support n r r r
equality e fix trace common x implicitly involve sdp inspire spherical scalar sub vanishe approach arbitrarily problem sdp method slow next show simplify variable primal strong duality z lagrangian sdp seem efficient interior fortunately eliminate simplified solution simplify simplify sdp dual constraint objective necessarily twice sect relationship optimal implementation l bfgs
poor relative similar mixture analytic regime appear cluster considerable comment far cluster penalize size observe gain relative structure follow parameter train test reconstruct false positive correlation negative false positive negative respect summarize quantity regard value well algorithm precision cluster assignment identify maximize agreement assignment calculate classification indicate true text graphical penalty km km bar error non penalize np could size size positive except outperform regime train competitive bic train regime bad sample sufficiently valid score cluster
intervention ideally base dag super estimate dag conceptually start dag multivariate additive fitting additive mention original correspondence index edge dag avoid compatibility nonzero sufficiently obtain property high hypothesis test nonzero obtain applie estimate implication inference intervention distribution calculus intervention intervention note screen property replace full thus causal dag statistically full dag present restrict permutation search tractable statistically deal perform neighborhood idea versus additive ideally smoothness variable intercept emphasize ensure neighborhood variable structural restrict permutation compatible neighborhood set regressor precisely estimate j kx k
score limited normalize weight influence influence semantic specifically term contain evident medium consideration corpus view attempt estimate imply believe consideration analyse characterize usefulness particular corpus successful idea supervise semantic exhibit superior utilize label question issue indicate corpus show random ordinary however observe corpus start performance label significantly corpus initial top learn wikipedia question make corpus learn semantic yet corpus consist span scope contexts book sentence etc context say relation context entirely context affect contexts tradeoff interesting solve text individual one extend literature interesting extract supervise preference optimize task good method follow similar setup utilize target utilize contribute extend accommodate context passive interesting active sample preference sort algorithm perfectly bind several agnostic interest rank difficult short ad search query review supervise belong share stop approach bag train clear necessary text feature consideration term consider relation two relation relation name two would assessment relation three summarize task involve nlp consider term argue
proposition spectrum problem var study literature highlight difference available series structure regularize like lasso theoretical var beyond establish significant portion analysis devote bound depth dependence finally var fit aforementioned paper inequality vast exist literature step extend penalty group lasso norm penalty induce satisfy x pn show center subgaussian thresholded version uniformity stability measure introduce asymptotic consider decay uniformly sufficiently eq body line unify procedure regime penalty scad mcp argue hold subject scad mcp suitable restrict strong convexity rsc sup penalize loss rsc spirit verify discretization argument present large mcp applicable review unify decomposable penalty nuclear incorporate drive theoretical crucially condition restrict eigenvalue deviation argument var probability present lasso extend stable var reader correlate error note align perfectly estimation stationary process component simulate univariate apply depict leave display error rescale size
hilbert inverse gaussian pde solve bayesian exploit namely observable square covariance operator gain dimensional pde consistent infinite rank observable discretized dimension candidate location configuration sparse subset candidate sensor assign observation choose vector indicate sensor place allow take allocate norm approach e use penalty successively design time initial condition demonstrate weight find improve typical dimensional insensitive sensor although dimension gain nearby typically beyond prior observable pde svd candidate sensor computational numerical number quasi newton svd rank moreover square employ present component description problem initial condition time numerical finally conclusion discuss extension background require dimensional hilbert inference formulation finally comment svd trace observable term bound solution pde application inverse full lead random let appropriate collection follow dependence real view observation update
generalize dimension combinatorial joint detecting costly basic review therein follow nk hadamard k nk statistic include additional corresponding center insight center introduce expand population center obtain z k equal p p z z dp xx dp xx yy p p whereby z k identify schmidt f hilbert schmidt tensor z supremum take ball respective op center
barrier proper semi convex strictly write formula reduce input r characterization co norm analysis hold convexity exist self order expansion quadratic surrogate around full newton solve generate sequence point sense highlight operator problem moreover follow obvious subproblem take place various introduce c solution practice case give direction method multiplier subsequent development proof indicate use eq provide per inexact full proximal newton fix solution follow characterize contraction illustrate contraction furthermore distance convergence end decrease varie inexact perform closely ideal e theoretically subproblem solve exactly
loss detail linear use probability margin estimation elastic net excess bound condition stress later see quadratic example net adaptive restricted consistency furthermore theorem eq corollary show elastic sample size arbitrarily large number relevant finding literature merely tend comment result corollary case away standard zero classify differently net possess zero elastic net remove certain threshold threshold detail omit yield consistent asymptotically setup shall elaborate stage since bind estimation individual sup norm provide sup norm alternatively eigenvalue convergence sup magnitude upon request elastic omit recently method high observation briefly terminology translate example generic define elastic net coefficient corollary provide prevent knowledge point cover cover cover outline present concrete recall sufficient
policy curve policy suboptimal stage identical gain sense sparse signal th sense chernoff quantify snr resource sparsity snr budget derive well budget devote decay vanish fraction snr sense budget quantify approximation optimal predict analogous policy recovery refined policy stage extension multiple non incorporate current stage allocation hence lastly limit bound along oracle probability moment likewise invoke bernstein chernoff
boundary thus bind numerator obtain theorem formulation thus comment apply f l numerator discretization yx q expansion h eq hence l finish hold skip detail connect dim riemannian manifold view bundle generality form orthonormal action eigenfunction either odd irreducible real distinguish type eigenfunction resp resp odd eigenfunction know heat family map odd eigenfunction eq heat eigenfunction operator orthonormal point eigenfunction separate take odd otherwise since odd odd eigenfunction q hence neighborhood exist argument compact embed embed construct nash start cover odd eigenfunction open account resp h z j p ip iv lx xx iv full symmetry rotation embed onto meet axis note rank cover claim construction symmetric kb z k xx yx yx ix help find define linearly vector smooth map linearly xu away induce scale properly scale ci dc v embed z v note transformation modify set choose embed simultaneously eq unity manifold admit smooth remark ease notion definition map indeed direct calculation give embed
cascade model ic heuristic sp support exponential edge furthermore scalable synthetic network hour fit ic infection infection essentially ic model lt income infected source source set b show window perform vs influence accuracy influence mean fix fix source vary window quantify influence world select use dataset cascade among medium cascade learn exponential transmission function discrete learn infection probability ic follow cascade quantify mae influence statistically cascade cascade node gap estimation improvement long path improvement influence improvement confirm evaluate select node spirit influence true distinct infect select select vary select source observation window continuous diffusion million significantly
computed signal stability recover tv property sparse frequently appear recover compressed gain lot many medical imaging small number ambient compressed sense fact signal nonzero big represented wavelet
divide allow amount across thousand unlike deal present limit individual need global support asynchronous prohibitive optimize asynchronous dedicated address parallelization approach yield convergence framework massive scalability entity processor name entity link wikipedia text entity assign national team latter word return collection note news web page wikipedia content entity content entity mention distinguished drop consist english wikipedia article phrase refer entity wikipedia page link wikipedia detail processing document variable first topic assignment mention entity topic indicate mention represent single wikipedia must decide upon context document topic mixture characterize document put mass team although word assignment explain detail wikipedia topic content treat dictionary omit follow entity represent correspond
convex deal family member overall direct beneficial minimization practice use concept investigate posteriori objective concave estimation tractable logistic normal probably certain elsewhere beneficial logistic effective modeling induction wolfe fast maintain make map intractable practice sgd resolve non originally efficient deterministic algorithm able jump optima close advantageous traditional complement successful probable logistic resolve problem analyze notation bold face denote
alternatively cb although attribute number map online change attribute score four attribute well interesting moderately bad attribute parameter vary suggest attribute datum present metric sample ks statistic produce result pdf small strongly value concentrate construct realistic distribution attribute parameter method cell table updating batch interpret analyze opposed update batch mapping online depend fast update batch scale big inherently parallel som obvious two superior either rectangular grid full attribute spherical topology nature spherical topology compare boundary topology although map subsection spherical topology cell nearly equal galaxy training limitation spherical rectangular topology fine tune topology topology topology force topology generate area cell center naturally along produce cell match natural rectangular look periodic boundary appear boundary understand spherical topology periodic compare solid periodic topology periodic slightly som optimize topological processing galaxy periodic desire star separation necessarily map away region besides previously discuss som
rapidly observation common p parameter proceed sample two hyper independently conjugate hard covariance hyper enter way trajectory analogous gaussian regression input output dynamic slice straightforward implementation factorize complexity familiar introduce efficient gp gp gp prior counterpart still expect outperform version happen gp live space inducing conditionally induce mutually although
choose matrix sample previous bandwidth experiment test close experiment estimator estimator strong distributional derive particular analyze reason measurement measurement future assumption slow nn rkhs estimator supplementary lemma iid
optimal formulation account uncertainty model adjustment bayesian compete posterior probability dimension factor bayes similar correspondence establish mild
normal entry consider spaced solve difference figure characterize regularization parameter ambient ambient fixed decrease mean importance perspective design amp policy amp theorem definition corollary concern know randomize recovery measurement give despite theoretical analysis solution know asymptotic active ii behave employ new reliable approximate pass vector measurement matrix noise denoise literature algorithm publish area fall characterize sensing suffer loose constant quantitative seminal researcher analyse sharp quantitative cs despite major progress lasso aspect algorithmic
many crucial well may help call property component characteristic observation study network adaptive store analyze understand content decomposition dimension adapt impose characteristic many interest redundant allow belong problem record physical record environment learn factorization resp
error train match show db work hypothesis acoustic representation reduce snr motivate development extract adapt use error nevertheless acoustic ensure valid condition corner encouraging seek inclusion information entire frame concatenation add analogue change little meaningful component phase phase combine result relevant retain difficult determine initially choose consecutive frame accurate principle statistic segment standard within considerable use giving add amount assume independence component also one long feature correlation space adapt separately combine standard hmm context recognition representation segmentation error hmm assumption purpose duration gmm centre concatenation close along figure five frame five model correspond likelihood q number likelihood point class ms duration frame close correspond map segment impact frame sum log likelihood
different operator set minimize alternatively respect feasible operator kernel admit algorithm value algorithm combine operator alternate algorithm look tf iteration training proceed determine stochastic calculate observation computing denote dimension avoid save complexity
involve disagreement appear principle pac da set classifier use justify tight majority lead elegant perform transfer suitable section transfer marginal close unlabeled region auto
overlap effect figure arise simultaneously reconstruction preserve space information capacity redundancy hence encoder get marginal information reconstruction observation marginal observe account contraction volume transformation joint entropy entropy mutual information factorization mutual information seek control decrease thus minimize dependency inherent sigmoid nonlinearity decoder unable distinguish decrease entropy increase bottleneck add imply deep bit shift threshold dependence incoherent final
embedding provide yet reasonably good wish thank remark author acknowledge support grant ia dimensionality reduction become analyzing embed new neighbor embed test point simple often sc diffusion map dm average neighbor give noisy intrinsic representation describe examine challenging incorporate avoid cost offer optimal parameterization iv classical synthetic dm extension value forecast pattern keyword map machine understand work common dimensional
know yield explanatory separability analysis interestingly factor report separability way denominator exceed separability increase unfortunately two immediate interest monotonically increase true distribution might attempt yield unstable density beta
context arise condition product center feature distance spread observe feature
lead phone label frame experiment initialize gradient descent optimizer momentum iteration set dimensionality iteration early stage trick increasingly embedding increase hard structure less cluster contrast parameter embedding produce mnist computation object also mnist visualize present speed embed mnist digit correspond embedding present figure trade increase approximately value substantial
arrive large size true signal write expression last case ratio possible find range wish range great apply eq q limit define
biology represent background estimate median briefly calculate cluster set model encourage encourage direct enforce extra dc du get eigenvalue big name describe mention challenge thresholding setting simple generate multivariate element eigenvalue greatly eigenvalue correspond true thresholding bt constraint hard thresholding counting extra constraint tuning constraint convex clearly force identity factor enter close eq correspondingly eigenvalue estimate eigenvalue covariance trace thresholding proceed
liu setup li fan li bernoulli rate uniform variable sum absolute multiply diagonal rate absolute nonzero covariate independently p ij qp ij pp ij qp qp ij qp pp performance report evident superior numerical performance computational large performance partially overcome inefficient alternate obstacle apply analyze high sep inclusion
fix normalize e practice I variable condition wide observation great equal shall number sparsity go infinity literature represents associate appropriate onto mainly compute solve square estimator bind choose support converge establish estimation solely classical seek minimize interested establish condition accurately result broad measurement impose present prove numerical paper measurement swap r I minimize suppose say let corresponding want transition transition subsequently find step step remark realization fourth intermediate iteration three noise horizontal iteration vertical intermediate support specify desire level unknown zero boost set measurement pairwise simulate three different use sparse lasso thresholded
jensen start simple square clearly also value obtain op k loss expert constant normalize logarithmic defining derive weak expectation kl divergence define equation x ix ix f ix ix ix triangle jensen arm equality normalization difficult find literature loss state formally proof
exist algorithm always decrease iteration output always conclusion show throughout fact something strong never entire begin inductive step inductive since consider depend tt b eq ts ts ss complete b ib ib sum inequality obtain b eq complete partition constructive may polynomial time update
ml deal image viterbi iterate mode cut unified advantage well originally design multimodal mode tendency since fail well smoothness implementation image patch initialization estimating give segmentation capability move away saddle comparison ever make hidden model hide markov mesh model within quite different besides nevertheless execution probable freedom labeling sequence realistic sequence contextual output em ml term segmentation graph comprehensive toolbox cut kolmogorov author also code available continue toolbox framework markovian segmentation c c problem ml ml ml c filter ml c ml segmentation study filter unimodal filter ml ml c c supervise c c
denoise seek set reduce penalty iteration one pair logarithmic one seek reduce interpolation group indicate function necessary table supplementary material table interpolation suitable quickly also effective denoise g denoise approach regularization base minimize denoise mse free mse stein unbiased sure mse require ref monte e sure sure value ms divergence calculate sure accurate disadvantage sure complexity ref calculate tend fig mse mc sure illustrate continuous bound propose summarize snr soft maximize summarize top log equivalent set function regularize fig preserve group maximize snr compare soft thresholding snr
continue randomness certainly assume call feasible storage though network interpretation mention spherical perceptron present later concern view relate scenario seem important previous subsection clear conceptual similarity root combinatorial similarity essentially prove limit storage perceptron present capacity find feasibility course great exposition well uncorrelated spherical perceptron case study make place uncorrelated view uncorrelated stand become question interest assume entry linear course condition enough think dimension however large make writing easy keep normal recognize rewrite sign way mention word presentation normal mention assumption nothing topic rely extend conjecture let scalar small scalar ignore long problem essentially establish course rigorously confirm mention confirm conjecture theorem indicate feasibility curve storage
still distribute manifold condition q conditional distribution straightforwardly illustrative dominant left stand estimate previously accurately individually pay
option measurement alg estimator coordinate apply measurement produce accuracy present experimental result idea false positive top rank gaussian measurement excellent coordinate select lp decode top coordinate expect research project interesting use furthermore pareto random effort simplify inspire work projection projection improve generate stable exponential recent highly numerous estimator measurement baseline gaussian utilize stable experiment matlab reader reproduce failure length vector projection correlate improve science ny usa detect difference surveillance paper e sense cs recover gaussian design programming generate follow absolute
steady two eigenvector c small eigenvector guarantee connect social might importance form eigenvector modification graph unweighted weight eigenvector unweighted equivalent symmetric reweighted give reweighted degree diagonal whose eigenvector write reweighte surprising intuition equivalence process follow eigenvector centrality connecting capture newly epidemic edge pair amount scheme epidemic
turn rather mild fact fourier shift involve necessarily fourier matrix true load sample process interest recent development fourier efficiently subsample remarkably fourier perfectly recover true author consider image random projection objective use prove apply hypothesis match shift underlie
estimator mml distribute investigate robustness centralize distribute presence neighbor perturbation load respect model estimator use original perturb randomize centralized magnitude confirm robustness distribute comparison gain centralized runtime runtime respect solve centralized interior solver iterative selection runtime sequential generic estimator parallelization also however parallelization approximately four low high parallelization estimator continue almost core solve parallel without iterative message overhead concatenation mml framework graphical marginal maximization problem statistically consistent expensive centralized sufficient centralized improved exist illustrate sparse semidefinite set relaxed marginal neighborhood true therefore relax mml ml parameterize mml fisher sum square variance two subsequently argument neighborhood
people player adjust abstraction list main player application group application add several cover satisfactory spectrum intelligence generally adversarial infeasible pruning scenario player create collaborative express player agent player focus characteristic style act long substitute player matter replace player behavior divide activity way make action concerned predict action player game extent concerned two list attempt player location try player know attempt games instance valuable player position ignore resource try player knowledge human similar use one describe describe player knowledge example describe goal finish abstraction highlight say opponent ai separately explicit separate main source code implement attribute task identify cm online implicit implicit online substitution implicit explicit adversarial game game preference implicit substitution explicit game design online review complicated obvious chapter machine present apply define generic author discuss applicability chapter evaluate feasibility discussion regard player propose methodology able approach model preference position six phase define example thesis unsupervise discuss previously concern player aspect able obtain player knowledge instance game ai behavior scenario determinant must model task require regardless agent algorithm require different player preference behavior preference game platform chapter game game indicator match indicator resource game person game derive player player case indicator total demand indicator model player although essential stress availability vary among game type mainly extraction code modification extremely game attribute decision game ignore evolve preference need decide able previously player ml unsupervise player cluster identify example player classify thesis preference virtual agent player classifier perhaps preference simplify work concrete concept model predict level preference preference topic follow call multi believe main difficulty define preference know drawback strategy characteristic binary predict player belong regardless model reason multi would knowledge test far classifier chapter tackle generally sub area meta hence may apply alternative level computational sample valuable player resource feasibility thus discussion regard applicability requirement besides environment validate game ai briefly factorial generic action propose framework preference feature multiply variable determine ai knowledge adaptive ai previously
automatic point identically baseline demonstrate classifier cluster knn recognize label moreover begin filter correct still improvement misspecification generating variate gaussians model differ logistic label three model baseline robust assumption robust perhaps suggest model parameter regularize c c provide almost identically result explanation consistently occur change initialize subsampling achieve small ratio feature model amazon distant supervision co involve generate amount commonly distant supervision report annotation extract relation observe error could attribute annotated degradation quality automatic prevent perform fully despite aware moreover
encode regularization choose difference reconstruct signal since complex well example air conditioning temperature mass vary result air conditioning usage room need enter room aggregate dynamic loss since likely source air expect states encode difference operator subtract consumption example group smooth loss next
descent dimensionality require linear numerical pass stream structural neighbor randomly application front gradually stream must preserve distance structure consideration first non trivially linear gain compress lasso prove column sparsity mean long may seem first quite application consequence highlight qualitative arise work summary show qualitatively subspace numerical algebra square approximate leverage score canonical analysis regression point method many input solution svd show small yield linear rectangular matrix qualitatively reveal incoherence know work subspace incoherence incoherent constrain least square subject popular constraint xx accelerate subject show b ax one dense sparse suffice restrict show square sparsity oppose subspace range span euclidean x restrict rip isometry furthermore approximately approximately recover vector dx dx compress kn j incoherent sparsity rip yield incoherent qualitatively make sum include incoherence
view result combinatorial claim exist mp n mp mp program even maximum consist bipartite matrix cost assigning pt distance j j call assignment section briefly discuss criterion obtain risk restrictive could significantly reduce investigate study risk correspond average study former performance minimax directly hamming translate follow fulfil pack upper merely cardinality immediate getting appear difficult state nr nr nr b furthermore fulfil regime moderately minimax rate picture less regime small dimension bind factor unable claim
property allow optimize next keep optimum subproblem constraint satisfy augment must augment subproblem invoke ii monotonicity constraint index subproblem therefore respective pool subproblem constant since subproblem solution k examine lemma optimize optimize adjust monotonicity subproblem total optimize optimum apply subproblem subproblem optimize right subproblem subproblem property way case possibility theorem version solve subproblem solution iteratively theorem proceed label
repository methodology extreme sense partition dataset good extreme case competitive technique loss cm r letter metric normalize cut segmentation available color feature present fully label exhaustive adjust position one cut either learning improve matlab hour convex learn metric vs exhaustive difference test whose value cm meet gr search segmentation original gray learn metric unsupervise partitioning stream image partitioning trend segmentation e change co segmentation video video partition article represent partition section
constraint overcome difficulty transformation deep mlp propose add third transformation regular mlp second transformation propose really conduct design possible difference usually neural robust enable training robust effect two transformation center transform generalize context bayes genetic basic metropolis linear eqs often probability poor mixing long propose mix
dot label case usage red curve first finding trade usage performance significant degradation superior appropriate finding address unlabele drop rather velocity drop follow seem always performance degradation usage unlabele meanwhile improvement vary vocabulary phenomenon go label least accord visually show amount degradation increase pattern seem degradation whole procedure largely unlabeled classification selection well usually vocabulary number figure considerable vocabulary bias likely break
descent entropy prox initialize furthermore adaboost mirror classifier whereby adaboost adaboost bregman arise example variable mirror prox weak learner denote sequence mirror descent sequence normalize concern norm un classifier fw ta ta tw tw thus define clearly induction tw lemma log exponential variety via follow adaboost classifier adaboost weak follow bound separate separable intuitively theoretically
become weighted rescale unit activation free factor step rescale work previous work value well preliminary experiment unit work well unit example unit mix sum ensure standard deviation ascent minibatch stochastic ascent conditional multiply component gaussian multiply work broad one heuristic gaussians conditional arbitrary give mixture skewed tail could circumstance image datum laplacian variant mixture strong baseline task uci dataset
source surface medium wave scale fine scale qualitatively expression factorization avoid apply post post linearize adopt give observational forward distribution describe field build dimensional number aim ensure underlying additionally construct rank approximation inform explore high informative parameter space fact square pde ensures surely pose achieve carefully remain challenge infinite inverse computing discretize challenge govern solve space wave stem fact require pde take million discretize conventional monte develop method accelerate employ linearize linearization mean linearize maximize posteriori linearization posterior appropriately furthermore posterior evaluate forward uncertain parameter informative field hessian term operator sparse exploit construct dimension product amount adjoint pde solve formula express visualization posterior pde particular never store solving pde pde pde scalable wave scalable uncertain dimension record unknown parameter employ million work overview infinite dimensional follow present section dimensional describe wave inverse parameter inverse candidate contain give consistent present
truly sense control let back ask bold ahead theorem even requirement meet pg equivalent classifier behave uncorrelated totally figure replace relatively correlation like fail guarantee vote unless high south could misclassification error interesting binary
frobenius half reverse cross validation suggest keyword validation norm discriminant recently analyze dimension genetic datum distribute variable matrix curse dimensionality estimator therein covariance sense group thresholding eq tuning
receiver exchange information past small delay reach actor view result event communication twitter drawback traditional vector quadratic requirement maintain therefore necessary calculation view algorithm acyclic communication suitable topology structure bandwidth compute tailor formulation capture indirect communication almost small describe receive amount people meet never nan actor communication often actor update actor justify social communication system wide important distribute massive rather actor nature communication cognitive size two people meet likely relatively neither directly reach indirect vector update process indirect actually restriction substantially reduce memory million actor communication event far path refer indirect update ever take
introduce assumption borel lebesgue include e subset finite condition composite composite three composite score subset arbitrary neighbourhood continuously point neighbourhood vanish separable score bregman composite composite score practice integral dm gradient zero sake affine determine composite produce affine assumption assumption score produce affine assumption score older find imply imply characterize differentiable indeed separable bregman old show score old corresponding variable density concern density vector composite composite respect average regard loss score composite expectation empirical need probability composite express respect old invariant transformation bregman old solution composite provide estimator estimator score
projective closure since result l l l generic sum degree generic linear subspace call ed already ed generic ed play isotropic give explicit formula ed degree class thorough class combine projective variety isotropic ed equal inner apply matrix bundle normalize volume face follow explain ed third generic ed degree dual variety duality matrix formula formula derivation third write formula intermediate calculus slight ed sx qx dim ed degree variety coordinate bold ed unit ed ed wish thesis know ed degree ahead
return theorem moment condition comparison specialized specific instance still tail input confidence state distribution instance minimax boundedness replace subgaussian requirement become case regularize possibly hilbert factor classification expand section tail example subgaussian guarantee general tailed assume heavy tailed assumption implement metric banach geometric detailed comparison approximation question lastly give prediction one improper learning enough implement empirically simulate verify standard tail however restrict contrast appropriately determine improvement readily worth investigate great plan core achieve concentration demonstrate estimator explain generalization noisy motivate consider estimate heavily appear group think constant well group copy return pick chebyshev random within mean
sample number due limit generate certain criterion sum probably limit state finite variable p polynomial formalism consist pseudo compound independence variable gaussian variable indeed correlate ratio gaussian approximate error
discussion literature oracle slowly aware stage cover logistic easily multinomial effect covariate cover situation exactly retain structure possibly unknown set often conduct conditioning event uniformly poor sample p approximately purely possibly great exist exactly instead precisely suppose exist equation collect function combination standard analysis model yield typically give approximation term want mix basis mix partition log odds tx unknown basis x give place condition many implicitly example practice e enter part nonparametric covariate misspecification double robustness misspecification misspecification relevant error asymptotically allow salient grow depend treatment effect array asymptotic interpretation first tn think covariate upon proceed without covariate successive upon covariate think collect common purpose perform inference must conceptual treatment yield although model doubly robust selection sparsity q appropriately notice average although estimate upon randomness selection select set randomness add economic random lead regression vice versa randomness choice method first return parametric want chance additional randomness conversely covariate polynomial include reason vice versa additional randomness estimation exposition additional randomness case condition additionally ref bound underlie x tx nr mild dimensional application limitation naturally common array asymptotic condition selector restriction estimator label tx n tx tx
factor combination involve update case state monte ise relatively fig factor modify fig original much monte modify improve increase temperature graph site v ise
application p p always estimator covariance formal least positively square whose interest least give projection eq belong gram
transition trial portfolio etc initial policy adversary take observe round state policy similarly round define visit achieve regret variable learn
linear goal small may systematic one describe fitting model gmm achieve employ summarize sparsity projection sparsity j span square since unknown aggregate extended notion group fuse sparsity pattern aggregation inequality contribution result lead aggregation pattern suggest pattern prior define statement aggregate estimator eq aggregate probability aggregation additional condition weak oracle extend hold specifically absolute noticed et ball ball argue ball describe vector moreover surprisingly sparse approximated sparse fix convention proof appendix far main
daily fx returns daily fx time contain consist five return six price process eliminate price eliminate price time market series zero mean unit standard deviation usefulness vary predictive addition compare execution likelihood estimation finally study method return receive make log evaluated augment repeat repeat sequentially table
successful fusion neighborhood match direction ann ann leave pixel neighborhood overlap visual difference negligible snr favor truly since case ann image processing regression implement local filter compare sharp pass display visually fusion artificial observe appear mse fusion produce iterative version capture image gradually change towards one propose previous ct reference hand ann corresponding image neighborhood pixel correct ann pixel ann example conduct ann image compute perform namely image neighborhood radius estimation neighborhood overall neuron single neighborhood specific manual tuning display individual reconstruction figure absolute fusion higher compare texture close fusion correspond right additional figure fusion similar ann fusion contribute computational fusion reconstruction ann rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb
metric perfect support task analyze roc look distant graph variable angle curve irrelevant useful visual figure roc initial performance vision consider irrelevant classifier scoring perceptron mlp backpropagation possess generalization simplicity universal power necessity quick simulator network layer smoothness activation present balance training backpropagation measure error mse epoch failure break
iteration use smc intuition smc algorithm improvement sensitivity refer bandit intuitively arise give take unobserved bandit basis must check provide bandit vary cm circle mm draw node thick draw edge connect connect connect p connect p edge connect edge connect connect connect connect connect connect select accord ty fy due smc bandit evolve sample grow
state transition transition reward counter transition reward first task transition task perform decay visit specifie normalize initial determine average run negligible experiment three thing optimal drop td optimize domain constraint require tight might option implementation depict top reward discount stochastically agent
atom dependency ground atom respect safe restrict rule rule apply start prove twice go cyclic rule resolution eventually grind use instance evidence simple body atom atom rule contribute semantic hence omit process omit inactive encounter evidence conversely body contain keep atom give rise loop ground formula I become disadvantage actual correctness specify show program alarm rule inactive negative approach cyclic program cause influence friend p evidence ground evidence third inactive truth head rule truth value discuss convert rule ground program boolean logic equivalent purely formula interpret semantic rule semantics alarm acyclic alarm rule alarm rule alarm e alarm go completion three clause alarm last reflect boolean coincide complicated loop positively recursive rule presence loop completion correct formula simplify focus example person completion rule model stress stress program positive loop develop lp highly unable repeat discuss formula generate set differ formula rule rule result loop involve auxiliary rule loop convert write construct proof case atom collect recursive nested try loop loop operate break loop boolean formula return sense formula denote use boolean formula simplify ground formula lemma stress stress stress though avoid say start influence person grind break surface
represents skew parsimonious increasingly tool commonly base mixture see extensive gaussian mixture base past lin lee lin elliptical mixture mixture skew
resolution order j modify coefficient resolution detect imply work consequently respect thresholding estimator standard probability artificial spike sense refined term lower bind simultaneous adaptation soft thresholding soft thresholding procedure empirical coefficient well effect lead detail smoothness soft coefficient must n j l suboptimal soft thresholding truncation unclear extend proof heavily occur
average average calibration realization signal eq since process noise calibration average without know positive help get signal accurately want perform retrieve zero close information analyze guess rough average datum sensitive available note square signal bias calibration attempt solution increase initially phase gain consistent bias reliable average sensitive calibration instability slightly investigation switch done express conditionally assume necessarily happen regard unique non measurement g never illustrative read want signal condition ref minimize datum bar denote define
expand provide work structure place propose formulation available instead new stacking block reweighte reveal inspired insight reweighte signal signal reweighted function simplicity case weight pre series outperform considerable coefficient already versa assign small negligible iterative reweighted discuss section establish mechanism coefficient relate mind slightly rule reweighted follow unlike coefficient dependent neighboring couple neighbor minimization encourage reweighte considerably improve result conventional reweighted recovering serve experiment bayesian
research mathematic would thank helpful cm mm prop prop prop prop prop prop remark present functional process expansion brownian apply second expansion power rely martingale embed calculus stable theorem keyword observation power classification expansion investigate vast I weakly book comprehensive asymptotic expansion mainly deal expansion datum asymptotic consecutive converge span remain mixed normal typical year lot research functional brownian term remain asymptotic sde second combine expansion variation deduce asymptotic variation probably section devoted derivation theorem functional apply calculus central introduce denote shorthand th partial resp resp variable moment denote resp stand probability introduce notion calculus book exposition calculus set product derivative unbounded adjoint integral respect
node fix space denote dissimilarity reduce dissimilarity dissimilarity dissimilarity map since substitute admissible cluster method establish admissible theorem suggest uniqueness network symmetric ultrametric cluster linkage direct linkage extension linkage asymmetric back framework study start quasi equivalence build towards quasi quasi direct single linkage admissible quasi stability single linkage construction modify axiom among axiom axiom node resolution dissimilarity cluster long axiom axiom ultrametric p replace say admissible axiom axiom theorem compatible axiom fig whereas require output ultrametric axiom qp network ultrametric hierarchical ultrametric separation coincide surprising axiom axiom value impose identical loop alternative imply regular vice versa alternative cluster form influence consistent study admissible play role p study admissible particular hold axiom axiom property see follow network cluster conjunction translate requirement clustering prevent node rest development move axiom bind method axiom dissimilarity notice operation operation ultrametric nu x establish ultrametric I apply equivalent linkage ultrametric properly ultrametric ultrametric output operation algebra satisfy axiom define axiom see infinite reciprocal admissible follow axiom satisfy axiom axiom transformation inherently satisfy regular axiom axiom far symmetric consequence equivalent comparison statement corollary observe reciprocal fact reciprocal single restrict symmetric consistent axiom alternative axiom axiom dissimilarity axiom clustered resolution axiom cluster respect issue axiom agnostic axiom ultrametric eq since cluster original axiom axiom admissible agnostic axiom axiom paper method admissible explain axiom arbitrary denote reciprocal asymmetric axiom output cluster reciprocal linkage compact isometry endow hausdorff study analysis properly define hierarchical ask analogy reciprocal reciprocal early discuss network capture exist write point denote consider q map map hausdorff network correspondence transformation correspondence element pair conversely think network correspondence close possible eq correspondence distance hausdorff dissimilarity metric ask relaxation case network metric follow entail endowed distance network metric space properly metric restriction implement permit study output hierarchical distance distance original stability cluster stable coincide lipschitz include network define relationship consequence idea process stable notice stability hierarchical quasi section ultrametric ultrametric asymmetric begin show stability direct linkage direct linkage quasi define pick symmetry obtain combine true conclude proof reciprocal reciprocal outcome stable use theorem similarly may reasoning state illustrate conceptual step yx n output apply network go compare similarly stable reciprocal clustering semi moreover behave reciprocal sufficient state reciprocal output stable show reciprocal expand distance network particular nearby yield nearby important noisy dissimilarity ensure effect admissible axiom stable admissible method introduce turn network since top active xx dendrogram fig u yy active u yy yy network output correspondence correspondence general stable sense instability output arise due switch reciprocal imply proof theorem moreover method stable nevertheless method avoid quasi united states year year dc publish census dc census dissimilarity decrease normalization probability transform dissimilarity dissimilarity focus attention flow rather percentage hierarchical extensively white singleton cluster resolution proximity cluster resolution resolution outcome result dendrogram fig partition obtain four mark cluster together partition flow resolution consideration correlate proximity california neighboring singleton form pair join three york resolution york york resolution formation explain fact share respective area city york city york state correspond occur frequently suggest cluster form reciprocal dendrogram neighbor multiple resolution show join new appear cluster form york closeness formation make california anomaly four state incoming country united states york california country proportion incoming state neighbor neighboring california neighboring mechanic lack opposite california york form neighbor
advantage advantage flexibility policy policy gradient however calculate controller allow roll costly sample policy reliable flexible current section extend manner collect current consider parameter let gradient q systematically resolve problem importance weight call iw analyze iw consider gradient dimensionality upper inverse covariance inverse bound factor plain significantly take iw term therefore iw iw effective cope variance gradient iw normalize indeed trade biased promising technique iw baseline still constant determine minimize iw minimize theorem iw iw denote ph analytic expression
approach nonlinear autoregressive implement gaussian process process identification se maximize model complexity process rely adapt arbitrarily uncertainty present prediction instance operate datum report adaptive strategy may reason lack data present obtain ability uncertainty easy pre processing stage ac uk new nonlinear autoregressive model filter sparse process integrate pre goes process tune maximize probabilistic
adversarial pick incur unlike adversarial play loss play player reveal play special call action I call system namely player measure action expectation randomization loss either past action adversarial system moreover begin step viewpoint two inform observation system make observation system adopt theoretic interpretation observation arcs pair arc loop create ignore equivalently equal
background norm operator kronecker exposition closely relate outline space function hilbert space infinite gp map compact bilinear admit give norm dimension apply task recommender system hilbert norm frobenius regularize optimize regularize represent employ kernel row norm compute approximation basis transformation basis square trace trace variational approximation variational regularization factor factor difficult select claim dimension f factor model matrix prove hierarchical literature generate low non instead low constrain trace inference restrict process interest matrix index mn mn covariance determinant consequence interestingly covariance collect regularize square identity replace regularizer lead mean function
tight every metric distinct arbitrarily know achieve unbounde subgaussian measure obviously gaussian generally subgaussian inequality infinite improve free address issue lipschitz formalize weakly tool weakly range note yield contrast everywhere statement ease quantitative extension unbounded
open detection detect tight community network formalize dense subgraph realization os enyi vertice subgraph vertex regime enough sparse regime derive theoretic substantially technical method besides scan test triangle sharp able characterize arise keyword subgraph os enyi plant large connect detection refer circle friend co occurrence increase internet old science aspect closely partition survey paper detection mean something extract community undirected generality adjacency mean symmetric nan realization os connection independent identically pair node imply subset paper nan let anomalous subset versus consider subgraph small tend zero relevance somewhat statistic similar modularity evaluate detecting clique address previous direct plant clique edge neighborhood graph process show reduce close spirit study testing reader thorough introduction subject otherwise case nan anomalous test asymptotically powerful resp often practically speak asymptotically substantially adjacency hypothesis merge two hypothesis
supervise attain wherein supervision specification mixture broadly applicable comparison specie simulate keyword analysis finite fractional supervision likelihood broadly concerned partition allow likelihood middle herein model fix adopt illustrate easily remainder organize follow classification review theory approach specie suggestion future classification comprise mixture convex component proportion n j reformulate basis observe em plus miss practice subsequently cycle maximization ease calculate
necessarily minimize rather partial iteration recurrent problem approximately combine recurrent net recurrent never sample mp approximate inference mlp average include exclude take geometric contribution average mlp divided trick model field contribute multi field iteration main way recurrent way also build graphical action recurrent predict
result even use network costly web training improvement magnitude pointwise product gpu address challenge vision aim great complexity require turn order advantage benchmark thousand current million bring new environment
treat correlated cycle reconstruct phase fast bring subtle phase seven appendix note subtle e well near elimination period trajectory mode control dynamic periodic space suggest behaviour dynamic note periodic transform preserve information comparison curve fig colour code colour g trajectory mode plot correspond code demonstrate power subtle difference behaviour fig display enhance slow agreement show subtle difference fine portion low corner normalise one within identify region statistically determine location significant pdf difference indicate solid box movie prefer middle side profile preferred movement difference figure importantly hz smoothing fine tuning necessary note finding
integration temporal trajectory odd integration observation draw adopt assimilation distinction normal residual equal residual error normal validity analytic neither localization plain set baseline various enhance investigation scope note adopt lt issue adopt update constrained bound coefficient later note background change value general assimilation
interest impose topology bound arbitrary diffusion binary purpose failure similarly status link failure choose time neighboring agent random satisfied adjusting realization ki k underlie topology weight strategy topology assume combination coefficient arise source randomness topology adaptive arise allow drop phenomenon cause interference policy use save resource bandwidth source relate allow link satisfied probable realization index relate random neighborhood neighborhood network neighborhood asynchronous k event assume positive certain combination coefficient matrix stochastic matrix leave stochastic meaning nonnegative observe desire useful asynchronous uncorrelated size uncorrelate uncorrelated
implement learner choosing compute crucial variant column force translate add layer variant worse suffer overfitte may layer intermediate layer product inferior explain note first represent polynomial remark svd layer randomize exact svd expensive experimental theoretically experimentally collaborative intelligence ci remark input architecture compactly goal derivation sense decrease every iteration eventually present implementation well preliminary deep architecture architecture development machine network go transformation concept consider suitable task system extensively study early success eventually extent architecture linear represent algorithm function compute input large node turn learner stage analyze etc third variant enjoy guarantee generalize flexible practice recall architecture ignore capable may seem specify care vector identify
risk performance dual program thank universit paris anonymous remark theorem eq assume another another also offline phase successfully reduce identical pde approximation project original discretized onto well basis application rather solution functional orient try dynamical basis mode accurately interest output showed adapt great reduction paper computable reduce consider output evaluate reduce output correction dual correction
apply consider paper design mind matrix straightforward focus machine variety desirable difficulty decade aim accelerate low complexity step global convergence global scheme practical proximal newton method hessian state specialize hessian computation limit hessian coordinate subproblem elaborate bit paper discuss propose describe refer reason operator involve ensure consider prox quadratic search prox extend sufficient trust method proximal prox decrease improvement enable show recently exact approximation inexact proximal approximation positive hessian accept decrease flexibility algorithm finally complexity thus quasi subproblem extend
stay zero add computation next make positive work hessian low bfgs capture curvature help fast symmetric assume satisfy obtain eq bfgs often memory bfgs augment iteration become inefficient limited setting maintain correction pair remove add newly exceed modify change maintain also basic effective ensure line update become collection inverse compute would coordinate subproblem minimization
bp ar hmm clarity var likewise compactly index distribution I weight define conditionally pt var process behavior result bp hmm space I e one think bp ar spatio comprise continuous markovian specification realization draw vector transition ii straightforwardly independent place var produce binary feature emission conditionally parameter sampler view assignment ar hmms efficiently likelihood second compute joint configuration number active merge drive birth behavior likely method change seven distinct auxiliary bp hyperparameter hmm hyperparameter propose birth death joint merge configuration tractable discard propagate mcmc detailed remainder move article include expensive require run forward backward I routine number behavior costly step sequence require birth death basically involve sample merge primarily resample sequence sophisticated section mean need reasonable feature hmms computation parallelization scheme beta ibp parameter ibp ibp customers customer subsequent probability proportional customer derivation ibp beta supplement sampling denote indicator exclude behavior series share index ibp else present procedure entry cover emission indicator ibp denote exchangeability ibp bp sampling indicator
various clique core see enumeration matrix know paper base path forest rely enumeration possible forest call forest node part tree belong part many forest high boltzmann countable forest adopt physics framework cost forest occur physics parameter forest cost forest negligible contribution cost forest uniform roughly speak forest expectation forest speak index take derivative inverting depend assign arc survey
computational seed step cost computational deal balance index define reduce well overcome procedure highly scalable sized seed formulate seed optimally computationally tractable seed match matching seed subroutine ideally would want pick seed set ideally column seed matrix would achieve logarithmic seed offer insight set non seed maximally distinguish mathematically translate seed match across fix adjacency maximally distinguish seek seed contain seed adjacency repeatedly maximize across entropy inactive active entropy binary seed seed shannon vertex give choose seed natural detail centroid mild assignment surely block integer position row union disjoint set respective ie lastly success sbm graph may alignment induce say match graph indicator moment construct sbm independent adjacent success adjacent correlate vertex
constraint add
perhaps particularly train match boost plain plot relative train consistent boost descent sign test find boost several perhaps add systematic efficacy dropout remarkably proxy intractable geometric sub compare traditionally classification performance share member significant effect analogously ensemble share efficacy employ activation function quality evaluate geometric exactly geometric tie ensemble tie novel involve
reliable see considerably begin useful method see conv l svm k conv bit reliable svm var conv compare efficiency test highest inefficient plot compare envelope reliable reliable envelope mis mis plot plot envelope help reliability prediction good nominal nominal equal conv failure conv equal also reliable conv figure provide figure conv failure concrete conv effective precise conv concrete band envelope tight conv band var dataset mis model conv rank third var conv give ranking summarize row dedicated summarize inter table fourth upper cm fix k var var var conv var concrete var var var var introduce comparison schema nine benchmark contain predictor asymptotically suffer quantile side prediction effective drawback limit conventional quantile regression prediction limit model proportion identically distribute space predict concern model promise idea predictive support machine side prediction model appendix predictive commonly term error risk risk let square error square hence minimize estimator loo see approximately n constant estimator independent distribution fx tend large variance non tolerance interval side level proportion note need side confidence must sided merge obtain confidence denote proportion upper must proportion quantile statement two interval describe q mention assumption interval eq uncertainty equation mean tolerance bias guarantee confidence contain prediction response estimate tolerance desire conditional error center proof proposition definition guarantee conditional predictor present interval build tolerance interval prediction error query predictive give yield interval reliable interval interval analysis science use problem financial forecasting regression variable random different kind technique characteristic square technique design assumption kind estimate response estimate parametric parametric model regression build make currently context interval application finding confidence least desire proportion environmental exposure trajectory prediction security category estimate technique estimation gaussian quantile outlier suffer slow quantile cross least regression technique interval drawback point paragraph interested finding interval confidence point conditional variable tolerance confidence concept interval tolerance model side easily extend sided interval test interval context efficiency obtain variable bias validation local validate predictive interval capacity provide side predictive interval reliability envelope evaluation chapter interval less envelope almost envelope base find guarantee predictor value simultaneous interval state similar high contribution look simultaneous difference local instead test comparison follow tolerance describe state art efficiency interval propose describe tolerance interval find chapter compare square quantile dataset conclude remark sample random find denote underlie regression represent estimated notation training set regression variable predictor regression true variance point response coverage tolerance response content coverage tolerance distribution content predictive distribution predictive prediction quantile degree freedom estimate regression statistical independent pair follow define q mutually variable function usual everywhere idea finding minimize find eq risk estimator detail risk local approximate neighborhood locally neighborhood evaluate fit covariate change regression taylor expansion polynomial write vector weighted kernel estimate tx kernel observation close point big weight bandwidth scalar select scale q author include take distance near neighbor neighbor popular technique choose observation consume common bandwidth plug unknown quantity estimate section plug regression bandwidth weight weight predictor kernel parametric function vector locally problem convert weight interval guarantee confidence specify proportion tolerance mean deviation quantile chi distribution quantile error inter quantile variance use variable dependent yx show estimation estimation tolerance introduce coverage tolerance denote predictor side tolerance content factor express interval regression separately detail see parametric tolerance response regression work address parametric distribution unknown deal finite need inference review square use find besides address method learn explain application drawback paragraph emphasize method method obtain regression tolerance interval build eq certain find interval average deal omit regression interval model instead prediction test square extensive topic square sample tolerance simultaneous study interval quantile square technique x give fold assume unbiased independent finite size statement appendix prediction drawback band tolerance bias interval treat tolerance least square part level mean interval function desire cb create envelope simultaneously part response distribution describe q detailed difference prediction contain least proportion coverage create envelope contain envelope coverage describe quantile side two interval treat separately side sided least sided level far similar tolerance build quantile regression quantile quantile estimation sided interval interval level pair quantile side interval build must side quantile merge interval statement probability combination side square explain overlap side enforce dataset prediction need purpose building plot far compare effectiveness interval note content contain train small medium dataset schema mean inclusion fraction content mis sample deviation prediction sample reliable definition measure content mis mis mis mis mis mis bn mis choose mis normalize mis equal mis value compare reliable mis complicated mis prediction yield mis behind find equivalent predict see interval average obtain correctly response variable gaussian inter quantile q trade sake find final wide envelope mis mean interval prediction visualize big dedicated dataset compare whereas chart compare method across nominal distinct proportion difference compare nominal represent proportion represent represent nominal line value nominal represent reliability obvious failure reliable higher desire obtain respective nominal precise line reliable mean label normalized looking efficiency line inefficient ignore mis represent mis look figure obtain envelope model envelope satisfy constraint normalize mis mis chart chart goal mention response value locate predict interval display contain mis chart mis different gaussian deviation chart display concept predictive interval interval introduce rating measure frequentist interpretation interval sample level tolerance interval quantile mean interval proportion pearson independent content tolerance sample regression confidence induce include therefore confidence interpret chapter frequentist viewpoint true response variable obtain interval confidence interval regression quantile obtain interval true unknown concept query desire proportion long measure predictor interval interval content goal content predictive predictive interval change regression predictor verify entire distinct huge impossible regression make state begin variable approximate normal related definition observation v dataset result deduce formally sufficiently include predictive interval average simultaneous tolerance schema usually approximate level prediction content interval reject quantile reject nan hypothesis predictive significance test simplicity response inside content interval simultaneously proportion build express content content proportion prediction prediction interval training include arbitrary training tuning convert hyper parameter take least method predict give tolerance hyper index denote parameter content divide find also pass result satisfie constraint one fold hyper small mse find small mis space parameter divide tune hyper tune mis model cross give advantage exist package aforementione remain hyper model know content upon sake brevity part interval interval quantile consider construct knn hyper knn confidence coverage interval regression hyper knn svm regression small high guarantee interval begin like decrease interval size long constraint leave influence interval obtain tolerance regression assume mean error locally find properly bias tolerance leave fold error regression bandwidth order interval bandwidth tolerance interval bandwidth variable optimal non consist balance bias vanish sample tolerance interval far find desire proportion find tolerance interval response add add remove response tolerance confidence part context tolerance interval tolerance condition deviation formally let query bandwidth local minimize satisfie condition almost include neighborhood usually regression neighborhood next reader regression satisfie bias variance content tolerance tolerance appendix prediction error variance therefore biased estimator advantage tolerance prediction approximated tolerance error inside represent without note become residual neighbor content tolerance interval compute replace propose take near variable tune depend tolerance cross whole summarize obtain interval ix equation hyper tune interval complexity interval instance evaluation local neighborhood inside appropriately take bandwidth coherent satisfied bandwidth bit great bandwidth behind bandwidth vector begin coverage interval interval interval add iterative uncertainty assumption match yield partially tolerance decrease increase increase contrary interval size choose value coverage interval desire much near number instance tolerance interval min return min kk increase contaminated prediction practice small belong subspace different error serve restrict region usually regression regression neighborhood give small tolerance begin everything calculation phase describe local confidence tolerance obtain predictive provide predictive finding predictive tuning constraint interval interval hyper regression hyper serve hyper respectively interval variable predictive paragraph review application regression method polynomial point near bandwidth refer first interval bandwidth interval instead optimization tune high neighborhood pair great select fold tuning define mean mis neighborhood big tolerance high coverage increase interval little bit begins usually evaluate knowledge tuning first find tune come way find try decrease interval goal size inclusion fix precede decrease inclusion compute local density response dense hyper repeat iteration min max min min mis mis min min fold schema equation tolerance previous compute interval upon capacity content model compare reliability envelope describe variable selection outlier preprocessing organize five part describe part interval discussion nine dataset list find double mention systematically remove dataset distinct dataset name total variable contain concrete compressive strength concrete cpu cpu describe interval programming language specific hyper hyper name linear list method side predictive neighborhood var side explain sided package sided interval regression quantile function type percentile se interval explain quantile hyper cv argument radial set default radial cv non parametric tune cv small mis satisfy radial basis radial basis dataset tuning constraint l svm use sigma sigma conv conventional prediction use near neighbor fold validation training non list must tune var may different mention l svm conv concrete auto cpu attempt tune hyper serve
remove solution c satisfy guarantee iteration remove residual hold satisfied unit sphere nice follow algorithm f iteration iteration lemma remove ia b ic update formula expand term conclude lemma prove therefore q since I c step part directly tensor change ta claim deal difference sum frobenius equal therefore ic ic show true replace true frobenius bind w term product replace leave hand frobenius frobenius normalize version last inequality tensor applying provide vector bind chi q generate tuple guarantee potentially generate bad tuple provide specific output key time without assume case choose work let tensor w I denote hold definition corollary norm bound enough know j suppose initialization correspond tt exploit part prove succeed output center close compute previously find tuple tuple satisfy w jj already tuple condition algorithm rgb corollary criterion derivation observation time bold bold paper provide cp adapt tensor establish third dimension tensor incoherent recover overcomplete also arbitrary initialize vector tensor slice guarantee tensor tight perturbation give overcomplete decomposition popular unsupervised range mixture community certain fourth order sample tensor source separation topic case tensor order exist stochastic guarantee orthogonal solve orthogonal eigen decomposition symmetric second step limitation practice learn especially whiten whiten expensive suffer instability lastly unable overcomplete number orthogonality limit popularity overcomplete many domain decomposition tensor mode alternate method extremely fast calculate global guarantee modify alternate method make update mode update suffer ill whitening approach presence incoherent soft orthogonality incoherent representation extensively literature number context compress incoherent flexible modeling overcomplete signal parameter construct feature tensor incoherent establish guarantee subsequent work various learning cover cp alternate alternate power asymmetric tensor mode project update local incoherent due incoherence update even noiseless set approximation overcomplete incoherence remove run remove approximation consistently overcomplete slice initialization alternating lead global tensor guarantee tensor rank tensor mode overcomplete tensor tensor arise low transform simultaneous prove consider tensor slice update greedy rank perhaps natural cp decomposition lead recovery tensor np obstacle limit tensor incoherent error alternate update decay end random perturbation tight contraction also decomposition tensor square guarantee update orthogonal analyze extend tensor whiten early whiten lead require overcomplete overcomplete tensor challenge identifiable general factor matrix however dimension procedure limitation assume generic time procedure overcomplete tensor procedure recover overcomplete tensor high instance tensor utilize slice mode level additional dimension obtain fourth order slice simultaneous perturbation provide ica slice overcomplete consider overcomplete sparse tucker specifically identifiable tucker decomposition weak guarantee significantly one consider fall minimization provide convergence completion phase code involve notice asymptotic sometimes clarity notation factor convenience identify way pi limit tensor instance mode index arrange row tensor mode mode similarly slice fix tensor slice rd mode r arrange multilinear matrix tm vector eq multilinear multilinear slice write notation represent outer product cp multilinear particular column adjust appropriately denote spectral tensor scale style fill corner sep c initialization svd base power power descent width c line width corner leave algorithm dash corner width fit label draw corner width pt leave alternate guarantee recover alternate coordinate one procedure power asymmetric mode tv w update formula multilinear tensor mode alternate different later relation approximation denote dimensional sphere program multiple local show update optimization vector alternate approach converge initialization iteration unit option option initialization asymmetric multilinear member numerator formula orthogonality orthogonal second power lead incoherence encourage soft orthogonality iteration propose residual discuss tensor initialization true rank random initialization draw unit sphere option right procedure vector initialization initialization know one identify initialization successful cluster procedure appendix standard singular vector update tuple k update starting denote tuple previous power recover algorithm remove residual mainly descent unlike iteration immediately stationary inspire residual analysis step matrix satisfy bind column bound satisfied c update I procedure I advance store multilinear computed multilinear sample method mm apply tensor linearity variable quadratic update orthogonality alternate view least square unnormalize rewrite rao column simultaneously update current vector column efficient complexity show procedure throughout assume perturbation unit vector goal recover rank challenge overcomplete regime dimension loss w deterministic tensor convergence show satisfied uniformly simplicity assume main part also reasonable assume deterministic assumption matrix impose soft w upper ambiguity recover tensor unchanged vector mode sign vector mode vector convex tensor decomposition tensor uniformly appendix perturbation satisfy q weight bind initialization vector addition update formula convergence tensor decomposition algorithm output matrix appendix overcomplete incoherent assumption assume simplicity many derive local also interpret identifiability incoherent frobenius whole residual term two main removal guarantee step convergence tensor result reason provide column guarantee initialization rest regime residual state explicitly hold recover tensor cp propose main change adapt tensor change change slight modification consider tensor tensor power remove order tensor brevity guarantee rd tensor cp recover th mode modify change tensor setting decomposition constant addition step satisfy h number initialization exploit svd propose slice tensor guarantee initialization guarantee mm number w svd initializations unit rank arbitrary tensor setting efficiently decomposition overcomplete arbitrary constant simple svd base argument adapt lead convergence tensor power overcomplete overcomplete mixture moment high low label initialization svd initialize update generalize set overcomplete component overcomplete component addition q theorem proof mode mode arbitrarily overcomplete global employ svd contraction power remove residual break incoherence precise convergence enough estimate contraction incoherence crucial establish part number satisfy close argument analyze dominant bind experiment consider experiment three aggregated initialization generate I initialization vector formula option criterion iteration improvement stop accord size tensor initialization illustrate average depict versus initialization horizontal plot easy column setting linear need initialization almost recover start initialization ahead hard rate initialization ratio average run threshold several eq corresponding estimate relative estimate stop output increase computation recover overcomplete hard descent remove error iteration see initialization nearly rate svd computation expensive desirable theoretical initialization appear experiment additional room improve guarantee c avg avg e acknowledgement discussion alternate early use lemma support microsoft fellowship award award award nsf award nf award nf column remove rao denote norm norm text matrix randomly randomness decomposition unit sphere vector component incoherent incoherence spectral condition norm constant rank tensor bound eq weight factor define particular initialization norm many choice discussion provide brief condition normalize remove issue additionally incoherence setting impose norm tensor w h draw sphere limit provide bind initialization ensure iteration define contraction initialization local tensor spectral seem even condition imply lemma go set matrix satisfy incoherent uniformly unit sphere incoherence high understand matrix tool unit eq hold notice proof incoherence unit incoherence incoherence disk index particular entry inequality use outside last old old proof apply h old application cauchy unit sphere eq unit exploit overcomplete property matrix spectral bound know first triangle cauchy exploit follow independent gaussian eq bound complete union rao step remove rao product norm rao product satisfy idea matrix bernstein prove concentration symmetric
complexity add definition corollary learn computationally protocol efficient agnostic learning accuracy acknowledgement grateful first drawing discussion regard threshold bound interval thank suggestion contract nsf grant google notation convention let type random take follow entropy universe hold equal boolean shannon information sequence leibler entropy jointly recall characterization mutual divergence jointly support say bayes definition exist universe recall principle input uniform computing error communication follow deduce deduce technique distinct refer representation let representation margin boundary setting represent represent representation therefore satisfied ordering value obtain strict function dd mistake tree complete mistake depth inductive leaf choose free property restrict identical threshold interval mistake let mistake replace replace leaf depth depth df z jx v x jt binary min pair ax ax eq impossible send divergence hand identical eq output contradict joint distribution view point occur line intersect formalize pm let hypothesis class quality output follow differentially finish small select independently accuracy call mechanism use affect one privacy consider hypothesis mechanism private independence hx observe multiplicative chernoff similarly fx hold hold hx fx least part work universit paris paris mail analyze private notion privacy introduce provide agnostic study work start main differentially private randomized communication evaluation characterize mistake mistake pure private sample complexity differentially pac open know build communication complexity training individual sensitive medical therefore preserve increasingly mine pac agnostic private guarantee individual observe private formal definition achieve amount computation differential privacy agnostic study show differentially concept denote whether gap class output point bound non proper infinite show either distributional require label domain exist hx private pac representation dimension natural version surprisingly prove characterize differentially private pac c relaxed differential privacy use sample complexity private proper relaxed version leave open question proper resolve establish definition ingredient term function message bit use coin bit public coin share need least definition problem clarity omit c function discretized correspond communication view integer b combine whose dimension yet pac imply result communication sometimes give proof bind relationship private coin additional via prove first study communication complexity line plane complexity class finally pac differential privacy imply privacy separation concept substantially essentially thing know communication notion specific privacy extensive privacy machine relate area survey convert include majority formal differentially private seminal aside already differentially private show learn polynomial point commonly set specific learner learning relaxation know studied denote equal disagreement also refer duality packing cover notion characterize differentially pac agnostic private differentially pac weak differentially privacy imply learner dimension private logarithmic query database synthetic fx sufficient low sample proper agnostic good error principle tell function mistake label label concept subtree remark mistake include leave define mistake tree characterize small mistake online mistake bind relate communication complexity learn class input concept result concept public coin let output h boolean prove minimax principle imply induce every protocol public know know fx protocol event contain hx fx g cf deduce communication also private coin protocol majority communication function hold average one bad fx von deterministic suppose function exist hx fx c strategy player payoff minimax exist coin protocol use private output protocol private learning simplify bt I follow view integer note bt bx communication lower correspond distributional complexity distributional way great use public private coin simplify presentation eq equivalence theorem weak private coin theorem communication private pac learner know use hashing convert learn function result resemble learner provide private protocol probabilistic efficiently learn equivalence pac error notably boost private constant one convert simple repetition protocol back probabilistic show bound complexity differentially concept proof complexity index get get explicitly prove way communication adapt concept complete path root
sampling base strategy simply sampling correspond monte single unlike original need step sample collect bad case mcmc small efficient collecting already early operator experiment empirically whether computationally realistic discuss eq consider factorial differ interpretation early show deep special training naturally lead train contribution conditional combine randomly joint ensemble possible model one fraction step kind uniform sampling subset independently mix precisely choose variable obtain gibbs deep would mix slowly gain consecutive reduce chain provide adopt distribution resample particular visible low transition gradually reduce high mix initial clearly plausible mode step fast visible consecutive digit start another digit slow sampling sample consecutive anneal sample use successive sample realization digit collect investigate propose noise run chain chain increase importantly none chain able however generate sample quantitative instead compute take sampling take get autocorrelation successive starting collect collect autocorrelation speedup factor discount accordingly procedure conclusion really conditional compatible deep multiply computing cost visible markov cost run predictor unless variable trade computation control validate permit trade gradually low step acknowledgment thank frank com op universit neural autoregressive recently show successful modeling multimodal rely variant deeply train unfortunately deep corresponding neural predicting give connection train markov associate way compute sample speedup account consecutive reduce achieve use combine mixing sample thank step low unsupervised representation learn learn great success instance claim art recognition deep convolutional network supervise deep unsupervised counterpart challenge popular model model autoregressive early work model binary network computation connect graphical distribution pixel consequently exact since replace distribution propose yet variant deep neural modify effectively train unsupervised learn autoencoder autoencoder estimator autoencoder continuous unlike aim stationary generating distribution able possible boltzmann approach multi mp instance label special achieve state dataset agnostic cast theoretical explanation allow alternative deep mcmc rather describe description connection sec novel sampling call inspire empirically effect deep deep dimensionality q predefine index index order hide eq weight bias logistic sigmoid nonlinear activation train maximize denote issue original predefine limit capability model ask infer conditional eq visible intractable marginalization build architecture lot share weight parameter layer front predict one sharing unit sum sum plus associate unit available predict procedure train factorial objective intractable involve factorial instead variable regular feedforward firstly accord index set provide input length secondly solve issue original optimize suffer infer lack predefine ordering make regardless depth neural train simply usual training procedure generative network distribution although tractable transition operator chain mcmc whole directly jointly operator argue easier simple mode mixture mcmc powerful parametrization special auto corruption denoise autoencoder parameterize corruption stationary ensure ergodicity word match albeit transition suggest possible observation operator quickly find distribution start random reconstruction encourage burn sampler model explicitly chain find plausible order agnostic procedure sample
absence reliable rarely visit medical typical indicate signature patient stay home infected computer trend positive negative underlie rarely fully subgraph infected node set epidemic epidemic source phase epidemic nature patient hide connect initially patient algorithmic statistical challenge pose consider instant inform subset exact reporting unable report snapshot statistical determine epidemic possibly initial rather infected seek scalable minimal assume neighbourhood need infection epidemic member social network spread share technology medium well possibility medium friend website decision make local furthermore apply negative positive find contribution review extension node decision distinguish alternative arbitrary infected infection accord edge infect infection infect node reporting node represent positive reporting process illustration report infection probability report scenario node fraction ultimately infect probability report false go setting diagnosis work school epidemic distant neighbor independently like epidemic mass medium affect independently illustration epidemic report infection zero infected fraction two truly report denote positive report report easier truly report reporting describe report report report report epidemic report hypothesis visually predictive forward first work focus epidemic estimating topology infection exceed give predict future focus inverse decide question attention transmission mcmc question epidemic source closely model seek completely rely tool topology infection may handle number seed insensitive hundred spread come inherently essentially count node many high level triangle parallel significantly knowledge something compare computational running scale report infect control pose negative statistically resemble second infect large graph play role statistically contact complete two indistinguishable algorithmic regime infection minimal address contribution number reporting require minimal know local performance give epidemic particular succeed infected fraction node infection succeed experiment real world theory easily epidemic denote near indicator indicator radius indicator enough report immediate neighborhood equivalent correspondence indicator evaluate call intuition epidemic infection report infect classify infection cause precisely algorithm epidemic report scenario counter counter epidemic reporting depend near neighbor theorem make choice require positive detail appendix topology analytically hand suggest choose graph information encode contrast report reporting node parsimonious reconstruct knowledge report rather reliable compare similarly memory therefore scalable increase noisy choice topological constraint converge report irrespective truly report positive great every reporting contain report show correctness proceed boundary infect alternative infect infected boundary infect alternatively infect every interior node nod local infect difficulty increase set number false reporting show regime converge truly report succeed epidemic scenario significantly indicator report random graph key technical proof sum appropriately sum node truly report reporting exist kf constant depend problem infection regime succeed negative positive truly infect report detail defer appendix infection infect node condition satisfied reporting goes truly report correctly converge like epidemic report report report report nn reporting epidemic interior non truly report kf key evaluate hoeffding corollary structured correlation decay pattern graph variable indicator independent environment disjoint local also disjoint environment maximal adapt concentration epidemic scenario great interior show appendix corollary explicit include statistical substitute corresponding grid q function difficult solve apply correspond long local contain infect requirement fail mesh ball number ball indeed report mesh case constant even epidemic infection total reporting truly high infected neighboring pair indicator positively correlate uniform reporting indicator I probability set n epidemic apply type ii tend zero kn ii count infect infinity truly report node epidemic correctly source chance environment network model email network internet facebook empirical test focus scalability simulation demonstrate ease real require care extremely set infection infect performance regime false positive infinity test regime rapidly predict theorem report reporting node tend infinitely negative positive converge see remarkably bad diameter considerable random network infected mean false reporting truly report among report zero addition reporting tend false positive epidemic email motivated spread infection negative false positive infection seed algorithm affect radius medium social identification network cascade failure attack security epidemic email comprise number infect small truly report report positive situation error test local neighborhood indeed person incorrect near account network information inter decrease epidemic estimate distance network epidemic report infected buffer negligible report close reporting mistake node ball increase radius distance type topological noise plot distance equally line two node infect positive number positive become increase depend opinion identification process force single reporting map snapshot underlie network analytical wide correctly false negative infinitely positive simulate free graph facebook email chain size practice extremely estimate finally facebook report infected epidemic scenario similarly report report scenario first q tends set corollary zero tend otherwise q equivalently highly first likewise np tend word alternatively converge calculation condition reasoning scaling consider infection noise tend condition particular report correctly alternatively network grid tree noise reporting conclude random sum obey xy xy c replace note xy mx xy mx xy mx hoeffding inequality np xy xy x x xy parameter must infect report infect near infect distinguish two discuss infinite tree start root infected level deep tree infect consider contain infected node ball ball radius interior hence ball case
study simplicity condition strongly consistent dominate x var n normal also account obtain thus asymptotic interval law belong power strongly mean variance denote c ii hold consistent obtain mle however observe size mle shall focus consist iterate log maximize calculate accordance method consider st let simplicity shall notation lk model individual satisfy q notice although lk z l l need follow normalize straightforward notice depend influence maximize log word constraint entire st iteration determine parent iteration observe reach never leave calculate nonetheless include description shall deal case begin stop em k ne em verify log unimodal mle choose determine p criterion denote one unit reduced generation control identical em construct converge step eq n np I prove lk lk l control distribution need verify take normalized maximize subject eq final apply check expectation likelihood function estimator shall result simulate binomial distribution mean rate practice kind evolution presence binomial birth probability subsequent evolution notice distribution consequently consider unity generation take simulate individual material evolution control prove exponentially rate evolution individual solid confidence interval family go observe accordance remark estimate solid line parameter horizontal pdf var pdf evolution estimate right course line approximate dash estimate approximate interval dash line horizontal value situation assume number start per known try reasonable per use illustrate individual plus run convergence occur procedure assess consistency estimate figure show base sample width medium width var right entire solid individual dot line value line kind work information come environment control clearly justify parameter evolution plot observe respective value study unity simplex uniform open em sample insensitive choice initial maxima saddle give law maximization evaluate parameter greatest start discussion observe knowledge kind denote aic consider sample observe aic binomial lead little kind control satisfactory procedure would term model expect cn binomial column need attain procedure mu leave entire family bootstrap dotted line pdf pdf pdf left parameter process generation em sample obtain approximation analogously correspond right variable joint one observe distribute curve give calculate respective preferable assume understand great content former sample interest determine respectively lk z lk max nb associate row element probability row equal product analogously assume let storing store say order determine possible form dimension tree generate consider great lead study supplementary material give complexity indicate one need population available evolution leave sample binomial em iteration store store iteration require sample procedure seed evaluate simulation perform software log package estimation consider nonparametric control observable generalize law practice observe family tree situation observable individual generation generation observable parameter incomplete procedure make algorithm show encounter maxima different high although simulate show provide adequate reasonably store iteration simulate entire tree bootstrappe approximation algorithm contain variability acknowledgement author read comment would thank ia provide de ia de grant immediate base consequently eq fp lk maximize subject negativity kullback verified maximize take maximum make strong law number fix simplicity let k martingale difference account iv verify I sm nz guarantee iv vi ik k I kn last inequality ii consistency take account respectively consistent denote sequence I central sum theorem adapt brief firstly due k cite proof know proof immediate proposition iv shall I I limit denote analogue cauchy consequently pe enough follow q nj prove analogous argument jointly condition vector nu j prove argument consequently material follow binomial distribution variance respectively sample family individual generation robustness em observe maxima choosing unit simplex uniform overcome problem propose likelihood binomial control convolution maximum coefficient degree take em estimate great log start likelihood function versus start seed center width p p pdf pdf width maximize log figure ccccc cc cc p em expect paper convergence figure start seed expect determine value function provide possible maximum notice sample remarkable analogously contribution z b bb determine eq q value possible obtain go go value nan store file take account table relationship r max z max thm thm branching branching evolution size generation interest sample entire investigate secondly practice consider use individual problem accuracy procedure example branching control branching class stochastic characterize random determine overlap subsequent usual framework branching relevance growth biology cancer disease cell link etc structure add branching generation degenerate deterministic control individual practical population possibility affected binomial control would reasonable concern introduction environment introduction specie control branching branching see see state recent therein development applicability model frequentist model theory control square branching sample firstly
switch switching horizon curse dynamic number number investigation switch use recently however priori result point cost switching mode moreover I assign modification characteristic switch function fail solution problem development aim develop cost carry switch network nn weight continuity change satisfy necessary uniform method analyze numerically close literature subject maximum switching need assume state discretize c calculated numerically switch minimize change continuously calculate organized problem detail iv discuss implementation analysis conclusion give vi dynamic mode model nx respective mode active identify schedule system schedule cost compose piecewise cost mode horizon c cost therefore consistency start assume sign definite develop admit reward e time decision make researcher straightforward cost go final fixed depend word e different go observation develop solution switch cost active cause scenario switch previous active controller want current controller want compare see cost switch instant previous active time cost go dependency go consider concept mode operation play switch word mode another go already compatible looking mode active configuration position stick manual transmission car mode mode mode go step operate denote terminology instant calculate form recursive bellman mode give consider concern run select next already switch already active concern address minimization concern address minimize fourth concern however address term confirm go close operation switch approximate cost go linear nn approximated select smooth denote learn algorithm cost go incorporate weight input function subject proceeding training concern nn continuous neuron otherwise belong integer e function continuously versus unless system comprise select continuous used idea incorporation dependency give use function I number weight require train multiplied nn following need prove give continuous approximate set every approximated use capability structure prove main idea adapt go minimize state continuous continuity scalar piecewise identical point open set belong boundary limit continuity every continuity eq follow contradiction side q contradict q hold complete derive example repeat weight training detail cm integer repeat cm cm cm go back repeat guess converge domain interest set step step select pair qx cm go cm load batch backward resemble control cost store final nonlinearity hybrid nature subject utilize find see computational increase conclude note capability argument weight train state eq compare online easily firstly provide robust toward uncertainty secondly program nn local method least fulfil condition result trajectory domain train go valid belong provide flexibility desire behavior switch investigation code matlab request mode give horizon discretization system euler integration select cost switch tradeoff cost state switching dynamic compare convergence fast origin speak pointwise switching basis accuracy capability polynomial sequential iteration result plot dependency go method utilize become less eventually switch happen go mode switching turn switch controller provide argument schedule fig turn go operate entire conduct controller able dynamic lead optimally go controller optimally interesting active mode switching switching stay mode confirm go switch beginning becomes turn go latter case need controller new method switch switch eventually mode active eventually need switch e finally initial active mode depict history mode control system switch switch cc cc objective open half open square height lower low setup upper dynamic mode discretize decrease switching also preference neuron domain z select cost switching machine core ghz matlab simulated solution select job control perfect tracking switch three fig seen effectively switch assign switch decrease switch switching switching resemble utilize switching cost go approximated training threshold switch cost apply another controller switch reason process consider fig study threshold switch versus due fig go turn fig high potentially present switch controller switch analyze function operation cost along switch negative controller reward active assign require train controlling
solution contrary pick minimizer via inverse qp refer operator nothing indicate value actually crucial make satisfactory relate identifiability ensure proof guarantee asymptotically unique exist limit highly unstable expression enable summarize recurrent unbiased gaussian lemma state find instance theorem deduce expression asymptotically worth note initial choice basis besides construction entry problem step close vanish entry rescale acceptable final however original easier asymptotically turn consistent asymptotically limit variance improve one suitably possibly non q must invertible admissible clearly influence minimizer asymptotic variance include optimality argument reach lemma raise problem optimal actually plug performance verify reason suggest favor procedure regularity performance situation devote monte study involve scale begin distribution well result soon estimator replace counterpart scale deal arbitrary application methodology model third consider different distribution integer set transition repetition deal arbitrary space bernoulli rescale digits drawing gap iid distribution markov transition keep way conditionally build step geometric error report relatively remark row consider ht c c p c c c summary size carlo repetition state determine binomial indeed convex state hand observe realization consecutive nevertheless favorable conclusion regard efficiency appear clearly observation geometric deal necessarily q describe two size gap result summarize table http c c simulation matrix size consider square correspond standard deviation error repetition performance seem even dedicated show eventually perform situation allowing draw similar consider asymptotically case simulation confirm theoretical implementation consider naive theoretical optimal property technique investigate markov censor sequence chain observable iid identifiability partially lie markovian exploit set framework studied lie empirical observation close monte carlo various situation situation make imagine support recover support determine necessary condition identifiable optimistic current starting tackle question research work identifiable writing sum p deduce identifiability condition equivalent result point optimal sense know map differentiable converge denote q complete theorem proposition statistical transition presence censor situation sub gap iid gap transition consistent numerical simulation probabilistic analyze huge range field markovian give rise markov hide statistical methodology transition censor simple markov gap jump consecutive problem initial gap kernel identifiable identifiability transition know specific show full regardless element framework markovian exist finding parametric seem tractable find eigen element contribution build lie estimate explicit moreover normality asymptotic variance carlo illustrate situation jump process jump occur consecutive observation grid imagine situation subject application field recognize provide phenomena chemical financial market markov study disease process restrict modern literature contribution sparsity difficult exploit issue address considerably simplify nonetheless spirit paper approach well develop technique model section describe problem identifiability characterize build support study numerical monte carlo simulation proof appendix irreducible transition available jump observation iid markovian generator far gap identifiability transition space dimension choose assume slightly say identifiability identifiable every depend remark never intersection combination problematic
leave tight future research would take number implementation step pass else equal mu v take pass distribution initialization entry procedure standard singular vector alternate complexity theorem require remark row multinomial failure mention multinomial sample efficiently change distribution previous section entry input obtain bound inherently non convex might converge rank completion alternate analyze completion proof similar initialization routine show differ previous work key exist crucially incoherent avoid initialization accord w eq iterate obtain singular good rank minimization hypothesis r tt step th decrease geometrically vanish error hybrid style handle leverage alternate coherent new pass interested store two set web need query ad co occurrence multiply matrix give produce rank final factor intend already column correctness suppose wish calculate rank approximation proceed stage follow intend element alternate iteration accord produce factored remark norm completely parallel matrix element set weight mention fast overall particular ia bt r f multiplication matrix matrix rank however approach compare rank r compute yy application server rank iteration server compute norm norm server entry f k initialization cp server server ij u modern compute million dimension store matrix environment problem however depend computation amount distribute set assume partition row rest store th act cp cp minimize standard interesting row linear compute typical requirement complexity weak depend crucially independently need server row outside critical detailed code simplicity drop iteration correspondingly modify simplification compute server server locally independence initialization initialization compute right tr tr tr similarly communication constant round svd step alternate compute cp server row th computed message server message server size combine pca partition completion server cp number update compute exactly hence algorithm communication bind argue paragraph suggest total communication desirable completely contrast communication communication transfer bit complexity bind ij represent initially bit I f bit hence bit b incoherent coherent projection computationally incoherent coherent matrix line b direct stagewise incoherent algorithm clearly directly error stagewise incoherent stagewise computing approximation low error computing simulation singular incoherent coherent matrix incoherent svd incoherent spread whereas coherent entry vary matrix incoherent matrix correspondingly average iteration sample subroutine generally sample plot projection projection choice hadamard compare algorithm vary plot algorithm incoherent coherent figure b give use algorithm compute low rank set approximation plot incoherent plot coherent dimensional row space dimensional first multiply theorem theorem conjecture edu microsoft microsoft com mail edu randomize approach different exist literature minimization factored intend leverage yet approximation weak frobenius combine aspect generate spectral exist besides spectral approach extension interesting new efficient factor give small alternate instead small pca want low rank matrix array technique modern typically approximation residual run pass approach involve column onto dimensional svd bias factor low rank try approximate crucial ingredient sampling combination distribution element row naturally utilize focus frobenius run approximation norm pass approximation spectral however ratio first singular alternative approach distribute contribution low approximation draw execute algorithm factor intend pass independent problem hadamard similarly reference frobenius briefly review problem computing pass present sample compute give guarantee extend additive norm hadamard project hadamard compute drawback hadamard transform advantage sparsity time frobenius guarantee area comparison heavily low decomposition streaming
propose learn modular refer problem submodular relate know modular maximize popular flow represent problem accommodate find item weight learn repeatedly bring second conceptually simple explore face refer episode item regret quantity interest sublinear world simplify sometimes instead submodular function specifically entry word entry maximum let maximum entry find item decrease broken rule e weight mathematically straightforwardly generalize minimization exist view generalize notion closely ground q vector derive submodular increment eq variable many solve nevertheless solve concept concept involve ground flow network unit flow minimum diverse cover item popularity popular item movie title popularity action movie movie cover movie restrict feasible formally lemma act basis combinatorial nature basis suitable solution maximal basis interpretation recommendation cover basis unknown happen instance diverse popularity perhaps movie maximize modular formalize problem bandit item unknown unit cube assume associated arm contribution feedback solve observation several generalization bandit basis agent motivate specifically minimum observe cost contribute movie movie choose movie bandit choose observation gain observe te te non contribution agent agent equivalently optimistic et e nu u te te w te e te te te te te particular refer compute ucb expect item estimate expect episode episode finally item encourage often episode confidence exploit item continuous simplicity exposition episode episode choose item item regret section sort major episode speak part gain item heavily major contribution exist bound prove bound summarize result key analysis episode basis indexing simplify loss episode episode hardness item item contribute simplicity exposition contribute basis intermediate generate select suboptimal item definition say entry ki rest expect bound subsequent gap restrict ready main expect fraction item observe finally rewrite regret k sum sum item follow contradiction result must contribution item episode stress sequel gap dependent cumulative regret episode key idea select regret bound next gap order increase item et fact quantity far combine item cumulative expect trivially q l ucb algorithm ask tight bandit free notable theorem notion item differ improvement different notion follow let independent item item distribute bandit property gap suboptimal item return item gap tighter prove free bandit bound bandit bernoulli set let independent set item independently bernoulli bandit small solution formalize dependent say episode generality consistency inconsistent algorithm poorly logarithmic instance regret bandit eq kl bernoulli mean first follow gap integer bandit view bandit horizon bandit bandit state adversarial environment bound expect upper proposition constant therefore also factor tight upper bandit bind gap multi armed weight bandit major gap key showing difference expect sum difference gain two therefore contain maximum zero fact entry solution combinatorial bandit evaluate synthetic episode choose contribute world cumulative episode difference compare baseline first baseline basis baseline greedy algorithm episode set probability randomly item perform greedy policy flow show practical experiment dependent tight node capacity link next link source nod one illustrate define source flow consecutive third flow cost parametrize formulate minimize modular function submodular capture note sum indicator draw bernoulli independently function episode three suggest upper second consistent fact surprisingly particular time regret episode observe upper particular surprisingly never ccc greedy policy learning service span goal span lowest expect minimization formulate bandit ground experiment network forest compute edge q record noise range motivated network explain distance unlikely cause high episode second greedy episode cost policy table greedy policy network spanning episode learn quickly edge title movie american toy movie recommendation ten optimal movie decrease popularity movie highlight movie third recommend identify people million rating movie ground movie rate movie movie movie episode user episode user movie rate trend number episode movie bandit maximize modular analysis decomposition apparent combinatorial ucb algorithm solve regret latter ucb tight magnitude perturb geometric online mirror adversarial semi bandit offline variant efficiently optimal guarantee hull problem projection single step order magnitude minimum modular paper paper maximize formulate modular quite popular span gap tight gap upper three learn efficiently leave several match eliminate modify leave thompson perform straightforward thompson replacing believe
category system device resource cloud add category training return refine require intervention relevant automate system set capability sound training extract sound new statistically signal evaluate sound classification coefficient compute pick g irrelevant affect parameter code compute time fourier among related centre mass identify magnitude spectrum fall class various system integrate magnitude band amount express sub energy encode prominent filter purpose instead compute filter bank whereas feature envelope sound thus coarse record channel acoustic scene delay occur sound measure variation channel link spatial system harmonic harmonic correspond occur scene help discriminate feature frame method base extract sequence audio signal firstly frequency representation acoustic time acoustic vector analysis model autoregressive instant previous determine envelope model information employ autoregressive approximation combination whenever basis parametrize function contribute example decompose transform frequency index encode audio occur specific time also extract convolution assume analyse alternatively already unsupervise way machine adaptively train representation brain encode along activation use segment significant acoustic application acoustic salient spectral main use signature event important recognition scene encode probabilistic encode elementary property therefore parameter design extraction comprise operation firstly audio scene spaced pixel obtain extract orientation extract frame histogram signal training event car scene employ within training feature essentially perform acoustic property employ acoustic descriptor classify model audio event language test describe process derive quantity use method enhance discriminative capability feature linear non transform cite transform project subset identify direction property general independent analysis employ subspace score measure feature belong cluster near class feature separable computed frame discrete consecutive identify evolution scene extract stage statistical use category work decision determine likely generate principle basic different obtain category statistic close centroid use discriminative classifier interpret class specific feature model output determine hyperplane class training criterion discriminate allow discriminate discriminate belong class train combination evaluate separate hyperplane combine generative model separate vector former overall acoustic decided vote statistical list highlight technique various aspect statistical moment quantile modal express detailed present baseline benchmark account temporal unfold complex acoustic scene record sound precede sound move three normally encode matrix sound occur correctly unfold indicate probability connect sound occur sound negligible connect wrong sound employ unfold acoustic event dynamical theory recurrence series context measure audio scene output acoustic feed system computation community verification problem sequence vector derive dimensional acoustic probabilistic linear discriminant criterion determine sample decision generally type respective pair decision criterion multi use already audio frame scene accord category majority vote employ vary audio frame contain assign training whereby close according whereby model system mobile device area environment encounter supervised design running instance parallel use result decision rule signal discriminative example meta algorithm multiple copy bootstrappe lee learner category frame acoustic scene technique throughout compression forest output approximate compression normalise compression audio file audio base thought algorithm deal select combine run test optimal majority vote sophisticated weak see range signal processing context allow solve category indicate member universe system statistical classify firstly divide short frame choose frame ms depend signal frame sequence transform reduction aim member yield distinguished far transform clarity processing operator number occur reason belong category let describe note stage illustrate function phase complete apply phase classify return follow combine extraction modelling spectral special case call name whereby word occurrence frame follow ignore feature tb supervise acoustic despite effort benchmark tackle audio acoustic line aim research similar music publicly available difficult previous collaborative include environmental music audio condition repository would suit rigorous fair evaluation available database series general effect sound com series sound constitute challenge provide researcher produce environment record office restaurant disjoint contain long scene publicly researcher back challenge website http uk table list challenge tailor sophisticated recurrence quantification feature scene histogram gradient audio scene acoustic audio extraction frame z lee acoustic selective event detection sound scene machine classifier representation j scene low due matlab toolbox present correct david dissimilarity bank none majority vote descriptor energy frequency near descriptor auto maximum acoustic baseline patch energy moment vote majority vote selective max energy spatial fisher majority vote weight vote moment audio distance summary name statistical fed separate hyperplane reference discriminative frame majority vote weight majority cite method good system comprise phase challenge provide indicate environment test optimisation far fair evaluation private recall training belong depend available represent general problem occur bias produce office ideally complete historical office world bias towards rich office infer office environment incomplete use available label partition phase propose purpose challenge employ fold five independent classification run design allow signal proportion keep signal per avoid calculate yield correctly classify correctly sample th belong classified belong chance perfect confusion classifier whose belong private fold box perform differently definition vote classifier audio refer human accuracy interval display algorithmic comparable human see accuracy algorithm method central dot accuracy average fold interval assume fold whose measure evaluate fold bar ratio fold root fold gaussian interval probability baseline mean group exceed accuracy level significantly box indicate perform red file common method indicate certain independence incorrect classification agree particular label allow relatively meta performance far perfect suggest acoustic confirm confusion figure commonly misclassifie tb confusion obtain highest fit among fold every hold file file basis recall test assign define file correctly classify whose two misclassifie classify return equivalently correct incorrect imply correctly misclassifie performed perform pair test classifier reject fix grey box represent whose significantly dataset ranking bad choose accurate performance range say significantly high majority pair assume x statistically assumption tb distribution classification accuracy accuracy calculate acoustic bottom plot accuracy tail ten result carry understand whether individual evaluate fold signal total category scatter file observe acoustic belong belong greatly vary exception office contain sound multidimensional scaling dimensional pairwise see detail demonstrate achieve tend decision expect mistake aspect decision pairwise number multidimensional approximately distance yield sufficiently suitably place corner private testing plot appear together svm svm human participant ask classify public dataset audio record category office restaurant designing choose evaluate acoustic nothing participant label prior test people allow audio appear ensure file classify work advance test old device high care include participant participant achieve low lack motivation accuracy people outli accuracy box reporting interval participant locate note decide include participant classify accuracy accuracy would lot close median participant public category ask participant total occur participant classify individual classify positive derivative improvement hypothesis participant perform calculate participant right reject great tail fail reject expectation smaller indicate participant find reject hypothesis believe classify individual find tb row confusion insight human confusion commonly misclassifie category estimate belong various addition distribution represent average assess accurately benchmark figure depict public public private disjoint subset informative compare trend human oppose achieve accuracy human appear regular outli whose sophisticated algorithm important factor learn classification abstraction expense detail nonetheless think valuable insight gain result light challenge trend regard statistical infer algorithm extract signal class label moreover whose achieve technique classifier audio discrimination audio signal environment classification boundary discriminate recorded environment algorithm motivation attempt extract encode
ig x k important q denote th block e form q abuse notation knowledge require operation record use equality matrix updating eq let special complexity latter expect average spend synchronization parallel alpha problem proximal discussion concern presence output first accelerate alpha proper proximal accelerate arbitrary proper c yes uniform yes yes yes yes yes yes yes importance alg alg proximal c accelerate modification lemma convex contrast block recursively eq positive clear get z p p z recursive substitution combination deduce positive p conclude k deduce fact sum z z state let sample arbitrary sequence produce recursion expectation reasoning analyze obtain fy eq blocks function hold thus follow descent serial variant without compact span sample unified variant focus unified strongly extend proximal sequence sequence k theorem remark reader refer find classical accord alpha unconstrained attribute uniform hold ad hence reduce output satisfie particular explicitly state apparent approach clear sufficiently accordance alpha accelerate coordinate descent solve proximal choose generate p z uniform sample output corollary distribute coordinate present school focus dimensional minimizing sum regularizer alpha update subset coordinate remarkably flexible randomized method accelerate importance deduce complexity many variant improve grow realize approach problem moderate solution moderate sufficient learn science shift problem truly method reasonably work read describe need able read describe problem information contain type descent semi paper focus seminal early justification convex extend primal dual accelerated coordinate characterize perform operation advantage reduce subproblem theoretically practically accelerate descent latter acceleration parallelism proximal setup accelerate paper unconstraine constrain progress et asynchronous method develop liu deal update coordinate likely coordinate importance recently consider serial arbitrary investigate objective zhang nonsmooth coordinate gradient improve specialized algorithm analysis material optimization close main contribution randomize composite alpha pick value focus minimize zhang possibly nonsmooth appear alpha arbitrary expectation bound cover accelerate variant knowledge complexity randomize arbitrary dependence counter objective admit sampling assumption determine need determine optimize bind serial common conjunction serial lipschitz block derivative situation serial block update need intuitively capture certain smoothness gradient spanned coordinate systematic study exposition focus apply smooth accelerated variant analysis alpha remarkably stepsize alpha reduce focus alpha deterministic gd deterministic new cardinality think distribute computing sampling let choose subset block uniformly take set reduce similarly reduce lead specialized distribute establish robust many one force traditional whether alone often iteration row think put element alpha bind bind upon complexity coincide algorithm depend certain level closely obtain unified variant alpha achieve establish recursion arbitrary alpha differ block convex choose sequence reduce sequence vector equivalent avoid extract relation sequence admit weight say admit denote hold vector formulate observe bind tool design formulate study coordinate method tool generic establish randomized descent establish vector appear directly influence address paper uniform nice particular algorithm refer reader systematic admissible complexity analysis alpha unconstraine cover also different alpha explicit solution reduce sample z fy result alpha alpha proper iterate q particular optimal alpha accelerate variant accelerate proper rest proof theorem alpha general prefer establish proof sample identity notice positive produce algorithm fy line v last inequality recursion inequality convexity side expectation divide side take finally r last assumption flexible modern special reasoning case alpha sequence see modify sequence obtain choice deterministic randomized latter choosing constrain uniform ht characteristic uniform algorithm alpha eq simply norm define likewise euclidean block choice gradient indeed reduce I k follow apply keep reduce positive follow particular special allow arbitrary k fy kx k accelerate coordinate iterate method case direct satisfie serial coincide randomize
large scene height roughly scene feature count none take five see histogram match case bag come reconstruct count estimate case illustrate office scene image overlap extraction create iii annotation row simple view scene likely practice rarely access reliable instead hundred automatically derive infer counting grid different human label output detector image various oppose input window counting grid spatial pattern depend reasoning example image full office fig extend include help overfitte issue count bag discrete counting nevertheless attention counting property relationship bag vision feature combination image significantly simple unsupervised learn discriminative supervision integer continuous lda multinomial spatial relate scene spatial among ignore topic assign co occurrence word model gaussians train counting tie inherent datum bag point count corner window bag define learn region sometimes refer grid help guide reconstruct grid capture region infer feature distribution spatial approximately align relax former represent arrange divide pre specifie assign hand place center arrange flexible spatial configuration generalization level uncertainty image patch build consist pixel generalize small image synthesis transformation translation employ would work sequence represent map multiple independently hand entire single point model however turn mixture bag experimental mainly generative grid mixture bag intersection grid hide simplification bag extract inside feature codebook real value feature sift feature function retain establish thing informative bag count bag organization bag spatial bag bag create window bag boundary image solve arise constraint neighboring window completely determine identity column overlap difference depend neighbor window etc propagate constraint uniquely determined count bag bag constraint constraint likely representation lead bag many window window reconstruct least spatial bag window image g category bag imply help predict count bag extract new category original spatial turn constrain useful way count index grid e count follow grid bag firstly window place sum window count fig letter latent variable network fig give bag count overall mapping sum rhs perform count place mapping location bag th bag represent position probabilistic interesting mapping sample window index grid share share hand variational counterpart window particular compute likelihood datum variable perform exact procedure effect use optimize optimize keep lead simply minimize divergence count bag count appropriate window counting optimize bind involve logarithm summation jensen locations k window index index summation bag distribution think proportion type source window performing add bag q consider distribution feature count know feature map grid additional proportion window normalization reduce count grid optimize variational expand equation optimize full fig dirichlet prior appropriate inclusion influence trivial zero equation location window window represent corner finally location update prior surprising mathematically model consider dramatically cg e grid relatively window g though make parameter tie interpolation capacity mixture grid capacity word able estimate grid distribution aggregate window window position mask one corner zero elsewhere experiment prove normalize eqs constitute iterate summarize alg update z z p return symmetry break align bag local minima important however count accordingly prevent problem boundary need grow count cg cg eq mix express many vision framework choice window size fig count illustrate third column remarkably essentially image bag representation iterative start inconsistent direction lead minima deal image consist bag case region relatively infer bag window formulae bag contribute information bag reduce bag section low break count grid window discrete single index equal equality otherwise likewise discrete former characterize differently make histogram pixel descriptor great performance location alternative count representation hybrid counting conference prove successful also introduce grid focus feature generalization window grid large source highly shift slightly avoid appropriately relationship variation term step reveal grid utilize require convolution operation update sum position show slice compute efficiently cumulative sum panel set histogram count well feature efficiency computation bag window uniformly along compute shift contribute mapping histogram image reconstruction histogram section iv input cg color video classification cluster hundred location discretize feature experiment sift enough grid large feature feature identity difficult simply property count procedure run color patch draw matlab load sample drawing transform map point one color section bag bag separately grid window help grid visualize count location equal z color color weight count attempt reconstruct color information location image aware treat feature remarkably lot spatial dark go green dark discover sense among histogram image counting representation reconstruct scene remarkable left upper low z find reconstruct reconstruction iterate eqs count detailed map bag step patch histogram replace e find count grid eqs extreme individual counting become feature redundancy image extract count image recovery use category large generalization bag sift discretized feature sift spaced way image transform create fair implementation allocation feature unless protocol per assign test low counting grid complexity capacity sample roughly lda windows grid parallelism scene grid even basic spatial think count grid fig window give combination report different capacity report level marker grid allocation prevent hybrid cg comprise consider grid use probably abundance training report discriminative baseline counting grid set hybrid cg generalize mixture lda spatial pyramid annotation report count grid outperform spatial pyramid camera dataset illumination lot foreground object category house office environment location class task place validation bag scenario dirichlet allocation well counting grid limit recover perfectly evident counting grid significantly lda likelihood panel count allocation use information fine limit robustness regime mixture spatial pyramid image window overlap window try scene fit acquire camera largely outperform bar sequence datum represent scene actually recover compose acquire camera descriptor successful method combine concern grid use sift model compute place every use forward observation posterior hmm observation constant significant marginally move camera indeed piece recover single bag fine dirichlet performance inferior equally count grid interpretable counting really fig complexity equally issue window learn scale big copy consider worth collection repeat grid day camera
set embedding reproduce rkh characteristic distance measure rkhs yet large kernel perform distance kernel however ridge herein consistency break problem difficulty guarantee nature make stage result short term even verification condition require care reproduce hilbert question general topological endow work domain euclidean admit density string structured object space suffer embedding intermediate cm rkh kernel introduce discuss borel topology l reproduce xu k rkhs reproduce bound operator algebra product expectation lx lx supervised problem hand randomness wherein generate relation scalar simplicity distribution use regressor model rkh function paper composition result element assume exist regularization sample observable ridge quite new remark access tackle transition ridge arbitrary contrast choice algorithm excess e compare estimation consistency function triplet difficulty triplet set ridge make rather moderately since kk x xy cx short insight assumption concrete example boundedness continuity imply supplementary also choose evaluate embedding yield supplement definite many kernel compact domain old compact topology supplement kernel cauchy separable hilbert hence separability hence verification hilbert space reproduce w latter h old supplement boundedness factorize boundedness triangle page continuity compact universal ccccc cm theorem illustrate concrete consequence convergence result outline idea technical detail supplement take excess pt pt bound upper old lead k tr effective cm h excess capture difficulty class decay eigenvalue dimension smooth definition us class ne normal identical use k argument equal mean gram correspond gram decay matrix source manner theorem cb c b k rate task reduces imply material plug bind material excess bound multiplier discard complexity due index marginal row second dominate word lipschitz since exponent small recent upon keep focused relax capture class open ii obtain acknowledgement nsf grant laboratory department uk em em em detailed proof consistency k follow old I topological borel compact compact space page metric trick eqs identity term second decompose notation exploit consequently notation rewrite present derivation z bind series ab theorem apply trick probability xt series l z fulfil see bound obtain matter rate discard bound get lb b l lb rl nh nh constraint cl lb reduce triplet match convergence match dominant carry summarize first term ii exploit ii e term go bn iii dominate I dominate one bc bc ac bc bc condition bc ii bb ab ii ac ac ac bn bn b b b b b b h iii b b b bc bc b eq ii bc nh ac c bc bc bc bc bc bc thus condition satisfy ii assumption positivity bc b cb bc bc bc bc bc bc nh bc bc h bc bc bc bc bc bc bc bc bc bc bc bc bc cb thus n ab provide numerical ridge experiment serve compare avoid density estimation modern toolbox embed ridge smoothing free goal learn choose entry uniformly matrix select point figure display learn entropy compare root square confirm reason achieve estimation large challenge optical small variability iii ground base consist correspond include bag iii bag baseline expectation maximization achieve follow work train hard sample validation regularization robustness pick different kernel exponential student rational mat ern expression kernel explore increase summarize accord use ensemble kernel improvement study efficiency summarize kernel rmse drop set despite domain fall range fairly precise however kernel poorly boundedness excess em regression problem response little distribution inherent estimate consistency guarantee dimension simple algorithmic reproduce hilbert contribution consistency technique stage endow kernel total observation difficulty answer old consistency classical set al stage response distribution machine side thought side estimation hyperparameter close analytical exist somewhat definition consist distribution meta rather sample l patient might identify measure health indicator measurement one mapping health indicator hope perform mapping candidate denote good derive excess proving rl zero sample triplet appropriately rate establish prior effective input mean smooth large motivation fold parameter gaussian gram briefly focus mild back surprisingly question question
take question representation require query kb schema answer simple topic single kb triple stand language question absence kb triple question answer end supervision avoid manual intervention labeling make supervision indirect supervision question kb triple treating mark meaningful representation question involve triple mostly kb previous embedding good quality scale fine tune simplify separate information extraction question answer language answer highly triple million entity interpret natural initial attempt consider template enough addition triple researcher turn specifically semantic parse flexibility interestingly weak supervision setting indirect attempt tackle answer realistic favor hard cover answer negligible intervention answer broad human annotation vector kb answer embed noisy indirect supervision kb triple complete datum associate answer meaningful kb relationship outperform introduce fine title remove even embedding make lead convergence propose fine embed optimize embed lead rest discusse introduce answering section answer mostly via track question query feed web search subsequently extract top return page approach engineer transform open question answer kb natural huge amount supervise machine language large question robust evolve triple entity via distant indirect supervision connect language supervision actually recently tend require manual intervention effort carefully design kb framework question answer little annotation answer question limited open manner kb trend propose embed answer train supervision get language reach performance many task less recently propose connection extraction work build question answer supervision answer consider answer triple kb triple use remainder kb entity word size respectively consist question pair find answer directly rank rather prediction directly query kb logical question parse aim label indirect supervision automatically rest section create consists automatically create kb set answer give kb triple k relationship entitie broad intervention many triple entity numerous entity triple unclear highly database many coverage despite cover triple hence decide create triple automatically seed question display round triple randomly seed question note triple relation table similar randomly except seed triple noisy create training firstly syntactic human english sentence type triple person relationship name string simply fine many generate rich kb would lead well train intervention choose hand triple kb triple pattern l l c kb triple question connect kb language satisfactory modeling follow learn margin want triple question enforce train however sample triple triple chance member leave element heuristic create triple somewhat positive counterpart scheme g embed carry update word entity initialize triple ensure gradient enforce vector learning sgd update course part replace score two contain conduct sgd example create replace question another switching day core force keep architecture entity embedding around also stay scale sgd appear learn powerful however control properly stop pre epoch solve room able rank fine tune often top embedding dot similarity triple efficiently fix frobenius problem minute bfgs subset example example validation learn embed define dot product identity fine slight triple rank end improvement question test ccc recall gram embedding fine detail label need evaluating create identify question hold likely add question various triple total hand question pair evaluate version provide triple sort whether compute precision rank well full whole kb conduct rank top answer compute precision whole kb candidate embedding fine embedding discuss present various version result essential allow embedding encode rich kb word note provide word alignment try gram bring counter factor gram generate poor gram conduct variant try order fail well perhaps supervision allow fine tune top list grant carefully tune similarity improve score almost initial language gradually iterate come automatically acquire allow flexible variation grant come recall triple among display full kb candidate indicate appear close relationship embed triple plain discard decide filter candidate string candidate string noun phrase noun phrase string augment singular final triple make tractable matching greatly reduce model lot table display example near entity vocabulary tend correspond relationship entity close thank use entity string respectively kb l string match fine tuning match question answer experimental section evaluate learn question another learn set match return entity string answer obtain question model record entity compute question return question matching question performance good system report f design still almost manual annotation dataset besides entity name explain rank evaluation new framework answer supervision embedding indirect supervision previous answering question way embed solve promising challenge remain semantic due supervision work answer even word modeling carry encode question
box predict determine response nothing prediction review base describe incorporate acyclic graph dag hide unit feed membership unit belong define combination nonlinear purpose logistic give radial function combination output class letter letter indicate softmax represent estimate note force bias give accord bayes mx dl need rely dimension well raw suffer likelihood x I model belief take extremely statement quantify incorporate adopt specific ard originally neural cancer ard layer hide part weight normal prior zero control become shrinkage pooling prevent overfitte gamma refer hyper leave value specifically automatically relevant determine whether determine briefly overview carlo please thorough carlo method draw suit high net guide network update value network weight hmc hmc update acquire desire posterior level variance include ard closed form conjugate one draw follow inverse control hmc variance momentum update simulate hamiltonian surface posterior hmc create term energy potential directly log natural posterior variance hmc need momentum formulation momentum mean bad result momentum momentum momentum notation proposal momentum distribution reduce reach distant point leave return accord acceptance probability high becoming period great concern modify probability whose acceptance give condition parameter interpretation modify posterior capable maintain favorable ability ease range acceptable previous evaluation hmc much wise similarly operator wise operation neural network literature propagation posterior feed forward operation gpu likely burden impose bayesian neural conduct environment library code contain ard question ask j relevant framework completely response equivalent variable ard determine relevant nan test variable true simplify calculation additionally approximation posterior state ard effect wish great phrase nan significance familiar operate fully bayesian ard nan ard wish test side iterative derive induction random gamma distribute nn degree value definition chi imply conditional value stage ard network unit input accord start update gibbs iteration however iteration thus update simulation ard ard base induction simplification proceed baseline evaluation mention use use website package repository http project tree influence package degree effect approach test assess test pilot find roughly permutation evaluate method effectiveness hundred prohibitive divided primary analysis section permutation since compare snps two strategy allow popular amount rely perform previously genetic involve contain disease capital letter major respectively cccc aa aa bb bb bb aa aa bb bb bb cccc aa aa bb bb bb symbol table baseline snps allele disease status risk embed causal snps snps combination yield evaluation create language run recorded disease snps take snps power correct recommend perform iteration approach use parameter ard prior gamma ard hyper hmc cutoff bayesian ard testing discard sample inference minute minor allele right allele frequency leave bottom minor allele powerful excellent effect test contrast relatively small never high effect threshold tell combination powerful picture appear well remain genetic may instance evaluate examine permutation size accommodate generate used software specify status I sense determine trait minimal relationship nearly purely effect hard snps contribute trait status method scenario approach purely relationship analyze level range previous h outperform method wide margin may detect snps encourage additionally outperform scenario detect across parameter test mention analysis exhaustive permutation significance conclude method genetic snps scale sized data framework investigation technique study cutoff ard case cutoff nothing cutoff value everything cutoff control tradeoff specificity positive false cutoff previous properly amount rate trade cutoff change cutoff increment record figure size produce roc curve legend display auc ard achieve maintain positive bayesian network design genetic marker tb roughly subject classified currently infect confirm derivative exclude snps subject top cluster membership ard hyper burn follow take top five ard ccc rs rs rs rs snp nd significant rs appear snp snps currently locate gene rs report locate mb report statistically either remove study likely replicate capable contain bayesian network association study approach broad different genetic architecture permutation powerful show performance competitive large conclusion powerful technique association capability availability code implement gpu framework outline bioinformatic university nc bioinformatics department state university department university nc discover causal genetic variant genetic pose difficult assess marker trait status demand presence parametric bayesian analyze genetic accurately involve control status graphic build decrease interaction across broad genetic conclusion framework detect handle collect large number variant ability leave genetic disease incomplete presence gene gene critical account interaction vary computational vast marker typical marker experiment marker examine marker consideration million situation interaction modern genome association million nucleotide require examine half interaction sequence cope little technology advance million several type one last decade extension combinatorial consider select via cross exhaustive suffer previously iteration need rw make
believe corrupted feature hope scheme assumption begin main generalization bind proof build top type poisson write write bound random single available poisson poisson write bernoulli draw sample ingredient establishing imply eq q establish notice generality prove translate proof sum gaussian dropout see define objective term write eq monotone equivalent except write eq possible error note topic excess sub excess make optimal guess know q imply optimal guarantee topic pure represent proof e generalize usual rate dropout datum loss e single example need hoeffding generative define show center write eq equivalently sign assume hypothesis q check hypothesis topic topic generative document multinomial multinomial probability topic topic prop prop lemma prop comment prop science stanford stanford usa stanford cs stanford edu originally design deep dimensional language generative long document dropout empirical dropout achieve like learn perform corrupt preserve induce dimension training increasingly dropout commonly network regression document name entity dropout poisson effectively corruption example hard classifier boost example dropout merely create create training erm word independently generative excess erm multiplying improvement additive word document expect minimizer respectively dropout denote variant logarithmic document text get modular bound erm dropout excess recall classic vc erm number vocabulary excess behind half document whole document tail compare variation document compare stem dropout excess bias bias negligible bind ng exploit generative generative incorrect large bias closely logistic dropout rate range logistic regression naive tradeoff dropout improve generalization factor bayes framework prove decay dropout adaptive improves factor contrast leverage improve minimizer regularizer encourage confident prediction confident rare dropout adaptive dropout language dropout topic base restrict setup count usual function analyze logistic dropout train perturb instead word thin feature binomial time replace independently occurrence probability boundary weight dropout often dropout differ dropout setting coordinate reason choose binomial dropout random remain dropout equivalent throughout poisson depict bernoulli topic vocabulary frequency accord document across topic contain topic simplex mixture allocation although advantage poisson gaussians explain extra arise approximate set bridge generating foundation translate generalization dropout first technical every apply assume feature useful balanced minimizer rule balanced topic topic away random substantially separately overhead separate topic almost dimension center proposition turn feature assumption hold risk dropout generate linearly meta modular standard heuristic previous excess picture address dropout dropout view assumption probability numerous shorter dropout poisson document preserve bayes let topic topic topic dropout rule true class incur classify short pair train intuition affects also discuss difficult simple relationship g score classify play almost drive dropout turn give good rule illustrate tendency spread problem red hard classify open circle hard classify dominate red essentially ignore left classify play less driving dropout fine near gray boundary dropout poisson value poisson intensity remain word neither result dropout intercept clear tradeoff end roughly document appear nearly b examine
deep deep implement understanding help deep neural network standard process standard multi layer perceptron typical infinitely many unit form limit theorem function gaussian result surprisingly put weight positive hide correspondence unit neuron fill red neuron black pt sep width text center hidden black cm circle inner sep text cm cm neuron cm h neuron multiple unit hide architecture layer mapping fix representation unless flexible neural deep c pt cm fill black sep center neuron node neuron minimum neuron minimum neuron h cm cm h random nonparametric black node width fill green neuron width neuron size width hide neuron cm neuron h cm input network whose infinite hidden unit whose weight hide unit parametric activation neural distribute introduce infinitely wide basis architecture show top layer node alternate architecture interpretation substitute eq weight ignore intermediate deep layer direct network deep deep view deep bottom section theoretical characterize deep jacobian derivative ccccc layer parameter control control simply derivative layer construction half normal choose regardless magnitude derivative euler limit gradient approach normal even grow heavy tailed bound small everywhere rare jump behavior next deep draw derivative individual dimension exp jacobian independent ij row jacobian jacobian jacobian composition jacobian composition jacobian jacobian compose deep j jacobian product matrix deep common useful manifold useful argue good argue conversely representation preserve dt node color compute little manifold white robust direction representation tangent preserve layer characterize jacobian jacobian representation vary h layer normalize net deeply large direction manifold dimension cc transformation draw code point successively draw distribution singular spectrum analyze jacobian point examine jacobian identical everywhere net deeply singular dominate effective demonstrate arise produce density locally onto suggest width modeling manifold whose cc identity layer function direction move locally change suitable shape remain increase example circular corner illustrate code deep figure functional graphical connectivity circle minimum neuron fill fill h h neuron draw fill white fill neuron bend neuron cm two different architecture deep connect layer layer connect layer adjacent find visual function ccccc layer layer show layer point singular value singular draw deep compare mean output often connect imply mapping one move put representation depend jacobian recurrence jacobian deep show machine generalization exp interesting non local create periodic first basis rise kind feature principle composition close either fortunately square repeatedly feature exp input prior degenerate degeneracy connect compose feature layer input deep square exp kernel repeat cc input deep kernel draw connect composition draw recurrence function input tail architecture correspond feature implicit kernel enable generally network transform deep derive simply fact rise learn way unlikely useful recently method entail dropping order result neuron maintain overall activation make average dropping neuron dropout procedure examine dropping feature independently dropout weight variable central form apply perform dropout dx model put prior mass understand rate therefore perform dropout input class therefore input depend cccc st nd rd term kernel dropout dimension order consider nearby distant long value model help explain generalization deep network give deep gaussian first use develop analyze relevance
node cycle exact hamming successful nlp instead structure set present analyze structured prediction key finding theoretically error theoretical perspective external tractable implication context grid node rather edge evidence clear improve decode relaxation optimal understand property prediction parse understand score relevant edge proof bad correlate chernoff continue unchanged definition task machine involve simultaneous prediction label maximize sum specific intuitively term formulate classify vertex label ground truth low hamming interesting hamming positive wide graph common theoretically expect toward justification exact edge label consistently bad edge label arbitrarily admit ground truth go graph poor path theoretically theoretically algorithm grid graph important theoretically parse language speech entity set input sentence foreground sentence specify encourage take e encourage neighboring foreground state big feature space condition learn random field structure application quantify discrepancy ground label study hamming hamming inference namely py practice map inference namely advantage computational estimate test worst np inference computational even pairwise understood measure obtain complex perform bad work programming search cut branch obtain accuracy predict test hamming however fairly art unable achieve generative setting high prediction accuracy limitation heuristic indicate instance characterize computationally understand inference good much lose prediction hamming assume version term vision motivated vision grid non often marginal turn analyze hamming expect hamming evidence first use combinatorial worst bind analysis grid graph utility distinguishing ignore knowing sufficiently give rate node quantify observation need make close submodular grid computer foreground thus emphasize grid moreover multiplicative objective setting similar know provide ham good job motivation interesting result computation obtain ham graph relational protein web even constant empirical grid demonstrate recovery structure highly trivial iterate conditional mode poor polynomial obtain constant close goal set unobserved observation various setting important channel e zero augment error deterministic bit send channel version view appear check bit code see channel coding appear notion truth one highlight free input via semi process exact truth quantity every unless extremely small number pairwise inconsistent obvious translate objective allow complete share much motivation technical instead error ground emphasis give truth dense paper paper also time semi numerous variant specify consider significant cc work obvious translate stable seek polynomial time low truth instead paper assumption near objective cluster cluster positive state understanding open level recover ground truth study numerous domain exactly impossible clique detect community notable sort total ordering present work order random graph mainly graph os recovery considerably hamming et explicitly partial oppose recovery degree paper graph infinite addition physics connectivity constant avoiding ground approximate error generally meaningful merely ground truth object classify incorrectly optimization error recover truth relate way ground truth paper ground truth understand theoretic approximate error relationship recovery possible error close high several extension relationship study prediction label observe depict hamming observation nod generative follow observation independently good observe adjacent label bad bad edge another bad vertex whereas emphasize edge bad label node label function label e noisy labeling expectation error two stage expect extend stage maximize subgraph incorrectly label agree yield agreement higher maximal correctly inconsistent truth two statement half bad boundary bad lemma motivate bound give set carefully far error seem difficult analyze directly simple upper induce connect exactly exactly two side side contain side side category component call vertex type component single procedure fs distinct every sided entire empty thus non empty lie imply moreover correctly label type path bottom side incorrectly classify vertex component type bad agree datum half edge minus edge neighbor every error bad ss least half edge probability least event iid th bad probability lemma parameterize square f fu rectangle km km km task bound boundary dual minimal subset property extra correspond face include dual correspondence cycle cardinality part cycle cycle part b cycle choice start part cycle choice final twice orientation set complete intuition computation relevant lemma hamming bad derivation definition lemma size tight statistical physics boundary upper region recall attribute type region expand type f c ic area translation cycle go name self avoid finally show analyze second stage truth label choose via majority second completes start point analysis h cp h kb wrong imply constant misclassifie less well truth prove eq expect marginal truth symmetry error truth truth parameter need away flip coin edge label generate truth function white label white white white chernoff imply white white white give minimize predict white minimize predict white wrong several extension graph graph section theoretically recovery grid beyond robust grid graph fail thin constant analogously boundary exponential sufficient face outer proof computationally theorem graph precise depend structure relational predict protein interaction web page section prove recall analyze two output chernoff trivially bind fact least half misclassifie every lemma overall sufficiently claim discuss recovery noise one procedure compute would constructive efficient observation eq fp chernoff large misclassified call bad let good flip maximize score get b cd hence bad region none share bad observation
potential uninformative heuristic hand infeasible maintain parallel enable possibility ensemble convention effect lie formalize show horizon feedback reward reduce horizon agent must explain different td fast propagation describe exact choose maintain learner f reward adopt terminology al within learn greedy w reward reward shape devise collect vote action shape intuitive voting voting etc ensemble contribute vote maintain happen corner accord policy vote vote schema policy modify act preference interpret give result eventually arrive attention classical car car hill system position velocity position episode end goal shape informative rarely technique ten learn policy uniform time potential progress need first high position energy fig encourage position velocity architecture rank majority voting voting speed helpful likely prefer priori challenge interesting ideally like able two comparable pick well use tune select greedy step run independent run evaluation episode greedy allow reflect refer variant cumulative l variant height alone aid significantly statistically exception scenario performance significantly combination arguably negligible note even table policy framework give sound capable learn learn shaped car scenario well signal former scenario outperform latter able general expect benefit extensive future limitation requirement important expand effectiveness architecture hoc voting combine possible combination fitness r learn challenge happen doubly relate size scalability potentially thousand efficiently context rarely sensible could define potential different scaling combine static throughout roughly mdp go attempt learn well since argue et al potential grant ph grant foundation ac advance guarantee possibility technique reinforcement potential base ensemble induce voting mechanism happen real agent interact markovian agent learn setup allow arguably arising scenario popular potentially temporal greedy implication multiple task share stream sound architecture spirit fast devise ensemble reward reward idea signal instead improve formalism consider scenario latent often setup environment costly failure make imagine exploratory policy note reward part performance purely propagation section rl policy architecture capable sound realistic guarantee limitation general apply brief definition notation policy car discuss environment rl model set denote state reward action markovian depend discrete discount store entry pair maximize mdp dynamics mdp temporal td iteratively method rate td draw trace way al process converge standard approximation large continuous suffice one thought way scale rl difficult rl suffer long however prevent ng show potential ensure original mdp maintain potential auxiliary reward potential towards validate speed uninformative reward augment shape shaped shaped converge policy base process section motivate find suited ensemble reinforcement reward bagging solution ensemble rl extremely combination limit practical usage overhead learn speed lack learner surely reflect ensemble lie contribute reliable fa similar fa disagreement
mixture identifiability necessary requirement usual asymptotic ml condition identifiability covariate provide contaminate identifiable provide positivity avoid contaminate gaussian equality j n classical incomplete source classical component membership govern otherwise specific leverage reference cf denote ij therefore algorithm iterate replace two iteration calculation rl ce ie rv ie ii nj rp eq th calculation r particular algebra nz j nz ij detail maximize perform analysis facilitate code upon request implementation start algorithm constitute respectively select cm contaminate ij nj model probability step operational view thank contaminate observe gaussian consideration base criterion choose gaussian analysis algorithm probability fitting component distribution implement acceleration iteration decide reach e log estimate acceleration q converge analysis r estimate depend q base update write straightforward decrease formula write decrease reduce leverage provide residual effect contaminated contaminate membership leverage component expect arise convergence membership although leverage interested classification ccc leverage thus observation rich information eliminate observation bad outcome detection proportion could th group typical nz nz herein use update somewhat specify proportion outlier advance pre specify point realistic many contaminated characterize far purpose computation criterion well adopt section evaluate contaminate gaussian square mse gaussian datum contaminate regardless b additional choice fit mixture possible switch mse help mse scenario replication concern negligible bias practically finally interesting improve governed value estimator bias negligible result use lead increase estimator contaminate student student economic business although seven illustrative purpose height justified gender scatter labeling line gender fit classification gender gaussian bic lrr contaminate classification display regression line yield six misclassifie six consider misclassification section sensitivity local end perturb set type accordance ht denote perturb contaminate contaminate one contaminate scenario towards point top corner representative entire plot sake local contaminate local contrary scenario two contaminate locally leverage detect leverage contaminate degree contamination group component contaminate ccccc increase group membership regardless point point aim evaluate fitting modify datum uniform square modify denote contaminate lrr gaussian contaminate generally well contaminate term bic sake model ht point leverage detect outlier detect denote bad poor line term maintain misclassifie agreement group accordance contaminate importantly contaminate put gold analysis group approach robust point leverage impossible discriminant fact type supervision provide flexibility compete higher easily work facilitate contaminate interesting directly model state quantify model datum dependent base least mixture contaminate believe robust exception extreme fashion parsimonious imposing exploit development property test coefficient moreover work eigen decompose suitable simplifie contaminate integrate yield divide side contaminate scheme point j complement hyperplane therefore j jj condition identifiable implie imply j integrate yield therefore simplify simply arise identifiability contaminate require five hand derivative partial rv update derivative q nan rv j nz rv j derivatives operator nan nz rv obtain partial ij ij transpose partial eq nan update analogously derivative nz phone variable component response give respect elliptical heavy tailed presence contaminate introduce contaminate proportion outlier control leverage contamination specify contamination crucially approach furthermore fine intra leverage concept primary mixture contaminate maximization outline operational issue monte carlo weight base contaminate continuous constitute flexible powerful often assume original problem compose covariate mixture fail incorporate valid represent mixture see two mixture fix mixture assignment generate value datum mixture paper dependence allow depend member mixture regression weight assume across weighted density covariance vector contaminate observation affect accordingly detection development assume assignment analysis paper contaminate contaminate identifiability give outline operational aspect carlo estimator sensitivity
stream benchmark scope collect ease algorithm incorporate consider entity preliminary introduction feature may significant improvement term plan integrate adopt aim platform optimize stream message twitter media theorem cluster medium identification discussion topic recommendation streaming medium procedure atomic token carry tweet aggregate measure tweet actual concept topic various diffusion tweet variant forget old twitter week systematically able outperform content detection assume network show adopt suited social popularity high message device news discuss reach group interest medium use spam ability pattern great interest classify piece content real platform class appear stream labeling tweet training volume produce tweet well single tweet due brevity modeling help context character user external resource tweet contain content include user user attention tweet follow user exploit tweet topic start character twitter rely conservative discussion twitter content content semantic represent spread person learn potentially overlap leverage entity content bootstrap aggregate entity tweet text exclude entity remainder paper refer initial atomic homogeneous tweet relationship algorithm assign tweet point tweet reconstruct political static although consist group tweet aggregate together tweet represent sophisticated aggregation identify offline static case scenario reality medium stream fashion resource newly operate essential mind pass sort stream broad learning datum point various algorithm income close distance centroid take allow cluster stream content outline base stream summary contribution tweet share test atomic measure stream able require suitable online algorithms variant slide evaluate baseline algorithm solely tweet tweet content plus underlie social network result various stream media stream system direct relationship generate stream twitter google take case twitter platform user follow turn used receive news feed user address directly tweet user visible basic similarity incorporate recently way message big block natural unit aggregate broad tweet entity tweet hash mark leverage activity user user name symbol include link web link resource tweet remove phrase help capture message tweet mention entity way allow tweet use tweet share entity entity stream accomplish real time therefore extraction next various measure broad fig tweet content post tweet projection space feature let preliminary similarity content user similarity cosine tweet respectively similarity represent tf weight word document tweet tweet similarity eq user tweet author tweet involve network similarity cosine set obtain similarity measure weight allow parameter search combination cluster performance similarity measure pairwise two capture base good content reflect maximum formally maximum need importantly previous provide combination next maximum data stream amount incoming datum store memory evolve cluster emphasize represent stream base window slide window generality duration grow point stream use time exponentially decrease high high importance moment contain step adopt model ignore summary slide yet choose processing algorithm simple algorithm online start initial seed assign cluster centroid average member general account stream concept assign overcome suggest check centroid outli also cluster find row column confusion totally measure mutual reflect oppose mining like biased evaluation investigation confirm limitation measure report indistinguishable due negative bias toward tend adopt task event produce suited overlap cluster refer discuss implementation introduce aim assess drive bring social stream compare framework baseline operate baseline outlier handle text tf tweet compute similarity tweet aggregate configuration tweet base current art streaming tweet rely present dataset provide advantage also practical consume stream tweet similarity tf use tf implementation algorithm make algorithm comparison user present cluster time advance slide deviation mean slide duration minute across slide window yield treat ground truth cluster label remove evaluation score window refer period plot cumulative period axis represent hour slide window hour evaluation essential tweet truth therefore tweet explain empty remove old list cluster ignore cluster last separate list assess window perform baseline online time run due topic discussion medium demonstrate baseline statistically significant b measure overlap algorithmic ground truth huge truth several one assign cluster performance detail confusion matrix tweet solution confusion class truth number tweet compute evaluation period display maxima quality performance comparison dark colored value row cluster dark square confusion job capture confusion baseline ground truth ground able discover stream great classic hierarchical access availability full stream set parameter slide window affect overall cluster varying achieve intuitive heterogeneity therefore shorter small time social configuration obtain increment window grow difference understanding stream fails affect day medium production effective figure illustrate gray band day cycle period fluctuation describe reach peak accumulation evident benefit example cluster goal identify measure tweet tweet capture time scenario display whose consistently exhibit low value consideration generally short content produce mostly day hour oppose performance tweet fact perform attribute crucially term generation diffusion cluster social topic stream identification technique emphasize term signature method take temporal feature keyword physical rely well formalize social medium stream present track medium news short act signature identify small create news topic news cycle systematic quality retrieve provide focus news extraction base evolve
topological dag compatible trivially sort dag neighborhood topological sort compatible prove topological generality compatible write x k k kp parameterize impossible logit subject index parent dependent contradict contradiction large mass desire jensen prove function argument neighborhood share compatible argument jj p k triangle k p p dominate term cn maximizer omit pn similar first tr need tend b pn proof consistent therefore convert definition department california ca article penalize data intervention represent logit achieve maximize employ select group variable encode together penalize subject dag biological method competitive word bayesian graphical model encode independence represent acyclic dag recent see popularity medical science infer attribute function dag rely conditional test include goal dag function criterion minimum description et scoring particularly various applicable decompose sequence linear develop penalize dag give penalty theoretical high dag despite development regularization method nontrivial code group penalty coefficient consequently maximize log likelihood become principled sparse dags categorical know bayesian logit propose formulate descent iteratively establish penalize section report type biological section technical proof network dag e j multiple dag equivalence bayesian network possible equivalent dags observational dag joint incorporate experimental detailed causal bayesian please refer therein assume experimental intervention intervention experimental therefore remove point intervention estimate observational empty variable index parent total state px nonzero product multinomial assume level grow parent multi logit discrete development straightforward dag point h dd pd conditional distribution multi logit coefficient logit identifiable particular I ir jx p give total growth assume experimentally indice jx n j logit factorization although availability log apply experimental structure network equivalence order dag estimate see parent wish group subsequently group alternative penalize regular drawback inconsistent selection certain circumstance overcome limitation causal bayesian call bayesian computationally demand logit successful consider j simple minimize quadratic approximation taylor value add approximation log current give suggest hessian small convergence necessary tucker minimize apply approximation meet immediate consequence substantial enforce induce cycle dag cycle minimize induce cycle simultaneously outline algorithm discrete indicate constraint minimization initialize acyclic ir I I ir j j update completely identifiability dag say dag direct set say natural x dag causal path connect assumption world asymptotic interior logit regard theorem p maximizer pn consistent local maximizer pn confirm local unweighted turn construct achieve generalize develop order causal consistently intervention limit network observational node subgraph subgraph asymptotic simulate three simulate chain network simulate regardless parent logit group parameter j otherwise free network generate skeleton edge initially undirecte convert edge direction randomly sort topological sort network dag grid speed adaptive give comparable dag measure false fdr predict estimate dag cd estimate dag low average primarily predict wrong false suggest skeleton accurate three predict false selection tb network p fp markov report problem dag choose pc algorithm availability efficient observational datum complete acyclic take approach pc hereafter produce favorable direction edge intervention direct dag report base obtain solution cd dag edge pc cd pc note chain determine pc alone column correspond consequently count fdr chain column edge average cd algorithm datum flow perturb component simultaneous measurement since three protein observational among measure protein perturb contain measurement level group interaction causal consensus reach interaction figure dag qualitatively consensus cd predict false dag domain good algorithm seem skeleton mcmc dag node scale see
occur use heavy vertex removal remove result plant every boundary define condition short definition degree edge g structural plant th u edge otherwise budget initialization current u ii definition also vertice markov inequality structural du claim size prove claim claim bound x boundary count vertex follow number token send get right hand token send observe token send edge token receive compare red token get mind cover h v h make show property belong eq upper size edge graph apply switch around edge edge short nm dm df du property first fix integer vertex let term equivalently replacement show bernstein two h f subset rewrite imply set g v since balanced least cut balanced cut balanced require half map expectation vertex thus minimum cut cut go equal cut equal cut preserve replace embed net present previous work li du u du ex plus ex budget short span conv var sr sdp cut corollary theorem property section conjecture conjecture semi random plant previously partition cluster edge sample balanced cut plant cut combinatorial arise area science research science dedicate designing analyze make case exploit suggest life instance practitioner real life possible model provably outperform plant stable edge believe capture formation social sciences balanced balanced study bad expansion cut plant condition plant number take disjoint connect probability independent refer edge plant cut give edge sample invariant permutation vertex aside lie inside complex example model fairly large dense consider vertex partition equal permutation informally permutation identity label vertex know formal set cut plant permutation let give recall balanced balanced balanced cut balanced cut give minimize algorithm balanced w fraction edge plant constant plant plant cut state deterministic permutation succeed extensive plant ex model w recover plant cut semi arbitrary sample ex adversary impossible plant arbitrary permutation unknown plant cut theoretically paper high probability planted planted plant graph graph inside plant model except adversary current inside far find planted plant impossible model theoretically instead give plant hold plant model block widely use social science paper relax choice adversarial tractable opinion capture planted cut partition similarity real vertex within impose restriction cut assume cause error opinion model plant edge independently second many tie social tie friend people common interest think superposition tie network extent g tie graph vertex people tie people live region divide group live tie usually tie region live tie necessarily college twitter tie network social tie different formalize permutation invariant choose correspondence identify vertex introduce believe graph combinatorial optimization network equivalent rest vertex v h g r gr adversary arbitrary adversary choose nature graph polynomial exposition attempt additive random edge identically permutation graph identically succeed succeed high overview balanced cut cut sdp cut similar slightly one rao ensure sphere give sdp solution say sake discussion make sdp depend edge furthermore sphere every square euclidean ball small thus long contribute sdp sdp cut discussion remove decrease edge cut edge repeat vector sphere intended progress problem conceptually find radius contain cut cut edge iteration serious edge short edge skeleton skeleton constrain sdp skeleton whole skeleton long special remove skeleton edge skeleton many skeleton structural encoding encoding consist note value reconstruct encoding tell reconstruct encoding encoding less permutation kolmogorov small edge technical accurately overview work skeleton introduce semi partition similar iteratively remove heavy removal analysis structural ensure however geometric notion skeleton structural completely significantly removal quite numerous equal average assume generality iteration relaxation balanced subgraph vector pt height pt height pt solution orthogonal intend satisfie solution graph intend sdp cost intend sdp relaxation cut rao normalization rao work sdp solution subgraph cut edge set nu balanced present cut remove arbitrary lie otherwise vertex edge remove algorithm partition cut piece piece part step store budget budget keep edge remove cut vertex get budget keep sdp graph graph cut long heavy section heavy removal procedure remove edge remove cut long budget extra budget increase active remove loop complete piece use rao component height height hide sdp removal procedure update obtain denote cut algorithm remove pt structural e sum active extra vertex whenever edge word budget edge neither update vertex lemma budget extra allocate piece absolute piece return size separate piece partition already know optimal sdp heavy removal output cost analysis proceeding remove originally equal graph strictly degree r solution leave rely state prove sdp solution divide piece piece cut outline define feasible integral sdp assign vertex orthogonal vertex sdp partition balanced cut procedure cut ensure structural section part plant edge thus edge sketch structural setting satisfy removal heavy removal remove budget active vertex neighbor ball lemma remove control overview du show encode vertex respectively approximately neighbor r random exist uniformly short exist define b ready encode permutation record record restriction complement record number order element precede encoding encoding encoding need bit record record vector record restriction record matter encode encoding encoding permutation q satisfy theorem probability satisfie solution radius around induce embed need scale constant feasible sdp follow condition u sense heavy hence remove control remove remain structural around gm let let vertex assume otherwise structural degree place total initial allocate budget eq finally structural property rather technical speaking v h satisfie edge sdp edge cut increase vertex budget edge extra budget non execution edge cut decrease f suboptimal q since optimal corollary budget long whenever extra never active decrease vertex decrease algorithm use budget pay extra vertex radius removal pick I step heavy vertex find boundary remove removal remove ball union satisfy invariant budget heavy removal remove several invariant independently edge total budget total remove heavy vertex least need sdp pick range
many large coefficient cosine general know datum system consist signal call way stable preserve reduction address solve much e general ill theory estimation isometry isometry vector nonzero entry property rip theorem show matrix satisfying rip noisy keep q obeys depending depend apply treat entire unobserved miss come mean basis basis parent signal thus unlike conventional consequently subject try reconstruct observe complete em q define respect w belong subset find ks call iterate new require maximization maximum step expensive instead maximization subspace iteration iterate claim performance use em em initial check reach minimum close compare reconstruct closeness setup datum reconstruct reconstruct call population conventional residual effect start reconstruct process repeat get closeness naive possible approach closeness assess residual conventional procedure bar work well nice unfortunately attention naive impossible comparison time approach value procedure consideration sampling reduce plot average residual work take work conventional reconstruct high conventional sparse essential ingredient naive algorithm approach treat iid population setup signal may work combination linear combination conventional signal follow shannon compressive paradigm recover reconstruct compressive difficulty subsequently modify compressive comparison simulate huge variety sensor range imaging produce much often store entire sampling
fairly explore kernel aspect multiple reduce idea upon explicitly efficiently construct input classifier positive kernel define avoid instead matrix trick nice impractical harmonic approximate feature specifically shift rbf harmonic eq distribution simplify cosine see rbf correspond decomposition approximate kernel approximate scale projection find key far cf million million lead method induce plug vector multinomial regression logistic regression mainly assignment need address optimal tackle latter construction feature sum investigate report automatic empirical acoustic often use develop major strategy challenge multinomial method sag theoretically empirically descent sgd applicable sag design well suited secondly together partition multinomial logistic parameter combine spirit parallel leave parameter weight converge minimizer size softmax activation extensive empirical study support argument setting model advantage use kernel engineer select basis task hand specify function select popular paradigm latter mkl start kernel identify combine adapt scale mkl novel hadamard product advantageous material use neural network mkl present tractable large scale also function approximated linearity concatenation scale straightforwardly logistic scalable parameter hold fix parameter first general scalability combine highly individual kernel follow construct feature convolution namely need component number kernel additive product argument kernel concrete gaussian rbf layer parameter layer introduce perform pca covariance kernel kernel secondly implement fisher discriminant spirit multinomial probability largely due alone multinomial classifier build readily validate challenging computer vision conduct extensive neural network perform attain material yet equally report finding complementary tune adjust similar effect hyperparameter hide layer rate decay momentum unsupervise pre tune detail describe supplementary specific computational support observation learning counterpart direction understand learn grateful microsoft microsoft work paper california center high http intelligence advanced project department laboratory contract nf reproduce annotation contain herein author interpret policy either express fellowship f support nsf google research award fellowship award nf translation eq variable feature invariant p I ie product kernel p ie new simply use provide image main text divide rbf kernel pairwise select performance unit firstly pre bernoulli restrict rbms sgd algorithm learn momentum learn momentum decrease every epoch mini size stop overfitte augmentation learning epoch noise compare dnn deep net kernel augmentation rate h type gaussian h cccc validation visualization also pre softmax activation first principle mnist network seem spread kernel digit low corner right text like result use achieve feature model show median rate rbm layer three bernoulli intermediate layer cd adopt rbm tune learn rate momentum regularization tune momentum l tune momentum rate epoch early overfitte datum run epoch constant rate epoch momentum epoch momentum schedule epoch trained overfitte observe increase apply additive deviation pixel kernel method dnn dnn overfitte layer deep model c cccc provide comprehensive study automatic speech task model crucial automatic speech recognition acoustic learn predictive assign short speech sensitive acoustic window frame proximity capture b hour hour hold hyperparameter hold acoustic challenging well friend record mobile acoustic environment public place phenomenon background familiar challenging recorded well environment work gaussian frame acoustic feature value dense overlap state million million hold million million hold million report evaluation q attain hold tune often next performance speech recognition inherently proxy goal recognition measure speech recognition posterior probability label interested linguistic truth token token rate error entail perform costly rarely tune token task report train set speech processing area language speech language model currently decide broad comparison system conventional description appear acoustic provide power token use convert raw feature frame show work dnn frame dnn acoustic model contain output softmax nonlinearity correspond dependent cluster dnn connect dnn stage wise layer range criterion minimize descent momentum learn cross design another dnn restrict rbm training layer totally language rbm rbm epoch tune parameter rate momentum back propagation tuned decay momentum strength another epoch acoustic searching factor rbf tune pairwise median well random use range stable text acoustic text use average sag tune loose combination combine one combination train kernel reduction text dimensionality first c dim c dnn train dnn combination kernel multiplicative report hold separate metric across reasonably attain red colored advantageous good different architecture meanwhile acoustic random feature parameter achieve similar accuracy fraction dim million way intuitive convenient instead adapt report measure rbm dnn well perform dnn highlight need reach proximity relationship different certainly plausible space explanation different might combine individual dim dim dim inspire previous learn two preliminary dnn compute pre similarly label pre activation visualize display two scatter represent representation visualize phone label considerably around initial suggest cluster color form spread advantageous tool pursuit conjecture fan em science california em york york edu center ny com em france share question
useful presence miss interest allow slope intercept separate year teacher teacher persistent teacher score model teacher refer persistence parameter one special fix persistence scalable implementation scalable routine teacher model teacher copy cp teacher teacher year gp model product discuss alternative concern identifiability estimate largely structure pattern entry nest diagonal factored nest highly inefficient infeasible large technique newton nr routine square effect dimensionality reduction nest dimensionality grow user specify statement design matrix scale estimate tailor software specify also large model estimation correlate produce failure setting positive estimate accord cholesky root offer parametrization challenge dimension covariance estimation presence miss mixed treating latent miss behind popularity nr depend nr restriction need advantage presence furthermore em additional technique em algorithm nest model correlate effect refer initial maximization iteration compute maximize integration expression observe derive e definition appear generalized persistence gp equation apply g g let conditional remain likewise let correspond j j block gs namely calculation change unbalanced solve sort year size student year although treat table lr student year pattern observation covariance matrix addition denote contain include student let indicator otherwise corresponding observation follow step update unbalanced profile structure score present routine update appear definition appear unchanged derive equal diagonal g ig ig ig yield q although may em definite equation singular current teacher form distinction step definition appear case scalar persistence cp student link teacher year column year score function step solution update require result perhaps covariance relatively fast em produce hessian mle working correction observe score derive value calculate central mle suggest forward also useful r covariate effect year right eq htbp std year year year htbp cp program list figure maximum parameter r year identical variation effect r surprising correlation year good slow maximum lie nr failure cp correlation persistence model student persistence significantly indicate compatible pattern figure future simulation wishart credible correlation include specification distribution compare error predict calculate product em large standard error likely correspond gp interesting interval average standard error prediction teacher effect distribution model teacher residual independent student though violate assign student properly attribute student correct response analyze student status estimate teacher effect miss score miss depend student teacher history student finally relate later correlation usefulness measure teacher depend aspect teacher standardize careful teacher mind valuable improve year teacher student strong later variability valuable year effect student score percentage drop teacher year student assignment estimate progress level unit offer method inversion depend effect total observation produce effect propose availability well sensitivity implementation facilitate provide deal relative effect student case flexibility gp may need able structure readily error predict teacher effect teacher effect include standard distinguish effect although develop well unit contribution health sequentially affect outcome level unit algorithm package membership allow expand situation student teacher careful basic step would remain application pd pd specification step however even usually near space variance long model student correlate much common problem estimate mixed occur pd concern future teacher notice portion pd pd semi matrix pd acknowledgment national science foundation grant finding recommendation material author view national science university individual student flexible literature although structure sequential level unit associate unit random challenge limited availability em compute implement example illustrate efficiency work publication control mechanism reflect publication subsequently mix popular describe unit primary belong belong level level static level child care live family visit multiple worker student membership model complex correlate nested structure particularly datum develop nested case paper algorithm longitudinal association unit covariance exploit speed membership example patient student teacher motivate research come score intend teacher student score teacher expect student potential issue variety different primarily persistence literature gp teacher receive end student therefore belong membership gp add context model teacher predictor teacher teacher score teacher represent teacher teacher heterogeneity causal teacher distinguish current effect would expect student teacher test complete persistence teacher subsequent year student persistence propose teacher year student year complete software teacher student future year though effect perfectly correlate refer persistence teacher effect impact student year multiplicative year multiplier persistence parameter teacher teacher year teacher covariance special gp current effect teacher assume general correlation detailed teacher problem membership iterative carlo chain likewise method estimate school computation require informative covariance teacher avoid need ml practically infeasible point calculate implement software development make calculation model many setting problem teacher student basis patient membership arise network example membership persistence measure membership since similarity former application patient different progress paper organize foundation em algorithm set school structure level response unit response ig ii covariate contain also level unit membership row nonzero value conditionally also observation together form linear consistency asymptotic ml meet addition regularity associate assume identifiable j identifiable school fitting find identifiable student move one next school insufficient allow level unit characterize process valid propose joint indicator accommodate informative teacher teacher score school link student score present specific student year notation depend model student history observation student teacher estimate student year score represent effect teacher teacher current year effect teacher distinguish persistence former teacher student g contain could
strongly see batch hold exist convexity fy list range inspire original linearize loss ms list second bind proof similarly result practical routine aggregation refinement grateful anonymous comment valuable comment cm thm thm universit es paris place paris france new recursive refinement set batch version strongly procedure satisfy optimal rate require achieve deviation weight refinement deviation second regret stochastic iid multiple keyword average individual sequence consider set recursively goal subscript closeness address predictive eq loss learner produce time quantify risk learner aim find measurable eq abuse aggregation good problem ms seminal book instead cumulative risk environment thank use cumulative risk temporal properly time aggregation risk bind risk apply jensen iid lower call ms rate ready procedure problem ms achieve fast aggregation call bernstein define properly procedure ms explicit regular criterion solve quadratic recursive greedy one opposite online achieve rule batch coincide excess extensively study fast rate use refinement describe round exist different none ms refinement crucial refinement thank order refinement distance learner aggregation procedure costly upper risk standard risk previous application bernstein quadratic counterpart instead martingale convention variation bernstein bernstein inequalities armed penalize erm apply theorem estimate successively f tf tf regret equal motivation notion optimality batch necessary appearing batch similar achieve verify center second hold introduce comparable ml ml batch convert cumulative cumulative predictive valid environment extend direction trick rate adapt rate order solve c gradient procedure figure replace linearize refinement q abuse good aggregation regret bound probability online theorem try practice bad case satisfactory introducing parameter learn weight extend tuned procedure order recursive tf f r bound excess learning rate sequentially recursively q classical rate q linearize learner give version range unknown procedure bound theorem second loss article nice consequence batch excess assumption restrictive generality result cumulative online coincide predictive aggregation boundedness small ms come require see restrict loss strongly iid extensive work modify linearized second order also initial theorem fast result online convexity optimality conclude bind well price version batch excess environment next useful bernstein theorem recursive difference classical recursive martingale leibl distance basic variational formula entropy originally obtain measure gibbs identity approach exponential aggregation naturally measure conditionally deterministic provide recursively cumulative respect weight inequality recursive obtain regret r follow classical order aggregation procedure heavily unique property issue consider rate precede sophisticated respect argument multiple cumulative rate let random apply reasoning equivalently inequality identity multiply inequality end corollary simplified favor rate rate bad initial differently adaptive tuned optimally batch obtain individual consider learner expert f notice rate depend exponential multiplicative solve concern modification describe adaptive form rate version obtain order procedure similar convex non cumulative adaptive reasoning theorem recursive argument weight center expression precede basic combine apply probability end choose trick proof effective range linearize tune rate restrict positive learning regret convex respect tuned procedure argument theorem exception adaptive case range linearize error let enough consider adapt reasoning small updating restrict range define correctly range second hold learning tune loss recursive argument proof index belong formula formula prove argument adaptive study adaptive pay variability adaptive loss avoid turn observe recursively thank risk provide introduction reasoning second order order go back moment online aggregation assume sequence see counterpart provide predictive aggregation first note tf tf classical argument adapt recursive conditionally eq fact apply
function hyperparameter call relaxation concave hereafter adopt prior relaxed j get question choose appropriate q target marginal way select consequence evidence type maximum mean probable explain incorporate constraint reconstruction follow ease without convex sequel cost convex nonconvex optimisation concave lasso reweighte jointly convex convex differentiable principle iterative formulate jointly globally w order define explain zero stage optimisation actually also iterative estimate close iterate stop criterion reweighte reweighte sequel heuristic study admm closely admm solve assume dual dual terminate dual stop set relative detail across share consequence call follow approach share new variable derivation simplification share share carry update solve eq k find sharing problem dual arrive satisfied break solve block involve square step collect single lasso problem stop reach predefine iteration penalty soft procedure collect I ip stop break classical biology consider natural frequency coupling strength represent couple topology know exact reconstruct time phase consist identify coupling parameter consider base add column series non natural variance create simulated collect include note reweighted package specify solving call sdp solver solver reliable class algorithm admm admm distribute subproblem implement matlab calculations core intel ghz ram illustration different ratio noise snr range weight reconstruction normalise snr snr show test different implementation distribute algorithm matlab worker time varied computation experiment unit large difficulty report size distribute reweighte processor note computation reweighte small matlab computations parallel core lagrangian parameter iteration reweighte number reweighte distribute reweighted core reweighted distribute reweighte nonlinear series dynamical regression interpretation use normalise classic deal network comprise admm reweighte previously still properly currently establish helpful ac distribute focus nonlinear formulate optimisation sparsity induce concave iterative reweighte optimisation algorithm design multiplier decompose subproblem use reconstruct reconstruction series name nonlinear form problem expand nonlinear model amongst function parsimonious description paper nonlinear use priori candidate system biological chemical electrical time dictionary objective identify explain good cast reconstruction nonconvex optimisation nonlinear solve typically biological e collect big reconstruction handle measurable measure combination build model genetic network capture law degradation mass action hill follow dictionary function I assume dynamic k exponential
model trajectory rise picture brownian bridge e constrain end state phenotype string end impose string dynamic point acquire node node occur lead acquisition trajectory transition start allow probability biological modification signal observer signal ensemble thus string comprise string could responsible compatible wish compute datum calculate compatibility support observe describe dynamic network give rise compatible probability observe specie consideration likelihood concern likelihood ignore simulate trajectory node ensemble particular network compatible high preliminary trajectory broadly uninformative evolutionary dynamic aim evolutionary support describe acquisition trait step evolutionary trajectory build ensemble trajectory probability current amount yield trial calculate take likelihood perturbation transition w obey detailed state simplicity describe quantity consider move proposal propose move propose start investigation confirm posterior converge yield report infer compatible coarse abundance describe possess distinguish decrease bundle bs cell adjacent decrease ratio bs cell volume abundance bs bs independent present order assign datum cluster depict blue algorithm partitioning hierarchical cluster branch partition assign activity activity bs bs illustration phenotype trait yield trajectory evolve accord signal trajectory every possible evolutionary unobserved character yet specie red triangle trait blue triangle triangle represent whose deviation block know presence absence trait absence score cluster partition evolutionary event shown suggest assign trait pair trait remove trait trait trait trait robust trait bundle bs trait link assess modelling evolution start phenotype trait acquire trait next event acquire trait list trait acquire axis multiple trait link acquisition evolution perform compatible different b network variation embed principal axis variation less distinct department sciences united department mathematics college united independently pathway trait unclear evolutionary major trajectory meta analysis specie intermediate landscape experimentally markov predict phenotype appearance determine trait acquisition flexible flexibility trait evolution trait surprisingly camera trait mechanism apply highly involve leaf also find least phenotype explore experimentally ideal mathematical evolution landscape understand evolutionary event generate specie pathway million concentration modification leave leaf biology high repeat unit bundle bs fig bs cell activity alternate generate bs cell co b I I c specie typically classify type trait genetic mechanism evolution gene associate bs involve element act independent co mechanism specificity parallel evolution part convergent however molecular know trait exist cycle one history evolutionary analysis phenotype phenotype intermediate independent characteristic principal component pca perform specie represent transition trait specie represent space phenotype connect phenotype I sub type type data activity five cycle confirm specie trait specie present specie study intermediate specie show representation pathway phenotype pathway specie blue pathway compatible phenotype specie c trait present trait use assign fig bit string trait meta define phenotype combination absence characteristic novel bayesian evolutionary evolutionary trajectory fitness underlie evolutionary dynamic evolution convergent distant reconstruction convergent fundamentally acquisition trait acquisition trait node label phenotype characteristic label characteristic landscape weighted occur constrain inferential model hmms fig simulate chain transition chain evolutionary pathway pass several record network mcmc meta represent dynamic evolution characteristic acquire path compatible specie uninformative order acquisition trait generate mathematical material path generate require impose trait acquire acquire trait nevertheless able trait evolution test positive control evolutionary event pathway trait acquire simultaneously clearly assign acquisition trait link underlie fig simultaneous acquisition trait evolution trait obtain diagonal time four grain artificial artificial specie replicate miss meta generate artificial miss compare absence quantitative phenotype specie abundance white phenotype already grey predict approach indicate phenotype predict phenotype evolutionary trajectory specie calculate phenotype trait outside one trait presence dataset choose datum predict phenotype trait phenotype publish trait neutral strongly trait correct assign neutral highly produce phenotype yet describe quantitative real time experimentally verify abundance successfully infer evolutionary dynamic prediction verification illustrate identify feature evolution resolution evolutionary event probability trait combine evolutionary consistent subset bs specificity occur high insight event well specificity predict evolve prior primary cluster intermediate specie use acquisition trait evolutionary diameter trait acquire evolution bayes acquisition deviation trait order acquire early c bundle bs strong bs c cell increase abundance increase acquisition trait consistent analysis specificity verify robust score absence trait oppose fig hierarchical difference small trait evolutionary trajectory affect produce highly change phenotype subtract trait remove trait predict remain trait appendix increase deviation observe use might affect additional trait trait widely species position despite occurrence trait unclear acquire early importantly trait trait analysis suggest trait likely evolutionary therefore aim trait predict evolve link consider phenotype genome trait contingency subsequent trait increase detect multiple underlie multiple trait pathway connect phenotype evolutionary trajectory indicate evolutionary pathway c trait acquire capable produce phenotype material trait bs acquisition fig multiple pathway trait time investigate I entire infer transition reveal distinct largely infer full comparable represent event generate trait c propose constrain broadly evolutionary pathway probability sub broad difference pathway differ event generate c I differ primarily evolution bs I I convergent I traditional basis difference detect evolution example I density early evolution I specie trait broadly bs acquire pathway generate I trait majority fig consequence evolutionary response non furthermore evolutionary sub restrict phenotype space discussion bayesian infer landscape powerful conceptual transition trait challenge incomplete state report development able evolutionary highly stochastic trait dependent trait record also limit predict infer pathway underlie occur shorter disease cell decrease rate create pressure evolve strategy abundance occur evolutionary majority pathway al change within predict bs cell predict bs specificity acquire early majority bs specificity predict suggest occur recent evidence identify environmental frequency leaf nothing well mechanism key modification leaf evolution bs division appearance evolutionary pathway sub trait evolve traditionally cell specificity I mechanism evolution type therefore different evolutionary leaf c type detect difficult explain early determine phenotype type restrict pathway leaf development biology provide change convergent flexibility evolutionary independent range upon trait upon phenotype specify diverse molecular mechanism series highly complex trait show evolutionary trajectory trait appendix limit small subset recent leaf fitness landscape trajectory pass phenotype specie landscape mechanism generate abundance cell specificity
consider property design element sensitivity subset sensitivity cf estimator base restrict condition usual error independent zero sub row sub independent provide bound role diagonal diagonal diagonal probability least n tw tw tw find mu selector selector thank design formally almost imply imply mu selector selector importantly u tuning p selector attempt term precisely constraint new selector estimator sparse belong selector eq set addition constant assumption admits follow bind establish establishe realize eq lemma imply argument lead imply q next condition view eq literature inspection hold assumption satisfy compare theorem zero mu selector selector probability term mu selector whereas term selector upon original selector whenever additional estimator term selector obtain exploit additional finally go zero maximum finally event cone since due immediately term obtain combine selector follow unknown assume penalty compare selector also infeasible know selector use benchmark ccc ccc rmse pr c rmse pr table provide literature ignore issue lead worse see infeasible easy estimator uncertainty know essentially propose nonetheless fail eq similarly substantial reveal zero simulation constraint exploit optimization potentially constraint ccc ccc rmse bias pr ccc pr parameter seem kept constraint improvement small sometimes test value make perform improve substantially essentially constraint help severe helpful recommend keep case additional appendix auxiliary brevity minimization least use matrix eq selector lemma feasible consequently imply note imply selector probability n least hold fact use together remark view initial least since hold event least complete proof suffice last proposition remark different generate motivated consider assumption explicitly observational propose design applicable require moment observational applicable observation proposal covariate literature comparison convergence model arise error design unknown parameter assume belong set component estimator arise standard selector unstable literature boundedness q denote design set selector selector selector solution level sufficiently offset grow literature independent estimator converge rate motivated bias mu selector minimization problem diagonal accord rate selector selector contrast mu selector design small programming component combination adaptive attain computationally feasible cast mild estimator q sup minimax optimal usually vector know estimator relaxation depend sparsity matching focus subgaussian analogous consistent recovery convergence propose
runtime synthetic confirm simplification give specificity short short path start vertex path source shortest path general acyclic graphs single material structure improve complexity since read would respect describe scale linearly another rule together point inside inside note pattern message pattern extend dot dot see extension general u come interact pair figure locate message therefore define possible p u sg way computation case expression consideration complexity l p assume dependence g dependence introduce rule leave modify algorithm without show weight rule interval satisfy present start calculation message reconstruct find family base px xx need weight parse possibility variable approach likelihood computing turned compute expression replace accordingly turn division compute moreover compute multiplication lead sum polynomial appropriately modify marginal efficiently require difficult sum compute sketch material interaction interact computational synthetic datum solve shortest take depth contain interaction pattern distribute value take varied pattern part dominant varied step value start show dependence global sum crf literature general slow show namely may field chain free processing encode view crf model combine advantage hybrid paper analyze task fast label formulation observation sequence take value appear domain bioinformatics approach field j concentrated subset call pattern word occur define x parameter depend word crf intuitively sequence application natural process syntactic construction secondary structure angle associate configuration sequence suppose model local nature language I probabilistic sentence language gram equivalent represent sentence equal gram know language frequent gram sentence syntactic describe free syntactic structure encode structure equivalently accord give another identify rna sequence certain discretized angle nucleotide sequence label label alphabet local rna secondary model complementary generalization context rna optimally parse interact parse include weight model motivate symbol weighting parameter probability view way define encode long crf problem posteriori minimize polynomial complexity state compute polynomial pattern algorithm appear refined version handwritten character name text optical recognition protein string string extend pattern correspondence probably close represent correlation gram unlike probability successively model slightly mix apply rna secondary former equal show done ignore interested ratio model total hybrid probably intractable conjecture thing argue computational advantage hide require sum minimization subroutine gram correlation unweighted would like combine context language define language parse alternative choose directly complexity admit certain restrict third weight motivate kind interaction albeit roughly speak simultaneously overlap restriction inclusion restriction fourth focus energy depend cost general length pattern general present message standard parse algorithm interaction message compute interaction belong belong partial message message interval variable pattern interval plus change assignment message approach good parse start apply rule need know end fig pattern avoid set e value message argument intersection optimally upper equal count induction compute check step equivalently serve argument message definition compute message assumption via case correctly serve induction optimal optimal first responsible word pattern optimal variable message could segment value q equal message message correctness whether correctly complexity go function proper f lemma complexity dominate step describe interaction successively iteration base parse rule computation relatively update computed parse start divide piece equivalent short short graph correct depth message parse applying since rule parse fx c j option either u u c inclusion patterns intersection set pattern thing interval statement obvious serve k variation preserve conclude k u formula need orient suppose sp suppose parse start rule rule ss ks u rx kx proof order kx parse parsing
whereas selection orthogonal ability deal include operation linear unconstraine focus alternative sometimes constrained optimization bind frank wolfe fast project algorithm cg paper contribution derivation atomic take atomic cg explicit constant proximity arguably pool ball root projection organize atomic derive computational tool regularization proximity section cg project gradient optimization result bold column th upper bold component break arbitrary vector sort non sorting matrix naturally wise increase weight negative ball illustrate fig htb w w low eq chebyshev inequality trivial show negative regularizer set atom compact atomic norm induce convex c atomic atomic atomic norm mathematical attract considerable atomic sign permutation group sign permutation use norm form fig e component strictly theorem atomic see minimal minimal atomic b example case atomic general definition appendix full dimensional polytope decrease sequence minimal atomic atomic inclusion b n unchanged define atomic fundamental linear programming see function use symmetric permutation symbol fact ni x notation norm thus arguably focus proximity derive new simple three simple match v result fact proximity fact immediately sign thus compute eq projection recently show onto cone notice onto monotone pool adjacent summary compute coincide propose mention worth lead operation radius ball compute e previous eq multipli minimizer duality thus lagrange multipli impose function suggest find technique monotonically root adopt van matlab na computation dominate almost lemma omit proximity argument efficient sort clear comment g lead sort operation onto v sg standard formulation regularizer fidelity typically refer three sense solve adjust formulation adjust address efficiently proximal fista proximity direction multiplier alternatively gradient cg frank wolfe appendix b gradient algorithm accelerate project projection onto ball cg projection simpler cg well suited tackle ball free require b norm value coincide factor follow cg address norm f h solve reasoning initialization k kk predefine take trivial benefit duality accuracy iterate stop implement dominate sort total prove denote solution cg q define iteration require optimal like theorem loose follow duality local step duality precisely typical gap subsection accelerate accelerate iterative thresholding bb application solve implement bb backtracking approximate x k stop fast thresholding variant acceleration nesterov fista backtracking know fig fista h cg fista matlab window intel core ghz processor ram similar accord column radius stop iteration total evolution surrogate confirm surrogate gap surrogate duality iteration compare backtrack address formulation accurate tight stopping show backtrack cg sample standard gaussian observe fast fista problem cg h c breast imbalance yield randomly iteration repetition table cv fista fista backtracking fista backtracking cg formulation norm atomic cg atomic dual exploit cg regularize arguably proximity establish pool adjacent efficiently compute experimentally cg accelerate projected gradient show former work application namely logistic briefly basic concept mention fundamental polytope hull set affine hull dimensional state convex polytope theorem degree moreover since sign permutation argument triangular invertible x x rearrange since x permutation
layer deep network differ pattern appear detect reality structure distance advantage must filter unfortunately major capture resource wide feature detector detector hard train likely multiple prevent would train even scale detector capture scales independently share presence absence pattern convolutional network si scale detect pattern scale pool response locally scale invariant convnet architecture differ scale rather scale multiple scale variation mnist dataset multiple scale si scale variation scale complementary convnet make available research incorporate deep feature boltzmann rbms infer high model transform filter large transform densely retain inspire incorporate extremely successful convnet pooling un tie explicitly scale require learn un apply semantic convnet learn sharing detector share propose convnet layers fed scale invariance influential pyramid tie parse layer forward propagation apply connected layer align allow expense increase final restriction capture range contextual interaction scene interested capture invariant feature image unlike pool response subtle effect middle size circle circle architecture scale recognize layer architecture circle two detect circle expense redundant apply parse effectiveness modular incorporate invariance convolutional network feed feature detector layer regressor optimize jointly via gradient computed propagation convnet usually activation spatial pooling idea detector image pixel nearby strong detector extent get convolution operation tie weight location send layer single sub pooling invariance amount detector regardless spatial image invariant convnet si convnet output invariant overall two layer convolution bold box detector figure pyramid scale brevity across convolution scale normalize response pool obtain representation size b scale multiple pyramid come size align map inverse max pool scale spatial pooling multiple serve locally scale allow convolution linear image transform transformation convolution transform detector convnet learn whose scale layer match scale path win bold line pyramid filter analogous filter scale train expressive increase fitting image dataset convnet epoch test seed convolution scale parameter material propose boltzmann rbm protocol fold error deviation si convnet convnet hierarchical convnet hierarchical convnet slightly convnet overfitting convnet convnet convnet introduce pixel exist si convnet learn large rbm fact rbms unsupervise architecture extraction original invariant rbms comparable respectively rbm si convnet si convnet improvements good invariant rbm network convnet paper achieve invariance neuron input unit times neuron response input record robustness neuron arbitrary score neuron ratio report score neuron please step score convnet si convnet layer see pooling response scale si convnet report scale convolution actually si train wide network less demand si filter filter well many pattern h plot consistently outperform detector observe increase test convnet consistently convnet gap si mnist wide variation account si scale scale evaluate scale factor scale vary correspond away mean challenge convnet si convnet convnet worse mean shown verify si convnet outperform convnet end relative lack symmetry around digit human digit robustness keep training factor range si si low rate scale variation show redundant variety si resource h architecture scale representation convolutional net share across multiple scale single detector arbitrary scale feature pool invariance convolution incorporate scale fitting si outperform aspect order align layer write forward matrix encode bilinear interpolation vector multiplication toeplitz propagation size encode transformation response convolution kernel encode toeplitz equation element wise multiplication apply derivative signal stage linear weight way convnet error spatial accumulate layer discuss convolution convolution convolution compute invariant use large scale different scale quadratic geometric sum convolution process subtract range three unless specify si convolution follow another layer feature
element independently distribution desirable property embed approximate embed hamming pair th angle variant distribution easy sake easy bit normalize hamming distance approximately preserve angle unfortunately row make hard analytically understand rand analytical hamming hamming distance generate generate dimensional rand variance rand repeat whole average surprisingly curve variance indistinguishable bit rand intuitive locality sensitive lsh slight still identical significantly hamming variance bit curve distortion distance embed come recent type slightly number projection preserve high also sign dimension preserve distortion one randomize utilize propose learn dependent fashion minimize opt alternate objective bit comparable empirically try row uncorrelated help reduce redundancy code orthogonal vanish rotation distortion orthogonal similar include hard find propose optimize perform input time optimize dft lead element update recover derivation dft project norm preserve formally arithmetic matrix conjugate transpose trace furthermore conjugate optimize becomes decompose minimize polynomial solution hard solution cubic bivariate system minima consider overall guarantee increase alternate optimization run bit optimization kk k understand frequency make domain propose remain heuristic conduct three long internet represent imagenet imagenet represent third dataset imagenet imagenet contain image dimensional unit version embedding bilinear bilinear bilinear opt version bilinear embedding well call code lsh applicable feature long much space show relatively datum high scenario set query recall evaluate query instance define near base experiment generation retrieval experiment fix bilinear order bilinear square full bilinear projection bilinear ghz core indicate generate bit code projection bilinear time respectively compare time fix fast factor due storage availability highly optimize library suitable gpu preliminary test gpu speedup cpu fair comparison bit computational bit opt rand times bilinear opt bilinear rand hundred fast lsh time bit make opt rand fast bilinear opt bilinear rand hundred fast lsh bit bit less bit computational identical opt rand rand hundred time lsh three top show method yield well lsh bilinear code margin compare different rand almost identical lsh hundred opt rand bilinear bilinear rand code save set linear svm code show give randomly degradation lsh bilinear classification task lsh opt opt compare conduct bit dataset shown opt bit gap become much suggest extend incorporate achieve add distance pair supervise frequency optimization update fix q eq supervise version auc imagenet embed long code optimization degradation compare expensive time full high require preserve suffer binary binary project compare dimensionality alternatively minimize extensive approach give provide degradation bit become retrieval massive computer vision biology finance code bit binary approximate retrieval happen directly fall typically hundred thousand linear follow bit eq
view great pac bayes attempt gap development practice propose pac view analyse generating make explicit analysis tight main advantage template approach knowledge rely regret multi classification error assumption pac computable paper see assumption pac encode term study name recently dependent place encode inference impossible illustrate pac enable adopt gaussian treat significantly bayes view agree example first gaussian last centre formulation prior expectation unknown datum estimation finite rest pac svms give multi involve multi svms evaluate bound lie suppose average provide bind bound svm represent space induce function centre pac bayes ready svm bound return average stochastic bayes analysis multi view concatenation p agree identity view view feature example feature though explicitly employ theorem divergence complete contain expectation unable prior satisfy natural namely representation prove margin classifier develop estimation bound error multi semi definite function semi inequality logarithm concave pac bayes involve outer actually determinant equality nonzero eigenvalue inequality independent irrelevant posterior give whose locate still pn divergence moreover inequality omit pac multi pac pac formulation augment feature representation formulate form explicit svms scalar balance weight usefulness evaluated set uci repository include index index feature form view view datum original view obtain partition form form provide svm train view simultaneously svms adopt fold pac multi pac datum normalization feature representation test bayes svms tables multi pac bayes svm multi good far enhance pac trivial sophisticated scheme four pac classifiers pac bayes view promise fill multi view learn possible explain usefulness experimentally possibility pac view learn could far adopt expectation w another motivate pac multi relate support foundation china project service
iv lemma confirm limit note bf model b iii bf strictly bayes bf variable consequence bayes true go consistency identical bf observe orthogonality design give l converge belong l l I hyper prior consistent g dt dx lemma thus n define orthogonality expression derive z dt z dt constant go proof prior affine cccc n k z map act non p ib ap p ap exactly invariant affine problem order monotone function denote normalizing density c j function early stochastically j j strict strictly happen g integral finite density p integrable away zero equal strictly finite proper limit sequence posterior proceed k k pdfs ordinary rhs equality c ab g r k dt true z z lemma e x lemma enough arbitrarily small denominator enough whenever n finite c normalize constant dt f f arbitrarily small claim dt dt b limit complete argument bf r jj ta describe laplace definition appendix behave laplace complete remain laplace integral go go infinity belong must possible group non om om j om along translate happen since nr tm b om om om know one equivalence block om laplace bayes case reasoning converge mean square prior expectation shrinkage dt k k b dt show integral b dt k dt dt algebra make standard lemma expression k z b q hence denominator vanish b proceeding ok ok p necessity tr tr dt strictly coefficient column triangular matrix hyper prior triangular stay yy yy yy dt tr proceed q z b z ok ok necessity constraint tr dt p bayes strictly j k r f imply I j r thus ir explain variation k j j I I fix yy compare yy shrinkage near intercept dt dt dt come result limit bound zero additional corollary section prior traditionally sensible characterize lead thick tailed prior hyper limit prior paper new mixture reveal argue undesirable scale coefficient place hyper avoid provide sample asymptotic normal bayesian method regression central selection conjugate form due ease ease update prior posterior theory conjugate quick factor popular impose analog prior ridge selection prediction consistent estimator desirable concern nonlinear shrinkage place normal regression pareto gamma representation use place discrete model implicitly near hyper placing along investigate selection search hyper particularly mixture retain perform study limit insight substantially wherein inference least prior include suffer mixture prior introduce ordinary prior predictor separately investigate tractable situation fashion group block introduce behavior show poorly criterion hyper theoretical new orthogonal dedicated examine hyper brief discussion supplementary material response traditional include inclusion represent intercept error place prior along retain intercept traditionally transformation provide argument common flat match simple closed expression compare coefficient determination error least value suggest consideration review prior behind variety lead careful result thick tail prior marginalization mix mix estimator least square often primarily prior place proper suggest bayes hyper bf describe several undesirable commonly associate prior reveal take rely information limit hold old summarize anomaly choose hold approach irrespective undesirable hold follow g factor though grow undesirable avoid prior careful mixing provide hyper criterion study limit produce initially see qualitative description formal prior estimator limit limit drive rather mixture exhibit arise asymptotic accept connect behavior limit drive phenomena problem write element linear hold fix variable consequence consider drive produce denote element immediately hence several asymptotic drop subscript result otherwise hyper prior define provide appendix estimate situation conventional zero coefficient leave unchanged appendix hyper suffer call length irrespective least regression big place weight size grow infinitely result tend behavior matter zero toward attribute proper hyper prior posterior define ig material show describe exhibit generalize prior specify situation prior suffer irrespective robust suffer recommend irrespective material arise result predictor must affect similar explain portion mass nonzero coefficient behavior avoid multiple latent place approach concept local shrinkage normal include regression typically avoid limit concentrate hyper prior corollary center limit finite corollary material regression let eq incomplete gamma ordinary prior e hyper eq prior three notion consistency first seven criterion bayesian consistency directly particular apply model model follow consistency hyper prior hyper prior information size block correspond block arbitrary coefficient condition proceeding need notion include block x basic design true establish slight variation prior second bayesian ensure bayes model describe early also strong appendix hyper prior hyper produce consistent concern bayes square tn use form satisfy block hyper consistent proved show prediction regression model satisfy prediction novel prior behavior use scale common argue undesirable shrinkage coefficient presence avoid light bayesian component model comprise consideration prior suffer consistent aspect failure prior replace g hyper share hyper successfully provide derivation development theory raise practical question good select identify predictor measure likely comparable relate explanatory knowledge place correlate existence sometimes indicate previously construct scope investigation establish setting result orthogonality condition relax modify suitable block elsewhere mean dt z increase state g b k b k note finite number k second vanish limit z proceeding denominator b b ok ok appendix material transformation design generality projection represent triangular hyper upper triangular stay yy yy sequence r dt define q proceeding q exist b b z b later k ok ok supplementary material appendix
generalized nonzero order case website thank helpful suggestion p scenario regression pool adjacent operator application decay move away area idea simulated feature regularize parameter solve tune convex large add call result derive application time predict move constraint reasonable determine constraint paper contains order standard order real simulate section auto series traditional criterion degree freedom order work lasso constraint ny setup sense natural convex modify write use component absolute monotonicity strongly encourage interaction solve programming algorithm proximal subsection intercept illustrative purpose cc h elegant obtain obtain adjacent subject solve hence decrease expand operator minimizer compute k order adapt elastic net show order lasso generate plus coefficient colored profile order lasso job recover fluctuation coefficient observation profile colored profile estimate coefficient bottom relax ny subject encourage monotonicity problem derive procedure create extra simply place first outcome predict outcome outcome outcome henceforth continue omit intercept ik ik predict predictor j plausible unit back first write follow block hold block lag choose zero order predictor kp x series different lag x write convert section large detail q block correspond predictor row lag coefficient block kp tr x j time four lag predictor true coefficient order blue plotted job true coefficient average time generate deviation space four predictor leave figure square standard order give mse panel randomly thereby violate noise order coefficient monotone reverse true achieve much mse monotone set eight california divided figure curve measurement day order lasso achieve degree coefficient order order interpretable predictor time lag beyond estimate coefficient wind lag day time beyond time lag h cm predict lag fit time series proposal seem lasso ar derive property suggest autocorrelation lasso example contain represent auto fit validation panel fold behave lasso monotonicity give picture three set htp p
get conclude entry entry eq q otherwise conclude proof key proof purely extension count variation process eq j thompson inequality e e te l f last give conclude martingale use martingale purely martingale moreover mean non intensity bernstein notation introduce suppose hold q bernstein purely consequence case particular let assume process ds hold particular take homogeneous knowledge sub adjoint hence force symmetry z obtain technical proposition p jump entail hold cf successively gets give conclude proof martingale martingale whose quadratic variation analog line point definition case purely martingale replace let z v td cf u tm quadratic row omit subscript get easily eq result n get get optimize z low matrix precise goodness reduce prior connectivity user low filtering induce penalization trace nuclear consider procedure minimization square penalization trace consider process focus derive one stand differentiable subdifferential q give lead way fit h old together use subgradient bind motivate eq h depend intensity variance elsewhere cascade common keyword tag connection message cascade observe cascade counting ct ct user hence depend cascade along cascade reconstruct cascade user motivate intensity cascade decay function decay cascade fit functional model dt end dt matrix martingale write entry ct concentration depend norm algebraic structure lead control market transition finance laboratory universit ed subsection simplify drop index ambiguity aim decompose easy one index entail z whose invertible write computation eq z line obtain eq recall compatible along therefore shall entry j recall use lemma q use finally rgb proposition matrix case purely trace calculus appear statistical system study counting process component new concentration inequality around matrix version chernoff scalar result joint quantum entropy physics base stein extension scalar hoeffding inequality application particular compress simple completion large work concentration inequality see instance concentration however available tool calculus paper bernstein purely see hoeffding probabilistic appear naturally problem system network stand approach social fix connect link edge click etc consider network user action approach development cascade survival organize notation paper purpose essential obtain concentration section bernstein purely whereas study sharp counting quick illustration concentration rank strategy penalization relaxation trace reference sum sake clarity technical f augment nan continuous small value say trajectory x entry assume possible index adapt conditional expectation column equal size context stand square stand stand trace operator denote another notation kk jj cm adjoint denote moreover symbol stand
typical surface output series amount large similarly return occur colored prediction standard deviation uncertain htbp color deviation four asymmetric surface skew contour large relationship contour parallel nonlinear sort skewed quadratic asymmetric transition attempt variance make environment become intuitively except market environment previous asset period volatility section surface grid various band figure correspond section nonlinearity return look pass financial setup make observation short computationally simply learn predict author setting use compare combination setting likelihood n table outperform predictive four inference summarize little expensive financial dataset avg calibrate still expensive implementation inverting covariance hand cost cost gps gp time vary gp space nonlinear present online particle batch method generally financial improvement nonlinear intuitive clear direction relationship volatility price market interest learn pricing derivative speed attractive live tracking accurate limited variance learn overfitte address change gp distribution flexible develop main overfitte offline financial performance often exhibit volatility large return financial frequently period volatility phenomenon volatility univariate capturing autoregressive generalise far inspire variant extension find variant address volatility past negative effect volatility return introduce functional add asymmetric term fundamentally learn likelihood volatility gaussian flexibility unknown variance effect return introduce new parametric volatility flexible place gps furthermore explicitly asymmetric effect volatility evaluate series financial predictive functional automatically asymmetric previous attempt capture main gp work gp focus filter smoothing gp transition dynamic paper inference financial linearly time eq model flexible several limitation term flexible introduce negative return volatility extension asymmetric return asymmetric capture hide hmm assumption limit place state way hmm transition fix inference previous gp develop method filter dynamic dynamic apply em learn gp dynamic use learn similar much call gp real function gp function encode system dynamic return function output I output highly correlate model thick size black node thick connect thick f x connect x connect connect previous enable asymmetric finally variance however gp learn unknown denote challenge task fortunately introduce dependency particle avoid particle adjust chain origin quick filter unknown hyper standard introduce add artificial dynamic forward backward never later consequently distant gp collapse state near input adopt mcmc monte procedure establish framework hide additionally develop sampling markovian learn prior sample particle estimate parameter observe index accord propagate chain forward adjust eq posterior parameter particle part alternatively sample current conditionally slice drawn filter auxiliary filter generate conditional collapse smoothed trajectory alternate particle draw slice particle include learn transition synthetic recover hide gp series measure likelihood predictive finally performance term execution accord equation linear encodes asymmetric volatility covariance give state hyper brevity typical plot hyper figure filter particle show blue generate synthetic gp conduct dataset low daily exchange fx price total eliminate particular eliminate price market return mean standard deviation return training make evaluate
aic even safe drawn method issue make version quite surprising even constrained value see even draw distribution regard training equal covariate shift process goal bayesian probable aic cross validation lead aic typically lead unlike small goal aim predictive concern stress unlike model criterion rather dependence help get thus select comparable highly prediction select predict e frequentist imagine draw within usually predict unseen datum define far vary aic generalization I u throughout additive term divergence interpretation sample estimate independent aic asymptotically select minimizing select close truth kl divergence sequel omit classification two set may interested behaviour give adapt replace capital letter contrary may vary represent slight abuse notation fix logarithm expression square give attain another addition ordinary example linear form correspond error appropriate aic intend bias datum derive evaluate old datum intend aic article structure extra explicitly concentrate remainder focus discuss behaviour aic experiment contain regard prediction conclude proof supplementary supplementary section aic aic point fisher analogously assumption kl divergence trace unbiased minimize term somewhat simple bad aic grow asymptotically aic good misspecification derivation lead generating specifie emphasize aic derivation work apply supervise datum corresponding identically assumption problem aic equal draw unobserved mutually automatically satisfied take assumption randomness learn instance assumption setting assumption regularity material variance use asymptotically unbiased estimator extra error aic new minimize estimator evaluate depend distribution density depend additional evaluate variance one explicitly q equal n extra analogue similarly accordingly believe penalty concrete base formula aic except extra input appropriate extra prediction supervise might replace input computing computing follow aic retrieve special bad distribution recommend course case follow distribution applicable yield analogue nothing point contrary already model criterion implement away input complex discuss section input training extra inconsistent case choose model selection focus desirable input value give focused selection hard focus consideration focus recommend evaluating criterion usually continuous undesirable continuous analogue curve quantity penalty input property apparent exactly characterization follow design linear training set expression aic place possibility aic also show trivial extra mention mutually aic go concern focus aic consider input iid almost invertible final design relate random input extra bias model amount grow aic evident formula term depend data likelihood log likelihood measure independent largely value computation variance great aic compare reduction come price variance selection hold large aic similarly distribution apply estimator affect experimentally small correct version several univariate degree draw iid unknown select compute perform function input intercept variance true fu u additive test input uniform mixture distribution test input report input another label gaussian differ model weight average correspond mean version method aic variant experiment univariate decide standard selection counterpart jeffreys likelihood variable unstable large respect polynomial jeffreys bic attempt probable give predictive probability three like recent focus design good though use unlike focus variance bias global bias available learning function minimize model large correct correct bias test bic cccc spike bic cccc univariate figure square risk table risk weight average variant clearly visible experiment perform expect center select obtain stable center outperform model adaptively input risk multivariate spike overall aic tendency go soon true aic continue matter large sometimes parameter small complex refer well assessment generalization test input achieve risk tendency complex available pick much tendency apparent experiment small cause note vertical axis training risk experiment perform aic difference notable exception error potentially bic try attempt probable give conservative average put select weight away multivariate rarely bad instance spike section select complex model near seem result bias elsewhere one observe switch near complex risk univariate multivariate experiment expect clearly aic occur bayesian predictive nx give model reliable must summarize prediction likely lose instance evaluate loss minimize weighted posterior factor use capture rely consideration find bic affect
four kind fold hidden layer neural operator fold space activation fold consequence collapse subset identify mean region identify offer restrict axis shift encode activation mean bound base network parametrization map lf linear region contain region least preserve perturbation parameter get number particular uniform even small exist volume attain work study piece region investigate number examine behavior mlp fold way activation unit analyze behavior correspond piece piece map piece layer start visualization propose perspective activation unit result example input consider visualization input identify deep mlp analyze deep upon result tight computable deep directly unit maximal overlap ignore remainder unit th subset weight p th select coordinate scalar eq act namely coordinate linear follow interval interval map onto illustrated restrict interval consider vertical line periodic pattern hyperplane composition pre treat compute deep layer layer identify generalize deep hide region neural input th corollary asymptotic assume hide width compute region deep polynomially model improvement em em l reformulate term behaviour efficient maxout feedforward layer unit activation f k jk two maxout unit maxout collection envelope view hyperplane maximizer region envelope show maxout region maxout intersection maxout number structure maxout intersection diagram describe partition maxout region region diagram intersection diagram trivial intersection region behave hyperplane hyperplane mm region maximal maxout look maxout maxout layer twice result maxout layer region turn case maxout whereby positive ray maxout layer maxout grow fast similarly respect certain maxout argument sec complexity feedforward network focus find superior machine piece identifies exponential region compute complicated replication help generalize follow identifie image low input region region correspondence describe adjacency intersection combinatorial simple hyperplane region reference neighborhood identify first layer network equal input distinct linear computed belong different function compute region construction divide layer fold dimensional per unit outline unit choice drop remainder define unit sensitive input right activation fold alternate activation pre argument view sum unit output identify cube unit cube way identify region multiply discuss use sec sufficient fold interval need interval interval x k function total space neighborhood map unit hypercube output th bias hide choose neighborhood hyperplane sec divide equal bind give otherwise complete divide sensitive fed layer activation pass next illustration computed pair fold axis maximal computable unit growth region per expansion number behave behave discussion give imply deep em em deep maximal grow polynomially fast maximal region maxout layer input maxout argument sec maxout unit value linear piece hyperplane boundary input entry hyperplane region go region give intersection hyperplane behave polynomially theorem maxout seed maxout point direction forget hyperplane normal whereby slight rotation output region composition coordinate slight rotation cone maxout open cone identical image shift input neighbourhood maxout width identify region input identify region input rank maxout parallel hyperplane region maxout mention maxout layer whose intersection diagram describe intersection diagram difficult diagram hyperplane understand nice maxout input bias way unit hyperplane intersection hyperplane panel illustrate maxout triplet parallel hyperplane region mention maxout platform feedforward without convolutional unit value array convolution array convolution map convolutional consider convolutional fall difference lie corresponding weight convolutional belong restrict obtain one single layer unit dataset conjugate minimize run misclassifie two layer capture region sec piecewise fully piece piece input map hide unit unnormalize cosine template input map map compute find keep use multiply adapt mlp piecewise activation maxout e g response possible response point large provide mlp face dataset unit last train regularization drop unit column project sigmoid unit regularization sigmoid layer visualize row hide visualize interesting response response activation positive similarly visualize output unit layer show visualization seven fig look difference distinct region invariance four map learn interesting hide map linear normalize attempt behaviour convolutional unit piece wise visualization visualization approximately feedforward neural map minor difference actual implementation approximate simply approximate inverse region train test plot visualize actual map show distinct input three unit face unit corollary theorem universit universit e universit cifar computable feedforward activation linear deep network able sequentially layer way compositional function piece exponentially compositional map contribute new depth family maxout network improve behavior unit keyword
snp every snp predictor gene expression model snp pairs strength evidence every snp calculate regression snp loading substantially suggest identify meaningful association panel specific datum individual color across snp loading type datum first derive sampling conditional represent full conditional integrate collapse sampler full factor w posterior proportion operation element column matrix calculate efficiently obtain straightforwardly replace expectation latent statistical exploratory analysis structure observation couple observation put bayesian hierarchical loading shrinkage factor column shrinkage remove reduce behavior validate compare result apply two study different identify gene co two genetic variant jointly ability guide produce unsupervised factor structure sparsity canonical association analysis attention recently ability exploratory rapidly low representation project gaussian I kk jj diagonal factor latent integrate factorization suggest contribute observation correspond loading traditional exploratory method analysis canonical latent factor model extremely framework load mapping scenario loading statistic loading matrix element sparsity correspond contribution factor effect contribute observe expression loading impose induce prior latent classical shrinkage achieve shrinkage effect substantial mass around tail allow signal spike sparsity induce sparse tractable relevance determination ard induce active focus incorporate loading cca pair feature vector identify cca cca concatenation loading induce sparsity combine estimate couple I result structured statistic wise loading matrix across representation covariance large subset study flexible enable avoid curse develop normal parameter beta shrinkage high property load globally unnecessary induce behavior factor drive allow loading either sparsity induce dense enable correspond feature matrix signal curse discuss sparse bayesian hierarchical prior illustrate signal matrix substantial performance model world section drug versus group gene genetic variant publicly extensively use low model transformation vector assume diagonal isotropic probabilistic diag represent analysis panel bayesian canonical group analysis propose pair canonical cca seeks find canonical project maximize description common latent model distribute definite diagonal dependency among residual within load orthogonal building bayesian cca I k common variation loading vector load originally inter analysis recently cca equation low rank factorization particular observation w p error factor loading mean share factor factor zero relate fix block group factor couple I motivate multi model partition observation concatenation loading block structure loading limited loading loading capture subset subset relaxed covariance observation model vector allow loading factor zero achieve shrinkage effect penalty put prior relevance ard loading share assume give hyperparameter precision posterior achieve wise induce penalty include either element achieve capture share avoid model covariance subset maximally ard penalty encourage element load interpretable meaningful research either bayesian loading penalty still go carefully structure shrinkage prior loading encourage element shrinkage parametric wise selection bayesian wise loading selection method regression little structured prior structure conceptually shrinkage prior shrinkage factor property marginal prior maximum penalty coefficient know double laplace lasso induce distribution heavy substantial canonical spike prior zero flat often model spike elegant interpretability loading exclude include model come many possible loading mixture alternative prior generally mixed term near distribution ard laplace strong heavy prior directly show model separately enable extend normal level induce behavior beta freedom smaller great towards let induce assign scale become shrinkage behavior mode near encourage induce encourage infinity generate put strong shrinkage hierarchical representation flexible representation make ideal induce loading assigning encourage element shrinkage enable load f induce equation depend hierarchy shrinkage loading zero interpret across load observed loading induce specific parameter loading estimate wise strength across local shrinkage sparsity loading level allow column wise give non behavior factor jointly dense shrinkage component dirac variation technical effect g large loading effect equation loading sparsity column wise sparsity modeling loading wise two outcome towards effectively remove column share induce loading factor model factor whether remove put flat loading z w loading loading loading column lead column sparse behavior model factor load column couple h component variance application local hierarchy variance wide couple simplify section regardless develop variational maximization loading column specific estimate jointly fast load addition markov monte mcmc use mcmc update loading parameter start warm encourage robustness variational model orthonormal rotation p produce identical traditional restrict loading triangular carefully right multiply loading multiply structured prior put constraint sparsity regard zero desirable structure loading low practice load sufficient solution address switch identifiability address trivially label switching switch either mcmc estimate sign simulation follow arranged loading match loading sign match switch estimation element sparse loading simulate pair observation simulate compare relate pair simulation small reflect structured observation factor factor table factor element load randomly element factor generate column error specific dense loading generate vector context sparse factor specific last include simulation four sparse four represent dense vector cca ard ard cca cca cca put ard loading encourage wise shrinkage extension ard multiple couple observation run ard cca aim canonical direction correlation maximize loading cca classical cca matrix non singular simulation regularize cca add two matrix accord leave simulated loading I constraint transformation loading choose projection true factor couple cca maximize project original sparsity induce produce encode orthogonal minimized concatenation choose matrix u vertical concatenation transformation frobenius load recover column column switch splitting rotation well quantifie invariant orthogonal rotation switching scale zero extend stability index couple regard loading recover load distinguished value load five loading sparse loading loading load column dense loading affect separation sparse loading matrix treat loading separately recover component final evaluate number dense four run start factor identify number run run correct factor run recover matrix multiply easy loading model panel panel recover loading recover column recover multiply recover random c recover loading model panel recover recover simulated loading inspection among sparse factor ard ard limit sparse ard poor figure loading well covariance unable subset poor identify c well across size ard sparsity figure figure sparse loading ard recover loading figure dense loading factor include figure c reflect figure large well panel across four b comparison loading four size across loading large well recovery small indicate method b across panel loading datum study hour exposure buffer represent gene project quantile apply initial initialization estimate estimation datum recover proportion calculate explain total tr proportion explain buffer sample panel proportion factor type order display panel count scale covariance control share specific control level biological loading illustrate gene cluster run three factor sparse share gene go david buffer term david significant annotation gene treat gene gene annotation annotation annotation david annotation term
specification r cell avoid mark iteratively update step moreover since choice n temporal logic temporal logic mdp decision probably reinforcement logic specification pac keywords pac consider synthesis logic specification unknown environment probability build call model base probably approximately mdp methodology attain probability sample polynomially logic specification horizon maintain initially unknown base logic transition finitely satisfying predefined integrate learning allow complete unknown gradually control effective correct meanwhile obtain paper logic system incomplete knowledge probability planning robot operate affect action differ level robot model movement possibly number approximate actual optimality temporal specification thesis efficiently controller satisfy temporal specification maintain different action approximate exploitation stop either policy information logic specification measure learn logic verification use infer probabilistic deterministic game chain great quantity admissible policy restrict path computationally temporal specification efficient employ inference yet exploitation controller bias apply model may synthesis temporal logic share apply specification guarantee efficient specification improve exploitation probability specification tuple initial transition atomic atomic stability build proposition boolean always represent acceptance tuple run word accept appear infinitely give present quantitative synthesis specification j v ss ff si deterministic policy quantitative logic objective chain v v chain state visit reach unique path follow notation paper path start exact path eventually chain involve denote pair empty define connected edge graph denote state visit state structure give specification specification want end policy end component follow synthesis quantitative objective practice underlie one motion reinforcement learn model use knowledge update eventually policy success specification product horizon h v tf u v v value understand transition neither give expectation eventually step accumulate reward design output value overview full space know visit many time unknown maintain update consequently partition unknown estimation state compute target horizon state state unknown state identify learn result policy informally state satisfy specification less satisfying specification minus small policy encode atomic proposition omit product acceptance condition specification probability follow assumption associate time transition transition probability large variable extend label approximation share action construction approximate learn learn true model estimating policy unknown learn observation error u range second requirement learn achieve close therefore temporal logic influence potentially horizon potential influence suppose optimal policy directly horizon close satisfy eventually horizon mix let u ft exploitation strategy exploration exploitation make knowledge exploitation insufficient zero encourage agent nearly optimal notion confidence interval action probabilistic encourage state aggregate initially unknown product fig figure respectively near optimal product rapid exploration unknown statement visit simplicity gx gx write probability set visit infer xy u tu reach state one corresponding end visit infinitely sufficient become known use logic formula input mix let policy return firstly state polynomial step induce lemma policy attain unknown state visit explore efficiently finite step polynomial mix value obtain achieve policy need unknown specification synthesis product provably efficient learning eliminate detailed discussion elimination tm q ss h h possible state initial explore state tuning run motion planning implementation
wavelet tree widely segmentation denoise document categorization determination wavelet e state mention estimation page gaussian hide jointly wavelet wavelet provide flexible handle latter work well denoise detail exploit lower develop em bayesian state integrate demonstrate conclude matlab code available abstract transform edge parent wavelet conditional modify frequently application assume mean structure model view wavelet child else typically application wavelet state coefficient mass density nature obvious model generic unknown stress dependence consider independent kp finite density index distribution wavelet dependence structure wavelet page property page coefficient neighbor also page variance initial unknown usually variance denote level ps ci appear mixture model use wavelet parent tree log hide level parameter level difference moment wavelet discuss parameter composite consider full parametrization I structure variance recursively conversely parametrize ci ci appendix particular derive e node path assume level induction depend node furthermore em difficulty two marginal child wavelet calculation forward numerical limitation modify replace finite state integration notice gauss quadrature rule quadrature consider involve smooth em estimate composite likelihood wavelet parent child distribution handle sequel likelihood term full child th wavelet composite li assumption estimate denote vector log likelihood apply l rr obtain composite handle mainly conditional marginal distribution likelihood estimate meaningful replace make sense marginal return local composite return indeed satisfactory approximate posterior density w ps complicate handle chain monte carlo end view approximate bayesian transform corresponding show histogram along fit figure fitting vertical panel provide vertical level equally additive noise pixel work orthonormal wavelet preserve denoise wavelet wavelet white eq observation discuss appropriate state model approximate calculate calculate obtain scheme noise ratio image image level frequentist parsimonious image appearance model perform one provide leave image top panel leave panel original see top image posterior median either turn wavelet coefficient homogeneous lie quantify edge thereby coarse tree focus label wavelet label pixel first bayesian compute viterbi third wavelet arise hide first viterbi compute successively step finite maximization multidimensional log matrix additive omit matrix definite third iw ip specify specificity j use pixel associate binary indicate haar mention present scale exclude wavelet transform simply pixel classification positive comparable notice present wavelet direction model variant haar wavelet transform wavelet perform signal determination wavelet note method estimate obtain variational may wavelet omit far variational natural sciences centre geometry advance foundation grateful thank conditioning structure immediately relationship straightforwardly derive appendix moment relation general wavelet consistent estimate wavelet tend consistent equivalent moment strong usual variance homogeneous accordance thereby appendix apply start density density expectation maximum point calculate density product density rule g suited approximate integral quadrature node repeat whereby return together obtain next
equally cluster size experiment aim size observation cluster item correct classification trial mean sc cluster centroid sized sized cluster detailed size centroid part health obvious difference compressive unknown incomplete compressive sense directly b completion follow compare relative completion cluster correct reflect handle part different individual number imagine half bottom half top column mean random rate may handle cluster may offer tradeoff couple offer improve method compare real data data california survey large survey conduct maintain center health phone extensive health health health health health services health health one major difficulty analyze truth come eliminate datum consistent individual replacement miss finally apply completion spectral cluster cluster expect rate decrease monotonically decay regardless reliable outcome even reach group simulate preferable compressive sensing giving recover explicitly take contribution bring mathematics health verify traditionally future performance type prefer aid design health survey miss thereby reduce burden receive university lemma question theorem conjecture remark analyze cluster health compressive sense spectral health datum test low misclassification completion accord advantage compressive sensing near health datum vast cluster technique mathematic refer separation meaningful within group health research unobserve participant response package sensitive local identification model fit also drawback method identify relationship individual measure matrix entry individual compute eigenvector separable dataset sort large eigenvalue entry plot cluster datum randomly large incomplete imply incorporate likelihood analogous multiple x score miss raw estimate compressive fast grow mathematic cs application low rank optimization ij recover completion since np hard due underlying completion complete provably noisy generate remove frobenius frobenius slightly recovery cccc
real triangular factorization implicitly compute factorization refer distribute stable dimension section platform thin svd thin thin q decrease orthogonal thin qr factorization thin svd factorization dimension compute svd svd svd dominant sample implement parallel assumption implement outer key row reduce sum process aggregation practice gaussians processor across affect hull projection datum normalization hull streaming read column disk twice disk normalize column simultaneous norm column norm triangular normalize read idea experiment nmf present section factorization optimize architecture reference restrict architecture intensive several reason dataset eliminate algorithm pass loading disk memory file output significantly reduce cycle many analyze read store disk loading dominant spent roughly measure sort single iteration read normalize show norm combine algorithm value pair matrix may optimal tb tb separable near stanford institute computational engineering gb ram intel ghz svd svd follow greedy gp require representative set approach choice qr svd transformation subsequent matrix million permutation word extreme matrix storage distribute file system function separation rank converge column select extreme extreme select recover separable matrix separation select coefficient identical tb test implementation configuration datum simulation location row coordinate temperature location radius construct near variability responsible additional rank structure nmf select indeed matrix publicly residual separation rank small residual gp quickly heat value residual curve five one extreme different remarkably characteristic case extreme close small illustrate tb heat index extreme look tb fc labeling phenotype individual combination correspond pe pc cd cd represent data interest pairwise cell marker vs marker kronecker pair cell pair marker abundance error quite nearly column figure residual define phenotype marker biological phenotype cell marker researcher omit still recover complete preliminary nature depth multiple similar desirable show coefficient coefficient diagonal compose nearby tb nonnegative need efficacy separation nonnegative date insight massive set structure heat show redundant would analyze scientific test additional practical impose requirement explore explore regime column rough long fit mean machine gb begin dominate solve office stanford fellowship lee office technology stanford fellowship fellowship thank helpful discussion stanford david nonnegative assumption row component improve transformation preserve separability pass suitable streaming multi core architecture efficacy size synthetic world matrix scientific bioinformatics nonnegative factorization nmf value entry general decomposition advantage nmf column pixel coefficient reason nmf broad application discovery hyperspectral property massive hundred million feature bioinformatic many concerned algorithm advantage large scale new technique orthogonal transformation particularly compare community computation implementation easily see begin show correctly recover heat software online remainder review computing issue unfortunately rank finding minimize np assumption separability q index notation live hull extreme index separability tractable nmf near separability live hull extreme algorithm near separable typically base describe determine focus efficiently implement pass severe restrictive algorithm efficiency justification separability exploratory experiment ray ray advantage fact orthogonal factorization decomposition svd top matrix row information extreme restrict column reduce representation separate invertible preserve column transformation transformation rotation preserve technique exact separable transformation briefly describe column preserve transformation preserve select extreme column possess invariance reason orthogonal separation try value rank pick separability gaussian hyperspectral
review finite random exist goal pac distribution denote pac outline compute hoeffding hoeffding without hoeffde tight small worst develop empirical bernstein name bernstein bernstein bound tight version traditional validation develop similar replacement sx j direct computation tight bound binomial tail without tail inversion take pac compute set refer entity identifies match distinguish type batch query match request identify batch knowledge identify query cover actual match know initially bound algorithm independently call bind derive validation actual match match sample select uniformly replacement without replacement match identify classifier start bayes failure sample draw replacement precision failure bind match one random h pr pr pr use substitute rhs rule apply recall classifier classifier recall probability theorem actual match must complete result without desire substitute complete batch bind failure least accord substitute complete precision request match match identify algorithm disagreement identify set actual include select precision q one match match identify actual match otherwise fraction match identify precision node replacement actual compute identify match respectively failure least also uniformly replacement select sample without replacement draw identify include draw without replacement oriented remark need address present validate disagreement actual match classifier collecting collect produce tight bind precision different set match set validate example merge contact profile social set match choose verify merge validation measure query node fail identify match identify match algorithm rate validate rate match validate disagreement example name phone email address location entry contact occur document pair field entity aggregate field matching connect address phone case want email phone set set actual validate structured people set field match multiple multiple home phone uniformly replacement application wish develop develop classifier develop validation subsample subsample separately draw replacement similar without replacement verify validation subsample subsample follow intersection size si uniformly without replacement sample consist independent sample uniformly rigorous validate regardless method network validation validation matching extended match match evidence direction develop guide process iterative match iteration infer match iteration adopt match rely future match confidence independent distribution validation develop could distribution direction research absence verify match use match support place match datum would adjust accommodate might validate match verify identify match validate match identify node whose know know probabilistic able develop assumption soon add recently distribution work proof subsample select without size replacement uniformly without replacement lemma corollary identifying introduce compute probably approximately bound require level match correspond combine network rich understanding person business identify facebook biology different algorithm match similar edge definition set social business strict overlap business match person social represent person business strict rather person high matching node weight matching problem goal pair maximize set similarity neighbor insufficient effective incomplete people name weighted bipartite big prohibitive allowing match must seed match high degree different connection match connection match establish seed match
dense region graph naturally lead class stochastic instance far ellipsoid parameter due limitation guarantee irrespective true still trick robust encode q feasible linear probabilistic normally noise covariance probabilistic k write k cumulative normal practical interest fortunately far knowledge link hard scenario essentially satisfy neighborhood show hypothesis way attempt statistical learn see two rademacher pseudo cover sample draw restriction expectation q definition often unless bound constant within generalization generalization statement eq constant relation empirical rademacher complexity cover use number integral fx fx c upper rademacher complexity argument convex rademacher bound knowledge associate extend operational cost applicable set bound linear constraint f sr minus portion spherical formulae volume integrate sphere bind example decrease influence complexity measure improve generalization know cover multiply complexity rademacher hypothesis p vector upper magnitude expectation first problem sign fix distance scale second generally result lower tight half constraint space vector upper rademacher operation l tx capturing analyze decision extend set motivate define matrix constraint addition constraint manner rx bb parameterized parameterized jk k kp c element small q kk q unlabele give force cover polytope structure point duality rademacher omit assume ellipsoid intersection find rademacher lead desirable namely call ellipsoid pick ellipsoid combination original contain tight ellipsoid family tight ellipsoid quadratic define rademacher quadratic let semidefinite ellipsoid correspond ellipsoid correspondingly magnitude eigenvalue since original region ellipsoid upper tight axis ba ellipsoid program optimal theorem dependence x ta use hand example capture matrix value become quadratic behave bind ellipsoid eigenvalue like lead high nx form depend lower similar ellipsoid regard handle rademach complexity covering describe ellipsoid eigenvalue pick side bind problem involve minimization step singular call ellipsoid multiply diagonal find convex look result come whereas pick qualitatively ellipsoid eigenvalue instance ellipsoid sphere thin ellipsoid tighter although problem stage ellipsoid intersection original theorem obtain bind generalize ellipsoid family ellipsoid minimize side quadratic lead quadratic simultaneously tight follow eq sl kk requirement ellipsoid way intuition linear inequality way result upper arbitrary constraint instance pac concentration statement presence pac though result unlabeled point author introduce notion distribution example quantity impose level hypothesis finite hypothesis q obtain bind approximate disagreement classifier unlabele disagreement zero training assume randomization theorem serve result focus exploit unlabele exploit unlabele restrict work couple rademacher define maximization right great value equal side rademacher thus maximization inside operation dual duality upper us dual eq g similarly prove appear rademacher supremum ellipsoid positive invertible jensen linearity fact orthogonal write define transform become substitute gaussian upper constant bind gaussian upper l relation obtain result give constant function orthonormal li x b concentration function gaussian standard function omit lemma py alternate distribution py py substitute bind substitute upper element let feasible substitute feasible b bound rademacher dependence empirical weak rearrange inequality get term eq expression rearrange term scale empirical rademacher equal sum mean moment need two gaussian use ellipsoid n state n rademacher maximization objective inside max operation empirical write duality maximization equation similarly intuitive reason serve upper get rademacher right desire result upper see duality rademacher rademacher value ignore right lagrangian k z objective maximization z show rearrange complete dual minimize complete term value result remain respect get follow upper upper feasible obtain instead suitable feasible upper feasible give upper give desire use constraint bind know outline side help focus attention give deriving bound beyond traditional paradigm hypothesis study space interesting space quantify describe encode linear side baseline performance rmse root multiple ridge ridge use response type label keep aside number knowledge size incorporate knowledge side multiple construct constraint construct smoothness section sort monotonic accordingly constraint r vector use ease time train knowledge use knowledge test ridge dependence change example summary training obtain across increase impose true rmse value shift difference model side useful rmse legend learn knowledge multiple range bar plot training setup side mit school management institute technology usa supervise side lead tight space quadratic lead hypothesis side knowledge quadratic could potentially domain actually domain nontrivial algorithm use successfully variety learning property constraint nlp language various use work aim beyond sparsity smoothness keep additionally different set example label choose unlabele label hypothesis search motivate expert example learn predict example encounter type constraint linear constraint main contribution linear space arise naturally circumstance unlabeled example connect provide bound linear find section paper constraints family upper provide match novel bounding constraint illustrate arise ball coefficient dimension proposition situation intersection ball ellipsoid theorem set ball second cone helpful circumstance fully side smoothness consider truly low sample complexity improve selection effort true gain benefit motivate erm svm incorporate classification dataset rule generate constraint norm classifier decision tree knowledge incorporation svms review erm demand auto demand method output monotonic demand whose work multi belong class translate svms know regularization augment cloud unlabele quadratic regularize outperform svms overall formulate robust lead classification heart uci repository introduce improvement introduce imputation uci also provide experimental
highly efficient proposal mention care updating describe qr operation rotation fill transformation decomposition investigate maintain update topic derive specialized fuse lasso component node underlie write lasso edge recall arbitrary theory orient incidence simplification algorithm reduce tx cd orient incidence guess linear edge analogous sparse typically find call basic norm strategy apply future place nan readily strategy fortunately fuse onto form solution alternate expression quantity alternate specialize fuse simple norm dc transpose minimum linear system compute z step necessarily section compute difficult maintain norm cover must norm solution linear g work alternate form big term fuse orient incidence linear computed computation hard span efficient orient incidence edge logic onto component solution linear topic science nice solver use laplacian algorithm indirect iterative solver return approximate tolerance system issue explain extremely computer community reference therein direct solver let graph component express block therefore decompose accord fully subgraph exactly follow laplacian matrix graph eq laplacian solve submatrix exclude connect span solution unique prove column orient incidence graph column edge corresponding row repeat show message graph form last column cholesky component finish specialized fuse lasso dual repeatedly compute dl outline connect graph one change whether run first search path specialize fused incidence center first add projection solve w z connect package sparse cholesky decomposition reduce prescribed employ cholesky algorithm see therein unfortunately cholesky admit empirically efficient number linearly subgraph provide solve full dual algorithm specialize lasso implementation fuse orient incidence graph row eq brevity project onto onto underlying row partition g g define edge follow connect coordinate correspond averaging within otherwise component system laplacian connect component dd laplacian subgraph connect component jj discuss linear factored cholesky sake completeness say fuse fuse soft thresholding output fuse lasso lasso long perspective recall presence view dual dx fine generic penalty fast structure retain dx apply usual path result carefully solve step solve solve linear strategy apply solve nd typically give idea calculate solve dense trend fuse prove correctness implementation fuse arbitrary full characterized suitably matrix full give general characterize hand system apply logic rewrite multiplying rank norm dx dx subspace dx x dx rewrite computation compute compute simplification norm system eq offer offer significant hard system td step require involve projection first begin ever reach quantity serve freedom estimate therefore interest stage step last really system qr outline trend st span take consider eq fuse fuse onto orient incidence span connected graph subgraph correspond incidence therefore vector give q remain node connect node write appropriately indicator node section dual fuse reasonably large come report city report date occur spatially aggregated census group census calculate think proportion measurement randomly census block proportion figure task census grouping adjacent census lasso difference neighbor huber block difference optimization appropriate capable produce component total graph first fuse path specialize little computer freedom note roughly region city side city risk city since block incur picture qualitative census city lastly fuse compete measurement counter tendency method create equal sized cluster iteration complexity implementation generalize implementation iteration termination super notable exception fuse termination infeasible undesirable typically regularize visit toward path path vary three fuse cubic trend filtering setting period fuse square bottom third first fuse trend step fuse computation indicate problem hour trend interest section step census group denoise solution hence step one likely drastically algorithm trend filter operator lasso incidence handle well acknowledgement rt support nsf brief review square excellent column form surprisingly qr operation decomposition primarily minimize leave equivalent equation recall triangular look box nonzero entry indicate row substitute procedure square require operation total multiplication qr importantly vector compute decomposition operation permutation span column upper triangular visually look column least criterion admit solution infinitely qr eq first last variable decompose operation let hence least operation compute solution necessarily however qr decomposition need rotation cover key message triangular rotation take compose form operation ok operation write side utilize seek must advantageous problem qr change compute special km find concept except require operation finally total rotation transformation maintain triangular rotation decomposition add qr decomposition notation largely chapter main rotation amount rotation angle orthogonal furthermore simply onto axis inspection note rotation identity except correspond applying affect leave eq rotation apply affect row take row common look example th rd zero begin pre multiplication pattern give triangular structure matrix multiply rotation matrix affect column compute operation logic multiplication appropriately th column look eq rd rotation triangular structure cover rotation qr add remove cover reference decomposition subsequently want qr row motivation qr naive problem separately suppose mm ar triangular therefore rotation n n triangular desire procedure rotation therefore operation qr remove change cover denote th rotation basis let orthogonal qr om add exist one cover rotation row triangular rotation apply rotation upper triangular operation technique qr minimizer subsequently compute norm row actually different qr initial qr rotation matrix triangular avoid rank case g right eq k triangular complete qr appropriate minimum second rank nonzero rotation right side q triangular qr desire rotation require operation alternatively na row rotation qr qr ai decrease zero rotation triangular help nd rotation row rotation j proper qr decomposition operation one obviously qr obtain remove add remove qr decomposition strategy addition removal require clear order initial square problem path correspond p p corollary taylor efficient implementation full rank case trend filter fuse sparse fuse specialized implementation offer numerical solution use repository path qr laplacian computation outcome vector problem choice assume ensure value implementation generalize algorithm compute specialized implementation special fused trend filter implement repository problem early work trend framework brief think fuse row th e contain zero fuse lasso many exhibit subgraphs concept note oriented incidence undirected laplacian realization fuse component successive piecewise across work setup fuse lasso additional refer fuse fundamentally pure describe trend fuse trend filter penalty version fuse term explicitly filter signal estimation statistical begin discuss briefly review aside central convex value parameter solution function former desire number quadratic programming multiplier admm general specialized technique fuse linear string trend specialized admm finally fall category proximal utilize specialized describe algorithm regression also fuse flow track opposite start end primal perspective assume row fuse lasso operate unified allow flexible enough specialized take work give detailed comparison alternative implementation dual path generalize place column name compute help case reflect compute path path level keep track coordinate compute equal lie leave outline minimum first record compute next leave add remove leave record main lie word start set often obtain fully less proceed path second concern solver encounter step broadly speak solver indirect solver solver linear round error perfect computational platform direct exact indirect approximate may preferable system boundary across rely solution sense really stick propose describe dedicated evaluation section consider without contribute iteration update row add qr norm efficiently decomposition save order computational qr decomposition complexity
text take computer intuitively vector close language roughly cluster manner get language identify include sample sample table gram letter histogram guess accuracy incorrect system detection identify gram htb pdf cm table confusion language base letter predict corpus detection language keep track easily accommodate efficacy indexing language solely letter text letter also thousand algebra vector make multiplication addition letter english next advance design mind retrieve arithmetic generality text well scheme way address indexing language indexing identify material simple dimensional signal rise data work memory inherently produce amenable experiment paper acknowledgment discussion feedback computer california berkeley berkeley center theoretical california berkeley random indexing wide application variety accuracy demonstrate encode letter gram method implement require validity task short language achieve accuracy comparable human recognize unknown language course sound especially language say language give language text categorization model similarity various language identify language count letter letter compare profile general store various google recently compact detector profile corpus perfect popular processing word mean statistic semantic context vector ideally word represent vector explain semantic frequency index simple use way al bag locality sensitive lsh differ randomness thousand refer calculate useful paper present detection indexing highly scalable main random indexing project well operation keep high implementation indexing similar language create text latter refer letter frequency text know letter frequency language letter letter idea physics consecutive letter text block appear gram example rise b stand frequency letter block language text language letter plus frequency letter would gram track indexing text vector running create create sequence describe early example block calculate label half half vector sum gram text store language exactly text compare vector cosine unknown define cosine high text language cosine choose
x measure coherent moment learn distinguish continuous function learn possible motivate restriction continuous exist increase old continuous old continuity every another old lemma interval argument argument chernoff probability lemma step arm minimize low choose try estimate risk mean arm explore try exploitation repeat tt algorithm algorithm probability time regret different continuous e match bandit case cover bandit log exponential problem bandit function old demonstrate efficiency eq see front go comparison growth old function sequence example achieve regret follow let consider arm principle arm negligible precisely represent th arm point I estimate arm word ie ensure condition advance arm arm happen sure ie ensure arm algorithm use instead construct knowledge avoid construct small occur arm repeat decision receive time repeat least bound motivate example take take region case since arm hence compute I risk two achieve logarithmic prove bind measure condition logarithmic might even continuous sound follow goal likely problem general open functional class coherent risk could plausible intersect inclusion identification framework pure explore focus notion construct satisfy uniformly previous implication f let every similar minor follow distribution high event union one arm event combine three give focus bound everything derive stop fulfil ac minimize regret multi armed goodness arm present sublinear regret stochastic armed bandit framework clinical formulation one distribution receive arm repeat prescribe regret difference arm small desirable clinical trial armed bandit case arm risk theory learn field learn study expert risk pure propose use variance measure aim latter previous immediately applicability lot risk mean arm risk arm completely bound pac formally state model discuss open possible extension conclude proof main distribution learner
whereas slightly complicated recommend corollary factor besides concrete applicable step discretization rule rate tell perform concave state change might relax impact make tv quantitative characterize readily eq bind get horizon eq tv proof infer initial impact density p small obtained accelerate extent amount definite close h characterize corollary k guarantee strongly concave cf also density log density one former target continuously inequality function satisfied define expect close broad necessarily assume leibler eq pf use formula divergence eq exponential second kullback leibler divergence right claim tv derive follow result p algorithm op comment dependence acceptable get substantially concave dependence able simulate precisely repeat initial distribution op op get case replace properly certainly get replace tight involved formulae mcmc strategy define stop strategy scope applicability spectral inequality employ ergodicity langevin lyapunov since inequality langevin drift ergodic advantageous gap involve dissimilarity explicit dependence exponentially advantage ergodicity even convex langevin diffusion euler discretization section diffusion attain desire lead diffusion hereafter langevin monte carlo h pf accord matrix spectral f approximation theorem defer let direct last provide number prescribe level trivial reader continuous constant outcome also recommend reach desire op pay attention computing exponential versus preferable situation require instance paragraph use worth perform hessian much case first approximate establish guarantee scope remark warm reduce useful replace g density experiment value device rademacher distribution value draw pn gradient precision median employ recommend experiment aforementioned estimator trial clearly large increase although median robust sampling rather posterior median mle put bold frequent winner measure euclidean draw posterior median mean posterior median approximate concave log concave langevin monte regard natural counterpart statistic beyond good theoretical prove computational complexity scale polynomially desire level prove show evaluation evaluation available chi divergence polynomially value evaluation theoretical guarantee computable constant involve work generality result difficult implement get tight constant context metropolis hasting mala concave target important particular investigate compact derive density bound make establish guarantee little interest remarkable prove logarithm power behave dependence dimension warm available bad analyze evaluation build recent connection focus propose sampling come convex hope investigation aim establish recall sake completeness proof second introduce dt lead prove proposition continuously throughout shorthand view global apply subtract proposition hold inequality obvious complete lemma yield simple application schwarz invariant density schwarz give kf relation suitable constant corollary theorem argument diffusion process lipschitz continuity hessian provide analogous gaussian h ss l l complete ease notation proof notation v conjunction sum schwarz distribute last use v go back side conjunction e inequality condition infer inequality hand check pi claim acknowledgment work support various kind pa importance ingredient procedure resort meaningful log concave distribution langevin monte effectiveness process beyond density present log estimator close computing likelihood require impossible provide iterative variety algorithm approximate work gap sampling stand characterize continuously define descent precisely achieve upper evaluation feature logarithmic dependence side somewhat conservative quantity involve expression computable simple stop recursive situation approximately exist study theoretically tune grow maximize density necessarily gap even numerous similarity optimization approximate sampling langevin similar algorithm step recursion often refer center vector matrix identity small product total establish explicit quantity approximation refined version term summarize keep thing translate one large complexity gaussian iteration performing finding work column magnitude number iterate perform error column contain bad complexity complexity one iterate iterate warm set stand norm unless probability proportional markov behind euler discretization langevin diffusion langevin differential equation sde brownian sde strong p call spectral ergodic bound operator value fast behind probably influential probabilistic asymptotic avoid langevin ergodicity choice case langevin diffusion choice result chain influence subsequent metropolis metropolis langevin numerous impose couple continuity chain choose fact differentiable note condition nonnegative expansion view entail point consequence long assessment rate end markov vector upper process precise increment increment iid readily vector h follow drift
image assume soft object dominant specific assign patch relationship quantization number learn distribution sum corresponding patch utilize share compute codebook soft label topic change assignment refine shannon entropy setting codebook remain new refined codebook q estimate class summarize propose patch total infer learn rf correspond train base final real world semi order unlabele image extend paradigm rf feature extend except unlabeled scene scene consist pixel dataset classification evaluate class object dense sift patch size pixel ratio retain rf codebook histogram use dataset table notice though improvement seem narrow soft codebook compare significant final label training good model scene label art despite limited label method effectiveness forest drastically label rf rf limited mind topic generative believe hybrid explain experiment conduct gradually experiment feedback fully setting patch rf unlabeled training rf enhance training feedback mechanism b b effectively soft role visualize patch review effect soft label codebook visualization scene rough original besides normally image rf background patch reduce background patch rf splitting improve rf method iteration rf accelerate soft label feedback framework rf codebook learn image achieve soft codebook codebook reach framework paradigm investigate support university rp center edu united uk china edu understand important due bag codebook essential forest rf tree performance patch paper tackle novel way update codebook codebook base feedback feedback perform patch art rf codebook learn focus patch experiment propose task part base discovery successfully apply unsupervised object categorization scene codebook patch achieve discriminative advantage counterpart g ground forest rf decision tree compare codebook quantization demand rf show codebook object categorization segmentation heavily truth ground belong face however notice patch belong class g red face train rf codebook background patch rf codebook framework novel rf rf weak image patch feedback c codebook arrange development relate visual codebook framework show discussion codebook essential representation find mean tree employ recent research focus codebook use discrimination codebook offer characteristic popular rf feedback mechanism detector create location codebook feedback scheme variant feedback rf learn perform codebook learn classifier set separate feedback node study utilize assignment label rf codebook effective topic codebook popular serve mid group meaningful representation ideally part part represent represent background require image combine enhance rf codebook feedback main contribution rf codebook level superior information patch level rf contrary employ learn rf ground truth label treat build rf codebook secondly train weak codebook label weak rf enhanced train refine
finding iid eq domain assignment exponentially find assignment efficiently add unary potential rbm correspond standard basically I alternatively binary unary potential section feasibility use availability np mrfs efficient minimization propose suitable energy change without change energy intuition perturbation compare put distant training remove false training pairwise potential unary potential remain desirable potential higher visible bipartite interaction need noise bias perturbation potential basically assignment inf new likelihood idea minimization allow landscape perturb reach thresholde fast closely step inference current may deterministic chain mcmc intuitively attempt update parameter unnormalized probability function refer phase cd training phase configuration neighboring state take markov chain train maximum posteriori basic extreme map assignment perturbation perturb ideal approximation use provide upper maximize feasible submodular potential even step could relate basic idea perturb configuration discrete training rbm rbm field n example rating speech model deep neural architecture unit q e normalization constant due bipartite form visible visible rbm rbm local field visible seek maximum log calculation derivative negative require sample current may markov chain regardless form q sufficient rbm calculate term term training markov
discrimination adaboost slightly effort embed machine vision unclear propose repository challenge vision model significant detection haar paper develop briefly review adaboost algorithm explain fail sample adaboost general ti ti tx hx tx indicator variation algorithm arc differ mostly compute adaboost select weak pool take adaboost round error go improve overfitte adaboost margin theory give error direction error size training formulation adaboost tie discriminative whether line pass type red one however illustration eps initial thus sample adaboost weight correctly sample pass usually slightly small discretized classify receive fig round essentially lead back combination keep due adaboost sensitive keep weight pass weak embed switching focus operation way improve design allow case hard feature separate feature often lead fitting adaboost vice versa mix recursively combine cascade lead decision tree weak embed weak describe much long complexity eqn algorithm essentially logistic probability q discriminative upon weight make impact vision million thousand adaboost strong simple use classifier adaboost call however still linear difficulty fig positive classifier failure select subset complexity give detailed description combine operation di tx decrease output straight classifier operation classifier final positive regardless classifier unlike adaboost mis quickly focus solve show illustration eps fig feature weight step weak classifier however positive negative receive weight adaboost create situation weight sample di h tx di role adaboost negative positive therefore decide regardless later weak therefore weak swap positive turn operation complementary decision operation operation htb give di stop output htbp illustration eps cccc adaboost second call classifier weak keep check decision operation naturally embed aspect logic confusion weak pattern fig result operation positive classified margin theory al decide decision much simple cart worth one present happen add major issue concern classifier computer vision eqn desirable produce error low vc good vc margin decide small difference reality enough task none limit since kernel mining also ultimately modern particularly error classifier major scope performance widely generalization al give behavior adaboost margin margin weak combine arc try minimum adaboost experimental arc big error adaboost try find arc indeed arc adaboost decision uci repository breast modify category merge class upon belong sample test trial cancer training compare alternative different weak operation give good similar vision weak eps eps width different number conduct plot weak improve show dataset suggest achieve introduce breast suggest boost decision cart base table adaboost arc decision arc cart adaboost cart decision show arc cart cart tree leaf tree around complexity big massive thousand cart achieve greatly usage classifier currently widely htbp eps width slightly illustrate demonstrate segmentation detection demonstrate testing image label pixel body task classify patch center body negative positive patch haar response filter cascade cascade node select algorithm identical bootstrapping algorithm uci repository improve error nearly cascade well report report cascade adaboost fig among detector haar computer understand
incorrect concentration direction region version incorrect theorem theorem specialize class test ratio quadratic power nan discuss way distributional devote apply autocorrelation series whereas appendix result real vector loss generality identity reduce transformation assume furthermore relaxation general although state stand borel expectation respect shall measure borel say impose possesse everywhere else completely strong need even apparent testing fact distribution line proof theorem problem understand typically identifiability meaningful still beyond moment explicit identifiability assumption randomize measurable space rejection borel together rise region test shall I rejection probability terminology always real transpose span symbol onto complement rank z row orthonormal satisfy vice choice orthogonal denote corresponding eigenvalue symmetric order definite root form euclidean operator denote interior closure w symbol denote lebesgue borel uniform measurable operation composition function r course column eq main invariance w test invariance another invariance remark maximal group denote generally sphere satisfy property invariant obviously additionally borel continuous neighborhood unit sphere moreover choose satisfy claim author define whether r z due assume e remark power every additionally parameter identifiable sense z test far away interest I limit limit confusion stress throughout sample situation I motivate autoregressive order autoregressive power like sufficiently test approach intuition depend employ spatial autoregressive eigenvalue intuition suggest intuition incorrect test significance employ see coherent general structure non mention thus randomize crucially function equal positive model power claim follow exhaustive exhaustive exist neither observe satisfied framework additional remark invariant seem power mt comment notion mt obviously give symbol third symbol eigenvector small eigenvalue invariant validity condition empty mt explicitly statement reader implicitly exclude invariant region strictly precisely complement nan appropriate interpretations cf correct part claim mt discussion mistake generalization correct mt correct incorrect distribution concentrate subspace occur happen infinity crucial effect test appropriate rescaling enforce generalization third mt even mt cover intuition set follow assumption ne normalize converse example underlie weak assumption mt large converge limit exist equal assumption satisfy mt sufficient satisfied discuss converge every accumulation measure coincide measure coincide view claim mt provide end introduce vector say assumption symmetric orthogonal implicitly impose assumption condition arise satisfied hold continuous generally absolutely note family regression convenient avoid indeed weak statement ready present mt stated possibly randomized test invariant third claim mt observe light remark weak mt mt coincide mt sign third specialized rejection region invariant reduce applicable stand indirect region modify test almost everywhere test underlie general note example establish bt region cf see substitute second mt critical n ty condition expression power test ratio quadratic ty substitute assumption remark study invariant rejection rejection rejection assume ty claim say subsequent claim incorrect assumption nd paragraph incorrectly provide claim nan obviously ty clearly rejection region matrix function speak statistic take whenever define rejection region probability certainly family r ii maintain adopt regardless nan turn convenient course rejection nan lead alternative assignment easy understand boundary orthogonal hold constant entire depend obvious actually arise subsequent proposition region form claim provide part proposition slight inclusion region part need hold provide case kb kb provide allow imply mt correlation autoregressive exist root restrict necessarily verify spatial model easier verify root necessarily although independently assumption theory hold claim establish note l arbitrary w even thus condition singleton l accumulation ty te precisely necessarily singleton e odd additionally remark assume entail provide ensure hold appendix formulate equivalent formulation easy spatial ar family density precede mt precede coincide cf precede substitute incorrect mt determine limit accumulation general limit expression explicit accumulation precede decide basis exist apply corollary vanish test totally relation vanish identically show element complement closure rejection conclusion precede verify test statistic satisfied elementary calculation view vanish except b kb alternatively large eigenvalue although belong rule case eigenvalue rejection precede singleton simplify theorem accumulation expression nonzero surely since accumulation accumulation question examine explicit expression observe accumulation open accumulation obtain regularity distributional assumption exclude degenerate case test locally x also assumption give assumption mt sign impose corollary follow assumption would corollary note would furthermore remark kb kb corollary always exclude corollary weak mt cf neither satisfy kb b point occur locally good test literature cite next corollary belong boundary rejection theorem presentation concentrate subsequent corollary rejection neither b satisfied hold view assumption kb multivariate matrix furthermore region iv possess cover equivalent b kb kb b furthermore family b kb limit equal eigenvector accumulation interval u x bc x c e bc non odd accumulation bc bc assumption exclude case b kb empty corollary go matrix bc definite note directly extend version corollary complement part guarantee corollary lem iii obtain appropriate corollary power question critical occur essentially equivalent ask size region size arise least subsequence end rejection invariance quantity infimum size vanish along subsequence limit power proceed narrow discusse appendix unfortunately error discuss applicable structure subsequent see version power equal satisfied test region statistic characterize situation occur level restrict autoregressive order lemma improve lemma relate characterize equal present obtain reject pass subsequent proposition exclude trivial rejection whether suppose origin kb kb kb kb b p kb kb kb b b corollary repeat part satisfied never eigenvector eigenvalue corollary invariant locally guarantee power fall precede tell construct design avoid kb whether property way overcome absolutely follow b kb open possibly satisfie kb proposition concern theorem result precede proposition remark assumption subsequent vector hold neither hence assumption possess w lebesgue everywhere neighborhood origin nan precede neighborhood replace exist assume upper identify power course depend model alone hypothesis indistinguishable test fact invariant function invariant error theorem proposition regard test iv prove ratio matrix I kn nu define group furthermore every whereas mean nan alternative function g invariant symmetric test invariant invariant distinguish alternative invariant less I condition choice differ c I family consequence theorem alternative invariant multiple assumption alternative may invariant give rise corollary l multiple necessarily condition turn matrix non sign hold arise suppose see invariant test cite identification reduce parameter identifiable maximal invariant cf special maximal statistic follow suppose possibly space invariant probability invariant test automatically carry rejection observation part continue requirement apply part possess uniform consequence possess uniform reasoning show distribution weak way everywhere explicit possess continuous explicit difficult give equivalent choose everywhere vanish everywhere possess everywhere continuous everywhere positive iv hence condition always choose almost part part b suppose satisfy atom identity covariance may appendix argument underlie discussion gaussian without apply replace base depend consequently vi discussion positive case correspond mass origin invariant satisfy throughout viewpoint broad setting distribution atom precede rejection family actually probabilitie result paper apart concern invariant impose serious restriction satisfy interested large invariant follow test hold extend covariance accumulation certain assumption hold lag error spatial lag element denote zero absolute algebraic also unique multiplication model random covariance aa cf remark implicitly latter connect hence however implicit typically maintain denote frequent frobenius see e one choose identifiability identifiability immediate nan hypothesis hypothesis disjoint hold without distribution model mild well w absolutely satisfy main e theorem immediately spatial corollary purpose corollary provide fact neither cf maintain assumption possess almost everywhere neighborhood origin possibly nan origin kf f b b n b everywhere hence accumulation iv part next corollary region elliptical assumption elliptical general test give maintain assumption origin symmetric critical b random limit whereas eigenvector consider lead quantity strictly theorem theorem sense limit early power correct proposition critical exclude trivial discussion subsequent always consequence assumption n precisely strictly statement limit b strong proposition statement result maintain rejection neither maintain assumption suppose absolutely neighborhood origin except furthermore quadratic ii p nb lemma test power equal maintain assumption absolutely density neighborhood n b b c automatically satisfied represent less eigenvalue b w empty consider spatial mean random vector identifiable rewrite equation case typically long spirit follow maintain put mass proper e part mt claim incorrect provide mt fit become appropriate version produce turn degeneracy statistic part consequence maximal problem experiment recognize identification occur assume experiment appendix condition condition optimal invariant part proposition immediate contrast result invariant proposition elliptical symmetry proposition sufficient corollary establishes respective proposition precede comment importance weight q first remark precede maintain intercept generality n orthogonal every q together function maintain observe obviously intercept element consequently every onto map mean element say attempt justify invariance spatial lag sense incorrect incorrectly invariant invariance coincide show consider comment error covariance testing negative autocorrelation assume fix distribution depend assumption framework clearly cover gaussian autoregressive readily verify hold assumption show satisfy ii denote fact ii maintain assumption I subject square root critical quadratic x multivariate equal suppose p k corollary expense maintain could nevertheless allow parameterization invariance alternative substantial I discuss expression case notice arise autocorrelation test consider intercept autoregressive power closure interior region numerical contain intercept subsequently indeed intercept limit except degenerate theorem rejection justification one mention extend express ratio note additionally also autocorrelation easily read analysis regressor systematic investigation regressor already mt statement explain mistake satisfie impose page mt simplicity proof degenerate simplify correspond uniformly compact zero everywhere tend degenerate eigenvalue converge obviously tight concentration claim fact opposite happen mass claim claim speak construct mt claim limit power define rejection differ nan family dominate concrete normally without case without q symmetric definite observe rank section arbitrary region invariant r invariance sphere I limit theorem obviously satisfied since absolutely continuous hold mt incorrect start rejection generally covariance spatial rejection region argue somewhat artificial rejection ask mt modify modify boundary subsequent mt impose probability nan nan strict obvious slowly invariant provide limit claim regressor present somewhat b absolutely combine vi trivial power either provide hold comment discuss discuss context restrict concern infimum limiting vanish symbol definition refer hence make usage seem mind test region ty ii remark ty tf tf problematic reason statement region critical region proposition therefore invariant region boundary refer require rejection lead base incorrect implicitly continuity assumption function satisfied value statistic refer size explicitly denote pose absolutely denote correspond stand content statement author typically critical nevertheless case statement however pure read pure limiting power cf irrespective test eigenvector explicit understand assume explicitly statement understand degenerate choice specify test incorrect incorrect discuss lemma would without limit limit case test test reduce reduction argument limit cf necessary clearly establish regard clear precise interpret derive equal test limit operation justification interpret derive course justification however argument perhaps argument version strong statement possible lemma suffer lemma incorrect rigorous infinite dependence without provide necessary reduction precede discussion furthermore statistic case regard test seem point invariant exclude easily test unbiased issue follow event allow argument go paragraph read stochastically also assumption easily relax elliptical symmetry part claim regularity condition contain invariant argument reduce test argument could invariant test strict strict preserve turn regard invariant hold separately elliptical symmetry read also verification display use precisely general refer display hold hold almost concern locally invariant mention turn last degenerate power trivially importantly proposition several incorrect incorrect stand arise equal conclude version probably corollary checked converge form basis order small must equivalently sum I side converge support accumulation sequence subset weak accumulation mass origin weakly say give accumulation certainly subset consequence assertion subsequence weakly almost continuous continuous conclude surely converge continuity symmetric nonnegative square mp ml mu ml mu u proof
output softmax layer model objective consider interaction pair word two two issue individually detail eqn capture intuition language encourage representation language conceptually consist representation word two domain denote word weight sure language embed nearby stacked express q subscript respect source pair derive scale product objective vocabulary argue primary exist due softmax regularization perhaps even importantly strong introduce issue sample objective estimating word regularization word care observe idea optimize et take step representation continuous word use objective architecture specifically language vocabulary sequence bold finally bag represent one bag word representation embedding motivation learn represent target vector embedding maximize sum noise procedure typically entire vocabulary typically hundred word language relate learn language relate eqn proportional alignment frequency step training alignment alignment alignment translation utilize word directly parallel since alignment interpret alignment weight write eqn english word alignment pair sentence level know translation occur sentence make naive align leave advanced english length sample english sentence towards simply bag word eqn parallel corpus trivial embedding regularizer practice document approximate eqn parallel strong proportional eqn eqn really estimate learn cross distribute representation exploit sentence align training evaluate representation task across language also setup goal label language classifier language document vector document utilize alignment instead purely corpus induce cross language induce subset english corpora category market classification document select corpus third remainder size separate development baseline namely word align word language baseline document source al day auto day auto replica cross dimensional embedding directly summarize comparable train english corpus exploit nature english al versus original day demonstrate loss efficient accurate next art train comparable original state art en de en report train english current art also fast embedding task publicly task extract english online google translate service dictionary source individually english pair frequent translate near embed evaluate fraction top return specific method translation baseline et rank base co distributional similarity word construct count word count language dictionary finally word target language c word occurrence induce embedding english translation summarize baseline accurate english improve percentage english word translation percent indicate grain translation raw sentence introduce induce representation require utilize parallel representation advance objective computation training scale sentence thereby enable efficient document art minute evaluate english task relative word university cifar google word computationally train signal raw sentence align datum sample bag language cross embedding outperform art lexical code open language part name entity mostly english technique exist another trivial generalize hard across desirable syntactic semantic feature task interested unsupervised language apply wide language induce representation usually language embed learn training embed sentence induce serve linguistic syntactic nearby embed especially high resource language baseline well name entity recognition translation sentence obtain translation occurrence frequency train regularization frequency pair induce costly traditionally training vocabulary length hundred thousand fast exist evaluate translation pair motivate use believe time limit attack cost head introduce simple induce embedding extension embedding uses align learn training requirement text contribution consider bag avoid induced embedding translation outperform reduce hour several prior approach feature learn cross domain english embedding word phrase similar property use
gate gate boolean surprisingly define truth table bits gate gate define boolean circuit circuit bit output bit decision restrict choice circuit consider input bit circuit computation correspond circuit process circuit specify gate gate wish table row bit ef transpose obtain bit either evaluate classifier require evaluate binary evaluate operation significantly total importantly want computable evaluating circuit consider feature store save operation list logical c five truth gate input output gate product truth gate tensor truth compose complement bit evaluation gate multiple elementary bit million gate evaluate parallelism circuit simple circuit capable obviously simplify optimization produce hyperparameter algorithm circuit second leave optimization global greedy quite simple leaf take coordinate bit input specify greedy leaf bit gate choose truth gate behave circuit decision gate gate reader intractable optimization gate gate nothing gate good gate outcome input gate reasonable mostly gate category choose truth specify bottom take dimension truth know second move tree versus decide success criterion gate gain gate example gate produce ultimately create list maximize information gain set index gate variety range white image datum encode binary depth scale exponentially hyperparameter give internal leave example reduce evaluate small classification circuit hyperparameter algorithm compare net observe repeat configuration section hill leave quite powerful carefully severe quite choose one leaf well back course evaluate array logarithm leave tree hyperparameter need see figure obtain classifier significantly greedy produce tree classify significantly well state art report paradigm preliminary paradigm worth dramatically indicate well feed neural outperform net thing context although preliminary believe investigation need significantly cpu enable fast appear circuit investigate structure initialization leave circuit certainly several dimension eliminate average adding could fitting hill look promise feature select subset simply improve else autoencoder framework net use gradient thing like ham poor substitute future investigate optimistic result thank david thank helpful comment I month finally would thank unconditional experimental supervise state classified bit avoid domain box invariant example example tree update hill hill noise cube bit bit pixel channel bit notice greedy gave depend second recover otherwise describe hill algorithm efficiency experiment core intel core cpu amount trivially experiment code algorithm extremely ask truth preliminary result indicate thousand experiment perform gpu optimistic speedup practically example cube compose pixel white bit cube contain two surface bit cube overlap choose overlap permutation nature pixel data train train train train train train test train train c c train n test k c c test default hyperparameter train run take less minute factor approximately gauss dataset previous list integer bit integer one classify integer receive perform differ plot variance apart round rescaling affect graphical interpret versus l k train versus versus versus test train versus train versus gauss test mnist use dataset distinguish distinguish drive dataset bit pixel range form bit choose bit obtain behave mnist loose presence significant hyperparameter greedy hill l c bit train train train test mnist bit l train train test c c train train convex dataset group hard feedforward net consist image convex set vector rbf feed stack hyperparameter default hill perform c error thorough hyperparameter search expect run time long
difference strong weak evidence bootstrap outline take due bootstrap fold let variance auc train new set training although cross hand side instead difference auc across bootstrap bootstrap bootstrap contain auc difference variability combine set uncertainty train combine net net compare quantitative relationship past technique variety along successful application inspire win recent competition artificial multiple conduct recent dealing report superior structure study relate chemical property chemical mathematical could study desire without actual distribution numerous benefit drug generating chemical mathematical predict property encode chemical structure throughput screen ideal collecting training test generate molecular study encoding descriptor various chemical property interest machine matter prediction competition retrieve use descriptor nevertheless net allow win relative accuracy internal history apply regression neural forest projection pursuit partial machine successfully advantage partial making allow user able assess viewpoint inform assessment uncertainty model important controlling overfitte concern capacity requirement measure importance control layer selection reduce infinite network require cubic number case single issue past decade wide advantage deep network highly successful task vision along researcher train instead network hide unit neural layer thousand million parameter substantial weight unsupervise pre avoid overfitte fully mention differ variety net task operate multiple something literature aware neural net dropout overfitte access feature additionally different neural motivation behind relate task model particularly share task multi feature govern law broadly descriptor task various leverage recent development outline net multi lead baseline model feedforward artificial powerful reduction neural map repeat simple module internal layer feature useful value parameterized eq layer know output optimize standard output vector regression square appropriate train descent sgd momentum minibatch case use network train molecular descriptor activity neural regularize forest order neural architecture vector descriptor input separate unit compound multi neural net backpropagation unit whose compound case descriptor observe towards handle control training go minibatch case case replacement emphasize leverage call profile spirit net difference profile treat compound activity previously compound side descriptor prediction compound activity another net train deep network hide successful numerous notably vision speech recognition complicate capable extract input million thousand net date tend single unit wide deep hardware dominant net overfitte deep network recent network always perform one practitioner depth vice since architecture advance deep neural become capacity well train hide weight regularization wide net avoid model large expressive capable represent dependency descriptor activity regularization broadly subject recent unsupervised training powerful regularizer attention neural early base validation overfitte limited effectiveness little network high help overfitte sophisticated experience dropout network training effect add uncertain hide view dropout numerous neural net produce random net weight powerful avoid overfitte weight l net eight million avoid expert sequential ideally parsimonious evaluation construct function suggest acquisition exploration highly uncertainty job software implement particular version neural optimize range configuration overfitte validation datum neural investigate good statistical error train new perform closely neural net perform single counterpart relate one difference statistically closely net dataset leverage baseline surprising p minor variation particular nearly identical enough merge positive negative relate work nearly gain task single reflect gain task create testing multi net relate dataset multi net exhibit improvement combine second highlight display table single net statistically net since net model ignore capacity irrelevant task much well primary combine even make prediction net prediction treat primary task improve upon net often somewhat task benefit demonstrate sufficiently model still gain multi interesting arise positively correlated net negative r primary allow use prevent enable non weight penalty always dropout well lot include contain reduce dimension drastically although informative thousand net informative feature information gain descriptor typically produce unnecessary drop figure show auc representative descriptor descriptor effect bayesian run layer task second little auc consistent trend multi net deep although depth across neural net r layer c contradict molecular neural network unfortunately neither descriptor public cause discrepancy several likely depth multi task net train regression binary classification bit information effort try descriptor may exist data descriptor open descriptor software package could descriptor add result improve effective leverage potentially inactive decision virtual compound essential plan perform future work version bayesian software package practitioner long sophisticated network since small setting automatically way implement net well rapid progress advance team molecular activity neural minibatch backpropagation gradient objective net parameter objective current minibatch formula strength training list preliminary allow range
belong child node side split essential gps propose partitioning capturing design stationarity likelihood hyper gp regression express straightforward purpose marginal likelihood leaf leave aggregate level tree index start exclude return depth return list root tree associate leaf marginal eq implicitly assume node give factor along path equally importance leaf example decomposition depict inspire work correspond leaf path presentation empirical however straightforward approach infer hyper leaf markov monte leaf approach gps leaf ei acquisition function point ei efficiently gps leave employ iteration propose standard non objective propose comparison optimisation well optimisation approach mat ern gp obtain different benchmark discuss function first figure function exponential hand side whereas hand code please second synthetic precise flat without careful direct modelling paradigm wikipedia article parameter specify mini batch original restrict grid latent structure similarly tune experiment structure setting tolerance performance original dataset community optimisation approach available table exp svm mean deviation end iteration achieve achieve result krige modelling minimal observation krige closely relate acquire describe square south measure gold acquire observation surface make observation measure gold surface exhibit stationarity try highest please comprehensive krige construct optimisation try find historical record approach minimize early run fail advantage latter achieve converge global optimum evaluation exp exhibit fast illustrate shown structured svm converge eventually important lack state mining extraction significantly outperform deal depict figure optimisation flexible leave readily combine approach leave improvement robust range evaluate performance propose competition normal setting yield gain robustness box tuning machine finance mining optimisation optimisation expensive capture function single stationarity consequently stationarity optimisation automatic exploration optimisation minimum generally convex modal whose evaluation objective often via interactive environmental material network carlo experimental reinforcement inspire great automatically algorithm optimum objective optimisation setting whereby th round select return noise belief smoothness observation model describe tt derive acquisition decide next acquisition exploration introduction please refer aforementione tend address process introduce flexible adopt tree properly avoid split parameter observe improve brief maximum acquisition ei remain bayesian optimisation package ei close represent density improvement family one heuristic member project input extend dimension try directly covariance stationarity optimisation non stationarity input cdf
transmission subspace principal equal compare exist system dimensionality reduction initialize observe however distribute excellent close fast estimation htb pt paper distribute wireless network result exist scenario require transmission propose distribute reduce adaptive distribute perform dimensionality follow dimension distribute reduce develop communication overhead improve exist advantage strategy reduction distribute reduce sensor distribute strategy fundamental wireless network grid specific neighbor combine require communication dimensionality order reduction kalman consensus costly processing naturally context reduce technique dimensionality spectrum interference research exploit estimation attribute employ meet reduce estimation joint iterative normalize least rank reduce perform decomposition agent information low neighbor dimensionality reduction dimension outperform compete letter inverse operator complex wireless limited capability topology protocol incremental consensus base neighbor topology fully connect instant measurement accord noise variance measurement wireless fashion operator possible technique diffusion set denote cardinality coefficient eq neighbor reduce processing network strategy alternate technique htb depict depend strategy section reduce unlike auto perform decomposition process costly flexible low cost fast jointly fashion method lagrange consider arrive lagrange frobenius part set describe ki ki ki alternate instant adaptation perform adaptation step instant node start reduce ki ki ki ki neighbor node keep instant adaptation combine rank neighbor estimator conclusion reduce estimator propose htb instant n ki ki ie ki ki ki ki reduce estimator send neighbor node topology ki ie ki update keep ki kl li estimator keep locally final node ki ki ki ki
assimilation formulae framework instance correction say kalman among gain instance replace reduce reflect couple h h h assimilation ms l hereafter cyclic z I system variable divide estimation sub role call term mode plot ms l numerically integrate state collect brevity call integration discard divide estimation assimilation step trajectory synthetic gaussian white state variable integration therefore step assimilation ensemble ensemble state assimilation extra divide achieve treat vice versa parametrization motivation usefulness assimilation combine increase stress running estimation scenario divide assimilation ds joint ds da divide divide ds stand assimilation ds whole da adopt assimilation fig ds da divide start couple split sub slow mode mark act line dot incoming ensemble update counterpart describe ensemble forward assimilation conduct plain introduce covariance localization covariance localization adopt initial square investigate framework analysis update identical experiment ensemble observation repetition std state analysis carry report mainly precision computation depict series scenario ds ds joint da divide ds divide divide da assimilation ensemble scenario extra parametrization ds divide scenario therefore panel framework early assimilation become substantial meanwhile ds divide panel ds da joint mean panel panel panel extra parametrization always panel appear low scenario possible view forecast challenge analytic da joint da ds da ds da adopt left column characterize difference work adopt subtract trajectory da joint da divide divide da joint ds divide divide instant ease visualization box step stand scalar grow final increment set matlab band inside box represent end mark individually fig divide scenario trajectory assimilation box appear time move ds da gradually indicate time remain similar da joint da divide except period short scenario also histogram difference whole assimilation window th trajectory state ds joint divide ds divide da joint ds divide da da divide histogram difference peak support peak interval instead phenomena divide da ds da scenario peak tend tend wide fig seem suggest ds da ds divide ds da da substantially reference localization important auxiliary enkf enkf arise systematic covariance increase explanation extra numerical additive model always bad work introduce dynamical context alternative artificial noise e research assimilation development residual appear sufficient conduct originally covariance localization method conduct localization see localization locate localization localization degree half ms choose divide framework component circumstance localization aforementione average assimilation covariance seem estimation experimental hybrid filter relatively indicate scenario divide estimation framework close joint separately filter performance section framework largely challenge assimilation model illustrate ds divide divide extension possibility ensemble size fast slow mode want gain member mode ensemble size apply filter formulae size address discuss material performance plain factor mode filter tend plain take clear ensemble ensemble mode impact reduce dominate l comparing seem well simply fast mode ensemble fast mode dominant extra error scheme significant fig filter plain plain seem system couple exhibit background window could reasonable assimilation scheme slow due implementation significant interpolation enkf background historical enkf appear attractive divide incorporate largely motivated status challenge operational couple computational resource implementation sophisticated scheme enkf combine assimilation divide mode mean analysis parameter slow forward update eqs use historical ensemble eqs slow parameter numerical fast mode background ensemble propagation ensemble assimilation cycle slow drawing specify whose equal historical assimilation available historical fast covariance historical ensemble mode generate historical ensemble mode step take fig fast mode dominate magnitude fast slow distinction hereafter refer assimilation da divide plain set neither set mode divide localization adopt magnitude trajectory da da interval covariance divide localization divide around relative da da divide incorporate divide assimilation problem couple system tackle formulae consider sub separately bring efficiency assimilation scale assimilation combine option divide sub framework addition possible may couple assimilation background precision extra circumstance assimilation one reasonable current work service datum assimilation couple balance generation additional challenge assimilation extend study include limited complement light mathematical equivalence exist example development tackle aforementione similar divide couple extension interaction domain assimilation conduct scenario kalman formulae complicate adopt divided topic investigate future would constructive comment improve presentation science technology also like realistic research financial step expand matrix side equality line h h combine eqs algebra omit summarize chart ds da divide system eps std eps divide framework single repeat background ensemble change ensemble observation panel value repetition divide framework state variable deviation std eps eps eps eps scenarios panel plot subtract difference due numerical accumulated eventually become c error ds indistinguishable integration trajectory ds da ds da divide eps trajectory ds da histogram da eps eps histogram ds divide da eps histogram difference joint da divide ds da ds divide scenario reference da rmse delta lc vary lc delta eps rmse scenario localization adopt eps l eps l eps eps slow plain inf eps da panel localization apply panel mail edu sa consider assimilation couple interact certain way tackle assimilation sub system ensemble kalman filter enkf assimilation system quantity system
variable clear operator close keep recognize discover relation discover appropriate continue expand linguistic horizon syntactic primary search semantic end language lexical appropriate truly need structure sophisticated understanding seems take theory summarize appendix amenable linguistic aspect seem automated aspect structure aim automatically aside relatively effort topic categorization extremely recursive early syntactic level relatively abstract structure yet clear stage stage proceed aspect reality external world arguably lexical lexical name example lexical specify word magnitude lexical lexical semantic mean lexical thousand learn external object refer word also refer observer purely linguistic say entire singular oppose every time structure require world model word likewise many actor properly semantic capture partition say structure relation direct partition entire devoted decision statement partition sophisticated extract appear already identify thing read sentence apparent deep structure text like build also relation syntactic clean enough abstract unclear language reality act parse external checking reasoning consistency closure say external indicate may particular relation rate observation stop language turn something else something else hard external reasoning external align human construct external visual associated resolve word regularity external name relationship order syntactic create mechanism external world language language overall part aspect linguistic part decade item learn e background demonstrate existence far propose create less final working formalize mathematic remain regard suggest distribution prior treat fairly goal probability evenly minimize broad phenomenon entropy seem theoretical clearly correctly thus room ad hoc cut good coupling mathematical development abstract conceptual depict language general structural entity word entity describe section mutual example frequency unique name require work treat constraint make suggest list short description desire way group actual task grouping grouping distance similar thing grouping entropy cut list item add relation lead discovery proximity another know aspect language armed perhaps become iteration become visible clear know already new deviation candidate relation thus sort compose collection linguistic relationship linguistic entity reasonably compactly pattern observable linguistic property linguistic linguistic construct may find via linguistic construct considerable aid linguistic come language linguistic imply corpus datum alone linguistic relationship linguistic entity purely well language learn seem likely accurately corpus far special describe elsewhere key intelligence pattern token symbol part knowledge part thing represent symbolic token easily pattern component corpus usage linguistic linguistic factor come corpus merely particular usage pattern principle create system loop bias toward particular usage valuable break multiple instance loop focus couple learning loop syntactic linguistic lexical provide high linguistic relationship relationship lexical output result semantic loop feedback regard correctness extract confident result loop confident interpretation loop loop attack sort slow issue dag syntactic associate dag represent sentence raw tree feed input layer sufficiently corpus understand maximization example entropy word fair mutual entropy pair search structure closely resemble dependency show pair mutual mi tree unique unique connect typical modern language million link type viewpoint link link automate discovery something speech pair mi reveal follow mi previously mi obvious word might correctness grouping relatively straight act reinforcement correctness group importantly discovery come rule million unique pair small link connect define logarithm total complexity rule plausible mind maintain million hundred foundation type example noun type mn short name np foundation deal thing discover text relation also theory inherent part language theory appear discuss dictionary list perform span entirely two trivially indicator appear many sentence count exercise often mutual lexical another structure discover subgraph instead somewhat occurrence frequency condition logarithm probability entropy exercise mi appendix formula word precise proper rare linguistic less syntactic group common lexical entry part ask grouping grouping perhaps level previously perhaps perhaps back feedback step early refinement trial recognize direct application observation pair essentially identical flexibility language widely variety situation mutual carry poor english case appear relate term link link apply situation discover limit bit challenge manually maintain dictionary construction english flexible rule certainly pair insufficient case challenge occur primarily construction flexibility human sentence phrase case avoid work child syntactic describe way get restriction change implementation overall core language syntactic context consider triple frequency count mutual particular order link contain come link link parse version syntactic mi word initial word link link word category base usage likely classical use single syntactic large usage link category usage word category create syntactic link remove pair link associate link usage link exist link plus syntactic infer category link modify noun link subject possibly link particle head syntactic sentence infer logarithm chain syntactic type may indicate previous clustering return link goal link category contain one link inefficient language cluster relatively maximize maximize assignment large english hundred type unclear sort variant obtain implement basic purely syntactic integrate loop relationship generate word usage way versus link set syntactic far carry soon carry parse word parse reference corpus rank acyclic dag usually syntactic label different prove parse entropy generate span regard parse actual one probability link agreement cause issue one linkage existence assignment incomplete fail link sentence essence parse parse feedback refined treatment phrase boundary handle embed price list chapter design engineering challenge effort syntactic come semantic relationship close syntactic separate heavily influence heavily system map link semantic prototype consistent semantic network formalism implement code spirit proposal however assume mechanism linguistic content via corpus learning code specifically suggest discovery semantic proceed manner discovery relation describe unsupervised neighbor extract way employ phrase save offer candidate sentence syntactic candidate sentence relation syntactic construction carry comparison make string issue alignment parse establish context subgraphs essence recognize challenge recognize challenge relation understand lambda expression employ place partly trick syntactic construction atom form term algebra probability term notion term network field mathematical formalism graphical range domain generalize distribution evenly prior maximize general drawback abstract dense fall np much simple come lack proper generalize generalize instead get metric semantic similarity np hard maximum rapidly convergent extract construction semantic neither distinguish word relation sentence necessarily justify meaning semantic lack mention neighbor across appear strongly nearby word solve chain appeal affinity word neither language similarity instead strongly grained observable hard detect phrase lexical thus word structure reasonable lexical prove otherwise phrase different measure discover measure text occur time sequence occurrence meaning thus word co occur appearance essence word mutual word expand word occur word slope phrase eventually failure carry standard second usage different occurrence frequent word get rapid sigmoid though semantic variety fuzzy robust structure need one discover similarity mean markovian count likelihood unfortunately formula poor ability discriminate distinguish receiver operating word co trick come embed network completely ignore markovian sigmoid sigmoid serve single importance essence sigmoid say forward discrimination counter probability discrimination learn speed incorrect behavior convert build effort thing useful yes describe build approximate unlikely correction discard handle phrase discriminate semantic apply syntactic approach mind require refinement learn semantic outline corpus syntactic early corpus frequent statistically high mi subgraph occur corpus subgraph word allow subgraph syntactic semantic annotated semantic context combination standard subject sentence form word instance subgraph set base division similarity fuzzy degree fuzzy web involve platform far nonempty determine involve assessment associate intersection find graph association instance corpus syntactic parsing pressure reduce complexity counting summing rule relation class content language exercise entropy away relation never occur leave use express fact semantic corpus category link type point link point sophisticated method assign worth category appear graph symbol recognition loop return step newly semantic noted semantic relationship use syntactic understanding way include part use syntactic formation parse semantic resemble deep level make another link know refine relation build syntactic semantic structure basic derive else build give syntactic semantic bootstrappe need semantic text relationship text gradually instance achieve read corpus build linguistic able together conversely layer guide learn low layer seek associate eventually external persistent object action labeling provide foundation semantics encode semantic reverse later unclear direction demonstrate certainly description nonetheless believe mathematical advanced author idea prototype hundred detailed publish current overall project focus direction explore language language analysis comprehensive text mean text although primarily orient I believe meaning dependency think rule algorithmic implementation lexical core linguistic linguistic semantic dependency primitive atomic thus definition specify role essence entry mean theory mean medium compare properly semantic capture mean term medium medium assignment thus medium distinguish pre mean example medium roughly distinction pre suppose information thing fine specifie style semantic network text speech english concrete object external refer person partitioning topic say medium word increase pt form semantic also syntactic roughly dependency like representation sentence structure capture correspondence translate structure another rule think attempt treat implementation discovery lf rule lexical meaning lf normally lf specify noun lf broad lf narrow scope subject subject noun lexical sampling strongly stop half reason lexical like comprehensive view lexical unlike describe mechanism syntactic description frame aspect mean capture semantic lexical hierarchy raw sure take place style reinforcement must layer syntactic layer somewhat develop semantic layer guide seem appropriate framework must comprehensive describe transform language must lexical pick structure pre viewpoint learn treat learn black able probability associate relation simple work mutual generalization typically linkage type leave right observe distribution observe relation relation fact word certain ambient perhaps nearby word commonly position matter notation observe quite since within less observe word pair almost give give mi associate example subject properly frequent pr large unconditional pair entropy relation obtain simple working relation language information structural relationship notation hard rhs identical appear summation difference singleton likewise singleton mutual scale pattern may rare useful mutual conclusion conjecture exercise fully corpus build approach enable needed generation natural regard explicit code linguistic annotate corpora approach fully satisfactory substantial formalize human annotation principle coded corpus nlp linguistic content operate high abstract parse set address semantic content practice yield full success semantic means discuss scientific community regard linguistic text video ambient idea art automatic third list decade address linguistic issue difficulty arise implementation notable build phrase essentially point equivalent progress mention yet fair say truly progress automate linguistic content corpus useful incorporation system content create code annotate corpora linguistic large text generalization author believe body idea enable corpus gradually decade aspect aspect validate simplicity deal text bring complexity purpose document syntactic handle speech similarly closely finally stress intend conjunction corpus processing unlikely happen syntactic learning guide exactly corpus wikipedia might web certainly devoted provide rather basic rather getting lose important thing class thing step e return correlation almost mutual information strength thing thing hypergraph early iteration thing pair measure accomplish primarily principle word b group word class w hold form take word actual consider complex detailed considerable thing environment able thing pattern probably pattern piece piece final picture match difficulty may pattern input language immediately pattern inter suffice viterbi parse relationship say relationship become essentially combination keep likely really affect minute complex pattern sentence together coherent fashion recursion learning principle previous tendency relationship describe minimization complexity tend increase logarithm logarithmic discover rule confidence phrase book group noun phrase suggest book odd noun incorrectly classify place form deep serve refine correctness rule outline capable lack collection unsupervise fashion remainder devote might outline begin linguistic content characterize ai concept language linguistic approach drastically linguistic assume comparison convert dependency easier verify head word word word image learn syntactic relation normally write abstract fundamentally lexical abstraction abstraction low level parse string word dependency relationship extract case impose structural relation noun point linguistic terminology actual require inherently linguistic discover structure believe outside point system learn assume key list list learn previous abstract view merely back concrete task list link link connect sentence head english link
give exchange action action calculus present causal observational expand structure miss check identify calculus effect observational imputation miss flexible observational causal effect predict causal assume know utilize causal calculus systematically apply standard way statistical available validation fitting observational combine carry summation actual demonstrate dataset use effect strongly nonlinear observational differ causal setup far miss aim structural distribution figure causal effect equation include convenience represent represent differently data eq indicator response completely observe indicator variable cumulative likely value small causal causal index variable indicator individual select sample record mark solid marked circle covariate response indicate missing case joint association covariate associate response observational observational due combination value response path latent child calculus average eq derivation detail mean estimate causal effect estimate observational must address biased design figure datum miss random additive necessarily model suffice causal flexible ready make software exist message work require causal complicated miss external need causal care scientific make present encourage causal package utilize fold tune smoothing estimate perform situation different causal high value around zero replace convenience imputation handle miss causal causal b mechanism form datum causal causal causal causal express association adjustment needed calculate mechanism nonlinear density normal normal skewness contain observation see model directly overcome distribution assume joint allow causal effect decrease panel datum illustration difference observational evident integrate method generalize additive spline smoothing forest layer integration summation forest carry package addition utilize parameter fold parameter smooth hide causal see network fail effect forest design number predictor smooth restriction forest scenario cm major advance take causality still effect estimate observational causal challenging tool model causal imputation estimate causal tool keyword analysis miss nonlinearity structural equation major advance take despite still causality observational causality lead problem consequence effect causal relationship dependency see observational instead causality alone estimation effect suffice whether affect affect effect obtain information straightforward causal easily physics causal calculus systematic way effect concept causality clear practically definition whether specify causal completeness causal prove causal fourth framework deal design bias little attention causal causal use consider causal restrictive flexible framework concentrate narrow theory practical present start causal causal qualitatively nonparametric form
element ols generalize relate ridge dimensional elimination beyond broad create several create sensitive column paper demonstrate square various quantity ols ridge regression full rank term instead study estimation return ridge linear motivate function satisfy full define ols estimator define negative zero soft section dimensional svd orthonormal become orthonormal matrix decomposition multiply incorrect basis multiplying estimator correct contain lasso showed bound fix lasso relate relationship equation proof contain great classical uncertainty I penalty term nan hypothese model cdf package normalize equally variable column length heterogeneous length strength ensure leave right q define element pn jj jj exist inference carry select fitting ol test statistic value define cdf algebraic require matrix reliability p theorems regular penalty objective function control let let minimizer row determine statistically care change high converge theorem hold analogy backward elimination analogy backward g resemble forward selection sparse x sparse around yield moreover even minima long ii compute identifiable distinguish function paper lasso paper connect type ol informative p value suggest classical suggest two statistically backward procedure multiple ol remain neither create multi match become complicated forward classical model weakly sign backward elimination trivially satisfy g elimination procedure consistent procedure become ol incorrect hold define orthonormal orthonormal ol n ol ol element statistic define
logical structure notation object dimension notation quantitative fit qualitative collect metric boolean represent alternative boolean derive degree categorical invariance transformation wide human logical category structure improve domain describe learn structure instead take roughly column category task list first familiar whether particular category symmetry reader appear elsewhere stimulus meaningful slight consider fit metric depict table comparable boolean complexity well boolean comparison exist conclude human conceptual reflect mathematical complexity conceptual difficulty information complexity shannon entropy dimension human paradigm dimension general learner child set mathematical domain reveal connection dimension separable learner able yield learner predict metric experiment learn effectively mathematical predict behavior question cognitive difficulty concept six category call paradigm occur human learner characteristic shape order occur human classify characteristic readily g child categorization identification paradigm mathematically measure logical complexity logical rule order difficulty amount uncertainty remain subset base shannon metric minimal uncertainty predict canonical six type metric uncertainty order type six uncertainty predict well exist complexity canonical learner test sized problem type instrumental evaluation category distinguished rule category type distinguish accord type order particular pattern type learn speed order exist literature series reveal condition encourage formation researcher associate set across wide entirely different specifically separate order separate ordering stimulus theory mistake pairwise comprise analyze distinguish order reach circumstance well classify separable acknowledge reader seem familiar detail shown fully match generalization predict ease unless attention human learner match characteristic order type modify type report child type test difficult significant child accuracy increasingly type child explanation category implement trial trial learning explain paradigm specific order use formal metric logical specific put metric account hand operator shannon subset specify provide paradigm specific learner abstraction attention regard hand correspondingly learner unable logic detail among observable I heuristic calculate theory ability paradigm well section classification boolean successfully predict metric logical shannon provide metric explain component description system system categorization boolean type logical kolmogorov algorithmic short program logical describe categorization boolean begin describe dark circle dark heuristic eliminate remain logical also incorporate selective stand structure category ignore appear consider consider object category natural essence easier similar boolean characterize amount shannon information observer observation shannon explain shannon example fair coin shannon entropy random variable finitely self weighted outcome interpret observation mean suppose therefore interpret uncertainty valid approach probability surprising occur completely uninformative sure happen measure maximize event probable symmetry make event event make likely extreme event event total probability ignore mathematically calculus event maximize coin flip coin outcome tail mention fair coin bit coin bit information coin tail yield information coin less flip unlikely maximally coin whose certain shannon fact hx shannon mention encodes dimension specify uncertainty categorization way aggregate dimension formally classification formulate category function aggregated uncertainty specify consider partition specify partition let example partition second demonstrate subset element stimulus entry particular consider partition fall three uncertainty entry vary determined function amount uncertainty formally function uncertainty entry second q function produce relevant overall uncertainty suggest denote two represent element equally clearly exceed value singleton precisely whole observe uniquely category could categorization category uncertainty completeness metric aggregate measure classification metric boolean type ii depict define type two category imply categorization boolean categorization boolean complexity begin maximally reduce length claim digit digit represent claim claim represent element translate confirm describe type boolean simply compact total evaluate categorization consider ignore bind dimension call proportion consider dimension ignore stimulus category invariant proportion produce proportion proportion least transform structural manifold metric square functional order preserve capture essence case information complexity consider dimension denote dimension say zero stimulus category first maximal therefore uncertainty uncertainty dimension know stimulus category dimension zero third category uncertainty second third uncertainty associate trivial minimized divide subset subset stimulus maximize subset aggregation reveal version reveal paradigm assume yield way divide dimension difference obviously idea good level divide stimulus two dimension min contrast opposite ignore remain object look far though twice information structural reveal search degree operator naturally capture might boolean yield boolean rooted finding complicated category find expression boolean start maximally redundant category redundant could minimum path minimum would sensitive natural analog order setting rely notion play role consider task comparison collect paradigm subsection
feasible connect precede insight prove generalization theorem alternative p building sec strictly course normalize perceptron von connect normalize perceptron duality paper margin determine affine lin coefficient margin key quantity easily intuitively rank lin lin dot matter easily margin zero difficulty feasible see search restrict lin quantity clarity feasible distinction really think unnecessary see often elementary simple unfortunately behaviour unlike conv strictly vector maintain feasibility change amount lin orthogonal lin zero vector lin product lin jump instability margin von relate many know interpretation feasibility center dual cone dual cone cone interpretation conv hold zero closely particular connect margin characterize relationship ball conv exercise return note perceptron yield iterate center origin conv zero sequence end popular learn connection margin scope margin central quantity version radius euclidean center origin mathematically leave lin infinite dimensional space occur dimensional ball hull thing ball span column interior full ball expect conv read large affine ball inside overall matrix brevity highlight conditioning feasibility far reach radius latter singular analogous feasibility ill pose normalize margin perturbation row ill make quantitative alternative negative seem literature derive mathematically spirit precede intuition extremely might proposition seem hope generalization interpretation particular prove definition previous capture proposition similarly must spirit statement equivalently follow either w note one entire often characterize iterate example prove prove multiplier equilibria game name whose follow alternate notice one surprisingly spend simplify proof geometric insight brevity strength measure crucially distribution define substitution see precede omit multiple whose interpretation feasibility govern mistake proof prove angle often insight simplify amenable involve duality primal machine perceptron subgradient mistake track property iteration feasible step find infeasible iterate satisfy interesting feasible margin margin true perceptron variant prove normalization margin subgradient minimize interpret unit unit respect yield argue iterate towards optimum require perceptron step also elementary open surprising singular value similarity communication latter publish go von circle independently identical go geometry von conv say loop von produce establish von though primal like perceptron prove step question special private frank wolfe light problem see connection duality subgradient frank wolfe finding solution converge linearly strict nevertheless perceptron dual mistake machine descent coordinate ascent relevant margin correctness affine relation generalization theorem statement precede tool remarkable theorem explicitly affine margin compare classical iterative turn spend simplify presentation though behind contribution lead final clear clear realize straightforward margin recently universal depend tie take part could round simplify reach remarkably geometric intuition margin duality claim intuition impossible algebraic generalization lastly claim perceptron surprise provide classical seminal theorem rate theorem proof strategy array usefulness lastly classification integral analytical algorithmic idea behind margin choose aa g aa b
inf mh mask blue value show sampler perform poorly fail completely inform improve sampling cluster cell tailor discriminative lead inference proposal probabilistic set future vision identify dependence recently accurate generative allow scene yet trend world idea inform proposal manually readily vision domain sampling reversible jump mcmc method investigate describe manuscript believe computer graphic current efficiency towards inform heuristic technique principle rich generative emphasis aim create scenario basis inversion involve plan computer principle generative model aim adapt proposal exist straight would proposal draw identify time markov accept mh value independent accept use discard replace inform fit exist refer detail technique monte carlo qualitative informed inf still challenge already inform mesh posterior obtain inform sampling inf mh informed indicate red page analysis sampler low sampler individually difference acceptance plain standard update converge high modal take different chain sampler result temperature perform mix acceptance inform inf mh choose combination various mh temperature combination pt inform result mh result poor proposal ar standard deviation four compare plain optimal plain optimal inform inf choose acceptance deviation find proposal show figure select ar value pt coefficient inform inf mh hard variability texture accurately formation belief posterior explain intuitively largely fail difficulty posterior efficient believe usefulness generative vision exist even principled concentrate invert exist graphic engine graphic inform discriminative technology improvement conceptually generative physical formation interest presence position nuisance light generative think deterministic image observation prior inference generative fail variable reference frame bad generative use intuitive fair track record generative vision use heuristic objective function problem design inference therein computer leverage dedicate hardware system generative modelling motivate research inference computer graphic world test stem reason dependency modal forward process prevent exhaustive enumeration believe usefulness generative task argue overcome substantial challenge devise allow different novel scenario want maintain correctness efficiency leverage aid paper markov proposal instance drive method tailor informed sampler vision feature make informed proposal latent accept reject inform implement model sampler incorporate object ill carefully assess sampler investigate probabilistic estimate exist inform sampler library produce inform likewise inform sampler present baseline method experimental inform diverse reasoning estimate body conclude future stand vision graphic build vast computer vision generative mention graphic scene understanding pose pose many infer spirit use segmentation human pose estimation sampler highly domain yet technique believe task idea graphic understand root computer graphic inverse goal category formulate convolution understand deconvolution pose specification generative modelling try also program module program formulate hasting mh appeal graphic plain inform challenge another piece apply bayesian devise show multi invert code mention paper posterior challenge make especially correspond graphic despite apparent observe vision exist distribution interested availability infer accurate offline stage computationally accelerate time metropolis goal particular instance sequence markov repeat step propose accept mh technique differ proposal hasting state image inform stage parametric proposal mh valid call inf move markov inf mh ideally global move local proposal responsible density locally proposal every accept enough acceptance rapidly process density estimation cluster observe unconditional kde choose solve diverse kernel detail kde transition random forest approach map observation simulate fit initialize representation discriminative heuristic invariance nuisance main method across region technique summarize test advantage give need identify efficiently kernel reversible metropolis hasting combine hence reversible ergodic distribution ensure remainder demonstrate three initialize sample except note center metropolis sampler draw dimensional propose move chain modal distribution technique replica exchange chain propose chain work individually high temperature implement adapt initialize fit kde already mixture size valid find sampler inf mh mode sampler proposal inf initially hold dominate mh indicate move slow sampler mode discover mode inf even inform inf due acceptance sampler acceptance sampler h camera also test find variable fast enough median separately camera inform inf inf modal well inf inf mh mh experiment orientation image add bit look image cube location orientation angle label switch color choose resemble leave scene readily solve source previous increase proposal sampler inform propose jointly product distribution single proposal scheme sampler inf square boundary obtain inf discard feature cluster kde observe image kde determine inf empirically inf find inf sampler fail proposal state sampler acceptance around follow separately reasonable guess fail informed inf rate median inf produce localize uncertain show relatively baseline sampler crucial enable heuristic library experiment inform simple baseline sampler inform sampler improve speed baseline produce fit fast estimate single sensor vision human body produce mesh body allow size characterization roughly mesh human pose angle predefine part use parameter mesh component person camera viewpoint orientation pose hold generate mesh representation virtual camera create image person choose take gaussian create depth learn proposal record value height feature normalization feature learn kde cell forest kde discriminative try require reliably forest adaptively uninformative adaptive mean tree score tree minimum leaf kde train time regression proposal place kde cluster example observation overcome curse semi capture explicit parametric linear dependency combine explore visualize angular error ground mesh individually inform analysis sampler include supplementary material test approach standard normal summarize method baseline mh inferior mh low rate inf low acceptance decrease rmse regression inf mh rare
di take choice counting intensity ff ai mark spatially nf turning whereby usual must case may cl absolutely last reference l control mark density take form treat spatial cox marked indicate additionally assign mark auxiliary mark control former process mark connect naturally stochastic forward stationary mark cox intensity mark point small intensity correlation marks cox setup develop affect employ cox dependence mark may idea ground intensity mark note fall category mark discussion mark intensity next turn consider construction spatio look marked intensity look particular define temporal marked cox benchmark locally intensity bound intensity I intensity whereby correlation functional stationary due randomly label next relax slightly specifically process constitute poisson intensity whereby correlation functional g g l al f g mark cox process spatial recall ground locally direct constitute cox conditionally conditionally poisson possibly finite negative locally cox write cox cox whereby cox spatio spatio xt may connect simultaneously way drive random intensity mark translate definition process intensity conditionally mark density extremely development tool kind development recall definition temporal mark order cumulative e construction ground may mark present behaviour n I n ng g assume derivative refer regular write intensity nx x I analogous derive product constitute natural gain whereby component stage f b define c b monotonically locally become respect call mark enough specify accommodate purpose construction martingale intensity adapt process write order relation pd assume derivative coincide except set integral ground l ff sx dx conditional respect tx statement far whole consequently exist l f radius note extension since start recall construct constitute compound process note construction location dependent markovian mark bit mark note also sm simultaneous evolution mark underlie evolution markovian mark pm tt ts give n assume reference measure ng ideal mark mark else point density recall quickly density fundamental exactly specified location see imply sense regular mark give define ng l totally finite treat construction equivalent probability pn measure n multivariate point recall auxiliary imply f f finite intensity al support whereby define f fp mn f g ff I density way current context density class g g l depend measurable interaction markov may conditional intensity intensity exploit formula exact convenience possess spatio g process pt ks markov define mark situation characteristic functional mark functional mark multivariate mark form define spatio I expression play role since sample accommodate yx functional mark reference say brownian motion calculate explicitly density sample factorial moment n yu k derivative absolutely continuous exist assume imply consequence correlation g u g turn intensity expression x df sx f ff df sx well refer marked intensity fashion may mark u l finding point locate outside spatial auxiliary mark conditionally absolutely respect turn absolutely consequently entity lemma turn mark mark wish essentially mark spatio mark point x l nu u grind good account quick could proceed spatially see n l u x dt l sx g sx dt process function kx kx l n u f j respect markov pseudo ground auxiliary suitable conditionally give ft nn mark accord slight abuse notation dt kt dt mark influence outside approach simulate miss datum family ft n exist density nn product spatio sample mark certain situation however argue mark spatio ground process possibly auxiliary mark able evaluation l correspond treat observable time analogous scenario stand birth radius height location retain function reflect nature th remove marked density need mark find find kf xt I xt x l stand time tree acknowledgement author truly grateful idea author grateful technology u discussion education corollary remark gb point spatio temporal mark c mark spatio process indicate sensible connect field spatio boolean mark cox double connection discuss purely intensity tool estimation intensity mark reduce measure pair field spatio temporal functional marked point spatio spatio mark wiener treat arise phenomenon mine naturally table application represent occurrence event death incidence review mid th interest expand point point event volume emphasis spatial process methodology range field distinction area absolute coverage cite reference temporal distributed point name associate model history david cox attention spatio temporal location event origin incidence relationship worth traditionally view principle vice versa instance times though consist zero one represent process point interval see analyse contain mid spatial process spatio regard hence special spatially process lot spatio temporal ad hoc approach spatially naturally binary number event conversely spatially multivariate process common spatially sufficiently justify explicitly spatio mark carry mark variable several variable type mark mark mark interest spatial size shape treat mark concern analysis marked pattern analyse mark conditionally mark mark alternatively treat mark case mark value concept refer process one among know example able density local mark situation mark field field determine whether technique investigate interpretation mark mark conditional variance mark another serve mark another generate process poisson depend mark point history analyse function qualitative quantitative collective tree location stand moreover point community form spatial location local characteristic approach classify discriminate belong clutter belong configuration permit analyse curve depend describe model curve observation number theory theoretical analyse curve analyse develop temperature year analysis none new characteristic process number derive approach spatio temporal mark propose new spatio point mark generality ability accommodate different structure new notion spatio hence provide framework framework mark interpretation connect spatio intensity significant practical aspect structure give thorough section new c functional temporal interpretation motivate spatio section characteristic intensity need development section discuss certain within characteristic cox c mark process mark type spatio version first collection euclidean location value mark I e variable control type random event describe note extra start define continue space underlie inherent spatio temporal filter dx complete borel borel denote identical lebesgue denote measurable dimension lebesgue indicator cardinality context measurable dirac singleton sometimes dirac short understand spatio functional mark shall spatio marked process mark point ground mark turn subset construct process location common compact identifying construct spatio temporal interval point temporal occurrence g birth auxiliary auxiliary may possibly type let auxiliary mark mark euclidean metric consider whereby propose space borel functional marked array thing growth dependence choose allow mark take see space continuous limit function function metric ff ft du borel denote accordance element path process evy empirical f ft bc rd bc f detail c stochastic diffusion path space supremum set supremum supremum b equivalent metric differently temporal point purpose preferable scale may time whenever cross context g forest stand pn xt construction framework application surprisingly ordinary marked spatio temporal x give mark provide similarly I x spatio type mark slightly create mark temporal process constitute marked mark come spatio temporal random xt spatio z xt I nm ti field k belong hilbert g follow model residual mean variance location z jt th tx x jt stationarity xt jt clarity prefer expression follow call describe spatial among across entire scenario one spatio spatio field monitoring location framework constitute consequently framework incorporate deterministic associate section hence obtain exploratory analytic indicator examine pattern relate point regard mark covariance structure live pattern evolve spatio idea local spatio surface spatio surface dimension surface functional mark surface provide spatio structure rise substantial idea extensively mainly context mark process birth death spatial auxiliary holding time distribute conditionally mark ordinary dt gm j l jt individual growth absence individual hx j jt spatial interaction find mention reference application collective stand birth breast growth radial scale add mark eq independent diffusion coefficient would measurement simplify interaction mention wide scope root provide shape first factorial density start set ai density permutation invariant measurable functional partly section region mark dl measure first factorial I definition intensity functional refer note ng find observation probability product exploit assume g n n n np np n whereby n space absolute continuity whereby e find express density give factorial measure regular probability exist intensity functional turn underlie conditional existence whereby mark n stochastic process absolutely continuous I version furthermore auxiliary mark x ff nf valuable dependence spatial see play light q correlation spatio pair g extend measure r r family q functional fix write expectation interpret probability event reduce counterpart relation give fp accordance choice define
conclusion table incorporate trust boost performance recommendation accuracy demonstrate advantage trust aware recommendation social performance significantly discuss beneficial relation relation type relation advantage utilize trust look notice incorporation trust neighborhood base error significant social trust know publicly include trust consistency relation work particular examine correlation metric trust seem insufficient user generalize item works practice preliminary hinge optimization viewpoint smoothness interesting direction gain accuracy corollary network recommender crucial success service due application need preference despite recommender suffer exploit social relation along rating recommender stem significantly influence user web trust list user friend base respectively incorporate information contrast incorporation recommendation potential explicitly incorporate almost paper social incorporate trust relationship quality recommendation trust enhance enhanced counterpart respect thereby demonstrate incorporation recommender huge increasingly cope relevant one recommendation take account history interest network enhance obvious system success online amazon netflix guide goal recommender book news web interest widely broadly cb cf hybrid cb recommendation try give past external item description profile extract analyze recommendation cf recommendation popular method recommender system assumption user express rating past access cb propose cb essence lie neighborhood user user recommendation preference user similar cf recommender recommender representation item user main also cf consider combine model type accurate recommender system heart similarity orient similarity orient cf orient oriented overcome oriented cf usually profile behavior seek orient additional wise also collaborative filter method rating memory collaborative express rating user perform user rating effective rating user rating base correctly decrease recommender major based suffer scalability reason application search identify computationally scalability issue issue overcome limitation observe interpret predict past use predict unseen cf employ datum recommendation learn category aspect bayesian due handle huge method low factorization nonnegative pmf small item rating recommender although recommendation scenario fail rate item instance accord non miss become challenge sparsity problem recommendation prominent tackle problem media application allow interact video page group idea significantly influence connect user share interest life application accumulate would utilize goal recommender recommendation historical user trust user social friend enhance become great increase availability review many review product social trust trust review find online community million comment capable friend site share condition online review social website require trust person user come role product influence confirm exploit recommender long fundamental base recommender recommendation friend probably goal rating trust boost sparsity issue trust among online trust pass one member another social trust recommendation social relation regularization trust trust recommender last user trust relationship relationship also quality later integrate reflect word whose review inaccurate low therefore utilize boost recommender contrast trust great research disadvantage explicitly utilize recently attempt relation recommender system proper incorporation social recommender system prove manner consistent plausible naive modeling trust raise challenge incorporate give challenge involve enhanced recommendation particular trust relationship recommender increase knowledge work model relation trust time intuition behind one interpret relation user preference rating must incorporate deviation latent combine user effectively preliminary experimental demonstrate deviation way incorporate propose facilitate feature user minimum dissimilarity formulation agree item friend friend predefine recommender trust relation enhance friend factorization factorization algorithm incorporation trust system base recommender leverage type propose social network empirical investigation rating particular examine extent align rating exhaustive propose advantage detail trust enhance recommender system put work social recommender system formally incorporate trust discuss include demonstrate generate accurate discuss research recommender system directly enhanced recommendation successful past recommender enhance approach recommendation trust review major enhanced recommender explore user recommendation aggregate trust neighbor social trust aware recommender collaborative inform trust utilize social connection social annotation recommendation construct walk perform ed trust trust trust trust idea depth independent trust two cycle trust remove remove cycle every infer trust trust acyclic walk combine trust item recommendation noisy outperform exist approach walk trust predict rating item similarity great rating predict reliable friend away recommendation approach trust user item systematically user similarity pairwise trust researcher factorization technique user rating among datum divide two regularization method typically social social term friend model laplacian add minimize et build kind semidefinite relation factored rating factorization incorporate graph generate disadvantage user recommender recommendation process reflect make interpretability drawback cause interpretability name social latent user rating combination basic show basic exist trust handle trust recommendation network minimize disadvantage user model incorporate influence make feature incorporation recommendation totally recommendation start recommendation recommender lack set propagation formal trust introduce formal trust trust propagation capable kind trust extend propagation enable feature comprehensive estimation trust metric propagate trust score aggregation use trust trust score trust propagation seminal incorporation information recommender address particular enhance careful incorporation enhance recommender outperform trust counterpart rational algorithm employ address connect two trust accord introduce trust path aforementione predict trust relation social utilize trust combine trust direction examine relation social work factorization effective recommendation l symbol mean latent rating matrix user network user social relation social neighbor similarly provide formal collaborative filtering concern follow literature collaborative filtering assume click item user rate aim rating user rate correspond th item matrix partially recommender system utilize user item factor preference preference effectively recover rate singular method rating applicable fact rating matrix utilize observe formulation e subsection review extend incorporate trust information item user rate goal constitute minimize factorize term matrix respectively control matrix respectively would practical collaborative filter application well trace netflix amount success million rating rely matrix expect user rate important challenge recommender start user start life system rating allow prominent tackle factorization rating recently trust relationship rich side technique ignore trust recommendation usually trust trust trust generate recommendation aggregate user intuition user tend adopt item recommend friend trust positively strongly correlate recommendation trust provide accurate incorporate relation problem keep user user vector close user feature user rating specifically problem subset behind social idea every user similar friend use social friend assume weight relationship user social easy friend simply objective jointly fix one recommendation incorporate trust relationship partially observe present generate computationally gradient descent rating suffer vast majority recommendation trust ignore capable exploit develop recommendation network trust relationship trust recommendation evidence trust certainly propagate social influence basic user close separate apart interest incorporate idea matrix user friend reach desired trust directly replace utilize careful trust chance certainly away regard incorporate another behind stem relation relation consider user agree friend item friend reasonable margin enough friend dissimilarity user friend view viewpoint connectivity feature influence user connect learn map basic feature isolate enhanced social trust relation correspond positive reduce latent edge close distant inherent topology network illustrate behind rating feature friend margin figure example illustrate mention ease exposition consider obey goal prediction rating trust user depict viewpoint user circle latent friend outside dash circle circle safe margin friend impose triplet friend triplet force extract triplet ensure exist friend factorization mention similar enhance existence relation investigation correlation social relation rating include rating people neighbor empirically social support formalize ingredient latent exist social introduce monotonically user user behind I latent triplet function k k two assign utilize assess loss logistic widely loss extract triplet friend becomes learn latent feature consistent consistency reflect problem make hinge assumption write consist output utilize estimate user item compare discuss reveal aim inherent among objective capture recommender system description object gender age potentially description way incorporate meta measure latent feature pair meta datum specifically obtain meta diagonal pairwise similarity graph base meta rating return like emphasize triplet trust link item user add latent item generalize triplet link dissimilarity link accord tag associate tag trust otherwise alternatively trust dissimilarity profile improve recommendation regime popular operate stochastic approximation regime approximation regime large use approximate tradeoff reliable preliminary example gd sgd gd sgd solution gd gradient full discuss gd detailed objective convex minimize iteration fix repeat fix exposition triplet auxiliary except ki jj ij write apply gd indicator take value argument gd computation expensive note large triplet next provide issue descent sgd gd slow batch output gradient update eq problem gd gradient idea fix triplet triplets triplet strategy unbiased make iteration triplet intermediate sgd triplets select triplet gradient decrease enjoy light mini brevity mini sgd triplets triplet gradient set sample triplet unbiased full gd convergence number non smooth sgd much gd exhaustive experiment conduct experiment fundamental question perform incorporate trust social extent user align friend question trust role accuracy recommender system tune exploit social prediction affect recommendation extent lack trust relation trade efficiency subsection begin introduce employ follow discuss choose evaluate aware recommendation customer review website share movie write review rate helpful helpful rating whether worth quality review user addition trust rating present review inaccurate quality explicit negative trust enhance recommender ideal social relation trust full statement conduct come item total rating datum approximately rating demonstrate rating rate overall summarize statement trust well utilize recommendation different create increasingly example randomly select sample predict rating independently fair since hinge well rest stick however loss thank smoothness negligible matlab load machine trust number rating number rating average trust max user min rating user employ mae rmse propose filter enhanced mae offline mae predict compare absolute value prediction mae value predict denote factorization assign item rate user item measure mae larger even rmse valuable netflix competition reward rmse trust implicit relation recall normalize discount ndcg rank friend define relevant friend friend user relevance rank user ap ndcg discount cumulative sum rank user position discount ndcg ideal measure ndcg measure user rank ndcg logarithm ndcg scale normalization logarithm tuning parameter may drastically objective important much incorporate complete partially observe rating user utilize item rating hand dominate reasonable social recommendation grid value two parameter combination good grid parameter achieve validation consider triplet computationally remain perform repeat set algorithm choose recommender system trust relation recommender factorization recommender take factorization trust exploit trust minimize network intuition behind assume corresponding feature quantity user e add problem obtain optimization factorization trust algorithm stand propose literature exploit trust memory recommender predict user combination rating neighbor user already rate idea trust recommender limit neighbor user distinguish trust trust adapt rating relation instead schema trust relation binary social hamming implementation neighborhood information use recommendation predict rating adapt exclude trust integrate spirit strategy use relation trust relation contradict propagation trust exclude implicit trust range rating ndcg ndcg ndcg consistency implicit ndcg c rating ndcg ndcg website allow review product service review allow trust review consistently valuable review consistently inaccurate rational incorporate trust recommendation trust relation investigate rating implicit trust trust relation social important aim empirically investigate user friend rating user assign review rating write neighbor social social could supplementary rating boost recommendation rating review analyze also claim similarity implicit trust use complementary compare implicit explicit implicit investigate relation rating interpret user literature adopt popular refer pearson coefficient respectively relationship behavior user implicit
shift high cause robustness architecture function highly non linear code cnn obtain effect principled subspace dimensionality acknowledge ec axis fp adaptation adapt acquire search leave prediction learn misclassification alternate sub performance aim adapt acquire new target domain several task scenario attention search domain invariant leave stochastic gradient learn world exception think part corpora time subject vision particularly way background motion mention acquisition resolution artificial e filtering train final poor domain adaptation overcome information come target label extensively paradigm invariant searching representation source target aside reduce crucial adaptation induce distribution would perform equally labeling use domain learn jointly reliable source source depend availability annotation semi set provide besides large source method modify formulation train learn source classifier correspondence label sample augmentation case domain part recently propose combine cross domain domain encode kernel challenge unsupervised domain resort minimize sample mmd reproduce use domain nice critical lead poor goal define overcome reconstruction map target promise domain adaptation shift present mostly feature mapping cca couple principal eigenvalue instance space variance preserve distance transfer subspace project alternatively use mmd subspace matrix exploit intermediate idea introduce domain geodesic strategy extend intermediate subspace intuitive alignment sa demonstrate source pass intermediate overall domain invariance less attention dedicate substitute pls maximize unsupervised approach go beyond search subspace exploit source discriminative encoding source distribution shift previous work method fix adaptation adaptive method exploit annotation name rest briefly domain subspace domain shift conclude classification sample unlabele different hold satisfy labeling operate adaptation establish condition source perform demonstrate eq indicate error ideal suppose low low dimensional intrinsic structure source matrix subspace transformation modify subspace target simply measure frobenius minimize obtain transformation matrix unsupervise pls extensively study sa promise cross domain subspace adaptive par keep domain adaptive process separate focus aim domain divergence source domain concentrate sa margin formulation detail target aim representation combination hinge function trade high give importance focus role sa priori c da avg choose see dataset combine office class amazon resolution provide descriptor standardize normalization consider split provide two digit different specifically image normalize gray pixel use annotated image search image per category typically representation adaptation estimate mobile device period contains label collect period unlabeled period location evaluate predict mobile device modify define location set repeat consider target split random subspace domain analysis supervise description turn locality preserve option fair comparison none exploit subspace publicly available implement regularization linear geodesic kernel alignment modify integrate preliminary obtain pls less explain jointly maximize source minimize equal pca baseline also source learn target subspace pca represent target method final tune fold cross three parameter remark annotated option indicate tune baseline source target office exclude extraction modern computer cpu ram compete training slow g slow sa provide optimize besides reduce test phase runtime comparable adaptation model offline issue change high value drop source square trend office source target state target similarity reduce divergence sa respect case lda domain amazon interestingly consistently understand performance target vary source change figure namely amazon appear high stable source target divergence loss appear always red obtain square target analogous trend domain negligible find good domain divergence instance label separate linear final indicate compare domain shift application sa learn exploit vs involve per separate sa produce suggest main feature sa challenging domain want
identity square index submatrix back multiplying maintain simplify definition eliminate minor dpp marginalization formula eigenvalue specifically exponential reduce elementary symmetric order elementary eigenvalue except note elementary polynomial note recall eigenvector denote singleton eigenvalue dominate require reduced consider matrix eigenvector identical eigenvalue get dominate letting eigenvalue update step depend eigenvector eq apply standard rule subscript indicate eigenvector express entire similarly sort expand zero row somewhat relationship body recall background section identity dpp relationship start step long express plug sum elementary substitute change sum assume row derivative efficiently specifically diagonal dominate contain body ccc wishart moment moment initialization moment low set setting set start wishart example draw setting third trial lemma observation single h bar bar bar bar style ne nest marker nest engineering university engineering compactly semi dpp learn entry log likelihood entry thus focus row propose parameterization eigenvector bound expectation maximization world product recommendation gain naive likelihood project gradient ascent example typically choose product domain diverse chance user find define assign quality discover effectively adapt include marginalization arise dpp compactly learn example maximize np gradient projection produce degenerate partial scalar learning make item direction store dpp attractive property lose bayesian unconstrained non method propose differ assume restrictive eigenvector develop maximization style problematic naive maintain projection sometimes nearly interaction lead nature make ascent dpp psd index notice psd non normalization thank intuitively capture quality item diagonal item eigenvalue clearly psd also imply exact marginal convert learn subset section naive project ascent marginal entry eigenvector hide variable apply inequality section ascent low log constraint ensure psd put upper let rule project ascent optimization technique algorithm refer ascent psd eigenvalue notice project guarantee optima poor probably accept though step truncation result still improvement initial near well employ dominate operation projection overall runtime convergence follow em well runtime tb kk p dpp marginalization provide dpp broken eigenvector submatrix index eigenvalue eigenvalue dpp marginal mixture dpp algorithm mixture equation sense intermediate variable em hide introduce auxiliary develop corresponding gradient proper position us eigenvector see appendix step practice optimize take ideally repeatedly various objective exactly expectation size solve perform trial update enforce eigen projection poor optima associate sophisticated technique optimization practice second order hessian use multiplicative em previously assume average runtime come step search test ascent google em require kernel neither start diversity thus explore option naive incorporate statistic initialization wishart degree freedom identity wishart eigenvector unitary output fit correspond dpp place mass wishart tend emphasize unless employ matching normalize single item recall attempt match choose start recommendation task ground comprise product category recommendation account choose popular negative dpp use basic live recommendation build test dpp dataset consist amazon com product amazon product category split toy item filter product least discard category remain sub category test diverse appendix detail number initialization initialization wishart initialization gain advantage truly strong negative create check value gain exhibit investigate far set step near wishart problem close comparable likelihood average gain moment perfect instance poor category
place therein exactly main difference aim nature result therein extension would guarantee direct lipschitz assumption concavity imply combination aim instance nature contrary demand aim necessarily satisfied require need severe use set get remark rate achieve original variation term computational projection solution perform strategy round mention omit sake recall set payoff actually scalar payoff mix opponent eq compute whether indicate actually singleton game hard strategy need compute query convex polytope particular payoff asymptotically statement neither statement opponent determine minimal provide whether cost drawback superior refinement quantify exactly refinement work mention cost aim payoff possible adjust adapt assume exist continuity soon take constraint game linear scalar payoff cost value bound formulation correspond payoff notation distinguish way relaxed propose computationally already simple strategy could well achievable conclude quantify general gain quantify mostly lie material objective fact proof omit main body appendix indeed section path various completeness read understand result main putting optimality prove supremum norm thus induction confident payoff adversary respectively product rewrite eq short cauchy schwarz indicate assumption quantity bound everything induction provide suitable take bound relate time whose conclude sequence induction satisfied assume latter indeed expand hand proof performance self confident proceed force immediate recurrence something order claim von latter convex negative expansion supremum payoff opponent maker get reward product convex expansion aim minimize payoff opponent decision maker value distance supremum negative graphical expansion solid line thin solid assume contradiction achievable imagine choose equal choose possibly stage nature choose next entail stage stage repeat part identically play prove follow computation leading entail substitute supremum component display mean decomposition converse thank start refer absolute combination admit expression leave switching matter payoff regime function indeed round illustrate combination combination absolute indeed see give eq study consist decomposition get help mind main local expectation payoff advance next getting play could play lemma target already theory admissible achievable function infinite admissible show example actually result exist minimal achievable target function totally order subset indeed least element difficulty achievable compact payoff empty empty fix would cover totally addition exist together contain differently achievable existence lemma show target methodology therein strategy maker target choose stage denote mixed maker round payoff receive q negative achieve denote next stage denote action play decision average receive one hand thus equation inequality together symbol play arbitrarily nature round first round play round denote play decision maker round payoff round limit claim argument interval universit paris france paris players want converge try exclude preference relation target possible payoff oppose spirit action player receive vector round approach set require projection scalar perform episode online learn minimization decision maker obtain much perfect average maker quantify g sequential decision cast objective possibly arise field finance resource management many call optimization solution use offline pareto front optimize feasible weakly dominate front choose objective pareto see multi player value payoff player play want vector objective represent admissible state player exclude prescribed start determine task determine hard reveal standard decision online reward priori make game structure every decision reward assume game treat game maker specify approach value reward define goal list multi exist work use efficient maker advantage constraint special present similarly general comparison summarize propose online approach furthermore start game move discuss target family possible base achievable achievable sort individual latter never worse strictly devise achieve goal amount regret modify direction application classical sample path payoff consider theory average payoff asymptotically base know small game repeatedly play two player maker opponent player payoff finitely whose opponent round vector impose restriction lie pick action scenario inform round payoff martingale study conditionally know put differently formulate resort formally maker want average formal latter small target indicate choice take expansion decision maker decide wireless channel maker power throughput channel much ideal work throughput maximal power zero maker look axis throughput power throughput plane value throughput throughput expansion long hausdorff distance one give link consider opponent possibly random payoff round opponent choose rx opponent strategy empty close course meet dual expansion close correspond constant put therein related mathematical mind cost aim player average payoff small expansion constraint prescribe formally matrix payoff vector abuse work opponent maker maker get payoff admissible adapt exposition set pp general satisfy soon follow value fix expansion discuss start target aim intuition target rely parameter denote payoff expansion component infimum continuity lipschitz achievable maker ensure opponent non follow convergence set graph coincide weak useful target function error stage prove resort relaxation see ask hull define supremum convex decomposition I strictly target achievable part prove rewrite eq whenever convex e g denote section strict summarize fact individual achievable second response function theory explicit efficient achieve however calibration intuition target calibrate sense know advance exhibit target call complexity rely property strategy output know advance payoff
q n e mn md replace exactly statement generate projection eq q least conclude distinguish let assume give jensen inequality satisfie see therefore modal universit paris reconstruct entry problem recommender filtering quantum physics work recover sub take correspond recommender necessarily uniform nuclear penalize analyze theoretically bound kullback leibler tackle potentially dimensional attract decade consist random classical noisy entry real observe presence finite alphabet categorical survey survey yes quantum outcome recommender movie rating dataset item survey course incomplete proportion entry matrix completion ill pose particularly popular low constraint completion observation low approximately reference commonly square rank constraint nuclear alphabet among case depth recover probit example observation moreover uniformly recommender rate theoretically nuclear variation norm constrain penalize lagrangian rather first upper kullback leibler distribution general sampling absolute upon previous find last coordinate recently design possibly approximate formulation benefit value scalar order decrease operator give matrix hellinger distance leibl integer link let cardinality finite alphabet logistic rating function entry though multinomial distribution success probability give denote q control estimator logit instance error binomial score link framework uniform index resp resp exist ensure whereas requires sample associate reveal coefficient rademacher material kullback leibler true universal stochastic quantity control ease notation constant convergence bit minimize slow study max constrain maximum rate multinomial logit associate distribute problem consider significantly burden significant impact solution introduce iterative mind operator singular singular potentially computation see vector canonical stand tensor normalize linear real space cone key denote zero singular otherwise obtain conversely auxiliary map map step nonnegative denote dirac k store entry index total allow table give rough execution cpu ram cache evaluate uniformly unitary equal factor uniformly observation logit logit use implementation moreover obtain classical analysis contrary logit completion observe outperform gaussian leibl symmetric rating fold grid gaussian modal unable necessarily value probability probability distribution real calculate kullback leibler truth dataset test
sgd individual architecture connect architecture tradeoff compatible deep exploring procedure begin accord multivariate normal network output prediction derivation position indicate auxiliary attribute j j filter propagate computed channel error fig sum loss detection sum multiply transpose convolutional filter pair fig learn pose expression feature face face exhibit pattern optimize eqn logarithm finding become definite dynamic ignore eqn become f imply value stop task eqn tendency drop period length indicate valuable similarly tendency stop strategy addition need tune threshold decide stop sec model auxiliary attribute readily handle initialize share tune label dense representation pre unit contain layer pool connect commonly conduct overlap fully fourth task stage include failure estimate ground inter failure facilitate choose table facilitate criterion region influence addition divide face category rotation pose face corner task select rest image face challenge conventional dataset specifically pose variation face testing image severe partial face annotate annotate rotation select face depict web face interaction face annotate web annotate densely face face face consist image expression show densely l bag big open gender face head pose profile coordinate different convergence verify train dynamic task slowly stable early coefficient early scheme network em inter task wise stop covariance give relation square root correlation compute absolute five corner correlation attribute determine global pose rotation affect randomly choose attribute attribute visualize attribute normalize one visualize profile leave right exclusive positive face rotation five attribute average absolute intuitive figure head pose heavy positive attractive correlation simply task target term demonstrate effectiveness cm cm cm examine variant along auxiliary global result variant full model attribute simplicity variant cm auxiliary task beneficial task group pose pose global attribute auxiliary mainly capture local information main gain cause attribute fig produce trend accord face corner low face learn share describe corner similarly location localization remarkably face constrain likely pose show demonstrate effectiveness build full publicly image superior cnn legend method cnns cnns implementation cnn layer comparable locally cnn intel cpu cost ms gpu structures table run ms wrong transfer train sparse tuning various pose regression binary auto encoder compare tree jointly head response map face face method detector error image face result annotation report near grey capture follow protocol face set face produce comparison see exhibit capability handle difficult large head rotation representation fig show protocol face face set quantitative algorithm depict achieve detect face cm indicate instead detection detection heterogeneous task appearance expression head pose auxiliary deep share task utilize auxiliary attribute severe pose need cnn gpu techniques future work explore zhang receive b china currently work toward ph department research interest track department chinese university interest deep vision graphic change science research chinese university previously semantic research include vision visual surveillance technology ny ph institute technology research engineering chinese work microsoft interest include vision recognition receive good award conference computer vision conference computer transaction intelligence international computer pt ie edu study improve jointly optimize detection recognition heterogeneous correlate attribute gender appearance since attribute learn address novel inter employ dynamic facilitate learning task extensive learn alignment deal face severe pose ii drastically deep face alignment face semantic corner verification great field detection remain head pose traditionally independent approach include template base coarse fine cascade accuracy compare previous exist system architecture believe number factor govern intrinsic second discover attribute help detect corner accurately also small face large rotation source space alignment divide auxiliary attribute corner specific dataset rich treating constrain detection achieve optimize b average attribute blue face face pose effectively divide investigate main leverage information task head pose gender age attribute multiple appear model allow joint conventional challenging several task face inherently difficulty identify easy determine negative enjoy balanced recognition auxiliary improve procedure poor study optimize correlated signal auxiliary back jointly alignment nonetheless newly address aforementione consider effective deep equally assign weight auxiliary dynamically task show coefficient essential reach alignment heterogeneous task concern correlation exploit well dynamic task correlation learn automatically newly thank effective share auxiliary learning base alignment five readily handle configuration method challenge dataset specifically dynamic coefficient relatively effective heterogeneous objective jointly improve usefulness auxiliary improve technical evaluation conventional category base task task coefficient method difficulty introduce coefficient early network learn dynamic task aim rate convolutional cast detection transform pixel highly nonlinear five tuned label prevent fit general pre procedure former step filter normal face extract detection prediction auxiliary coordinate let face pre imply center corner fine tuning attribute prediction generalize model suppose correspond first additive task model crucial detection correlation therefore accord dm dm tm differently begin training training fitting auxiliary attribute assign coefficient adaptively accord determined sec pointing wise stop beneficial task determine empirical task early dynamically dynamic solution value update summary face estimate filter coefficient posteriori map
ne compatibility chapter part head head head head head compatibility theorem equation em compatibility compatibility compatibility compatibility true false em mu mu mu setting true mu mu mu mu align align end end end align environment use environment style style construct allow allow false tag tag science centre university college science interactive centre college inequality derive rademacher complexity unit dictionary play eigenvalue concentration rademacher complexity standard input pair I complexity lead uniform example member function machine function kernel kernel hilbert kernel feature map norm h interested sequel similar decomposition bound exist follow rademacher replace replace read apply trick lie linguistic specificity sense individual different type linear prediction associate dimensionality dominant bind vector equation w denote large overall example coincide eigenvalue kernel divide bound significant number large tr high radial small type behaviour typical method benefit datum see structured regularizer propose give dictionary reproduce novel include application weak important go concentration rademacher random complexity complexity incur little gain gaussian sometimes convenient simplify certainly exact reference whenever inequality book aware systematic application intend exhaustive appear somewhat derive structured sharing section new give application learn concentration proof elementary covariance proof remark independent variable conclusion replace theorem exercise calculus subtracting point v tr benefit centering raw pixel kernel c use empirical convert bound complexity derive structured norm dictionary hilbert space inner span norm include overlap projection rademacher ix ix ip mx mx mx parameter mx p mx improve tr feature subsequent give value q let rademacher purpose obtain uniform tx value unconstraine tuple real independently become highlight role control dictionary intermediate admissible class expression ti w sequel give another computational order refer eq q delta word extreme point applicable give w ti ti ti special ti tn ti already reverse sharing liu guarantee multivariate extreme point unit k ti ti ti third rademacher token supremum supremum weak ti ti overall depend second weak large sparsity disadvantage sharing penalty outli task norm consider let dictionary orthonormal effective compare bound trick construct finite covering dictionary eq k ti k ti jensen put everything take infimum get compare
variation exponential tail therein et distribution approximately integral carlo sample quantile follow ordinary differential ordinary series convenient recommend take interested find quantile find convolution numerically many light random al cell claim one tail partial partial independence tail heavy marginal iff expectation thus distribution derivation et al comprise joint maximally decrease conservative tailed triangle contain heavy bind result convolution say amongst tail case instance one et ij write tail approximation perform core study involve quantile regression therefore parametric choice quantile regression good quantile choice interesting exercise previously feature company cover convenience two claim amount trend claim development generally development trend skewness mixture implementation development effect skewed symmetric adopt skewed primary conjecture appropriate understanding quantile higher quantile critical derive margin model wide tail development variance al est functions second model set mean detail prefer incorporate year accord development describe variance subsequent whenever c est est est est fix fix fix quantile plot demonstrate quantile fit model indicate specify different trend development depict posterior quantile level respectively nonlinear trend development covariate increase subsequently quantile furthermore trend loss quantile level al benchmark model variance year five triangular heat map heat upper triangle five row study map trend development level indicate light around year year increase loss peak scale rather transformation gb cells upper tail far adopt calculate quantile light tailed claim analyse technical claim expect available cover capital measure calibrate percent projection gradually percentile dramatically percentile c gamma flexibility model popular involve opinion aim approach statistical particular percentile quantify uncertainty uncertainty utilize adjust traditionally admit central margin estimator deviation calibrate approximately percentile total loss moment method suffer drawback influential normality estimator skew utilize model achieve base via var represent total standard iii median adjustment tail traditionally utilize estimate consider risk margin adjustment margin make adjustment quantile statistically denote year capital state uncorrelated presents quantile al percentile applicable margin demonstrate demonstrate amount third party cover million cover period use represent cumulative payment portfolio variance lot drop original skewness data scale drop scale necessity adopt dynamic skewness modelling choice al allow flexibility skewness modelling nonparametric proxy indicate loss adopt propose modelling year reason increase uncertainty appendix h skewness variance skewness fit complexity margin perspective margin estimation c skewness variance year change skewness skewness respectively start year gradually margin ahead risk margin base sound mathematically consistent reasonable margin analysis model quantile reveal quantile finance heavy tail compare parametric five al pp gb provide investigate three regression function generalize estimate margin overall indicate margin offer considerable drawback particularly rare quantile precisely solution aware limitation cm corollary school mathematic statistics university email department statistical science college uk pt develop derive margin capital application utilize entire function quantile regression provide estimation margin capital historical volatility framework nonparametric quantile model quantile model include distribution power pareto case carlo strategy adopt proxy scale mixture facilitate dynamic applied analyze datum discuss interesting regression chain monte margin work assessment claim assess margin inclusion margin issue margin relate inherent practitioner non margin requirement general establish develop risk institute th aim highlight calculation discuss aspect capital article regard challenge capital require plus risk capital furthermore calculation specification recommendation margin several little significant wide technique modelling highlight approach practice assessment percentile traditionally adopt claim central estimate define range outcome however inherent arise statistically robust claim claim differ practice adopt typically eventually cover claim requirement risk model capture uncertainty margin provide claim therefore increase likelihood sufficient require gps regard worth note display heavy margin large stable portfolio margin capital capital method determine margin measure return capital evident capital require initial capital claim estimate return capital alternatively percentile quantile currently takes meet subtract predefine percentile bring percentile ability rigorous manner claim micro environment explain attribute principled manner shall allow argue percentile involve somewhat margin square enable offer mechanism study explanatory distribution center explanatory margin variation propose framework function regression provide apply range economic finance finance explain important quantitative extensively analyze assess monitoring risk regression construct autoregressive risk regression risk var quantile portfolio accurate financial capital level cover portfolio taylor percentile risk margin assumption sophisticated capture shape tail claim stable pearson gb transformation datum result log sensitive et family compound stable extreme structure could claim incorporate covariate multiplicative quantile rather quantile recently alternative skew heavy tail adopt function long outperform conventional gamma generalise gb tailed gamma pareto family provide convenient claim perspective pareto quantile pareto combination difference capture claim incorporation function fundamentally illustrate develop risk forecast perspective distributional assumption markov skew alternative appropriate frequentist asymmetric laplace al asymmetric bayesian quantile et fitting single bayesian information type zhang propose bayesian growth loss form monte carlo fold quantile propose relate quantile risk instead margin rich characterization tail impact distribution merely secondly quantile regression bayesian especially generalize software user specialize knowledge markov monte methodology shape estimate capture claim year heterogeneous apply parametric quantile organize follow section explain framework way use two loss set conclude relevance model novel analytical perform focus asymmetric al distributional family al demonstrate claim triangle assume development claim year year claim incur simplify year number development year triangular predict claim triangle xx l claim triangle quantile structure within predictive quantile estimation cell quantile section component distributional model quantile loss quantile quantile solve ij yu minimization vector parameter vector al give scale clear maximize minimize formulation observe response al coefficient vector study simply link equivalently link start towards alternatively parametric conditional coefficient distributional quantile quantile write cdf standardize incorporate regression scale suggest ij transformation ij ij fy ij parametric regression regard relationship impose distributional naturally framework directly link quantile quantile detail yu zhang realization laplace volatility asset allow al yu regression purpose development context propose residual inverse cdf quantile eq shape affect note skewness shape magnitude distribution skew skew hence skewness moreover risk margin adopt mostly percent rather fairly figure show skewness skewness second parametric pareto function combine pareto main tail pp comprise function u may valid produce power pareto pareto power specification derive u give solve treat location really implicit regression complete pareto parameter pareto plot demonstrate flexible skewness tail feature al modelling data support upon carefully fit interpretability transformation year moreover claim tail particularly tail business class family gb beta modelling express gb include gb link link covariate eq q generalization distribution via ij give beta accord widely utilize gb family relevance context estimation correspond sub family understand flexibility gb gb sub see introduce three shape link hence quantile within family restrict consider adopt model suitable explanatory quantity subsection explain distributional conditional quantile simplify regression possible classify explanatory e trend explanatory distributional consider multiplicative interaction regression aspect well consider scale consider apply pp effectively al addition allow covariate make manuscript differ regression comparison concern distributional relate covariate sub limit specialized explore sub space distributional adopt set year development year claim basis may observe label mean parsimonious specification structure assume trend year behavior popular know level slope trend decomposition level specification give correspond denote year respectively constraint first distribution gb support log al correspond quantile quantile
make datum basis modelling diffusion tree stick prior basically distribution relationship accurately intractable posterior assessment particularly generally tree substantial modify tree prior tree tree child change consideration simplicity generative flexibility generality nonparametric algorithmic towards develop coherent principled inferential form article systematically assess advantage theory abstract invariant limit grow tree model critical condition aim article fold generative model structure mechanism develop goodness simple model tree use goodness fit test class canonical variance consequence technique algorithm pay mention primarily convergence condition process regardless distributional propose goodness base cancer systematic classification intra heterogeneity crucial development problem detect brain heterogeneity group heterogeneity intensity voxel intensity approach summary skewness intensity account structural complexity intensity wherein image hierarchical branch dendrogram tree relate pixel cluster distance branch length carefully dendrogram asymptotically determine conduct look binary density condition model key condition restrict attention relationship purpose estimate condition model goodness approach conduct number test test check group experience short survival comment generalization proof simulation material root order finite amongst parent notion child trees tree non unit branch length vertex describe strict internal vertex child fit article motivate binary benefit construction leaf form homogeneous function add edge inter time choose distribution edge point choose accord length leaf order chosen observe branch branch length form branch branch denote branch term interpret assign mass length binary topology leave root topology branch length order root q arise topology tree exchangeability length leave attractive tree wherein leave detail product act factor make tree density capture total sum branch length th arrival density obtain lead family wherein branch length model sample tree efficient test generate leave characterize arrival characterize edge arrival uniformly length action obtain density homogeneous th arrival consequence speak homogeneous poisson rate function non always convert homogeneous homogeneous intensity process one obtain solely path appropriately modify edge obtain follow determine total suppose generate homogeneous poisson gamma proposition test homogeneous suppose branch percentile branch length generality th percentile testing leave statistic topology test attractive assess detail application wherein branch albeit flexible topological goodness simplicity develop test considerable extend binary condition process lead completely notation finite root vertex eq finite trees difference terminal vertex leave manner integer tree start root give child refer copy vertex two arise probability tree tree ensure address consider tree condition know combinatorial condition condition importantly condition view pick wish accord equivalently tree flexibility modelling perspective useful offer structure order condition success tree trees bin strict binary order unary tree unary tree tree vertex contain tree vertex contain success useful scheme branch condition construct critical condition converge limit modelling purpose strict child proposition demonstrate extend sub super condition model bin bin super behaviour resemble critical finite may source asymptotic branch within branching technique tree reader arise tree grow size banach make tree natural branch lie follow concern critical projection specification brownian need construct tree role structured topological branch carry computation consequence code describe root order uniquely code traversal traversal walk tree manner ease branch positive imagine motion root explore move continuously edge explore come clear twice evolution time root offer intuitive length branch length take walk path length relate vertex length unique vertex formalize proof map order ht span finite branch path branch tree vertex denote root preserve illustrate preserve distance root new original ht v leave vertex pick vertex context reconstruction work purpose choose leave terminal manner remarkably two way strict arise limit carefully tree condition brownian arise limit scale tree first vertex q brownian brownian weak height vertex brownian functional functional randomly subsequent tree condition random leave limit leave branch factor order density tree variance employ homogeneous coincide equation view marginal dimensional use density lemma density dependence manner employ incorporate notation terminal leave subset parameter condition goodness tree condition equip parameterized variance homogeneous process vary length identical topology therefore limit condition child topology consistently propose condition two consistent omit normalised subset leave path length estimator goodness fit test path use clear role tree possible extension condition need retain leave tree choose leave clearly define leave interpretability density exponential go detail upon branch gamma minimal statistic leaf test tree stand purely homogeneous setting model alone position develop see generate variety copy condition applicable article goodness entail generative wherein branch estimator fit test start leave order tree randomly branch asymptotic rejection choose subset leave accord leave denote fix path length hypothese leave loss numerator exceed testing every order follow condition develop inferential statistical perhaps transformation indeed establish probabilistic condition probability pose equivalence class weak consequence invariance experiment counting size law motivation weak prove equivalence regardless variance factor handle condition tree connect pick kb label end move implication uniform tree statistic construction construct look total variable shape principle induce b approach verify translate pick vertex test lead order path choose vertex walk ex seek give meaningful construct depth walk root traversal scale test path degree freedom proposition consistent variance loss numerator exceed denominator testing tree random normalise root normalise distance condition tree efficient simulate large tree expect code value random construction straightforward walk condition necessary vertex description final supplementary examine performance binary tree test topological test statistic identical construct simulation objective critical various check estimate examine tree correspond tree total converge expect path correspond condition binary hierarchical subtle rejection framework vertex trial reasonable second provide carry tree contain vertex interested size theorem critical length sum branch constant estimate compare histogram tree vertex ht cc l first histogram histogram condition tree tree randomly leave solid red shape report two size examine nan sample list exercise trial choose vertex choose leave compute total length permutation base test goodness condition tree bin see grow test valid distribution trial success represent birth death condition tree fact correspond tree ultrametric cluster application concern mr elaborate upon cc cc cone c bin bin permutation condition vertex refer tree death rate f permutation bin alternative normalise randomly value axis statistic vertex verification choose random experimental encourage tree appear large portion contribute estimator tree death tree poorly tree tree cc cone bin condition correspond death rejection goodness permutation tree distribution bin column utility heterogeneity cancer dataset objective relationship hierarchical patient confirm molecular genome database sequence image inversion recovery cancer publicly pre process mr volume follow correction visualization automatically medical intensity appropriate heterogeneity variety genetic molecular occur development classify base intensity increasingly amongst practitioner motivation pixel intensity spatial lie heterogeneity image classify heterogeneity histogram pixel intensity image overview image pixel offer recently histogram factor skewness heterogeneity cell cancer task experience survival tree binary dendrogram individual intensity intensity branch length distance representation heterogeneity image cluster intensity hierarchical patient classify survival time agglomerative cluster intensity group heterogeneity binary examine condition tree finite rejection detection heterogeneity rejection precede power ultrametric trees leave result pairwise tree arise clustering consider survival panel heterogeneity branch tree three tree indicator heterogeneity height branch branch whereas topological short patient survival intuitively patient short survival appear intensity rich varied small branch length scale dendrogram appear observe dendrogram pixel intensity choose cc note linkage small tree perturbation base hausdorff distance ultrametric therefore function agglomerative sensitive linkage distance tree appear linkage test unchanged linkage order prescribe size patient time consistent efficacy test choose nan reject reject reject
exhibit specific variable analytic dependency exploit combine mixed motivation mix analytic consider capability ordinal nominal response datum mixed background economic nominal datum model concern fitting take health cover people study though nearby date water suit contain supplement home collected explore landscape describe area analyze response categorical item ordinal nominal item asset indicator asset car ordinal use ordinal modern power among item datum asset survey derive principal group examine relationship construct asset index principal score observation base previous survey construct asset index upon index routine principal recognize whole collection propose aim factor md md hybrid ordinal data latent nominal combined md ordinal link detailed item ordinal lie observe response underlie conditional latent trait underlie item usually term discrimination parameter link two py view say nominal complicated response set nominal inherent detailed dimensional ordinal response analytic nominal response variable correspond nominal item denote nominal relative cutoff latent analytic mean trait parameter ij analogous discrimination section note binary could ordinal ordinal analytic similarity mix binary nominal denote binary ordinal item loss generality column item use nominal item analytic continuous ordinal nominal item collect set ordinal nominal lie analytic term loading factor ordinal nominal latent parsimonious latent observe mixed loading latent trait vector marginally parsimonious facilitate mixture modeling impose mixture md md model gaussian belong denote loading latent indicator cluster belong augment gd gd ordinal nominal three way define rise particular response detail bayesian parsimonious mixture md provide approach survey unify md novel capability nature nominal datum capability modeling item unified monte mcmc utilize interest membership mix proportion latent trait item md bayesian unknown specify threshold conjugate specify term trait multivariate latent indicator variable follow latent likelihood mcmc marginal mcmc employ latent variable sample exception parameter detailed derivation allocation trait ng gd gd gd gd ig nature corresponding condition detail ordinal truncated row follow mcmc chain prior also gibbs adjacent slow thus overall gibb propose candidate j rr select metropolis iterate md identifiability aspect threshold ordinal constant threshold ordinal item factor analytic many propose literature solution loading triangular diagonal order appropriate md model instead impose loading matrix sample post process reflect closely reference loading trait transform posterior fit cluster loading reference mcmc hoc describe understand landscape md asset vary trait consideration md models md approach within set md well small exist region md trace plot chain achieve satisfactory mixing jeffreys gd absence strong relatively uninformative prior note flat improper mixture assess indicate sensitivity appear thorough switching address criterion md place interesting assess exploratory three uncertainty bayesian paradigm distribution discrepancy employ truncate sum square pearson assess fit context cluster deviation evaluate frequently observe response md intractable evaluate response multidimensional truncation dimension predictive pseudo count set across md consider illustrate quantile median across md plot trait improvement insight well apparent assertion exploratory assess uncertainty plot cluster general confidence low notable increase higher randomly sample latent trait dash uncertainty model residual residual normal latent reasonably focus residual item residual residual correspond membership residual plot model suggest explore fitting component md trait distinct belong divide time allocate cluster pt yes yes yes yes yes yes modal response modal across group response analyze modern group modal likely possess modern contrast poor modal location type less modern status close player cluster small low produce trait phone item ordinal pt detailed picture box binary ordinal nominal binary response code response correspond interpret box plot nominal say item cluster plot ordinal cluster reflect cluster economic status type cluster notably response group plot box plot relate item sample dimension nominal focus mean follow significant high divide item estimate suggest region survey show observe conditional member pt total hellinger total hellinger distance group hellinger hellinger plot modern hellinger hellinger hellinger many pattern evident plot hellinger distance cluster notable hellinger item account hellinger item large hellinger contrast difference highlight cluster notably list table modal differ component group similar component possess modern differ modern possess investigation reveal group group hellinger distance group group see group concern modern convenience interestingly consist essence remain modal differ solution item modal differ group solution component separate asset gray line score colored trait explore landscape base asset survey interest landscape md economic consider code binary principal matrix construct asset raw show standardized principal standardized asset index color two broadly roughly low middle top principal choice rand solution md poor asset score md model preferable base approach type landscape region asset status survey md achieve economic status examine information great benefit various aid decision make development policy result membership interpretation aid future survey survey md model important economic health account study examine cluster landscape exposure landscape statistically principle hoc md provide mixed mix md capability individually usual varied challenge facilitate md formal trait would beneficial paradigm pose difficulty md bic alternative underlie bring difficulty uncertainty incorporate reversible nature cluster parameter md md extended time survey however survey extend appropriately longitudinal move md could insight indicator similar metropolis hasting rank likelihood areas md facilitate consist little md latent latent trait extension achieve unit variance probit covariate could naturally md effect l main yes material modern material yes area room yes house house report none modern report water source report status yes tv report status report player
atom describe layer wavelet per dyadic wavelet level review wavelet first level second atom frame case obtain cross frame relu architecture normalize mask concatenation frame match context dnn refer frame optimize network optimize mse architecture case gpu sir nmf multi layer outperform gain sir high separation discriminative significant improvement discriminative context regressor play well architecture improve resolution representation representation able long context remove uninformative variability relatively finding discriminative improvement separation contrary modeling remain unclear phase architecture believe discriminative promising comparison subject interesting explore good recent study source separation dnn separation improvement speech currently address architecture exploit assess multi role study facebook ai edu decomposition operate feature effectively duration frame modeling frequency multiple temporal pyramid transform convolution modulus first learn widely audio investigate resolution alternative neural fundamental speech processing rely capture different elementary atom become audio negative nmf adopt various audio source recent many separation nmf fail characterize speech increase window dimensionality limitation extension learn code temporal include occurrence kalman integrate rnn recently efficiency train adapt task become inverse completely problem deep range simple frame regressor sophisticate rnn separation music multi benefit separation discriminative role separation take resolution representation pyramid information temporal defines increasingly think generalization apply excellent classification source preserve much possible discriminative pyramid nmf dictionary level selective localize separation regime multi baseline confirm superiority discriminative regime separation work family separation dedicate describe alternative fall category set describe popular nmf discuss aim source representative report concentrate music operate frequency version comprise temporal thought operator typically define magnitude short fourier transform alternative temporal resolution duration non representation success signal shift expense inverting space normally specifically estimate source phase recovery efficiently soft signal resemble wiener filtering demonstrate obtain filter multiplication division define mask impose restriction receive lot attention literature negative activation represent speech ideally measure dissimilarity input channel discriminative compute supervision mild autoencoder generate treat frame ignore work integrate several scope replace capacity minimize fitness truth common space phase recovery study predict forward use dnn fix time context fail temporal dependency speech increase intractable dnn exploit rnn long memory section briefly wavelet conceptually temporal layer band localization wavelet quadrature center filter result bank per define low pass bandwidth wavelet smoothing kernel critical rate preserve locality possible complex modulus nonlinearity critical sampling bank localize trade sensitive local uninformative robustness signal wavelet bank modulus dyadic reduce bank filter discrete every produce non quickly experiment operator desire reach context increasingly produce map low resolution tree pyramid feature source separation pyramid layer transform produce mostly interested layer variability inverse leverage level source sampling carry code test estimate gradient descent approximate phase phase simplify strong
expand percentile student standard help student compare probably ok otherwise compute interval skew advantage narrow interval percentile interval suffer symmetric skewed add coverage coverage skewness matter size z adjust effect coverage correct advantage error formula may large problem poor recommend bootstrap percentile bootstrap short percentile like people bootstrap percentile sample large narrow accurate percentile population percentile perform bias factor allow variation interval avoid allow multiply table population bit red population never exactly multipli population yet continue help long distribution take sensible percentile adjust provide normal population simple adjustment multiply adjust bootstrap q adjust give n table coverage adjust dramatically adjustment skewness uncertain help modify motivated quantile handle skewness adjustment se percentile expand percentile percentile poor common adjust quantile well coverage exponential population mathematically mathematical bootstrap limit assume bootstrap quantile bootstrap q g mirror percentile reach far percentile sample mean percentile interval reverse percentile skewness figure thing discuss mathematical thing strongly skewed depend strongly wrong thing transformation bad everything sample percentile wrong skewed wrong direction figure reverse percentile perform percentile wrong reverse percentile wrong wrong namely say reality say positively positively correlate large positively skewed opposite direction denominator affect skewness show negative skewness figure show skewness base bootstrap generate table reverse bootstrap percentile quantile bootstrap distribution eq quantile vice versa inaccurate table percentile mean percentile percentile interval skewness skewness closely figure overall coverage statistic skew interval test statistic estimate bootstrapping order formula available iterate sample bootstrap bootstrap original total suited location sample mean correlation coefficient transform depend reverse percentile coverage population bootstrap percentile percentile expand percentile coverage side normal population section coverage percentile reverse percentile poorly interval correspond se skewness variability skewness skewness thing expand percentile much still due variability estimate surprising optimize bootstrap extremely sample correction bad ordinary se coverage confidence side population interval non code leave hard side bootstrap interval next skewness correction accurate poor good reverse percentile interval poor population type previous interval axis second linearly side coverage e expand confidence interval skew hard bootstrap interval skewness adjust second accurate require include summarize interval inaccurate interval size bias yes yes yes transformation yes yes yes yes vs partial skewness issue replacement plus another randomly random add extra exactly opposite attention copy another round correction bootstrap finite four thing permutation test discuss bootstrap name permutation test pick replacement label equivalent name permutation test infeasible independence fisher use effect include include report value zero impossible compute side one create great less center measure compute fraction quick use discuss deviation one value may equivalent result mean variance monotone h permutation latter together south gold united united er li china zhang china south south free score two program free score permutation partially dataset slope figure distribution correlation short correlation side add numerator denominator calculate one multiply sided testing conditional one permutation testing test test independent amenable without assumption procedure beyond scope article variance differ pool population positively mean naturally variance differ nan company investigate expensive procedure alternative large live match would appropriate regression shift could perform test wrong slope multiple hypothesis whether coefficient zero side quite narrow tend narrow permutation permutation test variable situation permutation confidence might bootstrap hypothesis relative accurate permutation permutation compare two sample sample way straightforward reject nan interval fail hypothesis calculate value tail permutation nan could recommend skewed impossible categorical keep weight convenient normalizing choose broad bootstrap testing relatively bootstrap bootstrap goal bootstrappe test statistic somewhat deep method show inaccurate provide broad resample article offer benefit concrete abstract concept student tool visualize nan error arise visible permutation student difference work student many finally median odd student formula check bootstrappe role play bootstrappe prediction interval test skewed datum central operate interval accurate sided probability intuition skewed tail outlier observation would allow possibility tail percentile section right tail percentile narrow like computed skewness expand percentile interval percentile wrong thing transformation people large backward percentile poor well interval skewness adjustment formula bootstrap need accurate side criterion bootstrap skewness expand percentile percentile percentile bootstrapping normally sample original datum sample general exception fix sample bootstrap observation conditioning dimension bootstrappe odd independence permutation simulation side fall recommend routine interval handle little coverage se skewness sample correction skewness skewness regularization something smoothly large procedure place zero nonzero skewness software package eventually standard current david mm figure inference abstract article bootstrappe permutation student bias distribution issue practice compare inaccurate latter confirm asymptotic resample alternative percentile cover sample alternative bootstrap permutation randomization test I use bootstrap test student understand concept I mind elsewhere broad application resample motivate include text resample one student behind bootstrap principle guide visual bootstrap sampling thing inference skewness transformation something cause odd bootstrapping statistic also inaccurate procedure skew broad change statistical skewness different discussion simulation asymptotic regression avoid permutation look beyond test finish short test section include r package reading box return may read resample read first permutation broad read skip notation come population correspond often sample sample I say sampling unless deviation r permutation approximation quantile bootstrap percentile expand percentile bootstrap procedure behind bootstrappe student tv channel one come tv pay extended minute half hour tv minute hour channel vs minute perhaps stand random half hour tv actually half hour occur really basic like occur time figure also right labeling give would rare chance difference observe right pool replacement another compare plot histogram exceed numerator multiply sided detail denominator procedure nice visual student look make nan true statistic student learn significance rarely chance student work directly switch like generalize means formula familiar difference formula answer resample bootstrap sample bootstrap draw replacement create bootstrap calculate times bootstrap relate sequel mean figure distribution extend center spread approximately indicate unbiased spread vary bootstrap approximately quick percentile interval basic channel channel percentile interval section distribution tv distribution top extend channel observe bootstrap sampling bootstrap distribution statistic se bias extend elsewhere bootstrap permutation test replacement statistic bootstrap distribution bootstrap percentile bootstrap bootstrap minus draw sample compare size sample bootstrap bootstrap interval include difference variation permutation basic nan center assumption bootstrap center statistic confidence make abstract concrete concept central interval student involve statistic variability deviation student use student confidence interval work statistic variety make statistic also understand student check student know really ignore later hidden formula bootstrap bootstrap bootstrap abstract standard role random familiar tool like plot bootstrap percentile statistic check answer formula bootstrappe permutation student group like student partition nice visualization repeatedly build permutation resample practice country thousand machine count query country query country count add country per query per word ordinary bootstrap generate poisson save instead user come seed generate get count across user try improvement probably user variability across resample lift describe early version design people question familiar see treatment nest people ad subject visit website actual response prediction gender covariate across people prediction person error theoretically difficult derive need instead people survey fold validation resample technique routine imputation backward elimination mixed stability calculate million question among probably behind bootstrap later inferential statistic estimate something sampling distribution principle obtain draw sample sample distribution distribution sample population collect infinite draw sample expensive know sample bootstrap estimate population statistic show world sample population statistic collect center plug something substitute estimate substitute take estimate plug whole raise substitute possibility consist distribution replacement may sample population may parameter substitution tell something discuss important immediately center population bootstrap bootstrap aggregating bootstrap improve matter center add instead people bootstrap think create bootstrap help something nothing bootstrap sample real use accurate regard formula twice standard use quantile bootstrap quantile sampling use estimate bootstrap estimate replacement detail avoid case unique bootstrap infeasible carlo implementation draw simple rule mention fix produce modify question draw sample estimate would actually precision similarly bootstrap testing another nan match modify behind bootstrap behind bootstrap population well claim tell elaborate series thing work well exhaustive bootstrap randomly population bootstrap middle column three five bootstrap center rather shape bit lot bootstrap well parameter instead useful additional fundamentally spread carlo quick estimation interval adequate however tail bootstrap distribution middle small center shape vary shape sample substantially vary monte carlo may apparent narrow go plug fx x x error basic depend severe bias bootstrap course ever department survey result recommend student spread affect confidence sample smoothed sd median bootstrap approximation sample continuous odd bootstrap always original near bootstrap statistic heavily bootstrap distribution turn heavily different bootstrapping work ok shape bootstrap percentile median bad odd percentile fall one value fall percentile right smoothed draw thing though great sampling distribution leave population spread mean middle top two figure like sampling right spread interest binomial distribution depend show bootstrap implication confidence reach long normally middle right distribution sampling parameter sampling chi square depend centrality estimating number mode bootstrappe application bootstrap distribution sampling typically bootstrappe overcome indeed sample well assumption parametric population trust spread datum bootstrap population work poorly depend reduce variability distribution fundamentally look ahead thing accurate allow fact usual interval allow discuss resample little good suggest carlo usual bootstrap distribution sample draw replacement enough confidence criterion computer much slow fast computer easy second random due prefer implementation formula bootstrappe permutation fraction exceed bit add bootstrappe monte bootstrap replicate like bootstrap example se bootstrap percentile interval monte se percentile carlo error package package use quantile side necessary sided chance estimate permutation exhaustive similar percentile confidence quantile fall bootstrap se variation depend z deviation normal approximately calculation suffice round variability percentile bootstrap bootstrap accuracy get routine practice increase carlo want decision implementation coverage coverage smaller sometimes roughly large like coin tail overall coverage way sided value chance estimate within monte percentile interval se routine recommendation decision depend confidence hypothesis skewness population look affect affect functional odd bootstrapping subject low pressure high pressure group likely cc pressure disease risk disease show relative skewed right skewed se risk bottom panel log desirable property invariance monotone transformation invariant confidence transformation equivalent percentile transformation another answer percentile interval risk contrast interval risk take transformation invariant answer zero derive plug principle sampling minus statistic biased summary another r bootstrap square produce bias b generally aware bias double make another kind bootstrap distribution sensitive transformation median bias three cause helpful third bootstrap important nonlinear transformation median bias near strongly denominator e cause optimization bias category high use population quantity give biased bias selection optimize lack bootstrap even quantify fit line obvious curvature section bootstrapping panel resample bootstrap lack bias poor bootstrap estimate functional subtle bootstrap assume solely statistic answer observation twice distribution odd bootstrapping sample functional probability look treat question odd bootstrappe tv observe se bias average conclude biased think say non functional adjusted smoothing procedure regularize se affect calculate solely bootstrap bias observe statistic affect confidence issue skewness skewness skewness york responsible phone service term competitive something wrong responsible suppose quickly customer various different period significance customer slow customer substantially pay test instead service positively skewed odd plot hour period work sd surely discrimination clear aside outli time tend comparable quantile permutation value cutoff test pool test four nan hypothesis solely denominator permutation pool permutation test permutation test absolutely sir fisher originally permutation test computer limitation permutation test computationally computer skewness size permutation biased permutation population close tie value quite wide primarily small center mean se reflect contribution se sample bootstrap skewness answer share question ask skewness solution answer wrong base raw already thing point deviation normality bad sampling alone question statistical effective really work even skewness observation measurable interval percentile limit scale perform permutation pooling draw replacement pool three three pool vary accurately reflect pool variance often side isolate skewness converge correct slowly percentile simulation moderately skew confidence interval test skewness procedure skewness least g bootstrap procedure discuss combination historical momentum simulation note unfortunately much replication prohibitive budget would momentum person software start usual skewed look bootstrap rather twice skewed opposite quantile plot calculate coverage material
behavior dataset good small pls construct regularization component loading obtain assign loading integration center cm cm center seven age irrelevant response point pls en forest concrete next sect principal pls pls select similar sect tuning method sect deviation
lead cite vi establish contribution look community intuitively two study straight optimality analysis exist development provide basis lyapunov present straight forward lead presence vi control also concern versus continuity continuity function address state domain work long entire controller remain valid discuss rarely policy hence apply policy evolve establish respective contribution provide vi vi paper presence investigate contribution boundedness guess control analysis apply evolve analysis straight forward compare available interested reader development author regular rest formulate iii iv vi conclude vi respectively integer dimension control space index continuous semi initial calculation control minimize control asymptotically definite put respective one k hx trajectory control utilize rl boundedness definite condition trivially select assume value assume continuity function require boundedness function lead respective admissible lead unbounded connect contain origin intersection vector set associate contain trajectory utility without origin value satisfie solution mathematically nonlinear utilize idea use either look neural select specific valid entire trajectory remain approximated value pi vi pi initial admissible control converge approximate calculate approximate guess iterate equivalently pi admissible satisfie associate one may compare remain respective control confirm lead require system stability system evolve critical vi scheme arbitrarily prove vi convergence solution guarantee require admissible come disadvantage convergence vi see pi admissible vi also policy guess sake vi give use iteration admissible notational compatibility calculate utilize guess stability theoretical pointwise induction result hand assume compare result consider induction proceed continuous lead issue initial guess vi give admissible control iteration monotonically value equivalent limit lead prove monotonicity h hx hand therefore sequence help continuity result limit may idea proof continuity hold generality applicable admissible contradiction end relation inequality arbitrarily close eq q loop initial continuously note existence one skip result enough index tolerance iteration convergence admissible compact selecting iteration uniformly theorem ref theorem monotonicity lead concern continuous within equal eq limit exists form continuity contradiction one continuity exist open point away margin contradict arbitrarily within continuity versus approach note set hold finally tool conclusion value function induction interest capability sample stage lyapunov compact r ix proof lyapunov continuous control definite eq trajectory asymptotic difference value function invariance close system I origin prove converge origin word time asymptotic stability every stability form find lyapunov another approach system subject stable every trajectory similarly replace left side small per repeat q repeating equation lead hand side bound converge therefore consider trajectory path establish besides address vi guess reason cite either great exist number negative state less present require hold uniformly restrictive detailed analogy establish vi optimal control cost horizon definite horizon minimize subject control horizon time go summation along bellman equation horizon guess vi vi identical converge case proceed consider resemble method control lyapunov utilize terminal respective horizon loop remain iteration converge optimal horizon analogy go cost control rest admissible infinite cf nature also invariant utility lead go origin due q horizon consider evaluate horizon great sequence compare result theorems literature close lyapunov correspond admissible guess vi directly asymptotically control hence restrictive study requirement existence take account control go similar guess simplicity proof oppose difference besides address concern idea establish evolve next x let control control subject value estimation region closed loop inequality next step trajectory hence origin vi hand eq simple parametric rise error read iteration right convergence vi boundedness investigate boundedness worth train control actor approximate give vi control approximation regardless whether value remove calculate minimization side control actor learn training system independent actor control actor hence actor error beyond scope investigate separately boundedness consider comment denote minimizer assume boundedness analysis assume vanishe origin word semi positive hereafter sense vanish I x x respectively value bound exact sequence generate recursive mathematical ix ix ix induction compare note resemble idea programming idea boundedness respective admissible control iteration boundedness actual challenging analysis vi presence monotonicity lemma long boundedness guarantee neighborhood iteration importance system lemma result define admissible word iteration k satisfie initially trajectory result c confirm assume lead complete minimizer lead along show result evaluate side trajectory hand second fx ix x system system similar lyapunov function low upper theorem guarantee parametric show ix ix lemma replace consider approximate vi eqs f conclude vi summation function evaluate state consider one guess ix q leave boundedness lead c ix policy q leave side analysis sign quadratic left satisfie make suitable result non last c desire stability theorem zero assume positive end system trajectory within within continuity lead function error control approximate asymptotically origin theorem vi eq along equivalently evaluate
assume equation fact separately incorporate affine term row later optimality subgradient expand nonnegative must index solution loss dissimilarity incorporate affine rewrite q row substitute constraint dissimilarity parameter beyond row order several follow index assume order index follow denote dissimilarity element dissimilaritie similarly use contradiction select feasible achieve function write involve objective triangle inequality eq write q arbitrary centroid element together obtain contradiction approximate nonlinear ds gaussians notice quite representative nonzero row representative proposition example pt finding center learn health signal pairwise dissimilarity target source set row regularize formulation relaxation regularization parameter show group group deal outlier dissimilarity asymmetric implement alternate direction implementation algorithm real categorization representative series modeling segmentation dissimilarity simultaneous recovery video scene recognition characteristic entire vision refer visualize web video increase interpretability expert describe small requirement work contain original efficiency oppose training help dataset efficient classifier pool extract sift frame video form histogram frame apply ds matrix preserve find lie subspace qr rank try correspond good submatrix find approximate assuming express linear problem rank find lie low low manifold live similarity dissimilarity pair work several ambient high pairwise relationship efficient work high live datum pairwise compute importantly similarity dissimilarity allow suffer initialization find impose relationship try dissimilarity point employ initialization affinity pairwise ap require initialization empirically categorization single point process fix subset set semidefinite diversity impose similarity single specific hard propose unsupervise subset dataset selection solution fall location extensively operation research literature dissimilarity ij right representative dissimilarity problem dissimilarity dissimilarity element optimization recovery formulate regularize minimization put trade closely relationship advantage ap require dissimilarity dissimilarity particularly advantageous pairwise dissimilaritie difficult dissimilarity instance distance dynamical dissimilarity encode base dissimilarity specifically dissimilarity come asymmetric triangle guarantee group show regularization parameter representative propose unlike solver well computationally propose use alternate multiplier admm result show admm reduce world improve art categorization segmentation representative find along element associate representative effectively b red accurately approximate manifold mn dissimilarity well dissimilarity denote collection dissimilarity predefine euclidean representation access dissimilarity social learn dissimilarity use contrast art restrict consist see dissimilarity code error consist even correspond hence dissimilarity kl select representative dissimilarity geodesic set dissimilarity asymmetric triangle inequality contain scene converse dissimilarity bag short sentence converse necessarily efficiently associate dissimilarity denote b euclidean row indices representative row interpret representative represent dissimilarity objective want dissimilarity encode via possible goal consider count convex relaxation nonzero notice program assignment range probability representative appropriate whose representative emphasis select close representative emphasis select value demonstrate select lie absolute dissimilarity illustrate bottom dataset gaussian dissimilarity euclidean distance material compute world source contain section framework effectively dissimilarity find point source notice outli correspond element addition reduce large help reject outlier efficiently element datum may explain efficiently model program enforce often detect allow encode outlier value program encode via outlier equal zero hence encode hence put penalty selection without penalization outlier optimization w weight figure illustrate detect encode notice augment time membership z assignment jointly partition dissimilarity determine would like cluster bi group efficient direction admm compute distance representative element close plot decrease put emphasis encoding approach arbitrarily small nonnegative word element select close show jointly program partition implication illustrate notion joint partitioning follow let correspond index centroid well represent target source solution find target partition correspond say jointly index choose index notion joint prove optimization program select element dissimilarity element index partitioning dissimilarity kk jj k dissimilarity partition regularization determine theoretical kk distance group regularization reveal certain partition optimization x nonempty single dissimilarity identical algorithm like pairwise relationship target impose pairwise relationship theoretical apply theoretical result cluster nontrivial group thresholding dissimilarity fail around dissimilarity centroid dataset accord become emphasis representative form identity representative subset minimize enforce program multipli establish program store build office room consider single finding program optimization program np hard relaxation let soft second row arrive show three cluster parameter notice increase value obtain observe demonstrate effectiveness enforce deal weight multipli formulation evaluate real world improve challenge regard since multiply divide scalar dissimilarity divide unless otherwise obtain result neighbor nn significantly memory demonstrate improve effectiveness scene categorization category class forest make rest testing ap point rand baseline initializations fair spatial pyramid pyramid bins dissimilarity pairwise distance pyramid histogram dataset class distance dissimilarity respectively test result obtain expect rand perform method come ap pass relationship include select ds close training use ds important dissimilarity word dissimilarity group dissimilarity group improve performance rand ap ds confusion classifier ds middle expect increase result confusion important select use training row confusion matrix show store use rand ap ds subject activity motion capture use marker per subject trial comprise walk run stand jump c frame activity sc ap problem generate among segmentation motion framework efficient identification dynamical series trajectory p kk important measurement given regressor define eq series segmentation correspond hybrid system give estimate dynamical form learn collect dissimilarity represent efficiently segmentation membership select long framework argument activity motion capture several marker measurement human body informative instant loss
vector signature attribute signature attribute probabilistic direct add unseen class variant explore gain often work unseen specify attribute signature none attribute unseen see impact stress attribute former focus mid classifier exhibit attribute pac mid level attribute common alternative exploit external adapt unseen class unseen image imagenet hierarchy label spirit label embedding shot category attribute control unseen attribute signature unlike shot generalize shot propose random forest model simultaneously signature label enable shot attribute attribute mid discriminative label however guarantee discover align shoot unseen semantic semantic post hoc discover pseudo idea handling concept explore propagate membership entail shoot accordingly domain differ train label unseen signature propagate signature shoot visual attribute unseen signature attribute unseen training typically association binary real grain unseen signature modal vocabulary discover expressive discriminate unseen zero shot signature attribute unseen time unseen initial attribute classifier introduce zero random signature signature shoot sec attribute shot presence importantly classifier come object need instance unseen category precisely n descriptor bag specify indicate attribute v must descriptor attribute presence per th scaling forest v v v attribute entail operate characteristic operating example instance next introduce zero shot shot decision attribute descriptor vector shoot available apply unseen signature signature train forest signature unseen signature negative build manner learn recursively split signature record signature signature follow training instance recursively randomize signature child indicator thing node split norm signature training forest discriminate grow forest depth branch reach per forest novel test signature test test apply attribute predict posterior attribute zero shot alone train unseen shot random main generalize recursive procedure signature path determine individual predictor expand signature attribute space build appropriate expect error choose split idea extension receiver formation process signature explain signature maintain validation datum sec recursively denote root let record occurrence signature signature split node receiver roc attribute negative negative rate attribute gray attribute threshold sample signature pass leave single signature reach signature estimate attribute stress two thing role sub capture split select split fractional signature child must mean split leave child properly dependencie threshold address miss uncertain building shot choose split information gain signature unseen signal attribute classifier prefer class reliable point remain omit split highly signature find validation positive signature propagate left ji jk explicitly make attribute inherently attribute classifier unseen zero label play important robustness perfectly attribute unseen course forest attribute classifier attribute framework instance attribute signature tackle indicator annotation signature signature fraction forest please attribute signature label signature select split also traditional training gain signature define reflect fact image like actual output require propagation recursively select combined control signature suffice training appearance shot learn forest dataset unseen class scene image material etc solely annotation split select unseen sec include color histogram sift attribute label roc attribute baseline amount report measure trial cross validation negative reach node validation addition state variant baseline train signature attribute shoot literature approach design overcome classifier prediction examine attribute see sec consistently well large plot learnable break minor attribute benefit classifier avoid uninformative attribute yu yu et al name discover next shot apply attribute commonly shoot recognition exact variant furthermore model attribute presence valuable gain especially per attribute less reliable attribute vs table help impact signature propagation full aspect contribute well unseen error attribute combination initial attempt regard include signature unseen negligible compare publish name name attribute decode achieve art shoot simple strength signature bin show image rely solely solely signature shoot baseline method unseen dotted signature shoot towards prior play blue select automatically introduce zero shot attribute classifier prediction unseen challenge dataset indicate fact attribute remain reliably future extension accommodate inter attribute correlation random forest test multi forest unseen contain shoot avoid sec discuss signature sec shot noise sec list class check account inherently avoid completely attribute negative node signature propagate ji jk plug far plug constrain sec summarize deal uncertainty class attribute signature modify soft indicator vector amount perturb copy signature describe equivalent latter perfect signature respectively term signature uncertainty specifically oppose annotate signature run term rh familiar annotation add signature perturb per expand among probability implement uncertainty common reverse association class may per class signature fraction attribute signature zero shoot recognition annotation signature type material surface envelope type closely relate scene instance reason localize belong marked attribute unlikely person category discuss simply shot result similar fig shot shot class trend similar synthetic
another case tight sufficient primary algorithm succeed learner immediately low meanwhile convert low bind learner especially primarily apply separately primarily immediate direction uniformity small although might question paper uniformity uniformity whether far know paper worst well draw confident one program consider learn benefit metric immediate suggest testing coordinate probability see uniformity thin learn within perhaps distribution acknowledgement discussion thank project finally thank estimation introduction survey field section body consider support entry consideration denote distance distance infinity instance treat slightly true specify distance tolerance access sample wish determine sample distribution goal correct call uniformity place relate support immediate exceed already prove want exceed want jensen vector conjugate treat lemma jensen eq back give side maximize exact conclusion inequality side minimize analysis focus coordinate meanwhile q sum sum covariance pair pair case hold triple triple appear one distinct count way sketch give intuitively slightly choose expectation chebyshev chebyshev make regime dominate know advance simplify somewhat prove uniform chebyshev use definition since next part proof u eq meanwhile leave inequality choose inequality side subtracting divide side reduce divide suppose q check remain term lemma plug inequality prove start drop complete rearrange n n failure rearrange drop point iteration least vote incorrect draw theorem q q around integer hold approximately minimize failure uniformity failure far uniformly give oracle output say access uniform family low family indistinguishable usually say usually wrong oracle versa specify random coordinate remain coordinate zero two family toward property n expression chance inequality property inequality show draw number meanwhile oracle observe entirely uniformly family thus condition access access member correctness correctness oracle family uniformity eq immediately unknown careful construction modify exercise bind apparent plan distribution draw draw sample distribute outcome let output eq let analogously line lower prove lemma slightly uniformly follow flip coin construct valid p p aa uniformly choice fix expectation simplify odd case ia ia ia sum ok inner back expectation convert expectation take apparent q precisely mini draw uniformity need distribution coordinate uniform probably never indistinguishable let note sampling say contain argue probability drawing sample similarly correctness draw arithmetic one case uniformity require family distribution possible put coordinate put distribution put let pick let meanwhile length bind drawing inner expectation coordinate otherwise claim probability exactly plug necessary q briefly regime check coordinate draw u outli number sample uniform case except group least every satisfie binomial chernoff bind g follow fall range direction union range suppose chernoff mn substitution substitute suffice recall chernoff group probability threshold suffice suppose least whose chernoff sample least simply chernoff eq slightly theorem failure run draw proving draw run draw lemma draw say sample well sample sufficient logarithmic empirical concentrate formalize lemma state rearrange imply rearrange give th change change change inequality state plug suffice draw sample suffice corollary suffice follow directly bound prove deduce distinguish coin require sample prove formally author distance regardless tight tight recall construct member size pack bind error pack point simplex point exist add simplex contain ball member space factorial simplex set meanwhile ball dimensional set size eq pick numerator next bind q consist coordinate equal minus binomial distribution drop slight optimizer coordinate may entropy determine particular bind probability drop relate success relate guess conditional prove terminology uniformly lemma follow q prove least choice plug get case always take distinguish coin distance distance simply distance bind learn construct follow assume apply construction coordinate coordinate size large every differ sharing distinguish identify coordinate sample construct take choice half precisely point half claim two distinct subset pair coordinate lie half differ coordinate differ first need one last include include include claim test uniformity distribution examine classic learn size sample test uniformity conjugate suffice seem upper support uniformity easier side coin dependence uniformity factor optimal sample uniformity testing meanwhile algorithm metric cl discussion full question broad whether classic whether estimate would except failure study independent practical imagine web company keyword give day motivating require distance uniformity show uniformity support sufficient regime size uniformity testing work choice theoretically uniformity distribution like understand seek address goal survey big sublinear depend support question nice monotonicity query however ask answer say datum suffice primary result general drawing difference depend desire find uniformity sample solve data conclusion something distance must light primary conclusion sample make sake fundamental knowledge coin come might think require support norm measure distribution piece portion distribution find optimize tradeoff immediate application instance use box utilize instance derive beyond draw conclusion less develop deep understanding space test idea address lead simple general sharp broadly vector application stream tool question study norm may refine develop next result describe conceptual uniformity discuss broad prior future proof omit though proof prof low bound uniformity give distribution specify satisfy output except test uniformity distribution failure upper low match constant uniformity intend reference skip key contain factor employ quickly know bound aspect devote conceptually surprising opinion section detail technique uniformity respectively p cc regime neither tight case prove theorem match n consider phrase actual complicated regime chernoff sample algorithm coordinate outli either correctness outli uniform outli chernoff sample group matter count number sample large note large compare uniform put heavy contain outli chernoff probably sufficient prove ix jx output learn satisfy except naive frequency sufficient proven draw sample coordinate elegant general order interesting novel failure suffice number sample note q p x term I let x I contribute contribution x x tight reduce technique suffice learn concentrated expectation well dependence confidence follow must sample draw enough expectation suffice result failure number suffice suffice discrete distribution number
shape simulated size estimate sample true correspond complete use likelihood likelihood aic provide moderately nan model true approximately reason choose aic great aic high general give respectively low could approximation specify low ks r repetition aic provide fraction repetition reject level provides reject ks successfully aic follow misspecification bias section explore branch mis specify case realization realization parametric mle type figure branching article make introduction allow estimation easily useful estimating background branching highlight branching discuss recommend alternative quantification branching ratio observe specification finish provide relevance attempt quantify branching ratio high fluctuation event significantly day em parametric intensity event mini fluctuation heavy explain allow helpful estimate branching level model rich process type cluster process extend marked influence allow easy additionally em introduce cluster associate multiple cluster allow dispersion evaluation enable treat branch parent point algorithm find study introduce quantifie self extend branching branching realize produce along form attract lot combine seminal spatio mark sequence individual review genomic dna neural train brain become frequency fluctuation price location cluster poisson location mle maximum time I identically time distribution dispersion index variance call instance flexibility extra evaluation impossible paper branching next severe simplification make maximum use cluster distribute termination consider event long novel branching derivation complete address already case em miss present process process algorithm minimal datum goodness test present selection branch misspecification discussion relevance define window instantaneous occur value ts intensity process call give mass branching branching inter event become branch construct point generate subsequent process introduce generation define process sum intensity recover intensity realization fig represent correspond generation deterministic intensity pdf necessary intensity function intensity intensity define cumulative distribution intensity distribute inter decay intensity provide exponential thus go realization process introduce dependence location cluster dispersion process simple general likelihood realization maximize intensity intensity evaluate miss relevant section occur last inter factor inter observe density unconditional homogeneous introduce perform mle simplify form use iterative guess parameter estimate parameter expect value complete miss datum f nothing proceed iterate estimate likelihood algorithm identify em give case unobserve process event branch describe matrix diagonal element sub point parent rest split inter event e density square inter relation lag lag lag child inter event lag lag support define index return index distant intensity never vanish account q start neither point include square branching distribute child subsection step step em account branching evaluating complete give estimate irrelevant determination introduce event either one matrix em branching structure write form denote line event derive exploit branching superposition exploit purpose intensity event derive unconditional incomplete intensity weight probability event time probability parameter weight enter iterate time vector clear hadamard vector complement probability discard remain obtain previous compute iteration maximize obtain rather exploit intensity intensity allow process independently parameter form step estimation process complete square event time z I jt event likelihood determination numerical stability become small intensity require branching branching maximize cdf branching expect slightly small total denominator occur parameterized mle parent assume piecewise inter estimation density branching probability inter datum within take approach expect take replacement inter unweighted simple inclusion allow weight natural potentially example smoothness solve variational memory j large within lag reduce take estimation branching clear however tail approximation adaptively em obtain alternative efficient process may estimate implementation branching complete separate value structure concern use specific need rather regard branch numerically part concern useful parametric example positivity lag sec trick combination reduce likelihood evaluate thus must show likelihood aggregate index exclude average likelihood value take description branching ensemble realization likelihood take carlo statistics sec simulate realization set treat follow probability taken select way repeat reach repeat contain possibly repeat index calculate likelihood value treat poisson plug intensity log transform incomplete computed computationally approximation likelihood careful issue encounter average logarithm compare log standard process process process generate poisson observe one transform kolmogorov ks generally ks statistic semi complete h nan statistic unknown vector semi complete monte approximation value discuss consistency sec mis conditional parametrize pdf decay imply exponential hand event weakly become density shape heavy shift index financial ensure markovian calibration outlier tail alternative density typical application account computational exponential function discuss sensitivity start speed complete em require em get optima thus select reasonable starting point understand algorithm speed towards overlap branching overlap detail comprehensive phenomenon insight couple illustrative standard high branching ratio model choose pure initial estimate ii realization low em initial parameter estimate branch ht fig density converge completely density estimation despite poor analysis start dash solid increasingly dark parametric
jointly train challenge learn classical example find appearance address initialize part alternate part thousand part diverse propose still pool use image remove uninformative produce elaborate part comprise initialization joint section part level performance part translate directly test improve art mit cnn contribution part I part informative counter negative initialization extract patch repeat pool discard part fix part filter idea recently approach stage discover approach different score high cover max goal part diversity natural function model another use mapping part contrast part detection share multiple another work visual visual category similar train word terminology negative part concept image information strength pick attribute class collection part entire scene composition scene region scene position pyramid contain location dot product response location treat maximize define image collection response pool pooling region maximize region simplify notation response predict high suggest scene binary often binary classifier foreground score combine classifier predict say foreground foreground intuitively usually response filter binary multiply non score function hand long convex feature capture part part response vs part classifier counter part score another challenge similar object factor contribute positive ambiguity detector head detector class category filter natural class part filter filter weight vector scoring multi select matrix score imply invariant series unlike restriction entry negative part classifier impact norm otherwise therefore classification loss affect propose joint training objective encourage diversity encourage complement substantial train simple multiple example part think vector multi hinge optimize structural svm reduce define line initialize repeat repeat w convergence output joint parameter optimize weight equivalent training represent solve use method involve maximum make minimizer bind sx w j hx sx z j u z I implement minimize optimize function memory joint objective part step find initial part part procedure large pool involve pick regardless image label whiten patch estimate patch background discard discriminant random result comparable method get alone part figure may part select pool entrie th uninformative redundant drive zero regularizer generate number monotonically important mit cnn per part base part maintain pixel extract multiple grid feature value maintain pyramid cnn hybrid hybrid network imagenet fully fc response part pooling pooling arrange mu mu final left get part randomly weight svm pool regularization train show feature mit space pyramid subset part comprise flip selection improve performance large achieve level performance outperform flip dominate compare part cnn mit pixel train obtain performance imagenet improve augmentation tractable term show effect surprising model feature improve cnn extract entire get pca coefficient pca perform part gap increase number part pool part improve select part jointly part significance train part much gain large translate section blue red random blue curve part red curve part part part filter lead initialize initialize correlate positively outline section computationally expensive optimize simultaneously share filter weight secondly part require repeatedly slow mechanism tractable optimize candidate location location pyramid cache cut copy mu require repeatedly image cache subject cache cache hierarchy among cache hard hard global hx possible configuration image x place vice versa z j sx j w I iy yy u old b latent follow w procedure outline benefit intractable solve auxiliary optimization problem start initial cache cache cache remove hard line algorithm input tc old old save cache update cache convergence warm trick call converge cache remain update happen close iteration trick e triplet retain cache update follow work mean c w old w w imply convex strict convexity w therefore since w cache configuration iteration c stop change due step theorem cache stop possible mechanism cache w local implie therefore key idea algorithm visit cache number plane method solve optimization treat quadratic start update direction accordingly cache entry optimize auxiliary objective proceed gradually converge objective price however tractable slack formulation maintain constraint e tuple cache entry constraint training constraint total ie unconstrained problem equivalent qp eq set qp add repeat proportional size qp dual behind let f ib w k relate gradually constraint enough remain rest process early round discard certain consecutive cache vector qp find fast qp optimize pool initialize optimize objective function direction version constant second mu plot result part depend increase adjust part complement provide figure joint subset mit determine category example distinguish illustrate part filter image filter filter filter filter filter part filter image filter filter filter filter part filter filter filter filter filter filter image filter part filter filter part image filter filter filter part part filter filter filter image filter filter part filter filter filter filter filter filter part filter filter part filter part filter filter part filter part filter part image part correspond entire c c scoring cm cm patch highlight green view c cm cm cm part cm window visualization match particular appear part selective color example appear highly weight appear detect row highly strongly sensible apart filter part location patch number class bottleneck test depend affect procedure first standard bottleneck filter nest loop line loop line algorithm cache loop line qp solver number algorithm separately use one cnn dimensionality latent day joint full mit full
edge everywhere else thus encode difference site odd signal express composition problem encourage sparsity odd constant jump odd structure site fdr underlie spatial odd dramatically difficult inspire technique compute ordinary signal spatially constant optimization treat fix separately section describe detail turn action panel raw toy example concentrate heavily true site thick black fdr smoothing grey mean ordinary attempt recover show toy simulate site arise rare elsewhere highly testing come allele dna adjacent site genome figure show simulate reconstruct prior fdr smoothing function solid curve across dash model fdr smoothing show grey favorable stability site estimate though truth mean site estimate average truth smoothing consistently exhibit spatial site adaptation shrinkage raw top discovery leave spatial pattern density locally high score area locally realistic fmri working memory experiment analysis describe detail panel score arise experiment systematically experimental difficult working voxel horizontal obvious cluster shape task study panel score false clearly edge region many spatially isolate spurious panel bottom procedure partition dense area contain significant area locally significant bottom discover fdr region significant signal plausible fdr locally significance region right bottom figure specialized emphasize fdr simply merely raw locally rather signal fundamental address detail fdr two reason nonconvex guarantee find even evidence actually yield reconstruction power likelihood simple augmentation lead maximization negative likelihood convex function log design stationary via conceptually odd e linear plug guess binary variable conditional site give logit sub expand taylor approximation weight least generalized term intermediate solved gradient respect evaluate iterate denote diagonal log separable penalize square work give follow taylor expansion complete computed overall statistic current complete expand taylor thereby form quadratic surrogate problem problem use augment lagrangian describe practice iterate far expensive use date long improved iterate computationally part fdr smoothing repeatedly chain equivalent problem final chain method multiplier original approach denoise express slack could costly step avoid slack unconstrained yield whenever slack eq primal scaled scale augment lagrangian admm update stationary lagrangian describe individually step parameter update value simply separable soft eq soft thresholding operator subscript update must demand euclidean projection w v r objective size underlie change course diagonal linear cholesky reduce front cost system orient incidence therefore symmetric dominant nearly solver system implementation produce iterative provide approximate know solver oppose subroutine affect convergence therefore cholesky hope exploit specialized linear systems dual k greatly affect discuss issue dynamically primal residual calculate drive ensure happen thus next dual variable likewise large value scale meet describe separately amount rough bayes spatial fdr justify appeal stage indistinguishable repeat argument estimate simplify spatial homogeneity odd density formulate fdr distributional describe test problem poorly describe parametric form order produce reasonable variance central shape histogram test near come mostly nan versus empirical fmri comprise proceed construct obtain second taylor scale mean deviation curvature z approach test estimate apply family show fmri analyze assume unknown deconvolution dirichlet use recommend recursion flexible enjoy guarantee choice large right crucial structure top avoid tune hoc fashion adopt often taylor fdr across decrease grid warm find calculate measure small enforce image approach aic maximize generalize change stein involve distinct contiguous prior background remarkable freedom error aware analogous situation plug degree calculate absence number surrogate heuristic seem good freedom automatically improved panel likelihood two panel aic due admm appendix trace surrogate freedom problem much aic place freedom dominate produce bic balance problem recommend additional practical trivial efficiently scale na I spatially separate counting appendix eight cross site signal configuration large gaussian convolution signal spatial discovery rate dirac mixture convolution grey blue eight scenario panel show nonparametric corresponding scenario sense predictive reasonable job deconvolution right signal square ambient simulation four dotted grey correspond convolution curve simulate set nonparametric recursion signal four dash small compare procedure fdr explanation suitably rich trick basis use basis grid baseline group rate eight desire fdr fdr conservative fdr technique fdr multimodal come bayes true power problem mode nan indistinguishable nan fdr come spline fdr conceptual essentially treat handle choose basis implicitly smoothness underlie straightforward smoothing path choose fdr edge happen coincide find require finally benefit involve dense fdr basis sensible fdr fdr greatly interpretable rate fdr fdr region fdr fdr oracle eight error set fdr smoothing true positive scenario consistently two fdr come close slightly fdr small signal fdr modern scientific many analysis exhibit ignore exploit fdr increase control discovery automatically identify strong area improve area first penalty lead slight equivalently toward ordinary adaptive concave problem present important future moreover feature prevent concern fdr smoothing conduct yield fmri three require quickly hardware laplacian gpu programming two approach plan choose model setting entirely principle tuning fourth fdr provide suggest area could fmri literature fmri specialized fmri generally place intend statistical lasso independently body summarize literature reader group analysis fdr effort beyond paper nonetheless r fmri set analyze section acquire process trial grid sequentially ms follow forced series one present presentation hard second block block acquisition versus easy fmri band tr ms te ms flip voxel mm slice orient back ac mm mb scan length fmri develop st motion
cluster robust contaminate estimation need establish global ignore seem remark separation prove consistent computing hold principle property mind justify strong improve weak compare desirable statistical theory statistical method derive example boost view stop overfitte path another fan li fan li prove oracle property besides minimax huber least simultaneous variable evaluate performance begin focus nevertheless reasonable framework asymptotic discuss find subsample reader van van limitation issue seem valuable theorem present selective broad many list number real datum obtain direction property recently yu leverage algorithm square aforementione local algorithm wang liu forward greedy kind estimation problem derive one appeal start li valuable begin challenge fan liu analyze big datum ability expect relate national natural china grant thanks wu grateful key laboratory chinese sciences van square university york prediction van new york fan penalize zhang york york york york annealing algorithm york york company york mit new york zhang h unbiased minimax penalty proposition section section well mathematics sciences ac cn global solution optimization well likely global solution statistical believe call optimization display optimization statistic widely study paper aim establish discuss various several commonly encounter problem subsample see reasonable indeed well property cluster outlier subsample combinatorial estimation separation notable maximize brief rely good statistical extension estimation huber obtain function nonparametric maximize parameter model fit minimize justify fit least square smoothing fan view nonparametric setting minimize good subset norm regularize regularize see involve analysis fisher discrimination sphere projection et al likelihood special besides determinant compute multivariate number design optimize certain factorial design wu design optimize li geometric criterion utilize statistical support machine boost function regularize commonly minimize select thorough indicate modern meanwhile anneal attain global handle serious due time yu difficulty rarely verify solution objective minimization two understood sense seem likely global statistical usually use believe well principle strictly verify problem actually decision ever complex author formally discuss whether fairly dimensional tell sum square possess well screen property asymptotically view solution constrain therefore fitting actually discuss exist experimental design help reason theorem theorem optimization problem possess separation look result maximum good subset optimization optimization perspective refer good inference presence determinant subsample perform outlier contaminate model statistic hold experimental design criterion consider eq variable subset specify design square estimate solution design lead well whose variance therefore justify conclusion criterion design discrepancy seem statistical relate desirable criterion construct minimax act correspond experiment sample thus sequential deal objective sample study space simplicity map measurable field theorem provide two decision estimation second cover variable subsection confusion situation case remark inference lie desirable subset contain solution practice take decision lie another contain bad say strongly separate separation separate property denote almost surely note imply strong distinguish two former generally convenience arbitrary sequence positive sequence number roughly speak decision property describe strong contradiction n strong separate directly imply condition separate value nn e p np avoid problem countable obtain separate statistic p separable subset example always countable dense sense interest optimization need sequence decision contain decision statistical concern lie another almost say separate subset say theorem write separate two condition weak comparison separate separate sequence countable nn respectively value countable separation property sufficient imply confusion depend situation estimation decision space paper omit write despite effective establish many statistical likelihood let density respect problem minimize commonly multiple compute concern lie neighborhood estimator go infinity decision discuss great likelihood view objective fx dx furthermore bx limit segment subsequence number may triangle restrictive require aspect robust inference correctly identify criterion parallel subset subsample exactly usually van e knowledge asymptotic subsample discuss statistical formulate decision integer subsample objective section attain type serve decision asymptotic ia finite measure subsample eq determinant look low determinant minimize underlie assume normal assumption separation denote sufficiently denote x x p fx fx subset consider dx fx fx g I r sufficiently large banach exist condition fx therefore hold matter away indicate outlier class minimize population possess robust discuss see wu subsample new kolmogorov function discuss property separate weak c subsample contaminate far contaminate imply kf unable distinguish strong prove hold similar letting consider na therefore assumption complete immediately far simulation method likelihood let outlier iii generate objective become correspond selector simulation time likelihood end outlier conclusion likelihood subsample iii well well problem base linear estimate high describe I nx I ip r take correspond corresponding estimator mr mr mr assumption denote fix definite n limit rx k covariance hold array satisfy separate na na n n n n lx n n rx n rx rx x rx rx ia x rx x rx h follow conduct model generate matrix matrix search generating size simulation fix repeat time ci selection van nonnegative scad fan li zhang become screening rule reasonable condition well sub satisfactory continue discuss fix linear without full submatrix bic correspond square x know minimize bic lead prove separation I ax definite asymptotic fu strongly separate ax n h ax h pa aa pa aa aa aa complete almost objective rao wu bic special corresponding increase fan screening specify stage coefficient stage possess screening retain fan fan fan et al
originally convert angle w z tangent see say dimensional multiply project specify project pn pn apart mode really similar shape general comment cosine fix distribution unimodal circular mean identity create positive cosine high axis vary flexible shape result distribution near axis increase mode axis bivariate series x ty hmm indicate state indicator time otherwise transition literature circular latent give relax independence circular get fx k bivariate let normal covariance build joint r circular regression formalize circular flexible component circular specify r see circular regression one propose circular circular cl k dependency circular cosine circular circular argue normal density pn pn simplify cl ki ig ig lead f b b marginal posterior mcmc precisely gibbs sampler metropolis conditional full multinomial depend entire k eq mix slow speed try metropolis mcmc decrease dimension marginalization decrease simulate employ step slow toward burden increase marginalization impact simulate simple carry gibbs estimation switching issue class hide state parameter inference output various propose recent tackle decide post chapter regime jump non approach however goal demonstrate cl hmm circular increase aic minimize among evaluate criterion mcmc maximize map bic aic f bic aic generally indicator suggest estimator carry simulation recover empirically unimodal circular ignore linear plan study cover scheme uniform shape circular separate dataset three model diagonal cl circular unimodal circular indicate cl time element consider scheme summarize characterize consider regime circular one circular dependent l scheme variable b circular figure cl circular unimodal ig aic criteria regime frequency cl cl cl respect cl model perform true former latter hand bic excellent exception may expect perform amount dependent circular cl latent high course affect estimate need regime ccccc ccccc ccccc predict scheme cl cl cl cl cl cl c cl cl cl cl cl cl cl briefly summarize simulation estimate cl cl lead suggest whenever cast cl cl circular distribution unimodal interval ci ccc cl cl ci ci ci ci ci ci ci situation randomly accordingly c cl miss rank probability circular circular one measure distance circular identical linear cl cl dealing point view dataset iteration need thin study resource make project education di computational hour hour reach finally cl wind record semi locate km datum record arise environmental study value record profile wind event south north north arise warm air low pressure cell move episode occur pressure interior behind pressure far wind usually associate flow link interpret wind regime aic component suggest decide two ar circular look value choice three regime provide separate interpretable state classification probability display expect transition essentially persistence regime wind probability indicate direct episode unlikely confirm wind event period ci cccc ci ci ci ci ci ci ci cl distribution circular circular circular variable interval ci ci ci regime regime natural velocity concentrate respectively circular regime north episode south circular correlation circular regime hypothesis statistic interval circular condition first regime interval variable circular relation circular cosine thin check tool cl bayesian explicit cl likelihood circular circular component project circular fairly model bayesian arise several posteriori evidence marginalization conditional circular linear make wind new circular interpretation inferential simulation circular concentration derive circular application consider movement modelling drive development include circular circular project interesting propose latter dirichlet
activation nonnegative firing update learn decay epoch back normalize column set example update procedure repeat one practical issue number dictionary assign produce rich stage avoid procedure encode activity mean close across activity correspond activity simulation provide style align center width cm style width encode sn encoding sn sn sn effect example extend alternate pick example ideally example dictionary close truth uniform heuristic attention part choice element apply goodness dictionary element goodness motivate idea large reconstruction ground truth truth select dictionary regard direction reconstruction example critical produce one level put limit observation noise collect happen beneficial measure snr rare salient define location exponentially make channel channel four selector choose goodness selector select example dictionary sort operation epoch simulation possible goodness selector initialize example max epoch good pt nn loop pt dictionary encode example ground quantify epoch minimal span permutation display positive investigate coherence bound coherence regardless high set comprise x dictionary relatively incoherent easy letter rotation sign artificial violate choose epoch activation magnitude sample snr pick set epoch conjunction goodness trend baseline selector across epoch encoding figure column selector stage epoch contrast surprising poor estimate nevertheless good selector soon establish activation exception selector work closely track selector design use map dictionary ht rl ca dictionary cb encoding order epoch distance end leave order robustness algorithm modify ratio db element original good selector across element suggest great advantage selection extremely complete rl gradient descent dictionary indeed dictionary epoch special success strategy contain procedure implicitly rely identify example dictionary sophisticated grouping provably dictionary inter characterize inter whose work generative spatially generation apparent thus case intuitive pick information dictionary sample benefit inference validate snr last epoch figure algorithm pick high snr correlation snr weak suggest factor drive factor contribute spread reveal selector pick select example distribution measure select example histogram tends dash weakly predictive overall suggest snr cb dictionary nd plan instance selection lead empirically paradigm suited stack autoencoder case interesting layer explore question future task font font font edu uniform feature datum active current accelerate inspire sparse activation work code hypothesis hypothesis decade low code idea dictionary starting extensively explain capture efficiently usually uniformly training resource relevant
available construct combination list converge one coincide rely violate open tend kullback leibler weight n n fx I fx primarily focus bayesian procedure aggregate h la impose primary work prior bayesian prior minimax convergence la able minimax aggregation utilize extra da able automatically exist optimality require tune open fall list posterior put proportional reality hope inequality propose novel interest theoretically discrete burden continuous avoid combinatorial potentially computational distribution widely probabilitie rigorous relationship concentration relate mixture posterior effectively extra emphasis study moreover component fix allow unable impose sparsity rest aggregation describe ca simulation theorems technical deferred provide detail aggregation risk base understood design truth possess expect rate estimating case sf f extend precede gain minimax la sparse la ss la arbitrary constraint approximate extend show risk aggregation structure pdf dirichlet commonly simplex priors allocation concentration display dirichlet change moderate small distribution concentrate small capture sparsity htp concentration characterize absolutely exactly sparse relax sparsity consider index sparse concentration property symmetric dirichlet consequence section simplex utilize process fact dp reflect concentration favor suggest uniformly sparse pattern true general method satisfy f put almost mass able concentrate around assume square integrable fx assume convenience theory generalize problem belong aggregated try element mf different produce aggregation dirichlet da symmetric dirichlet favorable concentration property near minimax contraction dimensional shrinkage sparse place dirichlet include power la design write represent training mp aggregation high special parametrization determine correspondence prior prior parametrization example prior induce mind double dirichlet dirichlet equivalently q gamma pdf dirichlet form study focus efficiency aggregation la constant example characterize ca observation specify condition minimal leibler put almost whose towards avoid argument behavior error deviation proof justify setup frequently f situation th make integer jx jx covariance uniformly cauchy bound uniformly condition use part study hellinger gaussian one mean impose without sparse suggest radius proportional ca special procedure iid prior da eq ff non uniqueness quantify la fast constant design since normalize condition condition aggregation linear b assume study consistency boost high regression sparsity also aggregation gain also constraint satisfy spirit identifiability approximate converge therefore sparse sample b q fast among minimizer suggest concentrate achieve explain sparse uniqueness happen suggest depend estimator addition estimating estimator strategy follow divide learner algorithm learner j aggregate sample learner give become ns ns dominate impact equally plot credible ns non credible interval nonzero dirichlet prior change analysis htp cc figure la weight especially model robust result recommend choose conduct aggregation truth I training size first rf neural nn bayesian ba super learner sl sl package implementation base learner burn summarize table root square error rmse replicate cm lasso cart svm sl ba second fit response predictor learner large covariate base use full addition compare ba sl voting learner subset base summarize sl ba apply ba datasets uci repository cart forest regression neural learner discard half burn dataset forest aggregated ba learner dataset performance base learner sl datasets c concrete forest log cart rf sl ba auto divergence accord asymptotic theory iid aggregation problem convergence put mass around section iy concentration become dominate pc imply cc kp problem iy concentration characterize concentration property dirichlet dirichlet concentration cd characterize characterize interest da volume lemma prior concentration property thus part da play role characterize da aggregation property double taking assume property second ensure convergence prior put show dd da probability space sparse approximate la characterize term cover number integer q next complementary utilize stick da eq joint expectation copy distribution misspecification design construct test la design la remark section uniformly bound bound type fix design assumption minimizer kl aggregation space la mean similar need provide da b jensen chebyshev combining yield converge cn lemma constant select sufficiently n n cd combine fact prove construct bn lemma similar help dense accuracy therefore construct satisfied conditioning sparse fa fa let cover net interval net enough ga argument point integer exist md mn rest generality nonzero since j denote gamma q approximation conclusion double allocate application conclusion adapt additional constant similar instead part result apply index satisfie sm imply prove first conclusion fa b lemma nonnegative consider dp unit dp stick representation dp index let combine definition k application markov inequality yield apply I yy df np mp f yield acknowledgment support national institute environmental sciences national health mcmc la idea conduct metropolis hasting update distribution bayesian apply stand old
broadly comparable lasso frequentist method recovery mode give immediate bayesian section bayesian posterior contrast result quantification combine contraction latter asymptotically bernstein new identity crucial separate prior density instance laplace density even prior uninformative contrast parameter bayesian infinity zero small essential bayesian framework organize prior nonzero investigate ability coordinate distributional apply credible show recovery defer section parameter true generic vector ambiguity refer let column prior bayes borel laplace usual situation however setup induce laplace large nonzero undesirable assume value decrease natural distributional posterior hold read precise interpretation regression error unity scaling follow case light sequence condition create prior allow extension precede example unit variance location replicate response regression refer fix input th sample exclude shrinkage situation sufficient turn prior constant dim prior rate decrease reflect coordinate mixture dirac zero laplace briefly comment one replace prior prior model value laplace density even necessarily definition simplify setup discussion concept compatibility compatibility give compare norm predictive consider nontrivial compatibility simplicity value replace denominator numerator comparison norm schwarz replace compatibility restrictive condition concern compatibility p minima number compatibility dimension define small singular impose cauchy impose respect whereas suffice reconstruction matrix division unity submatrix coherence maximum dimension interpret reconstruction norm relax pair compatibility number sparse closely inequality evaluate infimum away zero thus compatibility value certainly zero size multiple e mutual three index previously reverse compatibility mutual coherence possible restrict isometry extensive discussion refer compatibility oracle bound prediction norm contraction spirit contraction coherence analogously estimator supremum coherence way albeit allow norm verification compatibility preferable compatibility valid compatibility maximally design moment situation coherence number true possess exponential compatibility away regression small survey compatibility eigenvalue see column diagonal satisfy example satisfy cover aspect case eigenvalue rate equal final consider vanish block handle index include proof indicate remark misspecification show true dimension interesting constant compatible constant make choose dimension read proof also dominate contraction rate suitable appendix concern center compare estimator concern rate contraction relative compatibility number bound thus rate recovery vector bound zero theorem uniform furthermore every consequence theorem upon choose assertion instance design assertion large selector develop statement compatibility oracle oracle besides give quantify size coordinate put qualitative manner compatibility matter model unnecessary truly nonzero selection nonzero detect nonzero exclude theorem compatibility condition term threshold similarly satisfy true replace large mass sense shall contrast lambda include magnitude bias small lambda regime choice small nonzero lambda lambda regime simplify thus regime regime let submatrix consist column square I correctly possess prior would equivalent von normal distribution dirac measure satisfy sufficiently neighbourhood true lack parameter outside different eq projection subspace decompose x two lebesgue improper mixture correspond weight sg coordinate hand still weight bernstein von lambda regime improper select uniformly subspace dirac note define choose improper enough improve prediction interpretation projection computational auxiliary variable typically laplace mixture implementation unity present problem prior study setting consider recently last year model curse besides time monitoring concentrate low also apparent result make thousand exploit solution graphical genomic view correct move model hundred smooth posterior set coordinate geometric move distant density performance equal exact analytic limited spike decrease model spike likelihood survey setting find reconstruction empirical pseudo bayes approach spike show put present posterior quantity implement computation hardware suffice design correspond ratio support laplace density side bound prove laplace side display occur event bound denote inequality display surely ix integral decay power follow display cauchy possess normal variance I tail normal dimension prior satisfy ty measurable bayes follow old schwarz therefore value right combine display q see infer choose proof posterior support intersection event compatibility q assumption suffice x display calculation theorem tend similar eq first assertion x total distance measure restriction bound constant one hence asymptotically precede restrict measure connect remainder absolutely denominator satisfy second jensen logarithm display set view shall factor right q relation principal submatrix jj assertion event property yx ss inside standard possess freedom apply give ps marginally upon tend theorem imply assertion follow assertion enough sm ss bind ms precede display square dimensional span onto abuse onto shall term start note x x mx mx first cauchy schwarz sx sx j sx together sx normally variance order sx op exceed side probability tend desire positive reasoning precede asymptotically acknowledgement thank grateful helpful discussion supplement supplement state von lambda proof supplement regression prior usual choice regime strength center give von lambda op q approximate normal estimator bias covariance lambda correspond argue lemma assertion abuse notation write right instead c disjoint assertion limit mixture obtain enough coordinate coincide index draw law index law eq thank assertion sum collection contain index write leave multiply element constant statement indeed establish sr ir si last center note guarantee coefficient together lemma nr exponential possess cauchy inequality ball bound conclude small result prior eq vary j orthogonal projection j volume j sd expectation sum contain term sum last jensen hold product normalize denote kullback leibler divergence cs expand quadratic form symmetric term p tv tp tv xt maximal linearly subspace successively union let lebesgue denominator formula translation invariance measure jensen lebesgue weight definition vc vc vc intersect deduce v vc vc euclidean unit ball one deduce enough display upon hold
sect sect equivalence sect sect finding explore extract invariant local detector various incorporate cnns one build invariance architecture yet another mechanism learn rather systematically knowledge work perhaps believe quantify property manner sift cnns function notable invariance representation formally invertible hence existence representation predictor case cnn require input image intrinsic geometric representation capture practice affine image invariance act invariance regard goal establish image object image change study systematically possible representation transformation study invertible satisfie mapping cell direction cell component symmetric permutation one rotation approximately rotation implementation densely sift network sense obtain boundary effect result equivalence must discuss discussion focus deal transformation mapping simple affine g g choice restrictive initially permutation hence permutation empirical datum natural image amounts adapt particularly quite encourage row q reflect component induce exploit convolutional translation locality neighbourhood index triplet neighbourhood site close vx vx measurement site fractional neighbourhood transform site g v combine regression activate neighbourhood location neighbourhood sect histogram hellinger distance layer cnn orient well understand orient cnn train end objective preserve quality ground label train case image set cnns target pre train predictor sect class encourage convolutional say transformation convolutional reason learn implement layer cnn consist map filter neighbourhood sect intuitively purpose channel fall integer coordinate permutation lattice rounding distribute site bilinear interpolation study cnn equivalence orient sect predictor cnns train imagenet transformation loss cnn interpret convolutional representation bank permutation may sect study mapping sect move examine sect mapping apply regression style align leave none legend align legend rr rr fs feature reconstruction cell hellinger learn map different rotation constraint impose structured value neighbourhood size train discriminate cat gradually scale l rr r c second regressor array unless sparsity sc learn fs rotation scale support sect evaluate finally learn validate explore formulation predict feature image histogram validation focus predict small baseline array array avoid boundary rescale restriction later transformation objective square ls ridge rr fs per output fig surprising array f zero error rotation exact sect error fs remain em legend column legend style align none cell align legend fs fs opt fs fs em learn vertical cnn fs task orient reconstruction bottom classifier image good red report evolution flip oriented rotation rotation hard reconstruct suggest learn address boundary effect substantially nothing original mapping cnns conv fs less training orient compare sect substantially less task orient well converge much fs fs significantly small reconstruction depth matching intuition deeply perfectly transform layer transformation vertical next geometric horizontal vertical rescale transformation mapping leave unchanged cnn implicitly invariant vertical rotation however learn substantially generic good cnn sect practice never high invariance replace row evaluated accept relative corresponding channel large approximately result rescale cnn notable invariant conv second conv tend conv possible vertical rotation flip flip c top top top top top top mapping layer flip sc channel r layer top several cnn obtain portion convolutional portion look representation replace sect sect cnn layer train mit place train mit place identical top hybrid learn layer sect fact intuition make feature channel compatible slightly deep specifically conv conv fully conv conv code conv task bad dramatically chance compatible complement investigation far section mapping sect output label fully rewrite advantage demonstrate pose pose object pose cat rotation rotation sample degree rotation line cluster r n contain example gmm train affine mapping cluster center pre image contain fs sect cat version rotation consider scoring report cnn conv conv nearly mapping curve regressor conv conv conv ms report cat head pose direct regressor error degree residual rotation normalise frame method predict h cm rgb rgb legend legend style align legend align leave legend conv conv em curve rotation affine regressor c pdf pdf pdf pdf rotation bottom cat face estimate affine pose represent regression conv within representation several layer deep cnns transform image architecture deep degree specific addition tool regressor elegant science image encode task apart function assign incorporate useful normalise class irrelevant orthogonal feature irrelevant contain encode direct vector reveal object class viewpoint nevertheless orthogonal irrelevant direction achieve invariant linear manually argue task normally different task classifier regressor discard task give image rotation translation invariance alone always ability distinguish differ invariant seek however eliminate factor simplest specify rotation exact invariance closure transformation transformation example closure even pixel minus font legend title style macro despite importance image representation histogram convolutional network mathematical representation invariance transformation effect visual establish empirically introduce layer cnn reveal aspect include cnn certain theoretical structured output regression demonstrate image notable orient bag visual word code super generation convolutional popularity representation remain generally invariance contain r conv tf fs tf tf fs tf fs tf fs tf tf flip tf fs flip flip fs tf fs
improve replacing step conditionally log directly either imply cm conditionally maximize function van combine sense satisfy ascent property share em simple integral pdf thus large replacing give multiple miss em iteration secondary stochastic expectation use carlo em complete single log model structure draw single sample draw carlo sample per imputation estimation pick maximize together map pdf prior penalty general penalize motivated term usually thus solve map modify em framework prominent field prior knowledge prevent algorithm change model em exponential complete variable include poisson pdfs moderate information simple pdfs censor exponential complete data occur observer record datum common trial e medical testing product reliability analysis algorithm replace latent good maximize em space depend generalize nature function derivative numerical complicated coordinate decompose pdfs come population recognition random population proportion discrete pdf pdf pdf sub fy rewrite ease estimate population variable use derive mixture sub population change pdfs population popular area cluster identification map distinct distinct space specifie sample maxima dot contour maxima alternate emission construct emission emission image level chemical scan start array detector around fine emission picture pixel detector simply determine reconstruction set random model population distribution probabilistic detector observe detector detector record geometry sum poisson response individual pixel complete intensity parameter detector emission detector depend detector define go average emission intensity function em lot space use iterative approach incomplete complete fit logistic mm generalize optimization mm specifie function tangent parametrize mm algorithm instead ascent ideally easy em al fit modification constant shift shift depend ascent ascent establish em primary parallel ensure ascent descent property imply curve tangent require observe point precisely another mm use satisfie effectively create local quadratic every produce true tangent estimation argue design less designing complete extra flexibility publish wu optimize objective converge compact close mm global theorem solution interior stationary assumption closure follow ascent property boundedness minimization continuity em example mm satisfy closure broad include closure may slowly information em implementation get complicated converge maximum demonstrate address slow improve rate application chapter demonstrate speed improvement idea expectation speed noiseless show maximum surface positivity corollary em gaussian censor positivity simplify quadratic benefit distribute decrease review formulate behind noise noise improve introduce corollary present variant present theorem positivity theorem law large sr noise amount improvement mutual input early benefit physics biology early description benefit include ice snr improvement brownian particle chemical inspire nonlinear signal estimation describe screen exhibit necessary sr weak signal signal screening help explain observe benefit system benefit pixel monotonic curve non monotonic sr signature curve noise em noise benefit em unlike almost sr benefit apply cauchy model censor benefit condition suggest condition last noise condition em important time shape signature convergence ht mixture deviation normal standard gmm vertical bar noise benefit sr involve threshold sr additive depend independent weak benefit dependent em happen enhanced noise chain proper test converge fast gain benefit examine plausible statistic likelihood appropriate could ball depend ignore working realization alternate specifie experiment high likelihood pay alternative calculate bootstrap favorable likelihood result high condition describe likelihood idea condition sometimes pdf pdfs em positivity condition benefit modify surrogate likelihood maximizer modify likelihood regular likelihood maximize benefit occur noisy value noisy perturbation current noise benefit familiar express pdfs information incurred pdf pseudo noise benefit pdf well theoretic description complete regular incur low difference condition guarantee follow pdf noise differential entropy additive keep average expectation distance pseudo likewise benefit sufficient noise relative assumption finite q integrable integrable permit benefit theorem apply complete positivity user condition generalize noise proof iteration suitably close towards correspond algorithm hold positivity imply noiseless algorithm number benefit noise sequence inequality converge equal imply convergence noise anneal continuity k q fy guarantees exist converge fix inequality theorem dominate condition positivity wise condition pdf positivity condition monotonically pdfs still useful effect benefit model special case corollary corollary population likelihood eq corollary dominate occur gaussian gmm pdfs pdfs ensure dominate population additive noiseless pdfs positive expand q hold conditional similar model pdf noiseless pdfs last prove condition corollary quadratic q figure exhibit benefit standard population mix proportion standard deviation start comparison evolutionary iterative gmm noise count noise similar ht cauchy add bootstrap estimation benefit dispersion equation condition benefit ht gmm dim sample dim sample optimal noiseless mixture normal use normal deviation gmm reduce admissible noise support depend value goal q side hold fall sub population take falls sub population valid fall tend tail cluster sub location gmm extend gmm covariance sub dimensions fy fy gmm vector population gmm estimation multidimensional quadratic positivity geometric description exist fall boundary hyper rectangle centroid eq population determine case multidimensional gmm set interval singleton origin origin illustrate geometry line ht min min dimension specify side length draw geometry towards centroid sub action dimensional sub jointly population jointly population state notation eq product degenerate suppose df pdf sufficient eq hold complete finite model iff sub determinant simplify thus eq give ensure summation conservative analogue condition specialize gmm condition us condition variance condition benefit condition cluster intersection dim case green mm blue overlap mixture boundary product dimension factor component dimension way mm noise geometry noise scheme ease increasingly dimension performance em seem theorem sort superior algorithmic function evolution evolve function come price evolution raise computational sampling step complexity sometimes raw fewer efficient keep noise case still noise benefit ideal positivity amenable inequality corollary complete pdf noise zero random variable decompose product censor unobserve latent gamma give pdf interval gamma density hazard rate describe benefit censor censor speed noiseless inverse gamma distribution censor estimate mean bar predict noise replace additive corollary method modify noiseless average noiseless noisy expectation schema current depend maximum replace em give decay scale iteration decay factor iteration zero need cause closed decay fix even final polynomial factor logarithmic anneal application variant schema generalize maximization simple replace ascent noisy schema allow outside scope involve modification step change step pick subject variant procedure noise change em gmm derive procedure gmm intersection dimensional version generator perturbation fall degenerate another add perturbation perturbation datum schedule difference em bar across add deterministic perturbation locate datum far origin perturbation decay scale perturbation like gmm converge em average roughly noise gmm dynamical interference interference map value gmm use scale fit set square ht algorithm logistic map scale decrease gmm gmm gmm depend apply gmm eq satisfy gmm analyze still singleton go probability noise value decrease condition eq eq maximal sort nest nest empty bound close interval hold fail happen positive produce benefit write intersection second property sub borel fall number converge tight size probability one singleton value identically condition correspond case guarantee em identically limit fall ht identically avoid probability even noise figure model equivalent define event figure event blind differ depend blind use blind blind average blind blind fail blind simple perform ht use anneal converge fast simulate annealing model reduction deviation theorem sparse zero noise benefit improvement entropy decrease benefit data f fx fx pdf show benefit central limit analysis term form random I probable apply population law equal difference arise pdf law number mean expectation positivity state standardize distribution variable noise noise positivity infinity positivity condition fail suggest condition match sample large result noise benefit mean noise benefit available likelihood nonparametric careful convergence sufficient condition benefit signal probable em speed satisfy condition blind noise blind noise condition obey positivity implementation level benefit variability surface likelihood average change control concern optimal noise vary show family cause fast speed noisy noise distribution easily flexibility regular comparative versus interference em deterministic interference three demonstrate important incomplete model cluster feedforward em backpropagation recommendation google news news netflix amazon recommendation often rely centroid classify computationally slow provably speed cluster benefit general include em noise benefit classification improvement classifier compare classifier training use train optimize classify algorithm find pre limit benefit corollary gmm benefit report classifier parameter show agree normalize misclassification fall reach minimum reduce misclassification beyond optimal misclassification ht gaussian special benefit different decaying help supervise competitive algorithm fully theorem generalized version benefit algorithm attempt differ mahalanobis similarity algorithm assign sample centroid close centroid attempt np centroid centroid partition look minimize cluster fuzzy agglomerative cluster probabilistic true population assumption fold mixture bayes population involve estimate parametric estimation benefit derive framework cluster apply population positivity optima suitably em positivity reduce positivity correct centroid additive count square decay constant function give mixture pdfs population mean current gmm posteriori assign parameter eq benefit extend benefit relative em optimal optimal misclassification parameter benefit misclassification free em misclassification procedure satisfy positivity condition reduce condition misclassification decrease optimum follow iteration count eq reduce noise satisfie classifier perform converge less em enhance algorithm dimension benefit figure misclassification procedure partition centroid try centroid euclidean distance indicator arise near classification indicator procedure optima em gmm measure fuzzy much belong population bayes membership cluster membership centroid assignment cluster modify gmm benefit population spherical know proportion membership gmm reduce cluster matrix update equation em centroid em hard change sum hard reduce centroid diagonal proportion knowledge estimate mix proportion update hard occur em sub result probability equivalence confirm predict benefit resemble noise supervision circuit art interaction bottom activation neuron activation input internal representation art substitute supervision field system signal recognition exist fail category within specify match flexible mean art pre specify cluster art update characteristic learn art open question provably benefit competitive pattern adjust weight competition competitive train centroid quantization converge centroid competitive noise system use competitive like imply topology compete neuron fan vector neuron centroid distance approximate dynamic layer center winner topology competition win neuron fan neuron nearest neighbor pick win neuron fan close current move winner centroid little close incoming first win vector equivalence alone benefit second step increment prevent direct em algorithm prevent theorem centroid quantization initialization linearly decay coefficient winner update win quantization update win quantization modify version pseudo alpha rewrite stochastic pick winner closely resemble deterministic competitive neural modeling memory neuron input neuron field sigmoid competitive neuron likewise scalar competition function zero rise winner winner result imply square connection matrix band winner teacher know know fan represent winner fan winner class increment get minus rather sign winner increment context adaptive classifier differential competitive hybrid replace competitive win differential structure law notation activation rather differential neuron learn competitive neuron competition activation reinforcement increment blind law compare simulate competition time derivative neuron approximate competitive simulation competitive cluster simulation white training scale standard allow intensity entire decrease variance distribute three anneal schedule neuron q win quantization perturb version h noise anneal pick winner win quantization anneal schedule noisy winner win quantization four show figure improvement noise speed expectation maximization fairly competitive noisy expectation guarantee extend benefit method also two co co cluster movie movie database cluster semantic use filter document benefit benefit also co generalize chapter hmms probable region space new algorithm train sufficient positivity positivity sufficient hmm figure architecture hmm reduction number take converge likelihood confirm speech hmm observe speech effort neither speech theoretical review hmms em point present boost hmms test training corpus ht variable series speech biology vision hmms speech speech use hmms markov hmm single map state contain density gmm common pdf coefficient mean matrix tuning hmm respective sequence make difficult concavity hmm indicator forward backward function maximize respect lead condition definition pdf em converge positivity enforce simplify gmm positivity positivity additive observation index latent variable noisy positivity eq maximize gmm covariance differ noisy modify noiseless positivity satisfy condition increase h kt st jt st st I jt kt kt n n kn modify perform embed create large hmm sub modify suitably produce positivity variance anneal scale noise iteration decay dataset setup frequency coefficient compute window also second vector energy hmms gmm metric hmm percent improvement percent reduction iteration likelihood marginally positivity number small careful addition average ml hmms give sufficient condition simple constraint gmm per frame hmm develop noise benefit neural mathematical consist neuron transform input signal usually sigmoid connect biological transmission layer neuron feedforward feedforward absence backward self connection feedback neuron distance black inner sep neuron center node neuron edge input layer feedforward network popular many speech translation intelligence computer network biological network acceptable white feedforward borel measurable backpropagation feedforward goal bp feedforward ff example say ff minimize function minimize feedforward network offer insight task lack explanatory algorithm much contribute training error unit architecture backpropagation propagate signal via chain rule local error feedforward expectation consistent miss bp hide match noise positivity condition template noise bp feedforward backpropagation fall backpropagation convergence activation layer neuron noise examine stochastic generalization noise generalization add activation reduce section review detail backpropagation em training discuss simulation bp ml neural parameter layer network deep neuron layer consist neuron neuron activation represent value encoding neuron connect hide target neuron matrix backpropagation follow entropy target backpropagation ascent give gradient ascent log formulation use square observed output error call configuration higher later development backpropagation convergence linear replace value neuron layer value backpropagation assume estimate optimal square square estimate backpropagation partial derivative backpropagation backpropagation epoch eq epoch know entropy expand backpropagation backpropagation show intuition greedy probabilistic hide bernoulli activation formulate ml feedforward parameter e resort carlo sampling converge true follow bayes easy approximation independent identically distribute sample dimensional kronecker importance importance activation output neuron maximize backpropagation perform backpropagation bp fast positivity depend converge assume keep positivity condition form condition activation neuron consider benefit gibbs positivity ml training feedforward neural activation ratio sufficient condition positivity become positivity inequality log activation sufficient noise illustrate hyperplane change output positivity ml feedforward neural sufficient outside sphere network average ht noise sphere use approximation quality substitution quickly neuron backpropagation mnist pixel lie fed pixel neuron use neural layer output digit auto encoder three pixel three layer gibbs activation classification class carlo encoder show backpropagation backpropagation add gaussian mean entropy add mean epoch use monte square backpropagation backpropagation allow develop backpropagation simulation digit recognition square cross nn feedforward learn feedforward backpropagation backpropagation noise chapter application shorter important training take stack neural deep neural benefit present predict estimation describe previous belief bayes heart application spam belief oppose experiment difference two kolmogorov acceptable prior fisher pearson provide free bayes publication involve inference modern individual application estimation highlight frequentist find mle simplicity property intuitive interpretation frequentist bayesian depend frequentist powerful prior rest chapter chapter corruption framework thus previously framework evidence belief build bayes evidence update compete prior belief hypothesis give evidence occur depend converse unconditional disjoint exhaustive follow partition state measure evidence compete hypothesis inform belief hypothesis well compete correspond drop normalize sufficient converse probabilistic maximization inference abstraction hypothesis realization expert ultimately come data chi square kolmogorov part application determine come could expert source accurate pdfs application limit close small pdfs conjugate produce close posterior pdfs posterior come three normal display conjugacy relationship beta likelihood poisson count prior datum htb beta unit interval prior success binomial coin bernoulli shape beta conjugate beta combine binomial produce new beta posterior trial mean square square conjugate still negative replace conjugacy dirichlet multidimensional dirichlet conjugate priors conjugate pdf pdfs chi generalize pdfs sided population mean poisson poisson gamma combine produce real population mean normal normal hierarchical still variable prior sequential previous pdf become pdf conjugacy greatly simplify chain monte posterior pdfs statistical optimal point estimate question belief give new produce pdf measure spread base spread answer incomplete incur wrong decision parameter estimate incur much magnitude function optimality high average estimate calculate subject bayesian optimal bayes opposite formulation decision economic rational choice economic utility bayesian pearson basic testing decision problem determine ideally engineer square loss conceptually minimization loss prior possibly improper maximum likelihood ml uniform constant thus maximization invariant scalar multiplication mle take hold unbounde improper pdfs pdfs integrable ml minimize strategy address estimate conservative estimator typically exist alternate value risk compare admissible rational scenario pp form person game minimax minimax strategy conservative estimate need description specify parameter measure generality around estimate inter variability mode bayes estimate credible familiar frequentist credible subset characteristic confidence interval statistic credible unknown credible interval interval unique bayes belong credible interval credible credible mode chapter frequentist approach also address issue method issue point exposition assume pdfs accurate fail corruption incorrect next chapter model observable datum source confusion randomness datum parameter bayesian inaccurate possibly form corruption raise reliable data approximation address effect analysis posterior pdfs apply bayes pdf fuzzy bayesian express description form algorithm expert linguistic grow function law fuzzy tractable fuzzy carry fuzzy show fuzzy fuzzy scheme likelihood well approximate later hierarchical chapter carry prior model circle draw black distance thick rectangle connect parameter knowledge form rule fuzzy likelihood pdfs pdfs describe fuzzy rule close also adaptively fuzzy rule closed range prior capture expressive linguistic figure tune skewed pdf simple rule fuzzy set fuzzy skew fuzzy fuzzy description occurrence rule rule small small large medium scale cauchy fuzzy tune shift probable set simulation rule quickly learn implicit prior fuzzy reflect evidence inform expert fuzzy reasonably bayesian approximation uniform prior lead fuzzy approximation time fuzzy converge adaptive centroid compute observe probabilistic evidence process little guess expert source source additive fuzzy output uniformly compact learning cluster gradient descent fuzzy allow user add rule fuzzy effort prior likelihood improve review behind fuzzy show fuzzy approximation approximation scheme fuzzy doubly bayesian fuzzy prove uniform fuzzy hence fuzzy system fuzzy rule input interval approximation expense pdfs truncation leave normalization fuzzy output centroid sum scale mx set centroid centroid form ix ip jj rule drop scalar care simulation fuzzy value theory shape cauchy adaptation extensive perform square user fuzzy give corresponding fuzzy practice fuzzy triangle c b b give direct measure inherent uncertainty define current rule cover affect fuzzy affect extent centroid tune parameter depend unconditional mixture density contain rule multidimensional fuzzy suffer general rule turn fuzzy patch tends quickly fill rule extensive high fuzzy approximation extensive fuzzy dimensional iterative chapter represent closed exactly closed strong exactly volume fc ff structure rule bound pdf simulation set function gaussians centroid fuzzy directly available hold bound posterior pdf laws adequate practice train numerical datum use chi kolmogorov pdf figure prior pdf pdf train sufficient simulation approximate train learn prior available use tune part area law convex parameter derivative rule law centroid form law manner expand ht success sample fuzzy approximation time sample prior likelihood approximation tune dispersion parameter triangle cauchy laplace generalized tangent set example six fuzzy learn law law eq law q law law symmetric parameter law eq double generalized set law reverse adapt part maximize fuzzy give law eliminate partial derivative ascent posteriori law six fuzzy pdfs pdfs six law software approximation fuzzy program compute square fuzzy uniform evenly spaced part assign dispersion initial volume centroids pdf location spaced rule law adapt harmonic decay pick numerator approximation representative simulation cauchy set illustration gave simulate six set produce pdfs gamma prior side normal truncated likelihood likelihood narrow support prior fall tail accommodate unlikely setting prior strictly bound prior integral well behave fuzzy cauchy respective squared uniform respective square approximation gamma prior respective learning sample iteration error fuzzy uniform interval training iteration mse mse generalize noiseless noisy pdf histogram sampling pdf systems still pdf random bin equally space pdf beta rescale rescale pdf random central location bin figure second example deviation separate case produce random sample pdfs pdfs pdf gx fuzzy rule iteration figure fuzzy bayesian conjugate pdfs h rule iteration next structure fuzzy generalize centroid long vary posterior expense centroid part form fuzzy accord corollary computationally first special dirac delta arise become law centroid substantially integration delta approximation curve normal cauchy pdfs alpha stable pdfs parameter pdf dispersion go zero characteristic fourier exactly dirac delta equal right hand fail narrow unless maintain unity second system several narrow rule uncertain compare constant pre structure another tractable centroid occur xx gd gd discrete cover system lot linguistic theorem suit scheme approximate approximation free uniform feedforward ff network approximation approximation white ff borel function ff sample back training feedforward procedure fuzzy ff fuzzy ff network linguistic simple case ff ff representation use nonsmooth activation hard ff compact domain equipped multiplication rely approximation generalization compact basis spline bernstein polynomial unstable implement use doubly pdf pdf doubly fuzzy fuzzy fuzzy growth rule lead old iterate ht gx n fuzzy base doubly pdfs rule gaussian approximate likelihood approximate doubly uniformly pdfs mild derive approximation convex fuzzy bind proof require let likelihood dimensional approximate fuzzy approximate sample everywhere f pdf approximations function bound expand error almost everywhere derive parameter measure compact thus totally invoke extreme extreme compact attain maximum maxima minima continuous right attain thus q inequality theorem maxima depend value sample pointwise improve extreme ensure factor finite bind attain minimum assume h denominator error limit guarantee restrictive determine satisfie proof lead uniformly respective nf corollary reveal fuzzy sum produce centroid fuzzy mp jx jx centroid centroid likewise part set sum centroid induce turn produce min centroid centroid centroid even assume minimal centroid centroid centroid doubly fuzzy centroid sum since mp argument thus bound sum bound centroid sensitive centroid divide minimum become centroid bound correspond least informative centroid multidimensional multidimensional component centroid hypercube max r fuzzy fuzzy density descent learn tune might might prior lead fuzzy neural problem conjugate doubly fuzzy general bayesian iterative chapter chapter fuzzy put next increase theoretical complexity conjugate track bayes fuzzy growth conjugacy conjugacy semi conjugacy conjugate size restrictive conjugate nest put prior uncertain appear pdf another demonstrate technique common inverse gamma pdf uncertainty conjugate wishart covariance bayesian pdf uncertain parameter involve make represent circle minimum size mm thick rectangle black label label connect edge connect capture another unconditional pdf conditional describe hyperparameter conditioning prior add integrate remove dimension benefit flexible proxy approach lot marginal approximation fuzzy pdfs conjugate hyperparameter pdf gamma ig conjugacy truncate gx n posterior use fuzzy gamma ht like fuzzy give hierarchical triple build double prove triple scheme pdf input extend call quite depend polynomial function statement require notation pdf denote set qx gx gx xx approximation applie allow prior almost everywhere x f qx prior approximation bound depend expand get assume everywhere finite lebesgue extreme compact attain maximum extreme minima continuous domain set similar ensure maxima depend well arbitrary depend note continuous extreme bayes upper maximum extreme assume gx gx denominator error zero guarantee involve arbitrarily satisfy appropriate integrate nuisance away compact lebesgue eq exist therefore guarantee multidimensional extend fuzzy datum hyperparameter long unobserve uniformly dimensional posterior pdfs hyperparameter function dimensional additive model fuzzy multi fuzzy produce prior hyperparameter representation adaptive approximate tune use law learn eq dispersion law replace parameter law law square fuzzy case fuzzy allow user prior hyper extend bayesian fuzzy inference model prior triple separately pdfs approximate fuzzy conditional separate fuzzy pdf fuzzy conjugate mean deviation beta hyperparameter beta fuzzy approximate pdf approximation conjugacy bc x point fuzzy use fuzzy b gx rule prior point likelihood iteration bayesian update propagate fuzzy exponentially standard conjugacy keep define conjugacy fuzzy shape set corollary preserve conjugacy part fuzzy function corollary conjugacy beta gamma support ht laplace beta semi gamma laplace structure fuzzy bayesian doubly fuzzy rule fuzzy system fuzzy rule j j fuzzy fuzzy product part part volume centroid likelihood fuzzy rule rule represent fuzzy approximate pdf represent pdf fuzzy prior hyper volume centroid qx fuzzy rule doubly fuzzy define fuzzy system shorthand fuzzy system approximate fuzzy system fuzzy likelihood centroid rule weight volume approximate imply fuzzy stage rule fuzzy rule rule linear fuzzy rule iterative inference track involve likelihood prior system rule fuzzy rule rule fuzzy rule fuzzy likewise represent pdf set function centroid rule beta represent four set conjugacy structure doubly fuzzy system kf conjugacy part doubly fuzzy ib functional part form part set product conjugate n f conjugacy doubly fuzzy prior use h gx g gm eq use set hyper function self case eq eq fuzzy show part fuzzy corollary doubly fuzzy conjugacy beta set suppose bm h gx g bm g f g j j special occur conjugate q fuzzy form beta part conjugacy part set suppose gamma poisson set gm gamma q conjugacy laplace suppose semi special shape dispersion semi conjugacy conjugacy semi conjugate set avoid parameter conjugate beta gamma laplace corollary latter center special result parameter fuzzy inference uncertain fuzzy fuzzy uniform substantially extend choice prior close pdfs conjugacy priors user knowledge open exponential fuzzy system device growth algorithmic corrupt record datum also inefficient improper increase daily yet systematic datum corruption lag behind effort corrupt incomplete effect misspecification show improve speed general analyze corrupted rely benefit result affect hmms limit misspecification present show bayes uniform function pdfs produce theorem quality drive approximate interesting know many diverse help reduce intensive application area explore subsection htb c chapter feedforward ng ng hmm extension chapter segmentation zhang al zhang deep refer deep machine rbms perform bipartite structure gibbs energy whole ensemble rbms neuron energy lyapunov rbms chapter show backpropagation feedforward procedure formalism rbms noise rbm rbm visible layer neuron value visible neuron ht sep blue fill red width cm width line visible neuron rbm backward connection matrix draw pt minimum sep fill fill red fill h cm h dot visible energy rbm rbm pdfs hide connection bias visible rbm conditional pdfs logistic cd gradient parameter rbms ascent mean cd instance em backpropagation derive rbm benefit average specify configuration bernoulli positivity hold rbm
increase programming offer still necessary variational propagation first speed subsequently belief insight symmetry work largely choice message bp recently technique notion group solver framework marginal particular lift tree reweighte formulation far present first follow benefit log partition marginal exchangeability immediately lead bound log serve goal analyze symmetry depend appearance probability quality affect suitably symmetric work symmetric explicit compact optimize wolfe frank map appearance effectively algorithm span tree graph benefit polytope inequalities polytope notably random sometimes characterization polytope elegant call compare demonstrate supplementary detail begin reweighte inference serve discrete assume overcomplete probability approximate domain essential view partition polytope seek tractable polytope formally moment entropy subset sufficient statistic moment span tree entropy span belong span polytope denote appearance bind use polytope constraint consider appearance polytope optimize span step span several derive geometric program guarantee relaxation polytope span tree decomposition essential directly tree frank wolfe iteration follow latter optimize appearance direction find maximum span weight build variational introduce construction structure nevertheless preserve key characteristic exponential polytope log entropy exploit tie cell rise tie family partition formulate compact specifically induce restricted polytope supplementary since jensen substitution appearance every edge edge edge entropy node graph edge simple loop fig encode incidence graph follow element assignment ground polytope constraint term connect compute induction length base statement right front base return ground contain let must since since connect path must node argument must one keep disjoint follow log proof demonstrate polytope variable conceptually configuration substituting variable configuration summation configuration compactly similarly constraint pair c k arrive number way sum edge exchangeability condition precisely obtain equality obtain consider arcs model triangle triangle rx rx rx rx rx rx
relatively restrict square real target common alternative normalize presentation concern gradient base feed forward neural target vector non represent compactly pair target naturally equally dimension allow similarly sparse besides target intermediate denote corresponding matrix letter transpose transpose architecture linear activation usual kind amenable backpropagation e g k b activation equally input tie explicitly bias traditionally place I linear possible old trick component hide update descent need vector operation able gradient backpropagation activation layer row non zero need precisely operation write operation part propagation difficulty final incur prohibitive cost priori reason e sigmoid non would fundamentally extremely suppose reasonably sized compute residual incur cost output one update incur prohibitive operation prohibitive maintain matrix update representation order prohibitive intermediate compactly compute respect w complexity prohibitive rewrite maintain much square update prohibitive computational update generally neither difficulty note decompose update yu yu tu new tu u tu u use formula need thank compute one together prohibitive direct keeping date eq date update date update implicitly translate constraint far perform p p yu k operation state precisely update require roughly require propose change correspond factor speedup access whereas update emphasize simply ordinary prohibitive standard separate update equivalent update straightforwardly minibatch yield need careful keep reasonably minibatch resolve minibatch generalize extended function basically function express compute softmax practice something limited class deep neural language embedding predict incur prohibitive updating weight matrix need backpropagation handling base family function softmax backpropagation remarkably without ever yield speedup least order typical part computation dominate kind network deal large representation arise vocabulary user sparse input
improvement text english text remove token document avoid effectively text need pattern dictionary string token effectively like report dataset english collection political publish anonymous corpus interesting claim text claim text study paper write claim researcher tend alone analyze dataset compose text jointly compute describe perform dendrogram binary represent dimension algorithm report leaf document appear evaluation done cluster branch tree e checking text isolate cutting agree text text place cluster result hierarchical obtain compression algorithm document evident close text rise method concept describe quantify string rely notion perform document automatically text text rest close minimize retrieve author adopt minimize gram gram author gram yield former find gram assign incorrectly among base measure outperform obtain rank competition exclude take furthermore subset english test classification competition problem adopt allow corpus belong author classification criterion assign document competition document correctly recognize comparison meaningful per correct author respectively encourage specific task correct make public former violate thesis without properly eventually title media list page derive attempt analyze page fail separate page satisfactory reasonable able text page thesis show text instance picture come page leave cluster justified page refer work happen author consider page author describe close characterize helpful identify text source evaluate similarity text compression propose compression text document report experiment document write english advantage method apart yield justify firstly meaningful contain document full discarding concern employ dictionary number object drive typical maintain keeping promise collection text english propose dictionary effective dictionary heterogeneous style write different come historical period state art outperform traditional task give challenging document interpret machine style therefore machine one author review report widely employ anonymous decade cluster text universal normalize distance general estimate object concept use histogram retrieve carry outperform state support art report compression base superior reduce respect kind analysis meaningful string instance match report improvement traditional usage automate text satisfactory preprocesse document dataset paper compression validate base similarity classification diverse text define represent compressed version string share usually characterize share common well efficiently generality diverse include text application categorization assignment modify version dictionary author apply text analysis character mark subsequently token character account string successively send point successively encode repeat
hand side notational clutter view relation condition complete comment relate alternative agent reveal candidate agent must distinguish turn cost cost scale gap suggest scale detection average iteration asymptotically tend dependence state network characteristic agent spectral centrality let imagine sense want dispersion evidence favor allocate token adversarial aim delay central regularity recall definition centrality regular large element eq informative procedure suppose agent default determine neighborhood centrality gap assume centrality fix idea markov walk replace generalize modify average network exploit optimize modify large eigenvalue min drawing term occur intersection min network agent add improves assign connection introduce correspond remove add self consider communication definite recall apply eigenvalue definite reduce theorem keep tend connection imply roughly behavior elaborate issue finally positive easily impact interesting existence central preserve diameter scaling lie cycle diameter poor communication affect model distinguish rely private signal identifiable consensus learn truth exponentially learn communication adjust fix parameter gap theoretically proposition agent suffer equally impact failure connection discard network bi directional eliminate view decrease amount plot network behavior almost monotone monotone relationship gap roughly signal state analyze study versus counterpart cost turn centrality affect informative optimize spectral speed learn link failure side effect poor discuss round potentially costly alternatively contact informative enough state direction scenario generalize model appendix elementary keep self contain positivity implicit setting equation fashion calculate kronecker discrete linear therefore matrix stationary markov chain observe recall mind break part sample since simplify require eq get q obtain probability take state use simplify thereby complete follow entail jensen inequality right recall ratio schwarz last line jensen I right conclude axiom conclusion conjecture definition exercise theorem remark summary address agent agent receive private underlie state globally identifiable informative signal literature introduce kullback leibler centralized network centrality relative agent informative central fast optimize speed spectral speed provide recent year burden agent regard range network broad class need therefore exchange dispersion adequate big picture consensus protocol gain grow popularity decade early decentralized detection consider fusion center classical centralize collect recently propose framework govern aim recover agent opinion quantity observe stream private signal likelihood condition state informative agent knowledge build learn finite non propose average bayesian opinion tend truth mild dual demonstrate generate weakly exponential weight posterior convergence present distribute convergence entropy individual analysis rate entropy describe dominant factor time finite gain affect work agent equally expert learning compare substantial agent upper bind conditional enough realize agent endow informative interestingly importance design facilitate interaction demonstrate achieve rate chain consistent natural rapid failure less observe spectral translate diameter diameter make propagation match finding paper formal briefly application conclude transpose vector element simplex th inner norm variation eigenvalue I centralize formal agent seek probability govern space assume marginal uniformly reason bind provide let evident state perspective true globally identifiable state identifiable continue triple space correspond agent receive pair throughout report agent neighborhood construct entry th row th column nonnegative normalization stochastic assume path agent assumption unique eigenvalue magnitude centrality negative eq denote centrality take form w entail chain irreducible motivate distributed scheme introduce detection mirror algorithm scenario could centralized agent global state specifically observe signal reveal nature characteristic round expert simplex let depict uniform rate initialize form belief maximize marginal divergence jensen inequality iff therefore fact set observe parametrize directly neighborhood uniform belief initialize let signal belief measure distribute compare counterpart agent opinion expert total cost incur observe
length ambient preserve additive multiplicative error multiplicative mc curve integrate side also manifold preserve geodesic distance geodesic geodesic q briefly map preserve pairwise follow establish dimension provide since geodesic probability q ambient map approximately volume curve manifold discussion manifold linearization mc contain linear theorem normalize cover divide normalize firstly normalize call short shorthand normalize tangent control intrinsic dimension tangent bundle secondly apart euclidean terminology cover decide apart quantify tangent purpose let observation readily inequalitie q net net sphere let suppose let definition q final step tool smooth reach final inequality trivially satisfy dimensional reach cover respect mc cf far computation improve extend subgaussian map remove inherent essentially research compressive like I thesis definition section exercise definition hausdorff mathematics theory euclidean several obtain case improve structure sparse manifold hilbert dimensional various informally curse several I map preserve certain dimensional possess intrinsic dimension dimensionality seek random linear know direction classical show orthogonal distance simple appear author reduction projection attractive subgaussian let entry mean constant denote small formulate compactly manner general dependence optimal random easy component g set meaning require incorporate dimensionality mixture gaussians manifold match processing square regression compressive compressive match rely extension different quantity intrinsic show subspace smooth motivate investigate subgaussian pairwise distance isometry subgaussian stable manner subgaussian e g therein development isometry subgaussian various isometry subgaussian matrix act matrix use substitute isometry unify restrict isometry subgaussian matrix additive counterpart formulate master subgaussian map act new bind focus make non demonstrate extensively extract result concrete known matrix subgaussian class contain efficient subgaussian certain processing application overview include subspace example hilbert variety express polynomial rate fashion type embed union space dimension subspace set measure principal subspace upon recent discussion main deduce reconstruct subgaussian thresholding deduce reduction smooth length preserve uniformly pairwise ambient preserve manifold first result linearization dimension subgaussian error terminology space random center variance subgaussian subgaussian subgaussian call fix norm let metric diameter linear hilbert let cardinality write width semi play role semi I slightly theory generic suppose subgaussian increment metric eq special process metric coincide translate cover small ball call let q finite reverse fail even time analysis empirical early database densely generally readily subgaussian map dimensionality distance preserve restrict isometry normalize preserve multiplicative say corollary radius mc subgaussian particular eq sphere parametrization width refine result obtain make contain sphere easily cover relax map isotropic parametrization set subgaussian expect concrete technical work dimensionality space q often universal say cover unit ball know argument integer cover dimension dimensionality reduction dimension short estimate statement result suppose cover dimension cover subgaussian map ik corollary readily deduce net assumption use arrive illustrate vector rip derive isometry subgaussian result sense let restrict isometry note since subset optimal rip signal sparse linear become count accordingly pn l imply upon explain expect upper lead guarantee recovery subgaussian matrix rip new restrict isometry map property role recovery play sense information define constant covering first subgaussian map tensor rip extend norm metric let isometry notation cover corollary originally subgaussian four elementary net respectively generic necessary signal hilbert signal piecewise overlap prove hilbert parameter natural consider metric subspace angle principal angle define subgaussian apply suitable parametrization h q admissible
irrelevant address p ty classification model intercept induce desirable encode interpretability relate control interpretability relate interpretability specify interpretability iii interpretability training example formulate program ip let objective approximation rate model predictive interpretability attain trade interpretability model attain vice versa convex regularization interpretability produce linear model use interpretability train create state rule classifier coefficient numerical worse induce direct interpretability producing represent example parameter accuracy unit optimal accuracy produce interpretability attain possible value regularization restrict interpretability attain guarantee interpretability set yield equivalence increase replace hinge polynomial ip discrete interpretability interpretability high robustness meaningful training probability estimate case use loss significant digit produce feature order magnitude coefficient attention unit feature coefficient expert kind relationship integer index indice non positive index either define pos j j ip formulation constrain narrow feasible region ip predictive sometimes model weight consider wish yield also training contain drop value adjust use adjustment drop see adjustment ensure value interpretability tie composition grain term number tune composition input alternatively impose include encode complicated encode require leave vast world produce trivial heart classifier predict heart attack handle sensitive positively label adjust respectively generality benefit train weight correctly example attain level accuracy benefit specificity expert sensitivity specificity encode model shoot accurate error label optimization aim positively correctly expense constraint prevent attain high single shot procedure intervention budget find attain high suppose budget action train predict aim positively accurately optimization aim secondary attain intervention budget different kind interpretable produce pair formulation adapt loss switching constraint decomposition scoring allow quick prediction subtract multiply number assess medical outcome simplify value score difficult reproduce system expert le principled scoring eq produce create system restrict interpretability tune restrict value great neither interpretability influence term classifier minimize make add yield coefficient denominator regularization restrict set classifier give value restrict coefficient without function n p j norm variable variable constraint big formulation parameter score misclassifie margin implicit feature interpretability absolute coefficient integer fine grain interpretability user set interpretability penalty coefficient heavily exclusive interpretable iii parameter monotonically train gain require example loss penalty gain accuracy k u r j big identical binary coefficient interpretability constraint coefficient assign constraint interpretability rule classification compose feature convert rule threshold rule model data rule assumption binary categorical exist j jt j threshold feature need threshold place rule rule notation rule regular thus give rule benefit mathematical expression least rule rule fold cv create ip n c constraint big identical ip interpretability penalty penalty scoring version suit outcome medical patient intensive optimize real feature feature selection exhaustive construct fine grain rule classifier feature rule constraint rule ensure binary agree improve interpretability ensure maintain monotonically outcome system follow formulation p p j j constraints big ip section f interpretability include binary value enhance scalability consider generic framework would ip variable constraint formulation however cut scale recent benefit relate stand aggregate setup solver different ip solver individual loss original optimization iterative oracle piecewise approximation iteration piecewise linear aggregate solution computation addition I linear solver repeatedly solve present popular decomposition initialize proxy aggregate cutting support subgradient aggregate clarity add cutting constraint proxy create aggregate plane grey linear loss form lie collection cut loss let function notice piecewise true point feasible function combine lb tolerance gap optimization z lb lb k kk trade accuracy minimize attain definition know control accuracy increase train basic penalty include interpretability train classifier comparison train case attain training range unit maintain difficulty also coefficient train run run ip take decomposition classifier loss feasible severe time summarize experiment setting ghz gb ram logistic runtime compute cut multiplication produce optimal classifier second train loss loss impose classifier train classifier train higher imply classifier impose severe ip solver purpose htbp trade scalability convex ip solver produce formulate even loss decomposition benefit substantial control mean discrete interpretability discrete interpretability loss well suited scale compute cutting cut operation decomposition polynomially exponential cutting performance scalability add cut geometrically center chebyshev center procedure train robust give solve proxy determine example computational provide work filter train proxy objective set classifier proxy datum proxy proxy feasible enough f denote proxy f relate choose filter helpful reducing follow stage reduction proxy compute level set I I stage proxy variant proxy contain force classify obtain exceed upper set contain dataset know optimal original datum provide proxy loss use determine enough loss alternatively condition proxy htbp denote solution original solution train variant force way convex proxy ni n I train proxy large eq appendix satisfy level proxy function p optimizer optimizer proxy proxy follow c z satisfy condition require proxy avoid proxy reduction relaxation ip hull convex relaxation ip proxy enough satisfy let ip optimal ip inequality follow ip feasible solution ip thus use feasible solution ip determine figure demonstrate datum train dataset specifically filter level instance proxy convex relaxation ip feasible correspond value compute user guess e proportion filter increase amount filter high keep mind attain htbp reduction computation include reduction use preliminary procedure ip even conjunction situation proxy reduction proxy similar impose even fundamentally branch particular reduce feasible label branch optimality impose branch effectively exploit linear coefficient g training value margin resolution train large achieve loss equal contain classifier attain procedure attain low optimize directly simultaneously round choose solution produce discretization bound consider use margin margin ki large magnitude proof apply theorem show discretize easily motivate discretized definition discretize solution principle structural minimization guarantee follow important linear classifier obeys proof hoeffding lead motivation discrete loss increase indicate large significant digit notable benefit generalization exclude suboptimal hypothesis discrete minimize function feature principle generalization minimize bound integer n generalization classifier coefficient every minimizer dataset size regularization minimizer obeys translate generalization relate apply objective bound integer vector penalty refine term point classifier obey argument integer relative due classifier improvement rule small left generalization clinical tool demonstrate flexibility real world tailor part laboratory contain patient feature health patient patient upper significant imbalance pr would model clinical list cm high positive maintaining ensure enough explain short relationship factor incidence suggest patient risk appropriate model training requirement parameter limit maximum optimization section process constraint classifier would establish datum fold cv solve hour gb ram summarize table imbalance use varied sensitivity range see explore setting inherently method varied mix dropping requirement violate one fold cv among ptc parameter cart lin rbf value provide flexibility method table ht lr lr ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc lin ptc ptc ptc ptc ptc ptc rbf ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc cart ptc ptc ptc ptc ptc ptc ptc ptc three requirement give base cart rule unable produce method svm lin rbf ridge unable regularization achieve require unable issue satisfy requirement tailor model expect able max net cart monotonicity control net highlight art pt accommodate reasonable crucial mechanism adjust sparsity incorporate constraint poor possible control accommodate reasonable allow control indirect require free extensive find cart max figure tuning consider standard case fold obeys final unfortunately requirement framework encode interpretability ptc instance ptc sign lasso lin cart plot classifier fold right highlight produce cart tune satisfy max two method acceptable sensitivity minimize penalty surrogate hold sensitivity sparsity train figure also performance ridge attain fit small linear integer vs coefficient final unable produce align constraint interpretability benefit coefficient understand relationship easier validate example htbp mml lasso mml htbp index mass point tv solve minute run numerical various popular uci repository comparison method varied nature process mn include include opposite htbp predict breast cancer predict go predict year patient breast cancer patient heart breast cancer predict mail spam summarize setup matlab art baseline package rule suited design interpretability assess via assess interpretability mn allocate minute ip ghz processor ram thus take hour aim run without constraint free hypothesis use restrict roughly rule zero rule contain coefficient coefficient contain coefficient set coefficient dataset attain accuracy htbp tree default tree c default setting lars lars ridge binomial link lar binomial link e net value svm lin rbf scoring np call interpretability different coefficient lasso e mn rule lin leave decision cart rule box svm feature relate interpretability visual table plot figure correspond cv error median fold cv test cv max size variation fold vertical horizontal size ridge net box line coincide path plot figure sparsity mn produced use feature g accurate use set likely coefficient attain training htbp l ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc range ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc size ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc function issue mn rule dataset feasible minute setting far provide optimality train proof necessarily binary mathematical overfitte minute plot unable attain model accurate regularization mn limited dataset correct accuracy possible mn rule measure sparsity restrict mn model mn mn rules particular know arguably happen path provide focused interpretability nice comparison attain perfect accuracy mn cart attain fold cv note able attain omit scoring compare cart lasso highly compare lasso cart structure note round round produce good one round difficult one randomized round require rounding attain perfect rule rule g anti rule create instance point domain unify lr lr htbp lr lr model cv htbp mml point categorical model categorical variable interpretability crucial aspect predictive help practitioner control important show create type design scalability use concept lastly present extensive illustrate art major method avoid achieve interpretability approximation interpretability long dataset integer programming software sometimes thousand integer practitioner allow periodic improvement code pool interpretable include section less point remove ii follow trivially follow proxy iv proof z inequality look rh inequality use get iv rhs inequality incorrect plug lipschitz ii satisfied set invariant define wise difference margin follow case lie margin margin margin trivially margin calculation whenever analogous calculation I I whenever put feasible optimization nz integer thus contain minimizer I htbp produce cut aggregate loss overview decomposition converge ip solver provide feasible oracle second cut aggregate able unable set requirement htbp mml dataset htbp mml mml htbp htbp lift lt lt lift lift lift lift lift lift class lift heart lift college lift class lift age class lift age lt lift sometimes heart abuse lift lift rule lift rule lift lt shift work lift lift lift lt age lt ess ess lt sometimes
whose solve see far increment value one know theoretically important store decide majority often refer majority storing reduce increase size polynomially expressive aware expressive c require compute separation increase result strong exist size imply efficiently except product circuit theoretic use give separation natural simple beyond provide simplified serve technique advanced later section powerful able prove address pose notation standard index row index context communication tool depth b fx negativity apply separate expressive depth define define function compute layer meanwhile easily result interpret easily communication pose whether expressive former require answer technical perturb identity associate equal satisfy obey must layer prove use become efficient depth depth node child efficiently computable state input univariate consist c depth super previously expressive show depth particular know various layer counting boltzmann efficiently capture capture deep boltzmann hierarchy property analogous boltzmann sigmoid belief never prove hold trivial rigorously gain expressive layer make analogous slight multilinear circuit monotone multilinear arithmetic circuit multilinear arithmetic circuit compute original arithmetic circuit construct circuit monotone confirm author circuit serious multilinear arithmetic circuit circuit capability multilinear circuit certain imply thing issue value exploit multilinear arithmetic decomposable value deal hardness certain circuit multilinear establish multilinear polynomial proof theorem give polynomially sized c beyond depth grow multilinear circuit multilinear circuit size depth multilinear function transformation preserve multilinear circuit multilinear circuit slightly arithmetic circuit apply analogous statement turn issue adapt multilinear circuit monotone multilinear circuit circuit small multilinear circuit automatically property equivalence monotone multilinear circuit end size arbitrary compute size give distinguished circuit fan formula restriction expressive answer turn indeed formula input real c compute consist size polynomial slight prove context multilinear circuit value monotone multilinear circuit node multilinear formula compute arithmetic circuit circuit confirm author hold circuit try theorem prove encounter multilinear circuit also apply existence possibly unnormalized imply density theoretic major drawback unlikely efficiently density know tractable counter provably depth notably boolean circuit size simulation capture probabilistic like notably proof theoretic might distribution model like network turn arithmetic circuit efficiently approximate boolean circuit approximate simple simulation efficiently reasonable efficiently polynomial size prop net arithmetic circuit distribution indicator absence label particular take edge present span denote computing formula amount graph span tree otherwise decide span tree check connect exactly efficiently solve boolean circuit trick adjacency solve add neural threshold simulate boolean circuit manner similar depth easily basic computable constant boolean circuit capture small kl divergence boltzmann pr sequence sequence also sigmoid depth circuit surprising remarkable corresponding variable compute possible computing reduce span tree set turn directly reduce derive laplacian argument formalize corresponding value label span main result follow arbitrarily arbitrarily univariate size trivially approximate arbitrarily large denominator purpose simply decompose weak function show output theorem sum weak must exponentially thus weak fraction avoid span polynomial negative weak relatively equal sized suppose negative polynomial multilinear circuit characterization compute exactly compute show arbitrarily analogous theorem univariate different sequence polynomial real value negative tuple satisfying eqn particular must prove show term span weak sum non start simple negativity factor intuitively requirement input think span either value vote always go pair factor reach incorrect despite status span cycle might visible potential cycle produce incorrect yes force essentially characterize conservative voting showing situation span tree receive yes vote real value form describe dx hz particular devote edge red graph vertex triangle one color triangle triangle reason soon clearly triangle simple cycle determine edge triangle impossible neither gets jointly triangle triangle vote whenever edge vote whenever formalize property value constraint triangle rise triangle color graph red total triangle bound lemma triangle choice conclude least triangle graph remain proportion consider span perform walk add walk subsequence sample one formalize suppose span proportion tell proportion span prove however separate span tree complete capture learnable simple tractable separate extend natural way capture care limitation want fundamentally character much statement indeed worth expressive thorough available nature amenable use avoid various exist circuit capable efficiently circuit seem fall open question tractable allow strong statement expressive acknowledgment like thank run helpful regard multilinear circuit support google result transform proceed process child process node compute normalize process node constant divide incoming leaf computing divide compute density product normalize soon child effect sum parent equivalent divide output recursively recursive fail divide normalize leave unchanged multiply incoming processing note income fact denote circuit node product non root node weight linear combination negative polynomial base trivial depth circuit non child scope coincide make child product consist node compute univariate constant node modification tuple modification constant observe add compute replace evaluated underlie circuit node aforementione replace child node conclude complete decomposable modify exist add child dependency theorem multilinear valid multilinear happen depend dependency scope univariate finite choose jx I jx jx polynomial combine positivity fact say term term collecting q use zero polynomial polynomial finitely root see multilinear incorrect evident degeneracy node product expand yield polynomial expression whose term give possible multiply collect sum product positive expansion weighted sum union collect weight degeneracy describe appear statement consider non polynomial sum label compute member scope element induction depth scope suppose member scope member inductive hypothesis mean element part factor element sum node root child induction depth case node overlap least product lemma set say twice thus scope multilinear multilinear circuit hypothesis amount establish multilinear multilinear happen polynomial part least factor member contradict way fail multilinear none scope previous set contradict scope scope distinct scope scope part product case contradict scope identical multilinear multilinear multilinear inductive hypothesis establish multilinear case set multilinear factor might twice contradict way fail multilinear element scope statement hold reverse depth depth multilinear inductive multilinear root syntactic disjoint respective disjoint well multilinear member scope member product form union union equal scope equal scope part multilinear multilinear consist product member lemma union appear exactly similarly scope multilinear node element verify co complete extend informally idea boolean sized output represent satisfie add evaluate output output whenever assignment proceed formal formula associate boolean natural label node count definition input form distinguished give node note accomplish polynomial moreover satisfy construct value validity require rhs validity circuit implement vector node multiplication implement product child component work entry node take node row final working hard produce decomposable function scope process proposition respect transition definition state kind transformation vector elsewhere realize obvious identically weight univariate variable compute claim index index weight sum b fx k iw layer pair obvious univariate sum node compute second layer compute x I proportional rearrange diagonal differ satisfy plug fx I semidefinite root symmetric root nuclear trace unitary matrix root root unitary unitary q eigenvalue equal thus fact multilinear polynomial proceed induction input hold trivially suppose multilinear polynomial multilinear multilinear polynomial eqn inductive proof multilinear circuit product discussion equivalent decomposable decomposable decomposable contradiction depth multilinear arithmetic circuit univariate affine add sum circuit remain multilinear circuit agree contradict multilinear arithmetic circuit size compute decomposable univariate decomposable decomposable suppose compute multilinear arithmetic univariate binary input replace add formula circuit clearly moreover circuit formula multilinear polynomial polynomial agree contradict proceed whose acyclic subgraph proceed label connect edge label vertex span tree span take label disjoint clearly consistent hard spanning construct subgraph edge edge subgraph clearly cause cycle edge span span form eq determinant asymptotic multiplication bad equal tree maximum accomplish replace fan structure scope replace pruning note circuit decomposable complete empty circuit stop circuit polynomial remove express output node remove stage input scope analogously depend completeness clearly consist precisely theorem form non polynomial thus define eqn choose give start root path child size dependency current arrive scope child child scope due child scope never path must become scope let apply polynomial satisfy theorem eqn finitely integer finitely dependency one infinitely often sequence independent dependency subscript subsequence forward whenever replace subsequence I x real dimensional bound sequence bound finitely subsequence replace subsequence converge converge write function resp iy ix complete sequence value resp disjoint converge pointwise subsequence zero replace subsequence subsequence exist zero select finite form convergent subsequence wise remain many infinitely maximize infinitely infinitely often replace subsequence subsequence certainly since wise proof contradiction contain value agree contain constraint triangle span follow contradiction lemma triangle total triangle say triangle upper bounded form take edge form number triangle original triangle original colored attain n triangle arrive triangle tree uniform obeys constraint analyze behavior due sample iterate step vertex current vertex visit edge terminate exposition minor algorithm distribution particularly sake simplicity vertex produce stage stage triple allow call triple pair triple convenience encode triple vertex give distinct triple consecutive encounter triple easy triple visit visit begin visit triple triple correspond constraint triple neither visit condition choice algorithm pick stage constraint triple occur stage choice constraint occur stage upper bounded plug setup theorem circuit sufficient call impose circuit typically intractable generative amount capability understand multilinear circuit question depth various establishe existence capture depth capture generative additional layer distribution capture grow kind hierarchy property never prove contribution include condition sufficient validity various type recently class generative unnormalized density function circuit arithmetic circuit feed connection arithmetic circuit input special valid normalize deep crucial evaluation provably intractable valid practical perspective validity call c impose structural limit kind extent expressive course expressive expressive efficiency emphasize distribution capture super instance latter question paper capture simulate unclear capture exponential tractable computable marginal etc come marginal intractable nonetheless correspond capture deep g indeed hard practice capture efficient general point obvious question density capture deep complexity theoretic establish sense many attention tractable could c analyze depth characteristic exist arithmetic theory gain expressive efficiency c despite expressive also numerous theoretical propose definition connection multilinear arithmetic circuit exploit powerful latter section provide insight relationship validity validity merely also generalize validity co np base use constructive efficiently efficiency give short technique circuit theory go answer open pose leverage multilinear arithmetic circuit prove powerful regard depth expressive extra depth computable thus give strict depth next grow expressive greatly reach recursive learn expressive multilinear circuit efficiently compute existence capture even approximated circuit circuit logic circuit neural network instead operation operation arithmetic formal define type acyclic follow either every known incoming edge arithmetic circuit go child arithmetic circuit value follow rule nod child weight polynomial singular take circuit node define circuit essentially depend node depth path general circuit allow quantity compute computation write compactly give generalized sum take value f f I univariate integral tuple whose arithmetic circuit property monotone arithmetic circuit gain arithmetic circuit node compute polynomial view set tuple define member denote primarily give normalize constant also degree represent subset integration amount summation treat deep density machine partition function density marginal accomplish completeness say let I ia I jx ix jx decode say respect corresponding integral say integral range intractable valid efficiently integrate respect allow f purely make say valid identity fundamental validity strongly multilinear associate trivial essentially think without circuit note reverse example compute count measure multilinear inspection easy strongly valid interesting complete validity purely application paper prove equivalence validity note validity compute way count neither decomposable show choice equivalence validity completeness need degeneracy circuit positive size procedure degenerate arithmetic circuit degenerate preserve structural completeness remove remove node fan except fan sure node node output degeneracy validity consider compute compute condition never degeneracy many non monotone circuit root child monotone arithmetic circuit equal form note circuit polynomial set scope scope degenerate circuit multilinear multilinear utilizing
domain domain control trade solver sgd saddle saddle stochastic time stochastic feed comprise fed factor difference stochastic descent domain minimize loss update learn fortunately layer layer apart meta act subsequent pass precede implement exist object package multiply nothing domain architecture depict result model converge mathematically describe backpropagation identity matrix objective optimize stochastic implement domain predictor use predict derive adaptation define distribution depend particular representation space family enough pick concatenation predictor layer perceptron aim assumption easily train closely relate backpropagation become small cm cm ccc cm ccc sign source mnist source sign mnist mnist source train digits background correspond e correspond domain bind da cover five considerably big portion gap amazon l sa l l last art extensive image office dataset standard vision much data branch include reveal serve assume addition new dataset comparison da boost performance baseline principal target sa activation classifier descriptor learn domain train new adapting compare label performance remain office recent da approach publish cnn general convolutional layer pick exact architecture stick use loss train batch image domain rest comprise domain instead fix gradually schedule experiment optimize detail cnn find visualize distribution digit window digit orientation color variation manually however rather difference structure clutter background propose backpropagation work target sa accuracy adaptation task challenging mnist mnist gap adaptation stay epoch avoid learn anneal direction equally diverse appearance observe separation feed solely mnist probably explain improve mnist scenario see opposite direction adaptation unsupervise mnist unsupervise da capable adaptation sign experiment sign simulate increase evaluate domain additionally split train set part solely evaluation procedure slightly predictor target suggest thorough verification office evaluate office collection three distinct domain amazon unlike previously office rather spread amount available crucial fine cnn pre imagenet recent da exactly architecture domain work transfer task image per unlabeled abundance target domain set art unsupervise adaptation amazon scenario adaptation feed forward architecture amount domain da adaptation alignment accomplish backpropagation approach scalable deep plan usage evaluation supervise constitute work interesting autoencoder deconvolution effectively inspire lead update introduce minimization discrimination loss constitute special domain entropy predictor adversarial loss label gradient domain result rgb architecture experiment mnist inspire single cnn pre train office bottleneck domain branch somewhat adaptation attain architecture momentum anneal eq q progress linearly schedule optimize domain dropout train perform massive amount label datum absence attractive label domain propose architecture datum amount unlabele feature shift augment architecture overall implement deep package perform experiment presence big shift office feed architecture advance wide performance label training set large scale time abundance fully label training approach adaptation suggest approach mapping domain domain compose adaptation mapping target datum either fully annotation semi focus hard although generalize supervise straightforwardly previous work feature combine adaptation training deep adaptation process decision base feature domain distribution feed target shift invariance optimize discriminative classifier label domain optimize optimize minimize classifier domain classifier latter encourage domain feature course green deep backpropagation training proceed minimize gradient indistinguishable result invariant crucially process embed compose feed forward figure use layer loss backpropagation modification g sgd momentum generic add architecture backpropagation component propose rather trivial layer unchanged propagation multiply negative backpropagation adaptation office benchmark considerably previous accuracy source approach select seek map target way reproduce whereas axis accomplish modify representation geometric rather separability deep several approach source among sequence autoencoder source domain approach train domain predictor separate autoencoder feature adaptation unified architecture argue conceptually implementation considerably office benchmark approach domain target context deep feed architecture fine network train require domain quite network measure minimize discrepancy discrepancy finally focus domain adaptation feed network mean may regard seek tight assume work space problem finite generic handle feed handle exist refer
algebraic technique selection use dominant feature improve capability compare classic art scheme technique focus learner approximate classifier generalization base approach one learner apply train excellent introduction method randomize train subset selection scheme boost whole mapping literature forest subsampling model weak learner high author extend ensemble every subset proposal leave future randomization strict simplicity randomization bad accuracy insensitive see classifier virtue propose significance similar propose apply blind randomization learner illustrative inherent interpretation capability explicit classifier non selection apply leave research tp build train decision classifier restrict bag utilize conventional indicate process uniformly utilize object feature two feature next feed accord generate tree collection majority voting apply derive return frequent decision sampling column let nn svd singular column normalize define denote entry singular highlight slow generally strategy randomized score include section experimentally two variant case replace subtle distinction construction utilize randomness split uniformly depend htp publicly handwritten digit people namely case multivariate highly challenge point feature heavily usage benchmark impact leverage result svd rank bagging algorithm default matlab b report htp c accuracy training pair behind requirement superior superior small case leverage less training interpretability limit feature first increase processing time improvement accurate rf computationally depict accuracy moreover well need among predefine time second hour complexity memory maintain feature selection study strategy time efficiency classification indicate tree state scheme interpretability least study effectiveness randomize experimentally feature selection feature experimental evaluation naive forest massive information development scientific ever contradict high feature space proper description easily interpretability curse pose qualitative noise difficulty abstract q learn j j low case report literature classification random due accumulation poorly information removal gain classification fortunately important removal dna influential usually prohibitive storage requirement post snapshot available storage ambient
point fairly raise choice fair dimension kullback kl remain mutual mi remain mi fundamental theoretic via low variant hard easier let kl quantity data mmd mi suggest behavior kl mi stay aforementioned fair kl easy irrespective fair variety distance decay increase alternative characteristic kernel invariant gaussian kx distance choice median always represent biased keep choose power decay bandwidth choice interestingly median univariate laplace variance center choice keep experiment decay bandwidth heuristic maximize choice origin center gaussian covariance say keep number constant dimension verify keep constant encode really try detect accurate choice keep kl fig mmd vs keep calculate aforementioned example b compare polynomial kl mmd actually shrink polynomially bring especially calculation unlike choice direct early aim bandwidth prove look involve taylor simplify affect choice qualitative constant clarity corollary ignore residual population go zero exponentially fig median median heuristic polynomially mmd hope kl bandwidth polynomially mmd smaller demonstrate approximation actually calculate mmd bandwidth population mmd mmd separate previous median maximize mmd present exponentially mmd expression laplace kernel bandwidth accuracy verify use bandwidth median heuristic experimentally verify drop exponentially make small correct bandwidth optimal make bandwidth polynomial mmd vs laplace kernel panel relate aim approximation expression verified verify tr taylor theorem calculation mmd qualitatively vs kernel panel understand various proposal fair alternative bias independence understand popular drop polynomially constant zero zero modern dimension completely behave acknowledgement nsf grant look mmd calculation case characterization mmd invariant kernel laplace kernel translation kernel definition kx iw iw fourier invariant substitute consider change substitute get proof proposition dx example dx dx obtain thereby lead integrate part equality follow manner kernel decompose step substitute equation get bandwidth taylor q tr approximated b median heuristic optimal corollary derivation proposition taylor order get interpretable formulae result demonstrate compare mmd sample mmd mmd observe quite thereby previous unbiased mmd empirically prove decrease bias mmd bias mmd decrease fashion mmd observation machine pa usa nonparametric solution distance behave well high source give rise specifically hardness statistic zero fair hypothesis actually drop dimension fair light bandwidth advance test independence testing sample determine sample algorithm homogeneity draw sample draw distribution marginal r problem parametric class quantity kernel introduce approach parallel testing introduce far summarize subsection identify address normal mean estimate hard whether zero statistic behavior low independent nonzero get hard test fair set decay dimension current mathematical solid completely formally rkh correspond statistic kx kx subscript bias exclude call every statement qualitatively matrix matrix subscript suggest quantity characteristic test provide insight test test elaborate error equivalently power also dimension work well understood type people aforementione hypothesis test outline understand distance claim presentation word claim worse get dimension unfortunately contrary base method introduction lead dimension
combinatorial search proportional manifold dimension ask polynomial complexity demand manifold sometimes ask solver exist barrier recover work theory counter manifold low play role many use treat ambient low dimensional manifold manifold union signal dimensional manifold may make exactly essence completion unknown measurement random low synthesis model signal zero representation group column uniquely recover possible recover manifold however feasible practical technique subgaussian partial measurement number need measurement easily result guarantee recovery turn roughly noise synthesis compress signal low one abstract reconstruction manifold sparsity framework looks apply example vertical horizontal signal becomes represent note hereafter tv g vertical finite define follow addition mod correspondingly thus stack vertical difference analysis model assume characterize denote signal entry denote unknown submatrix row subspace recent reconstruct random position orthogonal row dimensional recover signal ambient thus natural matrix norm count recover np hard gap theory tractable stable recovery combinatorial noiseless utilize exist dimension unless manifold reconstruct dictionary vertical horizontal matter theorems suppose subspace theorem prove combine without huge implicit remarkably efficacy measurement exist reconstruction demonstrate size signal dimension proof technique operator discuss implication theorem hypothesis pack radius maximal eq p piece lemma last admit large packing reconstruct pair pack reduce indistinguishable classical number lemma follow radius pack finite argument begin signal finite q worst suitably trick packing pick estimator side lower minimize highest straightforward relate prove cone packing suppose begin rescale note restrict reliably packing otherwise estimator project word divide side finish metric signal noiseless fail characterize noisy notion idea provide indeed le many problem subspace know proportional geometry state recovery measurement dimension fast pack proposition htb attribute demonstrate look experiment present signal ambient combination interestingly instability indeed dimension recovery heavily soon reconstruct increase htb color attribute square reconstruction additive white standard energy color bottom upper exponentially number decrease become error function correspond minus value synthesis suffice percent correspond measurement behavior quite soon accord increase become become attribute square htb square error manifold connect gray generate image pick pixel start walk assign stop pixel visit result image
penalty value aic penalty number mse blue case mse penalty position similar penalty lot case mis look correct false positive parameter detect lead positive detect choose detect far thought look versus approach value show figure circle could point near also see segmentation segment look sensible illustrate firstly neighbourhood efficiently mis specify something change independent identically analyse let vary mis let slowly mis simulate value simulate segment set distribute constraint set linearly normal mean firstly calculation segment set maximum evident optimisation improvement run neighbourhood datum speed cost without similar general gain red dash neighbourhood bottom mis middle sublinear mis approach efficiently criterion aic detect range range simulate estimate infer within positive positive actual measure proportion detect positive divide positive detect evaluate accuracy segment parameter segment separately true
use ensemble read theorem proposition matrix call subset basically model definite proposition summarize prove dpp kernel dpp dpp condition answer give eq know quadratic eigenvalue positive definite definite q technical conditional positive definite definite matrix hard semi therefore follow definite negative prove positive semi definite positive point process ab corollary ab dpp claim ab b corollary kernel disjoint know q since summarize ab however clearly b product condition dpp definite positive c q process subset claim know ab ab ab cb subset true generalization kernel disjoint follow statement k thing j look covariance dpp briefly graphical process definite give result subset pairwise disjoint subset separate vertex consequence fact markov follow graph disjoint separate markov ensemble ensemble disjoint separate zero place give disjoint separate independent separate kernel
hamiltonian hmc use hmc mainly generate correlate might implement software run discard half sample element sale coarse comparable acceptance across different run take second proposal single tb third concern heterogeneous hierarchical parameter identify package single drug sale drug datum conditional sale sale often binomial poisson per week depend sale heterogeneous contact depend vector intercept population I weakly informative row wishart estimate slightly sale period trend change run baseline double averaging adaptively set hessian implement hmc code gradient playing chain period search recommend posterior initialize trace plot chain iteration panel appear converge autocorrelation progress summarize population final hmc million inference tb tb ht problem bad step reverse regardless hmc resource hmc million evaluation assume sufficient small collect proposal evaluation acceptance however time low mcmc implementation single mac sample collect sample draw hour core reduce minute discuss scalability processing collect parallel give could individually could collect even confirm converged draw inference method density hmc hmc chain density level hmc baseline standard quantile decomposition close convergence parameter align little movement chain thus infer effort tb ability parallel attractive mcmc present favor scalability grow analysis computational compute posterior hessian hessian show achieve hessian pattern independence heterogeneous homogeneous component level add summation grow subsequent number together might hessian namely estimate matrix ideally would code require code gradient would fast yet either evaluation posterior log linearly gradient estimate hessian grow gradient accumulate numerical large hessian estimate ad algorithmic ad refer put ad treat composite practical ad involve code keep track derivative compute log ad generate additional return order package also access option remarkable feature ad scalar five package storage dense format store regardless precision dataset model ram furthermore effort quadratic cubic extent operation mode find computational grow multiply multiply matrix efficiency hessian instead cholesky cubic source scalability sparsity system add since cost triangular system solve triangular would grow cholesky benefit sparsity square hold order one nonzero sparsity next nonzero grow element add average nonzero cholesky linear cholesky decomposition store week period visit single week covariate vary true weakly informative average across replication expect grow matrix compute cholesky multiplying triangular standard triangular tb table factor acceptance say acceptance dataset could influence expect fast confident overall acc data additional method marginal output acceptance consistent easy compute solely importance method pseudo arise hull define demonstrate although effort collect probability give proposal accept therefore equation rearrange available mean observe empirical support proportion estimator regression use truth conduct simulated number number covariate simulate include intercept iid density correspond plus per density number draw hessian proposal exclude proposal present sampling harmonic include mean remarkably improve draw offer negligible note comparable harmonic compute fall input algorithm importance sampling scale sd sd sd acc spend long hour deal mcmc utility alternative algorithm appeal mcmc converge heterogeneous conditionally sparsity construct sampling grow unit attractive practitioner guarantee generate perfect target concern discretization could posterior increase proposal expense computation possibly experience find collect depend density posterior mode mode might determine manual search find proposal little time proposal gradually principle metropolis however advantage begin contrast tune apparent substantial appear collect try many multinomial closed form carlo integration augmentation advance parallelization might make numerical integral year efficient augmentation kind multiple suffer weakly missing could treat implication require involve multimodal posterior find mode posterior multimodal normal idea find mode one mode remain unchanged match recognize guarantee amount practical unimodal care optimizer stop reach optimum package gradient number package language method package rejection take gradient user information author acknowledge suggestion comment date date integrate science hierarchical capture unobserved heterogeneity unit markov outcome develop bayesian parallel chain autocorrelation conditionally make applicable use likelihood little management practically application course picture impact method multiple natural capture heterogeneity customer type constrain source grow salient familiar parameter question idea popularity markov monte carlo sampler involve block unknown theoretically iteration early generate difficult reader hundred reference despite mcmc remain start estimation chain converge hierarchical heterogeneous unit customer preference cycle size data outcome multiple magnitude require yet reason procedure practitioner let day chain answer end processing system technology solution outcome collect core processor also question indeed ensure chain converge hand envelope rejection impractical small inspire mcmc sampling normal mode traditional rejection unnormalize distribution maxima derive conjugacy requirement share effectiveness broad scaling proposal draw algorithm sampling intractable hessian density determinant efficiency fortunately several project scalability heterogeneous bayesian probability observe involve numerically prior note product key issue relevant limitation may require independence assumption scalable nevertheless effort involve joint unnormalized posterior factored index mix serve level likelihood include factor probably kind optimizer trust mode substitution write target posterior restriction least restriction later auxiliary yu simulating candidate satisfy comes write differently involve sampling sampling definition get need estimate simulate proportional kind prior polynomial extremely bernstein polynomial effectively repeatedly proposal cdf cdf draw order proposal become accurate proportional partition segment fall multinomial weight continuous exponential sampling finally draw criterion save sample true rv marginal restriction negligible must accept high high density choice multivariate proposal kind density researcher density negative hessian asymptotic order multiplying covariance valid mode scale three panel potential proposal leave panel log posterior tail middle right panel multiply sample tail however proposal unlikely would algorithm stop try density little inference make believe relative case good application nothing implement manual selection metropolis hasting tail proposal multivariate fail mode quite rejection distribution proposal posterior proposal advantage exact proposal ratio extremely deviation mode acceptance accept discrete exchange non call direct remove concern sample target acceptance sampling bayesian mcmc term resource implementation however direct suffer another even moderately sized large et al consider allow conduct without conjugacy mcmc important feature method several direct focus shape concerned characteristic bernstein draw may proposal ideally already traditional collect parallel issue autocorrelation discussion improvement
break construction equal l evy gamma poisson bound limit matrix independently draw gamma stick sampling marginalization baseline negative task review york times corpora exposition recursive stick break like extension integral give constructive definition discuss gamma far normalize employ model model recent examine negative beta break readily beta beta stick variational though scalability problem disjoint borel subset completely take countable process algebra measure measure measurable kp cp c rate mass respective ensure beta sum finite mass base completely component create atomic measure disjoint normalization measure increment stick break break v break round fraction stick broken piece surely dirichlet stick like construction technique simple process stick break construction mark modify stick break beta gamma poisson process define derive marked process formulae stick break stick break construction beta random stick broken piece weight atom weight along somewhat practice gamma also prove correctness gamma may construct ij poisson probability superposition tell countable poisson measure denote ij c ascent gradient two document corpus loading count document put dirichlet column variational get multiply hyperparameter setup ascent describe first variational nk lower nk corpus initializations rate gamma affect indicator round indicator hyperparameter latent prior kt integrate technique sampling break monte technique g kl integral stick take normalize infeasible evaluate normalize value consider model document count matrix loading count poisson gamma dirichlet sampling element count kn z kn kn relationship multinomial nd condition conditional distribution numerator denominator sample integrate break improper sampling indicator round atom describe poisson posterior c calculate carlo fall draw exactly pc discretize carlo technique multinomial document corpora count document vocabulary model derive compare avoid infer poisson monte affect indicator indicator update prior hereafter ibp put symmetric prior column add variational variable hold time expectation warm gamma stick break atom simulate vocabulary use every measurement measure second minute learn refer likelihood burn though slightly count uci edu ml york corpora count hold truncation update keep representative fig gamma hyperparameter variational improper sampler likelihood initialization dirichlet learn iteration variational attain likelihood dataset hour edge somewhat dataset unlike attain iteration burn convergence second log likelihood truncation among good medium truncation additional competitive small log truncation atom second iteration take dataset fig matrix require update minute hour average find fast medium dataset large dataset measure hold less compare substantially produce novel stick gamma variational
attempt sensitivity generate feature small affine affine hull include affine affine discrimination via affine constraint affine enforce near maximum extend vector challenge unsupervised relaxation maximum margin hyperplane running experiment outperform hyperplane optimal label convex convert definite problem obtain optimisation cut plane speed extend multi hull comparison hull intra hull normally may discrimination contrary near image sensitivity variation balance approach assumption hull model distant combination variation image ns synthesis require remain synthesis must involve synthesis call neighbourhood noisy define point noisy convex affect divide multiple hull notice hull control number divide control extraction reduction set extract convex subset far divide hull variation b conceptual illustration convex extraction hull construct close indicate line connect set blue combination b illustrate sample away set hull cluster individually notice close generate subset point sample area variation solution apply extract however convex nc maximally separate inside point hull eqn nearest svm optimisation combine discrimination eqn eqn margin distant similarly approach local cluster reference cluster convex matching comparison hull match drawback extract individually capture variation hull expensive conceptual illustration reference c separately cluster divide hull grey indicate set contain assign grey close address reference adaptively cluster reference probe accord total reference ie design local sub cluster also adaptive reference affine versus hull distance hull affine hull hull variant eliminate select top image arc wherein reference extract query reference individually way reference query comparable perform approximate set face dataset consist subject pose illumination subject conduct cross select rest normalise histogram choose partly situation face fail test category category various select turn normalise hull reference cluster arc image normalise bar bar arc technique bar correspond consider red combine compare method efficacy choose fix arc select counterpart hull discrimination consistently local capture comparison evaluation vs extraction method number indicate number threshold minimal ptc c c c variant show parameter performance well propose approach helpful variant counterpart indicate cluster cluster performance cluster sensitive drop normalise image achieve propose replace arc arc cost compare distance reference arc evaluate indicate number cluster arc indicate strong set arc argument arc helpful remove data matching nn perform well summary good state normalise use dataset raw method dataset regardless variant average compare two table set convex slow hull extra however significantly lead local novel balance maximum cluster constrain region artificial adaptive arc cluster cluster set hull comparison extraction subspace effect acknowledgement lp department communications centre chen university school university technology point method convex affine artificial noisy sample significant intra variation extract cluster undesirable environmental illumination pose variation two close represent propose enhance near point affine adapt constrain local arc constrain query image dataset show technique information improve discrimination robustness variation pose illumination two method distribution parameter art model attempt linear subspace convex angle similarity single structure select classification row artificial middle third artificial geometric close close point calculate adaptively distance degree variation close discrimination show artificial variation combination distant recent embed representative discriminant patch extract model local acquire local ie exhaustive fix two example assume image person cluster second represent
tool library consider main tool train predict additionally merge finally provide include variety extra tool facilitate basic bootstrap sampling categorical splitting recommend original ratio positive negative subsampling illustrate software keep majority aggregate sophisticated decrease accuracy desirable base number average function set randomly balanced dimension trend training order svms subset subsample kernel overall ensemble fast twice ensemble experiment simple ensemble single marginally fall short train basic aggregation ensemble standard svm ensemble voting tool svms experimental ensemble model significantly maintain frequently train complexity know benchmark may desirable instance benchmark ensemble ensemble scheme currently scheme nonlinear provide quality intuitive svm software date incorporate promising request de de library machine package svm currently offer ensemble storage evaluation support share model experimental result show drastically reduce maintain available online support vector machine bag become practitioner constraint amount become particularly train accurately problem svm nonlinear requires run quadratic set training complexity aggregate small training offer bag unstable bag voting maximize instance binary classifier per instance approach class aggregating ensure evaluation heavy library training moderately system use
domain measure suit connectivity consequently similarity zero node voxel measure voxel construct graph voxel introduce scalability location patch connectivity local patch approximately reduce voxel number dataset use distance voxel tune partition functional connectivity measure patch form connectivity mesh learn connectivity represent local pattern seed patch positively voxel indicate voxel indicate functional k fc functional represent voxel lie bold positively mathematically speak define neighbourhood generate select fc j fc fc index voxel translate voxel em fc functional compute employ close neighbourhood sample operation replace operation near positively correlate operation voxel obtain operation fc exceed fc extraction separate connectivity deviation computation distinction mesh neighbourhood formation mesh seed voxel algorithm suggest neighbourhood connect mesh neighbourhood method identification difficult class distinct distinction recognize voxel cognitive process need within connectivity brain discriminate recognize class state semantic analyze pairwise class represent pairwise voxel unique construct semantic illustrate number relation illustrate connect mesh fc em connectivity em em em fc cognitive eq voxel cognitive advantage use discriminative fc discriminative connectivity figure connectivity compute htbp discriminative connectivity voxel functional neighbourhood fc std kp deviation voxel vary voxel cognitive determined voxel cognitive voxel extraction voxel different cognitive discriminative consider neighbourhood fc even amount class similar divergence exhibit correlation semantic pairwise carry semantic class divergence trivial affect also illustrate divergence voxel functional entropy amount scenario correlation respectively absence last voxel pair semantic high entropy matrix illustrate pair near voxel construct note fc voxel fc threshold nearest fc contrary voxel set fc voxel information fmri list problem period whether probe member old new period period brain activation relate probe activity pattern test whether brain cognitive processing total ten semantic category use activation collect encode retrieval phase train semantic category preprocesse stage pattern processing perform http uk quality procedure assess outlier slice slice global signal correct slice acquisition slice match slice correction interpolation functional datum space affine transformation along basis sample spatially smoothed shift across consistent previous shift account response lag encoding retrieval consist voxel temporal fc generate nn support parameter user always much fc order form practically performance approach compute functional connectivity connectivity introduce threshold user work employ give peak relationship activation scan specific scan percent improve peak correlation namely positively specify discriminative connectivity employ cognitive connectivity toolbox avg htbp avg classical mesh mesh correlation mesh functional discriminative std mesh discriminative recall discriminative cluster connectivity computation employ peak scan generation entropy table connectivity mesh improve classification performance considerably classify voxel mesh performance mesh learning performance nn respectively main near connectivity voxel study mesh classify cognitive neural activation test employ connectivity voxel cognitive brain mesh cognitive state information represent brain focus finding cognitive would insight success cognitive improve mesh drawback mesh select fc brain hierarchy brain pathway rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb edu tr classify cognitive brain acquire fmri form around voxel voxel mesh determine neighbourhood define neighbourhood similarity voxel form voxel mesh voxel mesh linear relationship functional connectivity aware mesh voxel seed voxel voxel mesh relationship seed voxel voxel model arc mesh linear arc aware relational feature fc represent relationship voxel finally fc type cognitive state type information encode retrieve participant study category make recognition neural accordingly machine successfully category belong represent brain memory encoding style circle drop font style style auto b node auto swap auto swap j instant three instant belong cognitive mesh cognitive model individual voxel mesh voxel arcs weight spatially voxel voxel spatial coordinate brain arc indicate voxel time minimize location voxel fmri measurement category minimize employ seed instant mesh p mesh arc instant represent construct memory retrieval stage far cognitive detail
room improvement prefer kernel kernel learner et work component feedforward network et generalize idea sensitivity develop extract classifier manner superior identify pair discriminative act characterize discriminative platform visually sensitivity map rise principal establish introduce introduce let may nonlinear map sensitivity write actual intuitive importance classifier input define definition sensitivity direction denote define allows write classical quantify individual input generalize contribution rise map classical distinction sign direction influence rich sensitivity information eigenvector vector define principal sensitivity grant rich dataset fig unless otherwise recall pca case put q see pca center visually sensitivity exist code covariance map mnist datum sample add common template fig three step bit template picture gaussian intensity truncate intensity sample feed ten dataset dataset conduct gradient adopt final posterior purpose transform represent class standard digital logistic mnist describe way base sensitivity map compare sensitivity map obtain common neural train pixel character region edge st assign opposite whose crucial characterization edge characterization sensitivity rest opposite sensitivity extra sign rich counterpart benefit sub sensitivity st extra information benefit visualization classification problem sensitivity map particular class rd classifier quantify perturbation input look visualize deal binary visualization aid another fig st distinguish rd alone confirm map capture character binary st thus finally sensitivity depict map note pixel assign pixel localize understand discriminative digit dataset hand object orientation mean vary sample make visualization classification brain area necessarily face likewise translate may region digit mnist appropriate later dataset choose familiar summarize standard able character c stands test correspond verify highlight pixel clearly orientation effective template straightforward must template pattern elegant visualization remain decompose space assess artificial visualization least aspect dominate sensitivity classification classifier solve problem sensitivity drive attain visualize may decomposition variety problem medical identify biological region helpful
ultrametric node relation notice imply identity quasi ultrametric imply inequality definition guarantee boundary imply candidate two x show scalar property eq satisfied finally ensure continuity trivially dendrogram increase consequently dendrogram every ultrametric prove identity respectively former quasi ultrametric network arbitrary node ultrametric dendrogram influence either appear node merge single dendrogram arbitrarily identity concept concatenation point point define quasi valid ultrametric link strong triangle chain maximum exceed maximum cost chain xx axiom pick arbitrary node axiom satisfied point transform chain yx reduce dissimilarity chain eq far exceed follow eq q expression axiom separation dissimilarity useful show pair whose exist contradiction satisfy partition find partition exist compose represent node true dissimilarity conclude chain case cost repeat partition dissimilarity recursive correspond satisfie observe construction contradiction assumption incorrect satisfy return main ultrametric showing act admissible I network notice dissimilarity axiom substituting obtain valid ultrametric arbitrary direct chain block u vb b dissimilarity reduce dissimilarity map dissimilarity equal dissimilarity map increase separation since dissimilarity proof hold admissible usual projection say indistinguishable chosen notion distance clear check correspondence finally definition hence triangle correspondence correspondence show pick must element triangle define correspondence furthermore element absolute less absolute noting proof checking imply follow fact correspondence choose way contradict manner must two guarantee exist immediately particular force conclude correspondence write fix yy xx sc influence turn influence thus qp pc support service service sc fr totally influence sc two singleton describe preserve higher state definition dendrogram form node influence resolution influence edge mp rl mp influence cluster resolution service mp join co five five cluster hierarchical fr rl rl main influence singleton keep resolution sc cluster influence quasi partition resolution define define block partition relative importance influence mp except resolution totally resolution comprise mp observe three cluster co mp influence cluster top partition resolution mp red coincide total depict blue vertex height anchor center thick gray gray gray gray specification dendrogram resolution method dendrogram dendrogram component fig b quasi visualization quasi instead ten previous mining construction trade trade inf finance service service care service dendrogram influence resolution resolution dendrogram form singleton formalize dendrogram edge resolution b combine influence remain influence concentrate service whereas qp partition influence business service influence influence mining extraction influence apart influence definition quasi dendrogram reach trade trade service health care form green influence quasi partition include green one resolution green clusters main compose seven span secondary influence singleton influence quasi singleton join one induce correspond quasi interpret ordering resolution min resolution merge together red arbitrary combine red former latter sense edge quasi dendrogram cluster resolution dendrogram dendrogram fig min seem play clear large increase structure quasi merge resolution original dendrogram exclude form influential tendency anchor minimum height width anchor center black fill black fill fill center width pt introduce quasi network preserve quasi ultrametric linkage cluster stability method fulfil quasi study united network dendrogram family partition index arise resolution node group community dissimilarity differ determination difficulty formal whereby admissible respect asymmetric uniqueness former admissible admissible method cluster asymmetric decision derive symmetry dataset stage network relation dendrogram output generalization refer procedure back symmetry equivalence define quasi quasi dendrogram nest hierarchical clustering quasi regular partition contain disjoint include original block proceed respect axiom transformation axiom state quasi transformation dissimilarity quasi quasi cluster axiom linkage equivalence generalize quasi quasi stable establish quasi power algebra united year quasi exploit asymmetric california west result contain supplementary material finite node possible dissimilarity non hierarchical disjoint cover represent induce induce equivalence hierarchical partition nest collection term dendrogram different clustering increase start form node resolution quasi cluster concept node start connect edge consecutive node give dissimilarity link dissimilarity different endowed dissimilarity close asymmetric difference motivate structure partition equivalence relation search asymmetric symmetry binary necessarily relation hold quasi relation term order emphasize equivalence define unweighted self vertex edge satisfied pair b minimum height black height minimum black minimum height cm node blue vertex size minimum minimum cm node height minimum cm height node cm pp height minimum width fill height minimum fill pp height minimum width black vertex minimum height width fill blue vertex node minimum node width black cm node minimum height minimum point height thick bend thick bend thick bend pos edge thick bend right b thick bend pos node block vertex partition influence edge vertex quasi block direction block vice versa latter separate motivate direct influence edge likewise influence influence none influence accordance qp influence relation meaningful lack qp requirement qp qp definition quasi equivalence relation quasi equivalence distinct quasi conversely quasi partition induce quasi relation partition generalization datum far regular partition partition generalization hierarchical dendrogram nest partition nest index recall quasi dendrogram boundary resolution node separate cluster large cluster hierarchy x xx x continuity exist large cluster ever cluster join stay requirement give resolution except merge technical ensure correct definition cf dendrogram imply give quasi dendrogram dendrogram set vary nested recover dendrogram quasi equivalently quasi regard quasi empty quasi motivate cluster quasi vice versa quasi method axiom facilitate asymmetric version exactly node long cycle qp node qp qp qp imply every quasi acyclic dag dag construction quasi theoretic partial partition satisfy identity strong ultrametric regard quasi ultrametric dissimilarity ultrametric follow theorem preserve xx x either construct quasi ultrametric xx theorem dendrogram equivalent quasi ultrametric result allow cf quasi ultrametric apart importance since mathematically indeed paper term quasi quasi illustrated dendrogram ultrametric influence occur belong block conversely ultrametric class quasi ultrametric less contain distinct ultrametric equivalence quasi quasi ultrametric network ultrametric dendrogram dendrogram equivalent bottom first obtain dendrogram component node merge resolution ultrametric cf become resolution node belong vertex conversely depict value ultrametric edge equivalence resolution exist imply moreover one equivalence blue vertex blue bend left node bend leave bend bend leave pos node bend edge bend pos vertex scale blue scale bend bend pos node blue black draw black black encode axiom criterion axiom impose impose quasi arbitrary network axiom direct dissimilarity output direct axiom transformation relation reduce tendency cluster decrease axiom dissimilarity quasi ultrametric justification ultrametric quasi admissible axiom want method admissible axiom direct formally quasi admissible quasi ultrametric axiom turn unique admissible quasi axiom linkage admissible axiom undirected metric general asymmetric network lose axiom recover uniqueness asymmetric linkage clear non uniqueness asymmetric asymmetric dendrogram dendrogram uniqueness linkage cluster asymmetric develop study leverage admissible quasi asymmetric establish cluster space sensitive effect require weak axiom uniqueness aware effect admissible successful space network similar study finite formalize analogue hausdorff denote define metric supplementary material regard position express distance input network nearby yield nearby dissimilarity ensure supplementary non ultrametric outcome dissimilarity function original change dissimilarity quasi dendrogram influence arise original change supplementary material interpret dissimilarity correspond ultrametric quasi search chain infinity construct operation perform power algebra regular sum replace product quasi ultrametric power ultrametric triangle follow argument one furthermore inequality finite quasi ultrametric space building ultrametric network ultrametric operation algebra indeed exist cubic clustering relate instance take input asymmetric complete might modify follow quasi census state plus fraction come supplementary percentage come asymmetric dissimilarity application extensively ultrametric computed ultrametric dendrogram analyze dendrogram dendrogram influence
bound alg pruning step branching step state em branch split evenly subset subproblem ed queue subproblem select implementation subproblem sort subproblem bad branching subproblem select split subproblem element column branch node split node singleton node distribution evenly state node order probable branch b discuss section bound energy yield constraint relaxation conventional sdp sdp large interior sdp solver scalable sdp combine cut sdp constraint arise relaxation sdp integer state per normalization constraint one node negativity constraint become fully negativity relaxation loose submodular marginalization pi pi constraint connect negativity marginalization sdp approach directly marginal polytope constraints cut polytope equivalently polytope extend comprehensive constraint cut polytope polytope triangular consider three binary polytope cubic binary cycle odd inequality express triangular inequality cycle define cycle inequality odd odd constraint polytope define polytope binary inequality separation violate odd adopt graph separate node non represent project pi ph project inequality define cut maximum initialize lower bind primal cutting problem dual meet step round generate discrete combine aforementioned sdp map intersection outer lp several sdp inefficient computational involve polytope connect inequality grow end adapt propose cut plane cope constraint approximately discard case perturb sdp perturb simplify u column constraint condition eq solution large dual relaxation numerical restrict continuously twice gradient solve newton dual gradient step decide simplify feasible dual beneficial stop subproblem sequel far follow yield follow ii search gap x energy branch sdp relaxation minimum value consider lp sdp relaxation result integer relax relaxation sdp drop sdp significantly scalable interior add impractical find add violate cutting plane next redundant impose constraint cut search violate add sdp add constraint violate negativity triangular enumeration violate cycle inequality find separation cycle violate odd separation cut implementation iteration arithmetic scale eigen solver routine robust requirement violate negativity marginalization constraint arithmetic violate cycle save node odd cutting plane omit corresponding memory requirement efficient classic interior need operation sdp tight propose bounding b pre binary state prove solution reduce size apply assign integer persistent sdp bounding procedure sdp sdp computational reduce stop value calculate cast energy computation bounding stop meet see global subproblem low step small warm bound warm cutting procedure cut plane initialize constraint zero dual variable first bound initialize subsequent bound speed bound zero disadvantage cutting size subproblem correspondingly dual increase iteration traditionally never remove prune inactive address issue cutting drop work discard current activate relaxation polytope relaxation fully without branch plane cut plane result lower produce solve approximately impose well low yield cp cp effective iteration sdp connectivity strength performance mrf synthetic problem affect difficulty mrf per unary compare unary increase map mrf example belief propagation experiment mrf pairwise energy sample unary energy I unary versa connectivity link curve concatenation gradient within bound round conduct graph upper low meanwhile impact amongst drop decrease well lower bad amongst synthetic perform mrfs connectivity unary potential submodular denoise mrf use validate whether c experiment whether globally input add truth contain unary submodular pairwise branching relative latter submodular mrfs good lb cp v tr tr cp cp deconvolution instance cp solve instance respect bad six instance within hour six instance respect convolution growth portion size image third demonstrate partial white color gray last column image energy recovery deconvolution cp deconvolution reconstruct convolution formulate unary submodular energy connectivity growth image deconvolution different size deconvolution reduce cp report hamming perform post table result within runtime minute hour give hour omit minute quickly respect size six within hour solution short time exactly recovered v within minute energy cp image perform mrf variable eigen deconvolution core demonstrate cpu hour cp benchmark show inference densely unary good lb l cp bb bb achieve minute hour solution sub category fully unary compare cp bb winner bound method run limit hour hour limit method hour optimal solution three algorithm lb runtime label graph hour instance achieve bound chinese character term cut show achieve likewise comparable within hour solve bad specialized note utilize complicated well method include bound lower particular achieve bound lp validate bind good lb runtime top solve large instance method compare hour graphical maximize modularity six absence unary fully connect lp relaxation odd method modularity six large hour lin kl specialized offer instance solve hour word bound runtime deconvolution inexact detection runtime inexact inexact modularity instance runtime inexact bounding exactly summarize instance deconvolution modularity evaluation present cut map mrf advantage efficient sdp main propose variety incorporate sdp sdp solver cut plane technique branch bind warm dropping inactive unary type solve almost exist bound experiment compare art superior performance lagrangian function u transform serve increase converge firstly know matrix fix trace spherical x problem p p p primal contain duality equal optimal primal problem primal feasible proposition strong duality optimal add optimal objective equivalent bf font bf claim remark definition van university sa centre united branch cut solve mrf core bound sdp cut speed computation exploit warm start constraint method par magnitude unary experiment art non mrfs segmentation reconstruction find posteriori map mrf typically np however solve problem approximately exactly comparative cut mrfs potential exactly cut obtain globally optimal submodular pairwise mrfs portion highly graph weak unary mrfs move swap employ local move swap move energy subproblem solve class popular message max propagation exact structure cycle approximate belief tree reweighte propagation minimum energy approximate prove binary submodular mrfs propose aforementioned lp optimize polytope ordinary max lp relaxation structure generalize cycle exactly achieve tree structure graph dominate cone quadratic standard lp interior inefficient problem exploit lp method descent standard enough map especially usually perform poorly densely unary interest long cycle de se ed loose adopt consistency marginal loose existence violate cluster add search high triplet cycle separation cycle sdp sdp develop np hard accurate approximation primal interior state sdp solver worst require arithmetic operation requirement sdp node exponent even medium significantly relaxation mrf matrix eliminate point solve need many show constraint conventional relaxation consider local consistency mutually neither tight consistency tight sdp high lp sdp able linear branch cut sdp procedure approach either map budget computational art compete main contribution scalable sdp algorithm cut plane minimize intersection outer arise sdp relaxation scheme sdp linearly sharp violate interior still maintain energy sdp bound embed mrf problem optimize bound branching include reduction warm start removal model demonstrate compete dense unary potential line focus metric consider mrf branch mrf generally bb lp relaxation sophisticated include ce stochastic bb approximate inference optimize semidefinite technique work separate dual decomposition sdp subproblem qp guarantee experimental lp sdp well paper integer real value semidefinite aim solver program sdp semidefinite sdp solver interior sdp interior notation list use bold letter column bold low letter cone semidefinite semidefinite scalar statement trace matrix vector consist vector diagonal element frobenius introduce basic section contribution
functional standard less systematic recently machine ml functional principle ridge interact functional free dft produce highly accurate density energy systematically additional ml capable dimensional pattern successful medical stock automate categorization ml apply quantum include fast accurate molecular energy optimize divide new typical challenge new drive dft exact interact reference easy additionally ml approximation thousand million exact dft positivity approximate none issue example accurately approximations ridge system ref various cross validation great euler additionally descent modify euler effect theory background interact spin density spin atomic symbolic energy interact spin external potential potential elsewhere hamiltonian spin eq potential system energy use eigenfunction energy potential tb ref exact ref half half test dft dft variational euler lagrange chemical satisfie constraint density drive density satisfy ground density free dft energy ks density cause functional functional derivative approximation impossible avoid nontrivial von ref poorly mae self bad local add modified expansion little topology representation ref molecular predict energy contrast norm derivative conventional derivative dimensional practice expand finite basis density calculation represent express product result converge truncate fortunately greatly variety parametrized parameter manifold density potential density potential kernel trick structure optimize parametrize form increasingly linearly space linear tb illustrate lie circle transform back belong map assume wish space thought must existence term need compute enable ridge version regression prevent training density determine minimize regularization ij k magnitude prevent overfitte calculation uncertainty identity parameter kernel length find cross sect design e kernel choice reflect characteristic ref choose approximate equivalent notation e norm scale relate ref grid define fig distance tb tb select kernel dash chemical gray hyperparameter randomize fold median small inverse numerically limit force directly center mean fig illustrate example datum unnecessary center kernel reproduce domain domain require center regularization strength e scale standard kernel laplacian wave em radial basis rbf behave broad problem well contour mae set strength see cauchy contour scale neighboring rbf yield poor functional become unity effect contour comparable scale kernel vary nonlinearity region mae cauchy performance middle dash laplacian behave smooth hyperparameter pick value hyperparameter must hyperparameter generalization future never give final model essential selection cross training validation ml set hyperparameter set never determine assess see dot density analyze randomly optimize mae training bin bin bins minimize mae total final hyperparameter validation loo special typically leave simple fold expensive leave intensive good mae test scheme similar mae exist variation occur cross validation generalize kernel dot choice dot global mae relatively flat validation minimum indicate well em wave finally use time optimize functional drive paper ref perform c em n set give error likewise density test give chemical ref systematically absolute mae hand wave increase indicate wave flexible enough note choice need reference energy grid ml depend demonstrate grid cross validate accurately accurately type potential limit energy underlie parameter needs distinguish comparable fine desire large able basis sparse reference greatly tb mean drive density validate hyperparameter mae reduce jump challenge density discussion evaluate density e functional useful must solve yield accurate derivative plot model display inaccurate huge apparent ref dimensionality unable functional information ml direction fig minimization technique interpolation information many orthogonal produce inaccurate derivative dimension produce since exist dimension create fig deviation constrain tb starting show quickly cause density minimization stay euler minimization elsewhere manifold normalization previous long minimize ground give reduce domain avoid confusion call develop attempt reconstruct manifold tb j consistent square function elsewhere implicitly interpolation show accurate red become unstable jj tn approximation manifold aim locally linear principal density manifold weight generalize evaluate locality come distance density density ref smooth weight pca average define density covariance eq eigenvector direction lose keep direction direction tangent first pc projection onto space choose pca approximation square tangent pca project projected approximation density guess density derivative compute project see take trading speed ensure remains weight iterate convergence achieve tolerance max density converge tb take along project direction energy stay density project report error density density factor constrain density data sample constrain magnitude particle density locality density orthogonal individual pca choose pcs mae pc n pcs pca pcs structure see space introduce fail search report constrain giving although generate parameter
lasso valuable rectangle text blue corner width corner biology york ny lasso inherent validate variable illustrate promising synthetic biological field year science engineering throughput drive mathematical variable analyze interest irrelevant lasso popular practice dimensional turn properly adjust aspect practice cross validation tuning validation inefficient lasso lasso lasso tune adjustment design proper simultaneously accurate computationally attractive contribution systematic development square tuning bootstrapping scheme finding response constant exceeds typically ease exposition sequel allow random design support zero entry dimensional regression estimator detail lasso recall tune least criterion parameter regularization influential reasonable choice look bind cf overview proportional satisfy calibration need aspect tail calibration design matrix approach deviation approach recall tuning parameter root similarly lasso determine intensity see square deviation readily locate denominator distinction lasso act inherent deviation make lasso adjust tail address incorporate inherent quantity consistent define accord estimator one one mapping square root contrast establish interesting path omit brevity tucker latter resemble fact estimator formulation equip tackle spectrum estimation prediction variable task bootstrappe fix majority bootstrap bootstrapping scheme bootstrappe rule already practice illustrative sample readily least square perform dual extension pair dual straightforward beyond scope subject currently finite root oracle properly yet type function comprise amenable invoke integer fit amenable tailor convex fitting solution several among effective guarantee fast excellent novel increase non smoothly mcp prove computable example synthetic inspire biological set involve production numerical matlab ghz intel core memory cross validate schmidt approximate parsimonious stopping criterion tolerance iteration evaluate selection synthetic minimal validate mean error cv generate regression inspire simulation sample vector normal error multiply sample row normal normalize scalability repetition thick colored correspond thin bar precisely report runtime plain lasso performance fix function show runtime cubic scalability least comparison scheme reveal fit result setting selection strong hamming consistently supplementary material compare provide excellent cv recently publish biological comprise gene experiment vary profile expression standardize production measure highly predictive production report runtime matlab routine gene list specifically stability coefficient enter runtime single computation approximately select considerably cv corresponding coefficient list majority selection vote list select frequently reveal key insight coefficient lasso cv solution cv rank cv frequency b select plausible locate co express runtime lasso complexity differ considerably often error report cv three error l equal lasso give equal lasso bt intensive omit observe square counterpart
problem maximize eq regime strategy word mean use attain due restrict consider binomial difference conclusion alternative big substantially independent parameter divide english word arbitrarily instance high english model become likelihood likelihood maxima log start per model merge language symbol compute give merge english per english english merging language become recall non english consider entropy accounting eq correction conjugate parameterized function vector clearly distribution document dirichlet pre simple probability zero top topic tend evaluate computed spirit document unseen document hold fraction remain unseen without change topic frequency unseen document run lda improvement let wang medical institute chemical biological engineering department physics database text knowledge require extract document assign document enable search statistical characterization lda state art systematic technique lda yield infer adapt approach community wikipedia reveal big collect store analysis nearly digital keep knowledge challenge language text database gap topic database recommendation digital spam filtering dirichlet allocation topic rely document cover mixture characterize usage address topic document corpus focus might different word primarily word equally application topic mixture crucially rely maximization linearly large known computationally hard landscape gain thorough implement highly specify control theoretical normally standard rough topology landscape exclusive topic enable search landscape document model count mixture topic english topic topic might english two merged vocabulary document symmetric fraction english big variational curve infer algorithm th generative theoretical grey area lda actual model practitioner large almost equally pose serious investigate elementary language corpus helpful provide realistic complex language fully language language document entirely create simple lda use two document step select concentration si language step randomly word document sake simplicity restrict document bag english language language language infer language alternatively merge two count english part likelihood generative model lda fact divide english corpus si increase english document overfitte merge log likelihood per document fraction great likelihood limit likelihood identify model indeed non negative kl critical depend document corpus increase lda si likelihood though great infinite infinitely generative landscape consequence define extremely vocabulary per si conclusion find potentially model regardless likelihood technique highly equally landscape technique yield across accuracy match among fit typically lda represent different language slice language test value overfitte overfitte language line initialization resolve si algorithms require assumption corpus reality need resolve test synthetic corpora corpus ten language comprises belong language class comprise eight language word frequency language validity calculate output similarity infer corpora enough dataset si estimating would lead unable global landscape si show asymmetric implement gibbs result landscape current improve performance build intuition landscape view bipartite network construct word language corpora language component find topic complex comprise unlikely topic use compare co document word dot product similarity distribution depend si significance word filter present corpus identify word exclusive topic si information decide refine asymmetric distribute likelihood actually wikipedia validity lda corpus comprise document web contain publish six economic process document corpus remove list word pre yield topic number topic topic split merge suggest small split topic find corpus journal science frequent bottom topic find big topic journal document assign journal publish infer generative fig perfect standard lda lda comprise si let six infer put paper journal paper publish science likelihood model si systematic different implement choose lda tune topic within corpora extent topic result corpora si test generative size right portion equally overhead guess small easily wikipedia document research limitation model able remarkable validity simple function guess guess obtain correlation topic initial guess propose practical make separability algorithm interestingly improvement degeneracy yield well create synthetic corpus dirichlet probability small document make fraction implement latent topic write topic synthetic generate make lda sized topic equal topic initialization lda describe measure distribution topic use topic permutation label easy quantify similarity compare since topic topic get versus st stand look make average st similarity similar assignment label similarity rand eventually measure model define ht synthetic dataset simplicity document topic topic specify vocabulary simplicity model distribution corpus aspect mixed document topic across word decide drawing probability see word compute bayes corpus choose value easy corpus mostly keep constant document however mostly use since realistic corpus relate code david discussion outline supplementary language language simple set hand find absence sort eventually treat detail lda entropy language sake sec eq since generative asymmetric lda always likelihood document asymmetric information document per likelihood conservative correspond topic argue log optimization cumulative relative difference log generative find log versus match visible accord language support conclusion discuss number topic fit illustrate big english science share music word assume word english write sake language english datum find find language calculation higher lda lda sec fit english use english vocabulary english tell go english merge reason big split lda science see display argue topic make small method first build connect term topic cluster like topic optimize asymmetric likelihood via variational bipartite word number network word dot generic strongly put term relate area filter dot nan weight variable occurrence word corpus rare event approximate poisson average ab randomly occurrence uniformly fix share document poisson precise large dot corpus comprise six document build isolated refine topic isolate word topic build provide belong partition recall assume completely discard time word always generate module word locate since hard module document time topic word optimize describe series move aim aim topic make specific precise move topic select consider independently topic actually come binomial topic small significant increment zero decrease accordingly repeat previous explicit dependency maximize refined lda closely main data document case quickly situation similar describe practice software every iteration well explore filter lda optimizing perform set change measure sec topic need run take run significant document topic might select significant document appear likelihood depend application small topic optimization threshold wikipedia software let remove initial select topic hold fraction corpus algorithm dataset measure hold model obtain method tend actual one lda provide fairly uniform assess entropy topic entropy probable compare topic versus effective topic number show effective dash black line topic select black panel achievable topic topic five method low visualization generative one dedicate measure topic performance lda compare lda synthetic yield achieve provide implement generic generative unknown measure perform poorly performance dataset color allow way get similarly happen language small topic together indicate two across horizontal bar divide proportional document sort prominent sized topic lda actual comparison corner obtain topic sized lda fed topic hard guess information get set reasonably get sometimes give slightly much increase number basic option parameter choose relative difference mean affect lda topic document seed whole dataset though initialization guess mean actual similarly generative get check performance close one define topic lda grow overfitte main decide web science sec lda test difference systematic test step topic provide highly heterogeneous section asymmetric difference probability topic variational language well fig structure dataset model similarly topic topic
impact avoid set expense running find affect speed sensitivity spc generate combination averaged combination overlap length except value also greatest small l l spc clusters entire hierarchical path spc path include number obtained drive explore different still along gap instability select aim spc simple decrease datum log big increase log likelihood sort difference number cluster proportion estimate line implicit behind singleton difference likelihood simulate accord ari relatively ari score demonstrate average ari term ari path ari average show close average range take simulated separate averaged solution large ari score solution path determine whether singleton cluster truly challenging reflect average score table overlap scenario ari score clustered misclassifie contrary select fast appear spc various k scenario cluster well noise spc dataset tf self cell expression tf activity bind tf spc general cluster include difficulty provide spc proportion irrelevant observation search solution adaptively irrelevant spc fine tuning input result set lowest long importantly spc require cluster provide estimate impose two merge stage however eliminate perform split soft thresholding operation discuss practice case split spc assume object gradually split separate cluster later stage receive penalization spc across simulated datum scenario penalization widely apply path surface selection bad feature spc majority exist framework convex sparsity need address singleton outlier need cluster usually high cutoff address future paper solution interesting every fix value cycle minimization reduce thresholding remainder coordinate modify incorporate fix give q recall minimize reduce define see globally sufficient imply note minimizer k ari nm extra date date hyper nm extra proposition remark research fellowship year fellowship support nsf grant dms author amount create discover large scale result problem grouping address methodology introduce regularization difference adaptive provide produce corresponding cluster solution optimization carry block advantage compare simultaneously separate grouping methodology various simulate dataset expression competitive collection discovery information instrumental visualization adequate enable discovery group association deep insight biological cluster phenotype help variety rely usually like hierarchical graph popularity cluster map support name cluster vast survey usually comprehensive general development trend mining approach datum contain amount irrelevant identify recently researcher noisy irrelevant interpretation new popular penalize cluster noise account mean algorithm challenge specification cluster exist solution rule different gain complex penalization fuse lasso method generalize useful large loss compute entire penalize angle spc include irrelevant singleton cluster algorithm solution specify objective couple coordinate adaptive path spc effect spc simple work well work penalize utilize gene spc assume select object author cluster impose center cluster center result converge globally severe cluster procedure handle weight primarily discuss weight cluster unified solving cluster select penalize penalty penalty author parametrize selection pre grid penalty remainder provide path spc several spc spc big direction matrix object cluster achieve pairwise careful arbitrarily minimizer important little cluster naturally separate noisy belong prevent merging simulation adjust plot see section appropriately continuity assignment mcp develop mcp define penalty regularization concavity mcp penalty form mcp concavity compare convex penalty group penalty explicit concavity easily rate bias drive minimax figure illustrate case special minimized minimizing reduce eq plot see difference circular light gray contour penalize contour black minimum force minimizer penalty affect center gray contour figure contour produce dark gray contour objective small value minimize cluster center estimate proper choice correspond penalty surrogate cyclic singleton merge appropriately correspondingly two suppose solution center represent function minimize force merging current step iteration number due concavity obtain q weight adaptive weight penalty weight distance vary whereas change minimizer penalize sufficient step thresholding spc never gradually computation work well necessary could handle thresholding detail soft establish penalty condition meet mcp separable apply objective meet fix mm coincide set consequently eq reach almost iteration controlling amount later degree concavity create simple drive rule parameter cluster start individual form singleton gradually enforce great reduce decreasing depend current warm later decrease initial low concavity decrease concavity behave considerable whether decrease bias ratio cluster behind certain center beyond concavity bias lead bad address evenly lemma simple lemma lemma provide mm minimizer lemma lower serve quantile regard approximate proportion point default need point obtain first denote merge suppose decrease z upper maximum object collection object merge merge case bind decrease algorithm cluster summary involve mm consequently demonstrate affected choice small slightly small avoid unnecessary recommend paper user calculation stand approximate near tight could value dataset well noise force initially merge noisy high dimensional construction describe full spc algorithm initialize assume form singleton warm start require input spc provide kk g spc k weight use adjust ari across ari assignment solution true take partition ari find simulation ari quality compete belong calculate ari suppose estimate assign respective see contingency ari count identify belong estimated sum misclassification sum sensitive identify noise datum noise except table consider plug mean cluster k randomized center input cluster comparison input result compete find stay along user target clustering rely small conversely small noise assign reciprocal region calculate ij categorization cluster categorization identify generate recommend near categorization size small detect perform choose near neighbor cluster convex specification weight neighbor solution path result package performance separate cluster separate overlap equal outside overlap cluster locate simulate demonstrate spc noise also define separate scenario merge output dataset cluster center cluster present true cluster spc solution mis report ari range g spc path note detect scenario scenario spc spc perform similarly well tight outperform separate cluster pre hierarchical cluster remain excellent spherical perfectly match spc scenario initialization c n comparison method scenario average top bar refer cluster find spc report well considerably separate spc merge overlap identify noise spc create big cluster reflect cluster mis specify tendency cluster close together overlap cluster tends produce satisfactory aside add cluster figure assignment consecutive spc separate noise perfectly two method separate one tight figure add large add histogram iteration combine four scenario instance converge decrease ht b simulate cluster datum distribute cluster high detail small merge initial datum cluster consequently though decrease increase along spc cluster misclassifie point cluster satisfactory comparison scenario axis indicate range range spc dimension
g piecewise approximation logarithm scad directly become approximation multiply add appropriate procedure original form scad similarly approximation indicate dc result state theorem except depend way right namely calculation compare close tend poor easy p deep study much context furthermore hold problem via prove norm appropriate vector b b u reformulate penalize penalty solution iff function dc program demonstrate equivalent suitable function iff feasible q feasible solution conversely feasible nu fx nr fx nr nu equivalently special equivalent eq q iff xx fx ft ft fx sect therefore ki li virtue equivalent discuss solution proposition omit ambiguity three suppose right denote problem give existence derivative define prove without generality equivalent equivalently write concave hold concavity r rt dc component k find previous work scad induce dc decomposition k svms I reweighte problem dc program solution dc I k x since update solve algorithm initialize x k form iteratively solve weighted reweighte sparse stage run character reweighte propose justification convergence I reweighte algorithms context linear log z next perturbation third exist reweighted type optimization I I dc program dc iteration dc k k updating initialize compute k I choose dc become expression approximation algorithm reweighte iteration converge cm ir sparse p reconstruction k seem address additional subproblem less constraint dc decomposition suitable dc program dc svm approximations enjoy formulation subproblem subproblem also possess dc program dc quite compute influence property update let ki leave resp k k take derivative variable help exceed know convergence possess kx know uci repository point test description site uci repository htbp feature breast algorithms pc intel ghz gb ram program zero small algorithms accuracy solution training test sparsity solution percentage second experiment propose scheme experiment value dataset five choose suitable cross validation good perform cpu purpose fair run update procedure stop update solve linear last column bold fs fs cpu fs fs fs cpu fs cpu fs cpu comment concern correctness gain three select well fast evaluation criterion update update procedure solution update fold two comparable chosen fold validation smaller confirm analysis subsection become difficult bad contradiction updating updating give global training updating dataset cpu second solution second section experiment parameter follow fold validation point exp lp sf cpu sf cpu sf breast sf sf sf sf cpu approximation considerably number select select good correctness term give train quite cpu study dc programming dc approximation algorithmic consider class dc norm usual induce function consistency minimizer problem original sufficiently large relate solution solve original scad problem sense usual induce formulation common concave scheme approach nonconvex dc approximation piecewise usual concern svm scheme finitely local confirm theoretical identify winner induce dc programming light sparse nonconvex permit establish crucial relation induce elegant way effect dc decomposition perturb algorithm reweighte reweighted specify deep dc model solve world nonconvex large corollary le involve programming consider dc include induce study resp global minimizer resp minimizer use dc approximation namely induce function analyze cover sparse reweighte reweighte well dc tackle implement feature selection empirical comparative various dc function feature zero denote cardinality optimization one refer problem domain finance draw increase attention researcher recent origin nonconvex x q regularization make want solution important learn text micro array analysis feature feature feature preserve discrete classifier classification determining lead identically sample compose explanatory response vector input look relation possibly relate model parameter eq loss take relationship multivariate form composite identically distribute explanatory discriminant onto maximize class I e maximize bs scatter positive class label sample optimal fisher discriminant refer reconstruct dictionary selection available particular amount way among call portfolio portfolio management want investigate sensor digital decade involve divide category convex nonconvex approximation nonconvex belong group replace suitable efficient chapter penalty inconsistent variable bias introduce adaptive nonconvex approximate nonconvex extensively induce penalty exponential fu smoothly deviation scad logarithmic definition use algorithm develop successive gray local stage zhang lasso reweighte reweighte quadratic category name reformulate author context program generally intractable dc program sparse symmetric problem category heuristic tackle greedy orthogonal pursuit etc approach involve regularizer problem nonconvex approximation deep relaxation produce good many local minima approach weak concave bound term show set approximate original nonempty available minima lack mathematical always challenge researcher learn issue cite approach approach programming robust scalable nonconvex nonsmooth continuous contribution firstly dc consistency link minimizer minimizer neighbourhood strongly concave optimal solution secondly depth induce suggest approximation reasonable via identify scad suitable moreover box exact technique interesting sparsity induce dc main show concern dc dc piecewise guarantee permit exploit elegant dc decomposition flexibility dc view perturb reweighte penalize lasso reweighte careful empirical dc programming brief algorithmic minimizer study comparative usual approximation discuss deep problem approach conclude equip canonical euclidean denote identify programming constitute global introduce preliminary extensively le original constraint program exploit deep result elegant approximate nonconvex dc program popularity rich deep rigorous robustness method adaptation problem real nonconvex development programming mainly devoted obviously dc high nonconvex programming dc form equal take dc dc dc convex q dc n wide life operation dc constitute sufficiently reference order leverage powerful dc primal dc also dc eq dc convention function nonempty finite dc program dc play algorithmic optimality subdifferential generalize singleton nothing programming dc condition dc program dc lie distinction local solution global develop dc dc trivially point critical tucker kkt optimality next case quite practice dc program critical local minimizer since differentiable everywhere say critical locally locally inclusion resp resp solve dc program also reference therein local optimality duality programming simple approximate sequence program iteration approximate concave correspond minimize convex generic guess hx note convex far solve important critical terminate iteration bound every critical dc program dc program optimizer dc whole convergence worth involve dc hence dc version dc infinitely dc decomposition implication robustness search dc important dc decomposition tackle scale try either explicit form computation subgradient usual rule calculate subdifferential efficient adapt handle generic sensible study answer structure nonconvex search dc point dc programming nonconvex program field apply science especially learn reference decomposition suitably dc permit recover nonconvex program global algorithm program dc program dc programming use program dc programming reader therein dc dc dc stand precisely h gx yx st idea replace lead dc function function u v get resp resp local
ignore influence form term decomposition extra impose combination basis properly point suggest subsequently brief dictionary environment extensive simulation effectiveness analyze beyond principle completely solve rather target completion liu nsf nsf fa li nsf iii nsf cardinality play role suppose u obey provide ac ie ie ie ie ie ie frobenius sense u ie kronecker definition ie aa inequality obeys constant u lemma lemma u u u I u u I I feasible standard convexity lagrange l gradient svd uv u ia uv ta uv uv convex optimal solution feasible increase x aa denote complement orthonormal cc ta l equality already unless triangle f u validity f last minus height depth computer science nj usa department science nj abstract complete rank establish theoretically may low capture property merely constraint specify accordingly even handle propose low impose restriction datum dictionary properly non uniform lead practical completion experiment generate dataset encourage application need entry general give adopt assumption latent want fairly low suppose ij lie th entry location entry scalable various g notable contribution probably ease shall tell meanwhile I exactly parameter free scalable sum space support besides completeness also completion rank strictly uniformly critical success low constraint low detail structure nonlinear hard point figure e probably reality might uniform datum uniform might fail area vision motion provably advanced rank matrix citation construct learn unlike minimizer fall back regarded generalization properly mathematically uniform provide elementary dictionary devise proper environment dataset encourage summary contribution completion modeling matrix furthermore term version idea replace product concept g regime affect behavior coherence point non capital letter accordingly denote entry particular symbol mu rv shall abuse notation span onto space abuse notation space support complement identity function norm norm I large singular frobenius square denote I sum singular letter coherence besides extra structure coherence hence promise direction non influence parameter follow prove detailed noiseless svd svd subspace numerical confirm dictionary ask imply equality elementary learn h e one everything else dictionary unit coherence recover incomplete noiseless reality true observation contaminate entry completion accurately perform modify noisy completion follow theorem svd ac mn give recovery theorem potential kind framework like firstly utilize computation solve equivalent provide contaminate gaussian moderately support location solve optimize rank solve svd normalize column I denote optimize summarize whole encourage well sufficient facilitate dictionary unit theory backward whenever already successful recover successful effectiveness algorithm randomly datum index create rank vary step step fraction result trial matrix successful recovery successful pair mn sense denote compare learn work area success white significance dictionary real motion incomplete sequence database dataset dimension vision
acceleration use unit neuron time architecture cnn particular slightly architecture cnn architecture stable behavior hence cnn architecture subsequent cross cnn deconvolution aid behavior cnn positive ct image cnn module sensitivity candidate label away positive cnn subset rate positive patch balanced beneficial cnn balance channel patch patch mm physical ct window times translate scale take run extraction ct volume take train cnn prediction candidate parameter receiver operating amount quickly improve fp per volume fp auc point fp perform cnn candidate demonstrate task effective fp building reduce initial scale sampling rotation around prevent overfitte increase cnn exhibit range fp art sensitivity recent fp obtain fp assume candidate generation note available moment material publicly convenient improvement joint coherent literature alternative classification aggregation fusion indicate quality prediction work investigate fusion cnn show variety high orientation cnn clinical center publication automate detection clinical diagnostic structure distribute state sensitivity positive fp shot haar paper operate preliminary generation towards sensitivity fp level patient volume interest consequently orthogonal view via rotation respect coordinate train deep convolutional classifier cnn employ probability final validate ct volume patient fp respectively drastically art segmentation node play many cancer become e computed image small diameter short axis ct slice fig play assess follow classify coefficient manual processing consume clinical system ct base feature pool haar strong classifier availability intrinsic structure particularly appearance patient fp moderately sensitivity achieve fp range part region clinical employ high focus reduce shot via multi fusion voxel prediction multiple descriptor candidate topic sensitivity fp image red map assign slice volume channel slice simplify jointly three individual separately cnns slice prediction image ct scale edge length number voxel order increase variation analogous datum augmentation translate translate orient random permit neural net classifying unseen average per candidate classification one image patch purpose decompose channel combine slice cnn directly burden curse
proof axiom bind name axiom type interface already release since library include release tracking treat new logical axiom take name bind body canonical definition name type axiom name type infer body definition prove progress proof type name strictly step environment add exactly internal proven implement kind environment tracking walk type body construct tracking note previous proof main parse language expansion phase store collect try auto contain language try level else third typically basic atomic dependency tracking available implement protocol user dependency message progress make dependency obtain look define algebra anti convert logic z bi proof machine necessary dependency apply evaluated set theory logic available concept divide implementation class symbol meta mode separate object long type system treat proposition proof type name theorem name name term name without check type sufficient evaluate proof average advanced pattern combination modify combine harmonic classification method implement tool external near fact conjecture theorem naive assumption conjecture fact naive proving feature formula independence assumption say occurrence occurrence feature predict relevance ng filter keep track conjecture perform iteratively fact fact score roughly irrelevant occur fact perfect score fact feature feature cover coverage proof dependency suggestion cover cover dependency precision suggestion call need whole large enough position dependency order relevance roc rank close rank use rank iff evaluation sort fact try predict dependency learn topological sort fact ai interactive proving proof theorems performance nn naive comparable library correspond symbol proposition system heuristic dependency need recommend proof auc k ensemble far overall encourage library prove conjecture hard system include technique could implementation work directly logic thank help terminology research support proof dependency obtain dependency formula proof come dependency comparable large corpus last decade interactive various system system light reach previous proof already available library system important relevant theorem library actually separately early automate ai
evolution universe lead formation without shape shape information human understanding complex distribution galaxy formation evolution universe shape view point must perform way free shape galaxy image approximate perform analyse dataset galaxy show general idea sparse code approximate combination predefine approximate dictionary adapt basis well predefine galaxy solve frobenius package choose image must eliminate image small standardized band method galaxy low dimensional image intensity paragraph dictionary call dictionary collection collection suppose pool dataset fit dictionary vector suggest mmd statistic mmd select two dataset see image generate randomly n jk jk jk cv ht approximate galaxy dimension galaxy help distinguish distribution future work constraint galaxy shape shape elliptical one galaxy pa
square exponential periodic define kronecker delta kind symbol define sx replace sigmoid simple application start equal expression propose search operator parameter optimize scoring search greedy search operator base c acc sp cp se sp mkl se lin material report maintain recommend read first review discuss analysis demonstrate temperature exactly call centre demonstrate many highly structured white automatic regression explore open discover explanation set treat consequence trend compositional language allow term rich domain statistical field rely machine researcher simple automate package little interpretable intelligence modeling automatically paper conjecture ai statistic work incorporate open expressive enough many composition world phenomena search efficiently span language evaluate complexity fit automatically explain visualize choose quantify improve part describe note know call automatic covariance discovery define gaussian process information evaluate compositional develop language show automatically report interpretability term find art series consist learn expressive complex nonparametric smoothness process jointly practice mean equivalently addition rich structure trend composition way particular incorporate figure expand amenable automatic extend table list common model e operator include material descent bayesian criterion q implement describe generate natural description language convert expression simplify sum product kernel different multiply multiply original rule write kernel different independently product separately describe contribution noun description justify multiplication remove long since decrease increase linear cause vary linearly multiplication multiplication multiply periodic formally ht l phrase smoothly periodic linearly amplitude polynomially act table product form noun phrase noun noun phrase uncorrelated smooth polynomial number way description head noun interpretability description qualitatively rapidly description include periodic period description extra linearly report noun choose head noun choice area attempt present first add component fold q convert q head noun description uncorrelated smooth correspond finally third describe periodic period period demonstrate ability discover material year cycle rest rare identify description summary see identify component component third term trend well slowly vary trend next model component accurately describe periodic white express description system learn constant replace capture offset product approximately periodic component heavily approximated span kernel significantly interpretability discuss rational gaussian express infinitely capture short one rational quadratic material component capture medium term trend short visually describe contrast separate medium term deviation mat ern reason ht xshift xshift yshift box right find periodic great zero semidefinite similarly product become qualitatively anti credible xshift yshift box xshift xshift yshift yshift credible interval anti manually composite kernel supplementary include dataset procedure choose similar paper whose fit composite interpret automatically manually model spectrum delta kernel spectral kernel table body construct rich mkl polynomial time express language form specify time covariance function search decomposition graph space define select criterion model differ automatically statistical natural output automate ht accuracy building evaluate performance list time library report supplementary material six spectral trend produce interpretable predictive include brevity experiment default express restriction restriction mkl trend spectral greedy procedure marginal optimisation advanced use method move average construct class series interesting future bic parameter nest produce
become huge distortion distortion equal gap couple run distortion effect distortion contour source decrease curve source plot figure increase increase correlation se section corner terminal observe wave start boundary proceed center recover gradually particular create wave boundary depict experiment keep couple wave wave stop proceed recover result spatially couple see result error gradually increase contrary spatially case wave proceed variable stop sharp transition measurement rate transition depict sc wave wave space variable write finitely q binary nonzero index unless similarly linearly deal case terminal joint try single terminal matrix two check message fix appearance om variable check message different therefore get obtain eq negligible asymptotically fact tend similar giving replace obtain similarly equation intuitive justification terminal amp matrix word additive consist amp follow asymptotic get theorem row similarly gaussian converge compatible similar derivation tt argument obtain imply give equation ks mmse observation similarly variable covariance elsewhere matrix whose provide mean let conjecture remark amp initially compress sense matrix extend multi terminal show terminal behavior characterize se distortion terminal source spatially couple rate distortion curve phase approximately distortion fully measurement distribute match se pass amp measurement coupling terminal pass enyi dimension interested sufficiently many matrix respectively take separately recovery terminal hoc environmental signal temperature etc one imagine sensor terminal fusion center sensor communication low processing joint terminal process fusion recover usually exploit redundancy particular scenario sensor device turn increase kind temporal temporal correlation result slow temperature signal correlation sensor densely environment precise result redundant energy resource desirable sensor environmental densely distribute sensor assign measurement case require distortion address terminal signal terminal correlation sample study terminal make connection distribute source please refer study prove regularity condition encoder negligible enyi decoder prove block enyi necessary exploit hadamard capture spatially rigorously prove rate source source feasible rigorously analyze terminal term code replace also develop multi terminal variant compress well behave probability signal define independent reconstruct source behavior se distortion distortion spatially couple distortion low eq x dx mmse eq q mmse limit eq lebesgue theorem decompose continuous part e enyi singular well weight restrict singular space brief overview linearly conditional realization terminal vector whose component joint se depend state choice mmse estimator point se point simply check increase hypothesis result case already traditional rate require rate situation code threshold result pass different threshold spatial necessary briefly describe structure spatially couple measurement band weight roughly e band diagonal denote row column indice terminal ratio n measurement rate terminal variable explain terminal whose index variable mmse spatially couple accord index take belong block column belong block terminal spatially couple output equation obtain asymptotically infinity base terminal case give follow terminal terminal q ensemble couple measurement separately recover negligible rate code role discrete code corner achievable terminal distortion multi terminal term mmse use se dominate converge converge step single terminal zero gaussian use linearly similarly possible achievable dominant face measurement achieve corner copy ensemble negligible region rate dominant distortion
use way gaussian different kernel object similarity generate case interpret kernel ratio show problem class mkl rt formulation applicable like orthonormal pls procedure guarantee problem experimentally mkl rt select reduction modal retrieval mkl rt well mkl dr formulated optimization discuss ratio present mkl rt conclude indicator transpose denote appropriate denote absolute two ratio trace datum transformation use prevent overfitte popular pls make solution correspond pair kl transform x early computer vision trace evaluation mkl rt label correspond representation use cross modal retrieval map modality dimensional concept modality highly correlate modality feature ratio trace modality xx pair two modality common latent suppose provide label modality directly extension label sample class way implement replicate without implementation ratio modality xx n z far mkl parametrize combination learn mkl ratio trace formulate nevertheless symmetric rank eigenvalue eigenvector let define optimal please supplementary material iteration compute successively restrict maximize verify optimum individually unconstrained program system eq h compute solve I update summarize algorithm ratio transform representation mx summarize rt summarize new linearly transform mx mkl rt trace explain covering application discriminative modal retrieval wikipedia modal rt trace define solve trace use correspondence trace mkl approach dataset use svm nn rule reduction compare mkl rt nn best svm nn nn kernels eq reduction use use number recognition category kernel various image available online omit different split average use per regularization rate split follow approach produce pt discriminative constraint mkl dr figure mkl dr training sample multiclass image per come predefined split moreover author brevity omit matrix mx mx predefine average regularization set various approach svm report use standard figure weight mkl rt dr third split rt rt rt mkl rt mkl svm svm rt rt rt mkl mkl rt mkl mkl dr experiment modality modal wikipedia article retrieval text design training test group broad art etc text linear allocation provide histogram orient consist histogram pyramid rbf sift descriptor order descriptor sift descriptor pyramid histogram rbf matrix pt histogram rotation correspond pyramid kernel simple generate pyramid generate rbf descriptor record pool score see mkl rt give dr perform poorly rt rt mkl rt respectively show mkl mkl mkl rt mkl select show cross modal pt image rt rt rt mkl dr rt wikipedia convex different modality common distance wikipedia dataset number average score text mkl rt give poorly rt rt rt figure mkl mkl rt whereas mkl dr kernels text rt rt rt mkl rt rt wikipedia non mkl wikipedia select object image test text linear tag rank provide various kernel rt retrieval performance mkl dr approach perform poorly mkl rt consider rt mkl mkl rt select kernel perform result much present query rt rt rt rt c dr rt paper show formulated ratio trace include like orthonormal pls provide propose ratio problem demonstrate discriminative cross modal retrieval rt non mkl dr plan trace line mkl mkl plan rt pt pt automatic mkl attention various research community mkl context machine explore ratio trace formulate popular procedure converge global experimentally mkl mkl rt successfully use modal rt perform well recently propose mkl many vision application often transform initial new representation application consideration transform different interested image query may want representation correlate plan linear transformation many trace whose solution extensively computer vision popular algorithm ratio trace semi supervise fisher locality
maximization equation require chain hope dependency detailed indirect identify conjecture conjecture maximizer conjecture give easy identification testing purpose chain period allow coordinate least test conjecture random create distribution rbf arise choice g adjust adaptively decrease systematically varied along unit dimensional attain bottom conjecture argue justify heuristic indicate global possible start configuration adaptation optimum computation maximize drive coordinate adaptation progress cd consecutive progress provide optimally wise sufficient coordinate algorithm wise become ensure condition rate stationary approximation progress algorithm strictly decrease equilibrium quasi deviation additive progress deviation term leave side accord thus small enough towards equilibrium equivalent assumption indeed fulfil practice approximation cd extremely step argument gain validate test clear fulfil proxy deviation result progress usually flat optimizer careful variance variant discuss software problem cd class analogy stop linear linear algorithm soon tucker drop establish value even compute problem logistic component dual drop straightforward indicator always reliable indicator coordinate correspond roughly zero lasso cost cd varie svm extension library solve iteration solver pick derivative set evaluation list separate parameter varied grid apply shrink carry class comparison similar shrink solver well considerably time wrong shrink decision shrink well drop shrink uniform instance news benchmark multi svm experiment experimental baseline value r iteration second class svm subspace descent parameter vary within implement logistic solver analog linear dual logistic regression shrink applicable apply l c fold cv iteration second well speed indicate mark stop day news center good cd bit slow case improve significant relevant namely result training time logistic cd order cd tend meet coordinate overhead adaptation value regularization run extremely soon start pay sometimes dominate outperform svm shrink shrink priori contrast generic speedup specific technique shrink result introduce coordinate coordinate coordinate cd aim maximize minimize cd coordinate obvious feature cd guess inform particularly coordinate inactive perspective coordinate maximize characterize progress coordinate property true provide well simplify assumption towards equilibrium understand markov chain primary goal application notable shrink set compare art implementation four new show systematically outperform fall rare case adaptation descent universit cd method support lasso logistic general cd coordinate fix coordinate frequency remove need estimate requirement usefulness arise offer speed art becoming increasingly solve task svms lasso regularize cd pseudo newton stochastic contrast cd gradient iteration particularly vector problem step factor point stochastic plain equipped computational e cd obvious uniform cd non consider probability interestingly recommendation nature e run keep constant simple quadratic objective difficult propose realistic optimization scenario proposal adapt progress therein need adjust run often run similarly trust online parameter local characteristic adaptation cd technique extend inspire coordinate close coordinate coordinate direction step maintain adaptation inefficient adaptation general turn technique refer follow basic coordinate selection summarize review machine method algorithm coordinate selection probability new problem variable simple value want allow index let minimized course handle technique cd present cd partial derivative compute computation partial coordinate solver optimally e search e analysis analysis find cd variation highlight selection coordinate prominent cyclic epoch loop loop coordinate coordinate visit always epoch predefine coordinate avoid equal attention view dependency structure epoch pick select simplest distinguished index random generator nesterov draw often iteration greedy manner knowledge inefficient notable linear svms working heuristic drop svm cd refer extensive cd old apply factorization linear prohibitive cd distinguished coordinate often machine choose uniform advantageous uniformity question often solve address nesterov derive runtime bound constant minimization optimization offer problematic coordinate problem priori lipschitz derivative coordinate tighter costly continuously importance run reason outline concept implementation machine exception shrink svm cd cd suit problem one advantage quickly often speed insight heart shrink svm often result regularization absolute selection operator alternatively dual result svms demonstrate outperform four serve course sparse start output label unconstraine either instance simple form et propose coordinate result empirical term restrict coordinate time step newton derivative distinction need newton step end zero costly step denote number zero compose input input find follow situation linear solve dual cd key keep track I program dimensional newton derivative cd read eq interval densely step arrive applie remove originally remove normalize technique common use online note coordinate matter robust result costly follow warm adaptation justify svm importance coordinate course outli move maximal fix variable drop become relative upper helpful online trivial svm different extension binary exist arguably reduce two multiple turn lee al piecewise originally svms perspective classification denote subspace solve either qp solver style default box treat attractive problem solve arbitrary derivative computation sub logistic regression smooth hinge solve efficiently cd logarithmic share property allow solution cd iterative implement shrink technique applicable represent subject simplex review technique adaptation value type technique drive extreme trivial aspect parameter online adaptation essentially case performance increase tune efficiently online fashion break parameter differ gradient poor machine descent unbiased carry sake minimization inefficient solve online adaptation al model function online available adjust give outperform sgd deep reference therein propagation backpropagation constitute maintain wise roughly gradient sign derivative adjust multiplication agree simple scheme refine understand fix grid adjust characteristic problem probably make evolution class randomize last evolutionary highly respective sample recent es matrix denote turn extremely poor parameter fix decay roughly online e classic modern es treat whole free adapt essentially filter maximum generate block despite satisfactory theoretic understand adaptation newton case suffice adaptation powerful optimization raise date apply coordinate ask indicate beneficial direction adapt cd optimum optimum course condition fashion property seem coincide progress objective value become independent tool namely maximize decide answer straightforward quantity average progress coordinate progress make progress coordinate small roughly monotonically decrease soon progress answer monitor progress optimum relatively little progress equation nearly progress formally speak add beyond standard cd algorithm progress turn case product step majority progress acquire avoid rely old sample coordinate record progress order progress average fw fw quantitative update many possible form fine right reason adapt unnormalized preference track define coordinate step progress formal preference adaptation record progress default parameter initialize inform available e coordinate p p ip l default ad hoc tune insensitive simple possibility ideally like complexity despite variable index list output coordinate coordinate next inclusion guarantee argument theorem alternatively terminology algorithm enjoy scheme g cyclic formalize section chain first distribution formalize mathematical show quadratic eq exactly finitely coordinate rule exact trivial relevant chain divide unconstrained quadratic relevant understanding cd sake twice optimum
bias sigmoid linearity branch model bag translation highlight share connection sentence bag sentence translate language representation align translate representation achieve regular autoencoder propose reconstruction sentence representation language vocabulary representation sentence encoder able language reconstruction reconstruct language x z k z notice share encoder language given reconstruct reconstruct experiment equally weight investigate promising investigation exploit task corpus mention word train simultaneously encourage frequently align similar network language whether look representation phrase mention useful linear separately skip language propose train neural network learn phrase segment phrase base model learn setup test corpus importantly label document quality word language embedding overall early stop training document word language selection classifier test set word document linear compare use step embedding english english pair vocabulary document embedding either version choice normalize result autoencoder worse report original might preprocesse datum category word picture visualization show visualization frequent language confirm autoencoder learn embedding language en test embedding train gr en en gr al english pair right near english meaningful without rely autoencoder word preliminary extension bag autoencoder bag model thank probabilistic assign act useful acknowledgement help also provide dataset big thank universit universit work rely translate align word language autoencoder word representation autoencoder reconstruct sentence encode extract translation english compare exploit prove nlp meaningful representation syntactic similarity small generalize vocabulary start look align across representation machine translation common approach word alignment translate sentence relate embedding without word alignment corpus align sentence usual model learn relate paired bag sentence word want language document language level initial reconstruct autoencoder set work bag vocabulary bag word correspond order sentence word autoencoder bag word representation embedding train reproduce word meaningful encourage original bag choose decoder representation care choice reconstruction decoder efficient design reconstruct document associate reconstruct decoder treat word multinomial q must ensure compute efficiently
problem show k ij inner inner unbounded conclude k equivalent define exploit common imply tight go iterate appear redundant restrict lie feasibility h every closed definition eq minimizer follow separability write eigen decomposition fix minimizer define kb imply return argument prox prox exist singular decomposition form prox typically hold parameter rapidly set discuss exist convergence penalty experiment increase penalty admm I n ik I I see process stationary k f k block non less serious surface precision stage optimization stationary briefly need covariance ss rs problem covariance optimization search range appendix parameter close form minimize show objective stage equal square exponential unimodal true univariate minimization iteration stationary ss optimization e block estimate compute block prediction precision provide simulated real allocate parameter prediction th denote algorithm th standard parameter also define square se covariance range replication standard replicate covariance however deviation replicate value set prediction obtain value admm rs rs scheme split block test ss rs domain ss rs scheme show model fitting cccc replicate ss rs estimate appear unbiased deviation considerably point though violate point contrast seem reasonable domain rs scheme per rs well setup result range rs sensitive ss moderate high scheme come deviation replicate ss rs test different review code code method generate isotropic square se value method describe input rectangular mesh select boundary freedom knot initial replicate hence provide therefore evaluate fair randomly initial replicate solve first initial solution learn standard replication htbp see estimate covariance degree substantial considerably alternative carry intel ghz gb cpu require stage fast method fast fast unable unbiased range crucial considerably bad air total measurement website analyze cloud collect imaging website matlab read format input software website use covariance deviation table htbp covariance rs cf rs deviation quite magnitude magnitude confirm alternative precision selection method provide considerable scale parallelization stage optimization alternate direction approximation distance frobenius segmentation rs capability method turn square compete surface block since second line investigation numerically matrice isotropic near corresponding work analysis asymptotic covariance problem realization fix investigate estimation simplify range correspond direction connect range pairwise block inner vector ij rd ij ji eq multipliers replace four nonnegative minimum solution optimization one line parameter field via optimize function solve point computational ml inefficient procedure process solve multiplier parameter magnitude approach precise alternative handle enable without approach stationary convex gaussian markov selection gaussian field include traditional rely maximum likelihood mle difficulty yield furthermore mle operation routine inefficient call overcome fitting realization sparse inverse likelihood use regularization precision matrix parameterized constitute stage solve alternate method admm covariance square problem consistent solve region fit result let realization additive usual without mean countable joint I mean point krige variance rely correct unknown simplified family c main lead matrix ty j f j I respectively correspond marginal require solve contain family e local minima evidence stage work one basically deal precision global small block feasible precision resource note estimation big six function spectral approximate covariance usually determinant expansion reduce computational process technique achieve split estimate formulation predict full relate addition class approximation covariance propose localize rest present method solve efficiently line numerically prediction method literature conclude provide remark propose four isotropic c comes study sparsity property vector propose symmetric pd matrix denote cardinality operator envelope growth interest decade optimization include work exactly markovian way line work include regular random conditional assume neighboring lattice index lattice prevent exactly elaborate markovian markovian precision equivalence advantage approximation e behind lattice let determine conditional independence set employ retrieve implicitly precision make computationally hard require motivation spatial utilize precision indeed function exponentially precision utilize likelihood h compose convex likelihood program inverting fit least perform propose k kn note predict block follow scheme consider
utilize n use theorem thm computer science engineering university decomposition randomize randomly sample form compute statistical leverage refer incoherence simplify incoherence bound incoherence row limitation full assume besides row sample randomly randomly compare advantage perfectly include entry addition rank even finally adaptive randomly db bb ab uniformly random entry index goal optimization problem recover notation rank singular vector svd decomposition column incoherence incoherence measure projection norm matrix let assume satisfie subspace span column ii function need analysis lemma us theorem assume accord theorem directly directly perfectly observation need note incoherence assume incoherence row assumption sample hence convexity e let define canonical theorem minimum row end use define nm I k thus skew skewed constant q next incoherence measure numerical rank incoherence constant easy utilize replace eq
evenly precision equivalence definition near ff node different j vector first continue unit assumption repetition follow distinguish algorithm communication bound bit bit end algorithm atom atom claim different output atom similar argument node select bit constrain simplex extend case diameter factor match dependency communication bad case two body loss locate alternate direction multiplier dual communication cost function total section none method rely gd sdca need iteration converge illustrate method partition q regularizer across iteratively subproblem prediction node current converge converge within iteration lasso dense iteration slow add hand feature create tradeoff number communication admm tradeoff study select way fw communication scheme readily match pursuit enjoy series experiment strategy tradeoff distribute approximate first baseline strategy select atom method node select subset random strategy fw sized objective subset batch compare communication decrease admm section evaluate distribute computationally carefully particular kernel receive computational implement connect cpu core label versus rest rbf kernel average point runtime different synchronization issue across exact see illustrate assign share large center atom balance I reduce runtime unbalanced training negligible way cost randomly drop suboptimal iterate show average node asynchronous properly bit slow drop version set future element focus communication overhead frank wolfe favorable communication quality experiment confirm support nsf grant grant fa award microsoft research fellowship contract nf reproduce purpose annotation herein necessarily represent policy prove execute ik execute moreover execute information part iteration require constant value claim frank wolfe I come moreover step frank wolfe stop k g communication fact use wolfe update communication case change objective bind extend two node apart put communication claim carry university edu frequent machine element locate address balancing end propose distribute frank wolfe theoretical cost combine bind construct validate synthetic world baseline compete method study relaxed fairly execute computer year become increasingly machine mobile device wireless sensor naive interesting fundamental perspective study tradeoff attract interest view lot therein practical problem sparse combination spread support vector adaboost formally broadly dictionary basis etc weight weight atom g continuously frank wolfe adaptation centralize counterpart able communication atom introduce balance distribute machine term depend family prove low deterministic construct implement parameter problem practical lasso distribute baseline atom criterion sparse compete iii real world centralize communication drop update review frank centralized propose variant practical example match review conclude scalar k k simplex j j frank wolfe fw constrain problem compact space say fw move linearization current iterate stop surrogate product let optimal theorem fw find curvature terminate satisfie fw solve minimization extreme exploit subproblem norm find combine show find turn bad derivation matching improvement desirable interest subproblem ensure iterate nonzero entry problem result distribute overhead f sum criterion k iv compute atom k connect edge simplify assume simplify atom matrix wise local set index efficient set first identify absolute node ii corresponding subsequent iteration iii fw update round nod termination node optimality atom entry show frank wolfe execute far allow communication dependence total appealing scale I strong theoretical linearly atom suffer overhead due atom cluster classic greedy repeatedly center run intuitively atom nearby center select lead gradually center essentially following result local index opt opt g gap radius node exist atom claim second claim follow practical pick proportional another variant highly unbalanced problem efficiently include code learn regression aim approximate sparse feature respectively subset training distribute logistic perhaps interestingly lasso come source multi view encode categorical n dy psd iy augment notice lie version machine frank wolfe direct svm h j adaboost formulation tune thus straightforwardly classifier wolfe update add base classifier define point currently misclassifie potentially family classifier learner
homogeneity x although since correspond unit norm polytope finite accord program convex depend magnitude attention sort hull belong none combination depict complete vertex sign configuration notice choice mention previous paragraph zero large sort large n denote obtain take derivation prove let moreover allow write group value split respect proposition obtain group group obtain denote operation averaging finally satisfy generalize permutation proximity efficiently solve fista alternate direction admm aforementioned algorithm operator state short decrease weighted sort fundamental namely email lx lx family regularizers sort generalize recently norm instance argument sort non paper derive sort proximal splitting algorithm introduction recent year devote sparsity regularizer variable grouping unlike tie negative simultaneously encourage sort multiplication q sorting replace penalize regularizer write sort relationship different regularizer comparison regularizer convex relaxation induce encourage induce regularizer convexity lasso consider special regularizer sort negative sequence notable give entry include follow notable case non increase sequence group form focus term dual regularizer manuscript aware formally motivate result paper work dual optimality square denote wise sort tie
find optimizer later optimizer distribution dl consequently draw eq speed loss optimal estimate denote draw another probability distribution distribution dd rejection help condition stage algorithm return mm odd learner oracle depend passive addition learner active learning require linear dependence open lemma odd p j show first relate suppose probability empirical average odd last inequality statement union prove appendix provide ready definition alg inequality lemma note cr nc require lemma noting follow cd active imply observe achieve learner n px p p contrast passive receive label estimate mean variance mp passive learner many open active whether plain ridge static carlo estimation allocation constant open type active stem slight condition lead require theorem side constant since convenient derivation follow label example standard completeness sample md xu lemma jensen inequality definition france propose parametric set improve passive setting passive nonetheless characterize optimal learner risk regression linear square error design draw I cost useful costly obtain essentially learner guarantee regression passive excess unlike finite passive study simply square relationship wise predictor rate allow adapt underlie technique common integration increasingly refined globally learn parametric active specify use design normality approximately noise design possibility adapt propose adaptation query potential consistent pool focus learn formal setting notion state approach passive support strictly label example without subscript marginal distribution denote denote predictor predict dy dl optimal dl dl dd drop class throughout integer round effect negligible expression universal whose crucially simulate cost select sampling regression explore mostly asymptotic regime bind finite hold use label cost example rigorously derive lebesgue integration follow l dd r np passive learner solution eq constraint lemma proof complexity reduction ratio highly symmetric general active access conditional learner knowledge approach disjoint subset class I
deduce learn understanding rnn help utilize recurrent recently research recurrent institute technology chinese ac cn com xu com edu wang microsoft research microsoft com microsoft microsoft com bin wang chinese sciences ac liu microsoft research microsoft com click fundamental click search ad yield high behave past ad ignore spend page ad observation recurrent neural sequential behavior click click engine approach click major business modern web account google yahoo search generate keep accord click user maximize engine crucial click ad click click extract historical ad element ad ad work click ad recent point achieve ad query conduct study type user behavior behavior explicit click ad ad view ad fairly query ad finding motivate advance art click prediction temporal click although kind model still hard identify explicitly ability various kind dependency event click user query ad thus natural model dependency behavior series trend periodic ad capable recent study long span massive language neural language rnn speech recognition much improvement feedforward capability specific leverage dependency click consider user intrinsic hide previously accumulate hide embed recurrent large search reveal click state art dependency fold ad relationship use network user sequential dependency experiment validate rnn verify dependency might affect click rnn click experimental study last future understand sequential click discuss effect collect click enter ad page stay click user order along effect click click ad previous click click obvious long user likely click next click second ad click quick observe give rise click user experience user behave quick click long month quick click click interval quick back click figure along click significantly stay certain tend gradually pass study effect system ad automatically query also order bind user query query topic click ad topic cause strong dynamic long enhance click big challenge manually necessary kind widely recurrent output previous try sample previous effort devote towards long context back rnn overall rnn base click view deep recurrent share identical input sequential dependency denote unfold network output click ad weight layer gradient layer hide layer represent wise error hide weight recurrent weight slight contrast rnn test sample feedforward recurrent weight testing process user feedforward sample current make prediction store current hide last matter unfold validate propose really enhance conduct click engine month th engine event whole traffic ad week click prediction week datum dataset ad ad train click ad testing dataset record click click follow click employ metric investigate effectiveness compare performance rnn include lr network nn study click rnn framework able comparison rnn leverage improve click auc lr fair comparison every model achieve include epoch nn unfold rnn good epoch hide auc three rnn click particular nn relative system click prediction increment overall model conduct help click click rate varie often refer analyze rnn ad position evaluation position lr measure position rnn traffic come click rnn achieve position ad ignore user click rare drastically lr nevertheless well still inference rnn far importance utilize historical remove nn ignore dependency auc severe drop information recurrent structure significantly part check history collect sample fed accumulation period continue sequence auc user long feed accumulation maintain long turn model perform good setting accumulation period
constrain hypothesis class contain regardless theory predictor bound fix would correspond lipschitz square notable exception smooth tight contrast implicitly take prediction excess manner magnitude learn tend norm predictor value remain fix lower universal return logarithmic predictor r forecaster give good algorithmic return bm imply even trivial dependence specify common distributional upper dimensional symmetric dimension lead third optimal dimensional bound distinguish distribution excess risk bind respect eq predictor either uniformly attain lower bind theorem leibl instance target invoke kullback divergence plug excess lower pick use construction type quantify standard dependent careful constant returning later uniformly index calculation notation inequality deterministic kullback leibler compose md moreover get kl divergence eq fact back excess eq value pick least pick expression case thm prove thm take thm otherwise helpful short predictor agnostic distributional performance machine statistic well linear standard parameter existence mm example consist pair instance respect hypothesis possible focus excess randomness problem uniformly despite unable result include one agnostic nothing boundedness rely mean work
u uniformly leave find asymptotic get pe pe u asymptotic choose way component sphere vector uniformly sphere e u remark get integrable note q euler last rgb rgb cm assumption asymptotic aggregate elliptical universit france universit de france science business economics university abstract asymptotic tail sum elliptical motivated extreme risk finance result rare calculation numerical illustrate order key word aggregation asymptotic elliptical modeling risk management pricing standard common log risk g despite derive behaviour sum risk appear contribution al normal underlie threshold parametrize importance vector one natural aggregated elliptical paper asymptotic elliptical index contribution precise quantification claim approximation positively bivariate obvious numerical derivation asymptotic univariate log risk elliptical risk b asymptotic impose restriction elliptical risk dimensional normal random singular diagonal equal speed decrease probability imply rest section positive variable function uniformly sequel satisfy far hold calculation accordance finding hold scaling see von fx e et al since radius satisfy far j u c u u u locally numerical mean correlation practical second asymptotic replace term equation approximation use monte numerical study first tables monte carlo monte approximations column first provide heuristic measure quality approximation calculate inequality fulfil improve still order hope asymptotic order asymptotic significantly asymptotic quite display sufficiently c mc ratio main result sequel positive constant fr give equal every exist
set method bag bag paragraph bag learn third paragraph maximize record time paragraph error produce desirable triplet paragraph vector baseline tf weighting perform raw result tf paragraph paragraph significantly outperform bag suggest capture average bag word bag bag paragraph vector table far dm dm alone table dm often well sum dm achieve order information validate guess many varying parallel compute paragraph set distribute representation paradigm language nlp parse translation phrase recent receive attention style model phrase sentence typically parse sentence obvious extend paragraph contrast paragraph vision know fisher kernel vector generative paragraph unsupervise piece text predict sample paragraph classification stanford sentiment art demonstrate paragraph paragraph bag text text parsing machine require text common bag despite popularity bag two ignore paris distant unsupervised learn length piece predict algorithm paragraph outperform sentiment g spam heart machine require represent length text gram often however lose representation use though bag consider word order short suffer gram semantic distance word paris distant strong paris learn text variable sentence name vector emphasize method text phrase sentence predict paragraph paragraph word paragraph vector unique vector share paragraph vector infer word vector train paragraph neural network vector average use neural language use concatenation input neural network try next close researcher try word level sophisticated approach word sentence operation average word word vector show sentence rely parse paragraph capable construct sequence variable text length document present dataset paragraph sentiment analysis text give discuss previous paragraph introduce word word every word map unique prediction word give word multiclass softmax q compute softmax concatenation extract softmax prefer softmax fast structure tree short assign frequent good speedup common binary word descent model commonly implementation code google com converge word map close powerful paris distant difference vector carry mean word answer analogy algebra translate word phrase natural processing understand statistical extraction paragraph word ask prediction word despite semantic indirect prediction vector paragraph ask contribute task context paragraph paragraph figure paragraph unique paragraph word combine change paragraph token act miss current context paragraph call distribute memory paragraph dm slide paragraph paragraph context paragraph powerful stochastic descent obtain backpropagation descent context paragraph compute gradient use prediction paragraph vector descent rest softmax suppose paragraph map word dimension exclude though update paragraph token model concatenation context fourth paragraph context act paragraph length sentiment dataset stanford sentiment length al consist sentence also task document query subsequently extend sentiment sentence movie review site consist sentence sentence amazon dataset achieve stanford label human label label dataset http nlp stanford classification positive task axis whether label sentence sentence label full al apply dataset movie review play set follow protocol independent representation sentence feed logistic regression rating sentence representation feed logistic movie experiment validate window window concatenation vector dm paragraph character paragraph less pre special model grain bayes svms na neural matrix rnn neural paragraph report table highlight table bag gram perform poorly average word fashion bag sentence compose g word recognize sophisticated linguistic phenomenon recursive require parse take perform recursive parsing grain classification term translate relative sentence recursive parse unclear parse many demonstrate et sentiment analysis review key review sentence movie review divide dataset unlabele balance http ai stanford edu sentiment paragraph paragraph vector label feed predict paragraph sentiment review paragraph previous task validate window word present concatenation vector dm dm paragraph vector special character treat less word paragraph see model perform combine approximately come variation work considerable go barrier
minimize kl divergence minimize descent fixing variable form update execute meet feed output bias softmax illustrate correspondence bias vector tb forward iteration bias give bottom layer chain schedule determine b update schedule update bias drop grey height update point special feed network bias tie equal pairwise mrf pairwise restriction feed forward viewpoint nothing stop restriction restriction mean field output relaxation discuss aim beneficial field specific obtain grow expect layer layer different thing getting converge connection pass fast add flow usually helpful layer allow layer connection create relaxation crf potential aim crf test equivalently output budget compute potential target train output kl compute back develop feed forward network pt eq gradient chain use output discussion inference discriminative model factorial layer via minimize output sx sx tf part loss indicator form step layer step discriminative hinge usually straight integrated paradigm relaxation different relate pass step train graphical fashion empirical risk back propagation optimize graphical compare feed network see restriction straight forward derive enable restriction like tie another briefly connection binary mrf paper compatibility inference inconsistent approximate long align algorithm time problematic compatible neural side people try neural intractable belief therein connection field propagation paper approximate limited share spirit background intensity white intensity english foreground noise pixel intensity add pixel noisy pixel two foreground consider crf posterior output image pixel unary pixel vector unary potential potential one horizontal inference initialize take unary maximize conditional approximated use marginal initialize except unary mf improve inference crf parameter train layer layer baseline test mf mf number divergence constant log kl improve significantly well mf iteration train denoise directly start three tie weight baseline mf achieve mf learn hinge field baseline
vision application interest heterogeneous sensor span classify image view directly learn view transform newly canonical correlation cca pls sample correlation pls consider variation besides mutual theoretic encoding narrow inter sketch bridge view discrepancy view label method discriminant instance discriminative canonical propose local learn discriminant extraction large approach recently jointly view transform view extensively encourage transform capture share view however agree indicate transform extract deep attempt learn via vision stack deep structure building include restrict encoder stack auto effectiveness denoise domain speech etc know canonical also widely learn method much flexible learn process set inspire deep natural build view totally modality suffer representation capacity view make infeasible two couple seem couple auto building stack auto encoder margin couple input denoise encoder modify maximum criterion kind etc counterpart add margin naturally layer network illustration see multimodal auto deep canonical tend layer canonical constrain deep representation great separate well view handle insufficient organize solution couple efficacy conclusion basic part discriminative encoder stack network deep fig circle pixel connection middle part whole project separability layer build stack couple couple layer insufficient stacking layer whole compactly represent set transform network gradually narrow gap discriminative capacity enhance discriminative couple train maximum margin incorporate training maximize learn discriminant couple two formally discriminative encoder criterion describe threshold maximum sample note representation attempt nonlinear transform project two discriminant respectively neighborhood separability preserve preserve denoise learn representation discrimination modify denoise margin criterion intra inter nonlinear trade preserve separability denoise auto error formulate follow version specify transform decoder calculate decoder representation tangent operation consist intra similarly counterpart view add rather characterize meanwhile class separability sample class formulate follow belong nearest neighbor satisfy condition function project common term show red work intra class penalty adjacent inter sample couple nonlinear transform transform view training process discriminant gap eliminate representation consistency world single complex real insufficient stack subsection network compactly significantly large transform gradually narrow gap ability enhance training couple nonlinear transform achieve canonical wise precise couple feed input feature stack layer gradually adopt lagrangian multipli first term constraint call utilize decrease far help prevent balance parameter local empirical separability call usually set employ l bfgs often nonlinear optimization problem large requirement utilize calculate differential objective achieve fast section sketch evaluate pose neutral illumination choose pose two subject subject dataset image dataset subject sketch subject train rest subject testing image without baseline art cca deep seven cross view jointly transform view utilize report reduction default dimensionality cca pls tune report cca adjust good cca strictly tune inferior reason cca training datum besides pls vary hide gradually layer pt dataset mean stack space limitation accuracy explicitly illustrate learn conduct dataset project learn common principal cca principal direction method attempt merge fail convert view diagram gradually layer describe come view respectively stack compare cross face acquire seven pose multi set experiment result show pose probe rate method supervise method significantly superior deep cca significantly superior
generate comparable good coverage generator mine fig original generate produce split train name test use auc fold average b b datum important build original test opinion indicator close generate would substitute original comparable generator try work generator less successful evaluation examine generator scale empirical evaluation repository great variability attribute class interface uci assume artificial original instance load evaluation limit original instance generator author htbp anneal cancer breast cancer breast cancer screening band heart heart disease vote diabetes primary heart new generator attribute produce generate instance parameter describe sect skewness per attribute exclude skewness comparison ks reject value attribute compare value hellinger response exclude similarity generator report average hellinger exclude ari exhibit report ari repetition generate use classification illustrate forest robust performance various come default tree set cross validate set htbp test match hellinger ari rand train test comparison applicable ari breast breast band heart house diabetes heart encode attribute generator represent nevertheless measurement core intel cpu ghz time instance report column label equal mostly happen primary tumor cancer generator contain unit activation consequently attribute gaussian unit column label average difference attribute individual ks test hellinger attribute label percentage k compare attribute ks difference average hellinger set distance considerable tool ari rand considerable high ari set low validate forest model train generate test trend original well set datum lose confirm generate mostly nevertheless satisfactory substitute mining namely build original test original large case overall conclusion generator semi reasonable substitute mining sect encode single make value report test statistical binary encoding nominal nominal include ari annealing screen band heart heart house primary tumor heart mostly large binary encoding case generator time create need compare significance binary encoding produce hellinger ari indicate original finding generator instances activate form instance overfitte estimate replace dataset anneal breast breast cancer screening disease house vote diabetes tumor heart disease compare nominal encoding nominal attribute pair proportion difference mean attribute difference original recommend safe try rbf test success rbf rbf compare forest forest successful auc package default forest produce accuracy accuracy side auc skip htbp anneal balance breast breast heart house vote diabetes post primary heart disease forest rbf networks rbf significantly tried identify success datum generator success difference model original difference factor difference accuracy rbf rf generator gaussian collect correlation pearson coefficient difference indicator rbf classifier generator turn rbf test scalability framework public available preprocessing already effort additional effort source turn work provide different characteristic instance proportion practically artificial successfully big generator exploit property generate tool useful adaptation use datum randomization ensure privacy simulation test scenario huge tool generator datum set uci able original generator success classifier generator rbf successful versa unable successful generator intend generator classification turn future extend generator module rejection base acknowledgment thank interesting discussion uci author support research definition university si expensive generator semi artificial similar original enable development simulation without generator learn generative generate structural similarity technique generator uci set challenge well know bring attention application opposite reason inherently rare business privacy record expensive require significant human interest long reliable performance development specialized tuning solve original similar great development specialized problem yet easy background weather context aside small purpose aware extract overfitte exist generator low mostly review sect approach problem construct rbf prediction consist instance generative overcome space attribute categorical paper organize generate datum rbf actual handle nominal generate statistical section present try determine working condition cloud generator data generation cover method problem group generate r support uniform normal beta cauchy multinomial generator package need mass provide parameter several generator distribution less effective generator multivariate simulate decomposition symmetric contain covariance decompose normally datum sect normal distribution normally successfully generate proportion requirement multivariate desire propose datum replace population iteration desire intermediate space capture limited datum clearly kernel estimation population make data approach frequently gaussian intend space copulas copula copula describe write univariate describe dependence datum copula knowledge number careful copula family copula type limit rbf radial function tool continue see contain consist unit class describe dimensional instance unit rbf probability multiply radial function center function kernel away rbf architecture layer must start avoid manual set center weight standard deviation solution rbf adjustment build encode process dynamically algorithm comparable rbf present hide rbf add training fully hide consist possible encode class winner take output unit determine illustrate fig threshold bind activation train good separability class achieve inner circle center idea extract empirically notable ability eq multidimensional give definite decompose r package pseudo generator encoding equal e would encode binary attribute binary encoding require line line datum unnormalize attribute generator generate training specify size function create ic instance generate var zero g sigma fix attribute jt span min l data consist gaussian attribute recall also specify generate control kernel width start create generate kernel list weight kernel ic ki width spread generate around line width line covariance diagonal diag width instance particular spread dimension generate take number instance generate diagonal exploit kernel check line interval attribute retain generate line transform nominal form transform generator dimensional grid attribute center red blue generator eight class illustrate generator fig location center kernel rd simple example well width scatter color generator instance fig considerable pair scatter aware generator capable exist attribute exist generator evaluate skewness indicator insufficient attribute attribute thereby overall account difficult also datum mining difficulty incorporate deviation skewness attribute generate hellinger kolmogorov ks whole process illustrate normalize attribute attribute sensible generate statistic especially attribute attribute
handle column nonzero determinant determined get another dimension add consider body euclidean space fr almost surely exist map let require k singular iid distribution laplacian observe equip state form readily moreover arithmetic fr unique secondly non condition satisfied number context interest index readily satisfy verify outside coordinate straightforwardly derivative express second derivative finally diagonal matrix v ki ij ij vanish indeed covariance give state appendix conjecture axiom grant air force scientific research functional project international resource lin suggestion year become summarize assume structural relationship difference network group familiar strategy object method apply challenge geometry high dimensional geometric fr motivate result functional univariate mean understand modern working say consist signal collection two dimension voxel build various form high representation employ traditionally naturally increasingly denote often consideration vertex image notion tensor association representative structural connectivity brain hand fmri association think connectivity together count clinical prominent research brain towards database compose collection database answer network nominal collection gender contribute finally say change network question network estimation testing fundamental practice dataset fact combinatorial object simply edge nevertheless certain natural euclidean combination tool geometry manifold principle practical framework analogy classical tool undirected laplacian matrix denote loop edge correspondence space subset euclidean either corner subset complicate nontrivial structural constraint geometry nice allow goal certain fr define borel fr mean similarly realization fr thus geometry define manifold able theory average test require nevertheless advance researcher mass field motivate illustrated sharing costly consume conduct discovery paradigm system release set access throughput scale lead finding functional subject locate center participant year year old old scan minute strength varied across center voxel plane slice center project medical school datum fmri automate labeling template voxel series region result two consider prove literature respect researcher impact connectivity human genomic profile brain compare specific observe research consider edge investigate focus specific global consider network extend characterize space sequel evaluate difference organization research question characterization inferential framework network explore report propose discuss entry database percentile address summarize compare principled device available summary comparison operation e symmetric difference broadly various hamming univariate conduct edge adjust fail draw whole multiple difference necessarily lead globally treat point mathematical formal equipped understand geometric topological underlie desire manifold particular shape fairly history back seminal work shape notion average nevertheless little seem study geometry derive fr also relate object formal characterization associate fall cone semidefinite psd lot explore notion choice geometry choice adopt motivated shape analysis play key space psd cone furthermore cone immediately latter necessarily discuss eigenvalue analogous riemannian formal date establish certainly none impact establish canonical g etc aware work characterize subspace psd correspond sharing rank crucial mathematical embedding involve embed smoothly matrix embed lee seem compare via g embedding useful g hausdorff literature embed onto technique particular reduction isometry geometry domain precise manifold employ describe exist embed space manifold preserve information average geometric variation affect geometry space probabilistic sampling equality I loop associate laplacian diagonal e far connect positive correspondence graph therefore correspond admit affine sum appendix practical importance usual notion curvature edge edge distinguish purely say correlation choose thresholded version theorem include corollary possesse manifold corner text smooth manifold lee convex euclidean entry non positive corner dimension convex space proof provide importantly indicate real distance space concept fr analogue well networks topology fundamental complex tend possess mark structural characteristic example edge heavy tailed suggest appropriate formally depend implication extend case graph connect component positive sum entry column proof intuitively graph community increase graph maintain characterized network inferential framework select derive average construction number correspond combinatorial identically might image necessarily definite definite place statistical power increase simulate base experimental design process rely generation network mutual matrix type order second second randomly firstly group whereas proportional grow secondly specify small world construct regular topology proportional edge diagonal small family topology simulate base label diagonal give adjacency simulation scenario thereby produce ratio distinguish type simulation result definite consequently definite default brain investigate main simulating stem absence produce pattern sequence draw process first scenario sequence realization random group scenario sequence time use identical restrict autoregressive autoregressive autoregressive provide subject network matrix interest either guarantee positive semi choice subject mutual respective mutual combinatorial association give every subject aa ds target combinatorial follow laplacian group matrix ds moment modify covariance estimation bar indicate proxy measure effect compute frobenius distance population network vertex group condition topology simulation size representative subject find study secondly fine region practice size allow power decrease power test effect frobenius two population mean varied thereby result difference frobenius discussion bar standard mean figure figure network roughly increase power proxy noise comparable material covariance behave topological subject poorly high likelihood b mutual define small size poor greatly albeit slightly resulted type consider fail hypothesis small result suggest estimation preferred compare subsample test pattern five subject exclude subject analyze york provide unique extract template connectivity laplacian subsample subsample nan reference hypothesis reject high partitioning figure highlight introduction influence brain reject database group accord age equal subject subject respectively hypothesis state draw nan use hypothesis voxel voxel mass univariate despite good effort may connectivity sample focus site international brain imaging york subject stating reject univariate univariate test independently entrie age panel denote significant significant small indeed paper usually subject last size order produce might sample conclusion hypothesis find subject whether mean reference different age subject subject site extract likely population univariate laplacian subject even highlight advantage context univariate fail framework average importantly mass collection purpose exposition summarize collect allow development analysis produce theory offer quite broadly direction briefly confirm sample beyond rate see sample control analysis condition subject current global therefore applicable however subset e analysis single low sample alternative difference employ challenging laplacian inversion matrix different facilitate modern modification sample covariance force wish use behave able theory however year although possess structural relatively common heterogeneous distribution subgraph establish importantly formally impose choice network geometry embed inside riemannian lead nontrivial matrix need apart simple manifold psd natural euclidean moreover measure psd fr mean course impose risk relation psd hence
rapid auxiliary intermediate previous current intermediate solution construct ff ft mild adopt way previous surprisingly explain fitting kernel svms solver n regularization search scalar use alg method trace regularizer corresponding regularizers g proximity regularizer variable u j g scalar tt alg close let htbp rapid dimension per variable normal start code toolbox rapid please toolbox manual norm implement fig comparison identical rapid empirical guarantee rapid big sometimes use rapid svms use optimization matrix sample svm formulate entry wise vector predefine rapid solve alg contain line ignore update variable alg alg fix fista test rapid contain dimension per randomly select rest solve rapid compare solver check f check rapid begin check value add future rapid result well rapid rapid ii present result rapid fista property alg arbitrary clearly definition alg alg satisfy lemma rewrite rewrite alg satisfie let eq strategy prove accelerate rapid speed introduce way construct auxiliary intermediate current small upper bound gap algorithm I summary rapid converge algorithm sophisticated edu proximal simple proximal variable intermediate solution gradient step upper current objective method fista converge accelerate attention area process convex convex non differentiable proximal gradient variable proximity operator proximity convex minimizer feasible return original classic alg size fista prove speak bottleneck follow aspect gradient could consume recent inexact allow approximate proximal method gradient fashion gradient decompose locally achieve proximal alg unfortunately line search decrease instance tune step gradually search lasso order convergence auxiliary intermediate solution iteration prove arbitrary consistently imply precision probably empirically apply lasso machine svms correctness fast case algorithm sophisticated solver include line search construct svms demonstrate empirical solver theoretical prove conclude proximal
uniformly dx cumulative integer parameterize hash u equivalent presentation step scale hash furthermore grid move present optimal depend ratio threshold threshold interested top therefore desirable hash threshold inner similarity contradiction let monotonically show simple lsh eq hash random mapping lsh every lsh hashing eq monotonically increase lsh desirable achievable hashing quality compare hash desire also hash bind tuned hash quality give dominate lsh well optimize recall hash setting high code hash function netflix procedure rating entry zero unobserved entry rank top netflix define presentation movie user user movie high hash hash code length movie selection sort movie ham hash movie hash break tie randomly precision curve average precision randomly recall item hash code netflix dataset suggest setting unfortunately whenever similarity show strong negative fortunately normalize pair transformation always cs mapping ensure h x monotonicity also modification search measure characterization asymmetric lsh possible inner symmetric lsh query bound vector universal lsh symmetric lsh lsh symmetric lsh motivate asymmetric lsh query database two actually third set asymmetric hash indeed suggest asymmetric even second set specific universal important emphasize asymmetric hash asymmetric hash one must normalize query database hash hash strictly identify acknowledgment partially nsf award advantage asymmetric lsh use two mapping use hash hash theoretical observation valid mapping alphabet universal similarity establish contradiction query universal c cs monotonicity max row I mf p indicator conclude max jensen rr complexity sign matrix margin remark claim theorem design locality lsh similarity argue lsh problem lsh symmetric lsh enjoy lsh variant setting asymmetric lsh follow maximum inner product search collection datum maximize inner query matrix svm score efficiently approximate locality sensitive hash lsh locality hash lsh object alphabet lsh hash word hamming distance word recent explore lsh different mapping approximate similarity similarity hash may enable obtain lsh lsh obtain well lsh tree search tree method impractical regime lsh superior vice versa yet lsh tree lsh considering argue inner similarity distinct query asymmetric lsh entire lsh thus show succeed lsh enjoy guarantee performance require motivated understanding obtain simple lsh conduct lsh crucial issue lsh entire lsh lsh normalize bound normalize symmetric lsh asymmetric lsh also lsh also enjoy well hash recently structure recommender study lsh property alphabet function hash distribution family study study two however assumption want l able assumption database lsh subspace lsh modification lsh inner similarity locality lsh hash lsh say efficient neighbor lsh object quantity lsh minimum hash symmetric require truly asymmetric space formally hash asymmetric deterministic make lsh asymmetric locality hashing asymmetric say lsh lsh lsh finding lsh lsh asymmetric help also hash inner similarity assume contradiction exist similarity consider sequence define zero triangular also set
sampler describe require sum computationally prohibitive efficiently key channel force initially compute pass indeed simulate conditionally employ normalize forward message top message column reverse draw define perform sample calculate message chain work message k instead change furthermore resample weight secondly interested calculate proportional function take capacity practice bias negligible dominate resample lead stability logarithm multiplier subtracting improve stability resample must add sequential modify propose explain sequel refer tree compare run algorithm time bar run capacity sampler give capacity channel bar approximately iteration error run burn scale size subsequently display error experiment tree sampler log oppose example tree poorly slow mix enumeration sampler apply compare width either plot versus perform magnitude tree seem gain noiseless capacity upon order furthermore obtain particle sampler improvement modern significant day rate source support contract suggest derive carlo capacity channel capacity run channel idea generally yield improvement computation ever capacity page orient storage constraint help amongst interference analyze theoretic channel storage numerically capacity channel utilize sample auxiliary target exactly focus propose generalization propose capacity problem constrain fundamentally sequentially state backward art algorithm propose continuous goal imply two adjacent bit lattice probabilistic underlying lattice graphical interaction x mass product encoding pairwise depth exposition refer reader constrain channel square lattice graphical wise configuration cardinality support capacity hence capacity channel unfortunately calculate intractable particular know noiseless capacity agree eight digit finite tight bound calculate noiseless necessary use thorough see adapt mean propagation minimizing adapt significantly reduce tractable describe undirected chain see specifically normalize sampler however target sampler approximate collection particle column point approximation kronecker delta particle mention adapt resample particle auxiliary square graphical well constant subsequent proceed hand simulating give particle decide particle generate particle drawing resample variable index particle correspond resample emphasis
parent boolean graph hamiltonian describe indicate clique consist whose second variable far assume dag maximum parent however situation desirable absence search hardware limitation prevent arc need accounting wish arc generality exclude free variable arcs remain substitution utilize reduction degree penalty hamiltonian purpose increase energy ground hamiltonian energy constraint sufficient weight necessary make necessity tight low may exist meet ground duration necessary theorem penalize quantum less remain property penalty arc concerned parent arc bit penalty great arc third absence bit formally penalty least achieve show justification appendix briefly quantity h ji associate quantity define fact monotonicity monotonicity removal arc energy iteratively iteratively I h penalty weight degree hamiltonian construct bit reduce locality general conjunction sufficiently conjunction done bit contain arc heuristic reduce bit penalty compute appropriately use penalty consistency high cycle less cycle encode cycle minimal consistency ng encodes cycle whose strictly k contain contain cycle dag dag minimal cycle penalty weight ij ng encode ground overall ij nh state maximize dag ensure dag parent however latter dag interest per se enable former amongst resource logical device embedding done often value exact great heuristic sa use present embed hardware wave deal nevertheless quantum state quickly anneal future advanced anneal code describe grant recommendation necessarily author acknowledge advanced support foundation award grateful david useful discussion decompose argument simplify trivially regardless value still eq calculation equation right reasonable weight necessary true ji ji ji construction min claim h I I h cycle contain prove show cycle h l graph cycle imply existence l complete arbitrarily assume r l ij ij furthermore claim modify bit q contain direct cycle h direct triangle triangle switch direct triangle j h r r h l claim let finitely triangle construct l dag proof contain consider cycle h h claim n quantum scoring equivalent enforce propose prove weight penalty logical mapping appeal instance give network equivalently factor distribution broad class mode learning specifically structure diverse discovery produce reasonable formal practice require heuristic quantum heuristic certain exponentially computer however exist complete provable speedup one believe speedup availability quantum anneal device wave determination whether generation quantum efficiently formalism mathematically ise interaction physical annealing device mapping develop lattice planning scheduling diagnosis training classifier computation number encode indicate arc pseudo boolean function encode necessary necessarily degree add pseudo result variable embed physical appealing penalty physical fix strong logical compressed problematic scale compress inherently strength prevent sufficient resolution logical energy utility mapping anneal method motivate anneal highly simulate limitation body interaction device respect simulated anneal gap present annealing still arc interact directly one penalty strong tend produce local optima annealing directly exploitation topology energy landscape make heuristic unlike undesirable solution utility scoring sub probable optimum inherently run produce low energy structure utilize perform averaging done average formalism quantum anneal find provide weight discuss useful bn would require ground encode optimal implement implement reasonable overhead construction encode specifically hamiltonian embed minor minor another disjoint individually subgraph edge edge whose map adjacent edge hardware call logical physical embed two logical physical describe physical hamiltonian logical distribute couple physical logical act minor minor practice use bit arc direct vertex indicate whose adjacency graph graph encode consist construction arc score structure direct cycle parent encode structure vertex parent numerical logarithm equation likelihood let score eq minimize wish ji q pseudo boolean multinomial head encode loss simplify parent I note include reduce require many therefore limit parent q value parent otherwise slack node define convenience generality take
variance weight constant one difference train exactly experiment template drop template accuracy template conclude scheme indeed small pose scheme specify necessary ease similar operator operator operator relu max capability powerful generalization interesting architecture apply operator follow view artificial neuron feature space perceptron output block locality generalization machine measure similarity rise generalize rise specification include kernel level convolutional express property statistical initialization initialization arise employ unsupervised manner include acceleration benchmark convnet single require around convnet architecture incorporate weight input convolutional incorporate weight unweighted similarity rise kernel building respectively hand realize svm thm building block carry highlight unweighted similarity rise superiority display linearly say template build block go hypothesis respect ability incorporate layer pooling locality template instance patch locality constraint support process spatially sharing correspond template apply patch locality sharing compatibility patch partitioning patch constant assign patch pool classify z learn fact coefficient add coefficient add character character extend accordingly mapping x v v v power v accordance correspond hilbert space v h dd dd h v v v r I rule eq interpret locality outside pool pool character constraint eqn enforce pool entry pool entry template index conclude express instance option learn locality eqn classifier reduce train instance character index subject locality eqn connection demonstrate share pooling svm illustrate locality sharing illustrate translate associate locality sharing play constraint proposition example university deep neural similarity family convolution max operator operator relu pool additional capability architecture input special machine gaussian basic abstraction iii use experiment capability achieve comparable much largely large visual task convnet architecture include thousand employ convnet capacity layer convolutional windows assumption image control form success still fall reach human level recognition merely size convnet obtain network increase abstraction motivated convnet change since early create architecture arguably success year ever compute power contribution advance secondary importance attempt initialize observed scheme little advantage carefully initialization initialization scaling capacity therefore architecture give natural initialization convnet paradigm completely take developed kernel suit flat make convnet architecture architecture body machines convnet introduce network lift convnet architecture something carry old learning decade abstraction layer potentially provide third architecture endow potential determine channel generate architecture operator generalize special role addition capabilitie cifar layer layer specialize comprise experiment preliminary capacity experiment deep extensive code apply scale operator generalize soft min replace convnet relu max pooling layer input weight stand negative similarity form mapping inner I pp architecture similarity height index patch template z channel width step patch horizontal vertical dimension linear mapping convolutional whereas mahalanobis template every pixel weight datum globally unweighted architecture template unsupervise initialization responsible spatially necessary operator follow alternative expectation I smooth illustration divide possibly overlap map run serve later convnet relu activation input layer block area omit l obtain process allow flexibility layer correspond conventional form layer omit block wide layer special possibility particular subsection multi perceptron addition process patch involve locality pooling operation convolution something consequence make consist unit apply template straightforward weight maxout attempt generalize notably unit pp maxout operation create fix unweighted p inner high conclude feature prove assertion express z indicate k lin x extension vector I neuron feature unit view neuron mapping turn straightforward extension include unit mlp signal set hidden similarity similarity hide associate label index activation operator follow produce classification rule classification template dependent operator combine template attract template template value mlp fed line derivation carry mlp work unweighted unweighted hold n rl gaussian reduce multiclass whereas classical multiclass svm generality offset svm summarize mlp layer rise reduce svm kernel replace unweighted order rise underlie special laplacian similarity rise similarity neuron feature abstraction category input consider r index template index readily condition first linear half space intersection half union unweighte l union conclude mlp unweighte qualitatively equivalent convnet set exponential kernel abstraction level consider fix tell governed view space region surface polytope shape region unweighted abstraction induce linear convolutional govern non separate hyper surface divide plane piece wise separate boundary unweighted divide shift cause less allocate template thereby around template weight setting expect weight abstraction convolutional plane panel template unweighted divide equally template weight portion plane allocate template locality share process patch locality template weight share create stack template map pool layer predict label design patch base locality realize conventional divide possibly similarity channel match template template local patch value ij li j l output layer implement map coordinate pooling normally coordinate window layer layer take max implement node run coordinate pool template coordinate pool layer node offset coordinate fig output employ locality sharing conventional make case kernel input first associate identity p w l l equality operator eqn template e patch pool kernel eqn denote ij mapping consider eqn concatenation structure pool detail proof give chain pool basic building similarity support svm realize form unweighted weighted weight applicable provide rich experiment sec validate weighting show merely build design decision determine learn datum manually chain new give rise encounter idea switch layer play pooling role finish pooling interpretation layer majority vote form final decision paradigm enforce constrain result rule b l range template constrain apply identical rule eqn estimate manually find outperform rule j scheme network initialization take select initialization initialization scheme date hardness typically design large represent true latter overfitte prominent one master properly scheme scheme effective local minima thereby reduce support small computationally validate show unsupervised scheme improve initialization similarity focus z application unlabele data template shape stand coordinate stand stand mean coordinate layer weight follow template order would output hold probabilistic heat estimating shape mixture unlabele patch follow learn linear patch come patch prior make template likely likely appear corner global initialization patch estimation shape calculate output location probabilistic heat statistic certain template unlikely template heat map aforementione region correspond fig correspond convnet correspond illustrate network illustrate c similarity reach competition implement patch currently report later cifar image process pixel template z use template make softmax descent sgd include momentum sgd momentum decrease epoch epoch decay template equal validation compare architecture convnet follow follow illustration comparison convnet learn parameter outcome reach convnet considerably comparison layer study depth cifar art template single pass sum svm produce refer triangle template mean produce c scheme initialization validation report achieves convnet et al superiority acceleration enable large deep meaningful benchmark deep convnet implement use toolbox sgd use batch decrease every dense layer convnet image array whiten accordance whiten
easy frequently life reference common non network structure address edge function edge community experiment follow detect assess available normalize mutual unless specify weight scenario detection propose iterative nmf nmf network visualize network site bi different color node assign community lot split artificial consensus community red entry observe bi select create solution around bottom solution provide c preference pt consensus consensus solution bi copy horizontal create new colored propose consensus notice base treat generator modularity provide consistently perform produce consensus bold number truth community stand normalize mutual modularity cluster consensus mod b mod mod b b mod mod b happen information network match play organize play division ground community community network recover blue team match play division bit manually figure red community modify row logical mistake consensus pt blue pt pt c seed pt thick pt pt sc sc represent blue division green node th mod stand modularity respectively allow community analyze aspect g modality evolve facebook dataset facebook observe attribute gender minor build modality different gender link student otherwise assign share major minor share share independently run modality approach allow coherent something multimodal without need consensus gender modality assign gender consensus modality exhibit inference might build perturb copy copy result perturb still good non chance network long balance perturb solution hence perturb network trial original one perturb c multiple corrupt connect bi context object might intersect describe refer sense describe model outlier intuitive zero set use threshold measurement noise goal ill pose standard implicitly impose recover pair tractable design example least formally perspective minimum number uniquely circle cloud element model define trial describe apply sequentially model impose find idea sampling seek like shift outlier rejection heuristic parametrization high element degenerate level conceptual model object cluster model propose modeling exploit spurious produce throughout iteration object element traditionally analyze row cluster vector art call j linkage agglomerative agglomerative linkage j linkage use merging process cluster object sample application address relationship object need find cluster I already bi section conceptual advantage object allow situation line intersect share object translate block diagonal technique assign instance compose exceed far one analysis discovery technique bi benefit efficient meaningful method discard bad contain object object interested consensus simplify belong appendix consensus false mention form ccc linkage assignment wrong assignment detect tune notice final process leave corner bottom green linkage yield return wrong decay cluster finally propose vanish detection application use segment element intersect plane intersection segment help helps reliably detect vanish point could consider length sophisticated vanish candidate might lead pt bi description detect bi cluster overlap rich characterization datum detect parametric concentration robust give element simple every pass central consensus many create completely consistent formulation bi since similar circle lie along line concentrate miss constraint along address formulation employ greatly ccc configuration lr dispersion threshold number dependency bi common parametric transform share clear properly value many present reasonably intuitive concern simply select application learn outli detection necessary uniquely characterize detail trivial quantity actually try discover relate establish sound relation know research practically aspect nature chinese brief discussion drive principle form image make use namely deviation randomness occur principle atomic image assume subset object share common orientation position etc stand follow uniformly distribute object observe finally ask probable play principle state assume uniformly multiple control proxy occurrence visual exploit twice preference adjust row accept bi configuration framework repeatedly arise dimension nonetheless play role assess configuration bring sampling would fact appendix mn stand observation might develop simple present connection geometrically actually possess mathematical characteristic result configuration bi intuitive nature bi mean configuration summation bi configuration reproduce low stability framework configuration mathematically formulate pr need validate intuitive conceptual reach consensus grouping grouping detection parametric pose group bi conceptually rich modeling powerful tune though instance framework particular multiple parametric pose bi highlight explain investigate whether hard actually instead work value object provide quantitative theory visual show research work fully perspective fundamental problem explore test sharp coarse avoid huge clutter bi would alternative could extend wide preference th would explore depth like stress limit present address formulate solve lagrangian ij multiplier descent successively fix recent value I update multiplier step svd object identically law present tight carefully probabilistic specific form already useful demonstrate capability framework need pass point line equation area bounding bound box lie band width define circle circle eq q lie band circle aa point plane plane pass plane plane write bound lie band around approximate line use segment define pass equation detailed segment root band width aa area bound reader description community merging algorithms bi perspective reach dataset formally pose highlight equivalence connection seek inspire event bi tune handle community bi paper noise outlier element describe outlier application group group general characterization example address traditionally broad perspective overview dataset see comprehensive obtain parametric segmentation name pool universe candidate group run grouping modality pool candidate grouping would discard prove task even modularity cut select pool combine pool attempt consensus problem say group pool subset mutually quality maximum clique extend pick candidate candidate new issue need principle field assess classify develop unfortunately community example exploit group candidate family typically goal consensus partition consensus involve recall hierarchical thorough way try decomposition relaxed matrix factorization negativity community address consensus within thresholded adjacency new step iteratively make build mention aggregation individual pairwise relation relation large group lose partition might poor quality involve prohibitive node reach consensus bi advantage keep relation small tractable dataset bi stress goal find function grouping obtain good dataset pose bi problem highlight attention behind finally formal visual insight approach section show community multiple estimation diverse experimental finally provide remark candidate universe represent element th group preference fig present simplify object consist form star object consist point four case group visualization take form uniform weight simply incorporate element object good object actual fig pattern discovery need bi formally connect consensus algorithm contribution address problem grouping preference intuitive belong analyze classical consensus work grouping problem estimation commonly go back analyze base need overlap cluster formulation base mistake translate characteristic common consensus way detect mistake penalize mainly due conceptual algorithmic interpretable association iterate criterion meet bi element indicator presence row suitable correctly de via use find extremely challenging specifically tune value motivate bi matrix positivity intuition negative sparse entirely suited analyzing result adapt solve work algorithmic loop norm correctly detect bi conceptually bi subtract enforce set successive orthogonality non negativity maintain bi share element simple control bi cluster row discuss encode minimum element bi encode bi cluster theoretical value show tune part experiment bi intersect enable quickly eliminate spurious parametric bi intersect set posteriori strict high value use decide posteriori clarity homogeneity could bi compression encoder select yield compression loss fit I break ij symmetric matrix approximation encodes method parameter discover bi allow construction orthogonality present benefit double act pool frobenius outlier compute robust median preference carry need ingredient consensus candidate feed consensus time bad bad group phenomenon extremely pattern assume general candidate nature group parametric employ simple testing eliminate vast candidate currently investigate non candidate negligible consensus grouping mistake grouping algorithm uncorrelate possible approach ideally mistake cause algorithmic decision systematically appear candidate group procedure network gmm c c cccc preference bi preference ccc ground assignment assignment em ccc assignment subset intra one key exploratory bioinformatics application grouping pool preference therefore fashion weight assess ground truth standard evaluate express f figure synthetic gaussian cluster pre shift size consensus algorithm approximate qualitatively visual inspection compare figure rank first visualize difference nmf nmf iteration average preference whole active bi candidate I approximate single nmf fit group active bi depict preference l help bi cluster ground tuning method first consensus correct self argument get parameter valid although really j linkage popular
efficient approximation possibly graphical across advanced flexibility view toolbox discuss take particular mixture conjunction simulate kernel costly warm branching tree computational assume work tree practice several decomposition present systematic self ask explore address strategy mix decomposition mixture interact run sir direct argument joint run hence detail leaf extension balance unbalanced introduction expense contain resample tree index live may simulate run write count inclusion transition sir particle recursive particle distribution sample additional indicate recall straightforwardly division simply correspond component numerator way convenient upon respect derivative identification implicit notational sufficient counting denote importance particle product h particle unnormalize consequently q turn consistency particle normalize binary denote capture argument henceforth dm finite everywhere dm relax simplify residual perform particle I extension simple set cover definition outer coincide moreover approximation semi cover disjoint finite algebra approximate q assume apply simple converge everywhere apply outside bad q everywhere imply px equal leave internal induction weight ht induction imply population lemma measurable without indicator equation f f lemma pick cb compose union cx therefore next resample perform dm resample particle unweighted particle plug quantity eq dm induction use ci c accord simulate ic n refer particle use reversible markov z z result ise text repeat show estimate mix leave correspond particle flip mh numerical inverse ise temperature result give figure sampler box flip axis dash run box mcmc iteration flip mh axes region toy consist field periodic boundary respectively potential variable py distribution sampler distribution batch apply sampling precision difficulty arise interaction note tend whenever relative pose difficulty site mh shall setting sampler use random manual note small ising log constant site fails converge posterior expect mcmc superior sampler attribute c sampler approach simulate multimodal account draw mode particle sufficiently c sampler suit difficulty population site gradually take account step application effort axis box plot axis posterior sampler site correspond run clear sampler much x correspond data york city year indicate meet level correctly meet school code student student school extract school remove character school support student cognitive delay school extract repository check example obtain four baseline particle correctness argument experiment create detail leaf otherwise metropolis propose proposal unit first within package collect time implementation turn sampler adaptively select marginalization kalman shape adaptation single bootstrap sub forest build precisely fix order traversal traversal forest parameter sir internal propose child describe text package show range top level plot support approximation row dc note std similarly configuration intel core processor ghz detailed architecture dc approximately magnitude burn take run exclude attribute efficiency method locality array non linearity run theoretical time particle implementation std related particle std particle forest std tree tp use challenging demonstrating environment emphasize computer core fast several computer connect advantageous resample communication less example alternative contrast computation communication main cost recursion location tree simplify exposition introduce set root recursively depth depth vertex assign vertex computer architecture connect depth particle particle strategy decentralize fashion library perform decentralize decentralize package discover automatically rough node specify value fail still york mathematics varied particle compute consist ghz processor x node note parallelism either level population speedup distribute scale pt remark university correspondence laboratory email sequential divide auxiliary structured turn inferential collection recursively applicable loop employ particle merge empirically outperform accuracy expectation likelihood approximation novel implementation option possibility hierarchical sequential carlo simulate collection particle weight particle generate sequentially particle generate iteration depend see therein generally much distribution arise shape graphical auxiliary decomposition decomposition contribution propose divide sampler tool bayesian variable set suitable easy parallel result merge discrepancy approximate exact sampling divide methodology repeat overall weight merge costly mcmc via particle methodology broad construct auxiliary create small connect sub assume interest shape indeed demonstrate methodology obvious structure artificial figure either exploit conclude remainder provide work build methodology property slight abuse respect anonymous equip borel density z point wise two concerned integral correspond observed computing expectation problem often arise formalism encode dependency probabilistic graphical structure model summarize conditional graph often random field abstract factor formalism reader convert graph write factor graph space convenient factor bipartite vertex depend throughout convention take sequential monte carlo algorithms section precisely z evaluate wise normalizing computationally sequentially sequence provide estimate constant smc population population weak expectation sufficiently regular expectation test weak establish example resample depth slightly recursive propose recursively easy problem interested sequential nature procedure return empty convention hence detail unweighted resample copy particle multinomial distribution n resample informally particle part state preserve resample weight weight particle sophisticated resample sampling line base unnormalized density unbiased practice particle degeneracy severe monitoring ess ess small execution arise recursive call show correspond illustrate dependency computational chain largely structure chain factorize sir employ importantly fact possible original subset common distribution local mcmc form suitable auxiliary substantial define admit thus original structure backward formally restriction reversible weight selection kernel present methodology appear key approach belief importance sampling respectively provide sense propose approximate model loop method component rely practical sense population particle full rather employ system interact smc particle system attempt degeneracy markov inexact technique numerous author purpose generally employ useful auxiliary convergence provide concrete aforementioned algorithm strategy apply direct undirected modelling scenario include cycle method extension classical framework chain section simulate subsequent chain describe execution organize necessarily specifically auxiliary structured distribution distribution first coincide target recursively incremental towards leave tree recursive execution detail subsequent root tree point computational nevertheless relate easily understand consider target intuition get hierarchical panel latent put prior node auxiliary use prior filter ultimately weight upon course affect efficiency finally far illustrate left topology exclude observe turn carlo target purpose present thought analogue approximate particle maintain particle merge indice iteration bottom auxiliary repeat resample closely mirror specify operation recursive definition equation constant equally weighted ic recursive population support particle obtain measure child particle population equally weight merging implemented resample description population merge extension methodology resample effective resample severe resampling amongst result require filter degeneracy particle employ lag setting extension degeneracy scheme improve challenging setting step target independent multinomial sampling replacement lead particle exploit capture variable simple see replace step simulate basic event step case problematic mixture possible introduce significantly resample possible reduce branching introduce tree merging gradually variable strategy something distribution proposal instance transition detail clarity sequential previous resample unweighted c straightforwardly sampler c ti particle system reversible argument complete particularly mixture use enable efficient choosing warm annealing anneal simulate costly sample seminal demonstrate use approximation proposal particle algorithm extend demonstrate particle mcmc essentially smc year resample sometimes resample adaptively analyze formally mcmc location appeal advantage concentration effort sampling adaptation intermediate subproblem start value anneal size comprise adaptation subproblem effectively significantly throughout population fewer simple represent adaptively adjust remain direction investigation useful markov illustrate lattice ise spin configuration x graphical near interaction periodic boundary original lattice simplicity continue recursively see decomposition c operate leave initialize independent population merge successively leaf define basic procedure three mixture section edge connect correspond merged method section edge connect anneal schedule severe later stage use mixture sampling flip metropolis hasting ess mix ann warm anneal threshold sampler adaptive annealing flip implement matlab grid list particle exception closely flip sampler iteration discard burn value result bottom flip mh box plot increase flip mix inferior ann ann mix exclude method ann mix ann flip ann mix ann comparable whether section anneal take sampler run site number ann mix ann mixture mcmc take ann turn half standard mcmc expensive cost automatically illustrate value anneal process warm c mix level computational edge add merge leaf anneal less figure able warm effectively anneal need mix ann particle report numerical another square lattice multimodal posterior result elementary preprocesse acquisition appendix path root leaf follow root school school come across five standard distribute
aims influence output tune hand uncertainty simple reference variable belief turn total variance hoeffde functional induce uncertainty consider importance sensitivity index write estimate sense index order hundred thousand output monte carlo carlo approach science community include pick index obtain hold sample replication combine input large practice hence dimensional index claim applicable context parameter property want efficiently sparse context method would input describe relate recovery exact decade therein estimation goal draw bridge knowledge draw dimensional model contribution describe index use give new exact thresholded coherence prove proof preliminary prove remain appendix appendix rather thresholded frame input know output want index present index number index remain small observe estimate eq identity lead q high index evaluation generate realization expensive costly inefficient require time good difficulty new estimation scheme let method stem generate index pick subsection define hypothesis one subset choose compute use encode estimation error thereby extensively compressed regularization lar appropriate one key often choice pick bernoulli method monte estimation deduce sized use binary parameter gaussian vector satisfy necessary high property design matrix context observed remark proof thresholded function correspond thresholded version address issue done lasso use clear randomize rademacher summarize estimation sized obtain obtain consider state suppose large real satisfies property pick estimation bernoulli rademacher matrix page test lasso ie plot figure design monte require rademacher two pick design perform fast bernoulli accordance perfectly active ordering evaluation estimate index identify classical estimation index interval k correction length hence show limited new relaxed note apply approach design proof sake give proof exact recovery thresholded adjacency rademacher great invoke get probability great probability constant appear invoke observe read thresholded coefficient elementary analysis fail applicable analysis lead thresholde greatly sup error rescale rewrite expectation thank besides inequality proceed define q denote random conditionally n standard tail statement prove subtract correction minor vector clear absolutely give finish q hoeffding union notice satisfied statement r side observe side conclude assume convex program enjoy assume n yield deduce recovery thresholded thresholded assume denote column convex program enjoy order optimality program r moreover entry devote proof thresholded adjacency bi set exist otherwise regular e exactly per unbalanced define unbalanced unbalanced graph parameter expansion satisfy property unbalanced degree quantity sake magnitude partition observe sub row cauchy expansion namely
data laplacian slide regularization inverse problem depict curve inverse regularization test newton inexact insensitive parameter state number solve inverse size perfectly cg degree freedom inversion report iteration cg newton cg total linearize solve adjoint newton gradient residual newton drop tolerance norm inverse inexact independent dimension surface velocity refined mesh slide refine horizontal also evaluate optimum optimum maximum curvature criterion instability field depict slide parameter surface ice slide field vary nine red value low slide ice flow see slide successful observe surface velocity year dark blue ice red bottom image surface velocity reconstruct ice infer slide inspection velocity field agree region pose nature noise field nevertheless inversion slide ice model datum critical region ice ht l pa ht fitting ultimately deal pose equip answer turn systematic quantify uncertainty bayesian ice tackle quantify previous keep discussion begin present inverse problem rank approximation hessian scale dimension compute hessian computation uncertainty adjoint solve ice space uncertain physics problem assign member rise discretized scale problem freedom parameter field combine parameter explicitly rise pdf note analog pdfs discretization usually impose rough component observable pde operator allow build solver operator variance prior ensures bound well pose infinite actual observation velocity measurement represent vector ice flow extraction ice velocity theorem n covariance take role formulation clear prior posterior nonlinearity pose challenge typical dimension surface solution numerical matrix completely question chain sample statistic expensive govern nonlinear ice flow slide number million discussion execute ice lead log gaussian posterior posterior know posteriori amount deterministic appropriately weight inverse hessian give evaluate adjoint involve derivative posterior expression accurate nonlinear direction inform approximately well space inform linearization tackle massive govern ice flow method prohibitive compare compare posterior ice hessian stand alone intractable difficulty large covariance map forward pde forward ice flow vector linearize pde problem adjoint like vector form linearize adjoint pde ill discretization compact zero understand mode strongly influence present spectrum ill pose numerous mode highly one effect effectively mesh decay illustrate rapid spectral hessian ice demonstrating mode slide parameter mesh exploit action rearrange alternatively symmetric inner approximation generalize q generalize generalize rapidly low approximation eigenvalue eigenvector eigenvalue linearize nonlinear solve newton linearize solve inexact outer cg product cg outer newton hessian linearize solve incremental adjoint tolerance find point total linearize solve approximate map low representation employ hence cost incremental forward adjoint linearize solve measure solve quantify magnitude less depict figure show slide gaussian bottom pdf gain slide across west observe large infer surface slide slow velocity velocity slide field intermediate interest uncertainty ice uncertainty subject fast ice flow region west variability infer slide unknown slide field uncertainty quantify make ready propagate uncertain slide parameter ice flow yield interest associate ultimately ice decade model ice body couple prediction ice steady ice describe scalable formally solve uncertain sliding result large mode quantify uncertainty slide expense carlo prohibitive millions solve curse solution map point q forward ice flow use slide operator give linearize prediction jacobian evaluate map action direction adjoint enable scalability uncertainty problem jacobian prediction determine field steady state ice net ice mass gradient respect forward slide solve adjoint adjoint stress ice infer adjoint resemble adjoint regularize functional adjoint solver purpose turn linearize velocity adjoint find expression without dynamically ice expression find describe section adjoint incremental solve cg iteration rank equation algebra product product straightforward ice uncertainty surface velocity uncertainty slide parameter ice prediction univariate low hessian adjoint solve forward solve uncertainty propagation map influence quantity identify inform observational require quantity interest adopt denote hessian linearize map g give influential direction prediction map q influential simplify mode depict bottom ice east sensitive difference row uncertain vice versa imply role mode sensitivity respect slide right bottom east uncertainty mode show sensitivity influence ht visualization plot ice east column ice mass present solve flow ice level observational infer parameter uncertainty parameter propagate uncertain quantify uncertainty section solve parameter state processor well exploitation infer space cg converge newton estimate inverse datum admit rank require forward solve also parameter uncertainty ice flow quantify pdf form hessian turn oppose parameter approximations pdf interest ultimately relax approximation result scalability remain subject work u air office scientific program office advance compute scientific advanced award er nsf innovation office de dpp empty rgb rgb rgb research science give broad question observational prediction scalable uncertainty infer propagate uncertain prediction quantify uncertainty end context ice effect ice observational surface ice flow velocity uncertain slide heterogeneous ice present day ice require forward ice parameter dimension processor despite size sparse exploit linearize observable randomized adjoint quantification problem approximation adjoint hessian nonlinear inexact newton ice flow ice year warm decade indicate acceleration great project flow ice current rate drive even ice potential accelerate cause ice drive conservative rise economic million risk projection ice play ice considerable uncertainty ice leave panel change th assessment state uncertainty ice level rise dominate uncertainty ice temperature ice model severe mathematical challenge significant improve ice include aspect geometry nonlinear algebraic discretization result widely vary slide range thousand phenomena great mathematical challenge lie quantify uncertainty prediction characterize field describe slide slide ice heat observe ice lead pose quantify uncertainty inference ice accomplish inference discretization take pdf belief express candidate datum pdf challenge evaluate pdf forward ice statistic use art associate ice confidence prediction nonlinear quantify uncertainty infer prediction ice flow execute intractable field quantify ice play significant future paper scalable measure linearize adjoint processor core posterior result infer ice velocity uncertain field ice yield prediction map ice flow slide satisfactory prohibitive ice equal dimension difficulty map inherently limited direction influence limited measure ice scale idea achieve low svd result framework ice adjoint covariance scalability adjoint ice adjoint ice scale number processor ensure uncertainty quantification operation term fix ice solver need scalability prediction predict day ice infer slide ice quantify uncertain slide ice flow model yield prediction ice mass quantify uncertainty nonlinear ice flow solver fundamental repeatedly solver scale brief summary ice ice balance mass momentum denote ice velocity pressure ice acceleration relate tensor invariant tensor exponent temperature determine approximately equation accumulation use heat pressure surface accumulation heat come boundary ice impose flow direction slide onto relate phenomenon depend ice water ice section discuss inverse estimate observational boundary specify stress must pressure water boundary neither interest deterministic description ice ice ice surface refined mesh element ice domain describe ice mesh refinement est mesh constrain profile vertical hybrid refinement quality mesh control element refinement refinement refinement accuracy locally refine precise velocity pressure finite pair tensor velocity pressure inexact arise upon linearization control minimized velocity pressure iterate iterate correspond linearization evaluate exploit tensor nonlinear linearization tensor identical adjoint adjoint iterative efficient critical scalability inverse algebraic approximation give matrix algebraic grid expensive jk add cost bottleneck scalability therefore construct element presence element problem low order discretization ice scalability base coarse mesh successively finer also slide slide describe velocity intend parallel sequence successively refine ice demonstrate algorithmic despite severe nonlinearity coefficient mesh refined mesh system iteration iteration count bottleneck scalability scalable refinement scalable nonlinear solver tackle uncertainty thousand solve section km refine discretization maximum aspect create successive element respectively slide boundary p temperature exponent c c core solve p scale ice algorithmic parallel p pose successively fine tolerance run freedom cpu core core iteration multiplication solve block incomplete robust poisson solve linear poisson pose ic algorithmic ice iteration linear definite homogeneous efficient scalability core count setup phase increase parallel inverse infer slide ice ice adjoint surface velocity slide parameter ice use derive gradient vertical negligible vertical gradient horizontal horizontal compare scale ice inversion ice equation fidelity ice quantify solution ice propagate prediction remainder inverse formulation brief overview expression adjoint ice flow inverse give observational ice surface velocity slide coefficient boundary ice fit square optimization velocity slide parameter observation ice field surface vary normalize prevent division regularization slide ice restrict field reference slide regularization smoothness typically add invertible normal definite need stem small sliding inverse pose sensitive reference inverse value classical reflect prior slide high domain three dimension inexact cg
heart show filter setting delay extraction challenge one device receive heart like design separate capture record add also external human dealing correct person heart record person utilize record heart case heart task heart body close obviously utilize filter device process correspond paragraph extract child body solution obviously heart go heart follow heart explain extract child heart eight try fail organize describe setting filter conclude everything show signal like extract consider denoise filter going minimize least mean square mixed child delay last assign element subtract mixed shown section find approximation lr correlation notable l measure size later converge fast slow parameter store frequency hz sequence test child available home page consider lr experiment filter first change mean refinement signal table several threshold cccc delay e e inf e e inf inf inf inf inf table achieve child l act versa I analysis finish discrete child different dft domain
normal particle filtering learn posterior query probabilistic approach limitation belong similarity distance kernel primarily inequality task case another measure benefit say utilize knowledge fundamentally experiment modularity separate modeling retrieval handle respective explore aim query multiple together dirichlet conceptually article also observation enable part corrupt describe create come cluster center retrieve irrelevant depend share map prior objective reasonably compare retrieval rank generate randomly experiment database query experiment likelihood prior choose dl q compare consistently ordinary notice posterior descriptor show form show variation performance two feature average posterior sample observe well store reduce posterior store weighted likelihood posterior maintain retrieval performance store objective improve retrieval precision approach informative training representative present elaborate weight preserve rank experiment cluster investigate experiment split grind ground evaluate number store posterior perform equally analogous generalizing beyond posterior gray correspond partition database show store toward corner contour sparsity retrieval demonstrate approach restaurant multi label map fig train probit regression dataset gamma posterior sample dataset query consist detecting experiment highly like split class separate sufficient due reason retain computer miss measure propose preserve ranking observe sample able preserve however store since consist customer categorical sense drop sample decrease collect essential rank experiment query state retrieve matching categorical annotation argue actual relation outcome learn measurement relation retrieval compute likelihood query learn measurement exist towards highly extend include rigorously model yet beneficial able sensible storing often express analytic store select informative present acknowledgment support centre coin energy calculation resource school science science project available web acknowledgement blind review institute technology computer centre statistics college institute technology university taylor ac measurement outcome retrieve compare annotation valuable incorporate retrieval employing utilize measurement argue metric sensible inclusion analytical resort store sample therefore informative load retrieval demonstrate efficacy simple procedure comprise independent outcome example wide explore association trait control variable genetic variable specie cell outcome microarray traditionally retrieve qualitative assessment document explicitly handle experimental datum researcher throughout public compare manual annotation suffer terminology annotation effort beyond annotation compare experiment toward acquire annotation information term assume researcher model measurement study utilize metric idea query experiment metric efficiently likelihood posterior issue store evaluate computationally demand paper deal select storage requirement maintain retrieval achieve individual likelihood preserve suitable weight compute reduce storage burden illustrate define collection outcome di di rank database relevance query experiment suggest eq previously retrieval capture keyword generate document marginal notice certainly query kk store grey dot performance pool discriminative computational requirement dot trade therefore contour dot grey dot define binary constraint formalize highly realization entry aspect matrix assign switch actually maintain balance e dd mm dd second third first figure triplet retrieve p q sign arbitrarily matrix block diagonal block gray solve label help likelihood normalize use library solve logistic interesting sample dl treat one I map ratio observe consistently sample come observe store high noise positive negativity naturally
probabilistic two logical predicate logical reasoning system predict attribute gender education location relation user probabilistic logical reasoning result effectiveness course never explicitly gold standard really promise perspective medium directly offer gold facebook contain preference movie book comprehensive different type information offer pt propose logical answer question twitter user new york fan building reason attribute user location gender network friend inference friend I likely new york distant supervision attribute education preference text proposition feed investigate show logical improve also baseline extract online social political public opinion evidence preference come preference like friend like popular help collaborative describe preference movie social combine share attribute relation may latent result user preference source user preference tie like twitter knowledge attribute infer preference twitter relational reasoning framework markov logic probabilistic relational logical combine inference like people work device associate perform rule probabilistic attribute preference stage extract user attribute knowledge describe although medium facebook sparse profile site twitter medium describe activity education system combine supervision structured profile text message able construct comprehensive list attribute mention dramatically increase extract explicitly attribute investigate possible coverage extract profile attribute mention logical feed framework logic infer rule user attribute predict evaluate preference attribute like system describe predict interest behavior medium life twitter technique general contribution summarize logical reasoning estimate attribute medium combine relation relation preference present relation introduce probabilistic logical framework stream extract relation preference logical represent two kind predicate mapping object another return object predicate whether object friend twitter predicate give predicate graph twitter discard twitter publish tweet million next describe extract predicate attribute education gender focus detail extraction goal unite infer publish tweet base tweet specific entity drop publish tweet publish location united entity match name job education attribute extract probabilistic user obtain feed google profile publicly education major name twitter account google account adopted take percent least friend google circle twitter account person percent job education google account name job entity publish twitter content education job entity tweet require user mention education job publish percent job education many framework devote gender twitter study whether high feature help absence predictive without name rather extent logical social implement national social security contain record annotate gender birth name highly gender assign gender gender least gender user user twitter people friend follow straightforwardly twitter al publish return indicate likely confidence project straightforwardly extraction approach user preference specifically predicate sentiment analysis social medium e extract sentiment extract object sentiment resemble sentiment use base manually manually collect tweet among massive people discuss tweet form collect semi distant supervision extraction new new new begin pattern seed like think entity twitter package tweet tweet distinguish tweet tweet model like token entity feature tag dictionary crf crf package context window tag entity tag pos tag word use model iteratively distant supervision distant supervision label draw external sort come treat relation sense hold publish express supervise heavily seed influence seed add example distant help training overview combined distant cm begin tweet label tweet newly entity add training cm preference stop optimum stop tweet entity tweet positive match negative tweet label label dataset view parameter tuning score algorithm c tweet model tweet l tweet entity tweet tweet tweet tweet label purpose without distant supervision naturally constitute employ decide token entity tweet predicate construction exist assumption different like entity category would connect belong category treat predicate priori predicate evidence instead node number branch let system clique major challenge miss example situation mention entity deal user preference preference report entity illustration express summing variable system optimize entity denote user specific entity would mention setting address infer attribute multiple joint along partly retrieve mcmc probable explanation infer attribute able size greedy inspire attribute logic attribute value consider iteratively estimate attribute friend standard confident prediction relation would decision round expect yield former benefit gold detector user attribute relation preference base dataset section value gender testing attribute preference global logical entire network improve baseline use state user tweet state evaluation approach gold standard location report prediction location precise set predict user attribute baseline include assign attribute unify assign usa california svm classifiers feature predict extract attribute feature encode presence absence entity job education friend value attribute presence simplify relation simplify rely cccc acc acc unify naive performance outperform detect network consider evidence logic capable yield bayes correspond people north half draw gold gender assign high precision security inform svm logic relation preference c svm even incorporate design directly gender entity write style gender nonetheless inference achieve accuracy network table give gender infer prefer prefer form select specific distribution positive relation make pr co occurrence classifier global baseline user view location assign location assign proportion value recall svms relation prediction leverage interaction pre ability preference complex gold twitter user towards entity actually know like ability detect say opinion proceed predict opinion task useful structure alone solve could combine sentiment technique begin scenario entity try attribute text experiment sentiment type user g friend sentiment toward entity distinguished york e prediction pr pr gold guess evaluation term prediction employ classifier feature individual attribute value location gender friend along network cf account popular recommendation recommend similar construct user cosine attribute entity base entity describe entity indicate whether entity see outperform cf believe try sentiment difficult entity entity construct total distinct baseline employ include popularity whole decision naive classifier decide specific express entity attribute detail performance evaluate precision table like mention extremely task tweet percentage entity predict one difficult require much kind access
fidelity parameter inverse sense additive noise structure measurement task assume sparse paper group limit include segment block group connect element address regularizer review regularizer recent year attention non zero sparsity structure lasso make group encourage sparsity group propose regularizer lasso regularizer group select make encourage sparse formalize goal generic fuse regression curve model group lasso firstly regularizer net former induce corner latter strictly create grouping effect regularizer regularizer fuse lasso compose total encourage successive similar able equality absolute pair variant sparsity regularizer fuse fuse pair extend fused grouping model net grouping neither fuse elastic regularizer ability outperform grouping fuse grouping ability net inferior focus group paper nonnegative constant control term become lasso regularizer ball divide two locate axis four fig depict solution contour induce contour lasso specification fuse strong ability operator proximity term give q denote stack column operator function convex minimizer split fista admm sbm detail experimentally fast result aforementione fast step criterion satisfy type regularizer bias common say support magnitude phase estimate produce conjugate matlab pc intel core processor ram algorithm assess estimate nd f nd error time matrix negative fig h l l cm mae yes yes yes fista sbm fista sbm stop mae recover quantitative report solve accurately sparse fast obtain conclusion regularizer group
achieve uninformative depict dash mmse plant show matrix see intuitive finite dictionary available large amp relatively towards mse uninformative illustrate plot amp depict diagram calibration static identifiable amp asymptotically dash recover subsection illustrate learn happen noisy sensing become phase amp signal surface sharp transition mse compare fig relate relatively pca evolution qualitatively section fig zero smallest sparse mmse counting bind blue line fig amp line depict line amp mmse separation correspond length signal long sensor case signal reconstruct also correspond mmse error remain neither rank blind early appear phase observation theoretically achievable system perspective development successful lead algorithmic concentrate theoretical performance version however way match predict worth discuss algorithmic conclusion parallel evolution see recent study issue compress optimal correspond belong avoid issue basically sequential pass small correctly size part function helpful explain compressed sensing however implementation explicitly pca able prediction moderate proper incoming bp message income bp incoming incoming se realization cavity cavity se se se notation equation acknowledgment receive research european union fp grant financial core program equilibrium dynamic soft matter department intelligence technology sup universit universit es paris bt france france correspondence shall send fr analyse factorization measurement product two original arise matrix calibration principal principal component cavity replica analyze bayes achievable computational efficient message performance term extremely promising motivate development algorithm give wise measurement factorization requirement like negativity appear many application completion computationally tractable understand understanding limit randomly result fix existence phase formalism amp physics cavity method context replica cavity widely present rigorously describe closely amp amp place derivation phase diagram study framework via amenable diagram establish principle amp find treat separately problem dictionary think kind interestingly two unify formalism measure information denote q probability factorization assume case various various identical know provide parameter blind calibration case measurement normalization partition infinity degeneracy probable prior define index symmetry symmetry break etc usually write explicitly general simplify necessarily expectation maximization multiply arbitrary change derivation paper mean deal happen assume evolution concern square mse original marginal measure square mmse explicit formula mmse achievable element non treat mse achievable message pass mmse amp need notation input output amp uninformative another random take initialization symmetry nontrivial mmse initialize iteration close result agree amp mmse amp mmse entropy section formula mmse example whereas within physics cavity replica provide situation partly establish rigorously recently important receive lot recently factorization analyse data compression compress devoted basis basis datum decompose paper analyse teacher student scenario learn iid elements zero gauss non mean noise zero variance delta goal infer noiseless observed want provide eq assumption likely hold analyze benchmark develop application satisfy spirit pass algorithm mean measurement kind typically look basis compressed mind measurement regime multiply row learning column degeneracy optimization formulation normalization information supervise use hence measure perform reconstruction blind blind calibration dictionary eqs work obtain follow variable variance zero element quantify know measurement provide compressed explicitly factorization rank know small fraction element one know element function precise arbitrary long hence large limit analyse paper keeps work work analysis apply low completion need order reconstruct negligible counting know element reconstruction generate construct keep knowledge principal well decomposition keep svd approximation however tractable variant relevant require rank component sparse approximate product teacher analyse eq still comparable region dictionary learning way hence work zero counting need eq blind channel measurement blind typically number channel sensor sensor unless variant close interpretation create analyse largely non noise element require looking regime interested another robust pca zero measurement noise counting satisfy completion counting element position analysis infer fa representative methodology mean fa assume generate term entire x fa factorization form characteristic site dependence variance normally obey distribution compact manner py z dx ff ff mf mechanism determine minimize certain determining employ fa coefficient signal dictionary negative easily impose student teacher scenario element algorithmic give exhaustive concern paper work paper message phase transition limit dictionary learning identify visual extensively overcomplete many svd assumption g view redundant closely problem principal component blind source guarantee another differ exist mention difference work concentrate theoretical guarantee work work analyze case arguably bad practical application case provide tight literature error signal diagram message direction find processing work base give many variety prove rigorous performance less often consider algorithm stand core performance bayes offer total derive course heuristic mmse rank note amp derive rigorously zeros dimension whereas literature consider amp use zeros result mmse fraction derivation message state amp analyze two amp replica agree give prediction factorization phase diagram blind separation completion summarize conclusion sec identity many calculation physics system identity basically identitie equilibrium configuration signal know identity type average double trial similarly double average dx df xx ff identity configuration average see nothing last step identity problem generation limit introduce paragraph hold realization size f f average useful useful self limit x remarkable concern left measure overlap right self two independent sample symmetry average matrix relation straightforwardly elementary expectation crucial certain strictly explain amp investigation greatly nine equation impose leave evaluate mmse reach uninformative initialization plant bethe mmse bethe entropy exponent normalization bethe furthermore uninformative optimal mmse hard algorithm mmse physics instrumental transition two mmse one zero mmse noise coincide poor mmse simply enough recovery tell happen generally case state barrier prevent attain mmse suggest limit system amp derive mmse subtle low blind informative uninformative initialization evolution may divide one uninformative fix free hard fix phase hard phase mmse achievable phase mmse uninformative initialization denote amp transition useful minimal achievable mmse mse achievable mse mse regime region make mmse amp eqs view complementary cavity replica cavity method amp experience three rigorous proof follow cavity belief propagation correctly bp randomly evolution iteration bp term large cavity asymptotically derivation fail replica break fortunately bayes optimal exclude analysis replica replica normalization logarithm saddle assume saddle class replica cavity replica lead mmse apart claim result exact cavity method amenable rigorous analysis describe assumption make obtain belief propagation equation factor fig income eqs message obviously locally tree assumption rigorously show however resemble compressed prove rigorously expect matrix bp asymptotically sense equation start order phase transition need iteration planted reach justify contribute assumption replica calculation self average concentration basically assume instance exchange top interested evaluate bayes mmse unfortunately prove basic replica easy factorization cavity alternative replica physics strategy conjecture evaluate mmse factorization path cavity approach eventually conjecture mention namely sense matrix apply extensive need prior match fundamentally simple major replica break physics computational many problem spin via pn overlap define measure symmetry permutation replica replica delta peak limit limit continuous symmetry peak simply multiply symmetry configuration respect reference compute free take field purpose physic self transition whereas hence delta plant identity converge delta allow inference decay sufficiently prove measure truly replica break happen trivial even result trivial instance compress prior practice even know often full variance precise amplitude learning straightforwardly include algorithmic detail implementation leave statistical maximization maximize normalization measure message turn update impose compress satisfied empirical variance channel time perform order condition straightforwardly within compressed nice pass back line maximization improve quite opinion bayes amount marginal resort approximation approach combine belief propagation bp reconstruction approximate message pass subsequently ht il factor depict fig bp iterative equation message argument graph bp trees normalization product bp exact incoming happen know rigorously success message yet system point propagation equation sense lead understand theoretic factorization transition bp iterative integral intractable definition scaling shall show bp simplify write involve parameter bayes optimal provide careful loose term limit rewrite square expand order definition without square q message introduce variable integral expand exponential integration function definition term keep term way distribution input auxiliary instrumental notice analogously use message scale quantity hand lt ip recall general belief simplify closed message means define iterate definition notice simplification reduce approximated probability define analogously message amp derive message exploit always close message physics equation spin limit asymptotically literature present factorization clear independent equation avoid might negligible one careful might spin reaction define order equation keep track order equation expansion iteration factorization variance familiar recognize equation compress three algorithm dependent amp similarly find dependent amp index iteration evolution miss big amp expression serious understanding eqs derive factorization principle iterate show way improve wide application proxy describe analogy call identity hold generate correct meaning square equal mean difference identity hold conditional incoming simplify play amp expression rely definition parameter analog mean reasoning quantity hence simplify far simplify eqs stress independent contrary variance depend index sense propagation amp normalization logarithm normalization bethe logarithm normalization energy physics bethe five contribution derivative bp equation practical mathematically tractable amp message pass equation express eqs last keep expression write form put piece evaluate amp bethe log decide amp bethe free transform fix identity verify bp message bethe free entropy belief point stationary bethe amp factorization bethe free entropy allow achieve bethe order derive form bethe amp check derivative derivative note nothing quantity l satisfy equation deriving conclude point free fixed rely iterate compressed entropy current quality sense trick useful bethe entropy simply eqs variational expression kullback leibler divergence prior distribution eqs simpler compressed expression generalize bi white depend channel entropy point channel last obtain simplify term entropy large expression fix point hence maximization expression algorithmic amp compress expression increase adaptively iteration increase bethe free read check il z il I bethe exact approximation field derive factorization element iid matrix iid random generate bayes distinguish output channel element aim recover term order pair index index adjust characterize channel instrumental hz w quantity equation product write average expand double belief exactly indice quantity realization expression proceed conditional incoming bp equation obtain many assumption gaussian variable mean distribution mean incoming message z zero correlation argument lead quantity asymptotic uncorrelated incoming message separately new second realization behave assume case true bayes equation similarly see non mean gaussian analogously integration analogously together equation limit variable satisfy absolutely essential convenience formalism satisfy prior match true simplify justify statement need hold state simplify gaussian integral choose come expression line distribution take initialization evolution symmetry initialize evolution ise equilibrium correct converge absolute value general phase satisfy hence lie line e satisfy reflect factorization evolution equal aim equal denote q analogously exactly get condition result get analogously satisfied write product perform part measure equation integration respect q thank integral integral mind remain analogous integral dy dp np however recall self average randomness average use factorization problem replica know rigorous approach inference factorization problem result replica fully evolution expression constitute statistical mechanic generally examine statistically current expect entropy converge unity expect relevant central systematically carry replica evaluate th dy utilizing assume form generality equal hold come trivial
always unitary require continue triangle zero remain eliminate desirable impose negativity example constrain remain variable symmetry break although problem eigenvalue avoid wish firstly optimal position latent prediction survival use standard p accomplish optimisation make prediction optimisation show log global correspond perform analysis top exist attempt start initial search examine behaviour different generate datum firstly kernel value gaussian repeat possibly value c te independent censor individual censor radial helpful compare retrieve true value purpose plot allow quantitative sample circle origin origin radial belong circle similarly angle circle circle angular point line total square finally note coefficient determination take property secondly rescale ability accurately extract dataset time describe patient induce project latent observe lie one manifold space split equally sized basis structure observe whereas manifold clear separation illustrate extract reveal survival intrinsic show example determine alternative offer explanation third variable explain three non offer survival simulation study include survival retrieval dimensional effect extract insight sophisticated survival hazard burden european fp ec agreement partial derivative optimisation partial derivative likelihood identity purpose drop clarity partial define second partial derivative define derivative begin simplify q kernel eq second q zero otherwise detail sum eliminate perform appropriately matrix outside loop hessian compute partial likelihood define q bind manually function partial derivative become second require practice q partial section convert high survival challenge primarily spurious relationship infer fail representation survival proportional flexible extract intrinsic combine source simulation study overfitte intrinsic increasingly research current experimental measurement hundred snp nucleotide automate software thousand high outcome available fail test datum noise great dimension covariate outcome proportional problematic coefficient uniquely covariate broad selection aim either covariate cox forest elastic net survival suitable association survival attempt redundancy redundancy parsimonious prediction reduce covariate survival outcome interpret overview survival one contain popular survival flexible probabilistic non reduction gp choose kernel type specify space variable infer consist covariate maximum posteriori principal component pca intuitively probabilistic subsequently recent regression successfully detailed overview find dimensionality explain combine simultaneously term easily extend account extension information discriminative class incorporate extract include output advance possibly outcome capture dimensional covariate survival outcome hope capture covariate survival dimensional freedom overfitte recently recognition face term image angle analogy think latent different addition survival outcome underlie yet share dimensional complicated likelihood involve optimisation choice via overfitte extract low dimensional brief section dataset column assume call individual latent space hyperparameter th product map I paper control characteristic length vary individual st primary censor primary indicator primary occur indicate censor hazard base need infer combine hazard latent variable denote survival integrate base hazard rate dimensional outcome use make data latent eq likelihood scale
possibly dimensional hilbert drive geometric model another one generalization near ambient hilbert space drive carry pca subspace factor literature encourage report room constraint underlie one lack snr another limitation subspace usefulness highly nonlinear subspace handle dimension art nonlinear objective regard aforementioned kernel subspace nonlinear paper geometric term mc union ambient distinguish force formulate mc metric operate deal third carry mc extension consider term mc map lie subspace feature additional close regard formulate avoid addition derive novel iterative mc learn complete final synthetic justify main geometry presence geometry investigate result confirm superiority art discussion point literature specifically treat lead alternatively subspace experiment confirm vector denote identity resp submatrix row resp index corresponding row column index transpose represent organize constrain union subspace mc mathematically sec present presence miss mc feature sec conclude remark mathematically formulate learning example problem subspace ambient begin characterization mc canonical model ambient low dimensional subspace simplify ss formally capture distance constrained u paper distance hausdorff specifically orthonormal basis respectively subspace easy orthonormal basis note orthonormal basis principle angle order mc model manifold exist ambient formal characterization collection likely n definition pose goal learn I iy force approximation setup closeness beyond scope cross variant training geometry unobserve miss pose learn synthetic real mc learn true manifold ambient mc long learn mc theory still program iy practice solve intractable dimensionality space involve transform evaluation kernel semidefinite function trick union subspace mc mc scenario scenario remain unobserved quantify mc average test conclude point lead find pre datum address describe mc nonlinear begin available begin center define simplification membership different subspace notice orthonormal collection basis minimize likely infeasible alternate minimize term fix alternate minimization orthonormal fix carry amount matrix optimize large address update keep updating show solve define collection reduce find close subspace reduce pca appropriate pd form eigen sa subspace basis I l orthonormal subspace mc point convergence however indeed require knowledge subspace generalization require loose collection basis point carry update iterative fashion unlike redundant subspace subspace assignment involve removal training subspace iterate subspace pruning update merging subspace merge pair mathematically merging involve find pair subspace define symmetric td find subspace merge subspace great greedy loose pd p l initialize basis merge move subspace estimate dimension subspace effort g incomplete focus level mle first use denoise project onto write average every e k order basis l describe study mc location set explicitly author sp mc propose comprise subspace assignment update subspace carry lc manifold note ta respect component step specific function short direction gradient geodesic direction respect step term subspace orthonormal basis stop w I I ta pd tu l algorithm highly trick mc function solve image mc discussion g ty ng ng centering map pre stage mean map map data nn tn product membership let write discuss algorithm solve trick far sn orthonormal satisfy mc involve kernel trick write ny iy inner n c center inter subspace kernel center number randomly n c detail alternate begin alternate strategy early represent independent map since linearly independent method refer easy subspace kernel ng w l te u p sa move assignment equivalently solve subspace write update careful belong assignment initialize basis ready subproblem e intuitive reduce subspace reduce follow set eigenvector associate c c mc case train sec assume index observe uniformly case clear mc mc complete data inner propose incomplete mathematically proxy I proxy underlie type begin result ij span ij z z corollary plugging replace distance iy kernel skip proof elementary special case find estimator z describe deviation inner establish high relationship correspond j odd inequality trivially counterpart proxy entry therefore guarantee definite mc eigen u predefine parameter mc feature mc kernel conclude call far space complete trick noise processing task feature space give visualize ambient onto many term mc address mathematically reconstruction state need z pre begin calculate square distance q describe p express nn unique odd reconstruct entry see kernel pre regard value estimate value solution z note separately define entry experimental demonstrating propose denoise test sample geometric structure effectiveness mc algorithms sparse dictionary ssc sparse clustering component use ssc variation choose case data ssc robustness every belong carlo simulation datum power synthetic yield denoise subspace dimension pca note every pca experiment ambient orthonormal basis random w control distance generate stack norm strategy white experiment range repeat realization correspond average carlo htbp htbp synthetic level svd ssc ssc collection sample union subspace stack learn orthonormal basis experiment complete miss experiment respectively choose performance ground truth basis pair avg avg l sd avg performance mc regard error te te reconstruction noiseless part simply te te avg see turn fig learn well increase range complete close omit synthetic examine goal know experiment exclude step term time trial fail contrary give subspace next pca evaluate fig method outperform sometimes high effectiveness propose real city image shown generate clean extract signal way add form experiment range patch mc learn parameter trial fair state reason return therefore ssc output coincide estimate
explain table phrase sample encoder phrase show five accordingly rnn decoder propose phrase look phrase overlap phrase encourage whole rnn encoder decoder propose encoder decoder task translation look language use embedding rnn project vector expect fig embed word matrix learn rnn encoder european project document rnn encoder decoder dimensional word phrase embed use visualize word encoder consist hide unit sigmoid wise multiplication bias fix phrase phrase start encoder word encodes source decoder learn phrase word phrase representation fig p universit de ca university university universit du fr universit find I network call decoder rnn rnn encodes fix representation symbol encoder decoder translation empirically conditional computed encoder decoder additional exist qualitatively meaningful linguistic phrase great recognition see furthermore many neural successfully use language process nlp limited language model extraction promise summarize successful feedforward phrase line neural focus neural system architecture network act encoder map decoder back variable sequence jointly maximize conditional hidden ease decoder empirically english english phrase part scoring phrase phrase reveal phrase encoder improve translation qualitatively rnn encoder decoder rnn decoder capture quantitative improvement translation rnn encoder learn phrase semantic syntactic phrase recurrent rnn rnn function sigmoid term lstm case multinomial code softmax activation weight combine use learn novel neural length vector length perspective sequence condition input length differ rnn read symbol rnn change end marked symbol whole decoder another predict rnn describe also condition input hence computed symbol activation valid probability softmax architecture component train maximize log output algorithm encoder sequence output hide motivate lstm graphical describe unit gate th gate activation unit formulation gate hide force hidden state current allow update gate control act similarly memory network rnn variant unit scale short dependency active preliminary crucial whether previous hide eqs detailed statistical goal source sentence first side see practice weight normalization constant often optimize introduce factorize translation probability phrase phrase well probability additional log accordingly propose recently neural translate sentence sentence input rnn decoder phrase score train rnn encoder decoder frequency phrase original corpora expense phrase table simply occurrence one exist translation frequency pair capacity decoder ensure capacity focus toward linguistic distinguish plausible manifold concentration plausible train add phrase phrase enter tune overhead computation encoder decoder rnn decoder generate target phrase phrase phrase present similar rnn use neural short phrase phrase phrase length phrase increase apply sequence handle propose rnn encoder suit feedforward predict phrase improvement priori train author phrase embed distance score phrase pair feedforward phrase phrase closely similar use representation encoder decoder network propose representation sentence one rnn decoder target phrase account decoder aforementione ignore close encoder recurrent translation convolutional hybrid decoder list gold english translation build corpus include news corpus side word count concatenation datum necessarily lead result handle relevant give task subset model selection word translation training network decoder source vocabulary english vocabulary token system build default system achieve test c rnn rnn encoder hide approximated matrix approximated learn decoder intermediate maxout decoder initialize deviation except recurrent recurrent white gaussian left vector train rnn encoder phrase three explain depth supplementary end de la es de la la la pour pour la pour la pour pour la united et es et des des les one un des des un plus des ca phrase communication communication communication communication communication ne des send es les world du es du r du les day le des les les des et la se et le le le cb character effectiveness scoring phrase encoder decoder try use target especially comparison phrase score part system add redundant train gram target corpus project dimensional fed uniformly train achieve random partial score computational buffer gram stack decoder buffer stack gram perform matrix source rnn decoder de pour la la united et dans les un un une des ca frequent phrase sample decoder communication communication le la world du des du past les le cb top phrase sort decoder try em baseline baseline rnn word penalty expect neural consistently baseline
rescale w q check requirement polynomial px px hence lemma dy dy r rr dy find enough enough prove immediately explicit introduce density inner product clearly z give norm density probability sphere du enough r follow fellowship theory research part google fellowship author valuable comment theorem counter proposition open present r classifier approximation ratio technique regression technique learner access require classifier whose namely good run general improper hypothesis extremely practical application extensively machine computer unfortunately bad perspective algorithm proper agnostic know hard learning return improper agnostic agnostic ratio show assumption agnostic various restriction natural restriction marginal uniform even distribution hard regression efficient show namely uniform hardness exact naturally fit expensive biology make experiment useful complexity detailed interpolation approximation bind exact run almost match question obvious question extend product concave problem intersection opposed return build combine algorithmic previously learn approximation outline technique present course tolerance technical exposition algorithm input efficient hard running grow hypothesis xx x w w rw description hypothesis return introduce terminology df df df fx error good motivation enable convex enough function efficiently number time collection function contain good find spirit studied minimize dm px classifier overcome ds dp polynomial naturally approximate proximity define projection polynomial order find find output less access distribution return return detailed appropriate depending run time outline must show assumption w lemma w xt th handle th first tp suffice find polynomial sign except area origin say invoke say sign regime find map resp move accuracy maps resp function x second find polynomial sign area use tail complement section dimensional show aspect case erm linear error agnostic good approximation naive exponential distributional uniform know concave present run margin except instance boundary time agnostic assumption uniform hardness hardness formula imply hardness every hardness agnostic hardness proper ratio concrete argument uniform first w dimensional orthogonal x rw x vx vx vx accord follow lemma lemma explain approximation every lastly complexity complexity proof step dd tt r ready
closure inspection limitation theoretical issue accurately reproduce probably inferior overall fit specific validity specify order goodness sometimes research know set solution accept aspect consider goodness fit selecting reject reject neither finally report criterion package gap tool limitation aic observe statistical case detail aic goodness sometimes want could observed tie alone could generate tie dyadic know explain tie describe aic former approach relate would two available desirable goodness fit goodness structural would property propose make explicitly direct propose assess observe fit choose make assumption parametric simulate r package network adjacency link node consider undirecte laplacian sum zero elsewhere order brevity eigenvalue spectrum refer eigenvalue edge great spectral sometimes associate extensively characterize structure complex recognize scope basic intuition structure note reflect zero magnitude tie cut network divide connectivity network magnitude relative illustrate imagine comprise totally spectrum contain connect rather eigenvalue eigenvalue add successively large eigenvalue successively fine network sub magnitude eigenvalue one correspondence weight series cut large spectrum describe total strength relative strength definition goodness fit normalize unity give let likewise calculate normalizing q change multiply change size shape result tie increase tie size shape change structural spectra b double bar notion spectra however otherwise spectra spectra say calculate distributional statistic context sum error euclidean propose goodness necessary natural density also know enyi network remainder adopt note place nan observe mean survey important degree model nan model generate goodness fit simply divide mean euclidean fit one q additionally network imply calculate percentile goodness nan define zero percentile useful model extend great narrow structure explain percent randomness distant spectrum percent fit like likewise improvement finally model distant nan r enyi random approximation fit consideration fit incorrectly structure ordinary sensible specification highlight strength exist ever formation rather plausible quantitative sometimes play role case consideration reject negligible nan tie meanwhile substantial spurious high nan even account include fourth tendency tendency toward second tendency star star star simulate accordingly perfect fit aic dramatically star nan clearly fit complementary fit structural dyadic hand assess aic structural dyadic explain case star illustrate aic adapt add health survey go modeling effect final differ type create test make significant improvement aic star certainly likewise aic indicate superior model star however generate alone model measure fit measure absolute fit tie finally exponential indeed aic generative generate compare aic qualitative gain fit visualization spectra euclidean observe spectrum fit spectrum second fit spectrum color spectrum lie spectra fit improved spectrum light green error model plot opposite spectrum explain spectrum indicate red turning see fit differ spectrum observe spectrum contrast distant nan observe visualization produce red area net rgb green red fit sometimes intrinsic algorithm theoretically plausible algorithmic network form initialize tie walk theoretically generate skewed distribution assess fit find implement add second exist member network network combination pair well add random walk tool identify rgb fit goodness spectral distance possible two sample hypothesis purpose certain favor construct material separate graph direct focus establish graph follow remark network differently adjacency treat transition markov chain vector leave eigenvector corresponding distribution strongly connect zero way finally direct feature view consider certain condition semi calculate error rather e eigenvalue clear priori fit certain study present statistical analytically conclusion size network mean calculate biased number likewise bias median quantile exploratory simulation recommend examine simulated establish seek conclusion spectral goodness spectrum laplacian implement indicator complementary table summarize percent explain measure relative say absolute fit htb absolute yes measure yes sensitive yes fit yes sensitive yes yes aic specific absolute network ultimately however exist specific structural tendency structural aic assess allow hope ability compare facilitate prior cite end
trial mathematically nk k stable method average dictionary k digits mnist digit remain split enyi site among sample centralized cloud level detail give paragraph dictionary class x cd average detection experiment see cloud local svd use local representation considerably low detection achieve highlight variation model term cloud collaborative approximate massive across consensus converge centralized efficacy extensive datum rely guarantee estimate obtain site centralize distribute remain end consensus sum absolute initial obtain consensus fix z long iteration make desire iteration centralize power power site similarly denote centralized mix consensus iteration eq nm th express site consensus inequality bind notice also therefore eq remain notice plug finally iii plugging expression note obtain claim accumulation power finite consensus iteration fact help keep total argue begin centralized method next need show purpose q furth elementary invoke induction eq geometric main lemma therefore recursively result argument follow code however code accumulate dictionary atom differently dictionary atom overview know bound know j k j r recall challenge source eigenvector k b distribute combine first bound difference dictionary bound sparse code iteration know derive error bind remain satisfied bind eigenvector r fix theorem nc k te f dictionary mean get n mainly get k f ready know atom also consensus p nc prove first decompose atom centralize denote principal r notice consensus iteration theorem eq I algebraic lemma replace follow consensus satisfied k coding eq reality interested te td get centralize sec notice next apply n ref write appendix notice proposition follow definition bind perform consensus nc c n c b due fact submatrix c I follow note algebraic support centralized main case b q k fixing hence thing need cloud svd centralize dictionary j j I thereby prove fix assumption c b te tc q induction argument fix assume follow carry one prove claim conclude apply error theorem accord theorem atom satisfied nc nc must nc show however exercise algebra brevity nc appendix collect support perturb assume singular reverse triangular sparse perturb version sample jj note also p proposition eigenvector let define comprise eigenvector denote vector u u conjecture remark author electrical engineering edu representation big assume number site massive contrast union cloud svd learn overcomplete distribute cloud require communication individual analysis deviation dictionary learn site centralize global numerically efficacy svd synthetic high ambient lie near dimensional knowledge success task knowledge great much often focus centralized available effort work work distribute entity responsible assumption lie near extensive communication entity ignore detail topology paper site massive geometric datum public network term raw among site constraint big justify local concern age justify graph prohibitive internal topology main paper work structure svd base nonlinear generalization receive community interest drive overcomplete approximated atom dictionary learn compare recognition cloud name popular classical estimation average collaborative main deviation dictionary site measure early rely consensus heart cloud initializations dictionary k arbitrarily svd long consensus involve synthetic efficacy cloud k usefulness local distribute processing date nearly decade kalman recent example localization relatively little drive geometric include consensus geometric explicit consensus topology development carry average computing extensive communication distribute closely relate study site oppose site setup fundamentally learn dictionary cloud svd atom collaborative helps rigorously cloud centralize analysis finally cloud cloud provide cloud similarity unlike perfect consensus leave analyze cloud also analysis carry iterative difference consensus average carry work nature matrix operator nonzero operation entry denote product denote submatrix column norm max cloud algorithm cloud svd algorithm remark section main theorem site possible sense system server collection robot mathematically site denote connection site next assume site massive express site distribute ss denote fundamental opposed structure global collaborative learning outperform representation suboptimal level fraction outlier etc uniform site collaborative fundamental structure study learn column specifically centralize overcomplete dictionary non although convex alone alternate solve centralized datum location site impractical communication big individual learn dictionary ever sample concern rigorous establishe arbitrarily basis collaborative iterative decomposition svd enable exploitation distribute purpose svd presentation term learn iterate stage involve denote state combinatorial complexity pursuit sparse stage move k svd lie carry dictionary iterate individually atom atom computation define ts ts location easy find u v leave singular update coefficient svd atom move stage continue alternate stop criterion prescribe learn svd understand stage end site proceed propose compute solving site coding greatly simplify justified long dictionary remain close establish next challenge collaborative arise stage singular reduce error r resolve propose k r k ti matrix locally eigenvector site analysis cloud stop algorithm learn consensus svd stop comment communication cost cloud cloud total give thing site function available site cloud svd big normalization redundant retain consensus averaging understand dictionary return cloud k obtain centralized address understand cloud include code update closeness centralized solution property datum begin cloud svd state term centralize svd thing need deviation cloud svd dictionary centralize sparse solve practice analysis following focus sparse cloud centralize omp assume code accomplish q centralize svd identically centralize dictionary cloud svd centralize initial cloud k svd dictionary cause cloud dictionary centralize iteration dictionary centralize property lasso analysis among collection large among collection define hold singular te kt comment property centralize svd unique code algorithm compute bad result svd centralize centralize svd denote centralized site iteration ready mix doubly weight eq inverse gap denote modulus time dictionary cloud k centralize identically addition cloud k carry iteration k p centralized method initialize eigenvector long method mix comment major implication establishe arbitrarily centralized show happen iteration certain manner call multiplication need significant big datum enter relation fashion finally consensus distribute dictionary learn remain close centralized require consensus averaging provide brief heuristic dictionary identical initialization k svd dictionary begin centralized step perturbation happen consensus happen perturbation therefore cloud svd start perturb deviation code cloud svd add source perturbation update summarize consensus average cloud top eigenvector estimate centralize site cause error r r e k cloud svd cause deviation dictionary centralize mainly two source result need theorem error dominant eigenvector symmetric obtain site consensus site cloud step stability distribute dominant eigenvector
discover network increase user create tweet new movement external connect news event interested event connect learn dynamic comprise activity question piece develop quantify occurrence intuition user connect exposure target link quantify post compare similarity new predict make gain event property paper structure empirically cascade propose whether piece briefly conclude section twitter removal effect tweet propagate month english speak user month overall subgraph million user investigate grained dynamic create cascade post propagate user reconstruct subgraph examine evolution amongst million user million million relation begin month remove twitter user edge month twitter highly think grow evolve add edge twitter event highly amount twitter network average user month follow degree note even gain month twitter skew receive month dynamic twitter examine information diffusion mechanism user scale thick thick thick tweet thick thick figure process user user user tweet happen post get exposure tweet decide follow tweet way create diffusion new form newly cause somewhat local current tweet frequency tweet cause dominate mind investigate phenomenon twitter examine activity tweet dynamic scale partially tweet frequently high network plot even clear relationship dynamic user user similarly user tweet tweet tend degree month loose flow dynamic user flow event think graph steady flow perturbation steady refer user tweet around hour receive number hour negligible increase consistently receive gain intuition temporal dynamic twitter several time plot arrival rate per hour tweet plot gain tweet month notice hour correspond daily activity twitter steady twitter hour receive new diffusion perturbation arrival follow occur hour arrival activity still receive nothing next two steady arrival understand information diffusion identify steady period user receive compare expect identify perturbation henceforth periodic fluctuation arrival hour day remove employ transform abundance prove problematic instead treat arrival month user hour month clarity describe exact interval increase hour hour month hour day decay hour user demonstrate however occur remove employ method actual second method find follow activity hour pointing primarily likely hour day receive remove detect proximity event change arrival new coincide cascade generate tweet tweet follow time standard deviation occur hour hour much normally tweet tweet occur hour explain user tweet tweet weakly connect component examine least identify million instance follow arrival frequent detect reliable focus exclude occurrence contribute evolution ask user similarity pair information tweet aggregate tweet month document tweet tf user document although robust aggregated tweet document account small tweet largely tweet similarity whether similar content similarity hour interval day normalize exactly across user show average significant user follow similarity tweet tweet course month become gain versus gain diffusion high tweet even tweet tweet cause similar tweet compare entire month spurious cause increase tweet separate run randomized action preserve user user would randomly select tweet user decrease observe spurious follow tweet similarity interest increase user homogeneous cause jump apart user tweet user tf tweet measure precede normalize tweet increase tweet tweet tweet become examine neighborhood weakly number subgraph discover user month cause increase tweet increase proceed case connect day community analyze network density potential value follow edge largely grow node follow tweet decrease fast decrease tweet day around type steady decrease behavior tweet density change user decrease connect tweet become related user likely discover b network address ask certain type content cause new iterate tweet create token occur tweet measure token increase decrease tweet filter less identify token violate follow probability pearson tweet cause token significant whether tweet cause quantify token tweet tweet create movement movement token ratio associate movement token table additionally tweet token time cause tweet token event cause spam token increase star datum switch team would lead huge old team would team create link far cascade explain predict cause spike process applicable closure create user exist one short follow edge result closure light user come follow refer user neighborhood predict occurrence follow need neighborhood neighborhood thick thick thick thick thick thick thick thick thick dash dash thick green thick thick rectangle blue thick rectangle circle circle circle circle green fill white fill blue fill white node white node fill white define steady instead potential discover diffusion user user tweet interest unlikely know user interest tweet content via intuition interest previous tend tweet follow tweet effective compatible interest compatible expect edge increase tweet user cnn news follow wide range user stay informed event low tweet increase tweet similarity tweet intuition tweet notice variability tweet user order reliably distribution tweet follow normal tweet quantify distribution tweet choose equal quantify way normalize day confirm plot say form follow edge normalize tweet moreover also explain user tweet new current follow way number follow neighborhood fact condition occurrence quantifie arrival new steady occur high steady follow user tweet compatibility reach reach set normally reach follow time tweet quantify well predict occurrence appearance event potentially twitter need order quantify predict follow simple experiment randomly fit likely guess high guess calculated area performance compare model baseline baseline likely precision baseline user rank neighbor tweet follow rank sort model outperform baseline score score perform baseline user l poor performance surprising user raw user receive predict current highly experience large compatible tweet degree user new could potentially follow circumstance follow user also order improvement baseline due follow phenomena number successful might lead main ideal occurrence user compatible tweet instead steady arrival user portion nearby yet discover focus work various useful continue
df statistic df df df df df df show association layer layer contain number subject survival identify layer primary result association survival identify sc c test statistic df min df association sc sc compare patient differentially region cancer datum apply identify identification section choose variance identify report mean identify identify cancer also associate identify sc identify sc tends identify variance contain cancer patient sc less run ht exact composition min var var na var var min powerful study hierarchical case sc compete simulate sc several likely set indeed even necessarily observation arbitrary difference method maximize note also detection although sc may improve apply scope particularly high dimensional big sc provide nan determine difficult however demonstrate frequently true return exist criterion sc accurate criteria future limitation beta require another unlikely interest correlation tend even feature particular violate sc spurious correlation preferable weight interesting variance identify small homogeneous heterogeneous merely advantage mail likely detect sc identify variance area research chen department university north hill nc email north hill nc email university hill email liu north hill nc mail live chen candidate hill nc mail live edu distinguished north hill nc mail hill nc mail email edu grant de grant grant grant situation feature represent homogeneous patient disease cancer example feature identify variance world publish predictive time mean unsupervise exploratory play analysis high sample express matrix correspond powerful reference overall many subset useful situation aim identify subset patient exist subset differ cancer aid cancer sub contain feature influence observation cluster one partitioning however solve may identify identify respect identify feature identify heterogeneous exist approach study set maximize individual equally important giving produce inaccurate problem propose novel sparse maximize represent observation iterative initially appropriate maximize respect unweighted soft justification imply sufficiently subset subset sparse cluster tuning force small observation differ contain possible observation identify list approach experience nonzero weight propose identify belong first soft perform nonzero motivation apply denote statistic hypothesis heat map artificial quantile plot versus weight expected second remain weight explanation weight kolmogorov step nan hypothesis intuitively choose line return feature either small recommend sc verify give belong increase least exist belong assume law wish datum identify one denote element repeat identify fail reject hypothesis know weight exist approximated repeat step time approximate procedure expensive computationally develop fortunately modify exact calculate assumption write sum square modify optimal imply difference observation imply imply nan kolmogorov distribution easily numerically nan weight subsequent unless sparse produce mean obvious choice differ feature however situation desirable useful procedure design differ wish high low example analyze dna exhibit variance reveal region genome differ initially cluster repeat observation randomly partition variance assign cluster half small initially cluster preliminary approach latter subsequent feature replace version initially respect apply procedure appropriate letting secondary one denote denote standard note alternatively chi distribution manuscript slow variation procedure cluster applying identify well variety approach apply improvement search approximate sum search submatrix reduction sum assume iterative search overlap computational fail valid define consist identify compare simulated signal generate normal focused identify comprise row represent standard rectangular layer background simulation structure fail identify simulated describe illustration heat scale primary rectangular middle panel sc white region matrix approximation overlap moment simulate set comprised entry entry follow overlap shape construct manner background layer layer background layer background primary expect partition evaluate heat scale rectangular block panel sc two comprise layer contain final two layer contain show one simulation scenario illustration single simulation panel show heat map bottom corner one panel region approximation exist remain assume approximately dimension although reasonable situation fail violate spherical dimension linkage job spherical see reasonable expect sc linkage compete spherical sc iid partition versus fourth simulation panel method sparse belong limitation capable detect point high datum describe heterogeneous ability sc simulate set overlap generate way assess illustration heat scale variance corner first identify white overall plot scenario table prediction valid table simulation stop rule simulation sc perform misclassifie across misclassifie low sc also except misclassification second simulation scenario sc normality violate low exception sc produce fail produce fail detect simulation identify comparable sc simulation scenario sc perfect spurious entry performance identify false negative misclassifie excellent sc exist first variance high accuracy usually although negative poorly design generally fast sc slow sc simulation pre rank time simulation incorrectly th identify th consistently comparison return return simulation scenario present simulation determine present simulation simulation sc sec min min min na min sec l primary identification misclassification
intermediate array dim dim double dim cell dim across k multiply dim dim j I double dim template numerical vector multiplication operator array class template place header file appear evaluate pattern pass object deviation interior produce sampler code quantile collect class member hold supplement first multiply evaluate value e analog array response development evaluate treat name pass across np appear pseudo weight supplement look false array array beta dim array double dim master double new dim double double dim I dim dim double dim std integration std dim j dim std std std stream create array element correspond array transform normal process without choose quantity pass master receive master dimension collective product process array process hold name private member feature pass master average process approximation available supplement array object array double pz pf array dim parallel pz pf pz pf beta dim pz double pz pf std dim std std std std array object generate differ object normal might arise create array safe location provide master parallel region assignment np cl np seq context seed force mc np across double np function worker package manner stream initialization list program approximation hold expect program produce change change stream carry monte ratio serial parallel times table five processor program package expect execute program ccccc uniformly spaced point quadrature product require scheme package stream parallel use share minimal use stream correctly accomplish array employ uniquely determine access context option create seed master memory latter inter processor demonstrate package build demonstrate within acknowledgement computation carry grateful anonymous lead software create parallel environment effectively program via way generator package example monte application appear final publication http simulation parallel sense without inter processor case occur practice however division multiple processor stream processor behave sense fortunately stream effect stream dependence see random seed x cx seed along distribution stream derive default generator stream construct apart component permit normal mt although stream situation arise message remain review parallel initialize spaced position generator assigning randomly generate seed overlap contiguous cycle sequence segment parametrization associate generator varied overlap stream processor seed small period mt provide long period feature availability mt generation repeat generator ensure result generator addition division come stream example generator guarantee overlap number generator exhibit modulus generator split may undesirable g quantify nonlinear generator show stream split generator generator period generator recursive generators cycle division roughly multiple generator large period generator combine inherent recursive generator good generator well provide generator good cite mt option efficiently generator advance generator stream make generator easy cycle division package context generator jump mt stream roughly time slow package produce process parametrization generator package generator fall stream several reason package quality stream practical tie library quite provide orient uniform overall incorporate property cluster provide toward parallel generation generator available generator capability software setting package belief contain fundamental foundation safe evolve practice integration item trait describe basic generator section connect similarity generator generator recursion seed via produce number length roughly less use parallel divide overlap stream key stream transformation initial initial seed generator carry multiplication prototype none double multiplication result array purpose take vector allocate contain anonymous name manually stream long I function support header appear methodology false none h long seed start np seed world std random double std std else long receive seed master world double world idea straightforward master conjunction determine stream seed initialize communication like result program alternative array formulation reduce amount expense usage generator example generator set align seed use generator generator element seed seed cat random six seed require integer seed store code however treat integer sign complement representation must add integer seed use generator example seed seed although language another access package instance generator stream return generator request specific option generator use random seed replicate stream previous generator package computing package base suggest parallel capability function package illustrate cluster four random parallel related confirm generator seed advance correctly window machine replicate produce seed stream expect call frame line mc core important specify desire additional core master core must window gain valuable option multiple frame label k
mcmc distribution modeling dynamic inversion mcmc different evaluation use function typically intensive g many query situation computational burden evaluate computationally surrogate inverse expansion projection reduce underlie applicable type surrogate forward solve require evaluate full space span represent basis quality rely crucially compute employ projection base innovation select sample snapshot integrate simultaneously process numerical adaptively reduce couple together accelerate approximate induce reduce order latter biased h collect posterior exploration build reduce concentrated space reduce build offline accuracy space cover typically drive approach increase information dimensional contour show sample compute approach region drive basis reason drive scalability offline orient develop pde constrain optimization simultaneously process reduce parameter trajectory remainder section outline inverse efficiency section drive reduction drive reduce delay acceptance speed ergodicity drive order construct explore provide conclude brief overview detail find discuss computationally intensive pn bayesian construct model likelihood function represent knowledge denote uncertainty without normalize distribution metropolis hasting use rejection sufficient expectation distribution carlo integration unbiased dependent integrate detailed wish improve cpu mcmc adaptation learn posterior online proposal stochastic newton adapt local geometry mcmc give forward cpu intensive major model effort mcmc require enable fast approximation algorithms transition delay acceptance couple ensure full acceptable error explore section drive adaptive framework simultaneously construct explore describe nonlinear differential finite difference discretize discretized system discretize discretize nonlinear pde parameter e realization observable model solve reduce span projection greatly full define output must efficient solution presence parametric low necessarily solve reduce evaluate matrix residual onto result element expensive unless dependence miss empirical term order selective adaptively tailor focus evaluate sampling localization multiple structure approximate replace order give approximate posterior normalizing drive adaptively select snapshot scale output current whiten compute deviation evaluation dual indicator exploration employ delay reduce construct initial step simulation step candidate full density value delay acceptance employ decrease correlation fast rejection stage delay rely approximate inaccurate order potentially stage rejection delay algorithm accurate acceptance introduce delay posterior state snapshot snapshot exceed reduce drive provide select reduction approach sample full full algorithm full length error threshold reduce define acceptance q mx n full posterior mx x gram schmidt chain step ensure post last posterior control length second acceptance spend simulate proposal reject dynamically evaluate indicator infinity describe adaptation control must update scale exceed exceed adaptation definition precisely reduce model become adaptation stop step model least coverage defer use adaptation online reduce adaptation stop delay prevent reduce adaptation continue simulate model large range adaptive variant newton computational evaluating also throughout reduce order lipschitz since constant continuity continuous ergodicity proposal adaptive condition ergodicity adaptation delay carry continue exceed reduce furthermore adaptation stop finite adaptation reduce ergodicity algorithm regardless adaptively analyze induce reduce bias monte error approximate cpu adaptation terminate reduced basis provide hellinger full hellinger translate directly analyze full construct respect feasible posterior feasible posterior complement feasible reduce hellinger specify region formalize proposition theorem follow feasible set assumption exist constant set z z lipschitz exist full induce z x x x apply constant certain pointwise model reduce feasible decay eq hellinger characterized adaptively update drive reduce posterior user threshold practice check mcmc heuristic motivate adaptation definition invariant step reduce decay refinement basis refinement hold basis exceed terminate recommend choose delay time adaptation model ergodicity ess number evaluation consider reduced indicator provide reliable reduce reliability refinement indicator evaluate acceptance update exceed approximation algorithm full sample approach approximate even monte estimator acceptable monte small amount metropolis algorithm remainder provide detail analyze carlo adaptation discard evaluation error exceed mean use line indicator model threshold order reasonable acceptance correction less adaptation stop reduce full accept reject directly use line reduce basis finite adaptation full evaluation proposal reach prescribe potentially guarantee length threshold threshold reduce order acceptance full distribution accept reject accord delay acceptance x acceptance reject adaptation stop acceptance accept reject analyze estimator posterior first finite effective produce monte ess eq cpu sampling effective speedup factor depend expense reduce interested situation rearrange reveal ess produce suggest single stage mcmc mcmc satisfies mse monte carlo estimator bias hellinger result condition square bias reliable approximate way specific govern rate reduce var mh mse amount regime reduce dominate hence factor avoid expensive mh var mh accurate efficient propose steady denote spatial pressure head represent pressure head field govern superposition gaussian standard center prescribe normal problem impose boundary equation condition nine experiment various aspect spatially project radial inference radial apply endowed proposal distribution covariance proposal run offer h radial radial basis weight independent log normal field use pressure head evenly grid signal carry efficiency target iteration evaluation order start begin simulation adaptation single stage proposal burn match reduce evaluation algorithm discard remainder algorithm algorithm build prior impact step define complement ex ex ex ex ex ex ex ex ex ex ex ex ex summarize evaluation reduce cpu ess reference set acceptance target show produce value first metropolis acceptance decision simulate enhance reduce evaluate dominate thus speedup choice approximately reasonably situation could simulate iteration full target produce dimension basis reference full inspection parameter contour distribution black reference algorithm blue red accurate simulation show visually plot suggest contour various cause assess accuracy approximate second amount situation large speedup advantageous full reference algorithm monte posterior complement reduce construction reduce estimate suggest hellinger finite dimension target draw sample affect achieve desirable posterior measure conduct lead additional basis add result report consistently adjacent construction asymptotically decrease basis situation minor increase computational load small full compare reduce reduce construct reduce proper plot
draw seed proportion seed geometrically decrease multiplication refinement cluster principle select manner post laplacian cut energy use factorization walk principle another factorization aim provide total variation attempt find optimization technique initial initialize algorithm random partition factorization seed increment seed algorithm large quickly seed algorithm converge quickly allow exploration high clustering reflect speed generally slow accurate algorithm dataset experimental text handwritten text remove stop remove occurrence occurrence nn cosine tf variety graph unweighted graph mnist extract unweighted raw preprocessing source c news mnist reach comparison report either news speed cluster calculate accord ir ground computed count represent run obtain l refinement mnist accuracy refine refinement refinement get graph quite implementation neighborhood benchmark c rand spam k k news exhaustive exclude ad matrix entry perform similarity main body author word similarity text dataset prefer truth matrix obtain slow news unweighted raw document less time neighbor cosine tf unweighted similarity datum extract page classify keep library appear less remove similarity tf source obtain simply cosine unweighted similarity word count raw document library stop appear near neighbor cosine format publication describe value indicate absence corresponding dictionary mnist matrix image nn weighted similarity compute euclidean raw vector contain raw near von propose basic seed thresholding seed randomly thresholded state cluster benchmark run achieve accuracy refine approach remove still unsupervised task automatically graph point vertice graph whose encode point popular use cluster vast literature graph area resample base partitioning graph contain reasonably algorithm intuitive implement scale number well experimentally benchmark algorithm order multiscale refine experimentally combination refinement maintain expensive direct heavily optimize one arise unlikely leave quickly provide work guess come label label come extent seed vertex partitioning clustering cluster seed algorithm combine approach cluster comparable size low assign vertex might good propagation assignment label context overcome seed vertex assign near cycle gradually draw seed throughout refer formalize idea weight undirected encode degree randomize iterative initial rv rr rv number seed successive iteration update current similarity increment rf mf f w fm variety choice grow circumstance implement strategy initialize subset vertex column outline routine seed small increment cost prove lie cluster generate ff f simple grow routine choice diffusion circumstance utilize give size extent usually indicator grow amount measure another burden grow routine terminate produce diameter allow handle indicator experiment yet modify useful negligible cost experiment combination turn outline discuss upon idea notion label point form heat heat outline grow precisely propagation first unsupervised label propagation seed work localize vertex partition instead algorithm iteratively alternate appear presence step incorporate manner resemble principle inspire upon vs clustering relate interpret maximization minimization kernel power weight depend exactly replace weight directly appear utilize generate embed graph cluster random main cluster primary incremental process process grow quite
private use bind differentially private must show essentially follow provide tb p work upper mirror nn nk dependence curvature bound come free give differentially private erm output incur generalization error loss incur privacy asymptotically think privacy come free generalization many generalization lipschitz dominate ball loss loss dominate privacy risk r internal randomness private mechanism private property risk function respect work differentially contain analyze differentially precise tailor differentially set norm nuclear view assume convexity far objective perturbation dependent less restriction function respect norm use frank wolfe curvature precise polytope private algorithm linear eq q code bind bind nearly tight private q table summarize bound combine tight tb op l nk n generalize weak condition smoothness ignore form distinguish two lipschitz brevity lipschitz call aware risk dimensionality erm width mirror build aware work make rsc privacy risk depend meaningful tight strong hold practice require dependent generalize provide notion complexity complexity closely formalize width show constraint measurement need underlie analyze descent convex norm get sgd noise necessarily polynomial dependence depend width noisy gradient attract asynchronous work geometry applicable differentially private mechanism context answer database reason differential privacy provide arbitrary auxiliary privacy quantify beyond scope randomize differentially neighboring record equivalently hamming distance review exposition algorithm closed define strongly within norm convex bregman divergence always strongly convex within follow duality pair dual say establish mirror design closely geometry early set precisely private mirror depend width instead noisy express expect drop proof private mirror take input set refer bregman differentiable sub md differentially private mirror descent loss b privacy differentially privacy guarantee composition reader utility useful body strongly theorem depend standard mirror counter role one convexity guarantee suppose convex set gaussian diameter strongly norm notice reduce time scale ease parameterization refer begin jensen excess sequence tt suffice next used step triangle third step expectation choice gaussian proceed term q assumption strongly imply sum round zero sampling theorem obtain maximum exposition convex diameter respectively r algorithm lipschitz randomness literature mean mention dependence remove convex section twice continuously main perturbation perturbation smoothness privacy analysis necessary towards affect sub optimal sample tight guarantee strong full dimensional think simplex set differentiable strongly hold special theorem particular correspondingly ball htb l dd p guarantee diameter function continuously norm utility gaussian suppose twice continuously differentiable suppose convex lipschitz ease j n definition algorithm true due condition q algorithms lipschitz task linear e ij op private mirror loose case effective frank wolfe frank wolfe algorithm move towards frank wolfe algorithm remark curvature q v quadratic frank wolfe frank wolfe gradient major first reduce solve minimization vertex secondly step frank take boundary intermediate inside sometimes outcome combination boundary vertex outcome example polytope correspond reason frank wolfe many machine useful polytope exponential version frank wolfe achieve privacy replace private polytope polynomially vertice much small frank wolfe algorithm high compressive apply able gaussian appendix frank wolfe polytope I hull corner algorithm basic convex hull vertex apply mechanism differentially exponential cover polynomial perturbation differentially frank polytope set corner arbitrary privacy guarantee differentially privacy straight strong composition wolfe convex polytope c algorithm curvature ease represent utility invoke utility frank wolfe private theorem corner probability plug immediately get frank wolfe private wolfe tight define loss minimize preserve problem variant regularization start sparse success lasso private frank wolfe check bound far programming x apply fw work assumption might realistic setting consider probability ensure bind nearly private frank wolfe ball differentially private exist differentially private algorithm prove code argument similar implicit theorem code remove integer addition produce dimensional agree row write construct database pair th row argument mutually consider px eq crucial small consensus subset three proof contradiction denote use w contradiction must lemma theorem private markov agree privacy hence complete mm lem lem lem lem lem lem lem claim part research perform microsoft li zhang microsoft research empirical erm minimize function consist natural differentially erm line private erm output loss fairly well understand private mirror descent lead error loss dimensionality denote width constraint improvement strongly private wolfe bound common regression lasso optimize match supervise learn select point availability sensitive guarantee individual rigorous differentially private algorithm risk minimization suppose reasonable empirical model overfitte algorithm stable constrain ball private
ever additionally demand hold def sigmoid log layer predictive seek detail observe hold slowly approximately second modern baseline dataset move improve stack gamma lead performance finally function find gamma outperform distribute normal train deeply poorly def embed double row replace figure click px item click indicate paper collection movie movie rating star user document fit two def size regularize mf test introduce previous section recommendation test user computational reason subsample observation mf report measure un truncate ndcg outperform mf table highlight compare deep activity ndcg architecture regime practical hierarchy deep acting prior compare hierarchical capture semantics interpretable collaborative filter appendix field tb model initialize randomly parallel pz l p l variational family poisson bernoulli necessary score function poisson function shape score function approximation eq normal variational parameterization q score enforce positivity avoid gradient unconstraine score softmax ascent scalar reciprocal root historical gradient show topic choose three group concept super gamma achieve fix poisson use iteration def shape rate def run converge hidden network family perform recent box variational inference combine pairwise recommendation datum go layer prediction interesting exploratory predictive performance model develop exponential flexible distribution reflect behind unsupervised cascade layer latent inner fine advantage probabilistic family many kind representation network graphical model variable def count product share document layer activation via turn condition super topic term super via inner product level article york style though conditional word article define cascade layer topic layer branch leave center def recover something deep factorization context computer vision fall feed forward differ away addition weight value music binary sigmoid multinomial choosing amount choose finally build traditional research literature def pairwise double def user item representation low representation exponential family property context deep network collaborative def exploratory well generally rich landscape modern article word family family exponential family important sufficient log different sufficient base sufficient statistic construct chain family together layer control parameter z n weight k top layer inner model call depict conditional dimension vector variable hierarchy control product draw layer px separate def model pairwise focus count likelihood count entry next return document put form topic similarly layer weight concept topic topic document depict compositional sharing neural family discuss family connect natural specify link moment statistic exponential moment completely family link via identity deep exponential transformation source linearity exponential poisson level log correspond conditional layer modeling represent super property early next layer function derivative equal link require positivity factorize distribution relation layer case softmax function softmax preserve relation mean product well similar sigmoid allow extension sigmoid observation number feature turn summarize gamma r kn graphical model neural distinguished history highlight result deep broad focus sigmoid exist stochastic feed forward sigmoid network dependency undirecte infer compositional boltzmann rbms rbms layer undirecte tie direct away independent become dependent make hard rbms force parsimonious conditional exponential certain infinite def tie equivalent tie family represent broad family rbms variable relate model family computational partition computation sigmoid belief network kind develop variational algorithm seek kl approximate share across observation approximate variational running observation factorize component parameter expectation general avoid analytic noisy unbiased gradient respect approximation monte compute carlo gradient able evaluate markov detail function primary computation latent top q gradient similarly gradient apply gradient sum normalize
accord latent depict except difficult part reconstruct em cluster detail rand fig leave reflect parent child root channel flip medium technique fairly continue ica give global guarantee find optima example grow latent attempt reconstruct theoretical method technique paper order structure uncorrelated variable input connectivity show top start connectivity ten describe end visually trait largely reflect type experience design various intend trait survey statement neutral slightly ten belong trait answer learn show question big trait recover perfect try reproduce confusion advantage none recover type exactly interestingly ica close behind among contrast search among minimize ica trait predict true take consist describe nucleotide rand ari substantially perfect match gender layer east china dataset different message analyze typical feature signal principle word usage recent unsupervise hierarchical linguistic ten frequent token datum latent level tree normalize mutual parent extent post report fraction coincide expect portion intuitive relationship relate cluster word perfectly structure user send encode match precision short token perfect match g elaborate signature identifiable variable perfect precision sale predictor discussion discussion match version basic measure appear context interpretation less connection share describing provide extra optimization present bottleneck extension bottleneck behind small typically label maintain compression relevance redundant store ignore transform optimize use optimization lagrange multiplier reduce drop clutter fix totally py py actually next proceed fashion detail j formal define py py py px partition calculate sum term imagine py py py py py linear combination weight consist requirement sum replace py px py estimate marginal term marginal optimization correspond guarantee elsewhere sec appear exactly enforce exponent instead contribute non objective involve share common variable begin desirable allow learn structure value force amount correlation smoothly magnitude fall reach set base synthetic bottom algorithm value label sample reflect approximation practice nearly maximum well fig help dna gender visual root connect reasoning learn structure channel integer deal categorical data form outcome miss fact dna snps miss sample look redundant miss information briefly comparison cluster gaussian affinity neighbor affinity neighbor bi transpose note clustering nmf try boltzmann single cluster input variable neuron dimensionality either nearest clustered look component none build version dataset unsupervised combine normally turn heuristic look single end file line obviously many signature format result strongly perfect soon collection letter character I frequent occur average occur word use reflect word train top start resolution smaller apply fine grain representation discard layer unit minus plus minus plus minus ex ex plus minus ex sciences institute california hierarchy successively objective search make meaningful structure source include dna automatically learn uncorrelated really view univariate cause cause responsible generate hidden cause reconstruct propose principle principled search condition factor minimize multivariate word simple explanation account correlation building foundation paradigm tractable provide rich insight begin technique detect structure succeed perfectly reverse dna perfect predictor independent relate gender text recover hierarchical theoretical connection future notation capital instance ambiguity cardinality take always subset index high entropy way total multivariate correlation write group conditioning explain correlation symmetric argument zero distribution explain encoding group specify dag appear redundant carry connection see explain let search solution correlation datum draw worse surprisingly difficulty perfectly generally expect correlation therefore correlation impose additional single group objective bind regardless give iid output give search explain end begin
indicate suggest provide focus range ridge regression optimize correlation highly less configuration consider preliminary suggest provide beneficial minus ex ex microsoft suitable store core file pass processing framework g strategy provide excellent iterative canonical correlation cca multidimensional nystr om find machine familiar application include semi representation locality involve unlabeled partially datum vast motivating need find common view subject form give cca projection kkt multivariate equation strategy moderate sized directly reveal qr decomposition qr decomposition technique solution possible analog power iteration block variable multiplication via furthermore require pass b proposal outline good nonetheless iteration attractive cca e exact ultimately utility simultaneous translate document extract european sentence level process sentence bag compose preserve hashing bn embed hashing projection find spectrum pass excellent approximation exhibit ultimately decrease comparable dash pass b ex ex ex ex ex ex find hyperparameter vary set free sufficient precision identity covariance run
box base say something significance order side information loss confidence click include observe increase case order overlap include click summary find hyper combine side alternate parameter think model complexity parameter strength sequential alternate would particularly yet separate bad hence properly landscape review combine collaborative information provide model elsewhere rigorously day click rate manner confirm albeit difference popularity medium digital finance service product ad attention click website specify page ad determine place potential bid pay ad page user page bid act key evaluating calculate ratio predict face context click empirical poorly click scope dyadic label whose binary label give introduction transaction record shall elaborate record display click click dimension construct record click number click click click view always unobserved log hence formulation click naturally able entry prediction confident view probably enough pair entity predictor model information collaborative filter netflix movie rate rating unobserve filter binary outcome object domain entity collaborative filtering build latent think try offer applicability reproduce extensively investigate web search content retrieval logistic technique maximum operate user reveal query feature click model framework knowledge namely ad direct query ad base much feature model explore area well combine explicit finding combine advantage weak recommender effect prediction collaborative dyadic demonstrate model side inspire collaborative filtering superior notably dyadic single per classify data page factorization page henceforth refer ij mf dx formulate regular logistic distribute linearly continuous index click involve click distinct small total click thousand click particularly click summation significantly jointly vice versa exclude minima help control overfitte suggest penalty factor latent batch bfgs regularize special take newton set really learn framework require regularizer skew non click click capture baseline effect row case page hence latent actually explicit attribute name etc page ad server learn overfitte discuss intercept alternatively want page bias page index encode dyadic prediction introduce dyadic introduce tensor follow factorization p si p ij optimize bad local minima section odd input rewrite hence feature fix learn model find insufficient obtain good performance fitting model hold fix leave work unique click regardless quantity initially potentially involve alone section click obvious predict click empirical number predict rate involve either factor feature experiment dataset extract transaction international technology due nature consecutive day auc logistic hold e period total day report label click observation feature include encode include stre weak user country visit day site top country user never mid mention thereby result feature overcomplete experience sampling intercept correction performance investigate particular usefulness explicit feature logistic regression side alone strength log use correspond logistic practice feature include feature practice side lr hyper tuning highly weight bias search tune hyper dataset reasonable consume instead success follow heuristic tuning logistic regression alone weight suitable run experiment different well hyper verify experiment setting including find along train first day worth initialize test feature dimension annotation mark except low order grid plot peak advantageous increase advantageous run far decrease regularization experiment bias b performance add dimension hard us conclusion preferable thus report day strength top auc intensity loss mark specific configuration day confirm trend peak performance auc agree calibrate auc seem confirm logistic rate intercept could beneficial alternate latent necessary confirm ensure performance claim serve general pick mark little experiment daily basis day production warm start run epoch fitting
pure model advanced could instability mathematic computer university cascade neural network architecture investigate thin e international conference artificial intelligence artificial intelligence volume intelligence title cascade investigate surface propagation page publish self policy department engineering department mathematics surface interface attract circuit device cascade nn develop greatly reduce architecture density interface coupling surface enhance device optimisation band chemical sense enhance paper literature propagation optical mode call offer control assess propagation phenomenon well establish material conversely wave due poor understanding propose novel neural separate medium currently determination suitable nn sensitivity novel weight term share parallel hardware amount put proper nn perform software characteristic field grow rapid due behaviour light recently investigate light surface outcome investigation capture cell main interest wide range frequency decay direction coupling field wave flat interface half space real fulfil wave interface medium adopt interface air fig simple order purpose paper investigate relation proper nn affected structure concern equation material isotropic isotropic frequency interface propagate optical frequency take finite free connect length propagation field wave wave geometry separate media package match layer external surface wave range many varying obtaining interface decay visible range correct cascade different dedicated computation say propose novel paradigm run comprehensive cascade separate training novel topology feed comprehensive cascade feed whereby stage new stage depend stage prediction phase output validate stage deviation spectrum behaviour novel topology gaussian fourier analysis ad act transmission purpose peak window training phase optimum perform gaussian window hide number window fourier module act connect layer neuron module fully consist module variation introduce weight connection neuron neuron respectively signal output send last validation module perform transform window module dynamically allocate buffer implement latter signal perform admit plan validate send input module allow validation relative finally global consist neuron neural module implementation asynchronous validation barrier mechanism overhead result barrier process therefore version overhead avoid much join construct produce reason use share memory communication overhead process avoid however require handle concern overhead propose parallel solution process care phase cascade core processor memory support nn cascade predict value input vector evaluate kind consider epoch output see activity execute cascade training intermediate third activity activity fast transform predict signal result pattern datum processing perform module act appropriate merge merge give positive neuron activity start execution new training stop soon epoch involve though performance error vertical execute
fix qr factorization orthonormal projection skeleton uniform sampling approximation column approximation column well bound onto span theorem apply term error
ghz go ram perform nmf quick remainder good exact exact nmf linear slack start strategy nmf observation develop initialization slack cell ms ms ms initialization strategy strategy entry poorly new effective random fact issue random interval generate usually zero feasible contain sparse start exploration away entry position four initialization resp resp single sparse explain initialization exact nmf nmf scheme order assess performance compute mu accelerate alternate optimize block compare real document interested nmf matrix identify nmf subroutine perform number nmf poorly nmf poorly entry modify subsequence discussion c g nmf remainder heuristic widely use simulated annealing framework briefly multi explore neighborhood initial fashion hope solution subset generate locally temperature solution next iterate keep important happen go temperature temperature allow solution final accept procedure crucial nmf slow j u tw e truncated frobenius step locally nmf nmf class slack h h k consist combine heuristic example instead computed refinement word initialize hybrid run stop exact find run see h c ms point start heuristic rather poorly compute hybrid hybrid compute fix hybrid practice recommend many fast slack fail hybrid find nmf nmf become increase complexity depend illustrate exact report slack matrix g size slack regular none heuristic find important question difficult likely newly hull generate use procedure whose circle arc part middle distribute angle point run hybrid run table maximum nmf ten h extension complexity slack matrix equality hold another exact find ten might complexity leave issue far rank interpretation nmf nmf rank equivalently nest slack vertex inner edge polytope dimension scenario outer coincide slack outer inner outer nonnegative corresponding conjecture take role inner outer hence rank small slack reader detail hull one correlation polytope ts slack exist instance index q submatrix slack hybrid run conjecture slack correlation polytope nonnegative rank would exact outperform simple strategy nonnegative relevant initialization suitable exact research include development heuristic especially difficult compute nmf good approximate document hyperspectral would interesting fine tune heuristic test would library heuristic library factorization still speak compute useful develop transform example done manually stress heuristic exact heuristic conservative g practice would easily parameter exact nmf nmf table sa initialization strategy sparse good initialization sa exact initialization difficulty exact c value temperature end nmf computational g value j j j performance strategy good sa nmf initialization fail c sparse observe get poorly reason tend similar rank explore c c c g g strategy note sensitive sa able situation hybrid strategy sa expensive g g mm mm mm corollary ac factorization factorization heuristic anneal namely linear slack randomly demonstrate superiority strategy heuristic combine heuristic insight exact nmf particular conjecture generic conjecture value submatrix slack polytope exact heuristic slack matrix extension finding give nonnegative nonnegative reduction successfully variety mining include image nonnegative nmf look despite nmf successfully many practical dedicate nonlinear try minima optimization improve initialize much aim well nmf tackle minima finding nonnegative compute nonnegative problem factorization nmf contaminate subgraph subgraph bc exact correspond nonnegative union rank nmf provide upper conversely g reference polytope lift exists possibly grow crucial optimization whether ideally linear equivalent formulation appear call polytope slack slack polytope worth nmf recently extension know reference show perfect answer open question bind nonnegative counter belief euclidean matrix standard linear conjecture motivate strong along nmf nonnegative closely nonnegative computation probability matrix check arithmetic polynomial nevertheless check nonnegative admit nmf polynomial I fix rely exact problem matrix show complexity later rely elimination translate deal unable either solver run number dedicated elimination perform exact nmf scale tackle organize class benchmark description correspond present compare initialization select nmf locally strategie dedicate nmf sa along compare strategy perform remarkably nmf class matrix strategy discuss problem finally discuss heuristic well ii slack regular generic submatrix slack polytope heuristics nmf outperform heuristic algorithm previously nmf nmf counterpart heuristic relevant first time nmf algorithm generate matrix concrete use heuristic nonnegative datum set use make hope promising show motivate nmf compare however linear fact distinct linear allow non trivial nmf slack matrix polytope vertice nonnegative entry slack th polytope ai introduction nonnegative slack equal slack matrix several class correspond nonnegative low pattern interesting slack therein also appear cover class build sparsity clearly admit nmf table slack slack slack slack slack slack slack slack randomly symbol value well bind heuristic extensively value rank nice nonnegative uniformly matrix precisely location uniformly ensure specify table fact nmf compare heuristic nevertheless comparison illustrate present heuristic explore strategy main heuristic initialization main e apply strategy nmf multi strategy locally final refinement exact also
common illustrate fourier series n n n polynomial prior differentiable derivative universal contraction poor contraction stem poor rate polynomial tool fourier series spline choose choice fouri negativity coefficient expansion appropriately target restriction moment technique wavelet wavelet series correct wavelet expansion n rate adaptation noise multivariate spline situation consider tensor spline number univariate function product apply theorem sm always present deal class clear remove optimality establish sharp precise loss situation logarithmic remove contraction latter allow mcmc moment conjugate coefficient frequentist rate contraction log spline contraction unknown adaptive contraction possibly consider induce function spline spline choose spline appendix hellinger distance hence bound small hellinger distance hellinger distance bound square root lipschitz lipschitz constant assertion n n posterior hellinger use identity density condition univariate relation spline true level different direction old smoothness kf integer great direction product jk nk n n hellinger mixture density restrict maintain rate put density observation stand k take nonzero interval involve step posterior variance sum index redundant bin weight series prior view develop similar product dirichlet correspond part restrict coefficient gaussian error contraction tensor basis additive ease consider covariate outline suppose satisfie nf ip constant compute get covariate bound satisfie entropy contraction parameter exponential regression positivity boundedness distance contraction spline view choose link computation b ib result contraction model functional contraction base give assume h discuss covariate response integrable formulate coefficient treat away satisfie posterior write ik dt expectation schwarz argument apply distance longitudinal covariate suppose hence contraction logarithmic calculate evaluating term get geometric restricted ensure truncation use default dirichlet unit interval gp normal though summarize calculated replication computation estimation estimation case perform mixture low evaluate become compute term survey result term pointwise credible band nominal von establishing interval dm beta gp e htb dash band solid right functional data http spectra sample objective chemical predict consist training channel spectrum functional use gamma insensitive yield generally result rmse analysis brief introduction spline continuous space always nonnegative scale spline function b follow show approximation b coefficient approximation adapt smoothness spline assertion corollary universal depend ct j infimum attain c know f b kk tensor spline less f j kk maintain assertion assertion product spline I ct univariate multivariate value multivariate spline tensor product univariate basis uniformly bound basis maximum spline give use respective isotropic value restrict integer treat multivariate decay part beyond use prior function coefficient contraction smoothness model contraction statistical nonparametric regression interesting canonical coefficient mcmc accuracy estimator comparable apply functional contraction series rate proposition contraction rate contraction estimate typically practice investigate contraction logarithmic hold adaptive estimation give secondly regardless result establish estimation mixture construct rescaled range widely spatial drive besides common prior put corresponding series contraction series recently univariate basis estimation use class general basis function posterior contraction normal put component paper contribution obtain contraction problem univariate setting arbitrary basis show basis posterior carry conjugacy without general property conjugacy abstract many accommodate support obtain abstract depend coefficient term metric compute rate relate way series expansion consist eigenfunction prior flexible alternative contraction establish use relatively elementary finite computation process procedure knot approximate give prior b spline conjugacy like represent posterior analytically size small moment sample term derive contraction j degenerate point indicator open old derivative pack respect hellinger leibler kl divergences kp p p triangular basis stage make explicit notation j jj hc respectively dirichlet conclusion belong satisfy every unify
pixel update representation regularization add expression prior pixel probability analytic update formula distinct pixel also step formula involve find material crucial good accordance learn parsimonious start later idea model shift version model residual difference train good contrast bottom grouping base composition start elementary structure part joint covariance training inference experiment confirm representation distribution htb two show template composition letter character use come character letter initialize initialize example converge vertical bar sample realistic cover principal first figure character motion necessary max minus min autoencoder boltzmann machine synthetic pixel grid divide activate activate black probability non activate task recover part minus converge visualize obtain template model denoise autoencoder restrict boltzmann machine run rate tune tune corruption compose template max minus sum ground visualize material fair comparison leave transformation template supplementary material contrast dictionary learning st learn th minus min entropy cross truth composition compact representation learn composition propose test alternative competitive interaction rule create affect update focus interaction value consider rule mean achieve intensity layer via multiple minus layer science il interpretable solve review way expert binary interaction suit learn propose composition vote vote attractive novel procedure correction year network drastically improve task use cascade learn lee upon effective typically class compact solve natural representation letter term horizontal bar consequently efficiently six orientation work learn representation little example apart intuitively appeal low description scene know compositional use traditionally location size partition image grid overlap window size point detector local restrict certain spatial hard part split stable deep consist expert learn large compositional usually knowledge area expert priori deep realistic base model expert describe structure representation explicit task locate strong key challenge network learn robust discover part correspond large achieve rate whole image patch restrict key define composition new particularly well describe composition process correction handwritten conditionally template composition template rule vote order create compose template counterpart activation feed formally composition apply composition eq opinion word image black state white pixel underlie able odd rule composition pixel responsible procedure show plot see part opinion state asymmetric symmetric template intuitive composition simple average compose template interpret template note composition impossible opinion restrict manually composition odd type composition boltzmann sigmoid belief networks link pca trait coding factorization opinion odd probability vote odd arbitrarily adopt opinion pressure adopt opinion time part likelihood competitive rule uncertainty rule however computational tractable propose redundancy among encourage vote composition possibility increase place multiple time use opinion hand normalize lead subscript negative composition extreme opinion low symmetric summation composition like odd max min log consider create white image pixel probability attempt
brief investigation mrf turn learn extend preliminary result future crf ising mrf discover several red throughput connection include copy expression connection level commonly trait model connection connection level available level sequence process ii level patient merge mutation indicate copy leave patient expression gene patient yield level converse variable order level field rna sequence wise poisson mrf crf crf sub crf sub mrf family relax positive wise describe stability determine regularization estimate figure blue gene identify type red circle include link indicate well novel several connection mutation link expression expression level breast cancer sub previously mutation link growth breast cancer latter mutation link expression sub estimate novel connection validate mutation link tp know tumor gene long breast link expand breast breast cancer mutation link expression know affect tumor marker positive sub applicability graphical generalize mrf special case field flexible dag mix show fairly knowledge density permit rich implication analysis involve mixed expand ise model model paper broad big beyond image national internet economic work provide fit understand datum discuss need proper post inferential impose relaxed condition practice mrfs additionally work extend variable priori knowledge assumption unknown variable remain still learn mixed estimate dependency theoretical build need several dag little know mrfs direct undirected area broad domain p equality side eq q exponential family expression hand side q similarly furthermore plugging generally consider triplet reasoning product construction respect connect decomposable disjoint respect means trivially extend proof shorthand suppose partition show unnormalized mass suppose c fx complete mrf develop crf crf infinite linear crf cm infinite one ty even simply result condition mix mrf useful restrictive mrfs depend response dramatically relax example poisson study mrf neither hold mixed mrf formulation relaxed block need satisfied statement crf poisson unbounded statement crf poisson mrf crf component label ise mixed mrf crf gaussian crf mrf poisson ise ise crf ising mrf crf mrf crf mrf block binary continuous poisson z generate model sample lattice edge estimate mrf mixed mrf find connection surprise dependent mrf dominate influence influence meaningful assumption l department computer college comprise heterogeneous variable skew continuous area image national internet effort computationally amenable multivariate direct dependency paper direct markov univariate conditional directed acyclic directed field markov condition instance scalable conditional simulation sequencing expression mutation datum storing become big varied consist refer mixed variable sample belong binary categorical ordinal skew among big comprise measure genomic imaging single subject collect varied call coordinate message tweet history internet history among effort collect internet history update history online among type variable dependent motivate throughput detail recent molecular mixed copy variation binary categorical functional comprise measure rna sequencing count genomic closely relate belong complex multivariate graphical task range important understand molecular disease type develop class distribution expression well influence expression genomic dependency mixed address rich dependence mix term mixed popular multivariate relate multivariate multi especially popular trait seek link gene genomic associate response mix type machine measure probabilistic model copula potentially non loss efficiency parametric regime forest handle modeling approach popular especially spatial statistic propose mixed count latent gaussian latent guarantee possibly mixed tractable statistical seminal early review next background random mrfs case discrete continuous include sufficiently expressive rich propose markov random field serve heterogeneous mixed organize simple class heterogeneous group group turn chain exponential family mix mrf call exponential family family exponential family mrfs seminal work include show weak preliminary mrfs direct acyclic class model direct undirected edge general mixed statistical model regime study simulation well throughput cancer field develop distribution model heterogeneous condition set covariate suppose locally neighbor x suppose condition sufficient statistic form consistent graphical similar covariate global neighborhood conditional notice provide ultimately tb heterogeneous variable partition heterogeneous set group setting could cause variable could interest dependency condition dependency among clique clique suppose direct undirecte among node solely edge show armed notation propose natural follow exponential crf elementary field intuition distribution assumption mixed graph mixed mrf distribution introduce analyze restriction several example mix mrf counterpart graphical literature specify mix undirected within arise edge mixed markov marginal undirected edge response conditional mixed purely undirected drop additional restriction undirecte ask classical specifically restriction answer direct node markov clique depend respect entail specify neighborhood investigate implication global property direct edge edge factored set mix mrf write noting covariate similar covariate mixed mrf distribution differ mrfs hence covariate write primarily consider form mrfs eq covariate mrf distribution covariate normalization consequence next distribution restriction characterize refer require ensure density ensure finitely integrable mrf conditional mrf crf mrf covariate n impose mrf shorthand partition mix pairwise follow normalization overall term mixed normalization identical affect class necessarily impose mrf log form form assertion mrfs general demonstrate several counter next illustrate implication broad model ise graphical provide formulation binary domain give conditional give respectively two binary primary specify formulation distinct modeling model log normalization graphical mixed ising follow discussion highlight focus mrfs linear flexible permit relationship mrf give another advantage mix datum distribution mrf crf give let node partition exhaustive direct undirected purely set higher exist literature order block edge within block recursive mixed induce direct acyclic direct conversely dag graph graph thus partial correspondingly elementary chain understand elementary vertex lead seek define construction direct field however theoretic indexing parent clique subgraph theoretic define markov q specify crf detail substitute joint distribution arrive result different block distinct mrf consist note mrfs correspond distribution arguably could cg chain previously must ensure notation within specify covariate remain restriction joint assumption vertex mix edge undirecte mixed clique respect markov solely neighborhood discuss give well mixed distributional valid crf individual mixed crf weak mixed mrfs analogous extension flexible graphical ise multivariate variable domain node conditional block statistic finally block specify denote extend node node illustrate joint mrf gaussian crf follow poisson crf covariate take form mrf crf crf crf exist permit dependency vector three model combinatorial way crf gaussian crf fairly variable type rich class recursive non dependency specify block determine edge block seem restrictive variable measure block block clearly analogously natural language obvious arise setting throughput snps binary rna sequencing via sequence genetic count marker continuous marker genomic variable interact fix marker sequence dna thus influence expression expect mutation marker point type conditional dependency undirecte precisely input partial mixed crf neighborhood setting variable fold unknown mixed graph task graphical problem mrfs normalization close form belong maximize typically intractable secondly heterogeneous varied lastly structure mix underlie undirected variable connect even solely direct graphical unless know accordingly dag priori assumption relevant area throughput assume partial dag mix completely mixed crf overall mixed outline x require therefore mixed linear function correspond clique parameter pair clique parameterize overall crf reduce estimate mixed graph accord independently follow allow side partition function neighborhood seek zero estimate crf univariate node intra block
bias unchanged relative envelope change envelope correspond change providing except determined binomial n pn probability except last risk z z provide bias line contains easily calculate analytical fig function get z z compare empirical complexity denote estimate empirical histogram classifier substitution risk plot vc case continuous feature maximal risk histogram analyze maximize risk fix minimize risk change maximal bias histogram probability stay increase decrease stop change risk second bayesian misclassification feature easily hypercube probabilistic need assign belong piece area hypercube volume supplement hypercube volume call varie vary distribution maximal risk histogram sake define level tree distribution distribution great bias examine distribution interval one estimate confidence equivalent estimating must hold q build superposition play role base interval estimate analytical infimum probabilistic empirical desirable minimal finite believe unable dedicated figure empirical terminal equal sample dimensionality hypercube py x ng define equal misclassification tree consider construct misclassification bias find formula establish bad distribution impulse similar distribution modeling decision suggest bias risk attain histogram classifier particular build estimation maximize risk derive risk histogram carry detailed estimate empirical area machine arise disadvantage complex size order choose need notion reliably training sample estimation carry interval well reliable latter require form estimation among devoted refinement risk approach approach already exist reason work paper equal nearly chapter vc make vc empirical classification shall estimation however primarily orient toward bad classification analytically construct worst realize practice quickly practical although however contrary bad estimate probability misclassification scenario key subscript usage notation decision risk simple misclassification n drop notation risk empirical leave validation etc sample eq given leave remove nearly unbiased taken leave unbiased get expect misclassification estimate risk v rv method fc fc fc possible dependency
molecular often size abstraction develop infer specie leave think molecular sequence correspond first ultrametric b form ultrametric species tree topology eq reconstruct dissimilarity ultrametric distance sake agglomerative ultrametric mutation population represent diameter correct sequence prove tell succeed molecular sequence suffice reliable use hamming modification crucial branch tree possibly mutation first define ultrametric recover specie usually distance clear arguably paper dissimilarity condition I restrict leave information dissimilarity form additive leave hold surprising molecular topology result appendix reconstruct dissimilarity define base algorithm like dissimilarity recall mutation diameter succeed reconstruct replace tree employ enough argument appendix tell like tree reconstruct root reliably note assume mutation gene mutation change work irrespective gene accord specie branch big none branch argument formalize able reliably reliable shift fact px kullback divergence even specie q mention achievable irrespective achievable provably attain require raise tradeoff quantity topology separately specie segment tree notational clutter write leave effective mutation rate correspond exponential check condition denote common also depict condition correspond leave ac use numerator next observe stochastically dominate exp observe distribution ab f cd substitute respectively let figure see condition let denote fact condition eq hold ac ab cd observe fig observe hold early eq tell reconstruction molecular recall dissimilarity call like neighbor return dissimilarity result algorithm reconstruct f side approach make exist leave union term inside summation second say ac q proceed follow clarity us ab ab definition use condition mp ab ab particular realization ab ab first end process hold claim tell apply leave pick conclude observe begin follow lipschitz similarly condition ab inequality condition claim q variable stochastically conclude theorem remark discovery department mathematics university requirement inference problem evolutionary set species specie gene gene specie tree analysis far molecular specie method account estimation gene devise previous regime sorting distance molecular history gene lead population thereby tree topology specie try reconstruction develop reference rely roughly generate illustrate explain little accuracy focus consistency access tree specie gene convergence denote branch length specie show agglomerative dissimilarity specie need agglomerative instead gene reality gene finite molecular sequence estimation level quantify estimation progress towards perform fold gene correctly therefore light require modification reconstruct particular overall sample regime secondly restrictive molecular mutation rate population branch specie extend previous beyond specie call algorithm distance molecular begin introduce paper heart evolutionary isolate isolate leaf tree assume branch assign branch time small branch length role interested vertex unique path connect length tree branch ultrametric leave restrict species branch associate mutation small mutation produce different genome specie figure thick tree specie standard leave tree resp parent branch draw gene gene copy gene describe evolutionary history isolate population represent branch draw exp interact give random gene correspond length branch accord independently time draw accord notice evolutionary history agree incomplete sort road tree refer reader model completeness proceed record fact exponential exp density tree focus species tree gene gene enter branch enter
trade bandit provably develop combinatorial semi bandit consider bayes least adversarial combinatorial bandit combinatorial bandit combinatorial bandit large problem weight approximately exploiting model independent achieve linear across assume agent know lie transpose refer literature bandit coherent emphasize agnostic well agnostic learning uniquely moreover coherent choose episode episode episode randomization necessary randomly episode learn combinatorial thompson motivated combinatorial combinatorial two control specifically close small exploration reduce hand control decrease slow quickly converge optimal kalman te e input combinatorial structure dt n te episode randomly distribution second compute base specify kalman point episode distribution satisfy obviously case agnostic three regularization decrease rate matrix specifically converge sub coefficient insufficient exploration slow matrix ta te episode step confidence ucb compute specify oracle update kalman filter bind bound detail bind algorithm coherent appendix regret logarithm second hold start episode bayesian conditioning conditioning sample combinatorial independently conditionally simplify exposition q inner conditioning two key briefly upper thompson focus adversarial reasonable since indicate hope hence due remark word work oracle offline speak proceed confidence set self normalize develop regret bad event worst associate base detailed proof regret low factor modify achieve still hold construct real world evaluate thompson ucb problem demonstrate suggest likely generalization agnostic outperform art serve baseline solve two episode expect cumulative return divide rather illustrative evaluate path grid path grid corner corner coherent generalization experiment figure trend learn per return episode return episode remarkable poorly insufficient respect episode time imply enough discriminate people music likely music website specifically grind recommendation user fm music tag dataset tag assignment tag assign tag na classifier respect person optimal algorithm algorithm similarly per return episode discriminate propose stochastic bandit linear establish bound item variety show scalable robust exploit generalization contextual either process adversary state item pair feature contextual combinatorial open several question open bound agnostic open combinatorial bandit generalization believe combinatorial monotone leave future prove I sample parameter immediately outline conditioning independently even fix conditioning also furthermore conditionally conditionally ga ga constant notice function follow tb full use cumulative function cdf base follow c naive notice conditioning inequality follow implicitly point v dd constraint uniquely consequently consequently fact x rhs choose prove specifically provide ta q arise specifically notice equation imply imply logarithm notice hx hx hence previous subsection write reader proceed follow construct normalize develop confidence analysis start useful specifically martingale bound gaussian far define self normalize notice base cauchy schwarz moreover know obviously immediately imply exactly complement event lemma state q last naive realize hence bad case conditioning event recall eq first inversion plug equation induction far note suitably regret scale eq solution scale subset oracle theorem assumption last realization conditioning eq lemma put together corollary combinatorial semi bandit problem choose subject combinatorial observe item receive combinatorial semi bandit learn call computationally efficient long offline combinatorial establish statistically efficient assumption develop sublinear thousand experiment robust choice baseline combinatorial combinatorial modular many short weight bipartite matching optimize need bandit bandit bandit adversarial combinatorial establish combinatorial estimate weight website view million user product internet formulate impractical expect movie decision condition item use bandit extend thompson ucb semi generalization combinatorial thompson combinatorial ucb efficient offline combinatorial solve major establish sublinear major problem dataset evaluate thompson usually ucb practice scalable briefly linear differ paper
location output location single forward full replace convolutional map size output forward pass consideration need influence matching fail assumption within violate prefer adaptively collect pixel comprise pixel begin construct cross intensity position spatial construct analogously arm know horizontal vertical arm union arm suggest support region right support region number average refine matching enforce image define energy q function cost term add neighboring pixel differ penalty neighbor minimize image np energy effect direction minimize energy many direction recurrence second prevent effect match proceed zero deviation hyperparameter final car drive around city resolution transform object vertical right camera dataset predict pixel percentage pixel translate depth tolerance object camera object camera train cross epoch decrease example million class subtracting divide standard pixel intensity method method rate dataset mc sf ss anonymous visually map method select example prediction bottom pt implementation gpu take hour predict evident prediction pass matching aggregation everything seem good big learn also beneficial suitable application robot improve runtime university york extract information image convolutional two patch aggregation error method refer horizontal location position know object camera subproblem reconstruction dense frame cost optimization refinement plane propose convolutional neural pair obtain match pair patch intensity aggregation smoothness consistency check eliminate perform final depict method contribution convolutional previous typical begin match cost position consideration absolute difference intensity rectangular denote interpret associated patch image patch image good bad match publicly attempt solve match supervise learn convolutional small patch match comprise image center true example center randomly randomly method aggregation good match near size architecture eight convolutional
observation generate normal follow apply typical strategy propose anomalous anomalous anomaly measure normal anomalous drawback generally apply quantitative ordinal transform group dense belong anomaly detection curse dimensionality suitable curse commonly anomaly exception work perform subspace cluster dimension subspace belong contain anomaly subspace similarly anomaly dimension subspace find drawback kind find subspace ignore could lie method intuition anomaly neighbor normal anomaly score outli instance local near radius center near call multi deviation record deviation local neighbor use micro anomalous record improve avoid unnecessary calculation calculate micro detect usually thus big suffer curse mention training generate anomalous anomaly decision fit focus salient desirable characteristic deal anomaly adjust threshold network tree network determine survey decade anomaly previous divide trend subspace periodic curve anomaly detection outli periodic modify algorithm representative new anomaly time close centroid poorly massive method alignment restrict periodic limit scope outlier light star end correlation light light curve unfortunately operational large cluster occur find light calculate operational cost unfortunately method light imply observational periodic separate star galaxy characteristic anomalous galaxy close develop one former mixed factorization unsupervise explore subspace also normal dimensional basis quite opposite anomaly outside subspace basis factorization approximation additive mixture differently user consequently heuristic anomalie probabilistic model generative mechanism score main drawback inference stage consider restrict dataset survey observation make subset anomaly anomaly score anomaly combine many observe variable equally disadvantage detect outlier space variable approach candidate set train forest class proximity value proportion forest terminal proximity measure create outli score decide suffer density detection method expensive slow big datum evaluate instance set train decision label decision classify main principle follow divide generate robust combinatorial create rf tree forest describe data bag bootstrap sample bag replacement use pick optimize principle individual tree create diversity classifier feature contribute improve rf furthermore subject minimum size allow classify new tree rule node reach rf tree vote proportion bayesian direct acyclic graph encodes among represent variable application bn joint pdf bn determine simplify probability bn describe simplify bn decompose product small factor challenge bn word bn structure probability direct acyclic modelling represent class greedy possible structure explore structure find parent per node checking create network actual number parent input evaluate probability factorization impose assign probability vote distribute rf vote multimodal consequently multinomial get rough capture divide set every fall interval use probability parent variable probability determine value parent combination parent outcome consequently follow expression q parent example posteriori calculate take purpose overfitte estimation stay predefine value previously situation conjugate calculation posterior multinomial conjugate dirichlet parameter act find parameter variable detail methodology illustration stage panel follow rf probability bn discovery pass bn previously mention manner analyze classifier confusion immediately model start descriptor light class descriptor rf label construct bag tree bag tree train vector class ij v use bin discretization perform end dataset rf vote belong variability want decide unlabele rf vote vote rf object store outli joint bn recall product small model vote object rf vote already rf associate already bn calculate necessity possible case choose calculate hope uniformly discretization bin parent three reasonable empirically different massive observed produce million star observe small cloud reader find description nm present table light comprise study long period collect composition object star rr period variable set exclude never see class star rf present leave visualize voting rf scale vote class classify rr color vertical bn rr child voting high star node expect completed perform train rf vector learn bn find figure show class object value top panel performance outlier top ideal result place see class bottom b magnitude rr accuracy rf bn run million light list candidate bn fraction easily period day bottom panel bottom day figure outlier obtain characterize day period year probably observational pattern intrinsic anomalous kind obviously light appear many remove spurious outli candidate day variable non light curve object candidate list consider spurious remove list visually candidate group obviously spurious example shape add outlier explain repeat expect step visually candidate list move outlier candidate candidate candidate either noise snr match know collection know example period variable collection long period contain ray far outlier identify nature object summarize match number appear number rare kind incorrectly classify many automatically error human involve bias present final outlier class day curve appear identify know uncertainty result consequently outli deal uncertainty topic snr actual amplitude indistinguishable variable signal light uncertain reason attribute lack perfect method classification heavily ideal quality feature start rare type never train discover recover object post cm c newton source rr classification cat candidate anomalous motion star examine candidate list type advantage population fourth group set light curve interesting rare class paper survey drop cause star system mainly magnitude distance valuable look outlier light curve cv unified characteristic include candidate list magnitude become approximately recurrent surface year quasi recurrent subject research interest show like blue unified light training variability star disk emission characterize star disk appear candidate fall example light curve location member source newton source confirm ray binary counterpart ray source optical periodic non ray emission w contact ray type ray particularly ray either star black together star ray emission cause study ray process e representative ray outlier star rich extremely cause result drop magnitude light class comment run candidate visually object show individual outlier notice belong locate field therefore star confirm reject hypothesis peak variation variation average since year reject class diagram outli high outlier score box period c class period day snr class class class outli individual right right right left outli survey develop automate star characterize detect anomaly classifier discover forest network exist method work process computational resource nevertheless curve analysis explore anomaly periodic star project identify belong error available find rare previously perform follow characterize identify object isolate construct furthermore aim survey help plan release software tool service na ia cat grant ic institute cat institute university usa surveys lead massive need
text dynamic pooling sentence operation limit max pooling extension type unit last distinguish turn useful later variable sized vs bag gram document gram svm gram infeasible also bag gram one share contrast learn text region intend task convolution turn neuron large never confirm convolution simple layer deep layer explore parallel text improve pooling region representation one produce region layer take concatenation input topic sentiment detailed internet activation descent word appear seq region word vector avoid cnn typically input case scale multiplying test connect neural vector connect net dropout classification frequency scale unit always binary traditional vector component corresponds test nb lm later modification exceed generate bag gram vector gram training hyper net hold choose hyper movie k character review consist amazon review choose movie follow test set one half review review text text review set internet corpus news article describe category hierarchy document categorization thresholding strategy algorithm model like categorization month period early size concatenation except regard nd level nd show rate thing note cnn demonstrate effectiveness look detail first cnn convolution seq cnn table sentiment choose region max pooling cnn seq pooling phrase strong sentiment great movie receive high irrespective rest sentiment classification categorization configuration cnn size pooling pool vector classification short entire pooling location predictive text pooling point seq outperform outperform seq cnn indicate region turn parallel seq gram frequent nb lm paragraph lm unlabele seq k seq method nb lm report learn unlabele supervise due use resource cnn categorization cnn outperform even tf micro macro micro average macro average categorization study train cnn report word train cnn training unable type need tune dimensionality word vector modification sentence modeling notably word convolution region cnn find resource demand task various memory hour gpu cf svm excellent performance sentence classification short sentence categorization time minute horizontal baseline linear take minute core intel gpu figure effective explain cnn look learn comparison gram gram great perfect perfectly bi gram gram gram gram svm heavily gram tend gram nb bi show learn cnn convolution region embed produce dim component th top positive value predictive note embed list large embed vector tend proximity relation target positive sentiment effect help layer c return totally bad fail easy I p super example predictive bag gram training cnn gram long good ever ever ever satisfied overall region contribute entirely gram show test partially bi gram yet embed heavily predictive thus certain pattern entirely overall satisfied need entirely ever adjacent gram seq successfully exact gram g positive cnn effectively bag gram fail show cnn mechanism effective text categorization embed parallel combine complement art classification classification achieve anonymous author nsf nsf ex plus minus em ex ex ex zhang nj usa convolutional neural neural internal structure study categorization namely word text instead low done directly embed text straightforward adaptation text employ word convolution layer explore demonstrate text categorization automatically assign type categorization different categorization detect spam sentiment classification determine typically review standard categorization preserve classification loss cause problematic sentiment bi gram gram gram text categorization general topic categorization add gram reference benefit order text categorization neural network categorization structure word successful image classification win imagenet challenge token pos cnn search sentence notably cnn learn additional elsewhere cnn aid vector train arise purely supervise really categorization essence text region size later sense convolution bi gram gram directly cnn text go sparse efficiently handle infeasible dimensional gpu turn simplify tune cnn text publicly internet effectiveness cnn categorization explain cnn cnn test seq cnn straightforward adaptation cnn employ seq versa winner conventional bag previous cnn text particular work improve extension combine convolution layer embed improvement gram method document review task seq image compute linear share unit feed convolution layer illustrate top perform unit small concatenation pixel channel red blue conceptually region region center set region distinguish layer weight sharing region compute wise activation component bias train learn irrespective location appear regard convolution output consider pixel neuron convolution dim vector region convolution layer pass essentially merge abstract pooling region merge average compute text vocabulary
neighbor rough appropriate smoothing theoretically common point learn learn irrespective know regularity minimax instance loss incur ball achieve regularity general introduction still want hyper attain logarithmic regularity use study contraction extend line reasoning multiplicative ahead latter argument prove desire describe together discuss mathematical conclude copy poisson cube underlie integrable every counting set stress expectation involve notation respectively consider intensity equivalently observation priori q gp index sigmoid section model gp hyper endowed choice hyper inverse spectral decrease inner product denote respectively parameter density constant instance common tail prior satisfy fulfilled gamma increase infinitely smooth condition fulfil precede quantity construction result concern intensity prior formula g describe around intensity generate data contraction depend smoothness quantify belong belong old contraction square root intensity hellinger f sufficiently large intensity hellinger convergence smoothness intensity fulfil sigmoid length gamma extend cf adaptation case write fs fs n away suppose b form commonly encounter contraction rate put intensity datum remain mass together require prior concentrate term grow posterior subsequent subsection theorem result derive like extend adapt deal intensity section define smooth hence smooth compact away vary hence q factor assumption second constant low proof easily complete gp background notion unit determine subsection since second moment give sequence conditioning pg pg bind eq term rewrite view precede remain approach enjoy favorable result show stationary process link room multiplicative contraction lead believe sub impose strictly speak matter open necessary generalization instance generalization prior analytic generalization work believe example definition cox process approach learn poisson
exist fuse fuse give former base dynamic programming programming specialize admm utilize optimize routine q specialize admm update run admm approach neither dynamic beyond knowledge solve high order admm far specialize admm outperform simulated wave three trivial smoothness homogeneous examine spaced order space q small penalty polynomial instance admm record achieve across scale case particular show arbitrarily qualitative behavior clearly routine dominate optimum qualitatively simulation parameter specialize standard specialized term first rough interpretation utilize subroutine difficult subproblem linear progress towards minimize criterion reasoning admm hard subproblem concrete explanation come admm invert trend filter specialized admm ignore admm think perform alternate minimized lagrangian parametrization difference correlate bottom scalar block regression triangular make progress update direction align think contour contour form filtering may update admm allow step overall special admm run start mean without share warm large second large warm absolute illustrated specialized admm axis iteration noisy right warm start advantage middle representative therefore warm worth discuss augment lagrangian parameter use associate filtering rather introduce admm admm optimum however rate stability set numerically stable across update appearance soft programming level intuitively make try adaptively heuristic stable recall paper selection optimization default state otherwise specialized admm write author put also implementation specialized implementation actually number barrier barrier barrier backtrack line choose former size interior point close original linear trend setting try vary lead performance take early time admm comparison achieve mind criterion versus time repetition combination except size repetition complexity offset scale versus large scale admm iteration time fast specialized display display relative absolute regime algorithm n high regime admm large admm statistically curve piecewise fitting figure convergence barrier backtrack meanwhile admm stable across hundred routine small trend desirable fit near specialized outperform admm fit converge regime trend encounter converge winner robustness case defer brevity discuss specialized show size moderate barrier backtracking present suggest case poorly condition core take admm suffer system important denote admm iteration form eq augment important buffer make possibly poor conditioning meanwhile drive across optimality complementary zero dual lie strictly inside buffer condition instability issue solve iteration many dual strictly mean input extension specialized algorithm highly tool filtering much generic evenly space change trend space recursively begin admm input panel fit location slow converge admm slow alternative admm design aside change admm routine general lie augment arbitrary input use motivate admm evenly routine routine try match practice adjust make progress strength solve readily adapt modification trend extension novel manuscript exhaustive list stage trend filter trend tool across input parameter calculation specialized q still operation per example htb suppose extension order act order penalty specialize admm naturally extend trend estimate detect component adaptively detect outlier routine operation iteration encode suggest update fit enforce monotonicity x specialize update modify dynamic fuse take time specialized admm trend leverage strength solver fuse lasso order higher regime interior superior accelerated descent admm finally major strength propose extension basic trend filter software specialized package around c package package associate helpful reading manuscript discrete vary straight polynomially size moderate problem roughly speak flat htbp nk like moderate computation linear trend tune level regularization trend filtering path track computation quickly intractable panel figure point algorithms problem descent primal solution run proximal gradient accelerate proximal accordingly iteration operation multiplication operator truncation onto ball select large still intend top leave fit curve piecewise acceleration acceleration clearly basis matrix solution lasso proximal iteration time computation dense map see acceleration output close corner bottom leave explore non order apply formulation admm iteration descent operation full multiplication exact part capture piecewise although visually perfect piecewise panel reader illustrate iteration special visually indistinguishable exact fact iteration specialize admm operation per actually solve admm begin show size order magnitude specialize fast converge serious difficulty specialize admm steady within important prediction trend relevant proposal package implement feature prediction describe filter nx fit evaluation factorial filter k trend underlying calculate factorial algorithm corollary department fast trend develop nonparametric trend achieve minimax optimal way lack scalable fitting paper efficient specialized currently robust interesting filter software relatively point across trend k derivative operator write difference trend fuse variation th order trend order input knot generally knot reader jump ahead next future need evenly spaced operator structure recursive basically broadly speak motivation argue filter strength spline regression spline polynomial highly minimax optimal locally adaptive spline relatively inefficient computation trend filter minimax comparable spline focuses derive explicitly selection choose concern computational want trend filtering mean course still helpful th trend fuse already use dynamic equally special two direct issue affect classical
integrable check density belong check large class small versa present result smooth variance represent easy define choice stein estimation depend contrast stein estimator long knowledge mild shrinkage applicability datum drive show estimator kernel define kx x nn kx kx exist prove bernstein separable space rearrange therefore space theorem eq depend kx kx since depend kx use easy h n dt e dt follow theorem consistent kx n kernel bound verify kernel hold kx stein obviously kernel unbounded exist carry chebyshev inequality cost slow explore stein set deal estimate view mean restrict df x know replace assume stein seminal shrinkage estimator dominate principle classical notably estimator distribute like see kernel provide shrinkage connection estimate minimizer functional formulation ask question note regularize pose regression consider regularize minimizer lie span g term gram depend set minimizer eq shrinkage nice shrinkage wherein shrink towards require ill since estimator propose estimator constant yield rate n verify alternate choice refer correspond shrinkage measure quantify differ wherein instead omit observation deviation omit observation show shrinkage analytically minimize minimizer give q define nn kx df yield easy show derivative positive latter show u kx see statistic counterpart relatively theorem stem evaluate score guarantee satisfie n kx follow kx kx proceed remark show proof remark carry schmidt norm smooth difficult show x consequently q unlike take refer let q compact operator effect contribution expand eigenfunction empirical operator basis difference shrinkage contain particular account allow coordinate direction value wherein small coordinate large reveal wherein filter generalize alternate representation relate smooth operator formulation formulation x n use leave side xx xx minimizer jk remark suppose bound linear xx I ix bernstein q xx make observation universal bf along hilbert schmidt follow universal universal consistent guarantee knowledge xx cn g gx universal n universal achieve technique convergence shrinkage r leave shrinkage na expensive provide alternate expression n n proposition formula eq proposition aa x n I aa n unlike consistency comparison comparison outperform empirical mean dataset estimator obtain empirical regularize parameter whose obtain use different estimate copy j polynomial degree rbf lin rbf analytic form kx kernel dd begin simplest mean shrinkage comparison shrinkage origin gain small underlie follow generative represent wishart quite dd depict rbf realization uniform allow depend shrinkage determined figure important substantial appropriate knowledge incorporate close origin factor probability improvement iteration improvement estimator copy end conduct experiment shrinkage rbf addition whose loss figure illustrate difference improvement fraction proportion suggest amount improvement however trade become amount decrease intuition part compute shrinkage validation b choose leave cross shrinkage estimator increase shrinkage note non eigenvalue find outperform tendency actually support suggest one validation parameter improvement risk percentage calculate vary size tend outperform performance depend eigen dimensionality figure intuitive improvement size surprising especially function substantial small paradigm r choose classification window zero implement powerful non learn class positive negative window cccc statistically close rkhs kernel mean resp expect window employ counterpart uci repository bandwidth cross employ vote multi experiment test report error classifier consistently kernel wherein fit density unlike experiment goal different shrinkage estimator well initialization return pair significance via large log degree kernel highlight involve discriminative probability representation embedding approximate non investigate shrinkage end categorization support measure anomaly physics rbf fold cross setting report area roc mean shrinkage estimator performance compare summary r competitive estimate rl cccc cccc r segment tp cc classical stein phenomenon improve square show propose estimator shrinkage learn data satisfie also shrinkage wherein exploit spectral outperform importantly accurate performance focus mainly covariance cross tensor rkhs rkhs covariance pca preliminary present numerical xx yy unique center respectively covariance rewrite equivalent idea call trick tensor compare reconstruction rbf kernel dataset different shrinkage center center r omit detail perform center give shrinkage covariance generalize shrinkage use hadamard clearly sense mean considerably significantly affect summary encourage application method feature thereby center write nx nx nx center test thus kernel rgb xx edu university usa ac institute com university college house ar united bs institute reproduce hilbert central kernel algorithm also step modern embed stein family demonstrate especially small paradigm covariance shrinkage stein improve reproduce rkh measurable integral endowed guarantee existence integral kx estimate commonly key show improve gain advantage account preserve operation carry product rkh lastly require homogeneity sample curse dimensionality hilbert embed many mmd embed hmms kernel rule rely component principal component discriminant heavily operator kernel cluster basis early mean certain well nothing construct extent stein seminal show though stein square stein square interestingly stein outperform ultimately result usual relevant parameter definition ultimately look together remarkable counter stein phenomenon receive stein follow estimator space stein shrinkage rely work fundamentally stein seminal radial rbf see throughout paper draw independently identically measurable kernel subscript notation resp quality notation q kx r r mean shrinkage toward
plot crcr color forget crcr solid forget table sep crcr blue forget crcr width height jump ylabel title datum color mark mark option forget plot row crcr nan nan nan nan nan marks forget sep crcr width axis xlabel ylabel title color mark solid forget crcr color green forget crcr green marks mark forget sep mark option forget sep crcr blue forget sep crcr r ex ex sentence fill circle draw black pt ex draw black fundamental special map classify label equivalence set measure finite empty offer joint abstraction multi cluster maximally probable related estimating infer maximally measure program solve np relation order practical three maximally probable jointly mixed program mathematical learning given decide every instance classified element choose precisely every example decision pairwise cluster feasible relation equivalence introduce two pair albeit conditionally constrain feasible maximally estimate infer maximally relation rank relation mix problem mathematical programming appendix rectangle rectangle set subset relation structure associate pair secondly independent albeit conditionally principle observe maximize invariant optimum disadvantage require maximally probable know mathematic art solve branch set non priori probable would discuss equivalence closely equivalence segmentation measure maximally probable know np branch property polytope heuristic order word linguistic assess solution linear explicitly concentrate vector l finite every measure unique approximation form contrast nonconvex deep relation I set every depict random variable variable random realization hence characteristic realization random relation namely realization independence relation separate likelihood equal feasible infeasible measure probability define distribution family alternative respect ab ab ab ab ab form generality one finitely class characterize zero unconstrained introduce fix precisely exhibit pair secondly maximally optimization technique implement open notably estimate probable relation complexity write convex optimal solution time infer form classification problem estimate finite non element non empty firstly logic secondly characteristic arithmetic uniqueness ab ab ab obviously establish rest appendix vector measure separate optimization cluster pairwise subset characterize relation corresponding equivalence consist whose belong aa aa cluster relation logic integer arithmetic symmetry equivalence relation inference set satisfy correlation exactly branch cut exploit lin terminate estimate relation total first logic order general instance exactly branch exploit order feasible find heuristic chapter formalism map equivalence report computation core intel cpu ghz computation unbounded jump log true xlabel ylabel relative anchor north fill legend align format cd precision white red crcr nan red solid crcr nan nan nan nan green table e green table row crcr green table sep crcr e sep crcr blue sep crcr e color blue dash sep crcr white blue solid crcr blue solid crcr firstly classify image handwritten raw multilinear misclassifie linear thank multilinear error reduce random multilinear feature bad misclassifie image fall deep encourage work multilinear width axis xlabel ylabel format precision cd white forget sep crcr solid forget plot crcr white green dash forget sep crcr color dash table row sep crcr white solid forget plot table sep crcr forget plot row crcr forget crcr color forget crcr color blue forget crcr blue forget plot crcr xlabel ylabel test format fix precision cd red forget crcr color green dash forget plot table forget row crcr crcr solid row crcr consider handwritten digits mnist image contain digit show digit pair vector digit set randomly mnist test partition problem misclassifie pair test multilinear parameter infer equivalence cut software loop separate resort cut instance initialize lin solution rand variation information see lin unlike branch mnist solution image incorrectly elementary sentence estimate word occur word english replacement sentence english wikipedia sentence every occurrence feature vector index word occurrence less second sentence test word optimal use branch loop inequality metric distance sentence sentence statistic appropriately account sentence horizontal indicate normalize metric sentence see estimate sentence sensitive abstraction linear estimate maximally probable related problem infer maximally relation np np instance distinction motivate distinction partial evidence estimate broad research state jointly integer toward aside np order understand continuous relaxation necessarily state integer exchange idea learn community define definite convex eq partial quadratic semi thus convex eq example one correspondence establish define correspondence establish trivial respect estimate maximize eq solve impractical distribution approximate multi form polynomial variate variate linear irrelevant secondly iff let j f b x ab iff solution say equivalence pair define ideally otherwise property every feature invariant multilinear polynomial multilinear form width jump scale axis xlabel ylabel cd cd color marks option forget plot sep crcr green mark forget crcr mark forget sep crcr mark mark forget crcr blue mark solid crcr jump scale xlabel ylabel time title cd mark mark option forget crcr nan nan color marks mark mark option forget crcr
divergence rao read family etc density lee lee minimum distance method robustness inherently possess goal simultaneously minimum g several power divergence technique model divergence smooth later et measure divergence small divergence family outli family divergence family al family allow divergence divergence density base divergence connect read divergence smoothly end numerical illustrate estimator discrete natural unknown frequency bandwidth simple need article develop minimum estimator continuous set minimum hellinger estimator along use avoid smoothed model density obtain estimator divergence prove normality minimum interestingly minimum rest minimum estimator divergence continuous approach section minimum estimator detail present influence general divergence support suitable simulation study interesting discussion divergence divergence divergence limit give family family family sense density measure argument distribution f ss discrete technique estimating xt sg whenever influence estimate contaminated mixture divergence represent influence simplify let density lebesgue assume respect lebesgue sample model discrete choose density respect divergence immediate discrete model continuous distance simply frequency nonparametric density generating suggest construction appropriate density kernel density usually assumption rest process substantial derivation normality property smooth various complicated impose kernel survey take propose integrate smoothing justify minimize data need condition kernel vanish asymptotically play procedure play get consistent even hold work smoothed version entry derivative minimize datum routine estimate minimum minimizing behind substitution family smoothed lead prove advantage ordinary data scale get scale normal get bandwidth phenomenon prominent little dependence issue pseudo sample selected analyze mean value bandwidth choice r remarkably variability sake brevity little effect substantially impact bandwidth power originally divergence family divergence minimum kernel fact empirical equation unbiased kernel minimum model smooth us estimator simplify q spirit becomes estimate asymptotic result show estimate become produce identical estimator consider minimum minimum relation influence derive gx degenerate contamination suppose contaminate smoothed g x derivative present estimator give true belong e turn interesting follow integration prove divergence correspond define smoothed equation section assume rigorous minimum general definition property say density parametric independent say compatible support integrate p satisfied place assumption identifiable definition integrate distribution x parameter dominate minimum condition also asymptotic discrete define lemma see boundedness without provide us eq limit eq dominate dct hence markov finite distribution next contamination distribution shift degree contamination heavy freedom brevity present contamination shift mle get contamination estimator positive bad mle estimator ignore contamination heavy member moderately ignore contamination ideally location minimum divergence robust contamination significantly affect contamination variance contamination imply symmetric heavy tail divergence estimator similar contamination ii contamination small consistent pattern recommendation tuning would estimator contamination finding overall divergence estimator second contamination except minimum density short determination angle surface angle mean raw pattern use outli maximum likelihood likelihood remove outli previous section table minimum estimate positive positive outli minimum divergence continuous similar minimum triangular low corner affect finding light datum table unimodal outlier previously many suitable fitting usual see minimum obtained ht automatically discount unlike maximum measure region instability corner table develop minimum similar fully term derive estimator continuous section power case smoothed model exist divergence estimating kernel coincide kernel minimum ensure special justify function corollary true belong model e condition divergence become equation prove corollary side get represent derivative th taking expectation respect side condition first integral complete smoothing mean thus minimum coincide kernel family asymptotic become remark although calculation normal letting equation see crucially although discover role similar limitation classical influence estimator discrete influence divergence minimum divergence examine role robustness contamination approximation bias provide second taylor predict get scalar extension straightforward scalar influence effect special structure measure model influence potentially unbounded cloud one zero close preferred counter balance effect first illustration mean calculation yield bound imply constant unless decrease predict approximation counter balancing effect zero combine property empirical finding present member solely divergence divergence discrete particular efficiency decrease increase however efficiency estimator member tune logic combine asymptotic couple moderately highly loss tuning suggestion window substantial variation performance tuning applicability propose surely enhance situation literature like estimate select tuning consider briefly
reverse process reverse copy start interaction action interact environment forward reverse artificial intelligence offer encounter offline state art control ease burden user perform primary suggest intelligence also feedback part biological contain operation natural forecasting partly encode hardware biology therefore artificial intelligence use take device interpret way result user way improve intelligence system contribute preliminary machine learn prediction action generate electrical load potentially environment user biological copy computational human robot feedback platform arm extra subject ax di data acquisition signal modify send computer interpret send robot angular velocity temperature load design four term locate person fig platform therefore feedback find common device subject ask give inform accordance contain stationary ensure arm position movement work outcome subspace arm leave right ask separate control ask reach load threshold maximum threshold consider arm away addition provide source prediction subject feedback subject music throughout task volume music increase level arm also turn right alternate fashion contact feedback current robot reach effectiveness examine final task participant sound isolated electrical load robot system train acquire prediction task predict load advance load change load threshold datum determine level threshold main study incremental previous make world system divide increase decrease resolution angular position distinct dependent far divided counter vector indicate position direction feature contain unit store prediction robot arm standard technique update accord instantaneous load load product temporal abstraction q step abstraction discount electrical load second cycle roughly hz retrieve predictive reporting threshold acquire update learn weight principle continue assessment experimental giving find reduce load case load trial predictive feedback robot subject feedback subject observe experience drift experiment subject contact approximately arm contact uniformly visible arm contact difference contact leave predictive feedback robot spend angular position bin approximately bin feedback user spend middle angular fraction bin feedback case bin b robot significantly angular load load bin aggregate aggregate five fig subject load subject plot bin bin feedback robot spend fraction angular position record electrical arm motion fig aspect control note work area approach form feedback expect might never sensitive threshold minimal impact consistently feedback comes delay observation demonstrate feedback case load device bias user another desire operational logical load indicate improve feedback though cumulative load matter load already reaction reaction operation load load varied trial indicate device see status device particular highlight difference feedback area bin bin bin cumulative load measure feedback successful device spend two beyond bin feedback less frequently load spend impact condition device feedback great load note sensitive boundary indicate advance much feedback setting qualitatively observe subject contact level contact artificial parameter adjust device achieve objective learn feedback behaviour feedback converse provide feedback load sensitive load mathematical position load purely clarity intelligence acquire update period machine set period fig subject use feedback learn slight shift result user side correct machine study technical algorithmic prediction operational offline expectation computationally nature learn suited environment prediction subject requirement domain suit change persistent real pressure texture temperature body research wherein interpret largely grow attention match proxy location biological substitution device thought device take action prevent separate choice substitution information device hardware device encode prediction hardware helpful operation device natural thing substitution feedback training note et need perhaps minimize modern interpret device act e g response however think term substitution intend window research intelligence salient internal biological device device decision domain suit substitution match room area use user load distance appropriate specific intelligence intelligence learn take user help
long integer choose vanishing appear hold driven choice price pay translate degradation low since similar obtain appropriate minimax rate change inside construction near neighbor paragraph theorem improve algorithm observation accord density formally n number spatial link tail statistical correspond near sequence slice interpret spatial bandwidth small stress term omit make minimax tail drive several previous general assumption classification satisfie satisfy balance balance generalization neighbor drive result clarity obtain low enough next assume balance use suitable jensen inequality normalize denote large meaningful illustrate result small enhance discussion nearest investigate margin rate whose cumulative value translation argument study illustrate reach near tail margin k tail illustrate power law neighbor low q classification rate thin tail close purpose phenomenon tail previous classifier margin near neighbor represent degradation decrease performance neighbor underlie distribution theoretically several law distribution provide reach rule counterpart improvement compactly support point tune neighbor neighbor kernel base process preliminary estimator include fast neighbor j nx n q law location location still strategy nearest training c c gauss cauchy power theorem rule increase growth meaning procedure tried modify theorem become increase empirical subject study address balance risk leave future resp resp proof present inspire minimax testing refer comprehensive deriving measure ix bx mb opposite use density ball later lebesgue measure leave circle cm cm circle cm cm circle cm cm leave compactly fulfil fulfil sake convenience law couple argument soon briefly observe contrary study deduce lipschitz q study one satisfie involve inequality contain ball case tail fulfil soon meaningful soon denote constraint constraint optimize n lead end compactly point paragraph tail recover fulfil tail necessary show long compactly support ball keep q possible arbitrary hence distribution exist hold classifier consistent tail satisfy violate idea ensure tail assumption new whose measure proceed way paragraph satisfied satisfied negligible soon dx soon sequel great support long paragraph obtain large make slow rate value apply decompose assumption yield consider example thus j lead j side use keep notation satisfy equation equilibria previous optimize respect choice conclude proof want minimal lead excess natural balance slice proof j tail thank eq proposition lead sum n see bind term value conjunction balance equilibrium reach conclude compare classifier concerned use hoeffding n state upper error design nx sign nx follow apply proposition clearly misclassification term belong bias concentration write use belong attention sometimes quantity bx bx bx soon version variable bx eq n conclude proposition attention sample resp give incoming set discriminant particular classical measurable prove define provide analogy assumption equivalent keep study smoothness margin assumption quite eq minimal principle neighbor introduce two near decision minimax excess lower bound immediately minimax model knowledge model smooth never appear analysis reasonable think optimal discriminant conditionally spatial aggregate label difference section around subsection make satisfy excess logarithmic remove inequality associate asymptotically random soon remove logarithmic paragraph introduce tail derive near neighbor association argument assumption step major conditionally long appear proposition become yield plug last upper bound discriminant exist e eliminate introduce variable build part pick realization recall aim deviation introduce poisson deduce x nx nx nx nx nx nx study term n hoeffding q nx chernoff poisson nx n follow eq conclude q nx x nx nx nx nx nx x k slightly modify write n tail yield q section give vector law long supervise give near popular flexible machine community ability devote neighbor situation attention margin consistency derive sharp near end core numerous contribution literature recent continue view feature interest paper provide x retrieve law technical appear assign label conditionally provide exhaustive associate among divide family erm select minimize candidate context boost depth maximize decade excellent performance rely recursive dyadic partition state introduce improve refer theoretically bayes overview within motivation plug property overview neighbor correspond plug attract decade seminal integer training also seminal contribution receive even core general identify importance concern influence excess neighbor notion penalize plug entropy mainly asymptotic therein see away compactly support provide exist associate appear law boundary order paper compactly low contribution break first reach excess couple illustrate show classification result binomial one margin second step investigate near w secondary encounter vanish non compactly additional involved allow position marginal near bound classification different tail uniform see link establish even situation consistency open describe attention near neighbor devote prove reach convergence excess mild support location include discriminant model poisson model particular case throughout couple admit lebesgue hereafter measure resp real supervise classification complete predict decision interesting associate possible well q decision rule low unfortunately function benchmark hence possible excess also appear primary importance paradigm minimax define infimum classifier classifier minimal mass near distance build distance context near neighbor near neighbor measurable term estimator regression particular neighbor worth amount neighbor large process satisfy consistency detailed compactly observation core worth fall density strong assumption admit lebesgue density satisfy eq soon bound strong minimal assumption proposition away mass assumption support low conversely include regularity q possible involve involve omit relationship minimax rate consequence theorem exist conclude minimax rate support already high increase curse low tool nonparametric primary importance sake refer size study decade discriminant alternative binomial assume independent sample income predict correspond come classification draw completely conditionally long discriminant conditionally neighbor rule rule standard provide near neighbor discriminant complete detail neighbor near binomial binomial rely sample cope yet discriminant regard rate discriminant set difference smooth discriminant ok situation binomial weak log margin obtain fast seem margin finally applies bound compactly supervise compactly improper setting situation
enyi pairwise comparison suffice consistency retrieve rank relative consistency eigenvector os pair retrieve see similarity simplify laplacian sd preference specifically comparison rank matrix q obtain normalize drop positive multiplicative affect eigenvector result bound perturbation local perturbation rank rank retrieve large conjecture still assumption rely inequality eigenvalue translate seek inequality perturbation corrupt let fix could seek jk jk jk jk bernstein get bound bernstein get deduce n c ik jk equation p use independence time perturbation anti symmetry part diagonal indeed bound rest hold seek high probability p construction laplacian unnormalized close get unnormalized laplacian provide robustness unnormalize worse unnormalized regime perturbation proposition expression laplacian want eq get k I similarly k kn notice sx ks laplacian define odd eigenvalue know eigenvalue laplacian experiment show grow towards symmetric perturb get perturbation relate perturbation perturbation simplify notation l hence deduce constant absolute ensure large hence symmetric cf rank sort going goes order preserve make result compare result true simulation able suboptimal factor proposition perturbation translate start eq fix mean use union jk f c jk jk j first term begin bound hence apply n main comparison sample probability least notice f l lf sf f sf writing triangle deduce f term separately deduce f use therefore use remain notice cn numerator norm perturbation rank perturbation pairwise retrieve argue apart indeed far apart quantify far notice probability use odd negligible deduce similarly w desire real classical synthetic dataset pairwise comparison miss give item similarity nearby rank retrieve percentage corrupt comparison scale deviation interest consist comparison pairwise extract competition pair maintain rating update competition benchmark performance figure percentage transformation refine metric consider participant appear ccc corrupt miss blue centrality green line vary corrupted left proportion top synthetic dataset right ccc top various l city city city city city united united united united united west west west west hull west west hull hull west west hull set enforce retrieve e ranking comparison outcome home away game pair top roughly rank satisfied rank team spectral enforce rank correspond last figure ranking missing comparison index miss define monotonic technique similarity matrix similarity compute miss score eq whereas still show finally single corrupted outside unchanged together retrieve indeed strict strict matrix desired extend index index miss remain proceed except shift give comparison item rank comparison index either corrupt condition remain shift divide comparison unnormalize ccc corrupt miss rank centrality dash proportion corrupt vary parameter asymptotically eigenvalue laplacian characterize eigenfunction limit differential equation enable eigenvector eigenvector characterize x md kx fy dy fy dy fx twice expression solution differential root degree nonnegative calculation constant unitary numerically digit normalize proposition eigenvalue laplacian enable index uniformly calculation comparison rank eigenvector form pairwise valid detail extend input comparison option power robustness describe pairwise comparison assign compare construct pairwise recover observe rank still comparison corrupt miss spectral rank bind observe benefit formulation real rank competitive superior compare classical comparison item back seek ranking comparison order pairwise comparison two science player play player mark formulation music preference term one e rank fourth practice large noisy comparison incorrectly inconsistent classical vary website web page web link express link preference measure adaptively iteratively extract parametric likelihood metric algorithmic derive arc problem pair player understand point preferred item preference information reciprocal provide simple scoring difference example pairwise preference maximum usually fix technique rank classical item order item decrease distance chain reconstruct underlie ordering pairwise exactly solve noiseless case serial ordering small practice perform similarity matrix correct order item organize rank provable robustness formulation supervise equivalent additional impose focused minimum linear excellent guarantee case albeit summarize pairwise recover apply either corrupt incorrectly classical scoring method comparison os enyi independently suffice suffice retrieve consistency eigenvector os enyi effect order retrieve result produce supervise organize rank rank noiseless solution exact subset analyze os finally illustrate synthetic dataset base pairwise item item item close place item decrease formalize say impose coefficient row column without monotonicity permutation put call identity reverse permutation permutation strict strict permutation consist permutation matrix permutation reverse matrix strict order pairwise strict ranking rank item decrease able true comparison vector strictly monotonic monotonic nontrivial maximal contradict strict distinct first suppose monotonic reverse permutation addition identity locally reverse order row since strictly section spectral produce rank computing call comparison totally assume tie sort matrix recover item strict combine deduce repeat hence permutation strict ranking order rank candidate ranking apply enough comparison item item pairwise comparison rank sort either decrease goal item first rank refine also cccc row strict r corrupt keep strict recover right inside enforce strict noisy comparison cause spectral recover numerous first corrupted comparison multiple corrupt provide perfect recovery begin definition row minus comparison item rank index remain item write write simplify similarity impact corrupt score whereas incorrect comparison induce tie tie write get
connect induce asymmetric flow agent strongly network triangular desire primitive network sometimes matrix likewise contain internal appear edge sub contain network corner network receive sub upper combination show h examine steady behavior examine denote block example stochastic primitive upper triangular block sub network network limit eigenvector connect eq start establish norm subset long entry therefore establish contradiction assume frobenius attain denote ensure eigenvector happen attain strictly conclusion entry identity conclude consequently primitive power matrix tends fact claim useful hand side denote total verify gr gs gs gr gr stochastic group reach token represent scale internal information within similarly subsequent involve power pareto solution characterize fully explicit size agent likewise vector denote scale eq accord pareto cost towards conclusion network collect pareto solution across sub limiting extend agent uniform within individual limit total agent collection point network limit point sub argument follow identify transformation show within recall sub happen limit say within converge readily group collect limit key relating eigenvalue non right eigenvector equal establish establish ss ss sr sr obviously block kronecker product claim next observe follow equation condition uniqueness another possibly subtract equation however claim generic agent discussion whether agent limit right collect within sub group agent hold substitute find dynamic network evolve accord statement function assume noise size hold q triangular vector independent group stochastic recursion study argument correspond sub group need evolves instead introduce canonical except extend quantity side recursion appendix start agent group besides moment strongly agent bottom drive hyperplane wrong run logistic topology weakly sometimes elliptical separate data class concentrate outside outlier belong locate away obviously weakly connected agent group outlier advantageous suffer represent connect green weakly influence agent employ term coordinate approximate weakly connect strongly connect strongly large curve weakly compare boundary curve curve infer concern limit point associate streaming datum via white weakly topology assume agent agent obvious get agent simulation illustrate profile noise agent db likewise theoretical agent sub theoretical value agent sub find simulated cm examine learn agent weakly relationship converge agent limit reveal agent topology beneficial reduce impact zero variable step b norm sufficiently condition substitute give introduce proceed spectral purpose norm last next verify establish eigenvalue compose eigenvalue associate eigenvalue eigenvalue large large eigenvalue algebraic must occur block algebraic geometric small matrix follow size eq choose claim replace rr let scalar expectation next error proceed verify substitute return bind use jensen inequality introduce combine use use introduce scalar step follow scalar lastly arrive expansion eliminate since last I sr sr rr sr b sr I expand stochastic emphasize jensen conclude simplified lastly consider recall recall square arrive simplify subtracting block identity agent define block product verify equivalently operation operation stack examine simplifie canonical recall primitive exception equal correspond conclude canonical decomposition strictly eigenvector expression limit decomposition relation proof order q since dominate small substitute start q one moreover early identity th clear agent performance agent group arrive equivalently lyapunov satisfie obtain agent need compute diagonal return step introduce exploit block height department electrical engineering university mechanism agent reveal interesting flow topology result exchange mask certain totally agent determine arise example due attack failure critical work connectivity topology adaptation combination outli connect graph exchange pareto solution cost global minimizer useful developed diffusion strongly path self loop technical minimizer large size learn ratio agent main article examine behavior affect topology necessarily shall examine effect flow consist one back setting important arise attack neighbor inaccurate feed back behavior asymmetric information exchange model second example context interaction regardless regime flow medium third twitter small subset bias asymmetric information exchange reasonably connect connect reveal relationship agent outside influence scenario arise attack asymmetric result failure critical contribution help strong topology combination weak connectivity beneficial namely reduce plain letter letter trace radius matrix besides denote derivation consist nonnegative edge connect agent scalar datum receive zero network say agent direction say strongly flow agent possess loop strongly emphasis agent neighborhood agent denote assume default include neighbor coefficient condition stochastic connectivity network imply primitive property primitive matrix eigenvalue lie circle right normalize entry add eigenvector associate denote variable collaborative combine diffusion positive step iterate adaptive advance far ahead diffusion g individual use convex aggregate unique agent mean
outli dense eventually subsection turn truth problem coordinate descent subproblem column optimize convex q second subproblem base give alternative descent algorithm descent initialization convergent sign b I production stop first show converge point would actually denote part could convex combination optimal global aim arrive bernoulli complementary matrix submatrix simplify outlier appear outli successful index outlier index measurement reflect acceptable determine entry apply usual formula set equally insight advance estimation measurement show suffer bernoulli accurate fortunately inspire way combine remove entry lastly apply succeed would let consider recover contaminate rank factorization base start derivation thing contaminate corrupt appropriately derivation th column appropriate fit regularization eventually public dataset dark line ground line line achieve high efficiency ground efficiency top dr precision poor f specific tuning assign dr pre measure dimension fast implementation inexact alm alm speed previous improve non increase matrix type simulation low gaussian noise matrix matrix gaussian matrix indicate bernoulli matrix generate random matrix dr measure q correctly detect indicate corrupt detect precise three present tune observe high accurate meanwhile cost conduct two video lot activity foreground frame activity illumination equally left scenario frame row box last row run video simulation strongly validate robustness regression detect outlier contribute achieve rank outperform art look lemma one form eq entry except obvious contradict feasible obvious eq chinese china ie edu wu technology china wu ac outli additionally outlier bernoulli help approximately solve high popular algorithm extend factorization recover component extensive like aspect realize accurate estimate computationally intensive certain prevent robust estimate paper aim estimate regression denote denote signal denote fact assume noise noise consider outlier totally occurrence assume bernoulli operation eventually bernoulli accurate fortunately find certain level estimate efficiency extend apply massive modeling recognition rank contamination recover component equal understand
mu selector appropriate alternative non knowledge bind entry show depend report propose yet pursuit need focus subgaussian show omp analogous consistent error variable setting satisfy remain practical enabling although suggest subgaussian less independent component remains minimax state however answer situation two question assumption order cone attain close contrary knowledge compute focus subgaussian matrix appear extend suitable deviation property check eigenvalue solve convex devoted selector programming course compare selector furthermore bound fast uniformly subgaussian subgaussian subgaussian product subgaussian deterministic random subgaussian random subgaussian furthermore assume condition selector gram shall form consequence latter indice complement cardinality subset cone restrict constant selector start follow meaningful term prove lasso prove addition sensitivity fast regressor appendix base computationally section bound knowledge appear theory take efficiently polynomial feasible set appendix main assume parameter assume risk admit appear assumption simultaneously coherence see show subgaussian row well bound extend fix assumption probability although theorem result spirit minimax give bound feasible inspection reveal valid provide bound selector next state somewhat probability constant base selector prediction design slow convergence solve issue specific obviously reduce however shall additional even program simple programming algorithmic therefore detail brevity write set note problem q tuning constant program justify penalty parameter separate h ccc bias rmse pr rmse l ccc rmse pr report selector outperform option estimator high bias benchmark mu selector selector nonetheless feasible reason case align property error report qualitatively display respect model selection separate zero h ccc c pr rmse ccc rmse pr high dimensional regression mu selector programming estimator time practice use theoretical namely optimal somewhat bound nevertheless reduce linear helpful support gene grant appendix give appear diagonal square denote zero follow prove n tw w n tw tw tw pp eq constant give let random subgaussian together independence random subgaussian union subgaussian random analogously use cauchy subgaussian exponential imply union yield b bound preliminary feasible problem two theorem event lemma lemma feasible consequently note event belong let feasible together eq exceed since intersect probability least corresponding finally lemma initial event probability plug throughout proof probability hold imply cone definition sensitivity recall q proof definition finally q collect property restrict constant coherence constant cardinality provide useful assumption q relate ss trivially collection convention solution solution solution minimum solution obtain divergence omit gaussian proceed constant include denote hamming zero see denote component equal constant obtain next condition example school es mod universit et paris centre en paris linear sample new estimator selector turn noisy regressor introduce sense efficiently programming estimator numerically model design noise matrix estimate example reduce linear regression error investigate size show presence severe procedure particular
concentrate small hence efficiency run chain mean give coherent selective spurious couple nlp property construction truncation motivate principle flexibility facilitate fully develop remarkable parsimonious achieve accurate validate pre covariate sample serial correlation capture multi modality may desirable remark perhaps readily adopt develop strategy general adapt generalize graphical em state k direct prior multiply divide cauchy continuous differentiable ac pg n arbitrarily continuity integral make arbitrarily implication direct slight respect finite dominate state n mle consistency either strictly mle density limit constant case generate kp note identifiable model long singleton nlp definition imply lemma linear nlp nlp take form rd cg n prove result penalty sufficient hence write let x z k dominate obtain eq show choice g k k give explicit product integrate dominate algebraic integrate prove continuous grow dominate adjust divide multiply r x expression density rearrange conclude l kx bn n n n mapping k x sufficient nm ng n algebra k ng k theorem combine k k informally point need trace converge satisfied grow converge k contradiction e increase hence monotone improper complete indicating without suppose second derivative hessian rearrange obtain approximation convex maxima occur maxima mode occur jj j kl incorporate assumption mle solve jj pn pn intersection contour length shrink evaluate generality jj converge n far pn mode denote generating argument lie add approximation kt nm third bound finite term leibler law number kt pe n largely mode ti factor case manner probability note pm pm pm follow denominator apply pm km pm te pn ie pm pn pe pm k whole regard non proposition desire iii adjust section exposition indicator e np ip n prior straightforward algebra eq il state eq proposition ii p p n orthogonality give q normal v ji laplace approximation around pn pn pn pn express minimize numerator decrease mass denominator go apply q survival equal discuss min md md min side finish notice mp continuous notice min side md min h min mh md mmd min md constant marginal multivariate function decrease integral plugging dominate apply I apply I md I sample I gibbs non inverse z sampling follow straightforward denote truncate ji z l jk adapt algorithm address exclude sampling write union fortunately increasingly set cdf monotonicity convenient log determine denote q monotonicity z monotone evaluate inverse strategy guess continuity conduct interpolation dominate hence drop equal z z z determine interpolation update either continue guess often quite research remark university department department achieve possess appeal dimensional use estimate spurious quasi depend spurious differ mle theoretical constructive practitioner perspective enable extend notable high benchmark hyper validate magnitude remarkably pre contribute estimation selection actually dimensional develop good challenge appeal discard spurious fast extra important consequence density dimension denote fix set nlp density zero univariate prior pm km nm nm tf minimize leibler kl pn contain spurious hence truly ratio shall show lp nlp regression fully probability decrease show induce asymptotic particular light yet relate strategy fast role pm consistency consistency prior manuscript characterize imply address practical justification add modality performance simulation expression datum grey top truncate bottom intuition vs possibly prior estimation grey line preserve set significance combine truncate exclude line assign truncate coherence detect prior mass concentrate around decrease truncate cauchy express marginal integrate cauchy go nlp mixture assign roughly induce shrinkage nlp k often always induce integrate likelihood nlp lp k dd nm kl assume singleton kk k tm integrate dispersion term converge require mle see include identifiable l n converge typically asymptotic proposition grow rd k n x kx bn x kl generate n n cc strongly estimate satisfy condition minimize kl divergence ki ki ki ki cn k ki pn pn ki pn pn diagonal ii quasi bayes mle generating prior ti pn priors eq residual ti pn pn set spurious become interestingly term affect spurious asymptotically correspondence simplicity omit truncation nlp manner give let truncation define nlp p nlp convenient later truncation define nlp truncation gamma inverse gamma obviously affect p nlp truncation representation p p p additional greatly sample component univariate prior product freedom set function normal truncation trivial cdf survival quickly computation univariate right nlp form behavior depend nlp h min min min bayes depend penalty give small show nlp h p multiple truncation tail functional depend solely whenever recommend significance give analogous predictive yet possibility unit regard draw analytical threshold section posterior linear simple nlp truncation highly straightforward algebra truncation may challenge apply truncate draw truncation representation provide adapt unknown dispersion set hyper multivariate rectangular efficient algorithm serial implement sampling package important property truncation negligible assign often negligible separately posterior eq square represent eq chi algebra lose variable require multivariate assume appendix mh gibbs p efficiently combine search token require os posterior corollary h h require algorithm k algorithm yet multi induce benchmark scad default assign beta binomial whenever adapted never visit benchmark prior parameter beta binomial lasso scad penalization cross respectively supplementary material attain method help covariate contour top middle bottom e e simulate bivariate normal first compute possible integrate function package full model rounding draw obtain mass shift resemble elliptical table auto correlation reflect
parameter algorithm notice sampler effectively transition stochastic initialize sampler kt kt ft ft kt kt synthetic latent ft synthetic recover short audio physical time play simultaneously different audio fourier transform ms overlap yield matrix hyperparameter nmf figure variational proxy component right great see learn structure activate implicitly reflect pattern activation performance sampler audio report run twice sir sampler partially due discover last interference assumption ideally regular method conjugacy directly mean field variational degenerate delta effectively map implementation detail number sir structured field inference beta process optima reasonably hide blind task par exact hyperparameter channel mask activation particularly leave factor desirable capture mask prior encourage mask mask binomial model binomial efficient incorporate structured field non factorization gaussian study literature nmf process conjugacy variational relax conjugacy approximate result synthetic propose reasonably hidden approximately matrix refer activation application domain music recommender system hyperparameter latent offer put component activation allow focus model negativity impose address process nmf binary approximation variational inference gaussian computationally intensive computational burden field variational process strong dependency among latent local develop attempt utilize address first nmf inherently derive framework blind nmf model kullback plug approximation beta kl hadamard properly spectra latent activation binary mask formulated set large easier introduce auxiliary random making auxiliary enjoy conditional conjugacy helpful algorithm inference divide variable structure distribution approximate activation factorize form completely mask correspond local gradient global parameter intractable gradient collapse make forward quantity define ft h jt burn per distribute ft ft ft kt kt kt
function cv cv outline use interior demonstrate every iteration give ip optima costly optimize extension convex hull method apply machine possibility relate rgb rgb rgb false pt false title r thm thm thm thm thm thm convention exercise question develop selection save practitioner tune manually particular implement test fold extend complex compare extensively ill generalize datum crucial regularization smoothing svms characterize follow convex program reduce l general purpose regularization generalize description aic criterion recent also discover develop homotopy svms tune selection classical regularization scheme constant choose minimize subject offer outline practical tool tune manually bring automated examine rather characterize approach simultaneously purpose write ridge approach apply benefit minima approach rigorous see solve value value think speed computation svd solution search decompose low provide constraint parametrize moreover form hull easy verify set hence true characterize entirely polytope together follow relationship relaxation solution counterpart prove bounded pass construction hence ip kkt fix response notation machine optimization set problem optimal among kkt simply solution set relaxation immediate comparison require grid step define high qp latter
landscape subsample anneal landscape factor langevin effect subsample anneal speedup simulate subsample although many linearly choose moment probability energy simplify boolean generate unknown probability generating ball ball hyperparameter ball conjugacy since indistinguishable project blue ball posterior multimodal beta prefer segregation time variational jeffreys bias ball ball effect subsample subsample langevin intrinsic limit red single pde drift depend intrinsic schedule fast mix schedule model total thus absence datum energy mode subsample linearly shall linearity preserve energy inference behave like subsample annealing sized colored stepsize generalize model naive mcmc size anneal behave anneal speedup toy around two mode limit slight limit energy infer rather inverse barrier assume within mode entire mode separate barrier proportional appendix proof interest time arbitrary energy necessary anneal schedule proof thus anneal exponential feature already subsample equivalently anneal difficulty significance learn combination easy infer anneal relate anneal subsampling dynamic equation extra fix subsample annealing exponentially cross categorization partition r categorization categorical mixture feature partition learn hyperparameter partition categorical feature inverse grid wide range hyperparameter partition hasting sensitive significantly value us census dataset dataset include value categorical train log score compare empirically subsample anneal well fewer bad partition subsample anneal consistent learn mcmc sequential initialization initialization toy empirically sequential initialization deep hyperparameter poor initialize state subsample anneal address slowly subsample subsample allow tradeoff speed stochastic schedule inaccurate accurate slow much simulate simulate wherein landscape anneal landscape vary temperature subsample anneal portion yield posterior subsample size employ van multiscale monte avoid site inference hierarchy compression phenomenon liu generalize wide subsample annealing anneal technique suitably apply problem proposal clustering hasting carlo particle streaming dataset addition grow subsample minibatch approach variational inference practitioner increasingly subsample annealing cope principled heuristic demonstrate improve real subsample anneal speedup subsample annealing improve cross categorization provide improvement landscape section subsampling offer energy gap moderate sufficiently sufficiently relative barrier still observable subsample annealing may rare anneal poor hope subsample subsample annealing provide anneal help dramatically intuitively hypothesis grateful red ball resp leave red blue removal symmetry intrinsic define moment scale inspection subsample anneal schedule anneal approximately scale version dynamic effective length anneal lemma system sigmoid mix state barrier height dynamic low inference homogeneous transform case state transform anneal schedule increase equation sufficiently bind simulated clustering learn subsample datum portion inverse schedule gibbs temperature e subsample final state jump energy anneal subsample annealing subsample improve million census rating categorization simulate landscape anneal anneal recently model categorization infinite model rapid structured result stochastic sgd towards inference scalable inference method find site implement applicable discrete contribution extension sampler easy implement yet model subsample subsample grow subsample extra schedule terminology simulate anneal treat subsample indeed deep mathematical connection indicate anneal approximately simulate anneal stepsize langevin prior like subsample move towards increasingly long schedule assume linear anneal section accurate sample draw restrict exchangeability index assignment assignment part point possibly index set immediately classic finally nonparametric increase crp polynomially become grow subsample schedule gibbs sampler respect strategy initialize assign sequentially sampler
profile player particular game player profile gain player time strategy adopt profile nash equilibrium ne change change formally speak three question arise ne actually ne iii ne reach completeness presentation interested reader ne remain player strategy decision game ne may pure ne finite reach optima agent moreover search local polynomial pls polynomial game interpret shift gain favorable strategy player optima detection vertex ensure nash equilibrium community function represent vertex modularity gain community gain ensure detail towards nash equilibrium confirm let ne furthermore ne reach possibility bipartite direct conduct experiment hand traditional hand four produce instability compute modularity higher produce local nash sound community modularity vertex reach ne modularity modularity except appropriate interestingly stability good result benchmark checking goal partition event accord known identify group bipartite top edge associate event eventually highlight red blue author three beyond present overlap overlap namely modularity event partition community produce could member early entity nearly author appear overlap event cause compare author blue green community design vary community compatible nash figure nash reach step display tendency prefer association another unstable modularity become community author detect community assign community community number first value column unstable horizontal observe horizontal modularity third nash reach one facebook account tag different tag tag person name link obtain overlap tag individual tag ignore individual display community pass understand individual life period link life g community member friend life particular contain individual first except person community include constitute partition would one community isolate friend community thank present limitation conversely thank individual finer find unstable community suggest reach begin modularity nash amount increase facebook quite visualize address facebook overlap figure situation reach equilibrium display situation separate color person color nash community equilibrium community community lot lot person potential situation equilibrium community community whole person modularity increase lot nash low automatically spread goal vertex nash equilibrium reach second computational bipartite scientific sized bipartite benchmark dataset describe display company capital member hold vote find whereas interesting overlap community modularity equal value scalability test relationship order library http www scientific paper paper extract second unstable vertex demonstrate unstable one achieve objective knowledge appear suboptimal several criterion define partition sound nash equilibrium yield remain still heuristic offer modularity entropy nash difficulty obvious seem visualization dataset like community linguistic hard distinguish regard visualization raise challenging hypergraph diagram visualization high community read display initial overlap need bipartite facebook must narrow display row vertex vertex constraint predefine number traditional like knn modularity modularity use aware article detection proceeding assignment situation situation take every problem great introduce quantitative criterion example number community limited part may address simplest hypergraph many mention particular complementary bipartite profile distribution community else distribution community team company community evolution quite body field interesting since highlight social appealing temporal evolution cause network facebook evolution scenario dependent service political community partitioning vote overlap network nash equilibrium less detection method contribution display detection stable solution nash equilibrium community unstable membership find modularity vertex collective research enhance herein limitation present take community study unstable situation detect community contribution provide external within would service reality operate usa social focus method aim optimize call modularity maximize community inter completeness problem heuristic optimum paper introduce optimum necessary condition nash equilibrium function simultaneously partition visualization interesting experiment either graph medium sized social field survey topic detail algorithm world overlap focus mainly previous article extract overlap bipartite use bipartite oriented partition call benchmark author represent partition address yield partition community modularity optimum potential clearly closely take vertex community stability condition equilibrium enhance nash equilibrium acceptable powerful semantic address partitioning calculation mathematical modularity connection community minimum link external community extract partition weighted graph comprehensive art report adapt formulae modularity bipartite classical spectral genetic analysis strategy often extraction use design combine spin interaction annealing optimize local treat dual involve weighted link rare focus community method extension clique representation remain bipartite orient graph overlap community property membership decide apply iteratively modularity function merely stop overlap community change stable result article accord strategy community detection problem vertex assign nash modularity bipartite recently several modularity bipartite analogy modularity yet modularity modularity hereafter let adjacency diagonal also modularity community edge kronecker equal hereafter consider graph weight transformation show modularity bipartite graph column ij formulation introduce edge belong modularity yet detailed node community characterize apply partitioning result without edge modularity modularity graph numerous researcher remarkable build modularity community replace vertex represent modularity optimum corpora dedicate overlap vertex several community
strongly conclusion theorem sdp ability include write mc c eq q w fail outside block large sdp weakly constant slightly define weak replace respectively q translate imply hence sdp consistent proof appear relaxation mle apply sdp tight relaxation coincide sufficient translate read interestingly establish sdp specialized result predict success sdp analog mention somewhat base adjacency call eigenvalue truncation impose degree though report outperform let collect general regard zero element sdp proceed solution use conclusion second unique generalize optimization maximize consequence statement order cf sdp sdp construct ij k k j extend ks ks ij ij j moreover bm bm dominate block consistency generality low concentration discard sdp sdp variant summarize sdp adjoint kkt like obtain cluster matrix primal imply psd equivalent condition unique e ks mb I e mu primal solution optimality condition triplet satisfied linear eq accordance rewrite cs degrees community subgraph row column word vector sum take p also ks b k choose feasibility translate detail remark involve let projection feasible least hold feasibility use verify assumption ks ks u sdp next hence infeasible carefully increase block kp construction work balanced plant start k cn mp bernstein around simplicity effect diagonal I n defer assume enough turn imply q e sdp nb operator accordance k k case equivalent act k expansion block later read equivalently b ks u note analogue indicator table satisfie strict verify dual feasibility j ds proof I chain definition distribute assertion assertion sufficiently right inequality sum hold hence take satisfy enough imply eq need drop corollary low imply complete get form replace adapt first method sdp reasonably implementation admm start language affine symmetric th everywhere else element equal everywhere else variation affine subspace derive implement project sdp admm easily projection onto eigenvalue sdp balanced ideally suit require estimate mean truncation ambiguity fix abuse compute sdp introduction enforcing tendency sized block ideal sdp enforce flexible flexibility disadvantage histogram estimate block estimator ni ei q eps eps eps eps eps eps eps row plot experimental sdp sdp sdp amount spectral cluster choose graph operate tuning parameter value drive hard decrease separate behavior four community recover output sdp relaxation scatter row lead eigenvector clearly provide sdp recover exact sdp geometrically organize figure space point superior eps eps agreement information monte carlo replication relaxation sdp dominate outperform eps eps eps eps eps eps sdp sdp go block block prediction theorem conjecture regard sdp always community strong fall however remain weakly replication sdp sdp boundary difficulty recover rd sdp see show f reconstruct sdp severe behave sdp exact difficulty reconstruct see error result close see eps finally show sdp adjacency estimator row order addition sdp block nearly equal result sdp sdp sdp true unbalanced equality one poor representation htbp eps eps eps eps eps eps eps eps eps k eps eps eps eps eps eps eps eps eps eps summary good block histogram flexible unbalanced block desirable balanced sdp sdp nonnegative relationship treat relaxation balance plant various define tight analyze show block respectively sdp strongly class also conjecture outside weakly remain sdp mixed community direction simpler plant sdp obtain adjacency around turn growth new rely sdp sdp hard dependence latter tuning lagrange duality make general empirically outperform adjacency spectral call reflect guarantee dependence seem inherently noise doubly cone equality constraint outlier alternative relaxation far another hybrid replace function explore connection unbalanced sdp relaxation zero surely mp mp c mean bernstein eq rest tool detection involve infeasible problem new programming problem sbm relaxation sdp relaxation sdp guarantee carry however show sdp recover community wider relax consistency condition sdp thus class applicable derive primal construction sdp suggest sdp sdp evidence cluster real relaxation tendency balanced sized ideal tool histogram popularity literature throughout relaxation connection community attract physics sbm widely community analytical connection challenge assignment art accuracy rely start method popular spectral difficulty help even accuracy semidefinite sdp sbm optimization hope like cluster likelihood way optimization easier analyze also relaxation make noise outlier see drawback sdp sdp solver sdp continuous advance relaxation sdp tight relaxation relaxation unify connection empirically derive admm keep reasonable side throughout equal obtain sufficient condition exact sdp sdp wide sbm previously literature current sdp relaxation implicitly strong cf whereas sbm requirement success previous sdp relaxations sdp sdp limitation success sdp primal construction suggest relaxation flexibility complexity successfully recovery see sdp relaxation instance work inspire trivial extension general complex sdp doubly nonnegative also sdp strongly sbm divide sdp relaxation strongly weakly sbm purely mix sdp focus additional assumption equal class sbm simplifie since q recall depend q albeit adjacency mle obtain sbm desirable consistency sense optimality hard relaxed computationally restrict otherwise induce alternatively derive relaxation relaxation admissible kronecker convenience node recall feasible note relax positive psd column compactly since propose sdp relaxation recently first xu slightly remark relaxation via since psd addition affine thus main focus relaxation replace sdp directly tight constraint separate affine break though
high shall derivative property value conjugate mapping use follow hermitian taylor scalar order note jt kt expand order shall equivalence hermitian operator transpose evident hermitian subsequently second term eq term expand term expansion augment presence augment important consequence augment hessian numerical hessian minimization hessian analytic numerical solution parameter problem method order expensive store view c jt kt rewrite inversion h q newton eq substantial simplification complement remove operate reader subsection derive calculus apply find minimize weight gradient become substitute vector see calculus also rule complex original base difference arise formulate give error minimize setting reduce system equation way generalize express rule hessian become error vanish equivalent real calculus hessian real calculus counterpart descent derive perform computation field product greatly simplify optimisation procedure serve complex optimization solution apart address newton acknowledgment dr take discussion calculus wise definition weight calculate traditional nature substitute eq proposition example real array practical solution often calculation analytic address issue propose novel calculus derivation optimization transform domain practice calculus chain correspondence hessian counterpart usefulness calculus simplify derivation generic corresponding complex design calculus analytic newton numerous physics graphic processing communication reduction procedure accord procedure typically rewrite real take variable treat analytic framework often calculate show save burden expression calculus field conjugate algebra solve novel calculus rule value enable derivation error carry field transform problem elegant derivative calculus instrumental basic counterpart invertible first operate augment propose obtain operate conclude enable technique traditional q q difficulty calculus chain rule derivative j rule derivative intuitive rule chain calculate leave derivative focus lot consistent systematic component calculation pseudo make derivation equivalent elegant approach calculus respective applicable gradient base term gradient prove follow also comprise novel eq matrix n jacobian conjugate jacobian convention convention j approach q
filtering minimize optimality sketch doubly random walk general jump cubic second boundary equal inequality interpolation constraint segment change leads estimate trend whose neighboring neighboring coincide neighboring prescribe fuse trend trend resemble detailed consistent recovery change change regime tend infinity keep bound magnitude slope show location specific change perhaps change elsewhere point interpret boundary within detection result slope lead order interpolation therefore choose alternate slowly boundary outside neighborhood variability lead note walk grow boundary away random recover discussion sec natural consistency sign idea change segment detect point therefore filtering detect segment trend filter new proceeding go recovery situation successfully neighborhood location spurious appear actually estimate change due proximity one blue line correspond estimate mean nearly dash solid denote signal goes stay slope purely expect presence trend change point segment change point change remove segment move close end remove presence estimate point trend filtering show far monitor scheme remove point trend piecewise trend noisy provide intuitive succeed build interpretation corollary trend total tv mean tv suffer detect objective paper suffer interpretation integrate walk avoid fused point trend dataset generate non stationary piecewise piecewise way via variation tv counting measure method impose penalty least maximum trend convex tv piecewise without penalty impose translate regularization balance sum simplicity univariate trend filter generalize spline detect piecewise method dimensional denoise fuse lasso filtering fuse rigorously
artificial trajectory cost expectation probability respectively cost go illustrate scalar estimate expansion follow simultaneous value parameter rademacher random rademach cost give sf gaussian sf policy respectively condition accelerate approximation scheme iterate gradient base rademacher go instant component variable stay within requirement descent direction along sf independent hessian invert hessian state matrix ct bt tx identity would denote identity dynamic policy constant w u measure distance th denote radius contain least finally expect build trajectory lemma bound bias least describe difficulty establish asymptotic difficulty bias contribute recursion equivalent update towards critical establishing assume negligible particular ordinary differential ode discretization converge equilibrium ensure remains propose none stack none gray plot area legend forget plot none follow dynamic define linearly parameterize policy discount truncation trajectory artificial trajectory expect carlo go minimize xlabel iterations ylabel col comma exp ylabel sf pos north east index blue table xlabel ylabel legend entry pos north east thick red table index col comma exp thick col comma exp smooth order grid run sequence algorithm run project project run curve sf approach benchmark variance sum discount cost go recent direction actor notable unlike carlo policy resort value expect sum constrain discount expectation sensitive mdp constrain formulate technique x denote solve enhanced variance cost follow ascent dual multipli cost classical estimate lagrangian primal ascent lagrange multiplier q project risk criterion artificial criterion bound lagrangian operate project search batch simultaneous hessian order sf simultaneous scheme difficulty establish bias evaluation future plan condition horizon introduce follow give preliminary state cx u cx fx w follow trivial continuity assume u continuity use lipschitz definition continuity plug iterate proof give artificial trajectory affect transition expect return give x I equation u cx cx I continuity l dy iterating bounding truncation add x bound b dy l u dy l l end lemma w w reformulate w n expectation w end triangle lead contain width c p n observe exist differentiable evolve continuously markov policy irreducible bias rl algorithm visit number horizon impose ensure negligible proceed return finite denote ordinary q proof asymptotically eq projection operator ensure ode stay set equilibrium regard govern theorem prove correctness rademacher easy rademacher easy true analyse set taylor easy vanish discretization ode ode lyapunov follow claim pp set converge asymptotically equilibria ode asymptotically stable equilibrium govern govern hessian tx jt mt tx employ taylor expansion derivation refer proposition lemma theorem arrive update true see discretization ode govern converge ode albeit similar use sf perturbation I j proof proposition rl action monte policy search policy order newton incorporate hessian cost simple continuous paper stand field control infinite discount decision address batch set trajectory access simulator formally tuple state action policy objective develop policy control attempt policy cumulative discount govern develop descent cost go obtain batch discount set advantage henceforth refer state action space go parameter use gradient possess bias well know simultaneous two first second popular simultaneous perturbation simultaneous descent sf estimate parameter sf usefulness stochastic action set metric space policy mainly reinforcement gradient rl least policy extend optimal batch mode rl ensemble rl see aim scheme approximate gradient perturbation scheme minima observable via irrespective simultaneous perturbation method perturbation functional differ simultaneous perturbation scheme sf hessian sf illustrated operate perturbation step cost go perturbation see value
mathematic apply mathematics south road south centre road cb extend divide arbitrary simple analysis deal case pair coordinate apply covariance scale expect change width matrix normally effectively employ fisher generalise straightforwardly include error carlo data include cast software matrix fisher become widely statistical low limit er rao estimate maximum given jointly determine fisher much compute distribution posterior describe distribution sophisticated experimental sophisticated forecast surface space implementation ti fisher useful proposal survey survey scale dark european space purpose basic fisher matrix formalism case uninformative taylor expand constant irrelevant discussion constraint likelihood third curvature fisher case deal variable straight ad hoc axis ultimately combination average fit axis slow new error extract population formalism formalism treat discussion straight line fit bayesian example remainder follow describe arbitrary describe formalism particular conclusion replace equation throughout paper formalism taylor derive generalise principle correlation formalism cover represent extra may measurement simple wish observe amount depend assume expand condition expand integrate parent gaussian main limit consider datum limit text affect concerned bias slope coordinate bias unless prior taylor essentially assume integrate z nz ii form covariance include element intrinsic scatter matrix give final define collect algebra marginal algebra calculation simplify look covariance compute use standard formula find eqn replace standard variable derivative model uncorrelated find correlation recover key observation usual matrix uncorrelated width propagation variance increase main interpretation make likelihood surface generalize straightforwardly replace covariance example illustration diagram apparent length correction colour act around include instrumental dark matter plus correction colour relate interest dark energy hierarchical description principled galaxy may large error colour correction galaxy mis galaxy couple error apparent investigate survey find correlation theoretical could couple arise make maintain error divide error error across pair term generalise modulus matter chain technique simple distribution range modulus look contour generalise good accurately bivariate good agreement orientation actual offset accordance depend coordinate analysis general
meet game theoretic learning instance engineering fluctuation thus wiener lipschitz strength player observation regularize admit strong mind derive evolution special decomposable convex support remain govern k open dynamic quite interpretation recover learn study variability player impact vanish sufficiently correction follow stochastic dynamic eq denote population specie environment fitness coefficient impact weather population evolution account account jump besides fundamental difference term drastically highlight contrast evolution reinforcement important first evolution map correction player also summation form drift determine wiener process substitute prescribed compare eq conclude constant dense continuous show trivially aa necessarily solution interior logit nash equilibria depict payoff vertex wiener reinforcement converge nash equilibrium analysis player payoff environment evolve player involve consider player action reward context compare player payoff payoff could nature advance player integrable stream payoff equivalently originally context agent past action overview reference main seek reinforcement notation focus integrable stream payoff wiener process noise assume player payoff extend consistent payoff arbitrarily assumption generating evolves induce carry primal dual denote terminology reflect negative strictly provide see primal primal bregman divergence express regret term grow consequence iterate logarithm whenever ft z hand wiener obviously w ft iterate logarithm coupling benchmark process begin formula therefore player play proceed sublinear hx hx directly lemma recall suitably restrict face span strongly readily yield conclude decrease control rate choice pick regret law logarithm identify section discuss describe remark specifically payoff player opponent cf play opponent response instead correspondence stochastically perturb play time analogue fundamental process elimination suboptimal dominate formally give kp strategy obviously mind play say pure strategy context dominate strategy regularize dominate strategy eliminate dynamic aggregate surprisingly condition dynamic exponential show variant dominate irrespective elimination dominate basic proof dominant drift coefficient away dynamic study suppose solution substitute recall obtain rhs infinity become virtue finally iteratively induction dominate show vanish dominate regularize decomposable dominate decomposable form eq dt complementary solution player sm problem interior decrease denote cf independence yield kt expand complementary around establishe run diffusion affect probability observe mean elimination let formula reinforcement equilibrium end recall equilibrium obviously pure correspond strategy vertex noiseless dynamic exhibit follow equilibria solution nash lyapunov also strict nash equilibria turn generalization noise interior nash equilibria sure rest reinforcement ordinary definition differential lyapunov asymptotic let say exist stochastically neighborhood whenever evolutionary show strict nash equilibria stochastically asymptotically across strategy show irrespective noise rely heavily logit generator thing approach seem xt x lyapunov nash nash equilibrium stochastically asymptotically contrary direct rely result regard stable point brownian admit solution equilibrium nash must kx v kx k event contain nash one dimensional wiener fairly provide measure respect derivative eq brownian see theorem follow note proposition strict nash equilibria stochastically proposition tolerance strict neighborhood mind sufficiently consequence fair also ks end change wiener fact conclude kt finite w kt kt conditioning conditionally adjust aggregate player empirical average principle converge nash equilibrium version arbitrary regularize deterministic setting show average aggregate modify cf remark average principle extend even arbitrarily large payoff error game dynamic score difference grow distribution play surely nash case sublinear growth martingale multilinear dividing yield kx xt xt xt proposition always assumption player game satisfy addition almost surely equilibria interior kp kt kt claim identically interior term sublinear finally sublinear claim otherwise strong kt correction growth require vanish alternate cover kx k kx denote regularize response player average solution chain arbitrarily break arbitrarily jump invariant proper development subsequently average deterministic reinforcement importantly show response let game deterministic player game player solve q vanish lie vanish xt x xt xt xt x xt track perturb dynamic thank conclusion full empirical nash average nash equilibria game constant game nash equilibria follow converge nash equilibria games logit response match fig nash equilibria display also take evolution fig trajectory horizon tune average nash collect coupling strongly interior induce penalty convexity constant fp py n well claim sake subsequence necessary hx contradict compact get let strong combine rearrange claim let face contain py hx tp xt lie interior sided derivative show compact visit infinitely imply fp claim n subsequence necessary assume pass fine need two thus let absolute eventually k fp nk hx fp coupling capture function dy definition third theorem conjecture example game noisy robustness stochastically perturb cl national france france fr tucker stochastic differential chain theoretic payoff noise provide unified extend game dynamic irrespective noise player strategy become nash asymptotically independently perturbation magnitude finally player nash equilibrium sum game acceptable state equilibrium dominate strategy dynamic widely action payoff probability score payoff score cumulative strategy reveal evolution govern population attract well lyapunov stable nash strict nash equilibria stable
within actually distinction crucial concern spread disease node status suit capture precisely core contribution include name core identify core network weight core decomposition uncertain probability exist biological model instance see core decomposition scale insight worth decomposition would rigorous os enyi center every style fill sep n scale rectangle scale auto center every style circle inner sep rectangle n core core high core empty center fill sep pt n short fold interpretable core theoretical property simulation study rely algorithm develop decade contribution core wide range application core find generate graph core guarantee graph fit self loop remainder manuscript let graph node context subgraph vertex often core algorithmic vertex figure isolate right every style fill n n n n auto node circle fill sep n n contain core thus statistic core map vertice negative integer clique vertex natural information second interest unlabele graph summarize histogram whose vertex symbol example respectively illustrate instance vertex imply observe represent function advantage theory family rewrite term compactly parametrization normalize pg model define list increase empty every arbitrary eq rewrite provide count node simple vertex copy copy copy vertex index vertex index center circle black inner sep scale auto center auto style black sep rectangle center black inner sized usually maximum mle one resort testing solve generality scope short study question often value necessary fix value address remainder consideration conclude interior hull implement initialize vertex remain empty process condition vertex order yield pre sort index increment condition vertex small precisely argument vertex algorithm yield ii increment vertex hope add know new could get unable happen problem modify since potential equivalent make adjacent zero adjacent remove give sequence sort impossible definition initialize initialize edge construct graph condition option produce positive comment conclude simulation randomly construct node unlabele evidence example discover call resort monte proposal propose accept pg swap however chains choose generally mix bad behavior remove markov hand reach value size record group network code b high quantify observe like simulate graph similar goal accord proposition mle mle scope instead illustrative impose prior comment skew degenerate distribution lead concentrated clique behavior node compare probability hand whereas distribution exchangeable large label produce skewed towards account balance effect graph summary distribution triangle centrality allow correspond observe value formal goodness heuristic evaluate well goodness general use markov usual plot ensure sufficient convergence plot consideration figure histogram compare notice histogram triangle capture effect quite model centrality core histogram centrality large capture edge much expect fact tend densely connect consider visit mode distribution include mode truncate result
highly backpropagation generative forward propagation approximate latent variable undirecte graphical restrict boltzmann rbms boltzmann numerous represent unnormalized potential summation integration chain mixing pose problem mcmc belief contain undirected layer difficulty undirected alternative approximate matching require specify interesting layer unnormalize denoise auto autoencoder match rbms employ generative discriminative discriminate distribution dramatically correct desire approach machine prominent extend generalize denoise define parameterized chain generative chain framework adversarial loop unbounded activation loop generative recent work bayes backpropagation learn generator distribution noise perceptron rather simultaneously train word value function eq present adversarial net enough formal explanation optimize computationally prohibitive overfitting optimize maintain optimal analogous maintain step markov part learn procedure poor reject confidence minimize function cm blue horizontal domain domain impose non transform region inner train discriminate sample converge classify unable gm p momentum generator like capacity infinite study probability show optimum net mnist database activation sigmoid activation net maxout apply framework permit dropout generator test set window parameter introduce various report somewhat variance perform advance generative directly motivate far c mnist stack deep net figure generator claim well competitive generative highlight direct deep undirected autoencoder inference need partition tradeoff generator approximate variational mcmc base inference markov difficulty intractable approximated approximated explicitly approximated design nearly extreme property differentiable function framework advantage framework primarily explicit much updating avoid enough negative chain boltzmann keep date gradient inference function summarize adversarial net generative aforementioned advantage primarily adversarial generator generator another adversarial sharp chain somewhat admit many generative predict net advantage inference generator conditional index training net implement extension mp net improve determine adversarial framework useful acknowledgment like acknowledge helpful code would fr ed need support would cifar provide support google learning like thank les corollary xu david op universit generative two generative model procedure maximize mistake player game arbitrary recover everywhere entire train backpropagation markov network generation demonstrate qualitative quantitative deep discover represent kind encounter intelligence natural speech symbol
straightforwardly correspond learn formalism boltzmann generate notice random joint estimation instead w log introduce log r j reduce infer maximize may employ give gradient result estimation assume increase increase coefficient zero method deal parameter crucial problem employ put zero fraction small work suffer detail ability difficulty pseudo fraction step value interpret refer infer stage maximization assign ratio stop belong pseudo log put log likelihood present pseudo maximize note decrease keep one inequality likelihood drastically pseudo lot early stage close wrong quantity vanish take stop student true ratio pair ability maximize estimation toward give descent guess go step infer shown drastically put whereas threshold show comparative estimation put choose minimize example test remarkable increase error value bar put quantity uncertainty put bar conclude good detect infer emphasize ordinary boltzmann kind ability second also process stop pseudo call maximum instance optimal case lead curve sharp fig point pseudo pl max pl detect due drastically pseudo result existence must decide preliminary addition number number ht likelihood coincide vertical line vertical large datum infer lambda determine zero infer advantage remarkable formulate item student boltzmann degree test characterize difference ability algorithm base show remain outperform pseudo function determine function suitable function decide terminate experiment desire author thank discussion perform grant work education student communication inverse ise method theory describe correlate development part science complex assume boltzmann form define ise pairwise complexity system boltzmann likelihood training coincide well bias interaction number sometimes structure could enable week vast expect deal present conjunction reduce training put difficulty assess keep applicability response specialized type test kind although set item student detect method correspond student boltzmann give brief introduction third good existence student infer theory various difficulty well specify ability express resolve problem accord problem logistic form express answer ability problem define express th answer pl item extend simplicity pl
play brain noisy make online adaptive describe nine within open software parameter adaptation performance test repetition variability decrease less suffice reduce across variable delay signal state implementation user calibration plug accuracy scenario source implementation sophisticated dedicated matrix inter variability promise less stability covariance riemannian three minor modification single convenient software development besides riemannian framework bring since rely know spatial variation metric draw wavelet basis adapt show limitation increase channel condition riemannian regularize big mean covariance decrease degree france work ph brain centre national laboratory france research include riemannian brain minor university post institute france france dr research centre national laboratory dr grant human eeg time tool blind journal geometry riemannian geometry good subject method online adapt information geometry far brain interface phase precede actual depend regardless necessity calibration drastically appeal orient cognitive patient limit plug operation consider requirement device besides training discard calibration inefficient proceed pose plug achieve completely generic parameter derive previous continuously experiment possess namely albeit user trivial filter bad work level high show lot solve geometry regular temporal signal allow geometry introduce way experimentally riemannian geometry enjoy addition rigorous elegant conceptually easy constraint thank present able calibration less minor modification type present new dedicated p dataset compare section attempt plug purpose filter unsupervised adaptation inter make property adaptation build efficient see none art requirement classification essence prototype comparison obviously relate hand definition appropriate trivial accomplished nature problem process able metric support nevertheless generally predefine loose generalization approach relevant extract approach process fulfil quality subspace separation overcome difficulty matrix svm classify adaptive implementation adapt eeg useful selection generalization subject separation filtering thank geometry field riemannian manifold field establish increase imaging strength geometry natural lead eeg mean high filter common eeg model statistic eeg application geometry zero directly eeg eeg riemannian definite matrix information invariance invariant inversion invertible latter consequence signal remarkable formulate effect spatial source etc essence working mean denote fr square distance expression manifold find toolbox geometric geodesic short matrix manifold geodesic riemannian metric geodesic could matrix interpolation compare eeg trial prove useful eeg relevant feature spatial rather source separation variance matrix task difference riemannian class adaptation subject subject intra one database interpolation geodesic non evolve riemannian equivalent combine replacement mean covariance filter adaptation class limitation interface p symbol covariance matrix show illustrated potential dataset record game brain inspire classical paradigm e probable single classification score switch begin previous repetition level repetition motivate signal filter th hz filter name filter method spatially filter factor aggregate build use method name stage size classified lda selection regular linear discriminant reduce epoch frequency laboratory pz hz subject compose use offline canonical test paradigm calibrate record area report subject ht performance difference vs contrary pair vs overall usually evaluate trial result converge need reach efficiency contrary reduce fast spend calibration phase accuracy repetition correct potential occur always stimulus delay hardware software well stress classification method delay delay ms performances auc zero subject delay effect test delay draw ms loss selection sensitive delay performance super average p use experiment show cross subject pair vs particularly matrix experiment come source kind come show calibration intra estimate covariance performance subject accord leave l cs cs cs
walk represent agent go adjacent node remain sufficiently mobile choose neighboring connect irreducible vertex add one index home elsewhere control small chance return home base node ensure h run target motion walk sequence cumulative tracking target cost edge short path current normalize state steady plot versus bar evolution versus show minus stationary grow cost strategy good baseline stationary run baseline regret adaptive thus give h initialize start total cost minus see policy realization sequence combine aspect control online theory several mdps mdps cost regret ergodic stationary policy yu et online mdps believe regret space et achieve involve efficiency open address attain online mdps mdps bandit setting learn current cost realistic online mdps state construct regret whether promise far apparent duality control special equation certain dynamic state cf set correspond plan introduction lyapunov criterion ergodicity entry f nx e nx irreducible state right eigenvector fp uniqueness solve irreducible strictly well irreducible frobenius say exist unique show uniqueness essentially function inductive markov proceeding substituting turn imply x xt markov property w prove immediately idea programming map express unchanged add show fix assume item prove proposition guarantee term g establish pick explicitly complete show nb xx e hold begin measure calculation hoeffding purpose state substitute involve prove follow strategy proposition proposition always know fp side last equality fp j fp set reach passive dynamic mh mx p bind proposition second prove proposition rearrange get form expectation follow hoeffding write simplifying use due rgb rgb corollary problem involve perform space action action kullback leibl agent next aspect fact learn construction computationally efficient strategy mild along simulate process kl control markov sequential decision make dynamic environment time agent observe system interest choose system possibly vary admissible pair policy basic assumed function transition probability offline forward effect past action practical degree advance neither reinforcement rl learning variant learn policy online rl operate stochastically expect environmental need ensure agent eventually framework seminal widely sequential effect model cost step reveal minimize incur single action contrast mdps necessarily backward looking incur past observe bandit construct minimize strategy agent cost reveal unbiased full fed strategy minimize regret information reader recent discuss al combine mdp framework mdps mdp observe current choose fix like framework function reveal take minimize relative horizon interest aspect policy brief statement idea later notation motivated machine artificial intelligence merely memory emphasis desirable formulation recently transition feedback law resp kernel underlie one deviation action fix default passive reader paper graphical model mdp action simplex markov system transition govern probability law cost consist cost give prescribe corresponding motivate situation implement free low actually desirable active perturbation prescribe attempt balance tendency cost strong inspire leibl divergence next prescribe property automatically online version detailed state arbitrarily cost determine stationary policy since usual mdps mdp state state feedback law cardinality mapping range subset simplex account cost case state state control represent map law contrast agent choose finite simplex freedom choose measure quantify leibl free external control kullback widely desirable secondly purpose control shape joint relevant system law correspond dynamical derivation controller programming similar nominal canonical bayesian filter variational entail space motivate sort track multiple passive specifie motion target quantify target location target possibility target cost attempt track passive motion rapidly visit prior e target tendency exploration tendency potentially target exploitation another example set brain interface position device passive dynamic natural dynamic absence assume prescribed minimum device execute intend want intend individual cost deviation run well nominal dynamic potentially include rational etc tendency operate nominal mode offline circumstance offer meaningful class boundedness agent ergodicity passive moreover computationally divide increase length apply average cost function precede take yu mdps action space advantage time horizon comment far sequel wide simplex subset hence exist applicable extend regularity yu underlie satisfie uniform ergodicity need et strong significant simultaneous exponentially markov chain space law verify determine mdp ergodicity verify automatically ergodicity function correspond recurrent possibly e matrix denote x ergodicity show supremum consider agent perform walk environment proceed l draw select knowledge incur cost incur agent suitable agent collection mapping f knowledge function cost q define regret gap could achieve use walk lack sequence regret markov transition make follow every yu chain adopt standard terminology consistent w law process certainly complete may outperform stationary stationary indeed truly online strategy alternatively interpret consistency horizon achievable term achievable cost reveal time passive passive irreducible former latter great ergodicity ergodicity coefficient everywhere equivalent frequently ergodic impose satisfie exist point actually et show make ergodicity kernel every policy ready consist r contraction precede construction strategy mdps section overview several recall general mdps action feedback r chain control state construction optimal equation q h function paper process solve describe informally simplex euclidean topology transition correspondence policy passive specify absence shorthand average eq average transition stay prescribe form side form see obviously quantity zero uniquely relative write multiplicative construct policy compute instance solve due boltzmann boltzmann various context physics deviation involve gibbs affine term term indeed minimize hand sequel cost whenever etc policy list explicitly assumption irreducible invariant f relative solution additive solve fix subset cone relative function subset exist smoothly exist map basic steady optimality similar yu et behind partition contiguous phase duration policy match reveal precede phase steady within yet short policy use successive phase phase phase duration give need growth phase phase comment form h mp mx mx end policy throughout phase evolution induce describe frobenius problem begin phase obtain determine stationary follow solve algorithm compute radius nonnegative irreducible perform outperform experimental general major step notion steady
entry kernel provide inner transform include bandwidth turn derivation gram center counterpart problem scope center keep two machine pattern use fold formulation seek axis well variance project lagrangian multiplier axis eigenvector variance eigenvalue I account variation note total datum get lie mean allow onto represent eigenvector I normalization scale along axis fix scale estimation apply mathematical machine learn enyi form analysis detail probability sample estimator q entry expression enyi therefore expression composition motivation eigenvector small term contribute pca eigenvalue contribute emphasize center quadratic study detail issue center way one mode singular center svd reveal gram matrix hard take carry outer turn approach inner scope complete extension center gram product obtain center examine need link center projection projection onto span te complement projection interested projection eq consider onto subspace center give property use gram counterpart equality reveal center subtract column add center understand trace measure due center counterpart correspond trace verify verify straightforward next explore eigenvalue eigenvalue sum proceeding central literature theorem let singular gram counterpart apply separation dr conclude gram proportion divide trace center eigenvalue behave coarse low state theorem theory eigenvalue increase also diagonal entry direct separately connection decomposition gram spectral decomposition product orthonormal show eigenvector eigenvector eigenvalue next apply eigenvalue low I large describe equality due apply observe characterization beyond relation trace latter equality bound furthermore worth state term block far gram follow zero eigenvalue equality zero eigenvalue dual j matrix center wise j j j supremum inequality combine theorem gram constraint one drive normalization box eigenvector relevant principal component analysis axis get axis axis axis examine outer matrix covariance moment dd expression eigenvector satisfy section provide essential eigenvector I hand simplify since correspond equality eigenvector get build entropy easy illustrate theorem simplify therefore diagonal entry apply analogy matrix consequence state associate first investigate consider proof large arbitrary replace side simplify denominator upper cauchy schwarz combine result conclude immediate result cosine angle eigenvector straightforward product namely give relation eigenvector large center counterpart non center theorem eigenvector obtain counterpart significantly simple likewise simple property comprehensive conventional issue scope pca datum investigate weighted consider derive neighbor reduce centroid detail machine weight data become map orthogonal need get analysis analysis long raise orthogonality projection proceeding relation light follow substitute expression become become unchanged weighted expression easy verify eigenvector non eigenvalue matrix complicated due expression matrix weighted mean several special turn section great machine variation optimization genetic machine method lose generality tn presents see hard problem solution investigation evolutionary provide elegant free principle adaptation distribution relevant region solution covariance latter promise fitness progress update matrix zero gaussian present insight firstly trace derivation covariance easily step update simplify angle diversity multidimensional seek preserve pairwise distance dissimilarity expand get inner equivalently entry column vector entry respect translation inner center double center relevant axis describe sample center gram positive definite next study statement paper thorough description conditionally positive investigate worth positive bias associate issue mathematical derive analogy diagonal provide new insight former apply show show variability I give end inequality one latter tight decomposition valid give describe large eigenvalue consider retain define gram conclude interesting completes describe j former normalization projection axis scale feature approach extend establish easily verify repository extensively recognition since seminal divide attribute gram derive show illustrate datum center datum shape use parameter kernel center center contour feature row principal relate mean obtain center experiment c c c c cumulative great gaussian shaped c c center center shaped raw center bridge gap center center explore pca gram center gram nonparametric several beyond conventional include embed address impact function center impact center input oppose center space connection theory receive engineering degree engineering ph security france associate system laboratory technology france interest analysis representation machine learn wireless sensor network paper award theorem france recognition rely empirical moment principal analysis recently researcher work kernel theoretic even center order bridge gap design machine machine conduct product center center datum several explore outer provide extension beyond conventional centering shift rank update multidimensional illustrate relevance gram machine recognition analysis machine singular svd seek axis give component prominent feature axis large amount variance multidimensional partial pls cca fisher discriminant elegant nonlinear rely concept initially introduce fold regression write substitute inner without significant cost perform matrix space property reveal version pls survey machines cca origin algorithmic center center way deal center moment centroid available relate second central issue center propose
gd gd sum correctly lag np lag scale double range lag scale least double point gradient store np long iteration epoch epoch else I lag lag add entry date amount lag lag k lag np pt double frame university team sup paris france sup paris france call spirit sag sdca recently sag composite proximal use sdca strongly strong convexity effectiveness remarkably advance provably fast expectation machine problem requirement strong convexity likewise satisfied form strongly proximal operation incremental contribution incremental prove rate strongly sag sdca rate composite additionally sag method applicable modification establish convergence fast incremental stem start derivative derivative structure inspired sag discuss make take store unchanged proximal size composite requirement hold expense geometric exclude prove supplementary step strong give theorem size incremental convex via small additional avoid explore relationship fast incremental unified brief property consider figure composite list times one sag sdca convex prox storage simple sc summary property question experimentally apply amount trick sag gd decrease convergence unlike gradient reference therein sgd constant get rate present call mention sag reduce relate reduction one estimator convex x yx zero sag past store sag update non bias explain sag proximal update able use sag relate bias every appear inside outer start outer iteration sag pick whereas update gd number iteration henceforth make trade gradient usage sag predictor gradient vector class prefer method iteration loop near practical tune prior non composite intermediate quantity nf unchanged quantity explicitly store recover describe size simplify discussion introduce interpret update consider change expectation identical advantage operator require store prove proximal big pass access order access empirical speed storage sdca sdca transformation sdca work closely pick index compute kf zhang require operation simply use search conjugate simple primal equivalent line search sdca variant line evaluate datum instead full store weighting storage requirement sdca class store heuristic average use sag state slow derivative update suggest sag ensure update practice strong quadratic amongst evenly implementation step scale scale standard trick problem supplementary code implementation expectation respect condition start constant equation lyapunov lyapunov expand quadratic value shall k combine fraction yield size note square verify together ensure constant set round ensure expectation constant explicitly expression index optimality effectiveness test mnist test suggest fast test epoch basis expensive per epoch basis evaluation epoch double sdca sag slow begin sag adaptive confirm discuss duality sdca bind include convexity low k x convexity f apparent compare k vector instead definition proximal sdca experimentally proximal say sdca among slow disadvantage sdca handle although use objective proximal disadvantage sdca additional let strongly continuous gradient gx substitute let forward fy f follow argument tight key trick convex choice available use similar theorem lyapunov different define k x true property begin use prove quantity property full
output less since report worker put payment effort never worker function value observe expert modify regularize ridge regression square include depend two mechanism know possibly minimization ii also unknown expect beneficial point advance still optimally worker problem accommodate mild modification separate bias variance term matter estimate minimization use unbiased estimator payment extra worker reason behavior know term utility depend worker reason although objective payment mechanism accommodate many variant modify minimization reflect unique dominant strategy achieve list modify problem accordingly generalization payment accommodate sum worker square budget payment update effort budget another replace total payment increase square plus impose modify square effort exceed function combine behave modify mit mit computer berkeley berkeley remark assumption rgb propose estimator quality low estimation range include also generalize include estimation subject besides concrete problem mechanism design worker label reality essence business science entity crowdsource popular phenomenon unlike develop related crowdsourcing recently treat subsection interaction rational agent pursuit solve item draw prior sophisticated keep area come back datum aspect reality employ context one quality crucially poor quality worker effort level effort worker center worker lack worker knowledge participant include create mechanism protocol contract worker worker act effort rescale effort worker nonnegative otherwise th payment protocol contract value never must depend know quality worker mechanism may seem optimistic go follow worker worker regression worker determined expectation define opt opt economic opt bind cost opt precise optimum small minus effort effort mechanism light seem quite surprisingly linear example polynomial construct way minimize extract important consideration end assumption concept discussion turn extremely worker dominant optimize payment minus else course design space mechanism design attain opt mechanism worker depend datum seem mechanism work entirely relate oppose mathematical justification root create accuracy worker statistical optimally mechanism include regression advance assign worker even broad include ridge extension besides accommodate objective loss plus go beyond mechanism crowdsource pay crowdsource optimum crowdsource contract expert crowd expert perform scheduling mechanism use crowd accord paper keep assignment treat bandit budget optimal reward optimally participant address concern treat regret use crowdsource mechanism crowdsource worker pay create competition worker enhance nature design individually rational agent produce bias outcome subset datum paper proper scoring rule agent ask level effort contrast final decide design mechanism want maximum effort agent consist effort allocate increase label allocate task besides fact care reward use agent whose signal report effort look except quality sample decide mechanism equilibrium could worker surprising contract chapter contract worker effort contract worker effort observable worker worker utility formulate worker effort appropriate design worker observable two technique difference apply apply worker payment effort observable still instead mean effort employ worker take randomness output ie produce worker payment able effort worker achieve need expect point decide assign discuss yield much apply mean stick development technique minimal modification objective accommodate minimize square subject budget minimize sum square arbitrary discuss eliminate use objective discuss able predict effort worker result payment decision decision worker worker familiar comprising subset worker effort eventually worker outcome able evaluate agent behave game concept game whose close behavior rational player prominent equilibrium nash equilibria randomize guarantee dominant equilibrium comprise worker payment unique dominant strategy equilibrium iff expectation everything word matter effort choose unique effort level worker dominant equilibrium game fairly trivial agent decide induce equilibrium general rare design satisfie constraint objective finally note prediction worker worker capture follow comprise payment effort satisfie restrict induce game worker dominant satisfie might requirement solution requirement individual contribution establish induce dominant equilibrium later several familiar iff well regression regression behave accord optimal behaved definition relax necessary discuss remove optimally solve behave well dominant strategy satisfy strategy equilibrium optimal achieve objective worker effort worker equilibrium induce dominant strategy evaluate individual individual worker combine would achieve effort property always induce dominant individual objective behavior worker achieve even main unique satisfie value equal quantity argue choose assign payment worker worker payment worker choose constant induce unique equality worker worker effort ie estimator worker rational maximize expect payment minus effort level response find maximization ensure effort decreasing make sure unique game worker regardless
hundred md home collect configuration gb molecular activation mechanism analyze reversible atom position experimentally determine inactive hmms achieve interpretability choose monitoring relaxation reversible pathway mechanism onto structural transformation inactive unfold highlight rotation highlight interaction portion overall protein fluctuation freedom largely process degree grey simplicity dataset find activation pathway field drug design effect protein common entire behavior tumor well function intermediate likely unique target protein b state project loop red detail activation pathway framework analyze dataset propose reversible hmms identification state interpretability physical switch without theoretical discretization integral formally control transfer enable quantification error theoretical guarantee yet exist reversible long markovian reason believe regularize represent analyze md challenge hyperparameter aspect reduce required amount manual adapt bayesian unsupervise linear may facilitate goal massive reduce complex thousand degree statistical hmm turn raw molecular function protein extract biology rational drug thank md trajectories r support acknowledge gm machine modeling protein approach hide protein via molecular motivated massive necessity provide biology drug criterion physical contrast improve implement apply home dynamic activation mechanism protein challenge relevant disease cancer characterization pathway protein fold energy surface protein biology molecular problem genome phenotype furthermore dynamic protein design molecular md atomic resolution forward potential quantum reproduce moderately sized million freedom integrated burden md simulation central challenge achieve independent purpose accelerate home computer utilize cycle google production science dataset major contrast problem goal merely md datasets scientific insight protein model physics physical paradigm chemical understand configuration often fluctuation dominant understand move state paradigm motivate base location unknown latent hide hmms thus mechanic symmetry respect version law call equal essential detailed balance probabilistic knowledge protein study describe long see therein substantial protein via shift together motivate term deviation amongst uninformative freedom furthermore pairwise transition state differ along reduce formulation reversible hmm introduction scalable fit standard framework md physical interpretability follow describe associated learn potential protein indicate direction machine protein discover protein protein fundamentally system offer insight concern computational study protein md primarily quantitative approach include movie protein structural number pre specify degree characterize critical biological quantitative method capture rich temporal dynamic model chain cluster md fully hmms recently hmms emission distribution employ notable lack characterize manual purely state lack introduction order inefficient sized regularize reversible generative multivariate continuous series series simulation ji converse ij ik ik ji ji transition derivative term bfgs aic bic selection criterion discretization support recall choose become correlate increase discard criterion propagation state eigenvector stochastic eq eigenvector collective dynamic process equation describe central molecular modeling perspective visible protein perturbation relaxation enough long change discretization gpu cpu across spend exp architecture parallelism gpu fine grain array forward respectively fully utilize gpu parallelism update specialized write kernel trajectory matrix accumulate rest speedup optimize standard implementation intel bridge achieve multiple hmm dynamic hmm unlike feature hmm diffusion govern brownian differential reduce diffusion constant process double well euler produce ten simulation trajectory two fusion state surface learn display sensitivity discretization accurate relaxation long unable accurately lag time succeed identify identify state primarily loop region axis fail state post process show protein protein intersection pathway protein human link would obtain dataset md million protein compose perform extract configuration hmms choose monitor relaxation penalty hmms correctly ease comparison
prove prior lead consistency reduce regression consistency lasso expression completion prior spirit spike selection section lead describe eq generate summarize finally calculation bit principle array common gamma row inverse derive joint therefore gibbs appropriate gibbs large variable hasting high dimensional make quickly bayes expensive bad large certain distribution optimality work iteratively update skip elementary simulation prior describe first optimal factor necessarily whose denote iteratively formula entry th connection bayesian certain penalty posteriori mode recover provide insight prior correspond popular easy interpret easy map penalization nuclear see page rewrite link penalization close essential number hand penalization gamma case integrate respect lin group map prior proof contrary scale problem hand give nice gamma column toy generate square ie corrupt study grow second sample quick illustrate figure take lag case iteration remove period gamma report hyperparameter prior fix gamma gamma four prior result line consistency grow result explain section discrete hyperparameter discrete stable result bad rmse well reach gamma distribution prior automatically note poor slow heavy bayesian available http dataset challenge vb instead vb netflix challenge model rmse quite end approximation hyperparameter rmse gs vb gamma vb inverse inverse inverse discrete gamma vb consume test review propose two gamma discrete real life tensor spirit proposition come end acknowledgement would like comment thm thm incomplete recently several recommender netflix behaviour behaviour gamma decomposition conjugate conjugacy interestingly nuclear prior classical netflix netflix science statistical community increase movie column become netflix reasonably pattern movie problem http first recommendation norm preferred computationally ground result ensure noisy observation recovery exact recent general trace popular multi task derive reconstruction basically quadratic well bayesian context prior learn completion computational rely dataset netflix know rating must keep mind application observe entry long completion netflix movie less sensible define must posterior
one cell memory ignore cell unit unit learn lstm per time relatively memory cell store temporal computationally alternative architecture architecture learn lstm architecture connect layer connect cell output layer layer non output n r I unit recurrent recurrent note effectively equivalent projection layer unit compute unit gate gate sigmoid gate gate gate activation output activation wise cell lstm recurrent recurrent recurrent unit height dash height em minimum width thick circle thick thick scale name xshift name input name name unit name node cell bend auto west node dot cell bend center east cell cell west xshift yshift recurrent xshift projection right xshift output cm north north cm leave west projection west east xshift west east leave west east near north east cm north east north output east dot cell block mm memory block implement architecture core gpu cpu machines network bottleneck operation eigen library implementation operation activation parallelization technique gradient multi operate computational multiplication rather time effectively sequence batch process backpropagation update use step e propagate activation propagate original propagate activation activation frame error propagate gradient entropy criterion error finally parameter weight accumulate state subsequence different shorter could reach next new dnn rnn architecture vocabulary million hour google dataset represent frame compute every cd network map ci initialize try architecture result phone acoustic hold frame system rate report million sgd minibatch graphics gpu softmax represent phone state stack window frame either frame denote lstm partition propagate backward propagate truncate rnns rnns recurrent unit tangent activation cell unit sigmoid forget gate recurrent activation rnn log energy window frames future frames decision frame frame ht frame name contain architecture state cell rnns recurrent rnn dnn configuration name layer low projection layer evaluate rnn significantly rnns training limit activation lstm rnns converge fast project architecture well lstm rnn architecture projection generally lstm recurrent layer ht converge speech lstm recognition state embed mobile phone relatively output figure architecture obtain accuracy depth compare application large vocabulary speech scalability effective use lstm architecture introduce projection recurrent recurrent flexibility architecture improve lstm architecture performance recognition lstm large gpu cpu implementation long memory lstm recurrent address conventional rnn unlike feedforward neural rnns cyclic model use task label acoustic contrast rnns recognition phone scale task lstm architecture make acoustic vocabulary recognition dnn parameter configuration lstm quickly give speech recognition relatively lstm recurrent neural rnn speech feedforward rnns cycle activation network make store provide contrast fix contextual windows input rnn history capability rnn modeling rnn direction make current input labeling recognition
variation identify often body method illustrate box problematic obviously detection guess location unknown common stage informative hard training identify initial annotation higher informative training key accuracy final issue characteristic automatically discover discriminative correspond wrong background e water occur image configuration particular configuration object occurrence patch positively label image patch discriminative covering discover patch part cover formulation cover density would strongly cover formulate independence submodular subject case part effectively take viewpoint demonstrate configuration observe combination patch produce accurate occur interest box informative negative short contribution frequent discriminative visual inclusion discriminative detect method tight bounding box annotation object reduce train detector binary presence effort prominent object image recent challenging multiple intra category variation achieve rarely negative convolutional discriminative patch contrary contain full merely piece full end mutually object far select negative object discover high object mid level patch foreground grouping e patch contour texture recent weakly supervise discover discriminative relate formulate submodular formulation alternative discriminative less scalable use geometric occur visual pattern improve visual recognition performance occur represent star shape among inspire supervision feature dataset work relate object phrase truth annotation full box annotation discriminative positively configuration address patch discriminative occur efficient configuration patch easily retain configuration identify similar positively box selective regardless label remain one discard neighborhood within image identify small representative construct copy near near neighbor label cover bipartite monotone submodular aim configuration must informative patch merely redundant frequently occur often modification treat identical submodular covering case candidate densely identical patch thereby configuration independence may pick together redundant cover ever identify whose neighborhood overlap overlap patch identical diversity phrase optimization problem constraint express neighborhood greedy bb disjoint first visit high list immediate greedy bad intersection axiom exchange insight intersection ground set k ce kk color may node say adjacent k check set patch representative patch different occur head person top visible consist relative practice preference maximize occurrence count configuration amount write find frequent inspire supervision least occur patch operation translation viewpoint bin b I I fall bin share location via ji position edge occur edge characteristic configuration enough occurrence frequent also determine localization small configuration discover frequent configuration well localization estimate hard negative let configuration part object foreground include foreground overlap foreground rectangular region box negative specifically rectangular bounding box foreground hard negative l f foreground result negative foreground introduce undesirable false negative detector adjust coincide foreground finally rectangular overlap foreground box negative uninformative overlap cover foreground select negative region configuration likely foreground object discover lead count compare stage frequent configuration discover patch find positive discover detector foreground derive configuration correspond region detector selective search retain detector section discover configuration impact discover hard negative employ fc feature proposal discriminative discovery transform shift vertical px px px px visually pair loose handle detail space figure qualitatively illustrate configuration box box combination upper frame failure object configuration bottom protocol use art weakly level annotation supervision annotation table baseline improve detection majority class note improvement person
connect unit connect ram roughly reach attention digit translation invariant experiment search big object aspect classify presence clutter operate full clutter learn invariant attention learn clutter focus image experiment task call translate mnist place mnist digit add mnist digit random classify translate error fc layer fc layer layer ram ram scales ram ram ram core unit perform suitable hyper train million good video attention track ball demonstrate ability learn policy introduce attention neural internal focus control signal environment unified architecture method appeal ram control ignore clutter image center ram architecture comparable classification extension terminate make classification take confident allow object fix extra action train policy procedure encourage video google google com convolutional linearly extract video adaptively select select high convolutional neural amount control independently differentiable learn evaluate convolutional neural network explicit neural architecture recently great success challenge classification object come typically currently reduce object second run gpu follow slide paradigm literature classifier object box independently thousand window computation come map entire least one scene human focus attention information representation scene future decision focus resource scene pixel process substantially object interest place irrelevant visual environment clutter outside naturally ignore role cognitive scene bottom play specific see review novel attention task consider scene general video module play recurrent location video build scene bounding box past number amount control independently pixel train maximize decision backpropagation train gradient learn effective strategy look result attention clutter image receive attention vision instance dedicate slide window focus primarily reduce cascade e e g propose window likely contain substantial may approach add cnn root window exploit past processing way class approach computer detector approach process salient identify contrast detector capture human typically propertie ignore semantic task observer model framework setup restrictive work attempt rnn integrate visual decide sequential rely architecture interact environment attention decision direct interact visual observe e full extract narrow band agent control affect true integrate determine act receive scalar execute delayed agent maximize sum reward diverse detection static image play visible game engine sensor operate frame frame game reflect detection static state environment environmental action would decision reflect resolution image extract location map independent another combine produce core take action internal state action recurrent fig sensor choose sensor receive observation environment agent rather focus band bandwidth sensor encode region resolution far refer resolution vector extract history past agent instrumental act sensor neural external feature perform via environment state choose stochastically parameterize formulate softmax dynamic formulation environment agent receive signal reward reward distant less delay otherwise rl process unobserve need map subject case outline choice agent network interact agent combination dynamic playing induce interaction maximize ps ps ps involve unknown dynamic technique sequence episode rule running sample sequence adjust log produce gradient define computed backpropagation provide unbiased estimate gradient reward obtain action expectation log action small type baseline reduce well unknown priori image total episode try good bad know detection output optimize correct achieve maximize truth observation gradient core network evaluate several describe common center location successive twice corner layer e nonlinearity network train location component network core core environment core unit attention softmax select search reward classify reward
distribution often ignore gibbs general therein lie sensible inefficient particularly true allocation recommendation hundred thousand em issue optimization dependence step augmentation marginal simple gibbs replicate state additional state marginal cause approach optima make temperature compete competitive fast symbolic hold problem define joint tie latent eq power original optima optimum peak often subscript write perhaps obvious add considerable parallelism factor method us order magnitude speedup sampler limitation sampler discuss factor hardware present experimental expectation graphical compute expectation compute work likelihood expectation require optimize equation em locally attempt kl estimate usually unlike em estimate common factor approximate factor simplifie make constraint vb nevertheless factor describe gibbs sampler perform joint slice sampler gibbs sampler difficult slow especially dimension slow variance hybrid expect em suffer may likelihood propagation class variational message conjugate family vb coordinate factor although estimation independently infer aggregate initialize z sampler standard sampler group ignore normalizing product conditional product product member represent normalizing imply closed adjust anneal lda sample rather taking multinomial among conditional parallelism sample replace fully capture sample fast completely integer long code increment lda word source considerable independent approach similar coordinate factor use distribution lda process lda parameter previous variational lda line dominant count step line eliminate evaluate factor implement approach vb acceleration source system system al vb implementation lda et collapse sampling lda parallel al lda implementation art gpu parallel fastest lda date fast implementation system evaluate pc equip single core cpu intel gpu gpu cluster gpu come gpu report gpu yahoo york word token corpus million k token million word convergence gibbs marginally beyond start beyond flat mini number per show validate pass pass gibbs vb online vb dataset vb converge pass pass vb converge pass three within usually take pass dataset per reach pass begin converge gibbs move runtime different fix gs runtime system benchmark time c runtime illustrate sampler take cpu hour gpu implementation improvement vb second performance art implementation use million article machine overall construct repeat news article run iteration find news dataset comparable take k gpu process k simply second thus gpu accelerate gs sample give benefit parallelism study schedule logarithmic scheduling max max tm max max total sample size configuration identify anneal fast number c runtime iteration paper hardware accelerate estimation accelerate pass introduce parallelism show hardware gpu accelerate gs sequential fast code com gs applicable explore inference factor conjunction full approximation
phenomenon whenever cause importance construct wise boundedness publish theorem thm hypothesis moment particle weight ensure expectation particle converge filtering extend derive require boundedness boundedness moment boundedness convergence boundedness fourth moment distribution moment hold boundedness leave use also perform unbounded importance moment approximate filter carlo approximate measure estimation inference system measurement modeling dynamic measurement density important particle exist g reference therein convergence unnormalize wise particle filter wise boundedness assumption particle derive moment importance modify applicable square importance bound spirit assumption main filter construction filter construction equation borel probability bayesian bayesian regular require equation model approximate particle approximate solution mean empirical convergence importance boundedness qx I unnormalize dirac delta iw aim induction combine assume assumption q need bound second combine tn use complete proof lemma n together imply generalize assumption unnormalize qx hold lemma assumption lemma remark proceed inequality assumption proceed measure assumption markov borel argument cox priori reflect brownian brownian intensity density respect lebesgue measure respect require include select gamma parameter importance particle importance lebesgue eventually numerator nonzero accord particle guarantee empirical use combine recall argument integer even negative square borel measure converge cox process right
lagrange time meet tradeoff iteratively regard ranking score code sparse regard dictionary popular meanwhile database ranking score important irrelevant yes explore paper joint sparse explore sparse bridge code neighborhood sparse rank construct objective sparse dictionary ranking learn ranking reconstruct combination element linear combination coefficient zero coefficient sparse code dictionary code meanwhile near base content retrieval rank accord refer ranking distribution point performance neighbor search firstly code code rank code use rank ask question relationship yes explore boost ranking code code jointly explore sparse coding however consider sparse rank distribution approximate code consider reconstruction error unified function code dictionary score code rank score internal explore algorithm optimize regard sparse code dictionary alternative introduce unified assume data th one query query learn rank rank top rank ranking score point nf th code learn sparse function learn rank coding aim learn reconstruct dl code minimization error point norm sparsity measure use code local sparse neighboring approximate score learn propose score complex local function tradeoff please rank score force close problem tradeoff please score predictor regularize code ranking connection solve optimization score code role repeat ranking fix gs rewrite ik kf local code please compose objective point objective minimize minimize regard regularization consider sum objective point rewrite th rewrite diag
ex berkeley edu computer california berkeley rp approximation program approximate essential since program quite cubic addition random projection useful usage optimization ratio constraint broad random hadamard transform project tangent cone constraint dimension illustrate consequence include unconstrained vector implication sensitive connection denoise compressed sensing optimize fundamental mathematic program solve prohibitive many may prohibitive dimension million concern sophisticated cone program program rigorous approximate program scheme perform projection constraint cone program program case interesting statistical problem formulate side belong dimensional include matrix set combinatorial shrinkage relaxation principle deriving method extensively decade ambient dimension context program solve g therein result generalize provide broad program addition analytic exploit analytical banach sketch probabilistic unified sequel sensing case program addition storage useful modern financial record medical concern store small solving preserve interesting problem trade set mutual set statistical optimization organize precise main corollary concrete close section devote appear conference international problem turn goal simpler via problem natural geometric tangent cone denote optimality problem feasible decrease move belong cone transform cone transform define banach main relation vector satisfy I matrix matrix bernoulli rescale sphere say sub projection universal example width scale statistical freedom project preserve sensitive program possible sketch retain mutual rescale entry quantity privacy sensitive discuss generic draw random many guarantee per vanish problem type vanish symbol straightforward combination show condition mutual symbol eq substituting claim disadvantage vector multiplication multiplication main apply matrix order randomize begin orthonormal ij hadamard matrix multiplication basis random matrix rademacher base choose hadamard fourier product scale linearly involve corollary many gaussian width rademacher randomize orthonormal system draw orthonormal size approximate dimension general pre example corollary follow potentially offset form product main convex constraint consequence theorem way simple lead square provide manner accuracy quantity small corollary confirm intuition guarantee least unconstraine corollary estimate hold paper overview qr approximate least square substantially qr require consequence theorem sub result refined argument investigate corollary simulation problem fix form problem generate datum datum give hadamard project perform randomly predict scale trial consistent regardless projection approximation become geometry tangent statistic quadratic unique number zero program unique cardinality eigenvalue sketch low solution optimal improve establish dimension order requirement linearly constrain solve another popular use solution iteration algorithm obtain let cone take sign triangle upper tail imply applicable set turning lower involve low corollary application third follow specialized standard size form curve ft solve radius hadamard matrix predict approximation ratio control increase plot qualitative note corollary one simply correspond function perform wise operation noiseless approximation guarantee recovery compressed sense realistic sensing imply well moreover closely precise summarize denoise projection dimension recovery conclusion hold sketch q version rademacher generalize randomize atomic type provide atomic classification represent collection label specify vector label formulation serve least amenable let b paper take correspond vector svm case coefficient lie machine version lead scale sketch machine omit corollary linearly conclusion ft trial place equal program obtain support either rademacher randomize repeat perform trial bundle curve sketch involve simplex portfolio return subject return let semidefinite typically give pair allocation give factorize whenever expect return constraint tangent cone portfolio turn operator dd general estimation primary relatively constraint typically norm illustrative wish dimension frobenius non negative norm obtained long accordingly replace program dimension iy rank solution provide sketch low rank condition bound probability sketch bound guarantee likely substantial version however leave b nuclear eq block duality nuclear result norm matrix follow example enforce group notion sparsity collection subset index note special group reduce usual norm generally group enforce group analogy group restrict size bound solution generalization ordinary indeed reduce similar maxima upper detail turn randomized system high depend central vector arbitrary quantity fix randomized significance ratio triangle q consequently use convex optimality add optimality right appropriately basic need bind long change universal prove et let sub universal subset theorem proposition particular inequality universal involve shorthand q eq apply bind imply three term calculation give denote orthogonal write consequently v follow substitute establish eq supremum decomposition set put piece rescaling appropriately begin state randomize define width universal least universal give complete lemma c claim rescale suitable immediate introduce q proposition unit inclusion thereby base inspection put together piece q projection along numerical fix diagonal randomness h event rademacher complement turn truncation level
cone element side exist prove reverse converge obviously kn sn paper tangent derive definition normalization use verification mention use definition theorem follow existence analytic value path easily modify prove analytic curve put analytic curve involve probably since exploit formula imply formula singular reverse state normal cone imply cone n also know structure tangent cone calculate turn moreover carry fact generate fx fx kk orthogonality estimate square rank large singular value nonzero remarkable priori statement critical semidefinite illustration relative minimizer make tell impossible since finish matrix representation involve sparse second rank huge never form rank well unlikely event fix feasible elegant backtracking projection hence less gradient completion setup matlab choose six ghz cores gb index generate generate normal kn least uniformly start guess choose approximation start start zero exact line smooth relative error visible inferior latter plot think plot perhaps explain fast entry relative error sample unable figure th confirm approach search elegant riemannian instance synthesis increase strategy grow inferior cg miss solid full error index nk explicitly point consideration difficulty arise unbounded curvature optimization real grow treat dynamical important subspace rank tucker hierarchical tucker rank take intersection low otherwise nothing also generality hold n accumulation pick prove limit put hold induction inequality project use matrix closure gradient relate cone project arise unbounded pointwise analytic k know result estimate curvature fact justification assume point method rank riemannian descent concern low thus know riemannian simple projection take tangent plane become tool low several equation lyapunov completion newton search search sufficiently alternative project discretized flow satisfy dirac variational integration ode euler method size also relate dynamical admit lyapunov analysis manifold ambient manifold break boundary might happen need lead allow size serious difficult priori statement regularization certainly convenient analyze closure satisfied principal search method direction tangent instance explicitly project easy corollary need map tangent choice aim attract optimization seem proved n sequence sequence possess satisfie possible strong one critical information rate advance notable class real analytic use wolfe step selection flow consider descent riemannian manifold local search like convergence regularity discrete projected integrate ode existence ensure convergence via second taylor would unnecessary term gets iterate bind constant one limit plan via involved formally reason establish lie order estimate insight repeat necessary impossible regular considered detect idea summarize singular likely generate real differ search thereby establish convergence successively turn tangent theoretical explanation outline part search analytic highlight result descent method line algorithm select backtracking notion tailor direction need identity sequence neighborhood analytic must critical short derivation compare tangent simple priori critical section consider concrete direction matrix result constant provable rate black box limitation work completion cg unless something else euclidean optimality problem far close cone since general metric projection may uniquely denote always cone equal necessary optimality relative optimality paper complement everything fundamentally assume cluster satisfie project hold original unconstraine parametrization basically real analytic analytic open open induced least neighborhood terminology exist subset derivative analytic map derivative cone without assume prof since may depend know convergence statement generic happen version positive hessian second hessian definite clear concrete line shall consider hessian likely treat constructive remain meta intend shorthand enough assumption imply convergence point limit replace assumption prove technical satisfied gradient add estimate behind manifold context norm distance reduce complexity minimization case low rank change besides give actually simple sufficient sense e unfortunately full rank later force smoothness know project gradient flow manifold mean plane property map respect however smooth manifold make tangent cone call implication algebraic variety exist arc implie make cone decrease much importance analyze search upper impose serious practically always remark define cf proposition mind guarantee search call angle satisfy equivalent euclidean good angle moreover n f pick size small point descent choice subsequent importance principle use backtrack eq hold minimum ratio converge follow eq since continuous adjust chance restriction need calculate formalize proposition choose iterate assume constant exist property follow obviously necessary step impose point assume whole generally estimate apply two assume otherwise x lead subsequence inferior notational convenience rearrange hold may disjoint eq finitely contradiction wolfe linear mean
module partition contiguous duration book compact unitary say gx g k k detail template book matter say group invariant close come specific hypothesis see specifically training video show theorem temporal simulation template consideration consider temporal association like think clutter experiment unconstraine face decide person task differ contain face position visual fold contain video transformation module template face label possible fully thresholde dot product section determine chance pool representation performance clutter present association drop clutter without pca model individual row experiment individual template book background low hierarchical feedforward module signature vector module architecture concern traditional single hierarchy face datum oppose commonly distinguish face face rarely clutter positive high response template activate face become severe face spatial pooling resolution selective template clutter act gate prevent face representation model use present module leave low template second class specific template plane rotation layer temporal association video unified architecture operating domain plane temporal video face theory consider plausible module template book arise naturally consequence face specific resemble recognition hierarchy cell face resemble face use viewpoint explain study purpose prevent investigate face categorization distinguish train video fashion training patch layer template video template generate template video face video video may thing face third speed video frame average complex cell overlap complex cell complex domain depend place video purpose none cell final biased long video video complex equally evenly video simple video complex video simple cc c acc acc acc align sift mrf al align observer far concentrate recognition thought depend thought learn natural video depth categorization angle invariance class specific follow temporal strategy explore apply categorization figure video also per category template frame performance drop pool part think object perform use face amount like regard contribution establish possible minimal supervision representation enough height width network apply image colored video frame randomly central plane rotation horizontal flip single patch frame aspect ratio pyramid scale ratio scale pyramid try level feature filtering size pca project eigenvector x image low scale pyramid pyramid scale pyramid scale layer template e pyramid separately convolutional convolutional network three scale template layer dot cnns dot dot product nonlinear cell pooling result pyramid template pyramid rotation stage pool location plane transformation layer concatenation encode second store template simple face feature scale location normalize dot store layer training template e cell adjacent frame frames cell dot product perform input cell final concatenation response perspective spatial domain domain detail image randomly sample patch rotation patch call single video preserve pyramid ratio pyramid frame layer concatenation face first template use template internet x pooling step dimensionality template term refer try modeling mostly explore cnns pyramid get scale pyramid template pyramid template separately cell convolution pyramid template pyramid rotation pool location rotation angle output video consist template column project template window eigenvector fast dot product reduce dimensional space adopt plausible test versus frame pool pt natural video tend remain tends rapidly effort temporal enable useful representation demonstrate e video visual representation perform computer vision benchmark big million example advance aim understand plausible feedforward hierarchy representation video operation perform normalize dot product pooling
robust noise ij adopt project update ordinary project newly projection compute reconstruct compute top singular svd suffer optimal helpful multiple cauchy pca recover simulated usage real corruption evaluate robustness cauchy pattern magnitude try recover sample entry call corruption magnitude corrupt pattern student pca gaussian pca intrinsic matrix cauchy tune trade large recovery true result pca show corruption row display high third pca pca corruption display reason quickly great rate great work noise become pca perfectly pca increase analysis suitable sparse kind small cauchy comparable third row cauchy small corruption large instance average cauchy suffer student pca similar pca row student nearly pca large second student bad conjecture reason em student thereby face recognition image severe randomly percentage pixel corrupt recognition adopt use pca recover matrix training project basis face face subspace assign near testing corruption define recognize face extend individual individual randomly pixel replace integer take replication normalize unit pca vector tradeoff laplace pca recover rank figure corruption see robust dense noise exceed pca stable laplace pca drop pca drop pca achieve recognition bad laplace pca face image heavily corrupt reconstruct severe laplace result reconstruct recognize reconstruction even original appearance fourth wrong successfully appearance recognize student show cauchy outperform pca gaussian pca possess comparable laplace real theoretical future solver cauchy cs edu principal component pca application machine text mining computer vision pca noise laplace assumption propose cauchy pca simple utilize cauchy derive pca constraint matrix regardless sparse robustness pca robust statistic view present singular experimental simulated demonstrate component analysis relate subspace factorization widely reduction compression image extraction visualization pca pca magnitude effect gaussian quadratic method probabilistic paradigm accord noise item fitness try covariance remove corrupt probabilistic paradigm difficult possible desirable treatment pca mixture factorization facilitate probabilistic technique robust student student heavy tail magnitude suffer new laplace pca dense laplace induce thereby dense avoid drawback distribution try probabilistic student opinion roughly abundance noise limited noise dense neither pca suffice reality factorization illumination optical tracking quite tag image attribute considerable negative popularity low mobile device million video publish share capture capture video contaminate video move noise corrupt contaminate large component pursuit corrupt wise magnitude infeasible application probabilistic cauchy derive unable handle dense formulate entirely corrupted assume corruption entry another pca student distribution observe generate student student infinite vary expectation maximization infer learn parameter empirically noise well pca laplace pca intuition noise compare cauchy value parameterize family distribution transform parameter shifted estimate maximize negative laplace pca general specify laplace respectively curve gaussian cauchy enable aligned location peak motivation put mode give sense heavy away center drop quickly laplace cauchy density heavy tail far reasonably heavy since certain amount thereby cauchy pca naturally possess deal location
predict persistence remarkably achieve curve operate characteristic week student persistence last week build modular framework iteratively rest organize organization feature make present technique present present present finding previously focus circuit raw click stream learner page visit server side object comment store note view infer click stream passive view infer stream example database contain correctness infer click stream include release raw receive include learner scale organized schema result schema design capture thereby utilize standardized schema report scope intensive raw database significantly disk gb gb normalization crucial entire ram enable snapshot schema slice require explanatory express explanatory due basis week module regular modular slice slice week week ht regardless nature take active interaction etc assignment course access course page stop slice exercise illustrate assignment module week attempt definition learner consistently course assignment week learner stop show week ever course learner stop week never never analysis another learner week week course learner week learner range never predict week week week ht predict week predictive label represent many historical lag ahead value diagram careful stop week word stop learner point include stop easy stop illustrate realistic platform could week week end week predict exist discriminative form covariate learner attempt week lead ht treat treat learner decide divide rough surrogate variable choose course specifically divided learner page four learner passive learner name passive view learner learner learner assign chart size learner use build engineering length sake brevity pt height htp stop duration spend length distinct problem number distinct number distinct number per duration per ratio total spend distinct time problem week event max duration duration time spend spend book duration total spend resource terminology attempt could therefore htp covariate response response percentile student week percent maximum week week week student week week student past percent percentage total correct average problem due date week logistic commonly covariate input z shape note range ht range function weights logit function rather arbitrary linear logistic coefficient suit fit covariate covariate predict training example datum training iteratively likelihood random accordingly call represent final step evaluate comprise covariate evaluate logistic apply label datum point decision label confusion thus obtain operating evaluate multiple classifier heat present problem different heat roc predict hard become historical enable prediction week change relatively understand feature useful assess treatment randomize logistic regression hour ht student early maintain every lead combination set outline chapter involve fold train fold dataset put follow determine auc evaluate validation auc roc figures receiver operate lead logistic high experiment predict week collaborative accuracy result auc diagonal represent experiment lead capable week fairly week across model high accuracy passive far result week compute ht deeply try persistence week week practically enable predict student finish week potentially reason content become student early sign intervention model successful capture student persistence remarkably generate auc reach passive include ability reach student stop become give rough student week hold true course content thing firstly remarkably sign persistence secondly student would perform size increase student student student course power include collaborative predictive predictive detail refer user week yield fairly auc student persistent counterpart week instead thereby four perhaps passive student attempt lag week week attempt week auc collaborative week week week equip reasonably prediction significant week achieve high suggest prediction finish course small week accurately train student summarize reflect may persistence randomized methodology briefly four importance summarize finding regression datum service call ever group mit multi optimize attempt find armed bandit search balance fashion confusion validation training run lag since create choose lag combination regression predict lead lag lag file pass manner give good test roc auc auc attain similar hmm model varied include stochastic support lead indicate power predictive note vary self self crowd change lead conclude focus significantly lead combination neighbor great deal size evidence relation identify dropout analysis context distance education relevant literature table list list axis intend purpose axis identify whether student dropout model completion reason excellent study actual course insight record single survey collect module however build contrast take would lag interval historical predicting interval first good knowledge systematically prediction week course excellent study concern could intervention error persistence receiver operate auc metric measuring efficacy error aim provide choose intervention receiver curve categorization capture behavior student behavioral student age financial behavioral self model neither depend behavioral age play especially far allow transfer behavioral common behavioral education use college study student behavioral student resource performance tackle challenge capture student platform argue enable identification attribute knowledge detailed orient variable derive exploit vary summary per minute per process g scoring week invariant time important factored behavioral study variable summary usually aggregated course aspect close course different survey play survey collect specific motivation reference describe student test common among collect student among mistake manual survey student response survey survey ask question fails accurately trace include trace fine grain consider build interpretation student htp contain comprehensive overview number year follow finding summarize study find htp interest completion study goal primarily identify research study steady progress identify source influence identify student performance student behavior entire course week form longitudinal study take longitudinal variable later success formation language economic explanatory final integrated behavioral strongly persistence three close week ahead paper predictive throughout emphasize click stream feature explain student success variable auc week ahead passive focus form frequently capture learner interaction resource work student derive learner interaction available subset third learner generate different lag build require detailed description employ operate characteristic advance difficult predict student end week extra effort trend rather important successful variety consistent yield superior consistent across make exception less notably crowd familiar propose highly effort crowd research suggest education inform crowd realistic effort make overall incorporate student problem result arguably student relate student trend predictive count involve class strict frequency collaborative e power project reveal modeling choice variety way model fed model numerous challenge turn high systematically thorough exploration never know successfully aspect quick conditioning create time manual manually etc ready think extract flexible way utilize crowd rich discretize additionally number model discriminative include enable scope building especially include framework possible run hundred framework resource investigate mind create scalable methodology source would apply due shared schema attention scalability support wide applicability multiple would review produce microsoft template web template camera copy page template format space acknowledgment inside complete please detail item entirely web process available conference web paper universal review reach consider publication abstract anonymous facilitate blind review identify appear title page format another publish apply substantially conference previously review substantially version paper currently journal accept fall restriction ability provide please sure file contain program like graphic file pdf version automatically format really format accept paper need pay attention produce pdf default behavior scalable represent document read fuzzy font something ps ps tell file file refer font alternative program straight avoid must font file specify embed file option statement continue reaction review take final paper accept publication format except course author format review international conference must appear th international conference learn camera ready head page except first title horizontal run head type style file title head package file title exceed short form header format reproduce without easily paper exceed eight include reference format herein reject review margin top cm bottom margin whether us letter final produce letter please title bold horizontal page letter rest title facilitate blind file may simply accept argument file include author review publish author refer phrase reveal e work show remove name exception choose copy mind supplementary accept final camera copy camera ready title name appear point bold type mail address author whereas email author address leave author author file package final address abstract bold type abstract leave hand hand margin space abstract self limit paragraph six seven reader understand pt bold type leave subsection pt bold leave please level within subsection line paragraph without relevant appear want
random possibility bivariate fit separately kolmogorov distance fit distribution base distribution computational address component follow bivariate mle recommend asymptotic mle interval adopt computational approach handle statistical inference satisfactory acknowledgement author constructive suggestion b b row represent base represent approach c united united united rapid rapid pt pt department university university deal strength component stress asymptotic testing asymptotic discuss confidence test carry example give complexity wave wave wave induce wave reliability life role density cumulative survival vector joint independent random survival stress variable reliability take exceed stress interval due point reliability stress strength attract estimation normally consider estimate determined bivariate type application stress collect logistic laplace beta inference reliability stress ml system consider pareto strength twice system study white count stress derive bivariate exponentially distribute stochastically independent distribution estimating likelihood pareto bivariate main random mle give different confidence interval step computational provide result illustrate stress survival function easily let sample binomial obtaining determine also observation observation mle size interval percentile bootstrap compute bootstrap estimate approximate bootstrap reliability approach cat use estimate sample goal suitable express obtain equation cat restricted mle denote artificial iid mle testing calculate alternatively calculate either great cat construct reliability take point cat curve suitable suitable smooth finally solution interval intend computational present behavior various parameter numerical confidence bias size parameter replication mle bias size satisfactory increase decrease property average length increase coverage small
choose make derivation simple derivation original square number observation large exploit datum often low low solve root lasso fast modification effort find sketch sketch take solve approximate observation singular retain validation fast traditional approach employ phase extensive body algorithmic include power random sampling nystr provide flexibility work problem widely different regularization parameter popular include warm algorithm strong incremental training employ technique learn develop multiple generic solver present dimension complexity conclude assume robust lasso bad perturbation frobenius rewrite elastic net directly design employ low rank approximation algorithm approximate leave analysis design leave elastic problem dual u term since onto give optimal unbounded unbounded dual let matrix original bad exploit complexity grow present eq without replace rank robust root application attractive nevertheless emphasis care take involve show root approximation tuning parameter approximation lose insight absence sparsity square root exist uniquely suffice show cardinality solution generality discard column provide feasibility optimality dependent j zero alternative formulate problem constraint simple case lasso author order barrier generic primal dual specialized specific paper develop specific root involve barrier function original root inverse hessian q w therefore rearrange b log function rearrange q ap p hessian therefore barrier plus iteration invert hessian inversion lemma cost original root synthetic real life set set scale table gb safe robust dense sim dense random dense experiment complexity model time run repeat time reduction focus classical show run fold digit second second perform leave become impractical even carry validation report life second binary sparse corpus pixel data evaluation testing require cpu need second set second framework imaging image analogous spirit leave one analysis imaging remove lasso explore topic query datum share answer query robust text corpora papers york table query computational query topic computer political research data query image error image student
outcome row marginal contingency table choose odd risk logit function obtain association measure natural logarithm cumulative follow formula cumulative ratio pn response q age age I k k k k otherwise paper support indicator n order logistic model jointly response covariate odd ratio become rich model ordinary sometimes parsimonious fit response configuration proportional odd fisher scoring algorithm either fail association approach penalty difference effect demand ordinary curve surface penalize limit study proposal application patient play role order categorical datum marginal base bivariate order completely solve multivariate strict appealing constraint contrary like discretized version due lack subject matter restriction strict ordering constraint appropriate helpful possible range penalization consider surprisingly penalization ridge ordinal computationally longitudinal association refer form smoothing spline response regression parameter score
remain difference minimize divergence evidence lower solve maximize especially linearity trace form extension term generate create initialize comment variational determinant operation multiplication weight equation assess vb glm fit process interaction canonical intensity center intensity parameter strict definition usually parameterize simulate range maximal packing limit intensity replace lead additional prior flat correct tight around wrong state expert practice experiment point configuration current gibbs package legend correspond row respectively error estimate simulation vb standard r show vb prior induce extra performance tight value concentrated prior high likewise intensity large fully intensity order expert knowledge consist cell whether trend component intensity fourth vertical figure pointwise trend intercept posterior trend trend clearly separate especially model figure question trend derive repeatedly simulate point variable execution pointwise trend call priori give function converge gaussian exclude self whenever conditional intensity back situation impose prior short interaction conditional connect intermediate interaction weight thick step smoothed gray line mean red interaction estimate line depict motivated atomic molecular interaction model show window intensity acceptable characteristic infer maxima mean range characteristic range extend extension polynomial depict basis function characteristic range resolution width correspond smoothness beyond exposition account intensity process decade communication superior use poisson pseudo connection currently estimate recall surprisingly little approximate use feasibility scalability value example trend pattern consume model involve simulation describe provide computation computational bayesian counterpart simulation bivariate replace community orient emphasis might numerical connection advantage author support foundation rgb rgb rgb attractive logistic method exponential spatial combine variational technique technique demonstrate gibbs point model interact location inherently costly normalizing constant likelihood technique act become pattern popularity amongst likely mcmc design costly loop probability simulation provide mcmc refer depict flexible approximated framework software describe variational
layer plot learn cifar cifar activation cifar decrease introduce novel activation function neuron compute parameter activation function along parameter demonstrate lead significant deep diverse suggest activation fit suboptimal acknowledgment fellowship also acknowledge gpu thanks uci edu research com science university california usa uci theorem artificial typically neuron piecewise independently descent activation neural network compose achieve benchmark mode artificial rapid engineering imagenet science searching component linear sufficiently arbitrarily nonlinearity deep network fast accurate deep network active easy train vanish another innovation maxout achieve benchmark maxout compute activation replace impact function previous effort largely attempt set strategy powerful parametrize piecewise learn neuron descent input experiment piecewise activation unit hinge shape piecewise activation hyperparameter advance learn unit total typical maxout piecewise equation assume theorem reconstruct constrain condition may input function eliminate segment unit slope boundary point special term slope elsewhere element summation slope elsewhere last term elsewhere almost verify thus activation maxout unit unit maxout tune train impractical maxout network maxout unit expressive maxout allow maxout reproduce coefficient one maxout unit maxout tie implement unit maxout would maxout expressive penalty improve file solver tree cifar cifar color subtracting value cifar convolutional pooling pooling pool fully apply pooling drop layer pooling layer drop fully layer softmax cifar pool use activation cifar cifar almost case cifar try improve dataset also try cifar cifar image zero image knowledge report augmentation activation relu relu value cifar cifar cifar unit consistently outperform relu initialization report deviation bold cifar augmentation cnn relu maxout cnn relu relu relu supervision relu data cnn maxout cnn maxout maxout selective relu cnn units relu relu high energy event characterize energy supervised physical decay distribution area operate
x txt header x serial saddle ylabel error font pos domain title header result serial header serial txt header serial txt xlabel ylabel error font legend north restrict header index txt header table header index comma bundle risk like smooth well specialized solver regularize admm present dataset use detail table term minimize experiment svm comparable outperform demonstrate serial quickly poor probably processor problem optimum converge solution comparable trend logistic dataset time publicly dataset recognition million entire sub sampling reduce room improve performance derive random r implement communication machine run eight core sgd also reduce accelerate dual initialize machines degeneracy svm restrict tt epoch update useful recognize order order processor update st epoch processor update partitioning eq suffice serial convergence f end theorem dt convexity concavity let inequality order regularizer rearrange sum derive eq get additionally get tool update q geometric term bound conclude q instead constant term theorem remark definition axiom claim massive optimization efficiently propose remarkably linearly processor verify empirical evaluation batch minimization risk machine give furthermore regularizer brevity recover machine svms regularizer change loss general minimize risk regularizer smooth algorithm bfgs hand regularizers bundle popular solver alternate direction multiplier belong batch algorithm update iteration well computationally expensive point fact empirical risk decompose empirically accept effective regularize stochastically replace gradient step compute datum fast gradient descent second batch method bfgs usually computation speed rarely execute component processor parallel somewhat hoc paper fundamentally risk parallel minimization saddle solve prove processor verify empirical svms binary regularize risk saddle point rewrite introduce multiplier eliminate likewise component duality switch maximization minimization conjugate yield formulation minimize eliminate obtain moreover minimize saddle f equivalent problem coordinate coordinate ij denote training cardinality remarkably component optimization define interested saddle stochastic step step surprisingly regular implicitly replace therefore approach stochastic algorithm key would stacking optimize partition depicted processor fraction red processor dark rectangular active area depict processor exchange variable figure processor processor epoch active processor either active processor key carry processor intermediate processor partition begin processor coordinate point work memory distribute memory hybrid architecture fourth across machine redundant storage linearly fact local rectangle red rectangle rectangle node red blue node rectangle green rectangle color area processor dark color leave fix partition pi j epoch inner inner communication processor execute st detailed stochastic saddle prove uniformly random distribute certain order would recover produce serial convergence believe technique independent interest differ convergence parallel point respectively epoch duality please understand implication theorem processor partitioning note perform update time subset inner per finish epoch theorem number obtain conclude linearly eventually dominate effective risk receive significant limitation unfortunately partial consequently execute serial focused working limitation computing gradient parallel update processor popular share memory latter
cnn single image lead suggest alone effectively table baseline understand iterate triplet lead consistent drop equation refer word sentence dependency consistent drop dependency advantage end gradient raw pixel additional insufficient amount training cccc search ranking devise rank recall good sentence retrieval level score sentence apparent retrieve top mention blue bottom right imagenet powerful complex attribute object generalize novel class learn sentence triplet triplet high box generalize grain ball limitation failure model sentence bag relation relation noun align people phrase play count moreover maximum people inside person spatial hard distinct spurious person many example compound become separate white two relation black rise careful sentence progress address learn modal inter modal reasoning fine formulate alignment traditional ranking improve retrieval previous interpretable future counting reasoning spatial move beyond li department computer stanford usa cs stanford introduce sentence modal embed unlike map common work sentence addition previous add alignment learn associate modality extensive reasoning sentence fine improve sentence retrieval additionally interpretable prediction infer inter modal explicit ability image automate image conversely ability retrieve query immediate particular set language description rank fix sentence vice challenging require understanding modal correspondence query water retrieve corresponding entity relationship sentence complex primary deep language multimodal sentence image object dependency tree relation common embed explicitly reason inter modal allow image sentence k dataset publicly available grow mapping sentence write automatically closely naturally allow bi align scalable quadratic limited sentence triplet scene field relationship relation fall modal probabilistic represent sentence boltzmann bilinear autoencoder closely relate et introduce embed common embed rank adopt describe put entire correspondence sized top convolutional network object complex scene neural connect raw neural representation domain computer vision state detection propose gram representation sentence representation paragraph document representation retrieve image query conversely sentence query training image neural score compatible pair feed otherwise evaluate sentence sentence sort score location list core insight complex interact entity intuition break propose object tree relation true inner interpret margin sentence score strong alignment compose aforementioned objective cnn green map embed relation embed box score box compute extract visually identifiable entity child attribute dependency similarity ground higher learn objective hyperparameter validate objective detail blue box score box image mention blue violate way triplet visually identifiable triplet detect lastly visual nonetheless incomplete alignment sentence interpret visual sentence incomplete define together otherwise intuitively encourage red along green objective red box member bag score accumulate objective dense play attempt infer sentence triplet bag precise mi minimize define set bag sentence return belong inequality states sentence objective solve heuristic score bag global rank sentence truth sentence thresholded score set range member helpful add smoothing cross validate image make dot product thresholding descent sgd batch momentum validate epoch initialization epochs cnn fix switch keep overfitte concern run batch retrieval image annotate amazon sentence normalize sentence simplicity stanford compute sentence hundred overfitte concern consideration remove occur reduce implementation imagenet detection gpu second discard prediction imagenet activation connect immediately detect training image sentence sentence image compute fraction top follow
dynamic top spectrum amongst differently overall represent compactly gain great field natural language speech dictionary recognition construct phone hypothesis hypothesis actually understand sentence store lattice lattice target tuple node alphabet element encode lattice encode every character string notation simplicity path define lattice node initial consider lattice string b b string lattice representation string share among element common edge instead twice lattice share common common save data complex viterbi viterbi pruning magnitude lattice input string alphabet objective lattice task number correspond computation lattice minimal size moreover graphical inference construct lattice minimized think transition factor minimize suppose character stop correspond character merging node construct thought share merging make powerful heuristic utilize graphical structure naturally lattice encode particularly frame lattice lattice vertex transition vertex correspond character encode string lattice graphical structure determine v pg j p iv n jt dl pg n edge stay vertex source deterministic state e pg p please depend edge traversal however take jump unlikely speed please note gm stream like go rather lattice benefit prune speed underlie gm compress certain datum discriminative convert within certain around peak lattice alphabet integer time overhead query show model lattice behave gm feed track number frame max value max decrease specify share rest lattice rest theoretically lower pr cb negligible show contain z fed vertex intensity act affect gm compress encode instance small compare instance intuitive illustration lattice contain disjoint path path separately lattice simple lattice merge share get would size grow task mass hundred within redundant may prune algorithm inference fit perfectly prune strategy prune frame space state original various single candidate pruning strategy still spectrum pruning candidate apply pruning space end candidate lattice jump depict jump good matching small rarely lattice allow pruning pruning tree pruning jump mention gain acceleration amenable tool cause well boost overall evidence let generative find ps wish maximize maximization much discriminative wherein simultaneously parameterize set hypothesis candidate within mass criterion define denote maximize respect approximate converge obvious dx I p possible candidate denominator make objective compute efficiently use stochastic gradient ascent ascent calculate regard vector correspondingly previous detailed practice begin generative move maximize denominator encourage improvement numerator incorrect denominator influence ns discriminative execute denominator calculate possible infeasible represent e hardness constrain peak graphical theoretical since unit theoretical peak value matter sure mass mass perfectly solve scalability lattice together strategy discriminative lattice feasible lattice general hypothesis dynamic denominator lattice even graphical capable encoding amenable lattice achievable consist spectra consist high spectra regard database find available supplementary order score engine set rather protein score choose specify error low resolution mass tolerance th whereas set search peak whereas search provide two benchmark neutral peak peak model absence therefore follow spectra work target identify fdr identify significance monotonic score instead define minimum fdr incorrect plot dataset margin though two method ii gap training representative high quality currently incorporation entire believe critical score arbitrary worth ms equal evaluate regardless flexible result build dataset peak alphabet effectiveness varie mass peak candidate window increase reduction effectiveness improve time lattice pruning pruning setting describe prune wider early prune moreover prune space absolute efficiency give original speed respectively record engine experiment ghz cpu cpu report test utilize run expense high confidence capability hand place unit along access figure low arguably important note prune seven influence choose intensity spectrum peak peak train mean mean well scoring model dramatically fold ability compactly entire framework gain also greatly allow future investigate way exact computed increase training train state effort encounter plan high simplify process throughput technology quantify produce assign observe responsible generating spectrum recently network rapid achieve variety return valuable regard work significant improvement widely use process score give spectrum database thereby allow share candidate demonstrate across datum introduce variant entropy rather maximum enable spectrum discovery datum protein separate micro mass primary thousand spectra ideally single arguably responsible generating spectrum identification problem database genome review spectrum database scoring spectrum identification dynamic correspond model viterbi decode align candidate peak spectrum spectrum peak peak axis training model observe spectra without axis current introduce improvement word make processing represent context lattice compactly collection mass spectrum viterbi allow lattice reduction expense range examine sharing among candidate dynamic programming spectrum exponential score compute ms via denote potential tractable sequence basic determine amongst last frame expand depict figure r spectrum frame spectrum
histogram reasonably criterion estimate mae suggest reconstruction besides mae rmse observe large h error norm mae rmse unobserved calculate come population mae seem display indicate obtained suffer unobserved mae star nan difference star reject indicate significance uniformity er von approach section result poisson describe marginal reliability observe panel reconstruction whereas display marginal diagram fluctuation value fluctuation b b b reconstruction different clearly table well even good c mae figure display power histogram seem value difference reconstruction reconstruction clear interpolation scheme display partition employ make tail adjust h table reconstruct mae error little reconstruction reconstruction density cc cc mae rmse unobserved display mae rmse clearly display reconstruction mae ks uniformity er present loss e confidence serve reconstruction confidence reconstruction table simulate set additionally absolute var reconstruction var increase list calculate replacement datum cc error ccc cc table var reconstruction belong empirical confidence table aggregate nevertheless apply allow compute individual probability laplace contain help determine uniformity reconstruct large datum loss cumulative cumulative difference reject critical statistic distribution asymptotically summary reject level respectively test ks difference problematic sure sure test kolmogorov test version ks point case cumulative cumulative ad von statistic maximum integrate distance sample variance distribution ks statistic uniformity great er von reject statistic behave von goodness tail al fs normal make powerful rely uniformity besides test provide autoregressive possibly lr formulate function associate autoregressive convenience nan hypothesis hypothesis respectively supplement normality ensure standard skewness second third central square freedom normal nan hypothesis follow normal reject nan affected skewness modification utilize absolute deviation skewness number total loss robust degree nan hypothesis reject confidence respectively test ks ad whether integral help detection reconstruction dependence also power skewness sometimes box test overall randomness lag et plot serve fit close additionally overfitte fit lie line curve diagram cumulative go tool marginally estimation graphical device versus cumulative loss calibration quantile comment business iii de present compound empirically fractional method reconstruction obtain variety criterion good estimation var important management loss loss side compound entropy analytic transform available observe determine distribution loss sum frequency probabilistic inverse know compound describe accumulate loss towards advanced capital determine work shall frequency loss compound distribute technique exist propose fall transform actually try transform determined analytically axis frequency poisson loss compound may compound numerical happen many regard loss follow transform laplace transform carry begin describe loss frequently amount business cause tail possibility large claim correspond loss company bank implication determination laplace calculate simulate loss infer fractional begin think identity order relate mass change paper knowledge historical frequency loss available possible loss could order remainder paper organize recall additionally quick overview robustness determine reference devote computation two measure could interesting operational loss capital devote role conclude remark appendix test method fractional variational inverse consist find constraint care natural requirement point maximize concave functional entropy minimum problem actually moment explicitly generic minimize denote scalar obviously another technique auxiliary determined numerically factor add normalization equation positivity determine seem improvement dimensionality view continue positivity constraint borel point coordinate reference search measure first restriction upon hull generate purpose respect positivity introduce probability close whenever finite routine setup generic entropy define version duality achieve case idea value reference product poisson measure unit dirac delta certainly integer notice minimize forget matrix recall determine interpolation necessary inherently explore make set add detail exploratory comparison tool like calibration agreement reliability serve determine quality proximity quantile close loss cumulative empirical minor fluctuation zero estimation see among bin histogram bin I data disadvantage bins rmse distribution versus calculate distribution observe al measure fit bin due possibility density transform back et transform integral interest sample deviation uniformity indicate reconstruction fail aspect uniformity visual inspection histogram autocorrelation plot ks er von al inverse joint normality independence combine normality use evaluate quality reconstruction comparison make simulated compound overfitte well help ability perform unobserved example compound process frequency period poisson distribute analytically deviation simulate
fx hx x element state control denote rl function compact continuity function concern scheduling note cost constant time become infinite run multiply cost step geometric utilize option utilize continuously versus vanish origin definite guarantee state bellman equation read policy give horizon end unknown vi consider several question arise relation guess guarantee I sequence converge continuous author vi iteration go compare convergence question simplify horizon extend answer proof though continuity necessarily concern address uniform convergence advance utilize state instead vi reason horizon infinity final horizon actually vi regard vi address three dynamic system perfectly iteration vi approximate analyse need set vector besides origin sense running function zero need reason trajectory stay dynamic iteration lipschitz lipschitz recursion one vx vx hold stability lyapunov stay stability follow negative invariance lead value approximation matter fact converge ic x next provide stability iteration give continuous ic eq vx asymptotically system eq inequality stability low boundedness guarantee lyapunov continuity continuity error approximation rich helpful boundedness function place offline scheduling eq toward utilize separate rl control learn dependent learning infinite let take q vi select guess learning call however scheme follow replace action fact require independent stage eliminate learning require also matter learn car action let approximated possible second initial select measure action dependent go entirely need identify e g worth fit rl condition conventional decision algorithm chance behavior system call online decision motivate rl conduct note still select never chance learn never exploitation concern concern still decision exploitation possibly result concern utilize guess work ref investigate stability present make switch hand result theorem extend basis due page constant access denote dependent vx step long inclusion iteration recursive relation admissible call analysis policy address already control controller simple admissible policy obtain respectively admissible within select eq prove action dependent dependent iteration converge action select ix result action iteration system however apply step fix stability system similar conventional require stability form converge finding function lyapunov policy however policy subject adaptation origin origin region evolve policy r policy subject iteration analysis stability learn present new sensor controller together respective utilize stability capability monitor switch boundary stability scheme respective discretized sampling fourth conduct least take computer intel ghz single control condition simulate real world force open loop fashion calculate assume result comparison purpose history open loop fashion plot call performance deal incorporate stochastic nature policy utilize drop chance force simulation fig show capability loss transmission try controller successful another important simulate performance free dynamic van feedback linearization discretize policy element h calculated guess act additive apply case plot communication entire online learn use respective exploration exploitation choose scheduling black plot history respective also see end fraction bandwidth compare policy designing policy provide control approach load call take several leave particularly online lead consider limit function equal hence prove integer since x w term non continuous continuous finite continuous characteristic ref establish feature proof result dependent induction result use therefore completes consider finite cost step history horizon remain finite early nature consider limit function action decision decision evaluate unbounded per set cost go horizon cost cost consider value infinite horizon great value fix prove continuity result theorem function v dependent continuous function continuous guess switch lyapunov initial admissible continuous definite hence induction every proof continuous v ix consider invariance theorem close ix kx close origin monotonicity establish let kx k hand since definite drop inequality hand non entire trajectory contain v kx v definition v kx r prove trajectory inside school rapid city sd phone email edu problem allocation unlike discuss develop infinite include zero investigate development extend model present optimal analysis unlike conventional system loop controller require measurement task load maintain example generator point common spatially generator power different control engine etc sensor spatially throughout unified cost facilitate monitoring capability change literature develop approach design decrease reduce load design consequence induce quantization error digital datum effective approach loss delay periodic typically monitor current system scheduling available hold measurement generalize controller design assumption cite paper literature optimally rarely investigate study aim extend application dynamic rl horizon simplify control feedback receive investigate switch behind advanced case delay scheme horizon stability lead load scheme controller sharing scheduling state respective conduct law policy adjust transmission continuously though approach case previously receive state controller one finite horizon design scheduling function minimize store last receive measurement store time state measurement network control looking dependency consider system suppose characterize
transition assumption continuously lipschitz main assumption big theorem take ts ts sequence dynamic achieve select policy regret treat trajectory include layer ts repeat ts ts ts ts ts ts ts ts ts get side notice martingale hoeffding inequality almost surely probability least union simultaneously least union fix let l tc l eq inequality lemma sorting ts ai optimistic function optimistic optimistic policy obtain proof efficiently maximization confidence possible transition complexity number transition side provide algorithm aware precise beneficial enjoy regard ts parameter achieve rich far superior free state concern policy I td control method mdps reward transition change arbitrarily set see difficult version reveal learner select policy expect dynamic regret pool sublinear interesting future mdp make assumption learn adversary future plan ahead markovian trivial combination idea principle sequence construct achieve setting cm cm study finite markov decision mdp clinical trial recommendation objective dynamic take process mdp problem clinical patient response past outcome collect side simple make would treatment patient model patient current regret mdp alternatively mdp problem change kernel efficient paper problem markov decision transition depend new previous side test option apply patient decision transition influence patient formulate decision principled utilize reward goal notation markov decision mdp characterize previous policy lp consider free transition reward influence transition choose give sigmoid vector reward parametrize feature episode rise nearly dynamic policy account achievable expect algorithm begin action space construct equation l algorithm principle employ online
scene composition texture rest supplementary request figure relation indicate decomposition trivial square provide powerful hope various scientific show computer vision code large image acknowledgement berkeley france berkeley program research centre theorem axiom department california berkeley chen berkeley edu unsupervise call code gain lot though interpretable efficient implementation publicly important scientific bring fast scheme active demonstrate computer task codebook visualization unsupervise technique widely used automatically discover underlying structure serve several purpose look exhibit interpretable example neuron activation population similar topic text collection vision unsupervise model mixture gaussian yield descriptor unsupervise visual recognition probably task purpose visualization call interpret provide prediction analysis discover unlike factor force point association association centroid centroid among interestingly popular nmf independently around time approximate factorization lot analysis address develop optimization active scalable exist implementation believe application bioinformatics processing perform code recognition second analysis database image analysis section vision conclude let represent factorial look vector approximated combination close coefficient simplex replace factorization challenge non relate briefly negative seek component negative analysis fix norm sparsity induce produce sparse formulation negativity aside main vector encourage become combination useful entry input variant code datum use decomposition element interpret anchor represent automatically differ propose variant huber often replacement scalar since grow cost section huber natural notice solve qp update rewrite residual coordinate optimization g tb input initialize initialize x x decomposition matrix strategy way efficiently quadratic simplex huber robust reformulate per fix optimize program vector optimize block guarantee converge stationary point tb input initialize I carry carry line various code matlab toolbox performance software package implement publicly toolbox qp solver package level implement original analysis intend method unfortunately software package severe report computational intel report result package order magnitude slow experiment qp solver study scalability package mnist potentially main limitation limitation share classical regard slow converge right mnist far image patch small pixel encode descriptor sift codebook pattern call image finally histogram occurrence yield powerful task typical method sift descriptor encode max yield simple bag benchmark dataset ultimately svm sparse code conduct image classification replace code categorization demonstrate learn codebook sparse code similar slightly involve recognition report perform well use class rest testing lc classification testing randomize even rule digit class test classify good eq q normalize want thus hull near mnist remarkable aa
panel seen estimate right panel single u rl filtering sequence similar previous complexity evolution outperform vector eigenvalue improve structure n nu summarize fig depict outperform competition twice n gaussian also regard three compact yield single stream filtering meet united union present focus algorithm future pass type pass enjoy convergence value product present ridge space minimizer j weighted definition rgb novel iterative projection reproduce hilbert nonlinear nonlinear component multiple permit propose meet hyperplane certain efficacy reproduce hilbert filtering reproduce nonlinear adaptive filtering investigate reproduce author kernel approach multiple subsequently propose multiple component ii high adequate amount unknown limited time vary adequate system investigate norm situation multiple compact representation efficient name affine seek functional gradient hilbert space rkhs build unknown implie datum selective new criterion coherence criterion introduce raise issue regard enter discard contribution though adjust coefficient dictionary algorithm systematically enforce algorithm considerable dissimilarity project hyperplane instantaneous operate e rkhs operate filter present reveal significant interest stream meet bl bl bl bl bl bl bl bl bl bl bl bl bl bl bl bl product bl bl formulation article filtering projection fig characterize superposition lie infinitely way cause avoid particular trivially I share cover important mean sum direct sum space derive rkh uniqueness decomposition sum hilbert imply derivation sum key turn another simultaneously nest structure cover case intersect trivially nonzero intractable product hand close formulate formula selective numerical example effective nonlinear low apply efficacy rest space show particular theorem complexity example conclude nonnegative matrix bold face identity denote function sequentially output focused case component nonlinear etc describe rkh associate denote element unique indicate sum rkh reproduce apply recursively kernel real hilbert w reproduce ridge easy handle appendix processing product close fortunately inner build adaptive kernel span n atom initial initial filter assume element section useful due unique reduce correspondence tuple hilbert equip example positive typical know nonempty rkh associate gaussian nonzero assume nonempty manually gaussian kernel within contain devoted corollary mind present possible nonempty see j elsewhere n c kernel grow pruning adequate grow start n criterion modification novel n time instant new measurement strategy accept normalize project zero instantaneous relaxed therein assume search size affine dd hold hold computation involve p lemma rkh definite jj normal eq gram matrix entry indicate inversion invertible kernel size determine unnecessary orthonormal case reduce complexity selective update kernel geometrically coherence p form approximate n coherence justify example subspace dictionary element follow compute subspace propose state although proposition rather slightly kernel arbitrary satisfy l l hence obtain w light share common product article rkh straightforward nonempty interior exploit characterization prove another appear theorem strategy case analogy adopt criterion use space unknown contain system element enter virtue argument translate derivation product space fortunately translate even therefore formulation follow emphasize write case hyperplane selective gaussian kernel appropriately nr build project vector euclidean obtain call algorithm except individual dictionary multiplication dictionary gaussian complexity dictionary size subset denote submatrix suppose use selective update coefficient matrix inverse partition inversion addition inversion need update demand number coefficient computationally demanding coefficient n n reason
considerably efficiency unified complementary capability advantage base approximation residual sparse residual impose conditional preserve efficiency remark paper argue strong refine residual exploiting perhaps scalability work utilize unify describe approximation relax conditional large section improve predictive leveraging advantage reduce residual latter kullback leibler subject consequently trade size gp rank parameter achieve comparable accurately represent markov spectrum approximation advantage sparse parallelization core great scalability perform traffic implement cluster speedup empirically evaluate real represent input realize value output variable unobserved finite gaussian regression provide predictive u dx limitation practical poor inverting incur propose scalability sx approximation sx input realize unobserve reduced covariance section representation include unified approach approximate matrix contrast refined covariance set input partition scheme evenly disjoint correlate key block matrix b impose process comprise illustrate ease block block recursive series reduce block diagonal e large rank residual covariance matrix approximate v v r five block band block nb specify band rank approximation offer interpretation impose far bb fall outside outside block approximate v generalize vary markov utilize u predictive uncertainty directly invert regression issue leverage associate block n use block band follow specifically spirit impose residual independent give assumption relax importantly assumption equivalently achieve proof though utilize representation unify residual matrix impose strong assume reveal dr closely kullback kl minimum r proof appendix exploit derive formulation amenable parallelization core construct tuple tuple construct summation master master construct tuple u receive tuple uncertainty u local recursive discuss parallelization show mention scalability b b centralize incur cubic increase core reduce centralized respectively speedup parallel centralize increase improved increase overhead incur cubic cost achieve desire toy local exhibit prediction partitioning evaluate predictive scalability art real dataset dynamic degree freedom robot traffic km segment road peak hour road comprise five traffic dataset relational structure segment road topology traffic dataset gps whose prior define ix length scale kronecker delta estimation support dataset change large spectral table platform via memory core core respectively compute storing respectively subset metric use evaluate error b incurred rmse test parallel core incur parallel average core predictive likewise data independence incur expect scale incur incur minute incur time parallel achieve comparable predictive process remark oppose report table incur cause dominate incur incur time remain stable structural result parallel counterpart instance vary core incur centralized centralized increase centralize incur centralize respectively hour centralized incur almost possible setting huge centralized block e expect incur huge entail scale operation huge cache highlight sufficiently support single cache incur speedup increase explain speedup appear increase core primarily cache incur cccc datum size core gray rmse right fig markov use incur order vice versa respective second second incur indicate increase predictive performance latter cholesky easily incur I second use second see core less machine scalability w input denote month day incremental day start day output mean level pressure pa setup platform node run ghz gb ccccc incur core vary size incur parallel set parallel insufficient share memory parallel issue incur hour summary experimental significantly scalable achieve comparable fast achieve incur centralize achieve performance considerably markov early trading support markov incur reliable alternative increase achieve huge cause cholesky factorization insufficient share dataset describe computational markov assumption utilize residual centralized plan automatically variant stochastic plan release http code google com acknowledgment mit technology observation block band observation r follow lemma necessary derive sparsity cholesky u mn upper mn mm b mr proof directly cholesky q lemma fourth last last fourth definition
challenge agnostic general hypothesis requirement disagreement base active open key contribution independent interest connection active rate allow error rate construct label query agnostic contribution rate guarantee classification low extend predictor agnostic setting rate general active consistent complexity show bound label lie label space example marginal denote access oracle label input vc respect h h ss data oracle oracle query oracle possible say agnostic active frequently use disagreement two hypothesis assign formally b r h hypothesis sense significantly finally set hypothesis connection rate rated allow consider predictor hypothesis rate predictor label ensure risk predict predict rate predictor measure performance proceed epoch epoch achieve maintain contain epoch select rate predictor epoch run label adjust excess generalization error oracle vc rate excess call generate rate predictor induce example set distribution excess confidence get minimizer label class rate predictor excess minimizer label risk recall precise explain oracle oracle rate predictor target confidence label still maintain rate guarantee predictor candidate hypothesis minimizer example simplicity distinct output k query epoch excess passive factor agnostic key achieve large excess fraction rate get generally disagreement active input oracle hypothesis confidence rate predictor particular return satisfie agnostic state adaptively find query lemma suppose excess h j succeed hypothesis set example target excess confidence constant algorithm succeed use give non oracle hypothesis class rate labeling draw query label erm follow confidence rate guarantee rate error predictor optimal interest receive likely contain guarantee predictor predict constraint expect disagreement assign goal fraction key program disagreement maximize equation solve program rate confidence rate predict wrong predictor version px set rate coverage rate predictor much coverage essential true label observe rate guarantee subroutine consistency oracle rate predictor target confidence bind subroutine set rate disagreement formally x h db learning label oracle confidence rate predictor exist agnostic disagreement active characterize region label disagreement bind simplify require q label disagreement base active learning contrast replace disagreement noise noise label class condition condition oracle hypothesis rate target constant eq provide c analysis dc comparable consider concave much lead log establish area demonstrate active rated potentially label thank nsf helpful thank wang introduce problem selective vc corollary vc bind pick vc due pick label consistent version pick copy joint induce rate confidence rate predictor datum satisfy elsewhere x combine lemma hold succeed example immediate consequence dense labeling target j assume h j hence stop equation j equation third exist turn make iteration ensure crucial noise necessarily hold namely meet lemma dependence query suppose exist algorithm event succeed approximate true risk first h jj combine output event iteration j n cn cn cn cn last fact cn cn cn j j bc j j suppose classifier set minimizer happen lemma second follow cn equation suppose exist iid let combine equation cn equation triangle c plugging equation get prove thus happen induction clearly show inductive know succeed eq get lemma triangle dx dx dx dx kx ix dx divide q induction event thus k equation q follow happen induction clearly q succeed target combine combine equation px px x px assign I exist violate generate could eq event equation lemma run follow assumption h yx third succeed last second fact b h dx h c yx dx yx second equation c yx yx yx yx c yx yx dx follow lemma triangle yx dx yx h inequality follow divide get succeed dc total immediate item concave homogeneous classifier exist exist follow algebra label excess follow example excess confidence v happen succeed h combine eq call input oracle confidence rate excess succeed input example eq algebra begin observe combine thus equation moreover equation rate x two get eq algorithm consider solution lp z z show feasible coverage h n union bind probability follow hold program k satisfy weight iid copy least proof fairly vc u n n jensen third fourth hypothesis
sparse regressor leave prevent overfitte consensus message come form indicate algorithm belief presence pixel job square probabilistic segmentation regressor perform effectively count stable inference stage train improve accuracy latent fig insufficient demonstrate inference capability message pass normal image leave product n pl pn pn pl pr pn circle south fill red red south south realistic face primary motivation recognition pose approach normal pixel fig formation infinitely distant prior normal light side vector side approximately face line code model rapid although model formation similar successfully reliable usefulness simply true formation accurately column sep row mp mp mp visualization infer map fig consensus type describe fig predict contextual message predictor information message predictor guess estimate image directly forest create behaviour contextual message median max patch around contextual imagine regressor performance experiment dataset contain illumination remove entirely leave around normal proxy code qualitatively assess inference inference map map message pass match closely produce reference strong region mp pass inaccurate illumination cast inference produce arguably improve cast cast suggest future cast fig ability infer task recognition estimate rmse strongly compare close reflect choose result experiment take cast light error cosine angle distance estimate variational pass mp perform poorly produce inference horizontal line consensus pass fine light pass fig mp bad demonstrate use oppose direct prediction help message well presence mis make east column sep row mp forest stem kind decade intuitive rational see message consensus form pass intuitive exist proposal lead speedup parameter long early dedicated inference pass define predictor jointly train system produce long distinction predictor within pass regressor random forest message work concerned message attempt reduce accurate technique message recognize see rough success depend forest completely take work generic broad building contextual like framework computer vision make consideration cast face develop understand application broadly major scalable model interpretation increasingly heterogeneous appeal complexity difficulty barrier goal barrier anonymous microsoft research wish map message challenge message special care methodology give incoming review forest approximate pass message represent represent message different concatenation call parametrization value leaf label previously unseen construction likely contain similar example leaf message multivariate term contextual message leave regressor greedy manner node incoming split correspond j set residual incoming capture contextual message root leave forest might predict tree however sensitive choose moment moment show map normal choose show illumination condition produce close illumination extend synthetic image image use superior baseline quantitative use cosine posterior superior baseline synthetic image h light normal anchor column sep sep mp forest forest mp consensus normalize row cast infer normal map node anchor sep cm cm observe forest forest variance consensus estimate variance pass consensus pass style circle width inner sep cm minimum height width fill black minimum height red generative model reason imaging reason conditional traditionally domain purpose message pass expectation ep message simple vision introduce modification message learn message guide variety significantly ep generative probabilistic applicable wide language computer vision graphical incorporate surface normal approximate symmetry make counterpart perhaps mix bad effort pass many difficulty cost purpose message pass pass simple model attribute influential pass meaningful observe top additionally variable property learn early experimental variety efficiency standard message implication add tool toolbox improve bottleneck restrict exploit vision aforementioned illustration layer factor notation variable case experiment vision normal variable intensity reasoning message desirable purpose could send possess message inter layer factor practice access oracle message inductive argument regressor sec layer regressor predict variable layer inference global loop graphical due global consensus send contextual message e cl cl south pt south fill south circle circle circle fill north north north south cl north south cl north south north cl south cl north south north cl cl south cl north cl north cl north matrix cm replacement cl cl x pt circle pt south pt fill south south circle fill fill circle fill north message pass aim inference message point pass would reach useful good message pass message oracle predictor way message guide point marginal except latent instead message point target would label even strategy experiment consensus challenge message distribution need take fact supplementary material review forest illustrate diagnostic square use improve challenging face predictor train second sample significantly preserving pass experiment use default tree leave pa x sum pc pa pa c red south begin behaviour standard message gauss message initialization significant speed demonstrate circle wish coordinate circle radius graphical translate latent finally observation model express net circle layer presence take iteration converge circle marginal iteration repeat fig dash black figure marginal contain
arrive retain retain execution sampling retain particle signal retain particle state child retain branch loop entry previously retain execution retain yet able align retained particle next probabilistic strength improve approximately version inference evidence operate usage yield improvement program use posterior correctness carlo benchmark distribution experiment large sequential emission distribution crp mixture class probabilistic implement particle language probabilistic system implement approach run metropolis choice execution simultaneous metropolis engine run particle core cloud amazon ec run intel processor implementation gibbs engine generate good order engine implement particle gibbs sampling particle run repeatedly draw contrast infinity recommend kl sample posterior fair reasonably band cover median mark slope monte distribution complete particle amount sampler effectively immediately converge individual sample produce fast probabilistic engine much operating system call os compare core across count ec intel processor tb frame l forward monte particle programming language standardized system share exploit intermediate language machine link operate library yield efficient target new hardware optimizing source transformation phrase intermediate language library intermediate representation language intermediate probabilistic normally operate system library parallel programming language leave readily resource writing program computer architecture illustrate forward employ operating program program via monte implement execution trace operate posteriori trace program reflect data language language model purely forward generative process program execution trace site mh requirement operate primitive also level use inference albeit include var predict mu return output semantic include transition static double initial static emission double program state mean predict course execution implicitly define interpret probabilistic programming capability library execution datum mark expression posterior make choice log pass expression library include macro another loop nonetheless show gaussian return log particular program function predict posterior value underlie emission mean nonparametric generative program mixture gaussians chinese restaurant crp normal gamma point double mu var draw sample double variance observation draw alpha alpha invoke mu var return proceed draw trace entire virtual memory address machine probabilistic operating system construct os create identical execution identical continue correspond choice program forward program report match sequential smc resample building complex particle smc intractable p identity construct q program program execution particle unnormalized importance normalize set execution trace program continue trace correspond lead bad concentrated single trace execution trace trace index execution trace execution trace tb particle barrier unnormalize serial serial serial serial continue execution parallel trace terminate barrier l serial effective eq statement form barrier execution current unnormalized execution trace arrive barrier take particle reach current unique execution effective number store memory reach barrier retrieve child child execution terminate execution normal outline step execute parallel barrier desire additional mcmc hasting particle sequential mh propose set sample inner substantially smc tb particle program serial serial serial serial continue program trace terminate barrier serial serial current
marginal exchangeable partition group inference beta introduce exchangeable probability describe exchangeable collapse nonparametric one convergence good modeling prior increment impossible transform problem marginalization dirichlet random chinese restaurant structure chinese exchangeable lead collapse prior despite significant progress decade modeling usually limited point beyond interest group point exchangeable group number hierarchical dirichlet hdp popular wide gamma process beta gamma none group marginal hdp chinese restaurant derive collapse fully collapse neither unified mixture modeling membership law partition derive partition share marginal group count describe column random count mixed membership stochastic important contribution several additional simulate exchangeable group collapse topic beta update straightforward implement produce representation size exchangeable know constraint express addition addition allow one dependent count column membership propose dependent group beta random product finite continuous measure define evy large one hence define analyze sampling th binomial binomial jk nr pf rp truncation slice recent binomial describe binomial ibp ice relate ibp binary ibp different paper focus count generalize develop truncation free collapse nature value binomial poisson poisson potential bridge count assign count disjoint borel multinomial unclear exactly random us model cluster derive exchangeable probability govern partition later derive unit mass assign borel size membership amenable marginalization analytically coefficient categorical jk n provides obtain describe partition group group arrive matrix th nonzero permutation count detail matrix appear direct calculation j identically dirichlet pmf n jk r rr generate poisson partition count follow lemma denote summation govern fairly complicated derive prediction group govern exclude contribution membership rule select popularity cluster whose govern simulated gibbs run sampling exchangeable partition different setting critical role row kn infer fix data point group sum represent nonempty ji term rewrite product ji j hdp mechanism fully collapse assignment collapse globally derive collapsed hdp chinese book keep link tb topic iteration lda b analogous plot corpus curve correspond www edu ss toolbox http www cs edu corpora restrict occur count corpus document corpus document total count evaluate lda bayesian word one order collect j jk r p n j hdp per word jk jk sm final used matlab cpu collapse sampler topic take per infer topic sampler hdp comparable complexity infer topic infer large considerably speed first mix collapse sampler topic trace plot column slowly reach right quickly reach hdp small quickly quickly smoothing left column lead middle commonly corpus evident hdp corpora corpus middle small topic often hdp topic multiplicative control topic whereas shrinkage lda tend lda comparable topic able predictive hdp corpus setting sample moderate support moderate usually prefer could hdp lda three suggest topic achieve hdp lda view count model differently variance topic collapse gibbs sampler gibbs sampler truncation heuristic hdp collapse collapse comparison parameter posterior topic topic smoothing topic display scale omit membership develop value binomial exchangeable govern influence group dispersion construct nonparametric exist one intuitive interpret fully collapse sampler converge art representation method group unique interest investigate derive mixed membership modeling value gamma poisson gamma binomial transform representation beta transform
nx v nh fu nh obtain complete admissible sm sm combine desire lemma remark proposition type consistency entropy consistency learn function entropy consistency consistency coincide surprising provide infinity illustrate play analysis minimum entropy enyi use concept entropy divergence substitute covariance inspire series minimum error later blind source comprehensive survey advance unobserved power decade theoretical complexity empirical perspective early utilize bandwidth motivation minimize require term bandwidth parameter imply unfortunately simple yes complicated try full establish relationship entropy measure power statistic model statistical mean output measure two setting produce goodness approximation entropy sequel need denote h set enyi rf rf b z entropy involve make look estimator dependent summation involve u asymptotic converge q adjustment measure constant probability entropy power power imply clearly metric consistency good approximation regression serve contribution I model entropy necessary coincide consistency firstly tend consistency consistency result show empirical bandwidth choose large lastly special consistency bandwidth give make analysis regression throughout regularity density function usual tailed minimizer uniqueness obvious trivial remark simplify statement learnable constant cover h h first expect unbounded second ensure learnable happen impose theory fulfil target target adopt relaxed situation admissible verify least rf z literature implementation e choice rate minimum somewhat version later respect regression instead consistency complicated situation otherwise say state model f n regression corollary entropy consistency relationship entropy state theorem entropy consistency complicated illustrate example consistency fail error consistency show imply consistency coincide model prove bandwidth form z look surprising minimize entropy approximate consistency infinity view empirical motivation consistency theorem adjustment adjustment ib main bandwidth positive situation order case denote integrable recall transform crucial univariate monotonically decrease unimodal nonnegative define set assumption noise family look symmetric distribution say evy transform cauchy median choose fourier distribution refer statement exist proposition combine two prove consistency independent throughout notation obviously maximize prove translate prove excess quantity last equality part take expression nonnegative f fu f x fu u transform ensure nonzero interval identically iv corollaries ii previous prove error entropy counter denote specify subscript dx r f error functional denote minimizer measurable function bound entropy generality lemma uniform fu follow integer translate orthogonal f f measurable corresponding minimum equal write e e f x f u last x x x see I u b u fx fu x fu u fu fu fu fx fu minimize condition equivalent minimized minimum f x x fu fx fu fu fu f fu fu v fx fx fu impose need fu fu fx fu x fx fu fu combine minimize give take f rf conclusion iv minimum value regression choose tend suitable depend consequence proposition notice h role empirical use algorithm tend least special subsection unimodal let integrable unimodal unimodal variable belong unimodal consistency immediately state second hold appendix recall fu dx find fu tell minimizer notice take fx fu f fx f fu fx fu f e ef dc hc bounded noise w h fact follow xu virtue
hold several application interval instead extend omit aforementioned contribution besides growth apply theorem seminal mention stationary process task sample section concentrate case center begin preliminary specify proof omit df scale mutually wu u df hold indeed constant argument n u j integer u large r hence establish therein satisfy argument x hold theorem hence x nt limit side follow center stationary process proceed four u probability u u constant imply thus proof covariance stationary gaussian grid hold convergence rv u u almost index let z vector vector distribute unit sphere satisfie may denote df u satisfy n shall asymptotic interval appear define hereafter indicator jt ij p ns argument lemma side e u rest line tt nu nu k hence hold x e asymptotic supremum process methodology seminal paper dealing process check technical assumption process otherwise extension many finding mutually generic rather stationary process minimum statistic result order li lemma statistic gaussian component li show goal li maximum nn statistics dx define maximum order statistic ij n b bn il thus complete minimum il il ij nn il u il I lt bivariate bivariate standardize u z n n grateful careful reading suggestion greatly improve rgb remark proposition grant mathematical pl department university copy process constant define brain mapping interest empty impose condition tend asymptotic supremum process limit stationary version surely sample mutually independent copy interest th central interest fix conjunction time interest empty relate least prominent concerned imaging fmri establish seminal therein euler characteristic high smooth discuss result non field derive exact obviously phenomena skewness orient field engineer environmental study emission collect concern brain model since calculation general contribution approximation result empty df let satisfying assume random point distribution constant integer suppose simple c validity stationary process show validity copy asymptotic generalized notational th condition concern process derive proof establishes li vector stationary
object side gmm massive cluster sharp right gmm draw undesirable classification prefer guarantee gaussian super object number super position function super dense gmm pick pick decision boundary discard continue realization figure sample figure bin adaptively use correlation map count x ray galaxy build map place one pixel divide subtract q superposition delta eq attribute infinitely delta way make realization cross spectra make make realization cross power spectrum cross correlation red point show cross spectrum green band cross map simulation classifier galaxy readily scalable lot determination observe galaxy without dimension modification position object another us depth coverage add obtain statistically characterize generate improvement manual neighbor replicate sample feature space drawback observe random nn always discriminative sensitive ignore internal decision object algorithm properly paper survey galaxy consider among possibility angular ray space mass density q overall normalize unity sum theoretical realistic possible monte analyse complex selection explicitly limit application expand include example find physical principle promise direction near future method effective thank point solve discussion machine thank discussion like whose comment improve package computation calculate take reference namely rgb rgb blue problem thorough usually realistic complicated process galaxy specific mass simulation dark matter obtain kind purpose put together call pick physical property determine deal combine observation task rule derive exist object observe express rule behind machine learn learn classify simulated dark like synthetic galaxy select ray ray measure target galaxy detect eliminate dimension feature ray cluster dark matter simulation look observe aspect introduction machine detailed terminology application recent classify survey image galaxy detect large survey accurately etc support vector dark book paper classifier simulation also galaxie statistical sample classification goal distinguish train outli available boundary target target chance class property widely density boundary commonly reasonably sample size estimation determination well constrain demand compute paper boundary dash white surface sample supervise surface show class simulate black member target red construction svms family dimension use function adaptively class decision outli classify attempt mathematical formulation svm example function whose determine target separate order make separate linear surface space need use introduce space choose problem kernel dot kernel radial rbf determine decision return minimal elsewhere separate region zero region translate region end need solve follow dual optimization need kernel upper regard belong target example vector margin task uniquely give offset svm figure rbf decision separate outli define property uncertainty parameter set boundary right look outlier everything else right black successfully separate population panel figure increase modify distribution quickly inaccurate surface green boundary use show style sharp boundary uncertainty surface classification gmm combination distribution covariance datum cross open machine library model one svm dark sub observe make beyond threshold generative gmm mass ray ray cluster combine survey reduce rare subsample cluster galaxy scale procedure convert mass see detail develop parallel patch structure formation form matter peak measured predict position filtering individually solely initial match much simulation carlo construction galaxy formation phase simulate whether body semi method peak light core body particle train polynomial separate part observable region boundary boundary corner plot nearby massive object object far observe select surface top sample massive always decision cross randomly contain decision measure call rejection setup score specific change small outli surface derive boundary boundary use simulation x ray target though sub statistically object gmm parameter determine criterion
next arbitrarily ergodic period thus observer become become predict various drop result stochastic merge relevant purpose weak focus closeness predictive distribution obtain merge example merge matter strong merge unnecessary context decision discount period usually law stochastic elementary object seminal theorem de representation exchangeable ergodic decomposition exchangeability temporal parameter maker ergodic posterior weakly decision belief concept difference see outcome tail generate dirac infinite belief concentrated realization outcome highlight posterior belief little agent represent way make sensible learnable prediction event become connection ergodic long stationary canonical decomposition however ergodic learnable weak meaningful sense trace cover therein horizon proof rely technique literature update prediction derive maker observe start observe space realization generic product element way uncertainty capture bayesian view process stage outcome represent extreme dirac dirac assign capture intuition fundamental dirac copy trivial discriminate admit well know convex topology ergodic belief stationary admit unique parameter set ergodic belief ergodic stationary belief block realization equal process equal frequency ergodic every define extend recover ergodic finite process special decomposition exchangeable distribution outcome thus observe consecutive number outcome configuration good outcome equally give equip coin represent outcome give first outcome ahead cover finite horizon introduce merging setup outcome period merging let belief realization horizon learnable case strong every weakly rare explicitly merge merging establish potentially bayesian belief say period connect every take outcome average period strategy nf course agent weak learning say weakly play sufficiently patient horizon period payoff discount sufficiently discount let weakly period period motivation learn calibration idea calibration realize empirical show weakly forecast pass calibration weak characterization idea concern quality near horizon common think consistency think estimator reasonable bayesian weakly converge dirac measure reference hold property realization future dirac coin observe outcome belief uniform agree indeed insight future learnable main weakly learnable implication underlie change unobserved period period hide remain change period outcome represent decomposition ergodic belief topology concentrated parameter prediction complicated prediction merge weakly general history period outcome consistency estimator give agent observation block assessment probability appear wrong still agent horizon event process predict coin predict frequency economic weak modification setup bad period last period outcome unknown parameter formally define ergodic borel belief markov last occur value stationary observe time agent keep track parameter outcome probability agent know randomize deduce observe consecutive period point prediction agent predict decomposition learnable decomposition learnable time index hereafter algebra subset measure space exist unique equality measurable unique measurable dirac decomposition sigma algebra borel ergodic induce algebra borel interesting let shift learnable sigma ergodic consider trivial agent observe prediction distribution accord finite history give history therefore stationarity shift invariant therefore limit maker generalization ergodic cover simultaneously let let n formalize intuition belief event far I periodic mixing condition finer learnable necessarily stationary every sufficient establish equivalence decomposition fine ergodic decomposition decomposition gap proposition tail tail sigma algebra distribution trivial property prove tail weakly imply ergodic stationary stationary weakly learnable lemma past learnable tail shift stationary set outcome equal induce learnable rely outcome comment admit process asymptotically reverse tail decomposition show learnable process asymptotic reverse asymptotically reverse contain dirac atomic extent theorem tool merge weak merging extend say belief extend tail infinite equip borel give every fair coin entire tail tail dirac decomposition however
sure strong result unfortunately couple argument however contraction contraction convergence contraction operator value need exact variant satisfy establish weak asymptotic give explicit optimal value classical operator variant assumption satisfy bound satisfied contraction weak follow argument derive bound fix give give short reader refer proof construct chain stochastically dominate rate value get construct chain structure markov zero show stochastically dominate chain concentrate establish zero mix convergence sample asynchronous version even asynchronous consider visit full action deterministic shorthand denote asynchronous operator operator value leave unchanged respectively visit least contraction visit full slightly asynchronous probabilistic checking progress sequence update turn cycle random introduce pick compute bound apply result pair visit asynchronous q online fast though guarantee convergence show result compare mdp cost iteration state pair equation code since step figure rate fast reach relative take speedup iteration mdps establish limit unlike classical scheme mdps actor analysis incremental preliminary experimental state action update would pick pick sure sense result case infinite even continuous action partially mdp method rkh average reward reward dynamic programming operator reward mdp contraction mapping however provably convergent reward mdps ode approach interesting mdps reward criterion part fellowship department technology support office nsf award proof series lemma control markov show strategy stationary define r k markov depend expect value coupling strategy k mdp two ss remark abuse common copy let w j analogous choose choice lemma get extend control homogeneous completeness sequence state chain state law k dropping chain probability control corresponding limit empty start govern contradict stationary irreducible statement non stationary pair mdp simulate arbitrary k k p q proof conjecture notation corollary example remark propose optimal discount cost process classical algorithm mdps converge surely mdps sure approximation rate preliminary fact asynchronous popular empirical c h algorithm popular use dynamic programming actor td recursive observe payoff slow underlie evolve effect average ensure scale design property incremental therefore evolve scale usual conditional averaging underlie control obvious advantage expect work rigorous simulation ball indeed reality theoretically construct coupling simulation finite yield look backward guess terminal reinforcement programming view refer classic present asynchronous extension present possibility mdp action let ps ps action mdp control control px ps objective admissible infinite discount cx infinite irreducible integer bellman q bellman tv tv find arbitrary iteration calculate contraction mapping banach though iteration extremely develop mdps value q exact transition generality mdp drive function uniformly algorithm empirical distribute iteration maximum iteration counter sample stop main result proof every rule step general convergence incremental noise horizon obtain look guess terminal cost precisely value guess look showing backward iterate immediate contraction unknown convergence formalize equation overcome difficulty define rigorously approach backward converge almost quantity underlie space forward control chain c time couple chain f backward notion proceed side infinite sequence negative integer sequence n transition ease independence give represent drop whenever value define strategy map action ks ks resp resp ks write condition mdp implicit use notation whenever drop since markov k prove couple z k two simulated strategy proof consider technique thought function initial offline simulation initialize compute stop else simulate composition start backward composition forward simulation composition successively transition generate state collection choose contiguous familiar backward strategy q kernel important iterate let kk induction note definition get k
bandwidth smooth factor estimator estimator rule package study step estimator unlike case function case f point tend around observation equality f property conditional asymptotically application bandwidth selection datum tend large tend happen often usually rule rough order narrow develop trick depend impose way particular bandwidth rule bandwidth clear left section investigate simulation confidence quantile compare end bandwidth package finally equally distant grid estimate grid quantile xx rule multiply ex ex ex c ex ex generate normal variable quantile quantile namely univariate grid heterogeneous take coverage cc nonparametric quantile perform especially result simultaneous band derive asymptotic hand homogeneous heterogeneous asymptotic mention cc probability yield bootstrap general nominal wider getting find sensitive bootstrap specification upper show coverage cc probability simulation surface asymptotic nominal coverage see volume ex c ex volume cover derive expansion somewhat confidence theory worth note portion uniformity grid homogeneous heterogeneous specification level usually univariate notation sense median point complete probability proportion finding firstly improve volume band increase heterogeneous compare proportion tend curse dimensionality bivariate bootstrap part coverage coverage though cc volume cc much table similar bootstrap size method regression nevertheless obvious stochastically dominate scenario improve dramatically great word low growth growth induce great picture stochastic testing treatment effect quantile consist treatment observation control group group year year common describe unconditional density distribution validate unconditional density treatment treatment tail concentrate two unconditional slight deviation distribution get unconditional group er von reject empirical lrr statistic kolmogorov treatment first year sample lie age lie avoid boundary effect sample year validate density package result treatment group validate validate handle level repetition regression quantile level quantile estimate lie particularly group exceed quantile group hand drop age tendency risk reduction old individual benefit treatment heterogeneous age weak quantile group level h turn suggest treatment risk education nonetheless group rise group certain level curve associate group cc tend potential growth spend school note heterogeneous treatment although heterogeneity education age analyse separately condition age covariate setting quantile cc group overlap extensively sufficient find treatment large surface inside cc upper boundary hence tend improve low growth program high growth reduce negative conclusion surface age treatment effect heterogeneous age interesting case occur year conditioning correspond school increment boundary control old material detailed theorem lemma intermediate contain incorporate positive continuously differentiable xx continuously moreover function continuously differentiable xx h continuously xx old continuous old eq b b assumption frequently ec sequence characterize convergence hold dimension smoothness ec relevant tail nd email em em definition proposition construction suitable interest series inequality regression demonstrate band coverage finally national article material bootstrap goodness fit quantile treatment effect nonparametric c analysis inference curve center covariate though even curve finance management tail event thing event conditional covariate traditional way tail curve extreme alternatively moment tail description general regression confidence parametric available corresponding view inference turn observation necessity form check different kind explicit estimate pre kolmogorov type employ deviation smooth display cc band consider technique construct extreme sup center quantile predictor classical histogram one univariate band derivative year growth literature spirit quantile curve integrate simultaneous cover estimation wavelet adaptive estimation bootstrap poor bootstrap density quantile progress confidence band multivariate expansion somewhat extreme portion small classical uniformity set multivariate band go study aspect cover quantile minimum procedure improvement asymptotic generalize goodness quantile treatment quantile work randomize datum participant treatment beneficial individual year treatment word negative evident old spend school unconditional heterogeneity quantile treatment devote investigate simulation numerical propose assumption theory list discuss reference supplement theorems section bootstrap construct devoted issue discuss reference independent random nonparametric quantile quantile xt xx immediate context local curve cc regression theorem constant xx purely suitable curse regressor via linear dimensional think extension offer modeling parametric clearly band influence subsection property point trivial task challenge technique nonparametric add give xx kernel however investigate seem nuisance parameter consider error replace xx function heterogeneity equivalent sample issue residual ig cumulative chapter convergence develop subsection difficulty residual estimator residual base obtain true residual residual bias less residual estimator conditional conditioning condition v nh n nh price pay converge plug suitably speed coverage error asymptotic
difference return completely balanced hardness design characteristic hardness measure easily goal repository provide meta ready meta one store set server name experiment db relational database curve integrate implementation incorporate access without learn schema store store instance result preprocessing store preprocessing integrate current white purpose discard experiment compare implement correctly store machine algorithm gain provide access learn store comprehensive storing well previous result important sense database easy access desire meta ready access researcher help meta specific database database store prediction aggregate instance diversity choose choose non trivial meta deal select set hyper previous although research focus focus machine specific domain lack meta amount resource differ slight implementation thus meta learn study aid problem uci repository refer set meta snapshot underlie user update experiment keep comparison meta set typical meta meta learning instance level study effect generally level important ensemble diversity classifier important create characterize misclassifie work also prediction meta learning unsupervise meta algorithm score meta set recommend behave treat individually training base level meta weight create repository machine information hope bridge prominent experiment database report purpose database experiment store result learn unfortunately curve inherent storing complexity difficult beneficial potential user additionally acknowledge maintain database add offer store meta learn provide set comparison provide repository meta datum ease meta currently algorithm algorithm database section detail give use access result experiment learn instance meta understanding complex least training describe information three file run allow compare example name seed hyperparameter seed seed hyperparameter default parameter practice classifier bp momentum tree differ distinguish hyperparameter setting backpropagation hyperparameter implementation include case unknown meta mapping ccccc parameter l h momentum separate file fold run include column test file unknown represent represent filter instance unweighted instance value represent instance split represent tool fold sense prediction datum meta hyperparameter tool hyperparameter table accuracy acc fold e different seed partition fold provide hyperparameter single ccc ccc ds acc access researcher practitioner meta snapshot history provide even database store feature algorithm traditional database allow expand database store new schema create schema database piece store database collection value pair collection represent experimental la value document information instance store respective collection visualize show set hold collection document output file name number correspond seed document store collection include contain snapshot allow learning evolve machine experimental file modify future meta include repository store meta commonly meta al easily future meta et examine meta represent affect algorithm performance use deal attribute algorithm incomplete attribute ratio variance small measure set identify set value fisher discriminant attribute discriminant attribute class expand instance belong feature overlap tail overlap bound return discriminative attribute region return attribute return previous remove return separability training separable class value extent training linearly separable boundary tree entire return span belong instance nearest instance neighbor neighbor class set geometry topology set create interpolation return create classifier center measure number provide clustered structure attribute modification et accuracy meta include list create
incoherence use mathematically recover informally underlie sufficiently spread low recover sample incoherence follow relaxed form consider rank name decomposition rank cardinality program name q nuclear norm convex surrogate prove probability spatial moreover handle noise incorporate compressive equality constraint relaxed serve nuclear give rx leave right singular name thresholding specific two proximal pg alm applicable pg likelihood comprises nonsmooth pg often nesterov accelerate use pg alm alternate multiplier mc alm subsection eq usually nonsmooth name nuclear theorem pg pg convex continuous gradient repeatedly easy simply gradient nesterov accelerate another modification please initialize k kt pg method solve via solve update denote soft pg fix also technique accelerate implementation augment direction lagrangian alm classical tool lagrangian read alm ascent primal fix marginal turn nuclear norm proximal efficiently solve soft thresholding repeat convergence variable inexact augment special alternate multiplier summarize initialize k alm equality alm apply augment lagrangian proximal alm factorization completion recently nuclear surrogate minimization typical turn nuclear consider nonconvex surrogate minimize model low decompose matrix therefore small rank finally low matrix factorization rank matrix recovery method dictionary summary basically minimize fixing turn least extensively study computer vision literature recovery completion adopt alternate follow efficiently additionally scheme accelerate nonconvex empirical accurately meanwhile theoretical show completion computer adopt high alternating alternate newton update via well similar algorithm reader introduction vision incomplete ill collaborative address square norm name factorization idea use ridge stability follow equality establish indicate minimization study optimization alternate constrain optimize manifold solve denote form name iteratively matrix square theoretical method incoherence gradient follow estimate square develop name optimize framework matrix three type factorization invariance solution underlie nature class trust space factorization structure perform work form manifold conjugate gradient algorithm name completion manifold toolbox name develop rank handle regard outlier traditional biased address alternate minimization carry linear iterative reweighte similar alternate minimization example robust factorization direction instead generalized norm improve optimization alm group sparsity establish address online process incremental tracking online incomplete corrupted cost observed solve square update introduce extend handle outlier robust replace optimization framework matrix architecture large name incremental scheme nearly proportional number adopt strategy implementation first matrix combine subproblem version shall treat pca latent classical pca linearly prior mle give eigenvector covariance large span pca dimension advantage helpful automatically choose inference imply consequently automatically automatic relevance determination ard machine similar treat factorization method handle miss consider likelihood set representative work treat following probability derivation posteriori map estimate turn interpretation correspond impose modeling regularization predefine determine introduce later full method pmf markov chain change factorization laplacian model prior laplacian large term connection propose observation give moreover hyperparameter probabilistic include etc probabilistic factorization determine serve model hyperparameter play automatic determination extremely drive category rank constraint optimization often greedy project constraint conceptually name propose use scheme project intermediate result theorem similar denote os turn os exist decrease small convex alm iteration notice factorization fast pass method vb competitive parameter curve shape remain high os unchanged factorization shift indicate attribute directly depend case existence factorization relative attribute relaxation introduce remove consequently especially prove figure decrease os os curve rate influence besides recovery ratio proportion rank completion recovery program achieve noiseless know stable version test minimization mention subsection variational overall require perform drop surface dynamic contour shape track move object intuitively low underlie hence recover low many recognition face person show intensity image face recognition characterize face matrix input image face person image low face could remove correct face component face boost dataset surveillance include perform frame background video detecting stand background underlie video camera unchanged illumination variation matrix compose image foreground clean traditional presence foreground illustrate image foreground background include moving reconstruct low show appeal capability modeling spatially contiguous foreground use smoothing segment trajectory foreground cause camera motion lrr subspace dimensional subspace whose affinity combination neighbor estimate eq encourage wise orthogonal point affinity learning dictionary dictionary represent combination claim intuition signal block column order rank minimize transform transformation estimate align difference compare represent single assumption character reconstruct texture camera character etc widely analyze track video seminal observation track camera e perspective perspective coordinate motion object across point frame motion addition possible low solve frame frame frame pose adopt art method shape reconstruct limited basis give measurement constraint tracking feature across frame motion segmentation track multiple feature track group track belong discuss track rank therefore segmentation formulate subspace divide track detailed please completion desire reconstruct lose text approximately corrupt formulate illustration completion image randomly sophisticated texture recover model sparse unknown detect corrupt adopt minimization nuclear norm video video group group share finally completion patch group low coherence image noise removal medical analysis denoise mr usually frame image diffusion image suppose significant component classical achieve denoise importantly threshold theoretically stein bring great original shape statistical candidate shape shape denote consist vector describe training datum candidate often shape shape model moreover pca shape model active appearance shape method build alternative make segmentation similarity shape nuclear minimization rank modeling imaging coherence image mr imaging concept separability ps mr image spatial component correspondingly tu notation coherent space reconstruct sequence basic integrate specific wavelet total meanwhile temporal periodic model dependent locally ps nuclear dynamic imaging imaging modeling multi image modality ct emission example rank model detection parse fusion image paper concept rank review representative additional reading reader factorization low approximation convex programming noiseless case exactly true signal noise shrinkage try nonconvex relaxation requirement repeat svd make computation svd approximate widely real recommender mostly computational convenience factor moreover cost function variable online process technique work probabilistic great real probabilistic real knowledge purpose extract remove etc recent sparse powerful framework technique rank expect acknowledgment manuscript image yu refer interest rank achieve success processing bioinformatic convex programming completion apply collaborative topic recent advance overview concept rank model challenging advantage limitation application context dimensionality great document natural customer recommender bioinformatic fortunately high mathematically big translate correspond signal intensity array reader real raw low recover conventional approach ij respectively minimization interpret analytically correspond vector
tail sum base devise label example parameterize compute discrepancy true label vector predict approach capture widely adopt many multi label form constant prove show determine rademacher complexity tight rademacher motivate objective eq large value control get multi multi label rank correlation label regard minimization may incorrectly zero contour unknown trace range propose constrain derive singular successfully discover norm fail start regularization approach far reformulate linearize regard approximation term problem iterative ignore trace solve singular thresholding handle let svd diagonal ml base set evaluate evaluate average rank large value error jx auc area roc accuracy auc compare label increase label appropriate achieve predictor important trace structure compare three implicitly design structure ratio summarize improve nearly rank discover norm low rank multi predictor encode multiple predictor obtain increase number local complexity tight generalization unseen example behavior dataset confirm conditional singular solve principle guide new erm algorithms discover sum multi predictor inspire complexity rademacher tail tight experimental label complexity risk minimization erm base trace norm instead tail use minimize singular predictor play exploit validate effectiveness rademacher example one conventional many categorization annotation gene straightforward decompose series binary label poor number different classifier margin multi learning dependency removal learning theory justify successful label dependent measure dimension cover number rademacher erm trace explanation effectiveness multi hand implicitly exploit correlation rademacher favorable subset drawback rademacher consider result rademacher complexity rademacher complexity seek rademacher complexity erm multi label sum predictor motivate tail value predictor rather I trace advantage multi predictor predictor exploit learn result function efficiently newly thresholde real validate effectiveness new training distribution function loss global rademacher measure complexity class global error estimation pick subset instead rademacher complexity reasonable intersection center function rademacher global rademacher describe error give every variance lie error theorem rademacher complexity analyze complexity multi illustrate motivation develop multi model
computing reweighted weight minimization exact method share determine automatically yield exact equivalent instead heuristic particular weak truncate nan subproblem solve combine bfgs cg problem solve hybrid bfgs cg effective penalty propose zhang quadratic numerical decomposition solve whose vector arrange give open neighborhood center section verify minimization eq programming constraint variational characterization original increase statement hold locally solution locally locally open vx associate arbitrary feasible vx e solution globally unique solution neighborhood x sign locally conversely locally optimal hold part cm equivalence solve solve penalty penalty develop paper study penalty linear program give optimal say sequence sequence imply disjoint q moreover inequality hold e e e nonempty globally solution single problem solution coincide need since prove contradiction present arrange conclusion clearly hold lemma proposition follow hold I use arbitrary last rx r I complete cm solve unknown advance problem nonconvex solution fix motivate propose tolerance choose cm otherwise stop go go solution nonempty introduction feasible feasible converge bl ba ix b yx v cm termination result terminate subproblem algorithm know q terminate terminate cm result end write nan satisfy vector satisfie condition step nan argument yield next induction satisfy note equality inequality show nan use nan use successive vector equal together iteration nan space condition devote subproblem involve nonnegative one order cone software however consume suitable motivated recent partial proximal follow partial approximately first subproblem via denote function r respect gap approximate solving yield ty kx k v problem minimization summarize favorable continuously q solution map jacobian satisfie hx x note follow continuously continuously gap primal result expression replace cm know form cl I bfgs yield scale continuously differentiable need bring newton find root nonsmooth scale problem conjugate newton cg given go next step cg linear seek integer hold set step positive direction always direction convergence reader may approximate experiment form remark diagonal otherwise subproblem bfgs cg alone subproblem bfgs good feasibility newton meet involve develop subproblem solve subproblem solve bfgs step r penalty decomposition method large suitable bfgs ty end k start use k cl solution account convex surrogate iterate feasibility sequence bfgs good find turn sequence unless involve store choose employ bfgs solution step l bfgs minimization involve algorithm testing slow terminate l advance turn subproblem unless default search subspace matlab windows operating system ghz cpu gb verify effectiveness compare return solution zero denote component iterate yield solver remove entry entry small nonzero test algorithm problem table magnitude lie problem realistic include six solution limit problem magnitude htbp result solver e e e e solver product involve relative recover record four solver bad incorrect set subsection sense number gaussian whose distribute qr element independently hadamard column dct large six zero normally law decay decaying whose matrix store explicitly matrix type store unless state sequel signal successfully relative original htbp htbp take influence four solver type took test solver four solver solver signal little well testing display six kind see type among solver take illustrate solver signal number curve figure average successful much high less solver comparable even six type comparable even signal noise test signal noise ax identically constant choose algorithm htbp take example test recovery error solver consider take randomly curve relative solver vary type signal type algorithm desirable solver little yield residual attain whereas average residual yield word yield type compute high three signal type compare solver type took randomly test recovery solver vary number signal little together superior computing subsection collection report desire feasibility less require almost problem yield problem desirable also yield good feasibility zero norm yield numerical comparison solver collection solver e e e e e e htbp c c e e smooth numerical subsection conclude even superiority collection particular require time much bad
proportional approximate method quantification approximate assess denote random target extensive next functional neutral dirichlet process simulation study consider survival moment explicitly evaluate compare exploit consider survival illustrative purpose coincide specifically beta set jt end section importance section obtain high exploiting moment true one shape approximated distinguished finally fit truncated normal density conclude instant numerically integrate measure beta exploit average nonetheless apparent incremental gain moment respectively numerical instability good accuracy ht combine characterization together gamma expression characterization independent gamma prior survival estimate equally spaced interval distribution us survival median survival interval principle functional step draw summarize hyperparameter posterior moment investigation reveal second sampler describe admissible latent set l l approximate n st sampler every exploit remarkable survival cdf c subscript devise cf side sum depend nonetheless crucial sufficiently estimation functional meaningful median st I mode pt credible divide part survival focus real median credible four observe observation survival plot investigate credible region true survival concentrate investigate performance methodology grow summarize credible credible around length interval reduce close h ci involving time patient time patient treatment star detail censor mechanism adapt right censor observation estimate credible figure plot estimate posterior different behavior mean optimistic posterior median worth must large censor different depend specification nonetheless show capture posterior rely sampler refer panel compare estimate underlie completely gibbs sampler detect credible significantly credible moment credible interval avoid credible survival great panel figure plot effectiveness treatment credible two credible plot credible line comparison black credible european provide integral eq gamma denote conditional adapt right censor observation notation posterior change censor censor time accord censor rewrite observe censor carry case right censor replace jump component occur function result corollary straightforward inferential markov variety suffer limitation functional goal present methodology extend hazard order inference survival limited include remarkable credible approach rely moment polynomial inferential performance methodology mean tailor survival adapt hazard approximation survival inferential rely characterize marginalization element define variable henceforth refer besides identification markov method approximate evaluation form typically predictive mixture dirichlet estimation hazard become well establish practitioner implementation stress relevance worth point suffer drawback easily posterior marginal suitably endow credible choose choice absolute median mode preferable median nice issue provide focus trajectory truncate stick present paper aim propose combine close method approximation posterior develop estimation survival hazard rate function cumulative hazard fs fs ft area soon devote class accommodate rigorous bayesian inferential survival mention neutral cdf conjugacy drawing inference also benefit conjugacy propose full specify hazard popular gamma originally generalize random hazard mixture recent allow quantity statistical marginalization identify see quite implement hazard rate survival mean sample rich complete understand exchangeable survival I pt random suitable instant along posterior allow straightforwardly moment indeed use integrate approximate evaluation almost turn one survival st set gamma continuous kind gamma finally posterior kernel ease exact extension case censor straightforward latent joint transform among display result let coincide hazard evy jump jump display jump coincide jump insight see obtain value conditionally thus point introduce point introduction combine representation alternative aim trajectory involve distribution trajectory approximately since illustration survival achievable trivial issue evy conditional latter address augmentation scheme suitable recall realization jump section marginal goal bayesian effort minimal compute posterior evaluation yield expression moment conditionally technique next hold kernel monotone hazard rise hazard property generality notational display integrate end describe explicit knowledge moment receive great motivating application therein interest motivate determine density distribution
index rx x functional dependency affect connect configuration unobserve superposition unobserve whose column unobserve configuration capture residual product loading identification sparse mean vertex affect zero I seek introduce lagrangian element infer network np combinatorial approximated method novel degree freedom invertible determine minimization full expect svd tp sense sparse use outside support relevant know exclude unnecessary explanatory variable base compute problem infer compute eq constraint introduce degenerate reduce problem still combinatorial heuristic enhance solution propose approach I define index matrix index compute approximate right small perturbation singular give small angle vector iteratively remove converge equivalently large index singular unit sphere sphere unit ellipsoid ellipsoid singular solving require move origin choose index choose small ellipsoid selecting eliminate remove component move small vector removal max ij j compute vector max max single one apply thus replicate choice inferred generally go least solve matrix absolute pearson recover less stability reflect connectivity vertex topology great practice direction require update kk usually work increase solution close old notable memory step compute previous computed inversion take impose variable latent dense demonstrate theoretically test numerically well absence scenario structure simulated random bipartite network encode unobserved variable simulation adjacency infer close versa two limitation permutation many instead choose learn small mean degree avoid algorithm recover test easy slow rest less law uniform advantage probabilistic rather response factor accuracy network observe unobserved fig configuration fig vertex increase poisson choose mean wide h degree case compare improve iteration improve fig scale algorithm hold speed range use analyze unobserved simulate infer assume element close truly unnecessary valuable physical corresponding yet correspondence outline practically situation state unobserve situation modification versus observe assume state measure number index enough determining follow pearson infer likely correspond physical u similar enhance interpretability physical outline rather element significance fraction know come quantify observe alone hyper geometric know provide excellent rigorous evaluation rigorous biological activity limitation throughput scale state identify apply follow platform unobserved directly adjacency overlap tf h identify common gene third column observe chance list go infer correspond network many detect limited experiment perform gold partially topology algorithm interpret penalize edge predict new paper comparison allow identify experimentally crucially dynamic logic gene paper aim infer sparse infer degradation g salient distinguish inclusion ultimately computationally compete sparse acknowledgment thank discussion extensive constructive david david work research engineering technology rgb rgb rgb develop case observe fa decomposition underlie novel accuracy noise scale efficiency method k svd component recover exactly unobserved decrease range noise increase analysis fa decomposition useful systematic allow interpretation variable fa problem
transition consecutive point secondly since noise scalar degenerate cope difficulty drive variable dt equivalently express markovian latent motivation latent result significant mix gibbs dependency pair remain improper prior infect figure half batch write bootstrap kernel current law forward system mean exclude hx hx establish induction equally draw weight markov particle b f b b b line since bound induction sl bounded statistics california berkeley technology tool inference sequential present analyst highly enable mix particle reduce computational burden typically conceptually backward suited dependencie markovian nonparametric thing systematic sequential carlo analysis dynamical wide scientific linearity originally indeed decade process markovian model various either marginalization discuss tool useful however combine sampler make kernel rely markovian stochastic relatively area finance build construct sampler trajectory trajectory effect reference leave target regardless particle however suffer serious drawback mix kernel path degeneracy underlie unfortunately degeneracy high problem applicability address generic add simulation sampler yielding denote considerably much problematic model markovian trajectory backward pass degeneracy modify thereby achieve sampling explicit kernel publish illustrate use problem theoretical validity ergodicity markovian illustrate example applicable severe indeed section direction work unnormalize tx draw enable carlo draw sequential nature suggest use filter start standard consecutive meaning let particle system weight proposal complete notational dependence particle resample particle define particle proposal weight give initialize assign x sampler summarize draw w generate explicitly I natural turn construction trajectory generate thus state review algorithm introduction serve trajectory informally think simulated particle pass sample particle implicitly event reference trajectory retain pass coincide reference trajectory sampler invariance precisely particle return property hold note keep reference result leave recognize degeneracy fundamental turn new idea improvement considerable implement trajectory range current history connect particle encode connect path assign refer form trajectory understand importance formal invariance outline shall modification mixing map stochastically family index trajectory set set w draw ix w k argue mix illustrate numerical option pricing assume observation give smoothing employ particle simulate path mix report reveal significantly well view poor rate degeneracy sampler process pg reference trajectory system clarity illustration respectively blue retain throughout trajectory red extent identical perhaps much insensitive understand system generate thick line particle index line effect informally reference trajectory broken piece point prevent path degeneracy particle degenerate something red probability substantially enable update blue line dot line particle line degeneracy particle grey dot panel trajectory piece degenerate toward something different view state show invariance violate sampling kernel invariant apparent establish particle expectation possible however treat auxiliary thus avoid intractable integration view trajectory index recursively variable function density refer intend extended factor important marginal invariant kernel kernel partially collapse meaning basically refer process variable gibbs sampler invariance algorithm implement partially collapse instrumental draw collapse conditionally construction respectively propagation expression use plug expression numerator consequently step analogously leave conclude procedure commonly within simplify carry correct dependency procedure collapse conditional full clear variable never subsequent sufficient furthermore collapse gibbs leave law procedure lemma recall x b procedure conditionally law particle matter reference position imply give desire ergodicity target obvious modification basically cover support ergodicity establish argument apply irreducible ergodicity boundedness assumption basically slightly exist ergodic condition ergodicity normalization discard equally analogously make integration important illustrate nonlinear gaussian eq wish latent prior seek simulate sampling posterior recognize poor autocorrelation block invariance draw n also maximum q maximize maximize auxiliary integral intractable involve particularly auxiliary make detail smooth intractable general recognize distribution auxiliary simulated particle summarize draw run pf non standard nontrivial density x understand bayes likelihood also recognize step highlight use pass particle replace line extract trajectory backward implicit denote turn case joint bootstrap internal particle proposition build bootstrap backward simulator appendix handle trajectory particle filter establish equivalence outside class sampler variable model eq share conditional independence property history specifically markovian implementation truncation motivate area classical replace processes markovian include regression non nonlinear successfully marginalization part rao result useful instance drive sampling transition many simulator transition exist illustrate markovian statistical among probabilistic model markov employ backward need bottleneck markovian hasting know step retain sufficient leave invariant kernel propose move simulate eq sample accept set keep weight whenever moderately large variable recommend construct ensuring would operation force move mh force prohibitive markovian decay past markovian weight assume sl divergence compute quite useful inferential truncation truncation increase converge experimental tv level exponentially decay remove requirement introduce design easier small rapid change well mix evolve take truncation stop evaluation early accurate approximation dimensional right dotted respectively overlap leave vertical show result scheme way replace computational simplicity truncation another use proposal mh suggest accept mh evaluating implement property linear mixing consider example markovian depend illustrate inference degenerate reformulate markovian initial unknown superior parameter sampler simulate system time step discard sampler sequence row hold sampler require drop much hand robust comparable ideal sampler consist clearly difference sampler degeneracy result agreement discussion leave top row ideal admit density problematic backward sampling degenerate backward first equivalently degenerate state measurable markovian simulate single joint smoothing non markovian possible backward truncation
precise previously element query observe query former process identical query support amount single minimal low pair pair mass preserve detail result effectively intuitively modification unless match formalize support obtain crucially significant aspect difficulty bind obtain regard technique decision appear conceptual turn bound family distribution query tuple sample identically last analyze query actual size intersect one front indistinguishable uniform set adaptive essentially give big query prove return query indistinguishable obtained consider separately indistinguishable uniform spirit follow guess reference upper obtain double exponential refined new idea supplement effect dependence know therefore yield low size al rest proof shall cover uniformity test upper support reader independently logarithm probability countable ss sx sx testing set reproduce conditional oracle choose oracle element si oracle deal choice situation always include put outside support hereafter et uniformly use tailor let call either output follow definition straightforwardly independent goal property deal precisely resp distribution allow possible focus draw get relation make sample et al show invariant convert core core tailor setting form atom denote generate capture something like summarize label information obtain ta call jk sample summarize aforementione argue algorithms access configuration albeit algorithm act internal randomness configuration query obtain j I query refer query specification draw random whose atom element require previous fully besides already decide belong atom put differently belong one stage stage either ever access label know intersection actual identity contain chapter principle deterministic draw randomly apply instead work analyze work importantly randomness external behave core randomness sample external therefore stress external way via element random affect ease observe intersection q apply intersection prove attention intersection show indistinguishable simply pick element implie indistinguishable like correlated concentrate around expect consequence expect invoke eq intersection simultaneously concentrate selection element ns ss satisfy q ns adaptive break recall query intersect receive thus establish support describe analyze shall help support enough probability reference verify inside support last check small significantly big actual query output exist access input either subsection stop compare keep reference doubly guess support meaning repeat call j j rest formalize rigorously conditioning indeed pass reach step bind return subroutine subroutine return claim query query call j th cost query similarly call overall query claim hereafter output meet requirement fact ensure significant fix nx nx minimum suffice rearrange work get enough issue indeed element care support want case support perform therefore use increase detect work give indeed estimate heavy end fine since support failure time vote j kt jt jt multiplicative chernoff call specify happen break analysis happen get call routine comparable e probability factor unless support mark discussion return conversely mass less point mark overall fail probability good call guarantee repeat majority vote success step setting repetition overall subroutine fairly enough uniformly conditional mass mass h access ks fall behave return latter least claim complexity therefore turn subroutine derive random intersect r enough repeat sufficiently whether indeed roughly repetition apart return vote call ms step outer repeat vote repeat overall query sketch derive upper adapt level perform precede identify great guess al find return estimate return intuitively time pick set element non adaptively increment call correctness outside support remark lem prop theorem question acknowledgement dot ref ref sublinear support cl property sublinear sampling low enable allow condition domain explicitly support size contrast require whether hold establish bind equivalence testing recently et whether dependence domain explicitly pose sublinear answer conditional interestingly qualitative turning investigate size doubly generalize weak logarithmic lower bind carry uniformity author necessary adaptive end probably far seem technique like conceptual nice without discussion author decide plan perhaps discussion loose end fundamental seminal al computer science particularly set decade explore well often complete property sample optimal testing recent year situation specify subset formal need thereby sample prove testing extremely namely distribute outcome develop closely uniformity testing hereafter oracle access oracle decide far decide oracle decide factor impossible trivial generalize least show complexity strong uniformity testing provide allow make give oracle access distinguish uniformity understand uniformity sufficient tight uniformity additional suffice namely task require polynomially focus query show even query lower open logarithmic answer uniformity question define access unknown equivalence study decade equivalence upper need open possibility uniformity admit constant equivalence bind equivalence weak conditional oracle restrict pose sublinear answer possibility show sufficient approximate et al obtaining require almost subsequent question establish bind multiplicative distinguishing knowledge getting factor version yield intrinsic equivalence testing estimation uniformity framework
eq fact eqs j dropout study weight drop unit unit probability drop weight iii drop input layer hidden layer hide activation function many use activation assumption full connect type I denote eqs networks network hide f element draw I I show complexity drop weight give complexity apply result interest miss rs cauchy rs ne n hold jensen rs ne rs ne inequality layer rademacher complexity irrelevant unit dimension eq k number layer easy hold network rs ne lipschitz give term empirical rademacher another layer respect nr rs hold layer prove combine b j similar complete real implementation develop aspect deep far dropout effective strategy reduce intuition adaptation detector work study rademacher complexity type result none rademacher polynomial whereas neural lead rademacher fundamental type current light way claim algorithmic implementation aspect deep interesting usefulness motivated intuition detector type dropout theoretical network dropout deep neural deep network wave recognition speech recognition many however theoretical aspect far complicated million risk even control overfitte long research various weight early bayesian success main randomly omit hide execute certain remain overfitte though encouraging phase dropout able improve reduce theoretical dropout clear influence rademacher dropout hide rademacher complexity polynomial deep rademacher complexity relate design dropout dropout whereas unit hide dropout dropout et regular inverse generalization dropout analyze present pac et al derive rademacher keep element dropout generalization dropout reduce complexity neural neural dimension complexity polynomial complexity parameter worth function complexity influence complexity weight irrelevant unit follow presents general rademacher usefulness different classification throughout restrict make neural respect depend network dropout unit necessary introduce different dropout drop weight network dropout give objective introduce performance output entropy goal minimize risk r draw rs risk try rademacher classical chosen rademacher develop datum task notational simplicity w I hadamard generalization mostly affect thus rademacher dropout however generalization thus generalize rs rademacher define easy rademacher rademacher generalization dropout follow sample dropout easily find dropout every n
product subspace similarity angle subspace cosine approach database product aim focus compact e represent objective include three aspect accurately query code evaluation compact code quickly approximate compositional summation investigate compact code inner approximation code compositional database short code propose approach exploit inner operation compositional operation compute handle vector approximation product approximation approximation property inequality vector query difference product inner related meaning euclidean query query approximation potential basic compositional combination quantization aim database application mean jointly optimize basic code compositional produce form call separately combination code source combination vector combination size write combination combination concatenation nm furthermore extend scheme element set selection selection produce composition differently element form compositional length keep represent code zero consumption computing become large scale increase increase compositional dictionary sparse problem optimize transform quadratic w algorithm paper zero acceleration separately subproblem dictionary subproblem minimize hard propose greedy dictionary determine source selection dictionary good previous issue vector similarity database approximate code nm construct dictionary search handling compute relatively compare search sift search sift similarity selection formulate equation constraint ni nj nc formulation extra constraint equivalent mean case summarize compositional transform one successively regard mean combination guarantee validate solution feasible selection dictionary compositional four another generally compositional dictionary summarize combination case kk small inner follow complexity iteration updating multiplication involve code whole respect code sift reach intel cpu hz also benefit parallel acceptable sift supplementary material quantization correspond source source dictionary lie comparison permutation order closeness vector thus regard divide sparsity impose approach sparsity sift netflix lm query conduct sift sift linear lm netflix sift sift descriptor database vector sift descriptor contain sift vector lm dataset around linear query image engine sample feature query rating give movie aim inner rating user give movie recommend pca description quality fraction retrieve among inner evaluating return result perform subsequent item neighbor inner compare database raw compositional code three search neighbor fix source encode number bit selection observe result search compositional group compact include product quantization eigenvalue allocation equivalent quantization random projection locality lsh design cosine see superior nn improvement close show sift consistent improvement achieve bit use indicate achieve hash hashing lm inner product aim find query fit mean relevant retrieve view soft attribute apply search large provide fast flat hierarchical tree recently see netflix rate user apply rate perform much well bit code little come code perform second classifier classification dimensional signature compression ram raw large memory pc raw thus closeness contain recognition conclusion table achieve svm kernel inner product table inner symmetric symmetric sgd algorithm really product addition preserve similarity evaluate search average bit cc bit score product compositional accurate evaluating approximate computationally future generalize euclidean rewrite present show approximation absolute product dictionary quantization rewrite center entry extend quantization space rotation optimize product approximation equivalent analysis product view source dictionary case subspace rank near closeness accord towards distinguished anchor order closeness contrast find partial permutation rather representation compute aim coding find basis code fix sparsity group code approach divide sparsity main paper dictionary code group update time update close operation multiplication multiplication matrix matrix transform om dictionary select selection dictionary contain vector thus time four dataset average approximation composition production error approximate near table observe vector inner error another achieve c c sift netflix cc c sift netflix cc product cosine similarity database norm approach without way maintain extend spherical clustering learn accuracy product addition hold cost evaluate euclidean code produce little subspace si equation show give database thus table item c inner orthogonal map corollary microsoft china address near
third order vector respectively application learn et al liu algebra b especially rapid technology tensor become increasingly multi image video work datum arise due memory call curse dimensionality underlie though due major govern relatively tool curse tensor order tensor small commonly take tucker tucker cp find tucker decomposition tensor decomposition approximately reconstruct hope tensor three challenge selection tucker base give rank involve tensor liu problem decomposition attention problem video reconstruction matrix tensor show capable advantage provide completion nuclear addition theoretical development guarantee partial reasonable condition al progress recovery trace decomposition trace require addition improve scalability relationship trace tensor tensor cast convex regularize high orthogonal iteration scale direction multiplier admm solve fast insensitive robust extensive notation detail th high mode vector ni index map ni two sized tensor respectively denote tensor j ji popular cp decomposition well cp compute high tensor tucker tensor core multiply along core much small tensor g tucker decomposition tensor storage decomposition significantly rank unfold particularly tucker decomposition analysis width tucker factor tucker orthogonal al iteration b good rank general well possible tensor goal norm difference progress cast minimization trace value regularization handle satisfy trace due trace difficult introduce equivalent parallel direction method solve minimize augment lagrange parallel updating multiplier parallelization modify variant parallel proximal solution eq et al solution proximal unfold proximal term show description algorithm use block variable n verify n nn nu n k let n converge solution rank tensor problem tucker employ solve fix imagine optimize consider well optimal value repeat alternate solution clear sized matrix k nr norm computational complexity nr convex iteration outline hence attractive admm adaptively strategy lin al initialize validate rank converge update multipli n algorithm operator multiplication proximal operator operator ni tensor synthetic real world datum experiment core tm truth tensor tucker decompose art et al et normally tensor regularization method tensor datum vary three four much accurate solution outperform efficiency size rank time color kind algorithm important relative regard depict plot tensor increment increment trial perform htb liu al approximately singular large time trial attain relative fast convex addition tensor analyze low cast trace solve experimental regularize method noise outlier tensor decomposition handle useful comment support grant microsoft thresholding processing reduction multilinear parallelization alternate multiplier n via recovery condition explanatory modal splitting lagrangian structured inequality tensor multilinear b approximation lin liu linearize alternate estimate missing value visual minimization rank decomposition h li trace recovery j l j spectral decomposition decomposition via tucker three truncation singular zhang u convergence behavior behavior residual k k iteration provide residual drop much fast iteration relative residual drop quickly estimate rank experimental size rank estimate rank unfold tensor ht plot datum tensor rank varied increment increment generate choose uniformly give tensor size iii u accord property nu u nu
idea tv place somewhat difficult employ place derivation vb go similar fashion tv fact r ni mode formula vb isotropic tv detail mention model issue pixel index variable issue solve numerically value handle deal issue lasso approximation laplace heavy tailed origin similar approximation positive tv functional gradient optimisation method tackle thresholde hierarchical student prior freedom since density hyperparameter present smoothness smooth distribution derivation tv sum write regression invert vb vb problem block structure covariance inverting domain use approximation could similar vb domain matrix nice iterative conjugate drawback step system precede author could vb ignore vb appear corresponding mode density variant quite formula usually hyperparameters done style solve em problem brevity laplace difference prior aspect could laplace discuss freedom gaussian noise image reconstruction present divide comparison penalty deterministic tv method transform quadratic moderate image mask reconstruction hierarchical provide mostly equal slightly optimisation whose multimodal maxima partly improper good test quite near study optimisation determine penalty formulate prior mixture property formulation tv parameter assessed mean posteriori derive method less flexible slightly test tv tv tv tv work edge preserve comparison reconstruction cm vb sampling compute intensive vb region develop vb solve iterative satisfactory algorithm might become improper improper prior even hyperparameter dominate dirac delta study simplifie although numerically comprehensive study hierarchical implement test although separate typical scenario method give also method tailed image relate fact generalise denote parameter unimodal skew moment use formula certain asymptotic formula reciprocal give gamma gamma define laplace distribution q agree parameter see convenience exponential curve exponential square root multidimensional inspire laplace proof similarly degree freedom x tv vb cm mcmc vb sampler opt deterministic optimisation tv image tv model tv tv laplace prior f tv tv tv laplace prior tv tv theorem laplace gaussian prior total variation approximation find bayes method alternate compute automatic preserve promising result difficulty encounter future model variation subject classification variation initially alternative non useful want copy total example differentiable origin bayesian tv information result assess inverse study study paper penalty interpret laplace tv set laplace compressive study variational use hierarchical also tv fast optimisation g solve tv tv propose fix optimisation auxiliary trial estimate inverse scale isotropic also inverse posteriori although fully method present laplace mixture encourage try mix penalty know tv prior tail useful edge preserve scale inverse gamma gaussian interesting laplace multidimensional tv prior model paper familiar hierarchical formula vb section technical remark work discuss appear classical linear discrete solve measure formulate nontrivial solution even singular numerical try introduce term ill pose penalty optimisation simplify formulation deterministic approach proper choose dominate term ill optimisation assess notation image column stack special notational indexing presentation stack array pixel notation pixel tv discrete total functional tv boundary boundary could tv allow tv version might easier deal origin version rotation invariant isotropic tv isotropic tv tv isotropic tv penalty link study pixel bayesian optimisation tv detail idea variable g fix ordinary character denote distribution information prior x usually estimate also depend hyperparameter nuisance think hierarchical involve example specify consider tv jump connected control conjugacy parameter principle available value specification appear choose could improper integrate improper improper posterior nevertheless improper later kind emphasize minimum trivial tv challenge discrete tv conditionally laplace random variable normal place adjacent pixel hyperparameter priori easy see formula r convention root take component wise direction consist difference discrete operator naturally basically allow satisfied acyclic show mainly tv generality derivation follow tv tr show indeed notation tv alternate maximize keep variable loop value derivative optimisation present formula simplify use value solve compute sx derive approximate posterior conditional density sampler sensible analytic solution slow typically base subject converge compare approximate posterior pdf partition factorize eq parameter
des universit propose translation traditional neural aim jointly tune maximize machine encoder sentence decoder generate translation bottleneck improve architecture allow part sentence relevant target word explicitly approach achieve performance comparable phrase english qualitative reveal find agree intuition newly unlike sub tune translation build translation encoder decoder language neural read length decoder output translation encoder decoder language correct sentence issue sentence make neural cope long show rapidly length sentence increase address issue decoder align translate concentrated predict vector target distinguish basic encoder attempt encode sentence length encode adaptively translation fix length cope paper translate achieve improve basic encoder apparent long sentence sentence english translation translation conventional phrase qualitative reveal sentence sentence perspective translation equivalent finding maximize source translation parallel learn translation search recently paper directly encode sentence rnn source sentence length already report neural machine translation term art phrase machine english translation add exist phrase candidate allow art propose upon build novel architecture align translate simultaneously read input sentence beneficial hide predict predict word probability conditional potentially rnn architecture convolutional translation rnn search r architecture encoder approach context word annotation map input word detail annotation annotation eq weight annotation alignment position state alignment feedforward jointly unlike traditional machine directly alignment whole model understand weighted annotation computing expectation probability translate word reflect state intuitively implement attention decoder sentence attention decoder encoder burden throughout annotation retrieve decoder accordingly usual read last annotation word summarize also follow propose successfully speech recognition backward rnn read h xx result backward h xx summarie tendency input annotation focus sequence annotation decoder compute eqs illustration evaluate english translation corpora rnn encoder model english corpus word reduce combine although possible encoder news usual frequent word map token apply ji arbitrary select among length type decoder sentence sentence decoder encoder consist neural rnn single maxout hide word minibatch descent together train update minibatch sentence train day translation approximately neural architecture c list translation importantly conventional sentence word consider propose basic limitation may encoder dramatically drop sentence length sentence performance sentence superiority model basic decoder fact intuitive generate row weight annotation position consider english monotonic differently english fig see translate phrase european economic align area european economic back phrase alignment oppose evident source phrase translate translate le naturally look see able correctly translate synthesis ask character coefficient alignment alignment predict increase weight annotation machine need translation drawback translation applicability scheme probabilistic neural however network largely translation system exist feedforward network phrase additional phrase machine translation recently report network translation traditionally target although approach objective translation consider work work generate translation sentence conventional neural translation approach length problematic recent architecture address basic computed target encode source sentence focus target neural translation long unlike traditional piece towards produce propose english translation reveal outperform decoder regardless sentence length source able align relevant annotation sentence perhaps importantly comparable statistical translation consider architecture whole believe architecture promise toward well translation understanding language general challenge leave future match art machine translation context acknowledgment thank acknowledge thanks hill van framework recurrent alignment describe experiment activation hide conventional simple short unit share backward easily vanish possible lstm new employ eq element multiplication output update represent omit maintain much sigmoid use hide maxout normalize one need sentence layer w nu h pre minimize detail code output sentence vector language respectively length source sentence recurrent w mu embed number hide unit logistic sigmoid rnn matrix backward annotation eq decoder annotation encoder language weight word embed dimensionality initial nc alignment annotation v nu weight matrix encoder decoder decoder maxout model word embed dimensionality maxout unit hour gpu matrix rw initialize weight initialize sgd automatically explicitly predefine minibatch update require time sentence minibatch every retrieve length split table statistic model experiment htp p cm right admit patient medical centre carry diagnosis status health care worker le le en un patient dans un un centre un diagnostic un un est le de un patient un centre un diagnostic un diagnostic en est un un patient un un centre pour un diagnostic une
hold side apply get upper moment q conclude remain technical ready uniformly lead identically need set assume supremum first follow depend excess statement surely last going subsequently employ hold great great great inequality minimizer set step auxiliary assumption great combine analogue centre department university university sampling replacement broad learn study theory excess localize convergence localize prominent case class localize empirical class ingredient empirical interest role theory theory sup heart advance localization chapter yield excess base theory implicitly many machine many case I violate test different point typical computational biology area paper example sample replacement population sample replacement unlabeled predict label naturally appear text recommender system effectively constraint inherently realize system world image categorization area recognition manual inspection label unlabele e image several bound still remain good knowledge rate bayes assumption restrictive contain localize hold hypothesis view analogue common risk inductive setting error take function thus yield excess achieve prove process go concentration tool instance arguably prominent without cross fold replacement pool help non asymptotic cross procedure investigation novel inequality outside scope paper lie protocol inductive draw space choose predictor fix hx nonnegative use setting sample without replacement output remain unlabele label denote respectively l u mh empirical datum reason regard n play role quantity invariant partition test use define hypothesis obtain tight bound paper another commonly obtain generalization minimization note excess obtain risk bound bound introduce deviation side relate sup analyze fundamental obtain probability variable replacement particular concentration sup author general present implicit error certain belong u also attain several pac learner algorithmic roughly mention concentration yield rate present without replacement result closely concentration base first suggest notation finite replacement countable countable refer page page sampling without replacement use random remarkable type version due refer understand lack inequality present without replacement also hold great replacement hold appearance might want expectation however lemma order control could prefer worth novel begin obtain setup theorem write inequality mn replacement say theorem material frequently bind compare bind constant thing provide drawback preferable use theorem question whether least significantly tight theorem far present appendix briefly outline proof supplementary material consequence although result instead concentration refer proof inequality reference define partition finite set set inequality mathematical learn validation factorization procedure learn localize yield complexity bound apply concentration obtain risk set nh mh u mh nh nh nh nh play f nh h hx nh mh sum note random center random learning boundedness immediately calculus material sup naturally inductive upper process quantity mh follow theorem repeat proof follow corollary define corollary convergence corollary us relate rademacher contraction inequality section result excess yield fast reduction loose extent localize bound modulus associate tool end sub concentration theorem convenient loss class satisfy discuss common satisfied instance n hx nh nh nh uniformly bound relaxed theorem unbounde unbounded depend whether use notion nonnegative positive let satisfied root point emphasize appear well one appear share instead appear bind satisfied sub modulus continuity empirical replacement large briefly outline part theorem section rescale center class ff variance obtain slice able also sub root definition obtain finally use great replace excess great minimizer repeat error intermediate obtain detail present key apply develop inductive though apart tight multiplicative constant point regime potentially bind fact leave excess satisfied variable ij n uk depend result direct section decay question state assume reflect gram evolve grow generate think possible adapt relate asymptotic counterpart broad nonnegative excess risk learn localize class excess risk
trait value show ari us group c ari parsimonious membership party membership table selection bic component trait party misclassifie mainly mainly response individual group exception issue slightly vs voting versus l response individual one issue group response variable variable concern aid group education b concern budget anti test aid mx vs b plot group well select contain two four question contain contain pearson test applicable count check fit show count attribute htb count f texture plot group separate htb trait slope variational parameter model draw sharing enable plot latter plot give term estimate mean latent representation covariance place useful structure application herein cluster covariance wish investigate alternative nominal mixture logit acknowledgement author acknowledge helpful comment anonymous review grant aid development grant university research grateful access trait recent binary latent extend incorporate common accordingly low incorporation block consider approximation exploit determine demonstrate simulate extremely classified whether circumstance record something think like table trait term ht categorical latent analysis trait categorical profile latent principled cluster mixture combination component dominate cluster wolfe focused mixture elliptical lin lee base receive little trait table binary anneal chance gauss require ht type likelihood logit annealing logit gauss quadrature logit variational probit order categorical integration logit order categorical wherein identify trait accommodate dependency approximation likelihood implement quickly approximation trait come latent similar structure trait probit numerical heterogeneous discuss use gauss quadrature distributional parameter mixture item trait quadrature integration accordingly analyze underlying propose trait cluster variable exclusive compose latent categorical parameter considerably slope visual representation model accordingly identification trait integral use variable provide low likelihood variational parameter demonstrate vote cluster propose trait slope present real suggestion future approach assume trait slope ng datum reduce trait low assume assume trait p ng ng ng ng p ng ng z ng ng ng categorical group ng ng ng datum cluster cluster arise design repeatedly homogeneous accordingly likely variable outcome specific explain dependency inter variability response equation ij write model closely relate mixture item assume latent level discrete block one effect multivariate trait parameter reduce take follow parametrization matrix density eigenvector normalize eigenvalue determine orientation rise four corresponding covariance respectively assume namely parsimonious make ht l free g g dd dm dd dd dm dd dm g gd dm dd dm dm g dm gd dm g dm dm list parsimonious group need component grow almost ex finite model variable geometric contour principal thus function ng response variable plot effect block statistical common characteristic categorical employ load analogous trait covariance analogous function distribution course mix equivalent dimensional difficulty version analyze binary latent trait class range key trait provide also dual closely trait obtain framework series expansion point bind likelihood obtain maximize fitting integral intractable variational em obtain latent ng ng ng ng ng ng improvement take ng ng ng ng ng ng ng ng likelihood log adopt lack stable application voting facilitate comparison determine acceleration eq detail gauss quadrature expression b md eq likelihood schwarz criterion observation component low preferable likelihood purpose calculate maximize log use gauss quadrature convergence attain common small check goodness adjust rand index rand ari chance rand ari trait discuss give detailed trait constraint determine free parameter identifiability hold rank error maximize likelihood identifiability study categorical assigning n g initialize approximation matrix ten initialization low variational exactly
classification significant modelling field model clustering give partition disjoint batch gp gp govern deviation behaviour model may layer hierarchy construct model e use dirichlet prior gp group inspire cluster time fail previously inference procedure sample agglomerative regular span development limited resolution subject technical biological variation occur clinical grouping relate specie take inference require gibbs need fast dataset resource avoid model key likelihood construct reduce make derive link conjugacy manifold gradient optimization method apply expression series several differential process expression proposal lead propose presentation conceptually gaussian poisson intensity attract lot gene incorporate time potential dp like make gp gp assign function observation noise gene differ far describe gps expert aim cluster propose gp rich replicate infer aforementioned considerably widely gibb adopt agglomerative suffer scalability gene collapse collapse share see yet collapse relate apply collapse variational collapse unable correct overlap mixture gp objective track reduce remove gp dp use free variational show process introduce gps structure introduce perhaps particularly publication space one zero everywhere covariance publication exponential kt length publication separate collect write critical vector eq covariance construct function version write conjugate interpretation gaussian covariance directly construct gaussian consider group group experimental draw draw gp development normalise log gene frame posterior subsequent frame biological group replicate solid area covariance amongst depend compound function j application assignment well popular involve infer mixture widely treat assignment estimate treat approximate posterior parameter mixture achieve prior approximate assignment vb find mix proportion variational avoid selecting though dirichlet allow density proportion cluster concentration break mapping dp concentration stick break length break proportion distribution dp gp provide though empirically procedure variational bayes use gp dp vary atom function hierarchical hierarchy high level know dp gp construct series break atomic proportion atomic draw gp atomic reporting perhaps infer occur probabilistic assumption lower serve assumption factor turn recently collapse specific variable analytically equivalent scheme show riemannian conjugate direction serve dp mixture model collapse break stick length variational similar parameter aside cluster allocation simplification gradient propose gradient move merge split cluster ascent direction relate information quantity empirically free infinite unitary element shall select truncation merge split metropolis hasting collapse gibbs collapse dp gp gene necessarily simplify exposition somewhat let collect stick break vector wish simplify illustrated figure mixture group occur usual hyperparameter omit brevity point vb graphical infinite connect gps dp dot process cluster represent cluster graphical hierarchical collapse standard gp separation analytically symbol description collection observe stick length dp breaking length mix stick stick assign component component collapse select collapse observed wish examine graphical representation would separate variational truncation valid softmax computation natural gradient shall analytically jensen likelihood trivially tractable conjugacy without integral separate complete square integral straightforward study collapse break integral first trivially stick break length beta beyond trivially substitute integral similarity collapse stick break et break length make lead variational tractable expression n g softmax problematic suggest write though avoid first inversion size since symmetry softmax gradient nk nk g nk divide many avoid compute divide obtain expression natural eq expression natural multinomial softmax correct optimization collapse posterior field exactly kl correct bind step length coordinate coordinate natural correct field bind recover unit kl correct correct simple deal compact optimization gradient mean field bind round update perhaps kl correct enable conjugate gradient computation conjugate fail recover merge suggest mcmc current one cluster collapse nature correct particularly helpful merge split deal natural move bind split select examine mass new cluster accept move empirically merge appropriately order move bind collapse contain deal arbitrarily log increase hyper scale span datum account account variational unless model synthetic set randomly function around randomly per select correlated offset add present development aside across specie replicate measurement every pool gp replicate account correlate replicate use hierarchy gene eliminate gene expression hour dark cycle gene periodic projection dp gp nature rbf periodic far discover structure show reflect establish gene provide synthetic dp construct hierarchical dirichlet infer concentration parameter cluster infer diagram synthetic indicate cluster grey square represent gp find correct confusion gp dp account structure model ground truth allocation infer structure amongst introduce component fail signal unable infer method variational trial condition parameter create standard log riemannian conjugate parameter merge routine reflect case variance correctly place formation
instance distortion training impose objective minimize coefficient value size update rate update exponential decay decay well decay iteration dataset consider throughout produce good preliminary randomly describe loose treatment similar slack separable impose remove simply regularization generality train must term adjust score inner although type sigmoid give range sigmoid equation empirically effect variant formulation supervise feature scale equation verify compute derivative become hold likewise convex irrespective form convexity depend upon sigmoid scale irrelevant finite although sgd empirically function parameter tune adjust range require variant equation convex respect sigmoid class sigmoid second derivative compute order derivative proof scale regarding among product appear inside thereby reduce number train name scaling convex feature train pass online dataset dataset dataset evaluate classification real purpose uci learn repository detail dataset attribute instance heart diabetes compare feature implement train baseline vector update instance encounter theoretically weight fast unsupervised train binary logistic scaling feature unsupervise feature describe weight vector train unsupervise dynamic feature section method describe classification passive binary predict pa passive pa average pa linear passive pa binary weight parameter avg avg fs avg fs fs avg fs fs avg fs fs avg pa avg pa avg pa pa avg accuracy good sgd avg fs fs avg fs fs avg fs avg fs fs avg pa pa pa avg pa pa avg algorithm sgd avg fs fs avg fs avg fs fs avg fs avg avg pa avg pa avg mention q three e number negative test instance diabetes dataset passive purpose produce high accuracy next fix parameter table suggestion instance train initialization see scale joint supervise scale pa average unsupervised dynamic averaging accuracy dynamic method unsupervise dynamic deviation instance unsupervise likely demonstrate accord pair effectiveness among report performance pass training become critical compare set compare method experimental keep plan dynamic method compare diabetes method average version outperform compare diabetes ability perform situation cumulative current pass online train classifier method obtain error cumulative total encounter misclassifie misclassifie avoid show dynamic stand low scale pass dynamically popular dataset experimental unsupervised significantly outperform approach improve several supervise scaling evaluated explore learn range conduct preprocesse problematic reason stage change effective dynamically adapt scaling complex several classification classification machine represent often range supervise train frequent extremely relative informative value feature algorithm typically scale range use approach feature reason manner label document task scale possible scaling preprocesse pass online instance extremely stream call scale third compute value instance text stream might factor old dynamically term refer oppose processing focus specific multi classifier main approach assign scaling algorithm pass online scale store stream training maximize memory evaluate online three dataset binary interestingly much unsupervised dynamic scaling consistently algorithms compare include online necessity engine online learning impossible fact new continuously situation train tweet case instance predict trend stream note iteration example gradually criteria example another moreover whether training manner unseen ever stream web pass convergence confidence develop online feature scaling supervise
combine pd lda share among gram pd across n gram phrase post lda topic construct performing pattern mining phrase four heuristic permutation test phrase phrase extraction apply social service twitter twitter specific topic candidate phrase extension topology twitter extend corpora corpus frequent enhance topic investigate objective overall quality phrase concept place lda investigate markov probability independence assumption sentence extra generative hierarchy assign topic sentence output extract rank list phrase phrase operate partition document method comparison interpretability scalability six collect publish paper token area artificial intelligence database retrieval language contain word token article token k word token review review token address phrase remove english stop discovery comparable pd art approach topic use bi add specific phrase post permutation merge term lda extraction speed frequent frequent phrase phrase demonstrate effectiveness framework detection option randomly phrase one top phrase ask select indicate unable task separate ask answer question evaluation second motivated extract quality phrase interpretable visualization evaluate phrase visualize list phrase sort expert science score phrase qualitative coherence homogeneity phrase list homogeneity ask expert coherence phrase list ask standardized h discussion demonstrate quality stem frequent mining phrase inspection suggest key phrase notion may aid believe quality occurrence phrase many hyperparameter pd two intuitive interpretation topic rigorous permutation employ phrase addition phrase induce phrase high belong evaluate predict hold corpus evaluate review demonstrate lda demonstrates validate phrase high lie addition see phrase quality model incorporating phrase analyze decompose framework contiguous take bag phrase output separately demonstrate phrase see scale increase dataset addition one phrase portion negligible topic model scalability framework runtime hardware dataset various art compete computational lead intractable requirement whenever author implementation special gibbs user hyperparameter optimization ensure fair phrase model runtime model pd intractable magnitude runtime runtime intensive permutation n gram method able full dataset scalability large pattern scheme make large long intractable short entire phrase word table phrase method text probable well probable phrase automatic perform post visualize phrase interpret phrase naturally news phrase coherent phrase believe may phrase good great phrase display sentiment emphasis poor cm cm day day day na runtime calculate runtime topic runtime great runtime language mining solve classification character association sentence propose stream genetic large gram genetic programming language mining optimization speech machine language object orient stream natural machine evolutionary feature selection execution series run sign mining recognition orient programming spatio objective program gram drug nuclear house aid environmental health year medical west test chemical disease gram department health care environmental west bank house medical nuclear organization united united house nuclear anti report drug abuse drug united capital patient pay control house member heart member test h gram room store good ice stay roll place restaurant great area great gram ice lot store front great hash great price chinese room dim sum pool area center great good price reasonable mac systematically allow driven estimate topic another scalability currently decrease stem efficient phrase mining investigate another focus pruning strategy similar phrase discrete structure top phrase count merging phrase well properly notice occur due filter principled enhance mining framework arbitrary phrase phrase phrase first frequent phrase mining efficiently aggregate score objective bottom phrase termination phrase phrase assignment phrase phrase computational phrase principled construct phrase post processing demonstrate scalability interpretability phrase nf office agreement nf national science foundation multimodal synthesis foundation fellowship nsf collapse derivation find second department laboratory md edu microsoft com mail algorithm model corpora interpretation rely inherent discover length either utilize phrase suffer scalability moderately approach effective solution combine novel mining segment word operates induce document high phrase extra bag word publication review news recent topic modeling become discover abstract typically model multinomial multinomial retrieval search engine support extraction document query question answering topic topic human interpretation exploration within text corpora qualitative list probable topic yet topic probable provide intuitively description term phrase clear organization exploration collection difficulty scalability attempt make phrase simultaneously create mechanism phrase assignment gram pd appealing incorporate phrase element overall word phrase topic guarantee model propose new compare phrase method interpretability phrase mean word phrase phrase mean lost insight phrase systematically motivate perform phrase scalability issue phrase significant phrase use frequent phrase mining text candidate phrase second phrase effective within phrase assign phrase word frequent mining significance frequent generation frequent current status future share advantage phrase mining algorithm phrase aggregate domain specific linguistic purely drive candidate phrase title frequent phrase determine whether frequent phrase title segmentation phrase incorporate need reduced phrase maintain analyze phrase review conclude input document document sequence token convenience token vocabulary throughout refer word vocabulary corpus corpus visualize phrase statistically characterize example research high speech characterization advantageous statistical interpretability depend member fashion phrase phrase help define terminology sequence contiguous token concatenation proximity restriction phrase phrase despite word phrase mean motivate representation flexibility segment concatenation phrase title token represent outline property mining list phrase demonstrate phrase valid human interpretable phrase principled manner efficient comparable specify phrase designing phrase phrase phrase construction naturally validate candidate phrase constitute human interpretability phrase important regard within frequent topic formulation list probable lda probable phrase give topic corpus refer token frequency chance commonly appear frequency yet english language consider frequency informative insight motivate necessity analyze phrase mine frequent mining satisfy yet intuitive phrase human interpretable phrase phrase phrase divide phrase mining constrain transform bag quality frequent phrase segment agglomerative phrase induce topic phrase phrase phrase phrase phrase phrase partition topic bag phrase input traditional token propose partition phrase collapse token phrase phrase mining corpus tokens interpretable phrase operate contiguous pattern meaningful phrase break corpus frequent candidate phrase aggregate develop technique quickly collect without phrase step great subsection frequent phrase collect contiguous draw upon mining frequent closure phrase phrase frequent document frequent phrase length frequent phrase length closure first mining frequent exploit property mining potential merging termination induce upon original create bag contiguous token h significance left phrase place contiguous token significance key left phrase phrase put track construction agglomerative merging operate merge two contiguous phrase merging significance algorithm follow consider newly merge phrase consider newly merge phrase single merge free phrase compare occurrence occurrence merge phrase phrase aggregate necessary calculate significance merge proper datum structure contiguous pair significance merge document terminate next meet merge meet significance termination natural bag phrase partition remain algorithm requirement phrase completeness corpus actual phrase hypothesis phrase phrase reason phrase consider independent hypothesis absence phrase corpus random number occurrence token corpus assume fairly reasonably normal count phrase corpus trial probability phrase phrase compose phrase phrase population minimum phrase significance quantitative consecutive phrase merging compare expect occurrence equation deviation away number occurrence aggregate candidate phrase efficiently frequent phrase algorithm significance generalization statistic identify check merge contiguous phrase merge phrase effectively free phrase address concern significance score rely naive testing guide phrase merge phrase merge overall corpus phrase frequent contiguous mining aggregate bottom agglomerative merging segmentation frequent contiguous mining differ transaction pattern mining bad search pattern contiguous reduce exploit text splitting searching phrase rarely algorithm frequent contiguous mining corpus pruning datum heuristic reduction lead runtime termination phrase merge phase segmentation phrase possess complexity experimentally verify section new I often chance token share latent brief review dirichlet propose gibbs develop optimization phrase serve collection cm topic phrase token token phrase token multinomial token phrase token phrase multinomial distribution word token assign k token k
compute digital purpose need describe ground essentially quite consume investigate one look shape cloud cross road hundred consume research matter need misclassification difficulty type ground cover automate entirely require manually tree great success manually classify evaluate ground cover need review procedure slice random rather unlikely approach thing point datum set big disadvantage course chance class assign class moderately million make enough sample mean require ensure survey human sample rare representative population rare guarantee numerous e population draw method use weight examine described section go setup subset area select point visualization contain give overview size ht lr estimate variability cart framework remainder datum sample cart outline classification use metric point misclassification rare influence feature accurately classify rare total misclassification set q marginal proportion metric section classification conduct series result size sample large size misclassification classification notable simple metric reverse picture poorly among provide cart prior composition turn great agreement post present remarkable misclassification working finding great metric per right sample sample quality metric put correctness bootstrapping result class quality achieve interestingly case cart provide precisely specify post lead composition improve cart become ever survey integrate cart rgb end improve handle big usefulness sense class ground aim supervise various size result project cover area handle usefulness sense cover experimental setup discuss conclude suggestion character change dramatically decade availability ground provide cover code massive amount cover vast handling
limitation scalable linearly pixel since modern second minute produce system oppose invariant meaning scale reflect level recognition task require way size typical pixel instance scalability searching location scale limit coarse image achieved achieve desire outcome contain interest image recognition neural train patch image test compose scale pool bank cascade operation promise location attempt building paradigm visual domain task propose learn candidate look level class predict next location feed another object across integrate geometric treat perform search well empirical able competitive result mnist handwritten digit drastically cut maintain improve accuracy part unlikely work relate notice densely processing may fully image resolution feed hide bias time dimensional vector vector softmax linearity produce since component vector sample image level output fed predict normalize coordinate resolution generality sigmoid training specify look next resolution generally predict width patch patch specify diameter fix feed patch sake neural network network call give resolution outline system distribution eq show weighting sum weighting yield uniform work produce subject training geometric resolution multiplying probability predictor agree object predict take un yield system fairly robust increase instead look give location look propagate describe solution incremental restrict predict latent upon depend compound parameter resolution location predict patch image positive first low assign resolution network term similarly train step possible current perform minimization perturbation nearby around center predict minimize descent fixing variable log correct minimization rather retain empirically confident class incorrect location force location improve location predict correct location nearby train hold resolution generally hold fix work location location reduce preliminary certain slightly far order achieve prediction effectively put around previously visit encourage different position never unless last effectively parameter fit several extract high leverage context resolution interaction take location extract interaction predict simple parallel proceed location extension summarize fix fix n fine tune adjust last minimize alternate find loss confident wrong see update descent relate prior map although analyze informative task rely method drive across however train discriminative method grid image inspire psd variable psd resolution drastically minimize local descent include update design function meaning reconstruct internal represent share module predict make depend location recurrent fed self oppose propose cascade shall depend input pattern classify resolution dataset dataset pixel place location pixel zero pixel digit random set quality training sake original method well convolutional time l c test resolution fine tune input fed pixel consider scale original take patch pixel grid spaced patch etc fairly hyper run stochastic momentum report rate require well connect baseline hide unit convolutional classification convolutional stage pool filter neighborhood add third yield marginal improvement moreover fine reduce system match performance however amount computation magnitude low convolutional row cost threshold classify resolution classify digit second final rejection regardless overall bad sample time use fully resolution method resolution term diversity share apart experimental validate effectiveness rate location look remove loss function cross right dramatically increase demonstrate low nearby perform output assess nature suggest would well trade computational efficiency accuracy compose exploration experiment prediction nearby suggest term tune reach report tie last different qualitative
sequel ingredient modulus ingredient affect boost restrict attention gradient depend smooth sag particularly minimize explore singular consideration however difficulty application might add pre dense computation tackle major question order loss boost efficiently exploit different newton inverse hessian address turn construct construct facilitate first value continuity scalar difficult loss logistic loss obey notable entire I bound positive constant rate motivate follow easily show I become matter far construct transform worth note similar whiten transformation transform whiten transformation modulus change discuss method elaborate number ingredient namely lipschitz modulus ingredient namely upper bound ingredient summarize smooth ingredient derive I bind discussion upper diag rank two eigenvalue decay c probability derive bind individual quantify incoherence incoherence incoherence theory correlate canonical two incoherence since bind state number condition previous condition ensure reduce number l hand maximum original loss function unknown value condition loss let g square assumption smooth loss original datum covariance indicate due mean decay easy logistic balance always check calculate unknown trial achieve comment theoretically r practice especially size sgd begin unstable ingredient follow functional ingredient naive boost convergence verify example sag solve least similar carry sag suffer much study indeed proceed third question need cost complexity expensive attractive efficient key construct subset denote q define computing construct carry follow theorem datum scale condition improve vary exhibit understand ingredient I ingredient decrease aware increase would pose strong effect datum sample construct first theory inherent e numerical synthetic datum matrix vector constant around eigenvalue polynomial decay decay construct value vs eigenvalue convexity modulus similar vs large number cifar namely smoothness individual inner suggest sag like unless specify optimize performance pre initialization size zero either suggest generate describe least variable sign curve sag boost justify straightforward limit supplement boost validate generate plot constructing sufficient regression cifar experimental four regression predict year song year stock return financial tf task pixel classify image predict forest cover experiment report tune sag epoch cifar use datum plot e datum sampling could yield could finally comment original per convergence problem method minimization characterize experimental validate theory condition lemma thm assumption lin science department computer science engineering management science exist boost I ratio modulus effect method minimize good yield bottleneck provide loss minimize practically random validate supervise regularize eq denote decision convex least square become light computation reduce full
show prove compact path ball radius enough ball cover say ball center suffice ball great equation wise thus equivalent c b ia vector path construct connect path connect path statement proposition correct density immediate x k c px statement follow hold proposition intersection conditional independence cx statistical independence hold direct additive causal relationship challenge science decade causal statement discovery assumption relate joint say acyclic dag imply correspond reverse statement class correct identifiable skeleton method restrict linear noise order acyclic intersection property know hold sufficient necessary condition intersection corollary interest weak strict positivity mention model require intersection identification positivity characterization replace strict positivity variable connect identifiability lead intersection property graph identifiable correspondence noise path aware method infer dag inference technique generic mention formally possibly metric space introduce regard existence part absolutely mass continuous contain path see throughout conditional independence independent conditional bx property eq intersection hold intersection property condition intersection assume intersection give lebesgue example apart existence necessity eq make arbitrarily summarize clearly bx ax mu proof important connected equivalent within component consider mu predict intersection formalize intersection minimum cm circle draw circle draw dag circle b area dark gray function value ten minus ten correspond important implication inference causal graph lemma hold positivity characterize intersection density condition corollary notion become important space closed set component wise connect equivalence union member denote treat figure path connect correspond another contain three equivalence formally variable take definition able direct consequence density weak intersection property intersection property set joint distribution characterization property intersection identifiability additive noise distribution structural q function parent direct acyclic simplify identify node eq require intersection since provide weak obtain result structural density density thus disjoint variable
later another generative choose versus use bayesian user specify false determine change precisely change occur change present network difficulty detect synthetic real evolve network digital recover generalize hierarchical attractive change naturally capture structure interpretable dendrogram binary tree thereby interpretability quantifying vary point connection probability quantify uncertainty generative compose vertex edge vertex nest relationship dendrogram vertex connect low density vertex distribution produce eliminate possibility allow compact illustrate email communication connection edge direct dendrogram approach set choice provide little room increase consider connection maximum drop outcome set quantify prevent become convenience employ beta distribution hyperparameter binomial analytically tree requirement spectrum hierarchical end spectrum contain single internal connect enyi tree add model binary leave tree reconstruction amenable classic technique instead tree costly reconstruction solve majority sample tree consensus leave occur majority binary tree contain consensus internal connect mcmc derive probability remain thus prior empirically observe count connection produce observation become uncertainty model close root far structure implicit prevent structural improve infer noise piece network change point determine whether change slide occur fit ratio framework factor ratio two different alternative hypothesis time rather likelihood likelihood update hyperparameter restrict consideration window network occur denote window change hypothesis set hyperparameter window conservative network let window say exceed literature choice must occur result block suggest technical numerically way exactly rather possibly hypothesis see desire calculate count proportion ratio high test choose say change pg w pg pg g bar bar offset time positive negative unknown change systematically different network control circumstance variety change network provide since hard characterize two add one formation p p single parameter control switch distinct state merge change merge single community edge comprise formation change community simple version convert scalar mean degree geodesic show change slightly time slide window quantify detect change formation detect later false among method positive match false alarm rate negative widely even across four size fluctuation merge notice around rapidly change experiment change enyi point rate different change type magnitude change evolve network mit proximity email external evolve human interaction interaction quantify term detection delay estimate delta otherwise actual proportion proportion event delay change mit network comprise proximity student record continuously phone raw edge denote physical proximity week dataset external public period detection three use result figure term well baseline approach close detect reveal external geodesic distance begin cluster activity majority cluster detect exception begin seem inconsistent event along week week agree well week involve typically shift work dramatically seek meet project goal additionally find point fall examine change infer fall fall establish pattern largely highlight additional interpretable within evolve email show detect bar follow statistic colored bar comprise mostly management company energy investigation company apply simple long window window size highly large suggest window operate resolution window resolution bottom window size bottom mit network poorly perform precision examining external event identify share fluctuation examine large structural occur formation take evolve change detection principle non stationary utilize test principle detect fashion large scale change network significantly change equally datum reliably furthermore community merge internal connection accurately split many community formation question technique eliminate whether add weight make easy say point like cluster yield rate structural poor performance result discard utilize apply evolve social good recover external network measure computational inference work ref propose change accuracy scalability believe yield good generative place model graph work detection
pick term entry observation construct constant etc basis ease presentation pick coordinate coordinate remain family space incoherence condition notice sign uniquely whose last uniquely free consequently solution symmetry column observe plug approximation analysis phase fairly straightforward one reliable estimate column measurement show norm translate inequality provide invertible sampling apply proposition observation q observation replacement unbiased apply rectangular bernstein inequality triangle inequality cauchy schwarz plug turn quite soon equality follow linearity apply matrix second arrive norm large assumption state collect scalar bernstein scalar vector bernstein adjoint dimension rectangular relate task procedure adaptive allow eliminate passive project algorithm rank coherence completely row enjoy exist improvement adaptive necessary complexity singular compute nearly coherence eliminate decrease significantly scientific application fail keep study network involve inference individual challenging word amount generate complexity statistical network modern statistical result inference extremely compressive paradigm presence acquisition severe setting aware adaptive outperform passive scheme matrix completion exactly recover fraction recovery aim low precise low observe sequentially feedback drive manner thesis sampling assumption analysis spread passive uniform sample suffice algorithm problem uniformity monitor anomalous recommendation popular item highly active user fail show contribution completion column completion term place simple complement low row space passive must demonstrate completion approximation approximate appropriately rescale good approximation coherence outperform collect definition section defer proceed interested approximate column denote column r r capital symbol refer orthonormal basis subspace subspace versa orthogonal complement refer projection deal subsample denote list coordinate vector form rescale dimensional subsampling orthonormal row column uniquely define depend nevertheless onto subsampling operation exact require rank mean exactly error relax rank approximate effectively interested find matrix satisfy excess risk approximation bottom require svd limited divide energy apart observation coherence coherence spread fairly column loss matrix analog see parameter sample uniformly informally mean capture salient sample scale incoherence translate uniformity decompose incoherent result stochastic uniformity aim line remove stochastic alternative rank behave quantity usual incoherence rank space vast attempt idea relate well well norm uniform sufficient exactly recover involve imply subspace must incoherent strong relaxed well weak prominent work space incoherent notion consider essentially thesis matrix goal preserve property main completion observe relax incoherence scheme completion input constant requirement essentially optimize objective series paper span column approach column form unfortunately unobserved approximate approximation also aware strategy recovery possibly structure passive method difference view iteratively discard irrelevant remainder rely extension rank signal community adaptive effort approximate structure related recover binary recover ultrametric matrix idea similarity impose reason useful develop manuscript suffice rank coherence complement passive adaptive excess uniformly replacement tu display stream direction maintain processing norm new algorithm add lie ingredient onto orthogonal follow x lead side identically deviation allow subtle critical element importantly ensure randomness logarithmic dependence test reconstruction proof defer rank coherence time guarantee vast incoherent column space restrictive exactly number row incoherence remove column match incoherent principal translate assume assumption mention paper weak polynomially coherence weak super dependence interesting consideration operate pass store coefficient represent column lead computational dimension dependence run matrix allow take read input standard algorithm alternate iterative coherent recovery low estimator function let set lastly incoherence account adaptive passive strategy entire exception entry believe bound success probability minimax risk concrete whenever passive sample complexity passive completion fairly form incoherence non apply incoherence recover relax incoherence depend coherence sampling take measurement absence incoherence universal achieve relax incoherence finally argument even require express polynomial right system since must many impossible nearly passive pass observation random replacement x te tr obtain pass pass frobenius second pass additional place rescale preliminary computed top show pass column square column norm motivate pass algorithm non measurement approximation although case incoherent assumption follow assume satisfie complexity defer serve compute run dominate cost dependence mild dependence translation bind bind bind well set tight mention assumption q give matrix improves imply weak relaxation similarly completion recover recovery incoherent draw variance appropriate dimensional signal ratio constant term ignore spectral gaussian r bind f positive recover long use close frobenius number sample bad fix term bind et frobenius square root equivalent significantly thus particularly suited uniform apart sampling soft versus singular value choice regularization amount capture replace thresholde soft thresholding first soft ensure rank approximation second guarantee translate unless quite thresholde frobenius norm guarantee completion therefore consistently sample notion boundedness incoherence agree much uniformity lastly emphasize adaptive uniformity miss incoherence give uniformity enjoy significantly sample uniformity similarly rank uniform column sample plot per rescale simulation figure behavior binary construct span incoherent row collection want various algorithm fraction sample exact recovery demonstrate fix number column constant per column plot rescale rescale versus matrix vary show fraction successful show figure similarly confirm incoherence rank appropriate capture correct plot algorithm adaptive show govern norm figure record maximally simulation coherent space pattern algorithm relative error target axis display set figure column space span vector whose also log normally length construct via clear column plot relative equation function fraction next dependence rank relative error decrease rapidly need increase qualitatively suggest different matrix matrix column sampling length coherence norm standard plot size plot decay plot rescale curve phenomenon proposition sampling threshold approximation column norm dramatically outperform confirm lead distribute provide theorem concentration measure reproduce
report automatic surveillance traditional confident diagnosis perform surveillance combine advantage medium replicate traditional limitation generally individual may result predict actual second surveillance train fundamentally detect strong pattern instead top measure address new method generalize disease traditional surveillance detect even user system user long number mining approach near user five many twitter describe collection twitter tweet focus additionally anomalie behavior social meta receive information health service individuals privacy know month participant mostly student expect collection university twitter receive account account tweet friend user month keep tweet twitter account multiple time hour look day collect profile query limit collect parallel tweet account account account recently update account friend total collect tweet account tweet follow diagnosis content tweet predict tweet divide month user tweet tweet c odd occurrence keyword set keyword name addition serve expert fisher occurrence user six seven keyword table select keyword first find rank information choose top list space character stop perform convert text naive classify user month tweet message shoot bag technique classifier rating tweet time tweet study tweet human extract text rating machine human rare accuracy health table content individual tweet affect tweet dimensional anomaly detection follow tweet month study discard tweet avoid user twitter rate month q mean standard user month user z significant kolmogorov highly biased cutoff roc currently give health status twitter user account follow account user friend consider friend analysis analyze normalize count character friend activity friend detect user friend stream far analyse strength measure roc variance examine source friend number keyword classifier follow user well tweet account twitter user news rarely account htp cdf cf detect twitter analysis tweet reason would detect aggregate meta show adaboost boost voting start classifier roc boost j decision boost voting evaluate leave cross roc adaboost high boost medium high aid
annotation take various image level indicate count bound box seed problem input crowdsource low box document may tag document preferable bias annotation difficulty category object thing annotation category I label number researcher recognize importance annotation semantic label propagate annotation training paper weakly annotate employ weak annotation structural learn incorporate loss consistent annotation different one impact fully annotate since less kind challenge include former involve satisfie annotation latter involve consistent show solve algorithm closely sequential annotation define perform loss augment carefully initialize iterate conditional mode train annotate consistent solution unlike train specific function simultaneously fully label allow cut base augment finally different weak annotation use inference decompose relate supervised area pass inference three high bound local convexity utility minimize annotation establish annotation apply annotation type label bound box seed efficient function mapping mapping express maximization discriminant depend call learn vector variable supervise output learn appropriate compatible paper follow margin formulation call respect instance commonly distance imply maximization discriminant function decompose r problem cut plane replace approximate polytope violate constraint determine run fully weakly annotated subset individual annotation segmentation bounding box segment intensity component segment seed make weakly annotated simultaneously utility formulation seen degenerate weak annotation therefore equivalent slack balance ignore show order loss augment weak augment inference side use convex approximately semantic represent group co similar represent pair node node feature connect correspond discriminant unary potential restrict pairwise nonnegative attractive potential maximize exist expansion weight hamming loss label number pixel weak annotation need annotation bottleneck combine annotation level label box seed arbitrary truth annotation ground truth image plain hamming full argument annotation ground ground area derive tight label wrong well full feasible full need make change per miss significant augment inference decomposable unary pairwise maximization inference cost accounting label box annotation consist box image annotation image bound additionally road certain define level bounding box type category bounding box bound box satisfy certain outside bound unary potential adapt class unary yet infinite guarantee least contrast unclear neither heuristic significantly well initially bound opposite guarantee segmentation annotation seed annotation particular case pixel label annotation locate weak annotation seed centrality infer seed neighbourhood seed bring loss inner take pixel central pixel whenever estimate eq seed loss decomposable factor sift training image label sift dataset database label category crowd original unary sift word build use dictionary histogram color uniform normalize joint approximate dimensionality triple pairwise share strength code unary appearance vector context triple distance case histogram segmentation accuracy recall correctly label per recall correctly divide total category exclude pixel rare see exclude rare sift one target sign car use define label category uncertain model boundary would label follow least pixel cm acc local strong strong ccc c il bb os possibly image hasting sampling try distribution count approximate training show accuracy recall various comparison scenario label weakly provide stable recall annotation full need discover dependency strong sift supervise weakly label rest label recall label substantially complicated randomized hashing connect image may get local label annotation train additional annotation specific tight bounding box object description thing category category divide list background water road include category image building background enhance tight bounding box seed available seed segment summarize seed box annotation give significant box notably thing category numerous overall bounding box inferior label accuracy per object annotation give box contribution support image ex compare dataset foreground kind believe easy background foreground latent objective label bound box seed weakly annotate foreground partial infer annotate necessary look label train consistent annotation train similar outer
estimate label correct construct base refined prior information solve count policy expect accuracy call attain supremum mab problem correspond decision instance arm identically distribute collect reward maximize problem collect reward involve final although intermediate decompose technique lead mdp since note optimize stopping stop wise attain supremum proof present stage reward interpretation accord expect wise take stage th collect label instance receive correspond remain use get reward maximization tuple stage possible reach element next expect reward define technique space budget action take state equal justification mdp induction bellman illustration dp calculate label uniform one correspond last stage table instance label second c although dp find policy intractable since grow exponentially accord develop computationally approximate policy need uniform choose policy decompose horizon mab horizon mab discount infinite horizon index rule cost require complexity decompose reward show discount reward horizon index heuristic hand side stop instance large note state reward instance reduce calibration space exact method require time index policy attractive knowledge ahead next reward policy labeling chance refer break refer many mdp fail assume integer let consistently budget go proposition provide label instance infinity address proposition behave sampling case undesirable subsection propose allocation policy consistent input parameter budget ta ii I output reward instance index optimistic opt opt optimistic next opt optimistic reward reward obtain obtain optimistic maker process opt opt accuracy e almost prove label obtain opt detail b opt framework budget allocation labeling base conditional adopting measure two small probability x q x equal problem opt policy could select large experience limited sake opt opt computationally budget policy space connection optimistic ucb particular ucb select reward side upper confidence optimistic reward arm policy directly fact opt exploitation policy utilize exploration exploitation take optimistic reward play problem also outcome instance consistently budget discuss presentation beta parameter practice easily incorporate beta distribution allow easy skewed address adopt w cc bc labels negative bayesian put hyper parameter hyper proceed parameter hyper prior beta please apply reliability next worker reliability correspond reliable worker poorly inform worker crowdsource worker maker could assign worker worker reliability introduce extra get reliable worker worker th worker pair worker reliability get negative label label provide worker implicit worker single label instance increase reliability get reliable worker reliable worker worker underlie soft label poorly inform always assign wrong worker reliability beta j decision next worker notational word j decision outcome vs decompose wise reward posterior sophisticated long marginal long beta make reward apply opt large scale approximate technique worker budget worker posterior set parameter opt need posterior distribution close monte since inference variational matching technique close highly compute omit discussion adopt approximate q match analytical appendix still new collected distribution stage get worker present opt allocation heterogeneous establish opt heterogeneous apply crowdsource worker section worker fully reliable share unknown reliability maker label pool mdp address crowd labeling briefly discuss two opt policy sake presentation extension noiseless homogeneous set heterogeneous section contextual contextual assume budget allocate among soft average receive label observation budget level budget explore less receive necessarily receive budget sufficiently instance receive receive infer instance assign soft label worker worker simulation reliable vary run budget receive instance opt simulate four opt red opt generate blue comparison different opt highly opt opt robust underlying highly belief reliability worker informed situation prior average different generate compare opt red line quite generate compare worker include index solve mab discount discount labeling horizon since computation instance vary budget report independently last figure outperform regardless opt inf opt improve sample heterogeneous worker budget note homogeneous inf heterogeneous worker fail reliability simulate instance setting set three report use figure regardless choice different section instance pair different worker homogeneous noiseless set diversity worker prior set decide randomly dataset policy times opt inf although inf perform solve linear could expensive scale opt much quick require comparison opt inf cpu level inf opt opt inf worker reliability incorporate belief worker perform lead accuracy opt policy small accuracy using label time little bit particular partially observe opt go experience phenomenon worker opt heterogeneous worker opt worker beneficial worker reliability address crowd label markov decision propose computationally optimistic mdp binary contextual contextual crowd labeling crowdsource several direction work great opt performance instance worker equally crowd pricing motivate work pricing crowd interesting dynamic third label worker worker useful opt policy interesting decision liu share constructive comment quality appendix proof final maximize expect condition write maximize therefore positive set take b beta ia ia ia ia ba ba b second equality ia b ba ba b b proof follow decompose incremental vs determinant second conditional gain labeling th last change next reward formulate proposition deterministic reward integer lemma present therefore ia ia ia ia ia ia integer corollary reward ba ba way proposition accord instance I expect label I break first instance policy expect randomize getting randomized stage instance randomly instance consistency opt first show exact eq plot integer lemma symmetric decrease monotonically symmetry prove symmetry decrease r bb fourth odd r ba r accord property opt many go label go infinity need assume instance never label r accord opt label lead contradiction therefore take label th label ss independent identically distribute law conditioning h path event k q recall select integer utilize basic summarize property monotonically monotonically visualization algebra lemma integer instance stage instance label accord consistently stage select single incorporate reliability worker text approximate beta assume posterior take p b ic ij ic z ij assume independence ij form approximate technique ij make hold assume stage worker far place heterogeneous worker extension utilize contextual feature budget allocation sigmoid ts ti ty log tn laplace method newton follow matrix calculate reward also dirac mean therefore place summation omit need possibility use bayesian logistic multi categorization use conditional expect accuracy tc side ic h ic stage value write q way entry I rs convert integration cx cdf gamma perform dimensional could monte accelerate theorem corollary em minus popularity crowdsourcing task worker internet services amazon crowdsource worker label amount budget labeling desirable budget worker instance label consider reliability task aggregate simultaneously allocation policy obtain dynamic dp dp quickly propose computationally optimistic budget usually collect digital tag picture landscape datum labeling group provide become inefficient costly thank crowdsource service amazon unlabele crowdsource big crowd label availability crowd raise many usually non expert crowd suffer crowdsource service resort redundancy reduce collect multiple worker particular crowd label unlabeled crowd worker ask aggregate collect raw chance true label raw label come pay worker pre reward usually correctness website total raw collect raise central crowd labeling decide budget decide knowledge vast provide avoid easy raw bring fall near boundary worker inconsistent worth collect boost maximize budget simply put highly aside save instance ambiguity reliable worker despite budget ambiguity reliability unknown beginning raw online fashion dynamic conduct optimal budget ambiguity worker reliability formulate horizon bayesian decision process distribution bayesian necessary bayesian mdp budget allocation however dp computationally intractable level policy gradient performance policy opt dynamically choose worker optimistic marginal general opt measure opt achieve superior guarantee well opt start task instance amazon galaxy general worker worker assign entire worker worker worker reliable noiseless ambiguity budget crowd bayesian opt prove converge mdp worker annotation team microsoft allocate worker worker finish instance fully reliable worker parameter reliability next optimistic knowledge flexible extend information web consist fold budget crowd mdp characterize computationally optimistic gradient propose address budget crowdsource organize first process label task fully worker motivate section via dp computationally policy opt heterogeneous worker reliability incorporate contextual multi dataset follow allocation homogeneous labeling simplification incorporation extension category model instance true label goal infer assume pool note worker true incorrect due ambiguity far latent soft percentage crowd crowd reliable worker receive crowd large characterize concrete ask person person old person worker regard person infer raw positive labeling definition soft follow sense
approach outline rather seek boolean identification unique causal remove estimated entire causal iterative bottom order however causal structure bottom approach algorithm find subsection table variable list mm mm compute measure output mutual adapt fit high possibility input mm output h note depict fig bottom represent sort independence sort equivalent fact practically frequency incomplete computable obtain probability less follow mutual sort represents sort available table possibility every begin compute follow proposition list predefine truth step frequency output entire table thus memory accord requirement note loop find measure aggregate thus loop find function conditional element truth function repeat time accordingly variable moderate show later also favorable efficient choose potential parent choose care obtain truth table simply generic artificial way generate respective assumption successively derive value obtain time various combination index evaluation adjacency represent parent child relationship generate estimate wise operation obtain performance causal order true truth value algorithm explain combination uniformly random choice generate summation accuracy hand greater higher sensitive affected strongly reduce indicate causal approach thousand initial stage estimation even amount give various accurately probability heavily proposition require extra strong dependency density true est direct true est direct true est est form experiment previous level pc frequency relationship true direct miss direct non edge count double penalty incorrect causal failure global number pc approach distribution accuracy advantageous obtain pc identify child exposure nuclear binary conventional college school study aim variable causality yes college high conventional candidate select status retain individual background maintain transform binary threshold high produce fig intelligence affect college influence college algorithm five pe external noise directly check simply skewness external operation generic boolean generate hand accordingly model show check mention subsection portion incomplete miss introduce completion issue applicability variable promise way overcome may combine base approach discuss approach toward issue structural noise derive identifiable skewness distribution external computational promising real datum base continuous discrete perspective causal accordingly mutually fact set complement accordingly equivalent rewrite h variable except every upper since pe pe exclude exclude assumption imply one hold exclude variable obtain take relation accordingly one iii mutually accordingly imply acknowledgment discrete project aid scientific thank dr valuable comment causal discovery skew discover intelligence discover causal model paper novel causal datum new causal binary experimental excellent causal structure artificial intelligence derive candidate causal model set assume narrow scoring multiple causal equivalence linear acyclic unique acyclic equation drive causal require objective extend unique bivariate post apply derive order applicability continuous function identify causal multivariate bivariate address identification unique causal multivariate contrast domain computer bioinformatic maintain recently accumulate stochastic structural set knowledge address principle practically unique follow give regard objective discover feasible next briefly review issue third binary exclusive skew term represent estimation fourth section novel causal base characterization causal network develop efficiently focus causal structure within search space second datum search possible dag variable issue acyclic identifiable relation bivariate identifiability base identifiability aforementione derive structure applicability additive model constitute need acyclic relation algebra addition need bernoulli distribution adapt independence order identification estimation characteristic present concern introduce acyclic among jointly boolean function define every external boolean algebraic formulae skew cover generality kx essential identification analogue
uncertain input define include variation fluctuation value design pose comparable density virtue unique need regularization make minimum specification improve control modify relationship variable far ideal system get close specification metric function scalable ensure familiar norm integrable assume bound ignore choose integration interval discretize differentiable approximate replace approximate compute approximate design argument eq element define write oriented row put need function gradient uncertainty available finite evaluate matrix product select bandwidth density use ensure differentiable infinitely bandwidth spurious exist select bandwidth optimal gaussian pdf estimate formula sample optimizer numerical matlab plug minimize compare skewness estimate indicate kernel point row beta histogram four reveal pdfs generally represent indicate apparent skewness relatively small error cc cc distribution sequential optimizer every choose search second subproblem solve appropriate compute forward approximate hessian bfgs feasible line iteration response bfgs gradient hessian fall dimension estimate need interested find joint pdf response interest match bivariate cast extension quadrature uncertain sufficiently may response pdfs surface pdfs quantification include employ place dynamic solver surrogate sample surrogate surrogate polynomial variable optimizer depend design point q eq mean similar bayesian inverse maximum posteriori optimization chain monte consider representing determine minimize distance pdf eq thus pdf distribution deviation optimization seek fig blue pdf vary yield thereby minimize distance quadrature fourth place nearly number observe analytical match quadrature distance quadrature kde example attack computations euler stanford design upper surface design amplitude smoothly amplitude show location select analogy flow solver mesh lift store design objective include pdfs choose number surface produce moment sample speed four simulation geometry begin fold pareto design inverse mean write eq represent optimization value mutation mutation yield call make optimizer early carry plot clarity plot x axis ratio include individual design marker skew dark positively skewed dark highlight low moment pdf tail variance happen yield value however always design attempt two moment take use explicit adjoint method put significant disadvantage solution still design suitable mean inspection necessary especially satisfied design uncertainty ideal select initial design optimization carry pdf numerical computation optimization matlab programming tolerance met optimize previously discretize estimate pdf quadrature quadrature value visible quadrature rule compute polynomial gaussian carry product design adjoint computation surface sensitivity lift perturb mesh gradient rule computation carry result lagrange polynomial surrogate adjoint plot sample form summarize obtain problem require call lift adjoint adjoint examine four distribution pareto front match exactly target lie target exactly optimization carry design optimization comment target pareto front front infeasible pareto front principle robust design require total evaluation yield initial optimization trajectory take iteration call optimizer see second second roughly obtain nearly variance skewness contour attribute arise adjoint surface spirit plot take evaluation take second call yield extremely target target positive skewness skewness magnitude design close obtain routine design pareto pareto front b second design target third difference take iteration cost attribute positively skewed target skewed target find closely target match solely well average order less seek distance performance pdfs smooth differentiable approach illustrate concept demonstrate effectiveness design low tail acknowledgement award physical sciences uk author acknowledge center stanford college author aspect reference reliability typically use even prohibitive limitation statistical moment mean high order issue computational uncertainty specification find probability density pdf matches density robust design quantification stochastic critical modern design strategy range systematically availability computer trend engine environment device operate static uncertainty uncertainty physical
compression exist provide triangular consider involve form entry inequality bilinear arbitrary knowledge concentration inequality literature symmetric treat provide defer devoted selection operator norm validation operator norm loss computational criterion unbiased frobenius general principles stein risk sure begin frobenius bias unbiased unbiased omit follow straightforwardly estimate frobenius risk give could similar bandwidth analysis bandwidth asymptotically need optimal operator estimation encourage selection large weight term bandwidth moreover class entry large albeit high contribution estimate exponentially decay estimate interval simulation first bad secondly operator dominate sure tune sure tune except one sure variability error study uv n ix therefore distribution equivalently similar derivation combine lead q bind op eq establish rewrite derive result incorrect therein quick eq prove derive eq joint proposition follow remark yield q let joint equation equality uv eq great eq proof matrix put net moreover select thank estimator suggestion paper name section lemma remark estimation norm matrix improve addition sure estimator simulation estimator attract largely inconsistent estimator low obtain nature ensure cholesky regularization besides examine matrix mention visit rate refer covariance covariance matrix define unclear rate optimal date minimax stein unbiased sure aim operator norm block thresholding simulation inferior performance estimator motivate establish norm exist theoretic gap novel inspire stein sure outperform bandwidth conduct propose compete detailed result show norm notation denote inequality constant constant show estimator exist operator lead contribution take close argument shall fact see equality triangle op high show proposition
speed provide implementation facilitate predict svms linear inner test equation support vector theorem vector label expand rbf kernel product induce large slow prediction exponential inner instance appendix enable product exponential per factor front summation inner replace equation eq weight svd need compute rbf kernel least square formulation approximation dimension induce yield appendix guarantee compare eqs translate inequality combine assess approximated check observe prediction extra cost prior case norm upper rbf yield form polynomial polynomial eq effect kernel rbf relate rbf kernel must contrast polynomial fixing approximate rbf relative second rbf support act factor polynomial equivalent add approximated exact overall decision equation benchmark exact evaluation make predict computation differ algebra library consequence ii predict approximation dominate follow use loop matrix modern heavily use algebra software platform algebra library approximate evaluate simple operation exploit multiple datum support gain speed speed problem investigate differ exact approximated gain accuracy list report exact percentage differ exact difference report discuss briefly summarize verification set available website make without extra preprocessing set predict various information two class instance handwritten digit recognition contain versus competition feature testing instance instance contain training benchmark list key set dimension differently exact approximate list g illustrate even though combination inaccurate error overall remain always high ignore guarantee regard assess exponentially bind experiment get acceptable approximation occur schwarz bad upper grow approach schwarz conservative input mnist default somewhat optimize much run intel support extension operation prediction model evident time number number dimension approach ratio ratio optimal time minute impact library column naive implementation gain result file benchmark ann model enable ann range factor grow complex unit straightforward propose amount prediction model consist scalar dense small significantly counterpart table illustrate approximated datum compression ratio would approximate square svm even kb kb mb mb kb mb kb gb mb finally subtle approximated sensitive support instance model approximate consist combination vector reverse currently consider rbf series svms wide variety gain benchmark list approximation application benefit generalize regard normalization establish remain approximate svm approach fast versus iii prediction neural network always quadratic validity approximation illustrate absolute second absolute enough frank use order series inner support evaluation speed relate quadratic number approximated prediction memory optimize gain set additionally acceptable bound task ability use trick trick operate transform result attractive trick favor less cost rbf kernel situation evaluation limited span vision denoise radial rbf know parameter q prominent take
normal spherical scalar g call elliptical distribution transformation spherical elliptical elliptical xx ec ec cc elliptical characteristic guarantee specify elliptical mb continuous elliptical k symmetric function fourier elliptical fourier transform entire partly grant aid scientific b definition section property claim advanced institute mathematics nonparametric kernel bayesian incorporation certain nonparametric call rule rule current bayesian deal novel base mb exploit distribution incorporation mb nonparametric bayesian flexible inference nonparametric combination mb filtering model observation transition mb additive extend general conjugate model develop hilbert rkhs exploit reproduce kernel mean expectation feature respect distribution map associate kernel kernel map distinguished kernel guarantee specify estimation machine application estimation hypothesis classification chain rule rule kernel refer kernel sum rule kernel combination rule bayesian entirely study inference propose filter represent sample develop capture relation restrict nonparametric probabilistic robot vision robot localization hide mobile capture robot position estimate image specific model encode probabilistic desirable model mb exploit include conditional aim refer mb exploit tractable incorporation mb rule model focus mb develop filtering inference use mb middle probabilistic additive sample simply non twice mb mb respectively introduce mb estimator represent sum feature mean difference mb burden g addition bias error mb regression determine knowledge reflect describe mb mb sect focus additive mb case describe systematic consider conjugate comprise rule non mb space observation summarize mb mb incorporate mb thereby yield inference model mb additive filtering algorithm process kernel transition handle arbitrary e sect provide definite whereas filter whereas particle filter observation propose mc filtering space combine nonparametric transition dynamic simple present explicit expression exploit necessary filtering transition noise mc method analogous kalman require paper next preliminary mb mb propose sect ground robot sect review unless state positive definite arbitrary nonempty kx xx semidefinite gram matrix definite shift definite hilbert rkhs nonempty hilbert function exist unique product triplet definite kernel rkhs measurable measurable generate rkh study mean property use p w iw n follow consistent consistent rkhs consistent kernel rule space marginal let p p marginal deal py e computation x conditional non I ij ng li operation deal q kernel rule employ qx py give introduce mb define combination mb infer sect define mb sect mb example show sect describe mb sect base include relation obtain mb deal k mb consider conditional manner additive noise compute mb additive gx gaussian vector constant even rx additive gaussian gaussian noise gaussian additive gaussian conditional density kernel fx fx substituting expression mb compute case output mean often require mb additive gaussian computed systematic focus mb estimator differ estimate combination feature whereas mb input conditional simply mb give regularization smoothness whereas mb tune determined reflect knowledge type mb combine let factor comprise marginal chain fig middle model mean mb operation ng probabilistic distribution non operation matrix g ij computing evaluation n g l I z k I previously consistency mb sect kernel express weighted feature mean rkhs r g lm iw mb section dynamic model arbitrary misspecification mb mb coefficient fix horizontal exact leave mb value indicate horizontal correspond sensitivity misspecification mb perform misspecification set determined eqs analytical rkhs x gx pz xx ia q analytical describe error estimator show type mb mb mb iii mb mb decrease reflect knowledge mb bottom time variant state filter algorithm dynamic bm u dynamic gaussian kernel regularization parameter base grid search estimate trajectory trajectory sequence green dot sample color color top learn dynamic process know kalman kalman nonlinearity report accurate mse due mb result dynamic additive obtain appendix example dynamic exist fig phase demonstrate deal bias error sample thereby demand vision robot position mobile capture robot state robot comprise permit domain definite localization contain mobile environment building angle represent internal e comprise employ mean model motion part database comprise trajectory predict current current measurement argument learn dataset spatial pyramid sift descriptor image gaussian kernel bandwidth tune base filter maximize kernel test experiment size na I dataset maximize kernel pyramid markov property near nonparametric near image training determine demonstrate incorporate size perform property method yield combine estimation conditional distribution additive gaussian mb non mb demonstrate consistency mb show mb filtering nonparametric observation additive transition mb elliptical comprise contrast mb contain mean dynamic mb inference partly allow kernel appendix describe model mb kernel mean infinitely distribution probabilistic map convolution indicate convolution positive convolution definite gaussian generally stable closed definite light degree distribution definite kernel gamma density positive laplace close convolution note kernel mb additive noise stable case let index skewness stable rkhs x function generalization dimensional stable measure unit sphere denote stable aa fx stable comprise sub location denote stable gaussian case stable assume additive stable rkh stable density gaussian elliptical conceptually general elliptical elliptical elliptical dispersion characteristic generator elliptical elliptical nonnegative distribution stable respectively additive elliptical function elliptical rkh elliptical ec ec follow independent elliptical vector elliptical normal mixture elliptical give scale characteristic generator give h generalize normal elliptical elliptical elliptical elliptical lx xx generate laplace lx x laplace laplace laplace density laplace rkhs exponential direct computation omit derivation conditional kernel rkhs
sparse capture outlier tensor model multilinear interaction hierarchical tensor hierarchical student hyperparameter independently fully bayesian treatment linearly works implicitly tune discover adapt prior various tradeoff maximum extensive many art completion determination condition pixel pose illumination show latent past decade g recognition processing framework tucker cp know different e partially tensor miss cp miss cp probabilistic exploit exist manually severe emphasize surprisingly limited straightforward give tensor np fact determine even bound ard framework solution point applicable incomplete tensor bayesian incomplete tensor propose cp infer multiplicative handle heuristic way generally show convex gain considerable completion seek optimize apply nuclear framework completion low multilinear approximation auxiliary exploit completion suitable worth nuclear since standard optimize apply straightforwardly determination still challenge outlier frequently robust factorization use tucker outlier nuclear norm lead limitation quite tune evaluate unknown imply exist impractical obtain therefore automatic appealing limitation robust tensor mainly overfitte especially prediction unify model group additive modeling outlier partially represent tensor rank specify induce share automatic determination hierarchical student element hyperparameter learn evidence vary outlier fully bayesian framework derive anomaly automatic demonstrate term robustness allow rest discusse introduce preliminary multilinear specification factorization summarize show robust beta exploit however jeffreys outlier laplace treatment employ model approach crucial high pca recently tensor tensor error knowledge deal within g capital letter tensor denote letter g ni tensor element wise product instance hadamard product denote hadamard rao reverse except denote incomplete denote index define tensor otherwise measurement sparse cp express outer vector shorthand factor interpret tensor small integer representation factor denote row wise factorization denote correspond multiple n affect factorization latent vector multilinear intrinsic structural dimensionality unknown tuning parameter challenge seek infer partially datum attempt minimize individual hyperparameter denote share mode gamma maximum dimensionality concentrate tend yield minimum induce place hyperparameter place q individual correspond element framework laplacian student commonly apply enforce preference hierarchical student student control improper marginal laplacian lie encourage keep fully conjugate possibility fully bayesian treatment setting specification place illustrate simplicity notation factor matrix map sec principle provide treatment infer predictive miss infer resort approximate inference latent involve advantage employ present derivation proof approximate evidence constant kl imply mean field posterior explicitly derive virtue conjugate fig mode message likelihood incorporate parent express shown factorize subset entry mode index update evaluate straightforwardly introduce sec I rao sum index imply interact intuitive denoting fitness word fitness information current firstly coefficient scale fitness matrix incorporate posterior gamma q rd eq evaluate I far update obtain c thus strongly zero eventually several explain sparsity component predictive miss posterior yield student eq parameter uncertainty cost denote much cost scale polynomially automatic component rapidly present sparse correspondingly fully observe previously computation posterior matrix need denote row independent appendix accord efficiently simplified incomplete hyperparameter cp approximation efficiently result random sec computational addition simplify present treatment gain method characterize automatically tensor require predefine completion need discover various outlier elegant characteristic tradeoff rank evidence estimation take regard solution deterministic algorithm empirically computational firstly rank I first component possess third component mode draw distribution realistic setting subsequently randomly utilize relative evaluate recovery match cp cp ard factorization carefully truth generally cp tune tune penalty contrast cp ard tuning fully vary type signal ard miss entry percentage magnitude tensor ard ard poorly within recover tensor gaussian percentage confirm capability adapt true outlier cp ard observe ard rank value multiple datum term high note base applicable situation demonstrate robust partially observe tensor completion outlier compete performance quite stable vary robustness miss confirm accurately estimate multiple tune good run tensor completion runtime different miss ratio efficiency anomaly real surveillance video separate foreground highly tensor foreground move conduct popular sequence extract frame background alm perform video necessary optimal separate foreground frame separate person stand capture person foreground except clearly obtain auxiliary contrast factorization handle robustness robust capture tb property anomaly conduct experiment drop pixel compare alm illustrate fig alm recover background presence pixel effect severe robustness miss factorization color background whose consume fashion image tensor completion face multilinear first use pose illumination impulse gaussian noise removal poisson ratio tensor cp tucker ard carefully ground tucker ard require initialize cp perform multilinear tucker ard show image people pose robust satisfactory visual quality cp tucker ard removal detail quantitative efficiency runtime significantly note tune parameter ground truth outperform performance tuning computation tucker another automatic selection superiority non gaussian achieve determination tucker ard cp cp characteristic tucker ard robust characteristic framework overfitte determination
compositional regression compositional consider compositional start application become greater compositional specify compositional proportion heterogeneous method analyze assumption normal see compositional since compositional modelling consider compositional compositional compositional correlation analyze compositional consider simplex real distribution transform ratio compositional transformation regression response compositional give ij g ny interval j j percentile compositional methodology unbalanced compositional unbalanced bernoulli moreover generate size number follow software perform square error confidence proportion attack serve game win team follow attack serve understand compositional adequate kind since multivariate sake usual analysis analyze separately present proportion attack covariate transformation cc cc proportion confidence overlap proportion component attack serve different regard belong win outcome analyze result impact analyze compositional contribution winner team attack serve opposite team individually multivariate interesting covariate serve competition need simulation cp discover cp stable near coverage real pointing consider datum sp mail competitive physical international player carry element paper new compositional methodology estimation compositional illustrate datum compositional transformation origin change rule evolution player interesting field question competition development technical strategy factor player decision way team attack opponent among serve point opponent analysis mainly motivation compositional attack serve opponent compositional appropriate indicator objective procedure example team assess home advantage effect period country water verify home advantage way follow evaluate specific difference among home advantage home advantage et investigate play home indicator score home loss play home home relate country indicator assess home away quantify variable block attack opponent involve game home game game apply identify analyze play team multidimensional statistic attack attack attack predictor evaluate serve receiver multinomial obtain occurrence et al effect match e attack efficacy select action classify match status indicator action serve attack opponent lead team possibility win competition initially normality homogeneity kolmogorov quantify among independent comparison loss point break set also test work examine technical factor identify statistically game kolmogorov win match point difference win match factor service context compositional error team
condition number upper theorem sufficient sufficient affected greatly analyze asymptotic complexity result bound recovery section recovery measurement correlate optimization suggest significant improvement investigate iterative reweighted noiseless iteratively step except iteration suitably close regular constant iteration reweighte estimate lasso place weight identify justify property solve reweighte penalty encouraging justify much well reweighted minimum penalty make get undesirable minima choose reweighte bind ml expect reweighte achievable bind successively still efficient simulation reweighte lasso closely vertical interestingly reweighte nearly fail snr comparable require large furthermore gap infinity imply sublinear regime vs compute see recovery snr correlation achievable bind reweighte lasso db variant omp omp noisy miss sense omp vs information theoretic noisy variance support omp perform noisy variance theoretic recovery much level achievable conclusion reach omp recovery omp affect correlation degree recovery observation matrix information central discrete support conjunction tractable nevertheless identify gap exist fundamental sample computable consideration compressive miss show understand parameter snr correlation measurement set show vanishing also identify complexity gap lasso theoretic bound gap get notation conditional variable distinguish variable variable w set collection size hold theorem slightly weak clarity difference binomial line refer reader separately fact u summation sum specify condition equal sum chain rule entropy follow random outcome follow conditioning conditioning follow second depend index size contain prove even though idea general discrete generalization variable exposition fix I denote joint variable ix ix nn cumulative density assume continuous random quantization value joint ml decode indexing variable strategy quantization level level since decoder bind upper bind quantization convenience furthermore boundary equally e x j last theorem calculus also increase small upper furthermore boundary space py third variable assume measure lead lead eq derive easy independent across gaussian reduce inside plug integrate leave long expectation exponent exponent p independence integral leave write integral integral integrate r I analogous incorporate bound p integral last replace lemma necessary q q readily condition bind straightforward manner go infinity ii I sufficient exact theorem become element incorporate equivalent asymptotically satisfied choose choose necessity mutual necessary note jensen therefore necessary hold show clarity consider initially third take give result probability difference exponent integration term also replace w finally outline prove condition relax condition w incorporate scaling go definition salient recovery sample restrict non mutual linear performance gap consider frequently arise number scenario dimensional compressive low formulate observation set salient aim characterize arbitrarily probability parameter signal snr illustrative correspond realization markov depend combination element give nuisance theoretic salient see testing processing identification formulate namely code overlap share h namely severe related overcome limitation iid relate recovery instead general inequality bit represent set represent uncertainty output quantify observation exceed furthermore sharp exponential identify salient linear sparse characterize snr sharing markovian sparse regression sense cs probit sparsity variant particular analysis extensive deal model square variable list contribution markovian formulation follow repeat sake exposition literature specialized tailor instance relaxed testing quantization descent noisy form penalization sparse conceptually unclear come markovian viewpoint sparse observation pattern recover rely sparse thresholded introduce unnecessary distinction estimation discovery well support easy reliable conceptual tool capacity tool infer message indeed resort one tool necessity pursuit etc cover require impose extra reduce discrete natural object discrete pattern design ensemble rip key set herein purpose sense incoherence largely model herein tight ml decoder tight sufficient condition compare information theoretic bind practical omp gap gap room improve solve recovery recovery analyze easily analyze recovery observation variable obtain bound expression exponent identification code extend iid latent observation conditionally non identify gap practical regime explicitly row indexing row indexing use logarithm base latent markovian sparse exposition variable iid variable arbitrarily indice random randomness error recover salient salient salient label index iid latent elaborate mapping variable incorporate assume elaborate conditional example eq extend different nonlinear boolean indicator item result certain e denote set conditional conditionally iid joint coupling appear restrictive describe arise across see correlate condition account several possibility meta iid identically j nevertheless note iid exchangeable de bind analyze choose ml decoder decoder assume decoder upper decoder average methodology deal set true I exist analysis characterization lead sufficient necessity recovery select decoder disjoint iid hold observation sample condition arbitrarily sample di condition hold arbitrary constant ix mutual information salient total mutual denominator condition numerator bit denominator information subset control account error necessity support change condition support x mild mutual satisfied
size cluster encourage motivate encourage law cluster goal principled infer complexity generalize chinese restaurant objective small value law incorporate formulation result spectral long inspire objective derive optimize cut algorithm guarantee converge optima view precisely particular gaussian extensive segmentation asymptotic yield line normalize cluster approach mixture asymptotic fail relate base adapt modify small analysis power cluster discuss segmentation prior domain cut cluster undirected vertex similarity entry represent cut within cluster several objective cut edge normalize minimize relative complete spectral eigenvalue normalize cut seek minimize objective fall generalize cut objective extend original datum objective degree surprising fact establish mathematically follow normalized cut definite equal degree node effectively objective kernel monotonically minimize kernel objective appropriate weight objective equivalent mean goal graph objective bayesian chinese restaurant yield cut cut objective nonparametric chinese crp decay crp customer enter restaurant infinite table customer subsequent customer customer proportional new extension crp power modify crp customer exist table table increase probability start explicitly crp formula exchangeable indicator cluster crp express original crp cluster distribution law cluster distribute treat assignment negative log objective normalize desire clustering result give preserved size tradeoff standard cluster objective propose normalize relax cluster optimize globally technique law incorporate regularization maximization turn impossible instead normalized kernel adapt mean problem derive regularize weighted standard section normalize weight monotonic law treatment equally applicable cut association objective indicator justify cluster representative cluster objective e regularizer update weighted indicator minimize mean usual mean step make less fairly objective exist new cluster effectively correction arrive assign em cluster go increase restaurant table computing get rich new cluster start immediately specification analogous convergence easily show monotonically regularize equivalence cut weighted equivalence vector cut objective simply replace term exactly cut diagonal whose kernel matrix use space node regularize distance unchanged expand write weighted kernel problem scalable application utilize objective trade repeat suppose singleton cluster regularize w distance cd eq points nonparametric refer chinese joint maximize yield precisely mean equivalence cut inference author law datum add function objective incorporate encourage follow author exceed trivial datum minimize objective law experiment demonstrate utility approach mean law standard cut method cluster mutual truth cluster block specifically first assignment block random graph create cluster process adjacency leave figure stochastic model stochastic gaussian entry law cut algorithm compare cut big nearly ccc mean ccc grind truth upper next compare perform cluster power law distribute select uci dataset law class ground truth randomly split feature validate algorithm fair setting average perform k dataset dataset except mean bad power whole validate mean split large ccc r part convert normalize cut graph cut obtain adjacency similarity vector adjacency weight section truth cluster cut true cluster cut power cut cut normalize cut finally qualitative image berkeley adopt affinity matrix cut normalize cut generate image cut tend
careful constraint program strictly sake efficiency scenario impose crucial allow fraction predict unobserve via inspire success ground sparse matrix yield loose uninformative suit broad class pairwise partial angular interior second unable practical first optimization admm program empirically amount produce desire round procedure make matrix entry wise ensure round detail admm round solve original may return fractional rounding maps proceed round map denote compute rr estimate ji ji match repeat next recover set ground even vanish portion succeed minimal soon randomize universe herein generate point independently coincide independently incorrect matrix uniformly impose primarily simplify presentation remark significantly need generate mean theoretical appendix universe equivalently rank allow match densely corrupt input reveal consider constant corruption obey probability arbitrarily implication follow randomized succeed pruning recover account outli tolerance equivalently regime almost fraction input corrupt highlight matter perfect many none recovery regardless performance report pca semidefinite enable correction condition error partially come coincide tolerance algorithm outlier reliably provide recall threshold p recovery nearly soon theoretic recovery significantly outperform sdp heuristic match well cluster apply matching sdp exact minimize sum nuclear norm enable dense denote ground sign set sign pattern highly non negative sign sign propose assumption cluster inter two experience edge comparison deterministic edge error recovery therein encourage sec result performance consider recovery well evaluate describe simplicity full object fix universe remain assess configuration carlo trial reflect blue perfect red denote failure trial htp cc c map wrong incorrect phase object majority theoretical recovery figure noise ability improve illustrate transition diagram unable allow dense correction fig benchmark building fig six benchmark house building evaluate show building contain different generating algorithm present setup previous house matching assess sparse map match neighbor raw building build map sift view sample remain distinct per house count percentage correct building evaluate deviation manual compute image necessarily manual pixel evaluation htp fig ccccc house house moderate initial truth contrast ground house condition rank quantify specifically q eigenvalue eigenvalue put yield probability sec prove analyze kkt optimality treat sub I space via submatrix corrupted version convenient introduce notation complement support complement respectively complement projection resp way index element sufficiently claim bernoulli copy appendix define reveal derive result j nc q additionally random n represent unnormalized laplacian small eigenvalue see appendix contain resp concentrate state constant pass immediately bernstein variable duality summarize lemma satisfy additionally theorem generate high dual optimality decompose incorrect encode construct produce p sufficiently set represent encode symmetric q l j l toy contain incorrectly illustrate fig cc b toy example shape incorrectly map contain check procedure furthermore q assumption ensure dual satisfy establish follow lemma universal mn yield entry lie il il justify thereby algorithm correctly recover ground tn mn constant similarly imply eq allow constant q sec moment method control nice specifically follow sum cycle treat ki occur exactly suffice examine repeat least twice relevant edge span edge add adopt divide non vanish determine cycle vertex cycle notational simplicity exceed obtain since exist complete adjacency except exist bernstein vertex exceed probability least eigenvalue kkt suppose complement condition immediately fact assumption allow equality possibility suffice establish since semidefinite take imply put compare ij besides ij contradict must establish claim ii claim claim feasible optimizer sec proof would bind operator random eq observe q suffice examine due entry encode incorrect correspondence affect magnitude amount within lemma affect entry bound row row affect distinct block ji bernstein put universal ij p n universal affected satisfying express p magnitude hoeffding proof lie convert matrix one disjoint diagonal quantify block decompose ij concentration state entry lie support norm construction constant since upper universal h augment denote translate identity sum eq expand n derive positive negative j aggregate shape collection globally consistent close certain provably advance recovery none theoretical corrupt mostly object demand partially view propose jointly pairwise match densely corrupt semidefinite truth program spectral match numerically solve alternate method round near ability guarantee work even dominant behave outlier furthermore succeed minimal complexity perfect matching achieve include example confirm usefulness shape mapping cycle partial relaxation matrix graph find relation across fundamental scientific field list rna compare rich shape joint multiple object object matching pick perform remain pairwise algorithm satisfactory practice gives rise question aggregate one compute order efficient manner object observation pairwise preserve relational compatibility consistency composition map connect recently detect outlier work experimentally inconsistent cycle corruption despite empirical work underlie ground truth reliably recover provide match several order accommodate practical challenge state provide theoretical rise applicability presence highly noisy source input corrupt match result dense correction information consistency pairwise map appropriately exploit ideal map outlier challenge remain provably dense good approach ground map scene different camera scenario pairwise map expensive unnecessary characteristic infer unobserved match incomplete tradeoff correction remain require match densely corrupted pairwise map paper aim concern object error fold evidence propose call constraint attempt compatibility semidefinite constraint rely total match spectral methodology essentially scalable optimization guarantee surprisingly admit recovery input finding near optimal error precisely fraction behave outlier besides incur exhibit strong nearly equivalent full scenario succeed reliably unobserve partial input soon map connect theoretically evaluate dataset synthetic example finding art matching matching shape graph matching focus biased isolated technique easily exploit noisy fundamental nevertheless none demonstrate provable accommodate recent denoise model another recover relaxation observe relevant specialized joint also enable broad inspire rank completion component analysis relaxation recover herein occur analysis tight highly need incorporate structure encode regard estimate graph nevertheless intrinsic property herein explore form doubly belong highly symmetric translate language inter encourage detailed theoretical comparison organize problem setup include recovery method follow alternate method admm round strategy also performance proof theorem introduce demonstrate summary setup matching partially algebraic pairwise formally define notion throughout match encode discrete paired particular map w partial input output matching describe input partial agree partially totally truth detect incorrect pairwise map aim propose tractable return
countable express exactly determine learnable minor index learnable complete determine vc drive force must set nothing determine deal issue definition computable say within index pac learnable degree part present show follow eventually dimension computable initialize require least segment look also add stage stage index path empty infinitely infinitely many set vc vc set disjoint finitely computable effective concept arbitrarily thm conjecture definition thm thm thm thm learnable class indice computable make precise cover property vc method characterize object carry machine learn least gold gold number computable function initial string instance strength determine recursion focus interest index learnable model correct much exposition subject two kind arise target learn distinguished arise neither aspect easily gold identify index computable segment neither randomness present model pac learn recursion calculate learnable subset call call every behave ask running call existence know boolean pac learnable truth assignment satisfy space learnable learnable arise exposition integer hull contain pair intersection half consequently show reasonable theoretic arise finite vc david denote distinct least every measurable behave behave pac learnable vc meaningful determine learnable arbitrary class narrow meaningful constrain enough usual framework result know give class computable subtree path subtree co subtree define equivalence follow say formula calculus regard boolean variable let construct subtree include check subtree intuitively fall enough segment detect unless fall uniform representation take real string interval computable set path include binary length effectively subtree define hyperplane computable coordinate represent subspace subtree natural space exclude exclude space requirement computable restriction absolutely hyperplane give hyperplane computable coefficient hyperplane hyperplane lebesgue suffice show cumulative hyperplane furthermore multiply many machine situation could certainly terminology concept term computable enumeration tree interpret tree index concept adequate need would like class computable reasonable class computable class weakly effective computable computable nd place see mention soon strong definition want computer something concept coefficient let effective concept computable set complement
label subset training unsupervise frequent tag annotation vocabulary average annotation image whole annotation compare directly approach use specifically rescale side keep aspect ratio sift feature densely sample extract patch fix sift unlabele visual vocabulary represent suggest spatial annotation vocabulary frequent tag section together treat descriptor adopt used layer architecture layer activation unit softmax conditional discuss since image sigmoid output probability pc unsupervised layer label base top hide annotation word feed svm belong average ap ap curve average class compute metric report connection follow find normalize input histogram rescale unit hyper unsupervise weight parameter etc overfitte adopt dropout maintain decay throughout gradient average weight put emphasis parameter annotation approximately annotation word section multimodal hide epoch epoch layer epoch present baseline map performance among epoch unlabele baseline epoch epoch epoch datum epoch epoch deeply hide deep beneficial establish explore property section annotation imbalance visual annotation influence weight annotation weight value hide epoch annotation equal annotation configuration annotation bad annotation increase get among annotation perform good layer epoch annotation value illustrate multimodal qualitative retrieval scenario image retrieve learn task adopt similarity correspond rest multimodal query learn image ability annotation ground truth annotation probable annotation modality extension learn visual moreover propose deep version model bag image extract meaningful unlike model model autoregressive advantage iterative confirm competitive multimodal modeling achieve multimodal benchmark yu zhang department university china zhang universit k model choice task deep boltzmann topic autoregressive demonstrate state extend multimodal datum simultaneous annotation first increase discriminative learn employ learn joint word annotation compare topic model deep version reach state multimodal modeling source increasingly model allocation lda generative great model share model extract infer topic word extract visual convert word bag visual py model use predictive leaf word use multimodal variant lda propose correspondence lda relationship annotation modality assume multimodal lda learning regression module relate multimodal document multimodal corpus annotation word embed annotation power improve heart topic generate produce extract observation solution disadvantage sophisticated inference trivial expensive must approximate variational mcmc yet simplify visual image approach visual distribute representation artificial bag datum text annotation jointly boltzmann multimodal achieve extension time generative approach document autoregressive estimator directly joint chain modeling potentially inference perform instead document simple feed value network text rs text retrieval deal annotation illustrate incorporate visual highlight discriminative objective result confirm lda extension deep discriminative word illustrate datum imbalance histogram fed compute discriminative global mention multimodal lda lda modality describe image globally label inside city etc annotation content margin formulation line extend extend context multimodal modeling topic autoregressive network increasingly model review autoencoder multimodal though outperform favorable reduce multimodal especially paper deep version outperform predefine vocabulary data image convert visual sift descriptor densely detail image thus bag word sift descriptor descriptor contain distribution conditional model learn feedforward wise activation bias respectively compute equation visual word address issue tree conditional logarithmic leaf tree model reach binary regressor multiply right choice specifically root leaf internal always binary subtree contain bias logistic sigmoid guarantee could attempt organization word tree assignment leave v I document document descent word bag average exploit conditional previous hide since regression total practice unit eq representation could fed classifier computer highlight jointly concentrate single discuss deep incorporate discriminative feature visual describe text annotation feature unsupervise lda perform directly use pyramid reason statistical might computer address variant lda supervise unsupervised propose attempt architecture regular output softmax layer label put regular neural crucial difference hide visual conditional discriminative term understand encourage explain visual word hybrid generative term hyper perform descent multimodal visual word annotation performance note notation word annotation section concentrate unsupervise supervised mention order many view connection employ across interpret factorial ordering notation joint order treat explicitly random stochastically across autoregressive conditional rewrite summation conditional split first index order rewrite note equation simplify perform training expectation expectation require representation annotation random annotation perform stochastic specify separate conditional conditional notice annotation vary value vary procedure prescribe exhaustive sum predict stochastic predict implicitly sum gradient share permutation show split two network approach mention conditional summation layer histogram representation word original training instead hide effect complexity layer feedforward neural bias layer equation obtain representation implementation efficient binary tree regression however go back softmax conditional preferable conditional softmax softmax softmax amenable gpu end experiment extension softmax deep specifically negative sampling supervise regularizer importance unsupervise annotation framework treat setup annotation annotation problem example annotation ignore huge visual gradient come word conditional annotation vocabulary annotation annotation histogram element hybrid rewrite replace weight annotation weight annotation pay annotation cause imbalance hyper select annotation heavily improve besides annotation embed global play multimodal datum thing complement specifically image possibility matrix specific understand bias condition word multimodal layer world test retrieval achieve code publish ability multimodal measure simultaneous image annotation world datum annotation annotation benchmark extensive lda multimodal lda performance also support compare pyramid construct set tool image city country evenly maintain image evenly test occur densely sift feature size dense sift sift vocabulary grid spatial position produce pair use annotation annotation
principle relevance discussion statistical employ determine order refer determine reject hypothesis monotonic consistency propagation infimum variable line follow similar enforcing complement sx dd dd sx f x introduce parametric version kolmogorov consider parameter identify refer tailed ks tail ks test etc assignment satisfy ks hypothesis hypothesis constraint monotonic consistency parametric line h sn sn ij jj sn sn x propagation enforce complement application statistical employ classical hypothesis non parametric application mean sample sample accomplish one h reduce significance reject range fact work specific singleton illustrate illustrative domain feature singleton inspection desirable inspection plan unit plan horizon inspection inspection j model inspection inspection inspection interval inspection inspection enforce u inspection consecutive day inspection carry month inspection plan assessment day horizon plan assessment interval black dash h use encounter statistic fit look scheduling designing management instance desirable statistical schedule order constraint stochastic pass priori policy stochastic constraint property instead operate assumption specify parametric constraint constraint statistical constraint instead set work spread constraint statistical exhibit statistical e median bridge link program nature inference identify assignment specific discusse enforce consistency span encounter inspection scheduling inspection anonymous valuable suggestion university scientific project university publication part research foundation introduce modelling constraint programming discuss encounter statistic novel inspection scheduling desirable informally assignment prescribe first embed kolmogorov filtering enforcing discuss discuss application span encounter inspection scheduling aim inspection desirable modelling world consist outcome triple denote outcome sigma algebra return function probability mapping set element event often transpose distribute may time replica generate variate outcome variable define adopt follow statistical cdf statistical also cover outcome operate observe determine adopt carry statistical select g generate data hypothesis datum formulate suitable distribution determine obtain extreme outcome highly hypothesis great evidence collect insufficient reject follow survey test test kolmogorov student hypothesis compare hypothesis student inverse freedom respective tailed test great test assume statistic pool variate size student reject note range variance test parametric compare reference cdf hypothesis draw define cdf supremum set target converge kolmogorov nan reject cdf kolmogorov numerically approximate tailed stochastic reference I case vice versa kolmogorov employ band band entirely contain cdf tail stochastically dominate triple variable map combination domain use kind logic linear constraint constraint constraint among dedicated able remove infeasible enforce g arc consistency generalise arc exist compatible domain filter repeatedly solver heuristic search engine solver explore partial assignment order
set sum fraction correlation nk k let structure distribution acknowledgment office correlation uniform assignment half proof subset node eq choice cardinality union value prove fact david laboratory information electrical science management institute technology mit paper result unconditional computational bind class et al construction equivalent difficult noise computational learning bind exhaustive significantly restrict class aside assumption tree paper property focus ise show interaction strength exceed result impact variety application high dimensional biology finance determine access identically distribute undirected graphical vector question mostly undirected model search neighborhood decade ng give graphical kullback present learn algorithm perform neighborhood recover graph scale guarantee reconstruct sample complexity well exponent run great deal graphical ask answer show unconditional et al understand apparent plant clique show exponential computation algorithmic include chain fairly follow theorem algorithm tractable writing form exponent primary importance think require restrict class interaction seminal liu model restriction generalization assumption distinguish knowledge require informally property decay exponentially fast decay high temperature ise multinomial exponential correlation neighboring variable ise set candidate neighbor roughly correlation set exhaustive search neighborhood mention cost algorithm ray similar number reconstruct ise begin ise connect graph os benefit non negligible dependency wu et remove generic base regularize logistic algorithm logistic provably certain ise simple optimization incoherence restrict isometry difficult ask follow second general even ise whereby prefer opposite state correlation algorithm base prune pairwise correlation decay anti improve first contain kk lemma end basically reconstruction graph degree give rule require know let star nod star center none edge loss generality call edge upper probability star include star center non arbitrary cardinality match share disjoint event note variable mutually discussion error successful function lower begin observe include define take map reasoning section consider anti ise real working configuration amount example recover think soft large modify core eq estimate structure determine subsection e probability possible define restrict analogously z define set particular map neighbor configuration z ab prove iii ii z hz algorithm run core infinite obtain effectively conditioning conditioning subset law g u u corollary conditional estimate conversely ingredient show restrict attention zero start compute equivalently monotonicity together bayes denote last quantity number obtain stochastically dominate inequality reduce remove node condition strong e u b integer ising model use sample argue tolerance consider h statement eq next choose corollary inequality last show complete statistical taking
curve ranking hash hash noted rank well hash sub lsh important next subsection actual four hashing scheme parameterize retrieval parameterized lsh generate meta hash meta hash form hash hash consideration scheme randomize lsh preprocesse design intersection affect bias asymmetric correction asymmetric hash hash undesirable lead provably inner literature experimental evaluation advantageous scheme require modification asymmetric weighted weighted intersection general value vector asymmetric weighted similarity use new asymmetric promise hash corollary definition department ny department computer science nj usa widely index suboptimal desire overlap inner set propose hash scheme utilize asymmetric traditional monotonic inner product provably traditional hashing product evaluation publicly available claim easy matching operation web popular technique big estimate set later sensitive hash spam detection collaborative linkage representation common largely bag power number combination rarely often absence lead search representation eq component absence attribute binary application scenario desirable measure instance description five york base common typical name new york suppose query five I record match record clearly size short record matching scenario undesirable typically big often unnecessary record lead ordering order order intersection near neighbor interest collection universe interested product problem search eq refer search practical significance heuristic problem notable recent among use approach index record web large huge vocabulary quickly extra size query instance computing set cover hard greedy heuristic heuristic usually practical query locality sensitive lsh successful efficiently neighbor suffer curse dimensionality indexing scheme provably sub search impractical due hash indexing stream ideal modern inner inner product hash heuristic general inner locality hashing scheme product hash elementary argument inner special provable efficient asymmetric suboptimal hence asymmetric locality product likely suboptimal like common indexing product existence provable lsh investigation reveal result unlikely hash hashing show usefulness transformation construct provable hash remove undesirable bias towards eventually hash scheme binary inner call hash comparison binary web asymmetric hash provable improvement hash product construction asymmetric hashing neighbor four real world modification adopt practice efficient neighbor partitioning turn massive due curse theoretically exact near neighbor propose adopt near near near report neighbor near term deal similarity near similarity optimal guarantee underlie lsh lsh property high hash function mapping hash family follow obtain resort sensitive hash nn lsh accomplished neighbor query lsh monotonicity condition satisfy lsh determine neighbor note ready lsh family lsh hashing lsh known view apply permutation formally hash hashing even really retrieval nevertheless popular scheme intersection binary product argue undesirable suboptimal asymmetric undesirable dependency intersection novel lsh family lsh formally choose generate normal hash dx cumulative cdf standard normal distance monotonically distance lsh lsh parameter sign popular lsh concept give vector utilize component I normal seminal cosine reduce random lsh interestingly show actually non binary significantly near motivate asymmetric overview asymmetric lsh product lsh elementary show locality hash lsh unnormalized product inner vector hash valid lsh fix allow hash extended framework asymmetric locality hashing hash locality hash family sensitive nn query collection asymmetric lsh lsh preprocesse structure neighbor start scale enough lsh concatenation provably bound query depend realize idea convert cosine efficiently well start scale sign cosine query concatenation form follow follow end provably efficient l lsh neighbor thus outperform lsh sign structure problem asymmetric transformation sign projection also inner common typically prefer hash asymmetric hashing mh suitable indexing intersection general lsh unnormalized product proof lsh inner lsh inner lsh construction lsh hamming ordering distance monotonic lsh monotonic binary extra trick co present hash sample independently integer give independently happen hash lsh inner sparsity ensure web run hundred thousand therefore word hash scheme domain observe denominator way lsh sample thus almost argue thing later provable sparsity likely handle bound number lsh inner product f hash remove denominator worst meaningful providing provide asymmetric name asymmetric hashing mh monotonic inner hashing scheme theoretically exist scheme inner define query transformation concatenation zero power asymmetric inner unchanged query eq nature neighbor instead denominator large set sparse likely thus asymmetric usually lsh scheme asymmetric lsh lsh asymmetric lsh family neighbor product want quantity difficult denominator get formally eq new become monotonic formal complexity asymmetric expression preprocesse cs hashing intersection exist intersection space give negative transformation add chance match plain transformation create hash lot time sampling generation transformation eq similarity therefore nonzero weight problem know hashing sign theoretical asymmetric hashing mh lsh unlikely ignore asymmetric lsh maximum asymmetric advantageous binary query cs immediately follow query explain retrieval product plain approximation curve solid sign dash term irrespective general inner product compare product sign projection value compare suffice asymmetric sign binary inner summarize convenience although perform identically clearly irrespective asymmetric outperform sign theoretical powerful hashing base comparison hash binary rank
differently fast train part c help filter third fused rule fuse drive contribute two share part contribution feature complexity make part jointly slightly sub use evaluate convert relationship cosine formula make derivation unbounded make cosine function invariant cosine widely many binomial choose binomial recognition fisher training binomial criterion formulate denote subject numerator denominator total eqn learn separate far compare eqn binomial classified focus similarity equally make likely fix cosine cost determine eqn cnn connection train totally mini batch batch batch fast training eqn learn sgd backward top contrary specific branch eqn describe assign label pair tune positive build among protocol shoot setting intra dataset train intra conduct illustrate basic experiment illustrate conduct training per come camera camera camera disjoint subject split number epoch test report camera camera camera subject camera follow protocol subject probe subject camera remain camera repeat time split split report camera camera merge generic pair assign mask redundant one probe evaluate use view important factor asymmetric negative besides widely neural double although trick performance compare image similarity final fuse c rank without augmentation rank dataset crucial network know geometry person person pose virtual performance direction identification year pair batch positive batch art remarkably elegant pose information segmentation contribute bp simultaneously similar propose outperform art method superiority reason pair quality recognition dataset coincide capture capture capture totally device people image capture image person illumination subject small resolution image camera randomly include recognition c tr improve c tr improve I train comparison outperform par rank improve significantly quality similar combine similarity score intra result remarkably performance cause big dataset raise dataset cross number drop train hard generalize property texture dataset diverse besides experimental also adapt another target improve research learning extensive intra dataset cross conduct person person cross identification outperform significantly improve moreover research train person engine good sub share therefore cnn gradient eqn derive see cosine similarity eqn eqn eqn eqn respectively substitute eqn eqn eqn define eqn although asymmetric experiment reference denote sample view derivation gradient support national science foundation national science technology support chinese sciences project technology support project degree university china receive degree research face recognition recognition learn article international develop face b china chinese sciences interests computer processing international receive national university technology china ph degree university security chinese sciences work microsoft prior associate research interest recognition machine recognition surveillance publish paper international books li metric field person compare propose way pixel propose jointly learn feature texture framework two cosine function big variation person image use similarity label prove compared set dataset person identification intra illustrate person identification metric category identification person subject practical two view closely camera analysis retrieval algorithm recent year person essence recognition good evaluate similarity compare person identification challenge due person image identification usually person surveillance work mode low unstable pose cause image surveillance two large variation inter class summary challenge come aspect camera pose variation unstable illumination person collect first video person identification training source test domain totally capture different different identification good generalization dataset important unstable important identification exist method sophisticated histogram discriminative discriminate exist usually come texture strategy contrary propose module together feature unify deep originally signature verification person neural assess connection figure carefully image denote connection share identification follow advantage metric pixel layer optimize channel learn color texture concatenation structure switch specific dataset evaluation par cross significantly exist setting conduct strict experiment person identification far deep identification four review feature representation person identification focus person color texture sift fusion contribution final advance aspect color histogram structure person improve color texture feature extracted predefine localize part method localization obtain significantly configuration explicitly spatial constraint large history face precise pose normalization naive usually achieve rank pls among metric stream discriminative task variation metric divide sample accord pose metric explicitly obtain high loop style identification improve drastically intervention reproduce researcher early neural propose similarity signature verification compose network verification group property neural objective end network metric similarity
hessian therefore variable pg surprising correlation extremely depend two informally speak distribution inference c z w quadratic form analyze correlation posterior joint pdf px pz pz pz z deterministic auxiliary variable pe sake clarity completeness px pe e x pe eq square correlation hessian hessian log pdfs child determine parent child parent hessian l pdfs parameterization distribute hessian equation variable child general fast log case dependency
feedforward hide discuss behaved depict applicable multiclass multiclass sigmoid binary label scale minimum pt sep draw circle dash line min node mm name hide hide name name observe name name observe name black h line mm red red width red mm red line x line width mm width red width width blue mm blue line mm e hope moment start order learn label score mild regularity glm activation mild regularity assume row elliptical project span restrictive challenging distribution hide provide problem vector network provide note represent nonlinear neural feature build stein stein second law expectation follow stein last chain stein random differentiable gx px px assume expectation exist integration part result provide nice form auto shown regularize pre use correlation score encoder explanation elliptical project even gaussian improvement present approximation recover retrieve identifiable pose row problem class degeneracy least derivative activation deep network sigmoid assumption initial usually million satisfied weight sparse word neuron net argue sparse connectivity scale nevertheless direction back dense scale back propagation need weight feedforward hide hold uniquely recover version efficient algorithm traditionally formulate equality scale projection deterministic matrix bs ij bs additional detail node depth multiclass softmax sigmoid function network layer stein nonlinear network uniquely recover stein use rule convolutional assume consist parameter neuron therefore prominent progress subset nonlinear hide go challenge learn full weight middle layer weight manner future hope investigate new introduce paradigm literature unsupervise discriminative lot investigation continuous although discrete difference interesting small degeneracy class neuron layer small hand degeneracy square matrix hence weight tensor highly range topic challenge microsoft fellowship nsf nsf award award award support award notation bold time california approach feedforward network adopt operate moment label show factorization provably deep practice output gradient feedforward stein paradigm challenge variety speech understand deep net non problem involve million view guarantee contrary guarantee back paradigm unsupervise tensor survey tensor factor employ network dag paper employ net sparse theoretical guarantee prove correctness stein statistics stein show effectively base analyzing learnt exactly result feedforward neural label stein result state expect stein essentially integration employ stein row matrix weight degeneracy matrix neuron dimensionality note rank significant requirement argue performance practice recover efficient optimization early topic model dag establish also stein training connection encoder approximately provide propagation learn result correctly span weight vector improve score improve kernel exploit
quite namely intra layer therefore lead learn connect boltzmann six four hide standard c mean top bottom zero mean bias similarly begin break machine point computed instance limitation mean although acceptable small quality quantum probability deviation unit bm qualitatively see wide illustrate fail notable contrast perfect state expectation seem indicate success systematically sample evidence instance state confidence somewhat surprisingly percentile state percentile percentile data fidelity qualitatively although qualitatively similar rbm value fidelity rbm arise case take experiment boltzmann inefficient important provide insight database handwritten digit digit directly require compute configuration coarse grained image resolution digit distinguish order confusion digit appear pixel divide pixel set corresponding pixel pixel threshold subsampling procedure optima cd ascent experiment relative difference optima vary percent difference vary half percent observe optima synthetic body evidence grow approximately linearly unit constitute exclude optima rbms train mnist experiment towards find qualitatively surprisingly job predict estimate roughly mass substantial exactly vast case unlikely preferable b rbms mnist number mean stable median bad nearly variation unclear small example law numerical distinguish two possibility scale qualitatively nothing qualitatively mnist correlation field approximations fidelity issue choose training find uncorrelated minimal kl divergence distribution main benefit mean rotation mean call function equation implicitly solve iterate jacobian analogous gibbs configuration efficient forward exact methodology case visible mean take expectation derivative easy product approximation kl note low field approximation graph partition accurate small boltzmann approximation reduce error albeit suggest probability state approach limit strength correlation vanish continuous function therefore h v b h follow field unique find optima success apply behind sample ideally proceed hide unit newly place sample gibbs cd work unit unit sample repeat probability average compute approximation gibbs step rather true gradient try approximate ml cd optimize become log absence asymptotically approximate although approximate gradient gradient yield objective inexact efficient drawback main drawback cd visible class take hour day machine wise training employ potentially sub parallelism accelerate training rbms extent sequential update quantum restriction information store bit classical bit linear superposition dirac normalization condition measurement entry amplitude live superposition representation note write product four ability represent superposition essential massive parallelism quantum proceed unitary evolution unitary quantum reversible quantum knowledge quantum gate unitary acting gate gate need gate complete unitary computation correspond axis gate unlike rotation approximate within small em deep intelligence quantum computer show intractable conventional computer quantum boltzmann machine rich comprehensive deep lead efficient boltzmann multi connected counterpart introduction quantum conventional art classical recent machine intelligence task model ai task several layer raw train detect car accept raw pixel subsequent shape next shape aggregate shape learn nested layer hierarchy concept abstraction encode highly complex training fall machine network generative boltzmann ise encode feature concept ise interaction represent dependency feature output call visible node latent feature boltzmann bipartite boltzmann layer rbms discussion visible unit binary boltzmann configuration visible hide unit gibbs normalize energy configuration visible hide unit visible take hide observe modify likelihood boltzmann training process weight bias ml size regularization denote bm derivative take form computing resort divergence know lead suboptimal train full boltzmann provide framework deep illustrate elementary accelerate process lead well quantum quantum analog gibbs machine formal description appendix exist clear evidence consequence overlap complexity along probability configuration fact configuration use mf configuration approximation good refine exactly sufficiently mf minimize leibl tractable equality achieve mf apply another let configuration eq multiplied desire operational add quantum state right success amplitude boost success number quantum evaluation number logarithmic linearly number edge visible constrain asymptotically contribute operation require gradient scale layer number bm quantum show often make parameter exist algorithm store otherwise logical logical computed substantially reduce exact always hold bm state relative however probability small reveal fidelity gibbs continuity fidelity state formalize energy fail theoretic unknown example unlikely efficacy mf weight cd violate operation execute unit algorithm form case via allow datum superposition superposition amplitude estimation quadratic gradient consequently improvement allow process quantum training think quantum subroutine quantum boltzmann quantum simulator store computer superposition train entire set oppose sequentially accurately quantum oracle oracle query oracle superposition convert repeat expectation measure amplitude directly probability string scale analogy arithmetic success constant boost probability reduce preferable almost cd preferable circumstance parallelism energy energy layer depth see depth execute via calculation depth depth depth price circuit divide mini cd depth cd feed question regard behavior differ substantially question grow restriction computationally unit visible h h dash line line confidence impractical four add set flip contain digit bias show visible substantially increase space dimension differ illustrate primarily mf similar appendix typically close gibbs appendix reduce roughly rbms show rbms second issue determine boltzmann rbms tend rapidly scaling take set suggest assess benefit average find quantum optima find deep improvement ml constrain nature make local cd improvement optimization model bm outperform term quality machine bm quantum train enable rich fundamental work machine notably divergence approximation quantum significantly provide elegant approximate gibbs state mf desire gibbs bm operation depend reduction full quantum processor address result encourage advance quantum assess ability divergence set currently deep boltzmann regardless deep perspective begin computer unbiased allow probability sample quantum quantum approximate rejection quantum numerical begin state quantum likely inefficient depend state uniform joint boost success acceptable quantum number achieve fix initial refined gibbs coherent discuss generalization refine unit boltzmann unit form string distribution approximation distribution variational approximation field gibbs quantum approximation define let satisfy configuration analogous appropriate visible boltzmann visible jensen show lemma give success analog boltzmann success state configuration mean uniquely field partition coherent rotation eq step refine approximation quantum circuit measurement quantum reverse quantum save normalize distribution measure projective unit remainder proportional state normalize note valid value replace exact protocol place place success visible bias boltzmann vector mean field h add ib h h hz generate quantum machine visible visible unit compute x ib ex x ex amplitude provide reduction need occur amplitude quantum iteration proof compute derivative respect algorithm adapt compute bias superposition claim quantum hence success also visible require gate law probability ensure error error repeat state amplitude compute triangle inequality pick require query assume mean circuit visible bias edge training vector specification edge calculate within p alg superposition use amplitude learn amplitude learn amplitude j pt qualitative sample provide amplitude amplitude create evidence forward amplitude randomize case success use amplitude amplitude success version original circuit infer probability step explain corollary visible hide boltzmann connect within scale scale corollary iteration search boost success amplitude success find let identical apply amplitude measure success probability equation unless choose guarantee inverse assumption estimate propagate large error general propagate derivative exist monotonically increase taylor hence overall require repetition circuit produce amplitude overhead cost claim gradient bias quantum call boltzmann machine scale component compare cost incur superposition compute expectation operator compute run time rather vector imagine set inherent value mean although assess cost implement computable complexity implement learn oracle quantum process require use store query database issue may preferable gradient begin randomly gradient comparable error since two cost case majority uncertainty come obstacle assign magnitude configuration example boltzmann excess gibbs root mf reflect adjust particular regardless perspective transition confidence mf update wherein essentially unchanged use difference specify method field uniform overhead substantially mf break quantify divergence algorithm body investigation rbms evidence body comparable handwritten mnist ml optima optima technique verify result lie optima optima many perturbation compare point say fix decrease cite value repeat process perturbation second find use optima analogous training ml step subtle consider comparison determination divergence gradient ascent ascent calculation optima fix converge running varie least epoch learn calculation introduce absence bfgs ascent derivative choose occur c optimizer optimizer ascent objective bfgs objective gradient ml estimate small device rbm proceed ml add zero training point epoch stop meet noise ascent highly suffice yield optima find error adjust improve reduce rate suffice reduce error multiple reduce error need negligible c average rbms qualitatively quadratic relationship rbms weight vary weight bottom ability depend choose derivative differ substantially analyze result rbms unit know strong strong correlation introduce weight normally boltzmann mean weight find train recognition divergence set zero unit suffice state error case slowly function number rbm deviation deviation necessarily problematic deviation practical extract scaling choose cutoff residual bias unit variance datum random rbms datum scale zero chance bit optima average table give mean divergence field rbms less kl divergence slack see
take easy h restrict theorem third course triangle change cloud separately svm learn refer demonstrate improve oppose use traditional pre scaling variable feature range feature empirical empirically deviation variance distribution justification pre method precision less error important although comprehensive treating adaptation empirical observation learn diverse computer etc pruning decision rule robust reliable give specifically consider scale available coordinate value center integral equal discretization bin distribute value belong realize although clearly give decision discretize dataset decision dataset balance specificity discretize handling without direction key space consist segment disk four axis line shape line part triple plus sign perform eigenvalue correspond eigenvector yield per record six reason six every homology triple length train example category category evaluate performance computed maximum recorded sensitivity specificity type report table low rate six exception triple discretization appear six low triple triple shape location within dataset homology consist triple largely death respective homology high around triple former cloud contain disk indeed indistinguishable another decrease attribute triple point densely sample segment synthetic point point line segment either line densely segment line second case persistent total segment reason behind feature point densely segment persistence homology segment datum previous linear category low rate alone feature vs example bin discretization lead feature alone employ slightly feature collect ground see purpose randomly group scale persistent persistent yielded type ground rate feature feature lead greatly compare feature alone result promise multi feature consistently real synthetic turning persistence diagram treat diagram image algebraic geometry advantageous advantage bin discretization discretization lead real rule difference test life often slightly different allows retain patch however strong typically happen irrelevant match suffer identify plan future rely feature define construct beta preliminary show rate utilize continue investigate combined summarize table experiment bin indicate bin c max h max bins h bin bin max bins max xt proposition section extract feature within dataset topological capture diverse subsequent machine operate dataset construct correspondingly extract dataset synthetic context typical classification problem thus relevance technique assess result classification measure sensitivity specificity error feature extraction processing feasible computation mechanism remove add challenging difficult approach extraction rule mutual manifold benefit stem shape topological machine performance technique cloud return direction output onto subspace span intrinsic dimension look course line notion intrinsic want thin look look simply true sample intersection return reasonable notion prove build topological example chapter assess among nice topological differ move deal information local concept one depend entirely often impossible scale address radius version differ exposition good knowledge attempt dimension dataset quite different understand drop information effort learn learn use construction road map reconstruction outline follow briefly review utility result experiment cloud compute cloud ball around process repeat multiple persistence extend extra infinite diagonal dot diagram diagonal diagram define infimum diagram exist infinite dot tend infinity diagram perfect efficient state persistence diagram diagram red circle diagram red red dot close bottleneck dot diagram function function space non integer technical theorem stability homology persistent version fix non integer sphere radius simple take point plane zero complicated circle homology depend turn change unstable homology radius choice point center gradually line rank fortunately persistent
determine stop point f automatically statistically bold entry average unweighted column label represent fully supervise passive entire drop drop set implementation method fact worth average annotation average early high cause mention average cause subsequently many highlight regard various stop parsimonious annotation stop largely except sp unstable sense major failure stop g spam way early amount e g ls protein always clear stop tradeoff extra measure versus annotation know last clear annotation high sp sc depend annotation tradeoff promise develop preference much conservative user pick criterion suitable sp gap seem conservative sp stop likely method provide behavior sp control point visualize perform representative axis measure annotation axis stop sp stop exist without dna range demand dataset table cutoff requirement sp enable adjust stop fashion behavior area precise expectation intensity intensity annotation control intensity level determined fold average dataset c annotation window annotation automatically stop display stop measure investigate sensitivity number annotation fold automatically display ls fold avg stop learner annotation bold statistically significantly bold sc method margin learner entire training stop random stop fairly steady plus work fold simple work think determine stop develop create density meaning example make sure another algorithm stop stop unlabeled pool maximally stop efficiency get adequate representation latter accomplish perhaps stop add stop sp widely table model conclusion experiment statistically sp favor representative fold stop action fold tc corpus range crucial annotation identify improvement stop sp address sp widely stable sp exist informative criterion also informative demonstrate rigorous annotation tradeoff area user enable user stop user pick tradeoff substantially behavior sp center university md usa computer information sciences university usa reveal annotation stop al address furthermore stop handle range annotation tradeoff despite exist dominate conservative little provide stop al provide user stop nlp considerable annotation effort al enable must mechanism annotation motivate stop annotation axis see figure generalization perform stop make wind human annotation effort stop conservative tend right conservative amount far leave try reduce unnecessary annotation automatically stop improve restrict use applicability exist lead tend find far dataset break stop stop paper present new address area provide user essential idea applicable save annotation user stop al discusses explain sp detail evaluate sp estimate probabilistic learner confidence generally table figure fall stop margin stop denote table figure stop equivalent margin show explicitly al evaluation sp method applicable base tendency confirm say stop confidence consistently drop pointed criterion however two stop report max stop unlabele exceed generalization min qualitatively translate measure allow point stop idea experimental separate development cutoff agreement chance percent adjust measurement agreement human receive context drawback percent take agreement account agreement metric agreement metric differ statistic agreement chance category formally compute estimate instance label find require move separate development cutoff work experiment current paper agreement cutoff cutoff cutoff intensity sp intensity cutoff see give user intensity cutoff sp behave another maintain sp conclude stop check stop exceed cutoff ideal happen propose average window call window average work table tuning default fold vary requirement
video outlier return open circle table blue line viewpoint frequently rank removal top lasso order competitive video importantly indicate dataset consideration reference reference come publicly live totally varied comparison internet pair show zero leave corner table outcome table lasso image database agreement pairwise voting vote example l method integer score c far suggest contrast inspection confirm show show ranking return table rank rd four appear unbiased lasso successfully bias rank winner col usa country david usa usa col supplementary material proof mainly change column normalization assume gaussian present self leave turn restricted side whole less give achieve c sufficient negative outli side mean piecewise sum min min consistency fx right bind cl min min root comparison spread crowdsource crowdsource outli detection become huber cyclic linearize bregman achieve less detection os setting detect support simulate promise robust scale crowdsource vision machine crowdsource pair robust outli linearize random statistical aggregation rating back area various voting theory economic machine pairwise spread crowdsource g enable collection large rating scenario must address difficulty interest data iii online streaming iv among able characterize intrinsic pair incomplete sample vote rapidly assessment quality experience setting traditional environment equip development mathematic enable instead quadratic pair pair connectivity infer pair loop complex measure ranking setting propose classic method together track pair rating hence efficient batch deal despite lack supervision crowdsource single long decision significantly decision global crowdsource detection circular triangle triplet divide apply apply pair pair incomplete miss detect pair robust sparse uniform outlier angular embed approximately recover score history fill present lasso huber robust outli sparse component interest meet challenge crowdsource develop yet easy outli linearize contribution highlight huber outli approximation projection scalable linearize biased lasso suitable statistical consistency detection establish huber linearize bregman iteration methodology simulate world paper conference problem tend cost prohibitive crowdsourcing linearize iteration outli inclusion fast organize review work systematically robust huber linearize bregman consistency case establish conclusion pairwise outcome element aggregate ranking study rank centrality mc compression image assessment fitting consider set pairwise algorithm optimal aggregate global provide ranking pair angular theoretic pair comparison flow flow triangular flow local harmonic pair help various outlier score graph problem exploit os r enyi random refer computer literature occur large instability outli many applicable pair appear literature outli study well combine square discover regression huber outli modern selection linearize firstly variational imaging sense know estimator always bias contrast variation denoise view discretization combine gradient soft thresholding soft g reference therein biased lasso tune choose well achieve boost save greatly thus suitable computation systematically rank huber lasso linearize first successfully widely lasso cyclic statistic linearize iteration set paired datum miss degree generality assume vary quality assessment scale machine surface follow pair comparison map flow gradient difference gram view flow scaling contaminate go gauss tell eq score however put mathematical outli sense element achieve different type huber robust differentiable derivative pair solution z variance bind loss hence note get leave random graph comparison os enyi surely optimal huber loss q regard regard bad one contaminate reduce rank huber robust comparison ij I outlier lasso ij huber partially huber package solve fortunately two group column complement involve projection onto complement orthonormal orthonormal provide precise split solution obtain inverse full svd hence via correct pairwise say pair admit particular harmonic triangular ranking call cyclic ranking e unitary orthogonal cyclic detection via decomposition outlier complete huber propose jointly follow pair play role suggest robust normally hence efficiently huber estimation prove application sparse validation highly associate outlier leave outli validation projection subset projection random thank os enyi graph position consistently identify projection validation cross fail become dense magnitude nonzero tendency outlier regularization path example like drop percentage appear regard drop moreover magnitude outlier outli development projection optimal scale score suffer prohibitive item remove bias contaminate estimation introduce ordinary unbiased discretization scalable imply remove linearize euler discretization rule meet size estimator sign reach see sparsity large empty early overfitte early update exact stopping meet efficient establish outli share property e outli consistency condition speak incoherence magnitude however limit magnitude exploit choose pair restrict os graph tend unique hold contrary fails assume outli sign necessary sufficient outlier uniform lb path strong enough k sign play play six methodology latter exploit world collect crowdsource pt sn op auc sn data p first total truth add paired subset simulate pair comparison possibly two dataset total number pair sn number outli sn exhibit sn op compute vary give tendency outlier outlier roc plot different level create sn op see sn op auc deviation detection return accurate indicate half edge rapid decrease random guess edge perturb impossible distinguish op drop sn tell could case grow compute scale experiment image pair contribute outlier effectively however could simple scalable datum reconstruction first use ground truth obtain
align imputation minimize reconstruction reference corruption cell partition residual depth oppose split scope study comparison performance term digit report project test eigen clearly separation corrupt imputation improvement accuracy set avg std avg improvements nn map breast cancer spam digit emphasize first localize complete phase free separate test affect attribute comprehensive cover solution fair comparative nn separation utilize instead imputation nn neighbor corrupt instance attribute node utilize imputation detailed corruption separation step split segment anomaly segment anomalous segment segment attribute randomly split scaled instance corrupt attribute attribute instance corrupt explicitly compare imputation euclidean since low nn euclidean common method split remain split train svm compute corrupt deviation average improvement std std term split clean test corrupt set clean randomly choose split corruption observe successful corruption imputation even gain imputation correspond improvement relatively cf experiment algorithm corruption separation nn propose nn method outperform nn imputation corruption even nn superiority issue nn corruption cover portion g would several false corruption cover portion g choose anomaly detection use insufficient detecting imputation insufficient load corruption propose search fast experimentally corruption separation capability lastly imputation asymptotically maximization compare dimensionality yield estimator neighborhood easily achieve cross map imputation perform imputation opt clarity imputation sensitive attribute near imputation e experiment near lower corrupted pick imputation imputation potentially experiment superiority become superior imputation g set demonstrate devise experiment unitary covariance generate sample attribute therefore size nn part original mse term accuracy summarize imputation coincide imputation imputation consistently imputation mse sense improvement original accuracy around comprehensive localize novel I jointly identify local corruption binary characterization anomalous observation combination assume model drive conduct alarm alarm detect independently generate set experimentally purpose corruption capability algorithm training phase condition edu tr department comprehensive treatment severe noise framework novel efficiently give instance corrupt propose novel rank deviation among superior separate split partitioning iteratively detect anomaly clean anomalous vs pattern tree characterize affected attribute structure binary rate test experimentally remarkable classification purpose corruption capability typical localize corruption anomaly variety process even severe loss transmission channel localize result face effect novel emphasize neither existence corruption operate drive manner deviation nominal corruption external factor outside nominal consider anomaly specific interval attribute localize corrupt corruption property variety characterize formulate detection localization anomaly detection introduce false alarm nominal clean anomalous anomaly identify generate organize tree correspond anomalous pattern corruption nominal multivariate distribution success coincide alarm test direct acyclic multivariate derive alarm detect constant require corruption localize replace attribute attribute map exploit local dependency encode map load extra computational utilize also rank generalization label anomalous distance compare standard conduct severe indicate achieve imputation typical also empirically strong corruption separation capability corrupt statistically unobserve counterpart know corrupt attribute readily treat framework datum study impose solution imputation tool replace attribute draw density certain expansion contrary either introduce unlike approach incomplete attribute I precisely localize provide target object priori exhaustive result manual inspection miss algorithm jointly detect well imputation framework generic imputation imputation imputation completion neural study image statistical approach valid attempt enhance globally localize operation exist corruption common phenomenon application regard corrupted previously study study visual imputation descriptor part descriptor weighted handle partial solution extract error template map significantly fail remain applicable source another study imputation corruption improve processing stage classification handle localize affected attribute alarm detect acyclic estimator computationally utilize corruption load imputation anomaly notion description corruption imputation conclude corrupt clean observation f nominal severe multiple suppose corruption localize attribute corrupt distribute distribution statistically counterpart corruption model datum attribute vision provide uniformity draw scenario realistic consider include generally model mixture density derive nominal corruption distribution instance replace nominal localize appropriate corruption create corrupted corruption strength corruption corrupt modeling pose incomplete corrupt irrelevant exploit deviation nominal detect miss end formulate anomaly detection draw example anomalous corruption alarm well framework without single generalization onto give description algorithm tree separation imputation false alarm localize corruption affect affect vast algorithm corrupt attribute anomaly instance reference anomaly detection approach novel distance corruption localization corrupt nan nominal corrupted mix anomaly nominal unknown set hypothesis realize anomaly maximize alarm purpose score distance near th neighbor score function anomalous mix estimator remarkably volume test false alarm improve training point detect corruption test corruption imputation contrary truly train high corruption property anomaly achieve alarm first issue list propose metric sensitive give corruption attribute instance turn distance include create ambiguity term localization exhaustive possible attribute overcome result attribute responsible corruption permutation attribute make precision anomalous check noise attribute less exploit investigate euclidean rank euclidean sense satisfied derive consistency estimating level density characterize present corruption corruption might check anomaly anomalous alarm propose attribute characterization separate corruption scenario split simplicity attribute r rv v r strategy corruption separation separate tree create course expansion correspond e check reference rank pre encounter expansion binary unbalanced instance emphasize binary creating need continue decide corrupted node circle square anomaly wide spread attribute anomalous regard corruption characterize corrupted corruption unless create corruption since represent realization underlie attribute map maximize posterior hold w arbitrarily small drop maximizer approximate map nonparametric neighbor knn estimation neighborhood near neighbor lebesgue variation unnecessary approximate true underlie corrupted tree corruption namely I test reference associate ii detect attribute limited derivation detect calculation addition support corruption issue use rank neighborhood possible desire recall exploit imputation meanwhile decrease localization trade imputation localization investigate detail imputation bring result cf node generate step alarm corruption detection false alarm dependency anomalous partition imputation certainly false alarm correspond occurrence alarm anomaly detection tree separation operate anomalous pattern anomaly present rooted pattern reject reason false alarm rate must anomaly false rate detecting anomaly output correlate section label anomalous normal partition direct achieve certain dependency structure derive alarm detect globally rate encounter leaf anomalous describe corruption localization well imputation capability propose corruption algorithm detect task alarm rate detection nominal data nominal distribution depth root anomaly child root tree observe binary vector ease exposition map corruption label equivalently complement nominal probability mass tree binary acyclic node knowledge label leaf binary assume follow u obtain cf u define note alarm analysis localize definition root anomalous anomaly due corruption phenomenon therefore simple fig straightforwardly tree anomalous label root collection associate tree node equation factor last expand derivation calculation require e let child independent dependency generating would like attain hand provide dependent introduce derivation generate probability function parametrization alarm practical opt simplify corruption root solely depth symmetric child simplify note termination obtain short hand algorithm unlike multipli search stop corruption local anomaly initialization recursion focused localize exception straightforwardly incorporate regard change corruption pattern corruption search eq recursion stay valid recursion calculate node recall false alarm corruption subtract false alarm corruption simplification conclude anomaly map false alarm corruption secondly even hide label identically parent child obviously plot alarm detect anomaly experimentally discuss efficacy represent moreover depth uniform partition depth model acyclic anomalous labeling rank distance fitness dependency complexity main building anomaly compute train matrix distance operating distance score computation sort anomalous image distance let si si si si position compute train sort sort node expansion define sort load multiply constant illustrate separate detect corruption efficacy step corruption imputation improvement digit task gray intensity corruption corrupted specify region randomly provide clean test emphasize
dual give section establish global linear property regard convexity lipschitz continuity compact compact boundary convergence iterate supplementary condition subgradient translate si x si point condition vice versa matrix satisfy exploit similar lipschitz however show compact satisfie iterate belong compact begin iterate hence inductive argument appendix positive assume bind positive iterate k iterate converge conservative much conjugate primal hx use gradient establish use lipschitz continuity two call divide approximate inverse multiple solve solution dc algorithm big divide light base speedup therefore utilize spirit dc alm algorithmic alm attain solution comparison attain nesterov regime alm algorithm use speedup strongly function c alm substantially show fair superior establish regard validation edge criterion regularization probability graph sparse model consistency mle leverage already consistency refer regard dimension recover underlie proceed framework rich formulation generalization bivariate structure numerous marginal quantity inverse define operate specific knowledge matrix compare covariance layer regularization process achieve methodology regard underlie edge covariance correlation equal either reference estimate partial graph interest provide graph incorporate term generalization bivariate translate constraint somewhat generalized allow mle break away framework allow regularization base domain knowledge primal formulate modify covariance interval third note provide problem convex set constraint ij choose linear constraint constraint might always translate g formulation penalty consider regularization easily formulation equality constraint covariance decompose alternate b condition z second optimality substitute optimality proof previous new belong valid also modify operator x useful prove generalize iterate f converge appendix synthetic convergence demonstrate convergence iterate duality slow number solve synthetic library compute cholesky processor available resource solve conduct synthetic procedure matrix entry uniform obtain desire percentage level use small well condition size illustrate heuristic point use start well large decrease iterate dual feasibility consistent display require start majority case method improve therefore iterate optimal machine intel cpu gb ram table c sp sp sp e sp h e sp sp sp e e duality termination solved require well duality gap require prominent ill benefit work highly condition compare three consist temperature algorithm varied get yield ill ability duality highly unstable reduce tolerance solver report gap report significantly gap subgradient significantly slow converge c c e e na e na na e na na na duality achieve tolerance solver gap subgradient high ill report experiment gap achieve duality gap unable subgradient suffer issue perform high fast almost around speedup problem solve time determine penalty require field addition uncertainty quantification optimization times solve financial regularize critical ingredient portfolio require repeatedly shall efficacy financial context portfolio portfolio weight focus portfolio return asset period price divide price covariance asset portfolio period minimum portfolio selection tw return define associate portfolio budget close assume stationarity return account stationarity minimum portfolio portfolio divide horizon consist time start portfolio solve horizon constant hold period entire period update portfolio hold j asset return hold period application study stock stock average trading day recently illustrate interval begin begin return trading horizon consist hold period penalty versus condition covariance method metric excess expect portfolio switch measure weight choose metric appendix c fig normalize trade five table return realize risk ratio standard period fig cc c htp growth substantially across advantage rise portfolio balance transaction cost growth propose inverse inverse literature fast order magnitude regime recently moreover condition converge extremely slow poorly highly attractive modern inverse multiple maintain definite tolerance attain gap duality gap terminate inverse criterion tolerance progress iterate primal k tolerance impose step iteration bb maximize equivalent solve k numerical acceleration might feasible satisfy descent satisfied three case reverse inverse entry contain argument satisfy iterate use ingredient c fix define leave hand side c expression jacobian h inequality complete repeatedly inductive iteration rewrite term c subsequent iterate lemma step thereby modify f accordingly easily modify prove use similar constant portfolio background definition take pt trading period eq realize risk deviation strategy entire trading period realize sr realize excess return strategy risk hold start th hold portfolio period I e normalize growth portfolio grow recursion transaction stock short short portfolio trade thank code section estimate graphical despite advance fast ill modern solve covariance approach several novel fast algorithms order global linear rigorously operate covariance
attribute include output without validation significance sign create centroid assign output generate centroid centroid select attempt mod portion nominal datum outline add set output set number centroid time centroid centroid output output na feature c output nominal I synthetic always statistically exploit contain output make specific chance outperform I nominal synthetic well significant uci set mod decision mod create uci set allow nominal derive nominal original act class act feature act scale output output output repository experiment choose nominal mode accuracy I bold statistically uci nominal feature output l c ccccc c total ni ni h ni h ni heart post op vote contain I output original significant na I time significant difference na I outperform I majority na I model perform well never potential model mod decision uci motivation mod stem business com business reproduce version three nominal value output datum instance contact business contact method outperform I see uci mod synthetic real mod representative little uci supplement sources mod na I I mod solve multi I mod consistently outperform model significance uci repository business future use mod problem mod new identifying collect mod mod problem know degree piece mod classification primary issue function readily apply near refine dependence mod traditional seek set possible output classification multiple assign structured prediction predict structured mod address output incorrect mod propose company generate sale customer customer switch company retain customer could sale customer offer customer express course incur certain customer help write customer mail company generic mail e mail person see problem important approximate traditional define output output still acceptable mapping relation give choose solution many induce per output give without induce tree produce output induce neither approach output neighbor perceptron mlp approach algorithm auto network unable handle input contrast hierarchical neighbor hierarchical I approach traditional learning make focus nominal feature mod problem model modify use output near examine although label correct consider problem multiple dependency model overview classification first problem single already define mlp output may output mlp adjust towards whenever encounter whenever encounter adjust weight adjust ever correct try solid curve training dot branch arbitrarily however cc learn potential possible neighborhood towards extra provide mlp mlp near neighbor knn final prediction prediction feature output either output modification knn dependency mlp classifier part knn majority dependence claim dependence output loose dependence consider scalar conditionally vector output output dependent py py contradict must vector mod different type synthetic uci repository world compare single I separate combined output instance return otherwise correct
entropy shoot restaurant event word dnn representative heuristic shot shot dnn embedding confirm shoot measure auc area curve train scalable lead svm well linearly input shoot base use amount query log learn embedding without access datum experimentally propose semantic classifier category learn engine log learn semantic supervision shot semantic demonstrate effectiveness shot collect system aim request predefine semantic category instance system might query method svms model produce state require amount label mostly manual costly applicability problem semantic examine see domain add easy shot semantic class typically least category domain input classifier must without knowledge shoot none see well automatically neural use amount advance network result search click reflect query property call shot embed shoot weak supervision shoot discriminative produce semantic notably reach feature next quick overview shot section present introduce shot embed semantic classify speech upon maximize formally variation may express information priori system hand interpret weather function aim capture binary likelihood gram express user traditional text categorization devise maximize shoot concerned novel training label class predict knowledge input euclidean vision might semantic predict semantic class value train semantic match nn semantic none shot see semantic classification find semantic label would semantic interestingly semantic discover without label name sentence choose essence fact classify easy belong wish replicate net axis phrase relate movie detail shoot observation function property procedure space shot find match formally shoot z distance euclidean reveal mean space map sentence relate classify close properly semantic capture sentence class framework follow language interested method lda success semantic useful semantic click query behind query hide layer click include user query send engine engine extract thousand query million daily meaning meaning website query website movie scheme sentence task space embedding like sentence deep softmax hide word format index website q hide weight bias give semantic property semantic category novel encourage learn semantic without label precisely cluster measure class cluster hence minimize measure semantic good click task category close relate activity sentence pressure know visualization dnn dnn right sentence color location class name improve detail supervise label even bit supervision simple task train sum measure require category low require label data semantic space category space low entropy example around category low overlap shoot discriminative combine embed entropy hyper strength validation mostly determination phrase entity location phone large space boost good candidate first method result performance boost base baseline show big click feature entropy learn conditional label available minimize avoid label minimize zero shot produce generative evaluate shoot zero shot month query click embedding restrict word bag use contain filter
feature relationship normalization assume know output joint inference discard unfortunately necessarily give instead introduce strong bias optimistic cause explicitly account measure work marginal predictor significantly improve method minimize risk n user quantify usually e surrogate upper prediction define upper empirical second jointly form supplement augment optimize model sum belief respectively optimize start q operator may eq temperature structure semi eq obtain restriction comparison sequel provide insight select evaluation objective sgd concave gradient eq augment approximated belief propagation distribution expectation belief sgd lemma supplement unify control distribution sub sub gradient substitute sub gradient bp bp tb output weight vector calculate I define calculate w concave convex transform optimization problem naturally difference iteration new minimizing linearize eq expectation iteration loop learn ph u ph x simulate world margin uncertainty largely outperform especially small datum mrf discrete q graph illustrate parameter indicator vector output sample test hide mrf mrf train hamming loss high noting obtain mainly due sgd converging sgd ccc sgd sub learn converge within gradient much converge slowly effect mainly hard make nonsmooth sub slow sub descent illustrate converge sgd transform objective easy objective fair range training data mrf chain show outperform largely outperform sample outperform experiment relatively toy generally difficult enough model eventually seem since likely relatively test supplement uncertainty hide mrf figure fix accordance default default package encourage tune result table competitive significantly uncertainty explicitly outperform current chain mm e avg label label grid pairwise entry outer miss range evaluate list method increase see significantly consistently explain improve robustness accounting categorization microsoft pixel pixel correspond boundary etc model outline center treat patch texture descriptor compute sift descriptor patch word k color take patch testing find category explain superiority moderate building car variable demonstrate state art especially uncertainty optimize function also include nsf grant united air contract fa program uci edu uci edu uci edu give constraint slack unconstrained derive cut formulation proof lemma omit lemma objective unified framework paper bind q definition second denote sub temperature result complete paper demonstrate outperform example across likelihood datum high necessarily explicitly loss relatively instance pt marginal properly propose art uncertainty result smoother significantly outperform field real world furthermore unify case practice structured svms tool structure computer handle image segmentation predefine semantic expensive collect perhaps region partially relatively label couple syntactic annotation resource year variable perhaps notable hide field svms svms practical training violate hand procedure assign variable prediction even well perform max accuracy valid location improve category
represented call architecture topology statistical cluster standard activation radial architecture variable probabilistic behaviour layer historical unit dot perform operation operation provide summation sigmoid activation output represent modify window order verify obtain expect express window width width depend mean square represent preserve capability function radial basis rbf rbf weight radial intend second identical sum receive hide layer summation summation unit weight weight summation summation summation remarkably output selector effectively act one layer selector input score attribute text probable author probability selector adjust product output layer us part processing layer summation nonlinear fact layer task neural text attribute input exact number pattern match text summation equal people interested recognition database nn purpose since aim expand database identification dynamically database bias imply properly ensure adaptive reinforcement desirable human training give figure flexible model add time external tuning count group count multi agent identify concern text attribute concern probable information spread certain could extraction study exclude automatically enhance classifier filter probabilistic text correctly attribute attribute person performance probabilistic selector selection positive black mark mark validation purpose accord identify excess text person lack attribute text author assignment model dataset allow structure blockmodel report model replicate unable child attribute class class multiply identify lead would membership specie concern non parametric relational entity set moment sequential relevant use mutual class storage classification purpose recurrent show neural study result line learn analysis text consist successfully probable text sample complement integrate comprehensive verification kind try write classifier mean reinforcement follow correction agent supervision cope continuously feed adaptation reasoning support project project mathematic drive radial year title drive use radial basis reinforcement volume policy huge essence digital retrieval gain trust drive learn develop preprocessing period analysis recurrence frequency text radial author lie semantic apply lexical domain without modification external tune self adjust text author security availability digital possibility essence method digital text last decade classification intelligence language process intelligence orient programming intelligence ci ec optimisation problem management agent intelligence advance create analysis ci ci network use electrical control start strategy kind hybrid nn say work form cluster recognition efficiently fail efficiency result complicated general agent machine prove promise research purpose build classification rule automatic text one promise approach lie semantic category generate successively typical topic recognition appropriate classify involve business text belong party text technical devise solution extract text characteristic express obtain abstraction create analysis rely input evaluation concern historical period etc text use preprocesse grouping relate tool recurrence text lie lexical modification order reference suffice purpose reason intervention thank comprise e extract meaningful text mean radial nn feedforward detail implement preprocesse modification agent report perform related background draw figure agent characteristic text database database extract properly perform identification additional agent dynamically new firstly analyse text group mutually build ad hoc relation text word group predefine database contain dictionary return group identify new dictionary occurrence increase group start load load group database load break
objective lasso c call value statistically gaussian k k redundant tend tend exist feature need space large base reduce feature extension call propose moreover justify lar establish screening efficiently solve lar nn lar via lar lar standardized unit formulation modification optimization negative lar lar point lar center lasso redundant inactive lar lar lar base lar iterate change high lar ordinary least infeasible add regularizer lar complexity feature cost lar grow deal computational issue om reduce map advantage parallelism approximation universal detect two delta kernel delta normalize kx width normalize width classification delta kernel nystr om om eq nb bb bb nb bb bb nb bb bb bb regression similarly approximate simply entire lar om approximation helpful nystr approximation high e also lar distribute compute compute store output output k repeat stream fast establish screen method screening framework screen idea covariate choose value regard express exist objective large solution contradiction suppose obtain contradiction correspond large proposition connection feature screen select redundant iterative redundant iterative technique lar review exist mr feature relevance mr usually mutual relevance mr yet efficient mr input relevance redundant select overall regression interpretability redundancy relevance experimentally mr feature correlation base regard calculate backward selection compare strategy backward tend produce approximated window small optimal advantage show low case need compute computational nystr om propose experimentally om high setting mutual also elimination selection accurately implement optimal feature relaxed solve bfgs point necessary expensive high problem regularize base scale sample recently redundant find term exist feature selection method tend expensive moreover sparse spam feature globally spam spam closely relate multiple mkl potential spam deal additive spam need optimize tends computationally path lar computational lar distribute om dimensionality lar fold world dataset scalability scale biology lar baseline mi pc since efficient qp solver available matlab code size selection slow generate accord x variate covariance matrix regularization illustrate b om ghz core lar increase grow lar memory lar large case lar distribute computation hour baseline restrict comparative study baseline classification propose set type ar small p regression experiment rest run training report since dataset logistic gaussian kernel width chosen cross absolute check whether lar select kl red correlate many redundant ar r c lar spam p p show observe lasso high average red lar overall non next evaluate subject focus gene cause value use rest time select employ evaluate selection regression gaussian width fold finish lar lasso mean feature observe good lasso measure instead perform b auc use selection quite big used demonstrate scalability result goal predict activity inactive label determine feature sample sample classification randomly sample report area roc feature coefficient lar lar simple mr accurately lar solid lar linear call lar propose efficiently exploit variant lar normalize furthermore propose computation lar large feature experimental demonstrate lar promise theorem corollary yahoo com way specifically propose lar lar normalize lar incorporate reduce framework high lar evaluate high selection
evaluate standard mean remark detailed column respectively x n partition sum distance mean correspond formulate dissimilarity measure dissimilarity square I generally operational definition state dissimilarity object mean naturally rewrite square cardinality sometimes convenient calculate formulation form involve parameter euclidean interpret objective contribute become serious drawback solution mean considerable portion situation phenomenon experimental cluster cluster neither offer intuitive select framework drawback question accommodate develop drawback classical k cluster clustering mean characterize function centroid sample long vary play role formulation implicit centroid centroid confusion cluster universal exist centroid simple example p p verify gamma difficulty root difficulty try way optimal consideration motivated condition partition define partition noise feature make attain value respect make cluster noise feature reasonable state notation sample indicator belong indicator distribution uncorrelated obeys define minus state generate furthermore partition comment existence notice sample direct new result traditional theorem ii feature gap fact define feature ii suggest principle optimal relevant significance equation reveal vary partition rather reasonable intuition introduce special deal penalty mean example penalty replace difficult analyze overcome propose jointly word cluster surprisingly analyze theoretically solve cluster framework come existence partition variable variable responsible valid apply alternative solve iteratively respect fix procedure initialize q f formulation come like specify k sequence order identical directly set component element select operation approach cluster w old w nd ii kn cost cost condition relevant problem portion mean support assess theoretical propose mean main mild algorithm generate namely seek select follow partition consistency ff relevant partition relax whenever consider possibility condition certain prove follow show equal positive probability thus follow remark satisfie proportional degenerate grow growth sample notice condition gaussian generalize subgaussian k justified reveal mean high problem performance mean mean concrete involve gap strategy four comprehensive comparison first classification use adopt I criterion zero weight yield correctly eliminate fourth nonzero relevant correctly select algorithm correctly exclude criterion conduct experiment statistic mean mean k mean related cluster method experiment respective uncorrelated design assess feature relevant feature figure standard see maximal small average estimated weight close estimate noise show gap htp mean mean assume element simulation experimental means mean mean standard always k explain mean keep completely fail keep coherent contrary detect mean price little well k short pca previous datum different consist sample feature relevant feature relevant show ccccc mean mean comment k reason principal include mean treat equally dramatically penalize consider feature cluster datum result relatively simulation suggest generally cluster eliminate independent life validate broad feature experiment among centroid relevant conduct mean perform experiment generate obvious strong capability k mean mean subsection capability respective apply develop voxel expression voxel gene record operation correspond brain voxel increase brain grow resolution slice voxel annotate brain manually h pt region mean k respectively set mean improvement interpretability keep minimal nonzero eliminate discriminative firstly investigate observe noisy k mean instance channel beta relate beta interact alpha go annotation include bind electrical signal transmission cell much reasonable feature function whole brain usage distinguish thus support correct also still identify gene database institute rigorous concept feature cluster problem cluster eliminate noise yield simultaneously realization suggest closed solution problem analyze assess experiment study main follow concept optimal partition rigorously concept grow number size definition cluster problem cluster could time acceptable real efficiency mean possess selection property theoretical success k ability yield mean interpretable many along establish theory may compare lemma bind apply therefore variable sub exponential parameter exponential next suppose know exponential decrease definition thus feature cluster proportion sample expectation cluster j n n x naturally easily valid relatively suppose ij obey standard accord q I moreover base decrease big complete standard reason term zero partition discuss first essential last interested reader small probability noise big partition estimate complete national research china program cb china zhang university old university helpful thm corollary lemma remark penalty dimensional
discard tc sec state upper show tight provably thm hold discrete correlation choice uncorrelated truly dimensional characterized separately consider construct decomposition efficiency ability bind multivariate information really entropie thm directly translate bound dimensional system decompose sec optimize tight finally estimate datum describe thm suggest way build optimally informative layer maximally explain correlation layer maximize hierarchy representation layer easily compare optimization obtain bound convenient probabilistic surprising optimization imply self consistent equation iteratively variable space tc lx iy look optimum expression omit information ai iy iy positive existence common dependent dag overlap information keep lagrangian normalization guarantee appear formal marginal remarkable problem say write term involve optimization practice limit iterate self fix imagine start update easily calculate summing take iterate converge point surprising rearrange value function call bound intractable share tractable rule principle order final sum intuitive unique present hide obeys previous equivalent add index solution incremental obey long update guarantee increase stop iterate quickly update discuss look see really define input depend independent say inequality demand tree structure connect informative latent introduce heuristic solution tree I py j py j py correct prediction set percentage approximate fraction empirically unique question diverse redundancy quantity somewhat typically except represent quantity sample require imagine give unknown estimate typically small seem estimate marginal simple chernoff bind exponentially specify see optimize calculate marginal require update label amount function marginal easily provide quantity raw along datum accord learn recover fig b within percent learn representation increase setup group cluster take quickly sec world look b variable word correctly increase synthetic scaling include real consider series take entire treat month iid representation learn edge thresholded edges proportional explain stock color accord classification standard clearly capture significant wide construct restrict boltzmann tc lx mean kind wise total capture market decade connection new latent form thresholded hierarchical overlap connect group contain department like strongly home improvement home stock two contain contain demonstrate maximally informative contribute construct representation complexity outperform state synthetic demonstrate promise diverse human biology language foundation previous enable overlap contain specify cardinality representation optimize trade include characterize rbms explore connection bottleneck combination domain agnostic foundation rigorous information correlate datum research nf supplementary maximally informative hierarchical decomposition hold kl expand entropy definition replace lx hx tc I write variable give next thm want drop lagrangian constraint reduce constrain p optimize functional derivative identity unfortunately functional indicate delta px py take identity py px p px x px perform sum py py py lead py py py recall formal marginal py py py py like appropriately constant calculate term iterative bottleneck iterate value functional px long optimize argument equation skip appear mind lagrange multipli ensure recover equation optimum objective concave concave include objective unchanged objective must optimum iterative procedure start initial compare initialization always boltzmann hide thresholded edge keep high node online setup pt definition pt ex ex ex ex plus plus minus ex ex plus ex em input variable bound informative lead optimization maximally informative establish new principle demonstrate representation deep becoming solve great challenge image recognition language method constitute usefulness instead making directly consider bound characterize efficiently representation optimize successively tight modular separately lead compete maximize define regardless detail generate usually mutual correlation distortion intractable lead elegant self theorem foundation theoretically representation framework recent correlation explanation introduce excellent diverse source optimize sec maximally representation sec idea world financial use capital domain whose
dot product cosine none inclusion normally measure triangle move continuous space embedding diagonal uncertainty function asymmetric inclusion distribution perhaps literature gaussian distributional potential space map region inclusion provide geometry discuss work present qualitative quantitative common concern people seven similarity task lexical also demonstrate training support new incorporate distributional vector distributional semantic language broadly probabilistic row give relevant bayes observe space distribution train use arbitrary asymmetric factorization learn combination metric effectively per fitting embedding apply fisher preliminary graphical pair quantitative linguistic semantic traditional count region strength vector popularity word indicate position onto generalize eigenvector variance word nearby map dictionary linguistic relationship precision vs distinction orient unsupervised token type nearby token word type vector vector context input parametrize pair high negative pair accomplish define negative supervision provide energy base representation token context often word skip gram word energy context treat context score word sample train context achieve desire effect word context type pointwise limited dot product depend treat rely surface absolute energy train rank rank positive negative terminology function represent pre train context word set represent context empirical estimator covariance mean practice necessary add invert note learn possess unsupervise inclusion gaussian context rank present independent valid dot incorporate covariance would model logical similarity behave product seem natural indeed appear probability history gaussian inner broad gaussian aim always quantity firstly ratio likelihood commonly work difference interpretable hard logarithm determinant covariances covariance trivial model store efficiently matrix intuitive geometric similarity measure distance measure mahalanobis volume span principle component interpret prevent mean encourage concentrated encode context directional supervision knowledge sensible lead vocabulary type skip baseline two output subsampling paper training constraint improve vs diagonal gaussian embedding large comment original use constraint performance examine query sort measure gaussian denote broad frequency word get word frequency name appear context fairly word word nearby mix mind top choose sorted measure great qualitatively source less mention begin section precision pick ap empirical kl learn e learn variance diagonal spherical symmetric cosine mean base learn embed variance pre cosine asymmetric measurement embedding bad cosine word variance count reasonable kl regularize matrix empirical cosine variance choice either leave make locate gaussian discriminative power examine variance note word model possible force commonly context well distributional move beyond purely unsupervised learn unsupervised manner text form lexical reflect phrase look appeal embed dimension capture hierarchical tree area simple variance parent child create tree come come directional else large capture leaf negative receive node evaluate embedding seven similarity art scope however match report dataset skip gram implementation much embed algorithm achieve distributional quality experiment variance spherical gram dimension diagonal parameter spherical overall slight edge embedding embedding plot spherical diagonal significant shift diagonal vector see spherical generally set cosine outperform cosine mean spherical covariance distribution never include dataset sg sg mc word type directly represent directly notion enable rich geometry embed demonstrate linguistic qualitative spherical combination rank matrix going enable keep semantic align capacity move stochastic warm g descent gaussian concentrate high dimension multimodal another future idea
plug obtain q strongly continuity sum convergent subsequence z j j proper furthermore relation convergent subsequence j minimizer take convergent subsequence eq convergent subsequence use function convergent global sequence choose point addition algebraic argument subdifferential relation moreover subdifferential inclusion relation relation whenever hand subsequence readily hand x since non together third relation constant hold trivially property kl exist whenever furthermore x proceed far since non loss concavity see claim notice increase apply last first comment proof indeed dr see third stay dr state semi algebraic satisfie derive dr examine exponent rate z last consequently conclusion case follow contradict happen positivity immediately I tt combine precede result existence give guarantee sequence boundedness splitting choose satisfy addition sequence series lipschitz relation readily boundedness boundedness boundedness bound consequently boundedness complete dr splitting feasibility nonempty follow optimization close continuity modulus inner expression follow infimum splitting feasibility termination algorithm computation classical dr splitting dr splitting compare point study author parameter dr close nonetheless dr feasibility nonempty x finish need justify q partial tv particular furthermore passing contradict see result result super classical dr splitting method function minimize subject look necessarily super super limit hard regard many consequently nonconvex feasibility find nonempty bound eq projection onto state dr general nonconvex feasibility sequence sequence exist algebraic set corollary definition subdifferential conclusion together consequently finally definition conclusion ii play quantify dr classical dr minimize dr splitting dr vs dr dr splitting functions convergent cycle pair corollary convergent consequently cauchy sequence immediately look initialize z numerical method nonconvex feasibility code matlab linear system dr splitting x benchmark projection proximal solve specifically close subproblem origin terminate projection dr splitting adapt linear first random describe report well termination failure fail value termination failure different threshold easy minimizer fail hard splitting fail e finally dr splitting splitting minimize criterion splitting terminate exceed solve average termination failure slow splitting alternate quality dr method projection r dr fail e examine nonconvex nonconvex feasibility introduce local convergence dr method explicit threshold sufficient boundedness dr numerical experiment indicate dr method usually outperform anonymous comment improve cm research grant dr splitting nonconvex feasibility study class optimization less nonconvex set direct proper smooth rate give boundedness splitting find intersection general minimize size computable split cluster point whole convergent semi algebraic function dr splitting method usually alternate projection finding take problem mathematic engineering aim find closed set call feasibility problem cast refer reader recent detail dr split powerful solve compete find proper close latter minimize indicator projection feasibility operation main splitting efficiently dr aim find two close heat splitting close set scheme examine popular proximal reveal explain therein dr apply sum proper example reader recent exposition behavior dr split moderately theoretical justification complete nonetheless dr splitting method motivate analyze projection alternate method despite difficulty important understanding behavior nonconvex exhibit affine regular convexity feature improve local dr splitting super regular specific feasibility problem seek convergence establish intersection basic perspective recall find closed interpret optimization length distance solve easy feasibility common study prox regular proper computable splitting give addition boundedness generate see introduce dr nonconvex split whose objective computable threshold splitting method furthermore convergent finally alternate projection dr usually take paper preliminary material apply proper closed analyze feasibility simulation conclude remark value function define never subdifferential immediately robustness fx subdifferential reduce continuously differentiable subdifferential classical subdifferential v groups resp subdifferential resp finally say modulus indicator limit close onto semi union fx cover nonsmooth property particular semi
semantic interpretation exhibit experiment lda topic distribution five hyperparameter number topic dirichlet parameter baseline unconstrained optimization baseline intuitively poor sharp deep neural mnist handwritten momentum layer dropout input optimize validation weight million evaluate directly momentum descent difficult tune various momentum rate objective measure reporting poorly introduce sharp rate momentum weight objective evaluate full evaluate validation chain minute core surface discover simple varied burn fix diagram integration show baseline configuration step yield proposal accept run choose perform chain choose correspondingly significantly constrained constraint observation formulate allow user tradeoff risk specify propose acquisition bayesian include meta constrain application product design meta mobile device speed usage objective evaluate possibly acknowledgement would helpful discussion experiment award center edu show optimization function motivate optimize dirichlet topic hamiltonian monte carlo pass black objective appropriate company design measure good want ensure therefore propose perform customer people people company general might speech recognition phone user speech acceptable material bridge subject margin use arise volume simple chemical synthesis combination cause discover discover boundary laboratory rather would specify valid naturally proceed develop global start likelihood conditioning treat belief next acquisition acquisition ideally relatively proxy evaluation acquisition objective spend well via well spend new model acquisition complete loop task high result meta acquisition bayesian address exploitation vs idea interested region high improvement ei acquisition strong b improvement ei objective predictive density improvement target ei encourage exploitation input exploitation predictive minimum expect constraint ei close differentiable compute maximize optimizer acquisition constrain ei formulation constrain acquisition optimization address problem previous constraint noisy probabilistic specify constraint objective predict require trial evaluation discover resource spend simultaneously evaluate would spend acquisition incorporate support expressive parameter space example total memory usage restriction encode constraint bayesian acquisition conditional ei ei improvement assumption density formulation encourage place propose pareto active pareto classified specify determine number objective constraint infinite feasible region infinity however limited aim classify find discusse failure terminate thesis introduce weighted acquisition constraint first therefore input noisy know accounting uncertainty problem return always namely contain uncertain whose estimate natural constraint condition concept represent function represent boolean indicate satisfied constraint may also constraint ultimately solution satisfied constrain remainder paper propose acquisition determine beneficial gps gp multivariate arbitrary gps see gps gps model need gps represent real satisfied represent transform lead likelihood posterior compute gaussian predictive marginal permit discuss type program nonnegative logarithmic g g imply corrupt normal choice convenience form constraint model instead link subject sigmoid mapping cdf binomial form sampling need follow mat ern model differentiable one characteristic length fully integrate markov slice form elliptical slice whiten procedure avoid couple hyperparameter give acquisition efficient acquisition depend conjunction violate acquisition function constrain line constraint full acquisition integrate q gp hyperparameter gp hyperparameter previous acquisition violate ei exist therefore ei acquisition intuitively probabilistic constraint ignore objective satisfy satisfied feasibility purely satisfy high either continue probe region drop lower indicate black circle minimum find next property constraint problem choose evaluate discuss identify box possible acquisition individually evaluate ei cause prevent region identify belief follow exceed likewise improvement belief objective become occur
parent free overfitte way help overfitte degree grid two aim exponential complexity challenge previous empirical suggest bound improve hold great burden model real search view adapt learn hardness score parent liu work mixed programming formulation solve formulation effective attempt great encode easy exponential might see mostly hope highlight avoid context generation clean solver additional previous formulation describe formulation formulation encoding elimination order node obtain solution program ij ji number order variable take elimination eliminate specification equally minimize clique might indicate converse constraint whether node order result elimination produces force guarantee elimination order also difference partial order bottleneck reasoning ij ji order consistent turn consider perfect value eliminate node manually cardinality denote dag consistent subgraph scope dag program partially topological variable constraint th parent exactly force acyclic respect topological tie break arbitrarily node order node constraint ensure arc appear graph constraint responsible inside constraint direct put reach follow formulation dag specify dag n iw formulation directly optimizer corollary resource employ branch cut formulation stop still solution quality monotonically decrease validate feasibility previously call reasonably uci repository detail run gb memory parent per experiment one implementation language use order magnitude fast able result much domain estimation error tw breast largely poorly cope contain become easy consequence problem load unbounded situation demonstrate empirically reasonable time number variable reach unable next limitation handle domain bound dag graph design large handle naive design bayesian rejection structure discard hard poorly ii structure discard score constrain fact report elimination order compute topological search parent set straightforward efficient property ensure superior option base node edge denote graph subgraph idea sample search code moreover mapping code code draw sample trivial element computable structure idea extend divide give function dag combine sampling theorem structure obtain bound initialize empty reach dag maximize time precisely hope say base unconstrained need unbounded implementation search give might boost explain main practical drawback version process propose per iteration compare define define path topological force interested way edge order represent node tree partial order dag ignore arc dag exceed affect correctness specify parent bind initialize arbitrarily clique call root clique arc create mark link sample arc create cycle unless run sample create cycle represent known decomposition iteration step place order time space version small would drastically decrease space version run decode theorem code order dag greedy choose exceed take cycle form although much region space compute hash closely characteristic unbounded avoid set breast letter hill empirically analyze compare uci repository discretized variable sample set column original data audio community discard audio different equivalent maximum hill nevertheless one parent pre run pre score consider memory limit gb three hour solution minute version sampling method sample seed version ten seed relative version find minute hour version score ten score ratio value well whereas converse raw table intractable top largely superior even tree probably allow much tree satisfactory set hill within minute hour datum ten outperform worth note version matlab formulation amount minute might produce efficient try suffer show even create new mix integer programming formulation especially problem network result indicate state art might fail large domain purpose propose double provide bayesian limit empirically collection linear bind certainly every permutation tree work closely appear exact alternative cutting generation improve independently partly support office grant grant ccc cc min median max letter audio hill ccc version max min breast letter audio hill default ji present structure method programming formulation consist graph subsequently subgraph structure outperform state fairly accurate describe distribution reconstruct variable practitioner refer known bayesian drawing inference probable explanation maximize inference inference hard provably exponential polynomial time algorithm guarantee quality raise consequence assumption provably inference reliable learning method resort perform belief inefficient great learning inference np extend maintain relative bayesian bound show hard develop dynamic learn bad combine heuristic learn address recently seem
transform remark contribution relate length precision common absence recall inverse prove convergence distribution highlight whereas sampler algorithm comprise gibbs posterior sequentially predictive distribution posterior purpose distributional property marginal moment exchangeable appendix focus pick measure diversity sample equal pick summing diversity shannon diversity gamma know induce diversity site induce illustrate compute closed numerically formal computation shannon index display asymptotic follow continuity path vanish variation specie prior vanish case use precision value moment diversity easily description seem hard achieve probability diversity assess goodness interest synthetic give site describe se rational quadratic burn iteration show diversity covariate triangle diversity space quantile credible predictive diversity line graph estimate towards true diversity grow difference attribute smoothness study numerically estimator thorough purpose replication draw time estimate gibbs sampler iteration chain inspection ht triangle covariate black gray quantile credible colour resp axis represent covariate ht square exponential set ccc consist operational unit conduct site factor diversity round mention covariate example full essential experimental sparse model deal observation linearly different covariate give c cm compute fully describe conditional metropolis proportional compare truncate left target describe equal obstacle example output test observe distinct appealing process joint output entry distributional trick follow site covariate resort compute extreme intuitively ht obtained transform square marginally continuous sup moment factorial reduce stationarity process constrain come stationary handle diverse result size bias dependent bias particular namely index probability indice distinct measurable biased measurable product averaging transform encode pick state mention look insight random importance distinct site resp follow eq run distinct element covariate one replace element permutation final step stick beta random knowledge reduce ii generalize covariate whereas marginally covariate dependent define completely approach marginally stick define turn function process type j sup topology moment derive simplify require computation generality proof biased permutation hand side double simplification distinct independence decompose group treat fashion hx virtue summing kind pick covariate universit e paris paris france model probabilistic modelling specie site site classify represent give species site environmental covariate improves use dirichlet thus stick break transform stem markov chain algorithm sampler experiment conduct study specie operational site composition specie site index covariate probability associate specie able impact covariate population measure index focus affected membership partially obtain observe covariate analysis compositional proportion chemical composition specify field biology physics economic health quantification mathematical mathematic despite parametric pre specie suggest drawback although often term specie applicable sampling nonparametric extension probability factor extensively three construction class chinese restaurant orient line collection second completely analytical allow elaborate study distributional strategy seminal success stick construction stem great hereafter weight dirichlet model beta transform model diversity could predictive yield discrete nonparametric specie observe draw conditional rare specie conditional organized discuss measure characterize model bayesian propose datum study study defer study orient sampling abundance diversity notion question measure diversity numerous way diversity group specie specie sample also specie later nonparametric distribution shannon index discrete observe diversity index value diversity indice diversity specie fix denote plug shannon vary covariate parametric undesirable size individual belong specie parametric estimate avoid dirichlet prior eq direction impose species aim nonparametric unlike sample directly resort last covariate diversity problem consist many entry index improve model specie describe covariate specie count site remark site specie index specie denote also denote specie site abundance abundance site denote number abundance satisfie equivalent covariate infer interested path covariate fy x n p jx propose probability moreover vector nonparametric introduction nonparametric stick break idea nonparametric marginally jx comes break sample specie observe site abundance specie site collapse variation site explain site exchangeable investigate extension factorize factorize posterior I v jx jx species factorize specie prevent introduction site prior expression permit across specie problem drastically burden require marginally exhibit describe introduce gaussian
derive denote intrinsic manifold depend yet rarely study validation non several yield improve justification geometrically inspire approach come manifold laplace datum geometry hence compare induced geometry choose build riemannian laplacian method guarantee graph weight bandwidth construct equation discrete heat follow riemannian endowed riemannian riemannian definite riemannian category whether refer mathematical geometry mainly integration ii original zero dimension coordinate manifold linear distortion map riemannian metric right application positive riemannian dual show encode conversely implement laplacian binary h il h h dual laplace encodes intrinsic propose capture data geometry must measure trivially laplacian choose maximize geometry dimensionality riemannian stand metric ambient space propose tune enforce identity imply mathematically find self limited bandwidth laplacian ideal object equivalence involve prescribe represent h would tangent subspace inefficient tangent reduce evaluate sample point express r result subspace serve chart pass consistency encode notion heat kernel conduct heat produce requirement must map row heat neighborhood implement design compute around principal riemannian one approach step improve minimal whether invert trivially riemannian metric unit metric perform fast make numerically robust show represent algorithm column accord result straightforward brevity generalize word project proper submatrix I algorithms enforce submatrix close unit review enforce exactly metric evaluate propose distortion distance distortion move spectral riemannian metric q volume element represent laplacian geometry geometry derive practically dimension heat compute laplacian nr step high data speed computation large complementary possibility distortion variance distortion mention working dimension focus mainly propose already mention guarantee relevant manifold parameter usually interesting laplacian self evaluate statistically principle translate vice behaviour rise subspace align manifold noise direction distortion variable curvature curvature point note high large dimension reflect exhibit show even dimension intrinsic dimension first datum plane find range space shall large use range grey seen lie detailed limitation range partially range find find upper illustrate phenomenon use supervise depend parameter minimizes illustrate range small increase upper intrinsic method cost low curve weak nine intrinsic projection one depend choose smoothing investigate choose noise noise form noisy embed obtain align embedding laplacian dimension value noiseless embed replication replicate truth along find low slight systematic tendency support manifold present theory supervise task choose attempt split consist set group simulate anneal highly non smooth proxy scale construct laplacian heat heat kernel reconstruction error r lx think confidence eps change denote geometric use dual te six te te cv depend split split c c cv digit percent six use bandwidth away te lead regularizer five despite variability outperform case dimensional rotation scale take cv estimate cv effect distortion case examine finding panel order magnitude even hypothesis well find laplacian encode geometry find add introduce become become supplement experience associate probably consider short lead tangent plane provide principled select apply parameter laplacian near graph interestingly parameter possible finite drive method impose geometric intrinsic mode many expect superior yet competitive cv besides experimental reason supervised label smooth severe require particular
symbol decision depend symbol state activation rnn compute conditional vocabulary indicator variable sentence although originally straightforward corpus paper train english encountered fix long track consequently decoder recover encode vector propose sentence translate rnn encoder decoder wish segmentation phrase confidence find good segmentation integer programming problem source compose phrase subsequence I encoder translate consider candidate translation reverse decoder target language confidence q phrase eq likelihood indicator include q ij number source phrase contain phrase contain totally make segment counting relation hold evaluate well definition ns describe segmentation approach clause unless source language roughly order english concatenation translate clause necessarily despite gain translation translation question issue heart purely translation drop present robust intuition multiple short clause unknown neural propose approach computationally phrase sentence phrase parallel phrase english translation word select news un two word website news news phrase first development neural english sentence vocabulary word english consider token neural incorporate specific without segmentation translation decoder train english sentence translate english segmentation score eqs conventional translation expect validate segmentation segment mean confidence ht random segmentation refer mean length segmentation score segment clearly agree sentence segmentation ht average score translation definition le ann es le le le de de les du de la les es le et la du la des es health les ann es de service le du le pick remove pick take de cr un ai cr pi et de il cr il de move focus make really bank build great bank united say segmentation move make bank building great bank united say des une une pour la le pour le la l les une une du without les en une le like extend line segmentation extend reference il la cr une force les il la cr image un de le tend les cr est specify respect confirm medium difficult segmentation specify star confirm difficult deal pr te de respect le pr la la star le pr de le un select overall quantitative observe decrease form clause additionally independently segment sometimes htp source request begin segmentation request il le pour il pr send il le de sa il il sa automatic segmentation solution curse sentence score base translation translation sentence translate sentence quality especially translate mark research translation translate acknowledgment would acknowledge cifar research universit de van universit universit cifar translation exist phrase translation system paper address automatically input sentence phrase translate neural network segment translate machine translate clause form
optimality pareto front develop datum mining sort pareto front query retrieval triangle first pareto hull set method differ also front concept pareto good knowledge widely similar pareto front apply rank utilize pareto multiple criterion another anomaly pareto depth dissimilarity pareto database criterion correspond dissimilarity single entry rank score item score list community also combine query semantic multiple set contrast useful outperform pareto front fusion tail document utilize front method outperform avg multiple retrieval multi usually although disagreement view view criterion however give may severe disagreement area multiple kernel typically pareto many include economic science sciences overview pareto front set objective problem evaluate possible goal find criterion combine criterion usually choice yield minimizer without employ search identify feasible find pareto every objective pareto dominate another item feasible pareto front denote pareto front pareto pareto front generally pareto front front retrieve though equally pareto front pareto point pareto rank figure linear hull pareto front observation deep pareto pareto query retrieval introduce notice non shape pareto semantic related query introduce front retrieval sample retrieval query query combine partially pareto retrieve successive pareto tuple dissimilarity dissimilarity vector jx convenience pareto definition pareto system key pareto front e sufficient retrieve query return middle pareto previous study distribution pareto optimal point pareto geometry cloud non due pareto call al cloud pareto denote pareto denote pareto simplicity count follow front exist belong pareto front nx kx kx nh pareto traditional convex randomness even pareto scale account minor pareto geometry pareto pareto say pareto draw sure encode pareto set characterization density yield cumulative density uniform proof instead quite involved completeness preserve random among hypercube constant almost surely complete recall proposition context log concavity theorem method largely demonstrate query retrieval pareto integral characterized solution substantially overview limit non extract indexing retrieval processing computer use sift technique pyramid algorithm let md iy assign score assign assign rank distance manifold sort distance add connected rank matrix force nearby term force query rank rank iterative ranking function repeat inversion graph rank anchor construct matrix datum denote column affinity final ranking invert matrix inverting database computational computing require storage matrix computation retrieval xu al provide update prior query algorithm rank final individually main pareto front propose give give rank dissimilarity vector construct pareto associate sample pareto retrieve front return point middle front relevance feedback enhance retrieval could pareto experimental pareto state algorithm develop query correspond semantic label many belong query normalize discount cumulative gain ndcg community ndcg relevance measure single relevance retrieve otherwise retrieval binary score relevance relevance score performance assessment multiclass retrieval relevance query speak multiple relevance cover retrieve retrieve object uniquely query uniquely importance query instance retrieve image effectively query rank query retrieval retrieve object let logical logical conjunction label respectively entry query unique relevance retrieve query query unique relevance query retrieve set relevance discount cumulative gain normalize ndcg normalize possible retrieve contain label query difference assign depend relevance multiple set uniquely ndcg evaluate video widely widely retrieval community provide manually annotate level correspond key characterize global texture frame image key label entry concept image label label belong exactly belong class label members zhang database evaluate randomly pair run algorithm compute pair ndcg use anchor graph run time ndcg experiment ensures avoid particular state retrieval figure compare avg max avg query classifier joint avg figure outperform note generating consider pair label image multi unnecessary separately take retrieve pareto pareto front adjacent front user front bar visualize pareto front relevance five pareto tail middle tail relevance front fix average pair figure deep pareto fundamentally multiple query retrieval middle front suggest version return point middle returning say hold front lead improvement certain choice advance available information decide simplicity modification two query retrieval pareto front user visually explore front pareto front pareto middle pareto point identify pareto method shoot retrieve front include code retrieval correspond query retrieve retrieve retrieval algorithm linear theoretical convexity pareto prove pareto use combination front iii retrieval query include semantic image semantic query combine pareto front state improvement concavity characterize concavity pareto retrieval manifold two problem retrieval image literature correspond image semantic concept possibly shoot angle idea utilize object query call retrieval technique multiple query involve average problem goal find image contain query semantic desirable feature necessarily make fundamentally multiple query image form average query relevant align query retrieval context word approach seem query time tend closely query rarely query
enough use also index long convex lemma conceptually latter exploit consider hessian hessian st update version express substitute result recursively substitute result substitute expression add repeat transpose product except transpose need determine third multiplication summarize recursive plus one gradient variation pair recursion constant vector element determination require operation cost compute link maintain rank common adopt variation q iteration attempt observe see diagonal cost product adopt numerical store scalar multiply constant scalar return product variation implement give loop step constant element loop compute loop second outcome perform implement loop yield multiplication likewise inner product multiplication multiplication multiplication cost summarize computation implementation step initialize estimate identity determine variation curvature matrix step property section svms develop engine tb l initialize cf variation cf cf subsequent convenient instantaneous association fact goal iterate prove result follow eigenvalue impose intend gradient variance unbounde rare progress towards argument stepsize elimination variation require rapidly decrease step strong linearity expression write convergence proof need variation lower instantaneous inner product stochastic furthermore ratio variation variation strong inner hessian inner stochastic variation include stay definite descent stochastic alone guarantee arbitrarily bound matrix oppose formula update relate per approximation use find give assumption trace uniformly time appendix write account fact constant recursion give determinant sum approximation constant eigenvalue imply respective inverse approximation approximation large exceed emphasize realization irrespective upper conclusion direction conditional method bfgs initialize aside sake relationship infimum norm minor nuisance take present matrix give infimum optimality surely e realization establish subsequence role proof roughly eigenvalue limit effect variation regular bfgs variation eigenvalue result estimate strong surely strong hold sgd theorem characterization bfgs define iteration descent give satisfy difference expect constant appendix sgd dominate gradient prove bad sgd theorem parallel adaptive strategy latter description refer former description set known find hyperplane separate set set feature class hyperplane loss measure support support term problem train objective sgd algorithms accelerate memory algorithm randomness curvature nonetheless alternative sag sag gradient direction performance sag five objective vector sgd sag sag sgd square feature class half belong component vector interval likewise choose interval class order advantage vector sag process represent five stepsize improvement stepsize individually since minor sag various individually result average sgd sag objective sag l minimum sag convergence sag attain processing respectively average hold despite fact stochastic gradient feature vector discuss value sag feature magnitude achieve matter figure figure performance performance achieve sgd sag however average sag sgd order achieve far even large become vector respective computational analyze l runtime sag sag sgd fast among acquire computational cost account respective sag order increase process contrary advantage repeat record processing target objective parameter use processing time objective histogram sag minimum maximum run sag stand mark sgd sag analogous histogram performance sgd respective convergence sag still magnitude sag sgd advantage execution large engine apply click search engine query specific appear user descriptor title keyword position page ad display specific include gender ad success ad display query user vector logistic regressor ad vector skewed benefit component total observed age structure gender position engine set select feature whereas parameter convergence rough parameter relatively gradient sparsity nonzero orthogonal average training classifier iterate sgd horizontal vector evaluation versus read index divide axis correction way illustration achieve stand illustration make feature predictive frequency click ad complementary click ad separate define ad ad classifier predict ad ad fall likewise ad prediction consider interval histogram ad fall histogram predict ad conversely ad frequency count predict ad click ads eps eps click rate ad click ad ideal predict probability ad click ad bin sgd acceptable ad ad histogram predict ad point predict interval click rate ad test inaccurate click classifier test sgd classifier predict click ad perform complementary click label imply complementary click rate ad classifier sgd point classifier compute predict interval inaccurate element ad minimizer log likelihood cost ad train label ad replicate label likelihood give ad implicit sgd times select ad prediction ad eps ad click ad predict ad click ad count would bin classifier make ad ad sgd histogram click ad complementary click ad show histogram click rate ad increase classifier click ad sgd less ad click reduce click ad vector predict ad frequency ad display succeed find limited version stochastic sure bounding trace matrix behave far determine stochastic limited ability smooth support develop result term vector process well execution logistic regressor engine problem present numerical test train less similar classification begin observe product equality hessian inverse multiply yield fundamental term except last multiplication implement three product third last product analogous repetition definition element likewise last common recursively equivalently write nest simplification obtain substitute repeating process repeat final instantaneous instantaneous hessian segment instantaneous definition instantaneous definition variation claim true time inner instantaneous begin bind update notation define trace trace trace substitute simplify already one derive appear bound recursive expression give go provide conclude substitute common lead bind make recall determinant define simplify notation determinant determinant product simplify know last simplification observer symmetric therefore substitute simplification factor multiply divide nonzero norm third normalize occur associated eigenvalue large imply coincide particular trace eq also bind right conclude index derivation need determinant curvature scale identity write follow substitute upper determinant approximation bind make recall initialization reduce analysis inequality consider step sum recall conclude less low inequality provide determinant determinant product eigenvalue equivalently conclude product inequality reference hessian write consecutive substitution take hessian bind right hand product state third fact second term low side low bind use statement reference construct sequence satisfy converge surely almost explicit q vanish subsequence embed limit infimum nan transform bound optimality eigenvalue expansion argument schwarz simplification since limit infimum state line theorem sequence objective proceed provide sufficient give rearrange induction hypothesis recursive relationship substitution maximum bind substitute simplify term formula upon conclusion assume prove convergence specify gap term derivation completeness hessian state take around fix hand whose find imply true yield gradient substitute result value side double conclude substitute rewrite hypothesis satisfie define identify substitute function canonical class machine hyperplane separate hyperplane argument gradient train large rely base purpose category newton ever large time regard sgd slow use gradient direction replacement alternative effort fast gradient descent convergence descent still practice randomness challenging curvature profile ill condition slow deterministic deterministic newton stochastic fact limit specific converge stochastic newton use remain quasi speed time without stochastic quasi newton online bfgs bfgs memory middle ground broad applicability irrespective structure extend gradient direction estimate generalization bfgs gradient deterministic gradient differ improve bfgs reduce try adapt curvature quasi newton problem hessian possible singular estimate eigenvalue progress minor possibility analyse fact introduction retain ensure valuable iteration limit memory theoretical main show argument contrast properly guarantee brief discussion bfgs bfgs curvature differ gradient reduce memory deterministic property assumption sample function determinant lemma bound hessian approximation condition sufficient realization important result ensure suffer almost fair emphasize convergence sgd regular introduce dominate bad curvature condition describe newton behavior comparative well number vector term sgd feature make claim regressor click search engine section
valid maximum follow easily adapt extreme hence systematic sign statistic systematic compute observation function location modify wu statistic iid censoring present sign likelihood version wu model function derivative nominal regression dispersion sub test value covariate size six sign test sample adjust test size rejection rate adjust rejection rate nominal present moderate value asymptotic p plot relative sign ratio well plot anti behave test turn note nan guarantee simulate nan test give correct rejection power htp c maximum extreme location dispersion sub covariate possible sign likelihood approximation size summarize sign size converge nominal grow test htp c dispersion specification location value draw distribution present scenario sign similar behavior evident wu notably htp c c ht ht involve table maximum wind measure minimum temperature day wind speed reach ht temperature wind maximum wind value dispersion likelihood nan sign test wu display accordance evidence favor r intercept minimum involve jump eq put present sign much small adjust conclusion adjusted sign adjust test reject nominal nan rely adjust ht r discuss jump put p versus modify sign case possibly unlike adjusted parameterization simulation result reveal ratio test nominal evident shrink wu test behave well clearly wu test behave wu good test conclusion sign adjusted test simulation practitioner adjust component analogously nr r write parameterization parameter observe parameterization instance location h r x v z adjust sign statistic replace formula pc mm derive adjusted sign likelihood approximation sign ratio statistic monte carlo compare sign ratio tend large shrink distortion real discuss word area reliability frequency environmental financial highlight necessity improve reference statistic formalize unify recent book reference extreme reliability survival moderate specifically dispersion regressor literature nevertheless practical contribution small make correction derive adjustment likelihood statistic testing side hypothesis sample adjust location dispersion parameter simulation type likelihood greater moderate sized test present similar conservative adjust case perform test focus sided sign ratio widely side scalar nuisance parameter distribution sample law usually lead sign test considerable sign statistic derive compare finite sign sign test suggest adjust modify datum paper organized section present regression sign derive different author sign extreme performance real end conclusion continuous random dispersion euler constant type approximation inaccurate suitable error significance test normal accurate function space derivative statistic sufficient function notation sign statistic exclude row correspond involve turn require determination impossible application concern model e derivative sign extreme introduce approximation require diagonal note always possible orthogonal parameterization sign ratio replace sign take log interest adjust sign covariance correspond accord sign ratio statistic base estimate adjust sign wu function derivative f cumulative first adjust sign present extreme
matrix hadamard define recursively fast hadamard transform time detail construct recover cosine describe play rescale parameter adapt relevance control rbf adapt type fast method equation behave diagonal specific treat free operation diagonal need learn composite parametrize backpropagation optimize advantage th simplicity backpropagation datum partial gradient respect note simply hadamard since consequently permutation matrix multiplication back propagation allow jointly deep layer greatly overall prevent extract feature affect composite layer considerable store particular decay composite think several layer diagonal control layer include understand powerful adapting within essential layer layer suffice within apply dropout mnist optical character recognition layer replacement second jointly train table implementation convolutional pool use train consist relu linearity arc follow kernel cosine rbf kernel r indicate parameter adapt vary mlp mirror layer network investigation layer convolutional transform jointly network layer adapt performance far reveal overfitte increase final densely connect softmax dropout softmax improve part mnist train deep convolutional layer less validation factor capacity control reference many achieve r imagenet joint ad ad reference jointly deep final imagenet class advantage considerable redundancy could network work convolutional author use regularizer encourage connect memory sparse store nonzero storage take representation dense compare author quantization layer quantization weight clustering represent column close ignore decode representation recover weight book decode actual singular l half svd ad achieve usage please deep convnet reduce memory communication constraint svd decomposition train directly imagenet experiment reference usage drop fully connect svd half svd decomposition drop half drop decrease svd half new architecture call convolutional reduction imagenet usage exchange convolutional derive replace fully network conventional layer low computational cost previous memory potential preserve effect multiple stack potentially improve even far moreover logistic layer far either capacity control deep thorough introduce acknowledgment google national probe http www probe definition definition remark equally fully deep convolutional contain reduce preserve predictive constrain environment embed device replace fully connect layer deep end conjunction convolutional architecture substantially reduce standard convolutional layer fully connect vast majority parameter evaluate imbalance address test separable speed gain additionally address approximation layer total storage require many convolutional network redundancy parameterization exploit represent call parameter jointly decompose memory apply processing contrast convolutional substantially network kernel particular method imagenet million since full memory innovation adaptive variant learn call convolutional able standard imagenet possible line combine deep neural advance previous doubly effective scale extremely imagenet operate jointly filter convolution feature learn kernel way replace neural literature architecture benchmark connect competition global achieve drawback difficult practice train feature imagenet tune motivate add linear adapt tune convolutional expensive evaluation recently usage apply optimization introduce connection remove drop memory usage since memory gain maintain structure overhead consumption memory also train main bottleneck scale storage usually store computing take important insight kernel approximate basis harmonic fourier density turn drop distribution
zero keep unchanged slowly incomplete deal online adaptive reconstruction involve one p tt collect index also likewise introduce signal since dimensional natural leveraging attempt datum minimize unfortunately albeit rank np hard optimize motivate solve th value convex possibly rank control scalable streaming effectively c become large costly computation see nuclear arrive c efficient online complexity storage henceforth dimensionality bound quantity accordingly argue later bilinear suggest span column c alternative nuclear eq bilinear leverage p provide towards adopt frobenius optimality recover globally global solver satisfy massive ability modern computer store analyze incomplete updating obtain recursively track subsequently historical placing measurement end adaptive weight distant tracking environment weight idea perform online leverage nuclear conference network traffic anomaly popularity factorization separation music name tracking towards recursive solver alternate adopt coincide scale justification suitable instant q elsewhere minimize respect fix row obtain via denote subproblem solve recursive define ty l term plus correction resort inversion inversion track incomplete p l p careful reveal computational burden stem iteration reduce symmetry operation multiplication overall typically cf infinite case far reduce tune resort heuristic apply accordingly one window effectively develop well landscape end retain identical unconstrained quadratic obtain cf show inexact show cost number desire expectation take drop matter update l nothing tune backtrack rule adopt whereby sequence geometrically quadratic filter sgd newton order l small nonnegative building popular speed penalty computational per iteration critical nesterov variant accelerate l k accelerate therein acceleration backtrack stepsize sgd missing clearly accelerate sgd backtracking case complexity mainly accelerate incur online memory adopt lie regard acquire online cf fact accordance boundedness natural impose acquisition extensive computer upon eq cost unbounded evolve identical estimator scaling affect note evolve increasingly subspace prior subspace solve g resemble quadratic equation l g whether importantly news nan fall batch worth note play satisfy semi desirable hessian l pc inspire online theory martingale detail proceed main establish convergent rely upper update namely g l l certain regularity establish convergence cost c c nesterov acceleration establish adopt basic surrogate coincide sgd convergence g q l expansion tf sense l iii locally tight q l cf condition eq outline carry reflect iii argument regularity sgd convergent subspace satisfy coincide problem accelerate sgd step proof outline claim establish far clear coincide recently accelerate sgd put subspace technique convergent instrumental online offer batch estimator exact subspace correspond via window require p establish next numerical attain p iterate satisfie normalize regard even parameter g choose grow upon hand vanish condition modern become increasingly structure index far time matrix mean data cube indicate interaction sample ii eeg represent frequency datum incomplete loose nuclear missing encounter many application capture call high order miss stream multi algorithm capable latent structure parsimonious decomposition sequel focus tensor exposition tt via iteration q recognize quadratic surrogate correction gradient stepsize tm diag b diag diag diag diag put observe iteration close reveal update updating incur overall tensor setting accomplish tensor limiting say remain closed require tensor slice low approximation tensor subspace learn initial offer slice formalize establish slice set I live ct tc tc c asymptotically coincide effectiveness algorithm assess computer simulation synthetic real carry sequel generate entry noise simulate per take entry examine strength validation optimal size apparent optimal nuclear norm attain per suggest matrix discuss essence subspace tracking capability algorithm figure upon scheme accuracy constant true rank exhibit behavior expect choice relative nonetheless numerically ls become ridge term stable term computational iteration table compare algorithms c alg relative origin large scale internet ip management study flow dimensionality flow periodic across massive traffic traffic traffic small flow measure flow traffic collect operation internet network internet measure flow spike anomaly end detailed subset algorithm miss flow track representative b versus true blue flow traffic th truth tensor slice diag column likewise coefficient n taking w accordingly acquire slice form x examine algorithm test imputation streaming slice adopt various accordance matrix depict evolution see apparent collecting amount slice observation highlight accurately reconstruct fraction also adopt tensor slice size main memory report amount lead short time need less computation matlab optimize reduction another tensor beneficial subset employ tracking scheme develop sake imputation entail variable entail online subspace remark r c real traffic monitoring serve image diagnosis heart disease clinical mainly hold time may inaccurate acquire consist resolution mind recover ground amount completion low intrinsic dataset test contain entire scan divide patch pixel slice randomly miss infeasible limitation operation store candidate imputation acquire illustrate reconstruct tensor subspace note assume dft fidelity diag stand linear fourier cc image acquire miss ip flow traffic anomaly due failure denote link completely represent traffic flow connect fraction traffic carry superposition flow rate namely experience anomaly measure link measurement count loss small index measure flow anomalous anomaly sparse partial count instant vector anomaly time fully count time matrix naturally th rank nominal anomaly approximation online fashion anomaly denote dimensional subspace learn diag slice matrix absence anomaly nonzero cf sparse absolute anomaly traffic tucker anomaly traffic section fix slice contain nonzero physical I adopt figure run average detection false alarm depict available become traffic accurately three anomaly depict anomaly correctly pick note p accommodate slowly topology desirable monitoring health dynamic network advance subspace tracking put stream incomplete subspace base nuclear norm leverage characterization nuclear complementary strength develop converge provably performance regularize estimator online miss beyond scope paper worth future accelerate sgd alternative real incorporation optimality change satisfy begin nuclear optimality form q semi notational accordance must q complementary primal feasibility iv feasibility dual readily verify last putting imply due say latent encounter big incomplete dataset pose major challenge benefit scalable imputation miss latent rank datum propose exponentially nuclear amenable complementary strength establish simplify technical asymptotic offer develop internet confirm efficacy superior relative alternative subspace track stream matrix site web internet device volume consensus economic life volume importance volume fact motivate subsample privacy streaming comprise noisy correspond traffic collect physical link movie netflix acquire sequentially deal online estimation equivalently completion solve indexing column modern dataset give rise array index miss analytic traffic medical aim capturing call order presence principle tensor resort develop first preserve array paper contribute towards analyze stream incomplete namely capable
implication weak apart instance fall formulate metric previously propose represent hierarchy contain represent subtree share leaf maximally similar leaf define margin discriminant project thus near boundary formulation widely root share implication learn ultimately hierarchy seek hierarchy poorly near root recover partially place tree correctly relationship require split allow splitting generate splitting constraint hierarchy fine grain flat supervise flat via margin en metric return explicit function apply outside cluster semi time semi pairwise link cl improve semantic clustering indicate dissimilarity cl train constraint link proceed subset uniformly split operate supervise unsupervised check availability require link carry semi check unsupervised learn form propagate tree iterate point child contain constraint constrained separate hierarchy reach accordance processing continue minimum back unsupervised metric learn straightforward feed track return combine splitting hierarchy node pairwise split formulation function hierarchical incorporate link unsupervised seek include additional margin joint label cl slack pairwise slack unconstrained pairwise constraint constraint high high scoring satisfy problem link hierarchy relevant decrease low hierarchy case trivial class wherein cl cl term significantly modify reasonable divide instead attempt simultaneously constraint impossible constraint subset constraint separate cl integrate subset variant optimize cl seek satisfied via formulation constrain constraint two replace constraint attempt maximize cl minimize ml eigenvector q return method near neighbor full forest likely empirically strongly overall margin unconstraine ignore leave divide data operation fully training give parallel ignore parallel neighbor test distance near significantly reduce single ignore candidate different tree much approximate neighbor moderately dataset well case quantify validate efficiency approximate carry comparison classification semi supervise cluster mid know balance x segmentation x handwritten cifar spread cifar cifar instance reduce metric neighbor neighbor grind near obtain precision compute report htbp retrieval cifar htb htb technique gb mahalanobis purely original validate tune use rule dataset gb exception stop criterion minimum dataset evaluate weakly training balance segmentation find test competitive perform experiment compare method near amount label datum metric drop dramatically though become metric evaluate effectiveness nearest semantic precision label weakly unable retrieve class drop particularly relaxed effectively discrimination many broad make ccc analyze metric order semi number train metric retrieve near neighbor return convert yielded good neighbor vary number nearest range spectral measure record name indicate demonstrate consistent significant datum base yield result competitive segmentation dataset notable difference metric much oppose much strong semantic semi distance metric forest construct relaxed competitive scale present approximate near retrieval greatly retrieval little hope well membership extend incorporate relative triplet constraint semi even long many application long dominate mahalanobis advance mahalanobis metric forest interpret introduce randomness hierarchy combine powerful robust nonlinear semi information unconstraine relaxed allow subset hierarchy algorithm benchmark problem cluster core availability measure ad hoc whether propose address traditionally dominate metric method primarily generally easy allow fast globally optimal meaning operate notably need video document representation kind unable true semantic linearity version mahalanobis handle limited reason alternate inherently handle necessarily learn modality early nonlinear metric resolve advantage datum metric technique expensive explore advantage structure train nonlinear transformation shift however overfitte area formulate pairwise metric yield implicit could scalability lack representation neighbor order overcome limitation metric advantage exist
hz h note coordinate three dimension reconstruct concentration eq choose save equivalent spatial bandwidth bandwidth substantially begin peak boundary apply small preferred right flat also poisson data time location bandwidth zero fourth multiply constant serve facilitate thus determine cross validation adapt study parametric model several smoothed facilitate population analysis default software determine choice smooth deconvolution procedure take approach design penalty balance fit error inherently problematic easy smooth estimate meet capture popular dimension functional dimension reduction essential impulse smooth random non increase eigenvalue b mt rewrite eigenfunction variability summarize eigenfunction reconstruct voxel recover automate deconvolution inherently problematic computationally viewpoint curve functional curve advantageous linear choice adopt parsimonious principal component analysis popular functional term many rate induce spatially multiplicative curve adopt modified impulse reconstruct impulse response impulse directly perform impulse however computationally expensive automate deconvolution inherently detail denote impulse response course measure voxel function neither thus assumption eigenfunction covariance non eigenvalue functional component deconvolution pose due positivity deconvolution perform eigenfunction considerable perform hundred thousand spatial voxel advantage note deconvolution allow deconvolution usual standard measure next balance complexity strategy many include deconvolution implement see eigenfunction need implement deconvolution reconstruct result bias smooth unbiased small cubic interpolation fast seem estimation multiplicative voxel mean necessarily expectation mean derive estimation estimate classic eigenfunction principal equation principal score number eigenfunction voxel eq l kt ad hoc fit summary simulation select reconstruct integration voxel detail course score select next truncate datum contaminate independent error toy equally space first observe curve matlab function eigenfunction deconvolution well utilize deconvolution work region interest spline deconvolution deconvolution sp suitably deconvolution within however deconvolution fold increase integrate square approach cc sp spline except input replace gamma pointwise context considerably simulation pdf eq gamma mark structure five region outperform outperform show preprocesse satisfied always close gain deconvolution misspecification situation region structure neighboring deconvolution account contrary robust relatively parametric deconvolution strategy competitive mse value ccccc region sp region shape parametric restriction curve perform focus see improvement study normal subject available main control build understand normal brain twice analyse applicable across truth relatively subject subject population quantification quantification relate directly well fit investigation subject acquire subject min mode camera spatial resolution datum reconstruct filter cutoff voxel mm acquisition event frame movement correction c promise voxel curve intractable consider scan shape three eigenfunction eigenfunction variation function among figure component need response function cluster concentration specific difference bias bias parameter dependent analyze population study patient tend bias model quantitative rely would greater evaluate cccc experiment v pool h test analysis roughly corresponding brain considerably flexible model turn considerable correspondence pool show level variation different density value less individual fairly subject level parametric deconvolution take single scan carry intel core gb functional approach mass deconvolution via principal component expansion realistic plausible modelling assumption good possible inherent course deconvolution deconvolution examine change methodology explore segmentation show segmentation marker spatial coherence clear identifiable situation identifiable suggest effect could replace decomposition show give interpretation reason concentrate note smoothed estimate yield possibility function principal score eigenfunction however asymptotically go albeit necessarily eigenfunction smoothed considerable raw curve deconvolution naturally appeal several candidate deconvolution modality fmri response function fmri could although would deconvolution nan operator non response approach applicable replicate curve allow treat functional acknowledgement author would ci part wang nsf dms dms support ep author thank carry simplicity location voxel voxel density integration practice laboratory university california laboratory email uk emission chemical change human currently describe strong deconvolution model functional principal relative voxel methodology goal quantification concentration brain emission process human amongst modality associate work design process presence throughout decay compound quantitative technique say fmri establish target lead diagnosis high something characteristic cancer indeed diagnosis body diagnostic usage understand brain camera involve complex many available system system control brain reaction disease response later take part role system great estimate throughout subject disease diagnosis treatment fmri consist lead million volume reconstruction process reconstruct reconstruct facilitate practitioner scientific clinical suitable modification introduce could incorporate come ordinary ode system abstract voxel transfer assume ode biological adequate voxel mixture cell lead addition stable voxel square linear somewhat order account fitting explore nonparametric deconvolution flow online purpose continuously relative measure sensitive produce deconvolution deconvolution inherent deconvolution methodology observation possibly
model highly whiten common preprocessing remove unit allow training gaussian component spherical model fairly used statistic whiten preprocesse knowledge improve success analysis component visible right position natural anchor therefore speed initialize scale first scaling scaling determine ideally show hidden bias much weight initialize experience work well initialization hyper impact speed successful training acceptable number big learning converge local place weight become visible restrict prevent divergence big twice datum hold even big even therefore practice matrix rather also momentum counter grow certain recommend regularization momentum add percentage old batch vary lot momentum use prevent weight converge momentum stage rbms describe approximation estimating ideally sample low stay previously therefore step increase rate suggest persistent persistent persistent analyze training convergence restriction pt well performance cd pt difficulty image several author various modification propose address analyze generative argue failure predict pixel model rbm unit dedicate regard covariance diag compare restrict although covariance argue limitation develop spike rbm split binary spike conditional visible diagonal gaussian shift along failure procedure show learn filter difficulty propose modification learning difficulty problem expert constrain gaussian representation insight capability blind source separation capable meaningful toy natural image comparable ica ica orthogonal representation success highly depend setup propose directly experience imply vice versa knowledge center integrate distribution start train usually successful network illustrate write n expert expert gaussian shift j unit lead correspond visible unit bayes formalize denote respectively jk sum vector exactly follow isotropic gaussian convention conditional visible unit component probability view pick hide location visible take exactly place order component determine indicate constrain independent super source dependency able visualize toy example source yield whiten calculate assess run experiment ica ht one code efficiently pixel scheme biological plausible natural empirically code grey scale training image mention patch trivial vector filter fairly filter
extensive subset obtain partition median part actual suffer report linear logistic normal correlation choose intractable implement np selection vary experiment pt htbp htbp htbp htbp clear message excellent case bootstrap subset mse median dramatically performance bootstrap competitive range small clearly average lasso median winner consumption measurement predictor predictor code categorical date subset inference square error exclude produce meaningful time stage later stage subtle performance fast exclude due see algorithm produce physics interest distinguish particle size rest parallel classification accuracy correctly predict test plot exclude quickly benchmark list model achieve propose flexible message message burden aggregation message performance simulation prediction theory describe concerned topological exceed k let select select estimator average incorrect take side b justification theoretical nonetheless insight limited attempt justify theorem alone routine yu part address correlate convenient assume feature sample subset article cause choice strong article conclusion simultaneously next might satisfy elliptical high dimensional set elliptical special spirit invertible assume elliptical alternatively hold proof magnitude iii set inequality w w chebyshev inequality therefore take immediately part establish chebyshev taking term quantify take need minimum index evaluate solely adequate cauchy schwarz combine q part consequently single machine size least procedure begin c x assume big continue strategy part need satisfied size alternatively hold subset simulate htbp htbp definition department pa david department nc commonly algorithm store communication challenge arise guarantee excellent practical general median selection subset aggregation estimator attempt solve parallel feature inclusion parallel scale efficiently theoretical consistency relative usual velocity challenge promise procedure parallelization partition full different process subset subset type gain calculation optimization algorithm operation involve likelihood lead computational amount gain across computer communication drive efficiency importance communication limit communication combine step simple issue communication free improve statistical slow entire simultaneously article focus particularly approach subset suggest use zhang subset well utilize median show sharp certain broadly useful inference regression feature current combine design detailed missing contexts imputation computational another bootstrap fix feature fix justification excellent computationally highly organize message detail scenario evaluate message extensive discussion family proof vector matrix error assume fundamental efficient subset subset carry aggregated produce two rich literature consider generalized criterion feature dimensional attempt solve regularity selection solve problem solve yu consistency lasso could ordinary square ol introduction median possess advantage feature aggregation motivated averaging hence feature interest variable selection recommend simplify inclusion include subset median indicator otherwise inclusion inclusion indicator true inclusion polynomial vector gain time heavy tailed influence selection influence outlier datum put aside average estimate subset spirit
one fall shape usual center raw datum process cf middle factored represent coordinate cf structure convenient interpretation xx xy yx yy first method subsection graphical lasso term provide rather contain along boundary considerable edge partial pt ccc use non positivity part precision point sign constraint severe perspective specifically indicate long high post thresholde interpretable fall behind finding prove future acknowledgment characterization positive extract symmetric spectrum resp irreducible note symmetric symmetric must irreducible symmetric cf remark apply eigenvector follow fulfil upper ball apply theorem state follow fourth moment spectrum matrix stay fourth jk claim constrained fall spectral function verify r x bound finally minimizer non negative negativity kkt consequently choose claim would q equivalently write order matrix entry hence apply equality yield sequel associate row jk right hand jk jj j thresholding I say triple note converse independence sequel global markov independence graph c u partition formula submatrix negative verify comment virtue non negativity successively ab ab cf ab theorem theorem remark sketch finite precision negative vector estimation precision constrain determinant treat size greatly simplify provide log determinant correlation b random role discriminant confidence interval inverse independence relation miss modelling aim parsimonious conditional independence graph receive considerable finance comparable size development inferential procedure try equivalently case independence purpose procedure base independence regression suggest reference amount penalize likelihood relate regularization scheme enforce propose enforce dimensional address cite consider semidefinite precision definite symmetric e partial correlation precision gaussian positivity precision attractive sub form inference specifically knowledge author precision element dominant equal positive identity likelihood discuss restrict impose curse partial correlation negative gene priori unclear misspecification side constraint establish existence uniqueness maximum likelihood case unconstrained thresholding constrain yet sparsity structure thresholde negative high absence tendency produce approach exploratory analysis sparse discuss application sign log determinant estimation subsequently develop descent convex extensive include summary proof letter letter letter use submatrix index invertible frequently arbitrary submatrix column permutation entry likewise result set denote trace block compose positive whereas symbol I realization precision bregman divergence induce cf henceforth constrain know sign diagonal omit minimizer unclear minimization make difference unless minimizer word unless exist check constrained determinant divergence though employ condition fulfil subsection view extend determinant positive semidefinite cone constitute constraint negative q lagrangian multiplier tucker q note duality derivative equality variable seek diagonal follow definite necessity possible find unbounded multivariate consequence mis specification fact see rather target constraint know mis investigate select symmetric cf diagonal conditioning covariance give hand partition must sub role complement equivalently observation remain unchanged combine variable negative partition regression non mark infer mark mark student achieve mathematics mechanic algebra minimization matrix precision appear adequate one pair exactly suggest increase drop performance divergence minimization behave mis specification substantial bias discuss issue level coincide apart leibler precision accord preserve ideally one maintain partial partial loss example item one section ask least preserve negative question choice q accordingly diagonal verify consider dd observe see partition inverse formula non entry feasible feasibility hadamard preserve sign accord ar process order entry even even odd diagonal equal odd kkt optimality condition p lm satisfie violate recursion stationarity inverse preserve tight appendix positive e guarantee recover set fulfil ar opposite mind simplicity effect result replacement covariance cccc connect estimation seminal interpretable sparsity among penalty approach prominent appear complement minimization penalty desire modification end sign version post processing combine precision cardinality correlation aim finite I scheme minimizer hard let definite perform fit constraint diagonal entry definite improve regard small cardinality sequel successful entail estimate obey wish element still enough depend classic consistency use realization random fourth moment minimizer determinant order less justify cf first observe empirically sample whose constrain divergence unique exist constant consequently single high decay tail gaussian tail identification sense scope though convergence available g proof regularization substantial see explain prefer penalization penalty bias affect yet diagonal thresholding achieve level aim percent keep percent entry absolute result hence common solution solve instance constrain determinant handle slow ten thousand thousand million devise solver graphical analogous recursively regression recursively least regression solver apart ease implementation foundation block coordinate exist establish sharp runtime increase call scheme jj jj jj block optimize keep repeat block criterion satisfied solver exist sequel routine indeed determinant q moment show decompose term drop well problem give definite negative third constraint long linear concave function minimizer kkt lagrangian multiplier substitute kkt resolve automatically jj iterate strictly provide respectively solve write hard handle turn one solver problem experimentally fast exceed thousand existence satisfie suggest support cycle operation principal solve terminate view quantify criterion dataset systematically positivity constraint element specifically cccc accord obtain know keep aside hyperparameter parameter use example encode set adjacency matrix grid adjacency grid grid percent entry previous long away k setup exception setup replication average replication large error spectral leibler kl cc approach include various attractive constrain determinant divergence minimization validation follow quantile small diagonal compute validation fit quantile pick minimized minimizer determinant minimized variant glasso denote thresholding estimator glasso describe via glasso yield manner glasso likewise denote graph conditional graph provide see sufficient try structure node whenever associated regression thresholding apply coefficient regularization regression grid accord compute covariance jk jk obtain jk exceed regard proceed vertex chain star pair series marginal perform significance level conditional independence run fact independence perform setting rather apply dimensional employ drop performance observe grid star optimal sparse figure stage glasso exclude c grid glasso five different trajectory different range reason run report tuning exclude step graph glasso publicly code penalize problem implement approach algorithmic project report reveal sound empirically one degree severe average run glasso hyperparameter measure kkt optimality time figure
iteratively image confidence reduce several mutually spatio temporal confidence recognition use image recognize place problem system give detect place adjust bin stand drawback input continuous call cause place advantage learn unsupervised hypothesis every description quantization translate technique allow parallel place model discretized signature integration auto account dependence step relation unique word main lie visual world represent figure recognition represent robot time place model posteriori recursive encodes account dependency call restrict unchanged node posteriori eq tt simplify place characterization algorithm discrete variable dictionary place give discrete tt nt word estimation word sequence posteriori due unobserve unified gram descriptor divide filter bank filter scale project explain descriptor spatial quantization perform line image neuron word parametrize map categorization task stochastic average time high close vector quantization visual image learn computed step result strategy visual first sub sample image subsample strategy replace word compression rate strategy call strategy simple online carry place database see place recognition sequence acquire human illumination laboratory illumination carry protocol hundred enough new acquire follow five class illumination similarly test perform illumination htbp office c training share uniformly among transition use influence varied laplace set small value laplace give note signature laplace interpolation result could generally several percent see difference bar bar efficient usually except strategy gram speak effect increase bit clear gram performance note drop performance high seem confirm intuition behind laplace right new temporal recognition model dependence gain seem quite high study simple combine sophisticated useful look pt pt language paris sup paris fr inspire field paper filter standard discussion highlight improvement relatively field aim robot high human compatible ease daily environment notably environment compose place correspond house place call
test step variable decomposition treat problem ensure subproblem involve sort solve subproblem supplementary grouping together induced point subdifferential behave handle grouping singleton group sort absolute exclude r join decomposition subproblem proposition reasonable smoothed somewhat addition give control slow correspond prior ideally expect particular encourage degree model hard approach use monte apply applicable address reweighted minimize regularizer encourage term continuous counterpart difficulty objective alternative convex aspect apply double loop em gives monotonically improve true perform regularize method admm glasso inner loop admm give identical array typically micro method dataset http www contain gene tune produce near visualization major connect scale subgraph center relaxation tight gene cluster reweighte produce great computing proximal point algorithm compute proximal input ba plot convergence rate show test clear subgradient converge slowly practical applicability admm slowly problem achieve practice decomposition method converge quickly requirement iteration method subgradient subgradient decomposition dominate sort dominate least solve square run rough ba graph vertex per error reweighte second submodular standard conclusion fall grow structured previous prior make tractable use reconstruction department communication digital prop university mm key determination reconstruct scale formulate sparsity induce function graphical efficiently improvement encourage scale reconstruction graphical undirected graphical independence fit fit context model problem induce likelihood body paper various objective development knowledge encode parameter link sparsity pixel explore recover structure graphical recover free network enforce formulation enforce prior envelope relaxed non pose challenge option optimisation operator experiment produce scale real bioinformatics relaxation superior undirected place graph mean bag natural family assign probability exponential graph depend statistic consider degree parametrization weighting encode distribution correctly choice rest see take form infinite put weight posteriori set property lead non function interpret increase add consider decrease note cardinality modular modular sum submodular concavity restriction ingredient allow enforce tailed place weight node aware novel rise convex envelope precisely cardinality problem weight connect notation sort natural ordering envelope q piece intuitive behave like additional edge problem case q matrix graphical rescale distribution cone boundary psd cone handle interior shown gradient definite optimize differentiable suggests submodular subgradient proximal subgradient simple optimize smooth convex function subgradient due piecewise primal method return intermediate limit superior convergence sparse proximal rely iteration proximal close relaxation solve minimization submodular proximal operator cut function algorithm slow vertex clearly propose optimisation method gradient proximal alternate direction apply multiplier admm optimize number advantage proximal presentation update proximal turn updating criterion practice admm fast guarantee restriction place step size degree
data line save save save file txt specify confidence use confidence use error bar macro mean test confidence know mean confidence well detail allowed level confidence across value calculate trajectory computation enable help performance performance identify name identify line generate external first external model model section performance calculate case classification contain file file test file classified name predict predict file file performance test accordance percentage wrong precision equally auc sensitivity specificity specificity tp class rate class roc auc area curve avg test cross avg test variance test file file micro macro comparison file square average comparison file comparison matrix lf lf lf lf lf lf lf lf lf lf lf lf lf lf lf indicate lf indicate well indistinguishable mean statistically different statistically accuracy big average know tail know statistically tail account read contain cross performance learn store roc curve name curve curve micro average average curve interval average confidence name recall curves micro macro average vertical confidence interval lift chart name curve lift class case cross micro macro vertical averaging interval confidence file file supervise test file contain name course correspondence name name true predict txt txt txt predict txt class file performance contain rand time time manually see performance file file learn file section provide example execute follow source code create test call library copy section set classification log set parent evaluate performance store file column name name execute validation specify validation specify validation txt specify file extension specify class name specify section eventually enable specify name classification performance fold load pre partition describe line execute nb txt rely two cv enable validation nb result txt specifies file file learn set specify partitioning partitioning replicate execute test txt file model compare nb file section synthetic trajectory contain assume make cluster use learn maximize maximum parent store file name execute cluster difference name specify name replicate em especially soft assignment scoring inference static method performance supervise front follow part analyze figure diagram represent interface define trajectory trajectory stand alone implementation class potentially specify implementation trajectory trajectory vc vb vb vb vc trajectory instant theoretically happen code column string string add time string vb stre vb new string vb vc string vc class management set accordance partition file classify class line parent set consider node probability line previous line file format define previously bn class n n class parent specify complete parent parent iterate parent network simplify interface define abstract implementation method implementation implement parameter specifie parameter implement search optimization search algorithm return interface implement see diagram search structural scoring define method score rely conditional likelihood maximization select scoring interface interface simple definition evaluate interface define local good neighbor I local search use generate individual interface code interface interface generate marginal score maximization individual maximization model new classifier boolean false algorithm I string string put tx prior learn alg alg collection double alg structural use marginal scoring learn string object put put hill false start ia ib b ic boolean false ia ib ic true definition structural string alg string structural alg depict simplify algorithm extend interface abstract implement implement assignment purpose algorithm rely interface define criterion em provide stop example assignment put tx px put false definition object string learn algorithm order calculate code interface define define classification class implement trajectory classify double double usually classification probable criterion implement interface unbalanced order depict calculate performance execute implement implement implement execute performance class implement performance class implement implement test approach double test use validation validate performance detail evaluate performance section hierarchy class provided generate simplified diagram performance depict figure argument performance interface generic aggregate provide definition provide section run performance interface cluster aggregate micro average micro performance run performance implement micro averaging implement micro macro average performance performance unsupervise provide class possibility calculate micro macro averaging performance provide depict diagram performance interface define run aggregate interface single micro macro average aggregate classification provide run depict simplified diagram interface define implement generate model test lambda double generate interface provide test test implement provide use provide execute return result interface performance generalization double double execution double nb performance string addition provide test performance front end front class due start implement develop array information necessary I private help specify validation cv fold validation line column I help enable analyze call public string pass link line link list parameter add code use program rely loading algorithm loading provide generation execution paper temporal stand alone performance replicate test description give library many development class one possible future possibility parent static currently direction focus possibility partially observable sl section corollary definition time classification streaming duration open stand library implement continuous classifier marginal score introduce include understand library help extension contribution develop paper valuable suggestion matlab make correctness relevance source stream engineering reference image video science system social stream trading offer analyze streaming model evolution analyze stream monitoring time diagnosis study fire streaming among hmms receive dependency suffer limitation discretize datum poorly represent rapidly point dependencie necessary multiple increase time ct cascade conditional ct cascade devote require parametric dependency significantly problem selection ct cascade overcome affect future event homogeneous markov stream continuous classification stream stream trajectory recognition purpose stroke stream period temporal address continuous include latent continuous continuous classifier discretization describe source library stand line interface purpose prototype problem score score marginal conditional expectation validation extend extended guide stand alone usage provide explain library section read cluster usage implementation possible step system time several slice slice increment always propagate consist evolve space become intractable continuous overcome temporal network exploit continuous probabilistic whose evolve continuously finite domain bayesian component direct cyclic node parent intensity intensity parent probability occur drug introduce indicate person depend yes one empty variable fully specify text center node style circle empty hour person empty empty stop empty minute hour state empty state transition state e continuous network variable inference variable continuous time network continuous marginal hold associate node fully depend part style align center style em ei si full edge pt pt right pt right pt leave naive bayes effort continuous naive bayes classifier bayesian address learn continuous max parent formally naive max max node attribute take data structure marginal learning account count q x x necessity static count consist run local optimal maximize attribute search possible conditional likelihood especially value scoring log obtain learn describe stream contiguous interval continuous q associate interval parent interval interval interval consist attribute fully evidence maximum classification stream py n nx rely formula calculated expectation sum contribution statistic occurrence trajectory contribution occurrence count without set contribution count statistic time attribute assignment step hard trajectory statistic calculate account probable maximization stand alone library read file gamma step file website library test site stand alone show section file possible paper collection free source code accordance file datum parameter address case file section section hereafter term path library file library suppose test make virtual virtual increase avoid increase dimension gb argument specify table summarize separately yes yes yes yes yes yes yes yes yes yes yes yes confidence argument pt p cm cm cm p cm vs x x confidence provide help ignore help test follow naive bayesian parent learn scoring stand model hyperparameter count relate transition default spend datum big error big learn structural add dimension ignore apply example penalty enable count max default maximize default enable penalty model file soon format see file model datum possible vs must validation test validation clustering activate validation random validation cv cross fold default hold parameter fold cross validation cv cv section cluster require complete set measure rand coefficient expectation soft cluster I default trajectory trajectory change default default threshold soft soft threshold hard threshold
capacity image recognition popularity current object benchmark support tailor image experiment continue grow rely ensure inform second thm pt em author author token university vector task part explore interaction svm connect extraction thing induce capacity svm pixel preserve locality demonstrate surprising expression recognition detector improve decade largely svms train histogram gradient feature upon high page visual classification layer complexity view weight margin svm add capacity pixel possible perform classifier filter take remove sensitivity figure type successful change try remove leave paper choose square choice great flexibility show becomes refer convolutional visual write input edge hadamard operator pointwise sparse operation bank orient edge response descriptor q normalization address piece show previously descriptor form selection convolutional bank projection bank eq q feature affine quadratic interaction ii prior form affine projection kronecker expansion weight unary induce prior quantify weighting therefore prior svm pixel interaction discriminate distribution however deal distribution covariance often stationarity translate pixel fall quickly improve conditioning interaction order set unary redundant redundancy stem pixel account compact impulse encode spectrum natural train pixel information one preserve illustration result pixel fail discriminate information separate distribution overlap local discriminate spectra image contain structure contour experiment show inherently interaction pixel exploit encoding capacity separate noise sample spectrum train distinguish discriminate pixel perfectly separate two svms pixel encode affine good however prior simply reflect belief absence actual inform decision assumption may detection comprise possible pose gender identity background change appearance detector intra class detector unnecessary sort require perturbation classifier learn specialized aim answer geometric reproduce effect network heavily learn number primitive capacity let learn could amount broad expression sequence neutral formation task discard different canonical pose error broad visual recognition recognition pose face heavily contain condition label ground flexibility amount geometric control identity example coverage accord aggregation quadratic pixel error collect region pixel face degree geometric introduce pixel pixel far normalization extraction component traditionally receive much attention literature play invariance contrast normalization instead normalize expression experiment storage requirement local quadratic local quadratic take implement parallel dual solver server use core ram depend grid search come converge towards variation span green amount
difference correlation sense need sample study adaptive set access correspond need factor instance correlation decision policy figure constant eq complete proof optimal policy error let analyze sr successive description sr round round next round sr arm leave output arm verify sr exceed budget sample use sr trivial ti ks sr universal constant assume event event occur rest organize two sr subset suboptimal last r k aa optimal k know sr n k kk kk k n se elimination description se dynamic sr se exist arm rather round sr se arms suboptimal se find set sample ta u ta probability least return eq event inequality enough event se terminate claim fact separately follow optimal beginning eliminate round suboptimal eliminate b sample r moreover arm eliminate thus arm complete work towards hardness correlate sampling sr se result known argument literature term suboptimal problem equivalent arm correlation perfectly correlate se challenge design general observed identification successive allow optimal unfortunately applied correlated identification accept early arm mutually arm identification remain arm impossible summarize sample chi random variable two absolutely value chapter recall distributions eq application r positive strictly constant correlation describe standard define two center conditioning distribute gaussian h w divergence h h department research financial university edu sampling strategy rely wide integer use correspond probability model make arm select observe value uniform agent subset arm solution subset furthermore suboptimal devise reliably realize outcome adaptive budget give ask subset soon devise return want explicit limitation stop regardless evaluate terminate expectation algorithms sr correlation call difference argue easy close feature classical attain difference base arm arm ir satisfy arm need precisely adaptive level idea suboptimal need budget adaptive way estimate fix set correlation exclude suboptimal arm component focuses correlate accurately subset mutual importantly contrary adapt heterogeneity describe correlation go see use allow largely wide correlation sx
thank constructive comment manuscript support centre national commonly pair correlate parametric test useful uncertainty certainly one clinical method use coefficient resample create multiple original method assume representative concern size however principal difference though uncertainty distance clinical clinical intrinsic distribution point course count attempt uncertainty carlo analyse importantly latter two exploit uncertainty consist individual coefficient sample rank eq coefficient test score score z calculate infinity go bootstrap resample create consist statistical new randomly g may assign set use probability two quantity simple correlation coefficient calculate width e deviation score also resample method obviously account uncertainty likely pair perturbation create new perturb randomly gaussian center uncertainty composite language real ray indice optical band magnitude apparent significance basic distribution plot may clearly simplify nonetheless uncertainty magnitude significance correlation take apply perturbation composite one find return wide value green fact well significance histogram plot composite plot method bootstrap test method general serious heavily datum point distribution perturbation composite size uncertainty regard distribution clearly uncertainty account uncertainty tend return composite resample understand difference composite distribution estimate sample uncertainty lack estimate correlation coefficient uncertainty population whole
scenario simultaneously take advantage inter competition study segmentation pixel object convolutional design operation spatial output weakly supervise predict output propagation cast suggest replace fully layer train pre train level experiment without last layer weight initialization fine act baseline quickly converge semantic background class classification none weight prediction produce correspond identify max coarse maximally score negative compete object set heat ignoring non maximally score key background simultaneous inter help refine intra pixel time image segmentation challenge augment held intersection percentage segmentation mask mask union classifier fine tune supervision common degenerate solution predict train momentum quick network converge iteration achieve relative baseline mean mean classifier b b output show quantitative union baseline preliminary encourage propose novel joint inspire network kind merely field super refine grouping could likewise segment instead convnet map encourage segmentation mining berkeley edu pt reduce annotation segmentation degree formulation fully seek learn weak train jointly pixel label convolutional input need offer exploit supervision evaluate preliminary challenge convolutional performance supervision progress segmentation annotate consuming collect supervision improve infer train supervision proposal problem max margin representation sensitivity
furthermore fully induction suppose true integer lemma connect either maximal adding maximal add new maximal clique connect gm gm maximal clique define find find span maximal spanning include leave end maximal weight span clique right everything interested opposed complexity reasonable come elimination consider gm procedure eq consider potential case leave already step proportional message correct exact know problem bp gm subsequent recover use order algorithm exact property group gm gm possible guarantee subsequent operation bp tractable operation per bp need graph message message graph last inequality prove unfortunately thing gradient like ensure stay version size q three case give entire hide variable interested maximum ml achieve binomial parent look set order ml amount index write use triangle size write write therefore quantity calculate learn q maximization possible direct likelihood read thus I liu rely undirecte exponential family denote observe write use descent formula gradient expectation question link penalize introduce highly trivial graphical day last distinguish stand variable parameter iterate parameter proposition france observation realization want end error equivalent call compute calculate much knowledge inequality make gm constitute tool allow solve alphabet joint eq q represent factorize affect represent conditioning factorize obtain graphical sense graphical correspond model factorize contain parent edge chapter realization type undirected gm represent edge dependence gm undirected conditional lead word remove subgraph link edge maximal clique mrf q clique equivalent n I equivalent prove two claim therefore show stand side take bayes rewrite hand depend value equal know write mrf way maximal q nothing underlie example factor pixel probability q pixel natural unlikely piecewise crowd use human difficult verify evaluate crowd assign worker collect answer worker probability reliable infer task worker ia conditional distribution know answer distribution consist hard core instance problem hardness maximize become quickly intractable exploit gm reduce connectivity gm number formalize elimination make graph gm fig colored subgraph decomposition gm calculate marginal require
function widely mathematic statistic provide brief definition description divide order end restriction continuous spline spline basis spline one interval nonzero spline knot approximated combination work split correspond spline product spline column vector number tensor spline maintain nice modeling every j k spline product spline smoothness approximated lemma power restriction later function j every element assume sufficiently satisfy fy kx pair observation interval apply affine otherwise co conditional density neither index conditional oracle knowledge prior covariate spline j kk component affect inclusion assign indicator construct put model let mass every prior value support active index term basis spline stand independently coefficient r j j x rx j identical induce coefficient positive prior include truncate case assume decay hold e index next remain necessary default independence obviously isotropic well zero truncate poisson default simplex contraction long polynomially fast result allow covariate sample stand know hellinger stand density true density say rate fix n distribution q establish contraction lie class respect minimax conditional situation take rate compare grow polynomially coincide additional logarithmic grow obtain rate hence variable procedure smoothness trivially fix involve contraction necessarily recover predictor scope contraction determine assign mass complexity ambient dimension contraction entire consequence function condition relax qualitatively different following propose smoothness likelihood kx fy view form denominator involve integral power whose differ co collection conjugacy dirichlet write certain take consideration let enter moment kernel jointly covariate spline since spline basis interval calculation term take histogram property smoothness save computational exchange smoothness choose haar prior range bernoulli marginal truncate poisson random integer part restrict calculation subsection generate randomly sum least square develop select variable replication compare directly sum l directly carry sensitivity choose covariate probability fall range max dominate lebesgue density hellinger respect contraction standard rewrite exist conditional call kf f fy variation j fix choose number cover k r j r fact concentration around restrict every simply h f combine regard f view distribution within vary n suffice determine sufficiently theorem outline main difference approximation result calculation concentration change calculation concentration j sequence verify stand number k product tail cut clearly hence requirement meet isotropic calculation calculation entropy eq k define j great suffice since look identical isotropic proof mathematically express hellinger joint density dominate lebesgue leibler rest argument way isotropic tensor spline lemma multivariate c fy kx particular b kx j kx analog theorem tensor spline expansion multiple norm form consist dual univariate spline uniformly bound relation define relation boundedness exactly lemma interestingly usually require related discussion model let function shape change possibly observation predictor construct product incorporate issue posterior adaptively level also degree smoothness predictor technique calculate moment chain monte carlo large sometimes receive attention scientific genome association focus literature approach obtain distribution exist assigning prior mixture generalize break transform gaussian multivariate generalization beta generalized possess conjugacy modern datum many literature subset penalization least screen sis establish include linear gain popularity variable efficient compare uncertainty prediction accomplish assign model combine extend allow sub contraction often require allow result largely bayesian recovery trivial covariate restrict isometry problem framework analogous
explore cnn particular convolutional much convolutional cnn order method layer scalar quantization quantization apply scalar well method structure quantization give additional gain explore redundancy parameter knowledge systematically quantization parameter make contribution systematically explore quantization cnn storage comprehensive quantization quantization product significantly perform ability compress deep convolutional object image great area art already human great interest adopt scene cnn hundred million huge storage publish cnn explore property cpu speed execution boost operation efficiently work explore matrix little little devote cnn vector quantization cnn parameter show accurately parameter neural parameterize redundancy compression realization prediction surprisingly decrease performance confirm finding dense connected factorization widely cnn parameter consider dense connect orthogonal reconstruct eq correspond singular svd control optimal frobenius approximated matrix matrix compression turn neuron turn geometric view hyperplane round neuron scalar scalar scalar codebook form cluster look reconstruct need store index codebook center bit encode use center need bit per cluster index assume assume codebook despite surprisingly parameter structure quantization explore structure many quantization assumption subspace redundant perform quantization wise several submatrix denote th codebook store thus reconstruct x particular quantization codebook negligible quantization quantization structure quantization basic center example every represent compute vector reconstruct center need store potentially compression different quantization km capture redundancy explore structure try explore global vector hashing among store rotation paper output filter grouping filter grouping dimension filter investigation find explore group default image object contain convolutional dense patch feed layer respective filter convolutional pooling relu network epoch start every epoch momentum accuracy accuracy goal achieve compression perform compression several cluster bit segment column change compression rate mention axis case segment align compression accuracy center size able obtain small method usually low quantization codebook account compression rate center always codebook example center e codebook rate result slightly improvement make codebook size next center bit balance quantization herein error tune segment compression rate km compression svd vary achieve center iteration lower result layer mainly two factorize still store somewhat surprisingly despite achieve km keep quantization far improve km high compression codebook size big compression simple work reasonably compression give km goal km km compression suggest considerable redundancy neuron poorly classification fixing layer report layer layer lead especially rate result present application compress retrieval verify generalization compress server number server limit able process image server compress compressed perform retrieval database process retrieval precision compress cnn activation layer retrieval cosine trend consistently work one surprising center high original special application robust reconstruct mean approximation report ccccc compression svd km center km center km center center storage apply quantization save unlike approach factorization method find simply quantization mean able obtain structured quantization method able address apply cnn embed device
remark paper new failure ps call failure power failure rate poisson distribution failure binomial power eps class ep special ability cover five hazard I unimodal shape several em algorithm discuss class distribution fit distribution real linear failure moment many introduce discrete continuous random say component exponential ep discrete distribution see family eps series complementary series distribution series combine propose special distribution introduce new compound series produce I new due representation system appear apply cancer activation failure model class iv class parametric monotone unimodal decrease failure common outline special expansion section estimation em algorithm conclude failure function cdf parameter discrete power q cdf survival hazard rate consider density hazard give cdf cdf number density know nx x x hazard eq use generate e dx tx dx te e dx dx dx te bn bn normal distribution therefore xt nc nc bn calculation xt incomplete case distribution distribution become hazard either increase function contain special exponential geometric exponential poisson distribution density power eps special geometric power series hazard hazard decrease put eeg monotonically density decrease unimodal hazard increase fulfil interior boundary remain valid replace use hazard element element binomial mle lie interval root k iv nx rhs root proof denote rhs expression true appendix call step tool handle datum joint cycle conditional likelihood n obtain therefore estimation root equation unique appendix unique distribution estimator observe bx bx b bx nc taking give bx bx bx c nc ic ic ax move v observe information square simulation give theorem calculate standard restriction absolute firstly em distribution value assess determined correspond information shown suggest consistently iv standard error sample standard observe close simulate ccc ccc bias sim std sim ccc ccc ccc std std sim ccc ccc ccc c sim sim c ccc ccc ccc ccc sim std sim em demonstrate datum study represent health state fit exponential
vector fact easy see existence recall interested determine guarantee nevertheless suppose iff corollary subspace precisely come immediately corollary observe iff row infinitely subspace e contain linearly row state subspace imply even subspace us setup one subspace many subspace h thus explicit instance easy despite column subspace want know dimensional subspace lie additional suppose surely h want guarantee column indeed lie dimensional want true condition condition guarantee question document analysis column lie keep relation surely iff comparable converse previous different subspace converse suffice correspond dimensional subspace tell column subspace conversely tell observe unless element show definition arbitrary conclude conversely size iff size tell belong subspace remain determine contain basis characterization come lemma contain subspace assume infinitely many subspace imply might even converse come direct converse satisfie exist key behind idea subspace must onto lie vector determine subspace dimensional contain lie idea give constrain since must focus block block eq even position elsewhere exactly always describe arbitrary entry function entry make linearly linearly orthogonal onto essentially projection simultaneously would project plane constrain determine contain one accord condition obtain would ia statement force minimal assumption confirm entry whose answer question concept subspace understand hyperplane hyperplane hyperplane align canonical zero entry affect able subspace plane could nothing plane little show low subspace infinitely subspace always trivial generalize straightforward iterate happen drop would matter subspace generic subspace section dimension would identify almost extremely unlikely exactly converse state almost surely projection dimensional namely aligned sense suppose z course measure zero set subspace lie align general course distinct hyperplane intersection e occur projection summarize incomplete lie fit partially incomplete vector sense tool derive consequence already brief interesting relate research incomplete behave complete one form uniquely determined precise say document incomplete validate really want say identify solely way would exactly subspace lie identify incomplete insight sufficient deterministic conjecture em really contain set set vector characterize concrete manner characteristic mention tool area corollary example corollary infinitely subspace describe relation hope combinatorial reconstruct like reconstruct arbitrary projection describe variable involve equation support support equation believe dimension algebraic ambient element index distinct subspace general belong row subset size subspace canonical restriction arbitrary lie onto lie union subspace arbitrary arbitrary arbitrary incomplete incomplete version incomplete incomplete entry draw draw circle font cm font electrical usa guarantee partially union really lie subspace deterministic sufficient certain partially unique characterize incomplete try try fit look really really subspace subspace collection indeed validate generic iff lie become arbitrarily subspace even counterpart really follow vector lie suppose vector span counterpart lie course without know arbitrary precisely e rest union appear task lie subspace subspace setup far counterpart incomplete subspace subspace counterpart subspace vector want make complete counterpart complete imagine also counterpart lie subspace way directly imply counterpart imply nice extra incomplete could set extra generic behave allow counterpart lie answer yes characterize incomplete one strict subset vector observe row characterization determine incomplete really lie verify row first counterpart counterpart indeed subspace essential subspace incomplete satisfied fail would complete evident fail satisfy dimensional subspace belong could happen word precisely need discover incomplete substantially vector automatically lie incomplete case generic position generic determine generic exactly deterministic condition guarantee indeed lie sufficient sense satisfie indeed lie conversely exist condition unique lie element lie uniqueness give essentially result document formally characterize incomplete allow final goal indeed motivation particularly consequence detailed exposition document result intuitive result generalization brief future symbol statement etc document alternatively index al end arrival challenge information quickly resource fortunately simplify thing achieve big likely incomplete fortunately subspace handle subspace give infer miss handling attract attention recent year detect incomplete fit even miss converse useful bold something sure reach conclusion conclusion correct chance outlier find something something many subspace fit datum really precisely task certain incomplete really lie subspace whenever subtle match missing determine fit using already know really lie exploit subspace fit subspace drop incomplete want really lie certain really lie subspace whenever subspace fit characterize problem answer motivation essentially completeness mention motivate give lie dimensional subset trivially subspace fit converse explanation immediately converse rank lie subspace dimensional generic condition completion algorithm detect fit provide check fit fit generic say already belong subspace require alone extend reason stop rank matrix provide check perform fit em answer important open complexity one see identify false circumstance datum subspace want know minimum iteratively minimal enough give subspace denote subscript index unless state correspondence keep get subscript unless run setup easy belong make collection set subset resp unless intuitively contain set convenience rather typically room confusion think row entry incomplete denote specify index room confusion equivalently I entry depend specify writing shorthand shorthand similarly example thing simplify greatly degenerate degenerate subspace iff form rank every subspace unless state example iff equivalently iff fit intuitively generic formally every fit iff notice would vector easy fit iff shorthand define say iff define iff every informally say dimensional generic formally every particular entry position span setup formally essential satisfy section unify mostly lie give assumption determine say subspace simplify generalize dimension could arbitrarily simplify easily generalize emphasize fortunately measure hold without lie precisely subspace belong hard achieve convenience simplify argument analysis working assumption notice nothing uniformly result simplicity notation fully unobserved always onto e easily generalize alternatively define say nothing row exist assumption motivation subspace fit certain determine really lie subspace non really lie fit really lie precise present lie ready advantage intuitive emphasize simple usage goal really lie subspace set incomplete characterization formalize kind dimensional subspace subspace subspace intuitive interpretation set precisely property converse satisfy guarantee generic intuitive em row one observation least example converse statement column belong none subspace determine dimensional essential uniqueness subspace iff exist notice requirement strong requirement satisfie know little case subspace guarantee subspace difference subspace essential far prove goal intersection subspace contain contain dimensional subspace clearly see satisfy iff study every hyperplane characterize orthogonal non easy orthogonal scalar say entry position zero elsewhere hyperplane characterize suppose zero orthogonal next precisely elsewhere
constant weight neighborhood necessity describe note property take strict strict necessity hold derivative constant h ie case xx consequence characterization strict prove fashion strictly fix slope interval convex suppose trivially reasoning bc leave derivative right thm thm corollary thm thm thm lemma proposition state university year function sphere multiclass clustering moreover modification un cluster variant geometrically interpret recover sufficient recovery hide convexity sphere convex optimization simplex spectral segmentation partition class base similarity important vast array recognition bioinformatics compression include well practical aspect refer eigenvector widely cluster simplicity simple spectral partition straightforward thresholding eigenvector laplacian matrix however hierarchical split use approach procedure spectral process construct look embed sometimes rescale clustered conventional spectral eigenvector matrix interpretation meaning explain relaxation cut map machine mathematic geometry space datum approach result embed justification exist analysis minimum actual output local propose second cluster point ideal perfectly separate spectral embedding use asymmetric weighted recover optimization still without modification broad overview identify sphere derive characterization describe admissible turn function hide convexity sufficient specifically sphere correspond function ica guarantee algorithm connection result think use structured weight briefly cluster spectral term theoretical discuss technical sphere correspond description necessary condition space thought unit weight description recover local thought simplex everywhere else continuously ref derivative origin strictly twice continuously differentiable local vertex maximum necessity let twice differentiable construct integer positive vertex canonical direction local geometric direct spectral work ideal vertex normalize something let embed exist f simplex recovery cluster spectral happen local maxima simplex focus cluster text arise spectral cluster denote vertex two vertex cluster set cluster consist whenever convenience vertex index index index take diagonal component cluster practice rarely consist truly typically entry row nearly diagonal simple suffice perturbation introduction simplify proceeding notation column vector euclidean vector product space angle angle domain denote projection vertex unnormalize laplacian help light importance laplacian definite equivalently consist connected index orthonormal possible choice class v invariant basis extend orthogonal simplex separating contain laplacian contain basis nj nz jx believe tp nx x j jx coincide belong ni z ti demonstrate unnormalized graph map perturbation similarity interpretation consist eigenvector low perturbation correspond perturbation row interpretation normalize version spectral cluster isolated define place generate eigenvector propose proposition applicable perturb similarity way cut vertex vertex empty minimize partition cluster cluster make plausible cut simply isolate size variant min min minimize cost alternatively way cut minimize cut see section whereas arise reference interpretation perturbation insight cluster cluster normalization arise walk interpret state eigenvector formally equivalent analysis nearly reference cluster set truly belong distinct proposition map aggregate simplex connection continue simplex vertex mutually orthogonal suffice simplex sign embed point line approach embed sphere term contrast equivalently random vertex simplex recovery turn form property recover simplex satisfy equation complete enumeration maxima maxima strict embed use clustering contain isolated construct contain scale orthogonal basis nz x spectral clustering let g complete enumeration local maxima simplify exposition orthonormal basis simplicity condition series structure induce substitution map simplex let extreme point polytope extreme strictly point translate everything make let strictly maxima simplex immediately u ti lemma u f hence correspondence relative particular relative strictly convex strict maxima symmetry enough understand behavior around take strict convexity piece upper pick slope piece slope piece piece strict convexity precise mh piece slope strict piece n strictly convex let w strict local show strict strict immediately main theorem give strict maxima besides lemma fail f sphere distinguish power cluster distinguish power spectral comes intuitively choose increase distinguish follow q asymptotically one become magnitude rapidly expect g origin gradually growth contrast distinguish power convex perturbation need new class spectral contain define either component eigenvalue hand choose u discussion maxima symmetric however origin equivalence maxima cluster vertex place local maxima belonging class maxima fu fu fu pt ascent ascent expect ascent perform unit expand f u order new find constraint imply orthogonal simplex center find maxima point second control cluster center candidate x u c pt loop run mn loop projection speed pairwise inner slow exceed discuss point factor objective make couple nice find always optimization landscape base fully deterministic algorithm however choose respect toy example point circle uniformly radius circle radius circle circle scalar multiplicative display similarity construct ij p j set convention inter similarity color color embed sphere maxima ray go occur opposite direction symmetry depict black white high similarity class display similarity class circle high similarity intra similarity two embed encode desire simplex maxima color embed local cosine please spectral cluster image spectral cluster image divide represent distinct various region result promising exploitation pixel label similarity pixel
around proposal distribution calculate ratio accept candidate code rough identify central pick sub randomly pick converge randomly pick sub sub region generate candidate rr rand rr rand strictly estimate note proportional evidence point locate simple toy two pair distribution bivariate two mode integrate entire simplicity keep toy concentrate validity assume given calculate contour fraction volume posterior locate provide reference credible region correspond cumulative toy calculation take diagonal matrix proposal probabilistic region identity toy model theoretically credible sample sampling green colour sorting mode two already sub density globally markovian detailed balance requirement fulfil concept enable generate mcmc method mode however require rough information rapid identify different form reversible jump thus odd efficiency modal global chain accept multiple obtain rough directly k mean apply leave density autocorrelation mixed method toy well separate efficacy sample pick belong credible region excellent toy toy mcmc practically investigate individually calculation toy toy mode ratio q write peak value posterior r generality high value aim valid long c big mode third furthermore keep algorithm sampling modal become markovian long explore global propose novel variant allow candidate markovian apply demonstrate increasingly range datum problem popularity ready availability advance analysis methodology set complexity dimensionality main posterior desire parameter dense grid medium high burden problem high implement method efficiently tailor explore distribution dimensional generally long sufficiently complicated inefficient sampler posterior unable isolate mode work novel deal mcmc explore detailed posterior surface retain enable communication different sampler jump require structure introduce discuss main property principle state space point already randomly specify solely previous candidate point accept therefore algorithm symmetric necessary relax number sampling sample chain trivially guarantee understand requirement I particle jump state strong requirement markovian chain proposal properly mcmc sampler become mode whole statistical bias modal thus motivate really efficiently novel modal posterior local maxima peak well global note require mode sample rough information guide absence
computationally demand sg new solving generate include perform form sample set follow relate via thing stand random standard quantity class combination stable noise symmetric sg transform symmetric sg purpose sg model sg model dispersion skewness random lemma position dispersion version processing literature design tail penalize dispersion select maximize acyclic graph criterion stable graphical contribution ie random variable parent separately network coefficient sample get therefore asymptotically generate sg stable graphical network transformation skew stable sg sg identical skeleton represent sg consider x z sg accord network structure sg term variable structure detail sg lemma initialize initialize order search db first direct acyclic graph every combination j dispersion dispersion minimum estimate dispersion constant p tb node parent tolerance co initialize ols co initialize buffer matrix regression regression vector change tolerance square briefly repeatedly solve weight achieve successive attractive rigorous guarantee several software package available least square though section manuscript stable implement numerical ignore constant term candidate estimate structure lemma shape sg initialize parent add pa pa fs parent add repeat optimum shape sg sg initialize order optimum delta score find accept swap update delta score search structure popular hill algorithms acyclic start order parent part subroutine search hill search add family least least ol explore order elementary swap score local optimum stable search numerical assess five topology simulate stable datum project microarray magnitude positive independently well standard consistently observe topology appendix assess infer performance ols show box basic estimate low estimate tendency dispersion stable describe microarray profile multivariate sg aim compare ol result tail sg quantify expression task popular method detect differentially usually profile gene assumption quantification observe quantify differential expression sg quantify within exist tailed gene ccccc gene usa china usa usa pre process eight group provide table intensity microarray quantile normalize original microarray intensity measure learn probe intensity intensity probe obtain transform ie standard technique gene expression affect center intensity assign probe decrease order stable transform select center log intensity rank stable de quantification estimate bootstrap replicate diagnostic assess heavy nature intensity plot ol network quantify gene center rank ol goodness fractional co efficient edge parent correspond negative test heavy b average fold ol empty also assess treat quantification contain sample curve ol finally discuss quantify differential sg model cross de sg change negative likelihood test training lemma guarantee equation report dispersion noise probe heat c population change coefficient sg log expression validate heavy effect learn ol recommend diagnostic assess applicability ol graphical quantify sg wide biology next sequence rna seq sg dna seq dna measurement seq seq finally mention model beyond biology particular processing instance relate region image fmri series brain network stable sg image skewed heavy financial promise lemma molecular molecular density unbounded variance generalization skew heavy phenomenon stable graphical sg represent encode major extensive lack density base learn demand theoretically tractable structure dataset five topology ol also apply stable microarray gene expression belong global group stable improve ol quantify expression gene expression model phenomena justification generalization previously bivariate sg multivariate stable density direct acyclic graph dag topology density establish criterion topology empirically sg improve parameter linear tailed noise motivation computational biology profile expression involve microarray intensity show describe intensity assume stable evidence model quantify differential microarray datum belong iii project rest stable section introduce sg challenging criterion dispersion sg symmetric density furthermore establish sg symmetric sg identical topology theoretical efficient combine implement microarray develop sg introduction model variable direct acyclic set parameter parent symbol acyclic factorization parent appropriately family normal use density primary motivation stable limit limit sum stable density implicitly specify characteristic characteristic exponent skew close analytical stable gaussian large univariate
effect due popularity parameter effect income point correspond quantify tendency edge normalize occur configuration edge pair edge make convenience observe configuration pair treat satisfied network hypergraph parameter hypergraph determine correspond e e e describe move dependent hypergraph balanced definition balanced reduce walk bipartite vertex precisely consider j nh q balanced balanced become balanced operation balanced balanced balanced observation balance observable observable balanced edge ss moves model generate move observable write graph skeleton e applicable close several need applicable balanced edge dyadic configuration return edge remove output generate coin remove pair correspond walk select move walk remove move walk output random pair choose great edge head simple subset random composition great one composition length edge head tail trivial move perform graph simple return illustrate type move move complete dyadic trivial move would stay place affect time many move return metropolis mix returning depend one research try reduce move combinatorial move applicable edge f sg return move edge walk lift analyze fashion statement state lift contain walk return step odd entirely contain similar observable edge random move combination primitive closed walk step walk connect tail choose ordering output dependent primitive walk walk share make part great edge q section take observed implement scale step choose chi statistic goodness fit make package graph produce intersection linear bad algorithm run goodness fit dataset reported burn choice chi mle distribution check explore sample undirecte depict enumeration consider direct edge undirected b b discovered start point discover discover step uniform th tv step histogram graph step sufficient purpose chi square convergence node provide distribution chi reach approximately burn collect survey vertex vertex send mobile depict individual social actor effect nan chain running return model edge indeed would show histogram chi statistic p number convergence paper specific work highly sophisticated rely source extensively fit motivated heuristic procedure goodness fit direct test depict also vertex web direct specie storage occur value significance set would reject histogram mid two year relation model study fourth social edge direct represent b perhaps seem remarkably chi walk explore broadly discover network analyze direct heuristic goodness fit direct network metropolis hasting distribution present infeasible contingency always move basis notable e dynamically hope ever utilize main motivation move combination contingency orient provide dynamic exploration rely move could move balanced edge hypergraph model algebra algebraic applicable move hypergraph situation hypergraph entire dependent derive relation hypergraph implement dynamically hope hypergraph dynamic goodness exponential acknowledgement grateful project support fellowship nsf award dms acknowledge grant air force office research advanced project nsf theorem conjecture theorem rgb rgb institute university technology significant testing theoretically justify goodness fit basis connect model entire arise algebra structure arise individual web representation amenable categorical particular bring literature access remain goodness testing count comparison observe author systematic compare structural statistic fit recently review various modeling fitting remain linear exponential goodness difficult asymptotic approximation test network realization combinatorial enumeration determine distribution enumeration goodness privacy basis sampling move random visit hasting carry argue goodness furthermore log equip unique markov basis literature two remain make challenge broadly address remainder manuscript challenge markov connect highly object unfortunately fast algorithm move arbitrary fast even hour less structural markov family end compound guarantee move connectivity move minimal basis inclusion second model goodness efficiently namely markov basis common take move connect entire handle suggest move ahead time strategy beta basic generalization os degree cast contingency table commonly marginal focus linear move sequential adjustment appear importance sis contrast exploiting allow extend sufficient necessarily marginal test statistic marginal presence methodology obtain part basis issue algebraic combinatorial network ensure connectivity sampling basis ingredient move dynamic fashion table walk way hasting implement whose desire illustrate methodology dynamically generate markov basis specifically receive link edge derive structural result remarkably ti currently fast software capable basis fit feasible traditional metropolis hasting implementation familiar network chi burn vertical denote chi dataset indicate fit histogram web value move chi square chi dependent organize dynamically mathematical develop network find section study family time quick network walk explore little move due walk minimal suggest basis hypergraph assume crucial hypergraph encode view edge suppose edge recorded hypergraph hypergraph vertex computation independence cf cell table column therefore equal multi hypergraph edge vertex vertice another edge color blue collection vertex appear red show move recall observe hypergraph degree vector equal connect hypergraph set constitute move study convenience notation cover abuse edge move connect edge need record simply add remove mention connect realization constraint presence bound zero move produce marginal arise world structural relation put restriction network begin allow edge per model clearly introduce another running walk observable occur basis realization reasonable expect pass cell usual entry statistic integer cell structural zero observable contingency hypergraph edge b appear arise connect observable literature distance contingency basis connect constraint move structural zero suffice connect observable case constraint rely move arise call element reader move never equal move minimal difficult another move
axis cs none fill sep axis cs ia accuracy ex art achieve step choose quantization insight sequential copy one symbol stream check know sum stream compare generate stream summation stream model stream stream generator symbol stream generate anti stream unique since technique compare hidden realization uniqueness anti stream problematic mis synchronization applicability show dynamic valid section distinct model identical estimating stream specify necessarily frequency behavior carry selective stream stream ignore read match selective distance source stream bring stream close font font axis style thick corner axis scale width height gray xlabel read ylabel align center yshift ylabel title eeg ylabel yshift xlabel style yshift title yshift font corner scale axis top color gray color xlabel symbol read xshift ylabel align yshift ylabel error name east xshift west xshift ylabel title sound xlabel yshift font thick corner axis top width height axis axis top color gray bottom xlabel read style xshift align yshift ylabel self name east xshift xshift ylabel title iii xlabel yshift yshift north west yshift xshift xx north west yshift north west south east xshift anchor west legend text font thick corner grid style dash gray width scale color bottom xlabel symbol xlabel style yshift style xshift ylabel ms ylabel scaled style format sep illustrate convergence eeg ii sound circuit show fig stream alphabet obtain distance length short stream font font axis top width height reverse title ia title yshift align font min meta rgb rgb rgb rgb rgb rgb rgb rgb rgb label yshift style gray scale yshift xshift width graphic figure thick rectangle green font text font align green font center shift font legend align style thin align font ii distance grid thick axis style top color gray scatter map xlabel ylabel yshift font ylabel style yshift xlabel style font ylabel style font scale format cs cs cycle legend legend open legend sep sep fill text rgb shift style font path south concept color text minimum text yshift xshift node south sep pt snp align center non child grow yshift south snp child concept text width grow south inner sep color minimum width xshift yshift south close child xshift grow yshift south align center xshift concept text white width yshift xshift concept anchor pt font yshift xshift current south eeg yshift every font thick width reverse xshift yshift align font min max rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb style style style gray scale yshift xshift xshift xshift graphics xshift yshift font legend cell axis black thin height yshift xshift align center font title iii thick axis color color white scatter color fill xlabel ylabel yshift font ylabel yshift xlabel font ylabel font scale font format view file heart yshift font cell align axis style thin title style align center font title ii grid thick axis bottom color box scatter use map draw black xlabel ylabel ylabel style yshift xlabel font ylabel scale scaled style format fix legend cell align legend entry style sep text font font font yshift xshift south root color black text yshift south text minimum yshift concept concept color minimum grow node south n child black grow child concept color width grow yshift xshift text width yshift grow node leave text text width xshift node south concept concept text grow xshift node text xshift yshift south width grow yshift xshift path south concept child color grow yshift path south concept child color text text yshift color width yshift south black grow xshift yshift path concept color south concept data anchor west fill inner fill width height pt anchor font xshift heart digital north label south yshift every style font font axis width height axis reverse false ia cluster title style xshift yshift align point meta rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb style yshift axis yshift xshift width font xshift figure axis cs thick rectangle gray font align rr title yshift align font north xshift anchor north west grid style black thin legend yshift pos outer north background gray box scatter map fill red xlabel ylabel style xshift yshift ylabel xlabel style h scale false scale false style font format legend none axis scatter mark mark white map white table p scatter scatter mark option red pt table anchor inner sep font label yshift xshift south west rr anchor north south shift font center cluster cycle child color text black grow yshift xshift yshift text font snp align child south sep text width font grow yshift xshift node align center yshift text path south distance identify projection identification star application dominate understand discriminative characterize phenomenon find characterize size subsequent learning algorithm additionally notion optimality quantify variation classic neighbor nn depend space heuristic information constraint reduction geometric space manifold infer initial heuristic make task independent universal observe quantify summation stream leave behind flat white require despite stream underlie establish causal converge distance process characteristic hide priori sample always self determine step give symbolic stream copy operation row selective check version pre specify generate stream inversion stream summation show length sufficient effectiveness quantization able produce well maintain discriminate stream stream estimate inter stream estimate circuit threshold pass sufficient stream identical prominent find causal stream induce metric zero distance lead deep insight generator similar phenomenon together operation scale linearly time stream similarity pass stream selective output test short stream ratio refer error self show independent converge four rate scale limit similarity occurrence time problem implicitly never find actually framework system ergodicity stationarity finite priori perform approximately stationarity example algebraic space particular existence unique ergodic see probabilistic occur transition guarantee similarly stream alphabet symbol rare difficult section quantization error alphabet demand imply observation limit quantization process quantization similar still identical stream test quantization synchronization stream begin stream symbolic circuit compute pairwise similarity quantization compute embedding summarize table brain electrical eeg consist eeg patient free interval potential alphabet pairwise yield clear close ec show difference rest I manifold contiguous segment brain provide picture classify heat sound digital analyze series ignore label verify supervision data rest mainly see b iv classification table reveal optical experiment supervise proceed start light curve quantization allow light marginally see projection embed nearly ask period raw yield beyond fourth visually eeg potential database object actual merely randomly subject unknown element library extent outperform knn svm database include possibly knn embed yield figure establish correctness theory process stream establishe stream correct discuss scheme quantization section specific notion often dependencie mutual stream dynamical reveal establish correctness ergodic distinct overview sake completeness denote alphabet symbol unbounde denote string empty word identity string denote stationary stochastic process moment long stationary moment formalize develop theory construction initial matter algebra string small induce stationarity ergodicity countable notational brevity string use extension equivalence string easy equivalence induce probabilistic state introduce notion probabilistic final initial mark mark probabilistic initial marked alphabet state transition symbol generation recursively extend impose state string nan state symbol similar probabilistic unlike latter lack assume remove ergodicity formalize marked induce measurable measurable immediate imply mark correspond mark generator relation extend recursively word choose imply hence denote conclude complete mark index equivalence immediately probability generator mark introduce remove dependence need transformation transformation transformation generation state symbol state associate probability state exist lead string unique canonical induce canonical construct transition induce canonical representation mark canonical sense exist set construction stay follow ergodicity state mark initial representation state see degenerate letting follow strong lemma mark induce remove element construction starting next introduce synchronization synchronization rgb rgb rgb rgb rgb scale draw black width edge bend loop leave xshift yshift xshift yshift edge bend south xshift east auto fill black text draw bend b leave xshift yshift loop xshift yshift bend leave bb b south synchronization determination symbol machine symbol current top bottom exist string notion symbolic observable symbol hide state symbolic string symbol symbolic string count particular overlap string count occurrence symbolic string generate symbolic refer symbolic derivative induce q use convergence read q imply notion canonical derivative correctness stream operation representation notion translate carry stream notion symbolic match closely generative formulation underlie sense establish metric strongly connect symbolic derivative probabilistic metric one metric almost identically immediate low immediate string replace sum noting complete algebraic material include sake completeness symbolic probabilistic state description equivalence symbol alphabet induce index explicit sequel x integer right p explicit encoding say perfect possibly realization neither encode perspective equivalence equivalence say sequel equivalent I general composition composition rgb rgb rgb rgb rgb rgb pt font fill draw text minimum text bend yshift edge bend yshift edge bend pos yshift xshift draw bend leave pos yshift edge bend yshift xshift c draw bend node yshift xshift auto font fill right bend draw edge bend leave h xshift east font fill minimum right loop leave xshift yshift b loop right xshift yshift draw bend xshift auto distance cm scale font black text bend yshift bend leave right pos yshift bend node xshift bend right pos xshift yshift draw bend right xshift yshift edge bend yshift f bend pos yshift bend pos xshift yshift bend xshift yshift bend pos xshift yshift bend leave xshift yshift edge bend leave xshift gx yshift south node fill text right pos node xshift yshift edge bend pos yshift draw bend right node c draw bend pos xshift yshift bend xshift yshift bend pos yshift draw bend left pos yshift xshift bend pos yshift edge draw bend xshift bend leave pos xshift yshift bend xshift yshift bend leave pos xshift yshift hx xshift gx east font black draw bend pos xshift bend node xshift xshift b edge draw pos yshift bend xshift yshift bend yshift edge draw bend pos bend xshift yshift bend xshift yshift draw bend xshift yshift bend xshift yshift f bend leave xshift xshift auto node cm width scale right bend leave pos xshift yshift bend xshift yshift bend right xshift yshift bend pos xshift draw bend right xshift yshift bend right pos node yshift bend pos xshift bend right node xshift yshift draw bend right node xshift yshift bend pos xshift yshift b bend leave xshift yshift draw bend pos xshift yshift font font anchor north south font yshift xshift north anchor font yshift font south font south thick dash thick hx anchor l yshift xshift summing realization anchor north yshift south node font text minimum text width edge bend draw edge draw bend east minimum draw left loop edge draw bend east cm font fill draw text black edge edge font anchor north yshift xshift south alphabet composition transformation structure underlie crucial account proposition next restrict subspace algebraic group group induce map small domain string restrict definition follow immediate precede discussion addition operation x closure existence identity inverse element x string string x px relation px px px complete proof zero element rgb rgb rgb rgb rgb auto node cm thick scale fill minimum width loop loop anchor north yshift yshift addition operation q composition probabilistic general g sum section stream main stream realization formalize pseudo copy pseudo invertible always matrix underlie realization imply generator since generator string give symbol stream generator stream symbol move read one go symbol input stream symbolic compute exact input stream infinite realization transition invertible pass minimal establish note execute state synchronization length bring structure see ref denote transition denote claim imply follow without generality state arrange bound minimal realization realization complete establishe realization generator stream realization pseudo pseudo copy distance distance copy determine copy stream symbol stream generator stream read symbol output move generator upper causal I loss canonical stream canonical mark symbol symbol symbol necessary hide output observe symbol point realization causal red thick xshift east yshift bend left bend edge bend bend bend left loop loop node draw node edge bend leave west east rgb fill blue text minimum text thick xshift east yshift yshift bend edge bend edge bend node edge bend bend loop loop draw loop draw bend leave draw west yshift xshift east node rgb fill green green minimum width yshift yshift draw bend draw leave bend bend leave draw bend draw loop loop draw g bend edge g gray text dash gray black thick black control node anchor north l south anchor xshift anchor l east construct jump cause generality alphabet possibly minimal string observe stream summation produce arc jump equivalence state whenever jump back probability back imply assume eq weight norm contradiction state none row row compute imply claim norm unity term establish bind bound establishe summation perfectly include arbitrary input deviation sum small sum conversely large proposition w g w w q arbitrary stream p h inequality corollary stream stream realization alphabet read current symbol distinct write output move position go alphabet realization let associate current stream symbol alphabet output symbol symbol imply alphabet symbol denote generation stream transition symbol arcs stream distinct stream synchronization initialization occur back distinct symbol jump symbol new stream effect next symbol jump jump occur transition conclude x note establishe conclude generator stream inversion stream stream copy symbol distinct read position follow stream operation stream copy summation proceed symbol symbol symbol involve stream copy inversion stream copy imply scale shift font anchor south cell align legend gray black corner top scale style gray height style fill xlabel size ylabel style align yshift scale format axis cs axis fine quantization pass self stream sufficiently specified rate quantifie stream obtain realization flat irrespective generate selective implie indeed generate scale next inversion produce summation observe stream state copy text copy output stream get stream I harmonic probability stream ss proof occurrence input stream upper maximize lower upon product estimator string white define partial white carry string causality stream denote hide generator string generate xx x imply complete establish causality string stream converge absolute rgb rgb rgb rgb rgb rgb rgb auto font fill text width scale text bend node b bend auto fill bend draw bend bend bend loop edge yshift bend xshift yshift bend xshift scale shift font anchor south align legend style gray font corner axis dash height axis background color bottom white xlabel xlabel ylabel ylabel yshift scale style format black axis cs complexity slow discuss scale stream due selective step occurrence stream hide follow symbolic specify possibly continuous stream symbolic accomplish symbol alphabet alphabet range quantization value belong incur small alphabet length alphabet consequence fact size stream stream inversion stream fall rapidly alphabet make since fine fig illustration eeg quantization symbol accord hidden occurrence fluctuation probability self observe stream error stream observe stream average compute tt tc choose state property maximum scheme symbol data entropy scheme alphabet slice slice approximately c slice contain entropy property mean alphabet fine discrimination self fig ratio useful quantization minimize imply font ts axis style thin false gray xshift xlabel axis cs ts xshift ts east thick style thin false width height pos gray background xshift style yshift outer axis style outer style xlabel axis cs ts xshift ts east black height pos gray fill style xshift yshift extra xlabel axis cs axis cs ts south west yshift cell align legend font corner style dash gray axis height axis black xlabel alphabet style ylabel ylabel align yshift xlabel style yshift format title title yshift east anchor legend align legend style gray font axis thick corner scale axis height grid style fill alphabet ylabel ratio self discrimination ylabel style align xlabel yshift scale format format title style yshift cs coarse coarse alphabet produce error identical provide stream self test two stream contain notion theoretic investigate extensively stream random mutual quantifie amount information variable auto node font fill draw black black text node red east font stream align font stream xshift yshift east align north font yshift xshift south streams anchor north font yshift south dash east west dash east pt auto font draw minimum width align center edge path east font stream yshift east east fill font xshift yshift align anchor north font yshift xshift stream north font yshift thick east dash thick east initial stream show stream near zero mutual similar generator significant formally py single vector mass mutual px py notion variable amount information variable mutual precisely know stream generate px px py sharing information conversely high sort synchronization orthogonal stream nearly anti generate stream copy stream inverse require fall see simple stream twice randomly generate symbol accordance probability imply stream stream nearly stream stream correctly stream generator significantly c mutual inf ex color style font height axis scale true font format name plot horizontal sep vertical title ylabel ylabel style yshift xlabel table figure dr stochastic value font label align center difference mean vs yshift axis south east xlabel ratio axis background thick gray white group name vertical ylabel xlabel min meta rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb label style yshift style gray yshift width font graphic figure graphic figure dr graphics figure dr north align yshift style yshift width height legend pos south east xlabel axis style thick color bottom color plot horizontal sep edge leave ylabel ylabel xlabel rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb style label style yshift gray width font graphic graphic figure graphics dr std label align vs difference vs variance normalize yshift south title yshift height scale axis pos south east xlabel axis background color style plot sep bottom ylabel ylabel yshift xlabel point min rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb yshift thick axis yshift xshift font graphics dr graphics dr graphics figure stream generate stream zero nearly finitely stream illustrate stream provide stream conceptually distance information stream easy tool dynamical meaningful measure consider deterministic widely primarily model accept trace see stochastic development assume ergodicity stationarity consideration fall apart x x consider parameterize reaction simulate system every dynamic change fig reaction reaction reaction combinatorial current population reaction probability terminate partially length imply remove restriction strictly value count might behaved parameterization third reaction index yshift east title yshift scale legend pos south east xlabel axis background thick color gray white ylabel ylabel yshift xlabel rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb style style width font gray title xlabel reverse pos graphic figure rectangle axis cs axis cs cs axis axis cs axis axis cs cs axis cs rectangle axis cs cs yshift center yshift center xshift yshift xshift center gray xshift yshift xshift simulation carry system illustrate perfect reaction death expect degradation probable expect large however truly oppose simply monotonic parameterize dynamical series generate series update specie map sequential datum stream positive symbol symbolic stream pairwise compute sum copy flat ai entry difference measure series iii notably ai trivial dependent stream normalization normalization bi normalize little ci fig change via rich would surprising complexity minimum simulation cluster identify perfectly monotonic meaningful categorization tool discover obviously function precede tool introduce similarity sequential model training expert tune consequence may specific tune strength lie ability heuristic big challenge cc underlie physics notion set identical look heuristic design expert classify star determination state might require fourier activity quantify source stream priori space symbolic always stream anti degree anti mutually application detection anomalous object principle exceed specialize hard quantitative stream anti stream tie stream stream reveal
accumulate calculate way either dynamic increase al regret continue play reward world understand biological principle underlie physics work partially advanced institute integrate national life institute technology couple sophisticated capability find stochastic fluctuation body validate statistical exhibit principle modern digital phenomena device band material noise law logical response complicated circuit require relatively logic effort division costly consumption resource look biological system couple physics physics analog paradigm constraint body decision making suppose example coin certain player give reward probability respectively play trial reward repeat armed bandit highest accurately past phenomena application diverse field communication cognitive monte game application originally describe essence thompson index class distribution class computing become et machine system dynamic inspire dynamic resource collect environmental expand shrink terminal inspire algorithm decision physics physical volume body entail among volume increment volume part show derive volume law well greedy modify good device physical physical law optical energy transfer dot dynamic kt body bar slot represent make player coin accumulate initial play playing weighting parameter evolve particularly machine play high left simplicity overlap area fig area incorrect expression must satisfy large easily confirm obtain therefore conclude rule popular update play probability player though machine bottom row table play machine note update play give follow reward transform expect reward coefficient term eq satisfied weighting estimate simultaneously dynamic machine feature rule derive reward advance simulation verify exhibit derive accurately dynamic essence generalize separate follow reward fact
training experiment change run concept validation model rigorous ease initialize encoder maxout network dropout regularization hmm network purely wise per frame hmm decoder development propose rnn remove transformation softmax acoustic use rnn rnn encoder initialize recurrent ease match input frame c basic independent adapt gmm search maxout acoustic end rnn search rnn rule mechanism save batch divide sort form batch minibatch network frame zero rnn end network mlp receive input frame hide layer adapt rescale gradient pass update procedure use stage effect begin threshold average correctly report preliminary require produce sequence process specialized extension machine aspect rnn rnn variant encoder decoder explicit alignment state alignment unlike generation generate handwritten character next input look sequence help speech subsequent similar frame relation search match probable however without time decode search width even narrow significance vocabulary speech decode stream put another directly search probable hmm acknowledgment like hill acknowledge cifar development university universit de universit de hide traditionally speech bi directional recurrent encoder recurrent stream alignment attention symbol subset demonstrate achieve comparable hmm challenging sequence acoustic short sequence symbol two length know building classifier predict signal recognize investigate work recurrent train alignment decoding model keep position attention decoding model input select frame summarize symbol achieve error rate comparable dnn error rnns however greedy search tune month work achieve couple hmm hybrid system act acoustic predict input frame target frame way minimize frame classifier whole acoustic hmm tune jointly sentence minimize decode hybrid progress adopt fully feedforward recurrent hybrid hmm require control contribution decode hybrid mixture generate force alignment follow train acoustic network estimate furthermore improvement necessarily translate minimize optimize path alignment base symbol sequence merge consecutive identical output symbol backward sequence prediction extension rnn acoustic compose part sequence symbol symbol probability sequence rnn token originally perceptron mlp combine performance dataset rnn generate attention text step correspond currently word compute match location sum interpret align translation assign high location allow long recurrent translation recognition soft narrow encourage mode attention move nearby near input give current relative position successive speed decode generate raw also annotation generate sequence sequence element illustration present rl output condition annotation conditional decoder annotation context vector recurrent layer gate keep track dependency prediction realize softmax context annotation mean select annotation selection score decoder annotation base mlp output mlp sigmoid output attention search input frame match decoder repeat annotation location prevent location important show encode preference input frame location automatically learn prefer future concentrated like constrain advance network monotonic add cost input already emission normalize monotonically increase alignment add sum penalty show dot along intuitively encourage select nearby input respect previously effect alignment consecutive without hyper coefficient work optimize encoder implement
handle class novel statistical relational output efficiency max procedure propose expect constructive applied overview structure output little hybrid generalization hybrid markov logic operator namely constraint hoc translation form amenable rely solver conversely design integrate specific solver expect efficiently logical g deal assumption make expressive arithmetic formalism restriction whether logical boolean logical assignment theory predicate otherwise fundamental domain reason operation include e g arithmetic bit efficient find underlie theory solver solver develop max requires maximize formulae formulae formulae expressive package specialized solver intel circuit biology counterpart goal solver problem structure generalize max output compatibility joint learn compatible maximization output train negative quantifie correct margin rescale training error complexity eq wrong output play quadratic cut plane infeasible issue constraint violate guarantee find qp iteration original rhs appropriate loss solver implement cp formulae include definition formula refine formulae appear formula value separation max solver cost function boolean numerical formulae clause issue since clause towards satisfy practical impact involve constraint environment robot planning modeling gene two flexibility expressive power formal publication restriction activity sensor instant activity task tv sensor home include video audio etc cast discrete mean world activity inter case factorial use training intractable cast scheduling logic event predicate straightforwardly translate express activity specify likely duration activity occur hour involve interact constraint scheduling activity soft interesting application consider customer planning build house company price design minimum requirement boolean rate distance public service quality distances max develop prototype interactive various customer encourage constructive could learn viewpoint interpret generate first logic language solver constructive implement arithmetic paper novel class rely language allow describe boolean output solver optimization perform max constructive method reason relational characterize soft first logic encode weight logical formulae maximum posteriori assignment predicate formulae play role maximum problem solver issue logic suit hybrid require predicate make I
scheme binary hash hamming implement tree structure neighbor hamming point near computation extremely order meet locality sensitive requirement function satisfy normalize un kernel normalization valid family hamming distance histogram hash rounding sample kernel representation commonly kernel gaussian vector rkh key computation accomplish solely consider realization covariance vector practice covariance must show center change hash evaluation limit question rkhs space base non eigenvalue imply could time performance solid resolve issue section simple powerful allow aforementioned issue dimensional may precisely lsh gaussian vector lsh project deep hash infinite truncation view inner ti jt I direction approximation come know truncation view product two project word compute lsh use point estimate estimate covariance center operation lsh project equivalent lsh central approximately draw project whereas lsh know obtain good algorithm explain avoid technical issue could retrieval performance arguably popular nystr om rank original computing similarity nystr om point briefly difference nystr om method kernel eq whose eigenvalue column representation diagonal column center format hash even though nystr approximate whole center especially empirically issue section present theoretical performance lsh perhaps importantly improve performance present lsh make truly refine approximate via central possible incorporate practice provide gaussian first formulate similarity refer want literature semi implicitly associate feature infinite inner eigenvalue explicit span empirically become normalize estimator fit lsh normalize eq cause instability issue optimal proof principal choice query dominate bind utilize bind ingredient km go get eq near neighbor retrieve observe always retrieval empirically namely lsh thousand expect result good choice lsh subspace may achieve project obtain component replace good lsh hash sensitive decay property rkh affect retrieval transformation explore well exponential eq ranking stay matter change decay entirely slowly reduce eigen carefully validate comparison nystr method report commonly use comparison million randomly represent million sift descriptor comprise size descriptor whose kernel bit largely make choice number kernel namely histogram exhaustive near evaluate proportion rank position indicate retrieve verify original represent measure direct neighbor performance hashing scheme semi improve variant comparison restriction ccc b rank nystr decay property good ccc color transformation sift show fix small confirm lsh performance however entire tradeoff discuss initially improve use decrease drop show sensitivity choice addition kernel examine nonetheless still recall sift affect trade different sift used make recommendation nystr om moreover obvious tradeoff rank observation indeed nystr om regard nystr full choose transformation increasingly monotonic change scale affect decay continue decrease decay speed decay drop usefulness largely function kernel sift original room summary absolute combine retrieval regard power technique interpretation locality sensitive hashing conceptual provide apply appropriately project first suggest boost large show choice monotone performance present piece least bind product complement principal map define I sample draw decrease space counterpart probability state lsh admit hash randomize use query time dominate computation ready lsh relate lsh k reduce decompose right apply locality hashing vision near neighbor reproduce hilbert perspective step project practical benefit problematic conceptual motivation formal performance bound reveal boost several benchmark standard similarity database play application object scale fast
mcmc nonlinear model proposal analyze tackle inverse posterior wherein log approximate similarly gaussians facilitate construction mean directly approximate combine inform forward covariance approximation global restrict treat complementary move infinite informed subspace use dimension independent work interest truncate collapse mean avoid error require smoothness decay entirely influence forward observational constructs model sampling capture description notion fundamentally reduction technique organized posterior matrix examine interpretation covariance characterize approximation conclude remark paper along technical consider loss generality q infer forward observation statistical likelihood fisher coincide mode note posterior matrix depend matrix prior similarly kk covariance prior low reason describe along prior might decay spectrum lie approximate set semi rank positive covariance advantage structure optimality definite matrix canonical invariant metric cone give f invariance moreover treat loss function kullback leibler distance aforementione addition differently class function element value tend loss distance find appendix optimality approximation familiar hellinger kullback minimize iff minimize hellinger distribution depend optimality hold state posterior covariance eigenvector correspond unique eigenvalue computing criterion minimum generalize subspace span generalized direction parameter hermitian eigenvalue result eigenvector accord analogous transformation notice involve pde readily direct eigenvalue reference implementation available rich literature g parallelism method factorization hermitian product induce maintain efficient implementation solver accurately refer straightforward efficiently discuss introduce variance covariance inform strategy use norm function approximation develop hessian alone draw kalman approximate approximate optimal approximation explicitly posterior following characterize direction minimizer minimizer precisely inverse generalized appendix define define minimize span j along value informative first span eigenvectors informative hence direction along furthermore direction maximize direction minimize maximize corollary simple theorem linear dimensionality problem particularly effective posterior parameter constrain locally inform subject inverse notice posterior project exact posterior square far frobenius matrix definite cone matrix matrix minimize frobenius direction large eigenvalue different particular solution eigenvalue maximize metric identifie maximize distance direction maximize prefer former direction frobenius entirely naturally perspective eigenvalue finally statement approximation use approximation numerically approximation natural limit closely effort dimensionality reduction effort discard remain discuss update datum negative likelihood decomposition eigenvector good rank belong original different base space prior take mean pair u actual consist forward suboptimal general project update inversion summarize hessian technique method take importance quantity account key approximation theorem illustrate condition interaction problem forecast gaussian observational bayesian computationally way solve system bfgs typically initial matrix scale bfgs positive convergent storage bfgs bfgs bfgs posterior use bfgs approach bfgs result rank distribution fast accelerate repeat inversion set propose efficient realization already accomplish art iterative solver function seek approximation consider class relatively single linear precision datum approach justify posterior mean instance thus offline strategy costly offline fast evaluation approximation optimal cf additional posterior inverse formula efficient bayesian mode equal risk error establish basic notation let loss incur estimator bayes distribution function mahalanobis account geometry approximation direction approximation posterior approximation repeatedly multiple realization eq consist statistic replace update shall henceforth either bayes matrix bayes approximation particularly analytical exploit proof develop independently eigenvector normalization bayes risk rank define realization posterior bayes risk rank approximation decreasing easily bound analogously interpret truncated bayesian readily triplet rank compute bayes risk optimal approximate realization dominate prior need approximation theorem yield precisely stochastic describe gaussian linearization forward operator bayesian result support algorithmic precise statement worth risk dimension define approximation negative low associate bayes include power great approximation accurate show estimator counterpart fewer low sense subsection mean ill inverse stop statistical say iterative equipped stop use result justify observe iterate denote highlight aa observation heart iteration algebraic I nearly generalize usually satisfactory informative assume beyond convenient stop similarity minimizer may perform goal quite concerned ill whereas statistically approximation mean framework mean adjoint approximation I input forward adjoint numerical precede continue inverse investigate negative semidefinite theorem hessian bfgs reduction scheme difference explore shall refer hessian think denoise ratio variance canonical great alone inform direction reduction hand variance direction alone determine reduction effective spectra important direction depend one distribution generalize full prescribe spectra case orthogonal lead decomposition schmidt standard discuss experiment run bfgs optimizer x bfgs covariance bfgs optimizer bfgs bfgs bfgs paper store bfgs optimizer converge optimum taking result numerical prescribed move increasingly informative distance realization row realization fix obtain bfgs flat direction variance bottom row move third increase quickly combine great successfully whereas remarkably configuration spectrum generalize situation spectrum middle flat direction parameter great almost towards decay restrict dominate hessian however generalize reduction prior spectrum decay slowly reduction either quickly decay bfgs theoretical htp value eigenvalue leave f versus realization bfgs difference f green update middle classical ray intensity detector object present synthetic ray enter instance www enter ray system move reconstruction image detector location cross basis carry model ray intensity detector object discretize grid grid cell integral cell vector length line though logarithm inversion vector iid setup rectangular discretize circular ray measure detector opposite evenly angle ct exponential fashion computing intersection circular gaussian discretized length computation well nontrivial define discretized pde white process identity length control prior compute first negative update first formally approximation seed due match posterior course show htp htp assess low shall yy datum reference top error rank low include confirm compare figure right panel snapshot correspond mean relative time require approximation regard realization divide posterior computing iterative scale inverse solver computational cost roughly report realization obtain roughly relative time approximation could take cost forward sparse different heat exclude iterative solver converge popular mean efficiently angle application cpu cpu compute iterative apply much leave unnormalized approximation realization distribution dependent bayes theorem decay two consistently nonzero offset datum draw figure approximation configuration wherein detector spread around entire object informative red angle configuration slowly angle configuration loss posterior update great rank limit approximation informative relative limited angle relative drop realistic ray extremely flexible device configuration normalize number cpu divide black green red panel htp eigenvalue angle ray center lead blue angle htp angle ray uniformly linear inverse initial heat equation heat heat initial linear heat pointing unit spaced dot figure infer space pointwise observation function evolve show discretized example non field truth gaussian dependent panel numerical trend figure encounter confirm good low rank perfect visually curve somewhat occur approximation eigenvalue relative snapshot example visually indistinguishable therefore accurate solver section cpu apply adjoint model heat pde versus apply matrix also apply negligible example cpu figure illustrate important characterize heat notice eigenvector sensor show direction direction update capture direction great mode concentrate around sensor great htp htp condition inversion heat htp cf fourth four direction space typical large relative prior approximation structure covariance negative semidefinite form broad loss function symmetric definite argue metric identify direction space posterior variance update optimality optimality hellinger kullback divergence develop approximation realization approximation minimize error computationally efficient low numerically variety ray observation condition localize observation prior already natural endowed understanding function current operator allow generalize possible research approximation technique assimilation linearize assimilation present evolution introduce tailor work support department energy
chain account introduce nonnegative symmetric preserve condition approximation chain satisfie initially enter part circle j c basis conclude statement statement fact work construct inverse multiplication represent induction let theorem proposition example remark fundamental graph theory access entry take work depth machine field randomness addition random learn sequential markov monte mcmc algorithmic unless significant require reliable design scalable sampling challenge characterize characterization formally performance sampling subsequent sample parallel generate subsequent sample total random obtain gibbs method randomized process gauss apparent important multivariate model vector analogy gauss non underlie precision perform significantly worst partly mathematical et al generalize dominant scale converge correct partition processor sequential gibbs sample old processor lead development nearly solver dominant parallel bad logarithmic parallel scalable extend solve graphical focus parallel field natural arise likely gain diagonal precision matrix worst generate sample random field newton expensive sample covariance framework find square root return efficient parallel implementation depth nearly step framework concentrate nearly depth construct analog differ generate generate random univariate sample approach main factorization preprocesse randomness generate motivate develop spectral graph theory give algorithm construct factor om tight matrix essentially ratio entry always precision need remove importance run depth parallel processor prior could symmetric dominant factorization directly field precision toward linear matrix wide matrix structural matrix relate diagonal matrix representation believe structural barrier random field parallel graphical connection spectral graph field large notation paper spectral radius large definite definite definite matrix definite equivalence graph weakly matrix rigorously carry matrix entry ensure ic guarantee multiplicative rescale factorize combinatorial factorization factor satisfy satisfie moreover depth mn adapt main construct break product operator individually reduce compute count couple multiplication algorithm major issue use error handle error e compose alternate decomposition factorization base half term naturally incorporate factorization lead issue half power idea expansion polynomial power eigenvalue resolve front introduction one dense key correspond walk average well existence technical theorem without specify construction approximation total com nearly linear subsequent improvement near propagation chain follow scalable construct inverse length expansion degree finally procedure multiplication operation complexity extend nearly dependency refinement order obtain polynomial length expansion expansion moreover approximation polynomial preserve start degree provide chain substitute expansion equation preserve
vector regardless appear compositional successfully series compositional distributional one compositional model description experimental present discuss prior learn capable model complex relationship input nlp capture sentence representation word generic form composition apply concatenation vector word activation compositional representation fashion sentence merge serve semantic intermediate instead feed reconstruct encode hide optimization reconstruction unsupervise fashion deep input word representation token merge evaluate represent word compositional network probable general methodology induction discover latent occurrence calculate vector create vector agglomerative sensible word meaning mean black corner mm width height black corner cm draw cm height fill minimum size sep em text text em I mirror mirror amplitude mirror black black round black right similarity et al phrase task similarity compute match human sentence subject object sentence dataset every noun l comprise construct ambiguity play specific aspect dataset constitute evaluation term neural composition implement additive model sentence take wise evaluation conduct way composite sentence apply specifically sentence similar use procedure classifier report fold validation list ht word ht word multiplicative c multiplicative multiplicative suggest bring learn compositional l carry subsequent composition encouraging choose imply generic act processing outcome word never decrease clear positive algebraic despite benefit processing work suggest compositional constitute certain although return approach report study try interpret sentence phrase length deep model deal text deal compositional meaning word benefit explicit architecture compositional ambiguity align concept result improvement compositional google neural compositional hope semantic produce fed compositional net evaluation deep
profile distribution elliptical structural illustrative employ model propose distribution profile look fractional variation image model estimator compare estimator intensive size three fold correct cox wishart correct estimator quantify list assessment monte experiment analyze structured complex wishart discuss cox correct mathematically subsequently compare exist literature summarize coherent entry transmission follow detailed complex multivariate essentially commonly look hermitian scale wishart complex complex indicate scale wishart distribution maximize associate wishart stack operator show order ordinary function close nevertheless account adequate application filter guarantee technique ml suffer convenient estimator look propose datum discard derivation address datum describe obtain improve ml cox base profile method refer density play ml adopt accordingly derivative direct information additionally mathematical cox bias possess correct ml q possess asymptotic modify technique two vector nuisance address often mathematically approximate likelihood problematic associate bias therefore profile bias issue modification adopt modification tractable wishart approximation likelihood nuisance aa cox profile improvement ml scale wishart cox correct profile mainly goal derive correct focus cox ml eq need simplify replace ml follow discuss technique scale quantity obtain replace yield column zero obtain moreover estimator consequence quantity mathematical derivation associate satisfy word linear resort newton look study ml trace cox three assess measure error cv iii subsequently separate indicated simulation suggest monte employ follow size look iii respective element adopt square window respectively affect estimate employ mean table notice estimator offer derive size mse cv also show present term mse cv evidence bias estimator yield procedure estimator correct variance value lead present size look decision consistently exhibit reduce bias increase look affected well looks propose identify estimator among technique c em em lr c lr em lr lr lr lr em em lr lr lr lr lr em htbp separate operate band nominal look present area visual inspection ii htb replacement display obtain small mse thus yield tend less efficient indeed report complex verify r lr lr region c lr lr em lr lr lr lr em lr lr show curve curve usual ml homogeneous htb notice table discuss possess improve scaled emphasis look advanced cox mean assess adopt figure coefficient mean result correct superior appendix wishart ii order bias accord cox third kronecker product expression yield q formula indeed expression subsequent discussion detail likelihood particular address subsection result scale complex wishart equation profile log yield approximation log q simplify hold sc statistic department adjoint interest matrix sm receive electrical engineering sc mathematics de de
threshold ordinal soon pass threshold implicitly main rbm ordinal specifically form rbm offer impose random user profile easy posterior ordinal observation behave rbms never level pose gaussian another consequence rbms e see persistent ordinal rbms model correlate ordinal typically inherently influence user specific user item specific rbm architecture long map visible profile world art filter public rbm ordinal ordinal presents ordinal rbms discuss relate follow ordinal ease homogeneous draw ordinal solely hide k machine rbms binary observation example fig bipartite potential utility binary form rbm translate deviation free introduce gaussian rbms whenever category automatically define low generative estimate cumulative restrict spaced free nice thing like rbms factor posterior conditionally since ordinal observation n exploit bipartite rbm run gibbs parallel finally posterior ph kn nh sn lead update recursion utility ph ordinal unseen integration intractable datum rewrite make either ik replace value define ik associated parameter standard sufficient ess truncate put rbm phase ess ess r learn persistent effective strategy update chain integrate factor advantage bipartite stability average phase maintain per store phase short need per chain discard chain chain maintain chain maintain otherwise instance collect gradient ascent threshold utility domain r read lower le use derivative ordinal scale c c il homogeneous threshold assumption example collaborative filter user role influence popularity item switch user item I instance column hide row binary hide incomplete denote incidence set factor di visible ordinal variate model product potential specifically g define except threshold datum e condition posterior factor likewise give di di di di column still rbm item although explore chain resort structure modularity alternate process estimate item posterior pg di di item mean posterior th ph di likelihood th improve likelihood e estimation posterior reduce trajectory posterior simple utility decay previous empirically learn progress treat example efficient rejection bias zero threshold space evenly ph ph shorthand well view discover profile people life social political country wide centre people processing day economic country improve remain heterogeneous question hide unit fit obtain ph ph ph representation project vector plane locality preserve reveal us china see world three public million nearly million rating nearly netflix rating user nearly rating scale star uniformity remove rating use criterion netflix ensure rating comparison mf mf cumulative ordinal assumption without item item neighbourhood rbm multinomial assumption protocol stop report rmse mae mf rbms map mae depict base clearly treatment performance dataset matrix rmse competitive mae ccc monitoring rmse c rbm rbm gaussian ordinal multinomial rbm rbm rbm partly research rbms variety rbms continuous ordinal work extend rbms input ordinal rbm treatment multinomial less offer generative mechanism ordinal ordinal datum well science especially quantitative refer
essentially perturbation sake without give formulation three assume svm issue task relate directly adapt extract compare adaptation svm sample example adaptation optimization integrate accounting allow space go less constrain impose interpret layer straightforward target domain compare available target domain minimize train individual experimental sect confirm employ quasi newton derivative base efficiently svm general structured ground sample run output accordingly correspondingly integrate particular must give apply part focus learn set decide latent latent formulation concave write variable object alignment base appearance part return play truth output concave apply write moment adapt testing time learn adaptation classifier adapt domain adapt source window divide resolution regard low high tend discriminative virtual world www evaluate pose consider evaluate virtual world domain world pool domain b multi detector supervise adaptation roughly available training respectively evaluation average rate vs curve setting time repetition ensure performance baseline challenge part challenge hold conjunction neither report describe da use image note criterion vs avoid classifier virtual mention pixel hierarchical extra pyramid pyramid two pyramid pixel pyramid pixel feature pyramid illustrate training resolution height bound virtual real real challenging apply pyramid finally combine c moderate easy da lp see accuracy single datum achieve train importance leverage hierarchy set additionally assess full multi resolution show demonstrate adaptation explain training learn multiple domain multi provide good quantitative capable detect outperform remarkable mention adapt virtual take minute approximately ghz full need number virtual building virtual build available knowledge adapt detector challenge complete challenge new worse quite clearly rest adapt experiment time actually final measure image pca histogram bin baseline summarize svm target domain example source rest domain optimization perform follow domain adaptation fig l avg tree avg dc c w adaptation cd avg adaptation locate amazon amazon accuracy target split worth totally one setting clear importance style method domain domain domain per domain example adaptation style w perform analogously ad ac potential agreement hypothesis underlie hierarchical domain focus see three hierarchy improve adaptation tree show adaptation good two layer well previous study similarity quantification shift show similarity measurement yield good capture underlie w adaptation tree agreement label priori sect mix remove recent discovery latent call discover b b qualitative category domain discover pr hierarchy indicate correspond classifier configuration operate domain label discovery point domain category unlabele target mix pr predict category category domain discover source process domain want compare discover sub vs distribute predefine sub domain discover fair sect c domain discovery give layer pr pr layer pr pr among possible pr accurate predict pr see pr difference pr pr pr equally sect target must perform domain sect pr discard category require label datum reference sect three latter good configuration result comparable sect reader convenience discover domain outperform pr discovery accuracy see due train domain hierarchy error purpose object example present domain adaptation sub domain adaptation key target increase structural classifier adaptive exist sample adaptation apply detection recognition application involve imply category show effective accuracy ignore focus recognition target incorporate aware sa attribute support xu grant grant rgb computer vision es topic loss produce recognize vision classification object hierarchical idea exploit difference svm adaptive domain together source perform adaptation proposal detection category apply classifier show adaptation ignore structure incorporate discovery object recognition classification increasingly problem make testing challenge variety method increasingly machine computer image usually adaptation target domain domain adaptation allow adaptation consider intermediate domain adaptation adaptation single domain multiple propose label adaptation cover much case leverage multiple target domain criterion build domain implication adaptation main illustrate hyperplane traditionally option pool target single one strategy single domain adaptation perform propose target difference multiple hierarchical tree adapt source jointly sub approach b difference leaf hierarchy hierarchy imply adaptation strategy domain worth reduce domain divide target would ignore reduce time fact require svm svm term method vision field category wide category case increase detection ignore target focus recognition target domain discover organize discover domain experimental result detection recognition respectively despite propose decade comprehensive vision broad base attempt matrix classifier transform jointly learn discriminative classifier concentrate adapt parameter often base projective transfer svm variant require
dd dd conclusion follow many distortion weighting weighting factor infinite q issue address rescale rescale weighting tend distortion prop implicitly one time causal finally concentrate measure retain associate eq contrast causal state retain function q maximize whereas replace reverse causal state since prop distortion measure I information satisfy prop distortion measure weight average distortion reverse causal state notation mathematically sum integral vanish distortion divide treatment instead consider retain trivially proof prop time reverse state infinity ref reverse causal state limit counterpart recover lemma prop service clarity shorthand leave expand measure theoretic elsewhere solution objective minimize coding predictive predictive multiplier lagrange lagrange multiplier code predictive multiplier main r first eq combine straightforward calculate thus finally calculate eqs divide replace since start abuse notation somewhat justify bayes r rewrite allow slight enforce normalization constraint mean partition meanwhile formal codebook map codebook codebook iterate codebook realization practically careful variety converge cm every edge black state line width font loop loop style loop loop definition color rate distortion process suffer curse retain resource grow exponentially process dependency algorithm fail dramatically underlying correlation memory curse distortion objective term mechanic demonstrate causal distortion substantially rate analysis keyword optimal filter mechanic predictive bottleneck device predict either guide strategy adapt environmental ultimately due resource limitation coarse partition store bit store grow perhaps infinite storing exceeds predict resource shannon introduce analyze trade encode rate without prediction distortion theory principle calculate achievable distortion give test whether identify useful feature understand natural current calculate distortion arbitrarily long avoid avoid class future one retain length yield distortion long address generally predictive algorithm distortion maximally identify alternative demonstrate maximally dramatically improve distortion associate discover hierarchy mechanic distortion introduce distortion forward reverse state illustrate summarize application entropy fall capacity shannon coding say exist encode regime introduce distortion system capacity free shannon certainly never experimental guarantee quantum relation capture organization many adequate capacity said positively reduce memory act focused determine reproduce extent future adaptive simple require coarse collective root identify environmental human identify variant theory state review finally ref mechanic ref main generate behavior x contiguous exclusive x past tt well indirect internal mechanism forward causal state sufficient group equivalence relation shorthand denote forward generative build causal give start since output transition effect difference index alphabet hmms mathematical ref note process finite causal constraint next know next symbol similar minimal group together shannon reverse causal statistical predict process htp aspect future function forward causal reverse state form relationship symbol generate identity entropy residual employ amount information vice versa store shannon various information quantity function importantly algebra information random atom pose capture correlation capture hide coarse grain block process dominate vanish excess causal complexity hmms finite process past parametrize equivalence length quality convergence mutual chain rule information causal review distortion theory refer ref detailed serve theory combine noisy channel shannon channel encoder biological signal natural information processing post htp past imply chain source symbol encoder require source evaluate distortion quantify long period lead distortion codebook code distortion measure code rate distortion desire require achieve minimal distortion possible process successive semi series distortion forward source variable information source output capacity codebook decoder reconstruct input capacity enough irreducible rate regime positive regime ask equation view realization codebook solve numerically minimize one trace precede measurement sec denote symbol arise distortion use less familiar square like recently though definition retain aspect variable assume markov distortion q distribution straightforward chain since finding maximize ib since decide information realization relevant markov chain measure rate distortion determine limit calculate maximize constitute refer distortion measure confusion ib explicitly analysis vary coarse identify implement zero causal maximal bottom illustrate predictive soft predict fidelity indicate relevant thm b length retain information length limit capture ec nm though store storing limitation current thereby restrict distortion topological practically limited notably practice account measure sequence probability gaussian function call calibration state ref uncertainty mix associate h hmm take use ref information whose correspondingly slowly capture figure contain pair turn translate ref consequence simple notable compute sec challenge even become extreme transition temporal correlation correctly calculate curse dimensionality critical concern generate hmms away prototype biology finance quantitative highly markovian rather gap process countable infinity state ref heavily short likely fail curse require structural reverse proposition result lead distortion information equivalent theorem intuitive appendix sec illustrate partly partly analytical insight inspire argue temperature lead rate distortion set state retain somewhat markov fig imply distortion function say equivalent certain type measure htp bottleneck notation fig distortion measure distortion though instance distortion measure actually temperature great ask arbitrary causal long statistic temperature limit forward ref state without distortion reasonable distortion measure difference apply estimate incorporate entirely apply replace future sequence sec certain store unnecessary see process almost finite well concerned reasonably function causal state unnecessary htp hmm presentation process htp intuition side process hmm approximated feature approximate replace infinite discover single feature nonetheless imply infinite classic aware latter ref instead time past b ref process transition second phase transition critical causal state discover codebook change break key phase transition detail qualitative straight line obtain code fig bottom show feature indicate transition incorrectly suggest phase ref curvature function scale information scale reason exhibit straight recurrent forward state odd last causal fig ref whether odd one successive know uniquely causal versa causal reverse state conclusion directly machine result argument periodic ref noisy periodic process periodic relationship prediction key feature fig order periodic cyclic process causal nothing iterate attempt maximize however recall c hmm maximize forward causal state optimal sum code first critical temperature htp state even memory increase reverse feature block transition new process causal form g generate entropy many irreducible word restriction easily block forward reverse state odd causal give one must restricted state map add may sharp transition ref suggest contrary variable diagonal gaussian would curve see curve expect difference matrix discriminate phase transition sharp transition curve noiseless limit often bottom reverse htp reverse reverse bottom forward legend solid line circle color various htp describe change prototype rip therefore know joint forward causal reverse state statistical statistical rip excess bit invariant state investigate reverse causal function put disadvantage function greatly rip reveal correct either forward reverse time feature curve reveal regime reverse feature show phase transition reverse add codebook state reverse causal state rip fig ref causal fig phase end curve remain forward codebook reverse look sharp reverse time codebook reverse state codebook phase emphasis select prototype apply gain insight symbolic analytically symbolic symbolic markov know describe htp describe elsewhere yield show coarse new function typically low top bottom reveal representation htp black guide state state identify mixture identify original identify underlie feature inverse temperature fig state compress phase transition intuitive move add forward time causal finally implication hierarchy hierarchy equivalent predict probability fig say calculate estimate calculate reverse distortion viewpoint step recall refer machine answer polynomial np hard problem relate ise infer np complexity hardness infer exponential np seem though complicated common length retain basic implicitly sequence suboptimal sec without leave histogram rather build also deviation predictive causal identify forward suggest study series long correlation calculate derive infer exponentially long temporal sequence need curse head unnecessary adequate effective underlie process either use order produce inaccurate markov dominate well interpretation htp estimating curse word severe instead derive predictive accurately alternate directly mind predictive distortion quantifying curse sec new highlight side strategy analyze event predictive extraction ref dimensionality rule yield causal benchmark performance predictive infinite markov predictive state deep useful grained aspect appear machine switching determine function perhaps importantly machine state
low matrix technique hierarchical decomposition exist calculation force associate analytical consideration rank hierarchical construct dense scheme direct determinant symmetric etc exist low modification modification cholesky I entire direct reader paper linearly yield investigation worth point cholesky factorization semi scale symmetric factorization update identity extend product low detail compatibility matrix section factorization summarize discuss extension hierarchical incorporate perturbation hierarchical briefly know identity allow rapid inversion determinant update matrix low calculated formula simplify perturbation identity perturbation identity furthermore rank advantage perturbation cost worth formula kalman filter square direct solver calculate determinant expansion operation decomposition cost reduce recently determinant calculate determinant formula relate determinant identity often serve normalize evidence determinant right determinant precise determinant spirit determinant enable factorization perturbation factor rank relate positivity positivity large semi definite zero note matrix invertible exist definite fact directly I tu criterion meet prove criterion furthermore choice therefore calculation require arise give factorization satisfie simplify ready immediately equation substituting address restriction symmetric factorization symmetric symmetric ingredient square factorization matrix satisfy root combine recursive divide next yield root useful corollary perturbation extend update symmetric inverse fast also ti tn numerical demonstrating contain formula factor perturb matrix equation storage cost vector significant interest scale associate dominant product step tm pl make perform decomposition indicate dominant htbp c decompose decomposition proceed internal formula diagonal structured division inherent tree compression technique article hierarchical deal matrix shall restrict matrix matrix purpose diagonal block level numerically frequently level depict actual form rectangle rectangle rectangle rectangle rectangle rectangle rectangle low eq block diagonal feature factorization block update level rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle zero symmetric form ii ii tn worth recall sub block right I size step matrix need write step right side rank identity cost repeat yield rank exist represent separable computational obtaining factorization speedup cholesky rank entry give matrix scale ta context give xlabel ylabel seconds legend anchor south west coordinate coordinate coordinate corollary need versus consider q cholesky factorization ti compare versus perturbation plot take obtain factorization versus htbp xlabel ylabel seconds style anchor south east coordinate coordinate scale xlabel system ylabel style anchor coordinate scale xlabel ylabel time second south coordinate rank middle xlabel ylabel legend style north coordinate coordinate rr numerical benchmark encounter fluctuation act point dimension obtain occur perform radial force manifold embed test priori vector order diagonal slight modification variant arise frequently nonparametric find q inherent measurement cube arise gaussian process model eq definite table summarize tensor frequently brownian radius particle translation coupling refer show definite root tensor approximate chebyshev spectral square tensor scale locate volume gray white cd gray cd time error tensor singular rank diagonal grow symmetric factorization cost symmetric scale manifold benchmark
field office surveillance record expect correlation characterize datum intrinsic structure kind video geometric sample multiple reveal original extract view video individually method include complementary view integrate video intrinsic human video intrinsic htp multi framework view firstly unify video metric simultaneously capture real learn project greatly video preserve intrinsic view specifically maximal disagreement minimization summarize extract receive considerable past decade devote survey merge metric among disagreement maximal involve trade aim find margin superior traditional criterion approach optimize measure kernel metric finding metric base kernel additional pure utilize disagreement criterion study past decade comprehensive dedicated focus tracking move overlap view fu systematically video surveillance video use hypergraph view view suppose level view k unified matrix structural disagreement loss control objective former part problem preserve datum original disagreement criterion instance supervise achievable introduce graph ds tx sx ig cluster metric therefore formulate disagreement one cca matrix introduce variable simplification cca lead classification yet disagreement base learn introduce tr metric implicitly mean accord metric cut turn solve descent via eigen decomposition solve quadratic convergence assume real semantic space latent video dimensional feature usage impose disagreement minimization criterion decompose record k dx kx cluster I summary conduct road office hold scene frame show important object highlight video view construct view cluster select representative ed space dm diffusion important event office define length run ed dm ed office summarize video run frame video five participant give normalize comparable part multi view baseline tp office run cm ed dm reconstruct multi separability view learn propose combination metric define trade similarity original one science ns traditional design summary
ann feedforward pass ann activation lf b l l l ji I tensor extraction result rigorous back propagation norm natural machine character mnist digits activation th instead deal derivative herein write u towards begin function pass apply instead next constant embed play edge embedding similar connect belong however soon herein key herein projection employ herein advantage benefit instance overall different sum thing distinguish herein discriminant minimize less write ann layer shall henceforth unless implicitly incorporate addition regularizer substitute formulation space ann widely output negative ann separated belong different wish negative result sign pre term wish make extract weight decay prevent become help prevent overfitte finally control significance objective achieve naturally arise computational feasibility appear equation shall see sum involve carry minimize term undesirable grow go heuristic sum point ann sum distance near class thereby rise machine extract contain n pd ann vector activation clear expect lead gain heuristic situation wherein kx kx heuristic feasibility heuristic constitute equation think well develop especially fact clear back derive write shall illustrate since computation begin third compute ij plug introduce j w q notice permit write q round give shall propose minimize respectively use descent style compute l ij devote focus precede form stacking column denote transpose ann bias go equation chain dot column go equation particular consider herein ann bias ask compare reader simply replace latter role ann aim try supervise write clearly ann bias side equation perfectly fit put q require partial arbitrary weight essence task w ij c c play output proceed ann bias evaluate give bias consider play mind aid rule try compare ij value w w write b equality highlight stand truly ij l b ij know true essence compute l involved minimize indeed follow thus u u ann u require save stock far natural second quantity w b turn w go thing require advantage write u consider perfectly w ij analogy clear analog combine equation essence u backpropagation quantity call backpropagation pass train current vector ann flip argument backpropagation role current activation ann play derive b w ann ann activation end feedforward ann activation network l ij la repeat rather rather thereby obtain play compute final w correspond ij w ij proceed keep mind write write less reader keep plan backpropagation computation notation first feed relevant transfer sigmoid depend show derivative arbitrary weight
conjugate conditional xy ts yy enable c df begin define partition via df proceed observe default assume center scale update surrogate gibbs full c posterior approximate yy expression hold model df accuracy comparison draw df converge excellent report coverage length one treatment group eq sampler new I kt gibbs conditional conditional specific statistic df df specific mean I nc define set add simulate model estimate c df similar df inference extremely sensitive estimate perform parameter estimate uncertainty table occur make joint way df approximation conditional coverage kt coverage credible average arrival standard df full depend yield mcmc correctness approximate sample datum posterior closed increase often df model regression close form variational introduce additional pass augmentation via probit conditionally auto overcome storage arise section provide excellent example df apply conditionally finally consider poisson augmentation admit quantity online df apply namely ar default u hierarchical good backward kalman computation strategy intractable time grow univariate key importance sampling less grow propose move exploit latent otherwise poor resample window shift estimate df evolve dynamically define conditional distribution quantity enable approximate draw df metropolis hasting df metropolis within sequentially b j hasting bc generate experiment report prevent df propagation pl density pl df df concentrate pl albeit variance pl coverage df near credible pl see df substantially require pl use df produce contrast pl spread window expense hand avg mse df pl tn h c df pl metropolis pl memory ram propagate pl report big df probit augmentation sampler follow sample variate full truncate conditional present bottleneck size latent df surrogate budget index final observation partition dynamic I c df observe conditional xx tn I z collection draw c xx xx z b latent index example first generate table inferential case although coverage I predictor df addition good estimate avg coverage df arrival scale linearly observation storage score full conditional storage h additive mixed effect predictor q I implement df close surrogate propagate make increasingly restrictive restrictive seek posterior ignore among joint posterior completely specify df specify replace fix variational df draw prior u update rw u cccc report experiment case choice map default low df competitive increase predictor predictor suffer dense df excellent estimation high report coverage predictive df vb suffer restrictive df error replication case vb df error case c interval replication surrogate quantity sample namely report simulate c c df mb mb mb quantity finally update quantity surrogate df df online linear variable predict action base relate status angle dataset relationship vb entire mode sequentially obtain df effective comparison show df consistent summarize table partition observation horizon test method competitive predictive mse interval coverage df variational bayes vb suffer predictor select df partition horizon df suffer h c coverage df vb df identify study class mcmc df present limit stationary deal class sample proof time take borel tt omit namely tv characterize kernel finite horizon kernel calculate c df initially assumption kernel stream partition p p df require approximate admit quantity conjugate kernel proceed gibbs fashion conditional available transition hasting df stationary conditional scenario model conditionals dynamic present admit although hasting c df accommodate increase admit asymptotic produce establish latter case produce excellent draw n n essence refer lemma ergodicity ergodicity transition regular approximate chain formalize outline state neighborhood domain almost say grow rate regularity stationary df distribution mean get primary easy sequence conditional filter novel streaming df produce draw conditional posterior rely surrogate sufficient statistic quantity gain massive bayesian popular often propagation ep particle sequential approach face substantial difficulty exception df demonstrate illustrative c df characterization df runtime sampling improvement support table relate use relationship tv use r tv tv compare note follow gibbs stationary identical manner take account obtain q complete generate one obtain continuity assumption yield law consider evident law algebraic q similarly eq tt motivating coefficient representative group mean j response lee generalize pareto bayesian via fisher scoring filter nonlinear gaussian tracking stream variational process inference filter f g particle smoothing sequential compressed f wang theory poisson chen mcmc cut metropolis hasting propagation mix machine p expectation linkage cpu gpu e regression virtual j variational framework fitting generalize west forecast dynamic wang dirichlet gradient langevin b monte carlo pt pt lemma definition remark em em depth david filtering inference df mcmc online sampling surrogate conditional need store offer desirable memory requirement improve state art demonstrate dimensional df target sampling proceed data keyword approximate big monte measure statistical increasingly curse dimensionality truly algorithm infeasible probabilistic characterization lack scalable inference guarantee sample gold standard emphasis monte performance size mcmc number grow evaluation mcmc accommodate adapt transition draw sample joint mcmc require storage sufficient face storage scale architecture likelihood another monte chen sequential hypothesis approximate metropolis hasting conjugate model efficient updating obtain density broad approximate approximate posterior tracking error propagation ep assume convergence posterior predictive uncertainty variational et streaming combine stochastic inference variational method uncertainty accurately except carlo online rely sample smc involve large prevent degeneracy expensive degeneracy particle pl degeneracy satisfactory estimate add complexity propose mcmc streaming df proceed sufficient surrogate point sample df enable df definition identify update demonstrate online apply df regression way sophisticated section extension probit df increase also variational storage mix art df dimensional inferential extensive finite mcmc figure appear observe nan j j j value sample sufficient storage overhead multiple structure create computer ensemble updating cause address propose statistics j c c df proceed draw distribution potentially expensive conditional fashion
one receiver operation roc receiver error decide generalization base cross problem method overfitte hardness amount problem address case reservoir address aspect issue experimentally attempt develop reservoir paradigm reservoir successful machine aim understanding work reservoir study completely satisfactory result vary power delay perfect past computation finally reservoir carry implicit represent dynamic functional delay resource error long dependency narrow line fourth resource resource delay achieve reservoir computing trade relation nsf grant surface fit fit fit statistics h correspond matrix minimize represent fit fit goodness given map reservoir size task training error delay line measure c training measure measure testing training measure test training measure test I w reservoir node compare paradigm computation integrate method power ordinary dl memory arbitrary neural step reservoir problem series square error broader systematic autoregressive provide solid evidence computation device neural reservoir reservoir connectivity represent rescale radius sufficient input connect reservoir weight later propose assign et use reservoir scale weight ensure also contrary reservoir optimality demonstrate experimentally affect study reservoir performance depend heterogeneity performance communication reservoir scale world topology significant performance song coefficient odd circuit study study simple reservoir node cycle homogeneous reservoir arbitrarily find address reservoir demonstrate deviation normally distribute reservoir exponent optimality dynamic solve understand reservoir computation power computational delay line signal respectively delay line dl allow delay version reservoir dl connect read dl architecture delay delay fed teacher use teacher follow row dl row teacher dl augment constant initially dl state set autoregressive neural delay hide bias function linear transfer train perform teacher output previous value architecture effect fix vary reservoir reservoir connection reservoir reservoir distribution input reservoir evolution reservoir discrete time nonlinear reservoir row reservoir weight target linear dependent experimentally evaluate model act storage computing dynamical performance power attempt create dl task map lag dependency compare performance requirement performance easy white require computation baseline move discrete time present nonlinearity lag adaptation stage scaling systematically show averaging result run combination nonlinearity sensitive change heterogeneous weight assignment deviation experimentally power result power power sensitive map behavior qualitatively division understand attempt size device memory theoretically computational input universal reservoir clear computational dl perform functional comparison dl normal create weight reservoir choose optimize performance non map pattern simulation test step average follow line change delay line network series hide reservoir five error five reservoir reservoir use show measure normalize task decrease soon system sharp drop curse teacher expand delay overfitte high line highest simple logarithmic axis behave differently test around begin characterize increase test low time test reservoir dl reservoir task sign layer merely system find report dl comparison figure task map achieve delay map line narrow delay average task pattern delay task h delay much resource achieve figure comparison map dependency network result similar
theory generalization extend probabilistic discuss wide class trivial design operating evaluation recursively construct variant probabilistic estimate initial triangular matrix compactly point construct naturally treat posterior arbitrary distinguish property taylor coincide term numerical xt p method order strong ode currently mark parameterized introduction standard mean px sx k close map recursive analogous recursively form incorporate xt hc hc final describe ode estimate share structure sum evaluation careful gp ode perhaps naturally choose ad hoc guarantee choice work square kernel give euler model integrate wiener improper limit gauss markov posterior proper process give interpretation gauss although ode distribution gaussian uniquely suited banach add reproduce issue put posterior find second force albeit lie associate enter gp affect importantly interval solve begin error order term fan chain solver rather try option question pose hard answer means return method degree answer bar proceed main gaussian integral wiener process shift conceptual prior gray blue empty circle mean third green respectively line method final proper intermediate ode integrated wiener evaluation give rise euler corresponding hold observe value evaluate directly algebraic straightforward integrate wiener move fortunately limit lead twice integrate wiener choose evaluation node rise family limit integrate wiener gauss turn improper hold constraint extend show find second limit improper policy placing wiener evaluate specific rise limit entirely analogously match u table posterior mean q final node regressor highly integrate wiener wiener surely seem performance gauss trading good high integrate process model integration leave open conceptual probabilistic accept solver continue first evaluation oppose euler form next interval three option suggest novel classic joint column hoc run I evaluation first remain gradient posterior go point produce confident global perhaps probabilistic framework else strength continuously establish iterative lot weak figure result approach I lead distribution function naturally global simple case course least operation planning follow publication form bound favorable ad hoc fig se calibration close true se bar cover confident advantage calibration due choice framework show se line deviation show interpretation limit gaussian wiener class return particularly ode solver add point open solver rd ode solver proceed stage acknowledgment grateful omit paper additionally website publication symbolic toolbox multivariate analogously method work value derivation supplement carry case consider model separate kronecker define dimension dimensional wiener lead derivation wiener derivative eq last iterate derivative kernel necessary form form derivation symbolic integrated eqs observing euler generality convention covariance formula throughout appropriate simplify formula omit state formula sec perform symbolic toolbox write covariance covariance distinguish theorem covariance value integrate integrated wiener process infinite list function rbf kernel euler choice yield euler predictive weight euler even third order insight everywhere contrast wiener process interpretable gps markov model cost cubic reduce rgb rgb department systems ordinary art return return define ode work construct output good remain ode light solver provide rich output dynamical
measurement illustrate fig death illustrative grey analytically birth death absolute magnitude difference mean eqn consider absolute birth copy distribution proceed quickly note birth death process ref relation g z z expression compute likelihood measurement copy probability limit increase later posterior facilitate comparison abc posterior analytic expression posterior give support fig parameter birth analytic assume posterior structure right posterior thick vertical line show parameter alternative likelihood employ square eqn refinement abc abc first within guarantee reject vary reduce mean abc match low resemble analytic kernel enforce acceptance substantial away prior include parameter likely enforce negativity restriction step abc latter yielded investigate yield density support protocol allow lowest straightforwardly computable responsible approach abc speedup phase inference fast test approach parametric analytic result likelihood population assume start division uniform represent rna may rna loss copy straightforwardly ode use stochastic give copy ensemble trajectory compute average copy series prior discrete prior wider ensure range enforce range exhibit uncertainty rigorously mean skewed suggest possibility original posterior increasingly converge acceptance trial reject first implementation number inference experimental model rna number induction subset measurement line inference centre parameter ref assume infer posterior support quantify threshold efficiently infer stochastic biological analytic form likelihood abc abc mcmc avoid region straightforwardly trajectory equally speedup volume trajectory approach include sequential conservative approach alone exceed distance half contribution trajectory mean protocol mean refined threshold exhibit rejection achieve conclusion include variability allow powerful question propose arise deterministic exceed exceed negativity deterministic magnitude mean reasonably assume expand measurement variance measurement validity assumption biology desirable observation mean variance quantity circumstance analytic form usage implementation abc powerful tool calculation computationally rely repeat stochastic simulation address simulate demand behaviour lead speed synthetic variability much recent evidence biological cell influence remarkable example include within cell cell drug cancer information mechanism magnitude measurement stochastic description fall inference typical simple biological system simulate behaviour datum biological variant often require therefore simulation noisy biological measurement individual mean variance statistic parametric abc perform dramatically inferential decrease stochastic approach gene experimental quantity biological interest appropriate assume begin consider measure variance quantity imagine cell measurement rna take individual measurement constitute measurement measurement time develop series system quantity uncorrelated meet analytic sample assume record statistic uncorrelated distribute deterministic assumed estimate overall associated individual ref specifically log measurement overall associated eqn assume underlie associated sample propose protocol approximate bayesian abc abc computation likelihood true posterior absence explicit measure measure explicitly write trial complicated dataset distance summary summary threshold record posterior decrease computed posterior leave metric observe simulate simulated rise therefore difference facilitate comparison multiplicative allow comparison different magnitude associate mean versa square include model weighting note always higher change exploration perform inference begin wish abc rejection trial simulation fall trial yield ensure yielded accept possible perturbation represent propose perturbation kernel optimisation rejection times compute simulation times condition step pick accord anneal search initial ref parametric heuristic adequate search reasonable distribution employ algorithmic stochastic contribute final posterior desirable whether reject biological analytically easy perform simulation advantage perform
manual tuning network across patch also determine subsequent add horizontal work significant reduction b notation cnn filter channel dimension output channel convolution assume one channel accelerate multi convolution rank cross filter channel vertical filter dimension filter separability filter rank increase component solve eigenvalue filter alternatively could restrict connection field separate upon apply convolutional neural modification feature define convolutional cnns convolutional layer convert bias equation clean important remove slow three propose model alignment cnn classification experiment environment handle gradient update baseline model perceptron cnns minimize double perceptron filter layer prevent adaptation augmentation concentrate learning capacity initially eight one cnn achieve result table dimension baseline bias add operator layer cnns construct describe perform convolution channel layer horizontal whose convolution channel view post process fine tuning need structural filter except bias replace filter result drop dataset commonly cnn predict reduction distinguish discriminate feature find removing imply toward essential set vertical horizontal set achieve standard layer consist stage decrease model depend difficulty reduction configuration baseline decrease adapt vanishing cnn layer provide flexibility help path accumulation cause decay gradient trend channel update gradient weight cnns convolution vanish gradient handle initialization initialization balanced pass yield successful heuristic initialization table cifar solid mean indicate deviation variation model small illustration proper weight baseline figure baseline decrease baseline early region accuracy seed result variation backpropagation filter cifar though layer convert reconstruct cross two filter necessary surprisingly color sparse penalty effectiveness explain comparable baseline cnns trade convolutional stage channel spatial filter dimension ratio convolutional left layer layer portion smoothly cnns decrease layer channel usually begin perform second though convolutional achieve term r baseline c technique layer layer parameter optimize consumption concern affect parallelism scale intermediate backward pass whose consider break convolution produce intermediate need na I loop optimize memory usage highest adapt scientific I exploit memory real cnns many baseline multiplication memory convolution layer parallelism resource feedforward oppose convolution baseline convolution convolution break number cifar cifar mnist dataset cifar consist testing contrast normalization whiten whole reach outperform baseline mnist consist digit training apply per whitening cross correlation cifar highly simplicity almost accuracie cifar baseline structure cnns computations cnns baseline feedforward backpropagation pass present cpu gpu check parallelism measure intel cpu gpu model baseline piece filter ht feedforward pass convolution filter convolutional layer pass acceleration tend overhead become negligible speed reduce computation acceleration efficient feedforward acceleration convolution process consider effort access imply cnn channel increase use reduce training cpu gpu backpropagation consist update access convolution feedforward however accumulation convolutional operation frequent access acceleration negligible gpu technique convolutional feedforward acceleration convert layer channel vertical horizontal filter successfully ten accuracie cifar cifar addition effort manual difficulty learn reduction accelerate model remain acknowledgment office grant pr edu neural redundancy weight filter convolutional extensively heuristic rank train consecutive dimensional filter across obtain comparable conventional convolutional substitute filter feedforward parameter effort manual recent success convolutional network cnns enable researcher network cnn audio understand security system mobile accuracy execute cnn
complement construction capacity pose question term measure cloud learning involve functional minimizer functional involve minimizer correspond limit functional set imply notion convergence extensively calculus phase material science convergence establish cloud symmetric decaying rescale particular connect assign weight define graph capacity variational description total point cloud typically total vertex obtain lipschitz boundary support consider weight give z connect order limit rescale scale variational sense contribution identify empirical point measure arbitrary denote borel q borel second provide convergence pl pf consider grid cell grid grid consider isotropic radial pt kernel broad coordinate vector replace expression side otherwise rest denote main connected let let tv variation subsection definition sum become write topology good topology theorem uniformly precisely relatively subset hold allow tv conclusion functional scale converge theorems point total appropriately scale converge usual appropriately scale notion pointwise convergence smooth convergence obtain converge pointwise scale pointwise pointwise directly theorem imply subsequence present point distribute theory probability remark surprising describe task minimizer minimizer example may setting dd n convergence minimizer approximate functional minimizer limit extensive exposition book classical functional convergence free energy phase transition energy term kernel consider system show functional study one work conceptually work elegant complicated discrete set functional include interpret limit functional rescaling energy consider get depend size van discrete lattice functional year develop task important desirable limit procedure limit euclidean pointwise work von von eigenvalue work allow converge pointwise admissible regularity requirement show convergence cut knn graph consistency algorithm algorithm involve cut total functional illustration context consistency primary investigate satisfy problem find minimize domain minimizer complement describe minimize functional consider extension constraint inequality satisfy subsequence guarantee minimizer depict resemble present minimizer take minimizer htb variation tv function support function function define graph variation function approximate map preserve precisely subsection rely little far need variation functional weight analogous functional preliminary convergence space fact variation list subsection subsection notion space introduce prove result convergence functional tv functional prove extension necessarily independently distribute point case correspond context write abuse replace expression finally weight throughout restrict bound positive open equal norm also surface integral belong finite measurable uniquely condition variation associate call check depend derivative distributional supremum continuous functional respect finish approximation open every open algebra borel give set marginal refer wasserstein distance minimizer distance reference weakly borel denote borel integral tx absolutely induce via lebesgue measure equivalent sequence map inverse give third refer composition note point want point mass let lipschitz boundary rich sequence seek turn infinity maximal map infinity match rich matching form regular cube volume point point exist meaning range distance os dyadic dyadic composition consecutive nature length locally hand globally behavior scale source scale conceptually proof result domain borel lemma imply follow subset boundary measure let sequence let map discuss notion functional converge metric functional inequality hold q collection point check f identify measure pd pd pd dirac delta absolutely measure closure pd introduce pd remark lebesgue induce plan deduce f f n pd pd pd pd attention absolutely lebesgue eq metric subset topology convergence lemma prove map measure class bound know note vx obtain prove detail mention end triangle finally satisfy deduce statement every moreover measure absolutely lebesgue measure equivalent previous statement claim absolutely map nd f metric slight abuse notation compact absolutely continuous lebesgue u last one light remark finish also provide follow nevertheless decide present canonical distance topology endow complete endowed characterization moment completion geometric lebesgue converge remark distance remark functional weight variation open bound constant tv follow idea specifically first inequality functional argument presence boundary consider function regardless regularity definition family notation number converge zero limit simply limit let arbitrary purpose taylor inequality deduce diameter set finally straightforward claim diameter zero go claim q eq deduce right zero imply enough kernel suppose u generality bound lemma plan function limit gain function support bx u u x x dx infer estimate second jensen chain inequality assume conclude u open set compactly idea lipschitz bounding case conclude desire indeed define suffice right proposition u compact support assume k last change note transform equality thank symmetric straightforward constant implie therefore conclude deduce support kernel variation function moreover continuous simple enough imply q dominate note establish regular open relatively compact assumption function bt change assume constant enough x dx px establish geometric dx dx dx bi establishe outside radius straightforward yield eq role small eq follow bi lipschitz fact second inequality proof sequence remark proved bound assume function boundary remark bi lipschitz domain bi variable assumption compact set ball compactly lemma set compactly hold union set compactly cover finitely ball boundedness approximate l convergence bound lipschitz positive use let hold complement two matter estimate take slightly radius match almost u u un deduce tv remark proposition prove continuous nx h proof piecewise compact denote function use step compactly satisfy constant nu nu lipschitz analogously function proceed analogous inequality nu assume lipschitz tt n bound true nx ny ny ny nx ny ny eq change step conclude map nz enough note deduce l corollary n n inequality lipschitz restrict characteristic follow take advantage formula energy measurable exist nu tv verify functional satisfy tv n tv approximate characteristic key substitute follow follow approximated characteristic measurable class specify volume argument remark consider distance theorem assumption able distance would translate
ideally alone devote density drop quickly mathematically say function rigorous density multivariate preliminary construction system haar dimensional construct alternative necessary high basis exponentially partition illustration haar wavelet scale wavelet eq denote vertical translate l l j together dimensional haar wavelet let haar construct expand respect haar wavelet size impose function wavelet control decay widely use characterize dimensional next want localize cube seven function still level haar coefficient display decay trend generally haar condition perform haar plot haar coefficient coefficient absolute clearly haar coefficient estimation correlation mode haar haar summarize law try estimate spatially concentrate likelihood partitioning detect unknown structure rate partition condition normalize support rectangle negative haar basis upper therefore density support reach achievable rate small theorem depend oppose agree acknowledgement author thank discussion department policy stanford problem introduce estimator constant partition learn analyze reach conclusion curse adaptation calculate circumstance fundamental inference natural straightforward develop however currently increase size great may geometric dimension like suffer difficulty paper employ still paper thorough analysis convergence method datum establish density design low approximate sensitive window good need depend datum especially multidimensional difficulty cause classic show rate smooth method parametric large density still slow optimal curse seek convergence large indicate order depend result establish perhaps basic appropriately origin width histogram bin allow bin histogram generalization geometry density estimate density learn recursively partition partitioning allow opt prove support variation posterior yield opt tree p successfully computational recursive extremely resolve base partitioning develop major distribution analytically asymptotically correspond kullback leibler employ sequential importance dimensional eq density kernel kde apply compare hellinger estimated kde comparison unknown density histogram opt essence flexible enough geometric overcome thus density partition essence interest lie area perform dimension class partition achieve quantify representation quality formulate error density partition demonstrate optimal meet challenge far reveal nature explicit respectively insight high rest paper organize main section devote spatial respectively density measurable lebesgue scaling unit cube f space restrict one partition grow precision idea integrate search partition unit cube step recursion partitioning region choose along range partition display possible binary partition support size constitute approximate space background induce hellinger hellinger distance say well approximate satisfying let convergence section binary become determine log partition incorporate select promise accord score likelihood ready study converge lie cover precise fact decay empirical one part ratio difficulty truncate truncate truncation maintain key likelihood guarantee behavior truncate exist fail truncation omit truncation log expect low truncate log likelihood ratio large n fy fy hellinger induce ratio inside hellinger let see apply deviation inequality change rely class far satisfied likelihood explain probability ball radius beginning zero converge function denote easy check converge determined return expression lemma iteratively achieve f f k I last need dy dy mn du du du n apply establish argument automatically similar du kn n n I replace satisfied kn n condition assume n kn kn exactly order small optimal order select greatly contribute simplify improve reduce overfitte formulate dimensional essence true effective section calculation compare kernel regard relie make exist question class haar density approximation true select haar certain criterion tensor haar volume support wavelet involve derivative improvement variation condition replace mixed h controlling develop approximated density accurate description notation haar simple wavelet haar scaling haar haar haar scale turn orthonormal product haar respectively haar calculation heavily respect haar hellinger calculation hellinger haar basis haar support special denote closely approximation
different cross provide function average distribution leverage common elaborate end predictor label label framework svms however label information base conditional give learn challenge convex hard addition incorporate generative lead improved discriminative extract change set suffice consider regression set derivative label label vary locally valuable contribution derivative use establish high derive call score incorporate subsequent correspond many discriminative model feedforward neural setting result present supervise unlabeled distribution mechanism assume model code unlabeled rich coding learn assume score new relatively straightforward transfer estimate unlabeled estimation transfer propose feature high semi unlabele form high order apply several thank support support award microsoft fellowship award nsf award award proposition derivation claim example time bold department california ca usa department science california usa forms speech computer novel value unlabeled sample efficient extracting label theoretical characterize nature labeled employ tensor extract employ rich overcomplete thus discriminative feature supervise score representation achieve machine domain computer natural traditionally engineering tailor towards task consume instead automatically feature framework component ica exploit vast incorporate prior typically model incorporate explanatory associated generative boost discriminative task approach focus unsupervised employ expect scenario unlabeled one framework transfer learn adaptation dataset framework extensive learn challenging computer vision huge unlabele label one syntactic semantic access amount unlabele human unsupervise purpose without goal specific task human extract general purpose capability design unlabele give question concrete answer value general pre label present leverage discriminative information spectral discriminative extract pdf capture local score high tensor rich value allow characterize nature extract input label input moment label discriminative task employ decomposition find tensor analyze suffer spurious optima convex maximization overcomplete representation argue overcomplete get advance discriminative framework scalar tensor continuous handle structured problem unify end extract pre present gray shape corner sep purpose moment gx find width width line pt line width draw dash corner draw corner green width c extract discriminative characterize score label vanish carry however label degenerate certain derivative vanish even derivative average carry function moment useful discriminative discriminative model challenge establish discriminative work spectral recover challenge discriminative generative discriminative feed pre classifier fisher feed behind information learn classifier prescribe label sample conjunction sample run converge good solution
experiment ideal change condition potentially create different indicate task realization parameter normal reflect sophisticated manner individually element dedicate psd applicable psd newly formalize frobenius norm use use kind problem formulation weight reflect intuitively value deviation weight introduce balance variation small subtle deviation psd width cm experiment eps color distance matrix roc method bar eps specific learning modification replace regularization computational estimate solving guarantee exponentially overview hundred previous directly hadamard duality relate primal problem conduct alternate direction admm method density matrix try discriminate state note element normal fluctuation limited state obtain depict nm outcome contribute convert add diagonal density fig experiment numerically pair reduce density matrix state color trace state dash line fig roc curves width bar real eps select normal experimentally performance dataset consist five matrix tune value raw matrix naive estimate introduce case ed fair remove statistical fluctuation invariant distance color fig understand discovery fdr give true without detect colored receiver characteristic roc vertical stand leave corner fdr naive indicate reliable quantitative use percentage roc curve fdr frequently study value value dataset demonstrate method fig raw naive th th raw contain bias calibration accuracy place mode depict typically physical bias reflect distance fig well naive method particularly fdr curve higher moreover experimental state matrix state experimentally respectively demonstrate performance experimentally generate computer simulation demonstrate accurately matrix fluctuation naive matrix statistical auc experimentally computer density show ed datum mine key broad area quantum state essential interested quantum manuscript try detect state anomaly anomaly reconstruct matrix cause maximally process error cause anomaly anomaly matrix able quantum physical system circuit anomaly detail etc applicable wide focused anomaly detection unitary acknowledgment quantum project foundation technology reconstruct technique accurate datum propose accurately deviation reconstruct contain intrinsic fluctuation check trace average method detection grow rapidly computation
generalize modify li two population generalize p devote side log normal test normal examine discussion give deal nuisance impossible nontrivial test density nuisance xx stochastically increase xx stochastically book ij chi generalize approximated simulation set generate calculate u mt lp hypothesis side consider experiment normal summarize generalize al study power close amount random model fit log value reject em example journal department department decade accept reject test normal calculation find analytically statistic illustrated normal inherently life application analyze biological medical distribution term log normal researcher also article
structure clean weight direct simple bit flip able flexibility match graph difficulty demonstrate flip process version match copy measure match correctly vertex vary seed seed seed performance flip decrease increase study simple system make map neuron chemical chemical electrical potential chemical electrical individual across hence undirected self undirecte neuron remove leave vertex match utilize dissimilarity performance unweighted graph incorporate seed seed run direct seed match remain chance run weight graph different year across time realization seek period common unfortunately graph order connect union two utilize seed run mc statistically tailor real http www www finance enyi robust difficulty space flexible real simultaneously simulated well cutting procedure future potential appropriate future plan principle heuristic seed working approach greatly limit big scalable essential application lastly justify dimension automate approach partially national security engineering fellowship university technology project air force research laboratory contract theorem open joint fidelity department mathematics md university nc novel incorporate paradigm optimization fidelity euclidean match simulated matching graph characteristic many give seek correspondence matching preserve matching document processing name efficient even easy applicability numerous algorithm excellent survey exist matching often graph know cut algorithm graph contain match thousand vertex excellent achieve modify weighted generalization currently handle number arise aforementioned robust effectively match herein joint fidelity algorithm flexible inherent performance simple potentially section graph problem datum example outperform handle match chemical electrical procedure outperform across ability incorporate classification realization demonstrate outperform match x permutation doubly graph seek find minimize minimize adjacency respectively seek permutation allow quadratic assignment hard graph know without seed seek restriction equivalently seek permutation weighted problem know matching begin doubly form utilize frank wolfe methodology efficiently relax finally relax onto matching procedure herein attribute excellent survey match fidelity task perform seek provide maximize fidelity general certain assumption necessarily let graph generality label assume pose classical enough handle match vertex match actor communication graph neuron true multiple match vice versa see task vertex newly reformulate graph match intuitive often amongst incorporate order simplify sequel also aim perform match multidimensional scale common space readily dissimilarity representation dissimilarity ideally choose dissimilarity dependent dissimilarity although address choose dissimilarity match achieve excellent sparse detail os r enyi dissimilarity neighborhood mark global shortest expect dissimilarity embed space preserve contain within essence embed match dissimilarity match though imputation graph dissimilarity dissimilarity increase complexity access full j equal additional unknown treat miss procedure vertex methodology present describe automate labeling embed embed point embed match capture poorly preserve dissimilarity capture fidelity closely fidelity separability separability error preserve across dissimilarity match embed target simultaneously control dissimilarity raw cost represent vertex simplify fidelity separability error embed preserve dissimilarity preserve seed dissimilarity essential algorithm multidimensional outline vertex embed dissimilarity involve procedure simply dissimilarity suppose ideally preserve argument ideally procedure preserve triangle match amongst embed embed procedure seek minimize stress configuration represent dissimilarity graph e sum neighborhood euclidean distance amongst vertex approximate vertex solve avoid impose generalized np example use many set vertex seek classic assignment solve via present lie appropriate original chose tackle dimensionality dissimilarity procedure automate spectral present principled choosing work initialize choose iii iv solve j output v matching amongst approximate reasonable significant boost
decompose e plus compressive scope overall approach global compressive acquire essentially matrix exactly location outlier far section numerical several recent work utilize technique salient scenario seminal parsimonious decomposition salient image salient identification examine image drive cosine transform demonstrate salient spirit note work propose salient compressive formalize establish compressive outline proof comprehensive experimental auxiliary bold letter matlab notation form extract index etc exception indexing note notation throughout exposition bold letter usage clear context sum singular column matrix denote transpose integer may formalize admit matrix rank outlier lie span span dimension let onto assume cardinality aside nonzero assume column aggregated nonzero criterion informative subspace distinction index column potentially case contain matrix task etc issue subspace rank incoherent basis rank seek column seek like spread subspace incoherence low formalize notion follow column incoherence column svd matrix say satisfy q basis limit element canonical undesirable implie describe single distinguishing assumption satisfy nonzero condition outline algorithm distributional distributional preserve length multiplicative factor least union argument row note randomly gaussian subgaussian whose gaussian bernoulli position proof accurate structural column measurement satisfy bind set identify salient interesting outli pursuit succeed subspace location satisfy analogous identify identify outlier interesting achievable appropriate parameter algorithm identify salient column comprise fraction compressive salient operate directly matrix specifically succeed probability n column outside comparison address matrix observation form measurement author assume nonzero column row normalize sufficient approach may performance effect simplify analysis use recovery component condition regularization simultaneously identify salient measurement great leave reader prescribed function way action intermediate argument analogous provide appendix satisfy structural fix distributional column incoherence property orthogonal onto succeed provide satisfy structural sufficiently close simultaneously satisfie third support set produce salient matrix satisfy distributional overall intermediate union bind conclusion hold hold comprehensive motivated map outli pursuit op employ optimization low nonzero column implement method use accelerate alternate admm inspire op execution op procedure implement pdf pdf pdf percentage increasing allow recovery outli experiment follow I matrix notice square norm sampling rate fixing choose row parameter column regularization algorithm whether step employ outli pursuit identify true rate assess recovery achievable parameter outcome regime examine fraction observation result interesting somewhat intuitive efficacy parameter keep move top increase matrix moving outlier recover trivial see outli rank background pdf match far compare outli identification simplify applicable high notice vertical complementary may column relatively favorable curve yield successful recovery small identify discussion section difference due large two inherently operate scenario permit combination entry albeit individually op entry execution op step process task arise vision surveillance identify map image salient object image transform color gray decompose overlap patch matrix correspond notice gray scale value input collect architecture experimental approach somewhat necessarily bit heuristic due may exactly reduction subspace span column learn retain small leading singular generalize salient norm nonzero heuristic select qualitatively positive three regime examine rate result entry benchmark visual method datum well op identify visually salient region image identify salient validate use plus also compare op detection perform still produce reasonably sample moreover acceptable map deviation ghz intel core processor run os execute procedure discuss overall fast consistency promise salient task pdf b increase rank outlier increase variance estimation outli performance formally pdf pdf pdf pdf pdf pdf investigate experimental methodology euclidean essentially level row column sampling fix correspond perform trial record albeit reasonable might perturb energy result difficult support scenario require well support choice normalize observe degradation notice level variance inference step demonstrate extension amenable scenario characterize underlie observe subset formally denote let location operate procedure operate sample set matrix subsample comprise insight composite subsampling express operation subsampling specifically solve identifying span low component analog orthogonal operation column available element row submatrix form index orthogonal span residual energy th j column recent examine subsampling pdf pdf pdf pdf bottom column sampling parameter right empirically subsample column subsampling row cardinality figure outli outli degradation increase approach compressive cs follow reconstruct compressive somewhat simple kind recovery albeit background sufficient identification insight exploit operate compress original ultimately successfully location identification conjecture procedure least additional structural column contain location limit approach compressive principal pursuit comparable direct prohibitive rate storage element implement establish row compression incoherence compressive approach compress suffice investigation effort complexity op comment briefly complexitie examine op solver utilize accelerate op scale step solver step along would operation projection summarize operate full additional operation step mn p mn n multiply factor similarly operation platform could effect form implicitly light camera overall may embedding visual application likely salient element proceed formalism dimensionality sense stable comprise part embed second lemma least choose value follow albeit different throughout portion turn embedding stable embed imply stable embed nonzero embed satisfie recall svd nonnegative matrix strictly incoherence state norm row true account lemma space dimensional span third claim salient salient equivalent operator utilize intermediate term interference adapt let subspace complement embedding complement result useful stable embed directly stable embedding embed ij coincide establish establish generate specify approach begin brief geometric discussion embed appeal stable embedding comprise take word establish entail appropriate approximately preserve length comprise subspace affine embedding linear well weak result though embedding subspace receive fortunately may former latter recall discussion establish establish strong follow column combination affine linear zero sufficient word factor square length union unique subspace union adapt denote subspace suppose comprised part two follow directly five arise compact subsampling specify arise value ease exposition analogous replace albeit portion main lemma outli pursuit whose satisfy structural span column operator onto complement span column shorthand lemma rank condition large since follow span bound incoherence orthogonal operator use idea incoherence follow recall comprise subspace span span finally hold estimate column column span part entail binomial tail pr number draw denote let pr pr pr note utilize small value within bind bounding since singular chernoff let hermitian incoherence calculation pr put large realization random variable identification accomplish representative cast context formalism without let stable denote unique canonical eq randomly generate embed probability straightforward imply least denote positive acquire draw replacement kn nk nm know success result exhibit predict tight population somewhat analogous
applicable arbitrary later derive entropy ball completely spline applicable trend verification rate convergence fast base step roughly maxima gaussians recall use apply next reveal potential advantage gain linearize linearize side special adjust possible singular increase incoherence generalize estimate average squared leverage linearize hold incoherence basic replace grow reciprocal minimum nonzero something reciprocal large strong fourier scan require singular assumption incoherence regular roughly network likely vertex exception choice fractional provide deriving rate style seminal denote recall estimate main motivation radius cover closely fractional error small guarantee become reproduce rate trend entropy well k univariate hence tune trend filtering early standard boundedness way univariate trend minimax match establishe entropy embed contain derivative variation proof unlike previous directly boundedness signal instead evaluation hard manner currently limit analog difficulty merely concern entropy strategy number rich purpose bound decay nice display reader aforementioned strategy regard beyond scope demonstrate capability third covering recover optimal univariate fuse atom atom respect meet concern univariate wavelet usefulness univariate trend filter alternative wavelet evidence laplacian superior wavelet smoothing basis understand future rate reach acknowledgment national foundation international centre office nsf google support dms assumption projection consider first quantity note write establish rearrange univariate difference h k factorial basis evaluate column prove result row odd dimension nan nan become one operator order eigenvalue laplacian odd k dl suffice complete difference factorial form difference define factorial matrix order evaluate mind expand read solution want hx p use discrete matrix factorial application inequality place linearize claim argue arrive quadratic root bind mean complete decomposition establish vector decompose eq fact bind term th incoherence gaussians put term together associate vertex combinatorial pair normalize obey remainder proof include completeness accomplish apply follow inequality claim kx x rearrange spirit argue thus eq plug q desire hence straightforward second back closely care apply first mm x important form entropy equivalently scale entropy translate scale author desire stick input form orthogonal onto span hand analog polynomial inner factorial function ff therefore sup norm result constant r td latter break conclude prove main content rest reading cover j cover ball original noting secondly claim n denote column immediately apparent ball center ball cover ball cover ball ball com edu pa department pa mathematics department california pa adaptive graph generalize idea trend nonparametric analogous trend readily graph trend theory trend local rich statistic trend filter filter trend adapt across stand laplacian regularization enforce smoothness globally much hand yield either else throughout computational trend filter regularize penalty nonsmooth enough efficient large computation trend possibly difference node trend filtering fall call framework define alternatively synthesis first construct observe basis wavelet likewise kernel laplacian analysis mix motivation denoise census pa arrange connect spatially trend filter signal smoothness peak panel figure mixture gaussian underlie noisy right fit filter graph smooth quadratic define ccc observation graph trend filter df laplacian smoothing df df unnormalized build effective degree df measure complexity model top df adaptively fit peak graph df substantially peak center df middle begin high neighboring smoothing perform poorly df affect quantitative assessment difference trend spline df smoothing due consideration trend filter well yet sufficiently regime demonstrate local flexibility trend filter article section trend cover basic filtering estimator look simulated trend discussion row extract complementary vector nan rectangular begin trend univariate role suppose observe input location evenly spaced order employ operator filtering reduce dimensional fuse recursively operator take difference th fit nk th polynomial evaluate location formally verify examine analog graph edge observe order trend broad matrix difference lie entirely operator difference difference achieve oriented incidence row sign construction graph trend estimate node recursion operator univariate multiply transpose square exploit nk univariate remove row recover odd remove row intuition difference polynomial graph section sparsity difference specific piecewise since correspond value interpretation piecewise might ask component piecewise define question piecewise construct many difference across note exactly property sparsity orient incidence piecewise linear structure number require linearly neighboring notion linearity require would linearity euclidean piecewise polynomial piecewise difference mostly likewise cubic second extend lead odd exactly difference illustrate node compute plot graph penalty explicit detail htbp penalty n trend laplacian replace penalty define laplacian smoothing graph lie high leave part penalty laplacian difference choose graph small analogy comparison filter spline smoothness laplacian strongly throughout community generalize variant mostly discrete functional continuous counterpart quite say may meaningful embed aware trend filtering estimate k follow filter odd odd incidence augment odd likewise admm iteratively linear system condition gradient method involve laplacian augment linear solver solve solve subproblem trend problem tv parametric max underlying promising employ stack dual trend filter adopt interior update hessian issue grow problem poor condition experience experiment trend order flow parametric compare moderately versus preferred max thresholding prefer run naive admm soft cc equation mse achieve spatial examine behavior trend filter stanford project compose facebook real facebook user truth evaluate compare favorable entry draw nonzero draw run decay walk assign send number choose noise favorable design favorable design smoothing adaptive estimation wide level achieve square summary estimate laplacian sparsity nonetheless competitive smoothing walk wavelet trend motivate rely regularization semi supervise goal write observe node observe fall trend regularization observe row class th encourage class behave smoothly last criterion prior principle act fix still interpret imputation perform unobserved large htb ratio misclassification laplacian imputation popular uci c c car breast heart ad misclassification rate imputation uci repetition draw serve pair case highlight case specification place regularizer think heterogeneous smoothness laplacian design might broad run machine repository near distance serve choose tune wide experiment summarize rate imputation seed mrf smooth alternative sometimes laplacian uci select entirely base popularity belief favorable heterogeneity label nonetheless broad illustrate regularizers pure trend filter proper smoothly zero observe represent smooth
mu sigma mu factor claim analogous explanation alpha symbol x symbol symbol expression alpha beta alpha beta gamma alpha alpha gamma alpha beta alpha gamma beta alpha alpha alpha beta alpha gamma beta beta alpha eqn expand beta use gamma expand gamma alpha eqn q expand eqn answer eqn prove branch branch branch recover p finally gaussian branch take sample like suppose deviation bound additive algorithm generality normalize true moment lipschitz chebyshev estimate affect thing mean last give cubic show case odd root root perturbation coefficient large root I large sign algorithm reconstruct get remark theorem ignore ignore n pt usa university consider upper giving moment pearson denote necessary sufficient estimate error provably logarithmic yet simple dimensionality far bind separate reduce apply learn strong previous gaussian additive among well naturally arise population vary gaussian biology economic deal case mixture collection biology pearson gaussian pearson empirical distribution th define sufficiently true mixture dimensional parameter identify pearson locate root candidate match moment among pearson pearson prove extended reliably moreover sample complexity quantitative constant sufficient interpret year arbitrary novel surprisingly dimensionality allow logarithmic show necessary important six suffice identify gaussian moment differ first moment suffice provide lead polynomial extend dimensionality within gaussians specify mean pick dimensional coordinate parameter additive hope component simplicity combine say algorithm indistinguishable separate overall weighted average analogous statement precise characterization estimate simplicity component variation ff ff proper parameter estimate gaussian parameter tight characterization sample guarantee total interpret also technical quantitative dominant ignore measure variation distance benefit nevertheless facilitate previous component rather approximate easy approximate discuss gaussian tight bound regime corollary mixture gaussians bound dependence arise th reliably separate deviation mixture gaussian bound also desire smooth corollary theorem gaussian away output n well I essence gaussians reasonably separation relative accuracy second mean treat theorem bound away simplify syntactic level polynomial dependence separation essentially inefficient mixture bind hellinger mixture variance hellinger satisfies square hellinger rule statistical confidence impossible corollary algorithm gaussian learn mixture require tight mean approximate variance algorithm use learn gaussian away note low gaussian small away learn notably essentially incur dependence bind quite notably simple extend different copy result variation gaussians variation dependence exponent nonetheless exponent polynomial dependence interestingly improve isotropic let gaussians matrix far exist constant learn survey reader helpful discussion prior polynomial bound gaussians result least prohibitive moderate mixture gaussian component variation distance improper learning gaussians component however unlike component impossible general state nonetheless strictly good also assumption strong aware improvement parameter outline start moment system equation recover block unstable arise exactly rather exhibit set recover gaussians gaussian add formally add subtract unchanged mean leave independent accomplish call inspire well correspond four regime regime know applicable regime variance apply gaussians indistinguishable parameter regime appropriate algorithm excess fairly depend unfortunately root pearson compute root solution variance get valid mixture moment choose moment moment differently bind match moment another prove small perturbation root nearby perturbation argue nonzero need excess think interested correspond match six moment moment suffice show root compact give differently set region equal high degree coefficient get extend simple straightforward algorithm pair covariance shape iterate dimensional accuracy additional tell associated know newly position must ensure mix simple work use pick valid verification work project four anti form gaussian far true matrix identify matrix give overhead match close hellinger absolutely measure let subgaussian identical subgaussian eq like parameter denote hence series term inner bound constant returning imply approach q sample probability appeal relation hellinger way least probability sufficiently small variance constant match alternatively numerically certainly yield something plug mixture claim require learn gain hard distance group independent probability probability coordinate mean least guess incorrectly overall give sample probability coordinate specify would claim gives extend low gaussian main issue input get separate mixture nearly formalize follow gaussian subject moment expect well separate lemma useful use polynomial degree magnitude zero show randomly draw set consider draw arbitrarily bind singular without determinant product determinant formal know appear constant determinant close minimum result exist mixture match mixture far mixture free gaussians means relative match gaussian constant free parameter gaussians kp pz theorem different mixture match almost mixture minimal singular z f desire low gaussian differ apply desire give algorithm learn theorem result one denote constant similarly denote gaussian simplicity moment away eq make use gaussian increase amount make relate design gaussian plus definition correct also define excess excess section estimation bound zero mixture give satisfy statistic true value estimate I thm sample parameter proceed estimate however work cause estimate good nonzero job bind improve gaussians indistinguishable figure regime invoke simpler weak return additive x p max pearson substitute clear denominator pearson rescale excess root multiple root five moment suffice uniquely identify moment analogously expression remove explanation combine excess moment say fortunately exclude enforce sign root cubic otherwise solution suffice exact excess moment perturbation intuitive excess inspection bound convenient lemma full generality normalization proof recovery extend claim constant max max condition hold either recover approximation getting normalize denominator low denominator trivial rescale p perturbation perturbation p recovery moment examine therefore q gaussian mixture variance excess moment I imply x f estimate decide branch take ideal actual ideal factor perform algorithm gaussian appendix simple dimensional show reduction need factor accomplish f multiplicative reduction assume learn learn mixture gaussian mixture obtain restrict ii order terminate put determine number restrict coordinate ji first k terminate put exist k k total successful variance accurate show succeed case describe occur accurate step coordinate I case far p additive coordinate suppose described step accurate output must variance indistinguishable matter ij invoke powerful let normal eq degree see conclude claim direct consequence vector matrix constant every analogous lemma anti constant probability reduction gaussian fc gaussians rescale mixture grid check use previous lemma grid assumption let net coordinate contain definition sufficiently mm terminate following reject let failure terminate choose p problem accurate permutation enough claim every element accept need constant eq sufficiently union accepted eq hence accept sufficiently symmetric argument hold finish distinguish case center span correspond large probability element gets accept establish probability gets reject reject union reject accept argument neighbor finish distinguish c center distance pair span cluster hence pt pi one randomness possibility gaussians sample theorem learns gaussian gaussians empirical sample tt sg sx good variation prove gaussian intuition work direction subgaussian overall normalization identity eigenvalue therefore permutation approximation since approximation frobenius approximation branch find generality correctly correct classification lemma result measurement complexity dominate improve covariance eigenvalue g
properly model include member phenomenon occur vary underlie among specification effectively address phenomenon rather example specification cluster selection probability variable yet place evenly predictor step let contain reach cluster far choose evenly share piece weight mr mr mr mr mr mr follow stop iff strategy proceed induction easy procedure inductive next end q p us marginal q iii equality iii predictor latent jointly define sequence random final mathematically event eq expectation take final decision event claim see p u put piece map ready mapping find final prior regression bf update numerically prior particular form generality zero place intercept setup show bf versus eq coefficient determination undesirable feature propose prior introduce hyper put correspond bf versus notation
ga see power take high control node country flow country I densely mean translate triangle united china dominant united china steady suggest country volume indicator strength change consequence relation major indicator look middle panel pearson product volume year strong connection however positive south find united take country term economic exception find large major material produce country gate major rather highlight network economic combine conventional deep insight economic total either quantity relation make economic growth development approach big economic live challenge algorithm individual balance payment account activity shift lead forecast centrality identification indicator indicator novel suppose provide powerful monitoring early combine network long portfolio financial product standard ahead derivative product investigate international major local topological economic development individual gate keeping power volume stand establish economic amount gain understanding ever acknowledgement like european open financial cm centre engineering study sciences department college university usa global european policy public aware strong global economic architecture partly central conference macro fail maker limited help go far face conventional link trade size magnitude trade flow logarithm country china triangle trade connect rise availability large quality system component vast cover merge current economic behind generate interaction financial interaction whole science couple main major agent market two change stock portfolio network achieve qualitative gain international trade country logarithm country link size magnitude directional clearly triangle trade china remain already view intrinsic study picture merge indicator fusion trade financial indicator link several indicator result relational evolution describe demonstrate counter financial playing design financial come back relation use economic indicator change economic growth economic attract grow explain economic asset different macro face substantial large e underlie computational complexity total year track flow position technique big predictive power begin apply methodology particular availability fulfil consecutive relational financial indicator country constitute part country eight aggregated show track year position focus country position total china china usa indicators adjust change network represent node allow take call capture topological symmetric direct require reciprocal connection account feedback kind ready single balance account network financial describe indicator interested find coefficient matrix number indicator intercept indicator appeal accord criterion super exponentially number indicator big enter much availability amount time regressor describe likely contain optimisation perform regressor generate model additional regressor order significance regressor factor condition normalise go back final number regressor accept crucially maximally general maximal regressor side balance reject availability cumulative eight together meanwhile union hold total unfortunately partly incomplete expect enough achieve fit great majority indicator see moment strict criterion order accept single test maximal final fit maximal test perform year shift regressor nine fitting forecasting indicator error median fit indicator connect meaning separate cutting tell indicator couple network cover indicator cluster could valid outcome picture globally behind united china indicator fit true node lie tracking capability eliminate potentially predictive capability stand alone country applicability approach indicator dot point regressor colour code capability indicator highlight position indicator regressor axis actual physical portfolio portfolio use measure structural global financial market derivative market early financial relate aggregated term market amount range financial derivative underlie formation lead international participant leave large remove strong link find connectivity contribute indicate threshold market financial derivative evolution edge market expect couple cross correlation market mechanism translate class lead mainly united role international financial market create dynamical view would channel increase cascade get couple good collective may outcome cover network report country country edge aggregated country country dominant global edge million remove edge weak change world year stay remain connected apply threshold mean global describe connectivity growth depicted large temporal evolution take value continuously expand node track aggregated counter product product lag behind rate use market scale linearity scale account arbitrarily square value derivative positive relation indicate derivative market net market availability amount relation financial high maximal threshold safe reference rv market product multipli setting exceed must reduce transaction hold especially trade soon describe amount derivative grey dash line consistent signal red generate product rv variety description derivative may present achieve description around threshold fraction set signal note range applicability applicability description derivative product detail description link derivative
lack historical work address content base recommendation available formalize pool available budget assign rating order optimum verify netflix outperform baseline database mining recommendation increasingly try choose read modern service service user preference database common recommendation collaborative cf g transaction feedback exploit popularity trend much one arise employ deal lack transaction history focus availability content information user rating item movie book evaluate pool available assign item period receive rating adaptively associate select characterization rest survey work introduce notation optimal algorithm common cf item user vector user hold latent translate mathematical new reveal item latent cast budget constrain stand expect prediction devise validate result simulate netflix effectiveness baseline turn problem item indicate high portion item notation denote seek bias vector item user intuitively strong preference vice versa denote aspect train rating term instance verify assumption employ significance formally pool available budget constraint allow rating budget formally rating notation user set translate inherently generate user item divide item give pool user reveal rank latent ground optimal detailed exist well baseline evaluate approach approach unify whole paradigm aim conduct wish select minimize seek actual subset mse basically optimal rating equation least estimator give user provide model assumption rating assume rating user whereas new therefore estimator seek denote concatenation shall stand notation yield column since invertible practice usually add sequel ease emphasize regularize formally notation adapt abuse notation vector whose column vector notation divide two term assume follow key observation optimization simple assume use prove continue mean isotropic invertible transformation item invertible merely simplify statement without compute implement I square estimator term inherent avoid p represent turn available user use vector follow substituting result within r noise equality follow equation equality optimum refer user user initialize j alg j state definition definite u f definite say monotone last extend correspond user alg b mse additive sub optimal notice dominate follow define whose column correspond latent show follow three equal optimum third left prove minimize monotonicity rise insight mse user translate rational marginal subset e eq psd eigenvalue prove specifically column derivative resemble albeit plug monotone operator finally generate substitute alg inequality rely heavily sometimes assume respect model distribution analysis advance rating estimator generalize correspond scale sequel r b identically equation generalize least subset assign rating substitute square rewrite motivation establish adaptation refer root user notation define alg b item correspond user albeit omit advance recommender utilize formally item rate estimate baseline baseline follow large movie dataset netflix competition dataset contain million rating anonymous netflix customer movie process rating henceforth choose contain million rating user rate movie rating rating exist root rmse metric prediction result model descent sgd detail regard evaluation omit offline netflix movie henceforth movie recommender netflix rate dataset set movie rating movie netflix coincide available user conceptual portion actual rating actual remain note task carry metric rmse monotone mse minimize rmse experimental rmse separately
gap order design counter happen finally bound main gap gap decompose cumulative two gap analyze equation similarly trivially gap episode asymptotic bandit bandit low item item draw I distribution bandit bandit item small gap formalize notion consistent number choose analysis loss generality perform low bound inconsistent claim partition bandit parameterize regret bound follow bernoulli variable separately regret sublinear practical match gap free upper match bind adversarial semi gap upper armed bandit comparable armed major gap extremely efficient ccc minimum optimal node bandit experiment episode select observe update environment measure episode cumulative episode divide baseline maximum weight basis notion baseline common internet assumption span formulate six table contain cycle record exponential expect tend unlikely cause high report trend episode outperform episode report learn network span tree therefore observe episode cr dataset bipartite graph connect several region return episode select assign assignment study family call bipartite leave vertex bipartite maximize overall bipartite handle represent bipartite united states constitute bipartite top bipartite handle episode choose maximize overall success rate success assign list policy success learn report trend approach episode increase outperform policy movie title movie american child popular optimal movie return movie episode diverse recommend movie likely movie diverse interest formulate dataset people rate million movie attention rate movie cardinality movie vector indicate movie movie movie diversity ie dataset episode movie list movie cover movie appear diverse suitable diversity report trend episode increase greedy combinatorial semi combinatorial bandit ucb chen regret regret tight chen tight adversarial combinatorial semi main limitation efficient need exponentially need project hull efficient special bandit combinatorial past year submodular monotonic propose algorithm suitable first specific weight unknown learn interact repeatedly sublinear world practical practical efficiently introduce case combinatorial generalization combinatorial optimally one idea quite applied involve lem basis exist ia constructive exchange augmentation exchange k ia main finally step set contradict item event counter happen follow equation happen eq combine claim due sequence therefore conclude lemma proposition fact remark com notion independence closely modular bring propose combinatorial bandit problem maximize modular problem prove bound free bound sublinear interest prove world application resource designing protocol modern problem polynomial fortunately combinatorial closely efficiency find common forest modular modular represent sum item vector unknown interact world span delay stochastic finding unknown perhaps explore network return contribution bring concept bandit bandit new broad combinatorial optimization solve solve explore face computationally efficient episode sort number sublinear episode linear maximum network network maximize third movie recommendation efficiently framework real adopt subset denote remain set cardinality negative solution design principle greedy find weight basis optimistic give begin episode q episode estimate item episode order dependent item confidence upper exploration episode exploration avoid regret extremely episode sort motivated work challenge regret basic notation decompose rely heavily
gradient nonlinear behavior example discretize enough acquisition behavior map several acquisition exist experimental ucb ucb acquisition trade ucb emphasis easy adjust see across weighted area space may trade code bayesian available degree position control mx front back control robot fast operating system simulator dynamic physics flat segment robot simulator ode angular govern periodic amplitude cycle cycle period signal hz amplitude cycle signal filter sharp angular send every ms third angle controller parameter numerous different purely setup type controller controller classic self controller design keep balanced least place repeat cycle static parameter reference controller keep cyclic orientation subtract angle actual important performance tend fail chance behavioral descriptor contact factor controller simulate record whether contact contact behavioral descriptor store cell behavioral descriptor space behavioral descriptor actual discretized behavioral descriptor robot characterize angular position robot proportion interval roll frame additional robot end interval ms second movement return argument exceed return discount around orientation exceed robot possible parametrize move pre behavior thank result robot adaptation measure mapping fast compare reveal measurement robot flip backward distance great adjustment consider behavior additionally inaccurate outlier supplementary substantially promise maximum physical controller robot unlikely decide worth well controller discover controller perform stop eq location behavioral predict terminate select alternative way robot equation event occur robot text solution criterion strictly guarantee stop bad behavior map test trial adaptation drop choose area controller parameter controller increment behavioral behavioral million physical extended fig robot release ball classic assess place camera tracking track color eight robot position control maximize heavy near mx mx mx ax simulated robot way dynamic version simulator ode library result map controller target position controller parametrize eight angle motion range joint activate drive target choose make reproduce highlight trial recovery advance realistic experimental trial controller controller parametrize chance change value robot behavior position behavioral position position measure discretized compose cell experiment arm behavior map step performance work location bin accomplished performance movement specifically minimize angular joint mean angle map accomplish measure create distance well descriptor target bin joint angle use create behavior controller robot distance external camera bin physical position controller controller position outside camera marker rare corresponding experiment experiment frequent adaptation purely random angle continue time minimize cost run independent algorithm replicate detect low auto trial contain trial simulate release bin adaptation stop bin cm stop adaptation within controller controller dimension evaluation million execute condition physical robot measure manually measure dash manual minimum bar text report section come robot behavior influence single affect magnitude test affect behavior affect value extend paragraph test conduct behavior affected affect repeat testing replicate dimensional stop adaptation pass stop text criterion increase change matlab range explore around dotted red large change algorithm rarely iteration occur many cover entire search fast risk promise area space minimum physical robot choose search already largely step map area thus avoid choose exploration experimentally conduct intel ram take across robot happen robot map robot simultaneous localization fast slow million per second powerful computer frame process computer accuracy computer step much easily arithmetic robot second adapt conduct robot overall physical robot second second initialize robot second allow measurement second identify controller time arithmetic column conduct select second second investigate physical text datum scenario scenario robot generate map run consist directly parameter time experimental control need trial model inform enough effectively empirically dimension policy algorithm trial allow highly illustrate performance variant search previously directly search high show error produce high space search initialize map initialization allow evaluation typically previous art policy variant improve still publish automatic dimension trial run original evaluation gradient bayesian powerful provide prior optimization component trial significantly recovery state environmental robot recovery flat supplementary create also eight map increment supplementary roughly robot perturb multiplicative trial design classic slope trial find slow behavior learn trial error outperform flat every slope angle controller trial reference controller setup map increment replicate degree increment trial variation slope trial error find slope find fast slow positive ascent algorithm perform trial cause controller sensor controller keep science course perform behavior keep vertical reduce nevertheless trial outperform median performance discretize map map point behavioral randomly location store intuitively far keep intuition understand advantage map controller behavioral descriptor diversity behavior average behavior map million evaluation robot distribution difficult supplementary median percent cell percent appear fig random discover numerous also million evaluation average cell whereas reference controller robot diverse search measure performance million behavioral descriptor dimension behavioral behavioral behavioral descriptor proportion contact create describe test performance alternative behavioral descriptor descriptor evaluate affect behavioral descriptor test behavioral descriptor behavioral descriptor contact denote boolean contact contact contact orientation behavioral descriptor characterize change angular measured proportion roll angle dimension roll angle robot movement return exceed return otherwise motion around angle exceed orientation angle negative dimensional behavioral characterize proportion ms intervals robot along axis robot interval second simulate movement less mm dimensional behavioral descriptor second utilize robot second movement measure simulator behavioral descriptor capture move movement utilize second movement deviation descriptor capture robot location robot straight speed robot center final axes maximum robot position dimension second axis robot expect robot speed behavioral descriptor multiply reaction force behavioral descriptor ground reaction force movement ground reaction force behavioral apply ground reaction generate average second simulate movement angle descriptor capture ground angle contact ground angle normalize low roll angle descriptor roll low ground coordinate roll angle time low ground roll range ground angle global coordinate average second contact angle behavioral descriptor differ knowledge randomly behavioral descriptor intend little behavioral descriptor quickly pick descriptor consideration instead generate list fashion randomly different select random without descriptor descriptor descriptor available descriptor behavioral descriptor physical robot repeatedly break robot modify simulator remove map million map behavioral robot generation store cell behavioral descriptor behavioral descriptor behavioral descriptor actual discretize eight behavioral descriptor map behavioral descriptor ten replicate descriptor behavioral behavioral descriptor therefore replicate descriptor replicate descriptor randomly perturb noise deviation fast behavioral descriptor trial robot require trial experiment achieve choose behavioral descriptor similar behavioral descriptor lead median descriptor descriptor roll angle performance discover descriptor significance remain descriptor additionally discover alternative choose descriptor lead well evaluation experiment extended trial difference behavioral descriptor behavioral three choose low angle median descriptor performance orientation relative relative roll descriptor remain behavioral descriptor angle descriptor case statistical significance statistically descriptor random behavioral descriptor descriptor descriptor descriptor performance significant reduce descriptor factor description choose behavioral descriptor behavior discover descriptor reference experiment show selection behavioral behavioral descriptor randomly median prior reveal algorithm descriptor trial robot video behavior produce map classic robot deal finally video illustrate robot condition video type classic form reference extend figure table france l fr leave environment however adapt variety think box behavior specify failure diagnosis contingency plan introduce trial adapt require self diagnosis contingency novel create detailed behavior behavior guide discover experiment successful robot way break broken way enable suggest principle economic notably distant deep obstacle complex environment behavior behavior behavior effective cope broken occur case robot straight begin robot type behavior automatically generate behavior behavior well robot rapidly diagnosis design contingency self monitor sensor possible situation fail diagnosis plan differently trial allow discover behavior limit occur impractical curse dimensionality fast algorithm constrain tuning require minute limitation adapt rapid adaptation automatically compute thousand behavior insight whereas start survey understand behavior previous enable validate store knowledge robot try behavior performance behavior end predict behavior discover quickly way without understanding occur trial behavior create robot automatically robot describe dimension behavior behavioral measure speed contact demonstrate degree capture behavioral fill performance search perform behavioral extended fig simulate million behavior need perform per robot assign behavior try robot drop select promising measure robot behavior nearby assign extended continue satisfactory idea capture gaussian approximate acquire search select behavior maximize information select uncertain exploitation select whose behavior physical record behavioral update update affect test whose measure good performance behavior robot need camera estimate supplementary parametrize parameter cycle joint supplementary space contact supplementary fig failure run independently behavior default factor behavioral description adaptation time independently generate alternate behavioral see fig lead search perform creating robot experiment space six contact behavior robot promise update performance behavior map nearby behavior confidence similar robot well predict performance fig behavior high overall behavior affect behavior c c central box extreme create factor behavior per supplementary method condition test create body orientation behavior box trial robot drop ball joint degree offset broken offset joint condition create behavior robot dynamic classic reference median vs suggest trial produce robot aside effective reasonably speed time vs vs vs demonstrate robot initially fast reliably physical reveal art adapting environment trial less second physical four include randomly descriptor b fig performance map standard bayesian work initially area adaptation work robot error quickly identify high approach robot arm condition arm behavior behavioral specify angle approach less minute second fig fig map behavior portion contact ground behavioral space discretize dimension color highest discover behavior behavioral dimension legend leave behavior leave dynamic engine simulator http www ode matrix pre adaptation map adaptation conduct robot right matrix discover circle represent behavior physical robot red discover amongst behavior find error predefine cope condition new body behavior work quick behavior try modification additionally optimization procedure similar employ human strong evidence combine prior bayesian trial error period idea day quickly improve simulator predictive exist differently rapidly adapt circumstance thank discussion european research european union horizon innovation agreement correspondence request material email behavioral controller far repeat depict newly rarely map expensive perform per robot create multi computer supplementary running behavior simulation dark green uncertainty predict light green band actual robot dash balance behavior perform try behavior acquisition initially maximal performance physical behavior performance uncertainty nearby robot maximum behavior threshold performance dark occur physical test expectation occur variant behavior prior equivalent map map none none bayesian al none gradient al one art et et al colored discover evaluation robot panel pool removal show require slope angle physical trial outperform black six scenario pool condition median behavior trial reference slope line color dash classic reference try median colored area supplementary experiment speed discover descriptor simulate remove six scenario pool across behavior descriptor contact robot instantaneous velocity robot robot straight vi reaction force angle ground without replacement descriptor design descriptor bold line color median colored area extend colored circle supplementary experiment color represent performance high map black circle indicate indicate performance versus reveal robot legend color map black circle robot behavior test performance versus panel point last behavior robot previous whether leave typical performance produce map behavior behavior angle joint arm reach tend reach behavior nearby robot physical robot replication pool experiment success replication replicate robot reach cm bin center performance physical condition controller behavioral space discrete behavioral physical performance encounter controller currently store behavioral descriptor reality robot robot user deviation gaussian major behavior adaptation adapt new environment behavior introduce map create search high
entry edge node stand lie define graph nevertheless universe cast adjacency sample triplet feature good label interact get easy thus partition four depend whether node involve resp resp predict four represent unseen unseen unseen family four undirected two prediction validation evaluate network procedure evaluate global adopt practical context tree ensemble presentation assume derive class apply sample concatenation straightforwardly new unseen homogeneous graph handle sample without constraint symmetric separate try around constructed learn model exploit combine arithmetic make node learn node r cn train symmetry prediction model try scheme model take max prediction lead improvement building sample conditional estimate set threshold proportion edge versus specialized homogeneous asymmetric still node could base classifier ensemble context briefly several tree tree feature terminal instance output associate reach instance identify split sample output typically make competitive term extremely select good randomly split root candidate attribute feature output instead method local global learning sample feature come tree grow construction leaf result rectangular submatrix submatrix figure illustration ensemble straightforward variant build tree output model subset build sample r make require tree output submatrix base cope miss us case global region root leaf input partition matrix profile partitioning pair submatrix furthermore feature tree case local grain another give interpretable rank local multiple provide ranking one row output separate therefore complementary interpretability prohibitive network million fortunately go separately relative relative explicitly gram computational complexity tree tree practice relate sample computational complexity multiple output complexity however output carry six biological homogeneous bipartite four find assess relative approach method mn ern three bipartite main interaction protein highlight feature datum localization use gene edge growth fitness environment successive feature ern bipartite tf gene tf connect expression drug target connect drug protein vector presence chemical drug absence fold cross robustness run fold cv assess performing time fold illustrate return choice discretization precision roc curve fold cv extremely tree highlight high node expect node involve want assess realistic usual evaluate degree pair pair degree evaluate protocol baseline successively homogeneous bipartite local output last curve approach mn protocol similar curve appendix auc l ts ts ts ts ts pair informative interaction baseline network highly curve informative performance roc approach mn multiple indistinguishable result line indeed multiple grow single able tree additional precision obtain ern protocol appendix cv explain difficulty cv replace fold burden randomization extra tree ensemble bootstrappe rp ls ts ts ls ts ts ts ls ts ts tf tf baseline drug protein high node pair prediction ie significantly ern kind prediction generalize tf kind equally due kind ern intrinsic difficulty generalize family four generalization protein relative number well baseline degree prediction generalize local ern multiple term additional several method literature ensure fair avoid tuning paper summarize publication protocol measure result cv roc mn ern cv develop apply local predict mn exploit performance local multiple ensemble infer mn inferior mn ern focus know evaluate cv ensure belong close slightly good regularize classifier one protein prediction prediction well additional comparison method competitive notice test tree achieve randomization explore ensemble biological train network family train single carry compare state intrinsic importance score almost nature reasonable computational requirement turn method less model approach term advantage possibility introduction one loose however possibility method extend local unseen step kind train prediction svms benefit reduce potentially compute well improve exploit potential correlation interesting sl local approach focus biological network evaluate method try date incorporate tree base ensemble method however comparison like family protocol term prediction want merge family rank novel confident question largely biological record unlabele negative notable exception unlabele theoretically account example acknowledgement bioinformatics platform provide resource network class channel protein nuclear similarity protein similarity chemical structure number protein size edge curve l ts ts
good candidate create corresponding rank candidate scatter size increase repeat increase size small candidate dendrogram feature default choice result candidate size auto instead true dendrogram still htbp propose simulate contain x p organize cluster control separate remain clustering linkage partition cut dendrogram measure candidate framework either candidate specific pre candidate average standard different setting average produce accurate affect cc another cluster specify choose true variable candidate sr produce sr indicate selection selection accuracy affect ratio affect cc average deviation auto candidate select candidate natural sr auto candidate may sr sr drop sr affect sr sr investigate r naturally parallel computing rank currently boost simulate fix true variable set combination setting linkage setting number different second figure conclude long computational time cluster relatively explain complexity approach computationally demand give scale benchmark perform complete linkage classical feature gene auto select gene auto specify cluster detail use also choose candidate number three case specialized gene level process step pre exclude logarithmic transformation datum array nearest neighbor miss value feature misclassifie case misclassifie auto feature satisfactory hierarchical high conclude produce accurate htbp breast breast tumor identify class er observation belong four publicly result mix auto select case misclassifie marginal conclude microarray gene identify type patient distinguish significant patient classical hierarchical feature misclassifie misclassifie poorly feature produce misclassifie subset default expand default classical feature high marginal mixed cluster thus however analyze breast publicly discriminate distant within classical clustering misclassifie high relatively misclassifie case set perform well interpretable less framework select cluster hierarchical reference flat contain collect store medical tool cluster flat coverage feature explain underlying hierarchical observation adaptively limitation complex sparse framework propose produce compare sparse datum exist furthermore interpretability use selection demand hierarchical cluster hierarchical observation dendrogram broad microarray imaging mining etc cluster brief survey proposal row additive unless equal employ version automatically attribute variable extension complex multiple note truly variable propose new criterion j j nj feature weight directly difficulty dissimilarity proportional dissimilarity criterion obtain sparse dissimilarity remove component spc column alone spc dissimilarity dissimilarity feature obtain classical cluster dissimilarity show genomic interpretable dataset limitation criterion reduce hierarchical hierarchical spc dissimilarity e dissimilarity calculate mean fully illustration issue example contain x organize represents gradually increase choose feature include dendrogram cluster dendrogram hierarchical many naive compare satisfactory feasible rapidly performance difficulty subset several number size candidate cluster loading spc spc directly transform loading spc relate low refer candidate criterion dendrogram result cluster accept split node dendrogram dendrogram leave leave terminal leaf height dendrogram repeat cluster label compare cluster base area knowledge specify default replace conduct whole datum result dendrogram main selecting give candidate choice candidate present subset fix size candidate reference assign apply spc say assign step f subset contain conduct hierarchical use dendrogram select good dendrogram label leave choose record rank candidate cluster label calculate use candidate scatter value rank local discard one create scatter rank increase decrease j otherwise repeat small go small rank candidate loading potentially variable different use spc feature different rank set
work surprisingly regardless degree expansion reasoning provide extension clearly treat input dot inner permutation next behave variance albeit dependence input evidence simplify term product begin square extension stack matrix defer subsequent pair decompose goal compute hence orthonormal matrix follow u identical cholesky addition explicitly compute integral moment analogously construction mean combine write computing correspondingly plug simplify claim lagrange plug act hadamard agree permutation randomly randomly subset fix k calculation act put claim denote map stack iid copy q average arise omit theorem show give block weak believe could improve considerable analytic say guarantee performance work practice confirm nonetheless immediately benefit let x dd demonstrate almost net use random projection function note rather set relative decomposition arguably approximate inferior direct hadamard weak conjunction matrix motivation end isotropic albeit purpose kernel dd case rbf dataset via spherical contrary kernel fourier concentration rbf kernel actually rr speedup ram x exact rbf rbf rbf ct slices n year forest n cpu feature rbf perform variant kernel useful tb par computation offer eigen algebra library interested go take around speed large problem evident confirm dimensionality importance expansion cifar dataset pixel accuracy feature achieve expansion slow total slow demonstrate expansion raw class use test offer overcome obstacle competitive problem require real prediction run include formulation practical tool offer method research multiplication near see ex mm claim remark definition scale store compute decision typically expensive prediction difficulty propose approximation computation exhibit property unlike hadamard multiply store propose computation kernel dimension improvement translation dot polynomial experiment achieve full less memory memory make set prediction successful range extraction heart inner infinite idea show nonlinear separation body literature hilbert rkhs norm penalty furthermore one interpretation via gaussian detail employ trick day ten state expansion finitely coefficient must fairly whenever effectively space exploit frequently solve expansion instance show number many problem linearly consequence expense grow method exceed instance large solver albeit limit kernel issue compare access solve reasonable almost average nontrivial fact subgradient associate nonlinear expansion expansion full problem cost subsequently discrepancy expansion basis exponent arise reduced operation storage subsequent aim find expand suboptimal evidence well basis function extract good reduce expansion encode likely dependency covariate projection project albeit method storage able million gb memory store obtain minimal recent provide guarantee expansion offer one efficient expansion relatively dimension curse tune localize rbf kernel promise compatible choose iterate data suffer membership observation optimistic summary function potentially promise exceed algorithm discuss promise translation invariant kernel nonlinearity storage operation reduce expansion show conventional rbf simplify code expansion offer convergence decay satisfy key trace kernel normalize basis function establish prove limit function relate via exist e class computationally kernel action eigenfunction kernel basis reason whenever find efficiently decompose irreducible decompose irreducible eigenvalue invariance dramatically simplify construction unitary orthonormal concrete kernel translation fouri unitary kernel expand particularly fouri transform many may fourier expansion gaussian harmonic choice fourier transform gaussians equality fourier spectrum express multiple convolution polynomial one rotation spherical term provide radial contribution derivation kx degree linearly homogeneous polynomial dimension coefficient expansion expansion l always since unit sphere expansion homogeneous expand accord prove correctness sufficient matching follow establish equality line representation key depend case may isotropic product isotropic vector accomplish construction denote obtain expansion linearly homogeneous variable uniformly able efficient dot kernel symmetric invariant kernel satisfy symmetry efficient mean rapidly case expand symmetry undesirable instance fouri undesirable deviation phenomenon distant observe desirable expand function achievable expansion likewise expensive possibly efficient alternative generalize derivation nonlinear multiplication computation symmetric depend norm inner provide operation special term well clear express product basis need obtain expansion approximate follow integrate conventional conventional explicit evaluate approximately random cost subsequent operation divide proper weighting spread expansion tail easy rather express kernel evaluate directly expansion exist case draw sphere price function rather tool fix polynomial degree respectively cosine x follow homogeneous rotation follow integral odd integral vanishe integration sphere via rescale exponent curvature g detail offer simple immediately without form commonly x p considerable sufficient fast expansion introduce begin yield following discuss previously average method operation seems really multiply rbf summarize two reduce avoid store take gaussian cumulative replace generator progress need simplicity gaussian general subsequently without generality hadamard hadamard dt candidate cosine dct main diagonal scaling compute store hadamard multiply hadamard variant hadamard replicate random stack enough feature prove verify computational cost storage matrix entry operation hadamard storage implicitly transform operation carry block compute basis cpu budget establish sufficiently row long hadamard permutation see amount operation also iid act isometry ensure hadamard incoherent store sorting
power consumption explain researcher resource consideration fluctuation price day name automated area machine reinforcement different state electrical prediction instead section literature behind section conclusion vi work learn demand consumption notice want play device device moment paper relate publication learn train world individual consumption reduce energy consumption purpose exist appear numerous introduction detailed subject behind learn section reinforcement decision process consist perform type agent aim lack knowledge environment affect future develop would action good reward stage reinforcement learn particular make correction give action action state agent action h initialize pair state episode choose action next update episode hour episode current hour day go state control decision connect dynamically early hour several number give moment thing place total amount ready pay state initial intermediate pay car section time come expect ready assume energy moreover amount price varie learn typical total demand customer day characteristic queue type update arrive among place application hour maximally possible speed hour must go whose remove array sort tie occur type word array decrease every jt detail gradually decrease number system newly manually dependent newly day hour whose origin describe reward q latter reinforcement approximate state q describe essence initialize initial exploratory state cost pay pay price want type represent family pay extra family sigmoid refer empty pricing energy explain wind wind panel average wind generator day file wind generation france sake consistency number wind france generation france amount hour panel generation generation assume pricing france website price half hour decision depict measured day result purely strategy easily formulation percentage evidence function medium rt take h try several
priori technique isotropic class choose volume identity shape structure gaussian find limited sample produce quick little expense number low computationally take subset datum block incremental storage whole package handle large datum fail sample cpu computational expense limit heavily overfitte information contain bayesian outcome homogeneous numerical integral formula integral lead prior respect treat hyperparameter uncertainty account imbalance imbalance imbalance maximum entropy observation subtract new transforming bayesian hyperparameter hyperparameter evaluate integral unit towards term integral predictive dimensional integral integral large limit variance dimensional integral line evaluate integral lead negligible get lead simplify simple propose consist classification varied increase dimension low last handle dimension mass move away remain separate class high dimension allow class peak move increase constant increase mean variance indistinguishable dimension class classification number class imbalance observation h version simulate dimension receiver roc suggest lead dominate continue dimension expect neither class improve give subspace although use illustrate notice overfitte overfitting demonstrate well mm accuracy ex ex ex ex ex ex negative breast patient uk record exclude yield initial diagnosis group breast cancer year aim patient expression predict patient patient imbalance h priori outcome candidate report outcome order coefficient rank reduce set shown perform cause accuracy decrease test although decrease per h ex additionally posteriori information lie order less accurate gene handle rank information lose technique priori dash line dimension dimension three formula predictive produce superior efficiency succeed overfitte dimension extension take account instead apply training outcome limitation expand approach heterogeneous also choose optimal gmm uncertainty hyperparameter could incorporate risk heavily overfitte remain discriminant dimension parsimonious uncertainty overfitte remain imbalance dimension take resource result formulae institute molecular college mm uk ac uk centre mm department mm outcome high datum problematic imbalance overfitte become computationally apply overfitte level bayesian reduce integral obtain integral use simulated datum due computational efficient bayesian integration outcome classify use automatically observation test assign model discriminative variety context involve conditional base optimal validation efficient set number arise overfitte combined outcome prediction know datum class class observation belong x belong assumed independently typically probability belong imbalance accord belong
fields mrfs must resort iterative chain classic start repeatedly pick sophisticated sampling wang principle space focus univariate simplicity univariate define configuration vary define guarantee sequence run gibbs total extremely make gibbs ise strength strong main distribution gibbs sampling update mix time systematic scan way establish example verify case require computation form interaction ensures see fx prove summarize ise fast mix spectral absolute imagine necessarily like another derive projection measure result close norm singular frobenius v obtain close ise parameter issue general dense project needs control provide take setting otherwise enforce obeys enforcing find close derive triple dot close guarantee mixing notion closeness vary convenience solve something ease parameter ise theorem probabilistic interpretation divergence perform also project strategy iterate dd dd calculate project case sampling sample chain gradient estimate dependent descent markov total number divergence avoid tend assign assign nonzero easy however slow mix difficult toy graph inspire tree tractable motivation define remove continue distribution onto derivative divergence compute use maximize flexibility select simply tree cover make kl place reweighte variational mf factorize tractable much theoretically marginal run long take long gibb rapid asymptotically comparison topology grid graph node draw strength mix attractive respectively average random calculate divergence subgraph tractable grid horizontal vertical subgraph also random span tree cover original subgraph stochastic divergence pool repeatedly single affect pool mirror descent intensive iteration perform reasonably gibbs pass scan account effort total euclidean projection none original accurate scheduling interpret rapid mixing weight practice still rapid mix property give original distribution cc high intractable motivated approximation tractable family notion obey ensure gibb rapidly rapid matrix strength intractable family onto solve iterative divergence firstly consider novel piecewise subgraph secondly kl generate project gradient experimentally gibbs sample approximation enough sampling original accurate onto give extend mrfs secondly project onto mix also mix apparent term interaction strength know threshold roughly know quickly spectral equal conservative tight mixing occur minimization minimum fa ise absolute pass problem lagrangian return multiplier enforce matrix enforce duality saddle exactly minimizer establishe similarly dd divergence firstly dd fx px observe thm university new south inference ise model high gibbs time guarantee divergence gibbs sampling strength available high cope tractable intractable attempt fully factorize kl distribution tree overlap cluster way propagation reweighte
isotropic kolmogorov scale analogue hold ball graph one ball check note ratio integral manner quantity isometry metric embed reconstruct non neighbor short graph call von set straightforward convergent weighted geodesic metric multidimensional us isometry perfect neighborhood size graph degree code statistic centrality walk von neighbor density path von fail converge level perfectly estimator regime example figure gaussian right track near black fails cope isotropic line figure probability concentration parameter estimator maintain dimension validate nonparametric graph ground amazon sale recommendation naturally asymmetric notion product category popular association sale effect closeness fail display multidimensional metric belong separation across notably history overlap expect music computer science book little serve suggest overall amazon suggest walk centrality identity stationary invert extremely rapid metric near neighbor theorem technique little world information test predict amazon product figure question graph require connectivity suggest regime perform perfectly suggest degree bind much density suggest combine may lead highly effective theorem thm conjecture corollary conjecture mit mit connect iff graph vary arguably ask recover underlie density degree least graph resemble centrality empirically perform well well even analyze occurrence cluster unsupervised recover unweighted direct drawing arc distance typically symmetric construction typical neighbor arguably friend property local density recover integrate show walk direct relate term enable may isometry near neighbor graph e lie region ball walk graph toward high density recovery primarily focus aspect believe offer analyze amazon draw co amazon extend shape simultaneously product similarity metric embed address von integration ordinal graph restrict near provide dimension apply strongly von similarity manifold example discrete neighbor latent prove entire trajectory probability let radius depend analyze unweighted direct edge observe specify latent possibly specify problem walk weakly process via convergence rescale walk closely approximate walk move toward region follow state control technical express solely radius center rescale necessary regularity discrete drift regularity time rescale converge space brownian motion equivalent enforce diffusion strong claim hoeffding borel converge uniformly hold regardless graph drift drift diffusion coefficient draw kronecker delta converge verify claim limit taylor expansion lie inside definition integrating result
infer subspace challenge specify dimension challenge embed sphere relatively specify sphere posterior consistency utility manifold characterize riemannian machine manifold intersection possibly space space manifold mixture special arise subspace mixture linear quantitative application communication infer mixture subspace reduction projection onto subspace go contribution fisher interesting statistical great address reduce summary summary combination statistic range algorithmic nonlinear challenging probabilistic multiple population distribution mixture useful model datum assume dimensional ambient often population degree number population address arise mixture restricted manifold offer limited subspace subspace penalize zhang significant difficulty address subspace approach require come jump immediate subspace vary sphere distance subspace remove occurs move tool posterior structure paper subspace use frequentist analysis bayesian prove procedure utility discussion notation subspace denote manifold column group letter normal orthonormal basis equivalence x article involve multiple copy even draw independent manner subspace ambient concentrated subspace basis state normal residual orthonormal mean estimate affine subspace serve generality diagonal observe ambient either equivalent convenient parameterization likelihood component specify entry entry give mixture specify parameter conjugate location normal term dirichlet triple inherent difficulty sampling want fix subspace prior triple specify specify conjugate von fisher operator von spherical conditional specify seem since assume change needs add close avoid raise subspace integrate nuisance specification subspace recall set equivalence appropriately dd embed ambient embedding embed nice sphere measure subspace projection sphere radius proceed compute matrix low triangle lie key extra subspace high aa half origin summary manifold embed section integer geodesic angle subspace embed projection h sphere coordinate first column diagonal act sphere coordinate useful provide projective structure distance sphere projective subspace projection invariant projective square everywhere optimization distance distance van distance maximize distance distance numerical instability sub easy principal computable value exploit euclidean computation simple coordinate embedding pl pl embed spherical place projection place low half subspace half low projection point image address pre subspace minimize subject compute equal eigenvalue point euclidean full low sphere correspond trace sampling subject parameter parameter sampler sphere lower efficiently difficult address sample projection matrix gibbs close obvious sphere efficient base loss th close subspace subspace put subspace temperature gibb orient posterior traditionally misspecification overfitte arbitrarily use hasting effectively sphere fact relation subspace correspond procedure correspond trace procedure compute proceed draw mm mp p initialize multivariate normal sphere normalize projection embed step dimension procedure z mm z mp acceptance compute step procedure k center walk respectively th analogous j subspace instead matrix inverse u ti ki ik kn k kk tu f respectively probabilitie latent metropolis adjust first stage burn decrease variance adjust burn period sampler autocorrelation provide procedure extensive extend trivial space datum neighborhood weak continuous hellinger hellinger neighborhood kullback leibler kl denote neighborhood sphere uniform von distribution project section induce subspace denote induce e dx f x posterior consistency weakly weak radius leibler result weakly need basis assign mass kl choice infinite dirichlet process measure model theorem open subset measure first large pick approximate finite sum neighborhood continuity k assign theorem enough strongly radius hellinger model element depend let fx prior q posterior consistency hellinger posterior distance j j eigenvalue give k cover ball center origin therefore j ii mn whose fall divide interval metric entropy bound prior
query nature directly study loss closely examine metric investigate weight loss discount next probabilistic derive loss base theory weight reverse fields mrfs pseudo propose mrfs piecewise upper bind mrf finally aspect site web yahoo question answer input functional functional losse multiclass logistic base approximate design multiclass logistic functional discover control properly another rating small lead frequent occurrence tie suggest functional rating group object aggregation group functional indicate max geometric maintaining requirement piecewise discount influential rating combine effective section present picture specification rank piecewise weighting describe approach loss design yahoo yahoo section provide aspect conclude ideally impractical I sort simplicity drop functional rank relevance score e query estimate lc ndcg rr position depend represent object separately modality useful relevance quality since transformation space transformation compute association use sigmoid nonlinearity option q functional reduce estimate conjunction parameter hypothesis certain feature conjunction emphasize ensure benefit unlikely impose pearson uv respectively relevance score correlation min arithmetic issue often presence relevance tie may useful operate tie object like object define aggregation function sensible necessary whose list relevance level individual function reason impose tie perform average complexity object computational g whose next clarity drop reasonable rank metric ndcg see position object rank position loss focus situation emphasize good situation reduce choice multiclass pr pr well sophisticated three wise sum metric directly metric sigmoid function drawback flexible sigmoid approach approximate wise sum weighting depend perhaps prove appropriate weighting ndcg pseudo likelihood logistic combine es piece depend study literature often losse fact unweighted logistic actually ndcg study effective functional effectiveness experimental design specification likely emphasize rating discount section function start specify query cccc accord rank object essence specify permutation clarity reference well respectively start joint distribution accord rule eq shorthand representation informally interpret object probability axiom choice proportional translate worth finally pl context typically pl likelihood however good performance put emphasis ignore suggest still representation part interpret irrelevant thought worth instead reasonable specification interesting interpretation object weight e instance treat rating variable clarity drop explicit factor accounting object e encourage rank graphical mrfs case resort approximate several approximation retain make easy eq proof reader general metric propose loss form evaluate case involve answer site aim experimental present empirical risk study also detail subject paper omit clarity prior limit like offer traditional answer yahoo collect answer question question answer answer test multi opt multiclass eq separate eqs answer frequent training answer functional conjunction multiclass logistic metric note per essentially multiclass embed space appear multiclass employ yahoo relevance perfectly document subset another pre feature fall combine whole set improve space functional weight pairwise table report overfitte since number free space scale number loss comparison c c aggregation mean geometric definition unweighted unity include rank report functional type competitive various rank functional report approximate metric ndcg assume object query performance comparable pairwise see n quadratic hinge ndcg ndcg score list appropriate weighting decomposition report weight however role generally help probable reverse put remove bad see effect chance object pairwise clearly demonstrates greatly improve quality rank list approximation mrf likelihood aim design rank domain aspect choose functional losse logistic approximate loss specifically predict good multiclass rank losse state history decade rather complicated pre compute reveal plausible assessment multiple useful detect evaluation confirm provide employ together yahoo space functional achieve different contain rating tie rating group score benefit operate level group much attain task optimisation metric complex convex make optimisation share message many piecewise locally variable particular rank discount weight influential pairwise rating effective literature notably net bilinear power scale bilinear combination fall work contribute place piecewise log loss weight element bind ndcg score ndcg weight investigate wider weight consider field hard pseudo approximation study approximate metric contribute rank tie little tie group tie group functional
sm represent cone span vector span half complete characterization example vector unit disk half space extremely grow sm sm semi coordinate descent optimize unconstrained least square solve dedicated nonnegative solve propose row therein implement converge exactly give although initialization dimension matlab semi nmf original nmf initialize mean column centroid column binary add modify decrease hence rather would initialize drawback sufficient nature sometimes numerical denominator exactly require product significantly slow observation coordinate optimize nmf strategy initialization strategy initialization construction theorem initialize compute via truncate flip motivation behind effect correction good frobenius directly fact algorithm generate decrease ht nmf good multiply b ij compare initialization strategy turn appealing point equal bad initialization equal completely solve semi nmf matrix properly zero contradict z column remove factorization nm mb yy loss discard influence interior generality fact orthogonal complement q hence modify prove since must note row since flip keeping ii formula remain strict rv r remainder sm nm ni solve nmf polynomial suffice check check whether inequality feasible check model sm sm polynomial form sm decomposition nmf approximate nmf good approximation contain truncate svd polynomial follow svd optimal time therein belong case propose although interval hence feasible feasible terminate semi stop soon smallest far initialize large ten choose precision initialization refine nmf locally implement use perform multiply optimization problem rv matlab tell semi nonnegative fact semi replace note strategy possible sm interior contain positive space matrix frobenius guarantee irreducible nonnegative eigenvector irreducible connect every vertex let irreducible nmf truncate corollary suggest semi unconstraine truncate svd challenge mean nonnegative nmf nonnegative really nonnegative encounter irreducible even likely exist nmf bring several author nmf nonnegative view optimization technique update guarantee however nonnegative initialization make impose would enhance per entry semi semi factorization sm sufficient condition algorithm semi nonnegative intuitively column column belong positive semi whenever nonnegative exist note mx semi nonnegative likely section positive semi semi seem nmf np equivalent since arbitrary nmf ball stein net equivalent necessarily semidefinite positive semidefinite let follow symmetric checking whether nonnegative co np reduce since nmf surprising nmf solve general nmf positive element matrix clearly nmf np hard allow negative problem nmf well question communication always example half however infimum tend semi nmf problem pose cone approximating require uniform matlab notation take cluster cluster add update constant stationary taking indicator seem difficulty initialization stable see remark initialize initialize optimal row irreducible subsection generate synthetic dimension synthetic initializations rd resp rd perform iteration rd initialization km initialization real error input nmf rank unconstraine far away percent nmf approximation nonnegative matlab intel core ghz go ram user convenience generate theoretical finding compute nonnegative generate interval note irreducible sm mean interval value display define perform initialization perfectly lead rd km rd km solution close nonnegative average nonnegative nonnegative semi sm normal distribution previous average absolute add proportional display box previous single ht cc always dominate km appeal seem much practical km slightly rd initialization seem beneficial case test quality good perfectly uniform half space take condition meet well meet surprising approximation rd km bottom well close nonnegative km gap bottom figure even rd km also example become vector nonnegative perform get far away nonnegative expand increase matrix point always within iteration confirm cc iteration initialization recommendation give follow nonnegative recommend rd semi nmf particular factorization rank perfectly try nmf take semi nonnegative sm imply see run solution match face datum set arguably use lee image nonnegative optimality uci use see illustrate compare nmf initializations rd algorithm quality rd km ten initialization generate average obtain run wave wave wave rd rd best km rd rd km evolution data initialization c strategy bottom leave right top bottom interestingly matrix although iteration would figure identify point bad figure lead perform nonnegative allow experiment rd well although difference significant face nmf lead exact contribution fold show error small approximation counterpart initialization semi initialization practice exact nmf solve give semi nonnegative
already finite excess towards within region latter large illustrate computing period return assume dependence return accounting st yield return limitation discuss direction historical record availability historical much uncertain systematic precision water rating curve systematic ii aspect systematic impact multivariate systematic unclear therefore incorporate explicit dependence alternative version historical high quantile generally decrease historical include inference however necessarily small quantile estimate take shape uncertain shape take existence joint angular dependent addition dirichlet allow variate probability poorly limit probability threshold range confirm multivariate period shorter asymptotically author thank part fp www project corollary notation com paris france national sciences pour et cs quantile typical return period face challenge historical approach peak marginal site site semi observation augmentation involve historical ignore historical highlight availability historical investigate significant asymptotic bayesian reversible jump variate problem maxima peak commonly model major site illustration record year whereas period additional challenge one complete understand extreme theory univariate issue site frequency analysis jointly several site extreme value record use site induce site ignore dependence observation site alternative use elliptical spatial beyond spatial independence compatible extreme shape parameter turn quantile compatible theory method record complement systematic measurement censor miss censor extreme location censor inferential carry partially family admissible extreme parametric restrict parametric logistic practical censor version readily subject modeling choice allow neighbor resort characterize write weighted parametric keep flexible combine datum threshold neighbor partially censor combine historical explore build previous extend multivariate variate maxima approach correspond record episode multivariate implement combine historical investigate scientific firstly relative quantile estimate existence historical nature strength systematic bayesian parametric dependence dirichlet mixture allow inference vary complete jump inference censor adaptation inferential censor practical advantage mixture need wide theoretically realistic moderate remainder dependence summarize feature describe inferential fit result discuss propose summarize finding france close period historical marginal extreme water record work simultaneous record daily course censor small error proximity four visually confirm obtain miss record censor leave censor comprise possibly upper period location temporal observation j j stand record arbitrary missing model raw daily term level threshold cluster maxima treat threshold fit detail propose identify size estimate index approach heavily inter arrival available censor adopt duration representative choose region censor exceed none position typically case censor intersect end marginal position temporal start cluster maximum special care censor duration temporal index coordinate variate scheme situation location maxima censor marginal upper bind prevent belong extract horizontal black grey draw consider location censor variate inter day namely inter cluster inter completely remain account classify homogeneous block day contain maxima together coordinate censor censor location study available available modeling another extreme occur corner suggest model mixture period exact gray superposition increase rectangle coordinate threshold description overview reader refer review margin location consider pareto threshold latter let possibly unobserved maximum water model empirical intra variate day marginal variate handle fr define transformation transform one extreme extreme radial homogeneity large away unit fr switch system angular component scale context angular call angular angle correspond behavior entirely angular variate set different angular distribution pareto density represent point segment blue weakly concentrated say simplex occur mostly contrary variate due angular mixture angular short dirichlet center therefore average q center mass center technical justification paper overlap censor miss write term poisson intensity precisely threshold observation extreme fr measure relate angular widely context take kind q region poisson poisson absence censor jacobian accounting transformation integral expression context carlo build censor involve whose assess option miss full dirichlet dependence historical confirm ratio value statistic assess add value account historical consider systematic total iteration require remainder obtain score together distribution concentrate indicate identifiable panel distribution frequentist historical confirm taking tend uncertainty shape return credible interval inferential framework take increase return
useful discover cross powerful flexible pathway framework dataset ht gene far gene fig lemma thm corollary thm pathway challenge knowledge throughput heterogeneous informative specific integrate hypothesis source include novel integrate attribute utility study mechanism rna protein pathway current pathway study decade discover novel component pathway find pathway development drug target pathway one major biology utilize high throughput pathway nature research medium reconstruct pathway hypothesis mind try throughput contribute comprehensive refine hypothesis characterize three feature well set heterogeneous second exploit datum reconstruct part pathway informative protein domain far utilize approach planning study assess biological propose extension pathway hypothesis demonstrate pathway feature distinguish pathway refinement methodology network reconstruction comprehensive integration collection protein helpful describe global mechanism pathway refine integrate heterogeneous reconstruction edge path pathway diagram mm pathway diagram expression sometimes datum bind protein protein completely protein interaction predict pathway cause relationship perturb effect nest effect dna individual pathway far utilize reconstruct pathway differ pathway pathway identify edge differ formal determine hypothesis contrary spirit exist biological support suggest probabilistic model pathway refine pathway graph three demonstrate explore content individually e pathway infer use pathway encode pathway except part adjacency observe edge primary adjacency although differ reconstruction informative hypothesis pathway set bayes pathway proportional prior sampler explore pathway edge gene product sample accord without complementary bind datum dna gene domain differential specific location uninformative likelihood essential exploit fact translate localization product process notion informative location additional map conceptual first protein cell second protein cascade factor protein enter gene binary pathway type molecular pair description collection five data index conditional informative protein specification discard carry little conditionally expression pathway aim read sequencing pathway whenever co similarly profile uncorrelated differently put emphasis beta list protein conditional differential relate neighbor protein indicate presence protein logit relate logistic pathway informative pathway pathway hypothesis pathway reflect protein interaction gene expression target tf domain goodness fit auto knowledge pathway extract summary pathway tf target pathway protein cascade tf containing pathway protein two contain sub mark grey target member indicate explore content analyze pathway dataset throughput protein pathway cover edge sensitivity see b tf target clear biological pathway gene show pathway type prior wide cascade expression tf target however informative explain complete pathway pathway observation valid pathway take sensible concept exploratory ultimately hope evaluate combine type support hypothesis pathway fit occur protein frequency logistic sensible default pathway domain pathway structure pathway initial also c fig strategy power separate node left pathway neighbor left edge leave assume encode agnostic presence auto logistic likelihood simulation hypothesis lastly figure recall informative poorly gene contrast high precision model able pathway pathway pathway specify relevant location consist pathway sub pathway interaction pathway currently pathway include sub pathway usual tf tf tf tf match pathway fix pathway specify node likely domain node domain reflect interact leave precision recall
see estimate pairwise parameterization effort specify importantly link correlation dependence covariate model financial application allow model tail dependence achieve copula parameterization particularly financial tail vary covariate allow link covariate prominent covariate without dependence covariate margin suitable connect link covariate allow parameter approach make interesting interpretation forecast furthermore reduce see detail specify marginal copula omit subscript first turn variable indicator informally express covariate variance independence intercept slope decompose prior normal prior intercept include parameter indicator one intercept put g imply intercept covariate technique two situation imply model parameter trivially intercept prior intercept copula copula logit intercept numerically mean intercept parameter case normally matrix trivial positive definite matrix normal identically distribute reduce experience shrinkage meaningful great explore comparison consider special dependent new put restriction copula tail inequality visualization parameter dependent independence decompose prior indicator indicator generalize logit beta beta logit extend logit two generalize function reduce logit link intercept decompose application link prior mcmc metropolis hasting truncate draw reject mcmc likelihood copula j jj jj u u component jointly joint update conditional metropolis hasting update indicator parameter tailor acceptance rule g ig proposal mode finite newton small matrix negative hessian currently exclude enter versa allow indicator indicator scheme copula variable scheme update show hasting proposal distribution model low copula copula implementation dependent variable indicator wise function update copula mc alternative independently margin margin see widely asymptotic copula fr initial obtain request daily stock return copula split margin copula daily daily stock market period include large establish subset cover small possibility capital great index percent total hypothesis dependence parametric nonparametric dependence little covariate covariate margin table covariate univariate regression mixture lp description week month geometrically decay geometrically absolute return l high measure decay return geometrically decay figure series volatility return financial depict empirical respectively fr copula mean copula appropriate usual copula efficient extreme near fr application conditional tail autocorrelation lag marginal similar link posterior copula refer marginal selection important inclusion low dependence part p margin tend highly risk margin select copula copula selection marginal respectively intercept include p index subgraph variation tail even contour copula financial compare hoeffding bind figure normal financial copula capture dependence model performance historical predict ahead prediction calculation costly forecast assume add multiple processor application smooth gaussian table commonly predictive covariate covariate copula copula well margin link covariate sample posterior daily return index advantage copula improve margin augmentation work visit department compute information science university present mcmc detail implementation great hasting embed metropolis copula cdf marginal link parametrization numerically dictionary upper section let link conditional complicated need substitute density obtain omit copula eq u vs q exchangeable derivative u direct derivative elsewhere cdf q function function calculate integral analytically advantage connect derivative asymmetric symmetric density binomial knot mathematic finance economics china copulas attractive multivariate density marginal particular area possibility dependence approach tail yield interpretable practitioner dependence function copula posterior method covariate copula dependence mcmc copula research term univariate copula cdf e g copula dependence construction copula widely survey review include construction tail bivariate copulas tail px u low tail bivariate copula explore copula bivariate copula another asymptotically fx testing detect event study vast extensive copula model article investigate cause many situation relatively copula model tail copula modeling obtain tractable practitioner general copula copula construction form copula stand copula tail dependence outline copula introduce copula presents mcmc daily stock fr hoeffding copula strong positive dependence dependence see bivariate modeling copula commonly copula copula bivariate copula bb copula tail tail density copula tail use copula autoregressive daily
run slightly feature correlated pair nature far design correlate feature predictive task thereby lack instability ensure correlate assign exploit clinical structure inherent purpose diagnosis record medical international code code event build undirected node incidence share temporal term iw graph rewrite simulate create model feature select finally list feature hyperparameter stability resample lasso stability fig affect auc feature bootstrap clinical fig ccc feature stability medical inherent cox heart patient focus solely heart similar study competitive gender age history clinical level rare result past diagnosis demonstrate entirely datum medical database integrate clinical pathway serve screen select future include investigate structure enhance mm clinical yet little attention cox clinical structure inherent record intervention hierarchical medical knowledge demonstrate efficacy predict failure patient clinical increase discriminative power report competitive heart failure serious frequent heart prediction variable hypothesis literature reach medical record aspect care history diagnosis procedure marker record time comprehensive deriving prediction unfortunately automatic particularly clinical cause instability estimation little investigate heart cox high improve exploit clinical inherent diagnosis disease structure edge relation sharing strength among correlate feature thereby stability cox train validate datum validate measure index exploit clinical structure stability regularize cox utilize structure cox database one sided bank pool feature database filter bank event time cox risk due failure future instant unlike patient
version choose choose view variant update every number time cr r r r super martingale follow inequality last eq resource allocate despite future uncertainty optimization extend general arrival permutation random competitive bid ratio et draw propose subgradient optimization framework application internet online resource new constraint modify program considerable online ratio achieve offline competitive online maximization case infimum many propose trivial competitive lp derive competitive lp often competitive restrictive permutation permutation formulate define subset proper technical utility impose optimization vector procedure avoid take online optimization solution algorithm tx na member literature follow simplify online distribution permutation simplify identically distribute lp online lp go grow example permutation achieve light dedicated permutation online lp briefly combinatorial optimization relaxation program important lp nk ip nk linear primal variable dependency union al give author name choose linear number constraint ratio adversarial adversarial paper aware recent similar also prove online program include general show achieve ratio propose lp apply general permutation give adversarial interpret subgradient call n allocation combinatorial matching lp online integer solution allocation online associate online allocation terminology equivalent server amount order pay view server utility associate order server decision fulfil total resource server make lp utility one online briefly review case thorough reader online bipartite n online bipartite special allocation et call achieve competitive worst prove achievable competitive online ranking achieve competitive algorithm study competitive receive distribution competitive ratio problem assumption applicable propose online show competitive propose explicitly primal concave version utilize primal problem bad fall many al ratio infinity bipartite generalization bipartite matching graph adjacency matrix lp without rule review rest ne ne simplify bernstein hoeffding replacement convex proposition replacement one sampling replacement see sub martingale chapter denote complement transpose denote function e e start simple feasibility let consider solution uniformly nx sx algorithm derive concentration sum simplify integer kn ni martingale maximal inequality bernstein algorithm sigma show super martingale small inequality give conclude follow generality competitive offline concave pair primal pair f np h bernstein close define ia f ta sx describe apply represent subgradient tx f choose similar analysis concentration sx nf sx define q maximal inequality conclusion k p main present define tt super martingale tf
mutual conditional find candidate check independence graphical degree search become much focused complexity writing importance exponent independent work propose efficient liu make generalization possible distinguish recent efficient algorithm informally graphical asymptotically distance pairwise ise multinomial survey possible efficiently away variety paper give algorithm require require base use regularize regression show certain incoherence careful fail ise term mcmc temporal mixing e conclusion informally speak generate ise graph converse generate np precisely know without determine pose answer ise model despite strongly problem algorithmic connection ise efficiently discuss state conceptual identify ise structural almost liu arbitrary node maximum consider ise exist neighbor information state actually consequence information provide mutual algorithmic ise arbitrary constant interaction strength complexity algorithm doubly obtain suboptimal section statement order node form neighborhood pseudo neighborhood spurious worth add crucial add create pseudo neighborhood range neighbor high many condition neighbor potential argument whereby pseudo pseudo easily remove neighbor connection introduce influence state lemma average correctness result proposition prove section discuss study ise name boltzmann approach boltzmann machine attempt find ise use network protein network physics rigorous base truncation expansion relate entropy message pass learn broadly statistic dimensionality parameter ill sparse underlie discuss optimization regularize objective popular approach infer ranking optimize fail far ising tailor often theoretical progress variety gaussian work ise function effectively learn soft joint boolean dependency ise model apply fouri nevertheless level total log exponential temperature range correlation assumption depend consider graph degree bound associated shorthand eq partition parameterize edge external edge satisfy field satisfy bound appear complexity alternatively think implication conditional write bound spin away statement conditional appear observe configuration measure zero meaning expect corresponding graph rather robust bad entire eq runtime use certain eq quantity obtain perform neighbor imply result show neighborhood thus work u ix replace version influence event influence within learn data step neighborhood accomplish high conditional I set neighbor use potential construct simple relating reduce neighborhood try correctness rely subsection terminate construction remove let suppose numerical probability return probability return correct neighborhood runtime obviously exceed state value give prove give correctness bind neighborhood theoretic quantity include kullback leibler divergence definition add node add x add pseudo term quantity prove lemma quickly runtime claim maximization pn negativity entropy mutual u x strictly large consist add mutual shorthand conditional x q jensen concave root inequality variation lemma conditioning add ix state use correctness graph parameter u hold pseudo construct node contradict line algorithm pruning pseudo state discard conversely I neighbor discard finish proof proof lemma guarantee corollary correctness event u prove proposition proceed adjacent x neighbor begin multiply gives sum last quantity give eq inequality appear proof proof randomness decompose concerned anti concentration anti concentration result os show anti sum variable os let decomposition lemma conclusion approach mention strength let z decompose uniform z mu represent think obtain choose hence probability placing minimize subject arithmetic average z prove lemma proposition subtract statement recall odd definition z apply assumption estimate concavity additionally monotonicity partitioning estimate eq subtract third q true follow bound second eq negativity low multiply plugging complete node configuration x application remainder hold bind inequality inequality triangle depend bind ab ab ab latter plugging hold early evaluate eq learn ise ignore show light large correlation plausible one
tackle nonparametric approach process aim infer infinite bayesian paper propose exploratory neither sample output summarize take function article obtain automate dependent case may grain pattern sub keep reduce post technique consists merge successively contain curve cost sum divergence merge create dissimilarity merge post processing technique hierarchical decision tool plot dendrogram pareto chart criterion number cluster introduce curve relate alternative technique artificial power consumption finally summary describe goal collection value observation exploratory goal reduce pattern regular shape low linear combination base simplification cluster use discover functional e segment find minimize constrained segment space similar maximum estimation induce partition interval partition abstract functional exploration build cluster interval summarize represent canonical draw organize multinomial fit nonparametric overcome unbounded computing approximate sampling chain dirichlet require parameter parameter significant expect reliably estimate approach much parameter available dp approach clustering treatment mode matrix contrast probable optimization secondly order one retrieve monotonic sense aim continuous model clustering curve realization discrete computation approach first use build size rank exploit experimental fine grain retrieve pattern principle detailed functional grid phase mining process time consume result rapidly reliably conditional supervise joint interval partition grid datum combinatorial curve store curve density discretize cross univariate triplet curve joint per curve joint tend group cluster discretization optimize family cluster curve cell cluster define curve interval curve point cluster collection curve curve dimension curve curve interval point curve dimension select distribution parameter choose uniformly dimension cell cluster distribution cluster give resp resp log unsupervised datum accord optimal value eventually subset kind stand way partition nonempty subset code description relate number specification partition cluster multinomial cell follow specification multinomial third likelihood line cluster follow rank resp heuristic heuristic bottom heuristic grain curve per cluster cluster merge decrease merge post step embed meta heuristic mainly different initial improve extensively evaluate true detect grain pattern approximate instance train accord lead exploratory still fine interpretation aim simplify information retrieve agglomerative locally exploratory highlight set datum successively uniform white distribution p x collection per triple randomly curve accord apply functional method subset size per display average point subset discover interval start curve method recover pattern cluster curve distribution despite retrieve point may totally curves systematically place grow retrieve show well pattern property regular moreover deal distribution cluster one consist consumption home consist give power consumption day minute aim characteristic consumption grid interval record day group discretized power measure power segment highlight characteristic day home represent piecewise line power consumption segment power grey grey read highlight within segment prototype locate dark power rarely prototype segment multimodal segment highlight multimodal extend ht grid yield characteristic easily interpretable consumption year may agglomerative represent dendrogram chart percentage consist cell accord percentage keep cm opt dendrogram chart concave keep information ht highlight retrieve propose four processing power consumption time four prototype red solid curve compute discretization consumption conversely make certain cluster segment common consumption enable period pm pm able term consumption period day curve locally segment prototype average prototype estimation highlight segment distribution consumption power consumption locally segment display density consumption retrieve power dense around unique consumption consumption function also peak translate interval highlight power around retrieve compete approach track difference color difference use figure retrieve certain indeed way group highlight link weather france period rest year consumption appear early may day classify interestingly home consumption cluster period ht ht color consumption day show retrieve cluster year power consumption day cluster consumption characterize day home intermediate show immediate track retrieve scheme hand user prior powerful exploratory thorough understanding locally cluster complementary focus exploratory curve paper point categorical curve cluster point select
derive relation arrive excess square series prediction satisfactory adaptive vi goal output mapping hypothesis kernel hilbert rkhs definite may control smoothness solution rkh satisfie namely f k f kernel calculate j solving square usually necessity input algorithm alternative compute gram express step ei upon time current nice sample fit current incremental adaptation essence solve square adjustment involve step assume gaussian factor define depend look close size look inner product unseen fall study recognize build current previous rkh correction efficiently change motivate component formalize propose sequentially optimize unknown product draw define minimize practice alternative measure course batch present sequentially across previous size size sequentially iteration initial size unchanged addition new use new vary iteration adaptation rkhs learn cycle center remain term rkhs jointly optimize datum I depend train independent mean square condition I error iteration size minimize optimization input q residual mapping iteration condition noise space denote size iteration size much size density theoretically desire mapping impractical solve optimization importantly usually develop without discuss minimize mapping current optimize iteration prediction optimize minimize instantaneous stochastic readily denote kernel size adaptation residual case follow kernel sequentially kernel size center old center remain initial manually roughly advance follow observation sign sign sign size successive contain desire desire mapping easily input adaptive filtering provide tool rkh mapping derive energy express residual rkh contain correction term nonempty interior consequence contain rkhs far side q energy relation form relation normalize satisfie monotonically square decrease far reach excess e two assumption another uncorrelated become steady state easily map steady state close ii v related observe neighborhood size little confirm kernel little accuracy influence speed reach steady case accuracy steady section present result static estimation output generate size see kernel influence rather speed work achieve final size almost still final fig evolution curve adaptive size plot value size map plot purpose well desire function visible desire detailed summarize speed still select see kernel table little effect steady training confirm prediction except run average list steady attain steady state case steady state convergence speed steady kernel steady size q pick predict current step fig simulation run segment series train iteration mse compute iteration kernel small mse final expect yield satisfactory interestingly desirable deviation quantization apply network experimental set quantization curve demonstrate obtain show size converge dotted network quantization testing mse c define role adaptive filtering radial usually default kernel mapping influence crucial square efficient algorithm iteration computationally base energy relation mean prediction automatically proper converge achieve sequentially optimize function idea interesting line optimize cn edu cn edu filter filter develop gaussian size still square optimization develop sequentially mse theoretical convergence confirm static short learning nonlinear inherent space space kernel thank popular include vector principal component kernel etc significant counterpart learn extensively statistical literature nonlinearity system cost filtering filter reproduce hilbert rkhs filtering nonlinear algorithm affine recursive square create radial rbf adapt simplest fast effective main grow increase requirement especially growth important datum accept approximate constrain compact desirable selecting address implement filter two kernel adaptive universal approximate capability stability normalize kernel
model free online unlike agent strategie continuity truth employ empirical ne provably convergent ne also question concept indeed stochastic game learn observe prove game extension discount stochastic tuple agent product space state action go player choose discount influence reward obtain agent represent various stochastic game xx ix ia stationary strategy suggest transition probability agent randomize write expect goal strategy nash stationary nash equilibrium game nash discount well result discount strategy shall stationary nash programming write q picking act distribution behind formulate add ensure feasible nash equilibrium optimization objective minimize agent isolate meaningful agent natural ensure vector feasible problem valid equilibrium useful maximized combination implicitly imply formally give ensure policy ne point nash equilibrium correspond discount work manner remark difficulty solve quadratic summation multiply inside constraint easily see non player game page every game requirement optimization apparent two player descent convergence minima nash equilibrium strategy two player game case sum player contain arise newton solve hessian objective infeasible require hessian necessarily present break subsequently stochastic sg sp agent along state ensure bellman tuple let bellman let ix formulated derive nash strategy sg sg sp feasible sg sp sg sp equilibria intuitively objective summation zero imply sub ensure bellman error turn agent combine sg sp sg sp lemma sg nash sg sp nash tuple optimization problem suggest imply nash nash sg sp nash optimization sg sp point nash feasible satisfy sg case agent derive sp sp condition sg sp lagrange multiplier slack lagrangian kkt corresponding kkt necessary sufficient equilibrium underlie game strategy tuple linearly sg sp condition impose additional independence ensure establish sg kkt kkt sp sg kkt sg sp kkt sp feasible problem substitute eliminate reduce feasible sg equilibria sp sp sg sp sp impose linear requirement introduction actor operate value operate along slow sg sp dynamic update tuple follow operator stay I sign project outside small around continuity technical help provide recursion slow recursion proposition g objective q taylor amount show second infer xx let matrix characterize recursion present ode actor evolution ode precise point ode far see infeasible ode asymptotically unstable asymptotically limit nash surely ne discount rectangle mm join height minimum cm draw minimum fill environment fill red dot right dot fill south node n leave start south south south amenable rl neither transition operate free illustrated represent discrete state agent localize agent decentralize spirit rl presents operate agent temporal td recursion note recursion operate similar operate variant know slow motivated proposition start size play reward respective value quantity derive light converge nash strategy number multiplication behaviour appear state strategy avoid typical multiplication iteration complexity confirm iteration complexity finding game result scheme rate scheme show x see update operate model td stochastic approximation argument early literature update recursion change operate model free hence access update track handle proof analysis underlie transition discount multi agent rl detail later modification comprise size ensure quasi static update appear infer rewrite recursion eq cn space one trivially track ii use theorem analyse recursion system column reward globally asymptotically stable limit rearrange identity ode eigen particular system globally ode converge start see euler give quite prove dynamic programming latter variant converge rl govern globally equilibrium lipschitz h origin asymptotically stable show unique globally asymptotically ode term assumption ode trivially satisfy asymptotically stable point update recursion converge correspond strategy limit bellman ode recursion infeasible limit sg feasible point ode partition unstable unstable gb unstable since since value long equilibrium feasibility sp return ensure point induced spurious xx operator suppose I iv I iv consider possible contradict solution result well result recursion project onto ode ode make continuous surely size satisfy bn bn ode compact asymptotically stable converge almost update slow scale rewrite eq assumption recursion continuous since continuity fact continuous spaced trivially upper bound claim iterate govern unstable observe converge stable equilibrium offset policy compute use every numerically convergence govern stable outline crucial detailed establishe td update converge involve consequence fact free access analysis govern surely globally point sequence write argument q assumption assumption govern globally asymptotically establish estimation recursion technical result aggregate recall theorem almost surely let update write n v ny j g r martingale square integrable ensure ergodic verify martingale surely almost natural ignore main paper treatment recursion x I converge estimate allow proposition I order verify since verify straightforward space quantity verify update converge algorithm two variant involve intensive solve simple payoff individual agent game pick constitute ne pick either ne stage payoff agent accord payoff perform length stage aggregate evident ne strategy tuple ne iterations b b inner sep simple sum discount game name stick short locate rectangular like stay precise description component specify agent rectangular product action agent neighboring action one transition agent agent function next action reward agent state thus show sized game sequence q correspond slow constant step lead fast also domain equilibrium policy follow offset pick xlabel number ylabel height marker font file plot txt xlabel ylabel avg distance grid marker legend style legend column restrict file txt txt file txt evolution go nash equilibrium evolution algorithm homotopy exponential practically infeasible evident strategy converge grid within imply get iteration get grid drive corner grid short explore grid exclude take minute take nearly involve nash equilibria iteration q variant implement xlabel ylabel height marker plot x simple full game assume reward state transition intermediate case albeit know particular reward position runtime suggest sg equivalence nash necessarily avoid minima play equilibria certain differential ode stable coincide stationary equilibrium underlie rl experimental quick future successfully state huge strategy tuple reinforcement mdps action space aid bring system use extension constrain game stochastic game additional constraint function tuple might arise application detail sophisticated pt nash equilibria ne game ensure bellman characterization nash equilibria underlie game sufficient sg sub develop descent descent avoid minima converge self equilibria certain ode coincide game game establish consistently outperform discount stochastic nash reinforcement stochastic approximation game single shot prefer several action intermediate stage one stage chain random system next popular scenario decision process mdps give allow suitable reward incur influence another merge mdps markov behavior game select receive agent game agent treatment game pay criterion like game several model game stochastic also markov cf evolve act simultaneously transition agent get know aggregate include agent individual agent objective maximization discount reward couple vector
q lemma lemma rl well part follow fact theorem corollary institute pa usa learner prediction address analyze regret neither benefit develop interactive learning extend reinforcement commonly suggest broad exist learning increasingly notably game ai easy expert translate behavior perhaps develop space g list parse understanding ground policy execution performance supervise require benefit leverage provide make error expert equally drive expert expert choose go learn poor reason agreement poorly mistake cost learn user intend task contrast impractical additionally policy statistical rather leverage sensitive approximate policy develop theoretical iteration despite regret stability broad view horizon decision policy action use action state induce state system typically sample learning influence technique learn go expert form observe uniformly explore observe perform action train dataset iteration interaction begin uniformly explore expert continue new cost go visit train learner iterate policy expert action explore detailed henceforth assumption contain policy good future go lead low observe action favor initialize mix collect uniformly start execute execute current execute estimate go start expert cost view online demonstrate problem policy policy randomize deal sensitive sensitive online algorithm descent description default learn classifier surrogate sensitive stable regret highlights standard indicate predict sample go cost state optimization weight machine good cost ns nn squared loss go regressor common include sensitive handle bandit must go inefficient feature current exploration carefully set traditional care bandit algorithm finite many show procedure interactive strong analysis policy policy competitive expert rely connect adversarial online property sensitive choose policy incur adversary go exactly collect provide n policy class regret policy n beginning trajectory uniformly randomly policy trajectory visit go hold infinite iteration collect amount policy sequence j policy sensitive classification aggregate dataset policy interactive unable bind linearly classification play sample analysis depend reduction sensitive ranking relate task action random reduction particular regret aggregate example optimal regressor regression use pick regressor regret relate task performance regressor regressor training regret use particular present cost regret aggregate go obtain regret mention case cost optimistic future fail policy even policy albeit policy driving scenario reach goal shorter drive narrow road policy go time enough collect policy loss example iterate iteration initially guess use go specify detailed algorithm initialize collect point execute execute state aggregate I online loss go current correspond sensitive algorithm online variational step td find regret cost thus find good distribution state limit provide performance guarantee result present single policy execution allow generalization efficient impose strong requirement regret sensitive instead cost learner class cost still online reduction regret performance strongly limited quality iteration mechanism distribution policy guarantee theoretical observation first perhaps crucially suggest approximate policy theory counter understand share ensure like rely future go understand heuristic wolfe achieve performance variant effective batch suggest instance approximate stable cost go seem counter intuitive however divergence approximate ensure many broad form piece grow picture analysis batch analysis online seem concern performance robust become dynamic execution concern method rely cost impractical collect estimate state action expert visit trajectory per setting expert heuristic analyze combination first loss minimization provide expensive provide class compete mdp remain learner trade must explore detailed detailed begin lemma need bound expect query expert fraction let argument integral px qx fx qx additionally rl fx qx qx qx fx px qx make lemma encounter collect continue execute expert td rl useful present policy let execute time rl v u reduction sensitive classification achieve class pick number policy q q u inequality similar argument rl well e follow finite explore uniformly sensitive loss go action e linear regressor regressor regression j use regressor pick cost additionally
present compare intuitively output face produce need produce negative weight weight without reaction target prevent must actual concentration adaptation adaptation originally distinguished adapt residual input signal could happen soon contribution input shown distribute input contribution specie perceptron combination input adapt weight reach update reach point beneficial formal perceptron concentration negative entire present manual would genetic ga cross mutation use mutation fitness reflects give encode learn fitness task force utilize otherwise ga tendency opt utilize capability input choose target safe region far allow weight integration proceed define performance chance generalize sufficiently primarily drop distinguish draw among function function perceptron constant zero fully eliminate consumption branch perceptron discard adjust act consume part output function fairly would impossible calculate formal perceptron question count weight concentration trace output error chemical asymmetric asymmetric signal perceptron analog learn feedback provide et al simulate base capable compare system cross besides input utilize shift general bias would oppose employ system evaluate employ fairly chemical plausible suggest carry problematic feedback need introduce variant separate reaction chemical automatically transformation dna circuit dna range art dna circuit oppose adapt integrate delay could tackle series chemical event relevant drug adjust cancer cell material grant pre determine manner feature compute challenge implement extend chemical analog analog asymmetric perceptron perceptron simulate capable specie actual dna perceptron analog supervise build program chemical machine change limit applicability limitation environment adaptive chemical learn external imagine million molecular broken system design predefine specie could inspire chemical implementation adaptation chemical calculate e chemical achieve desire previous work first simulated artificial chemical learn teacher chemical perceptron system input step aim simplify implementation employ asymmetric thresholding flip side perturbation structural redundancy real application subtle among achieve perceptron model improvement necessary analog asymmetric perceptron original mass learn environment modular number demonstrate learn nonlinear analog combination chemical line chemical reaction formalism consist molecular paired rate symbol molecular importantly reaction treat concentration quantity require chemical constant order reaction reaction define speed signal contribution reaction rate total perceptron input contribute output process dot product chemical weight difference require perceptron output calculation reduce concentration formal specie input since bias represent weight three oppose formal asymmetric addition sign concentration weight otherwise asymmetric specie leave negative represent monotonically increase initial specie weight share effectively implement replace decay specie embed previously encode complementary opt reduce half complexity underlie concentration limited qualitative formally act weight impose negative pressure weight branch contribute concentration branch concentration reach
topic vary result figure bind vary true truth partly change dataset effectively narrow vary model investigate provide direction analyze generalized mixture weight sum product analysis component order derive structure singular example conduct mixture component gmm assume generated spherical assume gmm method likelihood similar provide namely maximal integer similar detailed omit excellent work analysis lda present bind order topic analyze q proof analyze value moment thresholding solve inequality bind singular especially bound ij ij set least probability follow least element study proof proof gamma tool degree freedom proof tail r shape follow hold gamma chi r r v hold n n c overall intermediate lda parameter eq q omit ambiguity also representation th document diagonal separately section analyze large corpus selection set topic advance topic first bind topic text demonstrate use easily generalize recently model variant prove extremely corpus word generate mixture latent multinomial become text lda variational topic play successfully apply show large incorrect computational lda grow equally unfortunately topic lda approximate via mcmc selection aic bic though achieve success asymptotic run dataset hdp alternative select inconsistent topic amount theoretical latent model spectral outer vector correct topic topic lda analyze topic provable guarantee utilize tensor moment upper term result computable contribution valuable determine sampling spectral information true inequality regard constant organize main synthetic validity generalize powerful building block recently moment explore lead topic directly derive properly third estimate decomposition line discover empirical document multinomial topic collection th document document topic hyperparameter topic generative lda denote generate natural meaning learn moment q outer moment show outer product topic since summation linearly large singular value direct way access estimate moment large overcome obstacle study estimate infer estimating size large approximate enough pick threshold simply count achieve examine investigate generality namely also simplicity definition singular variance lda chain step semi defer ii matrix frobenius therefore expectation proof task discuss high relaxation keep dominant statistic examine document one pay hundred k matrix diagonal row order chernoff bind appendix great minimum random proper conclusion assumption utilize
split ghz cores gb image cat specie table observe surprising example assignment contrast vs second suboptimal learn difference cat image assignment art approach head box segmentation well image segmentation make ht ground segmentation head extract encode total denote improve extract segmentation dataset specie consist class already well improve segmentation worse annotate information confirm effectiveness training imagenet train report table ht sift template propose art mostly implement extraction share comparison operational cm method summarize take employ stochastic method metric come aspect become significantly class image separate contrast class appropriate method address challenge arise stage arise many extend projection address challenge performance significantly art approach plan combine segmentation approach feature nsf com com fine grain categorization basic challenge occurrence distinguish intra paper propose metric address embed different apart address flexibility portion however end subproblem computational benchmark grain categorization aim distinguish class classify handle somewhat requirement many subtle deal intra variation pose example feature extraction extraction localization choice include recent development cnn e imagenet state art dataset difficulty training dataset benchmark thousand deep segmentation aforementione classification exist directly grain strategy strategy grain vs scheme effort variation metric approach occur work different far neighborhood effectively handle tradeoff inter intra metric find test limited dimension straightforward dimensionality analysis pca projection problem unable take dimensionality suboptimal challenge constraint usually require avoid overfitte total triplet example dimensionality datum lead computational challenge optimization ensure psd intermediate storage save gb store complete framework dimensional dimensional divide original optimization difficult classified currently metric adaptively improve metric optimize handle subproblem dual develop enjoy random learn dimensionality finally storage copy randomized matrix process metric overfitte extensive efficiency section describe conclude direction many develop find two survey paper base triplet adopt serve address although devoted examine project rank rectangle advance apply directly less address assume storage suffer high cost propose focus triplet near dy triplet assignment tm problem significantly study loss appear effective hinge loss benefit l challenge psd projection first psd project end follow j triplet constraint dot product summarize reliably determine overfitte triplet number summation term solve help image category visually lead mistake address process stage metric triplet triplet optimization improve learn solve optimize objective stage obvious strongly chapter solution optimize finish metric optimize stage original problem number summarie dimensional subproblem dual technique simplify investigate analyze introduce dr triplet project projection double projection preserve pairwise variable low b learn significantly develop estimate although expensive impossible save save method suboptimal save popular avoid metric efficiently challenging address recover step express summation multiplication triplet correspond matrix verify second exploit efficiently eigen accord independent appearance compute keeping fortunately
optimized mention capable achieve bit image possess complexity arithmetic computationally dct future consider approximate transform dct high adequate datum frequency quantization restrict significant propose dft pruning time ignore involve avoid pruning discard apply design wireless communication another pruning method dft dct pruning originally propose wang consider generalize extended pruning technique terminology vision dct base dct theory advanced propose architecture dct architecture operation avoid pruning refer operation discard bit architecture address prune discard distinction terminology grow furth dct present introduce dct realization version propose embed video modify associate dct approximation particular successfully mean noticed concentrate image low image quantization frequency effort frequency keep transformation derive associated transformation compute correspond compression computational overhead since quantization graph relate signal signal set zero transformation mathematically accord input sp font lb lb lb lb lb lb lb lb counting multiplication assess obtain complexity state dct dct chen dct full version version retain coefficient technique transformation mention section compete coefficient transformation compress process evaluate degradation zero nz quantization nz translate long zero beneficial subsequent encoding code stage contrast adopt average table value percent value set significantly maintain qualitative comparison compress describe capability separate modify approximation hardware evaluation transform transpose buffer test matlab realize gate array device validate hardware loop interface approximation prototype use complete agreement verification hardware code nm technology synthesis implementation respectively flip ff critical delay area time operating frequency synthesis place tool file design frequency propose dct show area consumption dct synthesis nm reduction reduction normalise metric clear use dct frame hz assume rgb resolution ff modify modify propose video embed reference transform reference chen dct consist software show frame code chen fig frame db minimal degradation db computational point dct arithmetic chen dct compute rd sequences qp compute bit curve
induce cluster modal ideal population straightforward formulate informally mode population modal shift example include shift modal alternative version section population lie behind modal population reflect region high separate cc exp exp exp node exp right node split split plot exp exp exp node node node scale final set methodology identify cluster density cluster level correspond hence single increase reach splitting branch panel component core probability part assign depend point line difference equivalent clustering singleton leave branch two branch panel core component node clear splitting cluster panel correspond precisely formulation define minimum attain solid circle panel notice level core constitutes approach color cc sigma sigma normal pi exp pi x grid smooth gray axis cs cs axis generalize high dimension follow normal distribution natural separate density branch differential topology topology study critical useful application range represent indicate flow effect summarize follow enough degenerate enough time nan degeneracy point index negative critical write sign precisely example possible figure minimum saddle index ccc x name width view title name grid domain title name view domain title unstable manifold explain minus smooth integral satisfying minus descent water unstable curve start analogously integral note form manifold point stable manifold unstable manifold contribution modal unstable gradient maxima modal flow cluster clear saddle point index associate unstable maxima unstable manifold rw r u cluster apply univariate example maxima manifolds minima unstable manifold gradient curve satisfy unstable manifold become stable manifold could xt give example ideal modal look bivariate normal terminology iv mode range true population modal cluster cm contain contour triangle point plot mark triangle point thick pass compute numerically make thorough density location mode take mean different connect curve finally numerically value shift saddle clear kind essential every twice redundancy add partition compute thus interpret penalization cc node scale thick idea even shown match depend permutation component empty obvious minimum match match estimate replace measure distance clustering differ include transfer extend population counterpart minimal mass need transform connection clustering hausdorff note empty hausdorff equivalently ab analogously take consist set distance identify clustering subset hausdorff distance whenever hausdorff regard distance hard standard demand meaning hausdorff value instance clustering hausdorff mainly due picture hausdorff obtain wise minima add copy minimum far involve match solely analogue obtain probability lead consider distance clustering cluster understand methodology population clearly consistent mode stochastic convergence represent clustering define sensible clustering replace include nonparametric mixture fit unsupervised plug dimension study easy nevertheless important development plot exp right figure estimation intermediate line density estimator density close sense clustering suggest cluster easy density modal clustering density univariate support modal induce sequence almost derivative modal clustering surely proof state clustering boundary minima density dimension scope manifold estimator bandwidth necessary time truth represent try close specify depend notion whereas population goal like modal modal identify make tool partition maxima modal need smooth certain degree specifically time differentiable function extend notion mean smooth critical treat resort mapping cover book present play gradient goal clustering ideal introduction aim methodology approach consistency mild study minimize distance base measure method aim perform modal necessarily rely adapt estimate connection part project grant use finitely many isolated critical modal minimum sure interval hand strictly big critical since previously critical similar argument also neighbourhood mode convergent subsequence subsequence neighbourhood critical neither critical neighbourhood contradiction converge n j write minima minima theorem lemma corollary remark despite recognize investigation theoretical reason problem specify target seek population base cluster algorithm try aim theoretical focus ideal population modal region new methodology evaluate population mild estimator modal branch research rigorous methodology recently express concern paper contribute regularization state cluster even seem method statistical cluster solely notion gradually merge agglomerative cluster graphical successive group know inter merging linkage complete linkage linkage notice usually pre specify seek center goal certain function represent extended determine distribution state need statistical nonparametric cluster parametric mixture generating use bayes rule region separate low concentration mass sense mode function mode modal understand definition goal modal introduce datum connect usefulness advantage drawback population recognize author like matter think many show univariate phenomenon modal visually identifiable none three usual recommendation several value tool orient graphic function detailed explanation plot
soft display soft green intensity vector provide plot soft assignment location plot across likely chance domain cluster distance determine soft choose base boundary tend color mild implement visualization technique distribute isotropic cluster coordinate identify display visualization pick sc consist chemical feature region area find use chemical measurement cluster normalize select show gap gap move cluster cluster cluster connectivity apply point area see cluster dominate type produce contain cluster south observe cluster connectivity map mode reflect capture structure produce two observe edge connectivity hide interaction width edge degree dominate connect spread south north north west cp I om pp database repository protein different protein class cp om pp feature find standardized rule filtering sc plot cluster visualization q visualization panel connectivity confusion versus panel confusion panel third overlap protein e overlap third successfully identify apply uci machine database repository later camera inspection image pixel picture wavelet extract datum figure separate seed little connectivity panel seed high connectivity mode cluster select new visualization high also establish high cluster visualization chi chen grant support nsf number nsf grant dms need assignment gaussian variable unconditional soft kx pz data introduce latent representation soft representation infinitely representation density lead mixture thus depend upper define mode define closeness specifically mode l py l ii profile eq control distance boundary illustration method replace kde summarize contrast coordinate free soft data bandwidth contrast shift evaluate level py soft assignment less cluster correspond weight define norm scalar function ordinary consist mode must correspondence mode bound note eigenvalue local density less must mode generalize local mode triangular mode estimate mode triangular inequality estimate vice versa sufficient require need kde theory eq constant location location mode approximate focus derive generalized third eigenvalue away invertible pointwise nonparametric theory page mode rate thm pt proof mode chen department mode define density mode soft variant assignment cluster visualization secondary cluster method mode relative population estimate estimate mean algorithm choose depend despite advantage room improvement mode cluster hard uncertainty visualize cluster mode tend call paper visualization shift segmentation cluster statistic idea select bandwidth rule merging mode soft assignment estimate measure selection mode occur frequently grow high multidimensional example assume degenerate unique mode e path path standard integral intersect contain lead saddle local minima kde kde bandwidth easily using become assign attempt assign soft belong soft type population intrinsic variability strongly mode boundary associate soft sample level uncertainty come soft vector capture uncertainty remark distribution latent discuss mixture soft straightforward obtain soft idea mode describe method appendix soft way start diffusion start mode mode reach easy interpret eq define actually run diffusion correspond th assignment discuss cluster despite literature mode mode adapt hausdorff hausdorff hausdorff generalize non condition l commonly kde grow estimate derivative kde local mode local hessian modal constant hausdorff assumption condition eigenvalue mode consistency number consistency explanation fact kde mode apply taylor local gradient estimator hausdorff distance decompose bias technique soft among generate hard mode assignment connectivity high assignment analogous classification class might row matrix connectivity cluster useful summary overlap else bandwidth gradient standard estimate common square generalize rule reference smoothed validation reference slight modification deviation reason cluster vector difficulty interesting consistency consistency supremum norm mode consistency produce cluster call kde converge slowly create kde turn consistent slow example mixture coordinate order smooth gray four hand filter gray gray want approach deal increase merging may cluster quick simple curve merging enforce threshold merge large cluster recommend intuition recall well diagnostic sc display gap cluster induce know gray reference rule signal structure addition denoise remove persistent persistence extremely computationally intensive
problem without gradient programming method programming experiment conclude suggestion subsection convex application non implement particle particle particle movement reach movement evolutionary particle particle respective particle velocity equation velocity varied iteration particle well update velocity iteration position update interest determine shortest short path ellipsoid pose region case distance suppose known reformulate ellipsoid outside ellipsoid ellipsoid determine particle close ht outside ellipsoid dotted evolutionary reach initialization update equation algorithm modify addition evaluation away addition advantage computation however particle move counter velocity modify velocity varied iteration ellipsoid surface sphere form velocity vector direct f minimize present position split previous objective dependent minimize direction velocity update relate include reduce purpose direction shortest place determined region particle place space particle point path particle reach often go position update particle lie within search intend position particle propose multiple gradient problem solve reach consensus discuss share alternate direction multiplier one constraint solver format present randomly update position global store maximum iteration specify experiment initialize good calculate range update velocity position section reliability test quadratic programming lp constraint lp reformulate identity ht iterations lp except reach monotonically ellipsoid length scale reach figure ellipsoid ht two lp iteration c ellipsoid position x neighboring space carry neighbor high fitness become ht algorithm deviation apply generally suitable classifier choose weight use classify test pose solve show class mean equation kind data feature hyperplane hyperplane region platform implement svm svm neural hyperplane learn show layer perceptron epoch training determination hyperplane class mahalanobis point ellipsoid class placing evaluate close optimize reach boundary determine hyperplane hyperplane observe hyperplane place hyperplane bias method svm equal network svm test uci repository dataset categorical case hyperplane characteristic network estimate classification hyperplane approach develop bias move recall varied nature attribute integer length linearly whereas linearly physical property colour intensity whereas chemical property content content chemical eight parameter value age value problem dataset database contain binary vector input perform formation sample step eigen decomposition eigen perform dataset independently subset validation project class eigen correspond covariance project test validation use remain fold ten subset subset equation hyperplane turn classify average cross
standard interact worker dynamically offer utility know outcome value time contract observe realize receive feedback assign compare utility goal subset infinite natural example integer offer minimal payment worker time let value outcome order increase tie arbitrarily convention nan outcome cost production obtain outcome effort let contract function outcome non contract sample payment ix e worker utility expect worker type population density many effort outcome effort type multiple effort worker contract break consistently worker effort contract minor minor throughout compare many benchmark reduce good contract contract maximize utility outcome regime quality fine nan outcome study special pricing obtain improve bound monotone appeal generality instance monotone appendix far monotone typically contract reason crowdsource restrict attention satisfy achieve guarantee relative machine useful benchmark benchmark relative guarantee relative x optimize special unclear extend contract crucial maximize utility economic often amenable rigorous suggesting may rational pointing heavily regard worker behavior serve enable guarantee collective worker behavior property use increase increment payment increase enforce form complete outcome pay formally increment splitting half dimension carry version task discussion contract mab additional basic mab repeatedly choose traditionally call specifically round select arm receive reward reveal mab arm arm round regret arm define reward dynamic contract design naturally model stochastic reward monotone mab assume upper precisely respect supplementary mab set reward payment determine outcome determine worker effectively supplementary structure sense numerically call area action know state discretization space several choose treat space use monotone bound describe discretization monotone contract nan outcome non convention contract bound contract monotone increment representation lie contract necessarily bound cube convention weakly discretization align dimensional increment henceforth cell least contract cell candidate contract denote increment maximal increment atomic contract advantage problem essential composite ideally maximal difference utility proxy useful follow prove type satisfy consistent break composite cell place worker behavior development worker behavior outline cell comprise increment round confidence index contract half corner cell introduce round algorithm consider round round utility composite payment similarly payment round anchor accordingly estimate virtual deviation suffice high average expectation algorithm active cell confidence contract become uncertainty due express virtual width cell summarize initially contract post contract allow amount composite minimal near allow contract corner issue uniform mesh preferable composite maximal corner contract contract minimize minimal anchor corner go omit issue virtual write pointwise dominate consider weakly resp contract resp contradiction type denote generate I xx contract give contract combine obtain note equality contradict outcome contradiction complete place directly break weakly worker effort respectively assumption either rule proof payment increment expectation worker finish contract significance corollary compare parameterize goal respect candidate regret rt contract utility optimal x cell feasible implicitly overlap sometimes contract outcome parameterize constant practically small outcome exponential notation candidate policy exponential type prior dynamic pricing bandit approach tend arm also fine equation apparent regret bound bandit state covering subset relative cover number cover cell virtual width contain cover collection minimal small cell cover size feasible observe exist minimal cell cell easily yy literature cover regret precise polynomial performance bandit appropriately notion dimension capture problem typically shape notion eq equation obtain consider dimension bandit approach contract design mab mab corollaries loose bound rt rt equation apply factor may well corollary setting discretization interval bound yx bandit metric discretization significant advantage argue without know coarse probably aggregate accord predefine formalize hierarchy subset contain hierarchy contract splitting half contain shape aware similar coincide regret focus spirit slightly structure cover ball ball albeit determine bad nice problem instance obtain match bound aware aware algorithm information special basic nan worker whether reject effort contract completely price distribution nan outcome rich discretization improvement principal somewhat rich worker salient contract outcome nan outcome bring outcome bring cost break way incur zero outcome high simple worker effort outcome high pricing x function note maximize know reduce essentially price high low lipschitz continuous fairly worker small width regret consider low give natural contract achieve range rt mab contract worker worker effort receive payoff therefore effort effort call inside non decrease discretization dynamic pricing discretization contract result contract denote p conclude width need count cell increment virtual width small denote benefit axis increment contract also contain characterize virtual cell expect payment value two definition represent simplification count cell virtual large care cell therefore relevant know virtual version alg alg arm arm randomly alg discretization xx arm argument mab regret execution event clean involve probabilistic argument tend simple execution cell contract essential ensure execution round execution least concentration chernoff need careful number activate th activate activate round fix cell claim choose round event cell tuple tuple consequently chernoff let integrate precisely let multiply side cell bind feasible union proof sketch one namely composite round anchor select anchor notion technique establish event play sufficiently often auxiliary enable choice precisely hold tc equation rest execution execution contract b claim anchor clean execution contract execution round atomic composite contract fix part clean execution fix contract composite atomic unique contract clean execution cell round else recent cell contract regret next upper bound hand equation clean execution lemma execution cell depend cell follow atomic contract candidate contract easy lemma ever activate order choose cell cell contribute focus cell never active activate cell call whether leaf since claim therefore cell leaf parent since case claim plug make simulation bandit thompson replace change observation consistent pick composite within increment minimal thus effort worker market extremely diverse third represent ground market market market across suffer near consistent intuition focus explore region slow eventually achieve worse initially eventually suggest know advance tune advance confidence confidence play simply reward plus try well across three market constant long utility round various first horizon outperform version market advantage specifically example adequate large fast adequate focus promising region achieve know advance optimize approximately calculate advance small vs homogeneous worker market show thompson logarithmic converge payoff confirm decrease round utility run round consistently outperform two bandit different discretization market discretization know discretization still outperform finish run set run thompson round opt offline plot large bar discuss see dynamic contract family perform price sequentially principal item agree item derive utility minus round know fix know principal pricing contract outcome special one effort generality effort level non crucial simplification contract design discretization easily mesh case implicit pricing modification pricing bind translate pricing pricing idea achieve match low sketch low lemma contain contract summarize nan worker expect contract loss price price interval virtual reasonable choice dynamic pricing regret sketch key call former happen otherwise mutually cell virtual red cell virtual dimension argument conjunction correspondingly bind task pricing horizon achieve give interval cost pricing interval partition sub dynamic pricing achieve continuous follow cell virtual diameter least diameter diameter j p feasible virtual feasible cell diameter overlap ok moreover less feasible theorem case instance rt nice two interval adjust two discretization specific principal derive utility fix p complete consist argument rt imply rt rt paper contract crowdsource outline area extension principal contract classic agent whose production describe specify contract outcome effort stochastically outcome maximize utility contract observe outcome effort level creating contract expect payment make maximization constrain problem principal assumption type know exist problem focus principle principal offer choose reveal type problem principal utility principal agent restriction must certain range capturing must agent paper neutral instead set hazard space know characterize decrease interact multiple agent principal adjust contract author agent agent offer variety set principal agent effort level uniform contract version principal single repeatedly agent effort outcome effort effort observe contract work agent learn type design accordingly study online principal focus uniform discretization approach study set outcome issue worker verify outcome standard know worker verification payment bundle outcome assume relax adopt david journal recently encourage high work crowdsource explore award improve user generate encourage internet content motivate attention sided encourage crowdsource market crowdsource market differ worker effort close crowdsource pricing task arrive price worker goal learn single generalization pricing outcome examine financial crowdsource market potential explanation phenomenon concept worker cost complete worker task complete worker worker appear payment mind research run screen worker quality effect existence overall demonstrate crowdsource market traditional people market rational worker economic demonstrate collective particular decision algorithm decision multi mab dynamic pricing mab operation economic branch computer theoretical science ai economic survey work beyond refer mab background bayesian mab lines mab mab reward neither know formulation understand handle arm information similarity arm arm relaxation discretization use particular general template lack priori require mab immediately search shape explicitly reconstruct part direction mab setting principal offer price transaction principal formulation version note initial improve specialized unconstrained crowdsource market define round agent treat bandit conceptual aside adaptive assumption provably case illustrative theoretical dynamic contract design far clear provable structure need order argue optimality contract currently fine cell sophisticated sophisticated bandit incorporate cell sample deep principal primarily natural mesh open concern x mesh mesh interest monotone prove significance unclear would characterize scenario contract bad contract result latter scenario need much extensive special difficult access appear direction corollary bound optimize apart design derive case mixture belong parameterized family unknown parameter direction budget extend corresponding dynamic pricing difficulty setting much contract adaptive discretization conjunction general bandit bandit choose mesh price budget discretization bandit section provide suboptimal restrict non payoff I level high type subscript describe worker choose effort outcome equally likely choose equally verify simplicity assume worker tie effort favor break tie effort level favor worker worker break tie consider separately contract make worker choose nan contract contract break tie effort contract cause choose prefer effort easy would low effort case expect worker worker choose effort worker choose contract maximize value maximize appear maximize appear set easy occur plug contact expect utility strictly preferable contract fact unique contract contract summary exist effort level effort obtain contract payment neutral I payment agent principal payoff principal agent payoff optimal contract give contract
bottom propose paper start objective illustrate step embed cloud follow propagate merge body extensive topology transition study application conclude main objective unsupervise coherent automatic move body body represent voxel obtain pay attention representation suppose collection link possibly meaningful segment move segmentation place unsupervise human intervention segmentation entire surface shape consistent element adapt topological priori move recover posteriori consistency segmentation limitation explain spectral apply action variety compute dimensional embedding preserve reconstruct follow square computed minimize error q embed cloud matrix rotation bottom eigenvector discard ht despite originally unsupervise display geometric unsupervised cloud voxel neighborhood preserve dimensionality cloud property roughly live approximately map spaced widely separate branch cloud constraint link form roughly force radial direction space much compare cloud arm unlike base geodesic neighborhood relatively purpose coherent obtain evolve geodesic strict sense laplacian embedding form link neighborhood preserve motion neighborhood evolve affect illustrate pose transformation intrinsic effect easier embed shape body pose propagation much exploit devise consistent shape weak cluster unknown inherently dimensional embed propose step instant cloud cloud cluster branch intrinsic low embed segmentation segmentation instant go description map embed space employ mean segment embed happen adopt sequence shape segmentation automatic moving embed try branch look roughly mean distance measure triplet leave generally tuple propose notion graph pair nonnegative tuple measure analogy edge point tuple next weighted vertex hyper construct finally part hyper embed affinity embed cloud trivially back use automatically step form branch instant branch termination certain empirically point project onto cloud red branch termination side square g consider termination well embed transition move whenever happen accordingly return need segmentation change contact different part centroid initial seed cluster embed cloud centroid square left position old colored circle xt jt obtain embed cloud embed centroid embed cloud time colored circle seed cluster arise besides help contain body contact cloud possess way arranged belong sensible branch detection side suitable tool allow adapt topology instant branch cloud seed cluster branch seed yield rough cloud branch seed inside k mean branch termination seed branch termination wise seed get close distinguished make sense opposite impossible distinguish require merge topological move body validate change management summarize segmentation integrate section instant ht current data figure embed yield cloud branch embed cloud detect branch plus embed cloud cluster group section seed use branch centroid branch splitting merge centroid embed shape centroid map centroid back algorithm synthetic qualitative quantitative test set sequence person ground body link e simulate form plus volume solid black drift inside shape motion usually span distinct body part phenomenon trajectory segmentation visually show relation sequence frames b subsequence frame whole centroid transition management arm contact number accordingly proceed smooth viewpoint several high camera long capture move around cm typical smooth centroid trajectory separate complicated perform body topology person remarkable possible three cluster embedding mean ground truth voxel close capture multi camera phenomenon gap figure middle body head middle adequate score high method show remarkable compare drop brief trajectory top show middle gap affect quality segmentation embed cloud stable three challenging sequence apparent strong corrupted topology bring size neighborhood former affect stability embed shape along low notice embed remarkable general embed varie neighborhood body different anomalous belong distant neighborhood top right value yield neighborhood method use computer graphic tool analyze surface virtual reflect classified encode operator mapping vertex neighbor point vertex edge th trivially represent also operator point affinity laplacian w locally laplacian eigenfunction geometry underlie eigenfunction domain set eigenfunction discretize follow eigenfunction coarse partition interest affinity link associate determine cloud choose dimension result make shape arbitrary able visually visualize eigenvector color cloud determine top four brief dark peak different eigenfunction locate eigenfunction peak result may possess branch eigenfunction able resolve head separate head appear eigenfunction argue segmentation branch turn amongst illustrate voxel fair run product voxel limit natural segmentation head show top bottom stable subject consistency consistently body move top bottom point laplace sample produce distribute along regular voxel neighborhood cause instability embed cloud particular main index segmentation sake simplicity first eigenfunction w segmentation figure plot segmentation apparent pose come contact cloud truly consequence measure distance along body path cloud illustrate split merging segmentation preserve embed change situation preserve change recover distinguished compare shape body synthetic sequence plot exhibit smoothly virtue plot space embedding exhibit body perform geodesic spectral transition clearly advantage cluster shape embed choose sake change dramatically see building motion framework segmentation body look clearly smoothness track feature track figure model estimate classify one segmentation coherent segmentation chain output reconstruct rough model free motion fit principal axis position part voxel set represent environment hand interact virtual object virtual environment ellipsoid implicit representation time interactive object voxel note look construction identify remarkable hand evolve become isolated rough show comprehensive present dynamic segmentation cluster temporal consistency exploiting characteristic estimate topology transition algorithm versus k synthetic generation extension separate contiguous pose quite straightforward manner propagation different object exploit method remain preserve near future universit france order motion pattern compact discriminate context learn recent time motion capture voxel gain segmentation move entire coherent robust construct bottom move body track motion locally embed useful volume shape dimensional easier unsupervise coherent segmentation shape embed coherence merge accommodate body real voxel datum consistently totally unsupervised robustness quality series discriminate infer learn fashion time set calibrate yield gain attractive inherent invariance motion flow kind vision computer graphic visualization sub mesh segmentation application graphic link representation mesh actor reality texture map reconstruct may topological scene actor extraction obtain smooth suffer surface curvature sensitive shape graphic applicability visually acquire remark interest recently segmentation visually reconstruct segmentation static mesh mesh propose probabilistic method motion segmentation mesh advantage embedding mathematically voxel think describe shape connect vertex represent adjacent point voxel address partitioning provide extremely powerful framework allow partitioning laplacian suited space span eigenvector laplacian problem allow circuit partitioning segmentation mining method laplacian characterize perturb ideal datum infinitely link locally map attempt graph circumstance partition kind graph strongly make mesh segmentation spectral cluster number firstly difficult completely secondly eigenvalue symmetric matrix crucial finally eigenvector interpretation mesh domain view eigenvector applicable measurement estimate pose discriminate motion mesh surface volume approach availability set volume surface geodesic volume invariant pose surface sensitive topological often occur visually less mesh local mesh volume consistently partition surface pose community curvature derive segmentation reliable visually reconstruct consistently segment point wise rely use explicitly
spline outperform foundation grant f theoretic smoothing spline reduce describe curve remove outli spline dynamical result problem risk describe convex solve via numerical control spline widely process particular give curve limit curve well minimize curve theoretic spline spline control spline linear theoretic spline rich curve give robot drive draw smooth find expect motion theoretic spline trajectory mobile contour distribution estimation name application theoretic spline see conventional theoretic spline drawback fit crucial drawback conventional control spline outlier overcome propose parameter utilize norm absolute shrinkage robustness adopt norm assume tail assume study matlab computation matlab program effectiveness method organize review control discuss drawback spline draw conclusion dynamical na suppose sample data control dynamical follow cost regularization specify smoothness define second weight loss theoretic spline formulate control e impulse response dynamical define control spline control compute offline formula draw curve drawback spline noise adopt design spline error spline coefficient lasso p pp additive assume heavy account unlike spline solution represent within extension nesterov achieve still use efficient software seek minimize basis variable optimization curve control trade fidelity zero chosen error cross section example spline rate
derive corollary fix section eq ix x n kk x probability continuously differentiable statistic quantile estimator assumption probability scalar pac arm eliminate total refine upper k derive regret convex measure average law coherent express integral guarantee risk depend choice either quantile section theorem almost arm assume small q derive give recommendation average bind rt h functional quite language maximize bandit describe focus refine countable entropy denote number consistent entropy plug matching entropy function support let near estimator theorem proof result recommendation rt information functionals r enyi entropy notion shannon cf divergence predefine arm tackle elimination arm elimination particular assume efficient arm stop refined use management plug equally want probability return advance proof lemma well eliminate drop episode arm arm eliminate denote number per l triangle follow bind finally triangle equation chebyshev v p v triangle eq verify two theorem yu arm unknown arise entropy language new combine arm tackle achieve provably illustrate method number management refine stochastic slot reward independently randomly arm high reward budget accurately range identify medical trial transmission mab optimisation find high arm functional arm finance risk capture functional metric arm functional expect estimate consistent bias functional entropy risk background call identification optimal propose efficient elimination optimisation arm trial algorithm generalise successive elimination sequential theoretical guarantee refine scenario functional applicability management risk management variance functional functional widely follow contribution introduce identification bandit generalise arm propose elimination regard generalise optimisation theoretical refine risk management discussion bandit variety adversarial survey bandit type average pure exploration relate pure markovian armed bandit risk utility risk functional real maker variable arm line notion finance optimization bandit consist receive unknown repeatedly arm result sequence functional goal identify high within exceed observe round total arm elimination sufficiently value I efficient one sided property guarantee far functional elimination obtain wrong order arm wrong addition eliminate eq arm eliminate round eliminate main state main eq probabilistic strictly success correctly sketch denote arm eliminate conclude decrease follow corollary follow pac probably correct pac former hold sketch sketch let conclude result comparable arm design expect compare follow guarantee exist value indeed l recommendation straightforward roughly speak outperform exponential exponential imply sufficiently one comparison elementary algebra
specie datum distinct similarity ignore closed lr procedure bold column hypothesis reject approximation literature information gain look hypothesis adjust elementary adjusted level overcome assess whole principle possibly real practitioner offer assessment whose gain remark type application provide know advance investigate procedure family contaminate gaussian p eigen popular cluster year remain likelihood tackle use alternative flexibility applicability along way develop reference lr considerable separated accordingly generalize procedure assess model advantage via application mixture eigen test gaussian see sect consider assume class original allow popularity largely theoretical significantly increase parsimonious model impose popularity contain member member axis member flexible parsimonious family see sect ml sect relax family base decrease eigenvalue ml constraints sect lr statistic compare family unfortunately solely restrict herein adopt hypothesis sect approximation lr component sect model line parametric lr sect drawback discuss pairwise benchmark overall member preferable mind lr recent sect aspect lr sect sect procedure demonstrate advantage variate th j parameter introduce consider eigen decomposition scale sort orthogonal column accord eigenvalue element geometric volume orientation impose constraint right side parsimonious herein triplet variable write different orientation restriction k pp kp kp kp provide representation model denote algorithm work I come indicate closed ip however characterize state fundamental family sect motivate orientation r scale component configuration family motivated end sect estimation procedure log group scatter derivative preserve programming programming primal active apply update common methodology sect transform adapt way test lr regularity commonly asymptotically distribute freedom specify ht start kp k kp kp kp kp k k kp k kp kp kp kp k kp kp regularity reference examine aspect come shape orientation eigen necessarily fix regard bivariate ex note shape eigen volume parameter shape orientation far generate regard choose orientation variate numerical guarantee overlap adopt measure overlap take value overlap overlap take computed model ht c c arrange move increase nan approximate uniform gray result assume size increase provide small encounter practice method bootstrap proceed replace compute bootstrap turn repeat successive assessment distribution distribution accurate concern solely rejection specify replication approximate replication approximately assessment hence sect bootstrap mm refer approach regardless prefer procedure respect entire generalize lr completely label hypothesis play true hypothesis nan false true false true mm indicate restrictive e parsimonious bf df say lr restrictive significant denote report reject significant strong evidence interpretable comparison hypothesis powerful among strongly control probability partial false hypothesis lr bootstrap lr environment cf strategy ik soft way one multinomial implie initialize force algorithm hence bootstrap sample correspond
normalize node node unseen unseen class markov come markov chain canonical form describe describe unseen leave zero shoot predict semantic involve perform shoot semantic connect connection probability specifically image see visually use classifier extend canonical meanwhile extend see node write extend transition state extend semantic node unseen g reflect unseen probability compute extend chain inversion formula eq start canonical block whole dataset image store probability equal stack final pre variable image therefore method linearly unseen maximum paths semantic whole semantic stable similarity zero shot see class compute c roc cat auc good per indicate bold zero shot provide class source class alternative first direct base see class category model bipartite image new vector test shoot prototype near shoot training rather apart publish first unseen name train skip gram text corpus word unseen english concrete website gray word unseen use training apply toolbox semantic choose unseen search near construct subgraph class choose see subgraph top cosine ensure unseen connect code shot area auc semantic auc six individual class ten result direct method nn almost class dataset class attribute propose comparable especially note visual attribute category manually exploit free word linguistic basis manual annotation encourage visual exist method c vector attribute ds see value see image influence parameter also totally divided fold fold run ghz average number shoot graph visual category shoot chain effective stable bipartite via transfer explore unseen class differ embed space image free space former focus work focus relationship unseen semantic word unseen view incorporate one probability unseen effectively shoot achieve linear image zero receive recently via social medium sharing image build visual large visual recognize make connection unseen class unseen semantic space embed early semantic attribute approach embed define binary attribute similarity attribute share class manually intra poor scalability attribute alternatively start popularity learn language scalability adopt remain similarity datum shoot unseen class training class need option first datum learn representation image belong unseen intrinsic limitation learn unseen domain adapt function unseen class option embed use low semantic feature embed purely unseen semantic superior direct vs shot object unseen unseen semantic graph see previous see unseen model fig unseen structure ignore view exploration modeling relationship flat hierarchical compare bipartite graph semantic stacking probability together zero shot linear image approach give section paper conclude zero semantic attribute employ embed knowledge exhaustive attribute work automatically discriminative visual embed attribute overcome explicit recently extract basis wikipedia attribute prototype target project use prototype shot learn word bi space manual annotation scalable scalability differ transfer experimentally exist label contain see exist attribute variant take mapping low embed learn mapping space image unseen class semantic prototype mention early strategy level test classifier similarity
sequence subject target indicate presence confidence information need human visual exhibit strategy location presence target ask human take present learn strategy optimally reinforcement setup operate regime inspire little challenging ground truth region available consist absence image human subject ask action available begin brief localization evaluate discuss experimental information extent scene play fall onto person background human weak localization overlap segment bounding box average action search signal segment pool analysis segmentation global vs image index represent index belong image image bag correspond region human image represent bag correspond contain region know none region detection target wish simultaneously separate hyperplane positive one bag inside positivity encourage separation bag response localization target distinguish contain negative poor localization positively label label slide detector exclude ground greatly localization overlap threshold eq formulation hyperplane eq k remove label treat instance g ik g n change iterative alternate assignment subject constraint bag iteratively maximal remain line positive bag enforce label maximal instance bag always label negative line detect bag long iteration equation exhaustive weakly aim minimize load restrict methodology experimental pd td f te pe response encourage formulate final incur execute penalty reward model start set approximated sequence reward strategy one please supplementary material bag consist segment bag consist segment evaluate svm c set segment extract image max convolutional bounding descriptor box image mask rescale pass record fully additionally build consist normalize aspect final neural bounding descriptor train function segment remove constraint train confidence segment restrict image label investigate setup bag segment pool image return truth box object action within bound play head mobile phone fig trivial evaluate metric response predict bound segment fall truth report classification outperform baseline metric large supervised precision metric restrictive suggest supervise approach recover invariant pattern within least absence full extent actor topological learn substantial metric additionally leverage movement annotation classifier good learn supervision mi fail human movement task train seq bound bb seq case learn image alone optimize bfgs optimizer regularizer initialization run metric measure segment total use intel cpu require approximately half segment leave base exhaustive bb performance affect extraction optimize operate reduce search form novel base static box annotation topological constraint accurate classifier movement additionally novel sequential achieve detection extensive progress speed supervision institute mathematics science mathematics department se recognition detection increasingly therefore expensive manually keep run address issue weakly supervise segmentation detector signal provide localization system use inspire operate detection confidence exhaustive fraction constructing detector decompose confidence detection search set search suit train require manually algorithm increasingly need search practical rich supervision development hardware less expensive annotate acquire training usefulness absence additional annotation insight operate paper detector confidence weak manual image annotation contribution learn static associated movement require annotation box segment movement integrate strong supervision target box develop sequential operate improvement accuracy slide window accelerate prominent branch neighboring region use detector base vector image prediction formulation supervise svms see review movement collect video include face action successfully boost action segmentation availability movement human use object detector method employ supervise human reduce box play role annotation ground truth limit two recover ground confidence require availability bounding box visual idea problem recognition face novel fully setup show want list tight bounding box actor additional spatial
initialize ap successive coincide red ap similar randomly lower constant lower bind linearly norm lie extension make pair small state duality turn duality gap translate discrete linearly require q minimize separately value desirable reader running times oppose one exploit combinatorial algorithm frank wolfe gradient combinatorial run integer value function frank wolfe subgradient behave berkeley submodular variety discrete machine processing vision minimize pose challenge extremely linearly upper rely geometry submodular spectral recent submodular set inequality high potential submodular minimization q exist inefficient scalable optimization frequently possess admit quickly example kind function count joint term subroutine minimize modular box cut certain sparsity induce covering express submodular recent demonstrate offer benefit admit minimize separately manner decomposition decomposable close bring together yield empirically implement case performance heavily well alternative leave geometry approximation prior relaxation problem importantly associate submodular draw submodular beneficial submodular consequence ellipsoid polynomial fast integer require address decomposable integer maximal cardinality simple approach nesterov sublinear integral algorithm study convex feasibility good problem survey subspace characterize angle rate alternate arbitrary convex understand give condition alternate case unclear challenge rate uniform submodular largely study relate thereby obtain generality relate subspace connection useful submodular identify modular modular polytope many extension problem relax extension round continuous indicator amenable smoothness alternatively formulate proximal recover indicator discrete bf bf imply project project project projection projection simple live set note submodular omit dependence simplify bind uniform submodular ap descent start via implement solve analyze eventually pair ap convergence ap motivate approach simplify setup subspace span case angle slow higher relevant cosine subspace arbitrary converge linearly rate ap rate generalize consider pair show rate ap nonempty face generalization angle two part relate ap faces general ap theorem angle corollary angle matrix tool specific state bad rate ap ap result weak assumption join triangle cm color color pattern pt color color pt color pattern color width singular product bf rbf rbf proceed characterization face base submodular disjoint immediate disjoint write follow directly multiplication less remark less equal matrix index row remain view matrix symmetric weighted let index vertex laplacian small eigenvalue appendix bound hence appendix ap converge probe submodular slow augment submodular f r zeros cosine around optimal behave subspace pick initialization angle low exists decompose ap generate objective find define multiplication map fx fx translate discrete set small value give number submodular highlight structure serve work aid grid hard additional suppose nonempty subset relevant applie run future dr subspace converge oppose cyclic stochastic award award fa amazon services intel microsoft yahoo office research grant nf nsf need map close nonempty subset suppose eq follow similarly lemma part rgb round join triangle cycle pattern color color color width pattern pt pattern pattern circle fill circle node fill pt segment convexity middle angle inequality face face far point amount toward interval every either contain face whose relative face contain unique face state ap possibility intersect ap terminate terminate otherwise face j inductive ap ap along see result face face amount affect angle
generalization cover return encode piece encode second part possible construction towards label diverse want encourage change correspond assign label diversity indicate envelope take value index pairwise message bp diversity cut plus minus berkeley origin negativity achieve achieve space svm extract often slack svm primal primal w equation add problem equivalent enforce negativity primal solution hence expect negativity plane label diversity presence scoring contain diversity contain ground define item belong adjacent label transition marginal gain become eventually maximize cut look front show different diversity set solution understand different fs fs prove bad case let fix obviously submodular cardinality constrain optimum sized correction least monotonicity submodular maximizer I helpful fa monotonicity rearrange
equivalent subset stage algorithm add variable consider stage lasso induction lasso know sign support recovery hold lasso lasso abuse show invertible al know indicator function trace solution parameterize probability apply al weighted apply always assumed lemma et new probability combine force lar al lar force allow algorithm analyze x x lar lar estimate lar assume variable hold lar suppose correspond lar active lar iteration slight guarantee variable point w zero x strict contain argue induced base x variable select inductive lar intermediate since run lar stage variable add stage since intermediate w stage always follow add principle end stage lar know recall give partition g define constraint union decomposition permutation product permutation b b permutation unique sorting specify construction q b permutation notice tuple permutation group b consider piecewise linearity supremum set piecewise lie local minima tuple permutation induce replace x absolutely shows immediately seek boundary lie concave switch interior boundary versa purpose derive contradiction concave lie hold unless contradiction lie wrong minima lie concave path path b nonempty union solution b suppose absolutely unique however produce produce homotopy maintain identical lar modification note notational convenience carry homotopy
portion portion memory single master machine similar repeat amazon ec master machine core gb ram run hardware acceleration netflix row column zero report large want remove low running unit storage implementation ni miss goal row variance one simultaneously consider standardize via model standardized realization random would realize unique include likewise overall refinement attempt center complete issue learn center latter important completion reverse center prediction miss full although centering similarly conclude lemma place replace lead consider converse appear use say hand analogous true thereby use lead rate complete q assume proximity rate interpret locally eq rate property towards limit simple problem suppose limit k fa kb line follow contradiction e limit point lead problem much load want center sparse class introduce old rearrange get similar modify multiplicative symmetry get similar equation amount iterate four equation zero algorithm remark proposition lot largely netflix competition solving margin factorization bring together completion software implement approach environment index element preserve replace onto complete nuclear singular rank develop iterative solve step replace entry correspond thresholded operate replace element set solution matrix svd netflix storage number pose netflix use svd singular reduce svd compute alternate exploit piece hence store piece alternate step multiplication well svd time use warm start get warm start path use solution warm additional different use alternate algorithm solve separate ridge regression response predictor ignore amount regression remarkable tie rank block note orthonormal rank give draw idea step use fully ridge reduce regression use alternate amount shrink component offer rank property step recently approach see simulation remainder detail large superior include netflix highlight publicly implementation describe center incomplete develop svd adapt likely expert proof convenience inequality value denote suffice consequence von establish problem optimum appear fact solution characterize factorization consider lemma conclusion theorem inspire alternate algorithm reduce present ridge initialize randomly ridge simply coordinate compute multiplication follow svd keep solution represent need subsequent ridge trivial essentially alternate necessary naturally block many leave likewise plus multiplication determine singular correct evidence shrinkage accuracy bias case index solve initialize randomly alternatively warm start follow repeat suppose wish side current use importantly eq sparse dimensional problem multiplication modification step computation version predict multiplication rescaling row solution change c significant final solution soft svd ridge might reveal soft discuss lack warm almost package stationary denote play rate produce algorithm correspond problem derive low formal update lead derive description first produce thought style step upper loss recall objective ab outer equality observation suggest fa b ab ab ta bb bx ab x z observe procedure proof easily establish iterate never function monotonically iterate lead q aa put use argument complete previous derive elementary update every limit sequence point problem reach point respectively consequently fix follow thus quantify close stationary algorithm make improve monotone sequence characterize zero decrease converge quantity rate establish arrive theorem role corollary employ closeness stationarity respectively successive iterate understand guarantee remain across avoid appear estimate property begin notion point say order point follow fix update point singular tie stationary let limit point sequence subsequence sequence converge limit subsequence converge partial uniqueness point converge must converge versa generally technical bound leave update unbounded objective implication imply modification modification implement step idea modification also take choice hold decrease overall compare follow factored decrease nuclear sequence condition factor converse true estimate upon iff point satisfy convex stationarity stationarity rank stationary point see verification converse point condition stationarity closely nuclear operation constraint matrix task attractive plus low generate let limit solve solution broken forming require form require factor multiplication mention solve factorization algorithm instance descent separate ridge initial ij x section may decrease criterion slow factor experiment show netflix show result increase last least green svd svd iteration likewise exist instead fraction operating algorithm involve alternate ridge regression third alternate orthogonal operating perform model true rank k movie pick operating criterion coincide degenerate fairly consistent reason use execute early even though warm svd netflix competition user make missing probe subset rating leave netflix imputation shrink panel calibration light far dot competition compare restrict shrinkage rmse score somewhat win improvement warm
class sl sl neighborhood data sl method train resource table several baseline sl ne retrieval popular linear classifier sl ne achieve much accuracy complete reasonable amount embed training effectiveness accuracy attain sl close sl sl project sl perform sl ne similarity space may sl sl similarity function sl sl function retrieval retrieval curves sl ne consistently sl achieve sl ne b sl ne sl scalability visual challenge contain object validation test set sl train evenly pca prefer give fair short flat error baseline include svm retrieval two metric sl ne sl ne rate similarity huge sl ne improve lot fisher feature representation learn sl ne weak reduce indicate mm sl sl de sl subspace serve ne sl take sl de hour exclude retrieve train distribute computer sl ne day novel investigate neighborhood margin relative similarity training irrelevant ensemble scalability dimensionality validate classification achieve importantly demonstrate scalability method neighborhood efficiency potential direction neighborhood great complementary metric token lin research wang classify category huge amount approach promise study decade impractical handle paper scale image descriptor end discriminative similarity induce exploit dimensional process parallel learn quick validate scale thousand art achieve efficiency scalability internet automatic categorization become conventional approach versus paradigm large mention size web vast densely database result include parse face classification near knn successfully imagenet challenge measure characterize distance yield learn theoretic margin supervision comparative triplet grow polynomially modal usually single insufficient correctly throughout metric apply part datum space assign metric discrete parameterize extra besides dimensionality descriptor another scalability algorithm mahalanobis distance form computation dimension calculation cubic semidefinite impose mahalanobi save non reduce free feature scalability finding manifold sample local neighborhood try discriminative save focus original high subspace superior scalability datum validate sec efficiency scale thousand sec contribution paper embedding similarity scale reduction manifold intrinsic represent weighted graph laplacian construct refer dimensionality regard case distance learn label distance exclusive weight embed sample transform purpose similarity number quantify similarity bilinear similarity parameterize low place find embed cast weight encode commonly whether come partitioning impose pairwise sample knn triplet assign neighbor cardinality weight triplet hinge similarity hinge require measured transform solve eigen decomposition approach eq instead use gradient iteratively whole sample contribute gradient triplet update performed iteratively terminate upon triplet though strategy triplet speed accord optimize return triplet relative similarity sl ne data set normalize norm sl ne introduce objective base triplet constraint key scalable advantage sl ne sl ne computational progress benefit build dimensional impose low unfortunately bilinear simple quadratic computation ensemble low subspace projection learn function similarity ensemble project however scalability practice build energy guarantee sense direction efficiently approximated entail computation metric perform extra iteration projection attractive dimensional data similarity function impose similarity try summation constrain subspace explore combine scalable employ combine metric metric parallel computational forest distance binary output parallel base partition subset scalability sample learn ensemble function apply sl ne induce gain sl advantage distribute manner sl ne similarity way potentially reduce consider overhead parallelization function helps quickly offer resource carry optimize coordinate lastly sl accelerate metric sl ne propose sl ne version sl locality constrain code spatial pyramid grid dimension unless specify project space normalize metric sl ne sl de step sl sl open neighborhood neighbor search hash neighborhood sl soft voting knn classifier similarity specifically index class voting sample select entire testing sl ne sl svm retrieval generate neighborhood code publicly rough five
boost ratio supervise high observe improvement augmentation yield improvement convolutional auto train unsupervised fine cifar dataset method subsequently initialize call unsupervised fine highlight unsupervised auto learn input bias unit commonly choose include sigmoid function meanwhile back representation bias learn via descent constraint encoder tend impose upon structure form regularization include noise input hide activation model auto show relu cause activation use thresholde auto encoder regularization connect address convolutional cnns responsible neighborhood visible dense extraction pooling stack network learn aspect convolutional net possible extract localize fashion include introduce pooling map influence limitation procedure inference deep feed convolutional showing cnns show framework break pre supervise fine describe incorporate salient yet network convolutional layer bias describe architecture previous involve module follow module encode module nonlinearity follow pooling encode module decode module pooling filter decoder fashion reconstruction network would express learn representation fix bias convolutional activation auto knowledge train initialization neural network training relu relu set bias initialize draw patch layer sample singular value cnn account additive overlap hamming window intensity build weight decoder module module layer softmax module layer unsupervise supervised descent decay select duration pre center natural cifar color draw object category object benchmark learn relatively per class unsupervised contain per example cifar structure similar show modification consist convolutional layer size layer size pool full softmax output net train neural library state unsupervised supervised tuning report overall qualitative result bias convolutional auto perform encoder auto analysis regularization completeness report full visualize filter directly filter present convolutional auto layer indeed interpretable orient color center quantitative performance pre use augmentation experiment train initialization one pre train cifar cnn relu interestingly subset compare unsupervised experience subset cifar cnn bias supervised regularization affect specifically horizontal training zero zero cnn regularization cnn figure individually subset cifar supervised fixing unsupervise vary setup augmentation dropout virtual large unsupervise notable unsupervised pre augmentation dropout individual iii experience large gain observe sample figure effect iii supervise decrease elaborate effect improvement rapidly surprisingly unsupervise learn supervise cnn unsupervise unsupervised cifar worse unsupervised sample pre convolutional assess experiment beneficial design benefit network comparison network layer unit pool layer convolutional layer tune split consist class set accuracy subsequently average final recognition cifar train split highlight unsupervised increase extensive augmentation rotation color additional value pixel like six image modify pixel additional data helpful pre maintain accuracy convolutional convolutional hierarchical match task bayesian bias cnn cnn auto
relative primal dual tolerance notation r r r indicate version tv use acceleration nesterov fista manually fully explicit variant primal reconstruction dual manually primal cp e implicit variant cf section cp perform tv furthermore pre si however q problem warm filter whose effectiveness validate ray ct dct ii adaptive motivation initial choose fix strongly influence speed algorithm matlab execute ghz gb matlab build matlab limit single algorithm evaluate object via radial angular use dimension compute apply minimizer ground iteration influence inexact due may speed expect iteration relative threshold adopt individually reach truth show data measurement event cpu since plot performance cpu present evaluation backward projection nearly cpu indicator performance backward projection compute step consume thus projection evaluation crucial follow discuss tv show inexact tv problem lead also well accurate indicate algorithm trade rate another concern accelerate modification em tv tv actually improve em tv number iteration parameter increase tv result denoise tv inexact variable metric view tv error second cp cp contrast em tv influence inexact decaying observe affect yield convergence e suit achieve versa plot figure acceptable trade asymptotic convergence latter aspect case natural acceptable trade acceleration cp cp line size evaluation iteration right pre figure si observation tv choice trade cp si si evaluation cp si achieve iteration efficient lead tv denoise effort thus practically choose optimally balance proximal evaluation error second within tv algorithm dependent augment yield achieve see row cpu algorithms perform cpu get relative make tv evaluation tv cpu decrease problem approximate increase tv denoise utilize rough cf improve tv tv evaluate need high result slow practically evaluation term cpu cp performance cpu time evaluation decrease step improve remove single evaluation step increase number total table display second evaluation error performance cpu value coincide em cpu cpu em tv cpu tv cp cpu si best cpu cpu stability regard choice value smoothed smoothed table describe parameter disadvantage solve solve increase I lead cp si rough em tv scheme able respectively high carefully additionally accuracy proximal tv si tv across addition proper different parameter time require evaluation tolerance tv tv cp cp si base regard cp cp cpu tv tv cp si tv tv cp cp si cp cpu cpu tv tv si conventional ray ray intensity ray clinical practice decade carry understand valuable object distinguish type agent detector ct energy thus eliminate obtain spectra layer integrate detector usefulness image ct advance technology detector individually provide availability development lead image name material practical ct specific refer overview projection mean energy reconstruct material acquire maximum distribution number material consider accept integral reconstruct projection element inter compute decompose high fact exploit fully spectral ct proximal material penalty admm preferable ray ct g example apply employ ct simulation measurement spectrum response counting prototype ct scan current detector height source detector view per energy spectral decompose effect decompose via fisher information treat filter admm middle latter reconstruction material middle fully material reconstruct row good iterative exploit possess reconstruction strategy manually indicate preliminary advantage exploit inter experimental ct simulate six bin em filter describe middle cross decompose covariance effect right recognize difference maximal intensity reconstructed show frank university monte manuscript work ct university award research north theorem develop quite flexible allowing denoise resolution optical flow form eq fidelity depend often operator nonsmooth functional successful technique nonsmooth generalization norm coefficient reference therein efficient imaging structure choice like augment overview result computational providing discuss notation average operator discuss pose present ct notation definition chapter whereby operation convex semi continuous real fx fx subdifferential set consist conversely differentiable element global minimizer conjugate positively conjugate usual conversely norm digital common basically kind comprehensive overview proximal proximal prox define envelope straightforward ensure minimizer characterized envelope exist unique envelope continuously minimizer iv prox prox univariate shrinkage threshold know huber label node scale bend fy f proximal subdifferential function df rule subdifferential calculus uniqueness proximal operator introduce positive involve efficient frequently appear mapping operator empty rewrite consider next substitute lead see back substitution db proximal orthogonal onto x start similar describe projection completely consider operator q threshold sort large index simplex previous group norm consideration sometimes shrinkage let prox prox elastic set operator whole proximal evaluate involve norm unitary characterize positively homogeneous vanishe origin permutation analogous symmetric singular interested define x ij x singular nuclear norm find singular determined q determine construct solution diag diag diag diag u diag continuous e onto convex similarly prox prox prox f prox x prox prox banach fix contraction contraction property start continuous converge unfortunately follow example show operator obtain well x eq back name average yet operator space every operator lipschitz average respect parameter operator average assertion ii rx ix iii rx tx ty tx ty ty ty therefore ty tx ty ty rx rx rx rx expression conversely complete average composition composition average also average concatenation operator boundedness sum harmonic asymptotic mapping asymptotically q exist subsequence hand r j bt strict tx l result replace weak short proof simplify convergence follow regular iterate hence whole sequence converge e combine theorem main result converge theorem characterize rise proximal back since converge prox fx generalizations cyclic parallel lipschitz lipschitz e subdifferential calculus know minimizer proximal proximal splitting prox special function non become also detail modification extension lipschitz continuous lipschitz exist average concatenation sense norm hilbert algorithm function convergence indeed fulfil algebraic process arise solve time slow idea rewrite prox r modification convergence see minimizer progress know variational characterization proximal ii add main inequality q give thus single regard yield generalization algorithm nesterov nonsmooth fast iterative metric strategy replace symmetric positive definite search strategy mention approach overview cyclic ask rewrite lagrangian infimum saddle lagrangian problem problem converse primal constraint saddle lagrangian optimization alternate minimization know precisely note differentiable lx version type allow overview primal r r r new admm furthermore correspond theorem proof lagrangian saddle produce saddle lagrangian arbitrary consider g r subdifferential r h r p gx gx r r gx gx r x gx r gx ax r p r well impossible keep explicit p cauchy schwarz p p r r r r x p x r relation get sum regard x n r n convergent subsequence cycle concatenation affine operator operator jj saddle x r p n n j imply equation linearize proximal interpret admm variant proximal positive introduction provide flexibility operator induce symmetric definite admm also classify inexact monotonically analyze additionally allow inexact apply inexact use straightforward see primal hybrid bregman method process worker admm linearize bregman yy bregman interpret first order special bregman distance bregman shannon discrete generalize kullback leibler proximal bregman proximal read fy r converge initialization minimizer attain convex convergence objective x eq use equation imply split p obviously split approach type inverse fidelity regularization generally banach cf unnecessary bias contrast loss variation iterative pose saddle close inverse pose raise issue practically expect range reformulate standard paradigm iterative increase decrease saddle version optimality would abstract subdifferential know cf reason approximate iterative iteration topology ingredient primal variable use understand iterative analyze augment saddle superior imaging eliminate fidelity bregman together detailed consecutive bregman prove topology variant approximate linearize analyze however restriction iterative ill pose completely open future well variational step splitting dramatically reduce exist saddle decrease estimate convergence speed discussion aspect far heavily force rough algorithm classical imaging vision analysis application natural science area present emission x ray ct detail reconstruct worth emphasize proximal splitting algorithm broad application last decade mainly cause ability distribute big intelligence internet finance another boost popularity splitting regularizer problem formation derive quadratic
number due snps selection application linear big feature general np therefore resort greedy use sure screen sis select independently correlation redundancy correlate penalize introduce regression penalty lasso regression penalize method interpretation penalize penalize maximize double exponential exact intractable result relevance crucial inspire success computer study strongly estimator prior approximated reason ise posterior probability mean rapidly intensive validation procedure especially well suited modern often approach traditionally focus weakly effect assume tend weak regime penalty strong influential suited usually validation prior weakly perform well exceeds greatly exceed prior strongly physics derive series expansion posterior associate present simulation correlation predict c predict quantitative trait datum aspect many g describe gaussian generality throughout paper regression regression strength choice penalty include ridge maximum least instability compute l end exactly describe conjugate give close indicator simplicity coefficient ahead reflect identify bayes theorem error expression identity row residual error coefficient condition specifically prior bayesian penalize regression partition x variance directly relevance could maximize posterior distribution setting estimate expectation evaluate analytically computation often long expansion ise external field define pearson variable assume enough necessary detailed derivation expansion long regime strongly restrict matrix standardize relate covariance rx natural penalty expect order expansion reference map ising perform calculate ise approximation lead consistent field computationally tool approximate feature calculation correlation penalize expression depend determine usually value ahead feature selection variable n computationally eqs pearson aspect require storing coupling potential adaptive requirement even first relevant specifically therefore rank pearson sure first order approximation strongly spin represent enter couple negative coupling include couple redundancy feature first analytic expression affect percentage body small enough posterior applicability expression gene intuitively correlation demonstrate impact inter correlation selection tractable correlate pearson regression coefficient study expression feature feature stay path generate feature correlation figure feature red high clear gap separate irrelevant correct feature visible inspection feature beyond performance demonstrate relevant black irrelevant black vertical line accurate correlation feature point root relevant suggest posterior figure small calculation rank relevant ranking highlight couple reverse engineering predict expression consist team correlation blind elastic net predict response actual value parameter validation elastic regression feature highlight present achieve benchmark compare validate path prediction phenotype expression sample probability line net rank gray light gray outer quantile gray middle dark gray line mean posterior identify vertical compute posterior lead team challenge choose rank rather zero plot compare et gray represent quantile gray represent quantile gray outer gray dark inspection figure al high posterior among show percentage demonstrate generally feature probability although gene expression identify validate posterior range chance highly variable et selection surprising gene root correlation critical compare large ignore correlation relevance highlight importance genomic penalize regime call ise ise expression control analytical technique develop study analysis study bayesian technique practical efficiently path probability relevance dataset genomic study work ideally suit genomic feature much regularize regression relate focus identify mean occur transition regularize influence highlight accounting correlation assess statistical significance even small correlation cause huge reduction probability expression represent moderately likely implication assess ignore implication result correlation probably significance high dimensional manner independence screen rapidly low intensive procedure infer penalty suit procedure briefly review aspect entire argument detail main text appendix distribute without loss generality intercept regression function include penalize square least instability automatic observe use conjugate distribution p posterior specify flat simplicity ahead reflect identify relevant expression variable square residual error regression selection posterior expression use expansion x term ask eq calculate therefore constant thing together posterior thing little truly bayesian reflect penalize choose inclusion penalty help indeed maximum therefore combination definite series log posterior expand need place standardized variable go column converge correlation plug result namely
svm softmax cnn svm output combine eq loss understand kernel weight enforce hide sensible eqn margin hinge svm front notational eqn margin square note decide course overall produce output act hyper overall reach vanish role importance addition use simple enforce second vanish epoch although summarize follow want filter classifier feature map depend satisfactory classifier restrict part consider layer highly proxy performance eqn wise greedy perform overfitte state benchmark particular advantage capability train gradient parameter gradient conventional come layer supervision next formulation eqn ease advantage discuss function lead vanish gradient deep neural loose formulation hope advantage deep highly convex objective energy dl observe large often strongly step strongly function attempt convergence eqn flat convergence locally regularization refer reason two fold encourage layer directly prediction keep vanish direct supervision ground stage filter necessarily illustration b give loose comprehensive study name one w w j layer see illustration every mean lemma around make necessarily however observe flat ultimately really care gp directly see eq base assume readily start rate small strongly eqn use top show compatibility equation derive improvement convergence layer help discriminative benchmark mnist cifar cifar follow protocol et mini batch comparison illustration effectiveness match architecture layer convolutional error epoch schedule difficult engineering widely setup adopt scheme equip show boost softmax svm cnn softmax performance presence burden require augmentation cifar augmentation without average exclusive method illustration ccc c validate mnist handwritten digits widely extensively adopt class figure cnn softmax softmax softmax margin svm margin svm cnn svm show whiten augmentation compete train training gain softmax sample comparison generalization htp stochastic network cnn cifar consist training normalization corner average phase center table error augmentation add robustness layer fast rate burden hyperparameter tune cnn observation feature example cifar show map cnn cifar classification maxout network network error network network cifar similar cifar cifar make classification task setting boost consistently show cifar demonstrate advantage htp pooling maxout network view house number digits digit testing follow select sample class follow normalization augmentation table augmentation net view understand nsf award nsf award lin wang david proof example em plus height width zhang microsoft tu net make hide boost cnn architectures effectiveness presence vanish objective layer different gradient analyze evident exist e g art mnist cifar cifar network dl form gain amount deep technique hierarchical demonstrate automatically thousand million pattern concern fundamental current dl framework layer difficulty due vanish thorough algorithmic attempt make availability amount manual dl capable activity open sharing greatly adopt dl technique dropout pre augmentation enhance dl variety fine tune learn demonstrate observation initialization difference presence vanishing gradient dl make study follow discriminative display less hide layer discriminative train hide feature serve map make quality feedback able weight filter favor supervision quality
tracking approach update approach publicly second frame gb table four approach success rate show c boost phone chart report observe boost approach detection result tracking curve grow horizontal show track material clutter challenge perform change chart frame camera motion frame illumination variation frame object book contribution performance approach part mt exactly experimental phone chart book table find verification situation produce tracking capability separate consist present task output metric joint simultaneously discriminative verification learn model scenario structure complicated scenario discriminative enhance create benchmark video challenge video experimental correspondence address li support national science foundation china national basic china aa china engineering pt plus pt wu college computer science china edu cn computer graphic tracking spatio statistical framework effectively balance aspect manner coherence across frame consistency frames discriminative issue spatio task structure drive characterize adjacent model verification base feature intra inter separability three module simultaneously learn demonstrate due motion object augment ar object compression structural object appearance various cause illumination spatio structural property appearance construction variety descriptor encode invariance appearance sift speed extraction extraction descriptor effectively adapt appearance variation cast tree boost tracking intrinsic geometric frame address issue al structure track geometric propose tracking simultaneously underlie frame interaction coherence track joint capable balance important part coherence across frame frames coherence encode optimize frame propose explore spatial consistency perform verification structure output cross frame descriptor adapt situation structure discriminative power inter separability summary track task structure task learn structure spatial temporal present learning release video challenge video cover several experimental video tracking manually annotate besides provide track mainly compose prediction object structure discriminative feature compatibility feature train f hc hc nonnegative determine tracking result without task multi task task case rotation temporal model coherence stable tracking result well tracking feature formulate metric map replace enhance discriminative show example space frame template map another large map task utilize consistently transformation f feature frames feature j j jj frame correspond template norm regularization incorporate framework scheme approach form learn transformation predict maximize added training use frame track online adopt let kt eq convenience let tc w solve th trace convert diagonal element descent update current show supplementary file k eq material gradient k material sample descent supplementary algorithm material calculate model transformation predict maximize collect solve nine book
produce period duration prediction daily monitoring http www analyse daily data year daily record account effect assess water management discuss ni anomaly temperature cover west normalise mean level pressure south calculate anomaly another strong place restrict site obvious discover focused effect study generalise variation recent model daily range summary explanatory variable range year period location daily daily zero small day show stationarity site area section hierarchical spatio model statistically complex relationship observation introduction denote observe process three layer observe censor alternative model hierarchy represent intercept regressor stage spatially correlate th generalise define vary autoregressive model single writing value process follow temporal subsection model spatial vector ts weather ts kt x n spatially correlate x kt nn diagonal corner ts ts covariate ern gamma rate correlation control smoothness isotropic detect one spatially model generalise cholesky corner correlation triangular q generalise skewness introduce identifiability effect overall product routine calculation give spatio tp denote give compute likelihood censor censor observe information time carry regard know eq augmentation embed integration also assign prior uninformative variance prior spatial decay parameter prior set range km assign uninformative generalise similar prior sensitivity use hyper hierarchical model censor censor metropolis full conditional show global covariate result std ar skewness spatial decay intercept brevity vary skewness ci vary effective dependency km analysis significant impact posterior distribution check sensitivity truncate original uniform result show sensitive hyper brevity omit prediction period return return period event past calculate produce period weather previously sample spatial return ci period curve return period site period dash ci directly observe daily quantile observe year quantile graph site include shape important return period model construct extreme period daily datum weather present conclusion demand extreme daily return period might error model spatio daily closely worth note daily another use duration threshold daily box plot show match efficiently short weather dot observe number prediction produce map threshold prediction define study weather spatial exactly value represent show close region develop generalise prediction spatio datum heavy tail substantial vary skewness theory maxima sized threshold fit daily across south west propose accommodate volatility skewness dependencie future plan process coefficient dynamic spatially often address covariate vary whose generalise similar adopt censor modelling zero approach alternative mix compose use censor couple generalised modelling limitation first fail massive spatio non model assume spatial stationary complicated interaction model h mm h mm mm proposition planning produce uncertainty spatio skewness environmental generalise construct duration sized block fit entire spatial spatially vary volatility skewness drive covariate spatial inference temporal modelling process model require duration period model require short term realistic associate variation cause environmental often data resolution cover valuable flexible statistical accurately shape skewness reliable serial dependency objective unified move towards objective generalise process significant model build propose event unify model efficient involve finite threshold large contain contribute spatio model methodology adapt type potential city development planning assessment extreme return measurement denote recurrence period calculation occur past take return extreme assume occur occur find occurrence event period uncertainty duration aggregate threshold measure measure extreme environmental physical process widely coarse current often extreme theory model tail model argument generalize univariate let normalize distribution converge justification maxima size maxima maxima china south individual weather spatial period calculate spatial sophisticated weather daily maximum pareto threshold get parameter control weather extreme isotropic induce walk describe temporal smoothing environmental level distribution shape scale amount discard usually diagnostic show also ideally like spatio temporal law tail well centre meet generalised connection model lebesgue generalised say normal random generalise gaussian conditional scale skewness tail influence contain appeal property moment generalise eq suggest case many laplace
dataset breast cancer division stanford school treatment take pre gene measurement per patient copy measurement six correspond tumor area select expression copy gain perfect employ issue pre variance gene copy measurement circular algorithm smooth variation fused fuse lasso slow consecutive triple copy average copy scale follow q coefficient examine plausible select coefficient smooth mention nature objective often term box focus optimize solution equivalent iterate many package package subgradient respect divide substitute xy b proof penaltie solver iterate total computation try fit follow mean solve solve far build package run independent normal net take second augment version take second speedup come introduce model collaborative response framework decide suit canonical seem increasingly generalization suffer issue real issue performance acknowledgment thank comment regard development scenario observe deal canonical reach optimum supervised problem many study researcher outcome expression copy patient type come may need make future copy future prediction canonical correlation collaborative characterize penalty possibility seem case simulation improve prediction advantage secondary look sparse compare biological penalize collaborative regression covariate partition response response store length collaborative minimize seem basically want base easy calculus satisfy order find closed assume none xy classical way series instead happen substituting get perform onto space also expansion help build actually projection column column projection onto set onto intersection space case eventually thus shrink part picture parameter act shrinkage one nice aspect add penalty penalize collaborative regression penalty penalty know introduce sparsity namely know fuse smooth helpful meaningful copy convex also linear penalty lasso ridge penalty often predictor lasso fuse combine discuss penalty term penalty potentially available contain identify useful costly amount noise alternatively different brain hard information current patient perform basically agree would framework fit fitting look solution reduce ultimately decide problem appeal analyze correlation generate datum follow iid discover issue cca seek third supervise correlation dataset suggestion make optimization trying iteratively maximize rest long solve iterative people take coefficient ridge replace identity matrix solve ridge replace order adjust sure lagrange convex regime add regime sort variable selection useful gene pathway particularly value fairly noisy statistical follow offer desire coefficient penalty add negativity iterative problem hard high dimensional ideally optimum coefficient start believe globally logic sufficiently search exponentially start output criterion generate extent get dataset x z run sphere line correspond start penalize solution especially thing start default start suggest supervision threshold pass cca also use cca supervision three objective cca cca bound form enforce variance constraint cca characterize enforce note
ref eqs relationship q derivation seem algorithm formulation hamiltonian hmc require nonlinear metric lagrangian mechanic ref solve equation avoid hmc equation agree ref splitting differential discretize linearly implicit scheme geometric effect decrease speed region unchanged region must often barrier approximate location saddle direction curvature assume negative neighborhood perhaps barrier ref use switch define help integration elegant practice splitting gaussian dimension gaussians dimension dimension distribute simple problem mode especially sensitive dynamic implement efficiently choose ref integration sample auto time mc hz n practical eliminate system hmc variable aim show feasibility acceptance split ref ref free either impractical stationarity benefit self understand balance recover easy get transfer operator p f sample transfer self adjoint weighted markov process replace get eq similarly name eq hmc satisfy illustrate langevin sampler way impossible generalized conditional verified mcmc balance balance make modify detailed balance langevin formally balance splitting add mean variance equivalent show exploration thank contribution investigation work project definition one generate normalizing often resort prescribe configuration step acceptance metropolis hasting acceptance rate configuration dimensional numerical essence monte present volume possibility derive couple barrier dynamic base prescribe protein unbiased structure task whose multiplicative energy require chain enforce method correlate produce offer possibility correlation integration requirement theoretical justification dynamic reversible preserve phase literature ref ref volume jacobian fully hmc mcmc rely ergodicity volume phase rather requirement special weak expand design move dynamical make hamiltonian ref begin probability marginalization volume hmc one efficient unless article formula special occur ref worth derivation calculus geometry mcmc style highlight mathematic substantial chain clarity enhance mistake avoid avoid multiplication requirement remain already couple also simple potential benefit generalization develop molecular mcmc md set characteristic rotation design tailor basic hmc also introduce generalized hmc three modify defer stationarity balance importance detailed idea configuration unnormalize aim estimate discretized equation produce systematic markov acceptance systematic surprisingly tend efficient challenge ref move stationarity ergodicity chain ergodicity inclusion stochastic follow inclusion example give generate auxiliary choose simplify scheme aim covariance basic identical propose extension give mala latter reference asymmetric example diffusion slow moreover possibility flow marginalization hybrid stem change auxiliary become momentum configuration hamiltonian duration hamiltonian arrange introduce additional hybrid hmc numerical scheme piece system general hmc density density hmc stationarity reversible mapping hmc jacobian second difficult alternative ref alternative hamiltonian hmc modify slight important issue size ref theoretical generalize mcmc phase space partial variable next allow duration momentum choose determined calculate move choose course outcome go trajectory slowly possibility hmc generate extreme random walk discrete ergodic mc given eqs langevin tensor scale result integrate hamiltonian denoting obtain sample far develop tensor diagonal explore molecular ref use mass hybrid monte statistics mass fisher always numerical illustrate bayesian statistic general previously state density mapping dimension practical density may configuration space generalize reversible w preserve mapping cc cc choose require specify may dirac delta represent law enforce
restriction nonzero entry satisfy vector zero identifiability establish linearly recall form large I column row follow nonzero without nonzero row denote nonzero linearly nonzero since column combination column linearly linearly row contradiction generality entry column accordingly thus back appear note write certain canonical case linearly entry linearly dependent imply subspace closely relate completion rank entry span column draw absolutely lebesgue matrix submatrix uniquely entry complete relaxation column hold submatrix submatrix result behind simply canonical easy follow projection certain entry g incoherence method complete low agree aware entry completion scenario show show completion remark observe necessary condition corollary incoherence uniqueness randomly comparable condition necessary identifiability come identifiability draw connection subspace theory insight yet insufficient subspace identification bipartite disjoint vertex nonzero title column row vertex vertex label vertex vertex font edge edge r r c recall vertex vertex theoretic verify vertex title title title row vertex vertex width pt width vertex label vertex vertex pt font edge r edge c interpretation extend context corollary state connectivity insufficient identifiability adjacent one identify projection subset random scheme meet trivial gives verify incoherence sampling remark conjecture draw circle font projection canonical necessary deterministic identifiability new completion arise variety signal situation grow subspace incomplete projection contribution sufficient implication organization state main illustrate implication result present viewpoint column nonzero indicate onto hence without generality
indicator principal incorporate intensity dimensionality well identification principal principal refer result ol projection subject basis two slice slice slice gray matter white cognitive ability maximize feature subject list mean lastly classify cdr great c feature age white gray coefficient pca coefficient coefficient recall correspond perfect cross test subject repeat time report pca result increase suggest single set radial inverse gaussian classifier classifier decrease solely pca component feature recall classifier set reduce indicating achieve voxel severe make reduced feature attractive classify truly cdr patient reduce feature improve technique acknowledge grant rr mh additionally would ng insight classify moderate cdr support machine act reduce subject segmentation feature contain classifier linear training test precision correlation coefficient cdr component describe classify accord clinical cdr train open clinical cdr work separate ad mmse report recent identify structural work utilize subject contain subject raw image raw voxel value feed scan voxel million raw voxel intensity severe bias implement classifier subject cdr score moderate describe thorough diagnosis please radial function svm cognitive via mmse clinical rating cdr subject cdr rating rating none great gender normalize brain process brief post process brain
py box dot box probability dot box probability one step possible probability apply hence py bn compute detailed refer appendix b instance bi th instance depict figure newly py bi use forward approach swap step bag inefficient lead compute py bi b b swap initialize b py b b bi bn bn bi bi bn bn forward swap th instance conceptual illustration forward substitution overview substitution b b p py bi r bi sort cardinality contradiction low triangular structure lower triangular element computing substitution th th h bag three set exclude sort sort construct proposition py bi bi py l bi py bi bi bi p combine substitution algorithm substitution e step bag efficient inference instead bi b initialize bi py bi py bi increase function follow determine use backtrack backtrack step backtrack step transform original multi regression transform space space yield bn regression require rbf instance tune note version make step py bi bi bi bi bi bi b py bi bi instance prediction instance prediction predict instance know inductive mode bag inductive maximize feature absence label ti ti programming bag bag compute prediction union follow compare approach annotation maximum sim svm nn expectation maximization sim tune search set norm divide feature training phase confidence predict confidence access predict ti classes rbf nn letter letter bag bag create instance letter bag create letter uci repository two bag class instance bag letter letter metric annotation instance divide predict predict label sim method logistic instance accuracy upper every dataset without inductive namely hoc parameter bag note free scheme baseline method parameter inductive hoc bag present post hoc instance accuracy table letter sim whose performance comparable high outperform low accuracy low union dataset outperform letter dataset per bag avoid effectively sim max softmax ignore mcmc challenge stop criterion bag ignore useful constraint union equal preserve dynamic classifier work well instance accuracy since accuracy table perform bag measurement instead directly level follow parameter tuning propose understand drop tune case bag metric bag phase divide fold inductive letter sim respectively dataset outperform inductive sim inductive report accuracy union sim test level p p mm l dataset sim level sim l examine ability sim exclude hoc tuning bag accuracy percentage bag runtime training percentage value accuracy depict bag practice label costly training sim sim use amount clear gap less efficiently information achieve small bag sim indicate softmax instance instead come result sim information explanation em another possible maintain union bag bag small number class occur outperform objective section indicate mm sim svm ap union rl ap rl ap letter ap co letter rl ap rl ap rl ap level examine propose bag level letter letter compare sim compare logistic optimize evaluation hamming note assign output classifier bag bag coverage classifier output bag confidence bag independent confidence sim bag maximal confidence instance w ti ti ti ti ti svm nn report measure bag similarity representative bag dataset instance sim instance information consequently approach bag helpful predict include sure overall scene annotation annotation bag level nn seem svm feature instance iteration sim tuning iteration number iteration higher lead runtime performance set increase runtime sim report letter improvement compare annotation letter letter significantly compare classifier instance feature unchanged compare high running feature time long run kernel union k sim level optimal reduce approach consume bag term bag letter bn bag l substitution l bn b bn letter letter pruning remove bag contain large class dataset sort value percentage bag validation every bag due ambiguity set pruning runtime decrease prune use letter letter sort bag bag contain bag ratio b bag constitute letter letter cm letter letter bn letter letter bag function bag letter significant letter letter dataset letter letter time remove letter letter bag letter remove word affect bag constitute runtime maintain bag runtime decrease keep runtime decrease linearly keep high proportion bag number computational per substitution consequently depend bag maximization iteration bag label costly runtime compute bag new compute q since label subset bag size runtime iteration bag change bag select runtime bag letter percentage bag percentage sample bag runtime almost dataset stochastic ascent sample bag bag sample runtime keep letter b b compute call dictionary long take converge one lot result use create em iteration become consequently convergence propose accuracy runtime letter dataset create similarity instance accuracy runtime exponential factor number reality additionally several speed pruning reduce runtime accuracy unchanged pruning technique runtime bag letter letter annotation problem multi ed logistic regression facilitate likelihood focus challenge exact instance label program bag experiment dataset approach outperform art especially dataset approach sim tuning bag get around training bag achieve sim classifier approach extend pruning technique datum mr liu application appear b py lp py bi iy bi rule become bi rewrite rhs follow bn n b n conditional rhs n bn bn bn rhs macro labeling labeling instance bag algorithm infer bag refer challenging ambiguity regard label discriminative expectation inference probabilistic bag label instance evaluate world song activity recognition outperform sometimes state bag label annotation expectation graphical programming multiple instance carry uncertainty conventional training bag bag segment name object house contain list providing I bag classifier level classifier refer reader construct without explicitly reason citation knn knn training bag cluster hausdorff encode bag hausdorff distance apply citation knn knn focus paper label annotation exist resort bag score bag maximize bag bag small directly annotation example sim max score bag graphical annotation application inference employ maximization discriminative achieve lower first instance annotation propose computationally calculation finally superiority various domain song image annotation bag level multi instance address process dirichlet allocation lda know processing corpus graphical lda proportion topic randomly select topic hide bag bag label consequently instance express propose learn mapping preserve maximum formula dp model bag
instance compressive proposition purpose equivalence notion algorithmic equivalence namely equivalence algorithm machine illustrate analyze regression fundamental property object field several theoretical concept accuracy algorithm concern raise define characteristic could train smoothness equivalent formulate algorithm adequate algorithm exchangeable even indeed restrict optimization ignore set never rigorously equivalence learning notion particular sufficient generalizing weak related equivalence hold equivalence set advance sense equivalent study regularize algorithm hilbert rkhs particular regularize rkhs regularization variable efficient call concept relation ridge contribution formalize concept equivalence notion algorithmic sufficient allow transfer stability equivalence transfer equivalence solution weakly section banach reproduce space rkh notation take learn output discuss equivalence problem optimization function limit solution small constraint function unique optimization equivalence focus occur optimization associate decide equivalence support equivalent stability view imply underlie share learn relate question rigorous concept equivalence notion equivalence optimization definition convex onto q convex unique equivalence naturally equivalence set algorithm associate optimization solution solution guarantee varie mean pay algorithm depend crucial controlling widely regularize learn indeed value regularization define different index weak extension definition equivalence basic section equivalence equivalent assertion become strong say equivalent weak frequently encounter machine naturally occur lagrangian duality method immediate supplementary natural whether know weak question study consequence algorithmic notion equivalence allow algorithm present weak weakly regularize proposition follow proposition mean however assumption weak equivalence indeed vary make increasingly consistency whether weakly follow get algorithm regularize equivalence sufficient transfer stability regularization widely uniformly nz property decrease decrease stability important unlike equivalence case lemma stable uniformly stable equivalence weak equivalence sufficient ensure transfer transfer express assumption knowledge easily express stability hamming usual hamming z n z z nz ig nz z one move permutation proposition see generalized notion regularity function I lipschitz stability satisfy assumption admissible locally proposition extend depth case equivalence algorithm hilbert space rkhs k kx combine square regularization term solve exponent classical ridge regression recover propose practical solve pose novel generalize ridge worth become ridge analytic spirit following cm basis ii transformation reconstruct weight orthonormal let material fast accurate exponent show weakly problem strictly lagrangian weak weakly unique unconstrained equivalent eq problem weakly easy map f depend weakly satisfied stability supplementary subsection conduct real world algorithm compressive strength instance world repository additionally attribute dataset input uniformly compute x part equivalent positive showing regard algorithmic property way namely notion stability transfer concept equivalence algorithmic robustness aim far quantify efficient two learn weakly strongly equivalent algorithm consequence say curve ensure minimum easy sequence likewise generality third fourth use try exponent reader theorem go repository root eq notice q also due combine determine formula derivative since semidefinite depend definition need write
approach curve flexibility produce procedure assumption future non model mm receiver operate measure indirect roc population lead flexible account use close show frequentist roc algorithm mixture model operate roc gain popularity detection world phenomenon justify necessity performance diagnostic tool diagnostic roc present practical roc desirable reason theoretical many author way curve issue model roc two category indirect appeal construct population divide mention curve certain monotonic increase overcome obstacle population use smoothing reduce distortion smoothing roc curve assume population derive construct parametric semi parametric population obvious adequate cancer distributional verify assumption roc assume population variance distribution determine advantage curve attractive presence population solution technique generalize smooth roc curve algorithm category ordinal suggest diagnostic patient relationship directly model diagnostic population status disease probability use like lack issue semi confidence band population motivation give roc flexibility natural mixture population enable shape gm replica curve bands roc curve mixture flexible carlo apply absence form band construct curve idea method replica simulate accordingly q major estimation consist prior bootstrap mixture modelling score respectively suppose follow carry via maximization parameter generate repeat monte simulation roc curve similarly compute estimate empirical averaging indicate therefore band curve law theorem several simulation flexibility fit e choose evaluate auc visualize method discrimination examine strong moderate poor practically patient different stage disease replicate curve discrimination commonly curve furthermore case closely compare figure band curve average without monotonic auc curve htp htp htp htp htp two calculate auc relatively simulation
method svm also space baseline exponential amplitude smoothness hyper noise svm use hyper smoothness turn validation search svm validation guide competitive metric independent website text task rest sample testing text information use explore different description frequency tf extract descriptor codebook text representation architecture sentence construct codebook cluster elaborate sentence planning explore future dimensionality domain keep deviation outperform however clearly network due svm already couple difficulty hyper mind svm hyper separate tb domain highlight equal mean method small task collect search category description colour attribute class retrieve category provide contain gain year due capability extract predictive interested power visual domain convolutional fashion feature provide pca dimensionality keep principal attribute one small average rank support rank also statistically four still enough evidence attribute dimensional base classifier principal domain term baseline appendix dataset advantage deep feature semantic performance illustrate sigmoid differently example gain importance one importance test four evidence reject comparable slope way accuracy term advance network representation plan extend multiclass speed quadrature free propagation repeat image domain neural highlight r v cat cat v cat cat cat cat cat v cat cat v repeat binary task highlight cat cat v cat cat cat cat cat cat cat c technology attract additional treat nuisance training slope sigmoid knowledge svm advance provably many integrate develop certain prior parameter training rely adopt see prediction attract considerable within reflect increasingly situation expert train request customer order expert use information typical service inspection classifier processing play role build mobile device operate constraint generate increasingly data care diagnosis available drug trial subject obtain impractical noise final require aspect influence become interpretable training directly read slope shape example fast sigmoid try fit slowly slope put interpretable higher relate originally think turn need classifier slack lead improve several categorization setup subsequently improve characterization constitutes commonly use predictive long analytically deal markov carlo posteriori bayes gps reason section propagation classification self review particular emphasis latent latent elegant aspect intuition make test input noise latent classification specifie property function evaluate crucially input contrast approach term assume input task flexible numerous upon knowledge nuisance view flexible dependent suppose say reflect contrast say sec posterior solid easy difficult view context however classification actually sensible investigate equivalent clear transition high slope slow uncertainty likelihood sample uncertainty n hyper amplitude smoothness parameter unseen label associate newly datum decision large interesting bayesian predictive marginal analytically tractable adapt quadrature please
conclusion birth ol ols ols pl black relationship birth weight nonparametric specify fit age record partition curve age available cell component estimate fit remove panel response birth quite particularly birth birth quite dramatically gap gram birth older optimal choose generalized sided response bootstrap response among pl covariate pl estimate pl exhibit birth gram difference statistic black marginally effect concept consistency though nuisance estimate configuration reasonably nuisance result fairly stable profile ratio method note greatly simplify predictor partially insight tradeoff estimation estimating nonparametric nonparametric little component burden loss efficiency negligible burden bias generalize broadly utilize theoretical derivative cm discrete covariate category lie simplicity dimension mainly one correlate corollary imply follow proof corollary proof corollary least form need first analyze numerator denominator conditionally sketch analyze eq use ng positive lie total large small clear argument lemma combine complete conditionally j together give complete give thus equal combine analyzing show eq q ix j ix j thus mm pt remark utilize theoretical statistical modeling estimate partially consistency phenomenon model via balance burden model bias approach square little behavior test satisfactory study birth weight analyze partially categorical normality estimate dimension parametric regression term surface practice effect requirement determine functional misspecification estimate contrast form influence bandwidth local kernel essential versa suffer curse exponentially design categorical appealing alternative specification covariate part linear nonparametric assumption estimate affect gain great popularity economic science lot devote model partially introduce estimator assumption extend specification first investigate nonparametric comprehensive partially expectation subtract way expectation estimate dimension obtain root nonparametric curse dimensionality call theory correspond true every estimate desire nonparametric efficiently convergent bandwidth procedure unstable correlate partially problematic fan fan lastly structure example interaction biased model complicated idea inspire property average square classic property moderate estimator almost component easy incorporate covariate explore regarding paper follow follow review consistency regression parameter consist univariate categorical analyze offer method far technical refer statistical nuisance nuisance phenomenon terminology nuisance consistency phenomenon mix longitudinal etc discussion theoretical microarray q assume grow nonparametric error sample fan show estimate consistently moderately pay nonparametric function variable assumption sort realize interval interval realize density narrow sub ji reformulate term ignore almost efficiently express profile profile similar least degree freedom use formula plug update estimator readily extend several express categorical categorical subset categorical subset partition sub model profile square use regularity condition e least define eq bivariate extreme model consistency outline component highly eq fan fan smoothness approximately denote model regularity corollary result fan replace unconditional corollary defer appendix extent resemble kernel view corollary limit classic partially naive cost efficiency factor increase remark average resemble testing quadratic suggest partially eq nonparametric greater critical reject critical hypothesis testing test slowly hypothesis bootstrapping suggest replacement bootstrap use statistic repeat replicate frequency replication bootstrap quantile confidence band examine effectiveness propose model mix categorical compute test hypothesis power examine alternative method simulation package fit nonparametric curve fit generalize validation bandwidth I example sample produce package first illustrate relationship logarithm slope size moderately propose increase decrease result study suffer capture theoretical follow examine alternative nan size simulate figure empirical theorem bootstrap section sample behavior power value leave value similar confirm least statistic linear consistency h cc estimate cc generalize additive mean z u u go simulation example property test indicate outperform package sensitive sample size nan nan distribution also nonparametric increase also reduce power eq q equal independently bernoulli independent continuous sample purpose bivariate pseudo nonparametric nonparametric see additive nonparametric estimation np tries estimate bivariate simultaneously parameter study np hand nonparametric interesting outperform nonparametric wrong panel curve relative parsimonious nonparametric extent regard table sd propose cn ci ci ci sd sd sd sd sd sd equivalence two component z test estimate use formula x r package bandwidth smoothed estimate plug formula associate bootstrap
user accurately predict score less predict suggest predict missing use recall per user average user metric give value recommender system dataset website user review user social utilize dataset rating friend rating undirected rating item review specify trust utilize dataset convert direct trust link link movie rating result density user connection identity item user gp I trace pmf pmf gp pmf pmf show outperform pmf five fold find select vice versa match researcher perform rmse gp factorization baseline term highlight shown suggest rank dataset table mirror dataset model approach similar trend highlight similarly slight improvement constrain variant recommender dataset nuclear gp introduce predictive variate nuclear recover gaussian norm constraint association task recommender perform characterize necessary condition norm constraint explore interested gaussian nonparametric interested nonparametric constrain inference biological implication disease expert acknowledgement grant thank u manuscript constraint duality subject general recently constrain term banach duality denote define match theorem conjugate q pz z dual ensure normalization dual optimization often alternative separate optimize unlike approach density satisfy empty subject constraint constrain write minimizer achievable associate density consequence state pz ensure denote feasible density give pz key fully specify parametric family give expectation constrain follow specify corollary convert optimization hilbert prior similarly gp bilinear admit spectral decomposition value spectrum common induce define functional regularization theorem optimize evaluate e covariance evaluate modeling approach constrain optimize include noise function variance q solution similarly column parametric represent optimize gradient respect gradient simplify collect similar hyperparameter computationally challenge optimize update require matrix proposition represent interaction entity information vector column available interaction entity axis modeling model noisy generate construction flexible side information row association disease set observe association covariance network recommender involve prior experimental highlight performance constrain variate process approach domain organize entity row entity sparse primary represent encounter addition matrix include entity entity graph describe column graph matrix observe low modeling recent variate compactly correlation row extend replace g variate new row gp describe scalar scalar collaborative learning despite gp structure rank prediction low product reduce improve low rank assumption computational concern size na I scale linearly factor method approach empirical inference enforce useful inference alone constraint margin enforce constrain relative capture rank characteristic combine bayesian nuclear distribution solve constrain problem gaussian finite iii art disease association disease side recommender network side begin discuss background variate nuclear constraint variate low art specific disease recommender domain statement building process section case bold upper identity pp p density let index index ny observe unobserved include bound node auto ex z free variate doubly index multivariate distribute notation decompose row covariance scalar value process nz nm distribute extend arrange covariance matrix gp combine noise see draw z n z interpret latent task posterior z datum index covariance close follow gp scalar appropriately variable scale storage require I probabilistic involve give often achieve via however may one impose intractable careful alone approach via relate sample probabilistic thus standard optimization enforce density enforcing pose give vector let set constrain bayesian distinguish unconstrained discussion inference provide constrain bayesian constrain relative minimization maximization study language discrimination inspire combine document apply prediction result intractable require variational make appear without require simplify oppose principal variant extract bayesian model linear covariance non factorization rank must prior correlation design variate far nuclear automatic implicit ex ex r edge r rank matrix process low factor detail baseline rank model nn computed optimization problem trace statistically interpret one probabilistic gaussian posterior quite practitioner approximation utilize priori recently intractable slow inaccurate large dataset approach focus prediction exploit model kernel base apply probabilistic relational alternative focus learn additive nonparametric instead represent term amenable fix nuclear optimization q ex ex represent form solver represent avoid storage full estimate reader refer paper detail increase factor far descent singular singular sparse power factorization nuclear norm differ number factor requirement nuclear complete dataset association domain dataset study consist column diffusion prior covariance validation selection result constrain trace hilbert trace contribution baseline probabilistic factorization factorization pmf pmf pmf covariance covariance implement representation outline cholesky decomposition representation hyperparameter spaced noise spaced variance estimate overfitte explore distribution g regression away gp column matrix multiply computationally observe implement allow expense norm regularize optimization implement limited bfgs design validation discussion refer either disease disease prediction recommender experiment design evaluate randomly partition fold set hold experiment experiment wise e test segment dna gene identify human interact disease include disease genetic standard expensive method predict list disease significant interest reliable gene share know feedback include like train similar differ hinge respectively svm regularization also pmf baseline implicit feedback dataset recommendation sampling follow randomly disease item set combine metric association laboratory consume costly rank metric remove gene compute label sort score test remove gene precision compute relevant retrieve retrieve fraction retrieve retrieve length metric set separately reflect dataset evaluate disease association database disease disease matrix extreme problem large set predict e even hundred database disease disease branch mesh extract disease database positive association result gene association density interaction disease disease genetic genome include protein interaction specie gene link
number coefficient iteration amplitude spike curve meet reveal reconstruction except return seem constant considerably laplace handle different level regard fix spike fig fast due origin heavily summary simulation competitive fast provide sense review literature generalize beta mixture signal bayesian prior maximization cs simulation outperform art solution beta prior sense hierarchical amenable expectation maximization em algebraic update yield experimentally validate recovery art method level sparsity observe sense reconstruct consider acquisition noise signal pose polynomial replace recovery method use algorithms support sequentially approach compressive know compressive sense proper utilize reconstruct modeling prior concentrate tail work area lead introduction prior utilize large require suffer computational low another induce widely exploit hierarchical facilitate student assign layer prior later work laplace student concentrate near tail coefficient model amplitude paper suggest prior capability amplitude level sparsity choice free origin heavy tail sort nearly hierarchical inherent advantage impose reconstruction convergence summary main propose compressive perform art superiority wide compressive analyze induce section detail propose framework signal distribution laplace student much model large shrink like toward statistical mean white normal joint assign formulation laplace tractable address problem joint pdf determine induce pdfs p suitable example stage student hyper precision represent method regard decompose q impossible adopt q conditional calculate p eq compute maximizer likelihood obtain suffer drawback inversion require suboptimal suggest model add fast calculate single hyper iteration regard hyper rewrite removed depend isolated looking maximize likelihood respect eq equation maximizer plus induce limit infinity origin maximizer basis local either root happen case distinguish discriminant cubic root discriminant real provide negative root negative root positive root scenario case point side axis root include root sum conclude two base subsection summarize reach performance degree non amplitude select exclude procedure great update sake quite algorithm remark mention hyper substitute analyze art
regret idea reference generalise notion name justify key regret condition generalise algorithm mirror crucially potential expert game expert reveal prediction expert player aim player keep close expert round loss player independent round denote make assigns assign predict expert make prediction reveal expert tp aa round expert round aggregate mixture set mixture eq expert prediction aa condition find result concern regret achievable round aggregate converse loss work proper loss main simplex loss two follow key result divergence formally sequence observation expert play sequence notion shannon entropy corollary uniform interpretation initial guess expert price exactly far mass expert aggregate recover usual aggregate generalised aggregate algorithm update performance understand prediction range study aggregation loss variant aggregate loss computational weakly aggregate regret observation aggregate also discussion relate aggregation vector observation loss loss loss product discuss offline make entropy call notion coherent mathematical throughout v conjugate readily fact entropy shannon countable concave proper thus original conjugate motivated entropy entropy call scoring rule say predict way construct express entropy proper associated shannon log reduce f see substitution inequality become definition divergence update condition eq expand result generalise aggregate strategy repeatedly next show process simply dual update constant arbitrary ignore relation sum aggregate substitute maximal substituting theorem update x series note something even strong state condition hence exist similarity generalise aggregate market difference bundle instantaneous price correspond framework bundle formulation say observe must market surprise outline main essence term divergence original achieve regret terminology bound bound seem bayes shannon entropy stand loss regret curvature discussion address large large correspond bind choice reference matter constant directly counter conjecture aggregate provide via convex kullback leibler divergence naturally replace loss bound generalise aggregate mirror aggregation bayesian view aggregate play prediction evaluate
conceptual operation capability mapping main tuple light gray tuple consist tuple return return tuple tuple look pattern observe location tuple way arguably convenient computationally eq denote white empty encode tuple tuple return illustration tuple consecutive shaped sequence formal requirement tuple network tuple issue architecture location thus surprising individual tuple eight ignore result tuple use network play tuple example alternatively construct tuple place representative tuple thank tuple tuple table computer world tuple external tuple input forest set evaluation interpret evaluation complementary color play purpose employ two play learn play black white vice alternative player output select position maximal whereas select minimal learn black lead position piece play black select move piece white piece player use output architecture rand rand rand rand tuple architecture tuple architecture computational n tuple evolution strategy weight individual evolution mutation double game play run compare tuple network game play despite et capable per thank finish run ghz repeat evolutionary measure last statistical follow significance correction compare present rand inversion regardless player architecture confirm statistically also detail moreover visual inspection plot reveal experiment rgb rectangle rgb rgb rgb circle rgb circle circle rectangle rgb rectangle rgb rand rand inversion performance score obtain heuristic shape white black line represent range black dot outer main possible tuple shape rand rand rand tuple make impossible pt rgb rectangle rectangle rgb rgb rgb cycle cycle rgb rgb rgb rgb rgb circle rgb cycle rgb cycle rgb circle rectangle rgb rectangle rgb rectangle rectangle rand x rand rgb straight n shape tuple rand performances fig explanation three pair reveal rand rand rand notice difference substantial vs lowest good see architecture robust variance rand cf rand attribute initialization process intuitively tuple short two time weight well difference architecture plot performance run eventually seem learn gap small among also six analyze tuple architecture give player obtain tuple result high player line player employ allow format date player name tuple tuple tuple n tuple tuple tuple tie ff position evaluation since use inversion performance good player come website performance tuple player game output able evolve name tuple consist straight tuple first care double game standard move situation game obtain good general variety evaluate utility player computational evolve tuple good player compare good method weight et find weight rand rand latter obtain result state mutation efficient elaborate progress claim suggest factor dimensionality considerably rand rand weights nonetheless three obtain high factor architecture finally tuple operator mutation although flexible add make hard analyze network systematically tuple shape tuple originally consist straight tuple weight usually weight advantage slow surprising since capture opponent piece three black obtain computational study nevertheless whether evolution interesting question systematic network advantageous reinforcement difference learn network support centre computation helpful remark early rand rand x rand x performance evolutionary run double heuristic player format contain tuple expansion weight use put pl network successfully game connect effectiveness network architecture tuple location provide n tuple sequence effective systematically place straight tuple obtain curve yield date evolution strategy tuple value function attract ai due mathematical early shannon lot conduct connect bottom game constitute valuable computational intelligence playing monte carlo technique employ quantify state look effectiveness tuple tuple long generate tuple exist evaluate way
multiplication costly subspace optimization use newton burden multiply span direction multiplication store previous enable efficiency substitute direction coordinate often burden change particular coordinate column still dense matrix need operation one pass multiplication multiplication wavelet transform coordinate descent without move along perform analytically multiplication via objective along another iterate elsewhere quadratic obtain always step multiplication comparable ill far accelerate every affine current direction several propagation progress step significantly nesterov figure adopt matrix fit ill beyond aim additional iteration explore present ill pose type converge fast go newton boost multidimensional looking approximate newton expansion accuracy inaccurate problem one k disadvantage line ill behave model fit far away direction therefore around iterate adjust dynamically actual fold find minimum time find unconstrained therefore ideal trust ellipsoid euclidean adjust ellipsoid proposal use use computing invert therefore quadratic taylor expansion iteration tn accomplish line guarantee decrease effectiveness tn internal attempt replace cg stay match tn step pose drive involve available ever become increasingly inefficient incremental contrast obtain mini large redundant sophisticated recently partially adapt unconstrained method mini batch mode mini bfgs still room fast plain trust region newton invertible newton fast hessian problem accelerate lagrangian constrain non trust constrain minimum time expensive ideal kind euclidean ball adjust trust region point expansion correspond go infinity pure step newton motivated propose current line method include step subspace take direction effective conjugate method newton quite burden newton newton square popular g moderately feed neural mention tn method internal cg go resolve replace step optimization way cg break outer stay tn tn approximately minimize quadratic eq gradient tn truncate cg come original cg tn subspace pass inner cg instead minimize extend monotone descent tn monotone reduce several step gradient order tn optimization start tn iteration stage cg cg tn approximately cg last cg iterate affine span tn last quadratic cg tn tn subspace tn sensitivity process objective trajectory cg function independently stop tn tn converge inspire recent mini quasi newton bfgs plan step previous mini batch gradient restrict hope mini responsible resolution accelerate computation first solution coarse grid formulation fine grid much iterate fine guess approach relevant linear e level gradient converge fast problem happen grid step cg fast iteration optimization previous gradient coordinate go descent provide coarse grid fine good cg acceleration acceleration lagrangian constrain look saddle dual current augment primal separable direction multiplier applicable include share regard nesterov possess optimal bad express formula lipschitz incorporate smooth nesterov composite objective direction gradient term objective spirit use direction outperform fista hope develop bad fista conjugate well nesterov however could extended function substitute method constraint indicator simple feasible follow composite pure tn tn tn cc tn tn tn depend cg tn tn converge cg truncate early
minimization one unique asymptotic depend second form kernel denote whereas empty set reason estimation pilot threshold considerably great density calculate intersect solve modify paper reconstruct level histogram cell dyadic nonnegative define denote lebesgue bandwidth empirical minimax course plug estimation h generate density model correspond density present asymmetric separate jump peak addition threshold although show therefore bandwidth selector facilitate follow horizontal small vice allow detect competitive density mean order error colour accord algorithm colour compare bandwidth behaviour sophisticated density specific competitive simple unimodal simple behave however bc previous chernoff mode mode ms optimum large vertical axis nx hybrid centre complement proposal convex hull hull method convexity restriction convexity hull classic smoothing assume complement context case union radius obtain represent axis unimodal case convexity hull competitive hypothesis restrictive case convex h smoothing density component seven quite promising model competitive unimodal conclusion extract one exhibit competitive quite present disadvantage depend hull competitive present bad hull competitive behaviour provide density none unimodal horizontal axis axis horizontal science innovation statistical system science estimate plug method nonparametric estimator thus view recently specific priori geometry excess selection avoid excess mass geometric restriction like plug pilot open exist review two behaviour extensive large respect different set domain concentrate great effective play crucial role scientific receive considerable component appear related mode cluster survey estimate level involve statistical two density include analysis outlier review outli belong effective scheme ba ba broad estimation motivate plug method analyze behaviour drive reconstruct set compare multidimensional counterpart excess method plug section mass section kernel gaussian bandwidth thus propose integration solve plug inefficient calculate empirical consistency methodology receive considerable literature ba many select next review estimation ba ideas population tolerance let assume work operation draw
follow chapter hand simply tb asymptotic dotted case involve continue hold check go theorem section standard non coverage require however validity unit matrix th row vector object modify replace significance significance concentrate paper determine explicit orthogonal first definition show pp mode concentrate prediction split series step look difficult general set point fortunately exclude absence euclidean intuitively hyperplane outside express expect typical data uci datum hold projection matrix onto onto even angle hyperplane element coordinate label exclude surely check let give simplify prediction cf except turn least corollary last contradiction complete opposite least many r n ni contradiction last component ridge lemma hand replace side replace allowed degenerate lebesgue suffice side replace prove assumption ensure condition satisfy q simplest give exact normality formula calculate last equality w x finally imply normality assumption early extremely perhaps extremely completely ignore reason type front condition indeed general intuitive residual iid vc disadvantage ignore eigenvalue true regression little interval illustrate informative different produce exceed significance level theoretical quickly small research extend ridge acknowledgement grateful kolmogorov project terminology support grant ep institute institute technology science uk proposition mm coverage iid considerably restrictive useful case little whereas asymptotically interval discuss primal early name prefer name also figure correct theoretically purpose assumption independently obviously original happen resemble nonparametric classical test turn efficient satisfied develop almost efficient violate different rather figure serious ridge interval apply regression result somewhat counterpart figure little gaussian assumption change difference point predictor empirical discuss notion validity two section ridge interested g theoretically theorem violate recent theoretical validity predictor ridge predictor conditional algorithm likely interest advantage regularity conditionally valid chapter noise case bayesian e perhaps apply objective toy simple tolerance stand form tolerance interval define show lebesgue difference interval however deviation z representation expression later agree involve g david normal david two ignore contain interval asymptotically train valid object validity attain predictor summary interested attribute particular random parameterized unit random element intercept recover add attribute object ridge object th prediction quantile enjoy equal significance object finally possible conditionally produce special g chapter reproduce length ridge regression residual row neither residual make score confidence give test object q score closure notation later guarantee instead observation define unconditional guarantee ensure prevent rely iid object validity generally guarantee conservative validity lower bind predictor validity difference
people forget else surface classified become error memory combine successful machine component read memory component introduce implementation answer index potentially component incoming internal old new input generalize intend produce example action word sentence audio internal feature test distinction test time potentially idea etc component make pre processing parse resolution text could encode internal dense simple store select slot part variant store store slot huge need store entity scale operate operate retrieve subset operate right variant implement choose replace memory experimentally yet component responsible reading memory perform e calculate example answering find relevant answer condition conditioning rnn poorly network component neural neural describe relatively implementation basic module base sequence text store available original return empty module memory old subsequent core lie module module support give support score support candidate input support bag modeling input separate module need response simple return previously sentence true rnn word see match task figure module retrieve fact leave would relevant give office place drop finally would score score dimension role map bag support model differently weight desire support label training give input know perform loss sgd specifically response support sgd employ use feed give simple memory subsection basic office office office sentence arrive stream rnn modify segmentation learn take far look proceed embed dictionary margin sequence first concept something training mechanism store equation hash hash hash hashing word embedding dictionary sentence word I consider share mean word fall word well word match memory speed take account memory slot answering fact capital france answer figure implement extra memory assume memory slot absolute relative success candidate triple older old win step compare winner human read lot ideally neighbor word assume similar idea incorporate separate word store bag co occur feature context learn new kind dropout embed instead embed efficiently word pair add learn another way stay extend match matching occurs actually build conditionally match matching matching context classical method document memory retrieval answer reference create graph fact knowledge kb question logical query approach recently base memory differ extraction principle kb follow extraction memory store choice embed potentially build kb away neural memory reference type model memory location store network module learn successively reasoning read salient fact operation design differentiable work incorporate change use allow network address relevant related read experiment memory whereas language reasoning task focus sort network know whereas toy relate method translation alignment representation predict overcome poor long perform recognition dynamically determine character one back single character retrieval look document look schema pdf box evident see consider highly time component recursively look path read create kb www stack fact create event I think highly relevant consist statement store subject relation triple triple corpus combine pseudo triple book perform framework return answer test annotate wrong human whereas ignore label support end also try bag feature unseen word modeling memory fact cluster embedding try hash string hash hand lot share match embed l candidate speedup hash k hash character object character around pick dropping object text label question task overall answer pick k identical testing rnns short memory rnns entity ask try actor drop ask answer true answer g I appendix object support statement sentence find drop support statement usually ask actor actor actor question question difficulty actor actor actor actor actor lstm pick drop room believe take office back office office rnn language modeling optimize fix margin epoch training answer actor task rnn without question bad question bad difficulty demonstrate encode memory high level distance rnns expect sophisticated limitation mistake memory wrong pick question otherwise pick person hard indicate successfully inference whereas without rnn fail multi whereby rnn demonstrate test ability previously word word train simulate except despite simple pattern base drop generalize mean stage unseen completely fail drop take drop grey pick office come find become go simulate naive train scale simulate simply output high score allow simple capable general question example answer future memory future develop evaluate multi inference try complex bridge require causal sophisticated architecture explore deal task sophisticated management sophisticated sentence important dataset supervision answer fact rich detail specific variant network apply vision acknowledgment discussion simulation behave game within simulation allow ground comment want understanding kind world secondly release evaluate currently within answer task location task simulation please pick please learner object object actor drop examine object object drop object actor something place drop something underlie act simple random valid restrict drop actor actor text e go office drop example correspond figure ask office easy question convert look look variety automate get replace take drop discard put replacement currently ambiguity article add lexical variation join compound join later language model model compound noun seem easy hope add complexity control hard substitute generate answer fact input
correction datum categorical subset test call hypothesis remove report false frequent insight discover combination factor question significant subgraph mining trivial mining vertex paper subgraph multiple frequent propose detect subgraph order magnitude far I consider dependence mining concept statement subgraph correction discuss algorithm summarize subgraph vertex restrict summarize comprise transfer subgraph versus graph set xx h p k effective subgraph give collection graph subgraph significant membership datum two membership occurrence absence subgraph frequency contingency cccc occurrence occurrence strength association quantify present hypothesis true rely margin count observe one tailed type error significant setup database many hypothesis subgraph wise error one multiple problem deal issue significance level common test subgraph despite know detect truly subgraph huge subgraph test correct level small subgraph ever reach one correction achievable subgraph x tail test decrease require minimum significance threshold never membership graph occur insight subgraph increase exclude candidate subgraph subgraph database natural subgraph decrease subgraph task subgraph al use enumeration apply mining next apply challenge subgraph use subgraph subgraph frequent subgraph subgraph ratio monotonically subgraph subgraph coincide frequent subgraph follow root subgraph root subgraph long frequency detect frequent subgraph search significance threshold find frequency root frequency expensive subgraph finish subgraph easily sort subgraph frequency level significant subgraph search search possible reach et al frequency repeatedly decrease one pass mining high frequency significance significant subgraph pt incremental decrease newly propose increase termination know admissible frequency large able terminate soon subgraph whole possible repeatedly search expect work number admissible subgraph quickly frequency finish full early termination publish search arbitrary modular fashion subgraph monitoring terminate use obtain without correction thereby search root root set set terminate termination admissible incremental efficiency current examine significant terminate subgraph combinatorial subgraph subgraph far computational controlling fdr recently become lead examine force bf na subgraph occur assess result subgraph bf comparison find subgraph efficiency bf strategy art employ fast always test use permutation ghz gb avg max min avg ptc mr eight dataset protein chemical summarize table benchmark previous undirecte dataset except ptc challenge chemical include originally design prediction graph label ce se divide subset mr fr fm use mr since property classify inactive protein database six top ec ec classify ec ec ec ec protein create classify non relatively compare national cancer contain chemical classified anti cancer balanced set retrieve effectiveness correction subgraph improvement detect subgraph permutation label varied subgraph bind factor number subgraph miss mark huge computation correction much large circle triangle stable subgraph factor increase exponentially might tend moreover confirm highly correlate factor dataset subgraph subgraph subgraph subgraph size maximum significant maximum subgraph reason significant subgraph detect subgraph furthermore long subgraph become correction confirm correction large test get close cross mark blue effective green triangle bf time summarize root mean deviation fast also subgraph permutation figure clearly show bf average subgraph candidate contribute find subgraph whole search order pass search bf average slow speed one often bf around repeat reach incremental search quickly incremental search reason frequency repeat subgraph effective mention subgraph case mean feasible within size subgraph computation confirm subgraph graph stem fact subgraph correct significance multiple control subgraph runtime statistical power subgraph exactly efficiently solve subgraph dramatically reduce statistical far reduce correction subgraph several frequent subgraph retrieve subgraph experimental find subgraphs art result biology believe foundation follow important integrate exploit dependence test string subgraph sometimes extremely fast dataset maximum subgraph fast bf pass incremental acknowledgment aid scientific start grant mining von grant ar department engineering science university di department engineering problem find transaction test pruning hypothesis significant statistical truly real exclude power
overall sample expansion sum total quadratic kind utilize correctness x correctness prove large mathematical correctness obvious get convergence currently system break fail instability achieve performance machine rate reduce calculation many massive traditional demand framework widely use machine apply iterative framework appear iteration name library improve storage improvement develop base percentage always communication solution calculate failure occur present able performance efficiency master node percentage master rather dramatically high tolerance influence asynchronous relationship statistic section speed mathematic balance efficiency performance conjugate bfgs exist apply example master example master first quadratic convergence quadratic
advance compress sensing achieve line algorithm reduce tool differ base dimensionality explore use representation parameter field instead discretized express coordinate continuous counterpart representation proceed equip prior inference particularly dominate propose quantify mle difficulty stem likelihood denominator bayes previously analytically primarily nonlinear implicit follow exponent quadratic note respect obtain taylor nd address adjoint base readily include section work order reduction employ jensen inequality construct likelihood readily lower bind express kullback kl divergence divergence maximize low likelihood aforementione imply inference functional form minimize kl divergence equivalently field approximation look variational statistical expression section linearization approximation ultimately posterior latent enable identify approximate detail section pointing achieve semi inverse approximate p unimodal nearby construct account employ specification span rescaling resolve require r imply equation equation covariance diag induce natural ordering permutation entry column order variance attain direction second hyperparameter adaptive addition coordinate prior induce smoothness spatial variability prior jump value neighbor adjust site belong correspond adjacent voxel jump pair strength induce weak penalty vice versa boolean matrix produce jump one diagonal hyperparameter jump combine natural hyperparameter product absence priori correspond limit jeffreys prior part overall discuss parameter absence low log alternate optimize keep optimization zero purpose aforementioned linearization previous readily deduce mean adopt aforementioned covariance orthogonal eigenvector orthogonal principal diag reduce despite uncorrelated along implicit aforementioned derivation unimodal subsequently relaxed employ gaussian enable modal posterior approximation combine component would possibility far along examine unimodal reasonable imaging discussion expression bind imply argument aforementione simplify make expectation q augment orthogonality constraint address iterative algorithm prove cost setting employ brief skew base aforementioned preserve orthogonality use detail inverting call computation update iteration far depend analytical employ expectation scheme appendix completeness location probability minimal neighboring follow determination derivative order derivative difficulty set purpose update current guess equivalently assume guess denote value increment approximation term keep depend concave quadratic setting exact improvement possible term exploration summarize basic variational converge equation update update demanding call evaluate derivative increase equation attain presentation fix coordinate question address accord coordinate convergence reduce batch achieve coordinate use termination prior belief employ distribution dimensional measure coordinate p explain metric add coordinate consecutive linear elastic material nonlinear employ scheme whereby one reduce capture subsequent basis precision follow accord absence purpose elastic material stress tensor modulus ratio consist modulus value breast spatial unit employ boundary employ encounter static pressure apply depict horizontal assume solution yield configuration contaminate adopt top row total depict corresponding row one infer practically identical say posterior employ figure quantile deviation c standard indistinguishable imply coordinate basis comparison infer depict depict information call determine function reduce coordinate basis notice drop small reduce coordinate early practically indistinguishable full forward call forward call reduce account evolution objective forward discuss monotone call update reduce small coordinate depict capture correctly contaminate nonlinear describe several employ characterize density modulus cauchy whereas term material modulus play integration contribution modulus strong constraint model remain see circular material inclusion small circular material domain discretize solve condition bottom vertical load vertical point employ mesh corresponding employ also contain discretization include result mesh high snr medium contaminate gaussian contaminate snr material depict coordinate estimate obtain depict posterior firstly quantile ground credible large snr operating dramatically relation h snr medium snr snr medium converge figure depict snr behavior variance example exhibit quick small number call function effort begin call measure call number forward call depict evolution three snr h snr figure vector decrease similarity basis dataset clear variability associate variance large small level high capability problem propose much fully validation information theoretic approximation limit solver consider call forward solver operate hierarchy refined call fine also fully enable medical diagnosis currently posterior posterior arise frequently noisy datum represent challenge aforementioned line offer appeal possibility subspace associate approximation combine accurate maximization scheme completeness proceeding jensen readily bs diag diag dimensional calibration approximate appropriately select two firstly enable forward discuss evidence validation demonstrate methodology problem material medical diagnosis uncertainty quantification dimensionality reduction pose several model calibration assign proper mechanic material identification guide inform assessment system reliability identify material exploration without rigorously consider without provide quantification solution aim compute interest formulation offer unified framework deal uncertainty incomplete assess inferential uncertainty engineer parametric field bayesian scale size inference standard chain monte carlo generally impractical simulator advance sequential sampler difficulty modal pose computational identification biological material diagnosis measure sample obvious ease patient ray variation general lead early accurate diagnosis insight modality progress propose new imaging technique aim develop rigorous image modality project pre post compression image position external load diagnostic alone appear coefficient raw noisy e bad derivative sometimes quantification employ alternative indirect admit problem discrepancy minimize raw arise paradigm development size resolution expensive tool content incomplete constitute advance progress improve efficiency tool applicable basis vb task machine community recently employ approximate solve appropriately minimize kullback leibler divergence enable closed identification signature correspond attract application develop representation dictionary achieve material exhibit variability e care scheme employ frequently equation solve accuracy capture physics mathematic impose burden general solution linearize need eq forward solution cost report numerical experiment particular converge solution inversion subsequent available low experimental location boolean pick since emphasize generally highly nonlinear inverse nonlinearity constitute basic addition scheme computation pde constrain work scalar employ direct jacobian need solve node repeat
particle depend child previous particle way total generation particle process expect produce point produce child vice versa similarly child child use criterion particle order make branching decision initial particle decide continue particle particle analogously expectation cascade estimate fairly smc statistic denominator tie n k likelihood summarize initialization bound make resample random particle resample resample particle cascade correspond particle assume proof backward computational implementation cascade particle gradually propagate truly introduce arrive resample eq particle count ordering good situation order order completely preserve na I incremental resample scheme dependence order reach quite first stage well cascade address order particle impose permit run count consume efficiently particle impose hard limit particle simultaneously particle generally process ideal vary hardware make scheduling across queue equivalently weight queue live control responsible particle likelihood particle terminate queue queue terminate particle particle model use exact posterior compute design stress cascade rather branch scheme particle expect particle filter compare bad particle instead particle forward particle statistical quite thus speed benchmark competitive iterate iterate smc particle algorithm provide repeatedly particle efficiency significantly non resample suggest previously total impose per true normalizing constant ultimately efficiency record marginal likelihood show amazon ec core intel v processor much fast estimate compete model quick filter incur cascade initial particle particle progress end simple cascade hardware forward simulation increase particle possible also interested large small hardware comparison hardware configuration particle cascade broad applicability particle appropriate graphical particularly smc recently appear reference suggest primary bottleneck barrier synchronization something entirely boundary exploit parameter nuisance grow smc leave parameter se cascade particularly relevant attractive smc acknowledgment ep grant frank material air laboratory agreement reproduce purpose notation conclusion contain imply u air laboratory theorem axiom claim theorem frame department science uk uk ac uk call cascade barrier particle throughput memory unbounde cascade straightforwardly barrier synchronization limit costly term memory asynchronous particle efficiency resample throughput synchronization follow act queue barrier choose particle previously proceed traditional pf particle completion termination simply carlo merging set particle iterate carlo merging produce smc run close nature suffer fundamentally inner suffer directly avoid nature cascade merging share bernoulli branch exploration method use filter propagate cascade resample total generation allowed gradually decrease branching generally observation order choose appropriate resample increase effort scale advantage remove collective particle cascade arbitrarily particle budget focus keep proposal allow particle normalize particle continue generation rely resample state markovian dynamical random observe particle proposal density intractable conditional complex black evaluation costly capable simulate weight posterior weighted particle importance estimate suffer degeneracy wherein mostly moderately traditionally resample progress weight discard many scheme resample overview approach think draw stage particle equal resample introduce resample prevent resample add unbiased scale number particle parallel separate must normalize require forward simulation compute weight collective lead large memory finite hardware limit capable run particle move must memory requirement particle great ram substantial disk cascade address
blockmodel node row belong let probability block adjacency covariate expectation covariate blockmodel nc next assume hold motivate graph sbm nc sbm say block graph blockmodel gender diagonal producing within definite stochastic tend edge within nc covariate eigenvector norm finally mis membership nc sbm zero kk th row block population equal derive mis eq contain appendix eigenvector contain eigenvector iii theorem bound mis cluster mis recall cluster rotation intuition mis rotation minimize mis cluster bound mis mis assumption iii contain choice suggest base bound notice iii contain simplify key result ensure sufficiently sensible sparse routine differ suggest theory alternatively grow graph value low attribute sbm allow number covariate blockmodel block independent kl divergence covariate opposite correctly least node block insufficient insufficient compare simplify give cluster probability achieve clustering require investigation cluster result regularize cluster simulation bernoulli edge block opposite probability covariate canonical spectral utilize regularize rsc covariate sc effect mis block conduct graph method perform simulation mis change specific covariate h tends regularize poorly weight structure final membership membership bernoulli covariate long differs align varie agree assignment robustness misspecification robust misspecification mis show achieve mis graph identifiability require graph graph voxel brain voxel voxel treat spatial covariate contain brain graph range density brain label region treat covariate demonstrate effectiveness spectral utilize covariate help discover block ability covariate spectral discover cluster covariate brain interpretability covariate mainly exploratory tool insight relationship examine relationship brain spatial utility covariate partition graph matching ignore covariate spatial distinguish brain covariate spectral give homogeneous partition favorable degree utility flexibility cluster stochastic blockmodel nc well regularize mis cluster relaxed overlap strictly graph accurate mis misspecification nc sbm useful study assumption demonstrate useful tool obtaining priori criterion brain spatially coherent relatively homogeneous community homogeneous balance decide analyst relatively homogeneous cluster easier focus partition align covariate could value still computational eigenvector potentially reduce step direction develop understanding graph informative useful ultimately thorough relationship structure essential deep social derivation change major concern determine clustering eigenvector approach let follow note j eigenvalue position result lead ji position position lead optimal transition transition transition argument symmetric covariate transition lead equivalence membership eigenvector b h exception symmetric eigenvalue spectral population apply individually kk next bound spectral variance entry otherwise eq norm assume expand restrictive restrictive supplement probability three term supplement bind establish q q hence supplement result term consequently five give q lemma onto span choose eigenvalue condition close centroid ki give use result simplify theorem decrease span assumption tuning investigate mis simplifying assumption recall tm k minimize eq mis depth suggest suggest analysis check assumption degree check finally find mis cluster minimum occur agree eigenvalue uses derive low specific covariate first divergence divergence bernoulli covariate block assignment rewrite easy probability b condition cluster satisfied high cluster clustering remark biological consist interact unit intuitively graph reveal insight graph network leverage help latent statistical provide joint blockmodel mis result without covariate large graph derive location region membership covariate easy node brain blockmodel areas vast amount contain valuable gene brain region relationship essential solve feasible technique diversity understanding block contribute common pathway common form network become social biological science network discover insight characteristic extensively aspect include unlike minimization modification spectral cluster regularize network certain bayesian flexibility block likelihood provide quantify ultimately computationally globally partition diverse modern often represent network brain brain potential utilize covariate graph covariate discovery covariate procedure enhance homogeneity within filter partition covariate natural interpret rely ad hoc heuristic estimation broadly heuristic focus categorical computationally expensive discover multi block binary update algorithm include space point cluster relative node node similarity approach covariate spectral laplacian square parameter adjust relative covariate section propose type graph without general covariate variant previously minimize weighted cut spectral relaxation decide chose laplacian covariate recognize tune blockmodel type paper derive performance method initially motivate intuitive covariate blockmodel combine stochastic model mis accurate covariate method laplacian cluster canonical correlation perform nc sbm however canonical correlation fast tune single covariate without intuitive determining tuning consider provide optimization set knowledge brain set use location spatially coherent alone easier interpret align connectivity vertex node edge adjacency restrict study undirected unweighted small direct weight graph ii treat constant improve graph parameter average cluster correspond regularize spectral prior run use node necessary introduction spectral
feature well comparable base object cnn part multimodal fix image feature deep stage future ms dataset deep cnn internal deep platform non tune via k across ms average sentence exclude stage core test benchmark level annotation tc take around people image sentence sentence annotation training testing provide five adopt separation dataset provide annotation crowd annotation previous testing task current release contain annotation sample validation testing currently set dataset ms scores b b translate reference similarly sentence generation remain evaluation drawback description generate sentence please calculation task material evaluate method toolbox version evaluation retrieval measurement retrieve give top top rank retrieve k important retrieve sentence tc metric please supplementary improve publication update serve architecture rnn representation input conduct evaluation metric mention accord give strong language successfully capture without content sentence rnn generate low indicate consistent perform rnn retrieval since report score future comparison retrieve sentence rank retrieve metric compare supplementary material tb ccccc base rnn cccc cccc c sentence dataset retrieval k art method devise image representation use feature avg denote confidence strategy help use feature cccc sentence text r rnn devise avg rnn b generate rnn htb text r random devise rnn ccccc ccccc b rnn dim yes yes treat version method appear result devise search keep layer five recent representation recently recurrent relatively compete advantage efficiency input store recurrent layer perform generate greedy rnn c rnn method server select time represent reference reference importance rnn rnn outperform strategy multimodal substantially improve supplementary main near rank rank share hypothesis search keep generation sign probable treat hypothesis hypothesis validation set generating hypothesis image retrieve near detail tb try two pixel window corner pool ten second feature calculate multimodal see scale neighbor feature feature rich visual row contain old retrieve consensus share rnn shared server rnn share rnn rnn suppose get neighbor consensus score treat image rnn calculate similarity near nearest cross validate server consensus point consensus get test variance image improve ten hypothesis hypothesis surprisingly room improvement multimodal recurrent three task image retrieval sentence model rnn interact multimodal rnn image flexible representation sophisticated acknowledgment ng yu technical thank anonymous center machines award cs l rnn rnn rnn rnn embed input multimodal capture level multimodal predict validate train rnn rnn denote whose word multimodal thus two rnn denote rnn whose two layer word embed rnn rnn connection multimodal embed multimodal layer rnn word multimodal perform embed layer three image rnn layer layer connection share table show rnn indicate visual practice hard train keep dimension sophisticated rnn adopt metric retrieve sentence image percentage retrieve retrieval neighbor query image space hard description subtle sentence pick image sentence sentence sentence figure word multimodal originally machine first gram multiply brevity eq length reference sentence generate strategy whole reference sentence close shorter xu wang com california recurrent network model novel generate recurrent sentence convolutional image validate four benchmark dataset tc ms outperform state sentence art page code hypothesis sentence rnn sentence description early education image retrieval thank development computer vision processing progress brief method treat retrieval semantic sentence lack ability novel describe image object multimodal recurrent network observe paper topic acknowledge address sentence part multimodal language part word recurrent vision convolutional cnn generate connect rnn validate benchmark tc significantly incorporate deep representation image sentence rapidly vision computer convolutional margin image task imagenet widely vision design layer perform substantially language recurrent task rnn machine extract sentence sentence previous method embed sentence image object feature generate feature global category language field model field kind generate sentence correct description generalize retrieve category learn sentence topic rich sentence serve retrieval task work embed layer multimodal pre initialization embed random see word three multimodal layer rnn recurrent space add element relu success training vision differ adopt relu backpropagation rnn appear temporal work heuristic truncate step early efficient truncated layer multimodal connect three layer recurrent representation activation framework activation multimodal space together activation multimodal
score average sentence positive purpose supervision supervision group review average predict score classify review sentence art training emphasize infer label sentence naive classifier review classifier transfer ignore neutral neutral sentence manually label sentence either half report sentence tool tool pre train web interface th supervision require expensive entity sentiment entity di context review movie force predict sentiment specific getting predict sentiment phrase actor sentiment role movie figure illustrate actor movie total movie score phrase sentiment desirable movie di sort actor provide work advance deep consider similarity embedding demonstrate deep label individual explore classifier embed modality well development application rgb institute advance cifar ac uk infer use learn grain area voting group illustrative vote city public policy individual present aggregate say concern privacy vote application privacy technology solve arise intelligence similarity measure instance objective function create capture success rating sentence group sentence review review detect comment toward service present overview create sentence document convolutional embedding sentence review embedding convolutional embedding illustrate regularize learn objective transfer entire review eliminate high cost label sentence success computer deep effort language processing considerable decade interest effectively convolutional neural network representation beyond block notable vector extend simultaneously move use convolutional representation document convolutional experiment method multi instance refer instance powerful extension learn variety prediction categorization translation object privacy bag predict prior instance make connect formulation instance assume bag group determine xu contribute equally bag bag generalization bag term bag label within survey multi vast disagreement terminology close formulation deep transfer set group instance instance assign measure label example essentially invert label advantage assign instance goal assign produce training set goal construct objective propagation manifold label similar semi term alone label group adopt loss ensure term label simple relationship say label element term act avoid trivial instance regardless group belong individual carry regularizer shrink compete two value fall second bound therefore order contribution equally assign label unseen label goal instance test instance group objective item similar transfer green embed sentence document sentence intermediate representation require refer sentiment sentence document sentiment attempt sentence contribute positively sentiment interesting explore causality automatic sentiment deep sentence outline previous sentiment sentiment obtain similarity recent distribute work show work extend text create similarity q expect nearby embed correspond similar sentence appropriate measure closeness embedding particularly match level supervision word sentence purpose figure use review group movie exactly keep incomplete child regard I
bayes study allocation family datum probabilistic free model hmm state introduce literature well derive collapse derivation let define maintain denote negative eqs z n k review collapse symmetric gibb take specifically start part concern object posterior formulate conjugacy easily r k l k I n denote solely combine q denote beta plug cluster assignment domain iteratively posterior stochastically statistic result derivation symmetric omit concern domain denote eq practice assignment z hyperparameter put omit scope employ posterior however never number detect valid manner difficulty slow nature sampler million mix network datum model slow need introduce sampler make implement reason motivate develop fast work sampler variational bayes quickly collapse vb vb vb hide posterior posterior variational assume minimize leibl divergence posterior variational posterior speak vb analogous iterative vb low w posterior vb monotonically vb vb break eqs vb solution doubly posterior number number virtue bayes unnecessary weight infer I b describe eqs mainly compare original formalize introduction truncate cluster simply assignment variable posterior conjugacy posterior v k vb equation derivative low eqs naive vb interest derivation vb variational posterior assignment variational j l sum vb posterior derivation present final q eq vb bind easy hyperparameter derivative hyperparameter k l l inference assume posterior collapse ordinary vb posterior posterior local first collapse gibbs estimation reduce evident good derive vb employ truncate truncate reader present presentation k require way membership k n k z z variational posterior concern original vb form vb whole conjugacy inference form variational naive vb reason form hidden variable inference resemble collapse gibbs vb inference one put difference sample assignment repeat rule cluster representation replace sampler exclude computed datum denote expectation likelihood prior rule evaluate expectation derive inside part readily obtain ready conjugacy denote statistic variational cluster equal rule intractable computation inference expectation side follow fa fa x st take posterior approximation employ consider lda obviously nd learn taylor I computation expectation variance expectation variance count eqs k l expectation variance compute expectation j l l plug eqs domain update completely expectation l l j l j l j equation solution iterate local inference code derivative concentration dp role never rule technique follow current inequality zero update observation hyperparameter eq evident update eqs update rule incorporate count vb parameter theoretically vb monotonically variational solution iteration automatically sound monitoring unfortunately guarantee far expectation see taylor lower bind sure procedure monotonically inference important problem convergence literature estimation many empirically monitor couple naive leave detection automatic prove stationary non viewpoint highly preferable devise practitioner easy inference algorithm em prefer guarantee allow user vb automatic termination guarantee convergence stochastic leverage fact lda contrary rewrite way l l x right membership degree eqs inference limitation miss within result inference linear inference correctly existence missing need entry another algorithm posterior trick collapse way update massive parallelization collapse direct computation computation usefulness experimental confirm fact inference model vb relational fast inference linear scale relational compare baseline deterministic comparison collapse sampler initialization fair comparison update gibbs report completely manner value hard assignment solution weight cluster solution gibbs priori assess examine explain evaluate average test matrix exclude relational roughly held likelihood entry evaluation likelihood test entry initialization hyperparameter compare solution convergence vb posterior relative quantity converge state employ utilize keep miss reference gibbs iterate sampling procedure discard burn repeat collapse gibbs would million obtain resource collapse practitioner size mail use study mail transaction company member send member transaction contain record fm service list tag relation dataset time tag relation tag inference naive vb tag co occurrence count tag name tag tag entry binary tag tag size deal require observation exist dataset evaluation c modeling performance solution test conduct statistical dataset vb c gibbs dataset vb reveal inference significantly confirm well dataset potential inference vb inference vb specifically vb well artificial small cross still estimation estimation dense general face analysis informative iteration expect bad optimum contrary sampler collapse million well perfectly collapse gibbs sophisticated index sort group horizontal color domain domain assignment cluster highest sort sort time convergence report trend average aside collapse magnitude nd unknown vb third count maintain able efficiently cache fourth landscape posterior smooth vb concern vb truncate fast cpu convergence threshold quantity fast illustrate plot collapse case practitioner computation gibbs tag c vb user tag far conduct dataset typical datum goal perform datum assume test time convergence level five relational large dataset tag indicate rating rate regardless rating rating movie subset netflix degree worst movie million entry c cpu linear xu xt netflix na netflix rate na present times average cpu times inference enable n general dataset take cpu cpu please row cpu grow expect nine cpu dataset number technique cpu threshold relative become fast beneficial collapse gibbs linear well million estimation also collapse inference process detect datum costly contrast require practical relational collapse variational infinite convergence inference replace vb base standard taylor derive formulation parameter never practically open start examine assess effective annealing average offer anneal mechanism equivalent converge lower stationary aspect shrinkage implementation inference application result offer precise inference naive vb enhance speed possible stochastically recently lda examine parallelization collapse sampler algorithm advance aside assignment toward focused dynamic subset filter easy observe relationship average guarantee practically useful collapse bayes inference number include hyperparameter study practically two study develop convergence inference enable automatic expert difficult costly manual monitoring inference describe large relational show performance infinite collapse relational analysis link social service customer record scientific statistical present among infinite simultaneous bi row dimension customer correspond item modeling tool datum without need careful bayesian sampler guarantee posterior infinitely stochastic often computation computation thank factorization approach easy convergence vb due factorize posterior collapse estimator integrate parameter inference collapse gibbs sampler fast sampler collapse especially hdp lda paper taylor simple sense report yield vb exact collapse paper collapse gibbs difficult preferable report collapse gibbs interestingly vb easy reason vb perform poorly partitioning problem promise vb focus topic bag style set formulate derive relational fast naive vb derive automatically optimize dp nonparametric study c c l vb comment seminal fmri gpu noise filter extension fully cover inference two practitioner aspect theoretically inference exception use valid lda however problematic practitioner familiar try manually sense favorable behavior naive bind pseudo leave log likelihood empirically serve develop anneal technique automatic detection anneal converge stationary equally preferred practitioner want art apply issue seminal introduce optimality valid first hdp hdp solution hmm attempt hmm implementation square object product observe item cluster user inference make impractical relational describe computational especially solution item denote practically guarantee experimentally offer comparable even naive variational multiple synthetic real relational magnitude fast vb stable dataset scalability propose relational convenient practitioner automatic collapse solution use precise behavior solution simple annealing convergence inference object gibbs solution introduce vb vb solution present convergence issue annealing discuss devote final relational dirichlet dp cluster unknown dirichlet mixture proportion break chinese restaurant crp crp employ collapse sampler collapse crp crp partition let partition crp k pz hyperparameter equation show rewrite allocate partition object number partition exclude object membership object new randomly partition crp result construction
ai package binary node node observation fine grid tuning parameter panel grid node calculate consistently proposal color pseudo proposal brain university project process occurrence computer select large term gaussian among term student edge node control fix criterion select six indicates explain occurrence word provide explanation relationship htp publicly available expression gene cancer poor patient among associated cancer reconstruct gene among likely role identify gene brain target identify identify fix present empirical covariance result network display identify gene decreasing estimate interestingly gene know role variety gene highly connect use recover framework framework three accommodate sparsity connectivity c define equivalent theorem sufficient solution remain eq follow desire proof present prove matrix dual dual ii equality compact swap min equality ccc subgradient matrix suppose solve support subgradient imply eq q obtain suppose ij cc jj feasible p last summing use arrive contradiction result ji consequently combine use consider objective lemma assumption v f solve graphical evaluate contradiction must correctly fail node htp color study extensive admm ghz core replication display iteration converge function iteration increase run never without htp f update ai gradient descent choose bfgs method unconstrained line evaluate computed solving note leave improvement future work htp lee consider author learn implicitly accommodate realistic network convex penalty framework three widely graphical ise model multiplier algorithm demonstrate illustrate proposal set gene expression multiplier graphical wide variety gene presence indicate manuscript type graph marginal independence connect conditionally marginal marginally without graphical model number specific variable conditionally independence marginally independent marginal density ensure sum determine interpretable encoding encourage take graph place double consequently approach equally edge believe certain world wide web world node power law example include social network li towards hundred gene result typically free paper densely substantial number densely connect contain node propose convex penalty estimate contain penalty ise author propose graphical however arise free less author section graphical see contain node much graphical apply graphical ise apply gene discussion general accommodate matrix assume convex order estimate non instance graphical sp penalty encourage estimate contain entirely zero model penalty encourage sparse entirely figure edge via form eq parameter control connection convex combine depend loss network dense value closely relate overlap context admm guarantee consensus solution htp p stop f soft operator ij z detail rewrite augment primal lagrangian usual lagrangian quadratic complete appendix ai depend special network assume take form serves extend lasso accommodate propose encourage contain connect update ai derive minimize respect treat domain solution denote eigen admm update compare admm ghz intel core take minute admm extensive run solution column condition depend upon result literature fuse partition block jj c j k check block bottleneck eigen block compute eigen compute eigen computational per exploiting take second apply connect large node optimization tuning denote p condition reduce tuning reduce solution diagonal decomposition scope tuning place penalty minimize q unique zero motivated fact degree involve estimate parameter degree freedom proposal proposal enyi proposal graphical start index node graph cardinality set highly estimate jj equal randomly column proportional world take natural scale free distinction n standardized proposal gaussian graphical model package glasso problem involve three describe previous section display average simulated square tune fine tuning parameter bic minimize involve tuning tuning display figure proportion simulation edge equal iid correctly proportion correctly highly ii nonetheless outperform select use graphical bic large colored correspond graphical performance proposal correlation screen thresholded absolute node thresholded purpose estimate scale tune parameter partial use package space combine author claim proposal iii average grid specify element thresholded node fine value note figure b c use fine curve error baseline include surprising graphical since approach implement iteration edge comparable network intend perform intend network intend graphical figure type criterion color
learn generalize another discriminative input risk nn note neural give source natural use probability correct label eq internal representation neural consider unlabeled representation sample hyperplane proxy see estimate logistic regressor denote target enable adaptation term solve hyper implement trade source divergence hyper parameter tune quantity network domain regressor parametrize way source layer accurately regressor unable detect sample option alternate simple stochastic sgd sampling update crucially opposite follow consist except bt sample layer adaptation neural regularizer domain network parameter experiment split source label use validation risk mention previously domain focus mainly hypothesis recently become increasingly study notably principle base denoise principle adaptation domain indeed directly optimize work hmm representation use inspire versus sentiment argue optimize divergence rely reason idea learn auxiliary also explore context belong identify result minimax learn minimax work assume learn adversarial generative datum work share pca toy color regressor experiment behavior distribution label way keeping contain unlabele dot capability experiment train use propagation execute risk minimization train regressor discriminate toy experiment boundary compare graph relate graph part relate column four detail label class sample contrary perfectly analyze affect hide present component pca map dimensional hide project two nn pca representation point point visible labeling easier correspond represent graph clearly conversely point locate opposite corner pca target resp difficult locate cluster pca suit classification regressor classify otherwise regressor discriminate prevent explain train regressor learn domain discriminate target although discriminant decision surface neuron th neuron three cluster allow straight classification observe regularizer prevent neuron indeed neuron corner corner c source name nn book book book book book mm svm standard equation hyper search consist target amazon process four review specific book rank star product rank star adaptation task example book book source domain unlabeled use unlabeled procedure logarithmic procedure adaptation nn svm use part show risk report one conclude help suitable domain brief unsupervise robust feature representation input find reconstruct original noisy counterpart show optimize objective experiment subsection pair source corruption execute three procedure concatenation encode nn representation sound test table foundation claim representation toy confirm real compare proxy run nn recall obtain construct equal first subset use large range low firstly representation experiment hyper parameter lead expect compare representation standard nn influence precede lead low lastly present notice give great data approach help seem representation algorithm inspire adaptation behind encourage predictive uninformative extensive toy sentiment show effectiveness strategy notably autoencoder turn representation believe incorporate extension deep adaptation task beyond basic denoise autoencoder thm introduce distribution domain suggest discriminate source propose objective implement task uninformative sentiment unlabeled domain performance either even input extract stack denoise autoencoder obstacle develop exploit one generalize focus context
trade overfitte perform goodness fit criteria information bayesian cv understand equip aic cv standard selection theory new recently impulse response realization learn counterpart selection paradigm robust aic type cv crucially depend covariance kernel process variety semidefinite kernel nevertheless straight framework fail lack system stability several deal stable introduce spline realization implementation concentrate totally kernel point algebraic connect completion particular band alternative stable kernel triangular diagonal interestingly spline factor admit covariance burden scheme spline organize spline briefly review matrix completion introduce end symmetric resp semidefinite resp matrix vector diagonal diagonal denote set submatrix index submatrix model fed impulse white noise collect dimensional express whose input represent impulse estimating system impulse dimensional model sample continuous covariance independent empirical paradigm marginalization joint density estimate impulse identification assign class covariance stable information impulse surely first stable see tune correlate tc kernel q therein also concern completion pattern graph entry specify every great band center entropy covariance admits complete fundamental main get ij top left optimization namely entropy partially band problem denote value mx band feasible admit property bandwidth namely positive maximum state matrix give lag also maximize matrix pattern problem admit close compute specify symmetric matrix bandwidth call follow extension specify bandwidth step suitable solution factor central extension partially central band admit triangular stable rely introduce form order spline stable spline highlight specify spline computed entropy nest prove dimension claim statement k k central extension band hypothesis claim maximum likelihood follow ij j spline covariance satisfy equivalent moment stable factor form theorem immediate consequence sum stable spline resp hand resp diagonal positive determinant I thesis recall lie likelihood nonconvex hyperparameter observe stable spline
eq quadratic root explicit formula constraint check omit detail x rescale generality strong upper conclude note lemma thus display easy generality prove acknowledgement would thank discussion grateful point barrier ex barrier seminal nesterov explicit construction universal barrier geometry concave elementary families interior main recall specific barrier prove sharp isotropic log concave bilinear likewise derivative self barrier convex furthermore addition barrier xt gx newton approximately minimize theoretical nesterov barrier barrier universal convex seminal always self also set simplex hypercube prove exist barrier self mid also construction barrier barrier barrier canonical barrier universal barrier give barrier tool dimensional generating body viewpoint prove mass proving derivative seem barrier inequality canonical barrier inequality play barrier inequality fact body concavity somewhat effect complexity interior consequence local sign dual universal barrier analytical barrier side analytical center universal know point geometry proper convex homogeneous furthermore recall characteristic immediate barrier homogeneous begin connection barrier satisfy universal barrier universal barrier hull different consider universal barrier body barrier barrier characteristic cone particular barrier barrier self barrier barrier barrier introduce unique amp exhibit convex riemannian barrier canonical barrier recall unique volume ellipsoid perhaps homogeneous coincide constant somewhat generally canonical equal affine conclude comment generality cone focused self tractable barrier barrier essential obtain universal barrier barrier immediately effort implement gradient barrier practical important efficiently computable barrier convex interior one barrier sequential known describe history possibly randomness player adversary compare cumulative cost cost costs play g challenging receive limited feedback bandit player observe incur survey view feedback seminal role good precisely run mirror originally choice barrier sampling key barrier hessian barrier proportional inverse scheme achieved support ellipsoid scheme good universal scheme make much ellipsoid sampling via discretization mirror barrier scheme strategy introduce exploration one property barrier prove implication noting obtain numerical whose concave move part self become word show eq proportional eq interior satisfy exponential associate weakly operator norm verify smoothness support technical rely log concavity measure gaussian give proof lemma conditionally stochastically dominate condition law equal constant learn exist confirm use find let x x x x thank proof let measure direction consider smooth support key thank differential lemma proof yield
deal different american category image style piece work th style art tend year style style style unknown lot classification quite image category style ordinary classification two comparative conduct review semantic level intermediate feature fine use different comparative evaluation evaluate three model bag generative use semantic feature use discriminative model discriminative model capture semantic employ machine intermediate generative specifie distribution new distribution intermediate step label discriminative thus avoid comprehensive semantic capture characteristic color texture color histogram edge feature formal intermediate level apply descriptor sift level descriptor localize generate intermediate bag create image codebook visual represent capture frequency level semantic content water denote existence semantic worth note color texture focus intermediate semantic utilize generative model like topic capture visualize document example characterized atom represent level descriptor describe constitute represent straight constitute topic tree region concentration color water subsection detail bag popular categorization document matter categorization typical several representation descriptor descriptor descriptor encode codebook machine overall presence pre classifier generative dirichlet allocation semantic categorization localization scene categorization fine categorization purpose dirichlet topic represent characterize graphical image topic total setting minimum tendency hausdorff asymmetric period construct represent direct indicate high potential define influence multiple space indicate influence influence generally influence influence truth influence retrieve top influence retrieve influence pair influence truth pair detect influence sake relatively influence truth mean feature descriptor descriptor recall graph percentile distance descriptor table percentile column similar generally manifold curve recall euclidean manifold three top recall l l l recall l l l l recall graph achieve visualization similarity purpose short scale dimensional graph htp htp figure visualization influence projection code plot reflect ground truth style cluster together mapping modern abstract differ leave seem dimension much distance broad influence similarity yet way style list influence ground truth close mapping truth less coherent lie consistent mapping illustrate suggest influence htbp surface automate discovery study pose question find knowledge qualitative quantitative measurement study present comparative semantic good task distance manifold comparative give different central link maximum formulate hausdorff central influence tool similarity annotate diverse publicly task lot search similarity many principle way influence difficult huge valuable examine expert period find influence connection influence contribution explore computer influence set present comparative comparative problem review model second feature compare level vs low investigate influence pose discovery purpose map work use concept basic art shape color line general sense seven principle art movement unity subject matter attribute work art art influence connection art continue art inspire body art influence similarity inference suggest ever inspire unless sake consensus cite comparison study study x similarity clear matter computer advance develop object categorization scene etc recognize scene category historical look expert landscape look color texture complex concept think logical tackle methodology learn concept automate determine measure task mention way describe description translate automated way van element art finding different automate influence quantified connect keep challenge people art suggest towards measure influence besides various orient volume database internet task organization retrieval properly become classify category classification speed significance broad view level indicate line composition square see position conclude make mean completely symbol symbol word express meaning work need image describe list many semantic opposed color texture influence subject matter semantic importance find similarity become prominent essential linearity paper computer automate influence set discovery methodology study different representation measure study collect contain time period collect ground truth influence contain claim use discover influence discovery would evaluate require compare different detecting influence contain truth negative different resort classify representation good classification determine influence perform comparative classify seven detail sec conclusion study confirm semantic useful task influence right similarity illustrate detect methodology ed look similar close look subject matter detect similarity measure discover however clear influence time influence influence result achieve structured survey describe use section describe classification include describe methodology evaluation automate automate fine utilize level texture study classification style define signature low among eight focused discover similarity influence publish experiment image classification pose annotation present et al texture exploit effective inconsistent texture visual image al set annotation keyword describe annotation class pose annotation pose human image query contain improve propose local problem et unseen annotated class retrieve visually similar
fix power remove investigate novel tensor introduction mention practical aspect model among iteration analysis overcomplete apply propose guarantee subset relate study gaussians list gaussian propose moment polynomial general challenge model improve divided spectral method among mention noise noise moment spectral recover general without degeneracy condition guarantee power third order tensor recover overcomplete vector factor asymptotic member outer mode refer row arrange vector slice fix slice rd multilinear particular mu multilinear mode multilinear combination tensor slice rd form tensor section exchangeable mixture state convenient vector q basis simplicity argument even style cm sep sep draw name view independent hide variable conditionally denote matter observation moment unsupervised problem distribution pass decomposition main perform multilinear thought rank alternate run initialization well return moment mixture tensor update multilinear set maximize iteration power output center cluster cluster center propose guarantee state explanation condition column noise universal guarantee assumption crucial argue conditioning strength sure ensure inside vector next propose let q iteration initialization initialization setting mention initialization rd randomness recovery therefore update require use polynomially main signal recover regime noise thus order initialization condition c relevant satisfied iteration different mean modify third observe q additional spherical state state relate hidden recovering follow first change asymmetric among appropriately version introduce new phase phase show component provide incorporate result initial constant correlation component simplify rd tensor dynamic tensor phase eq update vector w constant dynamic condition constant h recover sign resolve sign issue lemma follow satisfied generality initialization h exploit inequality desire initialization universal corollary make initial correlation result normalize intuitively roughly hand vector argument show ensure proving show residual randomness distribution enough tight enough argument hadamard product entry multiplication vector matrix operator space orthogonal denote analyze dynamic rd iterative write analyze evolution dynamic explain early step analyze since update provide tight bind evolution careful controlling amount exploiting enable matrix break power intermediate follow power rank break step introduce intermediate unnormalized thus analyze randomly entry follow middle update evolution fact entry projection randomness row orthogonal equal exploit direct evolution govern conditional lemma iterative column orthogonal constraint orthogonal orthogonal remain exploit previous characterize middle intermediate entry remove decompose q intermediate residual randomness variable reference throughout iterative middle denote copy similarly conditional iteration variable intuitively residual randomness characterize concentration component maintain analyze dynamic power update iteration black following assume iteration induction scope inductive ty induction end induction provide analysis show correlation latent challenge regime technical small residual sufficiently enable progress generalize tensor support part microsoft fellowship nsf nsf award award award nsf award formula n intermediate formula unnormalize remove column recover bt b r see randomness equation update part leave u b tv let partition first rest make remove induction vector induction induction hypothesis true hypothesis bound initial state prove hypothesis end figure scope flow start start show induction hold iteration induction tb concentration similarly matrix I know hand eq expansion step random dy ty b eq definition establish expansion dominate tw iteration hypothesis hold random bring randomness formulate lem subspace orthogonal sum random high prove sep black w w infinity bind induction norm know high hypothesis hypothesis exploit contribution bound combine bound guarantee formalize lem ir bounded order last inequality choosing depend immediately norm hypothesis tx unnormalize q notation hypothesis desire induction early random component subspace represent norm bind argue expansion triangle cauchy inequality exploit norm involve term bound x analysis combine norm argue conclude lem induction also hypothesis early bind unnormalized argue observe contribute term exploit induction involve bind gaussian inequality exploit second finally argue hypothesis bound total term lem basis x x p know know iteration induction bound even large know inductive step inductive parameter polynomial constant q generality assume induction hold inequality inductive step induction step enough constant addition constant point universal get step side say inductive section prove inductive bind value value random concentrate norm even project subspace high variance argue probability random least value prove lemma lem vector vector gaussian high p direction equality z rp inequality step argue high square entry restrict apply part product orthogonal lem p triangle third cauchy lem maximum th basis leave difficulty prove treat lem specify expand hadamard bind type form bound lemma bound induction hypothesis hypothesis correspond eq v bounding difficulty treat entry hypothesis subspace q dominant cauchy early desire rgb criterion derivation remark em time
face challenge prove correctness require minor section intuition problem cost distance cost start open center keep service detect conclude easy without new cluster expensive denote point convention optimal consist center cost cluster average vector contrary let cluster eq total trivially bound bounding phase complicated phase phase open subsequent triangle inequality r f I r n r sum center estimate succeed b assignment optimal cost stage vector start incur sum start additional contribute cost bound k terminate round conclude execution p r observation online stream small must two distinct dataset aspect ratio create center make arrive algorithm phase first conclude reach probability round discover set adjustment operate entirely ad hoc large center sum square close neighbor evaluate execute dataset website uci letter e e e uci collection engineering sake learn datum raw mean decrease slow large significant classification observation raw news letter particularly low improve enable advantage phenomenon mean result rather cluster value every setting range roughly interestingly choice roughly average reader online mean center algorithm sum cost inherently center decrease nevertheless function target different dataset normalization divide monotonicity plot center identical figure something cost fix significantly pick center improve rate random choose center surprising compare mean run invoke use mean mean term pass online online mean bad note dramatically everywhere cost helpful suggestion theorem observation one generate logarithmic much operate strictly study point cluster center cluster eq offline advance access obtain provably difficult see streaming allow keep logarithmic stream center algorithmic idea streaming assignment online allow priori arrive algorithm open consist conceptually choice unseen stream intersection algorithm output time stream hard algorithm trivially stream trivially sufficient mass stream overhead independent length stream even stream act assign trivially assign one exist suggest read conversely part line scenario yahoo news decide belong act advance practitioner simply refer recently well mean ratio search design base optimal effort adaptive reduce pass
level show figure htb htb quantitative bold remarkable theoretically central great derivative noise figure fail root lack derivative order contaminate use order quality demonstrate method parameter demonstrate technique demonstrate success show general desirable operator frequently inversion consider regularize previously iteration size use instability converge process transform equivalent initialize lc noting map ms use datum form justify prior pick estimation grid rectangular surface divide generating contaminate literature potential inclusion regularization inversion initialize weighting find technique condition meet iii practical knowledge constrain interval project newton freedom hand regularization whether e g overall rely interpolation typical demonstrate case reconstruct case solution less useful mdp accord confirm error average error invert noise contaminate except invert estimating regularization require ideally prior solution prior require ms provide alternative central curve efficiency require implement order find solve inversion iterative technique replace svd development nsf dms novel sampling regime present pair also correct determine value ts limit low contribution seek corner shift use iteration hold filter corner namely corner parameterize equation lemma goal principle inversion discrepancy augment weight fidelity regularizer size nan space intersect newton find optimal inverse noise map datum implement scale generalize result verify regularizer approximate unbiased predictive focus smooth property iterative data noise curve discrepancy experiment efficiency context show curve principle general principle desirable principle ill pose equation discretization space vector assume contaminate discretization decay responsible ill extensive literature literature ill pose w tw yield dependent tw covariance white map I tw symmetric permit white determination research generalize validation rp principle mdp comparison criterion reference vary measurement mdp assumption apply extension effective consideration nonlinear inclusion principle estimating mean np np interval around development advantage root converge algorithm scale matrix iterative extension non unknown may measurement principle mapping approximate distance inversion sharp preferable example inversion similarly minimum support ms support introduce iterate stationary iterate operator update residual iteration force geometrically technique mdp approximation ms reweighted introduce analyze ms reconstruct inversion update regularization find often mdp curve apply knowledge follow development strong literature improve algorithm use svd parameter technique present also impact ms shift intersect realistic approximate typically w singular introduce ease presentation vector state rank respectively generalize singular invertible indexing generalize note indexing result variable statistical strong non condition sufficiently large limit hold p tw functional centrality tw examine yield first g tw tw tw tw tw tc assumption denote consistent apply obtain transpose adapt extended filtering replace obtain pick filter already note statement uniquely order hand particular order spectra matrix calculate ordering specific order standard p difficult collect estimation assess parameter review aspect section literature completeness formulae technique inversion residual valid adopt validation measurement set regularize formulation yield minimization associate objective flat create compute numerically trade norm residual problem corner difficult find curvature plot unbiased predictive risk successful use simplify
instrumental state mm hyperparameter assume generate covariance attain analytic risk recall goal therefore j follow use derive section assume submatrix row index use prove second solution dual write objective last hypothesis transfer accuracy kernel regularize forward follow notation brevity truncate q risk label hypothese j last jensen inequality prove
p cm full vs rw propose similarity two compute fix feature subgraph arbitrarily accurately sample increase process normalize histogram similarity two recent induce social improve subgraph normalize subgraph induce subgraph size histogram node generate dimension dim reason increase representative actually graph costly see importance subgraphs subgraph histogram subgraph computationally practice walk similarity number rich regard domain cauchy kernel two adjacency define computed close recommendation value eigenvalue worth dominant take eigenvalue eigenvalue adjacency normalize inner evaluation consist run svm standard combine fold th validation performing value svm fold fold act repeat fold acting partition error show outperform compete state art capture capable accuracy three range variation significant ideally tune easy walk sometimes subgraph expect count subgraph significantly interestingly count subgraph size except vs histogram loose computational counting subgraph consistently demonstrate superiority argue incorporate big complex along adjacency sound similarity dominant poorly compare eigenvalue graph different different eigenvalue seem right describe graph characteristic infer analogy explain covariance comparable perform key require compute dataset record pair similarity network similarity take summarize ghz cpu machine gb poorly fast rw method quite surprising rw slow cubic time complexity linear computing representation histogram counting subgraph costly count subgraph histogram sample subgraph graph rw magnitude histogram subgraph every histogram sample induce next step match subgraph structure graph sample solve graph start become intractable expensive count capture quickly loose count big nice capturing space positive semidefinite characterize graph covariance matrix adjacency indicate matrix naturally overall procedure edge superiority state approach balance representation tractable meaningful believe provide mathematical allow graph representation algorithm like semidefinite entry normalize vector adjacency encode underlie contain sub triangle addition suitable represent graph naturally similarity graph similarity measure state make practice believe empirical study adjacency characterize become increasingly whole gain analyze neighbor recently attention social network informative collection close center scientific individual scientific link compare dependency likely lot exhibit densely compare people reflect belong physics thus characteristic utilize possible discriminate recommendation discover citation recommendation network right different similarity measure meaningful mathematical embed structure fundamental representation spectrum world scale spectral density consist sharp peak mathematical graph compare common eigenvalue characterize comparable eigenvalue compare occurrence subgraph representation social inner product lead clear histogram counting capture subgraphs size computation know histogram subgraph size subgraph subgraph computationally require give capture behavior computationally challenge requirement map count count take alternate adjacency vector one generate argue covariance covariance adjacency covariance count histogram semidefinite covariance kind similarity multiplication compute representation outperform example subgraph perform poorly study power adjacency social researcher experimental scientific explore contain scientific work undirecte unweighted connect represent vector default transpose component two adjacent associated permutation operator multiply row multiplication e adjacency graph adjacency matrix represent structure vertex adjacency eigenvalue wise eigenvector path consecutive term I path path path whenever path hold exist contain triangle highlight quantity fully characterization vector show generate truncate order matrix fast web domain include page truncate similarity sufficient describe associate represent common basic map characterize adjacency associated graph permutation structure order entity perspective perspective turn covariance value spurious start compute later graph structure give first generate normalize since normalize equal ease ie h input adjacency initialize x nm tc symmetric semidefinite symmetric semidefinite permutation yield q imply converse theorem true hope small etc adjacency undirecte number triangle number distinct variance term quantification lemma path count twice length edge type path figure repeat ii repetition repeat path possibility contribution path total length path count path path explain contribution double count twice path triangle loop graph triangle b loop length node triangle contribute path generate many path correspond path total twice therefore eq add algebra substitute expression clear small count triangle along observation behind count path length disjoint separately extend involve computing term along sensitive empirically publicly available twitter consist user twitter edge generate graph add edge twitter value se network social high closure induce ac ab triangle theorem infer encode discriminate structure tell dimensional comparable structure fix common mathematical semidefinite property well notion two respectively summarize compute similarity adjacency matrix adjacency matrix semidefinite valid similarity semidefinite similarity semidefinite algorithm operate social later determined spectrum right graph consider fact semidefinite matrix mathematical detail c physics node symmetric semidefinite look eigenvector converge matrix avoid choice general recursively inside multiplication summation complexity computing graph addition require compute matrix graph computation argue value graph reduce costly step matrix multiplication proposal graph describe task publicly meaningful label availability structure create social
vote going apply hyperplane dimensional item formally unnormalized shift label directly general follow suppose item hyperplane know error bound worker aggregate factor worker one need give list corollary condition mention vote important special cover several simplicity case constant assume constant voting worker model meanwhile plug error theorem theorem hold well critical reliability weight choose us room weight detail far crowdsource majority error voting uniform meanwhile tight obtain letting simplification desire exponent second factor tight real crowdsource enough reasonable crowd similarly omit possibility control small majority crowdsourcing case tend infinity label majority I vote corollary tell quality worker worker random label correctly probability worker property majority voting ensure enough reliable worker available aggregate majority voting require worker section discuss method crowdsource model maximization posterior well know optimal reality model true way class build parameter apply rule call refer introduce em estimate might estimate start thus relatively rule know label map bayes classifier worker well guess oracle map help em map rule rate oracle derive prediction map visualize model corollary show voting shift item balanced balanced rule voting mean section understand infer ground label via prominent estimating crowdsourcing estimate map predict item study design study weight majority voting version rule rate ignore class corollary imply relax item converge suffer local error obtain naive voting treat gold sampling predict step th item defer label hard apply concentration martingale exponent important away accuracy guess average population well make worker label error decrease decrease beyond intuitively score worker increase make error increase e accord weight step way assign alg plug weight practical noisy worker linearize stable meanwhile compare synthetic experimentally art datum majority em public code average experiment pc window intel core cpu memory crowdsource affect variation item error reflect compare compare b run worker label three ground truth uniformly accuracy distribution expect worker match worker control worker figure error trend true converge increase vary confirm result fast majority voting clarity temporal omit ccc ask rate scale label worker around consensus treat multi labeling experiment dataset varied plot performance label compare majority voting generally majority voting bound decomposable crowdsource good drive iterative weighted voting theoretical rate reflect trend real crowdsource superior voting perform low misspecification want similar description correspond error rate labeling obtain score function complicate manner formulate voting em algorithm difficult nature simple em crowdsource thank suggestion comment thank discussion thank comment suggestion since prove corollary require put section present simplicity simplify worker vote equivalent aggregated label discuss simplicity notation bound set label specific depend value subscript come item say drop subscript item eventually assignment probability major focus term relation want bound voting bound apply concentration inequality far rhs depend inequality variance apply note definition sum concentration rhs get result hoeffding bernstein since rhs inequality c increase function decreasing prove argument straightforwardly far practical high hoeffding bernstein hoeffding lemma go hoeffde bernstein give finish bernstein chernoff show condition nd step easily finish several assume prove course chernoff directly step inequality prove theorem result map rule posterior oracle oracle section special map step vote every item meanwhile label hard generalize omit clarity prediction majority worker majority voting worker majority proving need several final proposition use prove convenience enable majority vote agree label give apply apply argument bind agree majority voting mild th item give match vote close number worker close close voting get since lemma apply measure entry quantity mean trivially focus non trivial column go put j j upper case take proof bind achievable bound say score voting decrease proposition derive bound step get true label convenience denote algebra condition want step apply get next derive lower expand I drop irrelevant two argument step upper eq beginning depend imply crowdsource become effective tool human computation since worker crowdsource aggregate label exponential aggregation crowdsource analyze aggregation majority voting voting posteriori map show optimize rate set iterative majority voting optimize approximate oracle version provable real par method computational around crowdsource expectation weight voting hard computer visual video crowdsource people call appropriately aggregate crowd yield could one crowdsource apparent purely complete label truth evaluate worker may answer question assign beyond worker persistent finish drawback reliable answer crowdsource yes voting treat equal worker majority voting improve upon vote back worker confusion distribution give element represent misclassification worker principle true label confusion matrix maximization majority voting put confusion simplify confusion progress true endowed concept worker area label crowdsource graphical apply infer principle worker assign worker budget apply extend infer behavior crowdsource system investigate error various provide majority voting crowdsourcing apply minimax coin mistake final labeling focus global optimizer rule find optimizer provide finite aggregation crowdsource motivate main rate aggregation rule worker rate voting majority voting gain insight design optimal voting majority oracle maximum posteriori optimize majority em algorithm approximate understand crowdsource propose drive weighted majority voting guarantee implement art simulate cost focus error crowdsource obtained analyze worth focused crowdsource labeling multi meanwhile crowdsource set real crowdsource crowdsource assume worker task cat evaluate assume labeling item represent miss however convention true th th item kk I label item item th get call configuration flexible worker item item gets label cover general worker adopt literature chance refer require specific get match worker task select worker follow represent match ambiguity depend either general context constant bound denote bernoulli operator number meanwhile throughout locally cover case originally reliability model confusion represent probability note worker label true represent label item modeling worker flexible overfitte worker impose constraint worker confusion matrix model probability item simplify probability worker matrix parameter reliability worker another item actually toy confusion model vertical axis actual class horizontal axis predict color different labeling item correctly three refer coin worker label noise signal binary special crowdsource convention convenience confusion parameter binary introduce ambiguity item labeling item rule aggregate noisy refined label item worker great labeling worker treat voting extension majority vote differently majority write worker behind majority generalize way input worker item predict potential label potentially base maximize aggregated score label decompose worker shift gain label label item reasonable contribute predict thus noisy final illustration majority special voting express approximated also paper sample decomposable bound crowdsource weighted majority optimization bound guarantee simulate deferred section sample error high expectation decomposable aggregation main focus specialized straightforwardly observe base probability accord assignment process decomposable introduce section mind quantity measure play quantity associate
detail modeling task linear quite general result mostly realistic assumption may sometimes behave preference situation agent class error good output definition compression imply section learnable agent behave fraction pricing always apply broad price amount bundle example volume arbitrarily unit non item offer demand learnable necessarily efficiently situation prefer preference option problem permutation vector represent hypothesis complexity h see p learn reveal preference bit optimal linear bp concave formulation tucker kkt xx dual respectively bp simplify price derive utility per spend good price per unit per segment segment clearly completely segment allocate dx dx follow bundle drive price achieve unit good utility bundle get kkt class utility together feasibility h figure bp since enough bundle suffice bit numerator next calculate preference determine preferred bit upper bind extra initialize bp else round else round rational learnable reveal preference learn length segment make jk last segment learn budget sure segment segment price jx budget appropriately extra p bp return bp else denominator else j j bit good learn apply procedure learn swap good call segment sample complexity learnable preference surprisingly behave h ax bundle amount show price budget enough preference bp like ratio evaluate get optimal give budget show price learn reveal preference suppose learnable query acknowledgment grant fa grant microsoft fellowship award review establish recent part every compression compression agnostic outline compression consist subsequence h class admit compression learnable agnostic chapter learnable sample complexity satisfy learnable agnostic follow hypothesis low hypothesis price bundle bundle good utility pp see different utility two stand segment utility segment length priori utility reveal preference length candidate note segment accord order bundle segment fully bundle admissible bundle bundle demand mapping l follow immediately know segment learnable efficiently preference argue computational remark admissible bundle admissible bundle bundle admissible bundle optimal bundle h bundle sort segment section ensure segment thus admissible optimal bundle bundle allocation last good bundle good segment like decrease allocation finally good allocate second bundle segment class demand yield utility vector utility learnable preference sample class reveal preference statistical observe bundle relevant standard distribution h learner output value learnable multi implicit learnable sketch note point w value exactly set subspace dimension collection correspond vc every consistent suffice new system utility length linear segment create example xx xx learn predict employ linear xx define class learnable recall ax equality least utility going maintain algorithm xx utility define prove successful claim characterize example bx minimize minimize side bx minimize side define ax get algorithm successful utility function contain utility b since return requirement x sample ax probability algorithm output learn efficiently query set bundle output utility query output function since show suffice learn h ax query enough decompose constitute segment length segment slope segment easy give learn segment last thus check segment else else maintain inequality correctness separately learn total learn query x function function every kx nu proposition theorem claim recent line start learn reveal past price produce agent line sample number class draw connection advance tight solve numerous generalization ability preference utility common assumption economic choose bundle decrease classical reveal preference economic begin seminal traditionally explanatory generate finitely many seminal work algorithmic construction linear monotone note agree imply necessarily recent line explicit formal agent constraint produce hypothesis forecast future agent utility monotonicity concavity probably approximately utility importance focus class concave commonly sample complexity include linear separable significantly expand establishe connection reveal preference learn advance intrinsic compression yield believe variety game context establish connection e recent computationally tight guarantee utility price reveal preference setting improve concerning actually reveal structure much specifically independent linear class powerful generalization immediately necessary important reveal preference theoretic term agnostic set target fraction accommodate include readily preference power instance exploit optimal kkt order ability desire exploratory purpose able predict price point analyze utility reveal summarize reveal preference rp omit previously c rp value set set price price say intrinsic th amount bundle price compute decrease utility budget bundle utility preference bundle let bundle follow problem assume optimal optimal break utility define vector function class utility type utility function paper price utility learning assumption simplest reveal query h learning response oracle reveal preference utility due certain predictor learn term linear class scheme roughly produce thereby section class preference cast hypothesis class w yx x element tie handle tie remove distributional support machine xx www compute reason exponentially constraint violate scan svm h complexity I w learn h complexity return w w find well hyperplane maximal maximize quantity maximize unit hyperplane q since utility cast linear throughout section generating tie bundle respect utility generating reveal sample show class imply demand learnable reveal preference sample linear bundle always bundle also essentially price bundle decrease call
form available inverse resort sampling offer alternate methodology posterior accelerate gain address problem discretization scalable main contribution dimensional formulation aim minimize expected structure pde infer coefficient pde elaborate sensor pde expression assess computational objective evaluation optimal demonstrate scalability adjoint pde dimension seek average trace prior expression covariance independent problem form available covariance formally would problem cope data trace covariance possible expression computation sampling expensive applicability posteriori functional minimizer map posterior map approximated linearization approximate specify pde determine map describe objective trace operator operator address bayesian datum collect absence sensor place combinatorial problem assumption weight use inexact cg vector gradient objective efficiently adjoint derive lagrangian formalism elaborate infer coefficient pde interpret problem pressure log minimize comprehensive optimal design assess quality bayesian inverse design design show design design sensor adjoint solve numerically scalability flow setup th material solution dimensional hilbert borel borel operator adjoint trace say dd paper field measurable pointwise variance satisfie trace proportional average variance formulation infinite law model prior strictly self adjoint trace differential operator laplacian dimension trace endowed element dimensional denote likelihood probability experimental require forward typically pde follow q thus bayesian posterior describe parameter condition relationship bayes formula side measure observable finite pdf maximize extend ball finite dimensional minimizing follow argument point deterministic description depend experimental challenge describe experimental sensor measure assign negative location w determine location inverse candidate sensor regard dependent uncorrelated diagonal n likelihood statistic classical formulation probability candidate e sensor weight large experimental repeat place place sensor place optimization solve relax enforce binary weight penalty trace trace trace monte accurate trace covariance description estimator use randomize trace implicitly operator possibility estimator vector possibility gaussian random standard entry computation trace operator justify infinite analog operator condition adjoint trace eq moreover appendix carlo q take problem map additive depend denote covariance measure minimizing denote gaussian low observable map evaluate pde control sparsity various option possibility use strategy present optimal design nonlinear problem tractable approximation likelihood follow minimize infer vector follow experimental still experimental experimental general distribute inverse parameter describe average observable additive namely inverse nonlinear close posterior technique markov variance statistically observable computationally extremely observable map consider gaussian approximation design realization hessian newton explicitly implicitly function gaussian approximation experimental admissible design trace integration infinite integration replace q moderate later draw model enter hessian incorporate physical mode insensitive highly indirect dependence objective enter directly ik map hessian operator I trace orthogonal trace see tractable design approximation formulation problem corresponding penalty assumption differentiable elaborate experiment inference surely bound ensure space weak read subsection variable inverse whose formulate characterize hessian section optimization derive adjoint equation evaluate forward pde solve weight cost eq require minimization pde constrain minimization variational approach optimality condition functional multiplier emphasize formal lagrange multipli minimizer variable vanish weak weak form adjoint equation side equation solve q satisfied practice require coefficient field problem field note pde characterize pde note order order hessian application leave pde pde constraint pde describe hessian evaluating satisfy solve problem efficient follow adjoint variable pde derivation rather defer simply hadamard hadamard obtain right side hand side operator identification adjoint coincide describe respect exploit computation problem implementation easily perform computation ht trace compute ik ik ik discussion qualitative evidence term forward adjoint incremental agnostic forward pde solver employ solver pde pde solves dominate algebra negligible scalability like pde solve sensor pde scalability dimension identify hessian hessian solve covariance represent hessian inverse rank parameter dimension independence observable map denote grows initially contain insensitive mesh refinement solve objective inexact inner conjugate iteration cg mesh invariance cg operator compact cg turn forward pde pde newton cg newton approximately solve cg system computational measure pde evaluate mesh compute pde solve pde solve cost evaluate like pde observe side hessian application hessian pde argument quasi newton make newton problem dimension briefly comment enforce use problem solve cope convexity function guess subsequently penalty proceed precise function attain section section refer pressure u flow drive pressure bottom figure truth velocity th pressure state estimate compute prior square point allow covariance three correspond finish inverse available sample datum trace configuration design configuration solve deviation prior field study effectiveness design respect sensor conclusion design sensor randomly design indicate sensor entirely inverse noise square cc height axis e ylabel style north east red mark mark tr cloud txt color pt scale axis legend style font nod legend pos north east mark cloud txt mark marks table tr opt design red dot design blue dot figure respect effective try recover underlie truth address conduct effectiveness design sample get fm average see problem assess base design indicate compute expect point scale axis xlabel ylabel legend font pos east marks mark pt txt color mark size tr opt txt height xlabel node legend color mark pt tr color mark mark pt tr txt expect dot blue panel examine method location increase specifically solve adjoint pde building method discussion cost inner inexact newton cg cost report total cg outer numerical insensitive candidate interior newton problem dimension see insensitive sensor axis legend style north white mark mark size inner txt cs black mark x scalability txt width legend style north blue mark pt scalability txt color black mark size outer scalability txt width height axis xlabel legend right legend pos east mark scalability txt height axis xlabel legend style font pos north east color mark mark thick table ns scalability realistic comparative description physical field slice three four production corner domain production pressure corner impose boundary circle homogeneous condition remainder boundary field velocity pressure obtain solve ht b denote choose black center pressure obtain solve equation construction problem assume point corner production well boundary compute regularize tb b inverse discretize triangular finite freedom grid candidate location compute draw give estimator six method residual reach maximum bayesian optimal sensor truth triangular fine mesh degree freedom record pressure sensor datum subsequently use solve problem sensor assess effectiveness sensor use randomly design note width height scale axis ylabel font legend north mark mark mark txt color mark mark mark pt cloud opt txt scalable design nonlinear problem scalable measure pde additional forward represent hessian outer determine sensor optimally pde sense pde derive adjoint enable require expression hessian require medium compute experimental measure pde insensitive limitation define field case map accurate bayesian inverse expensive map challenge fact inverse inner limitation indirect sensor configuration problem appropriate pay otherwise problem tractable requires pde discuss characterize hessian operator solve suggest rank approximation discuss contain important grain solve additional iteration reason sample goal
chain point early problematic imply mdp iterate average mix convergence center exponential approximation efficient state expand iterate td available td asymptotically average td step dependency least temporal alternative classic concentration quantify bound without mix transition mdp mdp action function discount denote instantaneous reward action bellman td standard provably curse associate high dimensional linear approximate incorporate td incorporate markov moreover stationary markov feature column matrix eigenvalue satisfy assume nature make td assumption counterpart see upper convergence however upon problematic choice knowledge would imply state matrix latter draw white minimum em fill circle node thin auto align fill red right align right update gray blue center leave center block fill center align line join large combine average iterate stochastic convergence without constraint size exponent arbitrarily although constant remain minor choice would bound thus average dependency convergence td exhibit optimal iterate td algorithm follow fix iteration solution behind increment order larger iterate previous increment x f td finite sum discount make policy centering use td stationary mixing term affect finally fix sgd center difficulty overall start epoch epoch epoch update q mm td conjunction iterate exponential let epoch ex particular policy follow mixing hold mdps mix exponentially fast second mixing dominate first rhs reflect know dependency would choose split involve iteration first state depend fact td arise provide proof f rs x bound require complicate td project reader refer order present expectation q inequality bounding theorem mean martingale step establish lipschitz constant detailed show find bs integral td evaluation asymptotic convergence derive linear function bound high optimal choice td fast variant td incorporate center rate fill rectangle height em cm black ns f rule martingale policy plugging mix jensen eq equality deduce martingale algorithm rewrite sum sigma lemma ingredient invoke rs ns ng ii constant return instant equality two expectation q consequently bs schwarz conclude martingale function bound lipschitz eq obtain note regime choice bs c bs bs comparison integral bs q equality page detail last take give nm center solution epoch proceed rewrite sum epoch note I recursion obtain prove pt pt pt provide asymptotic know probability optimal knowledge underlie problem employ scheme convergence knowledge furthermore center establish process mdp rl solve mechanism discount hope function approximately constitute e actor temporal td evaluation sample simulate td representation entry state curse trick linearly parameter efficient td even rule td incorporate approximation
outlier occur overcome robustness enhance nearest knn subsampling cloud preserve topological outlier denote near average distance neighbor density cloud extent filter cloud persistence tb subsampling cloud take sequential pre sample previously select scalar persistence diagram confirm persistent subsampling another cloud colored environment selection persistence interval distance rough signal let sort decrease power n l snr cloud average use measure snr great snr summarize however cloud construct homology homology birth death persistence interval extraction persistence length scale invariant th deal indicator space topological optimization one interval space persistence diagram correspond length environment result surprising quantile density outlier classifier variety contain either investigate performance use negative scale environment environment whole justify degradation third retrieve point cloud metric topological environment mobile sensor whose enhanced piece information cope uncertainty cloud extraction persistence diagram system quantification uncertainty also work extract feature cloud exploit precise email paper inspire network motion mobile node inspire motion utilize extract weak build environment spatial feature manifold environment extract dominant topological persistence interval topological improve robustness outlier density base subsample employ diagram provide representation strategy agent enhance sensor network broad mapping attract lot decade mobile flexibility adapt environment model biological agent equip mobile sense formation behavioral distribute distribute requirement response make capability rough provide localization would fail computational extract require make localization topological persistent homology qualitative cloud take compact topological persistence diagram specific topological coverage stationary sensor neighborhood mostly static physical node network complexity look investigate employ homology topological mobile mobile hoc topology physical model topological map mobile sensor network sense movement expert localization retrieve status proximity point topological base subsampling use point topology extract persistence low dimensional structure cloud machine integrate persistence inference study classification persistence use construction organize follow present section inspire sense overview propose framework metric subsample feature finally discuss brief introduction present tb persistent homology way object connect space representation rank topological number cycle component sample represent cloud finite equip method topological cloud build ball vertex base pairwise complex persistent homology compute value class topological death persistence call persistence persistent homology represent stop stop mode short stop long stop characterize time model sense inspire sense capability combine wireless receiver body well boundary within radius equip unique agent occurrence furthermore able status rw state summarize exploration environment probabilistic motion describe sense power environment purpose environment free instead deal directly information base process build construct motion circular region extract cloud computationally expensive cloud possess topological persistent homology ordinary interval computation persistent homology use robust extraction cloud visualization purpose scale projection cloud tb assign network coordinate represent tuple construct undirected vertex exist encounter time limitation available base assign rough connect tb due agent stop detection redundant place beginning proximity occur inside agent exploration cloud color resemble corresponding cloud justify due accumulation uncertainty long period make environment could figure produce collect help estimation make worst short capture correct impose
symbol calculation ignore coordinate central word assumption expression ease exposition even independent moment relation calculation mean u independent rewrite eq calculate integral definition distribution integral taylor expansion final definition kind let term section term consider term appendix second section variance derive bind calculate abuse notation summation split different calculation expand term expansion use calculation obtain bind fu du fu du believe mistake notation paragraph power assumption
proof since get q showing hand schwarz inequality consequently tensor link q q triangular duality line combine immediately lead consequence let q adapt argument sharp deviation event define eq contraction contraction principle denote duality plug partial derivative sequence eq similar argument get mn md n term negligible finite nt matrix probability distinguish two use see therefore function dependence implicit packing construction inspired consider test first matrix replicate block matrix contain matrix construct pack satisfie e distinct kullback leibler either function get universal take value estimate entry work completion recover unknown real rank investigate take output generate recommender multi classification guarantee nuclear maximum advantage knowledge nuclear unknown minimax claim class arise application recover observation entry course proportion entry set framework amount least program trace value entry highly unknown rank voting preference survey yes agree opinion much completion observation constrain sampling consider random unfortunately recommender popular rate frequently important application yield fast rate obtain bit completion likelihood uniform unknown slow recently exponential family knowledge unknown nuclear penalization allow consider scheme unknown previous difficult matrix paper completion introduce establish bound minimax logarithmic factor bit extended alphabet coordinate recently introduce experiment entry order way tensor function simplex hellinger vector leibl parametrize coefficient reveal denote ease write instead likelihood regularization exist positive classical row constant q yield kullback leibler divergence universal constant logarithmic capture differ general set parameterize observation vector negative likelihood observation section function factorize binomial multinomial function ne k ny eq kullback divergence universal define give observation q without confirm negligible burden separable achieve coordinate describe support except equip value role triangular imply eq therefore equivalent implement iteration nonnegative support singular max soft computation step loop iterative proximal associate nuclear soft thresholding singular propose another interest evaluate entry actually completion iteration top singular value us advantage loop carry bfgs execution bit ram cache completion extend class consider comparison potential gain belong alphabet report article assessment obtain upon author
recent attempt eliminate variety approach level translate derive term relate language eliminate mt mt method corpora require aligned sentence extract simplify learn representation pair bag informative bag encoder supervise nlp label able reach achieve report bag sentence fix within learn word sentence present nonlinearity representation sum bag word encoder function decoder autoencoder bag encoder decoder must careful certain section reconstruction convert bag representation obtain encoder word linearity sigmoid tangent sum representation least decoder form bias reconstruction sigmoid training reconstruction training mini batch binary correspond vocabulary typically aim reconstruct since million training thus trick bag assume perform mini bag word bag mini batch result descent mini batch still produce reduce stochastic reconstruct bag efficiency training reconstruct whole dimensional previous work binary input bag autoencoder architecture investigate architecture directly firstly encoder frequency notice nonlinearity bag validate moreover assume output bag trial multinomial efficiently numerator normalize sum opt leaf treat internal root root right branch node observe bag decoder tree decoder compute logarithmic course bad decoder assignment finally need parametrized decoder linear branch logistic encoder bag language sentence language align translate similar representation sentence reconstruction language specifically language specific language word word bag w w similarly autoencoder encourage language language decoder language decoder reconstruct form decoder language specifically reconstruct loss follow propose embedding learn correlated term specifically optimize correlation encoder factor ensure either bag reconstruction autoencoder le translation horizontal line across input hide highlight parameter similarly language word tf obtain document describe representation word train simultaneously encourage aligned embedding use form neural network language work investigate rely word learn phrase mention separately skip language align network phrase phrase base phrase level embedding capture cross word follow follow interested corpus importantly corpus learn language successfully language language embedding corpus contain unlike pre sentence align language interaction induce embedding english corpora document pre hierarchy economic market contrast learn interaction learn embedding corpus en de autoencoder embedding reconstruct effect procedure language experiment topic document topic document process use en de de pr pr pr believe office wish report shall month year agree microsoft en en de de microsoft microsoft markets competition competitive exchange business material procedure summarize follow train extract language validate train document language document language represent document averaged perceptron train epoch epoch contain word tf combination early cr error training use reconstruction unlike cr section perform representation epoch use word cr cr merge mini batch adjacent sentence hyperparameter select validation portion discuss like qualitative learn perform english word english word term embedding cr show english word actually also notice semantic word show supplementary visualization embedding language describe compare learn neural encourage aligned embedding mt test document translate mt mt default parameter induce embedding every class summarize result en de en observe tr comparable embedding language rely correlation indeed cr de classification english en en tr cr cr al majority evaluate effect vary amount training classifier either tr en en summarize observe cr remarkably meaningful embedding even size excellent merge mini single significantly word rely essential effect use batch cr cross surprisingly tr decrease use mini batch size even en batches de en cr english
find match summation procedure exhibit bad without generality w unique probably stay dimension dimension unseen kx I use assumption bind depend time oracle combine euclidean constraint minimize n inequality classical nesterov fy fy l x fy x variants b eq fw nf w fw fw simplify lyapunov straight note also step standard q apply n give q nf change w I f nf k nf nf nf nf I simplify change equation product simplify w sn w f sn grouping remain term nf sn f expectation product sn w expectation straight forward n sn w I f w w f w expand term w sn f notice inner product recall simplify expectation sn iw w inner define constant convexity eq sx l l fy fx fy l f note fy f holding
comparison excellent benchmark reinforcement ensemble appendix besides split thus value respective medical drug schedule drug pi despite success drug maintaining associate long attract optimize drug lot structured patient alternate cycle successful regard protocol schedule sequential decision action correspond type simplify formulation pi amount reduce pi load little possible follow describe identify validate clinical variable describe patient problem generate also follow author briefly base approximation greedy merge select action pick experiment varied round simply representative state begin cost kernel drug schedule large appendix state state reason transition simply figure drug schedule show decrease contrast solution case schedule correspond unable reproduce confidence interval substantial associate illustration run require fact transition tree curse hand hand increase gap time complexity average verify implement independently single agent determine representative uniformly random state benefit reduce cost agent vote tie break seem boost show agent confidence fast conclude mention experience confirm stable method solution hand solve transition line line problem involve sample computational prohibitive demand empirical evaluation use structure frequency effectively experiment fix equally effective across system treatment potentially replace regime scheme reinforcement problem generative develop base collect reproduce original dynamical validate policy brain slice associate linear apply electrical total transition result clinical policy apply electrical frequency hz hz hz run sparse modification make representative state time reasonable use varied overall characterization value vary penalty associate electrical fix latter name seem expense occurrence solution policy hz regime know date clinical able time method draw one characteristic apart complexity transition ever access entire sample modification additional reinforcement determine transition set undesirable memory domain impractical incorporate ignore usage inefficient limitation action split improve clarity suppose transition index eq q want know simplify write reasoning derive know update discard transition store vector overhead recursively far subset fully incremental computing step transition drop transition update ij w I instead policy learn transition thus integrate planning algorithm cycle sample transition algorithm counterpart incremental version recover batch transition approximate matrix state execute state update select base ia allow inclusion representative state representative state suffice application update dynamic inclusion state refine think way strategy proposition impose allow sample near representative add experiment incremental version build require tuple keep extend transition reach time value see note current may apply address issue bind computed transition process use add stop algorithm show value space action space discount factor similar apply application base show think matrix iteration solid theoretical state use let encounter q value triangle q value ts slight abuse mdp hand see apply write ts contain incorporate ts ti ti resort step fact action factorization implicitly possibility value either completion apply bellman short restriction somewhat circular look empirical task match store exploit scalability task balance performance reinforcement use model batch task action show result result policy gradually go sample intensity process important policy transition confirm demand memory small size approximation processed show overhead strategy normalize v transition collect see discuss balancing address simultaneously challenging figure balance scalability bar perform adopt control start triple balancing exactly refine incorporate sample transition grow representative state add agent set close representative balance adopt section fix benefit incremental algorithm sample transition problem always subset option limited amount batch transition approximation greedy strategy compute compute show triple balance task amount insufficient describe control transition batch note amount complexity allow large use transition take hour time detail allow success cf triple balance task especially policy direction approximately memory argue fair latter amount former batch exactly process computational cost train phase look opposite trend surprisingly performance computing grow process beginning grow fast visit datum magnitude correspond page cr one fix neighborhood experiment compute close representative guarantee adjust advantage adjust cf experiment compare quality policy broad reinforcement huge body literature broad overview book narrow attention start smooth kernel essentially implement approximation kernel implicitly inner two framework reproducing relate discuss smoothing reproduce roughly rewrite term product properly kernel mapping apply reinforcement value approximation alternative mdp slightly propose applicable weak reproduce attention technique closely function see similarity consistent assume approximate assume show function surprisingly form mdp transition contrast transition numerical demand see build method small bellman transition applicable reinforcement kernel operation grow adapt start exploration completion may computationally feasible resort propagate exploit use whether exploration overcome difficulty underlie link transition use kernel limit case transition subsequent ignore work later guide scalability first simple potentially reduce computational burden user suggest representative state among sample infeasible problem resembles define mdp representative algorithm come hard aggregation row rewrite formalism element infinitely narrow close representative easy aggregation practically place would sample row compute coincide cf property instead factorization trick contain build strength mechanic incremental make transition code line sound theoretical value compute underlie show arbitrarily small desire translate build difficulty arise practice successfully different present reinforcement list believe become valuable resource reinforcement possibility future algorithmic demand principled method procedure solely far think kernel advance elaborate exploration regard integration broad investigation principle amenable algorithm complexity sample see understand however equally ask whether benefit view sample need achieve grow exponentially dimension way avoid exponential dependency sort regularity break curse incorporate think may cast incorporate whether impact sample interesting question investigation one reinforcement build model sample resort potentially useful reinforcement x b satisfie exponentially assume great ensure set sample transition experiment obvious property let w follow suffice z otherwise provide intuition magnitude impose region make possible adjust exploit allow accord factor difference understand small threshold considerably even difference ensure sufficiently small second influence size term plug let strategy q know w k necessarily yield obviously inequality regardless though analyze show sufficient multiplying define apply let true remain show recall eq inequality otherwise rewrite resort guarantee hold guarantee make finite mdps ia column ij p ij ij element analogously resort obtain lemma favor course bind trick property first less depend contraction mapping draw derive theoretical generalize mdps bellman think resort conclude desire contraction map derive upper fix point bind valid think notice bind derive theorem operator vanish trick reduce assumption order whole behind create zero nonzero element small mdp suppose define equality long infinity world world model discount transition reward result near agent reach goal grid compose balance simulator thesis use task angle vertical plane episode reward fall past cart reach track locate step comprise equally correspond hypercube origin cover axis velocity cart angular balancing simulation use adopt version length mass policy equally drug schedule system ordinary euler action discount parameter numerical suggestion existence monitor drug day sample initial day later reward drug select patient report policy value sample day correspond describe dynamic generative develop task label five slice hz hz hz hz manifold turn give rise therefore reward event model policy episode start simulator simulator adjust parameter accurate normally keep agent position dimension break try several pick result task episode start position kernel transition largest large find near outside specialized avoid store list algorithm modify policy compute approximate table across action decomposition ensemble split general increase task consider cut particularly effect experiment varied adopt rl algorithm evenly trial preliminary fix define technical report school university discussion regard relate subject make simulator national da discovery mm mm mm b c j z bt bt bt z z reinforcement conference manuscript substantial approximate learn theoretical statistically construct grow paper turn reinforcement idea transition product swap factor potentially much insight build difficulty transition discard incremental make result reinforcement regime compute resource would set potential difficult world significantly reinforcement reinforcement learn conceptual long artificial intelligence construction interaction among particularly persistent obstacle recognize real must realization come across reinforcement incorporation difficulty last two decade collective rise reliable stand mean add always eventually unfortunately good theoretical property since transition policy become prohibitive burden limit applicability nice present algorithm transition stochastic swap factor obtain potentially small fundamental exploit insight contain word whose construct become structure extra flexibility approximation take account find construct transition memory number property make regime study real never also sound theoretical view bound compute compute also appear bound solution present factorization trick divide theoretical one control bring reinforcement single double balance drug incremental follow extend line triple balancing discuss present guide reinforcement summarize related context conclusion research possibility try maximize reward interaction environment discrete state must choose finite set action move select certain reward agent policy maximize return discount reward future mdp mdp tuple task hand action transition mdp search resort theory dynamic agent policy e notion programming perform worse know policy mdp one action represent become throughout conventional letter capital letter dynamic ia fundamental programming expression give dynamic reinforcement mdp transition environment transition reinforcement learning use finite solve continuous reader think also kernel finite solely give occur occurrence state finite mdp reward eq dynamic mdp width admissible suboptimal discuss dynamic compute transition state bellman computational want much dynamic allow follow section importance objective serve rest factorization mathematical explore briefly slightly modify useful element artificial artificial direction element transition word accumulate artificial similar switching probability compact version idea cccc represent big white circle symbol use immediately surprising fundamental characteristic recurrent irreducible irreducible regular regular insight stochastic factorization transition factorization property strong idea former motivation save resource possible trick reduce summarize idea programming transition matrix obtain mdp solve scenario convenient mathematically obviously apply stochastic factorization trick unfortunately computationally demand approximation mdp stochastic mdp norm ij upper theorem mapping identity write hand say development tight bind factor mdp version classical state deterministic reduce sense factorization trick rise basic mechanism trick number mdp exploit function adopt show trick reinforcement construction main factorization trick mdp long computational impose calculation leverage component trick similarly kernel list assumption suffice assumption mutually build ccccc ccccc b ccccc ccccc b b b ccccc ccccc ccccc b cc b cc b b matrix transition occurrence apply mdp strategy note easily illustration conclude construction expression mdp depend state recall interpretation representative illustrate formal mdp discuss transition define state dynamic degenerate case change change define representative particular rise conversely transition mdps depend representative state understand implement work value dynamic return compute simply input lr return key mechanic require bit version construct cost become instead application bellman complexitie linear computational requirement kernel kernel region avoid store result compute occur become reasoning look thing transition representative computation action transition occur formulation practical would generative transition provide interpretation construct action stochastic continuous focus define knowledge function computation training kernel weight compute
inverse covariance estimation receive community use especially dimensional arguably penalize matrix covariance penalize advance effort literature focus develop principled increasingly simply literature review seminal work recent paper conference maximization restriction address gap pseudo establish sense move practical recent minimize cyclic descent update hold hold wise algorithm converge minima equivalent much ten address important propose rigorous associate method lead massive extremely fast principled solver outside notation diagonal term propose scalable thorough advance optimization derive proximal efficacy model fista treatment investigate popular minimization iterative gain popularity seminal backward splitting nesterov thresholding essence proximal divide objective part nesterov accelerate extension combination momentum accelerate section composite part vector zeros k matrix matrix initialize j j wise soft entry algorithm accelerate constant iteration fista choose search reduce iterate section implement three feasible iteration heuristic proxy information gradient fista multiplication zero extreme use w ij os operation complete operation inversion optimize allow parallelization contrast fista perfectly multiplication dense machine high restrict like provide proof coordinate algorithm argument convergence essential belong base function state bind hence bound start semidefinite matrix simplify arithmetic ignore constant theorem depend ki eq go hence function positivity trace function value element continuity remain sequence generate backtrack k solution backtrack set backtrack fista iteration outline comprehensive numerical give comparison wide result breast make wise implementation algorithms library fista implementation increasingly large scalable library make work eigen library algebra c name various step dataset non edge sample size guess criterion wise implementation highlight comparison supplementary material synthetic two little difference marginally fista two coordinate perform attribute fista fraction propose method fast coordinate time coordinate behavior fista perform axis converge constant fast appear perform fast fista fista rr nz second rr nz c sec sec dataset arise physical science gaussian outlier hence characteristic assess breast et univariate cox patient survival gene breast reduce gene often algorithm wise especially due newly fast fast gaussian graphical estimation advance propose gaussian inverse rate fista thus far compare fista coordinate demonstrate outperform coordinate wise general outperform wise magnitude test set comprehensive examine effort similar appear one several thorough contain relative rgb rr rr second second rr rr second rr rr c c form kkt involve term definition rewrite
em vb miss result conclude method implicit perform outperform miss auc auc auc vb comment vb em vb em em vb set cp miss average run instead pick model negative consist start view global optimal solution factorization advantageous iterate find recently co factorization perspective naturally conditional approach approach lead improved department engineering ci ci university approach aim structure constraint successfully paper bayesian couple even exhibit several world miss factorization modelling advance propose lee one factorization meaningful modelling field include recommender bioinformatic suitable variational bayes vb probabilistic appear incorporate arbitrary rule give individual factor mm indice product collapse index factorization one step multiple suppose distinct index specify attribute variant tensor relational heterogeneous datum large class tensor contribution variational exact characterization conditional richer naive formulate entirely term low tensor predict entity try kullback divergence since analytical intractable bayesian acyclic dependency write px pz leibler symbol gamma factor preserve conjugacy framework degenerate distribution px x miss define mask observe missing handle smoothly follow notation short tensor value refer particular element generative fix update via em convenience representation force sparse representation write mm stand compact kl method em compare vb quantity average q integral several deterministic approximation approximation method introduce attain maximum log set induce approximate become easy approximate hide intractable resort propagation intensity rather formulate posterior distribution mm q variational log indeed regard different computation mm vb resemble large couple observation available couple tensor side sharing method incorporate knowledge additional tensor set configuration distribution couple vb sufficient expand log drop irrelevant z miss link factorization link performance evaluate tensor vb low rank tensor compare vb miss use implement variational equation vb vb scale extract include type entity activity equal user equals collect preference gps trajectory datum location respectively aim link side link contain link may activity number link collect show social news resource user comment news lin action comment explicit contact extract five user topic among illustration study comment r r user tensor relation compare comment comment user comment comment ex ai ai dm k ai dm en tensor array couple update mask sparse storage instead work enable specialized storage cost roughly scalability vb vb tensor sparse value array miss cp cp kl extract reconstruct solve times ten rmse second slow mm mm vb
svm f margin step use svm package include solver follow divide thereby example slack objective package additionally recent reformulate form example elaborate transfer interpretation transfer aim different analogy slack margin classify hyperplane I back sample slack slack svm effort svm effort svm mean ignore hard concentrate satisfy question answer oracle answer first second margin constraint satisfy ignore one return margin transfer low vice lie oracle negative regression slack come model slack function validate space margin transfer explicitly predictor sample hard classify class ignore pair sample amount every consider three different showing handle attribute bound present modality subsection analyze propose margin transfer compare ordinary svm access accuracy report joint choosing cross validate range normalize range fold fold validation multiclass complete training thorough couple modality exploit annotation incorporate description shape concept introduce attribute classified default attribute provide dataset class attribute texture attribute predict binary classifier image statistic time transfer image image cat versus versus versus cat versus versus cat versus versus versus versus cat cat cat cat cat cat versus versus versus versus versus versus total svm highlight blue significant confidence additionally figure utilize object svm transfer information svm able information bar coincide mostly margin transfer high regime problem check consistently high original translate transfer margin hardness modality fulfil box annotation design object image object level know exact annotation available ball activity class image ignore box image group ball ignore uninformative annotation one dimensional bound representation latter accordingly draw amount sample remain similar double statistic repeat pt image green green highlight significant utilize provide margin ball ball highlight blue indicate improvement utilize information transfer pair svm column see utilize bound box grain outperform baseline svm case experiment method exploit margin transfer group margin transform respect standard bad margin rely space ability hard scenario hand complementary turn word image sample symbol pair normalize extract image bag split second introduce variety product group broad binary per contain text description text advance word vector instead term extract neural skip gram codebook word convert sentence normalization normalize descriptor visual codebook c transfer reference text versus versus c c transfer text text versus bag versus bag tie bag tie tie versus utilize equal utilize performance transfer description capture mainly preserve explore utilize setup versus strategy binary sample sample class maximum classifier model selection cross classifier cross validate performance good task l transfer attribute ball additionally reference method outperform dataset transfer svm contrary dataset tendency outperform column reference column collect annotation easy hard reliable rather make objective lack variability distinguish easy annotation study set particular ask select prominent proceed obvious difficult compute score range observe object size human annotation look easy proceed transfer pair hard human identify correlation learn correlation sample hyperplane easy hard score similarly predict train space space complete user easy score hard original entry coefficient c transfer transfer human versus versus cat versus versus versus cat versus versus cat versus versus cat cat cat versus table collect annotation space versus see figure datum space human space closely explain human attribute classification versus strong annotation utilize information suitable explore attribute description compare annotation little blue coincide rather comparison attribute information like performance classifier lot misclassifie wrong hyperplane margin principle human study set handle improve utilize multiclass two approach margin transfer transfer model hardness guide essence annotate study future explore direction european european framework technology introduce learn computer computer recognize want computer fast expense additional image look scenario study vision able binary multiclass interpret hardness object train thorough analysis space incorporate information user study hard learn inspire teacher teacher new teacher explain answer solve teacher generalize machine introduce successfully apply attribute annotation digit description gender resolution source metric back help want category label object plus direct yet representation annotation image useful quality extend publication examine different information semantic specify localization image context modality understand test evaluate core also easy enable define identify sample easy incorporate encode hardness formalize technique svm new contribution transfer analyze predictor report additional handle situation unified contribution section method naturally conduct experiment object utilize method attempt useful problem common access sometimes visual represent feature texture modality assumption extract accordingly space classifier would datum possible vice describe characterize sample space case know towards generalization quality margin interpretation simplicity notation linearly separable support turn linearly separable hard svm call soft fully slack example increase margin svm sharp soft hard answer
distance stem taylor expansion note bregman triangle inequality square dissimilarity divergence risk error choose stress mapping bregman divergence deviation latent usual smoothing spline analysis spline endow situation inner radial metric spline non description manifold due datum hz prominent target ica prominent noise raw remove subspace method follow system perspective scalar observation latent evolve observational dynamical embed extract channel source describe describe use filter outlier highlight note observe point reflect signature original normality hence highlight operator euclidean apparent interesting use dissimilarity segment spectrum channel temporal ica processing remove obvious subspace dissimilarity prototype determine select select dissimilarity psd channel point locate channel locate axis channel present range small dissimilarity target present simulate take dissimilarity view potential improve provide augment automate system normal investigation dissimilarity approach interesting measure distance situation augment pt corollary purpose environment response requirement decision reduce measure water prototype exploit dissimilarity representation mapping system analyse map euclidean non make concept use realistic target sound extensively fundamental commonly ratio surface high produce hundred thousand worth display result conventional observation observation result unknown preserve structure original preserve projection capable mapping subject represent accommodate content ie integrate anomalous behaviour track measure dissimilarity assume isolated entity nonlinear metric represent need even dimensionality reduction seek purpose transformation dissimilarity preserve employ nonlinear network parameter adjust classical scaling method transformation often x prior dimensionality often useful f nonlinear radial weight distance euclidean may
formulation characterization symmetric handle minimum cut undirected half cut denote represent pairwise respectively characterize undirected graph characterize iff label equal constant characterized fig characterize undirected correctness due optimize v x u graph contain indicator represent variable cut otherwise characterize undirected conclusion introduce next fig general unary general besides denote node undirected convert symmetric represent capacity construct characterization minimize transform minimum know minimize go introduce important equivalent transformation study literature convert submodular instance x x unfortunately benefit energy kind characterization switch denote conduct indicator graph characterize exactly indicator set indicator second type mean characterization indicator capacity capacity change uv uv uv uv flip original algorithm existence design minimize submodular function extend general undirected provide convenient sum part undirected check positivity sup optimality efficacy possible minimize influence transformation fig equal dense efficiency equally paper I derive absolute maintain record ratio negative order flip accordingly repeat undirected iteration decrease exhaustive search influence part submodular test guarantee globally could produce labeling well propose greedy call equivalently replace contain propose iterative refinement initial labeling experiment consist modular permutation modular modular function part modular result st computation iteratively refine permutation lead minima avoid permutation situation graph label modular approximation generate large need simplify energy refine variable initial approximate st graph minimization labeling iteratively generate labeling convergence optimum besides automatic submodular enable global optimum except iteration undirected formulation vertice characterization experiment within small fed initialization code publicly due energy study computer vision potential necessarily energie primarily hardness factor hardness combination technique energy hardness future energy use energy factor systematically performance energy large energy efficacy synthetic energy rule purely focus evaluate optimization energy form randomly produce empirically hardness variable node uv uv respectively curve combination minimize variable random initialization initialize bp unlabele lp bp energy bp clearly method bp fast appeal bp attribute unary fig solver configuration maximal iteration solver make I high energy note thus obtain energy minimum comparative unary experiment thorough cover connectivity unary test plot different value vs curve six connectivity unary energy understand computer vision study energy see speed outperform initialization worth bp usually dense energy variable label tell always low much test configuration certainly label worth use label seek labeling preserve cc connectivity c weak unary equally initialization unary usually obtain unary reflect unary pairwise potential computer unary cause unary pairwise potential hardness chinese character specifically image train prior character uv uv x vx pixel labeling either prior average fitting minimize u fig show result bad energy find lp similar thing able energy long time initialization local initialization contrast much slow speed comparative help hardness future direction importantly promise perform energy summarize generally efficiently correspond properly namely dense kind energy great vision enable extend submodular solver labeling thorough comparative find connectivity unary closely hardness reasonable recommendation energie vision several extension optimality still iterative desirable analyze besides combine future furthermore plan finding available energy vision besides plan function computer vision mrfs program excellent national nature foundation china national computer demonstrate dense function solve problem energy none technique cut bp individually efficient mrfs potential comparative recent
life size vi split subset subset present require hidden act bit output require bit relatively reconstruction bit country popular rbms model categorical unit poisson beta rbms categorical employ rbm attempt present ordinal sufficiently especially seminal hand recent numerical modelling variable except treatment variate model previously statistic name mix variable underlie need correlate handle hundred scale single know previous include share capture probabilistic manner time know adopt category category rank introduce mixed variate boltzmann rbms variable multiple modality six information ordinal rank capable handling variety include demonstrate large wide plan multiple relate share posterior interaction rbm architecture able inter without go intermediate plan propose comment day country thing go think health education social know opinion pose greatest great spread nuclear environmental grow old last currently separate never simplification mean field define approximate kullback leibler ik ik complexity fast continuous recognition university modern become heterogeneous modelling modality option choice interpret particular aspect rbms variable naturally task include convert input latent posterior thereby reduction multiclass tool datum completion multimodal model large scale opinion survey feature extraction completion boltzmann attract task include multivariate deep rbm markov visible represent interaction extraction nonetheless rbms visible variable gaussian extension ordinal poisson good rank type investigate modality example survey person ask range statement answer response vs categorical iii choice g iv ordinal assessment vi answer response come person inherently american typical chinese concern child education however model among layer rbm suit undirecte visible thereby introduce variety away nature dimensionality reduction reconstruct predictive learn perform large international opinion involve people section mixed variate machine jointly ease include single visible variable bias variable parameter hide category category visible type continuous singleton energie th visible unit variate markov connectivity must moment g gaussian assume energy bias parameter datum energy decomposition assignment ph ph ph extract similarly follow q subscript emphasize heterogeneous equivalently functional continuous limit gaussian reader image functional provide binary u I I category rank thus simplification describe detail assignment person interested let category subset assignment empty possible moderate assign category indicate category independent indicator indicator receive individual I review expression treat simply ignore consider convert triple proceed variable ordinal lose use specifically define free set completion give parameter application typically perform visible variable belong family ph due space constraint empirical estimate expectation resort approximate eqs carlo sampler run specifically k use hand category I multinomial speed cd mcmc observe stop introduce estimate completion treat hide ignore paper follow condition variable variable translate conditional predictive learn effectively latter typically inherently easy strategy identical except e ordinal categorical exact evaluation optimisation gradient argue preferable effort may actually third generative discriminative objective one hyper control generative component manner ph learn predictive complete miss predictive inference unseen ideally unseen simultaneously resort special predict simplified categorical ordinal type simplification assignment indirect ia I ia I c I aggregate probability pairwise I ic I sort lead optimisation variable simplify q experiment large general world opinion publish period question obtain people subset question ask certain answer ordinal difference response zero variance detail particular user type bin u ordinal mi create baseline baseline ordinal category normalise scale
front box input work assign understanding segmentation build diverse crf train particular task even though nature front segmentation exploitation score crf base sense try segmentation move lie feature label image segmentation tree recently within pixel classification propose pool inter work employ advantageous decision recently slide fashion convolutional spatial issue outline introduction inter refine coarse difference unary focus close direction mechanism node crf cut ignore dependency treat crf drive study recently show efficient segmentation manuscript make densely technical respective focus problem material similarly evaluate semantic inferior interested herein publicly available imagenet dense segmentation spatial evaluation instrumental success dense cnn implement convert connect one original enough pixel score densely variation previously employ subsample modify introduce zero increase three first efficiently sample respectively illustrate wavelet transform framework add patch dense imagenet straightforward fashion last sum term position output weight target optimize respect layer test original image illustrate map correspond simple interpolation increase resolution negligible produce coarse score output force learn increase hour report train gpu ingredient size pre imagenet network typically large field field filter become dense score address spatially fc reduce zero fc gpu dense top testing sec channel fully connect dense rough less suited pointing exact localization deep model layer successful infer work direction localization convolutional employ super essentially segmentation successful recent recognition capacity remarkably address localization challenge produce semantic significantly speed sec recent explore increase boundary localization specifically max pooling layer filter second convolutional feature main feed softmax layer adjust discuss fine resolution layer improve crf benchmark consist foreground object original contain extra annotation term pixel class crf stage unary fine network pixel decay tune validate fully default validation fine parameter matlab refine size around round crf crf x crf crf crf crf crf crf evaluations set augment crf crf boost improvement turn qualitative crf crf features add improve fully connect crf denote crf fig leverage refine employ arbitrarily view adjust several kernel crf modification set slow improved reduce variant different crf crf latter input attain change crf match performance fast run large parameter crf crf crf crf ht quantify segmentation similar annotate usually occur pixel locate narrow fig exploit intermediate segmentation connect crf significantly crf state art paper able object extend excellent file web site vs crf model crf crf outperform art crf crf speed crf attain employ cat train tv crf crf x crf work combine convolutional neural random prediction detailed map advance challenging segmentation refine integrate train fashion plan apply source depth map video weakly annotation box far powerful challenge vision acknowledgment partly cs fp fp acknowledge support anonymous comment constructive feedback present reader crf performance add crf layer crf camera field view token google google california performance classification method probabilistic address task classification semantic segmentation response sufficiently localize invariance overcome poor deep response final layer field crf system beyond previous image task reach test show careful novel community response neural choice become level year computer array fine grained categorization manner result sift partially
point put contingency count cell contingency column denote fit observation column expect cell split fit elastic although default tune minimize observation adjust search response value pure stop run logistic elastic net terminal stop contingency cell fit observation column expect count c split split variable linear logistic space split take ideally candidate minimize logistic sub dataset intensive take distinct exhaustive candidate categorical split form ordinal select sub nominal search computationally cart due manually proportion success case well fail option keep split response split subset summarize sub simple regression l cx ia iteratively apply point depth minimal complexity prune cart terminal pruning tuning parameter tree nest fold subtree sequence predict response dataset probability predict validation calculate rule denote subtree q often determine variable importance term important adopt importance forest give grow accuracy update measure measure obtain testing measure important correction probability bias present allow predictor cart select much selection bias allow split split incorrect take guide separate square exhaustive cart allow split find preference categorical serve regressor frequency contingency categorical value chi bias categorical numerical possess degree employ bootstrap calibration selection towards variable increase numerical split reduce however close problem transform outer loop create multipli make equal detail sample response split candidate draw convert value numerical categorical numerical proportion necessary value value multiply variable great split effect bootstrap independent table predictor skew study categorical exclude fit simulation sample iteration iteration logit nan jump cubic assess simple regression bar regressor regression selection model see predictor categorical chance simulation cubic correct choose selecting quadratic bias apparent multiple bias select linear node unbiased select variable select split possess fit regressor select split much frequently third experiment bootstrap logistic regression option show spaced interval selection tendency variable display multipli always three table p p jump cubic se frequently accurately prediction apply compete various fold validation setting option include correction new stand prune se simple linear tree without correction theoretically recursive partition multiple default argument multiple fold validation se pruning penalty coordinate fold validation parameter categorical linear logistic set testing take categorical serve candidate internal proportion sample outcome predict probability fit index although scale free ratio range ratio prefer matter rest adopt percentile proportion misclassifie straightforward prediction area curve receiver operate roc binary probability assign negative collect dataset uci repository order total categorical cat miss removal categorical l range max private local pay people census observation education level education st th college school education education numerical max status description c armed force op service house sale description gender c capital capital gain min capital capital loss max hour hour work week country country origin c census unbalanced sample observation stack population among group categorical find country visualize united majority observation education group school college education majority employ private self people low relationship suggest majority white white top interaction categorical visualize distribution level combination categorical education degree consistently low education suggest interaction education show population worker among worker although actually high counterpart ht scatter capital capital loss heavily education level increase rest picture seem advanced beneficial explore census classifier use predict misclassification subset naive rate tree cm l error auto cn objective merely algorithm hence analyze census naive bayes dataset employ permutation measure provide cart early cart selection bias lastly offer prediction measure rank subsection analyze census develop modify exclude good simple grow cart partial pruning list validate htb l option terminal display structure root model split age go population send branch sale service provide high school education good chance make year capital regressor capital gain split variable circle indicate ht indicate red otherwise green circle furthermore branch capital capital include close branch draw terminal capital ht independent interaction depth grow tree drop program small se short option short tree regression leave terminal however model interpretation therefore terminal list multiple split status sale service estimate age consistent node mid nonlinear green circle indicate observation red otherwise htb c intercept education capital c ranking explore census display ranking misclassification capital gain education ranking score obtain importance present htb variable capital education age variable capital hour education relationship country tree importance capital gain education term determine capital capital nonlinear age list predict test glm default setting htb se se census glm perform misclassification naive naive solely misclassification propose combine partitioning call flexible allow predictor play role build tree option logistic model fit regression elastic penalty datum control selection apply chi square split exhaustive adopt guide bootstrap bias correction prune cart determine prevent compare compete accurately census dataset miss occur deal miss complete grow miss fitting miss algorithm affect factor predictor study multinomial ordinal logistic tree regression tree decision make logistic tree response nonlinear recursively multiple descent method estimation elastic net allow structure comprise graphical provide apply adjusted chi test exhaustive bias linear square elastic net penalty tree recursive partitioning model traditional latter tool logistic provide however logistic satisfactory fit detect nonlinear pattern space tree regression logistic partition split node interpretability smooth node build tree still tree stand split rest overview regression review describe split importance discuss bias make split correction term accuracy show application census suggest build method review build brief exist logistic regression tree widely response predictor relate code categorical categorical ordinal let ik rewrite mle solution score equation eq nonlinear solve iteratively log term calculate parameter procedure fit logistic coefficient odd nonlinearity interaction select diagnostic tool second check pearson statistic logistic especially follow chi distribution hard interpret disease dataset uci repository absence heart list gender pressure dl dl c achieve exercise yes st induce exercise rest slope segment number color reversible heart eight predictor compare suggest parameter indicator significant introduce transform regression simple numerical transformation selection result final eq significance due hard besides hand extremely inefficient method couple ridge least square introduce stand shrinkage selection lasso similar ridge regression achieve lasso coefficient perform elastic elastic possesse shrink ability like ridge method add elastic net logistic coefficient estimate elastic include ridge penalty penalty situation correlate predictor fast outer update quadratic descent least square programming package efficient coordinate develop computation recursively automatically capture nonlinear interaction response complicated category variable tree provide practice insight datum method automate aid fail rule cart develop cart improve aid introduce prune aid cart search towards follow cart employ split know one guide design cart stand separate guide obtain construct contingency residual discretized group split employ chi square test way contingency guide guide accommodate grow maintain guide heart intermediate branch predict estimate misclassification give heart classic mostly design set response classification build classifier guide se grow use cart find state cart terminal insufficient statistical develop cart logit fit terminal variable terminal node contain cart variable terminal cart logit logistic regression mix q fit hard interpret may predictor inaccurate terminal unbiased analogue guide piecewise regression perform split sign residual test guide regression employ trend allow effect trend comprehensive guide datum logistic option advanced separation separate predictor another logistic contain split nominal build employ regression cart pruning tree computationally intensive due partition algorithm parametric include similar guide separate variable fit apply stability fit variable instability pruning grow logistic tree split split logistic recursive pattern provide
use gradient variational covariance gps concave covariance compute variational requirement parameter compute gradient respect take diag respect compute p diagonal element negativity estimate hyper variational parameter fast variational computed maximize step respect maximum hyper take step logarithm summarize involve dependency variational maximize update maximize variational ascent guarantee complexity algorithm dominate inversion dependency improve performance inner loop considerable speed number dependency approximation output predictive test predictive softmax latent score label predict associate dependency also noisy rely result refine take label value dependency refine account successive use prediction separately assign q l lt ty l l ty l complexity algorithm convergence similar discuss sequence label processing base segmentation feature test compare field elliptical slice loss point hamming output table percentage average hamming various problem crf partition ham c c segmentation crf crf number dependency advantage provide good next give performance marginally compare beyond labeling set consider matlab ghz intel processor crf code language runtime table fast use dependency result slight segmentation base np labeling task arise process output set arise label assign common crowd ability capture large handle labeling miss variation various labeling measure accuracy subtract repeat plot miss label increase show label model learn make neighborhood extremely handle propose sequence likelihood dependency become computationally set scheme neighbor wide processing long handle datum label provide variational machine problem arise natural model gaussian pseudo perform labeling likelihood range component become gaussian inference algorithm neighboring label capture range dependency label label usefulness keyword likelihood classify input output commonly task name entity speech pos label sequence label account among machine community framework labeling fields crf crf markov field prediction pointwise estimate validation bayesian crf crf markov maximum margin solution label kernel crf overcome limitation crf gps learn gps labeling labeling posteriori map instead label gps chain monte field expensive dependency pseudo complexity grid vision effective propose sequence label long range become computationally inference mcmc suffer perform effectively dependency usefulness labeling problem arise natural useful labeling might miss crowdsourcing introduce gaussian classification arise process prediction different form label pos input component discrete component belong represent example n component presentation determinant infinitely function latent kernel smoothness scale commonly square se associated hyper single take every label zero evaluate input class softmax gp multinomial probit likelihood close multi approach approximate inference laplace test approximation yield approximate label classification fail consider entire class number intractable label separately multi gaussian process markov neighboring component capture dependency computationally difficult pl approximation vision sequence label long capture become intractable deal pl define pl address process pl dependency become intractable pl field crf entire capture long dependency expensive sized clique pl entropy markov pl suffer pl cyclic among depict cyclic arise labeling neighboring component cycle make inference pl discuss efficient label output capture output sake clarity presentation component different dependency relation direct component assume depend neighboring dependence component denote dotted dependency cardinality output take account dependency direct graph figure label greatly overhead amenable data label output ignore component label dependency model gp local associate f call define value latent pair input depict family softmax among output define gp gp size associate size collection dependent latent diag function notational simplicity evidence posterior hyper
ba ba bb bb bc rectangle grid node yshift xshift mm yshift mm node gaussian ba bb ab bc yshift cb cd cb cb cc cd rectangle bad slightly hierarchical descriptor exploit ad represent information color kernel specialize color gradient generic formulation se come tool neural scheme approximate om variant shift invariant kernel convolutional full image moderate moderate layer nystr om set grid pointwise pool illustrate layer finite positive control image map respectively act patch quantity admit activation map k follow linearity obtain network comparison cnn benefit involve hyper size later cnn start operation learn follow pointwise linearity point next lemma expansion mapping mc constant front integral need weight arise chapter different cm datum set integral sampling kernel e latter orientation typically explain prevent curse learn training importance unit produce sampling map w therefore dimensional operation follow resemble right exp xshift yshift exp xshift yshift anchor north xshift mm north anchor north cm build present principle datum patch formally patch p z early problem sgd infinite preferred point initialize fast bfgs ensure convergence stationary type convolutional leave experiment matlab bfgs rescale average always cross try always consecutive last layer produce map z prevent unsupervised discover natural patch field priori spatially localize oriented work attention know achieve cnns kernel even image patch learn filter perform obtain display among exhibit interpretable explicit relate wavelet dataset handwritten map patch pm methodology comparison otherwise four architecture gm simple patch filter gm gm use filter work raw pm detail architecture augmentation c tr cnn gm gm pm na na na na na cm cm cifar architecture cifar report gm pm work rgb patch color pm gm co especially despite layer require learn note either augmentation cifar planning investigate gm cifar na na methodology idea near dataset mnist cifar regard consist leverage understand theoretical acknowledgment grant project ce joint centre european research project elementary kernel p show z h pointwise kernel one composition p mnist cifar gm pm subsampling factor map indicate goal devise paper new convolutional neural either represent solve network learn kernel benefit invariant obtain network complex train literature invariance recognition cnns prove recognition cifar dataset accuracy state recently attention convolutional network large visual consist pointwise linearity operation pooling result empirically perturbation encode visual cnns capacity technique cnn understand recently invariance architecture characterize wavelet convolutional adopt traditional indeed descriptor reproduce produce multi representation main scheme convolutional neural interestingly unit kernel believe direction learn supervision vector yet achieve cifar architecture augmentation open author several related kernel cosine successive rely integral kernel arc cosine kernel neighborhood oppose representation manner visual parameterize neural function consist compute response perform kernel contrast propagate layer kernel obtain benchmark produce representation link convolutional neural produce sequence interpret non refer spatial map represent literature follow coordinate hilbert practical image coordinate represent characteristic neighborhood instance size blue map provide pixel value non complex terminology introduce kernel represent pixel convolutional z replace definite spatial correspond closeness place invariant show concrete example map difference characterize angle exactly descriptor introduce set location center patch visual encode color convolutional generalizing convolutional us hilbert build hilbert coordinate location patch word ii definite
correspondence biological typically reproduce particularly early expect increase structure type aggregate summary wide table supplement purpose estimate increase scenario case coarse correlation respect reproduce lead wide slow overview supplement width cm mark dark red triangle provide supplement bayesian inherent model uncertainty parameter fig predictive plot generate calibration bias e display distribution scale prior range forest reach specific width example choose correlation fig inverse metric quantifie model technical limitation choose expert deriving field goodness directly approximation allow likelihood observation give provide connect forest datum forest virtual assess uncertainty fit detailed datum abundance growth unbiased well fig aggregation v correlation indicate trade reproduce fit occur mostly extent produce growth datum growth maintain scenario include also growth growth account uncertainty correlation plot true high correlation aggregate type difficult visualize compare constrain strong correlation mcmc make interpret may opinion would mid specie due specific tree rate full secondly range resolution affect equilibrium slightly temporal output provide good relatively runtime intensive model encourage heterogeneous address technical challenge effort conceptual cost gain probably inversion practical far beyond considerable interest increasingly available practical problem answer source construct type would error challenge provide answer include likelihood weight account correlation conventional see pattern decide base statement include possibility rigorous statistical process testing subsection apply nonparametric widely nonparametric approximation distributional advantage equilibrium prediction parametric start parametric ensure acceptance reach acceptance would accept large correct late number mcmc parametric conjecture favor parametric encountered study promising parameterization traditional approach prediction mechanism mean approximation relative interesting model couple cost factor run difference secondary rigorously test understanding datum likelihood seem run dominant heterogeneous well standard like helpful comment review advanced grant research service open publication research centre department modelling http inverse scientific quantifie prediction express general inversion metric observe computational reason likelihood general assumption recent year likelihood method concentrate application rich forest inference approximation place conventional well virtual data sensitivity commonly result demonstrate simulation offer considerable conceptual particularly heterogeneous datum structure adjust include therefore provide fairly parameter derive physical physical experimentally range restriction measurement need output bayesian increasingly decade addition particularly likely quantify approach metric often bayesian datum current usually application make distributional vary around ad hoc mind variability environmental condition likelihood art limitation predict principle prefer derive understanding process output observer could complicated stochastic limitation reduce technique arguably attract well stochastic inferential likelihood estimate pattern likelihood approximation repeatedly rare apply parameter rich simulate aim base likelihood approximation propose infer virtual identify field examine aggregation statistic inference fit field line refer specie relative year apply leaf area index per area relative diameter height sd growth sd rate relative end growth sd growth diameter diameter diameter growth relative dynamic prominent dynamic lose factor tree gap formation create gap natural forest within gap explanation forest formation history tree light highly diverse specie type specie similar functional per gap formation forest merely predict possible variation composition use forest use base forest forest gap forest various location area cell assign interact position entirely measure diameter breast height tree dimension relationship calculate subsequently light act important describe specific growth dimension description scheduling supplement model gray type rate original uncertainty likelihood one carlo sampling approximation robust bayesian suited deal interaction expect likelihood short introduction brief inference posterior calculate normalization uncertainty compare likelihood obtain broadly speak say quantify chose wide type proportional prior give many uninformative purpose facilitate definition likelihood conventional study quantify model occur uncertainty situation usually interact typically ad hoc variability hence conventional likelihood independent mechanism description virtual field field total forest approach go deviation particularly dominate error forest gap formation explain variability expectation output conditional technique suggest prominent arguably attract recent year discuss currently abc apply similar principle suggest classified parametric approximation estimating obtaining convenient normality fundamental requirement obtain variability correlation output matrix supplement abc achieve efficient unfortunately generally accept good summary statistic modeling fit first aggregation count number individual size growth quantify forest subsequent metropolis check visually supplement detail evaluate parameter estimation burn chain supplement visual virtual three create advantage true
expert well expert output decision rather high representation convolutional layer build work compression great valuable information non neural average prediction expensive evaluate target correct pursuit carefully capacity network arguably complementary ensemble grain convolutional connect relu unit class evaluate million image annotation example ground entry fraction wherein look two expand internal google million span table successfully network multiply add required evaluation well perform imagenet imagenet class thus train comparison work general add train model consider result task assumption underlie extend allocate fixed regardless capacity cardinality group difficulty necessarily either obvious connect layer capacity discriminative contain fully connect appeal augmentation purpose specific augmentation theoretically potentially division capture wherein pathway activate task relevance gain strategy google op universit view google com neural capacity significantly descriptor label visually hide pathway connectivity label report task augment multiply perform exhaustive architecture expensive furthermore satisfactory architecture discover improve fully increase tendency overfitte setting thousand class prove challenge jointly grain entity often kind challenge visually belong two semantic visually traditional building prediction exist additional significant sensitive production must rapidly matter service level way improve performance computational evaluate however immediately compete objective add capacity train neural hold datum demonstrate recognition base less overhead held apply generate possible
spherical benefit ad examine cluster mean deal pg mean mean pg project kolmogorov ks examine hypothesis cluster mean rely split concern single employ accuracy ks modify benchmark uci repository generate exist unimodal sample unimodal acceptance hierarchical rely criterion recognize check criterion split checking continue cluster cluster usually transform section signature illustrate signature generate run figure illustrate top middle sort plot compact form sort small therefore around dense small signature signature sort absolute unimodal suggest signature sort signature expect cdf signature index choose comparison demonstrate step threshold split un ng cluster leave single behavior associate unimodal lie boundary therefore method split e htb two signature signature deal unimodal unimodal family simulation ks cluster default significant ks confidence addition computation ks splitting mean benchmark vi adjust rand vi ari cluster ari optical digits ari vi ari vi ari vi idea signature splitting signature represent advantage propose simulation cluster cluster signature propose specific application hierarchical statistic test cluster group family rely available challenge involve hierarchical cluster splitting criterion nan hypothesis dataset estimate pass improper splitting incorrect cause universal approach kolmogorov splitting criterion also cluster propose
graphic macro ltb lt lt lt ltb lt lt lt package conjunction terminal option explanation load graphic graphic macro ltb lt lt lt lt lt lt ltb lt lt lt lt lt bp cb set instance error etc task examine affect find attribute fr increase requirement complexity frequency certain instance make complexity prevent avoid algorithm add loss boost misclassified often boost instance remove filter instance amount frequently misclassifie heuristic wrong decision machine filtering discarding instance decision instance weight influence discard learn clean train predict attribute predict correct accuracy validation let compose distribution thus feature possibly train factorize figure distribution unobserve variable feasible b cb explicitly possibility I rather focus classification show generative differ mostly concerned training seek probable hypothesis map ignore figure use approach neural network decision possibility ignore generally handle probable simple prefer instance learning seek maximize model observe label motivation pass datum calculate trivial use outside hypothesis induce approximated assuming train hypothesis multiply though trivial distribution generative discriminant asymptotic diverse diversity refer hypothesis diverse create unsupervised algorithm different prediction hierarchical agglomerative default result dendrogram height line connect create representative diverse conjunction terminal option load graphic terminal graphic ltb lt lt lt lt ltb lt lt lt bp r mlp near near na I bayes learner forest prune cross validation induce approximate hypothesis weight mlp random weighted examine train backpropagation derivative activation sum correspond forest select algorithm track represent sum meet mlp forest uniform weight weight biased p algorithm produce kronecker delta paper instead induce feature approximate single hypothesis calculate score mlp backpropagation score large value value class instance set rule score number instance examine divided leaf neighbor agree reach node normalize percentage instance cover true instance em algorithm belong clustered clustering data algorithm cluster filter ensemble examine good would remove misclassified near neighbor repeatedly remove misclassified handle experiment experiment noise introduce label examine handle learn uci repository pair sign rank expert previous improves generally handle artificial add experiment row algorithm subsequent value case higher represent significantly high handling noise handling increase mlp contrast noise handle achieve handle achieve highlight handling noise work set handle beneficial weighting induce instance weight hand forest achieve filtering represent approach increase decrease rand l g examine handling generally achieve accuracy inherent however random forest robust obtain low instance artificial forest average nn noise handling mechanism beneficial instance filter class compare biased accuracy significantly outperform case case compete weighting nine diverse represent suggest show obtain r high ccccc ccccc nn c l weighting weight filter filtering weighting great accuracy estimate compare achieve mlp weight significant algorithm hand achieve accuracy four mlp high overall accuracy level accuracy except examine increase classification forest level achieve high achieve significantly high effect filter choose produce high avoid ensemble individually affect bold ccccc ccccc r f mlp examine learn base examine weight extra select threshold filter instance great effect algorithm instance individually backpropagation filtering artificial weighting use handle future equally outlier equally machine handling
cross early end rich lot error selection selection stability base false set predictor outcome signal set boost integer boost continue select repeat select learner pre value show control predictor assumption exchangeable original boost variable exchangeability see assumption noise assumption restrictive hold sensible stable propose choose little potentially e half fit select signal subset signal minor importance long parameter compute equality assume check resemble overview essential subsample derivation modification stability complementary subsample importantly bound derive assume exchangeability without original selection procedure drawback assumption control per family variable run size error rate unlikely select compute random complementary additionally otherwise subset select procedure selection threshold case tight assume simultaneous unimodal additionally practice third simultaneous r concave convex concavity strong concavity concavity support tight application refine markov distributional assumption inequality derivation aware selection assumption generally concavity may concavity hold distribution unimodal concavity seem wide bad selection low include e control rate case exchangeability special e variable strong correlate select control similar practically identical number positive statistic test hypothesis reject learner commonly per per without adjustment significance conservative control setting gene study use relate false reject give situation fix conservative controlling situation obviously specify g specify choose upper usually per everything adjustment bind upper boost conjunction stability additionally impact characteristic accord regression predictor independently draw design predictor observation variable vary influential influential variable predictor correlate predictor toeplitz setting vary bind complementary improve time figure scheme true positive depict uncorrelated rate rate average replicate circle increase uncorrelated predictor generally natural biased base learner yet error bind truly influential threshold figure enough appendix figure correlate false positive bound conservative median setting overall concavity standard control distributional simultaneous probability general stability conservative assumption simulation open circle positive grey horizontal line increase positive increase appendix influential highly stable number select per number positive contrary false examine control express pathway patient design single ability cell utilize mb media produce end utilize nm indicate peak give measure background replicate patient due supplement package biological replicate I e incidence per incidence one contain annotation include non difference measure patient control pm intercept group effect control irrespective effect replicate effect belong mean code formula group specific either control case constraint code effect coefficient differ specify effect offset contain height respect offset additionally interaction keep learner check pm total learner effect overall group boost differentially stability per boost choose related significance lead cutoff r concavity bind subsequently cross validation stop stability supplement stability find frequency learner base selection assumption term selection dash gray value concavity determine vertical gray line dash line value concavity confirm affect weight patient decrease unit logarithmic rate finding might sequence code code patient paper detect five level report patient notice either affect function reduce collect notice patient whole seven reduce five finding might consequence observation add variable keep conjunction complex generalize additive gene positive per conservative specify sensible control adjustment extreme stability control tight stability suffer different boost package package use store package fit boost additionally specify assumption concavity alternative method fit interface specify index framework function miss stability selection check sensible situation acknowledgment discussion fu chen genetic center analysis conduct present setting independent toeplitz replicate open red represent positive setting toeplitz assumption circle true influential toeplitz observation line together observation replicate red represent positive independent predictor toeplitz compute replicate
high versa balance action small aspect argue bad propose family function find tight exist policy evaluation nonparametric policy space show outperform art powerful policy evaluation fit policy outperform purely approach outperform rl computational depend solve cost batch set problem expensive medical mining complexity surrogate action sampling distribution solve knn sign basis function question loss approach investigate additionally action space effect choose change could restrict impose constraint maximum study loss let n assumption exist unless otherwise bernstein geometric mean least inequality auxiliary x x maximizer action likewise q q optimizer property imply apply apply n use lx q exist q minimizer policy choice r inequality apply inequality l n paragraph v l c k get upper imply convenience bernstein pn class range fr ec loss minimizer loss cause quantify loss suppose action empirical minimizer use mainly estimate regularity factor currently extend note similarity considerably different line unless n geometric probability focus event eq xx maximizer x q use likewise write decomposition optimizer finally use apply inequality imply n probability proof lx q expect least get last inequality property bernstein geometric obtain result get separate infimum c n previous desire analysis modify proof remain follow policy finite vc xx maximizer x inequality eq likewise optimizer inequality apply inequality appendix satisfy state include probability least choose loss policy space choice add lemma probability eq n lemma appendix z inequality apply recall l infimum use confidence get desire main remain vc vc dimension independent main rhs integral cover relate entropy r indicate complexity equation theorem originally chapter lx lx eq bind additional quite flexible derive benchmark car balance standard benchmark reinforcement flexibility choice evaluation tb approach car task value base choose method space policy implemented walk value iteration minimize neighbourhood suppose endowed integer kx tie knn pick knn lead rule implement dynamic car discount initial uniformly random policy collection length episode reach goal go step reach episode gaussian reduce systematically knn bar standard reach return episode knn sample reasonable knn especially benefit purely knn underlie interesting depict function regime small effect optimum choose properly selection beyond sequential tb balance policy pure value fit denote tree fit algorithm evaluation improvement policy action minimum note extra represent trajectory length trajectory choose use amount generalize balance successful show per average tree row especially size try solved perfectly confirm exploit compare exploit tree computationally costly practice choice collect style powerful science engineering claim theorem corollary proposition fix c g p gap indicate inaccurate state space mdp restrict qx hold qx summarize definition reader bound measurable w subset discount mdp r optimal value policy denote ax choose arbitrary manner function optimal introduce complexity control low mdp characterize simplicity naturally mdps mdp actions action understand action informative perform improvement base greedy ideally greedy policy action roughly action less likely action likely wrong characterize summarize action gap gap p q control action gap imply inaccurate measure mdp mdps satisfy mdps almost find mdp form show gap norm qx supremum norm hold qx distribution close policy compute include policy space distribution construct entail improvement iteration gap choose greedy action large policy projection greedy weighted gap e minimize weighted evidence value exploit function impossible optimal ideally use use sample change sampling loss qx qx qx n loss difficult action possibility relax surrogate weighted hinge approximate iteration whose outline present dataset classification compute estimate q bellman residual minimization fit iteration combination td exploiting intuition policy improvement iteration action gap accurate maximizer distribution greedy large measure accord weighted bad policy account region belong dataset simplify discussion weight loss action gap regret greedy flexible policy parametric space finite grow example latter method neighbourhood tree grow reproduce hilbert nonparametric sparsity basis achievable estimate generalize state g policy affect another choice match policy well ideally automatic note dataset transition generate sample generate early iteration independent extend action gap difference action qx qx qx theoretical computational solving problem since loss difficult one relax gap return kx ix influence minimizer derivation action point majority greedy action would base tree represent analyze policy policy distribution e enable specify behaviour algorithm analyze affect early n clutter identically I action define pointwise pointwise expect loss action value empirical w instead close carry analysis behaviour take care main relate loss appendix complexity vc metric localize complexity often use localize rl aspect interested study take care issue cause relate relate make notion choice vc metric etc g localize rademacher since lead moreover vc policy property rademacher quite scope localize rademacher analyze reader p rr ex quantify extent localize rademacher complexity define root subsection fix policy assume consist bind important term rich mainly behaviour point determine global neighbourhood dr cf rate considerably behaviour term result rademacher exist action regularity problem evaluation improve benefit guarantee randomness constant second behaviour point hand side existence uniqueness prove make define error neighbourhood complexity possible extreme everywhere single policy subtle aspect note conservative behave proposition also corollary behaviour estimation last important gap regularity regularity geometrically regularity often assume randomness q c quality evaluation quantify question policy error question proper selection algorithm mdp main greedy error greedy definition induce greedy error quantity dynamical mdps future relate approximate algorithm quantity coefficient integer state supremum derivative r discuss definition ready main paper sub eq k regard approximation error apply early discount implication resource focus later iteration beneficial though apparent value iteration tailor type reader refer discuss early moreover dataset nonetheless might upper g present gap value classification pure designing state consider derive policy report compare pi use example everywhere know uniform run policy vi design performance policy good solution achievable considerably vi search pi comparison vi involve policy space small pi considerably evaluate report use poor approximate greedy attention policy fit belong large region space differ optimal relative region poor belong chain difference everywhere reward belong result pi go quickly however vi pi performance optimal action function report pi anti fall pi pi differently drug despite maintain long attract interest community optimize drug scheduling strategy lot attention recently structure alternate scheduling action patient simplify formulation reduce available combination drug small interaction develop real decision
compact identify space accordingly complement partitioning designing proposal infinite effective simpler yield ensure discretization invariant next context mcmc great inform direction differ method example proposal exploit discretization scheme langevin basis construct adaptive variation hessian use proposal adaptively define gauss newton adaptively refine proceed follow construct use current value gauss newton evaluate operator approximate low terminate adaptation evaluation f distance successive fall prescribed step eigenvalue additive terminate use build theory ergodicity f step covariance leibler hellinger distance symmetric operator differ range finite efficiently project onto reformulate dimensional subspace criterion maintain full hessian high maintain dominant left structure case truncation hessian truncation direction benchmark two representative proposal exploit proposal transformation obtain proposal term numerical langevin proposal rw brevity rw comparison hessian explicit langevin data langevin sde hessian sde form langevin proposal mala newton algorithm instead mala burden hessian principle modify newton h langevin newton langevin employ whereas use pde infer proposal proposal use invariance refinement spatial coordinate term potential field govern pose superposition four weight bilinear uniform endow normal prior operator prior length make field prior field observation potential figure b sensor circle carry affect two magnitude signal ratio noise snr set proposal rw langevin proposal every hessian posterior weighted benchmark proposal half autocorrelation onto eigenfunction operator large proposal bias result burn mix along direction onto prior eigenfunction projection mode evaluate mode yield proposal autocorrelation choose projection onto st expect hessian weight proposal map brevity comparison langevin sampling start produce adapt langevin proposal employing denote account variation dominant dimension truncation threshold relative sampling snr snr show algorithm produce mix mix noise snr due broad lag improvement global snr global refinement global convergence grid diagnostic use adaptive diagnostic inform drop course procedure comparable three yield grid yield effect discretization model refinement also first level noisy path double drive force molecular dynamic govern globally increment brownian motion follow condition sde double model science perhaps notably molecular negligible mass fluctuation map differentiable continuous wiener framework euler dimension h efficiency proposal solution truth path conditioning quantile marginal enough length burn also adaptively construct highlight sampler summarize trace also dramatically improve mix langevin h langevin pde langevin reasonably proposal plot lag autocorrelation figure eigenfunction figure improve trace langevin mark adapt construct global decay autocorrelation mix measure lag general explore proposal discretization dimension early dimension independent algorithm new sampler wherein direction capture global identify strategy proposal offer gain efficiency certainly construction construct incremental updating strategy optimally variation hessian available resort active gradient available approach might covariance regularize estimate second weighted proposal construction approximation posterior difference posterior local value room implicit discretization locally independent provide construct provide proposal mala discretization langevin equation proposal extend development idea acknowledgment acknowledge financial energy office advanced sc mathematics capability law technology inference explore high represent introduce chain monte carlo adapt structure intersect develop introduce general operator proposal mcmc sampler local hessian distribution scheme langevin operator proposal adapt result sampler large pde inform langevin sde high represent discretization example include inverse reconstruction differential markov monte carlo mcmc representation unknown refine present enable build change local information design cast proposal operator proposal allow structure exploit overcome standard particular absolutely choose deriving yield yield performance mesh work integrate design finite carlo use building employ algorithm effort proposal space analysis improve finite setting research local geometry scale include stochastic matrix prior metric langevin implicit hessian log direction differ idea formalize lie involve result great relative also optimal covariance suggest infinite proposal differ preserve independence attempt operator employ version hessian single near noise parameter dominant relatively inform dominant likelihood hessian posterior direction inverse smooth dimensional acceleration mcmc algorithm dimension wherein concentrate goal dimension independent sampling allow gaussian flexible manner local covariance e bound adjoint operator proposal sde incorporate proposal mcmc either metropolis rest review formulation method well previous aim improve sampler enables inform global sampler pde evaluate sampler sampler highlight key difference performance brief remark inverse proposal equip prior geometry proposal separable give assume away unknown element banach assume additive zero space associate norm brevity possible drop assumption forward result locally condition forward hessian prior adjoint class full operator define define covariance refer laplace differential respect measure limit shrink q maximizer vanishing posteriori mh accept hasting define chain otherwise mh continuity mcmc walk adjust langevin mala mh mesh refinement carry reference therein mh langevin continuity consider langevin sde sde see reference therein order implicit parameterized defines proposal langevin walk simulate langevin sde provide approximately technical putting let rewrite invariant remain one employ desire autocorrelation increase
search corpus scheme illustrative pattern trend discover similarity corner topic universe social clique topic discover detection clearly group instance community another life genetic dynamic material research dedicate develop material mutual molecular recognition show research appear example illustrate use outlier htb topic community orient pure separate community couple represent cover range htb discover biology population related htb connect network second corpus comprise english portion contain span different topic execute implementation lda labeling labeling wikipedia topic wikipedia nsf grant conduct user wikipedia agreement perform significantly nsf grant elaborate label topic wikipedia construct node use wikipedia network rather illustrative major trend salient macro visualization figure wikipedia subject music fan basis salient group connection summary green daily medium discover wikipedia music topic plot write daily news medium study label performance great topic nsf grant corpus wikipedia many topic broad general summarize single perform albeit equivalently phrase focus wikipedia expressive produce address find especially mine technical document primary limitation text optimize article summary corpus work shorter handle performance grant extremely text cause difficulty certain wikipedia topic contain article description minor character text twitter lda design short investigation improve lda construct employ explore nsf basic english portion wikipedia text previously develop department analyze refer collection represent label network model base network show powerful characterize collection text connection set insight topic large efficacy nsf grant span year entire portion wikipedia interpretability insight popular topic latent lda discover topic represent explore domain user search unclear begin search user automate organization massive scale quite challenge large trend lda fundamentally return document absence typically probabilitie label explore text raw output challenge insight identify leave numerical trivial implementation train lda massive corpora big discover challenge unclear discover oppose thousand hundred thousand present work investigate challenge graph nod collection similarity topic effective building summary contribution area topic visualization labeling section topic visualization discover employ community network discover macro subtle discover extraction unsupervise employ topic large surprising visualization next represent efficacy automate wave address challenge topic topic light graphical exist several like make towards interpretability exist relationship topic document provide insight topic come subtle connection among topic ill understand document exception correlate topic association association represent incorporation reveal challenge exist base appear extract refer relation view subtle third scalability chen et show unable corpus document finish week limited text show scalable cluster domain cluster machine process million access machine cluster correlation structure among aforementioned issue scalability efficiency association fashion similarity topic exist labeling visualization tool identity node scheme probable word topic derive always expressive fashion unfortunately unable handle text corpus topic gap motivate labeling goal topic method set supervise labeling consume labeling must corpus consideration corpus wikipedia especially deal collection describe art edge technology subject reference corpora requirement prevent utilize employ corpora topic intend interactive utilize text document engine document comprise topic filter domain filter remain document sub labeling cope scenario document collection reason label couple topic filter characterize labeling topic label filter might repeatedly execute document collection labeling work appear easily well long short text aforementioned issue motivate development massive throughout represent collection topic document sequence element return like exploit trend topic employ use insight discover densely topic detect group employ heuristic modularity modularity fraction expect fraction link evenly initially assign topic greedy fashion assign community modularity efficient large topic computational network mark community topic represent powerful tool discovery heterogeneous corpora topic label mechanic dynamic flow dynamic mechanic flow agent equilibrium game graph graph combinatorial theory algebraic human evolution modern early year human modern human evolution water engineering coupling mode modal recognition orient protein protein protein protein protein protein bind research social social social discover nsf rank lda subset corpus greatly facilitate document similarity comprise purely take corpus way topic proportion transform mutually exclusive cluster employ well text htb label return term filter hash hash dy gain sort essentially topic naturally observe topic extent express noun expressive retrieval instance expressive alone expressive denote uniqueness noun report proper noun system technique principle noun test association contingency extract test gram phrase save label individual intuitively term appearance early frequently weight candidate harmonic topic recently document belong entropy measure p proportion document perfectly document gain high simultaneously rare algorithm sort small candidate select sorting might select label frequently albeit slightly latter two indicate specifically sort normalize combine comprise scalability document online collection long combination dependent deal reason straightforward multi core line instance map job identity hand execution oppose pass processor system
function rare occurrence et implicitly matrix shift wise local form term independent term word context objective may perspective wider different similar performance dimensionality enough well choose weighting context unobserved explicitly sampling take corpus rare choice word sake efficiency avoid remain pearson correlation r r iterations pearson depict iteration two band may truncation less see well iteration less weight objective shift like done explicitly shift share objective though completely weight strategy bias tend converge term may approximation optimize future investigation focus choice report also skip implement explain similarity two specialized function differently represent valuable context meet context context word embedding matrix vocabulary similarly context word context count
remove turn leave stage fig result general result stage sift contribute performance take spatial representation selective convnet feedforward mean k k boltzmann machine convolutional network column deep maxout net major challenge dataset lie capture object scale pose accuracy outperform several feature encoding fisher vector field discriminative spatial preserve sift unsupervised learn encoding encode nearly accuracy challenge grey handwritten digit training normalize center pixel use atom block extract sift scale voting field size give result deep learning achieve success digit misclassifie machine network maxout single achieve highly complex architecture misclassifie digit misclassifie human clean beneficial focus detail l algorithm k map good mean query mean employ retrieval feature subsampling break map sift final multi voting e view retrieval train combine stage nearest search fair task table experimental dataset compare sized bank triangle filter bank achieve small combine consistently mean mean k representation view achieve without supervision near copy dataset type image perspective etc type three pooling respectively retrieval robust outperform nearly encode term map bit layer neural term unsupervise one major advantage feature representation classification unlabele sift discriminant experiment challenge classification unsupervised feature learn bag word big dictionary rich significantly last far without train several acknowledgement national science china national foundation education china paper investigate feature unlabele network network attract researcher recently effectiveness feature mapping tend describe robust purpose call ball local density show efficiently programming problem traditional method representation ignore global representation descriptor also field multiple pooling extensive several comparable learning attract attention interest representative deep abstract among one typical dl convnet stack classifier variation convnet different vision great success layer time performance autoencoder rbm restrict boltzmann adjust consume practice motivate among mean cluster commonly unsupervise simply cluster parameter involve base number cluster practice capable superior gmm representation uniform cluster contain influential description issue key definition robust noise outlier commonly encounter application could actually tool minimal describe target class obtain addition cluster surface come support boundary far fig ball may datum feature representation align quadratic need hundred programming save amount extend adopt range part level preliminary feasibility method call verify extensively retrieval benchmark part organize section regard learn investigate performance empirically automatically useful rely learn utilize representation subsequent g object pattern obtain distinguish noise since think variation transfer knowledge relate domain regard kind unsupervise word aggregated descriptor fisher typical include step local filter gmm center cluster filter bank step patch encode vector learn bank input image follow brief feature usually pool local regard encode count bin coarse way encode alternatively count effectively robustness code patch tx tx tx uk tx simplify define signal dim encode rich object method connection bank feature advantage encode even rich whole procedure architecture deal code encoding activation centroid less note triangle essentially complicated g autoencoder hence characteristic different point believe make difference aforementione mean cluster since cluster latter present overview center means triangle encoding discuss extend typical layer component image map set operation contain pool map serve option bank pooling grid major generally speak big filter help representative accurately high pooling need dimensionality architecture aspect local texture actually kind challenge highlight importance encode second spatial name adopt architecture dictionary add sift post pooling processing procedure essentially project response robust representation contain center radius description spherical point avoid outlier face tradeoff goal minimize formulate slack relate th atom simple atom face capability distinguish b patch attract atom see patch attract fourth atom yield atom see potentially could however since actually contain three encode drawback atom corresponding atom major capable characteristic dictionary effective row explain compare counterpart c experimental bag patch preserve huge dimension suppose field input densely encode map filter filter feature pooling map pool p p window information lose sift sift descriptor computer sift representation sift descriptor densely cause dimensional descriptor pixel dimension extract bit histogram dimension dim preserve rich subsequent task exploit scale scale different context since characterize object appearance information capture lost level valuable complementary manually design descriptor sift feature unclear high pattern c layer learn scale information way information training atom correspond size fig learn learn typical convnet window learn example give orient window effective feature big pattern one useful learn train layer view size pool combine output multi let weight versus normalize output decision eq classifier coefficient mention extensive experiment include image retrieval whitening whiten operation linearly transform matrix sphere euclidean use unless note important subsequent default recommend ball q encourage default background ignore use triangle encoding direct counterpart simply furthermore addition method network mean denote l default conduct behavior propose image label object partition remain image pre fold image fold fold report across fold randomly pool fed vote range major et control unsupervised compare particular extraction
na trivial bind assume vanishe enyi surprisingly show specification behave os r enyi er use sufficient statistic entail generality er probability represent os absolute distribution os enyi n ig ci nd np ig ig np mp np cover parameter equal yield close er unless observe isolated node er k f favor isolate order vector n still observation regard identity n k g proposition scale version k n isolate node write n write imply p rewrite exponential family remove without isolated normalizing figure apparent isolate time iid node isolate g existence mle easy bi verify odd element bi length tends tends infinity conduct non ideally compute element vector illustrate department pa family statistic label graph science attempt model parameter variety see form example node expressive power perhaps statistic g literature property statistic edge see whose derive dyadic long know originally originally increasingly property degree summary among statistic investigate offer preliminary trivial complexity statistic case rely base dyadic challenging contribution mle include concerned dense os enyi os enyi appeal possess demonstrate finally define bi undirecte label node iid observation observe ratio number number observation case observation observation available extract observation precisely statistic family scale subgraph subgraphs sufficient configuration edge node configuration triangle subgraph subgraph three node connect adjacent node g connect subgraph e contain order convention ratio edge equivalent os odd reason joint sense uniquely converse move move enable possible graph base practice usually term different labeling well distribution asymptotic prescribe non time different em em verify consider imply pg pg calculation regard degeneracy reason change odd point na point odd even vector lemma possible consist characterize odd project polytope coordinate depict em em em mle mle even e even n odd le n l odd b k k nn l conversely n imply vice imply mle proceed one partially ie appear appear term denominator side assume model node plausible simply polytope would simply subset whose nonzero deal extreme suppose graph start stop
macro corollary california berkeley human comparative measurement study choice selection empirical evidence amazon pairwise comparative fast one less ordinal characterize minimax rate measurement quantify ordinal confirm ordinal ordinal knowledge non expert human domain crowdsource collect human low come price noise crowd response address noise study set answer crowdsource image student approach enter subject ask search internet entry alternatively compare item figure show image ask internet restrict pair item ordinal allow precise value ordinal even measurement convert simply order value suggest original ordinal ordinal encounter measurement inherent possibly conservative evaluation ordinal measurement recognize human allow evaluation perhaps cost lack clarity regard fundamental question gain measurement significantly ordinal reliable pair comparative preferred measurement invoke study widely theory show preferable theory theory tool comparison investigate perspective estimator ordinal also incorporate highlight fit ordinal ordinal choose ordinal ordinal approach enough ordinal topology compare argue always ordinal approach lead processing mean seven amazon show argument insight ordinal subsequent focus experiment subject version question answer subject ordinal version pair present seven experiment broad coverage audio l circle audio relevance ordinal clarity comprise box area circle paragraph people ask ten people audio sound key correspond sound audio internet show subject query ordinal form compare answer subject five seven experiment ground truth solution compute fraction incorrect ordinal convert half error truth ordinal answer process unlikely setting ordinal argument inherent subject human evaluation compare amount ordinal ordinal discrepancy complicate aggregate produce final answer whether address small aggregate popular two comparison presence additive quality pair entry ordinal identifiable shift analogue involve individual remain entry intuition scenario analogous choose suppose enough ordinal item facilitate induced distribution allow measurable sample semi minimax place ask evenly ordinal minimax error provide many regime enough risk f case time characterize treatment strong convexity informally difficulty estimate arise risk ordinal setting dependency ordinal collect overall enough summarize loose tight axis pair comparison treatment allow comparison second comparison simple parameter play comparison carry treatment extend rather provide central role comparison comparison measurement graph laplacian refer laplacian determine canonical example large evenly along align clique disjoint vertex unweighted get unweighted complete nk well possible hand scale bad topology consider error distance frequency execute ordinal uniformly replacement inspire practical pool pool log rest unlike ordinal evaluated element minimum knowing infer scale correlation put rest per ordinal ordinal well identify mistake per hence ordinal ordinal ordinal audio coefficient coefficient compare human system rely non expert human crowdsource mechanism critical argue fundamental ordinal provide choice observation question estimate either bound overall option complex one incorporate worker accurate threshold determine crowdsource setting testing improve collection useful adaptively choose topology aware potentially ordinal analyzing topology review present present collect amazon com amazon platform put individual offer exchange payment task follow experiment comprise task comprise question organized ordinal answer question task worker country allow estimate usa allow american move experiment rating product truth circle question random experiment age connect table section associate standard run ordinal square coefficient coefficient average list present possible distribution kullback leibler pair estimate kl family following give metric induce distribution minimax subsequently positive entry graph topology satisfy semidefinite lemma provide e vector I I vector normal location minimax see instance ordinal scale nd construct j give pack choose bound desire impose constraint hessian function tw derivative l say set hence note put everything together standardized laplacian gaussian distribution divergence p packing furthermore I give pack bound note issue remain verify fact final follow relate loss I scalar log concave inference allow n define minimize k last lemma I I say virtue p e put everything substitute b tn index vector
mean lead sigmoid among gaussian recent phase place learn benefit small batch appear get energy landscape range probit rbm unit generate binary bias threshold field average persistent collect field mnist digits separate digits classification hide unit probit rbm comparable rbm cd probit rbm rbm cd operate representation probit rbm smooth exploration mode main produce rank area hand rating arguably learn directly instead go intermediate ranking per variation item choice share tie necessary rate unseen item person fast maintain estimated markov maintain chain update step spirit divergence away chain ph rank unseen movie rank movie user supplement data movie encourage diversity list remove movie remove less remain movie rate movie hyper rest implement item popularity naive implement recent ranking metric emphasis rank movie rank unseen movie demonstrate winner cccc popularity mf shorthand mostly consist multiple fact g choice choice choice ordinal rank deal heterogeneity call numerical scale star comparison tool lose format convert survey collect centre people explicit interesting gap uk relative separation china rest feed multiclass logistic regression suggest capture variation widely category category gaussian cost invert gibbs dense overcome probabilistic principle main direct undirected rbms system boltzmann machine variable restriction place handle ordinal study difficulty model introduce model nearby correlate situation attribute add combination modelling distinguish thus model row hide row generic boltzmann type inequalities boltzmann machine evidence representation support boltzmann literature specify specific support assignment censor binary rank demonstrate handwritten digit survey specific potential compute define apply let metropolis annealing time posterior express constraint method would deterministic probability concrete kullback entropy decomposable second thus ignore since decomposable e ie k combine completely decompose divergence ie ix divergence know proper multipli ix gradient w finding let truncate recursive ik short hand cumulative function sample straightforwardly true inequality visible threshold active sample use sampling markov propose certain level slowly temperature temperature g collect successive p constant decomposable gradient w pair simplify px h undesirable chain tendency point flip due fortunately enforce node pd gradient p sx phase way way update estimate fashion incorrect exponentially progress error estimate describe start mode taylor distribution replace thus turn per category plays share location th utility th category ensure lx lx I pe px mf lx rewrite change variable pe mf dy categorical category particular assume must word offer stagewise pick also pick subset give pe e pe l l give model survey country people world china second last gaussian approximate univariate assume form equivalently taylor expansion truncate read eq dirac delta truncation recognition university edu introduce wide range view type observation impose inequality type interval censor ordinal without tie three handwritten social survey analysis restrict rbms prove variety original proposal mainly unit hide applicable detector design input continuous multinomial ordinal boltzmann permit type jointly model thus rbm multimodal social survey audio modality idea conceptual drawback layer specific handling order specify main thesis capture key model spirit one variable turn beyond category compete mechanism economic structure learning relaxed assignment indeed cover model boltzmann machine unit rbms bottom contain variable group type assignment bottom point assignment never fully observe support manner ever assignment censor category complete rank tie propose handwritten analysis believe complex capture form cross connection allow boltzmann energy decompose n w ik w column factor rbm boltzmann specific real generalise transform denote element evidence triple px cumulative simple rejection advanced section due transform specify conditionally box suggest abuse notation let mi mi mi need handle direction sign join multiple constraint refer often posterior g unlike rbms unless mean numerically update recursive ik unit activate normal density normal cumulative distribution reader refer supplement estimate pe general constraint however couple
within define l deviation member evaluate bounding bounding follow I bellman error section obtain write place subtract mdp distinguished move challenge expectation bellman mdp return expect law program bellman dynamic operator take difference mirror early bellman mdps infinite nearby might value violate satisfied lipschitz weak help future use sum reduce clutter global upon write dim dim e n shorthand n set reward bound problem remain contribution bound lose dominant together cardinality address posterior effort mdps fundamental reinforcement planning policy intractable approximate mdp require would whether learn complicated planning examine acknowledgment support stanford award foundation elementary martingale random adapt conditional generating guarantee early discussion f gaussian deduce early apply substitute least solution discretization bind bound fx fx sub minimizer obtain least desire result imagine disjoint without show delay update episode dim kx k I dim upon canonical recall q f fx trace pp I ij ij obtain bind q lagrangian v dual objective tr expression linear rewrite tr say mean imagine determinant say c bind eigenvalue equation bx f p say valid identify know let number exactly simple lemma argument function imagine linear controller reward reward eq semi ball semi definite x x equal value exclude outer high inner mean flat outline optimistic use confidence algorithm episode solve optimistic policy attain algorithm tractable possible claim van stanford university unknown parameterize bound scale cardinality characterize explicitly kolmogorov unify base reinforcement learn state several algorithm reinforcement optimize reward unknown markov decision mdp agent sequential decision cumulative environment mdp planning explore poorly policy environment exploit knowledge numerous statistical efficiency include pac approximately mb mistake know know focus attention reward link overview li broadly rl environment pac number plan exploration near bound attain grow cardinality space extremely even reward beyond nothing beyond parameterization degenerate mdp transition arm another assumption state linear quadratic constant grow remove exponential old regret linearity analyse intuitive introduce show finite exploit satisfie cardinality underlie problem extend previously bandit unify reinforcement provide art important consider horizon ap mdp define ms say mdp associate mdp mdp episode prior notation td dim notion specialized parameterized mdps fix state space future per equal shorthand kolmogorov eq definition regret bound parameterize mdps assumption section bound mdps control traditional notion dimensionality equation unconstraine constrain upon outline optimistic variant attain unfortunately approximate optimistic loose complexity potentially infinite mdp function I mean intuitively length know reveal dependent dimension sequence notion reinforcement
dimensional space method use valid many assimilation calculate constant lead power asymptotic measure long smoothness satisfied suggest remove random confirm small error constant exactly proportional converge infinity confirm asymptotic demonstrate noise method perform small organize theoretical technical asymptotic analysis derivation map depend subsection explicit analysis use formula describe theory read without theoretical put introduce scale theoretical rx dx fx q every sampler direct sampler successive independent evaluate may typically perfect would sampler relate weight recursively product grow rapidly convergence consider assume global degenerate unless state generality locate near zero hessian sampler equation relate behaved approximation direct use find monte independent compute estimator minimizer hessian consume lead term expansion depend large odd hope remove classical present remove map draw evaluate consider weighted formula un ensemble map identify use way pdf q see sampler symmetric proposal sampler expect nearly small regime oppose map review completeness name ensure except star shape straight set choose arbitrary jacobian jacobian gives vanish matrix immediately verify respect multiply map adaptation three use argument probability weight q concern recursive application call proposal roughly see theory scaling scale expansion satisfy rest satisfie drop simple constant conclusion small become order method method small corresponding simple map limit method advantage occur apply except must throughout expectation numerator straightforward numerator denominator conclusion expectation homogeneous degree degree give iterate identity derivation change calculation behind calculation justify rapidly expect expect linear map expansion expansion odd variance obtain expansion anti symmetric show computing see follow symmetry similar algebra symmetry lead formula multivariate covariance function value odd formula see e expand denominator euler give method lemma formula give expect expansion result expand map similarly large combination moment result though degree correlation use computational confirm may sampler depend dimension sampler bad assimilation problem later tie implicitly q cubic potential nonlinear part increment standard variable right hand vanish increment simple quality measure straightforward identity number factor possibility add simple calculation c subtract finally coefficient numerical experiment average computation instead logarithm use weight linear map save maximum normalize red line dot circle square slope illustrate line confirm expansion specifically numerical agree relatively measure moreover random map simple observe result illustrate scale tb show predict observe method estimate initial variable ordinary ode matlab ode condition correspond shorthand ode initial numerical sample vary scheme derivative hessian minimum fix collect become mode appear tb vary lm linear dot red slope blue slope six simple version small slope four perform perhaps modal shape however vary pdf functional analogous filter tb linear lm blue dot circle line slope blue slope confirm relatively analysis method random map regime become simplicity map limit noise suggest procedure analogous example emphasize concern sampler even bootstrap propose sample present sampler center posteriori take discussion wish analyze practice roughly require ode numerical hessian either formula difference simple require linear method require evaluation particle filter want cost cost must sample normally solve map exploiting guess
walk subsequence member arcs solid type graph constitute graph parent arc parent point point cycle line contain point preserve path semi direction shall semi point extend moreover path however text let v adjacent inner inner path apparent notice graph consist walk single walk section relevant context arcs section notice consist arc consider edge point point pointing point symmetric arcs line cg graph preserve least imply characterize keep otherwise finally walk identical independence graph disjoint stable conditioning conditioning purpose conditioning start generate line remove point turn line remove inner repeatedly newly subset cg obviously contain semi direction preserve cycle generate generate generate generate seven case except illustrate generate generate follow algorithm type prove preserve cycle contradiction generate preserve cycle unless path previous c cc b summary marginalization conditioning dag act graph parametrization graph parametrization maximal subject acknowledgement author grateful discussion support grant air force office scientific advanced research project deal marginalization markov property purpose chain mix edge class marginalization provide method generate marginalization conditioning structure start play independence arise study graph appear cg generally formal extensively specific come apparent tool capture independence cg exist independence distribution chain cg independence unobserved structure understand acyclic dags study graph order capture summary capture cg independence cg stable marginalization marginalization independence marginalization essential provide structure structure next section define definition section two read markov property stable conditioning cg capture conditional cg define mixed marginal marginalization provide generate marginalization marginalization conditioning graph marginalization marginalization triple edge two
flow cycle determine flow nucleotide flow flow use compact elementary nucleotide extract power similarly bivariate recurrence relation flow number recurrence length nucleotide flow recurrence relation solved solve extract appropriate notation expression nucleotide nucleotide flow cycle symmetric nucleotide parameter involve elementary nucleotide probability inner sum range cycle value normalization factor factor length alone interpret sum flow cycle show normalization factor expression moderate practically cycle small extra place clarity reason cycle availability make derive exact distribution length formula nucleotide flow etc nucleotide flow point calculation calculation precision expansion gp algebra point length cycle discuss scenario flow cycle cycle cycle cycle follow sequence variance flow need sequence mean length equal cycle reach reach determine equal nucleotide sequence nucleotide cycle sequence equal nucleotide show nucleotide nucleotide probability exact also continuous curve normal distribution eqs evident accurately nucleotide distribution variance eqs two equal nucleotide cycle determine flow cycle calculate tail leave compare order fix previous flow nucleotide discuss curve normal eqs nucleotide flow collective behavior deal nucleotide symmetry result eqs nucleotide individual appear symmetric nucleotide affect next section nucleotide flow nucleotide flow finish nucleotide expression respect nucleotide first distribution flow sequence last nucleotide different see mean linearly sequence distribution cycle nucleotide probability sequence nucleotide eqs magnitude probability nucleotide shift slightly confirm show curve mean variance normal multiply cycle previously figure perfect variance calculate mean length determine cycle nucleotide flow cycle extra clarity reason determine flow cycle flow nucleotide exact calculate continuous variance distribution eqs normal multiply normalization cycle cycle approximate accurately slightly long tail slightly tail distribution shift confirm exact calculate normal eqs distribution technique important sequencing technology case cycle cycle give explicit formula distribution accurately formula variance vary flow flow cycle target sequence long length big figure length sequence explicit useful platform guide monitor daily user generation sequence nucleotide number formula accurately distribution software development sequence generation sequence sequencing ever already biology concept sequence technique compare generation sequencing long read length sequence technology development software development flow cycle sequence fix sequence flow cycle approximate distribution probability recurrence length flow remain brief description exact present explicit cycle length accurately formula would useful practitioner protocol synthesis chemical protocol add four kind iteratively nucleotide nucleotide nucleotide complementary nucleotide
predict stock growth month time period availability business stock market throughout year representative operating account among display stock year determine performance market quantify respect yield positive gain respect stock financial individually financial becomes consider flexibility handle capability initially available stock stock currently analyze behavior discard stock financial sake completeness combine limitation comprise stock small stock price model trend often dependent financial far small market financial parameter critical describe use supervise classification datum financial address step eliminate financial stock explain eliminate stock miss threshold financial secondly perform stock year financial aforementioned company feature normalize begin normalize mean across stock year smooth financial smooth frequency filter domain ideal rectangular cosine transform dirac delta rectangular finally dct rectangular discrete width dct rectangular smoothing component robustness compute stock determine relationship financial describe stock vector denote corresponding financial year let relative date financial past year supervision parameter form year end explain stock current performance make year narrow stock subset obtain allow change supervision reasonable fix cardinality cardinality particular initially supervise near neighbor svm technique suggest svm accuracy svm financial capability financial parameter take form high exponential grid search gaussian kernel motivation work involve time series analyse support tucker condition consider acceptable minimal apply partitioned entire stock average average ratio stock stock prediction ratio stock stock find suggest technique stock evaluate portfolio comprise share allocation higher future return portfolio stock return stock market month optimize portfolio market mid long since analysis performance naive near error svms optimize develop mining effectively collect period art tool cosine principal stock dimensionality several space handling capability discriminate stock portfolio optimization svm portfolio month significant algorithm artificial intelligence believe fundamental room machine perspective broad range simulate allow see occur modify close finance perspective impact stock statistic portfolio build work assign important future
b perturb event optima contain thus continuous density distribute class stochastic insight execute version give algorithmic foundation sampling discrete case suppose build node index subtree root node root root max root read yield distribution inductive know max suppose like leave child amongst truncate exceed truncate partitioning strategy allows still make infinite begin respective set truncate yield think perturb energy eventually complete construct b intuition first bound lb lb lb mb efficiently without henceforth add log density draw recover likelihood add decompose tractable rejection procedure refine upper region lower come come eq region term execution illustrate begin dark blue dash split upper compute child medium node put region sample split produce blue bound great queue tree measure exact termination run partition region return maximization use need determine variant computation computation drawing evaluation region lead case unimodal child immediately without queue maintain provably relate global bind runtime sampling parameter appendix termination refine rejection appear discrete domain rejection maintain piecewise target proposal rejection reject choose split computed region process point accept perform identically parent choose draw volume proposal piece would piece rejection return effect refine unlike return refined blind computation ultimately dominate possible persistence inside refinement aim section bound second show line library automatically compare mcmc slice experiment region region side split proposal case unimodal function peak essentially average run clutter mean fix outlier put term log bound bound dimension half half ensure make grow draw grow exponentially reasonably tractable computation run alternative case instance likelihood allow implementation problem mean bound clutter term quadratic quadratic tight vary point important evaluation explain happen show width interval purpose programming choose inspire physics biology automatically sampler legend c letter parameter wish noise reasonable multimodal large difficulty five set likelihood axis difficulty evaluation rejection accept rejection large varie tie uninformative bound also cost evaluation plus evaluation heavy tail log set set equally mode near decompose put per appear compare refinement discuss early refine piece large split region per drawing range along evaluation tradeoff two rough computation cost experiment single variant trade computation computation representative operating refinement ask algorithm across early evaluation expense bind achievable slice computational budget sample slice across mode ever switch answer question case energy develop approximate perturbation think favor efficient computation approximation develop establish hope explore question high uninformative search region little rejection adapt rejection allow leverage particularly several follow estimator would like advantage might branch branch subset support thank early suggestion appendix detail shorthand eq cdf identity throughout independent truncate interested derive gibbs eq axiom theory analysis construction correctness top construction circle center g thick g north east dotted dash thick g color dot west north color blue dot east g north west north blue thick g alg blue indicate truncation run space form queue proceed effect top would distinguish value sort allow choice distribution top index distribution produce set give prof equivalence distribution small tree notice whenever omit induction realization realize max queue come tree parent indicator intersection subset complete get location partition subset independence max get entire partition iff split result section deal correctness termination decompose tractable terminate bound global analyze global closely realization return thing upper search let optimal least visit global terminate function lb lb k lb kx lb termination queue need simplify search alg rejection bound terminate iteration rejection terminate termination iteration terminate termination history nonetheless stream value bind terminate joint proceed first pdf cdf difference independence look eq equal basically infinite function multiply permutation cube complete correctness termination return linearity linearity bound termination return q stream obtain b correctness consistency requirement need verify b partition fine get different set realization split chosen thing optimal suboptimal bind explore produce reach explore region condition region explore proportional thus region
significantly improve partial filter pca pc generalise entropy roc curve pearson thus filter neuron baseline challenge unlike challenge use variable predict neuron method forest constraint potentially tree depth conditioning level pre use correlation average correlation outperform dataset important poorly term well nevertheless partial well limit promise work unable direction seem disadvantage respect opposite neuron interact predict network e already reach curve report imaging consists two I process neural peak activity ii infer statistic outperform method optimize prove simplicity constitute solid fellowship office specifically tune challenge description respect present tune dataset monitoring simplify dataset believe tune clearly simulator provide purpose implementation detailed presentation present experimental proposal section filtering compose signal statistic method filter filter apply difference peak magnitude remain hard complex filter separately method case process application term sensitive value see filter average correlation f display similar resp equal less competitive weight resp combine prove marginally improve statistic beneficial orientation heuristic activate delay activation neuron indicate direct observation could time neuron whose choose neuron difference modify orient association discover provide enough way contain matrix give use use try value pass filter average pass filter slightly ht c c undirected direct report challenge except connection connection per similar fire e highly topology apply confirm average full pc report slightly account average averaging method pc svd determine combination variable combination combine know whose activity obtain sort magnitude component explain neuron exactly challenge seem necessary exploit value immediate carry infer directly team private private winner r yet effective problem imaging consist step raw peak activity infer association neuron partial statistic lead propose compare respect partial human complex biological connect unfortunately direct brain feasible currently activity neuron produce intensity amount neuron time series include device slow decay represent set represent direct connection express edge indicate neuron term version win method correlation describe present property additionally win slightly short period peak long period raw process fluctuation fluctuation activity overall illustrate figure raw light affect quality accordingly filter frequency noise filter furthermore spike indirect indicator mainly filter model exploit although time exploit propose coefficient precision concentration assume I measure dependency therefore naturally detect association neuron spurious indirect partial interaction metric practically speak straightforward obtain improvement replace principal performance infer process
variable characteristic composition child status status education school health interest home missing survey status miss status unlikely variability neither miss solely complete case pairwise miss restrict vector separately avoid continuous spike participant decompose indicator indicator discard first complete status home since miss primarily account miss compare display proportion month also account apparent difference estimate differential display report regardless actually display quantity median accounting substantial notable exception estimate adjacent line tend gain effective show home home complete mi small interval mean complete mi relatively universe previously miss benefit amount size production conceptually suggest adapt incorporate finally model structural zero contingency impossible skip restriction among expect adapt zero adapt mixed depend value categorical vice versa area gibbs draw parameter outline exact collapse sampler adopt simple quite break missing entry full vector element entry equal categorical variable update chain dependency block get miss block first marginally accord drop submatrix q submatrix suitably component column stack vector vector iv v mean sample dx ib ib rt conditionals b bayesian mixed demonstrate repeat use imputation improve compete run counter model imputation appear default implementation method incorporate default difficult translate encouraging field production nonparametric bayesian continuous develop engine imputation mixture multinomial normal incorporate categorical normal component categorical component categorical continuous link complex minimal analyst missing select collect accounting miss dataset similar complete panel default imputation equation program united longitudinal public researcher policy census period four census change hope reduce improve evaluate census field give overlap draw production panel restrict state census construct comprise individual state assess collection result key among status approximately unless missing mechanism completely inaccurate conclusion multiply sample imputation repeatedly analyst combine rule analyst account uncertainty distributional imputation skewness combination categorical desirable use dataset nonparametric categorical flexible coherent engine idea mixture regression variable categorical continuous specify categorical variable induce assignment dependence couple dependence well attractive imputation help preserve suggest miss categorical si si illustrate distributional exist imputation include bayesian present potential improve default mi multiply miss subsample change accounting finally extension future characteristic survey categorical dependence distribution datum panel display work education level vary discrete usual hour eventually skew hour panel education increase education dependence categorical categorical education df df education level way usual child complex distributional sort imputation discrete variable infeasible cell imply impose often vector include certain assumption seem unlikely thus restrictive approach specify variable e fully imputation prove challenge effectively categorical outcome remain challenge continuous predictive specify conditional undesirable relate distribution typical equation however model distributional many additionally conditioning sequence could result sample individual let let use variable respectively valuable categorical categorical model brief exist model follow intuitive beginning multivariate continuous truncated dirichlet mixture assume eq prior break construction dp introduce adopt truncated version mixture ij I assume imputation model mi engine arise h make put burden normality dependence dependence categorical true matrix distinct structure burden somewhat allow possibly interaction survey component much particularly since allow component independent add restrictive example dx unable already code however maintain give assign truncate stick give r induce component indicator probability assign truncate stick eq distribution call infinite shape rate equal convenient truncation level moderate initial paper specify parameter default hyperparameter usually insensitive hierarchical inverse wishart restrictive relatively particular poorly estimate insensitive marginally large insensitive marginally categorical conditional still latent distribution cell table mixture rx normal represent mean encode encode within matrix dp relax within component well low level parsimonious somewhat assume much flexible factorize author grow due assumption formulation force strong assumption number mixture induce distribution incorporate predictor within model discussion suggest discussion overcome cluster coupling model assignment couple mixture variable encode weak within dependence analyst simultaneously avoid center multivariate dp dp separate conditional assign dp difficult interpret margin somewhat arbitrary evaluate imputation study take wave wave exclude record consist census select continuous categorical variable table efficient keep challenging imply table cell status home education worker work increment hour approximately ensure complete create distribution describe material keep conservative half practice carefully relevant parameter mean quantile variance complete run cluster mid sampler minute make much scalability incomplete implement software application logistic conditional procedure mi complete incorporate age year expect rather skewness make normal likely coverage width imputation superior repeat sampling half coverage rate great nominal coverage shorter bias confirm range small driving capture effectively tend additionally year increase function year standardized child child child home appear play home vary correlation age child log old year year regression
confidence inverse difficulty low parameter rely geometry estimation cone capture statistical inverse complexity rest definition review formally program gaussian geometric beyond relation discussion proof result notation introduce show instrumental characterize difficulty statistical estimation later section complexity program convex duality use denote unit euclidean ball confusion also transpose inner matrix k nc na vary place place high much lie sparsity noisy completion linear view without loss generality standardize notion collection atom denote collection basic countable illustrate figure atom geometry functional q element hull atomic norm cauchy schwarz ball atomic hull atom figure collection atomic determine cone cone cone illustration red black blue dash notion look atom atomic recovery atom atomic norm polytope tangent cone ball tangent cone difficulty sparsity geometric cone refine local inverse regression completion matrix atom consist nuclear spectral hull nuclear parameter scale geometry tangent cone recover atom sign vector cardinality hypercube tangent orthogonal recovery constrain case hull local share geometric geometric two width compact standard multivariate quantify orient introduce show noiseless deterministic example explicit p proposition tangent cone use analysis estimate set euclidean capture balance cover radius radius maximize determine measure entropy width geometric quantity convex volume following achieve linear geometric quantity tangent cone isometry constant isometry define cone isometry preserve tangent isometry constant atomic local tangent cone model high respect introduce lead serious directly generic atomic tuning localization depend geometry atom give atomic minimization complexity specify localization generic utilize selector minimization trace regression investigate section set important inferential point inverse consider feasibility satisfie tune solve version convex program estimator parameter normality contrast test normality investigate program unify treatment general tangent cone hypothesis include theory geometric problem let analysis selector nuclear strong tangent cone geometry analyse asymptotic constant capture extend analysis atom atom unit localization radius guarantee feasible tune gaussian atom arbitrarily accord attain choose operator follow gaussian ensemble convergence bound atomic upper bound statistical accuracy drive two dimensional loss quite key lemma first follow gaussian tuning value address gaussian call operator isometry constant isometry constant guarantee capture local isometry bind design around linear like geometric ensemble design bias program empty feasibility q interval low contrast fix nan standard contrast variance nearly information dimension require equation important tangent cone cone affect tangent gaussian design universal stand conditional statement basically two follow cone determine universal b problem quantity width estimate section apply general high illustrate defer assumption sparse recover reflect gaussian follow geometric regression interest sparsity probability inferential guarantee biased selector type asymptotic normality interval length parametric furthermore confidence interval recovery trace regression lead recovery nuclear assume sense examine detailed calculation characterize low recovery assume true constrain nuclear estimation loss inference theorem happen width atom rich use construct confidence interval perform hard sharp theoretically approximation sign sign norm ball program eq general lead sign sign constant recovery please detail minimization program interest orthogonal calculation rate apply geometric constrain orthogonal recovery model high universal formalize framework permutation plus rank completion permutation atomic corresponding atom nuclear norm view see develop section geometric gaussian carry concrete specific distributional essential shall design random control section choose make happen tangent cone rate local isometry prediction regard quantify local tangent isometry condition illustrate behave worth note condition distributional view constant radius calculate distribution feasibility follow eq analogous event choice tune estimator decomposition extend set distribution contain gaussian intrinsic difficulty unify design compact convex lower bind inverse cone q mean condition volume ratio see apply cone cone cone parameter define design theorem low isometry design ensemble design upper link cone compare upper mild upper left give section eq leave see local tangent cone sharp p reverse hold tangent cone vb obtain sharp please reverse paper unified characterization statistical inverse major tool process minimax low dimensional general packing derive generic arbitrary star shape case prove divide lemma easy prove prove bound defer fashion tangent cauchy schwarz atomic early eq complete case least event easy calculation know p monotonic property probability least lemma reduce probability feasibility indeed high probability column lemma establish relationship last consistency key know denote packing respect e recall eq assume standardize linear cone packing cover ball element would like fix think nk trick fix lower modify introduce closed p tangent cone observation q step require standardize linear generality assume standardize inverse cone packing cover eq universal combine plug convex cone corollary application denote refer width cone proposition simplicity rate treat mh sm width bind tangent sparse vector lastly euclidean enough mh h th tangent cone transformation unit frobenius proper enough corollary corollary cone euclidean gaussian front thus tangent cone orthogonal show euclidean preserve v apply definition school unify ill pose linear compress orthogonal computationally feasible program statistical interval framework inference difficulty capture characterization width estimate drive wide high dimensional draw mathematic computer electrical often fashion mainly analyses statistical interest nz n dimensional much commonly assume atom capture true inference
convex dimension small relaxation unfold u ki ks element n u ki k e least cover compact maximizer believe distribute tw ji k moment ki use hoeffding sample replacement entry proof tensor position maximally hoeffde obtain theorem spectral
mis specification secondly structure establish chen estimator four asymptotically numerically asymptotic obtain specification appropriate series expansion enable illustrate difference four asymptotic notably great technique tend replicate asymptotic former extensive mis superior term mean range mis specification produce mis criterion three estimator demonstrate mis specification criterion mis short memory dynamic true form four mis specify pseudo mean mse mis specification also lemma proof derivation text bound specification spectral estimate away set derivative j g mis model lag autoregressive assume root root unit circle assume finite infinite specification process already assume move absolutely slow rather exponential typical process accordingly provide e realization process generally objective mis say explicit mis produce specification could size consequence subsection estimator hereafter outline derive subsection alternative minimize define minimize frequency estimator minimizer subscript limiting follow exist argument imply criterion express explicit covariance derive mis specify log function estimator expand expansion furthermore operator q appendix q vector density satisfie respectively estimator mm I ahead would mean implication mis specify member value close spectral give fit parameter value predictor class square mis applicability distributional however form associated order well relevant derivation specify determine relationship deviation derivation certain deriving expression full model polynomial power expansion investigation normally nonlinear infinite sum determine expansion replace sum magnitude truncation numerical memory order mis ar arbitrary truncation arise computation spectral expression take variance operator ar fractional series expansion gamma numerical alternative formulation expand yield denote use namely rr function mm obtain algebraic desire numerical expansion numerically ni dd q n mis specify sr derivative exact ar derivative readily pseudo evaluate treat similarly depict contour function discussion choose coincide great perturbation version contour indicate limit neighbourhood parameter behaviour elliptical function close globally turn mis show result write critical nature deviation property range chen derivation estimator theorem chen estimator establish three case outline show converge proportional objective see detail present key point pseudo something mis rate great sum case explore sample memory type mis specification estimator exercise obtain method adjustment curve depict statistic bias mse pseudo mis specification value function value two mis result parameter lie interval mse relative computation need specification replication mis correspond value subsection document mis specification report term true form mis specification turn three list ease mis four estimation j k reference scalar mm define give bias adjustment tackle issue turn begin apparent general separate expression e four formulae scalar specification standardize plus truncate replication underlie limit use kernel formulae u expression evaluate formula necessary behind method estimate maintain mm particular table label n specification size sample increase location go limit salient note cluster domain estimator small dd label dd domain discrepancy distribution reasonably limit distribution case provide four plus visual domain domain perhaps produce reasonably supplement four pseudo true formulae evaluate subscript table additional key value set highlight bold table small bias estimator bias mse theoretical tendency sample value addition tendency value two tendency standardized cluster far estimator theoretical mse estimator increase indicate pseudo parameter overall memory increase nevertheless small uniformly preferable mis still perform bad result record table mean severe mse bias tend suggest mm ar cause mse mm due limit experimental suggest dynamic correct form act extreme mis consequence mis specification severe coincide nest match mis estimate example estimate mis mse lack mis bias mse mse parameterize specification superior assess various specification parameterized model fractional relative estimating effect highlight estimator mis specification mis majority superiority robustness square mis incorrect relative estimator estimation estimator combination ratio c mm l c record confirm time efficient exceed estimator across specification formulation obtain dominant paper theoretical relate mis mis specification four pseudo general demonstrate mis specification pseudo fractional finite estimator extent specify extreme mis specification frequency together former distribution closely mse calculation superiority mis specification establish necessity gaussian dependent process straightforward relaxation restriction consider seem desirable true upon interaction mm extension result anti persistent stationary facilitate broad circumstance extent cover prior possibility mis specification explicit offer approach unnecessary latter also relevant boundary practice previous stationary fractional offer sensible investigation conduct smoothed situation fourth relationship bias parametric estimator mse mse express e mis depend magnitude mse bias imply mis point raise question within estimator still value reference practical circumstance remain current proof even value continuously set partition ny ny distribution conclude variance integral g n q em limit third ik ty ny ty therefore uniformly mis specification toeplitz zero reverse respectively deduce tn limit
term account software potential linearly vanish cutoff continue linearly compare force field comprise inter parallel force schedule replica temperature gradually boltzmann ensemble neighboring overlap tail varied varied force field obtain accuracy receive physics ny usa university usa research division institute ny interest include digital brain computer interface image ph university usa currently physics college work employ bayesian characterization institute mathematical university laboratory california technology usa division ny bayesian application present along numerical technique processing reasoning employ bayesian processing physical science perform evidence often overview behind technique great excellent span decade spread great neural nuclear physics engineering statistic general measure quantify within topic logical enable logical context probability consider statement propose particular conjunction write derive basic boolean logic expression result consist evidence think rule knowledge posterior probability must assign may possess instead probability rule leave assignment result inductive logic inference indicate symbol quantifie summarize particular maximize refer probable map estimate value act give prior problem factor sum often model theory compete probable examine rule way respective case prior lead concept equivalently odd odd q ratio two bayesian minor factor selection special demonstrate vary information possess assign effective likelihood around attain product well achievable goodness factor width multiple volume compatible volume knowledge compatible datum new prior pay entity necessity increase model interval beyond achieve upon factor scale increase normalized unity deviation ignore factor involve factor fall outside decrease rapidly actual parameter single computing odd comprise factor ratio classical selection comparison importance analytically technique quite expensive large forward point reader excellent resource review recent find respect zero eq approximated markov chain attain extreme compute ratio evidence write q sample eq avoid cause integration evidence build analogy statistical degree govern energy evaluate compute high dirac delta function know system energy heat degree view evidence write knowledge way canonical evaluate evaluate evidence previous class parallel method fractional power letting vary smoothly bridge recent sampling method focus logarithm connect distribution path define relation canonical constant yield choose constant energy distribution approach sample produce along path evidence sometimes accurate combination compare share odd define share simplify draw mixed log odd integration open intermediate rely stochastic numerically posterior contrast nest aim cumulative mass contain contour evidence likelihood understand nest use prior likelihood since decrease monotonically sort sort mass use increase thus iteration already attain valid randomly state evolve increasingly give sampling focus high constructing nest restrict higher high nest state strongly evidence simultaneously sequence contour typically nest rely k easily serve peak drawback nest facilitate sampling boundary help arise determination expense introduce parameter focus detection analytically focus spatial sensitivity light sensor estimate select molecular mechanic protein evidence ratio analytically correlation detection amplitude filter originally brain computer interface detection record channel couple refer coupling nan noise denote refer signal state q symbol detector record mt relevant coupling latter amplitude care detect represent without refer detector channel deviation signal entropy assignment mean know equal signal assign parameter write gaussian involve restrict amplitude integral odd subscript amplitude odd subscript range expression contain information aid signal filter synthetic eeg eeg response response commonly matlab channel synthetic eeg pz channel epoch ms length comprise hz single remain eeg effect snr filter performance create varied db typical see eeg application illustrate target signal snr snr db curve correlation b roc curve snr detection oppose quite snr db sensitivity specificity mean specificity I correlation detection cutoff curve b signal snr db filter traditional quantify respective roc db filter consistently traditional filter snr quantify area roc curve snr consistently snr demonstrate select order gaussian model sensor accurate pair surface measure intensity reflect light spatially weighted sensitivity make light sensor place surface consider gaussian sensor amplitude center constant unity gaussian model consist gaussians require subscript index gaussian gaussian intensity figure surface illustrate record increment step sensor surface sensor corner break parameter nest nest iterate stop log value less four compete model evidence time probable model compare black make intensity show excellent agreement involve light sensor determination importance effect find currently method characterize involve series observation light come distant star mechanism specific system perspective variation reflect light throughout effective back distance angle observer connect star contribute side spectral response side remain involve star star center center observer star amount star approach decrease velocity two shift due proximity star induce variation twice since star change approximated exponent anomaly total effect bayesian effectively create comprised effect four orientation allow circular evidence determine whether present perform call period temperature sort star emission fact
lot attention devote multimodal consider fusion fusion investigate recent elegant come bayesian value low misclassification propose extension tailor indexing pairwise allow improve precision take diversity provide adapt challenge benchmark multimodal issue multimodal bring complementary improve image analysis survey modality group extract another order discriminate concept scheme one available merge see unimodal however heterogeneous fusion score fusion outperform give base decision min refer stack extra step stack classical majority vote I simplicity scalability due dependency view combination account machine indeed consider diversity illustration adaboost weak distribution introduce adaboost tackle take account diversity strong framework express program learn vote function minimization diversity learn aim usefulness fusion good perform layer positive base extension loss organize follow fusion image presentation quadratic weighted majority vote value machine theory feature label output size define choose majority vote low specific ib mm h introduce margin positive convention author prove inequality minimize counterpart vote justify elegant pac generalization principle sn nm ji nm h mm finally vote n minimize denominator show performance classification stand rank base pairwise preference discuss usefulness diversity key success combination indexing justify vote maximally uncorrelated diversity popular pair classifier correlation disagreement etc diversity disagreement pair rewrite moment mm objective moment imply direct optimization imply diversity maximally appear fusion separately accord hinge relaxation previous hinge hinge loss slack abuse however incorporation hard relax size stand framework map aim majority imply good trade maximally ht car cat c concept svm car cat average empirically usefulness fusion stack approach objective concept less unbalanced carefully classifier reason positive ratio keep could index sift local binary gradient histogram color moment gradient dimension vector image thresholde neighborhood monotonic gray sift codebook semantic sift codebook mm rbf svm classifiers first h unweighted vote weight vote mm increase diversity follow set classical stacking finally fold validation low select lead performance report firstly fusion correlate linearly fusion experiment clearly baseline high student pair confirm p student test statistically produce constraint hinge really helpful diversity expressive hand diversity vary layer achieve concept diversity average preference student pair value preserve keeping note cost lower pairwise approach showing without approach good classifier account retrieval diversity come feature variability classifier error rate majority vote diversity adaptation preference indexing appear naturally prediction train modality fusion set confirm fusion indexing task beyond
unconditional family multiplying preserve parameter condition sum count poisson multinomial regret universal maximize close greatly compression likelihood likelihood small distinguished role compression sequence count extension eq positive q simply maximize poisson value later great string nevertheless approximately approximately bit description beyond know ideal maximize poisson match multinomial alphabet regret bit parameter minimax bit price pay sake code simplification additional know total arise moment depend alphabet count alphabet total count advance na maximize much tail law sort distribution datum within set demonstrate multinomial count strategy minimax alphabet strategy computational normalizing computing conditional require arithmetic code symbol appear large typical mainly give asymptotic pointwise alphabet formulate redundancy symbol find form asymptotic model focus infinite envelope pointwise work envelope provide large alphabet code purpose consider large realistic code prediction unnormalized count match total entropy projection equality approximate conditional investigate unconditional compare gain describe arithmetic code arithmetic count count introduce iii give simulated detail introduction unnormalize maximize likelihood count count prediction reduce sample right hand probability specify count question leave count maximize term restrictive target start look strategy use code count poisson maximize normalized regret product vector count set count follow follow minimizer partial bound derive pick rest leave close list case multiplier numerically regret expression redundancy reduction bit depend expression bind multinomial distribution maximize upper within easily see flexibility parameter introduction uniquely symbol relative portion symbol sort quite skewed string tail occur subset n nf ratio give mainly remainder product define dimensional symbol rest regret denote distribution minimization derive nf treat code alphabet count alphabet regret symbol symbol symbol induce bit contain count extra flexibility achieve piece code adapt count bit work tail count symbol finally suggest replace string envelope suppose envelope I string distribution envelope distribution restrict minimax envelope tail need envelope match regret alphabet datum choice envelope symbol string discuss fast enough symbol arrange realize count maximize approximated method c similar lemma approximately bit average lead achieve minimize account string finite length string look count component maximize poisson condition maximize multinomial count multiply uniform string stochastic confirm horizon sequential log conditional conditional produce cumulative regret sequence accordingly simplify study large jeffreys jeffreys adjust symbol alphabet predictive hence symbol discount predict rule alphabet frequent symbol approximately symbol assign code book translate chinese bc book rarely code character character total character appear small character introduce maximize poisson count performance code close assign optimal family pointwise finding produce early distribution characterize distribution count precisely fact suffice fig solid hence area curve although unnormalized mid stand term portion half step rearrange upper refinement approximation sum integral also due fact moderately large well k aa know sum gamma achieve follow bind upper denominator attribute little algebra bind need lower eq since attribute step approximation pick numerator subtract yield taylor expansion term get large follow equation c definition q therefore distribution condition equal eq say value function redundancy distribution moment alphabet distribution total count redundancy minimizer redundancy therefore minimax redundancy magnitude part second resemble taylor hence easily partial inequality attribute pick envelope class multiplying exponential normalize na minimize depend envelope symbol eq approximation deduce hence
concavity elementary symmetric negative argument consider km ne inversion restrict column set annotation annotate dpp compute theorem theorem theorem corollary pt pt suit prove useful application efficient dpp learn dpp dpp even study diversity process dpp configuration characteristic recently play increasingly role diverse configuration spread kernel define span kernel associate decompose eq interpret point ability maintain via remarkable efficient fix open set dpp arise maximize numerator concave log determinant contribute convex lead non even assumption form parametric use convergence stationary function kernel bayesian dpp capture scale unknown inefficient ascent mle case differentiable occur limited scenario size counterpart dpp kernel mle technique inference sec modification accommodate sec derive dpp assume know explore model checking technique method dpp spatial diversity image discrete set application size elementary recursion continuous naturally operator appeal density give dpp continuous eigenvalue dpp generally include kernel quadratic show approximation finite represent propose sec consist dpp dpp dpp dpp log iteratively shrink ascent ascent guarantee discrete dpp straightforwardly polynomial material optima dpp present sum gradient continuous applicability maximization scenario dpp operator truncation gradient unbiased attractive hold optimize kernel dpp dpp prior neither markov highlight hastings mh slice although method employ walk proposal tune width distribution walk mh posterior efficiency tuning proposal conservative exploration need proposal use first first slice width expand interval accept otherwise become new boundary shrink markov propose accept alg way extend propose expand whether slice direction approach slice illustrative two dpp evenly spaced square dpp posterior mix walk slice sampling position slice indicate mix slice continuous inefficient infeasible even case eigenvalue seem suffer upper accept reject completely necessary immediately immediately far begin iterative exact circuit ratio alg supplementary tf tf apply candidate slice decide case accept slice keep newly reject generate new repeat propose proceed manner interval computation examine slice bound increase decision make adjusted interval procedure supplementary bound incorporate mcmc convenient truncation kernel arbitrarily dpp corresponding explicit truncation show sec dpp contrast mle even know prop combined eqs supplementary dpp eqs sampler challenge additionally tailor inference moment theoretical marginal eq case nm moment form certain case eigenfunction analytically moment define sec polynomial challenge eqs analytically available low quantity estimate numerically moment estimate large provide continuous dpp operator furthermore trace learn gamma supplementary material scenario use dpp scenario first estimate vary sample moment total point stationary allow many many interested quantify datum instead dpp material derive result parameter image study patient cluster diabetes phenomena stage subject average analyze highly parameter dpp quantifying level moderately analyse sec perform parameter leave evaluate hold sample eqs since preferred isotropic specify material slice sec fig clearly separate two class classify six six examine dpp quantifying relate diversity different category image search diverse search retrieve diversity human annotate six image total five dpp sift material via amazon present remaining ask experiment result spread evenly spread evenly aim human annotate differ six google extract type descriptor supplementary material category image dpp human examine consider probability add dpp category parameterize dpp conditional dpp kernel use informative prior annotate partial set different category annotate significantly feature human human human engine put keep highlight diversity ignore application combine relevance diversity top dpp kernel informative diversity tune accordance human annotate versus increasingly popular dpp infer parameter addition characterization continuous show check dpp study human diversity annotate gradient ascent ascent provide parameter dpp theoretical dpp straightforwardly example discrete gradient ascent
previous generate total follow leave ei predictive ir global maximum algorithm stop maximizer mean median ir function interesting correspond average median ir tail exact good es ei show well ei another series experiment optimize well fix ei hyperparameter es work fix use slice produce nb single hyperparameter section hypercube show ir random initialization nb es nb es advantageous approximation perform ei ei appear entropy explore iteration relatively greedy ei consequence multimodal one network optimize iteration return production term ph medium standard return series adjust daily return stock ba student portfolio truth noisy measurement predictive adjusted sample negative must exist fourier dual write consequently stack version result briefly observation stack easily predictive inversion lemma rewrite expectation approximate position optimizer order enforce location stack include note block kernel additional explicitly factor q multivariate non replace z parameterized compute ep start factor factor remove focus eq replace ep kl remove influence cavity set approximate perform form cavity set moment obtain normalize v identity factor constraint hessian contribution cavity q soft moment turn prior concatenation write q assuming covariance covariance associate eq block zero diagonal arrive university propose call gain acquisition term expect reduction approximation alternative bayesian hyperparameter es synthetic application finance show gain optimize find maximizer nonlinear function derivative furthermore evaluation challenge model computation minimize evaluation graphic another application drug chemical treat hyper maximizer provide noisy output likelihood sequential propose condition iteration final recommendation global maximizer latent describe guide gp prior specify jointly likelihood observation location condition past mean bayesian technique guide maximizer acquisition optimize location intuitively acquisition area likely encourage exploration search space recommendation global several acquisition example improvement improvement alternatively acquisition I draw green black thick dotted anchor cm height box box east optimistic posterior explore uncertainty expect global maximizer acquisition empirically evaluate real gain theoretic interested maximize location measure term differential entropy expect reduction acquisition represent predictive exact infeasible main difficulty computation analytical evaluation perform note equivalently since maximizer intuitively unlike previous entropy analytic entropy analytically marginals py ny term approximate sample approximately use code publicly maximizer domain restrict point dimensional probability write return element vector process multi armed could maximizer domain sequentially construct optimize however evaluate ultimately necessary instead sample derive existence dual feature consist stack inner mapping conditioning correspond finite approximately early approximate argument n introduce additional intractable difficulty definite far assume give force negative hard large simplify require multiply specific encode constraint briefly detail f incorporate joint multivariate derivative computation incorporate cdf integral form expression approximation propagation implementation ep gaussian process approximate non whose approximation I I describe constraint concatenation function approximation computation one classifier incorporate multiply give approximated normalization density cdf approximate variance py unstable occur multiply reduce produce prediction actual constraint acquisition acquisition formal treatment variance gp likelihood acquisition respect correspond integral global maximizer draw acquisition note acquisition respect information parameter approximate necessary evaluate independently input cost dominate inversion hyperparameter ignore derivative observation impose constraint less experiment square zero covariance normalize
signal turn dct compute dct exactly meaningful estimation complexity requirement prominent include discrete cosine transform series possess complexity transform imply transformation new dct quantization coefficient expect exact dct attention good dct approximation propose approximation dct scale round let round implement environment matrix shape attractive nan multiplicative shift transpose inversion make coarse dct presence scale scale factor minimize dct range compression ratio drawback orthogonality result method e encourage discuss adjustment computation diagonal therefore dct approximate possess assessment matrix introduce additional overhead dct processing quantization merge quantization procedure ultimately fast assessment addition multiplication bit count less additional good shift comparison numerically evaluate dash line ccccc assess methodology set standard image bank employ imply mathematically domain retain remain adopt reconstruct process image degradation assess peak noise utilize consider average may compression ratio moreover could outperform mid compression expense index square understand assessment employ image compression percentage dct mse lead clearly adequate high application bit additionally operate demand ratio recognition popular compression compression qualitative retain compression reconstruct propose reconstruct image via dct superiority reconstruct image quality correspondence dct low scenario mse possess constructive usual approximate mathematical series approximation take transform offer option circuit acknowledgment support de dct zero multiplication bit spectral dct adopt design superior sign cosine algorithm computational dct complexity transform compression cosine transform video image
account quantify covariance time instant time previous case would reduce sample problematic chapter issue divide segment correlation dft jointly absolute auto cross component dft stationary correlation great insight inference frequency procedure reduce detection detection exchange achieve accommodate frequency dft screening row column specify number characterize value dimensional specifically give expression detect threshold discover bound result phase corollary independent underlie evaluate statistical discovery organize definition multivariate establish spectral value screening characterize discuss complex screening multivariate framework triplet represent event represent letter respectively denote bold bold low letter cumulative distribution pdf pdf follow definition conditional conditional part real part value variable real part gaussian vector entry hermitian write coefficient matrix letter clear context entry x x order random ik ik time dft translation property toeplitz depend represent toeplitz write ik state series dft e asymptotically uncorrelated auto cross toeplitz give jt I n generality time zero ik z functional ik km I km I km j toeplitz tn tn tn tn write toeplitz matrix km n g km lm circular dft lm km lm km lm km lm therefore ik e km equation let give ik km g km km g kn ik km kn I ik km g I magnitude summation km kn I I I lm lm n proof ns therefore n conclude ik j ik use ik n specify real value covariance write ct stationary ar process r conclude dft ar sequel assume series jointly dft functional theorem independence component property absolute asymptotically time screen dft next discuss correlation screen value identify highly method apply discover frequency matrix definition result characterize regime transition refer integrable strictly decrease assumption generalize make screening normalize correlate component sample assume correlation partial correlation matrix matrix h apply discovery threshold I correlation screen number screening level depict match match pt arc cm arc complex case exist h norm represent unit relate section area x onto fix spherical define uniformly fall distribute generality x chi square v u fold average detail I I dependency coefficient fig screening expression probability discovery least dimension use generic partial j represent population matrix distribute cc first prove vertex graph ip inner summation set distinct l ip k follow sequel index index among sufficiently large p expectation p l lm l identity lp summation p p representation e f combining op sum chen define b j ki p k chen stein index care op line round join round pattern pt pt pt pt pt pt pt pattern pattern pattern pt pattern pattern pt pattern pt pattern pattern pattern pt pattern pt pt pattern pattern pt pattern pt pt pattern circle line circle pt width circle circle width cm width pattern width circle cm width pattern circle cm pt circle width circle width cm line pt width circle width pattern circle pt circle circle circle circle red color pt circle color blue color circle pt color blue circle color color node red color circle color blue pt color pt color circle circle color color white circle circle white color circle color pt circle color circle circle white color white color white circle color white circle pt color circle white circle pt circle pt color circle pt white circle white circle white white pt white pt color color white white color circle color white color pt white circle color circle pt color color white circle pt white circle white circle white circle white circle pt color white pt color white circle color white circle pt complement union red index complementary fig inequality term argument max remain multiple integral op apply relation pp op p immediate provide expression goes prescribe weak define corollary depend evaluate argument zero arrange ok discovery depend assign statistical corollary large sample reduce conversely number wise discover least exhibit phase fix sharp phenomenon motivate definition critical threshold approximate j number critical matrix real screening complex value exponent different small threshold degree dimension triplet small example one discover bottom show either reliable discovery increment sufficient critical enough bring value correlation stationary sample divide series frequency available construct partial correlation series magnitude correlation quantify statistical significance significance statistic extreme assuming screen value maintain degree assume illustrate equation help initialization degree degree select value partial significance inference problem perform aggregate inference straightforward manner example easily value series j degree frequency time frequency degree I l simulation confirm
compression ms refer generation initial dm compression cost initialization pseudo quadratic neighborhood alg computational nature sensible presentation effect cause effectiveness calculate dissimilarity describe ds alg set permutation obtain arrange accord achieve consider generic efficiency compression rs asymptotic case compression efficiency implement varie sec notably theorem compression operation alg clusters parameter parameter present estimator synthesis dimensionality sample restrict experiment choice use exponent easily cause representation graph certainly cluster optimize one evaluate input go complexity section update improve subsection dataset adopted discuss well graph repository contains consider finally first character characterize level image brevity essential reader ref reference discussion since dataset contain characterize vertex edge vertex measure none none complex symbol version sec sec version compression theorem variant variant however first one near neighbor nn equipped primarily herein previous work fast train four aforementioned variant therefore straightforwardly v meaning fuzzy nn summarize configuration execute genetic population synthesis fitness setup allow comparison genetic random mutation aforementione code genetic implement consider execute moreover affect h size preliminary test process time seed report report cpu rs synthesis test regular cpu ghz operate measure routine library refer estimator adopt specifie feature base operate ds function adopt system notably operate classification detail aspect become constrained scenario big entropy operate dissimilarity c nn nn nn gmm soft svm fuzzy soft le svm svm bayes svm c c nn report c v c v mm mm set c c c improve discuss characterization dm enyi adopt construct compression directly compression level explicit set proof consider technique proof study cluster generation system known equip arc adopt result fast less parsimonious concern rs nn rule yield accuracy serial cpu focus accuracy confirm effectiveness moreover cpu highly global bring close applicability big large vector dm row know mainly worth performance obtain embed dm could enyi neural straightforwardly develop like generic object focus bad case give th prototype good assume adopt compression bad compression purpose sequential sequence first cluster consider alg well possible preserve bad order instead odd order alg consecutive element maximum consider combine obtain compression claim single cluster radius spherical measurement evaluate define dimension value estimation quantity input maximizer monotonically relevant remain change achieve therefore normalizing compression side hand simplified provide singleton convention express parameter eq evaluate theorems theorem theorem enyi represent label graph become increasingly field intelligence tool procedure test label mining operate computational complexity viewpoint major purpose dissimilarity theoretic evolutionary focus key subroutine system compression effectiveness result variant indicator classification structural classification set considerable concern complexity offer powerful interact static scenario application cite almost scientific circuit network general label graph rapid motivated availability describe complex orient level operate label deal notable map classification enabling possibility adopt recognition cause intrinsic label topological semantic vertex design classification inexact operating prove offer complex dissimilarity embed classifier show art term synthesis classification theoretic interpretation dissimilarity dm practice characterization effective compression deriving demand improve version scheme rely formal computational synthesis maintain art performance elaborate scheme estimate enyi fast technique span formal compression experimentally demonstrate operating estimator comparable result underlie organize scientific section original classification system throughout primarily discuss bad efficiency develop comparison classifier dataset dissimilarity base min max conclusion direction designing concept key estimation mutual information describe quantify model quantification characterize system shannon generalize formulation interested generalization enyi call enyi enyi q subsection enyi technique sec introduce entropy eq eq simplify enyi plug theoretic datum descriptor entropy kernel zero mean kernel rise enable trade dependent random gaussian evaluate assume value input extent notably normalize evaluation cost measurement nd connect mean straight r entropy estimate accord end set span calculate term constant dimensionality approximate enyi entropy suggest parameter perform task definition sensible measurement account respective euclidean edge quantify involve computation well know algorithm cost adopt fast approximation concern dissimilarity representation element pairwise key nonnegative dissimilarity relevant set rs dm property adopt dissimilarity obey common metric say metric dm embed unnecessary computationally apply dm consists call ds embed fast represent dm common perform linear correct dm aim preserve euclidean mapping spatial representation selection play course role technique select essential feature extraction many image verification clustering system attribute tuple edge topology moreover generality cover broad graph order pair undirected generality nonnegative label face difficult provide quantify graph non difficulty proper effective possible vertex mining define mechanism component modern graph embed basic transform np term basically aim assignment vertex graph successively edge proper positive enable hence applicability whole e svm operate embed geometric information divergence distinguish main category define work extract characterize former g accord capability adopt core g restrict variety graph effectively representation adjacency laplacian reader therein represent input dm configuration dissimilarity although heuristic dissimilarity operate assignment vertex graph edge ds dedicate operation quantify dm fall well unique normalize evaluation range yield extent system operate svm assign test fig give rs fig synthesis fundamental parameter entropy entropy fundamental role compression cross learn validation global optimization analytical respect global allow make use hardware software implementation characterize arrange code entropy threshold label dissimilarity measure range characterize synthesis objective dissimilarity representation compress expand evaluate recognition validation account relate characterize capture dm increase spread separability class assume presence class would compression search rs rs compression define dm dm submatrix value vector measurement concentrate joint close systematic compression representative prototype evaluation first address algorithm complexity grow analyze compressed dm denote corresponding prototype concentrate value estimate become degenerate prototype extract graph notably derive search recurrent although try new searching subgraph improve primary computational first present rs advanced compression sec sec rs characterize sec operation synthesis rs characterize binomial operate cause unbalanced entire dataset use mutual avoid burden involve estimation basic algorithmic scheme grouping dissimilarity main sophisticated generate intra low deduce analytically consider certainly proof order radius maximum cluster close representative cluster dx ix th cluster representative together compression
self prediction immediate observation prediction coefficient prediction depict prediction prediction rescale normalised squared measure series approach normally distribute self assume analogously self respective equation denote continuous depict measure substitute approach discrete track cover pair possess title track another comprise cover average track furthermore perform base meta pre audio music use define query track average track seek cover member entire harmonic content audio implementation account deviation hz extraction describe magnitude apply wave across band pass apply window determine autocorrelation maxima centre incorporate bias towards preferred obtain programming average root euclidean component predict note evaluation align predict interval feature account key cover summary vector sequence correspond circular shift denote circular prior discrete aggregate across track codebook execute square method detailed measure list method compute measure matching case parameter computation obtain string map character describe algorithms string complementary string compression loss x average perform self align analogously predict distance string algorithm kl symmetric compute symmetric distance performance divergence jensen divergence symbol normalise evaluate delay embed radius preliminary separate numerator complementary estimate conditional numerator self denominator song consistently track l l eqn string eqn string compression prediction prediction symbol eqn prediction music algorithm scalability work linear query infeasible approach retrieval indexing apply metric retrieval track relative influence performance examine normalise display surprisingly contrary outperform bag approach account temporal structure outperform evaluate compression dataset compare use average codebook outperform although relative performance gain fig b consistently compression obtain compression compare gain respectively tb h cccc h l cccc cccc h h display c comparison cross yield map baseline determine parameter performance examine qualitatively observe interval parameter correspond baseline approach refinement base normalise histogram observe cross discrete value combine string continuous value outperform gain majority approach consistently value value continuous outperform examine disadvantage self base estimate cross relatively suggest limitation prediction however dataset result conditional self facilitate results state improve distance describe combine fig display score mix combination compare observe improve performance dataset gain combine obtain maximal dataset evaluation reveal gain distance combine report rank dataset consistently gain tc high rank display combine tc evaluate measure pairwise song identification consider representation series cross secondly represent prediction mean determine require applicable firstly propose continuous outperform baseline approach secondly draw cross prediction discrete song song identification million song use compression view measure value preferable discrete representation obtain measure argue due work song large scale base value baseline evaluate alternative series far involve reconstruction recurrent network term architecture evaluate detail cover method quantify cover song information theoretic measure series discrete value operating normalise time song comprise million song dataset continuous outperform approach estimate normalised compression string normalised alignment improve performance value distance refine song cover song normalise audio quantify substantial interest music technique track song identification distinguish require audio track query track audio specificity track specificity since track share song previously record piece music cover song mid specificity song correspondingly song challenge song base quantify music significance intrinsic sequence reflect consider shannon theory predictive form modelling expectation unfold stream sequential theoretic approach conceptual quantifying purpose song determine pairwise track work approach quantify shannon kolmogorov normalise quantify string successfully across music choose cover song interpret compare shannon compete concern implementation extend examine value million song use remainder discuss related song determine describe section conclusion similarity audio distinguish temporal discard retain former distribution feature approach unable aspect harmonic structure variation role al limitation audio sequential music repeat piece music song identification determine song song across band song determine similarity dynamic energy propose et compute matrix substitute alternative recurrence precede combine high temporal et al represent techniques cross maxima feature distance sequence propose key value representation et discrete sequence use spectral peak pick extraction lee maxima evaluate pairwise correlation sequence apply symbolic al pairwise piece music perform analysis symbolic audio audio obtain frame perform differential distance track compression propose representation additional hmms pairwise addition representation compressed applie recurrence plot individual track structural similarity piece music base derive representation observe measure al measure similarity video compression alternative measure extend comparison propose audio concern song large scale music collection contain track collection infeasible perform expensive comparison track collection et pairwise quantify threshold combine locality sensitive retrieval sub salient track thresholde encode integer expensive apply approach theoretic measure measure al nucleotide cluster parse compare sequence alternative consider building sequence compress et motivate jointly normalised series represent piece independent distribute process quantify dissimilarity kullback shannon logarithm quantifie represent divergence widely bag specificity music content temporal audio sequence string number encode string similarly denote encode concatenation string denote kolmogorov aic bit output short mean distinguish string aic quantify string maximally string closely examine approximation determine use similarity sequential concatenation approximate heuristic shannon modification similarity detail quantify shannon quantify uncertainty accounting dependency analogously joint quantifie amount pair source addition accounting source entropy eq interpret quantify average emission knowledge observation additive approximated use entropy eq denote shannon account string estimate
become prohibitive take day lack big hierarchy achieve trivial penalty combine tackle challenge novel strict extremely implement organized notation general implementation oracle reveal minimax rate simulation analysis conduct efficiency necessary jj na na e follow symbol g b p j n notational design matrix ba multiplicative numerical equally sized matrix generality center exist intercept q consist describe parameter hierarchical attain penalty impose come second model whenever maintain without symmetry guarantee simple focus work group nonconvex sparsity induce induce vanish without concern challenging computational matter vanish give scheme predictor role identity vanish construct set functional arbitrarily impose apply often predictor around amount type preferable section offer general variable claim idea main paper theoretical challenge arise overlap indeed appear fast scalable cf penalty constraint derive sharp challenging cf section think admm penalty getting find application design track use admm lagrange multiplier refer penalty play lagrangian solve converge value r package augment lagrangian behave well consider general instance abuse suffice necessarily general operator define version sa otherwise extremely hoc algorithmic provide universal choice large I I convergence establish every accumulation conclusion purpose suppose conclusion include net helpful handling favor fusion cause double see bregman generalize take suffice empirically avoid line accelerate method b relaxation convergence error adopt attain p g p e ambiguity investigate typical type penalty give finite apply yield lasso tackle global perspective avoid strength uniqueness sharp kind incoherence e follow hold restrict cone hierarchy parameter play major theorem compare type convenience g j overall control gm vanish bind large achieve error bias applicable signal lot norm design bound true regularization choice literature group size order light size reduce treatment section conclusion grid extended type noise remove version symmetry develop w w j similarly regularity define however e e g type requirement design simulation usually indicate meaningful recommend regularization sense toward consider hierarchy double e assume ij satisfied hierarchy replace give example conclusion indicator occur mild e minimax minimax comparison benefit show g exist existence effect associate situation behave lasso hierarchy neither practically lasso lasso omit rr compare type consistency efficiency design generate correlation follow interaction interaction effect obey j strong regularization large perform theorem experience variable selection model handle get warm recommend implement set otherwise true calibrate ridge fair repeat time square robustness report run fraction j miss alarm rate cost time pc ghz gb bit summarize result fa fa fa fa cm cm ex ex ex hour hour hour contain achieve compare use group behave fa error performance bad reflect varie notice high effect think day path approach offer gain efficient large scale conduct california dataset consist nine characteristic neighborhood california median respectively challenge nuisance gaussian add study full prevent get optimistic hierarchical outer validation cv run variable post calibrate approximately day take scale median variable predictor panel figure original covariate heat restrict panel correspond interaction successfully add feature nuisance heat however interestingly interaction term confirm boost age difficult interpret overall provide cc function b p p globally p tool fix kkt imply sequence satisfied point denote fix tucker exactly global minimizer proof first difficult theorem strict iterate point b ab conclusion universal constant occurrence global minimizer p x column j follow complementary e p carefully ga e ep ep notational index let j j universal brevity manner easy sufficiently convexity stationary lagrangian
unit b recurrent cell content activation activation long short term memory minor modification lstm make recurrent weighted sum apply lstm maintain lstm output gate gate memory add memory forget gate degree content input note traditional content lstm decide keep lstm unit carry information distance capture fig graphical recurrent adaptively capture dependency scale lstm inside unit separate previous activation activation gate gate procedure newly compute candidate traditional unit wise originally activation gate formulation gate read symbol allow forget gate gate fig illustration lstm prominent share unit additive recurrent replace content unit current lstm keep exist content content advantage existence feature stream important decide forget gate lstm unit gate maintain perhaps importantly addition effectively create step vanish nearly pass reduce vanish unit sequence set model modeling speech music music dataset symbol binary vector use sigmoid speech dimensional raw audio design look consecutive sample consecutive version sequence output layer recurrent lstm rnn sec primary compare unit fairly choose make avoid size lstm deviation fix gradient prevent select multipli validation case rnn outperform dataset music rnns rnn lstm clearly outperform traditional lstm rnn perform fig learn curve cpu dataset update progress eventually stop advantage recurrent unit compare lstm heavily epoch ht second dataset dataset empirically rnn widely long memory lstm unit recently focus task raw evaluation superiority traditional evident task modeling could concrete two unit well experiment understand contribution lstm thorough future acknowledgment would acknowledge research universit de cifar recurrent unit sophisticated implement recurrent recurrent music speech modeling reveal advanced recurrent unit traditional unit comparable recurrent learning task book recently report challenging translation interesting recent achieve sophisticated recurrent long memory recurrent interested variant short lstm unit establish long recently two task al dataset sample speech dataset lstm cpu update recurrent network conventional feedforward neural able recurrent nonlinear composition logistic sigmoid length traditionally hide generative output give model output decompose end generative subject observe train rnn capture dependencie time rarely severe gradient base
mc dissimilarity dissimilarity mc proportional logarithm realization realization nan mean realization frequency relational unknown text profile entropy word profile relative undesirable often appearance network write author expression produce author appear entropy contribution text profile belong author situation transition profile rare word explore laplace infinite entropy proceed choice relative entropy text true text function frequency text symbol window length accuracy length connect independent clause entail change generation window length pick appear high computational accuracy since position apart order include different approach methodology adaptive frequently attribute frequent repeat marker experiment select function illustration text length word pool fix profile solid achieve network compose text likely variation choose say reliability large vary text attribute author find adaptive common implement validation author know text break length author randomly pick attribute piece author utilize interval cross validation round text build trial node well text attribute vary depend dash implement adaptive static I text static true example true static suitable method good shorter analyze likewise change adaptive rapid text text profile effective cross validation correct show variation rate text pick corpus profile word description range variation likely randomness approximately profile gain adaptive choose henceforth fix generation sentence discount length pick adaptively perform validation group correspond author span american author average book per book book mark translate author minimum word maximum english span th average play per length author maximum solve author ask ten text five five text book page text generate profile describe ten text result compute entropy multidimensional ten unknown text euclidean metric distortion unknown text depict empty author solid plot half author perfect two circle close circle fall plane blue dissimilarity entropy number distortion minor entropy relative entropy small text text text book text profile entropy text fig profile circle represent text color use distinguish blue profile specify author different text attribute represent green empty appear blue profile besides principal profile text dissimilarity l l l nr profile thousand rand l nr rand l l l profile thousand rand total word profile fix vary length increment text word profile consider profile randomly text piece build profile length text text attribute contiguous length pick write oppose piece different resemble text correspond run randomly text form profile amount choose ensure every text state page book typical play author difference rest carry information useful overall text even reasonably corpora author word accuracy author increase profile binary whereas ten monotonically author text increase long g word text accuracy attain correct vary text text first correctly attribute opinion piece corpora opinion piece corpora candidate reduce word accuracy acceptable binary short text support evidence besides number text profile text similarity write accuracy word exercise yield distinguish dissimilarity write quantify profile entropy relative entropy dissimilarity pair dissimilarity result relative entropy inter dissimilarity accuracy correlation dissimilaritie pool ten text generate profile remain profile rate profile dissimilarity choose ten text repeat accuracy text inter dissimilarity accuracy inter dissimilarity dissimilarity average account result pair dissimilarity small author carry period composition illustrate carry build dissimilarity obtain since dissimilarity fig plot two eight profile build text represent blue star red dot author period heat inter entropy color represent small entropy heat directly relative entropy inter profile dissimilarity correspond author th whereas remain author profile text notice block diagonal perfect entropy th entropy author relative belong time com carry illustrate text play color small entropy along diagonal distinguish sequentially text computing entropy remain piece attribute inter dissimilarity tend dissimilarity two author form pick word inter dissimilarity profile close vice versa inter dissimilarity profile dissimilarity write write contribute text text author two text mid profile mid mid hybrid profile compose c mid hybrid author word usage infer gender author divide author five pick gender author gender profile contain piece text author author text gender choose text gender profile repeat art gender e fact gender cm x cm nr author nn dt dt ce text author early english play entropy six generating profile author entropy one short length text author word close author profile build accept entropy six entropy confirm construction profile profile build contain author correspond hybrid table profile coincide table relative profile accept profile accurate play repeat procedure play pure two entropy achieve achieve profile compose profile achieve rely appearance number time author unlike word common naive support word candidate pick pool author attribute repeat preprocessing minimize minimize consider degree method strategy nn euclidean two dt dt ce low error frequency method naive decision outperform aforementioned author achieve consistent number average compare traditional method tend frequency fact carry increase majority well naive svms e four author majority entail frequency relational normalize word adjacency entropies accuracy vary text profile heterogeneity regard gender corpora know text substantial text attribute long profile page book novel text act text opinion show classify write piece predictive gender applicability multiple collaborative demonstrate appear importantly frequency capture method test introduce part speech express relationship carry lexical stand word probability compare entropy parameter diverse pool vary text length word tend capture summary exceed achieve alone combine source aspect match text unknown one potential candidate quantify traditional apply availability advance interest base least distinguish word length length characterize carry relationship advantage word content carry information function text play another include vocabulary marker stable marker word build focus frequency usage consider encode adjacency network co sentence normalization describe word encounter encounter turn imply transition interpretation dissimilarity text relative associate chain transition letter word reason usage letter approach somewhat positive result accuracy various selection choose develop adaptive compose author implementation method analyze modify length text influence distinguish text incorporate difference time gender classification section author base alone information capture combination increase author text text
comparable histograms cross reject b scatter method contour svm lower negative performance false positive negative detect whereas univariate test make distinction negative generally cc histogram scatter contour fit gaussian overlap univariate test say comparable whereas separated hence multivariate reject reject decision test time cv result multivariate reject find difference percent multivariate reject rarely happen reject reject rf histogram scatter contour gaussians univariate find difference two gaussian multivariate test recall frequently assess univariate test multivariate histogram univariate test look figure scatter plot contour bivariate gaussians high recall nan detect difference univariate reject decision univariate test four comparison though agree majority tend reject find percent multivariate reject rarely reject reject algorithm hyper lead different prior machine bioinformatic case classifier ec set representation protein body among look contact false find kernel machine predefine measure precision measure discuss fold find eigenvector give fp tn see univariate difference projection look look fp tn decision give tn assignment classification variate knowing explain variance histogram well separate univariate pair sum predefine measure confusion eigenvector may multivariate new projection advantage h histogram b contour fit find histogram separate tp tn b cumulative measure behavior propose error repository compare compare machine comparison parameter iii learn rather pre measure real world bioinformatics literature compare lift precision break calibration correlate roc combine rate fitness unbalanced combine give source test measure specificity medical specificity test arbitrary find clique subset significant allow basic statistic way difference behavior univariate cv fold interesting never fold data set distribute use recommend nonparametric counterpart tighter concerned normality outlier use research acknowledgment comparison tr department engineering university keyword statistical design multivariate abstract statistical test classification misclassification test comparison positive negative univariate test distinction source variate similarly precision variate three univariate automatically candidate algorithm domain precision positive misclassification sum false negative source combine plot visually distinction false negative may able detect error check cancer false patient false positive negative calculate risk false positive negative roc compare roc value sum comparison multiple simultaneously without collect dimensional dimensional negative precision variate work type new organize compare generalize seven future statistical testing pair manner calculate pair hypothesis population test compare near neighbor scenario test sometimes different different kernel instance make distribute normal theorem small measure calculate hence normality statistic approximately multivariate mean correspond term covariance correlation hypothesis fold fold calculate tp fp tn positive false negative negative confusion precision population pair calculate nan hypothesis performance degree reject nan fp normalize mahalanobis origin hoc univariate pair source recall difference multivariate difference significant preferred discriminant train fold fold confusion fold population test x validation matrix test statistic freedom
able compute smooth global metric principle generalize dimensionality mahalanobis projection semidefinite cone expensive moderately task vector share subset grey magnitude study new perspective address basis extract fisher learn high learning weight regularizer element framework call compositional projection flexible apply wants learn global metric exploiting make metric share arguably metric take weight smoothly knowledge principled feature use scalable subgradient descent proximal experimental support generalization theoretical combination suggest empirically art strongly describe formulation global local metric support review present present compositional exist formulation lie distances represent weighted rank psd cast set key rank discriminative basis global set metric define infinite local metric smoothly later highlight mahalanobis psd psd enabling want show later together rest start learn discuss formulation proximal metric learning seek may construct unsupervise implicit feedback click formulation combine metric j classic hinge norm encourage sparse allow element linearity minimum paradigm exploit relate successfully build share unclear allow translation formally task constraint extract nonnegative row define column induce overall constraint potentially benefit basis regularize group convex mt address limitation global capture pattern metric vary capture semantic well costly often severe overfitte learn aim smooth function informally metric geodesic riemannian problem simplification metric instance learn mt however appeal computationally heavy overfitte large furthermore give principled propose weight simple learn rbf bandwidth median td ensure valid pseudo combine locally metric feature instance local denote concatenation subset interestingly recover minima nonsmooth induce involve triplet stochastic step proximal induce comparable improvement backward initialize unlike exist projection onto psd scale thereby problem analysis algorithmic metric consider np return non triplet bound learn basis norm asymptotic justification enforce notice appear suggest long remain mt relevant task refer survey detail method directly representative paper subject learner boost clear multi popular task clearly rank achieve high compete especially train fast fast consistently appendix global letter element section main global pick relevant generate new add current report number use overall use use suggest scale well dimensionality nice entire induce regularizer number poorly st books accuracy avg runtime n min min min sentiment dataset multi amazon review book type treat task positive review review large split testing compare euclidean st independently euclidean union tune parameter set union mt average error rate st dataset mt outperform st significantly mt task counterpart demonstrate ability unable capture solution st element element select mt evenly across able exploit meaningful compact metric segment letter avg dataset global learning target local number basis letter table give method enough fast poor metric discriminative local offer training especially train minute fast dataset mm bad global parameter result competitive understanding apply colored vector pca vary smoothly thereby robust unlike mm point metric variation consistently generalize select basis basis element eventually reduce suggest generalization bind theorem generally global notice outperform element multi particular local algorithm instance test principle way support theoretically generalization propose method research contract nf contract ap reproduce view herein interpret express imply c triplet build admissible define generalization pairwise reader briefly algorithmic robustness ability perform similarly test proximity space two example lie compact belong xu metric robust ns sn adapting show algorithm algorithm sample triplet draw least establish easy say learn approximately triplets metric suppose global learn number nonzero definition subset subset either triplet respective deviation admissible robust set training sample tc admissible argument global tt global
reason rbms way gradually specify intermediate p kk kp x assume gibbs leave w importantly hold importance extend k xt represents generate unnormalized begin reverse show importance mathematical intermediate initial q define anneal temperature annealing isolate two rbms bias evenly although evaluate mrfs intractable tend hope provably estimate preferable save report test log section limit infinite unobserved distribution rbms transition visible approximate v w kf ann proposal reverse chain start gradually x xt identity store chain update reverse implement require operator merely reverse chain stochastic low yield conservative mrf low possibility report test log frequently interest additional agree weak assumption tractable operator preserve reverse similarly algorithm h k w kf ann procedure rbm could interpret sigmoid insight greedy belief single transpose compute rbms view belief net proposal transpose perform approximate unit rather use field interpretation suggest rbm direct layer direct green blue speed example prefer number subsampling introduce rbm handwritten digits significantly depend batch variance reduction apply ideally compute highly exact mrf mrf easy total assign small significantly evaluate mrfs estimate estimate obtain exactly log strong handwritten digit long dataset character across many average two distribution base rate distribution visible bias pixel transition probability test sec probability full dataset rbm log chain match number exact gap gap train algorithm cd persistent divergence refer rbms hide train cd mnist rbms evaluate also comparison conservative conservative rbms conservative probability estimate intermediate hand insufficient accuracy often differ give consistently well method plot estimate uniform appear report high obvious discuss necessarily rbm rbm rbm rbm instance rbm rbm rbm base show increase bind inversion outperform rbm rbm model diverse configuration experiment rbm challenging overall case estimate suggest rbm function anneal intermediate distribution sample poor great rbm matching estimate rbm anneal bottom model use estimate two train hidden layer rbms run test unnormalize obtain unnormalize v conditional sum use give however make unnormalized merely variate estimate table mnist figure mnist quite piece evidence contrast give optimistic imply rbm eliminate mrf model gap right hide obtain unnormalized recognition mnist within give conservative rbm experiment model typically conservative suggest reverse log mrf typically yet rbms conjunction one agreement require simple practical test acknowledgement google markov fields mrfs generative importance mrf partition yield quite accurate wrong reverse original experimental result indicate agree rbms typically year representation deep appeal representation partly directly measure assign restrict boltzmann effective various visual rbm intractable mrf widely mrfs generative function perform practice tend optimistic whether log likelihood lead researcher generative rbms art modeling highlight optimistic rbm tend standard benchmark insufficient accuracy compute mrf likelihood similar boltzmann add write rbms unnormalize f h similarly rbms rbms building
use least implication right uncertainty far turn toward probabilistic equation independence element ms mutually appear right term apply hoeffde probability least union l definition individually mutual j use mutual equation mutual j e use law j use identity law conclude present principled approach four drive distributional distributional know paper show statistical learning guarantee quality robustness project come mit nsf grant theorem remark mit edu goal decision datum past create particular handle simply way past tool provide guarantee robustness robust optimization uncertainty work often needs create future good reaction bad uncertain situation question maker interested answer question probability bring useful address important detailed available predictive modeling technique machine uncertainty set uncertain historical accord possibly illustrative portfolio allocation construct return market advance portfolio solve make portfolio wish portfolio portfolio acceptable make portfolio uncertainty exposition reality return solve good outcome many central complex past predict return might include sale complex carefully different priori range portfolio know possible knowledge guide construct return jj ignore past use empirical could uncertainty portfolio allocation problem define past could hull past return ignore linear potentially assumption draw class intermediate set use portfolio determine fit normality additionally normality define robust allocation prediction interval around union good decision realization reasonably make assumption complex assumption limit applicability modern assumption provide way historical two approach approach tool minimal assumption construct set construct robustness guarantee specific conditional quantile prediction prediction I normality give indicator away two illustrate extreme set error single policy effort choose independent manner learn give guarantee intuition set square loss function element interval uncertainty illustrate percentile model percentile interval provide fourth second boundary create method evaluation every plot optimize boundary residual boundary good quantile boundary uncertainty important specialized order daily ice weather much uncertainty weather conservative costly would budget large possible sale middle attempt tackle principled goal like uncertainty machine propose value needs guarantee finite quality robustness use approach approach uncertainty theoretical design handle classification ranking apply section formulate uncertainty use learn theory guarantee guarantee many either make optimization literature interest empirical along priori probability specify drive set guarantee testing use three design ii goal totally cost policy create estimate theory feasibility randomness thus entirely make paradigm portfolio regression big design set historical bit various instance multi unlabeled available work use prior create feasibility describe set historical correspond approach introduction decision denote vector decision variance problem robust portfolio formulation list option mm nice natural transform relaxed robust optimization problem solve ellipsoid box ellipsoid solve nice element walks give feed set function training empirically start discuss dy represent let say pick ni generalize observation consider empirical minimization solution eq set set function simple instance union disjoint interval non way j slightly method estimate functional generic quantile quantile quantile quantile applicable prediction task yy j quantile example typical pair conditional quantile realization quantile j I regression risk quantile aim obtain true predefine parameter loss let construct definition interval involve two quantile conditional conjunction deviation capture use define large deviation capture illustration member figure equation depend residual solution consider general training input output high pick simple empirical rademacher equal average interpretation rademacher come rademacher covering shorthand result performance term result feasibility depend estimate enter well desire proof provide constant shorthand variable use use rademacher result prescribe output minimization risk sa rademacher range rademacher class capture set follow solution feasible ms bound realization ensure ensure belong high tell affect uncertainty set predictive rademacher scale quantitative confidence distributional study rademacher propose define source notion definition conclusion theorem hold equation use pac pac probably seek instead classifier uniformly pick randomize choose classifier risk r sg pac capture p q certain pre minimize normalize good model define big solve way prior pac thus guarantee nonetheless way distributional robust portfolio example distribute tx tx distribute choose base interval ellipsoid probability mass equation obtain interval solve ds us future realization hold fix unbiased tx mass need justify construction contrast make section weak many approach may efficiently option involve directly guarantee similar solution make class bb use corollary relate bb function positive semi support svms hinge apply hinge upper appropriate kernel particular use dot get consider fourth define I sl guarantee optimal hold individually robust equation j ms designing prediction quantile residual realization source insight quantity learn importantly prediction decision e g svms practice suggest term perform type sensitivity analysis vary set assess intuitively function insight construct algorithm drive hand quantile quantile quantile estimate produce quantile estimation simple close loss special differ lipschitz theorem rely mild assumption make model include account set underlie true sense policy uncertainty lead distribution lead small strong centrality construction regardless relate good expand method use proof result rademacher f l sf sl side increase random sf perturb case sf l sf l sf f sf rademacher trick l give
evaluation termination immediate b preferable optimization incorporate mechanism maxima equally apparent search region inefficient therefore uncertainty fraction would maximize typical end step uncertain next iteration region offer immediate begin express incorporate kind exploratory gain set b maximize set criterion great close deviation retrieve efficacy toy example expect terminate maximum whereas opposite score yx x ask find choose search hybrid local optimum evaluate uncertainty us find maxima problem score individual purpose instance incorporate methodology principle high uncertainty unity permit exploration space original yet advantageous closely proportional improvement high propose tuple cm p cm threshold forest optimization threshold selection virtue exploratory decision necessary maximum unnormalized terminate significant complexity table statistic h b framework thresholding find able superior yield compete validation hope algorithms application parameter hybrid mm edu college area regard hyper via work exploit notion confidence uncertainty enable efficacy machine expect distribution parameter tuple tuple induce fit allow behavior point notion arise form fit enable process optimize search framework immediate fold cross validation discrete fail parameter set exploit improvement performance current show interpretable provide alternative option et al implement mat ern typically target efficacy commonly maximize minimized ability generate perhaps network whether continuous use
rest generality uniformly finite weak demonstrate bootstrap main bind moment hold average family process get stochastically result stop give convenience use use increasing imply rest choice arbitrary fact bound parametrize consider proceed expectation pe pd fp pe deviation call state shift nonnegative q since thm pa idea stochastically independently write continue expectation condition prove stochastically analogous proof bootstrap bind stopping rely defer appendix convert choice analogously however space need define stop event eq q stop unbounded simplification thm tool extend obstacle establish easily geometric brownian motion tight unlike discrete merely manuscript totally stop time analogy great effect process metric covering incorporate variation open powerful average regime treat time weak pe emphasis normalization use stop uniform go notably conceptually idea describe game motivate somewhat different taylor expansion acknowledgement grateful stochastically time justify prove walk generalized beyond walk construction leave remainder unchanged determine average distribution averaging construction replace high uniformly e e result examine x yy attain initial condition remove extend hoeffding bernstein style show simultaneously lemma desire bernstein inequality regime account uniformity iterate reason relate appendix first mixed process write integral like peak refined account factor construction rest current lower close change average family distribution index q mix conjunction appropriate iterate furthermore improve decrease tight strictly extend finite time average proof analogous different initial restrictive closely within f decrease first proceed rest proof verify bind particular recover dominate lead define family lemma upper u bound idea stochastically space define product bind respect notation stop convergence restrictive concrete also whose u finally side lipschitz fast fix write stop prove proof fix suffice assumption last u use substitute give lemma event lemma q chebyshev define convenience particular mean correspond confirm q u monotone imply chebyshev know upon simplification thm thm thm thm thm remark main remark give bound uniform hoeffding bernstein limit law iterate finite bound dynamic arise particularly range concentrate side concern behavior finite manuscript address upper half concrete martingale discrete induce repeatedly write I distribute rademacher rademacher walk law iterate logarithm rademacher random absolute interestingly capture tradeoff dominate regime time uniform limit bind rate encounter strong walk time finite maximal exercise hoeffding manuscript fundamentally weak hold uniformly epoch proof view generalize u cumulative straightforward
rna eeg demonstrate propose algorithm single perform case framework imbalance make balanced substantially apply benchmark extremely observe improvement easily employ sn sp clean rna eeg acc sn sp letter yes yes letter eeg complexity consume component ann construction exact algorithm run big beneficial typical linear experiment near neighbor set framework several sensitive stop parameter perform fast rna similar slightly level loss significantly framework large set try small nearly impossible allow level stop parameter serve fine level main drawback search parameter become considerably expensive method hand running pair optimize parameter global simplicity cluster dataset quality classifier letter cc letter rna r dataset paper promise quality successful reduce degree skewness balanced type support computational propose framework scale solving obtained gradually refine multiple include hierarchy coarse representation update hyperplane computational without demonstrate machine balanced keyword support originally nonlinear svms consume task extremely sensitive apply storage rapidly dimensionality tractable svm parameter tune advanced method tune parameter total quality employ optimization focus qp popular qp scale efficiently cache typically still recently parallelization split subset perform assign different subset accelerate qp often problematic implement svms parallelization dataset investigation accelerate training optimum vector shrink early optimization time save substantial successful optimize completion size practice lead poor measure area include diagnosis bioinformatics class technique datum adaptation sensitive learn regular svm term fuzzy heart method lie algorithmic mf inspire multiscale main objective degree introduce local processing global solution data exhibit linear complexity relatively mf heterogeneity external appropriate refinement framework successful algorithmic create coarse training solve refinement easy parallelization superiority computational method less particularly set create balanced coarse effectively hyperplane let set label number feature subset relate determine map slack misclassifie nest manner dataset respect paper three learn support vector create level approximated selecting include result make make robust change refine classifier level input ni give begin neighbor ann ann select size dominate present complementary already begin eliminate next add already skewness balanced representation correspond correspond label computational resource separability fast processing difficult level refinement fine contain coarse extremely consume prohibitive fine much original much selection complexity apply svm near neighbor support run center apply pair directly exactly coarse approximated experiment datum coarse current line direct opposite contribute binary classification true positive negative negative classifier acc common classify proper performance mostly dominate use sensitivity sn specificity sp geometric
ei pi suggest line keep maintain section technical produce monte integration discretization eq leave computing minimizer easily latent datum gaussian idea meta visualization kn py meta criterion upon produce global gain intuition form th element suggest draw return match armed sampling continuous domain construct sequentially optimize would ultimately optimize analytic shift mapping consist stack approximated product allow correspond posterior construct finite parameterization meta use acquisition randomize strategy exactly extend continuously vary function use albeit acquisition summarize meta develop system factor optimization allow learn around choose optimizer ball width proportional smooth difficulty set optimization set due bad quite maxima hyperparameter let hyperparameter p posterior fully integral must sample acquisition internal every hyperparameter sample ei pi randomize hyperparameter hyperparameter selection hyperparameter occur simply add additional loop sample global problem portfolio portfolio rp portfolio acquisition randomly acquisition ei thompson ei three continuous final absolute evaluation pi expect expert outperform acquisition rather imagine parameterized addition two ei clear winner thompson option motivate function synthetic inclusion strategy portfolio rp expert box base method initial stage exploration enough exploitation actually beneficial purely exploratory candidate precisely propose empirically especially reach digit meanwhile rp expert select acquisition due expert horizon reach rely past robust vary single outlier achieve evaluation refer dataset consist finite transform via constant three south gold dimensional outperform ei motivation pi rp particularly albeit acquisition example thompson poor acquisition boost final portfolio eight control fed simulator robot simulator optimisation problem particle drop particle circular placing degree plane report result poorly rp perform meanwhile perform portfolio tie rp add demonstrate rp rp significantly affect introduce theoretic meta criterion expert particularly acquisition robust portfolio furthermore poorly perform portfolio principled slice acquisition thereby extend popular sampling wang w exploration acquisition clear superior principled collection acquisition function often past portfolio portfolio theoretic consideration outperform synthetic simulate offer acquisition surprisingly performance finally wide inclusion poor acquisition popular successful expensive find minimizer non multi modal technique interactive environmental monitoring extraction network adaptive monte carlo experimental reinforcement broad application area tuning query point construct probable procedure must select state knowledge characterize acquisition encode clear optimal acquisition planning often much acquisition long early probability pi improve see point lie pi greedy quantify correspond ei recently variety advanced technique bayesian acquisition strategy provide problem instance empirically prefer strategy stage process ingredient among analogous acquisition assign utility space utility candidate suggest portfolio predict instead unclear also acquisition subroutine sample minimizer thompson also jointly location condition work optimization visualization expert th query early implement past acquisition lower base content initial portfolio collect kt denote meta candidate criterion great reduction arrive
mac mac snp mac mac mac mac k mac k k mac mac mac mac mac mac mac mac k mac mac mac mac k mac k mac mac mac mac genome study steady accumulation imputation genome sequence study need scalable key genetic call phase information variant represent primary format address issue release introduce extensive parallelism equilibrium fisher many algorithmic accelerate handle large fit ram follow efficiently version offer improvement user analyse genetic come finding broad file format trait map genetic year final generation analytical wide heavily processor comprehensive update notable exceed functional unchanged association identity descent drop replacement case require easy like feature format file platform genomic introduce employ expect benefit problem remain core file format second improvement parallelism x bit machine binary file bit datum arithmetic parallelism loop operate element replacement loop bit parallel logic old calculation roughly every marker marker miss call file long marker block single bit population count bit discussion post processor evaluate quantity however thank count quickly hardware take previously refine develop implementation vector even processor might example marker marker denote minor coefficient easily express bit call marker loop dot encode minor call call call name correspondingly memory requirement sum marker specific marker seven distinct allele major minor minor minor genomic per marker increment minor allele final instead partial save seven increment refer sum several point substantially seven distinguished operation seven appropriate manner bit marker bit marker bit describe relation marker representation us processor act simultaneously increment could denote f f exchange four entry mb modern table snp equilibrium al fisher exact snp exploit likelihood contingency expensive ratio table contingency entry could avoid calculation partial sum likelihood super geometrically move away probable likelihood partial digit stop double precision see example mathematical straightforward modify termination snp property likelihood evaluate handle represent et early termination snp fisher test call mid mid discuss algorithm detail explanation extend method restrict briefly variant strong strong historical contiguous base determination analytic hill cubic likelihood correspond check cumulative likelihood rarely extreme full cubic reveal exploit establish cdf evaluate likelihood inspection variant massive table later pair variant process final spend cache large must entirely skip variant idea implement last implement basic coordinate descent newly integrate party upon implementation speedup logistic crowdsource innovation perform hill manuscript author case user seven operate three intel processor gb run mac intel processor os ghz intel processor ram bit ghz intel processor core gb ram bit denote four core processor ghz ram bit sp ghz intel processor core ram refer dataset quantitative synthetic marker resample p prune remain refer variant phase dataset refer variant snp snps prune disk run straightforward indicate runtime several basic table display execution one simple reflect overhead due use run disk complete cluster calculation disk linkage population employ population count clustering furth improvement case table execution three calculation count version estimate information genomic early standardized genomic os windows platform memory requirement linkage pruning frequently use analysis linkage table single heavily fast contain rewrite variant runtime association analysis flexible major improvement bit population solve make fisher test window build bit despite advance recognize still work genomic binary format file format capable represent essentially modern imputation multiple probability limited call amount serious format binary file variant within file representation translation code program term explicitly design library able convert file party library compression demonstrate weak compressive arithmetic computation directly compress genomic exhibit explicit compressed merely pack support true sublinear sublinear compressive reality deviation efficiency available program operate library layer file ignore work exploit compression software plan software handle limitation meet genome association study context genomic system across preferred size enough serious genome sequence detailed study variation clean sometimes type strongly software package seq file less expressive file management genomic subject type type extend interpret availability name code home page operate os bit intel window require restriction compete interest author software manuscript matlab prototype
prior variation perform series test central less series flat flat prior however ht difference mean alternative posterior different dot fairly plot figure plot posterior quite practical semi situation underlie unknown least tend poorly alternative aim demonstrate involve likelihood base outperform adjust implementation parametric package iii iv wavelet package vary early idea residual respectively cc observe positively seem significantly slightly biased method unbiased first full pilot recall case output two generating mechanism relatively b differential plot short long effect opposite end spectrum peak posterior largely uncorrelated process behaviour spectrum effect memory dominate distinguish consequently ccc green line red consider green blue significant arguably although clearly form scheme outline posterior generally roughly around generating probability usually consider context plot unknown solid reversible dotted look method former slightly level posterior appear exhibit present showing mode pick consistently b complicated short tailed series see density cc look complicated wide reasonably finally algorithm wrong marginal application obtain actually homogeneous datum rr ci summary minima plot density minima popular memory effect work handle typically mind accommodate cope heavy tailed potential scope finally miss augmentation relevant gap example datum record fit assume gaussian gamma choose posterior conjugacy demonstrate gx yield follow conditional reveal likewise p classical denote p ar uniqueness firstly pi special equations ik q ar block divide pt ptc also definite block clearly positive definite also positive eq ptc corollary efficient forecasting whether represent long potentially high potential mathematically stochastic exhibit precise able market public effect frequentist autoregressive fractional integrated parameter parameter short memory hypothesis testing ever methodology range auto reversible capable memory long standard definition autocorrelation zero process long rescale retain stationarity essential rigorously memory property statistical span semi bayesian probably difficulty advantage classical flexibility could offer ability model miss effect towards burden class process phenomena emphasis popular statistic connection fractional property separately model long short argue sentiment parametric model work often model similarity bayesian generally careful approximate process series extend rich memory extend unknown work work nonparametric memory contribution aspect present focus selection unknown memory effect focus long memory conditioning emphasis flexible class remainder paper require numerical likelihood classical statistical bayesian propose focus long memory extension additional illustration method potential time record process joint tr normal integer convenience consequently drop depend temporal motivate stationary refer lag normalise power iteratively k say sequence expand power series act white iid noise purely non refer coefficient write power additional identifiability happen two therefore model hence stationary invertible process process ar invertible say move stationary ar auto process order although close arbitrary long causal decay exponentially word correlation nearby distant negligible turn memory extra possess obtain restriction stationarity clearly stationary similar ar one particularly long constant simplest equivalently assume scalar multiple generally call full long solely flat series memory structure determine one f way call integrated move order recover practical utility formally generalise operator call fractional operator degree use domain connect integrate law relationship ar analog short placing parameter likelihood approximate highlight issue prevent context thousand evaluation proceed dd likelihood simplify shorthand whereby log evaluation require encountered toeplitz scalar exploit yield large practice efficient scheme throughout stationary invertible mean location call I classify three innovation short truncate drop x ar ar retain obtain follow possible write convenience appropriate bayesian auxiliary writing ng conditioning write tx return observe conditional past suitable equivalently evaluate depend augment evaluate via cost infeasible however suffice bayesian inferential ready describe help accommodate extension short begin prior choice yet encourage calculation early familiar specifically improper flat prior limit distribution place imply gamma work aware promise extra component subsequently memory ar distinguish create variance unit unit write general explicit calculation fortunately polynomial calculate coefficient yet space simple proposal orthogonal would alternative explicit jointly align correlation structure reasonable mixing describe suitably walk covariance proposal sort arbitrary large truncate hypercube unconstraine rw mode return mh ratio since term initial choose interval systematically although prefer proposal align find pilot proposal rescale matrix pilot expand short indexing adjacent acceptance crucial transform turn shape model difficulty encounter complicated nontrivial require root write unit serious drawback consider natural remove root realistic appear obvious way remove might modulus unclear propose previously recursively fit motivate show lag useful propose structure limit long nevertheless near full value model individually suffer problem simultaneous update perform distribution rectangle detail expression complicate special must model variety prior simple option would restrict say proper formulation lot complicated prefer poisson parameter non possible birth death neighboring value point body boundary corner transition probability abuse notation detail ar almost identical de replace simplify regard choose set match specify either final jacobian unity made prefer decrease scheme move depend choice rw seek pilot extend obvious matrix etc block omitted specify various propose variance dot vector zero conceptually analysis memory ensure unless simulate reasonable use begin bayesian alternative mcmc tune interest value systematically start spaced parameter varied turn sufficiently mcmc efficacy start long simulate deviation extract average rr mean std ci estimate away interval nearly symmetric posterior locally proxy credible test eight hypothesis would six indicate interval approximately sized repeat experiment generate give analog axis estimate I perform show around space whole
fast avoid search publication context guarantee study nuclear reweighte nuclear conduct plain svd valuable insight quantify simplification update analyze svd plain rw plain relative error nn avg trial quantile simplify adaptive relative solution minimum simply singular svd high aa bound svd solution log heuristic tight accuracy even small singular approximation contrast plain bad large consider signal decay power improvement decay error support nuclear plain optimal svd norm algorithm full simplify fix simulate alm plot via divide norm rw exact figure rw nn recover optimal establish simplified good error dramatically plain analysis rw structure practically entry infeasible quantify impose rw accuracy nn far bad infeasible plain frobenius rw solution provide error toeplitz toeplitz toeplitz structure modeling represent multiplication toeplitz toeplitz start diagonal toeplitz however reach rw infeasible plain toeplitz finally nuclear success solver heavily diagonal entry moderate level rw nn approach solution rw nn continue good accuracy good initialization set cosine angle average utility broad biology biological human human community cell interest characterize different state across combine readily cell distinct state phase division growth cell fractional change expensive impossible directly separate know indicator throughput dna rna seq relative different accurately chemical composition rna normalization accounting bias bias positive state different growth rate mathematically extend z enter transpose side move everything stack stand kronecker block diagonal column row block single rank structure trivial nan compound error exactly precisely explore experimentally reweighte heterogeneity quantification cell hoc express across grow growth cell phase agreement expectation physical measurement extend trend asynchronous analysis research heterogeneity obstacle experiment biological comparison rw non outlier low noise solver rw gain nuclear case use notation frobenius case heuristic solve solution local domain linearization minimum contrast soft thresholding phase across growth choose increase order error quickly soft error entry heuristic rich augment lagrangian reweighte nuclear matrix completion robust quantify heterogeneity measurement solution ls square allow field computer vision identification special dependent uncorrelated singular svd many realistic vary measurement know norm convex plain nuclear relaxation incur approximation effective challenge popular convex solver augment lagrangian problem measurement expression measure generalization square l explanatory problem well base measure come additive noise minimize xx try norm reformulate find close rank solve via many practical additional may exhibit toeplitz outlier huber unfortunately none efficient approach method reach principled relaxation rank use relaxation highly toeplitz frobenius relaxation successfully range involve completion robust conceptually seek low contrary theoretically experimentally plain nuclear incur dramatically weight nuclear suggest base multiplier alm update weight nuclear equation task rely pca study biology quantification heterogeneity problem block diagonal recover state notation combine matrix frobenius norm small error range compute small singular allow realistic exactly measurement noise entry toeplitz deconvolution signal unfortunately vector search one sparse general encourage signal weight plain ideally unknown norm large solve define weight see linearization empirical study weight provide tight try singular one introduce linearization nuclear programming sdp p rewrite nuclear concave log small positive relaxation provide approximation norm objective initialize weight nuclear initialize solve svd k plain weight nuclear formulation thresholding next augment lagrangian alm nuclear interior even lagrangian alm alm augment augment lagrangian motivation optimal minimum continuously increase sequence one allow involve
less contain observe user source correspond job traffic post find remove meaningful noisy movement cc analyze movement york city date show date middle color six tweet second cover tweet see fig localize addition temporal see cluster cluster union square ideally separate group related real six cluster cover tweet mean leave right cover tweet contain date summarize third detect area square interval moreover tweet mainly take place mention event distant cache event cluster time spread present example anomaly event twitter certainly explicitly event extraction multiscale recognition community last decade take wavelet transform enable work interest medium wavelet framework solution temporal graph approach account spatial generally handle scale explicitly multiple believe essential generic scale spatial dimension treat separately state good knowledge approach model relationship statistical analysis spatial twitter believe perspective contribute social multiscale medium understand temporal different separate way present statistical twitter insight influence investigation possibility generalize handle multiscale detection ii appropriate social media scalability mod research ed social media approach reality scale social explore property multiscale transform enable automatic handling interaction propose similarity detect event scale simultaneously furthermore present stream novel algorithm help experimental real collect demonstrate approach extend numerous involve decade see rapid development online social media user generate internet huge many detection one important area medium present several real social service public fashion traditional second amount content complete description large scale advantage attract amount mining event detection numerous literature social medium define real world similar location image video text temporal span discussion game month regard concentrate illustrate twitter event scale york city account scale challenge usually detect yet multiple space interact simultaneously two iii contain event interest understand robust multiscale objective cc introduce baseline localize serve towards multiscale detection towards scale relationship property wavelet automatically explicitly algorithm compute scale furthermore present noisy information stream notion statistic multiscale behavior paper noisy propose event datum world experimentally propose spatial hand believe model relationship temporal scale multiscale insight media framework domain involve event medium reflect concentrated dimension usually namely interval twitter great home non tweet constitute input interest data stream stream contain approach event take temporal paper cast graph tweet reflect similarity utilize tweet tweet generate share closely locate way measure approach constraint base effectively handle twitter homogeneous poisson model helpful event previous impose baseline localize understand event together correspond event similarity measure tweet minute spatial threshold locality impose constraint reasonably text tend refer text angle representation weight scheme tweet adjacency tweet define cluster expect tweet furthermore eq event localize space way function repeat procedure modularity attain suitable purpose cluster cluster usually event priori ii unlike normalized spectral clustering favor balanced clustering enable detection cluster relatively also base approach correspond localize fig correspond cube simple post likely meaningful world reflect twitter threshold sect step critical temporal spatial event discover event sufficiently event set break lead cluster cluster already offer grouping hierarchical obtain usually clear scale result cluster process experimental section event scale introduce novel base cc pairwise similarity tweet adjacency recursive retain post localize space novel multiscale event introduce scale wavelet scheme multiscale similarity tweet fundamental question towards multiscale detection properly localization illustration event rectangular span spread tweet spatial event shall spatial dimension compute scale actually span propose temporal follow two tweet share could temporal vice versa resolution essentially say consider fine resolution necessarily time area could span city place power area city relax event detection choose suffer ambiguity twitter stream either incorporate temporal distance might detail tweet tweet time share occurrence tweet appear enable temporal similarity tweet compute similarity time share keyword share use resolution occurrence cell belong illustrated cc propose similarity transform tool consider wavelet haar way scale specifically haar wavelet aggregate scale temporal resolution coarse series temporal measure coefficient level evaluate similarity properly turn determined spatial two introduce predefine scale coarse distant fine initial fine temporal cell temporal temporal compute level illustrate fig similarity tweet follow share time series choose medium piece informative although tweet share popular series pattern fine next remove spread similarity time take similarity helps preserve information recall retrieval tweet favor tweet pattern approach similarity meaningful rely look common offer flexibility event multiscale h tweet extract series spatial similarity correspond series parameter multiscale resolution construct adapt scale various thank adjustment wavelet expect temporal interval span sect number spatial choice full scale might lead unnecessary influence resolution determine along variability implicitly scale meaningful span determine meaningful level computation haar wavelet maximum temporal relationship accordingly observation suggest denote choose ensure design event example could really twitter influence event result employ filter tackle derive working tweet contain wish specific event tweet empirically intuition useful locate tweet day york city contain locate tweet middle tweet middle contain tweet locate albeit frequent collection tweet appear relevant specific illustrate location tweet middle one tweet strong area spatial plot seek statistic lack assess use complete randomness tweet homogeneous poisson overlap within denote intuitively area concentrated assess level tweet term select say tweet test whether tweet tweet follow edge similarity consider identification evaluate employ function assess spatial homogeneous process sd euclidean tweet area spatial poisson evaluate process paper standardize proximity tweet specifically standardized value km homogeneous minimum depict blue dash line respectively location tweet range homogeneous indicate slightly concentrate space homogeneous poisson possibly twitter user different middle lower spatial poisson explain homogeneous extreme tweet achieve term process number tweet distance concentration tweet slightly intensity tweet far homogeneous order whether observation tweet york duration day ten frequent km focus area avoid number computed tweet illustrate variance ten illustrate twitter follow poisson tweet distance twitter datum assess irrelevant tweet serve strong tweet event interval hour pm contain frequent uniform chi goodness interestingly hypothesis case result term analysis distribution conduct filter aspect next event influence present consider tweet time diverse performance choice bottom leave top respectively real span temporal diverse experimental first event random tweet span analysis irrelevant tweet namely distribute content tweet locate tweet york collect reference term refer tweet relevant tweet tweet select depend daily tweet create tweet irrelevant tweet irrelevant tweet scenario event concentrate area temporal choose tweet goal correspond appear least tweet choose unless go resolution method favor consider ensure tweet group cluster term criteria satisfactory link spatial threshold case threshold large chance grouping resolution representation aggregate time appropriately capture link tweet choice sensitive selection space one concentrate spread spatial event concentrate spatial spread handle scale scenario comparable experiment drop significantly lack temporal spatial reasonably well threshold cover scale experiment highlight handle simultaneous event irrelevant event tweet spatial generate tweet sect detect apply tweet measure cluster tweet correspond event want ensure tweet separate base sect propose standardize tweet term generate time different specifically link create separate increase form link tweet contrast recall positive negative
next differentiable intuition state formal globally strongly point sparse particular definition pair rsc constant locally strongly globally rsc rsc impose nonzero direction rsc form convexity vector past work rsc convex arise statistical illustrative x I link goal loss function case th consequently rsc extension linear observe corrupt lasso inconsistent study previously version past show consistent previous paragraph natural choice eq set semidefinite nonconvex concrete condition pair glm glm log family equation et convexity generalized suppose goal inverse arise let refer lasso take derivative condition satisfies convexity primal technique establish recovery previously extend optima norm function generalize gradient guarantee stationary point consistency recall due abuse slightly comprehensive generalized stationary key argument enforce feasible satisfy condition stationary point program support construction implicitly restrict convexity minimize subgradient note conventional constraint omit automatically greatly simplify suitable restriction restrict therefore uniqueness section follow primal support program unique state theorem concern concern rsc regularizer amenable concern guarantee support strict feasibility dual note condition incoherence usual corollary feasibility guarantee amenable regularizer discussion function function amenable eq feasibility condition size stationary primal unless proper choice exist corollary high cause sample rsc know take pn way guarantee also translate large consideration motivate advantage regularizer mcp scad mcp regularizer amenable discuss later remove mcp establish strict feasibility mcp suggest incoherent preferred variable selection incoherent scad mcp may simulation regularizer condition although theorem nonetheless confirm experimentally situation yet stationary appear unique feasible yet sufficiently examine dual establish minimum program support multiple strict zero certainly global diverse convexity concavity possess still simulation stationary global optimum agree result amenable set assume rsc restrict convex notation strict feasibility dt agree appendix strength amenable regularizer indeed beta min bind remove oracle rate corollary hand expression bound consequence concrete first focus ordinary regularizers mcp consequence regularizer result nonconvex scad mcp amenable fairly incoherence nonconvex regularizer recovery absence incoherence regularize square write semidefinite dimensional imply nonconvex program nonconvex family recover limit restriction prove amenable incoherence condition objective stationary optimum furthermore amenable nonconvex min bind b corollary provide comment consequence regularizers include mcp recall mcp translate regularizer choice asymptotically constant past establish consistency distinguish amenable deal past nonconvex regularizer ordinary regression interpret work zhang global optimum mcp regularizer consistent eigenvalue design optimum mcp strong design paper matrix incoherence appear sufficient condition concentration inequality inequality fairly fail condition corollary scad mcp incoherent function define corollary convex involve nonconvex nonconvex illustrate applicability dual nonconvex function recall corrupted covariate response corrupted covariate may constant suppose nonconvex stationary global familiar sample nonconvex lasso corollary nonconvex global optimum result indeed simulation incoherence hold inspection confirm conclusion simulation power regularizer move function likelihood linear condition may remove regularizer amenable regularizers glm composite positive somewhat nonetheless logistic et uniform categorical covariate relax goal r nonconvex stationary et require paper incoherence show properly nonconvex incoherence requirement generality apply case nonconvex stationary evident priori contain finally consequence graphical recall graphical lasso vector covariance subtle seek scale opposed nonzero grow connected graph cone positive definite impose constraint regularizer formulation actually selection consistency rather norm denote nonzero g scenario govern comment note handle symmetry treat program optimization take definite early oracle sub amenable also unique oracle appendix spectral norm restrictive assumption work incoherence graphical lasso restrictive much illustrate another distinct report result run order begin describe program eq stepsize simulation convenient eq efficiency ease analysis satisfie follow convexity smoothness regularizer optimum program amenable composite appendix geometrically tolerance error ensure iterate consistency optimum addition k iterate composite descent proposition satisfie form c n guarantee bind composite support describe simulation incoherence bound matrix diagonal everywhere else prove provide incoherence eigenvalue maximum eigenvalue satisfy easy run come ordinary corrupt scad mcp parameter logistic boundedness assumption impose generate agree qualitatively simulation show scad regularizer situation incoherence selection consistent I obtain incoherent matrix response addition generate corrupt run update panel correct recovery increase scad mcp recover correct remain put nonzero coordinate rescale horizontal axis match prescribe align confirm scad small mcp penalty previous regularizer consistent sufficient consistency parameter share curve agree et depend regularization scad solid mcp regularizers regularizers design incoherent plot error regularizer regularizers set line nearly align regularization set simulation uniqueness stationary objective setting come either ordinary least logistic unique multiple composite observe initialization appear converge correct slightly however violate composite distinct stationary show composite gradient descent come generate scad mcp regularizers panels b distinct incorrect still continue produce panel observe rate scale predict reach increase overall scad mcp axis panel panel ol initialization composite b scad mcp scad mcp distinct stationary finally analogous loss function equal panel plot plot multiple stationary panel scad mcp regularizer contrast run descent converge point panel simulation plot regularization may geometric threshold scad may empirically vertical cc variety regularizer initializations composite gradient mcp however large mcp regularizers panel develop extended framework nonconvex technique regularizer nonconvex significantly previously regularizer mcp consistent nonconvex regularizer recovery design incoherent recover regularizer future theoretical violate simulation justification scad mcp regularizers penalty assumption result convexity whether local rsc condition guarantee good proper initialization finally useful able rsc constant assign nonconvex amount curvature acknowledgment pl partly support foundation fellowship nsf pd fellowship study berkeley pl partially dms grant air force office research proof technical lemma subsection outline step construction appendix sn low interior step ii shift interior point rule subgradient construction iii establish note ensure concavity condition trivially verify contrary inequality assumption thus contradiction equation false prove guarantee convex rsc amenable regularizer program strictly immediately imply minimum prove support condition turn uniqueness assertion satisfy stationary point strictly establish ss prescribe fix furthermore rsc convex strict summing precede inequality interior hence lemma rearranging imply rsc inequality zero subgradient inequality rearrange second come need c note q imply put piece old claim calculus zero subgradient ss imply guarantee inequality establish amenable inequality follow global claim simplify equality strict dual feasibility require derivation corollary subgradient define simple algebraic property vi vanish amenable suppose strict hold combine regularizer impose strong ensure strict cs ss assumption hold strict feasibility provide inequality reveal scad mcp impose corollary appear rsc state tight denote I sub take gaussian conclude equation feasibility property turning write furthermore condition gaussian eq well q combine theorem proof extra assumption q concentration cs ss remainder expression corollary work imply rsc verify bind check parameter condition hold establish feasibility h applicability corrupt suppose satisfie deviation hold least argument union furthermore well note inequality q first second combine bound inequality inequality inequality conclude define strict feasibility succeed turn decompose establish establish rsc terminology particularly derivative verify inequality eq lie px sx cauchy appendix take eq norm unit sphere ss ss appendix show put piece high return ss guarantee triangle inequality strict dual scaling turn q bind term scale finally previous establish rsc framework provide adjustment remainder proceed simple program amenable calculation deterministic quantity depend convex feasible claim particular subgradient let j j redundant ps guarantee invertible point summarize nc parameter least exist p jk imply old plug back inequality desired define third equality bind triangle amenable jk global objective objective must appendix reveal satisfy feasibility inequality fact convexity uniqueness stationary exist straightforward see suitably inequality turn appendix imply bound apply combine primal eq conclude claim note also follow early norm detail n concern rsc relation imply require proposition inequality statistical result consequently write maximize similar argument equality v auxiliary employ lemma regularizer concave everywhere differentiable tt concavity function rsc amenable sample scale minor program q exist isolate local proof objective nonetheless suppose sake contradiction isolate exist sequence feasible extract convergent subsequence region close must together subgradient v kt k equation second inequality conclude well together inequality imply note gx x taylor divide take limit fx isolate collect kronecker norm claim use claim b triangle role cm ex
gpu computer effort ensure allow final require streaming overview imaging methodology parallel result reach conclude maximization current designing certainly seem extensively modern collect ray raw procedure discrete pixel inherent physical limitation ideally copy physical density theory frame sufficiently notably space obtain htb maximum likelihood intensity model record incomplete firstly measurement sphere ray overall noise process data volume noise useful aim produce way basic procedure step assign maximum full fix terminology firstly discretize generally non discretization rotation sampling intensity denote denote produce frame early em algorithm benefit free measurement scale around fourier intensity eq noise parameter keep decrease proceed sum observe frame rotation get require although formula understand likelihood use rotation encode rotation rotation identify discretized purpose cell regular whose boundary compose cell uniformly divide imply adaptively whenever increase go average continuous pair nearly continuous smoothing add step purpose latter say resolution operation working combination compression nearby rotation interpolation smooth value grid compression whereas consistent fall furthermore defer compare distinct parallel character efficiently probabilitie w inspection consideration nonlinear imply common gpu computational master successfully gpu approach section devise finally simple implementation gpu share core currently typical complexity algorithm stand likelihood rotation frame large space gb resolution distribute divide space implementation single node implement use closely logic briefly cpu overall stream gpu iteration intensive gpu discuss step denote contiguous rotation rotation computation e involve slice kernel care rotation step complex latter kernel normalization rotation division final gpu gpu execute gpu transfer final division gpu c htb configuration every portion implement pattern node normalize sum communication aim use increase actual experience highly slowly improve critical combine discretization hence large value sufficiently simple likelihood practice iteration achievable performance one quality process synthetic assess profile possibility large run gpu equip intel cpu core operate ghz I total gb gpu multiplication fourier bandwidth bandwidth cpu gpu connection device device memory copy bandwidth library use inter bandwidth gpu data volume image ray coherent source shape display sample pattern ray logarithmic table experiment reconstruct run consist dataset synthetic gpu implementation provide gpu gpu expensive consume updating distribute dataset synthetic consist explore adaptive single gpu run measure nearly perfect efficiency htb synthetic efficiency run really scale fully attempt distribute belong distribute inspection require table per per gpu remarkable loose fully achieve single gpu respectively benchmark matrix single gpu adaptive simplicity reconstruct criterion display likelihood execution per execution resolution whenever increase sharp peak successfully increase version quite remarkable run factor load execution iteration take cc configuration q execution time configuration scalability mean work program happen different notably case gpu step contrary scalability compare real increase prominent role measure case implement oriented depend linear compare par work well useful implementation site process stream fashion likely adapt dataset substantially dataset improve european become operational hz bottleneck hope keep ray enable biological protein liu european liu suggestion early classical method atomic analyzing pattern development technique extremely ray streaming energy ray machine associate inversion collection hundred expect synthetic ray currently successful quality problematic protein modern protein
method method per singular minimization subproblem accumulation generate addition compare outperform moreover subsection notation introduce study extend proximal establish conduct numerical finally conclude remark resp resp dimensional denote vector hadamard namely entry infinity norm denote denote resp square matrix denote sign operator also early lower stationary stationary point finally local chen recently point eq aforementione point natural extension proceed lemma subsequently post together multiply side finally arbitrarily choose let observe view stationary point give stationary replace otherwise eq b vx follow lemma post multiply obtain diag q fact satisfy submatrix consist row index clearly permutation together definition lead q pre definition conclusion em minimizer minimizer minimizer subdifferential v q multiply obtain v u mx hold hence local minimizer stationary point eq order hard relation section extend problem iterative reweighte throughout establish zhang unitary invariant function invariant set result class solve norm invariant singular optimal consequence subproblem solution offer solve subproblem decomposition hard equivalence extension method vector converge choose arbitrarily apply subproblem go state outer inner iteration fact termination inner method moreover accumulation suppose bound accumulation stationary entry I inductive argument view argument one outer clearly subproblem decomposition singular arrange q accumulation subsequence boundedness generality subsequence observe inequality show imply due along proof consider side use together follow contradiction follow eq relation contradict therefore relation hold argument first termination criterion section solve minimize sum lipschitz continuously function nonsmooth proceeding regularize matrix subproblem solve latter problem separable suitably efficient tool subproblem singular q immediately em proximal method pt end outer iteration termination criterion satisfy inner accumulation point stationary problem let stationary moreover satisfy inductive use use proof show theorem value arrange lemma accumulation point eq similar used termination criterion conduct test apply solve particular onto restricted convenience presentation name matlab matlab intel ghz ram bit windows service terminate accord denote apply process target solution define criterion successfully recover correspond relative subsection conduct numerical solve reweighted variant aim purpose generate matrix entry generate rank addition set figure detail successfully cpu generally three recover almost instance recover instance successfully six instance observe instance comparison datum subsection compare fill pixel pixel rank detail pattern texture decomposition decomposition testing image pattern six testing result cpu report display recover method column achieve three cpu time generally outperform cpu sr e e rest image recover regularize minimization introduce stationary vector minimization regularize establish iterative reweighted value subproblem show accumulation stationary solving establish regularizer nonconvex reference therein optimization moderately regularizer use develop lemma corollary proposition remark assumption zhang general unconstraine minimization first stationary introduce regularize minimization establish minimizer problem must moreover value minimizer minimization iterative reweighted propose subproblem show accumulation proximal solve outperform moreover minimization iterative reweighted iterative reweighte least proximal decade attract engineering
subgraph bound spectral clique limit detection analysis residual may detection subgraph extremely unlikely intractable detection anomalous continue field enyi give alternative vertex still activity background occurring still share subgraph vertex choose signal embed likely q equivalently simplify portion summation complete normalize principal eigenvector decompose component subgraph orthogonal subgraph yield convenience let combine equality use remove norm back provide embed eigenvector concentrate subgraph principal subgraph I subgraph occurs expect rewrite consider bind subgraph subgraph vertex eigenvector small may highly correlate magnitude convenience combination yield equation solve far relatively heavily eigenvector cluster onto yield separation observe expected signal expect result background anomalous subgraph embed change appear degree embed volume difference matrix strength spectral norm numerator ignore denominator slowly signal certain term dominate thus application interest vertex approximately constant growth degree norm approximate q equal increase ratio allow vertex probably occur constant threshold since norm dependent subgraph clique star substitute meaning vanish author technology office support dr thank early work dr dr schmidt anonymous helpful comment wolfe anomalous application connection graph diverse entity anomalous rest detection anomalous subgraphs agnostic graph base call leverage tool metric subgraph adjacency technique great detection bipartite realistic trend verify area great embed portion background detect highly anomalous subgraph internet traffic theory numerous entity datum organization website computer science know reaction varied entity incorporate datum represent relationship set vertex comprise denote theory provide indeed protein interaction represent computer graph focus community influential practitioner derive classify see research processing transform signal propagate along storage structure lack convenient perform analytical available relational derive detector signal subgraph network hard despite desirable notion small context computer discovery activity social recent subgraph subgraph statistic plant clique plant anomaly deviation description remainder detection relate community graph cast community broadly detection varied domain anomalous subgraph goal outli extremely edge insight noise discussion ratio intrinsic framework regression residual graph space eigenvector two subgraph work algebraic many researcher familiar analytically empirically enable algorithm detect anomaly organize subgraph detection brief detection outline present demonstrate finally open work subgraph size set subgraph set edge whose unweighted undirected connect vertex edge graph union graph work spectral adjacency associate order vertex denote edge connect adjacency subgraph symmetric norm unless induce norm eq absolute eigenvalue focus signal base edge deal vertex degree adjacent observe denote note degree typical background activity anomalous subgraph combine union give observation discriminate scenario formally want resolve hypothesis purely noise alternative signal vertex subgraph research assumption paper variety pattern common subgraph research anomaly history would static focus interest operate complementary outline consider optimal foreground embed via occur equal enyi set al possible occur probability graph occurrence applicability therefore require way count subgraphs observation target target sparse require detection random subgraph hard background foreground enyi situation observe edge test simple ratio require know computable asymptotically theoretic dense complicated model subgraph residual graph analyze widely separation community modularity modularity disjoint entire set entirely edge connection denote half half prevent edge proportion degree cut thing maintain vertex total deviation expect community detection literature numerous exist propose modularity maximization modularity adjacency divide modularity maximize solve indicate technique community suggest eigenvector compute community detection graph inspire amount work anomaly novel subgraph background activity subgraph residual observe variation signal anomalous subgraph reduce spectral residual residual establish theory eigenvector matrix see presence certain anomalous work within resolve framework computationally tractable variety scenario modularity baseline residual several advantage know eigenvalue eigenvector expect term computation eigenvector computationally expensive observe degree possible variable inter connection degree add covariate mention aspect processing framework assessment norm quantitie graph several context graph intuitive would detect algebraic quantity frobenius residual matrix square residual separately exactly edge vertex star vertex star concentrate cause stand edge place subgraph stand apart background work norm metric metric power subgraph principal anomaly appear foreground graph bernoulli foreground subgraph embed interaction residual subgraph submatrix background residual rest within complement identify least subgraph also subgraph stand residual matrix extend arbitrary may one detect anomaly rely subgraph eigenvector alone distribution thus subgraph stand eigenvector background small concentrated subset provide serve proxy compressed sense eigenvector subgraph occur relatively discuss eigenvalue eigenvector eigenvector normalize deviation small negative value demonstrate graph skew distribution cl vertex embed large positive deviation index extremely unlikely occur hypothesis commonly create deviation value circumstance occurrence subgraph detection subgraph connectivity background anomaly eigenvector thresholded subgraph eigenvector subgraph residual consecutive eigenvalue together stability unstable rely sufficiently change subgraph detection subgraph large rather eigenvector whose principal find space limit utilize large substantially residual put number nonzero however np hard penalize form equality semidefinite solve semidefinite eigenvector denote return subgraph vertex thresholded rely decomposition leverage number algorithm thus scale edge run control future technique anomaly degree model enyi simple enyi vertice equation vertex expect expect popularity give share yield degree expect degree approximately modularity fit background mat stochastic kronecker base sum fold kronecker ij np vertex add edge generator create graph mild present challenging noisy subgraph model degree complexity modularity matrix match formula mat structure mat generate approximately graph flip diagonal diagonal make cl expect edge mat er complicated note moderate mat cl vertex degree mat lack low corner randomness enyi graph anomalous er combine entry subgraph bipartite split letting er generate bipartite subgraph expect adjacency form demonstrate metric outcome carlo graph subgraph vertex bipartite statistic outline create discriminate mat cl er average cl foreground vertex vertice vertex er always near cl mat phenomena result confirm note cl extremely term bipartite foreground subgraph mat cl likely cause among detection improve chi improve analyze norm subgraph embed statistic spectral chi cluster bipartite foreground make aspect technique non performance improve subgraph improves cause fig histogram mat minus axis eigenvalue bin trial approximately eigenvalue localize around value yield mat rank degradation performance square statistic rapidly embed improve symmetry projection balance bottom similar exception precision foreground vertex identification use cluster vertex improve eigenvector precision performance subgraph separate necessarily via burden limited trial accord mat mat er equal estimate detection mat er fig much
problem brief construct column element column except denote th row ergodic vary norm define mml agent agent comprise represent interaction I affect edge function denote direct path vary topology strongly exist vertex weight degree weighted laplacian associate family model practical graph simulation random construct edge realization decision convex reveal incur difference well eq perform fix move environmental present unit regret decision incur decision cost run tx distribute objective network di decision nesterov dual turn inspire sub centralize step onto set define dual averaging sequence proximal avoid undesirable strongly agent subgradient p attain via local gradient provide compactly preserve laplacian access path agent strongly construct stochastic require associated direct graph selection diffusion communication edge weight expert associate expert select j consequently present applicable loss decision regret sub weight allocation agent associate embed agent ft f jt j ni distribute associated information neighboring information place link self preserve row matrix row every weight eq graph irreducible give decomposable communication switching set change dynamically selection topology topology positive strongly communication communication topology specify j distribution matrix connect matrix employ present remark assumption lipschitz refer factor standard weighted averaging regret sequence weight dual analogous thus follow algorithm present evaluation appendix impose upper associate highlight underlie consensus problem extend optimization provide employ weak ergodicity reason imply one matrix thus suffice negative matrix positive mp specify note must ergodicity markov product converge exponentially rank maximization realization sequence matrix element strongly connect topology node integer represent direct strongly represent row matrix case switch topology state switch ergodicity connectivity proposition present since q therefore test statement theorem convergence highlight topology diameter coefficient ergodic eigenvalue weight network convergence provide ergodic communication matrix construct diameter integer communication round subsequently since f r ik ec kt ks kt base conjunction imply bind algorithm tighter capture ki perform subsequently topology moreover graph regular graph neighbor uncertainty distribute process objective di manner uncertainty environment belong present quantify ergodicity diameter direct topology arbitrary sequence definition right eq thereby express base first side expand hand first g ix deduce projection respectively present impose bind note imply appendix impose last highlight examine agent similar generate tf tx regret imply statement hand adopt estimation aim estimate convex contain origin vector vary sensor unknown environmental factor sensor assume accurately observation topology sensor represent presence indicate sensor agent summarize q assume local reveal allow error uncertainty environment subgradient sensor neighbor cumulative tp offline noisy case centralize error suitable scenario characteristic wireless dynamic effect example dynamic eliminate framework rely datum distribute step local reveal select lipschitz th th h algorithms sensor scalar agent example also qualitative agreement indicate accuracy topology scenario sensor also demonstrate well tp adaptive topology assume sensor addition various figure result without regret node signal standard furthermore connectivity correlate connectivity designing topology operate uncertain metric scale tp kk network operating algorithm evolve use agent highlight measure average well demonstrate range mobile network failure moreover extension examine filter adopt system operate environment embed decision process lemma present u increase follow result decentralized algorithm step eq evolve thus q upper arbitrarily strongly topology node exist matrix entry represent integer path entry aforementioned induction row note positive satisfie adjacency algebraic literature thm edu paper agent presence uncertainty topology inspire advance base dual gradient link adapt reliability agent rate topology
empirical gaussian process popularity decade gp parametrize determine integrate away gaussian would process reflect natural intuition simple parametric form general elliptical student value elliptical place inverse wishart process form student connection similarly utility uncertain example student perhaps might detail question student wishart process elliptical precisely motivate wishart arbitrary analytic predictive distribution inverse wishart wishart inverse prior gp predictive covariance tp even covariance gp contrary analytic separate analytically gp find misspecification improve covariance distant kernel predictive covariance introduce inverse wishart student wishart covariance demonstrate process section section wishart choice matrix arbitrary wishart definite density follow n wishart parameterization marginalization principal submatrix make wishart appear attractive covariance matrix wishart suffer impractical bayesian wish expect semidefinite wishart however almost thus requirement wishart marginal nevertheless suffer inverse wishart inverse wishart place mass wishart submatrix distribution parameter wishart property motivate define wishart k kx wishart h xx grey represent gps popular nonparametric thorough guide gps provide gps practitioner use place student parameterize wishart prior analytically derivation material marginalization multivariate student kk nk kx tp elliptical sampling equivalence model tp student generalize gp limit tp prove control large tail tail prior sample draw extreme behaviour also dependence variable jointly student distribution notice dependency control gp tp multivariate analytic material kn n predictive intuitively infinity predictive depend train somewhat covariance importantly likelihood tp differ predictive tp kernel hyperparameter student explanation square distribution value large vice scaling apparent flexibility wishart student marginally n scale surprising integrate derive student process show integrate scale arrive process insight lead student student student process elliptical overview distribute symmetric unimodal point distance property want encode make elliptical naturally extend countable subset jointly elliptical density alpha process characterize elliptical density variable r elliptical elliptical either generalize process elliptical tp thus expressive nonparametric bayesian analytic expression predictive distribution gaussian process wishart positive matrix offer novel student wishart orthogonal square row orthogonal rotation volume definite matrix let exist iw careful tell uniformly distribute describe exchangeable affect exchangeable draw interpretation scalar variable analogous wishart unit sphere independently marginally exchangeable orthogonal sample symmetric symmetry distribution rescaling variate common practice latent gaussian advantage model sum gaussian analytic unfortunately sum independent analytically parametrize kernel add independent effect propose handle incorrectly function noise gp gaussian various behave tp student attempt tail inference analytic novel analogous process key hyperparameter include tp respect supplementary regression hamiltonian function sum exponential delta mse function noise test tp generalize superior predictive independently show mean since hyperparameter tp superior hyperparameter well amount record year clear tp much measurement dataset due attribute ph quality score learn learn optimize objective powerful optimize expect improvement ei optimum paper ei ard ern wish multivariate mat kernel represent parameter n note x slice similar use acquisition net n descent intuition acquisition fix plot clear tp prior mat ern plus delta bayesian integrate away slice sample dim dim dim aim minimum function minima tp gp tp come take function method x x initialize one minima corner cube behave
unique minimizer differentiable continuously fix l every imply uniformly lipschitz lipschitz uniform boundedness e hand subgradient bound uniformly measurable constant q imply boundedness ready justify quasi martingale follow quasi martingale surely q function converge stochastic process q condition proposition martingale limit almost consequently converge first utilize let another sequence take sequence produce derive convexity derive gradient frobenius uniformly eq fraction plot corruption rank robustness notably result relatively rank extremely simple pca perform corruption typically less largely corrupt corruption propose sample indicate suitable capacity pca corruption hard htb figure low corruption corruption interested tune rank corruption fraction investigate fraction intrinsic vanishe converge observe bottom row notice dimensional drop begin possible basis coefficient inaccurate however become rank exact rank pseudo estimation however strategy aforementione xu li xu li decade batch big due bottleneck regularize scalable consider apply completion factorization consist basis stationary asymptotically demonstrate encourage robustness decade machine community domain include collaborative filter texture name suppose ambient observation aim low problem many variant typically involve residual error rank term speak intractable tackle researcher suggest alternative relaxation surrogate nuclear max like induce nuclear formulate semi definite technique show filter margin program nuclear come compression world sensor background clutter corrupt case tool subspace handle corruption seminal prove mild condition recover corrupt notably guarantee recovery relaxation max norm regularize superiority theoretical collaborative example prove max schema nuclear follow cluster problem another important mathematical sdp scalable gap progress practical applicability bridge gap regularize constrain empirically promise result filter constrain max however require access batch optimization bottleneck utilize factorization max norm advantage online sample online fit interested aim norm fold develop scalable solve prove solution asymptotically work literature work norm superior collaborative filter hamming influence sample uniformly nuclear generalization nuclear norm superior important regularization scheme scale huge scale effective one mini total optimization global example scale dataset machine server collect return worker obtain consideration communication global try give protocol work showing require restricted rip random accelerate establish aforementioned assume aim rank component incoherent sample assumption establish incoherence exhibit thus liu li propose dictionary phenomenon study sparse incoherence place completely remove distribution follow global work decomposition relaxed problem alternatively without intuition benefit rank factorization optima guarantee local factorize problem convex code alternatively schema demonstrate hard learn dictionary analyze progress assume rip initialization provably minimization non sense optima recently pca algorithm comparable nuclear subgradient max insight max example norm factorization couple optimization technical appropriate factorization amenable online part idea factorization robustness contamination convergence magnitude dictionary constrain uniformly naturally begin state manner guarantee mild give norm conclude introduce bold row th entry set online underlie rewrite intuitively coefficient term fortunately dependency constrained aim recover variable replace note see need solution satisfy still contradict equivalent sample equip rewrite fashion combine indeed optimize minimize empirical infinity empirical converge online detailed noise alternate manner iteration give optimize examine tucker kkt condition basis define warm accumulation b alternatively implementation previous small exceeds w computation computing consider sometimes efficiency computed matrix empirically verify decrease increase increase value help next iteratively corruption perform operator coefficient bi directional initialize low search upper q update sample accumulation optimal subgradient row section theoretic assumption generate surrogate strongly particularly small definite small term problem enforce add solution stationary tends infinity essential tool analysis stage bound firstly turn secondly boundedness g main converge almost index p concentrate ball martingale almost central limit almost surely loss imply surely taylor taking vanishe tend conclude focus mc another index observe orthogonal onto span matrix outside th equal otherwise interestingly max cast introduce otherwise reformulate product tends reformulate mc implementation regularize mc difference element either two newly optimize completion describe sample access compute accumulation optimize surrogate trivially justify decomposition clean corruption correct evaluate fitness subspace variance ambient sample algorithm report htb effectiveness measure last intrinsic report observe corruption low pca tight number besides pursuit baseline performance algorithm corruption corruption agree fit much fast ambient dimension difficult dimensional pca achieve sample optimize curve basically pca algorithm example
roughly sample predictive selection small cross choose majority background dataset px px model proportion mix simplex eq since negative detail gradient partial hessian definite long therefore maximum strictly unique maximization concave vertex simplex element vertex simple evaluation iteration element objective vector maximal remain enough property restrict evaluation line f objective quantity notation current regret iteration iteration show suffice background curvature show background show twice differentiable become brevity denote right depend thus long maximal minimal curvature frank wolfe background simple tolerance number depend objective function step j maximal element step px dominate q computed come early train complexity predictive feature gene set complexity likelihood background model reasonably growth public query take iteration second intel core tm cpu ghz weight old tend citation fig normalization technique citation measure infection cell model capability large dataset cancer human cell cancer cell vice versa top weight datum drive direct indirect citation favorable modern metric precision rank engine list strength gold existence edge dataset publication top relevance table size seven clique four breast human stage remain three brain among collection clique profile clique experiment edge sample form lastly study illustrate support manual search topic gene expression database expression eight dataset manually table test well drive retrieve dataset retrieval fig reason annotation vocabulary description level specificity detail individual result investigate retrieval retrieve query false table retrieve dataset follow connection area interestingly e retrieve query outli human finally internal ranking seem health annotate annotate retrieve e partially sample type e sample people half old background know contain sample disease old health states institute science molecular biology laboratory european bioinformatics genome cb sd uk technology university biological pre set sort normalize essentially compute list sum score increase gene final procedure essentially compute weighted kolmogorov divide ks match scoring gene score value successfully use early select set standard produce active gene activity gene express core
measure I confidence add originally continue unless stop reach datum labeling supervise broadly group ii learn unlabele datum inductive contrary evaluate goodness unlabele supervise illustrate learn understand supervise unlabeled negative supervise datum unlabele boundary remain supervise shift supervised shift positive semi green dot green dot one fail label supervise generative ii type two task speech language solve two way classify learner attribute attempt learner language learner mathematically discriminative calculate supervise predict eq q ignore interestingly algorithm use induce class give instance generate semi algorithm reason completely ignore label provided generate key semi understand algorithm give numerator covariance estimate mean tune accord algorithm machine graph self discriminative broadly ii iii initially label classifier generate classifier initially unlabele classification unlabele reach newly classified remove produce second classifier apply continue converge unlabeled classifier stop reach threshold cycle difficult nonetheless self dataset contain number training create label datum generate unlabele confident add confident cycle continue two co attribute naturally however meet training view sufficient make good classification learning unlabele provide human label label remain unlabeled unlabeled oracle natural language base finding classic state literature domain parse text classification mining self improvement report speech successful execution training come source domain besides third linguistic attribute firstly produce secondly experiment article hold million article north american news corpus perform well contribute contribute author section level sentence section label compare baseline neither improvement sentence poorly however supervise sentence pool particularly require memory execute cycle self source limit success language spam form text spam author reproduce first message dataset besides message show training filter supervise fail good aforementioned mind comparative study produce summary author use four classifier relevance event attribute try combination summary surface content attribute combine supervise measure score score gold summary co training I summary although supervise summary na I dataset cluster come summary summary among author score well word summary cluster drawback attribute primary co co view secondly surface consider ignore na I paper significant effort secondary interaction help identify phenotype pathway classify sentence describe protein supervise tool first name brief description dependency name tree analyze path sentence gold measure similarity interaction protein supervise semi generate sentence classifier evaluate gold supervise version classifier near respective harmonic attribute classifier aim cb aim dataset contain protein cb compose protein interaction algorithm experimental outcome aim perform cb dataset classifier base attribute produce result aim performance satisfactory examine effect training aim semi perform poorly available start well cb report distribution aim addition imbalance affect limitation since rest finding parse text mining substantial supervise difficult unlabele behind success semi classification rather surprisingly investigation find classic suggest despite whether really supervise label semi dot classification gray dot old dark dot represent come conclusion much classification old abundance amount unlabele people nevertheless deal point suggestion classification semi supervise algorithm match hand cluster naturally co attribute value tend class paper complicated distribution datum investigate instance poorly unlabele highly assume decision false negative proportion unlabele choose proportion affect overall empirically effect attribute classification keep detect unlabele usually source unlabele rather semi email idea several natural processing supervise exploit difficult expert despite wide use classification empirically supervise study explore possibility limitation classification parse classification method label data model natural language processing text due unlabeled data process time serious effect supervise
compression reconstruction would explore practical consequence viewpoint combine namely learn elementary finer next require transmission stack result bind appendix theorem hessian respect generative derivative term appear diagonal newton algorithm asymptotic neural activate unit newton hessian computing determinant gauss newton hessian consider reconstruct yx layer wise gauss namely initialize output exactly cost backpropagation pass look backpropagation backpropagation derivative network compute backpropagation pass propagation initialize layer backpropagation derivative value activity incorporate acyclic influence model length backpropagation eq layer derivative iw initialization influence influence influence subsequent substituting influence connect condition non vanish v collect backpropagation know influence term derivative avoid full transfer set provide way compute note rate propagation construction summing yield forward sum backpropagation sc sc bx sc definition prop lem example exercise exercise remark similarity difference encoder minimize encoder decode denoise auto particular viewpoint determine noise denoise auto aim datum hide dimensional feature space simpler describe interpretation framework length probabilistic correspondence simple datum auto encoder short auto answer actually encode theoretic auto minimize compress base encode simpler tight minimize refine space sense arbitrarily close generative result appear also illustrate optimize auto encoder look optimize depend upper aim optimize aim make upper connection continuous impossible sensitive quantifying criterion denoise criterion one differently connection derivative somewhat though penalty frobenius norm row theory additional various variance noise level absolute variational already network situation encode situation try encode encode input encoding minimize reproduce cost include generative dimension feature encoder function go generative depend instance represent neural generative law infer thus previous case dirac single base alternatively auto view draw random apply generative viewpoint good generative give distribution point bit generative eq minimize amount minimize datum know encoding optimize work integral feature contribute significantly contribute yx give know conditional know nice obtain guarantee reconstruction feature variational feature quantity autoencoder thus moreover bring priori auto generative explicitly encode encode use define choice discrete sense section q difference substantial relevant two necessary encode feature perspective encode deterministic cost involve decompose empirical kullback leibler probabilistic cover deterministic dirac discrete two part part comment comment error case probabilistic proposition network activity layer probability value section continuous entropy optimisation kullback leibler possible arguably introduce regularization absence front value fix elementary bernoulli optimization elementary fairly tune kullback term close elementary model two encode space generate one expect save differ save encoding generate probabilistic small pick generate accord depend optimize around feature specific bernoulli gaussian lead interesting denoise deterministic feature positive expect elementary read average noisy reconstruction error proposition encoder set carlo ordinary backpropagation activation sample backpropagation layer factorize linearity namely average input explicit incorporate result feature optimal choice around taylor expansions optimization denoise taylor small value let covariance reconstruction error choice noise deterministic use matrix hessian minimize favor error possible reconstruction denoise optimize bind third compute hessian reasonable valid magnitude particular different way denoise criterion diagonal optimize computing theorem compute gradient backpropagation pass diagonal gauss proposition taylor expansion z h yy estimate choice case direct minimization symmetric decompose diagonal z reconstruction hessian respect newton derivative output auto denoise instead jacobian norm quadratic reconstruct elementary variance reconstruction approximate upper newton approximation dominate square derivative compute computable parameter compute costly gauss layer turn instead distinct stack layer start hessian hx yx yx xy get q gauss namely valid summing sign implicitly matrix apply complex involve hessian adapting involve choice elementary maximize distribution adapt denoise criterion consider minimize reconstruction focus divide bit work situation output usual actual reconstruct datum recover likelihood fix fix loss incorporate various jointly give optimize estimate proposition q square additive average dataset focus error
core analyze model much simple dirichlet access accurate need dirichlet place perhaps effort multinomial system calculation mean point gaussian mixture important natural model complex demand estimate dirichlet multinomial globally converge publish dirichlet go implementation simple produce initialization stay newton access mle dirichlet multinomial fix also matlab review new mathematical lead ng general newton provide fast build dirichlet newton method use property propose algorithm particularly count domain positive gamma function equivalent minus input gamma number n input log problem make equation k mutually exclusive outcome vector sum multinomial piece dimensional count vector appear distribution multinomial yield also dirichlet intuitive notion put take account relevant count dirichlet follow want mle parameter access multinomial sum multinomial observe far observable row count multinomial computation multinomial speed multinomial k represent dirichlet dataset maximize build relate sufficient separating sufficient maximize look newton method hessian dataset follow derivative function library link library matlab follow matrix indicator vector compute hessian derive multinomial start except represent time category row start set constant omit unfortunately require full exponential dimension formula dual first summation processor job section update bottleneck propose use write microsoft matlab toolbox base compare time platform grow start round fp solution algorithm eq multiple times recurrence algorithm bottleneck newton computation become less important xlabel ylabel legend style cells east legend west coordinate hinge shape linearly linear constant dominate xlabel ylabel second legend cells anchor legend pos north coordinate negligible add overhead implicit answer process hold run generate dataset xlabel ylabel seconds pos south coordinate eventually bad version circumstance many per multinomial already solution reasonable size high phase ever long constant around xlabel ylabel legend cells east legend pos north west coordinate constant benefit evident dimension run cause allocation structure xlabel ylabel seconds legend anchor legend pos south east suggest newton computing dirichlet row stop add run constant multinomial amount law sample possibility handle mle intuitively valuable away possibility desire split dataset part add hybrid approach would independently may keep track would row go powerful dirichlet multinomial produce multinomial come single look dirichlet allocation row could produce map build accurately large cluster mle dirichlet matlab multinomial repository repository except correction matlab extend library experiment keep repository categorical
despite empirical significant adopting scale problem counterpart albeit usually mathematical constraint slow gain machine paper xu lasso regularize machine many make outlier perturbation case problem complicated uncertainty svm counterpart paper question two meta algorithm guarantee uncertainty regard uncertainty uncertainty allow second allow set oracle term find formally meta robust counterpart np achieve efficient tool algorithm recently sublinear methodology oracle contribute notably give perturb work formulation semidefinite quadratic program build recently linear trust robust reader address several solution cut goal run rest organize present rest section simple problem employ subgradient step assumption latter problem find bad exhibit perhaps programming formulation formulation fix robust formulation vector constrain without assume specific symmetric relaxed standard feasibility binary current course first robust ix derivation online latter reward maker loss metric call give maker select henceforth robustness adversarial reward thereby online perturb maker predict accord euclidean projection onto step current next achieve sublinear reward upper norm gradient diameter algorithm programming assume function online tp uniformly sequence decision magnitude approximate mathematical adapt formally approximate subgradient descent availability oracle either return infeasible oracle saddle point interior solve solver robust make upper u v gx du input target infeasible ex ti definition algorithm comprise dual update update conclude infeasible terminate call oracle first return robust counterpart infeasible otherwise return infeasible online descent combine final hence imply structure mathematical program term oracle approximate u ix procedure compute g linearity seem subproblem give least q conclude inequality convexity imply mention namely variant capable approximate one analyze approximate program domain choose perturbation noisy much produce upper reward vector throughout shorthand follow correctness proof use unobserved imagine round right hand q put thing final old cube identical otherwise cube claim feasibility program efficiently case interest highly efficient lp case combinatorial efficient generic lose robust solve robust favorable uncertainty counterpart possibly lp unit ball notice feasible correspond robust formulation robust program noise control u euclidean program amenable meta linear noise simple call oracle statement robust lp uncertainty also notation hence robust f maximal use see euclidean counterpart semidefinite program order motivate avoid reduce program sdp approximate qp solver qp uncertainty similar albeit general uncertainty r apply certainly fall scope hold program nominal show claim note fx prove claim lemma demonstrate qp uncertainty mathematical also iteration solution f norm accord k imagine transform eq require notice maximize establish robust approximate semidefinite sdp q feasible frobenius matrix norm counterpart sdp general np uncertainty nevertheless use uncertainty eq u apply robust term present ix terminate call sdp allow schwarz therefore finally note factor effectively solve robust transform problem equip robust approximately employ essentially applicable worst feasible accomplished subgradient solve large problem original problem support robust become process interest
strongly length immediately hardness fact could polynomial decide sc proposition polynomial impossible factor hard construction also unless finally noise aware variant treating remain norm variant also complexity remain wish focus design performance threshold notably easy solve would hybrid author thank look attention anonymous valuable manuscript thm thm corollary definition thm plus minus minus computational observation decade automate overcomplete turn processing dictionary iteration complexity actual technical show learn moreover hardness solution give result dictionary sensor result compress compressed complexity cs exact decade denote recovery read error also condition greedy pursuit omp relaxation subsequently numerous tailor dictionary certain class dictionary verify signal admit approximation setup thus achievable dictionary signal sparsity dictionary instead structure successful include audio speech name informally dl vector allow representation formalize way dictionary far incoherence basis g dl encounter sparse recovery subproblem treat classical cs reference therein broad dl concern nature widely challenge formal hardness well hardness dl relate via obtain hardness result recovery exist widely assumption hardness additionally unless pseudo fully solve within thorough treatment complexity mention goal formulation dictionary usually capture priori column function express fidelity penalty regularizer representation usual e seen often require norm cf priori become trivial exactly column also hold variant allow minimize error require large intuitively certain efficient algorithmic atom justify similarly since sparsity coefficient achieve representation basis redundancy expect maintain hardness result show indeed intractable dictionary hard strong restrict ms rank rank elementary reduce submatrix ms hard cf polynomially cut input hardness sense obtain equivalent set minimal dictionary fact rank require discuss dictionary reduction indeed inversion strongly polynomial gaussian strongly extend discrete objective scale dictionary learning ms achieve obey valid know ms contain similarly associate given decide square problem existence efficient polynomial general impossible interpret constraint turn etc even cost know quality recent work start theoretical guarantee dictionary initialization efficient guarantee hardness approximation ratio size usually e admit quasi almost hardness compare present seek full right multiplication infeasible close inspection reveal reduction carry efficiently provably train hard approximate objective objective maintain hardness dictionary rank ms hardness hardness remark new dictionary row achievable e system formally dictionary permutation nice contain strong moreover remain relaxed mn proof strongly cover collection word employ recent exist strongly complete problem cf sc parameter
big gain power discovery pattern coarse often ref far complicated volume ambiguity content character theory decision impossible importantly category problematic many diagnosis disease consider individual type determine survival without take two wish prefer student discrimination must course reject modern causal examine political benefit new resource well aid reject early goal discrimination enter apparent seem acceptable although trade utility value concern account classify contribution bayesian list ref recent example allow combined propagation processing system pass data module final prediction allow track final meanwhile list build call question live south redundant interpret bayesian tree human advance know combine remain uncertain mechanism wish may way solve artificial intelligence upon framework influence learn forest monitor day hope causal provide researcher policy reason causal us progress still make informally system outcome whether desirable mathematical example mathematical algorithmic method propose eliminate possibility output category time preserve method output possibility algorithm recommendation distinct consider ref seek exclude input indeed order effect correlation calculation require category determine odd patient college list list partition list consist track potentially wish avoid minimize impose eq outcome category population knowledge alone allocation accord life surveillance additional minimize leibl decision maximally indistinguishable lagrange enforce enforce imply knowledge correlation kullback leibler divergence become ill lead different opposed care notion suggest accounting accomplish exclude social mathematically strict particular desire population trade group outcomes california information concern college relative question come case hard impossible rely recommendation remarkable power human critical increasingly become infer outcome world advance review recent popular account seem scope powerful partially provide management prediction make hard assess elsewhere provide challenge evident existence challenge challenge correct estimate individual restrict datum participant goal reverse causal present restrict progress algorithmic box upon reveal algorithm innovation method computer change perhaps importantly allow big concern discrimination previously understand domain tie question argument increasingly familiar nature reasoning sphere joint present university discussion fellowship machine public big made absence fail demonstrate loss enforcing near role interpretable machine mechanism reason well mind augment automated reasoning subtle explicit decision political make reasoning cause machine aid decision make make policy involve pose developed program question interference unable resolve public social category make potentially half united states discrimination color national title ix education line insufficient prevent avoid category north american wrong track north south line decision making may prefer physical fitness access status status happen physical fitness character prove base detail know neutral causal role particularly
need rp rx rx ols ridge summarize basic validate orthonormal basis respectively I vector r efficiently parallelism query functional rp function belong ellipsoid shall control allow finite empirical functional j optimally lead dr h regression finite vary mapping f state risk suppose u feature take consistency stem basis stem ol estimation empirically prove effective previous easily work unlike need evaluation make scalable simulation formation particle matter unfortunately simulation multiple magnitude time step single simulation body simulation lagrange simulation order magnitude inaccurate experiment body simulation mapping area come area na mse result cube note set particle cube cube cube coefficient represent density cross validate parameter ridge report truly come body table average fast achieve improvement come improvement widely time prediction like mining miss embed hmms variable train em aim predict attempt audio prediction segment vertical music particularly short audio piece segment music compression music experiment song sound hold prediction take music predict consecutive total datum instance audio ridge quantity bandwidth mse validate hide bandwidth mse achieve lowest apparent audio audio superior predict sound material missing long cross fast speedup occur motion total look position unobserved step series joint perform randomly miss series output response miss joint position correspond spatial position projection consider segment take parameter estimator validate lc good perhaps surprising prediction point emphasis method speed x use prediction conclusion triple perform scalable manner functional capable massive low function assumption order magnitude multidimensional series estimation see eq l function note analogue response function nonparametric function cover application type previous set achieve address issue estimator capable nonparametric massive reduction previous estimator modern instance grow massive number instance function aim functional response function hence immediately unlike typical regression directly infeasible directly function nonparametric set triple framework quite instance may pdfs financial stock output stock another interested inaccurate output pdf expensive essence inaccurate previously accurate foreground intensity take pixel position foreground pose framework frame unit represent interval predict application predict co occur function motion capture interested predict movement movement state previously infinite dimensional output instance instance million infeasible want learn mapping want resolve scalable since shall inexact observation noisy evaluation distribute I take kernel estimator useful clearly produce lead prohibitive set previous nonparametric response scalable estimator functional response response linearity mapping moreover specific output observe evaluation sample distribution short input onto basis rbf function ij fp ix fp wiener omit simplicity input estimate output unseen function tensor serve mention function evaluation index basis index typically
density careful depend prior able obtain give require even leave strategy wish finish introduce significant address open rbm nevertheless recognize theoretical interest one note generate require decide accept reject would unfortunately however cost run investigate example implementation prefer issue future relevant section detail fully drive brownian reflect brownian motion rbm extension motion drift coefficient take counterpart denote brownian major later shall simulation indicate rejection conditioning arrive dt dt three subsection piecewise linear approximate rbm aim obtain rbm topology drive brownian algorithm wavelet drive reflect computable system equation result map equal radius see satisfie interior strictly surely continuity rbm boundary leave match locally drive eq piece wise brownian brownian brownian etc brownian denote shall introduce brownian increment relate sample approximation pointwise perform possess overcome difficulty proposal ratio constant conditional piecewise later rejection continuity k section denominator accomplish involve exact complete explain sampler justify provide brief description piecewise motion sequence process eq brownian dyadic available dyadic iteration dyadic index l j interval brownian collective refer arrive mr generate dominate piecewise form piecewise shall approximation q dyadic interval construction dominate recall lipschitz topology solve reflect n brownian dyadic contain like brownian motion mention rbm stay interior boundary stay dyadic know indeed continuity observation everywhere q recall piecewise rbm initialize nj nb serve reflect find n b independently dimensional brownian motion suitably propose denote brownian algorithm contain brownian motion rbm boundary match say increment return know simulate add sample know law collection ease exposition brownian available index dyadic interval denote motion density close indeed sl brownian bridge interval stay within ready outside support support sample simply uniform make propose eq density ratio explain reject perform rejection ratio lipschitz continuous positive sl explicit expression present variable accept note output follow know mention comparison necessary know left layer generate far refined rbm comparison part z initialize n nj serve piecewise l solution leave increment proposal product ratio acceptance rejection grant explicit proving follow continuous bl immediate consequence lipschitz continuity derivative lipschitz lipschitz lipschitz bound converge lipschitz since immediately fact lipschitz continuity boundedness boundedness lipschitz respect lipschitz constant lipschitz continuous lipschitz reasoning cm cm claim conclusion corollary criterion example theorem convention remark reflect brownian motion rbm develop tolerance allow piece wise approximation rbm conditional acceptance eliminate suitably design information proposal contribution exact simulation multidimensional reflect rbm rbm turn generalized comprise server heavy traffic arrival definition contribution outline underlie tuple give satisfy driving reflect problem drive process brownian motion diffusion c f occur eventually decide reject make sure
dataset describe section gender classification block vs kronecker toeplitz kronecker tn number operator objective rt follow incorporate toeplitz constraint equivalent study norm toeplitz correct next scale improve make dc shrinkage e minimize estimate shrinkage test gender video field distance frame surveillance camera period wide weather wide often video front back view total work evenly gender video remainder testing demonstrating htb attractive invariance property use object detect box relatively uniform lack clutter detector track connect track rare box uniformity position spatial track dyadic successively split array nest covariance generate reduce mean covariance gender learning discuss ratio covariance block positive weight thresholded logistic advantage adaptively block oppose relate quadratic several trial divide track frame track equally disjoint extract dc dc overall svm evaluate relate kronecker window number logistic particularly use svm indicate regularization curse note rgb pixel box neither exceed coin flip rate htb frame track correctly incorrectly train relating specifically crucial head area gender physical size htb c svm htb htb application temporal behavior nonparametric modeling appearance temporal significantly baseline classifier partially support nf fa wish propose application em plus minus height electrical engineering university usa spatio technique temporal adapt spatio characterized mean spatial correct apply temporal feature box gender choose covariance classification performance accurate human characteristic gender etc video surveillance understand spatio apply classify attribute appearance movement gender challenge low resolution surveillance track spatial area covariance address deterministic shrinkage estimate improve spatio matrix reduce discuss predict gain estimator
mean typical purpose crucial take obtain kernel frobenius expansion sign gaussian definite none expensive far argue mathematically strategy invariant kernel use abstract point consider invariance ensure existence algebraic group sign give group group goal k mn make consider act invariant kernel converse true existence state convenient variant symmetric positive kx call universal let act space algebraic invariant space algebraic differentiable group way finite finite manuscript appendix admit variant non group compact exposition characterize act continuous positive kernel symmetric qx qx diagonal invariant part independent resp part invariant statement space explicitly therefore explicitly invariant map canonical intuitively invariant kernel always separate conversely want kernel certain invariance factorization imply diagonal canonical construct variant observe general invariant name mind uniqueness map idea outline time invariant allow avoid associated fulfil may however argue true common kernel kernel rbf polynomial sigmoid kernel maximum spline combinatorial many ii fulfil kernel unitary absence would imply let unitary transform unitary kx proof together isometry kernel basis kernel isometry kernel also unitary fulfil variety proceed exposition explicitly write invariant version slight act unity invariance plane space group v vx phase kernel analogy sign kernel invariance e multiplicative map obtain scalar definite correlation kernel trick yield x leave phase invariance kernel trick either scale invariance sign invariant kernel scale sign invariant apply invariant effectively trick scale plus invariance combine multiple repeat invariant trick compatible presentation point practically group act mr rr unitary act rise representation unitary row act one center characterize long invariant decide adopt closely inspire existence invariant mathematical differ opinion main two ti kernel invariant substitution require group endow haar term invariant haar invariant rbf existence statement describe kernel invariant universal imply potentially avoid integral one behave quite definition refer reference able direct application definition furthermore remainder manuscript relation invariant feature far see invariant invariant invariant kernel outline aim incorporate motivated element invariance sense pixel give rise kernel statement globally kernel simply concept specific without focus particular technique spectral cluster extend g entropy datum eigen identify axes r enyi concept describe sign matlab invariant gaussian invariance handwritten sign pixel spectral successful grouping I group cause panel cluster sparse wavelet overcomplete six correspond six measure panel block invariance matrix ica address study eeg ica ica short fourier approach enable identification recovery solve overcomplete six linear mixture six play mix apply tree select point dataset figure space six cluster equivalent blind sign entry sign invariant continuous decade systematic constructive incorporation algebraic know kernel trick inner invariance kernel negligible gain substantial framework code variety numerous manner limit demonstrate sign sign invariance validate remark sign sign group imaging practical mainly conceptual future devote incorporate invariance broad field bioinformatics engineering acknowledgement acknowledge national research foundation education technology research support fellowship provide statement corpus always contain group act invariant act canonical show contain imply prove claim algebraic act geometric analogue universal property formulate usual algebraic geometry outline algebraic variety act family furthermore countable algebra geometry equivalence act factor uniquely countable sub countable countable finitely happen infinite hilbert universal proceed essentially translate act algebraic dense take intersection algebraic polynomial function categorical equivalence polynomial field w gr z w kind thus obtain analogue pass therefore consider orthogonal unitary resp consist orthogonal matrix act ng normal distinct alternatively orthogonal unitary resp unitary
corruption ability transfer image exhibit scale pose variability reasonably image database imagenet handle extremely learn direction sophisticated accommodate pose alignment normalization bottleneck keep g stage dimension result stage fortunately replace dimensional convolution filter future leave scalable classifier applicable regardless extensive extract face digit texture baseline study advanced architecture image receive electrical engineering ph institute communications national rd technology interest emphasis vision hyperspectral receive china electrical engineering ph degree science k advance science interest machine learn processing technology china ph university currently advance digital sciences center microsoft research fellowship interest computer vision currently research advanced digital science include computer pattern scientific paper review master degree school university interest school science technology university receive mathematics china degree mathematics berkeley associate university microsoft areas david mention award award national foundation office associate information cr pc pc work comprise hash histogram employ filter follow indexing name network easily comparison simple namely topology learn lda basic benchmark different verification extend ar face well write surprisingly model highly carefully learn record task ar public dataset potential simple competitive object network lda handwritten digit object visual content task largely large amount intra numerous counter intra designing task hand representative binary sift feature object recognition great success datum task require since plausible example method attention deep multiple hope abstract deep invariance intra one key ingredient success image architecture deep convnet architecture stack follow supervise comprise convolutional filter bank processing layer bank stage convnet variety boltzmann machines rbm review reference therein ad hoc variation convolutional vision success usually justify arguably justification wavelet convolutional simply hence somewhat surprisingly fix bank utilize convnet convnet task handwritten texture recognition generalize face intra variability illumination change corruption motivation study resolve apparent convnet goal deep trivial adapt basic network serve people justify use advanced architecture deep come basic easy processing mention adapt filter bank stage basic filter nonlinear simple hashing pooling name two stage characteristic challenge convnet operation stage hash histogram extensive experiment simplification performance orient audio lie couple hashing histogram layer gain robustness baseline likely linear discriminant extensive experiment discriminative section somewhat extreme totally filter conduct fair type network study architecture baseline comparing architecture finding think comparison already par state almost face write texture achieve accuracy extend dataset illumination dataset obtain competitive unsupervised mnist state background effectiveness propose task method deep far vast literature thorough encourage one hand serve surprisingly competitive advanced design success confirm certain remarkable even consist amenable justification effectiveness lead insight seem deep htbp image size patch illustrate pca filter input diagram pixel take collect patch I mn k jj patch mn image layer orthonormal matrix map matrix training course stack multiple filter denote boundary like patch mn mn jj I l l k collect remove patch filter pca filter stage obtain output build pca stage deep architecture beneficial input value output l binary every pixel irrelevant distinct word block histogram input histogram either depend empirical experience suggest digit histogram offer extract transform sift histogram orient gradient learn bag model process convnet filter number block size inspire common filter fine performance empirically stage architecture also histogram translation invariance extract convnet patch local convnet operation center origin filter major addition pca simple auto linearity stage build absolute convnet modulus test stage improvement quantization output introduce invariance output one merge stage field stage perform stage face digit single filter single filter stage learn filter whereas filter totally benefit contain object invariance capture comparative hierarchical architecture field stack relate extremely basic us example form patch eigen decomposition convolution bit number histogram assume overall complexity verify computational burden training decomposition whose convnet sgd gradient solver training take hour cnn take hour exclude fine section network require two eliminate necessity replace filter translation translation direction plane rotation e face face illumination test randomly select generic gradually sample keep even filter scale near nn chi classifier since find successfully stage cnn face supervise size convolution max softmax classifier pre cnn cnn epoch method cross test inferior whenever illumination drop offer cross variation final cnn seem face included follow superior filter report network number pca deep issue worth face around pixel take half hour exclude fine tuning pose apply pca learn b extended face image individual laboratory control illumination contiguous percent percent locate square block image overlap block illumination distance table illumination insensitive robust person various difficult condition surprisingly pca detector maximum contribution would ignore pass yield robustness percent learn cope real ar illumination condition consist image convert gray illumination neural testing overlap compare use table illumination variation perfect dataset insensitive robust feature achieve extended test finally popular recognition capture condition year partition probe probe expression fc different take four year apart gray image compare fairly dimension whiten feature cosine train consist people list standard state list learn art accuracy database rich nature learn outperform ii remark recognition prominent message section abstract subject one toward wide deep sufficiently inter intra class variation probe fc avg face evaluate unconstrained verification collect web pose illumination unsupervised evaluate learn discriminative face pixel evaluation split view contain class inter subset block performance feature cosine four descriptor square operation see root boost square quite show propose effective face condition aware convnet verification work achieve set train convnet align face base image face com l method accuracy dim dim le mnist variation widely test list investigate ratio state vary stage stage overlap region block half face almost case number block overlap ratio ratio c convnet scale set number set variation convex overlap region use error fair method augment good result comparable mnist report achieve result remain convex table also outperform draw figure observe vertical pattern attempt capture become background l description test mnist standard mnist rand mnist background mnist background discriminate background discriminate exclude filter unless otherwise method nn k nn convnet stochastic pooling convnet maxout rand convex texture texture class pose illumination surface variation also
traditional obtain pc proof e nx estimator regression x b n l b n ba ba f ba ba ba n ba ba b b b concentration traditional
entry restrict specific occurrence algebraic point point difference prove induce family n n inner sep flat prove module thus briefly filter module irrelevant multiplicative direct extend module module element great ideal module flat prove crucial result induce flat ideal e module way deduce n relation ideal trivial generator define irreducible generator generator four irreducible generator us di two generators module assume dx remark x correspond irreducible point statement suffice repeat n consider surface correspondence know say nothing critical lie understand vector assumption theorem even satisfy weak determine I define lie restrict attention ideal coincide know next question distinct point split factor tuple critical degree coordinate coordinate critical root integer distinct coordinate appear recall time sum weakly decrease translate ml partition integer addition specify ideal equal coordinate rewrite ideal write term depend statement ip partitioning subset need determine ideal instance critical coordinate since look repetition entry action thick thick thin thick thick thin thin thin thick point thick thin thin thin thick thin thin thin thin thin thick critical point thin thin thin thick thick thin thin thin thin thin thick critical try explicit notation later set consider index every univariate automatically vanish therefore proceed iteratively generator polynomial tuple form formula length ideal fill circle dash sep dash w sep pt euclidean unity directly whether root vanish root notice lead root also lie eq obtain ml develop likelihood n ideal table degree determine projection apply critical point implement software find processor range cpu exploit repeat field different rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle thick rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle thick cycle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle thick thick rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle thick thick cycle rectangle affect degree complexity totally rely partition heavily affect ambient perform eventually two partition length ideal differ different degree substantially ml degree slowly depend list ht ml degree computation well complete within scale thick thin thin thin thin thin thin thin black thin thin thin black thin thin black black thin thin thin black thin thin thin black thin black content school exercise go thank point comment suggestion mr corollary theorem de fu critical projective formula curve projective projective ml exploit list introduction explore recently branch mathematic call algebraic statistical result uniqueness mle log model easily algebraic however instance ask critical lie question know ml degree geometry problem instance hide algebraic light degree projective computing degree exist look aid software degree projective point general consider derivative lagrange multiplier computation value answer ml large algorithmic tool geometry exploit explore nevertheless happen apply statistical algebraic concept tool require moreover algorithmic play key construct scheme ideal encode degree particularly define two solve second flat extend look study example dedicated formula application achieve report computational remarkable degree formulation language want multiply similarity need want lie hyperplane define equation hyperplane paper di ki critical look lie distinguished ideal projective consider equation projective standard onto motivate ml degree likelihood correspondence irreducible irreducible variety hyperplane likelihood function finitely degree verify interested thus adopt correspondence factor order determine degree correspondence polynomial polynomial ideal define q compute ml degree degree map computational get surely degree computational degree long mainly gr ideal determine degree resp little elimination critical lie distinguished
standard element element element integer bad case interior variable encode programming able program programming total synthetic section query set describe result absolute precision order bound discretization domain range l spectral conclude event ms ss notice span complement span eq corollary execute execute c dataset ccccc dataset number grid uniformly choose table bad reduce theorem thm proposition section privacy query say smooth specified order develop differentially smooth algorithm real appeal run privacy improve demonstrate mechanism differential database conduct contain sensitive take privacy consideration database differential privacy guarantee almost nothing one without individual output significantly differential attack individual recently learn datum mining privacy answer differentially laplace laplace add sensitivity typical query database mechanism limitation nontrivial user query limit restrict remarkable due mechanism answer query preserve privacy nontrivial specifically refer accuracy improve mechanism powerful answer choose mention different query practical appeal privacy almost preserve certain attack output modify please survey therein preserve privacy answer query compare good answer universe linear generally universe strong hardness differentially output answer general polynomial hardness recently interest differentially private restrict query rich contain develop fast mechanism hardness result serious barrier privacy al consider query set universe constant rectangle query specify discretize efficient mechanism output database query another lot universe record binary attribute individual database series universe general answer mechanism universe mechanism database data universe smooth answer database learn reproduce ability differentially output mechanism accuracy exponentially time size smoothness contrast employ guarantee well large run super size database please generalize preserve privacy relate mechanism private answer query user complicated numerical experiment mechanism medical record achieve good practically size attribute rest organize briefly describe basic section output contain theoretical result performance section universe typically universe call neighbor differ data differential input range neighbor database take preserve differential differentially consider universe mechanism database eq randomness accuracy high query author weak definition query database respect internal randomness mean use laplace add laplace distribute laplace private output datum universe different database number q mechanism query query nonnegative integer contain order norm bound formally function machine smooth popular function machine function characterize follow good query generalize mechanism privacy accuracy compare differentially mechanism main specify smooth function mechanism differential synthetic description mechanism let query universe preserve differential constant hide depend mechanism I l mt km n r n idea look performance first derivative smooth give simplify running time database roughly nearly increase run increase one want database roughly ccc accuracy db explain detail smooth basis radial differential privacy problem requirement quite typical linear combination correspond uniformly reason soon clear mention set f combination polynomial polynomial satisfy coefficient requirement point basis polynomial query query guarantee combination approximate easily private answer small mechanism merely need generate synthetic evaluate answer answer synthetic query synthetic answer first distribution answer must dataset requirement computationally intractable original discretized universe query involve formulate programming lp well distribution synthetic bad interior upper easy control precision round private mechanism private k query universe differentially absolute constant composition omit detail differentially private mechanism private study differentially output accurately answer analyze simple note set universe query contain differential size universe ignore clarity smooth problem universe infinitely many element must universe smooth see universe grid precision datum universe query proposition smooth query achieve super section note importantly mechanism worst differentially nearly query run suffer minor time lp grid polynomial grid restrict basis function preferred degree simple grid suffer formulate differentially small almost every concern requirement privacy dimensional ellipsoid private top eigenvector square radius private mechanism private uniformly ellipsoid database private privacy schmidt set mechanism differentially top tangent angle eigenvector span suffice private ellipsoid converge true column simultaneou increase kk eq q execute privacy execute sample differential appendix mechanism repository community combine economic heart contraction monitoring rate disease breast diagnostic consist characteristic cell cc dataset size summary attribute universe normalize conduct mechanism privacy guarantee experiment privacy employ combination possess property section linear detailed query experiment value smoothness measure comprehensive performance bad error mechanism query kernel infinitely bad query error relative therefore good relative informative necessarily bad performance present intel processor ghz gb em ccccc table linear column respect synthetic output database ten explain experiment except show accuracy differentially compare ccccc time database preserve privacy appeal practical viewpoint mechanism database
likelihood user item profile analyst unfortunately provably item profile moreover user rating analyst contextual dataset user privacy reveal gender age rating information mf suppose gender political incorporate mf adapt user profile modeling estimate profile bias jointly q special case extend profile dimension depend treat another though priori mf rating comprise easy sampling put gender mean among incorporate per simplicity feature discuss multiple scenario analyst perform mf dataset explicitly profile item analyst typically batch rating follow linear extended profile analyst infer profile b analyst infer pick small profile analyst rating analyst classification infer svms alone involve regression separate inference analyst gender separate logistic svms rating reveal albeit meet simultaneously see near front right cube privacy thm corner perfect maximal privacy user receive accurate rating recommendation ask benefit provide surprisingly beyond fact analyst address issue set comprise analyst access user reveal analyst binary perform factorization dataset extend item profile analyst rating parametrize extend follow analyst identify aid analyst privacy private design analyst salient informally conclusion analyst component analyst private feature learn extended profile highlight three protocol protocol exchange analyst rating analyst section b analyst e finally protocol analyst item user rating private feature fix obtain analyst protocol extend protocol solution fact among protocol constrain way completely third conceptual three protocol practical seek protocol satisfy protocol analyst learn non profile existence service ensure privacy benefit extent make publicly item profile construct describe last allow service generally analyst thereby privacy user limit exposure competition ask accurately learn little c definition analyst privacy analyst access profile mf rating private analyst feedback profile formalize privacy analyst fashion extend profile rating profile mf noise set profile analyst rating item clearly uninformative light denote know private know rating nevertheless consistent mf latent easily rating natural opinion item ad news article tweet video etc rating user user website collect sensor response rating notation define analyst profile program execute analyst user item practice user analyst describe reveal set program execute particular r r l combine subsequently reveal analyst profile analyst construct feedback vector execute analyst refer triplet note functional analyst know protocol extract reveal mapping protocol protocol randomization protocol begin intuitively nothing user value variable algorithm determine protocol strictly inequality scenario extend profile metric relate criterion motivation rating product estimator expect also motivate analyst user brevity omit recall analyst predict rating often error preserve protocol precisely hence analyst equivalent minimize partial protocol reveal much v r natural intuitively retrieve mapping differently compute output simple refer remarkable mp analyst shift rate reveal analyst feedback analyst apply estimator mapping bad eq summarize protocol remarkable privacy accurate mp mp less accurate prove third establish formally protocol intuitively statement protocol privacy preserve among scheme statement imply maximal protocol protocol bias note mp appeal analyst gap rating etc enable minimax follow minimax computed know coincide square estimator close trace r jx clearly privacy protocol accurate lower preserve loss first private variable newly output statement definition analyst improve protocol fact analyst choice statistically analyst profile analyst attempt setting square estimator noise natural item analyst feedback extensively context exist item profile user fact particular maximize factor optimality protocol analyst particular service ask rate could user reveal analyst apply mp difference reveal interaction still depend remain maximally express categorical illustrate incorporate approach incorporate category specific bias representation whose observe categorical incorporate analysis first rating non reveal feature construct perform factorization rating bias j jk privacy subsequently analyst analyst bias user r k categorical analyst gender gender dataset scheme mp round ia fa ss sub sampling privacy attack rating gender view gender gender protocol mp user inference non little dense dataset private happen dense nearly evaluate tv rating dataset include star rating tv gender make tv recommender movie gender user rating rate user dataset seek privacy fold cross split user fold fold user privacy profile factorization empirically item use private mf implementation classifier rating randomly select rating first set rating several baseline detail input classifier profile infer classifier square predict rating computing auc time report auc range privacy item bias half rating private factorization compute regularization fold standard method private rating multinomial na I logistic lr svm radial basis rbf lr nb comprise item well movie rate operate rating user number execute mp rating outside range rating recommender therefore variation mp rating integer two round probability expectation rating high truncate process round round protocol sub round scheme replace rating replace e finally ia fa ss also l age gender auc lr multinomial mp avg item avg avg begin evaluate dense user gender political accuracy rmse apply scheme mp privacy rmse illustrate mp attain model little remove strong failure especially add round world mp effect ia fa suboptimal prediction impact prediction rmse little left value item rate mp method item rate four insensitive reason ss feature rating auc excellent across privacy auc contrast ia fa significantly rmse stress dataset rate nevertheless far impact sub sampling number item analyst feedback ratio feedback report ss rating remain exclude rating least accuracy suffer overall mp world dataset privacy risk impact linear eliminate rating private illustrate gender dataset distribution posterior enable successful auc distribution become indistinguishable rating rating rmse auc rmse tradeoff scheme provide curve almost flat consistently auc pt pt factorization raise question extend task perfectly private privacy usual privacy literature protocol differ opposite mp protocol could begin define protocol define tuple function profile rating two step determine rating user output rating item user truly rate feedback user analyst reveal provide determine x r analyst protocol r extend straightforward regard expect preserve depend protocol rate r protocol protocol bernoulli preserve privacy preserve learn protocol begin establish auxiliary r lemma states protocol construction rate user cv cv preserve strict v item probability differ respectively observe v p p identically particular privacy equal construction couple dominate former privacy protocol rate iff obvious yield must property distribution ready privacy conditioned condition monotone protocol lemma protocol coupling pt pt rgb rgb rgb rgb rgb infer attribute privacy protocol improve rate strictly extensively protocol demonstrate gender age political decrease accuracy give star rating book article kind micro service predict user preference prediction g make recommendation ad service accurate fundamental importance service reveal movie news article inherent call rating movie private political analyst use analyst predict user rate item dataset factorization recommendation analyst learn private access analyst private rating reveal relate whether evident report successfully orientation drug ask happen desirable benefit analyst recommendation rating analyst tradeoff distortion introduce first dimension information analyst understand scenario analyst rating analyst execute news article relevant would analyst rating clearly analyst reason analyst service competitive edge trivially attain natural analyst algorithm natural information analyst privacy enable service make follow introduce study analyst predict rating determine analyst privacy specify analyst reveal user rating feedback rating prediction protocol protocol mp remarkable predict yield privacy protocol mp protocol mp bad accuracy extend handle common deal item analyst discuss analyst item repeatedly analyst protocol gender attain excellent wide array inference blind area auc privacy rating take analyst privacy tradeoff proof model factorization validate vast empirical item rating approximately classifier linear validity real world extensively infer tweet facebook friend movie rating lead linkage attack enable close show political view orientation drug use accurately gender political infer rating tv show release analyst context preserve see construct decision task user population factorization attribute sensitive setting keep release apply distortion threshold address work asymptotic broadly cast treat employ specify
limit set need dependence target handwritten digit subject clear behaviour similarity define image hence clear behaviour sample two would highly unlikely bad behaviour limit square error sample restriction dependency converge call q target mean draw variance give see weighting point weight differ optimal weight point translate constraint program reasonable condition show assumption consistency mean fulfil need behaviour exclude fast dimensionality distance datum intuitively strong contribution estimate reliably target linearly combine asymptotic single state condition contribute go consistency fulfil averaging therefore require restriction dimension dispersion grow slow otherwise sequence uncorrelated zero without consider class target possible use overview jj correlation target condition consistent assumption consistency l p target additional set fulfil identifiable see appendix state exclude sequence grow fast product uncorrelate quite weak analogue behaviour covariance limit remain go infinity consistency statistical fulfil consistency eigenvalue covariance uncorrelated identifiability standard shrinkage therefore shrinkage intensity intensity moderate accurately estimate quantity measure value indicate accuracy classification quantity interest serve accuracy estimate lda simulation behaviour large limit target standard sign random four setting set clear behaviour sample tell receive show intensity intensity target go shrinkage intensity reflect zero go constant relevant outperform finite shrinkage intensity range estimator identical accuracy data set eigenvalue log achieve rescale discriminant ignore pool take pool baseline sample scale pooled improve sample pool improve find figure spike multiply discriminative drop target useful original direction spike still discriminative put whitening help give equal superior accuracy interestingly well spike estimation intensity dominate shrinkage estimate intensity evenly average stable whitening leads improvement discriminative subspace high variance covariance normal diagonal shrinkage target differ large eigenvalue analog intensity dimensionality intensity go target asymptotically less result shrinkage intensity remain finite range estimator also bias multiply sample simulation additional observation set constrain angle intensity rotation angle shrinkage small rotation small shrinkage target superior apply preprocesse filter class eq common computer covariance rescale random target target leave show accuracy estimation covariance pool w discriminative optimal set perform well pool multiply exclude rotation strong dominate cause degradation affect world covariance detailed brain discriminant target vs art stimulus stimulus evaluate decision binary lda approach lda stimulus binary stimulus shrinkage distinct stimulus consider independent stimulus pool comprise subject classification accuracy estimator pool estimate stimulus pool class consider superior brain interface subject imagine measure eeg subject trial subject subject band spatial class heuristic apply log approach subject subject dominate shrinkage high spectrum reduce importance noise apply whitening shrinkage intensity principal shrinkage middle show accuracy average additional shrinkage intensity dominate good receive large shrinkage intensity show ten trial widely year analytic formula covariance shrinkage become alternative pointed case usage target formula improvement shrinkage whiten step real propose explore transfer domain knowledge framework label whiten extent shrinkage quadratic decompose directly eq rearrange define behaviour calculate addition rearrange op conclude ii estimate q ii multiplying minus subtract right side rate generality asymptotic behaviour behaviour asymptotic behaviour introduce eq mean ii eq sum sum conclude proof restriction combination conclude generality assume start behaviour obtain consistency type similar eq sum proportional third restriction additional biased consistency follow asymptotic invariance low consistency unbiased variance eq look term eq expression intersection denote take expression term therefore integer disjoint subset set integer bring form consequence take account get total analogue us covariance rotation therefore lead otherwise term ij eq combination conclude behaviour show behaviour compare compare asymptotic use behaviour show ij op op sum sum sum simplify use obtain part hence side restriction step true al tu tu tu de department science tu tu computer science tu stein outperform shrink sample matrix identity shrinkage concept shrinkage target include datum set stationarity datum alternative estimator serve translate estimation shrinkage intensity yield error specific optimality applicable large go large go infinity limit apply stein stein admissible estimator always gain optimize bias trade unbiased high variance bias low analytic covariance allow square serve shrinkage wavelet density approach shrinkage code standard shrinkage line connect intensity optimal estimate two estimation handwritten digit digit person mean shrink towards two difference truth target figure moment impose tail
seq seq person histogram plus accumulation feature evaluate approximately sr low table represent form superior consider endowed vision track replace singular decomposition svd eigenvalue c diagonal partly project grant lp communication centre university school technology advance matrix riemannian manifold increase performance manifold tangent space reproduce hilbert space shape structure effort contrast offer solution preserve specifically hyperplane rkh space lead texture several histogram relational divergence matrix employ image video provide covariance definite riemannian endowed riemannian tangent riemannian solve widely riemannian induce riemannian inversion employ computationally non linear line research embed reproduce hilbert rkhs former approach enable use learn cost manifold mapping manifold consider space geometry exist load impractical scenario contribution combine recent specifically design dataset image map wherein still preserve employ create preserve manifold geometry point stein hyperplane project point appropriate study efficacy completeness generating hyperplane address training datum identification texture continue paper brief overview function visual future sized covariance topological manifold manifold riemannian map locate tangent tangent shortest geodesic curvature connect stein similarity manifold form aim still geometry aid divergence essence generate hyperplane wherein hyperplane map accord wherein preserve upon dimension despite use computer vision distance mahalanobis metric make euclidean space recently distance rkh approach make space manifold preserve manifold via follow rkh randomly generate hyperplane embed end training I distribute variate whitening form ip pca project uv q define matrix one correspond term calculate step offline complexity query point factor project hyperplane operation number class hyperplane define datum relational second represent riemannian vector employ discriminant classifier later experiment fig generate hyperplane limitation synthetic enforcing follow choose eqn around radius matrix radius establish address geodesic riemannian metric tangent whose introduce say normalise geodesic normalised point geodesic tangent preserve normalise map back present arrive prove result geodesic generate person detail experiment set comprehensive space stein feed choose hyperplane empirical sample choose similar eqn e classification several approach tangent rkhs evaluate synthetic synthetic dataset middle texture person identification version capture move camera contain variation appearance sequence create vector colour laplacian colour texture classification protocol present nine splitting scenario times recognition image pose evaluate consist illumination pose test pixel vector response wavelet locality hash riemannian hashing specifically design manifold person identification seq seq sensitive riemannian hashing seq seq texture recognition dataset v b average table report texture identification task hyperplane rkhs offer classification task present follow result still three look notable identification texture dataset synthetic suggest contribute probably represent variation completeness contribution performance face cause skewed add distribution turn affect dataset number generate hyperplane improve graph compare
however correspondingly randomness capable infer indicate complex lemma bayesian factor potentially infinite also non parametric construct make infer experiment learn apply method certain fit structure hierarchical challenge datum thus determination impractical hypothesis structure bayesian assume construct layer cope challenge potentially factor stroke deep graphical permit structure hide influence factor layer distribution factor develop convolutional connect processing set construct build factor prior achieve data ibp handle case application various network unlabele classification bayesian advance researcher challenge still lie construct nonparametric model infinite hierarchical layer factor proposition infinity fashion hasting enable layer make recursively simulate infinite state wireless security mainly lie number greedy algorithm objective discover hidden factor nonparametric result draw conclusion construct component finite extend hide layer infinity generative use model tn dependency successive weight instance cause influence generation component otherwise hide matrix illustrate remain vector weight generate I indicate element product independently construct variable infinity column variance inverse gamma r impose effect layer much influence receive high conditionally verify delta generation variable layer layer datum level feature extract human face second feature possess cause robust binary bring close reasonably weight express recursive infinite number layer matrix ibp distribution q hide layer select select restaurant customer array try customer firstly th customer customer generative intractable inspire initialize express metropolis hasting approximate infer infer hide changed second infer layer iteratively inference converge infer efficient metropolis inference classic problem generate first candidate density state state evolve stay explain specify generalize infinite derive setting connect matrix model propose express completed iteratively check number column propose link sample probability hide approximated generating multiply return factor zero compute ibp proposal link accepted algorithm use infer matrix turn k integrate column independently compute utilize distribution
u k l lt st result another major advantage derive compute linearly substantially exact rate convergence entry resolve fw fw call frank wolfe thresholding proximal compare synthetic clearly efficient highlight fw additional iteration k frank wolfe produce retain rank execute e easily frank wolfe implement line solve crucial convergence depend maintain non scheme convergence weight max initialization u optimizer k u g step easily recognize fw algorithm establish primal independent n e therefore convergence result still upon duality suitable serve terminate algorithm consecutive iteration fw real arise consider separation surveillance video removal application robust slight setting choose fw fista partial svd fw noisy improve recover whenever fw conduct processor core ghz gb ram bit surveillance normally fraction treat stack wolfe iteration updating component computer vision relate great fw acknowledge class fellowship university office award grant dms first useful proof type recursive constant base claim diameter set fourth v convex rearrange lemma notational contrary repeatedly one due assumption fourth ba x contradiction l l f f z g f f l theorem calculation u l k verify therefore combine l lt sg l k r th l argument verify recall x g l r l x x l minimize l hold feasible thus apply lemma define argument use problem data fidelity sl proximal singular x x decomposition explicitly initialization regard fista optimal scheme fista comparable summarize fista h singular utilize burden svd package g specify singular determine th adjust dynamically compressive corrupt observation fundamental rich application vision certain solve polynomial compressive provable iteration large provable combine classical frank wolfe iteration mainly frank update low svd exploit step scalability promise visual matrix dense noise collect parameter nonzero tractable ij tractable sometimes compressive pursuit equivalently since linear subspace span onto rewrite possibly depend specification mainly easily extended operator consider reduce onto j linear subspace work theoretical mild produce accurate even measurement researcher solve include alignment verification variable outli robust big data application involve large provable work develop first closed form nuclear involve singular hence dominant computing substantially scalable interior solver limit practical applicability involve several thousand dimension full serious algorithm solve frank wolfe norm turn precisely scale scalable practice straightforward frank wolfe frank wolfe block use modify frank wolfe solve penalize incorporate proximal numerical frank wolfe fw general differentiable assume x frank wolfe proceed linearization simple step algorithm detail fw method recently yield encouraging matrix singular value substantially exploit comprehensive survey development fw update five decade replace update sophisticated q analyze produce iterate objective frank wolfe frank relationship special crucial handle use lipschitz constant diameter perhaps practice useful precise iterate duality refinement exist match bad case provide stop burden subproblem result inexact calculation ball operation solution describe part involve serve method easy computational cost essentially nuclear optimal value one matrix leave singular lead dual one modify frank wolfe projection onto easily achieve effectively frank wolfe step proximal mapping follow close extension operator scalable algorithm compressive q application frank although converge typical example make sparse problem proximal essentially combine frank wolfe frank wolfe map l f feasible compact feasible gradient compact frank wolfe generate expression linearize v subproblem solve exploit take right summarize result h q v v major advantage derive simplicity close essentially linear input rank disadvantage update I k entry disadvantage nonzero even fraction entry foreground background separation red practical convergence simulate entry matrix show slow k matlab plot fw p efficient recover drawback frank wolfe incorporate gradient projection wolfe additional term variable frank wolfe produce iterate retain sparse call fw frank wolfe projection fw h k l k k analyze frank wolfe regard set point imply iterate objective produce frank wolfe require invoke frank wolfe lemma diameter fw original fw method surrogate gap define q develop compressive
information sentence sentence lose bag model vision understanding pattern apply document learn identify sentence movie level sentence document sentence level convnet transform embed entire sentence convnet single train document embedding entire level sentence tie convnet abstraction operate document transformation convnet transformation architecture force sentence representation appropriate help show carefully type intermediate version convolutional neural similar network vision cascade adapt text model detailed figure overview right section convolutional model embed correspond process level embed build word vocabulary word embed vocabulary word embed sentence produce vector document sentence embedding sentence matrix convolutional filter bank f fw number map embedding convolution operation map embed along row sentence correspond document sentence map generate value feature along output map stack representation feed input feature word one unlike treat dimension generate representation representation generates illustrate substantially approach typical map image spatial read embed dimension leave model map map generate map embedding level make deep document matrix convolutional handle arbitrary problematic want max pooling embed keep discard fix size length max also illustrate fix length next convolution however pool desirable effective fraction pooling help long propagate learn convnet vision demonstrate learn single convnet sentence precede argue convnet substantially convnet scale mobile device ask performance attain tweet classification comparable set contain weak automatically presence tweet test sentiment assign convolution dimensional use operation sum pool close significantly well model error max bt bi paragraph vector error twitter dataset movie review second wang fourth le convnet review convnet sentiment classifier building solve show sentence classifier focus movie sentiment datum originally sentiment movie review divide review binary train label set word map convnet process convnet embedding model feature width max sentence dimension follow pooling width achieve knowledge encouraging convnet achieve good model achieve expensive since must optimisation paragraph embed feed forward train convnet salient document compact summary review activation addition insight use extract automatic summary text interpretable work obtain pass network fact quite carry approach non layer document adopt modify objective document invert feed induce greatest taylor expansion pseudo entry word document perform pass behind magnitude magnitude change affect explain technique clear perform taylor partial level sentence summary rank sentence produce train I movie review sentiment use classify summary tf weight I bayes I summary technique even keep train review drop create sentence review summary lose choose random sentence heuristic summary explain fact review relevant sentiment review summary short sentence summarie opinion movie review ignore proportion margin pick pick pick last thing complicate actually movie nice connect dot million middle truly massive massive character massive graphic graphic good graphic say go say might well people get topic pure sometimes look home camera great make tv movie sequel get background justification mean tag line influence break looks suppose look look help lead act movie inconsistent movie scene unlikely quite finally actor real throughout record never check cm
rough well hmc cc cccc c hmc function fix substitute rate equation point equation transform transition mix hmc length state case outperform hmc hyperparameter factor distribution ill final condition quadratic rough maintain reasonable require sampler majority momentum hmc eliminate hmc search condition demonstrate technique hmc hyperparameter computational figure matlab table running variation standard hmc complementary could present hmc hilbert hmc hamiltonian hamiltonian importance split behavior explore topology state transition though benefit replace identity space auxiliary momentum momentum swap operator allow situation momentum instead momentum randomization rejection though slightly random walk exploration reversible discrete trajectory map could mapping indicate momentum variable direction indicate occur exploration choose sensible acknowledgment thank member center berkeley stanford course like thank anonymous careful reading text grant support mm support nsf grant upon research laboratory office contract nf present hamiltonian rejection trajectory state reach transition detail balance walk rejection great release source code package probabilistic model increase protein structure neuron sampling describe typically bottleneck work require evaluating improves fundamentally sample generate hasting acceptance criterion suffer forward reverse transition occur balance detailed balance go go thus explore distance long draw proposal random walk distance sampler balance mix rapidly distribution space hamiltonian extend include auxiliary variable hamiltonian contour extend hmc able long update hmc detailed balance accept reject long step attempt satisfy detailed turn sampler transition discrete hamiltonian generate candidate hamiltonian dynamic back slice violate uniform candidate acceptance window rather state window boltzmann method balance discrete transition use balance satisfying restrict sampling greatly mix sampler substantially improve concept relate goal assume energy chain carlo commonly draw probability sample since order must space condition act result unchanged markov detailed guarantee every probability identical substitution side appeal distribution proposal metropolis acceptance rejection detailed metropolis markov forward reverse transition primary advance demonstrate balance hamiltonian carlo extend auxiliary momentum hamiltonian along contour expand state momentum physical system momentum mass drawing hamiltonian dynamic imagine trajectory unchanged flip side st discrete hamiltonian dynamic operator integration like exactly reversible preserve volume sign reverse trajectory momentum total energy discretization cause movement figure leave momentum randomization operator velocity operator randomization operator equation momentum randomization operator cause movement discrete transition occur state state represented way state hmc view additionally discrete short hmc typically implement step sample item composition markov move accept metropolis transition probability metropolis rejection discard long flip momentum back prevent trajectory reject already move hmc beyond hmc explore random walk momentum case momentum small improvement probably adjust fix fraction momentum ill condition ill condition surface demonstrate hmc balance still fix momentum hmc greatly walk prevent typically discard rejection hamiltonian correspond rejection section intuition update discover standard modification hmc
depend party party influential direction collaborative dl training generally speak rely quite dl find aware ever representation dl grouped make classification dictionary first classification specific dictionary representation include lc add regression consistency contain call discrimination fidelity term sub besides discriminative utilize fisher discriminative basis dictionary separate dictionary common meanwhile incoherence fidelity dictionaries discriminative despite learn discriminative approach enforce coefficient either feature denote seek reconstruction sparsity reconstruction formulate training class representation residual ir class truly contribute face recognition replace equation benefit close tx computed require optimization regularizer constraint induce competition among coefficient compute ir reconstruction dictionary seek amount reconstruction learn reconstruction discriminative fidelity detail dictionary iteratively optimize coefficient difficulty simultaneously explore dl consume term keep light big exist format dl q sub sample column term ensure reconstruct easy discriminative overall dl simple learn optimization dl iteratively optimize acceptable thank frobenius optimize denote sub respectively rewrite equation still objective dictionary therefore optimize without equation straightforward solution consists however converge soon worked learn sample cover experiment sample follow reconstruct test form td ir eq replace td conduct five visual nine public benchmark cover scenario appearance environment texture texture leaf categorization categorization color texture camera identification vary sample adopt identification investigate influence test totally diverse scope collaborative general recognition classification various literature generate version change feature factor split regularization concerned fair person good choice parameter important recognition acquire item different day instance take protocol third instance dimensional feature whole feature experiment three build aa camera view collect surveillance consist sequence centre image person result unlike dataset detector manual annotation localization aa big sample sample camera person e camera camera version name etc data aa broad pose subject subject either shot identification treat use treat desire correct top cumulative effectiveness exactly texture feature adopt choose dataset aa dl ht fdr fdr fdr ex leaf aa aa aa collaborative none completely outperform answer comparison superiority concern replace therefore indicate perform well large non predict dataset favor know necessary dataset therefore representative class important property influence intuitive may feature enable sample stay relatively thus easy collaborative discrimination minimum point wise randomly distance point treat distance sample belong dissimilarity rely fdr fdr top accuracy fdr however effectiveness fdr coincide enhance involve likely prove negative fdr four special mark variation fdr definite r low sum show learn influential complex dl lc dl dl dataset also outperform person identification boost dl identification probably room dl moreover dl dl dataset generally learn look dl valuable explore collaborative ex aa aa pt ht lc ex leaf convergence converge several consider dl run concerned fast testing comparable fast dictionary promise score dl superiority sparse collaborative art dictionary comprehensive dictionary acknowledgment support anti anti safe integrated education g etc etc et et mm medium intelligence school university mail mm media ac collaborative sparse simple effective solution get necessity enforce come primary show change computationally similar well unclear sparse know study issue classification select strategy feature dimensionality feature superiority understand collaborative motivate activity sparse collaborative call quite error reconstruct limit sparsity tend force large coefficient sample belong discriminative modeling expensive infeasible combinatorial approximate though consume due optimization ensure carefully per lack argue success lie sample norm representation well understanding treat regularization notation respectively regularize regarded birth attractive superiority representation report superiority people lack reliable problem contribute propose analytic feature dataset predict extensive
propose credible global function densely et orthonormal function piecewise present spline al credible band construct bootstrap strategy lee al yield simultaneous connection discovery al fdr generalize growth multiple view induce sample lee estimate across represent smooth estimate noise construct pc fourier basis response curve software require general mind curve ignore technique empirically wise regression second functional coefficient effectively wu iid coefficient spline study response curve grid representation spline penalty square ol compare covariance estimate estimation technique response within covariance step fit represent b curve work pc decomposition spline spline inefficient within ignore lin aforementione experimental design present model multiple pressure spatially correlate unit spatial grid properly error wise band specify function curve deviation assume strategy spatially car zhang car alternatively correlation function add response correspond iid induce design marginal component capturing example compound symmetry within cluster share longitudinal subject slope serial observe index distinction mix I vary functional correlation st n functional specify separate level product al issue way capture effect spline induce chapter spline curve curve curve induce curve capture induce manner combine response involve functional coefficient effect classic random effect like induced inference effect reference integrate variability provide smoothed introduce modeling level spline base nest curve group parameterization random curve sum iid spline level capture coefficient cycle restrict fully spline basis component smoothing introduce fix smoothing spline iid include curve like model determine random intercept slope spline part scalar white fit kalman filter approach pointwise confidence functional induce periodic et group within group random subject diagonal covariance respectively wavelet form prior account multiple function intend functional sample common wavelet project wavelet fit version assumption across wavelet work allow wavelet variance covariance capture degree greatly calculation allow fully quantitative et genome analysis lee adaptively adaptively regularize coefficient behave shrinkage curve deviation covariance tensor covariance spatially zhang even flexibility capture structure model functional index lee add level matrix consist spline basis predictor nonparametric functional additive lee al effect function variability credible band et probability compute fix control fdr miss et introduce use basis subsequent al zhang et lee method spline basis transform modeling function covariance level basis specific joint al special scale tail likelihood functional coefficient inference insensitive analyze spatially correlate al general error use basis point inference effect iid variance along allow apply regularization function functional growth nest function covariate functional covariate fix function cubic spline grid subtract pointwise pc project span overall pc perform pc wang base residual g ar coefficient random residual variance basis penalty update approach accommodate longitudinal index include slope random function focus simply subtract smoothed mean covariance general nest b subject level random effect level spline scalar curve deviation functional response grid within level functional model subject subject within introduce truncate hierarchy hierarchical polynomial plus al alternative basis polynomial group estimate use moment yield assumption update multi involve sequential residual variance yield also spline represent penalty curve score unlike non correlation project functional zhang functional car separable introduce project choose basis car separate car car spatial vary yet strength across correlation determine basis embed functional involve extend additive functional new include nonparametric mixed model effect parametric form random functional effect parametric model utilize model describe response nonparametric introduce elegant notation specify array function include smooth nonparametric predictor effect plus smooth predictor penalty consist smooth primarily penalize iid multiple level completely determine within covariance scalar smoothing entire limit accommodate scale e grid lee show bayesian accommodate coefficient nonparametric component smoothing vary vary coefficient allow joint wise coefficient functional g derivative series domain model project domain involve smoothing partition locally process tensor basis specification domain log wavelet regularization keep wavelet overall al effect domain effect level effect spectra subject spline common estimate band series wavelet et al content functional among function fitting functional discriminant curve functional within spline penalty smoothed group et al score function error discriminant li yu introduce discriminant analysis predictive integrate uncertainty et adjust enable robust version wavelet transform model wavelet inverse wishart allow covariance structure assume wavelet develop majority classic literature method choice function approach model curve deviation response curve suitable choice basis residual curve deviation capture correlation suggest across curve curve yield estimate independence spline accommodate curve curve choice penalty simplify smoothing replicate rich curve deviation accommodate decomposition covariance parsimonious somewhat assume deviation important affect functional perform discriminant grow correlate functional spatially correlated functional effect induce function simple modeling flexible enough grid mass imaging complex curve yet principal reliably response window os properly et response mixed lee version although fit basis transform inverse transform broken post module linear et truncation basis automatically produce include wise credible band point wise specify effect contrast linear apply functional al software scalar package package along fan zhang implement generalize additive fit additive linear mix r date functional function predictor denoise grid interpretable term account additionally curve random effect may take account basis develop approach respectively case contain discuss truncation bivariate penalty decomposition measurement pc functional coefficient amount pc score wu account regression coefficient chen regression observation estimator et et function expansion keep many pc p spline include scalar effect densely issue context two depend vary wu describe iid smoothed estimation spline locally fan zhang regression al measurement error function covariance functional effect represent spline basis penalization selection wu spline truncation inherent selection b spline within al longitudinal b penalty spline score constrain triangular construct iid spline function include truncation decomposition iid coefficient fourier spline pc integration domain model take correlation inference incorporate model structure develop response mix functional predictor additive pc like sense generalize approach predictor grid extend bayesian predictor effect function predictor function fit past decade much continuous domain method grid commonly encounter longitudinal setting broader yield quantitative euclidean surface function numerous feature paper highlight type naive analyse far dealing value year potentially manifold case inherent high functional application grid genomic et lee imaging et local thousand million functional regression software quadratic functional mind regularization use analysis open basis setting size eigenfunction preliminary asymptotic explore pc basis flexibility covariance helpful graphical another statistically principled approach basis benefit understanding adapt incorporation prior lead regularization various way article accurate careful importance functional difference estimation determine setting functional derive benefit inferential important relevant accounting correlation seem address study type functional encounter smooth function difficulty determine position within curve assess publish primarily compute pointwise band band test still need area need various rich work grant definition additive car autoregressive discrete transform additive model estimation fdr false functional linear regression mixed square p functional lasso operator linear p monte multi analysis multidimensional signal ols ordinary square p partial square restrict ratio smoothly space penalize regression search university md tx observation observe discrete development field accelerate past become fast area within function regularization receive attention follow overview response regression scalar development manner historical primary modeling modeling structure end brief area generalize spline wavelet decade datum increasingly structure rapid datum ideal continuous consist function population could fine across domain manifold euclidean largely longitudinal typically grid area produce spatial imaging domain genomic location dimensional single rather simplicity enable handle part seminal book behind describe display observation involve make population involve strength within function interpretability area attention article development role approach overview follow description functional development scalar discussion development manner historical cluster researcher model methodology theory computation employ section contain list publicly fit discuss focus highlight inferential approach discuss selection either modeling independently function advance nonlinear high domain highlight advance functional datum highlight discussion relatively moderately sized grid observation curve third include euclidean partial representative set functional article fractional fa corpus cca ms plot black ms control intensities part spectra control cancer spherical pressure moderate grid fa profiles diffusion institute scan corpus fa cca ms plus serial cognitive relate see b relatively simple moderately sized grid position three paper literature et al al ms fa one ms status fa patients fa cca ms testing whether differ functional mixed account status cca use alternative could association fa position could surface effect fa cca position marker cancer perform university md cancer center describe et mass peak protein molecular unit whose intensity abundance spectrum illustrate functional regression development flexibility efficiency methodology development functional al et frequently analyze perform quantification peak spectra peak algorithm perfect model entire spectrum differentially protein base patient assess cancer fitting spectra cancer partial manifold study able induce amount pressure continuously higher old close right yield principal location fine spherical manifold plot investigate whether coefficient change functional longitudinal measurement pressure linearity functional mixed capture age index lee within euclidean inherently inference either involve commonly replicate predictor information across discover concept accounting correlation design together size example mixed model model unit across make involve strength across regularity observe behind single krige analysis nearby involve smoothing imply observation internal especially similar distant regularization closely tie modeling apply capture functional believe exploit interpretable estimator time calculation potentially stable separately unified common involve coefficient individual separate grid convenient inferior unified jointly lead information across place unified evidence determine define induce correlation among loading magnitude establish strength allow inference dimensional commonly spline wavelet suited characteristic spline suit modeling usual fitting observational fourier function stationary periodic wavelet resolution basis decompose dual frequency make spike possess basis reconstruct suit sample regular grid principal component pc basis empirically eigen decomposition suit decay capture distant consideration development pc kernel weight suitable smooth spatially function lin et equivalence certain accomplish apply penalization basis truncation involve commonly pca keep pc fourier basis apply polynomial polynomial certain spline limit pre specify location truncation penalty involve coefficient smoothing lead ridge convenient penalty involve cubic spline basis knot incorporate penalty penalize spline penalty use knot penalty propose l spline penalization assume identically iid truncate sparsity coefficient induce prior involve stochastic variable al al penalty penalization wavelet regularization remove preserve dominant accommodate vary grid predictor natural cubic common knot zhang al periodic spline basis penalty robust insensitive introduce reproduce regularize penalty smoothing spline explicitly fitting spaced wavelet basis wavelet pyramid discrete wavelet allow grid mention wavelet suit spatially heterogeneous function spike wang al wang response bayesian probit denoise use wavelet adaptively al lag wavelet regularization scalar correlate wavelet basis transform truncation extend fit fourier cox develop group potential functional predictor sample predictor common lee basis outcome probit orthogonal introduce utilize general adaptive scad fan perform selection multiple basis response al introduce encourage accomplish via derivative piecewise constant mention introduce bayesian use field surface cluster pc spline regularization spline pc cubic spline perform decomposition spline transform spline square combine outcome transform spline pc spline replace pls basis response predictor radial spline ensure radial symmetry inferential bootstrap simultaneous confidence alternative pca perform b spline smooth result present walk stochastically al variational pc series penalty truncate spline handle functional score predictor functional predictor truncation keep number make primary handling dimension keep pc series spline present vs select among functional predictor et datum measurement scalar iid among replicate score et extension di al longitudinal separate score pc previously method li like full functional flexibility li cross product decomposition estimate empirical basis generalize al extend quadratic across pc extension estimation pursuit regression family model smooth function use cubic spline regularize include predictor incorporate li predictor complete orthogonal basis use single extend predictor spline use spline penalty single penalization perform across predictor additive al nonlinear additive pc score truncate pc general perform truncation pc instead penalization dimension et generalize framework function surface parameterize spline iteratively introduce grid space pc score within eigenfunction eigenvalue mean curve product walk deal
set conservative choosing extension exist design x show gray indicate reduction maximize away triangle open triangle iteration determine actual segment ray segment five black triangle optima green close element panel ray select candidate ray opposite direction since ten box full allow segment start spaced show xx nx ray search start design near increase successive search create could choice adjust fast fidelity ray design undesirable many context parameter spanning segment derivative leverage speed library routine huge set initialization towards nn mode post optimize exhaustive discrete provide onto minimize explicitly ray start location choose unless agree precisely return prevent location besides modification round combination green ray exhaustive demonstrate sample development utilize core parallelization eight core gpu exhaustive package however new leverage feature implementation describe design local probability design calculate obtain benefit nn limited nn candidate primarily report early limit ray search slightly unless note ray adjust synthetic paper x cm lrr stage rmse sd ray ray ray ray cm rr rr summarize involve location space since space slight computation experiment defer ray design accurate nn option sub poor add space reduce variability great calculate sized magnitude mean calculation stage would require exhaustive feasible via lead overall predictor low nominal poor rmse achieve nominal coverage achieve expense cover sensible relative time utilize core ghz intel core gpu devices exhaustive subroutine experiment one right reveal search competitive exhaustive subroutine gpu big gpu exhaustive ray competitive ray search nearby experiment lift involved dimensional capability library addition local number rr cpu gpu ray exhaustive search fast exhaustive search cycle unable fidelity experiment dimensional detail therein one summarize prediction design design set hypercube increase design sample increase ray search rr rr exhaustive c cpu second mse second second location separate report gpu cpu cores cpu shorthand indicate cpu core mse summarize compare column time slow core amount large run nine million year old look base give nearly identical accurate one relative small searching set intersect candidate reduce find good column show intel bridge run comparable exhaustive parallel exhaustive slow prefer small huge gps show big gain nearly allow hour ultimately accommodate experiment approximation fidelity rise gps rr rr subset sep pre scale c core second mse second mse second second include build subset separable final ray pre globally grateful pointing result global predictor subset datum calculation obtain design estimate separable function organize row surprising fit compare ray alternative uniformly input isotropic show separable along row final separable substantial input subset limit thus globally scale trend estimate exhaustive search hybrid appear partly round partly slowly setup carefully illustrate fidelity approximation size increase versus several function range follow product cause numerical instability benchmark excellent resource benchmark problem involve computer ht six pair top time bottom repeat training via two motivated comparison subsample separable base square common early variability sub design predict good sub version separable global clearly isotropic isotropic good figure average besides scale change vary report repetition number legend indicate gain normalize rmse even grow gp becoming generate paper focus local approach build local design nearby prediction study surface predictive design discover exploit suggest interest design result ray design comparable correlation isotropic locally global may still parameterization discussion highly global scale scale global substantial discussion rich family certainly design material local figure choice qualitatively design mix whether measurement context author argue lead explore example estimate relationship spatially surface appear unstable smoothness visually predictor finally search design design low search sample setup value computational increase improve may moderately problem disadvantage especially distribute search uncertain reduction deterministic runtime operate random balancing challenge try resource ray perhaps might acknowledgment complete resource valuable anonymous improvement neighborhood school particularly massive parallelization mean two previously ease burden study observe exhaustive subroutine building design rather search location alternative work ray yield problem krige regression neighbor learn big data surface reasonably smooth relationship regression tend gps accurately appropriate input pair implementation grow collection modern thousand sized computer decompose due perform hundred much decomposition require limit datum computer canonical model cope modern simulation approximate alternative lead fast capability magnitude prefer example capture match cluster dense led capability hybrid resource design hour literature focus induce first subroutine vast grid identify design identify site regular pattern greedy motivating scope site gpu computing acknowledge inefficient context global software package remainder outline follow regression emphasis explore design brief gp leverage localization literature leverage big method motivate stein pair gaussian noisy salient typical gp comprise scalar response n nf covariate special especially move correlation definite comprise choice determine correlation throughout thereby stationarity isotropic k rapidly experiment noisy robust stationarity deterministic experiment add numerical appropriate methodology herein gp popular reference likelihood analytic fast maximize student vector nx yx v shape element attractive high represent sensible new eqs require dense matrix limit reasonable hour sized say fast design predictive sensible objective optimal minimize complex proceeding sequentially build globally minimize sized depend implicitly equivalent maximize design advantage avoid huge argue sensible proportional dominating apply sequentially approximate huge search near rapidly decay substantially location close away model prefer choice remarkably quadratic correlation potential local suggest maximize maximize inverse compete minimize distance upon tradeoff proximity observation insight nature global optimal design recognize integral thought result aspect design limit define mesh value possible try minimize need predictor site somewhat globally establish criterion prefer
marginal consider posterior prior derivation present posterior b wu b cumulative ba density derivation equal accordance definition end appendix marginal g g wu box york normal utilize letter k short tail interval communication b w bayes estimate possibly sparse statistic w bayes wavelet threshold selection international review interval journal planning journal regression selection regression american review page cm corollary department mathematic la normally distribute specify linearly suppose uncertain prior frequentist confidence utilize compare interval short credible dirac parameter bayesian interval way nonetheless interval keyword interval credible information regression spike prior author mail address nn parameter suppose interest define specify experience opinion scientific background suggest example scenario include factorial replicate uncertain high clarity comparison interval var var concern uncertain utilize two way frequentist credible interval utilize uncertain informative infimum coverage assess length expect utilize uncertain information follow scale substantially expect c confidence frequentist confidence utilize brevity credible uncertain apart uncertain prior precisely credible informative lead credible deal improper infinite rectangular dirac delta spike dirac delta function function prior class end genomic aim outcome consider preliminary estimator estimation make prior density marginal behave proper specifie belief prior tail credible frequentist b information credible may consist interval posterior therefore focus tail credible interval square p frequentist interval scale length eq offset half scale offset extent uncertain prior section credible interval offset figure scale offset half tailed credible graph scale offset half shortest dash word prior tail shortest credible variant informative frequentist estimator similarity offset illustrate nonetheless substantial tailed credible interval conclusion credible uncertain informative I tm bx dd define interval word scale scale offset statistic frequentist testing hypothesis confidence happen strongly uncertain contradiction irrespective tail shortest credible approximate interval tail short credible depend frequentist numerical factorial experiment factorial replicate effect factor uncertain take let factor code factor experiment identically square square figure scale tail credible factorial bayesian credible context factorial density h short factorial express uncertain improper improper plausibility suggest density interestingly scale offset function follow g tail credible identical interval posterior tail credible scale offset scale length tail interval factorial tail credible context factorial credible factorial begin describe utilize sense introduction scale even function minimize look factorial function cubic evenly space subject follow confidence probability parameter square figure utilize uncertain prior introduction gain coincide standard contradict scaled confidence prior consider interpretation offset scale half offset scale half mark particularly scale bs knot spline place frequentist interval vice versa credible examine view frequentist interval examine bayesian posterior coverage likely favor credible interval frequentist coverage favor credible frequentist interval credible frequentist view lead instance difference frequentist statistical var
improve iteration increase historical exclude report minima algorithm vector bi vector explanation produce minimization therefore team decide use method implement run every network run widely year great win division seven straight game lead team second player vary head history school four four separate decade head north win matter lead least time know concerned decide five high year overall rank overall time method instance year top appear top list ranking rank input error record game incorrect entirely game cause incorrect source decide incorporate game regular team top team decrease modification ran maintain incorporate great negative run decrease minor originally behind place rank see model accurately remove add positively competition limited change final indicate separate team accurately player flexible relationship team easily compare effectiveness margin similar computer calculate could eigenvector centrality percentage graph von sometimes yield cost year array minima several array fact improve significantly time high accuracy result ideal analyze nearly determine unbiased rank college constructing analyze comprehensive team centrality margin pattern measure player create determine occur multiply team vector use map combine individual select would like thank mathematical modeling like school science mathematic work thank put competition north mathematic college question player ask college create apply across college offer college need metric reflect accurately college lose describe year property network team team team probability network occur utilize margin determine overall history simplify ultimately team margin assume team player throughout year compare reality change player throughout make game win good team win team intuitive allow eigenvector centrality total team team player player player optimize difference conference team worst likely run bring team match advantageous winning match determine make assumption play particular unable single portion source process aim merge team specific match name ex st manually allow merge college note college exist national current body nearly college establish college college college datum collect final score division college datum game combine match record data college college game name team generate record college interest greatly decade medium game range present accurately create analyze loss team regard make represent play year node college team team draw point information direction margin game associate edge construct winning list associate analyze nearly use visualize team base previous centrality connection centrality graph connection number calculate centrality centrality degree closeness eigenvector centrality centrality simple centrality measure connect node directional edge loss team use metric team however prominent play form eq power adjacency walk power centrality walk length walk intuitive eigenvector centrality utilize library calculate centrality game eigenvector centrality account good team follow team team metric centrality account rank centrality quite rank centrality ap division eigenvector centrality college ten eigenvector centrality north north st st six determined eigenvector centrality also ten usa model create ranking team rank clear easy centrality reasonably rank simple approach various one college calculate ranking wide historical centrality game division large eigenvector centrality low eigenvector value node give team team player player team team think multipli player team relationship factor life work regard separate influence team fundamentally different play arbitrary return win outcome draw win outcome effect enough player evenly match high outcome game curve look outcome call player strategy break otherwise match curve slightly team chance win even likely say express
perhaps typical variant em like unstable assume sample represent whole np computationally besides sensitive improve employ outlier theoretically optima unfortunately solver infeasible hour robust sample worth improvement exist accelerate robust subset hand boost norm robust error dominate generally large speedup acceleration augment lagrange multipli alm via demonstrate encouraging show fig accelerate reduce complexity obtain solver letter vector row problem set goal effectively entire solely could greatly computational motivation come coefficient informative search optimal author solve inefficient algorithm square presence complex accordingly thus select informative accumulate enhance equivalently rewrite format objective convex minimum system objective reduce entry nn simple optimum infeasible demand complexity day selection system solve factorization forward substitution amount total amount cost infeasible computational hour outlier issue result speedup dramatically save computational cost minute boost feature formulate discrepancy norm empirical element dominate eq balancing matrix informative representative obtaining sort decrease index effective highly acceleration alm beneficial lagrangian alm subproblems iteratively subproblem eventually minimum convenient penalty equality require cause bad introduce multipli require alm consist follow highly solver thresholding update lagrangian multiplier predefine remove irrelevant unconstrained subproblem scalar recently generalize shrinkage implement able achieve problem obtain q solve extremely acceleration solver remove zero avoid failure system cholesky factorization mean derivation save linear efficient e q identity side equal sign substitute simplify updating multiplication solver highly method initialize update rule overall summarize solver highly generally e accelerate speedup solver identity follow gap number view datum accelerate solver solver theoretically reduce maintain speedup solver optimum extensive empirical acceleration experimental setting experiment conduct server core intel ghz mb cache ram brief description dataset summarize complexity handle handle table ten candidate sample add method part vary illustrate code verify superiority method speed three sub show benchmark dataset mnist dramatically time consuming grow theoretical analyze compare surprisingly highly alm derivation computational significantly accelerate speed four summarize table show display fast fast acceleration dramatically verify acceleration fast mean take selection day display large algorithm I e surprisingly acceleration robust highly encouraging column table experiment ten benchmark representative accuracy table draw outperform second well compare well last loss suit increase knn svm shall increase consistent common view boost prediction select subset vary vary quality propose accelerate enhance solver via technique alm derivation reduce computational run
complexity heuristic manual gps calculation effectively verification latter automatically extract feed decision paper follow face verification constraint complex exploit discriminative information take domain computationally version gps gps anchor report recognition accurate human environment illumination human illumination conclusion face dataset control change date showing could performance face exhibit variation pose gender work verification apply face verification good vector number li analysis complex besides approach source transfer target transfer bayesian complex moreover transfer domain restrict wide application datum recently pose convolutional deep face utilize face affine transformation nine neural network although method parameter core gps good gps face verification gps vision importance task asymmetric focus task gp cluster discriminative improve net gps architecture consider gaussian latent mainly follow notable previously non method distribution heuristic manual gps hyper learn avoid gps overfitte excellent read classification observation row column impose function neither analytically laplace approximate acquire method unseen membership observation value small dense area variance good support separate explained point region vice versa dynamic equilibrium find equilibrium employ equilibrium point obviously completely determine row correspond datum determine latent position uninformative prior gaussian introduce obtain need automatically discriminative covariance take improve face verification include place position spherical discriminative prior encourage position verification mainly analysis kernel space formulation replace one direction onto formally negative feature positive negative maximize within n however rather position simplify calculation equivalent form maximize equation position write normalization scale prior gps freedom estimate conventional perspective share information task way distribution extend distribution entropy source describe detail source domain dataset source target write optimize accord learn constraint q model amount minimize multi learning expand equation item ip scale conjugate technique covariance derivation depend ignore constant q easily item inference problem store computationally prohibitive anchor speed put simply cloud kernel identity transform compute efficient predictive center identity describe two verification affine transformation corner divide pixel pixel patch descriptor patch image scale patch extract scale descriptor extract patch regard covariance pair person un image representation predict person cumulative solve call model regard extract feature image person regard enhance hyper function learn cluster finally denote variance cluster th number point refer equation variance regard codebook un face image first pair center also w final face new see differ image encode label call section conduct verification introduce source domain source domain contain illumination people age range contain image subject person image illumination condition contain approximately collect benchmark verification face image figure pose gender collect web dataset know benchmark verification variation describe verification conduct follow procedure source web life domain validation protocol test image mutually exclusive overlap two partition feed sd sd automatically learn reflect tradeoff discriminate ability select domain pair pair consideration method space adopt anchor take determine anchor select validation fix tune vary train trade practice anchor conduct five gps fair source domain regard significantly gps superiority obvious source domain since regard paper choose popular lr adaboost table demonstrate performance learn improvement improvement c svm lr adaboost number rp tree gmm also regard codebook compare three rp gmm generate cluster outperform effectiveness multi improve varie appeal bc combine verification bc decision state publish benchmark achieve level incorrectly obviously human emphasize center patch dense like utilize extract easy validity target split exclusive part training protocol mutually subset subset match pair even implicit among computer face verification currently exist computer base face verification however scientific contrast already face verification human moderate really difficult human specific point human verification contrast face verification familiar face relatively illumination pose face drop substantially change besides human face combination drop original test compare human verification human ask match people verification variation improve verification develop verification become useful subtle information slight human significant reality ahead human robustness familiar face develop verification task latent verification computationally multi constraint approximation propose different verification extensive experiment validate efficacy face
true marked factor simulation compare detect scenario suggest true major situation wish decide epidemic transmission event suppose let observe removal sir epidemic previous example add unobserved one total unobserved infection removal infection come assign density describe allow epidemic progress si ir order time exponential rate truncate necessary missing assign uniform density drawback markov chain never leave z st z dt describe update infection form detection daily infection illustrate simplify miss geometric parameter run distribution poisson mcmc appear evidence suggest time well epidemic poisson comparison certainly insufficient evaluate bayes although epidemic clearly setting drawback issue second miss likely insight expect missing distribution ex thank work uk use ij ik kk sufficient equation form column component solve say fm ix jx yield low attain strict simply assume yield follow ij show secondly specifically suppose var ix recall column column determinant p satisfy straightforward kx kx kx kx equation yield school partially avoid reversible key compete component applicability markov observe event time assess whether poisson poisson fit observation alternatively disease know sir remove transmission generate correspond special generic wish point model example bayes factor quantify factor suffer two practical drawback particular difficulty briefly method assessment difficulty neither entirely setting stochastic process involve epidemic criterion nine candidate involve typically involve simple must evaluate numerically reversible jump chain carlo method precise indicator consideration state factor give expression implement jump parameter propose go compete mixture probability proceed define markov parameter space approach parameter choose instead mixture upon difficult practice introduce prior problem computational method model establish typical consideration identically mixture distribution contrast consider partially structure contain framework detail computational contain conclude discussion ease exposition abuse density typical observe model vector common define mixture partial consequence intractable adopt augmentation comprise datum jj iy augment nan share element datum conversely necessary tractable jj iy miss density depend application probability directly summarie mutually priori eq eq define e jx yield divide numerator fraction rearrange obtain remain find matrix remove form suppose either require bayes express summary posterior somewhat solve find bayes factor yield evaluation three remark solely tool bayes assign straightforward describe third mild constraint mcmc repeat exercise yield estimate allocation suitable density however illustrate full implication illustrate marginal density proportional density explore conditioning difference set framework indicator target specify conversely target adopt become existence important potentially although set miss model always always bayes theory classical alternative example illustrate applicability know model assign nest several order method estimate factor describe compete prior distribution paper cite assign mcmc run correct mix serial population certainly serial mcmc population mixture illustrate agreement event birth birth far likelihood reference measure unit require bayes factor relative x gibbs sx gamma density run good b comment reversible jump require propose jump immediately obvious ideally sir epidemic g chapter close population individual initially remain period specify period remove play epidemic contact member time contact occur become process pair assume epidemic distinguish characteristic disease infection observe removal intractable infection process
diagnostic precisely projection assess whereas suggest language like manner datum positivity assess datum positivity constraint study language covariance produce calculate treat score measure covariance projection language project inner language unweighte alternate balanced observational design achieve alternative example unweighted back language indicate positivity capture variability show analyze simulation effectiveness positivity constraint identify vary robustness examine linguistic datum five adapt root combination simulation addition uniform zero node simulate example entry differ depend pairwise variable direction ht circle fill sep circle label label x label h eight node utilize version sign corruption long generate four eigenvalue would non separable assumption thus information gain resource require certainly investigate simulation design language relate note simulated basis retain construction used check positivity repeat explanatory htb notable amenable consistently positivity particularly third explanatory effective satisfy positivity far guide variance positivity constraint become reliable amenable identify range explanatory plotted range display structural similar varied performance notable indicate power positivity positivity still even first furthermore constraint positivity appear tree positivity underlie assess sample variance matrix status perform observation sample assess gaussian constraint assess tree amenable affect scenario amenable tree amenable amenable amenable furthermore satisfied amenable positivity comprise relate choice due diagonal ten amenable amenable provide amenable matrix amenable consider well respectively result across sample component suggest tree retain even tree non distinguish group highlight interpretation notably due express back sound although difficulty invert sound trivial however include acoustic assess functional capability positivity robustness moreover albeit interpretation particularly exploratory linguistic comprise provide result new amenable linguistic tree examine projection tree amenable explanatory distinguish first separate tree indicate broadly begin end effective finding scope small set word describe apply language pathway offer plausibility historical development tree constraint decomposable structure paper usefulness identify variability distinguish group circumstance common separable practice work separate language technique carry across assumption truly assumption nonetheless purpose consequence language covariance acoustic datum unlikely however sufficiently second violate evident copula marginally necessary scope shall describe particularly give nevertheless covariance circumstance violate valid capture separability often separate tool thus deviation efficacy separability perfectly determine order basis intend suboptimal capture proportion three hold may unlikely completely exploratory permit great aside performance selecting outline language covariance ten furthermore include wide analysis word across language tree enhance possible tree constraint point satisfied particular constraint give basis provide distributional exploratory could binary describe bayesian replicate wishart covariance matrix could formal testing potential adjust method tree language could linguistic much semi however even currently despite fundamental constraint mark diagnostic relationship application subject appendix c component explanatory tree amenable b c percentage tree individually q mm mm increase development range application image medical well great availability power analysis statistical attract analysis acoustic english language find amongst principal acoustic word linguistic way speech utilize cross language inference language suggest observation unlikely identically probable relationship historical language acoustic know certain would historical record tree reasonably support language american language relationship language long describe tree linguistic variable feature language quantitative use evolutionary language recently scale attempt reconstruct tree language european language researcher shift away evolutionary relationship toward somewhat structure assess researcher acoustic five language american exploratory adequate relationship question area algebraic statistic literature g particular g much fully characterize case analogue binary apply covariance consider component realistic permit observe linguistic tree amenable tree good explanation set preprocesse audio spirit object application language functional object description form language describe tensor decomposable reduction brief yet exploit acoustic language question section use tree constraint describe construction score projection acoustic section general effectiveness tree constraint investigate simulation assess accept entail tailor relevant address whether evolutionary relationship acoustic describe evolutionary model explore choose tree constraint acoustic linguistic language complex involve language origin component tree comprise audio language american treat ten language resolution bit classify gender share integer ambiguity make language straightforward ten many word language model become increasingly common study involve sound reasonable observe finitely along smooth e duration vary intra adjust known alignment transform take intensity though possible alternative duration dimension measure generic unit point hz hz store broadly indicate standardized period derive word encode word counter gender beyond frequency gender short acoustic seven european language identify suggest acoustic gender european language gender adjust macro gender henceforth gender adjust object interest word area statistic principal component multidimensional functional counterpart also functional multidimensional scale acoustic benefit reduction provide extraction reduce subsequent technique implement straightforwardly estimate produce singular must one approach loss project datum prominent aspect retain remains optimize efficiently mode technique linguistic semantic macro comparison argue construct priori group canonical technique discriminate start maximize subject component uncorrelated efficient fine detail consider within consider aim identify function variation express equation eigenvalue infinite discretized estimation zero number h datum use function maximally retain comprise interact standardize frequency simplification likely consideration say function subsequently challenge covariance separable separately respectively product ff tt solution although implement pca multivariate tool purely technique approximation implement functional setting necessary differently encounter tool frequency time approximation tend question dimension reduction row description independently affect notable concatenation visual find successive uncorrelated combination language language reduce perform eigenvalue find eigenvalue produce great language r theoretically covariance hold mode variability distinguish mention section decompose propose decomposable structure separable product separable structure elsewhere novel though strong basis separable number retain amount purpose decomposable overcome obstacle cause commonly encounter datum numerical theoretically observational set standard kronecker language frequency l direction treat frequency rank require usually relax kronecker eigenvector solve note separability hold less basis still far pca language maximize language approximation densely exclude frequency broadly similar result direction begin particularly corner similarity end word covariance diagonal suggest covariance albeit acoustic e use linguistic study figure show within component take dimension project sub solely notational clarity htb projection mean plot encourage project aspect know language distinguish language dimension separate language acoustic indicate discriminate operate simultaneously manner language close proximity post share particular acoustic examine matrix acoustic interest compatible identify language dimension combination model help assess plausibility algebraic mathematically leave note non vertex various considerable include setting concerned determine unobserve examine
short phrase generality linguistic describe four directly evaluate representation behavior format determine core inferential relationship architecture representation allow us model represent vector work system natural language artificial logical language three training logic define mutually exclusive understand logic cover reason logical reasoning statement construct third cover quantification plain poor strong simulate logical concept neural build semantic representation language sentence reasonably size training whether representation text address question put nn disadvantage competitive performance specific carefully task adequate case yet inferential approach nn tp name set strict strict seven logic set universe straightforwardly pair phrase relation exclusive neural linguistic say via composition phrase merge phrase form fix input tp tree depict process separately structure share fed layer generate two phrase output softmax seven relation sentence powerful nonlinearity apply output column node learn add add full rank dimension multiplicative interaction comparison nn layer layer independently learn nonlinearity use nonlinearity provide valuable study generalize strong structured structure vocabulary embed comparison tree regularization use descent sgd layer tune time use five cross percentage addition balanced report accuracy code generate review period table dot kind logic atomic proposition infer relation theoretic sound inference relation depict basic involve compositional successful reasoning compositional sound inference kind experiment train test boolean set entity domain structure entity proposition fig divide evenly test train ideal correct create function tree structure composition use recursive ensuring relation appear test tp reflect geometric relation recover relation nn fairly well question tune nn label potentially frequent relation example p successful familiar atomic symbol novel recursively show exploit fact testing string artificial system symbol formulae logic operator relational statement data assignment value tree model learn expression pass unlike baseline guarantee ignore relationship statement sum baseline test accuracy formulae correct approximation training cutoff gradually decay model suggest despite sentence quickly suggest tree structure generalization generalization composition many work weak regularization even class rare prove logic language considerably key model representation semantic natural interact lexical quantification functional since form language consist english artificial language contain noun lexical lexical noun noun generate sentence time directly kind describe infer word level complete logic sum baseline lack order information previous experiment potential single token logic almost perfectly sum largely find across suggest fundamental obstacle perfect architecture validate elsewhere experiment handle investigate model ability noisy label natural sentence sentence variant template sentence corpus train learn language label aware nonetheless show learn world adapt word initialize wikipedia since input pass recursive layer vector layer win corpus label train separate softmax classify datum source find quickly technique across replace collapse collapse finally regularization dropout input tp htp cl little play play water water break spread use amount available lack lexical show strong neither win include resource task well result test mutually exclusive pair phrase substitution little overlap annotate use neither sentence ten category syntactic configuration suggest encode relevant also lexical inference address none dramatically impact model like corpus compositional syntactic summing nonetheless substantially remain confident truly quality evaluate task clean artificial algebra logic structure quantification reproduce logical reasonably sized promising semantic remain fall short recursion whether overcome strong encode logical inspection hope reveal learn produce similarly model learn perform complex inference corpus language rapid make provide optimistic meet
baseline use directly pair equality prediction difference summarize pair predict treat inequality equality pair pair use train paper draw training width evaluation metric area roc auc roc calculate evaluate rank threshold rate false negative fp inversion fp negative opposite predict opposite roc visualize nonlinear rank deviation pick train model rank pick level rank clear ranking accurately recover rank contrast exploit pair rank close error equality varied pair general model decrease increase good model train good pt rgb rectangle rectangle rectangle rgb model rgb model rectangle rgb rgb rgb rectangle rgb rectangle rectangle rgb rgb rectangle rectangle rgb rgb rectangle rectangle rectangle rgb rectangle pair rgb inequality rgb rgb rectangle rgb rectangle rgb rectangle truth cm level model equality pattern x pt rectangle rectangle rgb rectangle rectangle rgb rgb cycle rgb cycle rgb rgb rgb rgb rgb rgb rgb cycle rgb cycle rgb rectangle rectangle rgb rectangle rectangle rgb label rgb percent incorrectly predict rectangle rgb rgb rectangle rectangle rgb rgb rectangle rgb rgb rectangle rectangle rgb rgb rectangle rgb pair line band rectangle rgb rectangle rgb rgb rectangle rectangle rgb rgb rgb rgb cycle rgb rectangle rectangle rgb cycle rgb rgb rgb cycle rgb cycle rgb rectangle rgb cycle rgb rgb rgb rgb rgb proportion pair rgb roc rectangle rectangle rectangle rgb rgb rectangle rectangle rectangle rgb rgb rectangle rgb equality auc set figure pair varied simulated maximum area validation roc mostly design equality also clear test auc advantageous optimization different rate people person rate point equality inequality consist person style major minor gender home convert simulation grid select generalization training test performance algorithm simulation rank high varied proportion calculate auc perhaps learn perform error term study machine formulation max margin relationship qp solver future interesting capability slack add function simulate directly model pair show rank pair advantageous pair work scale large smooth gradient acknowledgement ss thanks li helpful discussion definition theorem rank nonlinear algorithm equality label pair feature label element well geometrically segment panel naturally human movie minute actor goal comparison generalize measure one q indicator extensively study interested equality algorithm datum bold letter row propose illustrative simulate real item pair option comparison rank reject discuss connection comparison rgb rectangle rgb original rectangle rgb rgb rectangle label circle circle rgb rectangle feature feature rgb rgb rgb circle circle circle rgb rgb rgb circle circle circle circle rgb rgb rgb difference difference curve grey w classify reject option output tie literature comparison tie model paper guess extensively machine model rating ranking tie input input ordinal margin learn approach input database label learn limit answer model present explain apply linear cost program qp index label rank use yet thresholde comparison thresholded training pair learn threshold suboptimal issue rgb rectangle rgb scale rectangle circle circle circle rgb circle circle circle circle circle circle rgb rectangle circle rectangle inactive qp vector rectangle rectangle rectangle rgb rectangle rgb rectangle difference qp qp learning case consider maximum margin program lp qp difference comparison nonlinear ranking maximize margin define function small lp boundary boundary draw separability datum linearly perform change variable solve qp maximize let datum qp qp significant middle right panel feasible max lp comparison feasible rank consider boundary together margin take margin qp max margin lp general answer middle qp define however solution lp qp panel learn
interestingly minimizer closed appendix z entry wise prop update huber q proposition iteration update lagrange multiplier recovery inexact pricing perform novel topology test next real load benchmark latter comprise generator transmission network consist range generator offer c regard offer quadratic cost piece linear block cost reflect level fluctuation nominal offer table deviation load generator benchmark realization load realistic load competition load demand consumption match demand benchmark perturb fluctuation price solve min separately day ahead market entirely market infeasible occur primarily experimental line collect entire run ghz intel gb ram matlab parameter typically validation assume degree tuned degree give ambiguity normalize diagonal small proportional trade degree degree table recover benchmark evaluate real collect consumption scale competition site benchmark tracking test simulate grid infeasible yield alg initialize solution alg sublinear regret track alg initially yet interestingly available subject upon exploit complementary strength advance compressive proved grid consumption month enhance competitive market rapidly probe program rich could leverage heterogeneous characterize research admit unique belong subdifferential optimality implies trivially minimizer multiply identify prove prop imply q post multiply three depend back remark b edu edu grid major goal market competitive yet reveal information solely available market calculate multiplier economic lp minute matrix successive lp spatio price inverse strength formulate streaming datum input optimization compressive alternate direction multiplier economic price system offer maximize limitation competitive market market participant real delay market involve price demand generation look grid vision call market reach account increase fine design prediction operation grid extend preference attack grid task monitor security operator generalize tracking purpose attack information know transmission line could inform line could electrical cluster reveal influential moreover pricing characterize decentralize although attack topology using readily attack market characterize possibility attack explore attack detect identify attack design attack generally grid topology topological change study transmission reveal overcomplete topology spatio temporal grid rely phase line delay phase could grids topology scheme access quantity delay topology readily system energy lagrange multiplier constrain economic lp typically solve minute primal offer instance quasi underlie lp laplacian line exploit spatio temporal factorize positive novel scheme complementary strength constitute section one regularize thus simplify big alternate version streaming market distinct load validity finding letter vector one zero vector symbol stand respectively semidefinite regard norm norm frobenius flow market correspond topology capture via branch matrix grid matrix express flow approximated phase stack invertible resolve positive readily real market market determine demand price henceforth market minute adapt demand fluctuation collect horizon market grid stack tn respectively factorize product recovery problem find multi generality offer generator offer correspond sum generator handle similarly constraint remain simplify separately noiseless namely eq constitute blind recover rich recall definite grid light property equal grid positive eigenvalue diagonal imply invertible entry far expect prop write line transmission market period day california line overlap zero combination recognize satisfie pair inherent ambiguity unity satisfy leverage matrix counting balance enforce np advance yield replace diagonal value property definite nonetheless guarantee minimizer interior hand admit trivial convexity least low minimize norm constitute np q solver focus henceforth although could solve relatively interior handle hundred main challenge feasible former involve transformation intersection shift albeit relatively project alternate multiplier admm solve coupling admm assign lagrange multipli equality constraint iterate q copy multiplier partition admm admm update whose understood entry wise eq
speed la grant mat de amp set dimensionality accuracy suffer serious convergence context approach amp amp result amp amp cost coefficient signal large powerful two drawback necessity first require pass amp bp utilize sparse amp generalize reconstruction possibly specifically subscript notation individual column zero function compress cs amp efficiency reconstruction properly matrix optimization approach concern amp amp drastically move scenario lead result instability slight variation strict serious main reason issue identify namely sequential bp iteration strategy prohibitive entirely clear convergence modify posteriori favorable situation instance might modify spectrum prohibitive strong limitation third take step bp oppose utilize solve derive amp greatly improve amp careful amp preserve modify amp amp possess derivation next sec sec numerical testing improvement obtain amp cs value term statistical normalization constraint enforce stochastic examine bit cs model eq degree factorize marginal strategy amp marginal bp mmse implement message ultimately allow marginal message send subsequent message send node correspond algorithm current distribution reference parametrization lead follow sometimes bp integral without specify sequential pick update complete sequential bp amp pass send create burden possible term physics spin bp standard expand detail investigate one make close iterate relation attempt key derivation indice old one implication denote later describe point difference amp difference note also generalize la output channel one replace function depend term present demonstrate amp desirable reconstruction conduct computer processor matlab calculate utilize non fail negligible possible use require later effectiveness amp projection control give present fig amp converge solution meaningful namely experiment row element normal experiment robust demonstrate iteration amp comparison obtain minimization experimental molecular rare naive item
efficiently pair hypercube hypercube calculate implementation processor memory application parallelism discover probable probability although problem significant dp bn involve procedure processor set processor straightforward algorithm compute structural separate dp responsible compute different separately computation dp procedure failure greatly dp transform variant processing fill parallelization nearly perfect parallel time processor respectively extension discovery difficulty previously way score processor exchange non neighboring avoid transition spend communication novel fast network rest dp upon parallel feature conduct empirically capability discussion encode variable convenience dag specify edge indicator structural feature compute averaging directly dag demonstrate convenient formally represent specifie say consistent modular structural modularity independence modularity modular feature e indicator either example represent assume node il ig convenience family measure parent note bound need sum evaluate recursively effectively proportional size take therefore compute posterior compute forward backward fix joint eq evaluate correspond change time posterior compute recursively recursively db edge present serve parallelization dp find computation bn parallel hypercube compute difficulty responsible depend need need evaluate subset hypercube processor exchange score new coordinate computation exchange processor way respectively intensive finally integrate nearly load balance parallel facilitate mapping computation hypercube generalize describe parallel parallel discussion score dp operate lattice form set inclusion power direct node incoming receive compute sum assume correspond node send node lattice next processor undirected hypercube variable connect node differ hypercube edge parallelization hypercube hypercube algorithm denote processor processor time processor active time receive inverting compute invert bit step score manner receive neighbor neither processor summation require exchange compute new processor invert bit neighbor invert processor hypercube operate start processor similarly run summation subset processor hypercube independently processor take certainly algorithm score hypercube cluster time truncate formula lattice formula view truncate hypercube computer correctness definition encode variable otherwise hypercube use denote processor take processor responsible transform run line start invert perform encode processor processor dt dt st j ns bit string variable processor processor processor processor dt st mapping hypercube processor neighbor invert iteration necessary computation communication happen kt sr present hypercube proof specify processor compute line processor line subset thus processor among processor thus characterize k j jj k sum limit thus n kn truncate run hypercube compute line total processor processor characterize proportional k n n di di di di di di di responsible partition dp lattice specify hypercube part specify execution processor active except let hypercube lattice hypercube hypercube process string specification order formally lattice map respectively computation figure lattice partition process process complete computation hypercube hypercube processor complete computation hypercube processor complete start processor work feature prevent processor h subset processor assign processor compute processor operate reverse compute processor add score evaluate processor name encode bit string otherwise hypercube processor hypercube processor compute processor processor processor execute processor evaluate characterize run score line computing processor total serial time parallel parallel score evenly processor storage parallel achieve scalability compute intel core processor core gb memory core node use memory core allow regular user maintain core core serial synthetic set compute run serial run serial processor memory per processor collect calculated run speedup value dominate well speedup plot efficiency maintain successfully run second serial memory usage usage per compare show table time much well fast communication maintain core memory core able interesting peak gradually efficiency mathematically minimize solve yield e core consist provide piece evidence e core suggest quite examine respect core usage start increase core range usage core examine usage core core consistent per usage per core allocate storing require memory store program order run overhead usage per core total memory stay usage stay usage memory stay respect examine far observe usage usage per core memory usage test require gb core core gb observe core hour core half hour still far requirement bottleneck determine parallel capable probability efficiency computing probability feature demonstrate capability scalability processor algorithmic way exchange development develop parallel dp involve transform object network experiment limit future less possibility space chen exact discovery bayesian programming dp fast serial parent optimal processor run constitute take coordinate correlate exchange develop parallel capability dataset scalability processor algorithm bayesian graphical direct acyclic dag concern bn observation utilize bn prediction inference interested structural probable discovery aim identification represent structure bn decomposable fitness dag certain search employ minimize application probable bayesian extensively method one space compute average regardless possible super find optimal network np hard constant fast known programming dp likewise posterior probability dp algorithm exact probability space I parent node bound simple fast substantially slow complexity large dp computer
engine observation bandit reward choose tradeoff bandit offline raise relate reward choose evaluate offline evaluation causal research try reward causal policy draw round reward depend select end contain call action indicator otherwise choose q offline word even simulate test fast way guarantee long give reliable definition place probability ahead exploration adopt uniform choice experience know reasonably improve exist reasonably differ reality concern conservative procedure effective exist production randomization randomize precise problem sensible approach work well score meet system drastically variance overall calculation offline score offline sometimes correct verify offline experience verification turn challenge private randomization seed generator generator select multinomial final form contain offline verification verify seed reproduce check alternative pseudo random round statistical detect experience expect occurrence score hoeffde statistically significant collection therefore compare variable close statistical significance harmonic consider offline policy resort whether various concentration inequality confidence interval helpful insight result necessarily use due case suggest reality normal approximation take estimate interval component engine enable translate query error form rank web instant user correction web query absence new entity read person another correct query cnn really noisy channel model cnn user train set query query good rewrite merge rewrite correction loss real predict cnn desirable correction serve behavior furthermore query offline measure engine technique improvement search goodness concern lead offline place terminology candidate decide reward click page business reveal although goodness click process fraction user week yield datum use experience top must candidate send parameter experience metric affect benefit randomization reasonably include datum collection decrease relevance quality candidate higher tend likely choose collection towards promise likely run arithmetic harmonic major help issue lead eventually interval normally bootstrappe sample replacement bootstrapping compute unbiased offline click finally implement multiply interval size confidence therefore include interval offline policy online statistic could validate offline examine estimate day scatter online offline offline center plot bias offline offline large online ground reciprocal metric closely daily vary week offline fraction click position measure receive offline offline substantial fraction subroutine offline policy datum collect fall training label base positively metric rewrite target metric capacity prediction set necessarily correlate optimize eventually fortunately reliable offline select capacity offline week statistically demonstrate demonstrate upon medical include amazon product customer review compare purpose click find baseline appear match correctly appear whose name fail correction contain leave probably intend methodology research retrieval dominant relevance collection ranking mean gain successful scheme however argue addition alternative evaluation engine challenge user center system comparison provide promising engine people measure click serve randomize online recently technique identify winner require run offline approach expensive mention offline technique closely causal aim measurement change often intervention statistic literature recommendation formulate contextual interactive offline knowledge analytic technique head offline engine formulate contextual approach unbiased true real use verify reliability offline evaluation promise number action set list set problem exponentially large would see acknowledgement yu chen microsoft online particularly compute click key nature change engine result query infer reliably page impossible accurately metric feedback new serve baseline b successful expensive consume technique run many test log metric focus promise design information storage service evaluate cumulative ndcg approach highly successful offline metric human give high relevance score website com engine suggest opposite actor website like search sensible third experience engine module engine overall search engine challenge imply consider feedback evaluate user click infer effectively substantial increase optimize online metric
condition x j j constant since w ib last derive formula w n j number mle consistency sufficiently sufficiently eigenfunction et gaussian discretize first class covariance discriminant criterion note accurate propose also depend less investigate radius kernel order penalize discriminant b spline functional pls perform functional pca penalize hereafter method penalize pls hereafter code code spline website simulation category testing display approach svm bad rate regardless large suggest observe author mention article approach sometimes could simulation lda outperform ccccc gp cccc rbf ccccc pca fail normally class orthogonal function pca explain variance project testing display simulate average deviation expect functional discriminative perform sample lda code rate algorithm list table list misclassification lda approach sample become misclassification different observe rbf especially increase kernel significant finally cccc lda gp cccc rbf ccccc size misclassification functional neither lda cccc ccccc rbf misclassification database list table well lda become performance training size large improvement gp lda moderately proper list table rbf kernel kernel rbf order gp svm matter train moderately gp rbf ccccc misclassification table work good gp significantly selection dataset matter kernel share lda suggest dataset happen skewed etc cccc gp lda rbf ccccc fisher discriminant inspire smoothness translate prior mean posteriori probability fold theoretical incorporate correlation tuning parameter outperform multilinear chen song chen gaussian discriminant image extend fisher discriminant explicitly formulate smoothness functional application fisher discriminant pattern vision stand either grow principal pca discriminant lda directly must dimension reduction technique situation dimension pattern empirical fact intuitive theoretically applicable datum pca lda merely large specific obtain randomly sample discretize sampler kind scientific application discretization example digital regard functional include spectrum spatial discretize standard functional canonical discriminant etc vision classical method provide utilize smoothness fisher illustrate reason even violate hence performance evaluate classification fold propose bayesian smoothing accuracy modification utilize datum exist way require tune functional computational effort tune simultaneously less computationally intensive approach argue kernel discriminant vector etc solution idea infinite observe lie project point later empirical matter kernel choose article arrange review brief introduction bayesian smoothing penalize actually framework parameter require section remark draw direction generalize class characterize class fisher usually estimate still normality work assumption rich extend category ridge datum idea approach intuitive function observation lda smoothed smoothing computation effort smoothing filter tune pre classifier datum tune besides smoothed problem name penalize discriminant usually become noisy fisher solution tuning datum matrix audio dimensional manifold example al manifold regularize well regularize regularize outperform filter careful modeling moreover sensitive usually select cross effort assume smooth discretized datum represent project either predefine b etc analysis pca partial pls introduce wavelet classify approach lead optimum predefine observe good basis function remain provide representation hence preferable recent functional pca classify functional pls functional example pls preferable representation discrimination difference group represent well merely guess pls basis group pca basis far discuss fisher extend technique functional example utilize utilize functional pls utilize pca since fisher lda provide survey technique smooth bayesian toy example observe balance distance smoothness smoothness unknown norm order smooth norm norm describe certain assigning smoothness td log irrelevant interest minimize smoothing smooth smooth denoise technique follow technical smoothing expression resolution image th vector noisy phrase article focus interpretation kernel assume apply prior section perform functional knowledge bayesian formulation functional bayesian model functional underlie process unknown assign bayesian smoothing also inspire smoothing prior bayesian well posteriori address simultaneously covariance section suggest show
code fraction retrieve recall top scheme panel retrieve respect scheme panel fraction retrieve recall code retrieve plot optimal fraction retrieve recall respect scheme quantization projection sublinear recently simple quantization random offset lsh simple offset need code hash recommendation bin bin use code quantization offset convenient practical department computer apply sciences university usa department ny usa technical compare code quantization quantization utilize quantization random offset offset depend neighbor importance requires offset needed performance significantly depend build hash width say bit code sublinear time neighbor extensive reasonably theoretical code building table coding near determine code code similarity storage datum twice rank retrieve estimate practically store demonstrate improve similarity base bit code paper focus quantization projection neighbor search identify similar near neighbor search numerous etc neighbor early day compute near extremely text still square appear partly assume marginal machine normalize convenience e effective tool idea multiply projection classification factorization singular neighbor potential benefit bit project real convenient transmission suited indexing approximate sublinear near locality hash two quantization scheme recent intuitive quantization bin operation monotonically monotonically function suitable locality hash lsh well window offset write randomization fix accurate often one separately optimum sublinear neighbor suppose exist target distance neighbor algorithm distance square present presentation target exceed certain value lsh difference lsh characterize gap small difference optimum figure figure range confirm always gap region h gap good gap panel panel gap attain entire gap similarity similarity level high good comparison smoothly plot confirm replace might iii mention panel see gap much decay small code value see always large instead gap similarity gap well gap attain well divided subset another table result average query uci repository build hash table original use lsh independent appropriate hash compute hash datum hash belong repeat retrieve union retrieve hash retrieve datum substantially small fraction retrieve retrieve desirable evaluate dataset number experiment consume way evaluate lsh count retrieve exact positive avoid retrieve top neighbor retrieve datum ideally meanwhile hope keep retrieve present bin width small fraction lsh target basically
define ensure array almost surely compact array measure content theorem I infinite dimension application covariance differential index space tensor krige background tensor product recall infinite tensor range projective product nuclear space infinite banach nuclear fr hilbert dirac mass weight I ie may unit scalar encode ce ce ce demonstrate may covariance integrate tensor ci u closure one e e ni v ni reproduce rkhs tensor continuous definite follow structure value eq map kernel value value sure subtle kernel metric define ce ce f scalar metric f array continuous transform array ol estimator predict output space space map function array section assume describe law array let denote adjoint operator act array w ai ad e w dd tensor mapping orthogonal onto subspace ed define definite ol v possible compact ensure ol suppose strictly ol linear strictly definite strictly dense ol blue virtue gauss dirac let diagram encodes full structure disjoint subset different label indicator function design vector metric ci present operator ol value norm divide zero operator difference joint equality v continuous varie continuously argument omit continuity similar omit inclusion class define dense continuous hilbert choice diagram separate map diagram construction involve dense diagram dense subspace latter separable translation affine construct space construction external identity map prove agree subspace map center point center prove map composition affine restriction hilbert space part residual main translate v bound claim continuity let exist functional expectation unique prove h functional replace continuous respectively markov function lipschitz show e imply apply triangle quantity use inequality quantity vanish complete simplify expression vanish main combine fact equal rest term equal justify ff b b reverse uncorrelated p p array formal equipped map ei formal minimal topology dense tensor therefore ci ei vi ei k negative denote distance equal quadratic positive proving demonstrate convexity prove part hence hyperplane exist solution separate would separate distance prove part write vector compact support compact point convex combination consequently measure adjustment prove theorem thm thm thm axiom thm thm hypothesis consider know parametrize embed hilbert space extend construct ordinary absolute generality version uncorrelated consequence exhibit extend framework tensor ol define krige space construct hope article connection theory serve community goal topological space partial covariance unknown topological largely diagram may subspace restrict ensure projection hilbert inverse general ordinary extend full affine strictly definite dense e hadamard mathematical pose depend linear gauss ols unbiased continuity result joint continuity assumption parameter geometric familiar coordinate free though novel section stochastic add noise demonstrate ol continuous parameter example general satisfy satisfied often value encodes explanatory uncorrelated nb functional form preserve linearity construct unified v use theorem ols estimator b g express adjoint ol xy xy yy x x value mild explanatory law imply sense converge goal explanatory unified x continuous operator form reduce denote adjoint ol xy yx yx number even try simultaneously unified formalism linearity map structure array ensure represent covariance arbitrary construct general mapping krige corollary generalize krige predictor hilbert arrays finite index machine svm reduce present svm arbitrary topological demonstrate structural approach valuable formulate ensure essential consequence consequently unnecessary topological basis metric formal language readily convert program hope tool community author di david david la pe na schmidt david stein l nsf grant denote represent vector scalar equip topology minimal hausdorff space separate continuous separate exist functional ensure topological potentially describe c series topological fr array space part describe additive constraint approximated compact theory ignore exist parameterized space hyperparameter topological topology weak convergence nature parametrization topological manifold often equip encoding parameterization modeling correspond perspective noise hyperparameter dense intersection set full empty practice separability support hypothesis tail care take ensure strong v anti symmetric try classify operator continuity ensure p meaningful sense ensure allow proof define separable completion nan consist functional equal everywhere separability consequence separable hilbert ensure hilbert space encode dense diagram continuous equivalently admit factorization h embed originally motion construction wiener construct separable section trick wiener functional subtract continuously clearly isometry wiener map well measure transform translation deal case wiener deal general recent finite theorem still ensure v pl diagram leave space complete hausdorff separate point discuss introduction p change vs continuous also continuous v act linear adjoint adjoint map transpose purely algebraic satisfie separable hypothesis value finite vector continuous linear generalize hilbert continuous lemma abstract close subspace closure diagram exist identity diagram path imply hilbert respectively restrict remarkable function neither whole inverse exist domain complement similarly denote complement space main particular continuous hyperparameter map equal h composition domain banach hilbert space open space affine adapt banach corollary admit construct ols recall partial inverse ensure theory hyperparameter consider map ols define extension close impose existence ols ordinary least square unbiased map mean composition sequential continuous proof ensure ol across prove continuity ol vary jointly datum short suppose banach equip topology structure demonstrate ol regression ol conditioning uncorrelated problem error parametrize law gauss ol estimator unbiased since ol well define f define affine support gauss correspond hilbert space orthogonality achieve optimality optimality least residual complementary ol right main ol projection estimator consequently adjoint operator l f projection original explain residual variance unbiased theorem ol consequence q functional loss see function nice tradeoff gauss blue error possibly distance onto error arbitrary functional inequality direction functional nonlinear stein total powerful tool give drawback room accommodate continuity ensure robust strong hyperparameter mean p stochastic yy residual bayes agree conditional mean stochastic every abstract ensure p topological purely may stochastic define convolution agree original measure trivially uncorrelated theorem demonstrate admit continuous corollary problem map lebesgue measure reference euclidean generalize dimensional complicated continuous banach space map ol linear banach fr ols certain necessary continuity theorem condition spurious theory see see continuous g ol yy value residual construction define auxiliary v denote auxiliary operator hilbert representation measure p define integrable value yy yy non continuous ols residual p demonstrate uncorrelated next ensure correct every concern domain satisfactory respect proof ensure joint continuity ol mild assumption space banach proposition check integrable however construct continuous checking optimality equal measure convolution follow respect law define convolution measure respect justify prove v measure independent dependence structure equivalently satisfied motivation definition measure distribution law form let ol convolution light ols continuous uncorrelated continuous map found immediately admit continuous linear solve problem use krige representation value array arbitrary index classification spatial array predictor thesis formulate topological space topological space primitive assume locally hausdorff hausdorff array function key pair simply array compact open topology measure array array array fr arrays think array array fr compactly ai fr operator fr value fr r array index set call time space array consist limited time value fr spatio spatial field spatially jointly value medium indexing random admit scalar I definite kolmogorov extension equivalent ci co array compactly cf subtle kernel replace si ie scalar value section value estimate reformulate statistic
minimum criterion tree analyze finite correct model order optimal prediction optimal whereas estimate depth estimate build estimator risk predictor order user use building report report prediction technique solve exploit handle handle report predictive information optimal estimate cell one simulator around distribute physical group sub allocate band resource middle group symbol symbol call physical resource constitute group band band scheduling transmission band frame band allocate allocate whole technique periodic five band rate aggregate band aggregate convert comprise user simulator exponent macro simulator channel generic channel model realistic channel model system angle distance profile delay delay characterization channel extremely even different link make mathematically ease detailed parameter l cell grid inter site km power level hz user distribute direction inter interference explicitly model macro transmission scheme proportional fair full bandwidth band information band user ms measurement none loading effort partial loading exponential arrival macro height ghz inter site interest km total power level hz model fix identical uniformly inter site interference macro fair adaptation band band user ms delay ideal profile loading loading exponential inter arrival simulation request user every frame typically rate table look u main b traffic inter arrival load situation loading vary discrete full loading algorithm band time model characterize mathematically traffic sequence predict algorithm build discrete build propose apply technique appropriate modification build estimate active build variable use slide context current length allow dictionary word assign incoming character add word dictionary contexts repeat generate tree illustrative get incoming character follow step get character step obtain step tree repeat get character whole occur example look bottom see parent occur occur seven suffer difficulty word require ever increase memory store word unnecessary learn furthermore asymptotically due asymptotically save measure cycle long control channel know receive short cycle sense channel feedback channel traffic period sequence length limited require depth modification short recent observe symbol plan depend use depth depth depth model tree assign occurrence recursively recursion even zero might could occurred look example alone value look depth occurrence give frequency state indicate j j sequence use probability next number time occur occur occur future store build frequency evaluate see build user problem build frequency tree tree reasonable must capture tree must reasonably depth particular px uk estimate estimate correspondingly word might large fix sequence possible achieve complex require property explain detail subsection sake notational simplicity henceforth subscript characterize sequence call extensive sequence source hence entropy volume predict future mutual information physics extensive extensive component rhs sub entropy extensive grow extensive joint writing calculate require knowledge sub observe predict compute entire practical system joint good markov predictive varied study nature expect grow linear monotone decrease information go decrease past retain information equally happen interference decrease sequence possibility sub extensive despite simple process immediate q term metric function increase k learn sequence infinite require never form imply predict require user behaviour capture empirically seem logarithmic instead continuously understand looking express apparent argue pick value increase need distribution hence limit despite slowly prediction user significantly compare extensive gain substantial obtain increase order beyond increase increase order use current bind order order sequence sequence model th markov chain description criterion observation estimate observe user build discrete building logic parameter th maximum user use bind determined model hypothesis problem likelihood hypothesis increase fail estimating cost pick maximize try maximize first log penalty covariance ml increase equation ensure optimally trading implement priori try estimate determinant option use model eq model aic however sequence grow nearly sample correct correct aic derive detailed correct aic aic criterion ensure initially usage determination use predict give occur frame look sequence cycle minimize observed vary section fix calculated estimator posteriori pick predict possible since accuracy treat equally rate predict transmission loss throughput map pick wrong result rate choose propose throughput event pick numerous cost enable pick fail transmission assignment rate difference rate rate denote give observe calculate expect minimize transmission successful incur hand high pick bias map thus low loading load simulator sub frame generate understand behaviour loading generate extent variation greater adjacent value load user full loading variability loading hence issue loading procedure implement simulator frequency one access tree early see probability instantaneous estimate curve stop user repeat reasonably tree reach period sequence length truncate use truncate predictor henceforth predictor fm probability fm fm fm nine median median user table partial scheme median naive prediction technique fraction predict efficiency percentage obtain user reduce rate scheme improve fact predict complexity mechanism partial loading cumulative distribution mention cdf load predictor outperform fail fm fm users loss fm fm map nearly achieve cdf partial loading compare outperform user rate fm user fm user map technique scheme look graph percentage map percentage map predictor error treat especially predict feed high full scenario variation likely sometimes feed work well compare fm choose performance load partial loading loading require adapt loading traffic frame presence partial loading investigate propose prediction need cast aic correct aic model sequence provide minimization rate implement simulation substantial level gain
dataset make idea notice independent subspace independent vector specifie belong dataset class independent subspace concrete specifically subspace originally subsequently independent class class project projection belong linear cost case linearly separable datum dimensional compact preserve separability specific manifold structure derive result margin subspace application root equal vector element indeed random element aside relation adopt template projection initial template new template feature preserve template study require multiclass first cosine angle vector approximately preserve two present draw gaussian true cosine angle preserve additive significantly angle empirically value close cosine serve preserve empirically otherwise notice hence preserve preserve irrespective angle eq high term vector preserve less preserved term representation examine preserve preserve dataset need hold simultaneously subspace continue belong projection preserve denote continue span column span straight remark need hold structure preserve random error subspace margin close principal maximally circumstance project subspace separate let well relate subspace disjoint subspace subspace pairwise sparse sr various recognition security sr compress sensing equation overcomplete dictionary solution achieve overcomplete property overcomplete dictionary reconstruction advantage represent zero coefficient training term class sample ssc subspace subspace cluster datum recognition projection however cosine close preserve vector angle verify setting random dimension space empirical rejection vary indicator operator rp c rp pca arbitrary cosine vector empirical absolute cosine angle probability rejection similar evident preserve equation length arbitrarily great dimension rp see dimension rp hold suffice trade goal report comparative accuracy pca mainly technique less extended initially locality preserve projection embed reduce technique yield accuracy claim subspace rather separability projection well classification exploit well classification random projection illumination independent per person take illumination image intensity vector evaluation illumination database pose illumination dataset split pixel pixel intensity projection table dataset result times accuracy dimensionality significantly fast preserve random use projection explain low I lie surprising even major projection occur streaming change data lie preserve still hold stream template attack original template construct inverse recover approximation ii yield accuracy par original become essential highly paper initial step life security require dimensionality ensure subspace preserve argument hold show cosine projection preserve evaluation present irrespective hand hold simultaneously r r get direction least vector get rx rx similarly hence hold similarly union bind hold probability n concern security system problem template template many quality early processing present formal projection essential dataset preserve generate derive physical etc gain popularity decade systematic assign resource template physical template attempt security research area deal trade security state successful far generate course generate highly template transform template increase volume drive security since computational increase system employ volume vector highly degradation employ perform task generate template many fail
generate perform virtual alarm set able presence case resp identify case high snr due converge section performance hyperspectral university image pixel pixel nm dimension randomly available run select whole abundance matrix row point redundant row reveal thus redundant spectra determine algorithm several subset mutual redundant discard give assumption surface square abundance map scene cast n determine angle obtain outperform c c avg angle approach blind model allow determine abundance consider synthetic real demonstrate include adaptive change environmental optimality detail every prove address hyperspectral without assume scene signature satisfy increase depend hyperspectral multiplier address reweighte synthetic demonstrate hyperspectral imaging continuously grow area receive decade hundred narrow adjacent couple surface chemical image application include surveillance hyperspectral image pixel distinguish mixed pure signature material mixed mixed pixel abundance determine number extract pixel scene separately virtual among alternative part separation consider express linear combination fractional matrix abundance constraint place symbol column study assume scene index least generality mix reformulate represent abundance admit row zero identify exploit turn situation observation handle cardinality unknown dramatically affect mix valid snr shall represent whole scene blind self estimate abundance prior impose negativity order cardinality candidate model multiplier admm solve square study literature three free author norm rather pursuit order scene predefine negativity organize describe result admm namely minimizer usual model f projection operator find lagrange multiplier carry represent run tend converge dual q stopping criterion must call main admm splitting subproblem square positive lead addition constraint turn realistic restriction index respectively family depend kronecker presence consequence l approximation constraint term objective reweighte algorithm update calculation update weight proceed calculate follow reduce norm solve optimization augment respect lagrange multiplier carry exactly hereafter obtain set amount analytic alternate likely converge minima reason warm initialize loop initialize j spectral extract library mutual coherence spectra coherence eight generate negativity three
explain component much incorporate smoothness technique frequently subject worker focus baseline brain store array correspond brain begin template image number map template record subject specific template voxel represent image process use volume examine visit create image voxel subject contain result require pc material central pc show figure correspond grey primary tend differ grey approximately remainder primarily feasibility pca interpretation pcs section confidence bootstrap distribution pc overall computational bootstrap propose notation typically bootstrap matrix vector store solution rank ultimately arbitrary pc adjust section sample although svd fail handle svd appropriately adjust find detail material complexity sufficient lead bootstrap bootstrap pcs bootstrap express bootstrap b distribution pc variance create interval additional subspace option additional scale project onto pc vector create percentile pc hold work block break calculation low operation material bootstrap pc require roughly equivalent however sign know across variability principal pc cause sampling decompose interval include absolute variation pc adjust sign change column reason sign column equivalent sign require calculation element calculation simplification note whenever occur adjust multiply result equal cosine angle dot adjust angle range interpretation pc dot product adjustment choose sign method previously sign pc column approach rarely different shift material dot product intuitive find take bootstrap characterization property store b eq operator bootstrap complexity combination multiplication oppose traditional multiplication computation bootstrap transforming allow calculation calculations parametric bootstrap translate dimensional component construct pc specifically pointwise confidence pc pc interval calculation full bootstrap matrix simple confidence component moment interval ci e percentile normal across bootstrap attain calculate store pointwise percentile ci define percentile unlike moment percentile calculation percentile tend bootstrap sample moment interval e quantile tail moment interpretation quickly calculate bias accelerate suggest norm pc unit p pc function percentile note calculation geometrically dot condition cr angle note cr coverage create band band contain exceed absolute contain component pc subspace combine pc pc influence rotation whose depend pc rotation characterize variability principal equal pc manifold kp create norm operation refer suggest form variability cr also principal pc simplification adjust rotation principal orthonormal covariance pc pc approximate rotation pc parameter interest pc rotation lead suggest rotation pc sample confidence interval pc however adjust rotation might apparent coverage problem estimate clearly eigenvalue less clearly rate pc curve panel variance result alternate residual spaced pc simulation residual median across moment percentile alternate value generally scenario median coverage pc spaced eigenvalue slightly coverage plot row correspond right plot show pc row rate confidence component well spaced increase appear affect either fast bootstrap eeg estimate first pc exhibit minimal variability rotation pc column pointwise three pc percentile agree although percentile interval skewness bootstrap pointwise interval show band around pc band calibrate pointwise coverage simultaneously contain statement population pc interval somewhat ad hoc furthermore contain within band norm low boundary pc tight imply nd wide hour reader think artificial pattern variability third bootstrap central pc pc little bootstrap pc spike hour shift spike population pc plot us peak peak bootstrap shift tend third component pc variation second noting bootstrap due rotation second pc bootstrap variance display vector panel see pc pc place place weight ci plot figure variability pc exceed absolute surely pc note percentile rarely thought percentile create bootstrap first bootstrap see simulated draw eigenvalue know percent bias across bootstrap covariance eeg slight bootstrap percent bias pc show computational feasibility deep interpretation pc provide pc estimate variability variability fit pc pc magnitude pc fit pointwise ratio nan display rotation pc truncate analogous substantial pc rotation panel notable percent pc base percentile test measurement subject calculation standard ghz intel memory parallelization calculation attain pc conservative loading bootstrap offer speed improvement approximate force test demand percentile pc minute minute error force high cluster reduce parallelization attempt file file relevant block sequentially material bootstrap need sample show time dimensionality vertical axis log time show pcs standard bootstrap require calculate bootstrap outline fast bootstrap lie feasibility eeg standard component usefulness demonstrate ability bootstrap well bootstrap rarely theoretical well study study pca research pca verify bootstrap useful generate display direction bootstrap variability pc calculate also beyond alternative method describe pc variability elliptical pc rank result region describe sampling variability span observe b cr region q elliptical fully define direction primary axis fact precede want first pc lead residual onto resample alternatively remain pc resample residual project vector approach source acknowledgement national institute health grant number national imaging grant stroke ns dataset article solely package estimating sampling variability large storing bootstrap computationally infeasible fast calculation component eigenvalue leverage coordinate metric compute solely bootstrap storing component fast brain allow standard bootstrap minute approximately image analysis dimension reduction field image multidimensional dataset pca identify projection maximally call component pc subject basis pc pc population pc low expand discussion sparse sparse dimensional demonstrate variability pc variance confidence pc typically existence order interval bootstrap confidence pca bootstrap context determine challenge calculate store bootstrap exact order magnitude bootstrap lead variability pc low little pca largely drastically reduce dimension far bootstrap scientific application method directly determine bootstrap principal regression quadratic penalty remainder organize section basic pca fast discuss motivate eeg fast computation final pointwise pc bootstrap use coverage pc pca eeg present denote denote block notation denote element denote vector identity generally rectangular highly pca subject basis vector maximally basis principal pc subject respect call pc matrix subject center singular decomposition denote contain singular order value principal vector diagonal contain variance know variance pc construct scalable calculation estimate draw replacement sample pca bootstrap calculate variability bootstrap computation infeasible important pc depend coordinate measure coordinate along number exceed involve reduce parsimonious vector span still include transformation first step parsimonious computationally demanding number coordinate equal improve efficiency pca contain low subspace span principal observation sample span sample skip demand reduction represent parsimonious orthonormal lie loop translate calculation bootstrap bootstrap svd step resample b b b directly score value sufficient b bootstrap apply decomposition setting result use generalize suggest decompose lead project however approximation demand decomposition score imply align coordinate coordinate orthogonal rotation pc vary bootstrap unlike estimate variability pc pc variability magnitude pc rotation pc pc weight majority bootstrap variability rotation pc parsimonious dominant component drastically storage memory requirement
restrict radial simplify fix drop lead formula simply otherwise build x numerical building depend operation inversion dimensionality add term preprocesse relatively asymptotically rbf kernel metric modification repeat exploit calculation arrive kernel kernel build unlabele consequently fold validation separately separate feature apply active unlabele perform nine dataset uci repository briefly scale fair regular rbf almost dataset geometry detect k algorithm bank cancer heart experiment perform small include use fold mode start good accuracy table rbf kernel mahalanobis diabetes breast cancer heart cluster empirical rbf data scheme separate base meaning run train separate htb dataset rbf bank breast cancer diabetes heart clustering behave gmm vary difference perform advanced cluster second rbf conclusion cluster various choice technique internal validation part applicability reader mind different method gmm gmm gmm bank breast cancer diabetes rbf easy selection practice exist library table narrow rbf achieve rbf also behave simply base htb rbf bank cancer diabetes heart technique perform limited table well small notice range outperform great importance resource internal application active rbf rbf rbf bank breast diabetes heart rbf achieve kernel big htb rbf mean regular htb dataset rbf bank breast cancer perform parameter increase resolution opinion reliable finding well rbf consider eq measure result exploit ability limited grid applicability training time great ensemble repeatedly pair hard behave yield high justification phenomenon us question conduct experiment improvement competitive base simplify regard view lead correct transform multivariate asymptotically rbf obtain behave rbf however select typical svm empirically fundamentally subproblem reduce amount show conceptual distinction approach model worth though experiment see general framework section section example edu pl gaussian transform dataset mahalanobis paper gaussian consider local geometry dependent feature projection emphasize construction new divide represent construct empirically nine uci repository stability find learn property without exact geometry density method conceptually often lead decision combine form complex algorithm introduce building exploit geometry construct feature part give see change problem solve implementation svm give less complex compare rbf mahalanobis grid crucial classification scenario ignore importance index evaluation measure tune require evaluate visualization characteristic roc visualize structure describe issue connect comparative dataset uci year grow among mahalanobis calculate disjoint element still define thing construct infinite ii fix iii iv projection dataset restrict first worth geometry try incorporate inside numerically remain question evaluation benefit problem focus contrary modify gaussians contrary rbf mahalanobis rbf variation gaussians projection thing point projection gaussian sake
convolutional layer kernel third convolutional layer neuron layer previous layer fully layer follow relu layer relu pooling pooling unit space apart center location response lm layer normalize adjacent kernel constant layer dropout size data conv conv pool conv conv conv conv fc conv fc fc fc layer shape prevent apply augmentation extract patch horizontal image patch first convolutional filter kernel layer pool convolutional layer convolutional one convolutional size pool convolutional neuron stochastic example weight small decay regularizer insufficient sentiment training suffer overfitte imagenet promise fine initialize except layer regard forward pass divide epoch fine initialize deviation initialize neuron bias fourth fully connect layer remain initialize forward train model softmax server core e processor gpu training day test store gb disk space annotation pseudo ground truth top annotation percentage image pseudo truth label k accuracie tune cnns cnns fine tuning accuracy visual sentiment concept strong community usually mid sentiment performance acceptable rank per cnn greatly base gain accuracy top without top detect tune serious incomplete incorrect detect concept accurate ground correct accuracy top important boost base train multi binary suitable retrieval instead annotation section model rank precision noun show design cnn performance object detection candidate concept classification convolutional network cnn train newly framework deal biased training datum prevent initialize weight show newly cnns annotation compare localization cnns improve leverage high boost help improve build comment sentiment sentiment top tune em edu berkeley sentiment classification method deep convolutional cnns visual noun discover tag web utilize nearly million concept popular show great imagenet cnns train newly deal contain strong prevent overfitte model train imagenet train deep cnns annotation accuracy retrieval storage index social medium social among research effort sentiment visual medium attract video opinion greatly media communication education concept study extensively computer visual difficult impossible big sentiment tractable sentiment mid fill concept noun strength discover occurrence relationship tag image chen leverage sentiment concept thousand category million image deep network cnn able achieve classification performance improvement efficiency similar imagenet much capacity control compare stationarity locality dependency mostly cnn easy feedforward connection parameter slightly theoretic cnns capability imagenet specialized sentiment concept train gpu successful back propagation achieve large benchmark dataset imagenet competition win result learn deep convolutional small dataset mnist achieve success et unsupervised follow supervised fine tune insufficient concept bank paradigm prove successful computer learn label recently et domain paradigm briefly review sentiment define affect monitoring opinion corpus sentiment visual issue et sentiment consist noun detector level drive video retrieve extract million sentiment image ensure prevent suffer
planted apply three synthetic form hard condition network commonly bipartite community broadly say except genetic vertex internet movie actor movie actor examine level spurious detection sized non type community community degree community heterogeneous resolve community vary amount specify create whether preserve degree correct expect illustrate induce bipartite mode projection network performance projection unweighted projection bipartite obtain let share weight adjacency matrix diagonal size correspond projection type adjacency entry weight path vertex draw ability recover partition vertex sbm partition type vertex measure normalize infer correct treat observe group partition group group partition hx I degree matrix identifiable community produce community divide evenly bipartite sbm unweighted projection amenable method mutual information vertex partition vertex recover plant structure less easily identifiable community create overlap broad uninformative third community connect onto overlapping type community difficult community divide evenly impose give prefer plant bipartite adjacency normalize infer correct exhibit classic phase find plant partition extremely extremely inaccurate mode community sort sbm weight unweighted sbm without correction unable plant partition community sbm unable lead degree correct remain correct fail partition optimum sbm modularity unweighted projection fast modularity able projection variability projection limit difficult projection outperform design produce relatively high arise find frequently topic co word removal mask noise stop priori correspond extract actor movie broad actor interpret language movie match actor movie bipartite adjacency sort correct define characteristic movie group movie group consist language correspondence language infer responsible b correspond stochastic block create community regime show bipartite efficiently sbm bipartite community mode one mode community avoid projection discard create clique assumption problematic fail bad scoring either outcome correctly suggest whenever projection avoid bipartite evident substantially outperform method result general many system word contain hand exist contrast vertex produce despite connectivity sort language indicate movie language group movie per actor dense show similar separation edge existence brief aside mode commonly yet implicitly projection subsequently network raise question due bipartite whether measure correct sbm two sbm increase apply bipartite must learn bipartite cause bipartite known information utilize accurate subtle point use choice choose selection burden model bipartite network also selection compare likelihood choice control extra sbm distinct community generative remain area difficulty limit assumption bic produce incorrect decision promise result recently broadly principled advantage interpretability infer process ref membership community bipartite future hierarchical adapt explore beyond community inter centrality bipartite form structure sophisticated project award gm ac sciences fa ac air office scientific research project solely represent view health design collection manuscript source implementation appendix implementation author find set x green partition size plot type highlight correct show efficiently infer structure green bipartite common connect like counterpart bipartite choice information interpretability solve community detection bipartite bipartite include vertex may trivially bipartite block statistically choice interpretable synthetic structure bipartite bipartite vertex vertex bipartite appear specialized remarkably document genetic actor movie user mobile author common way find subgraph definition vertex likely two vertex never definition community suit block community bipartite pattern detection bipartite apply community mode share neighbor type reduce implicitly without construct bipartite scientific ever mode large bipartite measure number path length projection bipartite projection modularity moreover information bipartite identical mode projection even highly extension modularity propose broadly speak model vertex connect bipartite express implicit restriction maximize modularity partition independent yield type consist pure sometimes call co partition elegant structure network system network capable bipartite sbm develop specific overlap edge structure partition specify among generate sbm parametric give explain configuration advantage explicitly state fact bipartite directly apply sbm bipartite also sbm bipartite exhibit quality community formulate bipartite search network community plant background uninformative bipartite bipartite block hereafter correct vertex extend include correction begin vertex group group community bipartite adjacency instead connect thereby notation work sbm vertex type must pure develop correction expect adjacency let number world allow poisson calculation easy unlikely correction bernoulli enforce bipartite constraint restriction q model bipartite generation bipartite network lack lack subset sbm discover bipartite structure bipartite discuss generate symmetry group bipartite network seek practice easy logarithm parameter denote sum drop maximize assignment constraint pure community motivation derivation correct degree correct tend broad structure sort degree model denote parameter edge number enforce eqs probability observe adjacency normalization connect belong rewrite maximize drop constant multiply partial lagrange multiplier yield likelihood network dropping maximize correct respectively substitution heterogeneous pure vertex type partition classic lin input adjacency vertex assign type vertex index group propose type propose move move possible choose move decrease move help optima pass objective optimization many score among independent well degree correct model demonstrate community bipartite sbm naturally clear specialized model perform sbm mix vertex community nearly less vertex type eqs posteriori sbm constrain indeed know bipartite network complicated rule connect generative correct numerically counterpart provide mix ii vertex reproduce adjacency sbm remain numerically bipartite network imply identical principled algorithmic key understanding make vertice uninformative give hand sbm lack informative density observe community whenever sbm bipartite group sbm network show bipartite
setting capture position pair likelihood item order intractable contribution key collaborative ranking contribute suppose list item drop mention tractable scoring let ij scoring function agree relative simplicity thus suggest specific rank permutation probability first position suitable rank community extension model propose make collaborative interested choice user rank item rank position begin end employ introduce permutation adaptation trick unseen log verify either item rank item interpretation user principled item ahead choose item respect community user permutation sum denominator take recursive array start pz step fix lagrangian pz multipliers lagrangian zero maintain lead pz u close apply resort u z z want output rank unseen finding rank unseen item sort order new item order position item old repeat new item position rank likelihood list introduce new item new item place th item old list naive right recursive assume position odd notice z z odd permutation cost per pick permutation cost case item hasting method move learn maximum computation generate sample accord significantly per propose cd work instead run step enough relax distribution cd stress pass cd model one chain move graphical efficient regularity application attempt consider pseudo abstract pseudo mcmc technique mcmc configuration configuration log unchanged position require energy would denominator current eq q energy pass local order item new search position low energy computationally choose probable pass special factored parameter let start score factor ignore position monotonically attractive constant position prediction case nice interpretation u basically say subject much occurrence potential item distribution use dependent item user swap item change gm collaborative rating well recent author pairwise preference none attempt rank pairwise discrete often limited goal deal item recently attract retrieval setting item image compute collaborative ranking study assumption introduce community user base inference recommendation tie rank correlation collaborative filter information community permutation rank make successive item manner introduce account specific generative second rely assumption make learn inference method time collaborative service community service community exploit rank list research preference rating rate star user scoring assign qualitatively carry evaluation limit intuitive way express preference easy place visit assign importantly recommendation core recommend unseen address open preference require intermediate filter system item decrease complete user collaborative
share herein principle science aspect publicly reproduce mark anonymous comment work co national foundation university systematic sigma biology mark technology center center conduct herein cm htbp three differ prior model probability pm analysis retain prior sample population plot glm replicate analyze proportion simulation replicate divergence population adjust analyze strong true parameter exclude divergence e support extreme divergence simulate population distribution top column unit assume site six please summary dataset pair population u indicate distribution million assume site please support cm analyze average place prior eight analysis mean implement scale relative size term upper population population different time unit model hoc adjustment possible alternative prior sample efficiency support asynchronous nonetheless conclude divergence divergence informative dispersion statistic see plot perform single divergence thus utility whether share one infinity numerically reliable zero easier less interpretable one index divergence estimate code mixture analysis grey black model b exclude divergence appear valid cover plot project summary statistic two dot summary empirically inform pair population draw indicate plot distribution million per htbp cluster divergence simulate divergence distribution divergence give unit million htbp pair population draw u column time unit million site generation dispersion accumulate retain glm htbp correct simulated analysis relationship glm adjust prior setting replicate include population bp probability divergence proportion bin consistent divergence event population simulate setting black black com word support author date establish population occur process assess bayesian method pattern simultaneous paper agree prior assumption support distribution divergence lead marginal likelihood probability model alternative posterior analyse rejection unable rich reasonable averaging empirically inform prior analyse tendency cluster exclude true tendency analysis share divergence primarily hypothesis explanation bias paper demonstrate unlikely region support wrong history computation fortunately predict principle flexible accommodate place vast region frequently seek phenomenon split potentially event probability choice study method support divergence broad period agree divergence uniform spurious sensitive prior mechanism behavior ref align prior likelihood posterior divergence prior rejection space within computational toward sample insufficient exact support approximate exact posterior support simultaneous divergence across divergence wrong perspective bayesian employ seek phenomenon exclusive distinguish ability share correct sound improve markov chain sequential carlo rejection implement try narrow informed uniform sample rejection would produce estimate posterior consideration empirically inform prior bayesian evaluate approach potential analysis error mix unit divergence exclude posterior reach evolutionary accommodate uncertainty divergence without likelihood model temporal divergence surprise rich show jeffreys also use analysis make hypothesis hypothesis suggest share divergence insufficient likelihood time suggest narrow highly inform divergence preference raise paragraph prior sensitivity expand method inductive belief become parameter value p describe belief value parameter define belief collect bayes calculate eq likelihood elegant belief accumulate dataset belief datum uncertain valid bayes say empirical empirical study obtain favorable bayesian justification parameter estimate candidate bayes calculate estimation choice see update normalizing denominator long probable strongly informative prior incorrect look entire parameter normalize posterior distribution average likelihood bayesian much bayesian value integrate marginal posterior measure uncertainty justification twice peak justification inherently belief prior analyze rule analyze fail uncertainty belief compare confident confident nonetheless bayesian perform well particularly aggregate parallel information prior lead favorable group wise frequentist coverage across far support example estimation concern practical narrow empirically inform uniform strongly model inform thus average estimate time match mean divergence divergence across limit furthermore prior see preference consequence posterior exclude explore possibility way average narrow likelihood prefer less yield dominate model generate reduce mixing unit time si prior sample prior model retain euclidean deviation remaining retain euclidean summary statistic across twice replace differ likely divergence nearly prior exclude time strongly prior distribution analysis model time sample behavior clearly insufficient computation hypothesis straightforward prior via summary sample plot orthogonal axis analysis indicate model problematic model provide narrow graph sa figure sd well exclude simulate randomly identically average dataset simulation replicate bayes odd exclude model exclude exclude value replicate glm adjustment exclude importantly exclude figure factor glm adjustment figure result simulation analysis narrow prior obtain toward model exclude elegant bayesian inference narrow prior density dominate marginal likelihood integrate capture uncertainty exclude fortunately flexible greatly spurious allow uncertainty clearly risk inherent bayesian justify risk power detect temporal pair distribute prior approximate demonstrate consistently simulate sa extreme divergence h sg variance evaluate sa glm adjust sg overall average highly cluster divergence per site mutation translate million power analysis simulate replicate exclude proportion divergence sa importantly exclude one quite many analysis narrow prior per detect among random decide address rare enough certain unlikely divergence provide much insight infer confirm work generate random divergence g interesting spurious important assessment power inference event versus look information determine mechanism prior cause divergence establish integrate produce likelihood employ parameter average light fundamental surprising tend cause insufficient would produce sample divergence time estimate present rough arbitrary strict furthermore branch tree million whereas prior implicit generation year importantly arbitrary estimate without year actually densely large number narrow toward clustered magnitude case perform slightly panel correct panel hypothesis cause look make phenomenon example insufficient sampling hypothesis create variance analysis number draw furthermore insufficient prior toward less increase see sensitivity repeat analysis dispersion index figure trend error bias present strongly preference model generate simulation choice confirm model confirm model si
robot collaborative call robot switching demonstrate sequence training learn update reward markov process preference task predefine role human task execution human action place gradually choose step part simulated take accumulate propose framework plot accumulate robot demonstrate place action deviation increase policy human demonstrate sequence therefore expert manually reward function policy use code plotted line denote accumulate validation action informative robot online capability two box box robot box axis goal robot preference rotation position position horizontal vertical box angle demonstrate subject output user vs total human human observation plane human specify team execution human team person move robot human information box robot robot type leave surface behavior human introduce stand worker associate action execution automatically learn type subject task demonstrate accumulate reward compute take collaborative offline execution show performance deviation algorithm reason policy agent compute code expert model task make enable policy human user robot robust collaborative completely without intervention unsupervise robot representative inverse reinforcement learn wherein human variable offline include human validate collect subject person collaborative robot system operate people need human group adapt style adaptation human integrate robot principle present enable robot collaborative access take completely intervention robustness deviation previously behavior base human preference actual human preference human constrain choose formulation collaborative task much partially collaborative preference action human participant infer demonstrate action human unsupervised demonstrate sequence robot reward type learn wherein type infer offline online user set compute robot align new action validate human conduct person collaborative learn human explicitly manually human perform manual work people although aspect work use type unsupervise pose kalman method ng solve program attempt match policy approach involve expert optimize margin compete game theoretic art preserve theoretical performance improve type user include robot user policy human scalar robot action encode reward decision model improve interaction rather human consume infer dominant associate researcher employ tree search likelihood behavior estimate automatically human train play human set human learn work collaborative ai player framework partially observable system human use black simulator partially decision human driving task variable human model matrix specific rule interactive recently framework learn system yet plan efficiency plan uncertainty aforementioned task automatically framework enable estimation user offline priori unsupervise dominant differ previous parameter limitation require amount robot uncertainty formulation work formulation collaborative game ai automatically human partially observable allow computation accordance preference partially observable represent preference rather partially validate human experiment policy perform code significantly collaborative describe framework stage show stage stage human preference align preference human robot reason type assume access set sequence human collaborative use cluster dominate human partially observable human framework learn human type task robot compute reason human accumulate ask execute human alternatively informative human robot block propose dominate human collaborative demonstrate robot ability human initialize assignment repeatedly execute e step converge stable complete line transition update sequence line two assignment use require select ideal use bic range line multiple initialization often consistent likelihood penalty parameter since transition iterations bic determine mixed treat value associate correspond describe reward function optimal different human robot introduce align preference serve partially learn reward preference human reason uncertainty describe learn approximately human mixed robot accord factorization observable variable load describe set observable variable task step progress toward task completion observable human observable task observable human discrete task level action x fully robot r observable state fully give immediate action human robot receive action observation receive eq reward robot want robot choose action align must manually specify consume applicability learn human belong associated type fix human mdp tuple section demonstrate action reward decision human compute represent discount accumulation feature q expectation type demonstrate trajectory type empirical begin attempt mixture expectation expert human type terminate implement monte new guess solve subject I terminate use compute back result list count accumulate function algorithm second ignore first half generate human reinforcement demonstrate sequence calculate function partially type policy human structure expensive improved planning updating belief solver scale thousand algorithm belief allow satisfactory solution applicability human robot hand task collect human share task place available position phase human give robot task execute leave demonstrate subject
cause spurious digit image maximally choose vision theory visual feasible advanced vision minor induce bar bar restrict observational image pixel increase minor nature cause behavior top label observational train fully layer create iteratively difference time green indicate successful switching variable bar h bar fail red column cause induce horizontal percentage training fail time iteration six irrelevant image pixel train bar bar behavior plot ten error whereas stay constant text detail handwritten digit terminology dataset nature contrast change part visualization digits detail binary human observer contain observer contain either image modify become originally conduct mnist digit amazon error progress successive show neural close row digits plot successive remove target image link recently field encourage image grain classification network easily adversarial attempt causal purpose extract truly causal feature causal causal discovery meaningful entail unclear counter distinct problem account causal macro micro set well macro basic graphical methodology clearly candidate cause micro science economics specific acknowledgement fellowship pp support would prove causal complex part use simple technique subset distribution proper improper observational inspire compatible causal strategy observational refine polynomial compatible equation root usage constraint additional create useful class still vary assume simplify parametrize arrange create joint n n dimensional simplex proceed proof value distribution lebesgue constraint first note fix define say image hold hold write equation term define polynomial constraint omit equation simple constraint hold equation contradict consist measure since finitely lebesgue remainder direct causal prove indicator property application theorem equal restrict joint distribution among generative form fig main text induce observational induce appear observational fixing observational express apply auxiliary observational hold observational observational full rest proof observational refinement example variable model induce causal observational agree observational causal case fix observational partition align neither zero verify induce last induce observational shannon construction note pair correspondence far small entropy causal follow causal consist binary white analogous article observational namely observational spurious class would causal e causal variable causal must information component separate experiment start ten net validation architecture layer layer layer maxout activation training use batch dropout momentum adjustment iteration decay decay column stop validation network mnist machine digit one digit set machine algorithm maximally image start ten contain zero digit zero contain create mix five image ten digit question mark digit digit target majority vote annotate digit neural completion pt minus plus minus pt axiom computation systems california institute ca electrical engineering california institute usa sciences california institute technology usa rigorous visually human neuron construct micro causal observational minimal effort provide finally image identify visual human behavior traffic subtle increase face scientific economic communication cause constitutes cause compose million variable pixel rather present theoretical framework visual image cause pixel advance target behavior macro construct micro visual cause distinguished macro variable contain cause causal thereby variable prove causal visual cause effort connect method automatically cause illustrate synthetic code implement available online cause set definition intuitive equally extract aggregate micro market finance causal learning automatically framework causal consist raw pixel micro macro specify causal micro causal latter choose criterion plausible macro variable efficiently search whole macro approach derive mechanic support incorporate cause feature behavior plain observe distribution vs investigate micro macro variable avoid distinction observational literature learn datum generally aggregation pixel causal macro instead annotate pre feature case use external visual contain vertical bar bar bar bar whenever bar construct visual cause identifiable bar image influence call presence bar causal presence bar bar value cause presence bar image introduce bar occur presence bar bar cause identify cause ability distinguish among causal causal possibly example stand illustration general spurious identification bar model provide theoretical pixel configuration account feature micro visual principle identifying cause behavior image pixel cause pixel atom table constitute difference causal relation possibility without pixel probability pixel stand relation without throughout almost except lebesgue relation causal imply provide appendix appendix extend restrict distribution partition put distribution require false put trivial agree observational proof indicate visual cause image problem observational assume causal change within inference observational region say observational arbitrarily within causal belong observational contain construction explain observational within spurious variable whose causal spurious well range spurious macro construct visual contain causal macro shannon theorem constitute small macro fig relationship observational causal due unobserved cause irrelevant structural treatment causal describe set desire directly relationship causal separate variable pixel may whereas presence absence shape image intervention use underlying variable causal intervention macro macro intervention image intervention keep impossible value change special macro causal would physical space specification develop visual behavior knowledge allow return maximally desire causal function causal ci k di search close desire causal close note candidate image causal check spurious close desire causal heuristic want offer automate produce causal effect predictive learning pose 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identifiability subset restrict stay split problem j j j mean conclude second equation hold hence set supremum collection method several distribution distribution abc bernoulli reference move integration average sample inference gaussian mean generate prior gives mean see success beta q standard deviation sample poisson distribution shape mode eq unobserve derivation helpful form follow zero since analytically integrate unobserved vector gaussian uniform normalizing integration matlab integral standard variable call consist column diagonal determinant integration integral uniform posterior normalizing matlab sequential abc get threshold choose schedule abc purely quantile thereby posterior informative posterior acceptance hybrid blue circle purely benefit hybrid yield quickly polynomial brief care center day care sample represent infect thus chain transmission dynamic care care center assume review transmission dynamic care infect evolve h small negligible small equation describes infect previously infect infection co infection parameter probability infection model static infection day care infection determine yield uniformly contact transmission number carry transmission individual care care center stationarity exact long regime model three internal infection infection infection carlo abc distribution involve review simulate care center infect individual proportion empirical cumulative summary day center form empirical summary accept distance threshold threshold adapt straight ex plus minus ex part increasingly generation difficulty latent find simulated computation indirect address choose statistic enable objective discrepancy thereby validity theoretically bayesian inference real individual day care likelihood computation modern ingredient statistical evaluating model prevent use computationally principle rely particularly suited model unobserve likelihood independence easily model unnormalized unobserved unnormalized exist alternative simulation generally simulate moment indirect approximate propose idea vector observe critical rely expert make benefit easy statistic include construction inference basic observation distinguishing set panel easy discrimination classification accuracy discrepancy appropriate leverage solution similarity become thus point bayesian continuous series htb discrepancy parameter datum circle easily distinguished rule indicate area distinguish task solve chance chance performance classification parameter chance accuracy trivially obtain employ yield probable unknown analysis regularize combination material method approximation parametric kind size move see material far set average lag coefficient quantity independent affect correlation point single bernoulli bayes generate identify average exception lda reason sensitive correlation bayes outside learn overfitte actually fold fold observe datum simulate set measure union typically call example pair validate accuracy q use discrepancy htb learn bayes rule sample size accuracy curve indistinguishable svm max perform bayes classification assume true distribution discriminant lda exception move estimation interested simulate minimize next assume parameter condition give size motivate data parametrization compact follow consistency frequency law learn classification accuracy bayes h n generate classification mean set must equally fast condition bayes condition condition solve available good rule suggest abc comprise several posterior major simulated show discrepancy abc materials algorithm empirical density function count gaussian show contain red curve approximated numerical integration supplementary match supplementary regard time tend fast max evolution appendix analysis report iteration datum series occur histogram scatter plot abc deviation posterior posterior infer binary count series result abc pdfs bivariate scatter obtain result contour autoregressive indicate stochastic epidemic center observe matrix day care indicate particular infect day care infect multiple perform statistic hand appendix brief hence classifier abc transform reflect nature well standard deviation subset material applicability summary assess reveal infer epidemic expert average generation lda second second spend epidemic infer result four classifier similar expert simulate classifier appendix datum drive compatibility abc discrepancy able incorporate expert covariate classification expert abc expert insufficient expert statistic epidemic simulate datum dash curve pdfs histograms abc expert black plus marker circle marker use generate free methodology limit classification problem whenever classify offer reveal free appear actually connect assess knowledge problem incorporate specify lead consecutive application choice automatically propose allow even random inference infect contain relate property rank fraction variability subset choose contain one size row subset dimension extract matrix run abc random rd wish thank computer epidemic model centre coin author contribution research rd rd research write text attain parameter example variance series parameter mark contour panel minimal equation unknown variance compute mean squared plotted sample decay square size line circle outcome dash bernoulli report degenerate standard lda omit since decay estimator property material provide main posterior accord abc monte univariate empirical pdf red pdf dash scatter reference contour solid contour red dash binary infer count infer poisson posterior infer posterior series distribution lag zero move infer monte carlo together iteratively prior movie process max rule bernoulli mp mp mp mp mp gauss mp mp mp mp gauss var mp mp mp mp mp mp mp mp quantitative relative deviation infer posterior distribution comparison integration supplementary method relative rule abc posterior occur plot posterior spread posterior error deviation surprising estimate variance twice estimate quantitative curve deviation average detailed material provide investigate applicability lda always since consume applicability point two keep value eliminate seed simulate seed show subset row diagram bottom discriminate localize run abc usual seed knowledge black point marker qualitatively even tail result infer infer posterior still shrink simulation iteration pdfs abc concentrated expert abc projection diagram random applicability lda time mean cross mark location produce localize region l st rd posterior evolution pdfs scale histogram solution expert knowledge code blue circle red abc subset result abc sample datum fourth generation posterior matlab default kernel bandwidth subset circle simulate generation visualization generation random circle yield square pdfs posterior qualitatively blue subset red expert black marker show compare fourth generation classifier concentrated abc red qualitatively shift abc subset work real st nd rd pdf evolution pdfs scale black expert classifier abc datum fourth bandwidth posterior qualitatively similar
diversity start eqn measurement representation survival share enable eq conditional multivariate facilitate monte data augmentation eq density posterior distribution weakly mean tn posterior avoid affected individual jeffreys half cauchy nuisance seem jeffreys prior implicitly complete whereas last undesirable scaling prior coefficient seem flat introduce weakly informative jeffreys prior hyperparameter due notion logit implicitly scale follow demonstrate small show latent identify autocorrelation accurately behind true c sim sim sim b affect basic covariance simple comparison useful progress hand martingale always reflect gain covariance association response strength eqn assess interference add vector exhibit presence start noise yet association remain magnitude reach model robust detect association response signal true conduct sensitivity survival figure large curve support well favorable treat ht record censor performance table bias adequate future besides ar capture lastly sensitivity roc clear traditional logistic nonparametric consideration also sensitivity aim subject forecasting traditional method notably use automatic widely learn spatial hierarchy process describe overall capture fine variation hierarchical conceptually goal combine forecasting incorporate longitudinal time ar complex structure possible restriction definite magnitude joint fluctuation share view time perturbation increase specificity bayesian area dimension efficient especially mixture process hierarchical instead mixture scale condition shrinkage coupling noise enable estimation extension improvement longitudinal exhibit distinct hierarchy gaussian mean hazard away skewed generalize may incorporate direction improve acknowledgement development program number partial grateful foundation patient comment h se se age infection bc infection infection diabete covariate white infection infection bc infection infection cf diabetes pt pt novel forecasting combine nonlinear past error minimize survival event hazard bayesian objective shrinkage detection accuracy forecasting function key longitudinal forecasting need clinical without researcher property fluctuation development study longitudinal past help prediction latter provide vary behind spline limitation allocation knot trace difficult avoid bar address quite smooth knot keep fast robustness form increment differentiable major progress krige predictor successful magnitude usually measure interpolation inside monotonically prediction improve forecasting gaussian process longitudinal describe batch spline batch mean individual approach greatly forecast longitudinal collect survival commonly adopt inference longitudinal song recent development survival online recurrence taylor nevertheless remain joint baseline full recurrent survival function cox eqn accommodate event collect forecast multiple correspond adopt discrete cox relative cox prediction tractable describe late approach process longitudinal survival longitudinal enable share trend among capture deviation survival smooth baseline serve share set nonlinear population autocorrelation straightforward completely fully maximization remainder survival section simulation performance apply patient remark assume process notation worth multiple independent copy copy word process subject span longitudinal censoring begin individual share correlation covariance generate example exponential replace role knot avoid ar resemble forecast obtain conditioning eqn difference process benefit illustrate figure generate subject remove ar line adequate autocorrelation cause heterogeneity fit subject common prediction prediction trajectory subject figure slow result improve h b covariance use eqn recursive forecast series analysis estimator forecast cox cox widely adopt
space integer isometry rip rank rip rank approximate constant constant desire economic leave reader pursuit calculate singular iterative practice obtain retain rate singular pair iteration tolerance matrix pursuit achieve problem denote mp mp obtain closed form solution formulation k property get complete algorithm mp completion compete algorithm singular projection fast fitting accelerated boost atomic minimum greedy three solve low problem comparison coordinate method include boost method available online http www edu thresholding http stanford edu http www stanford software http boost boost http boost greedy component http www cs il image collaborative problem compete matlab external package system intel ghz g ram follow compete recommend candidate validation mp mp stopping criterion ground result pursuit mp mp pursuit fr mp evaluation conduct mp pursuit mp mp computational inexact experiment iteration singular pair iteration curve mp preserve practice run away especially mp c mp couple crowd couple crowd exclude pixel guarantee rank mp stop iteration control list numerical couple much use need high fail htp boost mp boost mp mp matrix include dataset collect recommendation anonymous rating user rating range collect website contain anonymous rating score step size range rating rating algorithm value six dataset time mp fast among compete satisfactory scalable matrix storage rank computationally satisfactory advantage easy understand scale rigorously linear rate compare art completion compete generalize function mp weight one k inverse property remark scalable pursuit economic introduce weight updating reduce time storage complexity satisfactory advantage easy learning rigorously version significantly well real world netflix compete achieve low attract significant machine mining reference form value approximate span vanish outside matrix attack cf widely relaxation rank trace generality sorted base singular al develop algorithm penalize ji improve number iteration accurate singular propose investigate property addition trace low rank computationally expensive singular svd truncate svd large accurately attention recent solve wu question solve progress ik trace problem et improve efficiency although scalable singular efficiently iteration algorithm et solve problem reduce without theoretical involve compute lie refine apply frank size refinement information slow update constrain optimize solve consuming sophisticated refinement lead computational cost algorithm type weight lead boost learn alternate drawback inefficient slow rate refinement iteration heuristic affect speed improper even extend orthogonal omp residual standard update matrix consume algorithm operation appeal propose orthogonal pursuit orthogonal sub extensive rate iteration achieve drawback store rank current weight update storage tackle economic version iteration restrict observation keep economic retain convergence knowledge fast among verify empirically problem netflix main contribution paper computationally orthogonal matching pursuit theoretically accuracy solution need singular pair storage economic linear rate version storage scale matrix version rank converge operator onto span outside equal zero observe keep inner matrix equal frobenius economic linear extend sensing present isometry property empirical evaluation verify effectiveness matrix unit choice original approximation zero eq nonzero solve orthogonal pursuit algorithm coordinate achieve unit current observe follow notice rank unit frobenius product see reformulate eq singular computed basis matrix available least vector row entry multiplication simplify incremental implementation desire stop terminate method rank prefer alternatively residual orthogonal matrix mp stop pair right residual output pursuit construct onto subspace obtain current lee et step first choose svd use svd rank expensive difference omp recovery property assumption I e involve isometry convergent achieve orthogonal one proving say one independent formula stop suppose full statement indeed conclusion induction hypothesis column follow fact rank next relationship convenience define view k l r property fact uk observe eq last triangular inequality easily conclude equal index noisy mp help remove study orthogonal matching omp right singular matrix well mp basis save storage adapt need weight matrix square weight basis ki procedure simplify
locate view view anomaly single view machine anomalous develop view anomaly detection cc dd view anomaly detection application behavior aggregation database information management multiple view language wikipedia view anomaly find document language item anomaly movie movie probabilistic latent anomaly latent share anomalous generate use anomalous assume latent latent view space share view projection view view anomaly consistent projection infer give propose step use collapse step matrix map estimate likelihood iterate vector use instance robust assume latent suffer anomaly contrast anomaly latent matrix properly anomalie security analysis exist anomaly detection two view horizontal anomaly view simultaneously low embed close together spectral location view hyperparameter constraint label anomalous sensitive label extend principled handling combine multi method anomaly filtering instance corrupt view anomaly instance different foreground view instance inconsistent cluster propose probabilistic canonical cca view anomaly share view maximize minimize reconstruction anomaly score non anomalous anomaly parameter anomaly factorize assume every shared instance anomaly detect anomalous latent anomaly anomaly anomaly view observation th observation vector view anomalous inconsistent across multiple variable instance generate view projection select latent anomalous view consistent assignment view anomaly view inconsistent propose infinite view th mixture weight denote dirichlet prior weight use automatically infer instance generative propose precision draw mixture nj assignment observation nd stick break weight precision share integrate indicate latent analytically integrate vector precision multinomial prior factor view integrate calculate q q q either principal view different every view view spherical describe collapse latent analytically integrate weight integrate need notational current exclude th exist latent indicator subscript indicate latent eq intuitively latent view inconsistent view projection logarithm maximize quasi iterate e assignment view latent assignment matrix score use use sample inference iteration cross dimensionality space cross validation predict randomly anomaly evaluate create anomaly anomaly consensus anomaly model latent anomaly score base reconstruction require value control whereby different view embed together cluster view cluster score remove representative anomaly anomaly multiple kernel gibbs sampling area auc anomaly average propose performance datum anomaly infer latent instance view anomaly auc detection instance anomalous anomaly instance inconsistent inconsistent view need datum view case view propose anomaly auc vector dimensionality ccc breast diabetes heart breast diabetes anomaly vector anomaly anomalous anomaly multi anomalous non anomalous respectively observe latent three anomaly average different anomaly anomaly auc class anomaly lead anomaly propose propose imputation multi view anomaly estimate average feature instance randomly imputation whose datum effectiveness propose anomaly breast cancer diabete propose evaluate propose model view anomalous anomaly latent dimensionality generate mean vector show error imputation synthetic decrease rapidly anomaly indicate dimensionality square anomaly anomaly anomaly show model carlo latent high rate anomaly anomaly rating movie view represent movie user second movie
roc replace loss pc optimization could motivate manifold equip roc arbitrarily wise tv tv generalize universal choose deal outlier type basis roc differ way introduction reveal outlier inherently cut residual choose pattern roc pca loss asymptotic bad study point bias trade tune non quadratic optimization roc pca easier expensive utilize loss sparse shift outli loss bernoulli hinge visualization sometimes pc order importance pure simply complete robust roc benefit free robust reduce roc roc pc yield roc pca repeatedly roc challenge orthogonality constraint alternate manifold iterative nonlinear reduce way instead treat introduce lagrangian multiplier view problem manifold manifold orthogonality update tangent p manifold begin riemannian f difficult show valid trial determine trial due lie pd lemma fast formula turn batch low proper size bb search result quick nonlinear scheme backtrack aware trial point bb solve f f solution behavior bb ensure stepsize small criterion recent recommend important start run convergence type roc ps involve sparsity induce algorithmic solve denote ds thresholding let satisfy justification continuity uniqueness various thresholding cover penalty couple general summary complete roc pca e roc output line bb monotone k bb else see k wants directly form roc similarly e roc unless compare number sensitive subspace recovery long supplement ridge grid simply say constrain roc share fortunately estimator subproblem thresholde q constrain pca problem gs gs monotone integer decrease empirically pca great establish difficult asymptotic wish robust challenge modern tool approximation roc pca r supplement ignore intercept outlier formulate eq ambiguity always evaluate assumption globally oracle inequality p q hold gaussian include dependency expectation high probability obtain see supplementary detail nature apply incoherence commonly assume sharp infimum en minimax conclusion sparse contain reduce attain roc outlier great give estimator p furthermore show essentially ij p rp multiplicative therefore roc pca essentially tend real life roc sparsity algorithm dimension review material burden fail principle widely perform svd reduction outlier occur dramatically long value exceed true outlier outlier relatively greatly course effect removed estimation roc well slightly acceptable robust seem balance c c c focus subspace roc pca plain pca review spherical pca already integrate inexact augment lagrange multiplier test presence outlier class approximation concern subspace estimation conservative h c c outlier skew direction handle outlier behave level occur roc pca behave extremely perfectly pc except give extremely pc affinity observe dataset due sampling trial fall chance computational report table roc procedure computational robust great outli affinity run although roc pca run superior especially c roc performance type fail c give three randomly entry otherwise outli pca good job affinity set careful result check subspace propagation much try ij section implementation take threshold pc affinity increase roc pca sample way four outli batch try follow pca standard roc computational roc roc pca segmentation collected extract seven hand window contrast adjacent pixel outlier assess goodness call adjusted account top mark first run principal seem panel relatively change yield intuition also plot e observation find class roc automatic intervention roc pca segmentation among detect scatter help separate majority pc roc clean detection outli pc pcs pca pca l sensitive may reveal observation mathematically rise roc pca come guarantee robust computation regularization reveal roc pca minimax topic include optimization high direction jointly investigate observation anomaly independent outlier skew subspace let wise roc pca fix remain minimizer manifold canonical necessarily however detail omit proof universal constant occurrence space stand onto abuse notation dimension j j p term general sub gaussian dimensional random bound bernoulli assume mean r cs submatrix r r j p p j lemma follow let p dd c follow satisfy np sr sr q clearly b q applying row follow cf exist b universal model kullback ip mn apply rr pn r cn r line omit advantage instead repeatedly rank loading vector lemma recently attract attention computer study apparent original novel orthogonal complement analysis pca combine enforce rank element guarantee roc tackle basis optimization iterative efficiency real supplementary big arise pose challenge computation structure tool pca characterize rank solution value tr principal function know call life statistical inference break result subspace estimate pca extensively robust statistic lot researcher statistic outlier norm zero facilitate wise variant application video e outlier original call major htbp analysis attention affect subspace toy outlier identification short interestingly subspace outlier though possibly pc handle later stage raw coordinate space panel sample point whether show outlier skew pc unfortunately check coordinate offer help reveal propose complement roc recovery exist approach roc pca aim involve enforce establish roc review pca robust section thresholding roc pca roc pca extensive numerical datum conclude detail issue pca notice statistic introduction five covariance method limitation review representative work complete pca robust loading eigen dispersion monotone function gain large robust give repeatedly estimation run largely keep value cost matrix dimensional matrix accommodate portion observation contradict robust achieved simultaneously identify worth pc estimating unnecessary repeatedly direction project obtain idea systematically pursuit algorithm computer c direction center recommend make reduction svd reduction text unfortunately serious time new pc projection estimation apply run fast class un moderately high trial rely may acceptable application first deal element outlier observe dimensionality subset direction pass two datum point far reweighte share property excellent simulation yet suffer aforementioned drawback robust handle wise anomaly may efficient svd may problematic lie restriction difficult accurately trial direction subspace contrast purpose direction direction fail spherical elliptical pca relatively center onto plain elliptical spherical preferable recover simultaneously spirit step toy example intersection angle pc however spherical direction bad pca
result obtain pa query complexity far uniform condition achieve condition super speedup open stochastic complicated circumstance appeal evolutionary include genetic algorithm evolutionary cover nature inspire heuristic estimation etc study rapidly decade analysis development theoretically investigate combinatorial measure develop cover budget analysis case general even application nearly general give conclusion domain run exponential bound general derive performance learn analysis notice share consist cycle building commonly reproduce solution quality guide sampling portion capture framework simulate sampling strategy evaluate probable pa pa count fitness reach solution close pa upper bound version pa incorporate compare polynomially search polynomially far allow super polynomial notice necessary version iv search paper euclidean bound close continuous exist sake convenience generality besides mean polynomial grow minimization continuous compact assume loss domain implement every achieve good enough quite correspond two pa fitness evaluation take reach quality reflect intuitive evaluation probable approximate pa call find em generate solution new short global heuristic solution framework start record far solution search cycle learn hypothesis mapping via iteration empty solution well space balanced region balance uniform tt ts tt ti tx x fx note concrete summary stage accurate could explanation rigorous illustration various genetic deal solution vocabulary element mutation select vocabulary operation mutation mx change commonly otherwise solution search x simulate ga ga circle area hx mx mx use approximate polynomially way behavior ga simplify ga probabilistic argue base optimization entropy unify building framework correspond sampling step particle particularly simulation perhaps sophisticated set particle velocity determine current velocity globally good particle initial distribution velocity contain globally learn utilize set hypothesis globally capture search leave heuristic general output eq xx sampling hypothesis need size implement sample naturally bind iteration solution sample event whole failure sampling learn expand overall failure belong tm furth term simplify employ stage transform set current putting priori knowledge performance meanwhile small performance usually baseline serve search word uniform search pa query much average rely learn investigate accelerate search approximation event event condition expect error rate numerator second equality error average learn uniform within optimistic plug u kl u lemma ignore logarithmic find exponentially use bad search ask sample worst keep query meanwhile n super optimistic note obtain face barrier search still sphere sphere volume c f n straightforward pa probability search meanwhile consistent feasible note td ti td td tx x x prove obtain pa approximation query use sphere volume algorithm simplicity error obtain letting algorithms iteration eq n accelerate uniform close acceleration complex sphere algorithms optima show spike spike spike differentiable optima pa algorithm search sphere cover sample label dimension cover thus member pa proposition approximation achieve query error follow uniform algorithm every query complexity proof error convexity severe like spike function significantly affect still show algorithm super polynomially improve question therefore proof way powerful improve I super error circumstance modification nevertheless achieve learn proposition super acceleration sphere complexity choose use sphere affect error require tm obtain letting meanwhile explore require powerful condition imply error inside cost sensitive mis algorithm note label negative false negative error side control refine eq since algorithm behavior result use complexity probability sphere result affect iteration want produce
refer probability refer unseen datum type iii gaussian mixture assign calculation error know select method convergence normalize marginal likelihood direct relation asymptotic expansion marginal allow error resolution asymptotic study likelihood variable probability probability bayesian observable analyze kullback generating analyze selection regular expansion maximum regular case method case derive type iii regular analysis estimation expansion reveal order iii regular result determine advantageous estimation type remainder organize estimation result estimation formulate latent variable express learn hierarchical true express refer maximum estimator respectively maximum unseen posterior posterior normalize true include define latent type estimation define bayes estimation q likelihood eq estimation kullback true error expectation ii error eq present publish expansion estimation method expansion estimation difference information converge variable true two refer since provide label converge analysis label avoid analyze prove expansion notation ii indicate respectively corollary compare I method denote nn saddle taylor obtain average last eq form eq eq base rewritten let confirm relation prove find asymptotically advantageous type estimation error bayes accurate estimation advantageous estimating latent observable estimation bayes method variant type ii target constant integer q q numerator denominator type ii let observable let target q panel show type maximum likelihood respectively advantageous lemma type function follow bayes type iii follow advantageous target previous subsection advantageous estimation follow prediction observable estimation likelihood estimation let give leave panel target target show obtain error bayes estimate observable expansion parameter proof obtain relation estimation use bayes numerator maximum accurate mathematically explain comparing find I accurate prediction variable write iii rewrite eq formal form type ignore likelihood method estimation bayes kullback decompose thus target term variable estimation target confirm accuracy method present accuracy latent type iii variable estimation indicate equivalent advantageous estimation type iii research foundation equation omit equation corollary theorem datum hierarchical often kind part measure investigate latent asymptotically maximum bayes variable estimation regularity satisfied indicate accuracy equivalent advantageous estimation keyword machine learn datum science represent underlying example observable consider unsupervised observable component unknown often way likelihood method
g semi separability important computational matrix operation energy separable term particular notation hamiltonian h well energy energy similarly h energy energy term key exploit separability describe invariant alternate simulate hamiltonian dynamic crucially though separable return per step discretization correction simulate preserve need mh correction preserve reversible reversible see reversible transformation emphasize discretized semi hamiltonian start joint hamiltonian point discretize small seem actually omit hmc important auxiliary hamiltonian one step auxiliary accelerate see word auxiliary potential hessian hessian approximate hessian hessian distribution rough main difficulty correlation bottleneck hmc target computational cost iteration beneficial cause bottleneck gradient aforementioned curvature strongly give nearly mix separable ess mse hyperparameter notice efficient level adaptive illustrate hmc hmc suffer hyperparameter narrow contrast hyperparameter variation soft effective per ess mix neither hierarchical hyperparameter ij th dataset logistic benchmark hmc group dataset hyperparameter gibbs fisher use I g much high stochastic ar observation px update sample approximately almost many ess another benchmark prior function j di mx j posterior update iteration general step update update extremely slowly sample latent hyperparameter consistent method effective hour tb tb ess min ess min ess latent hyperparameter ess ess hour version riemannian manifold hamiltonian monte retain flexibility difficult several model outperform hmc computation hmc minus plus pt pt minus pt plus minus pt hierarchical hamiltonian advantage introduce use design allow hamiltonian exploit mix fast simple sampling instance natural complexity control overfitte complicated allow pooling dataset allow little chain hamiltonian hmc hmc note riemannian hamiltonian aim exploit property computationally problem simplify decompose allow large move hyperparameter term previous sampler practical datum datum distribution ii draw hyperparameter group allow statistical strength cause hyperparameter usually control cause difficulty sampler illustrative distribution nx illustrate generate hmc ergodic invariant auxiliary gaussian three step simulate hamiltonian metropolis mh correction define h hamiltonian dynamic physics decompose discretize discretized identity often outperform popular slowly correlation distribution mix act recent varie position result call hamiltonian al fisher manifold hamiltonian dynamic long non separable system hmc simulate hmc hmc within gibbs hmc mix slowly huge variation easily separable pose challenge hyperparameter jointly joint
selection generalization experiment synthetic previously high dimensionality pose computational challenge fortunately expect sparse small feature weight classification classifier phase solve elastic doubly elastic regularization perform feature selection grouping correlation microarray expression fmri algorithm take terminate hybrid algorithm proceed phase use svm average second classical add regularize regression elastic net select e grouping enforce attractive text hierarchical regularization benefit svm history back inexact lagrangian function equivalent constrain call variable splitting iteration augment lagrangian lagrange multiplier converge optimal subproblem minimize augment view subproblem solve minimize split admm auxiliary objective optimize sequentially iteration follow update lagrange multipliers original decompose subproblem hinge loss soft thresholding lack associate interior method class qp need define c iy consider kkt svm svm optimal solution index vector term compute give svm likely dense whereas svm identity primal large argue svm p identify realize implement qp mind objective admm illustrative non converge approximately plot early remain offer sign terminate adopt relative surrogate evolution index warm admm solve decision although reformulate qp norm already enforce reduce helpful form comprehensive unknown population result unseen arise abundance feature produce accuracy likely identify domain knowledge feature include relevant feature explore associate information equivalent qp implication show utilize homogeneous implication represent add constraint p number linear constraint feature classification want undesirable increase convex qp hence incorporation ready novel combination combined framework motivation exploit good knowledge kind elastic net svm incorporation admm method unconstrained inequality hinge n tw norm obtain p p minimize individually constraint augment involve solve q ty ty ty iteration fact minimize augment lagrangian convex crucial efficiency novel introduce slack part augment subproblem system q kb full reasonable constraint solve cholesky factorization observe element constraint k p k yx subproblem subproblem solution lack summarize admm appear additional parameter four admm svm practice one computational experience fairly insensitive phase solve base decide primal dual reveal formulation slack transform equality qp formally experience knowledge counterpart e admm admm k test nine publicly two class article auto real seven set subset sample instance test instance profile breast cancer cancer fmri brain activity except fmri website summarize experiment clearly produce test significance time outperform size competitive able original c c support cpu cpu synthetic presented begin dimensional specifically variate matrix diagonal else training contain four entry sample generate block two close entire population test know value block membership l denote information precise confident negative mean exceed distribution domain tends often exact two train train feature want achieve explain expert knowledge hyperplane generalize entire significance accuracy support cpu phase solve extension solve demonstrate advantage prediction general enough classification cc x tx x ty ty ty k yx k b k k b k n yx yx k
invoke primal lp dual eq probability since return rewrite gs uncertainty generalize note cone satisfy either give hence empty interior pick enough hence equal counterpart measure conclude present payoff shall proposition regard one round game risk neutral operator rewrite primal equivalent redundant gs gs immediately trivially round amount return lemma proposition game uncertainty g p f unique neutral value arbitrary arbitrary statement linearity expectation write give q price prove statement obvious induction straightforward gs set u lemma round track backward exponentially dynamic programming computation option corollary uncertainty binomial go number interval decrease imply uncertainty black convergence price european option round game lipschitz continuous european option geometric drift positive motion recover price convergence condition guarantee keep corollary upper tp term fr weak triangular hold lastly condition f gs gs r integrable end f simple game equilibrium general round concave uncertainty u gs gs concavity preserve induction get bound follow analogous initial price payoff hand measure convex payoff option game concave gs te prove backward obviously contradiction existence immediately next u continuous neutral contradiction exist say exactly bind three satisfie impossible let move infeasible infeasible infeasible imply finite pp trivial see bind uniquely define exactly bind constraint lp infeasible contradiction may lp compute either furthermore focus follow length exist ir ir g tx program lp solved analytically let decide consider similarly continue get mx r r r r tx small g r r lx lx lx use r lx r lx tx lx lx l european options convex induction corollary neutral measure th second convex conclude induction p tx r induction define backward induction note proposition I I rewrite let x integrable x gs uniformly integrable provide price black model equation pde characterize demonstrate pde rigorously pde model bellman informally denote use control assume moment xu x follow pde pde sense coincide solution verification pde optimal control application eq denote expectation satisfy denote take control black write boundary reduce ordinary pde whose lie theorem payoff natural allow movement community jump motion asset adopt adversary jump control jump extend adversary magnitude jump price payoff adversary jump assume adversary small modification present need model function perform occur move return freedom choose sufficiently small sequel single round asset formulation q write measure concentrate enough w w round analogously dynamic replace jump diffusion study demonstrate appropriate return consider jump option price jump diffusion poisson adversary jump throughout round adversary jump uncertainty set ordinary assume polynomial require upper next choose lp formulation round similar neutral characterize feature neutral measure comprise ordinary uncertainty whether jump dual occurrence whether reduce merely dynamic define approximation scheme length backward induction polynomial achieve pay run enumeration set adapt easily highlight first manner continuous binding reveal tx tx xt convention similar use probabilistic arrive section prove uncertainty payoff subset problem integer count integer equal reduction proceed count sum european option uncertainty payoff ordinary neutral round price factor move let measure coupling tree trajectory round next express price equation different let corresponding price similarly reduction lower justify necessity price single round query payoff monotonic lipschitz difference price word explain purpose least point easily relax j u dual neutral complete argument section property invariance proposition definition liu department science pa address address need financial option game adversary price finance european payoff option construct pricing demonstrate introduction artificial neutral measure finance extension incorporate jump payoff price type option limit european american type substantially explicit trading replicate option american design option payoff regression non pricing degenerate dynamic contract asset european call contract stock york stock exchange price european call option stock price price stock abuse price stock convenience exceed rational exercise stock immediately market payoff single european option payoff asset depend contract american option exercise price option core economic market practitioner trading area discovery expand order option movement asset european option choose portfolio consist asset portfolio portfolio must european otherwise positive risk fair option original price underlie asset brownian may unique example stochastic option pricing control movement stock motivation probabilistic description always market see capability broadly speak systematic context answer stochastic finance community question question option european payoff acknowledge literature limited example et al et convergence european adversary convex payoff line limited option al consider option european american wide model price movement non imagine show case secondly non convexity american pricing algorithm major contribution whose lipschitz involve construction artificial measure error neutral commonly use kind appear structural american option pricing thought adversary allow online contribution constructive pricing analyze option payoff monotonic nature artificial measure way deterministic extend convergence american whose pricing thought adversary online besides contribution extension include convex black option adapt algorithmic rare jump financial market important smooth price movement financial market sort pricing work chen formulate pricing robust asset dimensional typically theorem assume computational feasibility issue al recently price constraint variation asset et asset adversarial maximize gain round price movement european option price black price geometric round infinity call market study leave replicate option pricing unless american option payoff call option option payoff algorithmic show major market cm cm cm p european yes hard new european yes american jump yes yes function result pricing general discuss price jump generalize american option present hardness section model equilibrium convex pricing payoff address generalization extension american option jump present hardness discuss spirit discrete chance specifically consider option day transaction option total soon decide value game limit cost market always asset market price asset round round freedom return notational convenience begin asset position impose capital shall soon describe option price illustrate interest result zero rate option game worth asset payoff gs option get option rational adversary outcome gs option price strictly gs option give gain arise time obtain asset th gs argue option price interpretation fall strategy adversarial payoff strictly price option option adversary payoff positive price option trade adversarial strictly wrong option trading strategy adversary payoff access payoff leave set later allow dependent simplicity highlight merely equilibrium contribution come center start give shall adversary scenario characterization general low merely objective analyze equilibrium game induction benefit evident linear allow american options european american option limit derive guide borel gs maximization probability satisfy neutral widely contain interval cover risk construct minimax rp see detailed proof despite result case payoff convex uncertainty able characterize hand risk neutral generalize result game coincide suitably complexity extend option analyze pricing algorithm result correspond price geometric rather present payoff function omit end full payoff decrease use fairly allow adversary arbitrary allow discrete multinomial recursion th round let b adversary lp option price discretization backward induction stochastic area financial latter regard adversary choose way move long round track round price could multinomial exponential overall due discretization grow algorithm portion multinomial additive recursion interpretation formulation elaborate resemble mesh option notably american asset movement risk neutral mesh backward monte replace lp explain multinomial tree price give formalize building block continuity prove suppose decrease lipschitz building follow assume view hybrid discretization characterization bind lipschitz satisfie rely neutral measure several backward induction measure correspondence artificial martingale asset movement see effective propagation first error second come future handle second characterization proposition share write become recursively asset expand martingale bind thus arbitrary multinomial price bind shall appendix long run essentially tight set multinomial uncertainty though parameter remain unchanged still generalize price american include neutral algorithmic european single round upper american option make adversary exercise otherwise remark adversary right exercise upper upper argument nature hence early exercise gs gs focus gs gs gs r gs single depict significance characterize solution term neutral european option maximization expression characterization round consider american option option european counterpart exercise detail appendix use payoff recursion
pair event use denote query point belong range strictly r write use put everything follow hand finish proof get suppose sample use condition depend return begin note continue argue occurrence choose equal one function least return ds free get suppose uniformly random mean exist exercise second claim argue check orthonormal written follow I lemma corollary proposition support grant science united science foundation grant center program center foundation grant recognize limitation size training example cost mostly lead non trivial loss parameter desire case kernel learn might consider label domain kernel minimize possibly reproduce hilbert vector machine hinge hilbert space employ hard polynomial efficiently insight span reduce define q convex polynomial implicitly predictor hilbert size lead learning attempt know algorithm fall specific dominant learn entry matrix attempt number require read instead full map kernel see technique exist question surprisingly knowledge learning example kernel maintain learn always pay finite match study give type matrix generally assume much small number low make method previously although assume conclusion informally impossible make kernel learn suppose evaluation sub method uniformly away away datum substantially rank low impact go kernel predictor nature particular evaluation constraint soft regularization attain corollary loss attain although identify evaluation budget hinge loss square unless ridge regularization attain role loss recognize key role discuss section appear loss highlight kernel recognize relate sparse nystr om row budget perceptron sequential early algebraic rank work kernel work access interested predictor learn study complexity e support focus organization organize class constrain term consider constraint regularization without discuss different type bound consider conclude open appendix utilize essentially permutation formally block entry column immediate hence diagonal mean unit ball hilbert focus generic hardness still induce quantify follow permutation linear also approximately kernel close inspection truly distant fulfil fulfil presentation boolean although proof contain technical intuition approximate suitable approximation relatively budget detect block integer return approach reduce evaluation need standard require learn demonstrate absolute average equal parameter coefficient require subsection satisfy bind parameter size budget different way phrase find formally exist universal kernel return vector give budget fast set uniformly replacement get j j sample set vector coefficient essentially original drawing enough solve guarantee loss error match corollary number away moreover small degradation ask study use reduce main interested draw corollary negative parameter exist target appear roughly change optimum thus seem domain matrix find eq partially get thus optimum regardless show loss learn trivial examine detail get corollary absolute loss exist universal constant target lower verify bind theorem former match sub sample minimizer convex note unlike corollary whether technique corollary move change regularization loss location may hinge loss location universal verify long assume get dd quantify constant error establishe range without since strongly scale lower emphasize attain hinge regime consequence budget norm want algorithmic require attain hinge pick corollary different evidence hinge natural consider smooth differentiable square universal target low attain absolute essentially kernel evaluation sub error learn budget portion readily verify lead lower consider pick mean equivalently b latter former complete discuss explain early nystr om approximation feature typically depend result low low ridge soft ridge operate dm rank exist return sub require sample scheme tight theoretic focus use various loss conclusion case attain well trivial small optimistic bound substantially although know weak low tight exploiting result stochastic believe applicable optimize risk respect underlie obstacle basic subsection close sample instance training sample question tight know rank extend possible extend query respect randomization loss discover research program machine centre de support consider define certain uniformly begin equal diagonal block one compose block correspond sized block proof intuition entry block randomly algorithm
paradigm likelihood enjoy mle intractable common heuristic unfortunately iterative minima optimum achieve common alternative heuristic minus optimization feasible take extra lie round find often focus namely program sdp pose orthogonal cloud orthogonal ia gaussian e mle minimizer mild orthogonal refer ta unfortunately exponential intractable note c dc semidefinite relaxation drop outli recovery nature infinitely impossible even remarkably even often recover mle camera motion understanding hold distribution analyze lie fact mle dual carry various plot correspond reference alignment treat certain observe recovery formulate support conjecture ia gaussian entry solution alignment signal observe copy sake reader happen level mle considerably positivity also recovery vanish
art ap contour visualization work contour detection fine technique effectively overcome limitation sensitive pixel require local interesting direction examine contour acknowledgment go edu contour detection classification facilitate efficient cnns contour challenge per per base patch contour effectiveness performance fundamental vision recognition explore intensity texture take structured forest efficiency boost effort pixel perform contour construct pixel learner adopt convolutional cnns approach subtle deviation cnns feature image distinction perspective pre per image cnn imagenet adapt new model pixel edge classification convolutional ensemble contour follow contour section detailed technique research effort achieve survey present picture progress two area emphasis early contour detection image amongst orthogonal contour detailed discussion subsequent identify increase apart detect local technique notable address independently patch analyze combination mid contour sketch token forest classifier pointwise mutual contour clean contour current contour deep architecture encode contour restrict machine neural fine mechanism adapt imagenet pre convolutional produce detection benchmark cnns cnns cnns feature exhibit hierarchical perhaps implementation cnns generic computer vision problem cnn image cnns cnns cnns sliding cnn deep art cnns recognition demand contour detection exploit contour explore per pixel independently generate cnn feature employ architecture effective technique focus yield image patch design appropriate investigate characteristic detection point convenient multiscale contour detection feed machine classifier multiscale pyramid extraction neural extract per pixel feature feature convolutional conv convolutional conv stack convolutional pixel convolutional level pixel illustrate pixel pixel contour feed include local contour structure well distinguished neighboring pixel test correspond image patch conv add softmax edge image train cnn conv cnn except plane pixel edge conv properly contour propagation label patch training edge patch relatively alone usually still address edge evident certain distinguish learn database softmax cost cnn prediction layer bias edge reduce log fine sensitive fine rather directly back convenient strategy create bias positive twice non vice versa cost sensitive fine ap train baseline traditional pixel tune tune tune ap conv conv conv conv imagenet negative fine tune heuristic branch bind coefficient capture aspect fusion various learn test berkeley assess tuning technique detection competitive also demonstrate de contour detection contain validation image worker average contour f threshold precision ap non tuning server gpu set imagenet pre softmax ten modification pixel tuning finish pixel require fine fine tuning pixel train positive fine boundary per negative various tuning conv carry classification softmax observe fine experiment pre fine architecture traditional image fine tuning pixel fine tuning improve sensitive fine fine negative possible boundary boundary
mmd valid unbiased mmd basis function size method repeat mmd permutation consistent test htbp mmd mmd deviation approximate unbiased mmd mmd correct evaluate efficiency varied fig number varied compete except method mmd gb ram gradually become efficient mmd exact mmd extreme subsampling mmd increase slowly trend validate efficiency vision contain object set toolbox extract construct bag word pyramid codebook spatial pyramid feature image mmd discrepancy data pc cpu ram compare mmd mmd repeat standard execution meanwhile speedup linear mmd mmd mmd std second speedup mmd several strategy employ strategy utilize integrate task distribution describe bandwidth mmd bandwidth approximately match scale variance distribution method mmd level bootstrap type error drop quickly demonstrate empirically bias mmd eigenvalue vary four exact gram matrix two former latter utilize execution comparable efficient except mmd method bootstrap mmd determine statistic base calculate invariant kernel advantage method theoretical explanation one side intrinsic mmd mechanism side metric include find threshold thank valuable anonymous constructive comment support national china program cb china project china mmd unbiased mmd reformulate proof equivalent tool mmd linear df distribution bounded f j claim substitution side function fx minimum iii finally ga ga verify fourier use mmd procedure mmd equivalent geometric explanation mmd mmd calculate fourier pi I lk eqn eqn convergence prove mmd virtue certain eq uniform theorem dr jk net hoeffding net ii accord literature eq claim reformulate combine mmd unbiased mmd pdfs random maximum frequency accord hold cm circular discrepancy institute usa university china discrepancy mmd abstract maximum discrepancy mmd recently statistic sample quadratic however scale accelerate mmd calculation mmd take sampling fouri time mmd approximate determine accuracy convergence theoretically unbiased namely circular understand mmd extensive metric assess experimental mmd fast mmd test range use accept reject hypothesis since unknown mmd design measure distribution embed reproduce kernel hilbert recent test successful application biological datum attribute mmd family supremum kernel selection albeit various mmd pair assess greatly speedup mmd year mmd subsample extremely mmd possibly mmd bring high variance mmd recently split correspondence exact mmd omit inter block change smoothly mmd experience actually coming efficiency throughout effort accelerate development speed datum development mmd constitute branch research attain task utilize subset however accuracy mmd paper mmd implement summary fold employ theorem mmd mmd scale moreover sequentially utilize sample mmd mmd result mmd variance theoretically prove unbiased mmd kernel viewpoint extensive metric available mmd shift mmd consider nx mean discrepancy usually ball rkhs laplacian characteristic p prove empirical x I space empirical mmd mmd accelerate especially invariant kernel classical underlie approximation borel fourier definite value proper gaussian p view multivariate identity measure substitute thing harmonic theorem amplitude combination amplitude calculate time fig combining calculate mmd also unbiased mmd uniform bias mmd mmd spirit kernel dirac delta function calculation still statistic quantile approximate pearson first three moment mmd geometric circle circular explanation extensive metric mmd eqn project determine variable kernel investigate circular fix sample distribution sample circle pdfs two mathematically modular arithmetic dirac delta measure closely mmd circular assess circular claim dirac delta circular discrepancy provide circular close mmd claim amplitude combine mmd claim definition circular dirac delta eqn sign clear two maximize two construct project unit explanation circle angle circular discrepancy aim largely separate sample show diameter zero circular diameter maximize htbp angle light dark color circular discrepancy spread gaussian uniform intuition suggest tend dirac point ensemble circular gaussian circular unit shift invariant eqn circular discrepancy approximation circular eqn define kp normalize eqn
fast frequent evaluation make bad period slow lack long sufficient stage implement compare prox sg proximal dual power prox proximal prox accelerate prox method fista search scheme prox sag version sag prox sag demonstrate prox sdca dual ascent complexity need show prox prox sag prox three perform well prox sag prox prox sdca prox sdca complexity analysis complexity sag obtaining iterate regularization prox sdca prox sag quickly follow full prox sg show list include prox hybrid prox sg switch prox scheme improve substantially similar hybrid sag behavior method prox prox perform parameter worse much slow sag prox proximal prox two average large general average structure extend reduction compute modify gradient reduce prox enjoy complexity sdca sag recent achieve improve component substantially write proximal associate combine fy fx rx py tx rx fx tx use collect product tx ty ty ty put together arrive prox define mapping k rearrange inequality drop nonnegative dropping left minimizing convex problem arise machine know method use multi scheme gradient converge rate gradient problem many simple interested case advantageous incremental stochastic operate single know least component choice include net nonnegative regularization class close formulate set e mixture soft hard possible present base rx gradient overall I exist large may come either although must step specify definition proximal gradient compactly view case accelerate variant component component also average prox randomly take kx fx prox sg evaluate prox introduce sg much prox fair iteration step iterate satisfy see interesting ratio prox need prox gradient evaluate find accurate accelerate prox complexity hand prox sg sublinear prox sg expectation prox sg efficient low solution vast sum prox randomize incremental algorithm typically require survey prox sg prox prox sg structure room several work special develop component evaluation significantly superior prox sg zhang conjugate exponentially size overall computational cost increase batch gradually reduction technique batch maintain prox sg iteration whenever update gradient modify index pick replace prox sg kx f fx I variance use prox imply eq optimality I fx rx rx last convexity prove respect I q fx k k f k kx k f kx f q ix last inequality apply twice convenience need lemma well close q low slight completeness lipschitz continuous define g x fx step prove g x k note prox still proximal independent k first schwarz inequality eq q sides inequality independent addition q sum l
large wide variety machine finding validate importance rare reasonable priori use training cross resample separate fitness repeat final overall efficacy resample cross bias variance bias traditional fold cross small research repeat fold comparison denote induce resample complete optimize resample split fitness relationship characterize final setting entire approach optimize generate determine historical assessment goal choose model e neighbor versus near focus accurate assessment manuscript focus model acceptable precision illustrate predict compound molecular compound predictor descriptor analysis count molecular surface area size assess svm model radial tuning radial resample use tune hold perform resample iteration roc curve roc fitness figure cost fitting peak reach begin become winner strategy sub area curve determine treat equal substantial highly unlikely yield parameter lead computation computationally drastically tune predictor resample step efficiency major time manuscript acceptable value parameter fit analyze simulation characterize efficacy efficiency adaptive resampling resample value unlikely choose avoid clinical trial clinical trial objective unlikely clinical involve pre trial substantial detect specify effect well perform fitness value particular concept applicable assessment tuning fitness use value unlikely consideration long setting resample continue maximum resample process would continue fitness precision nominal still consideration algorithm predict break consider detail fitness well describe assess resample resample contrast treat error interest statistical test strong fitness correlation fitness split tend another compare fitness likely inferential statement inaccurate estimating parameter model comparison adaptive manuscript side current comparison secondly attempt positive context correction nominal manner instead directly correlation iteration diagonal covariance compound within effect equation slope parameter cell current condition parameterization performance numerically equivalent reject rejection bad resample tune whose remove subsequent return sub occur roc performance fix compute whose resample model remove remove usual winner select quantify time adaptive procedure speed execution process fit svm result multiple tuning distinct drive linear assumption since curve unity resample leave skewed generalize multiple normal residual develop consensus characterize tune across set purpose tune decompose comparison compare setting converse true number tie handle team set loss usual manner tune interpret odd approach use associate reference consequence large may consequence estimate magnitude remove model resample use tuning side ability asymptotic interval generalize least quantile great eliminate resample iteration single reach svm ten play pair roc curve computed estimate good average area roc ability fitness model filtering eliminate value estimate largely bar confidence effect inferential fitness roc near unity rmse skewed linear assumption residual may skewness statistic understand study system simulate nonlinear model independent add set efficacy use feed architecture tune decay repeat resampling varied minimum confidence interval create hardware failure simulate model adaptive efficacy quantify speed procedure version relationship tune six validation setting decay competitive eliminate quickly depend adaptive procedure use nominal simulated condition match however resample resample choose base overall fully first small size efficacy set discard decrease range effect compute comparable value figure median least total occur speed drive training surrogate tune parallel many computer architecture loop line serial computation loop model configuration logical computation lead substantial parallel speed fold sequentially parallel offer parallel benefit processing parallel additional require svm worker process resample speed adaptive parallel study use combination processor task process previously sequential median adaptive degree parallel eliminate technology
noise realistic inherent dataset google house consist google million image x natural label dataset imagenet step imagenet augmentation x probability cifar architecture configuration file implement layer keep imagenet model architecture noisy datum stochastically change label change use generate training five epoch figure contrast operate beyond label one addition noise consistently achieve well also rate train perform show effective negligible tb cccc cifar label flip detail color mean indicate incorrect increase convnet greater incorrect cifar training level identity decay b noise especially show scalability noise imagenet imagenet scalability noise explore adversarial imagenet random case label zero location correct change confidence size consistently superior imagenet add outli imagenet category use imagenet fall release fix imagenet outlier add outlier train normal version outli use run training run perturb amount robust effect h image image realistic label internet outli cifar mix totally outli image fraction know cifar convnet error second model train layer give gain web image challenge around imagenet web category internet search imagenet dataset example image highly rank precise outli train imagenet domain reduce add imagenet consistent three ii flip large training matrix flip convnet flip noisy label focus type large scale former gain gain however minimal deep implementation little facebook ai com availability label allow convolutional manual datum impractical noisy e available image may accurate noisy layer process simple modification deep demonstrate scale imagenet image availability large amount imagenet label image impractical tag keyword contain training abundance understand consequence convnet contribution classification noise dominate explore expectation convnet significant degradation modification convnet enable level simply layer softmax softmax model handle flip outli noise conventional distribution supervision convnet library readily imagenet classification degradation noise noise input noise several type source noise cause fast amazon www privacy handle label incorrect remove correct difficulty distinguish informative hard effect knn cost deep incorporate noise single tune cross realistic label make adjust class parameter deep particularly relevant layer pre purely convnet train availability clean provide supervise either one datum require external force pick inefficient resource fraction near decision sample unlikely many even pick informative challenging rank search engine likely regard train convnet difficult human drawback impractical many method problematic dataset consider may drastically result structure problem class aggregate denote parametrize label capacity asymmetric oppose example cat likely tree data b initially base green start updating datum prediction combine parameterize train maximize noisy eqn label true quantify confusion sample true make identity perfectly predict noise confusion eqn measure reality label objective force label eqn p equality ik confusion noisy noise singular force identity combine parameterized force model convnet softmax layer softmax role softmax linear modification perform back propagation noise model accurately predict unknown us infer noisy noise constrain network update back entropy base gradient weight project subspace unfortunately follow confusion alone give base actually infinitely force force noise base encourage blind deconvolution act base necessary ill pose take hold hold minimize sensible although
truth multiple experiment heavily class never mechanism gold long question require worker neither depict baseline mechanism answer gold skip base figure require worker worker choose computer object task depict worker gold question show short text worker movie order worker search entire search worker question baseline skip confidence present evaluate worker answer convert upper answer error match short worker paragraph play prevent search obtain text searching line task depict gold question baseline correct gold standard skip base mechanism present worker solution answer true audio audio second comprise topic movie speech depicted amount worker gold compare correct gold skip base mechanism c figure mechanism crowdsource axiom mathematically surprisingly mechanism feasible mechanism preliminary baseline mechanism suggest additional benefit pattern worker difficulty question worker secondly mechanism post process incorporate information overall accuracy simplicity facilitate easy worker conclusion mechanism practitioner researcher engineer crowdsource acknowledgement thank many discussion thank read part manuscript first author microsoft paper although skip case skip simple offer valuable recall assume worker confidence mind formally worker worker answer correct payment wrong answer remain mechanism first question skip skip confidence greater indeed answer likely skip let order permutation question mechanism compatible payment maximized worker answer algorithm payment payment strictly consequence distribute question relation associate condition payment maximize confidence finally see every question worker choose payment worker probable lemma theorem begin piece represent expect payment give answer identity random choice gold question worker expect payment function must compatible compatible mechanism simple element small mechanism skip last question must worker indeed worker answer question compatibility payment choose answer payment choose value side represent evaluate ball thus coefficient must constant appear desire argument permutation question complete element element value worker skip worker attempt last x worker answer quantity compatibility apply value constant get proceed induction begin simplify apply variable side polynomial coefficient particular polynomial order simplify hold answer evaluate worker remain question get necessity use remain constant pick pick side get result function type gold standard gold comprise evaluation argument question gold question worker evaluate divide desire complete restriction threshold observe employ free axiom neither confidence worker worker skip question worker likely eq q desire compatible payment question incorrect question incorrectly question proof induction incorrect zero payment base induction hypothesis assumption consequence incorrectly payment induction question prove necessity payment gold incorrect confidence payment denote payment gold e eq fy l l linear lemma must also comprise fy g meet specify give finally budget requirement instance payment payment worker question payment payment strictly zero incorrectly remain worker question payment confidence first worker attempt answer desire skip payment gold first include gold function must worker confidence question worker choose skip question order worker answer free prove compatible strong free even worker show work first p payment payment pattern strong worker answer expect payment worker payment worker question worker p gm payment happen answer payment question question p payment result payment maximize desire finally strictly hence worker restrict applicable avoid worker question payment incorrect answer proceed answer strong payment payment answer answer induction fy allow payment statement proposition proof proceed induction quantity payment repeatedly unlike hold quantity payment rest payment correct rest answer gold question alone originally question argument constraint thereby complete proposition skip strong answer incorrect hold even incorrect worker level correct irrespective question contradict requirement payment free axiom axiom payment u zero function payment prove payment observe propose payment function behave compatibility prove uniqueness replace payment eq evaluation proof proof payment replace payment theorem proposition pt axiom theorem california berkeley microsoft berkeley edu microsoft com field range predict structure large labeling task traditionally perform expert expensive rapidly increase interest worker crowdsource time typically fundamental crowdsource novel quality worker proof desirable free mechanism additional involve observe necessity engineering dataset need build speech label th task expert student pool student limit labeling perform worker internet know crowdsourcing generating label crowdsource crowdsource overhead crowdsource minimal crowdsourcing gain popularity bioinformatics environmental management computer crowdsource create require large amount label crowdsource often supplement automate task difficult alone worker crowdsource expert label crowdsource typically focus post quality input process reliable engineering crowdsource service crowdsource amazon gain popularity task image answer worker show worker identify depict bridge typical crowdsource worker complete worker attempt question encourage worker skip reward worker confident feasible requirement crowdsource job comprise question payment mechanism existence gold standard question question perspective randomly worker confidence answer answer belief correct worker likely assume question worker aim maximize respect gold question worker note payment solely competitive market worker call payment mechanism payment crowdsource threshold wish worker skip smaller great select likely possible mechanism indeed wide narrow impose natural requirement practical consideration worker wrong question attempt payment compatible mechanism satisfie axiom amount worker total gold mechanism among gold question correct answer axiom requirement weak payment answer question would half axiom none strict impose gold incorrectly prove surprisingly mechanism natural mechanism satisfie skip discuss one require offer minimum payment worker denote payment free axiom make payment answer gold wrong free axiom amount addition equation illustrate amazon platform display ask gate bridge figure gold image question mechanism payment additional answer gold payment take worker aside mechanism pay gold pay answer nothing worth note equation conservative compatible worker payment impossible compatible free ask worker explicitly fine g level low moderate high reveal confidence payment mechanism generalize axiom indicate high confidence question attempt turn payment worker exist compatible mechanism free axiom present low confidence gold provide worker question incorrectly level question mechanism pre mechanism compatible payment mechanism generalize free axiom offer payment worker hold amount amount conduct experiment amazon worker aforementione among expect also observe expect answer higher correct payment take present result formally theoretical claim appendix crowdsource consider worker task correct multiple choice figure question audio etc question accord correct mind possible answer answer probability answer correct shorthand define confident every skip pre confident ask worker either skip answer question either skip high skip phrase moderately absolutely sure skip skip correspond mechanism skip appropriate fall corresponding question assume question answer evaluate gold question gold question question question payment payment worker gold question payment depend e amount individual worker positive non evaluation answer worker determine payment worker evaluation determine setup crowdsource non negative skip question incorrect payment confidence worker question incorrectly confidence payment mechanism attempt maximize payment sequel payment refer expect payment worker e answer uniformly question worker question worker perspective payment level average arise gold summation regard correctness every choose answer skip information ask worker payment mechanism worker question moreover question select likely correct simple requirement axiom answer worker gold standard payment zero would payment binary half answer incorrect weak payment incorrect payment answer gold set mechanism present indeed worker skip confidence answer confidence great correct mechanism compatible mechanism certain minimum payment make modify axiom accommodate necessity payment answer worker incorrect pay additional worker payment whose answer compatibility uniqueness payment worker describe mechanism discuss worker ask select worker ask indicate range answer skip level worker skip worker make skip consider special axiom answer gold high formally require specification threshold worker indicate skip skip set specified skip attempt set every fix specifie level option question worker skip skip choose confidence confidence worker confidence indicate high include level select call payment list worker answer correct set restriction worker threshold must coincide additionally compatible threshold post constraint etc paper propose payment input threshold evaluation payment eq show worker select level also give early mechanism worker else whenever confidence confidence mechanism payment mechanism algorithm compatible satisfy generalize payment worker minimum payment modify free axiom accommodate necessity minimum payment answer worker confidence level modify pay payment worker answer x compatibility uniqueness mechanism worker mechanism compatible axiom skip zero payment incorrect receive payment generalization mechanism impose zero answer gold incorrect question confidence even question wish impose strong requirement payment worker primary focus section skip axiom strong axiom g axiom axiom propose event payment axiom add extra worker unfortunately minimal requirement satisfy compatible exception gold worker least impractical crowdsource free axiom compatible strong free axiom skip list worker worker payment free worker mechanism satisfy free axiom try worker act mechanism worker strong exist compatible condition answer nevertheless mathematically interesting belief event prove uniqueness worker skip payment next thing answer confident worker answer question confidence answer gold payment worker answer worker answer confidence greater skip confidence small obey free free condition I none answer mechanism strong compatible even section consider set worker payment payment increase example expect payment aim utility payment make worker answer gold question payment aim belief regard correctness answer gold question recall free axiom payment gold question high question worker
dependence decay linear em stream competitive erm averaging independently provide regularity herein decay strictly range tuning decay undesirable start seek drive arbitrarily remark regard parallelization stochastic see subsequent quantify procedure implication streaming work algorithm iterate sgd unclear erm dependency specify essentially variance iterate distribution erm paper provide average erm herein work erm initial low rather initial special least guarantee global super comparable adaptation would alone suffice identical wide need decay many iterate competitive variety reason lead dependency error convexity solely term argue estimator applicable mis model asymptotic argument make certain restriction variance erm finite stochastic convex smoothness case bound square estimator mis convex converge notably develop become loss near sufficiently attempt directly erm pose linear new generalization order exist generalization demand pass scale immediately observe datum sum store erm mean convergent algorithm increasingly hope cutting fraction erm streaming would eventually essentially hold entire second proving succeed aforementioned conjecture tight regression suffice generally believe erm herein statistical pass algorithm contain erm fact avoid streaming erm polynomial consider special least focused solely computational use state art sum function convergent obtain generalization approach comparable streaming setting slowly strictly yet summarize algorithm erm corollary summarize corollary theorem provide provide performance guarantee provide throughout dependent erm restriction asymptotic instance whereas section size analogous rao approximation problem set benchmark dependency rapidly decay assumption analyze assumption low approximation differentiable strong q weak assumption number definition assumption imply convexity instance whereas ratio namely derivative self self weak condition equation condition hold eq standard assumption phase aside close minimizer quickly global obtain curvature fast stream progress away size random streaming provide sum smooth proceed stage stage draw draw sample opposite show achieve aforementioned generalize particular choose coincide use assumption oracle streaming compute draw one point guarantee assumption fix denote upper regression achieve erm furthermore competitive ratio near follow upper drawn iteration p parallelization implement linear count stage theorem geometrically remains spend average gradient algorithm fully enter phase enjoy even dependency emphasis flexible driving erm super competitive drive erm polynomial know approximately multiplicative let number iteration decrease know super polynomial rate convergence set specify drive ratio competitive ensure adaptively fast fast vanish quickly characterization numerator erm streaming focus erm follow regularity condition third derivative problem polynomial specify application benchmark widely study least square upper low erm extend generalized problem huber regularize illustrate erm performance streaming erm distribution streaming bound become meaningful large initial decrease square word recover mis specify aforementione model last ridge necessarily specify follow provide statistical erm erm suppose remark appropriately universal x bind erm suppose equation erm comparison lie set allow erm upper bind least define regularize regularize w analogous interpret mis fit rate erm straightforward self include completeness logistic self consequence smoothness instead assume smooth smoothness second norm suppose definition follow step function suppose independently note assumption eq q recall schwarz yield multiply yield finally bind progress conditioning take sum strong little term yield assumption progress assumption jensen result case proof utilize follow lemma convexity self self self convert another self restrict property monotone small assumption hold eq solving recurrence bind high lemma suppose hold eq recall random hand last balance lemma ready stage positive lemma q eq q erm sense third exist one regularity compact interior invertible exist constant neighborhood bound appropriately choose also universal bl infinite theorem lower erm interior along argument erm bound less function eigenvalue appendix universal small term arbitrarily taylor p p w w w w right side boundary inside enough bl lemma diameter convexity enough erm taylor enough thus taylor inequalities q use perturbation n w w w erm great define pz less term complete constant acknowledgment thank lee rf microsoft done institute support nsf fellowship grant instead hessian assumption upper convexity fix
validity however concentrate therefore generalization property forest first assumption feature proposition pf pf pf fp ff pf pf fp ff set follow basic selection forest case build randomly choose node independently base forest follow internal forest internal internal extend give reason empirically moreover might applicability sake strategy choose use subset strategy application high utilize image computer vision slightly principle event feature tree assumption proposition know get feature node total binomial probability compute select tree entire count clear count partition partition denote partition partition two list element partition compute compute partition compute need count e number return compute tree know probability get time compute multinomial internal tree strategy internal simple give component need determine frequency relevant non relevant nan hypothesis determine threshold get time provide hypothesis specifically choose false positive desire negative empirically much equation threshold rely already discuss important presence feature relevant expect drop non gets select determined predict thorough synthetic quantitative analysis map propose depend cluster bag subsampling replacement subsampling bag publicly software neighbourhood forest problem zero binomial ensure correlation different size balanced set problem spurious close reality false estimate nan dimension per setting record selection threshold prediction experiment strategy htb ccc strategy strategy dimensional positive rate capture various different strategy plot threshold observe positive rate gray line observe accurately capture substantial subtle surprising effect depend quantity demonstrate frequency hypothesis article problem htb ccc false model black curve probable relevant statistical label assumption relevant feature evaluate false rate observe bag addition experiment false vary present expect size make relevant importantly prediction observe prediction strategy threshold limit false rate situation determine equation false rate false power summarize training column code false negative mark fashion false positive give false negative false rate desire I proportion relevant right show false negative training strategy high increase feature dimension change sample one hand spurious become node higher false positive desire change seem observe increase due increase relevant feature false also seem believe importantly show share drawback statistical see trend change increase second increase believe competition increase correlation false rate expect false change part analyze statistical relationship theoretical feature per set train forest false repeat experiment time present simply false positive versus relevance forest compose cc strategy fix rate relevance tree set desire false relevant rate increase increase tree surprising tree weak chance tree get number increase third number cause decrease positive rate observe get furthermore large subset decrease false large precisely observe graph number surprising tree feature weak spurious high chance trade detect characteristic forest comparative study determine threshold testing aim significance permutation yield applied frequency rate bag train desire base marked permutation base use permutation relevant positive permutation adequate time value ii selection frequency comparison propose selection permutation provide setting false permutation limit desired propose slightly false desire propose false permutation testing testing computationally parameter permutation hour ghz gb ram forest optimize though testing load backward computation frequency cost propose integrate permutation comparison problem rarely depend complex approach derive type brain software software discretize study understand disease map false control map map matrix publicly open access htb generate mean correspond dataset discretized vertex extraction image denote mean decompose denote matrix generator easy verify estimate one create produce display map synthetic similar fully formulate disease motivate relationship label map local region denote vector figure region region entire set generation give diagonal pearson label assume equally experiment sample number size strategy forest determine repeat c strategy ii positive synthetic size leave column false high desire plot sharp negative competition explain previous overall threshold split relevant relevant especially threshold experimental demonstrate false positive feature positive topic spurious assumption actually sort get high spurious relationship feature tackle subsample bagging bootstrappe bag ratio ratio improve spurious statistic enough bag lower bootstrappe prominent cause rate competition hence selection rank make algorithm backward combine bootstrapping scheme rather non straightforward backward bootstrapping stability determine unlike integration difficult propose burden negligible forest segmentation medical problem compute space think feature apply challenge dimensional nature bin come effectively model forest principle false heuristic threshold computationally keep rate great modeling bagging bootstrapping effect provide approximation threshold author thank comment read carry whole resource center resource national institute national health share program grant number rr rr rr also support foundation well foundation datum acknowledge grant mh mh r ns ns ns medical foundation mh project process brain subject surface measurement subject surface efficiency consist map width well software request remark convert forest tool ability bioinformatics popularity forest still crucial ingredient efficient computationally researcher ranking article build feature process forest threshold additional synthetic positive false presence light need approach applicable selection datum growth almost scientific machine substantially main determine sense maintain resource allow constrain certain scenario selection eliminate entire set relate refer label trait particularly biology part routine great potential feature wide biology univariate forest certain advantage contrary detect et study second deal feature often need logistic problem computationally tractable phenomenon explicitly raw require dimensionality transform interpretability analysis fourth modification regression multi unsupervised burden variety phenomenon despite popularity success forest importance still lack ingredient false produce frequency permutation importance last propose variation ranking identify irrelevant determine feature relevant assignment forward principled threshold random forest heuristic know interpretation natural determine quantify number false positive threshold quantification desire level threshold quantification allow informed algorithm eliminate irrelevant build principled efficient satisfactory drawback motivation attempt determine threshold quantify score permutation label significance lastly propose permutation testing variable burden limit quickly infeasible drawback reason feature relationship variable irrelevant time hypothesis give rise principled threshold computational ii tree biology forest especially advantageous truth available approach set assume complex computed measurement share drawback yield false article overview forest different importance motivate approach theoretical conclusion forest rf technique year vision bioinformatic due property beyond capability predictor rf overview rf feature adopt covariate denote describe importance bag permutation importance rate across break prediction forest quality quantify importance measure statistical undesirable decrease score importance increase drawback test determine measure make remark definition study present yield similar ranking show basic form category eliminate et measure variation suffer drawback date determine threshold integrate simple show threshold positive would like frequency use empirical measure ranking surprising permutation selection selection frequency random contrary possible need bootstrappe desirable limit question answer select label answer question estimate selection principle relevant relevant feature probabilistic training subsampling scheme subset node optimize false positive threshold determine model node randomness
state maximize among covariance lag maximize satisfy independence far admit close start submatrix next give specify extension factor q let partially specify band extension extension correspond band central partially matrix admit factorization lower triangular main diagonal provide dc extension dc representation partially matrix n dc entropy dc completion start nest principal dimension completion extension claim statement k band equivalently find inductive eq adjoint k k k equivalence maximum ij dc among covariance independence solve problem toeplitz eq factorization inverse take dc kernel determinant third right hand side triangular diagonal equal definite tc special expression determinant particular tc kernel tc mention tc spline entropy stable spline family dc interpretation spline stable spline admit kernel matrix exponentially decay poorly make exploit proceeding recall factorization qr thin qr factorization assumption uniqueness thin qr factorization triangular thin qr easy cholesky q thin factorization orthogonal unit vector triangular partition function rely qr thin factorization whose unit vector thin qr orthogonal triangular note assumption thin way qr suppose hold thin factorization moreover impulse thin factorization compute create step qr factorization compare complexity algorithm part require accord create cholesky create scalar scalar computation qr require step require computation iteration solution marginal maximization storage negligible particular stable depend cholesky conditioning return cholesky b hessian marginal th finally computation make use possible adapt stable approximately large recently introduction family kernel correlate dc kernel entropy interpretation exploit completion problem graphical dc dc admit determinant factorization property dc highlight dc exploit associated algorithm section chen approach impulse response process covariance depend parametrization identification framework correlate kernel entropy tune correlate tc point paper property indeed extend whole family kernel maximum exploit conjunction dc particular dc close inverse determinant kernel highlight dc estimator system rely error finite model select trading selection goodness criterion validation understand nevertheless sample equip cv model identification recently propose impulse see learn e maximization order parametric paradigm case prove robust aic crucially process variety function literature straight system identification recently correlate assess bank kernel admit interpretation entropy unknown zero represent impulse impulse response model covariance call practice impossible impulse impulse system decay exponentially impulse response certain become turn hyperparameter estimate maximize marginalization impulse assigning crucially quality recently identification covariance class diagonal account impulse impulse response tune
intrinsic suggest autoencoder reasonable indeed dot train frame dot figure filter learn phase shift fourier discuss bi able exist lc precision activity movie probably lower random dot still propose consist test video action train video feature sub block concatenation super descriptor layer spatio plug performance autoencoder outperform localize surprisingly outperform unbiased linear inference experimentally scheme unit linear without arguably represent intrinsic summarize region response invariant invariance perspective increasingly increasingly dimensional affine response invariance superposition region reconstruction square define response reconstruct input multiply bi reconstruction define feature active suggest high linear hide acknowledgment award education research university david university autoencoder typically hide natural act negative role representation intrinsic autoencoder activation like regularizer regularization autoencoder deep network simplest minimize observation eq q activation sigmoid relu prevent solution large input contraction force hide activation penalty across tend autoencoder mention scheme show negative bias desirable learn autoencoder consequence autoencoder weight vector take part reconstruct represent select combine cf bias allow role yield increasingly outperform autoencoder regularization autoencoder additional show without negative bias state cifar explain hide optimal unit tend force align dropout hide activation reconstruction autoencoder shrink autoencoder regularization type rbms common autoencoder yield negative effect color bias rbm contraction strength relu cifar result histogram cifar view arguably autoencoder constrain feature undesirable consequence encode effect bias relu sigmoid act activity inner weight ie yield value significantly weight bias activation spherical effectively define radial basis datum effect activation gets overlap cluster autoencoder however autoencoder relu even region merge word able define multidimensional autoencoder function add sigmoid autoencoder autoencoder manifold restrict active fix relu autoencoder write equation solution solution nan space eigenvector unit unit eigenvalue although w active reconstruction sigmoid zero active autoencoder thereby density reconstruction superposition activation value come probabilistic support one propagate learn dropout back prop boolean figure product derivative relu continuity common minibatch non optimization truncate follow unlikely work well often par well autoencoder denoise truncation contrast negative bias relu view inversion well activation activation function square drop cc autoencoder variant reconstruction either active cone thresholded thresholde literature plot illustration activation unit encoding shall autoencoder long hold span weight relate multiplication operator operator may tie weight one tie minimize frame identity hold choose cifar ability various contain color pixel class sample consider invariant recognition autoencoder mean evaluation base whitening whitening normalizing divide eigenvalue train choose initial epoch train total momentum representation weight validation training threshold parameter try regularization strength observable discussion early autoencoder increase input space tend ht input number patch cifar evaluate
eq depict measure provide fine uncertainty call improvement strategy spirit associate ei select ei ei sampling use criterion sequential therein reduction acceptable evaluation dependency function bivariate improvement equation ei assume computation independent accord minimize carry simple advanced monte spirit describe batch optimization define krige toolbox qualitatively behaviour criterion ei depict search situation sampling region thereby induce isotropic mat ern mat ern may gamma kind optimization figure error distance small c line posterior solid credible vertical bc bc bc bc evaluation sample mat ern simulate pt real value bayesian approach optimization usually formulate consider denote good put emphasis expense maximizer classical ei associated loss assume tractable call numerical value location maximizer ei l g real continuous set choice sampling point strategy explain motivation proceed present implementation qualitatively numerical criterion view path rewrite uncertainty inequality
truncate relevant estimate produce produce posterior filter brief represent filter solution integration relevant integral specification filtering applicability method illustration density instance ng current directly integral description preliminary final likelihood evaluate multiply impose restriction marginal produce fine grid compare abc choice well compute estimate use likelihood associate discretize normalizing produce generator uniform euler ng filter method filtering view typical equivalent gaussian transition exploit conclude computational kf euler base ng technique produce main slow infeasible score order initially impact abc first report remain abc auxiliary match summary abc via report square posterior give estimate posterior c panel one approximate c euler score marginal abc ss c fp panel rmse match multiple prior panel highlight parameter score accurate four marginalization marginalization minimal summary statistic four estimate inferior euler inferior four panel three superior notably improvement euler explore application certain reduction consistency auxiliary equivalence maximum summary also dimensionality application abc problem integrate extremely comprehensive model marginalization integrate auxiliary model common relate frequentist indirect inference efficient number exercise principle simple approximation approach success statistic success subject go also front despite static nothing state accept filter smoothing exploit see use state smoothed posterior state average burden develop herein base approach axiom conclusion condition exercise problem early new approach state abc summary data exact mle auxiliary auxiliary achieve precise sense yield mle separate integrate curse drive intractable abc fast produce volatility illustration free space kalman filter volatility increasingly integration fan review technique evaluation statistic calculate statistic simulate generate information summary statistic use technique determine devoted way ensure maximized contribution statistic approximate observe indirect approximate produce intractable frequentist model partial paper continue spirit demonstrate fix size key motivate decision seek state space setting auxiliary model abc matching focus qualitatively auxiliary achievable technique exact likelihood function sometimes ii literature well true efficiency investigation allow via abc alternative summarie space force adopt inexact auxiliary emphasis possibly drive least discretization usefulness apply illustration reduction result motivation note asymptotic auxiliary typical quasi mle establish technique allow tolerance criterion mle define measure proximity yield mle abc achieve setting block marginal integrate principle avoid outline paragraph applicable auxiliary base right particular already thereby exploit whereby kalman kf achievable illustrate via approximate discretization true augment kalman evaluate likelihood general applicability implement particular approach number principle parameter yield inaccurate non posterior marginalization meaningful basic would statistic reduction illustration score marginal section feasible repeat sample accuracy exact posterior assess abc weight autoregressive iv reduction conduct assessment firstly key reproduce overall superiority remarkable yield score particular exercise concept abc albeit exact available proceed diffusion accuracy approximate role volatility adopt linear central chi exact purpose deterministic base filter ng produce posterior associate euler model score great case gain still gain certainly mark linear simulation accurate euler approximation draw distribution draw posterior moment accept rp criterion sort tolerance arbitrarily small proportion approximate use practice model draw correction draw markov carlo sequential carlo smc improve give choose close b biological comprise may information spirit attention feasible may characterize consider scalar financial drive computationally euler measurement apply base particular discretized model see recent express initially financial adopt e smc inferential infeasible contrast continuous constitute augment vector comprise unobserved define x x period simulate simulate follow draw subsequent conditional crucially retain criterion view relate reference content importance proximity comment highlight key motivate sample illustrate form expression highlight applicable useful inference cardinality abc arise distribution ef possess achieve reduction relative member ef due vast possible either member ef reduction ef even gaussian necessarily reduction familiar move simple dependence marginally normal ef sn familiar toeplitz matrix autoregressive ar order construct structure accumulate achievable straightforward increase accumulation across successive reduce ultimately need attain determine sn determine approximate sufficient sn reasonable approximation sn ignore qualitative characterize nest link sn lack accurate base arbitrary turn motivate asymptotic mle asymptotic mle model cox algorithm would draw posterior mirror demonstrate chosen see also likely enough parametric abc knowledge appropriate analytically tractable computationally enough yield matching show certain regularity condition auxiliary log positive assume form coincide true analysis abc evaluate frequentist whereby degenerate consistency appendix q generic draw choice approach limit irrespective select hence degenerate parameter consistency fact frequentist obtain e avoid specifically abc property approximation go decrease unless increase solution within drive look abc posterior le large gain score close choice definite weighting implementation magnitude approach approximate produce simulate technique specify consistency frequentist consistency limit irrespective maintain remain impact mle draw yield criterion yes enough equivalent estimate course parameter expand point scale criterion subject condition regard affect value equal irrespective weighting matrix mle matter true exactly go comparable regard estimator weight form preliminary abc estimate posterior yield negligible proceed operate solely score extract set level survey appear path tolerance within stay away give addition constraint impose necessity reasonable recommendation namely technique perspective value induce form error firstly fundamentally summary selection posterior exercise equivalent exact otherwise posterior necessarily error analytical exact density density error highlight less thing curse increase firstly bring need secondly bring certain full statistic technique estimate retrieve remain still dimension set see approach framework match diffusion natural discretize auxiliary mle construction select estimate posterior marginal produce statistic translate function coherent full vector technique et joint estimate marginal marginalization yield non analytically approximate score base abc example begin formulation random variable include jump exclude discrete produce either notational continue true discretization affect functional section illustrate true e error process initially diffusion use represent evaluate g include simulation filter smc computation technique numerically brief transformation calculate variable involve select accord deterministic sigma yield turn cloud moment implement kf weight sum sigma excellent introduction generalize non burden comprise update sigma point sigma estimate score method non set accuracy discretization accuracy function aspect discretized model order accuracy regard likelihood reference relevant g document order kf filter measurement approximation beyond embed specific euler discretization exercise abc method set summary statistic conduct firstly within kf evidence two increase exact mle abc space compare summary whether curse marginalization section volatility use score produce function discretized result relate particular nevertheless serve illustrate dominate summary overall approximate particularly accurate simulate sample setting sn summary statistic may sensible observable distance firstly euclidean across draw statistic secondly briefly produce summary statistic step procedure scalar subsequently simulate conditional calculate r denote vector statistic evaluate kf weighting mle evaluate mle score estimate integrate numerically kf logarithm performance figure abc statistic abc euclidean fp produce initially reject algorithm draw process sn marginal posterior normalize likelihood evaluate kf multiply fine likelihood posterior produce panel summarize produce kf percentile dot short per text panel highlight abc true notable remarkable marginal poor statistic base abc figure produce percentile extremely accurate fp parameter produce accurate still fp tend approach yield score
respect property magnitude greedy counterpart set statement introduce prove rsc low regression section extend broad hard compressive well q thought e f n see need property rsc rsc satisfy convexity rsc constraint satisfy convexity constraint except rsc define except replace project gradient descent select magnitude projection significantly strong thresholding formalize index oracle sparsity step p combine rsc rate rsc sketch critical condition f f third small element ts hard thresholding insight see gradient hard I singular vector matrix top singular pm diag von trace follow rank rsc replace projection operator fw satisfie secondly place consider restrict span variety datum result differentiable rsc respectively incur crucially scenario z label hold nc l universal constant put incur look specifically corrupt obtain replace case additive miss n rsc sparsity hold small hold high error fully keep iterate prove fundamental rip consider rsc fs input tf concentrate fully popular compressive pursuit far analyze rip least use section rsc base arbitrary observation thresholding objective function hold observe property rsc satisfy sparsity tf ts thresholding family iterative introduce rip sensing member replacement backward method backward similar fully partial hard thresholding projection partial hard project still perform small add rsc either suppose added thresholding element magnitude observing provide proof rsc iterate ex dimensionality variation running times level increase htp clearly scalable l different increase verify project recovery offer scalable choosing recovery run time keep one independent datum hard style sp implementation l project scale lasso within consequently shift time experiment record support describe sake clarity htp result sp indicate equally recovery htp gap whereas unable respectively slow htp moreover nature sparsity level figure htp take less converge offer htp large run verify high keep select coordinate outside support heavy draw level increase see remarkable ill size study hard differentiable nevertheless convexity rsc sp require rip restrict large universal insight relax support show follow rsc already establish literature variety put relaxation term order competitive arguably dimensional learning generalize algorithm analyse unify probably create atomic norm claim microsoft com edu linear project known thresholding offer fast solution extremely statistical tight match lower enable analyze hard thresholding htp sp setting extend fully statistical consequently structural work feasibility issue often end np problem sparsity constraint rank require deal demonstrate avoid strong rsc restrict smoothness relaxation suffer despite
calculated mind reconstruct reconstruct systematic calculation affect library unfold iterative start histogram inverse many reconstruct represent presence intermediate dimensional bin measure solving simply bin approximation connect measure preliminary true calculate usage certain systematic principal guess connection create measure guess comparison candidate attack unfold sample test measure equal bin distribution chance true relatively big stage try add another entry require good value time influence make add match sufficient growing illustrate correspond systematic translation monte carlo assume true candidate ensure come ill fluctuation illustrate add regularization term role regularization projection guess coefficient dominate bin range bin prefer constrain complex bin fluctuation illustrate smooth
visible reconstruct rbm term update become decay bring improve performance avoid interpretable large value reason decay find illustrate benchmark user movie mainly item movie appear come internet movie http www distinguish different recommendation application depend class explicit implicit convert integer value binary rating big rate big implicit prediction rating feedback etc predict observed rating take rating imputation implicit usually hold predict rating observe evaluate take recommendation miss zero become dense rating recommendation distinction notable analysis mean rmse evaluation know usually rate fraction trivial besides suffer rating mean rating either indicate item paper receiver operate characteristic performance roc classifier insensitive reduce roc roc auc mainly technique recommender three typical baseline aspect tie factor select discrete latent latent undirected graphical start problem belong aspect mixture rating pair fill movie generality suppose movie well movie seem movie one put rank really want movie miss cause come prediction situation difference need test movie know answer ignore prediction rating prediction replace suitable feedback item situation technique understand four know difference negative feedback task though give perfect predict rating miss value include uncertainty deal class sample feedback class prediction ignore test could deal problem ignore feedback train phase rating indeed u could movie movie correspondence actor representation actor actor similar movie actor distance correspond actor mean actor illustrate experiment actor movie example etc partition c actor lee berkeley david new could easily situation typical rating task comparable imputation give rating imputation task superiority give future firstly good secondly kind application combine deep recommendation want layer undirected variable energy number visible unit bias joint configuration give sum configuration visible unit unit q rbm represent rate rbm represent movie rbms unit different unit rate bias tie rbms movie rate people rbm bias
locate operation identifie plant surely subsequently discover known detect hardness work programming relaxation method also fail hardness plant clique widely theoretical plant clique sense identify plant tend researcher hardness plant clique problem prove approximate nash similar hardness clique principal submatrix detection note section impossible achieve statistically sparse significant polynomial semidefinite programming assume pn parameter clutter theorem instance conclusion hold attain minimax time plant clique principal different idea submatrix uniformly replace flip sign row independently obtain joint distribution identically vector scale suitable false exist convergence pn plant plant clique could identify also college second air force scientific fa mathematics information california research let final apply take deduce take set need immediate independent deduce use result set proof notational closely zero proof appendix cardinality ham every distinct observe u u l generalise kullback eq curvature q denote index zero deduce conclude require assumption theorem n consequently risk follow sequence polynomial polynomial algorithm identify planted plant tend contradict n u w rademacher entry small subset large coordinate absolute u u w let clique matrix diagonal adjacency associate bipartite let convenient variate rademacher independent rademacher let ny iy shorthand elementary variation initially algorithm observe p pn mn n mn bernstein belong hypothesis large let sufficiently reverse conclude hoeffde large eq conclude sufficiently contradict rademacher rademacher coordinate four subgaussian deduce hand binomial right similar fourth final hoeffding final inequality result eq exist u require write measurable kullback leibler divergence denote derivative generalise measurable function denote basis require r variation proof take value element arbitrary b measurable inequality subset disjoint set write b g eq I na mb f I f c empty f intend short introduction notion refer reference information much follow generate desire string problem denote acceptable string solution string collection perform task mathematic history notion define notational system call calculus finite distinguished transition machine access consist label component square string start head th operate machine move square stop symbol terminate finitely step say terminate computational solve notion calculus modern input string terminate exist solve class notation configuration replicate branch far replicate continue replicate machine replicate output computation string terminate replicate make parallel step widely computational relate algorithmic complexity problem exist g section machine non situation problem triple distinguished state transition head state probabilistic exist terminate say string however independent fs randomness problem suitably construct polynomial thm lemma thm dimensional simple lead eigenvector fast fundamental trade satisfy concentration time minimax polynomial time variant semidefinite relaxation show essentially rate algorithm project multivariate onto span lead covariance device effective small size pca encounter diverse modern area vector arbitrary unit case eigenvector classical would unit lead eigenvector covariance inconsistent asymptotic call design interpretability eigenvector belong give p remarkable author attain minimax treat sample particular subgaussian infimum moreover show rate lead cite principal estimation subgaussian neither polynomial compute naive search become infeasible moderately whether computable polynomial attain progress nan ensure asymptotic problem plant distribution minimal level polynomial test arbitrary test tailor sufficient thesis principal rather distinguished fundamental phenomenon occur formula statistical section precise satisfied subgaussian certain key importance subject mild restriction place semidefinite computable plant clique result fundamental computation section estimator recovery motivate step class sparsity therefore semidefinite relaxation drop fm mf fm u complexity implement interior show solver rewrite make amenable possible matrix onto decompose orthogonal diagonal projection j optimisation htbp pn nu point certain particular step number often considerably costly compute operation take complexity algorithm operation use algorithm lemma incur population see projection describe property
cell lstm slice pass lstm tb lstm start lstm start net feed feedforward net time slice show tb gate forget gate gate input gate cell notation gate adapt consist secondary annotate literature see use output hard encode description full far filter identity cb divide n cb tb h loop bend bridge model layer lstm relu unit skip pass relu concatenation use weight sample bias layer initialize lstm norm divide exceed gradient scale correct field lstm perform rnn use correct tb ensemble prediction secondary cb currently show lstm conditional neural field inspire recurrent show feedforward backward net structure future include architecture ss supervision ss read final version article acknowledge support gpu acknowledge bioinformatics centre apply computer science technical university secondary bioinformatics svm slide sequential network feed recurrent memory secondary use cb secondary problem feed short memory explore recently memory lstm network include machine recognition lstm protein secondary neural specific improve performance conditional approach secondary structure prediction non typically feed classifying naturally present slide prediction backward network include feed neural recurrent feed forward inside primarily introduction large model lstm paper predict past secondary elegant train separate rnn start recursion
variation recursive compositional rnn vector vector fix word embedding treat give compute role e rnn table discuss bag word standard bag yield model exist automatic regard tree proper value important sentiment automatic scenario result sophisticated bag model weight capture hold generate set sentiment dataset contain label phrase task could way fine grain task negative neutral coarse label embedding fix embedding label full sentence neural mention recently paragraph obtain sentence paragraph vector produce recursive performance paragraph sentiment force semantic mean syntactic orient decide coherence sentence corpus contain report pair article assume coherent example permutation protocol consider window concatenation representation adjacent classify either coherent make get score random permutation state art regard use illustrate model entity art model task feature obtain weight art task c recursive entity propose version neural feature token weight dl architecture newly paragraph model model neural additional extend deep minor adjustment leave emphasize evidence tuning acquisition propose neural automatically contribute compositional acquisition recursive demonstrate significant neural one type obtain representation sentence long dependency extent capture brief short sentence embedding token parent layer involve activation nlp obtain embedding feed optimize sentiment feed aforementione classify negative embedding sometimes capture semantic syntactic manually feature nlp benefit c model neural train implementation recursive please report search parameter optimum batch mini batch embedding word suffer intrinsic drawback us token movie contribute sentiment network flexible less token sentence consequence representation trivial refer specific architecture neural svm notable big word come advantage evidence brief svm sentiment classification constitute supervision detail learn embedding less influence zero compositional former mostly specific word embedding neutral phrase regard issue compositional enable composition rnn interaction vector g pos tag tag compositional approach extent compositional idea weight svm try incorporate neural neural binary idea involve associate final token movie impact token like sentiment task optimize propose capability compositional approach undesirable information nlp task brief distribute framework beyond token gram phrase recursive recurrent constitute type framework acquisition recursive g include sentence compositional within sometimes input token explore embedding manner capability capture depend architecture work short lstm first back determine information memory lstm partially address recurrent model widely translation sequence token phrase sentence denote word vector stanford representation idea associate additional weight importance current towards retain relatively information expect importance current whether sentiment embed representation convolution function enable compositional bias intermediate view use three neuron output project parent current importance embed concatenation lead around
yet matrix functional matrix lead guide select real carry sparse sequence belong ball arbitrarily actually accurate signal encounter g estimator fully functional estimation exhibit structure functional thresholde penalize likelihood optimal rate class functional intuitively estimate since one estimate together matrix estimate rest organize section motivating efficiency market capital asset pricing prove minimax large matrix extend estimate performance two let integer space contiguous integer write identity denote coordinate iff trace operator element square norm real define associate rectangular determinant variable may change motivation functional test first gene validity financial economic dimensionality large statistical due limitation well illustrated suppose canonical statistic eq power avoid large lead provably powerful statistic implement dependence mild condition hypothesis asymptotically moreover depend unknown statistic estimator suffer power rather may prefer thresholded estimator empirical could covariance estimator chen take relationship need statistic exclude th exclude leverage reasoning thresholde hard quadratic beyond scope observe difficulty take hybrid consist put scale let q hard asset pricing asset risk specifically asset express factor return portfolio see market correspond let follow q factor pricing risk return possible market efficient traditionally problem point estimate residual ordinary define statistic moreover expression provide estimation impossible motivate case functional rapidly norm diagonal unknown situation remove diagonal implie measurable appendix limitation class covariance normalization frequently obtain sample simplified assumption accurately regime diagonal element could vanish employ estimator covariance empirical eq j diagonal element rest establish interestingly quadratic threshold plug present usually functional optimal minimax measurable proof subsection remark theorem tradeoff keep trivially match present arguably nature give minimax keep light effect actually boundedness diagonal noticed meanwhile omit information present phenomenon functional fan setup low model control decay coefficient inherent functional clean logarithm functional matrix functional matrix maximal row functional quadratic functional natural quadratic outside wise q boundedness space low assume exist define infimum appendix write bind estimator naturally technique minimax plug constant proof indeed theorem yet interest prove actually taylor expansion term parametric second contribute rate taylor stem estimation remark combination minimax convergence make enough maxima sum quadratic price extra term present phenomenon wise functional functional simulation conduct evaluate apply dimensional simulate financial study estimator part end simulation auto ar matrix otherwise vary quadratic bs chen time base omit bs estimation estimator b bs dash log size top bottom leave htbp plot aforementione estimation use naive plug obvious four b dash well solid small estimation improve capture quadratic cause eliminate come work well proper threshold simulation choose employ cross proper thresholding consist sample thresholding estimator threshold test construct thresholded consistent estimator candidate threshold choose take apply matrix suggest functional apply splitting study test functional dimensional two group gaussian consider correlation rest functional increase mean vector choose percentage identical represent situation equally comparable prop empirical six repetition bs chen chen modify estimate thresholde empirical counterpart dimensional univariate nan employ estimating splitting evaluate correction correction family fdr list estimation method also take error bs first ignore well combine individual aggregate signal together statistic outperform identical individual combine estimator functional among correct bs bs indeed bs claim chen performance leverage sparsity capital asset pricing standard return index keep change stock adjust service database series rate factor return portfolio test model window previous suggest efficiency dependent reject less factor except financial reject decompose q beginning observe cauchy schwarz simple together jensen inequality separately lead theorem eq old proceed get next eq lemma inequality conclude complete eq right decompose bound rx side taylor lies represent equal ij ij summation q eq q I low order exist order due utilizes dominate since bind remain similar obviously far show random constant give bind facilitate technical proof ij ki q marginal q prove get q easily convex monotonically increase employ maxima last jensen side show choose due actually sub omit follow ij claim minimax detail sequel leibl divergence distribution go sample eq di inequality end together square employ jensen inequality prove bound employ reduce begin prove constant leibler indicate zero mean obvious degenerate comes perfectly correlate since use instead lead use technique display imply reduce testing matrix yield enough low fix recall support give follow resp collection
weak exist hold speak optimization iterate process induce rigorous pd proving conclude contradict indeed suffice independent express convenient note correspond vanish vector independent set vanish show inequality state summarize initialization iteration large among optimization radius absolutely adequate linear quadratic convergence generality motivate conventional performance denote complexity relate radius iteration apply arguably property know iteration see particular radius highly inversion iteration matrix inversion sum sequel extremely fact derive inversion function combine equation must invertible yield us initialization point coefficient matrix eq shall omit inversion coefficient optimization partition readily complement verify plug implication polynomial quadratic polynomial technique goal last chapter show polynomial degree definition degree yet polynomial much small much must lead inversion newton complexity invert regular read thus say approximate radius iteration would reason decide inversion fact balance many algorithm limitation induce imply spectral radius scalar one chapter obtain wide family inversion whose depend derive iteration lastly address sequel spectral evaluation real q chapter simple designing efficient attractive minimizer strong coefficient obvious fundamental algebra root denote root therefore complex eq first equality line employ triangle economic indeed equal polynomial big summarize coefficient bound tight requirement coefficient real polynomial although notice claim regardless scalar value spectral positive us eq polynomial establish get thus plug inequality yield likewise range regard assume rest tight monotonically argument harmonic plug inequality yield thus rate inversion matrix big analogous sequel convergence inversion well rate inversion matrix positive definite follow argument assume convergent eigenvalue thus accord scalar q prove embed sub previous section tight heavily rich coefficient attain thing get radius ball although class disadvantage general strongly disadvantage priori convergence factor contrast attain factor furthermore ball scalar inversion surprisingly match range chapter ball perhaps algorithm carefully remark mention well importantly latter oppose low iteration say prove restrict differently show iteration relatively section coefficient scalar inversion disadvantage ideal algorithm execute spectrum finding optimization inversion whose answering efficiently computed coefficient conjecture tight state optimization matrix coefficient matrix scalar pure polynomial observation base mention vast conjecture polynomial expression coefficient straightforward convergence question root polynomial polynomial meet prescribe derive correspond remarkably enough adjust function discover systematic turn particularly due nature let iteration say admit say say scalar inversion efficiency constraint obtain consequently iteration efficiently coefficient lastly inversion sequel shall closely admit denote iteration matrix inversion coefficient respectively lastly give show polynomial lemma attain radius polynomial enough nothing eigenvalue order arbitrarily coefficient perfectly match polynomial latter unitary diagonal diagonal eigenvalue equation correspondingly apply equation converge choose equation grow linearly properly major drawback impose task inverse impractical spectrum algorithm similar fairly kind correspond various include versus present previous chapter demonstrate superiority present section suppose comprise coefficient exist apart inversion exhibit appeal canonical recall consistency rule express definition substitute counterpart obtains analyze matter future designing optimization matrix scalar bad principle maintain analyze satisfy verify iteration exactly see rule eq bound preserve path polynomial achieve economic section namely hope root yield equation q eq multiply remarkably extract accordingly yield lastly see interval ensure eq heavy spectral exceed convex combination straightforward verify conjecture look optimization convergence rate bad barrier focus analogous state polynomial tune spread present demonstrate idea let q result optimization specify illustrated expect counter decrease along algorithm initialize iteration contradict exist suboptimal contradict upper readily complexity eigenvalue avoid later sort use nesterov sake densely discrete approximation decomposition chebyshev kind nesterov dimensions degree exploit shape spectra contrast clear optimization strongly lastly applicability heavily frequent application form world iterative perspective derive one algorithm mathematical generalizing question radius answering question unify many optimization outline smooth strongly design polynomial improvement spectrum gap adjacent spectrum easy vector might applicable essence one analytically quadratic function technique counterpart quadratic beneficial unbiased estimator inversion give rise whose preserve expand scope method regard inversion matrix scope sag particular replace less batch simultaneous really lastly characterization modern implementation refer enable algorithm minimization get field learn understand convergence task say inversion believe likely entry dependent quadratic unclear light one execute optimization part adapt part eq denote scalar assign notion science many brief elementary terminology result thorough value exist continuously mainly twice differentiable convex follow useful characterization kind twice continuously differentiable smooth assign positive definite matrix strongly appendix condition coincide surprisingly shall soon scalar convex characterize optimization process presentation sag paper strongly convex optimization however directly would involve expression apply equivalently rewrite transformation change let compactly define define consider asymptotic certain scalar scalar thus inequality logarithm side rearrange similar proposition exercise remark thesis develop convex optimization focus examine application turn powerful analytic whereby particular new natural nesterov accelerate employ economic rather interpretation early solid iteration low regime contribution summarize operation optimization iteration mention early attain convergence state algorithm consequently restrict accelerate heavy potentially natural present new scheme offer exploit exist bound new presence huge lastly tight suggest extension framework analysis stochastic sag accelerate sdca etc thesis dedicate whose would thank remark great unconditional year real value problem science economic express minimization significant hard structural well algorithm hilbert wide range make fast kind say solve function accuracy arguably prove attain answer question exact widely accept science analyze optimization part issue box assume acquire round carry thus show obtain smooth algorithm smooth employ order receive oracle number query minimizer exactly bind seem mechanism oppose huge admit structural hold contrast exploit assume structure consideration technique approach modern certain class optimization reveal number design concern optimization algorithm closely know gradient descent gd optimal suffice researcher lead discovery nesterov see slight gd q unfortunately strong gd primarily base sophisticated researcher field e scenario task admit remarkably way derivation idea present ascent sdca solve minimization great sdca enable derive rate thorough sdca repeatedly pick denote refer careful convert quality n q obviously take step coordinate variable process govern eigenvalue straightforward calculation normalize eigenvalue plug bind eq inequality distance less must sdca tight remark loss smooth require motivated major part generalize aspect precede argument work appearance chapter introduce terminology convergence low bound bounds radius corresponding elementary theory appear form chapter framework essence quadratic bind plug derive bind bit entry denote column last furthermore q compact conclude sufficiently yield grow linearly spectral asymptotic bit grow scope corollary statement statement readily theorem endow induce notion series specify establish property let iterative vector pd z fix convention product multiplication carry I linearity inversion accordingly omit take z convergent conclude convergence may together property q naturally radius say derive equally question arise convergent differently vanish interesting vanish denote hold km recall conclude access iterative method stationarity iteration matrix lead theoretical radius nonlinear section evident method determine rate spectral symbol order worth point manner perhaps direct proof eq equality let follow matrix q conventional see matrix satisfied assume form simultaneously invertible invertible eigenvalue arbitrary denote eigenvalue order consequence remark simultaneously invertible importance derive understand may
atomic programming treat symbol mean representation character mark character nlp improper nlp character modeling program example token c code refer double share character return code nlp representation token nlp type double etc representation severe language word token improper level problem token token indicate express explicitly compressed abstract node constant state compress token furthermore node program fact distinguish structural semantic example see nested loop inside comparison follow assignment implementation sort program code detection program task theoretically map directly composition nlp state compositional semantic roughly capture semantic barrier overcome theoretical formalize code build representation learning symbol similar symbol aspect refer factor define intuition symbol reference programming criterion via layer node experimental leaf p primary leave notice number hard dynamic pooling take maximal mathematically continuous bag pool totally satisfactory assign parameter position regardless treat child gray bar position model information code network closeness euclidean likely representation sample negative pairwise representation symbol training substitute symbol least large often set error prevent fitting add weight overall objective compound constant cast compound compound assignment compound compound break break switch constant root weight hyperparameter code gradient momentum l randomly sample formula derivative accordingly code error propagate learn node speed adopt momentum derivative current first epoch axis epoch cross compute q sample cv respectively label actual coding indicate effective program deep blue demonstrate initialize architecture gradient vanish poorly contrary representation serve initialize coding criterion cv decrease drastically epoch high performance unsupervise report rbms autoencoder generic mnist handwritten digit explore give train result report field evaluate analysis classification adopt logistic regression accuracy support machine radial rbf kernel explore linearity improve explore underlie improve accuracy experiment cc random guess logistic rbf learn evaluate empirically near neighbor program experiment beneficial supervise propose result confirm program evidence literature make artificial intelligence believe become various promising tree extent possible perspective treat adopt traditional usage representation atomic symbol nod statement lose neighboring pattern dimensional meaningful code program perspective feature suggest technique computer foundation cognitive interestingly inspire deep achieve high architecture network high human cnn specify explicitly neighborhood neighboring circle line detect convolution kernel feed layer abstract abstract beneficial task object domain acoustic add cost function slow local neural reasoning base though trivial property mathematical give false important program beneficial point include code learning question address remain unknown literature perspective program integrate program question apply field besides novel code build deep analysis feed successfully relationship term criterion building confirm feasibility program primary success address problem program source consider field intelligence primary learning near liu xu zhang edu cn com make field artificial intelligence ability complicated feature engineering etc impossible deep program architecture pure back paper criterion program representation learn reality qualitatively experiment code building program evaluate beneficial feed representation achieve support confirm feasibility analyze program also give primary success new analysis language conclude program rich capture justify learn become one machine variety processing speech recognition compare traditional deep major architecture capture highly complicated non efficiently real world human engineering interestingly unify deep achieve application program deep practically infeasible analyze neural back propagation gradient vanish architecture extract poor vector abstract architecture directly reality node value element certain analyze representation qualitatively criterion successful analyze feed accuracy feasibility program light code dataset project website future analysis source code contain feed network neural architecture good first program first analyze program deep field contribution include introduce technique field propose explore program neural code motivate explain experimental analysis draw section traditional engineering g consume specific evidence suggest may et nlp application construct expert automate human deep neural easy example deep extract organize local category human engineering moreover decision classification task approach automatically automatically recommendation predict code nlp short neural network capable capture complicated feature program interesting promise deep neural network capture complicated feature practically analyze program symbol discrete symbol symbol feed possible symbol characterize symbol also one code mean cluster direct pure back poor optimization poor deep alternative learn representation regardless detection fed benefit focus language programming language nlp improper language one flow always code indicate branch loop extremely read language programming notion concrete program source build node nlp algorithm flat build fact motivate research representation program eventually make analyze program consider literature deep progress deep neural widely technique artificial comprehensive neuron build neural network input figure typically compute activation non etc power insufficient world back propagation stack multi neural power sufficient boolean inefficient grow exponentially order complicated layer raise generalization single neuron circuit architecture deep organize abstract high layer architecture capture abstract efficiently make train year architecture al stack layer stack rbm feature energy stack autoencoder minimize stack rbms autoencoder neuron initialize learn initialization back specific learn result meaningful
cloud transformation desirable property consistency relative say follow transform relative transformation consistent invertible transformation sake completeness consistency ij attention invertible definition ii transformation ji reference coordinate tool derive frame eq linear transformation set note due hold dimensionality k make multiplication right see contain express requirement suppose empty contain decomposition rewrite state contradiction state invertible retrieve find let td give retrieve block create call identity point dimensional instead square optimisation rank nan retrieve svd invertible affine invertible affine invertible represent similar construct proper affine transformation simple transformation homogeneous row similarity transformation transformation isotropic rotation retrieve estimate affine extract factor use I isotropic solution onto orthogonal matrix scale retrieve align onto arithmetic mean jj similarity isotropic transformation transformation extract transformation ensure determinant equal generate compare error solve generalise problem correspondence assignment ground truth generate eventually transformation generate turn transformation j j truth dot transformation entirely particular factor translation part arbitrary allow factor da dd uniform interval random singular dimension pairwise solve one wrong correspondence simulation shape noise shape comprise dimension find similarity transformation good shape subroutine reference perform symmetric factor one shape randomly select shape reference shape reference select shape update solution first follow aggregate contain propose transformation enable simply miss wrong experiment shape miss point point discard wrong randomly wrong using investigate original shape shape average shape set j every additionally common occur frequently must order determined point shape base mean missing see increase amount slightly miss point marginally however reference base mean runtime pdf pdf pdf correspondence shape shape comprise point dimension column correspondence assignment shape keep coherent wrong green additionally point wrong assignment wrong wrong shape aware result transformation extent transformation shape reconstruct additionally shape assignment align consider correspond previously correspondence iterative wrong shape level different level wrong marginally method result use alignment iteratively bias reference completely depend underlie transformation retrieve approach permutation generalise transformation experimentally effectively reduce set pairwise generalise approach correspondence remark frank centre de frank sciences de alignment object play object solve globally practice well relative transformation free however wrong may fail observation free retrieve nan matrix directly present transformation demonstrate noisy transformation even wrong correspondence encourage alignment object transformation play recognition statistical shape shape remove difference order common shape cloud shape vast research field establish remove scale base unit dual reveal robustness method object computationally expensive align object object induce align object reference nature constitute noisy wrong transformation observe
optimize pairwise mapping membership within variance minimize identity dissimilarity dissimilarity som problem triangular major inequality q som optimize upper cluster orient quality dissimilarity practice quantization give dissimilarity triangular one prototype far thing prototype som quantization rather way make explicit way suboptimal point dissimilarity prototype display put som particular quantization quantification induce restrict point quite describe way address observation let point equip inner squared distance matrix classical combination sum directly keep coefficient perform dissimilarity square variant som batch pseudo euclidean derivation amount etc relational proceed iterate batch som main prototype represent combination euclidean prototype update online som equal notice preserve update tend less sensitive relational som computational cost relational som incorrect gain som solve som quantization induce point interpolation som availability bad relational cost operation dissimilarity coefficient cost batch stochastic som som approach approximate explore pointed relie provide quantization cluster try solve without rely scheme dissimilarity deterministic annealing introduce cluster update anneal role gradually increase practice soft mapping procedure annealing iterate evaluate reached increase evolve som implementation obtain dependent anneal scheme miss al analyze temperature dominant dissimilarity minimal critical temperature addition full update intensive note update reduce relational som clearly try criterion som proxy relational som easy deal enable machine leveraging allow implement method som fundamental enable batch som value combination som hilbert implement som mean map become trick solely knowledge trick som notice use need coefficient som trick enable som som equation sake completeness som propose vector kernel som suffer constrain make dissimilarity guarantee equivalence find compact som limitation som distances scheme propose relational som som median som som relational som som even relational one numerous dissimilarity som section opinion possibly burden relational som dominate median som suffer performance som truth capability prevent display gap natural cluster matrix visual increase som median som see well remain test nystr approximation study extensively relational neural systematic som som som converge lead note analysis random initialization well e initialization case batch som point anneal schedule strong batch som som obvious online som batch som epoch online som presentation point roughly som online small epoch complete cost som contrary relational som per prototype lead per report prototype place presentation epoch online som cost batch relational som careful relational outperform online initialization properly comparison variant anneal missing simulation conduct one sophisticated anneal deterministic anneal solution critical largely increase really comprise loop loop give outer anneal iteration order magnitude batch algorithms som also adapt effect final remain summarize opinion prefer careful som pair like validate explain dissimilarity attempt minimize prototype criterion classical shown directly problem k mean simplify give relational burden dissimilarity sophisticated combine art refinement could like would empty organize dimension obtain som interpolation median opinion main som provide rich yet possibility dissimilarity limitation space visually som variant result case dissimilarity u matrix display numerical cluster etc type visualization interesting mainly som dominating anneal version som soft membership situation visualization build som node dissimilarity cluster show som simple graph visualization result generic dissimilarity som somewhat compare som work need visualization interesting display review som adapt dissimilarity follow differ strategy identical relational computational aspect experimental opinion relational som couple nystr dissimilarity som discuss usefulness som representation som dissimilarity improvement output som serve practical purpose beyond elegant paris france numerous represent machine technique really dissimilarity kernel measure object self som review use set outline difference discuss advantage drawback variant actual dissimilarity som practical frequently elaborate form attribute relational different type instance online customer product customer several copy leave review say adapt machine consume theoretical level euclidean view need build generic share datum successfully fairly give either dissimilarity dissimilarity kernel type see mining dissimilarity generic dissimilarity generic typical near dissimilarity self som research operate helpful organize describe general dissimilarity dedicated som som focus modern som presents som extension annealing describe variant som provide som property remark insight som specify dissimilarity kernel classical som dissimilarity matrix dissimilarity point convention number negativity also order som notice function satisfy mx aspect theorem reproduce hilbert mapping hilbert structure build machine rely hilbert algebraic trick construction dissimilarity always dissimilarity som relate counterpart definite dissimilarity dissimilarity introduce som principle som clustered datum specify induce prototype maximally numerous possible function lattice point reduce influence som prototype som algorithm adapt close close goal prototype reflect vice neuron associate cluster essentially major variant vector proceed strategy stochastic online som select randomly update batch som unit obviously neither prototype step som turn operation involve dissimilarity
bad case arise ground state least still solver find low find solver make instance perhaps instance degeneracy gs tw tw tw tw time slightly gradient tw tw interestingly tw gs tw meta technique gs tw gs tw easier moderately gs tw conjunction explanation load color graphic explanation need graphic macro ltb lt lt ltb lt spin configuration construct intermediate series function depend spin configuration adjacent dependency arise precision store spin h hz know combination subgraph sufficient require size representation depend mean grow face relative spin z rescale fortunately range limited locally computable grow energy spin configuration along spin note also h choice subgraph calculation subgraph exponent actual quite omit example temperature certain typically subgraphs method find energy ground spin energy sum imagine e loop arise spin convenient description generally individual function purpose run find minimum energy give spin usual include ground know moderately search low dynamic define apply subgraph define spin state vertex range spin compatible spin configuration spin standard tree make partition choice devote family assume desire subgraph type update spin dependent update appropriate version independently term belief propagation description give cover provide produce I subgraph describe evidence section de observation consider restriction necessary purpose balance rigorous efficacy subgraph induce subgraph induce subgraph reason primary specific reference cpu ghz something platform dependent counting spin spin take fact spin flip spin comparison tend least concerned find execute subgraph sampling involve track combine exchange compare site update combine parallel well significantly good subgraph recall subgraph contain require vertex induce subgraph vertex ignore restriction induced spin pre update subgraph however monotonicity lose induced subgraph guarantee subgraph induce subgraph obtain guide subgraph try good easy solve exactly calculate gibb give subgraph trivial spin usual burn independent operation item demonstrate evaluate configuration sum whole detailed balance clarity tree take instance tp illustration show work though proceed vertex dynamic collection connect example vertex many order process successive vertex process note graph picture inductive contribute tree interaction quantity maintain vertex tree case vertex maintain add contribution vertex edge join meet construction move vertex boundary new choice old picture build practice compactly spin represent difference reverse pass subgraph care fast implementation straightforward method deal numerical computing generating loop rather slow avoid rescale storing aspect produce carry choose originally aim optimisation wave hardware quantum contain wave hardware aim graph arrange write bipartite throughout collapse highly fact embed applicable hand simply fully dimensional behave critical odd temperature complete effectively fact translate vertex practice temperature rapidly branch method increase explore choose randomly behaviour spin prediction show expectation ratio suitably temperature upper standard well tune space sort meta carlo monte subroutine use top iid assume hamiltonian intend undirected range choice decide upon probability adjacent require probability exchange take random spin overlap excess average average unique might expect take close decrease smoothly lack analogue tw collapse aggregate vertex spin relevant step process reach something close distribution spin exchange step temperature equilibrium carlo increase appear acceptable margin turn far elementary operation approximately compare discussion color conjunction terminal option package graphic terminal graphic macro ltb lt lt lt lt ltb lt lt lt lt bp ltb see load package terminal need graphic macro ltb lt lt lt lt lt ltb lt lt lt lt lt ltb describe problem clearly see fig uncertainty serve compare return assumption equivalence termination comparable parallel tuned way difference negligible fair careful control subgraph site try simulate identical shall mean example vertex shall single site whereby spin immediate though operate one spin outcome operating individually update immediate vertex size possibility external field state small change energy change rescale interpret number mention specific make interesting fair make currently stand method reasonable accuracy respective may fix match underlie try majority accurate optimisation technique great eigenvalue monte sharp inaccurate characteristic much would inaccurate considerably hard still proportion fix rate class consider negligible chance describe general simulation average heat capacity actual simulation lie maximum remain decide trial slightly low justification assume lie couple easy full determined set temperature perform temperature perform energy average whole form choose small time rely purpose provide state assumed estimate week side variance reason break parallel include exchange spin flip device process run computer stage measure consider time respective fairly robust vary much optimisation respective level flip measure intel flip tw tw degree optimisation tw far arithmetic operation simple tw necessary arithmetic loop due level optimisation fast implementation describe enough well tw r r size advantage tw tw useful implementation tw face large four name gs tw tw pt tw tw gs tw tw state finding
resemble pick subset stability correct high taking illustrate simulation section number score interval blind obviously pick asymptotically assumption concern score grow weak theorem apply true sequence unconditional interval law number well denote well must attain unimodal bring advantage covariate optimal behave randomized independence arguably simple setting might somewhat scope belong roughly mean heavy cauchy pareto reasonable instance consider distribute identically orthogonal eventually recovery much weak covariate proportion vanish illustrate simulation error simulate accord cauchy student degree freedom count uninformative score uninformative covariate reach large score joint distribution emphasize direct criterion criterion covariate truth know performance covariate q vector denote additional tailed material coefficient zero entry choose control situation specify covariate covariate correlation among covariate toeplitz correlation close covariate multivariate toeplitz grouped predictor toeplitz indice informative consist index interval informative covariate covariate follow loading coefficient realization covariate follow elsewhere target break dependence covariate adjust ratio section truly informative rank retrieval compare choice lasso method consider subsample randomization base covariate regularization method specifically grid regularization equation pick fair covariate l informative amongst rank rank estimate report average repetition systematically outperform lasso reason difference importantly evaluate lasso single selection coefficient separately successfully select informative coefficient coefficient noisy true informative covariate systematic appear set material possibility modify rank magnitude thresholded show property precision remarkably stability plain qualitative stability limited range toeplitz false positive average positive preprocesse step investigate whether variable method use image handwritten digits dataset external interface heterogeneous come generally contain goal complexity experiment art wish extension base mutual information select covariate iteratively select iteration computationally costly part replace select result updating score r adaboost mh report result repetition covariate covariate covariate adaboost mh number boost lead significant main conclusion small subsample final subsample less memory furthermore base observation cost parallelization easier disjoint small covariate prediction comparison stability overall relevance support stability concern subsample small generalize error insight subsample practitioner version need obtain false regime result positive appear loose far refined concern procedure covariate theoretical toy certain circumstance heavy tailed search increase particularly appeal practitioner largely experimental precision show whenever precise method analysis give dependence precise rule choice subsample size far hold choice place b numerous discussion author notational shorthand x ratio repetition simultaneously relation repetition observation disjoint joint thus variable binomial theorem relate upper bounding upper xx lead desire follow completely random would covariate conversely one wherein minimum conclusion establish end imply distribution slowly lx belong real deduce rescale slow variation negligible disjoint marginally converge distribution imply continuity value low eq indeed remain conclusion still similar argument extreme introduce follow disjoint purely formal sequel bind note inequality hold second factor term enough converge variable arbitrary conclusion eps fill line microsoft ec base repeatedly subset covariate effect benefit validate method covariate carry information selection aim goal identify informative insight outcome identify focus informative variable science broad field share drawback unstable different come vary significantly prediction stability selection half final picking exceeds expect positive guarantee choose remainder paper variable selection base box propose stability direction half instead draw precisely randomly observation extend approach stability theoretically simulation compare semi secondly empirical comparison randomization covariate base full generality covariate interest covariate false influence besides toy perform one investigation subsampling combine stability apply observation goal reduce reduce complexity linearly particularly base load small furthermore allow parallelization processed expect generalize suggest improve subsample large independent finding insight subsampling method empirical comparison subsample stability randomization base take covariate theoretical certain covariate subset randomization improve description algorithm present motivate observation investigate randomization give select informative work dataset contain observation covariate time random partitioning threshold need enter observation subset size base covariate stand stability pseudo note precisely covariate subset per initialization frequency lt various way idea subsampling replacement strongly relate developed combine subsample investigate bootstrappe variable cox data final machine apply data importance apply lasso bootstrap selection include measure combine give simplify assumption study practitioner complementary variant selection bound selection false negative complementary genome discovery covariate investigate decision build regard training classification test drastically cluster selection selection method give method use stability variable usually method might outcome subset statistical typically discrimination covariate perform simultaneously assess conditional base original aim covariate much fast selection globally maximize select covariate large mutual already latter simple namely minimal candidate covariate add covariate individually rather simple information estimator speed find solver variable zero additionally relevant lasso outcome solver analyse random section observation full unknown subset section draw disjoint equal denote subsample index obtain define performance need set covariate exclude false since box average sample method frequently irrelevant one draw independent random define use rank covariate relevance define uninformative selection definition base reflect subsample theorem positive uninformative false expect positive base equation correspond negative dp leibler bernoulli depend choice denote choose equation corollary formulate suppose noise covariate select base large informative l bind well bind allow merely specific allow
beta interval epoch frequency expectation turn truncate posterior component must around establish quantity hence truncate normalize sub interval around chebyshev iid property ergodicity stochastically exponentially decay give along recurrence cycle maximum reward rs final coupling section event c occur precede paragraph enough parameterize parameterization induce across reward observe transition yield useful unobserved mdp thompson parameterize derive frequentist general parameter space result suboptimal probability encodes term kullback learn interaction environment act notion environment mdp comprise state difficulty reinforcement learning stem primarily uncertainty essentially plan face environment structure g reward transition need accumulate current influence exploration exploitation efficiently modern underlie maintain confidence transition instant action environment mdps freedom mdps observe instant transition motivate control queue queue instant either fast rate govern service arrival queue cost service hold alone mdp example arrival conceptually learn structure posterior thompson sampling impose prior uncertain mdps parameter prior compute reward state transition prior rule main thompson mdps reinforcement parameterized mdps operate posterior cycle throughout cycle sufficiently space initial prior sufficiently large neighborhood parameter logarithmic thompson without closed path scaling admit rl constant involve kullback leibler geometry weight divergence true mdp mdps discuss detail encode thompson bandit set significantly improve space advantage possibility state queue dependent parameterize arrival appear significantly distribution like thompson difficulty encounter algorithm latter algorithm theoretically construct optimistic tracking confidence thompson analytically tailor often complicated exercise posterior quantify thompson evolve parameter potentially convenient manner pose exist thompson bandit degenerate special mdps rely property action close conjugate additional arise generalize bandit iid reinforcement evolution couple evolve make evolution especially challenge thompson scheme thompson study purely mdp completely former work establish parameterization parameter latter mdps continuous frequentist regret role merely use depend explicit mdp parameterization work overcome derive directly normalize kullback accounting adaptively self normalize concentration sum cycle interest mdp help cycle measure transition mdp initially parameterization factor action reward mdp extension control play every serve policy kind threshold mdp discrete stochastic process ta tr denote htbp space output action time stop epoch probability c repeat bayes end horizon mdp denote assume broken fashion correspondingly term reward random contiguous epoch turn version use time marker maintain denote time epoch use sample mdp update via bayes effectively epoch begin state need hold recurrence mdp ergodic chain mdp recurrence time log ratio upper primarily control divergence kl divergence employ average mdp reward merely ease exposition concern behavior state immediately follow policy apply mdp correspondingly important leibl bind parameterized mdps kl divergence convex appropriate mdps policy across set mdps average policy fix comprise mdps average resp correspondingly parameter resp mdps region time belong occur et n playing policy begin epoch instant measure express solely count weight frequency correspond policy ideal trajectory scalar truth true specifically decay weight kullback leibl condition use consistency think neighborhood exist decay mass around neighborhood play policy finite section space satisfy typical root top type assumption policy dependent constant interpret r main leave reader interpret game policy negative epoch play policy coordinate policy irrelevant far throughout eliminate sense record impose growth occur dimension vector time policy eliminate maximize final optimization thompson square scaling usual regret mdps hypothesis least rs compare diameter mdp follow mdps gain obtain mdp finitely finitely bind suppose assumption assumption mdp quantity generality main thompson absolutely continuous lebesgue ease exposition mdp theory finite mdps irrespective action transition mdp parameterized take mdp optimal imagine recurrence density cube mdp loss policy check setup establish mdp small hold consequently conclusion hold directly kullback measure policy improvement due marginal kl divergence factor additive suboptimal apart kl divergence nearly significantly simultaneously term differently bad like uncertain thompson counterpart like force explore essence thompson lie evolve exposition finitely parameter expression ct ei ei depend evolution replace empirical average respective policy tv ct ct simply environment approximation insight shrinkage horizon property mass truth imply suboptimal mass less pick thompson scale insight estimate time bad end c c interesting sample count policy zero ct ts path across arrive though argument coarse quantity rigorous indeed technical tool tailor thompson mdps include inequality iid quantity using leibl establish frequentist line example buffer customer queue action action slow fast service slow resp service queue probability empty bernoulli type instant cost hold gain queue service however cost restaurant order hold model importantly arrival parameterize use confusion dimensional rate curve constant valid arrival depict policy parameterize arrival simulation parameterize fix dependent uniform discretized result time regret across horizon increase advantage confidence visit htbp single queue mdp regret line demonstrate thompson enjoy regret bandit et al relevant study mdps true mdp space w prior useful arguably weak notion influence performance moreover episode oppose set treat sampling mdps investigate nature prior deterministic uncertainty rl frequentist setup algorithm maintain interval optimistic adaptively shrink confidence interval time state occur inefficient mdps parameterize al mdps discount set pac different notion equivalence approach learn mdps plausible model available suffer serious lack adequate low seminal paper lower parameterize reinforcement sense finite space though crucially sharp thompson favorable continuous bandit parameterize rl derive bound pseudo reinforcement would performance happen feedback delay application reinforcement particular regret thompson function mdp term correspond could variant thompson path material thompson decision express iterate weight simply mdps ct tc dynamic follow index policy generate stationary policy initial c cs thus early simulate mdp round epoch index next resp resp record henceforth ease concentration define empirical j transition u marginal frequency u conditional whenever virtual deviation mean logarithm empirical constant probability argument finite irreducible chain recurrence expect negative iid markov cycle satisfy ergodicity conclusion appeal maximal concentration inequality lemma constant sample iid analog iterate maximal concentration increment event moment neighborhood taylor around consequence martingale turn tail half fashion henceforth parameter x typical trajectory previous q kind regret sample constant time sample exist denote log write assertion log universal eq see complete proof n constant guarantee proceed I instant optimal result policy integrating give decay exponentially expect also proposition begin iterate estimate far conditional sum finitely take complete help refine compare hold usefulness stem hand help kl cs cs derivation right side inequality ax c x step find maximize far
replication correct rmse stein wrong pool seem somewhat simulated make rarely underlie effect size misspecification appeal among group gene level gene control gene cell control cancer gene large correct generate I performance size difference estimate adjusted large quantity correct average training split use c stein htp stein positively contrast bias numerator adjust k contrast numerator rmse uncorrelated positively rmse rmse oracle statistic effect statistic oracle rewrite show next assertion definition differ location index statistic lemma loss generality argument calculate frequentist q converge zero notation use throughout th statistic size obtain false oracle estimator equal theorem corollary remark interest gene dna sequence datum naive suffer chance alone size independent poor bias without simulation context thousand measure group reject nan goal feature effect study hypothesis practical true throughout manuscript hypothesis effect chance large effect size among setting dependency among expression approach case build bootstrap study among inaccurate one strength largely assumption proposal proposal broad normality make consider statistic manuscript paper review selection introduction marginal density refer correspondence bayesian estimation refer notation manuscript denote tie e index intuitively quantify small negative effect oracle estimator bias practice parametric make estimator dependency among later failure dependency inaccurate exist principle could exist accommodate dependence immediately obvious tractable account dependency provide framework k approach involve large tb datum bias small bootstrap bootstrap bias apparent adjust accurate data htp size size code red estimate code size bootstrap estimate stack along estimate histogram set difference effect parametric briefly assume availability generate htp estimate calculate observation generate step algorithm multivariate generate draw heavy rarely datum challenge approach perform know generate repeat dependency implicitly independent observation replacement calculate estimate base generate favorable genome wide association study nucleotide regression new resample observation replacement logistic compute snp properties oracle equation simplicity estimate normally relate bias among bias amount mse relate simple mse estimate mse square bias hold among correlation assumption throughout normally various emphasize covariance correspond order statistic frequentist statistic bias estimate consider scenario equal scenario variance explore size increase bias since increase adjust advantage approximately approximately adjust frequentist selection unnecessary proposal relative exist intermediate assume two cluster increase bias estimate b direct size within truly jk q uncorrelated lemma dominate assumption motivate correlation among correlate statistic study account correlation among statistic inaccurate effect simulation study show statistic generative datum compare proposal study bias know use equation correlation bias equation spline five nonparametric stein positive stein consider version generate generate datum correlate specific present context correlation among test adjust effect mse estimate numerator rmse value large one indicate adjust perform estimate respectively generate n p ar block ar correlation study htp leave middle ar block correlation contain element equal ignore correlation test lead inaccurate effect normal distribution use report large average replication ar uncorrelated outperform account bias around keep decrease oracle oracle rmse similar oracle lower spread true end bias less oracle tend rmse htp small replication simulation section generate set cc stein htp block ar cc stein
account bundle expectation brain choice already mention one illustrate study study activity decision activity area could difference event reveal model optimal exist rather reinforcement brain serve policy target depict brain put along mostly td actor value td latter actor algorithm tackle belief decision shape state much reward belief attempt novel actor release basically set space mathematically support choice instance could comprise ultimately draw actor since activity direct understand utility consequently evaluate end many remain ahead static essence section type stay pass base prediction calculation one future process tree aim study modify general actor future agent model action depend sort action outcome deal amount feedback controller design operator monitor actor choice experiment actor promising situation exploratory actor actor composite actor action transition current environment state reward account immediate future receive composite recursively next state although generalize represent temporal amount eq parametrize error modify take grid take move towards goal side shape associate action ask reward movement way come pathway bold straight line policy first time reward find pathway mid execution te algorithm inspire assign behaviour importantly transform action worth actor function execute make interaction distinct trend assign actor mechanism incoming represent process way probabilistic structure wish framework element find prior come pathway trial author comment aim find behaviour actor ever process concern finding suggest existence belief execute pathway sub offer modify actor light importantly resolve refer challenge actor lack task actor keep encouraging study particularly affect great deal field experimental economic cognitive social brain multiple understand optimal course mention reciprocal process cognitive provide great insight determine economic group uk
addition solution common correlation magnitude select covariate scale sense covariance residual sum significance newly constrain fix adaptively appealing impose reduce issue affect relatively regularization parameter suitable lead oppose covariance covariate model appeal covariate enter covariate establish appeal verify covariate condition require sure insight consider true sample row I copy error deviation range lar generate screening screen event strength sure estimator large probability increase factor screening generalize characterize consistency lasso model asymptotic intuitively put noise one example condition wide regularization lasso event implicit strong contain sparse lead misspecification apart misspecification misspecification inference recently reveal true play characterize impact misspecification misspecification ingredient admit nice asymptotic penalty lasso variability selection include scad mcp question statistic testing would possible extend lasso corresponding regularize question generalize gain insight let admits bind false vanish panel df example penalty combine model choose tighter sure seven enter generalized slight abuse reduce constrained case compare chi df combination combine become top panel case show fit panel conservative interesting seem latter combine interesting transition phenomenon interestingly phenomenon relate model high dimension remain development comment nsf dms valuable covariate selection huge grow devote different study
come np ic np ic x ic jt rt lie impose utilize global color carefully choice always segment coarse heuristic provide parameterized eigenvector neighborhood scalar dependent represent exclusive point merge tend member converge mode share bandwidth iteration trajectory get cluster initialization principle member trajectory trajectory might merge start trivial trajectory else current point include trajectory c basically member trajectory location location member comprise mode position indicate arbitrary neighborhood u refer member u u x u u eq u ne xu ne xu continue u u u pt mm subspace extended equation kernel domain need set shift sec analogue g ne ne xu u likewise mm agglomerative iteration shift compute neighbor iteration merge criterion cluster within merge merge merge bandwidth converge perturb bit equation neighboring point point reformulate local trivial neighborhood cluster bandwidth neighborhood particularly likelihood denominator normalize fix come u update bandwidth eq shift update bandwidth fix utilize fix begin employ shift moving mode soon trajectory significance weighted point indicate confident use cluster confident choose base evolve scale drive adapt local orientation become oppose scalar partition bandwidth bandwidth member point subset consider representative joint density represent structure asymmetric arise shift mode converging immediate evaluate use across basically localize fit numerical stability update decomposition fall trajectory mean trajectory concentrate conservative also h effect color plot varied effect cluster plot image ms color domain image quantitative vary except enforce see smoothing level intersect datum proximity trajectory could cluster eventually end converge local basically merge high mode cluster mode help function return implement space stability work basically distance contain direction divergence measure metric information theoretic like partitioned post step mode ensure conventional shift post additionally could adjacency add naturally mode mode ensure divergence merging variance mode proximity somewhat specify adjacency density location form additional separate adjacency close adjacent cluster remain representative influence cluster mm ref ref ref ref ref use image maintain segmentation segmentation method color ahead close operate merge operating domain indicate quantitative qualitative effect variation although show value give monotonically cluster remain couple rapidly iteration net happen point iteration serves offset computational bandwidth straight implementation shift scalar improvement near search exploit domain hash delta end merge fine tune shift point perturbation robustness salient result perturbation result label label label affect salient keep still place varied typical method reduce break boundary break boundary maintain segment simulate gaussian respectively position ms comparison show domain segmentation training image together completeness segmentation also indicate search segmentation well low salient test mixture vary reasonable local mode salient adapting set comparison domain isotropic bandwidth near neighbor kernel mean shift indicate lack dataset feature uncorrelated uninformative processing attain ms qualitative quantitative efficacy scheme varied leverage adaptive shift unsupervise adaptation shift cluster point normal video indicate issue growth shift focus agglomerative edu com adaptive shift rgb font color font font color font color font plus minus pt pt widely detection due adaptive methodology allow evolve adapt turn methodology space preserve due issue though allow effective introduction shift unsupervise reference establish utility low level feature popular cluster segmentation high video scene parse segmentation improve utilize fix scalar bandwidth proper heuristic flexible lack cluster offline automatic smooth affect
mind derivative entity intend proof physical find well ht considerably come baseline operating protein protein less classified classification mark tendency protein large discrimination correspond contact discover protein locate peak specifically address protein analyze completely extreme increase high density preference give simple raw observe contact scaling perspective topological contact graph structure perspective adopt discrimination let move obtain preliminary aforementioned ds analysis satisfactory weighting indicate substitution perform substitution ds denote rate ds drop ds dataset induce pure discrimination viewpoint ds stress obtain representation could technique ds test set drop consider system easy justify operate space worse easy operate basically physical arrange different pointed contact adjacency sound descriptor structure worth make contact justify also cm cm cm ds ds ds g regardless protein entropy deviation ambiguity g analogously protein although well discrimination fact protein ambiguity herein interpretation toward protein topological architecture represent ds dataset value show operate aim quantify discrimination full protein structural complexity descriptor achieve training set instance split ds nonetheless consider structural descriptor representation complex characteristic ht derivation insight protein aggregation drive paradigm protein pure experimental base pattern recognition capable geometric recognition pure correlation substitution cost existence largely physical code protein interpret transfer transfer demonstrate combination mutually root contextual balance aggregation mode dramatically shift single mutation ref parsimonious force represent component angle preference represent confirm picture protein upon balance contact recognize importance vs note demonstrate baseline discrimination domain organization describe constitute topological consideration among accuracy protein structural constitute evident error ds achieve operate discrimination ds obtain weak conclusion regard ds operate context use protein spectral moreover achieve ds tell interesting ds reasonable accuracy protein topology topological effective ec relate contact ec allow modularity ec align ec similarity complement pure topological ec information show significant reverse predict ec compatible result proposal refine physical pca impose evolution obtain discrimination power hard task odd et adopt contact adjacency obtain characterize structure markov complex reach stable preliminary similarity protein regardless degree devoted purpose discrimination degree point contact play protein play formula loss atom yes alpha end focus actually analogously happen structural formula much sophisticated play physical description opinion protein integrate heterogeneous derive valuable investigate phenomenon different pattern device gap orient classical approach architecture vertex quantification build relative entire cell standardize hardness superposition drive force intra inter interaction shift phenotype point come aim compare different protein contact able identify interesting general notably consistent contact discrimination present paper prove relationship description develop discrimination structure protein encode structure paradigm unfold experiment certainly paradigm principle matter fact protein specie require intervention protein correct drive force aggregation formation within interaction shift equilibrium correct interaction balance subtle single point mutation important stress protein physical aggregation disease aggregation protein great upon explanatory study author fair reading gene genome sequence translate protein cell free condition protein experimental intrinsic toward aggregation intrinsic give feature ph keep concentrate solely devise matter protein amenable al aggregate clearly aggregation aggregation estimate relative aggregated author bi modal consistent protein protein intervention avoid aggregation bi modal character perfect peak fuzzy boundary behavior trend protein predict sequence structure protein one frequent population fold beyond obtain good physical representation possibility physical obtain alternative protein protein analyze recognition operating pattern representation conventional mechanism input offer interpretable exploit infer drive possibility reach aggregation constitute progress evidence sketch hypothesis model computational successful vs able single important dynamical threshold ii highlight relevance contact discrimination contact spectral structured dataset protein representation start introduce basis describe framework experimentally evaluate graph represent protein present discuss statistical analysis demonstrate iii focus point bi modal largely protein report protein protein dataset unbalanced bi modal character make selection threshold protein interval protein dataset would protein respectively provide basically symbolic e character identify ht experiment organize consider balanced dataset protein high protein low degree simplification original analyze achieve elaborate paper protein respectively place protein considerably large protein consequently denote ds largely unbalanced concern datum protein suitably represent pattern protein ds character show ability sequence symbolic successively substitution sequence matching component derive descriptor dissimilarity protein assess mean stre minimum alignment operate globally operation quantify involve transform usually dynamic programming quadratic process e string general cost overall distance worth odd protein comparison limit successively try structural side similarity recognition system algebraic use calculate deal inner construct eq gram pairwise euclidean center squared evaluation evaluation high determine similarity input pattern converse obviously length use sequence e predefine hoc substitution characterize operation aforementione mathematical correction make sure regardless character process base different weighting scheme ds unitary cost substitution cost substitution cost negative position determine search specifically tune hand encode lying mutation genetic vector solution instance evolve determine genetic algorithm implement check fitness change ds process graph kernel coverage classification system ds ds sequence use among substitution huge difference contact map share modular pattern protein existence suggest common possibility rely homogeneity consider pure topological widely scientific measure ds complementary way enyi chain stationary ii computing recently ambiguity root fuzzy set interpretation concept provide interpretation state ambiguity instead chain completely equilibrium see stationary always unique chain define order enyi order enyi know upper distribution maximally tend converge degenerate associate visit ambiguity uncertainty fuzzy ambiguity fuzzy hypercube encode map type fuzzy membership consider basically vertex union disjoint among fuzzy fuzzy q membership fuzzy generate notably account give term centrality fuzzy uncertainty ambiguity compute monotonic non representation ambiguity value calculate combinatorial problem assume maximally maximally accordingly descriptor topology characterize topology classification accuracy result obtain herein assess relevance check possibility learn physical reach canonical describe physical solution globally variation pc pc
effect bar privacy increase add regression select correct plot analyze specificity increase become additional effect also observe decrease specificity explain candidate namely allow effect term perform regular sparse private penalize penalty small regularization bound observe specificity e amount regularization begin sensitivity specificity decrease correct differentially regression show condition snps bar recover term middle bar correspond bar main specificity simulation elastic respectively sensitivity specificity aggregate genomic collect attack propose differential privacy come privacy arbitrary extend end end differentially procedure convex include parameter noise sample regression focus penalize widely analysis identify disease show applicable datum privacy utility risk tradeoff identifying decide guarantee allow release information concern decade function strongly subgradient w notational extend include v mh h ng lemma differentially private successive private even necessarily function dt dl l singular value strongly nuclear f therefore define namely distribution sphere absolutely variable freedom unit sphere science technology publication attack considerable develop method datum preserve end private regression focus penalize widely identify disease show private elastic genome wide genetic disease typical information thousand nucleotide snps associations snps certain phenotype phenotype relationship multiple snps problem approach dimensional popular select phenotype snps association impose penalty competitive researcher statistic snps individuals participants privacy challenge publication publication attention snp database elaborate genetic turn interest development privacy database external selecting snps approach enable step participant involve elastic snps differentially perturbation mechanism penalty satisfie net way logistic heavily regularization differentially differentiable net extend stability select function penalize validation mechanism slightly thereby accurate penalize elastic differential differ individual differential private q differentially validation let also draw used loss differentially private budget function stability validation give I differentially incorporate perturbation procedure describe regularizer convex maximal singular differentially q differentially private noise always section compare follow function noise j ng invoke obtain ne bb therefore produce result previous derivative iii differentially logistic elastic regularization logistic exist well differentially private section phenotype snp represent minor snp contain snp case control simulate linkage correlation minor allele real snps minor allele frequency analyze allele frequency multiplicative e describe odd q log choose result size snps high association
global feature part clique tree return obviously opinion pool special model indeed kullback divergence approach close expert see weight offer interesting discussion desirable sub aggregated belong author model independently weight fine product unity deal interest manner abstraction model two layer dynamic field movement trajectory record annotated present primitive atomic activity label partially semantic rich interaction ph ph learn unseen adaboost ph marginal perform deep bp method network state node mrf instance mrf total take base fully grid take tree latter per bp adaboost mrf complexities htb adaboost mrf collect scene acquire background primitive activity complex activity comprise primitive activity state generally miss map result htb c tv sequence time specify type correspond transition potential level correspond define ahead look back choose primitive activity correlate five top however instant offer complex activity often period square room window avoid simple indicator overall perform mle method slightly cost slow adaboost guarantee generally mrf bp show generative model recognition justify bad flat bottom layer variant consistently accurate flat encode mrf bp flat adaboost detail randomly adaboost mrf boost capacity boost learner strong reach log iteration condition meet learner previously tree structure span tree attractive guide mle diverse mle share relax share tree rewrite parallel adaboost mrf update time version property wish provide express log pool create mrf perform capacity poor novel boost network adaboost offer efficient maximum tractable network apply new activity recently stress readily appear exhibit tree tree automatically tree plan aspect adaboost mrf wide sign iff inequality induction obtain iff scalar word sign iff proceed prove part trivially ii assume apply rhs yield substitute notation hold complete pair graph q q select q eq previous learner structure except structure boost mrf idea handle thus practice reliable often work adaboost mrf surveillance activity model grow temporal activity address aspect level abstraction popular model level discriminative infer model suffer error propagate direct hmms dynamic discriminative crf general useful activity potentially achieve generative provide surveillance end propose activity problem know difficult crf expressive representation encode complex however great challenge generally intractable exactly mrfs deal efficiency limit happen situation mcmc attractive impractical extremely involve efficient addition learn mrfs explore markov adaboost operate markovian optimum mild discriminative include adaboost mrf label application essence adaboost mrf multiclass boosting call adaboost mr round tree weight error hard span tree combine parameter select tree since method work inference guarantee mild adaboost mrf reach unique adaboost consider variable mrfs label handle evaluate adaboost monitor adaboost mrf maximum likelihood hide markov handle comparable flat paper continue discuss conclusion boost undirected graph simplicity assignment discrete observation specify clique support state crf define family weight vector inner function goal concave global often expectation respect intractable except tree belief take quantity need tree extract intractable bp bind log bp bp guarantee converge formal extensive converge besides mrfs activity recognition promise due structure chain field output structured principle example feature expectation past decade supervision indirect supervision arise crf conditional incomplete show associated clique equation inference clique review boost develop adopt boost give boost strong boost literature expect assertion suffer possibility rank smooth verify indeed upper assumption identical posteriori appear exponential boost approximate estimation mle another related boost boost seek address structure apply bp inference variable limited field term adaboost mrf label visible formulate objective adaboost mr summing visible numerical l l weak learner learner interest sensible general network use computation learner become span weak learner thus visible span incorporation h collection mrf markov forest example simple spanning later shall continue derivation loss method unfortunately intractable alone propose loss tractable use tree recall boost sum numerator special select span fortunately around summation numerator apply inequality numerator mild assumption rhs far simplify I see tractable evaluate
iv iv poor prediction crucially train satisfy iv expert put ensemble bag due voting stacking require sequential training meta predictor satisfy iv stack limited ability iv process enjoy prediction fusion weak uncertain prediction yield sharp prediction together close another start expert model conditional model expert sense expert assign pointed training general expert special gaussian q qualitatively confident less confident correct would slight misspecification expert biased combination rule confidence expert desire behavior expert predictive necessarily measure measure ignore bad generalize distribution focus supervise measure widely anneal balance degree freedom case cause mode dominate product cause expert small combine ignore gaussian suffice show power essentially th gaussian cm control individual reliably change posterior expert variance expert come shall cover variance half change physics result variance true carry potentially effective explore kl divergence cm bag training uk dataset gp way gp tree construction ball recursively subset build expert expert gp expert expert sum ard learn conjugate jointly change normalize sum expert combination scheme learn independently core independent second one time tree test commonly metric standardize log square bagging combination large score explore variant expert path point could define expert boost consistently empirically previous expert correction cm c c c expressive give expert superior sophisticated variational inference gp uk parallelization take although time long similar competitive superior sophisticated well extend emphasize benchmark naive result potential use automatic h rmse method name gp expert take expert gaussian expert many combination thing
outline green operation training user text rank retrieval network gpu three retrieval requirement dataset image gpu able matter second dataset million image live web system input rank recent convnet contributions convnet architecture gpu computation stage parallel architecture budget limit image every architecture begin comprehensive convnet category evaluation scale retrieval take medium large dataset evaluate two image equip pc performance dimensional convnet scenario hundred thousand less fast cover spectrum descriptor namely reduce last layer ii product quantization convnet convnet feature dimensional convnet quantization produce architecture retrieval scalable capable adapt vary paper evaluation protocol assess representation architecture describe dataset subsequent condition world object category lack scale retrieval annotation use common large annotation measure annotation category additionally imagenet train convnet evaluation carefully tailor collection base assessment exclude reason split comprise aside noisy tag represent snapshot popularity sharing site confirm contain image svm evaluate retrieval measuring goodness page retrieve critical evaluate basis proportion positive list class page retrieve large object category rank list evaluate adopt protocol annotation rank class dataset rank list generate complete annotation detail procedure avoid set annotation image object category evaluation train people rank precision evaluate different scenario remove occurrence dataset use annotation remove annotated occurrence rank list retrieve purpose retrieve exclude image precision rank dataset easy since contain instance class test google search image tolerance differ google image centre world retrieval practice sub different image google pool web query google pool describe annotation apart tag annotation contain ground truth rank rank paper raw rank fall within top image combine image annotate positive top annotation store annotated facilitate annotation share scenario annotate class annotation scenario convnet base basis describe image benchmark imagenet focus evaluation baseline traditional encoding fisher detail reduce convnet consist connect layer cnn dimensional feature layer factor use compression feature sub vocabulary cluster learn use successfully descriptor sign iff tight qr experimental table result major challenge decade perform even class produce top dimensional convnet feature retrieve positive constitute scenario convnet precision particularly challenging observation method cnn nonetheless challenging class rank performance drop convnet compare negative appear image comprise repeat evident explain particularly severe appear convnet base much image starts cnn cnn cnn cnn cnn bin retrieved convnet indicate whereas retrieve query track retrieve road several compression convnet appear rank convnet compress appear exhibit similar drop gmm codebook gaussians intra fisher convnet publicly toolbox convnet configurations paper imagenet configuration contain convolutional fully way classifier remove turn convnet imagenet image descriptor cnn setup linear learn hinge machine experiment aside gpu store gpu memory outline green iterative query ii periodic model category retrieval fully exploit advantage convnet experience requirement instant repository follow choice internet model image fashion current I allocate memory storing rank efficiently sect convnet indeed even accurate convnet gpu hardware without illustrate cpu front user interface internet gpu trains repository follow convnet image line use pool google search return feature compute store mb dataset mb pool feature experiment equip gb convnet feature quantization without significant degradation performance make gpu gb aside fit amount single storing repository preferable ranking place cpu memory typically user front end sample feed regular front rank list back end interface gpu back end front responsible mini batch batch equal size end pool positive training front diversity quick every mention store gpu compute gpu top pass back end within dynamically expand pool top return image first four step right outline red blue moderately challenging performance run measure simulate internet typical image return experiment four class challenge performance converge final evolve time order diversity convergence occur fast real term typical rank often architecture query return result background expand pool web present performance true rank position head image change feed supplement motivation role end setup analyse training final rank list rank image bias diverse jump performance image feed deal challenge degree intra appearance image suffice rank close retrieve fed positive retrieve rank positive third ranking suggest coarse train relatively tail ranked suggest image available interface list available continue refine tail rank restriction mention false positive outline limitation impose classifier train demand google live additional figure query novel much query hierarchy query challenge representation repeat thick convnet base robust effect return abstract concrete figure return positive appearance object retrieval build upon advance convolutional representation compress incremental learning retrieval second entirely single gpu cpu gpu employ learn diversity rank acknowledgement support grant acknowledge support use research attempt resolve object collection query bootstrap source latter extract identify analysis pre computation classifier dataset effectively bite
green gibbs black dot green top bottom gibbs static parameter computationally inefficient degeneracy approach deal fix lag work practice impossible quantify realistic free recently appear literature perform batch ml forward smooth filter recommend per implementation besides memory gain particle ml situation whereas time similarly particle crucial line ascent establish admit empirically slowly useful conduct computationally expensive computational available particle move degeneracy recent paper dynamic propose estimation inference suggest well move might engineering sciences research grant ep j energy ep research partly ep g research part nonlinear de g sequential model particle chain carlo discussion l e chains york propagation particle en york maximization em markov smooth asymptotic unnormalize chen liu kalman filter b method carlo particle markov monte update b al hmms maximization hasting hessian evaluation backward formulae technical stability optimal derivative markov condition general state hide approach method j f new york filter year efficient markov monte appear york consistent technical report department sequential smoothing augmentation model inference base dynamic economic filter smoothly economic working papers university department economics thesis department national http edu file pdf via simulate normalizing sampling path w target monte dynamic west nonlinear estimation self series space l dynamical system iterate particle filter comparison smooth smooth new york computational particle million particle lee computer science lee resample sequential monte carlo lee le maximization base markov g le recursive hide chen liu particle liu j west combine filtering liu york liu chen f et al university particle filter financial particle approximation information parameter e smoothing application j paris nonlinear non ergodic l wu neural particle filter property simulation r p g score en recursive system filter model static wang online bayesian estimation markov west new markov nd particle markov chain monte n switch state discrete carlo lee gradient nonlinear non state signal also monte smc reliable numerical state static sophisticated present comprehensive review limitation popular environmental formally process latent markov density markov q parameter component space euclidean space well popularity stem model asset return specie neuron response spike train datum simulation technique illustrate inference space line observation gaussian scenario kalman filter call filter term deterministic implement markov obviously successful popularity easy implement parallel implementation numerous g practical model depend line infer often specie infer extended particle parameter recognize naive problematic past year development estimation recently paper overview differ paper focus e comment attempt make discuss implementation reader original reference broadly classify maximum bayesian characterize datum line specifically framework sequentially organize challenge review unknown dedicated filter ml simple summarize main open ingredient q density equation score fisher associate inference intractable inference reasonably carlo score carry involve integral approximated challenge approximated want sequentially suit integration estimate sequentially line outline particle ingredient numerous particle denote task posterior filter filtering easy likelihood q class class kalman space example intractable technique approximate numerically auxiliary special bootstrap filter sequential resample let density necessarily negative nn rely follow importance eq notational omit weight remainder filter h n I n nx x recover far recover bootstrap filter filter density n intuition recommendation make particular e np nx practice one mostly filter resample introduce replicate particle discard particle weight serve computational effort promise region present simplest resample scheme resample scheme propose advanced particle sake present operate iteration regard degeneracy weight meet particle result give insight difficulty particle test respect reveal nothing regard behavior space typically target n successive population eventually particle literature fundamental particle accurately fortunately possess property optimal condition uniformly chain observation informative informative effectively hide find hold possible strong recent integer explain tool many one approximation unbiased variance great constant standard design recall additive critical implement parameter substitute dx np recursively time g hold grow least degeneracy still particle require point suffer degeneracy yield potentially variance development alternative marginal large collect information approximate expectation x particle mean resample component particle suggest practical choice take tp particle vanishing bias since joint p filtering transition backward recursion tx x x procedure generalize tn dx dx tn dx filtering operation note possible operation unweighted hence cost rejection one trajectory average approximate expectation costly acceptance small particle hybrid procedure combine rejection marginal n obtain filter stand approximate path lag high backward estimate fast approximation procedure interested computing recursively backward require perform pass step backward procedure simple propose easily eq give approximate explain provide access I I directly require consist approximate dx n perform much naive forward trajectory many lag vanish constant exponentially fast forward backward procedure actually regularity describe particle section introduce maximize ascent technique approach initialization get maximum see approximated wish ml carlo evaluation optimize calculate monte ease helpful distribution cdf cause small change result importance introduce computational suitably elegant method number result cdf due sort particle maximize log estimate sequence sequentially receive compute batch recursive subscript sequentially advanced convergence em establish state remain evidence version easy usually procedure approximate term form posterior framework lag smoothing approximation batch benefit smoothing estimate limited experimentally demonstrate line ascent algorithm particle significantly em counterpart particle estimate yield particle uniformly regularity ascent procedure justify prove alternative set assign density joint line approximate unfortunately design linear variable gibbs strategy see particle mcmc class mcmc method build high proposal manner particle metropolis acceptance sampler eq implement appear probability sampler particle particle run tp dx tn particles particle accept probability ideal acceptance remarkable admit performance large variance increase likelihood ideal sampler target component general theoretical analysis select variance typically linearly mean algorithm upon improvement sampler particle unclear emphasize contrary estimation procedure none degeneracy particle variance increase favorable approximate particle approximate density n particle merely introduce possess discuss bind degenerate successive resample diversity approximation clearly simple asymptotic polynomial approach artificial parameter approximate density propose estimate static transform slowly whose bandwidth artificial correction introduced improve recently much result artificial require practical filtering bring information process chain run sample approximately px ix increment run n l lag introduce vanishing present avoid introduction model lag originally propose consist auxiliary approximately distribute p n n n add particle ii mcmc step nn contrary ergodic ergodicity increase iteration would prevent use suggest practice exponential family type exponential scenario n extension conditionally integrate propose artificial dynamic advantage add diversity rely implicitly even necessary statistic degeneracy notice conclusion practical observe estimate lot run demonstrate come accumulation particle mcmc sequentially reweighted substitute running degeneracy reach step update truly approximate posterior instance assess focus numerically impact degeneracy particle degeneracy practitioner valuable simple exact use hence dimensional effect degeneracy panel variance panel panel column method particle particle middle investigate bias rely n detail second per sake approximations numerical simulate replication compute empirical particle separately variance mse rescale growth particle roughly mse particular low superior agreement result variance growth appear sharp h realization algorithm panel panel dot horizontal ml kalman affect see previous every case replication size estimation statistic fairly particle I achieve use say
induce v notational work motivate rank lipschitz dominant term right notation result never section popular constant offer factor proof obvious get first force contraction principle cover argument r good key theorem rely inequality prediction case build define constant loss consider well online guarantee recall refer algorithm gradient set exponent first thus encouraging approach generalize apply erm possibility derive tight bound principle scalar equal bind clear take aware contraction principle pose apply context allow lipschitz involve number loss bound scale scalar vector plug prove recall rademacher convenient refined version rademacher bernoulli uniformly complexity follow plug optimally uniform erm logarithmic computable least hold large great building intuition mention additionally suffer serious loss base begin need essentially provide constant basic class function smooth cover class cover appeal result rademacher bound place smooth constant bound probability upper negative decrease increase satisfie least prove minimizer elementary calculation loss constant loss factor aware bound document particular dependence since top version variant technique error bound optimistic derive smoothness loss lipschitz improve pass argument rademacher logarithmic thing argument much apply situation multi acknowledge nsf expression q yield shorthand denote arithmetic mean geometric give final appealing give prove proposition w give put bound lemma smoothness function play role generalization bound risk minimization affect error rank continuity become continuity enable rate give art loss contraction rademacher complexity turn contraction refer generalization bound smoothness optimistic learn argument question smoothness consider use default lead suboptimal well regret guide average expectation establish error uniform loss norm rank expect right optimistic result generalization erm prove convergence convex illustration literature discover lipschitz constant number document novel theoretical insight retrieval lot recent research sometimes call subset bipartite training
programming formulation execution ad hoc negativity penalty nmf focus popular lee present speed result rather implementation algorithm dataset library science multiplicative provide fast continue hold collection collection use roughly report error svd nmf come also nmf iterative know sensitive initialization old four choose factor algorithm dense random random generally centroid decomposition expensive nmf compute quickly least software svd factor text event svd dimensional centroid consume centroid decomposition fast random averaging vector completely initialization initialization centroid initialization inspire term initialization choose column generally column text idea centroid center co occurrence method co occurrence matrix form co expensive dataset expensive impractical summarize name random lee intuitive basis centroid nmf must run intuitive foundation centroid random slight occurrence large co occurrence expensive subset classify frequently occur document website number report relative computed storage centroid random occurrence sec already column average column quantitative qualitative report mining application essential rank precise solution table easy give report basis error global minimum remarkable closeness subsequent table initialization start table random maintain occurrence initialization suffer diversity topic cover vector easy nonnegative factorization basis classification classification two category classify previously report c c offer bank rate company attack stock bank exclude market c pac induce country fed aid create c c bank company offer bank stock create bank trade offer attack net net country trade bank interest company feed gpu share initialization c analyst bank analyst analyst analyst bank market bank market analyst trade market price bank bank mark national price trade bank price market nearly nmf simple run fix iteration expense convergence number appropriate frobenius angular describe paragraph frobenius exploit norm compute trace angular moderately storage intuitively appeal angle successive I iteration vector converge show similar simultaneously objective clearly angular convergence measure maintain descent require maintain angular measure angular expensive require note regardless choose criterion wise compute iteration period htbp recent lin criterion issue lin stationarity check stop instance stationarity lin propose criterion stationarity fit agree conduct termination present truncate however guarantee minimum must recommend type one paper poor lastly stop replace application iterate reach unnecessary especially case one interested qualitative produce cm good nonnegative factorization nmf algorithm sensitive initialization alternate square include present six four algorithm lastly appropriate criterion key word alternate text classification b department mathematics college usa phone department institute advanced north university usa phone nsf institute nc david follow collection store count time document stack document create document intensity create recommendation customer observe gene sequence condition many interesting create goal mining automatically retrieve interpretable within decade call information retrieval extend mining popular achieve use thereby create factorization decomposition linear svd work point svd interpretation work common user next however know nmf nearly goal nmf familiar phrase term frequently replace observation etc application processing store huge matrix sparse nonnegative decade realize replace matrix several advantage denote rank much element identify essential component attribute low svd pca ica etc nmf problem use svd retrieval k v call indexing approximation reveal svd particularly possible approximation nice fast algorithm uniqueness concern successive comparison need available truncate svd dominate mining appear storage subset matrix svd produce always dense many sometimes svd truncate understand statement say document vector basis sign interpretation interpretability issue svd basis sign vector maintain mean local practice nmf prohibitive sized problem different nmf nmf approximation distance unique orthogonality orthogonal space hand nmf factorization maintain factor easy application basis naturally correspond conceptual strength one become severe nmf issue different nmf local minima minima guarantee algorithm become application help guide initialization issue currently create sparse desire accuracy improve undesirable nmf algorithm avoid alternate least square wherein least use problem problem restriction add user however parameter user resort advanced provide intuitive bound alternate differently alternate solve must algorithm design least appear function matlab active swap basis still remain bottleneck result make ad hoc appealing well practical algorithm initialize w solve
importance discuss focus kolmogorov connect rate distortion information theoretic rate lie direction communication relate communication sensor step treat adaptive source code method mean zhang et minimax machine machine construct constraint place treat nonparametric method divide smaller interesting promising adapt al estimation trading bad exponent minimax risk complexity denote monotonicity satisfy equality q mutual obtain follow concavity since calculation joint eq conclude suppose prior q desire low remain complement cauchy formula therefore write decomposition characterize quantization combine together involve codebook stein satisfy geometry orthogonal third scaling factor front inner space furthermore symmetry spherical q pn analyse us b px previously deviation tail dimensional sphere fix unit logarithm exponent nm pn sphere let vector universal acknowledgement nsf grant fa amazon education grant author comment lemma remark pt university central characterize normal mean storage particular limit encode excess risk sharp establish pareto tradeoff storage estimation present storage constraint particular bit excess risk noise ball pareto tradeoff storage methodology decrease asymptotically together minimax bad risk estimation infimum estimator setting increasingly error computationally prohibitive heuristic efficiency incur place estimator eq require within minimax bound feasible estimation pareto tradeoff versus operation computational mean nature establish tight computation apart concern wish question risk budget bit encode motivated star statistical task send limit estimate cloud environment amazon ec estimate store amazon cost dominate cost much lose principle quantization motivation risk storage nonparametric model orthogonal sharp characterization pareto tradeoff curve euclidean ball radius rich stein phenomenon apparent reader consider require minimax distortion source practical advance efficient compression use perhaps relevant excess analysis relate work briefly result distortion produce identically realization distortion allow budget store datum describe bit datum image cardinality refer quantization join tag encoder auto font pt node decoder every join source code write decode risk many normal term function excess minimax pt pareto tradeoff versus bit bit curve pt bind exact define limit view usual quantization excess quantization zero mean illustrate technical supplementary material sketch estimation code follow fact n accord optimization iii prior independent low problem eq n dp combine prior source coding attain give observation encoder radius vector method outline decoder codebook nb sphere encode eq store b nb bits b n xx make several remark code asymptotically codebook mapping uniform idea behind encode magnitude procedure norm computational shannon codebook implement attain desire vector could possibly code desire low probability codebook distribution magnitude agree intuition ball base uniform ball supplementary material
presence use approach probabilistic approximation provide pr example fast true rate efficient efficient open clean efficiently exist dy dy dy p dy dy dy dy x far form target conditional probability p dy dy dy dy dy thus follow b u composition prove proposition feature feature converge rkhs rkhs e ne weak large target rkh e n properly modify map constant k hoeffding equation rkh nx I inequality kx kx f theorem first estimate discrepancy density ratio bregman fr subgradient sample numerator rx fr fr class fr fr fr fr fr fr fr fr fr asymmetric rademacher complexity exploit bregman divergence rate noisy ratio exploit bregman divergence consistent approximation target ratio approach sufficiently pr theorem bregman divergence distance induce estimate square rate bregman x combine equation iw compare hinge non convexity dataset mean var var e conduct randomly split training accuracy dependent develop empirically perform well often implement uci asymmetric baseline estimate use iw synthetic estimate estimate true c heart e iw image iw synthetic comparison tuning even need estimate datum show hinge bad uci occur two space inaccurate difficulty noise valuable approximately efficiency importance test gaussian performance show bold iw comparison uci benchmark iw loss iw perform comparison c iw iw breast heart importance learn optimal one empirical rate density ratio numerator denominator space hard depend feature centre computation engineering technology technology liu com label observe independently fundamental scenario surrogate design label classification noisy noise show consequently bind reach confirm efficiency crucially rely situation corrupt therefore inaccurate attract significant amount machine random independently prove pac learnable soon pac introduce follow query restriction statistical approach learn intuitively bayesian support multiple reader survey algorithm free important investigate classification loss pac learnable class vc dimension analyze risk surrogate report asymmetric model class loss unbiased estimator risk unbiased surrogate loss use dependent cost latter notable calibrate surrogate calibrate improve benefit classification design noisy weight convexity different focus open limit practical consuming good try estimator design optimize asymmetric flip probability probability exist clean likely noise minimal training mn adversary adversarial target class learn algorithm margin smooth unknown summarize loss neither surrogate calibrate logistic world apart calibrate empirically cauchy loss show surrogate directly presence unknown risk risk df f classifier minimization erm design surrogate algorithms df df df df hand algorithms erm manifold slow guarantee deal derive erm r df f presence still asymmetric address learn presence asymmetric minimizing reweighte ix ip x ip weight rate upper complexity rademacher complexity rademacher
application sense noise note trace dataset low embed high dataset view point locally parametrize nonlinear denote dim boundary parametrize illustration mention kind take direction work point uniformly due analyze clean contaminate tx critical beyond connection common neighbor near complete denote quantile trivial connection dim trivial laplacian remove spectral evaluate notation eigenvalue set sample eq eigenvector embed different parametrization dataset idea new coordinate dataset clearly matter meaningful detail might discuss embed dark red mean norm datum map remove remove shaped recover figure refer explain seem emphasize relationship surrogate norm dark red mean large vector connection choose circle connection remove outlier remove note outlier parametrization figure dimensional scale presence datum average directly diffusion map task need point diffusion base determine near since ground may check distance preserve near rank measure quantify rank estimate plot cdf variant benefit dataset clean perform predict performance well non negligible portion well proved respectively large worse well even investigate influence connection function discuss connection dataset contain randomly version set align classify kind shape etc solve coordinate general rotation resample interpolation numerical numerical dataset circle equally spaced radial group circle equally since image equally rotation act exactly choose surrogate kf kn clean contaminate sample build connection next assign connection neighbor denote w recover rotation connection remove diagonal entry top build slight description describe rotation could rotation complex th complex difference estimate rotation ground observe rotation visualize angle piecewise exist clearly entry mention connection way could dataset graph semidefinite component interested etc diagonal remove h build clean presence obviously bad graph laplacian suppose convex dependent satisfied independent interval width map indeed give lipschitz inequality second immediately find invariance gaussian simple clear display imply see naturally discretized hausdorff choose affinity might discuss symmetric complete eigenvector eigenvalue note act view walk affinity status move modify accord encode connection might give regard synchronization example imaging obtain scale macro scale imaging frame least synchronization problem realization synchronization group intuition synchronization notion vector status encode top eigenvector view status vertex status value connection define notice walk confusion eigenvector decrease special spectral find visualize want first eigenvector mention comment specific allow study system understand bundle simplify exploration refer reader compact dim smooth x dx uniformly simplify constant uniform find assumption collect independently geodesic practice access bundle value function assumption present field associate bundle function discussion note tangent evaluate comparison work indeed tangent mapping field evaluate generalization map geometrically map coordinate field function connection frame encode geometry underlie viewpoint angular replace function laplacian operator derivative convergence operator heat operator onto mention eigenvalue also eigenvalue associate eigen decrease increase able discuss function theorem status trivial function act independent manifold status figure root theory refer trivial bundle eigenvalue matrix asymptotically constant curvature diameter hold laplacian diameter combine subsection application geodesic diffusion dim hilbert schmidt become product interval guarantee finite map abuse approximate riemannian manifold eq q diffusion time tu li tm laplacian map dm diffusion choose yet point propose li variation mention global signature diffusion although different connection dm dm sample refer diffusion eq map define pair dm dd discretization resp abuse notation dm dd dm suppose spectral dd accurate allow geodesic eigen statement dim satisfy g ed tn truncate map show embed embed uniformly uniformly want visualize circle topology visualization guarantee topology counting eigenvector analysis preserve visualization berkeley edu nsf dms wu stanford fa matrices diffusion geometry f analytic technique laplacian idea appear study additive product modification algorithm increase illustrate simulation mathematic idea type object measurement datum wide problem image technique channel etc far us method depend undirected vertex affinity function metric lie sample nuisance component connection denote analyst image establish experiment application metric dataset present information see example laplacian lie focus give motivate projection macro molecular collect model ray macro molecular special ray transform ray treat analyst reader canonical vector therefore inverse increase play particular ray transform direction know would properly align snr projection image proceed common image ray unit conceptually ray rotation unchanged rotation rotation angle stand rotation angle rotation clearly nuisance want measure e j distance ray align think vector like estimate element equip macro interest turn diffusion produce good information rotation ray imply analytic em understand corrupted noise concept geometry image connection sphere encode graph geometric derive estimate understanding impact free aim impact interesting practically useful procedure concern impact way may impact three building first em near determine enough existence likely create near graph build clean projection instead neighbor noise although affinity invariant noisy quite affinity determine experimental setup free corrupt additive follow scheme graph affinity connection connection connection intrinsic draw turn estimation geodesic manifold refer scalar denote entry denote block entry block matrix give connection graph affinity define associate connection hermitian interest large equivalently correspond synchronization property study impact two main explain affinity noise suggestion method graph commonly one also main work matrix theory show notation repeatedly transform stand hadamard entry matrix explain presence apply algorithm give subsection general subsection study affinity connection put subsection subsection modification apply generally matrix q matrix well data analytic well spectral graph idea affinity think corrupt follow dramatically spectral eq suppose exist remarkably condition could completely magnitude purpose eigenvector noisy call scalar entry assumption lemma since interest however eq approximation turn significant consequence lemma standard computing handle work weight suppose furthermore exist lemma computation e significance recent mathematic unknown ratio allow developed tool much since dataset block diagonal q imply imply q far trivially turn average broadly accept contamination demonstrate assume observe version image interested unobserved clean pure object datum dimension dim domain denote accordingly transform mean transform act setup replace invariant transform take grid angle equally spaced point compactly support inside disk angle exact hand discretize act discretized view dim act object act mean act simplify discretized object argument q approximation conclude circumstance transform p n lemma get fact semi also eq argument case light follow p contains follow assumption natural refer reader page immediate consequence gx gx gx fx prove get weak study word furthermore translate example eigenvalue equal eigenvalue might due discretization pixel discretize pixel reasonable interesting namely situation grid depend evident mostly correspond commonly polynomial q ij notation complicate clean behave rotation minimizer another contain exact rotation ij ij ij canonical metric orthogonal group positive independently proof theorem suppose underlying rotation go infinity g tend infinity regularity clean computed rotation compute step ij minimizer hence indeed plugging imply conclude assumption near minimizer minimizer robustness tie clean image number reflect difficult reconstruct manifold precisely reach small normal highlight clean distant distant geodesic fine fine subsection since purpose current far refer interested subsection argument discretization would argument operation map grow polynomially quadratic form gaussian extend hold example entry particularly contamination corruption pixel variable standard argument lemma slightly thing large replace whereas large eigenvalue weak instance bound would change dependence
robust outlier relevant contamination subset implementation resource explain replicate figure display monotonically hard little lose discrete configuration bias hard nearby outlier distant outlier increase colored dotted percentile point leave regardless spatial outlier fit simulation poor increasingly increase show almost bias curve gap increase still influence correctly correctly qp across comparable bias setting qp despite configuration curve correspond fits th percentile fit median finding performance method outlier contamination resource plot outli misclassification principal focus loading result outcome nearly separate outli pca character scale set remove choice add use section package resource explain provide replicate implementation seed result seed outlier observation put scale since outli curse dimensionality relatively balance analyze wish online resource use parametrize parsimonious eigenvalue criterion minor model contain replication hand extract nine replication coefficient shape manually combine replication digit digit plot main outli panel dark blue light curve member visually vertical loading outlier majority assign exclude space examine set dna collect separated value henceforth non comprise measurement application measurement light non dark visually group appear distinguish vertical group difference variability reveal fit detect none outlier additionally identifies fit robust meet robust method outcome deal outlier fail change pursuit fit index pursuit criterion pursuit thus example arise nearly pursuit contamination easy real field widely establish situation enough art outlier prefer inference plan bad distribution corrupt consist unknown contamination denote entry finite context define point estimate sample original whereby cloud form lie definition simplex determinant index shift member depend zero capable satisfy outlier pca estimate break contradiction contaminate must satisfy index unbounded unbounded subset pp population index index position I numerator exist orthogonal projection pursuit approach fix q exist fix conversely equation equation side high give lemma n h em analyze pca fit therefore reveal pca exceed carry extensive systematically outlier keyword outli exploratory pca used explore new account variation need criterion like robust pca case exceeds shift estimate computable accurately observation heavily contaminate insensitive pca relate robustness concern surprisingly instances fail criterion introduce robust meet criterion several search free outlier minimal clean compute loading principal component loss robustness retain component begin draw integer specify least one default offer possibility diagonal eigenvector first loading next compute loading measure member use square direction remove denominator direction increase subset contain index observation value number set eq cm evaluate pca give direction optimal index convention measure member share direction member decrease denominator numerator increase overall grow equation consider direction select candidate subset give denote full space accuracy summarize I member subspace numerator denominator member member call experiment index rarely projection pursuit pp proceed direction pp computationally pp outlier fact select h pp appendix similar also select select configuration contamination small eigenvalue end criterion use scatter subset bias towards outlier criterion favor cause assess pca classify outlier observation assume normality normally one freedom index second pp subset base dominate grow discuss comparable around overall practice impractical nevertheless suitable application class number processor suit enhance ex package evaluate pca although pca
set generate randomly pick strong weak variance compute calculate weak strong agreement theoretical offer formula full scheme however thing exposure exposure full match study fix variable relationship fix effect strength base choose respectively correspond regression intercept poisson estimator rely form episode exposure median absolute proxy deviation max max size full efficiency match size value standardize standardized bias instrumental full matching low absolute little large full main manuscript aspect median absolute method tend higher expect nonparametric parametric concentration gap quickly bias covariate expect however match cubic exponential log root low logistic high variability method use measure type size design type error I type notable exception function contrast maintain correct linearity compare another parametric covariate design generate similar study manuscript look different strength code author unfortunately code contain upon author issue future manuscript ij ij ij become main manuscript within variance var proposition medical centre institute study control regression instrumental iv offer trait important instrumental stage parametric assumption account additionally square covariate balance blind outcome drawback estimation matching simulate concern child trait decrease per episode cell trait substantial ci clinical treatment age two every height list characteristic public health question clinical cause growth among child estimate occur child child key child growth intervention strategy implement net incidence surveillance body exposure fundamental limitation observational concern important example limitation control child impact child well episode suggest control future study likely child family randomize clinical trial aforementioned study account instrumental exposure outcome instrumental core instrumental associated exposure pathway associate see detailed covariate available plausibility conditioning affect outcome exposure write restriction effectively randomly set correspond exposure pathway pass trait substantial trait provide violate cell trait cell trait carry trait region violate cell study american child child cell trait area trait affect development support although cell trait direct united concern study consider receive risk potential trait induce child thereby increase go pathway tend child trait restriction tend increase assumption condition likely trait mechanism table provide baseline child besides affect plausible observe birth match match among plausible unobserved role match match assignment probability probability unit match set sensitivity possibility influence instrumental monotonicity potential outcome affect individual mr restrict severe weak disease evidence therefore unlikely influence transmission monotonicity individual exposure bring effect exposure plausible risk characterize cause exposure cause iv iv monotonicity whereby episode material q iv monotonicity effect change exposure individual exposure nan thus deviation material regularity condition normal interval estimate maximize specifically give response exposure match solve get ratio material supplementary meet proposition material effect effect binary outcome datum whole attempt unobserve instrumental analysis would impact free condition match latter ij influence study birth covariate pz ij pz ik th unit respectively match set child th difference could trait randomize match odd receive unit odd q child presence hypothesis effect inference significance reject quantify addition interpretation understand material match iv traditional iv conventional assumption covariate conventional outcome iv put outcome generate simulation exposure outcome supplementary detail quantifie produce affect outcome simulation five f ij indicator adopt five normal identity multivariate generate I predict simulate procedure two compute absolute absolute median respect simulation exposure write manuscript section provide statistic nan every ii ij moment eq nan hypothesis statistic derive bias proof supplementary material expect statistic notation lemma mainly I fourth moment growth get rewrite central limit distribution converge high entire normal interval spirit interval corollary present q rewrite rearrange get equation coefficient
stability include follow result inequality ensure exist denote lie rectangle leave hand bound conclude minimax sense recall focus online estimation generally decade knowledge minimax estimation spread study establish moment could localize clearly meet depend present estimator algorithm x minimax rate bound meet condition impose predictor define require burn assume start recall rate technique follow lemma provide matrix spectral radius norm corresponding modulus let integer appear hand modulus lipschitz adopt convention integer division middle jt derive provide compactly iterate recursive vary polynomial constant also positive dd result r tt define uniquely soon remainder independently hence appear respectively choice prediction remain setting give analogously respect get positive introduce jensen get eq xx xx xx xy small index decompose exponent integer get inequality consecutive hence ratio give obtain equality center variance imply q use hand bound center inequality suffice define assumption check index include monotonicity least statistic admit number value proof improve either satisfie adapt ty n aggregate plugging obtain minimal net choice improve condition stronger impose noise aggregated predictor aggregate predictor p pm line obtain observe improve cardinality net aggregate prohibitive acknowledgement acknowledge de en de de france period parameter initialization convention study aggregate rely essentially three uniform time vary linear condition linear moment sharp inequality adaptive time autoregressive aggregate predictor minimax rate condition aggregate adapt rate study aggregate applicable application high stationarity smooth environment model time sequentially track operation low storage prediction online normalise kalman rely smoothly statistical evolve along adapt practice exponentially aggregation tuning emphasize meet online computation validation weighting aggregation develop parallel learn seminal statistical game community prediction survey statistical set stochastic setting design exponential weighting investigate stationary recently inspire sequence contribution exponential compute aggregate possibly aggregate past feature therein recent view inference stationary application locally vary autoregressive index propose provide solution rise minimax paper organize provide oracle aggregate predictor stationary process section base set aggregation method proof oracle risk build predictor appendix proof additional aggregation serie non coefficient deduce instance guarantee infinite almost moment context representation admit decomposition say often existence often use vary extend stationary vary sequence suppose general process come sequence additional sequence derive appropriate sensible introduce sequence introduce approximate sequence rescale inference assume smoothness reference therein white see suitable derive predictor smoothness extension sublinear since representation iterate model aggregation additional negative negative mild basic stationary usually rely weak recent work assumption usual predictor evaluate period predictor wish sequentially derive new predict accurately accurately well present aggregate predictor simplex recent contribution find sequential label recursively remain predictor follow building convention later element ti n tt x k compute brevity switch distinguish strategy whole oracle error aggregate convex remain term aggregate control negative predictor l satisfy number context autoregressive generally tx lipschitz satisfy stationary hold satisfying label aggregated noise aggregated predictor obtain find section sense directly appear assumption assumption aggregate give aggregate use assume aggregate directly since require remain risk compare explain well remain inequality page prohibitive optimal order choose provide hence noise observe influence risk choice follow trust existence autoregressive let define similar vary ar coefficient continuous vary unique eq moreover proposition kind initial variation smoothness representation seminal consequence bound sequence vary ar coefficient denote expectation assumption admit standard lower satisfied set determine index refer time rather correspond study statistical prediction predict sense sequence predictor definition sup smoothness subtracting exactly observe addition accordingly provide rely bind bind exhibit smoothness unknown smoothness locally clearly control use provide combine show predictor automatically minimax hold see seem minimax predictor sufficiently minimax section aggregate adaptive assumption seminal minimax nonparametric presentation model maximal process natural practical predictor behave aggregated predictor behave aggregated predictor ingredient minimax refer fact build lipschitz depend predictor possibly predictor say predictor follow locally minimax predictor achieve rate proof say minimax predictor smoothness sufficient minimax rate aggregate one locally bound predictor satisfy aggregate follow assumption satisfy section limitation obtain optimize account factor see drop oracle condition require minimax predictor roughly require operation restriction strong correspond appear eq exp indeed case front remain three equation aggregation deterministic aggregate adapting n adapt unbounded convexity probability lemma distribution ta get log multiply develop successively x lipschitz plug hold get inequality independent apply take expectation jensen give appear remains decrease simplify simply set autoregressive density convention nonzero write henceforth proof
goal environmental science collection scientific reality stochastic may understand biology environmental equation notable infect sir commonly dynamical ode transmission disease population allow simplicity dynamical disease two deterministic describe however stochastic dynamical select observe context via criterion suitable dynamical biology methodology procedure datum abc abc monte sequential smc smc drawback abc intermediate detailed incorporate abc smc hierarchical perform estimation credible type appear dynamical e framework become remainder organize provide use section describe smc simulate dataset population state understand transmission several deterministic deterministic sde study division epidemic epidemic occur dataset include key direct transmission indirect transmission infect mass environment characteristic population model stage specify account several adopt transmission number infect total accumulate accumulate discrete generality consider model transmission infect transmission sde ode direct transmission add population unknown transmission per markov epidemic direct transmission ode equation direct transmission ode sde ode may simple transition interact direct transmission wiener square derivation transmission sde derivation complex sde indirect transmission ode sde let environment indirect indirect transmission coefficient infect specific rate correspond sde model transmission let denote increment length length add infect infection death small furthermore square identity euler sde wiener converge square system initial condition note choose beta discrete gamma describe section discrete ct depend integration infeasible carry monte unobserve approach consecutive observation environmental epidemic record simulate straightforward many numerical euler simulate simple numerical euler monte algorithm suitable choice model build basic datum accept simulated data rejection propose smc proposal avoids rate algorithm white tolerance index index sample else sample candidate normalize go smc proportion select abc context simultaneously bayes commonly uniform normal euclidean smc simulate include death epidemic indirect model st simulated observe smc five describe abc smc algorithm set describe record indirect transmission five indirect transmission sde compute factor indirect transmission sde model high bayes factor indirect transmission sde figure favor indirect sde indirect transmission sde high dataset factor bayes although always pick indirect transmission sde apparent favor model weak h dataset note indirect sde smc epidemic parameter abc smc distribution ode function simulate sde euler one month epidemic h figure favor indirect compare significant factor favor indirect transmission strong average forecast predict epidemic bayes favor epidemic smc l ode direct sde indirect ode mode interval table fairly uncertainty surprising transmission mechanism quantify still try mechanism transmission epidemic super epidemic indirect transmission rate agent assess cumulative indirect transmission sde epidemic trajectory predict reasonable datum count smc environmental process continue
choose motivating give point lattice pseudo study sufficient table report initial also table run modification different past sample subsequent stochastic probably instead however analysis modify becomes decide include modify research mathematic mathematic approximation estimator intractable adaptive monte adjust control examine asymptotic new algorithm resample reduce degeneracy combined resample small estimator impossible intractable spatial statistic wide range constant overcome model include paper prove normality adjust generalize version importance complicated use markov asymptotic motivating statistic family unnormalized density dominate maximizer constant estimate form instrumental density maximizer guess behind instrumental density defer subsection far parametric instrumental value update history algorithm put measurable c continuous constant trivially uniformly insufficient likelihood approximation converge converge iff maximizer sequence converge gx mx g mx mx gx converge surely prove inequality cb step sure convergence introduce maximizer maximize surely already pointwise accumulation unique maximizer conclusion immediately easy boundedness concrete log maximizer assume maximizer normality derivative continuous inequality fulfil asymptotically stochastically begin regularity also consistency automatically fulfil concave adaptation appropriately family particular motivate play similar together appendix position well need show q denominator enough asymptotic use notation martingale dominate assumption exponent lyapunov last formula sufficient taylor expansion consequently zero hold adaptation improve instrumental use instrumental maximizer multidimensional dispersion various determinant eigenvalue trace covariance trace equal asymptotic tr schwarz optimality theoretical might show exist mc illustration distribution odd ratio family scalar omit optimum instrumental version suffer degeneracy instrumental instrumental density degenerate effectively impractical practically eq computed mc may change adaptive algorithm compute history special letting exploit derivative conditionally efficient degeneracy instrumental parameter markov kernel preserve compute put ik ty uniformly family var pt close look reveal indeed q within produce complicated maximizer ty surely boundednes martingale consequently pointwise convergence compact resample purely instrumental evaluated variance also ty value statistic key estimator fulfil difference therein run concavity hold apply difference give discussion precede order derivative
exploit effectively intersect introduce directional space avoid neighbor cluster establish variant namely use framework hybrid model directional local sparse coding improve underlying structure corruption geodesic practice validate art euclidean notable test synthetic sequence brain image texture seed digital technology university fellowship institute review describe implementation compete compete sparse smc embed sphere adapt current smc map tangent map task spectral matrix riemannian metric apply whose page replace euclidean metric standard riemannian third compete scheme embed lie riemannian apply classical manifold embed embed euclidean space manifold pd sphere already embed cluster radius distance angle threshold notice change local tangent space large covariance precisely number gap smc accept publication code code comparison number smc remark weight ji ij ji weight unstable experiment collapse experiment ij suggest try notice ji accurate though advantage overall ii vi exponential use rest experiment vi level phenomenon figure set analogous computing pd logarithm map sample general riemannian manifold slow logarithm require column orthonormal comprise subspace need singular equivalently pd compute logarithm complexity major cholesky decomposition matrix multiplication equivalently find eq dot take gx riemannian image computation map computational point involve distance riemannian complexity riemannian cr cl computational sphere pd major occur near associate facilitate use neighbor assume effort small one second product necessary entail nn product task solve solver popular alternate direction method multiplier covariance compute operation knn nd nk affinity burden identify entail order nd nk zero geodesic angle reduce moreover affinity thus order become kn nk nn complexity sphere approximate neighbor total special kn nk preprocessing near consider specific avoid leverage elementary algebra eigenvector xx tx coincide small cost pick subsequence convergent approach infinity gx rr origin angle arbitrarily contradict measure inequality application bind er gx depend depend riemannian apply orthogonal appropriate change matrix word th entry satisfy rhs hard see constant later perform schmidt orthogonal use preserve sign induction sign assume rectangular express eq upper triangular estimate change row rotation next sign lipschitz function rhs result denote tw I far finally side simplify equation sign induction b ix r definition h direct detail exclude imply riemannian claim follow lemma work tangent space q minimizer map map via new express rhs bounded rhs follow argument plug fact let follow prove generalize subspace fix concave fact imply long lemma proposition theorem definition axiom edu electrical digital technology university mn usa riemannian manifold lie important cluster algorithm matrix already texture medical imaging clustering construct affinity exploit geometry intrinsic geometry encode code importantly directional tangent space establish intersect geodesic extensive validation real theoretical state modern moderate quantitative framework study manifold hybrid linear framework dataset model union whereas propose underlying try propose cluster apply euclidean space nevertheless domain riemannian manifold sphere manifold symmetric semidefinite psd moving model utilize extract low identify spatio filter pd manifold texture nevertheless strategy handle application advance representation dimensionality development modern rely assumption exhibit subspace embed cluster manifold generalization develop example illustrate nonnegative distance ms motion ms analytic manifold promise geodesic distance manifold solve cluster dimensionality work quite convex separate intersect closely locate accommodate subspace restrict manifold embed suggest also strategy include inspire algebraic method strategy code sphere coding encode subspace energy minimization local order pca guarantee multiscale strategy riemannian orthogonal group pd manifold address spirit tangent space logarithm transform tangent neighborhood integrate infer despite popularity generic scheme low embed space end provide euclidean intersect nontrivial clearly modeling paradigm generalization superior performance extensive synthetic dataset efficiently restrict applicability work analogy nevertheless apply since neighborhood furthermore model distinguished previous multi manifold careful incorporation directional local intersection ii neighboring cluster different algorithm neighborhood include multi careful choice formulate preliminary background riemannian assume dataset lie geodesic riemannian gs mx subspace w generate uniform two htb justification setting include kind noise review riemannian review recommend riemannian tensor geodesic minimize among curve tangent exponential coordinate around image cf maps htb sx ix ix jt problem support define key quantity quantify directional geodesic angle present solution well concept geodesic geodesic logarithm radius neighborhood sample rt I nd intrinsic tangent form bottom cf tangent subspace span top exceed see top top occur short geodesic connect tangent empirical cf euclidean motivation propose practical discuss geodesic tangent spectral carefully represent locally come sake underlie assume accord connect clearly intersection figure intersection demonstrate able tangent geodesic angle beneficial neighboring belong red local disk exclude red done angle w point figure arbitrary point estimate difference estimate geodesic angle reliable dimension neighborhood close intersection neighborhood away intersection cf connect neighbor linear algebraic intersection filtering procedure guarantee proof cluster geodesic tangent neighborhood radius threshold estimate tangent distance threshold set label assign geometric point f matrix eigenvector whose exceed angle affinity apply affinity follow achieve correct statement rely underlying simplicity geodesic geodesic smooth compact geodesic generate satisfy inequality correctly sufficiently relative geodesic tangent practical geodesic tangent choice differ threshold one drop affinity indeed indicate portion point intersection sense connect point intersection pairwise coding produce large weight point come point radius default angle default compute among ij x nn cluster affinity algorithm task use code multiply latter increase nearby local explanation cluster code tb detect use avoid assign far blue briefly leave many neighborhood choice r compute cr cl depend riemannian cl symmetric pd matrix cl dimension cl riemannian modeling logarithm discuss rather computational particular sphere complexity assess performance follow adapt smc riemannian metric review appendix labeling label maximal label label q six generate dataset iii iv pd vi white construction non draw comprise random dataset subspace comprise two pair group model symmetric random group generate entry normal construct dimensional dataset comprise lie parallel arc draw vector since bar blue accurate bar one mean step diffusion direction baseline dimensionality generate direction noise detail level brain carry riemannian pair randomly perform report accuracy clearly suggest cluster demonstrate among compete assess structure various image goal independently pixel e capture apply setting obtained obtain transformation database covariance texture patch size feature specific patch carry transform patch covariance belong texture pattern patch demonstrate database angle shift image shift patch draw affine affine transform image plot dataset onto euclidean demonstrate transformation average cluster rate report transformation horizontal spatio dynamic video action leverage average experiment spatio temporal database employ associate spatio temporal patch subspace cluster distinguish spatio study govern temporal rule frame video eq latent vector distribution respectively explain subspace spatio accord procedure arbitrarily mp dm th conduct subspace give rise dataset rise video database video randomly distinct per frame patch patch reduce ft estimate underlie consequently category cluster lie pattern look shift version visualize subspace project cluster contain video speed video associate video spatio database patch spatio patch previous set associate consequently subspace create random video represent motion procedure choose video database repeat clustering table achieve highest clearly sample nonempty form remain whose connect exactly appropriate parameter specify organization describe present undesirable event negligible theorem briefly simplicity sufficiently end set instrumental fix later angle second within estimate noiseless discuss generalization idea consider modeling euclidean apply general lie euclidean manifold basic infer ambient space riemannian local different system local directional tangent space uniform assumption ambient care map e logarithm refer complete introduction riemannian geometry map tangent space coordinate endow riemannian choose metric north straight shortest short geodesic connect blue connect origin point measure n give throughout noisy thus denote support expect covariance matrix compact w bx r z bx xt xt mt principal recall apply geodesic lastly stand th review detail geodesic definition geodesic riemannian short geodesic connect let compact riemannian neighborhood example gx gx xx local matrix neighborhood define event define cover sample fraction proportional concentration covariance ensure assume establish geometric property constant appear angle intersection last claim gx imply constant transform third expect equality slight depend riemannian metric arise observation conclude triangle inequality thus next exist define order follow distance span eigenvector st st
impact make represent view well assumption particle move correspond detection detection accord predict dependent weight denote modify express inside outside keep probability within camera term dash equip estimate multiple camera pair proceed describe camera calibration reliable estimation require knowledge central object derive objective section multi previous camera joint origin system align camera position orientation calibrate camera pair space camera camera position camera orientation velocity principal distortion camera camera space jointly state object scene well right left distribution camera q one impractical prefer single variate describe detailed relate generate express camera assess either spurious confirm status interestingly track camera group track similarity hierarchical estimation single object exhibit linearity conclude wish possibility right infinitely camera representation compose particle express dirac density camera gm particle implementation sensitive curse dimensionality need maintain grow exponentially calibration particle approximation make distribution observation update estimation target degeneracy particle update camera end experiment away camera pair resample frequently cope observation approach filter particle particle assess representation single evaluate track performance commonly employ problem return match assignment cutoff determine shift emphasis latter influence sensitivity outli mahalanobis locate plane axis gm detail six velocity metric appear increase induce merge prune false alarm propose show depth particle moving target object track camera configuration one consider estimation position orientation calibration particle calibration figure demonstrate object cover whereas distance axis component would allow high address camera image propose novel camera calibration relate move target sensor exploit space camera track scenario well estimate filter object correspondence object correct automatic develop advance sensor calibration compare scenario fellowship science grant ep camera calibration camera address unify joint object state problematic exploit tracking version kalman correspondence hypothesis density filter ability update explicit measurement association target discriminate clutter camera static move object determine sensor kalman statistically sensor away sensor passive alternative observation case position method datum consider joint geometry make proxy express enable filter logical address object multi object track camera statistical follow measurement simple single state estimation calibrate follow calibrate camera since address independently turn concept object correspondence estimation calibration image vision tracking majority algorithm interested object co nonlinear dependent inherently estimation statistical finding rao previously investigate though researcher transform estimate uncertainty thereby consensus uncertainty non complementary instrumental reducing despite reliable camera measurement belief camera measurement direct define reflect separation computer extract attention focus space euclidean projection two noise range consequently object assumption kalman example know mean minimum establish practical theory extend tracking refer position move tracking sequence measurement filter sensor due comprises step describe update bayes sensor filter dynamic noise kalman long valid deal mild kalman kalman motion case vast majority ultimately know velocity world system history kalman filter presence observation kalman poor particularly depth kalman variant almost track target description state bias filtering specifically design sensor characteristic successfully integrate vision track camera relate sensor come sensor system environment know false sensor rely measurement identify vision suffer robustness rely measurement frame image optimally development sensor practitioner overcome association integrate estimation object filter develop set develop multi location multiple sensor discard confirm sensor image sensor rate measurement necessary sensor object attract international moment approximation filter density object statistical filter provide mathematical foundation datum assign measurement correspond ahead object robust scenario filter complexity target static recursively use make calibration refer imaging view scene solve inverse accurately scene reconstruct ground knowledge calibration euclidean perform directly projective practice calibration hence extract appropriate calibration assume calibration turn projective update measurement certain correspondence possibility incorrect consider consequence input begin calibration perfect input formulate extension object fact calibration multi object inherently camera address problem condition turn camera simultaneous contribute self underlie pair illustrate real onto respective recover purpose relation real formulate projective geometry triple represent projective henceforth projective perspective relate homogeneous refer equality perspective projection concept purpose perspective must namely express coordinate resp plane link camera setup cf non camera assume camera form baseline henceforth respective u camera projective geometry linear projective camera camera pair express mean proxy convert equivalent via inverse allow camera right camera plane respective projection associate maintain one correspondence purpose state state object intrinsic behaviour tracking justify compare resolution position sensor extent observable shape camera calibration extent increase difficulty convergence model extend observation camera affect spread resort distribution another stem know noisy nature image generally space non raise although object circumstance approximate however serious limitation infinitely far camera dotted curve contrast dash highlight usual representation space easily previous camera via gaussian uncertainty back transform make uncertainty kalman demonstrate inverse inverse suitable tracking modelling instance velocity object coordinate velocity space velocity step model object dynamic raise relate decompose step sample particle result recover gaussian mean kalman approximate enable nonlinearity reason kalman filter non linearity observation fairly represent recover particle optimistic yet objective aspect distribution several case overall yet evolution case appear though shift evolution well particle particle capture motion associate object handle kalman observation camera covariance suitably augment velocity unobserved specify camera say camera consider receive observation velocity compute distance camera sufficiently enough whenever infinitely away camera consequence negative represent behind must issue general determined carry uncertainty challenge camera geometry available nonlinearity camera yet detailed beneficial prediction handle camera pair similar space camera camera property strong introduce left respectively say abstract hence produce camera projective predict space resp resp call particle mapping represent object another onto camera approach object together observation dynamical object two camera right camera velocity distribution
p concrete section time checking theorem depend start vertex write I p direct calculation side c condition negative bipartite generality imply contradiction verify state conclude first claim markov field definition proceeding condition see minimize neighbor equivalent constant next polynomial calculation already omit finally j gm total let let identity follow note eqs therefore condition c compute consistent indeed fairly easy want estimate formulae immediate generalization obviously asymptotic desire markov exist note standard tail yield different choose model logistic fail unless satisfie efficiently develop consistency assumption distribution show establish result use ellipsoid project related round various task I aware statistic acknowledgement partially grant fa fa grateful discuss prior publication self proof yield convenient extend simple next eq induction dimension claim prove boundary let exercise transform claim simplify assume return claim correct replace polynomial oracle point replace query polynomial call sequence project gradient drop definition intermediate I q hence approximation error term soon guarantee already require approximation guarantee oracle conclude mr claim conjecture claim proposition remark task pre straightforward advantageous simplify analysis contrary reduce theoretical implication technique dataset statistic representation subsequently widely adopt turn carry replace solely phenomenon entirely concrete shall da ib clearly notation nn standard prove entail information already n fairly sufficient conditional word able distribution simple example give section word argument base organize computationally efficient statistic approximate polynomial theorem estimator unless remarkably indeed discuss denote possible follow denote deterministic variable hessian add subscript parameter sufficient law denote expectation use subscript emphasize consistent description description two terminology motivated remark terminology discuss indeed consistent surely hence finally consider q polynomial estimator exist return devoted implication negative approximate intractable consistent fairly class remarkably consistent direct consequence polynomial sufficient unless property specialize present consequence present self reader convenience standard statistic notation positive following recall pp pc p inverse mapping concave position version second maximize set modulus gradient work reality oracle assumption efficient replace give project indicate orthogonal
deterministic particle many alternative usual proposal gradient involve particle represent show theoretically experimentally particle capacity validity mnist reasonable network part digit could individual expression base hope insight stochastic feedforward apply context dropout like acknowledge source classification huge variety class complex useful test gradient pixel epoch epoch give see exclude comparison bias follow estimator c test unbiased deterministic na deterministic hybrid computer universit e perceptron mlp give mlp type structured make general activation per lead fundamentally try estimator test compare training estimator feedforward multi mlp model mapping hide typically unimodal isotropic factorial oppose conditional simple distribution advantage conditional configuration produce output approximate multimodal mapping empty add introduce noise hide improve additional setting decision solution cognitive early mix poorly mean inefficient optimize work propose draw feedforward guarantee propagation unit additive dropout long way gradient estimator bias approximate back unbiased network unit provide show demonstrate face person mapping object base hide configuration offer alternate extreme unit back rigorously training method less estimator benchmark mnist study stochastic binary hide layer extension original activation activation perceptron mlp denote row sigmoid softmax probability product q probabilistic deterministic back bring criterion summation derivative directly problem enough increase sigmoid unit behave nonlinearity maximize expectation deterministic maximize output q achievable distribution deterministic game achieve performance look situation divergence negative conditional h dy bad job maximize deterministic ph simply explore share auxiliary network line paper simplify use generalize give rather role mixture notation unbiased estimator new estimator technique draw c empirically sufficiently close estimator hide layer average mini batch epoch train top bottom appendix estimator correspond mnist task train experiment benchmark feedforward network base handwritten database multimodal first digit preprocesse sampling independently grey different individual mean expression subject training datum thus gradient keep perform significantly experiment comparison feedforward deterministic network weight deterministic use way hybrid consist neuron incoming connection neuron use gradient neuron stochastic gradient mini batch momentum epoch epoch good model test exclude comparison notable significantly particle infinite computational performing task binary especially interesting also outperform hybrid output probability neuron learn propagate hybrid easier full
h exist onto q eq norm choice solve subproblem spectral singular theoretically bind norm full slow consider firstly project norm ball project row onto norm ball al logarithmic length matrix preserve sparsity dual variable improve tradeoff mix convenience approximate project descent discuss projection divergence formulate tb initialize kk divergence evaluation parameter divergence px arguably preserve marginal kl intractable tractable surrogate divergence tractable divergence minimize mean inferior support implementation pool step iteration project divergence result marginal focus aspect computation accuracy extra roughly perform accuracy mrf rapid compare marginal absolute marginal mf step wish compare represent norm grid test intractable reference evaluate various denoise berkeley gray purpose xy iy encourage ise wang rapidly truth decrease error compare run time approximately less condition mrf fast project competitive method well sampling mixing set include research centre program give dependency state start distribution sigmoid index influence great change definition previous inside previous dependency convenient form series relaxation convenient rather mrf result matrix mrf prove mrf section give result dual representation ij c lagrangian dual independent straightforward eq eq eq extra thm proof corollary algorithm extremely practice amount sufficient univariate mrfs project project various compare univariate marginal mrfs intractable motivate markov among variational field approximation factorize distribution find tractable kl propagation propagation propagation use approximate kl divergence approximation correspondingly simulate target inference draw markov chain principle mcmc accurate difficulty markov converge mixing inspire model minimize sampling converge distribution ise model mix spectral strength project mix project budget limit ise project first condition univariate mrf euclidean somewhat computationally previous validate divergence project parameter interest gradient pairwise mrfs potential jj x ix parametrization pairwise mrf convert include sometimes convenient statistic indicator configuration representation equivalent review qx qx stationary state markov always distance irrespective start run gibbs sample central dependency one informally change matrix except central rapid mixing theorem generalize informally multiplicative mix multiplicative gibbs n size variety natural ask give computationally norm additional adjacency regular induced p norm rx gx rd use tend dependency large mrf mix necessary tighter convenient field indicate include impact regardless univariate self zero one distance mrfs parameterize project distance apply dependency projection closeness
separately control reveal motivation put error jointly constraint satisfying put seem lead penalty lead return eigenvector outside diagonal entry elsewhere false construct sparse h enough primal dual pair kkt solution optimizer kkt careful analysis second construct unique make elastic fact essentially elastic control establish negative control signal carry frobenius bind false eigenvector sign nonzero actually generalize direction subspace dimension whereas assume require second satisfie diag bind hold claim consistent solution rank unless increase oracle operator error factor small lead dominate gap selection depend condition result insight therefore remove assumption identifiability assumption interpretation assume identifiability long valid true subspace develop free interpretation let interpret maximization technique seek dimension transformation maximize replace sense correspond versa corresponding word sum residual satisfying projection smooth shrink reduction notion sparsity measure persistence eq well imply principal assumption subspace principal subspace principal simple uniqueness point interpretation predictive covariance stable perturbation pca linear establish rate suitably identical sparse roughly speak pca assume q orthogonal correlate term thresholding usually penalize relevant eigenvalue rate selection open time consistently range testing least solve plant clique beyond barrier interpretation lead collection shrink effective give reduce tractable nearly predictive covariance validation lead drive tuning property technical appendix proof section primal start form write tucker problem additionally solution let variable support need kkt feasibility check feasibility nonzero feasibility diagonal agree principal subspace q establish primal elastic net max max z h kkt optimality dual version existence imply consequence uniqueness unique small enough contradiction control obvious view norm wise principal dual sufficient eigenvector entry imply assumption sign tb bb hence frobenius pt lead orthonormal span principal orthonormal f argue condition orthonormal apply proposition rotation cauchy schwarz jointly choose theorem constrain lemma unique verify principal part gap next principal subspace unchanged matrix twice operator noise small th exceed jj z jj come contain perturbation theory dimensional exceed part strict condition dimensional claim inequality invoke comment style graphic label http edu support nsf nsf dms theoretical sparse often implicit raise sparse variable select consistently say result truth investigate general condition fail sparsity dimension technique reduction variable combine central classic seek linear retain much variation set correspond span interpretability yield consistent truly decade pca low convex relaxation two algorithmic development include detection various explicit often implicit truth lead subspace raise sparse say essentially nothing independence second selection agnostic semidefinite program whose sparse estimate formulation perspective rather appeal subspace time develop method multiplier consistency population low plus identity wide include estimator select assume hold important dimensional theoretical mainly exception eigenvector leave open also address broad roughly variable correlate large interestingly min generalize direction paper interpretation beyond independence literature concern main set symmetric assume quantify may helpful think vector necessary theoretical assume semidefinite two probabilistic tail sample q bernstein sub tail constant absolute subsequent introduce tail proof edge adjacency absence pair plant clique everywhere find clique polynomial time plant clique give reduction plant clique pca simplicity presentation focus applicable graph follow parameter efficiently alternate multiplier principal encourage moreover norm assumption mild condition
limitation locality problematic observational uncertainty outlier interest locally graphical ii issue locally model graph improve inference utilize becomes computationally scale fully address crf specific function zhang constraint position encoding al potential pairwise al fully connect crf target output aforementione fully graphical define specific effectiveness key deep take different inference intermediate manner et introduce train conditional factor conditional field sum yu introduce conditional consist layer layer input result observational characterize implicitly limited aspect connect graphical deep structured graphical limitation two type model explore leave exploration unify could benefit improve boundary boundary observational uncertainty outlier fundamental fully due structure study investigate feasibility connect graphical encoding layer maintain deep graphical methodology behind inference segmentation problem section present conclusion observation conditional goal propose incorporate interaction preserve advantage long interaction connect interaction impose high make intractable issue introduce auto encoding make structure measure represent essence variable extract parameter reformulate represent auto encoding characterize auto encoding principle add encoding base determine structure model auto sparsity characterize high number structure auto encoding modeling structure random show configuration fine coarse fine depend configuration utilize next section tp represent mathematically formulate label auto layer multiplied represent label auto layer intra inter layer interaction fully among computed encoding graphical interactive structured computationally tractable interactive image segmentation foreground object background annotated fig tp region foreground background unary gmm gmm parametric histogram apply account tackle train unary pairwise mrf consistency utilize background foreground use layer deep eq train mixture annotate result auto auto layer maximize joint auto encoding represent structure layer characterize image certain specify auto encoding auto hand random adjacent auto encoding fully implicitly variable give encoding layer previous layer compute encode foreground state energy minimize try base datum pass summary encoding py interactive segmentation evaluation database scene use database manual ground single consist image manual ground truth image contaminate range total seed foreground seed q positive false implementation method structured connect also comparison perform crf comparison component unary number encode iii image segmentation encoding encoding layer node annotate sample layer matlab colour total configuration cpu ghz gb ram classify colour tp image ground truth unary f level single dataset table segmentation object noise scenario slightly scenario free noisy noisy noisy object preserve unary appearance fully j light post ground truth unary handle slight model seed water variation illumination texture capture water foreground well handle
often one option expect translation reward bad lead upper confidence scaling need range feed kb select exploit cc maximize confidence sensitive ucb dominate small ucb need negligible even possibility ucb kl advantageous range become minimization subgaussian subgaussian coefficient tc weak bandit finitely arm analyze moment know moment must replace mean modify bound condition moment finite denominator calculate n general cc fast gets get sublinear term note put thing get prove similar simplify inequality b develop every well bernoulli variable transformation scale conduct keep variance shape unable instance shape negligible cox denote differential equation fix brownian motion price option price option monte method simulate trajectory value option monte carlo standard variant use derive w ip multivariate py ix I specifically feature denote correspond accord logit parameter I train result posterior normalization integrate integral monte drawing target posterior suggest step additionally mcmc anneal step move metropolis hasting langevin hamiltonian sampling operator generate use remark via consider sequentially carlo estimator minimize strategy development estimator outcome strategy provide strong practical advantage monte approximate lead task difficult know call unbiased expect error formalize scenario interest practice unbiased method rejection allocation output produce combine mse analyze meta strategy formalize excess mse combine produce contribution reduce bandit identify arm square mse correspond bandit conclude carlo reduction adaptive allocation alternative provide finite cost estimate suitably bandit formulation yield aware estimation generalize monte bound regret performance art base select discover computation complementary adaptive design fix bound proportion optimize differentiable broadly base estimation even approach bandit sequential allocation agent choose step long term payoff payoff action unknown give th payoff action action kt kt maximize rearrange mab show payoff polynomially consistent policy optimal action polynomially payoff upper confidence achieve regret asymptotically ucb drop requirement fit weak finite action ucb regret bound j jt kt kt jt jt kx kt kt improvements ucb ucb v variance construct bind variance arm payoff substitute bind scale relate worst bad ucb practice recent ucb bind chosen increase efficiently smooth kl ucb achieve q term low ucb expect large ucb additional regret another approach receive significant interest ts randomly proportion posterior ts outperform ucb bernoulli indeed time ts payoff problem distribute payoff variance set thompson value payoff resample step obtain kt kt kk kt kt kt formalize finite monte variable converge unknown target different assume estimator previous procedure whose monte evaluate excess implicit excess multiplied sublinear asymptotically match adopt value estimator unweighted sophisticated approach weight sample respective ignore highly weighted variance sample come estimator tv kl regret arbitrary allocation mse estimating satisfie cm algebra reveal bandit versa furthermore allocation obtain regret problem conversely arbitrary observe choice mse bandit observation low x choose k let v dd distribution fraction establish mse regret mention simplicity assume mse use feed modification effect regret additionally regret ucb bandit factor immediately imply regret handle still payoff unbounded take time definition modify accordingly estimator produce identical notion time update estimate assume observation etc round etc time cm kt kt kt kt kt kt choose sublinear note value produce stochastic use correlate biased sum kt kt kt l kt ft kt average compete draw k use aim time suboptimal generalize let let estimate mean assume appropriate problem upper comment logarithmic attain concentrated sample restrictive estimator rejection sampler geometric furthermore moment sufficient variance simplification cost since x follow difficulty reward construction use use ensure number round dependence nevertheless need last remain small follow concentrate c regret allocation also estimator example show ucb suboptimal c ucb problem event time show achieve assumption concentrate subgaussian tail thus last rest conduct experimental effectiveness multi allocation differ evaluation bandit payoff identical variance detail ucb payoff low regret independent run scale optimal second plot suboptimal highlight dash red circle bar indicate ht particular standard separate permit bound four bandit ucb kl ucb prior relative approach ts scenario whereas v effective kl perform ucb monte carlo pricing financial pricing evolve accord cox finance detail assume follow price european option give naive estimating simulate independent simulation strategy introduce drift encourage simulation importantly importance space restriction prevent description mixture drift consider drift mixture coefficient drift proportional new generate next approximated price parameter price wide proposal result effective bandit suit allocation ts winner likely level option pricing surprisingly bandit allocation believe stem entire mixture remain area future allocation continuously setting important estimation challenge comparison training evaluation desire quantity difficult popular dimensional chain monte generally sequential carlo sampler combine transition slowly offer advantage considerable effort set step anneal rate schedule mcmc execute anneal appropriately tune choice effective large bayesian logistic sized consider estimator anneal anneal schedule suggest slice use entail dimension detail differ slice internal
limit proven overfitte yield solution result improve search small strength include directly need put density concept use non require linear direct possible possible apply map rbf preprocesse cluster seed position rbf network beyond scope research help library intel ghz well uci briefly breast diabetes heart focus regard chemical prediction implementation implement acc coefficient accuracy average four center one obviously linearly term symmetry maxima entropy result margin rule flat method achieve almost kernel benefit show lead however nature bad measure perfect fitting capability support regularization entropy histogram number threshold purely cauchy lead model capability dataset capability cv dataset dot maxima suggest pearson seem overfitte decrease acc breast diabetes heart capability across fold correlation confirm aim balanced suited balanced greedy optimization start consist point sphere balanced high summarize protein classifier contrary class active importance lrr rr svm ht ht examine score fold similarity svm classify fold one dataset could rbf although lead construct big slow number require thousand evaluation build ht speed aspect actual huge database check build good consider one place middle perform five lead exactly score achieve model protein ht follow regard result complexity magnitude one internal geometry chemical well might detection active inactive family method actual paper entropy linear term projection justification property invariance svm propose constrain open issue optimize outline also classifier study behave uci well behave balanced balanced correlation propose partially centre work discussion suggestion institute sharing regard would access make proposition section edu pl classifiers hyperplane method separate hyperplane model concept namely quadratic cauchy schwarz general invariance margin analogy broad aim balanced quality directly far confirm uci appear real data classification linear perceptron logistic decision class idea behind neural modification like extreme machine activation neuron play decision base set classification lead htb sigmoid polynomial second good ask answer lie way boundary thus question directly fact split proper linear decision division allow reasonable decide entropy divergence cauchy schwarz denote schwarz divergence computable mixture optimization consideration cauchy schwarz translation maximization boundary consist entropy add view project data subscript denote width rule denote schwarz perform common reason classification rarely particular simplest interested good sample obtain precision support linear density final formulate estimation contrary well occur threshold fact kernel window width single nine ten uci worth solution different svm fundamentally different knowledge detailed insight geometry activity protein describe internal structure capture group encounter show exploit simple represent specific group active positive sample worth linear advantage linear classifier maximize balanced imbalance require datum directly scaling try margin build significantly model svm parametrize free drawback complexity describe schwarz divergence maximize margin play practical consideration drawback conclude long receive much hardness answer basic question et open related support concept generally present svm conceptually show span classification criterion self cauchy author broad tree recently deep architecture aspect cauchy schwarz insensitive restrict search unit sphere next discuss go schwarz divergence cauchy schwarz normalize density density correspond rescale easily r rp since schwarz projection rescale maximization restrict finally invertible put assertion inequality notice analogously datum generate gaussian let multivariate eq denote consideration well formula observe quadratic us attain eigenvector density q therefore entropy vector eigenvalue one multiplier attain say minimal maximally intuition schwarz information class say crucial onto line coincide discrimination next analogous limiting set maximum tm tm tm tm minimum equivalent cauchy schwarz attain q interpret discrimination view know span construct maximize close sample opposite class simple linearly vector express reformulate class unit generalization concept lie maximize threshold size even four require modification removal separation formulation lead threshold dataset would threshold number threshold appear probably subsection large maximization margin try threshold typically often window distribution although choice nontrivial atom local lead limit linearly separate information arrive large whose start potentially overfitte maximize margin formally show behave gaussians arrive formula analogue result prevent threshold support thesis divergence contain information entropy class support section empirical evaluation often unnecessary high consequence show cross minima put start gradient sufficiently linearly separate discriminate separate eq linearly suppose assertion mean mean assumption obvious contradiction suppose separate unique normalize svm going limit svm maximization margin arbitrary eq proposition choose separate cross eq directly maximization let denote possible margin along margin point q result margin big lead conclusion potential additional construction classifier result threshold result construct lead life uci repository versus schwarz introduction entropy cause reduction analogue hold limit reduction number threshold put width begin window eq denote element equality apply obvious calculation p v q v v trivially apply twice yield objective serve purpose optimal discrimination classifying tries maximize margin minimize result threshold show generalization dependent threshold probability point training lead choice minimization
idea design kernel encounter walk coincide walk termination relaxation labeling approach proximity walk similarity constitute contact level contact whereas build propagation common induce graph string stream unlabele propagation hash structure approximation fed base kernel propagation propagation concept utilize walk graph graph random walk kernel graph attribute appropriate review propagation commonly level kernel endow possibly partially observe adjacency matrix k represent markov state indicate walk current probability represent normalize partially graph attribute unlabele fully encountered leave row kronecker simple never leave encounter encounter induction iterate map initialize distribution unlabele distribution introduce simulate transition assign probable scheme discuss label extension naturally employ propagation suitable type basic scheme attribute steady prediction however converge steady kernel obtained provide kernel entire encounter represent accomplish sum iteration rather next v I ie edge weight label graph determine node label graph graph structure node evolve propagation kernel contribution important feature propagation node update maintain throughout attribute correspond attribute propagation kernel semidefinite propagation valid valid semidefinite positive semidefinite number convolution semidefinite kernel propagation circle style circle draw width cm bin cm thick draw node white fill circle fill circle circle circle bin bin bin circle fill bin fill fill fill b node circle white circle circle fill white bin fill bin bin bin bin bin circle fill bin b font b font g circle style draw width circle style circle draw parent thick circle node circle fill node fill circle edge bin circle fill fill bin fill white bin node node fill fill circle fill node fill b bin circle fill circle bin bin bin bin font light assume graph perform graph node bin strength compute count plug base simple outer product count th graph count value exploit exploit computation graph summarize kernel eqs design propagation kernel compare scheme suggestion kernel briefly discuss runtime runtime count strength computation note aim basically along operation information time usually label propagation kernel introduce distribution attribute propagation restrict represent iteration quantization locality sensitive detail next row attribute kernel attribute attribute thresholded commonly node deal attribute product kernel respective dimension kernel attribute distribution node explain locality sensitive hashing use derive way propagate attribute propagation kernel panel b thresholded quantization approach implement kernel distribution attribute inspire locality seek quantization space space probably bin vector discrete appropriate attribute simply locality hash give function family real independently know standard draw attribute integer kernel case decrease choose hyperplane hyperplane hash hx intuition behind expression hash attribute element endow distances hellinger scale locality family direct eq square root point introduction vary insight powerful propagation scheme utilize algorithm kernel appropriate unlabele attribute specific part change marked label propagation database graph total label efficiently sparse efficiently due sparsity general propagation graph green fully label bin tw diffusion node differ iteration label originally graph propagation label partially essentially unlabele adapt label add relevant large label iterative goal pixel among grid kernel value arrange node capture kernel spread grid simply goal complexity edge medium sized texture patch consider neighborhood complexity would million grid number fortunately exploit flexibility early discrete convolution denoise processing image isotropic invariant propagation adjacent lattice node regular neighborhood ignore neighborhood ignore boundary graph regular note neighborhood common grid actually grid form regular square derive line two grid define node note specify structure center carry value neighborhood neighborhood neighborhood illustrate gray sep count count count count inner count count neighborhood derive define operation modify graph interpret discrete variable g computation fact process operation digital resort highly develop example compute fast bin iw convolution simplify notation grid unlike natural notation represent third tensor make exposition enable efficient convolution label probability kronecker delta propagation appropriate matrix circular introduce pixel neighbor equally space approximated filter grid graph algorithm propagation highlight green fast fourier time efficient adapt tensor virtue grid use circular neighbor rotation make attractive implement extension propagation ask sensitive respect propagation use propagation kernel compute classification answer diverse graph include chemical texture flexibility kernel diverse database attribute attribute image pixel use total node node dim bioinformatics label anti cancer cancer protein world semantic originally introduce image represent random illustrate b quick among connect adjacent semantic label human small dataset annotate ground label mode truth pixel semantic object fall remove solely class thick white inner version color cm matlab dataset except evaluate run classification experimental exist learn iteration split protocol introduce validation learn enhance continuous encode bin chose evaluate propose propagation kernel choice analyze respect accuracy randomly propagation propagation first learn full combination far repeat normalization normalize large value ex yes yes yes yes yes yes yes yes yes attribute graph indicate actual propagation quickly perform bad computation label art comparable classification give split finish method pair test l h assess partially graph remove accuracy subtree design partially graph variant unlabele additional another miss obviously baseline slightly bad propagation might beneficial large missing method compute via string implementation kernel scalability property begin calculate intermediate classification kernel successfully partially question derive propagation bold significantly pair l accuracy split finish within protocol parameter learn full dataset learn quantization neighborhood performance color color neighborhood baseline label gray occurrence matrix compare intensity label pixel graph show sophisticated art computer vision texture feasible feature commonly computer ensemble conclusion mining computer accuracy standard error fold quantization min degree version additionally claim table support propagation prove flexible thus ultimately base principled uncertain unlabeled walk discover share construction namely propagation kernel propagation kernel common induce iteration graph kernel close experimental accuracy attribute moreover tie propagation adapt development directly message probabilistic deal closely derivation propagation ex kernel computation perform machine ghz intel processor compare computation extent mean computation finish h h average accuracy error fold run randomization parameter attribute hash per attribute computed version perform unnormalized mark memory perform learn training unnormalized kernel memory memory lemma kernel monitor graph leverage early propagation scheme walk capture label attribute benefit many label unlabele direct attribute leverage informative propagation kernel considerably regular modeling video database exhaustive structured area research domain become diverse situation example annotate document content modeling goal represent structure efficiently popular kernel similarity classification several literature strong assumption information proposal encode encounter real world rich challenge information lead partially uncertain aggregate source semantic annotation partially available collect sensor consist coordinate detector possibly semantic annotation entity document entire providing amount surprisingly broadly challenge design unlabeled label kernel attribute gain drawback handle graph attribute flexible consume overcome problem relational supervised aforementione efficiently initialize uncertain partial
degenerate distribution evaluate filter hmm forward marginal exactly marginal demonstrate fairly narrow performance practice successful application particle comparable implicit instead relative bind data base parameter express distribution assignment chinese restaurant number prior assign number ht cluster color treat mode structure particle c c synthetic gaussian table simulate increasingly build prior clustering marginal accuracy cluster overlap variance greedy cognitive particle outperform particle particle ht accord infer model use sort important experimental researcher amount extract spike attribute neuron spike belong naturally motivate sort use particle filtering filtering achieve particle choice dimension wishart material detail well filtering use parameter comparison qualitative demonstrate spike despite calculate hold likelihood spike quantitative summarize particle filter particle hold smc smc l particles smc bl particles smc generate induce assignment assignment restaurant probability probability select assign probability proportional visit hmms matrix use concentration b hamming hide multinomial resample log bl particles smc illustrate filter multinomial resample quantify compute hide show show particle next world taken begin apply character chapter book calculate predictive log character compute state sample particle hyperparameter show predictive outperform total outperform field study apply illustrate fix study lattice spin vary variation run field result principle randomness introduce vary initialization field particle particle trade achieve increase accuracy examine field achieve number field large particle importantly actually achieve iteration use large particle introduce framework particle discrete practical optimizing empirically algorithm particle optimize kl select optimally performance particle monte carlo efficiency advantage deterministic error sequential resample conditionally particle diversity degeneracy show ess performance relatively narrow avoid parameter worth note diverse unique particle conditionally unlikely avoid happen without resample combination key particle limit distribution important future require proposal question incorporate proposal monte carlo identify proposal distribution model combinatorial factorial completely may desirable proposal monte carlo achieve superior widely particle filtering wide method framework stochastically particle play deterministic variational approximations monte particle synthetic monte stochastically sample method make success sampler often paper approximation suppose place intuitively cover target variational treat problem ascent particle minimize divergence particle introduce filtering experimentally overcome problem able produce sometimes degenerate eq index clique stochastically generate proposal weight important particle approximation converge sequential filtering apply dynamical index sequentially conditionally probability step replicate particle particle degeneracy concentrated particle parametrize variational relate normalizing identity thus equivalent unlike monte converge variational improve bound iteration unlike likewise potential update filter helpful particle single particle particle set associate resample illustration different space circle indicate indicate cell high time iteratively rest conceptually approximate continuous use particle approximation approximation sense use pass attempt capture
due develop accurately solve heuristic orthogonal orthogonal matching pursuit subspace pursuit pursuit etc support component iteratively current update solve also greedy accelerate thresholding pursuit primal dual basis pursuit find variety apply comprehensive overview greedy relaxation counterpart read control evident certain properly coincide minimizer however challenge convergent nonetheless broad iterative backward splitting convergence characterization global minimizer novel primal active set develop study note minimizer mild contain true cf coincide parameter extend sense mutual incoherence isometry rely evolution dual global organized collect provide minimizer convergence estimate essential analysis minimizer datum noise column appear last submatrix invertible mutual incoherence isometry rip rely mutual coherence sense small mutual coherence mc say satisfy exist constant small mutual rip nontrivial basic disjoint subset hence nonempty assertion follow apply first note unit diagonal satisfy identity list product column column coherence matrix series immediately disjoint give set iteration active variable active estimate rip estimate rip condition follow hold update triangle dual e hold characterize minimizer multipli lagrange counterpart nonetheless aim recover expect shall oracle derive directly equivalence minimizer nonconvex global minimizer minimizer minimizer thresholding equivalently active minimizer analyze small active contrary nonempty b hold separately deduce contradiction minimizer minimizer see minimizer perturbation local deduce alternatively minimizer hold level solution formulation minimizer minimizer deduce contradict minimize minimizer nonempty global lemma eq rip I I assumption contradict optimality assumption monotonicity rip deduce view appeal assumption together yield contradiction moreover support problem minimizer argument theorem deduce oracle due equivalence lagrange clear equivalence cf problem active property algorithm determine active solving square newton convergence guess parameter active minimizer active minimizer empty max inactive check assertion inequality eq converge finite inner lemma index stop reach mathematical induction iteration lie large reach terminate stop satisfied analogue g kb deduce relation lemma rip similar omit step converge terminate monotonicity line reach proof imply criterion algorithm pursuit omp pursuit htp rip sense analyze omp appear rip appear omp rip omp require active lie iteration move inside outside confirm make omp htp due primal set method active component primal dual component iteration naturally illustrate efficiency sense signal mean take specify three setting approximate mild reasonably value recover exact active greatly insufficient practice rough study variation observe c unless estimate htb cc gain insight evolution inside set contrast omp iteration flexible valid bernoulli dct htb b e dct partial p nj guess observe dct matrix attribute hence cc bernoulli dct six art literature pursuit omp accelerate iterative compressive homotopy algorithm e percentage reconstruction whose agree realization setup value numerical observe largely ht illustrate greedy cpu result dct table compute setup observe reconstruction computing scale omp htp omp e htp omp e omp omp e e error omp e omp htp e omp e omp e e htp p rr omp htp omp htp omp htp omp e htp lastly signal matrix square update solve cg guess cg stop cg residual one signal sampling apply inverse transform transformation nonzero entry reconstruction visually appeal excellent exact reconstruction far confirm reconstruction true reconstruction partial nonzero table remain largely almost reconstruction effort competitive b omp htp omp ccccc method cpu omp htp ccccc omp
net overlap make voxel similarity minor difference subject overlap voxel person voxel force voxel encodes picture voxel force encode rise large argue account similarity difference across subject code interpretable drawback glasso voxel rise cluster sparsity negligible voxel encode picture sentence consider select voxel encode picture primary setting look simulated function toy overlap vary level corrupt white deviation average group retain fraction retain regularization minimize latent lasso group matrix plot account inter group glasso active non pattern color within active reduce overlap group overlap group however account hence glasso lasso account structure poorly perform bad glasso explain introduction motivating arise particular tumor gene cancer gene pathway also pathway replicate data balance dataset overlap group standard ill h compare lasso one perform enforce use constrain solution generalize group arbitrary result overlap outline fmri biology minimal generate result general light overlap paper designing allow cognitive view grouping voxel spatially co locate significantly remains see whether motivated way voxel functional take since index q corollary sub index singular prove gaussians product equivalently write define multiply mean width argument index mean g follow similarly prove time quantity sake lemma play dimensional feature group subset many however structured selection group comprise set rich lasso framework generalize conventional allowing present challenge paper overlap automatically feature classification establish bound classification bound classification lasso group fmri voxel source localization activation microarray synthetic demonstrate play role machine learning application feature far number search sparse notion prevent interpretable sparse case group lasso group coefficient group overlap penalty one success structured selection arrange prior feature specific propose overlap group reflect example relevant play role gene pathway pathway pathway bold letter matrix sparse simplifie allow tool correlation realistic focus classification randomly true inner satisfie euclidean norm normalization quantifie label inner product error bound consideration yield constant fact consider product measurement model model somewhat often correlate compute enter bound follow structured user define depend assume feature hand group structure wherein group relevant feature localize union group localize subset armed state theoretical later within measurement solve statement explanation parameter suffice looking amount group overlap program succeed group hold fmri nonetheless belong propose overlap group sequel lasso lasso tool encourage less encourage identical accomplished define group pattern glasso consider interested paper motivated fmri goal cognitive activity feature shape neural structure neural somewhat across individual guide locate useful general useful voxel voxel regularization lasso across penalty account coefficient account common across motivation grouping overlap classification pattern arrange select also extension lasso group interested apply regression sparsity analyze feature arbitrarily accord also restrictive introduce activate coefficient contribution theoretical analysis consistency reduce know sparse high recovery extend arbitrarily overlap regression work overlap group logistic far rich knowledge provide unified bound non overlap group method mention suffer task undesirable effect many version toy experiment advantage group especially image gene biology regularizer encourage coefficient overlap generalize two scope level consistency lasso group overlap group make minimal applicable setting correlated design turn translate unified theory structure motivating fmri fmri yield low hold test interpretable domain breast cancer breast motivate fmri author consistency give derive biology notion result potentially theory overlap propose sparse lasso character recognition gene handle show admit proximal operator exclusive modification express strategy problem solve coefficient bound induce sample possibly overlap group overlap characterize program solve extensively characterize correct author generalize glm classification minimize maximization linear subject organize structure sparse selection regularizer recover pattern across group overlap result toy datum conclude future return main unit motivates optimize natural positively natural thing maximize quantity term define difference corollary group overlap sparse lasso course effect parameter g g get pattern prefer tend prefer value function two sparse select take account account group group c g show indeed exhibit group overlap consider see list consider instance solution zero group localize sparse finally fourth correspond lastly imply hence convex optimization convex program positive homogeneity lead homogeneity possibility optimal representation triangle inequality lagrangian version structure prevent value method expand progress take gradient shrinkage proximal overlap special case general structured point composition obtain final correlate vector note definition group setup ideally solve instead overlap group lead admit relaxation tight relaxation vector overlap follow iii follow ensure tight first non crucial consistency width theorem lemma completeness behind ideal interested contain scaled hull non scale outer end ideal follow fix magnitude manner ij fix small average exact substitute give construction nc ij conv nc width result ideal scale result width constraint side jensen iv follow note lemma away treat us inequality statement proof lead light overlap group complexity want would bound regularize correlate interested correlation fmri major motivate voxel brain exhibit amongst entail number matrix measurement matrix obtain constraint reduce variable enforce constraint lead generalization theorem correlate entry sample wish group vector condition matrix along prove perform toy recover sparsity interested experiment toy datum yield optimization cognitive biology group commonly encourage set select wish focus less restrictive encourage restriction correspond task relate accomplished define subset search solution common across figure interested ability well interested arrange column correspond g coefficient standard result glasso task overlap star plus involve sentence half fmri retain stimulus yield distinguish point stimulus exist expert partition discard average bold voxel within subject kind encode assess aid discovery voxel expert pre
method conjunction example belong discussion goal study possibly turn inference reality collection recognition concern classify category independent category member define approach category share available insufficient quantify object category major literature subject dedicated numerical challenge problem histogram interest category model empirical identify relevant intensive conjunction derive mixture briefly discuss quantity also overview challenge article quantitie category option belong end uniquely determine observation height certain country category belong consider try approach density density single mixture parameter describe member emphasis belong component represent guess respective category independent category adopt know problem determine quantitative state knowledge category similarity notion consider category coarse key group category replace represent iterate coarse category characterize distinguished intensive relevant intensive often challenge less determine extensive element moreover coarse homogeneity among category complexity hand choice describe find invariant coarse mass particle intensive shall intensive expectation law allow represent scaling encounter constitute harmonic precisely geometric harmonic regard although nonetheless might infinitely interested quantity maximum among define mean short find lagrangian correspond matter convenience calculus belong note improper non compact support functional uniquely moreover narrow broad influence uniform distribution consequence distribution line modify literature well distributions eps familiar arise example drop list demonstrate abundance sake clarity discussion law quantity nonetheless recommend one conduct assessment start outcome along functional density aspect often dependent get statement quite simple density determine become randomly h p rough knowledge
select summarize equivalent define confidence test framework screen respect algebra measurable distribute element distribute rank computation screen apply theorem statistic v partition sum unconditional conservative sense type exactly select subset sign specify hypothesis compute via previously f discuss fact decrease need smooth monotonicity truncate distribution family hence likelihood formalize immediate consequence unique interval relation e hold boundary confidence interval however interval correct whereas response see confidence exactly unknown diabetes diabetes patient age body pressure measurement quantitative baseline disease year statistically predict tx I assess unknown use fit adjust adjust cover nominal always broad propose hypothesis test screen particularly apply match pursuit square exhaustive selection procedure condition incomplete list ease construct interval fan recommend algorithm follow lasso select two selection follow stage event sign sign lasso screen encode test algorithm valid marginal omp commonly omp correlate residual residual omp selection encodes omp choose screen variable confidence omp non eq eigenvalue several sign negativity factorization network interval estimate coefficient primal pair iff complementary choose give kkt q encodes omp marginal screening hypothesis linear important selection framework screening apply marginal screening part nsf dms grant lee nsf stanford fellowship theorem definition theorem proposition remark framework marginal characterize exact model allow coefficient contrast statistic exact eigenvalue like assumption marginal regression negligible particularly suitable focus applicability framework broadly propose procedure include match pursuit non inference estimator commonly distribution test however square perform variable aic match marginal select response screen selection procedure marginal screening dataset intractable comparable lasso screening screen select truly screen combine selection lasso far extend screening utilize response confidence nominal coverage may long difficulty post estimate sense counterpart post selection operate subset distribution marginal represent event construct truncate develop although marginal propose clean illustration applicability discuss extension apply cover clean screening omp negative square high focus establish restrictive select correct refer theoretical property interval subsample interval pursuit extension leave thorough investigation position mean distributional sample high requirement test compute regression propose compute method paper matrix response tx screening choose high absolute notation distributional long test similarly hypothesis empirically test screen h vector select construct interval independent snr interval drastically nominal model equation hold conditional screening event selection select sign partition q previous constrain partition possible subset sign event set tool
build response prediction amount response three classifier building vector implement comparison obtain net rmse acc bayes auc acc auc acc different calibration single observe obtain table amount datum possible leave building calibration model save one calibration repeat calibration measure prove three justify post histogram discrimination measure extension dirichlet simulate show compare work investigate histogram mini mini experimental histogram width provide similar another paper calibration problem auc theorem theorem helpful reading expect bin histogram theorem rewrite recall identity apply identity write finally combination histogram rate auc calibration would recall hoeffding inequality amount high concentrate concentrated auc follow auc nice possible show empirical concentrated output transform non overlap bin base two use assumption auc method partitioning simplify calculation part summation convergence summation recall method sure fact rewrite inside equal notation term true concentration second summation part iid concentrate around empirical estimate construction frequency bin rewrite last inequality fact order completely reverse apply b k histogram calibration notice proof show auc point even auc histogram dataset department university computer department abstract calibration critical category classifier method model calibrate transform well histogram scale use paper introduce measure calibrated three use calibrate discrimination capability parametric extend parametric demonstrate outperform comparable accurate probabilistic datum decision task unfortunately optimize task generally calibrate predict fraction concept readily outcome reliability display calibration curve predict outcome calibrate tend probability low calibration line calibrate represent make uncertainty calibrate produce calibrate critical determining business decision area study nearly extensively level develop calibrate machine traditionally focus development model discrimination exist potential well calibrate learn simplify linearity another scaling improve intend calibrate show calibration objective perform solely post objective theoretically justified limit post process perfectly calibrate discrimination roc least good two exist post processing apply predictive calibrate use limitation rarely prediction parametric histogram sort partition bin find contain return fraction bin calibration increase calibration algorithm position classifier variation predict introduce measure evaluate classifier prove three use calibration capability measure discrimination introduce method describe section present classifier mapping instance classifier calibration intend calibrate follow notation remainder predict locate inside predict locate inside predict go infinity empirical capability calibration reliability diagram reliability measure calibration prediction sort partition ten bin calibration calibration error estimate pi instance post calibrate bin bin prove max classifier lipschitz decision boundary histogram bin classifier behave histogram frequency near interesting open histogram condition bin non histogram calibration method plug bayes rule follow prior dataset term histogram estimate bayes iy iy iy empirical estimate bin algebra calibrate plug classifier likelihood advanced density simple density kde bandwidth bandwidth optimize use validation bandwidth unbiased however notice kde report estimator risk smoothness target rate practical fortunately binary classifier calibration prediction classifier calibration kde frequentist approach frequentist model mixture bayesian building calibration lack choose collapse collapse implement refer kde auc acc kde rmse auc acc describe calibration evaluate performance run acc roc curve auc discrimination three root square error calibration outcome separable scatter plot train quadratic method simplify discrimination figure intuitively ideal quadratic allow classification auc c auc kde method perform svm poor surprising parametric parameter make violate perform relatively poorly improve
slack pos train slack negative example balance tradeoff pl explore al modify balance occur svm handle imbalance use address imbalance resample address imbalance resample analog imbalance train pl approach overall imbalance imbalance use tc svms aim train extraction aim perform system aim tc belong treat separate ten category hyperplane decrease pa name pa al svm hyperplane high hyperplane pa imbalance pa corpus imbalance drastically imbalance figure pa example label get skewed corpus systematically example random loop criterion meet x f hyperplane point pa al guide sample require confidence many college computation aim enable confident proportion proportion size implement point graph show highest aim perform begin area want stop observe identical al reveal nearly category slack pos classify situation model achieve train outperform hyperplane two annotation stop learn hyperplane classifying round approach hyperplane hyperplane train vice hyperplane predict svm scenario call pl address imbalance estimate overall imbalance imbalance utility demonstrate situation help show sampling pl instead modify characteristic exploration investigation center university md usa edu sciences sample data characteristic sample datum use passive pl explore case al weighted svms arise address imbalance cost estimate corpus imbalance imbalance pl recently interest reduce annotation sample characteristic modify passive learn pl case svms factor imbalance pl showing improved substantially address imbalance pl show prevent degradation date relatively imbalance scenario bring pl approach imbalance
batch occur time annotate batch label predict q otherwise batch evaluation tb batch label prediction multiclass explore size stream characteristic performance need procedure carefully seem provide repeat test across attempt run dataset hypothesis verify classifier effect baseline logistic look propose batch classifier evaluate run datum available confident measure expensive annotation cost however visual arithmetic modify select informative batch try answer give robust high show experiment good discriminative work complexity framework feature video stream surveillance scenario grant surveillance query place human resource limitation way software video reduce impose herein incremental evolve visual stream develop track consecutive update try balance restriction evolve camera drift address well non stationary environment show little decade surveillance spread rapidly area hour day year massive amount come evolve method reflect environment underlie datum refer change movement change dynamic camera angle etc model update concept new enter machine model need surveillance angle camera surveillance camera view often disjoint due constraint enter surveillance system track object capture camera around stream record various associate detect consider span person person side capture camera person person b person capture camera switching position simple typical tracking occur b identity person movement track identity suppose global necessary capture identity scene label annotation consume impractical annotation extensively explore un label visual environment concept drift evolution researcher mostly address active lead evolve video stream surveillance scenario body camera surveillance adjacent overlap whereas require overlapping view herein continuously stream label still setting focus tracking receive directly track object framework camera axis indicate period discuss limitation former incremental provide future work bp cm cm cm cm stream stream concept table sl constrained md sl unconstrained md sl unconstrained md md constrain assessment method denote dimensional sl semi recent surveillance object change video receive despite abundance concerned environment recognize resource obtain label approach video promising scenario identification semi label track limited environment classification address label perform fully method track therefore stream challenge real stream mostly appear mining constitute handle drift recent evolution learn generate dynamic weighting heavily cluster employ active although considerable stream view explore area stream come thus track appropriate require stream start stream drift accommodate partially review qualitative review scenario stream visual learning framework label obtain carefully point ensemble incoming batch combine weighted voting sketch tracking analysis frame movement generate stream environmental challenge illumination lack bad acquisition device motion noisy miss address gap cause tracking provide batch index th stream present slot stream track camera stream frame start potentially batch align stream frame correspond object representation bag word scope obtain slot name composite arrival batch try predict kind work suffer assign view object help interact human stay track initially composite yield probability batch slot computing batch batch estimate predict accept correct otherwise label batch multiclass integrated yield batch composite c slot composite update prediction section classifier design composite equal many numerical problem logarithm monotonic k pc batch decision composite approximation posteriori build batch prediction frame arithmetic prefer author arithmetic geometric presence experimentally option decide automatic prediction rather manual criterion uncertainty perhaps rely confident class define confidence however consider away probable margin since little batch margin label would help discriminate effectively relative confidence exact involve put forward probable modify pc pc product issue alternative confident write follow comparison denominator simplification strong independence aggregate end side rewrite trivial obtain characteristic four confidence every corner indicate herein class corner move inside decrease linearly increase corner drop like composition class least would triangle ensemble informative lie corner b slot obtain batch manually belong tracking mix identity slot batch observation batch slot batch posteriori given predict option find discriminant probability play design discriminant still estimate experimentally challenge automatically resort class also present slot class previous incoming describe individual dynamically update respect multiclass ts adjust update normalise slot choose weight normalise q conduct capability order stream scenario test drift evaluated cover scenario camera surveillance conduct experiment dataset synthetic dataset label suffer simulate situation change parametric table present equation generate gradually also process use scenario complexity stream drift drift appearance drift scenario gradually stream class stream class appearance environment evolution depict occurrence drift number enter enter stream illumination well capture employ automatic scene position perfectly stream object object descriptor vector curse system suffer set
inspire component precision feature community split classifier generic classification problem goal build analysis normal population boundary separate poorly pose encourage employ already number proposal lda ridge ridge ridge penalty interaction discriminant independence naive alternative feature presence strong paper span naive use across force precision place share sparse make classifier interpret refer within model idea identify applicable likelihood bayes partitioning several community community improve interpretability classifier quadratic discriminant discuss idea real conclude training observation set independent quadratic precision ki jk maximize q last penalty force pattern kx important feature tune shrinkage matrix rise naive classifier degree severe naive eliminate quite instability superior strong group force share lead produce sub potentially massive quickly solve purpose et maximize q additional lasso purpose goal interaction instead whether precision component similar rule optimization speed solution namely admit l entry skeleton symmetric symmetric thresholded admit component estimate induce exactly vertex induce nest component vertex theorem quickly check connected component rule different block simply solve block make certain impossible operate machine block feature mutually generalize set split estimate connect sample involve work class admit kk kx lx posterior cx serve equation fit classification posterior adjust intercept normalize consequence tractable infeasible scale modular naturally multinomial logistic regression generalize thus variance conditionally independent community apply average complete linkage correlation matrix procedure conditionally community univariate strictly kx kx different make identifiable preserve x ef j gaussian monotone differentiable denote define l exactly conditionally nx correlation exploit directly base base base class define define n validation community correspond apply linkage agglomerative cluster cut dendrogram linkage cluster merge one time knowledge true agglomerative cluster consistently vertex component ij ij partition result proof appendix community use element absolute estimate single linkage ij pg l pick study example achieve misclassification interpretability tuning result diagonal correspond covariance refer version regularize misclassification standardized tuning four performance follow validation respectively minimize misclassification case class class precision result deviation replication interaction situation favor give small deviation naive standardized decrease model perform class table summarize result situation term favor level increase increase strongly dominate misclassification capture interaction need cm cm cm al illustrate handwritten digit handwritten le normalize sample smoothing help filter select misclassification standardize parameter naive decrease dramatically naive keep range include noisy term unit standardize correlation diagonal band use regression regression logistic regression compare logistic lr logistic show prediction training tune experiment identity employ transformation definition lr use linkage cluster community transformation population class identity cdf cdf transformation skew bi summarize misclassification perform one introduce class conditional distribution cm email email spam
relationship linguistic word embed example architecture network language learn unlabeled deal several language al skip skip learn text close position embed base similar context although word become public introduce benchmark name source performance moreover take discussion research organize create report art word consist task generate evaluation tuple word whose close bc question correctly answer two category task include e syntactic nine give evaluation semantic rest nine syntactic question tuple word tuple unique word pair small dataset check tuple pair tuple question tuple publish reason tuple pair word tuple capital capital city california apparent rapid comparative think read past merge pair extract new pair english derive pairwise word evaluation introduce scope vocabulary extract knowledge cover vocabulary corpus snapshot wikipedia corpus million token vocabulary size besides phrase token release leave phrase pair leverage wikipedia semantic merge wikipedia page area find u wikipedia city accordingly name entity remove name take advantage semantic new rest extract candidate filter vocabulary statistic wikipedia word gold semantic syntactic task base merge filtered vocabulary much well word pair tuple http microsoft com en performance state distribute representation public public public site representation representation demonstrate reasoning table yield reasoning task training dimension likely well analogous different diverse note experiment question vocabulary regard incorrectly measure use recently embed relation relation type regard pair researcher list nlp list syntactic list new word build collection art word research topic future work plan phrase consider web knowledge basis thank xu evaluation state word embedding c pair capital city california rapid comparative greater easy reading past l word word tuple water exercise face city attribute depth sound entail pay c dim dim dim dim dim dim capital city comparative
determine movie positive rating full popularity keep high tf tf tf document term occur keep consist set world extract stock year label label use day one ignore minor price result website review mark label positive review label negative assign normalise appear appear tf rating popularity description website associate vote normalise apply dataset vocabulary document collapse variational implementation collapse collapse run relative change fold calculated response response proportion often goodness obtain perfectly calculate full prediction fold relative deviation fold small choose validation hdp place every iteration residual converge iteration number topic perform show use computation coefficient algorithm sample binomial take step glm learn label way jointly glm compare show well variational variational clutter perform movie outperform pick right pick drop yield good pick posterior make little additionally glm significantly jointly competitive dataset partly model increase number document one remove since newly empty easy change allocate topic number word allocate make difficult change smoothing contribution fact low predict movie review deviation score imply noisy influence stock stock change stock price movement market pick subtle sample maximum fold method fold plot chain calculate compare chain chain converge chain indistinguishable give plot significantly indicate well experiment directly linear regularization validation fold accuracy movie dataset document dataset marginally outperform movie review outperform popularity benefit l glm usage dataset coherent usage powerful extra topic enable top negative frequent movie review topic contain name actor flexibility nonparametric mean consist term allocate actor review poorly flexibility result group around consistently actor coherent even small topic term spread topic negative actor name topic see specific actor associate movie topic seem name focus contribution movie rating learn divide content topic concentrated actor topic algorithm train learn hdp like sentiment term regression rating cccc party instead call topic six company teacher nature winner onto topic suppose unfortunately lot flat air break ten cccc political bank serious usual face want topic cat top frequent term topic movie review term good topic consistently stock negative topic involved stock price cccc topic record publicly water hold c whole american china effect account c drop account range among topic stock price demand indicate grain actor specific trend possibility topic supervise nonparametric datum document popularity supervise hdp topic document choose model overfitte occur learn grain learn classification experiment world movie dataset experiment also jointly learn glm model hdp inference improve outcome extra add sentiment propose general typical sentiment dictionary wide response generative restrict kind topic previously patch patch similarly paper keyword google view complete research interest include reader school mathematics networks associate school microsoft fellowship research focus e continuous time image medical associate nonparametric joint method group world dirichlet generalise dp allow flexibility nonlinear dirichlet hdp mixture seen supervise hdp learn predictive solve allow learn structure allow model adapt data nonparametric frequently new increase group item analyse performance involve label input predict response dirichlet allocation group text document vocabulary e topic think infer topic mode membership document mixture successful collection citation database corpus wide include information retrieval latent learn model document dimension turn model regression collection possess categorical order type sentiment modelling topic dimensionality topic topic ignore response learn response response cause end assign supervised topic predictor document lda learn unsupervised topic orient predictive example topic learn consist term sentiment topic contrast topic line document prediction make unsupervised topic infer supervise perform must topic unseen observation opposite model topic relative contribution typically dominate capture leave component number run difficult method naturally handle flexible nonparametric hdp generative predict model contribution topic overfitte hdp number g rest briefly review exist work group introduction generalise later paper describe use real dataset consist response outline group amount exchangeability predictor response response predictor corpus encoding vocabulary document rating category kind group outline previous paper nonparametric gain flexibility flexibility process gps supervise logit generative covariate response model jointly mixture assume cluster multinomial logit response conditionally covariate logit protein fold classification machine mixture generalise glm explicitly dp glm continuous use generalise regression generalise prior coefficient gaussian neither dp glm predict prediction group learn however flexibility topic leave work regression label dirichlet process hdp nonparametric analog lda allow flexible modelling though variational bayes significant suffer lda paper extend hdp learn stochastic thought fact distribute wide flexibility posterior important problem dp glm document exponential family dispersion predictor glm document exchangeability assignment imply glm map inferential possibility symmetry break exchangeability topic assignment use sensitive broken process generation label choose mean alternative proportion document response constant estimation expectation method lda collapse use require topic topic consume propose necessary response infinite prediction extend cluster align response response model generalise cluster allocate need advance beneficial supervise unclear model model generalise linear condition number vary coefficient generative sample effect regression treat coefficient treat topic also coefficient categorical model dispersion range range document dp concentration document level topic corpus act base document prior density document consist word dirichlet likelihood observation conjugate topic integrate collapse keep track model response categorical glm coefficient process draw concentration topic iw draw I choose topic supervision easily aid response document document infer group publish group previously could set previously pick topic allow topic give control type learn topic particular allow use allow topic document entire label document dp approximation collapse common technique
kx rx I h hx kx h mx x kx bind together corollary imply complexity logarithmic minimax section express term exponential label speedup agnostic active gaussian learn study long allow hx guaranteed exist satisfy result hold allow generalization key definition facilitate agnostic hypothesis agnostic compression small vs f extend version define agnostic disagreement coefficient except respect hypothesis case satisfy case begin extension agnostic pf p agnostic x b finally set f vs c vs vs purpose agnostic passive original agnostic specialized exponential speedup low accuracy regime passive sphere additionally provide express term disagreement translate agnostic active algorithm improve still universal budget label request request label satisfie exist class linear request corollary complexity passive low bad long satisfied hypothesis flip intel collaborative research institute intelligence page version also completeness present formal proof trivially union probability least right let work exist active active simple showing sometimes aside interval z I mx mi mx km match low aside disagreement technique exist literature vc active immediately vs I vs vs vs vs vs give ii j h jx w w jx w jx h completely letting vs w w vs x ds quite strong gap require therefore include distribution unlike disagreement potentially eliminate replace measure g vc vs xx wiener run improved label disagreement lead quantity small tight refined linear mixture gaussians density compression characterization agnostic active selective sequential pac learning paradigm sequentially request pool stream active label request active perhaps advanced article base sequentially request classification attractive thorough numerous guarantee literature thesis label exception root relate selective wiener wiener instead measure learn article characterization disagreement lead wiener correspond induce set special improve upon technique factor either measure sometimes vc wiener active characterize interestingly also express specifically vc relate compression disagreement coefficient show always logarithmic factor axis align product compression relate compression arrive base represent result setting naturally extend new agnostic well observation wiener disagreement applicability disagreement formulate result disagreement general wiener denote arbitrary call aside set disagreement pm throughout discussion vs hx satisfy vs two name exact query idea mf x selective terminology terminology article formal notion minimal specifically implicitly remain exist disagreement active coefficient classifier ball center coefficient introduce also root disagreement active disagreement specific show asymptotically provide passive interested even passive disagreement date sufficient work show bound target pass continuity dependence little big vary also several disagreement perhaps unit within though disagreement coefficient disagreement term disagreement vs bf clearly x chernoff since prof imply converse show constant algorithm else attractive maintain obtain f count first follow provide bound establish bound allow dependent direct compression rely imply bind frequency specify completeness result begin line purpose compression collection measurable set invariant value hold remainder case let distinct n index I n suffice n total q due e expression follow provide result selective improve selective classification directly compression ms ns vs ms furthermore vs probability vs nm mt monotonicity union least q bernstein imply let also imply query let integer result n nm maximizing imply furthermore union nonnegative nm u nm nm bound primarily interested guarantee final element mn nm mn nm er er minimization large e relating quantity definition fix vs universal large monotonicity instance implication monotonicity imply corollary sufficient condition improvement passive minimization typically versa factor validity interpretation imply validity valid regardless interpretation choose stick throughout decompose implication implication strongly connect direct equivalence statement therefore mi x mi mi plugging reveal implie imply imply union probability fact appendix study hypothesis conjunction smaller know result specify classifier include member normal distribution covariance combine constant particular label complexity bind result factor reduce rank describe problem product x j b classify distribution align base slight cdf gx kx kx gx kx ki ix g equality monotonicity intermediate let hz ia g ib axis rectangle furthermore every monotonicity hx combine set point monotonicity former ia ig ig ig ib ib b ig ib ig ib hz ki ig kolmogorov monotonicity cdf second union imply probability realize uniform right side without toward denote ix ik x j event probability px ij b probability
ability since also non generate nn compound depict average shrinkage case vertical value regularize dot depict oracle bottom sample average dot vertical resp shrinkage set twice view generalization tend quite case similar play primary interest oracle scatter fairly scatter case shrinkage outperform estimator close identity towards identity detect signal receive represent unobserved clutter v signal signal receive datum variate example unknown parameter accounting channel vs signal problem follow scatter consider refer alarm rate match fact great clutter model belong alarm equal rejection threshold however replace require fix become scatter clutter rarely case retain although consistent inaccurate affect regularize estimator provide property detection pd nmf scatter investigate detector study target signal trial datum detector secondary estimate detector scatter clutter note figure reflect pd nmf numerically integral signal clutter db observe pd mc detector able pd nmf detector length secondary regularize natural scatter suitable penalize estimation uniqueness establish regularize estimator sufficient uniqueness choice matrix estimate provide study illustrate method express go infinity readily definite hermitian sufficient singular semi hermitian trace one j assume z converge need show nj pa l b exist readily continuously differentiable non unique ng k v proof iv contradiction show sequence next convergent ii observe equivalent find next show identity proof rely representation r possess derivation therefore omit eq gives resp resp expression complex real similarly real definition david department nj usa mail scatter lemma theorem insufficient constitute generalization scatter penalize pair derive uniqueness concept geodesic include uniqueness establish solution shape match sense iterative scatter compare counterpart support match nmf adaptive maintain time nmf detector convexity distribution scatter normalize match classic multivariate analysis technique covariance eigenvalue variate sample z I many simply completely inaccurate example occur medical imaging insufficient support normalize filter realize require key partly difficult conventional environment well inefficient estimator draw non datum pose difficulty scatter scenario estimator scatter covariance problematic propose constitute generalization cost include uniqueness utilize study regularize focused regularize scatter variate note problem correspond treat sufficient establish ensure cost regularize strict derive regularization matching sense usefulness scatter application match although generalize real review complex symmetric ml introduce penalize stationary solution type give uniqueness uniqueness estimator proof hermitian resp resp ij symmetric f unknown parameter scatter generator ensure tp cn reader denote variate maximum scatter divide emphasize estimate scatter generalization estimator elliptical necessarily relate elliptical function estimate ml denote minimizer covariance estimator function relate elliptical nevertheless estimator scale mild take estimator numerous interpretation huber tuning f chi square freedom usually result estimator huber estimator estimator additive introduce cost denote regularization enforce matrix real case penalty though enforce precision dependent scatter paper grow eq parameter well describe naturally notational convenience equation solution satisfy regularize equation rise iterate convergent proof regularize continuously differentiable estimating uniqueness follow huber correspond huber detailed subtle gaussian form n z show estimator differ motivation estimator view ridge singular large spherical towards scale regularize weight p hereafter use minimizer hereafter assume continuous readily huber play key role uniqueness previously utilize uniqueness scatter geodesic convexity positive aforementione paper wherein treat class notion complex differentiable riemannian convex say strictly geodesic thus concept geodesic enjoy convexity euclidean minimum furthermore set subset minimum matrix omit complex analogous cost addition span cost whenever include geodesic corollary see convex convex convex penalize proceeding review hermitian respectively view case equality hold readily follow consequently geodesic convexity side strictly geodesic strict imply desire scale invariant regularize admit interior show need establish penalize minimum estimating place sample require proportion penalize condition sample extend condition occur continuous multivariate condition equality cm inequality
principal evaluation analogue lie neither algorithm involve matrix validate claim experimentally principal principal vanishing feature use kernel vanish kernel scenario synthetic perturb uniformly iii handwritten digit pca circle span degenerate principal square correspond runtime order htbp manifold manifold circle circle example approximated feature see near quantile example rest pool one vs follow evaluation correctness follow feature generating sample degenerate repetition sub degenerate good advantage dimension many repetition grow distinguished priori degenerate random feature extract generate fast principal comparable outperform experiment show capable extract competitive considerably scale comparable performance european european framework fp grant research ex thm matrix span fact scale kernel usual framework duality vanishing indicate cross novel algorithm pca pca real synthetic extract novel empirically validate trick costly implicit efficient huge learning task overview many make kernel source efficient linearization computational bottleneck part kernel kernel singular subsampling sub art write seem consensus subsample kernel unseen kernel output propose algebraic potential explanation issue suitably empirically practically able address analogous opposed subsampling prescribe thus address advantageous statement matrix manifold obtain lie invariant coordinate consider help span span reconstruct data point contain span back membership work reasoning hold large kernel cross approximate potentially k z kernel assume span conjecture approximately kernel subsample method discussion introduction kernel consider lie generalization carry goal relate image row span identify polynomial degree functional cut polynomial vanish informally identify vanish ideal relate duality algebraic geometric duality set vanish introduce scalar product slightly arbitrarily convention natural well satisfie n feature dx yx rkh duality algebra feature vector subspace polynomial fx vanish vanishing ideal sense dx usual duality result section kernel adapt exact approximate save polynomial kernel discuss algebraic statement prove latter spectral throughout manifold sample span cut denote sample row introduce kernel mi jk dx dx duality statement vanish appropriate span ns abuse notation say matrix independent notice elementary z analyse choose characterize dx xx dx kernel span imply I mf statement turn statement let denote matrix dx z xx nystr type demonstrate ideal duality employ datum manifold precede collect observation arithmetic decomposition require linear exactly svd via cost recover common subsampling completely need subsample contain subsample singular characterize
approximation read summation take grid form numerically discrete fouri dft slice restricted estimation memory series use g similarly excellent fit high frequency together summation correspond approximation final likelihood restriction image scale concern evaluation numerically scale invariant process size historical construction multiply cascade two multiplier multiplier poisson multiplier lp lp result mark bold la lf mmse lf lf lf mmse lb lf mmse lf mmse lf mmse lf mmse lf lf mmse r cm mmse lf mmse mmse mmse lf mmse lf certain limitation multiply construction localization multiplier specific normal poisson multiplier vanishing ensure consider estimator usual reflect linear standard sampler iteration burn period preliminary simulation illustrate fig estimation plot red classical bias lead vice apply lf mmse estimator realization process strong discard yield commonly improper wavelet transform deviation root realization mark c r lf mmse mmse lf mmse lf mmse lf mmse lf lb lf mmse lf mmse lf lf mmse lf mmse lf mmse well bold r lf mmse lf mmse lf mmse hyperspectral overlap pixel mmse c lf indicate red b center white dot original dot half patch histogram discriminant lf lf mmse lp size mmse therefore due result systematically outperform lf second deviation regression important remains directly reflect overall fit poisson multiplier find slightly inferior log multiplier due arguably slightly size force choice gain lf term bias rmse particular lf yield world notably reliably detect lf sufficiently enable estimation parameter patch come increase computational cost computation time image large cost lf belong design wavelet coefficient study g go fast wavelet plane indicate lf report practically rmse outperform lf factor likely lf real world fig pixel channel hyperspectral acquire project pixel mmse c lf red frame inspection indicate reproduce structure texture spatially homogeneous visually texture fig bottom leave corner corner spatially coherent consistent strong display variability texture lf spatially estimate strongly mmse lf corner truth illustration estimate image quality indicator quantify image measure coherence fig index mmse lf visual inspection spatial coherence fig mmse lf conclusion variability lf mmse lf yield portion since necessarily fisher discriminant criterion mmse lf separate bottom superior lf bayesian procedure quantity analysis I wavelet yet generic parametric statistical logarithm account impose theory design process approximate metropolis wherein infeasible constitute operational applicable size assess numerically process improvement rmse factor enable reliable pixel patch estimate future notably hyperspectral image rgb rgb member member characterization many application useful texture current two dyadic scale image difficulty estimation construction wavelet exploitation suitable within model enable infeasible assess several range significant benchmark discriminate commonly use gain notably enable image patch recognize common texture texture invariance literature scale image invariance tie pointwise image long recognize multiscale study regularity play central contour characterization contour texture consist densely strength provide texture tool regularity location description fluctuation collection value standard image tool include texture classification art capture image spatial location scale increment wavelet wavelet formalism theoretically summary would handle entire signature fluctuation regularity primary identification stochastic self former class tie construction fundamentally principle magnitude quantify image class process refer reference seminal tie j c lead estimator suitably relatively pixel sufficient encounter severe transform logarithm number pixel scale pixel consequence analysis severe first analyze yield make discriminate goal propose validate address overcome limitation e parametric formulate lie exhibit decay remain date wavelet exception brownian jointly parametric propose specific however method rely strongly easily formulated estimation procedure parameter recently employ heuristic parametric univariate univariate parametric relation actually image develop require mean logarithm process multivariate inspire covariance induce nature cascade construction new radial parametrize pass pass filter define wavelet characterize vanish filter low filter g purpose discard normalize reproduce similarity formal detail wavelet transform dyadic center cube eight neighbor within fine wavelet reproduce old exponent follow cube theoretical require construct formalism detail class extensively theoretically cf strictly regularity assess practice number wavelet choose sufficiently hold imply meaningful relevant novel statistical logarithm wavelet estimate bayesian numerically selection process lp stand normal log analyze range plot standard log associate within marginal wavelet member self member note wavelet process confirm formulate trivial reason property even log wavelet marginal significantly cf row mm standard normal lp center logarithm f wavelet cascade covariance logarithm wavelet suggest indicate decay r
produce infeasible iterate optimization parameter default exploit gradient option correspond bfgs hessian interior trust admit instead automatically indicate hessian solver brevity table report case costly reduction exact second order evaluation interior point minimum also recently practice solve subproblem concave subproblem formulate order tailor stationary existence adopt matlab module carry library initialize intuitive comparison among solver report give set solver solver performance ps occur solver mm ip tr nf fit nf fit nf nf nf tc ip ip tr ip tr nf number evaluation nf second cc table comparable fit difference attract overall satisfactory suitable scaling sophisticated flexibility scheme identification lead nonconvex nonlinear bind optimization problem analyze propose especially solution present compare art issue application projection scaling result depict obtain summarize view propose constrain performance nonconvex obtain respect technique order view hessian matlab convenient strength capability good impulse response order especially costly good performance rely scale constraint issue address wider arise machine identification sparse idea need context learn rgb rgb l lemma di di via b di crucial address criterion asymptotic hyperparameter marginal maximization primary importance impulse stable strategy projection play role computational order design presence box extensive effective flexibility wide arise processing maximization identification concern automatic dynamic model building measure field broad spectrum topic hybrid continuous tool system attract considerable automatic parametric maximum pe whose attribute mainly asymptotic property restrict understand see fair modeling consider instance advance consume costly demand fast reliable automate procedure identification identification inspire jointly computationally control face strategy main induce smoothness early reference field system regularization convex relaxation variation norm induce framework regularization prior bring encounter automatic determination make shall address impulse single describe convolution unknown white see maintain herein framework thus model impulse infinite prior hyperparameter prior flexibility encode impulse hyperparameter mention variety impulse response hyperparameter nonconvex handle large even matrix costly extremely ill design feature simple structure usually negativity exploit projection whose basic stepsize scaling computed scale technique convergence classical convergent combination choice robust signal problem scale scale projection apply impulse plan derive optimization present focus split define matrix presence negativity box present effectiveness identification respect solver conclusion symbol trace possibly absolutely lebesgue lebesgue shall shall record estimator impulse clearly pose dimensional literature problem follow marginal therein introduce density arbitrarily impulse decay impulse rewrite impulse ill condition stress truncation problem en condition e typical example say vector integrate q hyperparameter uninformative paper call estimate estimate follow map compute symmetric unless strong available impulse solve impulse coefficient introduce year impulse system structural dynamical system impulse combination exponentially decay seminal kernel paper appear family discuss alone impulse response exponentially decay rate semidefinite role alphabet grid list treat noise use problem k fx solution view compute lead negative value formula occur iterate objective product go point well good performance adaptively keep stepsize strategy cm cm cm describe show bb spectral successively rule different nonlinear rule unlike stepsize review idea bind extend approach split whose equality gradient q formulation relate method well study capability positivity whenever several processing lee negative exactly multiplicative algebra multiplicative correspond scale consideration motivate scale stepsize ill ill nonconvex recall objective scaling driven consideration constraint consist find feasible e eq iterate devise violate direction end define index I u define consequence x k diagonal eq lead choice scale relevant burden satisfied evaluation algorithm rely implementation gradient side ill implicitly devise detailed gradient cholesky factorization factorization ks finally objective formula cholesky simplicity formula q account ill previous formulae computation reduce matrix latter case occur iteration sake completeness report compute cholesky factorization compute cholesky compute matrix without need additional product detailed develop formulae ij moreover g ij x I k require kernel task worth need stack element output snr snr ss ill concern perform fine grid impulse coefficient consider kernel tc tc ss ss condition quality evaluate impulse
encode understand three medium handle rely ideal randomization randomization technique stand enhance scalability partial gradient proximal calculation algebra randomization iii first flexible framework surprisingly approximation centralized communication communication concept offer benefit exhibit significant acceleration counterpart quality inherently approximate speed processor motivation linear observation encode perturbation noise entry basic law physic image nonlinear phenomenon recommender retrieval model dimensional process first study convex ls vector multiplication absolute control ls producing mostly hard turn signal critical lasso denoise geometry readily low r r n n dct key lasso dimension exploit implicit operator dct large dependence multiplication cholesky finding newton surprisingly possess structure provably enhance competitive accuracy point normalize standard normal dimension noise fractional access sg optimization choice exploit within simple namely replace gradient gradient obtain strong access conjugate require full highlight role solution quality solved nearly understand mid within sequel reader basic notion convexity exposition gradient several fast reach target information expensive non smooth fortunately cost drawback dominate matrix multiplication l single potentially short reach surprisingly simple analyze need lipschitz continuous twice method lipschitz iterate bad case convergence attain q lipschitz case iterative function evaluation hope accuracy nesterov momentum achieve typically optimization provably offer key benefit unique improved efficiency obvious definition e strongly term estimator statistical differentiable hessian l strong simply lipschitz minimizer improve instead obvious highlight guarantees lipschitz imply guarantee turn convexity accelerate obtain gradient exploit convexity ccc proximal denote convexity summarize discuss section numerous momentum parameter computational trade rigorously bad rate lead cf similarly kk solve proximal accelerate proximal descent accelerate proximal l formulation basic method behave dramatically improve accelerate allow accelerate describe smooth minimization nesterov smoothing continuity consider smooth function naturally image graph quantum seem efficiency first generic require reach slow smooth fortunately composite objective far retain approximation non upper proximal gradient x x interesting proximal elegant incorporate classic method constrain preserve map operator offer flexible signal prior combination atom representation example atomic structured sign facilitate perfect proximal quadratic explicitly calculate connection efficient formulation soft infinite number admit rank whose atom whose operator numerous exist feature obtains simultaneously estimate smoothness calculate step order accelerate take method accuracy rely adaptation compact wolfe since element method optimally exploit f interestingly function tractable prove useful many application cover far composite q proximity operator efficient enhance smooth lipschitz commonly poisson objective cover apply call alternate multiplier augment h h u distribute turn closely relate bregman penalty input iterate feasibility admm criterion admm step fortunately support periodic admm long h k z k drawback proximal difficult surprisingly inexact admm certain reader generalization issue multiple problem solve simple infeasible optimization calculation randomize example convex objective extension notable graph via incidence aim solve find goal square relax q include positivity reality simpler preferable obvious operation calculate coordinate modify idea capture essence history classic gauss cyclic linear systems coordinate illustrate coordinate key descent iteration amenable pick lead configuration seek hope magnitude effort justify convergence provably slow coordinate hope slow surprisingly randomization choose coordinate random surprisingly rate randomize salient descent objective necessarily smooth cost incremental update importance coordinate randomize uniform strategy improve algorithm accelerate composite version explore often preserve accelerate version contrast randomize minimize include elaborate choose j kx p coordinate problem analogously contrast unbiased empirical observation expectation sampling index indeed optimize minimization provable capability enable beyond decomposable sg decrease step unfortunately lead constant stochastic quickly non vanish gradient indeed stochastic tune recent result size optimal convergence set show lipschitz another stochastic convergence algebra decomposition singular value multiplication due relevant representation efficiency method uniformly instance value svd idea behind randomized representation fashion generalize benefit nearly hence modern describe accelerate proximity operator depend nuclear traditionally easily synchronization however error bound rigorous secondly gradient finally obtain first objective integer draw r r tr classical qr surprisingly nearly approximation small provide randomization rank keep rest deviation around spectrum zero rapidly algebra iteration reader benefit randomization method routine multiplication block benefit parallelization ht proximity factorization core importantly randomize parallelism accuracy indistinguishable raw computational throughput storage capacity mid law consumption massive storage resource cost seem ideally heterogeneous computer within broadly drawbacks specifically communication master machine lead consensus synchronization activity computer procedure synchronization version development first practical scheme parallel proximity reliable communication refer ideal parallelization split job calculation compute great computer machine directly processor location form final ideal x nx k k f beyond parallel smooth artificial technique eq form basis extremely optimization far
label truth label independent run bp initial know well model sbm node core degree instead political line leave show correct bp fall instead sufficiently division succeed division political minima free energy correct global large therefore accurate partition determine size break move core divide political analogous learn giving panel algorithm structure propagation transition fraction agreement temperature large transition critical create line qualitatively behavior two overlap critical advance learn algorithm energy model heavy tailed formalism learn node node use future grant fa grateful de france membership application recently discover phase transition put access fraction well diagram cavity find agreement hard easy hard algorithm transition jump end qualitatively network detection network many include generative variety establish cavity analyze recently rigorously group size node chance distinguish propagation factorize likely globally factorize point locally lie transition factorize locally unstable efficient achieve reconstruction threshold transition first know reconstruction phase bethe point exponential would accurate compete point correct three transition dynamical regime community principle exhaustive energy likelihood enyi exponentially unless label shift transition essence factorize value accuracy nod certain belief cause throughout terminate phase beyond important setting label consume conclusion later replica lead result many investigate cavity belief physic happen easy mathematic calculation trivial rigorous calculation methodology carry property inference distinguish configuration posterior equilibrium configuration formally understand formation belief solver analyze variable affect algorithm picture hope organize section block section stochastic find transition accuracy qualitatively block split group assign edge parameter label via bayes rule label message neighbor estimate marginal node equivalently ignore loop message bp reach node accord marginal bethe optimal since likelihood however comment optimal exhaustive instability factorize phase straightforward adapt formalism except node reveal message external replace know plant ratio node connect equally infer partial interesting label label expectation maximization minimize bethe initialize em learning phase transition factorize fix go stable unstable overlap chance order overlap jump factorize accurate convergence number iteration block leave hard overlap critical heat map smooth logarithm base showing overlap various transition picture agree qualitatively analytical show experimental overlap value overlap peak thus phase transition critical overlap note author predict survival strength phase position predict picture happen regime transition focus plant graph qualitatively graph phenomenon easy numerically color
difference post content learner within enable profile discussion implication education community principle model community advance insight improve design equip diverse body like enable well department education help insight share massive international web conference community mat neural schmidt negative signal computer science piece massive rd knowledge conference w student self program stochastic gibbs transaction machine intelligence nonparametric poisson count data st international conference science city automatic relevance determination matrix prior ik ik nh ki ki control importance observe row lie irrelevant community place standard follow kb u equation fast et convergence cd ik k row column soft membership desire rmse feature treat leverage matrix iff matrix negative hadamard scalability issue ibp subset real world root mean negative log determine sampling row entry remainder via burn I hold rmse na I benchmark predict hold compute I repeatedly evaluate ultimately yield reveal great predictive I computational second run offer model extract feature avg student business interact multiple sub aim interaction final project sub company business challenge google think google competitive technical google internet project sub question assignment technical explore characteristic another assign community posteriori learner belong procedure community learner initialization affect group step run high extract community outcome unique create comment communitie participant framework dimension select reveal contain community crowd discussion individual respectively act statement similar knowledge group participant participant pass member proportion albeit degree pass course notable level act participant involve group group group large group pass explain high people final project similar member view discussion fewer locate suggest play role motivate discussion may difference response topic rarely learn pass second pass similarly nearly characteristic student case people statement proportion high read group final participant people attain master p indicator suggest number possibility course limit simply preferred individually act location member htp project sub place discuss final topic help careful community project support participant proportion act view relatively low pass significantly likely group trend suggest community still pass participant act group view seem participant group answer learner pass course assessment process sub participant discussion project review likely sub group p partly group opposed recognition pass group act review learning post word p interestingly proportion project course yet proportion fail furthermore comprise focus participant goal course outcome participant distinguish virtual meet discussion relatively group participant group albeit significance suggest participant support people group people world participant pass group participant project outcome htp extract subsequently composition reveal offer way sub degree sub social instrumental characterize learner need truly open identify relate discussion participant enhance discuss participant construct adopt reflect cognitive observe learner collaborative crowd see leverage crowd way information group instrumental project support crowd learner enhance idea resource confusion expectation learner mix outcome outcome clear kind achieve learn goal prefer learn individually group learn group relative pass project instrumental ask project yet people pass request help trend consideration case contain
action add remove action easy number number neuron output action denote connection connection round backpropagation layer sigmoid connection network neuron matrix reward reward action phase play model phase use exploration update play action know high adapt stationarity achieve expect train exploration k w w choice adversarial justify stationarity coordinate choose q consider exploration list initialize exp arm play action action reveal choose action predict network action exp km kx reveal update exp weight expression instance less possibility run layer achieve computed prediction outperform contextual bandit similar tend fast rate regret class circular every iteration increase low non stationarity outperform first drift worst well seem robust one neural stay bad entire model adversarial bandit network non stationary trivially show empirically drift serious candidate address issue contextual linearity regret contextual present contextual bandit stationarity reward neural network reward know expert choose successfully stationarity reward formulation variant appearance begin solution contextual bandit structured variant tree contextual leave reward combinatorial limit et al problem without per equivalent naive approach ik elimination policy exploration exploration unbiased train contextual thompson achieves performance point stationarity landscape landscape change reasonable continue main various hide neuron advance bandit well compare
focused point trick fuse detect lagrangian consistent give first add converse subtract consecutive conclude obtain conclude perform change variable put eq quadratic simplify explicitly show notice lemma prove extend k instant quantity sa inspire fourth conclude min mean absolute carry conclude proof express notice include lagrange multiplier otherwise infinite simplified last three pose subject proof opposite optimality conclude proof theorem definition remark work partially lead receive european european framework program fp asymptotic variance focus stationary series parameter detection mean establish fuse consecutive otherwise correctly detect key optimality technique class estimating mean area stationary datum detect parameter segment diagnosis noise removal piecewise signal propose know special refer recent survey tv denoise current tv fuse basis pursuit denoise trend regularize vast relevant focus location signal recovery fuse lasso study literature next sample lasso focus particular circumstance interesting fuse duality convex structure give study improve include variance non stochastic assume relaxed often specify value function piecewise variability total variation difference fit square likelihood tv fuse eq tv choice regularization parameter balance fit case value replace norm q known know question efficiently wide interior software moderate sized size alternate multiplier nice property theory sufficiently follow linear reduce large neither sign change prove reformulate standard know recover pattern hold lasso asymptotic however recover hold consistency exact change zero tucker kkt first add respect optimality condition namely unconstraine point stre key view bridge discrete brownian walk change drift insight analyze property start dual define provide intuitive explanation use times transition subtracting expression q term analysis follow signal satisfy piece package specify convex program call ht estimate detect change correctly reasonable replace q shown explain optimal explain bias term change appear walk problem study proper stochastic see kkt kkt condition solution derive solution give optimality condition give location notation sign know transition establishe decrease sign appear reference transition notation value nice estimate sketch piece wise condition drift end increase take specific bias whether respective minimum case segment part segment lie bias increase segment close beyond point consecutive together location merely bridge intuitive consistency view call approach consistency problem know since formulate lasso reformulate convert satisfies equation x n noise vector construction exist body exact variant construction lemma normal formulation interpolation normal matrix lemma x ai ai condition recovery k piece wise signal notice basically linear spline knot change condition k every shape k bad two k necessary reader necessity lasso consider lk n x simultaneously lasso noise almost interesting distribution achieve exact focus possibility achieve matter change change respectively change na notice consistency change detection impossible hand difficult slack sign relation regressor equivalent regressor basic pp exchangeable need establish low probability low ingredient walk end take sign slack add equation follow pair sign sign inconsistent sufficiently choose small change since lie determine point fourth I provide slack dual segment min consistency n suppose satisfied n n ks n nm simplify presentation consider accord start change consistency level within constant interval level neighborhood reach require reach course accord equivalent without generality take sufficient due enough require q event next condition reach give intermediate reach former reach eq initial segment give event therefore establish proof convergence sign condition
tx product sign recall higher vary compare recommended sign well difficult get norm simplify optimal value space optimization efficiently space depend radius top recommend lsh product focus comparison base use collaborative filter netflix rate user movie procedure generate item involve characteristic correspond outperform recommendation item netflix rank top user gold base inner different hash item q subscript ideally item generate item consideration recommend sort list recall start top rank item rank top gold standard increment see precision recall balance relevant report netflix figure indicate inner product sign product implement algorithms hash return item query lsh point fraction inner number product hashing repository thus recall aggregate query operate advance unfortunately top near neighbor fix threshold ratio scheme minimize evaluation perform rigorous choice threshold scheme select run thousand choose choose average query target produce low compute recall curve compare summarize computation numerous database provide scheme sublinear present asymmetric sign inner maximum subsequently solve projection demonstrate part nsf nsf sign conduct especially lsh complete demand number implement lsh thank computing team well department server provably hash author transformation approximate near hashing provide approximate sign theoretical well experimental evaluation paper problem technical size relate three equivalent value norm many arise element review recommender detection locality sensitive hashing lsh popular efficiently solve asymmetric lsh transform query collection transformation develop hash popular hashing etc another solve show lsh advantageous approximate neighbor bottleneck approximate near neighbor nn dimensional construct near report near neighbor lsh lsh time extra preprocessing existence lsh translate provably sublinear locality sensitive hashing lsh call neighbor need sensitive construct lsh generic framework lsh concrete hash present lsh random scalar xy popular sign random projection choose vector hashing show locality restrictive hash transformation unnecessary lsh main provable lsh lsh requirement incorporate figure fix panel transformation norm inner provided convert search hash sign projection popular hash adopt cosine transformation convert search transformation hashing function new suited exist code projection already outperform lsh get assumption generality q term approximately another method show problem define give collection define hash noting attain z lsh terminology construct guarantee algorithm query time minimize convenience see plot compare sign present value parameter fix would practitioner burden change norm affect top
expression fit er network gene table er confirm share across network gene across er recover run differential gene sub gene er er er quantify level gene gene b indeed group distinct er status differential er gene co expression correspond centrality level er annotate tumor two encode er rank centrality activity tumor interaction encode member factor highly er breast tumor lead death breast three role marker tumor express triple breast control pathway response primary breast cancer develop include capture markov feature node undirected matrix gene expression share across across apply methodology recover co expression er tumor factor method include approach tool exploratory analysis loading type observe categorical integer g status age load identify individual specific use extract gene greatly specific use necessarily extract matrix perform linear category network recover gene unique specific tumor exist differentially co restrictive covariate interest projection sample projection rely post hoc covariate interested see uniquely extend approach projection inform extend parameter shrinkage induce shrinkage loading substitution write row multivariate jointly sparse generate dense variable loading dense bernoulli chain appendix follow load residual beta three shrinkage specifically induce layer factor loading describe beta making equivalent indicates loading beta bernoulli hierarchical component follow extend work specific em posterior probability log estimate form form conjugacy prior beta expect log take eq simplify related finally conjugate description update sample I equation k manuscript warm start simulation parameter load component element parameter sparse dense specific conjugacy bernoulli z pz z k across th th eq dense parameter gamma conditional conjugacy q proportion beta parameter sample equation sample accord nk k accord equation accord accord accord process comparative breast gene datum cancer maintain cancer institute gene expression five run setting correct run default produce close match zero true iteration cc set accept delta row run set normalization construct minimum set sim component run redundancy statistic check redundancy across run follow number non group loading loading vector transform non gene number identify co prior gene matrix identity prior explicitly quantify value sample discard variance significance store count number find time draw structure explore co expression gene encode locally model infer gene jointly relate recover across diverse simulation gene expression breast cancer gene expression recover network mechanism necessity gene mechanism consist within expression undirecte across constitute gene co construct undirected module detail pairwise gene relationship rich computationally intractable gene describe compute gene cluster gene algorithmic partition gene module undirecte partitioning create imply gene gene module hold impact gene disjoint connect characterize group co gene expression decompose gene factor loading assume costly level gene limit loading encourage loading sparse enable extract sparse counterpart limit robustness system thousand mechanism create cluster prove without careful recover factor robust proportion expression besides encourage vary induce sparsity non subset uniquely variation correspond loading identify exclusive address subset exhibit latent loading vector orthogonality loading environmental technical status heterogeneity adjust remove covariate signal univariate testing estimate expression expression level module capture control exploratory association testing base recover co expression signal essential effect cluster number call behind develop gene unique next simple network sparse covariance matrix reconstruct recover quantifying sample gene express network specific iii co apply approach without effect level measure express er er similar gene co successfully sparse expression collaborative comprehensive fall category category assume approach category small category identify sample hierarchical cluster third category build iteratively group identifying group gene induce laplace impose loading specifically bayesian build induce prior gene error priori error across gene sample mean multivariate gene variance k latent remove factor factor factor loading parameter flexible modeling layer loading sparsity pc run initial correction dense initialize sparse software recommend quantile match set quantify proportion recall refer recover cluster precision score loading model calculate recover invariant switching sim loading score figure truth although relevance sim show recover sim expense sim dense loading sim sim pc simulation recovery score relevance sim inferior cc gene ability cluster gene heterogeneous e expression level orthogonality design gene row bottom row sim column sim recovery axis axis legend b recover co undirecte build gaussian recover appendix estimate analysis rank estimation component regularize highly approach matrix quantifie pair correlation specify submatrix correspond loading component edge gene assume edge nan edge distribution edge practically network semantic subset carefully component component covariate rank covariate breast cancer tumor filtering miss patient stage breast old among patient disease signature distant survival focus network er er er negative er patient er breast patient patient patient er er mutation mutation start random factor change recent stability run factor gene
start distance population first measure dt ds kernel pairwise correspond center sum matrix covariance simple standardized covariance discussion invariant application li population function identically copy introduce real value cdf center distance matrix distance I I always effort implementation correspond correlation value step newly unbiased sample publish let follow center name unbiased follow q distance I I k need inner q statistic apply subset notation statistic size element fact arithmetic prove similar must statistic lemma later statistic statistic removing order hold still define statistic counterpart sum entry remove identical lemma statistic correspond inner see degenerate moment nan I normal consistent independence difficult similar deep value argue univariate start lemma counterpart denote formula sign proof x I sequence modification use fast coefficient adopt spirit result estimator distance theorem term per lemma compute right note compute algorithm hand reader description idea subroutine describe propose fast run simulation distance screen experiment regard evident fast implement dyadic matlab go solution fast matlab desirable call low language theory implement htbp rr sample fast visual core cpu ghz mb r size direct generate message compute pairwise memory method require memory illustration run minute trend method linearly size experiment size rr zero pearson advantage wikipedia direct implementation pearson correlation representative bivariate moderate bivariate curve rotation rectangle aforementione uniformly rectangle curve size case observe get zero correlation significance pattern evident htbp size large fig sample method pearson correlation zero even distance clearly reader particular interest clearly pearson converge fast converge counterpart correlation note case stay previous large li analysis dc sis study use direct li originally advantageous size screening covariate marginal utility utility distance correlation dependence measure sir study li screen magnitude marginal function forward backward potential research name independent rank utility indicator appendix bivariate compute algorithm average assess covariance enjoy cholesky know minimum active predictor quantile replication proportion replication replication close minimum well screening sure ensure close different simulation empirically examine effect cutoff integer sis require cover dc sis sis c indicator model interaction fan consistent challenging utility predictor sis model replication sis c htbp sis present sis sis linear model far underlie dc sis outperform little chance predictor counterpart clearly advantage evident observe dc find application lead armed find adopt correlation evident certainly make need em proof subroutine call input observation sort indice th similarly partial use follow recursive definition compute step algorithm partial input quantity compute triplet stay e th among recursive enable partial sum compute subroutine eq subroutine dyadic input assume dyadic interval assign fall integer compute simplification similarly q simplify nn j nn n n nn nn n nn ib nn ib ij ib nn q rx rx r write evident verify ik k n n b kn kn ik b compare indicate proof x verify proof verify j j j c rewrite
lp statistic satisfactory job deviation range desirable strength application idea discovery sparsity density estimation note copula copula define copula du discrete continuous lp copula square integrable copula density admit x equivalently proof expectation lp copula function maximum entropy product score recommend copula well nature function slice describe conditional dependence utilize lp nonparametric data us v fisher contingency result smooth copula v wise bivariate expansion expansion show dependence contingency formulation integrable singular u k appear decomposition comment interpret transform lp canonical contingency maximum number estimate v th dark black light medium dark contingency seek display column category correspondence lp canonical lp define lp profile column profile lp correspondence table lp dependence use aforementioned fig bivariate correspondence display association scoring alternatively copula profile display weight spectral sm lp correspondence multivariate usual normal analysis use lp theory discrete density correspondence copula col construction piece copula correspondence piece correspondence compare shape function row category fisher know ratio storage requirement numerically table lp correspondence lp provide large contingency spectrum provide gain establish connection lp statistical derivation contingency pearson odd first verify p x prove j decomposition fine understanding dependence dependence contingency comparative simulation measure second follow identity significant index present belong simultaneously correlation whose uniformity monotonic transformation representation contingency next property em bivariate form contingency table choice orthonormal switch orthogonal orthonormal polynomial follow polynomial gaussian copula note verify complete py lp significant element evidence coefficient linearity investigate em fisher smooth compute divergence number computation freedom degree dramatically drop applicability sparse component quantify effect lp coordinate correspondence insight density change level covariate illustrate understand age accuracy chi depend classical association contingency like chi sparse table many small chi statistic yield df associate group em em em high linearity also great regard age agree age increase find transformation contingency q power testing restrict attention vary contamination vary iii bad leverage point vary tail scale em shape winner pearson ex winner ex ex winner winner ex carry rejection significance power setting finding remarkably wide setting em consideration investigate scalability detector distance generate report run three write evident size almost infeasible computationally efficient r ex ex ht lp discrete marginal demonstrate conditional approximated jx hilbert theory express fx fy fy jx fy jx alternative parametric wrong specification notice flexible function lp nonparametric yield easy misspecification integrated aic coefficient generalize major definition version discrete variable chapter extend incorporate algorithm provide generalization classical order lp function investigate useful lp correlation quick purpose follow relationship bivariate appear rank jx special corollary bivariate interest comprehensive insight follow lp fy fy fx fy lp simultaneous estimation slice densitie various conditional quantile accept reject comparison approach seminal linear nonlinear produce curve goal child lp discuss spline display b density slice result conditional quantile curve recall marginal age highly skewed prominent tail area algorithm satisfactory job estimating argue regression scientific level c component show fig describe conditional rapidly density bi quantile also evident shift go beyond description modeling tackle tail polynomial interval u u probable satisfying verify probable value probable table moment lp moment discrete distribution poisson geometric ex student df chi df lp lp matrix distribution em em thm statistical university pa college tx abstract approach exploratory lp statistical science statistical exploratory tool concept along real illustrate statistical systematically tackle fundamental orthonormal hilbert exploratory big lp orthonormal estimation lp smoothing correspondence analysis contingency goodness fit height recent availability type nonlinear univariate statistical technology substantially execution get easy become availability isolate seek systematic automatic permit easy establish relationship various statistical attempt em unify fundamental quantile lp skew density lp broad range goodness categorical modeling correspondence coherent hope lp design outline analysis illustrate use example nonparametric general orthonormal precisely mid lp lp orthonormal unit mid px mid lp score power note lp score function respect continuous recover polynomial orthonormal shift give model score interpret statistic score function form orthonormal score always derivation continuous hoc avoid power express transform discrete continuous introduce concept generalization lp compact efficient compute know challenging problem define lp moment unit function hilbert identification treat lp illustrate datum rating skew lead goodness fit discrete scale multiple density lp skew choose quantile density mass call comparison du du lp skew gx em family estimator aic type du theory equal equal provide goodness two skew figure previously analyze select baseline density drive aic select skew suggest model drive aic skew mode reflect mixed review sharp probability review
score law investigate theorem instead hard example projective cluster cc compute om nc kk k algorithm compute nk nc observe residual theorem old simple explain attractive practitioner understanding via appropriate see effective rigorously effective power law corollary axiom approximate select correspond large leverage score surrogate provable proportional leverage deterministic provably leverage moderately power law provide evidence decay sampling match state art lot qualitatively reveal component pca sharp actual form low surrogate interpretability make attractive practitioner rigorously matrix utilize deterministic technique discuss literature suffice highlight contribution score define leverage first back propose sample successful theoretical lack theoretical utilize randomization identically pass replacement constant ok k two ii version implement admit open provably tight leverage sampling provably counterpart law decay provably obtain approximation literature decay theoretical decay leverage score state art deterministic low rank well art nm whose clear equivalently write define spectral ij respectively describe leverage section approximation wish deterministic leverage compute singular leverage score simplicity sort well score ensure column accumulate energy carefully control parameter choose prevent approximation ki generality let sort require arithmetic operation discuss modification innovation let algorithm least column immediate column extract requirement leverage decay fast intuitively leverage score extremely fast decay output score come expect leverage decay column offer subtle point leverage score imply trivial argue leverage good make intuition consider score find relative score follow require absolute law algorithm art expectation constant indicate bound apply power theorem spectral deterministic column notice relative column fix norm power decay world investigate amazon citation soft google ct slice image citation work leverage sharp decay deterministic leverage score particularly htb plot power law curve exponent list leverage plot red marker power good true leverage exhibit decay decay correspond enforce sample offer good leverage set co bag word row row take finding achieve k indicate sample plot move achieve agree early however moderately case column suffice indicate loose law decay power optimality leverage decay vs plot relative vs fast surprising rank confirm compare error contain exhaustive approximation l nucleotide email work c work c c c c c c c c c c c c c c c author near deterministic theorem author test build third leverage score replacement tool matlab approach execute repetition observe appeal almost leverage randomize well digit show set law competitive case fit profile near quick overview deterministic deterministic go seminal qr develop strong require arithmetic column score column sampling euclidean view seminal idea randomly simple column euclidean hold improved accuracy distribution volume fast another include row sampling orthonormal matrix result essentially kind svd mention dimension thin cc cc k vector respectively well k k I use eqn nk novel deterministic score theorem use perturbation eigenvalue cc conclude proof straightforward picking imply lemma let decay column convergence positive monotone eq leverage score collect leverage score hand first side hence q q eq preserve immediate implication highlight nk frobenius unitary transformation zero conclude step compute top describe improve maintain guarantee
variant r cost long approach run complexity condition large rather arise independent dependency hard nevertheless practitioner address propose call stagewise completion exactly complexity desire complexity block project gradient descent context precisely iterate singular projection matrix step onto set solve unknown result extend correctness leave preliminary showing correctness matrix albeit number desire wise try recover get finally stage iteration remain novel understand project consider completion general hard obtain svd spectral bound eigenvector norm analyze prove natural extension bound perturbation enough eigen gap due consider interest organization overview section warm technical capital letter denote entry respectively stand singular tail result satisfy n c argument symmetric pp update sample select decay completion input continue n n k step routine kx x next p desire issue desire stagewise prove st present run projection onto basic avoid step ensure decay obtain pp k n iteration invariant k stage wish frobenius update k invariant present proof prove every follow induction clearly suppose outline q hold n hypothesis n argument tell tell stage hand use sample establish invariant completion complexity good completion behind entail term taylor series perturbation good eigen technique context still suboptimal interesting handle definition pt matrix completion recent convex paper iterative desire knowledge complexity project projection potential bound extension good eigen may independent low
observe zero loss round lf mf leave try sub similar adaboost produce indeed possible network linear key assume implie tell since say small observe q mini batch goal sure batch eq function satisfy constraint exponent go go suppose convexity advantage ability boost plan challenging task plan generalize binary claim example unlike like adaboost artificial layer recent development machine show variety successful gradient sgd ability major disadvantage sgd attempt momentum reduce natural weak learner boost adaboost example guarantee boost weak guess iteration ensemble train major disadvantage adaboost ensemble classifier mean predictor paper sgd rate sgd constant boost set network adaboost maintain proportional focus mistake example intermediate intermediate output achieve prediction edge sgd choose random sgd finding tx gx tx next try define mistake crucial number iteration example
comparison nc comparison functional commonly ad region brain average voxel preserve patient pool fdr voxel organize provide drive conventional fdr procedure finite lattice voxel usually hypothesis ise normalize ts possesse ise therefore neighbor odd reflect together odd account nonnegative dependency voxel reflect ratio conditional bayesian mrfs index mrf know mc prove et compound decision theoretic framework fdr imaging nan hypothesis operate fdr drive replace partitioned procedure base fdr fdr follow pool reduce order homogeneity fail model distribution state voxel six boundary test individual probability oracle drive procedure imaging model estimate model maximize introduce categorical k function p separately constraint method lagrange multipliers w sl lx lf take second derivative equivalent solve nonlinear nr nr rather search together maximization term em backtrack decrease definite function recommend n sampler constant approximated reader normalize refer work back number voxel avoid degeneracy likelihood penalize et gamma finite distribution penalty consistency normality chen think inverse would formal auxiliary em equation remain unchanged pool set update estimate stop consecutive specify stop monte three carlo recommend practice satisfied step stop criterion dimensional data datum drive procedure compare group globally procedure ise drive procedure sampler et burn simulation cubic lattice replication ise parameter comparison fix average positive yield fdr fix fdr level control case low generally fdr lattice additional procedure control fdr initial distribution claim fdr set embed generate burn five ht voxel noise signal signal result summarize summarize claim procedure detect procedure find area ad surprising decrease cluster affect include al temporal gray ad et al proportion rarely disease high proportion pre claim group mix progress subject et et al surprising chen p longitudinal treatment false spatial signal false discovery powerful discovery multiple multiple testing dependency j scan disease interaction maximize generalized likelihood automate b york burden mm testing group iterative image mm r random chain topological mm chen mean false discovery fdr fdr mm b series large false dependence la longitudinal brain mild cognitive york simulate normalizing bridge relaxation thresholding functional rate false control p value weight l operate discovery procedure mm discovery thompson monte likelihood dependent datum gray r longitudinal clinical emission diagnosis approach diagnosis online lee patients g relative capability mr imaging associated disease w j association cognitive chen lee e categorical analysis emission image random field association smooth gaussians hide measure field image transaction machine intelligence disease diagnostic approach mm early diagnosis disease l de li reduce automate standardized diagnosis mild cognitive p comparative nd ed york li gender mm hide master thesis di rd ed york oracle compound discovery multiple mm x imaging ph department mm population nd ed york wu false discovery test diagnosis nuclear modality ed pp zhang field effect nan mail edu department ann mi email department university ann mi nsf grant edu analysis write complete find http pdf disease human brain change consider voxel emission imaging study purpose voxel multiple testing procedure mostly ignore neighboring voxel substantial significance minimize subject false three expectation powerful conventional comparison mild cognitive disease status increase normal imaging false maximization ise mild ad predict progress diagnosis ad extensively emission imaging et demonstrate patient nc early voxel common difference seminal false discovery rate fdr error control fdr fdr false accept hypothesis error fdr fdr fdr traditional rely heavily assumption independent rarely wu suffer severe ignore al independence procedure procedure modeling dependence chain hmc oracle drive follow al pool different prove procedure high li et implement genome study analyze et among know software parametric topological fdr al fdr cluster way voxel level
recommendation combine complementary effect another regularization soft constraint impose remain prediction change incorporate uncertain leverage auxiliary rating incorporate factorization correspond auxiliary uncertain rating recommendation requirement need recommendation incorporating constraint collective e fm g summarize work collaborative recommendation auxiliary knowledge transfer via collective constraint attention art rich collective enable also adaptive cause sophisticated table datum base user categorization table recent filtering perspective domain domain classic transfer transfer practical recommendation cross domain heterogeneous technique effectiveness natural heterogeneous heterogeneous transfer flexibility heterogeneous simple exist integration typical recommendation recommendation application type social mobile useful need learn rating item recommendation evaluation motivate sophisticated objective metric exploit recommendation source explanation generation recommend item even leverage auxiliary collaborative recommendation learn categorization recommendation auxiliary recommendation collaborative research quite big ai especially variability heterogeneity recommendation auxiliary different technique exist transfer propose categorization knowledge generic describe framework finally thank technology play role internet display rating usually exploit user circle preference behind collaborative recommendation service practitioner extract auxiliary order especially transfer extend exist categorization technique collective transfer rule regularization fourth representative knowledge strategy expect collaborative recommendation recommendation standard component internet provide service use recommendation content recommendation collaborative recommendation content recommendation focus collective intelligence prefer similar explicit due fortunately additionally user rating recommendation content time contextual information potential help reduce improve recommendation static static user content etc l user health dynamic context remain quantity item context etc user item relevance network share friend etc feedback rating item user etc item specifically target datum question extend categorization transfer answer dimension transfer collective transfer rule knowledge detail particular may finally conclude paper summary discussion target target feedback content predict auxiliary left user denote htb fundamental transfer transfer transfer collective transfer transfer type study specific via transfer regularization categorization collective transfer later expand category expand strategy style mainly transfer recommendation factorization mostly successful rating incorporate different mathematically regularization contains explicitly show representative work framework eq aim adapt extract auxiliary target transfer similar adaptation section adaptive transfer strategy eq regularization transfer implicit record rating extract factorization feature factorization work auxiliary dense classical learning particular note compare collective codebook transfer early regard far transfer behavior book level rating pattern auxiliary co entry denote rating corresponding codebook codebook constraint rating share target indicator codebook codebook expansion membership codebook recent generalize codebook include share supervise label information effectiveness document categorization rating user target rating pattern kind stable high thus available auxiliary collective learn share effect bi transfer richer similar representative collective transfer note knowledge learn simultaneously knowledge collective factorization rating idea item share bridge use link factorize variable model propose matrix factorization item rating user factor universal auxiliary feature topic item specific feature item item item movie description lf gp nb multiple user auxiliary learn similarity target auxiliary effective compare alone item user weight similarity share feature selective align collective knowledge star numerical item besides matrix inner share share transfer balance rule effectiveness heterogeneous datum transfer latent incorporate raw learning constraint transfer mathematically rule include raw auxiliary knowledge knowledge model rule b ii fm fm represent user novel vector rating triple item entry rating pairwise latent via prediction basic factorization inner vector fm
develop thresholded assume accumulate past eq thresholded simply converge eq stage accumulate per gradient accumulate rate determine determined behavior tend modification rate begin g dropout relu cause pseudo draw minibatch correction g I memory outlier update mnist maxout compare algorithm test sgd momentum decay summarize converge technique include use learn momentum sgd analyze use minibatch compared minibatch able almost log local htbp htp gradient insensitive tuning hyper tune sgd improve provide preliminary deep able comprehensive empirically mm thank computational provide partially support cifar research grant discussion read maxout use parameter learn momentum momentum sample maxout experiment implementation experiment lead careful learning amount noise gradient utilize tuning element curvature far new automatically preliminary neural well stochastic search propose exploit curvature learning rate deep paper use hessian cost parameter gradient dimensional proposal estimate option gauss newton estimations hessian suggest way curvature diagonal inverting operation track order variance reliably root mean statistic reduce keep newton order method transformation gradient quasi transformation basically quasi newton scale affine directional newton propose solve directional require inversion maintain quadratic order directional update unit vector algorithm element size option hessian write directional unit want estimate curvature estimator consider visit bias optimally trade bias incur use bias expect estimator gradient unbiased bias base parameter make average next minibatch find close strictly real basically correction reduce stochastic step directional compute numerically unstable stochastic decide numerically taylor equation always practice work average also decide step take large step move rule assign element gradient update scalar multiplication perform detect outli detect
ensemble tailor rsc ensemble classifier art gene filter set decomposition analyse provide motivation rsc overview ensemble literature technique rsc ensemble scheme classifier constructs represent explanatory call possible value explanatory response measure define attribute sphere stem sphere centre practice sphere tuple centre within distance centre close class union call contain example class pure cover problem involve find cover proper cover graph case np cover pure requirement introduce parameter proper cover requirement contain case retain sphere admit infeasible datum classifier parameter section rather constructive whose decision classify new key diversity ensemble broadly speak diversity ensemble employ heterogeneous sampling weight different modify classifier randomization method follow ensemble bag bootstrappe member boost weighting create diversity classifier boost bag forest bootstrap optimal searching subset candidate replacement attribute forest combine sampling rotation forest involve partition transform component entire set train technique ensemble form sort scheme analyse diversity link decomposition framework applicable simplicity restrict class expect entire attribute actual response expect classifier explanatory denote equally set loss bias cause error result capture underlie decision describe prediction I variability cause finite component loss example call average bias decomposed express variance variance incorrect net error variance decrease net principle benefit variance ensemble address biased combination without generally estimate perform ensemble conjunction designing ensemble design diversity ensemble produce interpretation single sphere informally repeat discard datum cover center find find sphere sphere great add covered case save sphere centre class description distance normalise onto htp sphere classifier create center case store store boundary effect sphere cover cover cover sphere target sphere target cover classifier class sphere close centre cover select close cover outli preferable area decision specifically illustration figure area dense area cover smoothing boundary set competitive classifier use base basic rsc could create diversity majority rsc ensemble member assess diversity make rsc cover classifier parameter sphere use number misclassifie within parameter advance cross however impractical automatic implicitly sphere close class dataset growth sphere hence rsc principle ensemble informally current rsc classifier case list add htp lb e gd j formal description give vote continue focus misclassifie weight previously misclassifie drive constructive outline member different attribute sphere build classifier attribute replacement original attribute set majority vote employ final hypothesis input lk j base rsc generalise hyper rectangle classifier wish rsc ensemble technique aim confirm rsc rsc base rsc ensemble except rotation forest rsc ensemble outperform dimensional assess performance adopt describe base nan hypothesis classifier alternative result nan reject post pairwise lie different rank classifier hoc diagram introduce bar clique powerful adjustment nan follow standard distribution control classifier simply perform lc dataset diabetes heart tumor uci repository six benchmark table example class attribute diverse continuous normalised use six evaluate subspace ensemble feature feature base classifier rsc classification measure four ensemble pattern addition notice accuracy similar classifier separate rsc table adaboost train adaboost bag setting train training quick quick selection rsc impact overall describe rank difference average rank level htb c bag image diabetes cancer rank bag diabetes heart firstly high nan hypothesis significant performance majority comparable bag decision base rsc bootstrappe support base set tie basis classifier classifier comparison difference indicate misclassification direct resample perform adaboost well bag experiment widely weight rotation subspace ensemble ht c c diabetes cancer rank rank diabetes heart train optimal value entire show critical diagram significant difference clear outperform clique forest eps difference subspace rotation rank forest reduce base cv ensemble size however many demonstrate improve classifier indicate cr cr cr cr diabetes heart cancer show combine difference diagram ensemble increase ensemble mean critical detect similar apparent htp rsc classifiers eps diagram ensemble dominant classification instance domain research area database offer instance ensemble think outperform help responsible number case learner attribute demonstrate overcome inherent fact htp l bag multi bc cn lc pr htp ab bc cn lc pr rank ab bag lc pr avg htp l dataset bag bc ct lc avg f broadly speak use scoring rank produce information ig select training adaboost bag subspace rotation well conduct describe ensemble adaboost bag default parameter eight rank high rank conjunction filter space produce ensemble diabetes decomposition diabetes heart heart image error var var rsc vs diabetes rsc vs heart rsc vs rsc
expression decompose inspire communication analyze amount require reproduce observe quantify content send shannon character entropy fortunately equivalent identify without generality input line constrain simply correspond computing problem allow simulate namely pm v assumption implication relevance minimum causal influence require explain achieve quantifie measurement go determine measurement via solve consider display involve bipartite scenario produce introduce hidden serve decompose generality similarly suffice influence nature identify real likewise distribution pa px py x usefulness take close indeed vx observable uniformly e measure measurement dependence ref namely measure reason numerically lp stem proportional distance converse obtain mutual measure bipartite outcome maximal impose ad constraint obtain relation conversely realization projective correspond upper depicted text insight pure projective scenario share protocol receive particle produce quantum focus reproduce independence particle weaker arbitrarily involve receive dag source assumption usual correlation extremely characterize despite non nature input symbol input function analogous usual equivalent analogy measure measurement dependence degree measure eq fulfil measure obvious marginal impose observe distribution reproduce write distribution determine previously step write entry pa b represent correlation matrix encode reproduce entry equivalent eq find must also optimisation minimum equivalently find sufficient product decomposition v maximum compatible observe specific input state basis assign marginal thus full determine range linear minimum depend correspond must relax reproduce amount source sketch could measurement parameter q optimize four free look free remain free probability discretization continuous confident non conclude test maximally state distribution prop observation prop prop theorem prop definition prop prop example prop prop problem prop correlation causal experiment impose classical explanation natural ask locality independence conceptual treat systematically alternative causal mathematical causality express variety range relaxation novel causal interpretation distant experiment share observe detailed physical setup arise consistently property language parent causal assumption choose observer influence ideally constraint verify commonly ask quantum locality relaxation observe question great relevance locality protocol constrain structure security often important theory several question attract considerable example measurement dependence strong resource impose possible correlation measurement source measurement reproduce projective relaxation locality bit distant correlation framework treat relaxation measurement locality several concept mathematical aim describe causal correlation community develop quantitative structure systematically represent dependency degree influence explanation observable cast computationally program demonstrate operational minimum reproduce observe measurement define cast quantify relationship discrete dag graph parent unobserve identity encode relationship dag scenario perform input outcome causal measurement generality encode together hide causal mechanism locality measurement independence express network relaxation need devise way quantify sensible introduce concept causality bipartite locality causal causal communication relaxation measurement independence input correlate common observable observable intervention value place influence keep intervention decomposition parent probability notion coincide maximally correlation consider locality relaxation shift cause fig situation highlight relevance measure note use quantify drug interested case bipartite illustrate easily understand much e independence probability relaxation compute program long variable variant usual quantity carry restrict explicit component component expectation modify arise application constrain objective appendix constrain minimization variable reproduce reformulate primal lp dual highlight another nice framework close valid detailed section proof first minimal influence require require simulate intuition precise output eight quantity last term represent regardless particular quantify communication quantify locality require shannon send message framework binary maximal produce bit communication protocol reproduce correlation locality interpretation exist set two fig compatible form polytope polytope pa quantum matter generate correlation locality outcome dependence resource simulate mutual small number measurement fundamental implication increase requirement find result leave regard either maximally state two state unable mutual bipartite input assume numerically relation minimum mutual I ref ref quantum require tight leave room detailed one go via dimensional primal dual lp formalism optimization framework programming reformulate sign respectively concrete consider call transpose program call obeys call hold lp admit primal least crucial feature programming duality feasible feasible problem eq convex convert lp arbitrary implicitly obey yet however convert straightforward procedure kind combine yield lp crucial primal albeit implicitly auxiliary replace norm non unconstrained yield desire statement reformulate simply include lp clearly lp non programming formalism particular relevant direct dependence cast associate equivalent value primal I I ki argument constrain non negative order convert lp space lk implicit negativity guarantee due relevant eq extend primal simply simplify variable u variable suggest replace constrain non negative bound inequality everything value value primal proceeding along primal resemble extend q form decompose upon correspond dual lp simplify normalization drop go infinity turn desire simplification proof accordingly go present derivation namely solve primal form dual finitely feasible lp bound vertex feasible unbounded direction ignore text negative calculate optimize optimal applicable provide least hidden part contain geometry program explicitly dual duality dual feasible inequality establish see hold simplify consequently fully denote aim contribute ignore problem constraint constraint guarantee contribute thus characterize convex hull however well concave polytope vertice relaxation dag fig minimal direct influence simulate lp determine analyze depicted fig analogously variation briefly end assume causal consequently possible hide observe decompose way define quantify observation move characterizes last equality b compatible corollary problem translate consideration b state
map element mean point phase point measure great circle distance rational project velocity video salient motion region scene map map video near knn undirected network comprise stability connect matrix self reinforcement point laplacian typical dominate scene select query scene perform detect rank query label successively positive label assign precisely individually c identity parameter range initial assignment iy jj final vector evaluate diverse crowd public view exhibit motion subtle comprise use test capability propose detect instability region scene detect detect scene corner scene accurately able salient interesting motion dynamic category motion cause individual crowd google aim individual crowd motion anomaly effectively sequence manually label public merely publicly difficulty perform comprehensive determined region score region salient clarity different interesting perform scene feature inaccurate due illumination rank produce lead mis demonstrate low level stability phase similarity indicator salient detecting source surveillance scenario importantly track model capable discover optical flow drawback optical extremely motion source grant fp h education pt chen signal edu chinese email ie edu place salient crowd attention surveillance framework salient crowd scene transform feature global global discovery intrinsic dynamic level identify eliminate public demonstrate effectiveness salient various area security growth public recent automate video law technology event hundred thousand monitor due crowd severe attention span manual monitor task demand cognitive attention therefore effort towards solution identify interesting salient region ultimately event security deviation ordinary anomaly rare interesting scene generally accomplish firstly activity scene identify anomaly region contrast exist motion regular region due scene crowd motion dynamic detect accomplish unsupervised contrast crowd motion feature global crowd motion plane allow intrinsic motion dynamic manifold perform iterated laplacian approach indicator salient motion dynamic unstable crowd aforementione purely unsupervised requirement stage experimental public dataset capability propose crowd salient source local affect build motion amongst source individual crowd enter leave instability dominant flow scene crowd behavior motion tracking trajectory require commonly track motion trajectory geometric structure source semantic detect anomaly principle enter scene semantic tracking tends fail aforementioned method certain extent crowd tend fail dense track behavior individual build crowd entire flow field hide inherent motion lagrangian crowd flow stability unstable motion discover field level optical identify interesting region use direction feature scenario occur motion move opposite direction cope motion area source localization salient dominant flow motion unstable crowd limited detection instability real world public crowd behavior show classify bottleneck sensitive detect salient accurately summary propose low contrary require salient region indicator represent crowd optical flow crowd velocity dense optical vertical flow optical field accumulate comprise obtain inconsistent velocity average crowd motion field broad crowd denote low level computation dominant flow crowd capture subtle quantification dynamic particle particle apply track velocity initial position unlike optical representation
feed mf unfold framework base model still feedforward potentially elimination incoming express imply incoming belief message aside compare message mf differently yield flexible see question affect unfold generalization similar bp mf consider interesting maintain objective edge weight implement deep maintain general activation rather optimize test time parameter discriminative objective form raise possibility train sigmoid change tie investigate performance architecture instability example interpolation sigmoid generalize easily straightforward optimize related computing message formulae schedule accomplish propagation add parameter could necessary check might level training connection unit reasonable sigmoid simple activation vanish activation generalization especially appendix even complicated sigmoid activation unnormalize bp reciprocal uniform update appear comparable commonly spirit spirit maxout similarity whether difference generic mrfs mrfs incorporate novel deep unfold nmf nmf apply many aim spectra together basis operate usually magnitude fourier domain source column time basic appropriate generalize divergence yield active subject negativity multiplication initialize reconstruct mixture general nmf basis combine train separation discriminative base task basis term bottom control objective account de reconstruct speech part activation basis uniquely nonetheless bi basis bi directly derivative convergence basis reconstruction train reconstruction basis incorporate criterion toward across call cast identify iterative train network recursively multiplicative split nmf part update variable eq propagate negative activation time th layer recognition challenge stationary child home development journal speech room impulse six db test set consist total noise training room impulse complexity evaluation target spectral magnitude window window feature slide window consecutive frames slide window reconstruct line nmf vector speech iterations nmf base room network use feed hide layer tangent activation denote activation index dnn element nmf experiment frame vector logarithmic magnitude nmf compression linearity clean take direct consider speech experience train function speech amount objective magnitude speech magnitude sequence mask spectral nmf mask estimation come dnn comparable solely context deep architecture db snr db topology x minimize momentum early stop cross development set prevent set use nmf nevertheless perform investigate different dnn topology hide per development nmf optimize shall multiplicative training condition basis solution basis sample basis determine deep nmf kl divergence update equation layer use evaluation combination perform initialize layer basis layer mean final basis describe special layer basis context layer train frame reconstruction consist row nmf whereas train parameter rr snr db avg k avg experiment table nmf topology deep strong improvement relative compare dnn nmf deep dnn result deep nmf versus dnn least performance nmf discriminative training layer consistently much gain one huge conservative time come increase intermediate topology need speed accuracy work factorial research unfold unfold algorithm propagation markov field variational conclusion general framework exploration space deep architecture difficult sigmoid could see unfold architecture approach difficult hope novel insight show sigmoid mrfs sigmoid mrf n ba affect posterior easy mrf sigmoid use drop normalize sum constant assign add form rather separate activation look index message belief uniform leave eq verify become sigmoid numerator denominator back term fact exponent exponent give rise various c geometric arithmetic limit improper geometric derive calculus power jensen assume continuity limiting complete proof form pass clarity exponent mf bp q mf source stack activation stack able stacked source reconstruction function rule get rewrite forward pass start layer give start train basis activation nothing prevent spectra magnitude oracle thus optimize actual also typically speech speech thus wiener filter include optimize reconstruct speech measure source l b w h l h h accordingly kk use notice kl simplify final analysis avoid obtain rescale normalize version determine intermediate layer derivative intermediate split h tr positive part follow k see never store quantity update heuristic split respect part use positive multiplication factor eq operation need respect split positive recursively set negative apply though nmf split eq eq carefully storage tensor reformulate operation matlab create negative part eq eq rewrite similar everything source reconstruction wiener kl nmf update without normalization normalization perform part respect recursively gradient use respect layer tie computation usa deep network successful knowledge build constraint expense deep way architecture unclear aim advantage deep optimize parameter powerful architecture within show allow field new architecture inference algorithm negative incorporate sound source unfold yield train multiplicative style speech show parameter advantage goal strategy avoid iterative analogous layer architecture use expressive inference advantage intuition david computational analysis incorporate base straightforward include world visual geometry subtle latent constraint insight gain modeling model mathematically intractable belief variational approximation derive latent despite greatly slow challenging deterministic define expression execute discriminative speed versus trade produce system well know conventional mechanism box method work dnn system clear discover modify consider art methodology address difficultie task design deep step derive probabilistic tool unfold gradient relatively straightforward case field mrfs mrfs unfold belief show architecture unify formulation generative level mix power despite non form unfold basic multiplicative preserve non conventional sigmoid require unfold code unfold back belief markov field reweighte belief bp train implement inference predict unfold original critical architecture network pass mrfs input propagation replace conventional sigmoid novel work also mrfs network schedule unfold inference sigmoid result sigmoid belief mrfs rely contribution deep architecture base unfold mrf network parameter propagation showing benefit speech discuss mrfs sigmoid pairwise vertex v n variable identify pair I h I abuse indexing mrf write function typically represent scalar argument function divergence true equivalently maximize fully variable product posterior preserve form sigmoid message comparison message note must maintain schedule avoid maintain update rate implement arbitrary schedule compute keep eq schedule implement message arithmetic message optimize update schedule message convenient log take inside example sigmoid mrf write fact twice notation
association rule side rule support association support frequent situation combination variable variable association issue one treat tree base refer simplified rule tree rule ensemble summarize order build list order rule satisfied outcome new frequent classification mean outcome variable default rule avoid overfitte add preferred preference shorter break tie data rule default metric update base default rule rule assign frequent original default rule add algorithm rule tree ensemble summarize rule rule rule variable transform rule ensemble secondly replace discretized version discretization deal rule rule classification without ensemble function example team choose team game team team player play team team ccccc data randomly select assign variable framework build regularize maximum extract extraction rule provide alternatively extract forest outcome assignment rf sub rule consist assignment rule metric r format execute predictor return n x c rule short rule error x rule select rule metric condition n furthermore simplify tree condition x else extract interaction remove prune top frequent condition essentially condition rule forest contain outcome therefore lead assignment frequency consider variable interaction sup c n c cl r auto breast heart simplify learner package use forest condition length forest also extract condition randomly sample sample testing perform randomness average error rate difference large lower divide pair test rate run two well table statistically difference statistically outperform difference great great may classify table high c aa ff c c account good none heart x good yes c b x x x framework algorithm measure pruning condition rule extract frequent forest generalize r ensemble process conclusion area leverage classification numerous propose summarize extract build extract build categorical idea ensemble forest accurate understand tree framework measure learner ensemble learner also form framework regression many random forest tree rule rule forest predictor outcome predict ensemble learner information hard particularly model background understanding software refer hard discover potentially huge particularly ensemble refer interpretable rule frequent prediction interpretable insight ensemble irrelevant redundant set redundant rule discover frequent new function framework extract prune individual one summarize rule rule extraction ensemble decision tree split internal ensemble e random forest ensemble long transform package first index leave right column split value current leaf leaf node assignment package version currently environment status point conjunction aggregate root split current node leaf ensemble multiple leaf node refer conjunction rule rule extract rule ensemble often random reliable assign stop extraction reach also frequent extract tree ensemble value benefit ensemble want interpret error remove variable decay value consider rule prune leave pruning apply sequentially decay decay threshold last last remain remain next thus remove rule rule consider large frequency however rule similar would desirable derive rule redundant selection select relevant redundant condition create let whether target relevant redundant variable essentially present detail furthermore condition assign condition length process consider condition regularize forest node empty root ordinary gain decision call regularize forest add
derive bound approximate atom dictionary approximate operate pre power accurately relevance approximate dictionary criterion feature examine feature mean principal axis investigate instant criterion measure relevance compare dictionary dissimilarity construct mutually distant atom similarity investigate outline propose construct entry discard belong kernel dictionary predefine otherwise efficiently projection spirit criterion distant pair atom condition use name drop error mechanism see criterion compress sense particular kernel formalism coherence large cosine angle coherence criterion restrict orthogonal include function note denominator expression atom correlation kernel include eq analogy deal j two definition atom notion gram elementary issue approximate span fold one discard hand approximate approximate atom latter duality discard approximation onto subspace span dictionary criterion projection onto latter correspond maximum inner get moreover projection therefore approximation investigate expression order proper discard sample discard low threshold discard approximate get firstly derive expression follow secondly bit give projection onto atom upper compare discard coherence coherence linear norm coherence bind discard give quadratic atom unit thank min theorem turn aforementioned eigenvalue gram dictionary conclude atom bind approximation atom span follow derivation empirical project onto span give quadratic approximation upper span dictionary due triangular cauchy schwarz summation summation term belong discard thank contribute latter discard sample derive bind criterion threshold atom fundamental use method visualization nonnegative base empirical onto span ni eq schwarz thank constant consequence n follow discard contribute light term threshold criterion follow study theorem identify relevant unsupervised connect sake component seek principal axis correspond eigenvector large norm axis expression eigenvalue call highlight connection criterion th principal associate gram approximation upper kernel axis large small say principal axis moreover criterion use n tight one indeed derive coherence dealing criterion roughly discard explore atom namely lower beyond describe detail centroid axis devise criterion argue criterion behave interesting desirable without note machine approximation initially provide eigenvalue dictionary completeness put well gram associate eigenvalue center absolute gram follow measure distant bound distant substitute gram coherent atom coherent bound expression measure receive degree university ph degree system optimisation security technology research associate system model laboratory technology nonlinear processing representation wireless signal nonlinear adaptive system award publish review paper many framework radial network order address several operation efficient connect criterion theoretical approximation approximate propose bound criterion class mean axis resource gram pattern bring new demanding address conventional indeed underlie available stay computationally tractable model subset contribute reduce bottleneck scheme aforementione machine instant discard criterion widely investigate literature gaussian least relevance discard predefine residual representation crucial issue model literature investigate distance distance network radial function distant advance compress mutually least extension coherence criterion criterion study conduct analysis cost criterion favor criterion coherence extended criterion unit criterion cross previously derive tight bound extend result criterion bridge provide approximate already retain discard secondly provide bound approximate approximate sparse aforementioned criterion include approximation feature empirical centroid principal axis picture illustrate remainder follow present present aforementioned criterion approximate distance reference atom gray color new generalize machine present study seek feature connect desire one empirical feature control fitness regularity solution increase quadratic loss logistic reproduce inner reproduce state moreover reproduce property unit norm kernel deal unit restrict principal classification unsupervised form q derive sketch monotonically increase available constitute bottleneck indeed set instant instant thus parameter consequence continuously increase overcome need order fraction form predefine expression atom paper analogy stress fold instant dictionary challenge arise determine optimal instant reduce solution elegant overcome intractable recursive determining discard since already belong
semi bandit open adversarial sufficiently happen often item increase divide regret associate event prior tight summarize prove gap evaluate gap section discuss tuple item distribution combinatorial optimization item ground th entry item number observe item agent environment goal cumulative regret feasible et nu te c item te te feasible fa tw pe te propose algorithm semi bandit compute item around se se call combinatorial observe weight th marginal second initialization compute follow item one guarantee terminate iteration one entry optimization oracle regret gap eq dependent upper asymptotically tight latter bandit definition efficiently offline problem solve computationally efficiently optimal semi bandit efficiently operation computationally regret factor regret match free upper bound gap suboptimal theorem solution present proof yet proof lemma initialization suboptimal hard bound claim event item suboptimal sufficiently later event happen least mutually exclusive claim item happen happen exhaustive exclusive show happen follow eq contradiction happen ready detailed time event happen bound gap minimum suboptimal q stochastic eq detailed item item order event total regret substitute gap suboptimal identical relax step define many mutually exclusive suboptimal decreasing establishe key define respectively happen happen sufficiently happen happen happen definition furthermore must ie happen inequality assumption contradiction result happen appendix number happen step finally apply gap bound suboptimal associate regret event large small regret combinatorial bandit regret bound appendix key decompose gap separately step bandit one gap dependent gap start point node mark path weight weight note design key equivalent arm return scale know path bandit gap dependent step inconsistent logarithmic semi bandit integer path proposition divergence bernoulli bernoulli low due qp gap integer semi regret path equivalent bernoulli bandit payoff bind adversarial environment problem synthetic demonstrate suggest bind experiment path problem ground feasible grid upper corner bottom right corner bernoulli mean gap number validate experimental report trend item chen upper upper upper nearly mirror perturb geometric resample recently adversarial combinatorial computationally efficient open combinatorial bandit efficiently bandit instance upper bind apply indicator learn observe feedback clearly bandit observe low combinatorial adversarial nevertheless set stochastic van semi bound upper differ frequentist computational efficiency offline inefficient straightforwardly instead respect thompson often perform practice straightforward variant thompson weight thompson resemble regret main work derive novel gap dependent ucb semi bandit achieve near computationally efficient implement offline efficiently combinatorial semi bandit efficiently quite mild choose sufficient purpose may large well leave derive suboptimal speaking match factor eliminate modify leave stochastic c outside terminate finally rewrite regret equality history regret condition base te te ec happen conclude remain suboptimal follow quantity sufficient number event happen count magnitude happen n suboptimal item item increase one suboptimal guarantee observe happen happen ng te tt happen counter item event bound trivially regret introduce event regret definition event suboptimal gap solution order
schmidt preserve column spline qr decomposition invertible solving expansion span qr decomposition illustrate dataset fista fast algorithm size solve exactly avoid theorem assumption flexible interpretable model combine allow nonlinear know challenge treat combine selection lasso additive outperform across broad spectrum high apply real half million excellent attractive generalize additive datum relate q fashion glm present extreme treat incur unnecessary popular multiple field economic major obstacle categorization rarely know challenge exclude entirely overcome challenge automatically feature relevant take empty spam spam sparse partially fine grain spam provide bridge penalize glm spam interpretability motivating assume spam interpretability manually reveal appear exactly variance feature memory speed make aspect bootstrap linearity author decide versus nonlinear exclude developed optimization alternative spam smoothing operator penalty smoothness scad purpose datum toward perform perform feature contrast scale formulate theoretical introduction additive development inequality without demonstrate setting spam thorough often regularize spam relaxation main linear nonlinear denote I coefficient basis regularization loss n convex hierarchical choose sufficiently increase get feature none set problem described idea modify proximal suitable step size reciprocal lipschitz convergence sparsity proximal dual descent pair value decrease suitably choose seek deep understanding section regression set reliable factor regime highlight spam oracle guarantee wide decomposable regularizer kind potentially strong design compatibility etc see difficult different inequality rate make despite name inequality rate standard assumption particularly difficult slow slow matrix expansion assume emphasize concrete choose control different feature relevant letting denote present slow prediction suitably choose grow like show grow linear implication performance achievable reduce slow constant incorporation develop error presence nonlinear spam special follow easily statistical reason prefer spam aside truly spam incur estimating term bias sufficiently penalty intuition whereas spam linear note orthogonal spam line supplementary material establish thus basis vector find prediction away regardless group parameter apparent correct grow spam grow serve verify intuition regard incorrectly investigation various scenario spam lasso experimental use cubic knot bx x choose hold report generate x x lasso spam build additive consider evenly optimal thus pure pure parameter deviation show c lasso spam ccccc spam illustrate spam component ground truth spam toward summarize validation nonlinear ground model spam outperform carefully control high spam spam level accuracy priori accuracy improvement compare spam spam reliably linear counterpart spam addition spam correct support cross mistake plot b visualize shape close perfectly comparison visualize spam red plot recover internal permit future spam deep spam generate point train three choose set term win show advantage spam lot since lead parameter trade lasso regime reliable spam lasso nonlinear component linear still lot nonlinear since regime effect surprisingly dominate linear nonlinear unable effect spam less biased pos spam real size summarize characteristic logistic spam deviation spam character email length letter letter allow error spam substantially compare regularize logistic stay spam problem digits challenge middle create force dataset remain spam spam logistic regression select nonlinear confirm small act performance yet effectively avoid overfitte categorization application corpus volume predictive although popular obtain transform suboptimal computer vision utilize dataset place web image match central test whether two geometrically match expensive pair filter image likely pass verification vocabulary visual word observe carefully control regularize logistic select sparse partially perform regularization formulation make practical oracle advantage thorough experiment demonstrate find additive improve popular
reduction common setup pca mutually retain projection perspective variable formulation axis formulation factor probabilistic component pca projection rank reduce large eigenvalue eigenvalue sample reduce condition invertible exceed sample discuss reduce rank rank vary dimension become covariance isotropic reduce covariance balance control estimate use information criterion determine bic maximize free control diagnostic assumption diagonal covariance correlation estimate latent proceed set var reduce estimator residual order covariance ar scenarios ar coefficients scenario occur ar constrain arise autoregressive see ar scenario ar express stack ar constant constraint reduce estimator p fit ar become ar involve rank covariance var follow unconstrained step estimate ar show ar coefficient reduce rank noise iterative iteration current repeat step replace covariance compute residual conclude setup interpretation rank setup useful explore dependence impact ik marginal marginal weight help unobserved characteristic characteristic series associate characterize position setup adopt row space find interpretation interpretation unobserve form behind multidimensional see e group model construct representation ad hoc space lead representation dependence via rank var mention estimator shrinkage reduce rr structure estimator three estimator structural compare reduce performance dimensional attempt shrinkage shrinkage propose see e shrink sample balance control tuning shrinkage ss two choice target dimensional covariance first remain increment serve case accord minimum bic shrinkage ss intensity analytically infer reduce covariance admit summarize frequency replication reduce small covariance estimator reduce correct reduce rank probability select rr reduce result rr ss metrics stein sl z dimensional metric square define z stein estimator standard stein covariance covariance mark bold stein loss mse simple estimator improvement sample stein mse covariance satisfied stein loss various size medium size stein estimator stein estimator significant see improvement mse stein rr rr shrinkage much significant improvement rank percentage sl reduction rr ss reduce concerned stock scenario constraint china scenario e zero ar coefficient setup interpret stock stock finance technology ratio consecutive daily trading display return black red technology pattern dependence stock purpose estimator modeling series first fit coefficient fit unconstrained return bic fit minimum panel display vary minimum word dependence stock represent display stock dimensional stock panel dimension phenomenon stock stock close far observe stock opposite along stock finance technology energy stock stock hand finance stock stock dimension provide separate distinguish among stock exception technology separation panel vertical separate technology stock distinguish finance diagnostic check display cross estimate variable exhibit auto correlation auto cross dimension consistent black red finance green reduced covariance estimator large lead scenario estimate refer covariance residual correspond ar coefficient zero reduce model fitting return reduce impact model aspect interval ar estimate forecast error ar coefficient ar estimate indicate time interval temporal var stable unconstraine forecast mse forecast var ar matrix ar panel compare mse sparse rank estimator step forecast estimator p concern china var temperature series scenario section var sparse ar stage introduce order choose minimum bic obtain non coefficient reduce latent insight actual position finding summarize display panel dimension compare pairwise rank finding factor dependence temperature since neighboring condition impact temperature emphasize position purely reduce estimation var temperature panel display estimate computed auto correlation cross correlation like provide temperature nsf grant research section proposition give isotropic analytical form plug column consist regardless eigenvalue additionally complete mse forecast forecast approximate approximate forecast part come var parameter mse plug estimation dimensional q noise zero upper sub equal replace thereby forecast rank large dimensional reduce estimator outperform compete covariance estimator large var fitting reduce estimator interpretable description var autoregressive k time vector autoregressive vector
identical iterate run also verify leave proposition unchanged bregman divergence hand lemma check condition coordinate outside zero satisfy u term result divergence know strongly lemma strongly imply inequality throughout turn implicit eq claim adaptive variant denote logarithm think sum random bind measurable measurable finish optimize need union start fix lead total desire finish combine union proof use find tw tw invoke b kb follow note schedule check cauchy schwarz eq convexity plug bind case depend let denote want fact root appear meanwhile eq gm th sigma take direction get simplify desire proof tw tu tw sx walk drift standard lipschitz next going stop markov never adaptive regularizer q tw subtract yield slightly condition eq establish probability prove state technical lead give analogue give sequence loss run mirror descent loss regularizer notation weight extend control assumption ensure excess adaptive mirror descent learn need analogue term regularization risk also strongly result proposition proposition combine eq begin argument st apply regularizers eq yield regularization schedule meanwhile probability apply gm inequality proof invoke give lemma meanwhile directly inductive start note immediately prove inequality lemma respectively fact provide together induction know inequality need b suffice theorems probability attain rgb proposition condition support foundation fellowship nsf bc stanford fellowship grateful descent condition regression setup recover achieve statistically quasi optimal feature meanwhile computational resource descent streaming sensor click throughput streaming store obtain parameter article procedure exploit streaming formally prediction maintain iii interested q square loss second classic note closely observe ambient population lasso pursuit weight encourage literature show lasso attain restrict van van require solve global streaming kind optimize database test finite online stochastic remarkably ignore sparsity regret convexity dependence update gradually informative word spam intercept select analysis run feature example contribution stream lasso batch one algorithm take soft thresholding require carefully support different epoch attain conceptually also empirical goal spam spam mail weight gets gradually variable example weight path move look brownian motion et size result involve gradient sgd streaming sequence response loss condition generalize limit x output tw algorithmic easy convenient exploit rewrite equivalent mirror mirror usually close form encourage advantage descent way induce update efficiently stream online also aim classic see proposal convex trade dual describe proximal make condition guarantee advantage bad exist batch restrict van van even assumption guarantee performance stream tt theorem restrict orthogonality feature uncorrelated noise suppose sequence point tf tw w ct match bound since regime online algorithm optimization lasso stream data simple classical stochastic descent comprise achieve replace long statistical excess exist sgd like composite stochastic similar namely regularize mirror like streaming algorithm performance derive heavy prior carlo beyond streaming set none large dataset pre remove dramatically decrease memory meanwhile show locally regression compare streaming algorithm screen subsample investigation start define theoretical mirror framework leverage section result obtain parameter assumption long defer assume draw sequence depend main depend statistical expect weight available gradient simple zero relax condition condition well understanding recall sequence meanwhile quadratic lead long next f tw b uncorrelated relaxed respect evaluate ii parameter control showing succeed uncorrelated second weighted average naive online algorithm loose algorithm output q strong feature exactly relaxed establish design van van main assumption make design similar condition minimax sufficient essentially condition strong convexity loss fact weak stem achieve see guarantee overview relate assumption algorithm yield condition assumption batch big example cross hand allow analogue hold weak resemble regularizer adaptive mirror descent result form regularizer loss immediately upper emphasize proposition turn many originally use hoc follow corollary linearize main perform ensure sparsity final term strong goal show overview indicate detail remainder enforcing scale inconsistent us sparsity restrict bregman divergence noise pair scale bind standard inequality result bind cost penalization meanwhile give adequate time none method optimize runtime taken rough measurement runtime expensive operation th power basic multiplication size step control control dataset collect trust genome association single nucleotide snps case diabetes population code snp allele else compare random dataset compute plot average slide length streaming describe mirror take advantage intuitively mirror sparsity result become mirror divergence small measure belong rest coordinate proof mirror adaptive relie use size apply depend sequence mirror descent regularizers replace statement helpful suppose determine st coordinate simplification regret optimize bind equal convexity tw tu w satisfy get rhs inequality u ambient dimension large analysis dual regret could worse work loss improve regret rather ambient regularizer oppose convexity advantageous common strong convex remove like remove entirely decomposition sparsity mirror regularizer tw remove still simplification sparsity condition one state explicit forget mean noise preserve unbiased random walk still although main absence acceptable term grow dependence hard restrict attention become specifically term regret exploit order achieve impose introduce statistical thus need scale ensure cut hold square indeed appear inside exactly provide transform excess cost error notice depend implicitly specifically excess risk signal pure independent response cost sparse could potentially thus give orthogonal formalize relax uncorrelated hold long thus main identify logarithmic sparsity loss assumption moreover combine control obtain desire convexity individual convex strong adaptive mirror present section strongly losse key technical strong convexity necessarily almost lipschitz respect get mirror loss yield analogue convexity convexity hold invoke simplification expect strong convexity mirror descent regularizer necessary main result namely risk proof far excess stochastic algorithm main implicit logarithmic bound must parallel convexity idea discuss extend analysis provide analogue extend correlated bound know give guess transform standard generalization batch bind
certain sparsity variational variation signal derivative variational view two different tool regularization mainly additive unknown signal version filter return vertical filter gradient square root directional second consider derivative mention derivative calculate pixel version column stack minimization relaxation approximate signal derivative signal tv one variational framework signal function introduce coefficient parameterization I represent length parameterization addition denoise case optical flow estimation force behind naturally describe piecewise impose piecewise tool therefore already literature commonly refer look vector block sparsity handle enable recover usage motivate elegant efficiency dimensional recover organization synthesis propose recovery polynomial technique stable adversarial guarantee mean white block present propose direction research relationship note know subspace intersect select close pseudo norm count synthesis span np solution property relaxation pursuit omp compressive pursuit pursuit sp pursuit htp framework subspace consider correspond span sub correspond estimating zero synthesis problem approximation analysis gap sp htp piecewise change use observation wise synthesis atom function know represent atom represent plus dc therefore coefficient sparse approximated sparsity omp extension recovery hard generalize order e ambiguity fail recover representation contribution treat approximate guarantee though piecewise turn piecewise degree employ assume find signal model projection constant jump appendix htb theoretical guarantee two lead bound adversarial denoise rely property case rip matrix isometry rip piecewise polynomial function jump rip treat corollary recovery optimal project optimal yield depend may polynomial perfect noiseless also subgaussian compress sense even adversarial case exist vector finite proof jump parameterization form energy common application synthesis model generalization trivial operator high space coefficient parameter coefficient block aim greedy therefore start gradually element find current guarantee appendix advantage theoretical guarantee high advantage relative continuous continuity absence jump important add jump impose continuity solve may add edge demonstrate order polynomial continue compressed sense piecewise polynomial compare use start linear compare outcome tv denoise draw connection piecewise jump dynamic white gaussian without continuity synthesis without continuity similar focus analysis straightforwardly fig result continuity essential correctness recovery indeed jump segment recover jump signal preliminary parameterization provide may good denoise mean square approximated piecewise reference therein piecewise polynomial note recover measurement close piecewise point problem figure continuous constraint well due restrict initial thus achieve recovery reason number location though gets happen also order piecewise linear impact continuity significant line clearly strategy without tv effect appear tv reference htb tv present tv reconstruction recover texture thus slightly inferior tv cubic tv recover perform compare polynomial function jump small column norm function difference omit normalize signal fig present recovery behave achieve nonetheless move say likely suppose significantly performance model texture e plane horizontal vertical discrete apply turn vertical one tv denoise fig contaminate additive recovery result suffer image optimize quality setup time provide average use result well outcome new tv notice tv act form add direction one apply also diagonal derivative focus task together segmentation compare cut htb cut segmentation htb segmentation cut segmentation texture continue use house demonstrate result suffer tv loose texture inferior tv removal texture denoise salient edge recover image texture segment minimize segmentation polynomial instead segmentation display piecewise image together cut comparable place behave though segmentation room filter parameter suppose truly leave idea suggest improvement segmentation scheme reasonable great framework novel representation thing solve problem
work derive rate al achieve parametric demanding large offline divergence difficult get knowledge density contrast require support implementation estimator divergence functional index functional divergence whereas divergence specify computationally currently asymptotic appropriately true density density unknown smooth show average density accomplish concentration inequality taylor expansion random uncorrelated derive central entropy empirically distribution estimate classifier bold face type paper density give divergence divergence nn estimator ik ik plug randomly divide part n common mse convergence exponentially convergence optimally sample key idea exchange constant index set basis parameter zero optimization trade use mse entropy obtain rate kl estimate ensemble theorem summarize function kl n I ng normalize assume let asymptotic variance construct sequence uncorrelated estimator ensemble complicated deal prove central ensemble unit variance random eq variable extension et unit I sufficient condition central asymptotically uncorrelated define lemma necessary denominator slowly numerator require l il il taylor expansion density sufficiently smooth power product density bounding power schwarz inequality lemma arbitrary let realization independent eq require grow functional case bound eight previous result cauchy schwarz sum use combine eq bind apply mse number non enable divergence strictly finite task confidence classification problem increase utility focus show uncorrelated theorem assumption smooth strictly also neighbor qualitatively convex divergence simulate uci repository central kl truncated density cube normalize linear relationship quantile normalize bayes class decision error average coefficient chernoff upper eq include optimally error estimate minimum three class bootstrappe discriminant fold cross validation interval rate class fact class compare linearly measure distinguish c misclassification rate paper establish ensemble estimator truncate nn give result divergence mse sample include simplify derive distributional convergence extend central acknowledgment partially nsf nsf fellowship grant density assume ix l ds assume ii iv include beta lemma prove unit variance simplicity denominator converge nonzero slowly denominator preliminary directly tackle quantity il il l l let l l q binomial series expansion density show uniform kernel estimator il f om truncate uniform nn well eqs l l l value density inequality truncate nn lx kl lx lx pr lx lx uniform lx lx kl inequalitie pr cx il eqs provide along let fixed realization throughout first complete proof define way truncate distribute jointly establish event relate relationship cauchy eqs prove splitting covariance case fall second hold dy r xy eqs combine om hand side result eqs note g surely apply functional assumption expectation z ik I ia q z c q z respectively condition g result independent since I give g lx om om cauchy schwarz get imply om om complete om consider numerator previously covariance general independence remain require follow partial let
htp colour smoothed fusion result fusion approach also smoothed region noise false e classified region positive htp apart frame detector problem fusion smooth automatic detect illumination show apply analysis drawback rely algorithm however researcher focus well gate hardware human method human illumination essential well colour solution apply false able cope moreover novel combine colour detector refine specific person wide illumination purpose qualitative quantitative dynamic colour play role wide processing human computer interest detection colour multiple threshold colour component pixel fall pixel colour large region aforementioned solution although successfully suffer false common variety complex illumination image achieve colour change narrow require stage understand performance require pixel collect web framework use rule histogram automatic stage secondly histogram density distribution product automatic strategy detect colour suitable colour space segmentation colour space normalise rgb colour segmentation al log opponent colour human visual opponent colour encoding colour illumination claim illumination factor detection system remainder give work derive fusion result conclude process pixel video colour human medium detect colour internet early tv news sake video automatic annotation retrieval interested reader detailed colour leave appropriate colour use classifier pixel classifier decision colour colour database colour colour belong false complex illumination colour people another dark robustness use invariant colour cope colour within narrow approach non et detector plus detector manually label et bayesian model bayes bayesian decision minimum pattern use build collect web reader encourage suffer tradeoff propose comparison solution employ eliminate secondly detector attempt employ fusion colour show first et adopt secondly employ calculate face histogram smoothed density respectively rgb colour space convert opponent pre elliptical mask illustrate elliptical face image centre minor axis reader htbp detect include non etc interested remove edge detection due computational detect smooth region image useful colour available image choose space model colour opponent colour colour human visual use opponent colour secondly colour illumination theory study certain never human opponent colour illumination illumination log code mean human colour vary appearance colour image affect illumination image camera characteristic learn boundary solution channel channel approach approach person share smooth smoothed histogram histogram capable describe shaped colour model elliptical define colour matrix mix weight satisfy htp gaussian detection angle position red dot angle eq center axis axis boundary axis axis therefore increase robustness integrate single vote pixel produce smoothed histogram result fusion eq fusion make tractable section space quantitative analysis conduct google colour consist dataset difficult popular web top
pixel noise dictionary penalty obtain clean expect impulse quality model peak ratio image recover huber increase impulse huber quality descriptor tag scale typical retrieve wide range difficult use tag furthermore automatic image noisy tag visually noisy tag image image tag indicate semantic error prediction system could include description topic learn visual tag semantic describe visual visual store novel image tag refined code tag cluster discover refined tag estimate assume descriptor measure reconstruction tag vector tag penalty penalty mixed obtain comparative annotation image keyword vocabulary varied tag randomly refined tag use refined tag figure entire residual huber penalty penalty provide recovery furthermore corrupted tag penalty lie union allow use laplacian code contribute provide opposed locality graph furthermore possible confidence measure cluster huber overcome clustering quantile allow gradually quantile previous study huber quantile model robust outlier order reliability clustering cluster uci breast illustrate cluster dataset flexible function enable drop move away perturbation dataset behavior attribute availability choose generative analyze complex require code use challenge perform warm interior enable fast warm start dictionary inference incorporate another graph penalty believe scalable expand applicability important topic model large content figure lemma new york city prior learn simple interpretable discovery recover assume availability sufficient datum sparse fidelity reconstruction euclidean domain motivate look conventional loss huber outli quantile discover structure representative consider linear huber loss learn dictionary fidelity algorithm convergence behavior experiment study function robust image tag refinement annotation confidence generate classical linear response residual shrink improve assume refer control trade regularization deviation observe sparse model speech blind separation supervise semi assume dictionary observation sufficiently objective may code robust modeling detection develop flexible dictionary code function enough challenge penalty penalty treatment act specify penalty penalty finitely general penaltie huber penalty quantile huber classic penalty section regularization estimate non parametric viewpoint follow useful squared penalty arguably outlier robust impose proper improve example company accounting year company event economic b pixel due residual differently well extensively response various quantile vary company predict various quantile planning management quantile along quantile particular company figure statistical dispersion free dictionary use heterogeneous may noise apply tag width height marker leave middle axis thick cm marker axis marker axis line axis thick height marker middle axis coordinate width height cm marker sample axis middle width marker middle x class measurement recently generalize approach full prove assumption classic alternate code dictionary bfgs length important method enable practitioner test kind penalty interface residual automatic penalty ensures coordinate potentially lemma calculus illustrate utility apply experimental evaluation huber reconstruct penalty case annotate tag since tag joint tag huber penalty piecewise structure calculus affine composition use underlie useful addition specify combine measurement affine composition building linear map action vector addition affine penalty coordinate wise different penalty minimization variable constraint obtain function respect seem complicated keep mind conjugate purpose conjugate write tucker system optimality condition kkt optimality advantage characterize wide nonsmooth automatically kkt nonnegative slack equation dual equation full present include code directly solve kkt direct optimality namely kkt system guarantee discuss nonconvex dictionary approach code penalty descent e dictionary smooth addition interested apply entire block descent accommodate sharp coordinate fail convex coordinate minimization generate coordinate point limit penalty influence penalty key contrast residual mean lot application demonstrate smoothed huber outperform standard quantile penalty turn smoothed envelope calculus envelope convex clear always minimizer prox f converge salient penalty close envelope capture convenience also member amount idea envelope huber threshold smooth huber idea capture conjugate calculus b claim solve implement block column close column pose row decompose residual square close structure
study sampling noise multiscale random technique carlo deals example want estimate give unlikely standard technique unbiased rare achieve relative see fix relative rapidly event standard monte carlo rare lead reduce popular method sampling estimator address process random environment process stochastic sde dimensional wiener gx stationary field interpret perturbation slow see rigorous mathematical provably carlo medium form eq estimator provably asymptotic interested hope formula deviation since important become motivated relate molecular simplified preserve deviation order magnitude deviation event dominate event issue poorly paper provably importance quantity deviation deviation principle particular provably importance scheme rare author work address design logarithmic multiscale environment absence fast importance scheme design asymptotic efficient noise without scale find presence asymptotic importance scheme construct paper see periodic fast motion inspire sampling investigate regime paper nevertheless able importance asymptotically optimal scheme rigorous bound ingredient gradient bellman modification account multiscale periodic medium purely partial differential sampling ingredient remark theory optimality motivated deviation fast sampling scheme model classical interpret fast moreover example variable asymptotic sg cx vanish relation interest ergodic simply rough see gaussian specific correlation structure assumption environment result review concept classical paper main randomness environment review example environment need environment necessary ergodic preserve acting preserve action unitary generator densely generator stationary define measurable consider locally field define ergodic relation cx x fy make assumption environment diffusion uniformly cx fy gx derivative globally operator literature canonical lebesgue relatively close unique ergodic environment useful reason write operator follow divergence form j almost eq hilbert equip expectation measure particular unique measure process tool cell literature consider lemma weak abstract equivalently average integrate average invariant section even prove allow known q conclude representative first case essentially correspond ergodic periodic say period lebesgue shift operator obtain periodic special deviation relate scheme periodic environment space equip fr borel algebra borel invariant shift particular dynamic ergodic invariant via wiener measure x strong performance characterize moment sampling scheme appropriately choose control behavior shall need ingredient equation crucial establish efficiency case lead actually performance standard clearly multiscale environment random expect something ingredient shall solution associate measure notion appropriate equation nearly complicated may scheme guarantee performance rare scheme difficulty precise way introduce recall continuously differentiable xt mt illustration several condition consider nevertheless condition expense hard establish second small sx continuous control x estimate since function neither approximate argument establish condition set point infimum closure infimum everywhere logarithmic choosing measure elaborate issue mention completeness merge paper multiscale measure good performance asymptotically plan multiscale control simplify recall unique process explicitly q limit state infimum satisfy q allow result control random infimum particular define display limiting result computation deviation lower ergodic control diffusion appropriately combine particular immediately moreover subsequently fact equation definition definition example slow component z explicitly unique agreement function eq effective drift moreover old asymptotically theorem depend field case immediately cover since significant simulation differentiable moreover numerical effective simplifie become explicitly interface change measure carlo independent control simulation realization field zero fine grid simulate construct euler sn respectively small practice deviation measurement estimate well pair simulation c change standard carlo computing generator discretization presence fast scale rare significant discretization weak precede euler decrease decrease quantify significantly computational highlight fast multiscale environment table simulate turn imply significant sde fast small diffusion ergodic interest monte carlo estimation event functional rare provably result scheme optimality gradient theory framework rigorously
individual follow give close two induce work accordingly try try find metric label take accordingly inspire deep deep embed approach triplet argue triplet strong approach obvious inspire comprised instance feed share fed intermediate distance denote embed word encode network sample different class task express objective correctly classify objective learn closeness label comparison softmax apply effectively create measure traditional sgd examine replace q regard triplet implement train dataset consist image handwritten digits correspond house digit fourth class cifar big augmentation apply zero unit variance instance image third epoch epoch fix instance image follow fourth linearity consecutive network configuration order representation augmentation similar work similar svm knn affect notice later feature respect know augmentation mnist examine image euclidean significant semantic space measuring embed three use dataset gain try similar unfortunately use network relate context leave conjecture h triplet net patches spatially perspective could patches rough unsupervise applicable consecutive frame expect frame take minute net may provide well past classification environment human well accurately provide comparative easier collect triplet attain annotation introduce explicitly way classify model triplet classification consider know insight deep leverage new acknowledgement acknowledge z gpu use computer science institute technology prove successful model mostly implicitly part learn comparison wang ranking retrieval learn immediate possible usage learn deep learning extensively deep require feature
cluster galaxy dark survey boltzmann ordinary early galaxy iii training recommend name ball sample top recommend get might dynamic medical capture achieve examine rate return simulate couple boltzmann element article like article like formation star galaxy formation deep promise thing explicitly take boost research partially treatment turn back propagation derivation u ip I rejection version show red draw region probabilistic version curve tangent line black dashed correspond line search optima gradient take consideration generalize back propagation collaborative commonly recommender system conventional recommendation rating sparse many application significantly recommendation content utilize collaborative topic appeal take source nevertheless auxiliary information advance input propose hierarchical jointly collaborative rating extensive three world advance art principle behavioral science service recommender rs increasingly individual effective many g amazon netflix rs extensively service rs roughly three collaborative filtering method hybrid content use description recommendation activity seek get content method concern generally difficult activity nevertheless cf method prediction product receive information user hybrid gain popularity year rate auxiliary divide category couple coupled process feature guide manual contrary couple allow interaction hand guide feature power cf balance influence auxiliary couple often outperform couple collaborative probabilistic lda factorization pmf interpretable result nevertheless auxiliary problem focus recently potential vision language although appeal content inferior capture integrate deep unfortunately model boltzmann machine instead perform extend item item incorporate crucial significantly model recommendation music recommendation cnn belief model boost exploit content rating cnn poorly rating challenge develop collaborative deep learning couple rs bayesian formulation call rating feedback allow although admit boltzmann recurrent summarize deep extract effective content unlike simple probabilistic framework besides derive propagation hierarchical bridge state rs nature incorporate auxiliary boost world advance art recommendation take implicit test item movie bag vocabulary size article library rating although movie recommendation plot movie consider handle recommendation tag recommendation play corrupt output weight matrix bias vector convenience collection bias correspond ready detail give bayesian rating feedforward neural clean input output usually I bottleneck input layer clean solve regularization tb clean generative network draw draw bias k l clean process generation input feature gaussian dirac sigmoid degenerate formulation layer act encoder maximization minimization layer column weight matrix l clean j draw user item r n j layer bridge rating representation capture similarity user computational graphical notational use tb could carlo typically incur primary fair consequently devise style obtain maximize posterior joint likelihood become content encode item take encoding reconstruct example layer perspective perceptron fourth term infinity degenerate two common corrupt layer positive learn directly happen go decoder graphical experiment greatly coordinate ascent leading rule ij rating user control learn layer propagation gradient find optimum several term completeness appendix let observe denote operation offset tb sparse tb layer extensive real domain demonstrate effectiveness qualitatively real domain netflix way different practical situation dataset mostly independently manually seed article tag user result contain rating rating entry two rating movie netflix movie first extract rating rate positive plot procedure text item extract article stop tf vocabulary item user rest setting repeat select average aware precision performance recommender system sort rating recommend item recall report recall list collective factorization incorporate source simultaneously factorize raw feed music recommendation mention section collaborative perform collaborative simultaneously collaborative vary use hyperparameter perform rating well achieve hyperparameter achieve cnn also good tune determine directly grid tb cm level corrupt adaptive regularization word representation show compare sparse baseline sometimes achieve perform due rating overfitte dense inferior reason specifically sparse dense setting go outperform sparse dense increase go integration rs careful boost number setting number layer exceed layer deviation table tb cm layer result somewhat two discuss information rating word integrate content handle sparse much tb dense omit previous item learn directly interaction hence suffer bayesian component latent item factorization performance get already close even bad pmf gain take user dataset profile match topic article return train might tag topic one correctly article system network rate article article
train model negative skip gram framework actually finish table performance respectively default predict embed embed order mean dimension c model semantic syntactic accuracy word skip gram relation relation fix relation relation relation matrix relation either single skip type knowledge word skip group five accuracy root basic unit word composition syntactic besides recall truly word precision truly similar letter even though recall limitation truly task skip gram similarity relation predict almost increment indicate effectively impact natural language high leverage especially rare since context information rare building provide effective rare rare knowledge away frequent word balance branch add conversely input fix skip gram matrix fixing skip gram input skip relation fix worse skip gram well sharing bring effectiveness propose branch branch aspect leverage keep perspective easily understand language noisy additional may leverage avoid knowledge coherent cognitive information branch help insufficient refine experiment claim effective robust contextual specifically two branch branch balance update branch united analyze learn embed especially word far fair snapshot wikipedia corpus denote process similar occur less million token vocabulary experimental frequent baseline demonstrate power see skip fail relate rare corpus near neighbor random see relation leverage enhance embed rare similar knowledge suggest share neither similar contextual refine tradeoff coefficient generation similar word final embed example table indicate rare word great favor successfully bring contextual distinguish useful manner framework contextual influence balance branch branch framework great e rely contextual analyze draw frequent word contextual rare word rely compose rely rely contextual task mainly rare give discussion two conclusion gram illustrate phenomenon carefully frequency contain word show take fix ratio gray represent fall index frequent word word word frequent rely information rare rely surprise ht embed setting conclusion skip gram combination matrix train frequent word frequent rare sum like overall ratio middle drop besides ratio skip matrix skip gram trend probably small size conclusion middle skip gram relation model rely strictly opposite achieve besides skip always skip gram combination rely knowledge rare word word rare update consider initialization middle novel neural high leverage occurrence knowledge branch enhance huge work plan leverage basis like representation bin liu quality representation e embedding address mining information retrieval processing method context semantic syntactic challenge handle rare insufficient word cognitive take essentially address particular novel call build word meanwhile refine accurate experiment reason word similarity enhance effectiveness author department mathematical sciences china liu microsoft st china china mining retrieval ir language nlp obtain representation embedding bag continuous skip gram skip gram leverage document transform syntactic lie context various couple include insufficient context adopt extract space look matter rare effective way instance sometimes build new root guess probably henceforth similarity act bridge rare word inspire recognition enhance word beyond contextual already skip take learn might word kind guess counter inconsistent similarity since clear stick effectiveness word embedding tackle issue finding cognitive reveal human contextual connection mind trust knowledge similarity discussion actually neural architecture embed consist branch contextual branch efficiency similarity similarity branch branch share word feed forward layer modify embedding demonstrate representation task similarity contribution include neural framework learn learn embedding rare knowledge noisy contextual rest review knowledge section embed time semantic allocation approach yield limitation scalability obtain embedding variety natural process unified architecture representation amount process bag word model skip gram skip gram amount text embed intuition context word skip gram slide stream sample slide word format train vocabulary feed map vector predict word back matrix train word embedding skip gram perform nlp quality embedding rich extra word embedding word knowledge quality embedding recent effort explore introduce framework yu et objective semantic semantic take empirical enhance work syntactic semantic valuable word obtain high embedding rare word leverage since unknown word popular one previous especially recognition letter dnn first work letter replace word input gram order small rich select feature tag generalize vocabulary nlp neural combine neural word representation contextual representation text leveraging describe neural architecture word recognition cognitive learn language usually gradually build base try several channel recognize sound guess study something know word guess know article quickly mobile device compose letter guess word retrieve association current guess mean context different context g historical hard meaning decode manner sometimes error share rely knowledge bring lot distinguish help contextual avoid please human powerful leverage contextual embedding novel knowledge contextual next architecture embed skip representation similar context gram follow denote slide window sum vocabulary impractical directly proportional vocabulary art aim probabilistic discriminate generate bilinear generate noise use include embedding represent distribution set power computing summing vocabulary become vocabulary knowledge beyond predict new introduce context leverage representation contextual refer want prediction softmax branch branch detail ht accord obtain branch necessary similar knowledge forward layer method actually th leverage top word high score connect connection update huge contextual branch branch I yield dependency frequency frequent easy collect rich contextual might insufficient knowledge little correlation word word knowledge even though contextual divide frequency way interpret embed embed couple extract similar take original overall weight back propagation process pair frequency update process take skip uniqueness branch branch four complete four knowledge quantify two string one represent word root stem split calculate eq denote four separately combine together experimental experimental regard effectiveness mainly compose part baseline skip effectiveness mainly rare quality word embed rare gain noisy word rare also give study gain insight balance contextual branch question denote suppose answer question find bc regard correctly syntactic similarity associate receive receive average
nlp amenable counterpart follow computer vision successful application convolutional neural nlp hierarchical convnet support document show adapt vision document evaluation automatic extraction document label simple classify extract system alone preserve extraction internet restaurant review site front page neural method compare convnet sentence sentence convnet transform embedding sentence entire sentence convnet sentence embedding model train embedding softmax train sentence level sentence tie sentence embed produce convnet pass show learn relevant essential extraction application specific sentence model interaction embed handle explain nlp form document sentence follow detail level sentence correspond sentence process embedding produce sentence build vocabulary embed use together sentence produce embed sentence document embedding embedding sentence convolutional bank w fw maps layer dimensional align axis level obtain location sentence stack new matrix feed wide feature sentence include sentence document embed issue convolutional handle problematic interface sentence max pooling apply pool along row discard length max pooling sentence single convolutional sentence ie tie sentence document convolution pool weight level model model create allow extraction salient contribution network activation deep computer work fact formally carry net extraction create map assign sentence give objective perform pass class invert feed great infer first order expansion function formally approximate entry easily perform single pass intuition behind indicate change embed write sentence whose identifying appear respect correspond dot th vector document sentence implicitly sentence huge write level huge indicating appear respect backward rank extract sentence rank use qualitatively however compare lack gold extraction reference document create extracting extraction choose subset sentence hand irrelevant movie sentiment originally demonstrate movie review label divide label review experiment label review break review sentence break sentence perform task map number generic symbol times leave word vocabulary proportion rand fix convnet word rand pick pick full percentage sentence label convnet show produce create show reference full review use heuristic sentence convnet movie data sentence map width nonlinearity weight tie document document input bank width follow pool nonlinearity fed predict positive extract salient review review I full several baseline embedding embedding apply predict document technique convnet model assign sentence heuristic last sentence informative content sentence review drop
remove cifar name optimization prove effective critical effectiveness process provide flexible various remain frequently machine often optimize spatially develop beta cumulative bayesian multiple stationary inclusion greatly improve state produce reliably goal estimate proxy perform promising ability routine advance ability class express accurate value input crucially main exploration exploitation bayesian limitation output assumption simplify regression realistic stationary present many inherently non optimize bad g classify expect generalization performance gaussian variety stationary particularly suited learn remove shape optimization straightforward idea space another advantage share structure experimental empirical bayesian benchmark method outperform parameter challenge consistently converge methodology involve model stationarity fundamentally component effective powerful distribution widely linear bayesian attractive bayesian uncertainty unobserve positive set predictive cross apply common choice automatic determination ard eq ard mat ern space model gps covariance function propose project multidimensional thin spline flexible gp spatial extensively statistic perform simple yet flexible address effective gain elaborate technique problem multiple henceforth refer domain input predictive surprisingly function transform produce task index eq directly parametrization covariance noisy expensive explanation strategy probabilistic expensive bayes observation surrogate determine proxy query posterior probabilistic acquisition tradeoff acquisition combination define surrogate acquisition marginal improvement criterion normal propose independent acquisition analytic utilizing train collect scenario task predict sequence sharing find function auxiliary task project hypercube hyperparameter researcher often space monotonic perform grid transform space take stationarity inherent stationary property input ideal unknown evaluation use transformation transform specify value th cumulative distribution shape determine cdf close non accurate statistical software package alternatively stationary function stationary choice beta capable variety monotonic choice slight approximately approximately contract outer expand center logit shape single explicit transformation hierarchical place integrate treat collection hyperparameter monte slice e arise various median zero identity center empirical analysis expect trained task account allow inferring effectively try task suitably model task versa rr grid base protocol benchmark run evaluation low benchmark far evaluation comprise distinct experiment compare benchmark show task evaluate standard expect ei follow treatment ern use involve tune deep cifar layer layer optimize two six regularization weight norm pool logistic show stationary improve convergence dimensional convolutional network bayesian optimization different dimension evolves become observation intuition transform weight figure weight connect input weight transformation logarithmic transformation effectively mean variation occur medium scale especially variety learn different dropout confirm set learn non trivial rate agree dropout surprising give hyperparameter unlikely expert highlight utility subset benchmark benchmark design strength hyperparameter perform bayesian differ use forest package input result improve well standard decrease worth note function drastically many interestingly approach naturally deal albeit fundamentally uniform explain discrepancy unlike forest smooth input mean ei locally via method select random define function absence marginalization apart deep convolutional average learn regression annotate layer apply
introduction expectation necessarily write expectation probability write respectively mathematical refer size oppose emphasize aforementioned terminology level indicate less throughout paper make evidence size h force reject decision via ix h reject retain statistic reject key u detail general refer specify say variable yet mid mid goal make retain may corresponding relationship randomize correspond formally independent every state report reporting implication theorem equal entire derive closed form observe important reject abstract randomized value avoid usual understood step decision bring discrepancy randomize characterize nature discussion additional contain usual skip uniformly u e x viewpoint careful inspection indicate lose report whether equal randomize reflect lose report alone information specification equal information procedure conservative otherwise minimized bias whose minimize additionally near bias indicate conservative simply rule additionally report rule slightly moderately conservative etc take define adjust randomize especially countable goal decide reject retain procedure mx section formally multiple function potentially xu retain reject typically false discovery fdr family reject false reject expectation respect u single one procedure definition ip mx hypothesis equivalently decision well define randomize wise procedure dominate single counterpart extend wise expand providing specify rank procedure u xu independently relax conclude remark assume satisfied xu right x integration could generate uniform compute record xu b binomial wish hypothesis far generate compute xu observe xu indicate reject apply formalize allowing reject xu size reject adjust xu decision xu refer adjust x adjust abstract randomized generate compute xu histogram adjust randomize xu computed appear fall formalize link xu xu x I independent practical implication adjust abstract reject hypothesis fdr equal hypothesis depend nan fdr adjust decision demonstrate choose fdr perform generate gene hereafter step abstract adjust group adjust gene summary step step u analysis step histogram reveal wu additionally allow mid natural section decision randomly discover discover specify mid indicate additional unbiased natural reject take previous statistic identical test traditional testing procedure automatically whenever approach outline generate mean portion utilize computed idea retain nan small value reject usual step apply collection remain refer step method table nan hypothese h sort result reject retain nan additionally hypothesis due abstract summarize low group adjust randomized expect randomly mid nan p n implement mid automatically hypothesis step hence nan reject computation microarray choose retain nan microarray address well discussion choice supremum bias ideal variable multiple testing define define unbiased xu randomized sense adjust variable additionally adjust adjust mid microarray via wu yield last row adjust reject nan enjoy nice though less arbitrary drawback vary adjust mid abstract randomized hypothesis show testing view randomized approach abstract randomize understanding procedure consequently report assume generate compute consequently analytically fdr procedure value independent ensure remain valid approach may mention recommend goal illustrate opt significant provide tool quantify decision impractical single practical complicate similar additionally multiple testing long microarray support value setting adjust generally adjust mid ensure value statistic identical function usual mid may lead test may histogram abstract randomize whenever upper histogram adequate mid decision fact certainly report randomize ultimately certainly choose natural mid randomize reject understand verify px follow claim definition xu xu definition second law independent claim c mid abstract statistic distribution provide extend testing version aforementione usual randomized decision nan hypothesis differ adjust abstract dominate variability tool adjust abstract example type adjust mid value abstract adjust randomized throughput technology force microarray nan hypothesis test widely accept hundred procedure recent year book review ultimately employ nan hypothesis arise utilize challenge test evident testing nan vs alternative equal reject retain take testing strategy probability impractical decision depend generate reject mid natural less equal advantage neither mathematical generating variate abstract example capital uniform variable interpretation mathematical detail focus aforementioned abstract information likely randomized yield decision consider binomial consider suppose randomize randomize
stand c fourth permutation loading pc z mix fix sense ip ip ip r ip converge distribution tell limit pc classification ip directly geometric representation normality simplify situation pairwise leading set resp far asymptotic condition generality correctly classify set misclassifie converge identifiability notion identifiability consistency dimensional context state na z order q second definite b last sensible compare plot classifier line color report comparison consider statistic r r rt ratio group distance method error size c lda lda cccc lda lead nan figure figure instance grow htb panel figure show theoretical quantile uniform inside conclude uniform distribution htb underlie group permutation test greatly compare update rate lda classifier exhibit notably decrease apply permutation test experiment performance statistic table test different test statistic c usual low performance classification pool covariance shrinkage estimator positive modify anonymous suggest compare modify linear discriminant analysis appear give classification pool fisher discriminant discuss discriminant discriminant distance bayes n thresholded covariance eigen decomposition e thresholded estimate majority voting denote majority voting classifier lda lda lda lda lda lda svm lda svm svm lda lda svm remark problem classification set observation classification rule label label perform set motivate free empirical employ datum variation principal component extract major multidimensional scaling conventional well result demonstrate related cancer bioinformatics principal multidimensional scaling em em advance ease semi automate cell image discriminate cell belong set right task image rule predict study much precisely problem suppose consist iy cell predict sample contain image characteristic datum make may may characteristic rule base vector x I regard empirical contour estimate normal label observation set distribution whose mean covariance represent contour distributional lead believe idea image mostly focus extract texture first simple model set propose context free method statistical principal transform empirical direction subspace classifier scale extensively subspace real classifier discriminant label observation extract multidimensional scaling formulate relevant conclude arise label class incorporate characteristic level membership hierarchical observation consider variable dependence membership hierarchical classification absolutely visually evident correspond matrix ii k k pa parameter covariance vice versa panel example utilize separate discriminant clearly distributional hierarchical long hierarchical bottom panel common play role voting web appendix experience vote show framework transform extract appropriate set cell problematic overfitte location distribution principal visually separate location extract extract orientation dimensional space value th ij ti pi orthonormal variance situation major direction use lead pc direction essentially eigen covariance large pc bad indistinguishable understand whole precise subspace web subspace important matter model drive argument lie manifold element conventional pca first mapping form intuitive canonical angle subspace span unit form call angle angle without generality matrix orthonormal l modify measure choose variance good b I multidimensional nm j euclidean geometry extract value point multidimensional give require optimization detailed discussion configuration write euclidean z fa solution minimize n gram consist product nonnegative coincide map pc classifier extend map pc fix pairwise distance I n represent point configuration point increase eq work th permutation large th pc web discriminant extract obtain permutation consider study discriminant vector machine discrimination minimum distance competing include classifier class size membership majority call weighted hand classifier lda classifier precise definition classification summary svm lda lda denote zero ij p p u highly auto ij von distribution parameter performance section misclassification rate test equal independently summarize repetition dimension exhibit misclassification rate compete surprising pc good covariance comparable voting method pc feature effectively discard hand pc relate result successfully pc feature correlate show comparable correlate simulation confirm lda hierarchical appendix cccc ccc lda tumor remove process analyze normal contain datum available http www edu user software image different leave fixing misclassifie together misclassifie dimension inspection orientation right point north east close orientation north cccc lda th svm th th technology processing cost expensive clear training choose set classifier summarize exception original set raw texture introduce classification selection procedure model classification method orientation possess adaptively use classification multidimensional modify nonlinear solution mapping observation multidimensional regression mapping modify solve extend independence relaxed dependency observation especially cell close high correlation greatly accuracy classification focus hierarchical classifier greatly vary greatly feature exclude set sensible properly accommodate classifier conventional sophisticated machine machine choose selection work classifier present potential advanced kernel web section code acknowledgement discussion relate set associate anonymous valuable partially nsf dms partially foundation
recommendation datum netflix million distribute movie million star increment movie netflix user item million fm netflix performance topic vector user fm fm fm tool default comparable fm netflix well baseline quickly conclusion vector perform since exploit history validate topic traditional fm fm feedback update fm result performance result confirm future com zhang liu recommender system widely factorization factorization fail benefit feedback rating movie rating prediction history influence rating dirichlet word feature fm implicit fm exploit result demonstrate fm collaborative gain netflix score score rate reasonably element problem focus factorization dyadic fail role factorization take long memory factorization fm engineering fm specialized svd work fm implicit user call topic base fm model fm model topic model natural language nlp area classic dirichlet allocation text corpus express several document use latent lda rating want user history degree movie document movie obtain topic document interest movie draw latent topic train fm fm build efficient implementation continuous bag skip gram simple learning huge big vocabulary million train latent vocabulary represent word instead word fashion fm fm follow fm fm experimental collaborative latent approach conclude introduce feature character item fm fm previous latent history consume contrary latent time necessary explain fm algorithm make model gibbs firstly parameter introduce element rate latent user item dot product vector correspond item model topic index cross dot product exist firstly train user secondly item item tell topic item topic fm besides cross term item item step fm history datum show htb uk il model method baseline topic base fm item belong exploit user extent observation word utilize rating regard
constant effective obtain motivate follow heuristic pass select among positive choose compute quantity b ng ta ta start small iteration surrogate function perform simplicity extend individually heterogeneous dominate updating surrogate simple several surrogate time compute code matlab source software package core ghz gb ram six publicly dataset consist challenge website pre standardize zero unit name storage size gb dense sparse consider q magnitude would use learn impact output additional present limitation toolbox adjust sgd schmidt et size finding sgd g accelerate propose automatically zhang language mark mark schmidt similar search upper heuristic describe suffer update storing result mb mb mb second fit extra scalar x duality experiment conclusion empirical sag sdca fast variant predict convergence significantly well good dataset satisfied mini sag sdca mini batch gradient descent sgd compare pass preliminary present show solver coordinate appeal smooth sparse present scalar vector stationary way approach p whereas surrogate five algorithm lipschitz scheme adjust initialize x natural section bad objective search significantly adjust asymptotically l five seem converge substantially point minimize asset probably applicability include non smooth asymptotic point incremental competitive state art solver large scale logistic store iterate function publish alternate multiplier future currently scheme acceleration direction result inexact proximal acknowledgment author would mark schmidt associate comment sake directional directional direction directional direction convex directional stationary convex present characterize surrogate function find appendix regularity strongly strongly smooth regularity function differentiable l p l successively widely applicable popular various scientific especially process importance machine precisely asymptotic stationary one expect apply composite obtain proximal rate experiment machine sample large enough usefulness penalty convex convex incremental minimization successively objective locally decrease principle theoretical popular various approach maximization statistic bayes mode surrogate inverse image factorization scalable minimizing function exactly intractable find stationary difficult motivated represent index context amount explain become machine large point inherent sublinear enable solution incremental propose minimize algorithm incremental method per cost problem soon appropriately result upper class surrogate lipschitz obtain sure convergence stationary guarantee provide remarkably composite function method sag schmidt schmidt zhang two work different sag sdca useful cut solver scale schmidt show incremental dc nonconvex penalties mi ni mi devote present basic minimization exploit function n mul exp add x mul mul nh intuition quality key arise require definition analyze variant hold error surrogate surrogate surrogate difference state analysis surrogate define h obtain classical bind proceed function precisely section iterate asymptotically assumption surrogate composition show asymptotic adapt proof gradient nesterov set sublinear convex minimum impossible instead condition mild directional exist stationary appendix condition see point assess stationary differentiable converge note critical convex surrogate strongly asymptotic stationary increase equality denote limit sequence h n non negative necessarily two accord define l sum converge h derivative direction therefore cauchy infimum guarantee exist algorithm composition composition monotonically grow infinity note make write proceed z universal smooth relation directional upper part describe note nature nesterov originally proximal adapt regardless nesterov segment monotonically simplify introduce consider lr recursively subsequently lr lr bound nesterov convergence convexity point show minimize make strong surrogate show slightly convexity assumption surrogate f convex successively prove classical convex though obtain well one example algorithm exist though bring new asset smooth natural surrogate smooth sum remark ni mi f let composite surrogate function smooth surrogate follow analysis admit definite surrogate surrogate appear learn objective without newton though quadratic surrogate rate practice introduce incremental dealing probably sgd update n consider smooth admit order surrogate schmidt randomize incremental sag smooth variant incremental sdca composite optimization incremental update sgd sag sdca storing iterate context incremental propose bound iteration approximate surrogate al surrogate initial number iteration initialization surrogate randomly pick update section study specifically start non stationary essentially surrogate sublinear case one rate theoretical surprising surrogate use directional surrogate proposition hold proceed tn obtain inequality g surrogate surely monotonically e evy representing result g f let h since converge also analysis composition ng nf remark approximation error almost surely asymptotic effect g proposition easy te h n conclude notably surrogate strongly minimizer proceed point conditional iteration incremental relation similar convexity la l summing e n f convergence rate strongly convexity convex batch time obtain share sag natural make section use store vector z z sag suggest even behave guarantee converge sag sag substantially sag decrease large unless condition convex sdca resemble offer sag procedure sdca involve step update appeal strongly surrogate within upper amount perform sdca resemble convergence independently refine
instance bag kb ik feature assumption implicitly bag define hausdorff also certain ik instance contribute bag single bag represent bag standard representation convert bag strong make object originally hide positive bag negative instance goal classify bag instance focus instance classifier combine noisy relaxed formulation bag specific instance still apply bag bag product combine posterior probability away relationship bag bag whole directly define bag instance contribute survey concern label bag instance desire phase bag mi si category empty something label possibility bag goal train classifier require goal classify classify bag many vice versa important reason prediction bag false misclassifie negative bag correct bag misclassifie similar reason bag classifier bag bag instance goal optimize optimize bag match mention annotate link deal weakly annotate annotate bag annotation bag classification process elaborate object bag label bag see multiple instance bag fraction fraction learn bag estimating first turn scenario instance bag however si si assumption section inside exploit improve difference describe bag name set object mi label straightforward bag instance versa done supervise near mean vote majority pooling distance convert scheme nearest similar obtain instance patch image level bag level build bag beneficial several test I good mi mi kernel subset bag subset share object bag instance bag output bag classifier class label permutation find instance jointly guarantee perform instance label bag instance diagram deal vector bag instance popularity motivating reason classification firstly great power logical entity secondly label bag costly bag lastly advantageous bag whole illustrate bag si mi si si accord whether instance si popularity scenario si mi base several problem incorrectly mi mi progress connection make category diverse many classifier extend direct type input si mi originally indirect bag unlabele instance unlabele bag mi direct bag indirect instance bag instance test mi si heterogeneity object mi si train happen label bag instance discuss section bag bag classify bag phase bag bag integrate perhaps si mi si attract attention strong raise minimal need situation scenario review bag learning gap easily context collection label several scenario adopt enable suitable thank kind lead anonymous suggestion classification formulate directly traditional set training vector vector label set well extension either training propose mutual difference map relationship field pattern problem difficult regular available classifier previously unseen learning scenario include instance base review scenario reason might choose reason feature restrictive object example drug classify effect shape activity therefore label costly consume computer diagnosis application voxel tag image region predict patient grain g voxel label bag instance face verification video consider video confident prediction label neighboring video classify goal possibility show fig si multiple instance mi si si instance predict activity mi mi object bag mi mi scenario mi si scenario face verification represent mi face bag si objects bag si object bag classifier necessarily relationship lead poor happen type field finding therefore believe understanding scenario field overview bag stage process provide insight scenario often intend survey classifier particular exist survey type furthermore focus cover extend multi multi begin application bag representation instance bag label associate category conclude discussion prediction activity whether influence shape fold bind number possibility set however active available possible assumption label positive cloud atom shape compare unseen logical atom inactive combination contribute bag instance pixel object medical contribute several inexact algorithm bag article email website histogram application assign document bag instance seem classify relevant go paragraph relevant classify document email situation social page security classify describe website feature label relationship bag goal unlabeled bag individual neighboring link detect failure account music retrieval spam filtering motivate failure bag occur correspond failure frame instances example spam spam normal help later proportion would discount proportion circumstance proportion instance receive discount rather address store problematic fraction label assess customer bag represent ik label instance bag interested find learn characteristic bag bag individual classified bag bag label bag example assumption instance bag lead subsection organize type l cm si si instance supervise collective mi bag bag bag bag mi
mild gain sublinear regime allow characterization perspective many show flexible practical compare adaptive perform former well problem analyze specific allow consider framework low bind characterize assumption salient give sequentially observation framework interest compressive sense cs extension nonlinear formula type inspire capacity channel similar adaptive look nonlinear adaptive method extension lower help match bound happen cs necessary sublinear sparsity mild gain body however cs little ref bit low bound extent knowledge generality look gain sum variable difficulty problem stem observation increase observation may uniformly simple cs another testing item item observation pair use integer decoder index present analogue generate bind away zero proper subset upper upper generate iid error sufficient recovery tight iid mild scenario possibly function function sample asymptotically define mutual term maximization sample bind variable choose specific testing bit upper cs index salient reveal element binary probability event start write give uniformly identity analyze identity completely binary expand remove eq follow second putting consider proper recovery inspire feedback fundamentally extra overlap term explicitly past output discuss implication adaptive testing bit testing item large outcome item include formally boolean test item test denote outcome outcome formula arbitrarily error testing kt bind author assumption identically author argument asymptotically match low outcome adaptive testing cs row support bit nonnegative measurement measurement noisy variant snr iid measurement increase asymptotically snr look length variance w equal iid constrain total power noise ix tx satisfy simultaneously ix x k due maximization since maximize subject power jensen submatrix dft conjugate transpose easy independent maximize constrain entry maximization exact adaptive kt I let infinity independent relationship characterize linear
operation transform frequency domain fourier fourier transform respectively find reduce find noise pdf obtaining pdf using state ip di calculus lagrange differential give prior find pdf bayesian favorable minimax without assume boundary obtain q chen use problem loss become performance limit favorable bayesian criterion estimation widely give limit observation noiseless discussion condition satisfy cumulative give pdf threshold pdf simulation terminology shall call value assume condition e fx proof require minimize expression conditional I condition set u u u proposition condition arbitrary conditionally independent dependent dependent identical follow sensor observation fc sensor single correlate sensor receive say design provide split region whether lie sec correlate design dependent consider work show sensor rate identical location prior differential condition threshold assumption conditionally observation future distribute plus minus height em chen consider estimation conditionally fusion center sensor sensor identical sensor capacity wireless sensor observation certain observation present location condition widely achieve relax conditionally counter distribute er rao parameter estimation distribute local sensor fusion apply sensor however energy traditionally simplify relatively little decentralized optimality identical identically distortion criterion fusion true goal bandwidth scalability robustness change distribute quantization besides aware limited bit center channel throughput address decentralized sensor constraint address wireless sensor work consider estimate optimality average distortion build major contribution summarize derive conditionally unbiased estimator fusion sensor identical design bit rate sensor condition sensor use gaussian condition satisfy regime binary calculus attain hierarchical dependence organize model paper problem conditionally sec prove sensor attention wireless sec sec location binary relax conditionally independent sec optimality condition sec fc pdf sensor play fc sensor sensor sensor receive local observation realization set assume distribution fc paper conditionally identically likelihood realization fc free quantization sensor fc along observation realization refer estimator possible strategy view similarly fc true argument expression herein optimality cost assumption independent use optimality know terminology fix mapping require write proposition condition strategy minimize give variable u form consider random solve person rule remain remainder specific mean q find conditionally conditionally motivation behind widely estimator posteriori asymptotically unbiased er fixed conditionally optimization estimator achieve apply therein necessarily surrogate identical conditionally strategy exist fusion mse restrict strategy strategy wherein sensor bit identical quantization fisher sensor contribution datum since fc give sensor conditionally independent output eq observe identical group solution unbiased incur sensor bit sensor quantization without conditionally solve constraint sensor fc instant answer whether sensor less sensor sensor previous sensor question address sensor bit channel sensor scenario capacity wireless data channel ideal allocation sensor local value channel carry bit bit constraint admissible capacity sec conditionally unbiased estimator sensor strategy yy fisher quantization binary optimal quantization identical bit optimal divide rule first contain compose quantization integer rule decision notice imply sensor sensor capacity sensor decision fix sensor rate constraint equality conclude sensor bit quantization rule condition example gaussian sensor sensor govern statistic single fisher distribution contribution single straightforward calculation give proposition exist candidate binary sensor binary fisher cumulative derivative fisher desired result necessary optimality binary snr bit
current health extraction extraction period code code hierarchy period code intervention procedure character digit use diagnosis episode would network entire rare fig display health summarize cumulative stagewise classifier sec risk eqs huber otherwise possible fast optimization bfgs converge unique patient divide subset remove potential patient specific outcome correctly identify portion harmonic appropriate macro mae level adjust imbalance behavior classifier hyperparameter regularization purpose regularization investigate sensitivity hyperparameter fig measure high outcome former overfitte sparsity reach right peak surprising force largely unless otherwise cccc cccc share moderate risk risk precision assessment risk risk month health moderate score machine roughly improvement accounting macro average mae resource resource case moderate high negative classify risk relative figure significance remarkable management average basic resource slightly less false negative table stagewise share resource negative significance social negative serious detect case examine whether machine improve run also alone predict extra lc cccc share moderate high cumulative risk cccc share cumulative record stagewise different size large framework examine stability sec result record fold fig index function ordinal instability issue drop include laplacian instability stagewise stagewise stagewise separate snr moderate attempt moderate attempt attempt due drug history number code stagewise sec width delay diagnosis classifier distinguish class top stagewise share recent recent moderate drug abuse home moderate attempt moderate attempt moderate attempt code history moderate abuse moderate abuse abuse code high attempt self student attempt activity self produce stagewise share sec delay diagnosis stagewise classifier offer rank separately rank stagewise aspect association outcome medical machine prior chance factor discover one well clinical economic issue positively attempt ed physical report hypothesis result automated study depend discover totally universal fast tool clinical work risk although gain study literature new also statistic statistic largely ignore mining relation prior result relation realize regularization generalization place predictive surprising lead condition suggest may correlate tree ensemble instability end stable predict difficulty impossible partly conjecture perspective rare event thus clinical health patient impractical concentrated attempt differ latter chance death contribute literature separate medical system useful comprehensive sensitive relevant distant history limitation framework external variety medical result far match exceed state clinical discover resemble factor perfect label collect health alone track ii diagnosis may perfectly due suggest conservative derive predictive health record explanation central criterion similar similar lasso framework information resource validate challenging predict risk discover health knowledge collect exploit predictive diabetes stroke health heart attack pose research deal situation modify outcome ann collections management helpful wide record present great challenge datum largely temporal novel regression conceptual temporal construct extract diverse ordinal weakly predictive stable employ domain specific feature introduce index generate prior sparse random framework margin risk health produce enhance medical offer great useful support patient predict clinical outcome moderate construct automate historical medical record patient within challenge effective interpretable noisy modality mixture static moderate dimension event disease approximately episode dimensional call unstable instability highly pick see next weakly limit unstable paper framework address challenge number extraction novel patient medical record temporal extract diverse class ordinal without specific risk cumulative stagewise equip sparse selection smoothness interaction include diagnosis branch evaluate feature rank account thousand health assessment community ten develop year response health service assessment population traditional risk provide benefit factor criterion predictive feature clinical stability health prediction improve high class case detect agree previously report decade extensive readily collect demonstrate efficacy contribution generic conceptual view temporal temporal extract risk ordinal introduction stagewise risk formulation relational ordinal list feature signal weight contribution stability comprehensive demonstrate method demonstrate data problem knowledge task adopt type comprise clinical history rating automatically relevant classifier paper organize present feature relational section section provide far follow conclusion two type disease present predict outcome current intervention plan plan allocation medical knowing help trial assess construction hand pick highly previous free mining utilize diverse type fast grow still limit thousand signal fashion stability keep interpretability model stable another stability could goal especially model logistic produce unstable highly correlate relate distinct change output condition feature issue produce similar consider weight stability index popular set represent use discrete problem include weak subset exploit aggregated reference quantify redundancy e exploit exchangeability group suggest context interpretability sparsity recent network primarily stability ordinal since frequently use ordinal odd odd ratio risk factor special natural consecutive separate idea study study generalization however could scale vector case whose linear predictor scale phase train medical often choose clinical quantifying factor rather build assessment quantify aspect relate attempt multiple risk assessment technique clinical aim interpretability application item attempt prediction record study limit poorly complete analyze use nlp technique event event gender patient several clinical serve evaluation outcome generate make use international scheme intervention process condition type diabetes hyper define response drop drastically beyond specify width event time wavelet like kernel detect trend describe ordinal model outcome risk lead risk share share relaxation discrete underlie random l l say outcome coarse divide determine py cumulative usually convenience unobserved logistic interpretation odd cumulative risk reach immediately progress stagewise outcome may attain level attain stagewise level current pass probability py pz py z pz accept level accept eq step distribution discrete nice odd fail subsection treat outcome risk risk qualitatively people never treat stagewise sec instead distribution training select unstable variation medical redundant lasso tend feature feature vary unstable feature change instability problematic clinical another relational model quantify draw x suggest ranking length select list term feature select feature regularization impose snr since draw set way subsample snr stay criterion probability individual snr natural criteria health behavioral due rare feature wherein scale parameterize delay code diagnostic structure distinguish time consider correlated fig network code use experiment let nonnegative encode share modify correlation semi compound laplace minimizing transform diagonal translate encourage paired feature hand go special rewrite regularizer treat relation probabilistic identity equally laplacian encourage prevent toward weight frequently weak without effect flexible extension specific e diabetes detail world propose risk health strong enough recent health service region ed patient period record case record increase year every care item cover aspect abuse service history record
base mutation fusion event interface h commonly tumor fusion copy mutation label leave survey distinct label red green paper major confirm et moreover al namely even isolate justify al al neither et pca ar pathway mutation al ar pathway lastly several propose assignment gene respective test gene region relate machine discover causal build causality model network extensively community researcher dedicate develop powerful tool tool open science derive abundance graphical ill task monotonic one solve devise fully noise particular address biological realistic intend efficiently quantify efficacy material contain convergence moreover variety level size closely behind virtue solely system causality disease experiment molecular interpret rigorously develop causality technology genomic future several robust infer interaction drug well drug comparison metric respectively separate recall highlight considerably slightly high experimentally edge converge almost bic recall left panel show panel performance realistic term recall panel panel panel include realistic recall panel plot size include realistic demonstrate efficacy correctness hypothesis type reject converge medium pruning effective optimization dominate parametric exponential complexity hypothesis multiply total cost determine help avoid nearly hypothesis effort probability encode counting correspond maximum likelihood ml score compute ml require iterate sample local score computation hypothese one node node parent size parent dominate asymptotically one important asymptotic define structure learnable learnable sample negative monotonic number true filter negative show strictly may appropriately type use fact row conditional follow never parent summation refer exploit fact row I reasoning behind summation parent denote parent parent parent n pa pa pa xx ix ix pa pa pa ix pa pa ix pa ix pa ix statistically generate maximize consist bic monotonicity grow rate grow number monotonicity grow score score bic know perturbation skeleton undirected skeleton structure relation parent optimize let optimize structure child correctly parent parent two undirected skeleton orient incorrectly acyclic create parent consist parent contradict consistency edge parent consistent correctly size relation structure remove include true graph still return structure filter convert computer room york ny universit di di mathematical algorithm genomic compute causal logical relation initially leave tumor tumor genomic gene may outlier gray project co occurrence encode datum causal relation bottom encodes causal among graphical outlier often spectrum cancer patient cause average collapse biological model genomic mutation seem occurrence example allow logical background efficient learn observational noise bic learn discrete usually use integer score bic nice mathematical score belong bic insufficient exact structured possible score et consistently modify program relaxation modification ml conditional great modification monotonic probability author mathematical empirically network improve rely priori available fact know limitation learn bic statistically consistent correctly edge monotonicity know behind score likelihood reflect course compute rely explanation conclude convergence portion causal causality distinguishing cause two event must statistical infer appear vice temporal several parent parent present score temporal structure parent negative true must e means mat three minor modification depend constraint main positive row parent event treat define singleton temporal number zero model fix parameter thus temporal causal necessary specify correspondingly low uniquely singleton row work connection exactly indistinguishable analyze structure parent spurious parent score parent create mathematically asymptotically mistake filter negative mat hypothesis filter free publicly bn network combination depth parent filter possible conduct experiment datum ten topology monotonic type sample topology ten optimization bic across realistic sample evaluate measured fraction
polynomial cell decomposition bivariate take produce root proceed way decomposition identical semi polynomial polynomial truth cell free semi define publication improvement summary year great proved refinement refinement could remove sign invariance polynomial invariance present imply formula extend formulae partial cell exist determine truth note symbolic alternative decomposition technology discussion eliminate projection en application project quantify one far quantify project unless successive may big program construct relationship complex categorization binary output main classification refine non linearly decade perceptron give robust regression assignment output supervise technology flexible modelling predict class belong assign new example class example concept map separate function ever instead computation experiment describe software consist two program fit parameter classify sample transform affine divide two far separate hyperplane margin measure margin correct dependent kernel decide note implementation interactive competitive execution partial projection operator conjunction constraint exist order heuristic choose starting eliminate variable occur eliminate small label construct projection polynomial select low degree polynomial label suggest heuristic construct select low root fail act simple expensive explicitly geometry rather complex suitable take order heuristic convention right project introduce opposite heuristic broken convention pick first reverse convention heuristic break convention fairly choose traditional number machine extract restrict reason feasible clearly engineer increasingly outside perform since quantify problem partial technique stop problem free experiment input problem course quantify building use problem availability split validation problem ht pt minus input ex ex go finish input polynomial go go three six variable admissible measure radial automate radial rbf algebraic heuristic rbf feature rbf besides svm correct imbalance set often classification account negative tn negative fp positive negative denominator sum attain grid search optimisation find maximize commonly value varied varied completion kernel give heuristic perform parameter score margin ideal result select heuristic case observe return result practice classifier return instead magnitude classifier use efficacy heuristic exclusive possibility heuristic select learn test heuristic ask heuristic variable yes indicate list cover least definition fail case occur quantify quantify problem heuristic select order many one heuristic heuristic pick win machine succeed pair compare selection selection approximately repeat calculation chance success pick heuristic machine one bad quantify n number quantify learn seem albeit margin performance surprising well never formally measurement machine quantify heuristic pick two pick successful quantify figure significantly heuristic heuristic free quantify choice although wide benefit machine superiority initial relax select dataset restriction block follow availability limitation machine randomly apply problem real overall work would see improvement selection polynomial connect beneficial allow key heuristic heuristic order variable polynomial combine narrow breaking besides order implementation interesting investigate order elimination thesis draw experience superior problem depend superior yield choose random individual algebraic could development heuristic certainly aware know heuristic involve certainly certainly amongst algebra algorithm algebra solve algorithm rarely entire make decision numerical rather primitive algebra encourage symbolic acknowledgement support china useful comment improve ac ac uk david bridge algebraic geometry elimination field choice place infeasible another fitting model measure select heuristic order heuristic geometry implement elimination robot motion programming optimisation field prove special use rational using often choice ordering dramatically affect feasibility class give number
metric build mahalanobis metric supervision encode attribute describe herein multi version mt share intermediate mt carry encode information preserve individual task mt network challenge mt two citation wikipedia articles circle google mt significant database mining problem model jointly compare learn use data task thereby well generalization benefit learn single popularity develop recognition correlate source new wherein similarly correlate scenario important beneficial setting world citation either predictive citation paper since content citation vary different former methodology sense citation pattern latter methodology electrical article classify utilize member enable friend fine specific circle induce subgraph contain social circle circle relate friend informative building suggest leverage correlation task significant attribute entity rich structural encode learn originally mahalanobis metric structure supervision distance attribute inspire task version mahalanobi jointly learn task share task specific information preserve structure thus prediction far mt optimize via stochastic mt world social yu nonparametric text categorization follow et version near neighbor speech recognition wang et help recently et help researcher study learn multiple graph datum various purpose document find embed multiple al jointly area usually apply multi relational great relevance wherein way improve specific second mt essentially learn attribute topological combine feature local share pair nod another improve predict attribute incoming node connectivity sparse modeling power feature lack difficult link snapshot predict link observe nevertheless well attribute naturally relational datum handling heterogeneity propose mt variation differ depend technical preserve derivation mt model represent attribute adjacency linkage information node mahalanobis semidefinite psd matrix define j structure neighbor denote metric mathematically preserve q simplicity represent refer iteration process formulation subject regularizer set constraint distance nod large distance node satisfy exactly slack variable many optimize network hundred allow I subgradient calculate triplet hinge loss triplets subgradient triplets complexity training iteration reach final update article make challenge little marginal drastically area pose word informative statistic detail bag dimensionality learn diagonal far carry cross validation stage article citation receiver area auc entire svm network structure encode existence absence represent score proportional distance simple auc include single train task train test pool capital letter naive pooling train test use learn task common decision boundary final task exploit intermediate sharing use explicitly mahalanobis metric include learn correlation pooling pool together simply st naive share inferior mt network engine direct use use existence prediction predict heavily snapshot dense link target fundamentally unobserved thing base example structure mt exploit improvement observation align intuition relate paper paper violate constraint violate time fig number violate quickly first iteration cm member network force relationship family member college friend friend associate formalize structure google circle friend profile friend entire attribute social assign manner social social circle exploit circle mt jointly circle social largely strong correlation achieve circle include gender job title last bag type adopt start st circle jointly mt circle exponential begin circle result combination use relevant task compare st mt circle inferior wikipedia entirely social social circle slight social circle number case circle mt combination mt consistently social percentage overlap circle overlap union see circle node circle quantitative social circle common node semantic relationship statistic mt show choose circle mt jointly st performance task show mt get st circle simple pooling bar
sf aic bic perform intensive validation aic bic aic aic comparable validation mse bic aic select bic selection exactly simulation recognize aic tight correlation half validation therefore intensive cross reasonable bic time parameter choose graphical correlation analysis x nx rest determine regularize liu collect regression edge drawback cross find fold detect dependency hour regularize aic bic gene database straight forward study positive dependency structure zhang rl size aic use roc curve positive rate fdr false ccc auc fdr auc perform auc band bic especially band addition bic band aic bic well aic bic respectively discovery rate crucial association bioinformatic general auc fdr decrease large bic increase purpose potential cancer source cancer com searching expression http www cancer genome http survival dataset probe common gene gene patient cox construct co expression gene regularize meaningful module patient survival correlation bic second identify htb gene onto http string predict include physical indirect association string expression co occurrence gene fusion neighborhood gene compare together biological association survival interaction predict gene interaction gene large six link col direct gene protein col col remain link col addition gene gene confirm include remain gene indicate degree col col col gene col involve biological go involved pathway protein pathway poor overall survival os induce cancer express recurrent col tumor col associate et al yu contribute may far explore biological clinical propose propose solution base ridge regression guarantee mild establish fix regularize outperform substantial margin regularize accuracy test especially computation intensive consuming fortunately regularize regression directly aic bic appropriate well validation therefore discovery fdr aic fdr sample aic bic discovery candidate costly source gene expression demonstrate pathway efficiently expression rna appendix naturally mathematically eq ideas manuscript general em em reference look identification sparsity j model ann ms gene signature survival chi nonlinear structure lin ss fan li selection ann tu sc sr molecular mechanism cancer implement genome validation cancer microarray patient lee j j human microarray genome lin ratio regression liu stability advance liu free reweighted liu wu selection via journal graphical statistics liu lin machines lp sparse penalty journal penaltie wang ac prediction meta patient cancer schwarz ahead thompson ms beta cancer tumor yu pn md hc lin lt lin contribute cancer zhang v wu base survival reveal signature predict journal chen j cells h oracle zhang elastic net zhang concave ann regularize liu li comprehensive cancer center school public health university california gene high bioinformatics computational engineering appeal essential sparsity measure nice natural expect np resemble propose efficient em natural lasso elastic combination cross aic mild method simulation dimensional aic identify zero pathway topic regularize regression associate small bioinformatic engineering irrelevant prediction selection regression essential include aic schwarz penalty extensively challenge exhaustive aic bic combination computationally infeasible convex relaxation regularize gradient equivalent minimize asymptotically model consistently experimental pose additional equivalence moreover lin regularize regression regularize lin solution classic solution effectively aim scad fan li liu et mc zhang though zhang continuous smooth optimize solve regression effectively deal natural elastic zhang liu wu time regularize aic bic method gene eq parameter nonzero relevant irrelevant coefficient equation reach vector equation optimize derivative wise division x r nr combine together approach liu calculate sd nonzero sd sd validation number indicate mse consuming fortunately lasso identify cross criterion directly pick aic criterion know advantage matrix five fold cross optimal range regularization log intervals j toolbox www com report ccc sf average bias estimate outperform select close structure bias maximal small lasso result never choose necessary estimation though bad predict regularize minimal mse ss ccc sf mse standard sf compare true
subsequently round penalty programming update equip correctness nearly separable restrict parameter runtime reason dimension implementation normalize solve fitting need tune specify grid always recover match perform compare benchmark profile interest dna chemical dna occur specific dna affect gene site measure site row site cell proportion take represent probability simplex recover cancer shift proportion frequently experimentally costly without recover challenging study small cell compose major sample denote f n outline quadratic average indicate qualitative recover cell fix compute column obtain subsequently linearly row r r affine arbitrary iff unique turn observe exist unique b p repeating precede generate b r b conversely equality hold paper factorization constraint follow yield line imply cf subsequently linearly obtain return solve output linear counterpart regard approximate eliminate vector always column place singular select linearly obtain u sketch form noisy suppose follow solve return corollary follow recall paper permutation separable exist mr fulfil canonical conclude rely seminal inclusion moreover suppose write canonical imply contradict fulfilled iff assertion natural ask bernoulli far away whose answer experiment matrix I trial observe except set vertex hand bad tt draw entry bernoulli present present entire concern two probability draw simplex standard vary draw entry half I bernoulli rest projection aa datum potentially h regard comparison report display well theorem theorem section theorem assumption remark de second addition convexity share scheme complicate combinatorial datum despite apparent et algorithm recover factorization use yield compact representation basis element depend application negativity I blind wireless signal inference gene encode signature case absence overlap assignment involve factorization discuss restrictive model present matrix important research fundamentally boolean factorization binary factorization scheme convexity factorization coordinate commonly employ lack guarantee beyond convergence progress regard factorization nmf al nmf non far complicated impose optimization appear computationally dimension obvious hardness contribution provably factorization linearly remain tractable long extend algorithm show superior heuristic uniqueness nmf alternatively draw suggest continue negativity factor usefulness signature submatrix form row wise concatenation affine hull symbol vector identity present uniqueness connect problem follow hypercube entail affine instead combination essential reason order treat differently otherwise drop factorization unchanged e column linearly submatrix affine independence column affine dimension reconstruct obvious solve vertex contain independent dimension check solve remarkably check irrespective vertex provide subsequently obtain return obtain system solve illustration geometry dot area right row crucial compact system substitute instead pool filter yield determine aim find possibly without handle right dominate cost check form nmf permutation matrix raise whether uniqueness fail broad insight question bernoulli question essentially study improvement crucially theory whose positive uniqueness pose may conjecture affine hull vertex exponentially empirical continue boundary cf reduce cf vertex vertex seem impossible cost indicate candidate column coordinate vast discard column coordinate rapidly candidate check substantial portion theoretical extension form I refer number continuous upper bound weight sum contain theorem third reduction check cf vertex condition apply entry yield r reduction weight pick successively row lemma suffice identify evidence continuous derivation tackle view lemma achieve requirement candidate satisfy feasibility program solve g branch solve check impose recover experiment pool could achieve small pool sequel extension handle particular mind noise change e distance positive singular r compute set
variable fmri bold different scientific question term whole brain usually hundred method scale brain regard big novel estimation fmri fmri come perform cognitive make publicly available detail bold due stop fmri focus voxel activation phenotype behavior stop voxel activation study causal activity predict study direction major analysis simple infer probably correlation indirect alternative lack identification indirect connectivity seeks infer directional brain region approach connectivity dynamic causal modeling voxel region criterion include annotation voxel despite attempt scale infer hereafter type probabilistic direct indirect connection variate represent relationship node parsimonious connection inference inverse zero connection fmri interpretable indirect connectivity reasonably region heuristic approach roughly divide accord major penalize full lasso fast polynomial hundred penalize network instance hierarchical bold hundred thousand area g system interact understanding nod unclear structure include leverage scientific finding advance unify conceptual hide share topological structure incorporate smoothness fmri goal voxel alternate simultaneous advantage demonstrate fmri simulate conclude discussion technical plot biological assume node include subscript index collect notation stand norm pm introduce formulation mean subtract variable center usually column observe disjoint variable relate variable independent introduction assume inverse precision represent precision exist challenge storage hundred could small advantage outline interpret functional voxel form commonly estimate hierarchical also topological role group observation eq extract signal fmri incorporate glasso penalize hierarchical ignore scale via objective objective np conditionally via update update conditional minimization assignment effective incorporate minimization glasso conditional conditional minimization eq minimizer conditional glasso minimizer purpose precision glasso exactly algebraic property fast enforce positive small summarize stop iterative update tolerance iteration way rule usually good alternating point close suffer converge optima suffer select meet precision glasso kt alternate parameter exist scientific knowledge employ scientific knowledge negative log converge probability estimate tuning search small parameter control group compare minimal choice choose bic fmri publicly open fmri consist subject kind stop go illustration author employ implement library http uk include slice correction alignment average template smooth half filter preprocesse linear voxel remove motion standardize glm residual retain analysis fmri residual activity voxel comparable unit variance choice roughly match usual result interpretation goal finer yield network vice versa recover study avoid minima assignment random start compute resource evaluation figure extent voxel symmetry though impose coincide classic grouping visualize voxel examine edge voxel color template connection row color show r region supplementary play stop arbitrary voxel cluster voxel contain cluster together locate brain partly perform brain connectivity inferior brain direct whole activation previous group term connection cause furthermore finding distant region area motion visual cluster ica study coherent investigation study possible connection voxel group average template number arbitrary supplementary middle temporal area inferior sc superior area assess simulation iid precision block column normal vector similar fmri repeat time initialize glasso sparsity grid assess receiver operating characteristic roc overall name clearly glasso sensitivity specificity average receiver operate characteristic embed solid glasso blue line specificity bic select circle triangle glasso estimator glasso idea consider result connection moderate use different retain large likelihood simulate coherence across run measure stable equal truth exactly rate run exactly connection motivate problem fmri interpretable
nonnegative way important estimation exist construct show perform select j k satisfy bound two consequence follow first large model allow define estimator ensure mean infinite give avoid allow constraint use least understand estimator estimate resolve value nuisance devise consistently nuisance remove estimate eliminate nuisance define subsequently show display property nonconvex show write relationship parametrization moreover exponential idea lp lift estimator minimizer result partition mapping invertible linear argue objective equivalence define lead mu mm equivalence constraint use counting inequality contribute last give count thus give optimization constraint polynomial mix formulation gp lp significance solve nonconvex polynomial also introduction easy restrict tensor satisfie risk favorable hierarchical contingency table tensor wrong continuous interpret count bregman leibler divergence bregman risk capable estimate expectation oracle partition function risk convex show optimality optimization space lead strongly q apply lower bind key lead equivalence convexity related show equivalent empirical risk sense eq q square risk function empirical noting show risk promise computational turn attention identify risk key interpret dimensional lipschitz loss respect predictor partition necessarily constant supremum easier bound deviation fix respect lipschitz structural class compose interpret still vector combine fix partition depend triangle need focus combine encouraging need rank oppose exist exploit multilinear np base approach need essentially quadratic away np need measurement lastly loss partition necessarily partition express respective parametrization partition ideal partition express ideal unique could use ideal partition strictly partition partition risk property minimum whenever parameter satisfy property possible partition imply almost ready iid make completion typical sample equivalent emphasize assumption observe assumption without loss q strictly entry write outer nonzero useful say parameter directly make instead use easier additionally statistic must characterization result proposition strictly also proposition decomposition converse q measure uniformly despite fact fix tensor condition incoherence inequality incoherence represents couple versus issue incoherence existence tensor condition apparent look quite incoherence tensor specifically write vector whose entry uniformly within sample order tensor construction incoherence condition incoherence tensor jointly belong entry tensor change variable incoherence property hold satisfied variable ideal consistent significantly combinatorial step subsequently indicate partition gap partition least type mistake index constant depend expression partition event low lie constant proof difference assume next occur type error study approximate procedure estimate ideal need approximate partition discrete clear low estimate tensor tradeoff small analytically tradeoff describe validation accurate nest u set pick threshold e leave suppose compute gap error select optimal follow leave cross assume apply term term term use twice return triangle last focus minimize similarly union union combine success cardinality use overfitte thing ensure sufficiently lastly validation slowly sparse structure selection low involve free use exploit equal match move formulation additional number inequality lp lift still low proposition analogous risk exact low partition partition proof highlight key refer pseudo case modify immediately point note soft thresholding tensor rate imply count rather expand lc none pn pn pn tensor estimate unfold along first second unfold nuclear norm slightly nuclear paper variant norm specialized numerical implementation estimator package implementation worth fast norm informally benchmark implementation code nuclear example present synthetic estimation amount pathway author leave tensor measure jointly choose ensure require though unbounded noise early error essentially reweighte version indicate tensor completion log error nuclear nuclear square nuclear partition production combinatorial nature maximize production pathway relate pathway completion pathway either validation respective element datum measure log show identical construct measure fig measure prediction small qualitatively closely value code describe model nuclear norm coefficient design amount threshold partition log nuclear entry specific hard select achieve low cross numerical data pathway low expressive algebraic definition element inclusion face hierarchical table extend partition complex give estimator directly similar measure risk result measure structure belong sub partition partition test limited instance complex valid extend principal pca consist matrix successively termination criterion additional assumption decomposition approximation tensor hard allow approximation tensor numerical mix perform completion tensor completion identify work challenge towards method tensor tensor negative result semidefinite challenge define generalization correction step positive tensor may study broad question predictor require specific predictor linear orthogonality poor conditioning sharp regard ensure study independent sum distribution distribution acknowledgement author thank lee corollary supported award interpret noisy tensor exist convert completion unable possible tensor parametrize linear amenable choice function thresholding estimate unable exploit structure hard computationally rank completion datum refer model purely predictor different th response specify possible define predictor belong exponentially slow surprising combinatorial curse try estimate value approach combinatorial convert numerical difficulty restrictive impact completely combinatorial principal define combinatorial model interpretation index combinatorial completion possible extension hierarchical rely upon generalize discuss
competitive study subsequently detail scalable thompson appeal thompson bandit pay online action specific ad multiple ad website regard ad uncertain payoff ad pay exploitation present ad increase likely overall exploitation balance course interaction formally choose reward policy reward bandit substantial theoretical thompson thompson bandit asymptotically thompson bandit competitive basic thompson select thompson sampling within past reward reward model parametric parameter action necessary suffice draw action high thompson practically scalable large complex thompson computationally might logit use thus require property scalability thompson limit thompson compute thompson form misspecification address robustness thompson thompson replace appeal randomly resample conduct replicate bootstrap replicate numerical replicate bootstrappe bootstrap distribution bayesian use bootstrap posterior robust misspecification prefer remainder simple competitive thompson also discuss subsequently analyze computational misspecification commonly bandit bandit action select time arm straightforward set round obtain beta posterior ir beta beta play arm illustrate present bootstrap true increase number observation take center true row theoretical thompson armed bandit start initial replicate decide arm play arm update bootstrap replicate thompson sampling replicate uniform bootstrap replicate retrieve arm j replicate could exploit large costly allow exploration armed bandit simulation arm reward arm examine empirical thompson replicate simulation greedy comparison new replicate construct closely comparable thompson thompson empirical regret replicate thompson comparison thompson examine replicate cumulative regret arm easily separable albeit replicate practical small replicate suffice become thompson tune distribution analytical show bernoulli highlight however easily example motivate partly motivation generalization bandit triplet assign become reward would general would resort computationally costly produce give thompson use conjugacy relationship formulation getting update contextual thompson scalable alternative besides kind misspecification liu examine setup simulation thompson error coefficient error include mean incorrect datum factor thus configuration denote vary create optimal arm arm thompson sampling thompson thompson summation ridge penalty compute select replicate select random play arm maximize simulation bootstrap examine full ignore flexible present reward thompson vary degree relatively misspecification significantly cumulative produce large difference thompson confidence interval alternative substitute idea thompson optimize exploitation competitive play thompson substitute double nothing point online parameter perspective matter policy bandit observation series effect analysis
determine period fisher minibatch gpu note actually store compute equation despite sometimes minibatch update factored fisher otherwise two use differ update product ti ti ti enforce minibatch fisher operation compute minibatch place otherwise eq gpu transfer derive quantity diagonal point decomposition compute compute scaling factor implementation save let later compute multiply gpu working factor diagonal exceed orthogonality aspect neural net parallel include text discuss implementation store disk enforce parameter minibatch generalized model explain introduce initialize dnn implement discuss adaptation machine sgd one gpu computation minibatch example minibatch job typically advantage core processor share refer asynchronous prevent relatively gradient minibatch averaging ensure make progress regardless minibatch minibatch limit minibatch understand follow minibatch size minibatch size minibatch sgd become gpu fast cpu aside give result hard access order access access try sequential read file neural sequential pre randomized disk temporal randomization order probably view sgd amount access break training block per epoch process per ensure randomized give nothing particularly disk disk compression prevent instability divergence change minibatch explain completeness single formulate thing minibatch multiplied enforce implement involve store product norm penalty enforce minibatch range exceed sum right eq minibatch tend initialization discriminative layer backpropagation publish reference something hour less dataset eventually become impractical beyond scope instead layer mean hide train report remove add hide short repeat hidden fan softmax layer find essential discard add prescribe versus layer bp another possibly language otherwise notice outer iteration immediately outer average parallel averaging compute train extra meaning collective objective function make whole entropy fix viterbi alignment easily frame speech term train dnn speech maximum objective properly variant inspire minimum minimum phone expectation compute derivative posterior parallel standard ng generally minimum phone procedure net setup exist setup item training describe epoch machine average lattice piece part lattice derivative order ensure modify ensure change geometric fix rate mention generate consist lattice something connection scale parameter combination output advance rate randomly training frame minibatch ng sgd apply sometimes continuously necessary neural e report mean adaptation know constrain gmm system although online thing gmm decode complex ideal order turn addition range characteristic gaussian mean parameter supervision audio case dimension vector train switch I current take input normally consist plus frame normalization energy energy transform transform feature order match statistic include per generally give train frame consist process prefer application convenience cross transfer framework speech amount gpu hardware agnostic way parameter machine method efficient ng seem improve training neural combination parallelism parallelism parallelism minibatch net parallelism sgd speech work combine ng attempt explain parameter ng empirically significance speedup neural speech introduce behind gradient discuss parallelism ng sgd background dnn classify vector time acoustic cluster per duration contain several ultimately top log cost viterbi likely probability course slight supervision label viterbi alignment word aspect machine randomized subset gradually machine across job epoch job outer epoch training learn individual rate proportional rate stay sgd job get average summing concern hessian reach close equilibrium opposite time away relevant sgd namely schedule less relevant issue appendix cpu gpu randomization generalize decode speech recognition decrease thing training mention report start end specify epoch range epoch tune ng sgd ng extensive schedule tune circumstance ng helpful experiment common parameter getting modify enforce maximum parameter minibatch tend early gradient positive matrix fisher speak riemannian surface path conventional compute inverse follow rate scalar decrease something g training sample minibatch replace instead proof keep eigenvalue rate matrix prevent systematically particular clearly idea learn oppose suppose continuous fisher outer log justification fisher hessian hessian gradient direction datum fisher hessian transform parameterization easy fisher qx qx still generally quantity analogous hessian change matrix speech recognition million inversion fisher impractical deal factored form approximate rank block explore per material analytically neural form kronecker natural accept newton inverse factor fisher weight fisher weight consider th block separate include bias term kronecker symmetric positive definite row whose plus multiple approximated factorize kronecker weight row ever kronecker show factor way factored minibatch hold surprisingly online distant significantly fast probably help kronecker training weight matrix act quantity modify factor matrix multiply fisher form processing training derivative minibatch gradient minibatch easy minibatch eq indicate modify quantity natural eq matrix minibatch separate term interface natural gradient minibatch minibatch fisher hold multiplication return minibatch ti ti ti gradient interface minibatch twice want prevent early huge inverse fisher technique rate un fisher slight proof scalar view practical minibatch ng estimate invert minibatch full rank strictly necessary contain multiple unit find follow quantity fisher use stop ever exactly relatively large suitable wide circumstance simple big minibatch size interpretation fairly direction quantity direction gaussian hard fisher correspond replace course practical believe change make fisher try co transform fisher around motivation work allocate effort regard factor think easily provable converge factor use together rescale rescale multiply inverse fisher fisher randomly identity quite true due minibatch easy rescale minibatch back instead sense objective minibatch thing natural describe estimate factor involve multiply weighted covariance current minibatch factored method probably analysis initially steady state minibatch equal confident stable give effort converge minibatch method would kind something modification sgd experiment frequently sgd momentum effective parameter reason momentum quite successfully momentum cpu method likely momentum prevent instability limit per layer minibatch another popular modification sgd parameter schedule theory reason unlikely speech firstly inferior decay learn believe true use essentially direction concern interesting type diagonal diagonal speech recognition english english quick hour half hold hour long cc dnn dnn word rate side difficulty quick method use gmm process space normal build state gmm dnn adapt gmm across context input dnn dnn dnn hide layer nonlinearity reduce explanation number train parallel ng dnn dnn online decode audio datum process input equivalent un normalize frame dimensional include show include compute intend decode cpu million sgd epoch exponentially hardware fairly r core ghz gpu single notation becomes locate gpu reporting take slightly optimistic figure fast figure color plot versus process final proportional learning job natural help ng plain sgd using show amount job slow get speed take epoch get epoch get speedup show simple job word ng plot time simulate multiplying take outer outer depend queue load outer plain sgd ng sgd second ng sgd circle mark c plain ng sgd ng describe gradient ng sgd experimentally versus plain possible parallelization across sgd enable train parallel even one confident experience true relu sigmoid activation final rate speed except prevent parameter training acknowledgement neural code kernel numerous mention aspect setup contract contract acknowledge grant award speech cloud development reproduce purpose annotation conclusion contain herein interpret represent express imply element minibatch interface element minibatch inverse fisher row core inverse multiplication row minibatch smoothing compute gradient compute efficiently setting column respectively smoothed matrix row result rescale intend sure denominator scalar hold rule randomly believe contamination bias turn modify properly sample purpose believe efficiently smoothed holding differ hold row version expand simplify minibatch great formula derive correction correction row differ scalar factor use scale work minibatch g overall large typically considerably overall inversion compute gpu online interface simple multiply fisher rescale correspond minibatch row simple run minibatch minibatch copy twice matrix net input minibatch column update subscript minibatch row online estimate normally compute ensure denominator rank approximation row introduce specify describe make top slow inspired compute think eigenvector scale precise sense actually square root eigenvalue put diagonal put eigenvector add reader seem straightforward decomposition way speed symmetric diagonal definite reduce orthonormal unknown desire covariance ensure dimension correspond inner order ensure f tr work zero representation describe computing multiplication typically expression implement equation scalar whole expression identity part store factor write involve factorization go appear convenience expand
f bx decrease conclude intersection increase bx b bx bx bb bf bx bb b bx bf x get q accord intermediate b dx bx f bb bx get intermediate exist g b f b bx dx bx complete give restrict bb point intersection f bx two x dx bx bx b bf x bx bb bx f get exist bx bf bx bx bf bx f bx bb bf bx bx eq g c bx f bx bf bx bb b gb intersection gb monotone solution give eq prove point start converge prove exist assumption easy nonnegative bb g continuous concave proximal gradient q sequence monotonically decrease stationary point since lead exist subsequence subdifferential exist engineering school technology science key laboratory school university edu sg com edu sg edu cn study singular thresholding nonconvex obtained denote bound nonconvex surrogate solver generalize singular basic subroutine low solve rank attention application background achieve use suboptimal loose approximation bring attention surrogate smoothly scad concave penalty mcp show nonconvex usually scad structure extension sparsity nuclear however suffer nonconvex different minimization reweighte nuclear minimization q singular continuous concave nonconvex continuous objective surrogate construct simultaneously guarantee decrease surrogate quite loose minimizing possible tight surrogate relax name method later operator associate nuclear know value follow perform gx gx x singular thresholding nonconvex still operator open whether monotone nonconvex simply perform singular unique need I p challenging solver nonconvex worth nonconvex none proof rigorous ignore prove monotone detailed work general solution correct exist reason behind hold optimal give rigorous bound compute type special show general solver reweighte nuclear synthesis experiment solve equivalent von simultaneously optimal denote von trace equality hold decomposition reduce eq conditionally share thresholding associate monotone optimality monotonicity nonconvex special choice rigorous since monotone prove find intersection bx popular denote x corollary minimum objective lead solve solve give find local minimum start search fix iteration nonsmooth nonconvex local candidate nonconvex surrogate scad shrinkage effect shrinkage operator proximal nonconvex nearly unbiased norm proximal norm norm necessity nonconvex singular function get solver name proximal nonconvex compute convex method sequence property point expect decrease since tighter verify experiment conduct note test logarithm suggest compare nonconvex enhance logarithm set dynamically decrease conduct two miss evaluate regard repeat lagrange multipli alm plot outperform alm nonconvex logarithm approximate datum outperform surrogate loose c plot curve fast dominate pixel matrix completion blue recovery recover achieve large relative collaborative filter preference similar movie movie movie entry normalize absolute
form augment multiplier parameter equal size scheme subproblem value depend proper reformulate speed calculation qr formulate orthogonal although optimal iterative scheme equivalent highly modern introduce definition whose understand wise r mm minimization close norm rewrite optimality condition give update soft thresholding lr subproblems respect respect formulate subproblem gradient admm scheme essentially gauss admm compute particularly attractive accelerate adaptively change iteratively algorithm easily directly moreover adjust k solve complex operator outline optimize augment identity update norm similarly converge k f stop rbf employ criterion stop current satisfied tolerance u v dd cause overfitte rank fortunately work rank relatively addition provide dynamically adjust parameter scheme I rank detect eigenvalue usually specifically v satisfied big jump adjustment present relationship k k k tm u u k lagrange meanwhile replace computation constraint algorithm guarantee theorem generate ks cauchy proof find feasibility optimality solution stop k addition multipli e k conclusion v uv mn feasible uv please theorems f objective generate algorithm mn f parameter mn value arbitrarily hence rbf main run svd qr multiplication qr multiplication rbf problem usually iteration outline methodology consider space describe try auxiliary completion write admm scheme completion problem solve effectiveness removal background collaborative run ghz pc windows gb conduct task remove generate image image text form outlier mask time true pixel bf alm rbf tolerance detection area characteristic curve auc image bf outperform visually detection significantly perform short bf alm rank recovery moreover run bf alm conduct component recovery rbf chosen grid bf alm achieve auc run fig rank rbf test rbf surveillance detection background modeling segmentation surveillance video video satisfy background frame control hence exhibit low property foreground spatially sparse rbf surveillance video bootstrap consist frame video first column collect background bootstrap input extract rbf experimental database run recover slightly see rbf times show rbf address large ht ccc bootstrap ht reconstruction incomplete face often decompose capture illumination image randomly miss entry reconstruct people present rbf perform visually complete miss word rbf regardless corrupt implement incomplete burden projection raw pixel randomly respectively clear rbf successfully collaborative filter technique predict user evaluate completion experiment conduct widely recommendation k rating randomly test testing experimental report compare rbf soft two manifold optimization root rmse define ij rating denote predict rating rmse three report independent see fix except norm moreover bilinear factorization method trace consistently factorization minimization soft bilinear ccc ccc c rank regularization rbf robust method vary regularization increase become increase slightly confirm bilinear factorization respectively formulation robust rbf side rbf unlike low address scale matrix incomplete corrupted real superior l solution svd accord uv know solution l find uv sd accord lemma uv uv proof mathematical induction q svd algorithm verify v kp nu k ks kp k k complete appendix sketch multiplier boundedness endow subgradient k sequences respect v furthermore u k procedure u k k k I next subproblem optimality lemma complement satisfy ty k ty ty k bound boundedness u k k k k feasible cauchy sequence lemma give obeys qx calculate calculate k k k k k k convexity calculate uv u uv u uv uv uv u uv uv uv lead uv uv uv mn worth nothing globally v g mn please matrix g g u u mn mn feasible naturally follow svd v feasible mn l singular singular resp singular small calculate mn f f theorem theorem rank incomplete corrupt statistic bioinformatic well solved relaxation norm certain applicability paper scalable provable rank miss corrupt incomplete corrupted compressive specifically trace bilinear structured apply alternate multiplier admm finally provide compressive pursuit rank recent recover receive broad different bioinformatics area bring great challenge analysis digital surveillance video text web fortunately intrinsic ambient pca popular tool face pca contaminate outlier miss set address compressive rank base rank filtering recommender miss entry arbitrary measurement collaborative motion click tag recover corrupted image corrupt classical address issue sensitive outlier outlier extensively semantic indexing video surveillance simultaneously recover globally involve suffer effort towards svd svd provable bilinear factorization corrupt
rank consistency separability yes provide separability nd order ex model share motivation extensively last decade yield review ref dominant trend heuristic mcmc demonstrate problem provably structural form rating method considerable ref ranking preference explore combine topic review star scope key summarize dataset formalize universe item share across population user pair specific weight ranking generative comparison user token n htb generative user weight ranking convenience nonnegative ranking row order kk dimensional column finally matrix item principal algorithmic ranking component follow ex similarly probabilistic word draw matrix topic prior ref form ex distinct vocabulary stochastic propose statistically topic prior note matrix infer directly solve approach recent work topic efficiency exploit moment occurrence comparison establish particular establish splitting row make ex item geometry solid approach geometric define illustrated fig separability rank separable order rank least item uniquely rank ranking separable order separability identify real world modeling ranking appear albeit implicitly seminal separability sample uniformly estimate factorization separable hull identify find novel identify estimate exclude ranking rank leverage solid angle novel pair angle indicate distribute direction topic solid use detect separable full rank motivate angle select pair large estimate estimate modeling ranking infer preference predict new see ex outline expand detail algorithm novel identify constrain regression follow column estimate round element binary satisfy jk htb projection matrix ranking projection ranking I ks k ranking I I b k j topic derive approach upper technical hold isotropic alg result specific gaussian separable user furthermore draw parameter b solid angle extreme hull proof material alg alg order compare ex synthetic satisfied demonstrate variability collaborative filtering application heterogeneity well ref movie rate widely public availability comparison focus ranking viewpoint literature ref ref ranking ground adopt ranking use hold likelihood topic consider optimize root rmse ref netflix ref specifically projection tolerance alg alg ex ex semi dataset validate match dimensionality characteristic world comparison benchmark movie star rating million rating user star follow rate factor star art filtering task movie sort score column movie matrix obtain ranking set suggest separability model follow prior dirichlet distribution comparison comparison align column base truth due ranking pair normalize error rp propose estimate recent consistency guarantee vary truth ranking rp ex rp world setting evaluate rating star focus rate obtain star user rating pair user rate star movie comparison movie rate randomly select movie rate movie rate user star rating tie ignore select set rating use training split convert rating testing testing independently log e test train figure result pairwise comparison fix total test high likelihood validate htb htb ex user original dataset training hold use gibb compare rp number comparison fix ranking summarize agree task ex ex behavior prediction recommendation train model comparison comparison objective comparable art rating achieve rating rate convert training convert ignore dirichlet predict predict rating movie rate rate movie star predict rating movie maximize square rmse rating factorization factorization pmf factor latent factor similar pmf rp come perspective rmse rating pmf rating integer real prediction near integer observe rp algorithm pmf fitting match demonstrate behavior design well generate strategy ranking aggregation distribute database web statistical efficiency centralize retain communication upon nsf award fa view conclusion contain interpret necessarily policy imply material largely track methodology new ref handle type setting complexity analyse algorithm account valid rank guarantee distinct note present angle extend direction scope column stochastic noting detail view special prior pmf user type user first indistinguishable comparison main fair appendix single main result almost generic establish individual convergence constant ready splitting comparison large j I apply proposition optimize set union claim distribution tie special isotropic nature ref special type spherical consistently identify ranking give success consistently novel novel prove follow th follow matrix rank distant lemma similarly q span proposition statistic latter algorithm j ranking asymptotically main detect pair start straightforward separable solid novel q solid angle alg vanish iid direction draw novel let intermediate convergence solid novel j norm j j implication ip consistency identify distinct ranking pe p w mn distribute first decompose union sort ki accord pe p proposition two term therefore pe consistently rank success loss generality ranking constrain row k surely row second rest j k round put remain post consistently moreover separable recover column furthermore eq fail normalize solid angle hull combine alg lead complexity
weight bit possibly example bit look index simply bit interaction bit recurrent current parametrization require actual multiplication next aim generalization scheme obvious regularization success smoothed gram maintain index sequence update example require index make correction simply automatically often activate activate regularizer towards examine achieve structure bit thereby return empty return unit understand binary tree weight leave use sum node involve per reasonable overhead multiply add number vector serve help regularization degree freedom computation increase rapidly grow much weight regularize keep interval weight lose decay coefficient correspond multiplying towards issue decision assignment decision decision may prop usual ignore unit training case accord long job space exponentially store hypothesis evaluate provide training gradient heuristic obtain back loss noisy sum select weight index unit active contribution control create outside length one great challenge expand applicability ability require future validate valuable language dataset universit art deep large computation available able exploit large improve generalization decision favorable computation deep increase neural novel parametrization exponentially parameter weight matrix bit pattern activation parametrization tree sign unit hide deep learning abstract supervise unsupervised review application computer possible feasible report experiment factor recent availability net handwritten digits traffic sign face point achieve human far general scene understand speech could correspondingly correspondingly cover category modality concept current still multiply much favorable ratio select pool unfortunately prevent like technique rely generalize way region mathematical cover svms theoretical suggest deep advantageous characteristic learnable computation well aim deep computational efficiency computation objective mind way discriminative ratio exponentially computation
every iteration equation policy turn description conservative discount stepsize return successive greedy operator way step policy distribution article stop continue stop derive analysis latter mention variation approximate step eq simple implement require explain trajectory start stop require underlie mdp implement step new evolve slowly natural policy originally problem variation infinite stationary policy make action r horizon begin build sequence non iteratively policy return approximate horizon stop step one may return start policy good horizon policy practice horizon policy loop shall algorithmic may prohibitive aim next present article describe issue simplify variation grow approximate greedy value loop infinitely kk consider suffer drawback interestingly another parameter policy store similarly stationary set policy shall run formally periodic infinite stationary loop iteration respect compound operator forward eq process horizon form forget policy keep step stop loop infinite stop guarantee algorithm done see set consider go v policy policy order introduce relate wants guarantee fine lack space set deterministic policy p ci coefficient finally small c thorough bound thing dependence discount quality match parent good comparative hierarchy direct implication important mean parent finite infinite guarantee make picture complete mdp though coefficient derivation done guarantee analysis though require scale relaxed improve still slow technique sufficiently enjoy rate express guarantee extra nice hold respect instead theoretically mdps give estimate grey region dark mdps error mdps deviation mdps error ease comparison display practical store proportional may require memory explain make nice guarantee performance identical control increasing confirm well becomes discuss baseline slow progress take million stop except make assess variation differ addition compute problem randomly finite mdps correspond application remain kind mdp encounter brief branch greedy exact noisy mdps run compute e display variable display variability supplementary series much slow naive conservative bad overall provide bridge close relative tend vanish dynamic branching lot difference side fact use k two bound write multiply side observe start assume simplicity get back equation conservative address deferred supplementary cd ic reasoning rewrite follow use begin prove result v matrix ir subset multiply definition show nice fast get corollary small satisfie turn straightforwardly precise greedy show ht top middle pdf middle experiment parameterize branching uniform give reward uniformly sample feature factor
mdp discrete parameterized receive initial reach let trajectory mdp trajectory assumption part quantity trajectory maximize ascent work risk spirit policy detail transition note simulate trajectory together correspond single trajectory realization variable estimate use eq update maker gradient criterion policy necessarily augmentation current accumulate simulation policy still sensible h policy satisfy h kk rl take covered paper add negligible vs policy return return leave tail final parameter encourage take return examine rl benchmark study extensively technique among gradient modify policy expect line game performance versus different characterize behavior well modify emphasize whether extend handle currently standard shape induce sensitive game score line addition limited step modification risk policy induce tradeoff line less frequent batch policy standard warm policy reward average return rx I algorithm converge return return mc high final observe low policy decision maker contain finance understand difference cell controller weight controller may version describe converge full supplementary lr style descent optimum domain beyond reach motivate simulation acknowledgment helpful discussion research european european agreement n finite therefore well know empirical write follow empirical bound integrable q observe thus crucial proposition gradient term even direction quantile problem importance general reduce variance monte estimate estimator wish random mc sampling dominate r ix heart parameterize aim f typically sample gx gx gx estimate sensitivity recall addition outline proposition since know advance eq empirical sample estimator var q need modify scalar assume estimation I suitable reward nr li monte suitable update parameterize obtain exponential rl select deal scheme help reduce estimator difficulty find modify trajectory modify mdp however require simulator modify mdp modify rl set mdp specify give yx mdp heuristic rl trajectory weight bad outcome difficulty define bad note reward action fundamental elegant selection term outcome current policy know greedy selection rule behind high preferred encouraging transition produce bad encourage value trajectory obtain policy extensively rl literature many task td result mention choose naive sort choose use trajectory policy significantly due estimation return assumption ac il ac il risk prominent various domain new form propose gradient spirit estimator local domain reinforcement learn controller extensive finance among payoff define optimization find structure deterministic solve various payoff generally asset many resource allocation suitable case recently optimization derive expectation stochastic analyze allow domain reinforcement sensitive beyond reach consider interpret sensible remark often maximize extend problem straightforward method another incorporate importance important capture lr gradient payoff lr financial rl commonly method extend lr perturbation style estimator formulae lr utility event risk interest return sensitive scale accumulate suitable horizon problem function mention work rate decrease determined single formula version scale stochastic time algorithm subsequent random c convenience everywhere var denote outcome change express calculate analytically express expectation convenience make smoothness gradient note technical detail standard lr lr gradient expectation eq rule z z finally trick multiply inside integral justified obtain lr formula see proof account baseline turn elegant portfolio rl order use gradient need sensitivity performance complicate variable rl system stochastic dynamical calculate probability usually sensitivity trajectory typically trajectory shall generalize rl let support value countable reward formula variable disjoint close l sensitivity proof spirit difficulty apply future next effective sensitivity describe reward yy r proof supplementary r bias continuous assumption satisfy supplementary separate bias use quantile may lr work
efficiency solver second example severe main capture capture meaningful social network theorem jj jj dt dt fan jj jj check directly classical multidimensional work distance dimensional euclidean distance advance unfold semi definite sdp representation sdp capable produce numerically suffer drawback slow datum euclidean establish asymptotic produce uniform sample logarithmic explain develop inexact accelerate proximal configuration cope distance optimization rank purpose euclidean representation high model strongly inspire several sdp achieve suffer drawback distance limit practical resolve theoretical uniform model social network solve accelerate proximal distance briefly discuss relevant bind subsection serve contribution notation social tie actor analysis end actor actor relationship role etc observe relational composition structural social e presence absence similarity actor concerned relationship relationship actor actor way euclidean distance see information matrix often vertice bernoulli obtain pair feature bernoulli random therein mainly bernoulli model measurement social network produce preserve highlight image located strength actor tie preserve reduction fit distance social space reduce low classical scaling section provide one often embed produce satisfactory profile dimension dissimilarity matrix close embed embed euclidean space create embed motivated number try propose use distance manifold unfold maximize variance minimum embedding aim eigen gap distance either sdp enjoy elegant short distance accurately geodesic convex follow density near ball exist guarantee obtain distance highlight short distance distance derive social mail social network depend obtain rely sdp limitation inspire euclidean attract group relate natural interpretation tool excellent seem put category attract community recover via sdp approach guarantee probability theoretical property rip matrix completion sample satisfy rip prove rank recover noiseless observation near adapt short recently group researcher include setting propose estimator norm establish sdp sensor localization incomplete distance positive define matrix aim minimize nuclear equivalently objective minimize embed assume center obviously contradict possible seem bind read undesirable ball embed noise point theoretical result difficulty face matrix additional error derive euclidean nuclear equivalent embed contradict variance possible hence excellent straightforwardly useful learn dimensional may huge difficulty extension distance space theoretically guarantee spirit theoretical briefly describe contribution major reduction building bound building point sdp vs semidefinite sdp primary nonconvex driving sense three first distance minimization argue contradict maximize term third axis approximate embedding analysis accommodate initial valuable available embed illustrate situation lead part combine make guarantee mild control estimator roughly freedom symmetric choice reduce subproblem explain lead treat object benefit lead difficulty difficultie hundred slack explain use cpu contrary fast inexact proximal thousand method advance develop develop theoretical good problem manifold learn embedding provide cast score viewpoint benefit model detailed interpretation report list inexact accelerate extensive experiment paper real trace symmetric positive cone vector vector diagonal orthogonal remove column obtain product entry single sum value contain three part brief review key go describe explain work summarize key embed small definition embed well center known center semidefinite write eigenvalue submatrix q upon distance point first eigenvalue absolute mis eigenvalue satisfactory embed e aim score use interpret lead justification score root show matrix center role orthogonal projection onto geometric onto onto result distance point geometrically center remove embed gram encourage reduction maximize slack trade maximize preserve satisfie empirical total seek improve try eigen eigenvalue rise eq deal correspond interested slack model sdp rule element identically nn distance preserve embed point employ matrix encourage score eigenvalue remain good result moreover deriving solution follow interpretation facilitate description platform subsequent sampling basis sample adjoint vector assume jj estimator always embed give orthogonal write learning use deriving model task task try nothing quadratic slack essentially embed come spectral term small dimension argue minimize principal maximize third initial span coincide term maximize optimization third force control controlling penalty heuristic extensive experiment model sdp subproblem combine sdp keep update solve sdp subproblem therefore justify go bind summarize rather sdp exist use notation fourth positive semidefinite orthogonal subspace q yield bernstein matrix spectral norm mean exist ready following result major bind second nuclear random sampling strong convexity suppose least nc establish explicit depend sampling tail bernstein magnitude magnitude nc q minimization choice fact choice subsection choice major message usually therefore rank rank model tailor estimate serve parameter particular minimization part propose inexact accelerate proximal model easy inequality minimum operator order extensively correspond come diag ap scalar twice continuously twice continuously jj jj nearly estimation error nuclear without loss consider convex quadratic q cone ji equivalent way experiment inexact proximal study sdp suitable fx ax ax start iteration eq fy x kx adjoint major solution fact want form problem compute near type efficiently third approximately meet criterion formulation inexact fista inexact similar omit save demonstrate effectiveness real visualization model graph manifold physical capable generating configuration quality extract physical raise also sdp solver allow test issue subsection four sn communication social communication th fix mail social distance dissimilaritie user communication count imply social employ measure social email network facebook social top actually much high dimensional much explain performance train visualize social individual train connection record actually place indicate circle person place embedding capture lead eigenvector demonstrate sn visualize consideration social distance measure dissimilarity figure mean top top two lead eigenvector capture sn link political around part use visualize social communication widely use social generality isolate remain concentrate near zero correspond gram high however capture two leave blue circle set initial nn describe different depth near neighbor accurately gram matrix indicate capture first mention return eigenvalue lead
angle stimulus part explain dependent interaction subtract term marginalization g dimensional manuscript activity vary stimulus neural activity vary interaction stimulus general decomposition material separate period separate stimulus decision time period trial reasonably period aim separate x st st st st interval stimulus interaction component figure angle axis orthogonal mean neuron component splitting component largely figure delay period two water without period hand interpret stimulus active period clear highlight stimulus fire capture come cell tend dot axis dot absolute due ease axis decode stimulus interaction stimulus line figure time period significant carlo hold one trial neuron condition trial remain decode axis stimulus stimulus stimulus stimulus value time project trial decode close trial classify result accuracy iteration procedure reality trial classify pseudo pool different accuracy accuracie iteration neuron number randomly assign cross apply classification stimulus result chance actual decode accuracy consecutive bin figure period accuracy dataset monte computation hour intel processor centre sup paris forest university school nc usa laboratory ny de usa neurons tune variety tuning introduce dimensionality technique component analysis automatically highlight essential complex population area population decompose component capture highlight dynamic reward etc activity behavioral activity hundred technique common study external action internal conclusion neuron hundred neuron complexity pose fundamental challenge severe area neural response heterogeneity traditionally heterogeneity ignore cell criterion stimulus fire pre activity fmri area ignore higher simultaneously therefore display mixed look single neural analyze use reduction specifically datum account nature train dynamical population response dimensionality parameter control account mixed interpretation solve inform adopt e fire task parameter help sort mixed activity dimensionality latent component easily interpretable control task preserve ensure valuable away principal unnecessary orthogonality recently fire focus guarantee neural activity linearly individual spike train obtain remarkably complex population similarity difference pca largely ignore activity relate activity task extract orthogonal shift around space component fire colour stimulus decision population fire fire trial experimental trajectory indicate dot size visual clarity analysis pca firing rate neuron decoder principal second reconstruct fire principal decoder encoder grey neuron proportion variance stimulus component interaction clarity behind first axis long correspond axis case analysis fire compress two reconstruction enforce two stimulus colour dot decode axis axis encode axis expand reconstruct standard stimulus fire neuron fire stimulus trial trial fire also decision stimulus trial joint trace fire clarity fire population neuron moment reduce dataset pca pca fire rate neuron linearly transform principal neuron interpret population decoder compression latent compress another reconstruct geometrically cloud axis maximize variance projection axis decoder fire task condition find encoder eq like component stimulus information enter new additional constraint optimally compression achieve mapping pca axis orthogonal intuition b label stimulus decode preserve stimulus lose vice versa stimulus decode axis projection decode axis project de axes axis pseudo novel detailed briefly toy matrix stimulus part separate decoder encoder stimulus case paradigm adapt principal component stimulus component third row last row line legend thick interval respective reliably trial activity vertical almost bar variance compose four stack bar different colour stimulus variance stimulus bar colored chart split triangle dot star mark non leave task require discriminate separate neuron method fire stimulus stimulus frequency pair decision trial always trial neuron distinct many mixed pattern analyze period period neural activity across single neural activity activity stimulus row depend row three category stress method quantification activity component principle retrieve system weight neuron fire non activity component amount chart capture trial irrespective independent show cell delay period activity activity full trial activity complex varied indeed spread capture previously persistence period dynamic delay activity due slight variation fourth stimulus monotonic three period period period period stimulus shift fire stimulus encode single stimulus component short slide stimulus remain several technical overall c capture impose activity note nonetheless notable zero represent signal work work memory centre location figure delay second square appear match delay target appear report format figure paradigm adapt principal line correspond condition legend colour correspond stimulus stimulus stimulus line colour correspond match explain individual principal split dot product first axis leave first analyze task proceeding time fire study eliminate trivial eight preferred prefer analyze trial fall stimulus stimulus interaction similarity work first separate easily chart stimulus rotation stimulus decision fire activity stimulus identical task case condition independent stimulus component period stimulus activity delay period component period one notable figure presence interaction stimulus match trial opposite stimulus capture seem existence second delay work memory allocate stimulus whereas stimulus component overall surprisingly summarize obtain decision tuning non achieve validate art classification report recorded activity start exactly display required pre stimulus interaction come one memory figure specific activity extensive discrimination behavioral crucial storage stimulus trial uniquely water locate water choose water paradigm adapt legend four variance chart right show dot product pair axis show correlation activity neuron leave decision trial two self trial align across fire rate trial see lead neuron exhibit fire mixed activity stimulus thick thin bottom neuron tune absence similarity large part fall first component distinct mean respective shift firing pattern neural across epoch agree finding study far prominent component absence decision come reflect movement position stimulus prominent nevertheless clear stimulus demonstrate even activity neuron fire could categorization use pure vary trial different figure choice otherwise furthermore reward format adapt component legend show thick line pure blue period cumulative principal chart dot triangle principal corresponding mixture presentation mistake exclude datum combination present additional component present b component correspond prominent somewhat throughout stimulus especially already period separate incorrect stimulus characteristic correct trial agree predict corresponding confidence summary point pick population population neuron correspond encoder component neuron strongly unimodal distinct neuron component neuron component confirm apply population neuron one discussion response researcher resolve summarize extract latent component cell critical pca fa greatly activity report compare neuron conventional wide spread count response percentage use original neuron show response monotonic tuning delay percentage neuron increase fire period increase tuned match match difference significantly tune neuron level conclusion qualitatively perfectly sound limitation conventional focus bin firing rate tuning highlight neural frequency work report significant tuning tuning curve dependent component curve tuning imagine neuron choose cutoff g false neuron distinct stimulus dataset component albeit vary strength analyze test unbalanced experimental analyze version fire window role first limitation limitation remain many vary firing simply neuron interest highlight full heterogeneity averaged thereby fail neural false response introduce perform fire interest axis purpose neural tuning achieve deal extension replace non fire approach classifier population fire rate cross validate accuracy stimulus match condition work analyze classification shape curve match match true important removed firing reduction instance lda dimensionality account whereas look label lda look projection maximize separation within related concerned reconstruction ill purpose material extend four interesting outcome concern strength fire rate likely overall firing task period influence instance body visualize period method stimulus component active period stimulus figure possibility separate component task trial encoding pair mark star figure though unlike pca enforce orthogonality orthogonality neuron tend one example orthogonal fall orthogonality dependent condition independent first mean absence tend course orthogonality pair component figure many express component end special give figure stimulus interaction particular tuned fall list stimulus case dependent turn represent independently randomly map space encode axis nearly case orthogonality population arguably ensure limitation must need find neuron work trial average differently neuron record across multiple therefore specifically properly trial trial future argue exploratory analysis suit overview population important exploratory nature enable analysis analysis code matlab datum brief description experimental result reader detail trial manuscript neural obtain record simultaneously rr rr frequency hz
appeal sufficiently since lemma arbitrarily sufficiently construct alternate direction method admm rewrite constraint lead recently direction admm study admm solve lagrange note explicitly two subproblem subproblem positive semidefinite cone second therefore initialize section alternate multipliers optimal presentation let stand present global constraint convergent derive admm tackle population utilize actually recover initialize unlike stop criterion set analogous population block size matlab code definite obtain fortunately addition say sparsity semidefinite error admm false significant entry ratio entire plot covariance recover population covariance stand right recovered stand quite close place none deep green recover quite dependent ghz computation correspond ia produce give obviously value desirable method relatively addition decreasing spend addition small recover propose fix leave generate method bad undesirable behave variable initialized table time basically perform stable need regardless time structure exact population j q take dimension c c table distinct recover desirable cpu time identically even start recover admm initialize fix point produce point method increase basically start create rank solution compare reason probably low approximately semidefinite desire c acquire semidefinite utilize nuclear property illustrate numerical simulation time national basic research china cb national china remark proposition construction claim department mathematics state laboratory p china achieve convex encourage favor estimator large mild sample alternate tackle illustrate estimation multiplier population covariance matrix multivariate many decade advance technology year field brain imaging imaging deal massive setting covariance poorly fortunately covariance overcome traditional research matrix follow viewpoint determinant wise constraint utilize alternate challenging problem covariance essence contrast newly simultaneously matrix theoretical model multiplier admm go numerical experiment section last part hereafter denote occur covariance say write goal estimate unknown fundamental population xx denote support
use construct estimate regression task arrive sequentially carry update recursively extend primary local message become processor execute speed advantage communication delay substantial message pass offer easier include failure uncertainty stress vary datum costly scalable request whole dataset already organize regression simulate ease exposition proof sequence distribute generic e form estimate chapter return entity spread processor type moreover besides computation processor conclusion computation nonnegative coefficient constraint introduction stand recent assume represent type bandwidth computation processor update perform effective accordingly processor compute network communication vary advance agent processor counter need analysis may processor major asynchronous consensus bound come demand communication processor communication delay requirement delay adapt agent avoid agent account processor ia ij ti ti operation r requirement particularly restrictive effect update access assumption require communication delay prevent processor sometimes refer network topology direct describe reach entail communication hold vertex finite interval time infinitely nature processor mainly refined requirement processor I update processor computation position exist nonnegative assume addition bound avoid ambiguity comment apart distribution moreover type message asynchronous nice consistency centralized counterpart worker new arrive worker queue perform average worker perform worker value queue turn mechanism define queue worker call queue memory message happen next queue queue memory queue process start decentralized architecture communication dataset worker notice significantly procedure hour calculation processor minute overhead finally compute remark typical er ed worker theory detail relative compare basic estimate online estimate second eq gain non term estimation degradation negligible peak design gaussian design model asynchronous present ts hold initial specify case delay zero delay exploit linearity scalar determined general nevertheless property array assumption assume true tend exist exercise context agreement asymptotic agreement value instant assume processor stop asymptotically reach consensus depend imply agreement recursion stop start observation study ik ii recursion function role ensure bound r boundedness establish iteration observe obeys convention product view divide firstly secondly show induction reveal processor instant convention nonnegative condition recall news constant nonnegative real number nonnegative adapt offer version check consistency nonnegative integrable j eq therefore take side ft satisfy iteration proves secondly set recall positive eq hence lebesgue put piece together toeplitz lemma aim almost fact toeplitz identity sequel letter generic change universal constant fact conclude constant accordance series vanishe assumption provide proof thus deduce constant technical universal dominate aim condition exist time part proof almost enough upon recall euler depend tend right technical prove one vanish immediate consequence toeplitz similarly write follow let grow infinity section corollary proposition gray paris paris france universit es paris paris france com flexibility accommodate ever massive issue computation develop type asynchronous distribute involve processor excellent method computation prediction online pass parallel distribute currently area activity motivate example massive fit computer
natural proximity measure partly artificial setting forest classification many illustrate performance realistic especially introduce methodology forest adapt argue favor replace probability loss example conclude intractable issue thorough review highlight issue difficulty unable overcome alternative compare massive collection pseudo avoid name turn generic form require divergence exact parametric argument outcome approximation even interpret randomized version h accept n prior distance distance generate practice rarely version rarely make budget directly realistic low thing dataset force resort large tolerance proposal fact counterpart factor summary generally quantile simulated abc software pair generate distance function say modelling adaptation abc choice approximately frequency abc posterior summary want produce straight explain summary statistic subset evaluate criterion measure entropy bic criterion characterize introduction regression abc square distinct per se aim bayes abc goal abc type summary characterize selection crucial technique bayes acceptable one extend model pilot adopt compete linear combination summary consider separate select model versus time fit namely choice model stationarity numerical posterior summary statistic derive density estimation nn tolerance quantile deviation distance whole summary simulate distribution show hyperplane separation since model axis second replication time dot dot model parameter generate prior choice statistic produce approximation loss produce choice summary purpose powerful discriminate database learn interpretable nn observation state induce consequence namely framework sample abc table paradigm still prior abc computing posterior probability implement machine equally strategy methodology machine forest bag aggregate scheme classification mostly insensitive presence noisy large exploit towards selection principle behind rely extract maximal selecting rely relevant collection strongly correlate interest performance evidence forest forest establish intrinsic original additive notably forest adapt without description random forest randomness resample cart subsampling predictor node cart cart separation index contrary cart pruning tree random forest make category specific predictor default sub bag instance prove sub implement feature subsampling induce forest aggregate drastically abc number simulate instance computer allocate statistic select probability compute respective frequency variability often loose constitute thus widely forest abc autocorrelation summary replication simulated summary quantity agree rough still forest qualitative part contain evaluation quantitative vs insufficient produce remain far guarantee deal zero worse assess variability require one instead distinct confidence model abc bias induce insufficient face opinion endow variability well report method classifier whole prior easily evaluate bag forest parameter performance parameter wrong overall though irrelevant point replace unstable pair predictive index answer center area interest nan prior distribution observation x eq thus represent jeffreys decrease maximal strongly assess posteriori rather solely integrate average model rather completely intuitive quantity remain paradigm frequent population study random forest approximation core table already produce abc indeed easily rely produce near neighbor distance proximity random forest overall constitute distribution predict true since discriminant lda I marginal na I marginal abc forest error rate associate summary table provide size independent seven figure separation discriminate properly reflect abc really reach optimize performance move two seven statistic degradation justify achieve comparison performance error abc reference construct predictor nn procedure forest rate boundary model posterior evaluation bottom knn estimation forest abc posterior scheme new dataset display demonstrate predictive rate slight occurrence optimistic report rate point compare abc establish historical population population synthetic nucleotide snp respectively discriminate split change tree figure rely version candidate old population third illustration aim make consider simultaneously population range introduction release abc process massive snp production population compete scenario provide figure parameter choice identically model three population scenario leave model center split six population population correspond source prior associate population individual number additional population offer embed ht reference axis evaluate top meaning appendix illustrate first green triangle figure situation recent balanced strongly source population red figure unbalanced source population sample neighbor provide forest reference contain near neighbor index new pseudo random actual derive posterior rate green ever allocate wrong triangle rate simulation pseudo observe considerably case green triangle figure favorable case illustration snp per produce appendix table structure summarie ccc I gaussian summary abc axes forest initial forest summary two lda axis I marginal discriminant abc summarie axes axis forest summary forest axis I discriminant lda summarie abc axes summarie random forest axis estimate rate snp rate typical quite triangle snp abc neighbors forest summary error rate favorable ht lda axes color correspond index black model test additional green red triangle contribute lda occur role lda axis build nevertheless contribution important variable summary statistic lda meaning provide appendix power rf methodology challenging realistic aim pathway record north analyze include five natural individual marker compete introduction abc rf rf computation discriminate scenario dataset software marker plus nine axis additional summary statistic gap section summary model forest evaluate summary bottom add lda provide forest determine relevant statistic summary dimensional setting nn show summary strongly outperform classifier good axis able separate standard abc axes low forest summary axis ccc na I bayes linear discriminant lda abc summarie abc axis regression axis I discriminant abc summarie abc use lda axes local regression axis forest use summary forest summary I summary standard abc lda axes local forest summary summarie axis three evolutionary scenario forest strategy agree solely high respective extract forest show forest certainly another evolutionary assess posterior rate greatly assessment posterior rate equal new near forest summary lda axes table totally choose totally raise confidence probability close choose ht reference lda axis color index analyze snp population public genome database www project population sequence individual snp deviation snp discovery use discovery simulator snp include encode genome chinese china east encode representative encode population usa encode snp individual snp iii distance kb order snp iv snp equilibrium threshold population remove snp median two snps single population european east population independent give european one give second suggest study e possibility genetic usa european east rate origin stable bottleneck effective population size introduce scenario bring insight human population history study genetic illustrate snp context evolutionary discriminate among implement different reference table report summarie axis well present snp seven hour intel x context observe random construct summary table reference table forest report importance confirm lda able discriminate na I bayes marginal summary axis local axis summary summary lda c I marginal standard axis logistic lda summary I lda initial summary local summary random forest axis classification size table forest selecting consider population field surprising scenario population european east genetic origin european select high indicate rate equal forest replicate neighbor summary lda contain argue forest range evaluate single population population china six differ historical event single give european east population two event give possibility european scenario contribution forest important index summary axis mean reference color correspond index black red gold dataset regard model summary vs summary imply consistency discrepancy remain eventually focused abc concluding rely well propose posterior estimate average posteriori space weight solely integrate conclude forest forest substantially large summary suffer curse performance intrinsic require much effort abc allow reliable early argue statistical massive snp dataset production argue considerable massive production increase within use bin yu visit paris grateful former feedback support department writing help asymptotic forest part conduct research environment acknowledge conduct appendix summary snp proportion gene two nan fp population
map fully fashion state noun phrase sentence feature whose direction follow technique rate gradient learn real application depth edu intelligence international usa deep architecture nested datum cubic time sequence length prohibitive length propose cubic linear rao time semi chain crf suitable nest markovian child theory model semantic model free bound depth main drawback formulation inference outside cubic sequence appropriate nlp limit say slice technique handle contribution approximation gibbs cubic time linear depth idea nest across secondly step trick rao course price pay degradation give observation admit markovian simplified assumption widely transition markovian tend persistent assume transition element necessarily idea semi markov complexity segment semi element state detail exactly like chain phenomena nlp character phrase sentence paragraph chapter price pay complexity property parent new parent child child chain must text noun phrase say phrase noun terminate noun phrase belong parent child relation state topology depict topology respectively child parent parent topology exactly parent htb c review idea g rao see method conditional drawback converge rao rao possibly easy suppose sampling yield specify topological length dynamic multiple I start ii role iii child return parent begin emission activate bottom end termination occur bottom level continue top possible fortunately nest explicitly specific e stay suppose time collapse efficient implication exploit complexity length cubic let transition state observational q joint potential rao sampling time since essentially estimation p suggest gibbs rao integration without expensive fortunately omit reader result obtain full passing duration htb cc generate parameter topology model length semantic bottom level symbol generative first learn generate perform task rao introduce inside outside burn discard sampler forget burn want examine accuracy occur decode first estimation measure kullback marginal difference marginal decode use maximal measure kl decrease slow word mode cc htb c experiment run time iteration total datum many linear run depict figure divergence obtain quadratic cubic time log kl
symbol obtain solve formulation obtain separate hyperplane near follow hyperplane choose moreover hyperplane multiply software package quadratic efficiently extremely set moreover optimal hyperplane early illustrate figure reason offer attractive finding situation width pt pt pt circle pt circle circle pt circle pt circle circle circle pt unfortunately reverse vector separability mean discriminate undesirable introduce modification trading error linearly naturally raise appeal impractical alternate approach introduce slack norm vector slack distance feature call define x feature distance dual observation w convert z I problem suppose let replace w feasible reduction achieve separation rather optimization meaningful linearly aspect false positive false negative connection accuracy etc discriminant fx thus assign sample assign ccc c stand sensitivity specificity three lie convex specificity ac se se ac letter lead star star decide star star patient cancer elsewhere brief patient cancer spread group node size tumor possibility half half star micro star micro independent specificity extremely precisely want contingency exact less machine biology vast besides model tumor branching emphasis topic well biology propose induce select set lasso point section available penalty overlap biological master advance immediate application biology base assumption isometry perhaps something order technique compress sense cancer biology modify measurement rip choice orthogonal actual author handle cluster theory attempt promise lead order far development besides orthogonal measurement matrix connection pointing rip achieve compare whether matrix rip sparse classification data justification norm svm guarantee certainly star would medical tx introduce biology turn student support report perspective well sense advance recurrence present briefly open objective advance discuss biology cancer cancer lead death united uk case figure lead million develop usa uk challenge researcher one cell cell come cancer broadly class optimistic accurate patient patient within group cancer consideration appearance tumor measure tumor decade attempt collect central cancer collect massive public project vast amount tumor standardized protocol amongst international country molecular almost clinical annotation become useful contribution cancer need course would picture advance mathematically couple problem facilitate machine biology understand computational aspect molecular biology analyze hundred genome study gene probe sometimes gene correspond amount rna raw transform take logarithm divide subtract divide number sometimes good gene expression feature micro level variation value presence absence mutation specific molecular tumor depend tumor recurrence drug patient beyond site addition ordinal medium ordinal value attribute previous correspond tumor recurrence treat one simplicity real value binary row th throughout handle two namely binary belong take label simplify sequel whereas approximate would prediction tumor consist level thousand around together tumor value lead reliable recurrence cancer disease highly particular key constructing use scope end tumor recurrence throughout regressor weight bias restrict attention regressor regressor understand regressor biological feature possibility still process explain early process raw regression eq gauss everywhere rank ls context matrix less least impose weight constraint lead problem formulation treatment topic least norm unique ridge however would ridge reformulate become lagrange multiplier block equal definite lagrange major ridge every component nonzero function make undesirable nonzero regressor familiar quantity component common even though find minimize analyze square interest simplicity discuss generality alone weight minimize multipli shrinkage depend multiplier technical moreover correlate biological pathway vary sample therefore column correlate case ridge assign nearly extreme amongst discard choose biological level gene pathway correlate undesirable discard rest undesirable would column call elastic net penalty choose elastic net whereas become ridge elastic provide bridge elastic elastic elastic minimizer center suppose always nearly many net number number often see desirable feature correlate biology open biological pure try choose distinct group regressor select element possible group determine clear depend relative size component yet group limit consist singleton group reduce standard variation group group formulation overlap biological decomposition gene acyclic wherein master pathway seek choose gene rather pathway master master gene namely ii pathway g ideally feature intersect pathway example present circle minimum size draw mm draw circle circle minimum minimum draw circle circle circle size draw mm circle mm draw node mm draw size draw date sparse pairwise penalty augment refer suggest sparse overlap continue penalty date address see group obvious hold namely retain g modifying hold reason nod clearly ii thought reveal structured group really truly overlap sg maximal together set biological sense every pair define induce type consistent new application elsewhere choose nothing difference reason unlike en several example en lasso relatively algorithm unlike en assign weight far remain carry cancer predict tumor recurrence tumor recurrence well patient day exclude analyze feature elimination figure percentage error value h elastic year group sense compressive determine take area broad decade compressed find paper summarize I paper randomness motivation discuss sense biology search cancer stand apply fundamental compressed sensing call paper note refer biological biological area ignore hope argument application treatment compress sense development area name several make contribution paper survey reference reader begin introduce give n k kk nk x z quantity underlie large component replace kx c isometry rip introduce rip property easy exposition rip suppose isometry rip hold column row column integer sufficiently rip suppose state know compressed compare rip z write differently rip corollary follow c corollary simplify near ideal lasso oracle compute oracle appropriate reduce constant mean universal bind achieve rip randomized sample construct advantage ensure namely simple proof rip construction rip result fail nothing rip measurement hold tail tail nothing hold essentially objective constraint raise question whether example place result display ideal behavior lasso analog replace joint author effect decomposable rip describe imply
c cauchy schwarz inequality satisfie restrict event combine result tucker example follow simplification restrict event probability lasso consistent theory always case event instance get prediction q last fact u j cf f minimum evaluate independent j indeed symmetry inclusion inclusion yield hand variable f j j entail view cauchy schwarz complete proof remain bind resort ct n j jj j proposition infer case x triangle n x hence put together eq case conjunction least replace replace achieve f k observe close f k j j l conjunction piecewise combine l v inequality far get relation x j imply complete appropriate j v infer u u belong supremum right side q foundation support grant corollary red corner corner pt france ny understand insight incorporation measure reveal moderately correlate performance irrespective illustration deduce considerable effort devote guarantee performance understand common already setting prediction understand gain broad avoid prediction identically read noise matrix exposition simple restrict gaussian noise fix extend conditionally covariate even magnitude tuning amount penalization crucial influence typically unique irrelevant follow strict convexity minima numerous lasso universal conjecture correlate span relevant lasso one correlate suggest number convex hull difficult incorporate computable covariate really small fixed rate irrespective correlation covariate rely assumption covariate low isometry compatibility bound irrespective tuning depend true vector follow extreme covariate mutually extreme permit mutually covariate attention topic loss explore throughout every respectively subset removing set belong transpose u equal subset denote span orthogonal onto n n asymptotic write sequence present show particularly impossible rate even small devoted sparsity inequality compatibility imply total establish quantity accounting within developed allow second fourth question introduction discussion place outline question state concern relationship covariate fast fast begin discuss literature relevant x impossible much small may signal part inequality inequality error loss value vector consequence note appear stochastic least square covariate vector lower usually recommend choose prescribe tolerance demonstrates state cardinality covariate span trivially follow third chi conjunction tail latter fact unfortunately oracle assumption surprising formally integer less canonical avoid unnecessary rademacher reason show covariate rate correlation prevent third fail counter analytically clearly fast without assumption rate note advantage complexity oracle inequality literature state discuss learn combine improvement compatibility fix parameter satisfie reflect front infimum equal refined lasso particularly consequence value get mention intermediate parameter satisfie np inequality frequently refer covariate provide rate order situation available design subset vanish span compatibility every entry compatibility weight compatibility factor measure relax previously lead fast fix tolerance main difference presence replace compatibility factor always quantity think covariate justification tune cross validation choose denoise processing penalty employ similarity problem grid noisy unknown value grid tv f penalize square hereafter tv despite popularity surprising tv asymptotic setting establish n f optimal piecewise achieve instance penalty vector reference whether rate eventually minimax notice tv jump satisfactory tv differently assess jump raise question oracle small achieve c jumps allow increase almost exhaustive entry penalization completely application example fix cardinality expensive risk circumstance index least adapt proposition fast strongly tuning order small deduce rate however proposition considerably signal denoise total penalty previous study tv piecewise tv monotone signal review topic context find find monotone tv risk bind apply estimator mention early tv assume span n nh nh assume interest gaussian k f vector tv crucial minimax tv dominate minimax logarithmic conjecture logarithmic require tuning parameter proposition avoided avoid measure total roughly mis specification monotone hold old function almost minimax particularly old theorem exploit know achieve minimax logarithmic set old f fx tv theorem improve achieve set n l estimator proposition tv risk nearly differ important benefit finite sample inclusion model specification risk mis specification constant use trick mean stochastic satisfying include identical column vanish column vice versa compatibility factor well idea tight theorem less however really substantially one factor remark lead produce mis specify u matrix proposition tell mentioned proportional slightly differ c generally contain look characteristic supremum outside amount c repeat satisfy ct spirit consequence rate design comparison b compatibility furthermore advantage risk bound lead choice rate prediction compatibility factor replace compatibility factor compatibility vanish moreover benefit necessarily rely covariate reasonably small necessarily small expect refined respect potentially flexibility parameter factor front another explore discussion replace correlation achieve replace strongly show lasso competitive procedure explain loss design contain column furthermore believe strategy may conjunction form theoretical section suggest set tight aforementioned consume alternative replace sparse indeed close span find therein understand compatibility factor factor nonconvex guarantee subset moderately large remain interval program parallel involve belong solve set note right gap divide least problem
function initialize smooth since give boundedness result vi ix great x c initialize approximate specifically greatest bind prove prove desire solution approximation establish boundedness resemble convergence behind restrictive prove remain bound presence solution control aim pursuit tolerance positive achieve vi limit arbitrary exact vi cite proof pointwise convergence small achieved denote online lead considerable load online operation use approximate minimization operation denote actor introduce next provide asymptotic subset region approximate approximation bound actor hold asymptotically compact r x n ix appendix actor definite numerator right hand positive require q positive function great inequality lead definite determine actor practice validity capability upper example checking validity lipschitz constant examine train actor find unless pointwise value actor former utilize union find stability admit termination approximation numerical analysis around circular state stay real frame motion field xu total force reference angular velocity control time step velocity velocity force dynamic discretize utility hand x ref prove successive complete iteration enough converge actor due smooth basis find integer actor respectively iteration denote relate policy denote actor repeat make multiplying element follow function random vector I learn select tolerance guess converge inner respective element versus take ghz gb memory windows actor shoot select may evaluate state function goal converge respective pointwise cx utilize generalization per boundedness converge exist reliability stability controller verification approximation vector quantify p appendix respective accurate consider actor approximation control turn negligible evaluating turn exceed bind remain upper stability initial result trajectory comparison loop direct see controller accurate besides go go loop turn cost go approximation suppose upper go optimality simulate ever therefore control however obvious desire find analyze select evaluating converge therefore utilize guarantee stability select belong vanish origin conclusion appendix controlling need pick sample state select discuss observe value lead actor expand origin long guarantee still get good result fig present improve network option rich rich basis turn condition extremely fig similarity conservative condition bound probably loop controller analytical dynamic observe finite stability comprehensive numerical fourth problem foundation improve mathematical field inspire value cost define k trajectory respect associate function otherwise sum trajectorie summation hand control summation control eq let prove right convergence summation evaluate trajectory max sup side note reason finite finite qx along idea boundedness per respective consider q note equation actor follow smoothness compact side asymptotic stability holding hold equality origin one consider hold positive serve lyapunov asymptotic follow long trajectory remain inside leave relation stability concern estimation definition inequality trajectory also set close namely continuity upper boundedness engineer south rapid city email aim answer consider next deterministic investigate error boundedness control assumption approximation optimality condition system result obtain iteration along derive approximation process make development verify stability dynamic rl researcher approximate mathematically great potential popular crucial cite nonlinear due phenomenon happen regardless would lead vi challenging appear good author function factor becomes investigate interesting utilize restrictive easily error write e uniformly denote approximated involve reader study success value include analysis aim pursuit mathematical control develop boundedness approximation check investigate stability term parameter presence error possibly iteration deviation instability outcome system derive concern trajectory controller note train guarantee inside however remain domain controller remain valid use reader vi guess stability lemma vi th iteration denote side approximate represent value hand difference cumulative iteration approximate vi mention train actor remove conclusion result actor actor control directly minimization apply actor even though actor simultaneously learn offline system result stability system due convergence investigate suitable operation minimize
interval motivation weight constant empirical distribution statistic cdf theoretical term kolmogorov distribution case approximated remark tend infinity instead appropriately center objective deduce mathematical practical output primitive distribute weakly without change used reference observe depend ks close monte generalize classical since statistic cdf value depend weight evaluation course gene pose false discovery fdr correction test test asymptotic apply carlo cumulative conclusion gene beyond necessary output test gene set database value gene set initial tumor typical make encouraging set whole mathematical algorithmic advantage test could informative organize devoted begin description monte test comparison notation throughout convergence bridge q law variable ex primitive obviously converge weakly brownian supremum continuous p index induce apply yield q distribution term motion primitive deterministic brownian gaussian easily covariance explain process notation let standard conclusion code implement make manual regard essential distribution else first review integral differential interval hence integral sum increment simulate gaussian simulate trajectory absolute return write simulate increment motion sequence discretize trajectory proportion propose return sided estimate conservative state cdf weight relation validation kolmogorov explicit estimate close curve discretization turn difference classical sample test ks panel discretization simulation value uniform gene logarithm ks plot p value stay test theoretical asymptotic cdf tend conservative set clear theoretical simulation case replacement negligible simulated extracting replacement agreement asymptotic precise gene repository test set use test dataset annotate give name apply gene gene symbol common vector two tumor red triangle base display comparison sake raw fdr adjustment mark dash vertical horizontal line significant cdf simulation monte limited line due monte carlo observe globally coherent corner graphic never converse point corner correspond gene set analysis gene gene significant gene compare center appear gene significant step function curve far enough global significant moreover already deal significance conversely gene significant clearly compare gene tend name letter gene cancer test gene state observe type detect among new testing proportion center convergence correspond kolmogorov kolmogorov major depend gene therefore calculate gene value conservative increase set compare test whole tumor gene encourage raw good precision rank agreement yet version gene proportion ks side test sign difference contain small conversely positive together manual following encourage tool biological pt treatment center derivation center carlo simulation kolmogorov fact asymptotic serve short repository comparison test new conduct mathematical informative test empirical weak
laplacian correspond connect minimizer eq nk achieve solution algebra nonnegative multiplier nk active first kkt inactive last correspond one n z kp graph tp separate component computationally separately condition mean practice category small enough arise entire never attract dominate take assignment practically strongly mean assignment towards simplex solution practical seem let substitute kkt trivially condition th item category expert assignment correspond category similarity qp condition kkt condition lagrange equality constraint kkt linear multiply q respectively column multiply left matrix inversion solution transpose verify formula simplify index number describe guarantee take direction multiplier combine alternate direction qp q positive semidefinite admm new replace indicator function q eq constraint lagrangian multipli constraint apply immediately admm iteration simpler obtain combine indicator function nonnegative entry take part update apply modify euclidean penalty lagrange estimate equality qp kkt condition lagrange multiplier iteration update crucially convenient direct complement leave multiply equation eliminate substitute first admm linear primal guarantee admm qp identify q single simplify considerably basic reason identical copy graph sparse constraint matrix follow factorization reduce fill much iterative solver initialize warm fast inexact write laplacian cholesky factor system convergence primal multipli triangular cholesky factor conjugate gradient available implement require line search converge affect iterate update eqs upon multiplier kkt upon k kkt eqs finally kkt change sign lagrangian subtracting stop estimate respectively particular project inactive weakly value except make add setup also sufficiently practice since high accuracy runtime use graph cholesky factorization take factor solver research subproblem cluster warm outer loop iteration initialize eqs k k initialization simplex next fact arbitrarily try system q since simplex project independently stop last iteration fall tolerance relative update stop runtime criterion test condition satisfy lagrange multiplier iterate need satisfy iterate infeasible project assignment simplex category iteration select fast eigenvalue laplacian relative asymptotically long short small c relative l provide assignment error scale iteration implement assume update nu li n r k nu minus nu break end z find assignment item category assignment reasoning could augment consume assignment point practical natural mapping augment free point dropping reduce quadratic program w point thus probability finite computationally sparse neighbor large use hash retrieve neighbor receive simply reflect rather training describe may relation different constraint unlike prediction solve fundamental supervision actual label item give assignment would valid assignment constraint redundant particular label item label first nonzero multiple category tag category partial category rest simplex give entry implicitly forces summary close assignment directly able transform assignment use perhaps smooth user force category obviously force wrong poorly would entry leave free similarity mapping coincide mapping addition give similarity coincide particularly informative nonzero small item learn nontrivial achieve indeed constraint semantic frequentist assignment proportion belong portfolio portion example include classification restrict belong category exclusive assignment may interpret assignment long history operation economic portfolio seek qp however laplacian model different apply assignment training laplacian item centroid entry w nk g k cluster point fix particular mark define similarity location label ccc positive label cccc cccc diagram ground label relate diagram category contain category intersect quite usual graph denote partially circle unlabele true give crucial propagate unlabele partially exactly minimize fig bottom unlabele obtain unlabele assignment smooth assignment sometimes assign wrong assignment ground truth easily label task sample digit similarity image category give near nn machine width select code assignment unnormalize improve valid assignment lie simplex category respective optimal leave unlabeled avoid clutter point template method use outperform runtime produce assignment topic manually topic pc hardware mac hardware topic belong construct occur extract inverse feature document randomly one topic five parameter optimally mnist document topic high assignment actual predict truth fully label penalty classification improve outperform nearly assignment benefit handle predict full assignment category tag test fill assignment propagate unlabeled sample assignment provide demonstrate select category word non empty category category image build partial give category randomly frequent providing stop unlabeled categorization select base precision f sample although tag similarity tag fig show versus svms sample category nearly always see improve precision recall white green white black draw white old computer window window ex draw old white coin white precision vs fix annotation sample tag combine complementary source crowd expert attractive impractical category complex structure item category incorporate supervision laplacian similarity direction multiplier iteration make factorization laplacian implement search rate represent similarity item since exist application mapping nonparametric affinity accelerate train cm prop thm observation example wang electrical science give encourage similar assign category item item encourage intuition give unique effective multiplier reasonable generalization learn particularly naturally belong multiple keyword tag major many rely label unlabeled laplacian construct nonnegative measure graph exist base formulation conceptually solid foundation graph laplacian widely vision area mention image spectral use surface iterate product laplacian concern soft assignment give take tag annotation consider conference keyword likely extent keyword tag paper paper science redundant large include biology paper keyword besides category assignment although extent inclusion tag item certain patient code category indicate degree item also consider source example bag capture laplacian example source imagine conference paper author author bioinformatics send word predict
bilinear operation vector multiplication slice tensor entry include add tensor part fine grain sentiment intensity class ordinal nature ordinal network intuitively sentiment belong sentiment belong belong class therefore otherwise vector cumulative nonlinearity traditionally multiclass firstly entry assign entry low whenever proper assign purely unit hidden tensor two denote result eq unfold layer hide individual simplification extract vector rather greatly instead word vocabulary extract matrix recurrent neural recurrent tensor handling correctly shift sentiment interpretation follow instead learn operator word word one operator natural word low besides space one learn vocabulary unsupervised would allow use manually annotate level extract convert annotation integer ordinal label denote sentiment multiplicative improve development suggest indeed powerful careful early initialize initialize vs share parameter word limit preliminary word vector embedding yield difference test experiment identity consistent improvement extra nonlinearity identity significant however suggest boost nonetheless nonlinearity marginally explanation nonlinearity nonlinearity crucial stanford sentiment baseline bottom show rnn conventional baseline network structural far recursive bad matrix recursive tree unlike variant explore recurrent compositional interpretation fine grain sentiment ordinal previous previous stanford sentiment tree effectively separation representation vector extend supervised slice interpret simplified word mean share space affect word matrix hand sentence tensor well update towards drawback rnn increase explore word vector would every operator act acknowledgment work grant fa view conclusion herein author interpret imply present multiplicative compositional meaning fine grain sentiment investigate case recurrent well recurrent outperform fine grain sentiment recently publish stanford sentiment generate recent network deep nlp numerous nlp natural properly large phrase dense nlp require properly approach compositional vector act assign representation long phrase simply matrix english parameter vocabulary phrase semantic structural composition operate child sentence represent different way lead recursive neural network nonlinearity use aforementioned matrix composition recently bilinear tensor multiplication composition capture recurrent network rnns neural capability phrase act memory result composition layer next phrase sentence accommodate suggest neural additive theoretically nonlinearity end compositional semantic level capacity sentiment sentence represent space rnn recurrent nn successful multiplicative sequential computation replace rnn compositional sentiment space multiplicative rnn represent instead dense network representation suited scheme recurrent semantic act view recurrent neural network show rnns comparable performance net fine grained sentiment detection absence tree put burden multiplicative rnn reach variant single token per token vocabulary high dimensional equal syntactic semantic similarity representation token value dense usually dimension generally representation learn unsupervised corpus wikipedia embedding capability might employ vector representation input word semantic transform apply partially idea act noun vector word representation long phrase compute generalize space plausible fine grained sentiment vector capture transform apply bag word wise multiplication composition ignore compositional distributional semantic semantic argument composition structural particular recurrent neural consider hidden phrase one determine tree space sentiment active nlp various word level try formulate grain approach explore ultimately task addition bag incorporate account contextual transform represent successive inside phrase word representation representation model constitute multiplication score assign representative degree freedom recurrent rnn network recurrent connection
require alternative representation let bag representation centroid centroid codebook bag bag centroid arguably distance use achieve histogram bin specifically two solution describe mass distance bag word euclidean centroid distance metric limitation computational al regularize magnitude al simple iterative iteration algorithm scale multiple histogram efficiently compute hardware covariance use neighborhood covariance classification compress drastically md asymptotic eq training copy input neighborhood assignment radial basis randomly near cross inspire input classify compress I I imply compressed prediction kl ideal minimize divergence compress set ensure cholesky j j substitute single j I mp gradient matlab http histogram descriptor aim compressed histogram dd neighborhood compress correctly via compressed share label kl perfect histogram r introduce challenge optimization remain optimization nest problem formulation eq histogram occur correspond dual strong duality primal dual formulation identical optimum dual consider fix simultaneously descent ensure simplex normalize change histogram positive jk complexity p gradient update select compressed error person object face material categorization scene real nn six benchmark image background object camera place descriptor texture resolution image surveillance individual therefore wide ratio filter class result image face gray face individual orient angle image popular ratio covariance cloud view object covariance information white set sift descriptor sift bin horizontal direction orientation bin produce descriptor via transformation select size hyperparameter dataset material material material pose condition cnn rnn allow surprisingly covariance reduce amount subsampling essentially learn versus label agnostic set descriptor discard speedup various ratio dataset mark learn exceed match nn remove neighbor gain increase error compression speedup describe compression minute compress get amount compression compression contribution potentially parallelization ccccc recognition texture classification technique histogram histogram compare accuracy achieve describe consist object background black take histogram follow extract context sample histogram log shape white follow extract shape histogram ground bin texture dataset surface feature extraction use bag representation sense distance bin pair error neighbor similarly parameter definition initialize appear outperform subsample baseline dissimilarity compute centroid centroid accelerate centroid compression descriptor text detail average deviation compression ratio dataset outperform match compression speedup classification compression possibly match throughout final compressed set rnn compression ratio lead arguably interesting baseline full magnitude bad time table minute compression especially solve histogram believe hill set reduction consider nn primary sampling iteratively add reduce perfectly technique neighbor notably rnn search result cnn additionally find cubic prototype generation create training set cluster prototype appropriate propose compression finally al algorithms network set histogram descriptor nn speed distance computation bregman ball tree euclidean tree bregman divergence technique onto bregman ball compression devote toward bind efficiently distance bin reformulate add unconstrained amount previously compress drastically neighbor large corollary university st abstract absence sufficient datum descriptor influential descriptor ii descriptor histogram computer gradient image may result visual descriptor definite diffusion suit task nn histogram neighbor histogram descriptor individually bin euclidean distance bin wise dissimilarity descriptor lie convex half embed straight euclidean underlie systematically descriptor incorporate distance improvement nn versus distance costly cubic specialized distance operate constraint require prediction geodesic manifold eigen decomposition need repeat classify test ball dimensionality near many method match classifier original contain explicitly
activation write weight via multiplicative activation accomplished drop individual unit denote scale gaussian test require scale bernoulli sub sub connection exponential visit however extensive sharing make regardless explicitly dropout inspire variety subsequent dropout version weight axis align take mask recognition maxout design new type activation function exploit dropout procedure order interaction among input hide variable connect multiplicative structure relationship input structure recent include network update input output fix formally activation factor input vector activation special z e en feed forward factor neural output multiplicative decompose multiplicative formally feed forward instead l n l hide activation bernoulli comparison feed weight dropout unique represent dropout test time mean secondary input define mean n feed effect learn learning order unnecessary base method neural convolution convolution decompose convolution operation multiplicative noise convolution filter network find restrict restrict collapse single layer maxout million free million cccc constraint relu et bernoulli dropout maxout dropout relu unit gaussian cifar dataset contain color training remain dataset cifar contain mnist cifar convolutional layer layer image preprocesse global contrast whitening maxout result cifar cifar parameter small consider mnist expect optimize tool hyper conv dropout fully conv conv net maxout conv conduct separate dynamic hide factor prevent overfitte continue decrease test linearity experimentally verify testing visit test output procedure demonstrate geometric though procedure relu arithmetic mean appear geometric investigation link weight figure depict evolution norm imply weight factored actually learn penalty dropout seem impose implicit essential network traditional form capacity early decay currently mean amount aggregate apply interpret allow testing make dropout order magnitude additionally restrict loading control recent closely redundancy parameterization architecture weight matrix way parameter c regularization neural network sample poor unseen partially responsible win entry university currently method efficiently number aggregate learn noise especially win entry imagenet challenge use network competition top partially attribute regularization dropout crucial dropout elegant solution number model predictor generalize test relu percent zero activation dropout although training multiplicative activation add relu activation distinction similar dropout method ensemble robust visit briefly review generalization discuss advance dropout connect introduce multiplicative
structure eigenvalue external external exponentially structure detail leave search use estimate remain fixing adopt estimate scalar argue search direction make sense prefer choice regularity already currently try describe span precede side explore confirm tend spectrum show generative process haar measure figure uniform exponential pd draw top eigenvalue elaborate construct estimate middle row figure stationary multiply estimate freedom nonparametric square scale error external external external external eigenvalue external external cg plot eigenvalue external exponentially eigenvalue external standardized norm exponentially external structured external true gray error estimate black bfgs standardize cg demonstrate example gaussian ph w mh bfgs cg method show row figure row far equation marginal gaussian show uniformly quantity estimation rule bottom bfgs standardize prior construct cg arise structural overhead much cg implication method cg remain open question member inconsistent nonparametric formulation nonlinear question regard covariance finitely result remain early appendix map see observe bs prior establish linearly element q element search rectangular generalization lyapunov apply present show constructive part full invertible write q write note eq get bs sx equation clearly similarity covariance establish simply ns algebra complete eq notation element gram matrix ib simplifie similarly simplify equal choice bfgs h search search search estimate irrespective invert inverse last symmetry I sum eq j complete acknowledgment like discussion manuscript particularly grateful discussion basis addition author comment anonymous particularly point ab da draw black white rectangle draw fill black white draw black fill minimum pt sep draw manuscript propose framework estimate gaussian belief conjugate gradient conjugate gradient novel quasi foundation optimization quasi probabilistic solver unconstraine computational equivalent minimize bx iterative solver address step big bar put estimate quasi newton widely derivation method extend maxima probability covariance offer interest estimate entire space family direction evolution occur widely bfgs rule part optimization include early less rule broad among subsequently refine update formulate cite update current estimate hessian call newton equation rule update inversion rule estimate interestingly bfgs update exchange inverse bfgs rule bfgs confusion text sense inverse sense update exchange make explicitly bfgs probabilistic objective ask kind assumption rise contain derivation rule symmetric rule natural perspective another extend problem cg idea closely cg bfgs estimate contain list search method class identical provide variant future aspect newton evaluation gradient certain structural restriction observe statistic probabilistic encoding assumption measure newton method new method map arise inverse point introduction author rarely attract numerical mathematic argument sometimes numerical randomness distinction lack arise precisely apply probability deterministic prefer another point practitioner apply budget numerical helpful point instability save say exact answer ask hypothesis far attempt answer linear problem point popular extend construct posterior construct element interpretation provide independent nonparametric particularly elegant provide prior symmetric frobenius matrix norm member bfgs member consistent probabilistic rule lemma sr give calibration bfgs cg lemma gradient computationally convenient parameterization uncertainty picture particularly calibrate family possible covariance construct finitely motivated regularity overhead conjugate gaussian probabilistic area brief text hypothesis real linear bayes posterior distribution prior dirac crucial operation bar mean link maximum quadratic probabilistic quantifie specific iterative solver shall maintain direct access inference kronecker matrix kronecker element observation solver gaussian quadratic kronecker update search probabilistic much frobenius norm frobenius w w match central role call weight rule family show frobenius regularizer base domain noise gradient dimensionality introduce update restrict class matrix involve derivation consistent framework aspect essence insight previously begin use linear act appendix ij e g element act effect carry symmetric example inversion considerably appendix linearly q posterior step member consistent unchanged transformation turn central define kronecker rise popular method argument favor applicable family rule solver mass theorems consistency solver arise gaussian general structure perfect correct search remain linear within direction hold choice long course good crucially definite desirable cone normal distribution prior cone statistic irrelevant normalization constant conjunction various sided find linearization wishart match second wishart hypothesis individual desirable choice equation one implicit give globally posterior choice manner sr bfgs implicit w ss analogous circular try part sr exception involve estimation adaptation data apply update ignore old direction inverse obviously include cost show though simplification external scale experiment thin line experiment thick spike cause ill condition arise full update sequence conceptual experiment definite generate exponential plot haar measure give projection draw uniformly symmetric bfgs equal one frobenius normalise little intuition exact application rule track consecutive make big relate full dominate hessian apply estimator construct show track gram probabilistic step perform qualitatively posterior covariance use gray line frobenius conceptual uncertainty close solid gray scale dimensional property bfgs uncertainty understand sum inference ratio error calibrate achieve apparent calibrate diagonal confident estimating confident element unit course still vanish e matrix fix degree prior covariance smallest sr observation subsequent one exactly away suggest investigation address possible construct calibrate thus error ideally major increase rank proof gram repeat update classic implementation result equal exact equivalent statement inverse update conjugacy analogue optimizer choose citation put bfgs definite search inverse gaussian belief analogously bfgs posterior establish inference cg cg starting choose pre define bfgs inference mean conjugate bfgs intuitive probabilistic framework bx mm cg bfgs gaussian probabilistic problem cg compact iterate inference b conceptual open extension cg thing direction gradient scalar prior bfgs property conjugate gradient converge fx x orthogonal gradient span span establish cg extend multiple rule bfgs cg establish interpretation cg solver collect extremely popular look calibrate obvious uncertain linear solver reasonably addition scale bfgs implicit cone matrix additionally
obtain marginalization accord recursive degeneracy obtain collection proceed alternatively operation weight degeneracy online nmf model smc sample bb calculate q smc see current make model contribute calculate exact online horizontal correlation need algorithm horizontal line present online em nmf observation exact smc separate process reflect generality I e formulate dimension realistic particle smc implementation sophisticated proposal smc improvement handle dimension formulate markov nmf sequential monte approximation sensor drop collect record amount raw arguably computation effective set meaningful scientific financial political purpose unfortunately classical deal restriction slow large give matrix computation factor make inferential goal precise analysis ica nonnegative matrix semantic indexing understand problem understand probabilistic interpretation probabilistic generative derive posteriori advantage interpretation enable incorporate consistent building fit probabilistic natural online pass algorithmic idea generalise tensor see formulate nmf maximum hide model hmms asymptotic hmm nmf online nmf smc decrease particle smc algorithm propose computation approximation convergence present formulation notably nmf dirichlet allocation view incremental learning empirical th element multiplication element division set nonnegative capital letter letter hmm comprise parameter divergence formulate formulation problem static random constitute therefore formulation mle change expect w law necessary first online sequence maximal surface choose calculate intermediate distribution respect estimate intermediate update find nmf calculate expectation belong family update calculate updating depend law write sufficient statistic form recursion expectation sufficient explain sense assume eq intermediate filtering regardless reason first z z sum recursion nonnegative value recursion claim verify induction start tx z kt z eq bm derivation estimate average sufficient
choice parameter succeed success repeat random iteration suffice terminate uniformly ball isotropic finally termination algorithm unclear excess extend tight differentially loss processing generic private carry output ensure differentially private run next exponential mechanism excess technique pure extend naturally gradient achieve excess carry localization perturbation algorithm htb perturbation parameter parameter algorithm give differentially let generic optimize theorem strong convexity convex run input radius output input output differentially differentially differentially private generic expect norm noise accord gamma satisfie hence hand private efficient version risk formally bind follow guarantee suppose replace efficient output derive low decomposable p lipschitz whereas lipschitz useful incur differentially marginal sake appendix marginal nd take differentially private clearly linear nd I nd excess low private algorithm must nd give differentially private let nd nd sake every contradict lemma exist paragraph p bind differentially algorithm random nd nd counterpart factor loss big nd observe hence lipschitz computation lipschitz lipschitz convex restrict existence clarity completeness sake closure literature know extension suffice remain show minimizer w convexity probabilistic discuss htb log concave guarantee cube fu u pf cube moreover conditional distribution p output vertex locate origin inside isotropic position cone cone cone integral less integral region inequality second prove output least denote event induce good good note efficiently computable exist membership membership efficiently lipschitz enable polynomial run perform step structure run directly p use boost multiplicative concave sampling bound multiplicative output unit since isotropic output denote condition I e q linear use finally plug expression give fairly version algorithm run algorithm decomposable loss isotropic namely differentially private input exp efficient concave convex dataset privacy guarantee yield differentially let let denote differentially output differentially lemma straightforward differential privacy hence private respect factor hence follow finally observe put complete grateful concave particular penalty reduce cube lem lem lem lem lem lem edu support systematic investigation private matching erm contribution lipschitz bound bound provide match lower know strongly convex separate differential surprisingly technique algorithm simple work contribution smooth apply previous previously commonly empirical erm function datum erm information record motivation bound erm contribution build start draw universe close goal q map define obtain variant restriction end collection datum point example svm formulation capture erm add solution fold regularizer dependent function replace come generality lipschitz affect ex know glm statement success namely output expectation convert technique see appendix another excess measure imply upper convergence idea range statement appendix know erm fitting square significant actual loss point minimize helpful median reveal one subtle svm whose consist erm attack notion work theoretical differentially output motivation discussion role several basic differentially private dimension constant diameter constraint rescale rescale replace risk rescale always convert excess bound general multiply replace p c assumption rescale get multiply simplify art loss asymptotically principle technique purpose section factor achieve match factor excess always match convex previously well know risk general function strongly convex several different restriction perturbation tight apply include case smoothing function expect risk descent well variant data compute estimate update appeared previously investigate noisy variant lie randomness follow without privacy step desire excess base gradient measurement risk bound knowledge excess risk privacy obstacle privacy step use net probability op low privacy inefficient since achieve excess excess risk efficiently continuous log technique require provide privacy issue work define appropriately cube output sample multiplicative use subroutine correct statement base technique function however technique mechanism yield strongly sensitivity minimizer hence release strongly euclidean strongly convex first estimate output optimal roughly define running mechanism improve factor mechanism n localization privacy take convexity quickly develop bind way privacy essentially convexity constraint ball hypercube objective bind nonsmooth smoothness privacy set nonsmooth effort design nonsmooth generalization comment implication unseen generalization c necessary necessity generalization error lipschitz modification privacy error roughly root however special generalize match generalization polynomial gap mention work rich seek characterize private bound regularizer unconstrained paper orthogonal though implementation mechanism domain vector obviously differentially study tailor also role development query release completeness additional vector bound tangent smoothness denote vector efficient differentially loss lipschitz discuss localization derive efficient sampling distance arbitrary convex bound private detail supplementary nonsmooth loss technique appendix localization modification guarantee generalization differentially private gradient descent literature utility rest randomness failure analysis guarantee run instead use complete guarantee factor gd lipschitz tb output step randomness variable randomness condition differential privacy addition see randomness randomness randomness least ensure privacy differentially ensure probability least privacy loss differentially private te gd expectation line notice randomness randomness guarantee descent tt excess bind strongly convex let gradient e assume computation take current idea batch sample oppose excess guarantee remove constraint tight privacy use empirical loss section mechanism show exponential major mechanism efficient private excess base guarantee differentially notice utility guarantee risk expectation risk every outside analyze individually argument risk machine allow extra risk boundary figure
sentence assume terminate token system become sentence way convolutional network long architecture elaborate neural work encode input produce sentence initialize generate neural translation rare problem accommodate word however name find similar beneficial despite machine translation address notable exception address rare track unknown sentence source word responsible word could source dictionary know unknown token word unknown token identity apply token easily annotate one unsupervised next link dictionary post translation pair en fr en annotate example token multiple target language oppose token assign repeat annotation elaborate word align source token model target word alignment align token annotation translate token translate word former sentence happen model tend vocabulary limitation motivated annotation include sentence single universal token token indicate denote align source annotate token en fr token sentence speed post sensible word motivate token simultaneously align source alignment universal token language annotate en de annotated token slow effectiveness task quality sentence comparable model corpus use intensive naive softmax vocabulary target side vocabulary use frequent english model treat alignment berkeley setting discard sentence exceed token hyperparameter cell embedding word source lstm memory summarize rate begin mini normalize gradient exceed gpu parallelization allow achieve source training corpus neural neural feature list end rnn k rnns end single lstm single lstm lstm lstm k ensemble lstm lstm ensemble system differ architecture size corpus either sentence text handle rare vocabulary improvement process accurate model origin word report score consistent work english language art system include art neural phrase translation technique translation individual point large performance gain still provide nontrivial useful ensemble compare usefulness depend correctly word source identify great processing employ strategy use one track treat input level result system translation rare rare examine lstm architecture strong correlation highlight translation rare follow et sentence average inverse frequency sentence within rare system standard mt system mt describe rare translation curve produce frequent word part bad sentence many word proportion score rare word win sentence frequent word frequency sentence sentence comparable rare score group examine different copy predict align predict align position train hyperparameter vocabulary good performance still analyze include baseline assume align predict monotone offer slightly english similar word language pair chinese word monotonic imply gain gain model word accurate translation prove limitation force align considerable contrast align unknown target word source post process strong translation suggest easy lstm short sequence increase consistent source provide roughly improvement layer stage include clear op la er en op la la de l er op dans de la le leave european head trade e en est en du des point et du ce e en et mis de du trading pour les la se la de show human translation process lastly strong quality measure performance layer lstm compute good translate word observe accurately predict distance highlight reasonably translate end source example reveal entry b alignment incorrectly align result sentence overcome main current system translate vocabulary applicable deep lstm technique likely necessary achieve demonstrate english translation task substantial system architecture importantly outperform mt acknowledgment member brain team discussion insight stanford
hamiltonian ref wiener filter uncertainty exactly extensive want ref htp color signal reconstruction terminology wiener eqs wiener calibration mcmc monte gray method derive address illustrative ref scheme ref device perfect field dependent exhibit sampling position measurement process align data calibration assume spectra ref length implementation get calibration white relate ref measurement calibration beneficial break degeneracy calibration variation switch calibration strength measurement absolute calibration measurement ordinary eq version eqs version deal differential fig reconstruction ht online calibration reconstruction relate error terminology use reconstruction perfect eq wiener filter assume correct calibration naive pure average calibration signal terminology wiener calibration scheme htp calibration bad result method advanced mcmc similar average classic improvement realization naive classic naive realization mcmc fig classic wiener naive still sufficient sample converge increase increase effort iterative involve ordinary differential ode ode exist stepsize control save significant case one could cope reduce stepsize elaborate suppose representation require bottleneck within due correlation structure contrary comparison naturally calculation derive signal consequently calibration calibration covariance priori knowledge successively account uncertainty solve iterative turn system ordinary equation example calibration perform extremely wiener well well favor new classical sec serve realize package find find ref read pt diagram correspond external coordinate depend external summing close diagram mx sx high diagram end line represent connect internal integrated represent vertex compare non locality coordinate integrate whereas diagram divide leave topology book theory derive generalization flow calibration include ad calibration become calibration replace calibration measurement hamiltonian max universit device crucial estimator reconstruct know calibration starting calibration signal inference equation correction solution thereby differential verify self scheme wiener serve keyword analysis field wherein infer response understand response independent calibration environmental time influence exactly result affect signal take calibration iteratively reconstruction improve reconstruction scheme systematic bias calibration partly degenerate guarantee improve uncertainty present ref theory end signal successively flow approach non review interact latter problem introduce derive main formulae sec reconstruction toy example summarize calibration familiar interact brief helpful typically infer challenge reasonably answer agree within express tuple signal linear response operation act signal contrary manifold scalar linearity transform observation scale medical imaging assume uncorrelated relate denote product bx condition covariance meet relate physics hamiltonian consider denote determinant whole physics permit correlation density pdf gaussian mean wiener vanish gaussian free signal meet dependent scenario treatment hamiltonian compose interact deviation term fully nm expand work wiener expansion expansion analogously theory eqs correction signal mean order diagram boundary equation obtain wiener address infer physical field external calibrate absolutely response measurement device transform signal statistic two point framework aim optimal datum without challenge unknown calibration nuisance calibration coefficient gaussian appropriate higher consider first eq assumption eq correlation result ss calculation analytically adapt carlo increase analytical concept auxiliary r b hamiltonian expansion small whereby eqs permutation fourth signal drop flow reason priori signal center prior interaction source correction quantity read mm dot external expansion place dependency correction diagram also break set accumulate information thereby formulate operator boundary couple valid lead solve simplify residual field field wiener hamiltonian hamiltonian
fill bic size draw color bic style mm minimum thick mm thick black bic black circle cm white fill color bic style color bic sep black circle size cm black fill color text n style circle inner mm black fill bic black style rectangle size thick black bic bic minimum cm white bic text bic sep mm fill black style thick fill inner sep cm thick color bic bic inner sep thick fill bic sep size thick black bic style minimum cm fill bic text bic circle thick fill color bic text bic inner mm minimum fill circle minimum draw black thick fill black style sep mm text black bic style thick white bic bic circle inner sep cm draw fill text sep thick fill style circle mm draw thick text black bic thick bic bic style mm fill color bic bic fill circle thick white fill text circle sep bic rgb rgb rgb rgb rgb rgb rgb rgb inner sep thick black sep thick black size thick color text thick fill black circle inner sep mm white fill circle sep thick black color text black sep cm thick fill inner mm white fill color black mm thick circle white fill color black style draw thick color black sep draw thick thick white inner sep fill color white fill color text black style circle mm fill text inner sep mm cm white fill text black style sep minimum draw thick white fill text minimum cm thick white color black sep size draw white style cm white fill style draw thick white black sep mm size cm thick circle draw fill color text sep draw thick white fill style size thick white black style sep thick fill color color circle sep draw white black thick circle minimum draw thick circle black color rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb inner sep size thick color text style inner thick style circle sep mm thick color draw black fill color bic bic circle sep draw thick fill color bic sep minimum thick black color inner bic inner sep minimum draw bic text bic inner sep mm bic inner minimum white circle sep white black thick color black circle sep bic style inner fill text circle sep mm sep draw white color bic text bic minimum draw text black thick style sep draw thick fill bic black bic style circle cm thick white fill color bic style circle size draw white bic v bic bic bic bic bic bic n bic bic bic v bic bic v v v v v v v v v v frequency small include choose bic never select select frequently small select leave number leave independence depth always trivially bic small depth converse focus depth pick leaf note equal generate dataset fix summarize marginally real canonical threshold bayesian computed tree main tree propose refine usual alone consider exhaustive forest observe variable selection structural em consider heuristic put paper bound zero proceed step recall product move nonnegative zero indeed first neighborhood sufficiently away zero view function restrict bound away imply nonzero turn application line proposition note nonzero coordinate intersection minimum intersection lemma applicable b hc I nonempty asymptotic equivalence completeness coordinate substitution jacobian computation compact take restrict q taylor around since claim nonempty form interval minimum translation equal small sign coordinate gx fact support sign change say loss generality restrict asymptotically recall multiple nonnegative expand drop mix note nonnegative since nonnegative jensen equivalent amount case contain change sign coordinate form reflect version prove origin point origin origin small neighborhood point zero substitute h hx tree space correlation first applicable compact compact deduce theorem remain onto appear sum recall forest partition set leave set leave comprise belong inner leave forest leave disjoint sep circle draw sep label label label label remove square observe connect take component sum pair distinct node edge square eq also lie lie obtain node let three let square origin single suffice exponent incidence leave word leave vector terminal leaf newton satisfy define path include span hull hyperplane define inequality claim polytope vector combination incidence construct ii iii contraction tree node induction iii leave path leave base induction pick leave tree subtree leaf satisfie path extend leave leave path path path edge transform path give include leave lie tree degree generalization match necessarily tree origin connect neither incidence hull newton polytope incidence long hyperplane proceeding similarly ray newton polytope newton acknowledgment partially support union national foundation dms security university reproduce xy section thm thm construction thm thm di gaussian generally forest fisher matrix become along compute log canonical bayesian provide development treat laplace integral whose phase sum canonical apply laplace forest tree explore exploit recent latent forest mathematical information criterion widely guide forest bic generally long share asymptotic long asymptotically unbiased estimator kullback leibler behavior inference gaussian forest obtaining form canonical threshold expansion formally introduce object index paradigm induce contain unique node path conditional compare leave marginal parametrization latent moreover zero parametrize correlation correlation everywhere theory develop property parametrization map inner node connect leave label definite matrix three sign change node identify uniquely identity parametrization finite correlation identifiability matrix every lie algebraic intersect reader familiar root direct model specification markov structural seen apply independent write marginal hold rational dimension greater equal derive variance pair behavior laplace formulation adopt canonical threshold state rely present concern general asymptotic exhibit recall forest comprise leave forest model assign edge path exactly parametrization distribution namely extend forest arise data distribution lie forest paper begin review connection asymptotic likelihood introduce tree parametrization via technique geometry selection criterion simulation alone consider parametric inference determination derivation bayesian criterion behavior true marginal tie integral divergence distribution therefore integral neighborhood q analytic compact far sample infinity explain satisfie use nonempty analytically since g f take normal integration take concern fact hold paper concern loss generality center density positive q hence neighborhood positive q compare tree forest q obtain jj discussion parameter generate variance definite correlation matrix behaviour marginal tree square result context u u u case vector support right generality moreover proposition define zero part hull inclusion last empty compact bound turn linear integral clear newton canonical early nonzero part newton ray span zero size sep circle minimum b divide coordinate vanish eq gaussian latent leave dimension edge component number degree nod forest forest threshold model proof appendix leave space share node example canonical translate threshold tree behave model suppose leave label degree variable calculation example apply forest isolate edge edge repeat edge contraction tree least original node equal motivate wish forest forest forest forest analogy tree appear union theorem connect forest model parameter forest smooth positive share node two consider selection forest forest identically criterion implicitly forest degenerate whose give coefficient difference likelihood forest little consider criterion average possible bayesian short describe compute contain forest forest forest forest give q proxy side model system triangular solve recursively univariate possible bic give nonempty bic motivated dependent differ strict belong I correlation forest bic order lattice empty graph isolate maximal select forest optimize bic likelihood estimate em maximize forest model comprise latent inner introduction tree lie forest respect well choice repeat time black circle draw label label label b b b comprise lattice heat choose label independence rgb rgb rgb rgb rgb rgb rgb rgb sep mm thick color style circle inner mm sep draw fill style sep minimum draw thick black fill black circle sep draw thick white style cm draw color text inner thick fill text mm circle mm size inner sep mm fill black circle text rectangle sep thick black cm fill circle inner mm minimum thick color text black style inner sep minimum draw style circle size draw thick white text black sep minimum cm black inner thick color style minimum cm thick color minimum thick sep fill text inner mm minimum black inner sep minimum cm draw text color black fill n style inner text inner sep style mm white text mm thick white color black sep minimum fill sep black circle sep draw black inner sep mm thick fill color text black rgb rgb rgb rgb sep cm black fill color circle sep draw fill style cm bic text circle inner minimum draw thick black color text bic circle thick black color sep draw color text black circle bic minimum cm black fill color text style sep minimum color draw thick fill black bic style mm color bic rectangle inner sep draw thick fill color bic style mm thick text circle black style cm thick black black n mm fill text mm thick color n style circle inner style white fill bic bic style mm size thick style sep minimum thick text black bic style minimum draw thick color text style fill style sep mm draw bic thick black black bic style inner size thick fill color bic bic circle mm white bic circle inner thick white fill bic bic
diag operational composition entity bilinear diag element dot additive highlight bilinear diag provide insight common compositional multiplication entity category entity significantly category result entity predict entity relation category one learning improvement linear entity train phrase baseline compare entity entity parameter dimensional pre phrase vector representation technique introduce word word initialize art popular base completion evaluate structure relational deep hierarchical input datum figure embedding relation reflect release close country embedding learn hard ht table neighbor relation frobenius relation vector meaningful retrieve production production production award award organization role organization job company business organization university location people person location capital neighbor embedding lemma corollary definition ny usa li microsoft usa real entity people place often multi model multi relational biological multi relational category relational logic relational graph probabilistic path large relational embedding relational knowledge entity tensor bayesian study neural network powerful tool generalization recently learn entity entity represent representation entity entity neural relation operator represent relation entity report promise unseen basis compare result neither representation carefully entity vector pre corpus idea tend syntactic natural real world compositional phrase name movie name mean compose type entity include variant completion task dataset present several interesting finding tend scalability play entity operation superior modeling relation entity vector pre boost entity finding inspire embed predict vs triplet describe certain output scalar high vector project input low project specific formally network scoring relation triplet write scoring score unify transformation triplet reformulate denote r parameter transformation derive c r r relational scoring deep semantic learn pair neural project entity e cosine normalize network margin objective decomposition comparison among objective beyond score relationship triplet high negative triplet triplets triplet one objective minimize margin q triplet entity relation consist triplet subset frequent relation result triplet entity link task triplet entity corrupt reciprocal triplet accuracy evaluation gpu batch triplet negative triplet corrupt subject entity entity improve mini batch entity vector epoch k determine rate initially five slice
distinct none origin imply hull eq possible basis negative sign case hull suffice equality expand reduce coefficient coefficient coefficient lie hull otherwise equality compute coefficient expand automatically lie convex hull roughly lie convex hull rest separate member family let set hull none member real lie lie clarity positive vector lie thus one contradiction investigate analogue namely norm great question norm norm equivalence body interior respect standard euclidean symmetric convex symmetric convex vc statement prove preliminary lemma vc infinite consider interior ball norm closed interval terminology give subset pass note hyperplane pass operation set interior three implication close slight make section particular open ball bound send open interior close send convex convex collection dimension origin interior natural take union result element base choose interior appropriate interior interior bound lemma bound interior complete ensure valid assertion concern interaction hyperplane wu point form lie claim equality sequence subset choose real addition base subset n possible inequality q satisfied lie equivalence convex equivalent infinite vc symmetric vc nest convex vc convenient increment index become body contain regard sequence lie subspace lemma condition natural number hull closure set close additionally eq since empty interior remain verify body origin disjoint point lie observe cone segment exceed lie convexity lie convex symmetric body natural convex subset expand translate lemma subset important vc ball also prove vc collection ball dimension norm vc vc regard norm short mean close open ball regard dimension norm paper vc vc norm vc remain norm least precise vc let subset subset set definition finite vc vc accord vc dimension supremum dimension close form number convention diameter include affect dimension cube distinct positive diameter hand exclude vc collection proceed stage vc say prove cardinality agree claim dimension collection close close may closed bind exactly zero may proceed attain maximum integer point lie contradiction every point list leave least imply vc precisely assumption cube note lie di condition lie hence dimension three clarity recursion base even dimension exist higher odd exist satisfied origin lie subset suppose leave lie thus cube diameter cube imply cube lie still cardinality c dc fc c dc intervals df replace every odd exist may set accordance extend cube origin give diameter interval equal set form imply let cardinality
test criterion criterion considerably classification consist step encode extract whiten normalization learn finally classifier comprise pixel whiten finally pass patch sparse differ patch comprise apply thresholding extract cf fill text fill text width corner blue draw auto block image patch pre block pre unsupervised line dash training extract numerous insight principle whiten correspond trade look whiten recall whitening transform patch change precisely whitening transform patch whiten patch change entire affected indirect capture introduce high say round narrow define large eigenvalue width spectrum serve supplementary column gram notion understand whiten patch perfectly particular perfectly round whiten therefore remain randomization randomization random matrix matrix bound standard sharp purpose sufficiently particular round illustrate function entry gaussian random randomization link conduct toeplitz matrix local typical nearby natural entry toeplitz k ij toeplitz lead toeplitz matrix lead find feature matrix increase illustrate perfectly suggest increase matrix preserve global want understand split set unsupervised determine contrast statistical whitening dictionary randomize whitening create dictionary stack select whitening patch statistical whiten patch feature different number acc yes yes yes yes yes c acc yes yes filter dimension sparse filtering first result norm matrix transformation filtering form sometimes norm similar far obvious need htb apparent choice feature influence filter interested costly alternative cross filtering compute correspond basically indistinguishable three observation accuracy sparse monotonically highly peak coincide set test observation stop early optimize filter compute training intermediate peak version report claim confirm finally note version randomness involve spectral promise provide interpretation wide spread whiten patch quite considerably spectrum intermediate feature specific extended paper planning convolutional network cnns cnns deal recently many cnn substantial acknowledgment row proof entry invoke orthogonality obtain yield desire display corresponding row normalization multiplication prove normalize first write normalization multiplication eq matrix yield second claim claim claim normalize matrix display numerical cifar splitting result conclusion htb width corner text cm corner
improve case bias dominate estimation answer systematic improve estimation potential problem functional bias region additionally yield minimax overhead dimension large mle valuable going consider finitely among influence theory compress benefit carefully analyze practical size demonstrate efficacy entropy mutual discrete wide fundamental decision shannon two involve insight various paper I technique develop mutual employ estimator application tree empirical liu significant implement mle iii bayesian empirical mutual yield performance recently methodology functional show mle far minimax unbiased suppose functional differentiable smooth fall fall polynomial specify order polynomial smooth regime plug estimator reasonably hand parameter turn towards functional close sup functional polynomial order scenario unbiased power tool various experiment show methodology estimating entropy compare expense bias dominate methodology reduce expense slightly utilize polynomial estimate specifically show estimator present wu independently scheme achieve require mle improvement practice space horizontal vertical mle entropy demonstrate optimality therein essentially mle finding functional interested estimating estimate convenient impose joint liu dependence precise write liu solving solve span liu mle mutual connect word node span estimate liu assign tool design graphical research biology extensively reverse engineering expression dedicate theoretical property cl edge maximally extreme correspond reveal cl ratio cl begin reconstruct perfectly continue fail maximally require transition sample ic construct assign test important learn class class attribute conditional al assume dependence conditioning class probability conditioning attribute label precise factorize precede parent graphical light cl augment cl difference mutual mutual information posteriori map label attribute rule demonstrate section algorithm base reasonable mutual information specifically mutual way increase implementation computational improve classification list identical since alphabet theorem indicate clustered attribute cluster cluster original attribute bold implement cross error record separately error percentage bold reduction modify scatter modify relative none solid circle error top eight alphabet observation e light remarkably mutual expect mutual begin fail theoretical improve consistently require achieve acceptable classification conduct classifier size training testing implement time display error remarkable reduction scheme uniformly size modify one guess least bad adopt reduce modify outperform original method demonstrate justification apply mle take consideration might lead direct two stanford nsf discussion likelihood biology thank suggest wrong cl original mutual write entropy take also argue estimate denote estimator estimating consistently obviously suffice consistent sample theorem stanford edu widely technique recent introduce construction functional demonstrate improvement particularly scenario comparable result theory rate requirement message functional performance mle sample present highlight statistical achieve reduction require classical liu mle improve liu apply replacement network modern form year series remarkable paper since estimation article book indeed response popularity year accept parametric employ mle likelihood letter fisher possibility mle perform poorly example significantly improve upon cf le excellent overview create part see perhaps systematic
obtain minor since appendix simulation available upon request bandit randomization patient report simulation patient represent population arrival trial treatment pick patient decision arrival feed week patient process patient come assign patient random patient true preserve among take negative reward fed algorithm patient multi armed bernoulli admit greedy tuned patient patient patient set hard patient rate minimal obtain level treat display per average repetition instantaneous plot half cc also report patient treat duration trial randomization softmax trial treat patient must treatment obtain treatment cite success test test independence clinical weak advanced adaptive value contingency allocation contingency cccc failure greedy cccc softmax cccc success failure ucb cccc total ucb tune cccc success failure total patient clinical retain fraction patient treatment clinical accurately method treatment represent bandit indistinguishable patient look well patient treatment effect effect patient day treatment divide patient bandit indistinguishable randomization greedy softmax ucb answer question practical bandit algorithm delay feedback minimal randomize patient decision outcome negligible constraint arrival dropout pose context interpret failure require fill treatment outcome patient clinical equal effectiveness patient identify patient difficult treatment still return weak randomization return superiority treatment bandit bandit clinical patient patient successfully treat interesting patient still trial day effect occur patient suggest patient trial much easily guarantee usually strategy limit bandit notably address issue study goal bandit aspect affect reward relative cover type bandit surprisingly consistently outperform advanced softmax least find theoretical analysis significantly identify perform perform exploit design bandit identify bandit tune future arm improve reward certain algorithm ucb may setting half turn bandit problem trial clinical trial motivate armed bandit evaluate clinical real clinical use strategy simple randomization base patient trial treatment confidence offer reason clinical multi armed bandit efficacy medical far rich clinical setup unable difference among field network practical exist limited find broadly hope study encourage apply bandit preprocesse internet table detail clear result decision since treatment response determine entry indicate day determine patient period receive day last last day patient entry exposure treatment patient calculate taking divide patient present per day average measure test overall average average always patient patient treatment independence normally five meet number patient ucb tune ht ht c number treat randomization greedy softmax p like cccc total cccc softmax cccc success cccc success failure ucb tune cccc success failure treatment figure effect curve rating result patient randomization softmax tune rgb rgb rgb rgb proof stochastic armed reinforcement effectiveness present thorough popular bandit heuristic boltzmann sound secondly algorithm vary dramatically bandit setting perform perform theory even exploit practice heuristic relative affect reward find may subsequent evaluation turn attention trial clinical motivate multi bandit allocation study simulate outcome clinical trial adaptive trial successfully patient reduce patient trial treatment identify finding current allocation multi armed bandit trade automate gain explore bandit clean exploration exploitation comprehensive perspective simple generally bandit variance initially player slot player view many turn player select receive goal hand value hand playing specify alternatively express random denoting play arm classical suboptimal kullback divergence suboptimal solve armed match bound establish recent fisher analyse loose strategy algorithm limit large theoretical problem arm generalize setting evaluate extensive far compare include evaluation become comparison investigate paper optimally may detail criterion interpret extensive conduct thorough aspect bandit problem affect surprisingly characteristic remarkably outperform sound algorithm similar strongly practically varie exploit study identify aspect bandit number hope take account study viewpoint finding need formal sound simple clinical trial armed viewpoint turn bandit clinical trial design clinical trial practical motivate bandit describe refer clinical trial motivation bandit trial capture balance exploitation look treatment arm benefit clinical trial sized treatment level confidence trial dynamically reason decade theoretical modern adaptive clinical several design trial stop base sample estimation patient trial drop hand drop naturally trial promise adaptive design family thorough discussion literature clinical trial design adjust patient treatment assignment favor pursuit family adaptive randomization arguably trial randomization ad hoc heuristic patient winner though literature clinical evaluate treatment allocation particularly surprising many simulation aware clinical trial trial drug successfully bandit aim spirit determine bandit constitute effective trial strategy generally wish effectiveness question implement world patient arrival long time statistical base offer patient thorough overview literature simulate clinical find would clinical trial adaptive randomization effectiveness criterion patient treat patient treat significantly end trial algorithm attractive alternative strategy outline section setup present representative conclusion implication briefly clinical trial arm simulation section detail discuss obtain six four exploration pursuit heuristic capture handle exploration tradeoff play average maintain arm distribution handle exploitation tradeoff heuristic aware empirical systematically evaluate pursuit reinforcement latter ucb ucb sophisticated strong ucb solve armed bandit optimally factor make former heuristic know maintain empirical picking denote greedy obvious generalization select arm probability report accumulate summarize detail handle third relevant situation suboptimal arm trial every experiment repeat average arm evaluate algorithm arm equal behavior benchmark number arm respectively reward sample equal every arm deviation contain obviously separate bandit characteristic high moment distribution triangular inverse variance normal algorithm empirical always optimistic initialization initial find choice result criterion literature well possible third optimize armed characterize reward second half demonstrate aspect bandit tuning dramatically affect report every variance regret numerical achieve percentage greedy softmax mm reinforcement ucb tune greedy pursuit mm ucb mm ucb tune l greedy mm reinforcement comparison mm tune mm reinforcement pursuit ucb mm reinforcement greedy pursuit softmax mm ucb tune pursuit mm mm softmax reinforcement mm ucb greedy pursuit mm mm reinforcement comparison mm greedy mm pursuit mm ucb softmax reinforcement tune greedy mm pursuit mm softmax reinforcement comparison ucb ucb reinforcement comparison tune greedy mm pursuit softmax mm ucb tune boltzmann almost similarly softmax softmax outperform except medium setting ucb ucb little performance optimal slowly generate regret turn reward suggest advantage pursuit important algorithm affect differently variation handle number arm variance much find characteristic distribution surprising normally observe omit present pursuit softmax mm reinforcement quite surprisingly algorithm counter intuitive hard skewed tuning incorrectly although case regret large surprisingly bandit bad reward illustrate regret boltzmann tune every ccc initially tune study tune every strategy account three heuristic advanced suggest practically heuristic boltzmann close advantage appear substantial heuristic development sound boltzmann exploration open whether sense theoretical need balance exploitation throughout bandit strategy ucb theoretical ucb base softmax substantial improvement practice dramatically poorly possess
selector contain gaussian agree notation denote value counterpart arrange set cardinality p overcome elastic regularizer behave elastic sense within elastic paper report support correlate wherein statistical support besides current method norm formulation might fail rest sp dual present fast exploiting efficiently computable exist square directly c sr small root addition exist nonnegative unique op kp search motivated search part assertion accelerate search procedure execute algorithm complexity c sp u selector consist address width get upper quantity notation bound norm generalized selection prove technique group lasso k follow minimizer theorem choose natural interpretation special ds norm therefore admm concentrated efficiency hence side behavior matlab four operator sp ratio illustrate accelerate nonzero entry equally normal response roc h result correspond ds selector vary introduce generalize selector norm encode norm exploit flexible inexact conjugate proximal solve unified analysis utilize prove trivial show inexact admm support structured last sound support nsf yahoo constant proof q lie rewrite since distribute surface sphere gaussian lipschitz lipschitz choose old next gaussian lipschitz eq statement sr inequality pair pair follow without imply sign order focus violate similarly cauchy schwarz unchanged determine optimality quantity inside exist nonnegative root minimizer actually care lead give obtain theorem need theorem decrease r first simply monotonicity mention mm focus case satisfy one generality let know minimize construct choose decompose contradict k part part part violate second must violate uniqueness satisfy make follow contradict minimizer eq assertion note operator minimization constraint consider restrict unable situation violate indicate corollary increasing replace still modification decrease cauchy schwarz decrease modification monotonicity mention satisfie conclusion obtain first norm atomic prove satisfie consider every non vector element combination whose none convex norm thus norm atomic norm cone cone support cone define provide ease contrary e statement mm norm thought norm utilize technique specialized state enable henceforth understand overlap support go order prove gaussian cone nonempty convex cone cone lie normal construction proceed c appropriately let bind follow comparison q eq freedom complement bound therefore maximum use lemma pt mm ready follow mm inequality let eq bind note mm proposition corollary support upper need first confirm selector ds regularize establish ds generalize decomposable primarily focus sparse notable extension structured model suitable aspect norm induce atomic estimation constraint atomic aspect norm selector propose primal interior method generalization homotopy piecewise multiplier admm linearize motivated ds general inexact primal proximal suffice side bound width ball suitable support proximal support norm efficiently focus ball set need statistical guarantee rest establish section efficient error experimental section analyse selector ds approach approach similarity ds norm general primarily focus notable group structure paper consider organize present estimation support experimental conclude supplement suitable ensure empty section inexact present consistency subscript due applicable alternate direction method multiplier admm augment lagrange multipli control quadratic term admm q amount
use restaurant stochastic process exchangeable likewise prior directly random investigate prior gamma gamma binomial marginal suitably stochastic integer count binary restaurant count marginal sampling likewise process binary beta bernoulli gamma poisson marginal gamma beta organize preliminary bayesian random naive document categorization derive prior hierarchical product define continuous space scale evy measure gamma process sum base two continuous measure separable metric space concentration evy beta bc matrix sequentially new subsequently add introduce add row meaning indicate across convention name three stochastic define almost surely count name underlie hierarchical stochastic stand matrix process construct poisson binomial draw independently atomic j j atom nonzero matrix construct count count care specie mass pmf sum define let sum logarithmic pmf number kind pmf compound binomial pmf pmf wise count vector count matrix highlight difference result without deferred infinite dimensional unconditional pmf count separate absolutely component introduction count mass concentration prior arise conditionally nonzero pmf derivation count may verify random count pmf distribution count row column exchangeable column exchangeable recall new row construct n row direct calculation yield prediction express familiar add count crucially new count normalize play key role combinatorial appear gamma binomial process calculation interpret draw keep unchanged binomial introduce iii iv iii iv iii iv iv column original drawing identically add column arrive identically newly distribution different multinomial example new column random count poisson simpler fix binomial count count augmentation negative binomial draw gamma negative compound joint auxiliary count matrix scalar pmf distribution detailed derivation verify pmf compound count via permutation column differently row random maintain exchangeability construction count draw customer table new column add q row exist draw q j customer aggregate kt original mapping new follow logarithmic implication order count distribution sum integer gamma whose row sequentially construct row similar f random dispersion pmf pmf outcome beta conditionally process define jk jk derivation verify calculation pmf generate multinomial maintain exchangeability count analysis pmf thus row customer j r ice column ice thus analogy ice section provide description ice number ice implication infinite many ice exist customer similar random matrix exchangeable dispersion parameter relate marked dispersion atom dispersion however beta column I challenge sequentially construct introduce combinatorial appear independently negative binomial process share argument focus single dispersion dispersion large negative binomial process count submatrix doe submatrix maintain however indexing column permutation row bring column arise realization normalizing term performance construct order constant column choose k combinatorial analysis evident random count I column construction relationship insight pc column c construct negative ten via generate unbounded use one mode monotonically decrease addition count highly count decrease limit unique one identify count encourage th column distribution build use employ two negative employ parameter instead new column n p use expectation variance count allow fine count wise employ column logarithmic count column n j variance c count dispersion fine number random row exchangeable random row dispersion variance count simulate provide difference prior count count count matrix small range whereas count significantly count close gibbs exploit augmentation marginalization binomial distribution count distribution bring additional argument combinatorial random naive bayes classifier category summarize contain count exclude count count j com vocabulary detection tracking corpus appear category consist dataset use compare document process category I correspond infer affect neither long iteration collect mean row count collect calculate expense document term th c I row document term count example word count unique significantly vocabulary whole corpus could fast consider vocabulary corpus principled document contrast traditional discard tb binomial document count random count row right count document term posterior display clear restrictive observe limited heterogeneity adjust count closely expect prior tail distribution highly row wise heterogeneity indice training classifier row exist count training belong count treat feature iteratively row simply predictive use likelihood constrain vocabulary likelihood contrast truly produce fit produce multinomial classifier smoothing q predefine vocabulary discard document categorization unconstraine vocabulary grow vocabulary negative negative smoothing plot category categorization multinomial smoothing document fit matrix restrictive probability count moreover mixed nb tail help oppose multinomial classifier word vocabulary normalize count binomial provide length vocabulary section derive mass binomial matrix I infer construct add row predictive nonparametric bayes classifier classifier share vocabulary outperform multinomial classifier observable derive call process exchangeable count consider simplification represent row exchangeable exchangeable accord theorem gamma express k k k nonzero count pmf count row nonzero count across bring combinatorial question carefully arbitrary labeling index atom conditional likelihood q mass continuous formula directly ne n prior normalization arise column two count matrix although amenable complete update inference except share prior process define draw employ specific hence conditionally row differently focus marginalization conceptually simple compound jk pa jx express j mixed logarithmic lc binomial sum le sn mix
perform non convex cluster method moderately large impossible simple subsampling would efficient subsample new discriminant subsampling classifying belong elaborate subsample cluster discriminant modification goal subsequently towards computational efficiency small large primarily accelerate spc solution include assignment spc introduce concave function parameter choose center cluster cluster spc detect singleton due fact cluster solution relatively pairwise center spc enable spc combine spc subsample assignment remain iterate assignment subsampling spc order spc accuracy effectiveness spc separate need filter successful small cluster dataset new subsampling spc path cluster effect solution novel subsampling cluster estimate cluster provide user significance path step assignment design proportion iteration partition remain cluster identify contrary review subsample contain proportion include spc subsample subsample size big inherently operation would systematically loss lastly spc raise satisfactory simulated spc assignment section discuss consideration demonstrate brief represent spc penalize center concave penalty mcp controlling concavity merge spc utilize minimize cyclic initialized singleton gradually build center spc path decrease cluster include singleton path drive select base property mcp please spc singleton large cutoff cluster knowledge spc cut relatively cluster spc tuning determine neighbor initial spc couple path allow develop noisy produce single describe spc full likelihood ratio membership separate show grouping introduce distribution spc subsample parameter remain membership remain subset respectively suppose replacement subsample training remain index point cluster select index point gaussian denote kp proportion assignment proportion hand overall assign cluster rule take note identify assign estimate parameter update calculation next point discriminant analysis mixture generalization quadratic discriminant analysis generalize extended matrix mixture assignment spc prohibitive choose subsample subsample probably able cluster full recursion assignment step spc repeat random recursion repeat spc short essential select clustering rely cluster cluster characterize estimate include significantly check become assignment recursion increase recursion describe estimate assume test cluster sufficiently design negligible treat degree sort cutoff discovery fdr sufficiently many small corresponding discard otherwise control fdr subset cluster critical generally reject moderately small thus control finally cluster discard spc subsample recursion fdr mechanism automatic determination dataset sophisticated simply large size follow test loose give discover number assignment recursion consist find recursion note spc b b ba km b iy ic bb spc define cutoff determine keep big impact long value reasonably practice hypothesis testing mostly along recommend many occur chance spc rely violate especially subsample typical outlier spc incorporate overcome acceptable point remove discard perform unstable discover increase amount test cluster probably cluster subsample amount cluster possible tight cluster spc tuning determine however generally iteration cluster outlier terminate demonstrate simulate spc ability quality separate compare spc accuracy rand ari develop calculate different ari misclassifie identify merge evaluate cluster misclassifie right comparison proportion cluster total mixture generate radius center ari score use choose cluster spc dataset due operating limitation hold big time input include bic hyper please categorization appear force cluster volume package six dimension order spc scale figure considerable order magnitude fast even magnitude subsample find compare see depend noise shorter amount effectively identify reach spc show computational complexity much spc algorithm especially cluster amount subsample fact comparative original spc figure demonstrate middle estimate panel dataset spc amount indicate proportion spc report example small size though somewhat inferior spc small ari score cluster vary inferior spc risk create large mostly accept due substantial beneficial quality consistently good proportion suitable subsample satisfactory see ari gray subsample able fairly cluster scenario exception cluster subsample splitting cluster subsample size spc order fast mention estimate mean useful spherical shape complex situation certainly understand ari panel bottom inferior cluster tend remove clustered calculation center variance force assignment background sequential step particularly subsample generate necessarily small subsample considerably large subsample figure generally great produce simulate beneficial obtain compact finally demonstrate purpose correlate rest four six point proportion run ari spc result spherical expect tend split preserve albeit amount misclassifie also score dataset reflect mainly due cluster split see believe useful spherical datum dataset tractable two e across select profile perturb simplify grouping involve gene consist expression profile es cell across experimental behavior dataset second gene subsample section two large incorrectly include exclude apply cluster group run figure misclassifie gene misclassifie figure subtle difference big cluster distinct go cluster several gene involve biological b cluster fdr em central development pathway em figure term clearly cluster high level compare particularly pattern novel finding induction seven factor eight es clustered study cluster attempt identify potential cluster depict identify hereafter characterize cluster overall display distinct expression h approximately respectively separate go particularly gene go term cluster fdr em e surface pathway bind activity activity response dna stimulus complex organization activity activity act e go term indicate discover category bind large cluster p cluster go term contain third yet annotate yet six go mostly small biological significant go development transition material particularly interesting high expression control find sharp establish role es therein bind adjacent site uniquely gene responsible development significant classified relate heart development none large contain go low
satisfy due page limitation network page limitation substitute derivative therefore analyze change simulation iteration network steady evaluate run range steady reach minima reduce vice versa make department electrical communication institute signal theory communication mail com es investigate year network experimental apply rest optimum due incorporation subset computational need propose sparsity expression regularization system real data process stream simultaneously instantaneous estimate system impulse sparsity norm prominent amongst coefficient algorithm deviation steady achieved obtain fraction use rule aware employ update refer agnostic provide maintain uniformity computational burden complicated exploit adjustment aware claim validate via regularize trivially case norm scalar q noise require mutually node adaptive filter desire e refine two adapt combine receive b neighbor originally weight adaptation node estimation exploiting add scheme term node constant combination rule combination define par homogeneous sense aware major result individual index deviation steady neighbor exchange diffusion input assume spatially argument matrix top col w n w col carry stack matrix product notice term steady let term e take occur minimize highly sparse conversely zero node
achieve result qualitatively h al model realistic snps spike performance follow distribution simulate phenotype h figure phenotype choose considerable estimate considerably h h h resample genetic bootstrap auc phenotypes bootstrap auc deviation bootstrap deal fix effect account p select base risk denote lastly whether compare deviation set highly relative risk relative individual x risk prediction subtle aspect population control r case summarize phenotype ht identity variance section identity software fast invert run infeasible realistic mean individual focus vector scalar compute overall spirit formula eq plug eq use identity gibbs sampling control study sample obtain scheme mixed effect assumption normality environmental effect environmental effect genetic environmental genetic distribution variate naive gibbs reference drive phenotype I f
tool package project conclusion topic project document discrete treat word corpus words corpus word corpus vocabulary vector vocabulary probability corpus hierarchy generate word possibility probability model map topic sample distribution dirichlet word represent assign proportion sample corpus corpus topic represent sample multinomial distribution document multinomial sample multinomial distribution model parameterize probability topic topic symmetric document value begin sampling document network lda represent leave generate corpus repeatedly expectation algorithm gibbs consider posterior strategy chain sampling chain converge target selecting begin next newly probability stand document assignment word assignment document iteratively initial integer sized markov chain markov chain topic run reach current value gibbs derive unique document remove tag time assign assign word assign document assign discuss data preprocesse lda package project input lda project receive email server server email clean capture email server help far cloud computing process consist language fundamental scientific computing offer package sort code platform processing package remove package package package format unique current document index vocabulary computing tool environment operation specify use option parallel calculate count write program program receive generate implement option input path finish count ignore compare calculate speedup send account gibbs sampling efficiently learn result precision evaluate initialization convergent appropriate convergent gibbs tends reach independent initial topic topic winner enter party color email contact action medium email people receive health http entry email pay support country stay word reveal email health topic new contribution news information lda model word list part measure definition htbp word list top word word match capture correct stand choose tf tf represent major text classification relevance corpus raw document eq high term low word size tf case tf topic word free package project word obviously tf another likely document analysis network social certain people service topic analysis visualize word com dirichlet allocation gibbs train free server com word generate latent topic typical political news instance project contribute speedup project high tf
unfortunately pd pd property bx ib h bx functional lipschitz prove obviously since p pd pd p x n follow immediately similar prove respect versa fix r rx h nr r rd pd evaluate contour lipschitz less prove construct qx map lipschitz frobenius q identity x follow map identity claim lipschitz claim obtain obtain acknowledgement author support nsf dms also point reference point university mathematics center mathematical university college md usa mathematics scientific college md call problem specifically nonlinear map hilbert lipschitz term surprisingly independent frame span hilbert space apply iff product nonlinear phase retrieval reconstruction problem analyze frame map entire remain space continuous lipschitz literature e bi f bi real inverse restrict image surprisingly minimal lipschitz organization follow section nonlinear induce eigenvalue negative symmetric restrict necessary phase frame phase constant eq distance norm subscript choice compact particular nuclear frobenius expression distance condition state result distance induce additionally embed lipschitz topology however endow particular metric endow nuclear norm equal identity theorem together previous frame bi lipschitz metric particular clearly space lipschitz prove entire factor phase frame hilbert exist upper lipschitz explicitly mean lipschitz bound
irrelevant feature mf n class random train efficient mf online forest accuracy remarkably mf competitive batch forest fraction mf unable split choose split mf limitation forest influential machine extension mf hyperplane split instead axis align split acknowledgment e bl part hold fellowship college newton international fellowship european union framework fp agreement smooth label point fall leaf xt appendix detail training distribution exist posterior multinomial context special sketch picture refer reader explain inference predictive batch chinese restaurant wherein customer table leaf table customer parent restaurant class resort approximation particular smoothing smoothing interpret restaurant precisely customer restaurant table approximation follow leaf simply internal j k procedure serve fashion kk j add affect count leaf contain internal root summarize informally discount parent probability single pass jk predictive involve traversal starting contain leaf extend contain extension lead confident analytically along leaf gray leaf branch else point along root branch split equal lie probability branch j child contain location branch leaf node mean simply weighted fact discount distribute exponential forest initialize ty j x k k j p jk jx discuss associate trivially process binary processing leaf posterior count overall datum n factorial rf complexity asymptotic expansion hyper factorial discuss version nj depend nj j x nj nj j nj nj j nj nj label nj extent remove nj j x j j e u dx jj leaf else depth forest leaf fraction point leaf parameter experimental setup table report forest train c depth fashion tree true similar forest key mf forest ensemble whereas average limit whereas mf still multiple hence mf outperform scenario decision insufficient explain experimentally validate performance package tree leave constant leave multinomial leave correspond learner number mf pass support dynamic forest mf test accuracy achieve dynamic mf achieve accuracy dynamic dataset dynamic mf indicate usefulness guide split mf mf performance forest superior dynamic suggest explain fs fs fs fs mid fs unit college department page ensemble decision task forest test excellent candidate forest variant operate batch great demand forest require batch counterpart comparable work ensemble decision call incremental fashion remarkably forest achieve competitive forest forest fast computation vs tradeoff decade forest remain due robustness excellent survey novel mf present efficient agree online forest fast forest art dataset performance mf section conclude label focus class classification methodology supervise forest forest collection next point make standard expectation ty tree overfitte increase seminal introduce ensemble model average online example another incorporate label n n n tree incremental fashion irrespective order none random forest sample efficiently depth logarithmic sequence computationally efficiently forest define purpose rule predict root binary except distinguished parent zero leave child child leave tree internal parent location split tuple leave respectively rectangular along dimension simple partition decision tree split split root denote block associate red circle em introduce additional denote denote denote label upper training respectively b x small datum point node family refinement study care introduce algorithms nj add nj nj j nj nj n nj nj nj j start rectangle exponential split leaf node big leaf else internal split take probability u child tree cart split control total split maximum iv internal node illustrate consider family distribution range point distribution process possess self sample tree extend exploit datum upon observe tree conditional represent focus structure tree nj denote gray rectangle big rectangle extend outside split option distribution extent e r probability e abc consistent partition add data extent root split new contain I denote leaf prediction otherwise incorporate one leaf particular extension confident hence integration analytically depth forest vast attempt brief refer review forest batch setting tree split split optimize quality greedy manner random forest I bag random batch set rf bag node subset chooses subset choose split location bag tree split choose independent term unlike mf smooth test perfect totally randomize although difference forest grow tree every leaf every list split associate candidate split leaf node update sub base leaf minimum criterion good internal child candidate split process repeat memory deep cost maintain candidate quality incremental cart tree forest mf incremental decision tree single typically specify mf perform discussion comparison purpose fraction ii training time divide mini batch compare forest mf batch forest rf training mf author report
sensor time activity sensor aforementione fire change state pattern unchanged period unchanged presentation pattern sensor random begin say pattern presentation eventually form item item think present notice presentation presentation exist structure subsequent presentation pattern special high level pattern recognize item appropriately strength accord memory eventually presentation basis mean basis item see predictive contrast complete importantly without verify multiply pattern item none item allow delay sample depicted correctness item fire item shall accomplished initialization state repeat sensor yet establish review probability item fire item pattern repetition presentation repetition presentation step sample delay pattern item item present time essentially create share achievable tree another benefit decrease accelerate remark upon presentation item necessarily undesirable alternative parse turning pattern happen height claim chernoff bernoulli possibly response pattern since item activate input symbol fire recursively item iff item set item pattern next pattern encounter create whose item accord operational rule pattern must item item create item add item item pattern item whose high en none parent sensor round additional round sensor tree I round item parent item become step predict therefore item sensor sample exist item parent create presentation pattern item form input item create parent another level item round pattern create fact either lemma slightly weak number item fix pattern sensor fraction sensor involve long already common valid matter form since sensor n sensor pair valid already parent child pattern continue level round input arbitrary order time top stable sharp valid overlap distance pattern proof must draw fraction create come valid yet valid fraction sensor sensor sensor pick sensor level total expectation claim create create valid come presentation create number add create valid level claim quantity r high independence implement binary checking measure memory traffic input item presentation run every item presentation drop less traffic pattern percent pattern per traffic without restriction difference create early stable create predict show average create keep explore sharing pattern pattern obtain coordinate randomly ham consecutive pattern perturbation varied produces perhaps predict decrease pattern unsupervise currently one considerably pattern slow current intend certain aspect join ability briefly could graph basic direct suggest song et work chance far establish whole classical graph pair vertex case graph apparent connection elaborate introduce arc round ignore direction arc probability round specific fit rich reciprocal possibility little reciprocal connection back unnecessary carry three oppose small length item say probability hard see root discover process soon imagine create require algorithm see step synchronization simultaneous another whose implementation match furthermore join firing create assumes implicitly signal respective root early step fire fire become fire item symbol time availability sequence look form b form fine formation happen interference item star enjoy favorable establish introduce primitive intend capture implement item input later recognize activity pattern recognition predict part direct imply certainly lie point spirit work invariance recognition pattern challenge cognitive thing view object angle context high object reference language thing plausible analogous operation seem related machine besides primitive place huge sophisticated reason success elsewhere thing accomplish secondly sensor circuit among thing feedback loop eventually modification environment seem theory characterize kind environment evolution behavior acknowledgement grateful reference les early table see part signal stable approximately pattern size go pattern randomly varied pattern percent traffic third increase input pattern ht p pattern coordinate member visual possibility two sensor state sensor remain unchanged presentation next similarly state achieve appropriately strength zero activity aforementioned firing change present eventually current presentation mean difference item complete prediction importantly next verify factor state unchanged maximum sensor item random say eventually far think pattern follow presentation exist structure subsequent presentation pattern special level item pattern recognize example notation primitive predictive join extend join reasonably complex pattern involve phenomenon namely base traffic human thing computer intersection science task complex seem brain advance understand brain yet mind network solve nontrivial computational fashion reflect brain solve motivate present believe play gap inspire work les direct edge influence fact brain recent year visual example mt etc connection fire traffic hierarchy e g student brain prediction example vision visual also active input rapid field seem make layer visual control area visual third connection seem random graph reciprocal connection length direction would briefly brain connectivity way neuron call platform receive potential fire go world neuron vector formal brain read graph random know neuron fire cause neuron item element fire majority concept show join item exist establish via item stand conjunction main contribution implementation new item predictive join operation join predict whose prediction may specification discussion variety actually appropriately sensor word architecture happen adapt kind live operation nothing else create connect repetition share pattern recognize special item act already short item allow pattern section use item simulate algorithm create traffic generate presentation traffic consistent prediction particular cognitive quite bit issue control running algorithm appear agree well theory brain item consider disjoint shall disjoint set simultaneous firing majority define item create instant item henceforth item operation capability simple computation involve local computation strictly action argue capability neuron argue realistic need response operational ready henceforth item strength retain strength operational executed every employ intermediate neuron result fire neuron operational operational incoming memory operational strength small upon fire enter firing period fire model fire follow factor subsequently otherwise fire item enhance variation join join addition enter state ready firing predict whereby informally join intended item join elementary task create bc allow execute randomize something could perhaps component shall choose half expectation comprise strictly disjoint implement enter total operational ready well part context part reason discussion explain simplify considerably operation take fact memory parent identify come require double neuron
contextual track track prediction n truth track video specify flow extract infer truth label slack pay cost flow track optimize svm cut plane number constraint flow violate constraint solve iterate constraint video valid cut plane subroutine video subscript notation inference augment correspond negative inference behave somewhat differently constraint finding flow tracking generate relaxation constraint fractional rounding track incorrect tend tight relaxation flow track hamming ground label q indicate estimate truth critical aspect tracking criterion false false true true birth death consecutive link simultaneously routine consider pairwise propose decomposable link capture aspect account localization link rather constant loss empirically careful specification crucial order let four transition false detection detection true identity transition link virtual virtual virtual depending overlap transition virtual virtual virtual link virtual virtual ground flow practice specify ground need map onto video taking scoring window label assign track run simplify dynamic programming claim edge transition birth death false weight intra frame benchmark frame object object evaluate category comparative publicly evaluate percent label car trajectory trajectorie trajectory difficulty train special evaluation remove correspond training mining negative full sized frame frame overlap subsequence track link predict location frame via optical give good specifically optical candidate flow candidate candidate frame candidate repeat observe many raw post trajectory fit cubic compare publicly code first baseline scheme appearance publish successive default another baseline dp table various baseline baseline dp also remove overlap learn transition measure evaluation learn lp lp dp conduct leave validation lp dp outperform state baseline attribute appearance flow turns properly learn produce comparable dp flow much previous attractive feature shown learn lp mostly keep round approximation rounding inference via relaxation average time find lp round dp run cost forward relaxed produce often within relaxed global significantly round mt ml flow flow dp c c ml flow flow lp flow c dp flow dp flow c mt flow dp flow flow flow flow dp dp mt lp round ml relaxed preferable train conduct sequence cross validation dp lp relaxation inference inference metric stand inference slightly train lp competitive augment well tracking pairwise jointly optimize extensively evaluate traditional lp relaxation greedy dynamic programming performance support award programming successive show find find reader detailed sort short node short path dag frame last accordingly reconstruct variant minimize dp dp interaction apply find consist correspond subtract pairwise cost turn additionally pairwise node entire direct node end ji ji j ec track simplify construct instead edge cost original add pairwise cost pairwise cost backward turn backward turn often dp lp slow slow round video object pass dp use finish whereas dp lp round minute validation dp dp run backtrack cyclic note proper hash link list cache array second pass propagate label eventually might look overall pass still promise moderate optimization style default style style style default default style vertex default style default style default style vertex default style vertex default style vertex edge forward style edge forward forward forward style forward style forward style style edge style style style style style style edge style forward forward style style edge forward vertex default default style vertex default vertex default style vertex default style default style default style default style vertex default default default vertex style edge edge forward edge style forward style forward edge forward style edge forward forward style edge style style edge select style edge style forward style forward forward edge forward style default style vertex default style default style default fill style default vertex default style red style style default fill style vertex default vertex style vertex default vertex default vertex default red vertex default style style style style forward style edge style forward style style style style style forward style forward forward edge style backward bend bend bend backward bend backward bend bend bend style vertex default style default style vertex default vertex default vertex style default default style default vertex style vertex style edge edge style forward style style forward style edge forward forward style style style style forward style forward style edge forward edge style style edge style forward backward bend right bend edge backward bend edge bend style bend bend bend vertex default style default default style style default default blue default vertex default default vertex style default vertex vertex default vertex fill style fill style backward bend left style forward forward edge backward edge forward forward style forward style edge backward forward style edge forward style style style forward style edge edge backward edge style style forward style style forward edge style bend right edge backward bend forward style bend backward bend inner white thick draw thick draw frame video interaction track track contextual min cost approximate multi target track two track relaxation greedy pairwise algorithm across category enforce intra mutual co relationship within detection tracking topic advance build track avoid period since often find low often min matching min yield solution somewhat traditional generative formulation track draw estimate object discrete frame association observation track face joint inference track trajectory implicitly skip utilize successive could approximation associate integrating raise difficulty pairwise potential perform allow rich appearance maintain purely group combinatorial number rich undirected combinatorial often min cost pairwise track typical spatial object track object method order technique prediction affinity association task crf segmentation translation knowledge unique discriminative track death relation art category tracking begin track association min cost equivalent individual track markov whose video framework incorporate successive frame video discrete site site scale frame extract site frame collection track foreground generate background site track assume track behave expression map q appearance track site track find take log yield flow transition site know satisfy thus solve exactly specialized successive short path even approximation multiple short algorithm find apply original network edge track dynamic pass dp nearly variant aforementione track always key showing track integrate pairwise flow intuitive example would overlap boost occur object interaction site video integer quadratic addition discuss solution section describe conduct inference flow track min network explore build alternative global quadratic auxiliary integral lp round yield avoid expense relax cost objective wolfe algorithm relax keep video qp quite replace quadratic term new enforce program efficiently lp solver discrete track relaxed constraint round relaxation constrain optimize standard linear frank wolfe eq round heuristic subject integer flow execute round inspire
globally country country sum engine attention title past country country rt rt piece information rating ordinal nc plot distance plot cf capture solely historical cosine actor create share actor weight actor create actor counter membership large figure existence friend friend friend co target train rf predict rf normalize sale instance sale correspond predict decay demand release train week steady state volume predict release uncertainty great problem company term obtain late life cycle predict week uncertainty time next discuss construct sale interest forest construct drive go varied actual policy week prior week train week b issue historical demand transformation censor demand score reasonable value demand issue score censor affect compare one demand drive access product make fair comparison product demand market dataset past henceforth difference measure week live decision say objective sale volume impossible perfect average location city note insufficient neighborhood cost compute develop explicit impose function restrict expect transform term norm nuclear appropriately incorporate objective framework great potential problem quantile rearrange service level requirement auxiliary cf ms involve general expect distributional conclusion square quantile analysis predictive approach universal instead difficulty restrict portfolio allocation must apply equivalently constrain guarantee outside dataset run affine dimension onto intractable solve extend region maintain convexity planning quantity e positive end apply restriction norm show problem rule inefficient consider unconstrained develop condition proof convex every subgradient polynomial guarantee derive proof traditional framework derive rademacher c appropriately begin generalize multivariate give quantity also multivariate rademacher guarantee suppose confidence involve multivariate decision complexity linear conjugate consider sample relax restriction norm combine idea ms specific use optimal decision ms motivate tractable measurable together grow availability auxiliary potential impact asymptotic proof sequence random joint finitely consecutive stationary marginal distribution property sequence look ahead head nearly nearly metric mix stationary sigma sigma algebra subsequence total set square integrable value mixing establish mix mix satisfy mix cf cf thorough rate explicitly cf situation take example stock market dependent doubly stochastic arrival iid everywhere henceforth almost henceforth iid sampling hold n kernel iid mix process na I hold absolutely bound cost e kernel asymptotically preliminary stationary measurable mix sigma f measurable compact convergent nz nz restrict subsequence n weakly nz nz limit eventually k pg nz exist nonempty suppose loss c nz nz nz x x z exist apply consider least subsequence contradiction limit contradiction yield convexity compact restrict affine low semi theorem consider nz nz precisely e weakly f z nz constitute let hold desire statement iid mix consider mix hold therefore lemma turn yield desire I necessary choice I obtain hold yield use uniformity square choice iid boundedness consider combined boundedness equal hold apply manner converge yield assumption conditional support independence separable space bring inside establish suppose lipschitz involve lipschitz iterate expectation hand choice let inner choosing take left rademacher complexity class fix function note boundedness lipschitz hence complexitie z kx w f f complexity rademacher complexity get conjugate exponent term jensen norm follow result jensen expectation body via ellipsoid call oracle produce cut trivial membership weak constraint allocation construct portfolio observation factor eq generate security noise marginally minimize conditional risk b example serve space evenly circle space evenly figure advance per last simulate example portfolio allocation example operation management science three quantity simultaneous associate recent google search company news review traditionally ms focus period generalization priori cf presence identically distribute iid extensively foundation learn hand largely usually univariate base large address optimal decision uncertainty availability advance game query office sale manner ms good good decision account uncertainty absence mean aspect solution idea ml decision optimal joint decision future task full key contribution way construct paper motivate specific construction inspire great variety predictive ml forest rf summarize construction traditional erm ml construction erm multivariate value encounter ms construction limitation proposal study general cost full optimum limit decision surely observe demand via sale introduce term order content helpful analogue determination management international entity million unique world scan trading year internal company spirit public online source google combine approach improvement account toward counterpart construction predictive broadly idea approximate locally great cart imply forest rf imply xy mx identity leaf cm implicit construction focus period realize one make uncertain quantity multi extension uncertain realize subsequent decision period illustrate make auxiliary section real portfolio allocation decision portfolio consist uncertain return security interested risk loss negative quantile write extra decision return eq product location stage unit product store per satisfy location unit production know portfolio security analyst rating volume google searches company planning past weather forecast volume search us leverage marginal ignore datum q try predict observation guess approximate drive forest decision portfolio factor location predictive demand evolve process order simulate average solution distribution true drive decision performance quickly latter remain observe upon study general asymptotic proposal convergence observe guarantee mild ignore use appropriate outperform datum drive past construction effectively eventually motivate rf notable bad dimension relatively well e world uninformative normal performance dimension cart rf largely serve approach focus ms commonly active approximation drive optimization notable approach decision make robust cf variant literature inform far cf iid ml supervise wherein expectation regression mode interest cf cf function square specified minimax decision intersect erm criterion ml arise erm extensive cf ml ordinary ridge quantile regression decision erm policy equal cost management consider loss result square useful general ms decision erm show limited generalize class ms decision instead erm mode nn locally constant method notable connection asymptotic rigorously local pg even recursive method often form notably great former partition average cart popularity interest minimize observe true specifie one fail conditional require way generalize cost similar problem estimate predictive construction motivate construction take weight decision understand case motivate tie break index rule computation speed f variation neighbor regression neighbor neighborhood unitary nonnegative density ratio particular lead predictive square content measure fraction explain take average perfect know involve estimated cost deterministic counterpart third drive poor serve analogue prediction denote useful significant reducing leverage purpose predictive converge grow instead optimum figure allocation example knowledge correctly take denote successful property optimality proof optimization tractable give different complexity solve completeness separation oracle fix subgradient oracle call ix effective nonnegative nonnegative weight polynomially present optimization converge full sample guarantee mild condition asymptotic almost structure assumption constitute iid strong velocity variety collection mean historical world process chain evolve market daily google topic extend present rest iid mention follow existence continuity regularity either yy condition follow asymptotic nn optimal kernel hold kernel asymptotically I absolutely bound cost result cart observe distribution arm international medium lead ask identity figure location cd home scan capital hold cost medium mainly production medium secondary capacity location medium display back store storage sale studying restrict attention video particularly inherent whereas period demand determine demand release much trivial new demand sale formulate index product denote quantity uncertain demand capacity effect per add regression fixing capacity replace via optimal regression dual optimal conditional issue sale demand censor approach correct datum threshold censor appropriately I weight conditionally share support cost bound share event observable conditionally give past decision make realize cost drive internal company public source public datum company consist year aggregate sale week period sale week censor observation demand occur develop transform tackle sale cycle home sale sale week release pose great sale demand include chain location country google interface item item title title descriptor compose information france item title fr imply may title public box office review item desirable
produce would term tree multiclass five become fine fine grain previously autoencoder input different department ci decision every leaf allow every jointly backpropagation like neural even allow make soft root power perform classification internal operate build function leaf binary return node decision subtree depend multidimensional output regression many attribute threshold split relax linearity regardless leaf decision perfectly decision decision step grow search split reach improvement entropy child keep leaf accordingly prune tree replace leaf decision tree build multivariate decision sigmoid tree split stop reach descent tree soft tree leaf node degree traditional decision tree end encounter continuity induction exploit continuity backpropagation compute nonlinear train convert recursively leaf contribution internal toward leave train soft activation consider activation remainder mean response convex combination leave softmax power hard threshold soft threshold thus partition part node extend tree child node locality leave tree control child linear leave independent local path intuitively layer softmax threshold imply number leave traditional hierarchy tree essentially subtree activate activation node leaf subtree hierarchy representation feature must together mse com fm nh opt quantitative task ten binary uci task discriminant tree baseline separate final remain
strength smoothness parameter dependence describe simplicity carry water period bottom row fully bayesian collect hour hour standard intel core ghz gb become cost design work operation store base carry computationally massive traditional non depend allow communication low fix methodology use store extend spatio temporal demonstrate world sensible however get likely refined complicated full dependent explore work natural multiple measure principle stacking weight big require due size become prohibitive process acknowledgment material upon support national science foundation dms sciences institute support nsf sciences dms make aware problem spatial helpful anonymous environment comment discussion derive likelihood write determinant formula z j j follow desire sparse k diag j decentralize kalman filter j j j server n diag v rapid often carry distribute fashion store physical location relevant divide case inference move central low spatial scale parallel massive datum exactly cost communication size extend spatio temporal methodology distribute spatio particle filtering water sensor spatial random temporal datum capacity thousand past decade transfer therefore tool minimize movement computation situation arise massive relevant give costly avoid unnecessary goal analysis around relevant problem location parallel achieve work memory approach repeat substantial goal computer individual focus situation modification environmental science frequently usually contain environmental store center throughout aim measurement advance national resolution imaging conduct spatio temporal call total water measurement store consist spatially fine recent parametric covariance situation low model spatial scalability show full propose spatial present inference spatial low apply decentralized surveillance rao carry massive server operation depend suit distribute traditional computational ignore spatial purpose science massive describe evaluation considerable movement literature filter prediction sensor collect aware individual massive aware previous analyze distribute case article distribute context would correspond might ignore dependence substantial coverage block dependence work unobserve e g belong effort spatial dataset exploit split suitable organize brief review distribute describe discuss inference present important describe spatial extend spatio section total water measure sensor conclude interested spatial j give note order arbitrary way spatially measurement origin assume follow low scale trend vector assign form assume measurement massive widely class spatial spatial discretized mat ern parent covariance posterior numerical likelihood base inference normalization low rank j j us frequentist inference use bayesian server evaluate central updating sequential sampler various exact result evaluate suitably move create calculate transfer central calculate j take advantage parameter inference carry calculation evaluation assumption server unnecessary server constant likelihood update z j wishart distribution gamma spatial data amount carry inference final estimate frequentist particle nonzero weight determine none prediction location measurement support continuous spatial throughout manuscript diag p describe coincide desire spatio spatio temporal vector autoregressive jt z jt call spatio temporal effect temporal low first filtering mean interested obtain filtering collect time obtain decentralize nest filter outer filtering essentially initialize prior time server matrix server j central forecast collect might smooth p also actually information server filter smoothing time appendix small unknown spatio low rank approach resample natural inference use algorithm particle weight smoothing likelihood apply spatio filtering inference
sample spanning assume point restrictive lebesgue density combine proof link example visualize plot identity set dotted line represents represent estimate figure fig separable indeed suggest divergence section outline divergence distance exhibit make useful relatively straightforward fact attain value divergence express f dd pf pt requirement special use estimate deriving notation introduce slightly easy pf simplify notation estimate theorem problem binary formulate find distribution probability rate combine result eq tight completely go rate base affinity appendix affinity measure special bc widely use motivate develop new algorithm popularity bc mainly bind provide bc separability class result relate divergence f pf pf expansion chernoff local equivalence since induce manifold term tight since surprising since bc bc tight ever comparison sample come mean distribution overlap entirely separate integration empirically bind display two bivariate tight calculate never bc empirically estimate bc due variance estimator divergence consider distance rate two domain correspond source represent us distribution decision true similarly target identify error difference measure follow assume data distance minimize characterize labeling target us mean target target shift exist label g assume rule attain bayes distance matrice q label provide insight representation error source distance heavily word quantify versus feature separation class high third classification alternative consist distribution error analytically bc bc furthermore assume require c evaluate bind outline monte simulation assume parametric mahalanobi close regardless stress expression perfect knowledge empirically estimate machine learning reduce prevent performance dimensionality densely prevent task adaptation separation selection two seek bad minimize scenario da optimization forward search alg use parameter determine domain machine adaptation set minimize tune minimize domain j f empirically speech record patient classify speech speech speech laboratory consist include mixed disease pd patient speech phrase speech minute record speech split individual extract sentence envelope spectrum feature spectrum p slow detailed reader fs feature draw speech ensure separability selection maximize build training evaluate repeat ten result initial fast compare bc roughly restrict set classifier success stay bc method bias distance estimator seem separability bc converge bc bc efficacy adaptation problem evaluate order select different make training comparison selection algorithm bc assume come fs account domain bc domain compare attempt prediction performance source lagrangian classification use minimal contribution target reader involve source domain separation separation resemble fs present accuracy yield top fs train table high classification accuracy bc low trial utilize domain type scenario normalization yield accuracy low generate accuracy display accuracy feature propose improve additional criterion select minimize tendency safe e prefer informative helps build top return da fs feature return criterion similarly application top return da fs separation similarly feature source bc divergence bound error training first tight finding criterion speech rate alternative future around analyze understand convergence size fidelity furthermore bias estimator apply improve feature space text rewrite eq link manner begin bayes q next show tv q simplify begin relationship vs harmonic immediately chernoff upper distance tight clear theorem tight less use cauchy combine begin fashion identify target tv using express play statistic improve classification
method try partition ideally bs outperform setup well storage every contiguous bs contiguous location per bs mention mention bs propose exactly row comparative approach bs convergence note express common notation table method assume randomize partition block block row minimum eigenvalue denote method c method c x randomize op pd p able rich check optimization program subsection subsection show specialized instance generalization unconstraine suggest strict function framework notation space linearly generalization guess column affine concern easy see guarantee always use point alternatively fact fact connection framework suggest pick know guarantee descent sequence appear possible idea obtain minima special say alone strictly unique point fact easy see update framework block set precisely minima q contrast framework indeed numerous observe column dense store entry minor modification minima confirm method greedy unfortunately run briefly mention idea greedy pick block goal block follow immediate let partition strategy eq iteration immediate amongst choose block close norm amongst iteration well every th th serious computing gradient coordinate unlike run algorithm tell iterative block bottleneck remain fx time run operation run time per entry main requirement remain discussion put partition end subsection describe give explicitly deal secondary storage ht preprocessing store storage device p p fx section brief greedy strategy method let method block block k large amongst iterate requirement feature proving partition input effect possess iterate input make diagonal block diagonal well obtain convergence eigenvalue observe occur e k iteration index suppose operation detail reduce reduce node setup divide group associate storage device node storage device retain memory running modification preprocesse implement transfer hence operation second usual emphasize serial rest lead resource distribute advantageous delay incur storage device device device spread processor second dimensional setup copy section give matlab wish approach greedy conjugate gradient experiment realistic solve experiment highlight limitation strategy randomized scenario equivalently intuitive behaviour also use partition implementation moderately specification carry gb ram gb secondary space intel processor choose equal gb gb ram matrix break variable rule mas store file disk generate require approximately denote store individually entire operation result file mb entry define numerically entry dominant store file minute solve descent nesterov randomize define method implement block block partition maximum choose iteration require read submatrix plot sd bs identify individual randomized average method demonstrate intend superiority propose matrix highlight crucial randomize place marker trajectory descent bs respective conjugate take roughly bs take roughly period bs result subset column store finally arbitrary method matrix memory step involve secondary storage read preprocesse block randomize choose square origin comparative convergence key observe mainly matrix randomize particular almost large submatrix irrespective lie fashion lie experiment choose manner experiment specifically choose input block keep row index block remaining row arbitrarily partition block make loose use start origin performance paper resource desire row partition matrix amongst reveal block partition input almost simulation method find fast however remain open question arise web social problem characteristic consist million describe program bring challenge concern management setup adapt resource central aim simple descent one suit verification however method prove bring well amongst novel deterministic serial parallel distribute variant confirm conjugate paper essentially quadratic solving equivalent eq share aforementione nonzero involve resource available hardware technology main memory ram processor gb device observe store double I firstly storage device store necessity slice store secondary device third portion processor run memory reading processor time secondary storage location add overhead point iteratively improve processor hour million running discuss across processor practitioner usually processor limited resource per improve solution quality cut marginally usually mention decompose applicable block lie dimensional dense pointed serial leverage single processor availability setup henceforth limited sized secondary storage device number enough store dimensional solve develop far exceed main size serial processor convenience setup processor discuss distribute multiple processor practitioner prefer entry reason setup order run view say never entire course execution store move secondary hard consuming require load datum reason disk contiguous contiguous method addition prefer method possess run solution good scope distribute implementation parallel survey exist generalization one prove way forward discuss equivalent greedy establish bound way improve section give finally conclude direct classical broadly member require consequently unless solve high see survey discuss research gap art direct begin svd factor finite take dense give popular member elimination cholesky lack couple fashion setup method solve repeatedly improve moderately appropriate modification ahead method solve broadly cut plane direct every iteration current close descent gradient modification decide estimate gradient cut plane method feasible contain hyperplane centre contain new idea repeat difficulty efficient centre hard ellipsoid overcome alternative setup avoid quality member solve method begin simplex transformation sequentially function slow simplex run iteration include also iterative line search never end subsection reason fundamental use row matrix row fashion refer method traditional multiple per partition block round iteration improve estimate notation update clearly two partition use work choose emphasize choice guess pick k simple gauss see select iteration round unlike match index traditional arbitrary traditional method choose one round fashion coordinate
use package bm bm package http www cs bm ref available http es software c filter wavelet thresholding bm bm review rest terminology denoise method amp result amp bm bm call bm free amp efficiently minimum mse strategy denoise variety exist estimate standard image convenient amp package algorithms standard deviation package tune bm bm variant amp package skip without self tuning package tune challenge early effective effective small iteration table construct artificial additive white choose noise simply choose parameter maximize problem look optimize code recall evolution comparison bm increase attribute effective correct behavior bm optimize among value test contaminate notice level look control within amp amp typically stop less threshold demonstrate bm suggest variance remain reduce variation decide bm amp run implementation except amp calculating review insensitive exact wide pick effective amp calculation effective amp observe performance bm define value estimate intermediate denoise house measurement amp amp amp within predict respective online researcher evolution additive white reconstruction bm db amount measurement might expect denoise base amp robust outperform db c bm amp cs bm amp amp c amp amp amp bm bm amp bm amp dramatically cs amp presents signal balance extensive approximate amp sense variation denoise amp amp maintain performance amp evolution prove tuning amp use amp robust amp plug match amp since design denoise easy amp different area amount develop amp rely assumption signal follow support evolution validation future research likewise subgaussian matrix matrix fourier direction minimax amp state notational simplicity useful proof interpretation prove w exist last large evolution happen hence establish interpretation favorable amp e least favorable return suppose consider ax vector incorrectly part prove consider n way result low risk hoeffde hoeffding sample belong define condition derive risk bayes word risk step note define conclude dominate prove corollary remark conjecture denoise algorithm seek remove perturbation error devote amount compressive cs recover reconstruction answer natural effectively employ develop pass amp demonstrate choice amp offer state cs explain performance amp analyze critical appropriate fundamental compressive reconstruct dimensional signal measurement compressive measurement thought mapping model physical interpretation camera inverse transformation reference refer compressive sensing problem represent determined search recovery formally pursuit denoise first perfectly cs deal program extremely demand iterative develop pursuit iterative hard compressive soft linearity residual iterative thresholding extend extra eq average vector correction term illustrate figure amp effective straight line gaussian enable optimal lead employ amp work many signal majority imaging dct show classic signal coefficient far seek failure researcher elaborate recovery include variation sparsity hide non self representation image complementary non develop enhance simple denoise whether explicit employ denoise capture complicated recovery scheme inspire design approximate use denoise measurement advantage summarize analysis framework characterize limit suggest use amp assume belong signal class close employ treat receive employ achieve derivation applicable wide signal class amp employ correction algorithm many formulation require explicit exist denoise intuition amp eventually predict amp track deviation amp extend evolution mean square amp characterize amp amp also enable amp presence practical concern amp employ wavelet amp local denoise call wavelet thresholde amp original amp last pass message pass extensive compressed paper belong pass calculate pass simplified message employ state analyze amp signal introduce develop elsewhere amp whether approximate rely amp noise extensive writing paper aware pass algorithm employ pass adapt statistic major broad adapt recovery many noticed sparsity explore explicitly penalty functional encourage initially structure wavelet researcher dictionarie art penalty functional fit way relate et bm impose solve add noise miss spectra bm bad test finally emphasize approach application amp come behavior remainder devote amp connection evolution state evolution explain calculate correction summarize main validity set tuning performance amp goal estimate extension hyperplane point hyperplane move orthogonal hyperplane obtain repeat two step move project correct ease notation residual call replace assume figure apply unfortunately strong observe thresholding passing resemble gaussian rigorously prove class prove wise discuss subtle avoid non pass estimate similarly demand fortunately message pass algorithm amp correction derivation mp algorithm amp finding confirm display effective noise bm bm amp observation evidence simulation behave property leave empirical evidence theoretical implication conjecture main amp framework setting capital letter column transpose scalar element random event respectively expect two variable denote expectation value belong family index take require proper monotone level definition chapter class signal generic proper also eq every every proper since onto linear eq furthermore freedom cx complicated example signal sparse vector family short mention notational equal therefore straightforward see desire combine every figure signal instance image bm monotone monotonicity standard group monotone straightforward simple inequality independent monotone statement non improve monotone ingredient amp evolution measurement amp predict formally amp starts amp compare amp bm denoise amp figure amp check find bm mean amp wavelet simulation post validity explore conjecture range amp base require evolution call monotone noiseless amp monotone well amp claim monotonicity every induction every q hence converge conclude value combine amp value amp successfully successful goal measurement natural amp address evolution apply section transition threshold amp noiseless devoted presence measurement proper amp noise e amp recover noise sensitivity amp sensitivity variance level hence satisfy x calculation interesting emphasize noiseless setting amp theorem mse number measurement decrease certain certain variance variance perform amp rely effective parameter regard extensive diverse sure stein unbiased risk estimation scheme amp free produce performance amp jointly different notation set evolution change emphasize pick ask q note produce amp amp soft thresholding seem turn amp find greedy parameter tune call minimize denoise amp follow result prove optimality greedy strategy sense tuning induction hold monotonicity hence clear discussion amp difficult denoise amp researcher area denoise art bm employ optimally tune amp denoise employ scheme solve amp family recover outperform amp pose sense denoise recovery employ outperform amp use different amp framework amp section proposition amp recover signal measurement question ask recovery signal answer amp amp uniformly amp denote minimum accord dimensional integer fundamental recover amp class consider image amp unfortunately therefore evaluate amp optimality signal signal find amp employ denote evolution quantity amp minimize family note definition algorithm recover measurement amp mean employ amp unfortunately amp text book signal noise expect minimax family amp order recover amp require since proof appendix optimality amp less unfortunately answer example signal denote rest confirm standard normal amp involve amp ambient actual require case amp accord proposition class characterize signal amp sub optimality class natural image recovery amp eq proof state evolution low regularization recover cost regularizer consider return value many case propose solve accurately amp heuristic heuristic amp evolution theoretically sensitivity amp depend efficient amp optimally review amp optimization amp calculation correction employ passing study carefully direction researcher year area model approximate state analysis seem framework bayesian evolution framework evolution work observe goal calculate amp purpose amp state equation emphasize employ framework family definition set support eq side general hold state infimum information theorem set far key yet address provide calculate correction review correction relation give output popular signal literature group signal block soft calculate result mention elsewhere help soft denote x denote soft b word magnitude toward notation result signal diag employ rank divergence thresholding yield solution high dependent monte carlo work show calculate efficiently monte carlo obtain estimate signal efficient
replace shortest many coarse true distance sdp know fairly missing find location distance translate scalable sensor stress miss eliminate local approach exploit distance consider low classic integrate property reconstruction incorporate expect fidelity reconstruction section thorough evaluation array calibration simulate evaluate scenario vary number percentage distance present assumption pairwise hoc calibration local connectivity present hoc array vary uniformly diameter pairwise addition miss miss vary distances distance due limitation noise average quantify define evaluate illustrate deviation er rao quantify improve mc position cm reduce cm although mathematical calibration configuration pairwise figure improve express observation get term cubic room dimension simulate position configuration generate alternative implement sensor number distance source assume miss white require position ratio miss distance missing effectively handle require calibration miss e calibration present pilot synchronization model uniform model additional distance range cm diameter repeat configuration calibration quantify illustrate effect increase distance measure accurately small study room room acoustic room rigorously method approach set circular uniform array diameter cm array apart room dimension cm circular diameter channel array diameter cm deviation miss due distance miss scenario list repeat experiment reduce eight diameter cm c setup datum performance room cover big window rectangular room moderately time contain locate rectangular database sample pairwise distance two window parameter fourier compute coherence function frame state reasonable estimate confirm regard scenario nine estimation use rest demonstrate map stress completion quantify result belong stress cm comparative illustration mc propose well propose problem distance array method offer illustrate half alternative map poor entry stress sdp close mc mc property completion algorithm enable array partial pairwise distance demonstrate relation calibration channel circular also locate calibration list calibration eq position calibration line analysis propose calibration hoc partially recover missing entry euclidean distance project onto array calibration confirm exploit iterative reconstruction theoretical applicable framework ad hoc calibration find norm squared distance structure hence physical norm vector circle probability include structure miss necessary depict low probability location circular depict locate right upper bind figure circle eq constant union grow ratio increase grow positive great eq big extract use chernoff independent science foundation research interactive modal information management I acknowledge anonymous clarity manuscript ht symbol symbol circular square entry square cone radius define observe entry estimate calibration logarithmic distance error bar error bar correspond deviation mean calibration logarithmic quantify versus bar deviation ht position versus bar one standard error versus source total calibration standard pairwise effect mc quantify position error bar correspond miss ht evaluate number estimation trial cc c lr lr lr stress lr lr mc lr lr lr ht ht array c lr lr sdp lr stress lr lr mc lr lr lr array calibration c l lr sdp lr mc mc lr lr mc circle rectangle draw size hoc array consist propose miss entry alternative rank completion distance cone completion locally structure measurement know calibration level miss thorough theoretical achieve art hoc hoc array calibration cone completion hoc array sensor spatially acoustic ad hoc processing acquire sensor challenge attribute asynchronous position refer precise source localization separation study source specific distance source distance reconstruct array geometry self calibration source arrival source negative signal source signal arrival cross two source prior may et acoustic measure al exploit distribute platform arrival correlation use location since robust al auto exploit asynchronous spatially distribute acoustic impose sound align arrival measurement bilinear approach minimum sound chen likelihood distance extract array geometry joint source localization linear along arrival extract refined position delay source exploit structure rank enable position source minimal exploit calibration propagate equal direction coherence noise pairwise reconstruct line compact sub calibrate coherence field sound activate position response source use coherence field pairwise estimation practical assumption distant audio requirement ambient many provide enable calibration play increase goal pairwise miss arise measure activate device may acquire sensor fail capture leading distance ad hoc array distance base sound propose theoretical ad array impose distance euclidean square pairwise past year devise scheme recover know et reconstruct low et show optimality exploit nonzero regardless applicability array calibration proximity coherence signal approach imply connectivity pairwise far construct pairwise unknown enable pairwise approach goal combination measure array geometry pairwise distance cardinality final remove distance become effect entry ij ij ij ij sub setup assumption exploited state absolute scenario distance quantify position robust transformation denote distance summarize exploit rank recover recall distribute recover collect measurement rely carry ambient intuitively degree column recover define completion recover distance estimating row twice row row dominate characteristic thus entry decomposition singular guess provide minimization gradient last completion rank matrix distance modify composite property property modify step version transformation make output element characteristic error iteration algorithm cone figure serve dimension cone project follow inequality distinct projection thereby cost thus denote thus less summarize use search classic completion whereas yield reconstruction position completion well calibration provide hand singular decomposition loss incoherent great provide great explain great relation
nan orthogonal small close al consequently near suggest another author innovation science china foundation introduction china university mining meanwhile like molecular chen lin yu lda recognition small sample pattern recognition introduction usa g van computation implementation discriminant neural network al fold alignment dynamic journal theoretical biology perspective linear matrix multiplication al improve discriminant analysis vision application cm project wu com national foundation china grant natural foundation introduction china mining expensive recently discriminant multiplication scatter arbitrarily orientation may useful discriminant lose investigate nan guarantee orientation matrix discriminant analysis lda nan discriminant analysis nan objective remove irrelevant datum intensive processing mining powerful disadvantage scatter dimensional space usually centroid centroid scatter scatter matrix moreover generality assume realize scatter scatter scatter scatter singular sample overcome difficulty nan class scatter essence nan h satisfy theorem et nan nan lda refer et orientation matrix incomplete let suppose x rank pick matrix orientation lose discriminant choose criterion method satisfied sufficient orientation geometric since orientation eigenvalue correspond nan diagonal thus eigenvalue prove follow notice span span c therefore theorem give column position evaluate practice n upper
edge valid adjust begin node begin relate graph construct prove path path summation figure path score valid always label labeling produce specify latent come index label least different labeling clique pass position position except position form clique pass pass different clique difficult base world concentrate distribution mass rank reason since gradient objective analyze trend people maximize function portion regard higher dominate ascent latent label gradient degree rich trend rich trend also meaningful discover confidence concentrate task another control overfitte potential concentrated world find probable conditional viterbi definition search indicate search rank label mp p remain remain produce short probability map dynamic forward probability generate latent labeling leave labeling detail viterbi heuristic produce backward style compute corresponding try label remain label probable unnecessary remain p k label exact label large p condition k right n implement crucial design heuristic heuristic heuristic top probable probable easy way achieve find top score normally current current current heuristic algorithm never heuristic guarantee want try enough concern monotone heuristic heuristic informally formal monotone path generate monotone compute lattice search heuristic position set start heuristic viterbi score variable represent value step approximate try intuitively first intuitive alternative speed worse optimal approximated viterbi inference two step search labeling derive mean traditional estimate token compute way label marginal position naive exact rough estimation refer posteriori aim likely configuration complement researcher solve approximate exact search tractable require characteristic bayesian tight upper develop branch search map unclear crucial unclear concern make formal show np hard disjoint assumption importantly inference naive bound able fast latent latent field special popular language processing processing decode inference unclear try decode np furthermore able exact analysis decode hide capture conventional classification parse structure refine vision structure structure example area model advantageous representative widely syntactic parsing demonstrate latent recognition outperform widely svms hmms syntactic parsing without conditional programming viterbi nevertheless latent inference chain structure unclear limitation try simplify limited real distribute top try show reasonable field inference systematically combine top dynamic probable training class reduction hard basic notion value hard super lower know np clique clique labeling inspire consensus hide establish hardness analysis index define clique simplify divided score score
robustness statistic parameter shall type g follow composite hypothesis I variable proof independent statistic composite hypothesis theorem expression statistic normal easy verify property type test derive example justify level power test consider quality shape scale fact apply x nx nan hypothesis simplify similar testing q respect represent note asymptotically calculation asymptotic give asymptotically chi square freedom contiguous test asymptotic chi centrality contiguous centrality increase contiguous level r influence therefore influence become note always zero several boundedness second influence imply classical solid stability result derive function nan robustness asymptotic power contiguous alternative q q chi visible b statistic next level contamination proportion nan chi square power robustness different important applicability arise tune obvious control trade test pure contaminate robustness efficiency recommend enough robustness drive hypothesis testing asymptotic contiguous regarded efficiency example increase contamination increase approach construct minimum paper influence analysis carry justify influence remain contamination contiguous whenever influence influence justification behind power influence classical result exhibit chi establish test perform theoretically overlap one context model x great density derivative take function second order eq q frequently get get obtain note eq get follow axiom conclusion condition conjecture theorem exercise remark summary mm grant statistical university robust classical composite hypothesis base density power robustness property chi factor stable whereas confirm secondary density estimator influence chi testing hypothesis statistical procedure although misspecification small misspecification hard individually real life fold level test robustness robust stable neighborhood consistent arbitrary produce contamination nan contiguous robustness compare mostly develop examine robustness investigate global reliability although extended concept see besides consider statistic study reflect contamination nice review test however idea robustness study except chi overall contamination namely test popular develop empirically robustness fill develop robustness motivate researcher theoretical procedure also organize section section composite derive present justify result develop discussion choose conclude provide dominate lebesgue count two density derive limit case turn divergence density inference parametric density represent density divergence denote maximum estimating unbiased estimating equation nan equation behavior intuitively apparent n w np centrality central chi random centrality shall parameter define restriction rr type q type test statistic composite nan hypothesis chi degree possibility contiguous hypothesis consider increase move towards substitute equivalence nh power test contiguous alternative introduce crucial interpret influence influence function particularly robustness statistic observation underlie influence statistic view study influence influence establish divergence q function hellinger model know w simple hypothesis show adequate functional take test test statistic statistic al hold model power statistic statistic nan influence eq easy derivative contiguous important robustness look contaminate contrast contiguous contamination
cell alone find neighborhood connectivity converge quickly probable type chain detail similar recover know circuit generative broadly rest brain cell connect cell cell approach connectivity reconstruct cell hour divide cell type well approximated fit type distribute cell source closely reflect determine narrow medium wide cell fig reflect fig middle neuron author highly fine type even close agreement produce probable cell sort structure dark cell plot block roughly homogeneous collection along closely agree cell extent agree know red method outperform simple connectivity compare link fig far get particular distance depth lead ari human spatial region space essentially clique type extent layer variance continue substantially encode visualize cell type block subset cell cell group broadly coarse identify previous predictive curve inclusion source predictive accuracy additional conventional depth area type human recover connectivity repeat early unlike concentrated various project locate differ separate one chemical electrical initial cell connectivity electrical cell adjacency show identify cluster cluster position cell type coarse text label electrical determine inspection roughly neuron close agreement classification outline white cell reflect design neuron neuron vb mostly head together connectivity cluster pure correctly place single type reflect combination head combine mostly db neuron split thus reflect type entirely type artificial structure style spatial technology circuit identify processor complex bit homogeneity mirror original retain digital state contain additional identify spatial figure show horizontal vertical allow discover circuit like incorporation link accuracy discover base probabilistic probable parse cell connectivity carlo initializations converge similar obtain optimum broad offer adapt precise solution probabilistic become slow increase probabilistic area recent method work large expect close agreement exist become statistically cell know connectivity divide hand monotonicity connectivity class cell never less know insufficient neural wide functional discovery manual cell day type change understand modern allow discovery circuit development identification quantification phenotype quantify neural activity neuron across connect neuron one way already molecular marker could important cell appear molecular selective visual allow comparative quantitative aid combine connectivity way towards rich connectivity relation type one experiment brain rich model become human huge similar molecular biology gene find evolutionary biology ultimately resolve link connectivity matrix define connection cell material extension connectivity exist number latent cell assignment connectivity cell latent hyper jointly vector indicate cell step per parameter preprocesse runtime thank read manuscript discussion manuscript review amazon web services education grant derive test run kk manuscript text feedback interest material email berkeley edu probabilistic model incorporate entity extend take define cell material assume cell belong connectivity base well function jointly posteriori map assignment global bernoulli spatial cell type place graph process assignment determine automatically cell belong class number new global grid learn depth thus indicate conjugacy allow depth depth contact contact mixture mixture representation distribution contact study posterior monte anneal burn transition construct ergodic ergodic sampling lack conjugacy explicit assignment motivate auxiliary explicitly represent duration scan employ transition hyperparameter independent slice component slice hyperparameter global parameter discrete tuple transition chain kernel together otherwise chain random visualization full proxy prediction accuracy compute fully potentially overfitte bias favor compete collection computing cluster latent one vary dimension reasonable accuracy clustering dramatically generative setting collection markov chain pool probability profile connectivity actually inference mix chain evaluate sample distribution area proxy truth appropriate figure ari total runtime change initially temperature runtime characteristic jump nearby run ari importantly regardless exact variable parsimonious largely connection serial yield place contact cell originally select provide ultimately contact cell area thresholde entry input center logistic publish isolated chemical originally cell position distance axis normalize direct electrical undirected distance poisson extract gate source unable consistently source ambiguity effort create six terminal example connect likelihood probability poisson explicitly count neuron distribute learn rate cell close cell per parameter source code source code run along figure available please contact publication access multiple graph handle simultaneously extend product likelihood grid space poisson space lambda set adjust rand ari clustering identical ari become negative
denominator resp bind upper take numerator term experimentally cc x whereas context variable absence direct pathway exposure lead improve lower certainly case relationship consumption relationship plausible analysis allow direct unobserved keyword analysis explanation study conduct cause law want causality discuss topic claim ann drug death cause observe hand medical create address happen ann take drug question ann drug assess cause prediction formalize denote ann drug code resp address query decide I prefer associated decision situation query drug would try imagine would ann take drug ann actually drug likely drug address long purely condition know nevertheless answer introduce response exposure regard exist choice response observable observable fact contrary address name triple define ann q ann knowing actual response probability response ann drug quantity suppose experimental tested take ann live variation proportion compute table make causal x experimental drug ann drug death ann exchangeable individual subject exchangeability identify datum never observe never dependence inequality experimental risk death case trivial exceed often assess converse false find likely causality ann refine pre condition original fail adjust ann ann conditioning ann even ann sometimes refine thus know deduce ann refine bound wider far perhaps conditionally assumption ann x able observational ann joint observational bound experimental table observational live thus find deduce ann take drug involve pathway exposure causal split effect experimental ann additional evidence refine formalize
approximation semidefinite fast importance sample hadamard paper extend show perspective improve notion bound convergence approximation understand kk call original nystr om low within matrix use score column kk interpret implicit set form kernel accomplish variant inversion algorithmic statistical adopt exploit generally compare nystr sampling answer open zhang et divide parallel prove evaluation require evaluation notice comparable om leave open fundamentally well indeed nystr improve obtain statistical combine approximation let cast minimization reproduce kx reduce optimization solve matrix datum sequence square estimate point shorthand g risk estimator decompose decrease matrix proof need cubic running approximate low operate small matrix contain sample nystr om matrix multiplied produce preserve desirable om trial also random perhaps satisfy structural behavior concentration one statement section nystr om begin upper estimator construct place nystr om process uniformly zero equal sample column find b easily change notation arbitrary deterministic statement property randomness via highlight adopt offer clear result satisfy dependent algorithmic improvement theorem k monotonicity replace suffice second random concentration notion statistical leverage use one art bound concentration among constitute tailor yield still result integer replacement accord hold conjunction prove result eigenvector square matrix projection soft truncation smoothly smoothing close analog expect ss exhibit sampling robust quantification tail much effect optimality distribution reflect particular tight prove leverage ridge nu formulae na compute ridge quantity statistical literature classical hold I base sampling ridge leverage information column uniformly small accuracy high leverage become essentially equivalent away performance point still ridge na compute matrix well roughly take approximation description construct construct th time cholesky multiplication approximate leverage thus run note involve everything compute dimension correct inversion compute formulate memory construction involve additive multiplicative quality nystr om approximation probability remark column sampling square extensively randomized gram multiplicative scale instead virtue multiplicative weak minimum eigenvalue denominator concern think like sufficient column sample need easy see ridge score suffice albeit number particularly subsequent completion preliminary eqn leverage whether iterative technique score rank regression real dataset consist database provide method goal fold simplify setting ridge h nb linear fm nh fm nh former goal dataset freedom result small degree freedom work synthetic sequence uniformly score nearly optimal fact prove distribute ridge score importance sampling beneficial see density unit center unit observation figure leverage point low approximation leverage provide nystr om ridge effective improve upon attempt achieve combine improve rank involve analog notion score sampling score run depend algorithm formation full dimensionality addition unclear obtain logistic finally well recent eqn acknowledgement thank zhang point bias inequality exist sequence iid p ik kx px give fact estimator eqn given hold standard bind approximation appendix get bind derive low approximation appendix finish low fourth multiplicative bind l ik
define characterize function commonly kernel radial rbf rbf exploit sensor representation dictionary space sensor sensor support mapping feature kernel formulate eq associate sensor incorporate sensor share pattern sensor integrate collaborative share sensor ms feature kernel need express multiply fidelity reformulate mi dot dictionary atom dot product atom coefficient class kernel represent dot sample c ms experimentally method signal significantly improve effective interference unfortunately noise interference time kernel integrating learn incorporate improve analytically show enforce noise interference structure benefit assumption describe interference transformation sensor test still datum type dictionary description adapt corruption similarly trick mm ms classification involve kernel datum conduct nine sensor consist four sensor fig type sensor human lead human person people people run whereas one people people group test vary addition ideally would discriminate human datum difficult researcher laboratory run two nine starting sensor backward separate run collect accurately actual raw short period test subject sensor addition arbitrary useful detect location physical occur identify maximum detection second physical sensor run nine sensor divide overlap visually demonstrate signal sensor test visualize sensor sensors segmentation extract segment first accuracy fig visualize observe distinct framework ms exhibit performance sensor utilize ms classification sensor interference achieve examine sensor close take close look understand obvious fig boost sensor multiple demonstrating improvement alone average improvement sensor range among demonstrate even sensor type different acoustic event always achieve collective fuse always close cm sensor multiple sensor combine ms ms data cm sensor multiple sensor ms ms svm ms ms l cm cm h ms ms ms discuss unknown component sensor optimize effectively dataset validate model ms noise somewhat ms truly discussion interference likely co turn problem throughout broad enforce complementary information homogeneous heterogeneous sensor enhance moreover ms well beneficial counterpart sensor broad incorporate carefully classify induce domain significant improvement ms notably well sensor utilize ms l always single sensor ms averaging examine yet interference structural sparsity ms multi signal effect interference propose various source different interference linearity practical collect laboratory reveal complementary multiple significantly improve sensor appropriate joint bring classification interference sensor induce tool understand adaptive efficient although verify specifically rather locate present brief firstly close take f derivation sketch optimal derivation skip include limitation point yet converge paper collaborative sparse take heterogeneous sensor interference incorporate interference low rank multi sensor co locate simultaneously record physical event test training efficient convergence guarantee collect laboratory effectiveness propose automatic sensor discriminate classification multi sensor numerous detection sampling simultaneously locate source physical event exploitation complementary feature multi fusion classification information source sensor improvement classification mostly two category decision decision di di collect sensor decision incorporate method visual acoustic sensor furthermore compare di di counterpart paper sensor datum category appeal sparse interest inherently sparse dictionary approximately significant signal bandwidth storage efficiency also separation classification recognition furthermore allow simplified structure thus effect noise sensor novel sensor effectively scenario interference scenario collection external inherently stationary collection normally appear interference manner interference sensor spatially locate thus interference external effect extension collaborative induce feature method structure sparse ms impose within sensor sparse ms interference ms sparse regularization integrate enforce row sensor trick ms ms ms multi detect organize introduce sparse interference efficient direction multiplier solve optimization problem representation experiment vi conclusion recent year signal recently belong cp sparse simplicity presence discard description though fidelity assume coefficient particularly sample vector classifier problem employ coefficient drive extensively investigate sparse provide approximate collect variety precise classify test vote answer nearby spatio compactly p sparsity support row sparse row extension aforementione observation show application e sparse regularization sparsity common representation fit system capture simultaneously seek exploit homogeneous heterogeneous handle sensor employ di sensor aforementione describe perform selecting source decision propose ms classification sensor collaborative illustrate notation dictionary contain call modality sample feature modality pc cc belong segment signal segment simultaneously partition sensor segment sensor reconstruct atom dictionary support correspond measure physical observed sensor approximated matrix row pattern atom index solve generalization representation use ms joint representation present simplify lasso decide residual associate interpret assigning collect environmental environment source arbitrarily magnitude dominate accuracy obvious remove clutter fortunately certain band environment coefficient experiment respect account entry arbitrarily magnitude nonzero represent clutter sensor since type scenario contaminate dictionary develop al recognition remove clutter take advantage retrieve form matrix encouraging row entry wise compute slightly account presence noise capable large term interference often external source sensor sensor platform interference pick corruption interference car pass nearby interference frequency many signal alone record contain also intrinsic underlie train portion record span especially locate hence interference multi sensor representation rank interference l expect tackle problem extract rank correlate information sensor ms component wise low jointly balancing note dictionary interference look practical pca separate sparse rank hence structure address ensure bring flexibility side restrict accurate segmentation key source separation heavily incoherence atom base active area year explore large dense interference sense recovery representation assume interference change slowly rely project interference onto current anomaly constructive able deal collective structure across capability extract inherently exist outlier depict cover flexible model ms collect band nonzero sensor failure rank corrupt certain zero column interference component location distribute consider rank versa presence low interference eq ms capability exploit multiple sensor coefficient enforce level boost even incorporate coefficient study theoretically prove well beneficial task rather come group coefficient enforce active inside force active two level sparse search among sensor interference term ms concatenation matrix label encourage wise sparsity group regularizer minimize account interference group sparsity parallel extracting interference appear coefficient term label decide form become furthermore I ms interference constraint case sparse sensor propose
marginal confirm relevance covariate confirm explanation intercept fit capture addition fitting show three burn profile proportion fall profile likelihood interval limit compute indicate prior little impact respective inferential purpose mcmc width cccc sensitivity distribution distance standard hellinger distance default prior hellinger distance hellinger posterior show prior hellinger distance hellinger distance prior posterior plot cccc priori result parameter substantially parameter parameter posterior prior default prior hellinger posterior change difference difference conclusion unchanged since change small random default std report analysis beta one emphasis place sensitivity prior index worker indicate company add intercept fitting criterion one gain burden attractive dispersion beta mixed hellinger measure distribution show beta dispersion insensitive slightly unchanged social da ep model vast area linear mix response poisson despite proportion situation usual adequate likelihood inference mix demand call integrated nested laplace allow inference discuss model compare obtain mcmc life collect beta produce easy handling discuss mixed model reason popularity generalize glm effect flexibility hierarchical repeat extra variability view extension probability variable poisson binomial majority find despite flexibility response index traditional base adequate bounding ignore display skewness regression model identically distribute beta propose follow link glm extend precision covariate explore beta regression correction develop genetic recently variable unity interval overall prominent beta implement package r statistical add correction mixture mixed model analysis beta series rate beta bayesian gaussian specification prior distribution parameter inclusion inference two solve effect presence approximate adopt carlo come overhead upon attempt informative prior inference mix attractive specification prior novel numerical inference integrate nested computation enable prior accurate guide describe bayesian discuss measure model flat aid issue idea default choice assess model straightforward analytically available carlo mcmc technique fit computational moreover implementation problematic user program software user nest laplace model focus marginal replace deterministic marginal implementation call perform serve interface usage density several goodness procedure specify sensitivity include goodness fit predictive detail develop general hellinger assess change adopt assess choice shift change hellinger correspond hellinger density happen whenever assign density assign vice see eight area health education health development attribute stimulus united unit close life quality conduct social service da life survey eight unit company first divide worker individual health education capability provide life relevant question
show correct solid peak wide peak narrow likely end intensity original curve correct solid basic seem bias interval attain frequentist spectrum section note peak I bayes invariant wide unclear high energy unfold empirical regularization quantification hyperparameter wide appeal classical strength validation discrepancy unfolding unfold yield happen smoothness value appropriate true intensity peak rapid potentially biased situation case suitable highlight interpret mind main frequentist quantification solution bootstrap resample interval serve estimate moderate attain nominal take confidence effective bootstrap part presence intensity bootstrap unable probe away bootstrap blind unable account confidence bias problem elaborate scheme one regularize quite surprisingly I bayes even former take likely bias bayes coverage serve useful especially interval nevertheless clear frequentist interpretation interval unclear interestingly alternatively perhaps link would significantly help bayes least asymptotic acknowledgement wish discussion de ed call unfold spectrum elementary particle measurement due resolution detector arise formalize intensity unfold principled quantification uncertainty inherent argue deal satisfactory attack unfold bayes coefficient expansion unknown employ marginal maximum hyperparameter regularization drive hyperparameter credible empirical bayesian interpretation understand use confidence intensity methodology simulation real large inverse physics uncertainty quantification monte carlo em study unfold produce european organization nuclear powerful particle order interaction elementary particle produce trajectory energy use vast experiment analyze conclusion law complex quantity challenge unfold particle detector detector could g production particle due induce stochastically version distribution ill inversion map unstable perturbation trivial exhibit spurious take additional plausible non realization unfold intensity represent detector hand inference nature approximately unfold furthermore account observation rarely use datum early certain energy physics value heuristic provably account effect incorrectly recently problem practice difficult physical interpretation impose early stop iteration unfold positivity deal significant strength quantification solution strength analyse scenario quantify uncertainty spectrum analysis rarely account relate work unfold characteristic momentum top production propose unfold aim principled main bayesian expansion selection regularization monte carlo maximization frequentist quantification previously unfold account poisson positivity impose curvature solution unfold emission discretized difference unfold unfold interested scale enable intensive one quantification physic rarely naturally discretized expansion appropriate well contrary parameter one unfold receive attention recent good choosing level noise approach use regularization become computer frequentist uncertainty quantification credible confidence statement hyperparameter interpretation standard moreover quantification explore albeit computationally construct intensity ignore sensible use place regularization parameter enable automatic bayesian quantification dependent methodology carry bayes frequentist need background produce role formulate detail form real invariant mass experiment conclude remark km circular locate unit physics powerful accelerate particle move direction lead detector experimental every ns detector per detector detector capable variety range discovery study detector source unfold principle experiment compact france diameter operation international matter new particle interest physics particle decay familiar particle energy trajectory particle record create mcmc produce estimate histogram particularly attractive strength comprehensive relate bayes g chapter idea regard denominator use technique maximum maximizer non evaluate monte carlo integration approximate question space region rough maximization em algorithm find approach originally later little apply unfold log compute hyperparameter constant maximize hyperparameter maximizer incomplete coincide enable find hyperparameter involve integral monte replace approximation metropolis hasting sampler monte involve eventually arbitrarily maximizer find hyperparameter compute intuitive interpretation produce summarize understand tune match well match become iterate monte expectation equation least correspond reasonable intensity prior behave mostly intensity plain equation considerably take depend hyperparameter constant plug available combination agree ability partially estimate rise bias discussion resample conclude section note bootstrap intensive procedure outline since bootstrap replication matlab parallel toolbox computation generally roughly fold setup demonstrate unfold simulated data mixture true process fs e ef noise variance boundary discard setup setup deconvolution problem discretized histogram bin discretize spline knot unknown indicate ill pose set paper ghz intel relatively sample start iteration start spline significantly ill pose unfold number iteration metropolis post burn convergence size whole repeat replication obtained resample minute run whole min bootstrap core core bayes unfold pointwise percentile ill intensity obtain estimation pointwise percentile peak despite tail moreover percentile cover interval true intensity plot na I bayes confidence long percentile interval bayes unfold sample size pointwise percentile I bayes hyperparameter converge easy sample regularize rate make component diagnostic component chain converge close boundary convergence bar interval straight add appear slow plot histogram sample lag corresponding effective regularization ill pose parameter burn autocorrelation effective slow mixing apparent trace mean base exhibit also depict negative mcmc behave one try fall experiment converge variation hence increase sample enable probe bias point take illustrate iteration increase posterior diagnostic plot final indicate mix intensity represent near peak bias correction correct capture true intensity always come cost visible boundary basic albeit price slight conclusion coverage simply repeat several observation amount suggest subtract intensity normalize na I confidence interval unclear length move intensity bayes bootstrap blue band consist pointwise basic bootstrap show na I curve curve unfold size independent integrate integrated fs scale straight indicate slow illustrate unfold invariant spectrum publish produce decay almost particle decay decay resolution serve unfold intensity remarkable precision detect energy particle two particle rest reconstruct angle track mass preserve particle invariant mass spectrum enable uncertainty mass often simply call width half near peak dominant source measure energy resolution energy principle mass peak ignore resolution decay
topology cluster decision merge big therefore topology entire step nearly time decision invariant high become nearly invariant gain enhance clustering choose proper combination policy recursion know policy exploit benefit include metropolis coefficient determine neighborhood tend determined recursion large block block recursion involve cluster minimizer big group section well simulate agent agent observe ki identity also cluster belong belong load two namely step underlying connect fig agent blue simulate grouping begin non topology plot fig metropolis steady decision time cluster merge big link active link cluster topology active link fig topology imply interference cluster network connect steady fig topology fig fig separate big collaborative involve obtain trial first red obviously steady form cluster simulate node initial five topology five cluster fig respectively curve average theory fig learn detailed conduct segment sub enhance interference prevent node normal furthermore adaptive objective step size technique establish sequel introduce complex multiply side vector independent hold I recognize similar recursion drive immediately follow examine covariance next recursion evolve recursion eq jensen jensen get property norm converge satisfie lyapunov identify jensen q rhs substitute I first rh substitute extend denote arrive establish stochastic recursion I condition expand gx gx around denote gx jacobian e real parts martingale difference moment namely asymptotically weakly mean unique lyapunov recursion positive obvious unique condition easy recognize assumption eq jensen eq weakly mean follow convergence region constant rhs b fact marginal rhs chebyshev likewise substitute since drop subscript mean c moment verify substitute yield process neighboring share common many agent belong objective learn neighbor ignore result enable cluster attain accuracy carry probability ii false alarm mis detection establish correct arbitrarily diffusion consensus adaptation learn unsupervised distribute agent applicable wide gradient incremental determination np even topology consensus focus size adaptation learn response problematic size consensus diffusion modify step would decay active implementation gradient consensus size grow unbounded network suffer problem regardless especially context change motivate diffusion proper consensus exist algorithm agent common correspond minimizer interested different work investigation separate cluster considered collect arise location separate need agreement important appear multi problem introduce different different vector adjacent assume formulation many scenario problem involve multi problem appear assume fully network square minimizer mse moreover know belong estimate adaptive fundamentally study mean square risk agent handle broad situation adaptation objective word study component truly objective avoid belong agent interested application sensor move direction share beneficial amount agent belong link neighbor agent make cluster learn well highlight quite label neighboring agent exchange may certain accordingly devise strategy allow agent result correctly attain performance intra letter letter letter inversion trace besides use semi associate assume minimize accord minimizer agent mutually exclusive consist cost minimizer I different cluster share common aim q j agent network cluster topology minimizer employ strategy collaborative cluster mean every consist neighbor belong cluster circle agent well cluster neighborhood neighborhood cluster split sub fig challenge cluster completely cluster already cluster introduce group denote agent know topology information five two group five merge group partially neighbor fall group fig trivial cluster member cluster lead agent access information neighbor leave cluster devise enable agent automatically time turn solve evaluate sufficiently enhance network adaptive summarize main consist cluster denote link total denote cluster one agent indexing rule generality accord cluster consecutive index likewise index agent index index accord indexing belong belong either agent belong formulation section aware cluster agent initial aware cluster information learn group share cluster group group start neighbor procedure grow enhance collect individual minimizer equality stochastic guide relate ensure minima pose problem relate gradient process approximate true vector gradient approximation unbiased moment regularity assumption group strongly low hessian j assumption relate condition purpose function denote j denote random aggregate noise across I write noise martingale difference moment lipschitz function easy minimize cost cluster available partition group one group information gradient minimize cluster formation include shall argue able gradually leave stochastic minimizer w indexing definition I subtract side recursion nm recursion indexing eq express isolate automatically network stability agent loop primitive condition appeal sufficiently error recursion sense network square recursion strongly primitive left square stable long dynamic recursion dependent random recursion lemma square steady approximated long nm albeit continues drive similarly recursion term size term mse recursion immediate dynamic term variable two detail reader detail motivate low turn e correspond centroids evolution evolution couple arrive need equivalently indicate prior primitive frobenius inside unit eigenvector associate eigenvalue normalize add p entry ahead denote block unit rank rhs represent contribution eigenvalue ahead centroid eigenvector stacking indexing rule dimensional compare p g matrix hessian collect group recursion stack recursion describe centroid expand manner accuracy sufficiently give via error positive lyapunov know within steady metric asymptotically representative steady sum give examine normalize error steady lemma mean original recursion low centroid steady norm covariance bound jensen normalize finite steady moreover apply positive enough let taking gm semi nonnegative ii large desire equation verify positive accord gm equation reduce rhs continuous lyapunov network steady error satisfie assess hypothesis recursion use weak normalize solution lyapunov sequel distribution variable f fact lemma weak follow asymptotic normality converge triangle dimensional sequence sense rhs vanish likewise lemma verify variance lemma term vanish vanish vanishe allow enough sufficiently small individual say I individual minimizer th block let r possess obvious k pair agent distribution joint need decide pair difference k I serve sufficient true otherwise hypothesis difference know available pearson criterion ratio unbiased covariance predefine mn mn q central f stochastic sampling steady state sample carry replace identity square available quadratic vector k appendix dominate step chebyshev
use assume usual provide base question reduction sum contribution use policy q h n h three term suffice use come entropy entropy completely prove prove relation unconditional expectation note iterate conditioning apply conclude apply information theoretic moreover reduce entropy must q outcome true indicate inequality strict object question pp f hx achieve low example notation example strictly dyadic distribution next answer answer collection support question provide among binary set indicate etc consider vector matrix code object location characterize vector must exactly locate thus observe answer question describe lem seed fix event rewrite two nc j jx imply explicit characterization immediate follow du kf du previous analysis greedy dyadic setting question suppose object locate uniform dyadic possibility suppose otherwise dark subset mark object predictive I distribution history external source useful expect dyadic demonstrate proof lemma respective kf verify equal complement use conditionally provide explicit probability let random distribution poisson denote probability poisson binomial put together characterization probability mass mixture poisson binomial mass non allow environment description concern policy equality asymptotic dyadic policy definition dyadic introduce useful partition ask time ni du martingale martingale converge variable nk direct precede assume almost prove q lemma surely imply nz dyadic ask object line single case dyadic actually greedy deterministic respectively process deterministic law large illustrate normality entropy dyadic nk present greedy despite dyadic greedy value provide outperform dyadic section inequality dyadic policy already question expect reduction might greedy dyadic although deriving seem impossible following greedy fix history recall borel subset distribution policy conditioning rewrite question dyadic question theorem f kf rewrite borel borel continuity construct union element construction corresponding mass exist sr attain define class greedy policy low take previous argument lemma dyadic policy circumstance first question dyadic dyadic question consequence interestingly simplify specifically dyadic answer aim multiple testing vision bioinformatic entropy two policy dyadic employ pre set thus dyadic policy relatively certain assumption answer answer paper know square entropy measure differently question acknowledgment like thank chen eventually lead nsf institute grant spatial nsf nsf fa adapt time maintain interval obtain choose interval portion create version sequential differ policy design rather design case object question noiseless aim object set characterize dynamic programming equation curse dimensionality prevent tractable computation first explicit optimal greedy maximize dyadic split fine dyadic dyadic easy relative greedy intensive robust possible numerical outperform divide benchmark dyadic showing distribution asymptotically sequentially subset noiseless answer study devise method question find finite question accuracy dynamic tractable bind minimal analyze method similar object game game person million query find yes allow lie time game continuous problem less probability least lie analyze among work consider probabilistic originally dyadic generalize policy object work previous multiple object subset game game tell reveal thus answer either otherwise consider problem answer count localization construction code search auto collection failure electrical computer screen state problem value represent location interest assume ask series take answer answer previous question randomization previous call choose question policy sequence subset ba ba highlight expectation simply distribution equivalently distribution compute proportional learn final denote policy question characterize optimization attain infimum partially principle via dynamic prevent force easily policie greedy dyadic step forward borel subset dyadic cumulative density dyadic I value dyadic policy one sized illustration dyadic set dyadic generalize object ready main dyadic binomial theoretic inequality second inequality trivial computation observe answer question present performance strictly dyadic last equality special dyadic policy illustrate question entropy posterior figure dyadic right policy benchmark benchmark dot line lower dash question dyadic policy solid greedy identify single question location question bit object set somewhat problem screen
feature method redundancy maximal relevance q measure pair already satisfy greedy feature redundancy double relevance modification information effective mi feature selection pairwise redundancy mutual greedy heuristic independently criterion share algorithm know search weight instance evaluate relevance redundancy dependence etc arithmetic operation apply evaluation iterate conduct compare construct top bind information may result select achieve conduct result four cs implement implement sense typical incremental matlab slow cs time consuming admit drawback try lp execution conduct ghz gb fig accuracie four ten x consecutive depict ht ht ht ht ht ht ht ht ht r cm c p avg fig verify superiority perform well particularly five vs kp synthetic control accord six select good feature illustrate superiority subset pairwise approximation apply begin come clearly synthetic dna fig select independence rather measure redundancy avoid optima perform bad redundancy perform superior consider feature kp seem effective reason pairwise redundancy analysis effectiveness strategy performs control inferior possibly many e suggest relationship dependence obstacle determined break cs dependence select feature consist inferior eight nine synthetic end select estimating g result inaccurate obstacle feature illustrate cs feature grow stage addition cs evaluate radial may output end radial technical performance dataset fig thus fit cs determine term super efficiency score candidate feature currently obvious super large mi based mention rf selection select feature take index cs super salient strong conduct relevance redundancy analysis selection individual cs explicitly candidate take output dependence constant super evaluate efficiency subset increase validate effectiveness four widely classification ten uci difficulty sample since conditioning dependence drawback whole cause world improve criterion take future tackle super apply evaluate input orient thus input evaluate consider trade output pareto measure enhance ram consider future make output thank anonymous constructive comment helpful foundation china fundamental central fellowship foundation science technology author acknowledge financial china gray gray gray gray gray gray paper novel separability cs strategy relationship handle class super rank via feature super conditioning iteratively eventually viewpoint empirical verify feasibility superiority propose classification pose challenge recognition get large type become computer internet amount type rate realize important frequently reduce dimensionality remove irrelevant redundant application acquisition result speak filter select feature classifier agnostic criterion g score mi uncertainty schmidt mi efficient although information search combinatorial investigate construct former discriminate np usually conduct hand accord criterion find optimal contrary latter perspective many belong kind manner heuristic intra inter class distance etc however sort relevance especially high never redundancy lead redundancy relevant redundant high relevance redundancy apply selection relevance criterion relevance redundancy denote select perspective manner representative make trade simultaneously relevance redundancy one comprehensive conditional original set typical relevance select effective mi mi feature selection criterion note belong magnitude dependency rather efficiency estimation hand mi research indicate pairwise etc mi eq parameter correspond example correspond joint may detail item multi index employ programming unit production structure recognition mining attract increase attention effectiveness order recently zhang nature give overall among rather arithmetic operation overall sum relevance redundant drawback redundancy every consider thus feature take index propose selection separability label apply handle super efficiency via conditional score remainder paper organize metric section super implementation measure experimental evaluate compare representative discussion present summarize conclude remark mutual mi quantifying dependence mi variable form measure sense random give variable interpret nonnegative super method concept use represent cs mi evaluation super search identify salient evaluation currently select greedy follow end newly select currently subset th cd measure concept carry irrelevant advantage attention redundancy conditional dependence frequently mention dependence metric classification class assignment predict class label accuracy reflect include exclude feature redundancy e may redundant class example mutual mutual candidate guarantee dependence distribution feature redundancy dependence e histogram label evaluation evaluate feature cc label label cc ct cc capture label word mi artificial feature illustrate new place class hence mi capture absence partially sample label measure sample keep nonnegative whereas g candidate magnitude I select candidate scale system constant candidate feature take evaluation execute show greedy search ht cc cc model simplification standard focus hence avoid focus gap make sometimes class often influence select separability feature ff rf p pseudo code loop step stop make mutual medium scale loop totally bad line super efficiency candidate solve complexity analyze estimation mutual process
competition aic cv criterion poorly contain frequency weakly identifiable stand setting model small correctly select covariate table perform identifiable bootstrap aic outperform base size criterion employ likelihood identifiable nonetheless size conclusion selection criterion generally lead criterion well among aic criterion accurate display good performance finite criterion considerably criteria select regressor mean relative dispersion table employ regression competition table major city unite propose scheme take covariate show selection regressor mean selection scheme contain code beta regression people logit link dispersion include covariate quadratic transformation assume regressor include dispersion std constant people people ht correctly residual plot see detail diagnostic tool refer standardized th figure residual slightly still fit similar envelope indicate envelope point lie envelope band relation mean number people interaction relationship number people response dispersion beta fit dispersion two criterion vary dispersion criterion aic expect likelihood criterion bootstrap cv quasi cv typically lead alternative mean dispersion application discuss acknowledgement acknowledge financial information aic widely aic measure sample aic bias tends select criterion likelihood cv performance aic variation dispersion regression discuss aic bootstrap validation dispersion practitioner usually interest select yield broad criterion information commonly aic maximum maximize log likelihood bias bias aic asymptotically correction expected likelihood aic correction expand cover regression autoregressive show aic analytical correction difficult model analytical well certain restrictive analytical obtain correction aic explore class criterion bootstrap outperform aic sample additionally expect maximize paper approach expect adjustment nonparametric bootstrap cv cv parametric bootstrap cv quasi cv modification autoregressive model cox beta model tailor modeling proportion dispersion regression generalize beta dispersion goal model selection beta performance regression extension new regression monte empirical density measure discrepancy kl minimize kl follows formalize candidate sample denote possible family class small contain family use fit collection fy fy fy k fy fy times kullback notice evaluate require asymptotically biased minus adjustment aic sample aic develop develop class extend regression autoregressive asymptotically aic aic estimate discrepancy aic applicable regardless derivation bootstrap estimator bias small lead reliable bootstrap recommend algorithm quasi cross treat cv validation obtain cv another variant give study field proportion beta beta distribution quite since shape value index function variance respectively vector dispersion dispersion beta observation regressor likewise regressor dispersion include additionally strictly monotonic parameterization probit log complementary cauchy discussion beta identity perform beta likelihood respect information respectively maximum numerically likelihood goodness transform ratio maximize maximized function measure correlation propose regression investigate variation simulation carry analytic derivative beta replication large value notice yield bootstrap aic criterion covariate logit evaluation criterion process dispersion parameter identifiable identifiability covariate terminology identify differ relate uniqueness small present result dispersion cv aic cv c aic l cv parameter dispersion size sequentially nest dispersion nest candidate select three
minimize aic criterion bic know towards modify cross aim aic correspond sure bic potentially consider practice lasso approach select coefficient exception cross validation rule formula remain determined well known formula word condition rarely application choose property important selector proxy tend bias true bias estimate bias shrink residual unnecessary less optimize aic poor study result simulation jointly asymptotic normality cross validation also lasso thresholde selector base resample like exist calculation criterion achieved calculate single appealing new rule aim true predictive wavelet smoothing localization massive carlo simulation inspire compressed phase selection phase estimator operator employ sparsity wavelet conclusion regression fully selector instance property indeed nan one threshold quantile connected power entry good motivate extension rarely except total former control I gaussian control discretized bridge thresholding asymptotic counterpart threshold specific structure universal quantile must single cross involve establish orthonormal parameter thresholding selection letting tend conservative nan concentrate compress rich interesting vary role carlo range nine region time identifiable lasso experiment massive study calculate characteristic coefficient thresholde oracle estimator include correct property oracle oracle include small model practice oracle fdr stability sl serve benchmark discovery number cv ss method offer fdr know whether impossible predictive performance test choice performance preferable performance certainly many gene repeat one set factor record coefficient select predictive square respective coefficient risk tend select pointing capture right consider conservative bic take set leave observation partial least cv ht explore rule perform simulation fix sample screen nonzero randomly index select finally conduct let vary sparsity ratio fdr cv sure sl median fdr change sl poor fdr concern ss lowest closely follow linear wavelet regularize show employ ill pose radial block bottom cluster radial profile consider encounter galaxy emission function see end intensity measurement line intensity radial profile tend zero infinity hence simplification q profile expansion ray eq figure center function bottom radial employ sparse wavelet rescale sake identify go along haar wavelet factor wavelet noise ratio vary bic sure method replication fdr mse mse select sure fdr fdr fdr mse expect show fdr conservative snr lead fdr selector selection seek controlling behavior nan recover like stem smooth thresholding universal good low fdr single technique cross support finding extension generalize author science estimate noise sense selector identify covariate thank yet selection propose quantile extensive high positive low discovery achieve predictive keyword discovery lasso universal threshold relate covariate level consume technique microarray identifying significantly reveal disease modern device covariate therefore analyze exceed concentrate relate model error estimate also performance overcome paper concentrate last estimation decrease bias prominent ridge principal square three estimator assume make reasonable regularization technique govern control trade
h operator concave e g easily lemma assumption indicate table usual satisfy penalty affect analysis penalty nonsmooth general nonconvex nonsmooth base motivate right side solve follow update dc weight nonconvex nonconvex norm easy lipschitz update still nonconvex fortunately w globally ii iteratively lipschitz computable backtracking estimate section decrease monotonically continuously lipschitz sum inequality equivalently accumulation sequence algorithm subsequence k g k j semi subdifferential exist w j exist jj algorithm follow satisfie hold loop generally solve follow divide singular experiment effectiveness repeat frequency success plot legend function performance alm nonconvex accelerate lie relative plotted logarithm mcp since globally gaussian real singular dominate recovered image channel apply matrix ratio task nuclear solver admm solve code admm try tune report real image replace pixel add image scad plot evaluate nonconvex situation truncate nuclear nonconvex surrogate rank nonconvex concave monotonically increase problem able general nonconvex rank convergence nonconvex nonsmooth limit local real demonstrate outperform interesting future nonconvex problem combine alternate multipli admm acknowledgement research foundation international centre office lin support china computer national technology laboratory school university com mail edu cn edu cn surrogate functions nonconvex penalty enhance vector nonconvex singular value enhance recovery solve nonconvex low nonconvex exist concave monotonically iteratively norm nonsmooth setting closed real monotonically solver observation aim nonconvex nonsmooth minimization problem singular nm ng monotonically increase nonsmooth eq lipschitz nonconvex iff vision fall square loss adjoint know rank many segmentation work prove incoherence nuclear near rank violate solution nuclear may approximation nonconvex norm norm absolute deviation scad logarithm mcp property function rank another norm nonconvex sparse dc difference convex function program nonconvex dc function dc program programming reason nonconvex proximal proximal nonconvex solver even q low nonconvex minimization relate square relax iteration general nuclear loop compute efficient solve exist nonconvex surrogate norm extend penalty table observe nonconvex
value response consider follow smooth integrable space real within value variable unknown order turn affect estimation note estimate density term flexibility stem unknown form drive grow literature nonparametric functional estimator functional distance local mathematical dt generalise product admits value real value predictor distance associate infinite dimensional predictor second kernel predictor theoretical semi metric increase explanatory non smoothed principal smoothed semi use derivative reader real predictor express dimensional value order gaussian binary consider discrete notice x gender case natural ordering include rating discrete express determined function unknown know smoothing always nonparametric estimation prominent divide part square increase decrease select balance square bias regression value regressor functional rv estimator functional design predictive certain measure curve integrated mean integrate optimality cross addition validation appeal affect optimal inferior accuracy instability error regressor residual obtain functional approximated leave q represent residual residual parameter approximate leave nonparametric estimator residual square since gamma ig density hyperparameter assign small squared keeping result hyperparameter along possibility sensitivity table kernel bandwidth uniform bayes express parameter estimate h independent correlation error metropolis burn period first iteration record burn mean observation kernel heavily affect residual may cause use study observation low region express b goal way compare estimated error estimation replication briefly describe simulated build draw take compute also replication x function generalise value real predictor binary categorical error average function nonparametric regressor regressor latter regressor accuracy improve discrete ht cross continuous value residual estimator bandwidth term integrate discrepancy replication discrepancy form bandwidth discrete continuous model un panel panel path reasonably mix ergodic credible se se density se prior ig l cauchy em language check diagnostic diagnostic pass replication I regressor regressor observe two value two discrete point regressor smoothed bandwidth irrelevant regressor smooth exceed deviation prove phenomenon functional focus spectra obtain pure protein piece grid e observe protein obtain chemical note split group display ht give member member protein help forecast nonparametric estimator original allow learn sample allow testing forecast correspond cross bandwidth estimation table improvement accuracy log coverage functional validation ig ig cauchy ig ig cauchy measure paradigm attention generic method output generic good accuracy speed three likelihood prior empirical probability iid replacement replication remain large functional collect penalty functional functional optimal regression estimating estimating regression nonparametric regressor investigate forecast predictor functional propose nonparametric functional type regressor give forecast error among investigate forecast functional good regressor affect forecast accuracy remain transformation forecast compute compute grid point forecast solid dot pointwise vertical bar ht estimate admit mixed regressor unknown expression exist establish mathematical bandwidth difficult bandwidth marginal use study simultaneously estimate density prediction attempt type regressor extend regression local improve take local well forecast concentrate extend regressor covariate dependent model kernel nonparametric admit mixed regressor partial admit type heterogeneous optimal crucial adapt type choose among semi curve derive detail course compute outperform estimation fix thus practitioner good semi curse way bandwidth consist introduce quantity show minimizer quadratic procedure h produce spurious aspect bandwidth good phenomenon unbiased contrast autoregressive
ignore science datum work medium text help propagate offline general rather online contribution seek close cm methodology google site collect extend upon early research level compare find hold significant management message set additional practitioner power propagation help value comprehensive use root stream cm keyword word google effect type aspect text help propagate message understand offer limited view gap investigate medium google site computational classifier sentiment seek replicate early effect notion significant social medium propagation google factor message contribution co media site broad management precise word propagation insight practitioner word help need number medium comprehensive google sentiment advance linguistic start poisson seek sentiment message digital expect management towards word result analysis finding offer conclude remark take production phenomenon capture production logic especially create attribute active active message propagate essence much great challenge find connection link effect role study identify narrow definition influence analyse influence interpretation text reader wide message message increase propagation conclusion message affect interest community around necessarily share nothing common community important co social medium understanding create simply news select propagate make like medium elaborate text mechanic social message form stream user connect form facebook twitter google prominent allow content add possibly news stream one company meet message propagate possibly community fan page readily social medium past find medium traditional mass medium consideration channel challenge hard seed plant medium reach take medium state word social medium establish current research draw inconsistent picture fuzzy potentially differently difference key message sentiment message message offer hypothesis message medium count receive answer question complete social medium foundation repository base contain name list contain localize exclude demonstrate website exclude profile check assign manual check google selection somewhat nine randomly google page retrieve extension package table page naive package ratio measure keyword sentiment state concept seek network analogue scenario higher able google google receive attention simple glm glm message represent model poisson summarize q side represents receive message frequency message day among identify medium day contain age post reliably number take include relationship extend allow comparison mix conceptually intercept slope graphic produce section describe regression sentiment message incorporate company difference mixed analysis message initially compute poisson start character hour step control receive test successively ratio yield significant appropriately baseline day analyze compare propagation nest compare sentiment type evidence random require contain incidence deviation value heterogeneous sentiment ht establish word could validate
indicate algorithm alg tracking improve additionally exclude loop far penalty cardinality figure agree come price experiment take around algorithm introduce model choice track line alternative would mcmc move refine show mixing normally enough joint tracking performance particle association result show upper half sample tracking burn mle histogram uninformative maximum mle mle available intractable obtain propose poisson derive survival beneficial figure mle data time mle properly step min target true obtain biased show black sample solid line show show bottom axis normalised axis black estimate mle vertical axis show horizontal normalise horizontal iteration linear develop case da run update mle filter proposal move transformation random root accord nonlinear approximate transform rd input nd include extend hide sigma obtain sigma description find approximation produce smooth q suggest sample accord description mcmc explore approximate sequence sample particle denote dirac locate time consist independence sample propagate condition sample resample multinomial resample w forward density gibbs valid hmm well mixing follow simulator accord w lx ba ac uk school mathematics university propose bayesian multiple tracking monte mcmc posterior target birth constitute problem comparison compete significant improvement mcmc tracking parameter sample target reduction continuous new approach performance da linear gaussian compete demonstrate target infer accurately track object fact number birth new target death false clutter may record observation record say jointly infer track chain continuous discrete comprise target death time exclude parameter small discrete refer da target approach da demonstrate algorithm linear exclude da technique state metropolis hasting indeed unbiased fashion tracking combine integrated known marginal metropolis context appeal inefficient likelihood take become product form simultaneously estimate acceptance elegant particle state tracking essential tracking single incorporate e track see ignore maximum method propose contribution several interesting comparison da reduce da iii track online tracking incorporate iv obtained build agreement mode remainder describe model target static propose novel da linear track static estimation tracking mapping capital case letter random lebesgue write explicit transpose denote commonly use q observation paper r capital letter small respectively resp value set letter set element target number surveillance death target birth time survival evolve transition addition target process target target target addition observation target clutter appear superposition clutter measurement precise target birth time newly target certain specify target specifically time evolve time target measurement amount identically parameter detect index non association detect target permutation decide zero birth death target detect measurement target collection unknown time assume survival probability mass lebesgue target state satisfy likelihood time miss relevant terminology hmm correspond time hmms correspond birth death irrelevant correspond mis clutter birth birth birth death trajectory otherwise take life transition convention contain irrelevant k ki otherwise birth time target contain clutter appearance measurement index want description illustrate introduce correspondence description evolve introduce name name name style name name b description unique description main serve depend novel posterior distribution interested static regard general extend assume density concentrate np association sample nz essentially mcmc however paper mcmc da place obtain run particle target problem slowly estimate association design sampler old avoid encounter apply particle accelerate birth death clutter set linear hmms irrelevant apply calculate mcmc da applicable unbiased place obtain target slowly estimate particle change order rule designing small particle accelerate mix use deriving include mcmc go notice association hasting algorithm applicable reversible dimension consider index dimension finally reverse propose jacobian reverse target random termination death I decide detect whole procedure grouping measurement contain kalman filter proposal step matching finally nz death reverse birth track acceptance birth move whose reciprocal corresponding death p thus move exploit distribution observation compare birth move birth move consecutive mis note assignment also birth choose target track either backward execute decide decide assign clutter probability posterior forward extension repeat reach extension e death observation information extend part proposal denote n forward approximation posterior propose obtain birth move base jacobian reduction pair among reduction discard discard probability reciprocal acceptance move probability propose resp extension extension move merely forward extension move make hidden add instead continuous dedicated change successive time measurement move modify rate move combination merge switch move measurement update move modification assignment diversity choice enhance introduce locally move change move old clutter sub move differ old merge new move sub remain link move become observation time describe noise mostly modify move reason proposal resp w propose one w nz acceptance state q locally unlike move modify move first propose mainly measurement propose state modification proposal density change proposal first move modify move n acceptance ratio move state move joint state move explore hmms independently constraint move first ignore directly live long mixing prevent fortunately mcmc particle efficient trajectory leave k invariant perform whose admit marginal follow backward idea second loop nz k k sampling order rule see conditional smc backward b obtain posterior sample parameter prior execute algorithm algorithm nn nj mcmc move algorithm explore move explore prior implement run invariant z pp z gibbs hmm posterior posterior hmm conjugate represent gamma commonly resp trial posterior beta birth rate posterior conjugate hmm state comprise plane target move constant velocity
process latter sample particle degeneracy whereby implement particle output particle filter idea filter within two metropolis gibbs marginal appropriate state value extend implement also common possibility use particle calculate accept depend new input value particle conditioning obtain run I I intuitive use within algorithm ignore approximation eq proposal acceptance unknown particle proposal likelihood correct particle particle sampler target involve iterate implement path model observation calculate analytically filter informally particle condition time condition particle filter simulation call conditional sampler stationary regardless filter mix autocorrelation act act blue red act particle mcmc pseudo variance respectively act trade include information carlo error become value alternative work correlation ignore simple adaptation particularly update ensure diversity particle maintain depend smc sampler particle twice ease keeping standard strong dependency slowly reduce particularly due uninformative chose informative particle pseudo particle row middle row bottom correspond approach sampler choose implementation inform pilot run particle particle particle conditioning ny ny trace improvement run long period reject output output able mix calculation effective roughly fold effective cpu conditional plot run bottom column column consider mixture population genetic possible present individual genome individual come unknown vary wish individual come example assume allele population x I unobserve variable conditional lx conjugate dirichlet population prior mdp number population present assign population individual population belong population particle particle particle subset plot posterior population analyse people cause filter likely plot overcome propose pseudo contain population individual give subset individual label actual arbitrary population individual uniformly change recursive proposal easily adapt particle filter fixing update purely run storing implement algorithm particle filter store information implement plot trace bottom leave hand run particle remove burn keep every substantially many substantial period run filter variance hence avoid estimate respectively particle suggest augmentation new model move rest careful substantial situation perform poorly volatility help break make slowly particle filter early likely inconsistent similarity marginal augmentation gibbs latent implementing expand mix correlation variable sampler augmentation flexibility particle mcmc key variable suggest variance posterior acknowledgement author engineering sciences ep k consider calculation distribution gamma pz x x n I individual write sampling give process prior individual take distribution poisson truncate value varied em augmentation mcmc involve process move parameter generalise beyond idea introduce move latent variable generic way latent choosing amount particle trading particle mcmc observation improve scenario enable keyword gibbs particle b carlo call filter efficient unobserved process deal within unobserved value move use widely area biology implementation move particle unobserved alternative inefficient particle slow move paper mcmc augmentation variable particle mcmc algorithms particle px
unclear train deep autoencoder fair previous train svm layer significantly regularizer autoencoder table noise dataset performance beneficial powerful lrr rand cccc cccc j u mnist rand compare training unsupervise affect unsupervised autoencoder train epoch validation training example similar joint since supervised case train true pre suggest joint beneficial supervise unsupervised joint deep appropriate joint initialization htb rand conclusion unsupervise autoencoder circumstance autoencoder could train jointly could view generalization train multi layer autoencoder stack autoencoder well compare highlight potential unsupervised success autoencoder superior deep jointly autoencoder platform investigate usage volume unlabele consecutive bottom long show mnist rotation mnist random mnist image rectangle convex generate denoise autoencoder scheme generate autoencoder training train diverse traditionally greedy wise employ prior training suboptimal investigate autoencoder view stack two layer autoencoder single global jointly optimize autoencoder layer act layer empirically joint training scheme learn learn high layer representation find usage achieve deep framework platform efficient usage type grow volume introduction learn various application recognition face recognition exception initialize unsupervised follow appear ingredient supervised give grow remain technique amount method latent prior deep network high variable local important architecture deep performance layer wise disadvantage distribution bottom representation layer something furthermore layer summarize wise focus learn auto layer disadvantage fail effective multi auto various setting joint objective input cope wise allow powerful view layer global reconstruction attribute make confirm representation learn approach consistently outperform pre algorithm demonstrate superior deep amount datum label consume remain continue improved volume procedure engineering challenge difficult apart layer value mean joint measure input layer readily respect post network decompose cover distribution tends assume learn occur hidden layer recursively trick delay motivation simple easy greedy wise optimum bottom learn preserve likely learn lead optimum reason therefore autoencoder make consequently possible burden prior background briefly review autoencoder variant expand primary autoencoder basic autoencoder layer reconstruct activation take put encoding decode input decode layer function choice want reconstruct actual denote encoding include input input however training fail around set meaningful denoise denoise autoencoder idea pass reconstruct force trivially objective equation formally illustration employ autoencoder layer try reconstruct back final autoencoder stack deep jointly final well represent simplicity exclude illustration autoencoder propose autoencoder achieve robustness small perturbation frobenius jacobian activation respect thus would prefer activation stay input vary frobenius hide activation change representation represent manifold prefer would cost deep try deep I eq architecture decode sequence follow stack layer deep autoencoder autoencoder stack autoencoder autoencoder encoding layer reconstruction interpret modular train autoencoder joint exist relate technique perspective regularizer dirac delta equation recover perceptron mlp decay exception mlp commonly unsupervise replace identical ordinary mlp straight forward see training construct modify train slightly rate follow greedy tune however domain behave apparent represent decode word note modify somewhat surprising behave special practical equivalent differently data interpretation model decompose decompose bottom respect regard tune together likely take fact capacity add later hide unit architecture tune maximum generative stochastic modal try approximate intuition mostly unimodal easy autoencoder corrupt one follow imply distribution true like denoise deep denoise autoencoder autoencoder framework empirically analyse train helpful recent regularizer autoencoder joint mnist dataset split mnist variation dataset validation shape employ shape classification classify classify example dataset validation also foreground visual tie autoencoder unit activation optimize prop factor sample per mini good validation deep denoise autoencoder contraction level deep autoencoder gaussian mention section goodness denoise autoencoder chain train joint training estimate measure window converge true number window generate dataset window likelihood test table method rr dataset l mnist rand sample mnist mnist consecutive autoencoder notice fourth consecutive show training illustrate long qualitative purpose show however train spurious log test whereas drop illustrate prior training model f objective focus reconstruction reconstruction testing achieve less case previous advantage
algorithm since estimation compare shrinkage optimal singular infinity signal remain asymptotic remarkable square rule shrinkage matrix connection suggest iterative statistically remain reflect deep phenomenon datum correspondence rather direct correspondence discussion serve principal correlation decomposition transform total count rank estimate formula use original probability q logistic desirable regularize parametric jj efficiently solve iterative interestingly contingency table obtain shrinkage correspondence provide principled comparative reproduce competitive benchmark adapt experiment outperform correspondence different exist reproduce simulation experiment accord four snr repeat autoencoder sa define iterate apply run appear stable truncate truncate asymptotically threshold shrinkage asymptotic soft soft stein sure suggest assume noise addition sa tuning parameter r r snr bold make strength namely mse snr high low snr conversely snr surprising estimate happen lasso select meanwhile shrinkage function sa flexible scenario accurately nearly indistinguishable move vector term mse illustrate phenomenon expectation represent three component concentration third concentrated corner adapt poisson vary describe baseline motivate value r r c r rv rv mse also table measure coefficient correlation iterate rank ability appear stable much align one whereas shrinkage though motivate regularization generic finally use analysis collect organize associated available correspondence analysis visualize association highlight profile well method ran build original perform classical correspondence rank well regularize report rv respectively coordinate population one rank parameter course work however sa set section make good rv rv sa compare ca representation figure bottom obtained contribute represented use package emphasize correspondence analysis use visualization appropriate affect look like seem know table thus regularize transform regularizer adapt enable noise create pseudo dataset want induce estimate stable autoencoder intuition concrete bootstrap remain whether bootstrappe extend discuss grateful helpful david stanford fellowship people w support stanford fellowship begin establish bias two q conclude separate rank constraint meanwhile add plug unconstrained q solution next get verify expansion meanwhile everything define fix general monotone monotone positive cone definite follow proof fix symmetric decompose eq monotonicity matrix conjecture prop prop prop example prop comment prop stanford stanford develop transform seek basis respect noise simple isotropic non method estimator iterate scheme tuning datum analysis parametric bootstrap low play role scientific include collaborative filter genome wide imaging motivated suppose noisy admit parsimonious statistical try recover observe regime accurately classical center svd form believe close approximation noisy improve usually make close accord several sure provable identically distribute gaussian singular order new start shrinkage motivate classical allow encode want recover perspective oracle approximation solve know guess result choice parsimonious isotropic reduce singular shrinkage always poisson singular around iterate fix iterative job underlie isotropic singular shrinkage strong property fact resemble one propose discrepancy perturb conversely signal induce want vice experiment autoencoder attractive sense close view useful outside induce estimator reduce shrinkage non stable autoencoder efficiently view easily solve
evolve covariance smoothness subsection need notation convex density parametrize first decay covariance decay couple instant constant q assumption hold infinity denote go converge topology useful introduce eq select step repetition course g q description subscript replace convention function key role q differently limit depend q start analyze first rhs combine bind expectation rhs despite nature apply combined definition insight expression eigenfunction orthonormal kx correspondence calculation jx rkhs norm rewrite rhs take derive compact bind permit virtue class operator jensen lead proof assumption partition compute reconstruct measurement need cast field input discuss numerical progress allow appearance relatively embed sensor system environmental powerful resource application environment even become reality although single trend adopt area attract considerable relevance try function service monitoring center originally review environmental control compute coverage treatment begin similar environment dynamic partitioning limited coverage communication constraint cite cost coverage problem assume function area high advance hard distribution area refer strategy propose assume noiseless moreover parametric approach assumption recently parametric approximation gaussian unknown robot guarantee function simultaneously coverage robot centralize work design find visit cost collect draw pdf extension replace convergent sequence coverage mechanism location agent move domain interest pdf allow current markovian happen often pdf ensure case reproduce technical detail give propose discuss section simulation let paper partition centroid partition partition centroid function introduce define fix local point centroid partition consider problem classic center initial cycle iteratively step namely increase converge generate pair radial kx move agent unit agent basic computation capability denote position q capability store take perform computation partition robot agent base explore environment estimate partitioning goal hereafter ec base collection agent take compute trajectory agent dynamic base centroid position receive agent assign reach central base new additionally position varie agent simplify way every suitable establish centroid reduce goal density uniformly irrespective instant measure concrete specific rule report random gaussian constrain support determine follow estimate use define determine direction variance location agent instant trade exploration exploitation tune level heuristic function maximum report begin domain posterior consequently certain threshold e uniformly update agent centroid phase algorithm perform unknown function estimate reduce much switch radial consider kernel exists give kx jt pr collect input location location thank eq statement exist agent switch agent centroid trajectory generate generate partition implement coverage team combination four bi posterior grid switch phase allow variance favor agent movement centroid example display contour plot agent clear contour profile maximum number iteration average dx vx minimum posterior time figure circle ideal computed circle estimation reconstruct establish agent movement trying see input markovian allow happen infinitely discuss also appendix good
first term component integrate evaluate conjugacy crp hdp hdp auxiliary alternatively kind dim view analogy drive customer group act restaurant cluster crp probability restaurant restaurant customer customer accord restaurant specific crp table assign represent atom crp crf restaurant draw customer restaurant crp crp form restaurant crp aggregate study marginalization result establish link nest suitably endow dp distribution formally collection measurable q marginalization dp base let formal ji marginalization sketch step either side equality marginalization marginalization demonstrate application model three world state model datum unknown unknown cc consistency context topic datum unique letter string international conference letter letter generate document document variable drawing word iteration content ground successfully identify string demonstrate observation observation recover overlap character provide appendix c c word word title author word word dataset dataset vocabulary exclude stop contain publication comprise testing dim bag sift dim bag tag text level use hold hdp require document hold score comparable use univariate tag author collapse gibbs iteration burn context informative aspect level context achieve information explain author additional hold suggest induce simultaneously beneficial induce value show example discover author supervise incomplete via fast likelihood dependency discrimination derive understanding evaluate work content information yield fall illustrate distribution discover context topic appear research year top search histogram close google discover time use google wide bag sift image tag exploit observation bag outperform sift tag sift evaluate class truth report cluster metric mutual rand baseline min standard propagation ap ap document euclidean hierarchical hdp ap run hdp content affinity similarity consistently metric wide cluster content onto clarity display reasonably separate missing encounter context document observation c report observation utilize big top bottom proportion demonstrate approach single utilize level discover collapse experiment world domain content level encounter real model applicable domain ingredient bayesian form prior thus establish interesting bridge product yield new present deep main marginalization property derivation inference marginalization measure move measure measurable endow measurable set measurable disjoint set let ns finite collection property union write easy furthermore dirichlet ga ga accord measure every n space space marginal measurable set rhs let form note h h n general recall dp dp mixture realization generate proceed nest dp measurable set ji marginalization whereas result proposition still vice versa immediate swap result proposition still prior I h arbitrary measurable integrate stick break atomic placing sequentially conditionally k stick break lastly stick break put thing display integrate exclude document document context index recognize restaurant crp count conjugacy leave conditional exclude finally last context ji v conjugacy j j l therefore evaluate q sample give make marginal conjugacy exclude ji argument restaurant ji content context j j ji use q jointly auxiliary hyperparameter hyper similar previous q q use al assume concentration hdp previously count z last variable kt eq auxiliary accordance cluster statistic come content topic word mean soon improve contribute rich confirm document affect repeat al evaluate utilize dirichlet type wishart binomial word assign context set count suppose gibbs proof theorem utilize context group dirichlet block construct product accommodate observation model possess nest integrate variable collapse extensive world utilize text content e g student group group group information grouped set diverse model public health consider analysis cluster group source cluster document information example sift tag reduction content form document word problematic due vocabulary sparsity occurrence typical hdp document utilize limitation suggest alternative document joint expect improved document predictive recent jointly cluster specify advance hdp adjust cluster none context content utilize product accommodate possesse nest interesting context mixture collapse automatically utilize work subsequently exchangeable group model reflect formalism dependent dirichlet level another dirichlet
dataset pca method size ari determine make noisy introduce spectra datum see pca although robustness speed experiment processing face crowd outline pose fall dataset image extend outli scale center point project sphere report center geometric median denote remark dimensional dimension run projection distance degree subspace figure subspace point face face evident version comparison performance order pca subspace close robust close subspace face approximation subspace slight low face estimator fact robust outlier aim non recent successful relaxation aim ability obtain truly complexity empirically minimize convex minimizer robust outlier relaxation energy even strong power success verify iterate synthetic project observation theory case nevertheless experience seem potential reduce dimension massive make randomized implementation recommendation dataset ari david help zhang comment manuscript helpful encourage define energy satisfie pca scale eq monotonically decrease easily establish let arbitrarily desire inequality different pf l gd assume consider prove equivalent versus try initialization initialize whenever initialize subspace write run face crowd outlier find display test follow describe onto subspace calculate truth ambient htb htb htb add add face perform dimensional proposition hypothesis mn fast consider around dimensional dimensional possibly portion lie nearby subspace fast median accuracy modern collect increasingly dimension massive analyze high subspace subspace capturing find moderate singular decomposition svd svd stable fast progress pca follow corruption sample underlie presence outlier devote develop numerically formulation good compute ambient least approximated address linearly high underlying minimization difficult nevertheless support prove energy minimizer relaxation rich may theoretical model careful indicate competitive particular method classic enjoy efficient last decade fast singular review pca et fast however practice corruption clear emphasis quantify study success instance show achieve achieve completely corruption wise corruption within datum algorithm adversary former often initial datum sphere dependent energy relaxation target relaxation accuracy possibly however non estimator minimize competitive empirically accurate obtain eigenvalue initial involve subspace sufficiently competitive implementation online algorithm suffer denote vector seek subspace manifold subspace denote gd dl angle motivating discuss minimization propose heuristic establishe convergence summarize usefulness approach lastly robustness next motivate last approach minimization problem among subspace regularization relaxation energy remark constant continuity minimization experience center geometric successful center geometric surface natural geometric median subspace domain relaxation sufficiently iterative procedure summarize notation center desire regularization default l uk find affected replace exact fast advantageous pca although subspace seem convergence see without stop iterate difficult guarantee desire geodesic guarantee future create operation find top scale empirically notice treat copy basis subspace various real compare relaxation notice difference lower report case set median r mirror descent md try algorithm outli competitive report though md important algorithm also aim directly step run use md pass size iteration ht accuracy versus bind subspace superior synthetic run compare art distribution within random subspace experiment restrict reasonable estimation percentage runtime error recovery subspace identity comparable magnitude truth subspace ambient dimension fix also perturb add runtime percentage generate dataset average demonstrate problem high end fast exclude percentage display
solution sm implement runtime eq log proportional theorem assume trivial random interpret burn distance analysis leave future period independent even initialize burn period epoch variance optimum different point take iteration starting converge bad heuristic datum stochastic motivated regime readily knowledge far ht hybrid plain line see legend f iterate generalize see legend comparison implement use several step random ball hard require perform burn display pca version iteration though tune logarithmic rate choose sub exponential leverage finite exponentially behavior occur perform similar mnist size pre process divide deviation square root hybrid run decay step initial hybrid perform mnist singular vector generalization display qualitatively simplify presentation use remain divide multiply step ds orthonormal follow place establish recurrence relation begin epoch evolve key suppose since epoch subscript denote orthonormal denote condition quantity whereas simplify focus epoch drop simply constant tell sufficiently small numerical depend fix evolve recursion certain mc ct stochastic assumption fact verify cc armed fact version simultaneously long expression statement remain prove cb inequality confidence c part analyze epoch ready lemma pick take upper eq apply epoch therefore accuracy size constant recall whose discuss apply correspond possibly numerical svd convergence assumption scale factor require iterative leave open dominate inferior second convex compare draw parallel strong pca problem runtime however factor analyze initialize sufficiently optimum experimentally satisfactory might optimally choose finally formal square relaxed dependence acknowledgment fp intel ci institute science foundation thank lemma early sm runtime code runtime sparsity theoretical exposition reproduce epoch unit n assume zero value dimension iteratively add n store scalar line store follow implement ensures implement accordingly principal component fast intensive runtime scale reduce gradient analyze apply inherently analysis fundamental wish singular identity prominent application principal pca consist specify possible numerous let write simple find later extended solve exactly decomposition runtime prohibitive common alternative power norm show involve I pass pass prohibitive dataset alternative deterministic algorithm much update runtime flip stochastic incremental rate know slow medium prohibitive stochastic pca provable advantage avoid hand scale involve applicable runtime well mention logarithmic factor perform single scan build technique somewhat different crucially strong unconstrained attempt alone pseudo code appear execution inner execution outer step epoch mi n helpful pca rule repeatedly perform rewrite thus zero project sphere give show convergence relatively slow variance stochastic inspire reduce change encourage variance stochastic rewrite begin compare eq type add pick step decay long rather control lead decaying variance compare remark
issue impose nonnegative constraint solve capture low representation relationship basis e nuclear e noisy corrupted add corrupt call balance encourage corrupted relax norm focus popular alternate however auxiliary linearize alternate direction method lagrangian function problem update iterate add algebra scheme partial differential respect complete tb z update lagrange solution subsection associate vertex edge set give graph reconstruction recover derive naturally rank guarantee come fall capture structure note feasible solution insufficient sr practice noise small interested structure normalize summarize normalize x I obtain solve column make affect robust robustness improve improve make show strategy datum sparse subsequent separately simultaneously learn want preserve much minimize plug construction arrive reconstruction simplification embed refer ef ef commonly optimize follow augment lagrangian fix reduce alm lagrange code find http www di fr tb parameter lagrange multipli update follow alternatively ef summarize get difference great problem evaluate currently popular semi implement matlab run state server intel core ghz processor public database line lrr empirically different set generality dimensionality experiment subsection carry classification database select method label sample otherwise propagation utilize combine harmonic optimize function laplacian function elastic fitness term laplacian combine graph framework select range labeling report result case extension ef consistently achieve error labeling cutting error extension compare ef improvement representation robust lrr dense lrr base inferior contrary graph lrr graph graph case property construction effectiveness semi take semi supervise discriminant recognition database aim intrinsic infer set run image image test recognition rate graph consistently lrr ef tb ef nn nn graph lrr subsection examine sensitivity include sparsity deal corruption emphasize property base percentage corruption empirically vary setting average rate decrease ignore sparsity sparsity low graph whole tb subsection joint propose embed datum apply fair keep ef baseline ef learn keep ef raw demonstrate necessity discovery ef well prove proper discovery ef htbp novel rank supervise low derive informative suitable graph help reveal embed support science science foundation national foundation china aim discover intrinsic structure semi build non obtain structure discriminative extremely good jointly within framework term embed ef extensive publicly demonstrate construction semi supervise embed category application recognition label prohibitive unlabele utilize rich considerable vision appeal success represent unlabeled affinity pair sample sample formalize regularize despite many mainly try accommodate likely normally care manifold effort way good informative characteristic high neighborhood rule construction sr usually characterize global datum overcome liu lrr weight hereafter lrr result dense construct informative dimensional informative representation point hull ensure enforce coefficient low embed facilitate subsequent improve separately learn optimize representation learn graph improve supervise contribution graph ensure low capture global structure sr base enhance embed learn datum suitable consequently semi extensive demonstrate improve multiple evaluation framework robustness conduct propose influence review work graph section detail embed experiment section reveal intrinsic local capture structure whole manifold neighbor distance capture capture
france superior de e high education hyperspectral sense resolution conversely spatial infer fusion availability retrieve formulate minimization preserve account different noise regularizer across hyperspectral accounting nature regularizer live subspace lagrangian shrinkage multiplier convenient spatial linear link outperform simulated life hyperspectral imaging fusion total variation method multiplier way world spectral image term represent scene find sense focus field air sensor context common hyperspectral difference application dependent high resolution near resolution correspond somewhat narrow em spectrum cover spatial resolution offer cover large result small band cover band red spatial resolution high extensively address fusion latter band image range resolution significantly factor ill pose fusion hyperspectral typically act computationally asset often cover large cover band dedicate trend underlie relatively number signature corresponding material scene scene underlie material call spectral explain resolution use resolution regression exploit fuse hyperspectral via gauss jointly rgb reconstruct rgb impose sparsity hyperspectral match pursuit induce song dictionary pair test band old signature framework bayesian prior distribution work foundation work publish expectation monte deal chen treat fusion joint hyperspectral simple around inverse used fusion optimization regularization spectral order method multiplier admm augment lagrangian explore inherent redundancy technique efficient hyperspectral allow hyperspectral literature fusion blind sense inaccurate blind assume unknown make support response estimate response correspondence band sensor work extend process clearly establish present remainder organize describe present deal sensor present conclude hyperspectral think three array tensor however notational convenience band contain pixel band bold denote e g observe hyperspectral band spatial denote hyperspectral measurement spatial hyperspectral sensor spread assume band circular assumption deal blind allow vary band blind relatively regard advantage fast transform inversion costly operation periodic totally experimentally find lead fuse reduction amount base admm correspond complexity image work column subset account subsample spatial resolution identically band band band straightforward model response hyperspectral large usually live I translate description dimensionality dimensional space original normally work hyperspectral band typical infer briefly physical give pixel linear pure signature spectral signature numerous algorithm address vertex linear singular decomposition h rectangular diagonal contain singular increasing truncate low hyperspectral discard singular subspace complex hyperspectral reduction incorporate truncate svd denoise try solve ill pose therefore need adequate th compute vertical discrete periodic total purpose mean transition edge extensively formulation isotropic isotropic hyperspectral band band band one normally band work align band denoise hyperspectral reduce dimensionality subspace span svd def formulate regularizer eq first term impose explain shall discuss selection section quadratic difficulty direct use transform involve deal consist employ unlike primal dual require equation primal method admm auxiliary call four notational define eq augment penalty ready admm complex much simple augment relative auxiliary datum regularization initialization k minimization quadratic block cyclic efficiently fourier three respect solve matrix advance optimization scheme kronecker product simple computational gain splitting seem describe condition column presence close knowledge alternate optimization simple without know much strong minimize response set hyperspectral image delta impulse summarize present square correspondingly important associated response channel band band band contiguous somewhat follow reasoning l h def b couple quadratic represent intensity ms patch imply leave except definite optimization uniqueness subproblem matrix expression parameter correspondingly scale normalize dc spatial hyperspectral describe index quality experimental comparison fusion truth simple shape compose material digital library measure analyze national visible red imaging capable contiguous spectral signature randomly build image create hyperspectral image direction color representation hyperspectral fig color material capture four band unless noise hyperspectral image snr db synthetic hyperspectral university image image span spatial truth hyperspectral dataset image paris observe resolution provide hyperspectral hyperspectral fusion hyperspectral image resolution hyperspectral image hyperspectral ground evaluate fusion take ground image ms se truth ratio l angle angle estimate denote index image paper report degree quality window denote segment segment band truth deviation index image simple computed code wang fusion hyperspectral access ground inspection experimental perform preprocesse step hyperspectral first band remove information band band manually truncate subspace preserve strongly varied normalize band quantile hyperspectral band normalize b spatial share since ten result topic adapt two stein verify experimentally parameter yield image influence quality regularization stop work always yield run experiment aim spectral spatial response life check b yield quality dataset method spectral response take overlap hyperspectral band since original apply hyperspectral fusion image mind image number hyperspectral straightforward hyperspectral image restriction number band quick comparison restriction resolution impose restriction divided substitution family band transformation well may hyperspectral gram schmidt adaptation schmidt gram schmidt resolution band band modify band band expand spatial way difference gs gs band result hyperspectral improvement gs fusion include gs space intensity principal component image replace method contribution bt pixel combination spatially expand band extraction information spatial produce pass operation fig various evolution fig root square rmse truth band c life datum snr db db r gs bt outperform except index bt find publish hyperspectral channel hyperspectral consequence publish access implementation zhang henceforth comparison implementation resolution therefore interpolation input method work method author decomposition transform table method restriction available work image pixel
probable process reconstruct space source auto feature extraction calculate analogously ar ar calculation classic dissimilarity retrieve shape database align signature task addition exist call elastic dissimilarity optimally fig center form accumulate dynamic programming recursively respectively except initialized deal series call dissimilarity usually take difference deal multidimensional choose euclidean dissimilarity e perform series backtrack variant equivalent unnormalized apply computation common parameter formula deal series length round two general carry constraint prevent estimate go beyond computational stand main benchmark systematically source require extensive rigorous assess could say superiority unclear pay aspect assess consider statistically significant sec turning observe challenge chen real value recent outperform formalize specifically function otherwise suitably two initially improvement absolute difference relate sec perhaps extension algorithm penalty dynamic time control sample eq well choose final dissimilarity difference cope include aspect metric triangular inequality metric characteristic dissimilarity retrieval main behind jump cost dissimilarity resemble bottom jump visit jump cost increment want jump jump cost magnitude consider fx similarly introduce series parameter control advance dissimilarity measure efficacy similarity commonly ratio understand classify item total measure use nn implement relate error suggest simple near neighbor publicly set time repository series set comprise shape range length detail refer properly assess classifier need follow scheme balanced item balance balanced error estimation regard repeat fold precise estimation avoid bias ratio split provide allow ratio implementation agree error ratio one modify algorithm interestingly rl fc last rank across data position sort order accuracy datum indicate characterize extract fc feature bad elastic intuition sample important order magnitude euclidean one several set match set superior rest fig next measure statistically difference separate apart comparison make aforementioned global choosing solely hence otherwise could measure utility ahead look ratio gain couple set kind contingency reason contingency table euclidean gain euclidean mostly sec look perform measure stage ranking stand construction rl euclidean majority give across peak perhaps present fairly accuracy robustness quality incomplete plot ar parameter range indicate reasonable choice measure potentially seen seem consistently range good combination datum lie potentially accuracy set comment large tracking path advantageous account small measure track may advantageous set conclusion derive ar coefficient choose ar low classify group statistically suggest sub sequence variant could potentially join group however seems consistently consider distance often measure take time competitive generally euclidean statistically significantly find fc measure course exclude measure variant suited rest sec compare mostly train test error unseen list notable assess vs assess regard regard step pre step well discuss therein book wang prevent low dissimilarity series inclusion assess depend consider sensible approach implicitly consider pre strategy z invariance phase invariance ar invariance correction instance become usually consider usually accuracy improve set far potentially reduce use unsupervised cluster candidate quantitative strategy series emphasis impact scheme empirical multiple important necessary get picture approach unified analysis tool interested series similarity cluster acknowledgement make repository time series similarity core many particular similarity ingredient lack comparative study rigorous quantitative strategy extensive evaluation series principle family available datum come wide variety accuracy accuracy meaningful conclusion equivalence accuracy measure finding follow methodology researcher consistent criterion inform baseline time series scientific indexing stock fluctuation medical g motion location shape effectively transform determine give resemble retrieval clustering exploit outperform elaborate alternative tree perceptron logic boost multiple classifier correctly deal calculation measure continue future generic purpose readily task last aspect highlight desirable measure type wang year pr fu nevertheless make efficacy apart interesting oppose straightforwardly time similarity control quantitative framework come various scientific newly theoretically attractive efficacy measure vast case difficult consistently remain quite accurate measure look similarity measure efficacy chen competitive none three initially generic behind relate specific dissimilarity string comparative deal classification study introduce usually corpus lack besides usually significance formal difference rarely impact field sure baseline sensible choice perform pool time series additionally storage issue approach restrict stage aforementione sufficiently cover decide accuracy contribution previous formulation evaluation sample rigorous statistical assess superiority accuracy assessment rest paper firstly outline application sec explain sec end sec deal similarity vast comprehensive enumeration scope present book wang book fu measure auto selecting avoid measure small set lead alternative measure consistently aforementione measure include way dissimilarity measure group measure compare temporal approach alignment dissimilarity measure indexing
mm fig image pattern neuron axis linearly scale template history issue template variability technique template consideration closeness method template light representation calculate dot give mathematically cross correlation think traditionally template feature thus equivalent kind template template audio broadly audio short duration short song type song entirely test notable tendency improve basis learn fig believe performance far rather unit co variety order combinatorial layer abstraction temporal energy pattern perhaps advantage large dataset explicitly template fig field broadly sensitivity combination harmonic strong spherical k method design involve secondly show partial consideration may influence mechanism measure song unit biological paradigm worth current automatic motivated scientific volume come source simple effective boost training within impose extra effort learn similar report principal volume effective confirm large volume increasingly become dataset feature whereas raw spectra much recommend species recognition dramatically across together achieve peak classification quality make demonstrate data volume exploit trivial domain demonstrate community publication collection least label publish open acknowledgment project research website sound source fellowship ep g early fellowship ep availability compose sound file file request sound specie list record website website automatic sound computational monitoring communication classification useful crucial ensure big acoustic summary information learn automatically often outperform manually transform manual yet introduce feature volume sound inspire technique prove domain experimentally representation diverse database forest classifier demonstrate limited bad conversely substantial boost spectra computational particularly notable scale activation substantial audio annotation interaction choice representation automatic specie classification useful number big crucial huge volume audio audio volume much sound hold sound digital scale without manual intervention manual segmentation song section lack segmentation since audio classify specie study least far survey often manually specie application unclear study limitation remain question intensity large modification robustness distinguish ten hundred typical advantageous label specie present project name provide stimulus research classification challenge report benefit make available standard focus choice outperform spectrum evaluate role aspect four enable perform dataset specie boost little cost dataset demonstrate boost label task amount audio explore reason follow describe experiment learn classification raw audio generally input even input duration audio would audio magnitude time transform audio duration sound indicate energy frequency frequency carry information content transform originally reduce dimensionality traditionally dimensional automatic speech spectrum keep coefficient approximately reduction advantageous manual inspection cope high see modern algorithm cope well available discard little capture specie design represent speech yet human differ production spectra classifier could matter rather design representation manually automatic feature topic aim signal compression procedure unsupervise operate unsupervise component pca pca projection create operate inherent machine feature despite method semantic relevance already make feature surprising without feature add however transformation reveal machine expansion one layer neuron connect intend instead simple scalable modification classic audio identify specie specific aspect patch representation delta sometimes cf study variant frame time variant automatic separately four sound forest systematically feature spectra coefficient pre processing decision treat full audio purpose divide short duration window produce overall configuration random forest classifier test binary parameter combination rigorous hundred explore issue recognition separate aspect character dimensionality test item total duration france frames uk uk frames frames represent large classify two consist publicly project per challenge private evaluation final partition half addition sound environmental sound duration minute annotate list median approach adapt website many specie cover specie specie include retrieve retrieve allow system song call sample range widely vary characteristic example typical sound file class distinguish label specie use separate label song call strong specie list audio file region implicit collection audio noise audio monitoring manual audio tp width mm mm discuss feature transformation drive characteristic inherent study first relate mean cluster centroid unit direction angular modify update input centroid euclidean move centroid cosine angle update spherical mean spherical find overcomplete approximated multiple discover simple representation dimension normalise pca configuration one spectral frame sequence spectral frame short temporal spectral number frame index frame offset fig basis alternatively think stack stacking frame give find mean require consider volume author minibatch update stream stream online mean apply centroid amount centroid adapt case spherical optimisation presentation learn true single pass pass reservoir apply pca whitening start window decision pool overall window second audio decision purpose training decision aggregate audio mean specie across window reasonable default motivation audio strong factor window audio cost evaluate feature combination stage auc auc systems property unlike unbalanced true example probabilistic auc tell negative always good probabilistic rank list lead evaluation relate rank difference auc apply mis position statistic rank l label dimension spectra spectra spectra mean frame frame two test statistic general glm use package test glm interpretation auc glm odd ratio experimental fold repeat version glm fold grouping interaction mode decision pooling duration whole audio testing configuration result pool glm effect exclude case estimate odd dataset annotation therefore explore source recognition quality strong overlap mean testing systematic example generally short mean pool feature training specie wider expect strong train possible set intrinsic dimensionality difference potentially give degree freedom classifier capturing run create projection feature simple form learn feature degree classifier run plot fold mm plot fold mm mm plot mm mm mm width mm mm plot width plot bl fold bl fold dataset mm mm mm mm bl fold mm mm bl fold factor vs ms ms ms ms ms pl none relevance distinguish auc measure ranking tendency outperform outperform feature learn spectra except compare switch learn effect strong feature conversely performance large fact deep aside feature dataset switch raw spectral feature boost though feature across give classifier reach map chance configuration reflect observation score annotation insufficient full mode dataset auc dataset small boost auc switching combine outcome effect aggregate find effect reduction inconsistent extent improve performance tp mm mm mm bl mm mm bl mm bl width plot bl mm mm bl mm bl cross scenario dataset middle datum boost tell firstly contain audio secondly include setup perform expansion audio annotation contain lead accommodate well attain training poor tp mm mm except train turn broadly preserve dimensionality improvement change ordering though part feature overall effect estimate random small validate map red binary model relevance layer first list attain two second list evaluate classifier entire comparison decision system audio one outperform variant audio mean notably attain notable subset substantially work volume peak reach auc actual develop winning attain report peak auc full private
optimal passive learn datum w w ks k w x w b k w w inequality tell complete sample k first w w due give list elementary inequality let first check k corollary ready prove iteration argument prove three technical get complete proof lemma want prove apply take union classifier respect particular isotropic say furthermore mean zero identity isotropic constant sample iteration also data I isotropic f k k w add inequality surrogate linear inequality theorem require lemma present large hamming bind separated set make separate bind reduce switching subsequently fact obtain follow fact taylor bayes classifier satisfies condition condition point satisfie last lastly represent label decompose synthetic propose request unknown subsequently active learning define divergence parameter depend furthermore sufficiently small follow set semi mp hold prove corollary easily depend jensen prove use concern separable theorem fix denote code length I hamming ready prove first comment always even always remainder shall bound repeatedly pick request case excess w corollary definition robust base active homogeneous pass origin analyze corrupted impose noise low achieve margin condition membership synthesis scenario stream selective surprisingly show separate provide insight increasingly make unlabele sample algorithm access capacity request specific hope label informative achieve improvement passive noisy polynomial two access unlabele decide point point stream margin show condition statistical make active usually also low one stream base set surprisingly algorithms distribution margin active detail homogeneous unit algorithm later base query budget bound characterize cf eq key parameter available robust disagreement bad discuss development convex variant near amount exponential risk optimal present developed concave change shrink like surrogate function hinge logistic algorithm bind factor stream algorithm exist satisfy excess bound adversarial manner unclear low apply low log separable show exponential contrast polynomial membership query synthesis section analysis datum distribution cf focus low low bayes optimal classification hyperplane stream algorithm active inspire stochastic connection active optimization analysis build base active necessary keep track classifier optimizer bayes shrink analysis construct adversarial fail synthesis active responsible dimensional inspire convex assume joint instance space draw goal loss indicator paper consider df linear w consist angle characterize hyperplane increasingly w risk base selective request manner formally stream operate point sample distribution unlabele accept request conditional finite selective operate selective setting access entire pool make query request margin active learning stream query set active learning introduce query synthesis query label request shrinkage iteration allow adapt value excess basic idea request passive reduce scope classifier htbp failure rate xx w final key depend query divide evenly difference sample budget previous tuning free active learn query optimize algorithm surrogate sketch defer appendix defining notation f b k respect acceptance technique probability good carefully iteration long possible exist phase phase everything behave excess upper bound classifier speak apply risk constrain appendix appendix density slight modification defer appendix fix suppose distribution unchanged detail active stream imply low stream selective synthesis margin base unit facilitate synthesis condition distribution speak constant eq classifier use synthesis remark eq deferred density imply establish angle output bayes membership query label satisfy excess bind omit dependency excess corollary stream fix denote excess omit section defer assume remain construction tt rigorous intuitively want distinguish budget imply kl condition lemma constant deferred fix design distribution construct th hypothesis illustration indicate actual solid dash green respect distribution point classification hyperplane hand must bayes classifier must hold data
obtain ise irreducible ise configuration zero boundary prove minimal ise configuration configuration singleton configuration technical follow form subgraph short vertex long htb rectangular rectangular singleton give singleton ise unique correspond rectangular configuration configuration singleton configuration prescribe lemma configuration one prove rectangular max singleton connect rectangular prove two configuration mean irreducible technical rectangular configuration max result lattice e exist q tb consequence connect configuration denote rectangle simplify connect illustrate large connected simple component rectangle lie outside depict component repeatedly scenario move indicate box position either singleton move outside vertex start scenario swap remain site swap swap instead singleton configuration remain swap site situation swap tb pt c swap swap swap swap swap swap swap rectangular otherwise reach configuration lie latter rectangular configuration let proof show dimensional dimensional lattice small expand space dimensional show section prove since lattice hold version example hyper rectangular configuration swap prove sufficient lattice sa analogous omit detail proof connect rectangular configuration connect component rectangular rectangular lastly rectangular analogue vertex adjacent neighbor indicate forward ise suppose show irreducible various nan either presence long interaction homogeneity alternative test statistic define normalize constant minimal sufficient ising define could correspond take lattice otherwise pdf analog sided indicator high adequate goodness fit ise two statistic use sided pattern form vertical consecutive ise choice indicator range interaction alternative q normalize test homogeneity square configuration statistic degree homogeneity expect statistic small homogeneity homogeneity dt nn normalization statistic describe ise result lattice analyze ii generated ising ise model interaction ise periodic simulation compare ise ise compare ise interaction three ise value way equilibrium chain configuration metropolis remain available website author chain iteration generate six statistic homogeneity test sample chain carlo models statistic depict statistic observe depict standard distribution none reject data level hypothesis homogeneity ise model homogeneity homogeneity test test reject test homogeneity show difficult interestingly count adjacent pair recognize ise ex pair adjacent pair b experiment chains monte carlo configurations plot negative value model depict quantile hypothesis ise ii rejection moderate interaction large ise count adjacent pair generate ise homogeneity hypothesis reject test b weakly homogeneity recognize overall decide organization count adjacent pair homogeneity test seem line pair adjacent pair ex application spatial super resolution highlight component density picture lattice circular ise configuration adjacent homogeneity adjacent remove step analyze study statistic chain illustrate figure homogeneity sample pair size near neighbor discard homogeneity configuration max singleton component swap size respectively size configuration singleton contradict write adjacent connect large decrease increase singleton become result configuration max singleton maximal interest q minimize equivalent maximum attain rectangular configuration bound since obtain singleton configuration rectangular give component box statistic thus let complete reduce sufficient rectangular noting tb write rectangular move figure move indicate htb need move type rectangular move result configuration rectangular side length b swap join lie repeat subtle lie type configuration rectangular lie move swap b swap configuration lastly configuration swap depict theorem thm conjecture thm thm example thm ise simple interact system originally interaction physics widely process usually fit lattice ise basis develop statistic goodness fit intractable thus develop monte fit avoid spatial organization ise mechanic appear ise thesis usually arrange interact play mechanic see interaction area e social ise single paper goodness model apply contribution series paper asymptotic approximation irreducible markov starting build spatial test difficulty guarantee markov discuss testing specifically ise model simple state preserve sufficient irreducible introduce conditional statistic set move pass configuration preserve statistic markov basis building irreducible markov perform find basis polynomial principle gr basis technique lot exploit computing basis intractable toy section lattice consist compute ise lattice infeasible gr basis technology observe network node collection compute method computing easily chain connect simple move goodness fit irreducible inspired perform relevant overcome computing ise contingency network previously algebraic notation lattice basis ising prove construct chain change ambient construct irreducible markov chain hasting move uniformly move ise acceptance markov ise algebraic consider coefficient index configuration ij set give subgraph restrict vertex addition let polynomial definition directly ideal model induce color diagram composition lattice algebraic software markov degree instance generator configuration configuration sufficient move three side configuration degree move gr basis grow lattice large
unsupervise information overcomplete latter section semi unsupervise supervised tensor mixture let error include tensor iteration initialization rd empirical uniformly sub satisfie factor vector therefore reasonable regime I q give unlabeled sample rip random remark rip eq supervise mixture column true w satisfy error arise empirical moment orthogonality latter large minimax label much number unlabele overcomplete semi furthermore sample complexity unlabele minimax regime brevity semi supervise high regime notice regime noise magnitude general tensor concentration unit dimensional sphere reasonable appropriate negativity assumption require assumption high norm assumption weak incoherence furthermore discuss uniform draw column sphere hold spherical discuss reduce symmetric unlabele semi overcomplete build initialization initialization slice tensor condition follow sample condition rd condition sphere output estimate column bound whiten base huge complexity mixture incoherence mixture employ whitening set comparison sample well regime sample incoherent factor spherical mixture mention result gaussians spherical gaussian whiten complexity analysis regime unsupervise ica noise ica view already semi ica initialization give let eq recovery give symmetric high initialization column uniformly draw sphere subgaussian nonzero ica output h addition estimate ica mixture weight ica theorem assumption weight ica see assume observe overcomplete ica efficiently learn fourth moment unsupervise previous explicitly characterize initialization svd slice propose setting unsupervise state semi initialization rank given consider estimate symmetric ti initialization rank uniformly draw sphere subgaussian variable setting permutation addition appendix ica suppose subgaussian probability nonzero precisely suppose subgaussian theorem provide unsupervise ica change supervise unsupervised guarantee sample mix rip remark detail condition ica sparsity ica one nonzero entry ica dense guarantee also learn incoherent alternate obtain handle arbitrary enough expense sparsity small high expense reduce incoherent impossible identifiability method incoherent dictionary independent arbitrary expense extend analysis bad section consider noiseless code condition assumption universal represent normalization factor limit sparse code dependent sparsity dictionary random let propose dimensional appendix dictionary atom level quasi regime sparsity time noiseless noisy sample section learn draw sphere impose component effect component notice tensor compute multilinear section discussion option fix criterion experiment subsequent stopping error constant random initialization initialization depict ratio recover vs observe overcomplete work theoretical room improve guarantee initialization l algorithm recover estimate error average initialization perform stop overcomplete model last number normalize theoretical claim square recovery ccc c avg avg avg weight e acknowledgement part microsoft nsf award award support award award award nf notation operator define tensor th hadamard entry guarantee provide prove semi convergence provide iteration good behavior asymmetric inner analyze rank generic generate rank perturbation weight initialization eq formula asymmetric loop satisfy error approximation tensor local identifiability result tensor local good guarantee present exploit r run perform mm perturbation condition generate guarantee condition follow define initialization semi supervise variable tensor guarantee tensor appendix result apply algorithm regime dominant noise regime observed complexity describe theorem global bound concentration case apply concentration theorem requirement change sparse prove ica difference theorem exploit moment expand multilinear exploit expand eq assumption involve odd norm impose apply prove I latent linear ica ica unlabele semi supervise enough matrix bernstein concentration norm tensor concentration mixture theorem expand difference claim combine claim term ne similarly set column partition set belong value apply argument restrict set inequality contradiction assumption contradict rip partition care product fix however fine partitioning kt furthermore constrain net define possibility location bound give net net rip satisfy property unit l lemma section separately like intuitively like adapt simple idea bound norm rewrite symmetry imply vector b random variable norm construct net triple sum introduce definition satisfy term bound bernstein exploit summation exploit bound bernstein bound rip property bound show compare prove norm bound triangle step exploit argument union triple net union triple argument net triple probability claim random satisfy rip vector partition consist rip noise e b rip let rip cauchy inequality inequality sample bind however bernstein utilize small suboptimal additional get bind small column bind consider net partition product definition let sum fall begin term form bernstein exploit bernstein constant net union net hold net ready last q variable remain addition provide net matrix product respectively inner empty construction expand lemma lemma summation weight sum bernstein summation bound apply bernstein take triple net bind triple bind prove ica th moment tensor eq eq proof claim term perturbation bound term claim bound subgaussian subgaussian later median distribution negligible get desire order prove bound median take order rewrite partition summation accord fall summation term show subgaussian use subgaussian bernstein probability term bind imply summation ready h nx h spectral net construct unit ball net subgaussian entry subgaussian subgaussian claim least constant bind net close net nd outer good apply concentration suppose subgaussian equivalently term xx w ab bernstein apply summation inequality indicator subgaussian indicator variable suffice summation q bound bernstein imply least set hold theorem similar perturbation version sparse ica eq ideas claim loss subgaussian dense argument desire net unit ball net uv would claim standard trick two h h difference negligible polynomial inequality bounding triangle vector claim product addition product empty restriction exploit exploit partition summation value exploit last like directly need analyze tail phenomenon give tail subgaussian recall specifie bound chernoff bind subgaussian subsequently tail case summation least range summation q subgaussian least summation use union separate nonzero summation subgaussian summation subgaussian bound term range second union perturbation nd sparse similarly q construct eq variable subgaussian indicator bound summation variance eq lemma conjecture remark bold bold guarantee model overcomplete regime dimensionality spherical sparse bound empirical analyze recovery set exploit label rough refine establish spherical initialization svd slice tensor overcomplete efficiently tensor unsupervise semi overcomplete representation tensor hide identify latent disease community observe latent lead gain speech vision largely attribute moreover overcomplete achieve overcomplete observe overcomplete know provide model overcomplete representation gain mostly domain obtain label typically large develop novel guarantee overcomplete gap wide tensor decomposition employ mixture markov guarantee requirement dimensionality drawback behind work mostly tensor overcomplete mixture code overcomplete pose latent exceed incoherence redundancy establish make enable overcomplete regime compress sense sparse paper guarantee code exploit tensor asymmetric update tensor perform symmetric power update highly overcomplete efficiently tensor decomposition moment tensor unlabele initialization setting require guarantee concentration tensor argument learn tensor method summarize incoherent component basically impose soft orthogonality constraint draw dimensionality supervised tensor prove extremely unlabeled label sample note minimax bind rank svd slice moment initializations ica ica constant component ica fourth moment number unsupervise ica provide learn sparse setting give initialization special ica mixture ica extend dependent sparsity bad guarantee overcomplete recover decay order decay point bias update alternate objective fit learn leave suited non negativity topic incoherence sparsity method believe formulation well suited establish concentration draw mixture ica concentration norm rely net vector sphere net loose fine grain distinction employ concentration propose vector classify sparse dense correlation large refined impose factor impose isometry rip rip gaussian mixture noise constraint vector rip constraint bernstein separately final logarithmic level geometrically therefore overall additional combine analysis somewhat mix one establish bound fourth moment assume involve hide activate case hide correspond sparse ica establish bind depend partition bind tight empirical mixture ica coding novel involve tensor concentration conjunction alternate guarantee recent establish local global incoherent combine concentration range variable learn complexity topic analyze iteration extend nonparametric work require overcomplete require whiten input condition tensor provide improved sample incoherent learn overcomplete challenge ica fourth third overcomplete slice provide careful perturbation handle two fourth slice ica mixture roughly low every dimension call style h name x covariance proportion mixture component spherical spherical generalization overcomplete parameter problem special empirical empirical assume variance overcomplete scalar hand setting also matrix ica random signal perturb noise latent independent gaussian noise depict representation ica circle draw style minimum size inner black hide x observe x estimate ica formulate j ica model ica constraint sparse dictionary ica study code briefly work concentration high tensor concentration norm estimate sample mixture concentration rd moment mixture describe respectively recover norm difference condition randomness hidden notice hide sample correspond hide matrix satisfy see remark introduce noise regime theorem regime dominant concentration mixture theorem whiten whiten step orthogonal eigen whitening lead result apply concentration norm analysis eq estimate singular factor bernstein result scale rip property sample propose rip adapt rip rip spectral model entry prove concentration net argument construct vector net good result small argument net incur factor complexity get bad key usual
successfully employ impact medium patient medical text particularly appeal label approach base transform dimensional dense expensive special heuristic novel learn adaptation structure learn dense feature reconstruct language directly low outperform adapt pos web occurrence source driving method domain adaptation correspondence al autoencoder well suit heuristic furthermore correspond subspace face amount occurrence information avoid tradeoff induce directly tendency nlp per template rather treat induce fill template embedding give dense representation embedding skip gram negative yet method useful template skip gram induce embedding sigmoid embedding embedding target dense feature vector template nonlinearity representation apply embedding concatenation vector pos pos several web review answer well use development sentence representation web target unlabele sentence pos treat svm add dense feature basic feature lexical embedding correspondence et autoencoder target domain aside distributional stanford pos development word template distribution word embedding set table slightly use principle embedding compare
overall cloud conclusion solver apply alternate admm noise convergent handle long exactly decomposition iteration count require critical iteration rough start solver solve hour switch formulation accelerate proximal improve solve size f alternate augment exploit separability splitting system minute partially approach apply fista solve dominant cost example video formulation half variant admm insight interesting split problem depend low maintaining context encounter factorize thresholding approach potentially present leave future work restriction necessary particular huber smooth arbitrary operator operator embed index include transform g main formulation differ functional fall problem study find datum case norm write set origin define interior example trivially non negativity exposition implement must solve simplifie denote simple square main computational solver key functional decompose denote canonical give ex dual explicit formulae ex tx also asymmetric negativity model formulation challenge fast formulation focus challenge advantage issue project onto define product ball efficient onto project median ex ex ex long straightforward efficiently onto form scalar sort conjecture median reduce ball arm state may projection onto mn alternatively inner onto well depend singular minimization projection onto ball reference intersection note negativity theorem efficient accelerate ex deal quadratic whereas hessian solve descent objective operator rx il hessian remove separability instead cross coupling term bold prevent nuclear norm hessian coupling potentially use solve non smooth take fashion expect trick algorithm code software need testing purpose variant solve program cost vary solver reference l benchmark design required pick stop plot rather table dominant multiplication core randomize since number value order calculation without convergence unfortunately involve incorporate set challenging routine test create rank singular haar measure uniform add noise equal entry exponential long tail noise capture partly partly give reference solve find advantage solve different formulation non norm normalize residual show extremely simple formulation quasi variational section sequence show jump start solver accord proximal make slowly accelerate coupling depend converge slowly perform reasonably bad solver wrong likely bad smoothing several test show test accuracy knowledge solver sample uniformly distribute white test vary range error term nonzero feature converge accuracy need try ht show achieve test poorly competitive use impose time two quasi initially long long svd interestingly explanation subproblem increasingly hard warm start easier consistent regard ht conclusion largely similar ht turn band point major frame camera issue together scene frame hundred application clean remove error great model approach remove far full span iteration randomize anomaly original pick remove camera camera scene cloud cover review formulation new denoise denoise formulation show propose newton state innovation fast method synthetic removal publicly principal pursuit source separation corollary remark ny research ny ny introduce principal component
incomplete unless properly property carefully testing establish result complete family necessary condition subset contain density differentiable logarithmic satisfy positive finite function establish distribution many situation open differentiable continuous definite exist density degenerate two relate routine derivation asymptotic model like poisson partially acknowledge lead manuscript conjecture remark hypothesis robust robustness intuitively extensive property test form describe derive divergence context class test illustrate robustness result hypothesis component statistical help systematically claim basis testing mainly decade theory direction researcher yet formalize later hypothesis tool widely practitioner criteria serious robustness misspecification develop property know involve density continuous motivated minimum divergence density divergence test nan true point simple powerful utilizing although concrete robustness property alone general test recently family pd assume asymptotic condition simply similarly condition al condition description overlap condition take keep relate nan end extended rest test theoretical derive remark theoretical numerical finding choice appropriate propose briefly generalization conclude remark recently propose popular divergence power divergence family read family case divergence respective expression equation coincide pd family read family therefore natural reconstruct statistic divergence nan equation parameter behind correctly nan hypothesis generate density consider employ ideal choice put coincide statistic rest statistic although vary asymptotic first prove asymptotic divergence require minimum test derive nan al condition I routine minimum sense replace minimum divergence asymptotic nan independent observe face value equation parametric offset asymptotic parametric study theoretical test also power require achieve test suppose al normal quantile tm robustness property statistic robustness statistic robustness statistic result cover develop absence robustness ignore test minimum contaminate contamination proportion degenerate contamination order influence evaluate nan function second influence influence robustness independent density power however unbounded mle mle contamination test robustness contiguous parameter order size also contamination contiguous one tend confusion nan alternative neighborhood huber contaminate influence begin contaminated density variate matrix chi centrality propose contaminate square freedom random prove let taylor taylor series expansion around simplify get probability ta employ theorem diagonal ii part series central central chi function derive contiguous contiguous contamination robustness independent important follow corollary coincide asymptotic previous give alternative truncation contiguous case power contiguous asymptotic hence asymptotic contamination form series expansion linear independent chi central expansion approximation many usage finite ga g u f g hypothesis asymptotically normal zero cg chi square exactly otherwise independent corresponding interval misspecification solely illustration study cg extent interpretation influence whenever value bound extent follow routine omit scalar u u contamination hypothesis univariate use limit thus ordinary consideration nan density assumption recall normal asymptotic compute contiguous alternative alternative plot almost whenever second divergence power simulated power interestingly power power case illustrate give picture power base approximation work produce value close simulate surprising approximation nature simulate power much close contiguous present table contiguous hypothesis freedom centrality contiguous turn upper chi square distribution freedom independent loss efficiency pure offset robustness extend present influence zero influence unbounded always use robust statistic value influence seen decrease influence power numerically figure significance robustness perspective power influence nature derive replication contaminate type contamination scenario empirical sample three contamination give fully size combination table nominal compare like size somewhat value close consideration h r power contiguous hypothesis contamination figure size respectively power size power stable power contamination scenario although present calculation representation power size contamination contamination almost power decrease contamination away true combination contamination distribution contiguous satisfactory alternative contamination finding result parameter practical usage th illustration implication chi subsection mean chi contaminated define subsection independent show present contamination proportion various early correct highly produce confidence interval even contamination remain chi theory discuss effect contamination present examine contamination proportion generate quite contamination slope slope chi factor bound contamination stability unbounde far robust illustrate divergence within
carry primitive neuron template store dimensional transform template module template illustrate simple signal product cell inner pool nonlinear function cell bin histogram could moment moment mild moment could invariant signature complete characterization moment suffice notable complex smooth histogram transformation histogram complex cell group observable within partially observable group pool signature compute normalization constant transformation template neighborhood signature imply main module recursive architecture build invariance factorization stack paper latter representation extraction layer invariance shift property signature compare audio track sec evenly level majority voting frame classify global label voting track discriminative strength vs rest multiclass classifier result window achieve long stationarity add modulus lose invariant stable art combine classifier spectra coding achieve combine multiscale base fourier ms audio alone attribute instability frequency instead instability mid invariance add pool template template audio template k collect moment layer concatenation moment template base notably nd layer local translation explicitly neighboring frame pool subsample pooling window frames operation cnns field cover impulse template reduce pool shift template layer randomly set although drop method question architecture speech relevant music transform template manner c build invariant module stack provable invariance usually strong assumption insight weak deep stability lc rd translation invariant translation binary theory stack invariant hierarchical network currently rather invariant representation stack question transform moreover systematic evaluation music audio representation towards limit audio signal capacity invariance end theory pathway unsupervised di technology representation stream module invariance propose module extract build hierarchical mid level representation signal projection template template induce transformation result signature guarantee unique invariant transformation stable constitute network aspect audio representation music convolutional music music annotation detection rely automatic speech recognition transform stationarity window ms music signal apart acoustic music shown require content identify scalability specific approximation analysis scale music variance leverage projection propose architecture learn invariant analysis invariant improve music classification unsupervise feasible store transformation deep network cnn cascade wavelet transform frequency cnns speech tune neuron representation primitive dimensional computational principle audio network many form transform convenient mathematical formalism difficult normalize haar discriminative high within distribution sphere follow er
chain compute execution speed observation delay methodology intend prior bring big term e compute last accept ultimately inefficient unless consider together address remark stress analogy delay slice decomposition proceed simulate constraint delay sampling conversely scheme slice slice prove delay appear poor additional processor dynamic processor mcmc make ahead walk metropolis say reach figure subsequent future convention odd share parent master evaluate na collect master serial core core serial run update fundamental requirement underlie driving reject simulated leave start acceptance trivially satisfy actually metropolis proposal elaborate riemannian far ahead time scheme limited processor worth branch quite substantial rao could improve efficient towards exploration exploration acceptance branch reach static define thus acceptance sequence store average iii towards exploration branch increase per advance reach thing determine candidate assign child add illustration chain reach processor force next line second static possibility symmetric say strategy almost processor iterate examine candidate take processor b notation eq candidate next processor add candidate rejection candidate thus three step last core exploit correspond next candidate branch top useful involve future computation step select tree complete core return reader describe methodology straightforward major improve acceptance delay instead reach extra algorithmic reference give depth add assign k assign candidate k computationally individual split acceptance bias directly u decomposition set strategy show large involve approximation basic acceptance product straightforward remark delay remark reason nonetheless actual poor actual example code cluster ghz core use communication core combination delay acceptance upon delay soon likelihood start bring unit number delay efficient mcmc represent logistic box quite hasting delay concentrated parameter particle detector experiment large search particle decay particle background team provide public reproduce behaviour make adequate combine delay acceptance regular burn iteration sample first logit covariance mle obtain burn core delay run counterpart average draw iteration obtained delay acceptance algorithm repetition namely relative ess huge concentrated distribution posterior clarity similarity mixture offer challenge reference fisher jeffreys readily available jeffreys past drawback posterior hoc correction rely improper posterior distribution jeffreys derive matrix whole establish beyond goal progress sufficient allocate dominate event remain improper associated exhibit implementing delay costly integral form integral analytically involve integration ratio costly metropolis acceptance jeffreys determinant time delay apply accord improper pick second valid therefore opt small choose second multiplication translate n I repeat value compare metropolis implementation metropolis hasting version rely delay without implement maximum processor availability reason graph report result gaussian hasting algorithm delay second conjunction sequence histogram remarkable particular high variability case aside switching occur mh sample delay acceptance minor balanced delay even reduce number draws delay reduction overall acceptance size time less hasting delay acceptance ultimately respective say reduce acceptance since require little since version delay acceptance broadly time reduce acceptance nevertheless suggest time merge overall computational advantage mostly prior costly gain term mainly exploit thank helpful acceptance massive help cluster fundamental partly paris bs paris paris paris paris mcmc hasting distribution huge cost strategy idea generic divide acceptance division variate consider part computer hand example keyword big mcmc acceptance jeffreys running mcmc algorithms execution algorithm direct illustration difficulty solution issue literature likelihood handle unit computer consensus prior evaluate approach acceptance rather rejection sequentially computation propose acceleration modification algorithm present gain computer realistic environment regression benchmark mixture jeffreys conclude hasting acceptance decide value decompose ratio accept successively uniform result markov sequentially stop mean costly final hasting algorithm metropolis acceptance value target test preliminary step detailed balance argument take arbitrary decomposition associate q balance purpose metropolis particle mcmc decompose metropolis modification hasting likelihood delay acceptance good probability original hasting
edge appear array importantly exchangeability invariance property place counterpart provide exchangeable exchangeable analogy uniform representation consider positive equivalently jump carefully evy characterize able evy activity sparse alternatively compound family yield infinite evy measure building framework able considerable efficient statistical procedure graph evy utilize hamiltonian dense infer graph thousand million enjoy former model whereas connection nonparametric interpretability desirable node tune law straightforward interpretability hamiltonian monte carlo efficiently rapid power propose bipartite undirected importantly prove formulation graph cast exchangeability non explore section sampler simply apply efficient computation large network organize background exchangeability array measure important foundation propose present background form building undirecte bipartite graph present exchangeability present section specific case dense empirical carlo computations extensive analysis variety structure network build construct brief exchangeability discrete array thorough exchangeability present survey abstract notion place table l structure arise exchangeability discrete exchangeable law exchangeable measure mix examine time associate exchangeable evy process focus recall arrays discrete consider special row node scenario distinct identity bipartite array likewise array jointly exchangeable matrix undirecte exchangeability fundamentally important concept model exchangeability network triangle star separate invariant recommender derive crucially assume adjacency exchangeability consider throughout point examine notion derive de style theorem jointly exchangeable definition exchangeability measure lebesgue also array jointly exchangeable surely poisson place within yield graph exchangeability adjacency flexible class functional refer exhaustive countable disjoint e increment poisson evy laplace transform evy characterize increment note jump evy infinite jump jump surely model finally throughout tail evy intensity q poisson implicit weight often person message associate count message tag could undirecte direct direct node due relationship much gain carry atomic illustration restriction ccc auto grid circle color color draw blue cm leave loop bend cm edge bend left node bend auto cm every style edge node atomic simply give measure informally individual construction imply finite primary similarly atomic indicate edge arise undirected edge self could person page equivalently specify undirected condition random ij ij ii yield respectively mass drawn measure model simulating graph direct depend total value form undirected consider correspond cox attractive practically theoretically use power law exact sampler dimensional practice activity normalize background random probability variable distribution partition exchangeable partition symmetric argument rewrite though bipartite let set allow atomic direct similarly atomic graph formulation introduce whose jump correspond bipartite slightly formulation general exchangeability enable insight depend intensity provide refined choice jointly permutation da follow directly exchangeability rate l evy tail evy evy bi inverse evy intensity undirected evy intensity analog formulation extensive illustration arbitrarily yield enable interpretability ji indicate expression application l evy tail evy q slowly vary function satisfy constant equivalence notation degenerate evy intensity follow asymptotic derivative edge restriction obtain poisson infinity activity sub infinity link evy intensity evy intensity vary slowly xx direct consequence appendix theorem scale node activity evy intensity disjoint group simulate undirected direct transform undirected one imagine simulate direct infinite jump approximate possible evy intensity resort applicable evy intensity inverse define sample weight method direct problematic must one poisson bernoulli draw instead cox direct edge simulate inverse evy iw z show scheme examine various link evy generalized process graph hyperparameter undirected graph case direct bipartite poisson increment dirac delta recalling follow equivalent os enyi lead dense graph edge grow poisson representation poisson either trivially empty interpretable remarkable know l evy intensity jump jump jump include special process tail evy intensity gamma display enyi b gamma c stable exact sampler mass stable plugging process variable give direct undirected challenging scope power law undirecte direct incoming twice almost surely behavior corresponding proof sparsity dense whereas nod undirecte almost infinite activity evy intensity proof technique analysis simulate undirected various os r enyi graph nonparametric model explore exhibit power heavy degree show cut tail plot number number growth os enyi cc er ba distribution various node lead dense graph versus note growth os base explore empirically follow interpretation figure relate slope distribution overall network law overall direct interaction large determine decay law degree pure small infer hyperparameter hyperparameter conditional restrict decompose corresponding measure correspond total remain point poisson distribution evy probability laplace poisson random identifiable bring information homogeneous derive also want assume improper prior evy pdf hyperparameter interest w simple miss poisson convention efficient propose hamiltonian hmc within log posterior case q total mass admit analytical base need metropolis ratio summarize sampler rest hmc metropolis hasting graph count hasting computational linearly step hmc mcmc iteration hundred thousand efficiently hmc collection bipartite posterior exponentially intensity particular gamma total stable additionally describe appendix l evy intensity preserve identifiability update metropolis calculate latent symmetric repeat detail model regime run mcmc chain use matlab computer successively indicate rather remarkable node degree degree displayed show method model os enyi run chain specification informative expect section weakly factor convergence trace plot converge node os star sparse base graph relate measure costly implement formulation describe graph prior aim report connection mcmc output social circle facebook political connection network student california protein power united act bipartite mat method network email connectivity link www pages nd range edge empirical l l name nb node nb ci facebook mat www run chain specification posterior credible trace respectively parameter fail provide small circle likely dataset connect three infer note top network dense evidence subgraph spatially may highly though community sparse capture future generative note analysis remarkably leverage otherwise discussion reasonably network l see tail behavior cutoff explicitly cutoff dense cutoff tail bipartite article projection create dense contrary construct undirecte count two count great edge dense issue overall appear well homogeneous law cutoff tail perhaps dense cutoff extensive work dimension overview produce draw edge product parameter similar rescale projective another edge node interaction belong dense latent node embed latent factor case edge probability possible extend approach highlight connection generate configuration proceed follow odd node connect edge obtain discard self loop repeat work place generative projective modeling exchangeability network tool theoretical herein represent important building development incorporate attribute etc thank derive feedback value stochastic relation slow grow grow activity activity equivalently follow homogeneous poisson law yield theorem combine almost almost consider infinite activity jointly exchangeable w imply conclude finally q moreover surely thus eq imply graph construct exchangeable symmetric array surely law large almost thus almost surely combine dominate statistic value symmetric hx x x consider martingale
situation minimax separation characterize mean enable detection leave dependency sequel contribution know unknown combine canonical distinguish symmetric tailor dependency propose minimax covariance normalize competitive relatively test base top eigen propose test moment achieve rate detection obtain projection skewness statistic suboptimal signal term mahalanobis mahalanobis minimax minimax detection express regime minimax detection rate top ep n top minimax rate r unknown low bound extremely test nd moment mixture support mean estimate responsible parameterize certain estimator consistent dependency left selection work dynamic constant dynamic set give positive integer large ps analogously replace contribution support maximize consistent minimax estimator nontrivial asymmetric show support consistent imagine cluster motivation selection meaningful accomplish indeed mixture methodology test nontrivial suboptimal variant bring nontrivial skewness normality motivation prove rate moment hard control nan issue emphasize table moderate testing share pca detection gap amenable procedure contribution propose test wise relaxation sparse table wise precise statement rate regime test maximal top eigenvalue precise slow optimal symmetric nd sign moment note mathematical technical argument derivation reduce standard distribution put chi detail degree control bound approximation gaussian random already cite dimensional none offer real theoretical difficulty mathematical area gaussian focus design polynomial identifiability optimize exception analyze canonical mixture gaussian center note sparsity spirit propose appear initial present publicly feature wise goodness slightly suppose nevertheless rate relate cluster mixture two gaussians dimension identity exhibit method propose identical method relate obtain specialized author discriminant comparable instead close literature component expression work iid center normal know closely work lead sparse relaxation see closely tackle selection context propose correspond step study method strong lead concern reference sparse diagonal important general covariance indeed eigenvalue covariance unknown special diagonal coordinate wise discuss issue sparsity covariance mixture proof defer low notation matrix principal euclidean denote inner etc appearance covariance minimax q fix test use knowledge covariance fix minimax usual testing reduce design low proposition apply know display thus respectively lead maximizer principal standardized variance competitive moderately set vector guide top ss standardized observation direction roughly reliably detect versus fix powerful nc universal simple inclusion argument test remark notable difficult compute reason leave implicit test proof proposition say reliably detect consistent let sequence asymptotically mean nc constant strong assume dynamic range small isotropic boundary roughly statistic show top principal eigenvalue instead powerful satisfying however proposition mahalanobis distance schwarz eq symmetric methodology seminal applicable assumption test variant case mean asymmetric set minimax minimax detection distance degenerate sparse dimensional setting deduce become minimax phenomenon occur problem remark reduce reduce note testing center covariance bring sensible hypothesis high set test normality normality reject general direction calibrate test substantial discrepancy proposition come moment heavy tail concentrate enough fourth central absolute moment aforementione large minimax boundary variable unable original situation mixture symmetric argument maximizer normalize denominator maximizer align motivate consider estimator consistent effective seem fail simple condition otherwise omit simple low bound term mahalanobis mahalanobis relevant dimensional reduction simpler sparse asymmetric meaning shall asymmetric symmetric due ability symmetric cover pseudo test fix setting sparse note interest normality make sparsity assumption along necessarily direction versus substantial obtain issue enough third leading eq critical c universal achieve note minimax substantially unable statistic analogy consider estimator despite statistic satisfactory strong refer reader coordinate wise support covariance unknown lead linear analysis testing matrix eq eq estimate covariance estimator place yield convention square diagonal matrix ij depend long diagonal bound detection denote sequence critical asymptotically powerful constant maximizer phenomenon suppose qualitative difference statistic covariance proposition involve mean away mean result grow situation value result size subset precise compute practically central concern consider section eigenvalue tailor sparse nature motivate method arguably test implement inspire arrive statistic correspond statistic n testing asymptotic eq universal universal adaptation stronger somewhat prove asymptotically c large spread method even achieve rate multiplicative factor incur analogy sparse recent work apply polynomial plant clique time definition covariance constant adapt work calibrate set significant testing level detection eq small range bound condition fix spread reduce performance multiplicative factor rate extent intrinsic polynomial come instead principal need relaxation eigenvalue learn apply detect covariance simplicity eigenvalue semidefinite semidefinite relaxation perturbation soft jk jk mdp relaxations time semidefinite program computationally require eigenvalue grid statistic universal level go ingredient valid find mdp tend nan hand follow tend factor sdp worth clearly algebra binomial random chernoff binomial chernoff generality step begin brevity place resp resp prove asymptotically vector l sr sr entry sum turning moment rely distribute size integration follow denote derive follow enough go zero corresponding expectation small expectation zero occur k last kp first compare expansion ad hoeffding u dx nr nr dx nr k nr iv leave hand x zero derivation hand side supremum negative away side occur imply sr ep sr iv simultaneously positive sr know test equal turn independent variable bound sum consequently w q q since decompose iii k rely depend ii ss fourth need apply iii entail line numerical correspond nr us fact cauchy schwarz line inequality either rademacher hoeffde eventually iv stochastically binomial chernoff start eq v si iv get sum variable proof fix proposition prior sr op sr x turn approach computation follow formula let n assumption taylor rely conclude comparison parameter rademacher decompose separately closely argument without proof reduce use observe tv variation due contraction space triangle translation tv r tv distribution mean calculation application schwarz go tv nh mahalanobis schwarz tv l distribute sum exist go infinity expectation p k thus line hoeffding assume standardized mean bound degree centrality e x wishart apply lemma turn row similarly case performance need control theorem iid normal thus conditionally central freedom non centrality get generality define event bernstein tend q independent conditionally equal satisfying comes chi plug max p large rhs go conclusion small simply tend constant condition rhs rhs w standardize nan n wishart matrix although hence probability go universal appear degree centrality q note work standardize go ss subset union eq tend chebyshev hence tend eventually eventually let maximizer j j standardized assume write control numerator denominator separately euclidean unit bind denominator dimension maximum possibly wishart union derive numerator possibly apply derive variable control proposition universal constant integer cardinality continuous deduce simultaneously together diameter opposite cardinality lead tend note chebyshev inequality ps powerful suffice numerator control chernoff since generate standardized assume simple define bind nan proposition numerator distribute z euclidean norm triangle schwarz cauchy inequality bind subgaussian metric proof section constant appearance since subgaussian packing semi diameter pack coming add nx come deviation tend one calculation chebyshev chebyshev inequality denote extraction us quantity soon development give thus need amount study sign elementary calculation function decrease symmetry powerful section introduce q differ definition show tend shall uniform control absolute center namely appearance control use bind constant hence nn u deviation follow subgaussian subgaussian apply maximal cn get tend last control second moment u explain c n sc control statistic obtain u development conclude proof proposition without work statistic concentrate rely universal cardinality surely decomposition deduce simultaneously control combine together range cardinality moreover v large term p use powerful converge towards asymptotically extract generality translation standardize observation except x iv control denominator probability tend numerator define sign I sign second sep going go remain control deviation combine argument proof sign ss valid go loss v proceeding early chebyshev inequality n elementary increase derivative indeed powerful focus subtle tend chebyshev working conclude chebyshev strictly q variable diagonal find tend moreover fact use tend tend denominator eq conditional variance j holds conclude consistency one detailed omit proposition selection prove argument omit positive imply concentration denominator universal since bound j consequence chebyshev w w controlling follow decompose apply conclude application chernoff standard define get k x back moment x sparse prove subgaussian deviation transform v u taylor get u v k k already lemma taylor true application chernoff yield desire result rely concentration bound rademacher rademacher develop n inequality converge consequently fix superior r hence euclidean w n triangle schwarz inequality n v schwarz iy n apply
node group section dedicated selection generalization sbm briefly review challenge heterogeneity latent model weight self loop however generalization often graph without loop deal characterize individual characterize may either admit loop model specific random characterize behavior would behavior characterize latent exist property study extensively probabilistic dependency distinguish occur literature may continuous network characteristic summarize dimensional block sbm review section explain computed size set configuration huge computational model independent corresponding form nice counterpart maximization latent raise latent optimization combinatorial complexity besides result observation chen yu perform variable finite mixture respectively latent general variable index panel observe distribution depend parent pz pz z pz py factorize get panel give clique dependency structure prevent oppose hmm graphical shape space probabilistic legend observe variable line inference chain former aim rely suffer limited network hundred algorithm latent handle size random appear surprisingly sbm sbm model position define graph two parametrization direct graph connection node parametrize latent vector kind direct distance replace normalize become q case recover translation whether restriction ensure total include social putting rely author observation argue w multidimensional approximate position fit form second step author acceptance rejection conditional automatically position stage stage simple second procedure cluster sampling besides may determine rely conditional latent define connection node sbm induce science network neighbor protein introduce dot respective propose power diameter independently identically uniform fix moreover connection node dot product latent interestingly dot infer estimate rely also provide position label namely label concern consider apply web page wikipedia mention much already quite people compare take impact already investigate describe space latent space suppose simplex namely use conversely penalize dimension view model extensively mention use interaction popular simple prescribed degree uniformly among degree satisfy ensure know view precisely apply world wide graph company limit graph independently discuss start node I unobserve view sum multinomial consist characterize relation undirected loop direct loop easy generalization loop distribution note matrix model social network biology protein protein world etc weight early version appear conditional useful gaussian etc distribution induce thus mixture dirac strength distribution restrict cumulative cdf continuous zero coordinate connection cdf dirac mass sbm truncate poisson multivariate simplify assume varie consider structure connectivity take depend whether assume induce exactly clique already unconstraine sbm cluster detection sbm subset behave existence start discuss identifiability parameter sbm know undirected sbm solve sbm non valid sbm sbm estimate sbm less sbm rest current estimation sbm binary sbm first limit binary sbm group sbm namely nz formula handle early relation bayesian attempt develop heuristic given replace simple decompose follow eq kullback leibler iterative starting maximize quantity expectation automatically factorize instead go back kullback divergence variational instead optimal factorize unchanged current distribution graph result sometimes respective mcmc sbm gibbs gibbs iteratively factorize distribution right mentioned approximation convergent regular precisely prevent sbm ensure procedure accurate see weighted network version variational approximation appear graph detail thesis one approximate posterior realize factorize approach sbm node note implementation weight graph package weight binary sbm model consider proportion drawback moment general consider propose suit binary optimizing whose parameter look pseudo simulation approach graph group connection within likelihood result node procedure iterate perform theoretical parameter estimation method explore binary precisely estimate proposal reference associate discuss rely degree fast concern estimation sbm concern sense iteration local empirical exhibit reference empirical procedure variational estimate variational nice variational consequence approximate increase maximum convergent parameter graph increase kind dirac factorize distribution possibly maximum variational proportion assumption unknown require fundamentally different sbm see sbm composite surprisingly limiting ensure amount proportion limit conclude regime go infinity next recover automatically theorem formal start recall community additional intra group large outer connectivity cluster posteriori result rely behavior variational binary sbm establish state parameter group dirac actual value configuration also result converge rate convergence sbm yet dedicate posterior latent neighborhood estimator sufficient dirac locate actual condition highlight case sbm recovery converge product configuration explain tend sbm related use different consistent sense tend sbm modularity community difficult consequence remain unclear reference establish procedure rely modularity satisfy sbm modularity issue recover tend separability author spectral cluster sbm group binary group group refinement sbm receiver motivated study graph except allow group grow size nearly restrictive note provide cluster setup degree increase node sbm computationally demand network even space unknown question introduce posterior evaluate undirected graph integrate criterion entropy z proxy term refer proportion whereas whereas observe study approximation criterion formal exist bic correspond around recently gap order aim default often use assign sbm belong multinomial assumption analyze network play overcome limitation node possess choose membership link sample fairly strategy simple network propose overlap membership vector respective probability class binary overlapping sbm present logit z much membership relationship chemical specie etc network publish contribute contaminate column asymmetric membership membership draw first context simultaneously variational inference recently propose toward sbm assume indeed situation connect extend value sbm take play role sbm control degree orient version asymmetric generalizing likelihood regular detection sbm partially explain accounting desirable well understand structure important species covariate edge take ip presence edge logistic regression covariate raise focus covariate propose probit depend covariate act membership author
maximize regularity establish point observe far different allow exist matrix uniformly leverage property establish propose measure time empirical straightforward section evaluate influence synthetic generation building employ generate uniform maximum large inside generate sequence baseline hazard algorithm topic standard hazard indicator mention tweet tweet survival hazard short update record indicator illustrate newton baseline hazard week worth interval size show mean vector underlie ht see quality quality give influence interest exhibit relative next insight measure generation htbp rank one approximately realization user action generation rely picture twitter link account draw account u financial post york increase activity acquisition twitter tweet extremely trace twitter active tweet back tweet table tend mention mean self activity pay medium account political media influence twitter utilize build public pass date account tweet nj avg http rt mass http rt huge http http co http stand worker w keep track pay rt thank I community http take hope nj http thank rigorously apply constitute many algorithm account publish create adjacency data score denote great r rank propose financial york journal cycle cycle next estimate analysis max china hold death frank run use raw consistently influence include explanatory table coefficient mean influential twitter successfully tend influential life pass instance influence r std intercept age df std error intercept propose influence age characterize influence scale platform information user comprehensive demonstrate direct correlated influence broadly direct closely community modeling inference detection goal adjacency media platform incorporate fundamental action social outperform topology perhaps massive volume estimate relatively straightforward identifying involve analysis message relate appendix algebra element necessary hessian partial respect expression cross baseline hazard partial likelihood parameter ignore straightforward go concave expand definition rate define replace hazard integrable integrable converge number converge converge notation employ condition theorem simplify want establish convexity dimensional zero expect entry derivative derivative semidefinite definite convex complete ten influential account period financial post prominent several rank financial table period table score significant influence regression transform score moderately std influence df std intercept r estimate std intercept df variable estimate std value intercept age theorem theorem introduction remark measure twitter counting extract medium twitter text interaction influential sense generate platform service web apply interact capture influence maximum covering year twitter member news business volume increasingly complex datum create insight area growth social public easy exchange news idea area business network analysis vast amount careful content propagation product platform importance business twitter twitter million twitter lag behind facebook nevertheless presence mechanic twitter basic communication account platform allow short mid message daily twitter follow receive whenever follow serve primary spread platform account tend interact channel direct way copy another tweet tweet name symbol mechanism mechanic user rise message together use constitute corpus enable search query build capability create flow interaction account influence capable drive discussion topic valuable twitter user constitute active business service scoring employ good message reason sophisticated widely use ranking result network popularity necessarily influence follow account million propose account twitter social media platform ability message action interaction account count action intensity basic underlying activity account reveal world member united house united states two year prominent post cnn e g white house influential interaction direct give tend e edge tweet influence twitter indicate closely importance purely solution organize introduce modeling propose influence synthetic concluding remark section presentation future development correspond node twitter whether symmetric vice versa principle dynamically static explain twitter platform account message account mention vast majority let total generate topic message capture response account mention counting process hazard use cox hazard specifically hazard process account positively
neighborhood regularization belief propagation square euclidean distance originally maintain next convnet sift descriptor sift bivariate alignment seed sift convnet feature nearest sift flow sift row version align sift flow second image target image cat convnet case sift instance alignment measure predict target small great point wise align correctness truth lie bound box height pick compute visible target show per category indeed convnet capable sift field table tv sift flow sift transfer convnet semantic parts image train classifier datum extract feature convnet rf lie place vs sift five activation pooling layer convnet layer sift come specifically testing convnet feature expect high train car cat tv sift precise understanding response cat histogram location include maximum lie sift seem sensitive location convnet fine field motivate final experiment base convnet localization beyond sift feature histogram response pixel take cross cat mean convnet plot feature despite large work sift sift annotation ground bounding inspire slide window part detector predict location cnn demonstrate deep investigate cnns scale window descriptor descriptor region cell give field hard mining ten close truth example dense descriptor sift eight consider within bin ground time bin negative detector nearest neighbor gaussian take neighbor find cosine fix standard output detector combine tradeoff validation high location define width prior detector train outperform sift margin knowledge dataset five set rescaled box annotation predict sift outperform sift satisfactory despite offset noticed window regression fine prediction criterion car cat sift sift sift five cat sift conv prior view alignment intermediate implicitly convnet classifier understand correspondence field convnet ones visual support program reference california berkeley cs berkeley edu convolutional neural net improve detection understanding establish fine pooling whole label success correspondence precise localization effectiveness correspondence evidence convnet feature fine field size perform alignment conventional object advance convolutional neural net dramatically improve specificity rely large pooling region job coarse localization extend fine modern able field correspondence pool task well suited hand provide convnet conventional considerable image alignment task face motion object alignment correspondence across variability alignment supervision require class joint model unsupervise method hand unsupervise optical match densely sample sift correspondence motion sift recognition pose fine grain categorization depend variation category challenge localization human pose pool input convnet convnet spirit pair dictionary average convnet feature vector associate patch field table number replace patch near neighbor database densely database one million every feature match rf sized neighborhood throughout region notable e cat replacement visually specific cat get replace differently color shape
mx unknown use sample n pointwise alternatively build set quantile confidence set include smooth ordinary smoothed regression mode issue tool htb q process limit assume smoothed discrepancy supremum gaussian process couple behavior bootstrap consistency f theorem limit confidence modal set level sample kde conditional ny estimate pointwise pointwise ny ny correct coverage nx ii estimate prediction coverage prediction select kde prediction subscript denote smoothing lebesgue uniform define roughly speak estimate manifold manifold bias variance manifold proposal select e minimize trade versus mark line display modal estimate uniform set display uniform local modal comparison smoother also illustrate modal regression method conditional capture main next population modal base underlie pointwise prediction lebesgue modal somewhat abuse lebesgue length prediction consider denote gaussian variance important make population reflect state define several quantity minimal center moreover q show signal modal small way modal know expert vast function parametrize variance usage assumption consider simple tool use inherently base joint mode without component mixture model role modal regression specify instead flexibility regression kde tune figure give linear modal package specify run eventually high bandwidth reveal regression trend domain specification carry trend across assumption independent volume cover extension would address modal conduct cluster lead proportion modal roughly analogous cluster mode apply modal path start cluster shift accord iterate update point arrive mode jx determination modal jj immediate running shift point datum example run mode estimate modal place modal regression modal modal seek modal former mode second investigate nonparametric modal kde points modal regression confidence prediction kde compare relevant message offer method develop construct usefulness practical treat predictor finish example normally distribute surface come modal green modal identify local surface sufficiently large assertion kernel theorem write mode unique empirical point eq divide side away inverse mode prove assertion focus follow repeat argument mode local mode local assertion bx nx yx imply big involve thus supremum maximum vc envelope give proof theorem theorem local argument integrate still omit technique define empirical process proof couple anti convert coupling constant constant recall see constant envelope let center ba verify assumption vc type envelope inverse closed mode thus pick fact coupling pick triangular constant nh nh eq nh essentially theorem current basic detail depend probability index density approximate measure completely take estimate derivative kde determine maximal space putting result prove prove mode w another necessarily distinct local mode mode eq since mixture component away bad scenario define since term mode obtain local mode definition attempt solve hold see sufficient combine conclude theorem three step consider pointwise prediction set extend prediction summarize four mode apply first uniform gp note prediction pick regression mixture center kx mx thus reference z require involve inequality apply equation hessian explicit determinant namely hold need eigenvector therefore former case correspond modal regression usual simple kde techniques latter kde behind modal tie class alternate regression response variable unlike base modal regression would modal favor conventional answer level mode reveal illustration example fail capture trend produce band improvement well rigorously modal economic formally mode simplification joint consist general smoothly change local behave surface call focus derive kde nx plug author thorough modal prove modal regression derive error metric set base plug regression select bandwidth kde draw comparison suggest modal ridge begin basic recalling previous describe item end simulation response set classic modal denote modal value smooth local set rely yx kde joint brevity modal eq subscript efficient computing shift algorithm gaussian mean describe kernel mesh convergence iteration straightforward update indeed ascent nx yx implicit step actually attain critical size property modal union implicit dimension illustration univariate htb modal factorize eq connect parametrization open call convention write nonempty mode twice modal manifold yx jx joint modal function manifold guarantee mode smooth modal smooth smoothness modal condition hausdorff omit classic notion smoothness hausdorff think theorem interpret statement continuity hausdorff modal manifold merge vary though occur contact manifold figure modal manifold leave curve close look slice unimodal saddle htbp
sampler draw center order avoid vb mh assumption derivation appear covariance matrix mh ground model reasonable mh return true see mh consistently variance ht retain datum actually deterministic retain prior unknown posterior look spirit classical leverage statistic literature covariance classical score impossible naive distinct covariance draw evaluate leverage effect manual leverage score plot manual great one assign retain leverage indeed assign component affect affect affect way complex chain correlation acknowledgment suggest response helpful comment berkeley suppose x linear collect proposition proposition approximation interior apply particular track write eq part remain natural eq zero dimension finally eq lemma target use covariance particular return block matter form next order consider characterize posterior posterior derivation highlight column invertible since stack follow point exactly log interior derivative follow matrix multivariate simplify sufficient vector denote kronecker calculate derivative sufficient statistic statistic interest effectively ignore submatrix variational variational property cc simply multiply v apply normal draw correction transform fisher information use correction formally argue correction variational require correction coincide approximately proportional might affect whereas us analogy parameter asymptotic multivariate take find mode normal em equation transpose complete correspond variance fix variational leverage datum first leverage leverage model score x formula denote scalar form stack recover vary variable analogous fit improper light statistic term correlate respect quadratic multivariate apply consider cc x tx calculated tx tx cc tx since diagonal new perturb observation plus z px p x I imagine unobserved statistic point since matter property expectation convenient stack statistic single covariance sub covariance derive mind notation shorthand body block ccc interested eliminate immediately complement matrix two group refer everything else give cc r r x r I first r r x quantity aid write z r r r x q use version term exhibit power perform analysis substitute thing plug r r final rgb california berkeley increasingly collection interested analyze ever fit old paradigm capture practitioner uncertainty across method fast major model uncertainty interact develop family model particular
pass boundedness finitely accumulation generality limit side boundedness kkt solve subproblem solve quadratic solve suitable write perform processor ghz ram unconstrained subproblem inner condition approximately eq penalty update discuss minimizer closely relate residual thus typically terminate approximately describe place exactly ease reference approach inexact exact randomly generate generate row space standard gaussian test report table nonzero use matlab cpu second termination penalty termination usually recovery slow method inexact objective phenomenon intrinsic c c inexact cpu cpu e e c c inexact cpu e inexact exact cpu cpu cpu e e e e e e e sparse processing constrain widely use minimizer always nonconvex nonsmooth study existence penalty regard local minimizer minimizer penalty solve sequence via gradient prove method kkt preliminary solution system appendix solve optimization continuously moreover bound uniformly method pt integer arbitrarily globally however globally convergent th termination criterion convenience one statement induction use hard use relation relation finitely iteration hold statement induction statement ii hold hence induction finally statement total inner execute one follow ii em solution iteration hold algorithm corollary support partly research grant author partly grant possibly intersection ellipsoid induce tolerance incorporate fitting constrain exact objective existence regard local minimizer minimizer subproblem solve prove kkt preliminary demonstrate sparse penalty proximal nonconvex optimization follow continuous q nh necessarily convex locally avoid suppose region nonempty flexible accommodate wide important imaging science processing variable study refer reader emphasize function nonsmooth nonconvex non enable various induce concern one popular bridge fraction penalty incorporate must application gray incorporate lead substantial flexible range solve extensively decade study scenario nonconvex minimizer construct optimization minimizer closely resort important exact constrain development nonconvex non objective bridge use hard must satisfied soft advantageous knowledge paper various penalization follow problem minimizer global minimizer minimizer projection minimizer onto feasible produce minimizer solution exact smoothed suitable scheme penalty minimize possibly nonsmooth smooth globally nevertheless globally include subproblem addition accumulation approach solve approach sparse solve suitably result rest organize present notation preliminary material study regard minimizer optimality propose update scheme penalty experiment conclude number entry sup norm quasi norm pi ii n r function infinity elsewhere center ba recall subdifferential horizon subdifferential mean eq subdifferential coincide subdifferential subdifferential finally separable reader explicitly throughout nonnegative rank well concern corollary explicitly case start q indeed pseudo last follow follow simple next reader finite upper exist bx bx assumption desire lemma auxiliary concern generality maxima attain optimization complicate next consider increase twice continuously differentiable check optimality stationary monotone consequently minimizer nonconvex regularization bridge equal happen equal negativity building regard minimizer find minimizer nonempty special possibly penalty emphasize little everywhere lipschitz modulus local local subsection locally assume except local minimizer local whenever I since minimizer apply b conclude exist whenever assume loss generality minimizer globally lipschitz continuity modulus assumption hold modulus inequality minimizer minimizer concern minimizer say globally modulus concrete maximum value let discussion fix continuously differentiable attain approximation show conversely minimizer minimizer minimizer minimizer admit minimizer whenever minimizer modulus eq combine hard e conversely study necessarily feasible minimizer old take optimality old concavity follow immediately h follow immediate suppose minimizer penalty optimality condition look model motivate check whenever subdifferential explicit satisfying use hard kkt solution assume motivate kkt kkt exist case kkt equivalent check sx constraint stationary point conversely kkt point comment stationary facilitate exist focus local inequality follow inequality follow minimizer low nonzero derive magnitude lower low except ax b lower bound minimizer finding globally computationally inefficient form nonsmooth solve sequence smooth counterpart see hard show form solve consider adaptation differentiable globally objective observe locally globally appear fortunately capable verify applicable problem hence continuous
carry assign eliminate average testing number node test spend cpu whereas cpu fig result zero also consumption interval satisfactory section conduct diabetes concrete repository one report speak rate rank output calculate generalize inverse rapidly possess acceptable robustness sometimes point sort transformation come symbol become treat resolve operation method sort effectively transformation keep activation function satisfy radial belong function include pose use effective department china china extreme promising learn single hide layer feedforward nevertheless random layer rank effectiveness effectiveness propose improve make weight rate prediction accuracy experimental problem classification regression performance feedforward machine feedforward neural extensively mapping natural artificial cope technique address approximation ability compact make neural field prove continuous feedforward activation furthermore systematic condition activation al bound strictly construct show almost activation method advantage tune network architecture algorithm accomplish minute large might learn possess generalization svm activation among one fast superior overcome input weight bias column sometimes train predict point feedforward network method overcome machine properly input matrix extreme learning machine use bias sort due constructive algorithm activation use radial basis sort radial basis activation diagonal bias output rank simple operation give weight bias spend short fast possesse implement make fast generalize inverse follow bias strictly correct theoretically constructive actually complexity give discussion arbitrary ii hide name sometimes cause overcome extra bias acceptable keep column add definition inverse di j jj da confusion concept follow constructive sort transformation dx dd w x em w w ik kn ik kn p w kn ik k affine sort sample order sort correspondingly weight ensure non network activation gx dx dd square actually gx gx gx ax select obtain b b therefore complete w n em b w ii calculate forward effective algorithm training x cm plus minus ex ex ex plus ex weight calculate output detail practice fast choose bias instead selection big less sometimes random bias singular difficulty accuracy section orthogonal instead singular one summarize extreme call new basis ib I correspondingly sort ij I x gx
fall value polynomially make logarithmic hard conclusion know parameter depend matrix section bind clearly condition suffice importantly estimate parameter actually precisely source literature acknowledge difficult logarithmic perform nontrivial operation truncation concatenation require operation require operation run require include multiplication follow take theorem proceed maintain inductive hypothesis epoch begin step index value vector goal next estimate let algorithm decompose help keep matrix set track angle matrix orthogonal maintain inductive epoch incoherent eq define everything hypothesis satisfied assume base handle indeed probability incoherence satisfied suppose break loop algorithm goal gap analyze statement hold terminate terminate line reach obeys orthonormal satisfie define line return accuracy may iteration conclusion step initial inductive hold round hypothesis hold lemma statement imply requirement favorable bind epoch inductive round occur lemma base summing show precision produce round approximation tt return subsample tt node tt tw tu indicate argue noisy constant statement lemma thing top singular wish show invoke theorem long union condition close return find sufficiently case long lemma establishe conclusion imply inequality algorithm identify gap suppose hypothesis value q let read large q first conclusion analysis gap particular must verify inductive follow imply eq know reflect top vector denote actual use theorem perturbation close eq similar computation q favorable imply appendix unitary eq truncation probability incoherent precisely q ball orthonormal orthonormal projection close eq furth gram schmidt eq thus eq incoherence computation line conclusion incoherent step depend condition hold final holds follow fast completion frank wolfe fw analyze also naive algorithm code fw aim diagonal spectrum sample low fixing per smoothing median implement frank wolfe parameter improve cost run change qualitative dependence eigenvector large wolfe completion measure frobenius error recover outperform metric one reliably converge find eigenvalue measurement basically progress happen svd converge small quite find fw svd begin still outperform like frank wolfe cite fw depict reason reason note fw converge sample converge matrix error illustrate frank wolfe pre specify spectrum run frank wolfe algorithm observe entry predict fw much fw fw fw never near entire acknowledgement institute berkeley take probability subset thing convenient begin choose observe let split l l u rt divide suppose return include independently let let definition show generation treat notice independent consider collect upon record q similarly perturbation similarly singular perturb recall refer angle principal decomposition perturb q q angle span matrix unitary two close orthonormal unitary v v row ir let orthonormal suitably eq index indeed restrict angle fix denote suppose gap collect repeatedly chernoff version matrix concentration subset entry conclude chernoff sequence adjoint median proof prove noise proof index coordinate expression drop identity suppose q probability apply chernoff proposition constant I markov proof choice appropriately inequality complete constant claim compute expectation markov inequality attention choice conclude least claim together small indeed union least show term favorable q polynomial rank matrix number well alternate minimization recover subsample amount processing partly due applicability recovery guarantee feasibility semidefinite program dimension immediately dimension research effort large scale nuclear solver preserve nuclear alternate solve solver alternate scalability minimization less despite progress number th target condition serious completion unknown approximately singular decay typical alternate decomposition alternate truncate crucially sub routine solver appear kind dimension running emphasize main sub time alternate work resolve question variant achieve standard alternating framework improvement completion noisy completion first incoherent certain compare sake begin completion result consequence subsample include nonzero result straightforwardly rectangular matrix state intuitively coherence standard vector formally orthonormal vary state formal factorization expect exponent minimize imply small several logarithmic near step run nearly reveal except overhead discuss apply matrix close typically capture arbitrary assumption incoherence euclidean state entry correspond coherence norm argument frobenius well show generalization recover statement compare parameter enter suppose formally assuming discussion relate overview overview understand understand basic matrix start iteratively fix objective typical repeat generality square exploit square update interpret noisy rough spectral initially ignore since like discover value order achieve number arise alternate truncate svd onto subsample behave run fix call run alternate minimization instead suggest et number dependence unfortunately see run serious noisy completion hope black imagine day spectrum matter rather matrix run error prevent converge far might problematic singular arise residual arise multiplicative difficult ensure intuition argument seem single alternating however dynamically maintain proceed epoch proceed begin epoch converge singular prevent say remain vector correspond perturbation ignore polynomial importantly motivate epoch approximation singular point identify block singular principal mean alternate factorization epoch crucial always minimization subsample original purpose prevent accumulation basic say gap care gap might small additive super pay able must make sure singular issue face coherence incoherence completion outline sure incoherent rough estimation preserve incur alternate handled build extra involve take alternate incoherent related number polynomial crucially build minimization black box note work sufficiently particular polynomially al nuclear guarantee however dependence aware fast polynomially another nuclear
model call suggest global graph previous separate package execute complexity exponentially variable np network problematic high dimensional biology protein scalability impractical restrictive either dag address reduce independence score computation core processor make core hardware algorithm decomposable meta heuristic say si efficient basic backtracking optimisation examine implementation backtrack gain variability dag alternative software framework implementation set suffer section among share ic pc gs illustrate first dag bn originally suggest consistency symmetry correct treat false positive arc dag parent child illustrate step separate separate set increase keep computation greatly know I exception skeleton enforce third arc direction equivalent identify decomposition arc undirecte identify contain multiple dag uniquely skeleton step step dag complete direct acyclic variable symmetric symmetry drop parent child equivalently give place undirected limited symmetric pair direction arc I direction arc still undirecte recursively adjacent direct lead undirected arc symmetry make translate enforce construction g x jx reduce test introduce positive false type limitation focus model overall acceptable causal model large set many describe version step inclusion inclusion case backtrack undesirable structure store backtracking compare implementation compare define give correspond backtrack node si code learn already bn contrast node si b equivalent learn would si pc information omit brevity package implement node pc implement backtrack instance backtrack vast investigate various alarm arc expert alarm message unit monitor device arc bn diagnosis clinical condition marker arc base physics united assessment plan recognition action arc develop context linkage large linkage genetic marker arc expert interpret disease physical dual core gb ram size fact backtrack variability display three step mean step si test si pc step convenient show perform test intensive part amount implementation pc different pair arc merge step backtrack require constraint user use master process generate alarm alarm cl cl counter generate pass output affect structure raw run backtracking perform cl cl counter cl r inter cl counter average result link size run parallel si follow add small never fewer complete instance parallel overhead range attribute cost well competitive bad performance scale biological si pc control count former predict patient diagnosis gene explore presence system biology bn genome quantitative trait use human disease dependence nucleotide publicly available miss number biology genome wide association si pc average consider test student correlation see observe overhead normalise running consideration make across overhead surprisingly overhead seem absolute overhead comparable suggest strongly depend seem little effect run implementation comparable framework create implementation root ic particular part execute firstly limit overhead keep part see backtrack improve backtrack different motivated suggest increase dags speed gain competitive parallel computer come least core outperform backtrack even implementation reference overhead introduce high scale efficiently finally important consideration gaussian variety structure several implementation improve overhead might dynamically become bn likely little benefit overhead parallelism window avoid directly bn partial update would also overhead scale bn average variable latter learn copy could avoid share memory suggest overhead never modify hand bn overhead various bn likely leave even allocation scheme bn dramatically introduce variable rough proxy bn depend impose sparsity tool overhead keep constraint use four biology modern core hardware preferable backtracking develop processor dag distribution refer arcs connect represent
class maxout percent classify clean training adversarial model appear deep build part likely process model behave reasonably thin pt network adversarial form bad perturbation incorrect answer phenomenon focus nonlinearity argue primary cause explanation quantitative set yield simple generate adversarial maxout make discovery neural adversarial correctly classified wide architecture train adversarial blind nonlinearity deep combine insufficient average insufficient regularization purely unnecessary cause adversarial fast make adversarial practical adversarial training dropout generic dropout model average significant reduction adversarial change nonlinear family rbf fundamental designing linearity design adversarial possible tradeoff designing optimization nonlinear demonstrate variety include bfgs reliably adversarial imagenet example example indistinguishable adversarial misclassifie training softmax however optimization modern machine determine naturally datum use euclidean regard indeed already design adversarial perturbation though yet maintain clean adversarial many individual digital pixel discard dynamic rational perturbation feature formally assign discard storage problem adversarial adversarial perturbation maximize subject max assign magnitude weight dimensionality activation perturbation output sort force closely signal signal amplitude dimensionality previous example suppose linearity simple explain softmax adversarial suggest perturbation maxout behave way easier sigmoid tune perturbation neural obtain refer adversarial computed backpropagation reliably cause variety imagenet softmax classifier set adversarial maxout cifar standard deviation adversarial example example angle reliably adversarial misclassifie favor interpretation way confidence imagenet whose gradient small bit image number consider logistic gain intuition adversarial train recognize sigmoid function derive adversarial perturbation note sign gradient regression somewhat activation training eventually confident happen adversarial simply adversarial good logistic multiclass softmax treat softmax single weight decay deep multiple unit decay necessary decay maxout coefficient cause get decay coefficient cccc logistic model adversarial somewhat unlike deep adversarial universal layer unit assign discover desire obviously specify adversarial adversarial somewhat augmentation usually actually augmentation occur way beyond dropout benchmark partially adversarial bfgs may work guess work use error reach adversarial large original maxout cause model slightly adversarial make original maxout stopping terminate decrease flat adversarial therefore early adversarial five generator train weight trial rate result mnist though indistinguishable fine tune adversarial adversarial error adversarial training show robustness error train adversarial still highly confident misclassifie learn change train adversarial perturb interpret play adversarial request case human replace copy nearby insensitive change small training box correspond norm zero inefficient adversarial dot case difficult fact case set noisy noisy maxout noise base pixel mnist adversarial fast sign clean point space considerably rbf invariant generalize view rbf unit different tradeoff curve unit direction rbf point decide quadratic rbf adversarial obtain error train aspect set adversarial often agree non readily account behavior capacity consistently label adversarial rational adversarial example subspace product different see contiguous sign explain adversarial misclassifie fairly probability misclassifie another explain adversarial neural network learn reference approximately stability adversarial deep maxout softmax rbf misclassified maxout rbf maxout class maxout correctly number drive rate exclude mistake predict maxout rbf component behavior maxout generalize significant behavior cause adversarial example reliably large classification occur thin manifold occur plot make train maxout show softmax example see unnormalized linear wrong classification region move input input curve positive box indicate classified input hypothesis adversarial example hypothesis generative training constraint cause model confident test get good classification mnist differentiable differentiable mnist mp sure adversarial rather non top model find rate remains alone cause adversarial maxout mnist network use seed initialize generate dropout select error adversarial example design entire adversarial design member fall adversarial perturbation summary follow observation property dot product generalization adversarial across adversarial perturbation highly align function train perturbation adversarial example like rational direction adversarial clean example adversarial regularization control experiment fail reproduce simple regularizer decay model optimize easy capacity adversarial adversarial perturbation observation concern class easily class rbf modern ai design relu maxout lstm sigmoid carefully fit adversarial correctly imply truly ask confident point confident often incorrect work identify problematic point ease motivate development procedure locally acknowledgment thank helpful thank article concept degenerate input human classify belong want positive input classify degenerate input something separate binary classifier want near would
artificial forest decrease hand ib average noise build mechanism beneficial next diversity instance weighting scheme compare use achieve compare method weighting outperform scheme case biased weighting weight weighting technique forest count average bold ccccc ccccc dot count count count count f filter validate filter ensemble filter choose algorithm achieve filter yet filter classification bold value table average set bold significantly accuracy gray cell l ccccc noise ensemble compare compose ensemble compose mlp three section high contain low nine forest high correlation significantly beneficial compare ensemble weight filter base filter ensemble bias filter handle significantly high surprising handling decrease see noise filter ensemble class class noise approach consider ccccc ccccc count count count completeness compare representative gain accuracy single despite accuracy higher consider surprising perform focus mlp high accuracy consider vote significant within mlp weight filter handle classification safe compare mlp base weight mlp mlp level noise mlp achieve represent weight important weighting investigation topic examine diverse examine voting find outperform less diverse handle knowing filtering achieve significantly low classification accuracy able outperform consider handle bad filtering statistically technique despite handle voting ensemble exhibit possess base classifier gray widely induce bias less effective broad bias diverse prediction instance instance noise handle technique algorithm across keyword filter weighting learn voting machine learn accurate generalizing instance world generally attribute attribute noise consequence summarize fr frequency know case task instance relate work examine handling approach theoretic remove misclassifie towards especially artificial efficacy handle technique give upon datum handling technique artificial ensemble base classifier ensemble class accurate diverse hypothesis however none explicitly focus classifier inspire principle ensemble select diverse algorithm prediction dependence hypothesis could two classify way set algorithms base classifier vote set filter technique weight voting base explicitly diversity account find diversity significantly improve filter demonstrating noise set vote diverse classifier achieve high classification handling technique organize section label noise diverse inherently examine class find class generally thorough handle fr learn algorithm design error prevent pruning completely change handle noisy instance problematic boost instance place single svms hinge misclassifie instance instance misclassifie wrong modify possibility maximization remove noisy weighting instance criteria filtering receive generally result especially artificial broad handling always classification noise add et al also filter examine predict efficacy near remove misclassified learn van idea bag theoretic learn noisy instance remove instance wrong low label filter potential discard significantly filtering consider special instance assign instances pair influence discard filter clean automatic datum correctly label train decision include correct value instance increase instead single exception set diverse learn algorithm explicitly handle label bias examine diversity effort diversity measure accuracy ensemble study diversity classified misclassifie distinguished class effect ensemble boost bagging create selection knowledge diversity presence voting handling approach instance heuristic bias classify determine noisy estimate e specific multiplying infeasible though sum hypothesis non trivial attractive discriminant classification discriminative lower examine diversity refer meta classifier distance make hierarchical agglomerative dendrogram connect representative conjunction color package terminal classifier output training mlp backpropagation score node reach leaf node near label forest leaf return cover probability however induce class misclassifie filter find generally produce biased misclassifie algorithm c rand forest ib count examine backpropagation forest tree rather count sum meet compare bias seven weight biased technique list brief cited repeat near remove misclassified near neighbor base instance least correct instance removal filter training continue long remove filter remove fold default remove instance misclassifie base distinguished filter base classifier choose filter three ib misclassifie majority remove misclassifie training correct correct filter validate partition equal time subset instance misclassifie iterative filter partition induce misclassifie induce filter original tree em belong class criterion clustering form default finish since filter removes finish large set set finish table compare count represent time accuracy bold achieve contrast represent accuracy significantly many handle measure previous artificial improvement bias nature handle broad handling datum noise briefly application handling set compare technique add handling significantly investigate important example highlight handle beneficial handling accuracy gray significantly l ccccc mlp rip count count filter count conjunction terminal option explanation either package package graphic terminal graphic macro ltb lt lt lt lt lt lt ltb lt lt lt lt bp r r ib conjunction see explanation use package graphic terminal need graphic macro ltb lt lt lt lt lt lt ltb lt lt
generally kronecker nmf kronecker basis nz equivalently change rest proposition stem enhance ghz run window code observation additive independent gaussian nonnegative multilinear rank sparse tensor entry tensor meet specify noiseless version world observation signal interference recover normalize zero solver version nmf subproblem subproblem run directly procedure update fit involve computation probably ill issue algorithm although seem speed consistently level base average algorithm robust without sensitive iteration quite helpful noise consequently improve robustness investigation essential uniqueness tensor datum core tensor generally db component factor failed also guess cause convergence simulation factor substantially improve essential local minima largely th object experiment pose simplicity category randomly decompose r denote k time local superiority technique pca extract randomly list table outperform part moreover core generally impose whole allow adopt computation propose analyze iv iteration fit algorithm acceleration c mi fit th c c face recognition face database gray randomly decompose training unfold feature matrix knn classifier distance measure run factorization relatively also discriminate analysis tensor considerably problem seen accelerate marginal version affect recognition database basically accuracy computational load guess unique originally physical basis know give show c face without need impose sparsity nonnegative tucker powerful multi nonnegative give localize representation multilinear order free learn procedure reduce subsequent substantially flexibility indeed well establish contaminate various discuss uniqueness nmf uniqueness decomposition justify propose promise nonnegative tucker tool nonnegative latent high often high major low multilinear tensor significantly cost function besides dramatically reduce run quite flexible well establish incorporate substantially improve uniqueness curse dimensionality tucker world justify validity propose tucker decomposition alternate rich behind task signal field component highly gain decade rapid development observation data component specific temporal smoothness properly exploit base successfully color often cause perspective model widely apply deal high widely tucker decomposition pattern cluster denoise etc great success nonnegative graph datum preserve regard nonnegative prove powerful analyze nmf interpretable nmf extensive area code matrix respectively element integer large set tensor element division tensor define kronecker rao product column wise kronecker zero uniform nonnegative tucker decomposition gain recent advantage structure illustrate give part representation face represent possess multilinear tucker decomposition lack uniqueness curse indicate fact core exponentially unconstraine tucker provide meaningful core sparse discover significant curse exist perform update rule exploit special multilinear tucker suffer term especially large quite develop yield take tucker decomposition paper unconstraine tucker tensor access big thereby considerably proceed overview present together investigation important section iii review flexible simulation section vi high propose notation mode fixing index matlab unfold column th I follow concern mode frequently mode due property mode product tucker core connection column rank factor nonnegative factor bring key purely additive often may effect factorization ability localize th core broad application processing understand th matrix tensor core tensor equivalently column sample combination basis vector extract feature regard eq tensor involve multiplication base counter multiplication version ir nr mu update choice term descent multiplicative multiplicative see rule manner update update improve total idea nmf practice unnecessary subproblem execute converge tensor tr multipli method th respectively mu core tensor analysis lipschitz provide initialize converge nmf solve roughly speak inverse fast stability sometimes component necessarily nonnegative nmf factor algorithm q similar generally guarantee many accuracy approximate tensor respectively exact practice entry hence intrinsic often incomplete weight miss decomposition miss straightforwardly prefer step weight tucker decomposition complete tucker approach allow entry dimensional deal miss randomly approximation datum partial subtle difference whereas select satisfactory framework scale govern tucker decomposition curse dimensionality lack uniqueness former increase latter due tucker limitation tucker although far analysis uniqueness still miss uniqueness nonnegative multilinear multilinear multilinear nonnegative multilinear uniqueness uniqueness nmf nonnegative multilinear rank simply r ns trivial nonnegative obvious nmf exist trivial n contradict essentially
recurrent input current vocabulary element follow rnn length recurrent connect backpropagation propagate multimodal neural rnn deeply simple rnn word layer word embed recurrent multimodal softmax input firstly network dense word dimension vector secondly dense encode word find calculate vector sentence use vector initialization initialization sufficient treat activation embed input embed layer dimension calculation recurrent denote activation vector unit relu training deep vision differ rnn sigmoid relu hard sigmoid backpropagation conduct rnn temporal heuristic stop step hyperparameter good property relu stop early multimodal connect model b layer recurrent image part extraction connect multimodal please add together multimodal multimodal scale force non process rnn model softmax generate layer vocabulary rnn adopt sentence set length denote generate softmax context calculate equivalent set backpropagation learn part embed layer recurrent e mention layer part convolutional feature widely previous multimodal use improve gradient multimodal layer image deep train rnn sentence retrieval sentence retrieval image generation straightforward start word word calculate next next selecting perform pick sign generate image treat measurement top sentence might probability sentence frequently appear look probability generate sentence query normalize ignore sentence annotation consist around location action annotation sentence annotation one adopt available separation work image extract five sentence annotation annotation adopt provide image annotation describe image annotation image sentence retrieval task automatic translate give sentence treat sentence translation sentence reference description correctly sentence reference sentence reference sentence stop generation sign adopt sentence retrieval retrieval measurement recall retrieve sentence task retrieval rank retrieve result well tc exactly evaluation metric plot match retrieve sentence retrieve sentence sentence description often find subtle sentence publish score htb b f base rnn generation table serve model architecture rnn conduct fair include context necessarily correlate section sentence word content might although rnn fail rnn perform well term outperform comparable publication section curve percentage retrieve sentence sentence image word third multimodal rnn outperform publicly report retrieve rank retrieved sentence htb cccc rnn state devise feature method sophisticated avg method cnn detection confidence strategy object performance even well htb cccc cccc sentence text r random avg devise avg publicly rnn base rnn well metric table show htb cccc r devise avg rnn rnn network retrieval query retrieval query rnn sophisticated explain multimodal university california com multimodal generate description content description sub sentence deep convolutional interact multimodal whole rnn validate benchmark dataset tc rnn retrieval art objective retrieval description become education retrieval blind thank rapid computer brief review treat retrieval sentence address query sentence annotation exist lack ability contain unseen propose multimodal recurrent network task novel sentence sentence retrieval formulate learning
checked still check computational pre gap determine simulation effort considerable simulation consider implement broken batch equal algorithm bm w batch batch condition require geometrically common choice application bm usual batch storage entire store clearly memory soon serious check updating prefer simplify end suggest plan provide strongly consistent bm specifically could nn bm record merge storage batch already illustrate plan interested direct technique checked examine criterion pre specify user previously batch accordance simulation effort regard perform modification implementation deviation yield measure time procedure new plan applicable storing entire iteration ess sample way define ess implement package alternative approach ess strongly ess calculation produce study relatively systematically correlate ess one termination combine equivalent ess specify set ess practitioner particularly diagnostic gd gd gd rule statistic spectral frequency markov chain gd diagnostic gd markov apply deviation bayesian application weather dataset implemented terminate comparative study several advantage consider temperature collect weather start r package illustrative record nearby weather model limit continuous temperature specification uncorrelated measurement transition temporal cs spatial specification scheme package use interested reader particularly interested ts upon simulation conduct relative standard plan set batch batch number rule coverage gd check iteration effort confirm cccc ess cpu gd table statistic criterion equivalent sampler available ess cpu effort term variance illustrate tradeoff estimate bias work relative standard high setting default plan total gd compare gd ratio ratio quality computational posterior deviation comparison ratio significantly concentrated one gd agree gd application imaging fmri study change activation bold contrast course single patient stimulus brain acquire activate region intensity imagine patient brain divide voxel lattice time bold voxel spatio structure analysis fmri interest introduce activity relate level eight four four attention subject picture four second second later stimulus star sign plus sign present four second subject picture period second sentence take second standard standardize snapshot voxel voxel bold intensity alternative conventional feasibility baseline stimulus activation amplitude transform external stimulus proceed brain stimulus preprocesse response characterize formulation turn consist gamma amplitude convolution measurement appropriate distributional temporal spatial article single summarize ti tv tp formulate notice voxel activation identification nonzero end indicator rewrite selection averaging square voxel effective inferential specify mass prior incorporate spatial mrf interaction voxel regressor two voxel article six immediate neighbor reciprocal euclidean voxel external incorporate reflect knowledge eq address place I density two wise design update baseline activation amplitude symbol selection problem figure visualize linear previously deviation memory issue add batch nominal coverage symbol jt summarize interaction small eight slice task symbol voxel activate compare gd confirm chain voxel sample stay inactive although comparison termination compare two ratio estimate deviation clear ratio gd require gd expensive adjust reach truly high automate fashion tuning parameter attain confidence practitioner routine modify relative control approximately deviation ess ess desire affect simulation result balance depend estimation interest single sampling summarize simulation requirement storing solve issue limit estimation quantile arise dimensionality multivariate adjust interval nominal require adjust one volume width multiple interval separately explore state mcmc dimensional setting date proportion type stay fix reader acknowledgement author paper grateful recognize winner competition second work dms remark pt university challenge terminate chain automated stopping terminate uncertainty moderate illustrate high simulation current bayesian spatially correlate involve weather fmri image sequential uncertainty well dimensional simulation fundamental determining terminate simulation often encounter inspection mean extremely challenging moderate practitioner resort terminate terminate stop applicable mcmc collect scientific temporal association dependency involves mcmc considerable time study modeling spatio temporal fmri develop selection brain economic public analysis assessment inferential uncertainty involve thousand carefully describe game report lee employ fixed choosing lead practitioner utilize visual truly randomly width rule implement theoretically justified terminate accurate scientific unknown markov high dimensionality associate create additional extract marginal width report terminate fall
simplify adaptive deconvolution seem treat deconvolution counting intensity bayesian method prior intensity aim show satisfactory first frequentist asymptotic behavior posterior computation obtain still theorem process adapt count multiplicative intensity abuse sequel surely every integer depend asymptotic follow illustrate poisson intensity analysis censor popular patient patient represent patient censor hazard iy finite markov path integrable transition intensity state independent intensity counting process respectively aggregate reader p aggregation intensity true assumption satisfy illustrative introduce right censoring model support concentration inequality imply process empty surely consider endowed f f f ft concentrate monotone intensity consider parameterization increase density mixture uniform prior increase intensity dependent study distribution belong families lebesgue eq cumulative go denote lebesgue measure transform neighbourhood define define q consist general posterior interest positive sequence integer x stand sup mass lebesgue satisfied assumption dealing consider process quite neighborhood sup neighborhood hellinger propose derive intensity nh nj replace nh right hand example poisson censor n n kn old mild derive finite empirical prior distribution let observe jump recall intensity function done provide parametric estimation involve satisfy advantage hence involve mass dirichlet strongly influence propose additional information correspond weakly apply three determination dt simulation true consider use appendix dataset correspond dataset intensity plot intensity second column strategy compare strategy criterion quality computational empirical fix comparable simpler hierarchical moreover thus even hyperparameter suppose influential fix turn conditionally variable accept observe ar dramatically automatically phenomenon randomly number approach correspond concentrated around credible posterior simulation intensity prior strategy lead generating phenomenon execution empty last case prior narrow prior lead phenomenon large impact empirical hierarchical cc cc fix block ccc empirical prior hierarchical empirical plain dot hierarchical line concentrate hierarchical line true estimation dot third strategy line plain line line confidence dotted throughout may line bayes dirichlet mixture ordinary measure nu mn u n l sn sn l j k accord inequality repeatedly sequel abuse l verify assumption f proof partition interval diameter complete k u n c l jx proposition f l verify verify difficulty n k jj u nu j proportional super case ordinary since rate suitable need eq f f k nu cover rd know lemma inspection reveal b piece c u u j nb np c reasoning proof u l p jx proof need indeed na exponentially bayes contraction wasserstein metric remove particular every exist path prove appeal property deconvolution mix approximate mix mixed density use os ss short super characterize regular kernel fourier vanish p k recall super case take constant super x x intensity still follow subsequently kullback leibler evaluate second q exist loss generality dirichlet uniform distribution index dirichlet mixture distribution base index base type e inside therefore decrease increase second control control poisson intensity enough n nh use combine nh assumption compact enough nh together soon homogeneous proposition constant proposition similar notation note markov inequality imply use test prove sequel empty almost use nh tt tt x remain enough n enough deal denote may change k n tt tm k ce cv k q exist n iterate set k j j b kb j tt j tm cm j use j tm n radius center lemma constant q conversely ns q choose result tool convenient function recall non pseudo c since implie non negative n tt u nd u positive enough eq u c end lemma argument assume ty n ty ty ty n ty assumption imply since soon v tt v q inequality true enough u nd lemma depend ty tt nd depend use since nd depend end lemma detail posterior process decompose jump jump artificial truncation slice sampler breaking introduce w I term rigorously dependent take account detail initialize initialize c w sample identity iw u nt k slice sampler slice follow prior stick breaking k know component component exceed I component parameter class follow p independent h class appearance translate perform directly resort simulate accept detailed consequence easily could theoretical number x value p get cm section posterior prior provide seminal estimation intensity poisson former also deconvolution latter provide posterior concentration counting prior depend posterior dirichlet process mixture process title approach principle independently task prior hyperparameter dependent fall bayes statistical sample family integrate specification say throughout drive hyperparameter sometimes explicitly case regard nonparametric case mixture variable another empirical bayes mixture procedure adaptive empirical literature method claim fully way frequentist asymptotic bayes give little know general bayes asymptotically close framework noise wavelet selection investigate asymptotic posterior condition frequentist strong merging maximum empirical take frequentist behavior contraction rate pseudo empirical concentrate converge probability probability fully far consistency seminal article identically observation iid observation elegant powerful methodology rate kullback type however take prior develop drive contraction distribution spirit apply mixture application dirichlet mixture section estimating process mixture tool posterior distribution dirichlet study minimax collection exist due deconvolution error model base drive choose deconvolution process density model finance social huge practical establish behavior intensity process extend result intensity monotone b e e b j e nu nu control iid lebesgue prior class density denote typical tail n mixture investigate prove mixture lead contraction h super key ordinary gaussian eq article treat rhs extension regularity consider instance satisfy old case follow hellinger tail rhs empirical bayes satisfy k
daily stress include size result stress distinction non data split testing test accelerate dimension cross strategy hence trait train subset evaluate significantly vs rest vs visualize understand space find mutual uncorrelated room exclude produce make interpretation complex turn feature decrease outperform chi average approximately range perfect maximal inequality maximum maximum bag common metric accuracy reduce feature pool make efficient mobile classifier algorithm support radial basis iv neural network find classifier rest forest tree predictor tree forest form accord margin h h training empirical vote vote error characteristic forest tree number tree generalization validation train number select validation pairwise among correction chance agreement chance simple percent agreement take account agreement occur conservative selection replacement fold fold cross scheme order prevent overfitte call software empirical risk solution overfitte work penalty balance distribution probable model estimation iteratively basis exploratory analysis metric dataset test metric sensitivity random forest show model performance test l ci acc specificity metric validation provide substantially heterogeneous fold resample ht min st mean rd indicator weather iii mobile phone trait weather condition combination trait mobile phone vi weather infer mobile phone report specificity accuracy run class label neutral label score include score size neutral outperform return accuracy specificity weather weather weather table weather activity alone endow pairwise combination feature set neither weather weather activity simultaneous usage classifier feature mobile accuracy period non period outperform recently report combination mobile however make video audio stress stress environment pose concern people reliable investigation predictor stress association trait contribute predict daily stress focused analysis mainly association important played also experience daily stress create stochastic daily stress recognition use report limitation user study trait recognize mobile phone regard weather wind predict regard mobile phone proximity select proximity interesting play interval seem finding relevant face tie stress sure investigation capturing confirm available association interaction conclusion message income stress increase pressure stress people severe stress may detect consider demand increase severe stress stress financial health year american low technology reliable tool recognition life trait call weather reliable stress essential scope area applicability people capable rich life transform stress target assessment treatment stress record patient stress access longitudinal identify recurrent significant environment stress become serious health employ early stress stress motivate people behaviour inform toward strategy mobile develop increase stress suggest stress management relaxation technique also student availability concern people collection fact study variability background subject homogeneity investigate people daily stress people activity individual trait individual suggest type source drop performance moreover robustness generalization together provide daily stress reliably necessity activity environment stable mobile usage pattern advantage privacy moreover stress mobile phone device apply several real world situation application device clinical application daily situation stress management semantic innovation grant stress life cause disease researcher stress require sensor continuously propose approach daily stress reliably recognize behavioral derive mobile phone weather trait concern individual person obtain class daily highly power recognition methodology behavioral sciences mobile million mobile allow access huge stream daily device behavioral location device physical communication phone call message correspondingly availability continuously grow huge stream interaction problem finance work problem stress know life cumulative stress play broad physical cognitive disease measure daily life availability early help prevent life feasibility sensor sensor heart rate setting exhibit person argue research opinion tool field behavioral science make sophisticated world work daily detect stress well activity include execute phone people associate level weather environmental argue act sensitivity finally impact factor activitie weather stress proximity interaction person turn might automatic recognition daily stress vs type activity weather people activitie proximity diversity proximity behavior weather environmental along trait internal stable people daily compare weather condition mobile phone simple family weather condition mobile phone weather mobile phone find obtain evidence reliably individual trait drop drastically factorial stress seven comprehensive approach feature weather mobile phone weather phone feature weather mobile phone body stress detection focus measurement infer stress see heart variability provide reliable stress level comprise sensor carry continuous monitoring situation speech production study acoustic research analysis speech variation example stress detection life environment sound quality public place reliable stress study analysis despite provide monitoring employ survey daily stress stress participant seven item neutral subject consecutive daily stress fig high stress stress score neutral variance high rl decrease accuracy decrease index weather weather weather weather call social trait stress negative examine name trait researcher daily daily daily people event tend trait student show less affected stress five trait measure subject answer version question mean scale trait raw trait relationship health weather weather show weather et al six weather wind power air pressure affect negative affect reveal affect effect air pressure weather temperature pressure iii vi wind metric source weather alpha weather daily source weather previous characterize mobile phone predict people behavior trait table proximity feature max deviation chebyshev basic day backward move window possibility past event current subsection describe detail feature fall phone usage behavior
start vote ensemble determine simple majority agree upon majority choose say result chart figure effectiveness frequently retrieval priori extract combine consist scientific different science word tend run slowly raw nmf singular svd principal component analysis normalization two decomposition quite reduce create dataset mean proportion document h input clustering ranges misclassifie misclassifie reasonable ask solution low curse distance metric dimensional clustering surprisingly internal like coefficient one must careful high clustering metric clustering agree without dimension consensus matrix simplicity without consensus cluster well suit representation provide consensus iteration common choose consensus single consensus much great average table typical across algorithm consensus consensus consensus cluster methodology consensus similarity chain consensus sum accord section row consensus mean permutation diagonal block diagonal block diagonal degree nearly dependent deviation partition represent coupling without merely perturbation dominant stochastic block dominant continuous block cluster examine consecutive gap indicate nearly structure consensus cluster suggest might consensus clustering matrix count number sometimes helpful consensus traditional plot fail picture may discuss look similarity collection display eigenvalue information look consensus consensus build ensemble pair prefer dimension dimension create method initialize initialize dimension reduction cluster cluster determine clustering although original available ng collection result consensus form mean dimension reduction convention observe plot singular ng datum indicate reduce preferred reduction principal mean proceed fast clustering step associated initial figure might cluster distinguish iterate consensus use consensus input figure either scenario cluster eigenvalue clear repeat change present document reveal single h superiority certain consensus compare accuracy algorithm dramatically interesting contain consensus matrix nature poor author come ensemble mean c c consensus nmf cosine consensus consensus consensus average cluster iterate consensus number agreement cluster column table run round voting accuracy table consensus nmf analysis collection tool analyst tool chance solution pca particular internal validation metric mean solution difficult compare tool consensus framework work difference accuracy achieve algorithm herein present combine multiple input exploratory cluster discover matrix build value often appropriate determining cluster algorithm agree example succeed cluster dataset refined iteration picture might agree consensus practice multiple together explore varying purpose approximate improve cluster consider consensus discuss individual consensus aim solution another emphasize agreement ensemble solution cluster equally relationship draw whether favor iteratively encourage agree upon similarity reflect agreement iteration cluster agree favor agreement definition framework consensus present similarity high datum refine algorithm agree nearly determine random determine succeed number cluster method consensus provide average hundred text mining biological datum without hundred thousand guarantee analysis aid tend separation cluster tool make informed tool dimensional hoc science hope quick develop individual task become many mining cluster stem vast number information fact question analyst answer determine consider separate answer group point agree cluster essence suggest herein problem determine determine final solution algorithm consensus form voting ensemble proceeding majority user year consensus researcher challenge ensemble generally wide variety result produce ensemble consensus minimize distance metric clustering define clustering clustering metric information maximize median partition heuristic propose believe bind importance clustering inaccurate share cluster away optimization solution act move consensus introduce impose reach accept object introduce notation since algorithm ensemble n jk data individual simple clustering ensemble record prefer think ensemble consensus cluster ensemble consensus clustering value reasonable colored circle clustering result consensus isolated expect truly give cluster break total clear far break break way block diagonal structure offer traditional cosine traditional curse space mean address entry consensus compare cosine cosine similarity range document among benefit consensus depth adjacency output available something know object cluster previously
quantify se rs ensure comparison tie differently function loss tie outcome summary difference rs instance rs capture variability value average value rs provide comparison rs comparison final extract interval gain generalise additive interval specification function default thin regression relate expect fit seem roughly level dot rs outperform primary quantify experiment al early thus single experiment overall address complexity assessment provide find e generalise pointwise confidence include fit output al analysis generalise reasonably q dispersion h name negative binomial thing method mis match conjecture work classifier mis match quality pool high mis classifier mis metric sub mis match informally mi ill suited match improve mis match great quality mis match bad fourth task complex decision match bad fourth conjecture al budget word increase label gain select pool range would practical early al belief mean length notable significant confirm exist analyse detailed disagreement se algorithmic compare result disagreement disagreement vote l classifier logistic machine setup way binomial fit reasonably se confirm al mixed confirm hard may behaviour se stage classifier se result significantly well svm good classifier help examine variety range factor conclusion overall fail often consistent largely confirm belief literature se show e enable recommendation relate determine al study complexity assessment methodology assess award learning improve two central al broad quantify need al detailed complexity assess performance experimental learning field need motivated concern label consider question help question answer enable researcher tackle difficult al benchmark method variability difficult example notably show negative valuable overview al suggest still thing understand broad include classification simulation systematically careful raise contribute assessment issue application assessed structure present classification feature denote label consist classifier unseen objective label image diagnosis valuable label image obtain systematic guide al set typical would pool budget request al receive oracle se informally se metric decision al label provide newly label initially train improve classifier variation effect study factor analyse nature classification difficulty boundary express decision smoothness since affect pool intuitively task hard pool decrease label determine performance open experimental evaluate combination performance answer question attention create mixture gaussian curve involve way varied b modify cluster smoothness varied transform another input label high require classify quadratic forest rf vector machine label pool figure trajectory al set label happen repeatedly create trajectory pool overall trajectory denote point increase amount label several trajectory score interval score process budget score outperform rs whole budget begin benchmark score terminate score detailed section al instance benchmark rs rs substantial trajectory figure score se comparison shannon vs difficulty contribute novel assessment complexity quantify al gain scalar summary address preliminary issue benchmark al outperform decide benchmark option large improve benchmark al rs variability evaluate rs instance interval thus benchmark trajectory context understand al benchmark budget budget iterate entire pool amount label grow minimum
proximal prox inf eq prox operator would restrict function solution indeed proximity restriction g require satisfy summary toolbox ball lambda parameter reason ball compatibility solve image goal would closest firstly know position miss pixel result assumption secondly image patch color variance simulate operation want recover mask relaxed rewrite measurement patch q mask play fidelity signal tradeoff fidelity measurement vice play note trivial affect convergence proximity toolbox object compute non provide operator prox structure function role operator tv norm implement tune setting select log summary display step indicator proximity infinite previous prox x eps projection suppose indicator implementation note lead evolution since select different solver observe present lead compute solver backward gradient consequence matlab gamma select define iteration define display reconstruct done follow prox prox lambda lambda lambda gradient well define domain allow solver forward backward solver solver converge might problem parameter regularization ball choose allow compute I original sigma I noisy I sigma prox tv original prox x eps lambda f prox prox lambda lambda gamma lambda backward gamma lambda ii close toolbox proximal splitting use specific general try stay mathematical precisely matlab toolbox design problem split compose solver operator file user implement toolbox convex lower assume nx problem smooth method generally forward algorithm fall term generalization operator unique well proximity useful tool directly try find sequence proximity survey excellent splitting relate toolbox core toolbox proximity quick common toolbox largely inspire toolbox develop file backward continuously lipschitz without use default assumption constant exist q stepsize also allow smoothness step proximal backward algorithm f start two matlab structure function take latter matlab vector matlab act stepsize control convergence default classical augment lagrangian technique
sparse noisy real success utility quality source rest overview detail framework year formalism transform machine area node edge characterize transformation heterogeneous node type detection within fall interpretation weighting algorithm set type multi work leverage share information across source overlap iteratively select edge approach aggregation ground know subset require ground validate section list intuitive need similarity computable rough overview collection learner whose performance well seminal form learner think weak present boost make arbitrarily good allow pure learner equally good graph graph representation quality application standard bandit receive explore minimize many suggest take adversary know everything make advance expert similarity learn potentially bad artificial reward care end seek care final knowledge immediately largely heuristic even bandit optimize guarantee useful nevertheless successful bandit hope graph technique bandit list maintain element distribution reward receive multiply control representation edge update next section sketch implement unweighted undirected expert vertex combine produce global give round four part produce aggregate measure quality quality edge round end round maintain non edge weight graph run cluster produce observe effectiveness application update weight define two use present use first call intuitive notion superior across tie idea consistent simply evaluate algorithm algorithmic agnostic measure cluster idea many neighbor quality cardinality neighborhood neighborhood vertex conventional mechanism neighborhood metric due stronger adjacent brevity index experimental use consistent utility edge plug guide key every inefficient give improve fix weight extreme pick total runtime time balance parallelization addition alternative edge roughly round relational stay inside boundary consider suggest reject suggest discuss respect edge union initialize copy p ip u vi mu h iw pp ip algorithm synthetic structure list vertex block occur block simulate global stochastic block draw block model within cluster block represents represent refer representation graph community well naturally extend formulation community er enyi example note er instance noise combination er primary comprehensive database research extract subset publish science topic field author graph title least three word common exclude paper summary dataset experimental couple community structure evaluate notion reflect quality source quality assignment capture quality representation representation community structure disjoint clique clique modularity capture modularity compare nan produce also remove cross perfectly modular trivial display graph fraction total quality capture weighting source community weight contribute equally weight quality metric graph converge round far exceed modularity tell able discard graph round type weighting requirement total edge enyi appropriately preserve amenable select title recover community share topic sense title well proxy capture division modularity induce snr appropriately usefulness outperform overlap ground clustering produce modular enyi converge weight r ec modularity sparsity weight modularity na na na graph aggregation graph respect demonstrate application community detection
signature large additional implied exist place sum number bound idea contribution weight bound hand formally ta line cauchy schwarz g normalization assumption signature assumption size definition let edge k apply set definition signature except negative completeness version great set negative correlate either choice signature expand similar definition expand signature signature expand expand signature set straightforwardly general signature note expand lemma adapt bias expand large sample analog hold formal statement step purely magnitude edge help contribute two sign argument use overlap magnitude recover signature almost iff deal magnitude recover case state formally expand output use refinement conclude number furth closeness order want analog large constant equal absolute nonzero equivalent I triangle omit prove bernstein standard deviation iff probability tx j tx would variation otherwise sum variance probability te j te te weight know focus jx j signature signature tc secondly property signature expand set expand signature analog claim w similarly proof use claim bernstein know expand signature remove element contradict indeed expand signature fs thm thm thm claim exercise pt coding sample unknown overcomplete heuristic provable hard find provable dictionary et al algorithm incoherent et handle guarantee design provable algorithm work matrix individually notion motivate limited enumeration try combination unknown overcomplete dictionary fit used machine feature recently code influence processing dictionary super resolution provable program nonconvex unknown know general hard combination decode regression compress incoherent matrix even sparsity satisfy isometry matrix recover hard heuristic algorithm widely design direction mod reference however provably recently al give practice overcomplete dictionary prefer provable overcomplete dictionary polynomial incoherent weaker incoherent fundamental limitation intersect dictionary regime seem deep regime dictionary matrix weight bipartite allow albeit slight real life dictionary probably raise dictionary current time enumeration similarly learn discuss dictionary nonnegative partly analogy algorithm traditional discuss exposition purpose refer coordinate though apply vision definition effect add could object one specific version restriction dictionary make nontrivial formalize property life instance reasonable life dictionary feature involve imply dense match speak observer know produce assumption one presence pixel affect interest et al incoherent pairwise product sense count fairly secondly one incoherent one rip matrix dictionary check weight think constant large practical purpose unclear ng neighborhood intersection pairwise neighborhood matrix dictionary random entry close comment check need claim return dictionary equivalent true large constant assumption optimal sense significantly violate intersect feature characterization nice long valid expectation equal normalization magnitude weight lot also variance intersection neighborhood two feature entry large contain entry constant simplify notation g say nonnegative pixel small mn recall dictionary seem try extract assignment assume nice property recover become easy recover via overlap incoherent intersect fail slightly intersection among tend mean value simultaneous unknown therefore look subset pixel aggregate pixel consistently predict value signature identify signature good signature size similar correlation signature separate signature set correlate size idea try expand column pick column result expand expand
copulas student direction initial log increase new evaluate correlation algorithm terminate yield reasonably good choice present maximum likelihood elliptical copula task fast consume advantage suitable copula elliptical copula copula tool dependence allow model flexible describe marginal family copula strength use decade mathematic distribution unique theorem encode dependence structure vector margin distribution u copula refer example elliptical copula copula multivariate elliptical elliptical elliptical copula dispersion depend location copula elliptical especially copulas normal student gaussian student degree freedom univariate student copula stress copulas correlation general margin elliptical copula correlation description reader close elliptical square practically problem elliptical copula student conjunction efficient student copula focus elliptical copula space symmetric estimator correlation elliptical copula copula exist method student copula appendix elliptical copula efficiently student essence less project kronecker delta maximization student true maximizer maximizer ascent would seem move hereafter fact directional directional trace diagonal positive define symmetric iteration project log method reference small although perform numerical also outperform simple size scheme describe estimation algorithm exact widely software package calculate solution inverse hereafter exact estimation package copula slightly execution estimation fix gradient execution time case dimension converge second always second matlab hour every copula generate test description generate sample copula estimate parameter original value obtain gradient positive h dimension fail converge provide plot deviation generate eigenvalue imply method especially correlation extend copula estimation procedure extensively fast technique latter moderate numerical student
use iteration arbitrary substitution variation would gradient whereas variation variation stochastic bfgs introduce subsequent instantaneous implie show variable iterate optimal argument proving result instantaneous twice instantaneous hessian exist q eigenvalue hold follow linearity observation assumption typical inner proposition gradient variation recall small eigenvalue instantaneous constant eigenvalue hessian instantaneous hessian fundamental instantaneous definition variation eq e inner product bind write curvature compute solution proceed recursively matrix positive upper descent general f whose average specify assumption eigenvalue state taylor upper bind hessian substitution side observe third term side norm eigenvalue upper yield upper use lead low second positive semidefinite eigenvalue small far note aside term sake define relationship imply sequence almost surely per infimum norm surely observation probability relatively minor nuisance care assumption true infimum distance optimality realization statement stochastic substituting yield since nonnegative satisfy converge almost vanish infimum sequence nan square low eigenvalue apply expansion optimal simplify cauchy schwarz substitution simplification infimum nan consider establishes convergence low variation gradient section instrumental proper effect curvature curvature estimate eigenvalue term dominate ensure progress towards argument complement characterization introduce define sequence step give parameter sufficiently parameter objective objective satisfie expect value imply convergence sense sgd convergence improvement mark experiment sgd problem small particular definite matrix instantaneous minima surely minimum condition control variability instantaneous large note optimum comparison iterate study represent optimality stochastic process number require certain text instance discrete run gradient realization distance much iterate iteration since correspond evaluation conversely upon function modify j realization sgd order text realization convergence time text advantage fig keep yield family well function ill condition condition sgd respectively cf function fig ill condition condition family average average family sgd spread small go moderate start moderate value decrease text stochastic compute section problem instance cf interpret distribution trend apparent decrease ii decrease go start go moderate deviation monotonically decrease stay increase thereby yield payoff pay evaluation good half turn actual time moderate suffice provide curvature interval moderate number comparative dimension method define value determine interpret failure sgd respectively fail rare realization realization spread eventually well median well fail exceed fail furth smoothly evaluation need stable implementation hyperplane separate set contain feature vector find hyperplane support separate point separate vector deal introduction measure hyperplane support constant minimization sum hyperplane measure norm desirable common square g give uniform upon rewrite substitute general explicitly attempt select instantaneous take loss training half belong class likewise uniformly random overlap range know less parameter progress gradient large sgd stepsize use performance process sample processing feature objective sgd process vector conversely method dimension process fig acceptable reduce progress difference time translate show build vector record percentage correctly classify repeat histogram sgd uniform test exceed classification far performance comparison suggest investigate difference regularize version bfgs vector bfgs regularization induce oppose proximity requirement stepsize value vector process value reach regularize stochastic bfgs jump permit consistently occurrence always recover regularization amount curvature occurrence stochastic behave sgd stochastic objective introduce corresponding argument sure behave expectation far prove show particular significance dimensionality exhibit target develop advantage improvement cardinality spread true lagrangian duality lagrangian eq combine lagrangian optimal minimizer must eq multiply rearrange inverse log multiply rearrange consider trace cyclic argument latter scalar observe log determinant opposite determinant inverse substitute determinant rearrange term expression order dual eq gradient plug lagrangian compute hand verify direct must argue directly term observe hypothesis must write conclude belong semidefinite sequence rate constant satisfy sequence bound time prove rearrange compare inequality true substitute substitution simplify equivalent conclude substitution conclusion combine lemma show stepsize satisfy optimality gap side gap recall standard completeness g assumption eigenvalue hessian take taylor expansion bind side argument zero minimize imply norm substitute side double f conclude form hypothesis rewrite identify substitute remark mm mm edu appear bfgs newton method problem objective require argument prohibitive dimensional order objective hessian incur computational descent utilize deterministic gradient determination direction gradient rate advantage counterpart upper argument bfgs develop use function value determine optimization determination instantaneous function average problem common resource wireless convex conventional determination f intractable unbiased sgd limit dimension easy newton achieve rely gradient estimate unbiased generalization devise objective quasi retain advantage deterministic counterpart quasi newton near see bfgs quasi method structure regularization avoid function brief sgd bfgs continuously curvature previous curvature bfgs retain modify stay bfgs bfgs make large completely specify resolve bfgs require close possible constraint matrix
different prox sdca starting find separable time indeed enough say influence tight describe use arbitrary update paper sampling unconstraine minimization strongly assumption special method besides close coordinate analyze primal dual step sdca sdca analyze establish gap exception novel primal specific obtain variant pick similar prox always pick complexity n match term choose nice mini variant sdca specialized svms loss besides accelerate mini batch method zhang detailed comparison well dual however propose coordinate dual sampling coordinate accelerate efficiently implement distribute environment variable describe non serial first analyze datum partition speedup serial sampling paper however reader uniform find variant exist mini batch stochastic dual certain mini batch drive use inform speedup datum illustrate consider datum appear factor excellent predictor behavior special uniform sag gd gd serial primal mini ms gd analyze specialized dual coordinate serial generality far arbitrary uniform accelerate arbitrary purely accelerate variant simultaneously primal algorithm interpretation outline update sdca compute several select proceed result deal specialize nice relate stochastic speedup main loss describe random value chance proper assumption positive always stepsize shall formalize notion smoothness strong constant brevity subgradient dual maintain maintain proper vector n p iw let describe convex subsequently way numerous variant allow option actual update entire process repeat interpretation fix iw decompose w pair optimality current primal prox sdca adjust choose option example special serial sampling dual prox prox prox sdca primal prox perform convenient notation nh nh iw nh elsewhere positive shall formalize compact establish average hadamard equal merely diagonal matrix equal especially set computing eigenvalue impossible albeit perhaps suboptimal identify identification read special ji inequality x h function dual possibly knowledge method strongly besides study stochastic single pointing serial oppose parallel update dual serial uniquely characterized turn serial give large small loading select cardinality terminology nice suited indeed processor available assign dedicated processor processor compute access assign major depend influence processing choose processor available processor lemma nice nonzero block theorem extension case straightforward matrix form rank formation time format loading phase problem work compute easy sampling direction reader non serial example group share two associate random stepsize parameter serial sampling separability assumption assumption satisfy ji separability form partition fix form index belong ji ji later variant analyze huge tb computer simplicity block partition set assign dual partition iteration parallel pick locally node compute update dual lc consideration sample distribute primal distribute number active equivalent case improve constant instead lemma partitioning instance without positive scalar assumption sequence dual q analyze coordinate descent specialize nice specialized serial sampling iteration dual random cover general rate dominant two term simple serial sampling seek quantity give improvement uniform probability dual variable use serial uniform dominant separability let fix lemma sampling cardinality corollary serial composite deal albeit separability serial specialized sampling specialized serial uniform sampling vary degree partition balance speedup perfectly balanced uniformity si perfectly perfect speedup factor nice nice combine table second line fully dense speedup obtain get fully ht fully fully nice assume study speedup nice nice speedup factor depend level problem quantity provide low speedup last involve speedup modulus achieve course regardless matrix frequently regularizer mini inequality give phenomenon plot increase speedup data speedup sparsity beyond ht exist mini mini analyze stepsize special author mini sdca specialized square complexity descent vary mini size extension mini equal consider table assume table complexity sdca regime simplify complexity sdca ht sdca n nn n showing x third follow monotonicity claim fact regime match linear sdca roughly big outperform accelerate condition sufficiently order already distribute involve depend partition variable lemma say partition negligible fact vanish pick average similar nice interpret analogous note first perfect mini condition receive nearly mini batch sdca name propose variant analyze much bad ignore expression dominant term strict lower bind clear gap sparse perfectly well measuring amount specialized specialized serial speedup mini nice sample particularly suitable unless big implementation frequently increase matrix way necessary understand machine ignore cost nice sampling contour speedup axis contour nearly straight mean speedup factor approximately well plot hence contour average sparsity analysis first strongly g write separable satisfy lemma bind term bound side eq therefore write q option case g ia inequality list problem formulation application include regression multiclass focus square hinge loss main message dataset serial sdca practice theoretical speedup term perform several sparsity dataset table option c size sparsity ph support svm regularizer example smooth strongly option specify linear support svm hinge define convex primal update hinge section uniform serial prox sdca sampling sdca three w describe set number
three signal directly generic open claim retrieval core emphasize potential question symbolic computation approximate algebra retrieval lead guarantee like question use principle article contain result independently question section describe phase algebraic retrieval usual variant retrieval major algebraic algebraic complex ground close include make algebraic treat separately algebraic obstacle regard field restrict derive bound accurate version nz usual mathematical generality amenable algebraic know take observer backward original reconstruct rank hermitian mathematically observer equivalent apparent assume hermitian knowing say algebraic closed include algebraic reason let assume symmetric rank thing image complex want symmetric hermitian algebraic problem priori image restrict variant fundamentally much easier amenable algebraic rise question explain treat algebraic namely write write reconstruct assume elementary computation equivalent original phase retrieval know rule field analogue treat allow range though specific final retrieval retrieval set map notational clarity analogy matrix determine possible signal less whereas set symmetric almost forward mapping projection pair phase notation follow n nz write signal uniquely measurement factor ambiguity avoid identifiability signal cf yield set namely back real instead consider order algebraic short technical technical map algebraic book logic know definition irreducible element map perturbation sense identifiability perturbation paragraph identifiability algebraic concept valid model model tuple kk measurement measurement formal forward fulfil section namely irreducible prove statement formal slightly technical identifiable statement identifiability remove remain identifiable borel open identifiable identifiable closed hausdorff condition open imply condition slightly technical fulfil case crucial translate signal remain whole signal important note iii priori introduce terminology brevity identifiable proposition principle exclude middle state identifiable perturbation identifiable hausdorff continuous positive hausdorff perturbation identifiable signal perturbation direct consequence zero identifiable different perturbation identifiable situation signal perturbation identifiable signal signal identifiable axis corollary signal perturbation signal regard property make statement measurement three generic identifiable identifiable three case mutually exclusive exhaustive generic perturbation perturbation differ set identifiable differ signal usual signal signal identifiable perturbation identifiable simple non matrix origin identifiable restrict origin moreover exactly open identifiable signal example neither regard identifiability notation call measurement tuple signal identifiable identifiable perturbation keep mind terminology identify matter whether show completely identify identifiability irreducible process model irreducible fulfilled case irreducible analogue characterization measurement identifiable identifiable open identifiable proper closed hausdorff subset open property open variety generalize property perturbation occur two condition measurement regime identify borel neighborhood identify identify borel map x describe prove keep complement condition call regime perturbation identify identify perturbation remain proposition completely identify context omit mind terminology regime identifying completely identify definition identify mutually exclusive exclusive identifying reformulate identify regime completely regime identify regime measurement measurement completely ia ib proposition allow analogue case identify generic regime identify three mutually exclusive exhaustive measurement regime identify generic identify analogy define terminology case keep call measurement identify generic differently happen theorem identifiability I n na nz yield contradiction identify summarize infer identifiability variety signal closure statement virtue projection identifiability signal complex ir restrict imply imply tool statement let nb nb generic measurement generic imply identifiability identifiability recognition one technical due involved complex xx yy yx xy non one imply therefore summarize infer identifiability yx identifiability xx yy yx xy establish statement closure signal xx yy yx xy imply combine unitary connection deduce original complex let nz nb unitary equivalent generic theorem identifiability n projection extend irreducible succeed complex retrieval bind threshold generic dedicated verify recovery generic signal uniquely determine orthonormal basis vector must recover may assume measurement determine generic derive j loss generality choose allow measurement generic global one pure measurement signal measurement linearly apply measurement identify signal highly crucial measurement phase algebraic regression mean principle tool algebra formulae identifiability retrieval algebraic similarity consist assume identify formal variable give polynomial ix x ib ideal fitting performing symbolic object work slightly formal projection rise polynomial estimation x reconstruct retrieval fundamentally real complex whereas part relate consider certain retrieval focus explicit inversion formulae compute linear system equation write invert singular explicit answer numerically stable main idea ideal ideal contain part use polynomial generator orthogonal scalar set z I z refer experiment detail notational would reader instead stress numerically explicit inversion formula recognition experiment magnitude measurement inversion formula ideal outline comparison alternative classical retrieval alternative fouri set deal semidefinite ideal cause limit number shall performance ideal measurement uniformly measurement also standard haar set outcome performance comparison depend exact range poorly inexact noise number threshold hence measurement ideal perform whereas accurate acknowledgement science algebraic algebraic complement topology variety write union algebraic algebraic variable let algebraic borel open irreducible call point contain irreducible proper subset restrict algebraic state algebraic algebraic field irreducible converse proper algebraic algebraic continuous union irreducible prove algebraic irreducible follow condition set iii iv apply irreducible closed proper hausdorff imply zero variety open complement irreducible equivalent borel open open neighborhood imply irreducible elementary complex appear lemma properly literature intersection complete zero nf h variety converse irreducible intersection intersection analogous complete ii closure preserve therefore closure variety irreducible variety part algebraic property contain put obtain check start map map relate projection matrix generic resp hermitian generic open dense onto dense notation irreducible observable complex similarly statement follow ii notation assume n image choice suffice proof main case generic identifiable identifiable three signal perturbation identifiable identifiable triple formulation
decay determined circuit iteration scale implication choice minimize setting clearly statistically keep primarily reduction need execute small occur acceleration virtue computation stochastic hardware technique neural back network demand necessity hardware technique enable architecture big execution mini dense feed propagation weight mini inherently sequential achieve level parallelism dense result mini batch well hardware perform fast classification deep digit mnist comprise image pixel digit scale hardware precise proportional bit backpropagation generative pre architecture convolutional conjunction hardware conjecture deep initialize layer perform feature stack autoencoder train fine batch qualitatively initialize randomly pre error drop notice achieve error control clear trend preference bit neural classification range extend include dataset neural perhaps acceleration machine purpose deterministic hardware circuit hardware descent run stochastic hardware noise procedure variation somewhat bit demonstrate deep accelerate propagation hardware framework inexact error stack level hardware com highlight design boundary practitioner stay hardware hardware software co methodology intensive trade digital circuit machine theoretical impact simulator recognition extract sensor dominant algorithm enable apart dynamic binding well presence unnecessary consequently technique randomize linear become software stack interface yet application continue purpose traditional unnecessary degradation overall system robustness noise approximate introduce reasonable expect corresponding energy per computation fast often less hardware hardware degradation term metric software prove conventional processor success dominate time execution kernel dedicate computation interact processor hardware truly benefit entail careful closely couple include optimize interface trivial design consumption cost also equally model hardware readily software cost typically substantial feasibility target common circuit usefulness hardware adequate development address observation computation datum machine use digital stochastic circuit approximate abstraction introduce abstraction analyze computation gradient execution handwritten digit problem train error rate circuit day period decade method processing implement multiplication main application perspective multiplication computationally implementation multiplier resource area hardware bit multipli stochastic circuit hardware arithmetic parallelism computational computation bit variable stochastic generate compare draw encode estimate occurrence arithmetic operation multiplication logic deterministic computation scale appropriately bit perform logical implement express bernoulli view large central multiplication use mean proportional bit variance number multiply describe matrix nd counting bit vector bite possible vector processing parallel cycle number uniform fed gate generate bit produce counter refine inner computation normalization multiply tune suitably adjust improvement interestingly stochastic circuit compute low circuit analog circuit circuit compatibility logic rapid verification low circuit circuit extremely bit sequence include generators parallelism address limitation circuit diverse come device propose device bit sequence architecture speed overall propose discrepancy quasi generate speed circuit effort circuit need bit hardware feed component build give research question regard discussion learn investigation compatibility stochastic valuable insight hardware design formulate well typically
objective design capability correct model quickly also world repository california position ct slice human measure position none case integration keep another would involve process gps differentiable might partly ct slice database learn repository learn learner goal slightly property parameter little goal clearly uncertainty suboptimal gain expect posterior latent notably bayesian design unlike extensive evaluation active well candidate selection try find apply like validation batch wide range scenario choose query success active successful reduce datum acquire active selection choose datum paper learn selection discriminate class generally early uncertainty primary relevant provide evidence performance work minimize discuss minima exhibit belief wrong significant leibler model capture evaluation relate refer generate remainder active discuss detail world random finally use successfully use field range show active however mainly minimization class aim select model set selection mainly criteria bic need approximation parametric validation statistically subset class among selection sample choose must try try version competing find sample within concern prediction disagreement multi scenario hard define kullback member focus prediction approach optimally shannon information expect expect measure gain traditional may design recently bayesian disagreement exploit equivalence order tractable mutual adjustment measure variant algorithm arguably classified finding amount information great change process introduction pt pt circle plain shape dash circle dot shape dash dash edge random variable indicate hyperparameter observe far input model arcs dependence model depend eliminate dependency iterative candidate generate next add approach predictive class predictive introduce criterion criterion log point expectation augment gps entropy point maximize predictive entropy readily shannon expect entropy way subtract shannon maximize entropy eq kl q insight mutual transform space surely empirically local sample minimize suboptimal confirm current model augment quantifie capture relative knowledge decrease pm pm model expect quantitative experiment design optimize whole iterative augment explicitly hypothesis exponential hyperparameter correct compete hypothesis correct wrong curve objective choose objective kl kl divergence eqs low divergence objective first wrong hypothesis happen ground noisy actually occur wrong already compute already support increase decrease actually correct maximize hand direction lower recover fast objective measure cross hold compete hypothesis lead minimal prediction discriminate help additionally uncertainty different
fix proportion scenario analysis subsection decrease go recover theorem conclude whiten uniformly spread close pure one one exhibit heavily intuition behind design good efficiently practice original noiseless noisy approximately pixel could plus arguably sdp solution see let easily posteriori sdp method implicitly add violate observe extract intuitively extract well assess spread alg alg noiseless ht robust kind surprising make robustness let base identify extract prove generality column affect singular g ai factor base allow robustness process post process orthogonal moreover process outperform recursively solution significant would w algorithms alg hull matlab code test matlab cpu gb subsection effectiveness take wrong show happen fraction properly ht cc right explain perfectly perform potentially entry middle toward hull noise detail report ht ht c sdp improve noise correctly identify slightly oppose fast example perform sdp hull hence fact simulate noisy hyperspectral table report remove angle clean spectral signature figure display abundance correspond ht cccc variant material properly oppose slightly pre condition analyze make pure robust whitening yet effective analysis aim sdp provably error pure hyperspectral preliminary hyperspectral plausible large large derive theorem image make pre process pixel algorithm sensitive outlier enhance hyperspectral practically influence pure pixel result paper successive applie need increase decrease infeasible feasible domain compact objective attain product generate contradiction optimality relaxation solution relaxed optimality relax multiplier sum equality give combine satisfie mm theorem design enhance pure search blind hyperspectral analysis focus successive pure recently robust resolution ellipsoid sdp high consume generalize sdp multiplicative fast contribution allow robustness pre whiten interpret solution robustness whiten perform optimal relaxation sdp extremely compete sdp several set hyperspectral whiten algorithm nmf hyperspectral blind aim signature material call signature pixel combination signature let hyperspectral pixel noiseless case signature signature abundance pixel abundance sum constraint factorization nonnegative blind play exist pixel pure permutation combination blind pure assumption therein difficult provably successive describe note nmf pure assumption refer separability separability presence near nmf image noiseless permutation illumination pixel successive robust pure alg ht project onto extremely implement operation closely generation successive simplex see provably q www satisfy pure algorithm sensitive conditioning beneficial robustness column apply result corollary simply noise view condition corollary follow column improvement especially denominator hyperspectral signature similar essentially course otherwise solve approximately orthogonal conditioning volume origin column ellipsoid via ellipsoid noiseless optimal recover transformation precisely prove satisfie full rank w w projection svd dimensionality see see truncate svd cholesky truncate operation sdp however use constraint sdp namely remove volume ellipsoid base sense build approximation whose error provably expensive sdp consider computationally alternative focus contribution analyze robustness prove satisfie let away give eq w solution third combine yield obtain fact near allow nmf rank satisfie apply identify column q theorem need high solution increase fast approximately research understand whitening noise whiten blind analyze pre whitening case pixel generative whiten r rr rr frobenius assume noiseless gaussian whitening keep recall orthonormal alg ht truncate svd alg alg simplicity equivalently assume already whiten square q observe objective constraint active multiply optimal satisfy optimality multiplier together svd obtain robustness pre whiten volume ellipsoid moreover factor solution feasible eq q identify w hyperspectral subsection whiten pre whiten robustness analysis precisely tight bound denote whitening condition yy sufficient subsection use q h h singular value see robustness whitening satisfie whiten plug tight fact pixel th column q whiten pixel match upper relatively spread whitening perform wherein whiten generative
inverse parameter write set regression express b inclusion regressor prohibitive entail resort employ fully factorize distribution approximate inclusion regressor form encourage dpp elegant convention specification distribution factorize guarantee dpp investigate effective versus variational posterior random vector ensemble e also indicator dpp field propose variational variational entropy dpp entail form show effectively learn approximate namely tu base dpp summarize u unnormalized linear variational posterior advantage estimate adjust sure initial sample dpp set inclusion element cardinality tr cardinality require close solution logistic computation compute linear process encode long approximated supplementary material benefit algorithm automate include extend learn relevant diverse easily credible predicting bernoulli dpp posterior adapt change stay algorithm conjugate warm greatly experience cg converge quickly dpp per iteration computing likelihood regression matrix element converge experiment cover section supplementary section diversity dpp map strategy failed map six baseline orthogonal pursuit generalize glm glm elastic net forward spike dpp factorize mean standard convex elastic orthogonality induce dpp approximate posterior parameter match select average stage different diverse also diverse e although auc method diversity basically show gene assess pathway involve breast cancer use five community within preferred gene cycle dna breast module cancer activate cell role cell tumor understand breast survival may investigate anti circle grid point suffer g california pairwise distance mse measurement method diversity relate well outperform close dpp method seem well grid particularly measurement construct non construct gp determine implicitly spatial site space grid observe also ensure broad grid cover domain center basis vector different scale spatially spread overlap diversity measure month locate united sensor method report mse report select perform outperform method good balance reconstruction area measurement trait interesting prior similarly diverse propose elegant encourage select item encode omp information model similarity experiment section dpp active computationally variational far variational rely dpp svd number hence parametrization condition number matrix conclusion quantification modal omp demonstrate dpp robust omp supplement dpp factorize posterior modify adjust solve ii ii draw current approximation py tp w diverse enable diversity covariance pursuit information employ spike variational dpp generalize extend field learn efficiently fast property motivate come bioinformatics tumor gene gene explore application spatial statistic diverse feature classic enable create compact interpretable feature promise interpretability prevent regression focus selection diversity set compact easy play fundamental application cancer increasingly recognize present heterogeneous mechanism task tumor select interaction exist explicitly diversity subset pre balance model complexity combination sparsity fidelity typically encourage diversity show unstable issue successively bad orthogonal pursuit omp proceed way wise orthogonal previously orthogonality implicitly maximize diversity establish omp flexibility define diversity product view assign approximation impose particular measure feature regression classification variational dpp appeal encourage define alternatively offer appeal map sampling interval unlike approximate dependency dpp approach conditional al learn variational require fortunately operation efficient regression marginal framework close work al suggest variation call uniformly propose determinant dpp prior posteriori select although prior choice make contribution propose use dpp decomposition develop family available include fully bring advantage I relevant view ii feature sampling iii define rather iv compute marginal inclusion condition feature set computational review diverse identify diverse gene tumor diversity respect gene finally optimal point process
truth structure researcher utilize hierarchical learn occurrence hierarchical structure organize multi introduce learn space finding multi label refer label training graph structure node acyclic dag graph two node correspond parent label large scale label space make propose tackle occurrence arise dense way maximize word rely fix length context interesting word representation reasoning introduce adapt log scale efficiently state make hierarchical multi maintain human source reduce space occurrence formally basic predict co occur label maximize introduce I correspond vector output label respectively label share occur softmax activation connect activation label neural network vector whose element hide label together label parameterize well update gradient code give variable length hierarchy short code output decision tree label label bit unlike tree denote take otherwise node label computation substantial carry experiment index child control library database child real reasonable cycle wrong annotation different abstract level introduce difficulty visit traversal never end unless stop follow pick depth detect pointing probability rarely see co objective limited right occurrence learn structure co occurrence inter relationship locate opposite group part health term leaf manually representation capture label occur close see figure illustrate together likewise effective treatment co occurrence show strong relation pointing leave learn representation representation occurrence hierarchy well occurrence representation example analogy qualitative upon reasoning hierarchy specifically regard label representation train hierarchy advantageous train hierarchy kind representation probable analogy hierarchy poorly type answer question something conjecture method one hierarchical term predict predict analogy answer learn disease post stress behavior cognitive rational brief disease post stress external capture structure occurrence expect hierarchy change firstly hierarchy originally child contrary child keep new type develop environment group figure modify previous learn co previous result cluster since parent observe around please co occurrence original explanation hierarchy learn representation learn demonstrate hierarchical structure occurrence identify occurrence qualitatively observe though intra still analyze limited example choose arbitrarily currently check bring classification pre interested extend reasoning relationship engineering department computer universit discovery institute research tu multi label underlie pattern paper
gaussians covariance component publication theoretically additionally interesting optimality support datum much distinguished dimension approach mixture discriminant addition assignment return complexity cluster ambient require assumption high dimension relevant extensive research supervised feature typically largely extract arise many cluster patient base characteristic drug etc cluster select assume even employ projection onto individual coordinate pre processing suffice relevant feature clear step suffice cluster mixture spherical relevant unimodal motivated spherical gaussian mixture high computationally efficient simultaneous primarily number feature feature feature objective induce penalty penalization learn thus solve np papers iterative paper exception latter consistency np paper relevant may another cluster pre screening feature marginally unimodal learning history particularly computer community emphasis assumption paper spherical ambient rely isotropic whiten multiplying covariance feature covariance hence line question optimal norm build apart computer community propose spherical however either approximate provide statistical separation minimax bound spherical marginal thresholding feature restrictive selection separation minimax perspective combinatorial search tractable base component assignment identical discriminant leverage estimation optimal hence correspond discriminant lda label plug sample classification rule mixture plug cluster covariance assumption cluster natural drug responsible expression capture cluster equivalently dimension work relevant coordinate coordinate occur matrix consider special case zero optimal additional assumption guarantee feasible use restrict property satisfie similar penalty assumption performance seek identify need relevant formally relevant unknown price direction liu linear classification step availability mixture mixture combine moment mixture learn dimensional exposition choice cluster tie estimate recovery n ji parameter univariate datum mean discard variance put skip small apply datum terminate failure invoke discard invoke return state rate normal regard identity permutation standard cdf appearance account permutation eq feasible optimization problem eq q I parameter least define section permutation
figure recovery respective run principle recovery roughly orthogonal tight stay next design increase canonical basis hadamard parameter noiseless iteration show predict theorem decay however suggest indicate dependence atom cc training illustrate noise dictionary hadamard coherence stability coherence training signal criterion generating generating dictionary ratio create decrease increase parameter show average trial reflect theoretical whole range stay quite gap hand determine conversely recovery mainly level noise noise go step run oracle iteration trial curve prediction theoretical stay level enough sized coefficient signal almost half hadamard slight perturbation therefore show theoretical translate algorithmic exist point direction future response dictionary learning generate coherent signal precision present show identification locally possible derive level roughly somewhat algorithm complexity compare sparsity project local iterative sign successful step iteration multiplication serious drawback criterion believe local maxima confirm preliminary local seem global near generating dictionary behaviour strong guarantee important want radius radius generating alternatively could extend important extend convergence algorithm arrive level version instead pure exhibit research gap interpret relaxed frequently sensitivity algorithm extend guarantee exactly reflect practical equally weight analyse symmetric coefficient integration extend k c soon original dictionary large assume perturbation dictionary perturbation I q expand try perturbation maximal inner attain typical p mean long c q absolute collecting objective compare need remain suffice say hand simplify symmetry coefficient assign increase component permutation x kx expectation calculate familiar calculate decaying satisfy satisfied almost surely expectation get almost surely almost surely I soon case symmetric integration argument constant permutation subgaussian noise generate signal local maximum conventional scheme calculate perturb attain case response inside outside fact sequence soon maximal response perturbation define fix maxima attain hoeffding z union eq subgaussian v e split expectation sign get symmetric either term last bind q taking see imply proceed soon analogue employ low conceptually combine split constant outline idea concentration cover admissible dictionary note noise replace averaging lipschitz expectation I n n free need perturbation perturbation recall perturbation ball know radius perturbation net argument signal substitute make sure expression split soon second choose ks large probability tried therefore follow sure right condition eq four choose c ks usual except probability inequality present stable identification overcomplete coherent possible training signal criterion criterion well signal ratio translate scale recovery achievable thresholding sign identification k finite criterion facebook gb gb estimate reach actually concept decade high every linear overcomplete size ambient component representation efficient processing scheme compress analysis learning address fundamental question eq unit column development well experiment start aspect theoretical insight separation predict expect approximate tool compression efficient dictionary justify dictionary tool source algorithm source identification er basis interesting overcomplete alternate locally correct dictionary aspect common successful identification order incoherent dictionary give sparsity usually signal sufficient global simple thresholding sign locally introduce present analysis give identification stable recovery sample size find identification implication identification collect letter deal abuse notation collect maximal inner restriction dictionary transpose transpose collection constant follow singular element unit frame g symbol growth aim codebook vector see extreme case dictionary sparse allow atom solve algorithm assign turn ask getting learn q therefore sign k assign atom updating atom normalise sign detail go formulation sparse formulation common mod except however onto atom eq maximum use partitioning large singular sign training oppose problem effective learn bring underlie give hold simplify random perturbation behave dictionary behave optimisation simply absolute response q local optimum signal perfectly quite local randomly sparse page foundation provide suitable identifiability principle asymptotic signal expectation coefficient follow also suffer first coherence respect decay unfortunately size incoherent dictionary sign indicate guarantee equally sized therefore identify dictionary incoherent dictionary ambient therefore extend stable task noise unbounded white noise want identification finite convenient consideration model subgaussian parameter employ typical gap generate support norm frame frame symmetric sphere let subgaussian x ix satisfy outline ingredient add substitute condition generating sign sequence actually accommodate relatively level perturbation find sub gaussian quantity iid variable q signal balance white possible expect eq noiseless identify generating local close small coefficient quality say thresholde correct even ambient fraction decay turn size either noise noisy normalise version response coherence unit assume c kx sx ix satisfying unit frame frame coefficient distribution unit accord kx non component almost surely coherence r ix respectively proof find lipschitz property mapping sum expectation cover admissible close
necessarily treat stability stand eigenvalue condition equilibrium stable conclusion hold replace example process stable recurrent unstable upper situation node diffusion obvious reality reflect topology recurrent impose difficult propose stable sparse stability impose transpose raw coefficient implementation penalty extremely focus introduce matrix helpful one want knowledge preliminary allow turn negativity directly step replace programming negativity variety package much start define modification refer decrease satisfy observe practically converge step value inner loop apply enyi like generate sparse follow number choose randomly rest node exclude node ad independently I first recurrent popular dimensionality screen induce ad hoc roc false simulate network accord early third system matlab sde sde stochastic ex cardinality collect compute averaged margin roc curve necessity penalization use network stability network ex parameter forecast time concern snapshot obtain one process error long forecasting extremely excellent term forecasting study cycle publicly expression periodic dataset expression record point cell identify factor connection factor target gene evidence comparison connection detail stand confirm connection evidence interface observation pattern stability correction average subject much hessian note definite w k p easy equivalent eq apply require solution position q sl property thresholding variant ij ij globally solution minimizer remain line theorem separability prove existence equilibrium show existence equilibrium determine verify well uniquely lyapunov dynamical eq lyapunov chapter stable equilibrium exponential shown omit modify global modify improve reduce obey vanish handle maintain decrease f result desire sequence follow operator globally lemma apply convergence p soft rule l evaluated give u ds von lemma corollary draw capability model phenomenon physical mechanism identify parameter extremely modern observation strongly rigorous variety sparsity recurrent direct control estimation recurrent network stability lyapunov integrate sparse excellent forecast learning estimation dynamical lyapunov identify topology scientific dynamical evolution level gene influence follow connected topology detect behavior dynamical great stock brain social topology underlie network devote develop appropriate model linear commonly human network equation change activation nevertheless lot instance strength unbounded combination widely system gene stock matter strong go beyond capture nonlinearity must activation proper world mechanism behavior exist feedback loop dynamic capture kind say effect recurrent successfully bioinformatic financial forecast circuit computer vision often I brownian direct collecting connection direct describe fundamental extremely modern big available observation available face addition analytical difficulty hoc aim identify complete dynamical well automatic topology recurrent interest many dynamical structure gene number point consistent compressive rely alone limit address network propose system real interest reason practitioner limit shrinkage algorithm rigorous guarantee form quantile recurrent screening control topology moreover recurrent stability follow introduce recurrent regularize regression recurrent screening topology identification dimension condition recurrent system stable detail penalize among penalty perhaps enforce relaxation take interested enhance nonconvex nonlinear penalty may nonconvex observation learn great meet modern response sl recurrent subject penalization notational w subsection unless otherwise propose problem matrix version estimate thresholde shrink fit denote jk update thresholding proceeding rescale associated j penalty penalty elastic penalty instances suppose nonnegative iterate l prototype proof usually span coefficient intercept e penalty intercept set prototype update column convenient integrate cf formulate rewrite minimize proper penalty network popular penalty scad penalty parsimonious matter ignore validation time consume nonzero meaningful prior availability resource control contamination add regularize sparsity shrinkage tune indeed usually sensitive many researcher fix similarly recommend add mild optimization fall penalize thresholding operator
large matrix therefore datum outli analysis allow large could rank outli rank use objective formulation transform derive operator proximal proximal low outli k costly implementation build neural represent proximal splitting train network fine tune proximal splitting either constrain dictionary possibility first dictionary network term outlier result decrease increase bad reconstruction output outli change result non good lie act outli regularizer applying would autoencoder sparse select array apply element define norm sparse sparse parameter apply norm allow store prevent outlier leave sparse code far sparse function new setting k process use framework something happen shrinkage element large instead sparse operator k sparse autoencoder kind manner proximal operator code proximal operator operator u u function need apply function soft sparse shrinkage derive look original shrinkage complete look descent match sparse proximal change also include come norm non since part mnist contain interested sparse compare error regressor classify digit classified percent code error produce representation suit classification change fine store value similar error code select learn dictionary picture one typical pca dictionary sparse code network case digits segment produce present usage completely prior due objective automatically
motivate hashing algorithm core use batch stochastic context neighbor provide scan modified locality lsh space hash standard hashing product typical variant equivalent analyze always note computing collection project hash similarity coordinate example consider convenience introduce estimate permutation combine projection hash unbiased inner assume projection definite definite constructive basically expand datum recall nonzero hash basically length except note package format adopt trick bit hashing example understand express inner product easy also inner consider difference put require permutation projection expect would heavy tail normalize actually quite small web th th algorithms corresponding repetition var theoretical estimator unbiased formula typical tailed datum text use learn weight tf simply well formula extreme variance raw become essentially theoretical dash count panel panel input solver core kernel section estimator product expand permutation location linear mean projection e inner achieve choose dataset combine achieve suppose apply loss th entry project obtain svm expand form exactly always experimental result panel expand th entry original middle panel kernel expand datum hashing accuracy reach mention hashing svm basically idea keep develop essentially equivalent sample applicable binary suitable metric appropriately unlike kernel often help justify type core projection permutation many extension currently rbf allow flexibility improve performance type panel core right bit hashing projection time feed expand linear another line extension apply hashing view inner product hash inner product potential compression context sublinear approximate search layer projection top store sign project sign bit provide indexing capability allow sublinear neighbor lsh popular hashing variant inner product scale linear e outperform sparse core accordingly kernel fed classifier confirm line hashing scale expectation estimator c moment regularize report accuracy drop uci repository view comparison test example need l test lin linear core kernel
bayesian update output calibrate predictive unlike hypothesis whether plausible possibility quantify plausibility use highest credible finally whether phase challenge reliability answer test prediction assessment along embed formulation statistical e discrepancy physical knowledge predictive reliable model organize procedure assess capability introduce idea predictive illustrate mass remark challenge propose real world process describe constitute prediction simulate observational available scenario necessary available system credible fact observational uncertainty prediction endow characterized uncertainty need raise concern ask prediction part solely represent highly reliable applicability however highly reliable augment embed reliable reliable reliable foundation validity idea abstract prediction discuss validation process uncertainty background discrepancy discrepancy model validation generalization use reliable make mathematically eq example mechanic momentum energy scenario scenario include parameter problem quantity mechanic tensor system implicitly map require relationship unknown fully define case indicate embed scenario parameter scenario embed particular setting entirely calibration embed enable prediction calibration observable measure experimentally observable embed simplicity exposition embed either incorrect perfectly error seminal true output statistical directly observable original treatment incomplete analogous calibration datum pose require would way validation making model representation capability accuracy challenge observational datum imply reliability infer extremely direct mapping construct alone modeling alternatively mathematical relationship construct provide test influence prediction recognize error embed physical represent uncertainty additional uncertainty even inherently additive choice must stochastic drive physical well consideration computation tractable principle develop physics uncertainty specification dependent discuss observe embed composite model appear structural uncertainty furthermore transfer validation process calibration embed embed assess assessment gain reliably prediction design illustrative broad validate unobserved need prediction interact physical phenomenon generalization embed model phenomenon model molecular chemical need embed mass momentum quantity close calibration scenario parameter hyperparameter generalization situation scenario depend new quantity characteristic experimental describe experiment model quantity provide direct regard embed experimental calibration involve additional embed ideally none formalize introduce description element additional embed denote observation express prediction model model model quantity must pose model generally general determined validity assess prediction associate prediction state generally model stress determine embed apply prediction embed express embed consistent need argument abstract appear composite q experiment form provide meaningful error uncertainty embed small preferable model quantity composite experiment avoid uncertainty model exercise embed embed pyramid experimental input low exercise generally control numerous ideal embed base pyramid hierarchy pyramid experiment exercise increasingly expensive commonly limit higher higher embed critical pyramid experiment system complexity provide therefore generalization predictive validation process simulation complex system idea example characterize abstract generalization composite likely g must calibration model calibration assess accuracy involve activity assessment briefly specify uncertainty calibrate e g speed acceleration least well consistent exist phenomenon inverse determine require output impose knowledge furthermore uncertainty uncertainty calibrate parameter approach uncertainty parameter serve predictive approach discuss however representation powerful rely bayes knowledge observe give uncertainty parameter condition uncertainty guarantee datum less calibration indeed ensure match may output experimental check consistent notion however much acceptable reality metric approach determination tolerance opinion consideration uncertainty acceptable discrepancy outcome uncertainty include cause available intend plausible uncertainty insufficient outcome uncertainty mathematically uncertainty use tolerance obtain process yield observable helpful acknowledge credible interval define particularly interval belong outside invariant undesirable mean conclusion validity consider observable period specify density define small observation tolerance less outcome set observation skewed multi modal multi modal region consist disjoint peak credible valid far validity confidence embed regime assess prediction fundamentally available justified agreement alone uncertainty potential important determining prediction primary need address sensitive aspect embed effectively inform calibration domain applicability answer central assess prediction rely characteristic instead subsection consideration prediction credible sufficiently small purpose important nature inform prediction discussion far assess embed inform credible reliable aspect assess prediction make consider example inform validation fact current context extent little unless reliable e speed understand chemical reaction alternatively could highly quantity validation sensitive cause embed validate example validity calibrate pyramid situation would validation pyramid sensitive composite prediction depend state recognize include argument pyramid test pyramid insensitive failure assessment scenario sensitive determination close constitute sufficient sensitivity knowledge approximation composite scenario suppose largely due embed discuss valuable assess whether represent quantity insensitive reliable prediction independent involve inform calibration outside clearly regime expect scenario composite embed structural response however scenario linear embed magnitude well model model range calibration matter check composite applicability way straightforward predictive assessment ensure sufficiently rigorous uncertainty perform uncertainty requirement decision determination requirement scope know uncertainty check requirement second know simulate involve composite sensitive situation able one identify insufficient could case issue reliable knowledge system model represent incorrect detect illustrate aspect predictive apply simple involve position mass act mass reliable must specify potentially embed force truth system otherwise exercise take velocity execution information system information exercise conclusion describe available several high fidelity model embed model force observational thus constant embed composite truth appendix exercise independently observational validation regard si si si confident physical highly accurate physics si forces coefficient available fail reality si linear adequate problematic si cause notice warm move energy assume temperature would temperature build position mass table position actual truth use actual variable different si si si si confident adequate thus standard calibrate use specify take denote composite perfect parameter alone deviation si si way represent reproduce variable randomness intend variability need lack knowledge value coefficient positive variability normal uncertain must however fundamentally si determined characterize uncertain alternatively si goal cover infinite characterize pose well dependent develop complex far eq model coefficient model predictive necessarily different separately action validation describe model inference solid pdfs narrow highly posteriori map true approximately value change calibrate figure htp figure comparison interval plus make difficult base give uniform distribution show tail alternatively distribution existence nearly si lead plausible prediction uncertainty contradict assertion important uncertainty error characterize thus prediction combination obtain equip well description valid prediction allow enable concrete hypothesis mechanism output confidence observe situation si assess scenario si specifically range variation coefficient energy study temperature govern competition quickly add quickly energy one temperature slowly conservative necessary truth conservative prediction qualitative phenomenon present validate calibration assessment assess marginal pdfs bayesian htp solid post pdf result dashed label prior set broad si value validation unlikely si reproduce show comparison quantitie si much uncertain agree less show statement calibration cc compare calibrate prediction available phase complete dependence mass assessment extent separate set uncertainty fit result vary large mass datum htp pdfs somewhat well deviation decrease demonstrate htp show distribution shift variability consistent move check informative si indicate domain applicability check calibration conclude trust calibrate si model credible need make htp general unknown assign validate ask uncertainty specify however simple uncertainty pose well validation process predictive building enable reliable reliable augment less reliable composite dependence embed composite model allow process require specification fidelity embed connect lack calibration use uncertain aspect model validation plausible uncertainty validation uncertainty finally satisfactory predictive view maker mass understand lack confidence modeling confidence know reality nothing generally present research development issue address application challenge outline model propose respect know quantity broadly applicable uncertainty datum critical concern dependency arise datum qualitative information important make inference tool construct qualitative difficult express tool need kind qualitative commonly regard physical reliable discuss qualitative also characterize applicability predictive validate scenario model dependent model applicable technique develop calibration datum measurement quantitie uncertainty need large domain applicability embed model allow well automatically execute associate curse dimensionality expensive naturally introduce knowledge physical phenomena address acknowledgment material work security award fc na grateful david many discussion discuss situation coefficient vary temperature determine heat support decision make need generally experimentally observable assess inform challenging validation observation determine consistency ensure predict quantity limitation dramatically reduce effort decision imply observation agreement validation prediction
situation subsample rather bootstrap amenable bootstrapping subsampling although originally forest bag forest analysis work subsample forest prediction much sampling forest one paper adaptive neighbor use slowly tree small recent move beyond black box forest prediction give propose rigorous asymptotic prediction forest amenable make forest grow forest number action rigorous overview study bag compute particularly formula time subsample covariance respect formula low status school education classify select test replicate otherwise line smoothing spline context forest empirically work bias correction analysis provide show forest prediction motivate classical justification solid spline freedom dot connect setup inferential framework visit classic feature plot forest median house measure status prediction bar distance spline note nearby bar relationship forest error end back term coefficient throughout build tree bootstrap enough carlo matter carlo detail identically distribute admit infinity regular definition subsample subsample exist subsample forest govern reduce decrease conversely large tree grow deep get result describe course put used start state condition recursive leave fraction split sense tree form device sure random forest become local meanwhile theoretical simplify exposition sometimes get effect leaf ignore without spirit tree adaptive neighbor tree use split prediction paper forest fully conditionally know arbitrarily bias align rectangle infinity find tree use training split cart tree consistent even everywhere cart however rather cart separate rest neighborhood towards say cart subtle enough affect similarly estimate forest cart learner understand cart forest library promise proof back develop normality briefly capture effect projection projection expect asymptotically tend projection automatically abstract chapter projection directly around definition argument incremental predictor argument proceed weakly predictor subsample thus back forest motivate potential near random meanwhile device cart predictor nearest predictor operate neighbor consider near align contain ty decision axis learner tree predictor always value often good get formally show quantity establish show incremental suppose feature infinity thank ready show incremental proof follow lemma regular function uniformly bound incremental constant show subsampling proceed classical analysis flow motivate let variance base pair forest base accord restrict moreover abstract forest estimate level forest theory inference forest theorem prediction subsample setting considerable monte subsample bootstrap correction ccc ccc mse cosine cosine cosine e cosine e e describe accuracy bootstrap replicate synthetic distribution construct rx k average analogously absolute divide metric average test rather get cosine relative mse decay predict appear error yet regime decay high bootstrap classic uci repository set due predict log divide set parametric synthetic perform despite ccc ccc auto e e metric accuracy subsample replicate rule noise lead box validation measure like study subsample subsample satisfy establish normality show generalized density involve showing hold see pair equivalent independently distribute try continuous tie standard get gamma incomplete eq letting expression thus quantity around index sum loss regularity probability converge obtain meanwhile stein suppose identically show expression also projection write decomposition quantity interest individual tree forest equivalently
consist estimating minimize infinity appendix inner ridge optimize ridge scenario j ignore gradient given attribute namely behind lasso algorithm let b directly recall always case attribute sparse improvement harmonic dd require may available consider moment exactly run modified use upper count run starting ignore analysis formulate output dm value become achieve improvement moment attribute prefer improvement regime regime reason easy infinity norm remain upper hold sufficient ridge join regime line plug k plug k eq use q plug section test analytical claim conduct set experiment control mnist digit design ridge regression consist moment attribute moment tries require ridge offline attribute utilize page attribute efficient use represent attribute quantify avoid normalize ratio prefer upon definition exact quantity analysis scenario define fold increasingly average zero error bar algorithm result start phase upper confidence conservative find split attribute evenly inner improved linear easily scenario algorithm define exponent decay ball attribute namely independent corresponding expectation entire analogous stochastic addition connection adaptive additional improvement future learner prove complement room improvement partially grant grant direction use full idea beneficial coordinate rather single popular along instead size uses th attribute learn gradient across arise paper discuss run simulation decay exponent datum experiment htb offline erm erm htb adaptive attribute bad version improve understand risk algorithm actually perform gradient enough estimator state lemma randomization eq convexity lemma first see tr calculation give recall state probabilistic moment nz prefer bernstein fast pay factor attribute first realization sample phase hold trivially actually phase part conclude similarly bernstein arithmetic conclude plug dm plug lemma constant assume therefore prove directly completeness second vector examine expect respect lemma us value c next relate linear lemma proceeding lemma fact induction hand q combine rearrange complete proof proof eq lemma bind randomization hence bt triangle q note dc ic equal yet di I assignment probability minimal value attain equation hold institute science analyze learner subset attribute training ridge regression geometry sampling learner probability calculate data improvement excess state large main knowledge knowledge simple amount achieve improvement complement analysis claim effective whereas life learner per medical diagnosis patient learner perform diagnostic cause physical likely diagnosis attribute whether site email address pay cost known attribute observation formally attribute reveal select attribute attribute reveal feature include subset predictor minimize target discrepancy generally loss expect vector unlabele particular problem ridge scenario regression one online gradient descent behind scan calculate unbiased attribute algorithm sample excess online attribute additional factor interpretation provide low establish fact improve develop manner attribute moment advantage utilize examine reach optimize principal able budget begin moment namely risk scenario summarize notation old ridge km km easily prove bound always previous previous dependent moment sufficient rate distributional algorithmic elaborate attribute coincide online full scenario limitation approach second moment advance computable address phase phase simple phase always sufficient prior knowledge attribute rest organize follow describe develop scheme case knowledge moment variant prior factor ridge improve experimental simulate connection scalar letter font indicate index indicate pp proper triangle expectation randomness attribute respect respect randomness framework learner represent goal learner weight minimize entire follow standard training minimize expect fit require norm schwarz old assume loss generality attribute scenario limited attribute popular budget budget total total exceed bind budget efficient scenario exist full attribute scenario imply full thus trade linear regressor constant regression gradient scenario attribute expectation perform ridge prove correspond regressor regression ridge scenario call ridge gradient descent build current step direction result project ball square attribute attribute sample sample attribute build calculation obtain unbiased reduce estimator minus building unbiased set bt mr ki slightly notation show analyze bind therefore develop sampling estimate bound every need solve multiplier yield inner minus use follow strategy probability prove superiority generate lemma moment attribute idea attribute formulate output run calculate equation recalling algorithm always well coincide however dd exact knowledge knowledge task case moment initially depend next unknown prior equation address split phase run moment train slight modification upper order return output order apparent use moment method never get estimation formulate value perform phase second turn km bind assume guess proof bind squared bound factor analyze proof treat join single estimate constant factor
suited exploit short spatio unable investigate novel cache block trace trace vector explore count address gap propose dp mixture capture covariance however particularly dp multivariate generalize lead cache step historical capture trace cache experimentally trace storage mining modeling categorical count block trace exist policy cache least recently use ahead strategy sequential pattern nearby correlate extremely short interval typically find correlated interval predict alternative spatio novel count exploit cache capture trace trace sequence request often span million interested trace spatio arise long access certain read capture aggregated view partition histogram slice span get aggregate slice count count instance request adjacent together rich temporal dependency aggregate vector aggregate portion small count dimension count understand spatio temporal common correlation hide extension temporal inherent storage trace suffice since mixture type model kind adjusting datum parametric variant study decade correlate count parametric multivariate gaussian expensive sparsity datum extension sparse modeling select aware world trace technique outside apply often count mixture extension exploit propose novel technique dp poisson dp methodology tractable discuss extension hmm cache aware address take first cache long spatio dependency memory experiment world trace show improvement particular baseline improvement mt multivariate complicated designing mixture unknown reference aware unknown truncate dp study amenable extension model immediate good another sparsity emission component parameter aware work mixture density specialize investigate prior count cache cache storage cache warm cache trace different long temporal cache serve study trace trace grain short correlation specialized type large cache performance predict event operate contain amenable trace cache application persistent medium disk cache medium request cache cache constitute cache application else cache retrieve disk cache cache high improvement measure cache cache observe part derive place cache operate trace term hour operational run much restrict phase repeat even exploit trace partitioning phase slice second let access request sequence count bin trace arise range access repeat albeit hence dependency vector correlation markov choice follow chain induce cluster count denote variability trace k suitable scenario motivate non technique dp cluster correlate capturing hmm exploit count hmm predictive ability hide map raw request various slice latent aggregated operating aggregated like choice block load cache happen hmm dp algorithm deviation usual viterbi possible much slice viterbi technique fraction prediction would consist loading cache correlate count model count temporal dependency parallel along mixture propose dp dp base hdp hmm mixture challenge design correlate exploit dp mixture challenge follow hierarchical hmms fix dp hdp data hdp discuss dp brief overview collapse hmm dp detail supplementary throughout notation subscript sparse latent aid hdp detailed collapse variable likelihood py z old hmm dp know arise know require sum possibility exponential supplementary summarize standard dp also computationally motivate modeling dp suppose derive j variable integrate alg eq l z z l kb hmm dp exclude u supplementary hmm dp active sample variable benchmark trace likelihood evaluate effectiveness available block trace commonly storage microsoft choice trace trace aggregated aggregation number bin dim slice later trace two experiment understand well next cache dp detail compute dp dp dp margin correlation aspect necessity model independence dp hmm inherent datum r r trace hmm dp dp mt mt mt mt mt mt mt trace simulator augment simulator block simulator describe supplementary appendix prediction improve simulator trace plain capture portion trace dependency ideally data train see explanation trace pick trace repeat fine bin memory aggregation count find baseline attribute sensitivity trace superior bring focus hmm dp term train hour run algorithm sparse dp finish trace sparsity latent improve efficiency time c hmm hmm name dp dp mt mt mt cluster avg outperform hmm sparse hmm perform hmm dp trace spatial model handle fit good bin dp without trace sparse trace trace baseline parametric baseline suitable application trace vary infeasible exist dp model focus correlation type capture range ahead hope show model trace trace long present trace predict understand read schedule cache size prediction play important role incorporate capture disk scheduling issue hmm dp structure far explore datum hmm dp lead efficient leverage block trace cache improvement world trace outperform trace perform experiment publicly block trace microsoft represent diverse trace comprise week worth allow study long range temporal eliminate trace write percentage focus read cache present remain trace validate result comprise collect hour divide trace vector phase operation r rd mt mt
unit diagonal diagonal learn inference hide convolutional inference generative model three convolutional layer filter activation fully activation generative replace varied final step detail ex step step hyper epoch importance monte carlo numerical report test likelihood gap mcmc convolutional inference step report mutually exclusive combine specification variational specification practical practical transition satisfy eq x optimally choose reverse iterate converge running follow tr allow way detailed balance improve posterior bind make practical balance ensure transition balance hasting rejection transition first generate finally analytically metropolis interpret reversible jacobian evaluating target carlo estimator variational posterior estimator rao carlo calculate respect short attractive computationally demanding path accept reject alternative omit acceptance transition operator reduce respect gradually reverse importance sampling transition p normalize density reverse look like old log ratio variational strategy specification base guarantee inference satisfy impractical far variational last consist multiple different effectively become take distribution put iterate chain offset add iterate inverse set take bound effective reduce potentially input suggest optimize mcmc step optimize mcmc sequentially maximize variational improve optimization boost iteratively unnormalized posterior new maximize local simple exponential form improve variational approximation accuracy advance perform variational approximation synthesis variational monte incorporate rich inference variational fast objective option trading computation parameter quantify observe datum relate specify rule quantify simple conceptually imply computation intractable resort method inference monte carlo explicit case latter nonparametric get parameterized approximation maximize bind eq maximize minimize perfectly variational start distribution rather mcmc draw choose outcome variable converge exact advantage mcmc approximate time practice getting long interpret chain expand set auxiliary free posterior marginal approximation distribution since close fit choice optimal reasonable specify optimizing special auxiliary like z rewrite subscript highlight possibility different chain flexible parametric approximation choice operator however transition inverse variational without operator variable initialize transition ratio low insight behind parameter estimate low obtain gradient application chain transition operator case backpropagation obtain bind solution gradient h initialization hmc low tv z tv x l l omit monte discuss approximation respect stepsize mass hamiltonian algorithm local improve add hmc reduce thereby use mainly calculate additional derivative rule hamiltonian variational step expensive reduce mcmc computational dimensionality variational hamiltonian hamiltonian tune hasting step reject transition optimize marginal assess technique find may posterior short rely theory hamiltonian consider estimate death large contain cancer city occur city count compare expect assume low dimensionality contain integrate numerically calculate approximation number step hamiltonian see hamiltonian benefit realize iteration
exponential bias bayesian nonparametric model conjugate beta odd representation gamma couple general conjugate family representation likelihood conjugacy conjugacy continue place broad class mixture limit exponential however whether similar notion conjugacy bayesian literature family include familiar name poisson name refer aspect underlie bayesian l evy construct property conjugacy parallel known conjugate multinomial bernoulli break biased obtain separately significant formalism posterior conjugacy nonparametric analog provide constructive conjugate prior specification size representation broad process trait us trait trait likelihood point subset trait trait make trait trait allocate new trait yet allocate trait grow challenge countable trait frequency prior calculate posterior trait frequency trait nonparametric integration three principal conjugacy marginalization conjugacy dimensional exponential conjugacy turn beta hyperparameter iid bernoulli conjugacy certainly popular cardinality arguably parameter prior pair generally exist though process include classical bayesian dimensional family construct nonparametric biased trait play role year allow exact slice particularly useful representation general show family directly integrate trait trait beta bernoulli show generate bayesian exponential constructive exist marginal build nonparametric bayesian nonparametric calculate posterior development introduce generate automatic conjugate exponential likelihood size derive conjugacy discuss view trait trait express recall measure trait trait index many tuple consist trait trait descriptor trait weight q th point trait allocate degree individual allocate trait trait degree point belong trait belong treat observe bayesian topic model vocabulary occur document might document document topic concern posterior conjugacy exponential fact especially trait order trait specify measure well algebra subset consider measure random almost surely particularly form measure completely random property without treatment follow part component component construct e location location atom finite infinity deterministic random since measure generality distinct independence random variable ignore conjugacy representation cardinality countable deterministic borel product countable subset generate ordinary component start poisson countable ordinary yield place component incorporate atom atom infinity ordinary ordinary typically part hierarchy extensively elsewhere affect point trait henceforth measure proper ordinary point attention trait trait atom distinct location distinct discrete infinity helpful component point specify assume iid locate atom location take locate atom may state form impose likelihood formalize restriction recall trait trait unbounde collect countable infinity trait require infinity location represent trait sense know location atom advance trait discover priori countable trait countable location atom ref require countable trait location ordinary component must countable atom trait frequency infinite mass ref implicit part allocate trait finitely thus number atom every correspond fix finite restriction atom restriction finitely nonzero particular atom countable atom consequence purely mixed discrete henceforth discrete write requirement allocate finite trait translate requirement construction form mark ref thus capture call ordinary measure beta component feature multiply real factor contain weight hyperparameter achieve mass beta improper pair process specify finite mean integral proper e restriction imply range like posterior fix atom distribution ordinary proper jointly weight dimensional prior proportional q theorem ordinary atom location well draw fix atom locate weight atom deterministic purely without knowledge atom generate note know ordinary atom return atom atom form mark consider iterate restriction satisfied calculate location atom put normalizing atom density third component posterior exponential rate update hyperparameter fix likewise show conjugacy hold desire next conjugacy conjugacy establish conditional location rewrite poisson bayesian nonparametric first atom weight rate ensure ordinary component characterize proper weight rate distributional location atom finally range ensure improper gamma either integral finally hyperparameter discover poisson highlight poisson process atom weight gamma atom gamma conjugate likelihood process odd bernoulli exponential weight atom support odd parameter probability successful odd success failure write emphasize location atom ensure ordinary hyperparameter range ensure improper beta require beta proper hyperparameter restriction summarize component previously beta atom conjugacy odd highlight result corollary odd bernoulli process atom weight atom process conjugate prior odd find bayesian prior build representation despite biased random hyperparameter satisfy stick break stick describe proportion remain stick break describe stick break stick call representation reason draw thought draw limit proportion atom atom choose representation useful familiar atom inference truncation constrain sum explore past notably beta though term stick mass sometimes refer stick popularity stick beta case slice general discover previously unknown representation general random q location atom simulate location weight come come cf ordinary generation countable demonstrate weight finite familiar case biased representation atom ordinary discrete atom atom atom atom atom atom atom moreover enumeration break enumeration observation atom location atom atom atom atom atom atom atom atom process first poisson eq take component posterior equal poisson finite total q atom find atom identically across atom independently summarize give detailed calculation thereby trivially conditional location component write biased representation beta derivation bias gamma poisson process summarize let fix weight ordinary conceptually focus cf integrate canonical comes distribute iid marginal form let atom location example marginal chinese restaurant prove since integrate dimensional generally marginal representation atom write ordinary jointly union weight new atom location moreover express atom location induction assumption weight atom weight mass line let agree development cover present atom atom location new atom location eq eq finite let repeat measure inductive hypothesis hold biased find prior exponential atom trivially satisfy generate fix ordinary suppose conjugate provide atom location distribution atom corollary beta discover new iid condition q eq summarize location distribution ordinary iid across accord location atom distribution construction atom atom calculate posterior general prior draw bayesian nonparametric model notion family allow specify automatic likelihood
second robot system action angle robot currently receive simulator controller current angle angular velocity angle velocity physical randomly choose action simulate noisy draw contaminate deviation probability dynamic density pair transition useful model three scenario right roll thus case joint summarize redundant aim draw video identify action vector shape ratio offset axis near reverse driving transition dimensional dimensional output collect transition summarize bottom method successfully high dimensionality reduction conditional estimate least dimensionality density mutual denominator ratio effectiveness extensive computer q let express proof derivative approximation eq substitute partial gaussian derivative cs ac regression informative multimodal preferable challenging dimensionality first dr execute dr propose novel single shot perform key formulate need dr method extensive various computer reduction however analyze informative possesse appropriate naive approach density kde kde nearby problem nearby separately estimate density decompose form p thus problem method mini solution efficiently compute cope aim input reproduce hilbert possess systematic model available overcome alternative sufficient theoretically optimal rate dr dr promise accuracy preferable regard paper propose dr dr include execute therefore density usefulness name robot method conditional dimensionality let dimensionality identically dimensionality expand form theorem conditional independence therefore reduction minimize denote represent relation span equivalent member class loss conditional negative kullback leibler member loss ce pearson divergence sharp include easily critical develop method denote coincide search geodesic gradient estimate entire estimate computationally expensive execute randomly achieve small derivation conditional density minimize even maintain th gaussian locate may subset center increase advance notable appear compute analytically similarly normalization analytically essential include denominator uniform reduction accurately well exist experimentally experimentally investigate usefulness dimensionality reduction maximize gradient manifold artificial dimension execute conditional density neighbor square cross validation least square inside behavior plain method normal plain well due dimension artificial right leave measured norm clearly profile much discuss ratio smooth density p p loss q kde
per person illumination individual image correspond illumination change correspond pose illumination expression database consist intensity algorithm subspace subspace generate normalize pair basis perform svd subspace vector next generate projection apply point separately green either claim see experimental choose different label pair project evident figure project along analyze separation give dataset dimensionality project separately matrix dot visualization face element dot white represent inter dot consistently dark separability project reduction quantitative result discriminant regularize random preserve embed dimensionality dataset perform yielded compare method make dataset class evaluation test compute projection accuracy deviation similarly rp run c c method pca rp dim acc see section rp dim acc rp dim acc lda dim acc pca dim acc result table report find degenerate hence dimensionality first class class dataset show table performance multiple subspace sufficient independence disjoint iterative algorithm learn projection reduction three world reduction example proposition datum union application preserve independence subspace trivial design dataset dimensionality compression theory face texture segmentation reduction nuclear simply image image datum think traditional subspace principal application datum preserve reduction although try preserve geometry datum try preserve dimensionality independence subspace disjoint subspace subspace line idea vector sufficient find projection aforementioned handle corrupt say subspace dimension subspace margin separate margin margin maximum dot either vector angle definition subspace specifically dataset goal reduce continue lie formally let propose subspace number require label vector require motivated disjoint unit orthonormal tv jj tv show symmetry lie need respectively thus say projection plane line lie along along line angle separate angle principal principal dimensional subspace disjoint subspace plane project subspace two line argue add dimension plane project subspace one project concern notice least linearly forward already would vector would computationally handle though label circumstance specify try label dataset attempt sample heavily underlie principal two subspace projection subspace pair separate margin subspace submatrix repeat state idea span close repeat cosine opposite subspace opposite handle margin subspace equivalently express identity disjoint local minima gradient w thus lagrangian principal principal pair local minima w r minima setting
r poor posterior show bootstrap obtain member return sample construct finite subset agnostic bayes bootstrap possibly infinite model hyperparameter build agnostic e reflect repeat hyperparameter ensemble expensive would train predictor always matter trick exploit observation accelerate construction fashion maintain contain risk update cost single multiple share detailed v h run updating behind run gps deal construct powerful mcmc accommodate probabilistic order predictor ignore nature determine comparative agnostic predictor probabilistic traditional predefine adapt ensemble three building select search rs practice often superior ensemble construct use hyperparameter mat ern kernel scale gp evaluate hyperparameter configuration performance test several collection perceptron win frequency converse sure outcome chance use sign derive posterior significant significant substantial come datum convert multiclass merging class benchmark collect collection cccc cccc rs r rs rs present win represent redundant complement add method conclusion sign colored dot dot report obtain observe outperform method outperform rank look outperform concern forest yield generalization performance elaborate method look challenging space vector agnostic alternating analysis column clearly significant degradation term speed opposite left reach significantly exception well gets quickly outperform cccc rs cccc rs cccc rs cccc rs rs automatically construct ensemble hand adapt hyperparameter produce ensemble method attempt ensemble uncertainty risk extra generalization dominant task hyperparameter fortunately progress sequential optimization method validation ensemble properly select extension automatically ensemble recently paradigm agnostic bayesian confirm selection important make machine expert science recently report success successful hyperparameter configuration learn increase improved configuration converge find good configuration well instead combine good win entry netflix competition variety different helpful comparable performance likely differently input produce error average hope majority well globally dominate average much however ensemble system automatic construction method thus method error exhaustive explore space recent oppose agnostic weighting effectively generalize space model hyperparameter efficient set confirm regular hyperparameter follow agnostic bootstrap construct ensemble discuss notation setup task refer member predictor hyperparameter training hyperparameter let obtain set assess quantify incur target task minimize hyperparameter selection hyperparameter list select good example grow hyperparameter yield outside well replacement search inefficient inform test address limitation automatic hyperparameter consist treat learnable learn hyperparameter must gaussian representation promising configuration gp assumption conditional pf nf r k ij acquisition success q equal cumulative function normal acquisition maximize equation perform ascent initialize chance global optima expect offer exploitation face fit gp test initially empty acquisition hyperparameter procedure mean function either marginal hyperparameter sample detail hyperparameter good predictor suffer hyperparameter preferable properly extend ensemble
evaluate color green green blue missing evaluate effectiveness compare admm try recovery completion demonstrate final admm admm try recovery numerical lr admm try try lr almost quality admm computation well reasonably extra find admm notation name admm adjust admm admm adjust reference deal matrix lr show dct operator stage two note second singular estimate stable merely large lr admm important use fidelity admm recover sharp play rule synthetic certainly sensitive course jump way true try jump singular low limited good develop foundation china k recover large arise image medical imaging kind formulate problem operator recent use norm minimization nuclear major limitation nuclear singular correspondingly paper besides completion method extend validate superiority etc approximately naturally decision nature firstly approximate rank norm I bind unknown low recovered optimization nuclear reduce sampling projection incomplete problem result nevertheless suboptimal application operator together overcome nuclear nuclear minimize small singular truncation correspondingly q way aim extend completion algorithm euclidean space frobenius variable projection operator denote transpose low rank optimization attract interest algorithm briefly review influential reformulate sdp interior scale project subgradient computation concentrate decomposition solve scale slow uv parametrization factorization matrix decompose rank priori dynamically adjust alm multiplier programming arise admm widely et solve apply value though nuclear replace nuclear convex solve critical note rx xu u r rewrite get minima propose later similar sparse learn specifically present support short number detection identify reconstruction support detection reconstruction advantage prior information true signal decay heavily seek try al propose special detection completion contribution particular new call solve introduce solve mention subsection conclusion give algorithm elaborate approximation key define variant recovery beyond problem model discrete transformation dft framework procedure start e recover estimate recover idea iterative initial fix svd update author study completion algorithm plain pure nuclear via initialization x return explain nuclear regularize base process large feasible try aim well approximately often decaying extend previous detect vector value nothing specific implementation support show repeat estimate singular thresholding singular spirit significant jump singular minimize false sort straightforward last jump prescribed value decrease last jump unlike propose apply absolute value value look example cardinality computed jump neighboring reflect stability large cut threshold heuristic admm originally norm form extend original admm nuclear deduce subproblem enough convenient follow conclusion satisfy max particular matrix decomposition conclusion adjoint reformulate constrain problem lagrangian lagrange multiplier admm decompose task small subproblem involve admm ignore derive subproblem easily scheme admm approach update eq iteration iteration subproblem elaborate subproblem explicitly solution equip form express admm solve subproblem form remark eq scheme study omit analysis attract lot attention task prefer noiseless norm accelerated propose among accelerate proximal line al extend completion paper solve general completeness short overview meet possibly differentiable continuously lipschitz add construct q conclude iteration subproblem q iterate framework subproblem close closed well also omit efficient solve convergence alternate multiplier condition result name whose subproblem kind mention could form stack contrary process put correspondingly function n reflects equivalently transform lagrangian lagrangian linearize admm prefer handle easily update proximal iteration update update calculate update computation solve elaborate solve form ignore concentrate obeys rule solve adjoint operator show side accord achieve eq remark begin omit admm subproblem solution efficient problem validate part effectiveness hand real admm refer extensive accuracy
extract balance explore consistency trial trial provide plot misclassifie trial use large trial suffice theorems markov discrete rely non ground truth bias toward cut circumstance evidence weakly connect weakly decay word regime certainly perhaps lie component geometric log together acknowledgement grateful part research ds grateful nsf grant dms support support nsf dms nsf author counter counter counter definition counter counter corollary counter remark pt pt pt von establishe algorithms sample ground truth minimize functional cut minimizer cut minimizer sample hold cut result scale connect nearby leverage cluster optimize objective quality partition separate introduction functional cut meaningful introduction balance term functional closely cut functional multiclass cut theoretically computationally utilize cut algorithms expansion approach cut truth sample converge precise manner towards partition discrete subsection informally consideration investigate consistency graph vertex weight point connect scale geometric average desirable resolution take increase represent consequently discrete work precisely take consistency hold cut notion tool modern study minimization random graph provide study limit minimization consistency consider linkage maximum unfold consistency spectral rigorously von work eigenfunction limit sequence recover go zero normalize cut minimizer discrete functional minimize specific set discrete functional prove well one von von normalize knn graph graph hold quite minimizer functional close functional functional balanced graph cut take numerator balance c introduce simplify remark balance term appear multiclass pair note cut multiclass balance partition functional domain way term weight boundary boundary smooth surface cut regular boundary notion geometric measure theory mathematically subsection area tendency balance term refer pair equivalent cut read eq consist point set boundary extract connect nearby precisely decay zero basically describe point increase investigate balanced cut minimizer balanced partitioning uniform present cut unique balanced cut rescaling surely converge towards minimizer notion partition precise let let n c definition convergence discuss conceptually rate scale lebesgue compactly associate optimal cut cut example optimality relevant machine minimizer involve hold dd graph cut determine still valid parameter connectivity balance graph cut remark despite graph minimization effectively fact choose appropriate initialization use bridge rely notion optimal min proof convergence carefully subsection property desirable minimizer statement cut prove cut balanced notion recall subsection total variation illustrate also investigate related main n expand introduce partition turn useful characteristic function ix characteristic need borel interest coupling measure second distance understand focus absolutely continuous respect lebesgue pass discrete formulate way borel borel plan formula exist little match make lebesgue statement convergence proposition enough find sequence map important map convergence occur consideration control norm ic eq convergence partition ambiguity arise partition previous give subsection discuss let represent graph f convergence make remark consistency consider minimizer sequence minimizer need enough compact enough conclude cut energy approach extension viewpoint namely absolutely continuous lebesgue end respect see representative discrete think flexible discrete restrict correspond notion analysis geometric measure domain need extension whole extension give function variation total characteristic hausdorff measure variation finite derivative chapter simple precisely area relate weighted weighted formula rigorously formulate precise definition formulate either relation measurable q deduce ratio cut cut indeed minimizer lemma close balance implie imply cut imply every fact state beginning minimize follow calculus variation minimize continuity converge semi continuity total complete precisely subset similarity context similarity proximity kernel otherwise main cut domain satisfy converge zero point weight optimal balanced cut converge one cut subsequence cut surface scaling establish cut utilize problem particular minimizer minimizer limit remark hypothesis scale must distance measurable sequence eq theorem prove outline rather work indicator eq denote suitable nx rescaled coefficient show indicator subsection show functional establish converge toward indicator toward convergence follow functional discrete function subset start nu u iy nb nb nu nu nu analogue prove suitably indicator function set minimize balanced cut proper subset satisfy sequence exist converge prove claim first balance n nu fact every change obtain nu deduce proof arbitrary arbitrary show subsequence assume limit trivially enough us map inequality hold want start know since imply n nu subsequence subsequence sense convergent subsequence suffice hold sequences nu balanced balance cut sequence follow q nu subsequence convergent subsequence subsequence minimize particular characteristic c imply bound cut nu n nu variation invariant translation assume either continuity balance measure region region strictly zero bound away zero consequence turn sense consequence subsequence u instead moreover subsequence subsequence converge theorem balanced cut analogous space set collection comprise indicator cover multi class r collection disjoint lebesgue additionally imply set lebesgue definition may equivalent balanced minimization balance cut particular want sense define want follow kernel sample satisfy functional converge topology way proposition way omit analogous argument subsequence minimizer finally argument subsection adapt proposition argument constraint perspective due smooth partition meet next lemma multiclass combine nu ny n nu statement iii statement dl proposition deduce orthogonality subsequence every almost k k side equal conversely contribute inferior involve due orthogonality convergence recovery remark due belong let c fact define finite empty intersection lebesgue assume without mutually disjoint finitely manifold embed start construct recovery piecewise dense set variable claim recovery partition define partition consequence functional proceed particular assume otherwise nu nn inequalities tv rr balance term deduce establish r piecewise existence immediately balance denote smooth exist induction hypothesis enough simplify denote hypothesis equality r contradiction bound disjoint subset satisfy mutually q subsection belong consider converge correspond symmetric let x b let smoothed eq symmetry produce q word prove open rotation every constant summation chebyshev eq claim lebesgue property smooth lemma example lebesgue measure measure countable nan partition hypothese lemma combine co formula imply everywhere particular subsequence lebesgue almost subsequence continuity formula subsequence along subsequence analogous hold well relation subsequence extract subsequence satisfy previous lemma complete cover exist imply contradict denote boundary combine convergence show combine continuity total variation disjoint complete
budget pac completely subset happen arm low bad bandit yield budget set arm still introduce successive bind error bandit permutation either propose good confidence inspection knowledge drawback share ucb bound identify complexity multiplicative analogy three low bandit theoretic permit quantity subgaussian propose tight bind complexity uniform sample close contribution fix bandit lower bind budget setting bandit bandit complexity first towards new two confidence budget setting mathematical result permit line sub increment iterate logarithm permit efficient matching kullback make relate probability model lemma technical aspect generalization minimization framework draw draw let model stop relative low fix confidence straightforwardly bind identifiable satisfie make family continuously parametrize mean let pac without arm order arm arm arm long pac apply total number monotonicity yield obtain arm lead low relaxation paper every denote mean prove perspective algorithm design exist twice order natural gaussian common exponential follow kullback leibler family sample kl complexity eq bandit popular website version present user page model probability user feedback user two equally whether armed match different algorithm match bind provide sample pac identifiable let uniform particular obviously consider direct c variance imply strategy consider quantity analogous chernoff explicitly property illustrate two exponential bandit indeed tight change use hand inequality change involve use cumulative modify order alternative low bandit good pac property obtain use satisfying similarly bandit performance guarantee closely match possibly term elimination bandit section armed gaussian bandit reach rule expect elimination two subgaussian cover bound subgaussian enjoy bound subgaussian bandit case introduce coincide pair sample normally pac algorithm versus type type recommendation rule choose empirically arm sample match rule satisfie lower prove among elimination pac elimination threshold asymptotic low point exhibit preserve sum I variable obtain non choose iterated surely achieve goal prove elimination pac illustrate significant average reach conservative allow rely arm base governed ensure end round schedule deterministic match low elimination reduce effect elimination suggest feasibility bernoulli observe particular little gain together stop rule quantity introduce section aim bind provide bind arm sample uniformly determine algorithm arm subgaussian precisely theorem stop respect exponent significant exploration ti ta drawback propose sequential stop stop consequence ratio two proportion pair likelihood denote display consequence I interpret relate kl close armed threshold use guarantee stop eq result analogy conjecture conservative exploration lead conjecture armed bandit obtain theorem confidence bernoulli comparison set bandit present obtaining bandit low yield failure consistent algorithm note previously confidence comparison family distribution fix budget consistent satisfy kullback leibler symmetric hold hold family arm arm upper side arm complexity equal model q recall always define theorem precisely check strategy budget bernoulli exponential static arm appendix every observation arm interest bernoulli model unknown exist bandit comparison quantity bandit strategy arm satisfie see describe approximate universal arm varie observe indistinguishable provide allow budget armed bandit bind armed bandit budget lemma state prove nonetheless able low bind spirit simple leave room improvement bandit model exist bandit bandit q gap modify statement blue focus arm experimental design budget fix setting bernoulli illustrate bandit difficult right budget star set circle probability logarithmic probability carlo purpose plain line straight log set report elimination form three exploration provably pac rate almost pac bold green symbol specify symbol function slope complexity conservative time comparable rate significantly run maintain failure empirical symbol symbol testing seem error sample introduction number line stop matches slope huge gain use use prohibitive bandit algorithm exploration rate elimination stop rule stop empirical exceed exploration plain rate rate elimination bandit rate mostly coincide elimination sophisticated strategy experiment pac relatively easy compare budget pac algorithm pac case green observe probability algorithm usually sample budget pac design requirement budget draw arm preferable counterpart much bad exploration whereas predict budget difficulty provide principled way budget stochastic bandit complete testing sampling match certainly generalize gaussian show stop criterion comparison complexity behavior specify alternative confidence ie sub dependent arm confidence set algorithm perform arm notably gap investigate analysis improve budget understand complexity arm greatly identifiability common alternative bandit introduce element relate bandit differ expression simple change distribution provide proof let show conditional jensen successively arm ratio rewrite combine inequality extra ingredient provide low sum error probability absolutely measurable optimal arm order problem resp algorithm absolutely respect write show apply inequality q conclude exist show induction statement eq gx initial assume let measurable nz hence statement let briefly rule recommendation arm sequentially number arm give generalized distribution identifiable bandit eq unique let time whereas likely happen thus little precisely inequality tt ta define display construction picture convexity family kullback divergence may bregman twice argument admit relate uniform confirm indexing natural bb equality achievable elimination chernoff tt show pac upper last inequality go go one dt upper exist one get follow whose help bind quantity follow implication true q proof suffice map ss ss ss conclude optimal find region independent random ts suffice hand side infinity small proof omit every subgaussian super martingale sd dx stop confidence separate mention note involve exploration rate bound conclude use stop q large event show prove conclude bandit event give note algorithm every let optimize possible satisfying bound static arm rely family g interestingly proposition apply strategy arm theorem multiply chernoff random direct computation therefore one show b expression x ng arm contradiction exist h b propose change arm assume bandit bad easily armed
small high normalize document target choose hold subject loading automatically test geometrically spaced loop sequential due fix resource could screen hence main sequentially read disk ram hold feature hold termination exhibit use automatically select step performance open one average automatically range plot screen key test structure allow quantitative student fellowship innovation fellowship fellowship university award city distinguish gold wang receive electrical high university receive ph university focus signal distinguished award city fellowship engineering receive sc b university ph electrical university wu engineering paper seminal paper system com university award interest process learning prediction medical fmri definition wang lasso problem combination vector variety dictionary screening quickly identify subset receive solution resource speed solution intuitive understanding screen illustrative dictionary screening respect dictionary heart many column field term lasso seek serve lagrangian constrain formulation extensively signal vision literature introduction prove application range recognition recognition speech classification recognition classification text document dictionary iteration bottleneck address context classification zhang et collaborative scheme improve speed face application xu collaborative superior representation solution considerable recently target screening quickly identify guarantee remove reject solution appropriately zero solution screen run mode significantly dictionary solve lasso second reduce often gain lasso moreover since solver conjunction solver heuristic select voxel statistic fmri fan excellent review base feature formalize approach algorithm similar screen elastic net logistic sis false seek column false spirit remove non bind screening test call examine focus close problem screen remove approach result intersection elliptical dual execute require memory seek strongly moderately significantly reduce lasso focused concentrate screen efficiently full moreover foundation apply regularization exposition within development feature exposition geometric developing emphasize architecture examine intersection half consume execute test space examine region test describe carefully perform study scheme screen line successfully size allow screening begin review basic tool especially interpretation screening detail region form region plus hyperplane sphere plus hyperplane show spherical iteratively refined basic test screen screen screen eq analysis minor throughout instance nonzero say feature target objective problem result without accounting define purpose screen use particular enhanced parameterization solution dual via appendix contain origin problem nonempty form half space fig feature eps eps dual feature unit two feature unit function seek set inequality contrast may unique call lie ss dictionary column primal solution satisfy primal partition feature select index reject vector solution screen virtue memory hence without screen normally metric select solve suitable denote solution reduce problem assume b inequality simple worth state active solution resp solve resp conceptually obviously solve screen unnecessary hold resp resp create partition dictionary logic iw region potentially implement construction partition compact convenient encode partition region rejected select denote reject special follow region reject feature particular form region bound region spherical arise half space define onto ball c know give screen simplify depend subspace span problem constrain increase test reject also consume execute subsection simple sphere test insight test bound closed block close expression tt r screen lasso also theorem parametric n st rr sphere well screening want computation performance hence answer outline dual feasible point radius bind fig solid require specification call example feasible homotopy algorithm solution path actual instance sphere sphere st spherical variety test quick place test exposition safe sphere assume improve default center spherical comment far test core center projection onto strong sphere notational simplicity r sr sphere false rule fraction advanced version rule rule assume solution form residual also rr radius yield false sis intend nevertheless translate sis sis dictionary marginal b sis criterion lasso decide default spherical sphere algebra sis sphere particular region test spherical illustrate brevity form hyperplane hyperplane sign hyperplane convention indicate simple geometry relationship ensure nonempty sphere hence require subset sphere half intersection nonempty lagrange primal solve screen consist half sphere area nm screen u lt dt cv uv theorem continuous check calculus test insight test situation two function ds boundary test reject extra rejection bar rejection discuss nonempty sphere half contain ensure proper test reject disk maximize select spherical simplify boundary hence hand feasible base intersection bounding sphere check angle fig safe specifically screen triple exploit employ provide entail specify sphere solve evaluate scale close safe favorable circumstance refine bound obtain dr radius smaller tight spherical summarize cd provide suitable half space tight bounding reduce select maximize current since proportional residual make one space rejection sphere examine examine intersection bound two half examine allow stand trade rejection efficiency h ic half space sphere form parameter ir ir I r half intersect nonempty sphere half solve yield correspond na q region q theorem correlation I assume provide half bound c seek half select maximize radius feature sphere simply alternatively solve yield must algebra inequality f side compare examine rate increasing impose test unit proof generalization result general sphere constraint bound special strong claim test complex alternative nonnegative composite similarly q call implement arise also include illustrate calculate product iterative execute product use complexity feature test test composite mathematically region test weak demonstrate intersection intersect compact trading ease despite limitation sphere st st construct spherical default spherical ball green region circle test implement sequentially key innovation base ellipsoid ellipsoid ellipsoid fashion intersection half ellipsoid volume except refinement rather tight spherical encode test dictionary refer shot screening test equivalent rejection primarily obtain bind alternative screening examine idea safe previously solve instance dual instance help form sequential value specific et performance sequential screening geometrically outperform uniform shot test rejection power propose homotopy solution solve homotopy potential use homotopy variant homotopy feature loop via solve sequentially instance help continue solution merely help sphere center fact bound open feedback adaptively sequence instance scheme robustness feedback diameter rule ensure stop decide feedback scheme dual regularization effectiveness screen v f ft ft nonnegative logical evaluate h I ht j j r lt k r ti k screening test describe require logical indicating reject algorithm fashion product hardware computation running dictionary unnormalize pass normalized recommend simplify set unnecessary point operation f select feasible half selection evaluation basic implementation keep
localization art inference widely quickly intractable truncation meta model latent accord function property interest result specific crucial underlying volume state visual general idea function property least connection idea neural furthermore context researcher inference efficiently implement model notably design suitable challenging require knowledge considerable implement particular major contribution paper propose approach completely specific adaptively simultaneously em emphasize underlie accurate importance preserve selection gps analytic shot base adaptive target efficient et meta inference generative gp number benchmark code code spike equal pick spike case linear regression posterior shape mixture regression expectation reveal map publish translation match straightforward implement truncation extension maximization em optimize step adjust maximize g become variable mean ps n exceed turn subset relevant truncate otherwise construct rank relevance try marginal probability correctly exist predict prediction ni probability state word relevant state sort relevant latent per mass latent space hand posterior summarize occur use compute posterior calculate wish generalize em graphical want free way datum flexible exact concept marginal use gp iteration em mean contain information receive simply one loo compute gps chapter namely relationship early posterior apply probabilistic hand sample inference approach sparse generative dictionary point multiplication g consider latent maximum center posterior yield g latent n h singleton bar repetition recover basis converge truth frequency furthermore flexible relationship reasonable apply cluster target n em previous rbf covariance visualization first gp selection cluster select latent show rbf selection assign rbf initialize randomly gp datum approach successfully easily find assignment quickly away identical converge result figure rbf assignment solution relationship input priori strongly role success likely potentially suggest flexible diverse column mask third mask predict location grind show model function figure verify graphical challenge computer cope object object location latent object rely invariance apply construction mainly location appear predict next reduce construct accord object limitation tune scan costly function predict possible component could maximally regression image example pixel initially seem model dimensionality perform fast original selection scan make distribute scene rough entirely idea go convert information object selection exclusive component vision background randomly appear image generate optimize em use truncate successfully object mask quantify learning location distance hand accurately location selection avoid explicitly whole selection gp enhance speed infer cpu core inference cpu core gpu parallelization propose achieve fast approximate relevance
proved prediction define minimal borel estimation another way hope sequence theorem individual predict borel suppose lie unbounded compact considerably argue future convex ergodic surely loss algorithm asymptotically extended version article sketch ingredient mainly state lipschitz perform well ergodic infimum borel borel make thank let variable defer extend article space complexity involve design priori consideration expert inefficient practical restrict expert cost loss convex lipschitz result long material tree omit main body suffice index range rr calculation eq proof substitute induction bin besides decomposition bin child conclude tree depth e one node leaf select leaf replace subtree root inner node therefore binary tree conclude simple forecaster hoeffding jensen substitution sum first entail continuous uniform lipschitz see borel exist since inequality conclude true subset equip every every proof ergodic begin martingale resort eq integrable entail right term infimum set borel measurable super martingale sigma jensen inequality implies surely state france online prediction ergodic process lipschitz bound ergodic time learner ask prediction next bound stationary ergodic process knowledge function consider limit strategy surely loss estimation try design review forecast vast majority like year neighbor estimation window expert adopt sequence main individual sequence observation ask step predict knowledge past past forecaster cart finite forecaster observe forecaster predict environment cm forecaster suffer clean layer advantage computational efficiency discuss remark square build lipschitz function occurrence part implement maintain associate deep simplicity set precisely lemma good x argument get sum conclude pt forecaster two time respect argument imply constant bad small estimate know individual sequence see expert respectively vanish average respect case bind instance loss upper em one consider bound absolute define calibrate online rectangle draw fill white fill r label rectangle rectangle rectangle rectangle maintain region space pair integer refer node convention child denote associate terminal thus step associate predict local observation observation receive leaf tree update two cutting give root first coordinate split go split third histogram near neighbor ergodic process unable deterministic neither neighbor manner histogram histogram divide strategy predict partition optimize number bin trick regret bind function computationally inefficient happen height distribute datum allocate space observation un yield improve need online substitute diameter splitting proof control locate index diameter associate em basically proof induction defer recall inner node splitting step h em total appendix article tree exactly nod depth node em bind dt th start exist inner therefore get thus conclude cumulative regret sum cumulative incur lemma loss incur satisfie jensen inequality conclude section later forecaster sequentially step ask forecast strategy form vanish increase regret htbp forecaster feed feed straightforward corollary integer combine basically form
excellent quantitative software obtain piece literature raw text source make list character number character one process string ease software preprocesse handle text text scalability classical text employ alphabet author also class standard subsequently implement represent believe document create choice text gram simply object respective feature specify gram construct functional gram frequency store dictionary enable text information character information must create add text frequency call elsewhere software package major lie automatic identification wherein author influence simplify problem approximately detect give capable representation expressive practice establish equal capable produce document correspond software seek subsequent accord stress allow specify two computational many relationship previously calculation relationship module allow upon object metric query operate search character document length document document document decide query part feasible within time formulate connection document obtain document document sample define size ease create document return one determine similar text similarity misclassification interest similar influence question put text rate use combination two text tendency text l l c misclassification classify text many true text close hyper text usually within hyper consistency hypothesis identify interesting result intuition training text differ variability author text due suggest vary greatly notion loose word word author write intractable effort analyze one gram coarse tool face classify broad grouping grouping categorization book remove direct variant classify classify preliminary hypothesis fact heavily rewrite variant sound explore fact identify direct direct variant variant negative broad analyze text gram magnitude small gram sized true style however believe quite due short test probabilistic uncertain viewpoint case preserve heavily method preliminary finding true origin text software perform demonstrate efficacy open field also recommendation several arise result develop hypothesis develop rely author analysis hope confirm efficacy text several progress exploitation yield upon train avoid knowledge support area classifier author highly area see author focus existence extent author influence many style establish piece text behind belong text claim may characterize text write author form paper instead propose text computer science result side computer insight figure classical represent subject apply quantitative brief historical seek address quantitative bc parent day early evidence suggest choose regard major play death interest play great work either lose corrupt play effort original copy continue much year play heavy ht claim capture attention induce gram representation idea sound word allow additional produce author pose capable heavily modify proximity death fourth death study reflect power closely write death european numerous particular consensus despite closely play leave mark history fall city great great span bc historical writing little actually life significance body record require historical author time rather art bring remain extent fact underlie attention believe interest ability fundamentally ultimately character capture text sound encourage importance build character character text letter alternative gram implement inverse frequency purpose identify document importance word receive high text corpus term corpus belong adopt probabilistic achieve representation make appropriate probabilistic able robust quantify useful author entire space definition discrete follow study mahalanobis metric standard average common text svm anomaly text author algorithm whether piece text write solve point offset hyper slack kernel length vector take lagrange multiplier use lagrange multiplier utilize rbf kernel degree small moreover rbf project infinite expressive separability experimentally derive determine text author author metric document computationally expensive match possible length character length document comparison expect time consume string query wish follow character demonstrate begin
tail satisfy tend word particular embedding mention recover embed multidimensional follow classical scaling scale estimate decomposition diag dd specifically tend pt hard see define formulated cone program solve practical scale instead show euclidean matrix take devise compute shall alternate algorithm refinement von alternate projection design intersection close sequence htbp xx q k projection intersection pos node pos pt node consider evaluate projection observe close alternate projection readily apply input evaluate specifically matrix th lead submatrix clear replace zeros efficacy numerical experiment motivated protein relationship counter activity mutation understand interest study derive infect follow five long sequence study alignment sequence substitution substitution fairly report analysis similarity convert dissimilarity distance apply left figure derive stress c noise classical medium shrinkage difference one run shrinkage multidimensional repeat bank symbol consist atom stress value report compare noise standard shrinkage medium shrinkage classical distance score atom distribution mean shrinkage distance reconstruct figure htbp proof equality follow together next trace trace n ensure imply minimum also exist follow basis word obviously contradict second argument point argument x follow f light imply statement pt nx nd r n characterize kolmogorov geometry conditionally semi orthonormal span form matrix v last clear recall last negative definite taking yield p fact imply na f r f nr n nr pt cm corollary definition recovering fill simple kernel apply shrinkage pairwise distance allow imply consistently number increase programming application embed euclidean multidimensional scaling regularization trace norm euclidean noisy arise context object amenable provide dissimilarity molecular standard measuring use al encoding insight metric respective employ method numerous dissimilarity successfully convert semi definite kernel play canonical multidimensional aim object dimensional euclidean object distance preserve al al chen others popularity extent embed reflect largely exploratory tool another interest reconstruct distance determination molecular nuclear short demonstrate chemical shift need three euclidean distance result observe translate location become euclidean distance occur goal graph euclidean embed embedding object domain molecular determination case realization score measurement call eq g stand convention identify suggest embedding obviously embed particular high refer embed dimension euclidean embed dimension molecular determination dimension similarly multidimensional realization q correspondence euclidean column regularize estimate al tradeoff goodness fit hereafter semi definite encourage low al many goal operate characteristic statistical define difficulty understand identifiable distance latter preserve translation estimate subsequently challenge notion kernel resolve associated kernel characterize amount shrinkage project pair onto subspace offer ability induce low characterization suggest use version thank structure et expense principal alternate explicit characterization establish discrepancy consistently approximate embedding rest section exploit duality matrix explicit characterization geometry cone efficient proof correspondence euclidean minimum despite identifiable resolve ambiguity minimum associated euclidean identify associated euclidean obviously q positive semi kernel matrix translate uniquely semi hereafter write embed result different unchanged reconstruction score avoid require embedding center embedding unique distance uniquely characterize among correspond trace embed pt restrict minimum trace map minimum trace viewpoint instead clear addition center kernel figure htbp column sep em rd rd similarly distance actually explicit distance whose diagonal projection soon onto
select user rate analogous establish item recommendation cluster hold constant completion accurately condition far hold completion eq imply recover rating hold show exist majority user name recommendation summarize necessary notion co co differently r rating similarity two u un u r clustered entity recover select user select preference vote e name high preference determine vote cluster co pair item majority vote recover step decide majority vote practice verify hold hold rich contain rich likely pick rich user select ii rich select cluster rich heuristic early combine three use similarity rich find directly user iii similarity user cluster identify user user compute similarity super super high work original detailed hybrid cluster recommendation hybrid user set select accord similarity user similarity set define super rating user eq select high similarity generality super user rating emphasize dataset phase transition user rich call ii denote select item modify select item accord define super majority item e select super loss generality super rating iv high similarity denote iv item rate mu un r voting reason rating item region rating region rating entry entry vote htb test netflix recommend user movie user word movie believe rate highly user movie rating movie receive comparison binary rating rating predict rating testing metric accuracy top movie model accuracy error correctly recover recommend continue recover preference view metric among metric restrict recommend whose rating give hide accuracy movie recommend error compare case user rate compare rate htb low significantly rate free un rating netflix noisy perform recommend error error error htb sparse rate error since summarize c c item item index cluster rate user user rich negative function constant positive rich rating rating rich user bound k theorem user cluster preference except rating item word otherwise separable condition change r equality hold equation calculate follow case rich p rating user preference rating case rich sparse rich pg define rich sparse sparse user cluster z chernoff consider user independently variable apply chernoff fact e p obtain e preference bernoulli apply bind cluster consider rating scenario rich scenario chernoff sufficiently large choose assume rich similarity z rich user pick user rich pick rich normalize similarity z normalize end rating eq abuse true preference item km km voting user q cluster occur voting produce preference theorem policy rating item cluster cluster preference agree entry except rating item verify user long change rating item cluster contradict hold argument prove correctly give cluster rating item abuse far rich otherwise define majority voting give chernoff verify hold co cluster collaborative filter rich user develop similarity entity item cluster recover rating matrix large netflix experiment majority user rate item basic remark furthermore give rating size correct thm lemma thm proposition wu collaborative filter website recover user user item cluster user item cluster algorithm noisy cluster recover algorithm well popularity friend netflix overall error recover importantly co cluster scenario recommender user rate majority noise add recommender recommend user example amazon netflix suggest like paper call netflix user large item discrete item rating view row item rating user goal recommender recommend may sparse mathematically pose unknown entry user preference filter recommender multiple predict user practically rating exploit solve item rating typically justify state write view user feature mathematical abstraction situation look movie actor assumption item group cluster provide item strong rating matrix column rank strong predictive power well emphasize actually verify justify study result assumption whose matrix agree observe know result rank problem popular heuristic minimization objective nuclear show condition nuclear reality operation fast minimization know portion mention know entry select repeat improve name know guarantee recently establish obtain assumption nuclear require purpose one view good date make apply rate one vote user cluster user cluster subset similarity user identify top user rate user vote optimal model think well behind rank simplify presence rich user item cluster applicable case user cluster cluster completeness believe dataset significantly item call presence sparse user well know rate movie rate movie good presence rich exploit completion contribution cluster rating information entity entity user comment apply item devise similarity dramatically improve find number rating user easily provable logarithmic cluster co achieve logarithmic mention verify rating even assumption clear contain user use exploit require user recommend combine similarity popularity friend netflix later propose low item fraction entry matrix denote user computation score computation computation total computation voting dominate regard since please section actual easily implement similarity majority rating notable algorithm rating matrix add score computationally rating basic find order preference user item assume level preference recommendation recover matrix cluster user u b separable user fractional separability preference preference cluster index assume receive preference hold fractional separability user receive preference observe assume inconsistent rating ask time generate illustrate pass channel create pass channel reveal true preference bias rating preference contain let user user information user item rich user
power weak setting get consideration power test outperform alternative test dense substantially test reflect discussion preferable capability maintain line case power test increase numerical result alternative strength gaussian long range strength nominal significance h alternative sparsity nominal significance distribute numerical test screen outperform test alternative spread structure maintain nominal good power recommend sample powerful application sample research large setting test generality nan hypothesis whereas let entry uniformly draw integer setting sparse magnitude entry diagonal pool weak follow use generate I v identically independent draw long dependence impose non generation mechanism study autoregressive ar identically j nj mt gamma z I impose structure simulation propose significance ex ex hc ex hc test hc nominal gaussian block covariance autoregressive process distribute model strength sparsity nominal block diagonal f level signal along test significance level strength nominal matrix result test summarize propose reasonably close relatively improve fails maintain dependency section hc maintain nominal small dependency maintain nominal significance reasonably control fdr identify set ns ns ns independently value procedure level type originally insight genetic also analyze illustrate focus patient group fusion employ approach scenario exclude set small number retain gene mf gene display biological insight development type clinical aim level sample test employ control significant matrix generate may error number set identify identify find gene propose testing discuss overlap within explain relatively large identify ccc ex mf screening stand investigate gene set test set activity disease category recognize development find connection type supplementary material list disease test screen fdr control association set may biological reaction test enforce structural assumption alternative extreme self whose extreme marginal statistic target guarantee correlation max moment unknown invertible covariance study may difficulty complex strong experiment perfectly correlate include power test develop screen principle pre ratio numerical superior sample screening yet maintain satisfactory power datum disease gene appeal preliminary classification setting feature broad kolmogorov discriminant powers supplementary supplementary material online technical proof result data pt section population testing procedure employ critical heavily rely structural therefore scope applicability enhance power test preliminary screening test gene practice screen gaussian test equality vector field particularly modern quantitative finance subject small furthermore measurement possess issue multivariate mean paper dimensional sample hypothesis control scientific low traditional extensively examine normality deviation asymptotic statistic statistic use test statistic aim norm detect relatively dense type preferable detecting signal medical problem anomaly testing derivation limit critical structural unknown impose guarantee w whose validity moment satisfy pc restrictive verify applicability asymptotic rely heavily structural expression pathway associate covariance real concern point extreme value maximum usually convergence although suitable concern genomic image extent least setting validity assumption approximation gaussian utilize however vector account dependence organize hypothesis test report assess datum supplementary material discuss preliminary screening propose throughout w nc nc n k diag diag n sample consist identically distribute observation set type form q refer statistic intuitively ns cv ns cv cv ns cv cv cv properly expect size cv alternative traditional calibration setting unknown nan motivate limit critical ns cv w ns critical cv cv ns w w cv f ns w ns w reject detail wide explore drive characterize closeness process test indeed test naturally sample statistic n nx k ns ns cv ns cv ns section follow quantile compute w two cv cv ns f test consistent require covariance develop test restriction grant wide applicability validity procedure entail estimator proposition include k supplementary material natural employ low coordinate uniformly r pn logarithmic allow depend heavy enforce wide scope extensive form long test operator still matrix empirical eigenvalue employ enforce valid much large value enhance power propose procedure feature expect irrelevant provide substantially power alternative coordinate construct statistic index exclude upon analogue ns k ns ns original statistic suitably select coincide probability power comparison test procedure maintain significance preliminary aim original statistic discussion advantage problem focus reduce h result ns k p f pd problem respectively asymptotic power thousand device efficiency monte mathematically ideal show nominal propose alternative mean marginally u unknown impose follow mild order th uniformly sub tail one assume establish validity test nominal significance asymptotically numerical base nominal sample small augmentation bind power test without screen eq agreement statistic region complement sample test screen test ns condition screen nominal either sample size distance test let hold sequence satisfy alternative asymptotic type propose base control significance asymptotically part iii imply two counterpart test covariance either condition nan part specify theorem analogously property pre screen identical omit supplementary material asymptotic sample screening test covariance assume nan h h simulation several evaluate performance screening ease exposition one problem test hereafter hc hereafter three hereafter higher
consider tail master tree worker compute master evaluate improve estimate report back master meanwhile master constant master assign execute path estimate worker proposal become likely worker unlikely likely pick worker posterior subsample depend track difference value master difference empirical approximate correlation high actual decision implementation require master worker master basically bayesian framework target master core evaluate core core core intel processor serial worker problem target eight gaussians model molecular real real functional experiment spherical distribution provably convergent adaptive study expect perform point evaluation phase early proposal reject approach target outcome inherently predict estimate uncertain incorrect divide batch subsample min max result mixture run sequence vary worker produce chain cumulative speedup iteration burn compute two chain table phase obtain speedup speedup achieve burn efficiency drop achieve logarithmic speedup sub logarithmic number round explain whole range initial worker burn proceed cumulative fall logarithmic speedup difference scheme burn overall speedup burn necessarily decrease monotonically initial region truly region speedup maintain system mm burn predict predictor burn burn predictor quickly correct predictor vary almost evaluate typically wrong eventually opposite incorrect speedup figure achievable speedup molecular gaussians evaluation accord convergence condition step drop logarithmic speedup improve switch resource leave present inherently serial often transition countable predictive use parallel require focus mh predictive predictor accept effective predictive estimate prediction respect evaluate correlated variance difference state evaluate higher prediction justify resource great execution I achievable sublinear core logarithmic achieve liu helpful award health lm google e large carlo exploit approximation chain parallel accelerate without subset available exactly equivalent serial initial burn serial core model modern learning appealing represent latent real rarely amenable require approximate form target may challenge new inferential target density examine exploit taylor process regression randomize development stationary low factor evaluate arrive attack parallelism difficult mcmc hasting inherently modal chain decrease achieve sometimes chain attack use execution approach sometimes attention past decade seem iteration stochastically reject randomness initial future chain think single tree immediate effective challenge correctness exactly serial treatment pseudo randomness serial treatment risk introduce scheme require core speedup improvement speedup acceptance rate reject improve speedup heavily extremely speedup something still scheduling use speedup relative adaptively adjust acceptance available fast approximation though learnable schedule increasingly far improve approximation schedule approximation insight error small evaluation evaluate large expensive current incremental show synthetic system parallelism speed serial unlike system achieve near speedup core speedup eventually logarithmic core hard chain evaluation incur evaluate determine acceptance move slice expansion focus case expensive sometimes many expensive arise easily decompose item e achievable cost aggregate partial accelerate source parallelism class sampling ensemble accelerate share chain parallel chain parallel implementation generalize elliptical slice mcmc algorithm use parallelism execution accelerate idea literature core evaluate slow result I scheme core tree respect core node maximize depth summarize idea static fix acceptance version context anneal acceptance level tree computing estimate mh alternatively identify perform combine source parallelism obtain mh core usually improve scheduling exact unlike stream mcmc incorporate estimate acceptance tree proceed use external number generality uniformly hypercube back operator hasting slice etc countable qr disjoint setup metropolis hasting tuple try mh delay mh create large variant slice sequence converge elaborate usual intend purpose separation generate highlight point randomness view separately generally case evaluate candidate burden mcmc observe pseudo function evaluate tree yet reach node eventually remainder straightforward point uniform proposal correspond highlight leave evaluation core speedup iteration proportional fall scale perfectly evaluate proportional making turn whether indicate
loss embedding sa word loss sa rd say th pair token call word frame word train model variant adaptive magnitude use accumulation past slide window hyper momentum precision small use setup though update thereby implement train dnn stack unit output evolution cosine show cosine train overfitte unsupervised system necessarily like unit phone appropriate hmms make phone linearly etc discrimination pairwise token token linguistic belong b category minimal shape keep center phone vary cosine embedding discrimination p share performance ignore remove layer vice discrimination base order identity fully dnn phone label task show correct understand nature encode detail unit within corpus compute take across phone phone category intuitively large encoding median kind phenomenon regard code layer phone unit layer doubly code third ie task reveal code inspection phone reveal doubly rather discrimination suggest layer example would inspection code red phone relatively localize localize network discrimination moderate information different type need complement phone representation european pour france de de paris de france dim et train task speech possible share discriminate second discriminate two theoretically linguistic plausible acquisition put language recognize discriminate construct representation propose neural word help acoustic phone embedding
estimate show excellent term demonstrate gene set counterpart rna seq form q library genomic region genomic henceforth gene many poisson datum choice marginal gamma poisson rate estimate model gamma conclusion allow see sample appendix group crucial component prior differential de distribution determine jeffreys seq primary trivial de great distribution also g gene close joint resort carlo cycle call distribution set conditional dispersion parameter metropolis hasting posterior fortunately choice integrate define tv far discard step step setup sample iteration ghz bit processor software require assess publish take rna seq count drop assume calculated coverage posterior derive sub coverage indicate accurate dispersion h red star posterior de show posterior absolute de run drop count parameter star gene heterogeneous gene know ground improve upon exist method detection underlie truth generate consist divide gene choose small gene gene de individual expression count expression set individual level consider simulated dataset compare package receiver operate versus threshold take approach express perform well de much rich thing distribution parameter gene addition output analysis next distribution set gene difficulty posterior indicator input group competitive differentially express gene problem gene count rarely determined attempt accounting difference gene cause uncertainty describe posterior henceforth ff testing significance follow fisher de gene set roc considerably ff g develop bayesian rna seq count proper inference uncertainty sample change mass detection analysis conceptually suffer bias frequentist demonstrate show detect thank several anonymous research center molecular joint integrate update measure distribution follow bernoulli distribution contain article closely stage variation article binomial also poisson type describe fairly typically conclusion gamma pp update heavily implement binomial possible resort update account fact follow burn phase propose iteration previous iteration compare section main indicating hence leave simulation setup model th unit effective ess ess sample amount mcmc autocorrelation measure effective binomial even run account high accuracy two ess min try negative binomial hold severe mix dispersion identify sequence differential expression sequence posterior inference efficiently appropriately uncertainty account differentially gene posterior excellent detect package interface rna become
refer cluster similarity shape hard specify past cluster able shape initialization lead decide specify initialization clustering remain challenge many well clustering exist mainly limitation firstly access feature low probable significantly clustering even ill clustering secondly cluster unified limitation ensemble crowd explore base validity measure crowd agreement clustering unsupervise manner treat aware triple similarity regard common neighbor reliability clustering linkage unify term gp comprehensive literature motivation provide propose term accumulation capable ill incorporate conduct several follow review work technique ensemble crowd source aware triple propose consensus term link cluster ensemble combination aggregation aim clustering base cluster member ensemble step clustering give step clustering consensus base clustering cluster different initialization via repeatedly via project generate base clustering I consensus important past consensus category co iii base partition maximize partition clustering median problem complete finding huge partition genetic cast find algorithm median median clustering embed perform convert take consensus clustering time base clustering evidence accumulation co association agglomerative sl co association li analyze co novel utilize normalize wang accumulation take category partition clustering hypergraph structure cluster partition similarity partition hypergraph meta cluster formulate clustering ensemble bipartite node exist cluster partition disjoint set node ensemble approach implicitly assume clustering contribute low clustering ill clustering weight base clustering validity et exploit validity index vi connectivity ci si index di assign weight ensemble extended weighting partition need vector suppose many framework li clusterings optimization process deal lin cluster library partition selection weight preserve partition ensemble weight accordance cluster cluster fuzzy ensemble aim partition hard specific different cluster partition intersect cover denote cluster refer base initialization th denote look provide partition final partition solution generally refer consensus ensemble base diversity dataset construct ill clustering may significantly poor clustering one clustering regard evaluate quality know truth develop wu validity inter cluster intra cluster al deviation connectivity quality instance cluster connectivity often neighbor utilize coefficient evaluate coefficient measure instance cluster whereas average instance inside applicable suppose formulation clustering utilize distribution ensemble crowd quality individual science crowd consideration collective opinion crowd expert ground truth truth suppose crowd clustering compare opinion crowd individual base clustering crowd agreement clustering denote base agreement crowd member compare crowd member crowd agreement crowd agreement basic quality collect opinion crowd hold mutual base suppose ensemble connect triple cluster reliability neighbor share common cardinality coefficient consideration sharing measure cluster base cluster intersect intersect common al utilize justify reliability view source cluster quality cluster quality base cluster consider correspond base propose aware connect triple measure regard reliability cluster accord greater big influence influence clustering I l coefficient cluster thus coefficient cluster common compute maximum cluster similarity define coefficient eq two consensus utilize deal ill describe accumulation link assign specific cluster cluster original affinity assess co occurrence ensemble clustering ensemble instance otherwise similarity pair occurrence evidence accumulation similarity matrix idea matrix reliability clustering assign quality co follow eq cluster definition thus clustering map utilize pair wise occurrence reliability member co construct agglomerative consensus clarity htb accord build agglomerative obtain consensus cluster clustering lack ability treat partitioning formulate bipartite compare ensemble base distinguished aspect firstly utilize crowd agreement see exploit among base quality clustering unsupervise secondly integrate similarity bipartite instance treat link common bipartite twice two instance link link node link regard link via section instance cluster use incorporate clustering exploit reliability via crowd estimation regard graph utilize node disjoint treat possibility disjoint lead come probably force link contain together clarity method initialization evaluate eq bipartite construct partition cluster cluster output eight baseline approach criterion parameter discuss section base section bit windows server intel processors gb ram experiment eight world uci repository use namely benchmark evaluate quality consensus information share clustering truth instance instance share scale link cluster influence big gp r suggest cm clustering base clustering construct base pool outli pool base clustering repeatedly parameter initialization cluster hierarchy clustering width trade paper cluster thus cluster pool baseline approach apply base clustering pool experiment choose base clustering clustering ensemble ensemble ensemble evaluate performance combination clustering propose gp partitioning whereas pair wise co agglomerative cl sl cluster sl drawing base pool ensemble ensemble repeatedly average cluster bad base cluster ensemble clustering ensemble fig consensus clustering clustering clustering dataset clustering win ensemble clustering totally clustering clustering run tie count win associated base consensus clustering number clustering cluster benchmark c gp cm cm true true gp cl sl cl sl sl cl sl cl best cl sl sl cl sl sl cl sl cm true gp cl sl cl sl cl sl
xy test derivation rotation axis value rotation align estimate dependence three degree dependence corpus relative language block combine expression comparative genomic qualitative phenotype illustrate relative b large dependence varied median pairwise sufficient although dependent great demonstrate due tight concentrated becomes predict powerful ccc e plot repeat draw dependent size powerful demonstrate world parallel european corpus documents es english da broadly language variable first statistical language dependence language group remove stop word apply tf feature representation kernel bandwidth per distance dependence find language cccc fr da language high group language ccc source da en fr en fr es es solid rate despite advance child survival year genetic depend location biological process tumor location order hypothesis treatment tumor obtain child newly paris organize block block category third contain segment characteristic variable median pairwise distance tumor expression empirical dependency finding literature support tumor location determine source dependent build hilbert schmidt strictly powerful order demonstrate test performance identify language determinant dependence wide dependency currently framework construct author fellowship european fp describe novel two determine variable second measure schmidt criterion measure powerful independent unbiased statistic favorable property quadratic time matching effectiveness real identify language corpus tumor dependent dependence statistical many context dependency research non string diverse covariance correlation covariance instance test ranking partitioning approach problem multiple dependency dependence present visual brain automate source language match one language respective learn basic statistical determine two variable dependence third hilbert schmidt independence covariance relative dependence take measure derive joint test utilize result hoeffding determine statistic variance statistical construct uncorrelated subsampling synthetic language identify relative compete determine whether statistically question statistically nonlinear relation however address influence closely detect factorization notion schmidt work independence associate covariance uniquely express determine respective like separable xy unbiased q ij sample write u tuple draw u eq uniquely reproduce hilbert source target xy xy r xy xy xy statistic asymptotic consistent computing simple attractive effort implement correlation result respective kernel associate uniquely reproduce space estimator u statistics respective r xy variable follow joint base test p xy conservative estimate quantile achieving counter integrate result first axis rotation conservative xy also measure implement variance consistent converge calibrate even compute form collect combine unbiased empirical bound everywhere negative sample give statistic prove population eq test relative two equal sized set denote drop pair first pair x determine space
time despite near dynamic computationally allow gp accelerate dramatically simulator posterior population individual conditionally give summary collection invariant simulator design sample accelerate first wave find good polynomial wave exploratory determine rough log set threshold truncation error gp take replicate diagnostic guide selection accuracy gps improve successive wave reflect decrease validation report predict region accuracy application wave model increase wave wave wave space figure approach gp approach require evaluation chain probably accelerate similar ridge difference two scale usually accelerate examine evolutionary biology specie divergence intractable demonstrate various methodology branching unobserve randomly consist acceptance cutoff make acceptance simulator estimate scheme due successful wave simulator replicate solid line figure could run abc differ red expectation plot various flat ever mass near dot posterior obtain local adjustment estimate substantially improve abc trend decrease gp accelerate knowledge impossible expensive perform simulator evaluation monte gp allow universal degree great many model supervision build diagnostic wave gp building poor poor model design gps raise gp number calculate produce time simulator gps implementation upon reduce carefully example number simulator replicate location detail carlo exchange enable bayesian distribution calculation process computational determinant accelerate continuity reduce require computation approximation evaluation population computation collection complex simulator model phenomena parameter return output simulator abc enable simulator require range primarily biological science nearly abc complex abc prior realization simulator accept tolerance return conversely tolerance resource key simulator simulator rarely simulator run simulator computationally abc require hour computation posterior moderate extensive explore mcmc sequential monte smc previous simulation know function guarantee learn gp accelerate abc method resource gps accelerate idea successively rule build log find simulator abc abc exact one intend replace step proportional acceptance kernel get uniform abc interpret believe represent relate simulator measurement observation simulator discrepancy mass believe simulator distribution approximate monte repeat begin build simulator output instead project e base summary statistic relate function estimate simulator evaluation indirect indirect auxiliary approach mapping accelerate abc majority smooth informative return mcmc simulator test likelihood vary positive instead estimate ia priori mean accurate inclusion prediction design take mat ern length scale likelihood variance variance estimate help avoid identifiability improper inverse prior integrate analytically maximum ensemble multivariate update train value simulator carefully order minimize point simulator need quasi discrepancy advantage extend number monte carlo smc support manner place region space translate prior priori design complex certain maximum order posterior accurately capable predict sequential iteratively region build threshold discount side leave gp likelihood still gp currently degree multipli trade accuracy use cause wave extend determine simulator gp draw additional new simulator new together build gp model prediction wave far whether fit
sampling condition number search convert correlation synthetic experimentally validate network distribution count topology strength quantify assess method pearson benchmark model american module come round round datum contain filter remove sequence total small zero require round histogram justification parametrize topology representative successfully association well whose architecture respective range control association relationship condition type size distinct instance generate maximum fidelity method range covariance selection selection refer design compositional baseline reference pearson neither robust estimating determine however interaction correlation include code case ability precision curve rank accord confidence pearson prediction rank infer star selection summarize condition sample topology dimension blue pearson random area precision recall vs different bar one sided test line trend certain perfect follow degree reduce infer scale free high maximum band scenario significantly pearson correlation outperform compositional limit recover portion test well synthetic hundred curve rank final date nonetheless high confidence interaction edge precision representative suggest outperform art network test scenario superior accurate connect component length abundance help cluster versus topology incorporate recover network regime prediction edge rank edge stability pearson synthetic edge true topology color correspond kl bar side degree define number edge figure show degree type scale characterize exponential degree interaction cluster relatively reflected measure predict size topological centrality degree centrality centrality short path unique four method agree core network term salient feature unity figure comprise edge distinct negative association dense correlation negative method comparison total unique edge scale edge prediction respectively network eight respectively predict edge property distinct centrality scheme interaction observation explain edge indirect due alone indirect edge explanation hamming suggest american attribute instead band free cluster type component infer interaction specie understand ever number sequence study strong environment diverse study alone develop interact specie environment throughput compositional interaction construct challenge estimation inference interaction dataset generator underlie engine know sparse covariance robust transformation context abundance realistic look dataset benchmark inference two benchmark demonstrate addition sample agreement band number also demonstrate direct correlation community rough inference assumption underlie experimental design statistical synthetic datum term water american inference network reveal observation network appear composite network evidence scale band like important advantage neighborhood ability knowledge scientific principled manner grouping relationship improve verify specie interaction context covariance neighborhood similarly network agreement empirical confirm free structure network globally scheme scale network include although interaction covariance key address question example design gene area reference therein interact sequence could incorporation understand evolve association structure development association develop might perturbation art inference rigorous addition flexible principled mathematical incorporate association improve prediction serve sophisticated modeling hypothesis relevance environmental acknowledgment thank alm discussion manuscript presentation work cm abstract rna environmental sequencing population diverse environmental identification require tool challenge unit dataset compositional count detection relationship spurious secondly sequence hundred hundred association additional sequencing address combine develop compositional inference assume reconstruct synthetic benchmark validate tool generate datum state synthetic scenario predict american project sequence interaction routine component experimental biology collection american project bring recent research biology aim objective community unobserved community model appear lead concept steady study environmental covariate relate context observation disease status new connection infection diversity goal interaction detect typically association measure measure sequence result read common operational quantify proxy population environment population successful pearson correlation sufficiently survey spurious count total community term classical fail method compositional correct permutation design compositional bias yet correlation association correlation arise connect expand point pose hundred whereas hundred assumption development great sequencing challenge increasingly dependent simulate influence via datum draw negative accord pearson correlation green negative red thresholde relevance edge threshold notably importantly underlie inverse sample symmetric approximately correspond non entry color identify true correlation induce strong e correlation node although metric inverse depend help potentially introduce generation realistic comprise compositional apply transform unlike seek concept independence informally e abundance graphical conditionally independent relationship explain alternate avoid detection correlate ensure detail undirected link node represent gain considerable popularity network biology recently biology art synthetic generate realistic synthetic network diverse topology date verify gold exist synthetic reflect actual strongly impact network recovery performance scalable engine I feature realistic benchmark network apply real iii invertible agreement theory underlying method american project likely membership ii topology section present statistical current available material synthetic datum module summarize key introduce describe generate realistic dataset h consist synthetic datum count topology statistical suggest fit marginal generate data proceed synthetic count pre ratio compositional select graphical mb glasso assume sparse correct subsampling dataset low variability select edge output invertible input discussion typical sequence w nm j raw j pm composition unconstraine simplex lie simplex restriction simplex application covariance covariance exhibit closure advance achieve simplex ratio study compositional statistical term statistically equivalent transformation unit compositional gx mean composition vector transform component covariance transform absolute j j j dimensional sample serve serve basis abundance add pseudo avoid association abundance dataset network association undirecte represent association formal unknown encode variable undirected field family entry term precision adjacency factorization conditionally conversely entry conditionally inverse thereby association fundamentally distinct estimate correlation though highlight biological b two provable dimensionality neighborhood solve maximum reconstruct optimization key tractable provide reasonably association comprise inference scheme selection introduce mb inference independence denote column follow tuning aim necessary local edge entry view choice consistent neighborhood present selection read element scalar tuning expression distribution graphical encourage sparsity diagonal mb originally normality distributional estimator large problem include inference count nonparametric approach additive model use association transform inverse pd matrix ensure entry estimate infer value sign diagonal inverse wise covariance approach advantage obtain associate subsequent edge discriminant parameter control final rather empty criterion criterion scheme ability star repeatedly subsample incidence retain overall stability star empty edge accord inverse practical advantage theoretical available characterize asymptotic infer topology neighborhood precision underlie recovery interaction network highly nod evenly neighborhood addition theoretical implementation infer practice scale grow advance increase engine rely star abundance absolute comparative scheme remain biology reverse considerably advanced understanding applicability gold experimental realistic context gold realistic synthetic generator outline generation fit count specify e topology combine normal generate user topology approximate structure univariate function normal correlation cdf
boundary element equation dense recursively divide base structure low matrix depend tree instead article restrict efficiently arise fast applicable characteristic distant though dense employ store vector summation oppose factorization represent interaction several arise green compute expansion chebyshev interpolation entry low chebyshev available rank find algorithm solution software package relate fast article black box online matlab com reader reader box note exploit sparsity operator run box compatible wide eq chart demonstrate scalability usually highly obtain chebyshev improved chebyshev product one illustrate compare kf enkf assimilation implement continuously track co result model pilot reservoir simulator pilot co depth core assume predict co pressure wave build flow use baseline wave co vary co velocity delay conduct survey every apart acquisition geometry integrate propagate reality ray assume line connect source receiver limit assume due co express vary location invariant represent cell co induce velocity delay measurement background step perturbation observation delay contaminate ratio snr eq noise realization equation major source walk dynamic set assimilation monte enkf parameterization noise structure uncorrelated kf enkf assimilation enkf give reliable quantification spectrum eigenvalue step kf enkf eigenvalue eigenvector insufficient rest tail yet kf effective covariance assimilation rank number need explain total enkf suffer insufficient ensemble information embed kf enkf kf expense cost assimilation method computation pc ghz kf hour minute tool fast kf storing overall operate comprise offline online part offline measure form compute monitoring normally form associate cross cost storage enkf linearly enkf produce gray propagation walk gray kf enkf kf enkf gray risk ensemble linear forecast forecast kalman filter quasi assimilation datum rapid adopt random forecast gaussian error generalize covariance algebra propagate comprehensive kf enkf select chebyshev interpolation kf within less computational effort enkf realization ensemble carlo spurious estimate cross variance drive quasi monitor acquisition network forecast advantage full forward rely less dynamical walk kalman solution enkf well enkf linear pde forecasting rely increasingly collect desirable kalman continuously quality monitoring material work support national technology award advanced inversion modeling set national mathematical award dr berkeley national physics global energy stanford quasi datum assimilation kalman quasi assimilation continuously movement challenge tracking collect advantage provide continuous high flow aid analyze impose monitoring assimilation computational requirement paper filter kf dramatically reduce storage kf produce practically take dynamical tailor assimilation problem apply co enkf numerical enkf demonstrate usefulness walk monitor progress field operation co track enhanced controlling monitoring provide sample monitoring use large temporal vast monitoring collect quasi continuously continuously temporal resolution co important exploit temporal result challenge analyze arrive high resolution flow time monitor use space process algorithm kalman kalman kf powerful processing arrive continuously improve dynamical kf give good solution kf represent quantification characterize maximum posteriori extreme quantification crucial inform decision data kf kalman filter size meet computational requirement high operate matrix computational limit kf coarse heterogeneity kf kf still reduce reduction particular work type kf project reduce dimension resolve approximate rank reduce singular seek matrix correction direction ensemble kalman filter enkf find low construct root matrix method gain popularity efficiency storing covariance reduce kf dramatically approximation carlo method approximation slowly size enkf statistical enkf computationally sample size ensemble although like version enkf reduce enkf fast fourier fast method accurate alternative reduce structure associate allow reduce rank near inversion krige covariance generalize matrix solve approach spaced grid realistic hierarchical incorporate develop computationally filter present employ walk widely medical dynamic monitor rapid accelerate kalman computationally enable cost kf accurately reproduce minimum mean kf processing cpu minute feasible implementation approximation filter tendency filter error covariance rest arrange follow kalman quasi assimilation introduce representation physical enkf respectively subsection propagation follow matrix subsection synthetic demonstrate kf enkf monitoring examine light govern value measurement state assimilation recover observation system govern behavior measurement relate matrix vector evolution simplification practical rapid assume subsequent equation forecast adopt specify state jointly kalman filter compatible evolution kalman implement ii obtain step measurement refine state measurement kf walk state give time operation gray gain nm k nm require step assimilation number major kf kalman kf prohibitive kalman filter originally carlo kf enkf
prior enter way explore metropolis suitable proposal yield hasting transition translation metropolis adjust langevin evaluation require differential pde draw density likelihood computational dominate forward replace typical scheme evaluation example metropolis hasting advance allow refine evaluation add grow outline spirit previous sketch previous effort evolution suggest connection argue lk acceptance repeat else previous rather approximation construct local nearby subset evaluation effort polynomial expansion advance refinement allow infinite number proceed depict set evolve become region approximation ever small increasingly change allow asymptotically posterior convergent metropolis kernel behave obviously sufficiently approximation local center might radius generally increase ball early sample sparse approximation relatively ball imply become approximation refinement refine approximation make useful explain refinement cross indicator explain refine approximation evaluate change substitute linear construct draw ball contain may square operator approximation empty omit option indicator assume gradient function lipschitz constant hessian constant parameter separate fill lie near quadratic long show compute geometric poor geometry consider refinement need rigorous bound approximation reasonable converge fall square unless geometry sample carefully design sample inner unity ensure rank subsequently decrease put less emphasis distant sample derivative process subroutine produce sample represent appendix numerical approach multiple output construct separate fortunately construct scale select separate portion section discuss need explain perform candidate choose symmetric behave identically treat avoid coupling whether move refinement refine fit naturally essential establish criterion cross error intend proposal true forward error leave strategy computing sensitivity produce whenever indicator exceed threshold indicator inside probability full variation leave computed acceptance reverse acceptance capture forward computable variety interpretable make user exercise either forward feasible mh criterion purpose ensure quick run refine criterion efficient may asymptotically position combination refinement increase cross choose practice require perform nearby computing evaluation radius improve geometry sample ensure maintain quality location ill ball point obvious simply cluster induce problem type design near new maximizer optimization initialize constraint ensure ball thus operator find separate quality reveal inner minimization simplify outside optimization likely global meaning build although produce set limitation might problematic easily add process surrogate natural application gaussian section process polynomial describe adaptation indicator compute jj compute distribution produce regression gps use exponential kernel hyperparameter endow gamma find correlation endow likelihood maximize construct gp neighbor use mostly include combination pure unconstraine later quadratic handle separate predictive summarize algorithm simple else start often find proceed chain construct state draw indicator accept reject compute repeat else target posterior asymptotically algorithm approximation via interpolation point replace fix modification essentially direction seem affect representative choice substantially concept require check one drift true hold empirically refinement sensible point check prove approximation notation throughout new let target write rx rx rx px lx lx collection time time distance satisfy proposal x p x briefly interpolation stability correspond weak widely condition geometrically ergodic density assumption lyapunov distribution inequality vx define useful hasting markov chain r yx important piece markovian process state say generally couple stochastic aa z evolve markovian process algorithm denote identity tx tx observation allow fairly naturally stochastic kernel note define tx extend tx tx fy despite hasting appendix suppose compact away envelope mention degree use proof difficulty far happen algorithm globally establish mcmc remain perform section describe three local posterior dramatically compute absence analytical estimate compute chain compose posterior produce model thorough standard sampler focus evaluation algorithm representative use performance type approximation infer parameter ode genetic circuit field pde conclude illustrate figure perform course must walk several nonzero approximation discard combine initialize point run contain discard burn evolution chain measure consist divide frobenius accuracy comparison figure mcmc cost show model mcmc baseline show reflect reference variance length small acceptance error respectively low high value increase reduce surprising chain eventually improvement accuracy chain consider predict show efficacy chain validation refinement accuracy reduce seem relatively insensitive criterion jointly decay decay measure theoretically sound increase improve robustness summarize accuracy impact fast quickly chain validation value decay indicator denote circle exceed compare figure observe change refinement setting percentage balance apparent interestingly refinement approximation become progress htb plot truncate clarity give previous approximation ode compact wish switch inference differential algebraic switch six observation endowed hypercube gaussian broadly highly inform largely gaussian adapt proposal use metropolis size algorithm mcmc figure reduce cost quadratic approximation proposal fall outside prior without run htb involve diffusion pde leave detail pde suffice purpose pde solve resolution coefficient define endow take field pde posterior shift significantly pde forward gaussian strategy parameterization relatively adaptive local previous switch accuracy indistinguishable demonstrate true model approximation regressor suggest regularity domain approximation htb dramatically reduce yet address time although store perform near search might challenge find problematic storing trivial modern finding set neither implement outperform run measure genetic switch pde hour spend qr surrogate spend near run run competitive expensive forward fix offset evaluation run demonstrate metric class construct local surrogate expensive introduce approximation metropolis hasting refine approximation result markov employ variation employ process regressor thus span use class significant forward pde problem local regularity log small although quantitative manner capture great decay almost evaluation grow process show bias decay quickly discrepancy primarily cross evaluate primarily advantage construction remain significant room local approximation exploit surrogate share forward surrogate correction use metropolis langevin mala hybrid hmc availability approximation finally far mcmc acknowledge scientific discovery advanced office science advanced award support thank additional scheme restriction quadratic regressor entry scale original local regression sample ball interest correspond scale magnitude define rescale rescale compute sample column desire numerically stable qr factorization may remove qr qr tx x lyapunov constant non vx satisfied tx kx vx vx unconditional x vx markov start inequality inequality vx combine vx vx conclude adaptive correspond condition requirement hold follow short fix compact support p cover compact within collection clearly clear exist base combine occur set nx lx remainder claim put together proof fix satisfie exist let x lemma draw bernoulli variable success r success sequence couple write I choose complete show theorem drift time away trivially event let complete lemma place long whenever set sharp care take mode recall briefly almost surely either drift chain exist p px p constant lagrange polynomial associate ix p I definition also denote acceptance kernel target show drift infinity tx satisfie proposal satisfie sense proof tx v dy dy x dy z since assumption claim x lemma eventually satisfied also large ignore ray also ray one element cover either great random add cover particular success affect take borel I part almost sure infinitely contribute surely put return infinitely compact recurrent satisfy lemma drift compact recurrent satisfie finish proof analogously follow item assumption claim follow envelope lemma envelope sentence compact lemma origin ready proof satisfied function lemma great surely inequality constant depend assumption show w decay uniform sample denote except initial choice fail validation event pass walk sample point converse compact proposal ergodicity whenever decay rate example justified certainly find
misclassification associate misclassifie bound maximal point svm likely classified bound identify act model bag influence influence leverage contamination original define base hyperparameter ensemble hyperparameter stability ensemble hyperparameter determine estimate bag account contamination hyperparameter resample potential contamination instance design contamination separately increase model experiment tune also obtain misclassification class base enable imbalance bagging hyperparameter parameter learn classifier unlabele label ensemble bootstrap increase bag provide intuitive mechanic semi art benchmark comprise label fully false positive improvement exist unlabeled instance inaccurate problematic negative acquire amount iii label include web page bioinformatics variant gene virtual screening drug share common final fundamentally instance precision since target biological recall anomaly go refer instance give unlabeled respectively contamination contamination true instance g contamination positive contaminate unlabeled difficult supervise approach estimate contamination distinguishing proxy distinguish positive negative assumption learn violate application various outlier performance contamination ensemble machine several split conceptual category label distinguish cluster weight individual change penalty misclassification training bag rt try negative inferential classifier supervise step convergence mc approach relate bagging svm bag supervise penalty unlabele positive penalize misclassification emphasize optimization kernel slack variable misclassification tackle technique svms bagging base training bootstrap separately contaminate bag svm additionally relative positive unlabeled bag svm significant bagging table illustrate resample contaminate subsequently bag advantageous learn approach computed decision approach potentially contaminate resample contaminate contamination use increase increase original contamination half instance due contamination increase converge contamination equal contamination decrease empirically repeat measurement expect equal contamination decrease introduce bag strong essential grow diverse ensemble bag ensemble model voting decision view bag interpret approximated instability success bag lead tree bag bagging explain instability relate intrinsic variability predictor influential bag explain influential effect resample contaminated insight mechanic bagging
disagreement majority label control minimize easy equal high target indeed risk could imply performance self optimize note connection da minimize keep every fold learn vote label fold risk correspond mean fold k ib semi svm learn domain self da algorithm divergence pac da labeling use nn nn adaptation da target weight real base minimize vote control disagreement da transfer label unlabele self justify self advantage label da generally consequence necessity lead instance description learn learn vote thus adapt corpus ii da direction self accurate distance imply closeness lastly usefulness g relevant machine generalization bound cm test datum key information specific weighted vote present justified target risk vote perturb region marginal appear study influence labeling deduce hyperparameter promise result bayes expansion supervise machine transfer survey strong learning task spam filtering adapt one receive different scenario call domain adaptation da arise model label da latter situation address da us deal covariate self labeling learn classifier auto label intuition measure easy divergence discrepancy disagreement classifier enhance disagreement control divergence much differ da scenario perturb pay attention designing label close special da majority classifier call disagreement risk vote advantage theoretical derive restrict classifier framework vote pac bayesian scenario mind supervise da labeling every self labeling help self source marginal close unlabele region self deduce original name well nearest label pac da theoretical basis synthetic review pac usual introduce bound majority vote set value call input space output stand ss sm sample value belief aim majority vote empirical risk real sometimes classifier domain empirical ss usual pac bind risk predict first drawing correspond risk accord eq pac bayesian deterministic pac set different respective marginal sp sm td majority low recall risk real every disagreement target reflect usual favorable da divergence small achieve promise disagreement usefulness pac bayes da remain vote regard state tackle drawback novel majority vote equation indeed source majority elegant non real extend da classical relation loose tackle tight relation notion define margin b positive convention h sp sx know express distribution marginal p numerator correspond moment b risk denominator moment margin disagreement relate disagreement counterpart majority justify elegant principle program learn weight measure minimize denominator disagreement fix classifier regularization performance j view suggest domain disagreement equation relate target risk gibbs risk deviation source disagreement tend majority vote da vote algorithm label rewrite label come tb b tb recognize true divergence true since label labeling labeling tight still valid da point one bind target tackle define labeling label close thus investigate rise follow label justified b ed st labeling pair marginal resp target source counterpart match self goal use maximum matching step unlabele thank belong affect true else construct actually region coincide algorithm indicate st good good da
c ccc train gold element target map vs correct map increase map correction similar test language maximum map adjust decrease trend brevity setup shoot least across space affect strong simple traditional query adjust availability employ incorporate different learn objective show correction setup simplicity future plan extent different kind representation objective affect work pose understand mapping start grant extract linguistic label empirically propose proximity map lead improvement realistic cross labeling image retrieval extensive co occurrence word corpus learn manner paradigm manual annotation bottleneck domain image signal must associate available mapping domain apply induced entity originally test decode function fmri activation vector representation training read mind shot label outside vision exploit translate word learn promising technique manual supervision encourage term return correct top less shot chance specific qualitatively map contain item universal map intrinsic reducing shot relevance classification setup severe map elsewhere leave affect problem setup get adjust bring post process attractive least square train mapping stand retrieve shot map retrieve often whole phrase near query happen take account invert convert score retrieve base score empirically rank keep shot domain translation pair language label contain training test ts ny ts tr vx stand regression source straightforward least label estimate vector source retrieve return accord similarity map common use cosine query precisely position similarity integer cosine q stand near brevity item search counting list omit subscript occur space near query return know similarity converge dimensionality cosine linguistic converge know increase qualitatively observed tendency become bad target map different source space compare map element vector english word english translation pair english item consider simply instead return solution target high alternative present formulation many tie want word rank cosine break tie translate cosine map query rank follow equation implement cosine break test method work corpora seed vector co occurrence context rely neural representation shot induce seed irrespective report seed word set evaluation word word representation try word representation word sub estimate draw token wikipedia tokens en bins frequency sort literature generally word medium frequency useful also use translation dictionary english test query entire report translation accuracy english occur one translation entire predict test standard method well map test instance use english improvement use solely map simply need supervision well improvement standard decrease add low measure effect add actually whereas affect improvement important medium although number frequency bin observe similarly many gold c ccc size regularization nn ccc train nn c cosine test vector left tend word realistic low might point cosine mean l nn translation correction case wrong
I impose nuclear norm perform svd follow project singular onto ball norm computed constraint put sparfa guarantee sparfa tag question set tag tag question define complete partially tag concept learner large tag tag pls perform compare sparfa unobserve real efficacy average monte carlo sparfa predict unobserved algorithm five course electrical consist answer answer question introduction probability answer answer university consist learner answer question consist answer collect see university dataset value model refer dataset dataset compare sparfa predict unobserved learner computationally efficient sparfa convex cf sparfa tuning run conduct carlo trial sparfa sparfa require min intel core processor reduce sparfa nuclear experiment collect school conduct amazon dataset value answer question response fully tag manually assign question sparfa dataset simplify geometry average bad profile learner tag simplify geometry tag learner tag percent leverage tag profile pls provide feedback learner strength resource pls tag recommend tag tag moreover pls average entire class plan sparfa incomplete learner question sparfa performance learner response significantly reduce b k q c x x recently analysis sparfa learn ordinal e correct learner response underlie term concept use learner concept profile association question difficulty sparfa powerful include difficult optimization la build algorithm automatically use sparfa unobserved method theory sparfa computationally content completion convex advance system learner provide learner automate education experience large sparfa introduce la learner stand analysis resource e video sparfa ordinal learner course sparfa learner response govern sparfa joint ii learner profile solely response provide analysis enable pls automate organization analyze course suffer lack principled concept reason affect learner determine interpretability concept pls learner expert manually approach intensive massive online sparfa utilize cross extensive sparfa run value sparfa automatically select analyze response course assessment sparfa recent account ordinal sparfa compare conventional sparfa sparfa learner perform la variety real factor analyze response factor value response achieve superior predict learner priori collaborative sparfa parameter identify extensive scale require author learner examine decay automate mc recover value extensively recently mc rank binary ordinal learner scenario typically binary ordinal next investigate applicability mc sparfa aim unknown ordinal learner answer let underlie rank denote model logistic unit contain response represent quantization boundary quantization boundary bin boundary unknown directly detail equivalently logit
synthetic set benefit stepsize rule big strongly perform dual formulated finding computation uk lead convergence problem solve dual detail equal accord admissible stepsize number number nonzero element e tb access ghz processor core support hardware choose coordinate iteration scale convergence iteration fast per average time expensive q case coordinate minimize strongly optimal convergence counter theoretical stepsize uk synthetic distribute regularize loss nonsmooth regularizer coordinate technical exist matrix regularizer relevant increasingly modern describing encode ram hand read among manner method exist strongly type arise frequently strongly encode encode box e strongly convex propose squared convergence method accelerate efficiently computable cd big computer partition partition computer parameter union sequence iterate accelerate output iterate store update way store computer computer pick compute scalar parallel use capability z k k z scalar step proximal backward fista compute one indeed computation processor element step deterministic scalar step reduce algorithm sum execution want directly influence u diag propose generic coordinate descent make extend accelerate coordinate descent complexity iterate satisfy lx k lx suggest small satisfying propose new stepsize matrix row partition empty characterization convenience denote compose schwarz equality reach feasible disjoint prove differentiable section fix df identity view apply previously quantity define although definition follow stepsize satisfied computation however operation power twice nonzero element quite pass instead easily computable run immediately much still satisfied computable improve inequality need check trivially plug side deduce submatrix corresponding term negligible vanishe increase complexity computable appear hence partition near know run enough bind partition apply reasoning need approximated computable tight upper view easily stepsize eq q ease reference call see pass compute discussion cm yes yes next small e quantity small let prove let lemma small h j tu generalise element vector idea sum think preliminary accelerated also twice expensive problem formulate correspond convergence influence compare different stepsize influence derive replace lem lem replace note problem big follow solve order benefit new dual htp duality htp evolution epoch epoch
plan two concept regression kernel give source space reduce mapping weight flexible radial rbf knn kernel audio audio video common cca form mapping v v nh correspond eigenvalue done retrieve video audio versa representation h update layer update minimize learn merge originally learn merged compute layer update embed method evaluate another perform read video label range embedding rbf report oracle train set simulate soft globally like learn transfer neural multimodal learn feature representation plan map maintain get regularization room audio recognition representation video challenge compactly represent modality address create short report compactly aggregate information variable plan reconstruct generate motion audio input audio improve would acknowledge contribution valuable constructive suggestion development like thank student course university fall deep framework transfer neural network fine tune initial semantic learn analogy preserve abstract learn semantic modality modality modality audio task multimodal progress report dataset multimodal deep modality pattern modality main focus modality multimodal multimodal extremely resource thus imbalance datum modality example label readily read video learn imbalance modality learn transfer imbalance selective transformation transfer well task moderately transfer modality intractable due drastically semantic transfer modality fully exploit neural specifically layer leverage embedding multimodal modality audio letter map read albeit space level knowledge within tractable transfer transfer speech read flexible modality parallel corpus formal report leverage improve multimodal dataset show language addition improve machine lda text svms often multimodal multimodal modality recently deep neural learn multiple modality single modality share representation audio video final multimodal infer method modality work availability corpus modality therefore address allow modality transfer leverage task target resource notable study dataset exhibit completely transfer apply transfer semantic level neural knowledge entail network top counterpart comprehensive topic modality learn make abstract audio video audio tune audio perform reconstruct new audio reading multimodal illustrate modality truth label input audio video lie concept build output h x h v layer net abstract representation fine tune input audio lastly reconstruct previously perform audio reading letter contain region represent audio contiguous example contiguous video frame vector
link linear demonstrate variety text superiority patch filter external database non mean goal clean decade problem remain fundamental one variety highly regard date noisy find reference apply unknown q example local weighted patch denoise denoise performance reference patch noisy patch former know practically external less expensive internal training patch image orientation search patch image plausible often fail patch patch patch regard internal work rare patch extent external show theoretical large developed sampling large external denoise noisy helpful database database database database contain external database obtain practical face camera scenario image ct concept external database tailor denoise bridge address suppose algorithm utilize emphasize early etc less reference patch may look extend internal external database patch likewise treat external video feed image problem force straight forward method solve utilize database theoretical denoise formulate drawback easy yet challenging database external image exist iii propose group minimization fix basis mse spectral operation improvement strategy improvement optimization patch present thresholding method improve detailed proof proof paper rest iv conclude remark vi foundation propose linear brief review highlight limitation denoise patch linear minimum square mse truth assume symmetric basis diagonal contain become orthonormal note let give patch follow column see give truth span wiener shrinkage achievable oracle filter question surrogate answer case achieve minimum minimum eq identify part problem choose dct basis pca however basis fully understand depend ground truth stack dct apply wiener pass estimate unclear relationship determine denoise discuss patch patch patch nn drawback patch may truly denoise task discuss improve without loss assume return first distance project zero energy denoise norm row zero similarly illustration show go back norm ensure orthonormal interestingly surprisingly classical component pca summarize observation practice possible user define sparsity minimization perhaps underlie enforce tensor bm slice see dimensional transform bm dct haar default set sufficiently similar dct location dct coefficient flat final haar transform sparsity stationarity dct axis essence sparsity true utilize singular recently stack dimensional array seek orthonormal array denote mode phenomenon tend bm patch mask adaptive subscript add consequently pca sa emphasis pca bm component arrive group notice play basis word noisy share dictionary patch train basis latter expensive adaptive cccc b patch patch formulate patch emphasis overall insight penalty claim equivalent important building formulation systematic nn equivalence nn understand closed line segment must either check versa correspondingly clearly claim know nn possible choice penalize reference problematic due patch similar patch similarity share concept order short path try patch regularize matrix way relax geometrically use patch one notational simplicity diagonal shape adaptive bm bm denoising pre learn apply project filter image desire component framework role delta bm assume measure uncertainty result become sensitive suggest additionally incorporate covariance provide estimate denoise pca assumption perturbation pca implicitly dirac denoise use generic database usage concept generic global cover concentrate mean local reference prior propose sample thorough justification denoise datum see subsection compute truncation mse variance trade reformulate penalize norm introduce penalty define section optimal solution ideal zero desire require similar component demonstrate effectiveness propose example refine result new solution consistently th ccccc f h db db db stand denoise stand denoise database bm bm pca four denoise method modify search external database iterate specific internal bm default window include influence window denoise denoise external comparison window identical external bm good patch method compute bm pca first threshold weight function patch reference patch default patch fair mention train database train database patch dictionary implementation external denote image denoise internal corresponding new database external mean deviation patch slide quality structural consider denoise purpose identical text handwritten signature bar code capture add external different font show denoise noise yield db benchmark insufficient database denoise method size conduct window bm patch redundancy exploited extend external cc window external test image level pca variety make bad level increase informative average yield use generic usefulness database learn training build train contrast propose fully th noisy db db database affect offer insight clean like database compute patch indicate noise level decrease linearly database distance moreover slow significant low condition consider camera captured suppose properly corrupt goal demonstrate help clean view noisy could simulate computer vision consist view add view visually compete area indicate method remove noise fine consistently db superior confirm database denoise denoise denoise capture corrupted facilitate recognition tracking denoise use simulate randomly image face database denoise row one face still generate plot average b database image denoise ccccc runtime sec sec implementation matlab runtime database image similar code intel cpu table runtime indeed significantly external runtime magnitude pca patch svd speed patch discuss particular perturbation answer font text view
factor ard percentage denote take argument frequency correspond energy scale control rule physical ii natural iii estimator fraction ard mask narrow predict therefore test examine detail prediction ml state analytical characterize low paper hamiltonian spin degeneracy embed localize spin continuous site interaction term positive wave choose width far choose center notation site descriptor machine predict function tr well prediction lead exact particular include approximate one numerically exact carlo ed weight size body wave note retain axis axis fourier detail ed interaction vary half bandwidth finally varied interval also include ed temperature lead maximal solution randomly test divide ard representation fraction green frequency green polynomial hence learn fraction write accurate coefficient available ed calculate code give q evaluating transform real smooth learn learn derivative evaluation transform function polynomial polynomial polynomial offer act statistical come direct well green time give fourier transform calculation large noise calculate devise fast chebyshev polynomial interpolation chebyshev expansions chebyshev free toolbox odd rapidly around different database sign security use easily replace odd sign odd however difficult thus use odd great representation either obtain small fine learn reconstruct therefore coefficient present full infinite aim band real frequency choose frequency ed rely temperature axis take unit twice combination set predict ml fraction since mention contribute use calculate ard show dot ard example dot dot result learn green frequency particle though see low predict good frequency qualitatively fairly predict green fraction able frequency physics blue dot dash circle learn length line learn prediction attain equivalent physical act prediction predict learn fraction inaccurate even perfectly relative visual clearly small number systematically correct see b converge length totally random introduce dense member combination descriptor database minimal prediction none component difference descriptor descriptor ever database half scheme result predict quite small enable closely behave really green try polynomial parameter predict polynomial derivation effective result show fig example discrete frequency also curve fig curve learn function correlation number slice coefficient green model test representation expansion operation long expansion superior learn improve promising material representation problematic thus serve intermediate interaction frequency qualitatively create even less drastically therefore ml really solve ed representation handling material model logical logical learn number highly nontrivial inherent example concern representation office science energy f l numerous implement thank critical reading laboratory office u contract de ac l respect cost set definition kernel fix example kernel matrix ed new hamiltonian effect approximate weight hamiltonian site reproduce hamiltonian reproduce match site equal infinity could representation require green half energy correspond approximately green hamiltonian specify set minimize several wave calculate eq procedure ground recursion orthogonal precision force orthogonality lose complicated numerical calculation eqs modify state normalize define interval write multiply side polynomial l ref also eq polynomial expansion green ml therefore put k ml two highly nonlinear von j laboratory il usa computing national laboratory basic quantum body matter physics polynomial number size machine dynamical theory full extremely demanding survey material provide complicated situation simplify perturbation expansion theory interpolation development ml explore complementary ml calculation interpolation analysis method solution dft molecular force molecular dynamic body arise application dynamical theory physics material science information material interact quantum define zero dimension solution effort algorithm accurate provide rapid preliminary range material refinement conventional formulation relevant quantum material density would body correlation strength must specify green state effect body implement determining correspond need whereas approach dft output total energy potential key devise term sized material tool optimize paper address know priori namely work ml consistent green follow ridge sec test sec calculation present type green sec polynomial shown prediction look summary conclusion regression exact detail polynomial present polynomial approach learn represent green self energy descriptor describe appropriate infer integrable nonzero technical often integer sometimes study axis frequency important hamiltonian correlate different coefficient expansion respect constraint predict spectral issue ml invert variable inversion principle many eigenvalue ill condition consider pose proceed posteriori approach involve made discretization define cutoff discrete orthogonal representation spectral set seem
typically computation require easily large setting rest organize include sure false surprising connection exist penalize random propose neighbourhood thresholding refer refer neighbourhood lead decrease certain theoretical property appendix introduce connect edge constant sure screen set false negative raise neighbourhood answer must first assumption eigenvalue assumption covariance quickly naturally appear let constant propose expected value define would decrease positive rate false asymptotic pp furthermore screen property theorem hold sophisticated obtaining propose perhaps recently result connected thresholding connect pattern lasso think stage set model component word consistently consider generate set partition sized ji precision via create eigenvalue identity rescale finally perform control investigate extent control practice diagonal completely successfully however satisfied reveal control large value investigate diagonal simulation figure vast majority furthermore simulation column block consequently identical monotone two h simulation large red vast element correspond sparsity publicly available data microarray patient high standardized control biological dependence give gold take equally graphical lasso refer quantify accuracy treat gold detail calculate calculate graphical agree uninformative result split summarize regardless whether gold graphical great gold size accurate obtain sure procedure recover sure framework set theoretical present particular ensure dramatically still contain tend unlike eigenvalue advantage approach sparse graphical operation require operation practice element tend range graphical expression datum acknowledgment nsf grant dms dms grant dp research fellowship first reproduce sake distribute constant constant p ia ib ia ib bound ia ib ia ib I variable together constant imply imply establish omit uncorrelated eq constant satisfie side eq uncorrelated conclude q b ap argument finally conjunction assumption sure screening show pp show q p c next pp false furthermore bn bn pp consequently expectation desire c f c f last fact pt obtain possesse illustrate graphical modeling interest graphical vision processing particular extensively compose thousand infer hundred gene expression consequently dimensional feature graph edge form conditionally equal model recover attention recover brief consider penalize likelihood incur non scad entail precision type aforementioned dimensional efficient high precision
stage sf discount discount parameter formulation easy projection project onto lagrange multipli ghz intel core cc post style domain xlabel ylabel post style mark ylabel gauss gauss plot mark none xlabel ylabel gauss gauss post upper xlabel ylabel probability gauss gauss xlabel ylabel smooth legend col index space average rs ex xlabel ylabel legend pos south col sep index col rs sf xlabel ylabel smooth legend pos south table sep average index col rs xlabel ylabel legend south east col sep sf col average rs sf xlabel ylabel legend east col space sf col sf axis none xlabel ylabel gauss gauss xlabel ylabel legend south table delay col point rs xlabel time ylabel smooth legend pos east rs alpha txt index col rs grid alpha delta txt ylabel east col grid delta txt col rs n x alpha txt figure discount cumulative reward discount total road user discount set sf rs sf throughput measure road metric delay road show reward present average sensitive long neutral variant discount average perspective sensitive outperform risk neutral amongst discount rs sf though computational inverting traffic throughput sensitive neutral setting observe policy parameter illustrate algorithm early confirm plot converge similar observation indicate rapid observation return novel actor sensitive discount reward actor ascent lagrange discount point incorporated sf gradient hand actor compatible feature proof traffic result neutral counterpart future would risk sensitive trajectory discount bound portfolio application solution good knowledge rate approximation actor corollary proposition pt sequential problem may minimize measure variability addition maximize among common finance discount decision first variability give criterion formula devise algorithm fast policy ascent multiplier difficulty gradient incorporate perturbation average actor algorithm usefulness rl actor multi simultaneous smoothed functional sf criterion infinite markov process mdp discount reward develop plan dynamic value reward gradient performance measure case refer representation gradient actor rl maintain algorithmic actor whose action whose actor address whereas policy difference prefer minimize risk usual optimization criterion criterion incorporate induced variability uncertainty uncertainty mdps inherent stochastic sensitive mdps g make maximize variance percentile unfortunately markovian stationary computing tractable although risk sensitive history operation finance attention machine mdps work reinforcement result utility framework base transform occur et measure et short actor risk criterion return obtain episode discount set algorithm additional actor risk measure discount summarize contribution discount reward variability return maximize return policy return see section definition lagrangian relaxation unconstraine simple show state operate underlie td purpose latter lagrangian simultaneous perturbation smooth functional sf discount sf simultaneous perturbation refer introduction function require evaluation parameter evaluation irrespective useful setting algorithm prefer also original perturbation certain hadamard sf vector use perturbation originally sf enhance propose sf scheme variability policy follow identify definition solve discount derive lagrangian discount require sophisticated lemma suggest simple alternative employ compatible feature action function parameter show advantage compatible develop actor neutral serve calculate discussion usage obtain square actor employ bias ordinary differential equation locally policy stochastic essence slow view fast principle td fast algorithm converge bellman operator multipli policy update track asymptotic converge equilibria ode saddle lagrangian moreover feasible I policy upper demonstrate usefulness actor formulation minimize behind control reduce variation road sensitive long discount high neutral cost neutral variant discount easily financial remark risk taylor expansion much easy actor limit require tradeoff take consider lagrange multipli tradeoff variance formulation know ideal expect formulation despite replacement formulation point author gradient stochastic path devise actor discount discount function every state mdp compatible neutral set short setting discount employ simultaneous perturbation estimate hessian propose unlike dual ascent optimize multiplier rigorous describe rl set mdp discount actor present actor optimize section algorithm discount experimental present result discount cost setting conclude remark future reinforcement rl agent environment goal long term interaction process mdp tuple action space reward denote probability state space act markovian action condition rl problem find optimize long maximize discount policy actor define policy adjust policy place make markov irreducible actor finally denote action stationary finite similarly discount pair define return sum discount encounter start discount variability reward introduce action bellman straightforward bellman bellman unfortunately monotonicity dynamic dp measure policy actor candidate discount mdps parameterized satisfie infer use randomization paper sr popular risk extension discount optimize lagrangian convert unconstrained lagrange saddle saddle achieved operate lagrangian objective unique saddle dual ascent tuple minima maxima r set gradient lagrangian x u gradient constitute discount fact derivative q last policy gradient provide reward help actor discount motivate stochastic sf far function initial simultaneous actor follow actor optimize sensitive perturbation simultaneous perturbation stochastic smoothed sf purpose optimal procedure parameter nest inner loop stochastic loop run parallel identity project onto compact keeps lagrange multipli interval analogous slow fast use td approximation equation ensure scale lagrange multiplier slow operate mdp particularly suggest usage ascent lagrange multiplier complicated simulation employ simultaneous gradient policy parameter classify include rs former use sf rademacher sf correspond rs sf employ sf perturbation rs n sf policy function initial observe next state reward draw specific lagrange return policy function actor illustrate operation involve loop instant simulation take simulation reward state temporal td value function value function hessian lagrangian sf update descent direction gradient multipli outer constant trajectory discount ensure enough describe td section present first actor respectively actor respectively denote th subspace approximate linearly project consequence diagonal bellman square functions govern reward transition weight project bellman contraction cf first aforementione see lagrange multipli rs hessian estimate algorithm estimate involve sf perform part twice sf perturb respectively recall simulation update rs devise rs j j actor gradient estimate n update last lagrange sf second work along outline reward state neutral reward differential satisfy mdps criteria variance occurrence action pair sensitive mdps sr discount convert discount l differential action square respectively satisfy equation derivative side replace thus rhs without change integral replace reward rhs advantage difference td differential n statement feature actor average mdps algorithm rule actor operator discount proof satisfy plus fast update intermediate input parameterize observe state reward although unbiased use bias bias actor algorithm upon td estimating consist lemma show rx claim put l ratio sr actor recursion sr variant sensitive actor td actor present discount multipli sr actor algorithm approximation variability discount define close variability measure discount average average reward average compare necessity trajectory actor risk actor use ordinary ode order rs modification sf sf second recall rs loop inner loop td evaluate square outer stochastic update policy descent lagrangian slow ascent multipli recursion slow static slow recursion rs g saddle objective step give td fix bellman utilize lyapunov recursion track ode asymptotic limit early constrain mdps recursion overall saddle td estimate lagrange parameter govern converge eq td policy parameter perturb surely fix perturb realization perturbation outer perform td recursion infer establish argument q far sigma field ode globally stable negative observe infer aforementioned sketch imply final theorem assumption latter continuous uniformly stable martingale integrable verify ode martingale td recursion td loop converge purpose recursion update rewrite due track ode equivalent descent limiting depend ode q operator ensure evolution ode stay point limit interior boundary boundary rs rs sf lagrange lemma sake completeness ode ode let perturbation solution rewrite td inner loop recursion recall converge parameter policy taylor expansion use equality see line discretization ode lyapunov trajectory piecewise interpolation cf ode step first converge evolution limit recursion define function q proof recursion vanish recursion claim l envelope economics ode interpret equation generalize envelope rhs ode time differentiable point maxima next evident actor recursion tuple local convergence saddle far also gradient differ sf theorem establish manner ode rest td involve recursion similar early whereas recursion rs rs sf descent newton limit ode see stable analogue rs sf lagrange exist almost estimate almost claim l j claim lemma first method separation converge td equivalent rule recursion recursion discretization ode rest claim sf establish employ rs sf claim proof claim rest identical ode good approximation actor algorithm true even actor incorporate risk linear scheme rigorous could follow argument quickly analyse recursion td converge loop trajectory iterate error td one fix asymptotic normality asymptotically limit variant aforementione asymptotic scheme score counterpart normality rs rs eigenvalue refer detailed unstable equilibrium possibly equilibrium situation include randomize recursion place
rbm family explore author mix propose realization simulate set regularize easy sample construction keep allow replica certainly computational simulate address observation achieve stack training jointly replica neighboring exploit sample long move applicable paper boltzmann machine deep key parametrization visible learn jointly accurately reflect posterior define distribution v simultaneously equation proposal mean analytically neighbor replica consider swap figure f rand iv ii v deep implement gradient state gibbs layer greedy layer share similarity consequence rbms parameter temperature e smoothed temperature simulating correspond rbm reflect despite difference accept reject still rbm rbm stack whole deep network propose common rbms ensure rbms unit adequate move learn rapid ensure rbms ask style traditional potentially important consequence rbms simultaneous standard rbm notable exception upper layer rbms reach move ask change rbms rbm phase use mcmc traditional reflect rbm sample distance jump correspond particle neighboring rbms rbms share maintain model acceptable swap ratio rbm energy begin reflect distribution rbms provide diverse negative rbms rbms share parametrization layer rbm replica share mcmc move dt bias well likelihood rbm configuration single figure show rbm couple ratio likelihood good layer curve train bottom field ensemble mix property chain facilitate get high world deep hierarchy mix idea individual hierarchy layer concentrate rbms interesting boltzmann negative draw diverse simply even layer model propose state describe auxiliary carlo sampling rbm wang random add simulated system achieve differ relate auxiliary layer rbm wang think dependency latent variable rbm op universit boltzmann rbms approximate rbms typically many computationally gibbs poor mix novel machine belief deep level hierarchy dramatically increase ergodicity auxiliary hierarchical conjunction asymptotically guarantee simulate rbm experimental confirm gradient boltzmann requires draw sampling preserve lead popular practice rely markov training allow configuration statistic bias decrease gibbs step offset ergodicity incur boltzmann temperature temperature space become particle quickly energy landscape configuration local minima original nominal distribution thus explore serial despite benefit remain computationally mini pt require expense efficiency jump high rejection possibly couple adaptive simulate cast relies mostly occur back success factorial enable inference gibbs sampling analytically un configuration derive qx mcmc expensive burn update maintain method modal mode anneal reflect reason author use phase pt work instead simulate difference parametrization temperature act scale low close uniform facilitate leverage fast mix replica neighbor move numerator denominator swap
influence reflect linguistic supplementary material group start n unobserved quantify term product dirichlet multinomial conjugacy remain q however sample collapse slice material foundation infer central use process temporal reveal social readily model interaction analyze discussion argument power take language identify influential content significantly comparative approach characterize influence compares demonstrate linguistic infer turn move beyond dynamic word bayesian ability argument open market demonstrate model pattern also influence network turn infer et linguistic investigate latent influence parameter determine discover associate nine represent format party represent ask sometimes united intend seem movie limited cast people entirely room combine movie consensus set ideal explore strength generate movie market open seven member bank must meet four time vote year financial person discard contribution post concatenation frequent remove stop relationship salient characteristic provide supplementary parameter experiment generative contain length generate time round duration token infer blue circle infer average bar indicate accurately value hold sometimes express standard evaluating higher comparable occur split predictive analytically logarithm via draw bayesian language provide probability supplementary set influence involve equally sized slice slice take probability use perform variational bound available set language lr lrr lr model synthetic dc united movie infer infer describe report supplementary argument speak minute infer linguistic reveal rest infer reveal illustrative case unite remarkably network influence infer network illustrate quantile material figure bar posterior ultimately side side ten infer argument first support pattern infer al pattern status comment ask question unlike focus movie discussion consensus reflect influence infer linguistic turn influence infer influence receive show top significant influence extent initially vote ultimately vote vote vote content similarly vote discuss suppose last three confirm influence et show influence vote influence position much htb exploratory range available length token divide correspond second neutral outcome result depict aggregated averaging subset influence arguably fisher notable policy result correspond pre continue sparse influence result neither finally relationship play role role much relationship result policy strategy oppose economic latent linguistic constitute research political influence take explore combine bayesian al model scale person et capture influence linguistic tie likelihood likelihood hold type supplementary network tie extremely infer suggest reflect informative investigate turn promise direction future exploration discover latent via linguistic demonstrate synthetic compare influence variant linguistic model meaningful potential social latent influence member market separately content word influence acknowledgement early part information nsf finding recommendation necessarily bayesian generative evolution language unlike infer focus via linguistic permit validate use capability influence market demonstrate social dynamic group social datum find use datum social group people interact another achieve goal extremely complex social take g content study social question influence political researcher infer traditionally analyze structural link network facebook however state link exist observe proxy infer social relationship concentrate explicitly turn move beyond behavior present dynamic capture influence upon substantial within indicate interact person increase person extent increase depend power influence drift closely accommodate language linguistic linguistic reveal reveal reciprocal language model idea language self mutually doubly point form mathematical foundation et al depend take interaction multivariate person return make interval extent event increase time decay person couple people via
key backward algorithm recursively forward pass conditioning force transition great draw state sample forward become full state backward synthetic hmm utilize true rate express temperature sampler instance zero intractable correspond ignore complexity tend towards sampler cause transition duration temperature beneficial term sampling observe respective prior sample transition sequentially chinese restaurant crf dependent table sample chinese restaurant customer keep track table sample customer customer simply care exist customer rao posterior hmm restrict represent cf dot index make tractable use gibb accord note concrete hmm map state correctly three dot white even though dot distinguished duration hmms hdp hmm able stick weight hasting row sample hyperparameter depend mcmc technique explicitly hmms specific state inference take place describe principle apply possible transition merge merge state forward must ensure merge weight state gamma stick break weight remain must update update accomplish beta stay break accord ensure incremental infinite delay duration rd distinguish short assign unique emission explicit duration hmms infer existence duration name instantaneous temporal driving due quantum macro system pixel complete entire quantify effect characteristic must understand hmms duration temporal change datum expert four characteristic correspondence surprisingly hmm duration treat proxy map mean hyperparameter give duration share give hdp variety state believe hdp hmm bias accordingly could specific right hmm mean mean top duration emission transition state initialize hmm score histogram distribution duration region consistent observe duration influence hmm therefore wide test hyperparameter find similar value number major mining analysis either two gamma mining make change interpretation one show mining set posterior location well concentrated year finding black dot infer state infer histogram horizontal show occur structured parametric hmms hdp hdp direct hmms structure hmm construction follow exist infinite encouraging construction minor avoid state problem prior encourage parametric hmms hierarchical factorial review area research practical persistent state hmms demand efficient review advance nonparametric construct perform infinite variant enhance posteriori generate right explicit duration parametric hmms e domain recognition natural language write biological recognition parametric hmms long recognize identify fundamentally combinatorial determine highlight seminal exhibit kind nonparametric replace introduce mathematical overhead comparison bayesian approach learn inference appear directly address state cardinality hmm name specifically usage hmm hmm confusion extensive hmm characteristic duration duration characteristic segment rapid short segment desire segmentation steady towards infinite hmms rapid state flexibility derive infinite hmms transition duration hmms thing bayesian leave hmms visit practice particularly tree unknown cardinality largely parametric hmms duration hmms hmms hdp introduce generative explore em integer indicator measure th x x discount prior consist state discrete time endowed emission distribution follow usually state distribution element latent state learn hmms machine learn describe collection expectation likely latent explain max viterbi state infer must hmms number maximum combination penalization criterion alternatively cross procedure single model goal reason alternative model placing encourage small imply hmm small subsequent total characteristic sparsity difficult intuitively might way interpret graphical consequence beyond bayesian estimate include transition hmms let hmm carlo monte use compute next single hmm hmm every might segmentation data step insight hierarchical canonical transition hyperparameter control specific transition vary many diagnostic try observable clinical signal want restrict latent reach hmms restrict topology restrict hmm visit leave right hmms kind topology restriction encode instance use hmm transition hmm may visit restrict topology encode explicit duration hmms suggest hmms tuple long order time impose integer latent state consist tuple remain duration duration transition duration observable endow placing perform inference hmms hmm large state conceptually consider hmms possess state transition bin specified additionally encourage dirichlet process goal infinite allow refer review construction addition hdp generalization extension hdp hmm hdp hmm offer issue dirichlet hmm hdp hierarchical bayesian review hdp infinite tie hdp link top ensure countable tend concentrate mass dp allow different dp stick break meaning namely state specific emission distribution state popularity similarity construction finite base sequence draw hdp encourage state persistence hdp extra self large probability hmm similar chinese hdp hmm hdp generate mechanism otherwise hdp hyperparameter count state et theory parameter heterogeneity persistence markov hdp hdp modify hdp state duration require transition hdp zero generation hdp offer hdp distribute hmm duration poisson negative binomial even phenomena interest novel nonparametric generate hmms explicit duration distribution transition conceptually closely hdp hdp claim duration heterogeneity issue partially add hmm hdp rise different future build unclear would model like hdp hdp hdp except construct structural zero use process construction structured state define slice snp gamma totally measure define base draw measure possible totally almost similar sum pg p choose one choose set number transition example restrict would zero normalize distribution zero snps transition unnormalize normalize produce base hdp hmm restrict formal procedure base concentration way let realization concentration discount h certainly case draw parameter stick prior distribution introduction measure critical snp state collection disjoint whose track uniquely satisfies gamma restrict pg restriction projection ps interested draw role serve draw transition distribution zero drawing allow normalize distribution yet choose encode hmms lead infinite hmm difference partition necessity construct atomic understood portion product space act subset namely eliminate space control manner restrict atom transition structure note random requirement duration transition duration nearly hdp rise maintain range transition simplify discussion give notation state denote draw transition perform sec explain grow method structure specify infinite enforce define restrict increase model region let plug hmm hdp hmm hmm process construction hdp set discount letting variance simplification hdp mathematically hdp hmm recover
non uniqueness prior knowledge section upon kernel introduce advantage little impulse system realization whose give smoothness impulse impulse apply estimator parameter characterize variance paper integrating highly large novel expectation iteration sequence computational effort notably tune kind parameter recently retrieve user parameter order follow use identification organize background base present conclusion end linear transfer I drive output corrupt assume sample know restrict write u px pp vector characterize evolution input input switch constant input switching collect vector entry need piecewise input room load monitor amplitude frequency full amplitude problem obtain impulse response sufficiently sample arbitrary continuous setting focus discrete know problem follow toeplitz input output available follow gaussian impulse response covariance scale amplitude identifiability issue describe usually draw kernel give call scalar interval decay velocity impulse recall provide matrix hyperparameter hyperparameter em obtain noise compute toeplitz close admit retrieve solve scalar solve domain initial randomly keep provide bayesian kernel em blind identification initialization repeat update experiment specifically pick phase equal large piecewise experiment generate experiment noise time variance output estimate impulse response compare estimator criterion ls impulse response base least quantity system kb estimator nb input correspond kb known mean fitting monte carlo toeplitz need result six group nb access estimator know carlo one degradation increase blind median approximately trend ht identification impulse response process stable assume unknown perform maximization elegant permit computation wide class shall attempt belong adopt note random variable function two address carry optimize write recall correspond maximizer plug back respect conclude se propose blind system impulse unknown realization introduce
discretized enable valid physical experiment illumination situation paper object simplicity discretization axis heavy denote use view around obtain express mathematical embed corner locate mf ff choose point kk fix compact format object care vector convert index convert define illumination abuse otherwise translate notation collect compactly dft everything stack matrix lk alternate ap general object nm nm note information phase retrieval amplitude solution find vector p project correction entry replace entry th preserve information recall commonly ap ap solution proper ap well easy verify sense nature constrained search close characterize behave like might different entry furthermore mention matter locate p length dash decrease dash decrease curve emphasize nature ap find locate theorem nonlinear subscript take amplitude lose follow frame subspace manifold frame choose due proposition range main count frame theorem lead unclear us analyze ap signal high frame fourier transform report unique zero operator inverse distinguish global phase theorem proceed immediate consequence note also n mr dimension know claim prove conclude unique phase emphasize p quantification p p p prove contradiction k contradiction emphasize imply convergence algorithm lemma equality equality r ie ie mb ie ie ie p ie I direct view dimension real positive ray le mb inequality mention imply convergence decrease relate monotonic lemma inequality hold eq hold since inequality eq evaluating hence indeed note please numerical case imply similarly continuous p I ip finally hold locate inside set converge nontrivial convergent since converge convergent converge show locate claim condition ap algorithm situation ap generic unless ap finite ap globally indeed note force suppose still clearly infinite would expect ap series imply simplicity ib simplify imply well clear ap converge author notice frame generic hold theorem mention metric projection successively project say initial intersection point nontrivial ap setup know generic frame intersection claim ap existence initial point ap understand ap eq z mr q transpose real restrict solution locate evaluate calculation first fact derivative evaluate fact c ie curvature direct expansion derivative gradient hessian ap I set theorem locate mp illumination window pixel geometrically describe illumination window via image require illumination window match overlap assumption phase illumination illumination maximize phase interact phase illumination window phase forget amplitude indeed show synchronization function relate pixel relationship synchronization study hermitian matrix illumination expansion diagonal entry overlap th illumination window preserve matrix overlap function I ambiguity difficult distinguish view affinity illumination q show illumination scheme intuitively illumination window overlap fourier determine overlap edge I u please pixel illumination amplitude want pay reconstruct idea maximize regard amplitude section relaxation discuss lead initial ap take affinity vertex relationship follow phase unitary transform indicate affinity recover relaxation relationship connection laplacian define affinity affinity purely encode graph denote next graph positivity phase mention generalization laplacian affinity precise take view generalize random status vertex modify encode vertex relationship I I jj ji ii contain evaluate synchronization framework setup converge heat associated laplacian eigenvector parallel manifold literature mathematical propose consider amplitude amplitude consideration truncation amplitude eq evaluate functional equivalent hermitian phase eigenvector call synchronization ps performance optimization play essential beyond describe illumination form dark sort harmonic form represent circular denote illustrated top denote connect together produce illumination illumination connect frame intuition behind synchronization data experimental wide second amplitude iteratively adjust circular limited experimental pixel technique row size pixel figure frame pixel cover first odd random fractional shift interpolation illumination experiment eigenvalue synchronization p p nm gold complex gray scale onto circle complex notice decrease monotonically good convergence ps produce start also typically overlap compare illumination two produce illumination start yet algorithm large increase field illumination among frame apart weakly resource result iteration ap start lead notice experiment new algorithm acceleration enforce improved frame wise technique adjust every exist frame wide combine frame synchronization start kernel synchronization large eigenvector q replace wise repeat step ps initialize synchronization conjugate maximum synchronization ps ps frame phase lead range synchronization across phase write z find large eigenvector expand frame kernel yield synchronization justify understand build illumination frame phase accord synchronization cg cg frame frame wise ps ap convergence figure ap ap noise simulate use proxy distribute simulate variance iteration ap linear reconstruction limit ap global phase check ap inverse transform retrieval synchronization ps construct guess ap problem synchronization ap leave mention least illumination accurate detector range response information per detector channel second need investigation since uncertainty count illumination position incoherent detector discretization etc algorithm ps synchronization architecture base memory fourth ap image uniqueness relaxation last cg iterative study finding phase synchronization scheme support basic energy science advance scientific sm wu thank discussion acknowledge gpu test band limited phase complex illumination illumination content ps truth ap start ps start illumination c ps convergence ps htbp illumination ps select value row convergence ap ap algorithm right ap ps truth ap start illumination describe select top ap ps ap convergence ap ps bottom ap ps illumination scheme ps select high ground convergence ap start ap ps ap lead wrong illumination describe ps set high ps start ps change ap randomly line low numerical empty lemma corollary advanced light berkeley national laboratory berkeley mathematics nj mathematics stanford stanford solution synchronization construct initial accelerate speed light far apply varied ray particle short
ie ie u u inequality uniformly pr py ct pr pr ar remain theorem sufficient unique solution show dual condition svd obey uv f optimal solution unique next shall feasible perturbation objective subgradient convexity norm long unless lrr u inverse u operator u follow hold I I I u uv right high individually distribution nonzero sign matrix lemma prove remain prove f seem widely adopt easy fortunately devise approach q ii relate relaxation consistency coherence third coherence define leave singular u demonstrate coherence approximately constant dictionary condition uv f aa uv tu u aa uv nc simplicity require supervise environment demonstrate effectiveness generate accord l create randomly value bernoulli dimension subspace vary fraction trial successful successful weak produce solution able meet learn specific exclusive nsf dms nsf fa li also support iii svd ground truth corruption rank second coherence third onto resp identity subgradient vector large singular nuclear frobenius sup expect variance subspace prove namely characteristic enough interpret phenomenon coherence increase necessary accurate characterize parameter extensive uniformly logarithm coherence proportional logarithm constant law law induce approximately law coherence sphere notation see large divided component uncertainty vanish similarly coherence clarity million simulation calculate ideally happen provable datum subspace equivalent block analysis subspace law underlie denote sign sign sign h e u tb exact alm alternate minimization update update update lagrange alm solve convert equivalent alm minimize augment lagrange f fix update lagrange multiplier convenient corrupt elegant reality strictly incoherent inconsistent natural grow keep accordingly lrr overcome lrr namely mathematically dictionary lrr coherence potential deal coherent obtain dictionary environment promise result often high massive missing method probably arbitrarily estimate consequence develop explore several decade robust principal build upon exploration rank rank low except restriction location cardinality nonzero either magnitude give scalable fashion theory tell recover follow besides theory vision imaging imaging theory powerful reality matrix perfect latent require incoherent condition hold reality sense reason low beyond widely quite may demonstrate whenever parameter cluster go coherence keep drop well handle nevertheless coherence condition shall parameter discard interestingly impose environment approximately environment lrr advance identity far lrr lrr presence extra sufficient advance low rank e dictionary coherent elementary dictionary subsequently effective algorithm utilize construct lrr demonstrate motion promising include paper problem recover coherent version regime widely typical example coherent coherent practical understand standard coherence assumption excellent relate nature insight regard lrr special dictionary understand lrr well coherent spirit long understand factorization useful remainder organize summarize notation coherent corrupted algorithm complete demonstrate capital accordingly etc abuse denote space column onto space abuse notation projection onto matrix singular nu six norm singular large denote frobenius singular denote nuclear sup ij low letter list notation reader physical raise coherent observation basic notice characteristic indeed structure excellent quantity property characterize coherence first standard basis coherence call coherence introduce r notice calculation analysis work see behavior different thus adequate consider individually prove regard successful considerably go column zero else widely coherence accordingly unnecessary indeed realistic interpretation subspace subspace domain texture rank behavior coherence cluster verify ease citation phenomenon phenomenon coherence please affect nevertheless could accurately corrupt high however usually kind extra appropriate modality structure mixture subspace face sharp contrast much devise identify success condition nature lrr show lrr main proof defer noiseless column satisfy numerical sense z column aforementioned need coherence worth note restriction probably requirement purely subspace ask necessary indeed elementary criterion dictionary lrr confirm lrr avoid rank unnecessary lrr exist everything sparse set observation noiseless reality dense lrr need modify consistently near property solution please refer proof noisy svd dictionary numerical lrr handle coherent ideally environment construct environment dictionary also satisfy kind supervision long rank form contain interestingly even given introduce heuristic coherent except extreme could achieve improvement straightforward firstly estimate utilize construct dictionary post modify design encourage condition indicate construct n lrr already exact fail recover produce weak great equal sense double although program much lrr assume fairly I notice use far dictionary iterative converge simplicity assume entry
issue generic gs summarize technical need characterize solution proof function scalar define technical slightly convenient sequence sublinear define suppose ready u vx xu inequality apply convexity combine divide side rearrange eq definition eq apply relation view rearrange term side rhs establish recursion gs easily procedure k convexity last convexity add definition subtract inequality convexity eq combine view definition establish gs assume satisfy q vx n vx u fact clearly inequalitie fact observation various option specify gs feasible limit compact q identity imply use hold implie observe simplified view complexity gs algorithm find bound eq bound view outer gs moreover iteration subgradient argument hold gs worth requirement use much easier happen nonsmooth present slide still iteration gs nonsmooth component oracle jensen sliding replace exact except modify remark noise note specification generic schwarz apply moreover last previous divide side immediately note outer algorithm ambiguity search generate iteration accordingly origin eq ready q bi saddle nonsmooth assume prox satisfied solve aforementione accelerate solution reduce properly phase sliding follow h establish consequence require stochastic corollary definition follow convexity take side induction follow phase perform observation subgradient previous stochastic subgradient firstly possess subgradient evaluation exhibit secondly establish light tail assumption shrink slide situation nonsmooth approximate linear semi problem easy convex write subsection application loss group regularization nonsmooth slide approximated class us dy c definition nesterov show differentiable gradient ready present gradient slide study property search smoothing slide replace outer iteration respectively view plug easily relation outer observe observation conclude inner reduce outer iteration access maintain subgradient evaluation note use b aforementioned access operator associate present reduce gradient evaluation show evaluation significantly reduce accelerate especially exist compute gradient many application bound solve composite slide generalization nonsmooth bilinear saddle point point theoretical property associate gradient slide practical certainly estimation slide bound gradient subgradient expect proper incorporation search interesting future composite optimization summation nonsmooth relatively nonsmooth order slide skip gradient component require maintain total subgradient similar composite smooth component strongly develop slide smooth nonsmooth component nonsmooth bi structure keyword complexity slide nesterov c program form smooth nonsmooth function satisfy composite appear many correspond certain fidelity regularization enforce property solution paper information subgradient situation subgradient many gradient need order find exist order computation require evaluation recent effort direct lipschitz bound show variant prox eq develop enhance accelerate gradient also observe summation together would expect appear unclear problem significantly bind subgradient evaluation bottleneck first motivate mention example many enforce variation n ax mb mr relatively nonsmooth sparse case arithmetic operation b regularize lx stochastic subgradient arithmetic operation need arithmetic operation computation box simulation evaluation efficiency solve composite briefly summarize firstly method namely slide solution significantly reduce total skip computation require idea iterative solve accelerate proximal slide secondly consider case nonsmooth oracle search refer subgradient gradient slide develop stochastic number evaluation stochastic stochastic deviation light return generalize slide class composite convex respectively nonsmooth approximated smoothing slide subgradient retain prox exist slide establish devoted stochastic slide composite slide situation convex nonsmooth conclude remark make notation terminology necessarily subdifferential nonsmooth differentiable constant denote integer denote denote natural provide review gradient apply subsection prox control development prox place geometry say modulus differentiable respect prox prox initially bregman reference therein prox prox prox constant prox grow multiply distance prox subsection briefly work nonsmooth e search iteration proximity imply move proximal method find method multi sequence build proximity sequence specify accelerate find iteration evaluation also aforementioned proximal type subproblem difficult nonsmooth address
sparsity formulation group sub overcomplete sparsity achieve regularizer write similarly collaborative highly reasonable rank flexible compare joint sparsity row sparsity neighbor non joint rank state nuclear ht f h sum able summation nuclear norm propose group pattern rank prior within across l gs lr train c c gs lr hyperspectral toy example hyperspectral assess imaging generate table label imaging contain band band ground interest label table gs l sparsity toy atom pixel green location atom coefficient belong blue dot clear row many row activate atom demonstrate clearly image region worst result prior due area joint give laplacian via outperform admm low group yield show time lr l significantly computational window size many narrow joint contain small region laplacian give prior review five structure sparse propose classification confirm prior flexible compare latter work well impose high prior hyperspectral pixel wise pixel assign predefine one hyperspectral represent small rather plausible compare incorporate structured appear far exploit spatial neighboring dictionary prior sparse classification also consider prior classification image ne procedure pixel label numerous surveillance develop rapidly technique find effective efficiency wide variety svm modification classifier rule solve task often art also rely hyperspectral belong low subspace homotopy solve insufficient employ contextual neighbor spectral test dictionary eq wise sub total class scalar determine sub dictionary operation belong suffer coefficient fortunately reconstruct either dependency neighbor inherent incorporate classification sort three neighboring pixel lasso rank lasso c enforce structural collaborative collaborative lasso structure incorporate logistic contribution assess conceptually call mixed pixel combination low group prior take advantage group prior encourage use one regularizer section investigate role structure impose norm structure discuss laplacian sparsity group sparsity small consist highly correlate sparsity assume vector sparsity pixel pixel whose neighborhood hyperspectral image atom neighbor pixel recover follow ii indicator operation belong atom coefficient neighboring pixel different even neighboring mention section joint enforce neighbor pixel fall homogeneous region use dictionary atom laplacian sparsity difference pixel belong introduce weighting characterize similarity neighborhood laplacian characterize similarity spectra possess enforce coefficient allow vector pixel become sparsity
optimize costly allow rely efficient recommendation bias world social setting recommend build analysis contain item per roughly uniform recommend estimated sampling order repetition analysis period day date note recommendation first effect recommendation recommendation sense sense agree recommendation introduce weight strategy shown extract reduce constant recommendation focus simple situation constant recommendation item recommend item reduction elaborate possibly complex trade solution hence item eq implicitly independent draw pi pi coordinate pi pi gradient compute coordinate integrate majority system adaptation process influence interact system difficulty algorithm historical analysis propose weight impact extract network recommender rank supposed interest moment daily experience movie book music match ad display website history aspect aside recommender system sort consequence system obviously order ensure recommendation monitor ad article record monitor one strategy offline click item would profile numerous variation range account rating factor influence historical recommendation time database quality associate offline production start user good generate action attribute model user especially recommendation end state influence recommendation production want offline item online cause offline item winner take probably unbiased strategy address literature knowledge modification offline evaluation reduce impact general weighting shift propose optimize discard reference rest organize describe weight reduce section practical world social information item historical recommendation instant user list item decrease ranking present product recommend time possibility item recommendation instant user decrease finally offline scheme reflect business user item exhaustive probability pair quite probability favor profile online evolve uniform soon modify procedure calculating take joint large system accord small probability weight point two moment directly fall recommendation give guarantee influence induce evolve influence responsible snapshot database online system discard importantly profile online platform profile example user via practice obviously illustrate htb implement roughly static evolve various influence recommendation recommendation recommendation modification quickly probability recommend decrease phenomenon recommendation recommend curve recommend second always historical particularly easy illustrate external way record probability subsequently probability offline user overall need keep offline computing select item reduce bias contribute reduce offline modify offline business without business propose weight covariate via weight selecting
mind eigenvalue exist straightforward gram distant completeness gram follow I equivalently optimum get theorem follow eigenvalue coherence measure lower investigate deal unit norm gram coherent norm dictionary since deal measure measure gram eigenvalue gram atom eigenvalue linearly atom allow atom independent weighting coefficient dictionary nonzero zero sparsity sufficient duality gram atom minimax consequence different sparsity atom distant dictionary bound norm atom coherence norm sensitivity resolution inaccurate perturbation oppose large ill condition pose instant reduction upper instant aforementioned upper gram dictionary dictionary coherent dictionary pose topology span distance preserve associate dictionary namely distance coherence isometry issue preserve space establish connect note atom uniquely represent dictionary provide atom theorem dictionary isometry property quasi isometry inner worth isometry isometry product extend exist equivalence less deal quasi isometry quasi isometry aim bridge gap isometry product quasi respect inner product exist isometry pair isometry expression satisfied inequality become tackle issue show eigenvalue eigenvalue take bound isometry constant w r inner product measure criterion unify condition establish quasi isometry induce connect functional framework impact topology future extend new projection kernel span function equivalently eq substitute yield degree university receive ph security technology france associate systems laboratory university france interest representation interest application wireless signal hyperspectral paper award machine signal past review lemma share approximation challenge end sparsity quantify sparse construct one one distance eigenvalue analysis share independence pose prove quasi isometry induce filter gram essential datum big reference therein process essentially model include support machine gaussian radial network neural seminal learning rely sample interesting enforce interpretation tractable within last development sense advance online bring sparsity process number need growth formulation literature collect call measure investigate criterion couple radial resource construct pairwise another approximation criterion explore approximate atom process kernel development compress atom kernel extensively dictionary comprehensive structure limit cumulative knowledge criterion introduce filter square affine projection ap recursive filter develop dual framework space update widely instance second framework estimate extensively algorithm overview point relationship connect feature space aforementioned criterion independence atom number associate bound sparsity conditioning provide quasi deal dictionary bridge gap framework section picture illustrate find banach compact consider error output fitness regularity solution hinge vector machine formalism hilbert candidate base incorporate use rkh reproducing call commonly optimization show functional estimation expression duality regression recent eigenvalue bound isometry distance isometry expression matrix vector relation equivalence unfortunately versus problem essentially identity preference small norm normal equation therefore minimax eigenvalue consequence norm functional dictionary tight bound detail constitute datum processing sensor recursively instant instant add drawback control growth instant expansion order instant fix investigate instant prediction paper restrictive unit instant select study detail former latter next throughout quantity dictionary stress th gram denote study section term construct since formulation denote dual explore impulse quadratic instantaneous deal approximated yield reduce insensitive ap propose present comprehensive filter dual framework framework eq available analogy ap formulation report drawback feed span dictionary current subspace lead implement formula independently online couple instant unchanged diversity exist several diversity measure arise unchanged contribute significantly therefore discard dictionary arise atom removal provide discard least diversity characterize dictionary distance distant correspond project substituting construct measure parameter criterion follow atom comprehensive dictionary composition dictionary approximate follow satisfied project derivation remove become approximation constructing investigate coherence characterize dictionary correspond correlation atom two analysis quality
variance scale dissimilarity representation instance define linear primal common weight determine dissimilarity individual classifier trend choice linear majority vote well comparison measure p bag total alternative subspace simplicity subspace default validation performance popular cover range recent paper dd instance dd dd point ratio bag near instance step maximize machine attempt bag label likewise extension boost instance boost round convert bag svm apply gaussian similarity instance supervise toolbox default unless ensemble base svm dd method base dd mi svm minimax dissimilarity derive dissimilarity used ensemble classifier subspace sort dissimilarity dissimilarity dissimilarity dissimilaritie informative dissimilarity bag support concept instance ht medium engineering technology thesis detect outlier automatic sort towards ph laboratory interest david physics thesis learn structure receive ph thesis university supervision work two year recognition laboratory development classifier alternative performance criterion graph problem elegant classifier university ph institute development recognition work technology laboratory pattern university multiscale multiple object bag feature instance bag represent use dissimilarity bag prototype bag bag prototype representation number bag approach combine strength ensemble consider subspace use state multiple many face weakly training example overall locate formulate case vector bag label apply image patch could present real successfully activity document categorization computer diagnosis category rely recover bag classify output bag example bag instance category often collective contribute bag bag classify bag instance supervise learner bag performance one object bag set reference dissimilarity dimension prototype th successful study training instance alternative demonstrate wide however dimensionality content bag preserve dramatically redundant feature prototype bag analogous general third ensemble train decision test phase dissimilarity translate subspace correspond bag ensemble dissimilarity preserve therefore potential dissimilarity ensemble preliminary meet expectation ensemble dissimilarity ensemble method depth furthermore result insight success dissimilarity ik n bag extension bag positive instance formulation fraction consider bag dissimilarity prototype take bag db b dissimilaritie bag instance bag informative access dissimilarity relevant dissimilarity might strategy instance dissimilarity informative case correspond bag minimize form classifier classifier weight norm typically non therefore influence large discriminative classifier discover informative redundancy find feature redundancy feature reduce individually dependency cross problem feature choose could argue redundant dissimilarity minimum distance exclude instance part neighbor bag dissimilarity instance average third mind particularly classifier strategy address aspect feature reduce classifier resample introduce I overall ensemble ensemble subspace base classifier vector redundancy influence diabetes irrelevant diabetes responsible diabetes redundancy select relevant decrease simplify possibly relevant classifier example successful image microarray hyperspectral furthermore many sample
result ratio optimization decay geometrically error decay geometrically study focus technique could understand problem paper extend direction three treat assume draw condition violate mis develop em concrete analysis em algorithm initialization pilot initialization gaussian plug particular recent initialization interesting acknowledgment grant nsf grant dms center us nsf science technology center grant agreement like helpful section relate analogue wide range ascent size smoothness gradient em step omit theorem begin proof vector shorthand iterate piece yield claim base em mixture previously equation take auxiliary central condition constant verify condition symmetry fact immediately bind contraction fact make elementary fact function place begin define expectation value devoted bounding fix let orthonormal diagonal diagonal entry define event condition condition note bound consequently standard gaussian combine piece eq recall yield apply term term return whenever function previously sphere denote sphere put piece suffice rademacher process triplet consequently contraction q satisfie operator discretization put together piece conclude generate rademacher sign show parameter vector sufficiently combine sufficiently small combine combine chernoff imply sufficiently state algorithm split initialization bind result splitting need iteration optimally dependence establish turn hoeffding inequality eq piece yield claim triangle interval hoeffding inequality probability put together piece complete provide proof corollary level follow corollary splitting update begin prove maximize verify condition use q note write z notation establish upper eq claim smoothness since need show suffice immediate lemma rescale weight vector orthonormal transform also rv rv place begin scalar guarantee reference note separate case namely eq taylor series integral see auxiliary claim since cauchy second bind next combine cauchy use piece complete turn argument event reference event constant event stage control event section goal measurable conditioning turn cauchy observe need x yy combine step return conditional cauchy schwarz effect schwarz cauchy schwarz vi five combine select sufficient select claim section treat taylor function schwarz step event measurable note event put sufficiently consists accordingly let yield turn recall lemma bind introduce shorthand long equivalently substitute upper three component constant measurable let z successive eq simple upper piece conclude decomposition equation claim probability state give vector singular hold consequently apply claim iv follow vi tail ii parts iii combine matrix write ii appropriate choice noise note elementary scalar particular norm claim eq independence note complete proof consequently sufficiently signal condition upper yield bind bind sphere q event gaussian tail remain apply contraction expectation moment sphere result imply contraction piece decomposition need uniform x consequently apply cauchy gaussian put together piece cauchy put piece appendix result present corollary splitting verify smooth strongly equation hessian fix show smoothness concavity hold scalar pattern claim scalar eq need coefficient bound corollary completing remain assumption need indicator one position ease understand matrix sub exponential condition ensure rescale sum matrix identity correspond tail thus I sub sub remain reference lemma random sub constant introduce shorthand show variable variable q use since argument sub gaussian since complete note obtain variable gaussian sub tail q follow bounding variance equation implicitly understand let one consequently cauchy schwarz since sub argument expectation schwarz inequality eq bound find c cm cm ccccc bin yu electrical sciences california berkeley prove algorithm em analysis divide part treatment infinite result update sample global maximizer likelihood characterization em likelihood ascent em perturb ascent leverage develop canonical incomplete mixture covariate high mle theoretically miss practice likelihood complex extent concern maximization growth incomplete model return optimum goal gap guarantee rich em work g introduce modern among establish work establish paper show unimodal certain regularity optimum behavior algorithm despite popularity em sensible little interesting em initialization statistically instance mixture regression empirically good performance initialization refine encouraging type behavior understand related goal address tool suitably sample em estimator analyze alternate case directly em mixture problem mixture regression follow exact natural alternative generalize perform em analyze case em concern population mle ball completely population version certain around concern gradient subset around remainder follow regression miss em introduce concrete corollary concrete give characterization initialization em complement theoretical confirm theoretical prediction em along suppose joint belong parameterize rather datum observe component component structure goal via namely observe variable population maximizer violate identifiable non identifiability difficult computationally expensive observed algorithm suit bounded holding successively maximize notation easy specify consist maximizer requirement relax instead find optimum closely variant ease compactly extension constraint arise project problem straightforward additional projection condition algorithm namely form number population level statistical observe eq equation expectation analog population em fashion analog em popular variety literature review specific base balanced denote density assume equally variable component draw sample variable form example operator close population analogously replace expectation mixture population em operator step analyze update mixture regression recent sample pair equation observation assume design regression underlie regression observe symmetric regression closely retrieval albeit weight update maximization operator form calculation em expectation analyze mixture regression canonical use algorithm covariate introduce covariate directly corrupt component involve fashion em joint gaussians conditional give assume em operator population counterpart em form counterpart eq return case maximizer converge sample converge result concern operator develop operator operator relate population em oracle verify section concern em sample base operator population result bound base sample addition per update stochastic begin analysis version turn vector classical must condition play concrete three section relate population update mapping mapping fix update virtue self satisfie maximize set close inequality condition g van de leverage consistency regularity relate condition condition involve fix exhibit triangle ii section parallel early repeatedly second sample begin quantity operator fix analogue population operator ball vector large enough ensure bind least large ensure sample splitting round identical omit since follow obtain establish population function variant inspire stochastic decay section recursion gradient compute denote onto center iterate radius iterate remain satisfie gs initialization update instance result stochastic order expect relate operator ascent operator algebra iterate choice q integral dx claim model prove population reader summarize theorems c thm strong concavity thm splitting concavity gs r thm splitting thm em well population level develop concrete class previously update previously provide bound difficulty mixture ratio form necessity ml quite slow researcher justification separate snr condition universal em corollary mle fraction rate involve large snr concavity apply conceptually detail quite technical guarantee corollary standard previously guarantee involve proof em split achieve dependence pruning reach snr weaker require oppose fig eps fig eps em optimization decay geometrically decay geometrically provide rough guide integer choice bind perform minimax error computable since quantity contraction iteration qualitative prediction test predict geometrically trial standard panel curve versus curve versus geometrically optimization decrease geometrically tolerance qualitatively appear fig snr scale ep value snr predict figure apply vary snr iteration expect geometric scale snr geometric error analysis geometric analyze em regression model apply condition suitable guarantee population operator locally snr operator contraction decrease function however functional
n lemma tell feasible proper give ls first ls denote operation follow l l l lp theorem notational lp present recovery recover lp go singular satisfies constant l nn n monotonically e notice right tend tend eq q lemma one see x function happen illustrative comparison property normal th row method recover experiment comparison include experiment time mse estimate tag lp tag give mse oracle tag aic give tag tag become curve lp oracle htbp picture exactly efficacy support choice portion successful trial conclude empirical small successful choose small successful exist technique exploit tuning cross method part parameter choose cross test dimension evaluation realization experiment perform measurement know set element period system row noise tn tw assume occur period recover display fig theory obey assumption proposition exist increase system parameter vector sparse method explicitly require open quantify sufficient guarantee recovery resort borel exist explicit characterization another question suitable proposition remark unknown efficiently traditional estimate recover lp thresholde de support formal derivation associate property true go consider formal definition obey consider unknown nonzero assume appear place material demonstrate find matrix sense different compressive sense sense compressive theory value satisfie state notational assumption covariance persistent pe wider require gauss markov unbiased noise gauss please raise perform property term scad smoothly deviation optimization problem later adaptive recently two method bic concern apply could happen please discussion possesse consist lp linear whose finally detail vector capital bold second though formulation selector point selector lie identity formulation point selector behave respect however behave due operation decrease zero advance try illustration path equal increase solution computational scad needs non suffer discussion compare step soft detect second precise description step need step propose solve soft thresholding also computational burden aic
likely generation number consequently likely optimizer dataset optimizer select access l wise qp validate dominate plan achieve figure gap dominate dataset low efficiency large machine cache dominate across fast amazon least fast higher slow epoch interesting lp amazon decrease algorithm try validate architecture report performance improve attribute validate dataset row b show bottleneck increase become converge fast fast b gibbs validate fixing good plan tradeoff strategy two assignment replica accurate region find converge epoch seem preferable run access neural speed classical detailed popular inference row bipartite gibbs sampling calculate row access gibbs establish achieve neural contains contain neuron connect across consecutive layer neuron function label goal deep maximize label descent de network al sgd seven million benchmark call mnist process use classical throughput baseline quality report mining memory optimization database include extensive related trend statistical processing database put system challenge processing develop statistical point tradeoff system goal study body aware range mining memory improve locality decrease cache mining cache temporal locality association mining net work consider hardware free execution aspect machine seminal use machine group discussion decomposition locality year language help extract parallelism two goal trade hardware recognize memory change landscape improvement et et li tradeoff advantage bandwidth new affect hardware aware machine tradeoff benefit prototype demonstrate tradeoff interesting current thank team team sharing acknowledge research project fa fa national foundation award office fellowship google finding conclusion recommendation nsf implement scientific describe worker different strategy worker affect worker worker need node try protocol rely operating system worker second evenly worker node worker replicate svm throughput second operate worker dense extraction text protocol storage advantage store require sparse allow parallel storage scatter improve cache throughput overhead operation synthetic sparsity sparsity dense vs tradeoff intrinsic design current optimize dense major column storage study data storage strategy conduct experiment major intel boundary unable pick access cache gets therefore always access store ccc target hardware et al et et al hardware efficiency relate consider increase hardware mining k mining neural cache frequent pattern mining cache include spatial locality improve mining structure temporal locality careful study type task system almost core study parallelism task parallelism frequent mining number pass al implement load memory usage locality tradeoff pre et al implement parallel memory optimize reference memory locality cpu none optimize affect computation consider worker technique least system design role dense sparse computation computation kernel dense dense mapping model beyond consider vs storage study community traditional database technique sure hardware modern hardware hope future language language pattern effective trade insight mathematical optimization task community look asynchronous recently establish ji detail fair tune run throughput use combination batch type statistical efficiency hardware storage compression na size batch size together experiment try size parameter contribute converge dataset two report overhead overhead scheduling tolerance make fair conduct scheduling tolerance impact claim gradient strictly epoch epoch batch slow cross different architecture epoch scheduling computation particular loss epoch second epoch second scheduling use calculate cause implement tradeoff hardware impact throughput totally seven parameter relate hardware parallel measure throughput find throughput music set speed surprising gb high throughput trend language help user parallel program experiment tradeoff help high performance quality implementation logistic regression music try locality try curve figure different surprisingly hardware e core apply strategy hope illustrate scalability follow et al create million page al page validate scalability randomly example create finish grows cause sub dataset whole model fit cache sample tuple equally datum tuple result linear leverage however call leverage specify tolerance acceptable epoch example score datum experimental set time compare music use tolerance music importance slow tolerance increase music tuple detail represent access neural illustrate graph run factor calculate connected factor assignment factor gibbs conditional proceed prove protocol theory aggregate factor figure b gibbs row non correspond get column sample throughput e generate general code modeling topic implementation implementation application illustrate stochastic discuss contain sgd fashion inside one invoke path different layer stanford edu tradeoff access first execute memory differ incoherence share tradeoff discover tradeoff study valuable prototype engine least sgd patterns architecture management machine via amazon ec result support google pick tradeoff goal paper system utilize modern hardware sometimes hope identify next system solve core system study configuration study preprocesse class machine traditional study statistical traditional efficiently execute fundamental tradeoff need step describe precisely minimize loss several complete call pass may explore current pick discover explore several store row wise brain access statistical different hardware tradeoff systematically tradeoff storage access prototype include supervise least different find converge differ access method access support develop select nearly study mechanism share explore sharing informally share processor replica three treat e event approach share architecture part visible core end epoch scalable hardware perspective grain communication core processor uniform memory google care hardware may communication dramatically find beneficial natural hardware responsible cache coherence processor coherence worth method technique batch dramatically reduce processor runtime improvement technique fact maintain effectively update total across processor partition aggregate memory partition may replicate replicate datum list conceptual treat machine system exploit unify implement modern architecture learning row call distinction datum read perspective paper distinction analytic method pass epoch sgd google oracle epoch read single read computation scan capture trend statistical memory coherence storage memory read execute read atomic access critical incoherent memory converge rely atomic modern processor empirically costly protocol popular include sampling solver al element memory distinct classical coherent prototype allow region share processor share per simulate nothing access distinct system column wise row prototype several epoch pass wise row row take applie update model use access include gradient descent order bfgs set typically use row iterate conceptually iterate read row method de approach order popular access briefly illustrate multiple node multiple cache quick gb data architecture cache coherent use machine name local amazon ec configuration sp tradeoff optimizer consider access versus hardware experimental paragraph execution provide initial model list solve method first capture argument index access argument pair index zero entry receive study modify single variable specification specification contain execution execution plan execution plan thing core operate model locality describe locality locality engine explore pt read write write replica assign tuple core replica one key synchronization frequent synchronization need converge find possible asynchronous version model separate together core reduce take execute epoch use observation strategy loss epoch intuitively replica information redundant phenomenon execute time finish show strategy svm locality incur request rule pattern suffer hardware replica worker strategy system partition replica avoid computation epoch partitioning row resp column wise access method column replicate replicate dataset copy redundant statistically benefit average low hardware read frequent dominate read point surprisingly epoch epoch illustrate show run use tolerance figure strategy give within epoch observe region cause execution loss hardware efficiency epoch choice epoch number surprising epoch process c music music forest ls music forest amazon amazon report deviation tradeoff enable speedup validate affect experimental diverse set vector lr ls programming programming qp include determination choose music forest forest benchmark analysis amazon customer google google art function take loss hour low measure measurement tradeoff request request request unit manual conduct second secondary storage memory four descent lr wise access implementation possible improve model variety differ cache machine tune buffer disk os tuning version try locality machine implement validate system extra tolerance task scheduling comparison protocol search parameter mini size report good result number local logical core machine logical tradeoff local always give time fast difference great lp qp order fast order choice tradeoff
dc express say dc dc clearly h ff dc function proposition dc dc dc explicit problem proof proposition propose performance reference therein popular dc program cut branch inefficient scale difference suit solve dc program convex obtain solution algorithm reach minima improve linearize convexity solve convex project stochastic exposition solve subgradient ht label choose update project subgradient solve linearize problem perform stepsize stepsize moreover accelerate minibatch instead minibatch traditional feasible solve iteratively constrain eq solution propose algorithm recall correspond dc p around current many criterion terminate terminate maximum terminate satisfied concatenation convergence verify change q last thresholding section evaluate digits image dataset section focus methodology break mean dictionary ii sparse soft compare last technique subsection encode able model unless state empirically worth validation soft approximation sparsity manual procedure choose equal subsample whose entry change set quite algorithm set denote iteration several value test small minibatch gradient iteration gradient first soft comparative study image learn atom randomly proportion randomly atom merge successively code solve optimization package dictionary optimize correspond function set generic neural use mini stepsize cross choose make three mapping cross linear train feature approach set task finally dictionary simultaneously encoder texture build texture take patch texture patch vs vs outperform size agreement author empirically learn dictionary classification task vs unlike crucial test dictionary dictionaries price ht classification thresholding fig evolution objective sgd solution sgd reach ht cifar dataset class rgb comparison restrict scenario later section illustrate classification report vs task thresholding soft classifier dictionary outperform size small dictionary reach reach atom illustrate show class separate reasonable fig b observe minimize whose class summary encoder traditional unsupervised way sparse code classifier dictionary fashion table largely near moreover mnist bad highlight benefit technique compare train optimize unsupervised pre architecture outperform outperform classifier dataset interestingly tune classification fast technique mention aware achieve mnist incorporate translation training shift version digit go class cifar pixel advantage deal sometimes invariant result high classifier feedforward rbf svm sgd last entry sign intra atom encode set cifar linear last relu net layer relu error relu report confirm superiority svm outperform sgd challenge surprising stochastic one adequate training give difficult report classifier use atom dictionary atom discuss solution generic algorithm complexity classifier algorithms complexity classifying classify mnist respectively require product nonlinear svms linear vector linearly practical cifar extraction involve multiplication control complexity order computation slightly scheme sparsity highly beneficial help data level mnist cifar dataset mnist cifar penalization mnist cifar exhibit predictive complexity precision code homotopy last computational classify test need test mnist ghz core gb ram early thresholding scheme commonly optimize descent oppose compare testing confirm whereby good minima unlike descent critical stochastic descent descent sensible stepsize choose stepsize sgd beneficial prevent interestingly stepsize solver solve solve intermediate optimization stepsize stepsize rule unlike sgd heuristic slow learn hyperplane dc dc significantly outperform experiment result consistently lead classifier compare near show handwritten digit mention encoder competition encoder act behavior competitive parameter network gain bad work reveal layer reach find network insight soft thresholde coarse negative mapping problem gradient proceed iterate recursive proximal operator stepsize nonnegative otherwise impose condition stepsize first eq precisely correspond soft soft coding go proposition dc dc dc recall dc p derive dc acknowledgment like thank anonymous quality proposition representation show provide many recognition major limit scale scenario consider yet alternative code extraction nonlinear tailor linear cast dc program solve dc conduct generic classifier appropriately classifier scheme train soft thresholding development efficient method vision decade volume produce instance internet technique large focus one task vision goal permit predict computationally focus research decade separate classifier feature choose convert heart popular popularity classification complexity scale recent nonlinear compact overcomplete dictionary beneficial processing denoise nonlinear conjunction architecture code work task drawback prohibitive vision power requirement map unlike vs vector scalar map give represent simple procedure nh illustrate classification implement vector multiplication linearity soft successfully architecture feature remarkable encoder couple result provide proper objective soft classification soft thresholding pose learn comprise control regularizer prevent overfitte dc efficiently iterative solver extensive image exhibit remarkable comparable sparse rest paper organize highlight section dictionary classifier algorithm extensive digits scheme highlight difference technique draw connection aspect share similarity architecture code extraction apply dictionary known obtain particular gain dictionary additional dictionary specific knowledge especially optimize code variant still trivial discriminative soft thresholding view coarse motivate soft extraction code efficient predictor parameter learn code approach soft close require solely accuracy moreover approach purely supervise often tangent thresholde single easy implement tie code appendix consider quite architecture layer neuron combination activation neuron activation choice logistic sigmoid tangent neural hide unit define layer represent connect classification architecture recent activation result classical tangent top nonlinearity plausible representation architecture differ architecture unclear appendix impose restriction neuron negativity necessity regularizer enforce sparsity scheme work choose thresholde nonlinearity coding provide independent motivation turn motivation deep simple particular architecture network generally stochastic descent directly exploit structure estimate classifier classification scheme incorrect regularization prevent thresholding hinge
cut label vertex common unlabele suboptimal bagging prediction bag ensure receive label extremely subset without replacement describe obtain average average probability predict protein genome level five describe interaction row column protein come absence structural dot vector combine genome database matrix indicate protein protein protein indicate know gene profile number isolate impossible sparse examine q connect time receiver roc vertex vertex edge label svm sdp result roc normalization sdp column show column field protein understanding biological huge class protein roc improve slightly deviation graph similar versus sdp simple topology describe uniquely suited reason operate protein biology describe thank thm topological functional category yield exist understanding protein slowly build sequencing characterize ease dna sequencing protein database would third unknown number difficulty shift biology necessity large identification protein characterization hundred major shift determine gene numerous throughput produce protein protein interaction protein etc picture vast amount protein protein protein represent natural vertex group graph simple contribution algorithm effective predict protein perform chen code locate detail boundary define sum sequence create small removing create informally n call order show order order apply arrive irreducible ordering irreducible minimum complement
different mmd test base probability let performance sequence distribution replace mmd error mmd base test rate case finite demonstrate mmd comparable divergence test test traditional test fr test mmd five test laplacian distribution two distribution laplacian laplacian mean variance test see mmd among mmd test well suggest mmd advantageous two sometimes mmd much mmd mmd use three comparison figure probability case mmd perform test mmd daily maximum usa anomalous datum day year temperature error average anomalous detect two place may perform apply mmd base set mmd plot error see mmd test fr test test error converge fr mmd mmd see increase mmd problem anomalous sample mmd free detect anomalous scenario reference sequence scaling go infinity develop scaling scenario performance appealing test demonstrate useful mmd solve nonparametric measure variable recall th remove test analyze anomalous anomalous follow quantity affect change affect change imply satisfied zero fast anomalous analyze obtain q clear satisfied exponentially computation multiplication analyze generality anomalous sequence np apply kernel multiplication analyze without anomalous anomalous notational stack define independent distribution independent satisfie divide component affect affect hence combine hence pick satisfied converge respect complete generate bound hence enough eq conclude clear respect include computation multiplication test generality anomalous constant sn q large enough therefore inequality enough np test include multiplication hence cm ex cm anomalous kernel embed h poor edu department electrical university nj usa department bioinformatics nc email ex anomaly detection totally anomalous sequence identically draw whereas distribution distinct scenario mmd mean reproduce hilbert rkhs develop test consistently develop show numerical demonstrate test perform competitive traditional anomaly detection consistency test maximum discrepancy reproduce hilbert study anomaly detection totally sequence anomalous sequence detect anomalous sample distinct assume test anomalous datum cognitive wireless either channel issue channel utilize channel improve spectral study whereas paper detect anomalous dna detect computer computer detect modify study assume priori detection nonparametric explore distribution arbitrary li decay case discrete utilize major challenge limited anomaly difficult building decay challenge approach tool solve g accurately propagate anomaly detection traditional distribution estimation intermediate fr well arbitrary discriminant use probability approach sake implement test base anomaly demonstrate test test specifically distribution hilbert idea distinguish distinguish embedding justify certain kernel laplace kernel rkhs naturally carry embedding easily rkh mmd mmd complexity test mmd approach density estimate distribution build test avoid propagation mmd sequence generate anomalous interested large regime sequence go motivated dna detect datum clear become anomalous large increasingly challenging detect anomalous correspondingly anomalous sequence regime go mmd computational complexity increase analyze consistency assume sequence increase characterize I study without scenario free anomalous suggest reference advantage reduce sample need help nevertheless lack reference exploit example anomalous contain anomalous characterize impact anomalous sample behavior order guarantee consistency theoretical traditional statistical approach mmd compare result mmd perform among well perform real mmd base section theoretical guarantee test sequence remark embed mmd anomaly detection sequence I arbitrary priori test sequence anomalous generate anomalous case priori respectively interested go infinity sequence fix applicable anomalous comment denote converge assume available reasonable collect exploit anomalous probability performance let anomalous index anomalous claim sequence say consistent large become increasingly e case anomalous consider priori test large anomalous sequence characterize anomaly reference sequence anomalous sequence far apply exponentially consistent see appendix equal sample sequence far threshold increase somewhat surprising threshold decrease intuitive detect anomalous sequence increase thus applicable value far away close anomalous base understand test asymptotically positive characterize anomaly anomalous unknown priori test applicable hypothesis anomalous imply value scale level lack extreme case occur order anomalous sequence dominate dominate also exponentially nan compute average complexity desirable anomalous introduce capture anomalous sample result anomalous constant unknown scaling substitute consistent anomalous sequence reasonable impact increasingly sparse anomalous anomalous explicitly tradeoff anomalous number anomalous scenario case study example anomalous anomalous fully compose sequence compose generate impact anomalous negligible test anomalous characterize condition consistent anomalous suppose bound constant appendix sufficient detection general similar role except pick consistent anomaly reference anomalous sequence consistent condition test consistent computational reference sequence guarantee consistency applicable role enough exploit reference consistent seem counter fact sample hence small become contaminate sequence eventually gets suggest test although reference help although without complexity sequence build satisfy e replace substantially consider less base understanding scenario build requirement pick anomalous domain knowledge characterize condition anomaly detection
receive marginal decompose aside generally likelihood importance alternatively within mcmc parameter conditional observe apply state consider perform parameter distribution estimate sample metropolis hasting accept sampler proposal generate mean covariance covariance pilot simple identity user simple case alternatively mala view define time particle particle probability value qx px sample adapt filter prior calculate calculate normalise assume user propagate particle k normalise filter particle weight detail filter place mcmc parameter use within accept metropolis hasting see focus likelihood iteration compute p stationary particle score outline idea particle back slight abuse denote path particle px step index particle particle time increase particle filter algorithm increase variance expense quadratic instead use rao substantially maintain particle follow replace discrete distribution obtain shrink user idea however actual affect rao calculate depend summary add ii f store vector shrinkage degeneracy significantly reduce rule reliable estimate equivalent auto considering use density unlikely reference point proposal bias estimate assumption resemble control degenerate behaviour efficiency analysis limit acceptance unbiased estimate article eq proposal q early mala assumption independent source variation log target independent interested acceptance jump n distribution define q density next consider overall efficiency term computational cpu additive justified number particle act noise variance limit acceptance optimisation mala optimal scale mala variance differ slightly particle rwm however important rwm increase scale mala rather mala sampler opt contour efficiency function leave panel leave panel variance scale jump variance figure rate considerably sensible variance sensible achieve acceptance rate inefficient choose noise acceptance estimate tune particle exact gradient would mala particle mala sources eq keep particle mala proposal view mala mala retrieve noise degenerate limit must mala limiting impose term mala proposal necessary mala limit behaviour notation hold non acceptance rate ii degenerate acceptance non degenerate relaxed expense variance lead three distinct describe limit overall scaling limit value approach iii corollary rate part I rwm study article iii efficiency scenario proportional possess therefore plot reveal mala firstly evident estimate without exhibit rwm behind bias term dominate rwm decrease proposal exhibit mala allow proposal decrease acceptance monotonic vary regime resemble rwm resemble regime mala fix let fix nevertheless acceptance change slowly value acceptance lie use scenario outline expect iii estimate particle linearly would estimate estimate gradient procedure unbiased particle eliminate still raise question trade see support tend infinity seem applicable finite filter fully adapt filter meaning attain mala discard burn estimate calculate particle alg update mala proposal increase mcmc sampler gamma distribution parameter sampler transformation jacobian term mala assess option ess give approximate h scale option detail left panel vary particle initially increase increase computational setting support acceptance panel regardless centre regardless scale predict panel rate reasonably predict consider mala compare rwm moreover method linearly linearly alg however particle observe expert probabilistic mixture autoregressive expert economic cycle nonlinear non noise record data g release significantly ensure expert expert expert growth assume measurement could sample vice versa see cause sampler mix slowly would efficient implementation whereby integrate mala rwm implement particle run alg discard use mala take pilot prior constrain transform hyper comparison mala minimum per simulation algorithm rwm min mala max table improvement term effective particle mala algorithm effective approximately mala however take cost proposal perform walk account order proposal cost present empirical particle mala particle rwm apply pseudo filter compare rwm particle mala establish optimal scaling practitioner mala proposal particle mala significant particle execute large reduce variance look particle mcmc markov nc x eq moreover sum central limit see second n n combine prove proof converge zero schwarz zero theorem separately therefore denote statement consider hasting ratio mala let term mala mala rd taylor mala double taylor expansions usual mala simplify examine refer multiply mala proposition lead term degenerate mala proposal condition mala scaling list satisfied alternatively rate proposition possess ix x nz nz u nz nz bx u I mala derivative derivative let also q form produce taylor expand term odd power integration respect target assumption provide form straightforward imply part analogous use bias present
latent entry major direct graphical bayesian belief undirecte refer markov network represent conditionally independent parameter consist graphical model convenient main reason encode distribution addressing need test speech bioinformatics rna analysis homology detection alignment genome identification nlp pos due structure store thousand maximize length problem handle traditional table direct acyclic dag represent conditional table show quantify relationship node completeness guarantee since constraint guarantee unique variable parent parent know probability l affect markovian bayesian causality make mrfs vertex dependency clear causal influence one node link dependency neither cause undirected undirected graph conditionally whereas mrfs assign positive value clique subset potential function function potential clique graph calculate sum integrate potential mrfs common language processing time model dynamic system whose output speech synthesis hmms whose satisfie long need factored hide notation mean take specify distribution evolve work annotation ms propose apply keyword purpose build semantic system limitation hierarchical categorization large interaction another hierarchical gene author allow aggregation simple complex small observed immediate immediate allow although bayesian network massive automate ms third macro protein scientific attempt integrated relationship clear accumulate growth term chemical difficult identification prove protein structure diversity along absence standard representation result database format use format format use major analytical automate ms analysis mostly complete abundance spectra mass abundance peak make distinguish database exist suffer produce nature irrelevant attempt process program produce incorrect thousand literature datum mining technique resolve great probability iv x l semantic allow learn gradually integrate attractive big age need include result use new used variable annotation ms predict evaluate annotated peak level ms probability well manually annotation suit ms automate integrated ms call ms annotated peak annotation ms layer node assign ms profile table ms ms parent direct parent layer child ms layer frequently child layer occurrence parent identity parent massive ms ms node ms parent annotation run ms peak ms technique new time record demonstrate precision train peak annotation tool dataset peak ms latent term provide com operate job extensive million job million search hour recommendation team want discover build engine query order relevant traditional engine tackle search represent job engine language search represent place class root place child node form term back user term store belong frequency connect term net vb se se health care table probabilistic similarity share parent filter technique search term distinct final graph node perform among ghz processor core ram term discover evaluate send review discover search return pair discover ratio discover relationship search use relate big business analyst software software big science sale analyst mining datum sale project master modern mining major issue probabilistic scalability set focus datum modern computer system sensor attempt scalability scenario hierarchical datum design regardless automate mass bioinformatic semantic discovery search large job test entry computer david helpful suggestion improve university valuable discussion suggestion thank university valuable time share annotation york scalability become crucial requirement graphical suitable represent massive arrange structure level expect million kind bayesian network represent level single hundred value ii level usually also network predefine top parent
nonconvex formulation local optima tune carefully remain quite large combine go around limitation matrix combination expensive regularization dimension attempt frank wolfe similarity formulation find eigenvector scale consider dimensional efficient solve lie word nonzero typically much small goal sparse scale pair besides instrumental additional motivation learn thus psd learn onto psd cone furthermore psd prevent allow project basis rank basis jx allow easily learn notice text represent bag count basis natural thought encoding term term topic parameter consist triplet build implicit click notational convenience degree hinge similarity aim minimize average margin triplet involve next f k frank fw compact iteration move towards minimize linearization minimizer domain fw enhance call describe basis basis move towards possibly new basis reduce active away determined line add convenient way compact memory away step provide completely algorithm f fw observe gap able find approximate appealing high algorithm update careful storing allow well identify ignore find time line search available bottleneck find forward indeed consider element take memory iteration variant mini heuristic expensive forward direction mini draw replacement mild assumption replacement probability deviation mini eq decrease mini find forward detailed avoid follow find basis result shall next dimensionality compete predefine validation contain proportion irrelevant word representation binary split split detailed split dataset train validation weighting minimize optimization gradient similarity random project entry draw bilinear triplet similarity learning except machine dimensional also svm nd text paradigm multiclass constraint neighbor nearest due instance per label tune heuristic tune tune c bold dimension reduce accuracy learn similarity early stopping learn nn notice perform bad projection generally outperform space generalization information pairwise consider weighting table svms variant early linear svm although outperform function negative entry good ability selection similarity iteration show incorporate may overfitte data characteristic ability diagonal nonzero pairwise score co occurrence fast computation superior make nn ranking etc also worth attribute extra capability draw svm investigate recall learn psd equal run space dimensionality assess nn performance iteration give compare projection similarity rp note tune separately size show early feature eventually heavily dataset sign sparse achieve form similarity combination operate frank wolfe world confirm robustness noisy rely similarity partially contract w nf ap purpose annotation view conclusion herein either imply proposition edu carry california many machine mining learn powerful scale dimensionality efficiently parameter decomposable one specific sparsity wolfe learn incorporate provide control overfitte enjoy strong memory depend high dimensionality many application processing vision biology dimensional ability score crucial ranking similarity measure datum
matrix mean random support allow grow grow associate square loss include x x mainly study compute specify estimate sparse observed tuning parameter regression appropriate omp algorithm parameter write mapping tuning refer discuss select let sample plot sparsity level sparsity level forward algorithm refer sequence estimate consider output dimensional level appropriately sparse however visible unknown turn sparsity reliable measurement matrix c present besides additional appropriate reliably identify clear evaluate compute stop fall threshold quantity decrease support compute sparsity line algebra write proposition say could furthermore ensure algorithm box lasso blue score two plot show estimate loss stop see performance remark select fortunately independent popular regression performance insensitive choice involve residual compute additional complexity assume stop change modify computation nearly lasso lar level tune one increase solution finally minimal loss sparsity implementation alternatively depend path solve increase compose stock return let zero use compute score want measure red horizontal plot clearly mainly vary narrow manner identify tune reliably cc say property various state follow entry eigenvalue block open analyze incorporate scale improved setup highlight propose output property thus reach equilibrium decrease tune omp greedy interestingly lasso start select parameter select lasso theoretical appropriate constant accurate estimation motivation scale furthermore lasso root lasso norm entry implicitly threshold empirically sl pl want sl pl show box pl sl range sl pl sl nearly score pl pl relatively insensitive choice compare sl advantage lasso scale lasso apply specify vertical specifie coefficient apply total solution may true significantly reduce real data attribute community normalize contain gene patient value present detailed reduce set tuning subsequently process stage require cross propose computationally select tuning parameter contribution agnostic sparse appropriate select grow thus asymptotically regression drastically solution problem significantly stability motivate future example interesting sparse vector thank cox paper grant nsf w nf unknown vector seek stop stop rest step use use follow equation side bind right rhs ii write fact next upper rhs upper rhs rearrange definition write tail follow combine use state definition remark call thresholding tuning prove specify finite setting reduce parameter domain measurement imaging furthermore basis several solve omp analyze require reliable vector comprehensive
neighbor converge relax keep close neighbor centroid classic relax mean soon prefer prescribe convert assign neighbor assign centroid let mean convert experimentally convert regular surprisingly score often mean smooth mean landscape minima q eq dropping mean get right present comparison value initialization fair comparison extend heuristic summarize contribution empty event heuristic cluster event happen heuristic well merge method bring expense third objective convert relax mean potentially many local optima exploratory minimum definition fact close cluster many heuristic minima heavily depend mean method heuristic take account empty cluster event tend increasingly occur mean round lot show seed center objective merge splitting heuristic converge minimum finally generalize center convert iteratively relax minima grouping intra cluster inter cluster point let cluster one yet cluster minimize minimize globally hard programming exponential yield separate copy equivalent intra distance sum inter cluster square distance heuristic propose hardness search heuristic heuristic heuristic partition good discrete yield contradiction p ie initialization singleton say add time close centroid online single mean initialization cluster close center convergence single cluster convergence close center mean say heuristic improve partition partition pn empty number partition heuristic cluster converse heuristic exponential open initial crucial clustering several initialization replace require initialization reach mean build cluster seed minimize point global problem euclidean apply bregman organize follow empty heuristic heuristic mean performance objective point associate close cluster convert relax experimentally contribution discuss mean start seed center iterate square centroid assignment repeat monotonically decrease guarantee denote mean perform polynomial point report optima mean may minima pdf centroid pdf second exception h nk number get center cluster random empty uci repository consist classify random count phenomenon notice dimension note tendency vary heuristic cc compute million empty empty cluster produce empirical empty toy heuristic mean demonstrate empirically notice rise avoid surprisingly show empirically give meet cluster exception current center usual method table allow minima table compare heuristic without dataset initialization observe minima mean build tend random minima provide single increase intra variance statistic mean point cluster heuristic min avg decide merge accept iff merge operation mean example cluster close obtain keep short since improve detailed proceed center primitive j center deterministic implement operation center find force hyperplane predicate compute determinant sum among point mean heuristic pick pick distance keep accept respectively stop iterate mean macro heuristic assignment move centroid correspondingly heuristic last stage merge merge merge split monotonically function converge finite number since decrease
concept return concept exist agnostic sample case oppose learn private goal private privacy pac hypothesis differentially pure privacy parameter note learner require requirement requirement neighboring matter consistent mechanism preserve sense similar return database concept every return capable approximate database close term close class concept database output description improper predicate class coin database algorithm pure set output computational improper transform except probability complexity et predicate provide theorem state database require big element bound predicate away recall operate database output state necessarily concept database particular imply always database always fix na close database private mechanism laplace sensitivity neighboring laplacian generate noise output preserve differential privacy goal choose maximize input database sensitivity exponential differentially private mechanism output show exponential sf generic private release solution much input parameter sensitivity two gap differentially private database sensitivity hold database differentially private query unknown trivial release round answer operate database composition elegant scenario care meaningful answer care receive privacy sensitivity query sa proceed round unbounde pure privacy differentially private execute algorithm output least pn chernoff concentrate learner concept class demonstrate learn jx jx learner prove exist improper class complexity alternative simple complexity proper learner consider execute quality return else random intuition whenever copy differentially fix concept chernoff least label appear quality big enough therefore hand fail whenever good need e number requirement however zero concept dc ax ax empirical typical straight forward application fail observe give concept unique every label every high execute privacy answer execute solution da contain subset cardinality learner proper private class proper learner stability label sample p concept hypothesis error change significantly hypothesis learner motivate construction simplify aim hypothesis approximately error second diverse two assumption make hereafter remove choose differentially private refer point zero differentially lot lot every many interval refer interval interval line dot correspond inner thick interval define five divide switch locate contain zero assume generality one close concept contain zero close concept argument parameter h frame sep min min plot thick thick define find privacy sufficient hypothesis explain interval differentially private return length length interval shift h thick thick thick thick interval zero lot one switch inside contain zero zero large suffice lot recall therefore interval preserve exist complete attempt one laplace specifically interval contain zero interval contain contain label sample diverse label hence one many utility noisy reduce noisy big indeed big length zero moreover contain one simultaneously operate differential set search summarize length good length choose start threshold reduce tool formalize analyze tool later construction learner notion enable concave concave quasi consist order database sensitivity parameter call quasi exist otherwise label goal view concave define correctly classify concept satisfie find inner frame max max target dot sample point quality quasi solve problem privacy preserve figure range quality database recursive call parameter choose return otherwise l ls label exist l l jt pt see label recursive range return successful call approximation one k pt partition interval let property w mechanism label fraction mechanism good utility rectangle sep dot thick identify upper decide whether good check define simply high start call let I iterative monotonically decrease n proceeding privacy observation sensitivity proceed execute sensitivity bind recursion preserve differential note sensitivity recursive call mechanism twice recursive call execution mechanism differentially private differentially correctness number call recursive call claim quasi concave non point exist nn ns recursive call ensure output perform recursive satisfy lemma call execution denote power inductive need recursive claim function step quasi next recursive appropriate lemma plugging get index quality bind recursion h two return recursive inductive ls r lemma interval might interval therefore contain therefore concavity quasi concavity quality quality rp left interval quality ap proceed exist contain point concavity quality sub length exponential ensure probability step mechanism output hence probability output sensitivity must exist gs gs establish prove output obey k fact lemma utility choose mechanism straight function database mf fail assume solution mechanism define choose I km km show must operate private separate database size private string every query fraction algorithm mechanism point define restriction otherwise input initialize let mechanism quality bb c mr b utility element mechanism denote begin event label happen succeed step choose mechanism case event assume contrary exist iteration mean appear contradict proceed simple input mechanism mechanism exactly interaction preserve differential interaction applying preserve differential concept query execute denote every privacy immediate ki c jx x j al get pure next approximated initialize empty privacy call convenient maintain decrease call database subset call execute either recursively small appear next property initially empty contain point quality j pt execute algorithm quality quality privacy parameter label successful return denote otherwise divide might z union quality recursive call database twice laplacian mechanism mechanism interaction differential call interaction mechanism preserve differential last interaction preserve entire mechanism differential c c define function step draw proceed analysis good occur coin fix execute initialize database label step define interval suffice probability mc pass occur recall plug inequality execute valid define define shift must b iteration define interval part give future recursive none recursive call interval range yet event whenever execute event happen recursive call none define point event happen continue assume occur begin show event denote step occur define intersect future total thus least none occur ensure add draw event consider iteration iteration define step empty consider iteration define occur particular contain interval mean execution whenever execute initialize ne event occur throughout denote occur argument triangle therefore q pure private al pure operate database slight different differently concept label differently concept construct I f fx q fs exist database proper otherwise solve let bind possible else second kind concept define construction separately database query sample private relationship privacy reduction private lower bind bind term use learner query complement necessary show arbitrary restrict via element show produce w r different hypothesis imply hypothesis small must start technical database exist sized hence chernoff hold case q good matching trivially fs show mention first step private show imply modify define dc label notice class connection operate database predicate utility c error happen return event existence obey every happen mc ensure proper learning able use task concept concept treat randomly exist divide laplacian set output step first output big differential privacy next analysis step execution denote neighboring let output neighbor identical change database mechanism f overall two private algorithm private private utility execution h note moreover I close assume case label c hence get c label tc eq predicate proper note assume efficient pure class contain concept prove low reduction similar appear show require proper learner learner ignore part maximal cardinality every moreover cc acc c number remove hypothesis ga sm least database chernoff every appearance good database contain randomness distance ensure big moreover whenever construction yield learner learner derive necessary require database exist cause twice every operate database lemma increase database lemma equation exist proper learner privacy reference necessarily identity reasonable privacy individual consider label denote private database label pac concept differentially private definition algorithm see correct learner constant learner learner private ns mb b concept add quality describe construct unlabele every realize use mechanism labeling property mechanism private utility event set hypothesis choose event existence thus ensure choose mechanism ensure event happen algorithm nc chernoff bind hypothesis happen least happen least see privacy model recall label private require privacy label scenario privacy publicly preserve privacy scenario preserve differential private semi class non learner specific privacy complexity see label ignore privacy task know semi private learner guarantee privacy case privacy guarantee private learner complexity et helpful discussion idea gray theorem corollary conjecture complexity private pure differential task differential call consider task observe quasi concave instance allow construct align private learner relaxation private label vc completely characterize learner label privacy constant privacy collection individual privacy private task preserve privacy task also differential privacy differential privacy privacy privacy individual require individual affect formally pure differential privacy one output whether database significant private privacy private private operate classify private focused pure et show construction hand traditional learner term learner exactly domain picture change private learner come must evaluate many complete complexity learner recently give show dimension randomize communication separate show improper private dimension class sample differential privacy significantly satisfy pure privacy observation privacy complexity separation pure learning give task computationally pure computable notion predicate private agree fraction give pure differentially bound partially support complexity difference simple differential simplify exposition omit variable length domain release immediately learner approximate differential suffice tool proper private axis point function threshold maximize sensitivity call growth problem define concave solution order concavity quality observe solution recursive solve quasi iteratively define
convex objective objective monotonicity curvature reasoning prove rich mathematical language repository solver use solver include solver source interior solver solver representative inspection concentrate involve affine language convert time produce framework model language solve modeling framework times l l package notably fast science fellowship stanford atom atom monotonicity curvature affine affine affine positive atom atom cone atom concave sdp convex solver l language sdp exp simplex interior interior c primal x begin minimize solve begin I subject e x p subject solve subject x usa david describe translate language tree representation global infer problem rule programming pass dramatically reduce verify choose programming software verification checking modeling translate user solver language gap mathematical form composition computation convert solver description dynamic familiar technical language matlab match orient implement depend inside project technical compute high abstraction abstraction generality toward abstract formulation mathematic code language separation operating benefit operate parse solving familiar specify mathematical constraint concern structural form example affine program lp affine call quickly convex concern devise division solver structural problem appropriate purpose associate language make formulate solver jump design significantly outperform purpose solver target program exponential frequently automatically solver automatically embed matlab use concern idea language notably specialized parse requirement include optimization convex problem solver programming describe represent constant another kind simple expression matrix z variable positive symmetric nonnegative eigenvalue semidefinite treat positive variable z automatically underlie derive change expression word expression parametrize compose norm expression hence call addition atom argument useful feature arithmetic array indexing transpose valid indexing multiplication expression atom involve atom call expression type atom sign top atom argument curvature curvature atom affine monotonicity atom monotonic method wrong sometimes example return define use suffice five implement variable result apply atom represent leaf node atom refer leave structure closure name acyclic dag dag evaluated already assign evaluate evaluate aspect head together child expression automatically unique identification place problem reduce pass dot rectangle fill minimum n n n standard strict strict convex consist minimize maximize constraint x object appear example construct property objective inf x notation display figure p program value annotate solver dual checking approach implement model range microsoft convexity determination hence whose convexity kind affine expression affine constant constant affine affine curvature use argument concave convex curvature need level atom atom affine curvature expression convex curvature affine concave infer say order curvature argument atom appear expression atom example convex convexity derive easy add atom allow user expand function recognize expression monotonicity quadratic observation observation implement sign monotonicity rule monotonicity atom expression multiple monotonicity type multiplication enforce rule add multiply apply child end return else statement enforce appeal code mathematic since time since multiple rather constraint expression expression affine affine expression convex concave objective concave sense satisfy form recursively problem iff call cone convex n x k specify cone extend problem affine constraint constraint trivial rewrite solver problem language list rewrite
see interested science sequence semidefinite base e setting guarantee thus meta guarantee comparison local descent g optimum reach search latter succeed highly optima local minima must minima ball likely suboptimal minima square minimum achieve good x mx capture guarantee correspond expand need way elaborate explain behind prove analysis polynomial x stem surprising extent notation terminology operator notation make low degree moment necessarily formal operator order emphasize often subscript degree system polynomial polynomial hard see definition imply point support polynomial write proof say polynomial equation satisfy equation bound degree degree polynomial cone sum square follow assume linear boundary convexity square combination polynomial contain contain cone algorithm mx theorem variant proof dual instead try existence sense author summarize follow solution degree seem come trivial good access wrong next match find plant let characteristic es minimize satisfying degree describe estimate constant let discuss language algorithm base eigenvalue proof true predict proof degree able prediction nature expansion rao however approximate set tend yield proof closely moment largely discrete variant language help match theorem characteristic set proven distribution quadratic moment iy ix remark produce large take constructive moment first assume eigenvalue positive define equal second shift carry least positive test satisfy ii expectation case capture heart claim imply scale scale choose random therefore need polynomial expectation spectral concentration ingredient linear constraint u p lemma sampled satisfie event establish behave polynomial result old sum satisfy b inductive iv triangle norm degree invariant used norm iv iv b together follow establish u md j moment vanish I big magnitude sample md probability hausdorff vector upper maximum problem replace hausdorff ax show relation u factor depend hard unit pa allow determine polynomial constraint u p linearity transformation conclusion lemma idea argument one column different latter degree even merely sum connection mention predict particular give perhaps candidate heart may seem priori completely subset expansion geometric notion expansion equal otherwise enough mass sx see moreover project allow sense dominate heavy x vector capture expansion span theorem know hard omit reduce question maximum polynomial design could resolve could question interesting imply I w norm simply dimension unit subspace contain subspace frobenius define equal trace equal use q hand inequality exist eq hence I x depend bound dimensional sphere require see reference imply tend zero want distinguish graph achieve game improve exponent whether could namely two degree even vector give every evaluation seem evidence consist several natural come instance need parameter value turn proof fact heart proof show constant reasonably degree mean yet question provide reference diverse method theorem theorem traditionally tailor development surprisingly necessary great problem predict class computational polynomial algebraic geometry control theory program diverse quantum verification recently game square particular tool bind obtain solution optimization square new guarantee interest possibly primary semidefinite programming theoretical understand solve efficiently one regular regular set let set large vertex leave start multiplicative hence computationally purpose often purpose notion graph people system understand linear vertex hard compute assume fact quantitative infeasible maximum independent assume polynomial tend hard arbitrarily approximation trivial discrete yield computable efficiently graph bound away isolated keep sophisticated come hardness efficient rarely match tight hardness computing already know give non trivial formulate body conjecture defer hardness challenge trivial beyond reach complementary mean result efficient one broad sense polynomial perform game find going isolate unified complexity meta predict technique quite setting time view common setting give hardness existence conjecture imply vast show class include meta constraint closely relate well correlation precise yield efficient though actually factor summarize hard capture concrete meta already improvement conjecture attractive question conjecture discuss promise approach potentially game latter beyond problem tool context yield meta notion summarize could understand instead game focus implication direction suggest probably survey least somewhat conjecture turn eigenvalue approximate expansion exist mention absolute hard give approximate reader also imply optimality merely become become quantitative relation surprising imply tight connection hardness problem constraint problem priori nothing stick survey unique conjecture proceeding skip thought satisfy property restrict two author size game hard structured structured conjecture round game name survey square application expansion relate hilbert yield yet false plant combination evidence evidence subspace much paper issue game excellent survey survey survey focus around problem explain semidefinite survey entirely survey mostly implication hardness meta hardness manuscript actually understand go beyond basic lp sdp description topic volume topic develop researcher nesterov meta programming
consider generate swap neighbor inclusion metropolis transition inclusion vector allow adaptive comprise real let tx produce draw calculate weight sample reverse proposal unique predictor swap initialize predictor treat priori example framework scheme useful sampling inclusion enhance stand add remove neighbor subject variate discrete draw independently define forward mixed remove define reverse accept detailed posterior lemma version move function algorithm follow draw define move consider ar swap construct forward r sm use define reverse remove neighbor move j use function define reverse predictor k r r r define reverse swap neighbor correspond reverse pair forward reverse neighborhood proposal pair move scalable linear g smoothness adaptive mcmc within section predictor rapidly move complementary update inclusion pair scheme pair divide explore predictor move n transition predictor nn cd n index predictor define pd pair swap empty p jk ps move facilitate rapid exchange active component hold component e transition proposal pair swap proposal maximize optimal empty pd ps index randomly select update move particularly predictor across replicate run real predictive consistently move table provide section appendix transition kernel preserve k define fix upper number allow component predictor little variation ratio remain active additional subsequent let per per budget neighborhood individual update initialize one inactive component vector fix budget active neighborhood budget k il propose version chance add move chain burn period convention descent score increase include converge equilibrium see stationarity maintain score pair select predictor form neighborhood function pair crucially predictor predictor move allow neighborhood subset neighborhood particular neighborhood initialization expect neighborhood mp vast predictor retain neighborhood maintain paired swap letting maintain importance score predictor neighborhood function reverse neighborhood recommend term square rmse initializations independent replicate complete minute sample thin subsequent every draw data dimension ard exponential predictor suffer accuracy function final outcome concern simulate fix x degree nonlinearity vary consider process component display empty burn explain active sort importance observe evenly example graph predictor active thresholded quantile relationship important marginal importance th jt across namely fraction predictor recovery identify important predictor inclusion addition dramatically see section active utilize appear inclusion graph important predictor pair proportion co inclusion well show two effect additive predictor bilinear effect regression interaction successfully recover interaction way predictor configuration explore persistent predictor component replicate mode difficult effect neighborhood subsequently discussion predictor identify mcmc coefficient produce component model histogram empty roughly univariate least half edge graph extreme sparsity nonlinearity additive structure inducing adapt underlie smoothness scaling interaction fail component model configuration median dash line figure rmse report replication standard interaction quantification modeling confirm mcmc sampler pair inter move provide reliable plot test consistent excellent value line band rmse appear average rmse lasso ard covariance regression important primary gp mcmc estimation ard predictor dimension development interactive reason two generate predictive summarize ard iteratively improvement ard enable superior even small rf remain remarkably increase predictor replicate rmse ard map rf plot predict enable band four statistic offer effect mostly high count vary moderate example validate satisfy budget move adaptively rf compete dramatically lasso clearly ensemble nonparametric good good ability accommodate degree interaction rf rule leave fraction compute predictive hold rmse average split appear pt crf across component sort sized model dash datum primarily capture component inclusion identify predictor effect mostly interactive effect compare variance partition dataset move real display bar b algorithm trace variance serve measure dash hold test mcmc move var active empty component size appear vertical line middle marginal right predictor edge co across gp marginal split predictor accommodate probit future package accommodate predictor e development approximation enable mcmc enhance score finally mcmc move inclusion tackle local share predictor section component move subsequently interaction drive force pair move sampler obstacle additive fairly e move follow component subsequent pair move enable second pair move separate enable move toward grants es institute environmental health st recommendation author reflect section pair move neighborhood efficiently introduce neighborhood preserve stationarity pair move sampler predictor adaptation draw approximate posterior draw update beta l ap cd pd ps cd aggregate scale covariance let stand vector aggregate realization lx kolmogorov consistency gaussian response sampling proceed draw draw posterior quantile wise credible addition inclusion configuration thresholding probability run configuration evaluate invert neighborhood sampler update inclusion vector require iteration grow pair control exceed iteration cholesky aggregated enable involve overview inversion cubic order select control stationary pair move proposal balance verify pair remove add symmetry check swap swap proposal choose probability proposal reverse remove neighborhood contain q swap restrict pair reverse case inclusion vector remove swap probability pair reverse section construction predictor denote reverse pair likewise reverse paired expression move proceed pair pair swap probability move neighborhood construct section empty km pd ps pd j cd pd ps hence comprise pair swap preserve stationarity pair sampler predictor importance consider plot datum plot comparison section plot predictor importance bar trace additive interactive offer minimax wide case predictor large sample predictor effect response bayesian implementation interactive efficient markov hyper specification light computational consideration explore inclusion offer improve diverse real platform regression keyword model multiple try metropolis selection weak focus parametric linear regularize shrinkage assumption often model naturally occur predictor response relation globally remain quantify effect parametrization adjust non linearity several smoothing regression predictor relation mathematical performance various setting computational poorly evaluation importantly nonparametric assume predictor greatly curse set variable small e ensemble learner additive dimension address curse dimensionality may actual behave black box forecasting additive structure univariate ignore predictor unknown predictor additive interactive resemble distinct add learner boost efficiency avoid overfitte contrast interactive high divide piece add together enable learner seminal work gp indicate specification could suit motivate recent attractive interactive set additive interactive away extreme single bound smooth include minimax correspond restrict component one predictor increase develop additive interactive abstraction offer match rate significance present detail hyper allow pattern maintain adapt stochastic try metropolis additional strategy well state art interaction diverse section provide evidence interactive attractive platform high propose chain sampler effective extension work cx definite mean refer continuous supremum pair additive interactive perform joint proceed sequentially parameter regression conjugate analytically marginalization obtain marginal multivariate back fitting proceed enumeration intractable large sampling metropolis draw diagnostic stability sampling scheme posterior inclusion update inclusion random propose flip quickly grow move look assume inclusion vector walk toward add predictor rather remove search search quickly demonstrate develop explore straightforward adapt search additive add neighbor remove swap swap
conjugate analytical insight base include cg cg distinguishing include conjugacy general numerical principle conjugacy instance generalization primarily inspire cg contrary preferred justify proposal light algorithm issue parameter automatically note direction bfgs cg see think extension framework regard see role conjugate chance bfgs update quasi newton anonymous eps fill receive paper dependent namely generate conjugate preserve conjugacy value obtain cg provide cg solution wide real year reliable solver framework either consider considerable specifically aim pde constrain framework often specialized reliable iterative symmetric numerical analysis context detail linear generality implicitly assessment system people work great software development literature analysis become address equivalently reduce cg suitable choice primarily intend efficient mainly inspire quadratic generation parameter scheme conjugacy cause precision computation intend numerical experience clear proposal assess similarly currently proposal outperform cg carry selective extension cg symbol indicate definite symbol sect review cg subspace promise detail relevant conjugate direction motivate member property class sect conclusion include table process conjugate generalize share say cg iteratively cg step stop else k tp r ap symmetric often application cg residual search direction impose conjugacy condition prove implicitly satisfy practical computation fail property lose consequence sect detail purpose table cg eq expression generalize exact fulfilled process method cg cg cg generate sequence satisfie yield inspire order cg recurrence correspondence may cg extent resemble recurrence direction cg cg recurrence direction possibly conjugacy cg offer cg preferable solver statement briefly conjugate truncate latter rely direction direction outer perform step cg approximate table form framework suitably call curvature direction optimization see conjugacy introduce cg conjugacy loss great importance hand rule assess parameter satisfy thus conjugacy solve accurate introduce cg additional cg latter cg sect aspect work effort since independent necessary address current direction impose conjugacy previous automatically matter framework essential control computational item iterative might generalization indeed proposal item respect cg latter user conjugacy direction precision sketch table cg cm else compute else compute cm direction reveal difference cg compute eq coordinate conjugacy specify detailed sect double recover sequence compute impose orthogonality cg result cg additional inner scalar table evident provide term recurrence conjugate condition involve computation hereafter positive cg relation directly conjugacy direction e ap inductive yield conjugacy property trivial lemma theorem simplify available relation remarkable avoid storage step require storage cg stop condition result class cg orthogonality hold ap ap k coefficient k r tr inductive q along r tr hand inductive lemma prove likewise cg iteration assumption solution reader may integer chance table cg stop else else analogous cg prove see suitable inverse far minimize ap span indicate ip ap bp cp bp assumption table ia ia I latter yield tr ap p tr I tr yield functional inverse recall direction observe geometry might substantially contrary kp k r consequence directly similar cg cg might possibly observe cg step storage additional idea store base example approach may cg may equivalently explicitly impose conjugacy pair implicitly impose cg cg recall worth hold modification tp kp cg indeed cg k cg necessarily item cg relation satisfy cg possible latter conclusion order cg condition latter table relation red else cm cm else r cg satisfie position lead scheme red table recall substantially impose unique alternate cg analyze combine preserve cg
send describe algorithm order get start expert knowledge quantify transform level dimensionality iterative quantify fuzzy rule linguistic test realistic environment different robot environment result different application play central tracking mobile quantify fuzzy genetic fuzzy mobile behavior motion behavior whose robot environment evolve range etc order environment operate environment internal properly convenient cope interpretability rule fuzzy logic represent design mobile preprocesse raw sensor obtain sensor usually high mapping preprocessing automatically level describe embed avoid expert input variable controller sensor robot mobile e g capable dimensionality meaningful description proposition kind expression set level variable expression low fuzzy proposition formal evolutionary fuzzy rule combination fuzzy logic genetic system aim balance interpretability rule conventional refer low tree describe learn preprocesse mobile quantify fuzzy learn unconstrained e proposal design mobile internal robot sensor structure learn robot sensor preprocessing embed able linguistic interpretability use validate statistically combination preprocesse tracking tracking move obstacle structure present advantage mobile learn show point relevant conclusion machine evolutionary neural widely genetic fuzzy even combination evolutionary mobile get type logic knowledge evaluation function evolutionary function genetic fuzzy alternative membership membership distribute expert knowledge rule reducing learn main behavior curse among evolutionary position learn sensor competitive proposal learn involve label unconstraine multiple approach adjust balance competition rule behavior comparable proposal output control category establish level modeling hand provide sensor relevance group significant decide analyze individual range since gap frequent environment usually consist measure measure right mobile low stage traditionally knowledge preprocesse expressive meaningful within quantify fuzzy proposition useful belong clearly set proposition conventional preprocessing stage high variable group fuzzy high fuzzy apply reasoning tb evaluation evaluation equal equal min check ex robot two individual sec approach produce individual context structure define compact v terminal symbol separate leave two consecutive group order level symbol linguistic prop ii linguistic fig prop prop measure linear linguistic linguistic label linguistic approach universe space uniformly var five linguistic individual example label linguistic membership mask label limit apply linguistic use measure finite receive triangular search mask tb velocity initialize two linguistic velocity fitness calculate th define meaningful output example maximum minimum regression several desire interpret individual code individual fitness population ability support cover final line admissible cover covered support calculate cover total combination strength generalization matching go define objective information proposition rule important individual initialization population operator generate individual proposition proposition proposition one select individual accord follow criterion linguistic high r b individual copy yes yes proposition combination take similarity fig take partial merge could rule individual individual proposition eliminate partial merge combine proposition done minimum performed mutation two generalize rule high value mutation high confidence rule cover discard select min example give similarity individual cover therefore proposition mutation proposition modification among possibility generalize adjacent repeat decrease proposition adjacent process proportional example low probability modify proposition modify modification among possibility velocity proposition strategy mutation finally mutation mutation part select fig membership close label close one tb mutation selection replacement steady state new population epoch rule criterion fig limit vary stop consecutive iteration maximum stop regardless ends add moreover mark cover algorithm line fig part select subset rule follow basis rule rule base code indicate th good rule rank fitness execute set last implement search threshold well objective controller suitable robot move high control player robot software environment also real robot range amplitude scan loss generality robot sized distance value input principal component extract variance cover total use preprocessing change percentage input configuration configuration preprocesse preprocesse tb statistical significance table fold algorithm mean deviation eq tb train c straight concave convex concave convex min sample fold cross example preprocesse validation low configuration show include able adequate algorithm mobile controller good validate controller environment different difficulty assess velocity path execution simulated environment fig indicator linear velocity linear consecutive cycle reflect along robot robot parallel five calculate environment present five symbol table path period time controller velocity form purpose reliable robot alg preprocesse min min tb hc alg cm home home office home home office home office min home home office home home office min home home office home home office environment hc alg min min environment c except distance perfect situation result obtain low like curve get adequate velocity smoothness robustness fail environment recommend compare result hoc detecting table significant environment velocity corner velocity change velocity robot value preprocesse c hc alg cm office office office office finally proposal rule behavior purpose comparison expert sensor weight linguistic local evolutionary orientation table common environment environment learn embed method rule linguistic denote robot describe continue zero velocity get robot cm typical number base multiple show term standard deviation input tb alg output min c c proposition straight min straight concave concave knowledge rule per thus demonstrate general basis mobile tracking application behavior describe literature guide robot predefine guide person show goal service service path team build surveillance environment operate area environment numerous author able path uncertainty control measurement robot static difficulty combination perform people track environment task safe must endowed implement allow order mobile must application guide robot predefine initial new people make necessary avoid return predefine quickly reference guide come close obstacle maintain addition robot track object move mobile behavior fusion track controller controller robust neither avoid obstacle obstacle describe obstacle side solve depend obstacle detect robot value robot establish distance safe tracking behavior angular robot place move object different robot coordinate robot coordinate robot robot negative indicate robot move point move robot move angle object robot angle move object track perfect robot keep htb colors code medium path grey move obstacle path dark grey f place move behavior validate different environment try reproduce fig track medium grey path follow include robot mark velocity moreover without environment place track grey path indicate trajectory robot avoid successfully robot predefine obstacle generate robot obstacle return predefine path quickly possible move fig light grey represent robot also medium grey trajectory object go situation controller execute execute corner close robot behavior move obstacle fig light grey follow robot track grey avoid follow moving avoid show
break tie combine enhance accuracy execute time integrate draw kernel execute membership partition ensemble obtain fig illustrate cluster randomly datum cluster consensus membership sample matrix similarity complete weight meta meta vector update accordance less mean algorithm kernel portion need store classical expensive avoid eigenvalue large solve optimization rate accuracy computational cost ensemble additional empirically sample reduce considerably binary random membership express give matrix u minimize n n complete add kk n identity special matrix first twice q mean approximation introduce trade speedup denote coherence adapt gap refer experiment examine different sample size range satisfactory memory vary cover table small medium imagenet mnist processor performance demonstrate scalability improve implement matlab matlab toolbox available house ghz processor limit imagenet pyramid cover imagenet mnist kernel algorithm imagenet consist million concept know sift descriptor extract handwritten available image represent dimensional training test compare approximate performance compare step algorithm find representative mean imagenet employ pyramid pyramid effective take histogram pyramid mnist neural range directly demonstrate naive class time matrix cluster initial final clustering calculate ari adjust rand lie matching partition algorithms table table list run kernel algorithm speedup mean take need cluster later execute pyramid pyramid play mean column ari algorithm mean partition partition show achieve kernel achieve lower sample significantly achieve indicate insufficient center randomly kernel mnist spend calculation simplicity size fast ari column inferior partition see amount except algorithm well comparable mean cluster c c time cluster imagenet strategy table fig table pre sort index execute non great high complexity sampling fig inferior compare also produce scheme uniform strategy strategy spend cccc imagenet comparison eps large type scalable use dramatically reduce requirement survey us forest forest attribute qualitative like slope distance use group type region resource information network datum dimensional seven class represent traffic represent traffic set currently infeasible store reduction take hour large demonstrate non scalability eliminate algorithm evaluate range increase cover employ pairwise similarity employ set tune optimal number cccc run mnist calculation compare cluster less algorithm effectiveness run algorithm fast achieve error set imagenet forest network cccc c take averaged run set show ensemble especially cover significant improvement cluster small thereby efficiency avoid kernel restrict center small algorithm yield well popular integrate propose algorithm enhance enhance scalability plan acknowledgement office grant foundation research association ci use matrix partition full extension include definite coherence coherence column contain ss provide positive constant value row j ss combine equation pt extend enhance tight kernel approximation ability various gain popularity iterative ease run term size large cluster mean cluster center well demonstrate requirement quality employ cluster meta advance storage year massive amount generate service audio one tool amount web medical set group hand kernel cluster employ distance object cluster ability capture linear perform distance cluster som kernel neural propose mean simplicity efficiency addition several equivalence replace euclidean function storage kernel cluster scalable thousand learn cluster adapt use low kernel name follow idea avoid center vector span subset portion kernel lead speedup approximate explicitly yield efficacy relate scale kernel algorithm incremental cluster implementation restrict requirement datum cluster center center though complexity memory match unless large superior kernel mean fact mean center combination word span x n center small small ii simple denote give similarity point cluster membership
text collection practical multiple leveraging share potentially enhance multiple news ad etc explicit social make available interaction user complete observed leverage correlated component affinity among type wherein entity low dimensional represent entity leverage share attractive often view interest capture recommendation collective completion additional collective statistically ill pose decay observe observation localize individual entry oppose measurement completion development statistically optimal collective previously analyze collective completion trivial recovery collective collective completion completion challenge collective extension complexity exist completion leverage low key collective normalization observe general collective structure joint may behaved enforce avoid case assumption relaxed contribution algebra collective identify assumption feasible tractable collective consistent collective subset collective exactly collective logarithmic program adapt collective matrix scalable algorithm significantly use tradeoff accuracy paper simulated besides relate probabilistic seminal rank collective al wherein parameterize share factor collective author collective factorization completion guarantee scalable algorithm probabilistic collective develop algebra analyze collective denote letter etc matrix denote singular norm etc affinity primarily represent list affinity entity type entity affinity relation wherein pair type affinity relation entity relationship collective denote entity imply either collective graph connect handle entity instance collective common entity view convenience introduce alternate equivalent collective matrix paper collective represent form entity statement concatenation list collective represented identify block wherein block representation collective collective provide j collective possess factor dimension denote value joint factorization exist collective convex extreme symmetric sec collective hull interest also iff atomic program collective collective denote basis far without noise e pose dimensional etc low impose ground truth collective collective entity see rest projection onto iff matrix entry significant analogue assume incoherence basis c onto factor dominant atom need pose subtle challenge collective undirected assume equivalently odd cycle induction verify note collective completion necessary j n si k cardinality scheme expectation learn give entirely consistent depend convenient derive collective rank collective use atomic suitably modify sense practice program collective algorithmic consideration state recover truth collective cardinality cardinality type collective incoherence requirement meet kn cn kn kn free collective scalable adapting solver collective atomic state convex program collective cast solve loss z z u propose estimate primal error curvature function iteration involve large eigen non converge eigen accuracy collective rank entity sample present paper factor collective low also impose component feasible component standard matrix estimate standard complete entity independently recovery complexity sub completely share jointly collective collective also cast standard complete block collective partially coherent fail strict exist task collective optimally share dimensional sample exploit narrow subspace analogous show condition exist adapt introduce al c vs supplementary sm km lemma follow assumption sec exist ty f proof supplementary material complete construct partition r sign satisfie follow analogous proof fp use inequality use section intend ground collective entity entity collective
extensive accuracy suggest greatly partially explain regret near neighbor knn latter simplicity category conceptually organize instability minimax section propose classifier achieves present comparison exist neighbor stability devote couple take regard label object probability borel classification define rule x py define distribution classification instability x distribution define classifier measure instability measure classifier denote distribute copy training instability classification instability obtain classification ease procedure knn knn gaussian setup vector identity versus knn calculate classifier aggregation decrease sum view balance knn classifier minimal mark slight lead improvement stability confirm theorems extensive experiment sequel study function nx first show derive two condition satisfie contain q integer say support denote lebesgue center radius lebesgue deduce function distribution respect margin condition worth note type addition hold newly minimax lower seen bind old marginal optimal set requirement large stay long take lastly slow increase introduce attain review weighted neighbor explicit propose novel call moreover theoretically sense fix distance weight ni ni reveal compact nonempty open ii conditional absolutely continuous lebesgue differentiable exist dx sx x ax volume asymptotic knn pointing condition see assumption ensure vector near section classifier special subsection classifier sequel instability asymptotically appendix sketch detailed include nx ni e dx nx last area proportional asymptotically large knn meanwhile instability serve develop term expansion call variance instability ready procedure regret determine minimize acceptable lagrangian minimize ni depend multipli lead knn proposition accuracy classification instability lead respect minimizer weight classifier rate define assumption procedure corollary stay away approach theoretically comparison exist knn procedure significantly improve regret classifier knn neighbor apply near neighbor subsample majority vote sufficiently particular replacement approximately resample lastly achieve minimal denote classifier knn difficulty weight classifier step knn knn ratios notable ratio merely dimension corollary plotted figure stable knn procedure large ratio knn equal less bag variability phenomenon ratio bag knn accuracy furthermore quickly stable vanish regret note great compare ratio reflect characterize corollary constant improvement ratio figure show ratio function fix dimension increase increase ratio get grow htb investigate improvement large regret relative improvement percentage expression show logarithm large confirm introduce simulation base validation tuning subset proportion misclassification parameter value lead weight calculate summarize subset train label classifier j step repeat training search tuning minimize cross validation preference weight tune select whose risk pre choose set minimal instability subsection form derive ratio least sampling region replication calculation theorem risk see minimize lead figure along asymptotic asymptotic one carlo although slow phenomenon theoretical corollary accord ratio classifier ratio indicate increase however appear value cause classification error estimate issue classifier classifier subsection neighbor tune equally space comparison equally spaced fall simulation underlie probability choose df toeplitz entry classification empirically verify comparison combine latter test datum replication figure classification procedure case regret even explain dramatically small particular improvement procedure procedure knn procedure phenomenon advantage prediction big summarize obtain minimal achieve minimal scenario large see appendix slight procedure minimax knn bayes bayes sim subsection investigate knn machine survival diabetes heart heart information dataset randomly procedure error specifically procedure improvement addition knn procedure knn agree slightly illustrate accuracy significant improvement knn error error acknowledgement like thank mathematical sciences massive part thank communication em independently bayes n n n ease nx x p last prove call hypercube mapping denote positive define partition satisfy collection hypercube absolutely continuous show variational specifically eq hypercube hypercube supremum rademacher corresponding p n lemma nx part separately show nx appendix nx nx nx x c properly ni n ni nk ni ni ni ni detailed discussion constant substitute du b du plug du iii plugging iv lead desirable conclude e valid independently identically sample without give norm nx ni boundary x nx x n ni n show analyze complement combine apply normal yield derivative respect lebesgue measure tx theory integration manifold since r similarly argument imply contribution ps nx mm w apply let ni imply term due modify expansion accord sr rs r r density normal tx x ax ns dr argument substitute difference du tx dt du iv
use collaborative item easily optimize factorization nice mainly representation new ml yahoo regard ask rating use mainly quality protocol simulate realistic process income proceed user user denote user dataset model remain validation apply subset ii answer rating answer question rating rating rating predict rating system predict regard miss explore quality approach use equation collaborative mf present item knn pearson cf inductive model train rating phase testing tuned gradient size latent thus parameter grid randomly initialize figure table evaluate split evaluate predict provide result less information obtain ability additive user four obtain mf score yahoo knn representation belong moreover knn slow unable deal beyond able predict user time mf ability realistic mf item benchmark select item item rating popularity entropy g baseline rating inductive coefficient cs user pca depend positive rating concern modification consequence concern norm smoothly start interact provide easily integrate translation representation recommendation I warm learn regularization explain learn choose incoming result set protocol new set illustrate add warm extension link warm promising percentage add american iii star episode iv star episode vi matrix private lose selection recommendation past user mix rating informative technique distinguished neighbor item similarity item infer factorization technique collaborative major limitation process ask paper comparative question criterion like popularity greedy minimize seed also consider choose fit user branch correspond answer present node allow contain regressor model one tree warm interesting usually adaptive user usually one rate collaborative filter representation user latent translation depend allow user rating recommendation direction certainly incoming process consist review item user build ask new investigate I acknowledgement article rapid approach collaborative filter rating rating approach handle user start context ask initialization rating present good ask efficient representation context start show rely extract useful highlight seem crucial go intelligence gain parallel recommender system field research variety application aim product facilitate experience recommend recommender implicit movie star item song post movie actor common recommendation additional context method problem representation compute user us user user rating proper rating item latent representation user representation denote classical rating dot denote predict learn sparse rating objective observe rating user representation item loss propose since item nature well adapt practical application item require unstable limitation factorization approach rely representation e must indeed mf cf base interact rating make recommendation method suit focus method rating ask recommendation approach context simultaneously learn rating building representation inductive rating latent allow contribution formalism user formalism build simple nature user dataset start quantitative effectiveness qualitative learn organize generic representation present discuss related collaborative filtering propose rewrite detailed integrate start consider item building collect set rating compose item rating opinion typically movie provide rating movie select relevant collect item user thus rating loss cf free balance simultaneously build formulation easily mf base obtain computation complexity idea concern build recommendation rating representation suitable cf
finite increment eq collection q prediction depend segment roc confusion roc define interpretation follow plot receiver operate roc curve join roc integral curve roc curve curve roc sample observation parameter roc voting threshold word parametrize forest parameterized nn well prediction true roc would generate since evaluate prediction area allow true disadvantage course classifier divide make evaluate discusse divide classifier denote simply evaluation set accord depend nn trees forest divide run testing usually set often repetition consider train explain follow allow classifier labels label assign evaluation testing iterate q particular fold one set figure illustration process disjoint cross validation run multiple parameter determine fold commonly use evaluate simple suffer possibly train often furthermore conversely disadvantage repeat consume chapter evaluate accuracy f divide training evaluate chapter explain methodology forest genetic dataset nucleotide snp thesis reduction come heart study detail algorithm cross heart file thesis patient control patient file storing simplicity two count list snps dna microarray patient determine three snp minor probability near directly disease transform transformation let minor equip domain nn goal section explain methodology classification genetic cross project genetic projection nn forest random genetic virtual laboratory increase increase snps classification merely threshold forest result obtain technique method forest able predictive accuracy area roc considerably higher nearest chapter conclude thesis limitation short thesis two nn forest dimensionality random term thesis nn highly desirable learning algorithm finite thesis justify distance supervise closely thesis compare approach genetic study reduction comparative selection random considerably nucleotide snps important selection classifier receiver operate curve previous area thesis science perspective although evaluate random correctly snps entire genetic important community validate predictive genetic trust ideally snps genetic snps label snps concern certainly improve thesis concept control whether included ignore thesis recall thesis snp assign find raw dataset normally snp high snp check condition snp satisfactory snps control thesis control remove snps start snps considerable remove question remove related disease experiment control remove demonstrate forest exactly snps score yet high previous evidence result dimensionality reduction quality decrease snps efficiently case quality reduction handle three snp thesis classifier give predictive classifier machine neural advantage snp discrete distance simplify pair snps common simplification assign snps theorem provide margin would continue develop reference study classifier margin limitation thesis new predict snp information hope address limitation thesis future far classify disease day accurate whether patient disease discussion science parent heart science subject intersection mathematic valuable large technology large magnitude information big practitioner computer often cloud compute big science notion detect email detect email spam amount past spam spam email spam give equip spam spam algorithm classifier information observation mathematically present observe denote know major challenge big dimension computational map label would simplify commonly know learn near machine dimensionality include component pca discriminant lda main goal thesis dimensional dataset disease disease heart severe lead heart attack variation repeatedly variation account prediction disease individual variation dna base call nucleotide snps successful diagnosis understanding behind variation snp dimensionality thesis explain random novel theory mass introduce thesis recognize science predict majority vote important theoretical consistent rough arbitrarily predictive ability become partition hyper belong forest generalize trees bootstrap label vote unlike forest consistent excellent practice dimensionality euclidean lemma sufficiently absolute multiplication variable training via nn project pairwise preserve guarantee second reduction mass problem natural marginal separation infimum integral metric simplify consequently dimensionality take value coordinate finite correspond two eq dirac occurrence observation mass separation dimension coordinate distance detailed comparative label contain nucleotide snps heart study nn genetic space prediction forest select equivalently snps dataset apply prediction predictive f area receiver roc curve result predict disease subset genetic area snps snps snps area roc high whose snps classifier contributions thesis theoretical contribution thesis consistent although result direct consistency nn euclidean three universal lemma sufficient nn thesis see thesis introduce completely mass distance thesis practical thesis nn heart study disease nucleotide apply practical contribution thesis approach apply achieve curve ever obtain thesis structure foundation explain concept chapter discuss genetic dataset thesis disease genetic single nucleotide snps genome association chapter survey generally application chapter near finite forest include classifier chapter discuss feature selection thesis outline motivate justify explanation distance three area receiver operate characteristic chapter method divide evaluation estimation classification dataset chapter genetic consider thesis include snps methodology projection nn random genetic list reduction projection chapter summarize first projection approach brief comparative finally chapter conclude thesis address limitation list direction throughout theory thesis mathematic science thesis orient intend reader thesis explain reduction genetic result biological biology genetic keep master thesis biology question provide thesis define predict label dataset notion dna level genome study genetic formalize algorithmic sample applying theory base coordinate label sample label pair pair merely classifier define satisfy false classifier classifier since proved know universal consistency important supervise prediction infinity define learn call size far define universal become terminology learn family classifier take consistency rule take sample thesis simply always rule stage train predict explain know provide understand biology behind thesis health brief dna nucleotide explain genome wide association collect introduce work literature involve united number death disease disease occur build heart lead heart attack death understand important care population increase show diabetes stress prevent accept biology heavily regard whether disease predict high purely individual thesis attempt answer term information nucleotide explain biological study pass trait parent trait molecular dna encodes trait pass human dna contain repeat together complementary commonly dna string omit determine individual double paired structure dna base base pair million organization person genetic genetic dna genetic pass parent variation sometimes dna precisely trait variation pair dna nucleotide base nucleotide snp normally variation position nucleotide snp allele less term allele snp dna possible major allele minor copy minor allele copy allele dna variation snp minor million genetic among human dna level extremely certain disease explain genome snp associations physical trait nucleotide association trait consider trait dna microarray individual snp trait correction study snp mostly disease coordinate dataset individual microarray due microarray possible explain study detail include precise master thesis consider genetic predicting disease snp genetic thesis science study predict trait result publish survey past logistic regression receiver operate roc explanation show associate publish supervise compare support predict diabetes curve snps multiple snps observation forest ms snps roc curve ability machine dataset disease genetic snps disease modify classifier accuracy considerably also study disease two area roc curve snps forest study fairly argue trait investigate disease focus certain supervise reduction explain chapter explain science near consistent introduce section universal follow space equip distance nearest classifier rule search near classifier pair regression x distance x normally odd tie value life application determine figure illustration nn generally accord define negative add classifier sort observation equivalent nn nn introduce cover develop publish euclidean list together universal consistency use euclidean section explain particular classical generalize find nn theorem generalize consequently theorem universal require proof classifier condition non finite q notion convexity jensen along lemma entirely base pair classifier accord estimate quick scalar function jensen proceed show indeed estimate regression classifier q measure theoretic jensen jensen hold lemma right arbitrarily uniformly continuous middle term consequently eq since x prove satisfie generalize lemma prove consistency version whose order induce depend covering radius respectively ball sphere open radius dimensional finite sphere finitely require figure construction note sphere ball need circular prove imply since since span figure triangular generalize first depend unit still near inequality mark x less universal nn classifier nn eq assumption chapter demonstrate metric consistent classifier explain another tree chapter introduce science decision train predict forest decision construct decision tree build majority classifier build binary predictor explain classifier implementation classification cart label divide disjoint splitting feature homogeneity two recursion terminate divide contain class long improve label decision force geometrically correspond possibly axis consider observation provide formal explanation decision notion homogeneity term along tree coordinate assume ij discrete label homogeneity maximal contain classifier subset weight entropy entropy entropy label coordinate decision training entropy training split coordinate termination meet class termination partition associate observation observation would accord calculate recursive tree model parent divide node sample split recursively termination upon termination become leaf correspond dominate predict observation pass tree predict leaf fall package programming life predict representation example build new sound feature great observation classifier generalize first forest bootstrap take bootstrap replacement construct q second selection find build tree new decision prediction forest prediction classifier training predict total forest depend split usually validation explain excellent effectiveness easily generalization forest justify show classifier consistent fact common distinction theoretical produce accurate life absolutely support excellent predictive run chapter supervise forest generalize former classifier detail chapter introduce field extremely discuss concept section introduce explain section popular selection novel mass big genetic thesis include coordinate nucleotide computationally expensive may constraint technique often dimensional simplify e space training feature extraction reduction project threshold map property hand feature reduce transform simple projection define expansion method borel reduction introduce far easily feature extraction projection function introduce domain assign mass method integer exist map visualization distance theory nn distance map project pairwise guarantee simple constructive finding extraction work pick training x x tx include thesis justify via random matrix multiplication supervise section probabilistic provide proof constructive finding multiplication tail sub tail sub sub tail variable imply generate constant consist sub tail entry satisfy hold cn theorem generate pairwise enough great usually practice determine optimal purpose relate sub constant sub tail conversely suppose sub tail combination sub gaussian suppose real uniform prove tail euclidean prove transform distance classifier reduce save analogous also either discrete explain new distance method distance wasserstein explain infimum measure think minimal moving probability distance infimum extremely difficult distance else discrete metric space eq mass simplify distance simplification mass suppose hamming probability measure observation respectively theory exactly x consequently sample induce probability probability thought estimation
month date place much scenario unlikely person c give name absolute month difference agree disagreement since modification compare string simply name may piece token name token token token transform mean total token disagreement reader detail construct disagreement level except disagreement disagreement moderate disagreement take nominal truncation classified inaccurate nearly truncation parameter inaccurate fix year collection parameter field amount prior priori extreme scenario month name inaccurate versa context point posterior eight concentrate gibbs sampler supplementary discard burn eight partition although way eight concentrate day inaccurate posterior concentrate record entity record result coherent indicate field inaccurate record family name think inaccurate month fairly quite get equal day month think inaccurate name concentrate name become distinguish record therefore record probably finally partition posteriori record quite record assign partition properly uncertainty record record emphasize example determine posterior evolution membership gibbs sampler supplementary material five contain large file depend file heavily influence explore sophisticated datum generation corruption tool contain tool field permit generation adapt default describe characteristic file simulation file either seven involve include gender file seven phone number gender name jointly table frequency name name set name source phone number eight digit two field include default generator contingency serve contain category eight interval create allocate randomly select assign accord poisson interval allocate contain uniformly possibility miss error string error optical recognition use finally possibly name family name age gender phone pt name family name age interval agree gender agree agree phone code performance amount file field synthetic per file create indicate file disagreement name constitute prior truncation carefully truncation scenario believe file optimistic run iteration discard runtime implementation part language second file include comparison ghz processor start long chain report wrong another source gets code name although collect region indicate occur potentially give six token example record overlap token token la pair meet constitute remain use illustrative name standardize compare supplementary material record level either year month pair introduce record present f indicate belief likely exact still expect field agreement year death ij death high still go truncation interpretation record c truncation remain field believe error become unlikely magnitude indicate probability observe disagreement disagreement example p ij priori year death death expect year two year disagreement specification probability disagreement year table finally day death believe expect report node width pair never appear together group obtain sampler present sake record preprocessing pair graph package subgraph clique illustrate trivially pair preprocesse step partition get color width pair appear group together chain never appear appear group together method ensure output partition partition unique record minus cell file contain unique interval greatly vary file summarize different region leave percentage correlate report relation panel show percentage year ground truth important whether take region death record identify treat label record ground decision check idea partition partition file also like result change choose alternative prior one optimistic sense one truncation table subtract truncation additional keep fix table point indicate bold recall sensitive truncation recall robust optimistic agree finding file number application balance optimistic issue pairwise component classify pair pi estimation posterior triplet vary gibbs surprising treat model hoc methodology decision prior show illustrative realistic methodology time indicate report methodology usage distinguish point partition population important file account future multiple file incorporate record linkage procedure acknowledgment document discussion comment suggestion ball green file help generation synthetic brief standardize name implementation application node nsf census research nsf keep accurate account record detection independent status pair record ad hoc fashion file pose file group record present file interest ensure decision implementation incorporate file available decision multiple united truth record refer entity file entity file miss existence file need wide health census quality improvement arm receive report report come degree detail file step keep accurate form united occur report friend multiply leave national front sign agreement united henceforth report occur focus individual country information publish family member friend addition name occur friend detail provide nontrivial file variability record miss datum challenge difficult reliable record miss process field file linkage multiple file usually collection process assume file despite principle article approach record linkage model approach train know record pair type decision status record pair neither decision record truly record record ad hoc recently detection linkage decision distribution file file therefore currently categorical continuous model field name address phone detect field often advantage base pairwise comparison record meaningful record linkage basis decision currently take account decision modification detection problem propose build literature decision partition file closely relate record datum idea disagreement field introduction file natural similarly organize propose methodology deal illustrative address detect times united truth conclude file record record file grouping record entity entity represent think record file call representation convenient nonempty subset article entity record representation computation consider eq contiguous file record represent inefficient record alternative arbitrary cell record file safe assume entity file assign label potential entity entity labeling entity lead partition indicator label relationship obtain specify notice element possible fix get record rapidly record grow practice file fortunately early stage inference file entity record compare agree field completely agreement field normalize distance see divide set approach disagreement field appropriately fashion field divide interval disagreement agreement include high complete disagreement record comparison record record linkage construct inference require range functional field build disagreement variable generic long question threshold build level specific disagreement disagreement extreme disagreement practice simple number obvious early detecting reduce inferential record translate fix matrix turn assign record group detect refer survey divide file record categorical field reliable unlikely pair record field gender code record appeal divide type expect would unlikely record predefine ideally check record event date date death date record true containing field fashion context unlikely among record distant naturally assess ideally comparison string metric complete comparison comprise file still obvious combination different disagreement inferential disagreement field age record meet behind approach distinguish record probably record pair unknown candidate comparison present record partition constrain record already practice much partition file heavily rely able candidate medium file comparison ij comparison though realization comparison array compose observe among intuition formalize pair entity regardless record pair entity assumption widely employ file record linkage intuitive formalize comparison leave observe comparison fix depend depend formulation candidate factor ss partition take I proportion pair assumption decision undesirable denote z z n measure label label notice structure prior appropriately investigation commonly encourage formation cell dirichlet multinomial partition compose cell describe criterion record common record field miss comparison pair record incomplete field record assume miss base inference observe decompose record ij ij sum probability obtain arise miss comparison partition scenario except th record model membership arbitrary labeling use present use refer entity ratio square hand represent testing refer entity accord record pz I take proportional label exhaustive state cell record partition ratio likely get cell material gibbs parametrization assume field take distribution record model multinomial disagreement rewrite conditional specification parametrization pair f binary model record linkage
perform quantization train classifier joint manner optimize simultaneously column encodes pair verify obtain also iteration procedure radius classify three classification rate classify divide result bar generate classifier close superior bar output sketch cc black curve naive sketch number use mean recover rank track essentially constant bar correspond deviation face sketch classical sketch focus broad constrained showing theoretic view novel iterative scheme know iterative hessian derive showing grow dimension gaussian cone take addition also evaluation reveal optimality classical nuclear program naive sketch behind iterative minimizing square subject problem norm especially data sketch base cauchy paper technique obtain solution acknowledgement support office grant national dms addition microsoft fellowship appendix verification first let singular invariance showing case pick matrix row use row balanced let jj j mp show claim packing semi conditionally x denote result observe statistical estimator bad square infimum suffice first k kullback kullback leibler kl divergence eq q setting sketch feasible program piece suffice claim bind successively iterate optimization update original adding shorthand vector belong yield claim error estimate star notation complexity set star shape integer small refer localize measure bound square constrained constrain square together illustrative convenient shorthand claim rip property imply use width since see hull low property rip property vector norm consequently conclude claim upper straightforward e constant theorem width recall minimum singular g put throughout adopt shorthand claim expect respectively shorthand u event return prove bind combined bind great claim lemma u inequality g final view vector lipschitz constant concentration q final definition put piece claim cm section definition lemma berkeley edu california berkeley electrical department approximately solve square constraint assess way minimizer approximation focus randomized square surprising sketch sub present original least square unconstrained constrain constraint approach real experiment past decade datum procedure interesting arise frequently vector constrain simple case unconstrained class include constrain program nuclear ball enforce line book random solve possibility study vector approximate version dimensional square involve new substantial substantially solution assess term solution example unconstrained least square random size paper well reference similar base statistical leverage analogous sketch whereas past sufficient cost approximation notion minimizer oppose prediction course cost bind derive solution sketch instance use al unconstrained ensure cost bound satisfactory normalize quantity grow sketch observation expect provide population illustrative order suggest order require undesirable regime sketch precede rough least sub poor behavior least sketch unconstraine red correspond curves correspond sketch apply projection correspond algorithm mean optimality problem standard square regime sketch nonetheless main square use size underlie hessian sketch iteratively refine chosen logarithmic background turn statement theorem least hessian section consequence defer summarize equivalently background class randomize include well orthonormal low solution observe motivate investigation investigation sketch serve iterative hessian construct optimal type randomize restrict attention matrix sketch I particular instance vector rademacher refer straightforward point view however disadvantage gaussian multiplication random operation multiplication randomize sketch define sketch hadamard fast hadamard transform random uniformly vector give multiplication instance give sketch sample row canonical different weight I lower balanced section apply kind begin consider ensemble square namely constrain maximum characterize necessarily large refer apply eigenvalue symmetric discuss lower also involve measure norm pack optimality appendix combine theoretic understand sub ordinary simplest imply low sketch undesirable proportional reveal surprising sketch accuracy match optimal sketch round see precise figure panel curve previous curve round use fair mse sketch relatively flat bound good drop square involve variant linear ball entry fix square solve randomize early unconstraine nuclear sketch apply return bind analogue hessian suffer namely require hessian building novel iterative hessian sketch match square reasonable sketch begin underlie summarize event hessian sketch long dimension large ensure hold give hessian problem optimum sketch optimization approximation yield sequence whose decay geometrically iterative sketch take return summarize follow sketch combine compound event tn iterate lower base implement sketch datum show proportional choose panel illustrate result convergence increase geometric sampling assume sketch choose least corollary immediate consequence omit illustrate understand improvement sketch sketch sketch solution guarantee sketch sufficient scaling size note square use randomize sketch conjugate gradient lead reduction however type specific least square least implement use hadamard sketch iteration total total width pair unconstraine width bound solve case say involve constraint interior solve g guarantee equal consequence particular square prediction minimal square random dimension iterate obtain accurate original square approximate goal quality show expect measure ordinary least square error consequently tolerance perform roughly summarize run algorithm bound great confirm ensemble behavior confirm bar figure bar height average error run use sample confirm green bar dimension finally bar run sketch total large bar run bar dimension red bar correspond number twice constrain relaxed basis pursuit user radius zero entry illustration constrain keep
surface range agent starting collect surface spaced entropy use entropy position continue overall maximize initialize random partition continue tend enter py py intelligence angle cognitive side great area many computational intelligence act intelligence multiple characteristic aggregate especially relevant recognition effort broad theory goal effort inspire physics agent act maximize propose path derive entropy way walk global work paper arrive perspective concept intelligence computational sense entropy facilitate intelligence prediction reality training set show meaningful simplify include apply presence sparse paper discuss underlie collect discuss implement discussion discuss abstract like aggregate complicated role play present theoretic intelligence entropy drive paper attempt intelligence accumulate take intelligence randomness environment definition intelligence environment follow draw utilize discussion implementation artificial intelligence computer even discussion intelligence disagreement typically school thought act think entail problem solve direct behavior intelligence processing event event prediction optimize ci entity agent together event provide intelligence discuss research key theory amount uncertainty random among interpretation much deep nature core physical reality central physics theory review formulation denote denote content expand rooted physical reality central concept everything chemical entropy map straightforwardly system give serve summation physical reality relate exact state specifically shannon information remain boltzmann serve convention concept important limit although wish much set intelligence member input map map converge intend reflect fitness eq definition minimize involve norm shannon take log minimize minimize repeat logic sign section intelligence entropy concept element find minimize satisfy whenever take state always energy transition increase entropy environment equilibrium due suppose decrease entropy rest entropy amount great equal area physic extend conclude intelligence intelligence process discussion first unsupervised algorithm shannon minimization behavior act consist element group like neighborhood entropy use genetic member reach organization avoid case one operating system source enter py please prototype concept optimize test comparative
eq negative pointwise light make appropriately outli phenomenon order phase transition phase plausible robust correspond two component show outli outlier remove negative margin real occur indicator tend remove case margin concentrate margin indicator open interval relaxation non margin take variant classifier svm dc us validity numerical use outlier contaminate original contaminate decision svm outli parameter maximum function plot panel leave show respectively panel denote percent value inequality agree violate accordingly unbounded kernel robustness confirm region right panel get setup contaminate kernel work poorly bad case beyond violate thus learn provide useful mm ccc plot cb compare generalization robust robust dataset present library language negative running learn standardize zero deviation randomly split robustness chose label change add label robust svm accuracy learn five cross select often contaminate outlier express I kx iteration element hull decision present decompose lemma ii inequality violate index assume assumption ii lemma let contaminate dataset index outli way define note cr c imply ji k hence boundedness gram contaminate contaminate contaminate make possible outlier q therefore primal problem contaminate rational data sample negative outli obtain decision svm base contaminate define contaminate bf bx margin drop dependency permutation estimate unchanged contaminate non contaminate datum guarantee q contaminate non lead addition statement hold assume mean hence equal great negative number due ranked lead ii resp margin express long result sufficient theorem reproduce rkhs inner boundedness kernel become empty empty therefore index assume prove dominate eventually slope argument proof em vector successful popularity serious cause outlier deal robust outli bound investigate point contamination still give contaminate show regularization algorithm formula guarantee grid validation work candidate experiment explain svm world data misclassification balance maximum problem separate plane reproduce space variant svm generalization study svm drawback remarkable separate mainly misclassifie sample misclassifie significantly affect svm outlier svm penalty hinge loss misclassification convexity cause unstable outlier unbounde put one way instability replace simple hinge loss statistic statistical long kind mathematical analysis influence measure robustness variant robustness convex rkhs influence rkhs estimator yu et function convex study learn provide deal standard regularization provide tune introduce variant learning remove relate robust svm main contamination information contaminate inequality conversely prove inequality violate partly boundedness one desire help grid conduct study assess kernel consider property outlier paper setup review devote svm dual intuitive great robustness evaluate investigate statistical order generalization numerical conclusion appendix let notation natural finite set express reproduce hilbert space rkhs denote rkh resp resp I produce output test sample accurate training take rkhs endow kernel misclassification penalty hinge loss precisely svm decision threshold interval range avoid preferable input coefficient support provide support infinite space reduce quadratic rkh non parametric statistical pointed speak average permutation decision express minimize margin svm obtain hinge svm regularization instead svm svm make provide appropriately variant svm parameter svm replace outlier outlier variant influence outli indicator intend ratio outlier formalize rkh margin zero difficulty robust robust detection linear classifier solve semidefinite problem level margin include middle difference learn method refer robust emphasize robust rkhs kkt difference dc use algorithm programming efficiently obtain base derivation dc dc objective value finite objective argument robust svm loss programming terminate consecutive phenomenon convergence define compute sort component multiplication unchanged decision outli indicator picture geometrically dual lagrangian slack negative multiplier geometric expression training bound non rkhs reduce domain label dual distance estimate scaling rkh evaluate robustness measure evaluate influence estimator training sensitivity case refer quantify outlier contamination necessarily large amount contamination contaminate take family contaminate dependency drop space let express boundedness rkh decision boundedness contaminate condition contaminate dataset retain indeed unbounded regardless since prove rigorous proof omit rkhs great conversely trade ratio result correspond outlier reduce svm necessary greater rational svm strictly robust svm formula label negative integer note guarantee point decision less boundedness bias positive ratio rkhs
assume variety far objective machine generally rigorous guarantee investigate issue human take axiom mild axiom unfortunately interestingly reasonable crowdsource computation crowdsourcing involve perform generally computer human machine learning inference typically employ process worker infer algorithm infer solution design dependence response objective likelihood negative worker em optimization minimize problem obtain optimum extremely successful function meaningful computation appeal convex study extensively performance rigorous setting thus reasonable human convexity provide vast body two natural crowdsource convexity inference crowdsource mild axiom computation ensure interestingly one modelling indeed human select ability ability higher represent line number use ability worker regard ask crowdsource worker either notation choice say look procedure answer worker receive optimization base relax associate question inference x subsequently every ask worker worker ask question represent worker matrix element value ask answer worker worker response sequel optimization program axiom answer completion round step execute round infer leave soft answer jx towards natural axiom recall function axiom manner think maximum worker scalar distinguish lx lx lx w lx w informally axiom say report worker report less model highly crowdsource pose major axiom incorporation axiom suffice solely make continuity objective inference exist crowdsourcing aware fall subsequently constructive absence requirement model three axiom list translate worker observe ask scalar upon axiom list worker identical axiom axiom present example crowdsource show list two axiom denote receive crowdsourcing assume worker answer question worker incorrect answer parameter response else identifiable simply scale define likelihood consider subspace worker ask reduce list mean hence obey thus p coin worker model answer correctly true question true worker answer incorrectly else simply attention obtain coin additive assume worker interest worker f perform early restrict attention asked verify increase thereby property answer else entropy principle therein impose must form j minimization subset verify attention present property respect satisfy role axiom axiom crowdsource axiom strong property property hence axiom jointly multiple ensure satisfie axiom construct plot claim good use crowdsource objective crowdsourcing model reasonable permit framework throughout complexity alternatively problem bayesian non impose convex weight make objective albeit perhaps expense capture objective satisfy axiom present continue apply certainly complete absence indeed model theoretical instance model logarithmic appropriately may scenario minimax well result importantly date crowdsource alone convenient although theory extensive unfortunately exploit human computation paper incorporation
focus lead interested additionally convert free bind comparison interested quantity relate ex state theorem ex proof appendix implicit completeness make include proof ex plug bind reduce splitting reformulate fix c free style implication bind follow immediately completeness term low base argument obtain use appendix dependent star also survey variant robust noise additionally complexitie splitting case vc reader quantity analogous notion characterize complexity membership name characterize choose label point extend function classifier minimum sufficient quantity characterize learn return objective excess rate target define quantity prove low purpose star hx h exactly classification equivalently respect td prove label active source precise basic observation additionally refine thesis state relate quantity star obvious reading find extended way star versa learn familiar extended dimension equivalence formally td td previously instance become another remarkable set state lemma proof include appendix specify star center simplicity discuss replace though directly term bind equivalent logarithmic however imply extend classifier q u prove relate proof combine immediately imply lower involve appendix bind recently space compression q see u uv u uv v u v u h u specify vice immediately imply xy active budget return arbitrary bind analogous aforementioned bound purpose free independent take possible combine find match factor h follow quantity xy complexity passive show passive algorithm achieve xy though go idea xy though typically loose question case hypothesis though bad star range specifically proof h show gap quantity specifically p low sometimes within universal discuss appendix show sometimes tight aside divide interestingly process establish relate disagreement coefficient hold measure inequality quantitie passive aside minimax active model offer well discover unknown minimax label active vc class passive express combinatorial dependent equal maximized choice derive sense express bad sense express distribution label label complexity marginal achievable complexity exploration aside factor important minimax fix remain challenge open deriving consider kind guide restrict focus dependent bad bad free present result begin proof tn ta h gx follow collection vc union collection measurable vc integer ax algebra universal let vc xx x likewise interested theorem useful vc collection universal log xx hx hx x least h g imply g x h note log straightforward net relatively universal n mi x mi denote vc lemma let take g ie eq definition equivalently complete union variant lemma specifically exist universal xx variable x I result xx mi ie ny chernoff give least eq number sequence I q probability define n ny I I nx ie element assume x x kt py rr verify lemma continue improve follow immediate implication k ti tx trivially hold modification execution request return x k k xy tx h py p behave er er th er er achieve latter guarantee probability achieve er er probability prove exist hx implication behavior vc establish law number van result effectively guarantee high random make address fact strong obvious data turn relax classifier instead partition measure least partition property every constant final claim markov article proportional term cover bad value case let imply intend nontrivial simplify log proposition thus final follow xy xy xy xt choice instead least er complete bound part straightforward establish log technique finally third part technique analyze disagreement active technique modify technique place factor label eq give return xy h xy xy er xy xy er xy xy upper attention prove bind net u technique learning query treat instance specify label budget mm td j g gx ref jj particular let see return satisfie nx f agree return u vc u f nx un log least xy xy er xy xy well reasoning finally establish xy h input j label gx v eq chernoff integrate satisfie execute step particular denote define event conditionally k gx fx I gx since hand k satisfie event probability j k xy xy k k kk k define event e kk k kk induction imply upon reach setting technique three part also simplify proof let applied sequence simplify notation budget argument return else let else return subroutine label datum counter request let partition second commonly significant directly partition query repeatedly majority original discard point identify end high rate effectively distribution favor remainder xy xy xx x xy appendix range q xy xy return subroutine denote event least imply eq proceed characterize behavior subroutine via sequence exist j p law law mx mx p p measurable union exist I xy j f xy xy first statement jensen second inequality every j x inequality assumption inequality must establishes establish second monotonicity let hx hx union right hand side side side eq mm lemma agree subroutine lemma hoeffding inequality total value imply k recalling xy imply hold xy likewise one two clearly subroutine reach condition occur value q xy xy xy xy sign xy j f k measurable chernoff distribution probability law hold apply behavior algorithm begin budget retain run event xy proceed induction clearly xy v mf xy x f imply hand xy inductive hx principle er xy xy apply law union event v next k imply fact note budget question request algorithm produce budget budget question address lemma budget chernoff imply consider budget trivially remainder note term right chernoff imply lemma term note li j f section imply well li k g k martingale theorem apply probability k j k hand chernoff conditional distribution least event least hold let g furthermore ii last inequality ii plug iii iv combined bounding iii iv bit reveal q plug hold event sufficient reach return therefore guarantee run budget request xy er request execution subroutine represent running request budget subroutine ensure step ever condition algorithm request number budget take suffice reproduce budget value obtain implie summation therefore fix combined imply plug appropriate already appropriately final budget budget return return budget execution exceed infinite execution take infinite budget execution er xy xy er xy theorem note noise parameter dependence selection discussion far leave issue theorem know x least produce er xy passive n size universal xy therefore xy two bn simply correspond give statement theorem slightly factor log turn lower take technique contain lower bind bn bn prove bn bn bn bn bn case begin fix budget size produce er xy er xy note k kk therefore either particular furthermore dd xy note purely statement note log range recent base establish term remainder xy x xy xy xy xy verify redundant therefore plug complete aside logarithmic refinement bound away prove apply insight improve rather merely known star instead bound work modify variant exist constant size xy see thesis survey relate therefore prove sample passive match bind simply prove upper bind x x x markov x imply xy er q lemma budget produce xy er imply log log bind turn establish recent survey contain prove theorem imply otherwise q imply log term bind simply examine argument also strong expression state large state upper leading bind fix follow b variant universal least produce er er upper relax log also helpful measure z ex h xy ex ex xy xy h ex ex ex xy xy xy xy xy xy exist fact hx r lemma distribute random every I gx r gx r hoeffding union particular let xy xy xy xy therefore since arbitrary xy unnecessary purpose ready ex ex mh mi h b ref xy ex ex directly ex furthermore suffice support base fix furthermore z rr z r hypothesis q fix inductive handle nontrivial distinct z x hz gx r inductive fact imply j b hz hz z z k hz hx hx hz gx hz hz gx h h r b z r consider complement j hz redundant include clarity z hz hz ji hz I g hz ji z j z h h h j characterize since g b z r h z well case b ex ex ex due supremum specifically nonempty therefore take eq q usual xy xy ex ex present ex ex ex lemma focus finitely probability distribute imply union event occur let fx gx q qx gx gx hold ready ex implicitly original proof q b ex ex ex g fx gx q ex lemma repeat finite f gx inequality side divide immediately follow imply would thereby complete end denote g gx gx value entirely x fix finite mx gx complete proof inductive nested base nest inductive nonempty q x qx yy argument follow mi x ji x j ji define k kk ix k ji mi k mi j mi ks I k I kx kx I k z inductive r argue prove star k k z I kx z k z jx f h f z jj kk kx h x h argue imply induction give minimal specify fix specify respect star center u x j j v j u ts v j jx jx kk match star star lemma immediately ready fix let td monotonicity maximize monotonicity td remainder establish proceed sequence mh inductive hypothesis u u u u u specify star center since star centered specify inductive follow induction next subsequence distinct u last mh u r ks element h u g g ir u h u v v h h v maximize proof fix x h mi ix ix ix h ii x e modify proposition log set let denote chernoff hx gx I integer hx gx hx I establish g gx hx gx g kolmogorov gx h fix recall maximal rr see kolmogorov kolmogorov I gx g gx pe x b classification realize furthermore classifier r upper classification disagreement see divide maximize universal logarithmic factor bound theorems x sx vc star argue increase match logarithmic let ip ip furthermore define ig g p ig break tie also ig ig therefore reduction learn distribute j j j p algorithm execute request simply purpose request request label attempt request termination valid active learn budget request internal randomness return ann p iy p inequality linearity least plug strategy produce valid budget choice choose specifically j internal random conditionally distribution variable sequence ij er total von von probability independent plug bound noise construction proof star low already upper bound theorem bn ip bind prove maximal star appropriate therefore dp ip x ip ip px er pf establish proportional lemma apply hypothesis x h recall x ip x k x p dx upper sometimes logarithmic factor upper sometimes near logarithmic already consider proof last low p imply marginal k px px additionally since support distribution disjoint thus I lemma logarithmic establish upper sometimes sometimes tight choice furthermore theorem sometimes tight logarithmic factor tight first lemma x h p ii active xy er xy guarantee xy index subsequence hoeffde inequality bind subsequence run subsequence return subsequence empty return classifier note request label label request er er er evaluate complexity imply er er xy er strong sequence disjoint may j complexity law event er h er xy xy choice space prove sx star star sx fact factor specifically set define pa pa pa p ip ip together since distinct classification without request return note request every either thus take x ii equal negative denote return union x xy xy xy xy else xy xy xy regardless request xy er suffice q constant dominate existence tight measurable p pa pa px ip ip p bn every every contain star imply similarly respect low sometimes within logarithmic within bn bn tight slightly involved fix match logarithmic case x f yx trivially ix ip I hypothesis class imply star eq claim always proceed include bind p p x p yx x yx yx sign p yx p trivially ix ip I ip therefore dimension star vc imply plug factor sometimes tight within logarithmic factor since bind conclude sometimes within logarithmic factor specifically liu establish minimax label vc always passive good regime label low regime measure call interestingly previously learn star active strategy nearly complexity design learn condition sample machine primary annotation protocol sequentially point initially access unlabeled consider able pool label datum process continue must hope sequentially select effort already thereby require produce predict learn often significant observation result survey capability request sufficient desire substantial question gap quite case question specifically interested case request low model reveal establish propose active significantly smaller active get key study set applicable arbitrary range minimax complexity depend structure hypothesis roughly complexity minimax complexity find g classifier minimax passive nontrivial interest fall category minimax complexity essentially fortunately improvement passive datum focus various well learn place unlabeled minimax label thesis thesis upper minimax complexity achievable complexity achievable effectively measure capable behavior complexity active result reveal improvement minimax passive value bound beyond initial study recent literature develop provably advance seminal agnostic active thesis thesis label intuition inherently hard passive significantly passive pattern scenario passive learning scenario improvement unlike show accurately capabilitie nontrivial class vc reflect survey passive although minimax passive learning unable confirm truly really gap close work reason range noise regime reduce match basic surprising introduction particular dimension minimax complexity sometimes basic argue characterize combinatorial reveal label complexity active fact minimax interestingly dependent star maximized distribution include star summarize main active select estimate imply prior upper bound reflect minimax passive learning non possible imply hypothesis vc roughly minimax logarithmic factor imply literature upper bound complexity measure refer complexity star star study exhibit hypothesis gap upper bound demonstrate gap aside factor result vs respective minimax complexity passive bound base analysis query determine highly modification innovation net cover low largely directly combination survey incorporate note focus result reveal low aforementioned strong low assumption restrict leave work characterize marginal article introduce follow model work define combinatorial star lower bound complexity include discussion among scenario complexity correspond star maximize relate star concept star write contain section section follow section appendix rest make formal simplicity x b arbitrary statement similarly condition classifier independent active individually require scenario leave question number unlabele label particular indeed xy x jj iy proceeding initially access select index request index request continue round label formally active conditionally independent ready specification learn respect denote produce base distribute define classifier vc classifier h hx gx px proceed additional notational help simplify statement proof value function write equivalently express define define remark van van mention issue implicitly event study set corresponding specification collection xy f bn xy xy xy xy xy xy h xy xy xy er bc h study correspond optimistic case pac study name noise slightly discrimination form state passive however weak assumption condition equivalent imply thesis contain within related study admit distribution widely literature complexitie learn bound express slightly prove typically factor proof refine aside believe optimize bound additionally relation model since low theorem contain study log likewise case bind theorem simplicity explicitly include theorem variant basic point determined sequential test course present repeatedly able distribution give resolve argue cell vast majority knowledge datum discover vc give point partition label point majority cell effectively majority note simply apply repeat strategy distribute would determine label partition number sample become note less agree noise classifying excess er xy learn effectively favor discard become gradually agree every pass combine provide fraction classifier bind decrease decrease try classification able reduce request component essential case introduce namely value active disagreement request classification disagreement recent separately section relate star dependence state comparison protocol label passive minimax learning passive classifier er xy request run determine return one
subproblem principal seek outli anomaly seek detect outli magnitude seek recover nonzero coefficient outli variation problem deal relax contain properly hold table lead outli numerically efficient far unconstrained call merge discuss signal outli signal ad backtracking analysis try ii measurement follow satisfy feasibility belong optimum theorem detection graph perfect achieve b perfectly note theorem factor smoothness upper signal trust prevent influence outlier similarly deal relax merge combine anomaly provide provide clean use clean admm implementation graph outlier stop several bridge indirect bridge monitor opinion dataset dataset classify political either label correspond bridge feasibility indirect bridge structural health monitoring bridge system build acceleration collect bridge put bridge collect acceleration mass gram gram simulate acceleration near neighbor node represent eight represent acceleration shift symmetric undirected directed allow direct graph weather united record weather per day weather geodesic pair weather represent weather represent eight close weather graph weather graph shift weather normalize direct contain measure norm rating represent near node eight user signal I acceleration algorithm acc mse rmse absolute mae acc mse mae ground indicator split training validation part choose validation appear include completeness describe laplacian restrict undirected shift symmetric large classify adopt describe label thresholde classification labeling remain underlie filter also require sophisticated labeling unlabeled acceleration acceleration assign acceleration algorithm remain show average test ratio adopt mse obtain use collect comparison identification labeling matrix apply propose expert opinion classification propose include factorization describe minimize similar different factorize nonnegative low internal contrast hide fair non convex miss predict temperature matrix temperature weather adopt dataset temperature describe pick day case signal temperature pure well label increase algorithms graph mae outperform completion cc rmse b mae completion recommender dataset task predict rating incomplete rating purpose signal temperature average test advantage exploit rmse rating completion mae rating many opinion expert image opinion combine formulate solve classification ground simulate expert label opinion labeling mistake opinion matrix expert classify content split easy hard expert chance expert correctly easy make avg sign avg method opinion part entry tuning parameter report ground improve result scheme completion provide application robust apply robust signal identification contrast manually part algorithm graph signal validate detect label feed accuracy accurate classification cc semi supervise describe robustness randomly label feed algorithm together correctly acceleration compare signal provide labeling ratio general multiplier show signal matrix subproblem solution validate identification opinion acknowledge support fa well institute technology award lead improvement follow principle page reader decompose introduce residual capture rewrite equivalent form f operator note move function put equivalent augment lagrangian thresholding aim solution aim solve solve thresholding setting satisfy update lagrange multiplier mm electrical engineering pa usa pa usa com graph signal recover corrupted formulate multiplier principal analysis relate theoretical bridge recommender recovery completion semi growth generate source include social citation unlike image novel lead extend signal underlie graph signal generalize signal transform classification detection processing generalize series concept tool denoise classification root definite real edge weight approach root build operator elementary generate shift shift graph arbitrary real nonnegative within problem time deal undirected graph graph assume smooth corrupted assume neighboring cast signal optimization solution alternate direction multiplier recovery theoretical new graph completion validate bridge temperature recommender expert opinion work review exist recover observation technique wiener filter wavelet thresholding lose part image video representation take number recover low originally noisy decentralized recover rank corrupt measurement separate image two background foreground corrupt relate signal recovery include interpolation signal uniqueness graph recover random contribution novel analysis novel graph nuclear novel anomaly signal review graph recovery section subproblem completion section recovery recommender opinion briefly domain line commonly graph signal supervise node connection node quantitative underlie dependency pattern representation assign vector graph fourier transform expansion invariant simplicity complete expansion eigenvector form signal content weight coefficient graph qualitative underlie signal quantify use magnitude normalize guarantee call symbol shrinkage define signal multiple completion denoise graph section propose appropriate assume outlier outlier distant variability measurement magnitude small magnitude furthermore certain large graph denote component graph section counterpart completion minimize variation anomaly robust recover sample total access task signal true signal variation signal variation quadratic graph cyclic matrix combine recover subset subset typically rank recover rank model assume subset index recover condition also graph represent structure matrix difference corrupt recover low index associate identity detail part appear completeness signal seek entry incomplete noisy signal smooth formulate case exist iterative measurement derivative close invertible merge objective trade solution set zero invertible denote try recover error part condition smaller tight smoothness technique smooth bind smaller close estimation graph subproblem general seek graph column signal case formulate addition completion next call constrain problem solved project split component formulate proximity iteratively iteration feasible differentiable component convex proximity
measure infer year connect graph appropriate temperature trend correlate regardless reliable information observe experiment give observed temperature signal learn quantitative build graph reflect term connect two small compare learn comparison visualization focus top comparison edge graph consistent confirm score quantitative comparison infer topology graph learn score value trace recall verify result disjoint learn measure obtain laplacian spectral clustering learn disjoint cluster fig dot mainly blue flat region especially centre lie mean algorithm record information cluster together capture information measuring confirm record california per month learning would like infer similarity measure variation cluster compare result overall metric though challenging indeed accord description cluster obtained learn base measure c cm move national vote support lead like infer capture vote neither obvious relationship preference partition spectral blue cluster speak speak five red primitive consider among conservative cluster membership fact agree close european cluster political voting behavior cc confirm b national mass year cluster seven percentage support great largely speak margin confirm demonstrate topology smooth enforce smoothness signal gaussian impose factor numerous appropriate capture entity furthermore focus present paper impose analysis leave comment dr ed de mail meaningful crucial success handle especially process meaningful readily particular desirable graph processing application admit certain signal variation topology adopt graph impose lead signal propose enforce property minimizing learn demonstrate graph topology laplacian processing datum signal vertex set weight undirected signal vertex represent entity weight reflect pairwise vertex carry observation measurement numerous world structured currently graph domain e graph intrinsic entity desirable topology entity present ill pose associate topology topology pre define meaningful structure graph edge capture value temperature significant variation represent interest friend represent vertex interest datum graph representation graph multi view scalar potentially define fig signal however obviously objective topology speak would edge signal bar point represent negative respectively bar reflect graph smoothness unobserved term latent traditional transformation topology joint signal consistent latent specifically impose prior generalize factor obtain principal signal smooth signal base uniquely signal smoothness datum laplacian central process new operator latent iterate graph variation minimize upon graph world infer topology art closely idea estimation importantly framework processing perspective approach rigorous framework signal property classical insight signal numerous entity amount signal effort process generalization wavelet dictionary processing inference domain capture whose importance processing represent topology e loop usually consider enable generalization notion frequency fouri signal develop matrix signal laplacian operator central kernel graph via regularization process laplacian permit real graph signal represent behavior condition converge intrinsic laplace riemannian manifold may manifold benefit application view zhang et eigenvector optimally build priori domain devoted graph topology metric evaluate smoothness signal fitness learn valid topology fitness property statistical manner help understand adopt graph linear brain principal correlation distinct consider region behavior perspective link explicitly particular amount community graphical gaussian graphical small singular another consist know graphical exact zero entry partial log therefore correlation emphasize mention precision usually zero row however result precision matrix global property graph signal rather correlation straightforward convex order rather work learn topology adjacency classical regularize rank laplacian infinite basis natural impose together fourier assumption lie use degenerate see precision laplacian generic much graph laplacian commonly precision assume degenerate precision recover noise free classical lead principal component probabilistic interpretation highly successful analogy signal propose interested specifically map scenario quantity laplacian quadratic confirm gaussian similar scenario component signal graph ready introduce framework impose come joint change accord noiseless version observation recall quadratic eq usually smoothness laplacian signal propose follow objective zero frobenius respectively acts permit trivial valid laplacian furthermore trace function diagonal entry similarity elastic impose adopt alternate scheme solve solve cast convex minimizer symmetric triangular main therefore low triangular convert rewrite problem subject point computational graph instead split alternate method multiplier form second hermitian cholesky factorization compute alternate summarize htb input output laplacian update section graph present compare learn visual quantitative comparison existence graph evaluation criterion commonly namely performance partition pair vertex class graph package stop reach absolute experiment namely finally weight framework propose adjacency precision determinant regularize determinant precision regularize laplacian diagonal loading interpret priori consequence problem case one experiment similarly carry synthetic vertex base euclidean distance follow generate vertex square radial basis rbf width r enyi edge ba graph ba experiment add exist degree exist er graph science former network ba er graph unitary laplacian normalize generate show signal visual comparison laplacian random lead quantitative paragraph choice
require experimental excellent dataset convolutional network demonstrate key convolution layer serve seminal contribution improve deep convolutional deep convolutional machine model pool contiguous pooling robustness variation shift reduce move maximum map block pooling average max stochastic probabilistic max generative deep highlight deep start map multinomial distribution feature analogous impose pooling demonstrate yield generative statistical readily implement jointly use bottom bottom layer refinement phase jointly may readily goal obtain maximum find unnecessary expensive attempt parameter view alternative convenient learning dictionary deconvolution layer convolutional simultaneously nonlinear nonlinear testing network approach inversion test still aspect deconvolution operation hierarchy dictionary joint leverage generative model top operation hierarchy map imply contribution employ beta separately via layer proper top allowing mean feature map deconvolution experiment convolutional representation gray analyze jointly dictionary hadamard element shift w ki z ga b b may look complicated conjugacy admit gibbs bayes order pool use employ feature move layer learn stack upon refinement pooling proper never tackle yield discuss present pool closely model start sequentially stack parameter layer serve sec parameter layer jointly close stage input dictionary element view entity discuss spatial dependent perform layer contiguous part block location block one pixel pixel stochastically large block hence process proceed impose question bernoulli give multinomial model bernoulli follow multinomial statistical latent denote multinomial imply equal entry I element block zero first position zero phase datum learn use activation sampling multinomial correspond gibbs stack pool continue learn element via continue top layer top pool generative constitute refinement learn excellent initialization subsequent generative process l l multinomial map corresponding block block convolution w equal block multinomial multinomial unchanged discuss refinement via multinomial size refinement constitute initialization refinement understand element visualization layer associate multinomial show upper image capability deep convolutional well activation expensive issue filter explicit deconvolution follow though step must learn dictionary framework model accelerate project datum plane test perform top element map plane strength subsequent classifier dictionary map datum plane multinomial top dictionary plane element dictionary convolutional shift pool shift pool pixel layer deterministic retain deconvolution deconvolution must infer element plane detail aspect material conjugacy component close efficient gibbs supplementary detail convolutional accelerate operation update pre discard burn sample burn refinement ml collection sample spirit yield posterior select sample discard burn dictionary view plane refinement trial use hyperparameter hyperparameter perform matlab execute cpu memory refinement minute deconvolution acceleration via realize recently convolution mcmc batch vb layer train widely mnist testing digit layer dictionary layer initial dictionary large value pre discard indicator element summarie classification compare second top send support vector kernel multi classifier via fold cross layer dictionary element computation close simple rate learn similar refinement step testing report examine learn visualize dictionary map observe qualitatively refinement average
generalization rademacher finally analyze representative affect behavior comparison domain keyword adaptation deviation wu david research types adaptation learner receive source domain know adaptation zhang without concerned representative adaptation datum one adaptation cover multiple domain combine paper previous exist result regard case generalization representative factor affect discuss choice meanwhile representative cover target include additionally quantity follow integral quantity entropy rademacher generalization representative domain adaptation asymptotic representative support finding brief conclude appendix inequality proof jk domain respectively let stand respectively differ differ occur n kn k denote combination empirical empirical empirical approximate precisely adaptation david zhang david david david integral exist recently give investigation integral sign trivial domain rewrite form quantity discrepancy formal briefly quantity introduce exist one summarize follow upper david quantity recall condition place restriction contain function task discrepancy quantity mention quantity match instead author upper resp minimize upper summation note address paper condition next aforementioned discrepancy definition integral relationship integral probability three possibility differ occur two difference difference measure distribution moreover another quantity trivial labeling integral discrepancy distance show simultaneously specific labeling meanwhile though set definition rademacher generalization achieve incorporate complexity covering refer entropy class covering cover metric norm derive bind situation adaptation know source domain long applicable adaptation free clarity presentation notation sample z k z n k norm easily omit z kn frequently class random either rademacher complexity q version take base entropy bound domain adaptation hoeffding derive hoeffding deviation deviation inequality base present hoeffding type representative domain adaptation bound bind tf least risk respect three coincide hoeffding deviation incorporate expectation variance type result type type domain tf bound hoeffding two limitation affect satisfactory analytical inverse trivial since tf use lead compare strong bound affect representative second type alternative hoeffding type next provide detailed bernstein type refer hold kn convergence moreover observe rate varie especially imply become become contrast hoeffding bound affect adaptation detailed representative rademacher generalization representative domain class function give domain rademacher source domain derive adopt domain adaptation coincide assumption domain match replace derive type inequality bound follow notation take define match k omit hoeffding tradeoff rigorous tradeoff discrepancy measure achieve discrepancy w entropy tn part infinity accordance process event representative domain risk hold process distribution coincide match kn kn theorem generalization decrease small cauchy set minimize side hoeffding result imply fast domain process choice essential tradeoff number kn fast convergence relatively large representative domain adaptation kn kn lead type accordance analysis well experiment verify generality domain target tn n tn tn sample source n regression combination coefficient repeat time average increment choice big fail become big recall big means datum situation adaptation accordance present analyze representative support finding fact discrepancy fast convergence slow far away fast rate rate slow accordance finding domain source david uniform source condition classical multiple source know domain domain domain extend target source combine meanwhile analyze classification task introduce condition vc capture extend rademacher property learn source target particular metric domain provide mechanism domain theoretical paper study adaptation setting previous work also representative term apply type generalization representative domain adaptation uniform entropy respectively point process uniform convergence rate finding discuss hoeffding type hoeffding type result representative type complement cover adaptation include zhang generalization bound result theorem base obtain result adopt martingale hoeffding type concentration obtain generalization certain specific hoeffding source k hoeffding result result suitable coincide domain hoeffde generalize domain expectation compare inequality suitable inequality coincide one domain match explicitly reflect right completely satisfy hard analytical incorporate parameter multipli type deviation cauchy affect result
extension pr pr polynomial establish pac number target unlike standard vanish treat define notion hardness hardness result furthermore ergodic system termination trace construction structural output playing strongly consistent ergodic map generator structural construction lemma synchronization role synchronization specifically transition synchronization necessary infer cross distribution distribution first process synchronization ergodic minimal recursively b b symbol specific use definition cross give stationary ergodic generator minimal yx assume equivalence note yx b yx yx note synchronization stationary ergodic string yx string projective stationary encode sense definition yx complete proof string solve lemma suitably note establish string string ergodic exist inference derivative symbolic string occur follow immediately symbol string string number occurrence string symbol imply derivative derivative string respective cross negative summing unity bx stationary respective string x analogy describe seek derivative string stream history explain symbol assume history former stationary next cross occurrence row vector yx induce illustrate admit furth average predict weight choose strategy nan statistical causal strongly positively keyword education positively causal directional causal infer education search frequency education reduce immediate full search correspond keyword trend node degree arc google trends http www trend search normalize entry indicate sum google website trend long strongly political keyword table google trends education freedom ex music ex keyword easily expand theory new topic note interesting search datum correlate illustrate datum series keyword education keyword correlation positive education causal environment lower seem plot causality carry unique interesting causality series neither full keyword symbol stream symbol colored new existence dependence stationary source cross sufficient broad causal causal flow development open mechanism diverse intensive scientific pt north yshift north east cc use statistical relationship branch really causal causality hard construct difficulty causality practical operational restrictive dynamical trivial nevertheless computationally evidence dependence yield tool calculation causality symbolic stream ergodic precise compute stream linearity specific dynamical structure explicit sufficient fairly propose pac probability search google trend choose keyword causality keyword illustrate fail insight correlation causal correlate dependence consider past value maximally apply contrast future would reveal carry unique prediction causality early preliminary text causality statistic expert largely sound operational causality notion lack consensus causal relationship perhaps mathematically causality concept know know attempt influence statistical universe knowledge universe denote contain forecast however expectation simply causality future future past contain redundant exclude within definition note applicable causality say cause set intuitively cause unique immediate notion causal address primarily interested obtain mathematically lead algorithmic encoding universe series available consist fx j extra affected necessary j universal cause n say identically mean far structure causality discuss commonly employ causality find series self function cause implication dynamical specify ordinary differential able perfectly construction imply may concern causality require cause additionally cause causality necessarily require cause imply sense induce causality particularly white process cause general fix spurious two common causality mean easier satisfied incremental one ahead operational variance forecast cause bivariate causality cl dl side lag lag operator root mutually individually constant use determine significant predictive power nan strictly illustration notion generate ergodic alphabet b possible produce respectively class string class mapping symbol alphabet edge conclude end linear restrictive structure bivariate analytically limitation nonlinear autoregressive wavelet transform heuristic allow quite pre suppose causality show sensitive non attempt completely causality series integral similar nevertheless absolutely bind integral additional stationarity separate variant quite nonlinear stock factor return stock price trading volume despite application parametric test limit beyond financial interest detect causality specify obvious hand box advantage influence variable parametric system dynamic leave parametric although nonlinear causality detect nonlinear parameterize behind residual remove additional influence origin causal priori dynamical objective indeed infer heuristic influence linear appearance dynamical well absence assumption source sequential variation ergodic stream explicit early generator stream probabilistic stream model causal represent generalize refer infer machine structure nature causality logical influence stream identical additionally show causal existence trivial direction independence ability find dependence carry causality require ability base prediction impose produce generative suppose structure therefore carry addition stream identify stream stream symbol stream symbol stream value indicate strong thus causality quantify immediate future stream stream importantly directional see merely infer existence establish significance passed fail relate build state strong connection indicate past completely test inference impose latter perhaps approach ergodicity test process absolutely regular mix certain minimum regularity way one essentially stream nearly ergodicity figure class prediction pac infer asymptotically well investigation parametric test go beyond binary testing quantify causal influence observe pre particular influence show impose stationarity pac rest notion elsewhere completeness difference exposition section presents framework cross introduce causal directional infer self stream sake pair data stream causality multiple future section causality source trend list keyword upon literature formalism brief overview completeness alphabet possibly unbounde denote identity infinite denote stre length denote also value moment calculate strictly generator ergodicity able sufficiently algebra infinite induce ergodicity extend countable notational brevity class string equivalent extension string relation induce equivalence string right invariant equivalence induce equivalence construction final mark marked alphabet initial recursively extend impose probability unity symbol probabilistic however lack state additionally strongly remove dependence ergodicity formalize generator mark unique probability countable immediate imply mark corresponding initial extend recursively nan finite imply mark imply generate yield initial generate initial mark probability whereas mark canonical representation remove state dependence representation I non entry row mark associate stationary string lead begin unique canonical mark uniquely induce set construct stationary use induce include representation mark representation contain exist begin stay copy mark strong initial state minimal initial mark strongly exist map l permutation transform encode realization correspond represent index generator space represent ergodic state connect component correspond strongly node label iff initial us equivalence strongly right terminate immediate map contradict ergodic hence generator possible distinct exist string contradict conclude connect argument valid label unique state encode associate relation z initial state equivalence refinement identical realization synchronization synchronization state analogous context identify stochastic generator translate synchronization machine history determine top finite history machine bottom string always synchronization remove arc trivially state string hence generality x induction construct finite x ij tx contribution arise simultaneously state nevertheless see qx n satisfie arbitrary string search string order string alphabet find entry follow string scaling imply string computation symbolic symbol symbolic derivative specify alphabet symbolic string count occurrence imply symbolic symbolic thus probability refer symbolic recall satisfying reading unknown string imply class single state true state loss strongly q complete long extension specifically long non establish state class describe identification string observe corollary establishes string arise inspection geometric structure construct different derivative sl geometry sl hull combination string hull consider string probability derivative hull corollary number drop factor case kullback kl probability pr pr pr ergodic evolve assumption require dynamical specifie string process specific alphabet strictly algebra denote space map ergodic algebra map consistency additionally xy b effect segment vanish cross induce cross cross stationary cross tp ix derivative stationary ergodic assume recall equivalence derivative generating em b capturing dependency capture dependency evolve letter alphabet evolve letter alphabet capture dependency differ former specification could alphabet probability symbol generation symbol empty see pr dependency need formalize define appropriate call cross probabilistic equivalence relation ergodic clearly forget actual notion map string output alphabet identical respect formally finite state alphabet alphabet alphabet possibly parameterize mark marked cross finite noting extend recursively denote argument ergodicity drop without unique minimal figure transition generation transition alphabet alphabet alphabet different input alphabet next investigate may ergodic ergodic equivalent essentially evolves independently string affect symbol distribution string simple state copy ergodic step shift case figure calculate follow stochastic process ergodic single k kn nn assume canonical ergodic dependence ergodic respectively symbol b b b immediate minimal process alphabet represent possibly large graph specification minimal realization force projective vice involve remain transition state conclude complete proof b initial distribution causal definition claim one string string string lemma conclude next symbol claim symbol compute string imply state respectively b sequence induction hypothesis vector pr pr pr pr b w pr b w directional stationary ergodic establish capture well suited directional flow quantification directional introduce representation represent direct direct label composition strongly encode get merge equivalence let encode denote connect establishe full q equivalence imply map imply complete projective version projective strongly projective corresponding note projective operate operator alphabet second second operator indeed additionally projective preserve project choice equivalence choice state distribution invariance encoding alphabet state composition stationary denote encode state j states matrix establish stationary ergodicity stationary b symbol well process look coefficient entropy bit letter observation symbol ready define causal definition give causal ergodic finite causal dependence ratio due absence b entropy discrete process produce alphabet inner stationary lemma noting equivalence class correspond class string q coefficient x entropy establish statement x x b converse independent minimal x complete computation
generalization value let overlap meaningful sum average fully average gain map binary transition probability less refer factorization unbiased realization process generally markovian proceed parametrization minimize hamiltonian vanish find transformation play correspond read shall hamiltonian ise field map reduce limit variable overlap eq clear bs hmms infinite operational regime order transition transition alternatively intensitie intensity simultaneously objective straightforward model factorize overlap separate domain admit factorization uniquely proceed regime weak irrelevant regime maximize gain spin focus two verify refer regime domain end correspondence configuration generally domain support uniqueness reflect entropy consist inherently relate exponential degeneracy intensity indeed observation linearly thus indicate degeneracy regime three look positive old stay end positive form pair rule super spin play role regime odd build positive super regime small pattern implement separate domain spin advantage implementation already spin incorrect estimate one iii calculate approximately spin opposite map remove uniqueness overlap calculate spin determine close analytic form calculate fluctuation separate domain write normalization checked even spin likewise originally spin need seem surprising symmetry expect overlap gain define differently agree gain active large thick opposite recognize correctly change decrease recognize domain scenario one fourth spin case mirror symmetry respect overlap gain supervision lead overlap gain meet previous gain correctly flip overlap fourth spin spin see possibility possibility exist possibility gain fourth spin reach remain inferior one describe domain case active realize relate mirror symmetry overlap gain furthermore straightforward calculation gain scheme domain supervision one domain spin flip semi recover one perform simple suggest spin negative supervised overlap gain confirm thesis gain even domain still budget spin positive spin employ cf small efficient spin inside spin map configuration active estimation regime separate spin converge picture estimation unsupervised map odd gain spin yield maximum thereby gain supervise remain aim analytical prediction focus trivial regime high noise intensity exponentially many map observation recall domain run find sequence apply finding focus domain see domain figure show belong domain fraction inside rather instance belong domain solid error plot figure use infer align overlap infer hide gain gain noise simulation near perfect intensity switch eqs interestingly branch agreement remain near perfect intensity however agreement branch give assume start gradually break carry correspondingly way employ write error original active yield low whole intensity intensity slight curve analytical active inference symmetric hmms expression relate within approximation specify prediction active bs hmm observation correspond map find domain odd inside domain domain one focus filter separate weakly domain extent section always assume apply heuristic suitable measure maximize domain reduction mirror symmetry fourth scheme spin domain fourth spin relate unique principle domain since spin spin find wrong spin sequence however spin configuration beneficial another discussion uniqueness wrong appear configuration extend task assume joint problem might bs hmm extend robustness strategy extent optimal hmms answer examine imply viterbi return several candidate likelihood energy intuitively analogy physics domain generalize apply instead ise random ise system situation dimensional type recognition vision human heuristic regime relate tend confirm rather
person tv fc ap fc fc fc yes yes cnn acc baseline fc activation improvement visual third object map supervision localization comprehensive recognition task representation visual recognition convolutional cnns rich straightforward approach method fair activation aggregate activation scale wise essential replace activation significant mit use task task introduce scan data outperform method descriptor devote representation bag design descriptor representation kernel order major descriptor invariance property advance visual convolutional cnns jointly whole class stack processing million power training recent imagenet contribute scale cnn extract independent successfully apply generic combine activation show task detection fine grain attribute recognition image activation generic way response second response geometric variation common random augmentation though use prevent average multiple activation help geometric invariance cnns activation activation achieve invariance characteristic recognition fed patch pre cnn activation aggregate fine scale introduce activation paper discriminative robust geometric figure utilize cnn activation activation state fisher scale wise demonstrate scene classification pooling activation demonstrate object confidence map localization label object bounding box meaningful mechanism pooling scale performance representation fisher neural activation review add kernel visual extend bag model descriptor respect although across possible classifier linear dimensional descriptor aggregate descriptor intuitively direction descriptor kernel improve additional follow activation cnn fed network patch extract scale densely inefficient redundant perform extract dense activation replace fully connect image large feed modify output cnn thousand dense level extract extraction second per c scale activation naive sec fc activation cpu gpu image generate pyramid minimum size feed scale activation activation merge pyramid descriptor aggregate explain cnn activation descriptor adopt kernel cnn introduce modified pyramid contain scale local activation extract apply aggregate activation fisher merge pooling cardinality since concatenation improve fisher framework finally overall illustrate scale characteristic traditional activation cnn activation represent aggregate perform scale wise fisher important fisher label horizontal denote scale number obtain fisher give activation pool fisher combine activation accord patch take traditional kernel visual descriptor sift densely descriptor encode gradient detail mid cnns fc fc represent level posterior visualization region activate cnn different property fisher sift sift cnn wise dense descriptor framework encode performance fisher accord demonstrate clear sift activation come fisher perform sift properly low level aggregate cnn activation poorly activation ccc pool label horizontal pyramid possible aggregating scale cnn activation however dataset activation contribute balance examine pooling pool perform experiment pool five number pyramid superior rapidly finer involve activation fine scale level dominant form fisher vector exhibit increase cnn activation evaluate scene type rather use quite class precision grain number cnn compose convolutional layer three perform top validation evaluation henceforth since nearly cnn henceforth simplify five convolutional layer connect convolutional mostly cnn compare demonstrate cnns seven scale default seven scale cover procedure representation pyramid scale pyramid resolution define feed reduce pca activation consequently versus rest mostly implement library framework cnns perform comprehensive method compare state descriptor pool summarize cnn activation cnn dataset fc perform ap ap improve improvement regardless augmentation baseline sn activation scale representation ap gain representation activation utilize activation baseline exploiting scale baseline verify multi far naive fisher kernel pool significant multi activation encode option concatenation without raise proportional outperform pool far representation recognition pyramid representation construct pyramid region middle time difference rich sp redundant various activation perform compare outperform possibly superiority fisher seven quite way suitable aggregate neural record dataset fc complementary discriminative stack stack performance complementary summarize augmentation multi perceptron mlp ground box representation use pre augmentation use box annotation gain adopt well cnns source imagenet task pyramid pooling cnns fc slightly compare fine fine tune nearly stack low augmentation fine tuning believe augmentation truth bounding box major pre cnn performance low art perform among class class object demonstrate benefit activation fine handle pt description fc cnn ap fc pool ap average fc naive multi scale pyramid fc concatenation wise pyramid fc
compare discrimination large eigenvector u u k maximize sum unweighted variance without difference pca direction second p distance direction affine span anchor remove effect locally anchor show visualize maps equation anchor metric furthermore manifold anchor straight experimentally approach fisher outperform manner mahalanobis space instance differential fisher exactly cosine finite discrete transformation intuition anchor similarity transform thus distance anchor point speedup learn low linear ensure finite discrete large importance unlike margin triplet constraint triplet near neighbor instance neighbor neighbor matrix loss cosine problem use project sub triplet simplex study symmetric margin cosine cosine fisher metric problem manifold differential instance need mass discrete distribution interpret define similarity base geodesic geodesic form straight local metric would metric allow geodesic distance follow parameter similarity fold inner angular pairwise distance point separately similarity svm inner angular select inner cv use triplet triplet constraint three class learning evaluate statistical student ranking schema find point difference b manner eight mahalanobis rkhs significantly four bad eight statistically significant well score good predictive follow svm explanation score understanding performance speedup anchor select low reduce anchor fold inner train split cross significance accuracy achieve accuracy map instance similarity induce metric induce robust discrimination anchor distance unlike psd interpret experimental svm acknowledgment wang partially support education research innovation number award ap award coordinate smooth coordinate approach gd r gd gd term accord definition r gd similarity anchor fisher information semi definite significantly crucial role learn task metric euclidean address satisfactory manner lead last global call single computation instance follow project vary locally flexible local address limitation metric smoothly learn metric riemannian metric geodesic computationally expensive approximated geodesic form straight line along unfortunately distance flexibility first hilbert rkhs global mahalanobis space define mahalanobis rkh space induce semi transform psd kernel psd keep similarity similarity anchor riemannian density bias region low dominate learn riemannian direction orthogonal effect locally irrelevant remove knowledge first algorithm flexible various similarity moreover local metric algorithm distance form evaluate dataset metric instance dimensional vector type manifold manifold metric probability learn different type simplex etc manifold similarity differential map compute similarity define induce riemannian riemannian space intrinsic instance category density metric large discrimination anchor orthogonal manifold new distance remove locally irrelevant dimension anchor remainder terminology denote exist define coordinate contain smooth np interested variable otherwise fisher metric manifold distribution replace give explicit statistical otherwise leibler kl divergence fisher probability fisher approximate hellinger cosine importantly manifold g equivalent fisher distance smooth tangent riemannian induce pf f nd pf p smooth coordinate jacobian function g psd metric follow lemma relation metric map geodesic endow geodesic endowed fp appendix assume geodesic form line approximation learn anchor k differentiable non similarity distribution outcome I map similarity define discrete learn instance intuitively instance onto base lie ignore reader anchor give anchor empirically cluster norm control
smooth experiment evaluate precision baseline evaluate standard instance learn preprocessing normalize bag average bias implement suitable computing near neighbor object detector recently window annotation fine tuning annotation detection method detector note besides bit annotation bound annotation meta annotation annotation annotation detection efficacy recently construct window neighbor instance proposal feature mode account windows method intra background sign onto metric set contrast protocol scoring detection table show detection dataset method per object detector optimize develop object set window initial model refinement detection supervision object source code website thm proposition supervision vision since costly image submodular automatically discover set positive window formulation leverage quasi provide improvement art classical paradigm object instance bound exhaustive labeling costly dataset massive annotate visual different weakly without box object detector goal supervision learner label object access annotation box start object million selective formulate discover contain detector smooth recently proposal detector prior weakly supervised detector improvement achieve relative weakly weakly discovery mid visual number object formulation level presence absence present challenge image implicit correspond initialization early effort focus center simplify clarity focus design initialization helpful bias work detector generate box annotation challenge focus design shrink mid use discover visual occur discover element provide discriminative mode draw connection shift challenge segmentation object address pixel co submodular share submodular idea rectangular window detection however level label classic multiple think bag rectangular specifie contain specifie category instance label typically find convex mi practice heavily initialization focus extensively initialization initialization method approximately greedy initialization refinement produce detector however far improve alternative mi optimize auxiliary objective bound novel objective solve unconstrained bfgs experimental improvement bound selective search proposal box ultimately box box neighbor box neighbor optimize occur iii multiple graph ii kb image connect implement occur positively image equally close negative image consequently box neighborhood neighbor box equally box close picking box green highlight neighbor box box neighbor cover cover redundant relevance maximize box many neighborhood complementary large additionally cover gain close neighbor fs fs ft ft thus thus finally sum submodular also submodular obvious algorithm factor say intuition special covering filtering minimum merely single smoothly sub relevance visualize class experiment might mode shift htbp htbp review enable unconstrained smooth optimization analogy object binary want learn typically bound box amount find bound box contain image result exponential choice solve scalar svm formulation eq
datum show bss leverage competitive unsupervised feature bss unsupervise well bss leverage score run leverage score primarily feature heuristic svm support bss large dataset support bss select closely approximate singular perform namely contain task regularize svm formulation since pre multiply repeat experiment five around randomness bss experiment bss bss increase right projection extend provable guarantee empirically bss score comparable method guarantee full datum construct appear progress make full provable advance approximate dataset direction see svms sp support nsf theorem corollary science institute ny usa computer department institute ny accurate svm deterministic respectively supervise supervised prove feature bad case set worst thereby ensure pose world often well state art provable linear support svm theoretical result svms numerous technique work feature selection supervised preserve minimum relative case thus open supervised setting select preserve margin support vector error data label separable primal constructs maximize geometric distance hyperplane separate separable soft norm lagrangian formulation soft quadratic regularizer hyperplane construct relate result lie hyperplane width bound sample monotonic provably selection unsupervised guarantee margin run deterministic algorithm svms deterministic logarithmic margin sufficient linear margin margin margin solve suitably support prove margin whereas get strong deterministic select select optimal optimization datum prove within within effective dimension combination pure preserve non trivial svm practical heuristic bss allow main unsupervised score unsupervised elimination sampling qr method leverage come provable supervised art heuristic empirically provable empirical survey feature weight formulate perform step method formulate sparse svm selection rank radius bind work include doubly machine penalty involve bottleneck formulate fisher fix show al projection margin preserve bss leverage select regularize learn identity matrix decomposition svd contain contain singular matrix singular vector spectral consist replace rd whose lagrange multiplier determine dual imply entry datum ir cm relatively simple margin obtain full score say comparable feature bss rescale margin matrix optimal eqn feasible solution eqn combine towards difference rewrite get z opt opt combine geometric margin supervise leverage margin svm say feature ensure comparable bss feature sampling rescale part replace result follow result leverage sampling leverage score rescale margin radius ball radius sample subspace bss consider minimum equal b n n b b ball n center minimal b point radius clearly bss leverage svm output leverage fold scale like bss approximate bss scale time offline matlab r intel processor gb ram bss comparable suggest pick column satisfy choose high among column euclidean never compute quadratic program dataset feature point point relevant varie construction run select fold synthetic pick supervise bss feature repeat five set select ht supervise unsupervised l music education read iii uk bss read education
generalization locally optimize factored distribution essential step factor factor iterate place jensen logarithm q furthermore since equality contribute additive though entropy inside maintain factor form invoke optimize entropy rather conditional seek marginal begin modify multiply repeat iteration section compute efficiently multiplicative lee must compute multiply divide multiply multiply avoid storing array implementation total arithmetic source mathematically make relate correspond typical alone nmf learn represent dictionary discard st evolution activation learn fix divergence nmf measure contribution bin common magnitude fourier typical approximately reconstruct separated multiplying consider output mask audio take arithmetic operation per array sensor audio signal value enough wave array enough issue wave linear square bin fix geometry problem bin take single design matrix bin parallel array interpret frequency treat marginal source allow tie together source require direction account sound source choose appropriate multimodal come true desire projection begin factored model force finally factor number time mask source symmetric source priori environment source multiplicative update reduce resource requirement example eq multiplication compute multiplication divide multiplication multiply similar intermediate memory input output never array even mass f f td df simplify since nonzero denominator sum define define get arithmetic multiply take operation sum multiplication multiply resource requirement arithmetic factor supervise nmf use use clean audio still resource cost traditional separate db mask mask nmf instance confidence instance available version paper ghz intel core gb ram demonstrate random sentence construct two different array mix file delay relative separation directional less directional nmf consist f nmf two receive channel mix audio audio clean algorithm source directional nmf source directional nmf two directional close direction fit speech background directional speech center distribution location source applicable well array rigorous geometry work particular separate closely spaced acknowledgement thank paris david suggestion regard theorem assumption method audio sound propagation separation greatly remove nonnegative factorization method audio source sound provide potentially arrival bin form frequency direction tensor source advantage much traditional supervision clean source resource traditional array extend technique audio literature apply nonnegative stack drawback decomposition gain post cluster
dynamic equations follow eq let predict diagonal error enough make linearization accurate eq compute density substitute substitute membership along switch likelihood maximize posterior label arrive posteriori map current drop thus substitute f apply linearize temporal logarithm ignore diagonal large row sbm local initialize membership node change posteriori procedure employ initial membership spectral initialization prevent getting begin priori time step calculate multiplication inversion size dominate apply neighboring assignment visit substitute invert log inversion matrix assignment local search reduce multiplication search note search algorithm specifically visit neighboring execute separate core reduce inference four covariance noise relate prior form edge initial x diagonal state mean observation observation state assume invariant dynamic sbm relate advantage plug hyperparameter denote time furthermore diagonal affect entry structure estimate assume neighboring index exploit function propose hyperparameter non survey hyperparameter linear em maximize note propose procedure make involve density gaussian binomial rule binomial reasonable sbm correspond recall small approximation linearization approximately linearization approximation kalman filter filter filter often well computational argue pose linearity observation taylor negligible matrix entry denote bar generally suggest simulate dynamic sbm investigate synthetic generator class state evolve construct snapshot sbm since proportional variance term suggest sufficient square filter use pf g actual pf tracking confirm sufficient pf pf limit approximation network initially split class time randomly assign simulated network baseline static stochastic time spectral baseline propose gibbs anneal applicable posteriori tracking outperform slightly track slightly less second fig method achieve low mse priori posteriori bad priori observation proportional perform extremely poorly set evaluate adjust rand index adjust perfect expect accuracy adjust posteriori offer class achieve accuracy estimate true utilize estimate expense computation core ghz intel processor able outperform computation sensitive sensitive hyperparameter probability conjugate distribute rand hyperparameter posteriori ab note choice fig extremely choice certain rand index close assigning recommend maximize modularity strength partition ground modularity class correspond community dominant modularity extremely mse apply set suffer significantly evaluate baseline number node priori require second posteriori number search v denote increase hold utilize temporal suffer poor recover state show time class number notice magnitude expect al variation membership unlike space sbm result high tracking fig near require inversion covariance could achieve significant noise space rand tp mit reality phone activity student mit year construct dynamic physical measure nearby device exclude begin experiment week participant serve excellent network aggregated network physical first year business school student work compare accuracy posteriori show membership posteriori agree membership heterogeneity within community heterogeneity participant spend proximity time posteriori fit actually demand edge sbm adapt change edge accuracy compare email email week step correspond send cc addition role within company available use class place remove send unlike truth experiment task comparison link link link new edge current remove latter address static sbm alone operate individual predictor move combination link individual evaluate receiver characteristic metric undirected ccc sec posteriori dynamic auc alone priori add membership advance method method roughly auc magnitude obtain diagonal logistic interval tp examine reveal trend increase week inspection content send week confirm cause normal week week correspond event role another highlight fig select confidence show edge frequent discussion six know role begin increase fall notice peak align three week reveal peak email activity event increase volume identify edge across internal dynamic evolve furthermore temporal estimate would fitting characterize dynamic stochastic static either priori posteriori utilize inference procedure comparable accuracy apply base email trend trend steady edge financial investigation examine temporal class reveal examine send predict email propose evolve would provide source code anneal kalman toolbox matlab grateful particle thank comment paper iii effort development analyze network represent either snapshot observe time rich phenomenon dynamic static dynamic manner extend kalman demonstrate monte demand network estimation kalman interest complex biological phenomenon range protein interaction formation naturally network research represent snapshot interest aggregate literature complex phenomenon social researcher examine aspect shrink structural include dynamic social node correspond people edge correspond presence indicate occur characterize network utilize dynamic first propose combine type state commonly static social evolution model become sbm block increase employ present kalman augment monte yet accuracy demand mcmc true state analyze dynamic email interesting trend identify aggregate total send invariant time model applicable datum array social fit lee proposed attribute multi
user easy security access special htb familiar oriented analyst capability part worker difficult adjustment factor multiply produce technical factor get call formula environmental ef multiply f product ef final adjust calculate take calculate support machine implement inductive obtain pattern suffer drawback neural minima rather minima fit mean part suffer either drawback goal map space dot term cost point measure target deviation call basically I kx tx j goal target ignore implement support regression em string regression look em penalty value range default regression range different value rbf sigmoid default calculate actual effort train lastly default parameter predict eight four article software development improvement various technique scale e b f effort software em collect project use input scale element value calculate scale x predict selecting divide learn select optimal step fold generate validation criterion operation find validation select response check error rmse magnitude prediction accuracy test indicate visual comparison step effort various result use effort present test purpose validation partitioning training validation validation remain cm cm remain learn partitioning fold varied range generate operation ht model cp polynomial validation validation ht c model sigmoid rbf choose error validation sigmoid choose base validation c finally train test testing effort use error calculate cm effort test observation q square calculate divide rmse deviation actual effort datum implement software effort follow generate square squared squared mse cm coefficient cm square mse cm regression cm mse cm evaluating strength relationship actual effort kernel correlation point predict minor result predict effort sigmoid plot variation effort correspond show little hence dispersion predict data model exhibit various method effort rbf base htb actual effort estimate data htb comparative accuracy accuracy relate c table display section I less value use effort develop orient estimate effort develop optimize use study comparative assess compare result obtain outperform similarly obtain rbf outperform computation membership available soft particle optimization genetic ga science currently science technology interest software department since interest software engineering engineering management international member usa job software estimation early software detail improve vector getting map nonlinear kernel transform output software approach diagram effort project optimize keyword orient software development several concept abstraction play development effort model project effort effort line paradigm human effort early stage software feasible benefit use point diagram effort product help accurate software measure number actor multiply factor case actor determination simple complex number transaction widely decade limitation limitation effort software effort weight outperform base ba propose machine model public software organization usage produce software effort effort software extension point uk effort
software computational block sect write matlab net represent cnn potentially operation please detail basic neural block serve complex implementation train imagenet stock matlab http www imagenet imagenet mat net load imagenet mat I I I net net normalization describe preprocesse take intensity range layer language I image name convenience matlab score class extension feed average network encoding compute derivative propagation basic implementation gradient example example cnn cifar imagenet probably scale imagenet gpu adequate cpu disk highly recommend imagenet suggest imagenet imagenet convert image height imagenet manner every forget ram disk copy setup ready cnn enable multiple go able describe interface matlab take input return array pack map image arbitrary shape implement work backward direction well order pass third return derivative block parameter x take specified property w multiple matlab rest describe focus analytical refer matlab help implement compute map formally bias filter oppose subsampling array implicitly zero convolution fully array width use filter index various array usually field affect output later connect instead former output handle additional flexibility channel bank w filter group grouping use stream filter slow dedicated block deconvolution implement transpose filter output tensor imagine reverse use bank obtain convolution transpose transpose softmax channel convolutional location softmax exponential normalization operator ground apply across summing combine numerical stability compute vector location option eq q cnn mapping entirely enough hence output filter subsampling sample fall window stay since wide input generate sequence convolutional start signal width sequence signal recursive seem operation obtain approximate exact without filter input signal call determine affect level filter quantity width input layer discrete usually case operator odd delta continuous discrete sample matlab convention extent signal centre support operator coordinate hence offset application centre calculation convenient express convolution operation form I extract storing matrix eq I express expression eq formula derivative array likewise formula use implement convolutional inefficient fast approach allow leverage implementation understand convolution transpose convolution rewrite matrix happen index derive convolution transpose input may outside range convolution expand formula infinity recover involve filter likewise fairly tight depend element possible uniquely instead tight summation refined finite pool output element usually pose max relation exist binary order normalization eq q relu vector sigmoid output compute channel process derivative respect indicator bottom derivative follow note take evaluate simple divide numerator denominator eq simplify obtain array eq softmax array derivative little rectangle em width thin black gray true ex false convolutional toolbox simplicity flexibility cnns new cpu gpu imagenet document provide cnn implement give toolbox toolbox implement cnn document start cnns list build block combine cnns technical one discuss view direct document translation local toolbox contain thank modular create combine new one usually parameter result output learn suitable cnn architecture train thousand million conceptually train point result vector update minima derivative rule use capability default solver top library require iterate vast important large particular gpu reasonable integrated gpu capability design cnn layer software relu operator building cnn sophisticated several world cnn back matlab code architecture computer vision cnns contain fundamental building cnn convolution b filter bank bias derivative cnn suitably implement topology chain block current classification look start point implement cnns descent cpu gpu mnist cifar imagenet state cnn obtain connect image input real array dimension index dimension last dimension represent auto node f east west north south formally stack height width operation identically dimension ability operate batch dag network output output l x f block right node right block right west node west f east dot dot east west east south north south south simple l cnn interested effectively auto node distance datum dot dot w w z right east east f west east west dots east west loss west east south south dag work chain derivative work pass symbol derivative derivative block f composition simplicity drop subscript compute derivative auto distance f block w west east south east west derivative fact first derivative element shape beyond storage derivative storage fact compute latter apply recursively block chain section suggest modular programming interface cnn parameter message cnn derivative block
different unlike rnn require high rnn unable hence current slow fast rnn consist connection hide unlike partition module module module module module period sort module period module propagate leave slow module module standard output q activation time step activation simplicity bias rnn mod execute period period module period recurrent partition block module time period evaluate part highlight period part contiguous matrix triangular forward pass execute evaluation retain output step calculation illustrate ht retain speed module focus provide speed module error module execute activate module activate add module speedup rnn neuron exponential setup detailed derivation rnn lstm activation approximately rnn period initial weight deviation value descent sgd nesterov style momentum ht approximate much accurately task train target whole create ms sequence point interval input linear network summary epoch square decrease set separately keep find rnn rnn crucial forget high encourage nine rnn fail seem improve show big rnn far par rnn get output network five average generation lstm rnn second audio speech dataset arrange order make technical critical recognition competition example partition ms ms window emphasis channel normalize mean architecture softmax layer hide layer whole momentum input stop training epoch lstm forget gate neuron divide evenly exponentially follow give substantially lstm irrespective rnn rnn generation well lr lstm c rnn rnn learn inherent module period period intuitive option period back propagation would alternatively evolutionary closed low period adjust frequency set module size grouping module hard provide superior speech standard approach speech first translate rnn recognize module detailed internal taking place understand class reinforcement rnn assume rnn total neuron connect recurrent rnn half module operation step exponentially period per less recurrent typical evaluation rnn conservative acknowledgment research foundation grant reinforcement learn fp challenging identify distant recurrent ability theory cope virtue short connection long modification standard architecture rnn rnn partition processing input computation prescribe rnn rnn improve preliminary audio lstm rnns recurrent feed connection classification prediction rnn train difficulty dependencie sequence vanish specialized neuron backward order optimization preserve inform random allow training momentum gradient modification rnn error back performance sequence contain term dependency dependency solve different module rnn hide different discrete rnn rnn train module number slow module rnn test supervise using word preliminary outperform lstm provide simultaneous sequence deep variant state result neuron order principle time grow network add recurrent connection connection attempt enable rnn handle dependencies neuron activation bit technique technique use serial rnn neuron decay
psd soft element solve trace psd solve sdp need tune way single switch st projection tuning sparsity penalty solve algorithm trace relative run outperform spectral baseline superior report whereas sequential convex norm tailor rank motivation formulation inference first well understand formulate super using allow notably trace norm observe use support limitation work investigate support future nuclear optimize prove claim nan term large coefficient decomposition permutation decomposition obviously systematic enumeration possible right orthogonal show singular purpose express svd write primal check subgradient equal must satisfy ij z admit decomposition z equal claim decomposition svd disjoint support decomposition attain convexity decomposition contain prove claim consider positive semidefinite prove optimal lemma norm ab q equality maximization convex variational formulation infimum jointly convex elementary analysis also symmetric positively homogeneous j ij prove uniquely use characterization subdifferential characterization subgradient atom ba g norm z reason orthogonal vector nuclear norm ab inclusion middle equality therefore atomic induced atom rearrange eq k ik conclude start ij ij I I b g ia g one similarly op attain right give operator span show take u ia op g op I inequalitie g j I op j op lemma third j dt dt dt disjoint random chi square moment take intersection I jt let ab k q norm fail universal trace notation working norm notation decomposable norm norm point define subgradient norm j jt b lead hence lie cone follow enough ensure n let start th common cdf let denote pdf fu fu fu du v assume jensen cdf error inequality due deduce vector standard check euclidean cone cone index large absolute denote I otherwise ks subdifferential subdifferential let w rewrite coefficient expression subdifferential show mean statistical dimension use normal obtain take plugging show well hand strength lead eq statistical axiom rgb rgb pt electrical engineering stanford universit e paris est imagine des france centre computational france paris france paris france atomic number nonzero factor bilinear slow bind statistical formulation algorithmic scheme propose leverage promising range machine prediction phase dictionary sparse rank factorize sparse factorization allow storage interpretable accurate situation interaction highly overlap admit generally matrix explain superposition principal component sparse high genomic view convex instance note solve plant clique heuristic solve problem lead procedure right factorization optimization hardness generalization mild semidefinite sdp relaxation principal successive investigate coarse investigate convex nuclear investigate investigate naturally relaxation element basic however norm norm sdp find principal favorable thresholding work formulation guarantee new regularizer low multiple provably norm pay np resort procedure solve theoretical gain contribution norm factorization involved rank nonzero right surrogate build upon characterization nuclear support problem bilinear pca formulate norm however compare first slow upper insight trace cone dimension norm superior task factor gain vanish norm vector norm scheme approximately regularizer bilinear quadratic consist provide solution principle numerically focus one simulation linearly decay overlap integer index number support e set index vector entry elsewhere inner matrix notation stand number norm standard frobenius norm trace nuclear singular dealing form j outside allow formulate start define rank quantify introduce atomic tight relaxation operator norm construction constraint wise factor section relate norm norm establish define component concept solve problem notion incorporate j recover share rank follow proposition might collection sum inequality problem sparse consist symmetric approximation want relaxation aim instance atomic norm introduce atomic definition rewrite plug usual singular usual relaxation simply deduce expression follow trace singular generalize call sparse share number usual strictly large leave singular next differential subdifferential b restrict motivated matrix define define atomic atom atom whose element polytope coincide polytope q norm cut atomic norm atom alternatively instance recall norm characterization norm formulate nuclear infimum norm show nuclear induced atomic norm induce atom nuclear induced atom norm nuclear induced support completeness norm vector sort unique theorem find hull scale unit constitute see nuclear norm interesting factorization know nuclear norm nuclear elastic net briefly norm involve noisy low noiseless simply generally matrix priori input observe mean small involve feature convex instance combine retrieval note rewrite form transformation view parameter well feature map assume cluster point cluster low space design mean form row mean exist block sparse matrix dimension try matrix low although wish psd plausible suggest formulate component variance natural relaxation although psd ia follow proposition psd rank psd psd matrix write matrix may interpret successive explain less replace impose replace atom definition consider atomic precise formulation psd expand formulation proposition psd matrix psd approximate although norm formulate guarantee solve let optimization span symmetric np convex involve third heuristic approximate involve commonly recursive manner lead sample possibility force orthogonal component orthogonality motivation dense consequence thresholded pca clear motivation relaxation pca name direct sparse aim solve eq smoothing regularizer trace norm matrix community norm construction norm suffer conditioning atom build tighter rigorously interested depend bring support assertion compare enable avoid aim find sparse component optimization portion unit ball inside psd cone compute proximal map theoretically benefit new penalty low factor discussion building technique recently penaltie norm derive deduce square denoise interest rate norm easier rely denoise wish observation corrupt additive penalty norm control set study random noise order derive estimation provide expectation norm entry expect norm consider oracle estimate immediately follow control estimation error oracle eq derive upper error different call single ab ab ga ab ab ab immediately plug upper bound estimator respectively respectively straightforward matrix atom upper suggest denoise comparison table trace column magnitude instead penalize reach change norm enough conclude superiority trivially incoherent obtain decomposable trace rate still incoherent weakly valid rank low upper statistical norm closely relate powerful asymptotic geometry quantify nonsmooth regularizer penalty essentially point denoise sense quantify measure width intersection concept cone statistical induce theoretical exact recovery norm convenience linearly norm scale norm constant briefly related matrix tangent cone closure cone I standard normal result iid probability soon addition phase situation large situation corrupted assume noisy satisfy z least subsequent section compare technical let simple number actually regularizer statement hold prove satisfy follow informally trace scale nest meet matrix property norm meet nest tangent consequence satisfy hull decomposition c follow hull fact red belong plug show atom good good use norm upper norm probabilistic expectation fact inclusion tangent inclusion ball statement recovery consider norm realization exact recovery support cone cone vector support easy show computation substitute go appendix er add reference ok tangent suggest norm provide improvement recovery performance present specific characterize performance norm turn explicit estimation vertex immediately plant clique estimate coefficient recovery support depend signal atom ab ba compare note match logarithmic dimension trace combination table result vector trace kk p k km mp norm counterpart element plant notation mean norm atom bad dimension atom statistical decrease alone bad trace rate equality ab bring improvement bad statistical dimension statistical trace norm theoretically regularizer lose support specific follow upper dimension sparse coefficient entry sort absolute atom strength upper bind reach case hand atom bad standard never raise utility lasso complexity different regularizer elastic net tangent point half cone elastic net always dimension proposition improvement note degree freedom match logarithmic term aim improve proposition situation generally proposition match low statistical note cone equal tangent cone tangent exact statistical similar vector sort equation active say element surprisingly dimension proposition number degree element match consider aim perform unclear sparse matrix norm symmetry yield ab map fail recover universal appendix equality ab must grow ab dimension support small decrease upper norm bound tight sense suffer ambiguity become sensitive reader many involve sparse rank work column subset let optimality often component use working solve grow sequence zero throughout typically useful regularizer notably group lasso also optimality subdifferential writing ij j current approximately subsequently violate add initialize previous solve descent iterate proximal modification solve minor amount replace
large dimensionality play modeling typically produce candidate involve covariate compare researcher selection former kl deal devote extend chen chen liu al aic frequently use tuning mode penalize wang wang et zhang al fan fan inconsistent grow size implicit fix dimension case recently liu selection misspecification principle generalize linear lead aic bic prior probability motivate principle generalize bic liu dimensional counterpart misspecification misspecification high answer question gain motivate response functional size dimensionality regression criterion oracle working consist aic bic ignore misspecification reasonably select work fail model selection model newly suggest work significant establish misspecification high expansion different principle challenge technical justification incorporate misspecification set prior connection chen chen fan organized introduce misspecification present key quasi provide selection main technical supplementary material entail large denote deterministic practice work data misspecification generally occur true choose generalized link work db z contain value nf n observation vector define kl work close true two play role selection misspecification f asymptotic expansion kl divergence principle list prove property constant assumption establish standardized response liu major setting converge neighborhood wide dimensionality allowed impose normality condition normality dimensional glm liu next introduce additional principle hc n n n n norm naturally accommodate mild ensure restrict expansion kl divergence grow shrink except require lipschitz entry wise mild sensible bounding prove set principle mle expand lead aic compete drop quasi hereafter tending generalize liu high substantially due dimensionality correctly asymptotically demonstrate study aic substantially misspecification expansion latter introduce contrast characterize impact misspecification provide accurate criterion f assume hold liu aspects contrast previously justify liu simple plug enjoy consistency misspecification crucial practical implementation reveal work misspecification setting compete nonzero vector correspond bounded md locally zero posteriori model ease quantity quasi likelihood condition tend nc replace model reflect effect misspecification correctly specify reduce bic probability subscript candidate motivation far away glm larger sensible complexity motivate exploit extend bic chen chen show additive hold asymptotic new term non polynomially grow tend n np theorem consistent penalty fan side view sum misspecification counterpart dimensionality whole asymptotic expansion divergence principle introduce high dimension misspecification investigate selection bic error multiply effect misspecification involve covariate five function true regression result table latter supplementary term two confirm necessity fan specify multiply selection inclusion probability oracle prediction regression logistic pn rest response success section choose regression logistic interaction model argue oracle correspond regression five since replace table latter available supplementary show phenomenon section model dimensional multiply oracle error rate gene expression set positive negative nb set control project consist positive trial set fan exploit screen apply retain choose time median median table table good l median classification worth parsimonious expense effect model misspecification generally real suggest involve misspecification expansion selection percentage nb despite misspecification misspecification newly factor dimensionality complexity establish consistency contrast capture misspecification general adaptive correctly fan consistency criterion sample misspecification principle additive problem beyond current topic present theorem save technical material notational throughout proof specify order state constant notation response convenient euclidean main event calculate continuous full column definition n continuous concave concavity log positive entail event global maximizer must belong interior neighborhood hereafter condition herein due grow taylor expansion q line taylor expansion rewrite take derive eq negative product n entry respectively next obtain bound sub tail condition exist see notation let tail derive last th obtain norm dr thereby q choose hc choose stand omit subscript sequel establish require possibly n specify grow intuitively understand try put restriction nm large ensure mle coincide shrink hereafter mle unless recall e ne complement equality follow definition term expansion around evaluate tr n b regression last inequality verify order I n n n n n es n cp ensure show establish supplementary e cauchy schwarz yield entail yield similarly q c asymptotic use derivation restrict conclude proof view expansion square arrange increase large eigenvalue side n p result norm
reader development contain trait may importantly handle different govern unobserved p depend arise multivariate brownian realize follow probit formulation outcome underlie continuous threshold threshold alternatively assume order state map relative threshold identifiability non order trait value adopt observe dimension determine finally monotonic transform example distribute brownian diffusion give rise element node multivariate distribute unobserved trait variance manner characterize trait trait share descent trait integrate trait recall density function one equal weight short path node I nod augment trait latent convenient factorization factorization augment truncate normal illustrate four include tree annotate trait realization along code trait realization trait state figure modified package freedom specify aim learn mcmc development computationally kernel exploit scan metropolis scheme employ metropolis proposal full enable problematic tie also attack evaluate metropolis acceptance appear high form nf repeat limited algorithmic idea computationally sampling illustrate pre propose order traversal post traversal proceed imply internal root visit compute conditional proportional normal traversal precision characterize reader f ff distribution hasting approximate full conditional remainder distributional far collect wu conditional quantity normalization u u traversal integral solve n multivariate identify u pre partial vector precision scalar f toward must latent trait trait ic ic ic cc partition correspondence trait matrix generate possibly augmentation example manuscript range explore involve try metropolis simulate f hasting acceptance proposal start occur valid become mcmc mcmc chain towards probably employ proposal center assess relationship trait look pair wise correlation non fall great strictly less scientific lie comparison involve pair possible identify diagonal structure trait evolution demonstrate example order trait factor likelihood straightforward high adopt integral estimate possibly present estimate marginal likelihood efficiently comparison different path q parameter sampling employ numerically path natural path since require lead guarantee normal independent corresponding univariate whose match large trait cdf trait univariate analysis threshold map assume open interval dimension simplicity multivariate present wish assess type evolution report wise correlation reveal trait trait analysis additionally second trait highlight change feed trait order state model outcome position instrumental determine infer state bb regard adaptation examine compare trait bb bb bb formulation order latent bb bb inverting order sign trait marginal indicate bb factor bb bb bb bb bb share evolutionary estimate latent distance present compare account share history notice correlation pairwise orientation orientation contain evolutionary weak orientation account history surface provide rapid drift challenge drift site grant insight analyse site b protein site vary period major allele frequency suggest limited contain variant trait latent without generality site assess structure latent pairwise zero estimate site site contiguous suggest include drift present credible interval include positive range site trait see association evidence correlation coefficient site drive site latent assess use discrete biological problem show structure tool general threshold latent markovian argue trait vary spend state univariate order reconstruction simulate perform trait consider already comparative biology correlation requirement account lack sized interval would discrete correlation credible interval constrain prevent recover example two trait continuous trait root internal motion lead significant improvement compute successive post traversal effectiveness multivariate motion evy traversal improve regression gaussian latent perform integration build dynamic programming truncate conditional though truncate accept highly find rate become reference state model dimensionality mainly improve identifiability symmetry interpretability trait represent entry trait despite different choice change link briefly determine outcome common simplex make trait interpretable alternative evolution model tendency investigate whether identifiability relaxed explore trait branch comprehensive analysis acknowledgement lead result receive european fp grant agreement agreement trust health grant ai author acknowledge reference service dt provide constructive manuscript orientation length length control orientation orientation site page en sn ns dy dy dy sn ns vi ns sn ns sf site site align code correspond latent trait site belong department mail school public health human david university california ca usa center usa trust trust genome united understanding trait modern evolutionary biology assess type simultaneously control evolutionary molecular us trait trait trait state single along history trait history framework finally em evolution phenotype interested assess among trait genetic trait determine link alternatively selective environmental pressure act trait outcome trait affect pressure aim comparative purpose trait simultaneously combination type discrete outcome discrete outcome also tool hypothesis regard comparative trait trait control share evolutionary history dataset markov trait allow include trait assess transition
impose construct unique diagonal incoherent note fraction perturbation suffice make impossible condition l converge approximate guarantee exactly rank special guarantee noise n q present key standard argument detail iterate incoherent perturb appendix symmetric iterate sparsity ii iii repeat fold recover significant b demonstrate conv art solver experiment experiment real foreground result average pseudo code knowledge instead th tune conv incoherence cccc see interestingly remove step arguably dynamic foreground keep step take fast restaurant frame resolution moreover visually extraction well corner counter similar background video require non pca projection method match method experimental interesting match model recovery result improve noise need investigate decomposition beyond structured sparsity pt acknowledgement aa would like acknowledge nsf grant microsoft fellowship acknowledge grant grant acknowledge fact question recover rank unknown support project matrix projection establish require input run need run require per contrast complexity exponentially iteration synthetic establishe improve exist convex alternate projection principal pca preprocessing denoise carry pca implement sensitive attempt force outlier overcome pca reconstruction topic community detection input seminal work solve relaxation elegant expensive run large poor require carry complexity drastically pca exponentially accuracy gap singular rate global convergence prove minimization recently completion work method grow match subject sparse perturbation reveal gain relaxation run low rank thus rank nearly match pca sparse technique constant zero theoretical enjoy art inexact lagrange multipli time level accuracy real foreground separation visually separation establish contraction set sparse projection perturbation vector suffice establish correctness next hard thresholding inspire similar eigenvector enyi exploit characterization taylor reveal perturbation eigenvector adjacency subgraph thresholde contraction argument contraction case alternate stage alternate value argument perform rank hard procedure need hope perturbation convex pca robust past seminal work incoherent relaxation eq nuclear nuclear typical solver involve convex set soft spectral domain non incoherence upon match requirement recovery sparsity incoherence yield additional exact recovery entry rank robust would plant problem additional work weak assumption incur xu et specialized exact provide tuning relate multi alternate multiplier consider multi step multi block random work non method pca hold still intuitively project onto appropriately section robust formulate find lie project onto one ht sparse rank initial remove perform matrix initial hard thresholding progress subject large perturbation alternate computing perform thresholding entry certain gradually decrease proceed reconstruction naive extension condition matrix singular singular perturbation progress propose proceed stage projection thresholde run stage lower singular
continuous regret hand another relevant rl action mdps reward sub policy finite whereas policy htp iid primary simulation result demonstrate correlated bandit feedback mdp optima around optima possible equal step maximize experiment optimistic bandit alternative space predict bound per big identical empirically though complexity empirically fig requirement scale predict thm near optimality brief requirement grow polynomially usage iid regret whereas require order node iid create mdp function upon take value environment agent receive state priori reward rl optima rl policy mdp optimize optima algorithm design set mdp succeed find evident converge stochastic approach computational optima requirement suggest benefit approach online mdp global knowledge policy search current version learner unknown iid example require simple dissimilarity cumulative smoothness notice dependent reward policy partially mdps introduce new regret bound simulation feedback broad report iid introduce notation indicator create depth internal e leave nod n h need introduce last time time step expand node expansion expand coincide phase bound depth construct depth tree iid expand expand summation obtain solve high expand within empirical estimate depth application inequality upper depth e term choice eq recall definition sect term confidence interval confidence confidence interval bound probability first regret trivially suffice never imply remain term sum union imply combine ready bind tc step far instantaneous regret bound martingale difference probability proceed measure start characterize actually event node node immediately select value iterate parent node since cover space least include maximizer hold expand side high inequality definition p tf regularity node always exist leaf see parent select far simplify provide preliminary instantaneous select refine parent event rely rhs simplify need depth binary node depth twice node expand also recall parent select parent cover combine second term cauchy schwarz total focus summation child sequence rely time rely cover internal cover sum invert deduce notably term choose lead regret bind eq lem union final analysis assumption inequality iid random episode consecutive episode first select episode horizon arm episode arm objective build q notation episode condition time see definition residual follow bound martingale proceed arm episode total episode far rewrite sum lem grouping divide need need previous number episode step start episode notice unchanged begin apply result lem lem episode previous episode except episode large termination episode become large time episode horizon episode thus invert previous obtain episode probability lem statement simplify side objective achieve homogeneous h q lem previous case bind hoeffding high probability confidence estimate interval event concentrate arm could argument reward therefore inequality concentration eq arm event hold complementary step lem term definition step bind definition sect decompose depend event interval instantaneous rewrite interval confidence regret iid hold bound expect ready regret step difference step decompose instantaneous regret lead regret unlike sequence difference extra need derive follow hold definition nn every last episode coincide episode event result parent proof arm entire episode sequence simply episode instantaneous statistic episode immediate put bound lead final combine lem prove final nc hold reward generate store correspond branching time regret decompose depend since expression easily bound lemma rewrite total generate bind inequality expand fact large twice invert bind need probability event leads I c plug together depth lemma optimize statement online confidence algorithm bandit regret bound dependency challenge reward whereas reward generate iid process art well weak smoothness previous reinforcement sum reward sequentially optimization arm objective cumulative relative global focus immediately condition identically contrast bandit armed bandit relevant internet online game policy mdp mdps paper sect introduce first policy mdps build advance iid regularity e smoothness guarantee linearly number rely heavily iid feedback introduce sect explore arm space tree optimistic part arm insight achieve necessary expand optimistic sufficiently accurate even iid ergodicity mix sect correlate match sect iid dependency define require require though supplement development iid structure complexity runtime make scaling meet improve space iid sect benefit sect mdps arm formalize possibly relate arm reward context time arm dependent reward since infinite refine setting generate ergodicity ergodicity regardless arm time average follow mix finite exist mix stochastic reward trivially iid maximizer exist denote maximum learner observe differ contextual contextual bandit arm immediate reward function input contextual context next may reward current reward problem maximize reward sect reinforcement reward differ regret prove discount reward seek minimize binary cover detailed covering root cover index convention area region partition overlap arm algorithm whenever assumption dissimilarity equip diameter open ball bx x exist constant bx b coincide lipschitz arm require characterize optimality dimension set arm sake clarity dimension near optimality tree confidence iid arm tt tt tt tt tt tt tu correspond select framework discuss variant iid design reward arm iid reward arm alg ht tree ip ht tt hierarchical reward algorithm keep track optimistic begin episode episode episode valid I reason accurately bandit feedback assumption reward correlate reward actually lem mechanism expand node obtain mean feedback general iid variant reward arm full episode arm iid use complexity proof supplement reporting depth generate threshold guarantee fact estimate expand number grows depth grow report regret iid condition event immediate mean reward supplement perfectly match check show structure expand result require iid discuss sect although proof mostly literature move iid call technique main issue average different episode concentration inequality lem supplement episode bound technical derivation hold generate accord iid aspect iid iid major w arm iid mdps boundedness phase lem depth also coincide case node episode reduce selection cost cost node boundedness depth node still cost time unlike extra due truncation provide space space scale observation increase factor
coherence depend acceleration result noisy corpus stochastic mix channel set noisy clean impulse noise partially consist room mc corpus source near circular array diameter task error gmm enhance order noisy feature compute dimension show development test dnn acoustic achieve gmm negligible lead acoustic train clean noisy multi yield confirm coherence exploit dnn frequency resolution reduce improvement show necessarily exploit signal front input propose cloud speech recognition geometry device adaptation oppose sense arrival recognition achieve spatial dnn speech recognition environment dnn real multiple knowledge direction arrival bin feature acoustic rate challenge extract enhance spectral recognition automatic markov gmm hmm wide extraction employ extract contain signal efficiently acoustic neural network neural acoustic learn manually transformation outperform amount structure trend replace stage implicit array spatial channel feature gmm exploit signal aware noisy may principle noise estimate dnn acoustic exploit inspire towards spatial information field acoustic speech noisy environment treat coherence surrogate temporal variation aim dnn acoustic describe instantaneous coherence speech integrate dnn task outperform consider speech record th component short letter frequency axis auto f ratio coherence desire characteristic complex mixed sound bin convenient computation cdr diagram extraction signal domain correspond extraction term log combine power compute triangular filter apply show extraction enhance multiplication gain describe estimate weighting extraction apply sound field amount noise dependent expect estimate neural network enhance trend acoustic replace implicit use coherence coherence characteristic sound array may therefore array spherical array require acoustic model coherence system reaction change coherence change ms window frame shift ms transform dft factor triangular weighting cover code feature speech training corpus speech highlight
diameter bx side pick one optimal slowly bound see condition boundedness extension meet oracle action chosen long grow problematic controller automatically turned keep design robust back state safe system consideration controller rely controller replace available use controller come leave safe input safe initialize controller controller pick controller reasonable controller prevent exist happen theorem safe q consider family deterministic markovian smooth parameterization regularity running concentration trajectory addition optimal controller immediate r action step illustrate mdps mdp feature dimensional zero indicate mapping state transition probability state prior dirichlet nd tp ts ts action pair ts nd dimensional show frequency ts ts action dirichlet time choose next parametrize gaussian share similarity generality subgaussian gaussian without generality compact assume column corollary performance parametrize problem controller available parametrize suppose time satisfy resource management problem control resource chain describe evolution find class admissible action compact leave conjugate imply get propose algorithm thompson thompson distribution thompson compute propose choose time unlike thompson implementation thompson computationally may subject largely mdps sublinear consider finite horizon policy issue arise policy set significantly set approximate computation policy discount let result limitation trajectory follow policy policy general action obtain mdps space linearly still computationally even planning purpose simple server control control follow next web book book http server incoming queue connection assign process drop request second denote control long service usage server bound determined load usage operating point operating sequence gaussian diagonal deviation operating http server measure provide control purpose cost algorithm optimistic maintain find attain loss solve policy play objective solve consume regret avoid repeat time deviation horizontal axis amount process round change frequent bad prior regret right bottom prior horizontal mc vs chapter fx non inf admissible state action pair assume mapping variation norm sign measure admissible pair contraction banach substitute transition set negative inf admissible action transition discount discount policy exist satisfy bound additional assumption gx define one thank hold eigenvalue one large eigenvalue inequality definite matrix plug bind part semi nonnegative minimizer loss hx bx trivially fail proof decompose regret bind map deterministic let control oracle second follow change number last change multiplicative use old last cauchy schwarz second collecting inequality change together continue assumption replace along trajectory reason get cost thank check define x mx ax mx thank apply corollary satisfied na dirichlet p e thus theorem thm remark university general smoothly markov problem design posterior maintain unknown reduce importantly analyze show performance computation tradeoff method web design control randomly controller mass produce production pattern maintain good control rather appropriate knowledge transition dynamic cost would history apart policy resort suboptimal suboptimal compare optimal question computationally add reward estimate horizon discount sampling first reinforcement mdps failure regret factor course apart interested excellent unlike allow infinite subject regularity cardinality secondary compact phase begin compute draw algorithm keep uncertainty important element allow nonlinear allow go scope quadratic linear linearization policy achieve long run average loss resort measure dependent slow policy sublinear converge get
position contribution assumption make lipschitz logistic error convex hinge loss nevertheless smoothing algorithm framework still continuous ill condition usually larger add purely regularization purpose well complexity loose useful algorithm e find gradient require g pass form dense contrast operate complexity far take iteration fair incremental term pass expect precision divide complexity example batch batch complexity gradient complexity exploit average coordinate batch weak dependence method present batch saddle saddle function I presentation convex saddle assumption strongly saddle point primal accelerate coordinate complexity also mini suit present first apply saddle still accelerate uniform sampling batch complexity define much discuss method coordinate update extension primal recent batch complexity feature sparse norm penalty computational depend compare art optimization include batch sag coordinate comparable tx uniformly execute I randomly pick execute update analyze dual method idea quite saddle alternatively maximize minimize since dual maximize expensive reduce computational picking maximize computational iteration update primal give instead directly quadratic strength specify theorem auxiliary rule step acceleration fast present introduce mini natural mini coordinate mini mini pick index achieve first disjoint assume select add mini processor update coordinate compute accelerate batch operation ignore delay take basic surprisingly mini single since basic mini convergence choose mini establish iteration complexity mini batch assumption obtain e ensure equivalent inequality denominator recall corollary iteration complexity mini batch achieve batch less iteration extreme batch primal see discussion relate iteration pass mini batch lead efficient prefer choice mini batch batch approximate minimax specifically meet requirement complexity need extra exist either lipschitz smoothness guarantee px run minimize hence imply substitute old suffice hand q x denominator proof first hold second employ unnormalized bound establish addition fail saddle function method case formally continuous saddle scalar modify saddle employ mini add become saddle perturb effectively minimize assume convex lipschitz continuous mini px shorthand inequalities function vi saddle lipschitz lipschitz continuous px establish inequality smooth strongly smooth strongly handle omit obtain sublinear use perturbation complexity mini drawback convergence specific unnormalized norm batch r propose achieve accelerate sublinear rate extend complexity conjugate coordinate ascent efficient method coordinate ascent sdca coordinate pick update increase objective zhang sdca batch complexity ill vast coordinate nesterov randomize lot activity analysis composite variant study batch sdca sdca zhang propose accelerate mini batch sdca primal sdca mini show sdca vary size sdca ill condition problem zhang develop accelerate proximal sdca achieve outer outer loop primal loop regularization contrast straightforward coordinate recently lin accelerate coordinate method convex enjoy extra primal pass splitting equivalent condition formulation whole propose admm sublinear rate complex regularization mapping update sdca efficiently combine method dimensional operation computational per iteration exploit structure dimensional case penalty cost per iteration depend case update ta ta k p jj denote pick value update update delay imply update value iteration compute subsequently compute j x simplify similar assume calculate combination consequence basic algorithm solve problem update accelerate quasi bfg adopt adaptive scheme improve bfgs suggest gradient sag sdca conduct three compare simple quadratic synthetic generate ill condition standard consequence lipschitz cc horizontal pass vertical algorithm output pass entire global regularization coefficient relatively condition sag substantially fast bfgs fast sag sdca notably batch sag sdca name news classification obtain reflect news form hinge e bfgs result substantially stochastic batch decrease l become compare sag sdca opposite stochastic relatively fast get close comparable px fast method relatively ccc news focus maximized strongly minimize property inequality accord randomly specific event happen old sigma field generate define expectation q representation sum index divide I define ta I ta update characterize step derive strong convexity function inequality last definition relation ta term eq absolute eq inequality recall assignment define recursive implie eliminate hand side second equality bind prove relation establish sigma variable substitute average relation inequality q row expand satisfie plug assignment notice q inequality recursive calculate examine equation close equation q
prohibitive alternative efficient wishart p convergence element location conceptually draw wishart scale dependence reach within moderate novel posterior baseline reversible sampler show may build jump sampler way doubly intractable wishart calculation acceptance ratio newly graph reversible jump provide substantial avoids ratio invoke wishart nonetheless edge ratio cholesky double reversible auxiliary convenient acceptance remove flip edge consideration algorithm employ sampler wishart use sampler follow graphical accordingly usage direct double computationally scheme decrease proposal essentially accept mix chain poorly introduce birth death removal birth death change death independent death birth death poisson death analogous birth birth stationary birth death observation birth death use double factor exchange auxiliary novel double continuous dct current g create permutation variable accordingly compute accordingly draw time event accord probability death event birth event validity propose subsequently brain double reversible novel conditional independence follow scatter construct enumeration show expectation three matlab p execute double jump double expectation calculate discrete quantify leibl consider performance true contrary good find fast apparent finally efficiency substantial increase dct whereas time slow kullback expect precision visit double reversible jump mse kl model e dct bayesian assumption underlie structural simultaneously estimate connectivity functional collect subject reader acquisition preprocesse step volume correction derivative filter hz result region signal voxel standardize brain dynamical change produce direct region express correlation brain suffer drawback indirect alternatively partial correlation capture correlation matrix coupling must reveal connect word connectivity dct execute discard burn algorithm identical edge kullback leibl probability connectivity majority high probability show functional right correlation indicate functional salient expect correlation direct pathway well unobserved high partial interestingly edge associate weakly couple away algorithm direct result birth death continuous accurate estimate substantially fast functional connectivity simultaneously work improve sampler introduce move single edge correspond contribute efficient graphical acknowledge economic education innovation van david acquisition fmri macro bayes factor center prove prior matrix doubly partition development direct wishart estimate infeasible propose direct efficiently approximate graphical metropolis algorithm substantially art structural connectivity use fmri area amongst example gene amongst dna segment customer connection population link cognitive gaussian zero precision fully gaussian prove conjugate restrict decomposable wishart monte carlo wishart resource due wishart bottleneck wishart scale wishart fit dependency graphical wishart goal cognitive understand population couple pathway population connectivity correlate pattern population connectivity connectivity
article multiple sound source denote source vertical concatenation let band time please implementation two q originally human study prove part ratio space equivalently due phase nearby cost section model affect experimentally validate frequency refer acoustic wave signal express respective interestingly sound complex relative relationship may self none source correspond contain source common speech binary threshold value average activity sound source frequency sound frequency sound direction mind central entry sound variation variation mixture sound source aggregate information hundred speech white spectrum theory white noise power spectral density nan entrie sound acoustic vector sound mapping feature sound technique feature direction mean white principle possible direction main apply estimate direction firstly input space high training ill secondly nevertheless sound white predict accurate sound direction dimensional role corrupt hence parameterized regressor estimate estimate low sound advantage regression instance matlab implementation available generalization gmm analytic sound direction condition expression mapping train one transformation plus locally transformation reconstruction source spectra consequently eq assume mixture view low dimensional provide affine lie space summarize em evaluate posterior respect step optimally partition minimize affine thereby capture acoustic manifold justification dimensional single source localization affine covariance instead dimensional regression one full covariance datum localization speech already describe value activity seek condition assumption direction namely whose respect normalize predict general sound localization sbm sbm number source code available include sum variable give covariance p pz notation lead inversion conditionally denominator depend neither develop side formulae bayes inversion mixture proportional kp tp numerator simplify term expression localization evaluate source setup acoustic head leave resolution vertical one pixel horizontal vertical horizontal pixel relationship convert degree head camera place middle room computer fan well training room room room room room train people front device associate ground visual detect fig allow face localization method localization plane error localization manually correct pixel position sound evaluate expect sound convert degree accuracy vector fourier ms window ms yield window hz feature typical two head although head room robustness room validate different green testing training manually place position lie plane parallel front long white corresponding position record training refer straightforwardly importantly localization source dataset generate select mixing position source mixture ground truth refer live head camera distance vary scenario scenario narrow field camera person count english person speech whereas people count overlap language consecutive narrow view camera people count language english remain quasi position paragraph live source train live particularly variability emission distance head etc people head carefully noise segment along align video frame segment generally supervise mapping sbm sbm apply segment acoustic sound video mapping sbm training per white camera take pc sound histogram histogram pseudo probability sound length obtain horizontal image sound literature localization dataset result method correspond outperform matlab comparable binary single error localization standard fourth time propose number component choose degree decrease angular decrease localization seem significantly room decrease outlier none comparison baseline algorithm dependency image dependency model white right number maximum approach perform baseline histogram key propose sound rely sound solely base correspond base take minute standard compare sbm histogram peak mask initialize previous regressor horizontal pixel coordinate sbm sound localization variational estimate binary mask train histogram strongly source source dominate frequency bin train white noise mixture pair sbm show calculate outlier cc ccc ccc c sbm use pair type white speech speech speech mixture source db distance horizontal localization degree localization source speech sbm outperform term source sbm expect yield ratio db mixture though aggregate frequency plane high introduce activity reduce average white speech sbm yield demonstrate prominent sound source localization respect scenario critical correctly people party poorly well white sound sparsity white accurately map mixture error slightly sbm possibly use component sbm use affine angular cover transformation sbm times sbm mixture matlab pc suitable application iterative sbm em fast expression component sbm choose bring localization second localization increase localization suggest dense record minute source pair supervise count language circle find row successful localization typical localization full fr scenario square detect available team fr examine test source completely mixture localization sbm yield speech fail intuitive source locate however fact unlikely frequency match localization heavily second speech segment source experiment type overlap mixture plane perturbation vary source spectra section ms slide analysis position segment necessity sbm frames sound positions sbm frame participant sbm localize two source correspond number localize last column row localize face yield fig sbm single number return source position actual abc source include number algorithm position source another fig source supervise simultaneous sound localization require segregation point view addition camera base implicitly start train use white localization noise source single relatively frequency inherently scenario test make sound direction correspond pixel location numerous use mix sound corpora third audio alignment jointly face localization light experiment reliably sound source sound advantage explicit transfer parameter turn room position room cope position room alternatively additional factor variation investigate devise map scale parallelization reduce live experiment plan use detector automatically adjust window take markov grateful anonymous serious highly comment suggestion receive sc sc mathematics engineering france specialized research computer graphic vision universit france ph mathematics post communication interest learn sc electrical engineering sc engineering ph computer de position national en head team interest audio processing area member associate international conference computer project receive ba degree physics electrical engineering technology member department electrical imaging laboratory computer physics imaging vision modal foundation award student special distinction fellowship fellowship fellowship sc ph signal national et physics materials audio laboratory deal aspect model code synthesis audio visual speech separation computer science associate member team france france electrical localization linear address audio multiple location efficient prior neither segregation start gaussian directional source extract measurement length white reliably enable realistic audio fusion namely speech signal onto align audio modality thus enable discriminate face release novel corpus room quantitative evaluation localization sound accuracy art method source supervise regression visual fusion address sound acoustic head robot analyze interact environment shape setup phase band signal spatially narrow relative directional form sound sound direction source mix direction spectra assume tf acoustic power dominate source simplify tf relate direction valid extent mixture speech party state assign grouping select peak accumulate channel iteratively localization expectation intensive segregation estimate dominant account vast localization along simplified sound information must identify head either several source encode use train single source multiple source compete robustness map ccc sound head camera place head head isotropic filter responsible sound localization sound audio compose marker front head device location marker record red circle location train sound circle square face detector view need sound approach recently artificial network feature infer unknown direction advantage stage acoustic spectrum accuracy condition setup room position room etc rather simplified method segregation devise directly simultaneously source strongly scene although inspire mathematical
follow present large knowledge standard split compare split reproduce publicly available wide book collect used purpose seed sentiment explore effectiveness handle either hybrid annotate dictionary annotation strength amazon dictionaries corpora review sentiment context sentiment customer review amazon bias part speech build sentiment classifier field annotate twitter target review syntactic feature build svm classification build boost corpus twitter hash back review customer review concern sentiment problem sentiment english sentiment propose expand english translation scheme build business review build seed sentiment effect preprocesse normalization removal sentiment simultaneous system independently summarize contain movie review collect divide division neutral star rating consider positive multi modern sentiment review collect wikipedia page system sentiment considerable example publicly train research publicly tweet wikipedia opinion review wikipedia book review review categorization largely category sentiment neutral three category reason ambiguity token internet tend positive rating entity opposite language complexity language language different standardized challenge language name entity sentiment compound phrase work provide baseline sentiment analysis set sentiment per user number review book review token review review token user user review avg review book book median token token review avg token token sentence review dataset review rating negative neutral english show star notice review review rate positive neutral review book month review book book non book book perform processing step tag multiple dot dot character heart symbol character character character compose filter review release format review book review user median review book book review book token token per token number review much large review believe review book book rate book review positive review book datum include review color red represent review example review sentiment rating sentiment rating ambiguity review neutral review review book rating mean rating rating mean review rating book user review vice versa figure review review notice book user review negative review review book review per statistic review rough sentence count token sentence review token work dataset sentiment survey classifier sentiment classification add sentiment classification neutral moreover effectiveness number category unbalanced class category datum mini compare three work neutral review neutral positive rating map neutral neutral important reader review neutral set review category size class unbalanced equal proportion collect data review balance unbalanced count unbalanced notice unbalanced setting exceed pose challenge try feature c c balance unbalanced neutral review part test set balance unbalanced show feature explore two review neutral rating two wide balanced unbalanced gram range gram range gram contiguous degree show number review unbalanced table gram range bi gram range token tf token document gram tf normalize exist remaining define document frequency word frequency word use area sentiment benchmark library use default classifier nlp bag bayes binary term occurrence bayes describe linear select maximize margin online hinge order positive margin alternative improve cope advantage use machine optimize multiclass versus cost cost pattern feed forward linear neural account simple distance majority neighbor c precision recall neutral neutral unbalanced recall svm unbalanced tf feature show training number evaluation perform task sentiment inclusion hard confusion neutral class label write human get mark weighted accuracy class review positive review class q q positive negative c passive logistic knn indicate tf weighting bayes naive gradient knn near evaluation perform weighted accuracy c c passive perceptron knn table task five set unbalanced contain much compare unbalanced reliable precision unbalanced test tf despite unbalanced well dataset unbalanced evaluation proportional good overall svm consistent passive perceptron automatically difficult compound training compound english c passive perceptron knn sentiment classification compare table indicate manually compound phrase permutation combination extract seed utilize useful svm inherently sort gram gram negligible end zero automatic ordering weight classifier select weight positive sentiment low sentiment remove gram gram n gram operator sentiment idea use table sentiment compound phrase effectiveness domain specific sentiment experiment goal stand alone previous negative stand several million lead
reason claim pool vast bagging stack generalization technique propose stack generalization context subject stack divide collect trial trial come train cross dataset classifier train portion learn second level classifier create ensure diversity success approach test combination pattern observe subject specific combine stack way dataset covariate shift weight logistic regression great plain stack generalization pool sg inferential purpose within basic simple shift trial trial draw analogy decode across stack stack sg simply sg cs extract difference baseline sg reach sg show decode subject decode single subject reach motivate brain decode experiment predict group train trial unseen extreme difficulty across address across subject formally show belong sub account datum ensemble stack generalization variability across aim vs compare across subject propose consistently predict state stimulus brain record category stimulus denote decode build predict activity evidence information light neural decode subject frequently accuracie discussion ideally trial meaningful group difficult structural subject inherent environmental variability trial generative practical common classifier subject provide empirical propose solution problem across subject purpose early create across extent subject divide three main deal training feature space paradigm present brain computer interface eeg data device stimulus subject stimulus subject completely image fmri example multi identical task test propose formal definition decode subject instance transfer assume differ aspect transfer acquire label contribution solution enhance dataset motivate ensemble learn decode efficacy stack generalization covariate article decode across propose learn covariate shift stack experimental section across transfer necessary stack briefly standard basic application across subject category subject channel record moreover binary stimulus marginal record predictive domain task face task house trial record target record definition assume task example face face house transfer aim help target transfer difference domain decode available subject domain case decode face available set transfer call accord transfer learning aim train identical availability divide category share probability differ category name solution brief review record target convenient happen also convenient importance sampling minimization dataset test penalize
b stock variance helpful stock detail quantile stock stock predict particularly stock return quantile general significantly zero large quantile relationship return additionally lags researcher capacity entire financial consequence approach tail claim cross tail financial prominent include quantile daily market index return financial put gs belong investigate cross stock index show bootstrap confidence quantile replicate lag gs reach market trend risk take reach market exposure stress test reach peak market gs peak meanwhile market reach peak influence impact way figure either exposure wide manner change impulse function require partial economic index economic state variable highly persistent integrate quantile quantile generally remain individual economic cross interest management table size rejection frequency box test second lag box column rejection tuning rejection frequency box statistic rejection column lag critical value rejection self table material include theorem figure axiom theorem conclusion theorem example exercise notation propose measure apply directional limiting nuisance employ consistency bootstrap normalize use detect stock excess use stock return provide predictor stock return supplementary material quantile stationary hypothesis set series prediction whether unconditional quantile compare pointwise band literature statistic paper et several advantage directional conceptually appeal simple base heavy tail consideration allow long lag apply quantile approach stock return exchange rate issue limit nan limiting allow nan absence even structure look useful interested measure degree across quantile strong limit long run variance quantile conduct propose bootstrap valid investigate carry efficiently normalize statistic whose bootstrap methodology explicitly mention version result fact cross cross autocorrelation lag stock stock study apply cross risk paragraph derive let series density quantile iv consider serial dependence arbitrary quantile quantile serial dependency quantile single time become process moment invariant monotonic construct sample analogue unconditional quantile consider deviation quantile directional directional dimensional entry possess usual interested testing absence directional x k location normal accommodate lag test lag use confidence interval special sup p small improvement cross contain use among mix coefficient satisfy function kx derivative interest rate chapter ensure uniquely quantile ensure differentiable describe weak nuisance estimate may slow convergence address bootstrap block strictly sequence geometric scalar positive denote growth original I pair observation procedure take use estimate solve asymptotically negligible finite lag confidence interval maintain use length vector cross procedure sample nan eq repeat b bb b jointly directional percentile following provide statistic fix directional alternative vector directional range quantile lags test quantile follow alternative normalize necessarily idea lee call chen improve divide normalization self asymptotic framework et al construct replace population quantile lead analogue eq element follow asymptotic nuisance nuisance employ bootstrap normalization technique bootstrap x self subsample recursively normalize impose follow density control strictly strong mixing assume assumption hold v continuously differentiable weak z z z v partial k z suppose finite partial alternative argument theorem self power performance process identity kx commonly model volatility economic literature reference therein median quantile finite bootstrap save case table rejection box statistic critical replication bootstrap critical replicate adapt white later size property rejection frequency nominal case median median close table however rejection examine performance self normalize setup repetition report nominal quantile little size quantile show moderate size nominal size power period self size lag low bootstrap apply directional economic variable stock extensively consider stock return return forecast economic predict stock return relationship whether economic quantile stock represent tail return ols return certain quantile return specifically stock return conditional stock return information et al inferences quantile regression regressor unit analyze show predictor stock return point lag application lag stock return predictor however predictor highly persistent price autoregressive root motivate work establish bootstrap strictly persistent leave future stock predictor autoregressive coefficient variance unit stock daily
steady theorem da da c approximately virtue conditional eq also update sample become multivariate distribution limit equivalent asymptotic gaussian learning problem problem accuracy adaptation concentration estimation class choose clustering fully approach show cluster apply five lead alternative computational adaptively asymptotic severe overfitte tend ht communication message receiver bit transform symbol receiver perform decode probability error quadrature amplitude alphabet db measure channel quality successful receiver know recognition detect data point choose corrupt snr db cluster show fig detect lr imply new characterize db use reach decoder grow dirichlet gaussians motivate propose complexity drive concentration parameter number digital communication observation assume multivariate precision parameter wishart joint definite cone conjugacy us class posterior conditional class expression probabilistic conditional obtain posterior hyperparameter recursively would greatly simplify rule multivariate obtain update inside complete square integrate obtain recognize wishart th become eq ease wishart interpret update equivalently integral within expression inner determinant I h sufficient eq separate term euler divide limit q result term bound divide take limit lem law p base case trivial give q particular hold desire corollary remark proposition assumption sequential low mixture easily computable streaming assume asymptotic dirichlet grow rate limit digital communication show optimality bit error datum dimensional process make however variational optimization approach effort fast require pass adapt sample arrive author class label algorithm greedy selection fast impose heavily model incorporate account discretization initial analytically stability adapt adaptive adapt asymptotic call basic idea greedy novel parameter adaptively greatly logarithmic asymptotically cluster asymptotically behave detect digital communication alone error rate number organize review sequential upon sequential growth class adaptively experimental nonparametric component denote th latent search summarize completeness distribution eq observation consider calculation iteration within assign count parameter growth number experiment sequential even critical fully specify conjugacy recursively compute hyperparameter I inverse wishart interpretation positive definite likelihood iterate detailed remark allow concentration parameter show number class grow step follow update innovation toward thus innovation manner assess limit see appendix theorem suitably gamma form class alg asymptotic concentration need discretization tracking extremely innovation previous innovation n k mixture model choose innovation use kn initial choice sec q conceptually currently mode reasonable good model
precisely one type forest maximal vc cube latter forest complement complete cube possibility argument follow vc class binary maximum vc remove vertex binary cube vc let case vc cube claim vc class necessary increase vc set anchor six edge type sharing complementary vc coordinate cube close section associate example vc class yield scheme function vc boolean form sum maximum dimension function value coordinate cube symmetric associate mapping binary symmetric vector notation discussion coordinate class match value prove novel argument basis dimension choose function class hence coordinate class distinguish vc exactly next collection complement contain novel boolean exist vc ordering order complete anchor single coordinate equal anchor anchor coordinate put give iterate number iterate reduction contain coordinate anchor cube iterate form leave possibility coordinate overlap pair face happen coordinate order coordinate iterate reduction coordinate obtain leave coordinate face meet cube maximum dimension collection boolean maximum start cube form cube order sum distinct vc cardinality sum generating vc projection maximum vc cube hence consist element cube cube collection sum clearly element cube onto maximum claim chapter main vc vc dimension therefore collection vc compression scheme embed class satisfy conjecture attention vc place vc exhibit embed vc maximum develop generalise bounding class bounding face also believe bad offer reduction union develop three may boolean vc university electrical sciences berkeley usa theory compression class equivalently date statement possess cardinality vc positively class super vc dimension embedding compression scheme complex maximum vc class vc class vc possible embed maximum investigate recursive procedure vc vc class binary class recursive embed vc lemma discover system vc diverse computational empirical road automatic verification former bound cube meet equality cardinality vc view collection unique coordinate form tree important increase complete vc study class conjecture compression immediately conjecture determine converse finite sized compression scheme beyond provide deep notion vc maximum class conjecture bound practice date towards later david existence maximum follow recursive dimension class coincide recently form compression scheme scheme sufficient expand maximum conjecture technique embed vc dimension relate count vc low complement dimensional edge face show uniquely meet first present consider incidence dimension provide closeness vc maximal maximal class cardinality class compression scheme come establish vc vc project cube cube class secondly cube application vc class produce collection vc vc embed class vc improves embed compression compression via embedding recursive vc class resolve must demonstrate possible compression cube class class binary cube classify boolean vc classes cube sect contain complement vc sect develop new class sect sect vc demonstrate maximal vc sect sect cube sect conclude chapter consider cube call terminology derive evaluation interest concept classifier class concept point support outline number family concept vc exhibit combinatorial vc concept word vc number form binary vc extensively process equality concept without increase vc call trivially maximal definition maximal class canonical convenient type nc ci ic ic vc cube iff iff complete complete vc maximal iff properly contain union maximally equivalently iff maximally convenient complementary concept projection cube show maximum vc vc vc proceeding invert reconstruct cube place cube splitting along produce maximum obtain series predict concept admit compression mapping k unlabeled sample vc little rich class unlabele compression class embed positively conjecture without vc maximal maximum vc focus argument prove maximum section integer bound meet equality prove class cube maximum word string cube must count partition layer contain vertex layer bottom vertex zero correspond bound graph use prove lemma connect vertex edge consider edge orient norm class tree necessity converse euler characteristic euler characteristic number forest iterate euler iterate since way choose euler define iterate cube coordinate vertex conclude iterate consequently rewrite maximum vc cube theorem conclude iterate tree structure iterate follow minor iterate color color color anchor differ class collection iterate integrate geometry bipartite cube edge cube cube whenever former subgraph contain iterated direction embed immediately preliminary interest projection maximum first complementary complementary vc procedure find vc class embedding class vc class contain complete argument correct vc cube composition projection induction vc vc complete complementary maximum cube complement cube contain vc concept dimension strictly correspond cube map onto direction reduction cube claim claim relate binomial vc binary unless vc follow direction vc iterate iterate come reduction class long iterate iterate long complete since union multiplying cover contain direction direction maximum assume binary cube repeat application reduce hence projection imply vc example vc moreover exhibit contain class negative show dimension vc vc cube pair cube contain vc origin cube proceed coordinate respectively roughly complete string zero one majority coordinate immediate vc less vc complete collection cube anchor exactly anchor element value anchor consist coordinate majority one give contradiction conclude contain vc dimension tb abuse pair intersect dimension zero vertex one form zero class maximum vc take original maximum class well contain small vc cube deduce iterate meet moreover consequently structure iterate cube anchor anchor anchor majority see require majority anchor contradiction coordinate entry clearly belong must show
describe processing describe calibration result discuss advantage calibration conclusion future bayesian parametric binary generalize histogram calibration possible propose challenge score programming classifier th index induce calibration model specify motivated variable discretization calibration marginalization close assumption class equal uniform closed solution total locate bin class instance bin interpret define prior partition boundary bin contain boundary boundaries eq training equation bayesian mention call average calibrate total number instance prediction exponential tractable apply dynamic describe section dynamic discretization classifier output define subsequence high model correspond respective score bin compute bin score composite decomposition give decomposable choose composite subset repeat process derive well possible complexity programming procedure programming programming particular method use assume one correspond denote score optimal model cache analogous backward highest low correspond respective cache property equation use bin prediction remarkably since specific equally space map store calibrate retrieve section evaluate run experiment logistic lr whose calibrate make tailor lr outcome linearly separable test well instance show three real problem predict whether person dataset real categorical feature remove instance instance calibration model test uci well calibration application diagnosis single emission patient classify instance equal positive test instance instance calibration dataset lr na I allow comparison tailor na I classifier achieve discrimination usually well classifier frequently real contain finding g sign laboratory outcome community acquire examine patient predict patient outcome medical total patient divide classifier calibration testing I bayes discrete transformation dimensionality exist unstable previous performance calibration discrimination due lack linearly separable method excellent measure acc area roc discrimination calibration error statistic diagram partition ten fall expect calibration bin fraction bin mean post bin empirical fraction instance bin low calibration model comparison evaluation show bold see superior superior reason section perform real real generally retain acc calibrate important calibration sigmoid recall function restrictive produce calibrate fashion near separate plane limitation include bin calibration another calibrate calibrate one choose monotonicity increase restrictive boundary limitation however limitation monotonicity see violate relatively poorly discrimination capability secondary classifier recently introduce application tie ci bin dataset calibration performance histogram post achieve among discrimination accord limitation first single datum calibrate bin select around train prediction table base simulation appear promise outperform make restrict unlike disadvantage algorithm search histogram version use remain algorithm perform thousand show complexity method train respectively method bin binary calibration complexity calibration nonetheless efficient training thousand particularly calibrate decision analysis plan explore average extend finally calibration margin auc acc rmse lr auc acc bayes nb acc auc acc auc
want trace synthesis program would return would consequence language language synthesis engine start initial guess next guess return stop guess element synthesis guarantee terminate iteration language monotonic fact correctly thus identify program put restriction bound example far increase synthesis result synthesis program similar argument show inductive synthesis dominate formal program restriction type countable set minimal terminate terminate decrease synthesis program correct program program program synthesis dominate theoretical inductive synthesis automate synthesis speed identify class technique synthesis similarly two variant history whether variant enable synthesis beyond minimal interesting kind synthesis technique first towards understand synthesis inductive synthesis technique guarantee terminate program perform analysis inductive technique investigate whether space correct synthesis mistake synthesis power investigate whether use program inductive kind history inductive relative synthesis technique synthesis power technique history bound dominate science find optimize critical loop purpose program specification give specification advantage specification specification automated verification proposal specification automate synthesis iterative inductive synthesis technique kind validate program produce intermediate subsequently inductive iteration refer synthesis synthesis conduct examine impact nature successfully use synthesis controller set synthesis raise consider predefine example engine synthesis minimal inductive synthesis return significant produce aid conceptually localization synthesis minimal specifically second produce engine see example define technique history synthesis validation engine program localization accurate notion force produce engine previously whether increase candidate space program terminate terminate correct successfully terminate increase decrease power program program successfully power none none strictly good mistake bound enable program synthesis synthesis input integer output specific program space denote integer discover synthesis engine consider radial ordering order synthesis counter start initial always produce discover boundary one terminate arbitrary rectangle still paper question even termination infinite question synthesis synthesis widely study extensively receive limited good knowledge theoretical nature inductive inductive generalization previously learn string formal task formal language iterative inductive string string language procedure propose formal learn language formal language algorithmic classify include learner memory grow infinitely bound communication learner arbitrarily response query membership algorithmic paper gold inductive elsewhere infinite learner set theoretical inductive gold language identifiable limit identify language use stream learn example call learnable language negative example term learnable learnable learnable language none language identifiable text include regular language context language also class infinite learn example simple example vocabulary string form vocabulary string language guess index see use identify correct example language vocabulary language fail fact positive language currently merely presence negative learning begin guess next guess language survey classical example input noise string target language might detail survey present inductive generalization synthesis engine step synthesis design response store stream inductive generalization arbitrary intermediate synthesis engine algorithmic restrict learner synthesis engine rely availability explicitly memory intermediate respect differ learner synthesis engine engine teacher teacher analogous synthesis engine technique use kind query verification contrast restrict verification query investigate produce bound verification meaningful powerful verification provide simple help design trace differ trace source enable synthesis analysis verification use aid section preliminary notation natural minimal natural range argument assume tuple tuple computable recursive language total complement language convenience mapping language distinguish different program language number index element trace order language string operate numerical tuple order order element language denote minimal order language say language correspond brevity intuitively define component encoding program inductive synthesis consist synthesis identify correct language index language overall synthesis follow let candidate program correspond language target synthesis engine set synthesis correspond candidate program inductive produce iterative produce intermediate conjecture program develop useful section section definition target formal trace language denote synthesis technique employ formally formal intuitively return language way presentation non otherwise engine language predefine represent guess correspond intuitively along form language converge finitely p il language synthesis inductive arbitrary engine n language represent synthesis language say p kt il next synthesis history arbitrary generating case generate synthesis engine generate order element language element empty mapping l engine engine trace could synthesis language similar follow second verification engine inductive vary rest investigate prove replace verification engine arbitrary verification engine minimal inductive synthesis system non intuitive synthesis minimal summarize theorem synthesis arbitrary trivially intuitively simulate converge simulate two phase language return phase need trace cache language formal maintained store simulate iteratively multiple micro simulate micro storage component minimal map map language simulate minimal program program map know map mapping intermediate program know intermediate counter record part already variable initialize map know make one synthesis candidate program need find minimal
case negligible case tv considerably always interaction contribute way prominent initially optimize confirm find interaction separately interaction dependent tv tv novel novel interaction interaction good several outperform traditional model model specifically ht far early dimensional improve factor entity figure recall interaction model clearly pairwise slightly within traditional way model lag member rapidly model improve fast outperform importantly practical limited stable context independent model practically exclude pairwise pairwise recommendation perspective interact influence user inactive status affect consumption pattern argue context similar focus recommend direct interaction argue independent recommendation hard slow aspect dimension fairly kullback know state execute totally context different band little dimension explain perform pairwise ht cp sc c sc epoch cpu easily time practice scale linearly feature practically useful depend operation preference complexity accordance pairwise model method qualitative pointing factorization although advantage outperform flexibility regard also quantitative comparison algorithm machine fm machines fm factorization rating prediction explicit subsampling rate rate preference pair dimension determine two corresponding build sa basic therefore composite drawback training lot unnecessary partitioning basically result exclude certain pairwise learn sgd monte carlo mcmc four fm key fm fm pairwise dimension preference fm feedback computation implicit feedback either fm build sa extensive et fm basically either user item sum vector aggregate feature word pairwise keep drastically fm interaction drop experiment leave accurate incorporate explicit implicit use loss model model although pairwise rank strategy differ basic context comparison pairwise respectively appropriate preference special case factorization aware problem way much importance allow novel without quantitative factorization machine context aware factorization implicit require item occur fm fm factor converge fairly optimize fm optimize h way traditional way interaction include outperform fm case fm include context advantage measure similar test exclude depict twice fast interaction subsample feedback train twice time core cpu time multiple core lift impose sa attribute fully multidimensional attribute entity g tag include dimension tag apply entity transaction accordance attribute analogously interaction property irrelevant variable represent strength attribute assign compute entity transaction feature entity multiplication w property property feature vector property update fast assign high phase normal compute remain phase therefore stick direct optimization combine outperform g show include extend outline create attribute indicate item item token item create feature token compute treat recommender user preference change great help currently refine recommendation broad interest news transaction visit actual transaction exclude actual item context target outline entity thus dimension consist transaction associate attribute actual omit attribute attribute item occurrence item item event assign feature note different matrix incorporate gap less minute stre filter rare token normalize entity experiment run usage justify simplify compare classic interaction basically actual actual replace cf aspect item user interaction item improvement summarize need predict suggest information improve perform place basic start incorporate recommendation importantly preference implement separately optimize implicit feedback feedback datum demonstrate usefulness aware certain preference user well traditional composite refined well never model interaction recommendation novel model generally able multiple incorporation additional recommendation several path work well ignore great connect nonetheless characterization context usefulness help easy current maintain scale another could meta learner context acknowledgement receive european union fp eps fill rgb com economic aware recommendation algorithm focus recommendation topic gain lot feedback optimization context lack tool allow dimension space preference importance largely propose factorization dimension easily experiment model aware recommendation scale life circumstance framework explore preference aware real dataset implicit dataset increase model outperform one novel outperform art factorization multidimensional incorporation framework information great capability propose factorization framework feedback recommendation flexible dimension allow point important recommender system filtering tool relevant content factor gain popularity user preference practical content item explicitly user retrieve instance use type implicit feedback unary preference feedback explicit familiar negative feedback inaccurate item click direct preference miss interaction typically user aware item unary infer negative preference consider feedback spend distinguished feedback highlight importance recommendation aware system refine extend additional may user recommendation briefly latent factor work feedback strongly applicability factorization minimize preference usually necessarily rating iteratively optimize preference sgd optimizer dot optimize rmse user item dot user item state optimize monte e dot product every e context optimization strategy however preference explore method interaction former preference dot interact entity dot interact entity item proper argue proper due flexible experiment create factorization preference feature matrix allow recommendation task new research preference property work aware recommendation restriction preference restriction applicability real world besides implicit address weighting weighting enable dependent weighting miss hypothesis scalability interaction number make life recommender follow build basic introduce usefulness context aware clearly art incorporation like item etc potential briefly review representation aware rating straightforward attribute similar relational database usually atomic nominal attribute discretize attribute transaction combination attribute dimension item dimension location item location classical recommendation scenario individual present distinguish item property omit item item attribute attribute value attribute contain simplify major factorization fm attribute dimension limit dimension refer single sa ignore grouping attribute conceptual attribute dimension information grouping assume extra interaction interaction complexity simplify limit prominent restriction mean attribute svd require single user additional attribute binary item dimension attribute binary entity descriptor descriptor token rate basic sa extend multiple inspire latent cf approach preference estimation framework rely sa main user dedicate dedicate contain help preference context location interaction device preference etc epoch sized rank sized km ei yet bias add weight model implement cg small dominate transaction run operation compute member transaction co equation need change ds show factorization mf feedback predict user item aware factorization dimension predict preference tm tm tm tm w tm classic mf explicit weighting preference item rate include svd however recommend increase training demonstrate usefulness novel without feedback set evaluate tv user pair set period least depend mml mml month tv week music day day focus recommendation configuration rank predict primary recommend recall live recall well important metric recommendation proxy recommendation accuracy offline world available rank base metric map ndcg test comparison query metric optimize measure test epoch case epoch time epoch feature practice effect context recommendation find generally aware consider sa dimension context item transaction item improve implicit context state context transaction consist transaction tuple contain recommender exhibit first expect repetition aggregate offset aggregated need bin possible band length event band week week people differ hour beyond scope music item item complementary sequential type introduce use item information pattern sequential consumption sequential transaction set information test context test event would result preference accurately dimension put dimension usually interaction mf item tm interaction u dimension preference remove contain potential
blind extraction propose employ predictor blind source blind source generalise mixture blind separation bss extensively past base order class canonical cca matrix signal sum autocorrelation signal maximization autocorrelation generalise free mixture free mixture always traditional cca noise noisy successful accordingly cca blind source structure effectiveness generalise cca bss sec simulation show section instantaneous mix bss source employ spatially source delay solve cca vector maximize e proceed maximize correlation combination shown apply cca cca become bss simplify q multiply side follow generalise eigenvector cca recover proof denominator cca mixture white uncorrelated source signal give identity cca problem add white white e variable normalise noise correlate correlate component depend estimate generalised bss extract simplified next brief maximization source presence ss ss normalised eigenvalue draw maximize successful extraction normalise extract remove next procedure proof matrix different lag robustness denominator maximize function eq reality likely positively recursively technique online update updating case normalize yield predictor extract closely approach predictor implementation source show extract instantaneous predictor function give respect successfully numerator denominator propose noise predictor similarly note follow predictor blind ahead correlation source lag signal delay lag reality property meet requirement cost indirect assume ss matrix diagonal numerator autocorrelation clearly provide early draw successful extraction minimum normalise signal normalise estimate preliminary
cc demand history add file cost unit consumption cc optimize cache traffic take account associate place file cache request file cache instantaneous file denote cache content period choose cache initially instantaneous file store cache instantaneous q expectation file associate storing file cache total amount traffic minus cache focus find cache horizon factor bandwidth depend popularity solve initial cache content follow period horizon switch ignore expect immediate reward cache np branching exponential bad solution relaxation cache add file sequentially start file popularity cache full file cache discard partially file cache existence popularity output main instantaneous reward file obtain information cc want file discover storage capacity file static cache mab feasible cache feasible combination cache content cache instantaneous file cache instantaneous demand iid reward know popularity divide part due know due policy period reward optimal file arm reward probability find reward least iid reward policy period notice cache empty incur sum switch linearly mab play depend arm ensure sample reward arm positive less period regret bound time play switch arm play computation cost period period switching switching period period play switch th arm arm play period play h initialize cache file bt f f bf ft additive square grow decrease play reward file file popularity profile algorithm arm switch notice algorithm avoid popularity bound definition good opt opt opt say occur counter update bad draw bad switching n behind apart period bad period increase guarantee high reward include play zero cost switch two arm combination l l switch count switch eq growth rate study additional depend switch regret bound function number switch bad combination rapidly grow slowly sub linearly logarithmic imply switching imply switch cost period turn imply regret tradeoff switching grow regret cost fm complex cost logarithmic order knowledge prove bound switch g notice every period every cost remove logarithmic uniformly arm period period arm extend consider algorithm period growth iteration cache content iteration demand file period cache efficiency horizon cache traffic popularity cache efficiency popularity profile low cache cache close cache demand cache due cache replacement cache cache cost file user request popularity skew cache efficiency reach notice due term follow trend albeit cache efficiency profile cache cache study cache cache behaviour slightly cache efficiency cache cache size cache cache algorithm due profile file popularity rapidly file cache file cache file cache size file cache ht instantaneous slowly depict algorithm negative cache efficiency switch high compare traffic small negative bind confirm content population area hand cache small request period replace file finally file impose cache always available popularity skew wide peak cache memory store file popular file cache cache efficiency contrary fact cache user request constant occur file replace ht content cache content pressure popularity unknown storing cache cache file cache model combinatorial mab switch trivial cache content file popularity profile well know account switching network bring benefit cache efficiency skewness cache chernoff inequality hoeffding bound support realization ignore clearly consider period bad period counter update play event solver output combination time period bound j l n j j b n b j j n j arm play period update period fact counter play arm update monotonically period due f j solver fail period theorem n notice mean bad period l j bad period f prove plug opt opt cost relate switching switching switch constant switching bound r I j n f n u f f f tm b l switch maximum switch switch bad arm play sum play fact zero obtain period consecutive play finally b j state theorem j respectively moreover monotonically I b j jj b since second increase increase apply line separately sum j prove bn b j rate property fm fm fm lt use use find plug power side rhs subtract four great prof complete prove induction induction bb b j nd uk uk traffic wireless terminal cache store cache controller store popular content cache traffic practice popularity profile file observe instantaneous store cache demand associate place cache cc gradually learn popularity profile cache capacity cache measure amount impact file cache skewness popularity show popularity profile quickly parameter represent portion internet traffic delay video streaming growth wireless continue fraction traffic locate traditional traffic become grow pressure bring content end user part content receive great bss form edge content news cache locate wireless bss bss reliably user without business edge mobile operator raise exchange traditional third party store g video rate netflix file popularity advance cache even scenario popularity locality bs require significant take relevant popularity profile advance popularity good strategy instantaneous cache file internet request user cache request overhead cache file file cache request request call cache cache content cache popularity storing cache instantaneous cache capacity good file cache traffic popularity profile advance observe request file multi cache management contribution paper address content place new cache formulate mab performance propose popularity cache file provide extensive impact content popularity cache file lack popularity compare file profile rest survey background section present popularity profile bound performance wireless introduction limited capacity wireless popularity consider huge content decide limited space cache form leaf cache node cache optimally place cache consumption study wireless transmission code transmission storing end device user several storage cache studied show user move across wireless slot cache code content content cache content popularity result place content cache providing mab maker action balance instantaneous consider slot machine time instant arm
switch tailed pathway subject classification collecting observation deterministic deterministic nature phenomenon deterministic thereby random go decide appropriate show cycle specific type cyclic monitoring year slow several local peak situation minus convert observation make part phenomena production consumption residual output production output reaction produce particle may situation certain span period medium yet produce situation create triangle proportional another adopt distribute simple situation type sum independently variable density identically elsewhere act scale dispersion scatter suppose repeat successive sufficiently apart graph like take simplicity location generate location close maxima spike production cyclic pattern arise scale location happen residual input strength contribution come negligible contribution q within spike divide combination laplace create point govern mixture b cm asymmetric laplace case laplace behave graph type eq sometimes sometimes tail due area develop author another gamma consider normalize since integral correspond density gamma gamma cm constant eq figure gamma cm simplify convenient specialized well constant simplify computation cm switch three form density current stay generalize change extend beta go gamma capable switching pathway pathway family pathway note situation tail cut close cut origin go go model beta gamma pathway tail cut tail gamma brownian boltzmann stable ideal physical neighborhood stable form cm statistic extensive mechanic also derive unconditional density see various variety situation start author mathematical reaction situation non tail etc integral aspect since integrable evaluate convolution convolution evaluate reaction correspond moment gamma statistical integral eq structure ratio transform transform integral pair integral author early pathway generalize integral pathway consider keep negative positive eq integral integral integral generalize beta form different could whole model know transform know also integral cm paper author recently integral fractional operator kind side fractional integral operator convolution pre function arbitrary type beta arbitrary right give derivative cm reaction result reaction diffusion differential solution situation come equation sense integer cm thank department sr centre mathematical cm nuclear rate reaction nuclear detect cm super generalize lead reaction science arise fractional science science
yield policy reward lie multiplicative round multiplicative distribution decision eq take absolute value substitute modification monte carlo update use value set round maker get dm accord distribution dm next weight sr observe epoch give follow repeat avoid uniformly sampling cast nature context slight modification bind impractical game play discuss link idea mdps procedure approximately calculate policy seem potential inherently amenable solution method suffer specify belief solve stochastic sum work problem interact computationally efficient manner definition markov tuple action horizon utility additive take mdp utility utility mdps distribute horizon write mdp mdps finally define depend maker policy bayes belief convexity utility obtain reveal view mdp draw expect nature select wish q policy policy belief vice game couple well fact set action set policy policy mixed set mdp e u deterministic history policy know exist policy piecewise achieve state outcome fair policy deterministic mapping marginal action observe h b optimal mixed game horizon distribution robust define generalise achievable maker distribution oracle probability equal statistic cumulative particular policy choice optimal mdp policy define explain minimax bayes optimal previous finding optimisation oracle good response zero asymptotically via even expert literature weight majority maker
vary bayesian monte carlo method computationally intensive difficulty lag selection var turn shrinkage impose restriction reliable tractable early take perspective use intercept depend context prior incorporate belief lag informative lag lag structure coefficient lag decay lag coefficient lag toward shrinkage incorporate consider lasso need exhaustive space explicitly encourage lag lasso penalty lag attack lag selection force correspond shrink toward lag coefficient lag enforce lag desirable fitting datum order lag corresponding allow flexible computationally study advantage forecast lag vector regression intercept noise uncorrelated series square fit minimize convenient express var compact procedure minimize denote frobenius norm appear model challenging unless sufficiently space tractable small author use lasso building assumption performance even large structure arise dynamic describe lag structure notational convention lag define define maximal lag particular numerous lag simple pair easily lag structure equation maximal imply hierarchical lag lag structure add series lag informative lag lag hierarchical lag long self hierarchical illustrate finally completely c propose aim shrink lag introduce tailor group euclidean norm encourage nest hierarchical sparsity set multiple interaction estimation transfer estimation covariate decay lag aim lag increase group towards lower identical addition influence lag influence thought ensure structure hierarchical lag occur flexible strength across expect example expect objective unify solve appear detail differ simplification observe row kp solve via proximal view method nonsmooth f differentiable proximal gradient cf operator take nested euclidean operator l backtrack iterative fista minimal implementation three middle period ahead forecast forecast method observe three lag scenario row row use magnitude depict figure show scenario r lasso lag lasso walk perform weight lasso similarly total coefficient greatly lasso exploit exception orientation toward modeling behavior suffer lag estimator create manner lag lag I magnitude decrease simulate view autoregressive next row lag lag magnitude scenario lead method forecast slightly bad perform lasso r var walk lag scenario allow sparsity pattern magnitude var lag row method interpretability matrix matrix lag order define optimal prediction procedure modification intend tendency select regression standard favor parsimonious approximately sum table report relative mean good benchmark fact lag good performance follow var lag scenario incorrectly constant lag selection estimation series indicator information include stock price exchange full list variable nest basic dynamic stochastic equilibrium modeling plus consumption exchange medium additional aggregate variable plus consist primarily component production etc initially focus forecast code approximately comparison convention one ahead forecast summarize least square random ahead indicator medium lasso lag become realistic economic application core include forecasting table perform period forecast hold application component lag likely lag vary lag economic activity lag series period lag lag economic several economic activity product taylor hence aid forecast rational rate year serve see growth causal price economic analysis price appear exhibit degree three row series component important activity taylor suggest aid david university fundamental component increase quickly tool number incorporate structure var traditional attempt low lag among component short assumption universal order information lag relationship base approach notion propose selection regularizer lasso autoregressive inherent focus three computationally component forecast lag highlight improvement var seminal var widely number parameterization intractable system var infeasible except lag order model scalar component dynamic impose regularizer adapt specifically inherent lag
sum probability substitute inequality q hoeffding sum validation union select drawing example near hence empirical using theorem range side draw partitioning subset nearest near outline statement I drawing q subset imply treat complete proof slowly expand exponent convert exponent q research bound practice allow directly stronger open solve leave estimate bound use positive term odd versa range much less equation indicate requirement contribute neighbor nearest neighbor interesting paper node sometimes node yet add collective adapt local draw neighborhood neighbor relationship challenge classification local node setting refer show sum partition return input return generate sample example bound cube center origin label odd odd depend nn classifier example expect partition near integer depth error average test deviation estimate difference plot statistically figure figure curve small bound validation count sample decrease beyond show minimum exponentially neighbor exponentially increase practice replace length bernstein hoeffding reduce range remove coefficient tradeoff truncate term tend require neighbor near condition separate union alternatively combination subset rewrite q use probability define q sized validation optimal value derivative partial optimal sum simplify symmetry straightforward close well approximation small binomial expansion q bind valid convert corollary thm near error integer good decision correct error develop label develop perform goal evaluate want generalization want classifier focus nearest classifier input determine close input label near possible use validate probably pac bound actual rate effective pac small failure pac bound vc likely assignment give overview comparison type bound sometimes condition hence distribution neighbor average show converge twice affect neighbor see book classifier difference classifier half bound classifier remain call sample disagreement hold classifier plus disagreement classifier bad disagreement draw replacement sum disagreement classifier one near randomly disagreement minimize disagreement use set together rate classifier use bind full cause range must disagreement cause near occur select produce bind range bound disagreement combination combination grow sense size classifier validate come bound discuss develop inclusion let draw input draw output draw example bind base average draw condition let validation union validation example index otherwise call quantity wish however expectation example subscript error validation call rate inclusion sample close figure illustrate vx sx sc validate subset r f sx sx ss validation set index classifier set agree classifier illustrate rate decompose term condition near r g sx index validation illustrate sign label apply apply definition g sx set area pr pr pr system diagram result theorem later sample error near randomly rt minus sign upper bind rest bind probability rate wish side range rhs tends make close selecting
gaussian proportion ensure identifiability many within mixture mixture skew partition density concentration mixture distribution lin lee mixture cluster skewed mixture parameter focus mixture lin distribution suggest concentration skew extent concentrated affect tail mind contaminate skewed specifically mixture contaminate skew herein variate vector parameter definite skewness k modify third index covariance clear write mixture contaminate g bad contaminate membership otherwise introduce indicate bad bad mixture contaminate distribution carry expectation maximization variant incomplete iterated likelihood compute maximization expect maximize extensive algorithm likelihood contaminate eq extensively literature work log likelihood detail appendix employ skew skew normal distribution definite scale skewness pp generate relationship normal write furthermore show frequently eq classification compare adjust rand index rand index chance agreement perfect agreement partition account random ari rand perfect ari skew two appear ht component skew skew mixture fit contaminate fit contaminate skew normal mixture skew mixture fit good average ari ari belong body sn record chemical physical package available discuss seven specie bank second component pc principal implement ht mixture contaminate distribution contaminate skew distribution give ari ari contaminate correctly contaminate bank contaminate mixture contaminate contaminate mixture artificial contaminated skew mixture mixture contaminate skew previously base contaminate contaminate skew extension mixture skew mixture detail give method good performance skewed concentration introduce early award science grateful provide normal herein corollary contaminated base asymmetric well spurious contaminate shift contaminated controlling proportion spurious outlier contamination specify outline
incorporate reader setting comment consider ar ease variable four distribution reflect independent modify curve tail approach word reflect information maintain modify uniform gamma prior I application research likelihood close statistic force summary sufficient marginal beta univariate statistic summary statistic influence implication simulation portion dataset table generate extremely acceptance second mse repeat simulation offer observe correlation abc mse denote distance preliminary simulation variability variety adjustment consider statistic small capture dependency z eight describe size tolerance together dataset run mse interested observe behavior tolerance prior set dataset mse result know simulation present abc attempt broadly sometimes result rate heavily tune htb setting third add scatter parameter bias effort increase prior implement dependent variability clearly bias present reduction variability improvement mse similar conclusion table decrease expense increase mse require similar result mse furthermore require substantially overall summary quality estimation improve introduce bias reduce variability observe introduce additional individual correlate e p e e b write context simulate bivariate must cell numerous define absolute mh tolerance abc ar accept mh walk proposal deviation respectively yield initialize auxiliary accept b gamma prior guide equation simply near desire carlo estimate choice allow exploration effort estimate yield albeit cost ar require seem suffice iteration result see computationally demand near surprising positive ar ar mh result difference observe ccccc observe cell count accept abc ar compare apparent see decrease stem ccccc total correlation end partial table abc bivariate prior subject correlation specific monte run ar mh summarize though variability original perhaps would suffice carlo lower accept correlation ht ar ar mh introduce provide grateful two anonymous valuable improve manuscript second partially c bias mse mse c mse c c c bias mse iteration bias bias mse mse mse mse bias mse c mse mse mse c bias mse bias mse mse mse mse mse bias department california edu several beta bivariate allow latter accommodate however come expense intractable research carry case prior tolerance real serve binomial comparison bayesian bivariate reject bivariate become use variable incomplete use reader extensive bivariate beta along bivariate continuous bivariate beta contain bivariate marginal beta limit flexible negative correlation flexibility simulate closed form maximum mle refer modified maximum approach approach distribute estimating equation final obtain via expectation unstable zero propose result difficulty free approximate first describe element abc algorithms generate candidate base generate auxiliary close candidate plausible accept parameter accept approximate posterior notion close decrease mse presence introduce select know exist finally make application bivariate beta beta binomial beta serve proportion serve abc hasting base follow beta selection summary study finding define beta easy marginal beta parameter parameter construct beta pair construction variate density parameter technique promise combine marginal suppose n respectively set equal theoretical solution negative yield negative equation parameter influence discuss bivariate observation ar sufficient table summary set distribution heavy c evaluate due impossible calculate condition markov knowledge nonetheless likelihood spirit class algorithm perform sampling possible fundamental
region visit statistical extract information visit duration profile recommend information contextual however recommendation learn exp describe document high estimate observe reward open refer make randomly first iteration select greedy call decrease strategy document estimate select except select adopt ucb price reward distribute reward mean add additional confidence interval number document high reward encourage computationally contextual bandit recommender context reward document contextual document maximize click reward document author bandit combine greedy dynamically value set uniformly initialize update click begin technique describe model bandit situation critical situation situation exploration consider situation exp risk aware yet recommender define reward total reward uncertainty first parametric instance process mdps propose sensitivity term inherent stochastic model state develop cost applicability exp propose greedy introduce situation contrast indicate criterion adjust study risk study none mention new handle semantic concept express risk associate situation help adaptation environment exploitation resp line ucb focus introduce situation external semantic enable specification human behaviour nc consider dimensional preference preference click document spend reading structure situation model contextual bandit include situation bandit algorithm proceed discrete trial situation compare metric concept similarity depend relate compute path node root observe trial recommendation one great document click algorithm observation situation document obtain reward depend similarity sim current situation preference sim base predefine compute document time recommend ucb select confidence uniformly choose correspond return element exploratory adaptation ucb alg level compute situation strict lead document multiply allow maximum exploration avoid document rs rs rd ucb rd ucb system user critical perform exploration decrease situation increase environment directly situation risk art risk variance approach similarity state describe use detection h aggregated concept approach semantic situation situation variance reward click aggregation click click recommendation normal eq clicks threshold constant accord gauss time click document recommendation situation concept risk weight associate arithmetic mean level associate concept system permit situation situation situation threshold use situation increase centroid risk situation aggregate eq risk recommendation feedback idea risk concept situation computed idea make give result company company time application situation contextual location social use spatial place paris finance paris entry user click situation illustrate entry analyse situation manually depend risk level interval h take age gender click depict heat situation h situation mostly office mainly home situation situation level level situation click suggest content management click considerably compute threshold opposite impact end situation overall performance obtain identify value situation similarity take different figure click display insufficient exploration consequently fail click lot click well collect step randomly algorithm select feedback simulation converge goal evaluating period time retrieval exploitation work ucb ucb ucb decrease exploration exploration axis axis parametrize test ucb starts reduce every iteration small regarding decrease ucb algorithm converge exploitation ucb static exploration interesting dynamic ucb algorithm average consider uncertainty r ucb ucb factor baseline r ucb improvement come exploration exploitation consider finally expect ucb ucb risk approach r take advantage cs give good critical cs different test algorithm different risk level describe comparison first r outperform exploration exploitation high risk ucb come exploration base size visualize comparison fig refer level notice decrease significantly ucb exploitation begin ucb parameter lie exploration beginning base line promise r ucb environment time participant hour split three week record recommendation week equip group system run ucb run ucb easily follow user rs user usage usage compare new recommendation impact recommendation comparison week week visit document document week respectively significantly week visit week introduction document would chance recommendation appear recommender list recommend first without find exclude recommendation week exclude recommend new week group discover exp visit document recommend document discover recommendation benefit discovery ucb ucb ucb look recommend spend figure spend group exploration trade impact spend use document exploration
alternate slowly employ direction derivation propose admm significantly formulate affect score remainder hinge find justify devise solve leverage align matrix leverage non show provide assume sum side leverage score indicate score count index find matrix less expect penalize find uniformly row leverage hinge function descent coordinate access exact score obviously completion problem help adapt solve score observation begin algorithm hinge size close theorem series row matrix coordinate desire order follow suppose desire leverage descent hinge precede property hinge hinge hinge reason hinge provable optimize get configuration detailed comparison optimize essence descent need provable score leverage rank truth rough svd leverage norm approximately score score observation superposition conduct leverage score perturbation completion motivate leverage sample additive perturbation ensure leverage score additive identify rank uniformly enough theorem ensure necessary provable row weight leverage leverage within consider leverage mn leverage give leverage row sampling ideal leverage score row theorem I gradually analyze except take accord tb c ki index ik practical optimize hinge know leverage directly non uniform sampling estimate score except score step pick index notice leverage additive index kind weight provable provable objective objective decrease condition turn increment score increment leverage ensure hinge nonempty iteration iteration nonempty corollary indicate greater well great provable hinge loss select r hinge weighting theorem theorem desirable number much practice fact get sequence figure condition time n challenge seek cost reduce svd basis replace n help rough svd use matrix way drop practice gradually leverage leverage leverage may appear additive row weight ideally attain even accurate leverage purpose heuristic score gaussian noise entry method tune attain figure noise intensity show coherent fail large number entry observe method strictly succeed free datum heavy rank matrix matrix know plus show coherent recover world superposition rank observation surveillance surveillance stack background foreground robust perhaps tool recovery define l recover coherence low rank coherent weighted matrix l matrix coherence low weight compute input input solution increase completion weight input compute finally weight ht sparse set entry algorithm input vary result figure clearly row make coherence recovery accuracy relative recovery way subsection error fix error matrix highly coherent fail weight accordance achieve high accuracy computational ht column weight adjust leverage recovery well model describe discrepancy score leverage score algorithm leverage score algorithm objective condition problem coherent matrix complete weight nuclear coherent apply uniform matrix quality leverage score clear accurately score non sampling leverage application non law leverage uniformly weighting power real equivalently slack augment minimize maximize contain otherwise term update update ascent ht hinge optimize row region configuration hinge nd row hinge coordinate descent st hinge decrease happen scale nd row leverage score nd decrease toy compare generate synthetic use three kind decrease loss lead fast decrease fast small hinge problematic local hold define hinge write th lead decrease loss score increase decrease contribute loss know r I hinge loss equality scale loss increase follow directly decrease hinge term hinge loss mc u simplified pt leverage score additive leverage omit leverage score value inverse great increase since theorem ensure objective directly empty r basis leverage score respectively basis exist let r c c u dr leverage zhang zhang problem uniformly underlie require incoherent model practical weighting leverage score become weight effectiveness recover matrix whereas unweighted method free completion extensive world collaborative completion nuclear minimization model variant solve portion seek recover incomplete entry cardinality great factor sample coherence requirement active coherent column world impossible access miss column impossible demand user pay coherence eliminate assume independently sum obviously restrictive use previous reverse adjust let diagonal diagonal instead complete compute th proportional dependence coherence eliminate score motivate none previous offer potentially sampling setting complete add e observe number unobserve complete coherent estimate provide leverage near interest derive admm weight nuclear minimization machine recovery component recover heavily noisy rest completion nuclear model solving complete perturbation apply weight practical empirically evaluate synthetic dataset apply
density would parameter choice class hide show represent hide visible bit hidden exponential unit compactly feedforward reveal intrinsic prevent think particular combination feedforward net combine recurrent temporal rbms study focus binary proposition distribution statement item statement strong statement appear disjoint contain form complement subset develop jacobian parametrization jacobian z pz jacobian dimension span kronecker ij kl l operation input rbm remove without span vector low parameter space partition region piece wise map thus piece represent function state geometrically namely visible preferred row multiply indicator classify positive classifier hamming ball disjoint hamming ball time plus remainder block minus dimension column number statement contain point ball radius hamming sphere choose hamming center hamming apart ensure contain maximal cardinality code large constructive target adjust obtain hide adjusted hidden jointly input dependent difficulty construction successively compose lemma proposition two finite probability build support disjoint vanish represent rbms term hadamard model precisely strictly word hadamard multiplying distribution rbm product strictly n sharing take strictly two hamming consist immediate hamming distance coordinate intuition vertex ham corner unit cube exist probability idea realize add rbm sharing step ball dirac delta positive conditional let share take joint conditional share distribution support hamming ball center contain sx x c qx word restriction product hamming proportional vector entry implication strictly joint conditional share conditional dirac delta vector consider dirac delta enumeration start dirac delta th step ty sharing whereby equal verify satisfy condition specific note row get trivial sharing measure trivial joint transformed share conditional share depend sharing construct algorithm generate accurate sharing intersection ball center initialize leave leave row corollary sharing strictly joint readily evaluate algorithm sharing step length star pack star intersect star call packing star packing input star length star packing show state star pack certain length size bad star packing sequence illustration star packing construct star packing procedure define pack site star far star sequence initialization split set star radius ball star dimensional set share iterate th terminate initialize create branch produce split generally branch star split branch total star time whereby number branch create precisely illustration packing sequence figure branch show dash green clarity star highlight star branch translate highlighted distribution well whenever give universal evaluate expression coefficient appear yield except universal c r evaluate universal explicit ir rr sr monotonically leave direct obtain yield ir evaluate evaluation million indeed remark proof strategy select adapt restriction unit restrict conditional want conditional care case pack ib xx xx rigorous machine family without dimension denote input finite conditional intuition conditional arbitrarily conditional exactly imply distribution family point point jacobian map zero universal latter family p universal conditional consist v fx exponential family contain k k product integer partition analogous correspond set collection share lemma proof binary number qx qx generalization present consider rbm visible unit unit observable rbm element x w first cardinality choose appropriate repeatedly conditional give first joint desire conditional approximate probability proposition analogous compute deterministic feedforward threshold network number deterministic arbitrarily feedforward x z z w yy x yy xt v bx statement precisely note give feedforward marginal regardless strictly build combinatorial deterministic approximate policy well entry wise mixture py py w union py eq decrease arbitrarily maximize proof directly function parallel hence fact feedforward hide unit linear bit lemma policy approximate arbitrarily fix composition linear threshold nz b policy arbitrarily input know policy thus bound mn n small share policy learn initial work observation mm expressive power machine boltzmann machines undirecte neural hide network parametrize interaction bias prove restrict support universal maximal contribute investigate restrict conditional restrict boltzmann universal kullback leibler dimension restrict rbms bipartite interaction visible infer distribute network connectivity rbm define boltzmann states network weight bias expressive attract study numerous treat particular universal address expressive exist theoretical work analysis rbms rbms bias influence input bias bias visible substantial distinction rbm power theoretical focus lie class possibly represent fix hide unit give rise desire distribution derivation incomplete generalization discuss reference list depend concrete follow map distinct small suffice universal tolerance select well unit extend non unit organize formal definition subsection dimension distribution universal deriving unit purpose analyze assume derive minimal unit suffice tolerance theorem subsection natural ability distribution way defer nonetheless fair detail entry set denote style transform circle inner sep cm minimum dot distance cm divergence conditional whenever hide unit within plain unit one next ask class conditional compactly distribution familiar distribution express term allow develop second part specific nonetheless contain certain conditional represent field main idea appear universal approximation field form n output arbitrarily field define rbm random rbm architecture represent
learn capture relaxed dictionary ensure note vector atom atom dominant eigenvector initialization process mp I accord r code manifold helpful dealing manifold practice high reproduce rkh value perform code explicitly work like solution text achieve let p trick principal matrix considerably row pick problem embed eq code tb initialization q q I ti l manifold problem rkhs dictionary update atom atom independent code r maximize orthogonality constraint form define maximize orthogonality give eigenvector eigenvalue want load kernel dictionary manifold initialization ni r r classification atom label use determine query dictionary learn discuss video model subspace video order simply demonstrate information follow sequence appearance image video frame represent like svd specifically take account image frame however capture extended image formalism vector covariance speak one appearance spatio feature vector feature index two subspace angle extend give finite video span orthonormal gram schmidt manifold column see discriminant hull discriminant discriminant intrinsic gender recognition scene maximize discrimination image image point affine characterize affine feature kernel discriminant maximize measure inter intra consider manifold local similarity dissimilarity sparse locality code extension respectively preliminary define determine classification query discrete dirac function measure recognize gender human gender constitute individual angle show recognition gender video capture result select individual table consistently big margin well burden sometimes tb lc dataset image sequence class perform subject primitive obtain medium perform setup experiment relax tangent scenario tangent space experiment euclidean perform poorly compare log euclidean tb code sc intrinsic code experiment sample normal tangent reflect scenario class tangent tangent le sc classification face texture gaussian code fed classifier intrinsic analytic extension conjunction discriminant hull recognition still extensively image popular choice image face video resolution create face extract region region describe histogram local linear order ten report dictionary percentage dictionary tb video dataset pattern action hand leave turn movement describe gradient descriptor set set split six tb example dynamic video move certain video human comprise contain training video train time accuracy use take dynamic video video length frame histogram descriptor compare two design spectrum learn dl see component volume capture view volume dl descriptor descriptor well discriminate texture overall obtain recognition tb dataset tb dl load code logarithm size multiplication thin thin require add reader computational efficiency experiment assume constrain expensive unconstrained tangent geometry randomly table run intrinsic core coding manifold manifold show code locality perform problem manifold dictionary atom use coding manifold linearity classification gender recognition scene analysis recognition texture classification show achieve notable discrimination art discriminant embed code learn minimize necessarily benefit propose reconstruction manifold interesting devise solution geometry induce acknowledgement department communication digital research centre discovery dp arc fellowship proof symmetric u v p x sufficient x n diag u u diag pa kb kb w matrix distance form analogous rotation nm date date hyper extra date nm hyper nm extra date open figure token school university representation notable various riemannian deal aim bridge coding manifold space enable extend manifold furthermore algorithm atom atom lastly linearity code dictionary embed hilbert task gender recognition face texture considerable improvement state art discriminant past decade term compressive sense suitable basis overcomplete nature decomposition notable visual recognition subspace develop theory code linear vision example art matching video spatio include filter adaptation tracking despite wide appealing property subspace lie riemannian conceptual learn represent accurately develop analyze image sparse signal topic like efficiently superposition subspace code video datum coding manifold study opt intrinsic sparse code riemannian intrinsic exploit due complexity logarithm manifold sparse code demanding term base logarithm later logarithm analytic manifold contribution end manifold preserve accomplished devise dictionary atom furthermore linearity coding manifold version dictionary embed space linearity apply computer vision recognition scene tb conceptual diagram conceptual work represent code manifold green red triangle query combination geometry curvature manifold manifold red triangle geometry color geometry provide technique manifold vision loose emphasize word capital letter bold letter one euclidean manifold grouping admit right orthogonal consist orthogonal furthermore thought element form detail element specify order column span write riemannian formally product tangent bundle metric shall concern geodesic manifold allow many definition manifold smooth geodesic short curve manifold embed may define consequently length path length curve give distance short equivalence geodesic small angle angle column principal subspace recursively word angle pair second subspace logarithm map switch tangent space logarithm close manifold map paper however logarithm map previous code notion query combination satisfy constraint may express small alternatively constraint restrict combination reconstruct dictionary atom combine reflect energy encourage locality yu wang determine total cost q good dictionary structure dictionary write jointly minimize choice coefficient solve alternate treatment similarly generalize code general space riemannian manifold metric n every affine metric encoding shall concerned dictionary manifold embed default natural choice point q manifold tangent hand code q notation tangent refer step manifold follow terminology definite geodesic distance true riemannian manifold dictionary ty elegant intrinsic approach tangent bundle tangent accord root dimensionality riemannian encoding write extra trivial turn learn riemannian euclidean code along update tangent represent x fx reader manifold logarithm interest work manifold propose dictionary specialized vision dimensionality manifold dictionary alternate atom admit method intrinsic logarithm manifold code non linearity think possibly experiment manifold matrix subspace embed form smooth embed riemannian metric path isometry riemannian curve length metric geodesic work represent action projection embed x note geodesic use term establish link underlie concept use interest code dictionary code address combination way element generalize prefer generalization n point rely verify multipli minimize affine combination manifold metric mean mean geodesic metric metric contrast mean weight close I term geodesic metric closely furthermore give conceptual illustration slightly call code step reason later expand explicit manifold see store encoding similarity atom offline symmetric common package specifically let code initialization processing dictionary I ip manifold unit sphere albeit subtle code vector space result solution f x conceptual diagram sparse address work surface four atom red square describe query circle atom green step atom space might unit favor locality locality vice versa show sparse however free parameter neighbor fast locality coding wang q
code indeed much generative aim query model conditional fixing building focus code present force impose name believe building model co bilinear word loop learn usage notation motivate capture hierarchical structural bilinear traversal combine natural bilinear incorporate reasoning efficiently far outperform nlp previously code b programming focus specifically decision readily available recently c easy data processing challenge building process motivate representation terminology throughout code sequence token serve syntactic element code flat lead inefficient description token increment body fundamentally compactly represent loop instead code process abstract sequence code token correspond syntactic internal node token subtree primary source code example determine reason generative distribution generate root leave repeatedly tuple parent leave token first traversal tuple independently rest independence assumption produce weak contextual lose name construction dependence people limit see example write nest loop name outer loop inner loop dependence come code program evolve sequentially traversal variable distribution child tuple initialize stack stack root element stack line child child stack line token node fashion traversal update internal evolve produce internal traversal desire token distribution prior traversal variable child condition node joint equip stack probabilistic first token particularly suit traversal right leave internal circle token circle traversal stack state computation tuple indicate conditioning brevity encounter uncertainty generation child avoid child tuple use bilinear simple log bilinear representation pair child tuple energy tuple child normalize child tuple observe child notion index value denote tuple similarly look representation pair sum represent variable matrix child log bilinear grow high traversal exponentially extension certain traversal depend arbitrarily element rich type let combine traversal variable tree token latter traversal deterministic traversal generative traversal satisfy tree replace variable traversal inference explain unique compute type give token generate elaborate deterministic object let node take value problematic cardinality annotation annotation uncertain choice value evaluate bad token decrease annotation improvement token source great child parent token name build language keyword currently scope signal program variable variable scope vector string along key tuple vector string token decide whether token accomplish internal binary proceed global token token smoothing device although scope token pattern scope logic three include method available option many scope select token child proportional I normalize currently scope stre token probability straightforward second latent traversal allow traversal use set traversal traversal computed token total log learning problem production stack bilinear generally traversal latent traversal traversal variable couple across tree simplicity restrict restriction deterministic token correspond depth traversal algorithm adapt learn bilinear detail supplementary describe exist child token terminate discrete traversal child equivalent english many variant explore annotation annotation aside traversal make special weak bilinear widely language tree traversal bilinear novel believe include traversal bilinear parameterization logic general token inductive traversal rank gram effectively analogous issue factorize similar way nlp recently explore sophisticated non program repeat applicable language rule sophisticated language specification programming com program k line code program programming program identity validation overall split easily interpretable divide token report token choose strength smoothing epoch stop validation setting gradient stochastic dimension test token child unobserve assign locally smoothed tuple low log detail smooth additional material novel scope unobserved zero baseline bilinear model gram additive smoothing hyperparameter smoothing choose performance bilinear bilinear parameterization traversal result bilinear equivalent dominate gram allow context generalize appear train gram gram gram gram ccc valid seq augment traversal node parent store upon reach sequential token variant hierarchy hierarchy alone perform alone contribution r ccc model em ccc scope scope traversal latent traversal consider latent latent result gain train bilinear bad tried add traversal training slow step scope model train scope scope use list sort de index appear sort understand room improvement experiment value total log report contribution incur generate token seq properly cost token supplementary material far good reporting come parent kind cover local next model token seq qualitatively drawing loop ask simply token token initialize traversal reasonable value scope scope source code file supplementary loop capture particularly organization learn subtle thing like variable often square largely build appear key leverage great result yield improvement quantitative baseline qualitatively produce realistic many challenge notion source structure relate statement naive sophisticated child tuple apply compositional scope tuple would extend scope model handle call level piece briefly great potential properly find simple generally focus modeling popular generative extract might apply argue probabilistic code rich potential hope help acknowledgment grateful helpful work supplementary material generative model code traversal latent deterministic traversal traversal union firstly compute become use forward backward algorithm free energy brevity drop term e forward emission log bilinear weight handle bilinear correspond add unweighted sample step experimental validate minibatch initialization subsample set manually
parameter train learn learner basis able reconstruct bootstrap identity bootstrap nearly reconstruction may high bootstrap implement pyramid pool omp predict mnist tune summarize mnist perform baseline provide bootstrap bootstrap cca bootstrap suggest mass commonly class column common expression perhaps expression art suggest useful supervise weak label build pre full bootstrappe heuristic annotation large confidence drop come confident location bootstrappe confident modify top bootstrap curve figure bootstrapping end imagenet propose apply way network predict image proposal deep classifier region post classifier describe section l baseline baseline bootstrap baseline hard bootstrap bootstrapping detection datum mainly bootstrappe develop training weakly multi output method engineering effort purely supervise improvement even simple suggest move research attention achieve gain collect price label scale extend agent promise unlabele label benefit consistency table token edu google ca usa google art use purely depend label assumption often label may localize general labeling work generic noisy incomplete consistency similar deep computed substantial robustness mnist handwritten digits label case label recognition achieve art face modification challenge approach image output currently recognition purely dropout overfitte system account miss label annotate large image complex object image localize recognition human agree become noisy become argue vision noisy incomplete labeling usual objective consider notion incorporate consistency world match incoming output learner justification label effectively accurate lead label clean carry balance experiment robustness several mnist handwritten digits robust corruption case achieve benefit challenge improve single shot network improve discuss describe probabilistic supervise vast key paper paper weakly semi supervise deep bootstrapping unlabele build seed iteratively unlabele extract seed expand repeat algorithm recently co similarly pair iteratively additional label identify training object detection weakly self demonstrate comparable system work bootstrappe network share motivation label loop incorporate directly noisy annotate network robust handle label share motivation noisy label semi encourage notable beneficial image similar generative dimensional extend language learn language label newly collect training effort build robustness noisy deep rbm use hybrid generative deep machine multi network simplify training model enable backpropagation much deep supervise generative unsupervised imagenet image still behind term way benefit probability observe q label perform discriminative ascent purely discriminative bottleneck develop rbm multinomial energy conditionally energy lead hidden multinomial unit arising give via divergence prediction learn j assume rbm generative exact due mcmc generative certainly feature binary make rapidly exist activation exact descent analogous version rbm autoencoder approach consistency objective feed version experimental develop explicit reconstruction dynamically target result target current model improve prediction label incorrect eventually highly inconsistent predict coherent ability consistency noisy bootstrapping bootstrapping entropy target mini bootstrapping bootstrappe directly target show entropy regularization entropy regularization encourage enable semi learn bootstrappe mini stochastic step estimate target parameter well predict bootstrapping instance model target recover bootstrappe two operate may noisy structure state object system annotate box image label however category miss annotation modify object approach box cluster centroid prior predict object bound bounding appear location proposal enable efficient quality score attractive section predict objective detail write cross entropy note section
representation image variability preserve geometric function contour image probabilistic traditionally distance sized transform close contour template variation express interpretation give parse parse generic object infer fine pose human rich prior graphic simulator handle extreme variability domain ease denote range rotation express give popular graphic take define cross axis extremely due variability consist flexible profile graphic simulator mesh object simplicity circular cut along axis beta distribution span cut since hyper kernel gps point gps pass graphic simulator mesh generation reconstruct amount calculate formalize truth show box super buffer color consist result depict roughly viewpoint collect image illustrate challenge suggest object beneficial compositional gp infinite hierarchical shape human scope program compositional mesh body group graphic generate result mesh prior center center mesh underlie axis mesh part fashion ta mesh whenever define smoothly mesh illustrative inference invert resort markov model variable mix noisy invert graphic simulator propose hasting affine rotation mesh belong affine transformation affine mix proposal discriminative despite minima sampler make latent variable since time couple exploit hamiltonian denote model obtain baseline figure outperform pose detector set contour stick result failure primarily resolution fine contour mid texture descriptor improve get seem reasonable failure show infer position contour section utilize strength bottom aid pose explore act generator pose feed pose pose dataset pose local kde generate proposal rapidly reasonable leave fine effect proposal discriminative number prior fit pose detector image stick result treat bound pose detector retrieve datum give kde get proposal kde close posterior fit via kernel run speed world vision inversion probabilistic shape address computer graphic category appearance handle compare mid vision handle monte standard site hasting move yield quantitative human pose reconstruction compare computational additionally proposal synthetic pose detector inference research seem result handle incorporate shape decomposition mixture augment integrated potential practical build image rich appearance illumination via rich state modern neural many generative probabilistic graphic program programming system vision long go bottom vision yet understand scene perform recognition good offer rich suggest potential produce obtain good quantitative thank feedback singleton fellowship partly google ai project recently formulation computer graphic natural account via realistic generative seem intractable inverting version computation evaluate address show solve world generative output probabilistic program generate plausible affine place scene likelihood similarity base mid vision site hasting proposal hamiltonian discriminative datum proposal datum pose achieve quantitative qualitative formulation generative graphic attract single low comprise shape make heavy use temporal continuity mit edu microsoft com mit edu accounting object appearance via graphic primarily graphic engine lead identify engine seem challenge prior propose evaluate address challenge solve challenge vision generative model tool bayesian geometry probabilistic engine environment stochastic sample prior affine scene mid formulation rich model object shape reconstruction image quantitative baseline mesh parametrization discrete couple graphic computation site locally hasting proposal hamiltonian monte proposal learn discriminative successfully strength generator define affine engine express change scene prior generic
popular preferred interestingly also network backpropagation indicate design topology neural three loading table x necessity layer three neuron eight represent order idea cross neural stem datum name step intelligence important relationship model suitable purpose framework well exist dominate economic highlight way like school economic provide dataset mm mm ac cs ac modern far carry show offer experimentally result flexibility offer elaborate framework discovery neural lot community explain nature question factor affect answer latter reveal diverse factor economic deep factor mainly reveal economic deal characteristic real possess make unable incorrect limitation raise regard accuracy guarantee conduct consider quantitative prediction apparent economic benefit variety computational intelligence offer neural capable remarkable successfully linearity flexibility topology step datum numerous tackle relationship combine discovery real neural economic work evaluate data transformation deal high advantage topology novel topology derive technique assess improve great classification prove therefore work serve comparison great network mining method exploratory extend real characteristic nd predictions rd briefly together perform identify present proceed analyse conclude section amount separate logit suffer low way model factor predictor exhibit probit around surprisingly achieve explain build estimate whether regard finding model exist receive argue linearity learn mining handle lot ability handle large number performance interesting measure sale usefulness exhibit predictor capacity able encountered application outperform economic successfully stock possess topology network concept logical among neuron exploit result order include derive network purpose many achieve support economics model dataset attribute year order overcome financial category interest age status status person status financial car total service total total total total self employ detail outlier time tackle aforementioned difficulty series transformation perform beneficial unsupervised homogeneity map categorical datum coordinate together financial attribute item reduce attribute remove outlier provide interpretability nine transform coordinate discriminate financial factor financial necessity cluster seven characteristic characteristic dataset objective derive exploratory classification level employ employ old average old status linear simple explanatory variable explanatory try error input straight line estimating coefficient estimate parameter ordinary try aggregate create tree sample construct decide build aggregate vote specification forest simplicity allow overfitte node input layer predictor layer arbitrary node intercept fed activation pass activation non activation tangent simple neural perceptron n output easily generalised take parameter weight randomly mean model try learn backpropagation try difference calculate difference output adapt weight accord specific argue subtract gradient reduce predefine update big update keep sign tend raise concern common avoid validate fold get fold choose layer topology designing extract unsupervise perform neuron idea behind neural factor factor variable factor depict factor widely input incorporate hand characteristic variable introduce extra class combine relationship final something class fuzzy neural name define economic context neural amount rest reason whether series incorporate develop validate fit fold compare rmse fold method evaluate model neural representative take perfect fit root difference actual I well rmse random contain miss linear calculate equal initial choose order hide produce ten case cross evaluate one good appropriate transform create four build test transformation cluster table iii original network classification design check quick indicate forest clearly almost backpropagation networks rmse perform regression seem improve build transform specifically transform attribute dataset especially train backpropagation random forest around case forest around backpropagation neural classification provide backpropagation network forest backpropagation compare classification interestingly increase proportion explain backpropagation closely random backpropagation neural exhibit big backpropagation build transform verify suitable purpose neural economic traditionally dominate computational intelligence broad tool technique use combined framework forest beneficial preprocessing despite return case provide combine
draw normal covariate make incomplete randomly imputation imputation scientific rule mis calculate cf exclude variation interval calculation proper completely apparent information contribution degree freedom eventually variation conventional rule total simplified pooling coverage percent conclusion illustrate situation essentially observe precision estimate imputation inference find scientific field sciences big useful application study evaluation decade essential multiple infinite population simulation properly compare account miss sampling make generation much multivariate van imputation blue van multiply infinite unit cover population exist pooling conventional rule situation standard lead amount result imputation simplify pooling implement essentially study address medical lead bias incorrect inference straightforward approach imputation datum complete dataset analyze combine pooling multiply sample infinite unit rare play yet observe affect precision situation confidence interval long statistical
learn subject number know target exactly ensure try try fit matter represent predictor merely know large architecture excellent reality even large property make question back understand play base hidden increase behave capacity capacity play deep understanding inductive analogy understand analogy regularization infinite sized bound demonstrate implicit decay infinite give rise convex net feed find minimize input linear learn soft entropy correct cross write otherwise cross margin deviation negligible always dominate label training increase necessarily decrease loose estimation tradeoff train size mnist stochastic descent expect initially decrease network achieve continue predict control mnist attain allow well error add beyond generalization go cifar momentum test phenomenon artificial decrease hide cifar train agree think censor represent exactly still continue decrease reach force overfitte add datum network fit figure percent significant error continue decrease increase past achieve cifar explanation implicitly try even huge furthermore thus infinite control want explicit regularization modify drop weight pass convergence zero try identical get vast many final try add form still increase network help go simple feed forward single activation model capacity limit number sensible computationally last decade much instead constrain example frobenius norm use eq norm lead regularizer trace network high contrast constrain local trace norm factorization justify sensible bias ensure trace realistic factor suggest inductive activation reality explain light perhaps target unit implicitly toward inductive bias really fitting view fitting infinite common matrix norm model capacity purely indeed improve sized start decay regularization weight approximately implicit regularization neural network top regularization aim instead try match deep g simplicity focus e network regularization layer hide unit example arithmetic geometric mean attain input rescale h v v reason mapping piece piece rescale finally since rescale establish learn hidden connection convex represent part net regularizer unit layer nn even discrete support equivalently versus select limit unit merely select norm allow decay equivalent hide regularization equivalent decay implicit regularization stochastic descent equivalence network output regularization regularizer indeed activation feed forward relu perfectly provide enough succeed trivial polynomial network super sample correspond description return learner
value miss sensor specific diameter determine node euclidean sensor relative local positive couple neighboring near distribute inaccurate analyze distance invariant slightly application near neighbor predict iterate input feature property specific rich use dt resolve challenge reliability identify critical corruption build optimal tree construct chain unit radial recognize due computational requirement network management centralize network learn solve challenge localization network angle distance measurement receive node measurement receive indicator difference arrival node value coordinate reference therein neural big example use classify point label detect use give observation point feature part I read classified gap include optimize network unconstraine security e discussion please algorithm adapt uncertain datum belief give assess investigate statistical process wide structure available outcome k algorithm recognize learn widely cluster resolve node centroid cluster b close membership stop valid g predefine perspective centroid dimensionality aim set new orthogonal component order first correspond discard content uncorrelated linear combination simplify problem thorough theory important note pca interact action maximize reward experience reinforcement reward value state use learning determine fast easily seek maximize challenge memory sensor change failure management adopt challenge wireless energy processing aggregation designing protocol various challenge sensor provide memory bandwidth traditionally wireless represent set represent channel model reach span vertex leaf node child np hard machine sensor previous dynamic benefit summarize path dynamically divide simple problem consider achieve efficient meet method sensor network span path exchange figure machine learning neighboring node procedure decide assign transmission prove near sensor problem require communication single exchange machine protocol comparison protocol imply large scale network wireless sensor protocol adopt c regression limit sir flat som limited hybrid yes flat multi moderate good yes moderate rely measurement node execute facilitate refer framework exploit fact sensor overhead detect structure serve develop wireless linear utilize sensor intelligence sir som illustrate sir modification short path learn second high accordingly neuron update match pattern highly execution run hybrid som take account requirement throughput cycle process update neuron weight overhead set som construction sir throughput enhance hoc basically guarantee reliable resource allocation mobile hoc heterogeneous capability addition maintain network join forward join backward create learn overhead search energy efficiency requirement need e g consider communication dedicate band ghz spectrum technique reinforcement protocol reward use technology detect equip device moreover use simple maintain location benefit achieve acceptable solution pr enhance select high rate past period protocol importance constraint pr bayesian next node transmission introduce reinforcement novel source message disadvantage reinforcement highly environment sensor inefficient pass local cluster head typically works discuss head head process classical node network incorrect operation technique node cluster aggregation cluster extract node sensor machine efficiently head head enhance aggregation cluster architecture working compare protocol intensive energy aware network mechanism moderate yes high yes head yes low yes sensor moderate moderate yes som moderate moderate yes online compression acquisition compressive sense yes transmission consensus pca high moderate yes moderate surveillance moderate low decentralize datum yes neural network target efficiently transmission service head critical iterate input cluster centroid cluster hierarchy gp parameterize probabilistic base gaussian regression regression focus energy consumption broadly speak process preferable smooth function however scale space low lee self classify win neuron represents number win network traffic network lin adaptive quantization retrieve sensor node historical pattern code book transmission original reading compress crucial disadvantage use aggregation neuron far away competition develop token set use pca aggregation traditional explore original simple compose step expectation em function fix expectation cost method estimating produce compressive ability spatial correlation direct transmission collect distribute technique execute combine compression likelihood cb eigenvectors matrix pca cb method consensus predict hence global communication cb cb tune provide quality adjust increase consensus round increase pca transform collect time cluster head cluster head compressed eliminate achieve ignore throughput cope dimensionality collect keep important reduction li address track collaborative signal environment additionally track multiple target nearest surveillance collect massive surveillance complex introduce therefore mobile surveillance wireless mobile sensor enhance surveillance cluster cluster mobile sensor idea appeal straightforward implementation sensitive outlier role free wireless sensor clique perform clique enable node achieve combination address topology locally central control efficiency network transmission overhead energy consider requirement sensor introduce schedule human intervention classify drive query drive event fundamentally machine offer restrict area assess processing mechanism benefit efficient mechanism requirement storage resource event machine development processing query without machine learn assess controller spread intend node detect node sign area attention research rely define strict phenomenon recent query process complicated develop advanced query processing solution present processing solution detection detection activity recognition nn low query space enhance drive real distribute detection dt drive detect environmental phenomenon manner decentralized percent result important correction bayesian summary correction enhance present activity accurately body initially spread body detect sensor axis negative hmm sensor sensor rely informative description naive maximize human naive challenge solution yu detection processing aggregated maker beneficial environment core interpretable introduce highly query processing technique develop nn aware location k search region correspondingly knn process neighbor bind within snr refine primary concern memory collect delay develop tree event recognition sensor network area vote traditional overhead dynamically detect I e dominant set step language request attribute send database management management system query component optimize wireless sensor attribute mac mac enable base localization acoustic specification gps support localization network feasible propagation limitation gps develop system surveillance monitoring sensor surface unit central recognize site application spatially system delay major request moreover achieve failure ambiguity sensor collective employ regression predict mobile computational complexity execute give introduce som thousand propose execute som layer connect neuron formulate spatial anchor coordinate disadvantage equally spaced location introduce localization algorithm som develop limited resource require gps centralize central processing adjacency node similarly lee provide localization service node algorithm som propose network unit transmission overhead li develop reinforcement localization call path mobile management mobile mobile mb aware movement number brief state position mb cover sensor message mb run save resource fail mobile transfer mac protocol pose challenge wireless network energy consumption cycle fraction sensor node therefore protocol comprehensive protocol provide recently enhance mac protocol adaptively cycle transmission history able predict channel energy consumption design mac transmission concept mac protocol mac security able attack brief mac protocol review synchronization protocol assume indicate ability handle node comparison mac protocol mac dt hybrid present mac protocol active continuously medium bayesian learn channel allocate save network protocol sensor mac mac mac mac division protocol employ periodic frame access transmission change wang transmission schedule fuzzy distribute maximize length mac prevent attack type attack huge traffic useful case limitation capability neural prevent network traffic investigate network request consequently mac layer exceeds predefine importantly site design employ mac mac protocol basically mac reduce usage increase throughput mac mac rl mac transmission schedule frame mac determine slot length cycle active traffic load bandwidth new base mac inform achieve benefit simple resource frame nod transmission map slot node attain slot allocation transmission demonstrate initially initialize upon transmission slot update update rate upon successful transmission reward equal fail transmission certainly example three employ medium access reinforcement appeal requirement may initial phase share collect user mobile challenge communication service requirement propose adapt layer design mac switch mac protocol mac engine mac current inter e requirement energy usage traffic mac pure mac mac though mac environment introduce design mac architecture functional specification operational capable date comprehensive machine advance adopt security highlight effort specialized researcher improve learn security anomaly implement security limited resource constraint moreover attack observation network figure present monitoring sensor classify read region inconsistent attack consider anomaly phenomenon monitor system employ attack detect basically security adopting save significantly expand enhance reliability avoid discovery convert often intervention attack explore various address security review indicate summary wireless sensor machine detection belief outlier nn moderate distribute outlier detecting attack selective attack outlier detection online outli centralized system distribute adaptive analyze som som behaviors bayesian node temporal correlation infer observation collect evaluate branch detection near replace value nearest nn base black attack request message indicate accordingly source discovery assume drop attack classifier capable attack selective attack information bandwidth use origin requirement could sphere distinguish anomaly minimize communication overhead design svm svm outli detector inspire biological body chen detect algorithm furthermore zhang temporal outlier main svm scalability requirement address detect attack wireless ad map determine som attack large sensor network service event potential query node suffer energy furthermore issue couple topology important reliable art requirement review review effort machine achieve advantage recognize stream thus need flow aware guarantee detection depend network service handle ensure resource power review column service datum dynamic low link assess reliability moderate processing sensor converge aware power management management low tool grow performance estimate availability reliability failure dynamic capture dynamic behavior effect propagation idea requirement link inaccurate unstable variation interference wang link quality method protocol adopt offline method learner indicator feature classification tree receive signal strength indicator size load forward receive reverse communication experiment time method present handle assess provide iterative experience distribution environmental mean historical update consider sequentially collect adaptive sensor base learn technique throughput observe learner able site stage mechanism conversely power management energy capability base aware management level capability employ attain reinforcement mesh cc structure tool basically adopt reliably might examine impact load whole present constraint minimize consumption sensor intend consider application fig fundamental reading receive incoming put must execute network static schedule node go move take knowledge management environmental monitoring employ accurately behavior inactive movement velocity advantage implementation since design consider predict device storage power network measure air level effect air quality neural server computer server radial rbf extract measure fundamentally control digital crucial affect develop scheme compare wireless although machine open maintain several management percent compression reduce transmission hence traditional compression extra consumption requirement tradeoff transmission efficiency even extend basic concept compressive meet resource decentralize compressive please technique include device centralize enable rapidly tune current reason learn processing algorithm include adaptive weight improve ellipsoid soft develop efficient two communication protocol mac protocol design detecting activity first mac protocol discuss survey technique study enhance second focus minor energy I machine management circumstance basically sensor broadly cluster technique cluster network temperature monitoring cluster combine hierarchical spatial temperature monitoring energy activate hierarchical balanced wireless protocol tool consequence wireless sensor energy aware scheduling localization datum machine wireless sensor network table summarize adopt machine learning challenge challenge sensor network several extensive study adopt wireless network resource pattern numerous open effort hierarchical adopt learn resource management wireless lin school computer engineering wireless sensor network environment dynamic either cause external sensor eliminate unnecessary solution literature review period address wireless sensor advantage provide comparative aid suitable machine challenge wireless localization cluster aggregation processing medium compressive sense wireless cost node nod forward unit base sensor equip various acoustic chemical pressure weather optical diversity building characteristic develop scenario aggregation aware scheduling security shift robust last decade machine extensively task area bioinformatics vision come mathematic definition essence development acquisition enhance develop machine describe exploit
velocity consequently differential quantifie coupling measure wind wind take wide present challenge research datum develop method reproduce velocity measure wind speed non fluctuation power prediction energy production wind reflect wind contribute determine model well understand production focus fluctuation wind extreme anomalous responsible load additional show derive wind end wind power wind north show approach ref apply describe detail comparative sec conclusion topic htb velocity illustrate normalization three wind full month wind wind n measurement rotation angular velocity operating rotation hz hz rate measurement exist hz analyze protocol scientific requirement wind measurement accord velocity fig together right fluctuation responsible wind wind see increment statistic time wind hour increment particularly power several quantify wind velocity increment plot shift vertical axis visualization illustrate coefficient drift equation framework propose develop range modeling medical eeg stock market ref therein method wind ability properly power characteristic langevin briefly one full period extract apart multiplicative conditional namely dash interpolation correspond plot instance fit range offset moment evidence stationary langevin equation moment address narrow wind condition represent wind velocity section range wind velocity diffusion top drift diffusion velocity low value velocity lack range check fig drift diffusion depend linearly well see functional coefficient wind euler langevin wind integrate reconstruct plot fig together clearly reconstruct series real measurement increment time scale b clearly conclude ability langevin evolution langevin time reconstruct rate wind drift text fig increment fluctuation within lag second unit deviation wind also increment notice condition necessary langevin evolution still fulfil condition langevin stochastic evolution drift coefficient first order correction consider drift diffusion point evolution velocity velocity measurement less couple wind langevin straightforward forecast nearest properly neural
interpret learn function decoder bias e tangent project onto activation free matrix tie bias order function base error choice depend typical xx appropriate e l dimension dimension encourage learn underlie interpretable simply overcomplete kind autoencoder autoencoder among reconstruct attempt corrupt solution learn distribution noise gaussian noise corrupt probability apart representation regularization corruption procedure move must project corrupt manifold useful pre especially stack recent variant locally characterize generate corruption process predict form corruption markov autoencoder distribution sec sample generate generate spurious region around determine corruption allow place amount spurious mode large amount noise naive divergence reconstruction define series reconstruction random subsequently reconstruct final spurious mode manifold autoencoder relational autoencoder extension pair give define learn involve indicate indexing encoder store infeasible weight cubic roughly restrict project quadratic weight need factored model w develop neural training denoise autoencoder apply although training typically examine corruption procedure sample interpretable form alternate algorithm spurious define converge generate distribution argument x make geometrically datum like correct manifold multiplicative learn class structure share class digit example tail may relatively share tail correspond weight importance tail interpretation factor factor class label generative conditional mnist database intensity image scale thresholded mnist unit unit relu visible activation epoch via mini descent initial nesterov gradient noise apply pixel corruption average reconstruction markov example sample conditioning depict define chain zero corrupt begin generate chain spurious sample digit expression pixel gray intensity intensity experiment relu hide sigmoid epoch descent epoch nesterov noise train factorize activation variation mnist likely size training provide unlabeled generative way learning separate share light act practical apply rich explore sampling autoencoder operator unimodal limit capacity complex order extend work autoencoder purely train massive much attention last generative autoencoder transition markov theoretically empirically capacity datum
run step burn cluster probable generation diagram upper region ht capable mix tree performance real example first sampler examine breast cancer repository originally subsequently utilize numerous breast nine clinical covariate uniformity cell uniformity cell shape size miss outcome discard log wu majority suggest homogeneous contribute misclassification assess performance select set make remain ensemble calculation repeat ten separate different seed low wu support gibbs rapidly heterogeneous illustrate percent force continuous measure patient clinical visit via penalize cubic spline inspection aic short use longitudinal datum subject entry utilize clinical gender pa bc status longitudinal roughly carry concern several variable since importance covariate value covariate choose form tree importance quite variable forest rank decrease purely result forest gender role empirically group ensemble review tree random forest boost average interesting enough machine resort bootstrap calculation size bayesian averaging method demonstrate ensemble capability help self behavior average tree show important provide worth compare model component unimodal importantly modal construction fitting imagine gibbs sampling rapidly fit grow chance develop empirical possible greedy randomness implementation difficult user exist cart package grow cluster mention acknowledgment author grateful foundation patient comment breast cancer university author thank availability supplementary forest pt utilize ensemble cart subset similar heterogeneity aggregate approach develop cart classification breast cancer regression patient key bayesian cart heterogeneity regression cart nonparametric binary intuitive relation covariate aside simple cart affect cart conditionally independent simplicity preserve cart derive generate bootstrap tree utilize aggregate bag boost stochastic create generalize cart tree sum difference multiple create diverse fitting therefore combine source variability utilize hope tree fit nonetheless rather bootstrappe control tree tree performance three setting breast study benchmark heterogeneous patient record outcome cart assign record region identically independently distribute predict one origin probability th estimate impossible calculate later define conditional mixture infinite correspond dirichlet node unit wu child node node simply integer part child node node one leaf therefore least one splitting correspond draw element leave iterate leaf node follow q bernoulli multinomial distribution small partition guarantee distribution proportion certain constructing time utilize name change dimension create include explore jump model auxiliary new stick process gain popularity decrease burden stick break dirichlet straightforward illustration infinite indistinguishable large number slice slice sampler carlo sampling lead slice sampler rapid assignment scheme iteration grow clustering allocate rapid change update provide find choice grow metropolis mh grow yet grow tree therefore scheme conditional mh restrict result step compare use micro every node convergence force wu efficient facilitate jump mode tree change useful prevent switching uniform truncation effect keep posterior analysis marginal costly allocation joint type estimator ensemble estimator former assignment latter define assignment posterior estimator often know observation capability single partitioning
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v v v v v stroke v v v v v v stroke v v v v v v v v v v v v v v v v stroke v v v v v v v v v v v stroke v v v v v v v v stroke v v v v v stroke lt v v v v v v stroke v v v v v v v v v v v v v v v v v v v v v v v stroke v v v v v v v v v v v v v v v v v v stroke v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v stroke v v stroke v v v v v v v v v v v v v v v v v v v v v v v v v v v v v stroke v v v v v v v stroke v v v v stroke v v v v stroke v v v v stroke v v v v v v v stroke v v v v stroke v v v v v v v v v v v v v v stroke v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v stroke v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v stroke v v v v v v v v v v v stroke v v v v v v stroke v v v v v v v v stroke v v v v v v v v v v v v stroke v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v stroke v v v v v v stroke ltb def rgb exch exch def exch def mul roll exch exch sub mul mul sub mul mul def mod def ifelse ifelse ifelse ifelse ifelse ifelse def constrain exch exch ifelse def mul exch mul add constrain roll exch mul constrain roll exch mul add constrain roll def exch exch exch roll exch roll exch exch mul exch constrain roll copy mul exch mul mul add constrain roll exch mul add exch mul exch add roll def eq rgb ifelse ifelse ifelse gidx get gidx add def gidx gidx gidx def gidx get mul gidx gidx get add gidx gidx gidx mul gidx le gidx gidx def def def def mul sub ifelse pm def pm exch def cf constrain exch exch cf constrain ifelse pm pm def ifelse ltb stroke ltb v stroke ltb stroke stroke ltb r stroke ltb stroke stroke ltb ltb stroke stroke ltb r stroke ltb stroke ltb r stroke stroke ltb stroke stroke ltb ltb ltb v v v lt v v v v v r stroke lt v v stroke v v v v ltb def rgb exch exch exch def exch def mul roll exch exch def mul mul def mul mul mod def ifelse ifelse ifelse ifelse ifelse ifelse def constrain exch exch ifelse def copy mul exch mul add constrain roll copy mul exch mul constrain roll mul exch mul add constrain roll exch exch roll exch def copy mul mul add exch mul roll mul add exch constrain roll exch exch add constrain roll def ifelse ifelse ifelse gidx gidx gidx gidx gidx sub def sub get gidx mul gidx gidx mul gidx gidx add def gidx get le gidx gidx gidx ifelse def def def pm ifelse def pm def stroke pm exch pm constrain constrain constrain def ifelse stroke pm pm exp ifelse ltb stroke ltb stroke stroke ltb v stroke stroke ltb stroke stroke ltb ltb stroke ltb v ltb stroke ltb v stroke stroke ltb r stroke ltb v stroke stroke ltb v r ltb stroke ltb stroke ltb stroke stroke ltb v ltb ltb ltb lt v v v v v v v v v stroke v v v v v v v v v v v v v stroke v v v v v v stroke v v v v v v stroke v v v v v stroke v v v stroke v v v v v v v stroke v v v v v stroke v v v v v v v v v v v v v v v v v v v stroke v v v v v v v v v stroke v v v v v v v stroke v v v v v stroke v v v v v v v stroke v v v v v v v v v v v v stroke v v v v v v v v v v v v v v v v v v v v v v v v stroke v v v v v v v v v v v v v v v v v v v lt v v stroke v v v v v v v v v v v v v v v stroke v v v v v v v v v v stroke v v v v v stroke v v v v v v stroke v v v v v v v v v v v v v stroke v v v v stroke v v v v v v v v v v v v v v stroke v v v v stroke v v v v v v v v stroke v v v v v v v v v v v v v v v stroke v v v v stroke v v v v v v v v v v v v v v v stroke v v v v v v v v v v v v v v v v v v stroke v v v v v v v v v v v v v v stroke v v v v v v v v v v v v v v v v v v v v v v v v stroke lt v v v v v v v v stroke v v v v v v v stroke v v v v v stroke v v v v v v v v v stroke v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v stroke v v v v v v v v v stroke v v v v v v v v v stroke v v v v v v v v stroke v v v v v v stroke v v v v v v v v v v v v v v v stroke v v v v v v v v v v v v v v v v v v v v v stroke v v v v v v v v v stroke v v v v v v v v v v v v stroke v v v v v v v v v v v v v v v v v v v v v v v v v v stroke v v v v v v v v v v v v stroke v v v v v v v v v v v v v v v v v v v v v v v v v v v v stroke v v v v v v v v v v v v v v v v v v v v v v v v v v stroke v v v v v v stroke v v v v v v v stroke v v v v v v v v v v stroke v v v v v v v stroke v v v v v v v v v v v v v v v stroke v v v v v v v v v v stroke v v v v v v v v v v v v v v v v stroke v v v v v v v v v v v lt v v v v v v v v v v v v v stroke v v v v v v v v v v v v v v v v v v stroke v v v v v stroke v v v v stroke v v v v v v v stroke v v v v v v v v v v v v v v v stroke v v v v v v v v v v v v v stroke v v v v stroke v v v v v v v v stroke v v v v v v stroke v v v v v v v v v v stroke v v v v v stroke v v v stroke v v v v v v v v v v v v v stroke v v v stroke v v v v v v v v v v stroke v ltb stroke r r show five plot vertical help side denoise distribution plot five super laplace pdf logarithmic behave plus signal neuron scalar theoretically approach variance corruption denoise scale step readily closely serve statistical modeling translate connection help high level relevant component independent source however span source limitation scale distribution leave sample five five sub autoencoder zero hide nonlinearity denoise simplify essentially mean mapping matrix denoise assumption ica incorporate denoise covariance gaussian verify mixture scale apparent loading contribute hide unit load product measure contribution recover activation scale length dominant loading row angle vector dominant source correspond dominant source sign determine super sub source unit preference source represent structure perform happen miss unit pressure possible mode operation primarily span covariance align independent happen align principal pca ica hide retrieve ten loading ten loading without mapping around std turn require time converge ten loading loading network tendency extract see span eigenvector whereas note pre whiten autoencoder recover component allow perform principal normalize information shall expand dependency hide ica usually impossible produce truly statistically projection even activation lack normally high dependency example activation correlate activation represent recall q activation variance translate connection strength unit activation flexibility layer network representation connection allow high unlike ica whiten cost function dataset linear source ica source determine word high source variance source source independent source distribution level contain super pre mapping observation ica perceptron mlp encoder q activation operate separately make def dl mul mul add def vpt vpt def begin title subject plot author ifelse def def def vpt vpt mul mul stroke show ifelse ifelse ifelse vpt exch def exch mul vpt vpt mul def dl solid ifelse bl stroke def exch def lt mul exch def pl stroke def lc lc def lc def lc lc lc pl ltb bl dl def al mul def pl lc dl pl dl dl lc def pl dl dl lc dl pl dl dl lc dl def lt pl dl dl lc dl def pl dl dl dl dl lt pl dl dl dl lc lt pl dl dl dl dl dl dl lc dl def pl dl dl dl dl dl dl dl def vpt vpt vpt vpt vpt def pls vpt vpt stroke vpt stroke stroke copy exch exch vpt vpt vpt stroke exch vpt add vpt vpt stroke stroke copy mul vpt v mul vpt def star pls stroke exch exch vpt vpt fill def mul vpt mul vpt mul def copy vpt mul mul vpt mul stroke vpt sub vpt mul vpt fill vpt add vpt vpt vpt vpt stroke copy translate stroke translate fill copy arc arc fill def bl copy vpt arc def bl copy vpt arc vpt bl copy vpt arc bl copy vpt arc fill vpt arc def bl copy arc vpt arc def bl copy copy arc vpt arc vpt arc bl vpt arc vpt arc bl copy vpt arc arc def bl vpt vpt def bl arc vpt arc def bl copy vpt fill copy vpt arc fill vpt arc bl copy copy vpt copy copy vpt arc vpt arc def bl copy vpt arc fill vpt arc bl vpt fill copy vpt arc fill vpt bl copy copy vpt arc def bl copy vpt fill vpt arc roll exch sub exch vpt exch bl vpt bl bl bl copy vpt sub exch vpt def bl copy exch vpt vpt vpt def bl copy exch exch vpt vpt square bl exch vpt exch vpt vpt fill def bl copy vpt exch sub vpt vpt fill def bl copy exch vpt exch vpt vpt vpt fill copy fill def bl copy vpt sub vpt bl copy vpt vpt vpt bl vpt vpt copy exch sub exch vpt fill bl vpt vpt exch vpt vpt vpt bl exch vpt exch vpt sub vpt vpt def bl exch vpt vpt sub vpt vpt copy vpt def bl copy exch sub exch vpt vpt vpt exch vpt vpt fill bl translate def translate stroke translate stroke stroke translate stroke def translate translate def stroke translate def translate def translate stroke def translate translate translate translate stroke stroke vpt add vpt v vpt vpt stroke def stroke exch vpt vpt vpt stroke vpt add vpt mul mul vpt mul stroke stroke vpt vpt mul mul stroke def translate repeat stroke stroke vpt vpt vpt v vpt vpt v def stroke exch exch vpt add vpt vpt stroke vpt mul mul vpt mul v def stroke vpt vpt mul v stroke def stroke translate repeat stroke stroke def fill exch exch def exch exch add mul def add def def fill roll add add translate mul get mul def get translate add mul ne get mul roll stroke def ifelse lt def ifelse def l stroke l stroke exch l def exch def l stroke exch l stroke stroke exch def l stroke stroke def pattern landscape ifelse def pattern landscape ifelse def landscape ifelse pattern landscape ifelse def fill ifelse def pattern def pattern pattern def pattern def def ifelse begin ifelse translate rgb exch exch mul roll exch def def sub mul mul def mul def mod def ifelse ifelse ifelse ifelse ifelse ifelse def constrain ifelse mul add constrain roll exch add add roll exch constrain roll rgb sub exch roll roll exch def add exch mul exch constrain roll copy mul exch mul add mul constrain roll mul exch mul exch mul exch constrain roll ifelse ifelse ifelse def def gidx def gidx gidx def gidx sub gidx gidx gidx gidx gidx gidx gidx mul gidx gidx mul def gidx le gidx def ifelse def def pm mul ifelse mul def rgb def stroke def pm cf constrain exch constrain exch constrain def ifelse stroke pm pm ifelse ltb v stroke ltb ltb ltb ltb v r level image def translate index f f e c c b b b width ifelse ga dp ltb stroke ltb stroke reflect loading appropriately dl mul mul mul def def vpt def def vpt vpt put title denoise plot ifelse def v def def fill def def vpt vpt mul show ifelse ifelse def stroke r ifelse vpt mul def mul exch def mul def vpt vpt solid solid ifelse bl stroke def stroke def mul exch def def pl def def lc def lc lc def lc lc lc def lc lc def pl ltb bl def mul def lt pl lc def lt pl dl dl dl lt pl dl dl lc def pl dl lc dl lt pl dl dl dl lc dl lt pl dl dl dl dl lc dl lt pl dl dl lc dl pl dl dl dl dl dl dl lc dl def pl dl dl dl dl dl dl lc dl copy vpt add vpt vpt vpt v vpt stroke pls stroke vpt vpt stroke vpt r box exch vpt vpt vpt stroke exch exch vpt vpt stroke vpt v def stroke copy vpt mul add mul mul v vpt mul stroke def copy stroke exch sub exch vpt vpt vpt fill stroke vpt mul mul mul v vpt mul stroke vpt vpt mul mul v vpt mul def stroke vpt vpt mul vpt fill def stroke vpt vpt vpt vpt vpt v copy translate repeat def translate stroke stroke def bl copy vpt arc bl copy vpt arc fill vpt arc def bl vpt arc def bl copy vpt arc vpt def bl copy vpt bl copy copy vpt arc copy vpt arc fill vpt arc def bl copy vpt fill vpt arc bl vpt arc vpt arc bl copy copy vpt fill vpt bl copy vpt arc vpt def bl copy vpt arc fill vpt arc def bl copy copy vpt arc fill copy copy vpt arc fill vpt arc def bl vpt fill arc bl copy copy vpt arc fill copy copy vpt arc bl copy vpt fill vpt arc bl vpt arc roll index exch def vpt sub exch vpt exch def bl copy def bl vpt fill def bl copy exch vpt exch vpt square def bl exch vpt fill def bl copy exch vpt exch sub vpt fill bl vpt exch vpt vpt def bl exch vpt exch vpt vpt bl copy exch vpt exch vpt vpt bl copy vpt vpt fill bl vpt vpt vpt bl copy vpt vpt fill exch vpt exch vpt square bl vpt sub vpt exch vpt sub exch vpt vpt fill bl copy exch vpt vpt vpt def bl exch vpt exch vpt vpt vpt fill copy vpt fill def bl exch vpt sub exch vpt vpt vpt fill copy vpt exch vpt copy translate translate def translate stroke translate stroke translate stroke translate translate stroke translate def translate translate def translate stroke translate stroke def translate def translate stroke translate stroke def stroke vpt vpt vpt v vpt stroke exch exch vpt add vpt v v vpt def stroke vpt vpt mul vpt mul vpt mul vpt mul stroke repeat arc vpt add vpt v vpt vpt vpt stroke stroke exch sub exch vpt vpt stroke vpt add vpt vpt mul vpt mul vpt mul mul vpt stroke repeat stroke arc stroke def density exch def exch exch def def mul add def mul add fill def fill def roll translate mul mul def translate add mul ne mul roll stroke ifelse ifelse def def stroke stroke exch l fill exch stroke exch def l stroke exch exch def def pattern landscape ifelse def pattern landscape ifelse pattern landscape ifelse def landscape ifelse def fill ifelse def pattern def def pattern density def def level ifelse symbol index eq def ifelse translate scale def rgb exch exch def roll exch def sub mul mul mul def mod def ifelse ifelse ifelse ifelse ifelse ifelse def ifelse copy exch mul constrain roll constrain roll mul mul constrain roll def exch roll roll rgb copy mul exch mul constrain roll copy mul mul exch mul constrain roll mul exch mul exch mul constrain roll ifelse ifelse ifelse def def gidx def gidx gidx gidx add loop def gidx gidx gidx sub gidx sub mul gidx gidx gidx sub add def gidx gidx get gidx sub add gidx gidx gidx ifelse pm mul ifelse pm color stroke pm def cf exch constrain exch ifelse stroke pm ifelse ltb r v r r z stroke ltb ltb lt v v v v v stroke ltb def exch exch def mul roll def exch def def mul def def ifelse ifelse ifelse ifelse constrain ifelse mul mul constrain roll copy mul constrain roll mul add roll def exch exch sub roll exch roll copy mul exch mul exch mul exch add roll mul mul exch mul add constrain roll mul exch add exch mul add constrain roll ifelse ifelse ifelse gidx gidx add def gidx sub gidx gidx get gidx gidx def get gidx add def gidx gidx get gidx gidx le gidx gidx ifelse def def def pm mul ifelse def mul stroke exch stroke pm cf constrain exch constrain constrain ifelse g stroke pm exp def ifelse ltb v r v stroke stroke ltb ltb lt v v v v v v v v ltb exch exch def exch mul exch def exch sub mul def mul sub def mod ifelse ifelse ifelse ifelse ifelse ifelse def constrain ifelse copy mul constrain roll exch mul roll mul constrain roll def exch exch sub roll exch roll exch def copy exch mul exch mul exch constrain roll mul exch mul exch mul constrain roll mul mul exch mul add constrain roll ifelse ifelse ifelse def gidx def gidx get gidx gidx def loop gidx gidx gidx gidx gidx gidx sub mul def gidx get gidx mul add def gidx sub get get mul add sub gidx gidx gidx ifelse def def pm ifelse pm gamma def stroke pm exch pm cf constrain exch constrain constrain ifelse pm pm exp def ifelse ltb v v v r v stroke ltb ltb lt v v v v v ltb def exch exch exch mul roll exch def mul mul mul mod def ifelse ifelse ifelse ifelse ifelse constrain exch exch mul constrain roll mul roll mul add constrain roll rgb exch exch sub roll exch roll exch def rgb copy mul exch add exch mul add constrain roll copy exch add exch mul constrain roll mul exch mul add exch mul roll def eq rgb ifelse ifelse ifelse def def gidx gidx get gidx def gidx gidx sub get gidx get sub gidx gidx gidx def gidx gidx add gidx gidx get gidx sub def gidx gidx gidx ifelse pm def gamma mul def def pm exch def pm exch constrain exch cf constrain ifelse pm pm ifelse ltb v v v v stroke v ltb ltb ltb lt v v v stroke ltb rgb exch eq exch exch def mul roll exch def exch mul mul def mul sub def mod def ifelse ifelse ifelse ifelse def constrain lt exch exch ifelse def exch mul add constrain roll copy add roll mul exch mul roll rgb exch sub exch exch exch roll exch mul mul exch mul exch constrain roll mul mul exch constrain roll mul exch exch mul exch add constrain roll rgb ifelse ifelse ifelse def def gidx def gidx gidx def gidx gidx get gidx def gidx gidx sub gidx add def gidx gidx get gidx mul add gidx gidx sub gidx gidx get le get gidx def ifelse def def def def mul ifelse def pm gamma mul rgb stroke exch def cf constrain exch constrain def ifelse exp ifelse ltb v r v r v stroke ltb lt v stroke ltb def exch exch mul roll def mul mul mul sub mul sub mul def mod def ifelse ifelse ifelse ifelse ifelse ifelse def constrain exch ifelse mul exch mul constrain roll copy mul add add roll mul exch mul constrain roll def roll exch mul exch exch exch constrain roll mul mul exch constrain roll mul exch mul exch mul add roll ifelse ifelse ifelse def def gidx gidx get gidx def gidx gidx get gidx sub def gidx gidx gidx get add gidx get gidx gidx mul gidx gidx sub get gidx sub mul def gidx le get gidx gidx ifelse def def def pm ifelse pm gamma stroke pm exch def cf constrain exch constrain def ifelse stroke pm pm def ifelse ltb v r v r stroke z stroke ltb lt v v v v v v v stroke ltb exch def def exch def sub mul mul def ifelse ifelse ifelse ifelse ifelse ifelse def constrain exch ifelse rgb exch mul add roll exch mul add roll mul exch constrain roll exch roll exch roll def copy mul exch mul exch add constrain roll mul exch mul exch mul add roll mul exch mul constrain roll eq ifelse ifelse ifelse def gidx def gidx gidx def loop gidx gidx gidx def gidx gidx mul def gidx mul def gidx gidx gidx sub mul add gidx le gidx get gidx def ifelse def def pm ifelse def pm mul def pm exch def stroke constrain exch cf constrain def ifelse stroke exp ifelse v r stroke ltb v v v v v stroke ltb def exch eq exch exch exch mul roll def exch mul sub mul mul mod def ifelse ifelse ifelse ifelse ifelse constrain ifelse def mul exch mul add constrain roll exch mul add constrain roll mul exch mul constrain roll sub exch exch roll exch sub roll exch def exch mul add mul constrain mul add exch mul constrain roll roll rgb ifelse ifelse ifelse def gidx def gidx gidx gidx add def def gidx gidx gidx gidx gidx gidx mul add gidx get gidx def gidx mul add def gidx sub le gidx get gidx get ifelse def def mul ifelse gamma def stroke pm def pm cf exch constrain exch constrain ifelse stroke pm pm ifelse ltb r v stroke ltb ltb v v v v v ltb def rgb exch exch exch exch def roll sub exch def mul mul def mod ifelse ifelse ifelse ifelse ifelse ifelse def lt exch ifelse def copy mul add roll copy mul exch mul add roll mul exch add constrain roll def rgb exch exch exch roll exch sub copy exch mul exch mul exch add constrain roll copy mul exch mul constrain roll mul exch mul exch add constrain roll def ifelse ifelse ifelse gidx gidx gidx def def gidx gidx gidx sub def gidx gidx gidx get gidx mul def gidx gidx mul def gidx le gidx get def ifelse def def mul ifelse pm mul def def color stroke exch def def stroke pm constrain exch exch constrain def ifelse stroke pm def ifelse ltb v r v v v v v ltb v v v v v v v v ltb def exch exch exch mul roll def mul def mul mul sub mul mul mod eq ifelse ifelse ifelse ifelse ifelse def constrain lt exch exch ifelse def copy add add roll copy exch add roll exch mul add constrain roll def rgb exch exch roll exch roll def copy mul mul add mul constrain roll copy mul exch mul roll exch add constrain roll ifelse ifelse ifelse def gidx def gidx gidx gidx add def loop def gidx gidx gidx gidx gidx gidx def gidx sub get gidx add def gidx gidx get gidx get mul add get le gidx gidx gidx def ifelse def mul sub ifelse pm gamma mul def rgb color stroke pm exch constrain exch cf constrain exch cf constrain def ifelse stroke pm exp ifelse ltb v v r ltb lt v v stroke ltb def exch exch def exch roll exch exch def sub mul mul def mul mul def ifelse ifelse ifelse ifelse ifelse ifelse def constrain exch ifelse rgb mul constrain roll mul exch add constrain roll mul mul constrain roll def roll exch exch rgb mul mul exch mul exch add roll mul exch mul exch mul roll exch mul mul constrain roll def ifelse ifelse def gidx gidx gidx add def loop gidx gidx gidx get gidx get gidx mul def gidx gidx sub gidx sub mul add gidx gidx sub gidx mul add def sub le gidx gidx gidx ifelse def def def pm mul ifelse def pm def g stroke pm exch stroke constrain exch cf constrain def ifelse stroke pm ifelse ltb v r r v ltb lt v v v v v ltb exch exch def exch mul roll exch exch def mul def mul sub mul def mod def ifelse ifelse ifelse ifelse ifelse ifelse constrain lt exch exch ifelse copy exch roll copy mul exch constrain roll mul exch mul constrain roll def exch exch roll exch roll def mul exch mul mul exch add constrain roll copy mul exch exch add roll mul exch mul exch exch add constrain roll ifelse ifelse def def gidx gidx get gidx gidx def loop gidx sub gidx gidx sub gidx get gidx mul def get sub mul add def gidx mul gidx gidx gidx def ifelse def def def def mul ifelse pm def def exch cf constrain def ifelse stroke pm pm exp ifelse ltb v v stroke ltb ltb v v v v v v stroke ltb exch exch exch def mul roll sub def mul mul mul sub def mod def ifelse ifelse ifelse ifelse ifelse ifelse def constrain lt exch exch ifelse def mul mul add roll mul exch mul add add roll mul add roll exch sub exch exch roll exch roll exch copy mul exch mul exch mul exch constrain roll copy mul exch roll exch mul exch exch add roll ifelse ifelse ifelse def true def gidx gidx gidx loop def gidx get gidx sub gidx get sub gidx gidx sub sub mul def gidx gidx gidx gidx get gidx gidx add def gidx get sub le gidx gidx ifelse def def def def pm ifelse pm mul def rgb pm exch stroke cf constrain exch constrain exch constrain def ifelse pm pm exp def ifelse ltb r stroke ltb lt v v v v v v stroke def exch exch def exch mul roll sub exch def mul def mul mul def mul mul mod def eq ifelse ifelse ifelse ifelse ifelse ifelse constrain lt exch exch ifelse def mul exch mul constrain roll copy mul add roll mul exch constrain def exch exch roll exch mul mul exch add roll exch mul constrain roll mul exch mul exch mul exch constrain ifelse ifelse ifelse def gidx gidx gidx add gidx gidx def gidx gidx gidx mul add gidx gidx gidx add def get gidx gidx gidx gidx def ifelse def def ifelse pm mul def def def g constrain cf constrain def ifelse stroke pm exp def ifelse ltb v v r v stroke v ltb lt v v v v v v v v v ltb exch exch def mul roll def mul def mul mul sub mul mod ifelse ifelse ifelse ifelse ifelse ifelse constrain exch exch ifelse mul exch add constrain roll exch constrain roll exch add roll rgb exch exch exch roll exch roll exch def mul mul exch constrain roll copy mul exch mul add exch mul constrain roll mul exch exch mul exch constrain roll ifelse ifelse ifelse def def gidx gidx gidx add def gidx get gidx sub gidx sub gidx get gidx gidx mul add def gidx gidx sub add def gidx gidx gidx mul gidx le gidx gidx gidx def ifelse def def def ifelse pm def g exch def pm constrain exch exch def ifelse stroke pm pm exp def ifelse ltb r v v v ltb lt v v v v v ltb def exch exch mul roll exch sub mul mul mul mod eq ifelse ifelse ifelse ifelse ifelse ifelse def constrain exch exch ifelse def copy mul exch mul add roll copy mul mul add constrain roll add roll def rgb exch exch roll roll exch def mul exch mul exch mul exch add constrain roll copy mul exch add exch mul constrain roll mul exch exch mul exch constrain roll rgb ifelse ifelse ifelse def gidx def gidx gidx gidx def gidx gidx get gidx get gidx gidx gidx mul gidx gidx mul add gidx gidx sub gidx mul def gidx get gidx gidx get gidx def ifelse def def pm mul ifelse mul def stroke exch def def stroke cf constrain exch cf constrain exch constrain ifelse stroke pm pm exp ifelse ltb r v v r v ltb ltb v v v v v v v v stroke ltb stroke r r r r r r r r r r r r nonzero top row change belong whose verify separate individual source model figure second mlp result square scale one generate appearance loading rotation rotation feed ica readily depict learn denoise plot belong behave neurons sigmoid neuron activation function activation readily activation learn activation sigmoid appear activation subspace sigmoid learn assume second reconstruction combine alone change high ten unit learn would also need detail recover lowest initialization network turn function speed considerably stage crucial combine proper success help turn network alone autoencoder seem reasonable autoencoder close sensible finding third seem despite whereas speedup speedup phase initialization gradually optimum cost extra term therefore gradually decrease could zero simplicity kept present verify hierarchy abstract efficiently although six representation abstract layer feature roughly nc promising result necessary far verify really deeply compatible therefore support discard information semi autoencoder another split high use feature decoder recently combine overall resemble autoencoder encoder approach extend include interaction interaction motivated autoencoder target propose include exchange information encoder mapping appeal approximation mlp extract dependency independent analysis tailor mapping mapping density make multiple round corruption denoise particularly ability useful make possibility corruption simple add support also involve completely type corruption possible extend denoise type information corruption denoise function might keeping previously mapping essential hierarchical line complex inference take kalman study implement dynamical give rise like fashion study already structure model elegant feedback loop unit top denoise learn connection reliably capacity stochastic latent hierarchical level information abstract invariant feature support unsupervise deep cost key contribute term receive training network also pay match prevent competitive additionally otherwise biased pca preliminary verify abstract verify claim acknowledgment thank parallel version manuscript certainly create work research intuition unsupervise importance algorithm pca rule ica would cm standard autoencoder network autoencoder corrupt denoise cm clean support autoencoder connection autoencoder analogous combine denoise autoencoder framework contribute produce level invariant ever publish hierarchy increasingly abstract visual cat researcher hierarchy stage artificial neural extraction lot propose learn deeply somewhat scheme produce exist seem obvious unlabeled information statistical structure image label carry bit compare carry certainly order argue reason able version supervise learning try irrelevant chapter fit learn base stochastic network continue learn discard leave detail represent approach relevant unsupervise come new explain add autoencoder give abstract invariant vertical path regular autoencoder decoder decoder encoder slow even since explain add hierarchy novel combine level discard invariant representation target high layer discuss extension argue unsupervised need unsupervise learning stand exception hierarchical variable model complicate simple alternative autoencoder feedforward promise candidate combine unsupervised learn autoencoder normally layer stochastic discarding summarize role learn variable model propose connection capacity role supervise learn unlabeled sample output unsupervised representation task obvious act pre key unsupervised important fully learn continue representation supervise start tune filter unsupervise semi argue early unsupervised discard find select recognize face face unlabeled find detector detector feature miss specifically keep reasonable compatible know supervise follow seem common variable latent denote eqs try everything datum go piece would many reduce alone orientation benefit reduce latent discard abstract invariant fix variable eq need represent everything represent high level abstract hierarchical represent inference posterior intractable inference amount intractable simple tractable approximation minimizing proceed would require involve approximation limited structure mathematically tractable layer propagation autoencoder mapping unit model encoder observation model decoder mapping decoder mapping latent analogous mapping autoencoder minimize remainder chapter omit like latent variable autoencoder together observation take define connect encoder decoder path new add add layer mapping layer feedforward actual hierarchical autoencoder eq hierarchical variable intermediate call require call variable matter prior deterministic network mapping stochastic latent inference representation receive path fix add bottom path high layer also combine abstract information orientation low layer l autoencoder complete representation fortunately denoise autoencoder add input explain abstract invariant seem task hand standard autoencoder information bottom path layer signal propagate activation top need lt picture show autoencoder layer need detail activation receive high layer represent autoencoder feedforward part far signal train fashion autoencoder difference connection chance reconstruction leave shrink autoencoder layer nonlinear slow variable share hierarchical reasonable try introduce combine remove rather gradient turn unsupervised study component utilize learn independent ica denoise derive estimate going show nonlinearity interpret expectation learn combine efficient latent tune ica development denoise source separation mapping operate alternate step assume fix reverse mapping derivation assume q depend noise noise approximate amount substitute estimating completely step equation yield algorithm essentially nonlinear simple multiplication nonlinear expect noisy observation crucial input layer contribute training nonlinear rule additional implement competition require nonlinear instead possible require denoise particularly useful denoise propose autoencoder feed autoencoder ask force autoencoder corrupt sample denoise autoencoder iterate corruption denoise sample original distribution denoise learn diffusion corruption force average sample density denoise start corruption denoise flow cause exactly flow sampling model corruption place network surprising denoise possible information relative turn representation energy readily reconstruct turn input corruption normalize probability denoise normalization factor particularly important feature denoise autoencoder representation include connection copy output input possible ready function denoise autoencoder recursively autoencoder internal implement mapping denoise alternate mapping minimizing add refer cost mapping layer consistent alternate recall framework mapping practically mean cost constant optimize autoencoder role combine essentially hierarchical offer forward cf gradient propagate backward path supervise layer output thing care amount error possible mutual predict long avoid go sufficiently loss assume matrix equal assume activation zero generality long mapping kronecker word distinguish viewpoint style representation fact eigenvalue former infinitely viewpoint sound content determinant eigenvector determinant logarithm determinant equal analytical applie eigenvalue power expansion eigenvalue logarithm logarithm element equation small content turn sensible zero relatively relatively analytical require computing gradient chain rule straight form twice behaviour close close chapter cost sure activation really mean input autoencoder high influence mapping solution projection pca type desirable ready collect cost use mapping batch quasi properly could corruption keep apply add corrupt activation term clean activation reconstruction depict computational going share mapping corrupt network add activation reconstruct activation computation observation cost along gradient cost direction along correspond signal denoise path path take denoise autoencoder activation require close gradient present key represent pressure layer allow invariant speed section gradually put hide add variance automatically eigenvalue hyperparameter bp dl mul mul def def vpt def vpt vpt def title subject ifelse def def def def def g def vpt vpt def stroke def stroke show ifelse ifelse def vpt mul vpt exch exch mul def vpt vpt mul def def solid ifelse def stroke def stroke mul exch lt mul def pl stroke def def lc lc lc lc lc lc def lc pl ltb bl dl al mul mul pl lc dl pl dl dl lc dl def pl dl dl lc def pl dl dl lc dl pl dl dl dl dl lc dl def lt pl dl dl dl dl def lt pl dl dl dl lc def lt pl dl dl dl dl lc dl pl dl dl dl dl dl lc dl def stroke stroke stroke vpt vpt vpt vpt vpt stroke pls vpt box exch vpt vpt vpt stroke stroke exch exch vpt vpt stroke vpt stroke def stroke vpt mul add vpt mul mul mul copy def exch vpt vpt stroke vpt add vpt mul v mul vpt mul copy vpt mul mul mul vpt stroke def stroke vpt mul vpt mul vpt mul v stroke vpt add vpt vpt vpt vpt stroke copy translate stroke def stroke translate fill def circle arc stroke def stroke arc def bl vpt bl copy vpt fill vpt arc c bl copy vpt fill v stroke v v v stroke v v v v v v v v stroke v v v v v v v v v v v v v v stroke v v v v v stroke v v v v v v v v stroke v v v v v v v v v v v v v v v v v v v v stroke v v v v v stroke v v v v v v v v v v stroke v v v v v v stroke v v v v v v v v v v v stroke v v v v v v v v v v v v v v v stroke v v v v v v v v v v v v v v v v v v
duality path dataset duality randomize gap dataset small size difference compute duality hand counterpart progress furthermore exhibit sampling lead stop degradation measure reflect accuracy phenomenon typical fw iteration suggest technique problem prohibitive medium updating exploit well choice european european union paper reflect author union contain project g base grants medical ii office optimization macro lemma ia electrical engineering frank wolfe recently gain community application however linear strategy sampling researcher experimental alternative frank hereafter denote fw solve fw cm direction vertex community show fw variant procedure attain convergence fw problem arise application find subproblem often easy solve impractical handling motivating stem cost proportional fu random never systematically study effort fw try identify avoid fix motivate approximation suggest pick least small fw algorithms duality gap applicable without entire randomized possible simplification entail tradeoff consider context solve impact investigate iteration exploit keep fw naive update fw conduct dataset code execute gb ram average run obtain tolerance computation recently point pt acc acc acc ex acc size first dependent fw fairly appear case cutoff considerably trend monotonically expect monotonic
particular get bind tell special provide explicit value distinguish direct undirected undirected neighbor undirecte large adjacent independence bandit view expert always view reward recover structure hold turn bad computing acyclic independence np algorithm tractable fix entail independence compute independence unlikely approximate moreover either trivially lead regret similar bandit set less issue exp rely efficiently computable quantity turn graph direct namely choose motivate one whereas second click thm clique namely number partition result thm early rely clique key undirecte graph trivial bad problem regime bandit setting provide low characterize feedback system open mention interesting feedback would prevent direct construction unbiased unobserved loss upon nice provide sequence generate inform strategy achieve bind identify achievable information feedback action period affect observation delay affect leave analyze finally assume I result acknowledgment grant core sm support european community fp grant agreement science foundation united foundation technology center os science integration section theoretic lemma throughout shorthand exposition condition later take take remove q exp sum moreover together rearrange give take side q finally expectation remove claim follow direct induce graph acyclic prove initially vertex minimize along incoming incoming iterate neighborhood vertex step minimize node arcs graph cycle follow use armed bandit bind non bandit beyond establish probabilistic adversary condition whose round condition consider sequence namely equal whose loss e iterate I conclude next lemma relate dominate set operating graph cover algorithm see bind graph remove cover algorithm arcs theorem iterate thereby dominate eq lift graph arc let independence number dominate appropriately upper k discretization concern version unique single node shorthand I recall give continue hand make clique clique size arc draw arc lemma clique satisfie recall ready derive upper statement quantity occur therein occur occur see throughout appendix denote I first condition history expectation preliminary importance p recall direct understand neighborhood arc set acyclic subgraph eq hand dominate valid assignment return linear might left acyclic contrary include would create node direct acyclic subgraph relations q shorthand inequality assumption conclude distribution compute round lemma put establishes next martingale difference sequence also positive ti tp tt round item get lemma distribution run eq ti item expand item simplify exploit eq assumption lemma q back follow round round pick get solve union key use sum result series fix combine slightly rearrange simplify thing asymptotic notation notation logarithmic ignore factor assumption least I simplify substitute simplify pick plug thereby obtaining claim thm lemma size adjacent node include positive hold nonnegative allow take convention contrary since adjacent entire putting repeat guarantee independent respect original configuration expression split adjacent adjacent neither way explicitly function indeed weight adjacent node early immediate corollary possible clique definition universit di microsoft research university institute information repeatedly loss relate know moreover set loss choose player reveal address variant combinatorial abstract weather forecasting need devise forecast day well forecast goal time model adversary round assign action discrepancy forecast player choose randomization incur regret excess incur compare round associate consider choose ad ad forecast sequentially choose well fix ad ad know choose ad display abstract framework observe pick refer bandit refer player observe loss goal bridge setting create spectrum quantify expert available action play round good regret achieve perturb bandit inf variant achieve bad switching expert crucial action setting get single rather round square setting receive loss example web bandit assume whether ad action ad information ad package display ad ad unlikely click sort capture online social reveal interest product friend connect select drive consumption linear could arise type expert action also system arc action action play action round loss round expert obtain complete action playing reveal loss regret trivial describe set consider side undirecte asymmetric preference person person ad person game situation modeling handle case easy feedback system choose action know ad relate inform might player feedback choose party recommendation social around I send contribution lie providing summarize brief algorithm exp achieve acyclic set independence result factor fix round exp attain feedback exp direct inform regret inform another guarantee turn direct case exp find approximately dominate always gets recover expert observe loss action bandit standard bandit logarithmic bound scale lie depend graph log base framework unbiased key challenge design scheme exploration small loss variance feedback key quantity combinatorial recently factor without protocol exp upper base acyclic subgraph case handle inform algorithm section bound demand high conclude text question claim state adversarial pick incur unlike problem reveal subset action reveal observe loss bandit expert action write playing sometimes play role notation v call feedback arbitrary regret measure fix adversary would measure player loss action player depend past section variant system horizon round advance easily goal actual depend actual begin step informed set select adversary make learner prediction convenient adopt arc I e order loop ignore number notable feedback system word playing reveal playing reveal symmetric define undirected arc direct cycle symmetry feedback distinguish direct symmetric depend play dominate set connected edge proper independent still associate ignore arc direct bandit setting dominate property remain dominate proper dominate direct graph denote dominating example action loop reveal action light blue dominating dominate bottom light maximal orientation bottom acyclic subgraph action compute minimum dominating direct cover associate system cover greedy large also lift notion independence undirected graph acyclic subgraph acyclic cycle either arc length cycle graph see investigate set action current feedback algorithm exp logarithmic correspondingly regret exp regret exp ifelse action accord distribution set I similar exp exp importance sampling divide observe loss observe action recall bandit recover exp precisely recover exp quantity view condition event analysis irrespective suitable add inform building regret set expert hence bind large constant take form equivalent exp regret point hold undirecte follow note non return expand issue whether powerful unfortunately graph independence number acyclic confirm total order arc bound achievable regret undirecte adversarial strategy regret adversary standard bandit play action bandit bind case whether exist exhibit regret exp feedback stochastically via direct arc probability loop default exp expectation occur r arm bandit r regret theory see fact low theorem something refine way
synchronization evident eliminate cpu calculation update work throughout execution consider expect cpu essential bottleneck achieve speedup heuristic show previous empirical dimension process shrink shrink heuristic shrink multi core machine version shrink section limitation evaluation name description evaluation handwritten dimension box dimensional white convert represent digit odd digit census web page collection handwritten recognition united service description correspond represent conference net experiment cluster dual run ghz gb memory node gb mix result single multiple node evaluation connect modern cloud provide programming array dataset previous shrink evaluate lin et elimination line shrink proceed shrink similarly read shrink half call proceed shrink throughout shrink name none single pc pc random multi multi multi pc pc default htbp name acc acc speedup x training algorithm branch selection active problem part decompose quadratic approach include optimization decomposition seminal svm sequential al primary method separable simplicity implementation choice solve classification problem multi system cluster brief overview setup al cascade parallel primary divide combine vector load imbalance finish individual sub target architecture research conduct create paradigm primary approach paper large restrict projection method solver qp leverage svm consider solve problem incomplete parallelization use active decompose svms approach elimination sample system address limitation parallel support machine shrink intuitive shrink art programming pass interface array design storage efficacy algorithm work involve shrink heuristic evaluation heuristic consider work elimination study shrink architecture elimination kernel cache cache intel fusion machine wide range domain finance identify care student college role social challenge designing scale core machine cloud improve adaptive elimination elimination case might reconstruction structure heuristic publicly available improvement improvement time baseline produce grow dramatically machine mining algorithms extraction volume domain finance social rely algorithm supervise learn supervise learn due excellent broadly space category use surface svm excellent core machine svm extremely albeit limitation parallel extend previously optimization however good entire special calculation though vector contribute calculation shrinking eliminate shrink core system literature address limitation utilize theoretical framework shrink speed format observation world sparse effect heuristic elimination stage execution approach art programming message interface array design communication storage scale cloud algorithms processor make analysis time category elimination efficient compressed kernel cache make attractive scale dataset multi large efficacy speedup execution parallel shrink organize work maximal separate hyperplane formulate generality slack allow possibly dimensional space convex introduce multiplier lagrangian wolfe eq primal maximization tucker kkt treatment svms refer contribute separate hyperplane remove small sample package reduce datum maintain essential relationship show objective maintain optimize explain section equation working selection selection step derivative refer address possibility second evaluate two loop avoid loop index nature eq threshold condition termination algorithm numerical user specify sample optimize bind sample shrink mechanism svm hyperplane eliminate decision heuristic eliminate belong one paper programming array global array model provide array load store semantic distribute array side global array programming domain sub array useful store easy array design asynchronous read array array use communication work tolerance avg row array begin presentation training organization introduce parallel algorithm present follow shrink pdf shrink prototype design shrink iteration calculate update require need structure lagrange multiplier early computation formulate cache several avoid cache prohibitive target cache low temporal pattern trend exhibit memory unit core unit intel graphic compute hardware support wide multiply add movement individual row across hence approach paper avoid cache organization structure critical dataset dataset less compressed sparse algorithmic relate reduction core several calculation datum among read organization structure significant improve cache rate leverage locality read write design choice job co load balancing process feasible contiguous movement approach semantic compress row ga semantics semantic system cloud algorithm show inner execute hardware representation inner line algorithm simple algebra trick shrink shrink present reasoning shrink must shrink parallel variant also section primitive operation process independently new integer loop expensive calculation require several locally operation update update
interval dark circle dash line reward although notice considerable variation reward mean significant slight difference game attain determined examine percentile subsampling subsample exhibit proportion run attain proportion exception skewness go see vary considerably game generator difficult determine generator find also normalize generator generator attain algorithm across generator percentile sample mean generator generator generator good generator frequently generator h lp algorithm good despite generators portfolio fail algorithm would interesting run experiment determine one generator game large possibility complicate slow decrease game generator generator algorithm reward intuition many generator reject reward lower exist negative negative negative kind generator tie reward make desirable game show reject support large game bad coefficient positive sensitive anomalous consider algorithm reward opponent play make star achieve opponent broad reward play profile appear performance issue portfolio avoid know opponent construct opponent percentile call opponent response assign overlap percentile claim apparent frequently response never p interpret single shot correspond algorithm mean attain algorithm mean always response confident get bootstrappe check subsampling game check algorithm dominate dominate proportion game show strict weak pure equilibria equilibria ever occur play occur involve restrict generator generator nash equilibria self nash equilibrium seven generator pure equilibrium game play symmetric strategy equilibrium arise equilibrium pure nash equilibrium pure strategy nash equilibria equilibria yield e yield reward game exactly weak nash equilibrium attain reward pure nash game dominate indicate nash equilibria asymmetric equilibria twice dominate never play performance opponent dominate dominate playing play trend avoid dominate play interesting specific observed ambiguity opponent tendency self play reward self play play run dominate self run self self play significantly low play percentile interval self run play self play despite occur whether play equilibrium offer nash equilibrium keep reader answer equilibrium convergence similarity achieve major likewise follow latter addition also opponent regret dominate particularly generator avoid generator connection expect reward largely support reward link generator induce make outcome reward reward regret run dominate positive occur fashion attain reward dramatically run algorithm nine phenomenon arise single generator opponent omit discuss base action frequency strategy interest right necessary strong record stability always strategie successful criterion detect criterion stable match stable self match play difference tend small resemble particularly self instability stability test change stability match step indicate produce good stationarity always profile occur discrete behavioral state generator different generators difficult rare vast find stationary run game equilibrium nash equilibrium convergence converge run pareto optimal nash pareto nash frequently dominate equilibrium likely pick left equilibrium equilibrium find whether converge pareto dominate non algorithm converge nash play converge nash surprisingly game nash look qualitatively generator likely produce equilibrium optimal ne converge receive play self failure high equilibrium use play equilibria equilibrium improve play maintain exploitation aim stage equilibrium goal convergence generally correlate obtain high proximity game generator correlate close equilibrium furthermore generator notable especially reward converge equilibria algorithm play repeat repeat payoff achievable repeat game profile determine profile game necessary agent meaningful payoff profile repeat nash equilibria build achieve payoff game nash examine consistent game equilibrium overall repeat worth profile agent agent like standardized researcher experimental implementation conclusion idea modern would repeat game environment considerably suggest area effort experimentally drive many include performance game size detailed investigation generator algorithm game line sophisticated differently partly explain set setting match important nash equilibrium equilibrium equilibrium also third dominate algorithm portfolio exist promising empirical see portfolio switch different opponent portfolio algorithm portfolio empirical play track situation portfolio improve acknowledgement thank stage project feedback anonymous helpful suggestion david coding provide appendix metric seed uniquely instance could either match play quantile avoid formally sample yield sample yield sample estimate sampling equation claim point albeit pt exist many justify term guarantee little empirical claim literature experiment new tool design facilitate remove baseline implementation many test equilibrium confirm piece conventional discover surprising agent outperform algorithms road system system see system analyze algorithm environment prominent example well high approach algorithm qualitatively reinforcement learn fundamental difference classical reinforcement learner policy attempt identify environment policy opponent way opponent action opponent conceptual hard claim algorithm tend aspect intend stand game nash equilibria case self regret performance basis property ability achieve generally compare comparison literature g newly design survey landscape consequence small consider make overall centralized standardized exist centralized public decrease achieve difference implementation publicly implementation offer easy reproduce platform run platform several advantage hope facilitate analysis offer connection explore sophisticated extremely competitive performance rich investigation compete distinct know reward reward opponent action nash computationally expensive game survey experimental evaluation describe broad disagreement aim order access reward opponent costly game property able compare capable upon choose player repeat restrict instead game reason game experimentally interesting play two thus mention aside generalization probably game repeat essentially potentially mixed opponent collect count opponent iteration need response assume payoff converge equilibrium game property see eventually extend way see provide known asymmetric game randomization remain problem solve adapt signal name simply adapt strategy game nash equilibria maximize play idea play variant instead action payoff nash equilibrium move simultaneously move instance action equilibria lead equilibrium every reward costly game useful like modern focus class opponent attention updating behavior assessment track opponent try play nash implementation equilibrium issue act fashion depend simple strategy style play nash history opponent action period stationary agent exceed security level situation portfolio algorithms style mdps opponent discount encode step possibly decay strategy straightforward adaptation single agent opponent payoff eq essentially opponent part stationary entirely work environment explicitly idea function learn profile mix calculate eq q sensible aim payoff worst note game actual fail reflect security modification play game nash something similar except choose equilibrium use observe action reward necessary ascent maintain mixed payoff updating depend high action opponent environment gradient learner make strategy compare strategy strategy make perform guarantee regret limit great nash action version opponent use proof convergence experiment merely assume reward gradient start adaptive payoff mean unlike update current nash equilibrium therefore complete computational power one nash equilibrium additionally self class game way ensure unconstrained probability map simplex reduce less aim primarily use scale performance algorithm survey repeat consider include mixed simulation see table additionally test play varied version version great baseline seven investigate come small game version limitation partly create diverse instance game generator generate set instance recent paper finally substantially iteration range repeat burn period phase algorithm behavior record final generally code tailor consequence instance special new experiment spend run call available platform player repeat game experiment include version introduce version programming matlab variety gain overall hope researcher repository setting upon game stochastic work hard make add list engine term order pair two asymmetric payoff player payoff column pair concentrate generator generator instance payoff order generator heterogeneous example distinct call play performance due randomization hold pair characterize algorithm play different case characterize empirical run iteration feedback reward action choice opponent mixed strategy separate platform step platform three configuration engine piece visualization turn algorithm must pick must step generate job desire match reference game file file job generate primarily engine implementation nash equilibria job cluster facilitate job create sample action reward receive metric file plain file specify calculate metric batch across cluster analyze visualize result make easy validation implementation action action singleton well otherwise break action still belief unit virtual count ne break high opponent see repeatedly nash high equilibria opponent equilibrium tie one pseudo code largely use game encounter nash game need decide would play equilibrium compare computationally expensive pick equilibrium implementation implementation implementation involve ten instance agent show quality use kolmogorov game dominate implementation e reward track however considerable implementation code unable difference period tt security stationarity threshold switch window implement pseudo distance even norm follow pseudo table step set draw original iteration simulate compare map operation distance variant assume play arbitrary know reward implement experimental produce formula eq equation play vector reward table suggest resemble update reward vector sample control parameter give show use poorly setting schedule discount keeps perform probability decaying drop period decay end discount arbitrarily future action take solve linear strategy see removal preprocesse check exploration discount baseline uniformly action platform conduct scale performance metric setup statistical algorithm setting nevertheless
dominate median c sec sec sec run appear decrease linearly ex objective plot residual sparse bilinear real world bilinear sparse bilinear multi test performance bilinear eeg competition concern eeg record arm presentation hand hz mark choose temporal slice train tune tune select grid prediction regression logistic regression sparse bilinear show accuracy roc roc tune regression logistic solve logistic fista observe logistic bilinear well bilinear logistic logistic logistic bilinear regression class one eeg cognitive subject three category try decision record hz use channel hz table consistently three bilinear outperform regression bilinear logistic far improve benefit bilinear content various camera image camera record plane image camera logistic bilinear logistic bilinear logistic regression comparison sparse bilinear bilinear sparse bilinear achieve bilinear video visual video dimensionality histogram descriptor figure illustrate vocabulary codebook sift descriptor frames video construct histogram codebook technique performance side side front pick discrimination ex logistic bilinear regression sparse bilinear bilinear regression achieve good bilinear algorithm method reveal convergence bilinear regression dimensionality traditionally curse component ica commonly preprocesse bilinear reduction logistic logistic regression ambiguity bilinear lead feature spatial importantly complexity improvement challenge minimax bi nature bilinear introduce bi boundary still bilinear follow factor form write interpretation bilinear bilinear equivalent due critical bilinear spatial difficulty pose bi convexity bilinear extremely guarantee empirically bilinear boost generalization binomial bilinear logistic generalize multinomial hyperplane model py n ic minimize set sample form multinomial bilinear take eq r corollary section xu department electrical introduce concept logistic explanatory common vision brain computer style factor study coordinate descent theoretical inequality sparse logistic history vision bioinformatics gene sparsity introduce logistic curse dimensionality explanatory correspond informative therefore lead logistic attractive logarithmic complexity bound recognition task feature transform histogram result histogram base eeg channel bilinear explanatory two generalization standard logistic learn bilinear learn boundary show logistic outperform logistic include visual apply bilinear outperform dimensionality principle pca bilinear logistic regression content separation improve nuisance orientation bilinear identify informative feature nuisance thus generalization bilinear contribution lead interpretability result sparsity bilinear demonstrate classification three fold first propose bilinear behind logistic descent solve numerical bi nature conventional block coordinate solve subproblem proximal provide estimate rate bilinear logistic improve generalization classifier various task convexity bilinear consider give label explanatory categorical variable seek logistic transform explanatory pp x category binomial illustrate idea h assume class conditional empirical function logistic decision assume laplacian reduced minimization problem regularization form logistic regression multinomial regularize development efficient lasso ce lars fista bilinear bilinear regression preserve explanatory construct block coordinate follow subproblem coordinate summarize choose accelerate proximal stepsize specify independent subproblem b proximal descent summarize solve reduce subproblem subproblem convex solve dimension large elastic term close component computationally algorithm attain sufficient typically choose b continuous dependent gradient constant straightforward calculation b n f eq cauchy schwarz therefore dynamically integer let integer inequality constant inequality update allow namely small previous define global bilinear logistic regression asymptotic establishe method update iteration result convergence follow accord bound least solution subsequence stationary give observe subsequence pass say satisfy limit subdifferential f converge boundedness intermediate assume
multidimensional reliable estimation shape check effect hour slice explain appendix period day surprisingly law illustration display thing conditional law behave long flow law roughly behave exponent law kind kind time believe conditional kind appear reaction event law wiener second comment intensity intensity interpret source represent dimensional simply h c c intensity ratio low percent directly event theoretical process one mid price jump event ratio provide mid one involve external market limit order surprising show thus limit order market order order mid price occurrence asset discuss precise kernel comment self book simplicity blue correspond interpretation quantity positive linear lead reliable estimation even bid recover empirically matrix shape homogeneous input input stock index expect discrepancy side jump slightly jump market negligible anti shape plot order attribute less diagonal kernel roughly law decrease mid jump behaviour price guarantee absence correlation price agreement link change influence price change dynamic else come impact price impact recall impact event type wants cascade whose precisely introduce proposition correspond event number display ij price mainly cause jump indicate price mid asset price move mainly move price asset variation event former study estimation involve order flow link cause splitting order description focus kernel market mid high display kernel appear behave exponent high kernel loose fit range though display stress behavior kernel rich law fact clarity kernel plot represent normalization dynamic process indeed proportion flow move mid price appendix one localize delay ask price move market average delay law scale negligible order appear localized price move opposite bid conversely move order change order bid oppose order order market order almost order significant order impact price limit order kernel impact jump flow correspond normalize mid order ask contrary effect far bid ask selection price want execute order ask bid interesting execute bid move move bid limit ask ask order bid order localize average long short second stationary state time price occur kernel correspond impact normalize influence order price kernel negative stationary far order time influence become proportion see impact proportion concern reverse impact limit flow flow kernel resp bid bid order trading rate resp bid flow correspond display kernel market dominant also short influence limit localize order law correspond influence order numerical wiener kernel behave significant rather dimension estimation natural interpretation financial micro allow type first occur book retrieve event impact influence allow rich influence type subtle event example limit order localize time thing probably purely event directly book spread arrive side spectrum book mention markovian flow believe reality approach market book stability increase improve different account finance field measure acknowledgement support market finance growth laboratory universit ed aim empirical choose grid l affine grid one good estimation scale one law scale grid point represent result h b b proposition remark modify simulation least decade trade limit relationship mid process estimation order impact price formation challenge modern financial agent stock price execute limit use limit order execute formation book market large anonymous great practical theoretical book intelligence purely book markovian book simple asset price impact market order recent account book work share joint price influence mutually adapt count example successfully domain finance arrival order variation level impact account price variation book causality concern finance lead law decrease exponent slightly wide range scale capture book improvement illustrate us kernel scale recall property multidimensional explain principle adapt slowly book correspond front comment event concern work conclude process briefly intensity jumps intensity matrix simply stability kernel one admit increment remark stability remain equation lipschitz interesting satisfied describe statistic satisfy covariance intensity dirac average q notation time see e build arrival birth appear individual type individual type type appear proportion since jump process also law measure link proposition wiener law equation wiener structure autoregressive see counter equation wiener admit unique respectively average covariance use wiener matrix definite theorem set satisfie thus thus try wiener course solution necessarily interpretation point wiener system predictor square contrast proxy explain numerically wiener nystr method realization fine enough grid time interpolation quadrature wiener quadrature get system appendix estimate use display display theoretical perform kernel section quadrature follow thus three empirical kernel quadrature theoretical green dot kernel instead error causality direct interpretation average cause try give intuitive method quadrature thank change quadrature quadrature captures vary quickly bad around previous quadrature quadrature capture let time grid adapt quadrature scheme consist piecewise hypothesis k k k kt quadrature empirical integral linear law quadrature procedure perfectly quadrature point estimation quadrature step curve correspond blue theoretical kernel quadrature perfectly match theoretical green dot blue describe wiener quadrature obvious inverse point use empirically model account event various type arrival future event classical impact g accounting impact market dynamic price market extend multidimensional event
fluctuation motivate distributional conceptually suggest additive shape way modify relatively consider version model state high decrease burden associate smoothing consideration product state incorporate model gray series regression covariate control building b spline additive predictor common generalize model feasibility approach demonstrate simulation exchange spline model instability pattern employ vary finitely control markov refer markov regression seminal paper state residual chain state stochastic serial persistent regime active long regime classic economic effect explanatory economic simple glm exist relationship form little investigation goodness predictor build strength hmm evaluation spline obtain arbitrarily flexible functional estimator target penalize generalized validation select control goodness smoothness model include parameter comprise possibility reduce case conventional parametric switching nest flexible consideration switch additive ms simply time parametric subject regime restriction decide present consideration identity link density dependent structured formulate describe efficiently spline functional predictor approach potential conclude switch nonparametric target interest value denote chain finally distribution exponential covariate link canonical family link via map state essentially model use shorthand additional depending specifie whereas distribute specify dispersion parameter conditional probably popular dependent model assume homogeneity desire probability usually stationary switching underlie chain dependence covariate efficient importantly irrespective efficient forward variable generic symbol recursive derive analogously comprise numerical large moderate notably density switch regression g predictor glm concern predictor comprise function one state express finite combination represent simple numerically piecewise fuse together smoothly cubic use spline determine flexibility basis function allow curvature adjacent spline need increase basis long impact penalty second integrate ms characterize function dependent reflect smoothness parameter increase emphasis smoothness parameter dominate lead straight line similarly difference nest directly advantageous functional way allow obtain constrain drive choice model observation clearly powerful software already give determine markov chain spline predictor estimate maximize calibration remain constitute calibrate calibration treat datum forward subsequently validation assess calibrate convenience fit stage treat calibration pre meaningful experience computationally alternative intensive aic calculate aic likelihood denote freedom product information inverse fisher penalize freedom result penalization smoothing consider one distribute target markov regime covariate functional display dash curve function go covariate run ms link fit optimizer spline implement cross choose smoothing fold validation integrate q estimate report aic f use dash green state transition obtain monte shift fairly value target variable chain display figure go draw run choice lead marginally counterpart aic criterion straight notably parameter value case curvature smoothing scenario dash dependent predictor leave panel predictor monte shift estimate encouraging lead overall scenario encourage ms clearly occur form circumstance identification induce run exact change autocorrelation series modify fairly autocorrelation bad run fail overall wrong chain reflect mean probability sample value use scenario dash red datum collect exchange financial make assumption relationship two also probable motivating nonparametric predictor function illustrate flexible ms analyze predictor lin ms lin nonparametric meet restriction mean htb lin pass region around residual switch formally ahead fit lin ms lin ms forecasts ms ms lin result
variability behaviour epidemic frequentist infection epidemic model individual kernel via removal infection period meanwhile infect epidemic parametric survival contact become infection removal removal model dependent author adopt find homogeneous removal rate affect removal contact focus infection enable assume removal within involve assign wherein use chain monte epidemic population three epidemic time epidemic bivariate transition infection removal transition sigma process population process become remove period play epidemic end population period infection removal assume removal epidemic end infection individual denote infection time density proportional left limit purpose least epidemic infection removal rate temporal disease consist removal time think infection integral high dimensionality trivial nature overcome miss become intractable choice augmentation infection inference infection markov monte relax parametric force infection proportional knowledge augment priori time infection removal exponential remain infection discuss formulate form total pressure course epidemic trivial compute priori poisson condition distribute prior alternative priori follow nd continuous knot I ht b spline order function recursively assume interior even need sufficient condition fix infection epidemic assume cover full spline coefficient much purpose unobserved infection carlo conditional n reversible jump hasting use assume make three update birth death change poisson rate maximum interval new ratio jacobian death probability inversion probability propose birth death take repeat improve mix move uniformly accepted q change height height distribute q priori previous mechanism update acceptance birth death change height acceptance prior j death probability change acceptance height eq involve infection initial infection infection infection propose exponential infection infection value infection accept update infected infection minor assign nd interior knot infection infection time propose infection infection time accept illustrate propose dataset generate mass epidemic start among uninformative martingale prior induce case around may explain infection slightly notice close infection show instead set offer spline dash line percentile infection curve dash percentile posterior solid infection curve modify vary contact rate epidemic start population number fit parametric dataset informative although spline work obviously action infection percentile posterior infection spline hour hour table typical fit dataset removal day mass action several parametric infection initial infection mass run quickly infection rate simple fit spline severe consist population track number epidemic sufficiently remain epidemic martingale fig day removal spread similar length period day dataset want death removal individual period infect tendency uncertainty begin epidemic spike spread super event infect patient cluster explicitly could indicate curve infection per flat fit infection curve peak middle infection may power large modelling could intervention intervention spread epidemic epidemic assumption instance incorporate place essentially
conditional hide visible parameter visible small conditional feedforward joint top feedforward feedforward network intend feedforward feedforward layer hence feedforward independently apply nonetheless resolve able marginal bottom compound feedforward pass represent namely marginal feedforward feedforward resolve marginal order feedforward transformation way bottom marginal desire tune depend bottom require arbitrarily bottom account fraction distribution regardless second approximate element strictly arbitrarily approximate distribution ss prove subsection study feedforward put proposition arrive approximate arbitrarily bottom marginal arbitrarily layer interesting feedforward define feedforward approximate probability visible unit arbitrarily feedforward ns approximated obtain mean rbms call adjacent every bottom marginal pair feedforward layer previous paper well tool flip state invert approximate intersect along p np n describe length string entry form existence condition argument conclude material paper deep feedforward develop advantage vs undirected architecture seem undirected train initialize universal narrow thereby intuition verification surprisingly long address rbms narrow feedforward counterpart narrow compositional present trick activity part regard feedforward network pass high multiplication feedforward layer acknowledgment grateful institute article th l l k k k l z p kp k l p hold versa map output assign unit take state unit divide visible input output give relation implication note universal map deterministic regard map universal map rbms discuss input joint distribution would interesting corollary softmax main article hold unit feedforward layer value formulate case layer state minimal universal narrow bottleneck narrow visible hide fact odd follow visible conditional approximate mixture mixture assign string one odd even without detail imply odd conditional string odd even visible narrow direct towards bottom except layer state layer although layer narrow stem essentially feedforward layer hide exactly kind exploit rbms rbm rbms provide minimal hide rbm visible narrow minimal sufficient form rbms narrow interaction weight restriction interaction weight backward activity product arise input model pass feedforward desirable obtain universal detail help understand take close look future pt pt pt bp pt pt pt pt bp pt pt pt pt arc bottom double white mark width pt theorem theorem institute mathematics sciences deep narrow boltzmann machines universal many visible layer within boltzmann feedforward depth various undirected show narrow boltzmann compact narrow sigmoid restrict currently available power neural compare network compare undirected direct connection respect represent reach endowed unit refer universal property various feedforward feedforward deep narrow undirected architecture problem prove narrow boltzmann universal layer visible machine undirecte boltzmann boltzmann machine whose pair interact layer visible unit conditionally illustrate appearance practical especially regard node line black sep fill bl dot cm right dot node transform circle draw black cm fill bl dot dot scale shape draw sep bl dots node distance cm dot transform shape circle line black inner bl dots distance dot h v cm distance right h node l transform shape transform circle pt black sep cm bl dot right dot scale circle black fill bl dot dot line width bl distance dot pt black inner sep cm fill bl dot distance dot right node distance h distance right h right h leave left style shape transform pt version rbms exponentially organize fix narrow result section compositional probability feedforward share section elaborate study perspective present trick distribution follow feedforward universal boltzmann probability l l l unit exponential embed strictly assign bottom layer visible unit right panel figure restrict boltzmann visible set distribution top panel provide rbms universal distribution kullback visible unit enough precisely implication property remain unit input output visible minus layer universal unit universal unit unit softmax unit unit visible next compositional feedforward look compositional composition wise hadamard hadamard distribution definition hadamard product element natural fs gr g style circle bl scale circle width minimum size fill bl dot right end dot draw black sep size cm bl cm v
vector latter plane meet plane intersect must q happen consideration couple interval coupling conditional simulate exact determine certain time step associate calculation section time interval method distribution bridge association compare copula mu matrix complex invariant matrix reversible symmetric obtain symmetric reversible covariance matrix invariant summarize ergodic time distribute bridge expectation argument give simulated process diffusion euler scheme satisfy first diffusion approximate distribution compare mean bridge level copula two distribution copula distribution two simulation bridge since bridge fit exact mcmc distribution approximate bridge therefore nice diffusion diffusion unlikely bridge rarely rarely likelihood approximate copula lemma compare marginal distribution simulate marginal level copula compare copula curve plot compare empirical copula copula draw q marginal dimensional approximate exact marginal curve copula exact draw exactly distribution bridge bridge fit marginal metropolis hasting essentially example compare copula time produce metropolis run compute second generally compute generating metropolis varied dimensional hasting exact curve copula dimensional distribution compare exact compare copula time alternative rejection produce large output algorithm plot empirical marginal produce alternative level copula full diffusion bridge diffusion compare exact time approximate copula dimensional compare extreme bridge bridge produce surprisingly bridge fit excellent tend become tend mathematically generality marginal level copula dimensional full draw restrict simulation bridge simulation give exact whether value burn bridge expectation geometric conditionally bridge variance constant reciprocal simulated slope check conclusion variance ratio deviation vary unlikely bridge bridge ratio simulation diffusion close however show g bridge eq contribution bring close contribution therefore approximate start deterministic unlikely bring depend likely constant bridge mainly contribution depend end bridge good bridge randomly draw multivariate characteristic multivariate third diffusion ergodic reversible suppose set partial observation full path draw need simulate continuous path conditionally diffusion path likelihood expression integral apply detail q exponential family distribution normal bridge sample two run sampler sample method interval posterior obtain draw introduce wiener wiener pair span obviously function clearly dimensional wiener process projection orthogonal complement wiener eq wiener independent wiener characteristic v quadratic characteristic I td give event event bp tb transition bridge establish calculation conditional trajectory joint density wiener wiener dominate marginalization lemma straightforwardly multivariate give start check z bt nz nx formula g apply mu grant national foundation two grant university cm lemma theorem corollary remark de sigma mathematical university mathematical bridge fundamental role inference novel coupling generalize variate set first accurate propose proposal exact diffusion applicable work length simulation usefulness multivariate inference coupling inference stochastic address mail propose applicable multi diffusion bridge motivation play fundamental role simulation include inference diffusion diffusion volatility end state start start time go process equal start diffusion meet suitably make often tend infinity obtain apply couple generalization two independent ergodic dimensional intersect go application coupling efficiency meet diffusion bridge repeatedly two euler implement diffusion process bridge algorithm diffusion pseudo rejection bridge simulate new bridge diffusion bridge exposition thought sampler try rejection acceptable rather meet bridge literature metropolis hasting proposal distribution force go diffusion brownian boundedness relatively complex method simulate path spirit advantage ergodic understand transform equal transformation refer transformation multi variate exists rarely require exact important advantage particularly length time interval bridge surprising order time interval coupling process bridge apart bridge publish work interval note mainly short follow challenge approximation estimator time long accurate density approximation kolmogorov pde numerically expansion alternatively simulation approach back seminal incomplete datum continuously process observe continuous miss either em sampler continuous path simultaneously realize several base bridge simulation several author bridge crucial diffusion time observation simulation inference possibly functional cover volatility crucial observed measurement idea bridge process simulation approximate approximate proposal diffusion improve solve point meet two point know approximation distribution diffusion except extremely surprisingly couple usefulness observe consider briefly diffusion stochastic wiener coefficient function dd regular ensure strong solution ergodic invariant measure invertible specifically conditional call bridge bridge go start start intersect meet bridge equal approximate bridge proposal bridge couple sense diffusion equal subsection bridge question correspond initial u equal distribution bridge process variable wiener play bridge depend depend start let initial base theorem simulate simulate diffusion reversible simplify euler increment drive increment wiener discretize simulated method wiener process simulate approximation bridge rejection keep copy couple whether coupling coupling interval define probability detect usual considered lemma hence reversible follow time reversible diagonal simulation usually diffusion proposal exact bridge bridge diffusion bridge continuous induce probability bridge dominate bridge differential e give drift bridge end one carefully must correspond similarly bridge diffusion bridge diffusion wiener intersect definition approximate bridge corollary give bridge b random wiener associate bridge associate equation give expression quality diffusion bridge distribution bridge crucial detail usually time simulate time euler discretize z increment simulation equal wiener increment process meet equal hasting marginal bridge bridge proposal accept rx diffusion mh produce diffusion simulated type idea replace ratio estimate chain bridge draw irrespective randomness simulate index diffusion result draw mh go diffusion bridge conditionally xx path simulate independently simulate conditionally value rx I exact
fast perform slightly detector outline subsection use integral detector slide improve adopt classifier arrange adopt approach detector pass gradient speed classical slide object paradigm discard patch propose selective order magnitude detector patch underlie idea detector generic detection modification original detector resolution aspect bit integer integer use training detector aspect scan per sp consist level statistic exclude co discriminative exclude coefficient co channel color carry parameter weak classifier shrinkage adaboost final bootstrapping final cascade soft cascade rejection node detector train table beneficial pooling increase robustness percent sp bad sp bt sp sp pooling sp original descriptor eigen learner propose feature train learner learner highly compare original descriptor report sp significantly detection original covariance window sp fair increase combine result sp table sp test bad par set combine detector sp sp sp display illustrate ensemble radius angle angle draw uniform asymmetric testing baseline cs adaboost asymmetric adaboost assign asymmetric restrict high partial good asymmetric vertical horizontal train strong classifier evaluate decision asymmetric observe emphasis part cs worse optimize bt marked classifier l svm protein bioinformatic consider protein protein prediction predict interact use type publicly protein interact protein label interact evaluation form weak learner baseline svm asymmetric optimize either attribute linearity detection report tb deviation datum set iteration repeat time average ccc face comparison boost adaboost lda post adaboost train tree algorithm repeat report cross approach choose evaluate digits pixel even digit odd divide scene set descriptor visual histogram intersection manner sub windows face face extract principle component preserve table us mt mt motion indicate head pixel weight learn svm fig b near contour shape detection vision capture camera mid city graphic format fully along axis patch collect care detector resolution feature channel feature orientation sp propose learner depth decision cross validate divide validation bootstrapping around detection greedy maxima apply website curve evaluate usa usa drive traffic pixel scale include medium far start training exclude along expand negative detector resolution magnitude bins sp sp depth tree bootstrappe weak previous experiment pixel height visible publicly compute auc experimental compare approach exist detector spatial visual feature pixel head apply intel detector generation place detector set svm show pixel white human contour weight detector vary detector stage vary region plot roc detector fig start reduce similar performance demonstrate detector discard compare region proposal stage exclude processing maximum stage usa table threshold result reduce improvement log average value classify fail soft soft low level visual experimental keep number classifier bootstrapping etc rejection cascade various threshold classifier perform bad detector benchmark window table cascade weak classifier coefficient repeatedly rejection threshold increase window time cascade small important come achieve average scan soft average scan scan bt c window avg avg discard per bt cascade avg avg scan approach effectiveness level object proposal generation optimize extensive demonstrate plan scale order acknowledgement centre part fellowship analysis algorithm learner weak learner primal value solution unchanged current another add master solve objective proposition guarantee subscript index project positive j objective reformulate iteration know iteration continue definition simplify objective bound calculated ensemble easily apply computer master interest pattern recognition machine research interest computer study university receive future fellowship van university innovation production medium receive law science vision fig eps fig corollary many object detection operate prescribe situation detector basis area roc full label partial curve method positive directly optimize partial auc object detection low spatial pooling spatial robustness detection structure spatially pool report usa boost ensemble pooling gain great attention past decade topic vision visible give gain set progress decade area due computer surveillance interaction difficult appearance author evaluate detector boost apply promise recent success pooling feature type train commonly evaluation compare operate roc illustrate classifier system threshold human researcher false area characterize need world vision fig positive would preferable moderate positive researcher report partial area range name calculate roc curve specify summarize detector often optimize approach ensemble area range area calculate accord upon propose ensemble classifier directly auc boost algorithm predictor ensemble mechanism pass unlike place emphasis incorrect ordering optimize partial auc summarize novel extract low spatial pooling spatial code generic classification descriptor optimize curve method wide roc proposed term tight approach share conventional differs method optimize multivariate structure conventional boost visual transform efficient training cut plane solver auc directly optimize partial auc arbitrary good detector experimental set effectiveness propose new benchmark work cascade optimize rate visual spatial single structured tight proposal generation evaluation detector careful compare report detector several detector part convnet mutual propose newly benchmark reader excellent framework section briefly cover recent work consider pooling vision system state performance benchmark scene imagenet method extract visual summary patch window feature sub pool form generally spatial recognition convolutional achieves scale max layer form invariance spatial matching sift significantly outperform linear pyramid max statistic pooling mean value process selective image descriptor boost recognition dictionary know computer vision machine miss negative exploit weight heavily report classifier validate parameter order cost sensitive boost boost cost descent address need carefully false positive several directly optimize bioinformatics modeling boost develop optimize exist building ensemble optimize criterion range knowledge principled optimize auc difference structural ensemble recently detector detector instead auto automatically hierarchy multi scale capture detail shape component result detector benchmark level aspect author low processing image major benchmark improve performance detect align frame camera motion feature five fold reduction false positive well apply object work capture favor cast cutting assumption quite form histogram texture descriptor descriptor texture adopt filter binary contain transition vice versa pool descriptor pool common strategy use window patch feature pixel within translation spatial pooling pooling covariance pooling pooling summarize matrix pool region refer extract spatially pool extract compute efficiently trick sp descriptor pool ignore geometry matrix stack upper pooling simplicity carry image rectangular likely normalizing descriptor patch whole detection coefficient correlation coefficient return gpu extract sized pooling implementation sp scale patch multi enable capture human body part patch pooling pixel experiment pool sp divide window patch extract histogram frequency occur well translation perform spatial histogram pooling region spatially pool sp feature implementation neighbourhood pixel extract histogram sp patch pixel although pooling differ unsupervised problem instead encode pre train sp pooling extract remove encoding advantage conventional much word generic visual word thus classifier computationally infeasible structured learning review concept ensemble build auc auc equivalently minimize auc false write train sort negative sort negative obtain j sample empirical compute rank prescribe zero adopt consider pair label j q pair instance rank consistent define ordering produce optimize score summarize ensemble project learner boost optimize false rate vector learner assume train j include new write dual variable lagrange generation ensemble condition apply column generation duality kkt optimality restrict work globally hence subproblem weak learner
mode ex acc mode current scheme simple superior cm lemma prop thm corollary definition section conjecture thm electrical computer california false cluster centroid mean shift ideally centroid pattern representative datum nonconvex algorithm mode combine assignment estimation centroid estimate shift encourage assignment nearby separate centroid find estimate challenge manifold nonconvex centroid representative unlike predict soft assignment representative assign besides meaningful centroid valid input space representative challenge nonconvex digit represent nonconvex pixel digit image digit mode bandwidth require mode valid digit shift centroid regard centroid valid minimize non euclidean typically slow besides often noisy c remarkably binary assignment high representative cluster kernel centroid representative nice outlier disadvantage use centroid find handle nonconvex shape manifold unlike shift nice property idea modify rule become much give alternate mode nonparametric like shaped cluster shift spectral give datum achieve work minimize assignment bandwidth gaussian define kernel estimate kde apply iteration start datum mode initial converge mean round low image segmentation kde centroid create singleton mode computationally mean shift per desirable force one mean mode sum kde separately alternate fix l discrete constraint fix optimization maximization z nk proportional kde truly outer mainly objective leave unchanged local loop step obtain optimize indicator interesting objective difficulty approximate mean partition process multiple optima laplacian qp also combination nonnegative nmf basis coefficient produce part term regard objective coefficient directly cluster directly optimize optimize rescale however multiple solution relate rotation straightforward procedure approximate data cluster walk divergence affinity matrix point move augment stochastic laplacian mode laplacian mode obtain issue equivalence onto simplex mode handle complex shape idea cluster g near heat add nk nn nk trade sum assign nm mn kb nk graph laplacian constraint assign assignment mode control kde note case become mean point code first force assignment purpose intermediate call alternate take concerned second therefore identical cluster mean solve separately step laplacian semidefinite problem qp standard qp provide algorithm lipschitz gradient proximal first project counter nesterov identification smooth objective indicator simplex onto program fortunately efficient initial ks mm accelerate projection laplacian power stepsize determine right laplacian acceleration maintain auxiliary improved iteration neighborhood construct number nonzero account account project onto simplex qp per independent step despite sublinear clear costly implement kde leave unchanged step accelerate gradient counter well step convergence threshold use iterative number precision cost empirically moderate efficient inexact run algorithm valid assignment laplacian objective optima nonlinear term homotopy homotopy follow laplacian mode homotopy optimum scenario algorithm hyperparameter automatically number cluster often uniquely mode intuitive hyperparameter membership smoothness value neighbor kde near well kde bandwidth neighbor hyperparameter homotopy slowly good minimum geometrically computationally supervised section problem efficient solve avoid drop follow quadratic program follow q projection onto dominate cost since laplacian average neighboring training assignment nonconvex kde term centroid distinct rule give assign nearby point assign define mapping useful mean iterate another solve alternate compare laplacian mode popular shift clustering mode valid valid nonconvex yes yes yes yes yes yes mode laplacian mode demonstrate laplacian smoothing fig consist point denote partitioning assign nonconvex shape achieve differently mode even mode centroid lie build near neighbor heat weighting laplacian perfect one centroid step show show colored mixture color contour kde red cluster manifold kde localize shape density soft assignment flexibility kde point vary kde shift contrast mode major mode kernel small small centroid achieve laplacian throughout explain spectral nonconvex point around perfectly plot mode problem cluster know create merge heat weighting mode homotopy hard centroid kde outlier perfectly area density plot color colored color blue assignment centroid boundary assignment fine grid use combine nearby centroid complex cut mode l normalize spectral intensity cause connect intensity nearby heat width kde laplacian prediction background negative fig narrow mode fix much segmentation
discriminant channel locate max low short connectivity carry tendency stay matrix improve fusion element match improvement approach ec compare general obtain perfect ec fig optimal channel ec match fusion highlight symbol channel pair connectivity notice rapidly vertical plot pattern short line single discriminant reveal symmetry ec genetic eeg activity thus highly specific role distinction influence environmental factor comprehensive analysis report detailed table obtain identification short connectivity characterize brain notably range connectivity file superiority long connectivity volume effect coherence measurement remove important interaction generator study volume depend electrical subject structure regard eeg instead trait exploit recognition claim recognition performance volume spectral supplementary file sum herein identification subject consider head match level outperform eeg notably present scenario eeg stationarity wavelet mahalanobi possible reduce polynomial regression classifier third fusion sensor place good sensor electrical contact come technology computational spend min computer aim element validation l f tp fc cp tp pz f f fp f fc fc fc fc fc p pz ec investigate brain past grow trait signature etc eeg measure despite classification performance recognize cause rely method extract eeg signal majority method consider brain account point brain specialized area continuously exchange information stable present exploit eeg sensor exhibit strong class invariant specifically fusion score level approach number subject record eeg obtain coherence improve performance standard notably recognition close open consider eeg region eeg base although connectivity much attention probably eeg eeg purpose automatic receive extraction feature activity brain spectrum possible temporal dependency generate extract brain couple spectral connectivity different subject ec condition notably ec integrate ec power take suggest connectivity effective improve eeg eeg spectral coherence fusion eeg provide related brain trait point isolated attempt discriminate people electrical brain activity perform community investigation eeg trait potentially eeg protocol purpose automatic implement range open ec state advantageous subject eeg reduce occurrence eeg activity range hz share support cognitive furthermore eeg genetic eeg activity effort efficacy recognition recognize protocol study eeg single complementary information dependence activity area region exhibit coherent activity suppose key organization tool statistical brain different principle causality frequency domain capture tool allow brain connectivity interestingly specific connectivity hypothesis eeg feature recognition compare eeg activity channel amongst connectivity subject obtained reach maximum cross correlation mutual regard eeg describe content frequency change eeg subject occur technical contrast univariate power spectral bivariate connectivity sensitive amplitude change eeg fact signal therefore play critical overall classification intra subject scale issue eeg tendency power spectra less include priori parsimonious certainly justify eeg technical consequently integrate element beneficial aim robust brain integrate score obtain spectral coherence coherence exploited distinguish epoch characterize psd element study functional calculate frequently intuitive coherence quantify level frequency channel coherence frequency acquire channel respective maximum psd improve eeg range epoch characterize feature approach identity observe belong analysis assume vector transformation logarithmic psd mahalanobis epoch consist simplification equal pool merge remove normalize normalize template represent assess cross remain epoch perform identification phase framework mahalanobis distribution accord class pool misclassification confusion recognition instance eventually run percentage psd connectivity separately channel channel pair group correctly try complementary activity brain brain activity suppose subject fusion score sum element channel misclassification evaluate describe n forward retain sort accord subset retain remove code score fusion find supplementary file leave element specific step relate performance compare
repeatedly pick expand discover stage algorithm regression oppose feature selection intuition scenario beyond product simple illustrate underlie progress towards heuristic rigorously analyze despite guarantee follow variant support current budget magnitude stage stage hence degree exposition total enumeration degree enforcing ensure case fall minor overhead good case succeed rapidly since really address implement note support henceforth henceforth conduct parent keep track scoring element parent traversal reasonable empirical evaluation assess large end associate repository execute good quadratic expansion dynamically store read disk consideration deal dynamically map bit online always feature large marked parent parent zero choice upon recursively expand parent product base example learn heuristic collection medium publicly challenge uci repository common resource tune rate reliable apart hashing square evaluation regularization leave aggregate diverse refer algorithm among baseline ratio aggregate various baseline plot cdf dataset entry plot replace heuristic show set dataset aggregate table figure relative large uniformly dominate statistically inclusion baseline example key baseline setting relative relative overall quite baseline exception extremely improvement slight cost dataset performance expectation finally implement version building repeat build run approximately example roughly task pass average baseline prohibitive intermediate per select promising epoch set union parent locally find across parent pass average base maximally expressive roc auc highly report albeit cost finish dataset c like generic statement behind neither make upon upon derive satisfy convex respective strong smoothness existence suggest progress distance core display analogous smoothness boundedness wolfe last whereas naturally iterate incur extra parameter desire smoothness establish induction adopt grant case complete simplify eq simplification choice provide simplifie desire within algebra information way expand lipschitz non dataset problem news binary binary census hard binary target letter binary three bar performance baseline present tell time despite competitive much average example coding include baseline view baseline color baseline b bb proposition comparable describe base representation design experimental show tradeoff ability compare observe simple superior possibly question execute large I baye offer simple real large scale computationally achieve superior start learn explicitly adaptively add high order interaction learn guide increase power baseline gram approach appeal avoid additional negligible improve baseline marker coordinate dataset bar algorithm heavily influence consideration computational fail adequate outperform aforementioned compare baseline propose give dominant tradeoff ability illustrative aspect amenable regret effectively grow feature exhibit enable nonlinear learn scalable learning starting improve computational resource g batch speed statistical run massive arise boost learner exhaustive search batch algorithm challenge parallelization alternative polynomial expansion employ kernel trick generally nystr suffer drawback implementation typically test scheme embedding product recently create example lead substantial complexity exhibit linear reduction result dense construction information primarily challenge I expansion sound run polynomial batch suffer online variant describe algorithm expansion adaptively define justified epoch receive stochastic restriction order highest low expand k x k regard gradient space coordinate finite proceed expand current order magnitude create care pick track high term grow computationally add expensive statistically corresponding entire update bound loss justify gradient expand bound tt substantial budget opt carefully frequently carefully pick small effectively allow jointly converge stage ask well adversarial unknown restriction place stochastic evaluating implication direct way tracking might epoch immediately possibility rate obtain use expectation conditioning round differentiable convex loss regularization definition fix remarkably dependence unlike sort amongst good
weight figure consist actual weight adjust give rate training example perceptron compute accord perceptron desire weight adjust distance quantum execute step quantum training simply classification follow consist perceptron quickly since neuron weight layer feed network however process miss block superposition set quantum perceptron superposition quantum perceptron quantum perceptron present simulate quantum computer neural research especially quantum quantum perceptron scheme superposition process acknowledgement upon research south department national foundation basic model activation output income classifier several attempt make theory quantum introduce quantum perceptron activation perceptron resource application quantum inspire neural assume state consist feed neuron input neuron neuron govern word step neuron k field artificial intelligence subsequently input compare adjust accordingly image reveal classify respective figure function artificial also multi see application two decade investigate quantum law exploited effort intelligence network approach build relatively influential proposal direct formalism physics namely k replace unfortunately proposal challenge procedure inspire classical k operator provide severe quantum system positivity increase literature remain actual quantum mechanic rigorous exception introduce perceptron evolution quantum idea superposition introduce resource nonlinear classical reproduce device learn resource classical lie quantum perceptron entire superposition block feed neuron activation map quantum perceptron circuit write normalise quantum apply return binary precisely encode j j digit phase indicate big quantum perceptron activation classical quantum perceptron initial n encoding value represent state hadamard superposition x jj j j copy unitary transformation front input add result phase useful represent give quantum perceptron apply quantum fourier exactly amplitude except accurately interested value allow quantum transform require need distribution peak perceptron resolution consequently precision parameter simulation reproduce classical perceptron precision binary digit precision resolution order deviation course consideration necessarily distribute around quantum perceptron quantum therefore precision grow standard deviation consequently increase number neuron perceptron resource multiplication quantum transform
select purpose visualization incorporate cross validation serve stock return relevance choice select consistency five error determine cross surprising observe follow instead model identical explain practice either near boundary nature intuitive sense augmentation preferable classification summarize year material care technology neighbor predictor evaluate neighbor company describe know double triple stock purpose partition value cross error first minimized search visualization minimize reduce nine stop grow forest substantial tree variable dynamic inspection termination however undesirable true force require testing validation fortunately methodology forest essential third decision naive voting create prediction present estimate true validate case fall grow achieve true approach theoretical perspective say justify individual vary second stock constitute stock vary error column achieve formulate l benchmark technology careful reader notice exclude health care discover operation explain individually particularly scoring differ individual performance case health care threshold result return period health care major division market area distinct motivated explanatory say full fail stock belong field algorithmic standard train include execute aggregated nearly factor complete second suggest runtime realize implementation advance gap execution aggregate partition stock accuracy outperform classification seem stock price necessarily certain financial reproduce produce forecast low effectively explanation reproduce near stem model achieve recognize superiority neighbor question concern severe apply methodology nothing l suggest subtle attempt return far beyond train accurately predict learn stock subsequent corresponding neighbor ensemble model confirm hyperparameter prediction subsequent parameter keep involve partition model discover good financial notice instance sometimes exceed achieve remarkably characteristic uninformative remain interval stock interval explanatory power return remarkably advance information case remarkable year financial cause stock ensemble case model forecast stock price far would also train forecasting stock predict stock price subsequent previous result reproduce care suggest stock price financial maintain approximately effective prediction consistently error present price consider denote negative model forest vector neighbor ensemble present explanatory range poor financial recommend explore learn price prediction directly weight boost furthermore attempt representation autoencoder rather rely input advantage represent stock discover preferable able stock return classification daily daily stock return encourage immediately year incorporate phenomena efficacy model author like thank valuable collection general project portfolio trading formulate algorithm field capability identify index prefer portfolio allocation stock characterize consist technical reflect market classification decision classifier relevance machine classifier ensemble reason economic forecasting employ network cluster construction incorporate rate area outside finance addition model augment selection efficacy field include material information technology choose range market circumstance accuracy advance uncertainty outcome historical record motivating behind explanatory stock train price important contribution financial forecasting first recommendation conduct automate allow important force accept explanatory variable attribute parameter time incorporate rank component approximate financial formulate scoring intend undesirable stock capability concept yet hierarchy preference stock return uncertain game portfolio benefit confidence brief classification incorporate intend intend merely identify whole reader delay machine classifier relevance classifier classifier meta boost formalize trading must relatively motivate ensemble model sense fast strong parameterized learner forest nearest parameterize present advantage still maintain ability individually learner subset ensemble decision tree key node split measure entropy measurement class label binary split something probability hereafter almost svm elegant resemble apply ideally linearly text use non support summation label exist augmentation choice radial basis rbf resemble matter principled intercept separate feature machine assume form input prediction essentially analogous bias svm construct iterative optimization process rather take sign sense boost rely nonetheless experimentally learn comparable grid efficacy learner conduct extensive choose subset exclude division use rate record good average meta h total basic exclude share exclude capital total net loss share core capital balance expense total capital per include item core eps core eps basic preliminary core preliminary operating expense common price close high counterpart tune represents intuitively extent linear often feature determine assignment validate train fine tuning work datum service year interval history relevance reflect extent learn circumstance explanatory work index reference description effectiveness financial material technology service discover certain possess divide numerical calculation tend low set successfully partitioning learn explanatory explanatory power vary across capability model
fine tuning epoch validation report combination epoch low number tune da grow layer sentiment review amazon adapt experimental review six domain amazon com whether review reason keep transform indicate presence validation label example label domain mix example number unit train raw obtain yield relative raw improvement da explain da test learn da da yield diagnostic check da sensitive learning method learn work denoise autoencoder heavily coarse grain corrupted fine grain start find result significant boost denoise supervise fine tuning model achieve ever cifar feature level span pixel background portion feature might topic big tv feature contain denoise autoencoder provide autoencoder version corruption parameter affect final digit recognition notice detector obtain detector part digit learn autoencoder tune understand network encourage train schedule corrupt force coarse grain low learn reconstruct fine schedule combination coarse grain grained idea neural goal feature learn learn noise experimentally autoencoder autoencoder supervise autoencoder train supervise fine ever cifar among invariant idea corrupt recurrent neural purpose layer wise representation appear intuition contain corrupt original input autoencoder hide encoder encoder analogy interpretation principal component encoder decoder amount corruption reconstruct corrupt encoder sequence gradient mini batch autoencoder result autoencoder subspace hide large autoencoder error denoise contrast force large indeed denoise autoencoder representation parameter transformation continuous paper common include sigmoid encoder decoder encoder pair sigmoid important denoise autoencoder p common use independently corrupt training level effect learn hand distant autoencoder denoise autoencoder map representation learn another autoencoder stack autoencoder combine good aspect learn denoise aim initial final level da e hold stochastic level take apply meaning mapping encourage cluster centroid conceptually learn network train task less early task achievable actually early high level observe cf fast cf panel start minima easier understand insight da match density noise implication learn hard da learn distribution much smoother easy hard tb c tb evaluate two cifar text data amazon product review material similar learn representation fashion learn quality unsupervised use corruption encoder decoder library optimisation mini batch though good noise performance learn low optimisation epoch datum high cccc cifar da become training learn noise learn initially schedule filter visually detector filter learn detector ccc hide hide learnt mixture various value put optimisation achievable learn da da contain detector feature learn diverse detector learn detector noise yield unit da train result da initial level da necessary investigate network da epoch large use work optimally optimally use use schedule little bad da train yield yield put optimisation observe start training examine whether learn contain explore train da set set representation representation within yielded classifier representation piece evidence hypothesis learn help learn help fair train unit da good confirm learn noise help hypothesis train learn intuitively active activation item total total eight autoencoder da result close feature learn da da da cosine feature da j da describe learn starting find learn da confirm expectation da c level subset define right cm denote corrupted level preliminary even level da perform par standard sgd update perform much bad version sgd extension justify superior ability diverse denoise autoencoder learn autoencoder single exploit global retained level unsupervised help set learn sigmoid
three day pay special measure measure include several commonly test kullback pearson chi express examine interval sample small empirical need important base mean cm r r cm r r consider construct normally take mention family simulation table divergence n see show coverage simulation accordance interval test discrimination kullback explain coverage likelihood test characteristic modify ratio another possible strictly usually happen cc conduct simulation subsection shift shift shift shift observation shift observation follow distribution table introduce shift interest less shift quite non shift pointed statistic robustness difference well shift dispersion higher close nominal likelihood ratio furth e respect shift comparison test agree conclusion obtain cm cm n r r r r r r cm r r r composite hypothesis rest proceed follow introduce devoted estimating equation new parameterization obtain e consequently become know must satisfy solve subroutine compare interval separately since continuous student size see read statistic divergence early section statistic empirical lagrange multiplier obtain e replace consistent unconstrained system take finally lagrange htbp htbp focus two continuous exact probability confidence distribution lagrange multipli statistic comparative purpose test satisfactory power statistic underlie chi statistic underlie read normally theoretical coverage empirical statistic empirical multipli little chi slightly superior empirical practice know empirical read test statistic table distribution empirical ratio test chi coverage nominal seem superior empirical coverage broad empirical composite nan interval think coverage presence shift power base sample insight tend yield interval empirical test sample multi currently work hope finding axiom exercise notation summary proof university department university nan two simple hypothesis divergence carry likelihood construct respective modify test robust empirical test presence hypothesis divergence function ratio empirical powerful currently widely develop introduce paper appear varied contribution inferential functional population purpose statistic testing empirical ratio empirical vector unknown dimensional unbiased essential difference adopt method moment let empirical give I atom maximize empirical log restriction estimate base apply lagrange multiplier subject empirical log function test maximum q furthermore nr fact degree freedom test testing derive refer hereafter divergence statistic nan empirical divergence derive power approximation test illustrative monte carry respect likelihood statistic likelihood contamination propose test composite test conclude remark measure sequel since clear equivalently x test refer empirical statistic statistic generalize important regard main testing satisfy observe know especially maximum asymptotic example assumption theorem eq eq central formula van page equivalently give probability vector I f f restriction consider subject exponential instance subject restriction member et nx e detail valid replace expression empirical estimator share regularity integrable neighbourhood integrable neighbourhood integrable empirical divergence taylor expansion hold accord enyi write present expression ccccc q nan square degree chi square freedom n hx taylor influence vanish van expression q second come previous one nan member reach divergence present reject alternative freedom test explicitly eq order taylor expansion eq thus asymptotic extend f n hx present rejection first less power analysis le attention neighborhood tool e le rely direct contiguous fix asymptotic statistic n degree freedom non centrality obtain contiguous extend e contiguous precede obtain power want take note use well illustrate divergence hypothesis pass laboratory mirror back contain
traffic group partition independently encode entirely another desirable collect overhead bs one replace penalty inside square bs bs represent traffic relaxed solve obtain coincide sec partition belong fully receive central unit cluster head furth latter case bs head traffic cluster proportional cardinality former intra traffic proportional regardless system static cluster appeal thank implementation proximity scenario cell within radius connect locate center mcp user interference help drawback overhead proportional number nonzero bss cardinality characterization suitable regularizer bb special singleton cut vertex comprise bss direct associate give static mcp solves adapt dynamically interest partition tradeoff feedback possibly however regularizer degenerate full inter unbalanced size undesirable intra consider cost size denominator joint formulate even np complete graph aim w note sec optimize undirected graph care problem ratio cut state drop problem directly except q correspond small find row initialize vector eigenvector small stop far bs access full gain nearby bs provide centralized implementation need collect individual bss central processor appropriate feed bss however scalable processor overhead decentralize simply bss bs independently mcp entire clearly bs consideration individual bss bss decentralize static mcp algorithm develop graph bss undirecte twice assume mean node bss sequel extend algorithm finally bs know gain bss start randomly initialize nonzero bb collect decentralize fashion bss run consensus average bs factorization subsequently upon magnitude root corresponding eigenvalue magnitude obtain table execute desire cluster choice ensure definite desire decentralize employ exchange raw among bs possess decentralized execute obtain line sec penalty comprise bss radius bss show triangle drop channel bss deviation db scale distance km path bss high long channel bss assume fig depict traffic varied ratio bs ms locate without accounting cell interference denote thus represent bss small instance traffic necessary attain mse full greedy greedy summarize greedy pick bs sequentially bss assume depict average distribute total traffic incur network snr db solid represent dash clearly see cumulative plot traffic amount curve greedy scheme comparison greedy size yield traffic adjust amount traffic traffic level cluster partition edge experience interference depict c ms inter turn inter cluster distribute per cell amount plot cluster curve range similarly marker cluster traffic inter plot traffic cluster bs member head without inter cluster outperform intra traffic improve traffic practical interest gain central per bs partition major portion mcp instance dynamic bi intra link obtain check cluster scale formation solely bs ms mark circle mostly cell user ms cluster cluster inter mcp exploit compressive sensing reduce sparsity inter formation formulate decentralize significant traffic edu department computer electrical engineering university usa mail edu multi cell overhead sparsity regularize multi design formulate clustered base form tight solve via decentralized implementation cluster scalability robustness verify efficacy propose wireless traffic due mobile device development boost cell recognize inter interference major bottleneck link quality service mcp multi cell interference idea base bss transmission mobile place link heavy interference boundary overall trial bss interference cell user channel scheduling speed tighter locate bss transform array symbol signal bss together jointly exploit inter link adopt expand achievable burden bss complexity gain mcp issue address many impractical practice token connect bss central limitation range localize uncertainty due quantization maintain synchronization across challenge mcp become prohibitive solution address mcp analog sample albeit compress pool overhead digital share capture capacity theoretical assume bss finite capacity link share bss successive interference limit bss traffic propose bss collect feedback bss bss receive filter cluster scenario bss form dynamically channel direct bss context distribute greedy clustering maximize clustered cell network bs transmission consider distribute theoretic mcp place bss conference decentralize clustered work decentralize cluster mcp isolated resource centralize piece bss decentralized implementation component group eigenvector discuss extensive verify user cluster introduce sec constrain sec static cluster decentralize implementation sec conclusion cell network bss bs bs ms possess bs ms set bs symbol slot band define receive bs represent channel entry channel th bs output express compactly interference network vector justify normalization ms power assume mcp bss receiver bs traffic estimate bs need receive bs square one
adopt conditional generate draw involve contrast conditional conditionally second conditionally independent conditional differ latter multivariate challenge constant scale q normalize present integer interest rejection generate table sampler initialize prior density indeed plot column minor carlo surprising among freedom prior chi square degree column concern prior column propose propose identifiable loading invariant intend possible default impose loading loading differently concern scenario loading mention latter situation term issue address arise generally preserve largely prior need sample square setup efficient sampler work derive spherical normal loading situation merely considerably scenario acknowledgment work support science foundation dms theorem loading matrix identify rotation identifiability take loading prior loading normally diagonal loading truncate normal associated loading order minor identifiable low loading maintain centered factor load comprise model loading exploratory contrast refer situation entry model factor see discuss load orthogonal rotation orthogonal exploratory computation impose identifiability loading restrict entry uniquely paper also triangular loading conditional variance conclude numerical example discussion section identifiability loading natural would spherical clearly invariant induce prior matrix permutation describe come identifiability assume matrix uniquely decompose triangular imply low prior joint triangular haar density proportional triangular distribution q tuple
compute give line propagate activation compute refine refined refine compute root square mlp description rating item description netflix rating twitter set friend last fm create focus art recommendation take hour netflix use validation contain rating movie item description binary movie belong use phase training equivalent technique collaborative filter mf recommendation use learn rating preference user na I commonly neural backpropagation also neighbor range mf mf variable regularization node range rating set parameter fold absolute mae bold within demonstrate item engineering difficult cccc ccc mf cv improve variable similar matrix factorization widely state recommendation compare mae power problem mf collaborative technique address rate suffer item rate capable address still factorization remove rate movie movie remove use item challenge induce create description rating beneficial mutual user item share latent quality receive contain rating lack utilize new represent level near neighbor induce neighbor fed train rating new item weight mode predict rating rate weighting number neighbor rate choose mean outlier result detail counter keep track rating near neighbor distance enough neighbor good description distance play item use great rate choose content rate help quality induce latent item variable line find close neighbor index line rate rate rate count rating help discount item rate index count mode top table represent mae recommendation produce mae suggest recommendation item score mae individually exception movie lowest mae previously rate mae latent create rating user use collaborative technique recommend emphasis produce item recommendation utilize power alg efficiency iteration second require use previous recommend use near complexity mf present item user call item description recommendation hybrid advantage collaborative filter content achieve similar result art filtering address start filter achieve previously item art recommendation item rate build datum many factorization may inherently future examine incorporate look item description input address user single department engineering recommend item user without outperform suffer item yet rate rate incorporate additional user description collaborative start present model latent hybrid technique address outperform broad content recommendation description hybrid maintain accuracy art collaborative filtering technique technology access abundance datum make find try product amazon recommend movie netflix recommender system find look commonly recommender system recommendation item user description user rating description predictive user item user rating infer rating recommendation description accuracy suffer recommend unless rate rate particularly domain new user newly movie rather old movie old away recommendation profile one address hybrid recommender leverage advantage recommendation system develop hybrid hybrid approach combine rating recommendation present address start use description train induce input matrix allow flexible order unsupervise induce input internal latent input variable dimensionality technique hold network language train input incorporate input item description address profile latent portion item rating item user fed network unit refine gradient refine item description description description backpropagation compute input commonly train presentation know express gradient derivative input intrinsic affect unit input value algorithm unit additional backpropagation perceptron equal layer single hide term backpropagation error network hide strict output integrate use phase train first phase compute estimate intrinsic second three train intrinsic vector likewise chance quality three phase phase produce together tb weight single layer perceptron value multi layer perceptron layer
learn remove prior stack variety objective recent autoencoder additional noisy autoencoder autoencoder work analyze characterize unit activation reconstruction multiplication corrupt version denoise activation reconstruction costly approximate corrupted test autoencoder due noisy autoencoder yield effect encoder nonlinearity perform order taylor encoder fw encoder effective multiplicative bernoulli use result approximate allow noise activation unit autoencoder decoder use square independently activation apply noise penalty input accurate allow relate autoencoder framework regularize autoencoder network autoencoder autoencoder tie gaussian encourage unit penalty type penalty learn overcomplete representation ica encourage noisy autoencoder building compute layer encode clean input autoencoder second layer activation representation hide representation autoencoder learn insensitive frobenius jacobian encoder fx f additive alternatively hide activation recover autoencoder activation encourage activation activation activation many unit activation experimental neuron dropout multiplicative effectively link adaptation neuron motivation generalization dropout computing h projective field shrink sensitivity reconstruction dropping layer noise yield system representation semantic intermediate learn binary binary code add fully analysis autoencoder show implement autoencoder lead segment digit input activation error initialize perceptron hide importantly activation h c dropout c c noise evaluate hidden input dropout noise hide activation mlp standard backpropagation use optimize also autoencoder error backpropagation noise perform perform backpropagation dropout additive dropout dropout low error capacity require overfitte noise improve help decay validate dropout hide activation extract cifar yield accuracy slightly high autoencoder accuracy different representation classification performance mnist cifar mnist model train dropout activation domain maxout dropout noise activation additive fix call shorthand sgd momentum error maxout understand dropout lead less noisy layer far understand influence activation type sparsity neuron yield dropout representation noiseless row spectrum slow row act sparsity propose principle namely robustness auto level internal wide different lead design supervise use achieve mnist full systematically huge deal lie ahead understand automatically make explore noise supervise interact intuitively deep ball unless class ball effective invariance properly introduce compressive projective map hide contraction beneficial stanford edu autoencoder unsupervise internal representation conceptually extend autoencoder additionally nonlinearity wide variety framework practical benefit strategy new internal designing autoencoder outperform denoise autoencoder denoise competitive technique mnist deep information representation noise neural dropping backpropagation performance neural pooling convolutional net noiseless explore recent layer autoencoder show yield autoencoder success input layer autoencoder systematically explore layer learn see unify unsupervised prior autoencoder call autoencoder derive penalty relate autoencoder
k v p relation hence vector relate bandwidth illustrate semi clearly bandwidth semi interesting benchmark scale determinant extend determinant dense extend denote row write matrix matrix determinant row correspond last triangular b complement dense I arise permutation matrix extend algebra recall correlation lie monotone algorithm rely happen ingredient sparse use recognize separable separable see set ai j ji early spread large exponentially sparse issue analytic linear equation eq embed obtain appropriately carry semi separable numerical appropriate q extended matrix numerical benchmark semi form apart factorization infinity residual purpose choose sort benchmark sparse factorization eigen sequential perform use method system scale linearly store triplet eigen row format exact benchmark compare eigen fix rank htbp gray c time taken take sparse solve versus linear relative log extend illustrate scale factorization increase separable htbp stage residual illustrate scaling add equivalently rank solve scale stage equivalently semi illustrate algorithm separable factorization semi separable rank article discuss numerically stable enable determinant matrix entry publication formally sparse semi implementation available university remark determinant semi lemma corollary discuss invert arithmetic introduce solve enable semi illustrate determinant solver semi large storing perform dense finite one sparse study statistic detail reader refer al throughout separable article term semi q upper triangular separable linear library discuss implement york david put author anonymous detailed
length buffer size epoch find accuracy several dataset buffer cut exhibit method hence intermediate necessarily accuracy offer progress problem good intermediate advantageous scenario acknowledgement helpful comment thank anonymous suggestion thank google fellowship lem lem corollary lem lem definition lem lem fact lem lem title lemma claim fig microsoft modern sensitive frequently precision offer grain time challenge decomposable solver enjoy nice amongst sublinear provable learning function popular loss sublinear regret novel lemma descent solver uniform provably converge known proof extensive method order cut plane modern frequently require grained prediction hinge mild imbalance spam wherein spam constitute task diagnosis sensitive include rank precision top rank recall specifically express point unlike area roc despite success domain nearly understand decomposable counterpart loss function lead deep behavior online decomposable popular online first contribution instantaneous penalty decomposable canonical principled way objective desirable online admit online satisfy offer convex surrogate namely roc indeed achieve sublinear regret involve structural sort continuity list might analyze property designing solver offline introduce variant provably converge minimizer hand proof style require conduct extensive real life cut fast dataset take achieve comparable decomposable auc demonstrate imbalance interest indicator dataset due measure seek arguably hand hand perform start seminal receive interest average formulation design solver theory also interested provide additive regret focus show however implementation learn pair dataset crucially dy label shall simplicity p evaluate decomposable measure auc surrogate term construct simplicity drop point predict score rank position valuable situation positive formalize top q structural surrogate calculate use modify consider label area roc allow range medical application detection label number express replace indicator hinge hinge framework decomposable measure gradient method function prove much large function several player incur point surrogate definition instantaneous penalty clear remain challenge guide process learn framework point continuity indeed crucially sort list product let ic I I j z k see lipschitz gives show prove recall handle positive negative previous obtain appear resp stream use regret bind adversary may naturally see bound generalize slightly point compose batch penalty l population normalize sequence point batch q note rt appendix guarantee model rapidly become infeasible descent non decomposable loss motivation mini amenable computing environment offer scalable assume access limited memory buffer stream stream epoch stream descent step buffer epoch describe computation batch pass batch collect b unable point epoch help loss life scenario label imbalance table store label exploit utilize pass pass label stream pass stream restrict non perform goal sn eq convergence common decomposable true decomposable bridge show surrogate exhibit proof require technique proof use arrive loss demonstrate arbitrary set predictor return buffer stress assume order see regularize formulation improve explore consideration instead look uniform convergence nature area roc curve surrogate result cover large family surrogate logistic rank introduce challenge structural surrogate exhibit uniform version extend performance minimizer wide variety performance section verify propose gradient performance auc proportion positive auc
require adapt behaviour cnn introduce deep selective selective cnn bottom top connection implement learn reinforcement usual enhance capture initially aim usefulness filter manual inspection separable evolution agent reinforcement learn etc million require cnns cifar cifar difficult instance de certain maxout network combine underlie object recognition task outperform convolutional network reduce favor maxout cnns consist stack alternate convolutional image width map convolutional parameterize filter index map convolutional wise pooling dimensionality layer take partially reduce dimensionality width maxout pooling layer consecutive map map keep maximum every layer form pool large softmax activation maxout reinforcement sequential decision reward signal receives agent state receive reward objective policy future discount space parameterize attention space close policy rl evolve use strategy black algorithm multivariate gaussian parameterize epoch update natural gradient fitness instead use theoretically substantially power attention maxout net augment allow filter weight differently pass image strength activation learn order sequentially attention discriminative change cnn result follow vector describe net already classify x policy image sample represent trial maxout pass set maxout would normally net activation output layer vector average activation meaningful allow softmax note softmax action regular ensure filter activation process pass pass correct constant loss misclassifie classified misclassifie input process assign fitness q classify pass network maxout output weight maxout finally update natural gradient fitness repeatedly fitness stop meet system visual level construct area visual connect connection numerous bottom connection think play primarily analysis response newly stage visual fast feedforward follow due feedforward basic orientation categorical role foreground salient support top play extract guess category connection rely feedback computer vision learn selective face also combine vision processing reconstruction face localization rl lead novel apply simplify aim state perspective feasible case cifar cifar interesting case approach already aim cifar compose color training assign cifar similarly rl enough step enough practical meet could serious limitation experiment l cifar column maxout maxout model cifar show several method improve state art reference connection add augmentation consist convolutional maxout follow maxout softmax input vs method cnn establish art classification cat activation de show probability cat receive feedback cat dramatically subsequently drop bit successfully layer emphasis filter almost map high correspondence lose code final layer simple increase map complex emphasis network value run figure training cifar peak reduce stay stable even step
recover compressed firstly measurement operator condition rank condition guarantee matrix rank guarantee norm prove isometry property sufficient quasi provide rip affine nuclear isometry rip constrain affine minimization numerous processing case aim recover arm collaborative filter quantum arrival affine equality formulate measurement operator measurement usually np solve relaxed version replace convex relaxation condition form superiority quasi replace norm pp norm nonconvex update recover rank constant eq though minimization paper guarantee quasi author quasi discussion show indeed provide sharp condition prove uniquely large rank exploit condition restrict isometry rip quasi minimization generalize minimization dominant singular recover generalization sufficiently recover singular measurement rip value organize introduce notation present devoted proof conclusion denote obtain sort p always svd n n exploit nan necessary successful reconstruction minimum condition gap introduce lemma sufficient uniquely equal large uniquely rank recover sufficient weak formulate condition restrictive inspire arm simplify remarkably rip stable sparse directly prove equivalence condition recovery nevertheless one essence rip vector rip equivalence utilize aforementioned formulation quasi rip use vector noisy integer small eq vector def integer show denote likewise rip f sufficient g rip g threshold find work cover accurate nearly noisy simply rank mean proposition organize presentation herein prop let constant corollary knowing rip accurate c solution recovery obtain fix respectively satisfy decrease guarantee tend clear passing original range
similar trick already reinforcement classification task well netflix demonstrate online exploration tend towards user bad artificial surprisingly perform approach netflix difference strongly netflix protocol top movie movie user ucb user suffer strategy factorization candidate item deal suffer regret favor item old one hence old recommender policy new user item strong exploitation computational additional square total factorization consequence factorization stay idea principled effective way conceptually large publicly furthermore extension currently study extend contextual might translate want regret large full work item good bandit point translate plan ellipsoid user item sum odd perspective lead artificial fr version ad music video movie book system cope user visit website item user exist estimate handle side available perspective perfectly aware utility side information consider item mix fit perfectly decision set traditional continuously though hundred arm million make bandit asymptotic approximation look efficient effective way cope web obvious strategy consist repeatedly present seem exploitation problem exploration eventually problem netflix challenge recommendation factorization square rmse however along heavy rmse make item rate rate user user well rate wants recommend illustrate rating netflix range make qualitatively different correspond rate rating real allow precise rmse item already outcome recommendation user information regard interaction leave aside average birth death often past item really rank rmse aspect handle recommendation address since history order perform information minimize recommendation idea paper original tackle start system cast play optimize balance problem focus familiar bandit contextual bandit hypothesis introduce methodology assess recommendation introduce matrix factorization approach introduce bandit sec new user item line sec face transpose row bold denote scalar letter sub compose element contain accordingly small line index matter notation dedicate rs evolve user user number drop bound number ever necessary represent truth obviously application whereas rate user size denote finite majority submatrix make row actually available item rate user likewise rate symbol user rating respectively thing let assume triplet item know observation rs receive stream rating rating netflix challenge click sake omit subscript netflix challenge interested reader done approach convex alternate problem value svd compute compute minimize user item respective rating limited rmse consider arm receive follow good arm parameter aim cumulative consecutive reward player want unknown except player accord arm vs ucb play ucb equation exploration exploitation arm tend arm extend contextual arm assume assume arm represent rating item introduction item user item recommendation rs rs never return obviously objective rs maximize context scenario approach rs predict rating user pure exploitation suboptimal rs balance rs recommendation soon compare along fast drawback describe recommendation aim item trade exploration order hold fix ucb I possibly consequence uncertainty word mostly express iteration fix system coordinate axis mid coordinate axis mid fill blue blue ellipsoid circle circle circle mid mid circle dot area indicate associate optimistic rating user scalar ellipsoid contour scalar product equal optimistic recommendation item strategy select amounts tune ellipsoid closed name stand alg presentation optimize clarity item ellipsoid use regularization regularization reward please matrix exactly ji ji ji ji tr proof care context matrix decomposition degradation trick observe order unbiased description fact instead uncertainty occurrence criterion ellipsoid rating user item presentation clarity efficiency item lead modify axis mid axis mid color red fill color red red fill red red color red color red red mid mid blue live optimistic item ji ji tr j evaluate empirically real greedy greedy always item ucb ucb consider reward word ucb information comparison approach highlight context ucb user recommendation strategy netflix yahoo item ucb item user random rate request yet rate compute rating item user user rate difficulty real
conditioning var aim paper context short policy minimize ensure expectation level stay bound solution govern recurrent horizon mdp separability risk constrain reason globally risk return mdp risk constrain complicated expectation govern optimization constrain mdp complicated tail reduction speed first propose prove converge locally principle mini policy gradient line estimate policy scheme base mini batch spirit propose proximal gradient develop four approximation scheme along conjunction mini batch gradient ratio gradient estimate fast scale negative descent ascent multiplier operate employ batch incorporate close well interesting come variable concern scheme trivial variant sampling sum contribution synthesis stochastic sampling provably convergent propose another idea simulate latter novel neutral mdps rest formalize constrain present mini variant later section previous conclude remark formalize constrained variable continuous var low drawback coherent measure coherent variant define coherent finite terminal feasible incur follow specify stationary state continuously differentiable identifiable reach transition state class parameterize proper outline constrain randomized specify constraint trick convert solve operate follow inner episode along estimate loop lagrange multiplier converge white rectangle minimum height fill circle coordinate thin cm simulation align green cm label align center block green align green fill red height align ex right thick triangle thick triangle update triangle triangle thick triangle update chapter lagrange multipli assumption proceed recursion close expression gradient moreover observable k hence policy follow section differ establish describe component var well input output pg sa episode underlying end visit state visit episode action gs cs likelihood tuple th episode obtain let parameterized hold estimate alone technique estimate mdps policy simulate since know approximate approximation height width em fill white thin align fill cm align n fill red cm align pg mb obtain empirical mean negative mini batch policy q output batch approximation asymptotically scale lagrange scale pg sa variant know existence lyapunov follow martingale suppose exist continuously differentiable govern contain involve convergence pg sa fast quasi var serve lyapunov fact size iterate bound recursion set iterate establish recursion plain average intermediate scale main argument govern stable equilibrium follow ode effect static ratio almost discretization lyapunov recursion stable ode lagrange multipli update step argument mdps general view almost scale converge equilibrium converge multiplier use suitably keep evolve pg sa minimum claim argument particular former recursion ascent require envelope theorem economic mini variant var gradient mini large estimate converge rest manner pg sa importance estimation scheme continuity translation eq variance use recursion ensure result function double translation var formally classic concavity differentiable write px ba piece growth assume control hx scheme approximation hx b episode let straightforward one density pd ratio episode transition dynamic pd scheme rule result recursion estimate var attempt estimation approximate part place tuple accord would horizon separability cost separability devise constrain optimize employ convergent locally policy paper novel policy stochastic motivated energy market incorporate along line risk
payment call denote similarly via portfolio abstract span allow discussion beyond scope consist process choose portfolio hold move portfolio decision portfolio selection trading market context clear write variable letter use letter functional without preference preference functional asset specific theory utility theory economic risk measure see introduce measure put detailed justification name risk measure understand potential certain asset satisfy eq understand monotonicity asset translation invariance map asset asset asset asset domain fortunately extend invariance eq thus risk generic measure combination high hold word encourage risk measure loss predefine risk moment function kl hold measure market portfolio prefer portfolio rational choose trade rational prevent market machine aim objective design market implicitly global contribute objective section build multi period trading maker allow trade maker maker introduce simplify market trading market mechanism market share security one want amount share simplify market maker maker trade security allow maker make pay maker pricing pricing market maker step functional different asset maker price market maker portfolio asset restrict rational agent choose optimal portfolio portfolio selection rational agent pricing multi involve maker market maker market maker joint trading since agent subscript distinguish agent collect bring asset maker agent portfolio trading maker agent keep asset like initial pricing x maker update use market agent portfolio communication maker pricing trading choose trade initial portfolio measure receive pricing rule w request maker x study pricing market maker class mechanism market later axiom price trade maker trade market maker maker follow natural total pricing rule market analyse market machine want market find describe involve eq part functional share multi market define machine learning problem pricing motivate meet pricing market share cf transform could relax could replace convex conjugate ready show market involve duality pricing rule market maker derive generalise substitute back dual match cf dual duality way market market primal solve interestingly could market market distribute environment extra benefit build market discuss significant past agent model market base market agent model equilibrium author belief agent event market mechanism market market progress market scoring rule market mechanism agent draw demand market implement solve partially certain also convergence market dynamic market belief justify risk measure asset standard operation expect abstract I agent additionally help asset portfolio highly inconsistent dramatically measure amount accept asset sensible asset example convex risk functional risk illustrate connection multi period trading market opinion pool opinion pool log opinion state opinion pool market introduce agent maker primal problem market apply proposition simplex optimal eq introduce market maker aggregate bias towards however sufficiently maker ignore end aggregation observation aggregation bias due biased market maker upper agent increase much market sign quickly aggregated aggregation lead belief close expect reproduce biased coin setting market market build define security asset moment generate maker scoring could market implement map update market maker univariate statistic clarity exposition care conjugate goal transform agent market mean convert market let interested trading th share security hold rhs define agent deeply costly relax accept portfolio well current agent towards portfolio rule could choose backtracking line instead introduce agent match logistic discuss prediction market instead analytical global objective connection market tool interesting topic condition converge come agent involve nature support microsoft rgb ed ac uk markets introduce market describe trading process modelling bring convenience objective despite agent additionally market sensible objective market analyse global solve machine certain market valuable direction machine towards scalable market abstract learner system market distribute additionally relationship market probabilistic spend building still market
dnn utilize increase corpus heuristic offer performance understand build acoustic dnn explore optimization acoustic model function difference across example whether improvement due network architecture technique concern systematically explore several improve dnn acoustic dnn component dnn classifier draw dnn many task dnn acoustic simply speech rate dnn acoustic unclear acoustic ultimately across acoustic far understand aspect dnn rapid dnn acoustic new speech corpora language understand variant various understanding principle artificial intelligence study component perform effect dnn task hour increase dnn fit include architecture choice convolutional locally evaluate alternative convolutional dimension dnn fitting corpora speech research corpus us explore dnn ten corpus also impact choice layer final compare across dnn architecture process section outline question address neural network paper corpus focus dense convolutional choice combine corpus explore deep dnn act acoustic recognition use approach system resemble speech mixture gmm acoustic overview scope refer article speech focus component approximate distribution span acoustic acoustic represent ms audio hmm cluster dependent state hmm use gmm network distribution network acoustic use rule neural distribution approximate occurrence count usually acoustic span acoustic difficult feature fix scale drop hmm term form acoustic acoustic model decoding introduce scaling empirically adjust introduce un normalize recognition hybrid modern component progress speech recognition acoustic work refine framework signal building performance extension gain task complexity purely capacity gmm acoustic parallel interest new apply initialization learning provide interesting forward acoustic offer path capacity dnn early recognition recognition demonstrate state innovation couple capacity yield challenging task acoustic gain challenge microsoft google factor attribute modern hybrid acoustic specifically total number number hide initialization modern hybrid researcher hybrid dnn weight purely nearly many beneficial hide layer size define hybrid system neural network build understand aspect building network acoustic understand define set acoustic hmm early neural context state dnn acoustic critical success generally modern system variant context try acoustic model create baseline hmm gmm force alignment originally contain assign label acoustic force alignment ground generate consistent force hmm hybrid speech creates force hmm gmm align use train previous dnn system dnn produce performance yield dnn start force repeatedly use forced alignment produce hmm build dnn acoustic structure use acoustic modern dnn use success recently standard hide gain architecture counterpart task hold deep obtain dnn time translate modern dnn hmm system continue performance fundamental neural architecture aside series densely connect densely network intend leverage meaningful audio input task addition dnn acoustic architecture densely architecture perhaps layer window temporal neural recurrent modern recurrent acoustic task task available term hmm continue variant model loss acoustic function also control default dnn observe standard classification dnn account system loss gmm acoustic dnn acoustic acoustic begin strong function step combine standard view discriminative acoustic act component choose additionally function regularization especially simple widely weight penalty develop area regularization effective dnn apply regularization dnn acoustic combine change quality dnn far solution dnn stochastic practitioner default optimize researcher advanced method yield dnn well dnn acoustic quasi sometimes processor recently like accelerate task outside sgd still newton require consideration dnn procedure utilize graphic hundred computer persistent throughout history utilize modern approach final capable time network acoustic design component hold baseline acoustic building variant difficult assess decision systematically vary critical variation baseline question neural hide corpus overfitte build ten total drive build dnn acoustic much broad locally versus densely dnn recent combine type acoustic generalize still feature share reveal overfitte dnn early many training acoustic towards improve large small processing dnn acoustic yet dnn acoustic task ultimately far encode change large deep question use hour corpus corpus dnn architecture address corpus use present experiment large size dnn depth code property corpus understand encode integrate architecture amount equation predict class architecture utilize cross apply regularization many research entropy always task specific serve instead acoustic entropy objective single training take entropy acoustic short acoustic wish minimize mistake word step conversely acoustic frame experiment always frame error metric dnn loss dnn series fully connect transform act conditional dnn layer follow dnn layer vector activation first layer dnn partial effectively layer activation hide apply wise nonlinearity computation traditional neural network unit hybrid speech dnn nonlinearity final dnn output final nonlinearity softmax nonlinearity output loss state dnn computation equation loss non practice apply gradient dnn dnn speech benefit neural acoustic issue dnn study connect dnn serve acoustic modern speech vision deep convolutional neural bank architecture relationship convolutional code shift localize acoustic evaluate specialized combine replace network dnn feed fully layer image vision restrict hide connect spatial control localize stationary vision location rather separately convolutional time connect equation apply move across produce meaningful activation map operation pooling act code slight localize connect contiguous region convolutional layer apply pool local region feature feature map contain activation select activation hide separately use pooling pooling alternative pooling replace pooling often consist layer convolution follow densely softmax classifier pooling act input dnn dnn input densely convolutional layer densely hide frequency relationship architecture dnn layer input convolution pooling hyper size neural architecture combine connect hide sharing unit idea utilize locally region architecture apply frequency feature unit use unit slight occur frequency architecture convolutional convolutional connect weight sharing united behave grouping pooling post similar tb pooling layer behave architecture define wish minimum gradient impractical gradient many convex difficult general statement optimality heuristic classical momentum probably algorithm modern momentum denote accumulate momentum velocity vector close expect information update ill condition problem momentum might actually cause fluctuation parameter turn slow nesterov issue encounter network optimization help past sensitivity avoid ahead gradient detailed optimization question cm acoustic establish hour baseline force create open phone dnn estimate likelihood hybrid dnn setup input context adaptation globally dnn overall setup evaluation report subset subset perhaps direct improve large add unit dnn increase capacity scalability interest apply large dnn acoustic model architecture ask question improvement large frame serve improve size vary hidden total number unit hide network size million output layer network typically study output often layer comprise contrast rarely fraction modeling work explore additionally context use nonlinearity optimization nesterov accelerate momentum schedule update pass stop cross hold development tolerance threshold distribute gpu capable training restrict parameter task dnn day frame acoustic baseline gmm show vary context substantially dnn dnn small substantial dnn absolute classification accuracy improve large window context overall frame dnn window frame always proxy final acoustic reduce suggest increase gain translate large set dnn model beneficial task acc train gmm understand dnn acoustic dnn epoch fairly dramatically continues nearly set realize epoch training implication acoustic beneficial fitting performance become increasingly utilize utilize finding epoch window metric gain translate provide training dnn tb dropout recently prevent dnn dropout randomly unit activation training prevent observe activation demonstrate processing use dropout acoustic hour news dropout additionally yield gain convolutional network hour dropout impact alone dnn dropout whether large train dropout training preliminary set generalization possible otherwise evaluation build force hmm dnn train dropout improve acoustic beneficial dropout seem insufficient selection critical poor preliminary bt layer er er layer early converge early dnn technique dnn acoustic network capacity produce generalization par well network back propagation early capacity analyze stop improve low test dnn dnn early stop dnn beneficial perhaps insufficient benefit large acoustic label dnn acoustic force alignment level labeling training partially dnn improve acoustic span label supervise force alignment force alignment system force speech recognition ability variation version independent corruption correct label outline improve dnn performance optimization exhibit dynamic early capacity exhibit capacity early phase combine dnn hide suggest dnn dynamic categorization coarse class generate force acoustic baseline acoustic model train iteratively improve ann hmm hybrid fully dnn acoustic system new force alignment continue label begin predict label much early save day converge completely dnn fitting alignment dnn gmm force far dnn hmm dnn force alignment dnn proceed dnn newly regularization dnn measure epoch experiment epoch result low quality model randomly initialize dnn bad dnn occur anneal schedule adjust newly alone without performance well dropout epoch beneficial dnn bad epoch early lead train dropout stopping make evaluate corpus dropout train dropout figure curve acoustic train early dnn briefly surprising dnn follow begin label labeling computing label change early change label change dnn five match capacity dnn train corrupt extremely capacity dnn perfectly dnn take suggest early bias characteristic phase dnn require benefit train epoch five epoch eight dnn translate day implement modern system force alignment dnn overall early technique dnn acoustic minimal additional bt acc gmm dnn dnn far dnn regularization deep train use train acoustic experiment facilitate across bank dnn five train accelerate smoothly increase momentum schedule epoch acoustic overlap pooling region layer second filter use select preliminary densely hide unit map unit convolutional map convolutional layer dense hide layer bank run filter along frequency region run experiment context report frame system acoustic bank improvement connect acoustic tie weight localize field frequency outperform indeed dnn lack meaningful flexible architecture able useful parameter well bank much leverage acoustic run bank feature randomly axis convolutional fairly experiment indeed leverage localize promise compare filter bank bank bank bank transformation may automatically gender specialized post appear neural superior complement dnn competitive increase amount summary replace reliable conclude acoustic hour limited benefit dnn acoustic modeling dnn substantially large corpus experiment explore dnn acoustic maximize task combine fisher corpus approximately hour accurate gmm acoustic train feature obtain frame current project dimension discriminant lda normalize mean obtaining semi tie feature linear estimate per fisher contain system train fisher fine back probability sentence preliminary build web page text gain derive language alone use system serve compare build alone rt evaluation train baseline table dnn hold dnn optimization five roughly free typical acoustic literature nesterov key hyper momentum decrease annealing preliminary overall anneal pass quite find epoch lead solution table accuracy algorithm evaluate effect evaluate performance optimizer optimizer important appear train absolute thus remainder use optimizer optimizer somewhat robust optimizer bt produce performance setting momentum report test contain evaluate rt rt corpus systems optimizer acc rt gmm cm performance keep fix improve hidden layer hide hide layer total free add hide unit show frame classification hide reproduce dnn dnn code dnn comprise facilitate comparison framework parameter perform frame evaluation experiment dnn lead gain limit compare dnn move dnn dnn size rt evaluation small corpus overall believe corpus challenge acoustic induce quick dnn improvement acc rt gmm n n dnn system keep total vary layer dnn architecture total model change priori reason believe heuristic dnn layer multiple performance gain deep versus layer hidden layer model hide deep depth performance
relate six function formula ft place middle remove plot rule partitioning form length cost cost ft ti cost ft tr segment neighbourhood section neighbourhood method segment neighbourhood prune introduce segment search neighbourhood search enable set c position idea segmentation shall update describe fully needs specify satisfie tn tt nc k nk fit prune strong c prune large overhead detect univariate prune implement compare extent insight method figure amount store pruning optimisation figure illustrate rarely contrast evidence prune fact pruning pruning state condition c solving optimisation use program functional pruning pruning also slow inaccurate round middle profile assess performance confirm speed speed depend number change cope change need look size dependency clear pruning signal expect efficient efficient range simulate point varied signal repeat code simulation repository see fast interestingly benchmark accuracy annotate region visually copy benchmark training annotate segmentation annotation segment set vector annotate compute benchmark small test introduce detect question answer whether solve solve optimisation give disadvantage slow particularly interactive solve constrain preferred interactive scenario know purely efficiency pruning always prune empirically difference pruning apply require currently even prefer always detect particularly large observe computational comment discussion support centre mm mm em mm ex ex em definition lemma definition proposition em cm de universit abstract accurately long detecting description formulate dynamic programming exactly tend least binary segmentation computational segmentation suggest true cost extend pruning method new efficiency detect variation keyword optimal partitioning segment series model effectively segment model detect efficiently eeg datum speech signal section dna copy microarray reduce relate cell area crucial classifying widely moreover microarray basis many estimate zhang formulate define call segment specific cost dynamic programming variation segmentation circular offer slight decrease way prune technique pruning technique functional pruning optimum pruning take pruning try former pruning inequality always suggest regardless number segmentation structure introduce constrain optimisation review exist dynamic pruning optimisation optimisation algorithm empirically theoretically exist pruning empirically simulated assume order trivially order attribute observation distinct th segment consist let statistical consider infer type wish segment infer consist datum q depend detect segment segment likelihood identically segment detect across segment normally segment simply error depend segment segmentation get location unknown approach minimum datum often monotonically either call calculate choice aic linear penalty optimal note solve criterion program neighbourhood optimal insight segmentation vary advantage incorporate model pruning pruning method two functional pruning discuss prune stronger hold hold solve optimisation programming optimal partitioning first reduce computational denote position early use simple recursion order calculate q calculate increase partitioning discuss exact equation never location hand value discount time update rule restrict denote update pruning pruning algorithm expect basic partitioning show computational bound show optimisation solve constrain optimisation segment neighbourhood search approach prune segment neighbourhood derive relationship thus recursion obtain extract segmentation position calculate segmentation calculate value equation calculation total developed technique neighbourhood call dynamic generic use segment search assume segment split store candidate recover far return allow split c interval correspond idea example change value need store c square criterion line contribute line interval correspond previously contribute formally interval recursion bold long give criterion analyse four introduce corresponding change middle plot function long optimal plot time empirically towards overhead segment neighbourhood search
weighted square label instance computation unchanged unchanged except wolfe gradient challenge video annotate annotation kind provide truth construct dataset annotate action movie manually add build annotation annotation annotation form video sequence end range temporal frame feature vector recall frames aggregate decide pool feature long center compute video descriptor feature restrict channel improve run aware yield hellinger feature normalize experiment split dataset supervise annotation set sec hyper practice contain annotation use cost evaluate carry frank wolfe union three set annotation constrain rest annotation please assignment five split bar may task use yet another prediction ground standard compare ensemble slight big annotation low annotation take averaged align annotation accurate segmentation long predict label within ground interval compare three baseline train use baseline scheme sec proximity appearance interval j x square distance cut use frank wolfe intuitively search segment chi replace appear adapt size problem instead minimize frank algorithm scalable method graph compare sl supervision completeness classifier square annotate interval square setup baseline except sl supervision illustrate baseline weakly temporal constraint signal baseline recover alignment annotate increase blue mark make lack order fully annotate expensive movie easy appear manually video necessary good supervision baseline stand b figure right axis supervise error bar baseline annotation whole dataset annotation sl learn low improve supervision quality assignment z instead quality hold classifier treat use compute section classifier hold make data one split error correspond compare sl classification experiment sl blue always explain fact propose weak annotation sufficient train et task fact access supervision constraint weakly semi supervised red sl blue recover consistently well fully semi annotate example constraint improve one constraint scale sup video annotate list action walk extract seek action formulate weakly constraint time assign annotation discriminative manner dataset video total movie recognition video year cast detection fully annotate boundary give exploit order video stream annotate video time label quite consume fully supervise use weakly semi method promise easy video poor localization movie localization ignore supervise set discriminative optimal assignment constraint learn temporal order action constrain kind activity video infer detector surveillance laboratory limit action explore challenge video length movie relate composite activity learn give supervision composite activity atomic action action annotation without order action explore form action priori temporal order individual supervision explore uncertain temporal annotation action movie contrary multiple simultaneously incorporate label movie dynamic match speech like improve pre detector discriminative clustering unsupervised partition formulation discriminative explore vision successfully apply co approach order work frank wolfe gradient minimize classical permit continuously differentiable domain receive temporal assignment address illustrate set short define contiguous frame movie divide video annotated order action consist stand phone assign original annotation list fig contribution model supervision order localization video propose efficiently solve improve new localization publicly discriminative video annotate specify action order annotation formulate assignment individual cluster parametrization assignment discriminative lead preserve predefine way transition possibility extremely algebraic constraint stochastic matrix define use notation lead w frobenius computed inversion concatenation objective rewrite matrix ridge minimize done close plug back yield matrix p center p find optimisation z share video recover form sec combinatorial replace hull appropriate define convex large admissible assignment kind operation projection tractable convex frank wolfe sec choice frank wolfe rather approximation edge convergence interpolation iteration counter implementation frank provide duality refer linearization duality relax tb k programming wolfe linearization current view actually minimize linear depict hyperplane seem shift argument prop minimization mind wolfe amount solve ta plain representation equivalent well temporal let becomes solve indeed k sp admissible recursion p kp dynamic maintain frank wolfe optimum z find nearby simple rounding scheme consists find
argue previously monotonic inner unless experimentally compare hash report well scheme normalize example norm set lsh impossible locality locality hash inner exist lsh suppose hash function product similarity exist event lsh inner possible great show function satisfy totally lsh monotonic basic lsh requirement lsh algorithm step candidate use identify increase behind lsh still preprocesse hash increase traditional lsh runtime locality hashing locality family along query preprocesse transformation instance hash recover lsh hash back thus counter high note query processing create hash table efficiently section asymmetric locality randomization hash query preprocesse transformation nn use lsh modification preprocessing query retrieve element table definition asymmetric lsh probability element original lsh query q one ready transformation qx fix constant suffice normalize distance un inner shrink instead interesting connect show become negligible purpose perspective since instance e transformation define respectively last decrease nature root second follow complete proof respectively explicitly rp structure eq sublinear algorithm guarantee parameter hash rd main query constraint grid search parameter like lsh depend aim know threshold threshold choose high conduct corresponding optimal show actual unknown reasonable convenience recommend use choice use neighbor way small choose small consider figure eq h tradeoff hashing transformation compare hash lsh neighbor euclidean lsh optimal un product indexing capability outperform lsh product surprising inner product interest two hash function top task gold hash vector hash hash query indicator sort vector hash function consideration lsh asymmetric hash subscript draw ideally hash scheme precision item sort list suppose rank belong top top increment count already see item vary obtain continuously look precision recall higher recall indicate choose user important parameter lsh largely parameter l lsh lsh lsh lsh algorithm hash hash code show big improvement lsh lsh indicate task clearly asymmetric transformation higher vary color lsh present lsh lsh label curve different lsh top vary solid red available significantly outperform lsh l lsh l lsh lsh achieve interesting confirm precision rest demonstrate indeed choice sensitive unless ii sensitive search numerous scenario collaborative filter task find normalize instead challenge provably lsh locality study develop lsh generalize exist lsh input vector repository propose novel space provably useful line high product similarity even due pilot computation find fast hash special binary powerful binary bit hashing hash propose hash projection hashing rich make fast improve runtime application efficiency scheme application interesting work consist mention detection svms department university usa department department science nj usa provably sublinear approximate search un find hashing scheme locality sensitive hashing lsh insufficient lsh asymmetric interesting mathematical phenomenon convert approximate near neighbor sublinear hash lsh provably lsh independent propose simple implement collaborative item recommendation netflix focus query interested search inner problem two value throughout scenario place variation control directly solve subroutine large structural prediction recommender past behavior past rating model collaborative filter factorization latent characteristic vector rating user concatenation recently svd rating outperform exist neighborhood latent item instance control characteristic wide recommend solve recommendation web linear scan image art object detection activation various image image product filter million test costly score activation identify collection filter high activation image top product object cut plane popular method cut identify violate current exponential heuristic scalable class predict basically class vector class fine grain class compute multiplication predict e costly deal massive make branch technique come provable runtime guarantee technique partition suffer curse current linear search dimension locality hashing lsh randomize near unlike technique well accuracy guarantee lsh way dimensionality make lsh large deal common day lsh make ideal modern focus hashing suffer curse lsh lsh efficiently approximate hash hard lsh lsh negative lsh asymmetric framework lsh lsh interesting mathematical phenomenon problem neighbor asymmetric hash sublinear un hash framework independent experimentally item filter hashing netflix dataset evaluation theoretical asymmetric hash product well function surprising break bottleneck commonly near space construct datum near neighbor distance near
hasting propose metropolis go otherwise stay repeat carlo inclusion individual base easily obtain conjugacy proposal exactly position simplify ratio implement correspond case theory two theory choice near boundary unknown modify residual square lasso lasso experiment present multivariate unit give practical justification correspond configuration employ complexity allow move space mix reasonably carry selection exceed sense summarize mean inclusion expect latter bayes across three discuss method method credible denote show inclusion outperform argue assume work lasso extension distributional strategy subset entry specify depend intuitively idea likelihood track datum application explore parameter mixture mixture denominator involve suitable great big calculation equal accord square theory proportional exponential state write numerator make inside divergence available since rank square centrality formula function chi square integrate clear resemble multiplicative put everything take proof idea equal expectation work separately old consist like bound integral corollary remark sense concentration distribution relevant bayes selection variable assume design recently considerable drive primarily challenging application indeed study trait subject feature consideration association trait confirm non negligible association reasonable zero give literature set variety method base loss equip penalty include smoothly absolute selector give selective perspective selection include review recently establish selection approach distribution ask true rate show frequentist include compatibility matrix work allow conjugate prior enjoy posterior concentration special posterior minimax high describe concentration bayes truth rate consistency identifying propose markov chain monte sample study compare key provably concentration strong often schmidt dimensional specify prior incorporate decompose non zero assign put require suitable concentration prior restriction support primary impose practical e seven insufficient rank admit equation number interpret reasonable restrict support prior least square estimator parameter center square center summarize datum bayes restriction hold clear dependence obviously assume subset sparse alternatively could rewrite model modification regular inverse inverse stick restriction likelihood ny x proportional identity feature center greedy closely interesting approach fractional center center fractional regularization tool variety dimensional theory arbitrarily close see make flat conditional fractional subset empirical bayes bayes fractional bayes q fractional prior bayes dependent explain rescale help namely track property though want asymptotic array setup something provide notation array size call problem follow call exponentially datum generate cardinality quantity generic event probability denominator useful throughout fix cs result characterize concentration mean though simple specify notion consider equation would term act concentration empirical remark provide numerator see intuition make big justification lemma argument lemma write expectation lemma lemma eq empirical vanish bayes sufficiently concentrate big condition concentration discussion admit condition inside assume next take rates concentration distribution ordinary detect adjust high bounding phase parameter interesting open question behavior plug phase condition satisfie claim calculation prior satisfy expectation trivial bind could dimensional formula sum confirm bound prior bayes minimax ordinary equation constant condition remark claim complexity low ratio inside vanish p putting conclude prior yield posterior consider constant calculation binomial coefficient q last easy confirm corresponding rate next mass previous example equation fall subspace concentrated question consideration let effective posterior probability vanish numerator lemma ns see sufficiently upper since sufficiently tail condition upper dimension satisfie effective proportional claim light second order summation bound particular summation clearly therefore vanish prove claim learn provide light tail concentrate posterior bayes neighborhood directly norm dimensional eigenvalue possible maximal diagonal quantity call scale dimension definition facilitate norm get present follow observation control formulation put trivially indeed result frequentist ready concentration prior hold eq follow last upper vanish deal extra place first compatibility important identify
ascent prox sdca guarantee quantity result rather optimization improve reduce specifically study sampling sdca importance prox sgd adopt throughout process algorithm employ variance prox sdca propose importance achieve extensive rate importance suitable prox prox sdca verify traditional sample coordinate sample although uniform simplify insufficient introduce result sampling reduce rate proximal stochastic prox mirror ascent prox prox traditional sample since vary improve convergence propose corresponding gradient estimator analyze distribution proportional gradient simplify computation use improve exist uniformly prox sgd special prox sdca traditional ascent sdca pick show sdca converge cyclic order appropriately importance strategy optimal sampling distribution parameter convergence rate exist uniformly prox sdca special paper organize review work importance list empirical evaluation stochastic proximal coordinate proximal stochastic study approximation asymptotic term finite sgd prediction study general prox achieve rate loss recently researcher previous average return average average may issue old polynomially dual ascent zhang enjoy rate loss researcher non ascent obtain convergence investigate sdca dual primal learning recently zhang provide show convergence rate duality sdca loss sgd prox sdca study noticed importance stochastic similar variant iterative equation select pointed algorithm objective mirror descent cover furthermore prox sdca addition zhang version stochastic mirror convergence sag smooth sampling algorithm shall mention researcher result directly proximal fold prox sdca paper rate duality rely structure primal descent distribution prox primal addition distribution notice mini batch sdca version prox sdca version use inner sdca therefore apply accelerate convergence inner paper focus effectiveness prox non online regard extensively sample selective purpose selective sample label goal reduce label need certain importance key definition throughout vector function respect function lipschitz respect dual strongly dual norm define differentiable value taylor expansion example usual strongly optimization predictor regularization regularizer problem fall optimum analyze rate respect iteration stochastic mirror importance descent proximal prox sgd abuse mirror descent directly full stochastic descent satisfy desirable however efficiently proximal iteration randomly bregman minimize iterate regularizer trade objective unbiased efficiently assume proximal mapping operation wise proximal standard introduce variance prox importance adopt proximal descent sgd importance iterate np derivative follow implicit rule subgradient attract indicate prox define update norm strongly np variance np np combine inequality p second side combine f plug equality side p term full f f inequality due equality plugging conclude next study reduce rigorous result value choose easy stochastic inefficient issue calculate keep change parameter still inefficient solution relax introduce right inequality suggest firstly suggest smooth finally summarize proximal importance update section provide algorithm present assumption well importance convex norm strongly convex take firstly satisfy assumption p p divide conclude smooth p p achievable factor remove property easy derive use strategy improve variance explicitly v plug part prove part equality accord sampling strongly norm fact assumption corollary n page conclude derive set n ig plugging conclude first lipschitz plug observe bound bound lipschitz n schwarz importance hold give follow hold page side conclude lipschitz tn plugging proof plug adopt r r r l importance improve r ascent sdca importance prox sdca proximal ascent sdca uniformly pick dual maximize follow however prox adopt vector sdca prox sampling pick th element pick coordinate prox sdca main interested optimally accelerate sdca introduce prox sdca r update dual simplify nr n therefore respect choice ns note therefore plug ds many easy accord maximize dual ascent solution p relax r I optimize omit set optimize inequality guarantee ascent set combine fact inequality I inequality duality gap optimize p il sdca importance sampling ht n nn np ti option objective option option iii remark easy option bad option iv loss option option optimize option iii choose optimize three option lemma iv replace option option ii choose proposition subsection convergence present follow duality n ni sdca suffice lemma combine equality tn furthermore n n adopt uniform I accord conclusion importance improve especially smooth sdca propose sdca duality suffice n tell indicate n expand n inequality third hold inductive rearrange term imply eq combine small also n overall proof remark replace valid sampling theorem I max convergence numerous propose list mirror descent task solve typical large categorization bag sgd proximal sdca regularizer loss convex optimal svm project iterative solution theoretical analysis still way easy get form ny hinge solution euclidean distance analysis set sdca r plug equation suppose interest optimize hinge improve interpretability still regularizer previous approximated approximated adopt
projection inaccurate task neighbor demonstrate bit outperform hash answer retrieval cosine prove lsh establish cosine term cosine similarity binary datum view cosine similarity clearly illustrate freedom fortunately bind purely bind high similarity upper low note overlap high handle interestingly six dim news every query query rank point plot median among similarity separately dash solid dot query list cosine similarity panel plot among similarity cosine solid together news match panel figure similarity train cosine number compute vary curve cosine clearly formalism approximate approximate near neighbor parameter report neighbor notion deal similarity near point theory lsh lsh uniformly typically distance analogy requirement lsh family link lsh us structure provably time nn family measure lsh lsh lsh lsh report neighbor additional lsh depend hash lsh hashing permutation hash hash I lsh sign random utilize sign highlight show comparable similarity measure preferable vice lsh gold x immediate consequence corollary combine lsh hash cosine output integer recently provide simple informative section theoretically datum bit hash three study nevertheless similarity comparison outperform case obviously replace go lsh cosine note bad case still gap confirm outperform even z z figure less competitive similarity expect analyze hash even bit performance bad conservative call surprisingly confirm outperform somewhat optimistic parameterized lsh hash hash table hash top every gold neighbor use cosine underlie measure dependent similarity threshold hash consideration task threshold vary actual ideal implement combination hash function mean gold neighbor report number need percentage recall gold choice plot fraction retrieve good retrieve total consistently case irrespective choice standard computing similarity favor lsh cosine similarity despite disadvantage outperform cosine confirm add mnist evaluate retrieval similarity place figure although improvement figure hashing originally detect page widely adopt numerous web spam online web graph hash substantially preprocesse cost make practical however take theoretically decide base desire cosine provably lsh cosine provide provable compare different experimental evidence indicate computational advantage lsh wide practitioner view ph student nsf dms support fa nsf less obvious step qp pf achievable rational cauchy rational continuity c pf
address estimate unified deal quantile nest randomly truncate support central limit extensively heavy soon nest relate mass quadrature multidimensional generalise divide subset twice first derivative bound estimate arise rare complex necessarily continuous failure writing go split property interact furthermore gain link process indeed connection nest remain unclear fill core tool value link general carlo apply carlo central limit stop enable possible generate conditional law identify nest carlo propose one deal implementation numerical study heavy tailed carlo non random truncation appendix consider variable case reaction coordinate assume negative common increase walk define recursively sequence conditionally great arrival event x estimator especially generate marked exhibit asymptotically rao compare naive monte c monte pt quantile manner bit interested refer detail value variable idea optimal build point simulate associate mark poisson process markov sort marked consider come precisely nest nest one use expand poisson unbiased proposition finite moment order especially heavy tail much weak globally monte one monte require choice estimator define simulate infinite sum nest sampling propose sum criterion unbiased stochastic sde path recently two address issue idea biased convergence reduce construct unbiased base biased one basically simulate chain combine final truncate integer random independent nest stop sum randomly seem use keep consistency notation aim estimate order reasoning give independent bring give u measure interest forward expect go relatively probable nothing go process furthermore sequence corollary xx ie bring proof end generalise imply light tail remain especially interesting computational call simulator chain simulation bring convergence n furthermore power expansion dominate rewrite bring together close shall far limitation behind effort show product non optimisation assume decrease pareto distribution context auxiliary show bring solution c contrary bring constraint solve optimisation iteratively q q estimator recall marked process necessary tail optimal exponentially low present exact optimisation pareto finally present framework optimal resolution furthermore pareto decrease decrease turn theorem computation demand computer play key sequence decrease shift parametric e n e argument function choose truncate one especially power optimisation become hence reach one rate argument apply consider sample context chain relatively big already nc start basically change account update ns procedure stop become time estimate simulation computational budget computational know advance budget end budget mean compute implementation evidence show error estimate evidence nest hold seem choose simulated remark twice infinite fact compare usual notice tail consequently sufficient become approximately nested approximate hand still perform nest enable increase variance give new heavy tailed identify often address index estimation tailed sequel give pareto analytic formulae ideal monte carlo pareto pareto variance third one clearly visible moment density estimator monte pareto central limit generalise central limit law characteristic pareto decrease state proposition decrease q n exact give asymptotic approximation growth rate pareto rewrite bring bring hand equal order result display comparison distribution monte carlo variance explain get expression instead denote derive value furthermore instead denote dotted dash good variance far implement truncation estimator geometric well combinatorial turn computational scope parametric much simple aim depend parametric parameter fig pdf budget also deviation chain require infinite call certain competitive solid parameter almost optimal monte optimal implementation soon implementation soon confirm tailed variable proposition speak condition result test result one nest pareto nz nx multiply parametric gain chain latter initial pick ht proposition mh call simulator call simulator cf remark give formula especially quantile perform estimate standard deviation uncertainty quantification triangular moreover reason coefficient bottom bank width interested quantify estimate water variable vector independent transformation analytical quadrature approximation bring nest stop budget nest algorithm stop simulation monte nest nest negative easy limitation chain run previous expect handle leave tail split budget tail b g
finish discussion communication protocol present constant family compact instance bernoulli take hypercube receive single observation universal machine proof estimation scale bind receive generate bernoulli bit fusion center fusion center compute square bounded note family gap describe minimax square interactive turn feedback interactive message machine freedom powerful begin uniform family p direct nearly uniform q conversely dc eq receive minimax centralized scale bit require centralized machine bit whether show nearly identical receive universal see slightly weak corresponding interactive bit distribute allow interactive attain minimax logarithmic factor bit nearly dimension gap solution compute distribute manner specifically describe hand nearly scale dramatically scale sharp require careful logarithmic minor open rate constant whether scaling differ problem previous though closely complex section low estimation low generalize probit fix machine store independent goal unknown vector small large rescale matrix linear universal classical e scale corollary budget achievable protocol logarithmic factor bind achieve separately solution lower allow low design semidefinite regression solve reduction protocol pair expression turn binary particular draw denote cdf cf universal turn lower bind show probit least regression probit linear response probit regression inspection probit problem error construction low shorthand point pack conditional x determine send pack upper mutual shannon source yield slight refinement proof somewhat variant distribution member define suppose fix hamming distance pack hamming index suppose fix find distance uniformly infimum range observation set recover testing flexibility identify hamming variant control chain uniform pack neighborhood control challenge allow bit tight inequality sequel receive sample note moreover condition independent quantitative ratio contraction precede quantitative processing context privacy chain likelihood ratio require valid protocol appendix valid remainder break namely information since inequality follow code put piece upper mutual lemma low proof let hypothesis construction finally machine demonstrating sufficiently sized throughout provide section first quantitative processing analogous conditionally packing denote iff leave indexing implicit state precede paragraph involve indicator fix lemma number pair satisfy p mutual proof prove upper mutual use theorem shannon simplify communication relation lemma le imply complete protocol packing packing invert local minimum coordinate machine accuracy round machine initialize global machine operation index update list bit I bit clear global quantization yield minimax protocol step machine bit bit send index j piece machine draw across machine addition sample allow interactive protocol message dependent measurable machine simply value say nothing require analogue lemma protocol sequel reduce multi dimension statement abstract bit specific likelihood provide indicator state building dimensional make concrete bind apply pair negative apply intermediate bound chain conditioning reduce turn mutual equality conditioning reduce iii analog entirely analogous involve minor setting follow paper establish amount problem theoretic rely quantitative characterize bit constraint information message question argument differ logarithmic would interesting inference protocol improve require insight believe perhaps support laboratory research office office research science grant support facebook fellowship body paper rely inequality contraction develop present proof lemma direct analytical lastly width graphical eps write combine pa bb bb db version lemma likelihood absolutely generality bb bb bb e bs express conditional shorthand correctness indeed bind recall figure product combine three display likelihood bb bb auxiliary negativity require argument build technical independence expand kl bind lastly argue message conditionally give shorthand role definition collect must notational prove complete proof turn prove conditioning consequence recall final follow end moreover assumption apply yield cf proceeding eq classical conditional kl eq prove inequality two variance consequence addition condition recall lemma challenge however regardless yield note q choose choice desire collect restriction measurable set mutual marginal marginalization q kl divergence ratio kl divergence claim bound q kl inequality rule mutual complete remain establish variable verify hold must independent conditional q give ki k hold satisfy precisely obtain condition everything event standard cf imply integrate lemma em zhang engineering berkeley berkeley edu conference often machine minimax compare quantity amount communication achieve centralized protocol channel interactive server message novel quantitative inequality characterize effect communication rapid growth modern set computer natural involve computer yet machine expensive slow power intensive survey parallel system consumption bandwidth limitation inter impose significant algorithmic important study require machine large class estimate unknown classical minimax rate characterize bad centralized machine intermediate processor fusion try answer minimal realize minimax g characterize range decentralize g substantial communication though relate formulation bivariate protocol bit classical protocol bit randomization theoretic guarantee contrast characteristic difference communication however communication imply setting decentralize communication low bound distribute certain conclusion message send rise consider machine bit processor receive single dimensional centralized study estimation particular work focus store machine encoding sequence rate converge formulation communication attain centralize say finite ask statistical rate paper decentralize minimax protocol single message pass interactive protocol must protocol fusion center past message depend protocol simplicity use indicate send independent enforce shorthand contrast protocol interactive protocol stage message particular message fusion public machine reading incur communication think may processor centralize message message measurable message interactive protocol define risk central achievable message classical quality protocol estimator
brevity vector correspond complement eq compatibility eq bound compatibility later certain law ready direct consequence oracle eq implementation ensure asymptotically problem et although need consider value p nc oracle corollary suggest method maximize although compatibility sufficient compatibility would correlate index index whose corresponding thus low theorem compatibility corollary satisfied size xy irrelevant parameter follow construct drop column index implement find objective x large optimization quasi newton prefer fast super necessarily algorithm optimization bfgs memory test smooth bfgs hybrid result implement bfgs cp eliminate remain minimize coefficient column adapt use informed involve unnecessary rt true example select elastic next positive elastic net well correlate overall perform application cancer et al sample patient I severe patient stage mass scan disease could protein fit peak deterministic peak algorithm al obtain z peak location severe reasonable intensity peak available www mixed stage peak peaks choose et al elastic choose peak peak al peak common peak exist sparse explanation useful implement without impose assumption notable restrictive number example fan al extended analyse use regression propose know art predictive seem high dimensional situation promise sparse toolbox valuable comment insight manuscript new variable variable mean exhibit grouping screening result theorem standard deviation performance grouping penalization screen dimensional great predict response set add least predictor wide achieve one essential goal regression identify literature area early high dimensional minimize penalty ridge regression elastic combination penalty include selector predictor sure root involve minimize plus call recovery require noise euclidean base norm gain group property elastic net distance variety particularly signal variable matrix regression matrix set associate whose observation could however reconstruction parameter solution independent challenge detect minimize euclidean penalty minimize distance minimizer function norm one combine manner net I criterion ridge elastic net combine features ridge penalty lasso like square root grouping transformation design response note covariate nonetheless positive number large minimization exclude exact solution denote angle global must complement index build objective circle distance highly estimate standardized objective enable impose restrictive concept simple relative component relative grouping relative f p theorems minimizer penalize euclidean consider important highly base relative minimizer standardize response tx jx ix grouping perfectly correlate special grouping detect
lexical relation rather design lexical need lexical cover division company sub part mention herein also cover lexical water death directly whereas lexical directional specifie entail hold agree part whole lexical issue break category category ten claim eight lexical cover water death water instance cause relation handle table believe argument category incorrect offer lexical category relation hypothesis semantic relation treat algorithm research even wrong reason treat relation use relation evaluate train lexical case lexical readily expand discover place system include module early typically use lin understand inherently asymmetric symmetric rough replace without change sentence specific general paper measure degree entail describe lexical lexical lexical include accept kind lexical exclude lexical context natural paper lexical body relation classification semantic part semantic semantic relation nine task relational nine semantic lexical paper relation important involve e algorithm semantic relation supervise although lexical offer elegant training issue noun phrase noun entail head noun cat entail cat little effort seem applicable pair label amazon cover wide range semantic measure value entail generate score give example belong ap measure value score measure accuracy precision distributional inclusion partly inspire precision useful ap retrieval system query engine return rank document degree relevance assume relevant fraction rank list document label ap range ap several ap typical typical document irrelevant emphasis entail example scoring challenge equal text sentence entail therefore ap precision respect pair manually dataset measure assign word sort pair bottom rank label otherwise let document bottom experiment increase sensitive top happen bottom list low ignore prefer top list poorly list ap originally design retrieval precision truly query system truly tradeoff precision cost harmonic design natural size equal sized difficult class well balanced ap measure depend class practice two variation weight average finally usual accuracy fraction percentage discuss describe introduction behind word survey development tune optimize describe matrix development context choose vector svms supervise svms radial rbf experimentally find similarity measure asymmetric measure lin achieved mean eq lin terminology terminology terminology algebraic terminology reader easily believe helpful connect view notation word matrix correspond word row raw occurrence transform represent importance negative survey association raw set notation word correspond think word correspond nonzero cell value range thus normalize range close interpret importance feature information retrieval context inclusion tend context tend broad term include feature q range among feature analogous ready originally measure retrieval consider ranking include otherwise word weight word seem algebraic lin lin combine varied remove ranking impact low pair classified entail describe support utility row row correspond gram whether appear context gram window percentage engine raw occurrence frequency matrix gram th matrix calculate frequency retrieve detail matrix truncate value truncate equation density context column sum behind sparse assumption false well smoothing vector label entail enable cell th word th experiment context smooth svd svd three matrix orthonormal I unit top singular matrix k minimize error k norm vector vector represent concatenation normalize need singular power familiar information retrieval use word matrix lexical semantic evaluation follow mention say tendency learnable occur contexts context tend give concatenation lexical minimal probability binary value class fit output regression try kernel dataset polynomial length lexical thing know lexical reliably recognize pair appear datum broad range domain domain matrix design measure similarity topic domain measure similarity similarity relationship usage function near word occur part varied work context combination domain tune domain function matrix corpus column correspond entry gram complex column aspect matrix remove row value angle likewise let recall difference hypothesis tendency correlate learnable difference tend sim domain similarity reference difference spatial death suggest involve whereas death involve suggest similarity perhaps similar similarity death perhaps death see make reference english represent english vocabulary english wide range concept may inefficient hand look example supervised development function setting normalization generate class describe dataset lexical semantic lexical report evaluate word pair label entail create add pair labeling detail lexical label pair agreement two three size entail every word time bank education state fortunately use describe entail label noun noun pair pair validate although balanced semantic relation word pair appear vector create measure relational package gold rating dataset contain word label convert label word relation ten level category five ten nine distinct type come add inclusion schema song phase song amazon worker phase nine ask word semantic nine pair relation example original lexical improve ten low rate original pair reduce pair word pair label example car object create car increase pair map entail mapping word pair belong either none map word label label balance make balanced dataset remove pair label pair interpret label see table f goal entail f text however temperature due nine semantic functional class inclusion collective inclusion collective inclusion car whole member mass moment stage activity item room whole object ice j ex car similarity similar similar sound contrast contrary contrast reverse directional right slow g walk attribute attribute attribute object attribute attribute act attribute typical live act act attribute slow attribute act object act example object attribute act attribute act act act object patient relation act relations act relation object speech relation relation cause cause cause object purpose agent peak g cause cause purpose time product activity drive location reference representation person create mapping agreement table consensus label consensus five class class independently annotation entail paradigm instance relational schema interpret pair light schema interpret category likewise entail pair entail wrong proceeding compare table table mention assume none pair label label relational lexical lexical table percentage agreement manual automatic annotation percent versus percent agreement versus percent percent percent lexical manual varied assumption word pair belong reasonable manual agreement level lexical discuss table manual relational support hypothesis report pair entail support relational lexical capture automate manual relational definition manual labeling accordance evaluate lexical first word equal test size maintain test ten level table class inclusion class category inclusion pair inclusion entail category class inclusion whole contrast attribute attribute relation total five optimize set maximized measure accuracy tune tune try try increment pair achieve setting use data tune domain value respect test three measure accuracy usually column confidence f acc accuracy ten category set substantially expect approach near attribute substantially g attribute relation category part similar attribute cause explore contribution exclude similarity reference difference different individually tune acc difference accuracy significant together fisher level space two base discussion general dataset select set matrix choose bold font seem might difference difference significantly pre f acc word pair way experimental splitting evaluation validation evaluation pair common supervise easy pair share fold cluster evaluation evaluation cluster ten fold fold ten share two give fold allow rarely pair fold clustered evaluation entail balance clustered step remove pair fold train dataset balance randomly remove label measure threshold four fold fold validation whole split tune supervise cluster balanced challenge believe evaluation realistic system module pair field evaluation come field usage acc standard cluster cluster balanced evaluation achieve accuracy statistically accuracy supervise standard testing tune threshold gap question qualitative training testing helpful qualitative gap qualitative gap challenge quantitative face past comparable evaluate recall measure set use word class way balanced setup already evaluation table evaluation accuracy fisher test f acc cluster standard use setup likely minor setup accuracy dataset tune give context accuracy setup accuracy setup seem dataset nonetheless accuracy summarize evaluation bold accuracy accuracy nine type accordance relational definition lexical explain design mind perform poorly surprising well lexical difference learn dataset dataset cope qualitative difference positive relation relation dataset reach qualitative able bridge argue entail also summary report instead put emphasis entail clustered setup bold difference similarity difference model lexical suggest beneficial construct lexical manually lexical tractable supervised effort involve designing indicate supervised yield manually design evaluate application module well competition room contextual substantial future lexical derive score show novel hypothesis achieve believe progress come wide hypothesis lexical semantic relation handle hypothesis reject algorithms three degree task combine one voting value focus idea phrase promise phrase table see density class whole strong lexical relation relation relation find particularly inference rely lexical find difference three suggest difference make feature recognize lexical lexical learn semantic result lexical build bridge lexical semantic hope field acknowledgement thank copy answering thank provide answer thank natural engineering helpful comment lexical r national lexical identifying entail one construct asymmetric treat learn recognize machine relation experiment strategy similarity context word second relation represent pair concatenation supervise feature instance word feature similarity semantic extensive three similarity dataset dataset past make connection lexical semantic language relevance question answer translation involve sentence determine whether entail gold establish text entail mean infer mean text typical interpretation text entail rich challenging paragraph many recognize entail recognize text entail entail lexical definition lexical useful lexical semantic apply lexical three matrix represent word context context consist distributional distributional hypothesis occur tend algorithm average precision distributional inclusion attempt context capture inclusion word broad distributional inclusion prefer call distributional inclusion
goal learner modern make effective decade great broad quality intensive design code feedback ii scalable ml base learner learner learner obtain content content question automatic author recently sparfa sparfa knowledge give typically term leverage concept value response correct incorrect sparfa estimate association via concept profiles ordinal sparfa sparfa enable first ordinal sparfa tag exploit often sparfa tag exploit tag keyword characterize exploit question framework new pre nonetheless response superiority ordinal sparfa tag method ground truth real ordinal sparfa tag outperform art technique miss ordinal learner abstract response provide question sparfa characterize learner incorrect response question I question association learner question difficulty extend sparfa underlying score learner order label response encode question model positive quantity uncertainty learner answer incorrectly gaussian learner answering question slack variable set learner order accord quantization bin satisfy upper equivalent relation x ss represent normal matrix intrinsic form furthermore matrix low bin emphasize propose original sparfa ordinal affect statistical parameter ill pose since ordinal three accounting redundancy assessment response live dimensional question e question chance score value good vice mm reasonable context discussion particular often question learner mm tag question association tag single significantly improve limited interpretability sparfa rely ad hoc processing provide tag concept contrast tag oracle explanatory predefine tag tag complete association predefine association develop sparfa ordinal tag enforce maximize subject account prevent norm norm freedom overfitte different preference observation problem problem hold third coordinate variable normal variable set iteratively factor hold optimize hold hold outer loop ordinal sparfa iterative optimize precision instead bin boundary intrinsic difficulty bin instead emphasize optimization straightforwardly via keep ordinal sparfa directly bin keep fista fall fista constraint penalty regularizer otherwise algorithm two py shrinkage step suitable size form aggregate eq singular decomposition suitable constant step value ordinal sparfa tag tag question tag tag penalty predefine predefine might support reduce discover concept solve analogously except part operate separately index index give ordinal tag necessarily converge global optimum result ordinal sparfa tag optimum furthermore close optimum sparfa tag optimum sparfa tag synthetic ground truth leverage tag constraint two real ordinal sparfa collaborative ordinal response ordinal sparfa estimating ordinal sparfa tag priori synthetic experiment trial retrieve ordinal sparfa scale consider concept arithmetic simplifying expression concept concept geometry slope concept polynomial concept concept matter water heat energy concept circuit force formation motion concept concept concept property environmental energy force synthetic test evenly concept question size fix concept impact size first ordinal sparfa svd vary corresponding learner fix ordinal sparfa ordinal sparfa oracle k svd ordinal sparfa variant sparfa know accurately impact quantization bin bin quantization need value outperform svd decrease conventional sparfa sparfa approach svd quantization bin increase superiority ordinal sparfa tag sparfa particular tag impose nuclear constraint matrix tag oracle provide tag algebra dataset school carry crowd multiple question cover geometry graph manually domain expert manually map bin follow totally wrong correct show concept ordinal sparfa tag circle concept question label difficulty connect line line association sparfa tag entry red dash green solid association entry discover ordinal sparfa tag sparfa tag concept enable interpretable learner directly tag learner tag profile use pls learners association tool expert tag association course ordinal sparfa tag analyze answer incomplete entry label tag sparfa tag nuclear match pre association association discover pre tag domain expert association school algebra pre specify tag indeed question interestingly eigen learner investigation phenomenon learner response collaborative filter treat ordinal number rely ordinal logit ordinal optimize ii bins learner iv nuclear constraint test train fold rmse demonstrate nuclear norm ordinal sparfa ordinal sparfa suggest consider ordinal enable accurate prediction response variant sparfa boundary identical emphasize sparfa state predicting response also interpretable key application bayesian belief analyze trace learner concept
q vanishe insight overlap neighborhood via overlap bias intersection bias conservative bias g guarantee e together factorize intersection r ss q v c f conjecture institute cs university sample great large problematic grow scalable planning factored exploit apply planning reinforcement able solution planning planning planning achieve performance gain carlo solving become intractable grow problematic system specifically observable agent view action centralize agent intractable novel online planning exploit structure base exploit mass agent interact subset value factor factor applicable important adaptive translate problem planning planning planning planning efficient non factor able effectively exploit locality bayesian plan every agent receive allow team agent act centralize communication free cost delay tuple state reward immediate observation horizon controller joint receive joint remainder section inherent many focus full specification determine belief policy extract rs pz space continuous scalable monte plan current belief expensive create simply action history root use trajectory visit search relevant action history reach time domain consider know effective rl call bayes ba planning enable advance plan particular ba utilize intuitively observation result take represent see action transition state actual count count ba ba extend yield adaptive ba ba state count vector reduction possible intractable sample planning scale suitable planning elaborate sketch locality agent problematic branching though theoretically use severe must often plan previous sample particle filter bad act independent action confidence joint exploration principled may completely try action individual illustrated amount water house action factorize approximation certain compactly represent interaction paper shot cg specify payoff interpret select cg follow cite suitable factorization easily identifiable payoff function exactly factor maximization perform algorithm apply advance necessarily require seek joint close maximum unknown technique contrast factor directly try estimate maximize action action predict mixture joint scope use maximization remainder integrate simply expert expert particular keep mean payoff efficiently additional allow keep track turn integration describe algorithmic factored planning exploit expert variant address joint second address joint mass achieve complexity factorization joint method beneficial factor remain retained maintain maintain set component accord factor u q history store value count style parent style thick distance black draw parent thick circle mm draw child edge style node circle statistic via application ba directly address joint possible realize limit call try overcome joint factor conduct maximize upper bound set agent relevant generalize maintain node chance produce particle distance node draw child parent thick distance child edge parent draw thick mm child edge style node draw thick level circle circle child circle parent mm black child style draw edge child draw thick sufficiently factored factor suffer local depend future policy well past include long ft monte ft practice problem reinforcement network produce high exhibit locality factor good complexity modify elimination complexity width effectiveness compare factored planning sensor house also spread house align along axis reward track agent nothing agent correctly break factor side break along agent sensor agent agent experiment number core ghz gb compare flat code difference use planning already factor fs factored agent form simulator set compare action poor bt horizon poorly simulation increase converge ft number simulation able near well ft fs continue reach ft see sensor outperform fs simulation ft low planning target position factor episode illustrate benefit exploit mass agent ba end episode state count observable environment extremely observation indistinguishable therefore hard compare baseline apply simulation proxy expect ba agent fs ft number ft learn quickly fs horizon due horizon ba factor outperform visible grid ft fs outperform ba episode learn effectively game game continuous action action available many also design observable approximation contrast promise branch factorization agent individual reward reward incorporate consider decentralize factored distribute nd strict assumption action past observation nd impose factored function restriction instead know action factor nd method mapping factor locality factored conditioning central information perfect locality factorization function apply factored fashion simulator upon perform factorization shot relate since replace obvious integrate focus minimize issue factor relevant receive centralized maker potentially relax exploit approach joint observation grow agent na I agent exploit structure novel factor factored tree greatly increase scalability planning investigation scalability four ba space self interested support fa describe start root empty sample current comment algorithm filter simulate pseudo factor comment highlight joint initialization select maximize action return component e n simulate e factor ft particle filter update computational
fact distinguish large necessary large enough anomalous object detection introduce mmd construction test present computational brief introduction idea mmd include distribution rkh mapping refer hilbert reproduce p embed element many gaussian laplace study embedding without distinguish discrepancy mmd rkhs shown namely achieve maximum unit sample given construct interval compute expect anomalous differently determined corollary follow characterize anomalous test successful constant threshold boundedness many gaussian kernel theorem imply minimum candidate anomalous interval exclude anomalous require event get asymptotically small corollary threshold constant unknown priori satisfie minimum corollary prior capability anomalous resolve anomalous length resolve big anomalous first dyadic interval dyadic let dyadic show interval dyadic interval dyadic union dyadic error occur u x go infinity test demonstrate unknown numerical two mixture anomalous object bernoulli distribution plot normalize see converge agree state ht ht test size change minimum size error although respectively test htb affect study plot probability error versus correspond similar first dropping gets imply guarantee mmd unknown mmd choose laplace likely threshold go infinity minimum anomalous suggest theorem run mmd advance set average unknown mmd demonstrate asymptotically agree minimum length mmd much fast case importance prior knowledge mmd paper investigate anomalous arbitrary unknown mmd embedding rkh anomalous equal successful infinity reduce mmd guarantee successful technique study believe involve distinguish compare test anomalous line anomalous interval distribution generate node anomalous interval sample generate anomalous reproduce mmd show go anomalous interval asymptotically reduction show interval problem goal detect existence anomalous anomalous network take anomalous exist object nan compound object may sensor sensor take measurement anomalous typically occurrence arise anomalous dna detect anomalous detecting existence node embed embed structure line node multiscale analyze model study scan statistic result inference mean mean node structure anomalous subgraph detect unknown network graph combinatorial geometric small cluster anomalous connect subgraph structure detect anomalous incorporate successful majority variable application case differ mean either advance hence anomalous interval priori detect anomalous although simple already study deal reproduce introduction distinguish evaluate embedding approach sample discrepancy mmd embedding distribution shift existence anomalous size become model anomalous accurately interval object successful goal characterize length anomalous node infinity order successfully existence anomalous summarize main nonparametric model detect anomalous interval network length characteristic must successful large remark anomalous artificial test mmd test infinity candidate anomalous interval scale successfully detect exist anomalous interval length depend mmd adapt efficient algorithm test mmd performance propose numerical organize performance guarantee provide numerical result remark future consecutive node length interval denote associated variable set candidate anomalous minimum candidate anomalous impose requirement explain remark distribution line random variable put context scenario noise activate anomalous arbitrary instead one sample practical system collect stage detection activate occur initial serve hypothesis scenario node detector
user item rank triple item tweet predict item receive past predict user f rank user prediction base learner multiple tree gradient adjust forest subset split dataset user prune keep tree leave shrinkage early additional validation round feature split rely upon implementation directly goal ndcg summarize feedback express rate simple recommender ndcg factorization plot rating relationship rating tend receive item outperform fm fm tweet rating alone engineering fm include baseline write place team challenge fm competition ir metric ndcg result collaborative variability find factorization item work well square also help improve outlier help individual ndcg tweet ndcg threshold selective sampling successfully knn model rate try improve collect movie profile g release tag actor find year release improve paper base dataset provide evaluation protocol reflect tweet receive compare user assess triple tweet receive triple identify predict level g require user user popularity twitter access characteristic velocity recently author make effort characterize exist approach introduce aware recommender additional contextual company people recommendation rich collaborative art context recommender primarily collaborative rank cast recommendation rank absence rich feature learn rank rich user tweet part ranking construct neighborhood collaborative collaborative user twitter twitter build feature tweet capture rating express tweet collaborative attractive tweet insight business success public neighborhood performance direction future datum city usa ie web service amazon netflix twitter recommend item present would history cf focus assess recommend recommender rather characterize optimize item cast tweet transaction rating learn scoring optimize user ndcg conduct version challenge effectiveness information storage artificial filter deal information user base netflix amazon twitter customer recommendation rating prediction recommendation cast among item page article book place retrieve list advance year supervise application gain relatively little potential reason rank usually rely characterize maintain knn prefer cf item collaborative extract user tweet create amenable leverage directly ir ndcg user task training tweet provide tweet interaction possible tweet user construct rank rank test sense ndcg test unseen ranking return rank tweet relevance preference tweet comment movie video give tweet movie tweet create relevance indicate tweet document triple training number triple rank value learn feature item tweet respect I history boolean rating friend tweet give movie otherwise rate ratio friend aggregate tweet aggregate boolean mention update tweet tweet tweet infer tweet actually use include tweet tweet additional observe count extract field item tweet extract user tweet tweet parameter ndcg optimize ir ndcg additive tree pairwise learn method capable ir metric change iterate detailed scope refer comprehensive dataset part challenge challenge summarize table note one respectively figure interaction user item tweet triple note user outlier
cnn convolution size convolution pooling follow pooling extend first regularize tie train similar rate momentum epoch ten tune get final cifar cifar view arbitrary become scale indistinguishable large equation scale activation canonical filter image describe section transform filter norm original take filter cnn report filter invariance measure q take random set row filter learn filter learn row pooling usually invariance filter accordingly size clear sensitive column transformation need consider back propagation normalization become apparent transform scale pattern achieve invariance scale comparable filter adapt consequently invariance filter get example feature map pool normalization original middle filter clear applying preserve characteristic filter scale fix significantly train scan activation pooling layer scale pick filter visualize dataset scale cnn cifar statistically gain cifar drop drop central scale cifar verify pick different central area cnn compare drop whereas htbp lc dropout cnn cnn maxout maxout network error gain cnn network nevertheless address goal maxout simply extra result encourage benchmark higher suggest take suffer severe error reach model cifar error scale cifar incremental invariance come current cost linearly column refine expect explored name cnn begin refine epoch name continue baseline cnn use build refine softmax small incremental summarize row half cost fourth although get well cnn maxout unit able reach result incremental help balance gain efficiency incorporate flip cnn learn column learn detection localization task preliminary nice trade training remain concatenation column wise column instead canonical column plan imagenet zhang microsoft microsoft zhang convolutional human popular make big data augmentation extensive convolutional neural design incorporate column column focus scale filter transformation deal experimental scale exhibit classical vision primarily amount training convolution network competition cnn human image localization trying learn contribute field allow recognize regardless pooling contribute slight cnn deal shift invariance capture plain cnn rotation introduce filter cope magnitude proposal deal size scale use explore observe filter detect adopt design column scale cnn unlike filter among column exhibit deal indeed scale become dataset well previous complementary technique find incremental refinement dramatically reduce organized present foundation analysis one scale free stack convolution filter build complex representation representation mean activation layer job exist cnn architecture jointly scale variant map unit field save filter learn popular inspire exist cnn scale independent cnn specialize one crucially parameter convolution stay augmentation intuition mathematical htbp column stack scale image feed max pooling conventional filter share filter column keep canonical call canonical filter detect architecture top softmax case discuss canonical canonical convolution image scale expect another transform generate another convolution property want satisfy easy filter invariance preserve layer reach feed separately relu nonlinear max keep recursively keep relationship canonical image column generate fit column scale fall scale two neighbor column end eliminate variance column convolution transformation give canonical equation transform filter system problem make easy equation
perfect portion energy try capture energy important tolerance solving unique solution full give desire framework base mode supervise include experiment design optimization category sampling notion dimensionality measure signal variation manifold laplacian maximize try relate zhang experiment design consider way embed approximate try whole datum local reconstruction generalization global relate note eigenvalue speak function maximize try ensure unlabeled cf section graph work node maximize maximize unlabele agree give justification cut also motivate partition base heuristic say node graph compare three active allow prediction implementation method good cut learn package filter method fix accuracy also randomly sample rate effectiveness circle toy figure comprise connect ensure connection x unweighted see circle additionally evenly select circle accordance handwritten digit letter computational complexity scalable dataset graph typical experiment semi supervise classification handwritten digits pixel select digit vector digit construct w ij intensity rd th heuristic additionally restrict e remove node near neighbor construct label report prediction use semi supervise repeat illustrate notable criterion maximize text partition experiment document graphic os ms mac hardware clean remove document frequent feature tf statistic capture word corpus frequency document document feature document pairwise similarity feature vector error inherently letter consider letter alphabet set alphabet create consider alphabet digit construct weight node rd neighbor neighbor criterion perform semi label chance select point class effect repeat accuracy remain largely unchanged dataset slight improvement result agree membership frequency well cut word loose cut membership fraction maximize tight pick capture novel batch active semi signal active uniquely signal lead efficient intuition try efficient conjunction term cut hope desire accuracy useful batch batch improve offline graph vertex undirected graph similarity capture sample graph shannon signal reconstruct base selection frequency effectiveness classifier design pattern many real available effective learning technique inherent expensive learner pick informative representative label goal ability give small label query propose novel advance sampling graph stream problem pool semi supervise datum crowdsource would label focus batch label leave semi formulation node connect weight feature task function scalar depend whether belong choose label conversely space membership e unlikely lead membership semi view signal many technique harmonic consistency graph approach quantify quality methodology capture low graph pick node unlabele node pick lead semi inversion decomposition matrix pose implementation active suffer give uniquely significant well establish processing contribution wavelet depend local interpolation newly develop unified perform provide uniquely subset interpretation numerically make semi closely tie theoretically justify large show test rest review signal section derive active summarize experiment present section conclude remark formulate undirected context respective degree connect graph graph shall laplacian set eigenvector subset index submatrix sake brevity scalar value discrete ease notation paper signal membership interest graph realize signal value sample vector reduce sample include membership upper signal uniquely reconstruct notion fourier indicate variation high eigenvector basis signal smooth pass frequency vanish define restrict space signal note frame adequate graph label penalize thus equivalent relaxation nature study empty e include node energy si tends maximally find cut adapt cut discuss signal linear correspond recovery condition solve least square square eigen expensive iterative onto propose reconstruct case problem constraint pass graph graph domain operator iterative actual depict non asymptotically regular converge pass exact complexity possible use truncate spectral distribute fashion chebyshev smooth sigmoid limited supervise expect slowly decay end improve slightly signal semi supervise propose edge membership illustrate experimentally vertice membership see cut vertex set recover membership active aim sampling maximum good set data amount reconstruct strategy cut frequency maximize detail multi class class true membership class indicate membership signal membership predict node label unlabele supervised summarize label set q solve add summarize label finally membership function convex set connect simplify
inequality partition many union inequality eq b nm formulate loss distribution obtain generalization probability substitute substitute c cat mt max bad attribute front open cover blue negative one ac algorithm relate advantageous work process subsequent jointly find task criterion optimize expect classification able discover task study label achieve expensive consume application categorization information several relate experimentally allow transfer multi learning correspond relate parameter representation concentrate similarity distance propose corresponding task lie prototype effectiveness several treat realistic outli task similarity task negative scenario learner task source show vision detection hand image categorization though domain scenario multi area relate propose decompose adaptation school suppose learn process meaningful student gradually increase accumulate learn inspire propose manner previously learn process datum student school crucially performance question pac theory prove generalization representation use quantifie solve world multi jointly reliably discover advantageous idea approach literature represent linear combination allow overlap far extend experimentally subspace base performance underlie feature grow type vision problem fast method experiment setup clearly baseline even require method vector original particular relax task achieve introduce graph chen method amount regard similarity sequence study mainly experimentally gradually lead similarly automatically choose solve optimize pairwise preference sequence user scenario label one read vector term multi task object predict attribute share share assume learner predictor performance learner expected propose task domain adaptation specifically process sequentially order symmetric subsequent task domain method svm computer vision adaptive svm optimization q standard simplify vector equally need subsequent automatically define beneficial examine average svms algorithm task predictor inequality sampling set harmonic z gauss function hand inequality learner however distribution unknown contrast bind right I x monotonically specifically separate close wrong corresponding may subsequent task ensure lead right hand hold fix require search incremental procedure perform successively minimize already include order minimize every low fit human intuitive simple proceeding every summarize input minimizer return order task join task within subsequence iteratively learner solve continue learner empty transfer find continue compare subsequence continue task please refer claim task manner use publicly dataset difficulty svm solve svm accuracy baseline semantic human annotation inspire diversity instead diversity regularization trade experiment validation experiment figure outperform mt case task advantageous support claim task effective jointly task equally baseline improve single baseline generalization perform mt method example expect explain hyperplane unable share mt hyperplane g lead task solved reporting finding figure svms differ outperform learn baseline task order class semantic par case hardness coincide opposite task machine task medium random learning per easy medium define human visualize finally baseline visualize horizontal slice reflect competitive well fix clearly learn algorithm term combination achieve never sometimes even order beneficial strategy annotate attribute attribute top rank bottom class rank see balance randomly sample descriptor descriptor baseline attribute information transfer subsequence attribute order option ht average result main see baseline confirm advantageous equally strong affected transfer unable one perform poorly sequence compare task row task diversity baseline however perform well baseline conclude one affected transfer task diversity bad baseline pattern relate frequently subsequence next follow attribute always often end subsequence attribute either two task transfer attribute half relate solve task influence overall theoretical propose principled sequentially effective jointly solve effect overall able beneficial limitation transfer solve plan pac generalization sequential learner observe sequence I sample task share output use task arbitrary fix posterior prior task predictor associate randomized learner task minimize classifier task directly compute empirical base bind quantity fix learn follow nm kullback
set correspond support literature state form lasso asymptotic author though assumption net author impose weak restrict special variation belong partly relative instance well norm composition operator instance whose fit framework lastly similar infinite dimensional recovery interesting finding compute degenerate identification illustrate mutual coherence mutual degree ill conditioning correlation low frame fine variant cumulative propose account vector recover introduce recovery norm complement view weak criterion derive subdifferential check equivalence sign proposition order criterion coherence elaborate discussion establish lasso indeed ensure exist degenerate indicate error whose equivalently non degenerate pre rank incomplete nuclear norm decomposable regularizer measurement non degenerate measurement matrix identify correct comprehensive jx jx sensitivity minimizer sensitivity seek ensure lipschitz assess stable manifold unique remain start work smoothness show broad function enjoy powerful calculus sensitivity convex partial closely decomposition fact minimizer restriction manifold hence sensitivity partly perturbation feature variation function analytic analyze small perturbation see important risk argue performance solve sensitivity involve sensitivity perturbation sensitivity partly reason smoothness additionally regularizer notable manifold non respect actually single smooth hope observation see characterize precisely outside actually set point locally minimizer motivate coin transition boundary respect perturbation check hyperplane crucial able write explicit derivative theorem solution mapping every restrict hessian surely goal show lebesgue everywhere structure function algebraic broadly various area wide applicability largely semi algebraic possibly function semi algebraic minimal instance section qualitative algebraic share big function minimal structure correspond sense prominent algebraic rational minimal formulate framework result section algebraic stable algebraic algebraic practical adjust statistic density though sake white realization quadratic risk solely nice risk reliably reliable instance risk show value mapping choice see mapping quantify complexity statistical estimation empirical differentiable lebesgue everywhere freedom define lebesgue everywhere get intuitive notion close valid lebesgue appropriate main lie show mapping fact lipschitz argument sensitivity rather formula validity lebesgue e subtle partial smoothness rule hold every turn empty reasoning regularizer precisely exist constructive build hand risk differentiable lebesgue sure turn stein sure together theorem stein algebraic lebesgue minimize hard problem remarkably extended risk prediction risk define section review relevant smooth semi partly smooth prove feature toward sure sensitivity manifold appropriate key put emphasis unbiased generalize thus extension sure exponential family extend well sure extensively various lasso read q hold full extend analysis formula projection prove extend treat case close give sufficiently formula group orthogonal heuristic prove group denoise norm derive challenging address divergence expensive approximation prohibitive may difference monte analytical serious think approximate sure recursively though problem type mention attention proximal splitting one possibly quasi newton detail regularizer intend non replace subdifferential assume finite value close convex guarantee assumption make global get optimization handle regularizer cast cone constraint barrier interior enjoy fast quite costly become prohibitive dimension increase study frank wolfe solve forward backward variant projection many solution subproblem structure rank total rank recovery signal processing see e homotopy minimization regularization regularization see crucial path affine lar accelerate homotopy compute homotopy increase monotonically compressed sense homotopy empirically ensemble threshold bad case thus homotopy take cost per homotopy like ad hoc large imaging solver medium machine homotopy need extension homotopy method change see five idea pass pass form regularizer nuclear comprehensive review behavior broad ensemble split iterative tailor structured essentially optimization increasingly concrete general closed g smooth proximity operator form proximity splitting algorithm possibly approximately individual g proximity operator operator separately iteration never sum function composition iteration rigorous guarantee quantitie popularity image sublinear scope huge research field instead brief popular optimal fista guarantee scheme good elaborate section minimize either space projective dr interpret minimize admm dual conjugate dr apply whereas initially close bregman primal objective starting read sequence smoothness whose result favorable case degenerate typical circumstance compress local exhibit global sublinear term manifold regime general manifold identification degeneracy result partly cover linear fidelity satisfy convex decomposable partly smooth general variety manifold author show identification manifold associate partly projection newton extend identifiable surface smooth non simple partly necessarily prove identification remain review work work see unify namely smooth one regularizer low recover analysis chapter list believe important focused fidelity regularizers result chapter extend proper generalization regularizers regularity need deal difficulty tackle instance property hold impose bottleneck result present recovery less non recovery synthesis set input stand trivial adapt dictionary extensively regularizer scheme vector space extend infinite di far stability constant bound scale regularization hilbert banach banach space measure stable highlight compressed norm synthesis regularizer difficult smoothness exact turn iterate raise hope acceleration study guarantee extend splitting importance acknowledgement european research project sigma would like unify universit paris universit chapter dimensional problem linear measurement w observation offer acquisition processing acquisition hardware imaging regression problem assume section explicitly assume rest knowing measurement much ambient general ill condition entail ill pose might think convolution camera spread account low sensor medical imaging operator possibly partial transform propagation sensor imaging amount wavelet impulse response approximate wave propagation medium column covariate argue ill pose inversion plausible solution include adopt though class elaborate chapter reader refer discussion school notion conditional introduce refer solving stand fidelity fidelity use instead underlie noise instance stress provide fidelity replace smooth strongly focus sequel quadratic recover play give brief overview tackle class optimization penalty counterpart chapter performance guarantee brief account non approach fidelity theoretical hereafter optimally automatically offer neither strictly turn existence minimizer minimizer exist mild constrain formulation parameter share solution view problem reference detailed valid chapter one literature value increase fidelity perfect limit jx transpose pseudo hull interior topology boundary manifold tangent say say otherwise subdifferential function jx normal support tangent set compact subdifferential reflect subdifferential differentiable singleton illustrative subdifferential value separability subdifferential I differentiable processing use collection structure datum manifold high collection idea manifold smooth regularizer detail turn natural unified description thus manifold element penalty typical simply notion underlie description recover noisy accordance notion key whose statistical theory see set inverse problem restrict inversion correspond jx sparse signal subspace I index support use combinatorial pseudo jx intend way instance piecewise manifold parameterize location thus selection literature approach bottleneck close thus operator alternative point hard thresholde iterative scheme consist instance reference algorithm manifold actually pursuit comprehensive reference therein chapter regularizer complexity manifold regularizer prove hull convex penalty design necessarily surrogate remainder give convex formally subspace terminology tangent property belong differentiable essentially affine illustrate formula subdifferential obtain sparsity partly function partly smooth behave smoothly move move hereafter finite contain smoothness around continuous partly manifold exist neighbourhood partly relative uniqueness around partly take example describe partly use image machine sparsity anti calculus create pre quadratic regularization check partly interpret prior restrict pseudo hull restriction pseudo norm unit norm sparsity trace back decade application rigorous recovery appear mid regularization literature pursuit name dramatically capture pattern non norm partly manifold first learn typical structure see reference natural image model audio structure channel partly whole ambient example impose partly separable smooth fundamental attract year linear image texture wise part cast texture solve variational favor principal pursuit decompose superposition component finite partly observation contaminate important recover assess decay present ensure recovered body vanish terminology solution precisely note equivalent convenient first optimality ensure stability minimizer inside subdifferential precisely non degeneracy previously degenerate establish valid regularizer without particular assume consider choice minimizer detail plain tell noise terminology dimension likely restrict smooth regularizer fulfil reason show slow uniqueness condition might find depend choose constant get close intuition degenerate source condition literature problem overview implication check degenerate trivial give compressed particular linearize partly smooth regularizer full constant fine show quadratic decay extension hilbert smooth lagrangian dimensional banach ill inverse convergence rate non degenerate degeneracy convergence isometry rip homogeneous result equivalent regularizer recovery compress suitable widely isometry rip restrict isometry small show exist degenerate discussion sense generalized frame sparsity completion quantity random instance entry rip remain subgaussian constant expense comprehensive rip give rip base uniform recovery wave rip recovery guarantee claim uniform mean rip gaussian measurement obeys holds depend gaussian low completion minimizing hold high lead bound inverse measurement overhead necessary handle norm imply existence closely call dimension aspect exact
thorough topic simple intuitive mathematical algebra technique svd apply world thorough foundation machine learn paper spirit rigorous mathematical appendix proof complete understand work provide idea avoid please contact I suggestion correction comment spectra appear unclear redundant problem fundamental obstacle complex web indexing toy physics study system mass ball ideal along motion direction explicit express let alone axis measure decide live movie interest movie camera indicate position projection axis camera arbitrary angle angle might minute big remain simple equation priori axis camera world system question record need deal world problem toy air toy challenge keep mind understanding systematically use basis express new reveal goal pca axis goal determine redundant precise definition datum individual record etc set point camera ball position column contribute ball position entire record hz record us term equivalently lie orthonormal algebra know vector unit basis orthonormal toy camera reason naive record position camera mean camera algebra row row vector matrix construct naive effective record another reader notice linearity simplify restrict set pca moment let matrix record quantity row column equation represent change interpretation transform geometrically rotation obvious writing explicit dot note recognize dot product row set basis linearity find appropriate change exhibit beyond linearity arrive section build answer question two lie matter absolute strength common snr snr noisy camera camera straight line line motion signal noise indicate line diagram cloud include thin snr bad direction large interest thus dynamic exist variance snr naive direction variance intuition indicate large good naive direction motion dimension redundancy evident record ask record reflect range leave panel depict apparent depict plot nearby plot clearly panel meaningful calculate vice versa response express variable behind reduction identify line generalize notion dimension sample number individually define forward absolute magnitude redundancy uncorrelated convert express covariance dot I slight generalize arbitrary vector additional interpretation one trial arrive element dot measurement summarize property type covariance reflect redundancy diagonal diagonal magnitude redundancy option want section state goal redundancy measure optimize diagonal say successive order arguably vector orthonormal orthonormal pca act rotation maximal normalize maximize save restrict direction direction save select order principal algorithm true benefit assumption notice gain importance direction order variance implication arrive behind linearity problem area extend regime see large belief sometimes incorrect simplification algebra decomposition technique highlight aspect straightforward understand important pca eigenvector measurement goal follow n identify last line recognize eigenvector symmetric provide eigenvector arrange degenerate subspace constraint orthogonality situation fill finish evident summarize pca entail subtract computing eigenvector demonstrate matlab code mathematically involve continuity name basis quickly derive decomposition interpret pca let square fashion positive singular r cl theorem say bit multiply eigenvector scalar summarize vector multiplication prescribe construct order set singular likewise additional orthonormal deal degeneracy issue representation piece fit form multiply side arrive motivation decomposition matrix orthogonal second express q scalar understand new matrix stack place diagonal position generate look like respectively solving solve equation thick orthonormal speaking span formalize precise column equation transform infer transform orthonormal column basis span matrix quantity orthonormal transform transpose span understand implication though information pca fall framework evident svd original dimension derivation equal matrix principal component matlab include b must orthonormal column svd also principal quick pca type calculate covariance pca reveal structure solution summary implement pca quantifying dimension describe set along mean compare hope behind employ variance along component type reasonable precise intuition behind reduction vast majority variation direction record pca ask fail remarkable feature completely answer come parameter record plug play feature answer independent perspective pca agnostic track point variable angle pca would fail deep require primary motivation explore unite run big
angle canonical angle need suppose part last inequality use theorem least unitary put nearly unitary henceforth prove diagonal decomposition matrix eigenvalue real discard retain derivative complex eq show remainder full denote bound theorem angle subspace span take appropriately second imagine isotropic must account slight multiply accumulate depth claim appendix satisfy recurrence error final suffice estimate matrix latter suffice nearly four term particular ty tx need failure small argue get thereby give high algorithm derivative large gap pick find randomness tx ti tx p expectation overall appear large recursive scale smoothly matlab code gaps family gaussian gap degree proceed clearly true grateful anonymous suggestion nsf section perspective performance recursive purpose implement use one blind speech run gb ram run matlab synthetic fully model pick unitary first gaussian column speech speech run unitary source simulate natural blind impossible make isotropic construct bipartite product also desirable property invariant column sign extension synthetic recursive run unable c isotropic source dramatically dramatically raw non isotropic difference fourier us recover author distinguish even achieve still unit roughly speak ica fourth exploit second moment information fourth moment smoothly second fourth conclude note experimental section comprehensive account consideration implement highlight practical construct parallelization split approximation pick algorithm type dramatically improve outside thesis school computer science school science technique complexity apply distribution give ica improve decomposition decomposition unsupervise mixture obstacle datum expensive recover unlabele become classic ica become diverse area vision recently deep net input vector unknown observation write unknown component ica estimate hope subspace span gaussian assume component distribution fashion common fourth nonzero gaussian ica fourth improve polynomial broadly tensor decomposition tensor sample analyze spherical extend corrupt state precisely major ica latter guarantee paper improve fully result fourth bound away give deal difference gaussian set rectangular algorithm technique hermitian decomposition decomposition powerful tool theoretical generalization tensor decomposition np hard tensor provable decomposition polynomial dependence conditioning impractical moderately find gap eigenvalue core need large gap subspace gap project ica technique recover assume model surely find sign satisfy I ia use singular value decomposition symmetric substantially previous good recursive variant proceed first isotropic eigenvector accurately eigenvalue adjacent pair e gap idea simple estimating gap shot group eigenvector either necessarily desire recursively need gap thus sample motivate point sample grow square inverse go go apply characteristic call function precisely eq tx derivative derivative follow complex hermitian hold unitary isotropic unitary random vector robust algorithm spaced mix anti sample guarantee tu large eigenvalue fall use low isotropic transformation every compute compute reweighted svd large else return decomposition block carry span recursive three part gap isotropic gaussian partition column subspace version perturbation accumulate gap define function large gap successive polynomial iid gaussians least repeat type rough intuition tail work differ firstly quantitative pick fix quite bit long easily analyse maximum gap pick independently pick pick replacement pick wise chernoff bernoulli pr proof trivial chernoff take union
singular value intersection space kind include quantile express probability density pass easily quantile depth assumption refer contour contour half construction mesh straight contour dash curve function affine median affine mean singular matrix unique symmetry survey uniqueness uniqueness case strict positivity point indicate show section skewness deviation angular angular wise median median th square random distribution solution quantile yy partial moment distribution strictly support imply set q calculate obvious use indicate refer depth contour kind technique elliptical axis proportion however depth set shape level eq technical e comprise mean satisfy implication scenario depth satisfactory multivariate quantile scenario scenario es relate risk consider skewed canonical invariance suffice skew skewed tail freedom function student freedom later tail characterize shape student recover multivariate result give appendix introduce canonical important skewness family equal cauchy generality component contour axis let denote two whose first differ space index sign define p contour simplify next contour circular distribution indicate quantile distribution u element st du observing euclidean unit unit htbp panel example e right illustrate contour contour certain st skew cauchy depth contour contour elliptical elliptical calculate half space depth use construction letting panel dot line approximate obtain depth contour contour panel panel dot contour htbp e ic ellipsoid measure skewness variable angular symmetry median affine imply affine measure skewness canonical also alternative skewness wise skewness curve increase towards deviation angular symmetry closeness angular symmetry indicate close result angular symmetry wise directional ix express ix I arbitrary conclude angular symmetric median median family explain follow representation tend hence argument apply constant draw large approximation ellipsoid misclassification positive misclassification intractable quadrature integral let small misclassification case misclassification next observation consider approximation mass report misclassification increase high case negligible misclassification reach quality family year figure bivariate skewness indicate skewness angular symmetry estimate misclassification low evident scenario distribution simplify one component asymmetric additionally represent skewness closeness distribution angular symmetry propose angular misclassification approximation invariance availability mean sufficient skewness curve underlie although misclassification dimension elliptical construct elliptical attractive financial risk factor stress application introduction skew close affine admit given skew skew many material skew depend existence moment consider elliptical corollary location namely center angular symmetry lie investigate whether depth use canonical acknowledgement student sciences university supervision k generalize modify kind constraint mm mm examine scenario skew distribution interest motivated set stress test financial set half notion ed ed set elliptical coincide region contour skewed equivalence contour contour skewness heavy skew make form skewness contour elliptical contour exactly elliptical skewness angular symmetry elliptical contour keyword angular symmetry depth school university infection health university uk department statistic university mathematical science interest original study issue financial management represent index interest exchange factor financial portfolio give portfolio derivative bank book compute information portfolio formula quantify factor value question distribution plausible scenario opinion set space presence multivariate short tail many outer tail one change year quantify portfolio portfolio bivariate fit contour plot line divide half space form intersection close point depth grey line boundary space
dna structural dna sequencing length ignore information section alternative simplify read map equally informative existence describe model structural introduce observed intensity read sequence coverage index genome read location map within local intensity function single location length become apparent representation technical role scan copy seq moment section specification alternative intensity map length examine detailed simplify certain notation baseline rate general raw scan index differentiable detect pair read differ read pair least map template map position plus minus map dna sequencing read reference assign position read may sequence mapping due inclusion dna match minus fail call experimental conservative pair reverse orientation easily broad notation narrow definition thing definition pair read read map let read either plus position mathematical convenience process intensity integral second account unobserve read marginal intensity simplify map read minus proportion width reference window partition non w u u pair window window minus read map map uninformative whether broad definition remain simplify read set separately come way contain pair mapping non latter read sequence due reasoning form log belong b signature specific respectively coverage simplify fs fs weight length modify u u u v replace minus eq read peak genome combine peak give peak small since alignment accomplish statistic must read detect little ideal true maximization shorter long well section cut corner function ease within segment ignore formula priori one try combine separately scan long correct simplify possible develop variability length map begin statistic beginning whenever read begin determined chance appropriate modification detect suggest base section approximate p value rely transformation towards scan window zhang cite ratio transformation approximation calculation nan intensity define still field k z simplify expression call local field p field eq q denote partial vanish hessian evaluation consider vector clearly allow set increment special case variance x statistic threshold zero local theorem propose use order marginal magnitude marginal transformation rely interest summation respectively note rely quantity rely localization zhang condition asymptotically q note local brownian former latter increment respect smooth hold genome location fix add brownian notation brownian smooth element equation omit genomic location notation introduce w df r df rt control detection amount may range boost sensitivity study zhang zhang specific function constant window alternative calculation equivalent define suggest various scan approximation simpler appear tail zhang rx purpose numerical value improve identically equal calculation rarely per maximize complicated r precede w x maximization become negative similar paper effect maximize adequate result implication function approximation scan see fix nuisance threshold maximize respect threshold define easily interest event put choose leibl inside particularly integration denote minimum derivative eq asymptotically w occur maximization simplification upper correction carlo evaluate repetition increase experiment realistic consume approximation use helpful evaluate grid base multiply unchanged incorporate monte effect use example poisson figure homogeneous apply slight expression sum instead multiply throughout approximation moment calculate numerical formula smoothly twice w dt approximately simplify justification may maximize g toy pair read score derivation formally consider dt discrete important family determine choose minor formal asymptotically let find asymptotic natural ask approximation precede expression fact place equation x var identity possible upper z var replace conditional dy maximize seem reasonable term usually small section use approximation procedure power four statistic maximum rescale shift amount correspond base genomic example appear statistic significance tail correction max produce threshold statistic significance opt power statistic indicate actually remarkably well c appear far powerful considerably powerful quite seem less actual affected hoc method slightly consistently less powerful drive poisson process determinant small value significance question concern power value threshold simple describe statistic approximately marginal approximately appropriate appropriate slightly example power level would number calculation report power moderately threshold seem try vary accommodate change consider detect signature practice scan combination power improve power individually adjust tail combination individual consideration false score score nominal except read align peak detect triple corresponding b gx case simplify applie range range change scan line incorporate behave toy scan distinction worth decrease substantial fraction span genome read detection score piecewise smooth path account maximization describe remark value require change fashion complicated examine power pair maximize sequence library influence length read sequence coverage read heavily probability mapping read nan nan estimate typical publicly last sample quality gold sample standard read length trend increase length shorter read map increase recent fall sequence study period widely across goal wants currently usually less example population dna extremely desire refer sequence experiment frequency examine scenario coverage first coverage sequence case occur seem additional insight toy base pair statistic true relatively detect na na na first case coverage power large somewhat well length read table show detect coverage unlike monotonically capture comparison power size whether preferable large surprisingly adequate power substantially short small read case detect sequence examine length threshold scenario scenario read detect power power relatively complicated contribute row would bring tail power read add would produce find precede add lead row read case lead seem require combination dominate extent power take adjustment significance significance power fall read different genomic would illustration scan european nucleotide read base length map empirical show bold density estimate dash normal dotted normal density scan step control family wise deviation deviation distribution map read stepsize threshold visualize massive quite plot dash represent score analytically model show score skew lie nan minus read bottom plus length place call support overlap score merge call call make call robustness statistic combination factor generate boundary read right region end call statistic evidence overlap concern inspection read suggest call show minus read end na scan length statistic read read percentage overlap end region iii two pass ideal read shift precede support peak region contain top plot map plus bottom read read roughly bp precede half yet significant minus like datum read evidence pattern exception rule roughly map read region bottom show score genome scan produce many region carefully ideally read scan emphasis useful signal generation sequence detection seq variant pair formulation significance framework embed family technique zhang characterize simplify read simplified read suggest approximations carlo also study picture regard assume homogeneity nuisance statement regard aspect fail accurately illustrate row well simplify variant detection pair sequence formulate incorporate read map log likelihood scan call long increase read depend detect read report statistic poorly map reasonably powerful sum although row respectively marginal alone read even empirical substantially library tend exhibit skewness increase gain substantial analysis assume coverage read genome score vary threshold genome position appropriate substantially increase amount computation conduct likelihood ratio issue avoid option read coverage scan threshold value simply implement one genomic one simple lost adjustment threshold sum score individually reason dominate incorporate contribute mainly individually combine weight zhang focus mainly error genomic false discovery fdr often mode multiple testing control boundary fdr describe zhang characterize genome sequencing genome content throughput sequence p j e h cox identification acquire cancer genome pair sequence zhang scan weight observation american resolution copy p linkage complete identity descent molecular trait linkage discover generation sequence nature approximate smoothed poisson maxima peak statistic j zhang poisson application copy next generation dna sequence unknown ann zhang maxima fields ratio zhang simultaneous ann zhang statistics trait three analysis field united support national foundation foundation stanford genomic throughput dna sequence poisson derive false calculation accommodate deal example pair compare power current illustrate application modern biology especially great deal research local signal al typically represent datum standardize neighborhood detection magnitude location field central search local achieve theory field control normal provide adequate approximation threshold especially involve contain condition zhang dna sequencing possibly homogeneous process involve signal detection copy sequence bind site follow sequence seq rna sequencing pair sequence brief motivating section also dna structural although context relate detect local scan precede cast derive ratio scan field study variant directly consider tailor specific pair read dna sequencing motivating application give framework field first variant
efficiently matrix describe matrix uniformly simplest fast bad propose randomized row normalize hadamard choose fast version embed enable linear map guarantee theoretically leverage without replacement error uniform replacement compute approximate obtain x u attain way error bound arbitrarily previous prove sake prove bound theorem define u lemma w inequality singular q inequality hold leverage union least technique two column matrix obviously hold finally u u two third set least cost column induce time space discuss sampling solve efficiently previous score attain probability find linear seek square compute cholesky solve big prohibitive fortunately one portion instance use datum base efficiently speak projection instead projection construct theoretically construct projection error hope without technique self contain sampling still simplest efficient though score true perspective score sampling uniform still study theorem uniform sampling attain
problem convex time convergence derive dimensionality allow grow required convergence maximum norm strong condition weak inference modify subsequently estimator limit reach piecewise mild assumption multivariate lead precision regression acquire combined form normality aim element approach present build invert tucker paper demonstrate linear generalize fully invert kkt condition consequently normality section main result contain proof matrix notation ex ex ex e zero analogously cone e ps p model throughout estimator gaussian perhaps correspond rest paragraph come associate undirected edge structure precision correspond estimation graphical non selection precision translate gaussian cardinality loop allow shall cf certain allow grow presentation define operator I second shall impose restriction index consequently define e ex ss ex hessian theoretically one could estimator lasso sensitive besides procedure consequence zero graphical derive assumption parameter appear influence base concern condition matrix covariance depend roughly speak show pre assume model tail introduce follow given define tail tail probability sub tail condition variable interest restrict random mean random follow covariance sub event tail give remark positive characterize tucker condition belong subdifferential evaluate inverting condition lead de remainder converge probability hold uniformly sample additional vector satisfy furthermore wishart shift sub asymptotic theorem individual fix keep dependence doubly index parameter satisfy suppose converge next hold exception brief graphical fully simulation instance graph star may maximum degree vary specify corresponding matrix one diagonal chain graph precision cardinality active star graph implement interval cover number calculate obtain respectively calculate interval parameter average confidence interval construction estimator pre big cc graphical lasso specify star mle covariance cc cc value definition cm theory correct regularization I suffice empirically optimize average roughly experiment kkt optimization read subdifferential side rearrange obtain asymptotic expectation remainder equality since ex ex ex obtain q mx ax ex condition ex ex ex ex u ex b ex b ex together give ex ex ex ex ex note ex ex ex ex since uniformly theorem sub hold claim scale converge distribute nk e nk nk finite sum sub since nk statement satisfy nk n nk n z n remains apply follow hand tail de c show term limit substitute n satisfy x p nz ij ij
whose transition mdps compactly several develop exponentially however factor factored mdps factor optimistic confidence heuristic factored mdps make exponentially upon aspect discussion reduction reinforcement approximate many effective planning method extend optimize finite horizon action mdp p operator indicate mdp episode scalar transition make learn deterministic function agent th episode reinforcement learn eq episode mdp mdp internal history random reward regret factor mdp structure formalize introduce factored define scope mapping mapping factor factor draw within factored factor lr rx ir ix individually transition factor mp factored mdp mdp factor reward factor transition write bind regret factor factor I span optimal factored high high factored factored factor z dm factor cd factored span policy mdp may hold optimistic formally replace analogous factored instance shorthand simple clean clean factored factored satisfy tight factored mdp corollary appendix confidence estimate underlie contain literature exploit additional graph bound empirical tn sequence confidence depend q shorthand interpret nan follow bound measurable counting factor j ts confidence j k j k td factored refer generally either mdp optimistic bellman return expect state law place write break add subtract reward agent clarity change nothing relate reward mdp near plan factor dynamic programming I factor hoeffding remain bellman actually h aim create deviation triangle build confidence set factor factor deviation z also j conclude apply factored transition lie posterior finite relaxation j td j ensure td use fact equal law complete factor theorem corollary substitute upper bounding factored mdps reinforcement previously important question access may prohibitive priori algorithm seek mdps reward value belong stanford support award foundation elementary argument concentration contradict x corollary delay visit imagine let length finite act start episode episode define notion radius tx w tx times episode set td kx h expression time repeatedly say composition act low integer finally complete kx ir kx tt tt tt complete suitable trivial look trivially certainly bound last claim van stanford suffer mdp space interest curse dimensionality possible polynomially factor satisfy near sampling learn confidence factor uncertain cumulative environment mdp
visualization comprise video environment category sequence action video select video datum division video multiple include testing extract densely video training descriptor video fig manually histogram dense sift histogram binary histogram optical heat maps cl ap ap validate kernel baseline svm selection mkl connection mkl define bin histogram performance ap assess selection measure achieve precision outperform experiment feature major paper selection interpretability select codebook fig feature discriminative visual unclear mi art method use recall localization overlap quantitative fig rs chi perform linear however video segmentation explicitly assume video get result category reason lie segmentation since motion good segmentation see pr result comparable segmentation car cat train propose select convex problem commonly classification task plan address limitation beyond classification well art feature region tool region future selection weakly supervise tool visual pt tr abstract typically extension despite success unclear video discriminate among answer visual answer question present region visual allow visualization feature region jointly optimize parameter classifier benefit approach linear additive intersection spatio video unify selection scalable method learn illustrate decade great advance performance although much progress visual exist labeling involve category discriminative interpretability mid car image contain car goal weakly discover region train car successful weakly rely build art space discrimination aim weakly supervise discriminative aim discriminate image also temporal activity fig discriminative avoid consume manual informative train due importance popular exist learn mostly cope allow would approximation linear generally inefficient region contribution kernel discovery visualization word spatio volume video work include experimental illustrate address et weight space classifier construction exist method jointly induce build et al feature weight linear svm add kernel computer vision approximation purpose histogram differently come weight impose unclear convexity hold address region mkl image model class bag one instance positive weakly mi svm series mi arguably popular bounding localization limited aim jointly instance np heuristic heavily use instance moreover essential liu region visualize method use unclear convex feature additive kernel bold letter letter g work sample histogram codebook histogram satisfie I dx ik jk weight x assign bin bin histogram feature maximum hyperplane margin correspond margin two space directly normalize consider nc training error transform property additive factor jk interpret bin priori concatenation ik ik variable substitution additive optimize optimization see section note selection formulation mkl rewrite simplex constraint drive bin differently bin mkl exploit non optimize reformulate set lagrangian primal variable identify dual k ik dual eq objective problem ensure negativity constraint reduce positivity constraint take descent method however visual difficult classifier discriminative video segment video region spatio encode codebook learn video assume region classifier homogeneity long selection allow image histogram importance q map index simplex sparse region trade model bag bag segment image negative localization region belong b ik ik ik region connection
word track research public available ground surveillance surveillance program section period event specify developed simulator signature simulator going adapt purpose base streaming appropriate scale data http com research acknowledgment project north operational national reference framework european ci acknowledge european project p cm dataset laboratory artificial intelligence university dr pt surveillance continuously daily stream indicator detection detect day clinical laboratory datum usually dimension recent method search multivariate signal alarm find propose bottom bottom search track detect change reveal art false alarm keyword stream surveillance goal surveillance public health clinical laboratory come kind event usually made event activity like attack epidemic like west etc event make identify early stage save prevent early event purpose system stream diagnostic health room counter sale school internet health simultaneously trace ht demonstrate surveillance see aggregated daily generate complex alarm straightforward anomaly chart impose alarm diagnostic indicator apply detector individual reject hypothesis false alarm univariate process base series processing base surveillance besides event category method pca hmm var multivariate operate look via area surveillance spatial statistic fluctuation operate spatial account scan statistic operate datum adequate surveillance mention drawback static effect offline group suit bayesian pattern event real update network main limited exist handle temporal stream baseline vary raw historical day historical environmental attribute use oppose network historical many surveillance opposed overall change track consequently present delay alarm bottom rule middle bottom top track level approach take purpose cause overall surveillance turn epidemic early even hour detection require problematic signal emphasis surveillance rarely take study alarm inverse effect detection study concern reveal people deviation increase besides oppose anomaly assume process occur isolate static surveillance dynamic environment attribute week weather etc affect whole illustrate individual kind surveillance detection complexity environmental issue surveillance false alarm detection contribution time apply surveillance problem dimension correlation criterion event detection introduce novel baseline baseline environmental rest organize propose datum performance analysis last conclude exposition present fundamental rely tracking unless match recent streaming setting dynamic environment item strategy take stationarity demonstrate illustrative method receive stream stream slide window size top cell correspond region slide environmental fed historical tensor previously slide window combine decompose subspace pairwise observe baseline vector principal eigenvector correspond eigenvalue receive eigenvalue principal vector solid alarm considerable eigenvalue close consider dot eigenvector vector describe present slide day environmental weather significance slide phase slide format assess slide window need utilize window previous another environmental combine take window baseline present apply high principal diagnostic however perform important high alarm happen article name production p stock market ten period selection automate matching phase eigenvalue eigenvalue baseline define historical eigenvalue eigenvector q deviation purpose want transform ease use explain decompose specific match principal eigenvector spatial dimension may kind change match overall change infection reflect unseen environmental environmental algorithm receive include current historical tensor instant environmental vector environmental baseline tensor search historical match item illustrative snapshot day environmental color cube compose assume environmental dominant frequent history figure include baseline occurrence dominant day receive historical setting find input unchanged day receive match time one rewrite element match baseline tensor match still dominate therefore baseline compose preference day matrix stay repeat environmental setting update add already matrix unseen add distance adequate keep historical environmental basically data clearly infeasible anomaly research hand multi recently semi automatic event ensemble background access background include simulated disease network simulator namely base factor weather manually mention simulator produce extremely publicly online set way multiple environmental setting distinct temporal day also environmental cardinality record sample record xy ne gender feature environmental week environmental weather environmental receiver operate roc trade specificity roc method evaluation ability critical surveillance positive heavy delay surveillance characteristic like surveillance proper evaluation monitor characteristic curve evaluate specificity method surveillance release occur alarm correspond period consider detection alarm release reality event release release detection delay day delay figure false detection alarm day release mark alarm mark specify alarm release one day delay optimum alarm indicate depend alarm alarm recent performance use totally temporal day use day day whenever agent second year year contain release slide move window match output alarm delay detection axis month delay close curve pattern perform false alarm bad delay curve make overall instance false alarm delay specify need area false area outperform considerably separate look detection delay see detect event release difference oppose subsequently less small slow ability dimension suffer delay detection runtime surveillance system process receive huge processing hour efficiency runtime superiority version fast factor whole track structure factor relate compute exploit tensor decomposition offline tensor decomposition baseline tensor temporal size tensor tensor three dimensionality develop instance dynamic analysis streaming require mr mn case I I fast track slide window element 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correctness counter result set choose vertex choose go game two player token place token vertex token player move token edge go player result call formally infinite specifie formally function represent history end vertex strategy player multidimensional payoff map edge th dimension path k inf resp sup vector average multidimensional boolean atom example boolean always multidimensional payoff win satisfy condition player strategy infinite multidimensional boolean formula occur determine sequel abuse sure mean determine mean counter machine payoff counter first one counter tailor state counter right left allow increment right since state transition change counter standard machine game one counter reduction consist figure state final machine sim player counter reach player player play dimension game simulation sim sim sim sim sim sim sim sim sim sim construction win side counter objective sim run player game graph simulate simulate describe role sure sim read definition section way right q whenever state sure leave left transition round sim role sim entire value counter current sim happen simulate illustrate follow stand g explicitly role transition transition loop loop want assumption side side transition qp keep sure testing test four note win win go eventually game assign way enforce player player construction give player player act differently game player correctly win sure stay stay state get remain either negative sup player positive bad bad g g sim sim sim claim win win give win player make sure state enough round fulfil test player fulfil otherwise negative least win satisfied player winner prove converse direction never win strategy win win condition trivially stay sim otherwise eventually win winner correctness namely describe win win case subsection extend two subsection round player play begin play subsection counter denote play g player sufficiently reach state since get player maintain left machine right denote show leave maintain simulate number play current simulate sim round begin begin current transition step complete every play proof sim hence complete player step simulation player assertion recall four assertion condition violate assume case invoke end loop go round violate claim indeed sim sim change versa sum exceed round condition violate analyze violate get observation begin sim remain value every round maintain infinitely condition testing proof leave symmetric show visit actual counter counter simulation g prove claim sim get contribute contribution visit contribute g x g c player right maintain c item lemma round play current sim ng counter machine winning win counter value maintain leave violate correspond stay immediate lemma stay violate negative weight hence simulation right simulation whenever leave invoke counter maintain negative every violate transition properly player since suppose number visit subsection c loop fulfil sim simulate loop player zero strategy lemma accord sim sim initially sim first item item sim initially simulation g use player play whenever visit number round play self loop round every round hence round proof proof ready win win strategy enough play constant distinct case strategy invoke many play player sim sim value change sim run satisfied consider always thus sim infinitely sim proof first never sim fact never certain round either infinitely often counter win c ii counter proof proposition hence problem winner payoff payoff game deterministic input counter decide end counter zero counter player simulation know test counter second counter
increase sufficiently follow show denote nonconvex nonconvex stepsize kl respect uniformly coordinate block iteration little extra computation new cyclic greatly save nonconvex fix update nonconvex smooth addition nonconvex increase mirror stepsize sg kl km k f k compare block gradient stepsize reciprocal lipschitz k test start sg independently follow follow solution time perform start randomly use compare another calculate empirical repeat average empirical table sg sg report well sg take less gaussian sg sg compare belong sample sg epoch epoch bilinear logistic interface visual recognition test eeg concern eeg arm subject hz marked randomly slice size start depict behavior epoch epoch reach run return logistic default run method run choose give accuracy epoch plot give however take eeg epoch also analyze nonconvex sg convergence clear sg mirror gradient method acknowledgement nsf dms grant lemma second first eq complete proof prove lemma q cauchy schwarz second projection first prove hold q secondly complete exist choose integer addition way therefore complete xu theorem remark apply mathematics university mathematic sg stochastic block descent update combine sg multiple block paper propose program sg block gauss update previously outperform sg latter small constant establish expect optimality case nontrivial convex significantly sg early near local minimizer benefit update especially engineering involve way sample keep mind constraint partition block sn differentiable sparse throughout omit without confusion logistic bilinear sparse sublinear objective convexity establish expect condition convex solve accurately calculate expectation subgradient risk sg method assume gradient oracle kk stepsize compare sg competitive problem utility flow sg deterministic deterministic require nonsmooth descent update variable much low iteration together find solve benefit sg size integer mini batch order block sample randomly update prefer typically proximal iteration see reference use nonsmooth constraint one certainly take incur begin block especially nonconvex sg sg become require sg sg numerical list method coordinate gradient sg stochastic deterministic block back consider concave programming original format coordinate respect fix recent nonconvex example work original understand propose method method randomly update analyze show smooth sublinear expect strongly linearly analyze case cyclic smooth also history sg become popular stochastic programming classic require convexity great robust sg obtain asymptotic iterate propose mirror descent stochastic convergence mirror accelerate composite sg handle term optimality relevant propose combine mirror descent approximation propose randomly update depend early update demonstrate practical intuitive sequentially processor update resource partial gradient randomly cyclic block gradient thus require assumption specifically boundedness appear system assume update row vector explain update variable block benefit stochastic form huge amount sg method establish condition synthetic world deterministic sg problem restrict euclidean norm partial subdifferential scalar constant history st expectation set eq onto without loss generality order hence define depend big difference make challenging objective lower bound lipschitz every namely lipschitz assumption bound k update boundedness proper nf I continuous namely singleton pf kp give last inequality method vary literature need analysis boundedness together partial boundedness lemma gradient mapping scalar obey large integer next algorithm nonconvex case convex sublinear sg nonconvex optimality analyze problem strong ergodic non k furthermore usually play exact differently phenomenon sg numerical test page biased partial see I become sg although result bad sg generally per sg read gradient partial sg computationally dominate need update rest little performance numerical summing together note inequality let last substituting let convexity iteration establish sg l assumption strongly modulus modulus impossible subgradient reference therein convenience discussion become modulus result
wise sample take integer encode variable query take bit set equal ar appendix material remark differentially private answer number high dimensional task however package hard define integer solver prove demonstrate experimentally netflix multiple become concern task often operate netflix challenge privacy netflix movie compete mechanism competition great success team improve hoc able lead subsequent query release attempt strong privacy thing prevent identification release answer private release extensively differential query many query safe version dataset unfortunately size exponential dimension run necessary bad especially produce runtime evaluation release notable exception thorough experimental find quite accuracy dimensional nevertheless seem attribute query scale feature partition query never critical bottleneck maintain universe quickly impractical record grow complex alternative algorithm rather object represent np step require private exact exist quickly practice part extremely efficient require practice time query like query release strong way marginal table demonstrating solve release include hundred thousand perform query release convert differentially private complexity numerous efficient problem parallel preserve extremely strong synthetic exponentially notable achieve theoretically exponential bad mechanism answer evaluation base inefficient heuristic multiplicative attribute seem scale family release base runtime mechanism algorithm family view synthetic generation propose interpretation solve game repeatedly well optimize player regret main problem record record differential privacy become record database requirement removal outcome mechanism database abstract possible record often consider bit database differ symmetric differential database differential query private sensitivity fundamental tool bind algorithm privacy private rely interpretation player present database close may player player action universe player set let payoff sum datum try query payoff distribution player play von intuitively von advantage go minimize force payoff force suggest well play opponent equilibrium interpret strategy payoff player game mix nash equilibrium eq query query place least player play equilibrium query player play precisely release need calculate approximate equilibrium update receive update normalize material maintain action payoff average distribution payoff multiplicative response nash multiplicative response role solve query release algorithm response round database sensitive change mechanism quality cost round composition eq differentially define advanced quality privacy privacy material form let let query define answer least note chernoff hold release game synthetic database answer query exact privacy associate actually state delta privacy first query query next approximate game find suffice response release game union condition event sample aggregate payoff accordingly convention treat plug response depend query discuss query try player minimize actually play query maximizer guarantee accuracy look precise query record binary universe mean let query integer though everything general marginal query include query sample eq associate satisfy many clause convert optimization problem conjunction form integer clause clause result program solve exist attribute netflix collection several census uci repository binary movie binary lower find algorithm error predict satisfy collection marginal several census repository movie wide preserve beyond frequently set guarantee evaluate netflix report query range run differentially private actually privacy small example also could differentially small use netflix heuristic dataset gradually netflix allow runtime query set million marginal experiment privacy parameter stable grow show remain mostly demonstrate perturbation laplace rate privacy query evaluate behavior privacy report runtime attribute axis include measurement overhead common answer query synthetic uniformly synthetic datum record satisfie realistic pick separate attribute equal way answer expense differentially minute exclude experimental implementation mid machine core processor ram experiment reach discretization binary discrete attribute attribute randomly marginal query sensible take different attribute query
express artificial change forward reference paper indeed observation reference thing mention david topic model people claim select separability define hull problem offer novel formulation utilize submodular later establish cone find cone row separability trivial time via backward removal convex matrix separable factorization inner dimension propose procedure remove row row convex row run output dimension achieve run polynomial dimension find constraint problem lastly cover large lie row index hull problem verify constraint equal cover separability separable submodular cover generalize separability general nmf find hull need require point point unchanged anchor ensure finitely generate finite another lastly encourage among cover finitely generator say algebraic ki k hull separability aim find anchor minimum hull find cone row hull define critical one case variable separability contain point fortunately additional separability assumption row two sum stay identify apply substitute eq element ground rank side uniquely model equal respectively select guarantee uniqueness avoid satisfy note assumption much limit separable nmf identifiability achieve vector inequality hold uniqueness variable achieve minimal hull gmm view feature observation triple observation respectively sphere l hull factor tx lda word occurrence interpretation index cluster real hmm solve normalize anchor reduce general matrix general minimum technique besides nmf mf mf variable deterministic assign maximum map separable nmf mf otherwise finite recover intuitively small prior exist dimensional subspace g angle sc various segmentation cluster model costly lasso row reliable general separability sc reduce general minimum impose n ik ix k group associate sc cone cluster cover nmf special separable cone might difficult dim show reduce cone hyperplane solve efficient see reduce equation moment probabilistic variable hull stand hmms conditional independence also ix third write matrix operator mode tensor outer product operator mainly focus recover moment moment necessary estimate training side rh unified space matrix example gmm rhs either rank column basis rhs leave rhs let tx ta hull anchor assign index solve computational suffer thank retain merely ok still successfully acceleration completion nonzero entry happen model simplex mean topic separability treating common text moment sparse general widely filter hide continuous distribution linearity similar moment hull hmm therefore equation case diagonal even fall discuss usually separability separability separability transpose e select let matrix immediately filter allocation bag vision semantic topic document firstly proportion document drawing topic conditional probability h j linearity lda number topic word number word gram statistic co occurrence matrix word fall latent separability assumption occurrence simplex occurrence document show recover anchor reduce mixture noting prove fast accord mf hull divide multiple easy sub extremely dimension separable largely rich problem design insight observation geometry cone project hull low hyperplane partially preserve geometry problem sub handle solver hull secondly pick without iterative pursuit significantly solely subroutine cover due cone hull cover cover hull generate projection eq since merely minimum hull hyperplane rarely return hyperplane projection problem bad flat anchor hull surface anchor hyperplane anchor flat face span adjacent still hull robustness flat htp ref latent anchor set ty nmf trivially hull leave critical reason converse uniqueness hyperplane could violate fortunately minimal hyperplane proposition reveal p htp hull generator minimal hull iff angle minimal hull separability lead identifiability case sub anchor right anchor anchor green marks intersection hyperplane hyperplane dim hyperplane h dim minimal hull proposition immediately angle identify hull region anchor identify angle verify angle computed specific subset flat increase interior angle intersection turn flat anchor right anchor intersection hyperplane anchor hull anchor lead approximation hull aim failure still rich ensemble random ensemble datum sparse ensemble bring acceleration projection point large unique true compare estimator ty iy therefore chernoff flat proposition suppose introduce binary eq anchor chernoff randomness random corollary guarantee success I factor matter solver choose sub learn note use projection reduce variant subspace projection still project gain original sub furth speedup divide randomization base unified distribute solver subroutine sub solver iterative algorithm pursuit address although solver hyperplane always geometry hull plane sub problem plane cosine find max max plane min max angle large angle close otherwise vertical horizontal plane plug extremely note large axis nonzero broad span hull lead generalization novel divide low hyperplane solver present solver cosine max subroutine check improve dimension apply gmm lda nmf subspace cluster show performance rich maximization factorization commonly produce posteriori estimate use wide collaborative rely update initialization moment contrast relate observation thus yield suffer large estimating poor moreover simplify recover uncertain column latent extreme hull obtain matrix separability represent call non factorization simplex extend negative generalize build identifiability generalization
entropy approach moment derive clear consider ise spin variable principle probability goal ise observed make inverse quite partition derivative spin system inverse approximate infer ise approximation isolated spin expansion obstacle overfitte affect inference noisy fitting moment exactly incorporate sample obtain provide good description dataset general become serious exceed effect example ising regularize add penalty describe fit alternatively empirical moment modify accord ij work total error square therefore predictive typically parameter perform randomly mutually exclusive usually fitting parameter predictive model agreement model contain third focus ise study ability reconstruct sample g performance inverse ise define ise various reproduce work compare different ise performance quantify agreement datum easily ph p angular average spin ise ignore act configuration easy calculate model consist letter see letter letter diverse letter mutually exclusive comparison ise accord letter indicate bold use case estimate l naive nmf isolate spin direct performing method include variational magnitude run ignore spend penalty parameter regularize negative regularize nmf demonstrate ise direct variational perform many variational contrast produce result performance train model good simple learn letter variational specific ij ss pattern describe energy infer energy pattern unconstraine transformation break initial letter pattern unconstrained nevertheless feature present art modeling include process fine tuning step network order nevertheless demonstrate pattern data variational variational outperform consist letter computer attractive practice structure extend variational inverse spin helpful provide award consist letter com image simple define intensity spin vector variational
discuss compare indicator algorithm gaussian output functional consider finite heat nine parameterize reference nonzero specify functional integral domain h w boundary ax depict boundary replace u contain freedom discretize space fidelity output reduce maximum set less replace reduce basis analyze type error u output regard finite element reduce supplementary compute surrogate error employ residual learning ex validation use relate set indicator true polynomial interval span purpose large interval polynomial font legend legend align label font align center xlabel axis align west legend title ii e index size anchor legend name title anchor legend title anchor north surrogate ex method depict ex ex comparison display remark arise inherent uncertainty ex interval include train indistinguishable trend large parameter attribute see focus norm dominant surrogate polynomial polynomial compare due superior validity surrogate ex uncertainty mean behave sample report validation label style font legend style column legend align width ylabel histogram west legend title bs legend name title histogram anchor legend title histogram west anchor ex ex ex curve depict probability table report actual lie infer interval within set within model increase discuss section infer moderately sized consist correction surrogate e improve low fidelity order surrogate reduce validate validation quality increase moderately sized training converge surrogate style width align legend indicator anchor west name legend error input anchor south east observe former relationship amenable construct process quickly surrogate nine curse difficulties ii depict error improvement error correct surrogate surrogate report expect evaluate distribution error surrogate depict almost always remain output alone hand expect always great mean always fact approximation improvement produce correction greater suit far legend align center align center width restrict legend title correction v basis gp error style width legend font cell align center style cm center v west legend name title correction anchor west anchor north surrogate sample red curve depict report interval test correction validate assumption surrogate histogram mean density associate interval surrogate correction align depict confidence validation remark interval closely number training effectively addition reasonably converge hand surrogate exhibit training use style font text width cm align center legend style legend column legend align style width align anchor title correction north surrogate gp point often actual lie infer surrogate imply surrogate eqs ex important quantify bound imply rigorous ex legend font legend column legend align font cm align title style width align center bs anchor west legend name linear anchor north legend align label font text width align cm bs anchor ex ex report surrogate compare plot state surrogate convergence small dimension close mean figure correction surrogate small however dimension require factor error produce large I conclude fidelity assess performance discuss output denote dirac delta separate surrogate error indicator computation dual dual offline infer side coincide mesh reduce counterpart generate assess ability weight residual see remark bases fidelity tolerance tolerance tolerance surrogate fidelity figure depict indicator fidelity first exhibit good label style font width cm align center title legend font every xshift pt anchor south west sep west middle title index size size anchor west iii dual figure necessity employ dual residual indicator actually accurate yield error order utility residual indicator style font legend align title font font align cm ylabel small improvement anchor name legend ylabel e title dual anchor west legend ylabel e title improvement legend name ylabel well anchor west legend log west name legend ylabel title vi reduce space report result infer confidence interval converge correct accurate surrogate confidence interval basis present modeling error quantification employ learning mapping computable indicator error distribution reflect uncertainty validate surrogate lead one exist surrogate bind general allow output yield uncertainty modify employ input indicator input error model prediction demonstrate dimensionality although characterize nine indicator validate combination surrogate powerful number future analyze bayesian different surrogate algorithm basis near surrogate acknowledgment thank support understand selection support part national fellowship national security science engineering national energy contract ac acknowledge office scientific research contract section reduce parametric affine parameter dependence detail discretize pde convergence ref read follow interest input express degree solve function projection basis dependent full span reduce p selection transformation provide pde eq reduce interest assume bilinear parameter bilinear form functional q dependent quantity quickly via combination f eq compute offline manner complexity dimension ref offline operator approximate measure q ex output functional eq analogously bound bound ex ex ex constant freedom residual representation residual dimensional pre offline surrogate dual dual lead error residual norm far brief overview exposition generation primal previously reduce accurate find low distance measure candidate reduce state optimally achievable manifold kolmogorov kolmogorov know manifold possible reduce often define finite solution measure bound allow construction following maximize verify construct converge kolmogorov width exponentially algorithm bound allow therefore expect rigorous gain constitute area investigation yes definition reduce technique regression computationally indicator introduce employ norm indicator numerical experiment near expected residual improve prediction magnitude exist curse surrogate comment correction email address quantification computing power system answer question guide becoming rigorously quantify uncertainty view uncertain decision measurable assimilation employ collect sensor uncertain input via thousand require fidelity model avoid turn fidelity yet rigorously incorporate quantification context quantify measure fidelity markov carlo costly high fidelity appear employ output output represent bias posterior map evaluation surrogate error practice employ fidelity fidelity statistical surrogate computable exhibit e introduce uncertainty numerically validate various surrogate surrogate fit fidelity order fit employ gaussian process high prediction associate query suffer curse access physics fidelity high fidelity model mesh employ remain physics correction develop primarily global fidelity first trust region center exhibit fidelity dimension order employ high fidelity implement fidelity model fit limited primarily error satisfy often I actual magnitude equip complex computational burden discretization fidelity model quantification problem stochastic useful correction often input exhibit fidelity correction non rigorous tight rigorous surrogate correction efficiency reduce compare correction approach aim datum fit surrogate mapping key physics computable residual rigorous discuss mapping process fit constitute correction indicator depict propagation construct system probabilistic specify htp split west align reduced order anchor south west north west split indicator anchor surrogate input output indicator bind quantity interest correct next introduce introduction objective choice particular summarize relevance construct surrogate rely technique statistical analysis basis dimension nine system input error reduce computation supplementary section high fidelity reduce surrogate formulation quantification consider solve define arise element fidelity system output first query thousand output evaluation aim reduce employ execute g trial basis capture computationally approximately trial implicitly predict state affine reduction decomposition require low residual incur become output output count reduce method linear pde quantify incur equip rigorous residual also exist quantify close tighter control tight constant accomplished difficult lower result various effort improve bound et successive method lower depend offline entire space time improve solution dual aim reduce method offline often fidelity implementation useful quantifying employ rather reflect knowledge error would would boundary correspond uninformative interval demonstrate correlate observe structure reduce apply pde logarithmic scale true error exhibit residual fairly section correction wherein nine input font text align legend font correlation bs anchor west correlation bs anchor south ex ex space ex ex norm map employ strength surrogate indicator addition output norm state training point construct deterministic mapping invertible logarithm interpret statistical model indicator methodology indicator practice g modify fidelity ensure interval employ error essence validation indeed behave distribution predict describe propose methodology indicator employ relevance merely tool accord validate class correction framework equivalent propose identity function mapping highly reduce dimensional ng demonstrate indicator practice exhibit scale indicator equip error strong well bound output error always negative treat error error even employ logarithmic lie within range p bind true expect affine gaussian capture employ transformation permit surrogate assume employ expensive candidate indicator simply computation expect indicator produce residual costly compute constant model variation approximated return depict energy log log accurately model detail sec experiment strong candidate expensive result behave include unfortunately applicable error output strictly positive error log probable might probability scalar section model dual weighted commonly adopt reduction adjoint accuracy adjoint indicator main drop approximate arise
hierarchical increasingly demanding field approximated field although computationally set work approximation mat construct mesh mesh choose extreme extent follow extreme model likelihood sampler structure yield prediction quantile extreme resolution grid covariate outline give explore paper raw hour site year datum base correct account effect correction observational basis accord solid daily wind temperature site set htp observational site set area choose exhibit htp improve predictive wind analysis european center medium weather take account dynamic water output contain calibrate five year information km km across domain km grid reasonable extreme physical spatial furthermore extend calculated knowledge lead quality refer hereafter grid point km km covariate observational site grid point order construct observational site spatial furthermore mean observational site tuning distance illustrative htp use grid decay cumulative parameter two interaction shape neighboring covariate observation parameter smoother tune site close distance calculate covariate use observational site field mat mat field flexible dense become computationally demand spatial field markov field precision predefine spatial address issue mat mesh partial spatial mesh linear mat mesh behavior extreme scale vary extent model hierarchical continuously spatial present model assume hour shape site distributional assumption generalize belong distribution asymptotic condition parameter conditional maxima everywhere affect apart spatial simulate year unobserved site year structure design one mesh product capture spatial variation random mesh triangle figure basis sparse mat matrix enter hyperparameter mat field hyperparameter variance location vertex mesh observational site approximate linear basis spatial project linearly triangle mesh mesh line analogous spatial structure implement scale logarithmic scale covariate dominate distribution field fix weakly relationship approximate mat half km exploratory exceed standard field lead effect capture covariate deviation mainly scale interpretability two spatial field figure deviation lack shape assume assign due mcmc make posterior oppose method converge slowly heavily inference split datum notational dot zero condition zero gaussian precision correspond eq type outline order conditionally q posterior denote extreme index site ai u ai ai logarithm conditional step outline htp f f symmetric eq proposal k proposal calculate calculate k spatially grid th effect parameter spatial triangle mesh however every grid triangle mesh th sample spatial grid combination vertex mesh serve mcmc run posterior particular standard regular location grid analogous th quantile calculate regular generalize plug th scale thus quantile main objective spatial log scale quantile mcmc briefly base iteration modern bridge intel gb ram hard hour calculation base four set statistic mcmc chain show location covariate figure hold correct moreover figure indicate convergence plot four chain evaluate show trace site autocorrelation plot covariate autocorrelation lag autocorrelation highly claim amount highly burn log observational site site axis site labeling show suggest average south low htp b quantile correct measurement construction mean sd sd density suggest effect indicate location time posterior yield point simulated extreme indicate extreme moment finite correlation point near parameter km scale logarithmic standard variation leave spatial might observational site locate stationary behavior observe scale indicate hyperparameter b b observational site compare correspond observation behavior due observational site lower observation correct parameter fit value observational apply indicate difference overall time vary correct row spatial raise prediction surface south spatial south part observational site expect standard deviation near away observational site interesting inside triangle vertex standard regular figure south side spatial gradient rapidly nearby top air root north middle country law see figure spatial arrange manner see raise south part south second row deviation along south spatial see discuss year reflect high predict year south lowest predict interior approximately datum west observational site mm data set b g panel correct htp b leave panel correct methodology beyond framework
choice mh mix approximate multiple computation could big diagnostic monitoring space univariate use summary robust looking room example future capture heterogeneity cluster behaviour allow care alternative would use euclidean square large dense try dispersion cluster non spatial fact mark remark probability region miss realistic historical cluster interesting try incorporate source seem interesting incorporate supervision grant ep name ex model support study complementary name complementary obtain hypergraph metropolis hasting consider efficient proposal develop allow arise careful convergence diagnostic allow dataset around ad without strong intra dispersion interaction organization complementary name college consist location ad fully kind form information historical role dedicate moreover expect form approximately context coherence organized indicate within tend involve variety figure indicate plausible dedicate dedicated name hypothesis typical cluster together list historical period lot neutral historical avoiding assumption help historical already particular work topic question mark realization type variate process available simplify requirement assumption represent role inference model analyze loose provide single visualization historical interpretation nonparametric inference flexibility specification complementary approach process cluster cox process cluster center process seek explicit inference partition method mark mark search mark seek spatial prevent extended interaction point would provide explicit complementary cluster specification point target association measurement track problem perform complementary type datum association interested assess cluster interaction quantifying estimate modeling aspect careful whether significant type common k cross complementary appropriate see section intractable express term find classical association assignment metropolis hasting obtain posterior overcome develop scheme allow hypothesis cluster explicit inference historical interest discuss material include extensive calculation plot datum preliminary result spatial make different list involve refer ref date ref evidence sp sp db db db stand location express survey os national os grid great letter letter location accurate amount c precise term column record see discussion count merge analysis concern datum process entail subject project place variable variation actual record vary amongst treat convert os assume locate os triple triple record see primarily historical interpretation whether separate merge change present merge km record cross great package fall approximately buffer km include inaccurate region investigate type point bivariate interaction type bivariate function divide intensity g type weight intensity classical rely stationary therefore use contribution couple function whether show significant labeling type I location arise poisson dataset spatially concentrate stationarity pattern nan hypothesis poisson intensity potentially vary realistic type additional approximate monte intensity figure multivariate finally deviation significance version km summarize single value gray area simulate pattern dash black red dash value independently strong test function preliminary clustering include advanced provide answer exploratory indicate motivation use moment preliminary communication partition model process discussion nx disjoint trivial subset cluster accord depend global intra dispersion thus nh j abuse exchangeable respect arbitrary cluster unobserved observation q l expect independently value euclidean sample calculation dirichlet dp random conditioning model enforce almost expect alternative inference poisson preferable cluster graphical structure section intractable meaning inference inference complementary move precise little hope solve satisfactory application two induce blue red green type set admissible connect partition hypergraph correspond remainder paper treat formulation color reduce give proposal without change balance infinity help rate derive ordering argument asymptotic omit favor demonstrate diagnostic complexity proposal increase poor need mix property require little derive try obtain continuous tune high acceptance propose proposal long scale move mh hypercube integer randomly sample scale possibility choose move propose l j move roughly move seem perform implement multiple significant consider approximation great cite truncated region grid proposal scheme accept move parallel fashion perform mh factor note bound region parallel would increase especially dataset require need various diagnostic assess indicate section p blue accord code available qualitative look plot sample occur summary match real run configuration estimate autocorrelation integrate autocorrelation ess real use package see version diagnostic overview particular version statistic context summary informative compare look association probability consider proximity consider empty mode mcmc individually diagnostic severe one summary none method indicate except ess step step acc sec configuration proposal multivariate scale evaluate software diagnostic agree mix note performance computer case proposal commonly speed g keep unchanged matching cycle configuration maxima reach configuration cycle move need configuration configuration potential implement local maxima complete case configuration nevertheless application complete therefore exhibit sufficient use simulate diagnostic arise maxima mh long move target cluster pairwise kernel mh group color eq binomial select color project color color configuration follow color replace point color number merge together move know move balance prove indeed equivalent never merge color proposal cluster induce point birth move propose create move application among meaning ij informed color computationally see performance two informed project subspace contain hypergraph state would extremely therefore difficult efficiently proposal choose subspace perform inform space uniformly mix poorly inform inform proposal expensive complicate one care long run convergence diagnostic reach mix high two nevertheless color properly complete section section obtain partition cluster complementary permit range parameter correspond section c reduce respectively lie support posterior association region concentrate fit synthetic without would figure reduce consider posterior km therefore fit approximately km accordance project coherent historical density
show bottom opt rewrite opt combination denote right procedure represent unsupervised utilize mkl combination mkl dependent weight red weighting assume multivariate perform contrast intuitively guarantee regularizer binary mkl classification path denote transpose decision b mkl dependent predefine regularizer rademacher induce mkl classifier g definition prove equation target hold prove mkl l learn lagrange multiplier binary label optimize wise opt equivalent provide follow path product node exist utilize update regularization order utilize therefore node solve fix multiclass fixing multiclass fixing fixing summarize optimize show learn mkl equal path go accelerate speed generate correspondingly mkl label algorithm multiclass mkl list alg regularizers mkl describe structure accordingly encode structure distribution convexity regularizer efficient world information scene decompose scene part pixel answer like useful mkl tool aim combine mkl strategy induce attention mkl mkl direct acyclic dag et formulation information combine account construct regularizer formulation connection node make rather research sum product describe combine kernel product consider mkl create multiplication negative kernel describe embed directly taylor series still kernel regularization path weighting encodes entire involve strong connection regularizer rademacher complexity update organize weighting regularization include comparison direct acyclic dag
ai ai n bi determine eigenvector tv nc nx ix eigenvectors note unlike machine application information eigenvalue dominate dominant step graph consist compute final computed operation overall compute recommend value usually constant graph treat complexity bad fact sparsity adjacency word spectrum walk short e vertex actually application pre five step independently preprocesse pair permutation invariant algebraic consequence path point observation component simple correspond length multiply normalization come among number length length interpret path difficult aggregated statistic path length section capture structure kind indicate relative kernel path path walk kernel common subgraph relative capture summary exploit functional expressive vector deeply beneficial application deal graph graph expansion reduce skew spectrum walk kernel short kernel choose benchmark consist size label mean around protein ec around active anti cancer node maximum focus remain evaluate capture label procedure follow evaluation run split fold identical fold validation set train c fold fold fold act fold act whole classification error result average partition stable accuracy unlabeled graph kernel short count skew optimize noted tune keep thing easy dataset dataset previous skew huge dataset accuracy perform short path perform kernel achieve accuracy representation capture structure kernel power consist skew spectrum capture expressive functional well compare outperform short path kernel representation expressive superior surprising base count common path subgraph small subgraph run see vertex compete except gain time competitive wise superior except success capturing preserve near probability two graph perturb version compare determine graph uniquely determined hard practice behave dynamic loose kernel permutation invariance node long require different behavior less adjacency usually small perturbation operation like edge perturbation kernel thus perturbation kernel graph verify dataset randomly evaluation edge randomly process increase perturbation plot clearly smoothly perturbation compare relatively big clearly perturbation jump value kernel functional significantly power expressive functional simple huge scope possible flexibility room derive expressive row estimator power kernel another explore iteration demonstrate provide interface deal gain kernel lot partially nsf dms fa functional adjacency functional remain unchanged handling form construct functional significantly dataset superiority approach methodology make kernel cubic becoming span bioinformatic social network search natural etc meaningful operating similarity vary design graph incorporate rich structural spurious transformation like certain additional edge annotation domain paper structure extract kernel graph harmonic analysis technique extract set dot product kernel alternatively design kernel graph walk count common shortest count vertex distance although path still widely adopt disadvantage walk count subgraph possible lead kernel subgraph node kind recently subgraph facebook count common walk path subgraph etc instance relatively embed graph represent expressive dynamical construct impose summary graph distribution power benchmark node compute methodology graph matrix node node vertex adjacency always unweighted otherwise matlab therefore invariant capture information dynamical embed simply iteration matrix power sufficient summary power recursively normalization input x x x tx x starting node row use difficult along fact require general force limited degree freedom compare intuitive imagine associate graph start tell sequence generate node go preserve unit treat node kind update
end simultaneous become tensor maxima thus version eigenvalue stochastic ascent discrete rule modify neuron modify fire one component triplet triplet mixture rule rule selective step stable show triplet mixture ensure neuron selective component modify triplet align poisson name emphasize interpret latent underlie input distribution slowly spike period neuron need spike I class maximum somewhat limited sample sequence triplet triple tensor triplet perform ascent expect complicated pre fire multiple interval order spike full triplet r subsection triplet low triplet rule q ball cr see subsection detail projection practice sufficiently projection rarely occur make p linearly mean triplet identical proof update go unchanged view triplet view triplet w stable selective mixture say e co linearly dependent poor k suppose selective intuition emphasize extremely independent must triplet regardless regardless often additional fact transform transform may rule vary bounding thresholding easily either domain produce useful rich possible research proof theory stochastic decompose part update martingale ode previously point lyapunov triplet follow martingale take triplet classical triplet converge slight modification unconstraine space actually lie may act point algorithm behave biased walk toward perturbation algorithm stable infinitely infinitely difficult check set ever small find biological limit fire neuron define lyapunov finite tend hx xx vx x x require algorithms compact region infinitely deterministic lyapunov completeness step converge continuity dot taylor series slightly go go zero fix open neighborhood v nr n nu nu na un smaller disjoint therefore must go converge start simple full ball projection q let rank open replace immediately variance boundedness bound increment requirement martingale bound us requirement q q requirement satisfied note stability zero case somewhat rank instead increment drift randomly undesirable neuron randomly span slight would slowly decrease increment stable expect update denote fact w process note measurable stable martingale increment behave precisely like extra increment control lyapunov note expect lyapunov trivially decay rapidly directly column shrinkage ignore increase variance increment stochastic drift modify arrange stable remain selective mixture however prove kronecker canonical matrix fact property follow trivially kronecker update stable triplet say neuron selective interaction neuron ht vector neuron ik pp computation p nk nd fire neuron triplet fire neuron fire weight fire connection assume l identity hand neuron hand neuron notation kronecker prevent stability stable conditional meet lyapunov neuron selective one neuron selective component iff depend vector fire calculation zero critical point unchanged stability connection jacobian neuron selective analysis occur selective selective eq kronecker semidefinite neuron selective expectation member network feed neuron connect network converge distribute gaussian network neuron triplet mixture randomly unit neuron converge selective initialization per even cause neuron encourage selective call triplet multi provably independent implement mechanism sliding maintain also triplet combine neuron connection tensor decomposition information circuit publication dependent thm lemma thm remark thm plausible learning rule triplet generalize novel kind decomposition substantial incorporate triplet sample spike dependent rather mixture distribution biological fashion backpropagation signal modify incorporate refer provably broad class learning presence interpretation specifically show classical function input prove requirement implication spike arrive spike train learn spike adjacent stimulus biological fully spike dependent however much posterior requirement issue provable form presentation formalize decomposition triplet rule show network triplet neuron outline triplet decomposition definition triplet finally article notation tensor tensor product denote application tensor application matrix rule fire fire fire slide firing rate formulation ht variant rule purpose article define step rule system input draw linearly convenient expectation stochastic
previous possible require spectrum decay furthermore check sufficiently note claim satisfy definition svd standard sketch appendix low necessarily sort sketch randomize see sketch rank approximation satisfy satisfies definition follow projection rely diagonal select proof satisfie orthonormal satisfy furthermore splitting ensure rank frobenius singular selection need compute want approximate diagonal reduction start show us condition together satisfy sketch understand subspace know error several family refer family write transpose sketch independently uniformly except embed position position choose except position sign column hash embed alternatively sample I bss algorithm guarantee family construction family follow family stable requirement ensure f frobenius matrix prove via lemma c moment remark family f frobenius meet entry row preserve preserve preserve k frobenius decrease probabilitie sufficient column norm entry norm family list suitable svd apply purely matrix preserve definition fast tradeoff cost simple avoid establish lemma without go follow thus r apply multiplication moment family note generalization follow case approximate multiplication thus svd sampling produce easy interpret sketch maintain substantial benefit perform obtain error first column subspace satisfy probability norm suggest lemma alone could allow additionally suitable identify nearly without formally data matrix orthonormal constant factor md lemma routine norm column transform column norm preserve give desire family norm runtime md issue computation requirement analysis svd guarantee produce argue frobenius norm trick probabilitie potentially singular direction square spectral newly satisfy frobenius sufficient define effectively singular norm put everything f f ok I connection sampling row norm project onto span leverage respect refer norm residual project first round shot avoid step recover projection cost singular sketch project dependence rather satisfy weak span column finally give selection sketch introduce however extend stable furthermore substantially reduce runtime produce ok ok overall technique sketch multiply project short way project subspace albeit choose give dependence multiply single matrix let satisfy family reduce ok ok cost preserving sketch simply rotation frobenius sketch show cost sketch projection complete low orthonormal matrix satisfy approximate svd follow require combine give column orthonormal basis actually alternative multiply find row onto let whose orthonormal row within span first project complement give suffice svd sketch frobenius norm multiplicative frobenius frobenius give requirement appendix give note sufficient completeness illustrate application spectral project sufficient approximation sketch size interesting use dimension let constant may achieve cluster clear insufficient constrain projection column identity cluster achieve project dimension selection column least cluster optimize give multiplicative substantially write b frobenius norm simply row center multiplying preserve distance preserve alone combine inequality fact decrease preserve preserve give algorithm scope black improvement dependence streaming give streaming computation aside immediate application approximately subspace row compute approximate svd wish give streaming process server row necessary bit streaming stream word failure sketch bit specify give streaming row streaming approximate give matrix arbitrarily assume able central require communication failure probability recent line seek apply svd top vector server locally use communication improve additionally projection result preprocesse entirely inherently pca stem amongst matrix server project proceed could non technique communication logarithmic dimension connect server clustering succeed probability bit ok ok bits lemma column ok draw family server ok orthonormal basis server basis rank proof server row adjust factor communication ok open sampling svd sketch approximate coarse even svd refinement eventually relative approximate svd exact possibly lead extend schmidt lee also david discussion support science foundation nsf fellowship grant grant fa decomposition necessary also constrained mean choose let first prove orthogonal place form cloud drop notation place cluster center simplex center centroid gaussian near choose cloud simplex optimal cloud cloud rather cluster significantly cost optimal low define slightly row cloud lemma value gaussian comment turn follow yield exponentially therefore lemma fraction cost cluster gaussians naturally cluster lemma prove theorem first projection one put cloud simplex cloud origin rather centroid cloud incur sum square gaussian f repeat origin argue incur centroid total square point k mn include claim cloud proof origin cluster course cluster centroid point well claim notice term cost origin gain origin cluster I high gain prove gaussian bernoulli concentrate eq probability contribution sum geometric yield remain bind concentrated around since number high carry desire unit vector product n enough union nc function prove extend analysis svd orthonormal satisfying condition projection kk km svd give equation follow lemma f definition remainder row f r triangle norm substitute gm result conceptually result rely frobenius inequality alternatively compute set section extend preserve frobenius motivation guarantee give f rank orthogonal use notation give give span rewrite cauchy schwarz give finally combine derive bound give easy version theorem draw family error probability cost requirement preserve sketch constrain width width title corollary definition email mit edu sketch solve mean approximation reduce accelerate heuristic many svd streaming additionally give subset cover datum give first dimension sublinear reduction attention fast algorithm reduce usage decrease multiplication rank similar tool heuristic provably seek accelerate reduce cluster original datum approximately analysis nearly reconstruct e start note problem problem nonnegative concept independent ensures solve approximation obtain preserve multiply sampling well focus heavily implementation runtime amenable acceleration underlie svd embedding inexact preserve future randomize significant year embed preserve column cost summarize show compare prior construct apply constrain projection prior c thm thm small preserve project identify improve rank nearly due expense suffice application svd method svd typically lack sketch spectrum dimensionality reduction would preserve useful unconstrained setting rely problem allow generalize work address reduction use selection approximation via sided cost preserve orthogonal projection f k lemma except lemma place seek characterize
input mse estimate coefficient evaluate broadly divide set second measurement mse correlation measure percentage identify inefficient efficient implement matlab code request criterion production process table criterion variation production production pc present obtain model variable pc correlation type operation accordingly robustness variable varied exhibit consistently fluctuation percentage identify percentage efficient correctly identify three pc inefficient experiment e production low production efficiency result experiment production dimensionality production outperform test production mse efficiency production weak production decrease performance improve study robust variation covariance input production fast choice technology three roll surprisingly study overview output likely likewise specification must concern report selection benchmark examine correlation production base obtain experiment evident benchmark method parsimonious selection envelope sparsity lasso group multiplier admm york ny york ny introduction seminal linear powerful quantitative management research single comprehensive generate decision year economic range song efficiency technology stock interested reader popularity certainly research despite publish accumulate paper web decade surprisingly attention variable literature selection often experience economic matter major concern irrelevant omit relevant negative impact misspecification instance al irrelevant position production rank misspecification relevant addition misspecification production space increase tend shift lead power essential limit include consensus extend lasso selection design group derive version tailor multiplier thorough measure parametric production program output output input deviation inefficient originally output sample time input constraint impose ensure estimate case output input lp orient primal dual output orient augment convexity account production return production basically differ assumption production technology radial approach measure efficiency radial assume change assumption radial additive take efficiency orientation formal dual production technology respective associate give however note formulation note introduce situation optimization regularization add absolute geometric e select variable although linear extend additive model respective sparsity solution shift entry one readily application model reader guarantee variable select care selection consistency across stack goal variable extension induce e correlated achieve regression regression grouping limit joint variable solve guarantee study extensively machine efficient solve unconstrained problem section tailor method multiplier admm additive variable apply element sufficiently bound variable regularization drop sign introduce slack transform write stacking column matlab low letter elementary writing ccc ccc x vector similar apply matrix variable likewise describe next alternate direction multiplier admm belong augment method solve lagrangian ax ax b term follow structure unconstraine introduce constrain lagrangian admm find desirable solve augment lagrangian sequentially subproblem simplify cholesky leave hand side cache substitution subproblem solution subproblem lasso convergence admm way splitting function full rank tucker pair problem one decrease simply stay constant optimal treat way rank apply admm selection selection method variable literature contribution measure principal bootstrappe four approach result among pca bootstrapping pca replace pc retain true curse issue bootstrapping involve computational four select benchmark variable efficiency candidate particular scalar quantify marginal measurement essence test statistical significance contribution mean consist selection elimination removal support radial technical population random cumulative density underlie irrelevant additional represent represent proportion whose change consider production associate change statistic reader respectively test estimate production include variable statistically give significance proper output candidate production add process repeat include test technical orient radial behind radial orient radial production mostly observe estimate production contain measurement importance matter overcome study use carlo production process production input represent produce role production production production production efficient intuitively importance production thing denote additive production distribution study half variance uncorrelate generate production show increase
cache probability word representation long input however work contextual nlp part contextual stochastic present possesse unit change slowly see layer gram change similar cache precisely denote unit rule note nonlinearity apply contextual hide decay bag representation trace propose integration force neuron evaluate result observe show big gain strong unit unit activation unit identity matrix size show structural modification constrain equal identity diagonal reason fix constant force unit allow weight delay precisely contextual diagonal diagonal element diag stay strictly help self language corpus division part art achieve combination language lstm language dataset moderately text million character split first character development last character report construct replace less token speedup finding model recurrent contextual allow representation history unit various cache hide short recurrent weight seem text corpus fix text significantly long term current play illustrate cache gram drop model hide show contextual result add contextual unit drop hidden hide lstm contextual increase lstm slightly versus lstm much significant actually lstm lstm paper perfectly introduce structural interpret quickly change short pattern slowly update context short term lstm gain recurrent tune similar outperform lstm margin practically model thousand hide neuron help researcher understand greatly simplify recurrent long pattern published reproduce none model nature store long symbol reproduce would become net controller need increasingly task com recurrent learn time recurrent difficult gradient descent vanish gradient long pattern language perfectly slight encourage hide state slowly part close form kind memory evaluate short memory lstm core variety recently obtain state automatic modeling mostly feedforward recurrent feedforward architecture delay usually time history make hard done increase architecture represent recursively recurrent layer previous store complex period memory theory architecture perfect memory simply powerful recurrent model widely vanish simple simple time simple network memory term pattern practically ignore long reason happen sigmoid zero partially deep relu recurrent empirically backpropagation recurrent term pattern architecture deal vanish long term lstm recurrent neural recurrent promise write recognition lstm fairly sophisticated make neuron information another interesting direction exploit vanish gradient non objective hessian well empirical partially solve vanish recurrent close hide unit behave cache long term model modeling dataset h cc character contain able token past see figure connection hide token token predict store token see sequence token apply token embed vector token max dictionary type architecture replace hierarchical hierarchy token word soft loss mention descent back propagation gradient practice rarely reasonable hyper detail strong nonlinearity appear world neural along vanish recurrent vanish gradient back magnitude quickly pattern difficult fail capture simple extension yield retain
mean mean particle directly accurate within deviation prior become magnitude indicate base well case one guarantee walk surrogate indicate physics tb panel via panel estimate far assume predict informative away evident feasible construct mc posterior problem dominate require pde construct gaussian quadrature parameter locate problem failure lack accuracy reduce suggest mechanism sampling show beyond prior require unless polynomial rigorous impractical constructing base find costly acknowledgement office technology department contract ac grant dms dms dms normalizing laplace indicate substitute equality eq term equality delta follow substitute tu berkeley laboratory department mathematics university california berkeley mathematics university expansion reduce inverse beyond assume surrogate posterior different posterior inaccurate adaptively effective compare parameter incomplete pressure approach yield pdf sampling see require evaluation repeat expansion representation problem approximate surrogate result sample approximate accuracy informative behavior sufficiently sampling limitation quantification inverse analyze surrogate small study sampling numerical summary proof derivation appendix describe affect represent uncertainty pde pde computationally bayesian prior combine give pdf simplicity throughout gaussian prior identity relax simplified nonlinear mc monte posterior involve computationally expensive cost truncate solve pde prior polynomial orthonormal I assume convergence depend regularity p remainder regularity quickly replace model truncate posterior nd kullback hellinger moderate expensive g introduce unless represent truncation surrogate method surrogate approximate wish interaction mechanism due truncation poorly must construct significant unlikely surrogate region significant locate polynomial moderate locally lack base inaccurate tool posterior depend well truncation assume surrogate inaccurate move regime introduce small regime assimilation allow rigorous situation pick choice eq grow small small grow derivation interpretation small posterior informative informative problem example geometrically getting get around obtain q similarly surrogate posterior surrogate different surrogate singular respect posterior surrogate large accurate truncate increase sufficient rapid become increasingly expensive stochastic quadrature point constructing estimate increase minimizer truncate e make small wise eq exponent far mass informative accuracy grow increasingly informative effect experiment inverse choose parameter estimation understand realistic integrable wish datum see experiment length element mesh pde symmetric quadrature solve discretize first eigenvector square multiply squared function rapidly decay spectrum capture expand use gaussian quadrature effect vary could equal decrease length perhaps realistic capture impractical global figure show prior approximation require focus grid eigenvector assume two deviation restrict finite element correspond almost
similar prediction straight method neighbor count serious index tendency similarity algorithm prohibitive completely call broad refer dependence pearson adjacency challenge mean lot sparse extract similarity long large similarity poor outcome high order path method combined method substantially exist especially begin briefly representative unweighted simple self connection measure suppose top exist way score node ref accuracy similarity neighbor cn resource local path simplest overlap method drawback obvious number number neighbor future representative cn insufficient base global ii literature q pure lot easy large denominator tendency degree node tendency cosine index resource pair directly assume need neighbor play neighbor case resource similarity eq q symmetric similarity aa replace although aa index form contribution aa take considerable heavily aa previous study common network iv introduce ref local consideration wide cn eq cn connect extend path uncorrelated fast exceed around positively short prediction node common neighbor propose calculate similarity node rank node similarity coefficient mathematically attribute directly go consideration set fraction link error realization independent short length p cm email area receiver operate auc miss give order list latter consume record time score auc calculate score independent identical auc high cm p cm c compare method four representative email detail set similarity cn lp prediction extract path order compare measure apply move probe link correspond auc fig interestingly advantage fraction path achieve cn resource allocation method enjoy dense predict link link method suppose dense validate dependence real improve lp lp indicate problem address actually reasonable method generally network consider local cn cn account auc well auc outperform cn path auc cn link know literature item well cn accuracy node largely improve degree way auc substantially increase change cn information accurately address node prediction auc probe link cn lp auc indicate connect moreover indeed cn auc though result generally path auc lp extracting employ pearson accordingly predict future common variant prediction extract similarity path pearson combine resource method outperform little new issue remain open compare study pearson direction coefficient study also show also valuable especially already recommender system semi improve recommendation important paper possible way salient investigation partially project link aim node miss evolution link prediction investigate far prediction node base high finally resource allocation substantially epidemic coverage certain model citation
nucleotide incorporation incorporation significantly cycle variance nucleotide incorporation probability flow cycle incorporation calculate variance eqs normal cycle fix sequence cycle eqs q show length number cycle discuss ignore expression exact distribution fix nucleotide flow nucleotide incorporation distribution distribution calculate eqs distribution cycle sequence discuss exact flow exact introduction incomplete nucleotide incorporation determine cycle nucleotide incorporation variance respectively normal slightly long tail compare normal distribution discrepancy find nucleotide incorporation situation ij mean calculate show discussion incorporation sequence may previous generalize account dependent nucleotide incorporation cycle nucleotide incorporate nucleotide correspondingly nucleotide incorporation function nucleotide incorporation incorporation eqs eqs seem form incorporation incorporation generalization nucleotide complete flow cycle exact various formula incorporation probabilistic incomplete nucleotide incorporation although thing avoid traditional bring increase incorporation high resolution region template individually potential throughput become biological software development sequence work support school sequence synthesis generation dna sequencing especially explore allow nucleotide incorporation cycle sequence synthesis incorporation flow statistical nucleotide sequence incorporation nucleotide cycle distribution generalization incorporation significant variance approximate handle sequence incorporation useful software sequencing generation sequence technology aspect technology many available development sequencing repeatedly determine complementary template incorporate usually presence absence signal step nucleotide distinguish modify sequencing read simple relation cycle equal reaction sequence sequencing rather length nucleotide nucleotide incorporation complete possible extension cycle ideally nucleotide incorporate complementary include statistical situation paper nucleotide incorporation dna sequencing technology sometimes nucleotide incorporation cycle nucleotide incorporation incomplete nucleotide incorporation mathematically generalization obtain previously nucleotide incorporation dna sequence development testing software machine dna sequence technology define derivation cycle length nucleotide incorporation sequence technology become employ principle sequencing unlike sequencing target single sequencing sequence significant synchronization identical lose lead signal decay sequence reaction incorporation completion cycle increase individually exist reaction adjust incorporation sequencing reaction incorporation dependent nucleotide define nucleotide illustrate lc cccc cccc cccc nucleotide expansion power detailed recurrence equation close form normalization sequence flow cycle nucleotide look example assume nucleotide incorporation recurrence refer understand q nucleotide incorporation solve however form nucleotide incorporation probability recurrence equation need identity transform solve nucleotide incorporation put compact form four nucleotide sequence incorporation incorporation nucleotide nucleotide incorporation nucleotide flow cycle together unnormalized probability flow cycle treat work obtain get eq obtain normalization fix denominator become denominator dominant part expansion expansion come normalization cycle stand derivative cycle part expression compare availability length variance formula nucleotide
correspond kk k pc original pc solution coordinate many iteration along less al algorithm panel minimizer f give transformation affect composite monotone pairwise query pc show constant great pc positive pc objective strongly convex gradient whole relax convexity twice continuously differentiable hessian f kp oracle ensure correct high repeat reliability pc x subroutine request use arbitrary require repeat query response pc oracle coincide pc find component direction investigate parallel conduct ghz core run compute indeed dimensional showed compare quadratic generate positive use parallel computation overhead accuracy line stop hand tend optimization stop limitation tp several moderate scale optimization parallel implementation original e require implementation line except search assign core core parallel computation approximately serial great practically overhead among processor may scale tp quadratic ann positive matrix quadratic assumption algorithm find simplex significantly standard method depict solid cpu indicate efficiently even pc pc upper efficiency parallel outperform serial implementation cpu communication overhead parallel conduct stochastic pc correct query repeat serial extremely panel slow iteration implementation fast cpu algorithm pairwise value direction search hence effectively large practically implement outperform important direction include kind mm paper provide unconstrained require pairwise comparison estimate pairwise us function estimate pairwise along computation bind find exist engineer field kind tune infeasible treat information widely decade algorithm search function trust oracle tell value evaluation derivative pairwise collect estimate prefer among alternative comparison information stochastic sign oracle stochastic early simplex receive namely reduction order close guarantee problem poorly show positive hold constant convergence objective pairwise stochastic pairwise binary f call oracle affect meaning change al convergence optimization stochastic provide choose uniform solve
profile investigate demonstrate capability multivariate system paper follow ii background family divergence end iv conclusion reduction sequential incremental version technique dimension datum range generalize factorization perspective rank vector value eigenvector rectangular diagonal value sort pca less widely probabilistic draw low large family example exponential family parameter distribution ensure integral pca dimensional happen lot bregman introduce quantification q family divergence equal logit case bregman place efficient bregman divergence approximate work mainly bernoulli random logistic hence logit thus optimize quadratic alternating define iterate call q version streaming every e update e base base gradient vector sequentially equation investigate full td l discuss mainly variable exponential random batch loss function relationship take locally solution define opt surrogate lipschitz continuous regularity thus martingale within within constant appendix theorem recognize sequential converge within however note probably firstly use simulate illustrate focus since principal straight update ht ct cc try equal show sequential step interesting finding firstly within stochastic phase period initialization phase characterize decay whereas phase stand secondly behave differently place hence another regularization summation loss function note unbounded ft could behavior last important mention many completely ignore building modeling end energy attract dependence model bottom work energy pattern individual generate efficiently characterize whole consumption tt energy size want small collect minute obtain pattern enough consider achieve reduction fig good consumption demonstrate adaptively update model interestingly give pair periodic pattern whereas probably result non ht online address streaming extend sequential optimization capability storage sequential application end sum rhs proof lemma n nc tc want decay similarly prefer research berkeley education building
replace sigmoid good available cifar configuration file well classify mnist lstm introducing model objective decrease smoothly resolution investigate follow match begin end want parameter parameter sgd early sgd point coordinate span plot vertical axis objective vary much tell far sgd shape plot dimensionality walk plot residual norm converge similar geometric dimensional different maximum whole fig keep mind give information plot subspace whether behave path direction explore point primary investigate line explain sgd factored fig predict neural curvature curvature solution sgd connect globally neural equivalent rescale parameter multiply divide factor shape linearly high middle manifold kind achievable via span factor lstm feedforward maxout narrow text local saddle fig sgd pass point early trajectory seem explanation sufficiently avoid saddle analytical sgd descent simplify hessian view time gradient hessian taylor gradient go big encourage curvature visualization objective function necessity interpret visualization rich function trajectory instead multiply subspace intend side reduce trajectory circle point almost variation cost subspace sgd trajectory intermediate mp axis allow circle department electrical engineering stanford stanford com involve solve scale non minima motivating however modern achieve negligible task technique network optima initialization never network generally regard optimize train theoretical nevertheless commonly successfully art result variety simple roughly involved neural training intend quantitative answer enter minima pass variety saddle point answer question suggest exist could single break sgd behave subspace main text review case evidence saddle point suggest conditioning neural examine add training sgd ever act stochastic approximation could examine remain cost due induce seven model examine fully connect supervised feed model analytically factor qualitatively outside remainder qualitative factor competitive interpret sgd neural sgd structure training consists initialize extra momentum reach early high trajectory visualization simple learn repeatedly sgd rapidly minibatch gradient remain long period way technique qualitatively analyze line parameter objective behave line search job consistent work begin neural feed forward connect dataset maxout adversarial momentum see specification solution minima saddle fail minima solution break saddle rather fundamentally perform neural network feedforward network verify advanced convolutional network barrier network initialize correspond initialized weight barrier reasonably detail look behave barrier mp model purpose good easy business secondary visualization trajectory sgd pass learn mp may visualization technique explore area e interpolation lstm regularize dropout see experiment convex appear cause difficulty recurrent mathematical deep mathematical network deep form transformation learn transformation expressive capacity factor dynamic fit deep non deep suffer saddle point vary quality link interpolation carry analytically regression problem square error qualitative network interpolation
lp reasoning coincide differ solve impossible column privacy enough constraint time satisfy impossible sensitivity column need accuracy general program program classify private affect program efficient natural private program g semidefinite program multiplicative certain crucially compatibility projection seem differential privacy privacy strong privacy record database range differential record database output definition function database differentially pair record differentially private use laplace laplace differ laplace draw differentially tail laplace mechanism laplace scale exponential mechanism discrete value exponential output maximize mechanism differentially private suppose mechanism combine mechanism composition composition private consider constraint easy negativity approximately repeatedly feasibility unless attention privacy find whether lp roughly private database scalar constraint lp neighboring want satisfie differential notion equal equal additional algorithm operate select action favor write multiplicative maintain dense weight roughly algorithm project action dense distribution step point approximately satisfies arbitrarily instances define bregman let dense multiplicative multiplicative combine ht dense weight follow measure distribution represent hence public feasible independent lp oracle concrete oracle fractional set cover section oracle program maintain pick intuitively loss lead violate lead point feasible taking full least multiplicative pair constraint see projection db ax multiplicative density run point union succeed least condition sx ty ax q let contradiction make depend private point final point since public one privacy parameter oracle neighboring sensitivity private whole directly composition oracle private add constraint lp know project check neighboring satisfy bregman projection reproduce completeness identical respective bregman dense follow constraint privacy fractional though argument private width cover packing covering wish collection cover person fractional select whole set cover degree cost degree least variable degree constraint exactly cover otherwise optimal goal wish individual cover constraint cover contain person valid cover people cover constraint people constraint private solving since vertex select vertex mechanism suitable oracle adjacent return eq sensitivity neighboring database sum neighboring contribute sensitivity contribute differ lp formally randomize input sensitivity normalize multiplicative oracle loss feed laplace mx normalize run private oracle private sensitivity operation operation private private oracle exponential eq since mechanism follow private loss multiplicative weight distribution loss satisfy solver lp program feasible produce point let exponential choice leave x tp bind satisfie take union loss event guarantee independent desire unfold definition like result quite row private entire change neighboring differ private database objective pair neighboring constraint trivial randomized input vector column row slight b mx oracle find private private dual oracle neighboring satisfy private mechanism dual oracle eq sensitive low column sensitivity differ laplace suffice mechanism choice composition let exponential mechanism oracle find point proof nearly everything previous tight coefficient differ leave tight row amount privacy randomize solve private objective simple response solve throughout neighboring change randomize input vector sensitivity concrete lp change laplace solve get exactly lp optimal solution lp objective private solving perturb lp q privacy composition accuracy single laplace bound eq lp perturb add optimality perturb find exactly feasible detail consider various sensitivity turn section show high solve exception constraint private relaxed low reconstruction attack show differential reconstruct non fraction reconstruction key due differentially q restricted entry round zero also desire impossible private lp neighboring database neighbor lp non q likewise say eq bit change private feasible round exactly privacy zero bit objective similar private arbitrarily private find exactly feasible lp objective find objective place mass share private database zero column lp coefficient set coefficient private want satisfy private find feasible public consider find reasoning coincide e produce correspond two impossible possible allow however time produce satisfy single accuracy impossible column relaxation privacy linear program program affect give program approach multiplicative solving feature algorithm use crucially compatibility extend extend rgb rgb keyword claim systematic program privacy introduce several class private program incorporate class program give solver differential differential privacy strong database belong randomize output differential change single record database database private record private basic differential privacy laplace database record mechanism sensitive laplace mechanism differentially private follow tail laplace mechanism mechanism produce element range exponential approximately maximize let quality mechanism proportional exponential differentially private satisfie follow suppose score private mechanism combine mechanism composition theorem composition adaptively mechanism differentially private consider b negativity lp repeatedly solve search restrict feasibility feasible want private database scalar private lp neighboring dataset except solver vector constraint private standard approach algorithm brief algorithm operate action loss perhaps favor aa use multiplicative maintain dense place multiplicative project action probability neighbor give bregman distribution define dense combine multiplicative guarantee arbitrary loss subset public bind eq lp concrete fractional cover see give find present dense multiplicative pick satisfy intuitively lead feasible program multiplicative weight onto dense approximately ht ax loss via dense weight accurate point union succeed step event sx ty ax define eq constraint contradiction first depend final note minimize since hence public follow q private neighboring distribution except row entry density remove lp exactly except neighbor satisfy st reproduce identical respective bregman treat clear divide except privacy example fractional though example packing cover cost select relaxation instead whole cover decide set degree open fractional weight cover program variable cover cover goal fractional wish individual approximate person contain person valid cover people find people private since point lie vector I zero vertex exponential suitable vertex oracle return sensitivity neighboring database extra neighbor since take neighbor contribute since source contribute guarantee show select probability fractional exponential find constraint constraint let unfold applie q guarantee demonstrate fail constraint imagine cover approximation guarantee guarantee output implicit set cover private interpret weak rather differential apply turn adjacent database individual grow simplify form constraint private feasible note rescale find get kind first multiplicative receive loss update aa dense multiplicative maintain response approach ht db mx dual oracle multiplicative accuracy running find point linear public side private map vector neighboring database think decrease guarantee lp feasibility implicit matrix sensitivity scalar generalization offline private weight influential express differentially private solve differential privacy throughout private neighboring database norm look private accordingly oracle private private run private differentially private private query release appropriate vector differ neighboring private oracle private combine private sensitivity private guarantee linear feasible low private sensitivity private quality desire synthetic query universe privacy neighboring far differ neighboring want single sensitivity private neighboring technique equally matrix assume feasibility leave vector vector low sensitivity private basic primal selecting fed weight give ht mx private loss I normalize sensitivity operation operation private private whole neighboring private mechanism distribution follow private analysis regret multiplicative loss satisfy sensitivity lp program solution private sensitivity probability mechanism find oracle hand event tail laplace mechanism take union loss show hold side noisy exactly multiplicative independent exponential mechanism small since feasible desire remain unfold private entire neighboring differ differ neighboring row decrease formally input vector sensitivity private slight modification algorithm ht mx set compute normalize exponential mechanism dual oracle neighboring satisfy distribution dual distribution sensitive private private sensitivity differ norm add noise differentially privacy step private choice
evaluate point large population prohibitive exchangeability agent agent exchangeable invariant permutation main utility depend compete un agent outcome outcome rule realize report behavior h array initialize jt enable pair exchange pair one two medical one test sensitivity accept reject pair issue however pair exchange currently operate centralized exchange mechanism resolve exchange fit paper follow assume mechanism former whereas agent randomly assign mechanism per month usually round pool pool patient medical test easy thus report report pool patient mention compatibility medical literature respectively assume perform exchange study pool report compute mechanism apply along detailed define patient truth behavior adopt armed specifically try maximize pair match simple track utility play pt internal pair tt inference game theoretic observe agent report multi bandit band respective causal round armed believe trend point simulation causal take mechanism make distinction goal experimental collect interested term prior game prior former put exchangeability payoff matrix case obtain expect utility would average match mechanism iii strategy show ccc utility payoff proceed describe collect effect long ground simulate run agent report informative behavior separability take around report show method theoretic former inspection perform informative agent likelihood exchangeability bias long method center clearly biased towards payoff able capture evolution system extent underlie behavior report practically report respective figure histogram estimate method weak separability empirical center empirical overall difference theoretic method equilibrium towards evaluation mechanism challenge dynamic challenge strategy outcome former use report distributional assumption likelihood report agent equilibrium improve multiple way good however ignore aspect agent game theoretic crucial practice base calculation principled hyperparameter third assumption realistic substitution I case agent switch mechanism long independently agent sharing require sophisticated section theorem section economic allocation interested agent process type analysis operate orient good evaluation usually ignore nature raise outcome interest interaction interestingly methodology effect use equilibrium mechanism exchange improve ignore agent causal inference system effect mechanism determine allocation online determine appropriate report designing mechanism good mechanism appeal agent report desire resource price face intuition agent affect report mechanism high high bid initial bid even mechanism property practice typical ii iii model iv interaction get desire participant item bid truly round participant price light design outcome ad want able change decision economic property whole population adopt notation causal mechanism agent agent randomly assign view treatment report raise technical since outcome agent observe report reach interested sensible data consideration body work outcome study assignment potential outcome agent round fashion potential mechanism outcome strategie distinction whereas outcome realize realize potential mechanism denote show causal mechanism strategy compare option median summarize mean also dependent adapt strategy round long capture dynamic evolution agent literature inspection one challenge report observe outcome omit brevity report mechanism hence depend justification main estimate equilibrium accurate economic illustrate compare imputation uniform fully approach serve
increase property streaming encode single adaptively send code match reconstruct optimally dropout neural training iteration drop network dropout main drop unit drop sample subset result importance particular unit rely choice exponential architecture resemble parametric composition encoder lie via assume drop structure autoencoder representation truncation vector remove equivalently truncation function truncation take truncation subset contribute distribution truncation nest mutual representation truncation distribution p l mutual nest connection choice establish intuitively dropout idea index long autoencoder assumption index proof allocation unit autoencoder encoder decoder rigorously property nest dropout autoencoder subset class introduce quality second class characterize restriction last constraint eigenvector magnitude arise input order contrast autoencoder pca linear encoder apply encoder omit clarity similarly define matrix whose consist composition decoder semi autoencoder seek denote frobenius add continue truncation drop truncation define truncation truncation truncation let diagonal eigenvalue arrange respective similarly eigenvector arrange eigenvalue magnitude truncation place autoencoder prove invertible bb correspond eigenvector reformulate notation observe semi connection autoencoder great linear include rotation permutation identifiability undesirable nest problem assign seek dropout justification begin appendix dropout autoencoder problem lead inverse minor inverse combine establish tight lead principal truncation row let truncation inversion element nonzero couple effectively non add rotation nest feature unique optimum solution discuss dropout deep specifically deep autoencoder million image dropout introduce challenge proceed strategy overcome image process subtract conjugate select wolfe relate seek encoder feature motivation unit epoch independently element layer dropout nest dropout minibatch mask virtue decaying become index training phenomenon vanish curvature raw mean slow call stem example word latent unit index upon gradient omit speak iterate neighborhood cardinality decay terminate retrieval pre specify terminal marginal retrieval reduce retrieval retrieve fraction dataset retrieval complexity independent share consistent produce demand study property autoencoder visualize neighborhood query nest variation loss train dropout autoencoder invariance choose retrieval retrieval hamming scan database mean semantic perform neighborhood semantic great force scan addition increase likely query order retrieval carry terminal plot retrieval terminal neighborhood size similarity bit retrieval well representation continuous degradation degradation message give rise quality combination correspond continuous degradation property appeal digital video estimate bandwidth receive pose minimize formulate give minimize online streaming signal quality attain advance various seven different definition select order offer utilize high variant needs order advance length transmission correspondingly fashion minimize distortion qualitatively degradation compression autoencoder dropout cifar reconstruction column represent represent quality image row look original bit reconstruction code autoencoder architecture dropout apply truncation approach truncation un order bit optimal remove decrease influence second taylor units disjoint training compression quality low reconstruction suit image study spectra highly image lose content quality unit dimension autoencoder generalize deep enable learn representation adaptive truncate shorter order retrieval cardinality idea approach knn competitive optimistic combine future insight practically spirit variance idea complicate grateful partially award appendix proof every nest necessarily optimal solution autoencoder autoencoder recall nest dropout different truncation truncation pca decomposition exactly minimize nested dropout nest dropout dropout mixture particular truncation autoencoder problem lead corner truncation inversion apply mean zero nest truncation consider optimal dropout truncation proof hold must true equation give principal bb bb truncation solution nest set top order namely must principal minor orthonormal sake must identically theorem degree remove coherent linear autoencoder rigorously application deep number learn retrieval logarithmic independent allow long currently feasible avoid quality code perform speed order promise learn compression automatic discovery increasingly aspect learn consideration extraction often critical procedure representation enable feature engineering deep find representation hand analysis find low interest unsupervised discover code structure hash deep result encoder decoder autoencoder boltzmann code equally transformation parameter permutation give kind autoencoder invertible degeneracy pose due attain representation architecture freedom impose constraint learn include permutation propose structural specify dimension representation choice us representation intuition behind propose representation index pre decay dropout apply mask individual assign nested unit space mask early depend lead inherent ordering representation motivate order solution strict dropout
provide original split hold investigate typically help attribute semantic traditional descriptor indexing category descriptor reveal vs rank pick moderately domain shot learn invariant demonstrate e straightforward combine place descriptor want periodic variable like pose ii semantic observe acknowledge support style none text height begin ac uk multi descriptor framework semantic descriptor shot datum practically domain analogous domain generate descriptor demonstrate outperform alternative multi establish share domain address distinction subtle method distinguish relate capture device office amazon pose across multi individual category address domain un address knowledge neural address multi learn perform simultaneous concept descriptor exploit improve share classic descriptor implicitly task classic school pose student school school year group represent semantic descriptor tuple school sharing exploit variate semantic share know task paradigm address construct category unseen interestingly lead shot adaptation appropriate unseen suppose audio variety acoustic variety first shot address jointly discover various linear task another common column low encourage share share grouping framework task disjoint share low task fundamental task middle linear predictor think column predictor predictor decomposition ibp prior vector dp entity categorical variable study notice drawback task structure school year group school replace categorical variable impose variety rank suffer adaptation da study propose unsupervised mention typical amazon imagenet video despite categorical generalise continuous angle paper alternative formulation share adaptation knowledge way encourage knowledge direction exist review previous tackle classification camera problem versus eliminate time category construct mid information exist semantic refer attribute illustration go shoot shot see novel descriptor shot issue partially modality see domain despite title actually consider adaptation label domain task denote feature vector descriptor j effectively indicate task loss generality task two figure side start original descriptor weight train back propagation perform calculated ground every neuron try neuron neuron hide input miss neuron neuron align leave right output neural side inner efficacy approach middle length descriptor prediction next clarity task setting w ls notion keep correspond semantic descriptor categorical semantic descriptor constant domain available improve share simple state fashion contrast form task descriptor improve information share exist multivariate efficacy framework interpretation simple well share multiple task task simply descriptor task instance dominant task regressor descriptor word semantic shot match e present testing category turn category zero shoot f adaptation address rather encode descriptor domain effectively thus construct apply datum demonstrate help term relu place encourage model preliminary satisfactory task logistic lr four ii iii within verify original descriptor encoding setting except hold time hold descriptor baseline transfer lr fair descriptor vector baseline include plain tensor completion tc store model categorical always domain rank low rank dataset student predict note student year school year student domain distinct categorical variable domain leave one domain strategy base hold descriptor case hold performance hold turn outperform alternative h r lr
component simulated observation vary stagewise different top row frank wolfe frank wolfe computing coordinate descent update frank share stagewise one frank wolfe regression start warm frank wolfe uncorrelated simulation termination terminate stagewise interpolation estimate rule frank stop frank wolfe stagewise rule frank wolfe uncorrelated second setup count wolfe iteration case number meaningful frank wolfe stagewise frank wolfe match accuracy especially change frank wolfe converge compute default second stop frank setup stop situation frank wolfe overview completion solution trace regularization use implement proximal decomposition svd dimension generally expensive scheme alternate package truncate full svd partial svd roughly solution repeat iteration problem explanation emphasize proximal descent converge desire stagewise vector r computational example simulate add discard entry value warm start run stagewise step curve draw stagewise estimate identical exact solution suboptimal stagewise measure square large yield basically albeit square curve exact gradient solution stagewise estimate stagewise wolfe frank wolfe mean stagewise frank wolfe repetition frank wolfe proximal descent iteration across average quite rapid descent default moderately stagewise run type compute truncate svd become stagewise throughout rank current bottom average spend second compute second per stagewise translate square routine develop per stagewise reflect runtime leave standard implementation somewhat advantage naive stagewise specialize top big stagewise gradient collect project examine set movie rating estimate use rating error hold end warm start stagewise right plot stagewise solution slight advantage exceed stagewise error curve begin drop strongly continue size stop size reach slightly minimum descent compute average per simulated explain long second stagewise construction take second singular stagewise step beneficial stagewise frank wolfe frank solution value wolfe compute pair description frank wolfe frank wolfe problems particular implement frank wolfe algorithm start regularize use warm frank wolfe figure rule frank wolfe stop achieve proximal second stop square maximum value stop frank wolfe meet limit regularization frank wolfe stagewise accordingly step stagewise run less second versus message frank wolfe serious difficulty solution level proximal gradient descent frank wolfe run much compute second point stress performance termination actually somewhat mean begin figure rule cause frank wolfe warm trivially terminate take iteration trivially overview cast gaussian fuse total variation compute solution apply flow maximum flow elegant highly fast exist fuse fuse line stagewise trivially simple comparison require line code fuse stagewise sparse multiplication denoise example add pixel noisy display flow solution direct warm stagewise square corner bottom draw noisy record computer exact slightly towards end stagewise take stagewise roughly second majority spend solution mean square visually reasonable job stagewise surprising htbp stagewise step stagewise runtime maximum solution stagewise estimate stagewise cc noisy version stagewise step compute consider stagewise large front new color channel green separately lie noise pixel stagewise achieve rise red blue visually reconstruct remarkably noisy stagewise total recall produce fuse lasso discuss issue course less large stagewise fail path portion therefore across proper seem tendency practice monotone progress alternate back behavior encounter response decrease say continue stagewise htb uncorrelated case regularizer plot across size begin small step continue way end attempt stagewise offer suboptimal estimate completion problem current mention somewhat helpful estimate stagewise regularizer general appendix differentiable convex lipschitz pair stagewise value result stagewise estimate denote remark stagewise take approximate regularization still simplify e norm gradient optimization usually make norm utilize naturally suggest namely example square spirit extend regularizer nontrivial q one theorem go bound stagewise apparent stagewise repeat stagewise update case unbounded update constrain version theory motivate conceptual help procedure forward stagewise stagewise large absolute residual direction sign intuitively step inner residual inner change increment coefficient product variable residual component stagewise increase monotonically eventually occur thought decrease large variable seem unlikely especially variable many setting recover exact stagewise like therefore make successive update explicit backward stagewise routine significant amount arguably effect step stagewise parameter stagewise importance update towards even mechanism repeatedly achieve maximal absolute inner seem implication entirely speak frank wolfe step discard frank absolute e maintain section regularization stagewise wolfe trivial conclusion look regularization regularization use frank wolfe procedure practice insight gain introduce typical wolfe frank wolfe warm start frank wolfe choose iterate feasible minimizer far wolfe basically end run frank wolfe parameter prevent control place frank wolfe warm stagewise share great successive stagewise differ control history helps think stagewise increment another frank wolfe warm meaning construct iterate component correspond absolute adjusted say word frank entire start empty seem inefficient certainly stagewise step default give frank wolfe history depend distinction remain wolfe less history stagewise rely adaptively course wolfe guarantee solution arbitrarily stagewise goes also argue frank wolfe compare stagewise warm start consider specialized strategy provable achieve control duality varie depend frank frank iterate admit implementation run frank wolfe warm start sequence dense reasonably choice spread converge draw frank wolfe follow strategy stagewise subsection reason section run stagewise frank across variety summarize comparison follow frank wolfe adaptively start make efficient use previously guarantee stagewise wolfe algorithm speak solution solution predictor component predictor enter leave set strongly frank wolfe sequence contrast previously estimate construction one constrain additional type circumstance monotone stagewise necessary history carefully end sometimes momentum stagewise place claim perhaps surprising regularization frank set solution repeat frank approximate solution follow simplify entirely construct begin compute initial duality gap verify path visit property mean visit begin quantity serve gap representation construction mt q frank wolfe path follow case uncorrelated choose duality path mean exact stagewise maximum regularization solution interpolation wolfe stagewise leave square curve vary produce parameter value span path start path meet duality total iteration wolfe iteration stagewise cut stagewise frank tune performance frank wolfe frank wolfe estimate competitive square error stagewise estimate plot frank wolfe visit value number huge fuse setup noise display fuse stagewise path construct basically rough surprising problem limit path path ensure stagewise stagewise parametrization interesting implementation iteratively shrink adjacent component possible prove stagewise rely monotonicity fuse signal piecewise corner image fuse lasso grid adjacent format fuse stagewise see stagewise appear good emphasize figure stagewise stagewise build contrary stagewise outside generalize stagewise accurate path otherwise desirable concern want certainly low image fuse lasso usually underlie complex image stagewise simulated ridge ridge offer highly optimize coordinate implementation package ridge net utilize pure ridge compare moderate amount close choose example meet rough stagewise regularization iteratively amount gradient logistic implementation stagewise routine section meanwhile sophisticated coordinate thousand line code solve regularize regularize broad stand complexity predictor coefficient vector predictor differently entry row independently diagonal setup uncorrelated positively correlate predictor case run path warm run stagewise size figure look uncorrelated simulate draw observation sequence minimum path record early stage exhibit misclassification stagewise estimate turn estimate table share average draw record computer coordinate stagewise second stagewise second compute estimate uncorrelated stagewise stagewise estimate descent stagewise stagewise coordinate repetition simulation model dot show table stagewise perform uncorrelated computational require step produce meaningful right stagewise curve draw deviation stagewise stand uncorrelated represent close failure perfectly though entirely stagewise step full stagewise small show stagewise track closely display odd trend test error slow stagewise progress rough contour positively correlate contour exactly elliptical origin stagewise add update direction gradient thin pass norm iterate norm increase progress stagewise plot display distinct achieve jump may behavior square stagewise competitive probably similarity issue course stagewise thorough understanding topic future htb base serve feasible convexity helpful dual lastly rewrite stagewise q arbitrary express term old inequality third eq rely calculation prove stagewise update generality argument similar obtain bind separately apply lipschitz inductive use term b decompose term triangle argument assumption converge apply write upper together finish complete lemma fix follow limit hold limit verify taylor eq q taylor around generalize binomial forward stagewise start zero update residual interesting forward stagewise path furthermore essentially outside minimization differentiable stagewise even stagewise success modeling motivate stagewise apply broadly current stagewise structured completion keyword forward regression boost variable linear candidate sparse stagewise produce coefficient decrease inner residual precise description repeat q commonly element stagewise select standardized add possibly stagewise reasonable forward easily active assign small htb model predictor panel forward stagewise plot color stagewise algorithm axis path also stagewise path visually appear identical study intuitive difference stagewise closely name procedure iteration choose ia far iteration produce old early accord forward stagewise inefficient useful forward backward perhaps keep mind resource time modern present considerable benefit across stagewise regularize iteration product could trivially connection sequence stagewise review panel essence stagewise appear lasso counterpart step exactly consideration stagewise hold stagewise lasso yet situation former still point stagewise general convex follow stagewise regularization parameter vary initialize intuition behind stagewise algorithm update repeatedly implicitly adjust imagine solve htbp setting detail stagewise forward stagewise dedicate stagewise algorithm derive specific stagewise problem stagewise theory conclude discussion argument center stagewise stagewise procedure matrix completion nonparametric stagewise simpler stagewise highly actual though share actual stagewise component actual path competitive essentially case even actual path view comparable metric across setting stagewise proximity solution third favorable property stagewise estimate regression display hold norm square control tt tt zero path tt enough panel discuss early stagewise seminal purpose path limit stagewise path tt algorithm coincide stagewise lasso path monotone confirm stagewise estimate example regression detect truly less understand cast lasso favorable stagewise forward stagewise forward stagewise estimate path monotone level somewhat remarkable simple produce stand relatively sophisticated practice component rarely monotone stagewise theoretical answer know correlate enter leave repeatedly stagewise finding stagewise along lasso decrease sum rate limit stagewise speak arc length account entire history stagewise produce estimate tend lasso predictor lasso see point stagewise tool apart link loss fortunately stagewise beyond begin analogy step opposite component large reduce stagewise routine author path imply order expansion twice position path path cover loss position stagewise simple really advanced part logistic poisson outcome connection optimization encourage outside stagewise regularize backward stagewise forward general take towards would amount forward backward backward stagewise distinction result monotonicity path suitable globally lasso latter prove stagewise lasso along path forward stagewise path backward fairly extensive work connect stagewise stagewise still end degree similarity stagewise path mathematical much fact simple forward lasso next start stagewise maximal value norm q minimizer stagewise stagewise regularize stagewise direction minimize among stagewise balance small increase solution regularization intuitive aside stagewise regularization already present discuss differentiable stagewise procedure motivate stagewise small regularizer stagewise trade path iterate subsection stagewise special stagewise procedure stagewise beyond present make several remark initialization termination easy stagewise criterion general solution path successive upon reach iteration iteration last reach justification triangle g norm successive give justification step parameter modify replace linear taylor element modification different stagewise stagewise choose choose perform necessarily get taylor tight imagine close direction write subgradient norm admit come section invariance minimizer provide express common statistical follow stagewise update evaluate add analogy operator nonsmooth express simply use unbounded stagewise z stagewise modification stagewise problem initialize stagewise update lie replace eq modification g semidefinite work call lagrange probably optimization lagrange rather solution path vary respective necessarily mild nonempty hold visit versus focus intuition formulation relate paper reader familiar identify stagewise descent convex iterate minimize inner iterate one usual descent descent special meanwhile stagewise stagewise usage seek eventually nonetheless stagewise algorithm minimizer path minimizer stagewise iterate statistical property gradually balance path researcher method stagewise frank wolfe algorithm minimize differentiable project descent minimize approximation frank wolfe minimize approximations wolfe modern problem frank wolfe frank wolfe look general stagewise distinction frank wolfe rather iterate general discussion appendix linearization cutting bundle regularize minimization frank wolfe problem value one linearization history iterate conduct bundle stagewise though would another class boost iterative think stagewise descent wolfe vast review closely forward stagewise fitting consider setup weak boost factor select match scale unit equivalent selection stagewise express stagewise boost add search meanwhile stagewise loss boost greedy stagewise practically especially fact slight stagewise forward stagewise boost forward stagewise suggest stagewise universe weak learner boost transformation variable iteration add weak early stage dense problem specify universe straightforward completion image denoise learner broad stagewise offer learner intuitively reasonable group completion section lead boost arbitrary though form work apart method regularizer extend update utilize variable learner perspective stagewise regularizer regularizer similarity proposal stagewise block index partition group weight generic differentiable function stagewise invariant term computational lasso predictor variable admit grouping define group outside set see study logistic relate distinct consider regularization fit collect task write outcome task coefficient jj regularize default importance group stagewise distinction initialize regularize update w stagewise update let omit proof follow straight kkt simply block scale move coefficient opposite group visit leave identically stagewise match intuition coefficient nonzero value group actual intuitive examine kkt back compare solution stagewise problem stagewise step similarity path highly path behave stagewise later large give thorough comparison place group approach q stagewise problem stagewise compute follow cover recall self recall define update direction broadly stagewise update regularizer wise compute dual norm norm consideration nuclear sum singular trace norm partially observe define trace regularization trace multiple consider stagewise initialize stagewise procedure leave vector respectively proof trace stagewise stagewise singular matrix vector assume let method depend either recover multiplication computing method iteration qr second stagewise path completion problem stagewise fairly note hard path role clear coordinate path correspond interpret slight stagewise present large get effect square stagewise several regularization ridge regularizer component towards predictor estimate identity spline p express let spline basis cubic knot knot typically knot across ix difference set stagewise follow matrix stagewise check kkt stagewise quadratic computationally trivial reduce yield estimator compute system generally system across operator compute cholesky relate operation require operation certainly naive entirely desirable bandwidth initial cholesky operation successive importantly spline regularization case spline local spline care singular semidefinite strictly stagewise albeit slightly deal iterate spline difference projection onto stagewise arbitrarily must constrain stagewise check instead denote generalize computational stagewise rank spline apply computationally solve multiplication stagewise update p spline regularization exclude regularization display solution stagewise notably stagewise path surprisingly step number need stagewise large cover rough appear regularization stagewise path trend uniformly seem ridge really
describe useful exploring beyond neural autoencoder enforcing rely heavily perturbation generative stochastic traditional robust optimize output pseudo boost meanwhile child share member bag forest prefer diversity member output member average regularizer member ensemble motivated intuition robust perturbation perturbation input example produce programming seek perturbation optimization generally seek solution g perturbation several machine robust svm regularization correspond rkhs optimization support notion statistical appeal directly rather proof I closely ensemble work input globally optimize poisson input particularly relevant work effect quadratic ridge estimate several pseudo ensemble input layer input part use fairly network operate control distributional property layer output child generate parent unlabele observation variance activity importance act ix ix hold reasonably architecture dropout help prevent co encourage e hide activity remove local provide regularization kf never act parent accuracy perturbation layer regularizer sec ability regularizer reproduce support co empirical act perturbation fx influence dropout optimize logistic regularization indicate logistic penalty layer bring regularizer train network regularization apply penalty operate element wise far expand recall layer penalty performance test three digit mnist digit supervise dataset learn full implementation reproduce available mnist comprise digit test activation layer layer bias output bias inter early e measure state sde five initialization penalty parent network encourage match dropout course sde test protocol label set split set split denoise autoencoder pre penalty layer pre output latter gradually course bias layer sgd supervise table compare previous aside specific technique comparison nn pl regularizer respect manifold tangent manifold gradient tangent pl pseudo ensemble prediction unlabele treat label carefully training except pl benefit add I reduction label htp r nn pl sde sde manifold dropout plus protocol pl remain sde former agreement mnist label available challenge domain label cifar color unlabele comprise among neither class image target domain winner challenge convolutional coding pooling pool ignore convolutional pre cifar deep comprise max pool convolutional fed output remove hidden layer train domain layer dropout achieve training strategy pre unsupervise train feature activity allow improvement show adapt ensemble stanford task sentiment phrase extract movie review com label phrase amazon ensemble bilinear form recursively indicate indicate vertical stacking training condition full norm constrained pre train code publicly online parent weight dimension parent phrase computation significantly could training subspace time phrase tree process phrase tree perform weight phrase perturb implicit recursive recurrent may objective curvature ill training process code htp grain compact sampling paragraph dynamic suggest measure prediction fine grained sentiment sentiment fine grain class strongly sentiment similar neutral phrase negative class four test past noise past performance task propose pseudo dropout feature model conceptual ensemble regularizer perform empirically behind dropout success ensemble competitive real sentiment benchmark rapidly evolve line especially figure notion accord train child randomly parent examine
misclassifie otherwise would conjunction option load color graphic terminal graphic macro ltb lt lt lt lt lt ltb lt lt lt lt lt lt mathematically induce overfitte instance still instance impact early difficult instance negative hyper quality part hyper hyper effect instance induce loss support loss limit affect set hypothesis without hyper gray hypothesis bold instance reduce mathematically color conjunction option explanation load graphic explanation terminal graphic ltb lt lt lt lt ltb lt lt lt lt lt lt bp quality low quality lower search probable hypothesis searching subset result power training remove obviously induce mathematically instance induce package conjunction terminal explanation load package graphic terminal graphic ltb lt lt lt lt ltb lt lt lt lt lt bp r r fully search computationally infeasible induce determine way induce inducing investigate remove training algorithm probable classification interest maximize input contribute class start presence misclassifie commonly follow use diverse learning sum would hypothesis hyper course diverse estimate free weight hyper equal treat zero set run fold seed algorithm select algorithm use classifier output diversity different agglomerative default set dendrogram connect distance value py perceptron tree nn I rule learner forest repeat incremental pruning algorithm candidate filter set empty initialize accuracy return cross accuracy filter use entire filter dynamically allow set combination adaptive ensemble filter construct add binary also threshold discard iteration baseline accuracy approach without filter line stop algorithm increase add since maximize allow actually practical setting insight potential gain improve compare sign alpha addition report average percentage lr gd reduction acc gd set interested optimization percent likewise percent reduction accuracy capture time equal percent percent variance percent percent percent percent accuracy accuracy validation use search hyper selection hyper learn hyper considerably considerably hyper amount grid provide supplementary material high validation filter identify instance construct learning accuracy produce instance hyper six commonly nb memory hyper mlp ib rf rip red red red compare default classification increase hyper variance filter percent large percent also generally mlp ib rip red red red count red na count compare hyper learn f use default setting optimize adaptive compose hyper accuracy accuracy examine compose optimize algorithm hyper result gain exhibit less filter hyper filter filter gain filtering could accurately gain choose appropriate training datum provide motivation improve algorithm without variance compose hyper optimize increase na I forest algorithm cccc mlp ib nb rf rip mlp ib rf set hyper great filter bold base instance filter filtering previously examine efficacy filtering characteristic rule filtering include well efficacy filter learning determine set hyper instance examine existence hard classify significant hardness instance induce improve handle instance examine training survey noise classification quality improve technique instance misclassifie instance outlier induce negative beneficial discard produce model instance training seek clean instance valid weight instance discard instance consider binary beneficial corrupted correctly label noisy induce properly significant thus previous set grid manual type hyper technique machine combination hyper alternative approach discuss hyper mathematically mathematically hyper reduce instance process filtering remove completely effect estimate potential benefit also choose learn maximize filter hyper increase learn algorithm filter parameter optimize optimization reason requirement instance determine instance filter motivation quality dependent
uncertain uncertain link connect presence uncertain spurious reduce iterative propagation uncertain link stage designed formalize collective introduce uncertain connection exist uncertain network associate network undirecte easily label represent ease notation node integer set node special uncertain collective subset label uncertain label collective assign label bayes perform labeling belong adjacency incorporate uncertainty continue build augmentation classifier overall label rest refer give unlabele denote note uncertain probability particularly edge likely much description particular label contain use label node successively label far initially expand set adjacent expand label node propagation probability iteratively estimate propagation label expand repeat remain label label terminate remains perform uncertainty begin termination compute expand end end step propagation examining decide successively refine iteration examine edge fraction estimate consider edge reduce probability always prevent problem label probabilitie value default adjacency reasonably unlabele neighbor unnormalize bayes compute li li p incorporate sum determine noisy low impact classification example white must assign existence probability low htb unknown augmentation collective subset edge activate modeling adopt selection link would determine configuration training ratio aid selection check precise value set accuracy note avoid overfitte pick edge optimize start expand enable inactive probability ratio ideally want contribute positively unlabeled configuration goodness node value highest denote uncertain iterative good basic labeling particular vary evaluate efficient identify accuracy uncertain uncertain network probability node ratio network sample uncertain set high iteratively active edge estimate label probabilistic labeling sample begin expand graph configuration correspond frequency use conditional probability maintain third subset node classifier relational follow li collective equally linear combination classification discuss probability cardinality immediate unlabeled neighbor node represent node neighbor unlabeled probability require require sum iteration cost uncertain maintain list automatic decompose expand summing terminate cost simplify algorithm library intel ghz processor gb ram times average interval comprehensive record scientific relation co year author publish paper period research computer area author rest corresponding verification graphic human software bioinformatics security htb name verification testing graphic interaction software engineering bioinformatic compute security retrieval citation contain least edge category category report name node truth although manually category pick one two label word class algorithm robust lack name chemical computer electrical perturb set advantage test vary inherent performance edge edge normal interval parameter deviation large eventually control add edge remove exist edge datum retain criterion also know node label remove label whose noisy default equal assess use repeat sampling training label remain refer truth limited compute confusion count diagonal repeat statistically result sample sample node identify vary experimentally dataset algorithm remain variation sampling since trade run spend method method node probability thus weight relaxation version relaxation accuracy far algorithm deterministic sample estimate membership probability voting variety ratio edge report worst achieve dataset percentage accuracy worth note due high report eventually experiment deviation edge figure consistently nearly explain capture correlation different processing well ignore contribute overall classification set label default retain edge report figure perform well dataset slightly percentage improvement low sampling stress test consistently consistently retain edge uncertain automatic robust dataset percentage confusion confusion insight especially misclassifie cell report ground classified label label bioinformatic lead label network refer information example mining due class versa fact example chemical confusion c c c positive indicate bold true positive indicate accuracy vary control overall classification process always never configuration omit belong video category computer communication north american medical label chemical turn label improved answer efficiency perturb cpu algorithm noisy slow due automatic bad vary noisy see label complexity iteration contrary iterative become successive visit network propose high level accuracy time require execution baseline include remove iteration examine accuracy processing increase reference execute depict figure increase stop set htb algorithm graph becoming use protein network link collective relevant determining describe efficient effect probabilistic treat label parameter diverse classification conventional directly support fp ip project grant agreement author laboratory agreement pt claim real network graph probabilistic classification affect final link accuracy underlie network focus structure treat automatic show incorporation effectiveness efficiency collective node represent example real list co label network area desirable information classify interact protein movie actor actor edge actor actor movie correspond category fraction label collective study present many link well uncertain probabilistic
increment achieve parent child flat omit fig high code four compute root bellman block perform visit ensemble step accomplish third top way dag also dag nod compute analogously contribution decay linearly dag represent simple consistent flat dag include algorithm tree exploit child dag propagate top specific hierarchy positive prediction pass propagate structured assess effectiveness propose hierarchical cm cm di di universit di mail di range text biology label present tree direct contribution structure dag dag dag present accord hierarchy gene acyclic go gene flat prediction prediction independently classification relationship accord general step step connect learn basis yield learn machine learner provide classifier combine consider hierarchy prediction flat method apply protein categorization music classification annotation classification different structure dag contribution dag dag path design structure direct acyclic represent represent relationship parent child class unique dag add unique root add node flat eq classifier classifier example flat continuous classifier belonging case flat label set predict label true path rule say multi label score valid easy example flat classifier prediction without label class classifier hierarchical obey word propose hierarchical dag top level correspond maximum node root flat straightforward brevity I begin max bellman visit else v fig code root path end bellman use find sign obtain contain per visit row root process top dag ensemble first row bellman vertex graph dag version traversal step bottom visit traversal perform fashion traversal version top necessary true v begin algorithm compute bellman visit top visit
wang case pt informative screening mode result bound fan also assume correlation theory come screen eliminate uninformative mode later screen test excess test test multimodal uninformative mode informative present method interest provide cluster relate penalize version bic use pairwise none penalty provide consistency relevant relevant consistency np hessian denote denote closed review mode shift detail let feature population ascent flow satisfy iff mode function cluster assignment algorithm second hausdorff distance mm screening coordinate density estimator let mr empirical distribution feature reject nan multimodal critical want unimodal versus unimodal unimodal choose u suggest multiple bind least rejection refined test building simplicity available bandwidth scope accurate suggest rs deviation coordinate al note three finitely mode function critical point exist finally relevant multimodal particular slowly function unimodal cluster smoothness need sure appear set dimensional restrictive close axis middle show possible selection make assumption curve imply much version assumption well mass boundary furthermore derivative n b except error second low large vanish bandwidth pair near exponentially boundary decrease hausdorff mode relative mode separation tend hausdorff distance negative event close n false testing recall note lemma nx x bx extension bridge reject large previous include property mode separate eigenvalue finite take density supplementary material let mx latter gx xt bm et start mode lemma relative hold basically omit condition hence suppose already I cx jk condition word mean px hx expression b x nc near boundary third cardinality case hausdorff distance follow bound brief type false implement package figure show test fail value power appear logarithmic plot multivariate fraction combination incorrectly multimodal instance case surprising test conservative method dimension mixture correctly recover multimodal problem hausdorff signature quite strong selection assumption prove top loss probably boundary hausdorff think acknowledgement
letter modern advance thousand million impractical limited promising deal dataset relevant statistical difficult feature often redundant relevance employ powerful unified perform unified selection commonly unify unfortunately method monte bayesian describe model py notice flat maximize distribution maximize usual laplace impose penalty helpful perform logarithm energy strongly relevance computing strongly affect go assumption effect useful surprisingly likelihood mild feature strongly practically two rapidly irrelevant e feature apply intensive infer model consider variable furthermore assume denote specify position define contain definition observe throughout prior work factorize prior strength regularization twice minimize commonly plug subset dominate prior log extensive regularization much saddle evidence log likelihood physics commonly reason work strongly regime series expansion eq twice effective regularization strength spin ise proportional small weak model ise ise require expansion converge letter letter selection regularization expect distinguish bs informative label red part logistic commonly statistical model categorical data simplify notation extra feature variable always parameter take transpose supplement expression hessian plug except multiplicative constant ise use classify bs dataset diverse publicly computer use number letter comparable suggest expression describe logistic regression number feature square pixel b well visualize agree pixel distinguish b general approach vanish prior even limit show selection mean model ise model aside mild regularity independent inference new give algorithm many employ machine approach modern set potential number selection outline stage irrelevant variable reduce dataset apply comprehensive physics gradient model exponential exponential statistic notice denote connect glm restrict scalar write
performance dataset suggest difference composition candidate metric median percent percent percent candidate super compare competitive actually want one serve table super easy see challenging evaluation percent candidate candidate c super decomposition combination testing training dataset merge focus dataset merge versus testing significant standard training alone good merge significant benefit dataset candidate percent top percent candidate candidate super training include evaluate super explore candidate explore super explore much super encouraging section multiplication approach multiplication improve find super achieve multiplication achieve well top top achieve score super seem substantially account super explore decomposition approach due decomposition correct super versus statistically baseline train super compositional hope thus improvement super achieve take domain adaptation insight table super thing ability ability handle work attention noun noun consider pooling sentence generation recognize relation analogy rely section hand eight super way avoid use selection super instead first pass could use highly version super expect pruning reduce still super set future limit adaptation recognition unlike task avoid ability new beyond recognize extend distributional composition noun composition generate limit pseudo distributional task decomposition indicate considerably composition increase candidate table generation allow composition accuracy domain model training insight limitation achieve level expect super compositional suggest section may supplement standard main extend composition approach pass generation semantic meaning aspect component mean distributional semantic context consider decomposition tackle simplicity semantic noun noun noun semantic noun generate noun candidate decomposition pass generate initial candidate slow solution highly include top highly distributional semantic hypothesis similar represent extend beyond phrase sentence work phrase sentence sparsity noun explore phrase noun consist head noun noun mean head noun noun noun test whether recognize noun list recognize give list model recognize many noun noun example head noun composition promise noun decomposition noun composition seven candidate noun choice task candidate different generation avoid choice resource corpus page university text corpus approximately gram gram include gram majority majority noun phrase composition target fast unsupervised high scoring slow supervise super top candidate top dataset unsupervise score list top top combine concatenation every head list candidate slow super top candidate variation unsupervise learn super supervise main contribution together recognition decomposition dataset evaluate super describe present composition cover discusse relate look work composition noun thorough survey generate knowledge focus method much effort technique corpus distributional phrase distributional hypothesis phrase occur context tend consider shared phrase compositional phrase phrase power vocabulary possible gram word phrase excellent suited phrase eventually diverse phrase sparsity compositional include baseline noun phrase context composition measure similarity noun phrase noun calculate cosine angle relatively although order since representation word remain order meaning composition select context word vocabulary composition operation word seven compositional element multiplication word construct frequency train example map vector context vector distributional phrase propose extension distributional semantic involve algebra tensor product proposal similarity context phrase conversely distance product frobenius capture search tensor wise instance follow various composition composition similarity mean equation build call composition sentence unsupervise autoencoder softmax classifier dataset composition question split question training show noun seven belong question heart disease heart seven noun question unsupervise training question yield get correct suffer create noun phrase pair phrase noun task distributional noun double star binary word electrical noun composition evaluate seven noun rank candidate experiment noun phrase either noun noun noun phrase modify thus believe noun phrase noun phrase noun noun noun phrase noun comparison also noun production human ask short target understand scoring answer measure semantic gold reference similarity create noun gold definition show four definition narrow child child dataset term label evaluate classify definition noun definition must class wise attain noun noun noun rank list candidate attempt combined noun experiment create package derive noun test table dataset include compositional force non compositional difficulty problem decide make dataset easy avoid find compositional compositional contain noun character five character act noun neither compositional force well field head noun occur character match five character word force compositional extract choice noun check compositional pass composition solution composition divide testing whether test seven decomposition extract seven noun solution check compositional select target standard decomposition decomposition divide whether target seven noun question effort pseudo composition pseudo true idea red composition standard analogous red although trivial three correspond construct target dataset match size four solution give successful guess red red good major stream price room target decomposition composition five type pseudo argument return five fs group description log pointwise domain five cl variable description right fs power measure ds fs frequency zero frequency corpus cover pseudo experiment hash google web frequency pseudo never argument web dataset much ram store berkeley rapidly pointwise mutual measure strength association positive indicate negative indicate useful semantic improve force logistic sigmoid shape follow positive infinity infinity rescale infinity logarithm base yield normalize pointwise store detail gram correspond mark approximately gram correspond pointwise mutual observe side word normalize treat stop mark assign zero phrase contain look phrase rare phrase relatively phrase likely phrase likely mark never correspond opposite never pseudo domain ds word domain row gram near gram gram corpus phrase process part speech noun close noun leave row frequency column word column noun context density convert process svd svd domain specify truncate singular generate raise singular power zero factor weight similarity extract correspond gram cosine tune wide use since domain gram pseudo extract calculate similarity design experiment except nearby hypothesis role pattern nearby word function row pattern context functional function compute extract gram cosine present super use refine initial super build take input rank super take input rank list generate list scoring consider domain fs negative let zero noun noun general head noun example thing red functional every sort high scoring package rapidly score calculate quickly process candidate work candidate show follow experiment trial composition training come super factor domain singular exponent value function target parameter list use score head noun like similarity fs term candidate corpus red corpus phrases score preference side appear high scoring scoring candidate head define sort score highest design candidate occur frequently follow four set trial decomposition super description factor space exponent domain factor exponent target equation explore candidate consider target decomposition fs million target considerably web gram split form pt pt triple super refine input super view triple range target generate super triple range triple vector represent triple regardless triple come super let triple super possible ds fs order pair matter cosine symmetric kp feature except log positive pointwise super triple train first target appear output leave training triple adjust target standard candidate solution triple label triple class incorrect candidate imbalance target triple randomly triple class triple every triple triple triple triple apply four dataset lack class minimal provide output super target candidate super standard setting four description ds ds fs class super supervise construct super use eight include computation time spend calculate super speed accuracy advantage target red composition solution mark candidate score rank term candidate blue candidate red rank super candidate answer rank median rank row label top percentage solution percent candidate generate possibility metric rank candidate percent top percent percent candidate composition calculate rank include rank define super answer target make median rank confident six broad preferred percent percentage answer super work super vector treat pseudo optimize smoothed element multiplication section point element multiplication contain ds truncate modify form element multiplication target target vector tune approach form merging optimize smoothed matrix show domain multiplication noun recognition decide multiplication output apply vector approach percent multiplication percent benefit significantly confidence level super super percent top candidate candidate percent top percent percent candidate candidate super compare baseline approach target vocabulary composition test section learn expert like past noun vector dataset investigate apply composition dataset use construct ignore rich table super composition subset size composition target target standard dataset challenge dataset testing metric candidate candidate percent percent percent percent candidate look performance combination column super training merging distribution candidate median percent percent percent percent candidate c training carry standard achieve significantly fisher standard testing achieve benefit merging algorithm train good merge noun phrase super noun merge dataset remove appear noun target noun phrase mean candidate median candidate percent top percent top percent percent super noun noun noun compositional noun dataset heuristic seem noun phrase noun noun copy evaluate seven noun dataset column dimensionality nmf value dim rank median candidate dim nmf nmf nmf noun addition introduce wise
modular organization adopt confident correctness modification david school sciences markov carlo probabilistic failure outline write code modular unit implementation mixture gaussian often justify correctness mathematical ultimately implement software arise correctly implement mathematical specification outline correctness type monte sampler believe widely factor test correct reason mode often matter correctly work conv deal cause mistake still sensible look prediction problematic mean might yet high number percent researcher far concern job simply prediction run inner affect vary change hyperparameter setting focus sampler partly good challenge involve highlight specific also implementation distinguish kind test check correctness small piece test possible fail overall behavior implementation test whether different interact produce kind discuss test check distribution review integration modular enable run example isotropic gaussian q toy analogue strategy state alpha alpha k self self mu self pi pi know evaluate def mu self mu sigma log np pi sigma self sigma np sigma conditional probability def count alpha def np mu return evidence h np np h sum size else def self self gibbs sampling routine sample mu x section way unit test code must modular modular happen automatically define keep modular unfortunately encourage program neither modular student update make easy project box fortunately formulate principle modular organization formulate solve purpose function model fed library check finite implementation optimize computation optimization code separate purpose sampler update rule routine mention sampler conditional decompose directly correctness undesirable unit instead recommend eq wrong would fail value verify suitable inference reason suggest write mass function conditional routine replace distribution number exact modular organization support also project modify sophisticated lot use elegant keep separate main additional def return alpha k pi pi sum mu check mu np mu model substitute self b np z enough fail fail sigma n file package file code testing py run fail failure sigma test ok mistake unit unit subtle test interaction sampler chain two goal somewhat mathematically fail good algorithm mathematical correctness powerful technique testing mcmc algorithm simple generative want posterior mcmc resampling operation sampler correct yield variety two indistinguishable frequentist hypothesis account determine significance less simple could ccc pass unclear figure synthetic representative output indistinguishable pass middle unclear
dot os demonstrate hessian weight choose voxel fit equation acceleration theorem applicable os lack acceleration practice show reconstruct scan acceleration acceleration early os appear still well os momentum os use standard os algorithm window convergence support tolerance th os os subset dot os subset baseline electrical ann mi supplementary material detail linearize lagrangian inexact update additional composite q proper inexact linearize separable quadratic l scale lagrange multiplier al show inexact linearize solve consider convex denote inexact linearize method linearize al linearize minimization split corresponding lagrangian eq al definite column iterate yield identity admm iterate linearize convergent linearize convergent satisfie proper saddle since inexact linearize method kk j inexact linearize inexact choice substitution inexact inexact linearize inexact inexact applicable solve inexact mapping mapping inexact inexact linearize reduce proximal update inexact inequality side sum primal gap
haar rotation union put together q putting together hence find correct minimize polynomial constant prove find neighbor fully statistical step limit depend fact choose valid eq complete eq replace last equality coordinate manifold equip metric follow jensen put triangle inequality claim theorem conjecture edu university mail proposition subspace cluster near subspace recover identify subspace estimate subspace geometric linear many specific paper new algorithm statistical affinity subspace weak standard datum demonstrate segmentation simple cost subspace classic dimensional ambient like linear contain jointly unlabeled setting phenomenon separate application cluster motion segmentation face system application point person illumination move lie dimensional subspace mix model subspace reader reference many theoretical performance two find cluster neighbor subspace contribution devise new two near neighbor point subspace subspace neighborhood conjunction main statistical subspace considered cluster weak exist always provide art much simple fully random semi neighborhood lrr intersection ssc omp neighborhood intersection none none none dl refer reader formulate canonical axis arbitrary mostly community result justification provide theoretical dataset name curvature good theoretical guarantee remarkable algorithm call ssc ssc find match pursuit ssc call lrr norm nuclear present construct inner product use spectral paper focus subspace exact ssc ssc omp correct neighborhood exact cluster intersection exact see denote point lie point near set subspace letter scalar frequently denote index span v indicator construct matrix spectral near spectral greedy fashion likely step lie subspace span describe neighbor dimension j u n neighbor dimensional span point close th collect lie subspace u ok pn implementation describe ssc lrr share orthogonal pursuit recovery pick dictionary correct assume point currently sparse close omp theoretically empirically one work provable perform collect neighbor lie lie natural close span span true subspace lie span subspace recover subspace subspace successfully n n matrix contain subspace obtain schmidt give store basis compute projection onto near already subspace consensus select point collect receive analyze noiseless algorithm nonetheless case use subspace subspace iid arbitrarily iid basis subspace eq principal two subspace identical every define lie draw iid uniformly random unit subspace lie dimensional subspace polynomial constant constant explain consistent find neighbor subspace look lie note subspace distinguished increase become subspace compare condition require literature correct neighborhood become worse guarantee exact wise correct neighborhood table semi general subspace choose cluster semi condition ambient lie subspace depend close difficult distinguish explain intuition easier guarantee importantly condition improve performance algorithm neighborhood fast lrr ssc fix compare term ce incorrectly label cluster index disagreement label performance generate uniformly subspace norm fix figure ce average figure theorem however subspace believe tight correct theoretical computational algorithm motion segmentation video use individual illumination raw exist code table ce average method algorithm ssc omp significantly c ssc ssc omp ce ce avg ce ce c ssc spectral mean ce ce time sec mean ce median avg time sec ce ce avg sec ce ce ce median ce avg onto norm implementation reduce find projection square norm u th number ji k u w j step spectral consecutive number group subspace lie subspace subspace also intersect practice extend step main theorem whether correct neighbor exact subspace lie good subspace subspace one neighbor point lie subspace pick subspace probability step consider correct union fact establish cluster neighborhood trivial exact fully find surely subspace form fully subspace find neighbor fix replace find neighbor subspace every proposition find correct neighbor subspace lie construct subspace true projection correct lead subspace index oracle whose success easy concentration random provide iid let draw iid ball row exist constant prove
without share well click display rank position try unbiased click model user behavior click almost click simplify user interact sequentially user examine model analysis click engine user interaction confirm assumption contain click click order click xlabel consecutive click xlabel near ylabel fraction clicks ylabel click conjecture scan small page track linear search apply search click model click page search short word display font look might page contain want user somewhat click distinguish estimate attractive query information accurately present novel statistical reverse click separable model search position past click contribution summarize user multiple click click furthermore share display response user click post derive click empirical engine document ad search click separability examine view independent independent ad factor product separability cascade scan top user ad eq cascade click cascade click extend click modify previous click ic model relevance bayesian differ relevance click click incorporate click document display essentially number display naive bayes sharing user interact skip click incorporate behavior distinguish post click allow kind transition probability depend location examine decay current click document page certain display recent try click ad show position affect click context recommendation basic account modular gain score affect relevance sake brevity discuss deal reverse click rank document query multinomial denote click c I select position click satisfied stop yes west stop east east node click choice return click document user find result bernoulli random satisfied turn ps ahead search click expect decrease click increase multiply pre click instantaneous click document ad click click previous click relevance user continue position click document vector click influence position next click capture transition allow click list specification htbp fill black inner right ci draw ci draw sep ti vi inner si draw sep right si ti vi ti si si ci vi ci inner circle pt ci si right vi circle fill inner ti matrix capital click describe follow user capture click well post click relevance estimate efficiently pass click novel method display three description display word exclude commonly occur stop appear title compute occurrence occurrence click week training query frequency great restrict word time occur due normalize share information response likely relevant query method give able word copy collect large use second week filter retain display close commonly split particular retain purely statistic query four click baseline parameter via independent click click position click position one click position baseline click section position attractive click click click mainly motivation study click since model post relevance dynamic predict click happen dataset accuracy click ad gain understand outperform gain substantial hard learn h xlabel query xlabel ylabel ylabel symbolic east legend pos north bar width table bin forward click table bin click txt bin table bin base click txt clicks txt xlabel query xlabel near ylabel ylabel symbolic anchor east legend pos west bar click txt bin clicks txt bin clicks txt clicks txt base forward click txt xlabel xlabel symbolic x label anchor east pos north west bar click txt bin click forward click table bin click txt bin clicks txt xlabel volume symbolic style anchor east pos north west bar width bin click txt bin clicks click table bin forward click click txt click sequence click sequence click predict click click resp resp pm accuracy predict click sequence fraction datum although unable predict solely click play account click pm figure consistently click prediction click perform well query tail top well overall focused predict click correctly rank actual likelihood permutation click sort sequence likelihood check actual click summarize see rank click click sequence click rank click sequence click frequent click click low query xlabel query xlabel ylabel click ylabel symbolic style bar anchor north bar bar bar bar black click txt error click table click txt bin forward click scale xlabel xlabel ylabel ylabel east legend pos north west bar width bin click bin clicks txt table click forward click txt clicks txt xlabel xlabel ylabel ylabel symbolic x style anchor pos north west bin forward click txt bin click x click table click bin click xlabel xlabel ylabel ylabel symbolic anchor east legend north west bar bin reverse click txt clicks txt bin reverse click txt scale xlabel xlabel ylabel symbolic style legend pos click txt bin rank reverse click txt bin clicks txt xlabel query xlabel ylabel symbolic legend pos north west bar bin click txt bin reverse click bin reverse click txt reverse click bin reverse click txt xlabel xlabel ylabel ylabel symbolic label east legend pos north click txt reverse click bin click txt bin clicks table reverse click txt ignore focus click click table predict top accuracy h click pm click observe reverse click table click predict location click position click click expect prediction reverse reverse around multi click reverse click click observe reverse click pm c click rank c click pm click user interact search reverse click extensive empirical
neuron fully connect neuron keep image architecture image resolution convolution exception specifically convolution layer max unit layer indicator convolution layer difference x convolution third convolution layer size pooling layer convolution size size perform convolution resolution except max pooling third architecture simplification convolution second convolution grouping overlap convolution layer architecture follow modification fully connect layer layer remove response normalization layer convolution layer worth note convolution architecture study convolution fourth twice exist package network dropout rate momentum intensive study network range heuristic us experiment understand configuration validation image great cost set diversity sensitive resolution reflect recognition image hand scene level recognition category entire nevertheless resolution always performance convolution depth impose due pooling conv conv conv conv conv x conv dataset claim human obvious visual recognition art usually imply computational cost grow investigate visual range either show consistently relative convolution bad performance image resolution pool otherwise extremely bad degradation resolution stem purpose dataset design object dataset tag tag concept may nevertheless increase consistently well enable usage deep resolution different add layer would possible additional contribution minor deep grow monotonically depth layer map depth layer top convolution layer number convolution turn yahoo fc conv fc yahoo fc fc conv fc conv fc training conv fc conv yahoo fc conv conv yahoo fc conv fc yahoo fc conv fc fc conv fc fc conv fc conv conv yahoo fc conv fc yahoo fc conv fc yahoo fc fc conv fc fc fc video static dataset subsample million mix dataset new static video video dataset convolution fig video successfully ignore ignore examine mid initialize video unchanged convolution unchanged learnable avoid convolution regularization learn pattern line corner unchanged table pre train update fully layer avoid problem identical connect totally convolution therefore tune convolution layer dataset different kernel change fine dataset helpful dataset different domain combine initialize image network video benefit additional precise provide supervision enable indicate benefit supervision target supervise recognition annotation entire intervention annotation overhead unconstraine preliminary requirement sample overfitte important video hard overcome train transfer video image corpus weakly medium rich sample video frame even domain transfer training supervise pre collect meta performance image resolution computation result indicate resolution image always yield object scene additional helpful sometimes bad select meta future facilitate would unsupervise current far eliminate level important topic great past recognize concept unconstraine show annotation video corpora robust network obtain datum video image corpus pattern pattern ignore video video learn image weakly process less visual lead enable video deep transfer learn complex event video much research recent year technology real popularity device sharing site generate video internet become need video management recognition specific recognition video come movie record event require various concept various spatio capture diverse work learn deep convolution suffer video trivial overcome problem weakly label collection learn image video corpus help learn feature appearance corpus intervention extensive effort make image capture appearance information frame aggregate addition effort develop video overhead empirical show image may spatio complex concept design video remain video recognition recent recognition video show domain natural speech etc great imagenet attract far make apply difficulty extremely imagenet source image collect ground corpora static effort million video benchmark irrelevant video annotation noisy content video impose pixel scalable meta svm meta search extensive requirement grid meta infeasible previous setting overcome lack avoid overfitte apply approach knowledge static image improve unseen video perform correlation meta basis heuristic video competitive previous basis aggregate frame meaningful improve recent work frame benefit verify efficacy transfer semantic image effort fully human annotated image corpus contain supervision collect internet boost recognition practice dataset annotate frame annotation effort much image contribution transfer visual recognition systematic study g resolution knowledge network review propose describe use study convolution show learnable knowledge work evaluate architecture difficulty deep train complexity depth large pre overcome pre fine tuned capture pattern lead transfer computer vision deep architecture multi perceptron mlp manually architecture mlp series transform follow gradient descent extremely matrix illustration essentially response tie connection field tie number learnable convolution visual layer reformulate entire position show small part small dependency mlp learnable learnable enforce share still learnable overfitte unsupervise pre fully technique therefore easily moderate recognition come imagenet report significant improvement traditional architecture significantly group acceleration partially explain develop receive recently power learnable except learnable layer include layer manually fix layer pool previous usually convolution locally pool purpose local certain degree translation handle shift pool output input therefore translation invariance reduce cost pooling overlap size convolution operate consumption pool number include architecture initialization specify depth detail great configuration popular architecture research configuration far successful architecture explanation lack configuration provide high motivation provide systematic experiment justify architecture factor optimize field convolution layer empirical hard layer evaluate various recognition network architecture provide information exist network significantly motivated static grow video intuitive video spatio convolution short extension successful benchmark perform well event fusion million event suggest frame similarly complex recognition frame exploit image properly frame wise static image combine wise feature static propose approach motion appearance information motion frame capture optical process frame spatio volume capture usually semantic concept use top high concept boost frame pooling frame exist work learn approach image help learn case collect solve domain network belief network stack sense necessarily backpropagation capture supervise domain transfer transfer train video frame video learn network overfitte learn video equivalently train frame dataset intermediate pattern additional help well middle transfer learn learnable use train consider supervise pre analogous network visual pattern outside convolution image convolution natural image dataset dataset dataset despite learn overlap kernel visually support image share fine tuning optimize especially high capture corner appear network lead learnable pattern corpus suggest process previous focus pre representation parallel utilize supervise pre address training problem work complementary label truth video rather also conclusion property static architecture boost video recognition certain include target process recognize semantic video
exploration walk proposal popularity method year limitation hamiltonian problem involve streaming instead estimate datum hmc surprisingly stochastic introduce langevin dynamics validate provide task neural hamiltonian hmc powerful chain monte carlo mcmc term parameterize momentum hamiltonian dynamical system enable proposal distant discretization continuous system need metropolis attractive property hmc rapid hmc popularity limitation hmc necessity compute simulate hamiltonian million inference recommender ever scenario massive batch infeasible since utilize entire big datum development maintain desirable attempt apply langevin langevin crucial momentum hmc explore space hmc big enable scale rapidly hmc assess noisy long dynamical system one noise mh costly computation entire practice mh correction acceptance deviation hamiltonian efficiency recent hmc momentum update appeal analyze enable maintain stationary noise discretize small fix computation tradeoff material hmc ii stochastic hmc incorporate langevin finally standard effectiveness suppose independent hybrid monte propose metropolis mh efficiently explore state introduce sample hmc generate simply discard result define identity hamiltonian energy momentum hmc hamiltonian dynamic concrete imagine slide ice energy height momentum mass surface move velocity positive decrease energy hill increase energy direct whereas momentum artificial construct rr u hamiltonian dynamic define map importantly reversible showing leave invariant likewise preserve practice usually simulate system outline alg introduce discretization mh rate long however acceptance tend development make setting turn sampler method propose tuning simulation hmc geometry enable sampling attempt hmc direction potentially implication implement gradient hamiltonian scenario observation appeal theorem stochastic depend parameter abuse introduction random accord multivariate accurate small wide consider minibatch hundred limit hmc hmc introduce momentum continuous differential return order property analogy sec imagine ice wind wind theorem nonzero long invariant dynamic govern vanishing infinity furthermore vanish entropy since semi intuitively noise preserve entropy reasonable fisher increase toward dynamic proof material must introduce correction step consider discretization dynamical entire argument hmc datum splitting likewise consider simulate hamiltonian dynamic importantly result full rate reduce gain fig noisy system minibatch herein different high resulting provide deviation poorly behave extremely intensive mh short run rate mh step rejection variant alg future use sec modification hamiltonian gradient invariant continuous hamiltonian require frequent costly mh alternatively run low problem remainder omit analogy imagine play ice introduce wind surface prevent away decrease reduce type dynamical refer physics langevin use view second follow gd symmetric follow decompose govern supplementary verify calculate furthermore stationary show invariance original dynamic second langevin momentum partial langevin partial momentum show greatly hmc case demonstrate crucial gradient refer previously discuss relate langevin particular dynamic langevin demonstrate large much rapidly case fast lead stationary b serve decay gradient langevin dynamic momentum hmc hmc gradient gradient conduct standard hmc alg without mh correction hmc gradient alg mh correction finally compare mh sampling see imply negligible unless correction add finding validate theoretical maintain distribution stochastic gradient correct costly mh simulated hamiltonian noisy scenario associate hmc illustrated consider hamiltonian sampler fig trajectory path significantly dynamical add correct resample momentum though fig mcmc naive maintaining well behave hamiltonian I million hmc correlate positive five decrease million per calculate sample average absolute autocorrelation versus five stepsize autocorrelation inefficient autocorrelation sampler indeed explore distribution sampler momentum instead move contour handwritten digits classification instance split remain instance classification network hide sigmoid four sgd momentum base regularizer network fully place weakly informative regularizer resample sampler discard burn mcmc iteration burn report momentum converge converge backpropagation dominate computational show scalable collaborative filtering application predict movie music pmf due rating matrix versus recommender system severe issue conduct pmf million rating movie compare base approach rating update item minibatch neural large pmf difference consider hyperparameter use model discard burn rmse sgd result online pmf key mcmc online set high distant build hmc costly surprisingly natural gradient hmc poor address langevin term effect maintain target modification next explore technique broadly technique herein enable scale bayesian acknowledgement fa intel discussion material sde sde nz evolve upon hamiltonian position momentum r p evolution govern two free hamiltonian change vanish contribution equality assume vanish p statement immediately behave r also fisher full note conclude increase langevin dynamic follow temperature usually set apply verification give particular substitute imply compact desire generalization consider case depend problem adapt simulation correction dynamic govern stationary reverse associated r r tp generator generator reverse q g sde reverse tr together detailed balance allow backward property hmc symmetry rely balance efficiency trade efficiency case nonzero fast relate choice relate sampling inaccurate stationary indicate indeed sde correspond p correspond inaccurate divergence distribution evolve divergence govern decompose change inaccurate dynamic size mix rate bind correspond unclear leave bind relate time process prove refer reader detail sgd momentum analogy momentum learn momentum equivalent estimation
iteration yield incorporate ensemble maintain persistent round team cascade round introduce new warm start classifier ten five weight iteration five train variant private evidence utility benefit reveal comprehensive empirical cascade regularization em em causality author common significance physics solve close manner moreover improvement discovery significance complement derivation significance weight classification cascade derive maximization optimize measure challenge let w w n represent assign label b quantity g n w g employed equally increase differentiable close convex make conjugate hand fa fa ac fa c lemma expression representation representation significance apply f e minimize strategy hold optimize carry support furthermore optimal optimization scheme consist series weight cascade optimize illustration weight cascade progress ht input minimizer classification weight classification g ht minimizer weighted procedure whenever achieve small respect monotonicity characteristic maximization h unnormalize analogous derive optimize divergence section turning support maximizer procedure couple effective ensure adequate generalization describe team incorporate
stochastic rich history begin thesis loss information geometry li convexity take hypothesis er chernoff method control generate particular value tool inequality exponentially erm later er chernoff erm let copy since apply show excess constant must take value subproblem excess pair constant erm prefer conduct nice measure measurable space general moment term let g equal choose need let form curve segment clearly equivalent program exist need satisfy result stochastic find stochastic let element value apply corner perturb nearly nearly perturb erm would pick slightly closeness erm pick common setting random yx random maximize previously xy xy oracle various vc class logarithmic class polynomial exact oracle vary excess let z f excess random hyper concentrate show exist arbitrarily arbitrarily empirical great risk hyper erm replacement function latter apply recall erm select hypothesis purely inversion erm excess present vc type class definition cover minimal ball cover far constrain ensure stochastic argument localization result class vc subset contain union center separable function cover bound net long point present finite vc compose oracle l fx minimizers x bb jensen question necessarily minimizer sense bad minimizer minimizer arbitrarily ball minimizer x stochastically stochastically stochastically z decrease minimizer show loss index sequence slow size consequence poor connect good constant target measure poor bernstein rely bernstein f fix excess high erm mistake e modify n q straightforward result excess certain amenable localization control individual straightforward extend result open result vc result class bernstein condition question bernstein show loss bernstein offer problem whether bernstein constructive bound loss condition minimizer true bernstein regardless classical motivate great interest discard assumption ignore metric loss difficulty extensive wise function understand concern mild thank initial chernoff visit without thank serious department communication research centre g value interior moment objective pick yield replacement q definition q condition small either compute eq eliminate sufficient hence far pick constraint minima root substitution root root increase eq consider yield c arrive suggest attain fix verify true eventually limit put regime exceed objective bound z z v less increase arbitrarily arbitrarily consequence separable satisfie ball loss loss bound select large large taking zero trivially setup large proof center follow jensen inequality appear equation convenience countable measurable function positive g separable admit inequality every f take control control state current yield put concentration incorporate step avoid issue operate assumption step expect nd define sense radius life make step state term careful inspection reveal argument cover moreover rademacher process convenience let sub countable paragraph concluding apply result rademacher l step jensen follow assumption thus everything replacement amount yield set coarse bounding yield approximation separable consider countable slightly number observe cardinality hence cardinality probability function analysis factor term particular observe set f cover union imply rhs q inequality exact class convenience begin notation abuse next introduce play recall norm f concentrate excess nf apply concentration theorem hence er chernoff imply failure obtain last since failure guarantee statement erm erm excess corollary thm empirical minimization erm statistical accord n excess exist result joint property prediction expert phenomenon entirely role notion build bridge reduce special fast rate erm phenomenon exploit old suggest recent contact area statistical online include unified bregman
linear complex introduce process idea observe transformation expressive successfully function gps enhance capability include robot contact modeling acknowledgement department college european community fp rgb rgb rgb assumption complex space often learn lead novel feature gp regression gps process nonparametric encodes assumption hence suitable stationarity common smoothness violate overcome limitation approach combine covariance transformation covariance one implement transform input transform subsequently stationary periodic reduction transformation heuristic suboptimal overall regression base devise expressive gps gp space gp function space property incorporation model uncertainty attempt learn objective combine unsupervised supervise unlike motivated discover within experimentally validate model scale ground challenge g full would integrate analytically task discover subsequent learn mapping review use provide gaussian jointly regression representation input dy function latent mm integrate often common mapping consecutive learn reduce dimensional representation simple high input discriminant reduce curse case nonetheless gp unsupervised mapping nothing input reconstruction ica unsupervise learn insufficient regression optimize objective necessarily match unsupervise maximize marginal likelihood guide mapping toward representation overall intuition mapping jointly objective figure gps probabilistic jk measurement use square relevance ard ff weight possess select negative likelihood thus relate gp model show manifold decompose overall overall objective marginal parametrize map input e kernel operate valid gp test distribution gp matrix construct x covariance function train jointly optimize mapping gradient objective equation compute parameter feature dimensional gp parameter approach deterministic multi layer number neuron layer layer backpropagation sigmoid se ard covariance function blue nn dotted sigmoid dash capture well se ard discover input non mapping world demonstrate function assess datum cause gp embed gp capture thank transformation uncertainty still assume require care effect example encode easy space jointly underlie well gp set c se ard ard use possess length anti map I substantially model horizontal slice spectral problematic ard assume learn hyperparameter length need trade frequency preference short scale generalization standard se ard gp ard point sigmoid transfer use show outperform evaluate believe transform intensity transfer intensity map sigmoid transfer notice tend initial transformation frequency visible log sigmoid spectrum non transformation transfer translate superior model physical especially good force evaluate model inspire covariate regular interval covariate leave angle right two signal contact sensor remain five consecutive datum use structure inspire sigmoid real data se ard data gps either se ard nn show gps se nn predict angle area movement occur due regularity degree uncertainty angle fully preferable trajectory smoothly per point set method rmse se ard sigmoid gp ard gp ard learn representation neural successfully feature gps feature discovery feature often similarity neural network gps unclear exploit good deep neural network gps stack also regression framework similar
attention core figure core varied core speed lda yahoo amazon set number core machine dramatically outperform memory disk version yahoo solution time lda number document topic appropriately update handle propose asynchronous framework lead core result yahoo lda able handle million ability stream document disk yahoo idea black conjecture cs edu university edu amazon com california edu meaningful massive document collection contain million token challenge deal topic scalable efficient way paper novel simultaneously handle appropriately modify moreover change computation across processor asynchronous inspire lda significantly art massive topic topic provide way vocabulary corpus topic dirichlet allocation one meaningful massive token challenge typically second need across multiple resource develop scalable tackle package yahoo recently award win distribution effort towards computation across processor early effort work processor word vocabulary partition processor subset document synchronization fact idea lda count across arguably effort scalable recently trend towards asynchronous algorithm synchronization iteration large tree datum allow multinomial item time topic count order computation across novel asynchronous collapse technical key various single processor present tree encode sampling maintain lda type modeling utilize avoid parallel asynchronous communication moreover scalability method million briefly allocation lda number denote vocabulary denote document topic include denote draw hyper generative draw collapse accord technique lda collapse bayes follow wider unnormalized many sample initialization compute generation first p arrays construction scheme generation generate comparison head tail shape style normal style head tail text label grow style level label label child b node child normal child child draw none densely corner thin fit transform every style mm normal style head blue black grow style distance label head node child node child name thick child head draw thick edge parent draw thick none start west east densely dash corner thin transform mm style center style distance distance cm child child node b normal child name thick label head edge child label parent parent draw thick draw e densely corner thin describe multinomial sampling initialization f maintain parameter use accelerate lda without simplify generalized sampling update regard version leaf internal leaf value two binary internal representation use index store child node tree use addition carry simple traversal z consider f go right child ensure half remove cost time toy htbp ti efficient routine deal slight support routine single th simple tree f carry leaf delta update q see procedure deal normalization construct table clearly update method generation procedure operation htbp tw www tn td tw f tw w sample td tw tw td tree yes yes yes yes apply tree sampling step current document current decompose implication sampling sampling fast element increment time tree document document document document element change switch one word document propose cumulative initialization sampling word lda word fact level element change maintain occurrence generate required word performance lda document document expect consider sum dense non zero third implementation yahoo lda follow change blue rectangle node node node node node node node rectangle rectangle rectangle rectangle red rectangle rectangle red rectangle rectangle green rectangle green green rectangle blue blue rectangle blue rectangle rectangle rectangle rectangle node node node worker work area begin scale red green node node node rectangle rectangle rectangle red rectangle rectangle blue rectangle rectangle rectangle node node node node red rectangle rectangle node node node node node red rectangle green rectangle green rectangle green rectangle rectangle rectangle rectangle rectangle rectangle rectangle node node node node node node node j process worker rectangle rectangle rectangle node node node node rectangle rectangle blue rectangle rectangle rectangle rectangle red rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle green rectangle rectangle rectangle rectangle node node node node node node node node node parallel memory processor partition split document corpus unlike document worker grain split correspond occurrence worker illustration split denote occurrence word block big partition rectangle stand worker asynchronous computation suffer work aim worker maintain job queue without synchronization characteristic worker update occurrence access guarantee worker access keep th difficulty parallel execution overcome difficulty token token access resource token share resource access token token dedicated token pass token token tuple token worker token mean activation result guarantee always date access token far successfully keep token pass update require make depend base summation deal issue token copy worker always modification arrival delta arrival worker local illustration unlike case parallel mechanism distribute yahoo update snapshot synchronization conduct yahoo central server server communication yahoo avoid expensive network yahoo sampler significantly could follow copy machine close utilize variable completely concentrate completion graph need processor core show performance lda core demonstrate tree sampling handling compare approach distribute amazon among bag uci repository many paper fact capability implementation demonstrate scalability algorithm amazon amazon million product review amazon project home review typically stop processing discard review review leave process discard result document corpus collection process process stop word follow approximately document amazon parallel platform advanced gb job node core yahoo deal experimental evaluation yahoo scale fair yahoo
table report none semantic cardinality symbol occurrence unbounded equivalent unbounded equivalent offer define schema name impact call consider library describe kind item book art define element publication report book paper article name publication element name title type title name element name element publication element article publication publication thank want global element prefer shared node schema relationship element share loop concept consider library publication additional material like software prototype suppose specifie point site contain specify material software program use paper specify publication present conference element specify title number web prototype paper dataset experiment report name schema element reference additional circle pt inner sep thick scale auto publication b diagram offer definition schema source schema target schema attribute way schema schema assume concept suppose file specification book example generate book book book book share book book book library student schema http www http www library http www library book library element book element sequence schema describe may opt include new schema schema reveal semantic two share exist book ideally style adopt reasoning discover semantic shared challenging template reason goal author receive schema could state schema element high tool degree two state two overlap nice definition author introduce concept schema repository scenario want able query encode schema rank repository schema second researcher work approach appear instance heavily exist attempt one describe solve outline template applicable template schema element regard approach component template act enough template domain indicate set set one processing instance convert step representation definition template match er template tuple format domain return element ki k array receive two schema return real play introduce similar schema resp belong confusion symbol schema denote resp aggregate score schema one convenient aggregate similarity extent two describe piece next subsection template observe handle generic review module management mapping coincide document onto array call array token design deal kind token token token element token specify hierarchical schema approach map array token pair array root name aggregation direct acyclic direct name sub attribute approach name element sequence name parent sub child name element name element schema similar schema map tree sub element presence repeat share graph loop rule loop schema tree map schema parent relationship report type cardinality constraint map name approach onto graph root node uniquely schema tie encode relationship like aggregation finally procedure onto direct acyclic original schema path schema direct uniquely associate relationship attribute al label uniquely node follow return return iii return parent return v returning element say label schema element encode function process detail accord define fall like name element call group concept context element element context context belong pay special attention ability implement implementation merely linguistic match structural principle extend type metric first category metric classify name name last two schema name purpose gram provide overview string e implement receive first end phone exploit receive string compute short character transform latter result call syntactic distance string character character different gram character string house gram use approach rely gram distance gram schema gram report experimental string matching classify like elimination language metric element name parent contribution child two schema element let define child child resp coefficient element similarity higher share induce penalization since element parent q resp parent language dictionary context adopt case semantic formulate schema semantic former neighbor exploit naive bayes look tf score schema far capable consider include third category metric similarity category cardinality similarity discussion compatibility usage cardinality usage language constraint tf elimination yes yes yes yes yes distance yes elimination yes yes name child elimination context node represent resp close closeness exploit interpret element behind element context similar schema match furthermore popular classified approach ii approach approach tree coincide similarity leaf leaf classify child similarity child two child child compare coefficient leaf similarity subtree similarity non leaf compare leaf leaf node array cosine node place node pair high matching pair similarity similarity particular node root node correspond refine efficiently encode structure document map onto vector apply combine leaf classify sub fashion leaf schema element highly mutually similar consist attribute element link leave root element semantic formulate reduce tree matching problem formulate node word word suitably formulate deal complexity dynamic often schema model structural like converted schema neighborhood whose less threshold take integer schema resp neighborhood iii exist syntactic reveal maximum matching solve corresponding threshold similar parametric semantic solve schema approach structural semantic constraint feature schema element sufficient poorly reason similarity schema element multiple score similarity global aggregate partial similarity score improvement aggregate score propose suggest classify aggregation homogeneous see iterative human expert line expectation adjust threshold decide similarity function remove modify aggregation introduce belong interval belong say aggregate aggregate aggregate vector equal totally agree similarity two schema aggregate return monotonic monotonicity agree pair equal similarity way aggregate sum interpret confidence correctness produce system configuration weight exploit adopt option max high similarity available return lowest consider non rely appear similarity approach among similarity accord partial score score equally contribute latter term partial threshold add mean similarity coefficient instrumental interval investigate definition later therefore approach apply linguistic schema linguistic score similarity similarity obtain weight sum similarity score semantic used map onto detail appear involve collapse map reduce two detail consider two list list filter element similarity recognize proportional however happen think describe university describe correctly handle normalized degree computing simplify onto mapping use mention implicitly primitive operation call change node subtree cost sum form consist cost operation uniquely equation tree coincide q th input map map onto map j review tool handle schema handle microsoft server template system specify discussion aggregate implement element vs match external microsoft name yes structural external open element source structural name source relational structural structural researcher base prototype mapping support schema match nice company either source worth observe conjunction datum service subsequently acquire case source file also deal provide use represent match column table focus operate structural discover structural cardinality server discover discuss section specify detailed discussion offer capability semi automatic fashion simple technique check schema share matching support external apply string pre procedure also interesting observe server candidate function heuristic former lexical similarity latter implement aggregation exploit aggregation max operator discuss end people manually specify schema element advance provide report line ask line element mapping format support enhanced interface visualize report auto mapping core intermediate output variety help user datum integration two integration server effort area schema put issue address properly strictly match deal discuss uncertainty match cluster task traditional integration technique source unfortunately span multiple many service handle country social public health service education clearly virtual schema virtual schema heterogeneous ultimately clustered domain domain whole cluster abstract represent repeat represent service schema proceeding schema classify kind conceptually involve ii cluster significantly small source iii activity effective schema classify strategy mapping approach low similarity belong operate stage available linguistic reduction activity map mapping perform array array belong accuracy preliminary play tree linguistic conjunction user schema arbitrary require reason substitute degree schema target one obtain involve approach schema engine search repository one query keyword schema require repository candidate likely user query report degree element schema produce finally combine candidate rank multi affinity keyword schema filter removal schema construct pair coefficient keyword schema uncertain introduce consider contact detail book contact mail phone user contact phone mail assume semantic similarity uncertain preferable identify absence select yield want contact detail student book depend obtain contact mail loose contact phone opt contact phone retrieve user contact mail address schema map score discover matching state study uncertain relational database analyze join et extended e count refer reader book receive knowledge approach et issue source schema schema author element correspondence schema correctness number pose storage fact mapping suggest unfortunately require management graph still demand cost bipartite graph recursively partition merged mapping store mapping mapping high overlap consequence share mapping store call node schema involve block query uncertain mapping purpose recursively query take tree fact opinion fact highlight exist improve overall effort require existence standardize observe schema offer business popular define semantic match similar explore emphasis repository repository initially business actor integrate fashion domain match consider next compose detail perform composition public business life consider standardized usage constraint relational schema analogously key pair schema reference multiple context recognize explicitly model connect argue schema model structural schema differ kind match open computation schema similarity observe year computation schema similarity integrate recognize compute schema similarity look implementation implementation consequence approach generic reason hard achieve different approach theory schema open problem next strategy combine produce semantic step ensure multiple detailed depth construct specification available diagram think hierarchical schema element introduce describe main component role interaction template compare popularity focus finally relate cluster collection management despite schema match perspective external domain dictionary play schema handle expert intervention correctness advanced tool auxiliary employ great system deal hundred schema future plan analyze scientific anonymous thorough suggestion quality manuscript top white drop xshift ex yshift color bottom color em white red draw red color draw version proposition corollary schema discover intelligence many largely database relational field researcher match semantic well know originally exploit schema research describe impact schema template template template introduction template useful future implement introduce related source uncertainty schema schema management de standard representation exchange wide scenario widely scientific domain like order make exchange easier wide web increasingly advanced language content schema schema build schema schema schema schema availability schema simplify exchange procedure software program impose schema exchange interested capability name attribute name denote semantic attribute name content document identify schema despite name share semantic long artificial intelligence schema community alignment vast review survey format represent model diagram subsequently grow researcher hoc schema match offer advanced capability usage schema research contribution survey schema regard narrow schema match describe extent schema hierarchical diagram provide call implement popular help template act appear totally act discuss template classify challenge schema management survey schema match aim section summarize notion schema template systematically schema provide cluster management conclusion schema context like integration distribute answer matching issue bernstein recognize relevant originally application domain se classification researcher work exist development schema decade suggest list current schema excellent schema therein cover survey perform schema evolution schema merge survey devote adopt assess tune optimize discover problem valid specific matching survey several respect show exploit broad area management schema web huge impractical manually automatically classify schema source web organize group task integrate belong domain basis relevance benefit answer get likely sound correct query irrelevant semantic focused alignment matching resemble schema relationship schema match review book find differ r relational flexibility level cardinality hierarchical organization available r secondly schema specific piece reality exist relational schema relational schema generally human expert vocabulary become decentralized effort discussion task resp relational match hierarchical matching discovery big axiom semantic trend schema match uncertainty schema excellent uncertainty schema book various aspect alternative representation schema uncertainty narrow area namely source area publish survey survey exist agnostic survey focus schema match develop subsequently influence schema match problem detail schema business pay discuss management specific schema match impact uncertainty wide range world schema match effort deal specific schema match schema find semantic web literature schema also alignment refer user exploit next subsection size become business imply explore pose relevant challenge surprising design capable rely idea filter element form match one schema recursively partition partition schema base recently entity resolution research size call author framework prototype entity resolution recent approach incorporate describe interact query interest analyze frequent attribute two attribute frequently co query likely match aim attribute attribute co occur aggregate several resp attribute exploit find attribute similarity score database scenario search engine quite traditional schema source time bad quality matching behavior ultimately behavior handle limitation adopt user query available semantic available poor activity extent schema throughout sake simplicity resp encode resp matching matching encode resp subsection encode show present far find usage part generic r diagram relational schema convert internal oriented schema database
initialize message empty line satisfy iff incoming message neighbor strict disagreement original ii reduce bp inaccurate marginal example fix apply bp give true false false markov inference recover probability q markov certain unbiased become correspondence gs variable gs update sample message bp gs require idea bp gs perturb message message bp gs linearly get final iteration bp gradually linearly change summarize initialize message tx marginal I combine gibbs incoming message strict inherently avoid contradiction bp bp repository complex extensive format use dense factor remove instance discard instance represent form remove many factor bp probably instance representation perform variable initially bp fail apply reduced attempt perturb start factor failure repeat bp final perturb bp number iteration iteration bp compare bp result appendix perturb solve bp hundred efficient iteration average run fold iteration bp perturb bp appendix detailed report study combinatorial related spin physics follow hamiltonian spin resemble problem interaction allow sp neighborhood analogy extend relate dynamic behavior solve spin translate dynamic phenomenon focus geometry space rigorous algorithmic rigorous rigorous cavity confirm picture work instance instance situation characterize constraint select control col random col generate sequentially select generate equivalent distribution analysis several phase phase increase reflect r col symmetric r blue regime neighbor one solve belong replica analyze characterize solution distant distant member phase dominant roughly bp converge regime valid bp transition identify phase finite total picture summarize replica symmetry break geometric perturb bp message initialize message neighborhood resort perturbation message bias towards solution continuous message focus short ensure absence message remain equation completely existence several point fix point bp quasi solution uncorrelated result incorporate equilibrium specifie weight implicitly back true true false false distribution cluster uniform solution construction represent bp bp message bp cluster requirement distribution make practically infinite simplify bp limited apply bp solve product eqs message use max posteriori high assignment I represent message message initialize ix ix incoming trivial define allow assignment correspond consider false false false false allow assignment particular update max message large sp j denote sp update equation aggregate sp message marginal perturb combination gs perturb bp iteration perturb gradually increase perturbed sp reach implicit marginal advantage perturb apply single search perturb perturb solve perturb bp solve factor knowledge general sp sp tailor various sp second use sp search iteration different instance power use sp report col satisfy different portion satisfy help break variable use threshold variable perturb bp perturb sp per fail increase factor attempt sp iteration iteration instance col row report bp local show fail attempt sp time figure instance disk iteration control close chance require computationally inefficient bp solution instance large sp trivial I allow col point bp sp similar col sp success attempt sp search bp easier see bp result col col support advantage bp perturbed sp perturb instance factor cardinality variable col col sp perturb sp impractical dynamical besides report perturb packing cover clique cover min max col success rate average successful transition pt pt l pt pt bp sp perturb perturb avg avg success avg success avg success transition transition l transition n transition n transition n n n corner table west table south check produce bp sp eqs product bp eqs whole assignment formation distant bp focus reduce bp sp bp well reach ignore analysis similar effect variable form fix assignment loop lead alternatively long lack marginal perturb bp non avoid adapt choice variable unable backtracking attempt sp simultaneous message towards region bp valid prevent formation correlation experiment bp sp bp fails exponentially update meanwhile negligible local accordingly sp limited applicable perturb attractive experimental conclusion produce assignment hard combinatorial col message pass perturb sample perturb bp solve tractable factor perturbation sp produce sp outperform sp anonymous constructive center sr technology use compute resource compute benchmark report iteration attempt assignment fail series perturb instance avg satisfied avg avg geometric aim c c c school c n n n n book job c c c c c c c c efficient message pass perturbation belief propagation bp propagation satisfy smoothly end solution perturbation sp bp hundred perturb bp compare state sp bad cardinality sp outperform make incomplete solver regime science neural physics code pass successfully solver constraint produce suggest assignment subset sequentially marginal give failure guide procedure back track backtrack branch rely solver purpose propagation solution well bp fail survey message bp typically convergent message hard produce single pass avoid alternative bp apply bp gibbs gs update message perturb start end smoothly change produce change bp marginal bias bp bp sometimes bp fail random perturb bp sp difficult instance sp bp sp perturb sp experiment perturb sp sp assignment bp particle perturb bp directly gs bp introduce perturb gs compare bp bp fold solve geometric review order replica break perturb sp present instance discuss
non fully model al include clinical interpretation fit etc survival log logistic researcher parametric analyze cox et issue prominent grow survival observation censor attempt obtain covariate wang life power along automatic estimator likelihood inefficient provide pure datum develop estimator censor covariate need law central limit censor suitably covariate note consider usual semi regression cox proportional cox robust model respect etc develop robust propose accelerated failure location without parametric survival censor stochastic covariate brief background parametric general censor propose context censor performance illustrate property density divergence suitable remark tune propose section end one derive iterated number central etc mainly work wang wang theorem assume life censor censor denote give limit assumption life censor respective whenever life covariate censor precisely form strong distributional consistency integrable mean atom write define coincide consistent far strong hold corollary measurable function length consistent estimator replace definition respective assumption strong assumption censor see write statistic censor censor biased censor covariate move like cox regression strong efficient see fully set paper deriving result propose power divergence inference quite day robustness high parametric smoothing density two dominate measure parametric power minimize datum equivalently correspond coincide mle drive apply suitably lee index lee extension identically distribute generalize extend censor efficient censor generalize estimator parametric censor covariate set covariate note early focus give f estimate property mle know drawback lack robustness previous density efficiency pure common motivate et suitable estimator joint optimality give objective nothing generalization routine u estimate simple form substitute equation divergence datum simultaneous root equation objective clearly inference objective particular estimating extend concept define however make censor estimator note estimate may suffer root need technique define examine simple response generally variate variable response unknown regression coefficient symmetric distribution group reliability distribution response incomplete censor observation inference belong consider frequently robust solving work example family family distribute covariate p auxiliary variable normally ph px dx function random numerical technique simply minimize function simplify particular px simplify see simplify estimating simple subsection life science model censor short support identifiable become need say suitably cover routine medical science widely popular science variate multiplicative ensure positivity life application exponentially covariate exponential error belong robust normally covariates ph n minimize ne tx use equation simplify erm form equation simplify linearize natural logarithm scale model log covariate density general extreme distribution early censor section minimize objective objective easily highly robust presence accelerate robustness comparable alternative advantage et al censor normal previous simulation exercise scalar covariate simulate response exponential censor consider censor censor keep expect censor exponential censor censor respectively study numerically bias bias mse mle contamination compute total along absolute censor proportion efficiency absolute mse increase censor repeat simulation contamination covariate censor covariate simulate observation total small ignore outlier generate inference prop prop prop prop base censor correspond define empirical prove consistency normality result replace far coincide replace finitely finite proposition consistency asymptotic integrable parameter distribution wang normality result censor extend present censor response censor huber routine application wang assume distribution estimator property wide equation population root eq strong estimator extension wang page replace hence simplicity presentation assumption satisfy estimating sequence converge I complete censor data covariate estimate fall strong really root numerical estimator strongly attention normality regard first proof similarly continuous neighborhood continuous real sequence set singular converge multivariate value yield q follow convergence fact distributional convergence estimator convergence wang extensively appropriate censor complete present case wang replace particular derive particular however particular closely assumption censor score existence second moment condition assumption lemma asymptotic normality require rather assume require strong simple al property extend many researcher censor relax family define parameter function equation condition support parameter space differentiable integral f derivative finitely singular third bound assumption sequence estimate tend fitting normality part show tend local interior tend respectively present get tend combine proving equation sequence root well complete routine assumption although particular estimator namely divergence estimator general suitable equation optimum class huber fact class estimating location odd wang optimum might similar censor presence reason variable belong family asymmetric censor
train radial regression validation follow optimal kernel forecast approach calculate forecast conventional ap integrated move ann table fig summarie numerical ahead power ann svm hybrid svm conventional ann smoothing ann base machine prediction evaluation machine forecasting ff ann sa ann sa performance indicate rmse mae optimization ann stationarity wind note processing field modification ensemble component original hybrid algorithm employ system forecasting series hilbert transform algorithm random tree technique examine rank model employ hybrid forecasting employ radial support introduction follow approach forecasting power hybrid machine forecasting parameter fourth describe variable experimental flow price wind speed direction two major market lead inter trade neighboring region source system often non limit efficiency effective market participant whole past decade power effort novel short flow wind forecasting forecasting addition difficulty power researcher publicly make delay consequently pattern possible novel approach power combine effective stationary series machine organize follow forecasting hybrid machine short forecasting power system balance maintain consumption may power scale intend forecast forecast refer short forecast forecast trading term system aid analysis aid intelligence purpose include machine machine svms random forest etc forecasting note often forecasting show hybrid potential include ann wavelet ann fuzzy expert system reader characteristic account influence everything price often information scientific series impose forecast performance consider improved series task initial investigation improve forecast machine preprocesse forecasting collection merely two mode hilbert transform ht intrinsic block random bt rank ann employ obtain forecast ann svm forecasting system hybrid forecasting increase forecast limited forecasting wind major power need competitive technique wind power energy price decade algorithm system forecast system success success forecasting forecasting mainly reveal close reality plausible important cause forecasting algorithm predictor believe researcher identify subset another separately take value root split item binary node lt reach tree constructing maximize decrease forest decrease proportion reach use examined decision tree tree decrease introduce modify machine realize develop demonstrate forecasting pool market wind hour wind wind year decompose hilbert hour calculating illustrate wind machine forest frequency wind speed comparison exclude train construct test rbf rbf phase rbf scheme rbf network neuron briefly svm xy svm regression g svm determination reduce follow capacity trade deviation parameter loss base fx condition negativity represent case validation follow rbf feed ann candidate ff ann time wind times wind direction time normalize ratio validation set hour search algorithm network ahead wind use pruning involve define maximal intra neuron connection parsimonious neuron use bias backpropagation htbp candidate hide neuron hide neuron predictive inter activation
consider admissible model size nonzero entry later consider subset theorem implie subset respectively subset complete cm pt lemma example corollary generalize glm link assume subset proof use necessarily similar kullback risk include correspond exist equal p repeat difficulty prove generalization g
fast though potentially evaluate face five uci demonstrate cluster consist unlike exist completely setting scene camera use different people leave background acquire feature result face stanford different feature affinity via kernel dataset dim diabetes gene uci measure via kernel cluster evaluation represents assign cluster ground assign alternate determine truth define entropy completeness result harmonic homogeneity completeness member harmonic mean homogeneity completeness weight measure active strategy method number variation baseline multiple active variant active reduce gradient without gradient parametric baseline comparison include active pair constraint fed seek seek nearby cluster use guide constraint query base query maximum reduction value compute uncertainty multi active label request constraint run variant list compete parametric set uci coefficient lead consistently particularly notable minor relatively consistently meet exceed performance scale consistently perform par well particular validate show combination scale way selection problem drive compare technique uci reasonable binary outperform al competition generally plug cluster uncertainty learn clearly superior seek global impact uncertainty idea query consider entire active visually appear applicable present overall compete method far exceed dramatically match also run constraint fail somewhat competitive nature limit usefulness method winner though nonparametric leaf diabetes reliably work active rule make dataset amazon experiment query rate dataset slow experiment overall active passive clustering actually discover cluster effect face certain unknown result initially converge discover test indistinguishable novel sample online active select pairwise query problem estimate expansion decompose scale two entropy pairwise query uncertainty support demonstrate state initially burden adjust iteration naive select uncertain redundant adjustment active powerful particularly crowdsource grateful grant nsf nf ap finding reflect david seek cluster side via semantic side randomly require could redundant unnecessary semi maximize human human great impact select proceed principle taylor decompose step assignment result uncertainty sample different image uci validate show superior noise number cluster semi uncertainty play machine top external effort pairwise face supervise cluster categorization surveillance grouping person particular location may problematic recognition human label might realistic probably image action make context human crowd amazon probably specie even semantic cluster image visual identify specific apply semi supervise clustering rapidly approach large redundant expensive improvement circumstance overcome explore constraint constraint base method interest query constraint select propose intermediate explores initialize grow cluster large semi active min criterion utilize exploration also seek min max informative parametric provide encourage however complexity semidefinite sdp limit constraint process xu et al wang problem suit multiclass recently seek prove meaningful criterion method cluster suffer drawback offline select thus incorporate cluster decision online overcome limitation novel base online great pairwise query base user cluster cluster discover human interaction proceed section yield great identify uncertainty component estimate perturbation eigenvector assignment uncertainty formulation baseline active cluster dataset face leaf common uci gene see show state art cluster ultimately relationship thus relationship pair sample may highly clear relationship relationship decision ambiguity relationship semi supervise remove ambiguity relationship reduce assignment uncertainty measure contribution local entropy detail uncertainty propose uncertain uncertain uncertain beyond limited presence every inherently complexity advantage naturally via proceed pairwise result increment active active typically control generate domain output face dataset initially available application conduct evaluate method selection encourage recall proceed iteratively informative sample pairwise similarity matrix partition eigenvector via effective whenever constraint modify produce new affinity pair link value proceed define dataset similarity current therefore direct uncertainty remove ambiguity cluster uncertainty result consider select sample greatest though estimate must simulate answer could predict oracle would expensive worst require iteration active adopt estimate impact sample perturbation entropy select present briefly describe select pair base query certain generate sample within close l record record sort correspond connection certain sample relation create certain sample add certain regardless correspondingly add reflect newly discover certain select human describe use eigenvector query eigenvectors taylor expansion decompose eigenvector ambiguity represent ambiguity eigenvector result ambiguity reduction reduction first spectral change ambiguity approximate represent incremental change jx iteration reconstruct eigenvector correspond
ps big ml single memory pool machine read simplify distribute program ps systems interface arguably interface challenge application ml program stochastic subsampling dependency parameter parallelization work introduce relaxed read throughput promise consistent success nature play relaxed relaxed synchronization throughput possess relaxed affect ml stability improve progress iteration throughput execute per recent focused system various start principled insight ps ps consistency angle read impact stability learn insight design outperform issue new ideal gold progress ml problematic bounding amount synchronization different empirically scheme parallelization develop attain guarantee provide deep particularly exist ps theory outperform distribute carefully throughput ml still mean optimal answer convergent ml converge environment enforce reflect quickly lead frequent consume synchronization limit speed parallelization therefore synchronization parallelism closely approximate sequential execution inconsistent algorithmic imply still happen read carefully explain trade introduce asynchronous parallel approximate strong magnitude value single view represent worker method worker fewer visible worker update update worker variable condition vary worker analyze algorithmic broadly speak either within base sgd worker operate ps popularity later well sgd expectation optimum successive implement update worker pose worker condition need update structure difficult achieve design consistency provide correctness ps implementation impose work algorithm worker assign initially zero operation share parameter store ps ps worker progress fast worker parameter achieve fast update make visible crucially communication meet implementation propagate beyond require reduce average age axis observation count factorization minibatch experiment bar computation communication show reduce priori predict complex empirically read ps draw conclusion consider worker worker could behind ahead worker last cache update tail profile salient advantage convergence tight bound base analysis strength achieve comparable excellent ml topic show throughput thank exploit scheduling affect place ml descent popularity prove sgd involve miss follow product constant update put access introduce computation produce apply order worker start upon update different worker server network view update I update r worker mild worker sufficiently bound drift global schedule expectation via component size tf measure expectation use speak execution bound assume var var optima capture randomness condition imply argue amount synchronization motivate analysis noisy system noisy update view worker start condition spc decrease suffer theoretical threshold usually unnecessary ml gradually execution read close threshold use grain update implement inside server one process type distinct logic key interface system increment worker define library locally parameter fit memory request cache cache read request send server mean worker worker generate update send server request time server make read request server server advance get row read request cache call exploit often convergent server explicit request cause accumulate usually parameter regardless specify threshold burden server update batch separately request quality collapse gibbs factorization robust additional use gibbs sampling gradient interface minibatch call measure quality mf minibatch record convenient step mf topic york token topic netflix dataset use run core connect via lda connect iteration leave consistent progress pointing help much less robustness tuning slow problem stepsize distribute aggregate deterministic dependent mf investigate profile introduce produce low improvement parameter addition per provide reduce reduce chance update right speed per second lda server state art consistent parallel ps frequently big commonly ml framework yahoo might special generality machine factorization small world lda lda vocabulary size memory lda data scale speed count matrix mf kind ccc header cc split header header software tailor single scalable roughly group category library framework purpose tailor category constrain mf yahoo lda topic primary solver restrict application program improve code communication synchronization protocol careful design consistency iteration specific category distribute ml consistency benchmark fair match algorithmic benchmark ps practically ml framework specialized algorithmic tailor purpose framework ml way purpose enable application research unlikely solver many nature ps propose implement present herein enforce assume transmission could delay read delay inconsistent parameter wider popular framework ml application sometimes knowledge superior ml framework ml alone solver salient consistency support bound consistency side ensure program yet give via update term fact divide gets desire answer search x dx l divide tt eq hessian close expand use take gradient bound invertible optimum var capture condition tf similarly variable condition represent
error occur representation multiple concern latent dimension mean social phenomenon contribute express space help decision feature believe easy simple appropriately world seek follow characteristic repeat learn similarity member generalization community view membership satisfying requirement stream walk originally walk combination denote walk root k vertex walks content recommendation sublinear structure motivate short walk tool extract community information desirable machine explore part secondly rely walk possible small learn walk sub linear walk primitive capture suitable method capture power law observe appear walk follow word frequency behavior law walk text contribution work idea language symbol distribution community law distribution language grow representation appear formally vocabulary maximize network representation language beyond goal present language stream short walks walk think short sentence phrase language analog give visit walk goal distribution occurrence latent vertex later walk grow language modeling turn head use secondly compose right give word remove required appear offset give vertex optimization find relaxation social representation random walk time problem representation neighborhood citation similarity machine learn walk representation social exist encode intermediate adapt change topology b walk vertex softmax factor occur variant require walk vocabulary know walk ahead walk update window walk initialization iw sample vertex random walk walk neighbor visit walk experiment return advantage practice specify walks start start random think walk pass order strictly know vertex generate representation use accordance language word sentence appear window line vertex representation figure give walk use model logistic huge could million resource could whole speed iw expensive leave tree maximize vertex identify root speed assign entire graph maintain enable web web c label interest overview dataset use reproduce website figure pt network relationship topic category website label user popular label group validate baseline generate laplacian utilize eigenvector assume cut generate representation modularity eigenvector encode modular modular graph partition use mean adjacency vote relational neighbor neighborhood appropriately iw surprisingly real sensible frequent micro r micro macro r r nodes micro majority macro majority sensitivity several facilitate comparison method baseline node use repeat report micro result vs logistic implement classification liu experiment bold perform consistently label perform prove much still datum macro micro label b vary approximately label outperform baseline micro additionally micro word baseline perform perform improvement size prevent run much close real result present table scalable baseline create micro improve macro increase lead end micro macro experiment benefit classification order change conduct experiment label sensible emphasize local walk start determine impact available figure vary performance quite dimensionality dimensionality stable accomplish start walk second consistent interesting various show walk see increase start dimension amount datum initially result effect quickly learn walk machine easier dense thing method dense walk extract place limit hope walk distance way good emphasis effectively hope limited difference summarize representation statistic centrality extend procedure collective kernel scalable local information feature relational link collective np solution approximate guarantee converge relevant add nearby relational substantially propose learn inference include representation relational approximated complementary encoding directly method concept propagation instability decade recently compute allow growth distribute diverse speech language novel learn walk learn encodes variety different task cs edu novel representation vertex representation encode exploit generalize modeling sequence use truncate random treating walk sentence demonstrate multi classification task result baseline global missing compete outperform scalable build trivially broad class anomaly mining artificial pattern network strength sparsity hard application detection prediction must able introduce prove successful develop stream walk capture membership dimension generalize generate neural language semantic syntactic logical b representation exploit generate representation community vertex color input take produce output apply well study force show beyond linearly modularity graph demonstrate real scenario
proceed note many variant momentum em algorithm update repeatedly perform dropout parameterization would z dropout hereafter optimize dropout example differently hereafter wise dropout mask unit layer set dropout test validity optimization problem vector informative dimensional informative obeys informative informative irrelevant variance performance evaluate work variable approximate gaussian q dropout rate optimize dropout optimize dropout describe properly decrease dropout inverse rate dropout dropout rate dropout dropout optimize maximum several study except theoretical state cost propose dropout maximize hidden variable parameter infer infer pd interpret bayesian solve nothing assign achieve well assign suffer one one look discrete value parametrize light improvement standard dropout posterior distribution adjacent parametrize freedom expect discussion parameterization obeys also match consider var number quickly distribution diagonal reduce hyperparameter structure intend parametrization allow resolve sophisticated consider conventional dropout overfitte modify interpretation enable dropout beneficial encourage dropout neural kind task language likely fail huge difficulty one important overfitte researcher study perform understanding explain kind dropout think dropout solve input hide train accordance optimize weighted marginal learning model model dropout benefit likelihood close already paper dropout learn training dropout dropout dropout involve feature selection note dropout machine input unit output respectively activation function sigmoid w optimize selection describe model unit mask diagonal sometimes z mask determination determination correspond problem may rewrite represent explicit architecture redundancy huge challenge optimization good mask binary mask take stochastically mix possible stochastic summarize follow initial mask determine every dropout rate decrease properly denote increment termination output denote mask matrix bernoulli pz pz w pz pz fast apply also weight sum independent upper denote transpose unit treat approximate utilize lyapunov improves show beneficial idea artificial corruption dropout adaptive artificial kind regularization viewpoint regularizer generalize linear equivalent transform fisher input transformation regularizer cost dropout function function bring dependent also typical treat extend dropout train dropout bayesian share dropout depend variable mask input whole try mix work clear optimize discuss treat variable infer dropout hyper mask mask take omit simplicity introduce call trial log kl q kullback trial negativity kullback leibl trial respect maximization gradient descent lead dropout mask sample mask explain assume pp dropout dropout determine dropout also output infer intractable calculation summation variation consider trial last explain precede mean
ed mmd mmd rbf mmd neural net mmd word count tf domain represent bag ignore review split task target label source without target domain unlabeled prediction evaluation hyper average prediction random cross domain adaptation connect hide unit relu mmd penalty gaussian mmd mmd source domain train descent gradually decay rate count mmd get boost tf mmd method feature domain adaptation even word count baseline taylor hidden unit dimension dimension recover auto mmd dimension feature expansion equivalent variant contraction tune mmd hx hide momentum mmd filter tend localize variant b distribution feedforward relu units final sigmoid unit mmd sample edu transfer learn develop factor new domain learn readily salient factor important unbiased definition bias bias discrepancy mmd suggest mmd representation apply across formulation include domain adaptation invariant insensitive autoencoder generative suggest formulation transfer focus deep formulation learn scenario task
dot add nb worse exist nb find method compare statistic accept small performance mean sided rank significantly small level set discover attribute datum switching rule test side significantly nearest way control instance inaccurate unweighted datum attribute discrepancy weight whole sound theoretically improve correctness make eps stroke locally naive classifier li chinese university li department electrical engineering stanford consequence strong violate bayes nb classifier favorable size relax nb local ignore weighted special unweighted weight intuitive handling imbalance learner naive bayes base near show parameter seven keyword weight naive nb computational competitive let class nb violate independence categorical call realization label unlabeled whether comprising instance unlabeled instance test instance nb estimate common laplace modification application nb achieve violate behavior observe data nb multinomial characteristic nb confirm available correct classification nb classifiers assumption set attempt network average dependence use estimator extend replace naive classifier parent add paper modify fit locally weight impact neighborhood remain apply nb call seven appropriate choice conclude respectively weight random least weight reduce assign call avoid vector weight estimate weighted relative probability I frequency weighted classifier weighting instance instance datum make compatible b depend compatibility weighting classifier failure sensitivity condition former estimator hybrid associate approach nb partition axis surface utilize test propose compatible nb call cell weight realization hamming total hamming weight use convention weighting cell cell multinomial nb multinomial compatibility hx mx probability encounter laplace law unweighted laplace unweighted desirable sample weight weighted importance effective size importance scale weight weight multiply I choice constant label number weight multiply make total common training class search dominate unnecessary l cm x size training select calculate multipli class q return look candidate appropriate choice exist nb augmented average weighted dependence v vi locally naive instance local uci repository website description summary training attribute kp letter breast cancer breast tumor diabetes heart rate cross fold miss unsupervised attribute preprocesse percentage correct balance breast diabetes heart heart vs kp letter tumor l degradation analysis comparison large size bias towards impact picture statistic interpretation mean list second average rank large method mean pair bar chart arrange bar chart permutation bar lie performance reject significance nb likely inferior two appropriate value choice conclusion side lie method rank differently rank commonly rank
variance scale ok gradient summation term expectation contribute variance undesirable scale number independent variance assumption generative structure energy weakly correlate discuss nature respectively whereas correspond configuration large variance additionally variance former later variance marginal likelihood completion pixel recognition reconstruct recognition iterate respectively computation imputation write miss transition model constitute see eigen marginal immediate consequence apply fundamental practice complete pixel norm stationary recognition q marginal sense apply obtain true fix variational specify bayes consider use compute gradient variance mini batch energy objective maintain remain use size use variance recognition describe appendix show joint explicitly separate deterministic view work transform match provide view clarity interpret formally time determinant jacobian co transformation visible generative model equation explicitly consist single gaussian layer linearity use st international conference china cp author derivation theorem integrable twice gradient eq identity product integral term evaluate support line differently general exponential base parameter show bx also lead derive rule would simple search section self distribution rescale base distribution propagation
costly proximity require gradient dominant well gradient dual operation proximity step see subgradient could fast proximity stop desire example method necessarily obtain desire formulation provide applicability penalty gradient technique valid hilbert problem useful machine machine svms motivate step reflect singleton step past affine weighting still subdifferential let variant extensive literature k attain reference therein gap estimate order mirror variant algorithm view consequently computation become q algorithm domain imply quadratic continuous lipschitz formally assumption every simple therein norm penalty norm regularization composite composite penalty well remark smoothing involve smoothing envelope define smooth smooth proximity u gx property g proposition control tradeoff hybrid adaptive smoothing lemma smooth smooth view conditional algorithm algorithm besides also algorithms lipschitz objective method primarily nesterov smoothing envelope connect nesterov smoothing unlike nesterov function proximity subgradient single dominant singular feasible large power rate bound convergence objective exponent term regard mostly side similarity fista suppose descent twice obtain yield add theorem finite eq convexity iii ax theorem obtain every add lemma assertion every easily computation depend translate iteration rate convergence flexibility penalty standard involve term become method impractical rate smoothing method proximity problem matrix subgradient subgradient require full decomposition addition build parsimonious fashion solution simplify desirable parsimonious dependence square loss denoise prediction proximity norm soft operator sign multiplication absolute matrix symmetric matrix may u k value k share convex relaxation q sparse prescribed indicator bound subgradient eigenvector input k u rank semidefinite propose semidefinite initialization example fall penalty prescribe positive semidefinite prescribed arise fit favor norm norm total hybrid specialized show subgradient etc large compute latter problem stop match penalty dictionary pursuit gradient absence term similarly extension omp penalty hierarchical yield scalable variant omp jj z j simultaneously aim proximal scale recover simulation random draw entry uniformly denote rank corrupt matrix fraction entry solve simultaneously frobenius ij n constrain equivalent comparison intel gb memory evolution sake comparison apply cpu include singular termination termination algorithm change refer per iteration figure time whereas entry observe simulation efficiency uniformly uniform distribution q nesterov optimize matlab core sufficient tolerance rescale order keep nesterov smoothing computational change optimize optimize verify hence run nesterov smoothing optimistic observe nesterov study composite optimization example problem norm penalty nonsmooth technique proximal order operation exhibit unlike proximal benefit advantage advantage matrix trace penalty relaxation acknowledgment lead receive union fp agreement european fp agreement support grant kp grant project structure decomposition g grant policy fp mc theorem proposition lemma question paris fr study frank wolfe programming much optimization formulation algorithm past field control statistic computational currently gradient problem example involve sparsity use learn inspire chapter method solve convex focus function continuous lipschitz domain available proximity subgradient conjugate particularly cover closed term present review gradient whenever composite conjugate many interest alternative smoothing proximal nesterov smoothing involve approximate besides modification show suitable choice objective claim theoretical recent hybrid interest applicability denoise sparse pca penalty trace rank require subgradient computation whereas require expensive computation vector practical mean thus though exhibit rate gradient backward scale large chapter space endow inner convex take constraint
project validate contain validate comparison pass involved loss appropriately website available website acknowledgement helpful ep via centre training pt detecting region property behaviour possibility present potentially model show enable independent distribution motivate application variation individual subset bayesian give evidence copy variation pass outli segment behaviour time kind behaviour could change correlation work concern proportion number dimension detection segment recurrent segment application relate image take detect copy copy dna show account within cell log r probe probe normal would ratio away give substantial noise genome detect cell pool individual complicated observe datum portion affect series segment individual research method subset dimension less variant segment dimension change able rare whether region recursively statistic point region region able estimate number possible simulate posterior describe partition interval segment normal want tractable location observation follow come likelihood define hide full assumption segment segment likelihood form likelihood normal segment likelihood draw segment segment independence completed find practice numerically need specify segment follow model normally also study present calculate give calculate challenge conjugacy numerically integral calculate numerical value reasonable task computationally develop hide start segment filtering eventually full posterior distribution enable calculate widely markov two b p equation segment kt filter c term computational storage cost store filtering thus calculation prohibitive filtering many negligible remove point without much filtering potentially keep greater remove resampling done store approximately posterior straightforward simulate simulate assume simulate repeat go time simulate segment early posterior hyper use hyper parameter monte em rapid look section fast initially segment method detail posterior easily want estimate loss mistake seek segment overlap segment detect small indicate overlap impose generate affected proportion detect accuracy false positive ex pass pass ex ex especially worth proportion detect ex consider clearly mis specification accuracy positive pass apart detect pass mis specification position segment seven segment intensity value normal varied randomly dimension set mean ccc method proportion false positive ex pass pass ex pass ex ex see still pass positive robust misspecification draw keep mean normal indistinguishable replicate segmentation distribution segment plot simulate either two geometric fit take partly pass potentially segment long occur cdf straight line quantile propose fit generate actual think segment datum suggest reality tail distribution distribution shift take dimension take affect give simulate measured observation segment histogram mean segment simulate varied affected dimension set c ccc affect proportion pass ex proportion normally
random technique decompose ica ar ar ar information identity mutual cross q optimal dedicated solver subproblem thank flexibility sufficient switch solver entropy quantity base one base meta function member cm unify formulate solve theoretical whose template aspect scheme spectral ii computer million ica element extensively test file work alternative source toolbox interested user mathematical available example project european co agreement ac computational centre university college house ar estimator free open platform toolbox mutual association measure distribution modular support additionally combination ii theoretical application prototype central problem subspace association analysis extension modularity matlab platform machine entropy provide quantify measure offer tool define distance central objective party relevant ica cluster ica state scale prove iv observe nonparametric dynamic recent exist package quite specialized fill gap come modular free platform toolbox package estimate kind association kernel offers construct exist optimization problem overview toolbox capable numerous quantity complex generalize kernel generalize schmidt shannon quadratic base copula dependency multivariate version hoeffding distance approximate kullback leibler shannon jensen jensen k pearson divergence bregman extension center
operation maintain library hope benefit community least way practitioner elastic net researcher facilitate performance ss program china education grant u ns nsf grant center grant author suggestion chen chen university st arguably past decade rise demand implementation classification easily voxel genome million meanwhile availability gpu multi parallelization however easy convolutional neural machine svm multi although originally utilize ascent drastically lasso outperform optimize single core implementation imbalance parallelization handwritten truly utilize graphic intensive also software elastic net special design recent bias reduction result vast svms immediately elastic non trivial equivalence hinge equivalence relationship elastic net box square loss world set eight four fast date efficient across almost bold scalar capital bold matrix scalar convention contrast refer remainder briefly elastic net regression real value response normalize sparse minimize elastic net constraint encourage effect strictly convex unique highly correlate stable large squared separate hinge denote please separate hyperplane pass origin solve dual duality directly derivation mi formulation connect inner rescaled ij remain run formulation commonly achieve decision boundary help trick product matrix hardware therefore peak hinge svm simply elastic formulation state constraint substitute rescale entirely follow negative represent negative rewrite non please long l always tight large solution constraint carefully classification p p p classifier denote elastic scaling solution add become equality become constant drop remove affect solution difference design without obtain highlight reduction highly summarize refer mention primal formulation svm complexity choose fast versa dual implementation default trivial remove experiment svm formulation small running depend implementation adapt would allow recent solver might exact practice elastic net many effort parallelization dominate core implementation elastic mostly language strategy know net extremely hard propose run include memory constraint transform solve newton popular implementation library optimization tailor svms hinge modern gpu acceleration update operation tend recent contribution svms extend trivial net soft margin validate gpu line match conduct extensive experiment set brief online gpu core cpu cpu cpu baseline core al solver implement et processors ghz ram core gb different solve path slowly evenly spaced setting path select particular procedure compare implementation setting identical tolerance gpu paragraph original evaluate gpu eight clinical volume weight matching budget gpu eight dataset set gpu markers diagonal cpu gpu baselines elastic net eight concern mass predict scene car area whose financial report tf figure depict cpu gpu correspond comparison gpu path training gpu budget marker correspond run fast marker diagonal gpu observe trend across eight gpu marker transfer parallel even cpu fast baseline baseline gpu markers slope
color ground produce effect relatively popular mit bottom row comparison mapping ground produce collect dataset choose image image enhance participant range little experience static website image enhance enhance et al right enhanced ask vote image enhance enhanced choice enhance receive vote category produce enhance user verify whether capability enhance statistically significant conduct effect section ask join design left enhance image produce ask assign enhance enhanced image look visually ground receive score participant discretize range look ground truth score enhance conduct pair significantly enhance desire effectiveness mapping layer descriptor descriptor contextual descriptor build top parsing conduct include conventional propose able automatic spatially vary adjustment parse object contextual challenge vision recognition propagate contextual affect adjustment show foreground contrast incorrectly increase scene parse rapidly develop failure effect semantic label area incorrect semantic labeling highlight correspondingly area receive incorrect result failure mit group mit learn adjustment ground high distance dnn train adjustment semantic object treat correctly system spatially vary transform exist choice dnn layer activation give rise consume search dnn architecture dnn behave box predict neural grateful discussion suggestion support research general adjustment neural paris yu image email address edu cs microsoft microsoft enable invoke consume advance beyond automate alternative manual face many rely subtle content spatially characteristic exist limited cover subset challenge machine learn unique motivated explore deep context automatic adjustment descriptor account semantic experiment technique semantic yield qualitatively ht enhance deep adjustment region effect digital device social medium popular try exposure invoke visual even traditionally well extensive study color wish novel automatically enhance several reason adjustment empirical relate color enhance image need process quantitative relationship nonlinear especially style vary nontrivial capable relationship accurately learn computable scale semantic pixel meaningful human type improve appearance likely appearance region would semantic challenge semantic image specific content automatic accumulate speech semantic motivate explore context cast adjustment neural highly spatially enhanced dnn arbitrarily complex continuous scale design key issue dnn sure learn color color design informative yet descriptor pixel feature semantic contextual global descriptor whereas descriptor understand image semantic advance object detection pixel semantic incorporate novel context descriptor automatic adjustment achieve superior framework yet discriminative image contextual descriptor exploit multiscale improved region effective choose subset test deep use standard learn possible descriptor demonstrate effectiveness comprehensive descriptor system traditional correction adjustment google auto microsoft office automatic automatic operate manner content consideration automatic first include salient determined well exposure code achieve style shall inherently achieve effect especially practice technique outside global semantic category handle deep provide universal style receive much assessment actually adjustment al process mapping statistic method image adjustment context spatially local wang approximation semantic contextual automatic interactive soft infeasible automatically enhance collection propose scalable search within combination work neighbor challenge create slow thereby mid level color sift high semantic difference impact adjustment cast regression dnn represent pair image mapping function color image enhance color pixel complex color parametric feature color training function regression connection blue backpropagation start line force frequency noisy relatively tackle color color basis color space use transform l I ia I ib I ib vc vary frequency pixel much minimization pixel dnn multiple architecture sake network prove able acyclic neuron hide input input output layer map color transform precede neuron relu activation function tangent relu induce unit neuron layer neuron output linear input precede neuron output color span neural architecture architecture classic useful capability contribute backpropagation typically actually enhance pixel color transform dnn smooth descriptor pixel serve feature represent contextual entire detail reflect high resolution vary represent neighborhood position within practice attribute intensity enhance image feature representation six feature give contextual car result region pixel semantic scene parse scene train highly parse good labeling category shape texture detector category characterize appearance foreground two semantic parse fusion parse semantic feature descriptor annotation scene parse algorithm set scene parse road parse obtain parse map pixel receive label indicate cover category state cover predefine type person predefine object high fuse map pixel predefine threshold since vote image merge frequently segment image segmentation map map final contextual feature descriptor pixel multiscale point nest I sensitive nearby consecutive eight semantic histogram sum histogram nine small contextual descriptor concatenation histogram multiscale descriptor inspire shape context unlike shape descriptor either facilitate calculate descriptor contextual complex spatially adjustment contextual simple region large contextual able local adjustment ground adjust color transform three pixel potential even specific local million largely risk finish neural medium hundred nevertheless train enhance neural stage first scene parse object segmentation obtain extract centroid every apply color transform within adjusted compute cover enhance foreground effect enhance effect enhance pixel remain testing image wide operation adjust include select object area variation tool contrast foreground salient background foreground salient region selection production effect foreground object visually make less three enhanced refer material training enhance generalize popular tailor category scene parse object profile series color channel adjustment name tool region apply image region within avoid boundary adjust color profile profile style profile additional minor heavily show effect enhance effect ask try style also apply foreground foreground foreground create layer small foreground background together mode complex color force network enhance testing example transform pixel calculate look visually produce enhance rigorously simulate visualization color enhance successfully effect important make practice find helpful involve always define style category scene parse consequently semantic action tool semantic transform apply color truly spatially vary verify collect pixel region draw scatter pixel image able visualize vary transform clearly transform differ build road region successfully spatially complex conduct approximation due contextual adjustment parameter input enhance enhance enhanced enhance close image map local plot semantic region scatter plot plot horizontal coordinate vertical capability base adjustment mention early far number use thousand pair show image significant image top appearance bottom configuration car people different despite dnn adjust input example effect enhance visual demonstrate contextual subsection calculate color image ground truth reflect magnitude image result contextual third contextual test enhanced mean foreground include contextual feature error indicate necessity input enhance enhanced simple contextual pooling enhance contextual obvious enhanced enhance contextual feature cm truth te foreground effectiveness multiscale spatial pooling design simpler intuitive contextual descriptor pooling region contextual helpful reduce take drop multiscale feature multiscale region rotation invariance histogram visual contextual region enhance might severe color transform help frequency variation dnn spatially nonlinear part highlight benefit use color transform train different dnn color directly dnn similar enhance increase indicate beneficial cm w transform foreground dnn primarily layer inherent complexity dnn sufficient able learn exceed novel datum training show feedforward single regressor layer vary inherent easy small training error deep neuron hide layer dnn keep hold validation vary train repeat experiment report inferior deep exceed layer execute
exploit besides persistent localization variety desirable e due runtime streaming time requirement task arm robot weather online gp constant field plan algorithm richer high like camera localization gp exploit gps krige lastly gp method employ improve scalability full gp marginal equation gp online likelihood compute maximize online detail future mit r localization simultaneous localization determine environment past measurement map representation know field evenly grid grid cover access environment become map know problem localization firstly sense representation information robot reduce summarize environmental grid need maintain secondly environmental field since point exploit assume give localization difficult costly information robot need maintain national mit technology edu sg mit central robot exploration persistent environmental characterize paper exploit field measurement take robot gp observation online capable achieve feasibility persistent robot localization dataset robot focus develop modeling spatially environmental characterized measurement monitor phenomena concentration span light concentration field affect towards environmental different overlap wireless field environment operate assumption widely gps device gps robot exploration environment usually probabilistic update robot state measurement take robot preserve impose location conditionally violate environmental strongly issue integrate rich filter model field probabilistic gp uncertainty gp persistent incur cubic computational availability exploration localization assume location current past measurement gps datum step localization train relaxed hold easily violate environmental consumption permit relative large environmental spatially distant making train gp uninformative motivate fundamental training localization environmental spatially contrast work spatially take robot rely gp capable believe towards demonstrate feasibility employ gps persistent robot localization empirically gp outperform localization gp field environmental realization environmental realize unobserved latter define common choice exponential control intensity noise measurement diagonal controlling similarity horizontal field kronecker delta environmental capability perform robot visit gp predict unobserved location correspond uncertainty predictive pz mean component full filter persistent robot localization poor require invert incur improve scalability gp measurement matrix yield redundant turn rank exploit set utilize predictive measurement variance sn reduce either invert approximated matrix incur matrix employ incur grow increase size impractical repeatedly train localization conditioning field take perform location visit realize field measurement take denote tx robot maintain possible denote column measurement time step track robot realize prior robot posterior bx pz bx px robot moving location describe likelihood preserve efficiency bayes filter impose markov robot action past action exploit learn work model parametric offline former extensively discuss latter already say realize field pz pz ease discuss impose restrictive observation robot normalize constant robot location location visit robot pz z gaussian provide derivation computation constrain motion integration denote location sampling motion past action ensure entire path motion accounting constraint spend ignore constraint theorem exploit considerable poor scalability derive incur persistent impractical incur per gp gp newly slice summary slice span size slice far slice n measurement take robot slice slice tuple manner sx localization offline c c n definition memory keep require independent summary regular step online sparse pz n c theorem equivalent formulation offline incur offline compute incur slice incur time per equivalence structural offline measurement conditionally time slice frequent large slice summary improve slice summary motion motion time step draw belief maintain motion interval particle belief update consequently motion often cause sample locate occur independent offline generalize online gp generalize point gp online variant offline slice summary slice e update robot little predict current summary resolve incremental thereby pz transpose computing pz incur theorem online incur constant localization simulate produce access throughout intel berkeley research km h road network speed km km mobile weather fm office localization road road speed direction field model relational previously whose exploit road hyperparameter gp field gp interested refer technical filter particle represent belief mobile take gp train use action mobile move another learn use along generate localization performance robot location scalability step method possibly unobserve location localization exploit evenly compare localization employ access table error run exploit field well perform poorly area capability produce inaccurate gp achieve small error explore densely relatively make localization localization robot explore area highly
uninformative block uniformly moreover identify estimation enhance close negligible interpretation count improve reweighte outline sequence form identically cf data bias sequencing specie introduce spurious functional part sequence sense protein family partially weight pass value elimination sequence score contact ranking pair position strength mention interact one direct di mutual support information follow frobenius interaction column gap symbol cf reach di achieve good clear interpretation indicator compare gaussian di result gain field achieve l di direct omit moreover matrix position represent characterize interaction interaction precision positive definite marginalization exploit formula block inversion matrix kullback algebra recall help tell invertible factor l l l equation identity notice constitute latter observe identity equivalent substitution denote necessarily scheme introduce computation convenience pre similarity weight average average pair constant refinement threshold neighborhood sequence identical carry protein count least factor mean use estimate eqs cm carlo science sciences human foundation universit es universit et paris biology paris france centre national biology paris france mail author protein remarkable constraint aim extract constraint rapidly datum thereby infer protein recently global direct coupling towards prediction successfully prediction however due nature variable require protein efficient need propose multivariate gaussian modeling coupling problem superior mean field coupling signal implementation website biology sequence complex property protein empirically evolution distant along contact work review recent lead precision sequence alone evolutionary analysis find valuable insight specificity protein interaction progress algorithm principle naturally lead protein initially recently evolution behind global occur two multiple alignment couple show couple aim indirect evolutionary empirically inference challenge biology infer protein homology model single protein highlight predict auto functional protein well subsequently confirm experimental ray guide protein prediction resolution structure molecular protein biology cite signal system constitute way sensor protein signal rr rr typically act external pathway pathway avoid evolutionary pressure evolutionary interesting infer reflect physical come interact protein rr obvious protein evolutionary analysis understand allow sampling simplification allow empirically correlation approach share mean field simple allow comparable aforementioned result briefly material section fast implementation model gaussian sequence briefly come highlight support input protein align form contain alignment gap multivariate bias probable produce observe turn provide infeasible sequence space approach constraint number bayesian introduce convenient prior normal wishart conjugate prior choice posterior parametrization result analytically uninformative interestingly pseudo correction term amount inversion one strength direct predict protein candidate interaction gaussian contact rely matrix rank numerically identical field test di introduce norm di mutual information direct expression support prediction hand therefore contact context score yield however di invariant physical therefore assess contact aim original publication protein recently evolution intra development efficient wide availability like co protein whereas pass limited require hoc protein ten efficient explicitly computation relevant coupling di likelihood analytical formulae analytical major advantage run include algorithm analyze pf sequences di pf intel core cpu computation pair model blue gaussian describe count field information pseudo correct mutual arithmetic thin deviation gain aim first prediction intra fig database family select allow statistical average average sequence cf determine di di average correct mi computed approximation model pair pair apart pair evaluate proximity protein structure cutoff heavy overall note well mean di underlie turn see surprisingly also overall pseudo count explore optimum di code except check insufficient positive mean family list thin deviation cccc pf pf pf pf parallel matlab r version code test run data score show marginally infer accuracy slightly predict positive size fast order magnitude candidate protein structure produce software panel predict contact map obtain predict green last grey positive contact occur predict visual inspection reveal bias protein first pf protein use rr curve rr apply contact prediction report positive show use di substantially true prediction specificity highly little improve gaussian interaction protein use external complex signal mechanism strongly mechanism rr protein pf rr pf closely relate interaction pathway correct recognize rr belonging pathway co localize correct call rr protein isolate genome signal rr major system identify act co evolutionary pass cc bs experimentally interaction cc correctly obtain co evolutionary scoring signal bs interact visible evolutionary experimentally red green dot overall method cf equal log odd infer rr interact infer independently family set rank fig mention present clear cc neither able interaction prediction interaction kind b great protein display red maximally rr evolutionary whereby cast interact interact protein formalism parameter major advantage distribution analytical likelihood posterior efficient demonstrate test gaussian comparable superior field interaction comparable furth example di could kind use suitably design informative relevant enhance prediction prior notably advance interaction mapping interact interact inter protein material multiple domain protein database successively sequence generate family statistic family experimentally assess quality purely length family pf profile list structure l l c page page benchmark together discard gap large choose position alignment gap remove processing improve identification directly summary detail datum hmms domain pf rr rr find code gene sequence rr protein family row align
mostly english word vocabulary character substantial lm lm lm character correct occur lm lm demonstrate train attain without rely lattice good list speech dnn acoustic whether recurrence directional recurrence essential evaluate recurrent compare roughly architecture bt parameter dnn recurrent substantial recurrent report dnn parameter worse fitting total free conversely bi directional recurrence single recurrence recurrent speech recognition present language train decoding remove hmm system find pass decoding demonstrate capability space lattice result recognition system create multiplication dominate suggest recurrent bi recurrence help recurrence together quality hmm science stanford university stanford stanford university stanford ng department university stanford vocabulary acoustic hmm system recent feasibility discard hmm sequence modeling extend way neural level modify pass speech journal corpus competitive error bi directional recurrence vocabulary continuous speech modify model sub hmms carefully design complex difficulty modify isolated advance demonstrate speech use predict audio approach modern favor treat speech recognition train network sum able predict character character journal yet exist system speech often heavily upon decode lattice hypothesis list introduce factor well additionally system final language decode train neural space pass decode enable rely exist word lattice remove speech enable network act acoustic model acoustic dnn place hmm sequence dependency reason handle temporal dependency lstm lstm network architecture originally design prevent vanish gradient recurrent neural architecture amenable graphic unit gpu hmm system without vanish train letter sequence acoustic single maximize full exposition function character fix define character audio feature function basic dnn dnn hide activation matrix wise nonlinearity choose layer character output subgradient parameter dnn utilize integrate temporal extent extend form representation propagate backward backward via recurrent forward backward entirely nonlinearity obtain representation recurrent aside change recurrent layer length network character alphabet audio character character ts map language propose capable incorporate acoustic input seek language language attempt find maximize bt string character alphabet character extend character incorporate language constraint otherwise extend incorporate active include probable probable list nb ta b nb pc nb tp nb pc nb p nb nb maintain first audio maintain time never probability product word probable sort give variable character sequence character character language include propose character word act constraint character string consist hour news available hour transform corpora subset system instead drop
generalization many domain include novel closed scoring examine exponential scoring construction restrict maximum give take rule q pareto score agent belief th lead follow score gamma factorial measure real number interested gaussian density rule stress score construction generalize rule depend weakly second moment density expectation instead rule write agent scoring agent belief market belief seminal market scoring appeal thin thick market section adapt market close function agent belief claim approach statistic interpret payoff function security security portfolio share hold share occur inner example outcome security outcome occur statistic security payoff amount contract literature centralize market maker maker maintain convex collect vector share portfolio give neutral payoff equal portfolio move market vector way choice reveal risk neutral incorporate market information budget examine relaxation later remainder focus arise market course exactly family sufficient statistic recover indicator statistic portfolio share give economic interpretation correspondence partition interpretation market maker family state parametrize share belief score agent report expect report portfolio share reasoning rely assumption share c c recalling leibl family bregman divergence kl rule market entity share consist security q share security stay market maker enforce property share see moment volatility payoff log natural parametrization effective possible share exceed however arbitrary amount increase market dimensional sphere lead alternative dimension security outcome entropy statistic von refer function quantity positive parametrization von rule expect component several price perform aggregation final price expectation aggregation agent take account form belief require belief agent exponential suited reasoning px tt sufficient direct belief maintain conjugate family map partition think prior size observe empirical random agent respect base px dx x dx line recall log market exponential utility belief natural market trade move agent trade maximize equivalently strictly argument private grow agent make reduce exposure stay market weighted market adjustment centralize maker allow capture price adjust parametrize cost c price mean share need reach price transformation know perspective adjust mean risk neutral mean q high share must market price scale follow result respectively theorem accord agent market state rather share update directly right section analyze pose future exist portfolio belief incorporate portfolio reason utility parametrize share first market market give move vector market maximize market subsequently market behave utility maximize belief exposure market financial exposure understand utility pick market state compute draw result game regard strategy family market equilibrium utility utility I final equilibrium convex market belief weight consider market budget multiple market vector measure standard log amount market experience eventually unconstraine together market suffer ill inform also make informative run budget suppose budget share move state want budget market final move market budget maker result informed entity instances market thus exposure market maker loss share px market uninformative budget round round move tx x impact round thus incremental change budget round market evolve never fall round interesting loss quantify characterize budget informative market parametrize move market limited belief differ market function equal expect net bregman divergence base expect belief need belief belief formation growth positive sequence sample belief state accordance belief budget limited notice move utility utility case market exponential utility adjust payoff least dual supremum mean exponential backward mean parameter px px nice dual negative entropy multinomial market maker supremum natural mapping px result nice essentially distribution maker thus exponential bound market maker special case alternative market scoring market learn connect payoff associate point expect likewise entropy equality dx dx equation market scoring kl security quantity particular cost initially imply market price distribution incur give expect kl outcome outcome dx family exponential dx unbounded entropy unbounde derive conjugate primal definition supremum may rewrite q achieve market outcome sufficient loss market maker fx follow expression distribution px dx recall dual negative log term mean market maker maker sign observation design proper scoring scoring case simply different implication scoring rule agent agent expect fundamental statistic variety suppose rule precede space rule provide broad pareto density one mapping alternatively parametrize mean give follow proper scoring pareto stress rule know pareto density e exponential outcome parametrize score median eq q median expectation highlight circumstance subsequent market state give movement px x x follow rearrange share current ct c expectation exponential parametrize thus choose share dx utility maximize exposure behave identically utility maximize exposure market exposure change belief achieve argument belief initial function market aggregate prediction informative want informative receive draw belief able initial market prohibitive market growth budget eventually move restriction idea dual use recommender section belief cost payoff impose market requirement budget market restrict movement payoff market budget cost movement budget market maker maker directly limit allow would market represent belief enough share market market budget state market c move market move optimal trade rational move case move close maker might entity multiple exposure market maker several use loss initial budget define share hold eq prediction share security market make report assume outcome reveal receive market track budget tx eq q incremental budget active capture incremental market evolve fall cause expensive market budget increase expectation prediction input move result net theoretic interpretation informative market increase budget round belief informative result market market informative belief parametrize budget limited expectation belief whenever budget differ theorem belief payoff payoff last market exactly belief positive sequence belief eventually every accordance without limited require combination maximize utility market parameter theorem trade respect optimality update formation growth extend true introduce bind prediction set share payoff first note px tx move market write px aa utility parameter payoff conjecture thm thm example thm market mechanism template market price agent analysis market price equilibrium assumption exhibit aspect budget constrain behavioral artificial market aggregation mechanism market assume private belief payoff market probability sequentially private sense market aggregate consensus forecast prediction focus heavily mechanism compatibility market fluctuation name literature correspond price interpret equilibrium market underlie aggregation price bayesian incorporation posterior classical tool market statistical attractive interpretation relate via family market price
theorem solve consequently least confidence strictly satisfy prove first statement generate number theorem probability let get statement remark numerical practical difficulty require define approach issue convert define augment large potentially exploit solver solver natural angle major low minor minor high residual angle low major alternatively deduce remain pick remain orthonormal na k component half make know poor potentially dimensionality reduction versa combinatorial combinatorial effort introduce slack polytope dimension row argument bound consequently dimension general appear satisfy inequality inner fix combination preserve inequality minor number iteration path different matrix result problem generate first table residual angle however map residual monotonically dimension residual decrease furthermore angle interestingly number minor high major iteration quick notice uniqueness solution representative report table representative carry section solver solve solver directly point compare minor total report obtain final point major additional minor report solve high initial point notice minor solve minor high problem recover solver compute motivate big reduction feasible approximate form appropriately project solution solver thereby exact numerical author dr research systems engineering technology affine compact feasible randomized produce dimensional choose quality subroutine generating solution lower appropriately choose solve solver original recover validate substantial time collection datum change interact store report accumulation concern challenge large computationally polynomial whereas context rather quick approximation spirit essence accuracy large focus generalize convex optimization saddle nash nash game amongst affine region include quadratic feasible quadratic subproblem affine competition game dimensionality high affine substantial high derive solver low error satisfie get project made unity choose inner product translate high deterministic high norm quadratic program remarkable recover approximate require deterministic exact generate solver plausible run nature emphasize assume plausible improve theoretical computation appear try considerable ambient may conceptually speak work exploit convex embedding metric result obvious convex analytic euclidean embed inner optimality preserve follow introduction present background formally introduce concept require proof discuss low conclude f problem solve amount ensure range solution vi nash equilibrium game simultaneous vi equilibrium capture besides contact continue equivalent k quantify rare solve solver work solve variational inequality structure reference therein direction splitting decompose large small subproblem deterministic work probabilistic introduction subroutine science community research control two aware first zero differ rank vi solve nonlinear whereas linearity formally interested seek expensive involve low exactly proceed projection embed deep mathematical study limit operational involve lie dimensional need type randomness introduce construct study uniformly realize orthonormal call manifold distribution schmidt surely formalize random form normalizing column uniformly zero subspace probability induction follow produce r manifold invoke lemma value standard qr uniformly rank column paper automatically multiply expect shall ahead multiply depend upon random arbitrary dependent projection call think vector distance improvement show project multiplication pairwise preserve original distance preserve expense concern preserve differ exact fix argue random lying sphere fix projection coordinate deterministic depend project uniformly random operator easily convert apply union bind finally project norm phenomenon product preserve mapping vector try norm pa pa p since fu fu fu fu fu result result finitely simultaneously random section mapping fu union require random projection high inequality feasible region vector generate solve approximately ax kk dimensional polytope norm linear vector final main compact low claim b k sufficiently give sense inner range least computationally run invert hence deterministic solution second produce correctness part establish show showing solution prove approximate proceed analysis lower low x vi polytope n generate combination
efficiency randomize fourier accelerate large approximate arise approximation integral shift invariant kernel approximation evaluate discrepancy measure discrepancy adapt explicit minimization empirical offer mathematically well model wide machine learning series modeling testing rather general non associate embed feature model provide input representation construction dimensional however elegant generalization assume complexity solver requirement kernel nonlinear counterpart however require involve gram none particularly conclusion apply algorithm rather appeal setting adapt strong hypothesis empirically potential large data domain necessary method intensive improve kernel recent randomize approximate distortion approximation kernel complex inner though real technical exposition simplify adopt scalable solution back technique successful accuracy characterize definite function positive definite point dm borel measure henceforth scale shift put one notable member shift invariant kernel integral subscript denote goal quality work need observation density approximation approximation kernel mc discrepancy quasi overview theoretical provide clearly demonstrate superiority analysis potential subscript rely use scalar k nz see mean randomly typically unit cube definition rkhs reproduce possess reproduce f reproduce equivalently space functional informally hilbert space nice I word control pointwise scalable scalability identify dominant construct either randomize map kernel classical nystr om nystr om deterministic seminal harmonic map invariant considerable effort extend technique feature deterministic random wide class group invariant suggest random intersection generalize kernel gd gd suggest product classical result suggest feature kernel show subspace observation feature laplace feature work effort fast original construction devise random random fast convergence amount apply distortion make map rather try scalable scale method year start early day optimization gaus specific expansion optimize svm objective draw scale reformulate broad restrict subsequent section excellent computing follow integral computed respect central monte carlo carlo converge rate discrepancy illustrate dimensional graph scatter random see empty little region empty space lack uniformity fact design correlate avoid phenomenon fast integral theoretical designing form variation integrate dependent measure uniformity remarkable classical sequence variation star discrepancy measure actual volume discrepancy construction discrepancy notable example decomposition construction detail however mention notable sequence also mention star discrepancy decay base convergence rate improvement past integration however notice integration literature leverage discrepancy classical measure variation direction provide reproduce behave term detail follow set kernel g cauchy applicable integral cube integral form generate discrepancy drawing sequence convert integral unit cube cdf low yield procedure summarize map analyze tb made develop approximation show q unbounded integral unbounde new characterize integral throughout convention interest behavior reproduce space relate interested rkh derive integration particular integration bound proposition rkhs rkhs follow vector admit integral associate support transform fundamental shannon sampling product space wiener constitute wiener admit inner rkhs kernel notational discrepancy suppose write univariate function box star discrepancy box discrepancy notational wiener state integration unfortunately member directly discrepancy measure integrate function uniform spirit similar q explicit multivariate important equal discrepancy minimize decaying go hence pairwise separate distribution unit cube drive cube tradeoff compete expect decay behave formula follow density zero subscript analysis density characterization measure typically sequence discrepancy behave box discrepancy needed leave future unlike box function formula candidate low specialized via optimization proposition discrepancy namely global greedy pose discrepancy q rank lattice rule integral generate local box discrepancy non greedy recently present notation series minimize p ps equivalent restriction unclear restriction restriction hold compact case since report sequence learn discrepancy examine behavior discrepancy fouri digital implementation www com lattice digital net publicly people generator low recommend literature net introduce randomization generation compare sequence around net long long second report essentially work gaussian fold favor monte fundamental exact examine randomly example dimension subsample exact reason lattice cc trial clearly classical sequence gram lattice yield sequence may mc whether sequence yield high sequence quality gram use build ridge mc summarize cccc lattice mc cpu census truth execute deviation list behave lowest significant almost follow digital mc sequences model regression worth analysis connection nystr om gram kernel examine normalize box box base range inside far box value scope rule discrepancy bound yield plot discrepancy box error part box concentrate predictive see quality discrepancy cc sequence box discrepancy provide proof concept sequence describe demonstrate sequence produce well gram matrix running sequence less experimental flexibility adjust one long shorter force bounding box scaling feature turn applicability variety generate learn sequence number dominant term scale learn name global adaptive set optimization initial use gradient examine unit use mc see mc concentrate near significant expand integration give sequence control box adaptive sequence make ccc examine adaptive metric gram plot examine behavior various plot various square squared norm error evolve perform dataset evolve iteration examine optimize box discrepancy initially go box box continue go plausible explanation box entire increase box actually concentrate handle box concentrate try improve reasonable box hard subsequently discrepancy translate monotonic metric gram however
essence measure utilize hypergraph underlie relationship propose hypergraph partitioning characterize intra inter cluster separability maximize vertice maximization pairwise nn hypergraph algorithm datum hypergraph show pairwise two near neighbor display hypergraph structure highlight color consist left hypergraph incidence hypergraph hypergraph similarity three hypergraph partition affinity hypergraph hypergraph pairwise hypergraph reflect relationship vertex nn hypergraph neighbor cluster among type construction mechanism explore relationship easy exposition denote mathematically graph correspond return affinity measure word easy cluster similarity correspond hypergraph hypergraph equal traditional hypergraph compose many mathematically hypergraph incidence ne order measure belong hypergraph pairwise affinity hypergraph represent diagonal hypergraph vertex hypergraph near nn define indicator hypergraph assign th e similarity nn compose centroid vertex near vertex similarity hypergraph n n capture hypergraph characterize n nn similarity nn hypergraph similarity interpret cosine similarity feature essentially explore nn hypergraph nn hypergraph construction hypergraph sample right display highlight encode vertex often vertex mutually effectively discover community propose hypergraph vertex vertex fig case belong mutually influence context loss vertex community convenience community order incidence indicator vertex similarity relationship index similarity hypergraph formulate cross within vertex reflect affinity relationship vertex th therefore hypergraph view cosine vector similarity obtain group information order hypergraph matrix diagonal similarity matrix hypergraph draw hypergraph hypergraph close truth hypergraph combine type aware hypergraph encode local hypergraph show bottom part hypergraph capture manifold datum sample accurate therefore aware hypergraph keep easy emphasize hypergraph associate hypergraph hypergraph context aware hypergraph partition hypergraph partition hypergraph hypergraph partitioning aim disjoint eq diagonal th diagonal element point typically relaxed trace trace maximization decomposition capture optimal hypergraph partitioning eight pairwise hypergraph list experiment dataset configuration c face dataset comprise person near face convenience face image dataset service handwritten digit subset handwritten digit constitute digit ten dataset trajectory shape dataset uci repository contain face digit mnist corresponding descriptor computer trajectory fourier dft uci repository j nn hypergraph exist cluster spectral cluster vertex cluster refer issue iv sec ground cluster configuration pairwise kernel refer accordingly nn hypergraph construction cosine similarly cluster hypergraph cosine partitioning eigenvalue newton complexity partitioning therefore spatial method lie hypergraph incidence hypergraph incidence compare recently convenience classic tune spectral noise robust spectral hypergraph nn hypergraph spectral actually special different evaluate context hypergraph together normalize demonstrate aware hypergraph make quantitative similarity hypergraph datum cluster nc nc quantitative introduce evaluation criterion configuration obtain cluster cardinality intersection cardinality dataset large truth th cluster clustering evaluate original comparison perturbation effectiveness cluster report accuracy eight accuracy regard seven dataset nc report weight cluster performance sensitive configuration weight perturbation additive fig bar choose high accuracy noise accuracy gain nc respectively element display performance regard outli corruption level corruption ratio consistently show accuracy corruption average gain nc hypergraph corruption weight mnist corruption configuration different configuration nearest nn hypergraph fig number community cluster hypergraph clearly see three type hypergraph capable intrinsic optimize hypergraph partitioning criterion intra inter separability robustness aware hypergraph similarity type nearest hypergraph high hypergraph capture neighborhood grouping vertex capable explore intrinsic vertex robust capture intra inter separability discriminative hypergraph maximization spectral develop cluster experimental corruption effectiveness adaptively combine weighting mechanism hypergraph mnist fig author li zhang laboratory institute chinese sciences china mail ia ac cn li school science york usa powerful tool analysis aware hypergraph type hypergraph hypergraph neighbor hypergraph hypergraph pairwise hypergraph hypergraph capture neighborhood hypergraph encode dataset affinity intrinsic dataset intra discriminative hypergraph spectral theoretical experimental propose hypergraph graph measure play important unsupervised range circuit load balance image motion segmentation video affinity aim cluster local still issue traditional discover number graph construction noise enhance issue design local scaling explore intrinsic laplacian discover mechanism cluster resolve propose via eigenvalue gap vertex intra separability sample video densely therefore map topological information cluster cluster around origin may separability hypergraph address issue iv hypergraph analysis affinity among construct hypergraph video
domain review large heuristic technique multi particularly bag label positive hx lx lx equal indicator belong bag bag distribution hand arbitrary bag statistically hard pac pac learnable side approach model bag manifold generalization set combine paper regression iii determine true realization bag bag I might bag specific bag several guarantee problem iv vi feature extraction bayesian adapt bag come probably hausdorff metric compact finitely exist hausdorff sensitive applicability hausdorff design minimal hausdorff instance hausdorff contextual hausdorff instance unfortunately lack task might consider however highly standard negative notation section formally define notation paper cm integer hilbert closure complement intersection direct set denote logical topology borel measurable product algebra weak topology b functional operator shorthand hilbert call schmidt schmidt know iv compact cl trace schmidt operator identity rkh embed reproduce canonical map average separable norm reproduce kernel let conditional k belong embed adjoint operator value kernel property hold cx xx requirement assumption see task endow topology algebra measurable I lx learn involve wherein sampling goal notational classical regression rkh random construct composition word map embed measurable regression empirical determine lk remark cm cm tackle stage difficult enable stage scalar equation bit generally analyse risk scenario specify establish f detail fit problem ridge focus stage present specify derivation illustrative example serve compare cm theoretically justified regression avoid specific alternative experiment concern section toolbox technique latter goal entropy whose uniformly construct rotation validation e select goal learn marginal display test typical estimate square confirm figure need density estimation sample pose optical prediction I image correspond consider radius sensor use bag bag bag baseline em achieve accuracy experimental protocol testing repeat hard validation set first linear pick ensemble polynomial quadratic mat ern smoothness parameter summarize obtain l nonlinear summarize one drop prediction decrease far set despite precise choice however poorly output separable topological endow study analytical allow parallelization parallelization difficulty well specify case assume model class case old kernel family expand table focus quadratic loss kernel distribution regression I relax plan question whether present focus section somewhat demand certain present detail concern excess specify proof concern excess eq without modification provide empirical xt make kk consequently bernstein q uv kf g cm ig boundedness c bernstein inequality tt bernstein see kf adjoint kf k cm kf cauchy schwarz analytical eq hence apply arbitrary ii triangle arrive bound let boundedness old hilbert schmidt self adjoint cm cm k adjoint operator countable f cl j exploit identity k bound supplementary technical detail derive k al ac uk college house department state pa edu school pa value fit point analytical hyperparameter entropy quite observable rely good guarantee distribution estimation step often perform poorly study analytically hilbert regressor output scheme stage setup mild condition answer old classical kernel kernel include stage ridge embed instance address bag distribution output learn statistic way case bag analytical expression entropy hyperparameter intuitive suppose meta distribution I I l health patient infer observation test health indicator hope observe test large mapping solve consistently work mapping analytical depend candidate analysis regression belong bound goodness high use hard excess convergence vast learn address response consider case nonparametric regressor act reproduce kernel appear throughout variance regressor consistency construct regressor rate regressor old propose handle scale bag set bag distribution kernel call multi kernel kernel average point similarity little learn introduction bag bag allow increase however distribution valid kernel mean hilbert characteristic use embedding reason embedding operation paper consistency mean distribution regression basic relate herein break arise difficulty cm specify case bound excess prove parameter triplet obtain large particular large process obtain occur rigorously intuitively particularly process easy regression rate guarantee open topological domain endow p compact dimensional string separable value suffer work estimation step proving sample consider use focus short upper verification care embedding kernel rkh alone take invoke fundamental challenge cope combine rkh technique construct operator associate distribution section convergence specify section although illustrative numerical give idea supplementary
bc conduct well bold nmf bc nmf vs domain work conduct preference user click comment system reasonable scalability recommendation propose transfer pattern pool together rating domain rating pattern co cluster capture propose novel probabilistic recommendation art recommender recommendation belong movie historical item preference record recommender system suffer case item even large rating matrix prediction content number user domain treat item acquire domain refer domain recommendation show domain recommendation learn study rating pattern codebook cluster codebook codebook call combine codebook expansion membership matrix existence codebook multiple domain share common rating diversity common rate improve strength cross recommendation learn simultaneously enhance across flexibility sharing capture specific rating prior item show domain offer evidence suppose rating item multiple isolated rating datum rating matrix z r z r cross domain collaborative predict miss rating domain knowledge across domain world scenario item simultaneously example movie music classify movie science describe music region movie affect result movie domain rating rating specific sharing cluster item clearly co user belong cluster user belong exact membership previous represent feature z domain item describe specific item cluster user co rating rate common cluster rating rating element expectation user cluster rating define function rating cross domain specific rating cluster rating recommendation combine cross domain balance weight cross define rating prediction movie result introduce pool rating expectation ease loss k kl tu v equation compute pool rating rating pc pc l simplicity pc update eq em name adopt get model rating also compute accord rating rating q predict domain examine rating domain recommendation recommendation nmf matrix single domain nonnegative method factor model single single separately rating matrix transfer rating pattern multiple experiment evaluation dataset movie rating scale rating movie experiment million user movie user rating normalize scale contain scale book rating scale comparison examine respectively rating start user keep ml vs vs discover different domain conduct repeat mae mae predict mae use good k pc c manually report setting observe show mae vs domain performance clearly show cross recommendation
derive forest tree tree hierarchical convert randomly height take forest kernel algorithm interpret collaborative filter movie recommendation cluster movie two view cluster splitting similarly word understand kernel select forest sampling forest train leaf explanation binary necessary tree height give random toy stationarity distance hyper normal phase gps svms replace forest computationally unsupervise nothing inherently train overfitte supervise dimensionality plot fast pca cluster kernel fast partition cluster center assign largely result piece wise note fast easily training tb compare forest radial basis without automatic relevance detection uci repository train evaluation likelihood posterior learn variance covariance kernel cluster outperform kernel dataset graph scale discrepancy often simply mse forest improvement improvement standard prediction kernel kernel value low random forest perform nearly well whereas might convergence wise matrix fast square exp execute entirely fast partially execute intel ram log gradient predict scale fast processing point dataset theoretical comment future scaling worth fast random forest algorithm trivially present connection kernel intuitively trivially algorithm forest cluster kernel show excellent regression dataset gps rgb rgb section demonstrate kernel construction forest fast show consistently outperform kernel problem inference vector often cite success world algorithms kernel never single task hope kernel intuition commonly kernel generally derive despite practitioner periodic radial spline rbf far away almost kernel argument datum towards smoothness may leave operator automatically search structure difficult general intuitive originally design find partition partition trivially kernel real simple svms regression semi psd indicate machine choice view implicit issue element gram operation find solver solver gram store offer great quadratic form common use free solution svms generate take partition representation allow partition eq partition induce two cluster constitute eq psd psd psd psd far since psd psd valid however partition possible evaluate fortunately approximate partition bernoulli kernel definition use crp ibp party affinity naturally naturally evolve people join good state class efficiently store space operation use analytic require
remove choose probability crucially equivalent select cl la w nice markov event low constant denote intersect intersection hold contain form e np n substituting solving complete proof g c suffice chernoff upper q intersect complete shall together bound primal constant geometric e addition universe keep exposition self discrepancy shall sum set original subset axis plane half etc behave primal dimension system set suppose plane follow dimension belong discrepancy discrepancy match size less discrepancy e throughout apply vertex get notion discrepancy reason sensitive discrepancy theorem set rest partition colored discrepancy continue process constant depend use inductive recursive bind eq try eq triangle constant imply proposition packing additional answer question another sensitive bound system imply size begin primal play primal nm om system grind exactly member even dimension system look bound subset hamming cube packing vertex tight bind packing packing constant packing probabilistic look vc dimension book simplify theorem refine size size onto primal sensitive dimension set separation could remove make paper packing packing primal scenario thus extra remove imbalance system discrepancy space set let set big generalise size sensitive primal make set discrepancy considerable case pack discrepancy primal recent improvement appear discrepancy conjecture low entropy constructive see constructive proved discrepancy constructive discrepancy relative system proportional subset inequality approximation tackle range previous recent primal sensitive constant geometric section upper universe wish minimize technique fraction leave low colored universe colored recursively develop major extensively yield discrepancy imply subsequently constructive lemma say set let recursively remain element choose sufficiently round denote discrepancy final decomposition refine separate clearly exist member member contradict notice lie hamming denote close call sensitive refinement truncate chain truncate close assigning chain sensitive family denote later apply
admissible value real appear blue initialize text nm fm pixel pure art estimate inversion step perform constrain besides extraction denote namely fm nonlinear nm inversion achieve nm fm taylor inversion achieve subgradient base scheme gaussian generation initialization algorithm stop successive function algorithm evaluate average spectral abundance square measure table first algorithm two namely improve pure analyze flexibility various scenario ability kind nonlinear preserve analyze mixture hyperspectral discuss hyperspectral availability truth acquire visible imaging water remove lead band range bandwidth nm compose report image consider acquire france range consist sub image mainly compose additional unknown plant hyperspectral divergence divergence noise assumption physical support divergence measure grind unfortunately public hyperspectral perfectly spectra abundance way select ability unseen datum require truth study nmf interpolation remove hyperspectral image pixel various pixel uniformly image describe fit entry outli miss entry outli entry datum identifiable use minor mm pixel reconstruct complete initialization set pixel performance study standard hyperspectral kl come optimization equally implement technique image consider divergence spectra abundance depict abundance visually description regard explain display residual component regard pixel accurately mainly locate pixel probably correspond water water confirm image rd regular vertical surely post present mix hyperspectral datum model denote extend standard include capture effect nonlinear active hyperspectral require specification simple provide penalty specify illustrate various acknowledgement feedback manuscript receive engineering degree group et de laboratory university department technology company paris la universit nice interest generally process separation member technical electrical france sc institute ph post associate computer ann mi since national institute university associate communication group laboratory also member laboratory sense hyperspectral several signature generalize mix handle rely mild assumption regard constraint spectral constraint impose nonlinearity factorization fidelity express take kullback leibler case minimize minimization result data state nonlinear hyperspectral factorization analyze hyperspectral datum comprehensive various sensing monitoring consist hyperspectral propose commonly provide observation result interesting inaccurate nonlinear need multiple scene lead take account bilinear bilinear differ impose nonlinearity incorporate interaction demonstrate nonlinear feature consist supplementary major drawback choose nonlinearity limit practice nonlinear detail build supplementary account nonlinearity merely motivation valid number pixel contribution observation reflect pixel impose sequel article organize coordinate real hyperspectral preliminary conference generalize use square general update obtain rigorously minimization rule choose weight efficiently finally propose pixel observe lk outli term accounting formulation symbol dissimilarity section abundance coefficient sum hyperspectral nonlinear well bilinear constructive introduction general pixel become energy define objective nonnegative defines nmf problem appear nonnegative free spectra refer feature fitting square regular article hyperspectral regular robust entirely next take divergence scalar introduce auxiliary separable w r thank order inequality lead detail brevity lk kp approximation problem eq may lk kp lp lp lp lp kp kp lp definition tangent hand r concavity root write essentially replace quadratic tight involve effect within square result lead exponent value follow constraint induce extra optimization handle multiplier set special corresponding leibl resort approach nonnegative turn approach unfortunately able long resort function ensure nonnegative descent resp positive experimentally decrease value denote update simply kp kp turn update implement operator operation fraction bar term matrix tolerance function
combine obtain prove generate algorithm maintain condition divide argument q k substitute expression substitute inequality maximize obtain k complete proof induction condition relation indeed f q estimate indeed third corollary function strongly let write optimality condition cl p concave g g parameter eq k g maximization side finally estimate inexact augment lagrangian divide function smoothed sense q estimate kf k k estimate write k lead eq start e plug estimate pt pt ed de primal dual algorithmic rigorously efficiency analysis structure fashion primal method instance choice smooth lagrangian alternate direction multiplier case primal primal feasibility gap iterate cm alternate separable convex minimization programming concern constrain convex capture surprisingly broad sequel rigorously characterize assumption efficiency limitation eliminate onto understand smooth minimization barrier point method use smooth unconstrained simple overall strategy curse dimensionality well numerical formulation medium simple function despite approach capabilities numerical alternate scalability rely three structure stand among say set f ii pn parallel implementation hardware architecture convex problem pose significant numerical smooth nearly constant canonical feasibility algorithmic iterate many smooth proximal function possess enhance efficiency highlight aid design momentum parameter full rate composite minimization complexity unfortunately penalty approach block ideally characterization solving value iterate feasibility primal significance primal gap since constrained time trivially demonstrate far ideal ergodic averaged iterate feasibility reduce scope applicability rate dual primal residual feasibility necessarily feasibility convergence function function necessary parallel admm rate cm decomposable gap decomposition decomposable admm decomposable dual decomposable linearize decomposable gap primal augment lagrangian inexact decomposable k development algorithm special scalability away foundation flexibility restrict solely unfortunately decomposition often complicate backtrack computational selection efficiency well handle solve proximal characterize primal primal feasibility separately positive solution still exploit favorable exploit decomposable sub optimization trade residual feasibility gap crucial numerically numerical dual smoothing technique optimization primal dual rely duality saddle point monotone approach mixed use develop smoothing technique replace non smooth bregman augment lagrangian technique lagrangian smooth property solve norm bregman smoother rely proximal nesterov dual solve nonsmooth unconstraine characterize combine three unified convergence mild theoretical primal constrain cover decomposition prove variant cf theorem framework residual particular inexact algorithmic case subproblem maintain control appropriately proximal class characterization different importance well bregman parallel manner feasibility act consensus practical trading feasibility front well numerical synthetic real state art source enhance performance trading gap gap result advantage interest unfortunately impossible comprehensive expand reasonable algorithmic framework representative method subgradient provably rate e sensitive rule overcome difficulty instance augment smooth study establish nesterov accelerate recent feasibility f separately several primal variant hybrid primal study several ergodic leverage instance variational also belong tailor may come offer convex define lagrangian produce global k accelerate scheme method variant alternate admm recognize split separable cover problem use nf admm solved iteratively admm update subproblem except interestingly notable completion sub entry differential computational difficulty use one efficiency significantly penalty guarantee well convex drop term forward backward splitting optimality inclusion approach idea structure whereby preserve moreover formulation algorithmic study augment lagrangian bregman smoothing dual problem primal dual boundedness simultaneously odd four objective function hold constrain feasibility gap oppose primal variational formulation bregman also formal function property section main solving specify connection devote implementation provide bregman sequel close n f denote lipschitz constant notation nonempty nonnegative prox bb prox smooth bregman project prox diameter range b distance write base lagrange call continuous nonsmooth numerical weak duality guarantee duality require nonempty either assumption dual strong dual solution lagrange goal solve numerical specify approximate give accuracy say solution feasible onto absolute objective residual mf tucker gap indeed augment lagrangian principled smoothing technique choose center nonsmooth bregman convexity center projection primal characterize smoothed smooth function lipschitz continuous smoothed simply smoothed function specify note case choose g augment dual augment lagrangian concave gradient lipschitz continuous constant augment smooth short smoothed smoothed smooth summarize define diameter respect solution gradient ml bf md hold condition form dual gap nonsmooth smooth add distance however however overall gap function optimality explicitly goal become follow gap g k scheme nesterov function call nesterov smooth convex note gap basic analyze allow show find sequence definition definition k g smoothed bind objective design primal update template scheme subsection assumption metric might lead nonsmooth require replace linearization follow q decomposable whose defer k switch primal dual new k update maintain whose respectively choose point g follow obtain lipschitz residual primal feasibility simultaneously exploit introduce gap primal dual convergence residual primal feasibility gap inexact lagrangian objective feasibility nesterov accelerate scheme case characterize feasibility quantity drop close result corollary strongly procedure ff f c estimate corollary theorem conclude choose standard knowledge algorithm multiplier minimization multiplier admm linearization alm instance separable primal subproblem alternatively many scheme primal indicate arbitrarily residual feasibility gap scheme k instead obtain primal function boundedness set consider still feasibility stochastic case set result closely relate look author joint constant result combine feasibility gap take arbitrarily value residual primal feasibility separately joint primal algorithmic prove criterion residual trade quantity primal discuss enhance observe enhance parallel distribute implementation bregman propose option c employ bregman euclidean variant discuss point unknown k bregman simplex b decrease smoothed decrease might improve increase primal feasibility gap ks default option define correspond figure link request neighbor ij iii pi link number number coupling end problem coupling constraint separable constraint bregman distance either pd pd main need primal scheme step separability solve precisely subproblem form dual k neighbor next iteration request send need store copy dual matrix neighbor feasibility consensus communication approximate operator expect characterization important future present section extend simple slack transform modify update eq indeed variant solve numerical simulation several machine image compressive sense numerical mac os ram terminate algorithm feasibility state variant bound basic group solver k theoretical performance randomly iid ii use algorithm tune calculation twice variant iteration empirical feasibility deviation increase improve adaptive proximal suggest outperform hundred illustrate vs augment technique augment lagrangian smooth fista true suggest actual basis close primal feasibility exhibit strongly convex elastic select generate randomly iteration configuration enhance version backtrack converge ht relative line variant one relative correspond compute theoretical plotted figure line ht cm corollary variant iteration basic require procedure end verify justification done use pursuit indicator compute variant adaptively obtain cm figure variant relatively reach hundred variant smooth variant deconvolution eq isotropic oppose tv additional coupling constraint solve result implement per point lead method case tune recent admm surprisingly periodic condition norm solution subproblem fouri hence class problem illustrate resp done suggest exact code use move cm admm solver admm rule calculation choose common imaging problem norm self image regard result inexact computation optimization variant paragraph solve subproblem background logarithmic fortunately ht approximately prox inner randomly randomly group error n show many time profile noiseless computational rest inexact fista slow step linearly system inexact present basically noiseless group increase increase converge happen penalty significantly follow eq norm suggest perfect video surveillance camera video matrix tuning compare source admm inexact report algorithms k svd operation admm reach value gap admm many plot algorithms ht plot object human low similar solver reformulate probably basis generate generate tune lagrangian admm admm tune smoothness work three paragraph subproblem matrix carlo fast cf ccc size multiplication prox multiplication
rademacher random observe coin identification obtain I accord generalization coin identification let coin identification q position complete reduction identification task coin identification subroutine need input clearly hypothesis return coin add q hand complete sequence recall return set function vector da di lk inside time symbol mean standard basis ab goal subspace expect sample attribute low bound involve variety application recognition instance measurement correspond outcome medical relatively attribute patient measurement partial show example let subset find rank distance subspace pca general arbitrary assume usual learning learner sample allow reveal squared analyze number particular set sample propose three subspace matrix stochastic strategy attribute upper sample algorithm inferior really inferior analysis pca regime small bound observe regime family include bandit low show balance suffice follow small optimal log dependence mention hold achievable accord subspace formal example think partially entrie one strong come free guarantee completion treat subspace problem estimate correlation partially rely resemble bandit obtain partially attribute g matrix main difficulty stem example vector therefore think prove single attribute give correlation build control exploitation subspace bandit formalize rx x subspace function every dimension exist call satisfie randomness follow summarize next section complexity bandit bandit learner namely learnable family learner pca bandit inner maximize learn full replace pick learner eq eq call matrix independently r v v approximate rely follow spectral surely translate complexity subspace pca mention part bandit bandit descent gradient approximately optimization descent descent replace implementation require implementation convergence sgd essentially completeness getting convex combination set guarantee detailed index independently ss iw runtime accurate comparison use complete correlation loose nothing runtime observe sgd recall xx conclude short difference sophisticated strong subspace learner include learner complexity deduce dyadic describe zeros learner learner maintain attribute bandit dyadic dyadic learner assume prove bandit subspace advance random sample observe distinguish employ low information sample partial case provide part bandit q divide part rely suffice satisfie close pca c I run combine surely along every trace old maximize return section subspace partial argument sp u uniformly let subspace allow attribute order return fix subspace accordingly denote computation show let environment recognize recognize output subspace successful use identify I mention attribute learner distribution since attribute sample expect least recall definition concrete distribution view denote output obtain l l task environment role recognize reduction maximize identify r zero negligible distinguish zero draw subsection successful subspace learner distribution concrete make learner equal zero subspace prove optimality complexity information eq first idea behind draw random coin integer th coin successful learner bias coin identification subspace statistic bias lower follow free attribute whenever distribution know interestingly also e
vertex hence generality assume henceforth index set eq follow chernoff tv v inequality index index ball bin stochastically follow chernoff tv tv tt tp ta tc let prove stochastically dominate eq denote scheme c straightforward suppose sake contradiction randomized polynomial arbitrarily follow moreover definition contradict holds plant dense dense random vertex plant approximation least notice plant subgraph plant exist solve construct sequentially let replace connect vertex take unit run bind type ii bind bernstein plant subgraph chernoff exactly vertex plant dense bernstein inequality follow pm notice trivially assume separate depending value probability pm pm ms mh stochastically fx px rest shall intermediate dominate stochastically last function satisfy proceed toward end definition pm ed pm pm bt mp complete hypothesis chapter edu detect plant enyi graph within exceed assume hardness plant clique community exhibit grow become exist computationally intensive procedure hardness recover dense approximate often community many edge vertex numerous science exposition therein work study detect community random recently detect new event theoretical interest understand statistical algorithmic community model formulate plant dense enyi independently plant subgraph include connect connectivity elsewhere plant subgraph model deterministic dense subgraph detection henceforth distinguish hypothesis difficulty intuitively subgraph decrease decrease become recent obtain condition plant subgraph certain unclear whether procedure show test maximal subgraph relaxation highly suboptimal limit problem sharp admit test vanish conversely detect plant dense subgraph reliably community factor rest graph adopt approach parameter hard plant clique parameter regime plant clique vertex form clique henceforth refer distinguish plant clique extensively state solver pc pc require pc constant pc hypothesis hold require detect enyi asymptotic regime dense subgraph either pc depict detect reliable detection thresholding subgraph solver plant subgraph h font right cycle hard hardness transition moderately detect procedure graph detection total regime therefore surprisingly linear base edge procedure reliably lead polynomial term parametrization boundary beyond reliable detection analogous plant succeed satisfie sophisticated spectral method succeed hardness result recent community scale linearly density scale resp sharp regime slowly achieve exponent demand plant clique hardness plant subgraph ensemble bad clique hardness well particular plant subgraph subgraph constant light plant dense subgraph research e follow randomized polynomial appropriately previous highlight technical theoretical pc hypothesis g approximate nash independence pc hypothesis investigate incur complexity principal submatrix strong pc positive pc use open work theoretical computer science literature hardness instance certain reduction pc generate feasible prior space establish hardness pc problem reduction close start dense arrive whose pc tradeoff graph rather complicated enyi problem refer find subgraph edge view hardness np hardness subgraph vertex fraction subgraph hardness comprehensive discussion bad result average case behavior plant subgraph scaling plant dense subgraph recover polynomial simple region approximate within bind subgraph high average hardness detection consequence scan achieve vanish probability succeed scan succeed cardinality intensive unclear whether exist solver question plant problem pc reduction formal problem bound problem constant equivalently adjacency equivalently resp draw draw close reduction vertex vertex parent child number vertex cardinality uniformly remain give ideally want construct edge clique unfortunately provably nonetheless accomplish long nan distribution match core match node distinct parent plant desire plant though desire average negligible let consequence follow show induce solver probability theorem establishe hold satisfy polynomial test type error bound hold regime computational barrier limit incur significant phenomenon line noisy submatrix submatrix exceeds plant dense subgraph plant dense subgraph size bipartite plant computational implication hard dense recovered subgraph result plant recover green regime consequently conversely polynomial otherwise density pc subgraph plant subgraph hard open scale font thick right left impossible cycle node plant subgraph hardness deterministic deterministic enyi graph alternative vertex dense subgraph distribute entirely clear extend approximately lie variation hardness extend subgraph monotone imply whenever obtain intuitive likely plant dense contain scan monotonicity statistically also monotone scope bind hard solver plant subgraph least interesting bound without restrict test monotone conjecture establish bipartite deterministic plant subgraph intractable regime statement bipartite vertex plant plant subgraph vertex refer bound use bipartite pc test bipartite plant plant bi clique refer constant succeed analogue bipartite low graph much verify variation hold computable regime carry limit distribution conditional since chernoff inequality subset copy
preference influence environment face perform environment identify appropriate satisfie preference work video perform task internet user preference encode trajectory optimize use robot arm robot alone human environment oppose cost function drive trajectory preference show activity human environment prefer arise human activitie tv prefer minimal block tv agent activity cost generalize environment activity challenge spatial crucial activity distribution plan planning trajectory object work limited planning tv activity informative move propose planning develop web service short video rich environment human activitie feedback segment video bad neutral previous feedback require environment come cost preference room environment generate environment feedback aware validate use pr human environment section plan overview complete discuss evaluation evaluate human pr initial collection routine future human implement object extensively wherein preference learn task preference preference preference human human object prefer minimal agent share environment user agent important second preference object object act agent human tv music prefer minimal environment object rich environment understand plan accordingly social tv stand front planning trajectory action human differently object instant people book people room state execute environment successfully path environment pr preference rich environment planning serve tv object serve agent environment capability previously human way exploit design bridge interaction provide strong path planning tv behind human front robot understand tv label planning learn jointly user environment exhibit depend demonstrate tv read book though environment unchanged rich multiple interact tv room human interaction design trivial relate trajectory perform environment distinguish preference base object activity mean plan differently around object e tv object trajectory indicate undesirable robot configuration learn expectation trajectory number particularly environment human short robot video ask reveal preference interactive interface setup non feedback segment come tv preference activitie cost trajectory location object human activity plan rich environment human activity paper e context rich human object activity robot interaction trajectory design expressive accurately reflect preference environment ccc object away trajectory cumulative cost preference activity separate activity denote cost along multiple associate activity human activity activity define membership allow decompose activity activity trajectory preferred short trajectory latter tv discriminative trajectory human interaction stand behind move multiple human robot contact human environment human human object arise produce stand behind multiple human interact robot contact hand code environment become arbitrarily introduce human therefore learn human activity environment goal robot plan robot robot configuration problem challenge configuration state hand challenge environment object spatial learn activity context environment define human environment correspond environment mixture per graph preference likelihood design environment arbitrarily human interaction drive approach collect rich parametrized family take learn advantage furthermore adapt available heuristic set arbitrarily human object heuristic tb user iii preference build easy preference wherein robot maintain human activity robot observe environment distribution trajectory cost present engine user observe feedback robot book color interval carefully avoid human colored expert user rich principled achieve drive task environment trajectory drive collect preference trajectory cm distance normalize interact preference symmetric activity prefer approach previous reveal preference approach accomplish expert preference clear instead demonstrate demonstrate relevant approach iteratively multiple preference consume expensive collect preference environment main preference environment approach specific interested situation large develop video video particular keep user provide segment pass tv label segment neutral trajectory b good neutral effort reveal environment multiple activity capture learn trajectory human environment currently database trajectory environment interact activity reveal preference idea environment three generative preference activity give user collection easy neutral good segment pass affect activity incorporate give follow activity vary activity prior obtain environment consider bad use solve likelihood solution calculate activity assignment update step keep posterior activity consist von von von first moment von provide detailed derivation supplementary expert idea planning path approach interact trajectory collect preference goal optimize heuristic optimize guide work approach planning room leave reconstruct environment type often human view human pattern recent human joint differ activity preference arise ambiguity maintain cause e robot study effect entity value train relate action work plan preferred trajectory environment interact context work scene planning robot et al user specify robot human et al gradient trajectory function learn rich cost expert similarly preference understanding whereas preference preference perceptron rich environment either create environment google reconstructing correspond activity environment six activity activity generate video receive preference feedback video feedback trajectory trajectory segment assign trajectory mcp near trajectory trajectory stay away human base human activity rule hand encode opinion plan trajectory map euclidean heuristic show heuristic baseline learn user online set use trajectory feature apply trajectory cost preferred quantitative assign feedback ground score minimum score segment segment trajectory neutral ground score score metric quantify chance incorrectly trajectory trajectory trajectory grind truth assign normalize number trajectory quantifie execute sort train good bad learn heat match plan good reliably good trajectory bad trajectory strategie train room vice test room bad order incorrectly misclassification baseline testing improve room human activity environment converge optima rank environment training environment preference well design also observe model robot discriminate trajectory bad one present trajectory choose accordance misclassification rate cost encode planning chance however drive well learn improve room bar cccc activity object heat heat activity heat interact work heat reach activity spatial activity interact human unless contact preference front human critical human towards object however activity move activity human object reach book work spatial work human region spatial front environment multiple planning map use activity planning map region
gmm allow different deviation contiguous center mixture belong intersect bregman solver com compute scalar disjoint interval extend choose model selection illustrate refine bregman maximize complete cluster programming mean divergence primitive homogeneous mean seek minimize intra cluster one solve heuristic locally hardness center well center centroid center etc surprisingly mean program seminal dynamic dp optimally element contiguous maximum appropriate dp require correspond cluster problem refine dp optimal rely area table bregman application identically iid maximize use exponential family series bregman mean optimally dp gaussians graph intersect laplacian belong family space totally nk nx contiguous among contiguous partition ask minimize function intra cost calculate inter counterpart etc sort I ie matrix find contiguous indeed say contiguous recurrence q store position top yield denote require require recover storing index cluster solution iteratively retrieve index also note satisfy partition may potential auxiliary look solver contiguous optimally fact far run consider add constraint add non empty th great constrain balanced obtain kk appropriate clearly costly monotonically depict explanation choose cluster compute good entry column range regularize last avoid computation check entry last row range center store prototype center dissimilarity prototype intra cost rx l j rx dp return contiguous cluster cell prototype dx potential induce dissimilarity diagram display illustrate refine bregman bregman bregman triangular asymmetric bregman variance bregman bregman prototype lx j bregman area table contiguous cumulative time preprocessing evaluate cell x p since bregman diagram cell cluster contiguous directly bregman center exactly memory element bregman solve often mixture dominate lebesgue simplex indexing px space usually locally maximum need proper reach iid amount maximize q maximize dissimilarity prove proportion solve
often roughly tie emphasis interesting distinguish effect standard illustrate issue normalize equation bayes choice reversible jump consume view follow compete random label evidence q around section density normalize integration estimate g g g integrate routine evaluation manner factor enable model previous suggest systematically prefer computationally intensive promising composite recent composite likelihood currently acknowledgement acknowledge insight centre science foundation grant support science foundation grant mix integral simulate network auxiliary network exchange close case flat therefore two reason simplicity random extend heterogeneity framework yield model paradigm estimating factor develop feasible calculation bayes mix analysis statistic publish article development refer introduction field represent adjacency connection vertex symmetric triangle equally matrix cross distinguish explain existence model first model edge replace see unit principle actor consider heterogeneous heterogeneity observable lie generalize mixed estimation software network modelling call exponential number star see term graph therefore early approach advance fully develop interpretation focus node change statistic deep discussion exponential equation equation model possible heterogeneity actor include homogeneity lead want exercise latent heterogeneity exponential form term likelihood statistic vertex fit node effect accounting heterogeneity fall family random unlike specific latent author effect issue extension bayes calculation suffer numerically calculation selection extend fully develop routine package http package organize fully model routine deal bayes factor result propose mind infeasible calculate small numerically impose prior independent identically distribute accordingly use denote unity prior q prior choose flat hyper note doubly intractable firstly possible marginal secondly infeasible normalize draw entire vector drawing augment proposal accept p normalize easy direct step detail follow
goal rotation independent branch mathematic distinct suggest section reading function measure independence function say hoc present idea theory close independence generalization information multiple reach independent example ica estimate underlie multi therefore ica reach zero middle example matrix form rotation variable calculate multi zero statistically imply rotation optimization appear abstract validate source rotation recover unimodal gaussian bottom add integer minimize practice interpretation ica entropy amount sum entropy entropy joint employ relate determinant rotation rotation information constant rotation maximize statistical connection mention calculate ica strategy interpretation distinct equivalent interpretation ica solution equation find transform kullback leibler variance minus sum rotation maximize assumption statistically interpretation permit turn reconstruct component rotation angle multi plot rotation angle recover source rotation respectively bottom grey marginal distribution rotation degree unimodal quick summary datum optimize code ica mix recover assume statistically toolbox find popularity signal processing perform mixture basis independent component source interest party recover independent selecting music recover music success analytical approximated numerically inherently minima objective equation estimate quantity optimize extremely challenge entropy identical analysis bar part employ remove ica employ rotation remove high order pca inference understand observe apply whiten distribute covariance factorial whiten pca achieve factorial implication ica try ask prominent correlation high small pca recover independent source empirically measure correlation check rarely issue minima optimization rigorously calculate procedure overcomplete exceed middle underlie source lie superposition exclusive reader recognize ica readily highlight reader ica back source clearly suffice mathematically intuition match exceed overcomplete discussion employ additional form regularization latter perspective situation multiple ica blind separation focus handle arise sparse source representation popular journal ica relationship elegant achieve performance writing experience I hope ica assumption behind technique please I write orthogonal transpose orthogonal associate eigenvector let eigenvector proof part eigenvector independent matrix eigenvector complete part e degeneracy place column therefore little provide complete distinct eigenvalue relation unique eigenvector symmetric orthogonal final observe require covariance datum note popular package employ notable speed attempt direction algorithm although elegant demonstrate demonstrate ica introduction linear algebra intuition ica principle behind reflect thing piece measurement reflect city etc sometimes clean arise distinct identifiable subtle measurement estimate source corrupt fluctuation additive white instead distinct source broad topic separate source name source bss writing solve arbitrary bss often last decade ica solve blind ica filter reduction retain e person operation retain project filter ica g eeg biological e micro array audio ica naive treat technique black box one ica appropriate believe necessary ica write thorough goal algebra topic pca truly concrete example building intuition mathematic insight please I comment correction blind separation bss intuition signal interact world make measurement addition make bss entail intuition well bss sound fundamental physics sound linearly record pressure multiple source amongst background person signal party sound environment highly applicable base idea process bss remove image due camera try camera steady pixel sensor record light integration intend camera motion recorded light original camera pixel image pixel original require camera party de ill additional vision highlight signal interaction world complex exist recover individual combine bss arbitrarily generally progress interaction superposition problem ica bss address ica party problem ica mixed music likewise arise solely depict source record term component whose amplitude weighted magnitude magnitude source formation top row unimodal bottom middle component blue figure might correspond imagine party record music axis record figure happen music play look alone play datum would lie music since music fix record must volume volume sample panel arise middle panel note audio merely reflect source basis label ic would sound add depict panel color accord contribution give diversity bss middle row axis namely unimodal appear orient row situation analyses solely sum examine piece arm direction extract desire music benefit visualization become difficult might salient salient middle figure might middle technique fail find understand recover basis goal mathematical highlight type distribution play understand framework sample unknown e keep thing interesting variable observation party amplitude behind underlying source component ica find inverse construct ica appear black point constrain number observation challenge hope find essence red correspond operation composition operation divide strategy problem simultaneously rather piece fashion cutting divide piece svd decompose simple operation axis svd parameter infer diagonal matrix figure provide familiar recover piece individually exploit rotation appendix successive stage order independence ica consider failure concrete suggestion might appear motivated provide strategy emphasize reader general explore three operation equation start covariance capture correlation capture dimension outer product discuss plug linear exploit property arrive choice independent extra equation look familiar student algebra aside tell decomposition refer operation multiply orthogonal orthogonal stack whose covariance orthonormal basis compare equation matrix decomposition underlie ica purely propertie matrix symmetric behind
variational work core computation supervise also term analytically rewrite scale transformation element analytically gradient gradient contain efficiently gradient conjunction sgd adaptively training procedure experimental algorithmic minibatch variate distribution layer average classifier neural complexity complexity form label gradient stack generative algorithmic provide complexity make appeal since encoder low model fully wide range inferential query approach semi h I figure please benchmark supervise vary label ensure number create use confidence mean repeat draw construct hide activation use layer unit pre deeply still optimize solution semi machine label svm obtain reason space generative variable image comparative semi near kernel performance generate discriminative model original present supervise classification cccc knn knn cccc knn initialize objective optimize minibatch ascent momentum bias momentum normalise pixel intensity image weight decay prior introduce extend full variational provide ground perform selection particularly supervise image task network current model readily exploit general connect promise future exploration limitation expensive potential reduction use truncation combine truncation mechanism code label approximation supervise generative amongst competitive currently hope supervise classification model much scope grateful e experiment google google ever modern label significant datum supervise develop label scalable deep inference exploit recent advance consider subset observation interest wide search language parse entire ask datum improve decision classification accurate label datum answer question building advance scalable amongst scheme additional obtain confident prediction repeat termination reach poor svms aim ensure approach difficulty extend efficient open amongst popular similar information label eigen laplacian scale unsupervise feed forward classifier additional penalty auto encoder manifold classifier train learn manifold lie follow train invariant local perturbation manifold use combine amongst currently problem hide successful mixture either process scalability variational explore small set generalise scalable semi supervise still gap new framework employ rich parametric estimator form fusion develop inference optimisation scalable demonstrate approach qualitatively generative intra allow fashion variety dy omit index clear subset class exploit label alone model separate relate latent feature limited auto encoder generative able generative form linear transformation essential choose network variable regression latent dimensionality embedding form linear approach performance propose probabilistic describe generate latent treat latent marginally digit separate writing transformation label unobserve integrate data inference perform inference miss see mixture two learn subsequently supervise instead generative stochastic deep exact intractable variable allow tractable scalable advance variational approximate variational principle derive likelihood form posterior variational variational
fast star asynchronous performance performance conduct another master perform loop master obtain require subproblem master information back need master master obtain master allow increment ensure master level synchronization master decomposable conservative master step master step master atomic increment preferred period case vary stop three add artificial delay complicated setting show speedup fully method suffer star asynchronous method suffer achieve although speedup star topology second vs parallelism randomize base dual state sequential cd outperform descent selection rule randomize clique svm use clique introduce sparse box constraint describe one coordinate descent however expense computation without increase maintain basically increment increment increment fact ij prove equation induction ij prove choice last statement therefore statement induction substitute prove definition third last descent require solution intermediate find vector formally prove vector element iteration small let absolute albeit ni ar decrease least terminate termination otherwise algorithm always pick continue multi manner remain part essentially similar detail completeness step recurrence relation form simplicity line consider gradient orthonormal column g use less rewrite q become clear preserve preserve wise gradient denote minimum zero singular note however operate variable much much conditioning operate tight compare bind technology cd constrain scale separable couple knowledge cd iteration constraint present four key convex smooth separable asynchronous detail illustrate coordinate cd conceptually unconstraine study greatly root successful statistic therein cd randomization generic randomize cd though optimization nesterov global iteration complexity improvement nonsmooth term randomization cd cd problem allow constraint develop problem differentiable lower coordinate separable necessarily lc specify see however fit cd framework nonsmooth part separable usual alternate direction multiplier latter treat familiar special fuse constrain recently study set assume feasible ensure descent maintain feasibility scheme well convex problem recently generalization separable block cd obtain present dependence study couple cd gauss apply cd linear present light background rate randomize block case tight composite sum asynchronous solve contribution exist art proof claim lc prox parallel yes yes yes yes yes yes yes cd gain full refer reader thorough analyze author family gauss like analysis gain combination randomize frank wolfe though frank wolfe approach projection cyclic descent global lin recently cd nesterov prove linear sublinear slightly refined pure et rt problem processor update speedup overlap strategy study read fix liu al prove convergence dd allow inconsistent read bc arithmetic cost well similar stochastic advantageous size closed form stepsize tuning duality gap handle inexact linear easier parallel minibatch distribute delay minibatch sg nesterov parallel minibatch subgradient sag asynchronous also read asynchronous vector store asynchronous fashion rate speedup control max processor able impractical delay notice contraction method essentially linear general assumption decompose communication constraint node present exchange denote restriction column column identity place typical coordinate lipschitz gradient make block critical composite also pt ready asynchronous stochastic variant idea maintain ensure pt begin nonsmooth pick minimize around iterate maintain formally involve stepsize bound present result update always however condition update space constraint group typical size apply describe smooth theorem nonsmooth assumption result I k kx kk solve iteration inherently disadvantage develop asynchronous smooth difference processor solve subproblem asynchronous execute without require asynchronous presence non ensure feasibility throughout require sequentially consistent swap increment update despite convergence asynchronous base gradient iteration bound sublinear gradient parallelism asynchronous conclude discussion asynchronous algorithm extend nonsmooth function suggest maintain feasibility respect constraint convex iterate retain feasibility future loss many separate add separability innovation addition randomly pick also update since calculation involve gradient random random integer later price convergence size generally outline describe hence lack key constraint loss generality rank rewrite form transformation ease exposition experiment assume also unconstraine introduce diagonal induce quantify far take reader impact laplacian graph consider simple update fx prove attain generate algorithm convergence iteration prove follows desire decomposable worth improve upon involve towards read context like assume enforce asynchronous expected value define equation sketch ease exposition analysis carry manner iterate iteration existence consistent read fx k u derive prove mathematical method expectation large stepsize decrease rate convergence rate oppose iteration factor tradeoff algorithm slow believe generally algorithms careful analysis nonsmooth sum assume separable write coordinate totally impractical set uniform algorithm graph clique linear improvement theorem form oppose
tb set pose quality report mutual result table embed cauchy score reach accuracy base mean furthermore k matlab demonstrate benefit tackle tb accuracy utilize locality hashing perform video projection low hamming encode key neighbor query time sublinear contain hilbert familiar euclidean kernel ability approximate superiority previously cauchy keep kernel open future could give answer equivalence pl embedding follow definition intrinsic metric curve l dt intrinsic infimum length intrinsic metric induce metric define respect intrinsic metric length conjunction conclude proof token li act communications digital centre program image set prove beneficial incur arise fact subspace type riemannian leverage develop support subspace study hilbert make positive unfortunately two kernel none show introduce positive superiority code embed positive definite subspace nonlinear euclidean video visual subspace prove vision despite success suffer drawback euclidean subspace lie manifold consequence popular euclidean space reproduce hilbert rkhs exist euclidean recent study report manifold intuitively attribute fact geometry manifold capacity capture nonlinearity manifold rkh preferable rkh cauchy former ability approximate universal well new include universal kernel two embedding cauchy pl embedding yield distance distance conjunction analyze kernel ten summarize evaluation demonstrate benefit gender categorization kernel bc rbf ex r ex laplace universal k binomial bi bc universal bi p universal logarithm log bc ex k review property paper use capital letter letter column identity trace space euclidean note become projective orthogonal column pp dd denote slight abuse notation whenever represent subspace riemannian length short connect call geodesic geodesic geodesic give let subspace recursively principal angle principal correspond addition geodesic metric similarity metric manifold hilbert us real kernel nonempty symmetric p kernel pp arguably radial basis radial replace euclidean unfortunately symmetric define verify counter digit nevertheless manifold principal subspace bc successfully problem hilbert space receive little bridge discuss space help devise kernel pl concept algebra alternate multilinear vector space g copy multilinear slot alternate g multilinear copy k generalization arbitrary projective pl space describe pl dimensional close embedded indeed pl taking row pl pl inner importantly meaningful inner need two realization correspond point like dimensional hence computing goal compound arrange compound cauchy rectangular matrix pp hence pl coordinate suggest pl indeed linear kernel verified may sign issue problem design define induce principal angle invariant follow show pl nice relate geometry scale turn well pl embed projection embed one differentiable smooth choice subspace induce discuss projection isometry length curve reader thorough discussion embed see order induce space induce pl actually exploit kernel derive pl embed cauchy kernel define kernel create consider kernel readily kernel often crucial impact pd learn express importantly kernel arbitrarily sufficiently develop universal negative definite us kernel nonempty symmetric kernel definite form example nonempty inner function f fx I therefore hilbert important measure half designing cast probability kernel euclidean rbf dirac delta discard scalar rbf embed positive neither prove theorem employ laplace pl embedding extend kernel binomial kernel translate noting see us kernel bi important positive formally conditionally kernel nonempty conditionally relation kernel study kernel translation equally instead kernel invariant position origin svms separate
degenerate prevent directly apply recursive summing u lemma get vanish hold eigenvalue zero write equation reduce equation iff condition lemma asymptotic asymptotic involve quite remarkable penalty extend idea yx x l e center spatial let f update spatial estimate yx cosine bc b cauchy schwarz inequality estimate choose upper bind bx ax x apply p ip j ib j cosine rewrite equation estimate bind ax bx ax equation l enough take expectation variable pseudo initialize ix g implementation master momentum initialize batch image batch training center precisely local worker general use datum read memory map file raw always send worker request worker request different datum data cycle file worker receive request datum reach start file worker rate summarize rate explore htp initial explore rule initial rate divide decrease htp experiment sgd htp figure log computed experiment always begin experiment start average show compute discuss dependence trade exploration exploitation learn rate experiment observe test fluctuation worker conjecture high lead impose opposite learn rate interestingly train lead bad performance experiment trade exhibit get energy avoid decrease figure seem sensitive increase period crucial result experiment seed fast package interface gpu communication htp experiment various communication decrease gradually base run load experiment whereas process run whereas negligible large ideal communication leave right correspond experiment summarize need worker level method time level time level counter though large learn meanwhile outperform achieve consistently achieve level h bar denote achieve htp leave never achieve section lemma facebook deep environment process worker elastic force link server enable worker allow variable reduce communication demonstrate many optima improve asynchronous asynchronous variant compare stability asynchronous asynchronous convolutional neural deep compare baseline approach large use descent deep consist devise large yield run gpu variant method interpret idea worker among elastic link store update move time worker contribution provide convergent simultaneously reduce worker maintains measure deep organize asynchronous momentum analysis conclude supplement contain environment master parameter distribution reformulate assume worker refer center equation fall far communication master worker emphasize highly trivial resp descent resp gradient rule center take become local equation choose elastic symmetry rule force influence stability small allow center worker exploration master differ set asynchronous worker master center maintain update master update refer see whenever worker master request center worker entire capture send update communication worker trade exploration exploitation rate communication period randomly move randomly x g ix v momentum capture nesterov worker equation momentum overhead computing parameter explore asynchronous momentum variant asynchronous realistic study show dimensional lead analytic show analytic stable supplement property strongly defer minimax master worker master round lagrangian multiplier give let fx tx become lagrangian multipli ascent update center multiplier equation choose convention multiplier update algorithm similarly worker activate perform follow equation focus state dynamical compose map simplicity write map simple absolute batch momentum moving rate start examine period comparison also outperform figure supplement rate explore table supplement time well small achievable sequential training center different value conclude achieve error small unstable significantly outperform also become tendency characteristic algorithm simultaneously htp local worker convolutional neural achieve compare relative error equal htp layer decrease factor loss speed simultaneously overhead communication explore different experiment experiment result result experiment converge fast lowest achievable potentially explain exploration supplement exploration exploitation period section advantage supplementary loading communication supplement achieve training quickly stable plausible communication method behavior momentum future work acknowledgment li implementation helpful l valuable feedback focus center local quadratic generalization strongly quadratic observe noisy definite eigenvalue strictly positive covariance square center error center variable stable verify root iff iff symmetric due relation stability asynchronous elastic symmetry substitute rule local worker eq q lemma explicitly linear rewrite capture diffusion eigenvalue satisfy eigenvector projection eigenvector therefore last expand recursively substitute
prior superior test histogram mean percent percent gmm percent histogram bayes nb histogram eight particularly poorly censor believe due issue h flat mean ci ci gmm ci rate gmm sbm indicate superiority well gmm pair methodology wikipedia graph graph represent wikipedia page either page neighborhood algebraic vertex class article label available analyze label exclude isolated induce adjacency pair adjacency spectral figure red green figure indicate wikipedia sbm nonetheless wikipedia class green blue adjacency wikipedia adjacency spectral wikipedia illustrate bayes generate bootstrap embedding depict gmm embed estimate figure adjacency spectral common reasonable justify sbm gmm provide wikipedia induce constraint empirical color class gmm membership depict prior yield statistically pair sign empirical improved gmm represent statistically significant absolute misclassification bootstrap gmm statistically significant perhaps improvement chance performance optimal misclassification class gaussian yield nonetheless gmm neighbor indicate conditional gaussians clear despite real dramatically stochastic robustness formulate empirical methodology motivate theoretical advance regard embed blockmodel within gibbs algorithm block latent consistently outperform gmm alternative notably dirichlet wherein wikipedia graph though sbm extension interest automatic eigen case sbm embed justify adjacency dimension bayes misspecification importance dot model extend methodology challenge sbm position simple bayes conclusion adopt empirical bayes approach estimate membership blockmodel embedding assignment acknowledgement work support security science engineering fellowship technology advanced project fellowship corollary apply mathematics stochastic blockmodel interest statistical various diverse social citation brain network etc dot product position formulation stochastic normal adjacency spectral theory bayes membership vertex draw blockmodel practical utility posterior inference conduct within theory carlo studies blockmodel wikipedia blockmodel sbm position use various diverse may indicate citation network neuron connection vertice edge indicate connectivity statistical statistical become area sbm vertex edge depend membership edge give membership entry represent blockmodel memberships important approach membership maximization maximization modularity parametrize latent vertex particular dot product dot vertex absence dot vector blockmodel define dot vertex share common dot step often estimate position position position consequently describe latent truncate eigen adjacency dot latent position adjacency multivariate gaussian mixture sbm identically distribute multivariate paper mixture methodology block memberships blockmodel section formally blockmodel dot motivate empirical prior methodology estimate blockmodel mcmc implement section experimental demonstrate bayes methodology discuss conclude vertex may adjacency symmetric undirected edge loop multi edge let dot product random product latent vector furthermore condition position ij tp decomposition u p nu ds diagonal along introduce obvious identifiability loss generality suppose adjacency embed dimension blockmodel dot accord semidefinite blockmodel sbm block distinct mass standard purpose block membership stochastic blockmodel assign memberships sbm typically specification include special often sbm advance describe empirical position prove position converge gaussian express follow corollary motivate eigenvalue second moment principal row corollary setting suppose theorem likelihood position membership probability inference membership base gibbs behind vertex calculation assume block latent gold prior position represent corollary center theoretical spectral correspond gibbs thus position sampler gmm metropolis distribution k denominator use multivariate limit adjacency gold thought present empirical bayes membership probability unknown posterior simplex gmm choose conjugate dirichlet distribution q block follow marginal metropolis initialize utilize metropolis prior eqn k compute special parameter alternative flat gibbs sampler identical present metropolis position proposal however initialize provide modeling initial k point k k illustrate various blockmodel sbm dot stochastic blockmodel three wikipedia graph block via compete two percentage vertex calculate inference base calculate assignment times blockmodel parameterize probability entry membership graph sbm parameter spectral adjacency position subsequently gmm cluster embed membership component variance prior latent position know scatter replicate color denote membership symbol cluster membership curve estimate gmm gmm scatter estimate carlo replicate sbm denote true membership vertex sbm
length dimensional employ normalised radius randomness normalise uniformly know provide unknown use classifier model independently select li form surface fraction reduce surface pdf introduce follow dimension unitary dot product examine concludes select uniformly independently surface di ip second determine circle matter cdf general expression give neither pdf formulas sphere radius coordinate eq hyperplane distance large point lie sphere hyperplane cut distance small portion cut hyperplane consequently height height q htb maximum height recently prove li surface hyper surface angle beta angle sphere radius follow cumulative sphere length eq stand proposition refer radius replace length dimension sub htb length dimension cdf recursively formula sphere second term right cdf reduce similarly always score length end radius eq approximation suggest q eq propositions infinity figure length around order quantitative difference show table length sphere interval length around distance intuitive change dimensional sphere fix north point concentration expense high area ideally almost lie mean within radius point dot product centre q distance dot matter unitary dot reformulate correspond symmetric around odd moment among formula length estimate pdf basic property start dot two unitary definition theorem study length end
truth truth underlie subsequent particular suggest condition exponent assume initialization circumstance uniformly unit sphere normal zero instance obtain random amount heuristic condition additional available information unfold unfold cf gaussian necessary sufficient tensor power initialization suggest unfold magnitude suggest procedure tensor initialization power natural tensor within machine main tensor assume away vanish consider characterize convergence bound allow unfold message amp prove successful reconstruction retrieval appeal dimensional characterize technique evolution develop tensor focus publication sophisticated iteration unlike normalize multiply simple expression conclusion two side signal amp remain number theorem evolution exact sharp characterization location next summarize defer journal publication denote tensor follow finally produce decomposition proportional uniformly recursively state coincide apart iteration describe power depend apply amp power show consequence obey evolution require information optima say optimum surely converge local large special optimum numerically note emphasize practical suggestion tensor unfold superior tensor unfold produce improve performance warm decomposition power approximate simulation describe unfold tight compare side dramatically simplify product semi psd suggest cone vector belong cone ni mat eq special however rigorous efficiently project onto psd cone compare correlation n unfold psd psd power initialization psd initialization line confidence consistent theory tensor poorly approach unfold dimension psd principal slightly plain unfold initial unfold component either recursive unfolding performance simple algorithm close behavior heuristic argument tensor random initialization work unfold plot superposition correct experiment concern simultaneous tensor noise fix addition noise varie experiment triple lead report predict apply theory otherwise appear superior capture difference already matrix green pca acknowledgement partially grant fa fa introduce operator eq ai symmetric f recall packing cardinality nm b g q symmetric tensor objective eq function dramatically whose prove lead quantify phenomenon characterize growth rate minima lemma far monotone negative strictly informally exponentially maxima value obtain last indeed maximum next maximum theorem let unique asymptotic eq term variable get asymptotic follow obeys turn show around lipschitz modulus euclidean tensor modulus notation proper n consider tensor loose except case matrix loose factor optimality modulus ns kn enough vector exist triangular inequality borel note eq gaussian prove exploit symmetry argument process process follow x aa op q imply prove induction assume inequality fx conclude notice inequality note imply inequality strictly positive monotone prove satisfie order statement parametrization algebra discard point read continuously differentiable calculus stationary maximum point imply latter inverting parametrization since evolution mr lemma conjecture claim corollary arbitrary spike rank noise establish sufficient computational resource turn soon remain dimension tensor power pass idea unless none succeed fundamental limitation tractable initialization tractable initialization statistically finally unknown signal allow iterative replace approximation understand pca tensor exploit include collaborative context information hypergraph hyper find bottleneck np last greedy bad perspective know regression rank resource arise whereby unknown noisy multilinear precise observation read notation analogous immediate np summarize observed tensor recover therefore natural dramatically unbounded computational resource probability threshold sake well polynomial tensor unfold pca provide heuristic unfold factor odd conjecture confirm conjecture rapidly argument necessary iteration substantially observation warm power initialize output unfolding appear unfold variation unfold method pass amp compressed sensing estimation behavior amp qualitatively fail computational barrier weak information measurement relate matrix noise amp threshold amp rigorous require tensor unfold random theory paper insight believe insight throughout paper defer low ordinary nk n nk frobenius euclidean furthermore maxima develop picture translate guess algorithm initialize section intuition popular estimator perform operation unfold vary expect affect qualitatively summarize sake amount unfold succeed heuristic argument tight expect generally unfold essentially construct achieve remarkable behavior point method construct perform introduce refer mat j relaxation problem natural expect signal latter minimal phenomenon integer universal bound minimize gaussian modulus mat op concentration standard triangular standard mat q consider identity since concentration
correlation obtain section mt jt serial long z describe argument r formula dim criterion pc bic ic max dim true max restrict loading nan role specify argument simultaneously however convergence slope iteration control trend modify df lt unobserved individual perform consumption price balance column consumption price dim pc price dim pc dimension number inference error choose corresponding argument presence serial correlation choose realization appropriate inference correct call r consumption dim pc slope std e code none unobserved factor degree report factor effect log price sale effect log well graphic factor reason explicitly effect see hand effect consider follow time effect order restriction variable eliminate effect eq x n restriction affect effect procedure variant model appropriate control formula refer intercept purpose model ignore illustration continue section concerned existence factor merely specification appropriate datum rather existence factor classical dimensionality hypothesis dimensionality test statistic reason simplification reject test function test consumption testing interactive song factor consumption price presence interactive effect h equal level model significance level intend outline interpretation panel estimate convenience choose attractive form unobserve heterogeneity beyond great individual valuable literature stochastic leave additive variable right panel common factor common importance reflect loading convenient quantity share loading variance loading explain two rl red middle figure difference effect factor loading parameter explanatory visualization vary visualization individual effect model propose eq usually cover rotation factor interpret appropriate rotation scheme set rotation package preferable loading parameter instead factor introduce package function package available elsewhere remarkable package usage function demonstrate many helpful comment paper research science trend package procedure dimension heterogeneous procedure complement heterogeneous factor case common whereas focus bound factor additionally range dimensionality criterion unobserve remain model difficulty time appealing advantage panel heterogeneity classical try unobserve heterogeneity structural assumption e unobserved heterogeneity time within unit apart trend fairly availability panel model focus advanced panel individual heterogeneous time trend basic explanatory time vary individual vary specification package individual loading consideration intercept intercept vary center intercept individual center classical panel individual effect choose individual loading factor identifiable rotation ensure uniqueness require ex replace certain sign consider factor strongly auto correlate well stationary way semi identically error allow weak iterate stationary vary stationary deterministic trend rule stationary integration moreover usually package refinement package criterion factor dimension compute estimator see allow test classical fix panel section function longitudinal package panel exhaustive package publish panel toolbox possibility heterogeneity effect well package criterion publicly code ng demonstrate explore american price g sale real price per year dataset htbp price devote short common usage recently discuss well relatively common panel effect parametrize term asymptotic rely second difference discrete series restrict purely functional interpretation estimation first common vary semi component use describe eq time continuously differentiable denote spline imply possess expansion spline basis st see rewrite formalize kt dt follow usual cubic smoothing spline specify explanatory allow specific around intercept common also possible factor factor define eigenvector factor normalization individual loading ordinary square regression lt il crucial testing vary consistency obtain determine smoothing propose cross observation unfortunately computationally costly determining factor advance overcome disadvantage discuss follow theoretically cv cross validation costly explain specify critical get quick objective updating formally calculate start initial proceed step advantage approach inversion update moreover rapid routine formalize goal rather factor cv note smoothing dimension explicitly dimension argument devote argument r formula dim dim cv convergence restrict factor restrict loading symbolic specificity balanced panel replace processing imputation logical dimensionality compute ask default maintain argument adjust dimensionality logical computed default function discuss restriction alternatively loading variable store necessary function variable equal library l consumption r r factor loading default inference slope consumption summary consumption residual pr intercept price code additive effect none unobserve r price real sale significant summary individual effect provide summary factor right effect figure six estimate panel figure correspondingly eigenvalue obviously extend effect logical argument dimensionality criteria consideration slope propose follow test statistic lf l lt significance begin test hypothesis reject estimate rejection dimensionality stationary tendency weakly correlate criterion cause formally fit propose specify residual condition impose restriction proportional requirement fulfil bic large practice bic error cros correlate arbitrary kind case problem criteria ic ic q improve performance ic ic strategy different tuple value grid refined criterion ic abc ic respectively modification affect maximize criterion er gr theory pc bic ic abc ic ic er gr stochastically number panel introduce threshold distribution dimension iteratively ed eigenvalue argument pc pc pc bic ic abc ic c ed er seq seq criterion character default tuple appropriate construct sequence sequence give standardize matrix imagine input consumption call pc offer selection procedure give automatically compare procedure consumption criterion er gr consumption criteria pc gr ng pc criterion er gr criterion user method display percentage criterion er gr come criterion dimensionality ask consumption price dim estimation ng pc pc ic ic ic abc ic
also per somewhat simple projection challenging interestingly also compressive course seem target influence main exact characterization equip result fairly perturbation first analogously characterize bernstein inequality adjoint dimension self adjoint resp onto top eigenvector capture informally say principal provide appropriately small bernstein write deviation suffice setting two high rank measurement draw uniformly invariance think project onto standard ba geometric argument angle orthogonal uniformly random subject basis vector orthogonal sphere compare increase dimensionality bottom see measurement column target although play role plot careful explicitly lead exist result lastly error fix improve qualitatively data compressive insight independent operator preserve practically appeal simulation bad finding secondly mention compressive column justification immediately would understand show compressive column lastly fundamental compressive measurement column achievable algorithm achieve achieve hope acknowledgement research part nsf award nsf research fellowship eq dd expression rational first plug principal large number compressive number suffice principal insight exploit average effect projection preprocesse wide signal processing application give subspace capture vector large singular minimize reconstruction dimensional situation obtain motivate different theoretical study range column burden concern acquisition motivate line datum compressive completion focus approximation adaptively entry compressive column translate usually project principal strategy use sense approximate projection phenomenon compressive per column column exist require rank proceeding mention motivate analysis inferential observe point scientific measurement overhead extract principal sensor network suppose sensor record make cost compression share synchronization acquisition avoid need synchronization compression good proceeding recover principal norm x subspace span eigenvector th st eigenvalue principal subspace large hard matrix large compressive vector unit sphere equivalent projection column connect two column span compression kk call conceptually algorithm vector covariance top estimate span eigenvector appropriately normalization necessary mention algorithm streaming sensor early quite appealing sensor observe I communication overhead compressive need projection acquisition overhead two vector make overhead achieve good property approach even method main turn remark order term active thus term dependence relationship since compressive suffice sharp
compression try apply autoencoder solve sound input autoencoder appropriate source music signal encode distinguish separate unknown source description autoencoder fourier audio frame index autoencoder rectangular window spectra rectangular autoencoder set frame axis design autoencoder unit respectively source mean whole window cluster window also classified source reconstruct propose separation carry mixture source speech music acoustic music generate audio autoencoder frame ms window decide connect node output speech third speech six compose straight line frequency music signal mask middle music peak fourth htb htb autoencoder separation time use represent spatio temporal autoencoder source main complete effort national foundation education technology science department advanced institute technology ac unsupervise deep source signal properly autoencoder coefficient investigate representation audio domain coefficient layer original reconstruct mask speech work extract noisy
later assess effect longitudinal treatment potential outcome tt history make identify dynamic potential outcome assumption outcome correspond actual outcome assumption within randomized manuscript law definition markovian drop subscript discount factor immediate consequence reward specify inferential take severe taking say however say high regime action equation inner quantifie quality policy interpret reward treatment would optimal ensure optimal also note action estimate turning recurrence update density large infeasible explain section need estimate represent parameter vector pair accordingly discuss first bellman science account feature multiply depend transition observe q sometimes solve deal take similar technique objective manuscript improve minimization algorithm generalization w optimal treatment respectively estimator make assumption list statement asymptotic estimate consistently replace expectation estimate objective non process often fail incremental stochastic greedy tool point forward gradient descent gradient introduce show objective introduce toward minimize start individual trajectory obtain tuning step continue size rule distance consecutive constant empirical inside update simulate diabetes construct treatment consider manuscript include reflect treatment extract patient treatment drop take treatment simulate consist individual start decision option continue decision bernoulli augment interval soon treatment variable generate multivariate pressure continue although ideal raise concern feasibility need treat treatment regime patient treatment control whose treatment patient continue take augmented patient either treatment depend variable treatment add c bernoulli augment treatment treatment pd pd pd take treatment avoid effect percentage effect similar effect regime bp treatment operational definition otherwise reward help identify efficacy treatment turn recurrence relation rule rs function summarize similar chapter note transition usage large cat bp cat axis cat respectively axe percentage treatment action construct radial gaussian specify function satisfy list multiply triplet find transition approximate benchmark depict treatment discretize oracle discount average report vertical side leave horizontal axis represent plot propose classical moderate fourth plot efficacy specifically present optimal value size indicate error theorem interval decision tool another important asymptotic whether estimate whether contain adjust option interval result state suggest sometimes dynamic regime horizon priori decision assume base temporal asymptotic property study work raise derive distribution practical provide violate may type may major second try minimize alternatively regression diabetes cyclic point decision happen decide visit patient request easier require visit patient discuss visit award da institute drug abuse content solely author grateful satisfy follow deterministic radial basis quantile variable second first third quantile trade decrease decrease estimator besides b b b part prove complete normality vb vb normally distribute continuity cauchy ga ga listed van show result assume full decomposition eigenvalue satisfy van small condition automatically enough sufficiently definition fa ga ga thus van decompose part n w n b rest follow subtracting complete eq enough identity hence optimal regime simulation scenario discuss manuscript suggest effect treatment get large policy long effect immediate effect present proposition em minus em cm pt title pt email utility manuscript inferential residual dynamic regime horizon priori treat throughout life large necessary inference diabetes third wave national health survey examine
design feasibility z follow application algebraic extension proposal directly regularize proper initial univariate unclear directly mention justify hypothesis interested extension chi take subsection match group structure projection result rotation scale subspace essence inferential rewrite eq match e approximately projection g condition moreover g g g convert elliptical mapping g consistency statistic testing least estimator course need relax requirement orthogonality k relaxation theory true term g moreover standard freedom chi square projection proper upper g motivate summarize suppose provide geometric insight mention equal call try sp projection complement addition g follow immediately remove g g v k somewhat moment feasible feasibility describe subsection iid subgaussian row let orthogonal space well supplementary let r iff r k kb k k v since ball bound n k n inequality complete discuss g g ty subgaussian g v orthogonal g g project indicate g g g g remain view due group theorem prediction weight may use define u j bound view event eq os result eigenvalue kkt j rearrange h g kkt multiplying event g h n follow optimization scale account individual could proportional literature see iterative convexity joint limit give provide freedom practice discussion optimization characterize derivative profile objective minimizer l involve claim scale proof gaussian setup theorem ii matrix cone fix take certain corollary upon prediction g fold j prediction mix obtain lasso g j g bb g g scale proof suppose j kkt h tt group estimator theorem rescale error imply due pn limit fu n fu fu concentration fu e result section lasso simulation experiment design independently true zero design regression wise jj scale lasso penalty replication lasso dot fit replication setup deviation show plot convergence correction frobenius simulation scheme simulated group sparse specifically group nonzero provide base variable group contain suggest group size normality group ps author blue blue theorem remark support grant dms small projection group dimensional estimator chi inference group develop chi inference sparsity condition group benefit inequality scale group q p py unknown inference test approximate version procedure potentially group dimensional possibly low projection regular sufficient asymptotic normality group design match efficiency efficiency imply large group condition expansion moderately large certainly beyond chi type statistical scale group combine extend idea lasso group estimation consideration attain efficiency super characterization informative adjustment min attempt select regularizer done consider subsample consider conservative alternative project residual adopt correct generalize treatment effect consider precision matrix upon brief discussion literature index effect regularize prominent g group variant among many lasso assumption develop reference follow statistical feasibility subsection work availability work verify subsection formulation variable subsection provide subsection solution use paper vector norm u jk denote column indicate column complement span additionally coefficient vector inherent pre group
index x header header header txt index header txt table index header index plot index header txt title td xlabel ylabel ndcg header td td minor xlabel ylabel ndcg header td txt header td txt xlabel ylabel legend pos east x plot txt header minor title xlabel k ylabel ndcg legend pos south east true plot txt header minor title xlabel ylabel ndcg header plot txt index title xlabel ylabel ndcg index header txt table x header true plot txt estimation initialize seed blue line ndcg truncation though bfgs reliably solution try line td initialization give baseline identity replace identity medium dataset million song baseline td xlabel ylabel ndcg k legend pos color thick bar cd title k xlabel ylabel ndcg legend pos south explicit thick mark option index header txt thick mark black mark option index header txt scale minor title xlabel ylabel ndcg pos east color cd mark plot txt thick color mark option solid mark xlabel ylabel ndcg pos south blue explicit error color option solid header plot mark option fu v fu fu fu u vx fu fu contexts partition mutually exclusive exhaustive subset partitioning find qx fu fu fu fu fu fu divide mutually exclusive exhaustive partition let previously iteration instead define set parameter access every progress fraction guarantee optimum accord expectation stochastic optimum objective function retrieve partition way parallelization therefore linearly default com ex definition conjecture axiom assumption department science university west ranking observe close metric competitive extension feature across machine require rank learn literature perspective ranking namely ask machine logistic outlier capable observe requirement standard metric try evaluate discount cumulative ndcg metric therefore algorithm metric show ndcg loss maximize ndcg convexity robust easier suggest reliably converge though non optimization bfgs use large efficient latent collaborative retrieval unlike rank stochastic attractive characteristic interaction therefore scale serial popularity computing service amazon services collaborative million song record rank objective interaction amount art lift metric training attempt vector predict denote dot mistake call function mistake difficult solution machine optimize x easy optimize hinge upper outlier intuition large negative x solution decrease outlier expense transformation follow loss function much slow derivative rapidly loss behave classify ii therefore base function difficult optimize often successful intuitively reduce fraction rank example movie recommender movie subset document set adopt literature context aim learn important feature use write function see g top word motivate objective weighting factor xy reflect metric discuss sum upper bounding function minimize although objective user pay top desirable gain quality top intuition discount metric ranking achievable metric rewrite rewrite objective function monotonically increase bound logistic apply construct loss logistic enable model give discuss transformation difficult transformation generate type eq q twice minimize regularizer add norm conduct standard small follow rank source yahoo benchmark consist fold validation split provide parameter ndcg value test dataset report optimize implementation bfgs advanced framework minor title td xlabel ylabel ndcg header td txt index header td txt index td txt header td index header td minor title td xlabel ylabel ndcg x index header plot td txt x index header txt x txt td txt plots index txt index header td txt index td txt header txt truncation space td plot seem insensitive truncation ndcg hand perform truncation note ndcg consistently outperform inf ir list dataset list dataset td ndcg table complete period assign context item difficult pair collect nonetheless may realistic movie recommender movie somewhat relevant lot user netflix stream leave rating movie express preference consist feedback attempt without euclidean latent embedding otherwise score summation inside instead range avoid overfitte regularizer add frobenius norm become evaluation stochastic optimization pair still summation however stochastic nonetheless guarantee descent attack introduce iterative minimize code algorithm calculate fu fu easy stochastic calculating exist fu fu uv computation spend step linearization trick enable across space detail appendix subsection million song song go ndcg applicable evaluation scale machine across second cpu linearly processor line exhibit speed scale mark xlabel number legend legend pos south east mark header col comma header col sep header sep comma plot txt table index header col sep minor xlabel ylabel west mark option index header col comma header sep comma plot header false col comma txt false col sep txt col comma plot txt minor xlabel second ylabel legend north solid comma txt header false col comma comma plot table header col sep comma txt header col sep comma plot computation art optimize objective compare fast solution code structure librarie comparison identical grid figure machine competitive convergence clear consequently significantly objective basic optimize ndcg objective metric immediately relate construct upper bind improve convexity also nonetheless ndcg metric propose optimization collaborative explore attempt objective advantage differentiable therefore gradient algorithm bfgs convergence guarantee since natural linearization trick necessary guarantee discuss unclear insight binary scalable optimization task collaborative retrieval large care experimental learning collaborative retrieval experiment arguably towards derive parallel averaging gradient machine dataset dataset well therefore dataset principled appendix could constraint similar comparison provide run ndcg obtain library name ndcg c td td yahoo yahoo name td report ndcg medium song report precision htbp minor xlabel ylabel ndcg header txt index header plot td txt plots td txt td xlabel ylabel ndcg k x plot td txt x index header true td index table td header plot txt minor title yahoo learn xlabel ylabel legend pos south east index txt header txt index txt header txt scale yahoo xlabel ylabel ndcg legend pos south east header txt header txt index index header txt index minor title xlabel k ylabel ndcg pos south east table header txt txt index header txt header txt table index header true plot txt minor title xlabel ylabel ndcg legend pos east plot header txt table header txt index header txt header txt title xlabel ylabel ndcg table txt index header plot header txt index index header plot txt true minor title xlabel ylabel ndcg k header true plot index index header txt table txt header true txt xlabel k ylabel ndcg header table plot txt table header plot txt plot txt header plot title xlabel ylabel ndcg index header plot txt index index plot header txt index header plot index header inf ir title td xlabel ylabel ndcg table index index header plot td txt header plot td txt header td table index plot txt x header td index plot td index td index header plot txt header plot td txt title ylabel ndcg header td txt index td txt index td txt index header plot td txt plots td txt table true td txt table x header plot td table index header true txt minor title yahoo rank ylabel ndcg legend south east header true txt table index header txt index header true index txt true plot txt index index header plot txt txt index txt title yahoo xlabel ylabel ndcg pos
guarantee optimality converge exactly rank semidefinite alternate numerous suitably provably range dictionary prefer naturally architecture reference therein reduce present differ way consistently regularizer approximate generality us loss regularizer moreover perspective enable extend idea consideration broadly first must misclassification categorical say predict handle regularizers nonsmooth infinite generality consequence give algorithm variation minimization old pca minima organization first variation reader notation factor return heterogeneous technique abstract type dimensional algorithm row example column example boolean encode false consecutive encode categorical variable represent principled deal number principal warm variant seek approximation square solve k square square factor variable product use rewrite interpretation associate think use compressed original map onto objective na iy form xy reduce well obtain compact svd give orthonormal rewrite rank approximate diagonal well rank svd keep orthogonality u k x pca statement find svd value value familiar svd solution unique form pca analytical exist technique computer machine example iteration find record iteration converge mention method solve readily extension hold minimize hold fix guess na ij l ij condition unique objective iteration iterate stationary solution lie span vector column span vice versa orthogonal arithmetic iteration see appendix matter alternate minimization work minimization ij parallel ease exposition update compute gram outer compute streaming split entire matrix computation operation trivially worker add form cholesky factorization row row factorization worker require scale j entry find matrix ij entry regime miss simply exclude instead affected row strength regime relatively missing become rank surprising recover noisy certain incoherence minimize nuclear penalty fit rank regularize large nuclear norm encourage completion like predict customer consist rating customer vast majority customer product miss customer rate analytical alternate alternate quickly satisfy recovery value carefully hand sample alternate none use achieve minimization logarithmic reason plausible expect alternate minimization recover regularize pca admit interpretation course interpretation regularize feature vector use interpret think space example example row capture maximally might profile every example represent combination give coefficient th axis simple interpret representation intuition example cluster group might rather consist noisy miss column column represent combination might redundant row example matrix discover provide explanation summary building pca matrix generate normal noise sample posteriori explain recommendation simply map observation encode back good auto encoder bi linear impose low rank fit pca bottleneck interpret solution generalize critical ensure match error need get offset exactly whose extend pca arbitrary row form regularizers pca reduce pca regularizer express compactly matrix notation regularization separable infinite enforce example impose nonnegative depend regularizer satisfy vary choice regularizer wide regularizer turn pca convex bi fix regularize case minima however column parallel ix minimize I regularizers problem convex regularizer convex many analytical nonconvex see concrete subproblem program quadratic discuss factorization variation detail clarity strength course mix match regularizer different different indicator define indicator factorization np hard analytical specialized code replace easier interpret small understand penalty together wide variety could enforce entry column vector define denote nonzero regularizer pursuit branch bind reduce small relaxed convex regularize know component choose impose due matrix orthogonal conversely column mutually addition nonnegative eq let geometrically row ray pass nonnegative along orthogonal entry penalty norm matrix define equivalent require sometimes bound rank completion rank factor bound function encodes cluster assign minimization well know update update assign solve assign assignment indicator approximate subspace also interested datum think generalize assign centroid frame pca indicator nonzero solve block define correspond compute th subspace square sparse provable understand variation label predict procedure solution regression make sure model use supervise regularization regularize regularizer encourage column uninformative sense say feature regularizer feature dictionary design representation supervise learning dictionary linear atom represent atom fit solve pca pose notation usual dictionary regularizer one sparse nonnegative eq factorization go pca nonnegative make subtracting introduce allow offset r j trivial case offset included regularize slightly offset term modify regularizer extend regularization regularizer penalize first q form l j ij reduce problem bi alternate still speed subproblem discussion extremely problem alternate minimization direction algorithm allow generalize nearly detailed objective importance fit rank hard despite sensitive outlier replace least less problem interpret robust component large use rewrite huber huber place loss huber small result outlier previously matrix decomposition huber make ingredient observation huber log error regularize pca transformation replace huber obtain use similarly huber vice versa function loss th percentile may interested matrix less logarithmic may loss nice fit note least solution formulate exponential give family family pca kl correspond poisson model loss function property loss loss mean audio fractional error generalize recover loss limit recover kl divergence datum datum subtracting matrix preserve loss importance automatic scaling time loss nonnegative easy generalize column sample median deviation column form offset may trick encode offset regularization simply generalize table consider column draw discrete represent entry ordinal incur feature value give function ij variable regularizer convex bi separable minimization objective subproblem suppose observation low surprisingly accurately technical take regularization sum minimize minimization repeatedly name margin factorization also logistic measure quality loss fixing minimize label xlabel ylabel pos pos xlabel pos framework generalize low poisson family differ ordinal variable encode wish rank give ordinal hinge loss generalize ordinal ordinal loss loss encoding degree agreement question might fit ordinal increment bad agree bad neither agree ordinal learn ordinal determine agree neither agree suppose tuple denote interval miss sometimes map equal loss huber make sense abstract must still think boolean pca approximate boolean solution boolean say boolean boolean fill compute minimize lie domain losse huber type general boolean imply value use original datum lie interpretation pca interpretation think capture embed understand intuition may arbitrarily complex plot cluster generalize convert vector think non give agree say form value map representation back imputation think discover good benefit approximation latent good explanation generalize generate accord matrix random take entry maximum posteriori solve matrix encode auto encoder impose bottleneck auto fit bottleneck reduction giving maximize bottleneck efficiently store previous offset describe function note scale instead type much little adapt regularize alternate explain fully run alternate model identical normal compare square boolean truth round close show error run regularize show boolean pca draw entry hinge advantage boolean htb boolean htb boolean pca boolean consider boolean might customer medical disease negative unobserve positive rank draw let constant positive observation consist entry matrix draw random fit model ten observation usefulness predict top unseen figure path range generate see error reach improve increase since compute insensitive shrinkage introduce grey identify significantly particularly round auxiliary axis line leave width xlabel regularization ylabel legend north west view axis xlabel ylabel axis cs boolean censor grey line probability entry boolean column map thus correspond ordinal heterogeneous also fit produce fitting heterogeneous ground round close third regularize pca show boolean misclassification define notational convenience let truth draw truth heterogeneous mse heterogeneous pca mixed recovered block boolean removing describe truth block round entry close fourth show heterogeneous similarly perform censor compare difference ground ground truth table misclassification ordinal h miss datum missing regularize generalize represent abstract example permutation ranking loss embed loss approximation column formulate regularizer vector entry section suppose categorical q fixing optimize per separate label sometimes multiclass optimizing identify svms accurately predict categorical boolean miss entry project onto span simply interesting column lie interior column restrict use classifier categorical model corresponding function function imply acyclic one vs separable mean feature function fit boolean example combine rather perform well sophisticated perform nonconvex example minimization cluster rank see mode ordinal embed onto scale ordinal much similar infer label integer multi ordinal optimizing hyperplane separate level hyperplane fix optimize place appropriate hyperplane simple ordinal loss loss index generalize find permutation permutation pca example interpret deviation level strongly choose set ranking observe ranking modification literature correctly rank loss include area weight pairwise order allow scale loss american community survey fit heterogeneous population unite economic exclude year variable home boolean boolean house boolean water boolean health school currently school boolean high education high attain status categorical force boolean worker boolean year ordinal work week look categorical huber hinge hinge ordinal categorical categorical parameter offset select two minimize exclude group water california intuition hour work per week work per education similar california water hour work work education feature generalize rank computationally global optimum low rank matrix completion optimization implement per alternate notice variant likely saddle point explore alternate sense see discuss strategy provable sometimes show approximate show extend alternate generalize ny naturally algorithm loop may lot optimize iterative minimum sense early replace rule towards empirically local perform computational rule write fact switching role still execute parallel example might replace size rule globally subgradient value regularizer take say represent use proximal method indicator onto set globally optimal fix sufficiently small technical step update globally critical mild f proximal prox prox prox minimizer proximal gradient global minimum prox prox admm initialize objective see prox update make iterate admm mean subproblem alternate allow step fast progress towards convergence ensure size motivate grow row step choose objective decrease decrease reported size increase increase decrease check prevent poorly decrease adjust compute operator sum update per iteration job onto replace gradient among ij less implement prox objective exactly update take ease invert gram prox prox objective gram prox prox take advantageous many form example nonnegative square project alternate converge objective example explore serial alternate regularize entry draw initialization normal entry come trajectory convergence factorization zero plot objective optimal initialization trajectory converge fast somewhat substantially ylabel xlabel coordinate coordinate coordinate alternate regularize pca htb ylabel xlabel coordinate coordinate convergence proximal alternate converge converge differ significantly initialization problem discuss scheme extensive provide provably completion algorithm minimization show value choose carefully initialization consist svd datum give construct motivated key insight zero construct take expand categorical column boolean interpret scale propose insensitive sensitive ordinal make encoding initialization mean whose version change column decrease truncate row th instead triple compute triple time compare initialization initialization census detail six five initialization iid converge substantially behaviour indicate random view width xlabel ylabel coordinate coordinate initialization svd mean initialization scheme initial centroid centroid choose minimum previously choose centroid quadratic even alternate expectation randomization initialization initialization center arbitrarily bad proportional loss distance metric important non convexity argument one find factor nonconvex regularize efficiently sdp rely subroutine lagrangian problem rather saddle nonconvex consider global present solution compare z following constrain factored svd u xy x use inequality rely third taking orthogonal frobenius note orthonormal still solve solve optimality globally value least problem solve subgradient subgradient nuclear equivalently function nuclear rank result particular globally objective pick globally suppose entire frequently regularization achieve initially fitting value norm fit correspond call regularize huber draw normal entry draw uniformly huber quadratic normalize error eq fit decrease reach interestingly second path view xlabel ylabel normalize error regularization path needs specify regularizer rank domain expert intuitive fit hand regularizer choose consideration model generalize unseen consideration balance discussion regularizer table compression pick give ratio low observe error well high low error rate achieve require analogy aic degree freedom rank compute example regularizer transformation number degree propose dimensionality observe perform cross observation small entry true contaminate may choose l distinguish method dropout linearly eigenvalue increase decrease parallel objective compare fitting model draw cross simple apply denoise leave entry leave present understand explore wish similarly loss consider fitting miss neither discuss resample validation cross indeed distinguish rise chosen row case index make sense reasonable rise scheme validation resample row matrix far resample bootstrap residual number resample generate pca reference therein example explore vary draw draw outli draw pick huber result draw perform validate qualitatively interestingly difficult rank xlabel
theoretical precisely rate worker recent line generalize allow subroutine coordinate optimizer result different update moreover variant entirely trade choose optimizer internal procedure duality fair discuss empirically loss convex possibly regularization class ordinal problem become iteration access understand associate conjugate define dual conjugate come convenient dual primal optimality configuration duality serve useful criterion form prove primal sdca make completely software package prove suitable primal superior defined stop loss problem worker dual efficiently merge worker disjoint h distribute k k kk arbitrary apply local internal try observation worker state compactly represent single ever machine dual initialize sdca single h subroutine local round dramatically worker outer suggest dual coordinate sdca optimizer practice partition across worker write na rescale locally worker iteration determine trade block optimum geometric alone scale block coordinate outer iteration machine let dual objective q satisfy eq eq show tight special interestingly subproblem optimality let serial main g recent generalize distribute practical internal descent dual coordinate inner notion sub address without local round communication result solution rate full loss experimental encouraging yet provide quantitative gain communication efficiency compare analogous experiment section first parallel subproblem solve essentially rate journal quantify show local improve potentially interpret include update mini datum example iteration mini update add worker study naive variant essentially identical mini descent difference define sub suffer immediately furthermore average instability illustrate coordinate sdca difficult choose especially parameter range many convergence safe convergence method linearly per worker practical order small mini one consider local author compute communication matter accurately subproblem machine sometimes communication online divide inner perform outer early framework update delay insight study sum directly case spirit implement communication process coordinate descent analogous time communication machine require communication experiment large convergence strong sparsity understand communication round classifier mini ascent stochastic locally mini batch sdca mini mini batch update iteration update size mini batch sgd machine scale tune dependent hinge machine implementation amazon ec instance though smooth analysis remarkable converge regularization r imagenet compare analyze progress primal compete size performance locally size process batch batch mini batch sgd averaging become quickly account correlation spend process number indicate accurate ability avoid still computation communication node describe communication also affect convergence attempt scale small mini communication framework dual ascent solve minimization distribute machine communication amongst costly show real world dataset rate update safe remain rate efficient describe discussion institute give block everywhere outer worker procedure local improvement function following rate outer concavity subtract side maximizer consider ready procedure choose improvement inner optimizer core primal structure coordinate ascent change restrict notation index worker us valid reader status active block idea constant observe duality th dual formulation local dual k optimality derivative inner plug back minimization write precisely dual coordinate duality gap contribution active collect iteration identical keep subsection h I k strong convexity hence q primal eq prop function smooth q local subproblem
algorithm concave fairly weight right center expect flat interval concave symmetric consider measurement north water describe capability low loss previously study amongst place sequential normal posterior related figure old left bin log flat modal density method center think component flat measurement log component flat symmetric concave next population study di interested identify isolated population population genetic homogeneity uniformity style valuable diabetes disease simplification isolate population gene genetic homogeneity individual relative environmental isolate necessarily large estimating model component alone concave component maximum plot relate plot around estimate multi component approach capture near include individual use plot center label center value need especially near center density shape make log component semi location unknown motivation choose totally bandwidth kernel kernel consume offer section enjoy easily active implement likelihood know mode enable put concave estimator concave mode constrain location mixture probability location estimate parametric find estimate log concavity would find maximum component impose challenge derive convergence involve expression unfortunately likelihood sophisticated overcome difficulty conjecture remain alternative validity introduction may tail appropriate concavity however development concave density barrier concavity heavy tail concavity sensible result present numerical keep reasonable introduction location identifiable identifiable symmetry certainly impose concave natural log concave argue non class hard price pay necessarily piecewise flat transform consider function piecewise clearly admit e logarithm mle necessarily piecewise flat maximizer exist vector n order u u imply u u put respective put x decrease big satisfied integer let small j j increase reasoning conclusion dt last contradict definition perturbation x x z f dx nx nx become straight line enough z nx nc pc denote fix convex simplify notation respectively g fact continuity imply converge nc u g nc g nx gx nc pc nx gx p denote g consider g imply complete proof b function decrease large finite show apply lebesgue dominate need support enough take infinity integrable integrable since condition large integrable respective mode component respectively restriction easily modify h c function monotone dx dx admit function decrease fix probability monotone generality increase bound case handle monotonic ga gx gb class fx nx hx nx many proof go proposition first restriction respective c x respectively h dx c dx dx admit fix symmetry decrease reasoning side large convex sequel k c find write event n u n q hence enough decrease include occurrence imply turn argument event nx u occur great event e occur great depend g x dx g dx use fact concavity logarithm log put g pm c proposition last subsequence extract subsequence integrable f inversion inequality turn dt indeed dt dt f inequalities dt increase hence extract subsequence weakly assertion proposition surely follow arbitrarily net second hellinger u u j h j u fact conclude complete k n u mle page section criterion claim concavity logarithm n g associate nr n gx gx n nr obtain fulfil density choose maximization take approximate probability let proposition consider j n occurrence n n pg r pg n enough hence hence put r note turn imply triangular rate normalize write pn proof triangle n p pn g pn inequality decrease symmetry f f inequality property dt u increase hand precede display handle give expression switch role take switch switch role ratio stay stress fact ratio remain configuration handle ft dt rate dt u u f dt dt pn assumption ft ft dt ft ft ft ft dt ft ft dt ft ft dt ft ft ft dt ft dt ft dt ft dt ft dt ft reasoning definition dt dt h ts use author calculation j j definition mixture concave nonparametric log concave mixed shift location establish converge distance location probability unknown package mode article symmetric concave mixing nonparametric mle hellinger location mix establish mle truth shift unknown use package supplement technical material concavity number theorem statement unified document independent identically cdf q common modeling part framework problem reason popularity flexible major difficulty work multiple avoid thereby leave question another add restrict model restrict symmetric focus largely many application restriction identifiability make mixture although identifiability detail main parametric parametric asymptotically however property build progress focus take constrained modeling concavity gain major require contrast nonparametric make smoothness successful inference tuning parameter statistically different choice mle tune difficult verify practitioner believe unimodal distinct assumption appropriate practitioner choose poorly specify parametric log concavity often surrogate log density unimodal log additionally familiar family approach unified approach handle mixture benefit concave estimator density concave turn distribution converge mean log converge log heavy tailed admit smoothness parameter impossible concavity concavity base concavity assumption effective require modification concave estimate component concavity consistently log concave sample global mle behavior estimator includes derive use construct furthermore programming assign symmetric concave log transformation log density log mode mode active set proportional probability em able mle almost symmetric hellinger supremum true provide mild assumption although use true fast kde concavity modeling mixture concave impose perhaps fundamental pure feed implementation maximize component likelihood difficulty already primary property rather make presentation add detail establish hellinger form however deal mainly bound maximum likelihood likelihood symmetric special compactly support stem exclude convergence concave smoothed version first section absence mode critical bootstrappe concave ratio trace log concavity end include cluster performance log concave algorithm assign corresponding great application conclusion proof base minimization distribution estimation symmetry parameter zero kde give mixture u consistency rate density writing fourier estimator mix q support maximize overview cumulative cdf explain follow dirac probability let imply location symmetric x minimize empirical simple equation system drawback three aforementioned consistency rate assumption show regularity condition obtain smoothness rate convergence estimator supremum location kernel estimate smoothness choose optimally specifically notation I nx x observation nb b symmetric concave nonnegative concave concave hellinger distance dx integrable g fx gx measurable come location subscript commonly fix symmetric mixture maximize class penalty proposition estimator estimator admit maximizer ready symmetric concave mixed density f pn pn three mixture consistent encounter estimator adopt approach pose package latter author david private communication description estimator u theorem identifiable adopt write associate rewrite identifiability equivalent origin u u minimize distance origin jt thus estimator mixture sm code indicate function use generate number minimize maximum n n decrease initialize start center variance symmetric n case moderate replace simply value mode z z transform concave maximizer z z I maximize stop smaller choose equal need method density simulation follow ny ny k independent u u knot dx nx dx x ny j ny ny j ny ny ny ny ny dx calculations n ny ny ny ny ny j smoothed concave mle choose smooth mle always concave occur mle characterization univariate extend explain bandwidth pass yield j dt u absence component definition mixture symmetric hypothesis exclude concave mix ratio statistic nan around denote statistic hypothesis reject use bootstrap concave detail usual statistic quantile aim nan underlie concave log concave density identifiable around mode possible density expect differently indeed assess standard beta give nan hypothesis ignore powerful detect tr whereas tr easier distinguish lr power especially less scenario trace anti conservative behave level differ size unclear cause normal notice log concave tr type lr nearly laplace concave difficult regardless counter distinguish follow g leibler divergence concave density nan follow leibler exist tend
previously categorical majority patient concept categorical dataset work identify characteristic particular time look yet appear history broad presentation component source manually massive chart imagine present raw infer factor rich result plan acknowledgement work grant foundation national health lm clinical support grant asynchronous nature medical away categorical modeling longitudinal function several infer event intractable duration paper infer interpolation twice accurate method regular finally demonstrate abstraction amenable algorithm medical pattern problematic record categorical get record contact activity laboratory visit stay event would like thing clinical contact disease often apply away stream stream stream abstract longitudinal intensity contact density process intensity raw standard previously unfortunately infer applicable categorical intensity monte intensity event approach find flexibility scalability datum event stream unable adapt flexibility good property datum continuous clinical process event assume independent iid drop iid add longitudinal intensity event model work gamma time q homogeneous interval think event time space draw well simple poisson specifically process produce clinical behave highly regular intensity ft square smoothness covariance generally must inference assume homogeneous similarly ft place uninformative prior set desirable avoid degenerate intensity parameter slice surrogate compute incomplete interval end challenge direct require certain log gaussian integral numerically smoothness efficiency bottleneck compute need instead time driving scale intensity interval work nearly medical cubic period fine resolution year inefficient bin poisson infer intensity neither intend suit form abstraction raw discrete hazard efficient intensity span stream intensity big medical range several flexibility exist refer integration avoid computing intensity shape function stream clinical event parametric function piecewise event intensity interpretable raw intensity compare kernel smoothing method assume poisson compare assume test intensity burn inference intensity shape efficient magnitude rt accurate true might sensitive available prior mode near tune slightly mode follow advantage next synthetic generate medical amenable event intensity gamma consistently accurate htb patient record infer marker aid clarity division clarity confidence lastly direct sequence represent clinical code five great mirror medical record arrange stream disease broadly level stream event include code division event strictly still group event informative intensity figure
sample value base reject acceptance construction enough sample large augmentation begin increasingly rely become convergence stationary figure I ambiguity branch barrier energy convergence initialize period check converge stationary converge check leaf grow monotonically internal node barrier monotonically barrier update initialization match match node penalty omit definition important figure decrease mixture two influence separability supervision ii behavior performance maximization wang synthetic mixture derivative landscape partial derivative computation restrict inverse gradient definite equal need restriction possible step gmm I matrix symmetric project space symmetric decompose ensure point possible range component upper unbounded sample gmm run walk location stay boundary gmm separability separability gmm overlap often measure difficulty true show minima landscape separability prominent landscape supervision assign ground truth label portion label separability become much label minima labeling decrease landscape supervise lm behavior separability expectation popular pattern separability condition wang cut generalize probability em step energy approximate learning include comparison synthetic run time energy vary em gmm landscape local minima energy em cut sample low separability good converge random almost always find outperform separability majority local high confirm showing inductive bias separability energy observable gmm distribution energy gmm encourage low gmm easy separability run repository gmm represent linearly separable remain linearly visualize minima minima energy first merge local minima overlap rd nd rd run percent label assign landscape figure scatter pattern illustrate template faces lattice denote typically sketch image template template template simplicity assume template energy boolean experiment propose em template template experiment take value use template represent face face align grid cell cell contain cell location sketch straight connect cell possible detect edge binary generate minima landscape heat map expect minima increase particular noise landscape convex local experiment face swap face head thereby template degree overlap generate various template degree minima overlap template face extract prominent eight filter different corner filter strong fixed correspond element dimensional response face cat equal version image model template template minima corresponding identifiable minima face across explain energy face mixture run close count minima display bar minimum face bi process bioinformatics e use find movie occur phrase bi multiplicative share element see graph mixture component conjunction co theoretical co show observation bi bi matrix bi matrix bi bi identify bi goal bi explain therefore instead adapt element row bi term coherence minimal proportional correspond prior bi add exclude entirely zero bi element varied random background point construct overlap maxima cluster bi mark red circle bi mark gray maximal marked green regime learnable bi minima regime dominate biased bi weak energy level true although approximately difficulty difficulty difficulty visualize choose landscape cluster convex energy problem separability level supervision level strength landscape algorithm worth explore repeatedly adjacent branch gradually structure landscape low supervision previous next conjunction dimension study challenge thank suggestion wu discussion acknowledge project statistical highly non difficult analyze paper leaf barrier adjacent mass volume correspond construct adopt wang dynamically classic mixture template ii study landscape separability cluster supervision visualize behavior k em step wang cut optimize research replace regression designing algorithm local property inspire spin model problem bi figure barrier adjacent characterize landscape energy intrinsic problem either task bi hard impossible regime complexity separability percent algorithm show frequency various minima find highly separable work well less separability htbp htbp illustrative component denote construct range visualize keep map landscape cause component like little finite show identify mcmc sample cluster minima leave minima landscape b energy work multidimensional efficiency notably generalize wang effective barrier bayesian segmentation belong fall example upper collect construction kb adjacent collect consecutive across eq next iterate barrier barrier ridge descent structure energy modify agglomerative initially leaf coordinate minima new parent whose energy barrier regard barrier merge energy merge complete structure clarity remove less energy
diagram bottleneck distance range individual yield wasserstein et linear speaking establish old wasserstein distance map apply operating string persistence induce kx kx yx lipschitz continuity particularly exist hyperplane perturb separate persistence diagram define half plane motivate persistence diagram possess persistence uniquely dirac delta dirac functional hilbert adopt unfortunately induce account perturbation diagram motivate dirac heat diffusion dirichlet condition diagonal equation persistence scale formula wasserstein differential sense rigorous persistence diagram persistence empty diagram linearity map partial differential equation replace diagonal restrict original initial closed simple q derivation visualization solution wasserstein wasserstein denote persistence diagram diagram persistence diagram assume achieve wasserstein leave decrease adjust accordingly influence cause beneficial question extend call diagram additive choosing say sharp stable persistence diagram non sec investigate conceptual difference sec texture sec persistence diagram banach intend computation space hilbert structure kernel induce wasserstein persistence diagram eq persistence range think let diagram point point move diagonal increase consequently unbounded mean persistence value response compute persistence diagram detail persistence diagram handle margin svm implement ten fold validation average ten validation split distribution result choice explain smoothness scale allow stage extent capability rely suitably choice carefully adjust undesirable consume even additional c performance synthetic shape evaluation allow assess kernel induce distance list measure query shape neighbor scale list similar classification specific time input induce another unstable top drop retrieval list synthetic rank top five entry topological persistence alone assess performance improve elaborate fusion c top training informative train normalize histogram response table evident perform margin gain apparent topological alone histogram persistence persistence conventional representation lead nature topological feature combination lead gain include assess illustrate curve selection cross validation discretization drawback shape could operator strategy would change contrast adjust c texture svm validate theoretically exhibit task texture tune parameter prove practice future would address scale include leverage error summation wasserstein useful persistence wasserstein persistence diagram lead topological topological machine area topological ne set nest nest offer rich source vision theoretically kernel pca establish design persistence diagram summary topological wasserstein texture compare persistence visual surface analyze homology roughly persistent homology capture birth death enable theoretically sound representation kernel computer vision etc processing task appearance descriptor sift high activation convolutional feed popular vector technique progress extract discriminative recently start become readily demonstrate capture characteristic often along study persistent homology birth death etc homology input lead change wasserstein persistence topological surprising persistence diagram wasserstein metric employ persistent homology obstacle typically hilbert space possible persistence diagram wasserstein main propose positive diagram fig feature idea theory map wasserstein thereby maintain stability persistent homology robustness applicability shape texture benchmark method topological vision medical group first identify utilize topological specific identify information topological input representative adapt surface shape drive topological persistence diagram persistence cycle extract segmentation chen propose topological segmentation category investigate surface contrast persistence directly feed discriminant control patient disease topological typically instance diagram regular grid eventually density kernel unclear induce bottleneck wasserstein directly recently bottleneck wasserstein distance employ correspond complementary persistence combine bag inspire focus development persistence propose diagram motivate algebraic geometry algebraic function death persistence conceptual probably close another diagram persistence design machine sec show admit persistence attractive alternative bottleneck matching review fundamental notion result persistent homology relevant topological grow sequence shape growth shape gap etc give rise point formally persistence diagram detail every diagram
proceed precisely eq old chain bind equality hence equality claim attention fix imply qp qp eq union one subspace bind first valid let put thing proof make small step shorthand whose z generality finish attention sufficiently q q observe combine noting conclude observe prove converse follow prove finish pick q hence arbitrarily characterize regularization uncertainty correspond parameter magnitude let except scenario easy compact non interior analyze another regularization variety rely decomposition leave singular natural question claim find equality inequality strict bound gap proposition consider subject generalized know similarly eq general regularization strict low proof proposition summarize function equivalence always qp rp proposition equivalence g throughout row connection regularization sort residual quantile despite hard property possible provable hour advance robust formulation structure nominal formulation recall common formulation remainder separability norm summarize p ng g c norm norm separability maximization form summarize particular norm likewise part complete easy duality therefore q observe recognize convex conjugate assumption conclude implication norm norm proposition operate analysis begin state main theorem choice exactly reformulate mixed choice norm linear integer program formulation mix order nominal use formulation nominal uncertainty norm satisfie reformulate program formulation formulation bit care solution maximization unique q direct tucker condition equivalently write reformulate programming eq reformulate exactly formulation substantial problem core development involve variable prominent completion principal pca common nominal uncertainty expand exist regression novel substantial class introduce uncertainty completion e interest constrain appear wide netflix preference give important parsimonious description user preference term convert nuclear convex uncertain f frobenius pca arise low plus well truncate popular application robust low rank entry pca form penalty rank spirit sense recovery small sufficiently explicit expression certain uncertainty additive uncertainty concern model uncertainty different assume uncertainty measurement ny ij entry matrix note nominal subject linear ij ij form direct analogy clarity interpret albeit cost avoid vector uncertainty induce induce many choice interpret hence depth truly directly continue theorem model uncertainty uncertainty note result exactly therefore remainder q restrict induced uncertainty begin therefore without note ng h ng case conclude theorem subsection follow implication matrix frobenius corollary recover completion note shown arise directly sparsity nuclear envelope ball nuclear directly may arguably induce detail penalty nuclear appeal convexity attention begin note f eq continue present linear observation combine imply uncertain pca entirely model usual replace surrogate respectively regularize uncertain pca appearing impose penalty summarize always regularization completeness regression state precisely proposition n qp qp bind strict long gap low attention non arise model plausible one column examine j matrix multiplication completion netflix treat true rating address within allow user rely early equivalence vector regression characterize modification theorem proof q jj p uncertainty possible regression variety long conclusion section equivalence q equivalence problem modern take directly understanding least quantile regression completion emphasis modern massive scale statistical modern work broadly rely estimation engineering science inherently desirable effective inform corollary conjecture office science engineering fellowship penalty beyond contrast appear plus regularization method reliably solution paper extend loss show regularization completion modern lead one especially nuclear minimization compress practical scalability due advance variety statistical linear early goal median matrix contribution include uncertainty consistent provably appropriate known restrict nominal appropriate method perform median demonstrate problem regularize integer principal characterize uncertainty regularization paper section background focus equivalence turn regression matrix variable consider depth conclude regression homogeneity dual transpose norm define analogously eq denote norm also frobenius spectral induce definition reference norm norm entry analogous spectral norm value eq contain singular true penalty precise summarize p n imply without replace continue another fix eq q q replace completeness eq subsection comment learn relevant domain
mechanism relation law many physical follow clear experimentally measure provide description behavior understand interaction material study phenomena e behavioral single hope nature implement successful search important regard automate typically law science find law demonstrate truly automate match study dynamic understand key underlying mechanism law law unknown grain description dynamic mapping interaction typically system one specify hard simple dynamic sir optimal proceed important center study national laboratory support grant laboratory program grant supplementary ref hierarchy maximum fluctuation never take away arbitrarily sufficiently search multidimensional model single predefine path suboptimal true fall even guarantee eventually dynamical done power system dynamical govern ordinary differential production degradation call rewrite variable correct e taylor integer value predefine space find model biological example outperform typical production degradation grow bound force selection discard negative network among biological often system show approximate variation rescale switch linear combination traditional advantage system existence hierarchy simply interaction consist add hide without connection specify dynamic variable ii ji g ii example form law x dt I dt h dt x x dt dt x h dt g g seven fix connection parameter vary time add dt x dt dt x dt x dt dt I full prior normal standard deviation motion mass evolve specific angular momentum velocity vector connect initial measuring distance unit rescale equation system represent exactly system result never construct adaptive input cover circular elliptical determine time fig ensure would hence requirement reliably motion mean initial time observation deviation equal maximum value typical adaptive fitting hide certain transformation leave initial parameter rescale loss second shift parameter perfect dynamic compare performance fit mechanic result network still generalize range contain range fit true limited multi site treat input measure single treat total dotted circle error circle vary motion include correspond know likely well show datum contain leave dark blue part divergence correspond right half blue part trajectory explore complicated biological relatively imagine five site arrange rate site affect state nearest neighboring site reaction specify total easily site log min measure site uniformly minute add noise min compare evenly space guess functional total hoc quite well amount well dynamic site example system typical performance fig class qualitatively correct close interval notation section metropolis carlo multidimensional hessian fit ratios monte step remove avoid condition every inference complicate originally automate dynamic seven molecular parameter value ref ref mm min min mm mm define solid line circle interaction right fit measurement circle indicate strength clarity self show c ic sample ic mm initial noise visible specie ref range range reference set cycle ref visible specie choose range list ic table visible minute mean measurement table input uniformly sample ic evenly actual separately visible set condition plot assume aside measure assumption relax assumption variation learn ref engine infer roughly detail attempt specie specie time costly integration match remain unconstrained ref fitting constrain dynamic entire space produce main range ref ability tested compare sample initial range note appear see infer sample twice range plot choose range narrow range plot standard symbol derivation set measurement aside prior constant normally error goodness square residual normalization constant pm p since fit ref sufficiently constrain limit dominate near approximated derivative jacobian measure generalization term constitute overfitte goodness penalty individual selection know general section residual parameter integrate parameter use ensemble routine parameter sequence fit number consistently decrease stop small calculate metropolis monte encourage acceptance ratio integration treat starting else default candidate space isotropic residual hessian previously less ensemble start member perform stop detect norm step member ensemble fit temperature temperature model full model member full conservative achieve use multi example fast evaluation search model calculation evaluation gradually depict size case evaluation scale model plot compare parameter number infer plot direction constrain prior expect stay realization dynamic drive network molecular understanding limited extreme hoc define propose construct coarse grain dynamic automatically adapt adaptive insufficient computationally dynamical variable software realization sir unobserve motion overfitte biological produce half interact specie unobserve mail edu biology field complicate vast amount demonstrate encounter physical datum cat success generalize insight resource sometimes impossible large yet unobserved molecular structural involve intensive easily challenge unlikely solely accurately dynamic bre predict response system perturbation disease agent system time early day natural engineering result successful continuous dynamic dynamic recurrent nonlinear common biology unnecessary approach possible effort especially couple move note complex parameter structural guarantee perturbation hope accuracy coarse grain propose adaptive attempt well interpretable possible account restrict hierarchy complete gain theoretical meaning able adaptively able happen search believe infer complex dynamic search space polynomially computational resource construct interpretable much effort experimental variable unobserve sir input intrinsic dynamic random system repeat condition series save field trajectory g elliptical familiar statistical representation form create gradually complexity fit sufficient ideally much one hierarchy complex desire material som criterion dynamical along two add variable study within two match time som degradation form power mass law class interaction similar reaction represent hide perform bad input show deviation adapt red fit second rely intuition manually parameterization salient see som exponential third create fig approach varied fit fitting bad prediction less subtle accommodate subtle axis b time model blue overfitte red prediction median dark line median behavior sample parameter som confidence site demonstrate sir rather true prediction qualitatively different infer reasonable overfitte evident detailed average complicated interest system inform knowledge pathway consist specie perturbation make ideal sir circle measurement observable
conventional cloud processing quantization theory interest detection operate clutter detection interference optimal pearson np sense np theoretic criterion kullback leibler design set perform receive place environment sensor fc capacity channel cope limitation communication sensor receive transmission fc operate receive refer system cloud cr mr tackle problem jointly optimize code operation receive adopt quantization fc base investigate seem optimization code quantization bold bold letter determinant random column denote symmetric mr system clutter receive fc limited capacity present accommodate capacity letter available communication fc receive capture channel share form code large control clutter sensor type return envelope fix interval target envelope observe clutter homogeneous receive match q respectively represent absence target resolution cell useful part receive target clutter clutter complex amplitude accounting interference distribute respectively fc e receiver fc receiver know whether target facilitate standard effect quantization quantization signal receive th quantization sake quantization communication fc form receive additive sensor n c alarm aim code quantization end sake resort theoretic detection capacity requirement therein adopt receive leverage zero n make explicit assumption approximate argue distortion discuss requirement mean theory operate asymptotically hypothesis evaluate adopt measure bit fc dependence mutual maximize metric vector covariance capacity formulate minimization distance n convention exceed ensure total transmission receive theoretic difficult solve obtain global locally accordingly th code iteration feasible mm repeat step still require resort mean purpose converge convex convex locally bound iterate convex function feasibility iterate local optimum emphasize use identify outer loop index discuss application goal end mm locally b h easily x x lead attain step matrix desire evaluate follow convex iteratively find equality vector use algorithm optimize jointly table throughout receive length code variance clutter n k fig gain properly quantization beneficial plot operate characteristic alarm versus probability bit evaluated implement detector remarkable gain
key gradient simulation adaptive sequentially mean noise signal model element associate canonical adaptive subspace dictionary update typically fix instantaneous instant coefficient cost coefficient ed n nd rkhs respect optimize optimize rkhs subspace short optimize filter span singleton thus derive follow gradient induce respectively inner element linearly definition observe j correspondence note correspondence follow q give ascent direction subspace plane gradient j n natural make impact selective serious selective work theoretical mse stability kernel multiply side nj rewrite regard descent q cross modify respectively weight guarantee make recursive integer natural spectral less say square tend state regardless initial condition complete steady positive verify tend mse mean mean stability remark simulate learn conduct generate additive parameter mse selective update parameter theory steady mse selective selective mse depict steady dot respectively simulate obtain average mse estimate steady computed theorem input correlate experiment consider dictionary selective generate dictionary element coherence criterion experiment depict validity multiplication natural selective see simply mean hence drastically mse selective exhibit mse drastically low stochastic descent method steady meet outcome serve basis present behavior analysis give ascent direction provide mean square include stability normalize initially chen et validate reproduce attractive rkh adaptive filtering classify perform rkhs ii normalize example steady square mse present rkhs counterpart natural natural distinguish primitive gradient descent theoretical eventually relationship two adaptive orientation give short rkh
accurate accurate term leave give value occur binomial square write distribute doubly variable draw must binomial network small activity unlike good nonlinearity though attempt reduce walk show vector give produce unbiased predict compute start well generate blue red variance walk predict numerical equation show nonlinearity average variance nonlinearity provide value due growth use vector show solid dash provide practice right panel scale propagation randomness world far initial final need adjust separately affect output layer summary walk scale generally tune entire far important training deep mnist equation mnist multiclass mnist auto frame depth generalization bias width actual value value possible deeply also narrow encoder layer middle pick increase total number first lead integral layer layer per compare depth varied ensure deep varied essence function mnist result agreement figure good agreement analytic calculation middle panel goal second assess error focus nonlinearity believe utility scale value tie place train auto encoder achieve demonstrate scheme mostly layer classify mnist mistake random walk initialization figure result mnist good training among test tie depth improve task examine nevertheless deep real world classification mnist auto hyper mnist show epoch hyper deep network curvature average discussion imply correctly network layer one simply decrease derive equation correct avoid regularization reason walk initialization different deep bias initialize always allow modify bias care bias quickly happen careful initialization forward progress schedule huge landscape layer optimization indeed deal improve nevertheless allow network go broken hyper zero extremely reconstruction depth whether difficult task use deep feedforward way apply regardless deep feedforward network initialization accord value sensible initialization acknowledgment thank le york ny usa deep learn difficulty decay feed forward difficult previously unlike recurrent amount matrix scale initial result vector compute make walk square vanish gradient optimize relate mnist since early neural suffer vanishing refer fact feedforward back increase final recurrent rnn back error exponentially vanish add layer rnn network vanish recurrent recurrent network back involve matrix repeatedly process lead produce lead vanishing achieve lost process go rnn suffer vanish gradient magnitude scale square network feedforward apply initial network train also address vanish gradient mathematical training analyze successive analytical hold propagation equation procedure norm feedforward activation transformation bias depth wise nonlinearity normalize scale network layer length far initially otherwise initialize define input assume back propagation eq evolution square back become apparent entire magnitude across network layer magnitude gradient keep order appropriately adjust adjustment experimentally gradient variable random variable product approximately log case fail mean interested avoid issue instead equation walk walk unbiased equivalently describe logarithm norm back back output like vanish independent happen back rather vector
definite attribute numerically unique transpose subsequently negligible loss numerical kernel normalize value call traditional inductive overfitting ec post ranking make symmetric average transpose algorithm compare ground apply commonly retrieval ranking return argument negative roc order class accuracy reason firstly unlike ec hierarchy account determine different ranking ranking rank count comparison optimize software use characterize important traditional kernel traditional ranking know evaluate bipartite ranking e ranking relevant area roc auc discount gain ranking convert bipartite ranking retrieve relevant ec section truth compute datum set validation fold use individually allow comparison query remain ranking average fold fold three every fold cross validation neither database demonstrate loop implement control contain power final fold train cb fp ndcg map ndcg ii cb ndcg supervise ndcg give global summary obtain rank difference cavity despite considerably hard fact certain site active site expert active site choose cavity mistake functional similarity hard cavity rank supervise cavity cause accord cb data though relatively well cb clear explain easily cavity representation information moreover maximum subgraph node loss resolution drawback though solve size subgraph bind site lead slight hand transform moreover treat every equally emphasize rank approximation optimize auc coincide latter make distinction ec use severe performance measure set auc close theoretical cavity similarity case curve apply cut digit ec scalar roc information ranking immediately supervise rank unsupervised rank former left corner show certain relevant high sensitivity specificity detect without mistake step indicate offset curve fp roc section slope hard detect unsupervised straight detection concave ranking score ndcg auc ndcg explain map I ec cavity require perform figure quality measure low ndcg bad top depend application might interest since bioinformatic vast protein reliable geometry fold regard alignment template also biological close return program despite powerful become nevertheless inefficient similarity score focus protein protein account protein protein network try neighbor optimization connect protein share graphical markov field conceptually might prediction base cavity site ec annotation substantially rank take truth ec contrast rely heavily similarity annotation account focus cavity sequence work meaningful similarity could demonstrate indicate highly nevertheless supervise unsupervised cavity similarity match query ec well preserve ranking supervise powerful alternative retrieval traditionally bioinformatics acknowledgement ms acknowledge university bioinformatics financial foundation cm structure biological science database usually look surface biological annotate ranking kind improve approach similarity active function approach annotate training outperform measure ec hierarchy annotate training experiment consistent improvement similarity measure surface modern throughput molecular biology generating ever annotate remain challenge hard despite automated decade service often tool rely notion vast measure calculation abstraction solely take fold bind measure able condition prediction protein sequence alignment reliable sequence comparable exhibit sequence reason structure become database protein gain increase attention secondary know biological contain miss calculation fold protein coarse often responsible show diversity show functional many local structural evolutionary appropriate surface know valuable information similarity help bind similarity particular drug discovery provide family allow protein highlight bind site approach relate substantially improve training build mathematical ranking construct demonstrate cavity machine often learn popularity bioinformatics discovery protein solely rely ranking without utilize annotated search annotate database make amount transition four cavity base base input algorithm improvement use learn ranking ec hierarchy commonly ec adopt detail importantly focus chemical homology ec number truth subsequently truth ranking annotate fair comparison traditional approach way evaluating characterize engine algorithm ec number work unable output nonetheless instead predict ec query ec provide generally ec encounter build automated detection storage bank chemical first predefine spatial characterize geometric particular spatial bind functional group protein structure rule mix pi interaction property regard compressed surface protein encounter consequently representation distribution chemical ec experience expert choose ec generate I retrieve protein ec set protein ensure unique set protein server protein pairwise homology filter result cardinality extract protein assumption large bind site contain center take bind site maximize ray protein remove result drawback bind centre determine bind site protein sufficient resolution select low rely expert ec datum set resolution structure condition eliminate ec accept ec ec ec ec ec ec ec ec ec ec specific ec ec ec ec ec ec ec ec ec ec ec introduction restrict measure multiple transform unfortunately boolean flexible paper specify way greedy programming unfortunately quite kernel gain lot realization less protein poor representative category measure spatial denote transform remarkably calculate superposition derive alignment apply geometric hashing several point unfortunately cope biological also represent cavity feature geometry cavity see subsequently protein experiment representative base also protein five different explain processing label directly transform another spatially specifically approximate superposition structure fix cloud second cloud well cloud match point cloud small fitness maximize fitness taken label suggest similarity representation label cloud transform label become chemical geometry consider adjacent measuring common subgraph define relative graph define recommend make consider large subgraph large common subgraph use determine means protein post rule derive use protein use comparison bind site transform protein bind site moreover feature namely label weighted label label perform edge graph represent protein bin pattern mean protein bind site alignment sequence sequence subsequently perform introduction explain service construct rank similarity annotate annotate database similarity
allow give known support know solve obviously equal require allow little tolerance distinguish indeed vs show version conjunction poisson binomial albeit moreover recent applicable vs interesting seem optimal evidence binomial observe identity single latter try solve problem whether answer allow subtle identity turn make big hope distribution enough identity identity proceed first learn constant decide restricted support identity tight support overall complexity look reduce testing test feasible binomial sample consider translate poisson translate poisson get heart perform estimate indeed possibility distinguish logarithmic total variation reason argue use solve boost repeat majority basic logarithm interval distribution place mass bernoulli distribution pi variation allow unknown binomial use output binomial deviation within standard deviation run author poisson estimate mean variance sample poisson binomial indicator chernoff sum indicator bernoulli variable poisson random obtain chernoff poisson eq binomial deal interest next translate part bound poisson binomial bind via let random eq variation translate poisson follow translate distribution provide start run theorem outline give sparse heavy variance separately first identity describe finite compute bind enable simple give output distinguish imply exist length check mass outside identity distinguish level plan argue variance mean translate enough preprocessing follow close closeness proceed within factor heavy know probability go good variance compute estimate give enough triangle give go choice ensure hold pre distinguish step explicit distribution albeit somewhat involve term hand term use cauchy schwarz show side suffice recall enough claim lemma markov exceed distinguish appropriately threshold claim conclude correctness heavy conditioning various correctness account algorithm continue pay factor algorithm continue heavy factor account factor account sample majority answer desire easy convert low construct uniformly satisfie succeed distinguish sample straight check whether succeed probability along even vector specify prove unimodal suffice equally point since behave varie smoothly expect typical lot mode unimodal eq suppose increase q unimodal simple chernoff bound consecutive location high along enough far distribution proof pick uniformly distribution low w length random number probability binomial q objective inequality concavity follow elementary calculus simplify get therefore unless distribution pick author thank helpful valuable close part suppose find run pick choose large use chernoff deviation almost surely let high underlie translate close estimate interval small triangle q schwarz inequality times appear sample shall keep mind large variance part also suffice q spirit distribution consist succeed distinguish least success satisfied use distinguish succeed prof absolute specify later prove pick since poisson unimodal unimodal pick equally behave smoothly expect typical lot mode say mode unimodal triangle prove mode suppose unimodal randomly choose string consecutive location show enough far unimodal probability pick generate r occurrence except concavity calculate elementary simplify last therefore unless distinguish pick mit mit poisson sample near access bind complexity improve upon follow applicable distribution whether test besides matter stem
improve coverage interval value raise bias underlie approximation euler adjustment sufficiently ignore multiple provide well neighbourhood additional regressor variance see specification recall dr relative moderate via pre filter assess bias whether attack nature determine estimator table record bias subscript favorable highlight bold adjustment estimator already analytically adjust bias gain evident size full ba estimator estimator regressor unity save magnitude regressor proof follow directly ba ba lemma standardize standardized bias adjust analytical adjustment low bias absolute highlight c mse bootstrap analytical ba mse highlight bold k analytical ba analytically adjust ba report close nominal highlighted c coverage adjust analytical adjustment lowest highlight bold ba mse highlight bias adjustment ba experimental analytically report close highlight bold c coverage justification center bootstrap long integrate filter preliminary parametric justification bootstrap adjust simulation evidence performance bootstrap analytical bootstrap analytical already interval bias adjust estimator nominal reasonably adjust measured bootstrap correct analytical secondary log local long strongly process come play wide characterize decay absolutely rate stable dependent popular long memory process interpret via q short stable coefficient mean process model run integration particular impulse reference behaviour empirical seminal article area thorough fractional finance notably financial return highlight paper issue statistical memory method asymptotic gaussian normality mle fractional provide sample usual however correct inconsistent autoregressive average incorrectly specify parametric placing near semi estimator short broad applicability parametric asymptotically distribution place parametric asymptotic slow true despite asymptotic semi parametric exhibit particular substantial see correction area explore log hereafter least square ol n regressor local estimator denote cumulative estimator monotonically locally assign small whereas entail bias bias present seek reduce example memory broad band range zero even power frequency regressor approach present demonstrate usefulness adjust estimator particular bias semi correctly expense correction think parametric semi scheme rely parametric odd estimation nonetheless require capture salient importance series employ find poor coverage probability one side compare achieve attractive alternative bootstrap whitening fit basic property linearly minimum invoke likely although extension number past minor autoregressive coefficient kronecker delta mmse order unknown suitably process autoregressive model infinite validity regularity condition prove covariance estimator coefficient asymptotically aic commonly employ bootstrap asymptotically efficient sense select via plausible parametric raw bootstrap fractional reproduce realization adopt adjustment bootstrap standardize ht h tt mass h tt realization autoregressive uniform support integer crucially fractional index formalize follow regular stationary assumption place proof bootstrap see regularity achieve rate anti persistent regularity range filter series specifically employ modify wherein preliminary value filter value approximation filter apply construction filter short coefficient fractional binomial preliminary filter filter ar approximation generate filter pre filter bootstrap distinguish draw produce raw show choice shorter induce accordingly pass draw short memory proceed bias adjustment justification adjust choose proceed preliminary process evaluate estimator correct bootstrap copy magnitude surprisingly dependent proximity employ namely autoregressive use datum say main content bias correct asymptotically distribution admit expansion normal expectation original space realization process use pre filter sake integrate bootstrap step approximately strictly derivation produce report proceed construction obtain approximation bootstrap estimator expansion process reference precede require bandwidth estimator see term assumption estimator choose closely make induce quickly e n e provide justification estimate expansion c c j similarly construction yield algebraic give obviously depend bandwidth entail correction ar mmse coefficient derive apply preliminary approximation eq conditions express preliminary autoregressive proximity preliminary employ true choice optimal aic estimate aic mean model sense aic increase behave deterministic autoregressive appropriate pre filter value require give guide choice exactly n borel converge pre filtering sufficient convergence need nevertheless case basis denote ba probability pre simulation experiment initial actual adjust latter perhaps necessary consistent detail proof provide paper expression adjust common concern bootstrap adjust serve valid role pre iterative adjusted remain severe biased counterpart propose correction initialization assign tolerance go ba step ba notation tend bias adjusted estimate bias replace value b k value iterate determine accuracy achieve add iteration criterion correction think move estimator current b bootstrap filter conditional plus infer variance successive recurrence formula var nb iteration linear asymptotically evaluate percentile point similarly criterion accumulate correction tolerance level percentile convergence criterion decision probability next reference test conjecture iterate strong little change prefer terminate evidence substantial correction therefore go iteration follow ba ba comment far discuss follow process operator component early highlight set calculate modify estimator bias ba adjust modeling yield optimal seem use bootstrap autoregressive base square comparative purpose also analytically ba improvement analytical improvement via analytically bootstrap nominal interval plus interval bootstrap tu coverage estimator coverage calculate proportion cover true interval replication coverage bootstrap estimator record bootstrap produce two stop criterion criterion whereby iterative retain one iteration ba result estimator include omit brevity record bias mse bootstrapping adjust relevant ba subsequent table favorable highlight bold column modify key
consider repeatedly use decomposition case q observe na enough term argument enough enough cut coordinate cell informative variable sense theoretical make coordinate support european valuable suggestion lemma pt minus assumption minus universit es paris paris france fr universit paris paris france de france fr university centre f france paris france paris france forests combine decision despite usage randomization consistency regression random forest sparsity high dimensional forest randomization forest construct tree publication seminal become practice apply tune aside recognize ability size complex forest involve air win object microarray extra forest quantile focus version step subsequently forest explore year consistency perform simplified move ever practice prove consistency online forest forest difficulty nature procedure subtle analyze forest bag hand cart influential cart individual cut select optimize cart split notion bag cart theoretical done simply ignore bag cart split protocol leaf terminal individual tree pre specify total leave call difficult analyze rigorous mathematic author focus simplify create gap practice motivate property context forest theoretical guarantee consistency cart grow regression function proper number forest analysis adapt ambient carry organized notation technical framework goal predict integrable response aim prototype respect forest query denote variable set prior grow individual successive candidate direction splitting forest study forest limit infinity justify sc sequel forest rectangular cell form select sample leave resample resample word random datum choose maximize stop reach therefore contain exactly nm level level replacement optimize cart cut split call cell level xy resample different resampling bootstrappe point replacement replacement mathematical induce bootstrap offer establish precise forest regime forest occur fully consistency subsample make cart split properly cell position along th limit possible cut x cart independent variance extend univariate easy interpret provide calculation year play dimensional analysis successfully involve lasso various aggregation infinity random forest eq hold play tree sufficient forest cart noise control situation replace variable term turn account let examine regime e tree subsample complicate z nz finally coefficient whenever follow sequence surely technical statement h interpretation understand show h mean influence probability connection tend zero random vanish enough case satisfied verify noiseless partition strongly unfortunately whether let knowledge bootstrapping subsampling theorem consistency forest far independently thorough mathematical force action also interesting behavior ambient small constraint model assume loss generality informative independent ambient believe representation value proposition set split informative convention strictly forest select along variable everything happen forest upon variable support good difficulty assess forest process individual tree local averaging estimate chapter upon thing complicated proof adaptation theorem tailor forest result partition rely forest variation cell aim control let forest proposition offer control forest separate analysis regime allow standard thing subsample requirement term requirement every tree probability become datum inconsistent fact reveal forest small small probability ensure connected forest diversity difficult analyse subsampling come price assumption know theorem practice towards forest imply tree consistency require terminal tend infinity forest individual tree provably architecture also interesting pointwise diameter highlight forest inconsistent particularly mention extend context sake proof notation notation represent recall cut cut generally consecutive cut build understand build particular tuple cut proximity accordingly cell similarly cut later x forest follow eq large almost surely cut cell forest cut forest divide change instead cell point impose theoretical stop fix random randomness cut cut tuple equip clarity firstly lemma theoretical theoretical cut establish consequence proof within cell fall theoretical aim tuple cut pl x ensure stochastically denote k let stochastically lemma cut ready prove fix sure eq uniformly still need notation achievable accordingly terminal z hence finally index subsample tree estimate eq work tailor sake completeness equip theorem let n iii arbitrary pick h eq terminal assumption complete q contain thus may jensen large enough term double product handle lemma recall large subsampling observation cell subsampling combine enough inside satisfied equality thus surely statement surely return recall h connect select subsampling procedure
manifold set extract latent topological diffusion process start compute numerical subsequently feed diffusion amount classify subset high case bag get assign stationary except low domain distribution subset great enable diffusion part domain graph gender form decrease average appear function retain construction diffusion use procedure classify df fix metric token intuition project amount domain graph compute heuristic use select correlation g correlation graph gender wise wise graph suggest compute vector generate make robust vector get subsequently show gender well reduction bag subgraph extraction compare establish elegant conceptually principle improve quality context believe useful detection many domain domain construct df detect shift study co network online df profile df enable universit classify directed start collection reach topological original successfully apply get art pathway illustrate present classifying leaving make helpful context combine way either item walk provide explicitly matrix merge direct diffusion dimension token text undirecte make propose construct kernel graph walk collaborative walk user recommender system work set operating enable deep insight biased random walk diffusion reach time finally vector present formalism successfully extract pathway illustrate collection free association association define compute different appear call cardinality necessary consider corpus document consist token stop omit token position token fix omit association token every position token every token token successive occurrence occurrence still token appear conduct measure distance two token frequency frequent change somewhat want matrix density certain adapt weighted bring direct reflect association diffusion particular give document fit node df process start represent diagonal compute df recursively call define document default property well study nothing prevent bias random walk vector extract pathway idea adapt throughput biology quantity produce dramatically increase decade interpret result whole represent simple bipartite link draw chemical reaction case specie species chemical relation refer species context pathway transform source propose develop rank connect source node accord possible protein first pathway random walk pathway connect search short pathway mean path reach pathway whereby total pathway df explain call either annotate pathway weakly nan degree pathway general pathway reconstruct set source df df highlight hadamard multiplication direct application pathway connect effectively short two propose direct elimination degree arbitrarily fix relevant characterize path keep node account use full reverse multiplication division intend wise boost effectively pathway connect increase subgraph connect weak pathway large entry belong connect find pathway extract annotate chemical apply pathway minimal short target node pathway reconstruction pathway scale xlabel ylabel coordinate plot coordinate choice tune generally surprisingly variation parameter behavior explain constant pathway opposite pathway independently neither exponentially decrease source short pathway preferred one behavior explain geometric influence infer annotated pathway number annotate infer pathway dependency free appear bioinformatic relate walk pathway database quantify simplify nonetheless even since diffusion start cardinality whole annotate pathway cover result geometric give accuracy diffusion give terminal pathway compare select pathway analyze therein version df pathway reconstruct value database difficult publish inference df dedicate apply df binary problem give gender node form connect direct diameter scale xlabel ylabel red densely marker densely reach part line mark start decrease computation costly random average accuracy adaboost algorithm classifier comparison vector note moreover domain xlabel ylabel accuracy coordinate coordinate projection solid red note token split select token
exactly difficulty candidate detector show fusion sensitivity measure predefine build ensemble detector detectors result ensemble appear small circular dark detectors way channel extract enhance characteristic coarse extraction rest detect fine algorithm usually remove false former detector candidate apply preprocessing method technique slight increment false decrease voting detector combination exhaustive quantitative superiority prove competitive screening preprocesse section propose discuss discussion candidate extraction component comparison preprocesse dedicate publish yet select candidate characteristic image unlike generate noisy histogram medical processing preserve summary preprocessing aim gray max f max intensity enhanced transition popular technique make salient part visible split region histogram apply boundary eliminate bilinear complete remove base part fill removal preprocesse illumination pixel intensity original intensity local appear enhance also consider preprocesse formally operation l histogram removal near removal illumination aim show characteristic principles brief overview involve preprocesse add candidate future performance measure candidate extraction accomplish dark radial follow application match threshold candidate representation grow implementation publish et circular obtain detect circle circular circular extract candidate construct accomplish coefficient standard maximal response thresholde pixel wise cross directional map assign height describe map thresholde fp diameter transformation circular transformation matching framework ensemble preprocesse preprocesse preprocesse extract ensemble contain distance small centroid ensemble perform evaluate ground truth predefine otherwise ground truth candidate ensemble currently preprocesse combination would resource demand anneal search find proven procedure latter evaluate configuration function competition predefine positive rate preprocesse candidate pair ensemble phase namely detect fusion candidate build final decision output thresholded confidence combination candidate ce p dc h good evaluate detection dr present capability propose competition detector publicly available overview database competition dedicate roc mark compressed image set image marked expert database consist receiver curve average false sensitivity false level thresholded output detector ranking serve ranking addition calculate normalize calculated way likely auc uncertainty high also evaluate dr determine image presence contain indicate yes provide database train quite strong measure circumstance publicly database compress image range clinical patient dr r mild severe appearance classify contain sign dr detector measure specificity detector level thresholde candidate use follow correctly recognize receiver operate roc auc database present dr include row preprocessing list removal illumination rank result roc competition current performance ensemble see ensemble auc individual algorithm ht team auc htb publish database yet ensemble htb sensitivity specificity measure detector different fit roc detector area auc fit curve recognize case sensitivity specificity perform circumstance figure use removal miss absence thin preprocessing create diversity member multiple diversity ensure diverse different mistake false receive confidence voting detector outperform high flexibility ensemble high database roc consensus expert fp database level achieve good thresholding detect dr r recognize dr affect detector sensitivity severe recognize appropriate specificity suggest specificity dr screening threshold value specificity achieve level specificity result database dr dr case auc minimum sign
distribution omit primary univariate follow induce use query well query estimation artificial test discard serious setup first good exponential mat follow likelihood process optimize distribution demonstrate methodology copula distinct task copula primary middle secondary merge result toy omit identical performance input increase input root mean average number query significant cd ni ni f co krige induce l time ni cd ni ni ni ni ni cd provide completeness rough baseline contain type chemical element lead region primary sample secondary hard expensive divide sample primary cd primary ni secondary variable furthermore use mat ern cd ni square model marginal extreme mae task input less prediction omit gaussian use less induce process approximation second completeness l rmse mae opt show deviation concrete water fine cm day compressive compressive secondary ern generalize variable interestingly happen optimization optimizer sometimes well change process extremely different inter show copula learn address derive furthermore experimentally synthetic public learning centre centre rgb process ex plus minus ex ex minus minus plus pt em em em spatial monitoring address convenient dominate non gaussian likelihood copula process elegant handle capture structure cumulative distribution rather task use prior hold task expression copula model compare artificial publicly resource many environmental fusion problem advantageous dependency appropriate base tool problem gp predictive incorrect gp comparable root flexibility informally cumulative function cdf couple handle variable distribution help appeal address cost copula multi task problem analytical process copula machine fundamental process finance copula stochastic volatility propose heavy tailed robustness copula field krige process share approximation individually predictive depend location demand memory computational divide process representative approximate later recently residual mixture problem computational handle grow significantly simultaneously consequence copula decompose marginal cdf actual copula map though call integral transformation copula meet joint copula possible create huge marginal copula analytical cdf cdf root density create distribution get though leave toy txt index marks black dash mark toy txt toy txt scalar aim task broad improve secondary situation occur environmental extend gaussian copula task gets reduce inspire krige convolutional cross result kernel task mat ern process ordinary merge input different output univariate usually parameterize way denote advantage standard going minimize log explicitly minimize simulated require numerous numerator cost dominate rapidly element task introduce attack cm axis axis xlabel ylabel sample pi densely mark densely pi table index toy txt toy txt index toy data txt vs height x bottom axis line xlabel ylabel legend north index toy coordinate width height axis ylabel
application semi relationship deterministic inductive unbiased otherwise constant scale small singular ground ground informative subspace inductive completion underlie recall inductive optimal solution deterministic solve inductive clean eq case matrix expect related transformation continue extend inductive bound row typical proximal bx singular use power fast procedure thin order rewrite solution store residual computed remain computed solver solve million nuclear sufficiently trick thus descent cd scale link hard constraint relax bound cd complexity synthetic meaningful effectiveness real theorem link consistently orthogonal set linearly deterministic setting zero fix fix interestingly decrease linearly decay plot poor behave range inductive completion include recommender system semi supervise latter demonstrate usefulness semi supervise problem low norm recover biased dataset sample feature present inductive inductive pair minimize classification ground vertical rate q well approach motivated modern application completion bridge even bit quantization well sided measurement setting similar recovery insight bias past theory effectiveness evident link real principled select exploration want apply ij ij change chance fact want rademacher ij q universal mn minimizer therefore eq q case hand q ij r therefore get need hand side side convenience argument get bind ij ij take trace l discuss element exist value enforce constraint equation complete claim bit measurement zero modern application recommender system social network learn positive unlabele binary classification positive entry reveal assumption provide recovery guarantee shift subset propose completion recover binary denote sample scalable procedure effectiveness consist node million link semi recover arise netflix predict motivate rating observation low variant completion underlie one bit quantization modern completion link recover snapshot social consist pose recover adjacency otherwise negative context call unlabele short study completion recovery paper completion answer minimize observe solution popular treat good positive motivate positive completion sufficiently nuclear learning bit binary consider completion non deterministic show estimator square estimate motivated shift obtain scalable reveal low end scalable differently inductive bilinear row extend two contribution paper propose inductive recover imply insight completion efficient scalable simulate social consist million million superiority link establish hardness describe give extend inductive completion describe synthetic world last work completion remarkable low observation also recently motivate domain recommender heavily draw motivation recommender seek case matrix albeit sense field closely completion measurement bit quantization consist sign remarkable prove assume matrix independent state completion problem straight goal recover sided basic however world unlikely setting special non generality normalize partial randomly precisely uniformly let denote deterministic specify q bit apply subset uniformly bit unobserved bit completion satisfactory completion subset assume obtain substitute recovery error recovery high dense drawback completion complexity make large moreover average affect vanish deterministic indicator subset uniformly impossible entry recover therefore underlie good completion trivial way deterministic completion observe fix incoherent suppose location sample recover matrix deterministic matrix completion bias deterministic matrix proof defer want lipschitz traditional trace one use instead unbiased formalize mn interestingly rewrite follow want
meet pair visit infinitely learn outline repeatedly sample cr cumulative measure receive denote discount select maximize secondary exist queue length existence strong interaction use q learn state computer wireless center communication cognitive organize cr user assume cr cr user queue buffer queue moreover queue channel empty assume transmission policy cr arrival channel cr energy primary optimally cognitive usage utilize cognitive cr users environmental exploit reasoning dynamically communication diversity communication attention cognitive primary secondary cognitive e investigate cognitive cr cognitive terminal queue maximize stable author cr terminal primary spectrum inactive assign queue cr admit fraction fraction secondary incorporated management management allocation horizon throughput wireless programming online average arrival channel state cognitive energy g markov mdp secondary policy spectrum cognitive terminal explicitly involve queue author cognitive throughput region author investigate service rate secondary capability add consist cr user investigate impact node traffic well protocol randomly begin slot sense maximum throughput queue propose cognitive select duration activity throughput queue network utilize spectrum inactive one decoding outer secondary cr second primary action take slot exactly sense inactive cr end slot unity reference make queue capacity assume length nonempty e furthermore arrival arrival identically queue arrival bernoulli generic apply adopt share channel channel cr low inactive energy queue queue fig buffer incoming datum denote buffer energy token cr user store traffic store primary store environment finite length precisely queue maintain duration slot second bit arrival bernoulli process process evolve accord queue assume two fig queue slot slot queue slot slot assume variable queue queue time gain node gaussian random read primary secondary link cr cr user perturb model additive independent channel specifically slot slot channel unity link channel capacity know channel gain slot primary channel send primary dedicate narrow phase spectrum medium control begin cr channel begin activity record binary inactive cr queue queue cr user slot correctly cr decide accept negative decoding drop primary queue cr primary primary overhead negligible size feedback receive accord description distinct action begin slot cr high early cr four cr optimal slot follow slot cr user energy energy queue cr receiver service term unity queue empty queue empty nonempty cr queue cr access channel receiver inactive arrival queue queue primary queue nonempty queue channel arrival energy queue either either queue queue process energy queue channel receiver queue accept follow whenever energy queue primary queue nonempty occur queue size begin slot queue argument rl maximize value payoff user cr adaptive accord rate mdps framework uncertainty bellman many dynamic programming mdps discount immediate discount small investigate satisfie bellman discount cumulative agent cumulative beginning maximize sum service rate
time decode half measurement count sketch demonstrate measurement major scan sketch compress sense entry instead design propose decoding utilize estimator absolute theoretically analyze estimator estimator ratio zero ij happen motivate nonzero tie long make detailed need easier particular false certain still practical preferable see general estimator estimator introduce theoretical exploit prior estimator recommend estimator recall I residual major computing absolute iteration positive irrelevant care express ij ready lemma practically convenient numerically understand substantially hand convenient theoretical lead complexity convenience set convex upper resort poisson consider define confirm poisson two small basically away perhaps choose reasonably news practitioner confirm ratio suffice express datum construct absolute sort value examine difficult ix measurement error easy suffice recover nonzero nice property reveal nonzero recovery consider need pm eq q kt e compress crucial theoretical page see count sketch l available achieve count half sketch show contour plot decode count recovery compare become replace dense sensor cost perspective decode department statistics department science university nj usa department nj usa sparse projection compress sparse recovery design fraction major scan coordinate absolute use minimum combine practical decode exist scan algorithm decode nonzero entry positive method l compress sense become popular field computer science mathematic recover number adaptive nonzero location coordinate database naturally formulate compressed sense sensing may back paper compress moment linear lp
width axis style col comma auc accuracy utility dash red fill red col sep comma bb smooth thick table col comma auc bb nb xlabel ylabel recall legend legend pos east font sep comma sensitivity utility thick style red col comma bb sensitivity accuracy utility col central bb nb prior plot set well impact thin another indicator recall indicator sensitive htbp classification return classification behavior available reason aim close wide thank knowledge group one safe class prior one safe meaningful vs vs vs fraction instance recognize safe instance drop close accuracy safe perform dependence phenomenon know observe detect cross checking prediction induce model quite done recognize safe dependent qualitatively yet independent eventually recognize dependent note include accuracy xlabel ylabel legend legend pos east every font table col accuracy traditional accuracy quadratic function respectively denote compare accuracy valuable classification grow close figure follow assign classification utility high situation become fully reasonable robustness function risk differently yet rank despite explanation utility assume costly miss presence assess art uncertainty yet set especially small deal single sensitivity instance version represent valuable however variant matter partially cost error interesting analysis detect check automatically opinion acknowledgment grateful center application master student partially nsf grant support project moreover grateful anonymous show ib contain include set instance belong definition address pc j pd includes minimize maximized interval k j numerator polynomial complex interested solution together boundary solution constitute maximum candidate retain beta beta bb treat prior include covariate beta pm b explain beta k beta give combine uniformly lemma remark empty di di data result address prior sensitivity building regressor characterize show tune probable return compare variant presence recognize dependent predict outcome categorical basis several call include covariate plausible drawing conclusion conclusion average inference model sophisticated model yield inference specification conclusion especially often report sensitivity present model characteristic value return single class safe generalize thus combine classifier firstly naive bayes express weak classify posterior sensitivity identify namely probable depend prior predict presence environmental covariate slope analyze logistic analytical condition near moreover present preliminary presence consider yet knowledge class prior previous new variant expert publish paper specie master also extend two informative uniform prior expert informative also another posterior inclusion organize algorithm study expert covariate covariate denote I covariate train th covariate address combine inference weight probability model give respectively marginal respect model parameter log number approximation viewpoint adopt bic parameter probability huge approximate summation computational limited covariate often interested binary include otherwise prior ib ib covariate denote include covariate probability obtain assign informative viewpoint covariate include ib far flat flat prior adopt bb ib bb less bb recommend handle bb treat prior choice probability uniform analytical derivation formula ib distribute bb covariate eqn distribute compare ib bb htp xlabel ylabel legend cs anchor west draw every font col comma beta bin col sep comma differently specify inclusion covariate generalize prior inclusion probability thus recall call nb nb ib independence inclusion covariate set model call discuss generalize induce ib induce nb ib ib specify vary ib apart sharp remain identical word prevent ib prevent ib exclude condition prior follow inference probability compute probability sensitivity probability minimize covariate covariate covariate otherwise analytical nb describe nb specify permit collect us admissible value covariate want use one represent get vary curvature near posterior varie compute dimensional complement low inclusion covariate include analytically package interface symbolic solver far assume prevent sample strategy space discuss accommodate inference formula whole summing accord interval dominate class vice generally class lemma class deal equivalent prediction probability probable class upper probable depend consideration induce interval prediction match regard collect distribution explore area cell cell consider range introduce covariate third piece aspect namely maximum aspect temporal covariate concavity namely cover digital mobile use huge reason average cell ask inclusion report publish paper specie l master belief expert table label expert assign belief expert covariate two expert skew inclusion htbp expert aggregate two way firstly use within specification secondly take represent knowledge later nb among slope expert different inclusion appropriately represent substantial central inclusion covariate point uncertainty characterize ib informative bb informative informative call probability covariate hull report consider three variant originally represent assumption covariate equal configuration wide variety prior third partial knowledge inclusion low hull expert belief inclusion covariate prior probability inclusion recall binomial prior include reason outside slope curvature consider huge remarkably probability curvature recognize estimate inclusion discard covariate interestingly achieve assign curvature upper thus conclusion sharp bayesian inclusion depend yet inclusion exceed much comprised showing repeat great create comprise presence set shape ylabel post legend none west major
explore mode mode whereas penalize mode benefit hyper penalization leave bayesian method tail hamiltonian carlo restrict mode focus report investigate fully elsewhere lasso choice cauchy fit relatively stable therefore unnecessary greatly increase mcmc tail hyper play role shape structure lasso laplace illustration technical investigate report apply method microarray set article logistic class integer feature vector datum column feature look consider infer prior article genomic however rare df gene regression model find gene marker great mcmc clear difference laplace freedom well high express stand inverse shape term number generator parametrize penalty superior dimensional prior cauchy uniformity notation prior article table description generate generate look huge look tail value figure move change moderately tail great well belief df tail heavy either tail htp distinguish redundant feature moderately heavy tailed separate path constrain maximizer ie contour find shrink contour tangent df generate map contour origin axis path go laplace look also find divide conceptual illustration datum contour figure penalty two end contour explain divide correlated make selection among correlate automatically within large contrast gaussian penalty constrain middle contour explain absolute discussion difficulty problem minor unstable require see full advantage choose mode mode describe technical letter index index letter subscript index denote integer integer collect class case take integer logistic coefficient hyperparameter define convenience useful predict provide control variability variability ig equation bivariate assign coefficient shrink signal assign half cauchy cauchy half various induce propose coefficient assign exp mean obtain form notational simplicity call name coefficient well towards study confirm little regression rejection adaptive ig example hour hour denote recommend fix contain empirical stable choice especially heavy may avoid treat hyperparameter capture fix hyper differ greatly freedom consequence stick region likelihood sampling accommodate recommend fix issue model coefficient identifiable add class identifiability implication may inference class common identifiable equation c markov naive sampler long chain symmetric transform identifiable transform symmetric transform distributed indicator likelihood parameter integrate prior element normal variance hierarchy asymmetric cause difficulty useful label normalize deviation select look standard deviation alternatively sample joint put conditional alternatively transformation deterministic leave invariant state state still reversible transformation compose transformation valid full explanation sampling procedure page gibbs full last sampling ig alternate pt carlo transformation leave straightforward prior use differently distribution concave sample give sample key high hmc hmc greatly walk due common regression long trajectory sampling prior local redundancy sampling correlate probably fairly hmc chance ordinary move contour mode dimensional thousand bottleneck challenge greatly important gibbs coefficient detail trick notation setting recommend iteration compute rank discuss pool mode true feature appear frequency markov correlate ranking correlate totally note recommend useful feature light choice mcmc number discuss alternative generate predictive across nd therefore class pt run ie ig setting chain since take estimate lasso package see give lasso second stand path see estimate stable bias large marginal skew absolute explain predictive feature pt compare path predict importantly predictive contrast choice weak conversely validate even nearly difficulty look follow equally draw mean feature within group class draw absolute differential differential class however relate feature higher generally select discard obviously million run prior choice choose choose hyperparameter reason automatically n large chain setting prior validate lasso set shape stand group point horizontal pt separate times nd weak recognize another useful discriminate believe useful base consistently stable apply almost large separate absolute set think hard tend discrimination entirely coefficient tail flat tail therefore heavy tail good likelihood tail table summary ie relative prediction feature choice note moderate chance weak therefore choice critical figure see prediction performance measure attribute nonzero pdfs without statistical degree freedom almost complicated logistic regression hand negligible high problem chain take whereas chain take merely posterior rather ig large however possibly eliminate well microarray relate cancer set website look f statistic rank statistic use leave fashion always standardize lasso gene take hour prior prior use top pt run perform compare thousand small omit top rank gene rank narrow around value small rd check report predictive substantially method test performance prior statistical figure improve htp r r er pt use case select plot show useful joint skewed absolute fairly classification slope hyperplane function indicate necessary hmc ordinary mcmc weakly redundant separate gene different absolute gene clearly indicate redundant multimodal prediction prior result gene figure probability label shown think biological see gene separate gene pt pt statistic introduce bayesian prior investigate microarray demonstrate feasible high hyper retain selection hyper similar superior appear also problem choose demonstrate light hyper modal average particularly group highly coefficient fit divide markov look separately list markov coefficient totally skewness demand development simulation future drawback still slow penalize room computational difficulty lie exist transform feature mcmc simulation crucial step solution devise transform li support engineering foundation li discriminant vector produce estimate suppose leave log ie independent p physics momentum particle qp way qp qp give discard transform qp hamiltonian move keep unchanged crucial hamiltonian metropolis hamiltonian dynamic discretize stepsize transformation several alternative transformation I transformation independently denote property nearly hamiltonian series add transformation reversible jacobian leave exactly sampling transformation qp reject algorithm carlo current step independently transform qp p time transform qp qp trajectory connect along call decide accept qp last implement hmc appropriate hamiltonian poor may slowly even low rejection hoc close reciprocal square root nd automatically account adjust factor adjustment choose hmc beyond work thing hamiltonian trajectory nearly adjustment appropriate phase sampling value frequently make look problem fairly well another initial trajectory run need value hmc
always primal unknown ax center know potentially simply invert find solution unknown treat vector regressor imputation one among alternatively compress interpret accurately e concept cast task name partially concatenation position entry noise name reconstruct decay strategy capture target ac imply elementary circuit circuit target vector say circuit circuit correspond dependency regression circuit scalar multiplication let minimal circuit pick define circuit circuit either circuit circuit formal circuit circuit formal associate circuit cd emphasis circuit circuit circuit vector circuit respective circuit circuit see differential accurately efficiently circuit circuit let call contain circuit circuit rich circuit vector equality span suffice circuit statement equality one circuit circuit algorithmic benefit treat circuit set circuit vector reader think circuit term circuit think notion make advantage describe close short circuit affine equivalence short circuit general circuit element additionally important particular particular circuit circuit regression circuit generic uniform special generic generic circuit particular circuit support rank circuit require name cycle homology compute completion find special circuit combinatorial non behind technical circuit generic non kernel span generic vector hypergraph construct evaluation form view outline circuit special circuit decrease regression unknown want circuit estimator let circuit expectation equality circuit converse optimization major formulation circuit involve inversion nothing gain inversion usual circuit circuit analogue induce circuit circuit circuit circuit matrix circuit entry circuit bilinear positive variable cholesky observe definite let minimize kx slack term minimizer satisfy exactly give independent minimizer algorithmic good generator keep track circuit vector compute highlight advantage pseudo matrix oppose pseudo inversion naive advantage scenario estimate depend choose choose circuit circuit circuit needs change bilinear j variance error circuit circuit circuit matlab concatenation circuit circuit circuit calculate computation huge circuit small scenario increase row negligible stay algorithm compute multiplication circuit evaluation circuit covariance return estimate circuit covariance variance bind return matrix circuit matrix column compute multiplicative column circuit basis fairly near consideration circuit submatrix believe invert merely structure imply small circuit problem simple setting much circuit well attempt circuit e solve choose circuit row combinatorial property completion concept graph homology may local neighborhood also example combinatorial circuit identify circuit sparse dual highly structured analyze network circuit basic identity regression circuit element square multiple observation copy circuit contain regression circuit type copy copy general circuit prevent multiplicative suggest pool observation single denoising occur circuit observation row denoise occur row improve merely observation difference basis orient node characterization circuit circuit show form contain circuit one edge circuit circuit therefore element graph homology edge sparse sum row graph assignment regression circuit path length contain circuit vector start circuit cycle length circuit potential search homology provide cycle efficiently write cite concatenation entry unobserved case sum scenario discuss always neighborhood principle apply search homology arbitrary tool equation structure measure matrix scenario reality matrix format case recognition hermitian potential scenario circuit circuit phase non symmetric product see otherwise include namely circuit disjoint form inversion exponent estimator blue variance unbiased row relate furthermore estimator drop compute without estimate provide employ important add circuit improve incur error conversely add circuit estimate system make relate inclusion wise maximal conversely c proposition estimator unbiased imply blue analogue gauss estimator way complete minimum optimality estimator need information signal noise first gaussian statistic respect I remove contain sufficient virtue follow thus see transform prove statement prove complete prove write ax follow straightforward exponential similar conclusion signal f bf immediately imply statement px n px px include
schmidt difference projection operator similarity involve w mixture relatively separate measure whenever bound mixture relationship translate mixture recall set simplify clear suggesting remove important relationship stem kernel square root concentrate tight spike illustrate colored ratio color latent analysis limited population infinitely relate version laplacian embed sample certain geometric call reveal I subset draw nu notation embed label pair distinct almost illustration establishe embed cone additional parameter mean low decay decay finite perturbation root close difficulty parameter previously require essence small notation apply finite angular depending hold probability normalize embedding right display laplacian embed orthogonal component overlap result high illustration way proportional small depend dominant simplify tail dominant increase increase tail truncation literature chen von de restrictive hold add increase performance practice performance standard apply embed h normalize ix work proposition quantitative apply embed notation random initialization enough latent fall within angle angular occur close let initialization cone close case q single step fall angle angular level provide perturbation approximate invariant recall analogously three ideal overlap mixture generally perturbation long schmidt relative problem long sep apply follow hilbert schmidt mx spectral consequently material combine moreover theorem remain projection operator write give consequently continuous calculus expansion n put piece shorthand subset orthogonal angular structure subset element diverse tuple break step number tuple diverse construct diverse construct select form laplacian embed copy diagonal nx lie follow tuple q hold return explicitly claim end combine establish finite sample intermediate entry satisfy n q aa transform involve write note therefore find inequality b bb mn expectation generation selection diverse least v condition auxiliary result set simplify whenever consequence kk tuple thereby complete remain hoeffde bernstein control put piece nx ny laplacian matrix principal principal prove relate namely bridge intermediate e k supplementary material lemma must handle fluctuation denote reproduce similarly satisfy condition k x lemma triangle identity return term note schwarz inner product logic bernstein nc e q eigenfunction eigenvalue note eigenfunction form orthonormal therefore r g ig analyze spectral context nonparametric mixture level square coupling parameter kernel undesirable necessary guarantee identifiability follow sense rich mixture component component mixture conversely function representation could optimize cone symmetric find family building characterize normalize collection take structure component almost angular angular perhaps fact minimize believe bandwidth distinguish vanish however project shrink population spectral mention provably shrink leave bandwidth cluster interesting acknowledgment wu helpful discussion grateful associate suggestion manuscript give overview background material symbol nsf grant dms grant dms dms grant nf information technology agreement cluster many area spectral eigenvector normalize laplacian recover label finite nonparametric difficulty label overlap divide compare root embed cluster past decade spectral learn information attempt answer spectral np hard partitioning cut spectral cluster machine application clustering set decade normalize division back modern cluster eigenvector laplacian apply embed cluster ng al well separate recently expression analytical eigenvector perturbation away ideal laplacian study manifold primary convergence limit connectivity reconstruct eigenfunction sample analyze laplacian laplace von embed much laplacian provide proof convergence part unnecessary kernel nonparametric mixture study characterize difficulty eigenfunction top eigenfunction eigenfunction integral sign fraction laplacian embed nonparametric normalize laplacian operator component overlap span square characterization result certain nonparametric refer mixture orthogonal structure perturbation theory remainder follow separate component result support distribution set hilbert schmidt supplementary material symbol introduce result discussion consequence involve span square root provide description mixture give play important intrinsic difficulty kx cluster index symmetric measure overlap respect density precisely distribution analogy kernel component coupling laplacian decompose mx small measure difficult split splitting identify since ambiguity define one measurable shorthand kx dy infimum measurable subset mixture coupling illustrate triangular function correspond triangular similarity
sum rank component svd decomposition term analyze detail decompose rank unlike tensor dimension fall less overcomplete regime regime rich option extract early base incorporate discriminative subsequent show discriminative model feedforward far unlabeled function label present mechanism relate assume component ica rbm conditional correspond reasonable rich element element previous work estimate new transfer estimate proceed due sample domain vision framework supervise transfer popular domain nlp vision list extensively study sample bootstrappe label dataset train label main general tie investigate transfer use various work transfer unlabeled propose fisher score yield argue learn mechanism score cross moment extract attempt discriminative feature consume feature learn ica machine rbm argue overcomplete latent representation latent dimensionality good develop guarantee ica code various information transfer argue incorporate act improve learn label dictionary learn unlabele label sample coefficient fed framework score review probabilistic model base score score function involve partition intractable compute compute far nice maximum presence rbms extract superior learn auto encoder input encoder map penalty see review special case establish score zero argue employ establish yield derivative derivative stein construct familiar multivariate polynomial polynomial polynomial review semi assumption establish describe setting example general label contain specify cat assumption unlabele label input instance human task cat cat gain another access imagine set image human label use classify transfer original unlabeled instance contain label unlabele image internet framework conjunction score feature setting self draw mild incorporate discriminative probabilistic randomly mild regularity variety unlabeled g mixture boltzmann distribution unlabeled cover assume learn previous g early contain internet observe correspond share probability draw observe image include random marginal represent datum show mechanism explain semi consider unlabele limited learn task unlabele many challenge general work extract entirely extract conditional useful variation gx e derivative derivative w expectation picture vector depend derivative component question score score th perform manner sample see mx estimate score empirical moment investigation work propose task show yield differential operator discussion formal lemma yield derivative fisher differential start stein building original version stein regularity scalar usual part introduce decompose derivative derivative tensor gx order principle eigenvector high tensor array framework tensor represent outer tensor decomposition computing form high similarly perform power multilinear form definition different vector clustering ensure converge vector tensor property guarantee dl multilinear tu update remove center algorithm base tensor tensor orthogonal analyze decomposition rank unlike overcomplete tensor large dimension incoherence impose soft orthogonality component non dimension application power perturbation whiten tensor multiply whiten tensor orthogonal decomposition discuss framework connection score expression distribution exponential family energy let vector know order give kernel joint kernel gaussian manner score posterior center contribution mixture center score attempt encoding decode call unsupervised unlabeled argue approximately learn go describe framework match fit analysis let good score minimize two sign equivalently laplacian nice interpretation fisher divergence relate leibler px sense robustness note estimation ml kl density regularity score minimize change amount function introduce derivative minimizing operator function sign score px form efficiently close compute nonlinear density compute ti self self unlabele label assumption sample unlabele information model joint function use assumption unlabele unlabeled unsupervised way characterize notation include convenience restrict identify array rt limit tensor also multilinear tm j eq multilinear similarly multilinear combination slice rd tensor rank cp write tensor th derivative fx permutation states stein score yield stein lemma function continuously integration scalar scalar provide recall expand early collection probability integrable along line px x dx map derivative characterization stein characterization definition function high order score function denote denote score recursive differential relation induction prove score stein high generalize order order operator let random suppose function consider continuously gx regularity gx px prove recursion stein identity see relation induction q yield parametric exist thank nsf h support support microsoft fellowship nsf award nsf award award award first regularity inequality iteratively recursion stein lemma formula stein identity entry appropriate mode tensor prove permutation step affect symmetric mode gradient lemma derivative apply necessity argue gradient tensor permutation put last tensor require last prove score induction hold show hold substitute rgb corollary conjecture proposition claim observation example bold time challenge vision processing extract train label sample high establish theoretical characterize nature extracted label employ tensor advantage employ value rich discriminative information form employ discriminative feature good critical achieve performance domain speech vision language traditionally engineering carefully tailor towards task automatically feature various framework deep sparse coding approach unsupervise thus vast incorporate almost probabilistic incorporate important explanatory input incorporate boost discriminative behind input unlabele learn label one self framework transfer adaptation involve interest mainly due natural language process syntactic access unlabele human learn common without goal human design capability unlabele general extract relevant answer class pre unlabele present pre discriminative nature consider high derivative pdf local manifold pdf input rich access feature characterize precise nature work extract moment main expect word moment capture informative employ spectral representation moment algorithm suffer spurious optima typical problem backpropagation construct overcomplete representation tensor argue overcomplete crucial get spectral method extraction scalar discrete handle regression classification multi present efficient end extract overview figure fill shape corner sep sep purpose score mx moment derivative discriminative pt line width width corner line dash corner green label extract supervise discriminative equal informative indeed derivative vanish input distribution carry either nearly degenerate vanishe average scenario cross contain useful model mixture moment recover guarantee challenging contrast approach incorporate generative discriminative feed fisher feature feed behind unsupervised find learn leave generative discriminative prescribe sample unlabele discriminative decomposition tensor
initial random subset zero otherwise ensure adaptation pick interested pick serious use dependence say disjoint subset intuitively dependent probable pick pick pick important negative chernoff independent general approximation base adaptation two negative weight unweighted modify bind material detail state guarantee relatively ideal optimal bound reasonable trivial regime cut bound edge need approximation cut intuition find directly linearity subset number weight unweighted let minimal cut edge edge let weight mc intersection empty prove reasonable graph cut small low cut generally cluster precise vary algorithm scenario spectral graph spectral towards graph consist matrix element assume imply assumption relatively weak connection cluster laplacian simplify angle way measure theorem approximate show obvious supplementary theoretical guarantee guarantee use notion nothing make partition cut separate small assumption basically inner relatively observe edge separate cluster cut proof chernoff bind cut supplementary material proof provide depth analog theorem structure weight constant bind think vertex clique connect sensible algorithm connect clique take try however sampling connect clique probability connect need scheme connect connect component component wrong right small probability connect sampling uniform intuitively distribution edge approximate structure original non motivated leading distribution weight estimate weight scale without probability replacement iteration moreover sample negligible modification suffice hold unfortunately find unstable bad clustering initially mix attempt weight pick unseen probability graph modify pick uniformly pick biased scenario toy early pick edge wish use desire discuss early mix appear initialize zero pick otherwise cluster pick connect ij aware algorithm somewhat nd drawback specifically design nd laplacian eigenvector extend cluster require partition suboptimal eigenvector decomposition costly impractical cluster single edge add eigenvector although couple another option several make test inferior spectral perform surprisingly inferior cluster cluster percent frequent weighted circle comprise four gaussian unnormalize weight half classic easy bad dataset dataset cluster image dataset suggest test unnormalized previous hoeffding chernoff dependence approximation change edge quite change proof paper pick equation change zero first second small unnormalized eigenvalue prove show two small unnormalized min spectral inner fact need main tool matrix chernoff without consider hermitian define replace requirement without proof function result without sample edge connect cluster probability laplacian matrix chernoff p en prove theorem use cut cut weight prove extension trivial round good minimal edge weight small n original graph finite n increase pick gx x bound drawback minimal unlike situation prove cut c chernoff hoeffding multiply set pc pc completing get simple sampling bind bit see weight observe cluster cut mc cut separate cut cut inside mc small great cut cut need show cut separate consider cut separate need dependent chernoff mc finish adaptive well theoretical uniform cut pick approximation accord pick pick accord pe view subtract suffice bound give cut define choose nonzero dependent ik get easily verify monotonically finish proof prove concentration output weight chernoff replace independence markov induction union finish show chernoff bind bad adversarial bad clustering graph pick least besides edge choose uniformly
contraction equal k implicit td td affect contrast implicit td implicit td benchmark td td evaluation compare size alpha method method alpha implicit alpha experiment perform library fouri final step may see alpha size stable cart td algorithm justify evidence implicit td td great td preliminary many ahead error exist result td implicit td surely td implicit td future extensive evaluation several adaptive compare proximal update successful td rl actor policy instability kk equal eigenvalue q eigenvalue b algebra solution ready td eigenvalue replace e discriminant lead contradiction thus corollary remark usa technology department statistics usa reinforcement method implementation td choice step empirically high instability cost td evaluation implicit typical td td state art wide applicability rl fundamental td linear make pair td successfully apply scale domain drawback td empirically study try solve stability stochastic sgd algorithms asymptotically significantly moderate sample stability implicit sgd motivate connection proximal inspire enjoy standard td explanation introduce maximal td suitably evaluation implicit td within td convergence implicit td markov finite probability underlie transition chain denote reward time weight value vx discount brevity sequel calculate sample state transition standard td note td introduce td discriminate standard implicit td implicit eq solve inner td td expectation take suitably td td thus implicit shall implicit
list notable tuning practice time sort diversity weight maximization basis item generate recommend item fa list recommend list objective function therefore treatment technical allow overhead optimality argue problem modular submodular diversity independence polytope problem dimension greedy solution item sort order weight section interpretation aspect finite topic movie diversity measure unique topic cover highest covered belong item belong item list another item must item item cover increase item objective diversity gain view cascade scan list top stop item due diversity gain second last equality interpret none early item expect recommendation list depend diversity consider diversity item result item return sort utility mathematically diverse useful recommendation maximum choosing idea diversity suitable user may redundancy recommendation interest assign q recommendation list characterize function diversity return recommendation item utility moreover length prove contradiction item choose list item contradiction topic among test value diversity number add allow controlling length irrelevant rule evaluate offline compare optimal solution evaluation list diversity time amazon mt separately recommendation mt worker movie match relevance movie study mt worker movie list finding offline fine assessment create preference profile evaluate diversity utility recommendation application frequently rate rating assign movie maximum utility I maximum include popular movie restrict movie reasonably movie evaluate mt worker intelligence ask worker interest generate list three list choose worker ask movie list match address contain worker movie ask movie recommendation choose address cover movie list mt show recommendation list movie contain movie movie recommendation list differ worker movie inherently recommendation adopt want number movie put disadvantage generate list either utility diverse table complete worker mt worker quality work ask complete average second second evaluate permutation highly hypothesis low quality worker evaluate result present compare percentage find movie match choose percentage movie movie match movie worker match movie percentage time match firstly percentage find match find matching list well perform method compare equally answer worker generate reject nan secondly time movie list perform significant time good list second perform compare hypothesis answer worker statistic less likely nan list generate ratio matching movie find imply movie likely recommendation due high popularity movie ground practically recommend difference real outperform movie movie match however insufficient cover five movie list movie assign diversity cover relatively movie cover likely user dark action dark dr dark c dark action action diverse movie cover parameterize ask worker go take prefer movie four recommendation three none mt preference recommendation time frequently rate movie rating assign setting generate movie diversity randomize suitable neither percentage identify recommend list complete worker ask guarantee complete worker second second list movie suboptimal answer baseline statistically permutation test recommend list suitable compare equally observe likely nan permutation interpret unlikely worker rate randomly reasonable evaluate dark action action action dark dark action action action action dark dark dark action life cover four popular assign insufficient diversity popular item happen movie recommendation movie similar strongly dominate movie assign diversity therefore movie movie less popular movie sum outperform topic utility low utility utility item item goal evaluation recommendation multiple profile million rating star user rating create profile recommend movie movie rate split randomly ratio factorization predict predict utility three report split use create profile whereas along utility profile movie list step belong create multinomial popularity movie rate rate create user preference recommendation proportional preference term diversity utility compare rating movie utility b rating list reason predict keep evaluation recommendation rating movie metric metric consider diversity choose first evaluate diversity first metric combination avoid compound metric diversity different metric distance recommend intra diversity list movie diversity exploit accumulate utility recommendation discount list item user user assign movie ndcg ndcg ndcg achievable item utility score compound metric utility diversity intra list discount avoid use c compute every recommendation list setting metric user figure utility rating movie utility movie recommendation step factorization trade mean diversity ndcg diversity situation diversity superior regardless score high ndcg high highlighted utility simultaneously also superiority compare figure observe improve hence recommendation various recommend utility operate diversity curve intersect diversity metric setting significantly utility start importance diversity take objective superiority balance diversity goal parameterization combination diversity system focus composition recommendation list diversity list pose utility diversity important propose diversity maximization modular different aspect user aspect cover study utility movie diversity movie cover study find baseline maximize diversity diversity result execute individually moreover diversity superiority objective combine modular submodular utility orthogonal maximize modular submodular conduct variant future item respect consumption user diversity apart diversity contribution item list investigation recommendation furthermore tolerance redundancy item agreement opinion redundancy item opinion future remark united com yahoo united yahoo com recommender diversity recommendation list replace diverse work method recommend diversity user optimal study offline incorporate evaluation superiority baseline popularity recommender system social network service item user typically recommender interest list top na I score item however sub recommendation instance recommend popular appear recommendation target topic recommend potentially diversity recommender diversity deal recommendation list cover single recommendation topic increase chance answer recognize recommender heterogeneous term interest music incorporate diverse member music diversity recommender system interpretation query desire query come unless topic reference aspect document formation virtual together maintain diversity come account decrease item relevant objective modular diversity find free maximize item subject diversity interest cast find modular list interest represent high utility primary concern maintain avoid redundancy interpretation suitable diversity conduct extensive crowdsourcing list baseline diversity superiority list superiority prominent variety diversity utility recommendation successfully overall analysis recommendation diversity priori tune two fold computationally aim improve list maintain evaluation online user offline demonstrate support propose element instead furthermore instead represent refer approximation relevance metric combination metric trade utility diversity give rank item create choose maximize relevance minimal movie also cover preference movie select recommend movie list substantially user movie recommend movie list great another movie utility ccccc utility x movie select first recommend movie recommend movie movie ground movie whose movie utility movie utility movie recommendation irrespective chance idea notion diverse recommendation contribution movie movie place diverse movie higher
particle practice instance infinitely solution matrix transform transformation mode behaviour parametrize general set er rao low parametrization major jump linear express sufficient eq parameter write use detail verify inner indicator compute smooth calculation tu indicate sa sufficient argument maximization last product particular lagrange multiplier maximum algorithm property rao matlab author identification jump markov mode n backward trajectory initial value mean run transfer n particle black blue particle use plot new rao significantly compare mode r low pass white noise initialization rao pick parameter particle particle figure run estimate plot figure multidimensional tb error mode type smoothing expectation solve leave smoothing key introduction filter maximization step could obtain term particle idea smoother great outside jump model something worth indeed possible turn contract contract department uk jump consist identify jump challenge derive expectation recent solve inherent conditionally discrete thought different variable hence variable furthermore zero measurement define via interested identification jump number mode formulate problem measurement input eq c unknown static throughout input challenge maximization type strategy separate first nonlinear state nonlinear solve monte particle mcmc particle mcmc systematic strength need mcmc estimator jump markov model inherent rao rao particle approximation recently switch exist jump approach stage segment approximate hybrid recent relationship smc approach mode dimension bayesian nonparametric iterative maximum em maximize maximize iterate solve maximization think select estimate likely involve nonlinear intermediate intractable force solution e step still intractable carlo smc particle general arbitrarily q view smc carlo algorithm space kalman expression filtering f tp relevant pdf normal mcmc kernel smoothing make ergodic markov define fold jump simulate limit initial ergodic allow smooth arbitrarily k particle important particle similar approximate accord draw probability proportional trajectory kernel map useful ergodic admit detail requirement fulfil rao fusion kalman filter particle rao somewhat process markovian handle rao rewrite adapt cholesky etc includes find rao discrete draw linear handle implicitly define algorithm estimate construction ta ia ji ni I ip
variable shrinkage summarize topic tuning penalty ridge value lasso mm penalty red square blue mark rectangle versus shape seem pure rectangular contour illustrate matrix lasso look equivalent derivation set solve present start lasso iv plus positive obviously decompose positive continuous mixed sign twice disadvantage case real disadvantage intend often two sign gradient interface solver tight obvious decomposition residual yield absolute solve derivation constrain equal derivation tucker lagrange multipli hold extensively section incorporate row analogous condition scale define solver quadratic model qr decomposition h green derivative sx contribute similar factor projection contour straight slope ellipsoid conjunction every lasso matrix penalize quadratic lasso penalty hold e huber aspect mention iterate compute lasso sign restrict square e g b calculate numerically qr solution lar coordinate descent model relevant incorporate advantage via qr svd interesting library qr svd algorithms quantitative qualitative path identify outer decomposition bridge regression ridge nlp augment penalty uniform b variable ol augment negative nlp minimize nlp minimizing solve nlp
similar computational aspect review resort non whose global difficult remove lasso square second hand notice bias contrast total denoise automatically remove early stop widely linearize bregman extension boost descent method solve problem linearize add linearize gradient ascent apply lagrange dual solution without setting early regularization necessary recovery basically nearly lasso stop without loss differentiable bregman linearize iterative soft widely different name literature q move shrinkage linearize generate path easy implement distribute store distribute choose load balance variable reduce input matter independent unit nearly parallel truly scalable implementation divide block row column recently unit communication link center long scheme multi party regression internet incur introduce denote complement submatrix generalize adjoint inner similarly omit reason discuss right throughout rest property bregman bregman generalization linearize bregman proof preliminary summarize section piece regularization iteratively ty kn ty go piece ensure existence uniqueness uniqueness bregman differentiable solution addition uniqueness define case uniqueness impose I involve must impossible argument hold f continuity nonzero remain either stay continuity kkt identify specify time strictly unique strictly strictly unique lastly column existence uniqueness linearize show follow existence linearize bregman right unique solution become lipschitz theorem ode close piece sign remain reader sign oracle least subject sign sequel question remain continuously differentiable continuous uniqueness path necessary noisy bregman restrict strong x restrict empirical linearly say one variety name exact recovery e covariate effectively check support alternatively mutual incoherence eq hold hold sharp translate kkt relation temporal bregman lie support incremental add drop distinct projection mean piecewise incremental sense bregman drop bregman omp fail bregman gets plug side ensure mean kkt split trick lasso part integration eq reasoning q incremental process tt q lose dropping happen lasso mean bregman statistical consistency path let false time moreover signal e reach sign sense statistically mean bregman path light incur path bregman path averaging cause bias tell always dynamic pick select incorrect reach bregman return path false strong eq probability unbiased strong consistency ensure square root assume subsection linearize section take omit linearize bregman linearize bregman give establishe consistency linearize path time magnitude eq c n sign consistency guarantee bregman rate monotonicity appendix x statistical linearize bregman linearize bregman big enough satisfy path enough x nk k linearize bregman ensure noiseless analyze differential associate restrict evolve potential fast potential study dynamic oracle continuity multiplying obtain dynamic induce oracle one restrict evolve outside subspace true strong decay reach goal decay nonlinear differential case gr bellman lead tight generalized right strictly ensure concern stop time reach equip generalize consistency say potential multidimensional dynamic drop ready sign discrete find support lemma depend noise depend noise thus datum leave stop q bregman stop minimum magnitude path along sign bregman must one stop bregman full addition rt comparable factor bind arbitrarily present proof theorem show probability bregman stop sign consistency algorithm stop bregman sign high residual x mean stop stop p x experimental relation lasso linearize experiment identity three lb lb big curve goodness roc receiver regularization lb level positive roc curve auc roc large auc indicate pick path repeat deviation noise reasonably become big lb decay h c lb lb noisy signal dynamic bregman linearize discretization lead widely linearize stop regularization achieve selection consistency unbiased estimation linearize bregman linearize stop bregman direction rule setting q operator mean false positivity consistency directly proof differential inclusion path matrix integration side obtain follow take directly union inequality end denote eq noticed mind continuous convexity generalize min min min min min dx min let notice sn n cs sx sx lemma false positivity lb monotonically upper suffice b negativity ensure cl big min min first version linearize bregman lyapunov subgradient side iteration present discrete q hold x recover continuous step size discrete end continue lb monotonically lemma admit monotonically
elsewhere establish completely analogous instance theorem computation must uniformity repeat seem intensive maximal since interested check contain ball intersect radius equal center contain exist sample center check uniformity rejected determine mr uniformity reject compute performance see addition value maximum cycle b r smoothing parameter convex estimation estimation mention table illustrative really estimation rs estimator theoretical convex hull estimation reference multiply table support estimator h r h support estimator last benchmark multiply size rs rs benchmark multiply rs mm mm estimation rs result rs present behavior hull provide result case estimation real estimation mainly see significance also discuss outlier provide risk satisfied select smoothing rs estimation error criterion large accord rs increase would closed ball subsequence converge subsequence converge subsequence denote contrary addition contradiction exists converge since necessary guarantee existence point boundary nonempty satisfie necessarily unique aa h pt pt aa aa relate existence coincide onto onto contrary condition line accord q pt let condition proposition without generality prove impose restriction verify accord auxiliary necessary first nonempty converge clear define ball accord prove take account notice arbitrarily give exist meet nonconvex nonempty let exist convex ensure exist open show contradiction satisfy straightforward convex nonempty accord loss generality replace ensure r bc imply nonconvex parameter converge r theorem consequence proposition ensure theorem hold converge notice proof criterion consider support project develop major science point drive propose select value hypothesis drive rate convex hull much flexible shape condition uniformity reconstruct practical reconstruct figure imaging make shape sort center see influential radius ball whereas large considerably ba ba shape respectively sophisticated priori instance convex contain analyze depth restrictive introduce flexible say center closely c literature produce reconstruction point approximately set although convergence see depend influence present however practically dot comment special emphasis especially give select aim drawback select available problem selector minimum span automatic see figure area ball sample calibrate maximal result uniformity opposite compatible uniformity back two usually set bound nonempty hausdorff hand borel measure hausdorff physical proximity measure distance completely useful shape hausdorff boundary evaluate ba section optimal establish drive achieve hull estimate practical analyze performance selector analyze proof defer reconstruct estimate definition define let nonempty compact nonconvex property verify obviously condition however take radius convex define gap maximal point lebesgue denote uniformity reject quantile instance applicable et select maximal region volume ball radius b dot show bad big value smoothing allow clearly uniformity hypothesis contain uniform must small idea technical section existence guarantee consistently compact nonconvex nonempty converge exposition infinity value reject behavior set prove guarantee go ensure rate hull compact nonconvex nonempty define define converge
consider determine asymptotic derive belong alternative assume condition normality determine weight maximize restrict test power asymptotically class estimator et unbiased pn tr tr tr tr show assume pn lemma adapt asymptotic asymptotic improved estimator derive expansion percentile addition percentile replace percentile compare sufficient allow high low local power asymptotic among test compare test test define significance level power valid normal hypothesis covariance replicate n point provide z consistent level table multivariate normality significance level approximate nominal reasonably tendency conservative power select replication size r ep summarize among show test test good small test always high close behave location asymptotic local asymptotic test stable statistic power conclusion recommend range author extensive discussion ms simulation numerical preliminary central form random diagonal express tr nz lyapunov lyapunov lyapunov lyapunov moment distribute diagonal tr tr tr tr tr tr tr al expand stochastically orthogonal eigenvalue independently statistic expand pn expand stochastically follow tr pn pa pn stochastically c pa tr tr monotonically find c derivative get tc power define pa pn h e h h tr moment relationship moment obtain coefficient relationship characteristic ct result use lemma therefore necessary test asymptotic test superior local asymptotic power pn pn pn result section ex plus location average statistic mathematical mail com school business loss exact approach du paper test comparative difficult local power new weighted statistic determine maximum local induce local asymptotic respect show statistic parameter power location subject dimensional hypothesis test traditionally n nan use determine important issue study maximum asymptotic power equivalence easy asymptotic size simultaneously go sample
challenge naive ignore presence set dramatically valid see reference therein involved focus target irrespective selector generalize crucial quantity confidence desire also quantity denote throughout selector standard explanatory precise eqs beta set explanatory obtain explanatory correspond regressor model call design dependent coverage target typically lead confidence irrespective design error cf explanatory optimality little relevance thus cover dependent target may next coverage target denote standard coverage infeasible cf remark particular target suffer discuss precede paragraph model interval cover coverage minimal nominal level interval valid irrespective selector representative situation extend confidence design active problematic select model component situation costly irrelevant component resolve variable maximize inactive width interval moderate extend seem consider discussion remark interpretation introduce model variable observe explanatory design confidence cover minimal coverage nominal computing interval report proof stress naive confidence interval selection correct target valid easily set confirm simulation model full identity matrix refer eq linear hence outline square observable chi square degree exist n p extra generality provide additional estimator possibly observable well full column retain relation use follow empty write cardinality write abuse square estimator inverse variance imply restrict square obviously estimator function depend allow procedure make allow case procedure restriction range power case consider universe quantity vector empty paper aspect force infeasible cf presence predictor post feasible infeasible target emphasize design apart cf infeasible discussion interpretation remark nominal confidence confidence interval design form depend depend interval zero reduce thus contain replace eq ci naive see numerical line relate quantity distinguish observation regardless post select observation component consideration observe follow general ci hand display neither p x consequently distribute cx collection ratio rx positive hold quantile universe subset maximum model rs consequence precede immediately interval general form ci proposition guarantee procedure post selection prop feasible compute select define replace hence via course depend want stress see arbitrary interval satisfy costly compute prohibitive sphere column convention eq distribution degree freedom represent expectation depend hence hold f quantile whenever obvious besides want stress full full full appear extension contain publish bound note counterpart reflect model possible interval event fact available confidence also cf section k immediately imply corollary arbitrary eq aim remark note coverage represent call full clear coefficient target inference depend dimension rise component mention amenable combination mx argue rather infeasible apply observe justification argue note applicable desirable problematic iii proposition post approach infeasible predictor sense nx draw otherwise optimality feasible typically force ii carry dependent provide universe ignore exploit expense complex sketch change degenerate x x x xx illustrative subset describe situation act regressor correct belong happen satisfy select overall interval presence want universe precede highlight require subsequent overall distribute possibility rise universe e present extend appropriate contain ol framework make become problematic model except cf framework continue otherwise continue proposition chi denote constant stress union bound conservative showing lem exist vector infeasible inspection also analogue replace random sphere column space close finally use k x bound improve marginally assume full column almost accord matrix moment denote maintain section distribute chi square distribute section conditionally section dependent conditionally shall estimate size furthermore result fix subset motivate target eq inverse interpret inverse justification infeasible infeasible call subsequent remark section design nothing post analogue predictor mx argue xu u additionally force straightforward computation prediction normally thus predictor depend square normally good predictor class depend shall remark interval section interval procedure mild typically procedure bic design sequence random weakly dependent moment also degree freedom course choose condition present recall interval model confidence asymptotically valid ci confidence arbitrary arbitrary sequence positive integer replace relation asymptotically target minimal choose theorem also continue replace condition uniform consistency property generally note satisfied post condition lemma consequence use interval remains valid allow also early infeasible term depend converge repeat result target continue hold independent ci continue consequence noted begin design dependent continue hold conditionally consequence turn drop instead continue cover rate ci generalize set employ provide statement concern probability converge difficult remain interpret statement outer establish presentation treat canonical coordinate compute proposition square estimator set show canonical decomposition section x replace precede analytically integration solve shall f freedom case calculate kx c probability obvious costly alg involve search solve search around one tractable one hour computer since alg solution monte replace involve range define decompose continuous analytically c approximated function suppose satisfie identically sphere calculate quantity choose hold solve uniquely always except hold uniquely r p beta observe except h furthermore r converge kk solution search negligible algorithm second equation run convention argument positive obtain version one integration negligible run much search compute universal bind constant negligible compare case uniquely determine uniquely determined exist unique exist unique positive case approximate equation convention constant remark modification square distribute freedom ii appropriately generalization alg tractable report quantile beta always reasonable numerically length confidence interval length confidence interval selection selection lasso matrix obtain exchangeable design obtain ci six replace either degree constant interval hold provide standardized contain ten explanatory term intercept square water square slope percent thousand surface index capacity water water hour period hour take except intercept explanatory peak choose length interval ease burden standardized length belong ten universe obtain approximate supremum monte follow monte step keep large evaluation monte algorithm second step monte standardized improve standardize constant combine increase standardized length indeed standardize increase standardize obtain decrease standardize almost standardized length say interval short interval standardize base standardized note costly especially length interval close confidence interval standardize length size confidence discussion confidence length yield sake brevity standardized confidence always increase respect evaluation minimal report costly would report sake brevity minimal estimate various aic bic procedure procedure explanatory intercept information constant universe intercept explanatory bic penalty equal aic bic aic minimize result intercept explanatory lar outline regressor regressor residual lar cross cv lars intercept model comprise column obtain standardize similar obtain exchangeable datum independently component nine row use target vector probability target investigation coverage estimate monte random gaussian column monte overall record currently confidence constant averaged small coverage first repeat carlo sample record small second confidence time estimate minimal coverage consideration coverage find obtain search upper coverage
multivariate say successively optimize hold solve successively achieve natural statistic netflix attention rating whose view coordinate recommend item user align every miss coordinate multiplied penalty introduce avoid typically quite impose fix decade technique attract much attention suppose penalty ridge convex recent year start mc hence detail scad readers original function interesting currently prefer penalize model coordinate coordinate solve fix successively lasso descent scad guarantee global make get inferior solution find get serious getting saddle introduce share perspective share perspective expand search slightly comes avoid possible thus proposal undesirable location search choice space upon traditional include space idea mc penalty stress motivate minimize point fixing thus see actually particular point able search would suffice main simple strategy undesirable observation follow run continue search sufficient strategy work process improve starting outline minimize key insight switch different undesirable location share perspective early alternate search component scaling adjust accordingly next optimize scaling begin somewhat helps scale view respective space interpret illustrate time search still much understand scale conduct improve well illustration imply equation suggest expand simply joint kind search view avoid full simultaneous illustration versus imply joint search denote fact would matrix solve switch saddle inferior solution minimize eq simultaneously bfgs search quite expect space effect expand scale type subspace propose restrict search factorization search item mathematically feasible item search important search notion informative describe two trying determine set vector large establish baseline space serve illustrate power search index incorporate shall report even sophisticated choice specific take still space observe improved context front highly would expect decrease lead depend negative selective correlate pre selective compute fit surface warm start penalty think fitting introduce challenge work fit concerned good pair surface solution sequentially surface warm start desirable solution inferior warm worse empirically common occurrence keep surface surface warm point surface coordinate use previous warm like obtain previous point warm keep actual surface surface strategy conceptually think easy surface triple computation experimental mc matrix demonstrate factorization million review dense subset million review rate item rate time restrict allow number item good performance wide test rating run absolute mae mse amazon baseline translate rmse mae baseline dense subset information item subset fair comparison cross show average show table optimal cccc scale subspace sec sec k factorization cccc sec k appear conduct greedy strategy fast factorization mae versus mc demonstrate mc predictor generate mean whose entry linear predictor space logarithmic spaced logarithmic small percent converge restrict indicate coordinate find relatively remain solution expand reduce small half percent decrease little large nonconvex average percent decrease point computed percent measure show strategy lead improved mc terminal regression comparison selection call conclusion article general upon nonconvex switch conduct different illustrate problem namely factorization mc penalty help search subspace carefully undesirable produce notable algorithm factorization regression contribution acknowledgment support explicit problem solve identify satisfy condition write hence
outer maximization f study extension show simple strategy efficiently output dependent however threshold achievable technique focus conditional consequence particularly set corollary derive general make empirical batch observation relate depend example explain prediction optimize micro macro false increase optimally thresholded characterization maximize relationship achievable score score discuss version result start valid lemma confusion tp ps ds fp ps ds tn ps maximize ambiguity rule conventional sensitive latter side replace later elaborate undesirable divide region except similarly besides add decision negative add simplification limit claim case output calibrate predict assign rule maximize decision calibration optimal assign eq simplifying calibrate half maximum assume intuition theorem save version result section label confirm length thresholded output define count gold standard total function ap achievable macro tp fp tp aa loop predict positive sort proceed stop number pick follow stop designing maximize batch observation uninformative potentially undesirable prediction depend distribution prediction exceed note exceed threshold never exceed predict instance batch assign great uninformative classifier calibrate uninformative thresholding maximize result uninformative seek maximize label denominator positive predict actually difference always positive simplification derivative whenever positive uninformative expect maximize optimally thresholded uninformative base close figure point context macro optimal batch micro macro micro rare additionally macro average uninformative imagine know base th label rare perfect actual positive rare nothing predict positive gold label thus single trivial system adjust constitute average calculate rare consider label trivial optimally thresholde trivial however improvement perfect common macro rare label macro uninformative classifier micro optimize predict macro optimize calibrate half positive classifier confident prediction micro macro thought threshold distribution assign low base predict maximize macro micro evidence study discuss thresholding macro undesirable real macro f consider assign tag vocabulary mesh article literature represent abstract bag vocabulary consist word tf preprocessing step word rare multiple square probabilistic overfitte approximate training value consequence rare lose problematic probabilistic classifier theory relate threshold plug rule micro macro prediction micro thresholded among mesh count human middle label base uninformative thresholding positive label macro system overall maximize experimentally rather ground maximize winner future phenomenon overfitte different threshold f empirical performance high threshold converge true higher demonstrate idea uninformative uninformative uninformative uncorrelated independent label accuracy convergence irrespective base threshold depend optimal threshold occur threshold close threshold consequence consider threshold experimentally sort order uninformative lie score include accordance maximize threshold f lower optimal far f optimize identify threshold whereas threshold identify problematic lead issue identify arise nonlinearity score treatment label choose label threshold maximize select plot predict threshold shift positive negative base low even number derive decision positive theoretical well achievable optimal make threshold classifier uninformative predicting maximize maximize macro uninformative contrast micro maximize micro potentially undesirable rare every desirable precision reporting alone sometimes practically optimize compete choose winner behavior edu insight maximize score binary context harmonic classifier micro average macro use value achieve optimum calibrate conditional uninformative classify example behavior undesirable surprising metric prediction commonly classification classification area supervise machine micro average average instance average use macro impact performance per experimental property performance minimization incorporates convert numerical classifier latter scenario alternative case belief produce thresholded
million point since keep gp use parallel computation achieve induce auxiliary involve al al adopt combine parallelization gpu acceleration experiment inference consideration construct likelihood induce covariance see practical application parallelization decomposable specifically write nm notice associate treat introduce gp maintain tight straightforwardly propose variational induce analysis unsupervised scenario namely w parallelism naturally distribute portion although computation recover computation collect node get locally distribute optimizer determine locally exist bfgs currently collect gradient global parameter optimizer rest computer write gpu write write scheme mention scale exploit gp quantity constitute computational bottleneck go something dataset make gpu acceleration hardware number small core efficient ideal advantage specialized architecture computation onto gpu making properly divide parallelization within gpu gpu synchronization memory expensive division difference try choice design assign block gpu computing assign block gpu intermediate share local write memory gradient depend division apply acceleration parallelism section integrate assigning subset portion gpu b gpu acceleration synthetic mapping radius rbf function recover representation gp reduction latent induce parallelization configuration processor directly run iteration time scale speed parallel implementation communication overhead negligible speed gpu core time show computer speed additionally algorithm gpu acceleration constitute counter gps applicable implementation plan model
section statement existence compact fix scalar determine c prove compact invoke existence establish iterate need classic fix theorem map iterate map every begin classic theory able proof arbitrarily converge mention concave whereby unique first necessary fix iterate fix show merely iterate repeat concavity unique yield rewrite introduce new scalar choice addition decrease large large respectively multiply numerator denominator first I p multiply definite substitute small rearrange write p multiplying span definite therefore equal let datum large zero small section one find imply converge large three decrease toward zero increase previous find large one case proof toward similar possible show ii end explain easy convergent calculate eigenvalue become eigenvalue small eigenvalue become multiply obtain small want calculate eigenvalue invoke proof convergence existence ml generality consequence ml solution exist still converge convergent possess structure subspace generalize present strong method highly outperform manifold technique experimental easy observe one choose become two variable first without fulfil firstly entropy namely expression knowledge surface sphere obtain hold expression average loss radial need radial follow square easy f radial expression average kl divergence expression follow log summing observe rotation axis kl increase kl become infinity fit substantially generalize modify fix parameter simply use counter em derive refined merge identify false division cloud cause minima practice iteratively find candidate merge split split merge merge one merge explain difference merge however entropy clearly k nk propose split solve need therein split improvement limit split lead improvement split threshold split stage find component reach perform step merge ml describe splitting small parameter merge typical optimization overfitte amount kn old measure bias test stop fine propose successfully alternative explain split recover cccc pt pt stage cccc different stage explain experiment show speed three bfgs trust also test like plot dataset size van dataset contains exclude etc extract image patch remain pixel intensity white standard evaluate mi mi bit pixel intuitive patch formally hx entropy pixel procedure mi dc component different observe mi change increase show simple capture layer reach parsimonious propose radial follow ica correspond deep boltzmann machine mi method explain emphasize difference mi mi improvement capture distribution claim effect baseline rate gauss spherical mi bit high different patch study elliptical existence uniqueness maximum modeling art remain direction study involve tool investigate non gaussian mixture gamma process complement process investigate behavior model rather gaussians hope basic outline encourage researcher rich author title author ac ir school college engineering institute institute biological de institute study elliptical parametrize tail peak rich ml nontrivial develop globally convergent merge expectation propose mixture modelling model analysis often capture express heavy tailed independence situation unsupervise mean usual gamma additional factor encode tail behavior gaussian pt leave display study stem broad mean successfully multivariate financial modeling pattern application survey application reference paper potential direction recovery multiple various field encourage nonlinearity easy implement work appeal concept brief mixture context paper make fix ml parameter handle numerically nonconvex despite optimality demonstrating gain obtain manifold variable dimensional random variate one variable kullback divergence zero gamma multivariate begin special scatter matrix exist
yield mini variant filter normalization convolution require modification incur negligible overhead different filter crucial version mini max pool convolutional two investigation imagenet dataset dataset contain million image image image assign category evaluate image train network supervise criterion work imagenet black box visual front involve possible benchmark involve digit involve imagenet deep mini network convolutional employ standard max pooling fair comparison case six convolutional follow connected way softmax throughout normalization connect layer type conv conv conv conv max dropout channel filter input x pooling baseline specify employ pooling accelerate neuron convolutional difference architecture specify neuron layer use mini max layer mini invariance layer accelerate layer conv conv conv conv weights texture report imagenet yield well find improved aspect ratio manuscript aware work report max pool neuron convolutional c max pool top quality propose imagenet feature purpose dimensional without fine weight new svm experiment preserve aspect per report different pool norm classify network follow set find beneficial convolution layer rate maxout art c maxout model maxout cifar mini mnist cifar comment property imagenet validation correspond et contrast max pool without quality somewhat improve paper representation deep neural successfully substitute consecutive convolution pooling empirically learn around well pool baseline challenge imagenet develop exploit amenable image reconstruction objective software framework get publicly source code file fully acknowledge convolutional neural recognition task propose replace consecutive convolution max model use mini small learn organize propagation learn assess classification benchmark imagenet convolutional network architecture pre imagenet mnist offer learn increasingly work demonstrate benchmark build like sift success interest learn computer refine aspect feature successfully employ deep successful image feed neural fashion propagation availability annotate dataset gpu partly build block deep image around convolutional layer field spatially share abstraction resolution max build around image translation aware representation position deep consecutive max pooling layer convolutional neural spaced patch across filter max pooling hand filter dictionary within input layer center alternative pooling employ mini mini enough desire position output mini maximum across mini use output per spaced within propose primarily train propagation mini achieve conventional pool convolutional network classification converge especially normalization accelerate convergence imagenet report excellent classification network build initially towards video modeling unsupervise explore mini variant sift learn similarly maxout critical difference layer hard extract value overlap share significantly reduce maxout conjunction max substitute moreover maxout perturbation create inactive contrary require regularization connect explore pooling avoid build paper unsupervise deep learn adjacent share area thus relate c mini max pool convolution filter pixel span channel represent element convolutional densely position also resolution map produce window position pixel apart map max pool convolution consist channel filter point attain propose convolution scheme replace spatial mini filter extract patch regular maximum position attain fix dual alternative filter max
different question date concern discovery approach practitioner expectation science brief end preliminary cluster achieve management maximize adopt specific extended pair base investigate assess potential learn continuous assessment sensor accurate use highlight really setup assume really hierarchical experiment highlight ball link different cluster learner identical imply learner previous behavior static fig south percentage cycle duration fig west mm involve duration maintain speed group goal increase stroke goal single second increase stroke receive arm arm arm suppose learner trial cycle period successive maximal equip angle frequency hz act individual relative define anti relationship series relative fig record cycle per trial cycle science view specific aim study assess behavior potential learner assess investigate possible existence behavior learner condition possible search group analysis point point entire cycle may highly beneficial discriminative cycle learner machine view tackle cycle describe feature continuous nevertheless preprocesse priori trial directly describe fix feature use cluster reduction stage cycle similar cluster observation represent observation label likelihood know observation fisher em principle mixture lie lower choose efficient medium example hold realization observation group indicate decide discriminative let complement dimension lie discriminative lie discriminative complement conditionally help deduce observation follow center ensure impose projected discriminative subspace generative discriminative parameter gaussian enforce gaussians combination iterative fisher discriminative fisher use observation computed probability projection maximize variance whole tn ik probability parameter ik computation efficiency nevertheless back observation original feature enforce fisher penalty criterion penalize algorithm cluster purpose informative sequence cycle clustering consist number bic level highlight result qualitative learn learner visit cluster across different lead pattern group analogy whereas use lead type practice gray bar level height bar nan corresponding latent cluster point movement fisher existence highlight transition cluster group high receive second cluster analysis allow highlight additional strategy preferred fisher highly correlate feature interestingly sparsity qualitative need term c
family r enyi entropies bound entropy shannon enyi nonnegative order shannon explore entropy min entropy enyi entropy min entropy relation easily extend class entropy end bound enyi relation section explore analogy analysis feature examine atom investigate dictionary topological overcomplete explore diversity measure zero measure diversity measure synthesis dictionary input study connection conduct illustrate learning gaussian process network diversity quantify overcomplete spirit recent degree receive ph optimisation security technology research associate system model laboratory technology interest include statistical processing wireless sensor process hyperspectral author paper learn processing past year review linear collect dictionary representation orthogonality condition yield overcomplete dictionary overcomplete elegant nonlinear define dictionary dictionary coherence overcomplete dictionary associate uniform spread entropy definition entropy map space generalize shannon sparse recognition gain increase popularity denoise representation atom formalism combination define basis predefine dictionary analytical wavelet dictionary widely dictionary deal orthogonality bi orthogonality deal analytical structure orthogonality year dictionary adapt call advanced statistic fall orthogonality increase overcomplete dictionary largely several dictionary problem counterpart overcomplete dictionaries relevant increase diversity measure quantify diversity simple diversity certainly measure examine either fashion thorough way characterize pairwise correlation correlation atom yield last pursuit dictionary also extensive compressed sense diversity overcomplete analysis framework pairwise distance atom neural indeed interpolation unit distant turn operate pair atom thorough measure atom combination machine gaussian online filter component aforementione consider formalism atom nonlinear give atom transformation conventional nonlinear study network nonlinear adaptive aforementioned diversity heterogeneity dictionary relate mechanic randomness generalize overcomplete diversity illustrate atom spread comprehensive kernel within entropy propose finally entropy derive depend diversity connect input space follow introduce nonlinear formalism present diversity measure quantify overcomplete dictionary throughout examine section space section estimator kernel investigate network particular l reduce complexity pruning contribute learning measure paper connection consider analysis entropy term coherence provide dictionary extensive space generalize diversity conventional linear model well conclude outline issue banach space theory give elementary estimating eq atom approximate span namely include transform cosine basis orthogonality overcomplete dictionary investigate representation coefficient representation sparse assume view seminal consider available drive iteratively alternate data posteriori probability dictionary correspond problem good dictionary subject method direction respectively detail therein worth difficult atom formalism elegant framework tackle feature kernel hilbert space product induce reproduce I divide category projective product radial kernel kernel radial norm paper restrict kernel conventional model nonlinear investigate rbf include take q deal easy linear consist residual rkh linear investigate dictionary determination difficult technique turn literature problem recently filter context element factorization j projective linear radial inverse j synthesis overcomplete dictionary grow atom heterogeneity atom require diversity pairwise thorough atom consider diversity measure paper connect information source alphabet measure quantify randomness need average store investigation aforementioned definition investigate r enyi entropy finally section extend rkhs studying quantify diversity dictionary diversity criterion dictionary diversity simple distance atom tight atom factor tighter say distant residual onto atom scaling take yield impose atom latter rely atom exhaustive analysis quantify capacity approximate atom combination dictionary dictionary say follow correspond residual project subspace cost coefficient q respectively study thus atom dictionary approximated characterize correspond large atom mutually atom two initially match pursuit union basis quality consider measure coherence give coherence correlation atom definition coherent norm atom I coherence construct dictionary enforce cosine candidate include coherence latter exceed dictionary basis coherence rely correlate thorough measure large atom way connect coherence investigate norm operator indeed gram atom explore connect latter sake simplicity construct dictionary include exceed threshold throughout rigorous overcomplete diversity attempt bridge follow theorem coherence dictionary quantifying dictionary vice versa atom exceed coherent also coherence coherence exceed proof fundamental analysis approximate exceed approximate dictionary nj ii coherence aforementioned measure theorem randomness give enyi entropy shannon entropies quadratic see deal random code definition yield correspond datum know entropy estimator overcomplete end initially generalize
efficiently well efficiently undirected dynamic mix correlation decay show underlying update dynamic runtime nearly process epidemic cascade infection epidemic observe broadly number chain include spirit show relatively generate literature noisy game author structure study infer observation rest learn section present give theoretic reconstruct ise graph assume binary variable gibbs partition normalize distribution edge vector simplicity suitable minor modification accommodate external field implication node allow chain natural time discrete version dynamic write configuration time start rate spin notably spin eq randomness later efficiently graphical plausible generating process check stationary stationarity family local chain include know slowly allow imagine access potentially information obtain spin purpose convenient heat version chain take update denote node identity first arise update sequence observe time expectation spin argument amount convenience set performance use minimax good triple tend infinity determine focus derive arbitrary assignment implicit dependence imply conditionally hand side sample claim decide suffice simple quantity justify follow combine give q generality replace emphasize turn possible sign depend allow contribution scenario select poisson determine two independent connection formula conditioning edge expectation event shown estimate eq q inequality estimate plug spin spin remain plug already side reasoning bernstein find implication adapt surely theorem denote bind imply kl k k observe z x ij ij bind take union see state inequality suppose kx kx bind suffice clique single remove I large removal difficult identical consist clique odd fix perfect clique cardinality edge value capture effect edge dramatically weak suppose section prove theorem use inequality kl divergence kl zero bound parameterized construction marginal clique divergence projection equal abuse project relevant clique measure keep update initial configuration draw index uniformly note configuration select symmetry construction clique consideration last symmetry bind probability likelihood later event give term mu mu l mu last equality expectation justify symmetry bound lemma eq times suffice uv uv show uv uv plug clique get second inequality take display main message paper quite setting underlie ise markov generalization observe acknowledgment grateful comment nsf grant office award nf david laboratory systems electrical science center school technology mit undirected graphical dynamic chain sequentially nod frequently additionally natural access direct low main work reconstruct binary pairwise theoretic might include stock financial user network markovian govern interaction interaction site dynamic underlie fit traditionally pose abstraction many natural generate graphical I sample hand represent focus low algorithm generate turn approach scale procedure observe paper node neighborhood search candidate node maximum prohibitive focused find
harmonic parameter secondary stepsize vary achieve stepsize secondary stepsize early result performance run quality cause slow necessary secondary stepsize run stepsize mdp stepsize stepsize outperform secondary suggest insensitive choice secondary stepsize line figure value range rule slightly performance value achieve secondary stepsize robust secondary figure rule worse slow horizon harmonic choice large consistently bad iteration good compete stepsize tune continue increase even slightly approximate iteration perform however become discount increase appear discount signal harmonic stepsize good perform poorly contrast insensitive parameter simple yield robust several stepsize mdp manner randomly pick let transition state state lead variety value stepsize upon state n accord next random briefly reasoning policy function make also implicitly stepsize stepsize affect affect visit affect ensure good visit practical algorithm quite problem outside scope stepsize policy state action run stepsize iteration evaluate follow find cs ns ss quantity iteration algorithm measure stepsize harmonic rule secondary stepsize secondary stepsize order good magnitude also order harmonic number visit see discount performance later half conclude yield harmonic tune sensitive visible secondary competitive horizon bias period behaviour run cs v ts n evaluate policy performance discount approximate horizon rule harmonic rule well easily horizon state rule competitive overall mid largely close harmonic finite horizon magnitude stepsize formula synthetic state transition stepsize identical last horizon apply early horizon stepsize observation propagate backward across time early period go exploration curve suitably calibrate last demonstrate conjunction present problem appear finance reservoir management abstract setting wish keep stepsize generic contain dimension amount resource currently price resource represent represent positive time increment action incur make decision resource minor change could demand continuous impossible furthermore programming post extensively state post pre obtain post state adaptively price discretize sec optimistic initial discretize continuous price discretized detail volatility spike pure choose process visit level generate random price stepsize architecture secondary stepsize secondary stepsize stepsize rule iteration decision cp ts horizon average path report maximize large require several policy consistently improvement figure harmonic horizon several iteration improvement harmonic experiment offer evidence new applicable complex post difficult expectation continuous advantage stepsize set encouraging sign propose mathematical stepsize single mdp derive new slowly application importance stepsize stepsize value approximation single state stepsize inherent prediction single considerably simplify estimate extended mdp horizon test lead rule stepsize rule harmonic tune sensitive stepsize test stepsize find rule conclude rule conclusion harmonic tune particular result strength adjust evolution function lack know converge equation stepsize bounding reach fast time differential approximated derivative step definition onto positive interpolation define recursion greatest less great writing weighted property omit increase derivative interpolation function eq dnn v bound bound increase across generalize define boundary condition differential equation solution order equation integrate dm cn dm plug equation imply require clearly similarly write n side observe q v complete inductive denominator show numerator inductive result n g n proposition rewrite subtract side side impossible contain denominator additional relation rule originally optimal refer alternate name adjust kalman filter processing sequence choose formula n smoothed stepsize minimize prediction term represent design general goal track scalar move signal general violate assume use rather bootstrap old approximation single construct reflect derivation improvement first explicitly incorporate dependence approximation updating handle bias would value use rise special c average period easy also secondary sensitive secondary stepsize conclude note may process observation construct inherently lead discussion fa fa nsf contract theorem definition prove wide care management algorithm dimension stepsize update operation research computationally intensive obtain popular parameter produce result tune stepsize prediction improve short insensitive new dynamic health care management management period planning period operation often require solution nonlinear integer quickly stepsize control merge horizon dynamic value state possibly action horizon describe take difficult solve curse bellman solve approximately name dynamic reinforcement however literature rule produce theoretical provide insight stepsize nonetheless stepsize slow construct single poorly remain part remain hundred poorly part stochastic adaptively estimate include rule stochastic bar variant additional challenge mdp work heavily tie whereas practical learning approximation arguably approximate view adjusted kalman selection study behaviour general dp stepsize bias tradeoff dependence model contribution computable demonstrate form easily stepsize stepsize account dependence tune general stepsize rule limit stepsize provably numerical comparison set sensitive rule insensitive importance allow focus strategy concern poor tune stepsize rule demonstrate action derive optimal stepsize set section new stepsize motivate stepsize commonly stepsize slow dynamic programming stepsize commonly refer refer inside observation albeit reinforcement gradient kalman filtering processing challenge close reason stepsize adopt adjust exhibit stage require extensive adopt different instead general close reduce equation independent identically prediction formulation simple general mdp tradeoff variance govern stepsize observation consider finite bias general issue exhibit complex behaviour subject insight issue stepsize capture able program allow high stand horizon mdp recall decision iteration optimal state steady illustration multiplicative bound free plot upper proof imply stepsize enough show apply proposition convergent subsequence subsequence find proposition know accumulation subsequence subsequence convergent subsequence follow accumulation immediately result observe vanish stepsize standard rule benefit rule design avoid stepsize valuable finance price management tight dynamic stepsize secondary stepsize choose stepsize become stepsize calculate optimal secondary stepsize require estimate period straightforwardly collect extend many action replace reward action depend visit reward however policy expect state system wide dp sufficient keep estimate similarly store dependent quantity dp example separate pair lead stepsize mdp generic complex problem store procedure initialize function solve let wide cs n stepsize x x x increment suggest stepsize avoid period mdp steady policy briefly mostly carry surely arbitrary
I irreducible two get call property ergodicity converge global balance gb condition stationary book mix strongly recommend book mathematically gb equal total employ gb special pairwise balance x py oppose gb db gb flow reversible markov usually numerically enforce mcmc gb db prior due metropolis mh generalized use algorithm first specify acceptance probability way new element total converge obeys db fulfil acceptance hasting mh ax mh process know converge section eventually interested average correlation observable spin transition equal ft ft ft ft fx go infinity converge stationary f xx omit dependence time subscript average dynamic autocorrelation describe correlation observable autocorrelation autocorrelation lag formula simplify ft f measure equilibrium exponential autocorrelation integrate autocorrelation inverse autocorrelation observable q autocorrelation relaxation observable place equilibrium inverse frequently eigenvalue lie disk large degenerate follow eq spectral define large yy expression tx fast converge obey db observe obey gb absolute part equilibrium call autocorrelation ft check substitution integrate autocorrelation control carlo average excellent another source monte algorithms statistical book mcmc detailed balance dynamic phase example lattice steady mh unbiased neighbor occur rate mh diffusion require step imagine sometimes beneficial momentum helps spread air mh slow suffer critical fluctuation sampling phase transition computer create implement convergence open discussion add cycle cycle steady practical way create stochastic impose skew detailed enforce adjust block transition state simplicity skew balance stationary turn essentially cycle start another west east west uniformly node turn need correlation inverse system walk visit site east west site random site vertical horizontal make autocorrelation see steady spectral site suppose ise vertex exhibit symmetry infinite vertex carry spin spin configuration run ground energy ground degenerate pointing point equally probable size temperature dominate tend align though still perturbation break mechanic transition energy solely eq e em q fm fm energy free minima degeneracy functional perturbation external functional q fluctuation proportional give vanish fluctuation explain transition give excellent introduction spin reversible e confirm numerically autocorrelation copy copy would result system make ise ultimately notice degeneracy external remain saddle vanishe represent ise blue represent mh red label mh online dots respective represent reconstruct fit large asymptotic reversible respectively slope pt initially system state ax new x periodic three converge reversible excellent explain analytically mcmc ise reversible skew detailed chain converge example control speed ise critical create change balance speed change size configuration gb convergence field model observable natural besides several similar propose mcmc significantly reversible variant detail balance violate db reader rigorous paper compare reversible reversible asymptotic observable increase markov chain notation mh chen improvement still review reversible bottleneck energy rare energy vast rely symmetry ise lead reduction equilibrium violate balance useful phase soft matter protein structure interesting explore convergence know complete center support thank discussion rgb tool explore property feasible physics reversible chain reversible physical relax detail yet balance certain case acceleration root improvement reversible introduce ise complete ise review applicability converge physics quantum field sum evaluate amount carlo square already carlo outperform point carlo extremely
value copy appeal weight connection change student reach algorithm extremely whose student critical high student sense many character teacher sure response teacher stop teacher teacher pass ambiguity information dark teacher second get seem within gd similar table gd sgd gd sgd systems also simulation non domain repeat spin rather case find identify quantify system lead support acknowledgement david suggestion simulation mnist acknowledge support gpu research institute ny city university york facebook ai york ny minima real function challenge whose agree spin prove tend teacher mnist phenomenon network finally descent descent level interest need surface question landscape challenge especially complex critical problem context seek find match learn describe form desire another come physics align equilibrium reach reasonably fluctuation machine spin mechanic connection attractive nice slightly exhibit stochastic critical minima spin interest critical point spherical spin establish name example spin high point absence contain minima level long ground matter slightly random random extreme external landscape polynomially count critical critical point surface minimum flat jump critical landscape exponentially lie clearly existence point regardless alternatively favorable performance context spin ask question address difference discover hope light future simple two spin spin consider interaction interaction total energy represent lattice geometry interaction strength weak field particle alignment configuration matter landscape easy achieve assume favor alignment favor picture drastically introduces simultaneously attain rather landscape spin discrete state sphere particle interest study detail critical use far find reason triplet move remark critical hamiltonian eq critical landscape explore continuous symmetric global point sign index critical find asymptotic horizontal analytic description function analytic minima show expect decrease beyond critical keep denote number increase random variable sum implication hamiltonian extensive scale give sphere low energy ground respectively level vertical expect low index probability finding confirm deep trying level consider centralized loss approximate I hinge entropy sample loss increase fluctuation life true approximate property expect converge necessarily test fluctuation paper first landscape really imply surface energy arbitrary drastically landscape starting point descent narrow moreover irrespective simulation spin clear qualitative dimensional surface surface practical aside reach gradient start fix variable start trial criterion previous section hold concentrate hope nevertheless asymptotic fully connect spin couple modify polynomial achieve version way machine view sphere point product hamiltonian instead iw normalize
maxima give whereas deviation considerable head table gps high small type gps gps gps cm type gps gps gps h cm gps gps gps gps gps gps plan methodology control establish rely base calculate take production dependent aggregate relevant operating characteristic numerically simulation work stage accuracy formula stage turn estimator well gps extension inspection allow investigation firstly remain aggregate extent stochastically lastly end many day arise measurement curve iv curve extension effort well beyond article acknowledgment thank sc read part grant b dependent auxiliary observation central limit theorem suppose variable detail proof expansion independent expansion obtain continuity drop virtue continuity ensure validity well recall follow argument q virtue asymptotically normal fourth moment see virtue first assertion estimator standardize available additional third fourth consequence p standardize remark g measurement module newly virtue er device ed array easy nb entail go along line independent may jointly calculate account namely conclude unknown correlation replace observe replace p thing arrive proposition corollary keyword acceptance dependence energy deal construction inspection produce item reject motivated output panel capture appropriately production acceptance represent produce distribution power non side usually tolerance randomness true mean replace decision thus critical account module item quantity control although away specification item module directly since inspection infer fraction acceptable probability acceptance fraction unknown production control module module customer production far pass later check requirement operation stage done avoid e acceptance second seem double differently acceptance statistic e say one control sample instant reject first lag stage fact inspection already sample propose item form necessary batch spatial carry general apply sampling well organize acceptance operate two two inspection additional construct valid cover expansion control normality operating control independent realistic dependent lastly simulation acceptance back seminal contribution overview study double robustness normality type likelihood estimate fraction short production compactly approximation normality powerful mind favor acceptance rejection discuss focused type normality sampling inspection quality continuous fourth employ theory sample distribution sample simplify find extend difference additional discussion production smooth estimator cross validate kernel density adaptively bernstein purpose estimation application extension side specification focus quality become area result certainly adopt production panel high throughput production process cell sophisticated regard stack pair ease movement cell amount cell optical filter ensure process let briefly work compound positively bind four adjacent fill meet energy contact move contact top cell internal due differently power load cell make physical material cell stack bottom top substantially chemical material process associated weather may heat absence cause serious physical chemical electrical degradation degradation module couple internal module micro arise module production site construction improper image characteristic serious impact long degradation failure several year micro experimentally heat test loss drive cell surface ground degradation surface circuit market degradation reliability consequence inspection combine production construction notion rigorously depend lot pair sample lot accept operate item specification plan two acceptance procedure lot examine production construction module take decide lot accept lot instant inspection agreement specification realistic variance measurement acceptance accept lot deviation take deviation replace j accept notice sum behind rule inspection pass quality control comprise favor respectively lot accept close inspection drop aggregate concrete impossible know underlie shall appropriate approximation operating characteristic allow optimal arithmetic average otherwise replace turn depend standardized assume regularity quantile measurement take estimator moment consider natural candidate order quantile acceptance concentrate degree define coefficient bernstein polynomial mse sense attain parametric degree choose control density closeness distribution result establish quantile inverting kernel unit variance bandwidth rescale nonlinear estimator result consistent consistent use function case quantile quantile stochastic process index interval quantile example bernstein attain density far refer decision rejection q operate explicitly valid particular cover generally accelerate module approximation expansion involve q expansion hold denote obtain overall approximate necessary quality inspection module inspection module rely panel design random draw panel analyze inspection module sequel establish take inspection paper aim aggregate inspection happen sample subsample item draw control item pair one already yield pair observation draw item lot stochastically dependent observation size satisfy subsection long valid standardized extension handle dependent share trivial stage satisfied expansion jointly stage bivariate normal variance say attain plan approximate stage apply assume satisfied satisfied plan without inspection acceptance methodology example lot rely likely stochastically production I put plan reformulate clarity quite spatial batch arrange spread module module site course observation affect wrong wind direction stress
kernel train statistic k n asymptotic retain normal limit define equation normal forest build randomization simply randomization subsample apart condition response implementation produce asymptotically prediction tree load select build randomization obtain statistic formulation resemble computationally mention appendix forest theorem establish parameter obvious determine straightforward share lead page equivalent select initial size training must subsample record mc carlo point final random mc mc calculate note identical simplify select fix subsample include subsample record prediction average select build subsample value situation iteration accurately depend factor course ideally estimation choose computationally feasible many sample necessary accurate tree correspondingly external estimation produce forest estimating parameter need inference begin parameter method outside could generate estimation initial subsample predict record average prediction add produce estimate mean estimate conduct building theorem use uniformly introduce procedure carry prediction provide distribution prediction produce prediction variance quantile formally bound quantile variance n recommend mention introduction interval statistic reject great quantile rate check within calculate confidence interval fail hypothesis otherwise expect prediction tree building consistent produce accurate prediction asymptotically valid occur rate build true underlie sufficient confidence however precision limit distribution way significance situation datum feature interested make prediction reduce feature mean full prediction utilize would whether determine hypothesis hypothesis reduce feature prediction test size take subsample give average tree build prediction tree finally difference function ensemble single test interest case vector q consistent estimator variance clarity obtain appendix prediction statistic reject nan hypothesis procedure though decide building contribute response repeat compare full dataset prediction commonly reduce feature trees us contribution prediction two additional prediction randomize reduce additional randomized feature final randomize unlikely due structure present illustrate limiting function regression visualization multivariate adaptive spline investigate response form limit comprise row bottom subsample histogram subsample title ensemble least splitting node parameter normally mean subsample build variance procedure interested distribution prediction estimate worth lead case figure row fit near alpha incorrectly nan hypothesis central build full reduce utilize hypothesis variance estimate setup none result nan alpha histogram confidence recall confidence capture though conservative repeat internal alpha level statistic internal shown build forest ensemble establish asymptotic forest histogram generate forest tree node random split terminal limit parameter take prediction estimate prediction new calculate histogram forest prediction dataset part science project report specie effort ask contain report characteristic united states national database predict specie abundance restrict specie far restrict little report either like abundance primary goal confidence abundance feature month month figure point obviously absence month report next month highly predict abundance report issue throughout abundance month categorical category miss value include calculate remove month consist size root build estimate build internal abundance show positive interesting high abundance observe observation report expect variance prediction positive nearly visual appear certain conduct test conduct month perform statistic internal estimate calculate statistic month highly abundance month statistic still ensure add prediction training statistic difference tree advantage generate month calculate statistic month significant predict abundance month feature significant month specie year significant predict training dataset consist sample perform method year mean significance month find follow manner randomize year prediction statistic month significant abundance add procedure supervise learner mathematically demonstrate ensemble statistic limit prediction allow formally additional computational cost traditional interpretation modern algorithmic primary prediction among concern formalize hope see something distribution focus bag forest learner supervise satisfies prediction procedure carry way reasoning modification subsample primarily ensure subsample extra computational small select subsample surprisingly remain repeat replacement statistic class also point subsample theory differently say different involve practice raise issue procedure prescribe fashion true building show consistent negligible careful hope address future beneficial select result could complex hypothesis interaction work grant nsf nsf dms science ny usa formal inference ensemble bootstrappe bag forest improve predictive aggregate bootstrap averaging build result statistic prediction allow prediction form incomplete result moreover internal develop procedure tool algorithmic combination variant bag base learner build demonstrate demonstrate allow regularity subsample provide consistently increase test supervise binary long predict oppose vote value additionally process inference result look illustrate consider contribute outcome statistical begin calculate statistic statistic interested field reject chance coin also seek correctly powerful clearly conduct prediction generate simple often enough reject order formalize plausible prediction course prediction combine hypothesis scientific probably approximately develop pac bind error hypothesis appeal estimate account uniformity minimize uniformity pac ensemble fit statistical modern frequently parametric test become increasingly subsample require extension tree bagging suggest recently may bag forest estimator employ confidence interval receive pointwise recently extend suggest determine relevance introduce allow interest largely devote study discuss partition chapter seminal book context prove certain bag individual consistent discuss general bag sample proper consistency forest behavior presence mathematically forest suggest prove forest investigate demonstrate prediction converge prediction distribute class building ensemble subsampling view consistent limit parameter carry limit section forest statistic explicit introduction thorough treatment eq generality symmetric argument size normal variance x subscript enough asymptotically normal subsample interested make build subsample write tree treat order pair estimator form independent thus
partition eq likewise recall write vector use transform quantity transform multiply eq similarly partition obtain q alone error multiply side leave obtain use collect follow primal dual dynamic connect primal evolve I I square examine behavior recursion compute side regularization ensure examine study stability derive error recursion already transform dual variable correspond correspond redundant transform n recursion orthogonal incidence simplify arrive interpretation diffusion seeks arrive different allow different metric square deviation desired extend evaluate argument verify step adaptation regime power argument conclusion result match actual small replace rewrite equivalent moreover kronecker derive analyze possess negative real relate concept stable let eigenvalue establish rely auxiliary stability follow possess full procedure correspond pg possible follow result regressor definite relative complement eigenvalue laplacian connectivity precede ready stability al write q lemma since conclude lemma stable conclude algorithm stable aggregate positive hand fact restrictive definite definite matrix enough exists regressor definite generally individually solve either small size sufficiently square partial similarly corollary restrictive positive I argument similar note stable non insight analyze eigenvalue assume definite matrix eigenvalue call upon demonstrate corollary u ml carry al although encourage performance al definite sufficiently simplifie q examine al large diffusion strategy recall regime proportional conclude diffusion topology laplacian case primal even al topology identical addition steady strategy close state slow bottom else less fig obtain steady convergence rate primal steady sort rate plot steady state algorithm steady hold constant variance choice convergence algorithm see scheme primal exhibit phase dependent second largely curve primal choose stability move observe step general guarantee guarantee furthermore assume converge guarantee large substitute al less primal diffusion network positive random converge consensus utilize doubly metropolis note design fig simulate bad furthermore far make match strategy important increase necessary find find network would enhance well examine primal particular analyze discover match solution stability limitation show match unfortunately increase fix al link modification al match strategy change stability restrictive illustrate unstable partial fully incidence furthermore covariance even incidence r straightforward spectrum let guarantee al guarantee see diffusion consensus converge simulate scenario converge scenario average surprising yet desire indeed experiment property kronecker observe size ignore power collect since q enough see define substitute theorem construction primal adaptation base discover performance conclusion exhibit necessarily steady consensus find unstable partial observation strategy step regularization show algorithm strategy less stable augment lagrangian diffusion primal lagrangian slowly solely adaptation rely update step become gradient network assess provide algorithm consensus diffusion strategy belong class strategy step parameter stability square reference therein broad literature second primal augment rely primal dual main deterministic ability ill conditioning solve constrain exist useful primal g shall examine adaptive static minimizer cost variant continuously dual determine explicitly long employ instantaneous approximate direction influence noise measure direction dynamic trivial surprising behavior primal comment finding version work useful assume exactly agent explicitly adaptive turn stability strategy al stream primal version construction explain anomaly al strategy exhibit update cause respective carry availability bridge regular network homogeneous capability variant multiplier steady node role lagrangian important conclusion refer agent aggregate entire discover fail recover become unstable allow arrive surprising illustrate analytically mean al strategy able range network able solve still connect consensus network disadvantage strategy range examine steady state adaptive discover processing employ agent achieve consensus al must step size value denote kronecker throughout column exception capital letter scalar letter aggregate across network formulation primal diffusion consensus later agent access random arise contexts application channel localization second process allow possibility matrix across definite correspond scenario agent node determine aggregate unique algorithm particularly stream prominent type latter superior mean size learn diffusion strategy sufficient enhance equation small coefficient combination satisfy stochastic use comparison consensus strategy important note state strategy source instability solution connect one agent self trust square error stability sufficiently small step deviation consensus strategy show doubly hold emphasize strongly connect stability expression doubly find manuscript focus compare doubly matrix across individual agent w notation quantity agree order conclusion strategy able estimate agent agreement solution furthermore guarantee long agent consensus implementation become unstable topology agent highlight diffusion execute presentation attribute present fact add encourage explicitly examine previously incidence exposition interestingly turn study deterministic optimization nontrivial necessary notable conclusion incorporate constraint limit primal strategy motivate incidence graph incidence loop exclude laplacian matrix whose agent hold incidence exist edge connect access column laplacian aware network connection node access incidence possible rewrite extended quantity constrain lagrange multiplier associate q regularization function minimize know convexity determined determine saddle lagrangian method rely gradient take alternate direction multiplier admm know result determine either consequently directly rely stochastic saddle variable vector evaluate eq amount give u ki I index saddle cost context reinforcement target reference consider employ decay adaptation step persistent end rely variable benefit iterate
note perform dimension remove independence feature em em actually posterior assign expert bayes em step rgb rgb lemma claim observation remark bold ff consider mixture model mixture distribution condition variational bad optima insight input moment tensor establish consistently recover mixture degeneracy assumption critical ingredient mixture mixture classifier score tensor hide employ model combine expressive latent predictive capability framework widely syntactic parse machine traditionally xu however optima slow rate approach guarantee moment pearson pearson involve observe recently highly successful unsupervised tensor community rank spectral decomposition tensor degeneracy tensor guarantee correctly recover method dataset effectively important assumption model suited rule high moment set high moment label information learning consider early challenge moment appropriate form amenable detailed ingredient ingredient employ feature accurate framework representation exploit superior purely discriminative many incorporate tensor high exploiting quantity differ high derivative density capture input establish label yield expect derivative input expect derivative nice unknown generalize glm e u establish employ tensor learn scale spurious framework classifier g xx thus weight decomposition cross moment input kernel yu complexity assume regime regime refer variance regime number classifier recovery method incoherent spectral compute whitening slice tensor assume value matrix recovery high guarantee method computational sample complexity learn weight technique maximization discriminative yu get optima employ learn construct feature classification framework svms learn weight construct discriminative guarantee input moment mixture specifically label consistently degeneracy mixture expert divide consider xu alternative carry usually hierarchical xu guarantee guarantee decomposition moment linear problem alternate convex restrict subspace gaussian hessian eigenvector order moment matrix subspace outer however fail moment vanish overcome transform result vanish line gaussian density stein handle bring feature use tensor individual weight vector employ tensor involve moment tensor gaussian class procedure drawback dimensionality eigen sir label project preserve eigen assume elliptical slice slice surrogate establish strongly monotone provably paper consider single glm monotonicity vanish derivative activation third utilize throughout denote order tensor I pi canonical rd say te ai cl set assume draw probability density incorporate generative mixture glm linear mixture activation although limitation classification glm employ glm u rr extend tensor specifically adjust follow stein identity weight method present result full vector tensor algorithm appendix follow recover vector scale bias estimate method handle violate constraint overcomplete dimensionality tensor recover incoherent detailed score compute j appendix gaussian scenario extend ingredient ability label derivative distribution result distribution use order derivative respectively th continuously gx mild learn general moment tensor cp component recovery mixture respect w guarantee provide section obtain activation know need fully propose far provide linear q similar framework nonlinear assume propose extend connection density ti equation z recovery svm function ingredient lie estimate estimation fit addition deep argue auto learn order score estimate multi versus analysis since tensor decomposition algorithm recovery need difference number require let u normalize guarantee first power line state satisfied guarantee input remove need power whiten orthogonal perturbation whiten gaussian regime appendix gx I discuss theorem appendix initialization normalize independent denote incoherent hold incoherent satisfie output satisfy since know function normalize bound output let almost surely bernstein I perturbation incoherent output employ learn mixture classifier expectation maximization discriminative svms yu discriminative since optima convergence local optima employ learn construct exploit direction variable work spectral general expert variable interest microsoft fellowship nsf award award award h award tensor multilinear form view multilinear form tm eq mi u mu multilinear combination mode multilinear combination slice stein lemma simplify first term give gx obtain obtain hand substitute complexity general empirical state take frequency perturbation translate bound see equation perturbation provide low order
mdp entropy sample visit region way always beneficial yet counter mode illustrate section admit action right source reward start middle know everything end probability terminate reward illustration belief ts end since coin take bayes htb thick thick might failure lack ts adapt dp optimal several episode explore problematic discount objective complicate material similar generate planning avoid integrate future planning deal inference rich probabilistic search adaptive planning guarantee lack performance sparse increase share ts combine less sampling filter forward need update thus horizon number root require belief reason choose forward supplementary material huge domain adopt strategy area bayesian permit carefully likely consider rich contextual bandit however solve planning say real likely motivate realistic dataset uci repository instance specie family attribute g color instance mdp ignore ignore incur illustrate initial observation indicate represent indicate component supervise learning ignore contextual bandit however unlike contextual bandit early reward valuable late one exploration dominate task parametrize context generate denote ss increment state update update cost key aspect joint matter uci unclear planning possible base inaccurate agent assume allow substantial underlie characterization evidence observe particularly parametric chinese restaurant pt crp assignment measure dirichlet assume hyperparameter inference scheme detail draw component crp correspond sample state infinite horizon generative label straightforwardly uniquely characterized context imply configuration stress really section somewhat realistic agent highly challenge particularly natural ignore lead neutral return run ts statistical result surprising result bayes adaptive agent obtain return despite abstract exploration performance investigate label free reduce uncertainty ts improve inferior adaptive et al regular ts outcome time integrate purpose simple contextual bandit apply ts ts bad crp base demonstrate large discount ts crp inference start step discount ts ucb agent control domain handle customer make decision different consider generalize share address actually draw plan mis key decision contain parameter generate dynamic denote either leave x b draw generative domain different assume know generative contextual model usual contextual arm option play contextual bandit exploit even extension include intra supplementary focus investigate performance sample reward material sensitive discount dependence exploration exploitation strategy ht concentration hyperparameter inference avoid maintain similar performance despite run simulation researcher powerful sequential exploration parametric emphasis planning factor mdps capture exist problem safe planning monte scheme depth branching factor limit benefit exploration deal mdps discount objective structure consider mdp combine bayes inference mdps specific domain unbounded state infer size state online search planning limited depth sized gps employ infer excellent however capture explicitly exploration planning address plan uncertainty reduction generally learn ultimately label concerned discount return fine labeling base attractive particularly domain carlo powerful optimistic planning severe avoid explicit planning domain benchmark uci bayesian exploration exploitation demonstrate feasibility advantage adaptive various planning roll interesting think function within tree open domain truly computation explore simple computation gain policy planning challenge model parametric amongst readily domain arm extension something collection measure expert solver figure mdp reader tuple sa discount component mdp mdp planning estimate line latent accord observe action tp uncertainty current inside augment possible tuple form bayes adaptive mdp solve obtain augment space readily action execute constitute action agent mdp equality degenerate support model agent go end hand aim right equally htb thick thick black c c b thick black action ts p v ts arbitrarily bad policy therefore choosing imply construct optimistic action across take present decide action result denote sample px showing add usually mdps decision put policy perfect first n c pn depend bad easily since policy action stress strong objective bayes severe pressure short horizon ts label free report agent fewer ignore show horizon sampler tractable concentration couple every simulation assignment sampler infer planning tree generate pool slowly htbp bs ar htbp restrict task contextual extension section mdp contextual informative motivated exploration site come know contextual e ignore site run type action type site model intermediate get site correspond except establish model reward binary variable ts bandit depend environmental ts act ignore conservative act large ts dynamic return sort execute explore cumulative ts solid line error theorem rgb rgb powerful bayes planning parametric simple planning thompson fully thompson leverage efficient adaptive parametric perform qualitatively conventional thompson rich inductive bias allow confident inference limited benefit control safe exploration balance act look plan partial exploration trade problem great unfortunately costly leaving might compare similarly computational cost treat tradeoff focus discover demonstrate despite sample base planning rich challenging planning provably optimistic thompson sampling fail risk perform highlight behavior uncertainty non way consider include material discuss reinforcement rl outline exist planning thompson introduce exploration motivate mdps finally adaptive
convex convex relaxation tensor connect well investigate technique develop address give utilize trace norm showed given improve latent regularization infimum convolution analyze rank tensor problem reduce efficiency expense address possess exist bayesian method study basically construct tensor decomposition decompose performance analyse collaborative filtering spatio gaussian decompose decay speaking obtain cp analysis favorable property without assume adjust rank priori significantly approach convexity convexity sparse tensor exist inner k rank rank tensor exist k u relation write cp paper investigate cp rank cp regression predictive q observe input observation completion denoise observational completing unobserve j j ia accurately tensor obtain sum standard task dimensional vector cp space show rank extension rich analogously gap envelope cp well envelope matrix norm np envelope present regularize assumption procedure compare provide u positive suppose gaussian k condition gaussian posterior mean square give rate define tensor pa key characterize convergence well concentrate around large bayes estimator truth location truth much specific wide possibility balance concentration dispersion trade normalize scale simplicity assumption assumption bind tensor technical convergence predictive suppose follow assumption constant appendix speed mass around eq outside show follow jensen strong state mean estimator assume symbol actual degree log number rate basically emphasize true placing estimate rank give assume convexity sparse lasso convexity require reason tensor practice turn accuracy accuracy error bernstein infinity mean tensor estimation avoid large tensor tensor completion recommendation apply large conditional accordingly sample proper corresponding truncate assumption bound front gap population simplified observe term analyze actual assume impossible derive focus weight tensor completion l px recover rate estimator respect max sample accept much well previous inside improved hand rejection recently analyse tensor work utilize tensor unfold unfold tensor th rank tucker general cp analysis empirical see bind achieve estimator k nice point automatically rank unfold minimum rank large bayes remark rate tucker cp suit apparent latent setting counter bayesian rank place prior author utilize strong convexity require bind utilize scheme concentration analysis course tensor bayes tensor investigate applie set element observational tensor randomly execute five repeat repetition varied addition actual manner figure scale ratio accuracy curve scale accuracy mean scale accuracy behave match predictive paper investigate convergence rank base predictive without convexity adapt rank describe behavior bayes negligible however experiment show behavior investigation thank discussion partially determine let eq k k eq combine definition unnormalized fix think random utilize originally convergence bayes parametric model technique denote da construct eq indicate test even eq check event fix hold q calculation rhs bound r yield prior therefore posterior rp rhs simultaneously pack packing unit ball bound
baseline add consist architecture probable viewpoint orientation fc bound baseline correspond tend table appendix ap clearly cnn one favor baseline fc valuable orientation stochastic momentum decay patch balanced patch discretize sec patch sec batch experiment select validation patch proportion patch selective iteration start avoid divergence divide stop decrease successive train discretization pose outperform state cnn explain training orientation increase interpretation train training imagenet annotation present increase considerably ap pt r car avg c c finding separate representation improvement detection orientation obtain pose explanation observe fine detection drop perform joint pose discrete detection treat continuous orientation detection joint orientation approach show cnn art acknowledgment work imagine des centre building partly support project c car r car avg search c k k h variant c c car train variant car v r c c car train v variant c c car avg variant car train c car c c c car rectangle fill gray center draw minimum height rgb study application detecting pose representation orient energy choice pose continuous variable object detection benchmark detection pose exist baseline benchmark performance cnn specialized vision optical availability increase cnn recently outperform less constrained vision seminal work imagenet apply explore potential cnn image namely annotation detect pose computer vision work contour match contour instance towards object category focus without take sift drawn section overview joint orientation real image rotation degree cope strong appearance object appearance intra illumination difficulty orientation tv rarely follow pre imagenet learn tune last idea pyramid manner output fine tune allow computation layer provide art classification adapt tune architecture orientation perform pose use vision attempt object category information representation orientation several feature pose handle pose pose patch benefit vision pose alignment alignment alignment recent one specialized pose handle intra category geometry simplify applicability real classic challenge dataset class average viewpoint standard metric evaluation viewpoint orientation use adaptation perform well cnn neural design learn successfully apply many specialized digit face attract vision advance improve vision problem object detection special easily box selective lead drawback candidate reason propose convolutional lead achieve slightly propose neural network available imagenet train imagenet art detection classification connect top imagenet unclear invariance encode could pose discriminative b b pose approach associate show subspace method network combination pose regression classification leave pose space state pose class roll angle slightly angle cnn predict pose pose set point define discrete feature point representation adapt develop pose predict probability orientation plus background contrary aspect associate patch circle angle see approach patch jointly class angle angle option function feature detail finally cnn comparable choose base pyramid pooling framework efficient testing good result pool similar imagenet pick selective image rescale layer tune function cover suppose training orientation dimension interpretation choice architecture pose bin k k category otherwise imagenet window patch extract extract map rescale follow softmax minimize log softmax sum minimal orientation framework appearance vary continuously discretize network suppose jump viewpoint vary appearance view one develop orientation unit enforce far circle positive local negative circle instead far avoid effect dimension circle live classification without softmax output respective negative natural loss follow property parameter indeed huge negative negative example clearly large radius circle probable orientation angle pose treat extend mutually exclusive distance category distance inspire consist divide orientation classification follow softmax pose
regime soon quantity dyadic multiplicative analysis single vary routine tuning issue quite surprising consequence loss value hold use adaptation hand well induction hand definition line sum combine rearrange achieve tune sum logarithmic working mention rate vary loss pick rule mixture k learning cumulative material give idea complete non sequence rate update tailor induction prove aim bound core update could handle inequality main notice work version quantify gap inequality precisely rate price vary also regret present indicate work analysis simpler elegant resemble dependency achieve order sequentially rate vector round wise nonnegative loss loss vector justify expert report refer expert differ round expert expert round weight k never empty regret account k expert confidence therefore depend bind also scale issue generic set immediate latter report detail essentially exist expert prediction reduction easily run modified round weight another strictly loss loss follow equal modify loss set q subtract side regret loss confidence second upper technique regret excess loss introduction plain expert confidence expert expert suitable lead expert leave prediction second bind already case key feature excess instead plain loss order eq cumulative introduction close loss gain ready symmetry nonnegative indeed regret satisfy loss translate scale canonical visible ta significant improvement worst realize nice affect loss even positive negative therefore expert substitute initial state substitution guarantee adapt non adversarial identically bound case least expect satisfy satisfy constant regret strategy form regret law number cumulative exceed cumulative order large bound theorem require bernstein basic cumulative deterministic factor let martingale v derivation study typical able consider variable sense play role vary learn instantaneous induction conditional inequality solve quadratic inequality yield claim bound q substitution conclude order loss fact body shown rely simple convex line consider value indicate low show induction hold round algorithm induction hypothesis inequality already equivalently trivial since expert proportional desire bound rearrange useful number eq first follow stem bind inequality term note hold inequality ratio particular square root apply apply substitute bound b alternatively exceed latter proof associate q part entail element diagonal fix eq entail instantaneous develop product equal hence lead inequality substitute finally impose increase consequence nonnegative q increase rewrite get fix rely variable pick q define bind apply far proceed induction side inequality application conclude show increase remain use bind put thing conclude specific section reduce loss trick unified reduction expert expert predict possible prediction determine function choose weight compete wish combination expert component forecast reduce confidence section reduce linear loss pseudo tx convexity inequality eq equality linearity regret imply compete set another paper expert selection selection expert converse couple expert like equivalent vector algorithm expert respect indicate although tune sequentially believe consider optimization tuning reduction suited expert report general entail therefore bound state
keeping allow split subsample cv frobenius norm adequate propose cv justification method memory dependence evaluate thresholding correlation functional connectivity datum multivariate covariance four px model n px norm ability recover sparsity positive define conduct sample range replication vary cv cv candidate range increment estimator correlation perform temporal overall perform ordinary ex hard soft r ex norm ht loss technical lemma second hand inequality tx nx ni eq nz ta constant sufficiently thus proof let plug yield unknown integer without q equality schwarz hold q r constant plug yield constant line dependent obtain thus hold convergence condition theorem constant polynomial constant key impose equation proof equation see complete proof proof follow line equation omit theorem inequality since eq q eq assumption mean norm frobenius similar exist yield inequality r corollaries general assumption set set thus acknowledgement wu principal van mh centers support center grant partly grant dms dms dimensional decay generalize thresholding convergence consistency impact temporal investigate intuitive thresholding parameter good method temporal implication fmri connectivity nx inconsistent overcome covariance regularization approach cholesky correlation researcher become particularly useful analyzing fmri assess connectivity temporal process traditionally impose overcome difficulty introduce recently hard interpret straight cross impose weak cross covariance extend weak fmri stationary memory autocorrelation rate much slow important example invertible integrated move generalize decay cross cross matrix simply dependence weak dependence decay temporal cover restrictive correlation aforementioned study brain connectivity moreover entry surely dependence condition pp extra care true replace sample mean unknown article estimation correlation consider series decay restrictive violate mainly focus rarely brain image intuitive show generalize thresholding keep originally develop matrix organize temporal dependence special temporal describe broad range result long thresholde cross validation method evaluate contain theoretical brief norm mx nc n xx x ij tm independent matrix column define product ij ij kt spatially average voxel within pass stationarity false fdr control approximated check mild rate dominate rate practice figure dependence seem fit least linear yield illustrate two brain clearly assumption homogeneous decaying time mild consider generalized estimator sample matrix correlation first temporal subsection detail memory temporal specify subsection matrix eq correlation matrix eq thresholde popular soft thresholding smoothly example generalize thresholding estimator thresholde c pm n replace respectively p I tend additionally probability tend replace without impose define correspond correlation consistency pf nk n q frobenius norm q r j eq would estimator like subsection temporal applicable long may
complementary representation constitute improve contribution entity augment semantic intuition entity play central heavily expression case identification implicit explore entity include role syntactic status despite solid linguistic foundation feature contribute word pair status entity distributional semantic throughout entity compositional begin phrase mean current semantic attribute distributional information linguistic structure distributional compositional supervision enable semantic include rnns rnns propagation rnns nod probabilistic entity propagate node entity semantic equation outside inside combine parent term style parse distributional semantic relation word shown utilize idea distributional word incorporate semantic text stanford argument parse semantic compositional representation entity semantic apply syntactic induce representation entity overall compositional annotation outperform previous relation semantic resolution annotation addition share parse edge edu relation small linguistic text automatically identify require argument subtle relation link level entity distributional representation syntactic work compositional representation compositional relation distributional entity system obtain substantial implicit relation characterize adjacent relation task annotate automatic identification art implicit parent one poor predict implicit relation fundamentally semantic may difficult implicit sentence seem surface unless annotate etc far address compositional semantic argument series syntactic predict bilinear compositional compositional operation classification implicit purely capture relation see make corpus sentence long appropriate preferred distributional almost unchanged syntactic representation span single capture relation entity role put meaning address issue compute entity capture play entire span account feed compositional combine pass structure parent tree combine bilinear resolve combine representation achieve classification outperform classification novel entity compositional model surface syntactic feature work prediction semantic relation indicate sentence justification sentence sentence refer sentence entity share entity relation sentence clear pair share entity relation totally change suggest relation sentence model capture semantic sentence semantic share entity relation without new composition capture entity sentence semantic combine distributional sentence composition part recursively combine distributional architecture illustrate start composition include share entity entity entity identify sentence implement stack composition show jointly composition composition composition work test focus outperform relation also discover formulate model sentence relation composition easily extend follow approach semantic clarity exposition extension section feed pass terminal syntactic distributional child rnn parent child element tangent composition matrix compositional find leave pre word representation sentence combine obtain feedforward little distinguish certainly relation one role semantic rather logical would parse logical representation role neighboring make pass compute composition recursive occur root procedure parent compositional maintain influence pass also feedforward influence node since pass feedforward node feedforward efficiently inside computing score fashion outside inside sum observe describe constrain tensor contraction involve parameter composition predict argument decision bilinear product parameter scalar entity share entity sentence root low apply reduce classification surface surface set representation advantageous experiment serious overfitte backpropagation present argument gold regularize square frobenius euclidean hold depend delta update similarly matrix composition every unified derivative form information also computation include compositional operator q composition word derivative objective compositional operator set topological convenient way equation graph illustrate start reverse edge trace review take train implementation use norm trick fix latent set latent dimensionality regularizer composition composition classification vector classification initialize composition initialize uniform train induced unit experiment pre give broadly syntactic binary stanford sentence syntactic sentence identify span branch automatic gold berkeley entity instance entity gold annotation intersection automatic gold line supplement use include four lexical feature dependency contextual mutual select feature three lexical level journal corpus annotate two argument identify challenge meaning focus challenge problem classify implicit temporal contingency fine relation specify contribution main evaluating relation explore relation binary build evaluate primarily multiclass however correct relation pair among second relation type exclude five relation annotate annotated instance ex cause account simply mean sum bilinear publish implicit relation feature lexical syntactic implement system enable comparison online method relation multiclass identification line outperform distributional improvement accuracy great distributional surface system individual prediction use sensitive significance semantic significantly outperform surface choose set figure accuracie narrow identification relation outperform prior distributional improvement choose development range latent entity line entity entity semantic without share therefore seem sensitive entity gold annotation intersection entity find inclusion entity
soft thresholding np np group index sparse gradient give build vector choose build closure operator value decomposable task positive reflect decomposable ode k resort two optimize semidefinite differentiable respect differentiable function algorithm reduce projected sdp lipschitz step pc onto cone semidefinite eigenvector non loss sdp eq numerical highlight novel framework ode finally ridge learn decomposable scalar summarize cross search trajectory model descent manually mechanic noisy equation spike potential neuron recovery assign prove numerous optima spaced add isotropic mean figure present learn figure top smoother estimate trajectory match true tend sharp curve represent truncate use space truncate depict smooth predict trajectory smoother learn peak reflect trajectory consist biological gold exploratory ode smooth well considerable uncertainty smooth true level approximately half automate time interested trajectory arbitrary true error model accurate ode non classic initial parametric well ode realistic comparison good highlight error ode ode parametric solver fail ode specify mse parametric parametric free ode matrix especially flexibility penalize rkh way address realistic ode learn nonsmooth help proximal also discuss approach presence issue theorem de division france universit de france france paris france dynamical view dynamic ode reproduce rkh matching approach ode smooth ode derivative nonparametric ridge ode dynamical physics attempt understanding eventually make prediction ordinary describe state account dynamic ode two choose ode noisy obvious favor choice rely test angle ode nonparametric issue principled parameter knowledge precisely govern differential equation dimensional dynamical ode value ode length additive point eq ode square subsequent parameter optimisation approach intensive suffer name gradient matching estimation iterate procedure iterate asymptotic enjoy consistency nearly work approach ode learn estimate learn differential play capture trajectory value assume value function want derivative reason contribution reproduce work subject rkh theory flexible definite scalar value svm rkhs value attract elegant structured supervise semi nonparametric scalar reproduce first endow inner property corresponding include sequence property rkh theorem follow empty kernel build let function hilbert want theorem ignore independently along act learn gradient p expansion coefficient matching stacking stack matrix literature sdp scalar kernel pairwise similarity kernel parametric use series come different non want nonparametric suppose propose nonparametric smoothness impose close estimate input multi strongly manifold regularization observe start condition learn describe minimized q stack value matrix ij ij rr elsewhere anneal g avoid average
metric vertical axis precision compare similarity instance similarity rbf change bandwidth score score similarity mle look incorrect similarity determine application least rbf dpp mkl rbf kernel exclude mle horizontal axes line mle mkl similarity deal appropriately similarity introduce dpp video dpp task cluster summary naturally want representative text understand conference testing time news article reference summary summary identify agree human summary oracle practice use algorithm evaluate human reference separately accuracy use package gram p additionally length character yield dpp task sentence similarity standard frequency frequency tf modeling similarity baseline enhance cosine method competition dpp decoding set real data subset achieve consensus consensus measure metric depend f actually flexibility dpp user infer output metric select necessarily diverse bias towards specific summary summary summary balance precision dpp summarie five summary one contribute gain package develop section describe frame stop negative summary achieve oracle summary able target independent oracle learn application specific summary selection result package summary number match view match visual color difference match frame vice versa develop number match experiment recall cf balance hamming cf text contrast neither mle able summary include hamming pair interpolation draw curve dpp generate high mle rise summary want turn dramatically conjecture u california science science california ex plus minus modeling application diverse subset dpp label make contribution dpp modeling flexibility propose novel trade error extensive contribution document video modeling kernel matrix balance imagine search retrieve retrieve image frequently cite need incorporate notion ground might contain many item ensure exact diversity point dpp technique diversity power set diversity likely diversity physics dpp find retrieval extension dpp dpp markov dpp dpp space dpp crucially square every pair ground reason quadratic impractical element necessary secondly document document annotation expert costly difficult many task dpp select summary estimation maximum typically underlie limit number reliably restrict precision two dpp label improve model dpp kernel domain fewer whole correct subset margin reflect desire measure closely selection error sample dpp principle novel superior video organize dpp work study conclude task large margin base approach discuss relate translate item unlikely occur subset arise matrix quantum nature determinant dpp select summarize rank search possess analytical margin analytical research restrict force dpp focus propose dpp individually structure model map resort algorithm extensive effort dpp explore surprisingly diversity dpp kernels mle approach note selection mle maximize joint margin discriminative dpp kernel limit successful large model explore outperform mle application handwritten character speech approach analytical margin bring modeling flexibility meet practical learn dpp testing stage recall review dpp excellent define symmetric semidefinite column proper call ensemble dpp subset matrix computable submatrix despite eq marginalization dpp marginal either case lead never item similar diversity subset diverse subset attain probability map l interested mode hard approximation investigate suppose annotate discover thus specify item unlikely represent share compute characterize optimize diverse subset attain rise estimate mle estimate dpp limitation multiple representation estimation function dpp apply large optimize reduce advantageous advantage optimize track component attain dpp semidefinite item ground right however application item retrieve image diverse sentence redundant also represent document decomposable relevant depend item encode contextual item feature sentence set descriptor represent item whether optimal adapt thus largely severe retain aspect gaussian rbf secondly base kernel annotate parameter estimation technique consist select frame impose dpp data dpp benchmark training subscript decompose quality ij measure reflect bag word visual appearance far compare mkl j turn mkl parameterize key synthetic question come learn parameter mle follow training maximum closely track error improve likelihood likelihood subset mle mode subset highly mode problematic dpp fall error approximate extract margin incorrect constraint loss measure discrepancy intuitively maintain probability incorrect explored structure exponential counting subset item unnecessary severe add trivial sentence type error ham function q item towards incorrect demonstrate real challenge deal constraint hard jensen inequality th diagonal detailed see undesirable contribute term hinge function tradeoff coefficient tune objective likelihood objective subgradient descent supplementary introduce parameter balance force fix descent project weighted hamming distance margin discriminative incorrect subset distance violate overfitte training yet careful reveal system people expectation document example reader may something summarie article usually piece online video summary mle merely focus fortunately large modify hamming hamming distance recall show marginal towards conversely make dpp model put effort item give rise high result course type discriminative meet dpp kernel form beyond quality diversity limited function kernel parameterization high summary learning maximize observe potential mle optimal test maximize enforce mode large py dominate tackle margin dpp minimize dpp researcher introduce dpp restrict dpp offer diversity adjacent structured dpp dpp generally hard guarantee alternative resort activity work explore popular estimator mle mis flexibility incorporate posterior minimize margin dpp make tractable additive large
forest goal depend furthermore write forest forest give large section compute quantity show term call primary tree forest bound paper partition almost show proportional leave say partition rf lead sake simplicity I compute lagrange c j x appear forest true c proposition precise bias infinite much tight low convergence smooth toy obtain piece formally uniform random forest key sx decrease section point make goal corollary get term appear toy let smooth show corollary inequality dx regular density framework surprising model regular difference regular randomly translate histogram regular randomly translate effect boundary highlight phenomenon forest suffer phenomenon e tree given compare notation define integrate therefore consider order constant leave tree precisely equal risk statistical risk build respectively x forest follow assume addition rate dx assume attain function whereas except constant infinite forest minimax next bias account involve rate reduce practical estimator need infinite forest independent random denote q far enough forest bias estimate appear decomposition true involve rate valid except avoid however conjecture toy issue scope paper small forest reach proposition consequence section corollary tree soon multidimensional purely forest partition piece piece make z split random step set tree whereas split model partition end average weight average infinite contrary xx plot compare p slow leave estimate monte appear bias hold corollary contrary toy effect approximation compare infinite forest term term q allow forest single approximation height infinity emphasize decrease slow cubic partition set indeed risk choose control control point forest bound proposition eq volume hence subsection risk assumption sx proposition leave point available l p infinite forest estimator follow infimum distinguish ii integer soon ii slightly apply imply p decrease tree infinite forest rate function model minimax split small split partition proportional function finally suffer lack next set choose would certainly would tight smooth following hold eq previous forest magnitude eq mathematical previous r section toy consider rf hold rf original tree extra consequence rf approximation tree take input function proportional range model choose toy remove realization estimation rate accord adapt significantly quite surprising investigate phenomenon systematic scope forest model forest improve regression forest equal forest compare toy forest instead toy tend leave stay leaf leaf contrary keep split leave forest precisely analyze latter partitioning set variable put node effect output weight infinite forest fast even quick study forest result fast smoothness regression combination analyse regression beyond research well rate tend balanced tree choose consequently reach large minimax compare forest approximate smoothness whether analysis suggest mechanism seem next justify reach minimax consist length choose uniformly practical forest get single tree build forest finally apply forest partition learn random appear forest particular mind rf define address research hold rf grateful discussion acknowledge grant detect prove conditionally classical q last term come convention sx integrating separately eq done separately hold definition appear prove schwarz conclude bind last toy ix proof assume occur x eq interval change eq conclude q k proposition quantity toy random variable uniform therefore follow eq quantity result gx yield integrate directly q gx gx p x p quantity appear gx gx get integrate variable define consequence v independence variable k conditionally since binomial every convention follow take resp x jx jx summarize k dimensional deduce deduce computation key appear follow since proves prove eq integrate integrate proposition key q prove proposition eq integrating eq taylor expansion direct integral section imply quantity b formulation distribution section accord px belong define every j hence x px l b px relative position point finish furthermore sub uniformly variable uniformly multiply p p j j every main summarize repeatedly proposition since I odd integer let integer sequence prove I additional q I apply prove take since sequence every jensen inequality get eq increase every proof j directly combination proposition proposition appear q every integrate yield every yield eq proposition proposition integrate partition induction clearly unchanged change one multiply uniform variable get end eq pi z p p z pp particular chebyshev every combine every eq take since function b fx follow straightforward upper integer u define definition remark pt forest forest order forest framework focus tree forest regularity bias infinite rate size tree single attain risk rate furthermore sufficient product purely forest forest rf henceforth machine remain deal rf bagging bagging posteriori rf rf neighbor towards theoretical rf purely henceforth simplify rf establish obtain partition independently first easy secondly mechanism obtain partition usually simple enough allow calculation theoretical describe try compare performance perfect ensemble rf randomize encourage good birth variant understand rf precisely model focus regression partitioning space base mechanism abuse recursive way leave tree element recursively belong decision tree classical tree partition denote obtain partition throughout leaf response forest sequence correspond aggregate important rf define recursive partitioning put repeat meet choose among find split variable split split crucial method forest heterogeneity quadratic put randomly choose split uniformly split uniformly put forest
onto area every crucial mean every solution k lebesgue assume eq integrating give equation derive point note cause nice involve multidimensional explicitly second critical development lebesgue calculate value interval marginal sum active analytically sample carlo carlo easily interest motivation spirit understand respect pp four figure define concrete rectangle integration emphasize tuning parameter estimate ignore regard sampling covariance n p na argument normal particular simply affine pa restrict bivariate lasso thresholding simple last obtain apply theorem estimate corresponding estimator example normal nonparametric method ni h h reduce estimation besides normality motivated perspective lasso draw use distribution assumption valid specify aspect explore introduce direct assume n draw draw residual routine form calculate expectation much alternative although method example high set reversible markov subspace dimension ordinary mh I new say p dominate measure standard mh involve mcmc distribution matching component reversible jump contrary assume move dimension reversible usually hard mh call sampler hold group accord follow proposal proposal j proposal symmetric mh simply efficiently especially remove add reverse mh analogously proposal one need ratio side efficiently dynamically detail proposal consume proposal consideration proposal let vector suppose ta mh mh mh input parameter numerical section distribution gibbs consume algorithm estimate interval approximate accurate monte conditional distribution rare contrary involve calculation tb proposal ease tf understand mh move design jacobian q coincide jacobian current jacobian account use view computational computationally tractable quite sampler conceptual direct linear transformation univariate size univariate numerical move analytically fundamentally relatively consume model greatly unimodal chain converge amount computing confirm numerically next routine initialization reach equilibrium totally remove detailed however routine numerical initial example coefficient give design matrix cccc b weight numerical lar dataset choose implement package determine estimate coefficient type error distribution correspondingly section routine lar package examine serve weight reasonable see routine notation estimate chance proposal proposal iteration normal plot sample illustrate subgradient away autocorrelation among decrease update proposal update proposal acceptance mcmc estimate conditional simulate sample sample quantity calculate accurate mse great mse ratio run around ratio mse estimate estimate estimate furthermore simulate sampler confirm serve direct simulate lasso routine parameter previous quantile standard compose practically use algorithm ground truth example accuracy variance across table model approximate estimate sd b c effort establish penalize high consequently positive eigenvector vector linearly independent augment estimator n augment row achieve constraint word lie augment restricted mapping denote unique satisfie constraint lie fix constraint determine jacobian simple constraint minimizer satisfied rank matrix n n dd accord n rd confirm continuous fix ready report estimate extremely order moderate around cccc multiple matrix kk coefficient true many relatively choose active coefficient along path summary well cover wide procedure therefore simply ht range value run value figure dataset confirm variation huge direct previous improvement ds tail bad variation bar give result dataset minimize section section augmentation publish extend definition proof q p eq immediate regard vector condition construct fix assume condition assumption lemma probability least assume borel depend give bind pr nr n satisfy sufficient least decay zero apply dimensional depend identical quantify difference let q satisfy satisfied fixing establish pr tend scale sufficient tend assumption beta comparable theorem one strong eigenvalue residual precise residual routine residual consistency spirit fix previous general explicit imply consistency recently number condition fundamental consistent stay result initial magnitude nonzero coefficient considerably generalize draw augmented become give set routine I reliable sufficient development mh design explicit draw importance expression unnecessary choice sample augment selection intuitive geometric respective active allow grow sign unique size active definition valid kkt via affine component consequently definition therefore consequently intuitive column regardless establish follow condition necessary bind example one c nd min replace vanish applicable posterior continuous nonzero bayesian lasso thus invertible assumption tn incur mahalanobis norm encourage sparsity loss e joint kkt kkt distribution n reasoning n discussion framework decision although improper confusion call estimator instead lead solve kkt condition therefore interpretation lasso vector therefore decision familiar correspondence worth loose sense interpretation kkt depend lasso monte estimator advantage direct show limitation augmentation relative augment randomness stress direct sampler similar way handle determine draw direct routine easily since proposal importance independently certain reach overall computational multiple draw sampler markov routine reach iteration computing assume access run initial draw sampler routine reach routine last assumption derivation routine decay autocorrelation always decay case component first decrease fluctuation empirically suggest direct need node direct sampling bootstrap density augment article mcmc method linear regression approach clearly method gain flexibility room idea augmentation use penalty study however difficulty uniqueness penalize augment mean manifold another future theoretically empirically finite coherent truncation pp p assume ii suffice q therefore since v thus complete proof let condition event equality e bb event construction consequently minimizer j direct kkt minimize give minimizer loss lead due lastly distribution event assumption least conditional choose see proposal index immediately proposal computation jj readily reject position proposal convert routine regression show determine numerically many distribution augment tractable low variance carlo draw sample augment respect concrete example offer regression obtain monte interval monte carlo regression design coefficient widely sparse estimate vector minimize penalize choose approximation however except special type complicate close covariance fail confidence develop orthogonal design limit bootstrap bootstrap algorithms lar homotopy time apply hundred time point circumstance modify justify develop linear several article significance asymptotic various lasso hand distribution useful selection penalization stability possible obstacle estimator sampling density interestingly joint normal regardless distribution simply marginal study sampling may accurately efficient another evaluate minimize numerically bootstrap furthermore mcmc multivariate locally remain article organize derive mcmc calculation inference provide theoretical estimate establishing include design selection interpretation sampling article conclude notation regard column set v b j
logical distributional two neighboring node pass parent si composition function bilinear product relation entity decision classification argument progress parse representation semantic compositional induce distribute argument entity jointly classification compositional operator base find font draw style edu linguistic element coherent text identify understand semantic link sentence sentence link element entity distributional tree key compositional distributional semantic compute representation entity novel compositional distributional also entity result obtain substantial improvement art predict level organization text adjacent relevant task sentiment coherence automatic identification implicit relation roughly predict implicit fundamentally semantic relevant may relation sentence appropriate indicate relationship however surface compositional sentence series compositional give argue purely expressive capture happens make change sentence original relation long hold despite syntactic address issue compute sentence entity mention play compute entity novel feed compositional combine pass syntactic combined help approach achieve accuracy identification syntactic automatically stanford ask whether might recurrent language preferable language resource whenever possible think unlikely key language recursive language processing semantic see strong evidence history natural syntactic capture accurate annotate language topic substantial differ substantially order language question whether leave recurrent extract language entity distributional semantic relation name distributional compositional entity augment distributional semantic pass composition distributional sentence entity representation sentence non syntactic distributional computed distributional child representation word
onto kkt condition w sufficient subgradient always hand side due hold condition screen al parameter plug define b bit screening overlap overlap bridge primal dual primal problem solution note path priori linearly spaced screening screen perform show screening screen base path omit proof theorem theorem leave hand valid tight screening feature goal optimization thus algorithm hand adopt minimization k group intersection kl determine compute set finally minimize overlap screening l minimize bind obtain iteration set process counter squared hand fix subgradient subsequently upper z k iterate side take root f overlapping include simple fast coefficient individual algorithm overlap lasso summarize mention fast lack algorithm similar various utilize appeal ratio lasso however necessarily small zero nontrivial compute couple simple denote technique summarize demonstrate speed rejection ratio discard coefficient zero overlap ol overlap screen image dataset genome image ad individual ad know predict variant code rna brain genetic information randomly color object image select dataset digit nine test induce locality output jointly image genome structure group consecutive group four root consist parent overlap group consecutive overlap overlap knowledge ad group nucleotide locate region start end site overlap solve overlap lasso screening likely except single report average performance solver screening implement matlab group structure size screen scenario rejection ratio structure figure rejection ratio represent dot diagonal rejection ratio ol except ad reject feature overlap hand dataset ol use ad size size ols increase rule ratio regardless size ratio single machine ol comparable measure speed gain rule dataset overlap give solver without solver ols solver illustrate portion run time speed without running screen ol dataset ols ols solver ols discard feature appeal portion hereafter observe experimental dataset structure group rejection ratio rejection ratio plot maintain rejection drop rejection ol group interestingly even tree group ol rejection ratio penalty htb ratio experiment use overlap group size change size ratio size increase group increase number test group left keep rejection size rejection decrease fixed window rejection start include make independently test group advantage test group ol screen verify screen tight develop various loss research overlap regression regularization determine sparsity primal q lagrangian optimization present author machine author author author author author author projection author illumination database author author author author author author proposition notation section theorem develop efficiently discard screen infeasible develop lasso arise group take
edge related maximization seed topic seed topic intuitively reasonable assume topic cccc seed overlap seed randomly mixture topic greedy select seed mixture percentage mixture intuition seed seed network seed topic contribute seed topic topic influence solve item mixed influence probability apply mean topic influence maximization inefficient choice well focus online mis compute marginal achieve competitive convenience budget algorithm value preprocesse idea minimize online seed mixture good influence call selection item pre seed seed preprocessing guarantee exactly deal issue spread use round method seed explain preprocesse value online enough maximization sophisticated rounding notation return neighboring basically round every influence spread round di output topic sub additive preprocesse sub every mean spread seed mixture spread topic verify network assumption could even layer edge odd layer topic structure believe real influence sub constant p p grain follow spread I p additive worst tight focus topic explore focus maintain reduce seed search seed seed topic preprocesse stage seed item seed topic mixture I greedy describe call stand small much spread seed find seed j ps derive seed pre seed select seed seed set motivate especially intuitively nod much whether mixed pure precise seed I separable topic case definition topic recall every seed let otherwise greedy also mi p advance sort mis round seed pre add marginal influence simply influence return seed I although mis original greedy algorithm topic mixture separable reasonable assume seed topic seed seed possible mixed denote final seed let fully set disjoint select topic seed set v j pi I uv conclude greedy suggest mis network verify well marginal sum individual thus mis verify real compare influence experiment preprocesse mis comparison aware greedy algorithm preprocesse aware greedy employ evaluation hundred greedy ic influence achieve optimize degradation spread paper degradation work ic ic use node seed output choose preprocesse ranking factor use degree high finally compare greedy preprocesse mis utilize seed experiment ghz cpu core server code write probability node topic scalability large www node author million topic practice medium influence practice usually mixture topic cover three dataset since case describe section maximize learn test greedy pre seed except algorithm equally distant pre core different seed test compare spread run carlo simulation influence seed bind solution influence spread greedy seed multiply theorem set influence spread influence influence minimum greedy seed total day day day n sec l sec ms ms sec influence show preprocesse greedy dataset seed table count cpu time need cumulative greedy suitable two preprocessing separate gap among algorithm small low average spread seed seed analysis first perform well aware observation ignore topic influential seed topic thus spread demonstrate mis fast seed three order report order magnitude aware sort baseline perform aware replace greedy mis spread indicate preprocesse reduce preprocesse table separate among utilizing greatly save online mis well small degradation seed spread small small topic overlap preprocesse still seed topic topic suffer mis online aware need least slow large indicate time mis graph run influence close competitive spread suitable large performance heuristic vary significantly minute long complete fast processing achieve time graph influence spread outperform heuristic furthermore greedy slow finish mis achieve spread mis use preprocesse first formulate cascade provide influence maximization large body influence study optimization number study machine g social influence topic influence topic aware influence et ic maximum aware computed seed complementary list introduction follow combine advantage guarantee world investigation topic wise preliminary bring insight influence guide acknowledgment provide observation lin seed seed node base certain diffusion aware model issue influence user dependent idea aware maximization topic preprocesse couple stand candidate spread preprocesse effort idea social connect maximization seed seed large people social network model direct represent relationship propagation seed influence maximization set seed network spread activate influence seed maximize wide text word decade improvement efficiency extension model information etc refer item influence friend influential influential influential topic aware cascade item topic mixture individual aware world task adopt propose topic influence maximization scheme topic maximization every diffusion preprocesse influence item mixture come seed study available aware preprocessing purpose analysis user relationship significant different topic property seed mixture come top seed topic influential influential category motivate explore preprocesse first minimize online seed ratio influence sort mis influence computation provide justification show mis conduct evaluation art result mis stand preprocesse spread either aware preprocessing comparing include world motivate theoretical seed mis novel complementary mis simple achieve competitive spread within need influence focus cascade ease presentation parameterize direct graph represent influence ic capture discrete activate activate chance inactive stay active activate stop activate define influence seed influence node diffusion computer mining like investigate topic overlap define topic p u I overlap topic fairly network apply among topic topic node network may research type movie cccc dirichlet seed overlap seed greedy come seed check source seed mixture seed topic mixture seed mixture seed separation even significant seed topic contribute seed network two present influence raw prior american site discover raw trace network represent user edge friend rating rate movie node direct topic maximum action trace topic influence probability due lack
problem especially organization hardness binary neuron unit binary infer pattern assignment constraint act constraint cause critical nonempty perceptron build complex learning memory weight weight hardware implementation perceptron thus rule structure mining code compression biology perceptron effort devoted solution case local density increase explain relate statistical past computation picture topic science machine physics recent landscape focus solution hamming distance element comprehensive description solution sample boltzmann equilibrium landscape reference potential state spin spin physical meaning cost temperature temperature work term entropy temperature physical insight understand organization demonstrate space perceptron isolate solution density minimal separate solution grow reveal hardness perceptron become exponential time require fixed finite learn binary potential replica conclude remark feed neuron perceptron association desire input pattern desire randomly actual nj pattern impose constraint denote perceptron configuration energy incorrectly e convention ensure order unity sake statistical limit perspective perceptron able extensive pattern capacity configuration quite nontrivial replica method analytic landscape perceptron densely connect learn idea select constrain equilibrium constrain angular partition constrain averaging quantity reference coincide typical current aa nm mx interaction replica replica mathematical identity average nr get constrain eq gaussian q saddle transform characterize lie apart hamming divide extract value concavity numerically point work move identify message pass word extensive barrier landscape always state solution problem sense make isolated solution separate picture report study moreover convergence solution replica symmetric introduce replica break weight zero internal apart result require analytic perceptron describe entropy landscape reference equilibrium independent reference solve saddle point equation concavity lead distance constraint minimal density imply compose one flip extensive refined picture organization perceptron understand connection study algorithm analytic analysis offer basis mathematical study hard information spike time neural classifier partially fellowship researcher grant core soft matter information current context equilibrium configuration temperature perturb constraint configuration satisfy free coupling configuration interest ground set equal substitute definition energy zero temperature evaluate value replica identity aa nm characterize average u q replica approximation symmetry overlap share average h v q deriving use gaussian parameterization structure pattern eq integral dominant saddle transformation saddle laplace respect keep consistent equation eqs characterize equation equilibrium affect system perturb depend get follow together saddle derive measure saddle equation way time perform integral retain covariance dp concavity saddle fixing coupling field method solution give derivative
may regularity fourier wavelet edge set translate penalization lead involve extensive inverse problem recent many mainly focus variation address optimization vertex incidence play intersection ball variation image follow display ability present significant decrease admm denoise matlab intel ghz core dual measurement original corresponding spin obtain solve j diagonal modelling transform ng redundant frame area background round subsampling strategy array acquisition maintain subsampling factor account reader therein reconstruction shown implement apparent subsampling matrix iteration require gradient effect absence slice snr wavelet primal learn audio management streaming network great computer vision dimensionality inherently noisy model aim explain usually interest relationship particular suitable methodology solving offer great dependency hard modular manner use powerful non hand discrete highly nonconvex np hard discrete unary encode order encode dual offer important advantage characteristic example provide problem theoretical rely schema art applicability mrf apply low level include view match tracking estimation image motion top bottom compare per iteration expansion form challenging task medical two domain grid cost project deviation system advantage complex divergence modality furthermore admit broad graph employ fig minimum surface matter mm result primal another application input right seek left problem measure difference patch spatially depth nonconvex exist obtain level convex relaxation discretize solve primal dual discrete optimization similarity pixel two image employ surface curvature dual principle fig especially np also minimization upper optimum approximation throughout assess primal extra practice correspond estimate almost review primal employ contribution perspective although prove effective extend also accelerate parameter speed parameter various technique fast method block distribute mention relaxation combinatorial np problem challenge e g high field label set appear main generally develop dual bridge kind problem blind deconvolution duality height em theorem corollary theorem theorem pt playing method problem drastically derive bring play idea important optimization nonsmooth emphasis present principle give method propose large discrete provide usefulness discrete duality computer optimization extremely paradigm constitute branch mathematic signal communication popularity approach stem characterize form uncertainty signal uncertainty noise often inherent perfect inexact application characteristic encounter area scale example computer solve low require least pixel image bad therefore exception similarly field due ease collected cope truly naturally situation arise application g network network constitute address tractable exploit problem possible take regard concern particular class method proceed task well formulation problem mention broad primal dual primarily problem duality nonlinear composition operator advantage method scheme iteratively compute subproblem involve scheme former gradient operator latter use proximity operator implicit implicit may easier exploit property flexible efficient dual achieve known proximity linear separately result expensive require significant last least becoming increasingly efficiently handle primal prominent vision labeling seek include task segmentation optical flow estimation matching mention image problem highly nonconvex optimize offer lead message easily handle regard estimate call principled derive powerful combinatorial bad aforementioned primal dual primal solution optimization background theory originally may appear topic allow broad class combinatorial problem relaxation certain discrete good solution encounter constitute source develop technique thorough intuitively principle behind continuous discrete advance place concern primal solving connection method alternate useful context signal notion duality duality duality programming notion subdifferential proximity section explain various devote admm deal schema well technique combinatorial dual base relaxation domain inverse field summary discussion definition use dual algorithm later hold space dimensional allow take modern discard search optimal problem component intensity must nonempty domain establish proper low f see nonempty fig htb subgradient proper differentiable subdifferential reduce singleton nonconvex extend definition subdifferential may reduce subdifferential growing introduce early work proximity define thus result minimization show variation p soft operation equal projection p operator view projection proximity projection view contraction engine banach proximity ensure algorithm proximity operator nonconvex guarantee uniquely define notion deal notion case graphical illustration conjugate fig particular subgradient necessarily proper convex conjugate express result always see subdifferential important characterization minimizer question exist subdifferential subdifferential provide property link proximity useful ss parallel draw multidimensional fourier transform tool process main kind nonsmooth encounter feasibility transform conjugate fourier notation define r dr denote dirac cm l invariant translation translation multiplication invertible jx nu nx j inf convolution element addition product offset define support nonempty positively norm norm compressive sensing variation area sharp contour convex set conjugate hypercube impose array express easy know solve dual bring information first basically solve provide primal precisely proper addition condition exist vanish gap equal zero sharing suppose composite distributed manner assign vertex reach consensus original rewrite space indicator orthogonal complement form u want utility saddle saddle relation actually prove tucker primal lp j nb mean formulation view property conjugate readily duality lp solution obtain lp lp convex sophisticated one wide q differentiable fidelity encounter respect introduce additional smooth may flexibility problem see possibly term inf convolution reduce trick section jointly instead focus find tucker mention quite simple algorithm admm direction multiplier view belong since possible saddle point augment lagrange lagrangian lagrange admm split gradient result perform potential advantage parameter norm nonetheless exploit generally comment formulation benefit split objective example operator quite operator introduce conceptual fact account role differentiable strongly convex I point amenable architecture successful problem dual method idea parallel dimensional mf mf projection separability proximity note consensus derive version simultaneous multiplier even onto simplicity instance involve formulation allow propose manner mention another image primal express l lx nk nb formulation model broad hereafter encounter lead principled approach find np call much extent reasonably obtain make former relaxation relaxation relaxation instance integer define relaxation much easy quantify hence approximation lp relaxation relaxation tighter interestingly tight lp relaxation involve expand feasible mention derive relaxation use maximum objective relaxation allow nonconvex relaxation discrete naturally primal lp powerful relaxation exist many number grow especially semidefinite second cone relaxation observation primal dual focus principle powerful heavy underlie lp aim lp fractional try much whereas schema technique optimization worth start exact program initially problem tight probably go back maximum max max essentially schema branch minimum span case schema drive fact initial iteratively total guarantee optimal since integral end notable primal often lp series combinatorial unweighted minimize complementary primal use instead algorithm np hard problem admit formulation applicable tool combinatorial cover network feedback location domain vision analysis schema broad np integral complementary since lp could schema turn relaxation need primal compute optimal due relax constraint give attempt solution primal rely explanation primal feasible assume show approximation optimal e dual principle rely fact sequence primal lp duality optimal exceed lp unknown guarantee dual feasible cost gap cost cost thus approximation principle heart impose directly dual convenient viewpoint generate program latter duality primal applicable duality gap exact relaxed variable check hold j relaxation give hold clear follow ny feasible satisfy complementary satisfie principle x base yield dual employ follow solution update apply primal schema strategy maintain primal complementary satisfied introduction measure degree minimize opt maintain relaxed complementary condition infeasible iteratively infeasible primal solution would improve feasibility dual ensure feasible matter strategie gradually bring primal close together complementary primal versa fact improvement lp lp compute problem relaxation costly primal schema purely combinatorial algorithm notice require primal schema np element disjoint subset assign cost subset cover minimum cost see determining include cover ensure least set obtain replace boolean relaxation mean number schema relaxation call
number uninformative voxel carry category include informative possibly redundant focus feature fmri voxel often redundant completeness stability feature discover possibly redundant accurately mainly aim uncorrelated reveal discriminative voxel cognitive credible three category filter embed score algorithm use score feature subset use embed typical embed method voxel fmri classification treat degree voxel proportional value component selection identification nonzero component consider challenge focus voxel motivate however inference plain sparse hard interpret voxel portion voxel result potential trust success plain column condition correlate noisy subspace thus plain incorporate structural brain imaging segregation achieve reliable interpretable hypothesis voxel discriminative group cluster correspondingly strongly use elastic net try use voxel regularization deal highly recently penalty add correlate besides penalization total penalization use simultaneously voxel tv make activation spatially voxel fusion successive certain smoothing plain regularize group structural correspondingly explicit segregation structure plain enforce structured solution voxel group either drive group agglomerative cluster grouping sparsity discriminative voxel completeness voxel result selection uninformative voxel year important method selection voxel bootstrappe behave disadvantage plain select even bad instability informative set information major advantage positive obtain expect addition stability recognition brain fmri plain regularize characteristic focus ensemble idea apply randomized tree discriminative voxel often spatially contiguous common clustering run subsample selection add help result rescale implement stability voxel fail clustered voxel paper via constrained subsampling voxel wise fmri voxel subsample classical sensitivity discriminative voxel stability achieve include control positive voxel new summation stability structural case structural rough voxel highly subsample scheme help shape numerical perform computationally organize section introduce new voxel selection real high specificity short future give denote fmri voxel idea follow difficulty effort detail voxel intercept voxel discriminative plain enforce structured correlation way grouping adopt grouping compare main belong sparsity feature prediction interpretability provide grouping allowed incorporate redundant therefore help voxel induce plain regularize due grouping reliable either voxel obtain incorporate discriminative make choose simultaneously like plain choose regularization parameter sample effective way reduce difficulty voxel connection control positive expect redundant voxel correlation explicitly consideration paper probably discover aim integrate stability selection common one sparsity subsample former likely yield latter whenever pure regularization subsampling easier extend subsample combine structural sparsity first explain subsample baseline returning feature special except stability sample subsample version drop important randomized structural sparsity incorporate structural consideration fmri structural sparsity general concept implementation depend fmri datum propose specific implementation name subsampling adopt replicate subsampling subsampling voxel group voxel lie cluster subsampling reduce negative size partition quite correspondingly solve group part select voxel subsample either drive select subsampling constrain prior selection able shape discriminative flexibility voxel belong area voxel though might aim voxel association response label part response would pay save lot correspondingly baseline subproblem prefer average idea apply variable positively correlate feature yield fit variance voxel pick block subsampling lie single voxel variable greatly result recovery much therefore greatly improve pose compatibility apply th large magnitude pick voxel lie cluster average simple boundary structural drive partition voxel patch algorithm mean spatially group cluster reduce fraction total large come constrain block stability stability subsample perform matrix subsampling select integer voxel notice run result small cluster expensive procedure input classification subsampling term score voxel brain perform voxel perform row block voxel calculate pick voxel pick voxel basically stability mainly observation point subsampling guarantee false positive adopt method plain finite bounding ratio expect false feature adopt sparsity base pointed term incorporate interpretability learn voxel functional relevant might bring bias intuitive explanation thorough study subsample reduce constitute cause stability rescale improve efficiency cluster large proportion paper algorithm univariate voxel voxel pattern include randomized logistic selection random feature implement logistic implement projection software logistic great software code otherwise might inherent allow block match easily positive negative block probable prior knowledge performance synthetic fmri experiment respectively chinese problem block repeat nine break block state block subject acquisition preprocessing mr west china china fmri image te ms flip angle head slice without gap centre uk www uk head motion institute imaging template brain three isotropic mr fmri high pass hz noise fmri cluster algorithm figure show brain base score different thresholded visualization meaning zero zero noisy localize brain identify visually recognize voxel balance control false false negative general identify control positive control strictly need threshold filter wrong confirm knowledge carefully threshold positive brain notice threshold via small positive achieve nearly apparent false control brain extra region likely brain science pattern successfully region indicate work cognitive default human task state stability selection method identify brain functional structural part randomize logistic common slightly concern make distinguish false positive compare advantage fmri size subproblem h public face object consist eight group ms ms inter stimulus interval brain fmri record volume stimulus volume fmri acquisition previously fmri house cat consist evenly spatially constrain software number evenly use sample select limit map first score thresholded visualization thresholded algorithms requirement fdr control method validation svm candidate code threshold correspond high except sensitive voxel potentially build logistic discriminative voxel select table extra voxel voxel could relevant voxel explain validate discover discriminative viewpoint rand tv rand maps phenomenon study area setting contour quite similar thresholded select stability maintain control false sensitivity compare voxel region unlikely positive cross validation voxel tv large discriminative voxel voxel positive even estimate discriminative voxel around positive test positive share common logistic conservative control false positive setting positive false estimate however reveal contrast voxel keep false positive result table algorithm rand tv select positive rand false look voxel pick randomly consider accuracy among combination achieve high randomized logistic reveal discriminative accuracy predictive specificity feature voxel degree prediction significantly high window matlab run intel eight processor processor base frequency ghz gb parallel involve master cognitive smooth lasso svm logistic take short notice write computationally master master test test spatially constrain take long time tv l directly software efficiency present algorithm rough rand logistic rand master voxel selection drive extraction implementation variate correlation multivariate mostly empirical understand limitation spatially well alternative method distinguish false positive address tv l integration thanks ga team provide randomize anonymous many constructive suggestion greatly support program cb cb project science foundation china specialized research education china z z slide volume voxel incorporate voxel stability effort fmri give brain brain proxy brain make brain fmri level bold grid know fmri compose series bold voxel high segregation area differ global integration consist voxel whose spatial voxel organization subsequent identification perform set allow discriminate brain localize discriminative due directly interpretable growth activity order multivariate argue determine sample distinct whether mean differ supervise analysis training svm training build set marked belong two category new common svm notation section regularize regression categorical variable regression necessarily score predict dependent intercept scalar regularization adopt intercept elastic hybrid elastic generate reduce value outperform follow intercept scalar logistic regression randomization stable brain imaging traditionally htbp htbp region accurately stability less contiguous block voxel positive positive sparse purpose receiver roc versus specificity different support roc selection ground voxel base positive specificity rate tn respectively practice use spc roc large false see voxel select two brain combination voxel select two test ground unlikely
parallelism simple effective big lda strong model topic million word parallel implementation face difficult challenge access explicitly access input categorical assignment topic lda token position token assignment topic topics lda reformulate collapse fast accord perform iterate analogous analogous lda schedule rotation scheduling mod return empty w p f b return update sufficient f c parallelism identify assignment statistic speed sampler worker one subsequent schedule amongst worker subset partitioning divide token denote worker worker subset schedule worker gibbs assignment worker word parallelism observe assignment choose sampling sample exactly fast gibbs sampler fast lda machine parallel gibbs sample unless conditionally disjoint document worker conditionally except dependency extra end worker denominator thus induce ij ij word proxy copy cm total token must lie plot error wikipedia refer topic machine processor core total throughout lda exhibit parallelization benefit consider collaborative predict preference give preference tend rank interested enable decomposition pose challenge purely datum sgd address divide across worker mf incomplete item preference discover product use miss user formally observe index column cm schedule row index disjoint index set compute analogously merely supplement motivation counter prevent dependency candidate next represent naive scheduling execute worker partition worker submatrix partition manner rule accordingly worker th iteration u illustrate h frame lasso schedule scheduling get new draw c safe u j j partial sum f frame z aggregate demonstrate implementation lda mf size baseline fast baseline partition baseline implementation distribute lasso rr alternate square implementation mf lasso rr random scheduling choose popular distribute mf two machine effectiveness parallelism hardware cluster lda cluster mf core cluster contain ghz cores gb interface cluster contain ghz cores gb ram via interface use size unless otherwise english wikipedia tokens token token large publish demonstrate topic large create extremely rating movie vary exceed paper feature world add noise otherwise reach run mf either handle converge quickly attribute fast parallelization lda reduce synchronization requirement mf nearly limited quickly dynamic figure schedule cause second mf achieve well parallelism fast trajectory machine fix confirm fast machine big parallelism scalability efficient memory partition invoke worker parallelization correct parallelism study enable scheduling aspect parallelism static partitioning parallel coordinate abstraction direction cost star topology eventually become bottleneck issue wish asynchronous parallelism parallelism also want machine logistic theorem computer fit take long natural processor naive ml make inefficient proportional develop parallelism explore schedule ml speed inference correctness demonstrate efficacy versus implementation model factorization digital storage medium improve massive scalable machine numerous heuristic principled big million deep problem attention world challenge efficiently worker access large k furthermore parallelism important demonstrate suited big subset allow partition memory allow variable parallel tackle big exist framework show variety crucially framework automatically decide next choose next user automatic scheduling convenient offer grain subtle graph structure moreover allow user criterion schedule lda rotation scheduling collapse gibbs topic scheduling descent dynamic scheduling coordinate sample framework parallelism schedule execute aware way dynamic parallelism schedule specifie specify individual partial variable specify aggregated full primitive ensure distribute automatically execute user utility implement schedule popular ml implementation enable solve programming modeling vocabulary matrix application j send worker update j frame u u worker return frame update worker schedule basic signature primitive parallelism refer parallelization model high change quantity ml iteratively iteratively use strategy like parallelism parallelism allow problem massive machine memory advantage model memory machine machine algorithm whereas strictly constrain small practical consequence pair enable topic enable user systematically parallelism framework map paradigm user ml repeatedly create iterative figure schedule primitive vs less
n j paper one notational ease efficient preferred paper get difficult large effect deal due adjust root parent hypothesis continue child adjust factor denote cardinality incur fine effect effect specific marker partition region marker form elastic allow non zero coefficient final operator individual marker score apart involve genomic kernel parameter relate net post process standard performing typically hold explore detail hyper parameter choose accuracy improve informed opinion reflect resource aim depend number marker marker hyperparameter estimator allow control value model kernel flexibility detail locally test importance genome toolbox seven trait date md trial year line evaluate model display genome wide use replication split trait trait available divided rest line evaluate fashion display level accuracy association diverse genetic diversity target also lin locally trait whole genome length trait display trait genomic trait individual individual select angle angle width data marker national day accuracy region lin score display figure width angle propose although seem population single kernel additional advantage utilize score use marker occur context advantage solve matrix matter dimension involve matrix memory problem marker loading marker genome miss outlier matrix separate region marker genome divide linkage proximity calculate code sequence grouping trait allele frequency marker absence marker guide incorporate membership marker subsequent set code file acknowledgment award section example property ph genetic additive genetic lose argue line genetic map additive effect parametric mixed genomic relationship design lasso performance explanatory genetic semi parametric mix marker successfully predict study genome marker always prediction value bi population express design effect eq kernel hilbert connection recognize model implicit input dimensional space refer number symmetric k k exp taylor reveal kernel regression extend additive though option marker genetic incorporate additive marker implicitly incorporate study empirical prediction gaussian however increase lose issue overall effect additive potential argue kernel estimate genetic article argue marker effect marker locality model additive marker since incorporate view semi incorporate contribution article genetic whole genome local effect genomic snp marker easily reach million hypothesis focusing argument segment genome building block schema predict complex evolutionary mechanism fitness tend well structure building block example fitness genome lose segregation would build block parsimonious genomic utilize counterpart genomic product remain section briefly literature notion similarity come source literature include method commonly component calculate component perhaps genetic gene interaction kernel kernel learn building stage subset divide marker subset genetic process local fit training step obtain marker nest subset marker conclusion subset annotation marker however capture genetic possibly nest genomic define marker linkage genomic region inform illustrate root whole genome genome width hierarchical allow region coarse divide detail availability nice coarse fine hypothesis tested error keep scheme
calibration approach structure sense calibration estimation phase compress sample sparse shannon number measurement denote transpose compressive recovery retrieval imaging optical imaging vector fourier reconstruct measurement retrieval recover essentially vector definite convex pl trace low among one acknowledge retrieval study theoretically large measurement therein iterative describe propose measurement compressive quadratic pursuit addition sparse magnitude need provide isometry condition reconstruction signal pursuit solve class compressive define contaminate cross hermitian semi definite signal recover program become identical though determine sparsity structure constraint joint induce objective induce improve constraint know norm suitable therefore ambiguity recovery problem value system investigate perfect generality compressive room improvement numerically pursuit problem basis convex empirically pursuit provide bound good recovery performance insight sparse determine perfect show eigen operating element eq projection onto equal represent order imply strict assume bc show constant choose ensure eq phase w convexity x lead convexity space definite local convexity optimization imply eq solution q equivalently suggest since satisfy consequently satisfied provide combine satisfied similarly condition true set p tight transform non nature constraint requirement simplify criterion tighter guarantee perfect straightforward sparse whether perfect recovery respect non entry measurement varied level complete perfect signal lowest highest select among figure increase input signal mainly recovery broad majority upper bind feasible perfect recovery result tight estimate recovery display phase diagram compressive different phase demonstrate accurately report observe match simulation evaluate performance tight perfect http advantage simulation evaluate method determine possibility successful
information reconstruct coverage map allow robust resource allocation combine mobile user balance wireless resource performance scheme far enhance addition predict promise like discussion corollary institute department electrical address reconstruct map evaluate adaptive alternative offline algorithm tailor iterative subgradient user quality estimation measurement process simulation realistic algorithm world application mobile estimation coverage attract attention challenge enable network wireless service reconstruct importance device device channel extract provide device reliable decision make crucial allocation decision resource allocation present future propagation quality service mobile information user utilize resource allocation media streaming buffer mobile reach area avoid failure decision traffic traffic load transmission cover problem estimate two loss ingredient reconstruction coverage development resource scheme application maintain coverage prediction base user measurement path quickly adaptation coverage reconstruction need requirement important arrive continuously online nature ensures take account continuously quality addition exploit enhance information understand information concrete location exploit incorporate measurement input due various wireless user usually position measurement control network non uniform sampling significantly handle situation prediction overall despite area kernel aspect paper enhance provide coverage aware base adaptive algorithm incorporate simulation attention realistic publicly real trace spatial short work online loss user kernel estimate map side algorithm time enhance weight underlie wireless network challenge researcher wireless empirical path decade measurement measurement mainly machine neural ann aside technique recently krige apply track map capture spatial correlation scheme spatial available prediction scheme model measurement wireless author fit correction real wireless university building type subsequently update corresponding path refinement measurement refine iteratively study propose mechanism map readily without need environment rely project subgradient iteratively generalize subgradient easily tool machine affine successfully network image recovery real respectively letter thereby element element product trace frobenius inner possibly dimensional notational convenience notation respective space context infimum nonempty closed set uniquely point singleton reproduce rkh property ff reproduce calculation carry trick high calculate f particularly suit design minimizer specify model outline important basic side scenario problem pose reconstruct path map use availability wireless wide low cost device physical method input information trajectory technical notational clutter section deterministic good performance mobile sequence receive signal th measurement relation path measurement unknown measurement arrive give operator finite quickly estimate function computational online algorithm keep improve measurement arrive easily investigate highly base subgradient technique cope operator technique feasible computational limitation practical algorithm grow measurement relevant relevant sequence call estimate projection proximal gradient smooth function notational dictionary treat computer interested reconstruct path location devise computer define construct path future location expect visit attention outline section model selection scheme correspond specific application belong contain account unlikely contain enough provide reliable precisely set contain expect point reasonable obtain problem feasibility possible example able store whole sequence n practice recall able soon come back estimate finitely many use set however mild assumption estimate ii approximation I propose variation project subgradient detail set approach choose continuous constant solution produce solution note average averaged follow fix adaptive hope obtain tracking time comprise differentiable function compute easily forward modify end comprise structure iterative proximal step base lipschitz mapping time vary two measurement add column euclidean close remove irrelevant discard refer algorithm weighting row enforce cost employ weighting compressive first stability order keep weight row enforce inspire minimization algorithm connection aim minimize induce enforce convex absolute vector one fix obtain follow incorporate omit constant sum address intractable minimization detail convex concave g additive form q variable become find monotone try many norm alternative formal approximation need scope approach element completeness literature reweighte enforce pointed numerical suggest minimization recover report good frank wolfe type closely wolfe construct move correlation scheme regard increase roughly factor numerically propose estimate noisy uniformly parameter variable error mse communication loss realistic city measurement subsection devote numerical except kernel common applicable kernel free l projection weight ci width l reconstruct planning network wireless channel assess real evaluate realistic city language create meet realistic precise city loss datum ray base momentum appear provide path valuable mainly capability algorithm reconstruct path loss long sufficient specific resolution underlie location low server assignment cell base estimate define base assume poisson far randomly end realistic movement trace conjunction simulator tool simulator generate movement trace realistic limit intersection mobile trace enable processor interest area parameter bs pixel run capability propose figure simulate respective show sake similarity original path estimate loss user system estimation location dash dot solid green path trace area mean time instant compare evolution predict note one frequently available measurement completeness account pixel belong define sufficiently evolution outperform speed value report location incorrect measurement variable absolute offset e pixel usually gps device relevant error sensitive path inaccurate achieve drastically small dictionary figure simulation show evolution size observe sparsity algorithm tailor raise experiment select concern impact factorial choose indicate subscript value table l result accuracy default figure choose small gain achieve bad versus steady capability set dataset project receive strength measurement collect university area trace measure measurement take particular find remain performance gives compare low due
signature element generative employ foreground histogram foreground image contain essence appearance foreground complementary body symmetry complementary aspect person appearance extract salient feature weight exploit principle geometry capable provide meaningful solution distance video separate automatically give person large diverse camera view comprise extraction descriptor riemannian stein stage detail subsection reduce vary pixel image generative use advanced pixel locate xy xy colour channel xy colour g r indicate magnitude channel rgb colour select relatively certainly thorough beyond several noisy sample iii correlate symmetric definite interpret riemannian manifold space handle manifold non due largely challenge manifold take function even riemannian structure consider riemannian somewhat size define logarithm demand essentially eigen furthermore prevent conventional riemannian embed tangent distance distance give start discriminant find intra distance scatter similarity map classifier refer relational method person dataset image person cover aspect represent improvement caused evaluate stein conjunction classifier ie manifold without create stein capture real arrival video image normalise height overlap illumination person probe stein method outperform also conjunction preferable stein direct base capture move camera contain variation randomly select image rest commonly set propose considerably outperform pls direct stein par create apply cluster ensure signature close stein pls appearance person comprise represent covariance matrix foreground treat riemannian manifold similarity class aid stein divergence iv discriminative vector final classification traditional latter result inaccurate modelling manifold person identification dataset recent histogram accumulation communication centre shot person relational box university school identification particularly challenge due view people signature effectively illumination pose camera appearance space application model non end represent covariance matrix interpret riemannian manifold manifold similarity bregman divergence stein similarity classification similarity vector manifold tangent space suffer comparative evaluation identification obtain technique histogram partial accumulation feature surveillance identification person match overlap camera diverse location within context surveillance large candidate pose body illumination variation make human person approach typically use entire challenge person separate camera view challenge change body illumination variation appearance person generally argue geometry end recent covariance interpret riemannian manifold embed manifold pairwise tangent shot base identification manifold tangent space require adapt recently riemannian similarity similarity aid form stein divergence task manifold convert vector discriminant analysis method separate continue follow person
primarily web message phone comment movie precise ratio aim rnn hour list epoch newly pass learn ensemble hide apply normalize training example total consistent spaced filter compute window ms audio file training set divide global input describe train phrase character phrase alphabet word keep token speech benchmark situation create web dl speech benchmark range hundred benchmark hour speech rare utilize expressive recurrent synthesis approach vision find convenient speech properly present base scenario environment gpu synthesis strategy build system noise combine speech rely processing stage believe continue increase dataset size future acknowledgment grateful dl speech forward project also ng ai art speech recognition architecture significantly system rely process traditional tend poorly environment hand background learn concept key well optimize synthesis obtain varied data speech test speech also widely art system rely compose stage paper speech call speech stage language achieve network rnn learn directly specialized setting corpus system speech heavily process stage input acoustic hide expert deal model introduction algorithm improve speech acoustic deep play speech noisy robustness deep recurrent advantage capacity learn performance sufficient power however pose build must train enough effectively utilize challenge handle text speech address enable neural meanwhile rapid et demonstrate speed gpu aim vision large scalable rnn complicated vision partly lee al techniques map novel scheme parallelization quantity speech collect learn robustness take idea suffice build end traditional task achieve new noisy speech recognition system error rate remainder recognition begin describe recurrent follow gpu synthesis experimental deep conclusion core recurrent rnn english sample vector audio denote th audio frame sequence character hide unit recurrent frame frame non layer compute eq relu activation bias fourth layer hide recurrence backward recurrence note must sequentially must sequentially recurrent take backward softmax character probability slice denote column weight bias respect output character point back nesterov accelerate rnn considerably simpler relate model literature long term memory lstm circuit one disadvantage lstm cell store multiple neuron forward backward bottleneck homogeneous model computation recurrent activation relu output involve highly gpu wise nonlinearity length expand fitting reduce several dropout feed layer recurrent employ computer vision feed version beneficial translate audio ms half bank propagate use rnns output speech character character rnn external plausible english example occur never appear set speech language know impractical integrate gram language train huge corpora language section corpus phrase support vocabulary train n experiment rnn weather right weather rnn character probable language character word aim objective validation n gram maximize objective use highly search network amenable speed execution homogeneous network implement optimize call fully connection experiment feasible multi gpu accelerate two parallelism multiplication prefer many example prefer gpu wish support parallelism across separate minibatch iteration parallelism parallelism implement multiplication resolve problem sort similarly sized inspire format accelerate rnn parallelism train minibatch face example update improve yield also inefficient minibatch spread example minibatch gpu scale partitioning parallelism due sequential layer layer comprise backward perform unfortunately split go depend less model half dimension layer trivially decompose along time series assign gpu another gpu first gpu begin forward activation begin backward activation mid intermediate activation swap gpu gpu forward work run recurrent rnn recurrent layer library deep require abundance record correspond english public extensive consist hour comparison dataset read fisher read corpora publish linguistic expand potential training even synthesis context primarily environment capture speech environment way audio generate superposition noisy speech audio track audio track noisy speech track audio necessary form together realistic audio risk hour clean create hour speech track span roughly say repeat since become recurrent single hour shorter public video source separate noise end ensure match synthetic datum datum reject noisy effect encounter recognition environment effect overcome collect ensure induce play background person record induce thus allow capture experiment system use dataset character level fed word yield highly researcher split report new alone full rate
aggregate name bootstrap dag overall entire ensemble structural hamming dag node fitting thus positive implementation hill regime package keyword skeleton hill positive consist edge direct node interpretation dag dags relationship intelligence genetic learn quantitative research field body topic topology graphical structure learn indirect system nucleotide gene practitioner expression examine conditional would conditionally impose constraint parameter learn future instance dag sample huge exponential lead challenge commonly dag tackle challenge utilize search acyclic check sample gb learning procedure highly drastically perturbation tackle use achieve reduction consequently ensemble dag base aggregated minimize family metric aggregation perturbation approach inspire name study procedure false positive bag initially build recent year successfully perturbation aggregation previously dag learn propose measure confidence graph learn stable log perturbation locally average learn though graph acyclic aggregated dag propose rest brief overview model hill structure aggregation strategy dag aggregation section brief overview direct acyclic reader direct acyclic edge node dag graph figure give I x pa set set show graph obtain discard parent seven skeleton middle edge due connect highlight red node sequence node meet head neither meet head node figure separate next relate dag model say admit probability formally dag admit recursive factorization specify local conditional namely distribution dag df dag respect admit obeys set I say map obvious independence vice versa say perfect map hand compatible dags dag say I encode follow skeleton edge structure equivalent dag converse true relation partition dag equivalence bottom right since separation separate clear equivalent x x ccc impossible dag solely equivalence formally dag structure distribution dag structure p brief overview method dag score dag commonly score early exponentially nod exhaustive search therefore greedy dag include search dag dag view independence hybrid method structure test restriction focus implementation hill search performance dag structure procedure score discuss decomposable hill dag propose hill search make dag structure identically map goal learn map denote moreover dag define score define dag space depend node parent hereafter update rest subsection include matrix property negative negative estimator decomposable residual sum dag reasonable model candidate bic decomposable score aic I set score pa mean node hold minimize strictly score add edge imply ii eliminate hill update distribution term dag want find among dag mention due space moderate heuristic employ algorithm initial subsequent operation operation maximum operation able decrease score exist operation result cycle acyclic check operation apply major hill acyclic check potential cycle calculate score note operation one operation decomposable score change neighborhood score update score neighborhood involve operation graph lead change operation nevertheless score costly operation step moreover take stop e greatly high efficient hill search information facilitate score acyclic current illustrate idea operation current hold operation involve result score cycle cycle cycle operation operation dag search dag dag aggregate dag hill subsection metric hamming conduct much set dag learn dag aggregate crucial aspect dag dag aggregated subsection metric score theory vector need convert dag hamming lead valid hill search edge dag among specifically edge always operation j distance unit lead generalized ij define j j I j correspond correspond compare dag true study penalize skeleton edge reasonable orient support aggregation dag tend edge well sf dag ensemble p additive function individual edge indeed hill simplify input ensemble dag sequentially sf sf initial graph current sf pass acyclic check edge add proceed stop set cyclic hill decrease moreover addition hill conduct cyclic edge ensemble dag node dag minimum proposition dag generalize direct dag aggregation q constant hill simplify proposition sf search depend sf orient edge penalize extent frequently edge appear possible selection sequentially stop empty pass edge acyclic add stop cyclic edges hill score selection frequency reverse large lead hill reach proposition conduct extensive simulation examine several dag learn sample independent mechanism variance deviation part consider namely standardize score replicate dags size http snr graph dense graph extra large skeleton v edge table reason report skeleton structure table skeleton edge edge learn number skeleton edge identify correctly identify report average standard less fair quite identifie besides facilitate search skeleton skeleton edge graph total versus across end result stop package package independence test drive run series draw trajectory curve one method structure aggregation effectiveness empty method number positive graph omit graph node size figure skeleton point indicate stop presentation stop number skeleton structure skeleton learn figure aggregation color non black curve skeleton demonstrate superior moreover aggregation implicit model result aggregation aggregation reduce positive edge inferior well curve stop early demonstrate penalization discrepancy efficiency figure sample aggregation false aggregation able performance main list detailed learn curve table snr aggregation curve aggregate skeleton detection detection snr graph curve poorly lack curve range stop early aggregation induce positive skeleton curve difference snr term difference lie stop study procedure snr vs much aggregation aggregation low false mainly edge bic method graph infinity conduct demonstrate skeleton curve size size false give effect topology study topology focus ease figure
f il n r compact z trivially restrict neighbourhood z b neighbourhood z x numerically calculate rr desire b x software integral describe expectation decomposition boundary define restriction interior cube map parameter sufficient claim boundary parameter closure hull polytope polytope therefore cell relatively open cube cube give note parameter note cube translate map equal convex vertex properly separate meaning lie space generic face lastly follow closure use space induce decomposition boundary space figure face highly behaviour right also begin show boundary cube duality relationship euclidean eq similarly say image become despite dimension enough give enough kf sn kn sf sg sf kf ss sn sn f topological relatively face dimension association relate closure closure closure hence face lie induction hypothesis kf tt sg sf f g essentially g sn k n prove induction ss rs sn establish theorem say face boundary say image approximately approximate large relative parameter generic theorem function continuous ideal one prove connect recall generic lastly concentrated intersection hyperplane numerically volume every show jump large reflect degeneracy let degeneracy subset row row generic jump z n endow sum square open define converse volume generic always degeneracy enough rr dominate generic matrix compare q approximately see since rx denote hand approximately generic element argue b ib z jj combine degree degeneracy let reasoning last equality perhaps logistic novel finite jeffreys volume simple elegant version give model therefore parameter though arise principle boundary space logistic map cause implication behaviour exponential deep duality lastly acknowledgement author would thank interesting discussion remark prove generalization classical finite volume interpret volume generic matrix less nearby way tend prefer arise general principle lastly topological boundary draw bernoulli canonical riemannian transformation geometry behaviour concentrate geometric volume prove consequence regularity jeffreys interpret theoretic measure maximize natural datum minimum description length parametric model previous logistic study prior place principle hyper code adapt logistic remarkable geometric matrix complex nearby tend choose matrix behaviour though fit equal fit derive mild full covariate approximate volume compete volume choose small maximize meaning surely go principle giving volume lastly linearly raise possibility generic relationship decomposition boundary open closure boundary approximate radius divide spherical hyperplane decomposition dimensional boundary dimensional become behaviour interesting right implication computation rest model cube embed calculate metric unchanged rescale cube show apply image processing generic topological section generic conclude remark though consider component realization model q odd odd odd odd interpret column family sufficient correspond family riemannian likelihood hessian expectation assume regularity distinguish fisher independent map sense natural odd q eq riemannian manifolds riemannian manifold turn either euclidean cube space model isometry open cube light euclidean identity dimensional respect matrix prove lemma euclidean odd domain partly establish logistic regression volume invariant real unique odd consider matrix real substitute family natural fisher know prove publish author space function differentiable function jacobian back natural odd recall dimensional riemannian metric tensor local notion integrate jeffreys prior determinant volume density strictly everywhere rank everywhere invariant recall subspace column de theorems model let parameter model row real follow design natural euclidean isometry onto euclidean volume hausdorff inside embed around circle monotonic use volume logistic say onto co matrix sum subset generalization de binary cube unique obviously identity complete proof bound sharp degenerate particular volume upper q show generic close interesting trivially identity follow column range x horizontal integral continuous would would close compact effectively restrict fix recall derive theoretic statistical behave logistic regression model space suppose countable set model logistic design choosing minimize turn choose model large case interest principle likelihood x moderately instead use eq valid note observation regularity condition valid exist exist derive begin section generic rank mean full converse unless covariate lebesgue e component row probability generic fisher iid random number region integral region arbitrarily error assume hence give say limited computer experiment hand lastly minimum degenerate assume generic approximation fact row e row row approximation approximation selection give criterion choose datum result show criterion almost goes note n consider difference true enough sparse reduce information however bic approximate volume application processing application partly problem particularly process black white pixel black picture signal noise pixel pixel follow interpret white black subset pixel column represent segment pixel contain approximately lasso implement regression fit parameter validation
node skip infect cascade always zero whether network cascade cascade regularizer la successfully network estimate yes estimator cascade network survival hazard concave hessian sum outer matrix table hazard hessian capture co cascade check impose restrict identification hence definite diffusion cascade answer question crucially incoherence diffusion sampling cascade way child relation co compare commonly naturally satisfy specifically population take cascade hessian city index parent complement index ss min connect co reasonably cascade incoherence exist ss jk node neighbor infect cascade neighbor cascade hazard norm n appendix cascade lipschitz condition boundedness feasible evaluate absolute remark state strictly condition exist remark depend observation study income node easy incoherence condition ba ty node source cascade cascade incoherence direct condition consider star edge transmission like incoming leave long cascade incoherence incoherence condition remark pairwise satisfy analysis cascade need polynomially node ga mi million parent compare network individual edge cascade q exist constant follow regularize unique uniquely specify incoming node incoming edge sample improve remain remark co parent union bind provide co child remain largely large parent node primal previously ise good knowledge technique optimal share pattern far unique prove primal subgradient vector moreover hessian strictly unique next primal dual furthermore construct kkt pattern deduce incoming primal ne tucker kkt regularize primal solution condition substituting construct state kkt hessian lift hessian satisfy eq kkt condition hold construction step provide use expansion remainder entry eq algebraic cs ss small help lemma incoherence converge incoherence condition holds apply lemma condition hold cascade pn min lift dependency incoherence impose population regularize implementation network satisfy natural incoherence exponentially decrease cascade cascade record future cascade confidence infer edge network exist window value incoherence condition satisfied finally activation allow missing activation nsf gm fellowship song survival concave hazard pairwise hazard express sum semidefinite concavity survival concavity hessian positive definite semidefinite cascade order cascade within cascade hazard cascade index cascade matrix triangular sort continue sort cascade source position infect across cascade order break tie cascade infected never cascade infected index node cascade order similarly column correspond cascade node infect early finally remain assign cascade order lead desire spectral problem constrain constant convex constant across primal primal kkt lagrange negativity solution primal gradient since primal since optimal strictly strictly convex respect sg deduce segment strictly radius bb value segment inside entirely end detail function regularization neighborhood ci series separately bind reverse bound remain challenging introduce quantity q proceed n k b next lower set kk c convexity z j kkt respectively hoeffding start hessian proceed condition lipschitz continuity difficult q first condition continuity boundedness cauchy term inequality condition eq select regularization difference nuclear cascade jk express fisher cascade ss ss ss reasoning start incoherence score vs cascade hold prove infinity jk jk z jk prove confidence proceed final bind incoherence easily apply ss conclude eqs ss cs cs c success incoming canonical fig super cascade use cascade one super node window lead line well probability infer incoming kronecker neighborhood cascade transmission window lead edge cascade transmission method outperform cascade polynomial cascade need success exponential number cascade information across network structure trace cascade kind cascade cascade despite increase availability cascade infer ti un question literature continuous regularize mi cascade na tu condition framework structure ty cascade maximum parent soft consequence alternative pt behavior disease temporal trace give rise information neighbor influence observe person get infect cascade model infer unobserved underlie attract attention predict path spread sale stop focus evaluate experimentally synthetic real network go analysis enable answer condition run recover network cascade cascade ba li ty direction view identify relate interaction cascade make depend network structure sampling cascade recover occurrence parent nod cascade use li cascade rate cascade finite sample guarantee cascade theoretical solve especially suit scalable find sparse soft thresholding demonstrating outperform state art inference cascade general net recover network cascade cascade par discrete cascade instead diffusion validate author network continuous independent cascade source uniformly cascade network cascade additionally study bound degree source decrease polynomially cascade incoherence network cascade infect cascade generative cascade introduce transmission parameterize contrast associate differ discrete sense cascade iteratively round transmission continuous
practice learning term popular iterative nonlinear aspect fit directly term via hyperparameter penalty neural network help variance hand fitting optimization performance hold use optimization proceed loop iterative outer hyperparameter perform rigorously application search computationally train light effective systematically principled attempt good evaluation optimization train quality rapidly useful even loop complete explore hyperparameter effectively fit goal iterative procedure bayesian framework hyperparameter favor start partially complete old maintain train theoretic one continue loss roughly decay towards nonparametric decaying characterize temporal gaussian able partially train hyperparameter many different ordinary bayesian optimization space definite process prior gp gram applying form form gps become ability characterize come computational grow invert gram implicit hyperparameter characterize behaviour ern infer constant prior integrate slice domain generality hypercube optimization receive develop tailor hyperparameter sensor optimization proxy determine location choice determine utility trade explore uncertain exploiting region yield result acquisition ei eq q density correspond minimum far posterior variance probabilistic ei minimum choose input location minimum represent minimum acquisition location yield entropy expect minimum simulation ei sample natural curve develop curve specifically develop decay fix integrate allow region mix basis function use model factor procedure row draw independent gp prior learn curve partially train curve gp colored training color surrogate require computation I model training gp would expensive computational incorporate independence curve draw global generalization curve specify column vector generative setting gp another constant mean mat ern gp prior illustration restrictive machine block omit derivation require gp marginalization property gaussian give eq size use lemma efficient distribution likelihood gaussian new curve omit space absence gp pre time total practice keep increment monte simulation observation use condition observation carlo equation select run gp develop create system decide user fully train one low asymptotic discover reflect hyperparameter setting curve maintain represent model entire ei use standard bayesian ei task become choose ei favor one iteration ei similar acquisition pick location minimum method search outcome unseen input consider subsequent information common hyperparameter report low epoch empirically validate art method logistic task allow report epoch run epoch training procedure run five method stationarity specific provide optimize train descent popular hyperparameter include weight minibatch dropout optimize hyperparameter latent lda wikipedia topic implementation optimize rating penalty optimization procedure distinct hyperparameter curve epoch color bayesian optimization promise promise start show specific epoch clearly experiment significantly art due dynamically prominent online predict observe negligible small additional incur explicitly gain rapidly reach visualization optimization pmf empirical analysis
greedy greedy require priori hierarchical prior sparsity priori analogue et factorize user author estimate column induce low induce generalize low reconstruction induce low e leave singular left definite determinant precision diagonal wishart generalization gamma naturally log q relevance iteratively update maximize r give estimate resemble weight one problem unbalanced balance eq balancing remove limit computation parameter numerical p leave vector equal right kronecker precision computation thus far reduce corresponding want complement decompose variational bayesian become I likelihood iteratively since convex iterative converge objective accelerate computation uncorrelated reduce block iterate block time point operation iteration reduce numerical measurement matrix vb block know toolbox nuclear completion matrix variational bayes vb nuclear norm varied db vb varied method figure accelerate less dimension sense take combination draw normalize vector accelerate low nuclear nuclear small nuclear see figure sparse kernel machine singular vector prior construct call vector machine iterative design priori complexity accelerate outperform outperform relaxed develop bayesian reconstruction relevance singular machine accelerate computation reconstruction problem numerically relevance low reconstruction sample attract considerable generalization compressed sense greedy method hand develop heuristic provide refer development non
attribute field color map represent proportion market relative distribution job statistical perspective job besides advantage discrete plausible form encoding form quantization embed highly summarize job naturally lead public policy important job attain job eigenvalue find coordinate figure sub along directional gender plot strong correlated flow sum plot explain due job augment represent economic latter use intervention depict panel suggest tend region seem specialized gender observe diffusion gender slightly worker corner characterize proportion corner embed specific corner economic force towards end show situation outside flow corner blue right panel overall seem theoretical view direct vector field plausible general take make manifold moreover result node spectral graph knowledge extend generative possible set value component result coordinate recover recover albeit theoretical embed framework check immediate currently since manifold education force core ball integrate tangent integration change approximate reproduce explain origin locally flat normal along orthonormal tangent orthogonal laplacian coordinate finally denote polynomial remain arise derivative along use radius meanwhile differential come volume drop term take interpretation variable ball around form expansion important global field substitute integrate vanish meanwhile manifold change term ultimately come pick come amount divergence field implicitly xt asymmetric em height em consider graph directional direct diffusion endow kind direct graph estimate laplacian type construct highlight strength advance visualization foreground laplacian element infinity put study embed foundation lead elegant lead development manifold map laplacian undirected graph social alignment international citation naturally asymmetric type affinity work propose contain popular cut principled way similarity asymmetric result cluster successful adopt purely concerned actor affinity assume statistical step work explicitly manifold field account relation direct sample link determine overall connectivity laplace limit recover manifold local intrinsic pay attention tell extension also help previous method depth denote node generative compact make undirected graph node asymmetric similarity kernel define vector assign top leave corner aa ss embedding preserve generative process recover distribution practical process increase laplacian diffusion map answer infer aim undirecte direct laplace eigenvector eigenvector eigenfunction thing laplacian principle directional geometry know scaling increase embed converge original manifold preserving embed detail start observe asymmetric affinity part unique em symmetric diffusion essential add correspondence skew part good family interpret something retain radial smooth field orientation worth note domain locally appropriate field pointing though seem rich describe form omit define density transition represent reader recognize operator give normalize denote correspondingly eigenfunction limit definition study limit operator like one limit convenience operator manifold field combination operator eigenvector counterpart next core follow exploit obtain algorithm method graph transition close operator asymmetric integral asymptotic expansion expansion origin help come vector limit remain curvature tangent interaction follow mention assumption obtain use form interesting operator briefly present mean x derivation apply differential calculus drop specifically general operator complex eigenvector nevertheless play role extract meanwhile instead generalized q discuss add theory become author procedure part cut well cut criterion eigenvector symmetric play precisely represent affinity diagonal dense small natural equal normalize graph laplacian derive limit immediate symmetric kernel correspond pde describe right act source worth point absence pde field diffusion cause source flow recognize flow motivation theoretically separate manifold whether embed direct thing generative manifold locally laplace diffusion describe step discretized version eigenvector constant embed coordinate coordinate simplify every task reconstruct gradient plane exploit simple serve coordinate unit use component figure component run replicate step operation affinity n ss q ss order right eigenvalue embed n aa tr
cc robust cubic spline knot cell cubic knot cell variation discard gradually decrease shape objective converge variation middle datum upon amazon actual partition curve figure show solution rough similar fitting regression indeed spline linear peak transition curve present narrow cluster differ peak middle left contain curve less narrow peak large middle look flat cluster spline regression help addition unsupervise hand profile present attribute scheme pre map som apply piecewise notice two fold indeed som som raw step infer observe behavior narrow narrow peak contain increase follow slow decrease cc spline knot cluster knot algorithm converge variation show gradually variation objective partition middle mixture optimize penalize likelihood world application curve degree always cell degree good solution retain datum clearly spline cubic cubic smooth kind continuous approximation take second fitting number model I mixture fully regression conditional mix proportion expert proportion softmax direction fit expert hierarchical expert number expert sequence knot respectively th spline eq basis ij l b mx ij spline consider find w mix proportion lagrange multipli function multiply summing finally update proportion figure toy toy figure universit france universit france widely however crucial em perform also mixture criterion pre fully carry em spline regression mixture approach unsupervised proceed criterion accurate initialization apply curve propose robust result real confirm propose practical one successful estimation discriminant focus model density density component estimate e estimate vector model mixture density achieve datum functional concern paradigm analysis curve dimension observation curve series mixture spline modeling arise assume generate spline regression widely study analysis algorithm initialization number cluster case criterion choose estimated candidate provide gaussian multivariate focus cluster unsupervised consist likelihood expectation maximization polynomial spline spline regression mixture propose proceed rather standard brief maximize penalize criterion model formula propose approach conclude cluster generative mixture formulation assume multidimensional model set mixture gaussian matrix estimation likelihood base curve compose density way regression mixture approach overview model em mixture assume draw suppose realization polynomial corrupt mean noise polynomial degree tp ik represent correspond denote conditional density polynomial regression parameter vector iteratively via em expectation likelihood log complete z nz value otherwise em regression start initial two current update maximize mix maximize lagrange multiplier consist square solution know solution cluster estimate component fuzzy represent fuzzy observe assign cluster high summarize standard mixture htbp randomly partition mean initialize equation output ik extension previously polynomial mixture spline spline constrain mp derivative spline continuous piecewise write spline spline matrix version nearly singular spline thank matrix spline spline finite support everywhere else b spline curve regression polynomial use polynomial em notice standard em regression sensitive might estimation mixture initialize drawback em start em cluster know criterion issue mixture mixture separately among regard automatically estimate em algorithm algorithm number attempt indeed call minimum message mml penalize negative observe penalization control start cluster penalization proportion discard simultaneously estimate initialization still become serious dataset number concern datum observation assume reduce adapt individual structure rely analysis lead problem analysis model limitation functional mixture attempt limitation mixture curve cluster propose regard proceed polynomial spline spline derive robust em fit present regression spirit extend functional spline fitting mixture furthermore adaptive similarly mixture adapt proceed integer maximization r consist maximize analytic update proportion estimate cluster represent hard partition compute maximize denote estimate initialize regression model parameter middle stop code summarize algorithm curve regression mixture mixture curve converge nk use equation compute em equation discard cluster proportion q ik may simple data linearity even spline adapt choose spline knot spline use cubic spline spline twice kind piecewise spline adapt order piecewise constant concern knot knot space range technique location cross knot place knot determine either sufficient knot easily type selection knot much fix knot paper sensitive location knot knot sufficient dedicated propose approach simulate world implement develop code available request em spline spline perform estimate accuracy misclassification simulate real world course simulate arbitrary curve curve simulate nonlinear curve cubic spline spline regression estimate interval actual correctly classify cubic mixture mixture knot quasi identical accurately estimate middle table normalize mean one classify actual cluster function one actual em simulate number iteration cluster rapidly majority discard iteration see penalize likelihood value objective algorithm middle iteration iteration number precisely iteration obtain regression also accurate result rapidly provide elsewhere consist curve h h mean unit temporal interval generative cluster spline correctly spline slightly well polynomial em robust cubic spline three misclassification absolute cluster retrieve misclassification slightly regression spline em objective model highlight evaluate datum original one describe namely construct five five dark iy dark water retain frequency consider discrimination
max game adversarial net conditional model generator extra label data modality perform conditioning noise combine adversarial compose mlp player minimax illustrate conditional condition encode vector net dimensionality within hypercube relu layer hide relu generate mnist maxout piece piece layer maxout layer piece fed sigmoid architecture critical maxout unit typically batch exponentially momentum dropout generator likelihood validation show log mnist draw sample detail adversarial approach include adversarial net efficacy exploration hyper architecture match exceed non generate condition one label generate mnist adversarial adversarial site vocabulary adversarial model work experiment tag tag image tag repeat tag generate top cosine similarity vector vocabulary annotation tag generator receive relu layer vector relu hide map vector relu hidden image layer piece join finally sigmoid unit mini batch initial decrease also momentum generator hyper mix manual albeit limited annotation people tree house water net interesting thorough analysis tag multiple generative hope achieve obvious construct scheme suit specific acknowledgment would helpful author acknowledge vision production frank yahoo train model adversarial net simply condition digit condition label illustrate modal preliminary example adversarial intractable probabilistic adversarial net wide incorporated produce realistic generative control conditioning conditioning could construct adversarial net demonstrate experiment digit one modal despite network challenge accommodate predict category issue date focus mapping many instance label tag human use different describe help address additional natural language corpora label geometric make g predict fact make predictive generalization
frequently model apply mathematic mining twitter tweet analysis pattern collection tweet extract twitter start combine tweet relate cluster topic tweet algorithm give nmf prove explore result tool mine world group relative text cluster analysis collection text extract twitter contain game start begin tweet tweet english keep tweet contain user political news twitter security able search could country certain range size research algorithms factorization mean twitter center choose space assign centroid assign close minimize close centroid continue centroid metric cosine magnitude distance contain give distance however magnitude sentence contain world contain cosine purpose research cosine deal matrix distance tweet thus tweet might consider short tweet word cosine tf document result range non value range initialization initialization random mean different disadvantage order algorithm world sometimes difficult many algorithm dataset create together consensus running mean times parameter case b nmf topic non non topic text pointing direction vector multiplicative alternate constrain aim multiplicative rule converge local greatly one multiplicative costly depend initialized sparsity element long mine remove start path least speed initialized matrix initialization flexible create multiplicative disadvantage lack replace negative replace square advantage alternate add matrix sparse fast since nmf base spatial form mark noise removal h input dataset dense radius point dense radius remove parameter drastically vary variation tweet tweet twitter call much thought tweet decide identical tweet preliminary exploration tweet tweet tweet remove eliminate original tweet much vocabulary stop look collection tweet want tweet tweet closely figure keep removal create run varied could tweet cluster remove drop consensus tweet consensus matrix drop row sum another tolerance entry consensus average remove noise decide tweet noise distance use cosine tweet cosine distance dense sure dependent create tweet run tweet point also create tweet decide keep cluster apart decide distance keep tweet create classification decide tweet mark point consider noise unique remove cluster frequently strength combine look represent least mark tweet remove keep tweet decide diagonal entry row consensus identify gap eigenvalue topic topic broad choose gap eigenvalue figure cluster major text mean widely algorithm highly consensus give tweet tweet text file tweet word cloud visualize overall throughout use mean tweet know tweet help cluster tweet cluster visualization tool discover word
infeasible instead take approach imagine represent individual entity datum record categorical advantage variational posterior model data vocabulary challenge particularly inherent clustering model process dirichlet non proportion change clearly entity resolution cluster record constant behavior record record record complete every miss record database make database field lda noisy record record database regard record latent capture let th kf non trivial assume th categorical putting assume individual record latent individual length dirichlet assume vector draw dirichlet encourage split merge record take special record hybrid assignment record split merge health care database record hour database database million hybrid take wish million multiple census contain million record database country million record model show entity approximate generative show full appendix generative amenable lda word model vary latent contrast part allow proportion topic key regard mixture model entity cluster size would order simple apply via strictly datum may finitely finite infinitely dirichlet case cluster without grow theory cluster cluster natural assume record individual rather traditional must ask regularity inference draw cluster infinitely dimensional grow generative proportional solve derive follow ascent variational q new much fast since make simplify assumption incorporation realistic assumption record moreover remain individual grow construct wish solution address entity broadly domain grow bound proportion tb berkeley fellowship nsf grant dms grant gm independent conditionally independent give independent within notation database record field categorical field th record index define dr model record database latent separate whether full value aggregate likewise record divergence maximize concentrate field parameter parameter approximate recall ascent sometimes variational first eq zero proper together database minimal represent latent bayesian allow generative share across principled quantification query final resolution mcmc modern database contain
pc motion human acquire marker system fig pose position gp metric see latent periodic pattern motion riemannian geodesic show geodesic straight pose comparison pose see geodesic straight reconstruction stay reconstruction straight drastically length geodesic match truth h pc train dot green denote dash straight line pc pc euclidean original different difficult operation metric latent distribution smoothly point local metric short straight interpolation new gp expect uncertainty long force avoid latent desire behaviour worth riemannian reasonably track riemannian kalman classification potentially metric entire although low less understand g almost surely metric manifold lead curvature worth investigate influence acknowledgement author great european project project ref foundation education rgb rgb rgb rgb section figure figure box intuition appendix chapter computer university university uk structure dimensionality riemannian geometry tensor treat variable riemannian expectation distance expect lead represent dimensional potentially represent nonlinear mapping metric metric uncertainty space metric capture thereby provide low useful illustrative display representation underlie rotation want analyse datum insufficient space go datum raise choice euclidean still meaningful question approach generative reflect intrinsic recover mapping metric observation space riemannian computing distance short path riemannian metric natural trend concept riemannian paper overview state art reduction introduce extend probabilistic metric finally path experimental discussion latent embedding surface manifold learn interpret manifold underlie basic geometry surface surface surface dimensional surface machine terminology correspond chart straight intuitively chart illustration q jacobian surface chart integrate riemannian riemannian symmetric smoothly product point riemannian metric smoothly change suffice curve geodesic q show imply unique starting reduction provide probabilistic dimensionality define unobserved joint variable dominate mapping associate feature account model typically choose advantage therefore dimension principal component analysis probability datum write independence nonlinear basis tensor compute short path differentiable embed section explicitly compute riemannian metric jacobian eq define conditional jacobian follow naturally row central wishart number degree freedom centrality equal central wishart distribution joint vector function possible lead formulation gp latent notation differential derivative gaussian long derivative jacobian mapping compute every partial latent jointly gp model observe define support observe embed differentiable jacobian mapping metric jacobian independent jacobian form tensor compute metric tensor imply uncertainty curve uncertain metric define gp exploration way dimension generative map explicit example latent colour proportional space endow riemannian compute short path short problem space result grow exponentially latent quickly infeasible geodesic differential equation geodesic independently dimension nd ode st ode matlab gives solve repeat derivative illustrative provide specific square eq
g equality covariate relation accord j p set sum identifiable since pair identifiability complete q transpose q furthermore take mm mm mm parsimonious family cluster account attain eigen impose component sufficient condition identifiability compare show accounting dependency regression offer use last decade arguably methodology parsimonious cluster package typically variable insight gain accounting dependency cluster incorporate perform response whereas weight logistic often utilize represent investigate statistical paper discuss mode e distinct linear various parsimonious univariate eigen covariance response response currently correlate recently model decompose covariance investigate deal response eigen parsimonious recently response parsimonious eigen impose parsimonious schwarz family hereafter organize basic summarize section recall identifiability likelihood issue assessment base conclusion idea response framework covariate disjoint multivariate density normally matrix normally rewrite density variate variate single parameter necessary decomposition eigen yield entry sort constraint eigenvector accord geometrically determine orientation group split model align orientation belong volume free spherical spherical align align axis align align g g covariance decomposition lead refer sequel usual asymptotic theory estimation cf multivariate mixture generally class finite function q identifiable sufficient general follow identifiable variate parameter via unconstrained nn incomplete note incomplete ig complete ig maximization expectation update closed form detail q equation update family choose fit among family model criterion bic asymptotic development well practice extensively incomplete commonly additionally mean component equal occur note convergence maxima unbounded surface comparison package make work package user initialization facilitate initialize section analyse dataset hereafter covariate correspond component group respectively lastly matrix htb family result bic choose fit result bic choose estimate model contain mixture analyze concentration body fold percentage composition variable response summarize htb ari vi vi comprise vi ei yield ari estimate second bic quite freedom pick result ari well grouping package provide width variable ari difference bic select freedom ari together bic choose ari model ari result ari two family ari assign essence choose ari implement pick component ari pool htb contain blue highly width measurement take variable colour know algorithm run select ari bic pick ari htb ari parameter model ari ari table estimate model lead good membership agreement estimate ari ari basically put component pick ari model utilize also note surprising multinomial logit multinomial condition covariate density assume give result ari respectively bf om account multivariate correlate response covariate allow eigen decompose response covariate component structure identification parsimonious unconstraine
carefully transform various naive would dimensionality complement nuclear convergence epoch stop epoch line prove convergence worse careful lead outline objective log loose weak fully apply former latter dimensional batch well loss within I n loss locally since gradient bound derive somewhat weak lipschitz provide guarantee additional denote log determinant let j limit need guarantee bind contribute directly property however condition utilize result note epoch improve factor compare lipschitz replace satisfie intuitively condition initialize consider network observe zero mean multivariate gaussian distribution vector column diagonal allow limit cm sep fill circle mm h observe name name name h joint rank among triplet parent child term presence additional edge convert introduce parent direct graph remove undirected case loss fm I bind singular bounded method previous sufficient walk efficient guarantee online slightly reason scenario ball good initialization ensure subsequent involve section approximate applicable somewhat weak local limit bind contribute radius log reason q hold local depend factor reason intel gb ram see design nevertheless accuracy within addition epoch high fluctuation therefore projection good method st e e st admm reason st ccc run inexact alm design epoch work epoch specific epoch delay cause compare reason art inexact alm direct eliminate inexact alm reason reach useful error reveal projection far either reach projection expensive svd projection alm inexact alm thus omit high multi multiple regularizer optimization problem reach factor component propose modify multi admm algorithm sparse optimization matrix decomposition outperform particular accuracy consider decomposition provide address nonconvex additional addition descent acknowledge detailed discussion thank valuable recovery author discussion point regard microsoft fellowship nsf award award number minimax decomposition component provide guarantee p match minimax low respect latent admm annealing consist projection ball sparse reach high accuracy rank regime stochastic extensively uncertain involve scalable scale contrast traditional technique far operation work alternate direction multiplier admm scale employ many g locally via augment apply update admm solve globally regularize since natural encourage optimal g sparsity employ admm problem regularizer regularizer illustrative set result simple modification inexact anneal huge implication rate dimension instance scale projection certain ball anneal introduce dual constrain obtain begin epoch average pass estimate projection also decomposition admm update sparse project nuclear norm admm dimensional problem problem scale minimax linear guarantee minimax rank guarantee matrix literature comparison framework convergence dimension noise well compare admm decomposition inexact alm method recover admm improve expectation per rate table contrast admm admm rate require function regularizer contraction whereas sparse study anneal achieve one derive capable variable incorporate online dimensional admm matrix poor low impose batch fairly convergence low model note weak condition matrix suffer e noiseless set rate rate however worse fix however epoch follow establish derive modify epoch size batch combine epoch vary epoch additional ensure estimate trivial different careful enable first online decomposition set match guarantee many interesting model change stop c method st st admm condition admm sc e batch df reason r p q generalize result setting optimum solution regularization nuclear later optimize inexact admm employ base set estimate epoch constrain close go expect provide admm extend involve block admm access epoch length prox radius rate impose high efficiently implement discuss update carry set consider matrix model access update estimate impose desire propose detail algorithm reason recall assume update linearization use inexact proximal follow project epoch constrain matrix impose rank impose nuclear initialization nuclear encourage rank entry note impose assume discussion projection perform efficiently appendix auxiliary eq efficiently projection step approximate efficiency projection stand compute initialization epoch epoch prox shrinkage initialize I provide reason efficiently dimension efficiently need follow addition fm fm intuitively constraint control low separate type sparse low f require jointly nuclear dual update step total guarantee least appendix improve vary epoch max scenario lower bind provide bind convergence rate match scaling factor attain discuss otherwise intuitively bound term need two batch average sample efficient concentration observation conjecture set provide incoherence constraint incur even noiseless identifiability error decay online list ki k set well mean gaussian entry another independence diagonal obtain q batch entry e matrix assumption connection variable efficient set match conditionally variable long since compose inverse precision express
stationarity ridge convenient could stationarity worth linearity nonlinearity linearity linear work nonlinear work linear nonlinear datum nonlinear indirect causality causality extent without extent compare side transfer extent side spurious causality good causality yes transfer yes causality test augment lag somewhat sensitive promise somewhat sensitive somewhat sensitive lag lag histogram kernel parameter lag crucially financial measure causality dependency aware first research decomposition decomposition asymmetric causality causality third suggest building intervention causality describe prediction becoming successfully build u obtain gram intensive perform cross calculation kernel validation validation point whole select appropriate use calculate testing datum recall still significance window experiment measure perform believe strength measure reason parameter optimal nonlinear optimal employ cross choose parameter learn embed split subset th belong create range kernel scale dual dual weight calculate particular average correspond calculate prediction optimal mention undesirable college uk discussion valuable feedback comment max device social centre es study collection university college bt centre economics political science series concentrate causality causality causality schmidt transfer examine attention ability nonlinear causality theoretical benefit dependence set generate nonlinear dependence bivariate highlight month sp rate circumstance research series causality financial beneficial causality distinction intervention causality answer patient survival answer operate intervention involve tool analysis causality direction knowledge financial model intervention use causality past expand must lie joint observational need distinguish capable discover cause causality describe causality method independence useful causality characteristic know well case financial stationarity exhibit nonlinearity become broad provide review practical aspect methodology synthetic financial application contain supplementary material causality appear wiener predict use past one concept introduce winner economic autoregressive cause occur unique information contain variable include causality mean concept deterministic say signal time signal simultaneous instantaneous coupling expand publish feedback instantaneous causality instantaneous coupling causality instantaneous side quantify paper measure therein crucial place strength causality one causality measure definition series subscript understand random variable accordingly causality cause natural stand cause couple bivariate case way instantaneous coupling alternative optimal instantaneous coupling definition assess formulation square many introduce process generalise multivariate model way linear allow everywhere restriction quantify usefulness causality instantaneous eq q x later linear quantifying machine causal machine perspective initially independence causality last become method become search meaningful requirement individually pairwise kernel function interpret also suggest causality ridge surprising introduce clear property way alternative causality square schmidt space permutation quantify kernel concept hilbert schmidt normalise independence explore method causality please hilbert create definite kernel positive definite name semi henceforth kernel define provide feature identity fact product space replace dot section trick causality nonlinear show reproduce causality standard univariate good linear infer causality alternative particular four model past new lag represent reasonable assume lag typically causality early involve look value depend simple univariate well drawback poor sample size address least square solution write primal q notation w tw px x kernel form inner explain combination representation result dual trick allow kernel element gram denote linear allow operator prediction way square whole denote fit analogously define index causality instantaneous coupling use way causality covariance use analyse pointed assess independence use measurable finite correlation appropriate importantly equal even maximum attain nevertheless machine completeness schmidt product element covariance analogous cross operator covariance symbol tensor product definition operator two covariance follow denote reproduce hilbert rkhs k topological field expectation ensure random expectation covariance normalise conditional operator normalise operator rkh dense supremum norm independence independence appendix normalise conditional independence denote hilbert schmidt operator use normalise marginal normalise operator information schmidt normalise square schmidt normalise conditional q hilbert schmidt operator straightforward good behaviour define use cross normalise cross covariance inverting next construct construct estimator schmidt necessary inverting introduce alternative information theoretic measure provide comparison method measure sense transfer entropy transfer popular among transfer develop entropy bivariate nonlinear causality improvement max review causality present perspective design state bivariate omit side causality transfer prove variable causality decompose shannon well mutual information mutual assume random probability uv shannon independent mutual information mutual lack mutual direction natural extension directional entropy previously stand transfer entropy generalise define hx calculation already joint impractical serve comparison couple deviation generate dependence consequently need assess measure assess achieve permutation test permutation test significance causality time since causality rely create keep permutation causality surrogate small causality quantify depend permutation hypothesis causality level significance set overlap window latter useful stationary case believe kind subsample beneficial concern world eight relatively lag instantaneous try lag eight subsequently shift relation lag lag lag lag network show lag causality occur data correlation testing dependence cause variable ts ts ts ts ts ts ts ts c ts ts ts purpose source matlab transfer matlab toolbox causality open access matlab measure negligible comparable range code cross grid histogram performing incorporate write accommodate implementation permutation test permutation median inter lag causality occur causality lag four lag allow analyse effect range lag different lag present conclusion measure similarly dependency range causal direction short range well lag two direction detect causal coupling permutation acceptance fail spurious analyse lag inherently reason inefficient range lag need couple mutual lag entropy report zero value relevant causal direction fail direction lag instantaneous coupling correctly impractical range lag range handle higher correctly causality correctly report lag instantaneous value lag bottom measure transfer te lag retrieve c c ts ts ts ts ts ts ts ts ts ts ts ts ts ts te ts ts ts ts ts ts ts ts ts ts ts ts c ts ts ts ts ts ts ts ts conclude gaussian lag misspecification seem transfer direct direct causality introduce refer causality test four degree acceptance rate direction particularly linear detect exception causality detect spurious case effect different lag three possibly turn test eight describe size play crucial causality however kernel work type example present causality demonstrate distinguish indirect cause gaussian zero know effect indirect calculate assess causality repeat experiment time lag causality measure obtain linear expect causality side take consideration indirect causality pick kernel dependence h equivalent causality z blue face calculate gaussian face calculate hilbert schmidt normalise independence transfer allow achieve face year economic around scientific causality formulation field methodology successfully field characteristic distribution generally finance economic tool devote mostly information reduce dimensionality g subset factor causality structure help relevant forecasting causal parent future financial characteristic biology physics finance long biology though dependencie many researcher stationarity usually kind clearly direction rate considerable concern real interest country causality analyse namely index economic country reflect contrast set interest reflect ask whether economic indicator statistical sense run similar gaussian kernel median investigate windows significant interpretable observe long window clear direction short window considerably often dependence window lag month lag report value lag show scale causality series month assess causality direction accept reject level causality translate roughly explanatory somewhat pattern separate interest fall consecutive explanatory interpretation causality h hypothesis red lag chart scatter set causality price index one month model lag scatter plot causality set month lag scatter hypothesis causality month red lag scatter causality lag long clear lag rate time direction causality linear result direction strong separation lag lag reason perform well interpretation causality conclusion aspect detect causality transfer causality u month transfer entropy significant direction lag often stress might please refer direction one blue line way red lag scatter causality set u month lag scatter causality measure carry trade six exchange index sp daily period sp exchange volatility information logarithmic return measure window length day window method case employ result separation direction especially sp causal consistently sp indicate period period sp effect cause sp red obtain regime purely lack causality series test specific lag explanatory lag permutation causality autocorrelation introduce bias also high datum correspondingly instantaneous causality transfer appear bias similar side causal sp trade causal effect include pattern exchange rate lose explanatory sp distinction direction h hypothesis exchange cause sp volatility blue relation ask method quantify causality context well develop firstly often field science economic lack context secondly question management question cause loss tool quantify causality currently develop quantify causal inference understand result understand enable reader set part describe comment testing direction conclude nonlinearity causality assess causality efficient causality good dependence financial normally exhibit arguably analyse requirement causality causality bring causality distinguish indirect consequence reduction repeat indirect causality introduce notion indirect whole variable concept indirect causality measure distinguish indirect follow compare conditional causality indirect causality explicitly build condition cause intend notice sensitive call partial transfer indirect cause spurious causality cover cause spurious causality wider indicate causality infer relation introduce add spurious exhibit dependency none dependency instantaneous coupling nonlinear causality reader causality domain numerical mention high sensitive causality bivariate good significance expensive regression layer calculate expensive unless
svm find margin label justify consider voting behavior statistic proportion york randomly instance individual proportion sample bag form instance population chernoff proportion population proportion enough bag sample bag generalization b conditionally bag title group training instance spaced scale large far formal learn recover real datum predict census label cover subset census instance binary education status week bag people feasibility form bag divide testing bag label propose solver bag proportion validation report first simulate case draw instance assume instance bag instance sampling bag relatively test feasibility training bag bag likely solution simulate attribute group assign simply base perform random replacement time select group figure l country education bag error world application bag predefine rather bag new instance grouping attribute large predict predict elementary education negative proportion elementary education assigning bag education individual education grouping performance baseline bag form bag bag bag redundant novel analysis individual label proportion affect bag proportion bag mild bag proportion b c proceed conclude denote use standard concentration inequality assumption easy verify latter negligible obviously complete contradiction put assumption equivalent prediction come label classifier misclassifie ne p several define involve differential access small database substantially satisfy differential database q coin function adjust differ eq laplace database database draw differential privacy differential privacy preserve extra conducted application mechanism privacy differential notion different proportion private mechanism serve paradigm construct overall private reduce step partial publish guarantee often proportion decision tree leaf label item access explicitly time structure different attack privacy often item feed label proportion later publish differentially way laplacian standard construct differentially structure know private algorithm point give privacy guarantee make count projection loss generality assume object instance label differentially private mechanism count sensitivity count consideration disjoint parameter one output differentially private mechanism n nx thus proportion close version explain differential scalable tool label proportion instance use differentially publish training bag task predict instance political answer fundamental match proportion analysis vc bag bag sensitive bag show mild together formal guarantee label set application proportion paradigm privacy guarantee demonstrate feasibility base real world predict census information individual proportion proportion proportion disease public predict recent setting call bag work available individual show combine capable correctly predict acquire census lead promise privacy sensitive attribute label proportion namely optimize match proportion formal mild assumption recover learn bound generalization proportion bag sensitive bag word bag possible proportion unseen bag conclusion get good proportion bag predict disease certain predict rate department mild control bag imply point good proportion concern increase aim private preserve world demonstrate census seminal work estimate label proportion conditional exponential restrictive instance conditionally label learn generate consistent label proportion show know proportion provides learn related bag boolean bag positive draw single easy sided label world arbitrary sample tool generalization good predictor bag consist attribute bag bag inside bag bag bag bag proportion proportion learner bag proportion instance unobserved th bag denote instance low prediction erm instance however label therefore try proportion define h select bag set compute framework immediate label bag proportion show sample bag proportion sensitive bag mild bag give bag proportion generalization bag possible unseen bag basically bag hypothesis smooth loss proportion term proportion measure utilize relate proportion main intuition bag instance hypothesis formally adapt class number generalization bag bag bag give appendix bag sample size grow bag bag create fortunately grow sensitive result bag proportion bag proportion predict bag proportion discuss provide insight proportion already proportion section bag bag bound inequality either instance error control bag proportion learn justify bag least summarize assume bag least correctly bag nr fraction bag learner bag extreme bag label proportion instance learn provide fail bag study insight way bag conditionally bag lot sample individual individual bag bag bag consider draw bag firstly pick note utilize bag generate proof straightforward bind close hx probability bag proportion small match monotonically generative strong result mild express consider generative assume instance assumption restrictive proportion prior approximately match adjust bias prior cdf want curve monotonically control
approach process implicitly match gradient observation requirement look remain since uncertainty know integral generally form expansion avoid explicit case process see eq xt linear gp covariance observation time boundary gp whose covariance gaussian formulae yy yy globally give cubic evaluate much small ode solver smooth approach approximation gp derivative gp interpret assume subsequent outline practical approach subsequently retain measurement form gp fx value subsequently compute continue gp right indistinguishable line standard gp retrieve integrate densely complexity gaussian much solver use solve ode solution experiment use squared covariance procedure substantially bad accuracy reason function namely similarly predict experience extend drawback technique alternative many direction solver outline three solver plot derivative stepsize plot indistinguishable small follow obtain eq simplicity deterministic ode forward future requirement ode derivative derivative requirement ode subsequently f nx repeat formally procedure differ significantly firstly ode requirement also condition complexity cubic say limit complexity approach bring complexity demonstrate despite excellent comparative computationally x n draw iteration xt perform compare solve number iteration gp calculate curve requirement ode derivative variable condition final variance term independent equivalent problem suitably constant optimisation derivative rapidly converge principle carry optimisation solution solution therefore move pass experience perform implicit plot approach window stepsize indistinguishable solution benefit conditioning derivative jacobian order derivative gp conditioning set option order novel solver ode sample correct derivative estimate gaussian ode operator direct solution xt joint define euler method approach recursion x ft k accumulate generally method weighted combination carefully define future value require algebraic red x red red x x derivative infer tx ft add observe derivative collection obtain integrate draw may curve extension version note drawback unclear make practically suggest collection ode solver implicit date carry limited believe promise exist gp approach explicit gp method analogous ode solver experience forward evaluation sophisticated implicit gp numerical van outperform explicit ode solver implicit explicit comparable implicit though require
draw independent py exponential ratio use kullback divergence drawing substitution interpret monotonic decreasing tell rgb evaluate
large arise privacy concern researcher decentralize social since design implement focused movement gradually fact date accumulate million user establish discovery constraint observe whole whole motivation service towards literature decentralize privacy preserve discovery matching problem attribute interest social activity way profile common primitive speaking hold two protocol compute either party raw scheme hardware generic construction common protocol protocol adversary security efficiency drawback social connection topology heuristic precision discover result span towards extend discovery community topology privacy preserve set party possess social translate preserve protocol circuit construction impractical world tradeoff privacy efficiency contribution propose community detection largely improve recall topology discovery decentralize transform walk preliminary result community preserve privacy extension variation end widely discovery topology graph mine term community closely community mainly scenario contrary system one decentralize execution work exchange much exchange adjacency privacy formulate omit interested reader survey survey intra dense linkage review classical centralized decentralize formulate privacy make formulate problem graph partition vertex detection try maximize minimize depend remove get overlap artificial surrogate community necessarily tractable via modularity classical truth decentralize scenario observer view whole partitioning tractable ask observer formulation observer stack community restrict overlap encodes outcome application privacy adversary passive adversary node execute protocol single adversary capture connection otherwise begin knowledge community one guess connection procee even without preserve detection even scenario researcher use tool incorporate specific heuristic perform heavy tuning amenable community base graph model encoding connection community number matrix illustrate protocol main involve truncate live initially rw record accumulate reach intersection answer intersection pairwise privacy intersection scheme reveal reveal reveal intersection size adapting follow existence primitive extra decentralize preserve truncate community enough enough intersection come proper ensure protocol adversary exclude community priori protocol limited assume adversary use average challenge omit summarize follow protocol adversary successful base example strategy information end advantage successful rate strategy problem community community edge protocol rw negative false advantage preserve privacy proper heuristic optimize repeat protocol design formulate privacy paper multi protocol truncate walk thorough protocol meet objective protocol suppose protocol exchange cell adversary know infer intra community generation different guess link measurement network size set represent node rw infer think hash define w community identity prevent adversary guess protocol version know nothing indicator weak widely variation adversary know intersection set sequence community adversary potentially exploit protocol problem preserve define security privacy preserving scheme topology new connection find community update protocol cause normal since walk preserve requirement decision minimum party propose community
cancer reach paper resp easy discrimination vs cancer dataset performance vs good evaluate discrimination systematically roughly evaluate correlation plot cancer positive normalize model also note numerically roughly report mrf discrimination cancer benchmark discrimination sketch namely discrimination involve spectra yield list site code mass spectra two respective frequency event within focus site site highest low select fit distribution average percentage correct discrimination decision set accelerate vs reach signature identify ratio table retain involve clique list statistic table margin interval asymptotic normality evaluation margin sm quality datum evaluate yield respective quantile signature discover discrimination either score score coordinate list clique involve correspond discrimination score signature clique mass spectrum activate score belong cancer otherwise promise algorithmic acquire select strongly link specific group use automate far incorporate protocol interpretable signature cancer stage tree artificial mass spectra acquire cancer patient efficient discriminate generate black box biological rigorously fit parameterized spectra dataset acquire efficient signature interpretable signature group variation observable homogeneous acquire systematically field classical spectra several hundred thousand strong peak potential spectrum binary status distribution study dependency realizations automatic spectra spectra reduce two point systematic discover explicit signature enable code peak fit achieve quality signature discovery experimental spectra acquire three stage cancer publish acquire cancer patient final computed leave one performance good performance report concrete advantage signature interpretable key ratio thank sciences university cancer thank institute spectra acquire mathematic publication none state literature mrfs configuration space result implicitly element proof proof space length parameterize vector give generate maximize fast concavity pseudo likelihood strictly function reach empirical pseudo concave surely notation resp hence quadratic vector positive since form take configuration condition binary strictly obviously reach prove configuration derivative let conditional specification random index identity index conclude normalize asymptotically positive determined equation see section outline technical concave sure vector tw formula expression sample hessian q conclude hessian large formula situation normalize variance normalize vector computable via hence hence gaussian zero covariance conclude particular asymptotically vector z conclude probability recall percentage must true hence systematically precede pf joint inequality norm use bound similarly imply nd imply resp compute correct resp achievable discrimination arbitrary small find computed formula large q explicitly elementary eq result prove eigenvalue become variable last impose eq percentile proved immediately combine immediately apply prove equation spectra acquire cancer patient technique automate discrimination stage implement new signature lead interpretable signature model homogeneous spectra parameterize markov random present detailed theoretical successful discrimination acquire cancer well cancer patient broadly technology study protein present biological identification cancer stage mass enhance mass mass intensity quantify ratio acquire acquisition modality range specialized software machine artificial forest box discrimination level group variation develop software tool spectra automate discovery signature power easily interpretable automatic combination quantify impact presence paper interpretable signature spectra co markov mrfs recall mrfs dependency interact discrimination achieved study successfully mrf discovery acquire patient spectra acquire process clinical http home cancer group spectra spectra spectra reference range z ratio newly publish cancer patient provide science mass acquire research institute usa include spectra early cancer group spectra cancer spectra call accuracy ratio signature spectra mrf implementation performance mrf benchmark discrimination describe acquire cancer cancer vs cancer processing mass spectra remove acquisition affect intensity peak could relative acquisition hardware raw processing normalization extraction baseline removal peak detection outline spectrum step peak detect peak could potentially stage peak list ratio spectra site binary index activate detect peak spectrum code vector site activate mass generate observation binary set systematically unknown identify site site call field mrf q describe system clique recall pair site clique gibbs clique naturally parameterize space denote coordinate see binary code spectra generate binary vector length cancer fit reduction achieve site specific site correlation potential clique order often still respect seek enforce moderate clique clique discovery set clique seek impose whenever parameter force precise achieve fitting classical benchmark example fit introduce play spatial intensive configuration fast site clique zero coordinate pseudo likelihood brevity restrict theoretical impose coordinate constrain specification q binary configuration coordinate eq principle seek vector estimate observe configuration denote hessian gradient pseudo hence non linear q existence follow likelihood concave strictly concave proof supplementary concavity standard stop inferior benchmark discrimination roughly around order site clique automatically select among ghz computing spectra small second group full focus gibbs asymptotic proof precise asymptotic supplementary material configuration asymptotically observation normalize error computable supplementary proof material provide tool decide coordinate approximate appendix n explicit whenever interval estimate descent implement iterate estimate benchmark accuracy coordinate display desirable acquisition phase study involve model quick show automatic discrimination main criterion goal signature enable discrimination principle quite strongly nonzero patient mass spectra simultaneously reference large cancer processing yield fitting spectra one want achieve statistically high set must select presence absence provide set spectra resp frequency ensure significant select one two number site benchmark study site easily justify site weakly small number equal focus site fix positive integer low site select binary systematically restriction generate binary datum binary site compute frequency four contingency table quantify stochastic dependency significance dependent retain event within typically quite study mass spectra acquire dataset size moderate estimate dependency achieve reasonably clique hence include clique successively inferior optimize far retain precisely clique high pair fix clique clique integer moderate integer set select site site determine clique parameterize set technique explain section error margin different force mass spectra formula partition sum heavy task approach justify law partially prefer implement function subset q robust plane separate estimation affine quickly compute discrimination discrimination repeat regression discrimination ideal unknown classical leave cross spectrum eliminate outline correctly classification evaluate range leave costly fortunately unnecessary discrimination maximization immediate access base reference involve constitute explicit indeed compute restriction site classify mass absence signature mass add present jointly sum yield actually equal classified sign performance develop margin frequency decision frequency provide rough obtain maximization provide rough good compute leave quite subset verify constraint discovery reduce hour maximize properly normalize leibl kl well know iff thus recall fx view formula expect typically statement distance account weakly normalize mrf separate dimension conjecture easy compute maximization fast discovery
aspect bound identify true describe conclude column also pm tr ix pi covariance group group overall correspond eigenvector normalization uniqueness population classification define mahalanobis project discrepancy optimal classification come distribution observation within group tn correspond show canonical express orthogonal transformation directly decomposition unique express interpretation group exist base population proposition rather penalty choose suitable function deviation respect transformation define discriminant correspond exactly common context although penalty feature eliminate canonical word vector individually element preserve imply overcome orthogonal row penalty refer overview choice fact preserve suggest sample make arbitrarily instead regularization quite encourage accord substitution preserve result convex bound eigenvalue analysis literature take otherwise equivalence method set belong view correspond choice value group problem equivalent proposal affect optimization coordinate descent interior reader overview sparsity induce choose coordinate advantage range fast convexity solution kkt row lead matrix subgradient far lead jj block ij optimization subproblem solve block algorithm proposal selection serve selection denote support aa bound v motivation rely normality normality possible supplement establish variable coincide population rule p evaluate alternative refer report concern sample structure take definite estimate dataset dataset bernoulli base scenario consider structure range error value structure setting perform well knowledge terminology programming discriminant within sample need simulation requirement affect misclassification misclassification replication report component population canonical truly perform bernoulli comparable feature rate misclassification difference small bernoulli reveal suggest significantly unclear optimal positive identity autoregressive consider group structure structure group though popular versus versus discriminant vector ambiguity consider discriminant group case find canonical nonconvex mean replication table row matrix zero truly scenario comparable feature tends select good tradeoff oracle identity autoregressive bernoulli positive autoregressive package value precision level treat canonical restrictive produce package adaptively final respective fold run detail follow method program solver much global biological genetic integrated genome environment direct analytical advance possible hundred parallel perturbation researcher platform compound study demonstrate identity investigate dr seek lead compound identify throughput response patient replicate follow patient misclassifie replication split training test report term misclassification misclassifie lead perfect significant group achieve selecting whereas less substantial replication sample far canonical use illustrate figure perfect group choose one tuning replication validation tried project though variation replication variation compare publicly analyze author patient gene free discriminant follow independent split contain sample use misclassifie number split report well select comparable misclassification group addition tractable perform feature selection canonical penalty vector possible propose canonical vector regression context propose size perform enhance interpretability addition result effectively low modification direction research extension manuscript aware consistency sparse proposal case however future appendix follow n x nx first group nx tx tc g unbalanced denote row h tx c n r r dd h r analogous population analogous define classified group last matrix eigenvector statement q similarly ab note follow definition r since group n furthermore expression far simplify remain constant rewrite eq cat aa proof directly eq constant rotation tw aa triangle constant follow c hoeffding proof corollary mapping imply gp text corner
autoencoder purpose use ht onto pool product response mapping linearity sigmoid mapping activation first reconstruct transformation encode likewise second obtain reconstruction turn shift span class component absolute quite relational look relational kind hide partial view subsequent dynamical sequence assume unit explicitly keep video motion seed able video first multidimensional dynamical subsequent view derivative high order second relational pyramid first relational analog partial derivative experiment layer support view frame video describe relational autoencoder modular order construct module relate mapping refer mapping mapping accord module transformation rotation sum response yield detector angle delta angular acceleration frame directly way reconstruction adjacent mapping see compute infer amount prediction transformation assume train minimize minimize reconstruction supervise guide mapping representation image help ht transformation violate transformation look ahead iterate ahead prediction compute compute amount relational step prediction make order feature describe follow relational describe prediction multiple ahead repeat inference prediction e one compute next low activation activation seed frame gradient ahead training dynamic video vary complexity synthetic accelerate transformation transformation whiten work descent momentum reconstruct subsequent mapping evaluate yield transformation reconstruction video transform berkeley video shift rotation set uniformly pixel divide sized bin contain filter unit choose search good performance filter unit model epoch learn momentum mapping input logistic classifier experiment set accuracy reconstruction objective generate explicit transformation content shift train training sequence sequence shift rotation image berkeley initial angular angular scalar angular sample angular angular sized accelerate velocity acceleration pixel acceleration shift mapping perform grid train rate epoch epoch layer infer first frame accuracy use mapping descriptor layer mapping mapping predictive accelerate shift concatenation mapping descriptor show bottom prediction predictive bottom predictive layer achieve significantly high predictive concatenation mapping angular base simple shift acceleration transformation improve increase concatenation relational show improve capability evolution explicit state generate three seed frame filter pair datum figure set figure introduce change author object instance instance frame video frame long train performance stop improve prediction reflect well frame datum dimensionality overfitte localize dynamic number number mapping show model capture ball prediction frame major sequence model deal range correlation deep address well input learn representation temporal evolution input aspect input allow prediction future interesting predictive analogy take frame new analogy task target may relate model datum relationship play crucial acknowledgment work support education grant google award bi feature way frame reconstruction frame previous encode transformation inherent thereby encode motion bi introduce encode frame transformation encode structure bi show natural way commonly force evolution input future achieve
anomalie nan period expert together acceptable exploring range numerous hundred indicator indicator link score easy presentation source least presence absence anomaly class desirable discriminate anomaly source term however one binary indicator also state interpretable cart hundred indicator coverage parameter expert maximize anomaly seem obvious redundancy choose feature reduction limit information operator redundancy excellent high difficult early sign tend remove sign could detail record early sign anomaly close huge datum goal validate methodology justify label methodology describe case assume therefore distribution observe model notation accord signal anomaly therefore change slope observation increase point choose uniformly th balanced corresponding anomaly anomaly htbp slow deterministic randomly period signal amplitude shorter difficult model anomaly degree distribution choose randomly anomaly type anomaly anomaly explain expert position present window test conduct shift signal occur window u shift two test variance parametric window define sample signal test detection complex binary way build classifier one indicator fraction test observation two consecutive window minus indicator consecutive change take consecutive window window recommendation original use simple average configuration lead indicator subset indicator vector signal shift classification et compose keep three divide estimate put report random acceptable see performance confirm reliable fit forest satisfactory generally difficult interpret indicator allow perform indicator rough global decision fitting argument reduce mutual base estimation forward approach indicator acceptable performance indicator take account redundancy indicator test summarize median white bag accuracy quite indicator performance expect around indicator accuracy circle subset summarize dot inside white accuracy bag estimate result general expect difficult expert particular distribution achieve performance satisfactory aggregated forest table indicator average ccccc window ks indicator correspond good classification indicator length window ks selection induce aggregation indicator understand operator methodology engine health monitoring build expert parameter hundred cover introduce turn diagnostic indicator understand decision automatic choice indicator interesting methodology decision model illustrated working methodology sound reach predictive performance selection behave instance fulfil univariate set complex anomaly extremely important note forest easy cart simple indicator majority voting probably simple indicator health firstly health monitoring involve class imbalance secondly cost asymmetric fr paris paris de paris france fr paris fr engine collect large engine help optimize cost article study build sign anomaly detection final idea generate indicator anomaly score expert scheme lead reduce indicator tune illustrate method contain sign reliable thus operational event produce jointly per hour rate take availability nearly engine monitor external monitoring among typical message status overview engine send anomaly early sign degradation anomaly automatically expert anomaly confirm recommendation send company operate consequence measurement sign degradation delay despite case inspection prevent availability avoid general build automated decision algorithm build hundred sign sign health monitoring indicator interpretability base human decision operator health detail propose methodology dedicate monitoring base acquisition equip sensor physical quantity pressure temperature mention etc engine good health engine potential detect diagnostic make diagnostic send company detect change sign overview consist recurrent obtain analyze operator methodology article kind traditional engine monitor expert survey sign drastically operational partly indeed currently design integrate nature partially via logical probabilistic monitoring help decision present health monitoring engine produce multivariate engine addition critical time turn know engine phase behavior anomaly detect anomaly major informative long indicator capture failure partially indicator approach couple experience coverage transformation monitor standard classification indicator decision decide whether anomaly engine type anomaly responsible potential describe indicator early sign anomaly display numerical real world shift slope instant roughly center expert typical situation source
wave wave intervention passive quantum upon depict system measurement update accord adopt version measurement correlate cause trick splitting causal connection distinction two distinction scheme intervention passive observation show problem completely passive scheme performing determine restrict specify draw scheme system experimental scheme passive observation quantum equivalently understand wherein restrict experiment implement circuit fig structure vary create h denote vertical measurement pair swap gate mode detect representation circuit fig take one produce preserve cb hilbert must circuit fall channel formulation proposal motivate least goal describe causal circuit formalism multi formalism formalism object describe suitably generalized understand causal quantum latter field causal present causal bipartite purely describe cause describe state bottom probabilistic cause direct give even general cause cause contribution act c pdf reconstruction scheme observation three causal show cause maps cb array represent positive red negative cb b identity reconstruct state tr passive cb tr slightly suggest experimentally quite match reveal causal finally expect find fidelity scheme appendix real hilbert measurement span output complete set allow conventional process achieve causal describe causal constitute scheme include scheme obtain implement scheme result average fidelity reconstruct map constitute description grain instance probabilistic common mixing causal show extract probe common direct passive probability common colour logarithm least narrow correct thereby causal square deviation passive unlike scheme passive map find therefore although span operator span nonetheless signature demonstrate turn one distinguish unitary pure maximally bipartite state probabilistic nature process bipartite thereby inference choice implement mixture remove aforementioned ambiguity obtain display reconstruction causal map passive average fidelity result well passive implement context alone pattern explanation cause observable measure perfectly positively correlation correlation pattern constitute universal gate three identity correlation sometimes row explain correlation signature causal discussion example make coherent channel common cause separable state direct cause mechanism implement break channel measure correlation causal case conclude causal appendix purely common cause relation scheme extensive markovian act cause system help correlation lead process several future infer observation mechanism measurement passive intervention produce embed centre nm component match nm wave place horizontal upon exact set wave maximally separate light mirror pass quantum fidelity consist h h half wave wave setting wave completely would extract desire leave gate directly swap fig distinct phase difference input implement shift implement swap probabilistic switching control three wave path implement path pick gate implement phase gate pick effect gate swap switch hz swap choose proceed mode pass another desire light send gate output gate mode detect detect ensure produce detector rate hz acknowledgment valuable research part innovation institute innovation develop project experiment numerical calculations author show completely specify describe circuit box main article limit purely cause show generic reduce bipartite process respectively value onto dt dd article transpose normalization express assume convenient essence type act maximally mixed scenario outcome measurement imply hilbert schmidt certain operator reconstruct component provide complete basis refer component wherein zero correlation system subtle express hence wherein expectation outcome subtle trace calculate schmidt inner reconstruct eq scheme reconstruct causal bipartite process case purely relation neither influence causal cause consequently consequently cause reconstruction yield eq purely cause distribution state meanwhile tr consider possible causal causal reflect underlying observation projective outcome measurement setting measurement outcome causal case relation common causal bipartite follow quantum note marginal expect produce serve measurement bipartite operator call ref emphasize causal encode infer certain sort affine ref analogue law like state semi completely composition map cast alone direct purely cause map output tr b therefore determine entirely facilitate comparison common scenario operator ref conditional trace preserving q positive transpose ultimately specifically cast operator mechanism indicate measure cause summarize statistic positive admit purely cause form admit explanation direct cause common explanation note ref cause observe passive scheme contain apply describe system interesting common cause respective two connect cause cause case set fact direct allow trivially conversely wish study restrict namely maximally eq q condition possibility set observe give rise ultimately restrict maximally mixed otherwise common cause direct cause relation marginal direct assume maximally prevent marginal bipartite maximally common cause summarize inference problem eqs possibility statistic nontrivial problem wherein signature passive quantum cause unitary common pure maximally previous causal rely cause express term state define bipartite rise cause maximally constraint maximally sufficient maximally describe unitary achieve causal case determine refers apply euler formulate representation effect sphere correspond scale maximally representation easily sphere ellipsoid code colour ellipsoid instance image anti several analytical basis encode offset centre ellipsoid channel maximally unitary channel maximally state vector index encode ellipsoid show axis length context channel ellipsoid coincide know imply channel implement pure unitary meanwhile whose ellipsoid maximally simple rotation sphere colour green sphere describe distribution improper rotation follow conditional change multiply main article ellipsoid image input denote anti unitary green identity leave maximally state mixture two cause height orthogonal realize experiment point mix channel produce point direction radius plane input lie connect centre span probability turn state effect origin angle decompose image alone rotation image lie remain meanwhile coincide offset plane along associate fig transformation therefore semi along degenerate semi axis root direction oppose ellipsoid disk dominate axis image lie plane magnitude q ellipsoid combination straightforward extract angle semi axis oppose still read normally follow scale plane similarly cause contribution ambiguity rotation take convention whether point axis ellipsoid generate possible solution rotation ambiguity implement unitary process maximally bipartite mixing angle previous show process perfectly passive probabilistic object causal without observation causal channel state separable ambiguity identify ellipsoid al separable channel break ellipsoid define fit sphere order condition mechanism channel common cause mechanism coherence conversely coherence causal purely separable partial transpose imply explanation process break rule purely cause explanation identify scheme reconstruct observe passive observation statistic statistical fluctuation causal square determine causal close passive analysis section frequency outcome value run model causal eq causal probabilistic mixture direct cause ie trace satisfie predict simplify rather appropriate normalization subsequently parameter operator unnormalize seek parametrize frequency express infinitely unique close passive common cause cause recall scheme state channel combine unnormalized operator write count passive seek minimize represent mixture pure maximally bipartite unitary aim type remove ambiguity consequently fit close realize impose maximal
instead optimization current bit depend bit well alternatively eigenvalue loose lead inferior modular formulation inference search large group block optimize block time let cost block denote optimization modular block hashing leverage similarity easily block meet modular sub modular word block modular z prove sub modularity method need variable optimize h code bit train hash bit usually one loss surrogate loss exponential adaboost boost coefficient tree train threshold minimize feature time speed summarize highly recent fast conventional implementation feature quantization largely consume linearly feature apply weight iv apply splitting summarize hash alternate iteratively code learn binary training encoding test retrieval train precision c cifar n comprehensive experiment image method retrieval specify tree depth hashing way retrieve example denote area curve dataset cifar contain scene category cifar truth tag annotation image annotate identical keyword portion allocate image training retrieval aside cifar split randomly select test query employ codebook soft thresholding patch test codebook feature code step much less perform much objective relation inference time outperform set compare different binary tree function spectral codebook retrieval fig type hash decision tree able code rbf data rbf svm training compare hash low codebook extract cifar resolution dimension dataset include hash semi hash codebook consistently feature result train dramatically increase solve eigenvalue expensive compare train vector sample large codebook order magnitude retrieval codebook feature plot retrieval c cifar cca pca cca dimension reduction codebook compare combine dimensional reduction train whole except cifar slow improve decision tree hash time performance hashing unsupervise hash lsh spherical hashing poorly preserve margin bit high feature large length linearly l increase high run bit high bit length training increase train outperform margin retrieval c train time precision map lsh outperform challenge scene codebook set bit subset training almost intractable short bit challenge dataset bit whole example weight example splitting may training due less margin usage contrast tree method involve comparison easily advantage retrieval significance many image cm van david hashing aim map original ham hash function demonstrate advantage encourage price achieve linearity suitable hashing modular hashing binary code inference solve hash decision precision especially order hash compact code hash fast search table hamming ranking code extremely data storage retrieval preserve distance hamming supervise try similarity locality hash lsh randomly hash cosine hashing learn affinity iterative approximate euclidean hamming space hashing manifold take intrinsic supervise hashing preserve take hashing increasingly hash kernel step embedding world usually hash despite hash interest may demonstrate dimensional example codebook remarkable thousand exploit advance feature desirable able deal efficiently sophisticated high hashing leverage efficiently incorporate hash however could feature supervise contribution ensemble decision hash hash number high thousand map general hashing decision efficiently binary code inference decision binary modular formulation significantly outperform retrieval high order training employ function inferior loose dimensionality decision propose method inference
test stochastic block network vertex define bethe block result negative sign curve recursion isolate bar black axis leave bethe hessian eigenvalue decay towards axis informative reach decay top finally information interestingly decay bottom informative eigenvalue choice bethe hessian generate informative one informative eigenvalue straightforward eigenvector backtrack must relevant inside number argue regularizer backtrack claim numerically compute building efficiently solve bethe straightforwardly carry vertex fact bethe certain weight unity reduce argument generalize immediately relationship backtrack non backtrack ise spin connection along spectrum bethe hessian backtracking operator index remarkable efficiency backtrack sbm uninformative disk lie outside real precisely real community eigenvalue bethe hessian notice eigenvalue correspond bethe definite prove e circle theorem eigenvalue course phenomenon takes translate negative eigenvalue adopt q outside circle radius come close eigenvalue spectrum stress backtrack matrix correlation small setting refine guess world choice informative eigenvalue infer membership standard bethe theoretical ise distribution parameter control strength analogous statistical physics approach machine bethe moment belief restrict bethe goal justify independently uninformative eigenvalue eigenvalue bethe spectral delta peak remove vertex degree belief propagation recursion cavity formula marginalization graph solve lead density locally limit dynamic iterate excess pool updating justify analytically eigenvalue bethe linearly around exist introduce equation rewrite jacobian backtracking operator identity square contain derivative invertible around implicit function contain exist around show jacobian eigenvalue modulus strictly long continuity respect exist recursion real prove reach far regularizer fail cluster region backtrack bethe propagation bethe systematically backtrack operator illustrate spectral propagation optimal measured overlap true maximize bethe systematically complicate run community bethe count number eigenvalue feed real graph illustrate show block bethe identify several detection large well backtracking consider identifiable particular eigenvalue backtrack case word matlab reproduce result bethe synthetic bethe bethe gave combine real non backtrack oracle answer tractable parametric perform optimally reader file matlab impact wide clustering expect impact spectral value similarity oppose backtrack promise arise use generalized maximize e modularity else eigenvalue carefully choose could solution give relaxation np discussion european grant agreement grant triangle universit et paris sup paris france paris de france approach node low adjacency recently argue symmetric operator detect simpler know bethe hessian combine performance backtrack theoretical limit computational advantage symmetric cluster community range biology benchmark sbm create matrix infer concentrate algorithmic case concentrate equally size refer connect group important conjecture rigorously prove also detect community meet perform optimally stochastic block cluster transition far detect transition pass propagation fed well limitation spectral adjacency matrix remarkably version cluster suboptimal detect backtracking
accuracy bp vary bp theoretically capture solution principle design formulation express rely linear corresponding indicator totally underlie constraint matrix combinatorial use technique acknowledgement european grant tu group otherwise tu text convex feasible start leave look sum simply child leave ball tu text convex envelope envelope use linearization trick tu let envelope eq definition structure sparsity tu structure relaxation polynomial programming framework sparsity introduce argument important reduce familiar p pn typical absence impossible reliably learn nontrivial must exploit knowledge impose structure sparse theoretical broad generalization rely convex establish sample algorithmic obtain describe fortunately convex encode within effort description inherently restriction coefficient structured review find tractable surrogate capture combinatorial end combinatorial issue arise quite simple summarize encode identify whether totally tu verify investigate notion derive combinatorial convex relaxation illustrate tu description popular norm hierarchical show tu description support relaxation fact tu lemma specific induce induce complement submodular modeling go beyond tight norm novel theoretical group group sparse root lead description provably totally exclusive lasso overlap scalar letter letter entry yx I pp identity context introduce definition sequel submodular f g totally tu every square proof omit see supplementary material encoding combinatorial support hence task find surrogate determining envelope low condition computation f pt p ball completeness conjugate otherwise last tractable noting without necessarily lemma restrict sequel unless general conjugate hard numerically generative submodular quite popular know checking approximate light three allow tractable convex relaxation regularizer fact equivalent submodular minimization minimum empirically run recent solve may non zero magnitude make sense combination continuous envelope dual p seek combinatorial approach satisfy sparsity encourage simple via inequality structure tu admit relaxation tu us simple template tu tu penalty support feasible modeling arbitrary tu extension envelope tu envelope tu tu lp still despite non note simplicity need tu interaction tu penalty weak hold penalty definition besides tu capture study tu relaxation structured naturally therein group together collection support figure represent group set iff graph iff pe intersection two iff structure group cyclic black fill thick sep black thick pt rectangle draw white thick inner sep transform auto label label leave label v v v g v shape fill label g node node node group typically seek express non decrease submodular sum penalty express ig sum weight force group select tu entry zeros condition envelope intersection penalty induce application union seek minimal bipartite representation correspond minimum cover problem propose potential tu penalty admit convex envelope worth note latent homogeneous envelope g lead tu relaxation structure matrix acyclic shown induce acyclic tight far induce level variable enforce two lead non surrogate give tight group tu penalty surrogate q x propose group surrogate group otherwise provide formulation enforce sparsity signal signal bipartite actual minimal f small cover seek signal tu tu group lead tu envelope eq result convex program lasso case material hierarchical organize rooted subtree model wavelet tu model circle draw inner sep inner sep pt shape mm child scale mm mm mm child selection norm hierarchical model incidence iff tu give group hierarchical norm envelope far encourage implicit within speak opposite within sparse opposite form model exclusive prove tu structure tu partition exclusive lasso actually relax tu necessarily group tu matrix tu penalty tu acyclic tu trivially tu dimensional see period two eq tu exactly exclusive constraint exclusive actually version relax desirable
miss occur arise perform principal retrieve spectra spanning hereafter l principal well describe remain hereafter subset retrieve maximized computing component first surprising iteration since write dominant dominant eigenvalue uniqueness depend obey dominant eigenvalue eigenvector failure check satisfactory iteration similar retrieval ghz present principal idea intuitive flexible generalization algorithm exist give behaviour usage assess already author science policy office national foundation office science web site http www iii laboratory group de berkeley national laboratory physics new university york university de universit b li straightforward analysis weighted method retrieve principal amongst meaningful weighted principal retrieve component illustrate usefulness digital spectra measure shorter benefit fast component pca design huge datum set principal arithmetic mean corresponding variance coefficient coefficient principal interested reader deeply wide variety digital spectra point describe hereafter spectra cover problem require pca decomposition limit tool assess long nevertheless limitation weight inherent classical pca difference variance come come limitation focus problem miss case factorize deal none account orthogonal optimize describe explanation give maximization include observation well individually datum come fact finding well individually take em order current alternative algorithm simulate extension use bold I th denote element product hadamard contain retrieve reference matrix I within ii potentially sense mathematically minimize regard generality like retrieve component number ie decomposition orthogonal variance minimize covariance differently explain order clarity pca become solely drop orthogonal transformation maximize vector purpose choose practically already deal pca component equivalent minimize cf rely latent hide iterative procedure optimize fast pca fulfil follow expect pca algorithm converge still relative change give design general approach dimensionality spectra specifically attempt equation part smoothing strength non negativity constraint reflect interpretation spectra drop negativity constraint restrict concern optimize comparison deal ignore constitute major drawback implementation result go approximation solution solved step clarity matrix orthogonal straightforward gram schmidt retrieve secondly principal eq state begin regard retrieval matrix optimize equation consider hold equivalent preferred step retrieval decompose observation combination principal q moreover single iteration regard huge lead good insight component fit individually hypothesis reasonable since component account result implementation one apart order cross suppose retrieve principal q retrieval whose last retrieve orthogonal nevertheless manually check finally end though find orthogonal decomposition suitable set none component reconstruct idea weighted result principal variance relevant identifying necessarily variance variance weight convention straightforwardly definition write fulfil accordingly constitute implementation observation variance fig unable suppose purpose observation point explain maximize variable consequently zero retrieve dominant covariance eigenvalue dominant q maximize sake clarity equation reference may exhaustive proof hereafter implementation fast iteration recognize nonzero direction dominant eigenvector unity round inherent number eigenvector associate algorithm subtract find section method nevertheless design algorithm scope fact understand associate normalized unit converge regard point minimize principal retrieve real distribute happen correspond small observation problematic principal regularization factor expression equation result weighted allow strength regularization go behaviour rare conversely assess regard namely come fact fairly competitive test observational consist basis take shift period schmidt evenly interval provide amplitude uniformly value discard contiguous observation latter assess performance weight retrieval simulate perform observation realization various retrieve five principal compute follow follow chi quality use completeness decision estimate really strongly depend section deal quickly study account preliminary need describe set twice number give refinement miss top middle average fit various fairly algorithm somewhat high dispersion set show mean increase miss datum average difference presence miss still parameter simulation moderate increase miss average first latter dominate show reach unable converge maximum detail average clarity plot remove solve explain already suggest choose retrieve component noisy maximize variance release either uncertainty determination insufficient spectra frame spectra variance inspection show fit variance spectra mainly attribute spectra thought cause signal retrieve previously algorithm assess various assess variance amongst amount presence large perform retrieve keep assess quality fig ccc
imply equality family follow dot express definite assumption symmetric definite reproduce hilbert norm pe dx dx imply aforementioned fact statement kernel condition compact suppose result respect exist sufficiently detail requirement theorem inequality kullback hellinger distances bound positive bound hellinger depend guarantee moreover part note imply also enough full correspond bind admit disjoint credible set source base small simple subset alternative view approximation random measure approximation median combine small size credible performance rate cover hellinger exist particular typical theorem yield moreover hellinger bound k condition hold satisfied recall geometric f k let hold discuss method respect misspecification distribution primary probably p pp conjugate course sensitive achieve concentration k ks nz sn moreover besides proceed mean conjugate l correspond chebyshev concentration event subset result comparison way particular posterior exclude simulation evidence favor thresholding rate acceleration viewpoint modern see discrete w tb w ji two previously comment choice theoretical guarantee many interval acceptable size g available yield suggest pick among posterior computational achieve run distribution goal compare run parallel tm approximate degree tm n refine way advanced optimization reference two improvement achieve magnitude outli posterior simple univariate gaussian outli linearly increase index replication flat jeffreys prior q I repeat replication datum representative posterior fix compare consensus posterior credible calculate replication empirical contrary consensus overall posterior across length identical length wide interval posterior wide absence outlier ht value hereafter mean robustness grid gp standard convention standard obtain equally grid location algorithm draw across case subset correspond employ posterior median location represent band correspond quantile posterior replication extremely sensitive shift truth outlier coverage band location gp produce true unstable instability matrix avoid work approximation massive subset posterior subset gp computationally stable contrary great subset gp computationally carefully depend computational regression promise massive datum general social survey capital consist contamination small survey answer question incorrectly use process dp multinomial probabilistic response detailed description generative include appendix divide sampler account remove atom weight associate posterior posterior mode case tend subset accommodate density density estimator around mode slowly however heuristic approach explain remark follow space occur imply cardinality part proceed event occur triangle inequality part complement wasserstein take hellinger hellinger follow eq every satisfying exist note chebyshev finally recall note q let proceed support conclude numerator hence large put bound numerator denominator together c generative generate stick break construction represent response latent latent generate stick break hyperparameter shape fix latent sampler obtain analytic conjecture ex author support grant es institute environmental health sciences support grant dms provably computational technique evaluate measure propose measure equip distance quickly efficiently practice evidence improvement datum pose general challenge cluster storage contaminate outlier identify remove place statistical literature progress point method understand main propose provably scalable big allowing implement parallel split part implement markov chain carlo another draw probability properly section overview exist literature explain goal aim introduce main proof remark constitute study robustness indicate research robust study sensitivity uncertain uncertain typically heavy tailed likelihood usual assumption e g contamination large place outlier removal also robustness misspecification progress scalable design distribute subset machine local communication machine optimization approach distribute limitation dominate subset posterior sequel knowledge rigorously justify combine posterior major evaluate return likelihood master appropriately combine conditional repeat among sa successively learn batch sa hamiltonian langevin dynamic method parameter learn variational posterior see excellent well substantially uncertainty fall avoid communication independent chain posterior combine variety simply draw alternative density call limitation applicability justification model unlike method provably inspire median technique apply framework key fact throughout distance pp pa trace totally space packing number structure ball hilbert k form see application median discrete characteristic sign borel integrable definite transform characteristic compactly support almost always question favorable upper corollary wasserstein well hellinger absolutely lebesgue density explain theoretical index absolutely hellinger metric eq assume let value vector define inference borel algebra observation borel assumption towards mean surely address happen arbitrary usual concentrate corruption description propose construct distribution integer divide typical prior subset depend disjoint median evaluate geometric identically define fix choose grid mesh principle prior discuss possibility practical application weight properly rescale overcome approximation subsample unstable metric measure improve set coverage often numerical note
apply meet iterate one end process construct formulae smooth es es imply guarantee one exponentially satisfy gain tend fast formula quickly standard similar back increase circumstance norm view asymptotic formula sense incorporate thus hereafter idea final normal ensemble square produce associated root exactly normal square root filter kalman jacobian matrix eq gain computationally expensive take purpose residual implementation evaluation reduce computational cost element may magnitude less evaluate preferable improve take hybrid enkf give instant minimizing ensemble way enkf enkf perturb assimilation ensemble smooth ensemble complex user therefore analytic jacobian end stochastic example relatively real accurate expensive correspond inverse calculate factor consider jacobian relatively relatively requirement may deterministic inverse rule residual let inverting factor spirit one experiment adopt method adopt otherwise transition trajectory discard rest respectively truth assimilation measure odd element fx observation error assimilation observation integration normal initial background ensemble way previously whose residual norm reach may experiment extra investigate localization experience suggest default experimental beginning member however presence process equip covariance localization norm descent purpose conduct investigate relative normal randomize lm iteration fix constant aim cost essentially choose minimize instead necessity circumstance conduct comparison worth lm lm normally construct enkf though toward reduce illustrated test cubic section vary factor width ends suggest enkf nonlinear report norm panel lm cubic figure background call hereafter solid line together dash line residual background assimilation indistinguishable opinion lm iterated average member criterion update estimate continue eventually negligible background almost reduction apply odd assimilation error run ensemble member step nonlinear apply error observation two observed scenario odd half four ensemble ensemble observation variance error take even increment average size frequency panel ensemble rmse monotonically increase relatively appear increase large tend tendency may b observation nonlinear observation figure case clear rmse exhibit shape achieve low possibly observation observation c observation panel suggest nonlinearity hand panel b half observation panel nonlinear incoming pose consequently freedom constructing tend state toward scenario pose solution well observation ill pose small low show panel instance linear covariance background introduce localization circumstance rmse however certain value half localization even guarantee residual effort impact localization investigate also assimilation window consist step nonlinear investigate ensemble setting show figs variance panel size frequency case indicate variance relatively seem opposite variance error affect conjecture occur combination variance background ensemble step variance necessarily effect rmse decrease examine mis term experiment term true variance observation test variance report function sensitive mis specification variance possibly cubic may increase linear estimation mis variance might achieve certain find certain error improve assimilation overall uncertainty work concept assimilation derive handle observation residual implement iterative ensemble numerical handling achieve reasonable term mean assimilation realistic conduct resource future iterative filter thank anonymous constructive comment suggestion project realistic financial eps mm time residual rmse lm cubic operator upper background solid cubic panel assimilation solid reduction example reduce toward upper final analysis assimilation window exponential function time step one half scenario p panel nonlinear variance calculate panel rmse legend panel assimilation mm mean observation panel magnitude iteration scale horizontal observation visualization scale international gate kalman enkf assimilation circumstance enkf author improve stability enkf adjust able accuracy extend nonlinear operator modification iterative suitable assimilation illustrate iterative filter enkf nonlinear operator various kalman filter enkf variant implementation kalman finite ability scale assimilation problem enkf receive assimilation influence enkf enkf relatively estimation covariance adopt auxiliary call localization improve enkf covariance increase increase robustness enkf various covariance hand localization localization suffer certain circumstance especially error assimilation assimilation residual residual norm one show circumstance assimilation equip stable well operator suitable nonlinearity observation operator main gap adopt filter assimilation cycle use residual suitable convenience refer linear nonlinear cause confusion organize introduce residual observation section aforementioned method extend modify conduct stability conclude work observation state instant instant project space respectively zero discussion drop end suppose observation certain instant difference space set residual find euclidean hereafter reader behind prevent combine combine call inversion original state work residual proper explicitly extended reader formulae ensemble consider slight generalization kalman positive scalar analysis ensemble objective general suffice conventional gain resemble kalman gain enkf analogous multiplicative covariance residual satisfie denote satisfie ease inverse transpose give follow b number directly alternative obtain computational cost omit brevity reader refer restriction need relate certain discuss kalman kalman covariance kalman variant variant put emphasis robustness estimation kalman reader therein nonlinear complicate since explicit analysis residual may long operator generally satisfy large continuity state assimilation relatively readily iterative framework aim long enough low process solution least square problem aim solve remark residual intuitive term hereafter minimize pose inverse uniqueness inverse theory assimilation introduce e b aforementioned avoid presence derive behind bayesian state observation gaussian interpretation situation may dynamical reality often evaluate e state scale fix estimate purpose value combine iterative enkf construct enkf optimize work state assimilation
good standard get less grows adapt life bias estimation really assumption static need window static variance window big bias contrary lead bootstrap simplify run experiment static yahoo take portion experiment ground obviously news compute average ucb evaluate ucb ground truth use acceleration interpretation htbp close please tend batch enough recommendation evaluation realistic focus offline estimate reasonable prove asymptotically convergence counter intuitive introduce fast static accurate publicly make yahoo server present acceleration highlight acceleration issue evaluation desirable property context risk estimation bandwidth extensively study kde safe controller company recommendation behave certain collect tight replace ascent policy notation apart notation modification experiment non contextual exhibit importance sake algorithm history triplet efficiency learn set triplet hx ax rt pp recommendation news recommendation recommend come challenge recommender set seem solution purpose evaluation rs live avoid offline evaluation option thus trust method evaluating literature nonetheless satisfactory mainly fraction limitation bootstrappe estimation latter risk online proof superiority compare various name become common activity web think movie recommendation netflix amazon news job recommendation application yet profile order attractive serve item recommendation piece software rs item interaction click recommendation train predictor click user past recommendation implicit paper recommendation recommendation recommendation continuously replace new characteristic sometimes dramatically web news find yahoo tv example item recommend item movie contextual nonetheless recommendation recommendation predict offline argue idea require continuous effort item picture static greatly political movie star rs portion engineering effort able offline rs recommendation computed accepted community nevertheless give dynamic offline fairly yet use argue web issue evaluate explain previously bootstrappe theory empirical clearly term bootstrapping allow estimate sense especially evaluation decide algorithm also synthetic detail publicly motivated order deal dynamic precisely contextual bandit framework recommendation problem also arm variation thompson variation contextual tuple reward pair reward user action recommendation game round player choose round game reward reveal whose score player important game reward action offline evaluation typical learn try exploit therefore player face exploration exploitation either uncertain improve explore perform believe armed bandit study ucb deal upper bind contextual problem study additional without although basically normality action estimate linear bind contextual triplet choose action say maximize reward convenience per click output simplify systematically drop bandit live fact period understand impossible concern region different potentially different thing evolve equivalent playing go reality likely acceleration contextual look lot challenge evaluate yahoo news chapter work contextual record acquire yet deal problem evaluate dynamic model line understand protocol suffer stem bootstrappe thus dataset draw bootstrap underlie yield converge bias concentration speed recall bias want map history policy map action appear efficient implementation would also contextual triplet kx bx bt b protocol bootstrapping bootstrap dataset subsampling allow classical purely policy obvious datum formal reflect together interaction estimator estimate deviation expand last contain expectation expand dataset prevent amount context neural avoid overfitte practice network online bootstrapping smooth smoothed bootstrap kde sampling kde bandwidth get smoothed bootstrap kde core loop henceforth analysis evaluation denote recommendation algorithm generate recommend asymptotic series independent moment explain admit realization actually adaptation bootstrap convergence result introduce produce dataset evaluate expectation mean estimator convergence allow evaluated sketch respect consist guarantee gap subsampling order index estimate dataset policy chernoff denote inequality obtain probability admit expansion recall q admit thus
model share representation consider work criterion salient model tie salient share component framework improve propose e feature salient use length document message dirichlet optimize work concept model sparsity proportion specific one two subset salient follow occurrence component machine specialize derive na penalty term interpretable effective paper organize lda topic parsimonious section derive bic joint bic objective corpora image dataset conclude remarks corpus dictionary unique word topic follow document document indicate whether specific word originally extract generative process topic topic variable algorithm briefly review variational log family change dependency dirichlet value document determine leibl kl low document log next variational parameter probability meet proportion control corpus optimize proportion document lda model document essentially estimate topic hard assign maximum document lda develop new fundamentally differ treat deterministic rather maximum bayesian hyperparameter hyperparameter parameter topic document topic develop treat model deterministic bayesian set alternative approximate approach overfitte corpus select function pmf pmf indicate present possess topic generate aforementioned together specify structure full model constitute double product switch th word constrain pmf nu likewise topic satisfy pmf assume derivation sequel seek bic moreover dependence bic respect equivalent accordingly parameter maximization em element treat incomplete increase iteration consist complete step hide current parameter origin add normalization estimate maximization topic proportion derivative satisfy normalization constraint respect achieve multipli multiply side sum word e initial estimate assess iteration next meet estimate share principle take respect optimize estimate initialization hold show quite work count jointly optimize bic fix bic alternate locally learn computed sizes taylor integration derivative thus taylor negative taylor mean evaluate topic document specific share fix irrespective approximate usually uninformative description treat deterministic minimize tradeoff negative proportion number q hessian information approach bic block q become penaltie different type log derivation parameter interpretation equation particular sum estimate total lead generalization cost penalty configuration topic principle invoke uniformity across topic generally topic shannon accordingly mn topic across define configuration select uniform algorithm jointly determine switch global bic generalize step guarantee minimum note applicable bic apply monotonically likelihood toward substitute log complete taking term bic complete datum datum incomplete term iteratively step iterate expect datum form incomplete datum term complete fix term bic dependence minimization complete log step via visit current change respect bic ensure descent bic repeat occur predefine reduce current plausible remove mass repeatedly reach corpus respect class include whose probability compare implementation com approximation log fitness divide hold keep part disjoint expect proportion topic proportion correspond also compare extract topic exhibit coherent covering concept agreement expert occurrence probable contain similarly document percent specific topic top coherence lda coherence initialization validation topic accuracy create document topic document mesh document label document word removal four model topic massive show bic curve bic std show lda train complexity trade label dataset distribution count ground proportion respectively label label class proportion assign label high criterion discover divide ground number correctly label assign criterion different threshold report area auc parameter topic document unsupervise lda auc lda well entire auc curve coherence lda compare occur least topic average specific total topic word occur document corpus suggest well class also number average indicate great overlap topic salient topic std c topic topic std document standard removal corpus initialize topic two show lda std lda likelihood parsimonious performance hold topic single label compute label document consistency fig good class compare topic coherence lda plot lda table small topic although add occur corpus label unique specific share table separately share word topic write first topic cm model patient bank health gold year share year plan patient treatment bank gold lda come law ga price share stock month thing price disk window hard record write post program offer unit problem file record compare include unique removal elimination hold curve hold set compare consistency classification across minimum bic curve class order high coherence report measure lda small compare top sample topic lda report section report comparison class unique extract sift slide grid collection give learn center sift descriptor nearest cluster perform mean pruning cluster represent length text corpus accordingly equally model reduce topic lda std hold lda order classification corpora fig consistency consistency lda sparsity proportion however word probability fashion four hold order achieve hold bic minimum lda increase much topic moreover average per label per document curve fig possible achieve class recognize unsupervise salient choose huge lda label topic word interpretability hold class consistency model process core processor execution smaller comparable lda typical document time lda running time lda step em improve nevertheless run nsf consist parsimonious salient give word derive bic objective penalization jointly topic performance log agreement ground email department university university pa parsimonious topic corpus model even salient explain universal share document document bayesian goodness interestingly identify size minimize specific text corpora test ground design reduce number free covariance cluster grow number parameter across parsimonious less overfitting rich various document bag introduce topic thousand word probability topic expect topic model dirichlet vocabulary lda mixture corpus proportion model topic lda intuitively many model universal principle every proportion cover allow nonzero proportion document method identify word may
sequence stack v instead comprised frame frame vector notation express multiple proceed error assume real subsection experiment relative error second experiment compare propose algorithm true sparsity algorithm sparsity change estimate sparsity give rough initial sparsity level five evaluation sparse index nonzero gap simulation carry xlabel nonzero row ylabel style draw fill legend align leave color solid color solid width row crcr color width pt crcr color solid row crcr show sparsity root square fewer non zero estimate twice true estimate support height xlabel iteration ylabel draw black fill legend cell line width blue width pt crcr determine total time configuration ensure carry number measurement experimental threshold mean less term varied actual support plus level experiment repeat time experiment measurement vector measurement vector achieve row zero increase decrease row vector row axis xlabel ylabel recovery rate fill legend color mark option sep crcr color solid mark table crcr color solid line width x option sep crcr green solid width option row sep crcr different calculate twice success experimental repeat three multiple recovery rate multiple determine rate height xlabel ylabel legend fill white align red width pt mark option crcr blue solid mark solid crcr red width mark mark mark option solid row solid mark mark option solid sep crcr mark mark row sep crcr blue solid pt mark mark row crcr choose recognition comprise audio people sentence person contain head audio sequence contain view short sentence office video camera video image resolution contain face loo evaluation person compute recognition lower large cccc norm heuristic experimentally sufficiently actual experimentally find value twice sparsity sufficiently simulation multiplication converge art use propose technique matlab website email www ac iteratively randomized exponentially weight solve measurement measurement solve common modify model face recognition project gradient synthetic confirm project system speed application various area process art row randomize reconstruction band prove converge algorithm solve equation term square solution interpret I observation explanatory useful ideally know observation overcome solution lasso try sum error solution solution well greedy approach outcome address algorithm solution similar experimentally fast almost equal multiplication imaging solution e measurement stack column sparse requirement sparse multiple measurement propose modification problem propose algorithm recognition video compare sequential almost sure estimate max j cx I ts element show support sparsity initial index heuristic follow optimization follow call number multiple measurement decompose single problem initialize row large choose support vector index lx x change
bandit pre process collaborative completion solve relate observe mixture learn sample produce set reveal rating recommendation reveal two set rather item classify efficiently support nsf grant office award support fellowship derivation clear context indexing writing reproduce ease presentation lem en step appear begin ensure user user chernoff explore user bad explore thus jointly item exploration z z iv change constant decay potentially assume bad event lemma occur hold high suppose item rate least user neighbor lemma verify bad neighbor enough jointly explore rate item pair user proof four less bad happen tell provide suffice require lemma tell neighbor suffice since satisfy inequality preliminary upper probability user suppose rate item jointly rate user neighbor vi yield guarantee inequality subset cardinality shorthand user suffice good neighbor arrive rate neighbor bound item note exploit n finish show comparable recommendation popularity amongst friend dm friend popular rating user friend dm beyond preprocesse compute online dm item simulate recommendation system movie rating netflix movie rate collaborative real recommendation rating simulate item address issue dense vs top rate receive rating dense rating initial reasonably explain source movie rating effectively user structure user item could reasonably movie experimental rate star star less look subset rate corresponding simulation upon mark thus item long recommend netflix dataset top result nonzero netflix nonzero item average reward recommend item reach I movie movie simulate recommendation item feature vector dataset reveal provide thresholded rating thresholded rating dm estimate datum next user rank number movie rating simplicity search choose set achieve high reward netflix dm wide dm expect roughly around time mostly item fact mit despite recommendation little work especially recommend address introduce recommendation cast item recommendation learning analyze cosine user either common probability item user distinction bandit item goal maximize recommend time establishe collaborative filter know exploitation step type step explore explore user recommendation vast tailor prominent amazon netflix movie collaborative decade news amazon recommendation netflix win song million song dataset challenge recommend recommend already movie movie separate case item recommend user good recommend different item success development justify effectiveness gap paper online bandit cluster bandit impose user cosine similarity collaborative filter key inclusion two type exploration space different type problem setup near nearly logarithmic exploration recommendation system collaborative give guarantee section overview consider item recommend simplicity immediately reward system step recommend formally item recommend time rating indicate rating yet objective q item item maximize clearly user recommend focus maximize aim recommend random whether maximize former user music prefer music like user low find diversity experience merely item maximize equivalent challenge preference item preference user rating broadly inference possible preference relate paper preference type user identical item preference type heterogeneity ease exposition assume belong user user correspond latent user movie model clustering relate version bandit armed infinite solution armed bandit determine keep apply cluster seek capture bandit available adversarial combine collaborative aspect bandit dynamic availability impose strict greedy bandit explore item far vote greedy exploitation use similarity exploration item ask rating item randomly choose let fill user rate recommend item rate recommend user rate maximize threshold either ask explore recommend user choose describe exploration space decay exploration decay user score indicate give yet reveal restrict jointly precisely cosine cosine overlap support user neighbor cosine propose collaborative respect state reasonable necessary condition establish item noise user preference u uv item classify incoherence cosine example suggest incoherence reasonable allow rather generality divide recommendation independently pool performance pre input define latent proportion recommend initial step algorithm recommendation oracle item recommend recommender would item meanwhile give period scale step simple incoherence consider user rating type produce probability random variable probability inner product rademacher standard incoherence previous choosing source vi vi scale suffice example user assumption event proof focus neighborhood event neighborhood event enable argue initial rating bad part exploration sufficient neighborhood event hold user think enough user type decay thought yet correctly neighborhood accurately item user hold exponentially decay term thought classify last two decay thought cost explore proof combine appropriate constraint user lemma specify user combine lemma corollary lemma generality meet bind ask great depend statement detail appendix simulate online system rating netflix rating rating consider recommendation reveal rating rate issue simply item rating latter top vs item user rate item receive rating dense rest rating rating interactive online beyond reasonable dense look behavior across dense top movie
know cauchy conclude definition institute ac optimisation gain popularity parameter machine model hyper optimisation reasonable advanced fail optima surprisingly little bayesian optimisation apply acquisition hyper optimisation become development machine recent medium interactive environmental monitoring combinatorial optimisation automatic one quantify uncertainty illustrate parameter fall range integrate use monte carlo advantages sophisticated treatment knowledge introduce estimate hyper symmetric goal unknown exploitation exploration process optimisation natural function mathematical function evaluate massive bayesian whereby introduce encode smoothness rule derive carry location process flexible place reader detail process covariance input jointly eq convenience type kernel mat ern theoretical general type bernoulli presentation focus gaussian noise refer reader introduction evaluation point marginally mechanism update datum turn crucial must computable optimize every exploitation exploration acquisition ei default popular optimisation package ei write closed form density case improvement member member cause stochastic mean seem alternative choice ei scale enable gaussian distribution despite statement discuss must smooth assume reproduce intuitive review rkh proof rkh property f rkhs theorem consider inner non positive clearly f k cauchy cauchy precede rkh converse theorem condition eigenvector eigenfunction eigenvalue expansion rkh therefore rkhs suit present consideration proof appendix material discuss could restrict kernel condition without regret obey analogous convergence ucb agnostic ucb detail rate convex kernel power decay mat kernel dd dd proof idea sketch consider instantaneous challenge gp quantify way cauchy dedicate inequality separate reproduce hilbert concentration combine aforementioned challenge relate ei quantitie easy analyse via build bound turn instantaneous regret variance sum subsequently regret maximal say gp accommodate
optimisation problem good problem use discussion approach close form eq cross avoid either inverse treat unknown qr speed become multiplication discuss enable practice avoid calculation principle matlab qr inverse pseudo inverse inverse package exploit cpu core modern inverse significantly magnitude hide become multiplication require time acceleration describe incremental streaming offer insight offer output large potentially training double gb ram typically modern pc problematic identify follow key activation sum activation similarly simplify key introduce column write way keep memory solution ram expand runtime rather face memory limitation batch subset implement iterative output describe typically uniform binary small make possible unless typically product albeit dot zero row occur orthogonal aim weight match statistic ideally datum could sense focus primarily selection several introduce layer bias dot product weight publish al call constrain third neuron rectangular visual call rf although machine aim limit visual convolutional rf superior pass rf considered follow backpropagation adjust weight simultaneously train weight maintain backpropagation backpropagation follow repeat iteratively convergence rf mnist ht symbol vector symbol multiply vector weight difference field rf neuron random rectangular field image motivate backpropagation operate weight sum weight define term argue reason strength bias conventional backpropagation mm normalize subtract dimension divide deviation layer neuron block train sample class size block random sign multiply transpose product block normalize row unity input vector sample sample different space addition propose overlap shorter near sample largely unnecessary implement elimination select mm distinct difference weight difference sum randomly normalize row input weight difference select solve blind percentage advantageous storage rf resemble tuned preferred region tend contiguous visual frequency aspect biological image create rectangular influence generating start input integer coordinate mask mask small discard repeat two zero vector correspond matrix hadamard term multiplication weight unity find beneficial exclude mask mnist database ensure exclude first small note convolutional beneficial diversity layer pixel towards class sparse provide unit specific enhanced performance combine shaped weight either rf rf follow obtain weight follow hadamard normalize input unity length rf bias difference rather weight use rf c mnist benchmark combination combine output albeit neuron ten output ten first label quick hidden neuron note consists therefore combine multiplication increase network combine effectively middle layer development report learn unsupervised train error solve capacity weight capacity able suggest capacity however return layer specific backpropagation tune hide introduce possibility well understand mode backpropagation neuron solve weight equation backpropagation mm whole indicate matrix weight continue desire illustrate solution test maintain six method describe row input unity seven randomness inherent input train hide plot marker rate figure actual training result occur rate training improvement fit verify cross first classify point point training train layer marker actual combination three part total therefore rf outperform total number unit rf produce second illustrate hide unit overfitte error continue note train seven method iteration backpropagation using infer still relatively give converge hand improvement surprising backpropagation backpropagation iteration backpropagation describe mnist achieve backpropagation hide rf cp former backpropagation describe unit time report slow shaped weight outcome input backpropagation outperform backpropagation time percentage apply mnist backpropagation marker error improve method little impact runtime matlab ghz intel core os gb ram plot exclude load memory file matlab exploit four cpu core negligible consume formation solution square large none depend conclusion runtime individually minute runtime mnist comparison backpropagation minute report second unit second hide rate benefit train various size train testing exclude load mnist file backpropagation apply scale linearly backpropagation trace iteration classify handwritten digit improve rate preprocesse apply affine elastic sensible way datum comparison problem nevertheless train able rotation scale elastic date training increase runtime reason point large significantly multiplication systematically mnist enhance expense neural network benchmark network define hidden simply find one combine accuracy comparable publish without augmentation preprocesse denoise network effort belief network rf input input sparse close weight implementation part pc require little highlight avoid calculation moreover possible set iteratively compute streaming application every inversion principle hide engineering framework utilize large model potentially boost course hard argue entirely mnist cifar accuracy cnns
empirical different order suboptimal satisfy slowly propose implement specific implicitly guarantee reject graph g center matrix xx update xy lb b lb l ji l j j qp l sf lb run purpose get convergent instead terminate usually network modern come make fortunately structure early complex individually idea appear concentration decomposable refer dynamical reliably capture topology see whole getting detect exact block contamination possible treat identify robust refer effective rely estimate may identification different purely estimate perfectly decomposable noisy datum tend remove deal robust decomposable write conduct fine fashion screening learn smoothly drop case reveal alternatively enforce sparsity set take stage constraint drop instance screen reduce fortunately fall framework f show search ny popular graphical similar screening handle coordinate worst come stage complete scalable lie fine implement matrix separate package topology consist example sized measure ij ij ij ij accuracy model I xx matrix scheme I multimodal run comparison whole screening parameter throughout spectral existence minimize validation evaluate screen tune quantile show I I infeasible suffer simplify fail frequently conditional dependence sufficiently seem try correlation topology low true connection sign shrinkage surprisingly degenerate job rate quite infeasible take minute run intractable design nonconvex accurate efficient stage remarkable exception comparable validate unnecessary successfully examine network decomposition index obtain membership come cluster true define true negative auto sized element network estimate decomposition consider screen nature figure setting show decomposition rather great ease practice network experiment pattern comprehensive cost consist equally sized generate manner computation computation without graph screening slow path sparsity report decomposable large computational conduct resource infeasible analyze stock keep record stock stock take transformation decomposition cluster place interestingly obtain varied systematically base stock category comparison clustering decomposition quite seem category reflect design decompose network nine isolated reduce tune package huge graph thresholding mask truly exist inaccurate correspondingly course lot fact transition matrix conventional mse window past e use estimate repeat end define n h synthetic bic b category relatively even offer comparable forecasting ccccc category category consist large stock market collect price stock remove consideration consist sample segment consist get idea cardinality graph connection whole take particularly node com intuitive three large service technology technology group product share connection negative condition differ although cause purely causality fortunately link screen perform examine detail scheme investigate forecasting include intractable horizon transition segment suggest existence dependence beneficial take correlation forecast stein work reduce search fine finding cccc segment dynamical sparse structure direct transition second order undirected dependence topology stage synthetic current state contaminate multivariate include stock describe evolution translate matrix order translate notion association joint edge screen small manner refer screening reduce problem fine stage world identification dynamical shrinkage dynamical study stock market stock interact evolve dynamical resemble random infer topology ax finance evolves translate causal observation conventional fail identification perspective shrinkage sparsity prefer produce network influence graph perspective nevertheless totally network even observation node ideally structure capture gaussian attract lot desirable unfortunately directly challenging substitute mean comprehensive picture necessary experience big infeasible ordinary pc make reliably network topology accurately energy stock motivation graph short correlation isolate helpful graphical statistically speak similarity exist joint regularization isolate index also notice brain connectivity network network detect much possibly course decomposition connection propose jointly undirected dependence topology identification dynamic association screen decomposition scale identify remove unnecessary link search problem reduce enhance successful develop decomposition estimation mask screening decomposition describe call section algorithm graphical screening screen network exist dynamical network behavior point component previous multivariate gaussian characterize node translate cause node translate undirected conditionally nod second topological dynamical topology perform exhibit dd decompose completely mutually regularize problem extremely inefficient moderate computational performance boost propose framework consist stage identify structure group induce group element form stage maintain sparsity package operation possible popular parallelism bb sparse another covariance bb impose feasible motivation screen computational algorithm nonsmooth unknown result depend appear algorithm efficient asynchronous line odd unbounded soft thresholding throughout sign define thresholde define general general nonconvex discrete rule guarantee universal convergent refer point equation satisfy rule practically nonconvex cover important role follow form choose group
qp alternate direction multiplier qp otherwise write lagrangian lagrange estimate constraint modify admm nonnegative entry nonnegative part finally admm apply apply multipli inequality consist success crucially rely minimum apply qp augment lagrangian step quadratic unique separate minimize whose see fig order apply argument auxiliary consensus move involve search compute positive step fast stop time form cholesky compute operator qp originally need high possible constrain qp costly solve efficiently cholesky cholesky factor solve linear cholesky factor solve linear permutation feasible relatively dense cholesky add iterative solver gradient warm start inexact update problem cholesky add much thus first slowly admm problem dense take pc subproblem large warm start outer otherwise stop fast use matlab follow implement use cholesky program qp inefficient runtime converge code qp separately implementation b r rt rt break end lemma proposition observation example example
evaluation aforementione easily full system first write fashion pick row position q multiplying give parallel orthogonal j tx q proof substitute present take expectation randomness st iterate randomize I I tr tr invertible differ essentially consideration column mention imply substitute set converge square verify upper bound try proof wrong hyperplane hyperplane project hyperplane alternate span confirm hold optimality x tx term optimality iterate randomized modification full early easy imply norm claim behaviour start span orthogonal span converge important since update never early go consistent however span component orthogonal unfortunately opposite iterate converge mathematically convergence proof carry hence convergence residual getting convergence iterate preferred reason prefer view suppose positive want pose update basically lipschitz update randomize psd treat similarly psd treat follow rule pick proportional normalize uniform distribution normalize column must norm rule exhibit ridge style randomize descent psd system update look pick contrast randomize descent xx take update lastly count evaluation kernel ridge maintain function rkhs different parameterization make ridge ridge subroutine machine application nonparametric major issue involve scale formation gram matrix style get issue never form avoid form great algorithm receive instance view stochastic easier understand perspective sgd viewpoint opposite unique extremely direct however system consistent converge preferable inconsistent prefer unfortunately preferred update exploited avoid explicitly form invert potentially randomization technique help scalability thank point class correction version manuscript lemma conjecture iterative convergence row prove linear randomized work direct relationship often stochastic examine discuss ever store form recognize encountered topic limited randomized algorithm involved work column represent dimensional one want minimize residual consistent sparse ridge extension represent coefficient regression depend stepsize coordinate update like stepsize depend randomized descent lot like perceptron know gradient descent algorithm difference descent stochastic derivation bring traditionally present manner kernel ridge perhaps subtle manner connection explicit reader build understanding first aforementione minimum assuming row use solve stand minimum return situation section understand situation deal two focus specific thorough relationship analyse setting proof direct inconsistent square solution iterate preferable hard mathematically solution explain linearly iterate
manuscript base reconstruction interesting hope drawback alternate tumor tumor copy allele frequency incorporate maximize single estimate make thousand simulated tumor population reliably sequence use correctly simulate reconstruction alone base patient tumor previously manual deep finally state art breast tumor advantage correction tumor reconstruction possible enable automate reconstruction medium depth sequencing read throughput variant read variant allele position allele reference population depend sampling allele population affect fraction cell variant position dp prior dp generate frequency pa I infer group occur furthermore nonparametric evolutionary structure rooted stick break height unique frequency multiple node constraint tree infer monte frequency non great enforce rewrite observation explicitly result use auxiliary root design frequency via child construction ensure population appear posterior distribution auxiliary variable sample frequency new copy reconstruction allele reference allele copy proportion population change available one possible absence able region site determine relationship allele frequency tumor population allele frequency half model population pseudo represent binary mutation read uncertainty frequency reference read support allow region relationship cell expect copy lie proportion read mutation copy allele allele sequence reading contain allele vice versa proportion read contain reference allele looking population un look population population copy potentially number copy evolutionary relationship infinite contribution first find population five population contain contain population occur occur population contain cn occur ii rule occur cn iv reference occur cn copy copy occur branch calculate observation cf circumstance place nearby genome occur genome easily multiple tumor regard applicable simultaneously structure share main model lie sample evolutionary satisfied tumor metropolis hasting move mcmc burn fix hasting factor package convergence complete trace autocorrelation increasingly sequence genome result sequence error different sequencing mutation define tumor allele frequency frequency introduce principled copy improve diverse population expansion reconstruct insight present population frequency increasingly whole sequence tumor automate method reliably reconstruction attempt heterogeneous base frequency solely nucleotide know read genomic explain copy variation read depth current proportion cell mutation magnitude typical preliminary evidence number decrease read size region reconstruction reconstruction need population new copy copy status population share resolve often reliably resolve unclear automated open question overlap reconstruction population impact allele overlap make mutation neither place important reconstruction genome sequencing unlike method appropriately region overlap enough five thousand previous method probe read depth absence still automate less copy overview evolve tumor result process tumor variant allele frequency inference iii show evolution tumor time grey blue tumor tumor circle reference genome mutation indicate low case letter mutation also mutation contain mutation include cell increase mutation even division define rapid expansion large acquire selective population drive indistinguishable frequency noise sequence panel tumor mutation analyze tumor copy case population present point tumor evolution tumor b tumor exist always every attempt cluster mutation identify without reconstruction overlap introduce mutation tumor prevent overfitte balance fit versus parametric iii cluster recover appropriate cluster mutation set still define tumor panel ambiguity one evolution powerful site evolutionary tumor evolution perfect persistent subtree tumor rare compare genome nearly valid incorrect reconstruction alone permit tumor principle require many actual application resolve ambiguity small therefore select maximize tumor ambiguity figure validity establish condition branching occur false either assign mutation mutation weak guarantee whenever identify handle multiple tumor n site multiple report reconstruction violate strict sample carlo mcmc posterior consistent rule sample area determine major read mapping quantification change low applicable region genome overlap occur affect region infer value negative integer infer average copy always allele upon resolve attempt high also know tumor cell tumor population attempt affect computing know independently occur affect copy number computing require know copy figure illustrate information would interpret two cause region ignore allele method change relationship allele frequency infinite associate describe automate properly account comprehensive first provide brief explanation incorporate pseudo perform illustrative permit tumor effort quantify relationship accurately recover apply simulated read next application real sample patient single tumor assume already sequence estimate two first population read imply simulated variant reference count contain evolutionary ignore incorrectly assign population infer reference count run run copy produce identical integrate tumor sequence order structure answer population count read depth per tumor read read depth number complete relationship runtime log plot core intel mcmc hasting runtime decrease implication three single intel k complete complete remove identify result show read depth recover true population population subtract first read estimate relationship characterize population increase decrease number intuitive sometimes demonstrate stick breaking eliminate ad removal cluster leave read depth experiment correct x need resolve six accurately systematic account imbalance precision recall curve matrix cluster construct co cluster matrix sample burn co average well predict co compute chosen presence imbalance plot result relationship cluster population line per infer read provide qualitative user example infer matrix co population row correspond co cluster probability tumor normal free depth time read depth apply importance incorporate compare see relationship precision curve plot result overlap variant examine consist various proportion publicly file variation bic seq find result read read collapse read two take intersection previous verification tumor achieve run bic seq output require see varied return nearly composition decide rely copy simply remove seq identify leave despite still able change composition content run infer benchmark patient extract supplementary tumor treatment equally collect simultaneously read examine mutation gene proportion variant read variant cell copy location proportion cell imply expert manual nearly exception assign child leave expert generate deep sequencing allele five analyze datum coverage tumor analyze analysis genomic status genome region affect copy run normal copy normal perform correction manual identify assign performance look panel continue
necessary formula np suffice univariate proof dependency et al base matching literature notion distribution equivalent moment match polynomial support behind program match dual equivalent likewise distinguish concave match string moment ask consider project moment hypothesis generality unit expectation distance concavity convert cumulative concentration distribution explain yield upper univariate technique upper bind moment canonical density moment moment integrate observe factor exponent concave factor moment attempt theorem depend side yield vanish vanish actually fail vanish classical polynomial dense weight kind sequence make little bernstein excellent polynomial apply prove dense immediately result dense assertion continuous approximate marginal polynomial since turn concave arbitrarily confirm conjecture even arbitrarily polynomial normalize formally exist follow roughly say derivative sign try lemma markov survey exist exist polynomial force therefore idea let must case enough yield show value polynomial approximate impossible product density specify polynomial approximate arbitrarily polynomial univariate polynomial latter let w exponential law uniformly specify get dominate factor grow law dominate generalization power dominate question tail bound distribution reduce namely discrete amount binomial truncate apply weighted approximation know origin degree grow imply must introduction find follow program formulation question fourier character tail tail program program upper shift univariate question polynomial upper appropriate optimal capture prove barrier may weight binomial distribution upon distribution degree key move polynomial evenly space sufficiently degree rearrange anti well entropy eq proof eq integer interval whole interval complete infinite determine interval origin near origin th chebyshev kind property rescale prove formulation enable sake q applying give q contradiction learnable distribution rule task polynomial nothing work show combination hope non polynomial obtain boolean work least hypercube polynomial suitable concentration inequality focus distribution et give sophisticated generator nearly seed thank comment know completeness lemma immediately let require start let wish bind moment pick term without increase change odd rearrange prove moment bound tail theorem uniform degree substitute r question theorem conjecture polynomial approximation generator limit technique prove sign agnostic model fact concave approximated polynomial ask distribution show polynomial concave real strong limitation secondly chernoff chernoff sum schmidt et establish variable independence tight factor study well various distribution area classical area approximate simple computer science application capture quantum query algorithm polynomial approximation measure agnostic polynomial et polynomial distance show idea yield bound computer science establish al tight characterization amount wise chernoff fx otherwise back building include pac perceptron difficult problem agnostic concept agnostic even restricted class uniform hypercube sphere information theoretic hardness result learner hypothesis np hard moreover arbitrary pac open problem high dimensional form linear compute program subsequent certain approximate learnable approximation much hope circumstance distributional assumption technique address approximation exist namely absolutely distribution laplace distribution threshold arbitrarily possibility classic approach reference therein give threshold give log good exist extend coordinate establishing various study restriction learner capture besides gaussian elimination regression fact agnostic hypercube limitation agnostic algorithm different hypercube time leave determine learn fix constructive seed give short concentration fundamental objective replicate study question namely linear generator generator seed use suffice wise r small new seed work whether tight strong tail independent independence hoeffde like independent denote show essentially previous due low support independent theory good knowledge indirect imply existence wise independent tail bound idea force wise linear maximize kn
pass pr source match additional source resolution improve matching source source imply movie likely movie informative netflix greedy similar pass close similarity movie fundamentally similar greedy design follow real datum already experiment next weight output weight match calculate pair movie movie movie netflix high comparative multi six source bipartite display message greedy weight approach six source source pass get weight low suggest matching movie match far maximum matching far publication show match vs compare improve create noise vary primary greedy truly competitive message pass movie represent generate represent dataset point marker movie noisy experience approach message pass operate far greedy finally perform equally value along approach vary message dash gap figure even gap small gap severe least message perform message pass pass depend need converge examine empirically efficiency message pass approach entitie message total change increment candidate threshold figure iteration message threshold message graph entity pass approach converge message cc total conduct empirically compare efficiency message pass much slow pass source increase computer intel gb fix number per fix number greedy message pass increase acceptable hour movie source around increase increase integration pass greedy motivated source ratio latter conduct movie recall leverage message slow message direction area formation connection surrogate precision zhang ad team twitter microsoft facebook ph dr publish numerous retrieval database pc major computer science berkeley database security major area work microsoft google yahoo research production system entity degree dr world life author wide area life video database microsoft product include false definition usa mail twitter com zhang principled explore optimization graph arise entity resolution integration perform structure typically proceed record block record match similarity often match member statistical record linkage bipartite appeal natural global improvement unfortunately bipartite max matching inference algorithm theoretical latter world literature publication result quantify exploit matching discover complementary recognize explicitly entity replicate entity would copy site rely netflix rating split multiple product business page amazon page maintain uniqueness database record linkage community er system employ leverage natural lack initially benefit take significant quantify reason score poor er however impose one one local kind bipartite matching combine community widely applicable integration source little multi er community np community maximization successfully er requirement pass combinatorial optimization approximation lead statistical record linkage extend greedy bipartite sophisticated passing enjoy easily bad competitive ability leverage perform economic service drive customer world services state unconstraine experimental recent cite magnitude measure benchmark typical paper crowd conduct publication generality vary main contribution principled factor pass constrain greedy approach sharp bad example sequential matching sufficient matching demonstrate precision second real world publication datum message enjoy superior discuss paper source entity section setup entity record linkage investigate resolution web negative linkage approach base uniqueness attribute value many constraint entity actor name attribute census record however resolution use entity systematic record linkage source match source instance exponential prevent principled pass well entity resolution problem million entity approach pass principled tractable bipartite maximum solve polynomial come weight extremely special imply fast run principled weight presence endowed competitive entity pass bipartite graph prove desirable finding bipartite loop design message match bipartite graph differ max programming weight tight compare pay attention application entity tune experiment finite number size score entity etc database appear mapping involve source represent mapping entity source entity together one tuple show experimentally leverage one source yield recall linkage match netflix movie netflix focus exploit global particular global source simultaneously source individually equivalently real naturally due argue significant netflix crowd site copy site rely suffer recommendation make attribute alternate past quantify raw largely publication source rather source perform source match heterogeneous characteristic individually variability source overall source sequential entity previously sequential resolve poor poor propagate possibly fashion reduce global demonstrate weight source true b b c truly bipartite preserving achieve global local optima quality contrast look ahead develop strategy approximately maximize co iteratively match entity locally greedy discuss exact np yet principled loss generality illustrative max algorithm reduce include message eq minimizer combination combination observation replace similarly derive source show message keep update iteration reach optimizer similarity reduce gibbs optimizer optimize entity one entity matter optimizer keep update optimum optimum always computation need converge always getting begin optimum entity entity choose combination among optimum among output round less max follow formula configuration final message therefore q employ omit general entity need least message decompose source optimum candidate require time sort candidate favorable pass message leave explore natural multi pair discard order far pair match resolution already entitie clique examine entity clique entity singleton entity derive different clique merge clique merge clique include resolution clique merge clique would sort selection extremely implement max weight generalize competitive greedy duality weight matching least max matching primal note enforce ensure value lagrangian introduce bring thereby form unconstrained optimization ij uv uv appropriately maximize lagrangian g ij uv variable constraint lp original dropping dual match weight lp lp greedy duality feasible primal demonstrate fact sharp max behave practice usually achieve much well experimental movie aim application publication add data synthetic stress multi main data movie meta movie vertical note order production entity source entity source netflix attribute netflix comprehensive list strict one thousand movie netflix score exact normalize discount well among release year release year cast count cast member five name cast match divide short feature score tf perform inexact understanding matter focus entity accuracy regularize score score pair train logistic model evaluate entity matching label truth movie source hundred movie source ask human matching exist match movie assign matching movie source pair share protocol publication source detailed title author publication pass pair
learn dictionary polynomial store centralized task acknowledgement van de provide quadratic objective function set particular row row stack vector column objective constraint express affine inequality vector definition desirable ability specific implementation dictionary signal challenge incorporate intrinsic datum structured particular signal combination overlap graph pattern dataset dictionary learn competitive learning algorithm dictionary localize manner processing task compression classification dictionary laplacian suitable modeling structure live domain signal network simple example sensor traffic city illustrative interested meaningful representation capture signal weight graph overcomplete class combination atom additional challenge design dictionary graph geometric characteristic euclidean domain dictionary often adapt implementation dictionary direction mod therein realization signal given learn costly apply processing task dictionary wavelet transform overview structure dictionary realization represent accuracy numerically train impose structure dictionary structure generally desirable implementation list reference generally day b day c day benefit analytical dictionary incorporate signal combination pattern describe localize evolution similar incorporate underlie graph encode atom concatenation parametric adapt signal representation graph graph necessary understand learn describe synthetic real signal discuss overcomplete dictionary past restrict design signal approach mod signal learn neither fast structure meanwhile dictionary signal overview reference signal transform wavelet wavelet sample frame vertex feature pre implement generally hand two diffusion wavelet try bridge transform numerical dictionary learning algorithm propose learn structured dictionary graph topology close necessarily lead take consideration explicitly dictionary author laplacian code smoothly along dictionary none able provide class exactly compose laplacian overview definition graph find vertex edge weight laplacian diagonal degree define throughout eigenvector avoid large power laplacian matrix nonnegative signal function vertex characteristic eigenvector signal live notion laplacian fourier vertex frequency inverse transform besides harmonic transform translation convolution center vertex allow interpret operator act spectral localization around vertex smoothness localize around translate localize atom diagonal note power localize topology atom center graph learn dictionary signal combination overlap pattern learn dictionary capable use definition translation learn generate form main directly support detail dictionary pattern translate give vertex localization atom representation spectral impose semi upper signal consideration component cover impose eq constant particular prior behavior prior incorporate problem modify frequency certain spectrum choose flexibility derive dictionary c ny generalization leave schwarz combine would tight atom learn summarize parametric graph represent signal atom equivalent cast signal objective ii stability optimization solve alternate code fix parameter q orthogonal pursuit dictionary omp dictionary omp atom dictionary sparse code remain method pursuit soft thresholding fix coefficient dictionary parameter step dictionary learning dictionary converge local dominate computation laplacian enforce constraint line split well example interior method htb target sparse attribute kernel learn performance svd depend size blind unable pattern particular significantly slightly training signal dictionary show much stable well atom neighborhood tend poor contain localize signal respect svd fig localize area course signal translate pattern training dictionary translate version learn dictionary instance translate pattern appear intermediate solution svd form rather continuous evaluate contain pattern graph since necessarily complex implementation discuss structured term generate c study signal polynomial even though signal preserve true fig combination atom different level omp graph polynomial noiseless testing scenario performance polynomial divide spectrum four band concentrate four particular atom band generate uniformly entry index band zero atom vertex localize randomly atom generate match generate spectral band generate fig kernel localize notice approximate band generate dictionary similarly behavior fig atom atom topology smoother particular fig atom spectral kernel atom concentrated frequency laplacian associate smooth generate exactly performance improve polynomial attribute atom localize neighborhood reason explain generating dictionary flexibility smooth generating achieve nonetheless learn dictionary efficient observe dictionary learn svd graph structure generate polynomial behavior entire concentrated band generate two band construct generate atom atom accord signal linearly order polynomial learn support frequency band illustrate exist training since concentrated part generate signal support particular spectrum eigenvalue examine synthetic localize graph world take represent graph construct assign two location distance shorter threshold daily california data dataset throughout major area california connect distance euclidean gps weights proportional bottleneck persistent drop length bottleneck active maximum learn use testing normalize respect energy fmri acquire five brain contiguous subject measure state completely movie treat brain euclidean coordinate centroid determine short edge dictionary atom use dictionary signal validate learn normalize norm dataset synthetic section adapt clearly atom dictionary signal six omp decomposition brain dictionary learn polynomial note polynomial dictionary consist localize poor localization clearly ability localize
segment rating segment netflix dataset million movie integer ordinal log user mf vector ask whether movie typical per figure indicate quality possible enough store main handle exchangeable cause subtle likewise similar mapping rank respect focus produce individual contribution inference explore exponential space demonstrate empirical suggest art collaborative university department edu ranking arise group item movie properly subset combinatorial approach procedure explore discover variable large collaborative filter considerable machine community rank datum cast generate document arrange decrease relevance compatible object object likely group contain complementary movie beneficial recommend compatibility group pose group subset situation need stock store rank rating example quality package rating somewhat relate learning produce label input inherently subset introduce situation pattern object partition scheme partition partitioning order former exploration metropolis hasting iteratively possible way partition order consecutive two consecutive merged propose term order latent fashion machine rbms posterior hide visible use collaborative filtering g group rank list unseen public dataset competitive art rest section together main introduce extend collaborative filtering review follow conclusion merge middle b conversely split represent preference order causality modelling depict group subset direction look impose distribution difficulty partition thus al careful partitioning allow care property group order impose capture relation compatible order linear potential account parameterization allow flexible possible take split position subset remain unchanged reverse e appropriate guarantee entire armed sampling procedure stochastic give object denote rank belong furthermore collection object partition subset usual partitioning among subset denote notation write order subset wherein element grouping complete govern recall divide partition pair perform partitioning consider give know super fast log size standard permutation encode compatibility among encode property potential effect relative hereafter refer propose mcmc inference evaluate mh mh proposal sample move accept define proposal ratio intuition local partition cost change walk slowly move singleton split take subset guarantee possible configuration illustration singleton distinct chance drawing subset merge back ratio compute potential depend order rank operator merge subset merge subset recover merge operator ratio metropolis hasting present initial l sub iii acceptance probability iv accept move consecutive subset merge keep subset unchanged evaluate acceptance eqs accept procedure introduce temperature uniform anneal repeat computed z configuration linear f ax bx ax need unfortunately inherently chain follow iteratively e b far introduce latent serve purpose collaborative choose ranking reflect discover partition g cluster datum conjunction activate thus boltzmann potential admit x capture relative hidden model figure hereafter refer posterior indeed shorthand ph ph representation configuration generation involved need explore order alternate straightforward remain px potential jx kx eqs product product xx ix x j except rao straightforward alternating usual simplicity similar modify assume share specific ax bx ax bx statistics statistic ax equip gradient trick parameter respect application application preference item rating star item user often rating thousand million create sparse discover latent user factor individual limited partitioning order unseen item rank want reconstruct complete ph ph rank let approximation j jx resemble ph due rank task think intractable approximate treating completion fast simple assign worth grouping factor contribution item compatibility first compatible worth order item
section volume direction chance practice visual process volume double check euler surface triangle close surface sphere adjacent triangle share face total number characteristic check mesh satisfy binary volume surface exception topology heat construction lb sphere find compare iterate diffusion surface study representative z treat measurement surface smoothed comparison smoothing root rmse error surface heat eigenfunction correspondingly bandwidth effective sufficiently size iterate reach figure b square iterated kernel mainly localize iterated kernel heat kernel comparison limitation iterate heat kernel smoothing converge heat diffusion heat give iterate diffusion smoothing heat gave sufficiently iterate give original one visualization actual replace coordinate kernel discretization error converge heat discretization converge increase smoothing solve discretization smoothing heat bandwidth know image perform small snr shape surface almost region surface differently signal take ground variance mesh vertex black group variance measurement mesh vertex add region measurement group detect region heat sensitive I simulate substantially snr iteration necessary smooth empirically heat eigenfunction determine performance test threshold detect iterate heat addition heat diffusion incorrectly visually figure discretization scheme approximation step higher simulate functional substantially group figure smoothing bandwidth heat eigenfunction detect however heat smoothing diffusion smoothing sensitive snr raw iterate smoothing due heat negligible region minimal well substantially snr diffusion iterate perform well analyze growth ct human divide group iii main biological localize growth acquisition previous surface alignment surface surface subject affine f template map construct choose old identify template template remain template remove subject remove difference surface perform template metric framework metric construct transformation differential ode vector constrain sufficiently integrable generate lie field reproduce satisfy template connect define application employ approach template template simply initial subject template template template individual subject initial template ii ii row growth direction mean difference color top row significant bottom row group level correct bottom iterate use growth ii iii corrected determine significance length much easy length assume smoothing measurement make smooth heat eigenfunction less heat response testing group ii show degree freedom comparison iii statistic map black finding finding simultaneous growth also perform iterated diffusion result diffusion size bandwidth split smoothing figure iterate snr perform mesh vertex kernel diffusion number significant present novel heat regression framework analytically weight laplace weight expansion relate isotropic heat diffusion validate discretization regression parametric establish equivalence diffusion wavelet growth growth identify quantify first decade life overall growth small currently ct grant dc ct study develop institute grant center child health development clinical award science associate thank university wang comment edu present scalar eigenfunction heat formulate new bivariate expansion heat kernel isotropic heat wavelet validate characterize localize surface ct surface template kernel laplace eigenfunction diffusion medical surface represent triangular surface process likely noise mesh widely reduction technique surface heat surface brain subsequent involve random isotropic isotropic iterate widely solve surface surface smoothing brain smoothing spatially heat discrete tangent manifold heat linearly heat bandwidth process heat analytically eigenfunction lb avoiding use although introduce heat vision heat kernel descriptor manifold heat machine however heat never framework machine kernel wavelet surface wavelet map sphere local wavelet however wavelet surface serious metric subsequent less parsimonious surface intrinsic lb expansion spherical transform graph primary unified wavelet coherent mathematical define manifold apparent provide theoretical extend heat kernel surface diffusion wavelet explore transform mathematical equivalence explain growth surface identify show significant localize growth surface snr sensitivity surface continuous fashion assume unknown signal estimate square integrable area image eeg filter boost surface filter isotropic form laplace diffusion control amount green cauchy follow dirac differential initial isotropic various numerical smoothing need discretize discretized fp mesh triangle share neighboring mesh angle opposite contain adjacent otherwise triangle diagonal matrix adjacent diagonal ordinary discretize estimate laplacian row euler scheme need iteratively weight heat surface mesh heat break iterate small expansion heat bandwidth concentrate vertex sufficient iteration numerical use eq neighbor truncate localization circular angle north outside band laplace operator spherical laplacian spherical degree order use spherical harmonic expansion spherical least sphere use heat bandwidth harmonic expansion severe localization wavelet phenomenon close eigenfunction laplace surface numerically solve among many medical formulation global individual implicitly consume amount memory surface thresholded visualization eigenfunction numerically parameter estimation basis minimize residual q square method coefficient harmonic mesh mesh eigenfunction condition numerical mapping template surface use study statistic mesh vertex comparison heat datum smoother enhance snr integrate inference level signal normalize fashion bandwidth I e previously smoothness quantity along mesh surface bias often smoothed sufficiently small motivation develop heat interested determine significance note consider continuously index underlie level heat kernel subsequently often comparison correct degree manifold functional euler characteristic ec pf cumulative freedom second bandwidth incorporate propose ct model
evidence toy test equally spaced measure toy htbp reconstruction handwritten goal lie center split example digit dataset root average scale lie color give outer test image tune unimodal bag context training solution impractical large human dataset apply neighbor software motion capture frame average mod true pose view consist pose camera use half frame test pose marker coordinate frame pose pose pose vector theoretically affect affect step optimizer step prove change power theoretically motivate role cross converge dataset suggest purpose role purpose cross cross range practice cover set initialize prediction regard table parameter share choose justify result sm sm l notice toy improvement toy figure tuning perform well pose report give toy refer toy toy conclusion outperform argue b htbp sec sec sec another toy example slightly perform least dataset bias towards input however learn towards output justify sm powerful divergence optimize however member tune conclude result report knn knn knn five comparison toy dataset knn neighborhood see compare outperform knn knn knn propose structured sm divergence analysis understand sm part argue could finding kl understand perspective cover analyze instead cover sm base maximize correlation test highlight computationally sm complexity reduce complexity equivalent sm major contribution practically achieve structured tuning sm perform task experimentally observe generalization experiment outperform toy example dataset tune validation would indicate hour dataset hour grid significantly decrease time enough validation like save instead cross validation experiment prediction name depend interesting future measure efficient form sm divergence main section number operation compute sm operation operation practically achieve structure perform intensive experimentally adopt work propose divergence report new cubic time simplification straightforward gradient prediction theoretical parameter generalize name propose yet computation quadratic cubic structured causality extensive validate finding pose two nsf award appendix relate etc b sm number calculation calculus follow q invertible matrix multiplication hard final dd simple expression sm gaussians ignore multiply e xy xx k yy xy xy factor sm gaussian lemma cholesky operation ignore operation ignore cubic contrast require decomposition compute multiplication need appendix require could efficiently sm accordingly time sm sm write x xt dt xt dt xt dt proof analysis function start denote xt notation sm equation px py px py z compare xt k k indicate proportional k theoretically title journal volume page number year book title page year proposition example ex ex minus nj usa computer present generalize divergence divergence mutual measure enyi leibler entropy divergence kl insight divergence experimentally framework result offer big since lot probabilistic shannon powerful mathematically development communication lot physics science reliable divergence kullback leibler divergence use machine texture negative pose lot connect turn view measure information r expectation jensen gap equivalent uncertainty lot relationship alpha divergence investigate consider machine entropy sm later entropy parameter converge shannon entropy al suggest sm equilibrium sm harmonic similarly sm mutual enyi kl domain sm close form sm motivated set utilize sm kl particular sm process structure structured gaussian divergence sm sm divergence study context probabilistic specifically presentation community generalized base sm divergence subsection simplification divergence variate cost evaluation subsection theoretical sm subsection experimental sm toy example rest sm multivariate gaussians gradient present theoretical perspective discuss simplification sm framework sm could associate subsection sm toy thank understand concern review perhaps look kl sm community address valuable prediction similarly start adopt share regression perspective cover miss theory analyze kl cost cover kl claim theoretical sec material write line line paper sec proof b valuable also present computationally derive require determinant require matrix computation simplification sm function computation cubic however simplification sec straightforward new illustrate agree comment complexity extensively evaluate various method cover proof show correlation test argue another entropy sm generalization rd ref sm reference refer sm distribution sm divergence sm divergence enyi kl sm enyi divergence enyi divergence sm originally recently alpha enyi relate divergence boltzmann shannon information sm divergence reason generalize suitable structured possible consideration work entropy sm divergence close expression main entropy affect structured analogy motivate study physic enyi entropies generalize extensive present non extensive linear enyi interpret quasi arithmetic sm generalize linear mean enyi limit trade quadratic expression introduce lead sm gaussians write vanish form expression apply sm expression divergence new cost xy xx yy xy x multiplicative ignore ignore contrast py px determinant stability minimize factor depend constant yy xy xy xy xy xy quadratic improved closed form sm cubic complexity gradient complexity expression context compute cholesky decomposition decrease significantly quadratic cubic expression time fast need operation need proof determinant efficiently sm conclude equation identity multiplication ignore need time need less need compute appendix indicate sm speak prediction unknown discuss detailed subsection property subsection interpretation subsection size marginalization extension marginalization term matrix determinant distribution elliptical eigen eigen hence determinant notion volume elliptical orient eigen one could interpret scale look closely decrease new close point eigen eigen value term maximize small I e produce could think make input however discuss uncertainty extension subsection detail sm straightforward think space equation kernel lemma xt xt dt sm sm compare since xt dt xt x p xt yx prediction xt dt k x maximize k px x py x xt px py px claim achieve x xt k py p xt px
variable lag z solution order autoregressive multivariate time similarity correlation time assume derive solve approximate marginal transform sample linear piece wise fit piece wise case piece divide domain interval coefficient pa k I doubly bivariate order moment moment get approximate piece transform invertible give piece wise invertible monotonicity cdf thus piece wise process accuracy call naive z z jx iterative naive r difference r j r h x iterative require require solution start coefficient close desire case transform bivariate increment gaussian correlation amplitude moreover guarantee monotonicity monotonicity interval large threshold close safe practical purpose monotonicity check interval number search accuracy closed expression know feasible marginal bivariate feasible maximum feasible correlation deviation matrix difficult condition feasibility procedure check feasibility feasible solution correspond correlation semi definite gaussian checking gaussian positive lag modify slightly positive correlation comprise correlation change eigenvalue identical differ solution structure eigenvalue slightly positive value component second repeat entry replace entry back cause definite repeat work step coefficient var equation var express uncorrelated determine stationarity root reverse lie unit plane equivalently modulus positive stationary process possess typical piece approximation former marginal possibly correlation piece marginal display thick clearly due correlation var marginal bias pass correlation monotonic occur sample differ theoretical correlation monotonic marginal transform match example make var monotonic vector match even show monotonic transform var black line grey realization black line denote fisher confidence row cubic thick report computational efficiency single correlation non correlation correction derive close marginal density monotonic three pair uniform legend indicate correlation relative match evenly around mostly concentrate large somewhat rmse matching attain approximate succeed slight output figure fast pair three correspond line power relatively bar standard shown respectively indicate generate continuous transform correlation respective e skew number purpose converge reach approximation marginal always obtain unless e time moreover make doubly could expression time iterative transform demonstrate piece integration investigate insight close practical derivation need consume dimensional numerical obtain proper matrix definite directly correlation contain introduce iterative turn match marginal process auto correlation lag serie moderately say auto cross however study system system var practical generate proper time use randomization marginal transform encounter matrix eventually extreme positively skew strong time series randomization nonlinearity truly comparison multivariate nonlinear dynamical system future generation realization multivariate extension univariate process transform autocorrelation piece autocorrelation transform determine vector autoregressive demonstrate marginal autocorrelation gaussian randomization u series consider fit generation marginal simulation randomization particularly model dependency among constitute multivariate give marginal internet temperature randomization testing dependencie nan nonlinear surrogate test distribution compute series preserve distribution surrogate nonlinearity field investigation dynamic linear structure surrogate series arbitrary call autoregressive modify series rely numerically double product two marginal computation simplify approximate marginal pareto surrogate nonlinearity approach developed match marginal randomization marginal transform refined transform autoregressive process call transformed form autocorrelation autocorrelation use numerical parametric originally piece series simulate multivariate briefly univariate multivariate simulation start univariate autocorrelation lag equivalently spectrum problem q sufficiently spectrum frequency without series solution solution transform transform eq variable standard density cdf objective random though constrain realization test surrogate nonlinearity another onto generate amplitude iterate realization attempt identify generate give realization two approach decompose step marginal
foreground incorrectly predict foreground background speed get intuition incorrect encourage accuracy stop policy prediction obtain evaluate compare ap alg let exclude location since evaluate part method evaluate class vary ap decrease incorrect grow evaluate part due report positive speed require mistake policy evaluation experiment table ap grid ratio significant accuracy negligible versus log versus training car cat person tv ap car cat person tv ap ap car cat tv cascade speedup ap ap car cat person tv speedup cache pca cache full cache full cache pe cache cache pe cascade versus baseline cascade cascade ap part relative speedup experiment dataset publicly cascade process label additional part decrease visually score agree intuition correct location terminate part posterior reflect score ap demonstrate reduce irrespective feature sec show negligible recall via ap second cascade feature location location projection low discard filter fair adopt similar selection policy schedule filter pass select filter make slow summarize discrepancy cascade second second individual stage filter evaluation full significantly slow combine fast cascade slow dimensional high blue bottom visualization part pixel color color leave car heavily selection accuracy unlike pre specify threshold schedule obtain potential include image position detect simultaneously assumption problem edu object optimize describe response formalize schedule classification look base cascade detection optimize negligible model powerful appearance accuracy demand cascade branch scheme response location future introduce art pyramid part decision confidence approach optimize apply part part quantifie false mistake threshold utilize idea object name part phase learn scheduling inference policy image image probability update sequentially response suggest terminate evaluated part contain original score proceed foreground maintain location sequentially response low learn filter round color policy stop location confidence foreground evaluate versus approach art evaluation time cascade make detection filter evaluation use detector achieve without additive use sift point refer detection optimize representation identical inspire acceleration detector cascade however evaluate cascade select next maintain foreground ensemble classifier stopping optimize branch bind bb search location easily search test yet give location early bb constrain slide window transform sequence object test orientation minimize total coarse classic cascade adaboost study introduce cross cascade response classifier exploit prediction cascade optimize pose refinement cascade structure pose resolution filtering pose state emphasis structure scoring pose recover general feed maximize use close filter evaluation accuracy optimize filter cascade still part e parametric representation pdfs annotate emphasize hold simplify likelihood conditional part stop assumption expect learn joint pdfs fidelity indicate g car cat person train tv example place score bounding filter record place score positive obtain negative example pyramid collection smooth fig likelihood discuss order detection proceed round apply root part topological ordering next part past take location ts zero everywhere part uninformative independence score bayes seek choose run plan sp repetition admissible termination formalize make error hidden stopping error label small infeasible relax lagrange multipli lagrange interpret incorrect elaborate cost choose incorrectly cost incorrectly introduce mistake solve use compute proceed part apply force terminate bin term stop choose alg summarize step sec bins store
function spherical j mm lee normal skew discriminant pattern recognition york york independence distribution normal development foundation www mm skew instrumental gaussian direct unobserved b discriminant function pt pt theorem question discriminant mm mm normality time practical situation index modelling consequently discrimination method skew elliptical study skew normal quadratic family simulation word skew elliptical skew normal unobserved variable goal rule describe van several researcher normality study flexible discriminant discriminant normality quadratic discover skew linear discriminant discriminant robust estimate generalised rule van adjust skewness skew distribution skew elliptical lee al linear regression longitudinal method sense al recently perturb distribution ml interesting application skew elliptical distribution case classify give skewness skew classification generalise extended skew elliptical see skew skew normal accommodate skewness heavy tail model mle place class unify skew elliptical distribution start extend skew elliptical section explore extended discriminant obtain find know proper selection pdf al x x consider mechanism disjoint classify belong fall actually assign misclassification group assign classify decision classify know assigning equivalent classification q assign large extension rule elliptical distribution elliptical elliptical distribution al elliptical introduce skewness perturb screen unity specifically elliptical variate generator word elliptical h h k fx h q univariate induced generator write random skew elliptical se denote two distribution hence group conclude set equivalent large selection classify se elliptical otherwise generator convenient popular correspond normal multivariate normal scale elliptical density generator extension skew al variant al dimensional location shape variate cumulative unlike linear rule ax assign minimize linear rule normal classification whenever multivariate skew mahalanobis distance two normal index variate population namely complete depend assume rule population u eq jointly yield cn ab replace consider propose assign cn maximum discriminant em well complex comprehensive know ht iteration assume miss k ij ik eq ml depend th depend propose work parameter correspond location scale al conclude likelihood appear satisfactory simultaneously remove discussion case classify discriminant simulate discriminant accord carlo consider proceed auxiliary representation multivariate em obtain discriminant rule indicator parameter generate individual use ht ml bias ht allocate total group simulate development team cc indicate fact overall accuracy classification value versus data classification derive skew classical property distribution focuses skew potential skew elliptical well research reference mm pt mm normal j mm r g unified selection mm b shannon mutual multivariate skew distribution extend skew unified skew elliptical skew normal family skew related york multivariate skew mm
help day c c test arm arm boost c upper body detector order width detection window scale factor train scale detection window percentage correct pose tool calculate correct body correct ground truth present low head part set achieve part get bad result whole mis subset predict bound box achieve arm run code full per denote get training contain increase cost day need model compare arm factor ground predict position th show figure accuracy large accuracy bad strict suggest estimate pose exact location full improvement rl effect multi weight table poorly weight detection significant increase task low error gradient dominate case greatly guide enhance generalization detector datum feature among seem feature detection test convolutional operate reflect neuron sensitive expect detector rd mid find neuron instead neuron middle input neuron receive backtrack filter feature neuron map local one optimization patch contribute test rd body detector maximal activation map occur visualize nd rd convolutional head mid feature detector head arm fig localize body right level body fig c two band horizontal window frame could useful identify context location filter cc b cc paper heterogeneous task pose consist task pose slide jointly learn generalize testing visualize mid maximally find neuron selective pattern localize pose combine unsupervised pre would like extend video object china acknowledgement city li edu liu media city computer science city university edu pose neural regressor slide window part architecture good empirically localize body pose computer vision application video retrieval pose depth format mobile device equip camera estimation general classify two part method regression part graphical structure find configuration match pose estimation globally expensive two definition part part avoid orientation part appearance capable capturing complicated slide window pose hand allow multimodal rapidly pose view regression pose good information currently approach training calculate prediction expensive deep success computer task network popular computer vision fully model convolution large capacity hard train network generalize task pose pose window detector heterogeneous detection train benefit greatly local activation selective localize multi training task review regression relate heterogeneous encouraging regression share joint find feature pattern force heterogeneous task good convolutional scene labeling define category classification define slide window window contain detection task window train sensitive neighbourhood neighbor feature pose location regression stage predict prior region increase refinement improve performance network task could length convert stick annotation indicate absence window eq body portion upper annotation indicator map body several window minimize truth probability combination detection part image weight base consideration share task motivate learn task helpful second sharing parameter generalize body part predict position translation position preserved context sometimes difficult bound box part long arm lower distinguish look include neighboring part help detector part detector box image human network rgb task layer pool activation information consist weight share neuron previous pool linearity integrate neuron convolutional layer regression layer neuron neuron receive neuron activation neuron relu show task train jointly train regression global back propagation update image task calculate gradient
call formulae kt represent number know empirical characterization natural invariant parameter always alternative type adequate currently describe limit deviation efficiency analyze asymptotic new regard theory calculation test kolmogorov nan hypothesis therefore applicable coincide supplement research power efficiency integral asymptotically degenerate statistic degenerate calculate nan x x k px ps side px remain calculate term j x summing follow calculation degenerate degenerate statistic considerably simple calculation family test describe attain level exact follow hold p always leibler nan quantity alternative kernel non kernel see asymptotic large nan hypothesis deviation degenerate see state moreover accord law also hx present alternative eq gamma eq negative see alternative exact leibl distance similar omit q variance q regard asymptotic analytic moreover eq slope sequence state satisfie alternative table use alternative well alternative gamma power simulation replicate htbp alternative kolmogorov type projection f distribution variance complicate case equal elementary besides limit use gaussian find simulation type obtain c kernel sense large supremum family degenerate result equal deduce hx gx get tf se te e satisfie calculation therefore considerably exception probably relate kolmogorov statistic get te e figure degenerate describe use case impossible find limit statistic family supremum degenerate sufficiently moreover alternative alternative projection section te e te te slope see previous table power four alternative htbp gamma maximal nevertheless favorable sequence sense describe local efficiency relation hold give density hold easy consequently local bi e schwarz constant constitute simple density alternative gx e kolmogorov hold schwarz inequality constant alternative class facilitate presentation e gx gx x two family integral many favorable asymptotically power test closely order correspondence criterion recommend try statistical test test local efficiency turn rather common alternative power optimistic change regard probably closely intrinsic however virtue especially favorable deep send
behavior sequential involve hide generic mapping condition relaxation follow two condition algorithm associate automatically attain prediction write likewise second monotonically admissible relaxation relaxation enjoy bind claim version enjoy paper admissible forecast relaxation enjoy regret offset rademacher schema derive non eq prediction derive schema enjoy eq estimator derive simple admissible relaxation enjoy problem online notion alternatively play standard know schema design admissible enjoy regret study work value minimax hence expression fashion back minimax supremum supremum denote since linearity expectation write eq arrive supremum prof first expectation denote jensen eq upper random conditioning ensure proceed proved except end worst along exist easy leave subtree contradiction depth tree sign choose splitting block last stay close need require mean q low may change examine illustrate let choose delta optimal clearly view discussion initial trivially check note check expand loss rearrange eq jensen eq supremum admissible far ty enjoy exactly notice closely notion derive since offset relaxation derive relaxation initial condition hence apply condition expert rademacher soft arrive inside hence give obtain round write eq hence dependence tree may rewritten conjugacy relaxation relaxation view inequality arise use conjugacy relaxation ty relaxation square notice forecaster final bounded online regression introduce optimal transition analogous frequently sequential match rate generic forecaster establish design computationally derive exist expert online arrive stream forecast square datum leverage law learn aim develop prediction formulate regret family notably upper require past progress include study rich nonparametric class function regression partly motivated remark appear obtain approach forecasting aggregate exponential interestingly respective property view online since algorithmic main algorithmic aggregating mention aggregate beyond supremum underlying remark arise pac bound day require notably long recognize cover potentially entropy growth empirical characterize loss cover combinatorial rademacher regret correct behavior minimax logarithmic loss partition critical radius minimax employ ball aggregate integral main difficulty precisely sequential minimax decay low logarithmic real mild behave noticed regret completely many situation relaxation framework provide develop characterize rate relaxation admissible constructive generally feasible relaxation dimensional make slightly regret notion suppose converse minimax regret extract get handle minimax regret inside round describe range range minimax notion mention scenario thank online scenario subtle binary entity capture end definition depth complete root binary root label child th level level word cover one complexity tree form function sense polynomially parametric behave necessarily main technical statement normalize constant sequential cnn cn parametric logarithmic uniformly growth uniformly modify entropy growth cover priori additionally optimistic cn np regret bound optimistic slow obtain rate subgaussian tail upper factor state play style control parametric coincide cover evaluate growth scale choose extra yield particular finite use small normalize rate lower optimistic bind obtain entropy infimum evaluate extra infimum evaluate take optimistic follow recall rademacher eq rademacher minimax fail rate regression give complexity extra fluctuation complexity give critical parametric irrelevant g purely curvature loss help square phenomenon occur risk observe enough sided concentration statement rademacher complexity side minimax complexity minimax response range tree range tree complexity upper bound entropy analogue bind rademacher crucially choose scale value depth optimistic offset arise depth argument lemma value eq also compare approach compact subset
decompose pc ij w ij de rapidly increase also confirm error pc pc reality work see fig issue pc w cross element trial error choose asymptotically adjust bias acknowledgment helpful domain representation vector search matrix minimize datum regard embed multi include multivariate canonical vision paper propose code domain single version augment domain cross discuss illustrative domain get image datum vector image typically hundred like retrieve alternatively retrieve image strength theory association association image unlabele color classify remain match true association otherwise ij project single transformation transpose trace diagonal block domain eq minimize weight supervise let unobserved weight transform look common perform across domain formulation matching graph multi domain similar popular recently vision formulation multivariate connect code indicator call canonical discriminant correspondence cca let paper code mention similar code vector domain vector domain represent get solution embed graph embed minimization classical model illustrative example parameter reduce matrix easily compute say cholesky us error subject minimization work regularization properly data domain vector ed matrix objective domain solve spectral graph express consist simply graph would match final code element diagonal nonzero code weight maximization kind pca code idea find whole vector input e association cross matching reduce assume matrix specify coefficient cross matching connection regularize show case array de simplicity correspond matrix factor simple generation repeatedly define element generate vector standardize variance grid
network ignore machine support generalization I train consequence due numerous example towards address dependency form study machine effort general relaxed elegant classic concentration example improve early weight well naive ignore illustrate law classic structure example learn derive variance concept theory several weighting section task improve conclude summary contribution future work introduce intuition however share get sharing make introduce formalize problem fundamental learning hypergraph hypergraph set zero hypergraph vertex hypergraph vertex group number abuse example tuple tuple special case example member network contain member share binary relationship object friend network prediction vertex two type hypergraph hypergraph partition disjoint vertex example tuple movie rating person illustrate setup denote object suitable usual supervised label member contain set may wikipedia want active target whether friend feature interest education friend movie rating movie actor etc movie describe contain amongst gender rating give movie hypergraph hypergraph label hypergraph tuple hypergraph vertex alphabet alphabet label labeling labeling object assign value form assumption dependence strong perfectly make I assumption still believe several way explicitly dependencie dependency detail work dependency information hypergraph label draw assign vertex target value sample possibly identical choose training vertex draw assumption long possible training test movie example may may hold rating preference movie experiment movie participant member ask movie would hypergraph partition kf iy assign independently relational logical direct relational dependency template
interesting color relate transformation model affect layer template layer configuration variation seem effect segmentation million million foreground augmentation augmentation simple sample heart single neuron neuron kind create layer feature map layer final position filter edge channel color tailor neuron activation allow neuron forward propagate correspond zero produce neuron keep viewpoint input look like much visible vs achievable neuron row neuron fc viewpoint transformation representation viewpoint much realistic semantic activation layer fc row activation actual generate single neuron neuron edge like result hence must fine spatially neuron c segmentation stream fc fc class fc fc stream scale correct size normally image analyze layer look modify observe approach already material level neuron smoothly activation extent neuron feature gradually go almost level outcome clearly sharp observation layer correspond empty rd filter explanation pattern frequency fix size regular figure happen map bring pair c source previously unseen learn previously unseen view cost separate source remain number transfer train knowledge miss angle transfer visible interpolation fine preserve start view pair transfer produce satisfactory interpolation work reasonably view bottom row fine lose euclidean transfer dramatically suppose bottom row simply view find similar neighbor match source view miss view try distance rgb descriptor although figure performance suggest learn linearly remarkably object meaningful representative show bottom naturally row intermediate look like however intermediate supplementary cnn appearance optical consist apply optical flow concatenation optical flow connect optical flow refine flow numerically test ground ask people manually mark first total pair ground truth average manually annotate validation set tune compare sift pyramid baseline intermediate error pixel spatial pyramid sift human performance training network standard task generate image task merely smoothly meaningful relatively fashion parameter additional state manner show powerful applicable modern expect elaborate involve kind might beneficial probably extension apply predict depth future variety different people add network realistic generative object viewpoint network merely heart find meaningful allow similarity network different approach cnn successful vision task predict depth task supervise problem cnn label network mapping input output object position work stick supervise cnn image train capable give color etc neural network image backpropagation enough network perfectly learn heart reconstruction behave input observe namely capable transfer object interpolation describe detail internal network unseen track accurate model typically datum representation priori infer procedure prominent rbms boltzmann rbms build encoding solve direct wide range consist attempt forward perform generative image latent second try discriminate contrast approach image generative rather give reach approach label image problematic uniquely identify significant noise unsupervised incorporate label form semi unsupervised approach variation include rbms rbms rbms digit condition autoencoder restrict structure generative formally refer convolutional turn composition layer fc build share three fed layer neuron stream process connect layer fourth fully fc connect split stream fc generate object mask share fully fed layer filter layer follow relu nonlinearity layer see convolutional layer pool conventional cnns span oppose shrink replace entry block corner elsewhere width convolution apply deconvolution use parameter train euclidean error reconstruct segmentation mask omit weight
kernel adjust scale individually gaussian ard optimization exact become use expansion kernel mixture gaussians describe marginal rotation invariance novel piecewise radial scalable gaussian individually location heavily gm model gm equally describe learn section initialization initialization pick minimum marginal continue signal deviation dimension scale pick random optimization run ard standard initialize initialize ard close parameterized controls origin ard initialization rbf distances uniform rbf width radial follow ard initialize cluster basis expansion rbf ard kernels htb compare method accuracy leave score large reciprocal performance across htb also ard rbf kernel small medium hence gm take kernel place r datasets rbf ard gm cancer hardware auto stock energy road song compare htb time basis gm function component flexibility gm become although sensible combination parameter tune give allow large parametrization allow setup wish emphasize gm adversarial order gm ard gm method hyperparameter commonly rbf ard gm ard rbf entirely expansion gm expressive ard rbf popular vary gm continue increase function gm fitting basis extent mean true gm seven method normalised predictive training consumption score low time despite require similar expressive ard good memory show clarity plot log supplement gm gm greatly outperform accuracy valuable family parametrize function expansion minimal diverse runtime consumption additional gain group short simultaneously expressive hope work flexibility method sense flexibility scalability problem expressive rgb rgb rgb dark medium blue great representation compare scalability introduce flexible learning basis expansion mechanism learn spectral expansion class speed consumption entirely control represent product typically flexibility whereas expressive large rich representation spectral arbitrarily basis conversely recent offer expansion kernel priori hand issue might indeed expressive scalable novel radial expansion mechanism automatically likelihood optimisation adjust learn parameter basis computational frequency computationally allow great group require interval four evaluate advantage accuracy consumption describe background kernel include basic tool kernel contain evaluation fouri flexibility expansion expansion fix recently incorporate scalable parameter enable parallelization jointly proceed descent achieve instance optimize frequency expansion spectrum formalism gain scalability expansion expansion computation dimension mixture gaussian require hyperparameter kernel generalise incomplete grid learn large naturally enable statistic flexibility make ideally suit dataset datum partial efficiency gain consider expansion weight expansion expansion property inductive purpose novel radial flexible require denote domain label joint hilbert schmidt speak semidefinite key kernel allow product high implicitly mapping desirable might solve combination beneficial amount create approximate manner exploit translation invariant ensure integral product kx suggest carlo approximation fourier transform popular normal distribution use rbf refine approximate admit expansion random randomness allow efficient uniformly zero expectation define recursion dense subsequent hadamard reservoir sampling uncorrelated diagonal draw result iid gaussian encodes draw define length via straightforward change adjust computational remain adjust rather generic additional la keeping introduce flexible learning allow translation accomplish additional sample moreover learn translation basis individually optimize frequency still computationally expensive fitting undesirable optima particular want enforce also scale location frequency result parametrize section describe assume expansion next piecewise radial scaling use formalism introduction process next derive kernel general purpose particularly expressive choice objective assume gaussian drawn feature parametrize involve infer mixture model access component parametrize mixture weight integrate away solely denote design parametrize covariance index minimize negative cross training inspection simplify expression immediate storing provide predictive accomplish via formula require rank approximation projection computation kernel formalism preferable kernel yet rotation violate rotation leave follow term fourier choose frequency symmetry rotation invariance linearity transform expansion translation approximate gaussian density estimation fourier approximation fourier amount directly amenable fast expansion provide kernel form insight shift fourier accomplished multiplication x inner operation multiplication multiplication original accomplish preserve translation otherwise preprocesse random group product feature dispersion likelihood describe individually less fitting local efficiency retain rotation able adjust spectrum piecewise radial recall instance rbf inputs rbf design integral radial analytic remain two transform efficient parametrization provide explicit piecewise range parametrize eq basis piecewise piecewise via explicitly invert considerable cumulative pick particle substantially mm automatic relevance use radial radial component piecewise require basis partly approximate expressive efficient algorithm optimize likelihood matrix rbf adjust vary radial procedure obtain marginal kernel hadamard retain preserve modify main multiplication uci flexible scalable vary intractable propose competitive intel operate ghz gb ram marginal objective group rbf ard gm test grouping datum rbf ard partitions every average rmse partition fewer tractable rbf ard kernel hyperparameter ard kernel relevance determination scale individually large expansion gaussian mixture gaussian describe regard marginal likelihood rotation radial sparse process individually optimize location frequency heavily case model gm section optimisation section sort quantile initialization minimum optimize signal noise initialize pick well multiply ard draw uniformly standard gaussian covariance assume scale initialize ard function parameterize control ard technique rbf random bandwidth rbf kernel kernel radial ard specifically variable fix expansion ard htb accuracy comparable compute take reciprocal score large accuracy ard rbf five dataset gm basis medium take typically intractable ard gm cancer hardware k protein ct slice road song assess respect compare function htb time leave gm gm become valuable many although sensible combination fine parametrization wish gm model adversarial order gm comparison rbf ard gm hyperparameter commonly implement ard gm large exact ard expansion gm expressive ard popular figure rmse gm gm indeed optimize gm model investigate compare average normalise testing lower correspond gm alternative runtime expressive model ard third testing runtime time efficient consider log transformation supplement gm gm scalable expansion parametrization collection accuracy runtime memory consumption gain group short scalable expressive help scalability flexibility flexibility expressive ex example conjecture axiom rgb dark medium great rich statistical dataset neural network scalability introduce fast flexible parametrize expansion mechanism expansion wide alternative consumption entirely represent tradeoff speed flexibility function class fast adaptive expressive need rich flexible require arbitrarily basis lead restriction conversely offer expansion kernel priori address priori appropriate form expressive purpose radial function frequency expansion mechanism automatically optimisation individually adjust frequency flexibility free lead fitting limitation spread location method computationally number control many basis function furthermore special four kernel range advantage speed describe relate background basic approximation evaluation discussion approximation kernel carlo great flexibility consider kernel learn sum learn parameter separate enable parallelization jointly stochastic achieve promise acoustic instance deep optimize location frequency expansion formalism gain scalability expansion expansion basis flexible kernel mixture arbitrarily basis combine modify generalise kronecker product incomplete grid statistical representation kernel enable ideally suit domain partial gain mixture weight expansion expansion several learn spectral inductive also novel piecewise radial kernel overall perform learn regression likelihood additional innovation denote domain finally schmidt speak semidefinite key represent theorem might infinite instead kernel eq beneficial amount create burden expansion explicit map translation invariance yet invariance violate rotation representation fourier choose typical pick frequency symmetry linearity transform inverse component universal expansion translation approximate density density fouri theorem domain hence directly amenable expansion provide shift however modification allow efficiently form key insight fourier accomplish cost multiplication induce accomplished order preserve translation
volume construct hard rank decomposition achieve k ok high overview approach explain start tool connect opt obtain sample row adaptively row sample sample approach ok far approximate svd choice costly method bind carefully matrix adaptive lemma additionally sample turn opt section column near immediately play orthonormal obvious orthonormal desirable would c ok ok row desirable row step next seem would really want apply ok inequality require need additional construct ok opt k sampling argument lemma derivation relative error call design matrix relative opt ok span opt ok sampling column sampling pick condition lead desirable address primitive know include bss e primitive find employ sample primitive address primitive row summarize algorithm error deterministic issue ok ok primitive primitive primitive column primitive row primitive sample primitive algorithm proportional implement time implement sparsity already section know rest step develop tool design version combine idea input lemma combine idea algorithm possible within subspace restrict see matrix construction product matrix embedding view implement deterministic tool version obtain adaptive body computation provide kernel square low approximation area discuss mention construct problem frobenius relative method select cx theoretical column build theorem error error subspace sampling construct subspace sampling probability proportional leverage row constant see discuss decomposition trace norm relative error near method along zhang eq summarize table ok ok ok k ok ok c ok ok nk nk ok ok ok ok ok ok k k row use lemma twice state article validate careful comparison step score unknown particularly helpful algorithm limitation rank wang zhang et bss column orthonormal ok ok step within numerical skeleton rank svd bound summarize know provide perturbation rank rank opt decomposition k pseudo n rank costly speedup make svd matrix considerably sparsity time omit se instead approximation running describe deterministic relative sub arithmetic deterministic arithmetic construct obtain column assume k result correspond exist randomized compute arithmetic operation plus nk construct summarize exist literature bss ingredient dual ns nr ns equivalently introduce row n jj rr write k r q residual approximate sample adaptive improve column approximation theorem n ip arithmetic operation r np ip q operation connection transpose section nk nk algorithm qr nk simple corollary nc prove matrix construct well want proportional address good good within inside specifically requirement denote detail discuss subspace embed preserve well geometry sparse call I entry choose subspace dimension choose sparse embed opt opt theorem dimension remove review method preserve element b r problem opt opt kk prove combine subset selection tool om next sparsity version c first probability satisfy omit probability proof continue repeat idea although take construct careful projection routine operation ok r ok ok qr r equivalent algorithm run convert truly decomposition un rescale introduce carry version unchanged goal fast simplicity potentially randomize algorithm fast though difference algorithm closely detailed finally theorem analyze precisely input c ok ok algorithm detail choice time three row iii analyze describe lemma next operation om kn k om om om r k follow prove eq q intermediate matrix approximation precise lemma satisfy least suffice notation algorithm k recall argue offer column low argue strong norm follow norm value failure lemma probability argue satisfie write notice fail guarantee eq immediate lemma follow lemma similar satisfy proof satisfy ready implie expectation take expectation failure bind probability sparsity convert truly keep un rescale scale factor carry unchanged closely complexity ok ok qr step input rank input matrix ok ok k algorithm closely sparsity run optimal row intersection construct detail section lemma pointing replace implement detail use next analysis arithmetic need nk k sparse subspace om km main quality theorem implement via column make algorithm argue offer factor rank column indeed choice term offer probability prove combining relative relative follow prove identical construct opt randomly subspace equivalent opt immediate combine satisfy opt opt opt opt opt opt opt eqn write lemma failure eqn follow matrix drop optimality furthermore follow orthonormal failure probability q deterministic convert truly un rescale version introduce give complexity algorithm input ok r ok closely specific implement run iii third construct refer lemma ok ok ok qr r r k formula arithmetic om need find om asymptotic algorithm present prove might independent implement via preserve necessary low first argue satisfied prove term norm connection spectral follow hence offer column base combine relative eq immediate lemma result lemma proof ready prove construction bind follow fact overall unless column outline k ok symmetry corollary approximation observation consider factorization statement theorem three assumption contradiction error ok ok ok ok ok continue give basis look k lemma lemma ok ok integer look like matrix rank n later small error occur approximate cc result possible describe diagonal precisely copy row select ready q column sufficiently q least symmetry immediate corollary symmetric ok ok q extend symmetric mention symmetric appear choose intersection david institute berkeley work like acknowledge advanced project air laboratory contract
furthermore equation impulse still write gaussian depend vector system estimate compute remainder effectively joint impulse parameterize hyperparameter introduce represent flat improper prior accounting positivity factor accord variance choose map solve hyperparameter state assumption recall hard reason scheme end obtain iteratively iteration hyperparameter indicate iteration latent ease notation define iterate step compute employ guarantee subsection scheme key iteration em hyperparameter use accordingly e predictor time instant accounting impulse vector posteriori diagonal element define impulse accounting hyperparameter vector iteration method employ estimate rule student hyperparameter element remarkable establishe solve sequence scalar crucially depend posteriori impulse update admit close hyperparameter principle admit objective function available robust identification obtain experimental em attain output repeat posteriori total residual impulse response update compute operation posteriori differential impulse energy relate posterior recall impulse coefficient parameter flat become accordance conversely must student outlier parameter detect outlier desirable automatic scheme estimation treat aim integrate precisely follow include scheme assume iteration selection criterion satisfactory e consist fast process could reduce integer integer hyperparameter adopt update rule remain average pay principle choice group accordingly reasonable expect force converge estimate nominal behave section numerical ht estimator outlier perform performance monte carlo run generate order gaussians eq way outlier variance generate probability equal noiseless ease trajectory impulse response true impulse response carlo estimator new kernel adopt laplacian hyperparameter except noise degree section opt student model introduce take fit operation impulse thus practice ss ml impulse model see outlier accuracy robust ml presence outlier estimator em opt slight degradation price pay parameter suggest anonymous employ situation scenario find misspecification end perform experiment student noise variance show monte carlo simulation em offer estimator ss capture feature student r heavy around box estimator section equal noiseless carlo runs estimator ss estimator laplacian instead estimation value variance value section different share note case correspond force white plot fit variance degradation em accordance ml method identification laplacian description via gaussians monte see detail impulse gibb noiseless report score time burden lower fit avg em gs average motivate quality score absence em ss regularize identification nonparametric method constitute particular kernel stable spline kernel limitation paper heavy tailed laplacian exploit joint noise hyperparameter efficiently kernel base rely closed show effectiveness outlier compute introduce assume correspond compute expectation respect make proof compose convenient rewrite rearrange position since recall follow first proceed adopt laplacian minimized deal plugging obtain kernel note eq rewrite due hyperparameter depend minimize partitioning adopt two form se recent development system attention type compare classic respect novel method output heavy tailed probability density focus laplacian gaussians cast identification require hyperparameter overcome difficulty posteriori hyperparameter solve maximization method outlier experiment currently history series regularization strategy impulse response compare parametric approach selection usually require method establish aic validation set square estimate machine tc autocorrelation process depend context usually available interpretation impulse process whose hyperparameter marginal integrate dependence impulse retrieve rely measure experimental outlier output describe impulse feed white first signal measure estimate impulse right panel situation outlier output much impulse outlier novel denote identification impulse namely adopt paper allow theoretically impulse implication sufficiently define toeplitz matrix problem make persistent require ease assume laplacian
perform rank curve illustrate regard skeleton principal curve curve fail consistent connect besides fail order curve strict present strict monotonicity function dimensional piece monotone ranking rank like monotone list tangent another pair follow attack attribute performance account always produce acceptable object rank list work rule meta assess ranking list second serve rank perform skeleton principal produce list embed principal cubic five meta existence principal highlight contribution design assessment meta rule unfortunately rank observation object rank link rule theoretically attribute object reasonable list rank approach improve construction et monotonicity consideration improve ordinal domain also capable assess manifold able unsupervised observation object curve serve rank molecular surface bring preserve principal explicitly one cubic strictly monotone corner box avoid confusion end end point control cubic red shape rest formalize next rule namely principal prove five meta function cubic carry rank indicator vector ranking point I I rank list totally require ordinal thing partial proper x easy verify subset eq ranking vary task prefer help rank rank score order order monotone preserve requirement partially totally point state rule indicator life quality one country numerical eq order monotone mapping strictly order converse readily monotonicity respect define differentiable eq strictly monotone monotonicity equal monotonicity monotone decrease versa monotone monotone mapping monotonicity origin exist theorem assume smoothness output value rank score point list sort verify list rank five reasonable list level level rank five feature serve rank able rule rule namely translation example thousand range translation order strictly meta rule rank ordinal problem monotonicity object classify class require strict monotonicity ranking hold indicate high otherwise example refer linearity relationship nonlinearity take rank task score case rank task meanwhile nonlinear smooth smooth yet continuous derivative guarantee rank two line rank approach interpretable parameter allocation ranking characteristic design biased propose perform principal cubic rule summarize dimensional seek direction explain maximal cloud project regard project point order rank skeleton order smooth express skeleton skeleton produce discriminate project parallel horizontal line pca extensive application comprehensive recall principal extension pca summarize indicator appendix principal assume principal cloud project curve ranking unsupervise still rank skeleton instead curve pca score noise measure error influence exclusive indicator score remove correspondence curve inverse take numerical five rule correspondingly minimize optimization determine residual reconstruct obviously achieve means associate rewrite q eq pseudo computation substitute always ill condition optimal intermediate would thereby employ column converge maximum respectively solution rarely root method respectively consider root design curve datum learn vector adopt summarize perform rank task numerical minimum vector unchanged scaling perform end point procedure automatically b curve occur begin therefore find infimum decay stop rank produce along summary unsupervised rule weight ranking cost little assignment weight expert proportion indicator whole rank completely rank five meta rank meta capable evaluate hand rank guide dataset ordinal information determine relation include object influential select rest still observation nothing formulate error adopt observation object object order aggregation b denote keep aggregate well rank ranking suffer monotonicity ranking list information model meta detect ordinal illustrate object dimensional show order respectively since rank object rank list remain table curve give table ordinal candidate unsupervise attribute object significant journal illustration index reflect aspect list evaluate comprehensive propose attack provide skeleton task evaluation open source software gb memory list list cccc cc united life year per control et people indicator indicator visualization fig illustrate shape include linearity nonlinearity indicator begin bring exceed person increase little decrease matter hard evolution control list bottom space point table three curve depict skeleton fig al center assign country country take understand principle parameter size understanding present five meta explain good life quality interpretable easy carry four cc c inf uk mis en learn web journal citation report science social sciences report citation article score remove try science artificial intelligence application illustrate rank fig journal high indicator linear like take frequency transaction rank high transaction bring therefore get comprehensive place mean one whole list account indicator different behavior human positively ranking activity challenge greatly rational unsupervised critical truth unsupervise rank candidate observation multiple domain rank motivate five rule invariance monotonicity linear nonlinear smoothness assess principal formulate cubic b restrict control interior hypercube computer science model rank indicator rank feature principal summarize pca ds ds originally define manifold distance denote wide e variety principal curve perform try smooth meet expression interpret curve formulate bring make interpretation even hard curve rank white box monotone regard totally hold one monotonicity set subsequence converge uniformly sequence converging assume converge eq
economics management university long range bivariate strongly mean expansion full reduction limit process available process stable move average memory application goodness goodness kolmogorov random variable rv cumulative strictly event nx know ep non either weakly dependent rv range constant paper study properly stable orthogonal polynomial discuss cdf rv cdf framework analytic expression testing purpose generate stable rv rv paper approach provide reliable method generation framework key ep discuss reader excellent review general specific relevant discuss bivariate non bivariate expansion average study devote non worth follow provide discussion show essential moving obtain sequence ep concern goodness test organize follow argument ep rv final presents need rv expansion ep provide define sequence stable rv random copy unit lk stable rv characteristic consideration analytic see connect equation rv detail possible notation let g proposition rv transformation specifically finally admissible parameter discussion stable rv well know box transformation integrable density polynomial denote du measurable polynomial v converge explicitly dependence refer transformation otherwise technique cdf r rank polynomial indicate nz mn mn md md expansion ep exhibit bivariate principle weak result construction multiple wiener integral weak equip sup multiple wiener integral copy gaussian integration indicate exclude integration gaussian normalizing introduction non would result z z simple bivariate consider reason satisfy exploit section properly normalize case discuss exploit outline formulae derive formula formulae formula function formulae derive formula compute formula induce discuss detail prove recall need highlight write one follow g z ax need denote integral concerned detail g last obtain concerned bx cx note expectation definition x need integral q transformation turn reasoning reasoning yield z cx proof z parallel reasoning bring follow expectation transform recovered reasoning one far coefficient computation transform recovered reasoning result hypothesis stable previous rv satisfy rv obtain worth appeal ks
marginal learn inductive useful unlabeled identically name optimization globally analyse well art mnist rest paper review multi analyze equivalence equivalence later next input stand confidence rate soft negative class equally approximation definite pairwise hypothesis sign theory define underlie available distribution k k equivalence geometric confident accurately body alone introduce associated orthonormal geometrically origin center ellipsoid angle hypothesis principal crucial iv conversely lie small equivalence multi fit unknown subset pick reveal label unlabeled unobserved account modification suffice set set generality label multi knowledge former study latter study fully partially label label datum possible set label set measure cardinality classification unique work volume soft concern em possess possess th class jt negative label predict label form stacking pairwise provide positive matrix due kronecker positive equivalently e g consequence geometrically origin ellipsoid sign nc nc nk geometric stack soft argument frobenius equivalence class conversely lie constrain nc np identity necessarily domain originally nan propose volume method develop analyze define entry depend whether ever appear label otherwise label loss measuring volume like eliminate scale region become although carry axis subsequently dimensional multi class counterpart fourth fourth loss identically constant undesirable issue discuss thus version rewrite use stacked origin fortunately could fundamental kronecker qp ij ci qp objective lagrange stationary eq locally theorem feasible globally plug sort z nc globally minimize globally proof root eigen summarize fix find case stationary v eigen algorithm dominate computation sort find third problem multi recover cost like comment firstly employ loss latter loss optimization eq would compute eigen complexity secondly eigen decomposition sort eigenvalue hyperparameter setting finally complexity improve fix certain stability bound matrix bound ground truth guarantee fact meet assumption note unconstraine bind train label optimization unconstraine ground soft firstly ensure implement large secondly already correlate correspondence label unlabeled position close unlabele completely uninformative recover notice need even multiple totally number possible globally solution unconstrained optimization theorem appendix instability bound constrain section numerically directly prior latter see impose detail class regularization belong cluster belong specify influence class closely unconstraine motivated propagation viewpoint constrain use generate ratio ground uniform distribution result mean classification rate plot task performance decade cut regularization state art call report similarly standard repeatedly consider another cosine define neighbor involve select cosine fix result cosine similarity local similarity singular exception often state volume approximation setting convex theoretically justify analysis derivative increase interval
ball goal square square assign whenever ball four square shot assign assign point take shot otherwise goal player guess goal square matrix vector encode belief game develop indicate independently distribute neither knowledge guess complex subject belief ball position likewise non likewise type reward perfect perfect correlation mutual deviation lead vector accordance weak symmetry singular present procedure recover ard reward ard result evaluation term column recover original experiment perform combine mean construction player leave infer benchmark reward show red corresponding mean case htb whether algorithm goal receive square receive square stem infer ball exchange affect change calculate result figure conclusion actual reward goal recover player toward consequently observe substantially player shot position distance use quality response situation notion learn game environmental consider discuss typical want policy reward sum minimax policy simulate c rate rational minimax reward minimax reward recall recover reward assume reward first c develop reward rest game comparison episode outcome column ht base c vs vs vs vs vs vs mm vs vs sm vs sm sc vs term win draw outperform tie reasonable b know truly ard ard small win notable winning ard become drop implication crucial infer setting game seem challenge demonstrate difficulty person ill good know observation fortunately many problem additional prior important inverse observe player observe deterministic infer finitely observation action strategy often infer many take game reward likely appropriate generative include perhaps person challenge primary equilibrium possibly nature equilibrium strategy general equilibrium player specify assume equilibrium player player uniformly equilibria reward another player drive toward equilibria player inverse acknowledgement international proposition theorem conjecture axiom term inverse formalize game markov optimality mdps mean nash uniqueness equilibria reasonable inversion equally problem establish foundation competitive person sum reward play follow context abstract extent quality reinforcement aim reward agent behavior environment subject extensive expert demonstrate preference graphic motion controller human decision action understanding experiment move quantitative evidence inverse planning predict goal inference solve agent jointly make rational significantly lead multi propose conceptually rl simplify former treat system environment agent game involve agent propose agent payoff depend action adopt concept agent choice concept weak condition observe agent reward know weight distribute note converge multi planning involve exchange consider include payoff play action take formalize game mdp game bring mdps equilibrium nash uniqueness equilibrium give equally sensible paper study appear present reinforcement multiple compete key consider simultaneous game solve non system decentralize active player want infer third party observer system person sum broad deeply help examine prefer yield tractable computing apply dynamic numerical relationship quality space game play agent equilibrium learn win game terminology definition basic later work reward reward reward play success conclude remark work two player play begin finitely select finitely hence reward sometimes player make transition dependent jointly select action repeat horizon discount reward specify person r kk provide rule player follow loss generality mass refer play state follow player player accord know formulate optimization adopt assign learner make complete observable point reward reward know observe give effort focus determine function optimization generate function formally model let denote denote observe maximize observe selection select minimax reward minimax select minimax mass eq minimax countable normalize reward q minimax posteriori maximize equivalently maximize provide detail use focus gaussian prior reasonable represent nominal reward add leading covariance probability nonzero minimax reward function function otherwise compute map computing map computing maximize prior reward solve devote feasible static minimax person sum payoff theorem direct inverse special goal minimax constraint stage theorem value minimax must constraint relate function relationship game player reward select action expression let express formulate concave equivalent problem solve person sum class reward depend depend action stochastic reward satisfie note expression second policy take action policy inequality finally develop demonstrate
especially sgd significant improve performance speed convergence combine sgd quasi newton rate hoc even work implicit sgd definition statistical comparison furthermore would require solution multiple exploit efficiently implicit efficiently root finding find narrow exploit monotonicity transfer nb r estimate identical implicit optimally tune stability covariate continuous glm n nk n limit sequence lyapunov inverse operator map sgd define decrease sgd iterate typical sequence satisfy assumption form asymptotic optimality averaging scheme impose weak constraint appropriately correct rate satisfie assumption consider matrix n assumption show asymptotically leverage summarize suppose asymptotic explicit estimator n n asymptotically explicit sgd converge sufficiently become implicit stable small variance subtle sample show asymptotic variance definite asymptotic satisfie variance I way compare assumption need prove normality central exploit regularity n satisfy sgd loose compare mle dataset variance n quantify achieve leverage experiment order achieve comparable efficiency gain equivalent minimize eigenvalue need recently extensively tune sgd estimator bias simplify remainder I initial decay explicit limit method however explicit procedure inspection magnitude eigenvalue dominate stability setting convergence stability price convergence implicit j critical sgd eigenvalue less one finding implicit specific consideration avoid etc misspecification learn implicit sgd asymptotic extend introduce assume write parameterization extend sgd sgd eq multiply generality notation statistic generalize result hold regularity expect scale explicit implicit method asymptotically rigorously several notably refer geometry argument prove aforementione impractical consistently inversion sgd transformation sgd new rate complete sgd parameterization asymptotically optimality appropriate estimate optimally invariance parameterization efficiently parameterization correspond parameterization inverse transformation thus jacobian thus eq mle demonstrate particular perform follow experiment conduct running core ram gb technology implicit poisson poisson log sgd unstable rate result implicit preferred equation concern compare function mle procedure mse compare package mle much r package develop elastic net aforementioned implicit sgd vector sgd counterpart machine study additive air public health design fit scale demonstrate leverage theory develop explicit computational devise strategy work uniformly illustrate variance analytically n variance particularly contrast sgd unstable order implicit quantile deviation sgd procedure modify avoid explicit understand determine learn take substantial effort implicit iteratively glm optimal estimation experiment implicit dataset normal binary row determine experiment generate sampling replacement second generate nh slightly algebra eigenvalue sp bs thus use derive theoretically optimal compute numerically range estimate implicit run regression approximate runtime mse well size memory model row observe come loss average mse implicit glm simulate implicit sgd thus almost sample mean square roughly se se implicit sgd fitting computing task view project project update incremental qr decomposition requirement simulated roughly efficiency sgd satisfy run package method hoc file report computation exclude technical unlikely e optimization component utilize regularization efficiently iteration achieve component experiment experiment package first pp control experiment outcome generate tuned experiment generation time average replicate implicit sgd consistently fast sublinear section affect update sgd method regression normal model replicate sgd method maintain scale linearly contrast affect covariate slow sgd significantly slow numerically covariate cross sample report second line median ccccc cross averaged repetition ccccc mse c direct comparison net parameter produce aforementione elastic implicit use therefore situation indicate trend big high sgd outside family procedure compare standard sgd typical benchmark author variation regularization complete observe remarkably misspecification sgd small affect mean rate however maintain stable implicit log sgd air aim risk public health daily roughly g concentration health variable separately city due dataset procedure qr decomposition gb construct datum identical observation implicit sgd fit entire second almost fast home estimate random mb implicit mle replication aforementione reveal suggest limit moderate key efficiently property reveal implicit subtle fitting implicit obtain implicit iteration thus explicit factor fisher information sgd procedure incorporate involve implicit sgd combine efficiency extend implicit available least quickly equation computationally sgd generally n either expect available another sgd estimator easy situation general exponential sample assume parameterization similar f b n unbiased monte implicit monte approximate implicit sgd apply carlo accord suppose satisfie usually normalize pg g triangle normalize generally summation graph use fix independently normalize constant mle assume edge sgd edge binomial edge iii variance obtain p iteration various rate confirm high bias rate slowly bias estimating parameter depict line replication explicit trivial would modify sgd draw show quantify variance theorem invoke asymptotic normality create testing normality justify theoretically consider bootstrappe normality plausible construct consider chebyshev paper algorithm massive term asymptotic bias theoretical principled common achieve theory exponential family suggest term sgd model procedure suggest explicit stability concern procedure outlier misspecification shrinkage combine computational efficiency first experiment suggest implicit repository contain code implementation repository formalize implicit defining procedure adapt clarity unique start define approximation furthermore implicit counter intuitive observation current effect e possible implicit depend future implicit suppose implicit reference definition procedure q furthermore substitute term equation term carry completeness let term converge construct series construction identical definition find requirement since fulfil monotonicity requirement glm notational convenience subscript let fy give fy part multiply get side become solve finally implicit correct h u n straight wish point monotonicity furthermore nr algebra n search successive map recursion recursion n b see therefore collect hand side b e ai ai e n finally substitute second part recursion identical definition recursion n identity obtain line b lyapunov apply n ni ni eq side n side use q explicit method asymptotically implicit n establish explicit notational convenience variance side simplify rewrite remainder mapping q variance since side n mapping application corollary also proof q obvious even formula stability sgd covariate n positive definite variance n asymptotic carlo parameterization inverse ij n corollary efficient descent popularity task approximation contrast implicit version iterate amount shrinkage depend fisher latter implicit hyper gradient affect asymptotic contrast rate agree fisher information bias estimation efficiency maximum avoid careful parameterization estimation stochastic demonstrate real stochastic descent superior efficiently efficient procedure popularity estimation stochastic term next iterate context formula estimation show relate implicit explicit shrinkage observe fisher information never benefit scalar hyper guarantee contrast require agree efficiency suggest careful parameterization imply observation methodology exponential class method compute underlie theory extensive involve compare maximum posteriori however widely fisher iteratively scalable modern hundred million hundred thousand root insight develop naturally extend generalize produce form log assumption er rao traditional running range newton compute mle procedure mle inversion per method scoring replace positive definite fisher similar two algorithm actually identical quasi newton calculate mle quasi hessian update iteration algorithm favorable estimation iteratively expensive estimation massive time complexity roughly sublinear optimization notable sgd simple carefully define stability sgd sgd appeal expensive inversion single furthermore single observation essentially convex parameter I keep simple naturally regression variance regularity theory j sgd procedure write motivate true n procedure derive close explicit sgd converging implicit determine act exactly procedure stable misspecification indeed cause explicit sgd insight stability procedure regard efficiency n n optimal generalize concrete optimally sequence implicit sgd derive aforementioned
jeffreys perform considerably believe interval paper test focus test hypothesis type composite e invert behave interval theorem brownian bridge scale bridge generalization brownian bridge scaling determine force estimator improve substantially bias simulation brownian brownian motion brownian bridge wiener bridge scale brownian alternative brownian brownian interest brownian bridge brownian brownian absence transaction follow stock tend option example phenomenon however price market start early brownian bridge brownian bridge stock bridge suitable order stock show phenomenon stop bridge possible include brownian brownian brownian attract considerable community brownian study focus mle study finally proof well bridge self self extend brownian suffice quite substantial particular mean market market bias correction inverting expectation correct observed posterior prior proportional fisher jeffreys tailed median seem unlikely prior information might preferable informative bound support estimate realization rectangle estimator compare qualitatively nearly unbiased mse thereby upon considerably jeffreys biased correct bayesian estimator towards nearly reliably market heavily correct mse recommend one reason bias preferable jeffreys jeffreys bayesian market situation frequentist financial decision
experiment synthetic answer template scene dataset multi human st question template category test experiment human remove turn demand incorrect answer wrong association difficulty unseen category train uncertain semantic segmentation create world severe drop automatic segmentation class part serious bottleneck world prefer human last generate fact observe trend high counting language substantial incorrect attribute segment g question answer world visual despite uncertain program indicate bring idea scene parse symbolic reasoning combine multi computer vision language seem bring open challenge h answer accuracy c accuracy human baseline de pt minus method automatically answer advance combine discrete uncertain prediction represent human realistic count list answer benchmark modern attempt vision technique like object increase full scene often understand correct labeling term annotation prediction inherent visual propose learn question fact world infer interpretation correctness fact core address answer real world formulation different interpretation scene date substantial serve answer test chain relate ai build test vision early day ai argue answer task step direction combine real world symbolic reasoning framework answer dataset question produce human modern visual benchmark advantage multi approach challenge ahead source different inspire answer natural language supervision rely manual annotation logical form contrast never language goal connect sentence image physical world constrain domain object instance object block build diverse world scene relationship moreover beyond scope reasoning impose challenge answer scene representation deal language efficient spatial reasoning alternative learn language bind unclear integrated execute restrict execute restrictive scenario system color green table probabilistic database entity recognition segmentation apply visual answer visual scene interpretation visual scene account picture confident possible denote although confident opposite benefit wise spatial object coordinate build answer part latent variable formally world perform semantic tree logical logical world pa recursive evaluation relationship subtree template template string predicate search tree model parameter engine answer logical form linguistic phenomenon engine detailed exposition refer reader operate world correspond world overview fact automatic semantic image purpose art semantic recognize position color position represent max define minimal axis object coordinate schema release purpose predefine relation table association class answer consider fact segment different probabilistic database possible interpretation visual scene segmentation answer question semantic segmentation logical semantic segment assignment category segment bind set tuple practice draw every show parallel need synchronization cost sum parallelism question answer end answer pair contain induction consider fact million predicate bad image induce fact batch fact space batch boolean variant tf measure training test dataset website tb description image many color room depict room object image object image color object bag room object answer dataset v annotate consider canonical view depth define spatial uncertainty visual technique prop spatial content collect question answer dataset work annotation synthetic answer pair template template fact answer ask house participant provide answer answer color number set answer impose question question believe robust human jj mean
pde account model physical stochastic pde extend gp vector develop pde physical make adjoint regression application process obtain estimate state bilinear gaussian process regression standard gp finite base pair method incorporate albeit relating combine greatly system nonparametric method functional noise nonparametric bayesian inference gp regression different view think functional define functional inference functional inference theoretically interpret nonparametric bayesian avoid construction present introduce pde demonstrate remark nonparametric bayesian appropriate space pde model state v functional pde describe physical practice knowledge pde find paper shall drop indistinguishable measurement differ output additive distribute acceptable validate good close trust behavior many good due source geometry good produce process proceed review assume training vector vector aggregate output add perspective encode belief account observation mathematically eigenfunction joint conditional joint distribution predictive combination covariance explicitly function predictive implicitly depend impact likelihood see choose gaussian provide among family choose hyperparameter different term model characterize us function regression maximum must describe output linear functional regression knowledge carry advantage model able far disadvantage standard mathematical I differ knowledge characterize output model capture various source model namely first operator operator characterize account observation play operator functional process stochastic pde pde accounting formulate functional adjoint q functional bilinear pde output relate characterize relationship characterize introduce training wish given emphasize collection adjoint determine collection joint observe function give q notice describe share similarity important way difference regression gaussian process ex ex ex n n nonparametric bayesian framework gaussian bayesian framework linear avoid operator introduce family bilinear hyperparameter covariance crucial ingredient bilinear operator form symmetric specify fortunately process allow determine hyperparameter use depend convenient hyperparameter maximizer determine operator compute function subsection functional sense light combine determine describe compute pde pde model basis knowledge pde choose arrive system j follow gaussian spatial ij note term evaluate functional matrix showing grow figure ex compute compute ex compute posterior least square q adjoint coefficient follow mean sum adjoint mean pde light output observe covariance bilinear adjoint equation recall delta adjoint equation complete differ exactly whenever expect state good show optimality state assume bilinear introduce lagrange multiplier constraint iv ex ex partial two adjoint give adjoint knowledge adjoint complete covariance bilinear square also posteriori functional discuss know least advantage whereas mechanism pde heat dirichlet boundary satisfy eq source easy serve everything boundary space observe knowledge replace counterpart problem element element specify functional iv vx chebyshev interval functional element mesh grid free remain observation model furthermore bilinear operator likelihood utilize scale likelihood c xu u mean prediction posterior gp knowledge state figure show considerably prediction heat gp regression square root length exponential see table small gp use good model functional whereas posterior albeit slow regression indicate standard rigorous estimation show region panel confidence region mean measurement state error remarkably rapidly increase confidence see figure poor prediction rigorous summary simple functional gp f statistical improve incorporation prediction state superior propose conclude point nonlinear pde adjoint depend pde preserve due nonlinearity extend oriented infer output rather management mit discussion fa mit framework describe delta functional eq functional target noise I output formalism need without generality diagonal diagonal always diagonal work obtain normalize substituting eq capture weight prediction functional posterior give predictive however fortunately equivalent recall formula inversion obtain need attractive observe need l follow
block imagenet cnn precede operation follow protocol image reflect patch time net imagenet epoch top ht testing building net cifar imagenet imagenet mini share layer parameter model mb base cnn ce cnn ce pc imagenet hold category cnn fine tune decrease mini batch build nevertheless top wise cnn mistake make build net fig collect building net fail receive mass fine layer category protocol edge perform average refer detail imagenet imagenet build category hierarchy overlap gpu exploiting parallelism fine k decrease memory fine classifier directly compression cnn obtain imagenet imagenet cm layer dense imagenet dense net network multi testing net achieve evaluation layer net cnn error net cnn slightly outperform prediction net cnn architecture net building net future extend cnn architecture theoretical image separability object highly demand dedicated convolutional network cnn flat effort leverage structure deep cnns embed deep cnns hierarchy cnn separate easy class coarse category distinguish coarse term category cnn scalable visual cifar imagenet benchmark experiment cnns cnns ht cnns visual scalable algorithm cache batch huge volume training well recent year dataset become object category becomes come along separability category hard cifar belong coarse category cifar nonetheless deep model make flat structure adequate intuitive alternative attempt deep large model impose hierarchy inference dedicate category classifier category cnn classifier cnn slow consume principled hierarchical architecture image task step coarse easy another class fine category adopt principle cnn build upon build currently rank cnns cnns benefit progress cnn fine paradigm fine category block correspond cnn increase contribution novel classification organization coarse fine cnn fine tune logistic category consistency term cnn category cifar imagenet class art cnn integrate hierarchy main cnn cnn image detection parse considerable interest nonlinear either improve expand capacity building building hierarchical deep cnn recognition vast category structure build hierarchy class predefine learn imagenet utilize trade specificity leaf hierarchical achieve speedup certain attempt insufficient various label relation encode hierarchy cnns category mainly scalability constraint exploit hierarchy novel cnns cnn follow notation image respectively category hierarchy fine category end end illustrate comprise namely coarse multiple category left side layer receive pixel low share layer precede block layer component configuration building cnn produce coarse coarse aggregation layer coarse fine coarse one coarse purpose prediction category thresholde enable fine category component coarse probability bottom layer classifier category prediction fine component category fine prediction category fine category layer build final fine common reason first precede agnostic level corner face coarse reduce execution significance last decrease success cnn side receive fine category coarse produce weight coarse image coarse category fine stress layer configuration flexible modular design hand building category grouping category dedicate fine category employ hierarchy hold set evaluating net matrix diagonal set entry discriminate category spectral cluster fine category coarse result level coarse coarse coarse category coarse classifier mistake correct ground label implicitly separability coarse add fine coarse category fine prefer add fine coarse category hold aggregate category branch full set category hierarchy coarse prediction update coarse classifier category linearly coarse amount train training hand training mini batch fine mini gradient fine category large mini decompose multiple training complete sequentially component fine building block cnn precede coarse category cnn coarse fine independently classify category use coarse precede layer initialize keep layer except last cnn coarse category fine tune cnn tune cnn hierarchy category focus classify fix fine fine tuning semantic category predict coarse category keep category category conventional learn mapping coarse category image coarse within mini batch use fine show size mini batch regularization cnn memory execution linearly visual develop execution parameter technique necessary weight negligible conditional weighted accelerate use parametric fine classifier evaluate layer category grow coarse category memory product quantization row store cluster center precision achieve compression hyperparameter benchmark imagenet implement software test cifar natural e whitening size stack cifar cnn building block fine share precede account independent category hierarchy hold coarse category visually layer rate iteration fine tuning mini batch factor corner cifar cnn testing improve net hierarchy coarse either disjoint investigate fig occurrence coarse dataset affect category coarse cifar dataset category occurrence overlap category consistency cm average net double cifar cnn ccc share layer memory sublinear fine building cnn category cifar build net without compression memory execution
know shot shot strategy utilize advanced human annotation stage base eliminate achieve comparable four cifar class exist object recognition zero learn image classification base video surveillance et al humans able class object detector million likely come shot learning paradigm abstract unseen available training recognize unseen object intermediate cascade enable recognize example use semantic relationship reference unseen approach extensive human supervision attribute tight unseen class training propose replace supervision relate loose relationship hierarchy start bag image herein codebook employ probabilistic latent base signature topic unseen class object visual object perform signature representation publicly cifar effectiveness rest show description representation bridge transfer shot issue class attribute categorization attribute indirect approach image subject subject rich language supervision description commonly though attribute shoot intra belong relationship replace order human supervision need shoot attribute topic attribute free jointly semi latent space modal attribute motivation attribute focus object learn topic unseen eliminate consume human annotation replace topic latent model latent dirichlet allocation model topic representation object class shoot conventional image tag insufficient shot attribute redundant many evaluation effective tag codebook concept utilize stage concept common cluster deduce specifically integrate image namely coarse hierarchy topic among approach attribute inter inter similar shot et model wikipedia represent object object million document wikipedia visual embed devise al concept class unseen classify object relationship computational knowledge instead relate unseen class class learn shot attribute direct use shot concept inference random rf decision recursively split training shannon probability codebook codebook occurrence collection codebook joint document infer unseen zero shot paradigm handle concept discuss infer class shot learn nested figure broad coarse narrow visual concept coarse concept share conceptual fine coarse relationship concept device water house codebook example forest example tree tree build tree difference codebook codebook similarity codebook fine coarse fine substitute shannon utilize codebook codebook drastically belong therefore j fine codebook rf total tree similarity associated histogram codebook bin joint coarse fine c codebook shannon yes yes yes summarize property still rf tree splitting codebook handle tree utilize rf splitting split zero see unseen collect associate class belong conceptual describe introduce novel namely associate create relationship unseen class unseen similarity could relate signature set infer shot predict concept unseen unseen rf codebook use either codebook j codebook codebook build calculate signature pick relate test employ public cifar pose visual challenge term illumination classification perform unless cifar extract pyramid histogram pyramid bin instead put codebook rf therefore descriptor globally nature codebook shape well patch rf codebook public face face random extract similar attribute feature identical combination find nearest unseen class choose relationship propose accuracy compare al increase system histogram expect unseen c feature relative number number unseen category perform test test comparison consistency handle intra variation annotation c propose beyond unseen concept cifar class unseen use testing major exist people dataset resolution pyramid feature learn method compare outperform unseen achieve unseen computational low employ also see class drop class indicate robust capable cifar dataset collect codebook learning strategy utilize rf c unseen electrical
play move eventually move player however since belief condition b probability hypothesis weather sum hypothesis subject player make move move possess draw action prescribe odd underlie correct low correction identify transition state state low subject assign every transition lead remain show reader mathematically conceptual motivation arise account device player correction causal subject belief transition save hypothesis weather htbp term belief illustrate nest vertical axis axis carry conclude calculate belief unchanged observation indistinguishable action read subject weather analogous calculation purely observational virtue action conclude weather weather model intend replacement framework provide common abstract contain minimal causal detailed limited countable axiom causal event causal causal dependency dynamically cause come force suitable cause gene restrict causal le stress causal seem view boundary boundary maintain identify removal tag event attribute subject self subject unity life identity classical cut distinction similarly control artificial intelligence distinction indeed intelligence describe construct plus develop whenever intervention intervention belief subject probability fix induction equal intervention belief along plus measure possess original measure contain trace intervention form action causal intervention suppose intervention basic treat event observation learn gain distinguish world classify classical theory hypothesis explanation light translate somewhat statistically generate external world subject term regard play word numerous reason choose status efficacy study conceptual framework intelligence great modern idea otherwise prohibitive literature stress mathematical agnostic reader indeed compatible interpretation idea forward causality writing wish anonymous suggestion manuscript abstract causality thesis biological principle grant office research modern theory use way provide outcome experiment mechanism bring quantity choice probability phenomena measurement reveal discard generative subject aim observe choose distinction crucial identical stage either stage randomly decide exclude experiment odd right odd last colour swap yes pick colour white figure set protocol player name flow htbp carry auxiliary device fair select prescribe previous odd ball give stage probability space eight outcome measure figure eight probability cccc pick colour black right right white yes white yes yes contain space plan outcome swap colour construct generate outcome colour swap perfect accordance law experiment obtain plausibility black pick belief probability outcome instance first paradigm instead let observe determine take turn randomly leave stage ball protocol swap colour make probability subject belief last stage behaviour stage interaction familiar analytical case sequential order swap change calculation previous exception carry stage experiment whether hence statement swap yes swap attempt first regime information specification possible drawback belief axiom equation tuple attempt add auxiliary whether swap form treat another however fundamentally extend swap extend swap swap consequently would variable indicate whether infinite indicator conceptually swap experiment situation specifically pick right independent follow law odd pick albeit probability outcome new law list experiment swap pick black leave yes yes white yes black yes thought reveal requirement change probability reach consequence knowledge familiar probabilistic correct concrete recall pick infer plausibility would choice first experiment conclusion probabilistic context learn conclude derive game brevity exposition connection game player variety mechanic move chance move game root leave player illustrate internal take move terminal label pay notice strategy walk away empty assume white make move pay bring replace player move strategy choose suboptimal move fig drop pay description white previous know time move game game player von information belong player semantic reach correspond extensive specify player decision partition htbp loop around set omit brevity fig result contrast strategy knowledge see good hard von restrict game result thought precede part causal dag htbp dedicated hold status abstract empty iff member iff neither follow say axiom see set tree nest subset root limit reasoning impossible order partial endow correspond intuitive depend determined nest possible event partition recursively branch reach leave termination class namely union potentially say collection subset consider two turn think possibly exclusive base representation exist axiom induction respectively v n cv n must choose false member prove equality intersection member member axiom place unconditional probability probability connection measure view henceforth drop disjoint q causal provide support skeleton construction visually fig figure htbp measure completely exception introduce observe subset take aforementioned uniqueness two measure stable intersection either either member next lemma measure agree agree prove previous ready define object space serve dependency event causal tuple contain information represent probability say immediate consequence essentially compatible equal importantly unique causal say give causal rise crucially differ causal additional causal serve relation functional dependency outcome direction interval define q q interval previous close closed context two sub process initial different define former instant instant determine causal course notice drop subscript close start instance consider interval unique point discriminant unique discriminant lead contradiction since must true similarly partition prove part part always interval capture split exclusive causal branch give identify respectively process generate representation iff figure member instance appear htbp member algebra generate theorem countable countable repeat construct representation respectively must member similarly imply countable arbitrary countable follow analogous countable ready define space definition intervention define event mass intervention algebra say iff intervention think remove transition root uniqueness member intervention nan intervention check definition axiom contain intervention intervention member algebra compatible intervention eq intervention contain moment branch lead remain intervention branch generate iff gain critical formal measurable value member collection algebra measurable learn far specify understand causal one expect necessary space respectively say picture illustrate also converse representation x measurable converse ss next define endowed causal intervention algebra intervention abstract intervention variable algebra intervention causal interval contain define share perfect accordance far instance space induce member root leave value assignment tree path necessarily measure variable illustrate possibility model dynamically causal succeed causal dependency control obviously causal critical highlight intervention pick direction interested bring intervention critical mutually exclusive circumstance causal pick intervention change theorem successful framework reasoning define notion theory must theoretic theoretic inside extensive game player example induction bayesian causality modern development political shift collective notably axiom put think everything else proposition fundamental subject operate heart economic system education subject sciences inter experience scientific recent advance massive capacity internet improve process scale history medium stock user aim system know hold responsible progress pose novel old one range investigate basis learn make understand question address adequate mathematical subject enable discussion program argue implicit basic concept counterpart study conceptual second I forward probability need causal measure theoretic causal intelligence economic back early discussion idea follow trend fundamental concept seem several dominant see subject study discussion firstly theory secondly abstract nature subject entity unity vary different account speak acquire separation inside rest early belong instance distinction term crucially subject divide belief belief know distinction latter constitute interpretation terminology aforementione describe subject self word responsible reality image symbolic pre structure language entity linguistic material language thought detect possibly cascade crucially association establish related logic computer symbolic subject experience namely perturbation subject pick symbolic thereby belief finally question incoherence consequence point chain randomness entail post detect pattern mathematical prominent theory formal synthesis lebesgue kolmogorov modern start degree hand observe action reflect belief probability logic limitation account subject statistic world belief update govern bayes theory capacity recall theory broad include thought subject throughout life involve randomness form algebra subset operation comprise complement assume pick nature device something phrase pick conceptually algebra universe question proposition aspect extract via symbol complex collection potential hope furthermore three typically next play conceptual constitute associate ground symbolic reality however require symbol boundary model symbolic flow occur particular respectively causal intervention causal relation relation static symbolic analogue theory systematic appear idea causal intervention ccc symbolic space flow intervention establish economic term hope aforementioned suffice summary fully mathematical draw logical inference subject shape necessary formal thorough requirement defer synthesis theoretic interactive subject summary main idea therein always central aspect explanation discussion year receive attention partly express prominent figure logic recent decade computer causal rigorous thorough exist arguably causal draw informally intervention hold choose acyclic influence mention tree capture rich causal although aforementioned definition scope degree ultimately mathematically much later
speed feature drive involved contour simulation construct aim connection cell horizontal preference stimulus intra connectivity pattern find across specie include difference specificity axis orientation affinity matrix fundamentally orientation prominent visual stimulus velocity seem play orientation selective prefer orientation nearby motion direction horizontal strictly order segment align environment segment dataset depict assign instantaneous embed r velocity describe motion affinity understand extension provide improvement velocity change unit velocity way fix maximal velocity assign belong velocity vary random bottom row space r velocity producing contain stimulus magnitude orthogonal velocity apply section use fig affinity velocity evaluate grouping compute number stimulus number point incorrectly noise incorrectly recognize partition correctly obtain repetition change stimulus calculate kernel grouping region proportional percentage repetition stimulus kernel vary stimulus velocity assignment performance orientation velocity separate correspond stimulus compose element low grouping curvature contour stimulus path minimum approximately error dominating indicate width fan stochastic connectivity contour separate worth coefficient mainly length element distant element induce recognize distant give kernel potentially distant contour grouping correlation stimulus contour grouping mostly contour generally contour grouping tend together element order contour connectivity diffusion contour successful property image discrete add curvature scale numerical connectivity argue curvature scale concept deep aspect address analyze non velocity inspection contour curvature orientation however significantly completely eliminate random influence correlation detailed parameter length arc assign explain velocity choose contour diffusion constant grouping capability preserve result comparison obtain spatio view rely affinity temporal assigning point spatial operate instantaneous high indeed eigenvalue level level partition result row element bar stimulus geometrically affinity integration quality angular diffusion kernel coefficient kernel stay value subsection kernel perform connectivity consideration effect velocity contour visual study visual grouping contour trajectory set motion contour level trajectory describe two parameter sensitivity change velocity reflect connectivity high composite spatio surface relative bar level separate level random element total grouping result random noting retrieve fail contour thus partition though inspire carry grouping understand mechanic segment temporal position orientation connectivity propagation direction movement shape v mt measure stimulus orientation solely tune stage subject movement direction speed refer continuously example orientation selective position contour velocity contour tangent paper model construct grouping spatio capability geometric low vision mechanism already indicate spatio play important suggest concrete allow primarily spectral clustering spatio feature space velocity present geometrically isotropic belong contour form spatio surface visual grouping affinity stimulus position detect capability stimulus analysis show velocity affect low percentage spatially extend grouping time extend previous affinity instantaneous affinity position activation detect cope causality evolution work asymmetric one connectivity segmentation spatio realistic assumption behavior like model neural aspect visual task address mean delay dynamic certainly matter aspect understand physical observe implement spatio plausibility accordance framework generalize extension depend extension lead modality discuss tune visual behavior mechanism apply rgb rgb definition well understand property stimulus execution global visual segmentation concept visual task underlie visual experiment present orient patch align along continuous study phenomena stimulus patch recognize co linearity co compatible functional primary detector range link orientation directional operator specialized organization naturally model integral curve algebra plane within seminal therein global movement velocity spatial stimulus motion path level specialized cell spatio temporal spatio visual neuron experimental motion integration lead extension stimulus provide geometric contact position time detect feature structure purely association mechanism extension capability trajectory task spatio temporal completion accordance aim capability spatio temporal grouping address refine structure properly describe dimensional spatio weight whose affinity connectivity grouping laplacian simple deal hypothesis algorithmic focus geometry temporal connectivity stimulus capability visual system implementation principle apart study address problem association motion grouping grouping co circular natural basic actually neural dynamic visual role indicate strong motivation segmentation property could grouping plan arise spatio architecture detail connectivity graph basic principle adopt affinity spatio geometry different constitute spectral artificial feature mechanism spatio temporal segmentation contour shape also relation intra european fellowship prop european community architecture visual contact diffusion admissible take compare behavior layer pathway visual stimulus make reconstruct filtering cell spatio selective cell basically local directional stimulus uncertainty hand cell dimensional eq spatio frequency spatio simultaneous worth capture tune depict subset optimize orientation fundamental stimulus consider fixed spatio temporal image space filter v interpret spatio directional stimulus derivation maximal direction smooth level always orthogonal vector field coordinate q text wise organization primary spatio temporal image hyperplane two represent orientation velocity present induce surface tangent complement hyperplane call contact whole name contact contact several contact among geometrically field concrete along aim contour propagation along force diffusion frame streaming differential eq brownian motion diffusion field mechanism aim move contour spatio spatio segmentation apparent propagation force diffusion process density eq process assign spatio stimulus diffusion worth static consist along associate provide stochastic connectivity appropriately connectivity normalize interpretation process increment see decay represent propagation replace transform another identically consider reach evolution intermediate depend evolution uniform keep track length path notion connectivity worth diffusion visual curvature simplicity treat address outline numerical compute connectivity lack notable obtain several approach flexibility differential carlo discretization use loss discrete covering subset path eq value number pass divide multiplicative result connectivity compute deep numerical reference vary track connectivity kernel stand diffusion diffusion parameter projection connectivity kernel together e vary counterpart contact horizontal curves orientation association extension association spatio orientation velocity see relate fan curve motivate role spatio temporal neural path leave kernel projection variable projection horizontal curve spectral graph previously geometric task broadly problem locality literature huge address reader therein cognitive spatio temporal visual grouping interpret spatial visual three dense embed throughout system normally object lie environment dash embed field segment random orientation stimulus rise line quantify formalize set task rest consider know easily purely cluster branch devote development spectral technique address issue property symmetric affinity construct locality preserve embedding set project affinity matrix segregation step input datum algorithm group geometric construction associate spatio visual vertex originally real vector basic grouping partitioning recursively separate foreground argument improve minimal essentially affinity see symmetric affinity reversible row normalization matrix normalize give eigenvector node edge connect result diagonal eigenvector piece constant function affinity matrix version possess binary spectra purely make pose ideal case affinity point weakly connect several normalize affinity relax cut nice probabilistic real choosing possess relevant cluster look maximum particular cost g adopt fix cluster eigenvector block spectrum order decrease smoothly fig ill pose case facilitate suggest sufficiently close threshold spectrum see transition walk eigenvector row thresholding parameter dynamically assign background table h build affinity upon affinity order decreasing define integer belong fix join less remaining partition fig endow isotropic affinity choose cluster result decay gaussian intuitively describe cluster kind similarity visual second affinity fig perform assign remain noise worth many mostly element orientation position orientation together affinity try separate boundary contour additional suggest consider point contact besides purely presentation whether plausible responsible visual discussion connectivity constitute step motivate connection neuron income heavily incoming discussion possible implementation field symmetry break sufficiently eigenvalue input eigenvector connectivity model eigenvector locally unstable hence generate activity aim reproduce combinatorial principle unit stimulus eigenvector since relative weakly stimulus concrete step already step almost neural eigenvalue magnitude population gamma aspect mark substantial computation proportional activity artificial correspond spatial spatio temporal stimulus different specialized neuron kind nan feature stronger weak locally consider value simplification capability deal feasible represent synthetic generate feature indeed seem connectivity one stimulus affinity connectivity compute compatibility contain one indeed kernel markov generator adapt theoretical setting couple geometry carry reasonable neural model cell spatially feature connection selective connect kernel hermitian mean fundamental fundamental solution angular rotation transform kolmogorov symmetric hand angle degree turn angle process properly reason deal modify cell detect rather geometric connect along angle orientation apply differ affinity depend role connectivity spectral stimulus stimulus segment position center
study lattice state early consider size design complete lattice miss miss disk mcmc lattice disk report mean deviation average conditional contain true parameter latter size lattice average depend increase lattice complete similar design lattice iteration incomplete disk iteration incomplete quantify regression base fit model increase like root roughly lattice iteration mcmc lattice lattice lattice lattice time consider em efficiency profile conditional ts independently maximize note expectation dataset monte ti generate use substitute function maximize generate complete lattice miss disk three parameter close density carlo method prediction also e spectral implement give random disk disk disk multiply table em estimation method replicate note definition difference approximate mle mle comparison mean square define design parameter inaccurate largely negative histogram unimodal fairly notably strong compare posterior deviation ml error maximum note substantially method km km relate fine scale variation vs estimate standard agree quite estimate identical method mle exact mle mcmc error bayesian method likelihood method algorithm find incomplete cholesky composite stein conceptually require updating challenge periodic embed lattice well c plan publication multivariate spatio report reason sampler unobserved embed lattice require infeasible simulation simultaneous updating plan matrix block fourier namely multiplication quadratic summarize inverse multiplication respectively vector multiplication involve vector multiplication require algorithm solve symmetric positive solve equivalent relative tolerance matrix inverse imply block likelihood q nm zero one define block conditional form j j j approximate ignore constant fully lattice domain store unique write sum pt google propose miss likelihood miss surface composite approximation augmentation markov spatial lattice environmental science realization stationary process extremely value make likelihood exact lattice spatial widely lattice value composite method spaced approach maximum spaced field krige process need base recently stochastic function increase feasible solution e one converge paper propose new lattice view periodic value periodic augmentation realization periodic efficiently markov monte maximum carlo em simulate practice compare complete lattice miss approach full probabilistic composite recover surface method application introduce process lattice embed likelihood illustrate extensive stationary isotropic z goal n spaced likelihood computationally require rectangular lattice miss toeplitz toeplitz block incomplete lattice rectangular likelihood require lattice datum incomplete domain lattice consider periodic length embed block augmentation unobserved location embed compute embed simulate lattice lattice highly assume evaluate lattice periodic rectangular matrix fourier simulate random variance eigenvalue random field embed number positive problem require large prohibitive embedding cutoff limit use isotropic modify cutoff compact ensure periodic provide certain covariance definite figure extend scheme lattice miss lattice disk shape panel minimal scheme embed lattice illustrate embed cutoff size miss disk embed lattice lattice periodic complete point inverse matrix advantage multiplication compute excellent stationary grid efficiently embed covariance unconditional fast fourier generate periodic covariance ignore transform product multiplication operation observe datum observe unobserved location denote lattice observe z nn conditional observe infeasible store conditional matrix cholesky substitution direct simulation avoid cholesky proceed two simulate complete field unconditional approach note solve require iterative solve compute exploit form u u conjugate system system system original appendix stop iteration tolerance tolerance multiplication former partition compute multiply also element solve multiplication two fourier transform multiplication cost ideally criterion low include block block incomplete cholesky decomposition generate part imputation maximum expectation consist elsewhere infeasible approach conditional complete parameter avoid computing covariance matrix determinant therefore initial algorithm proceed maximize complete use newton conduct lattice generate isotropic z exponential ratio contain squared covariance still effect variant cutoff describe cutoff quadratic zero approach definite embed allow parameter choose trial em algorithm change size variable reversible use simplify throughout trial simulation lattice function lattice cutoff behavior become dense lattice
quadrature loo handle integrate loo integration make deterministic equivalent importance parameter quadrature also importance loo propose quadrature approach robust tail original maximum easy case numerator close bias difficult towards truncate truncation raw idea limit avoid tail capture level deviation far already show usefulness truncation quadrature loo loo loo well loo asymptotically bayesian loo provide refer originally mean density density mean training density optimistic interpret interpretation correction gibbs change describe criterion posterior quadrature integration series prove loo examine loo apply around expansion leave predictive density expansion expansion match expansion loo expansion loo negligible contribution show gp low observation posterior close log predictive thus clear happens depend accuracy marginal instead could series loo desire accuracy cauchy concave high limit monte quadrature eventually loo numerical drop dependency loo cv approximations handle importance weighting level full style posterior review instead reason experiment property review loo list four datum one available internet classification likely skew often approximation classification set affect difficult loo cv select often analyse survival result report probit probit probit probit logistic censor square function separate scale except probit use censor toolbox tp loo loo loo fact fact la loo loo classification datum performance approximation take length gps flexibility difference density interpret fit relative length get small full loo marginal loo happen locate corner ep loo use cavity marginal cavity distribution look loo estimate large quick la loo ep loo start fail vertical tp vertical dash line flexibility vertical flexibility show combine loo loo weighted hyperparameter work unweighted map importance tp loo loo la loo tp l loo loo show la loo loo provide fast distribution prediction point distribution global quadrature loo give accurate ep loo ep combine accurate full ep loo ep loo fail relative likelihood group multi class low grouping loo loo acknowledgment thank acknowledge resource project response loo derive leave express mode loo solution computational difference treat taylor expansion give two derivation classical example remove change remain variable quadratic linearity likelihood define collect contribution give recognize approximate coincide introduce approximation second account explicit removal likelihood leave simply divide square obtain response equation log likelihood non define obtain remove term mode ie ii side indirect due last contribution linearize laplace get mode derive linear use article model laplace propagation validation forming accurate generic predictive leave laplace propagation cross assess bayesian include bayesian leave loo validation laplace review approximation leave computed cost form explanatory variable variable notation use denote focus scale latent joint prior covariance applicable posterior generalise gaussian clarity interested application expert simplicity logarithmic application specific logarithmic bayesian cross future q approximated validation shift estimate density interesting may reveal influential straightforward validation posterior leave replace fold cv approximation computational form full comparison already show likelihood want usual practical map ep integrate latent change substantially integrate whether improve predictive base continue well predictive additional expectation propagation paper good property linear review notation discussion prior mean covariance characterize correlation zero q variance multivariate gaussian pf analytically need la form latent unnormalized approximation propagation approximate non normalize gaussian site pseudo normalization ep likelihood ep update site site first remove cavity marginal analytically cavity form leibl distribution moment moment use dimensional site approximation match single sequential ep site approximation update parallel ep laplace construct taylor mode posterior marginal laplace write leave marginal brevity drop exact likelihood approximate marginal represent except local locally different marginal approximation cm explanation la la py ic global py ic approximate ep ep use marginal expectation ep denote variance pseudo site simply simple improvement ep ep laplace write site order way cavity response ep la new gaussian predictive account also term approximate laplace ep marginal consume global marginal correspond marginal intensive find taylor method refer la cm correction take taylor mode la discuss value interpolation model fit marginal density ep ep cm factorize term improvement la practically global slow ep approximations small approximation previously posterior integration commonly method marginal posterior approximate posterior narrow may negligible marginal density loo section review loo list base loo importance weight quadrature integration truncate quadrature integration la loo loo loo ep obtain match term loo taylor loo drop bayes add correspondingly remove integrate loo analytic monte version affect result loo unlike leave predictive density often easy sometimes leave alternative remove impact furthermore produce integration gaussian equation leave joint use matrix result predictive integrating loo ep explicitly cavity form site observation loo posterior loo obtain product ep ep loo likelihood analytically generic quadrature method usually use converge cavity cavity cavity accurate loo visually marginal marginal improvement shape moment loo accuracy response theory loo also consistency derivation cavity approximation write unnormalized cavity compute loo numerical loo response theory compare loo equation restrict obtain carlo posterior approximating drop importance proposal loo importance form importance weight explicitly leave loo weight reduce presentation tail obtain loo mass loo towards full loo cause harmonic unstable harmonic marginal harmonic sampling loo effective sample q normalize furthermore detect variance weight loo truncate truncate mean towards provide use importance
lasso paper receive claim topic focus topic explanation centrality natural choice merely report area detection discuss undirected community analyze b undirecte connect think union subset call stand nothing across community community belong I community network cluster approach profile pseudo key modularity lead eigenvector idea first recursive approach method chen profile thousand slow al profile aim improve speed price ignore make tractable score spectral ratio community adjacency associate simple matrix community remark degree remove ratio propose undirected analyze type extend score score network citation remark different measure adjust rand ari information vi large ari vi predict vector component figure plot suggest run record vector label largely inconsistent vi ari vi focus vi ari ari method moderately inconsistent size identify community dot white circle represent four community north community researcher north north researcher parametric parametric compare major lie fan score fan community north explanation fan tie suggest instead assume method inconsistent reason score north follow include branch north cluster connect regard result score meaningful differ several small branch two branch fan htb htb component large dimension reduction node sophisticated theory yu research li help meaningful evolve university bridge connect de berkeley ph berkeley group ph department university start li large west stanford include david etc quantile experimental group ex ex di david ex ex ex j ex ex ex ex de ex ex david ex zhang ex ex l frank ex b also hard node consist call primarily detection present apply assume finding compare correspond ari vi somewhat surprisingly inconsistent ari vi show substantial reasonably ari label ari predict agree three community identify interpret arrange size objective bayes researcher range size variability community triangle north include ann university dimensional size three range include researcher variety area high g bioinformatics community r community identify worth note mostly university ann behave differently either counterpart score identify size community compare panel top community fall community score second fan think belong community branch north cluster north community member b group bayes b score citation direct network way additional network detection citation network htb direct cite citation usually focus component connect citation network edge cite citation community network spectral modularity undirected network modularity however properly direction represent therein method undirected adjacency network think split disjoint community bernoulli heterogeneity degree heterogeneity motivate community network adjacency let leave singular define node edge node cite common two cite least network respectively note separately restrict cluster assume community community partition group th community citation node assign large sophisticated citation n illustrate citation panel cluster suggest community section lk citation axis axis column index show blue bar dot identify multiple citation node weakly account restrict attention associate respectively detection present plot inconsistent score briefly identify scale testing node researcher include bayes berkeley stanford groups david david fan lin li zhang taylor lin yu lin spatial nonparametric short discussion figure consist hard interpret end restrict network I ignore obtain component parametric statistic david li nodes stein parametric statistic david wang communities present htb order understand community network column connect semi parametric spatial c dim exp var selection test among citation semi parametric sort wang chen lin li zhang lin david group north tie tie group score section score assign citation network remain network theoretical stanford strong evenly three community citation former weakly connect citation latter focus component comparison first community two part part researcher close tie lee selection community network seem many node chen li consist total lin yu lin j wu david zhang reasonable yu lin david zhang community part include wang david high taylor lin third node nod david david david cox subset non testing community citation wang second part large testing additional insight testing part consist subset community second close another significant researcher bioinformatic htb bayes citation parametric hard researcher model functional etc sort lin david rao dimensional high sort fan lin community researcher nonparametric david mac stein vi vector community moreover community score observation properly detect investigate centrality community exploratory tool sophisticated tool network present array interesting paper spatial objective machine also set collect limited paper publish year period recognize core limited science science economics finance science recognize one david period bias present serve home serve network space focus paper discussion brief set provide array centrality score detection network underlie method analysis also sometimes detection inconsistent happen light theoretical framework strength interesting issue mixed relationship national influential work popular future citation information study research trend community informative abstract studying report finding pattern trend author suggest per year total number paper publish average number paper result largely year paper paper decrease drop ten collaborative area people enter area increasingly difficult view top wider make substantial also present paper author author way count divide approach way cause insufficient author contribute axis paper author approximately look straight distribution tail htb approach panel suggest contribute top coefficient dispersion paper physics community seem publish evenly another span year set four statistic period usa range fan david high degree author degree suggest investigate time panel year see network community mathematic community physics usual moderately author citation per largely cite neither cite receive highly skewed highly cite paper receive citation coefficient suggest highly skewed observe pattern return favor citation early among distant period proportion decrease roughly proportion distant slowly probably communication increasingly easy blue cross leave probably effect publish cite publish include datum delay citation later overall proportion self distant respectively confirm appear online website department early publish overall delay e publication paper mean distant year quickly overcome focus publish object journal number cite raw set paper remove item book review correction etc title remove leave paper information citation directly every collect author keyword abstract journal etc challenge online strict eventually overcome science little source find good paper could serve format resource paper paper substantial successfully identify web paper combine citation relationship information one mention paper effort uniquely author paper interest consume publish name middle author cause wang wang wang wang second name list consistently example list li three california ann li internal none user also people try hard author spirit introduce however use program use author name g may manually author additional e email address file reader www publication file review correction remove review correction author author rule name cluster manually define cluster name author txt list author txt bipartite adjacency txt adjacency citation txt citation jj david fan li partially grant network paper published analyze focus centrality community pattern trend cite meaningful community group statistic well one machine find author distant suggest increasingly collaborative competitive finding topological ground relate social frequently interest area scientific topological researcher understand useful range researcher research topic citation network convenient address question hand resource e google convenient collect citation network provide community many aspect also study help assess study recognize people tie researcher effort pay collect aspect network know community structure community truth analyze effort collect citation set base publish half statistical journal american association journal provide social truth set theory understand topological structure last project collect cover long period analyze network centrality area collaborative
small consist document article simple document layer multinomial visible see minibatch set th h x derivation bind h h expressive belief inference none approximate propose non sampling inference jointly maximize estimator apply variance reduction sigmoid belief network show outperform mnist achieve powerful globally normalize deep fairly counterpart system highly latent challenging difficulty pose tend suffer simplest difficult state observation update provide alternative method efficient approximate fully factor variational posterior expectation analytically however highly expressive one expectation simple variational difficulty variational sigmoid belief net optimize log fit often derivation propose combine feedforward implement inference maximize gradient although network apply technique result feedforward compare many exact highly independent much store observation training discard handle discrete continuous latent variational complex dependency employ sophisticated variational bind primary naive practical range applicability show train sigmoid well capable effectiveness scalability state intractable option variational standard variational serve exact simple easy kullback leibler divergence distribution maximize variational distribution well variational observation variational approach feedforward distribution architecture inference architecture deal architecture approximation locally maximize scalability estimating gradient variational simplify notation parameter involve involve intractable special carlo objective anneal schedule convergence however heavily section gradient behave pose scale inside potentially result slow gradient next practical estimate useful practice eq inference effectively seem want distribution turn distribution affect see depend affect equivalence evaluate price pay estimate suggest subtract simple make adapt systematic subtract observation affect expect depend elaborate implement though account magnitude baseline improve incorporate baseline contrast elaborate baseline distribution easier center trivial magnitude dramatically variability fix center run normalization rate stop signal great computing update provide supplementary material make assumption however advantage property noisy instead global signal involve remove term signal layer latent posterior naturally denote layer learn variational use law iterate rewrite use drop without signal signal simply layer signal expect within structure apply inference network whether factorial case far yield leave explore idea training approximate variational go back derive autoencoder encoder inference respectively however gradient model infeasible feedforward perform initialization boltzmann machine recognition match marginal involve limited inference call stochastic bayes feedforward model perform approximate train optimize considerably use model handle inference dependency benefit treatment value converge fast relate sigmoid belief framework feedforward approximate concentrate deterministic handle latent thresholding unit ignore thresholde variational absence generation feedforward year optimize use unlike traditional learning network fully factorize field share enjoy scalability applicability wide algorithm introduce machine augment analogous updating phase update eq recognition generate cause model distribution optimize optimize bound recognition easy estimate training network high estimate naive section improve see reinforcement learning depend output update output reduce thus serve considerable baseline reduction rl likely contain intended demonstrate handle generative randomly use early base configuration see input dependent hide unit superior anneal report performance preliminary experiment rate report dependency within signal instead report variational bind considerably evaluation perform benchmark generative handwritten test training model center subtract work layer baseline normalization dependent baseline appear compare two gap combination exclude reduction effect
run current modeling approach extend naturally work datum accurate look infer simulator produce tractable detail free set eq vector modeling simulator generate indicate simulator pseudo parameterize control approximately infer delta repeatedly act slack however prefer improve unfortunately trade approximation large large rejection sampler marginal sampler iteration mcmc simulator accept parameter denominator carry marginal unbiased interestingly view lead sample sampler suffer eqn attain mix away sometimes denominator numerator procedure convergence mix lf marginal interpret fluctuation propose approximate uncertainty acceptance fluctuation repeatedly produce distribution clearly delta confidence allow local discuss pseudo sequential monte next discuss introduce pseudo parameter e replace simulator order also eq analytically give satisfying simulation imply bias use similarity motivation extremely sampler accept result see analyze decide sufficiently confident accept study general case eqn explicit sampler accept probability simulation would similarly significant replace normal expression randomize error either condition integrate unconditional error cdf accept mh probability actually probability accept error unconditional monte carlo analytic cdf hand tool adaptive start mh fine draw user threshold accuracy mcmc simulation higher uncertainty around mh confident usual actual another close nevertheless remain serious expensive simulation mh section simulation improve consequence eventually eliminate input c eqn mention introduction simulation extremely expensive mcmc unless store accept perform provide gaussian purpose simulation conduct simulated able away simulation marginal us conduct confident accept decision synthetic likelihood frequentist favor nonparametric literature surrogate surface surface directly independent process well model single joint co although may assume robust full gps experiment gps well enough bivariate distribution covariance input kernel evaluation th acceptance evaluation eqn gps abc mh eqn bivariate step expectation take carlo mh input force current step analogous acquisition implicit goal mcmc simulation training training output hyperparameter may key gps frequency simulation surrogate confident region gps uncertainty introduction consider reduce ingredient gps abc differences procedure aspect gps abc approximate gps abc proof approximate step chain stationary distribution mild bound abc fit proof approximation stationary add two adaptation ergodicity gps abc acquire decrease adaptation resemble ergodicity gps abc latter experiment toy stochastic biological system experiment correctness synthetic henceforth gps gps abc demonstrate illustrate output independence demonstrate along parameter spirit simulation run marginal abc unconditional mh additional vary mean different gps run one simulation great involve gps illustrative gps statistic generating rate parameter exponential vector draw distribution observation statistic simulator process exponential e generating experiment seed experiment show row abc histogram abc abc magnitude simulation bias bottom right gp figure algorithms kernel abc abc likelihood gaussian posterior dash line discard run sampler gps point run sampler algorithm use kernel abc toy pseudo sampler rate abc slowly note smooth measure may per adaptive simulation gps significant kernel abc require simulation gp model mode shown dash interior training red circle uncertain axis mode indicate line uncertainty indicate population compete population use simulation produce setting model simulator replicate series along generate series generate q acceptance total significance maximal maximum replicate base log series degenerate etc simulator ht last require simulation call gps call compare use gps abc abc abc twice gps control step mean gaussian deviation dimension scatter plot abc interesting relationship inspection mode abc due gps abc run estimator force gps full interpret approximation difficult determine estimator influence enforce likelihood lack abc call abc million expensive versus heavily gps abc gps abc top posterior similar covariance posterior potential gps apply cell cell terminal begin initial update statistic log consideration show simply arbitrarily broad figure experiment allow number change bottom miss value solid curve distribution draw three predictive whereas shift towards value row row case predictive large compare distribution remain demonstrate checking algorithm reason choice check pre part observation simulator simulator b n
c profile equilibrium eq q player nash player game finite player player type player transition resolve along infer opponent adopt odd game game similar characteristic corollary game various economic field importance political biology describe solve rational interested model opponent player behavior infer move player motivation present introduce whose game try know repeat incomplete describe game state know actual nature player actual past payoff action know markov stochastic player actual state transition game recurrence game lack bad formally repeat player player probability state type introduce game payoff know know type action transition know solve player hide compare way stochastic process exactly observable state observable know idea get weather help office office day weather weather possibility depend worker observe people come weather know weather state consist state transition store state store observation state produce array store state htb transition label describe game inspire player hide payoff opponent player player state case player know opponent opponent markov game player transition sequence behavior opponent due tool game aim game chain play accordance game bayesian inference accomplish markov observable equilibrium infer player array storing finite array store state nash store opponent playing game observable great else play frame versus situation strategy server try open receiver action server type choose strategy player interested receiver need opponent reduce loose obtain game hmm opponent choice use opponent payoff c open center open profile scenario original observation every observation game behavior hmm stay hmm e picture odd bayesian hmm stay increase odd train hmm close c bayesian h two take transition hide information think type know opponent game use infer type markov use method quite work present solution special lack
analysis resort construct graph adapt use rate movie movie rating movie rate rating rate movie take star share distance become natural weight w ij distance fast one nearly identical around weight fast distance increase star transfer star distance find well nn explanation user positively recommendation essential emphasize paper access movie business cross validation evaluate vary test result plot level clearly outperform observation seem nuclear regularization alone green dense nuclear reach combine observation sparsity nuclear message improve row optimization matrix offer recommendation combine collaborative solved pose iterative admm scheme system real validate recovery suggest usually people completion construction furthermore completion propose algorithm way uniformity matrix partially special weight nuclear non movie influence quality correct improve firstly scheme scalability deal matrix big dataset qwertyu qwertyu b qwertyu qwertyu qwertyu qwertyu qwertyu qwertyu qwertyu qwertyu qwertyu qwertyu I qwertyu qwertyu qwertyu qwertyu qwertyu qwertyu n qwertyu qwertyu qwertyu p qwertyu qwertyu qwertyu qwertyu qwertyu qwertyu qwertyu qwertyu u qwertyu qwertyu qwertyu qwertyu qwertyu qwertyu qwertyu qwertyu qwertyu qwertyu qwertyu qwertyu qwertyu qwertyu qwertyu qwertyu qwertyu qwertyu qwertyu qwertyu qwertyu qwertyu qwertyu qwertyu qwertyu qwertyu qwertyu qwertyu qwertyu qwertyu qwertyu qwertyu ed universit ed de miss recent although low hard exactly matrix completion proximity community make recommender receive encode graph manifold constrain order force pose iterative study propose recovery completion namely sense recovery actually possible reconstruct appear recovery problem low sparse important world cast matrix completion problem include recommendation completion indeed become movie recommendation netflix amazon netflix see collaborative filtering widely use infer recommendation pattern miss entry recovery show perfect refine reference therein study consider propose column assume completely netflix age education etc movie release year actor origin country advantage movie rate movie recovery new literature induce factorize low good knowledge graph cluster along methodology collaborative content user idea force solution manifold movie standard graph user movie uniform achieve laplacian heat diffusion manifold non convexity challenge optimization problem recently exist completion collaborative recommender netflix popular rank denote element replace surrogate semidefinite incoherent sufficiently minimizer coincide minimizer observation contaminate depend type square frobenius mask notable representation column smoothness row close linear eigenvector row column laplacian overall simultaneous outer product wise laplacian efficiently success choice introduce follow handle constraint constitute success fact require sub rather augment lagrangian x close strong duality primal saddle augment lagrangian e find saddle iterative admm prove approach fast sub x k v singular c g h h k condition ba symmetric kronecker product conjugate gradient complexity dominate nuclear cg nn evaluation recovery synthetic netflix like model dataset two assumption column graph movie recommendation integer netflix rating movie graph group connect within neighbor edge belong likewise movie group type graph cluster fig movie often noiseless performance compare particular reconstruct report observe green line poorly nuclear line nuclear obtain alone discretized laplacian tendency try reconstruct nuclear fig quality green nuclear norm smoothness green line dash add noise connectivity green dash wrong benefit low regularization solid therefore graph graph observation observation
n final equality homogeneity exponent location totally always challenge work spectral closed form exponent equation density combinatorial dimension increase rapid respect situation expression lack close block chapter compute one prior define function true datum generate produce likelihood generalise need available importance sampling produce appropriately use observed weight algorithms order importance algorithm w useful computationally prohibitive procedure evaluate procedure develop bayesian operate parameter quickly general precise credible dataset retain conversely unlikely discard draw repeat exact yet approximation intractable principle heuristic make precise generate generate reweighte generate metric mahalanobi discard correspond receive weight retain sufficiently close reject sample draw posterior auxiliary sense quality kernel get dy dy otherwise accept candidate parameter exactly reproduce unless discrete unlikely practically typically great extreme dy dy show weight abc approximation distribution give clear fidelity lie closeness inferential low aside approximation practice commonly great quality abc univariate taylor expansion substitution h dy du function form estimation likelihood interpretation limitation abc know suffer curse dimensionality arguably impractical appropriate univariate measure unlikely reproduce observe dataset dimension simple statistic manner image summary summary possible posterior give p ds l due consideration give summary sufficient information sufficient loss precise abc precise reduction kernel offset information summary input vector simulate statistic h generate work avoid draw come abc pls pg intractable illustration block year central notable extreme cause historical flow world human economic previously daily mm maximum daily record open univariate approximately model generalise specification easily obtain posterior markov chain carlo gold specification dataset three parameter analysis choice practice abc curse summary model dimension prior implement algorithm distribution improper situation explicit set exploit know scale correlation estimate need approximately correct identify area statistic convenience smoothing kernel scale summary note definition line strictly first however procedure draw determine abc practice e posterior left centre dataset solid four centre abc decrease apparent generate independent small decrease course sufficient even close quantile million abc approximations dot line however beginning improve quantile obvious although achieve match likely occur due dependence likely element summary appear overhead vector line broad even quantile enough use slightly well posterior information seem quite increase decrease far statistic practice regression adjustment improve abc determine true unknown see abc intuitive procedure evaluate summary highly parameter price pay discuss suffer curse purpose number still sample great posterior explicitly regression correct simplest least adjustment variation regression adjustment ridge adjustment qualitatively different adjustment improve conjunction adjustment describe approximation estimate panel solid statistic abc line regression grey figure illustrate distribution focus vector statistic approximation grey line adjustment indistinguishable use analysis adjustment effectively initial relatively top bottom occur effectively unchanged varied large obtaining adjustment rather identification huge research provide comparative classified good indirect approach include abc reliably single well good currently list sampling generate monte circumstance variety importance importance sampling construct markov augment auxiliary target ls proposal l result always section tractable term detail hasting g monte alternative abc density estimate production clean term production consideration size important inclusion two dimensional inclusion great measurement threshold inferential unobserved cross slice focus extreme problem mathematical process inclusion mutually conditional approximated generalise distribution scale shape standard accordingly spherical inclusion inclusion associate unobserve inclusion diameter observe inclusion great chance term large inclusion adapting problem numerical difficulty function treat unobserve latent assumption inclusion independence plausible family addition poisson consider specify generalise define large diameter uniform ambiguity measurement principal diameter inclusion spherical analytic difficulty extend family inclusion approximate specify implement total million summary denote quantile interpolation distance either identity analyse uniform scale sample approximate dataset bottom row correspond inclusion solid model approximation posterior dash indicate identity mahalanobis grey version subsequent adjusted marginal bottom inclusion dash dot grey adjustment solid black dot marginal spherical clear approximate dot fairly identity dash high variability namely likely highly happen correlation take closeness substantial improvement clearly adjustment inclusion marginally show suggest require initial available qualitatively conclusion good accounting correlation statistic agree posterior difference reflect ellipsoid principal diameter ellipsoid intersection inclusion predict inclusion inclusion analysis suggest depend strongly likely strongly assumption computation factor datum slice inclusion choice minimum weather weather weather report pre weather contract undesirable weather occur low weather reason weather variable payment event interest threshold payment payment excess show mathematically weather positively correlate financial outcome weather derivative positively consider payment derivative recognition essential max process stable generalization extreme hold subset process stable stationary poisson intensity replicate max fr margins stable convenient interpretation center independent take process extend stationary intensity measure eq fr margin bivariate underlying correlation family correlation ern realization mat ern correlation range summarize simulate realization bivariate sample term location get location bound coefficient complete dependence think independent location consideration pair eq observe bivariate show write coefficient explicitly transform fr coefficient eq take coefficient grow rapidly dimensional statistic prefer add detail triplet value draw fr margins location average produce absolute deviation measure parameter quantile maxima daily take location united record locate west fr spatial parameter aim candidate process location credible parameter space shape shaped shape function incorporate parameter solid posterior line either evaluated numerically write form various model complexity intractable require simulate suitable summary former trivial subtle accurately simulate continue
force eq besides also effective return pareto behaviour true normalization enough correct behaviour return return normalization guarantee cover critical area difficult objective case parametrization force pass limit show without concentrate center distance area normalization correct behaviour slightly seem try convex combination obtain normalization without try produce behaviour lead dominate converge correct completely choice figure among return pareto accuracy pareto reservoir model water store reservoir control release transition depend reservoir objective problem reader consider objective immediate reservoir threshold min due water discount objective different phase episode use policy objective radial convergent parametrization force pareto capability decide simple indicator start parametrization report start pareto covering approximation produce important policy release reservoir phase modification unnecessary penalty try add return mixed insight use thereby set precise pareto cover indicator penalization term monotonic previous take section go metric propose idea dominate solution enough figure behave mean use dominate solution dominate short pareto solution mean increase use wide achieve cover circle circular fill draw thick document material multi pareto markov decision policy pareto differently algorithm optimization perform gradient ascent run step improve idea exploit optimize get close pareto besides metric assess quality pareto finally propose empirically evaluate pareto publish conference artificial supplement complete provide smooth map volume r bb directly volume direct indirect derivative expand kronecker bb permutation last term expand I differential give hessian reservoir study paper addition present normalization take pareto area integral unitary eq combination area function idea obtain dominate area cover pass iteration trend objective study discrete quadratic follow dynamic column semidefinite couple policy extended objective particular
bs electrical engineering university institute fellowship frank fellowship paper award conference electrical college engineering college interest recognition datum storage book technical apply associate security technology award filter object track traditionally use fourier dft correlation exist account fact circular previously design truly optimal criterion eliminate ensure optimization circular benefit diverse challenge associate version http net project filter correlation filter localization transform signal processing recognition tracking possess attractive make task localization classify scene shift invariant object recognize peak location object attractive stage template refer spatial domain image concept temporal template design sharp origin image peak peak scene may large template peak object detect peak peak peak traditionally typically metric minimize correlation actual correlation frequency element fourier dft image algorithm multiplication multiplication circular correlation circular fig circular part add influence localization cf circular avoid template resolve ignore carry correlation cf use exist energy cf design design present localization obtain example circular correlate appropriately zero linear output accomplish dft circular correlation output generate dft signal part correlation effect affect design mostly ignore paper force template correlation rather circular zero consistent criterion design correlation capability offer cf span decade introduce filter extend subject extension scalar efficient descent base numerically comparison demonstrate superior main convolutional convolutional neural potentially thin thin green cm inner sep pt rectangle fill corner height fill red distance height em dft dft dft node zero p image cf template dft dft cf cf cf c p zero correlation cf template circular guarantee correlation original criterion organize summarize relate literature circular issue tradeoff discriminant filter filter margin applicable discuss design application face recognition automatic localization object detection detection detection action couple recognize image recently interest bind face track use detector image face use appearance localization object prove detect string video improve wise search recently base correlation pose video detect action motion combine activity apply vector optical histogram orient transform sift ignore problem filter spatial circular domain handle circular recently circular image circular originally cosine effect great detail undesirable fundamentally discuss fall eliminate circular effect practice image take dft although true stage train design eliminate must stage template coupling constraint optimization correlation result get cf next correlation design circular design filter illustrative notational ease expression rest cf design aim gray scale formulation accommodate g sift bold transform operator act channel independently complex transpose typically filter signal design interpret mse signal template cf express mse desire channel template origin peak center dot signal template column whose result solution filter minimize correlation equivalent closed form solution filter output sharp however term precisely multiply together circular correlation spatial circular attempt circular obtain dft result circular force zero eliminate extension explain intuition benefit template generalize template domain mention expensive begin mit long expert main peak signal class template mention zero dft conventional filter design index template effect eliminate formulation demonstrate train template length template template element template signal compute formulation term template energy circular imply peak fig filter note conventional decrease simply adequate considerably become equivalently template ensure template template whereas conventional differ significantly bridge zero circular filter filter three image person database face database face computational reason dimension template output template ht plot template conventional template outside template size template display value template ht filter correlation filter mathematical detail incorporate cf perspective group cf dot image specify unconstrained shape actual inequality constrain idea well derivation template eliminate effect cause circular template dft set note aggregate channel modify formulation minimize enforce peak force template filter notice improve requirement inspire early experiment see notice rapidly evaluate although implement impractical perspective algorithm solve descent unconstraine filter equality filter idea optimize impose constraint spatial impose spatial template project onto signal respectively describe design f iteratively allow satisfying impose unconstrained template tail I transform template outer width height thick blue drop minimum cm thick node xshift yshift xshift yshift amplitude xshift xshift yshift xshift pt yshift black xshift yshift amplitude xshift yshift sep cm height thick fill drop minimum rectangle thick cm minimum rectangle node pt xshift xshift yshift amplitude xshift yshift xshift yshift amplitude xshift yshift xshift pt pt mirror xshift yshift node xshift yshift amplitude mirror xshift yshift node xshift yshift amplitude xshift yshift yshift amplitude xshift yshift black xshift pt yshift spatial constrain step update extract portion template close template desire peak contain signal p loss descent hinge solve closely accelerate along gradient objective newton filter conventional cf conv conv prox accelerate proximal descent conventional prox conv prox prox accelerate descent filter prox conv prox conv prox prox proximal peak spatial form constrain advantageous save memory resource need system complexity fast efficient scalable demonstrate benefit descent comparison close form zero proximal method design terminate use face image platform matlab window intel core ghz ram experiment compute close solution near limit fast show perspective amount require large prefer proximal memory times filter close horizontal image recognition formulation observe counterpart original design account circular fundamentally performance realize cf design performance apply task namely automatic recognition localization object baseline cm cm cm apply database face face database face subject image cf subject result image one correlation repeat calculate energy measure compute solution accelerate proximal descent version comparison demonstrate effectiveness close extremely high computer therefore available filter delta function desire close form also true remove less vary subject select build cf subject rate rank proximal simplicity output achieve performance video database video frame eight video range note make circle long allow frame fig general one template template classify variation select manually background select frame testing frame template plane assign image class classify localization peak e half truth horizontal correct correct cf template training add false positive help filter observation traditional localization recognition performance conventional help development image base base face important accurately determine location bound face face detector experimental setup localization face detector transform translation factor rotation image people partition rest testing evaluate distance ground truth leave respectively human patch suitable great design result design descent learn version localization cf template
intuitively make additive easy false future simulation indicate irrelevant positive dc zero independence coordinate concentration replacement argue probability control false negative hold restrict regression let boundedness strong additive full proof follow theorem hold ambient attain reflect scale relevant smoothness achievable convexity demonstrate condition draw additive symmetric vary ambient diagonal combination set use ac dc mark ac dc plot exact recovery variable reader two minute ghz intel cpu gb vary extent b diagonal case design dc generate figure dc even true additive third use correlate see design moderate correlation ac dc covariate detail repository dc regularization range indicate lar ac dc consistent finding fourth variable select ac dc component ac stage zero dc capture shape agreement spam validation describe ac dc slightly select fold dc figure ac ac dc find similar ac spam optimization optimum cccc additive additive cc ac dc paths mse estimate ac dc framework estimate convexity suffice carry variable additive model convex coordinate quadratic programming establish result scale ambient analysis work building model convexity concavity program concavity grant grant amazon education learn grant convenience impose solution restrict kx coordinate ki optimization omit boundedness constraint verify item conclusion construct variable satisfy complementary c exp nc claim begin define gx support likewise minimizer follow concave analyze likewise version dc index convex ac dc argument risk error small reach risk risk th enough theorem theorem restrict cb proceed empirical risk bind entropy say empirical plugging thus lemma remove bind w f corollary least h lx verify uniformly h l gaussian vector least put c number furthermore union multiplier plug another union substitute concave step bn nk theorem step term proceed identically concave least condition np statement convergence plug fourth likewise term take dimension complete constant possibly dependent set absolute verify hoeffding union bind probability likewise plug equation norm balance choose n n decompose bound take union twice dependent kx fx second bound away bound function shorthand prove term derivative additive bounded suppose sake contradiction x positive imply gaussian scale inequality least enough least list proof take exist constant result absolutely define line px px extend convex component clear f l uk lk l direct union hoeffding inequality make expression easier suppose fx cc additive distribution verify guess verify interested constant need satisfy section lemma example machine department pa department university il sparse subset quadratic convex follow concave sparse additive smoothness appropriate setting yield false negative efficiently advantage effective screening shape restriction monotonicity convexity concavity natural estimation constrain understand minimax multivariate rigorously problem regression sparse dimensional significant statistical contrast general fit convex sparsity penalty concave residual procedure population mean result negative density second simulation analysis estimation arise reconstruction theory interest assumption natural tractable recently increase activity shape analyze convex surprisingly estimation mle latter estimation lower common convexity estimation knowledge variable variable selection adjust local estimator minimax isolated advance provably scale hold penalization theory technique hashing scheme problem show adaptively scale achievable smooth intrinsic give showing selection study perhaps working induce penalty formulate convex smoothing hilbert space tune additive smoothing regression naturally use finite convex additive convenience actually selection scale dimension intrinsic polynomially exponentially summary technical include give program reformulate efficient scalable grow full appendix provide high description technical detail precise main identifying variable convex false negative population quadratic program throughout vector denote one restrict denote shorthand convexity finite quadratic program equivalent quadratic support hyperplane dimensional quadratic formulation curse subgradient sparsity effective reason appear constraint regularization group small additive additive component appear convexity selection play approximation error component approximation additive additive approximation assumption x support satisfie discuss condition population set convex boundary property twice differentiable fixing prove show zero maintain show screen sufficient additive additive fit series concave notation additive component boundary convex continuous set respective univariate depend result naturally suggest two stage stage convex additive shorthand ac dc ac fit additive function residual nature dc stage process optimization convex function analogous multivariate represent hyperplane subgradient point equation impose support hyperplane suffice univariate characterize subgradient monotonically observation constraint solver descent sparse involve package establish variable consider proof technique manner separate rate irrelevant ac negative primal theorem differentiable involve signal make additive procedure make stage regression assume future boundedness new hold ac dc np n achievable ambient estimation reflect respect smoothness mark irrelevant mistake inherent population example convexity assumption begin additive mild integrable eq stationarity solution result use result build stationarity state e strictly convexity guarantee turn fix eq uniquely optimality convex therefore kx x fx distribution let fx particular fx example uniform mark therefore expect intercept slope pt additive understand capture call intuitive function differentiable depend suppose eq say imply population yield general support weak satisfied property boundary density follow support twice differentiable present fix slice show slice convex together still maintain convexity fx f shorthand likewise integral boundary zero integral necessarily hessian must zero column gradient respect play variable function additive twice expect condition find direction imply depend zero additive case additive component parametric function similar next certain convex dimensional appendix additive close easy additive violate concave function numerical arbitrarily mixture gaussian uniform distribution square square boundary closely approximate distribution additive integration true nonzero zero approximate htp gaussian importance boundary although condition additive define reason convex prefer free approximate abuse notation use represent function variable fit x satisfies continuous function let differentiable depend kx h f identical replace x let concave center close k kx twice differentiable away large second must eq center kx first xx conclude imply uniqueness constraint objective strongly uniqueness screen population concave population mark irrelevant fit separately second procedure ac straightforward construct screen encourage sparsity penalty estimate penalty component appear dimensional equivalent show reformulate ac additive describe estimate scalar offset support hyperplane I impose constraint suffice since univariate characterize subgradient scalar increase monotonically lead nk sorted order explicitly kkt condition additive constraint qp relatively notice couple error term optimization vector nr ik r ni k introduce impose penalty subproblem package use www com cycle covariate convergence cycle admm qp input dc modification modify inequality reformulate
brownian motion k x mm specie group specie record level tend recorded aim remain variable improve group mass motion conclusion refine well explanation study body mass brownian motion parameter check brownian motion size size analyze remain front end relationship df log aic aic bm ht remain front end notice value bm aic homogeneous unknown body suggest trait homogeneous currently available parametrize describe laplace motion comparative relax homogeneity present investigation analytical formal couple process type evolutionary author centre f foundation scientific research education mathematic author comparative multivariate adaptation date page work article author title model road journal page article author title comparative volume page author origin species article author author author title journal date page author author title comparative identify shift character journal evolution date page article author title comparative journal date page title quantitative character date author title journal volume page article title selection comparative adaptation journal volume page f title assess trait comparison journal evolution date article author author author h comparative adaptation randomly evolve environment journal evolution volume date page title evolutionary science page author title detect character journal date page author author title generalize journal page title adaptation journal date page author author author title continuous space journal date page article author author process option pricing journal date page author title g return journal volume date page article author title testing rate trait evolution volume date page author title account comparative adaptation journal biological statistical volume date page article author author title accounting journal page author author competition journal date page book author team title foundation statistical laplace majority assume evolutionary process homogeneous offer rather expensive way preliminary whether offer establish species specie take point specie come sample due history due specie trait notice birth availability genomic allow comparative challenge comparative run observe trait currently branching mean way increment trait label motion xt trait specie mean trait brownian major drawback variance estimate discuss correspond however consider complicated brownian motion comparative vast comparative limited allow sequence well something never majority laplace application consider laplace motion variance hold non interest idea laplace relate put formal mathematical evy parameter em exploit laplace treat length variance miss comparative equivalent change branch
bandwidth trick batch performance achieve sdca propose perform except sdca dataset prefer rather sdca good feature sdca sc sc preferable sophisticated decision utilize could computation requirement huge costly computational cost efficiently sdca operation prohibitive ccc mnist cifar imagenet block compare convolution filter layer specifically filter dimension apart predicate bandwidth median trick batch dimension random mae neural nets chemical predict molecular database trick batch million contribute method scale artificial randomness doubly stochastic machine successfully meanwhile achieve compare several dataset comparable sophisticated support nsf grant microsoft fellowship fellowship nsf gm nsf edu school institute technology edu science cs edu general scalable learning try hard novel concept rely express solve gradient number function income reproduce hilbert readily regime net achieve competitive net million energy million handwritten digits feature pt pt pt ex general scalable scale theoretical remain incomplete enough neural net exploit sophisticated architecture virtual deal invariance bottleneck scale storage kernel usually store attempt effort algebra kernel require operation basis nystr om incomplete decomposition follow strategy observe percentage loss performance regularity kernel typically nearly kernel good ability impractical dataset preprocessing memory feature directly approach point compute operate generalization need approach often feature well procedure solution obtain straightforward issue kernel coordinate coordinate iteration incur strategy thus computation memory requirement however serious drawback keep test big classification summary want inspire novel call rely rkh descent make functional behind property long unbiased random generator fact initial exploit memory method interestingly possess ridge logistic different type kernel adapt generator optimize flexibility method stream random computation key generation seed point complexity memory doubly allow prohibitive streaming keep program sample pre seed need guarantee nontrivial analysis involve newly recurrence relation might outside rkhs optimal introduce contribute rate kind independent interest regime net net handwritten digits mnist material million imagenet suggest method replace neural net large scale world nonparametric remainder support kernel kernel positive definite pd process play design pd x relate process kernel shift characterize fourier invariant pd kx b kernel distribution ball explicit sphere random feature design dot additive homogeneous hellinger kernel reverse kx another reproduce hilbert rkhs rkhs exist kx exist pd dominate inspire work concept issue potentially subgradient respect rkhs try linear direction kx stochastic point inspire process additional doubly doubly outside since outside b associate functional source randomness development scalable meanwhile also create analysis deal carefully mm seed mm seed key intuition behind randomness another artificial kernel intuition feature doubly functional determined seed suffice task proceed functional gradient algorithm summarize perform feature maintain collection feature align obtain need well modify convergence analysis respectively iteration simple form doubly functional size convergence mini random empirical covariance potentially expectation high rkh generalization compare gradient estimator rkh inequality estimator ahead exist optimal solution loss generate x mm present theorem due sketch proof appendix q lm eq probability fashion technical difficulty rkh construct intermediate difference f term due apply rkh randomness recursion slightly error bind lipschitz continuity jensen rate classical strongly convex classical speak convex feature contribute therefore still able log achieve adopt classical refined sophisticated reduce sgd batch tuning trade desire error quantity number memory achieve prescribe require rank functional sdca sdca sdca om method run dual similarly combine stochastic mirror nystr r htb cc preprocesse doubly sgd sdca sdca sdca r one method r sdca dependency factor interested random procedure sdca clear refined term iteration memory requirement c doubly sdca r compare medium net yes last virtual mm c cccc dim ridge yes forest svm yes yes imagenet compare seven method kernel adopt criterion purpose stop entire sc stopping criterion design motivation bottleneck ability advantage within time sc preprocesse nystr om
extract show approach efficient flow chart figure htb publicly dataset http roc auc disease relatively compare technique rest organize processing component system present detail experimental methodology section extraction classify specific reliable decision serve segmentation propose field tangent field enhance connectivity structure classify disjoint simple texture descriptor train classify image information representation reflect geometry structure signal technique extract filter representation supervise sign appear red dot efficiently preprocesse primary sign occur shape also detection combine preprocesse well figure related image htb refer vision automatic dr roughly center location define incorporate circular shaped structure area center correctly reference e system aspect appearance certain position indicate advanced rare serious like detection computer reliability separate dr detectors section basic concept literature concept formalize base dr definition vector correspond respectively function majority voting weight assign weighted majority voting classifier follow classifiers dr know classifier test select dr search classifier select far end formal dd member remove description ensemble one search classifier member ensemble contain good experimental study publicly database compressed image pixel range r clinical patient stand serious appearance proportion r r http fr feature extract also table assessment number image severe lack part screening describe detection find confidence represent normalize divide center regard feature diameter fm negative scalar confidence dr large dr description feature result dr severe dr confidence level pixel confidence euclidean fm dr ensemble alternate decision knn perceptron naive forest svm selection q false false negative classification apply realization classifier one give respectively energy ensemble fold cross energy function database sensitivity specificity fit receiver evaluate strategy investigate sign challenge minor visually sign dr advanced r r table specificity fusion dr table accurate bold c vs c forward htb dr dr regard perform ensemble specificity use fusion dr dr sensitivity specificity achieve search see table energy vs dr r similarly ensemble dr three energy function fact r bias balanced dataset vs specificity accuracy dr specificity backward search similar specificity backward strategy htb c vs sensitivity specificity vs r sensitivity specificity accuracy forward fusion effective aggregate confirm however suggest strategy sensitivity specificity c fusion strategy specificity ensemble search method recommend automatic screening group evaluate proportion image dr completely different meaningful area provide regard comparison lack sensitivity specificity auc dataset comparative r dr also base solely confirm necessity component htb htb vs system sensitivity specificity auc htb htb dr dr specificity single propose ensemble dr opposite art level create extensively publicly area outperform system specificity close recommendation association dr european european grant nk develop computer system office research technology contract om european european inf ensemble method screening extract assessment pre fm
paper challenge vision perspective camera view overlap helpful appearance camera angle illumination camera miss low individual camera camera view view pair assign structured bipartite simultaneously potential match fig camera view probe set entity red matching require similarity entity view learn instance manually formulate pose text encode word priori text learn way test weight word visual similar word text ambiguity issue word significant variation appearance due illumination handle distortion base occurrence visual word weight different occurrence statistic motivate co visual behave similarly view occurrence see observe image large view co pattern instance statistically speak white color camera camera light camera negative pair novel occurrence capture important first encode sufficiently word result embed kernel incorporate appearance locality sensitive co occurrence interpret transfer illumination common comparison change camera visual camera method appearance across learn visual occurrence statistically camera identity co occurrence robust co occurrence contribution structure match deal shot shoot unified appearance visual word co outperform art benchmark efficiency testing receive seek probe broadly focus focus metric learning aim spatial video sequence discriminative representation viewpoint attempt match attempt learn appearance ambiguity distortion aim transfer template matching contrast metric learning attempt positively appearance base shot shoot entity image literature et al good discriminative mid describe descriptor entity redundant shoot wu et locality regularize image boundary al representation overall content deal entity entity probe meanwhile handle shoot multi shot accounting appearance adaptive function fundamentally base learn entity impose semidefinite enforce consistency person camera person view person match camera pairwise camera contrast testing approach bipartite priori structure liu utilize structure integrate metric color ensemble individual classifier feature pair feasible unlike approach appearance express basis feasible appearance change occurrence decision function co structural induce ground truth enforce globally assignment occurrence partially people recall paper align identification benchmark level deal issue person track association world object tracking association temporal lead totally goal aim correct entity test appearance information associate appearance adjacent locally prediction detail list report camera common sequel overview location let camera match depict scenario entity image shoot multi shot entity probe green exist reason pairwise structure matching intuition binary match probe entity bipartite graph account ij probe probe correct among scenario unknown arbitrary similarity model function similarity goal build intuition collection document probe tuple probe namely w training instance similarity obtain v v match bipartite truth bipartite constrain rather need encode visual typically match challenge appearance camera similarity co occurrence pattern insight appearance way static pairwise occurrence visual ambiguity motivate entity camera view view along analogously likelihood co visual match instance function empirically estimate visual contribution location visual occur visual simultaneously parameter along truth approach let bipartite matching entity represent two probe insight structure narrow space structure graph knowledge node probe entity match entity probe correct match among however nod test enforce structural probe entity probe match likelihood constrain help avoid entity probe entity graph test probe node accordingly degree set degree degree calculate degree bound narrow match structure bipartite probe view predefine probe operation spatially close distance map smoothly co probe multiply location guarantee descriptor insensitive entry indexing occurrence descriptor match probe image sparse appearance descriptor simply p u otherwise compare computed make however multi shot paper visual multi shot three spatial q denote image patch shot entity location explain accordingly base person spatial control locality image utilize represent ignore relation turn outperform art spatial quite truncated gaussian filter definition predefine illustrated euclidean much make question probe view bipartite formulate structured weight predict bipartite denote basis function ground truth predict vector constraint enforce match structure ground substitute eq construct utilize knowledge chance subset slack svms cutting training alg violate feasible set add current resolve search adopt ij indeed efficiently thresholde trick test speed alternatively sample trick widely demonstrate without notable implement strategy extract vector patch encode pixel centroid patch map visual model descriptor probe descriptor extract dim color sift feature pixel patch visual randomly sift per camera sift word euclidean pixel encode weight pair view person pair black contribute employ pixel cross standard cumulative curve recognition match solve far follow rank entity probe ask solver three single shot multi shot comparative try figure comparative either necessary know report trial consist view follow describe randomly training capture camera per pair form probe camera entity image dataset overlap camera view camera scenario single shot people shoot pair sequence vary people test camera view form probe ss ms shot multi except typical camera pair always negative indicate camera view compare white weight unlikely black camera compare within region white pair occur contribute black contribute mid filter mid level semantic super super fusion color semantic single fusion metric colour ss discuss shot list comparison dataset rank curve comparative non fusion method aim combine metric overall fusion well always comparable lack interpretability fusion mid filter utilize discriminative mid filter powerful utilize foreground comparable outperform method perform compare utilize occurrence integration explore previous structured reduce rate ms colour ms multi shot learn definition image person list result art rank shoot also improvement shoot latent kernel shot shot entity shoot visual co shot improvement robustness miss scenario different purpose utilize different metric I learn similarity image apply sm short utilize similarity simulate match comparison structure help improve performance general among summarize rank sm fig since sm general rate however much compare result demonstrate probe set respectively probe sm display structured matching match incorrect match structured matching match simulate probe set occur time randomly match randomly probe positive match negative non divided entity matching help improve two issue world descriptor calculate three descriptor entity matching similarity structure color since record storage probe roughly speak visual
r implementation two crucial vast array collection correlate principal know factor component capture successive remain precede commonly core dimensional largely small cca compute projection maximally principal random variable mutually order amount explain cca modality ability modality available training keeping application cca extraction finance biology processing speech computer pca cca reveal relationship variable study limitation extension cca autoencoder neural network tend rather high complexity cubic theoretical separate research help reveal basic regression incur loss achieve complexity cubic feature amenable contribution variant cca sum dot analysis rely extend effectiveness randomize world state deep canonical information scalable autoencoder neural numerical validate lastly implement stream research dot kernel development hadamard randomized component hadamard transform appear twice statistic canonical projection nystr map cca achieve learn presentation recall key nonlinear sampling nonlinear version fit class seek approximate minimize risk true convenient nature insight independent obey ultimately nonconvex optimization remarkably theorem claim let function argument take fit operation computation yield interest help connect kernel feature normalize invariant transform kernel approximate fourier may kernel matrix scalable method exploit extend straight feature nystr om superiority om aforementioned experience section random store orthogonal variable uncorrelated computing transformation nonlinear variant know em pt reproduce computation principal operation autoencoder artificial train representation transformation autoencoder bottleneck principal propose nonlinear randomized pca may equip kernel nonlinear load pca pca loading approximations nonlinear belong function consequence theorem approximate tend infinity feature dot converge evaluation study operator spectral grow analyze henceforth let exact counterpart express define shift approximation q counterpart define bound triangle inequality note eq invoke thus upper bound error divide please aspect spectral clustering use spectrum applying mean therefore easily variant cluster analysis cca multidimensional cca basis stand correlation refer canonical vector act cca speech audio pair transformation particularly view available use kernel trick nonparametric nonlinear cca exact variable deep transformation correlation mapping autoencoder used autoencoder use nonlinear randomized equip shift invariant understand cca pair randomize mapping computational complexity cca perform pca interested characterize counterpart grow avoid spurious characterize approximation random shift kernel approximation analyze first analogously individual eq q recall hence deviation expectation q deviation follow apply old twice turn issue variance term take follow norm jensen eq argument show definition yield worst bernstein inequality analogously bound k take maxima concluding extension linear discriminant analysis seek separable paired therefore nonlinear canonical copula matrix apply section introduce tool train autoencoder set describe paragraph section compare nystr om set form evaluation first projection form cm value norm equation vary agree effect feature close modality cca cca fourier nystr om unable run cubic show inferior replicate accumulate canonical canonical unseen mnist representation mnist width height cca validate simultaneous acoustic speech frame yield measurement vector use validate cm cm fouri nystr om minute cm
suit recover perform source direct state orthonormal wavelet domain explain couple db invariance wavelet synthesis db natural source reconstruction nmf confirm effect nmf peak also adapt capture structure wavelet local peak smooth enforce sparsity sharp peak structure peak width smooth conversely extremely synthesis interestingly experiment visible perfectly case parallel domain wavelet width several kernel wavelet conversely version outperform wavelet domain appear structural provide source complex correctly especially source regularization preserve peak correctly reconstruct large scale smooth component lastly handle sparsity base synthesis scope blind separation already inverse generally contrast synthesis regularization retrieve decompose limitation possible validate use regularization bss regularization superior separation separation performance active entry model approximately source take example wavelet effective noisy display namely gaussian show ground mix matrix observe part head visible picture sparse exhibit similar perform retrieve correct separation special treatment take overcome produce author introduce author propose iteratively soft thresholding opt call reweighte consists choose amplitude transform less amplitude usually source update source use source estimator base absolute mixture image reweighte display sake regularization square estimate reweighte square snr sir reweighte provide denoise snr decrease sir greatly regularization reweighte sir snr apply simulated spectra introduce synthesis formulation wavelet yield improvement db figure sake synthesis constrain benefit see improvement high regime may wavelet mainly paragraph synthesis reweighte imaging set standard coarse coefficient relate coarse account wavelet mixing wavelet use b source reconstruction noisy analysis synthesis redundant wavelet reweighte retrieve clearly direct source provide analysis exhibit lastly reconstruction low finding turn enforce synthesis lead significant article cope blind source transform end novel enforce synthesis formulation enforce transform negativity direct recently calculus blind separation propose enhanced separation performance neither direct correctly finally synthesis formulation arise imaging generally sparse improvement define non lagrangian optimization tucker eq consequently projection negativity follow projection ball aim compute eq column proper eq von min min finding relationship value rearrange term straightforwardly projection matrix positive everything analytic implement synthesis converge w proximal proper continuous relationship computation synthesis analytical reformulate notice variable straightforwardly proximal projection constraint blind separation bss refer factorization different retrieval nmf domain negativity together sparsity transform simultaneously deal domain article impose negativity along sparsity transform domain formulation present reweighte version blind separation encounter scientific sense usually spectra mixture elementary specific physical entity source recover source underlie still way blind source separation negative nmf spectra notation number source bold capital matrix call measurement spectrum mix measurement account instrumental pp multiplication hadamard n orthonormal wavelet domain mark subscript source transform provide several separate symbol bss linear mixture notation form negative matrix nmf source arise mining audio hyperspectral intensity spectra instance necessarily relative physical entity I nmf update least square hard minima beneficial minima desire display feature exploit nmf recover therefore carry multiplicative sample smooth tuning toolbox implement smoothness difference signal energy non large signal nmf sparsity descent update update efficient recent automatically handle available datum fidelity non negative analysis author use technique order automate describe implementation online explore enforce direct favor constrain go perfectly algorithm continuity none aforementioned nmf domain depend see basis capture representation signal enforce transform domain problem therein nmf enforce sparsity deal domain impose transform limited impose orthonormal clearly performance nmf enforce introduce preliminary detail show yield provide however tackle transform either redundant tackle sparse synthesis well never nmf comparison mixture spectra type yield estimation bias generally dramatically major present reweighte prior carry show propose yield enhance normalize design extension tackle formulate subproblem solve regularization separation transform different synthesis formulation synthesis unknown minimization denote aim reconstruct linear possible dictionary call synthesis build atom synthesis domain directly carry multiply synthesis aim synthesis formulation indeed invertible redundant dictionary advantage redundant dictionary come atom available redundant wavelet translation redundant transform formulation show behavior minimization space analysis latter w orthonormal transform extend synthesis begin k indeed backward apply generalize forward two semi proximal proximal operator close orthonormal redundant improve reconstruction particular tight operator still analytic proximal generalize proximal could redundant pseudo synthesis algorithm synthesis admit convenient analysis transform use transform carry synthesis proximal yet proximal analytic need compute subroutine computation need subroutine transform code experiment redundant transform iteration multiplication forward projection negative multiplication transform length order mostly redundant wavelet transform sample multiplication note indeed update multiplication physical identify make mixture mix spectral acquisition spike laplacian measurement naturally however benefit wavelet polynomial wavelet redundant wavelet
solve minimization iteratively square new dictionary sparse metric metric datum matrix nmf address et online example sequentially svd offer flexibility simultaneously dictionary aim minimized problem iteratively metric fidelity preserve svd atom entry simultaneously svd convergence learn tailor member complete mean symbol stack cost noise adjust sparsity result versus fidelity solve adopt alternate wherein guess coefficient current start guess typically begin iteration update n x n denote entry diagonal entry denominator stability choose result package available threshold sparse application specific propose interpret follow prune ccccc iteration iterate scale u u c c black red laplacian input report average five realization update atom atom absence denote seek z whose index ambiguity restrict key idea respect set eq solve description algorithm expensive train validate conduct experiment metric rate recover atom consider recover exceed unit near atom compute recover atom measure closeness indicate recover closely truth create generate zero subsequently contaminate noise initialize training example iteration repeat optimum find variation average trial facilitate fair omp svd k svd one superior k term one robustness laplacian attribute simultaneous dictionary execution respectively execute platform system gb ghz processor decrease plot recover atom suggest conduct denoise laplacian similarity output image svd adaptively noisy noisy direction noisy extract dictionary initialize iteration repeat sparse svd form experimentally noisy follow hard patch fraction experimentally clean patch generate gaussian report svd comparable high svd especially noise denoise svd job preserve structure range comparison image contaminated laplacian indicate improvement ccccc error apply motivation metric fidelity achieve robustness texture edge image svd update dictionary flexibility superiority counterpart svd ground superiority example limit denoise k input svd algorithm turn
specific several moment processing deconvolution convex application communication often intractable sometimes devise tractable nearly equivalent formulation differ across principled convex notion simplicity atomic norm application application choice simplicity constitute hull atom atomic norm operation atomic ball induce serve regularizer atomic norm well effective regularizer atomic hull often atomic follow constrain norm euclidean ball lasso overlap processing show atomic application communication sensor atomic atom sequence support methodology wavelet graph biological network application atom define subset hierarchical modeling fmri atom group atom regularize use penalty correlate variable interest decomposable rank atom consist unit tensor deconvolution component atom typical include atomic set sparse basis sparse signal loss representation observation subject constraint impose besides generality aspect improve fidelity dramatically existence ambient superposition small atom origin additional cardinality ambient operator use often sometimes slightly iteration refer column induce eq convex hull collection equivalently representation atomic atomic dual amount argument achieve norm know atom atomic rigorous remark algorithm optimization interior impractical difficult formulate scale instance often scheme popular scalability find machine find atom optimize objective atom basis perform include gradient iteration prox method fista accelerate method completion prox part computation lead singular structural cg latent lasso extend signal employ amount overlap increase prox intensive cg scalable solve problem oppose quadratic program projection prox solved retain guarantee interested solution sparsity usefulness scheme atom may ultimately contribute acceptable quality atom remove recent iteration truncation simply discard alternative remove current least nature atom limit seek completely basis atom backward opposite possible linearize direction new improve property contribute sparsity backward greedy consider previously omp extend set basis seek norm type square method warm perform simplex equal maintain without organized specify parsimonious atom section describe compare apply scale algorithm deal deconvolution problem major element backward truncation discuss construct alternative characterization acceptance threshold output step f forward step linearization current specifically solve argument minimizer attain atom perform efficiently shrinkage prox method search exactly discuss option backward truncation amount sufficient decrease criterion modify close frequent removal expense seeks expand quadratic prediction removal atom approach removal iterate frank wolfe subproblem relative objective accuracy lower check similar spirit inexact approximate linearize fraction solve signal problem test eq atom sign reduce randomly construct I measurement check performance run value count logarithm vs cg step well convergence conditional cg figure quality cg pursuit hamming trial trial corrupt cg choose pursuit variant group require atom step amount accelerate pl approach consider size index take add htp prox overlap observation element first top singular area cg practical approach rank approximation range multidimensional latent tensor tensor fold product tensor reveal wherein ft atomic form approximate efficiently implement basis via toy various rank tensor always component tensor stop plot entry h c c sample give observation find frequency infinite step operation formulate semidefinite program since well high limited implementation form initial frequency refine add pair select control accuracy discretization simply select negative implementation backward step algorithm frequency heuristic nearby frequency multiple adjacent spike cg sample signal length recover indicate small spike recover role play result report cg right spurious frequency cg grid blue spike circle solution circle solve comparison include signal combination arise physics kind atom define triple atomic infinite atom choice superposition limited sample wavelet gaussian characterize much learn key ingredient atomic mention solution via discretization shrinkage atomic norm formulation define atom standard deviation vary perform succeed recover form sense express mention section adopt optimization drive outline arrive describe informally start choose nearly respect step respect backward step proceed unless backward beyond repeat termination rank type consider graph consider simple undirected graph adjacency superposition interest sample wish graph note neither edge class cyclic corresponding information cyclic permutation yield grid atomic adjacency set cyclic permutation cyclic order weight fix canonical atomic permutation observe full deconvolution norm
nonconvex expect algorithm globally structure possible algorithm global formulation optimize subject constraint formulation program convenient suffer behavior geometric optimizer orthonormal guarantee plant exist follow plant eq orthonormal produce optimizer quite nonconvex next heuristic near stationary round technique recover sparse blind separation however develop solve slight variable penalty huber make optimize close soft k easy recover nonconvex unlikely produce dimension extremely ambient initialization purpose normalize program plant g g z n I bias probabilistic gram schmidt condition bias biased global optimizer biased direction optimizer suppose global optimizer matrix global optimizer invariant row optimizer output prove algorithm fall radius recover equivalently linear prove recover describe succeeds plant orthonormal q pi provide constant suboptimal optimality theorem demand aside guarantee low mostly lp round succeed rather iteration illustrate main detailed deferred invariant generality work go sketch nearly proof simply argument note vector favorable analysis view fix numerator independent q study process round general q inequality significant portion sphere move direction observation one bind numerical imply iteration allow feed scheme programming rounding whenever input magnitude claim enough round return produce synthetic plant learn planted generate sparse basis gram schmidt operator regularization use initialization p repeat simulation five dictionary observation row pair nonzero construct u repeat plant sparse present successful phase transition seem beyond linear sparsity regime whenever plant sparse gap future direction extend vision appearance approximate low pixel image person illumination dimensional subspace subspace select row orthogonal sparse continue subspace figure different illumination htbp manually experiment show interestingly concentrate differ etc first experiment reader think discovery cast vector concrete handling meaningful sparse adopt subspace extend believe paradigm work preliminary gap result likely bind place cover potential structure despite mention start hope provable nonconvex approach estimate various structure setting dominant dictionary geometric algorithms initialization plant sparse thank foundation partially grant nsf compact notation index denote scope always context probable dominate safe rounding theorem theorem theorem subsection edu problem dictionary machine convex target exceed nonconvex alternate direction provably succeed assume plant target embed challenge contain arbitrary efficiently numerical algebra control structure spectral dictionary graphical manifold contrast relaxation optimally understand np hard computational surrogate nontrivial
delay message delay delay system identity respectively certain adversary observe infected subgraph source adversary likely metric protocol spread reach scale fast achieve perfect infect immediately solid message trivially center infect subgraph contact tree infection even randomness node center infect subgraph attack infection right illustrate break combine warm spread infection insight spread maintain center virtual infected novel protocol diffusion provable guarantee protocol inherently distribute message adaptive diffusion reach scheme pass neighbor contact source perfectly user infected source among infected class graph cycle numerical showing protocol nearly warm protocol tree combine insight approach analyze empirically discuss limitation discuss contact warm protocol protocol message source protocol fail broad contact network line protocol develop contact reveal high large two develop contact source high novel call contact network protocol source spread neighbor scheme trivially identify center infection add randomness random time infected protocol study estimator scale give source detection vanish appropriate perfect insight infect node likely origin infection figure equation adaptively choose away infection source likely spread infected node infect boundary summarize goal anonymous rate contact start spread line protocol location expect infect bound expect fast deterministic line deterministic delay detection source source estimate size infection achieve illustrate protocol simple diffusion choose random spread accord protocol detect computation appendix example distribution message send source source line protocol provide detection many path center I match tree size infection contact analogous fast protocol neighbor fast trivially identify infected subtree infect subtree balanced leave diffusion infect infected source hide node present protocol infect keep infected subtree infect subtree leave infect subtree illustrate protocol regular message one refer virtual virtual infected subtree infect balanced neighbor virtual source message make tree notice equally follow form protocol source infect subtree significant infect node contact start spread message protocol tree protocol adversary location use property number infect least estimator expect source proof explain protocol protocol perfect large contact network line two line independent infect subgraph approach node neighbor node combine show infection adversary protocol provable line graph tree line tree protocol source away tree protocol infect balanced source close leave protocol perfect regular tree line diffusion protocol infection function example partially illustrate contact illustrate ensure infected depth call virtual true regular subtree another illustrate infection diffusion source start infection pass virtual token I virtual spread infection balanced depth root pass virtual token virtual virtual take spread infection balance root node consistent depth root infection virtual symmetry underlie contact virtual neighbor virtual chain virtual h v figure show leave whether virtual token construct appropriate subscript contact example virtual right pass virtual leave fully let transition represent virtual source pass one virtual symmetry equally likely infected except virtual source origin statement precise together ensure choice show except virtual equally contact regular start message protocol adversary estimate use likelihood infect source protocol diffusion step source neighbor virtual source pass virtual source virtual source pass along choose virtual source leaf infect subtree leaf virtual infect subtree infection subtree pass happen message spread choose remain virtual pass virtual token randomly exclude previous virtual thus virtual receive virtual message get pass virtual cause infection one subtree leave subsequent virtual source message asymmetric infect symmetric virtual panel infinite regular contact cycle degree realistic contact ht contact source infect select tv uv infection tv study underlie finite network degree still apply diffusion graph challenge first immediate degree maximum adaptive diffusion sensitive long depend discussion approach odd adaptive diffusion virtual computing virtual path since virtual source compute virtual constraint diffusion pg v tv v p v virtual introduce always choose virtual virtual token virtual specify pass virtual token trajectory wish compute summation valid path exposition use contact spread snapshot infect infect subtree path understand spread compute give node assume give large source close leave subtree leaf node two regular virtual identical regular long infect subtree virtual source infect subtree virtual candidate v g ta v infected equation still equal infected virtual v ta pg tb leaf tp g v g protocol efficient ml naturally computed message pass algorithm gets pass virtual leave every message degree virtual source start pass continue node turn child divide reach leave discuss depend leaf regular give leaf adaptive tree degree fix illustrate random degree trial number infect represent underlying value tree expect infection source likely boundary infection successive illustrate probability infection underlying legend indicate choose degree average size whereas degree suggest perfect average infection adaptive cycle connectivity network facebook facebook eliminate three friend guess friend spread diffusion pass source estimation could preserve possible constrain infect node also adversary undirected infection explicitly identify pair spread infection subtree contact adversary source message expensive find trajectory depend whether virtual one ml describe graph cycle denote time loop cause varying candidate ml leave likelihood long exponential problem virtual source loop virtual percent also distance small say informative connectivity graph induce adversarial adversarial example like adversary protocol network activity adversarial attack chance source design anonymous protocol extent adversary exploitation difficult anonymous protocol ensure contact infect subgraph contact infinite infect boundary infection message current naturally contact infect subgraph select neighboring message virtual source eventually infect subtree protocol procedure infect virtual create virtual decrease stop infection ensure true source infect virtual choose virtual state contact regular tree keep infect subtree start set node set message infect keep state virtual source infect subtree center integer subtree example leaf height subtree depth node dot move stay claim prove therefore true assume px px px px k infect subgraph chain chain leave write line protocol hence infect sum random accord probability adversary infect subgraph moreover know bad performance assume pg maximum node source whenever could reader verify whenever put distribution contact finite put prior location fix large remark posterior infect normalize constant ensure probability remark infect infected subgraph g g kt k formula uniform first protocol exception root even root whenever odd follow immediately protocol start claim leaf leaf evolve ensure equally leaf prof indicate exception depth even therefore statement tree exception child depth whenever odd probability root eq lower infect virtual since pass token virtual iff virtual exist virtual evolve protocol path path virtual candidate claim observe regularity symmetry pg pg tv tt design equation combine infected virtual get pg pg virtual virtual stay case
bold response shape shift remain total description inter stimulus activation pattern epoch random dataset bold voxel plausible course precisely use subject variation process process simulation instead use mean corresponding canonical outline allow inter stimulus repeat epoch set description simulate construct outline except relate design stimulus randomize trial description simulation dimension voxel wise standard related spike stimulus spike second stimulus single randomization scheme outline upper corner fig correctly specify glm profile various degree order direct standard ol ordinary multi subject fmri stage begin coefficient subject across subject component subject coefficient variance method estimate population inference determine significantly comparison fdr control inform university drug pre reliably apply rating fmri device ii fmri compatible mm diameter calibrate warm moderately tolerance heat period trial participant ask stimulus point analogue fmri compatible weight bold image tr voxel acquisition functional minute correct slice acquisition delay adjust head http www ac high image voxel tr ms collected run manual check adjustment ensure alignment institute template avg image pass filter cosine deviation voxel slice z cover temperature high heat basis fmri subject combine run fix voxel alternative outline control control fig simulation map thresholded glm hierarchical glm ol delay within square upper left corner dramatically duration point et left corner handle large deviation see misspecification rise autocorrelation though white simulation clearly irrespective shape improve improved sensitivity specificity simulation activation however voxel activation method particular previous datum difficult detect use glm canonical derivative impulse logit il show comparison new activation estimation multi fmri model determine forward second researcher use therefore potentially ignore contain vary subject population benefit inferential perform test voxel also canonical canonical field diagnostic purpose canonical situation population activation flexibility latter present activation detect canonical temporal dispersion impulse use date manner common voxel amplitude currently extend datum consist spatially neighborhood brain resort exist g drive see decide discussion present include canonical temporal canonical temporal derivative impulse set version il change bold substantial among term bias derivative shift shift il amount bias il amount examine handle large suggest purpose encourage balance specificity none addition il activation extremely signal reason propose towards effectively multi fmri matlab implementation propose methodology available request time fmri datum moderate parallel voxel fmri estimation solve matrix multiplication definition slow voxel small voxel programming problem solve arguably necessity inverse algebra regressor acknowledgement anonymous remark grant omit index notation identically n n lag minus twice augment n term r get optimize entry l r compactly take element voxel aggregate estimate voxel carry justification sampling estimator estimator rely whose property consistent increase number reduce estimation sampling effect pilot shape nan pilot step derive least average except assume apply v kn n negligible argument n tt convert paper simultaneous estimation shape response fmri allow vary across subject provide activation inferential allow activation test shape validate application fmri researcher fmri activation certain date accurately assume many fmri attempt magnitude example glm arguably towards time combination component test whether typically assume priori focus analysis obtain across assume constant subject rise assumption relax several function within glm stimulus canonical type function brain use set response impulse response consist parameter cognitive arbitrary brain canonical temporal shift width choice basis cosine radial functions logit critical glm subject analysis analysis fmri glm provide assess group predictor status behavioral problematic truly self contrast compare difference number issue notably basis entail corresponding derivative nuisance canonical fall apart shape begin differ coefficient derivative create subject enable voxel population magnitude offer flexibility basis simplicity inferential allow test activation canonical idea towards impulse constrain gibbs later number suggest spline random draw level voxel across parameter include canonical model simulated set study flexible outperform glm derivative smooth model logit approach propose simultaneous group suffice estimation inference assume acquire bold voxel scan right nuisance sum stimulus whenever nuisance typically cosine basis head heart nuisance subject specific strength across subject represent suggest knot say basis desirable spline I possibly reason immediately interpretable inference greatly computational shape stimulus determine shape determine amplitude stimulus orientation considerably reasonably maintain one activation voxel across previous autocorrelation kronecker delta specify structure partition domain suitable ar variance e te j spatially characterize smoothness function subject voxel express convolution k v amplitude j j v variation variation summarize notation subject index voxel voxel subject fmri population voxel voxel deviation subject voxel voxel matrix nuisance nuisance voxel effect voxel subject effect voxel matrix outline voxel mathematically generalize guarantee select good start pilot objective final describe subsequent pilot residual estimate obtain voxel pool suitable median voxel voxel generalize square pilot voxel first square pls q nuisance signal canonical temporal whose contain determine closeness cross validation randomly rely j v j consistent smoothing v latter voxel activate experimental step voxel residual consistency j j voxel take voxel temporal estimate temporal efficiency estimate number voxel aggregate suitable median voxel usage first iteration good start value likelihood section voxel write multiply q fix concerned likelihood voxel assess step optimization detail linear voxel variance variance residual subject quadratic
formally indicate function specifie belong worth region restrict define wise act model w capture region linearity fundamentally important convex loss regression wise model formulation linear optimization region sparsity induce structured function space two region principle prefer representation motivation utilize interpretability although rule hyperplane region empirical wise model well appropriately optimize special global employ wise linear active rewrite integration global determine affect globally always return regardless f residual although convex wise propose let vector zero formally introduce induce constraint effective partition enforce natural practical preferred interpretability optimum indicator make region complicated standard constraint constraint penalty follow equivalently rewrite term decrease modular index pair confirm envelope modular envelope become convex respectively derive proximal tool problem iteratively convergence gradient acceleration iterative shrinkage fista incorporate th decrease empirical evaluate step regularization frobenius backtrack avoid calculate step employ convergence fista proximal matrix follow solution confirm wise depend step improve efficiency norm computational determine node large size alternatively paper employ idea decompose proximal first map map perform decompose proximal discussion q subdifferential subdifferential norm subdifferential hull max value max derive soft operator decompose feature derive include condition regularization group wise problem projection efficiently proximal describe map single utilize gradient gradient partition practice stage step dominate operation dominant sort ordering become computational derive value width previous step consecutive terminate derive proximal initial considerably affect initialize initialize among random initialization initialization etc initialization empirical local present generalization model discuss sample relate overlap lasso rademacher condition rademacher expectation value rademacher almost surely partition wise reformulate basis special assumption third norm apply give residual straightforwardly discussion uniform uniform global residual model satisfy function within eq partition candidate term practice summary able candidate fit wise linear comparison linear classification binary uniformly dimensional space rule first feature add noise e sign feature nearly output candidate logistic iteration residual illustrate learn red line weight apply line piece yet exist structure make capture benchmark art candidate calculate partition value categorical active feature yes value several regression dataset census internet energy community breast nh bank fm summarize specification cccc cl census cl twitter cl cl breast cl internet cl energy energy nh bank local global local sp discrimination svm kernels rbf kernel note region linear compare respect stop early low cc sp tree svm census breast cancer internet table error global consistently achieve rate dataset sp census breast significantly bad census twitter internet partly partly initialization svm case obtain census internet stop regression compare cart rbf performance depth validation experimental setting use summarize rbf well global many c cccc nh fm comparison warm start warm rate warm calculation become bottleneck technique promise respect incremental incremental minimization despite stochastically approximate gradient direction parallelization parallelization calculation importance direction greedy solver matching work reasonably advanced partition take locally analogy know anchor considerably advance propose add piece train applicable model boost approach partition generation concept treat candidate sparse sparsity induce notable define hierarchical partition structure structure understand structure regularization directly technique optimize sparsity induce structure optimization thank dependency derive acknowledgment majority central school science technology application highly propose model partition key assigning region linear combination region globally formulation make proximal map sparsity inducing demonstrate model well competitive art divide sub assign well linear understand interpretability advance specific prediction challenge convexity inter optimize region arise bad local lack generalization analysis propose model distinguish help convexity propose partition wise divide possess weight apply input
integer write hold theorem degenerate triangle let consecutive supremum improve subtract multiple affect digit base independent uniformly distribute case diameter discrepancy triangle suffice study boundary boundary segment area centroid triangle place centroid segment segment neither subset great discrepancy eq attain pass centroid row triangle case discrepancy pass centroid row centroid pass leave right else discrepancy increase two disjoint outside band case sign line contribute discrepancy horizontal passing centroid left triangle result discrepancy area similarly portion sign discrepancy contribution fact record line summarize contain sign pass centroid centroid discrepancy table triangle horizontal play important role analysis sign discrepancy contribution line centroid pass centroid triangle hold triangle line base include exclude centroid figure line meet centroid sign triangle line include centroid right low relevant show discrepancy column show sign discrepancy discrepancy line line meet meet leave sign sign case contribute contribution discrepancy irrespective sign column discrepancy empty triangle sign sign discrepancy exceed second discrepancy intersect great discrepancy arise triangle configuration exceed fourth band intersect discrepancy triangle one line great discrepancy suitably copy lattice angle cube begin say natural integer perfect repeat representation set angle subset side angle horizontal axis say exist list angle give satisfy take lattice angle remove point arrive inside right angle triangle eq side horizontal always convex side angle axis angle choice write integer denominator perfect finite angle integer lattice angle lie list constant hypothesis point yield set point point linear linearly triangle attain different angle scale discrepancy either lattice run sample satisfy lemma map subset contain end rotation rotation point lie step onto latter leave relatively triangular triangle integer respectively satisfy figure angle already range discrepancy roughly parallel cite parallel discrepancy triangular lattice angle solid may add point exactly use average dividing sum randomization usual find book triangle triangle let integrable integrable k modify integrable bound everywhere sign respectively parallel side subset sign set sign next panel set either integrable triangle proof attain discrepancy construction van construction continuously randomization produce root square kronecker result construction smoothness triangle digital proper corner average project back arc acknowledgment foundation dms restrict band band outside another attain end band result search triangle carry quasi focus graphic quadrature triangle paper present vanish discrepancy van sequence integer angle tangent attain discrepancy indicate discrepancy construction available construction accuracy construction also integrable triangle require quadrature problem numerical triangular quasi monte integral arise quadrature method correctly survey classical rule poorly choose attractive integration weight inequality star discrepancy point sense mapping cube equal hand use approach difficulty composite suited version simplex variation simplex simplex vanish discrepancy also integration simplex mention discrepancy function variation discrepancy measure bind factor cube simplex point neither simplex vanish present construction digital construction van exploit partitioning resemble point kronecker construction rectangular grid intersect retain combine vanish well digital amenable construction vanish also vanish believe construction triangle along describe twice former whenever first triangular van copy choose keep lebesgue measure volume lebesgue low exclude list distinct counting discrepancy measurable sign discrepancy absolute shift consider bound finitely extend take ns understand simplify omit box star value sense inequality continuous variation sum integral face simplex indicator corner extend corner vary spatially simplex inequality discrepancy triangle corner real value let illustrated vertex triangle discrepancy study list
task work focus prove nlp ideal lexical lexical across induce lexical agnostic set lexical tailor explore formulation induce task lexical embedding predictor lexical learn compatibility lexical linguistic refer predict probability noun sentence like capture compatible complexity lexical embed obtain employ agnostic embed retain lexical along line confirm task vocabulary denote noun relation relation paper semi fashion exist unlabeled simple distributional contextual word skip model access pair e able query unseen relation word lexical essential word minimize use regularize constant controlling regularizer interpret query highly predictive interpret original tailor relation matrix relaxation penalty common setting identity inner common evaluate semantic projection noun noun conduct initial lexical configuration experiment six syntactic noun query query always unseen pair report active need query candidate vocabulary distributional word dimensional skip embedding window thus main skip embedding compression window prediction non dimension embedding test unsupervise noun relation speed regularizer similar low start bag skip embedding three former relation latter conclude agnostic useful necessary retain nuclear norm representation embedding table present scheme rank relation relation l city membership membership law law code law show set
reason sentence time reliable quite sentence argue help optimal solution notice training put aside backpropagation analysis bottleneck layer ccc bottleneck autoencoder refer reconstruction ability highly size bottleneck bottleneck autoencoder project near adequate bottleneck estimation bottleneck layer comprehensive autoencoder model differently focus training text metric allow identification critical metric autoencoder fine back orient autoencoder characterization linguistic phenomenon carry ei fp texts c project multimodal es research star edu sg autoencoder model differently explore construct autoencoder level novel text reconstruction capability autoencoder critical bottleneck dimensionality language lose text space deal usually document thousand meaningful association try relevance explicit reduction dimensionality reduction inherent latent semantic probabilistic semantic allocation topic document reduction prominent similarity language association document language broadly linear technique compare principal component large dimensionality projection multidimensional representation language project unseen projection matrix reduction autoencoder da autoencoder structural autoencoder reduce document deep biological study task issue give find deep reduction technique nlp entity unlike retrieval similarity representation text question reduction qualitative assessment reliability reliability capability space comprise metric autoencoder capability distortion reconstruct autoencoder analysis explain provide adequate critical dimension carry dimensionality level assess autoencoder finding rest autoencoder autoencoder discussion critical bottleneck dimensionality estimate future datum brief autoencoder autoencoder sub describe autoencoder detail model differ way autoencoder presence document softmax deep autoencoder count similar input approximate reduction map hide variant autoencoder multiple remain add pca stack multiple deep architecture truly powerful space stacking restrict boltzmann machines rbm bipartite visible usually unit vice versa primarily bottom rbm rbm layer rbm base visible layer sigmoid non visible document hide unit state visible unit bias weight eq sigmoid rbm softmax vocabulary variable define count term visible layer softmax multinomial visible unit word count recover distinction bias way document visible unit autoencoder fine tuning cd rbms train one rbms take rbm autoencoder train rbm epoch cd cd rbms autoencoder show activity replace value entire show word input vector length layer stack rbms upper rbms take lower create autoencoder tune propose use subsequently comparative analysis projection reconstruction unfortunately refer poor neither detail reconstruction preserve justify bottleneck autoencoder dimensionality text quality task modeling estimate poor decide whether metric intend aspect autoencoder capability distortion semantic metric datum compute cosine similarity data eq q distortion attempt capture reconstruct measure reconstruct strength normalise accumulation dimensionality reduction play role document literature autoencoder
asymptotically optimal change point generalize kullback leibler yet communication sensor stream observation present point represent sensor system presence fact although place physical case monitor traffic opposite may correlation environmental wind appearance interference dependent happen sensor curvature field sensor region produce matrix literature include stream observation correlation time treat general correlate stream partial post change stop character alarm partially know stochastic matrix stream post drift rule delay examine upper similar methodology efficient behavior derivation sharp dimension handle non markovian methodology alternative dimension although prove formulate exist result introduce stop rule establish detection delay rule section paper complete correlated remark section omit denote filter p stochastic differential th process treat however sign change latter case assume know positive hold singular correlation brownian cover instantaneous correlation arrival pass sensor place subject yet formulation even capture realistic scenario observation high ti stationary white arise full detail subsection facilitate canonical n derivative comment dimensional brownian motion trade delay alarm ultimately follow performance bad bad detection delay take rule detect regime delay detection delay impose follow problem false alarm describe acceptable alarm stop detect concern know change brownian observation optimality criterion page process stop incur negativity bad delay occur process optimality stop rule know drift type stop optimality process rule stop rule h chart stop rule semi respect small independently subsequently alarm central fusion namely center instance sensor easily see difficult devise rule achieve rule detect say second optimality examine stop rule present detection delay upper delay stop ever detection system subsection detection threshold upper dominate delay introduce detection imply q uniquely instead choose able computable bad detection delay q last stop law tuple essential side definition strong function monotonicity give square integrable decrease measure fact integral set conditional expectation measure g threshold result rule alarm robust respect drift treat general case assume monotonicity slight abuse derive mean false suppose threshold stop alarm proof integrable decrease together expectation equality side get singular stochastic instantaneous correlation choose threshold delay rule bound big hence proceed rigorously heuristic argument process satisfy mean alarm kk argument one similarly sum side complete result stochastic instantaneous equation rule imply optimal delay detection delay ik false alarm define similar q see imply sum immediately result proposition drift correlation threshold imply delay stop rule strong rule optimal detection accomplish derivative tuple brownian tuple exponential define brownian driving find stop threshold clearly assertion similar formula argument stop tt exponential martingale drive ex integrate expectation support delay alarm argument proposition theorem non py py inequality yield optimality either hold hold examine detection delay alarm increase proposition delay respectively determine either optimality assume drift stochastic choose asymptotically detection optimal detection q optimality result drift singular instantaneous stopping equivalent stop asymptotically finally know partial ik assume ik singular instantaneous rule define asymptotically delay stop upper stop asymptotically optimality theorem give optimality proof asymptotic know decentralized communication suppose sequentially employ asynchronous fusion fusion want signal stop sensor change dynamic distinct point suggest health monitoring fusion sensor occur sensor alarm implication center receive take decentralized setup word distinct rule asymptotically detection easy device central fusion center processing efficiency valuable problem instantaneous exhibit optimality tradeoff bad independence optimal delay fold design stop stop trivial exist robust unified rather analytical treat change multiple especially acknowledgment grateful anonymous comment local martingale ready
status influence tweet logarithm status tweet cross datum show solely twitter acc table percentile partly situation considerably expand pool ccccc svm c svm rbf dynamic mm project city political characterize micro establish differently depend active frequent attract project act project match new project extremely important common failure work website input project recommend list twitter extent study focus mm google european fellowship social computing thank say valuable comment yahoo bring market often option name internet support project post project medium site mainly look project way set propose way twitter project drive analysis finding recommendation good accuracy list potential twitter account key insight behavioral sciences g internet crowd project successfully community fail project cite propose automatic matching mm description period recommend tweet project behave differently depend site project one support project tend interest pay less attention aspect project upon quantitative potential twitter percentile baseline order list predict twitter derive random conclude discuss practical implication finding start site successfully million project usa goal reward offer sign cd exchange success video visually connect dedicated account small traditional capital flow attract researcher economic computer science example economic yet tend come friend focus predict whether project successfully category existence video capital friend project project accuracy base predict indeed find powerful success phrase mainly principle evolve series tweet duration conduct behind contribute fail potential fail project cite leverage online conclude automatic project carry three step hypothesis behavior collect twitter hypothesis finding match project another family individual project extent distinction behavior depend formulate convenience active behave frequent good mistake look fact good segment frequent pay management project translate frequent update make e project reach goal receive tend realistic goal reasonable project goal video tend frequent since friend tend expect project project dispersion project reveal location city convert city coordinate location dispersion live close dispersion live frequent familiar site able quickly project look project interest concern tend support interest keep track project topic classify topic twitter build run project page art technology check project page change project eliminate outside project project category project proportion mm duration final number tweet mm period publicly twitter tweet project tweet project title project page tweet project report report project project project successfully meet goal success published retrieve rd oppose successfully financial goal goal dataset day however take previous representative mm mm mm mm project project match frequent span extent project span find largely span project frequent high span coefficient support recommender system project match confirm hypothesis find tend high growth contrast project happen majority project limited activity project growth hypothesis twitter run lda tweet description project represent twitter interest cosine project project interest interest project project variety tend stick project correlation activity cosine similarity well frequent frequent project goal grow interest instead decision thus infer considerable act appear friend member happen facebook successful project characterize considerable friend probability facebook friend project facebook friend moderate partly previous recommend specific project support basis situation recommend potential twitter twitter user project twitter project twitter project matching initially formulate predict predict whether prediction project twitter likely project include project interest twitter dynamic growth project dispersion comment datum add project pair construction evaluate regression fold project subset project project repeat result resort acc recall characteristic auc train pearson correlation coefficient update lr include feature c ccccc lr dynamic linear static static static rbf suggest point separate static good acc dynamic slightly type slightly feature individually g matching ts exclude number comment accuracy individually behavior category similarity improvement inspection give category red bar project active bar project goal case balance set create unbalanced find obtain yet acc binary classification problem project return rank evaluation resort reciprocal reciprocal ranking order predict project project definition formulate percentile project list list rank project score highest rank rank rank c lr static dynamic svm static cc rbf mm mm activity
illustrated reveal current currently possess stop second state immediately start immediately progress degenerate allow write make path piece connect utilize conditioning expand macro window component annotation enable iterate alternate draw macro observe consequence parametric form respectively monte ease motivate independence simplify leverage expansion express evaluate explicitly categorical constrain evolution obtain four pass similarly shoot attempt shot yield far thus pass shot discuss context difficult require integration trajectory evolution time path characterize secondly play second detail independence marginally denote markov process generalize time semi markov denote visit record term transition combine transition markov embed state rewrite system deriving actually homogeneous ultimately much smoothly evolve useful making time section hold prevent marginal estimate enforce interpretable checking computing meaningful computationally tractable full integrate ball acceptable describe evolution process depend rich homogeneous interpretability model level high resolution simplification resolution coherent require resolution distant shift intuition shift approximate begin describe simple assumption specification computation encounter model parameter specification model full discuss repeatedly draw macro sample formally stop move call shot attempt make pt employ state shoot treat red pass receiver state intend future represent possibility pass annotation induce draw favorable decision specify possess possible option attempt without generality correspond pass event attempt event begin complement implicitly express term aspect decomposition type location intermediate process critical connect basic draw blue draw player location resolution black loop degenerate event pass player correspond pass shot attempt nontrivial shot shot resolution prior shoot parametric lastly factor essential computational formalism let simplicity restrict consider optical snapshot tracking exactly high include coordinate player ball game situation time etc annotation occur pass shot intuitive path provide available tracking second value time expectation take end evaluate amount integrate use lebesgue annotation component dependence quantity imagine different achieve potential resolution player meet consistent aside datum value model guarantee consistency interpretation stochastically curve derive stochastic evolution consistency require sensitive grain without spatial configuration trading potentially methodology current process choose resolution expectation contrast estimating combine level player movement chain exact portion require map markovian plausible summary transition represent meaningful event column style anchor east column fill xshift pass c pass player player shoot shoot pt p east player north east east player south north east north east player north north north north player north shoot bend shoot north shoot south south south north edge loop pt north leave north west west south east ne west player east se se ne se ne cycle corresponding group together player discretize player position gray transition represent player bottom figure three associated value whenever player possess define possesse live represent colored diagram reveal discretization indicate annotated resolution currently transition air pass shot pass progress progress list transition pass air shot attempt pass design notable discriminate pass lose ball marginally semi embed markov chain associate matrix specify full conditioning define additional possesse attempt shoot state pass leave transition history label
gap exceed threshold detail c let number choose indicate tt word mod hold optimality gap wireless use aware detailed estimate appendix confidence performance particularly receiver time action incur choose arm within arm translate overall appendix theorem overall cumulative overall acquire discuss mod strategy reward high confidence theorem duration explain worst well predict environment theorem communication environment receiver pair scenario code base code encode message combination linearly successfully message message recover scenario high help decide successfully wireless receiver achieve indicate prevent exchange application video transmission consider length symbol say need affect least achievable sufficient duration knowledge choose difficult slow present ucb inside instead ucb evaluate reward explore arm associate mb detail later benefit numerical along ucb scenario receiver vary study adversarial reward entire action however pick adversarial strategy fashion interference wireless channel understand fashion strategy wireless channel un fashion avoid interference randomized strategy regard continue irrespective strategy learn communication irrespective strategy level wireless widely allow optimize action regard employ arm environment certain might assume employ random unknown choose manner strategy I mention predefine level include section select strategy learn repeatedly interact regret scenario r indicate cost take employ appendix incur incur case derive case due lack adapting strategy duration typically wireless system employ track discuss discuss receiver employ static compare symbol receiver scheme receiver receiver feedback learn strategy choosing enable previously via know parameter optimization optimal strategy contrast strategy expect try db db use various discretization show fig fair initially assume discuss receiver db agreement use learn db use phase offset learn performance factor instantaneous algorithm achieve due result due exploration phase discretization discretization initial take discretization greedy explores try exploit e use unless optimal strategy strategy perform significantly bad novel algorithm see discretization achieve satisfactory close scenario algorithm sub incorrectly know performance behave coherent unknown derive indicate performance consider wireless environment observe receiver performance receiver optimal strategy db remain h ensure maximize objective predict theorem optimal extensive ucb loop inner loop discretization arm thereby learn scheme converge step simulation send instant employ choose uniformly every instant level range employ employ scenario assume algorithm db strategy track change may important history achieve slide environment use adapt strategy round step step term passive slot term slot frame take active passive slot slot frame slot per frame however passive frame update ucb index frame start every mean reward action passive slot frame estimate used slot frame exploit exploration phase splitting horizon quickly strategy please window randomly across employ conjunction slide user adapt fig fig level track strategy vice versa successfully capability subsection scenario strategy depend factor wireless ignore consider error feedback different user fig learn signal vice versa agree complete set strategy user level compare mechanism reach maximum overcome interference cycle window capable track satisfactory mean receive several mu case allow receive rather spread improved expense etc worth applicability wide cognitive type without knowledge novel algorithm bandit optimally receiver pair capable coherent signal either asynchronous commonly algorithm capable strategy pair confidence successful theorem department electrical engineering electrical email edu adapt adaptively optimally receiver armed scheme power duration present learn efficacy receiver term rate prove optimal fast particularly dynamically change wireless characterize static pair inherent wireless medium make wireless largely adversary passive adversary wireless try infer attack active adversary order transmission hybrid attack adversary transmission agent attack receiver traditionally theoretic theoretic principle major disadvantage pair gain practical match strategy strategy receiver pair contrast develop learn learn interact receiver pair act approach environment canonical example reinforcement rl agent success feedback transmission action specifically learn optimal repeatedly interact wireless channel receive feedback action bad take meaning depend specific consideration throughput cost address anti multi channel go guarantee critical severe consequence opponent none work environment exist mix guarantee action novel arm mab algorithm communication offset paper know choose level duration scheme propose receiver wireless unknown exposition theorem scheme power alternate send enable error average instantaneous energy characterize fraction reward formulate mab mab action power duration action propose bandit learn interact receiver receive observe receiver pair estimate symbol
source audio sound meta like influence attempt modeling geometry take modal heterogeneous modality audio generate combine symmetric dominate challenging song infer music audio tag modal music infer modal evaluate retrieval similarity similarity relation index word w similarity latent dirichlet incorporate distribution modality multinomial explain object outline object strong feature modality model inspire improvement section estimate take expectation respective dirichlet define hyper optimize point topic asymmetric parameter respective modality symmetric already similarity document focus solely document measure kullback distribution product cosine candidate introduce dissimilarity proportion document log document similarity hold topic proportion fold style parametric similarity nan statistic permutation matrix exclude diagonal column maintain examine similarity subset song track compose last fm tag tag audio modality audio naturally occur count word audio approach total pilot topic combination modality list style similarity set fold cross split combination hold fold calculated figure correlation permutation nan complexity issue similarity approximate randomly model gain insight result similarity seem provide inter significantly positively possess label increase topic model link describe modality model conclusion modal predictive extend direct evaluation correspondence work
gaussian much solution boundary correct boundary minor case one never result resample make mistake assumption hold dataset uci machine repository characteristic name object source cross fold nine nine unlabele rest unlabeled semi supervise curve unlabele datum repeat determined curve figure semi outperform lda measure test significance four semi l dataset outperform significance semi minimize could hope empirical improvement loss applying relate goal unclear margin negative offer lda step principle semi supervise open question extent converge unique global proof objective lda canonical assignment self self weight control influence hard parameter help bring perform supervise counterpart semi supervise seem learner rather supervised learner partly public research laboratory university technology department computer university department molecular medical supervise pattern variant work expectation discriminant analysis new principled supervise linear discriminant implicit sense expect improvement misspecification unseen real recognition task expensive obtain document image unlabele object easily web unlabele semi classifier decade improve learn add additional datum effect unlabeled lead unlabele offer robust lda implicitly constraint unlabele compare expectation study supervise principled semi supervise implicitly lda comparison supervise version discriminant explore expect offer improvement variant term organize discuss several discriminant illustrative toy benchmark work semi supervise later refer closely relate maximization generative unknown likelihood maximize relate object assumption usually manifold smoothly manifold low separation support incorporate unlabeled leverage rely unlabeled object dimensionality normalize lda propose accurate subsequent semi adaptation theoretically practically successful introduce discriminant supervise semi procedure additionally design distribution function biased object find new q employ assign high semi straightforward adaptation supervise known bootstrapping classifier train classifier new classifier done predict underlie treat possible add integrate unknown hard supervise objective expression log em em sum jensen maximize update obtain practice sum label unlabeled object e bind effect self instead self learning suffer label update approach idea certain constraint parameter use feature alone lda instance matrix link covariance covariance latter accurately rely label point mean accordingly hoc update estimate unlabele label alternatively force objective numerically hoc ideally implicitly constrain intuition train true unknown would outperform classifier two problem one label safe know
good validation probability make available follow procedure unit preliminary regularization need single layer interval epoch fine epoch hide layer hide choose test log l rbm mask mask mask mask mask short refer table layer h ensemble achieve performance comparable belief hide layer mask confirm auxiliary mask input improve epoch suggest long variational help mask model train b outperform corresponding increase bad network consider effectively train start decrease fig input investigate test training case seem show performance drop fig continue seem phenomenon show digit digit argue task restrict variance mask c mask clearly filter random decode evaluate sample compare mask mask optimize hyper mask experiment hide regularization mask without rbm rbm mask mask h h see table addition rbm outperform rbms result well iterative neural extend conventional neural maintain original intractable boltzmann machine belief network extension inspire probabilistic boltzmann rbm deep boltzmann like multiple hidden infer perform number compare utilize model able generative come sophisticated future instance theoretically empirically well see testing fully sequentially efficiency sampling ensemble confidence corner miss confident reconstruct digit correct digit acknowledgement would acknowledge universit universit cifar estimator miss learn improve reconstruction reconstruct step combine compute use engine boltzmann machines competitive art two test machine potentially mcmc estimator mode autoencoder simple corruption paper distribution neural deep training selection train backpropagation observe room miss imputation criterion three recent training pseudo autoencoder corruption function generative generalize disadvantage sampling train resampling imputation inference cope cope vector reconstruction next previous paper inference inspire evaluate agnostic start define factorial conditional denote observe component conditional give permutation propagation parameter draw variable x one train deep experiment reconstruction replace equation agnostic consider special see sharing additionally mask initialize miss see one use mask probabilistic task imputation probabilistic variable get conditional often p factorial qp one posterior parameterized eqs rbm boltzmann
minute estimate viewpoint mala consider proposal merely point study insight method note calculation list question insight reader note matrix add computational overhead require step covariance geometrically correspond curvature could understand inherently property global geometry author relate research simplify manifold mala term large calculation correspond manifold place make appropriate lack cause geometric ii subscript tangent assign tangent set vector along thing along basis directional tangent case derivative consider along onto derivative linear derivative basis drop basis satisfy product direction argument partial directional derivative surface vector normal c know th riemannian q repeat usual manifold chart symbol q geometric employ monte wider diffusion euclidean connection carlo commonly physics idea geometry highlight method introduce manifold adjust langevin hamiltonian since article idea consider detailed unnecessary hamiltonian scenario interested reader review necessary markov chain riemannian geometry minimal measure inform reader prefer skip section provide derivation langevin diffusion riemannian manifold intuition langevin challenge geometric manifold discuss literature instead question research practitioner distribution density measure measurable appropriately construct chain invariant posterior briefly concept overview markov define x aa pm call admit equivalently write stationarity invariant chain certain see useful sufficient relation eq integrate respect reversible stationarity relation primary invariant ergodic markov chain analogue law element estimator estimator reversible provide first chain first efficient clearly arise intuition sort chain assess chain far need measure choice markov literature q informally difference event distribution admit density write proportional imply typically unbounded distance often inequality geometrically grow provide qualitative central limit assess stationarity average suitable mcmc visual simulate devise chain distribution suitably relative perform iteration also little practical exist chain choice hasting accept reject case remain focus step qx x behaviour move change zero many reject remain place autocorrelation see challenge balance proposal acceptance transition result markov chain way invariant reversible propose move reversible limit simple consideration broad researcher proposal mix large discuss extremely simple choice denote acceptance simplify intuition move accept typical structure conduct property rwm show proposal target propose former autocorrelation move walk sometimes proposal result acceptance autocorrelation ht show rwm markov chain choice efficiency increase rwm exponentially light tail necessity geometric ergodicity mean fast require demonstrate tailed pose far markov almost proposal remainder discuss choose superior rwm proposal sufficient benefit property specify continuous markov provide introduction langevin dynamic govern differential brownian motion informally imply change define part e often description evolve differential drift later smoothly drift typically form whether class side find volatility diffusion user basis metropolis langevin describe dynamic molecular eq suitably regular exist construct mean commonly encounter langevin way metropolis adjust langevin mala whereby euler used candidate tuning offer langevin proposal deterministic shift towards relative part part dominate vice versa opposite though hasting propose large enough accept optimal acceptance form differ factor efficiency result highlight mala geometrically ergodic typically result tail tail converse true offer away rwm also fix quickly far away neighbourhood reject strong mala also condition rwm proposal tune invariant idea information geometry successfully widely geometric insight common differential method diffusion turning discuss geometry make property evolve tuple number additional chain mala advantageous impose metric draw current position occur reduce reach efficiently attractive constructive dynamic calculus riemannian understand define notion euclidean purpose dimensional riemannian way chart exist invertible available sphere challenge coordinate sense say differentiable inner product aid intuition think curve think pass define give straight line agree geodesic space always give orthogonality thought flat since straight manifold level think always define product define curve velocity lie denote think local define dx riemannian purpose define coordinate restrict still use convention manifold object euclidean sphere lie ambient nash embed rx x seek idea concrete example graph embed coordinate euclidean linearly canonical partial use curve r although start object riemannian distance coordinate also explicit knowledge nash essence manifold define suitable high euclidean induce define map vector trivially volume riemannian coordinate follow area jacobian case manifold riemannian measure manifold local actually mean diffusion desire diffusion sphere produce brownian motion draw surface sphere piece flat brownian image brownian define euclidean langevin diffusion object appropriately technical reader wish motion manifold increment move along manifold tangent generator euclidean local deduce stochastic denote laplace operator trivially laplacian value brownian motion operator generalise unit neighbourhood also derivative provide directional onto tangent vector lie therefore field manifold seem coordinate ambient tangent define e generator brownian motion diffusion familiar formula drift integral rule ordinary calculus mapping typically drift simply ensure correct lebesgue q putting become upon simplification diffusion require map onto distance map hasting mala eq tuning parameter drift turn sometimes switch direct posterior distribution choice define distance parameter key explore rao year although measure often fisher tailor fisher combine negative log style understand mcmc proposal conditioning method match structure locally hessian long match discuss previously unless hessian globally definite et may scenario efficient procedure set metric problem ensure likelihood contribution positive globally provide log mcmc geometric start viewpoint appropriate absolute hessian way computable name negative decompose remain act value close fisher expectation tractable effort mala proposal long way possible one give diagonal fall difficulty numerically positive definite hessian accord metric well induce simple style difficulty tail take long variant mala hessian may avoid since major take proposal tail mala choice need
leave cardinality uk technology framework sequential via sequential distribution space auxiliary use able unbiased monte order construct algorithm involve capable efficiently tool intuitive underlie area application construct utilize carlo last year unbiased partition normalization method importance estimate partition decade first nonparametric propagation message send monte carlo method propose provide normalization constant another branch model build sampler tree subsequent add accord anneal particle field discrete hold somewhat inspired produce hand particle underlie clique clique node circle right node distance scale rectangle right edge edge subset graph consist random factor simple toy undirected graphical corresponding graph decompose approximate sequence probability recursively update weight typical model sequence observation limit applicable sequence target marginal iteration intermediate target arbitrarily intermediate sampler decomposition amount simply distribution iterate factor add construct artificial albeit use sampler probabilistic graph clique define emphasize need factor anneal order practice affect order decomposition auxiliary target function sequence intermediate modify target set sure establish unnormalized corresponding normalize subgraph circle style main style circle draw right cm style scale rectangle leave edge edge b distance cm main node scale rectangle leave left edge node style rectangle edge edge edge scale node draw distance scale right cm scale factor right edge edge edge edge target collection particle empirical iteration conditionally independently smc adapt resample target increment density particle index particle assign smc w r resample propagation skip importance apply comprehensive collection result limit sampler adjustment multiplier choose sampler say interesting likelihood statistical mechanic energy capacity channel partition discrete simple function contrary provide normalizing eq obvious unbiased due also offer sampler besides asymptotically theoretic capacity channel inspire art solve p k evolve reference propose play construct exact construction change asymptotic fit straightforwardly make kernel index leave joint detail implementation instead setup illustration enable useful general autocorrelation however context ideally sampler simulate systematic random amount simulate subset gibbs get directly conditional make facilitate gibbs sampler let construct depend throughout sample applicable sampler useful reason set decomposition subgraph partition subgraph sampler three illustrate additional result available reproduce mechanic xy sequential resample confusion choose second scoring toy markov mrf potential decrease xy model mechanic ising spin periodic order normalize constant xy adaptation proposal von distribution node right add towards site update linearly start geometric importance design fairly evaluate performance order box sampler match cost spend give r algorithm order depend temperature option easily implementation anneal step node cm main minimum circle width fill gray topic latent corpus often conduct hold w learn challenge since procedure sequential decomposition graphical decomposition include node order rao variable reduce exactly propose sufficient case original learn simulated simulated compute keep particle mean demand perform plot real held particle sample run bar estimate bootstrappe logarithm show see simulated could degeneracy long tailor toy illustrate incorporate deviation gibbs tree sampler moderate particle admit correct among result hold model gibbs sampler gain simulate moderate particle scope beyond model fully drop zero fast surprising latent jointly reduce autocorrelation strongly note iteration new propose see improve acknowledgment thank kind provide lda project contract contract research additional main direct state note result new provide proof state member mechanic ise spin describe angle lattice periodic condition individual site spin temperature hamiltonian describe size target
truly mix task example unlabeled set accommodate different penalty annotation intuition entity confidence explicitly label example indicator total training high replacement entity tb wikipedia point distribution label wikipedia regardless choose improvement stable language maximum language wikipedia create bias entity canonical mention usually throughout remainder belong belong single bias reflect name entity tag link inside entity link tb stage annotation word entity article annotate tag tag tag link exact string matching mention entity appear link exclude frequent vocabulary improvement add improvement especially improvement recall tag p english heat ba pe en v pe name change france want european de twitter star win di circumstance result chinese news public security east bring death throughout english build house david thompson cross en subsequently team record home operate pour le successful l sa rl da short da arrival city ia di di york york chinese party become head china security france unite several organization denote error label acquire google translate label translate annotation analyze produce qualitative efficiency solution wikipedia annotate example language correctly annotate mistake show system name language example system identify entity scenario robustness stem vocabulary language sufficient contextual error group category word appear hard system tag error occur include confusion tag entity chinese tag company name lc english corpora dataset english wikipedia evaluate alone competitive outperform english apply rule appear wikipedia evaluation contextual include english dataset name map vocabulary embedding rely tailor preprocesse step difference approach scalability notational rely tailor preprocesse wikipedia suit human annotate training domain train wikipedia well annotate scope evaluation far seek translation tool preserve count annotation language aggregate sentence emphasize indirect annotation quality translate name entity language preserve name entity hold language set entity phrase belong category category belong source language measure entity entity sentence translation pair language english wikipedia annotated sentence pick sentence detect translate sentence translate language calculate entity positive language large wikipedia article english vary category english origin source annotation benchmark highlight language evaluation skew metric general find translate translate language translation english affect count translation original english investigate effect count wikipedia wikipedia attribute coverage wikipedia average versus wikipedia article negative hand wikipedia entity versus wikipedia entity versus wikipedia article language feature language annotate comparative analysis translation language wikipedia languages author subject keyword nlp extraction massive diversity language introduce ir write language family build massive intervention build name wikipedia language resource parallel corpora therein agnostic competitive language automatically wikipedia finally linguistic performance annotate gold community content process nlp number language english internet correspondingly text surface form address aspect work build name system entity know text phrase pre name essential processing nlp retrieval ir systems extraction base address rely supervise drawback human second design linguistic require language building work address language technique wikipedia automatically language c preprocesse stage yshift language language dictionary tag corpora annotation dependency language yet comparable encode syntactic language internal article name entity link entity include anchor link address propose matching avoid annotate standard language machine tag constraint appearance indicate base mechanism semantic syntactic unsupervised successfully develop embedding acquire huge amount raw language capture co syntactic semantic abundance embedding specifically language embed range feature language investigation train wikipedia without label frequent representation consist embedding objective word cluster name proper word w I f tag neural hide unit hide tag
knowledge guess mix community dendrogram partition dendrogram enable systematically remove community root level dendrogram tree subgraph statistic iteratively select subgraph high statistic value since remove likely member community step nod remain mixture statistic location mix idea dendrogram similar removal summarize mix mix follow dendrogram set amongst si es complexities complexity complexity give network complexity exhaustive virtue mix method outline probability derive procedure tail field differ major aspect rely variable normal distribution depend complicated eq second derivative variable expectation respect special upper es mix community case relatively false community set contrast false raise alarm calculation http www edu method delay es mix sequential exhaustive es hierarchical mixture mix sequential ratio es complex quick community drawback community mix incorporate dendrogram decomposition theoretical mixture demonstrate focus paper community accomplish subgraph high mixture extend outline bernoulli variable large central theorem unit variance use equation tail function variable recover expression keep keep integral desirable combine equal use since replace sum adapt series detect model graph form es mix run length detection polynomially exploit active community h mix dendrogram decomposition analytical mixture threshold approximation determine mix detect community quickly es detection variety cancer often consist differ characteristic detail often clique detection realization true observation divide dynamic shot observation static concern sequential increasingly network either structure change latter community due real processing approach many datum previously stream usually base heuristic recognize theoretical community therefore cast framework community network count inside detect graph adopt differ sequential statistically property formulation three exhaustive es h mix es method perform exponentially complex size know polynomially fact raise active community mix address impose dendrogram detection procedure method numerically verify numerical explain mixture contain numerical matrix community community illustrate figure represent adjacency red form problem nan th alternative subset node baseline case either know define run minimized expectation define member interact typically represent anomaly replace setting replace community assume test likelihood possible possible window window limit grow exceed exhaustive search size recursive statistic u statistic formula calculate
around fails skew normal skew well fit assume show covariate small skew numerical normal method skew et omit probit skew coincide poor lack dependence correction confirm little situation logistic show bias reduce apply covariate probit regression appear widely r r regression skew logistic skew carry assess effect type consider normal assess degree skewness assess normal latter derive bivariate skewness type respectively usual fit omit unit skewness scalar covariate one omit solely df et al regression parameter log variable covariate addition treatment throughout approximation provide close covariate covariate apply apart binary treatment indicator clinical variable category include predictor randomization covariate treatment covariate analysis adapt randomized treatment continue e partition factor denominator extend arbitrary number rather restrictive correct treatment arise allocation statistical model would perfectly lead trial calculation odd ratio course practice importance element zero penalty asymptotic however issue relate account logistic expense investigation nevertheless help inform analysis probit analyse logistic probit correction essentially correction estimate treatment less probit prefer logistic false approximation equation false probit maximum probit presence factor e equation skew expectation logistic expectation give denominator author trial randomization take assess summary bias arise omit randomization accurate asymptotic normally covariate omit compare convenient form insight apply additional asymptotic logistic probit trial use linear important baseline omit randomization ensure relevant omit treatment necessarily carry identify also important logistic asymptotically covariate author randomize approximation treatment omit general scalar exposition article taylor restrict whether fit varie article skew logistic obtain false covariate omit taylor approximation excellent form apply usually take treatment arbitrary number result wider compare skew extension allow additional covariate require assumption conclusion draw suppose respectively e assume tend false expectation take distribution extend skew scalar dispersion mean distribution multivariate normal principle could dispersion change analytic case partition outline detail derivation even I variation fit repeat probit trial fit omit expansion imply scalar covariate fit normality make author series expansion omit fit unconditional expand exact analytic normal increase close change correction slightly reduce solid line dash line dot adapt fit include addition give still th limited alternative multiple probit regression distinguish analyse natural probit tx estimator essentially denominator place although result slightly see appendix
understand immediate thus minimize use q form original definition thank fx x fx fx fx finally induction conclude thus conclude fx come second hypothesis show nesterov accelerate descent vary sequence initial smooth nesterov accelerate unconstrained obtain q multiplying obtain remark put get u induction show constraint behave euclidean ambient meet gradient technique dimension rate instance euclidean fx mirror minimum ball mirror descent devote mirror alternative presentation inspire chapter intuition situation forget dimension observe project arbitrary hilbert situation banach make formally dual update point point space lie outside need way project back mirror map chapter compact convex fx fx bregman useful closure mirror boundary mirror mirror point lie notion precisely map take ii write result primal point outside one project mirror projection bregman associate precisely uniqueness projection follow lemma bregman essentially imply immediate mirror let xt illustration thick side inner token label token label token token token control map strongly satisfie claim one lead mirror conclude observe mirror mirror view mirror try linearization move far away measure bregman mirror simple mirror descent mirror furthermore mirror project descent recover early project descent write equivalently bregman map kullback divergence bregman simplex result know one mirror achieve case von gradient update write equivalently exponential logarithm projection trace non show easy word simplex consider descent true advantageous situation mirror descent extension mirror averaging replace ask mirror descent average w x x x proposition x putting display schwarz obtain would computation latter hypothesis mirror chapter mirror attain mirror prox mirror prox smooth mirror descent mirror prox make mirror instead rectangle side inner token right token right token token token token label node control mirror fy fy x separately mirror obtain term schwarz sum straightforward computation mirror mirror satisfie come learn see mirror descent saddle calculation setting satisfy prox arbitrary eq dramatically already always function optimize globally specific structure smoothness smoothness overcome chapter description interior rather oracle know simplicity clear description inspire recall want minimize assume quite natural locally approximate n gx fx iterative shrinkage come elementary computation fista accelerate fista easy fista fista metric euclidean composite mirror obtain detail quite smooth see structural attain nesterov smoothing find subsection saddle computation proceed descent chapter warm powerful mirror prox next compact convex yx explore algorithm produce candidate solution duality gap x key similarly eq z g x view mirror mirror map later field sp saddle easily md imply norm thus use mirror prox mirror algorithm sp saddle point mirror describe follow tw z mp satisfie light suffice field r introduce sp md mp minimize mx mf il x l sp order mirror obtain much box iteration non smooth let denote zero correspond obtain attain furthermore step sp mp dominate multiplication get nash sp mp separate look generality replace column find write sp solve iteration vector multiplication fundamentally far type q objective self empty interior fix enforce live subspace span modification algorithmic consequence newton stay word equality lagrange multiplier refer chapter explore optimization randomness key go quite computed progress optimum small gradient correct vanish picture case long step order oracle take point output possibly assume need smooth stochastic oracle follow interpret loss convex find minimize stochastic draw distribution report second describe want minimize report stochastic quite situation access want oracle pass indeed expectation use point contrary pass mirror md thanks ft x mirror map strongly md satisfy generalize md size q let strongly oracle optimization basically exact bring acceleration acceleration square sharp exact descent thank next use smooth mirror map convex r furthermore assume fx cauchy schwarz strong obtain thus yield sum conclude modification mini batch sgd conditionally query oracle convergence mini modify stochastic return thus obtain call mini mini batch sgd call mini batch two situation iteration sgd multiple processor central unit processor gradient independently serial mini particular calculation several estimate examine detail unconstraine context compare basic independently everything else gradient computation improve sgd computation low reasonably gradient positively sag sdca coordinate ascent require gradient computation descent sag natural one nesterov question sdca rate sgd typically center oracle precisely center also convex f ix ix ig ix particular ix ix write respect observe yield claim center rarely idea lead epoch initial everything else convex clearly simplify follow dependency upper thus one obtain fx fx fy x fx fy note fx finally section simple optimize denote everything else draw md lipschitz accuracy next greatly differentiable maximal bounded see smoothness independently preprocesse context basic descent attain improve function attain potentially iteration obtain I fx fx fx putting obtain computation potential acceleration la directional assume attain let strongly follow elementary convexity elementary calculation fy fx proof theorem hand randomness computation true sp sp md stochastic saddle point assume x r r satisfy md section column quite oracle step sp md index take overall nash equilibrium sp mp dependency instead subgradient modify note ij unfortunately turn free specific situation briefly relaxation randomization study let entry dissimilarity two maximize total weighted adjacency rewrite follow turn optimization problem existence combinatorial difficulty stem replace efficiently eigenvalue power randomization hypercube clearly furthermore immediate implie sample uniformly hypercube arbitrarily approximation even display know sdp relaxation sdp find solution point follow state solution sdp relaxation lemma v j v quick I euclidean remark use fact one l l x positive expense constant interested sdp relaxation sdp show b semi latter taylor denote matrix entry one conclude gram randomization naturally section assume could prove generalization cut convex draw uniformly particular isotropic set isotropic one near isotropic position replace method isotropic one obtain chebyshev nn ensure randomize center progress constant fraction progress would unnecessary isotropic x n least isotropic explain convex picture direction run enough distribution total uniform randomize good distribution correctly like ellipsoid informally center need random either step walk overall need call oracle walk chapter chapter corollary chapter address financial engineering edu chapter chapter study convex segment algorithm write compactly precise constraint machine problem interpretation cost simplicity I mf iw detail origin one take hinge obtain svm row lasso capital letter observation unknown matrix complete way result sense result extensively separation guarantee immediately support hyperplane support hyperplane notion set result essentially admit existence fx fx fx recall obvious g f fy g rescale interior build subgradient hyperplane exist let infinity enough imply conclude differentiable fy interest empty yield interior affine generate notion particular function exclude sometimes extension notion algorithmic around subdifferential information another instance global minima global minima minimum happen enough algorithm alone justify optimize surprisingly admit excellent describe aspect argument already many learn logistic regression formulate conclude extension constrain sake simplicity optimality trivial decrease black assume resource objective input oracle input subgradient interest many sufficient minima reasoning need box allow derive theory matching limit resource course pay box early reference recent year algorithm popularity quite high explore detail chapter chapter chapter dedicate noisy discuss cover free black section seem extremely objective globally structure address ultimately optimization extremely interior technique fista mirror able efficiently program direction lp consist problem semi frobenius sdp section sdp quick specific hard summarize emphasis present expense make practical lipschitz know also tune reader adapt potentially reference text consideration denote define depend whether subscript positive rate iteration dim non ellipsoid separation lipschitz one smooth nesterov smooth fw opt convex lipschitz one smooth nesterov fista gradient prox sp mp md md newton empty interior chapter box solve let let stop query center side comment optimal center query well need fast rate follow require attain digit double comment concern turn computationally carry general section center randomize method turn theorem geometry center prove point otherwise would proof clearly obtain e nr x rx convexity fx ellipsoid convex geometrically semi length eigenvalue simple lemma heart ellipsoid method x furthermore one ellipsoid computation derive show ellipsoid ellipsoid norm quick picture sense ellipsoid would one inverse semi axis principal axis look half look ask write eq latter quite naturally minimize ellipsoid show maximum attain elementary computation ellipsoid ellipsoid c h c x ellipsoid ellipsoid formula conclude ellipsoid access separate hyperplane separate tw x h ellipsoid stop observe remove ellipsoid ellipsoid much bad indeed need call call situation case derive basically always intractable instance context oracle overall interesting property ellipsoid give radius back simple minimize differentiable function point fix behind make minimize local descent see complexity feature attractive section chapter operator follow study rectangle token label token token otherwise chapter simplification contain euclidean note subgradient inequality schwarz make modification gradient exist importantly make update onto iterate prove token token leave token label node project subgradient satisfy elementary identity x plug value directly recall convexity reach need call oracle sense complexity ambient dimension quite center ellipsoid put differently could hope ellipsoid dimension explore restrictive optimize complexity computational bottleneck often case admit analytical think combinatorial algorithm operate function finally recommend depend iteration practice undesirable vary one next indeed go auto say continuously differentiable explore improvement theorem iterate previous smooth gradient descent eq represent integral cauchy smooth give improvement next lemma inequality smoothness smooth fy denote show display imply thus together constrain come constrain q gradient iterate fx gradient smooth optimization gradient started decrease see case expect may cut lemma progress constrain x still prove result convex give show decrease see g describe compact convex introduce perform direction illustration perspective key replace case token right token side turn gradient descent former precisely fy x follow r norm true easily arbitrary convexity induction inequality produce iterate finite set definition know iterate vertex dimension regime see vertex representation interesting property together rate prove simplex minimizer clearly function schwarz one norm iterate efficient example inspire problem open signal dictionary way q dictionary instead constrained situation nonetheless reasonable want assumption polynomial problem meet combinatorial dictionary span lin lem finally correspond seem take save admit minimizer discuss exploit descent study compute needs derive smoothness fx fy mean respect effort descent significantly first follow inequality immediate verify strong convexity curvature instance lead rate optimum step optimum course smooth careful tune
anti nominal interested na b quantity compare asymptotic otherwise always write easy integration determine I must q figure error nominal range asymptotically conservative variance choose depend large line point conservative rate increasingly formula solid line anti conservative nominal rate conservative small error feature suggest critical value hybrid bootstrap critical asymptotic examine detail penalize ridge statistic let I nn smooth ridge interpret assume write write q penalize testing effect penalize whether fix solving ridge penalty know regression lasso pattern standardize group understand appropriate penalty penalize implement package acknowledgment thank provide valuable insight state prove additional lemma need proposition mutually ij sub gaussian follow sub jt kt union hold otherwise depend convexity ball expand enough write lemma hold satisfie without p b tend minimizer argue show hold add subtract c tending condition write multiplying give recall central q distribute dominate dominate corollary interested association assess constant variance formal method likelihood ratio set test low case q condition identify value interval formal test remain year technique formula bootstrappe expensive involve variance suffer zero recently produce feature become parameter produce test nan hypothesis variable lasso however give individual conditioning framework permit regression interpretation datum place alternatively inference dimension invert stationary lasso regression gauss markov unlike knot lasso start decision make need diabetes measure diabetes patient list remove covariance sake regression outcome feature propose penalize section choose validation htbp l lin tc age refer refer code author test refer table ease display order decrease stop observe stop upon reach covariate produces variable interpret presence model association accounting trend interpret discrepancy answer question broadly however propose regression feature separately lasso decision association outcome result related lasso rest method penalize score general penalty consistent special test extension induce penalty bold font matrix font k procedure notation way also hypothesis assume score statistic large appropriate reference apply set multiple regression variable feature wise select penalize serve penalize non statistically hypothesis discuss appropriate since interestingly pattern lasso tucker lasso choose mixed effect model effect assertion precise connection explore suppose log l regression bias incur inference penaltie classical vs however treatment depend throughout scenario regime define parameter stationary association effect note concrete parameter include associate extent concept fix choice theory special regression come briefly greater penalize penalty vector proposition appendix depend may distribute p need central limit large quickly require c constant condition zero distinguish particular require inactive residual ensure grow correlation inactive grow quickly boundedness allow quickly convenience tail slow zero detect condition condition hold approximately hand approximately n variance use test estimate residual option fan rely replace appeal interpretation light test adjust turn pattern regression apply conservative nan surrogate classical unbiased relationship unbiased related condition test consider connection variable selection penalize score variable penalize score interpret shift classical score score take eq order give differ statistic center variable penalize consider case omit ease exposition statistic testing know advance order support hold condition among thing connection recovery support penalize statistic close statistic reject correctly test meaningful result score depend se expense freedom spend explore numerically score proxy test small behave generate matrix outcome rest zero size average replication dimension penalize turn sake regression result use residual simple obtain test truly associate positive unbiased nominal rate give function choose control examine produce non consequently behave multiple hand include feature demonstrate test penalize test penalize test respect type measure ph formula penalize multiple plot summarize replication simple regression marginally correlate whereas relationship nearly surprising identical simple decrease dimension power decrease enter correlated typically high htbp indicate six correspond penalize bottom panel line true average bottom panel close nominal rate beneficial assertion support show
quickly whole segment consumption orient purpose ideal good entail coherence clarity summary compose part song may issue produce summary summarize music automatic human redundant performance rely portion audio compare music contiguous truncate begin middle song music comparison purpose present review review similarity describe detail introduce classifier report conclude remark music propose music extract person automatic consumption coherence requirement summarie centrality similarity sentence page text another social focus improve speech sentence sentence previously select another technique segment select include summary segmentation aim extract meaningful segment segmentation change detect repeat segmentation boundary build contain every apply cluster output first similarity change cluster produce final strategy label belong call extract long piece filter building lag embed find position summary term lot segmentation use simply allow look use important part aim meaningful segment song later include summary lead combine generic review algorithm choose approach segment duration song later song aggregate whole song second keep distance pairwise centre sound select build similarity value feature calculate sum column row since similarity accord desire whole song length piece index start maximize evaluation whether summary song evaluation average summary clarity diversity summary produce summary take select sentence maximize model metric sentence previously select query sentence relevance diversity sentence centrality rely pair centrality google web page list rank one first cosine step create accord weighted calculation convergence successive vertex guarantee vertex unweighted edge vertex summary length reach determine score sentence mathematical text reduce dimensionality element times occur sentence global sentence singular diagonal sort use correspond calculate column extract sentence iteratively singular equal increase sentence never summary sentence introduce half ij music impact classifier classifier song music solely classical used song concatenation song energy base hz hz range low frequency hz hz frequency matrix frequency along min peak peak max distance peak frequency contain string content distinguish half dataset encode hz microsoft file request cross calculate first begin middle end summary algorithm extraction extract also operation frame overlap vector generic need additional music audio frame piece k vocabulary song frame vocabulary segment allow sentence depend sentence cosine sentence apply summary tf calculate iterative picking length reach type implementation operation segmentation concatenation sentence singular small score explain rank weighting combination frame overlap word size widely music present try interpret overlap g
complex extract together represent adopt code sequence site follow biological great challenge area system biology main focus theory biology protein study individual application biology p tumor important study individually measure system compose however despite great theory limitation lack classify redundant measurement mine theory important bioinformatic characterize gene extraction relatively information succeeds even consider belong category extraction genomic sequence complex network use among sequence highlight difficulty recognize predict present genomic adopt genomic graph theory connect formalism real property sciences physics thus complex connection internet group topology represent biological characterize term possible complex list shortest give length minimal determination characterize short cluster agglomerative network two friend common define kind centrality pattern find maximization start site dataset various specie work contain sequence extract genome remove code segment merge reverse reverse genome select code region genome database network information global relationship genomic sequence total maximum entropy sec networks accordance approach occurrence propose consider next consider genomic consider complex nucleotide network network measure standard deviation node genomic measure evaluate extraction dna coding method machine adopt ten fold cross change radial feature task perceptron bayes notice indicate genomic figure show roc possible effectiveness classifier propose genomic network occurrence identical possible identification sequence correctly correct genomic pattern combine composition genomic sequence potential methodology achieve classification addition obtain complex network
hyperparameter compete decomposition one sparse representation optimize hyperparameter automate also effect desire micro min macro min low representation document combine present advantage supervise eq expression depend concavity term inequality relaxed update rewrite matrix table table summarize matrix division one indicator il il corpus tag readily lie utilize variational factorization coefficient discriminative tf representation label yield pattern inter represent convenience inter categorization latent supervise essential machine cover research principal analysis factorization well upon formulate entirely application available appeal use dimensional preserve discriminative incorporate fisher discriminant purpose lie mathematical offer graphical family along semantic baseline mining modification probabilistic know robust prove denoise missing property formulate labeling relate syntactic word simple intermediate word term result frequency term approach specific term document measure tf normalize rare entire corpus document document corpus document contain approach mine know space pattern frequency relate rather learn perhaps good formulation assume inherent nmf decomposition worth nmf kl minimize objective term intermediate nmf decomposition decomposition general extent model divergence impose paper categorization regard collection represent tf organize semantic component word model superposition model assume gamma noise exponentially distribute indicator formulate mixture discrete eq indicator convenience notation variable represent expectation prior tail around prior impose additional indicator large activation constrain notation px il il joint p gamma additionally formulate sparsity decomposition sparsity share tag variable hyperparameter unobserve general bayesian introduce instrumental kullback rise bind density factorize q show update low conjugate factorize approximation expression analytical specify computational convenience include outline treatment b rather generalization potential classification representation test sort date multiple computational load reduce tf consequently classification place consideration poisson unsupervise tf learn dimensionality test optimize component learn gamma decomposition fix shape gamma distribute coefficient run hyperparameter optimize maximization directly bayesian varied hyperparameter optimize follow initialization fix burn optimize predictive k square root cardinality latent varied method nmf minima initializations metric macro average document belong denote belong split label accuracy macro average imbalance measure l originally context penalization take purpose column treat document component expect document belong model pattern support denote sum coefficient accumulate label document motivation behind exclusive pattern sparsity relaxed definition paper micro accuracies initialization vary semantic constraint improvement regardless penalization explanation though representation natural differ greatly label representation well boost result present decomposition experiment beneficial label sparse micro accuracy matching produce
prediction interaction glm glm interaction impulse response linear impulse response direct interaction roc supplementary material detailed outperform matrix predictive baseline normalize ten graph presence absence interaction experiment roc identify glm hold outperform spike homogeneous process standard glm power supplementary discover interpretable world interval sep price price event process yield trading course day daily variation incorporate periodic supplementary scale compare prior train outperform interpretable structure std ex bottom model stock nearby stock interact describe embedding plot stock list com notably energy tend together indicate interaction stock broadly suggest stock influence slowly vary may diagram bottom top interaction top notably suggest activity cascade fourth eigenvector drug perhaps encouraging report weighted interaction period cluster break offset share day per related underlie network occur mutually occur frame training cox rate baseline process uniform process community consider community separate group community cluster material process network latent interaction identity improve south allow area due overfitte insufficient potential interaction prior help community predictive high predictive log likelihood come cluster model distance suggest discover look figure safe gold buffer neighborhood activity might report west show increase rate consistent trend point wise add periodic capture interest previous poisson intensity maximization fully bayesian consider special case infinite exchangeable discover interaction process gmm identity prior suffers recently leverage conjugacy prior promise discover network perhaps inference discover tendency similar interact recent development nonparametric linear interaction consist purely outperform discover background unobserved process identity uncertainty fully bayesian noisy prior interpretable variety real generalizing beyond promising wish thank discussion fellowship text parent first induce process spike weight recall impulse response spike proportional gamma distribution rate impulse response dataset allow scale cox equally spaced background rate grid equally spaced calculate integral elliptical empirically period day year well know trend scale offset log set homogeneous event training able distance place log characteristic scale weak gamma place uninformative gamma shape typically scale know scale scale invariant prior root information generate event spike glm test second run chain simple cross correlation w thresholded point process model external external impulse response likelihood concave easily allow link direct roc homogeneous top component logistic impulse sample glm last second glm figure likelihood os model interaction glm compete average across collect week positive negative change price activity would generate stock ignore outside trading price short choose parent interaction high vice versa brief jump stock markov interaction last sample trading vary day peak market cox periodic posterior main sbm stock interact interact stock latent distance belief explicitly block model connectivity vector probability place prior sbm comparable infer correlated com h std net sbm year predictive consider iteration likelihood expectation illustrate main dataset intensity spatial normalize community introduce location prior belong encourage spatially cluster cluster os discover cluster prior localize model fail dataset intensity figure prediction latent model spatial likely suffer complete prior spatial gmm process name play central role modern analysis enable study part analysis edge many enable probabilistic combine random poisson superposition enable elegant empirically modern characterize via vertex infer critical pathway trade disease associate identify low representation e know entry literature concern directly perform implicit infer noisy dynamic structure financial stock market execute time stock relate infer interaction wide throughout discover interpretable pattern insight explore stock infer trading example edge attribute induce cascade vertex unobserved financial possible dynamic infer case identity spike activity mutually interact recently result markov augmentation efficient parallelism process set poisson canonical govern nonnegative intensity moreover subset set event poisson poisson poisson event interaction event known point specify history event add nonnegative impulse generative impulse draw number induce draw time I enable computationally intuitively background rate event attribute constant fluctuation share variation trading trend fluctuation log cox background periodic capture daily fluctuation offset intensity process recover constant identity infer alternatively occur cluster
lyapunov condition variance define lyapunov condition specifie demonstrate lyapunov note limit term membership hence lyapunov lyapunov convergence rearrange term show stack observation k define ab ab process state mutually linear term give kalman linearity filter estimate estimation enough linearization vector mean optimal vector membership item item address space alternate item search hill estimate sbm factor probability previous plug section recall form probability exist occur state initialize ml estimate initialize search cluster vector assignment estimate assignment compute predict phase two make mistake cause test real dynamic facebook analysis day step people less leave people tp initialization singular value snapshot adjacency actually community diagonal block snapshot enter quite regardless occur variation histogram edge facebook sbm sbm network step sbm fit histogram sbm figure assumption majority show fraction sbm propose provide fit well forecast notice network appear replicate effect add dynamic simulate edge depend whether step would forecast number create future author thank provide access facebook great recent model network inspire well stochastic sbm absence influence future sbm derive procedure reproduce analysis statistical network static consist modeling appear representation allow refer evolve target discrete network assume snapshot particular greatly simplify flexible replicate observation network dynamic static extension utilize absence time edge would version adjacency advantage property combination extended kalman dynamic ability accurately replicate dedicated modeling dynamic mostly several excellent survey extension static exponential model continuous extension stochastic relate membership sbm propose dynamic extension sbm model discuss markovian dynamic snapshot tractable realistic interaction interact interact facebook interact month hide markov influence occur influence weak incorporate future current currently propose satisfy latent static represent edge node square adjacency matrix denote edge vector static adapt adjacency static membership vector membership class identically I adjacency assume estimate interest focus difficult posteriori set evolve edge snapshot mapping index mapping correspondence invert remainder mapping time follow denote adjacency let dynamic stochastic dynamic sbm dynamic extension past sbm sbm parameterize specifie author simulate annealing place evolution govern dynamic apply kalman procedure magnitude fast relate conditionally previous edge tie form edge density particular undesirable propose edge long density time iid likely appear edge class respectively sbm iid edge accord two matrix denote probability accommodate node membership formally model generate accord stochastic transition adjacency static sbm node statistically independent time choose satisfy node scale probability adjacency matrix follow sbm transition valid finally provide connection sbm derive satisfie three must valid next present change step becomes scale property inequality meanwhile substitute combine arrive q low bound function accomplish choose argument current substituting value satisfie property derivation
loss classifier excess newton method yield expectation contrast excess learn learner access advance additional exponential loss excess function utilize complexity tailor exp sharp convergence setting sequential notion loss margin condition focus concavity derive study concave risk learn classifier follow optimization concerned optimization classifier step parameter receive n newton difference online smoothed version covariance receive newton gradient classifier online covariance examine several also vary focused excess classifier first batch introduce key note unlike hold respect ii make verify strictly convex property concave ii first monotonically therefore sufficient surrogate calibrate binary excess batch learn eq provide risk fashion final intermediate return indicate excess online general modulus dd function unclear excess exponential question batch achieve batch advantageous aspect batch second important determine step proof technical result defer variant improve also eq rooted concentration use key follow deal concentration e fix bernstein eq domain next proper net bind probability moment return theorem overall prove excess loss state fact combine imply plug excess complete plug turn prove base notion rademacher work consider rademacher small hypothesis set certain divide variable sample domain item side p rademacher rademacher complexity bind use next g together union obtain complete combine bound turn prove online exponentially theorem ii add bind separately start indicate inequality martingale lemma condition lemma complete proof address online show address remove excess concave careful open question investigate future improve dependence sparse accord excess risk plan sparse analyze generalization exponential concavity combine rearrange obtain I I conclude take optimal proof bernstein e bernstein inequality martingale denote variances martingale eq martingale kt kt kt kt kt r kt I bernstein desire inequality goal excess exp
tried achieve demonstrate base heuristic vs comparison multi task average test small specifically outperform plain advantage compare robust feature solve nesterov employ accelerate result superior perform convex formulation rule intuition refine estimate threshold intermediate alternative gradually improve synthetic effectiveness theoretical acknowledgement china cb natural science china like constructive suggestion task aim feature alternative present might performance adaptive additional adaptively determine expected selection algorithm embed come empirical world effectiveness variant limitation common incur potential unlike structure successfully stock exist relevant feature share task commonly share learn well common share convex restrictive suboptimal regularize correspond task feature algorithm achieve convex computationally prohibitive notice regularize formulation employ prescribe row unknown value adaptively determine achieve structure intermediate current stage threshold stage solution scheme iterative wang et jump rule threshold though method could apply prescribe feature obtain first conclusion scalar denote denote capital letter euclidean frobenius denote scalar brief introduction feature share learning row represent scenario share learn quality measure function throughout magnitude reflect assume establish situation commonly case unconstraine occur standard impose constraint obtain reflect example aspect really natural choice parameter feature base regularization I iw unknown unconstraine formulation balance loss sparsity individually make almost relevant another prohibitive computational burden make joint sparsity norm regularization th share relevant share certain share task regularizer type convex loose moreover employ may convex formulation base task feature learn task performance model prefer regularization th problem denote row penalty feature formulation multi algorithm regularize feature theoretically obtain proceed circumstance interpretation l initialize w jj indicator like multi limitation fix prescribe may optimal difficult well aim threshold refine adaptively last rule appropriate adaptively determine value follow q still function balance threshold dependent assume prescribe significant jump significant review correspondingly modify task feature threshold detail threshold accord keep update proceed eventually reach empirically j jj end though lead achieve refinement model regularize well reweighted algorithm compressive regularize non convex formulation gradually iteration proceed necessarily inherent error intuitive explanation could undesirable local correct decrease rigorous addition exist intuitively critical constitute original prove solution one return unique rule sort small amount jump set rule capable detect false fast decay heuristic kind matrix adopt method formula could try true many quality though case tuning key typically large gradually improve result intuitively jump partly magnitude one magnitude spread false one cluster jump compressive observe pattern importance rigorous value challenge effectiveness jump multi task row nonzero generate data distribution noise sample response subgraphs subgraph clear
euclidean empirically vary dimension perform validation neighbor six near synthetic weight weight synthetic fig point manifold euclidean base dot right fig geodesic distance compute function learn ground distance remove effect mr achieve poor preserve section world database experiment face vary leave image second datum contain category image set label query retrieval query precision recall evaluate result different algorithm relevant query score rank position ap ap query c mr le map average scope scope rank scope curve precision table map provide recall indicate reliable ranking list comprehensive improvement indicate manifold embed study field perspective provide characterize geodesic novel heat field learn experimental result demonstrate develop machine designing algorithm use heat differential show heat accord appropriate boundary dx dx derivative derivative self adjoint v derivation positive manifold give heat xt xt eq essentially heat field next analyze behavior heat equation primarily along connection heat geodesic dx p please represent dp dx lie line segment connect unique parallel dx field flow field around heat field field gradient heat view specifically vector field uniformly riemannian riemannian manifold desire function measure intrinsic distance geodesic distance simply call euclidean tool study manifold tangent smooth let point linear derivation vector totally abstract space tangent define derivative direction derivation write directional derivative denote use local chart verify coordinate tangent dimension manifold embed sphere dimensional plane denote union vector usually property field nothing point tangent derivation worth note tangent manifold field summation convention imply summation index might write next assign manifold inner tangent smoothly mean smooth subscript omit context write metric define close curve dt axiom positivity difficult give function riemannian field coordinate define direction measuring coefficient rely coordinate constant vector constant fundamental field connection vector field manifold compute coordinate use hold rule connection equality due since manifold symbol symbol curve chart field curve vector along curve tangent vector manifold parallel three curve might worth parallel field curve field along gd v along angle parallel field curve field suppose vector parallel along three general parallel along curve call geodesic geodesic recall derivative result also dt second partial equation geodesic geodesic constant parallel curve geodesic tell short path must geodesic geodesic short length measure minimal study geodesic tangent tangent exist complete every geodesic connect hold please fig example map minimal geodesic connect opposite long geodesic connect geodesic unit measure opposite everywhere except laplacian operator field field smooth field smooth tensor q tt lt x laplacian adjoint connection orthonormal notation function denote hilbert schmidt show derivative space open subset say represent appropriate evaluation differentiable define origin define therefore exactly derivative functional equality self adjoint due since manifold geodesic complete actually unique minimal geodesic inequality ds rp equality geodesic hold whenever geodesic curve integral exist curve pass curve pass first prove neighborhood point vector since gauss pass curve geodesic vice versa curve pass pass manifold complete geodesic connect geodesic uniqueness geodesic prove function connect note geodesic speed finally connect cut unique cut limit rx dp next compare hessian evident g r hold due fourth distance theorem section section ji center evolutionary state college university china manifold great learning euclidean learn preserve function manifold euclidean well paper directly characterization gradient field theoretical propose learn field whole heat gradient experimental datum demonstrate effectiveness learning pattern recognition desire retrieval depend label classify category unsupervise supervised point label unsupervise learn view space lower map euclidean preserve principal mahalanobis typical le diffusion try preserve original manifold reflect connectivity diffusion preserve preserve manifold learn distance original may euclidean preserve sphere embed geodesic geodesic distance variation definition property intuitive geodesic short short distance consume handle manifold geodesic pde grid ambient impractical ambient note tangent equal ambient field tangent ambient inspire field heat propose geodesic characterization heat flow field geodesic field unit learn learn initial local heat flow asymptotic approximately gradient geodesic obtain close normalize optimization linear system complete synthetic demonstrate riemannian goal desire distance provide natural measure fundamental follow briefly concept detailed introduction tensor inner relationship let neighbor approximate estimate tangent space neighborhood mean true mean component vector recall tangent abuse vector j jt tx jx please function block diagonal th block x ix j ic tx ij n selection tangent parallel worth note connection laplacian semi matrix normalize derivative via field normalize solve plug remove th side varie column field summarize dominate part search near tangent space
context dynamical manner interact consist whereby become replace step parameter gp perform dynamical ability state high use likelihood apply variational dimensional nonlinear system distribution gp mat ern particle lag smooth particle lag compare gp take particle ern ard identification offer substantial available test performance obtain batch time cccc rmse tr train time gp min var gp min gp gp min neuron ec region potential rapid little overfitte learn space crucially tractable oppose length series believe future work variational eliminate smoothing well gp prior equilibrium limit etc material side noiseless q transition independent induce interpret deterministic time radial deterministic lead opposed parameterized perspective analogous regressor undesirable variance away induce input exactly opposite behaviour would process transition emission augment variable emission straightforward latent matrix log emission induce trajectory optimize hyperparameter ascent compute uk successfully sparse process system parametric straightforwardly trade avoid overfitte main hybrid bayes fast widely success diverse finance popular series auto bayesian sparse gps encode continuity variational complex risk overfitte lead posterior prediction linear use smooth state dynamical inference accelerate learn scheme work uncertain prediction enough nonlinear create principled arise engineering finance instance adapt characteristic nonlinear dynamic new adapt control situation quantifie avoid risk flexible later extend parameterized process view tailor develop nonlinear find map latent learn issue apply much small could mapping state recently manner employ particle smoothing prediction article state lead representation space eq noise future another noise state dynamical useful time deterministic signal omit function mechanic application specify ability thank specify straightforward way e light place process process severe strong correlation variational formulation apply double supplementary prior convention return group hyperparameter note restrict particular distribution gp convention gp induce matrix argument equation doubly transition rich nonlinear hyperparameter qualitatively instance dynamic panel tractable approximation dynamical tb prior generate qualitatively linear panel appear limit cycle nonparametric necessary find distribution integration expensive since trajectory alternative aim transition series technique induce latent density induce induce omit conditioning variational popular making lead tractable maximize inference methodology induce distribution relate inside denote analytically variational ability location induce location posterior tight trade without risk distribution evidence eq mean optimal depend x smoothing likelihood augment tr smoothing markovian modeling strategy characteristic particular system severe likelihood heavily multimodal smc present auxiliary present alternatively propose hybrid variational whereby sequential carlo smooth discuss section characteristic dynamical appropriate smoothing particular respect instance covariance assume respect variational use optimal
nn sc nn mat ern experiment sequence video pixel ability dynamical variational vast allow video learn give frame pixel near achieve firstly benchmark processing video challenge create scene frame periodic contain video mat ern functions compound kernel periodic reconstruct frame original video outperformed nn demonstrate visually video material l reconstruction dynamical nn frames e reconstruct video variational gp aforementione recover smooth nn video dynamical var reconstruct dimensional gp smoothly whereas problem file ht successfully ard video training frame generate frame future frame amongst video experiment demonstrate ability reconstruct without frame smoothly generate gp divide gp latent variable allow digit bayesian approximate low bound describe classify determine label class comparison vs approach c misclassifie logistic kind dimensionality scenario seek find low gp temporal propagate uncertainty treat unobserved uncertainty full spectrum range unobserve fully unknown rise auto gp model extensively great success setting denote unknown however world application uncertain come gp extend aforementioned also model input show explicitly model input input free induce typically give pass subsequently generate form make variational variational probability parameter implicitly uncertainty prediction autoregressive manner uncertainty output denote subscript input output shift uncertain aforementioned way inference particularly consider around center iterative ahead future approach predictive make iterative step beginning include predictive point prediction immediately uncertainty happen predictive input framework consider benchmark represent see something dataset challenge train dataset create point future comparison firstly give model output give mention refer make ahead predict add straight uncertainty autoregressive propagate autoregressive gp robust handle something low notice method expect prediction part input obviously uncertain general row latent supplementary derive mathematical formulae expression low derivation complete square recognize definition hand detail quantity jensen use square equation distribution equation complete recognize derivation complete final put quantity exponent get algebraic use replace eq replace well replace choice equation computation kernel ard statistic analytically ard quadratic eq j variational ard function ard integral tractable derivative variational variational transform vice versa scalar diagonal single element software datum therefore derivative obviously rule write prior term involve involve turn depend demonstrate forget dependency mean calculation account result equation replace eigenvalue numerically task explain bind aforementioned illustrate certain standard formulae variational observation gp far expand term equation test second exactly augment observation version latent variational could approximation correlate set experiment optimisation base output explain q exactly phase product easy appear accord form project sparse distinguish calculation identity gaussian find fully denote row partially miss point replace distribution delta observation area area mean accord subsequently along parameter gp whose behave partially define initialize fix variational u train distribution prediction figure obtain experiment datum depict ard experiment predictive angle space mat ern kernel gp scale switch correspond initialize add part original line dynamical variational gp use body mat encode run separate subspace space learn train investigate information subspace consider employ mat ern covariance constrain ard project interact dimension separate walk regime value belong multiple latent vary dominant output row dimension belong region vary motion motion clearly different latent generative model ability produce projection dimension blue dot dimension point output background predictive variance latent position generate output ht rgb rgb rgb rgb center box equation chapter chapter sep font draw fill sep distance sep distance corners distance institute university k gr department university economics business ac institute university uk flexible widely maximize method introduce inference framework subsequently marginal method gp iid observation learn non dynamical show robustness overfitte automatically dimensionality nonlinear generic flexible extend purpose process synthetic machine benchmark dynamical dimensionality seed rand seed class recover simple pass present problem mu alpha mu mu axes position exp alpha pi alpha axis xlabel center axis p xlabel axes position text center option option diagram multimodal normalize appear use autoregressive uncertain uncertain computer dimensionality several dimensionality state physical manner nonlinear result density extend dynamically lead resample suggest nonlinear kalman transform representation space variational input gaussian combine process represent process dimensionality perhaps function low datum subspace map linear principal analysis component tractable place datum mu alpha mu mu mu mu exp mu axis x hold axis xlabel center ylabel plot end end axis pos pos pos pos pos pos position axis axis center text option false height diagram dimensionality reduction suggest spectral locally attract attention closely multivariate interpretation explicitly underlie mapping dimensionality reduction seek low precisely additional dynamic modelling incorporate algorithmic paper prescribe kalman step rather specific datum entire function restrict filter basis cast gaussian specify distribution gaussian dimensionality linear challenging extension rely posteriori principle desirable automatic dimensionality formulate variational framework propagate process marginal follow jensen infeasible inference build variational gp induce result define product distribution variable generic easily extend measurement input partially consider dynamical significantly extend kalman filter nonlinear markov place trivially spatial derive string graph deal model demonstrated conduct experimentally extension input fully unobserved variant prediction gp finally theoretical current variable specify review dynamic datum discuss map estimation characteristic introduce process determine property map traditional covariance rbf infinitely use covariance linear latent space appear covariance denote give equation input kernel hyperparameter form construction density term p come space discuss detail latent pass challenging fixing treat omit notation give straightforward latent suggest subsequently author notice gp structure simple observation fully factorize latent space prior incorporate discriminative literature seek constrain system temporal auto model whereas track temporal employ smooth path shall dynamical dynamic define component draw via datum follow covariance evaluate covariance fully factorize row function use rise markovian choice experiment find rely procedure optimize hyperparameter several drawback firstly fact could overfitte provide insight optimal add require slow consequence employ typically complex help latent automatic determination ard unnecessary almost benefit use occur bayesian provide rigorous limitation map variational demonstrate avoid overfitte automatic selection dimensionality detail simply approach immediately tractable auxiliary tractable variant concern outline extension method enable application modelling dynamical trick vast field treatment marginal associate joint latent integral prior nonlinear nested write complex infeasible progress invoke variational variational aim approximate jensen obtain low log eq mean field problematic intractable intractable observe integration appear analytically tractable bind close jensen expand auxiliary induce variable originally induce speed extra gp expand extra induce constitute evaluation augment joint take q marginal prior induce expression evaluate covariance induce induce illustrate augment ht node latent latent node latent node latent latent u variational gp node level augment value model hyperparameter induce drop expression variational approximate distribution form however proceed variational allow analytically key reason term derivation follow eq shorthand expectation easily gaussian explicit expression j see numerator explicit value trick explain optimally tighter long depend I q notice appear refer require tractable straightforward analytically worth decomposable appear pair respectively particular term associate one point speed computation parallelization computation suggest average separately constitute ard quadratic form aforementione appendix lower write q sum side essence follow maximize standard main optimize variance parameter gp investigate carefully observe term column closely resemble low contain variational finally restrict set additionally place parameter subsequently integrate apply gp property separate kl divergence easily accommodate form rise incorporate prior require suitable compute variant section resemble gp across variational prior variational appear result simply form variational bayesian dimensionality dimensionality resemble fully term optimisation many free reduce stationary diagonal denote dimensional parameter transformation newly inspire iterative optimisation scheme depend actual newly lead treat case jointly appeal improve optimisation indeed equation confirms couple according treat one optimisation readily dependencie different nature kind dimensional simply replace non use section observe take non gaussian extend let sequence dynamical capture underlie priori block sequence block sequence use induce approximation variational cubic gaussian reduce select induce gp handle variational thereby induce computational gradient substitute svd cholesky practically million video kind interest calculation density estimator secondly reconstruct unobserved overlap index problem miss reconstruct covariance discuss task dynamical task predict discuss write likelihood marginal variational numerator integrate newly construct ratio low appear denominator specify numerator approximate q exactly analogous impose equation mean well need decompose average extra minima sensible initialization associate discuss wish reconstruct thus totally previously instead approximate moment achieve introduce underlying free latent variable form obtain methodology marginal low quantity detail project sparse compute calculate describe standard solve dynamical specifically predictive input optimisation distribution challenge shall forecast empty fully gp optimisation find gaussian mean close save diagram clear rbf bias else load end fr height eps system eps fr eps eps load else height eps fr width gray eps fr eps fr height eps part eps fr width eps eps clear generation bias rbf load eps system eps eps eps fr height eps system eps gray eps fr frame eps eps else height gray eps fr height eps system eps eps width eps eps rbf bias else rbf body eps eps white else eps system eps eps eps eps reproduce mat matlab variational performance gp dynamical ard determine subspace construction present evaluate reconstruction class source video structure covariance dynamical world capture dataset able handle work raw video dynamical benchmark framework evidence enable bayesian proceed actual review mapping output nonlinear thus selection method infinitely dynamical mat ern covariance
algebraic copula generalise copula dimension towards nonempty subset function volume lie follow central copula nothing interested application consider cumulative otherwise uniquely determined conversely cumulative briefly bivariate copula empirical ranking object rank retrieve associate dependence give respective equivalent pearson rank value substitute rearrange seem consequence rank constant copula cdf empirical indicator allow express form integral unit unbiased monotonically increase nonempty real function box domain multivariate margin probabilistic reader statistical background real unit continuous cdf distribution trick marginal analogous bivariate tend theorem express copulas two vector generalize product configuration proof positive similarly inductive product similarly product positive inductive assumption way product observe configuration positive product half relate integral copula integral copula form change reduce change similarly induction multivariate vector marginal copula give argument j j possible way product possibility difference probability integral form function copula copula possible multivariate bivariate multivariate bivariate fr hoeffding minimum possible copula correspondingly rank object provide rank put access discuss far require rank recall rank range copula express rank convenient normalised expression copula interested ranking plug expression integrate product position derive correlation rank geometric capture notion rank generator rx rx generator idea generate rank order element rank rank rank generator show possible showing imputation reverse ranking imputation ranks particularly consider list belief discussion imputation scope manuscript task aggregation extreme core extend bottom rank common partially label generator close rank rank label weight ranking label rank like weight see weight rank list appear twice calculation interpretability dimensional optimisation decompose bivariate notation product observe rank correlation bivariate express term square rank normalise rank use algorithm extend rank convert rank learn weight transformation naturally scale consensus rank give order though bias offset least interpret rank interesting closely consensus close among st supplement test method challenge aggregate pre fold provide leave expert expert identify bad benchmark bottom potentially label tie tie evaluation website implement discount cumulative ndcg well stagewise recall consider fashion approach perform apart rank weight imputation assume ranking perform quite rank suffer arbitrary demonstrate importance st good fold fold aggregation relevance unclear form intersection retrieve contain order label list dataset maintain exactly cross split report note outperform confirm justify well importance choose ccccc rank aggregation rank solve estimation normalise imputation complete aggregate list surprisingly art dataset without computation simplicity model expert learn mining bioinformatics department communications centre program result may show express tie two denominator express express appear match term numerator expression numerator denominator together soon aggregation want worst rank extreme main parametric aggregation multivariate extension copula normalise multivariate propose geometric square list miss allow apply aggregation benchmark rank retrieval recommender bioinformatics one major normalise hence combine many learn lead formulation task rank rank aggregation permutation object instead association combination pairwise multivariate measure association informally value observation inequality concept capture ideal multivariate copula corresponding copula review mathematical construction familiar expert turn learn expert scale rank denote normalise interval square contrast involve complex sampling rest theoretically square expert geometric mean normalised multivariate target extreme rank
prove suit tailor dimensionality operator encode wise constraint respective alternate modification variant consider follow augment quadratic initialize follow optimize optimization step lead involve measure feasibility convergence know special aspect wise negativity constraint constraint property concern state converge optimizer expensive involve eigen decomposition limit applicability large modify call exploit low rank alternatively express allow k k see bottleneck store derive solver evidence experimental meaning rank idea invoke encode throughout iterative us memory problem dense efficiently whose efficiency nm nm turn complexity initialize k k theoretically ensure modified address iteratively lr lr rather converge state reveal lr optimal small iterative file marker gm cpu gap inf cpu inf cpu gap inf inf e e e e na na na mul update na evaluate lr benchmark solver mix k gm gm gm dense number instance run lr experimental evaluation three benchmark see table new consistent camera comprise representative category evaluation surface gm chinese character gm mrf gm match gm category gm gm character gm map gm category relaxation tight comprise category evaluate lr exist sdp admm present cg conjugate nonconvex interior relaxation serve heuristic propose lr cutting method incremental fashion small add mul sdp solver consider category algorithm applicable align gm label contain expand lf star na na align gm na gm na gm na na gm match assess program duality nc different report solution duality run dual turn remarkably scale compute point method provable scalable report nc large nc solve optimization local turn inefficient solve mul update lot perform expansion make max cut iv lf vi tree reweighte problem assess evaluate objective result assignment category report necessarily high percentage summarize art map category lr superior gm lr either poor advantage map em method exhibit specific potential explain programming relaxation moderate structural huge space get lr outperform dense high obtaining art gm problem tight gm comparable sense energy attain arise highly optimize design lr optimum issue result gm occur gm match gm combinatorial star find lr round preliminary matlab less gm label medium align gm run lr minute hour scale gm hour huge room improvement alternative accelerate eigen future propose admm solver sdp solver result confirm sdp various enable relaxation algorithm benchmark direction combinatorial formulate estimation light combinatorial acknowledgment nsf grant grant fa n award examine kkt begin involve detailed various orientation estimation construct kronecker condition exist uniquely characterize dual block rise recovery produce problem submodular constraint otherwise relaxation example optimal introduction lf object align lf gm na gm gm na posteriori inference discrete fundamental wide world semidefinite map develop accelerate direction lr exploit comprise hundred thousand compare lr remarkably commonly hold relaxation mrf evaluate quality demonstrate outperform art produce computing span scope scenario range gene estimate approximate various formulation surrogate semidefinite usually dominate formulation programming programming despite superiority obtain applicability paradigm purpose propose relaxation refer map pairwise undirected key marginalization programming semidefinite accelerate variant multiplier admm scalable problem pc state per node practically remarkably variety lr collection inference consist experimental computational well concern model scope reader refer topic convex relevant structure linear method convex formulation nm quadratic objective exhibit property non relaxed replace complement replace constraint sake relax constraint crucial sdp scale negativity necessary loose submodular state property
thm proposition red test accelerate address sparse regression upon idea worth computational dictionary preprocessing stage small believe even great iteration one may advantage computation iteration henceforth dynamically formalize screen principle inside number first order assess show screen thresholde focus numerical optimization problem consist fit induce dictionary fidelity regularization non smooth induce rely information gradient base suited method demand iteration fit application transpose application extend algorithm challenge fast multiplication bottleneck govern fast associate dictionary screen understand cost inducing entail optimum many aim zero screen test screen draw detect zero relation equivalence remove zero locate screen remove matrix atom column index imply solution zero depict upon screen cost take aforementioned improve exist computation overhead consequently dynamic use small update em illustration dedicated reader approach mathematic figure particular combine version safe screening screen effect illustrate evolution atom draw independently unit sphere dimension actual begin iteration dictionary dynamically iteration get second screening particular screening iteration equivalent state static screening use tt start atom new formulation improve algorithmic scheme problem adapt screen give screen detail discuss significantly reduce cost optimization definition integer indexing th sub element notation extend select respectively primal vector dual ar b dynamic accelerate term notation refer might formalize update compose primal extract update convex duality scalar characterize order accelerate family screening screen dual design screening begin line screening update show line enable inactive atom dictionary variable screen transpose thank successive share variable acceleration impact assess experimentally section np screening ti iterate screen order state preserve convergence minimum convex n dynamic screening inclusion solution sequence exist update general algorithm may various regularization associate describe may accelerate order specifie namely define primal aspect first algorithm algorithm formulate way computational optimization l backtrack fista backtracking make proximal beyond subsequent proximal description may embed correspond norm sparse norm eq respectively necessarily link eq define solution trivial simple screening may feasible maximize focus initially quantity base concept geometrically contain upper admit sphere safe et al formulation rely three screening generalizing thereby use present screening efficiency proof screen boolean value screen lemma safe screening st exactly except aim reduce screen atom form inclusion screen overhead mainly multiplication already perform algorithm acceleration dynamic screening implement screen screen stage evaluate screen I determine inactive atom fortunately begin computation already overhead zero finally algorithm may small first order experimentally dedicate assess principle deal measure screen dynamic screening acceleration extent gain datum screen static dynamic screening evaluate focus static dynamic run reflect acceleration screen dynamic strategy base computing quantity sparsity current iterate algorithm iteration compute operator operation respectively dynamic screening screen compute operation static separate screen number operation c n sub group dictionary implementation screening screen represent time equivalent check synthetic type dictionary I realization introduce draw I dictionary experiment build randomly atom coefficient active group draw I inactive observation sum ratio audio cosine dictionary adapt audio music image digit split testing compose digit observation taken address fista tf strategy algorithm dynamic screening several st pdf normalize time dynamic screening circle value account fast plot fista typical strategy acceleration parameter run range ability really normalization conclusion fast observe time trend fair group median run represent dynamic screening wide group grow inactive group confirm intuition screen inactive group trend discrepancy discrepancy computation screening test previous fista dictionary gaussian fista pdf fista screening screen acceleration improve static audio bring important safe static strategy safe counterpart atom improve safe dynamic allow acceleration show practically algorithm use induce strong acceleration make mean screen principle apply conversely apply modify preserve convergence first order answer anchor screen theoretical present whole study screen combine adapt solution successive computation high give exact examine dynamically iterative nice behavior orthogonal pursuit
impossible achieve edge label isolate node resort objective correctly strictly error least think always contain component label node thus enyi contain connect show fundamentally impossible two impossible attribute fundamentally correctly whether indicate need exploit encode graph work infer distribution community case positively positively correlate positively infeasible sharp simulate embed pick large eigenvalue nearly useful embed large eigenvalue plane show fig well ten color attribute coincides find simulate pick normalize square label square ij fig divide three correspondence spectrum algorithm satisfie surely surely eigenfunction get existence eigenfunction decay slowly bad bad display decay zero yx far eq independent conditional r therefore introduce notation matrix use norm norm root sum norm use usual denote sort norm overview scale font thick right theorem surely orthonormal eigenfunction ki sharp control spectral symmetric entry independent universal lemma perturb symmetric denoting exist eigenvalue absolute proof next proposition apply observe inequality ok n hand orthonormal eigenfunction v ki r main distance define fraction eq entail note constant hence right hand side successively go technique adapt label attribute root attribute couple two attribute agree agree coupling agree child child check agree grow branching branching well branching eventually bayes question remark objective microsoft centre community whose general randomly unknown latent infer partially observe inference conversely show well without provide spectrum eigenvalue law community network attention find numerous across various include physics biology statistic exposition reference assume network consider graph blockmodel sbm k plant simple node partition extensively provable guarantee real display cluster observe interaction rating question accurately interaction absence cluster answer sbm assumption allow carry come accord attribute label graph label collaborative user movie label one view label person person relationship either bipartite gene label edge label expression predict blockmodel formally seven endowed finite denote measure attribute manner independently draw finally retain infer inference shall statistically indistinguishable combine emphasize label inference without sparse propose computationally efficient spectral assign construct adjacency label spectral finite lead adjacency use empirical neighborhood space underlie true isolate neighbor edge isolated provide impossible make meaningful fraction impossible trivial threshold asymptotic specify appropriately prior work main theorem sbm spectral attribute space reduce spectral use underlie sbm rely edge rank establish compact clustering base attribute analyze assume attribute finite latent node attribute simplex endowed dirichlet sbm reduce sbm study community fit exactly exchangeable edge point approximate spectral eigenfunction exchangeable focus component phase identify sharp cluster degree rigorous community multiple positively correlate impossible threshold sharp tv throughout almost surely tend notation adjacency goal time spectrum suitably detailed description step weight label top integer step node spectrum construct label exploit labeling encode know priori irrespective indicator weight adjacency decomposition extract eigenvector unit let estimation give small define guarantee spectrum operator define operator act function zero see g eigenfunction decrease eigenfunction performance guarantee continuity appear exchangeable random graph uniformly modulus continuity fix integer characterize estimation positive satisfie give satisfy least second vanish simplify go go strictly successively sufficiently small measure endow stochastic block small theorem interesting fix adjacency precisely fashion derive expression operator lebesgue fourier series
series section series performance detect cs format mm p noun noun noun noun proper noun noun noun noun noun noun well detect fail detect significant window l table format mm game book amazon review tweet series word column value usage table current demonstrating change sharp frequency could attribute rise popularity movement popularity political false positive intuition illustrate google useful visualize detect linguistic shift last detect tag noun noun indicate meaning suffer high false negative rate linguistic however linguistic annotate domain false negative linguistic resource detect attention demonstrate detect seek point introduce understand acquire previous detect word detect second word detect well change continue production pre music mid significant word yet introduction flexibility popularity book start book change life consumption table start noun dominate common computer word company mid shift mean operating decrease corpus method require intersection field discuss linguistic point detection language evolution detect political frequent period study quantify linguistic tracking shift fine track language still able period entity rely parse construction method linguistic resource enable compare sequential embedding moreover fact web word learn mapping symbolic continuous embedding language outperform effort propose computation big word embedding capture fine structure prove range natural task embedding change area describe change bootstrappe establish significance outline excellent survey series internet language medium influence internet online form medium like medium focus usage implication mail instant I language online excellent internet provide include linguistic analysis medium twitter google different series approach statistically significant linguistic shift represent medium analyze google able historical tweet amazon review game book capability detect shift ambiguity system language implication semantic internet real change acknowledgment provide access stream gray rgb pt significant linguistic shift mean usage shift especially rapid exchange quickly meta construct word usage statistically change linguistic consider analyze complexity generate property use distributional infer word occurrence language word time unified coordinate construct distributional time series word linguistic demonstrate scalable linguistic change micro twitter decade review movie amazon google book pattern language medium information storage artificial language inherently evolve accommodate internet rapid idea linguistic shift media micro review book semantic word throughout mean propose tracking shift model temporal series per investigate series extract second construct word pos tag contextual co construct significance word compatible illustrate semantic semantic shift last observe semantic year construction type regard word usage difference frequency two google trend sharp united word acquire mean distributional observe scalable investigate linguistic across year micro decade review corpus movie review amazon books books corpus introduction product book aware web understand request detect semantic word contribution statistical speech tag construct series investigation first sound linguistic change score book review corpora time several change rest structured define language word evolution describe significant change language qualitatively limitation blue change mean quantify linguistic shift mean temporal create corpora dictionary e usage appear vanish corpus corpus snapshot reflect several calculate construct quantify significance usage shift usage question shift happen first quantify significance semantic syntactic usage syntactic distributional aspect evolution significantly influence type detect phenomenon immediate change frequency google trend google book analysis index corpora linguistic frequency word method track appear snapshot corpus language construct occurrence word snapshot information time jump tb metric calculate bias popularity entity could significant meaning detect significant usage involve syntactic serve word evolve syntactic category noun acquire speech proper noun describe company figure leverage syntactic corpus pos calculate distribution tag snapshot quantify corpora specific pos shannon divergence dark line tag noun dramatically popular company stack chart noun tag dramatically increase dark blue shift restrict speech acquire sense device categorization subtle semantic change deeply use distributional appearing context semantic space space representation appear recent variation track learn snapshot corpus track embed shift discuss embed construct time change snapshot goal begin vector representation context vector equation occurrence snapshot word word appear within equation calculate probability classification reduce cost normalization factor use stochastic propagation training epoch normalize force word experiment implementation model unless snapshot align embedding change complicate train alignment change distributional word snapshot aid alignment simplify assume structure assumption alignment model fail align properly linguistic near word tw normalize set shift normalize step shift observe determine series track shift snapshot relative occur across period capture linguistic distributional word embedding semantic introduction popularity constructed discuss determine word score threshold change preprocesse normalize series bootstrapping draw w b ip describe exhibit normalizing attempt detect shift variant shift analysis outline method normalize transform series score snapshot w
solve sum task column row stand traffic anomaly detection range source case pattern regularization weight certain feature task feature problem regularization nonsmooth gradient among trace extraction flow second address address rate capture mb mb anomaly field privacy concern use maintain trace repository traffic trace capture link united traffic start traffic comprehensive provide could achieve well derive header computer use ip ip protocol complete flow account variety select characterize flow include information deal time simple count content server number arrival arrival inter arrival inter arrival arrival lack truth traffic difficult label step detector analyze traffic report similarity similarity stand traffic strong community community measure reference traffic traffic anomalous anomalous stand traffic traffic anomalous efficient detector label reject none detector traffic method dataset process effectiveness task raw traffic characterize flow dataset table cc experiment multi feature package package svm matlab apply task top turn vary build classification svm evaluate data estimate overall metric selecting achieve high result multi observe improve classifier work selection lasso top multi improve detect anomaly employ real network generate anomaly detection generate evaluate multi plan extend evaluate selection network simultaneously wide h wang pt research area accurately detect anomaly traffic multi consist flow different period task different period perform detect anomaly anomaly area detect anomaly simultaneously particular task know base selection show accurate utilize simultaneously united state anomaly outperform norm anomaly behavior user become fast accurately classify behavior traffic sensitive prevent effective traffic anomaly inspection flow fourth learn state art traffic anomaly effectively traffic al traffic modification conduct compare seven commonly use feature accuracy anomaly mention traffic meet traffic task character disease diagnosis computer process traffic consist flow period task learnt simultaneously extract utilize multiple anomaly apply selection area employ effective preprocesse extraction step deal raw datum generating anomaly task potential
decay slowly polynomially kind tail treat unbounded regardless impossible general class surely underlying compact rather belong framework optimal rather natural framework outcome interest sparse compressed sign symmetric isotropic euclidean n rt x rt r section scale incorrectly accurately show erm absolute outcome incorrectly notably error become noise persistence explore great later poor outcome rather merely get picture suboptimal boundedness wise boundedness invoke tool contraction function behave interval argument put supremum concentration play essential exception supremum empirical rademacher average g reference role rademacher contraction careful realistic hope bound introduce totally barrier reflect nature insensitive level describe depend variance would distance problem almost tend among phase exact possible insensitive level totally cm entire contraction address handle must scale correctly view concentration argument tail suggest arise statistical article need erm title substantial sided state high low may concrete integer straightforward copy event variable well behave estimate obvious assume fix constant binomial obtain estimate immediate sided heavy tailed class heavy heavy tailed two sided take centre actually side make eventually lead sided concentration impossible consider different regime difficulty explain difference expect interaction complexity intrinsic regime another control intrinsic depend exact perspective localize scale deal significance obvious property agree mistake away mistake mistake erm business identify appropriate formulation immediate outcome function function proportion distance control interaction mistake happen turn interaction sign measure endow nature scale supremum measure maximal random f f definition maximal generic nothing happen bound contraction every dominate bound moreover insensitive contraction affect close right sense multiplier small description role reasonable splitting capture two concentration contraction mechanism every l l loss shift minimizer satisfie belong rather interaction quadratic solely side obtain upper highly quadratic early side type restrictive assumption lead term generic small minimal choose appropriate absolute notion ball concentration concentrate behave trivially satisfied highly contrast sort weak nontrivial ball see formulate convex set particular introduction way persistence reasonable target persistence probability error study fit outcome lead minimax whether technical scope fact class note characterize regime diameter estimate functional persistence endow turn diameter version lower indeed arbitrary target space diameter base show problem procedure exist therefore regime exception example typical version describe interaction target optimality gaussian result mild canonical without go subgaussian canonical gaussian index parameter regime estimation subgaussian subgaussian coincide optimality unfortunately extend somewhat subgaussian beyond scope turn heavy class member useful nevertheless class every constant immediate outcome see f constant cm pass smoothly sense let mean variance vector nc lead apply distribute put ff either erm eq fall scope rapidly decay illustrate difference compute independent accord mean playing assume surely r proof q probability erm big contrast abuse exist constant depend put q yield much diameter capture exact far rather iid relaxed tailed measure may technical reader begin class star shape around shape around eq sometimes also shape around example component let close give follow empirical ball sphere hx np function every inequality every vanish contraction process eq least eq star shape shaped consider q star shape imply star shaped end distance characterization metric projection hilbert assertion q consider fix star shape q claim present claim section prove strong statement th isotropic vector unconditional coordinate tail belong fix mean despite claim p functional possible symmetric variance coordinate may hold unconditional extend follow isotropic unconditional function attain ball set isotropic imply unconditional every depend isotropic unconditional z l follow subgaussian variable independent mean absolute cm first intersection ball radius equivalent convex increase z constant proof tail en jensen proof follow independent copy independent copy distribute accord subgaussian moreover constant range verify suitable fit member scale show high event claim proof use evident depend thus depend next multipli subgaussian standard lemma recall choice concentration inequality bound see relative endowed least
cumulative priori posteriori indicator put mass probability dp address issue dp name bb prove bb frequentist bb diverse priori view event previously report value large bb also remark imply strong reasonable side dp inference example inference spread answer characteristic choice issue worth explain model information observe specify concern unknown prior lack prior propose generate event reflect concern class beta span posterior inference reader one parameter call inference expectation prior calculate specify behave define dp class obtain let normalize vary inference satisfie calculate proof appendix respectively prior since obtain degenerate degenerate bound one let place e posterior expectation finite explain mean infinitely drawback view inference distribution unobserved consider posteriori greater insensitive tail finite measurable case statistic recently interest bayesian nonparametric dirichlet coupling may follow example population population one well response traditionally rank test population equal come population fact eq computed stress use limitation estimator nan although weak see detailed overcome issue common goal versus interpret therefore limited nan drawback test meaning value approach making practical much row e letter probability dp besides limitation extremely weak instead inference obtain lower accord inequality satisfied decide great evidence large value decision figure probability result posterior evident great say distinguish appropriate sided great prove derive presence draw hereafter simplify px dirichlet priori upper prior satisfied furthermore reasoning satisfied measurement posteriori I I n z ji ji bound posterior extreme low distribution bayesian test population compute dirichlet remain correspondence prior suggest decision additional information know posterior would hypothesis dp bb dp information analyst collect additional eliminate start bb dp test loss monte loss dp response evident bb practically coincide noticed case bb dp bb dp response run bb dp clearly accuracy bb useful analyst well available data situation bb dp issue well turn coin although bb dp precisely percentage bb dp large easy instance bb dp loss action significance accord criterion decision reject bayesian accept principled determine lose put decision format believe equal significance level adopt power figure evident performance test practically coincide verify random answer maximum run random lead conclusion coincide well correspond random value choose show test c also return response repeat minus increase test trend decrease together result see guarantee observe difference test gaussian distribution htb propose nonparametric extend set develop variable strength normalize vary prove predictive dp strength dp make computing inference exist thus avoid demand stick break base conservative nonparametric develop test numerical compare test prior strength go robust almost work plan statistical test infimum definition degenerate class exploit ds degenerate proof dp break dp write px f f px f equal always kind prior base equation exploit equality linearity result thus property ta te trace te te ij ij il correspondence posterior give give realization computation theorem goal convergence dp iw normality dp prior vanish state sequence limit low converge variance first rewrite numerator ta note equation tend u ij ij j var dp covariance acknowledgement work support grant em pc pc pt pc pt dirichlet prior information require consist dp set bayesian frequentist test commonly particular robust instance aforementione dp dp nonparametric etc consider prior statistic naturally arise extended function censor data dependence bivariate dirichlet dp justification classic nice promise result development testing package nonparametric relate dp well fact characterize prior parameter prior strength scalar parameter lack information bootstrap bb prior go bb quite since actually viewpoint base drawback bb generalize idea robustness review lack family prior carry consider order natural candidate turn set learn prior statistically useful suggest alternative behave inference e prior credible inference bayesian model already extend near post assumption straightforwardly aim dp dp class strength let normalize vary probability behave priori inference statistical show nonparametric test sum sum health status trial speak near present several advantage formulate problem evidence favor near allow prior view inference expectation close exist monte distribution specific dp come free natural expectation nonparametric near prior efficient effective practical share similarity significant come rank produce test translate minimize
accurate ng high I large amount none kind mistake ic kind ng difficulty ic get summary accord simulation attempt method ol basic choosing method elastic net always predict low elastic net ic ic mcp ic often accurately intensive kind ng get national foundation china remark technology systems chinese sciences china square simultaneous high scad simulation term present still helpful scad mcp elastic field business people demand variable square successfully decade popular lasso elastic net study popular concern practitioner paper numerical micro array fan propose sparse least focus traditional study reader rest organize method end paper response vector variance intercept expression ol lasso lm compute ols penalty covariate correlate ridge explicit tuning select generalize shrinkage ol multiply ng u solution optimization ng directly aic bic accordingly aic highly ridge estimator ng lasso derive fold validation put validation fit expensive mcp nonconvex scad mcp simulation implement scad mcp developed algorithm bic convexity choose appropriate mcp vector datum mean I estimation incorrect ic kind incorrect ic ic ic via intel ghz three configuration ii case iii htp ability selective perform well make show ic training ridge elastic behave reduce ic ic htp htp next sparse vary
unit logistic output network stochastic mini batch increase linearly final epoch entire million feature take machine intel core gb memory graphic processor software package use burn bayesian space possible six decay momentum momentum six unit scale initial momentum log epoch eight layer seven momentum mini performance parameter deeply explore good single decay factor momentum network deviation last root dropout algorithm stochastic activation difficult primary avoid overfitte david comment help optimization support acknowledge nsf grant google award think provide interaction power barrier pair tune training result layer improve performance even improvement discovery equivalent increase accumulate lead discovery collect come back direct mode mode branching ratio measurement consistent claim physics physics improve analysis small scientific learning area software package rely artificial result advance vision speech linear efficiently generalize difficult vanish advance deep network significance high physics operate time million parameter overfitte thus challenge select architecture detail letter deep worth tune parameter carefully maximize statistical systematically heavy form unstable decay successively particle decay decay mode involve large momentum direction visible particle intermediate observable generate particle difficult show process identical particle distinguish particle momentum direction visible particle examine perfect measurement mass state detector impossible calculate mass momentum sophisticated program produce simulation distinguish possible show model classifier detector high predict example describe network similar network train two feature type hyperparameter minimize expect bayesian optimization algorithm combination hyperparameter million example generalization rate momentum momentum epoch layer per constrain neuron space eight hyperparameter random train random weight deep achieve area receiver significance performance boost create classifier auc show network bayesian include hide number large network architecture bad deep ensemble subset level optimize network put practical boost discovery dnn measure need significance nn dramatically operate complete nn dnn dnn dnn dnn ensemble expect derive capture poorly alone train low level deep complete though perform complete deep alone high regard include dnn
work face straight manually build output representation become choice work build feature process unsupervise screening feature generate candidate direction difference cnn black box pair utilize stage cnn image representation map closely identity irrelevant minimize correspond especially keep discriminative power encode abstract another minimize design ideal close within uniform representation eq contain complexity express complex computation preserve information unsupervised method aim modeling unsupervise pattern relate task influence include illumination expression consideration supervise impose preserve requirement deep neural raw capable non abstract property image cnn face pair person numerous match pair also adopt face produce whether pair belong indicate whether image computation network stand encourage person learn preserve specific recognition intra person convolutional compose layer convolution operator q deeply large powerful grow pyramid cnn accelerate multi figure pyramid pyramid cnn cnns divided compose part share network scheme large region share first view pyramid cnn supervise layer share level network f relatively input filter image level train process image become greedy continue level train purpose coverage unsupervised target reflect share another pyramid cnn scale architecture pyramid naturally patch face fed pyramid multi region level computation verification complete dimensional detect improve enable fair pyramid scale feature position system acquire web belong outside pyramid take training representation outside face thousand benchmark slowly fairly system furthermore face system make mistake attribute face detector pair case human argue rely knowledge people comparative effect pyramid cnn pyramid networks pyramid pyramid amount low level improve expense slow room verification increasingly protocol reflect world application especially pair significantly match pair though case receive actually person attack protocol successful attack within attempt pt improvement recognition suffice million face match social rarely significant gap accuracy similar effort suggest pyramid apply area classification scale typically pyramid cnn significant challenge face image part object object believe extend face crucial face optimal discriminative numerous room face pyramid pyramid cnn operation enable pyramid naturally share face result achieve recognition accuracy extended pyramid system performance benchmark face network validate face recognition system image vector learn task verification search
tailor adapt ei couple scope much beyond facilitate handle multiple complicated constraint allocate meet policy requirement knowledge case despite region treat ignore infeasible utilize base lagrangian tool mathematical programming problem novel ei tailor constrain al utilize burden statistical optimizer subroutine mathematical condition derive ei guide subproblems numerical carlo alternatives importantly carlo unlike stochastic method utilize global convergence experience herein scheme number specificity paper problem constraint statistical throughout remainder describe synthetic statistical introduce framework handling combine statistical surrogate toy potential test give tb figure local optima global minimizer solution strictly hold bind bind local may present challenge design boundary set toy problem characteristic common notably objective highly response lagrangian implementation defer date since flexible help one option surface true deterministic objective since gps accurate conditionally distribution surrogate focus extension accommodate constraint often restrict valid region uncertainty capture change normal surrogate function variance together exploitation toward global statistic variable likely show ei analytically case standard cdf reveal exploration global branch later ei improvement yx yx yx ix ix weak algorithm ei optimum gp surrogate ei see one wide family radial surrogate local viewpoint computer rarely ignore loop fit distribution search slow local solution practitioner prefer global ei search nonlinear favorable property find local device method augment lagrangian serve lagrangian define stationarity problem reduce approach constrain unless considerable penalization introduce ill subproblem sequence parameter lagrange multiplier approximately solve subproblem lagrange multiplier inner approximately termination set optimization termination primarily computational section involve evaluation e cumulative could stop approximated lagrangian example threshold one stop determine inner solver dependent solver motivate convergence inner accommodate constrain leverage available section benchmark solver briefly al output obtain determine next direct generate spatial mesh size determine mesh limit fail find parameter take software recommend maximum mesh fine subproblem outer iteration progress spirit statistical propose employ local radial trust region method solver build subproblem approximately solve method design minima ultimately converge minima statistical surrogate offer potential simple directly separately model via predictive mean ei form expression special surrogate order denote number obtain inner outer f ny l ax ic approximate without maximal ei attractive option modular tradeoff modular software ei exploration al iterations inner search region several drawback apparent consider nature exhibit primarily boundary valid gp stationarity boundary region regime behavior gp accommodate schmidt change latter option pair public partitioning divide accommodate limited stationarity challenge change roughly align modeling ei treat motivate example unless inefficient address composite stationarity likely violate simplify many extension improvement surrogate provide c nx nx serve trivially surrogate convert composite calculate swap deterministic way choose composite random predictive ei ei yx include three modification asymmetric entropy uniformly multiple variation comparative discussion perform none poorly easily dominate sensible counterpart require al separate modeling part section involve via analog analytical alternative indistinguishable ei case ei total report al without ei ei separate gps initialize ten pair I start fast mle augment algorithm inner inefficient objective improve candidate easy rejection nice fix densely improve progress two consequence convergence candidate like ei address inner ten search time however find exploratory early search inferior local great explain salient consideration please implement package work toy summarize carlo experiment toy ht cm ei ei model ei ei ei ei monte carlo repetition track iteration distributional repetition valid value plot middle quantile variation eventually global minima five case ei fail instead brief near ignore method ei dominate ei mark behavior drop ignore seem help early sa bad evaluation sa eventually converge reach evaluation compare ei ideal simulation experiment asymmetric perform ten subsequently consistently motivate contaminate environmental increase attention year water prevent disease place water back spread site locate lead contamination site provide illustration expansion identify city website boundary include pose treat version rate operate subject let well objective operate solution element treat never boundary challenge treating via figure valid leave e g abstract stochastic may surprise simple sampling new replicate suggest implement varied search randomly early stage observe half trial minima global slow converge success illustrate clear exploitation early experiment repetition initialize randomly cm ei ei ei ei ei ei al surrogate worst good monte outperform run achieve behavior particularly regard moreover substantial proportion repetition valid comparison initialize contrast surrogate difference final solution well section expense likelihood convergence could thousand iteration attempt sa competitive ei progress rapid classical never quite one sa result amenable approach process method computer attractive augment unconstrained method optima conservative programming ei composite objective show sensible variation scheme leverage traditional usually still room toward success ei provable local global let direct solver toward front constraint leave potential cross idea prove format treat composite al one keep pareto strategy promise offer mc calculation important ei like loop eliminate yield entirely deterministic sometimes clearly usefulness tight nice setting good value rough way monotone case drop al attractive yield improvement inside region outside another slack variable return might guide lee gray lee extra perspective ei straightforward allow exist statistical software directly attractive relative mathematical programming statistical optimization practitioner package box hard imagine matching engineering capability constrain acknowledgment thank american institute
article represent advance piece inference hope piece ultimately activity fall article rare enter statistical nan lasso difficult tie essentially heuristic abuse framework account think sort formula quantity distribution comparison keep grow remarkable particular forward document naive routine nonzero first select assume statistic statistic nan predictor maximize explanatory proper illustrate stochastically nan way effect outline conditionally far test lasso guarantee arbitrary remain already include predictor version stochastically importantly never enable inference predictor proceed test nan predictor simulate exist value correctly account whose properly estimate first correctness adjust significance else naive multiplying approach conservative predictor method coefficient adjust treat obviously easy obtain correction test predictor obvious significant perform stage distribution one simultaneous combination inclusion stop indicate combination predictor select stand magnitude statistic remain statistic distribution statistically significant outcome indicate need power implication statistic although difference half sample eight ph free conclusion endow poor lasso guide curve connection test statistically jointly one remove remove model turn lead example issue test notion lasso issue hypothesis procedure hypothesis truly start arise sequential provide cancer significance carry add carry inclusion previous conditional six carry rejection article guide nan become tight orthogonal predictor assume value step show figure multiply value oppose fact kind behave try control fdr somewhat suggest aggregate series smooth repeatedly inferential fair author actually recommendation methodology read test range article discusse rule fdr emphasis present considerably author work arise meet practice insight comment response issue limit ahead observation assume situation anti conservative day experience rarely sparse stick like need make come exist test sequence test ultimately home message may help form nan distribution may live trade procedure random predictor comment post inference fdr would decide fdr sense mutually error predictor orthogonal error meaningful concept adjust statistical inference article necessary path large underlie test assumption misspecification misspecification place nonlinearity become yet effect inference due test lasso test test ultimately augment sequentially however validity whereby selection nothing realistic situation analyst try forward device datum heart mind face meta favor practice benefit selection meta accord subsequent big graphical exploratory
entry eq exactly place invoke theoretic result ij provide exploit semi triangular strictly follow nonempty nonempty lemma stay away influence eventually two positive respectively difference oppose lemma three every extend remove start near remain satisfy q proof meta proof triangle r cauchy note subset coordinate update algorithm convexity optimal convex strictly follow argument e e state lemma minimization uniquely uniquely minimize thresholding define eq note solution strictly turn minimizer define value infinity belong trivial problem use provide argument similar establish ultimately follow every assumption need deal oppose argument taylor fix arbitrarily convexity negative eq q follow obtain follow parallel version exactly suppose repeat q bound least limit point note arbitrarily q every show iterate appendix follow solution system large depend result approach et al provide constant definite minimize get follow hence hold definition iterate converge problem special applying stack precisely permutation z z p note I view kk verify convexity none independent logistic regression number tackle situation respectively define follow note typical necessarily cyclic variant appropriate choose coordinate minimizer method cyclic application convergence global minimizer proof cyclic matrix iterate cyclic satisfied every logit appendix automatically follow hence establish contradiction lemma obtain r e non non negative induction every eq definition j hold every definition prove require part proceed claim go every exist hence result part every contain absolute obtain contain absolute exist large occur observation hence induction establish non q x contain absolute large entry assumption exist conclude argument sum strictly subset norm appear modern associate optimize employ effective consequently crucial iterate cyclic regularize likelihood variable function establish ii establish convergence establish inexact iterate produce variant usefulness application employ twice strictly domain empty interior suppose also everywhere boundary let give column follow extensively particularly example typically log pseudo correspond like however modern obtain sparse solution exactly inclusion objective challenge convex hundred penalty finding problem scalable theoretical convergence form cyclic minimization consist minimize often offer computationally respective form involve dimensional numerically achieve high step hence understand cyclic rigorous cyclic minimization optimization iterate cyclic minimization stationary point thought descent function differentiable subset contain effective every follow inexact ensure produce consider way choose quadratic alternatively choose clear result variety descent hessian non entry many minimization cyclic substantially different block coordinate minimized iteration speak possible ordering coordinate cyclic suitable situation time amenable important convergence rate establish reason motivate random descent cyclic former allow easy actually non cyclic establishing iterate objective quite although function converge cyclic solve separable descent type trust detail propose extensively outline rigorous iterate cyclic algorithm build extend incorporate cyclic cyclic shall smooth challenge non trivial question provide summary assumption algorithm cyclic respectively convergence problem cyclic c establish later I provide domain empty twice curvature everywhere minimization empty h tolerance q r return stop claim contradiction x follow x attained f r contradict case lead contradiction theorem formally establish convergence produce theorem generate cyclic value detail successive iterate go establish square difference zero sufficient cauchy combinatorial iterate cyclic cauchy limit follow zero coordinate bound coordinate stay away influence eventually zero therefore establish immediately distance iterate boundary zero solution consider problem recall make application arbitrarily fix however separate statement two algorithm iterate set initial tolerance go minimization unique closed form establish produce cyclic second paper cyclic descent minimize converge compare method prove start minimization follow argument therefore omit eq definite open I differential version kkt imply alternative provide iterate descent appropriate argument euclidean matrix depend vector also continuously uniformly follow say say empty cyclic contradiction hold since bound subsequence follow minimize along coordinate contradiction establish successive iterate cyclic arbitrarily let subsequence around follow every every fix exist constant r bound satisfy sd since argument trivially establishes follow every consider second coordinate
learn create often overfitte interpretability expensive actually less classifier svm mean curse might also incorporate genetic cause might end participant gene assume gene cause end possible fit perfectly discard variable attain gene might actually produce cancer might need take low dimensionality well learner might become infeasible article broad introduction selection ten year passed state art feature cancer patient conclude responsible human pose reduction present overview modern justification take validate continue author identify strength build accord selection accurately overfitte might well treat valuable selection author gene tumor preliminary selection sophisticated procedure select top call implicitly feature uncorrelated ground absolutely select perfectly high would probably discard tumor denote tumor grow separately would ignore connection subset try reduce variance boost filter necessarily see figure correlate get tb approach proxy look break superior heuristic train depend atomic feature anneal branch subset exhaustive search either add way replace least treat necessary simple way classifier probable similar p take optimize distinction could step reason motivated selection small compare compress store express bottleneck recover low concept create place create intuition efficient code occurs often give representation much body model dimensional hide gaussian low dimensional map body track body shape neither fix training never approach independent identically distribute split identically take historical book pixel pixel perform well page split author author instead modify datum modify add variable chance ratio subset feature examine expect add feature call examine objective function evaluate necessary linear predictor variable variable model take noise eq pca case assume iterate natural show optimally score drop fix globally optimal subset find wang classify expression determine tumor think vector huge dimension chance require interpretable discuss feature embed describe understand separation direction maximal discuss information criterion descriptor minimize exponentially criterion n ignore influence one length algorithmic store descriptor give good cause variable paper start rank discard repeatedly respective collect seem unnecessary even overfitte discriminative autoencoder neural find fitting bottleneck transformation tb minima stack machine optimisation stack extract distribution somewhat sift enough autoencoder learn concept cat face supervision simple logistic feature concept sift bag human create variety discriminative possibility neuron single selection extreme benefit advantageous necessarily deal dimension computer actual pixel image document image incorporate heterogeneous evaluation useful instead thing whether prior useful example tumor gene could unfortunately segmentation causal proxy pixel smoothed limit big advantageous one variable rank bad classifier fast due allow area many still apply justification application develop drive advance high autoencoder motivate integrate usage principal goal avoid interpretability opinion integrating learn expect embed approach fit ensure treatment support vector machine extend incorporate expect usage
parallelism amount neuron activity neuron activity data parallelism per weight parallelism arbitrarily synchronization batch batch hundred thousand example consist layer property layer connect computation representation ask whether data parallelism appear attractive layer parallelism attractive connect layer propose explain mention nice convolutional net rely heavily parallelism parallelism fully worker reference pass like worker example worker convolutional batch activity worker switch parallelism worker activity activity connect worker convolutional activity gradient example stage layer activitie worker worker activity worth consequence big worker batch undesirable device big batch worker layer activity consequence much fully far advantage scheme worker scheme utilize major usual way forward pass stage convolutional worker pass continue usual worker compute layer gradient gradient responsible worker fully connect backward propagate convolutional computed example must operation I reason backward worker parallelism three stack parallelism connect layer pass replace six pass convolutional batch process fully layer pass backward backward pass worker layer weight gradient simplest worker accumulate implement modification backward propagation nonetheless run batch size notice backward pass wish backward pass extra layer kind batch algorithm pure parallelization long update consistent convolutional turn batch size fully minima question somewhat complicated question answer benefit batch size large widely imagenet million fall scale iterate many training minor winning consist map dimension equivalent minor arise convolutional another softmax layer multinomial final unit minimize perform easier require normalization multiply progress use w momentum decay learn gradient batch size hyperparameter plausible momentum big batch batch multiply multiply expectation constant adjust weight decay batch size like decay weight application batch weight batch size w w learning rate decay coefficient w net approximation neural net aside heuristic multiply multiply practice multiply instead decay patch compute error patch machine eight intel cpu amongst simultaneously gb express incur penalty big batch size greatly parallelization scheme well scale dense multiplication output multiplication inefficient size gb spend product gb scale size dense connectivity kind show gpu communication simultaneous batch connect parallelism convolutional parallelism connect hour table publish alternative parallelism parallelism parallelism speedup relative gpu implementation hour train epoch sgd gpu parallelism parallelism speedup relative gpu train accuracy gpu gpu cluster neural network worker spatially across neuron activation edge could potentially convolutional net
argument goal topic mixture vote resort approximation analyze utility supplementary vary proportion support maximize utility support brief vote bottom sometimes bad utility function present perspective support vote know side optimally write brief ip odd vote favorable marker actual simplify ii consideration writing style choice etc fully brief treat share author author despite tool generation research ip roll infer position vote ip multidimensional multidimensional study especially influence present interest study decision encode text text evidence behavior relate maximize look extensive structural decision ip researcher familiar algorithm distinction matter maximize utility domain improve vote prediction capture simple text secondly importantly interesting question write fact differently acknowledgement part fellowship sim nsf google award resource kkt th issue q first vote topic equal marginal value highlight ii large whose vote receive iii controlling care utility expect utility sample turn hasting public code national vast majority range year public dc management collective north collective action member preliminary matter title vi act discrimination old worker life reasonable impact claim party construction plain mean dc circuit year relation master interest rt act reservoir video circuit circuit fourth circuit instant fourth fourth probable agreement national bank rule act public international law international law convention united challenge school strict education discrimination school interest local political law final removal proceed proceed death death trial death direct new environmental water clean water master public material water school private school belief organization free circuit art matter act act drug act service high public law party private corpus award state trial ideal depend various reading law notable expect unless attribute california law survey rank vote ip ip influence suggest present ranking note encourage hypothesis l curve vary topic vote hence model favor prior interior proportion experiment inactive school university pa usa institute university usa idea piece maximize one utility make concrete decision unite past work quantitative political science framework empirically model decision incorporate friend separate decision benefit improve piece huge array behave take incorporate united states american far reach nine public organize group friend hereafter author know reveal explicit attempt vote language build establish political vote although incorporate analysis draw rational agent argument toward favorable outcome derive inference substantial gain importantly answer would change brief characterize brief different fact applicable goal behavioral response review commonly put argument party file brief response argument recommendation side necessarily conclude vote relate vote political science etc ideal ip ip time simplicity interpret spectrum ip model vote favor case popularity favor capture otherwise recover maximize opinion additional right vote embedding incorporate text evidence infer dimension build latent dirichlet popular corpora issue lda vote vote outcome determine mixture proportion issue ip infer mixture proportion although text serve incorporate label supervision infer address label multidimensional g multidimensional due preference issue fact fact influence outcome argue public group unite goal present argument u recently ct neither consistently attempt position strongly ic proportion ip influence form text embed rescale discrimination parameter generate relative vote hereafter b proportion infer support support share influence ip equally influence vote capture collective focus position vote suppose e vote favor outcome indicator policy cost increase fact control notion text carefully frame costly match unnecessary role outcome uncertain brief expectation incorporate ip maximize impose utility checking expectation make difficult constrain imposing prior negative negative write brief discrete choice relax precision assume topic expect consider likelihood utility resemble vote maximize assign principled manner rational expert variable ideal diagonal direction dimension case ip dash leave ip blue issue ip diagram ip non use text associate hyperparameter token text share similar dirichlet right fig diagram importantly note serve structure topic ip argue conceptual define rotation multidimensional preliminary issue joint vs stage latter relevant stage mcmc latent gaussian diagonal covariance target likewise univariate gaussian proposal utility utility hyperparameter supplementary topic case fact vote cast brief label manually label label support take content support g phrase give interpretable extraction preprocessing brief material ability vote probability vote find side multipli expected ignore non utility due specification ip vote vote imply voting towards vote distinguish identify evaluate actual vote validation vote regression topic ip ip na I vote use proportion baseline exhibit perform well adjusting suggest model furthermore believe insufficient slightly well paired accuracy qualitative
understand accord clearly preserve increase due step result td easily result text rank td exceed corollary assumption minus height supplement analyze omit zero j jj nj kn nr nj kn jj row product therefore add either conclude independent subset vector ready x full column column lemma main iteration q matrix definition utilize operation involve step show output td
surrogate multiclass surrogate define surrogate risk learn prediction learn make fx several minimize multiclass space tf condition surrogate minimize example raise natural loss amount surrogate loss square multiclass rectangular ignore generalize surrogate loss w imply w proof calibrate follow calibration surrogate mostly concern note calibration follow result straightforward loss f surrogate calibrate cx goal surrogate k end certain multiclass multiclass relate multiclass multiclass surrogate necessary probability call diagram diagram probability calculation l easily calculation n empty optimal extension note converge positive r set section give necessary calibrate r normal surrogate normal calibrate start derive happen positive intersection calibrate imply contradiction necessary calibration behave loss discussion strong positive individual contradiction sequence l tt l calibrate exist q empty contradiction include special direct surrogate look positive figure clear zhang give sufficient calibration helpful surrogate contain hold finite normal nr z calibrate calibrate normal surrogate surrogate calibrate ordinal regression apply surrogate calibrate additional provide necessary also convex calibration section necessary calibration involve normal order calibrate compute characterize set computing surrogate result compute operate number applicable u w u w attain minimum subdifferential subdifferential u u n computation lemma two surrogate operate calibrate another calculation surrogate positive surrogate u satisfie note satisfy similar p positive compare u normal insensitive insensitive u z r consider compute condition lemma p computation z figure theorem calibrate detail calibration surrogate calibrate ordinal sufficient surrogate loss calibrate w multiclass surrogate raise support calibrate multiclass lead otherwise cc result multiclass small let appear surprising surrogate surrogate calibrate loss however multiclass accurately accord probability show composite task helpful algorithm operate cc depend tight nd parallel hull column correspond vector u u dt u u r u calibrate definition normal probability q corollary l follow immediately hamming hamming loss bit representation r r therefore dimension loss illustrate section obtain existence calibrate surrogate surrogate space convex denote subspace p bind make feasible dimension p p v v subspace point us dimension let denote one identity matrix ordinal loss know theorem n class show theorem tight p therefore immediately immediately dimension rank loss framework rank together simplicity fix predict permutation document popular loss subset discount cumulative ndcg disagreement pd loss view multiclass ndcg loss relevance acyclic cc ndcg section ndcg result show dimensional calibrate surrogate ndcg pd et surrogate map ndcg set relevance say level permutation non ndcg view multiclass calibrate surrogate ndcg direct acyclic associate instance th document document prefer object permutation label g I pd multiclass term sum simply subtract column minimizer loss loss result loss therefore one show exactly et certain popular calibrate pd calibrate surrogate exist et exist surrogate calibrate pd allow go surrogate pd prediction dimension predictor permutation object label q bind result et al show surrogate predictor calibrate map fact one surrogate unify surrogate multiclass define multiclass cc multiclass possible calibrate respect analyze loss multiclass loss example loss surrogate must surrogate lee value learn implication disagreement pd average loss rank rank problem exist surrogate pd loss admit calibrate surrogate convex surrogate loss operate surrogate learn value value scoring cc tight class loose characterization cc dimension develop design calibrate surrogate operate according given design calibrate surrogate surrogate space form always possibility surrogate space certain loss issue contribute understand calibrate surrogate multiclass proof nr z z z claim banach theorem corollary hyperplane strictly separate w w calibration fix e n z jj p contradiction calibrate j give q l claim pt u tm tm p require relate feasible intersection property lemma contain get contain row affine nan p recall convex eq clearly condition far satisfy q let clearly therefore orthonormal f verify eq subsequence converge point orthonormal n satisfy condition theorem take limit thus lemma equation get z n therefore l b l calibrate eq claim follow l graph row linearly sake contradiction direct coefficient permutation b apply two eq definition verify column two give b j I show q since r true vector form removing dividing establish make denote rr rr pt linearly independent exist cccc exclude diagonal entry moreover vector intersect trivially column permutation together trivially expand recursion establish fellowship thank technology fellowship technology k z calibrate pt conversely see u z p u u p n p satisfy multiclass lemma calibrate p z z tm suppose calibrate h convergent still call say p tm contradict conversely calibrate particular consider mass f mx complete eq symmetry loss p p p p p u z set z z algebra q q lemma thm study function general multiclass problem multiclass notion calibration multiclass calibrate multiclass loss measure size space matrix upper quantity various loss dimension tool calibrate surrogate result et al certain surrogate rank surrogate loss interest surrogate binary multiclass classification problem finite prediction multiclass structure understand loss consistency unified study minimize surrogate loss surrogate respect target give sufficient multiclass matrix fundamental surrogate calibrate quantity difficulty loss class calibration consistency practically class value give quantity term framework tool arise rank discount cumulative gain ndcg pd practice subset relevance query learn loss sort document pd loss admit calibrate surrogate together sort operation convex lower bound surrogate work year consistency calibration give brief overview body risk focus largely classification example consistency universal show result boost zhang calibration binary particular seminal classification calibration surrogate yielding give necessary sufficient surrogate calibrate calibration enhance surrogate require strong early calibration theorem convex theorem tight pd additional ndcg example throughout minor improvement emphasis notation terminology example throughout formalize
result circuit bound monotone circuit size work function computational theory respect influential boolean monotone uniform learn consider various generalization arise monotone circuit obvious monotonicity boolean circuit small circuit access compute denote boolean generalization start alternate characterization boolean monotone chain increase monotonicity flip terminology position write denote alternate maximum tight quantitative inversion complexity ii motivate circuit circuit theory power circuit show circuit contain circuit circuit non circuit work circuit give result structural circuit establish theorem boolean monotone conversely yield every express either well know consequence shall circuit significant possible upper boolean must investigation circuit extension markov fourier learning circuit tn give circuit run essentially match monotone slight though membership learn monotone fact membership bind membership query accuracy thus give fairly answer circuit low match wide learn unknown membership strong make arbitrary query tool hardness strong task balance monotone hardness strong boolean function moderate hardness lift circuit moderate ingredient crucial final low detail extension theorem write denote write f I jt f simple observation thm fix x conversely boolean boolean immediate inductive express get converse observe induce possible monotonicity immediately follow corollary circuit express monotone every exactly circuit goal boolean fraction significantly suffice ff inversion complexity boolean approximated circuit bind boolean boolean th coordinate influence fraction boolean hypercube easy together prop lb must inversion theoretic showing learn circuit significantly shall cn sketch learning start concentration high estimating degree coefficient refer learn fourier monotone monotone armed straightforward extend fouri finish even boolean function immediate give answer subsection theoretic low limited number start establish membership bound bound example defer exist balanced variable gap monotone hardness learning alternating exist balanced boolean ii tradeoff range analysis query k require require function eq overview alternate get hardness moderate repeat moderate detail idea take base monotone hard monotone sensitive hardness must take constraint monotone alternate possible useful recall notion play approach draw obtain quantity coordinate left build sensitive give stability minor detail exist constant infinite g upper hardness uniform build hardness theorem deal run learn algorithm inspection proof query class boolean distribution membership query uniform learn accuracy hardness accuracy claim lower give completeness appendix hardness class balanced universal range hardness noise low stability bias proof sufficiently function variable membership contradiction learn membership query infinitely many membership contradiction mr error achieve indeed balanced trivially recall accuracy q improve give monotone family f bind hardness monotone hardness establish hardness hereafter monotone every bias zero recall boolean use function could use variable instead hardness function need scale odd rx middle hypercube chernoff layer sake proof define balanced sufficiently n membership query infinitely membership impose constraint impose lemma need ultimately show first constraint meet setting remain check constraint inequality
estimate cost machine memory factor since computational besides obvious handling incur centralized method point centralize adversarial achieve contrast naive division algorithmic machine significantly robust appeal division though division averaging helps preserve robustness bring prohibitive might practical moreover offer computation error instance machine many individually take less precise concrete distribute roughly require center aggregate suffer build framework us sample parallel take simple average robust corruption one robust partially reduce machine linearly sample finite target matrix underlie additive another problem covariate parameterize lr explain iid well many learning corrupt particular contaminate challenging fraction outli fraction briefly concept develop robust median call detail median importantly let hilbert close inner norm geometric define practice version admit atom later estimation machine median point eq median exist rather calculate median employ limitation collection strong presence robustness geometric median set median long least particular geometric median skew significantly away implementation example concrete division strategy core definition fusion step aggregate suppose ready evenly subset onto specific estimation denote estimation previous division propose separate estimation reduce average aggregation estimation many outlier machine lead aggregated estimation base sample communication error differ bad concentrate single point estimation lemma base provide distribute input guarantee characterize corrupted estimation present ground truth k basically even machine break either base communication guarantee final high monotonically monotonically monotonically account geometric concrete specific rapidly failure machine trade pca failure need trade machine real world series outli outlier machine potentially final distribute learning algorithm way exactly expense potential introduce robust point robustness point corrupt algorithm provide concrete centralized averaging counterpart classical component analysis pca propose reference therein cost prevent big first robust pca enhance efficiency ij product product product remove remain small select index obtain estimation covariance eigenvector decomposition produce estimation md perform eigen decomposition large sample n aggregate plugging distribute robust outlier simply arbitrarily span subspace eigenvector arbitrarily due limitation proof supplementary material divide onto large least eigenvalue denote onto suppose outli projection matrix small basically say fraction outlier machine corrupt integrate underlie detail covariate n p h pair kx k aggregate output reader detail similar proof supplementary depend fraction outlier guarantee design probability least straightforward constant outli linear regression problem whose iy p sample outli generate estimate conduct outli degree uniformly thus outli fraction outlier distribute instead outli machine similarly adversarial case design repeat simulation implement pc cpu ram take centralized second distribute cost parallel procedure negligible improvement performance fig division centralize non robust outlier avg pca offer high efficiency hold fig begin centralized fraction increase centralize blue division average green line still robustness well indeed compute break favorable aggregation computing besides result machine error machine user machine quality aggregated performance machine finish randomly estimation sign estimation aggregate averaging offer strong report avg far solve recently large scale around tag provided employ tag tag deep cnn large impossible gb store training implementation divide division robustness well provide lr achieve compare division average geometric median actually negligible avg lr robust different subset memory preserve learn addition bring node adversarial example xu department corollary remark distribute traditional robust learning order robustness property show robust precisely adversarial outli break contrast naive averaging node framework component efficiency advantage tag big challenge scale current
possibility non factorize partition assign consensus distribution modularity extent statistically structure converge neither factorize stable spin replica symmetry break word exponential bp jump obtain current marginal long bp spin retrieval modularity consensus many factorize compute derivative factorize magnitude like factorize fix random know stability census reconstruction show average excess number statistically phase spin converge factorize retrieval factorize bp converge thus statistically note fall retrieval heuristic necessary scan linear perturbation oppose respect backtrack bp eigenvalue network sbm assume correspond community bp factorize long disk complex inside structure isotropic community retrieval eigenvalue spin study heat mcmc hamiltonian cut measure modularity analytical transition fix spin spin sparse problem sensitivity bp whether converge fix spin bp expect happen phase retrieval free fix exponentially bp start message fail energy spin beyond replica bp replica symmetry phase quite narrow c grateful mark draw software http skew de work grant modularity community however maximize modularity modularity poorly produce address modularity hamiltonian finite temperature belief propagation partition modularity partition numerically work transition network generate work claim show recursively statistically detect hierarchical network method address physics try partition network maximize modularity seek modularity partition message treat modularity hamiltonian apply cavity analytically perform network method determine community biology connect many spectral adjacency statistical stochastic block variety review partition modularity partition label group node modularity network edge edge number neighbor kronecker delta thus edge community fix give modularity random community modularity even modularity exhibit amount degeneracy poorly small perturbation notion statistical hypothesis nan enyi however community true partition distribution block right regime correlate modularity right approach hypothesis appear work hamiltonian spin gibbs modularity search look modularity partition single one analogy marginal gibbs assign tie achieve call modularity claim modularity language marginal posteriori prediction informally efficient propagation bp marginals cavity algorithm scalable number group sense way provide community tend statistically significant validate work claim find obtain community easily model plant sbm popular ensemble community group plant connect commonly group entry ratio community become er call transition bp transition establish rigorously number group complicated mark hard time succeed easily behavior er sbm regime two phase bp converge equally replica symmetry break transition spin modularity return bp bp jump little bp assume replica phase sbm spin bp statistically significant retrieval enter increase retrieval modularity er convergence transition spin sg transition sbm r find retrieval modularity indicate statistically analyze factorize stability correlate perturbation cross analytic fig leave diagram retrieval phase retrieval excellent agreement find retrieval define plant I e emphasize optimal subgraph apply indicate subgraph network sbm er find state subgraph stop network large world algorithm repeatedly retrieval subgraph suggest political find two ground split eventually find hierarchy total leave fig level modularity explain split community explore report give level split community stop indicate remain leaf denote degree color group final division order partition algorithm popular statistically modularity group network find modularity algorithm find modularity community emphasize modularity find community show network apply sbm normalize rather overlap group plant show correctly choose appendix heavy degree distribution drop transition er find average base statistically community graphical modularity attempt marginal gibbs bp algorithm cavity next likely marginal essence look consensus partition modularity indicate opposed fluctuation retrieval spin correct bp correct modularity however em block variant clear optimize work sbm barrier community community difficult appendix give evidence barrier namely clique clique proposal determine group eigenvalue backtrack spectrum network political large deeply detailed generalization modularity weight internal interesting normalize cut consider run bp marginal fix bp marginal reinforcement external toward leave graph modularity partition nearly uncorrelated language material landscape optimal modularity hamming distance replica symmetry breaking jump optima focus optima contrast
different sign saddle eq intersection look like corner rectangular confidence confidence co fluctuation repeat previous confidence region unbounded geometric linear relationship comparison correspond panel interpret throughout subsection stationary remark co increase give increase actually conclude responsible european combination keep useful combination ga conditions equation equation ga li develop identity acceptable positivity yield co trade unit e co level formula show develop quadratic interaction compare relative relevance variable table complete variable co instead relation specifically arrive relation variation direct derive application thm proposition thm order finding optimal second linear second study dimensional region risk part model co various include mixed interact produce ranking start develop region comparison indicate part per million collect co emission datum list united member exclude present individual period former respectively cm co goal response summarize canonical respective effect decrease neutral type region region elliptical parameter recommendation optimal management emission trade policy order requirement country restriction second interaction determine eigenvector equation kronecker define computing eigenvalue arrive software precision overall factor eigenvector find represent bring form constant non use become conclude find degenerate point normal quadratic write find case cm width positive negative define interior right panel width
data absence train unknown noisy estimation know disjoint life non possible sample differentiable training ensure absence datum attempt system function available effectively point value comprise datum measurement uncertainty learn value system point within difficult task paradigm unknown function word bin bin value define try impose exercise translate density course identify unless construct subsequently model hard learn wavelet fundamental spline wavelets capture amongst difficulty particularly invert learn high arise variate uncertainty vector capable datum could meaningful big within practical determine paradigm lastly paper discuss hand section briefly model methodology subsection inference synthetic round density space available practitioner estimation parameter west observation current modelling error interest parameter another equation system high well uncertainty space treatment replace state observation embed variable partially thus include static addition address methodology present inference mass simulate galaxy mass project useful dark learn vector give comprise th light suggest f term however deal sub must project invoke embed accomplish rhs generate since functional relationship attempt upon impose place size equation variable cell equation cell unobserve vector general give compute rhs impose edge cell call identification mapping detail consider use incorporate measurement refine density advanced inference likelihood equation give hand discuss advance drive suitable version implement credible region learn value ask treatment estimation unknown measurement pursuit express inversion function bayesian method variate learn computation unknown test absence available datum e paradigm model functional possible yield connect unknown therefore alternatively learn learn learn instance becoming value address result share relation paragraph nature dark science dark matter dm individual observe system exercise fundamental quantification dark direct observational evidence dm light quantification measurable physical field I dark physical property temperature live play light field act extraction learn total mass matter subtract latter self matter matter reliable mass physical include pattern perform galaxy attention galaxy learn measurement particle particle refer particle star signature mark include old cluster state refer galaxy coordinate particle system velocity galaxy vector playing rise mean stationary aim physical velocity align line observer e particle come towards observer similarly particle plane x x datum k allow galaxy high addition measurement noisy typically datum typically incorporate measurement data matter dark well galaxy unknown mass proportional product function observable conditionally likelihood write application explore situation thereby achieve evolution space discuss tell evolution time td however particle inside star scale age galaxy tf boltzmann q state attain stationarity constant time bt correctly suggest stationarity inside write another steady equation invariant lie proof respective connect vector dt dt attain stationarity space variable dependence rotation cross product simple system along location r evident high circular circular path bt ready methodology section aim computing domain range place grid state application particle galaxy live inside galaxy galaxy particle attain e rr attain normalise maximal low light keep discuss bin bin range cover normalise bin make physics could available recall aforementione less less thus clear normalised extend bin width learn inference equation place centre learn mass lead choose discrete fix radial entirely radial learn express r v unobserved grid mapping coordinate turn function cell depend integral grid inside space consider rd grid cell cv circular lying provide particle bin follow ks lie bin semi minor lie lie implication equation give observe energy bin lie within circular extend elliptical extend semi minor axis cell writing volume area region integrate recover rhs multiple way region distinct overlap know overlap excess identifying allow numerical computation area irrespective allow bound area plane area overlap plane equation express integrate th grid cell low value recall discuss vector parallel q root equation attain value hereafter triple give contribute towards equation dependent within implementation metropolis hasting ensure value enhance integrated max equation conditionally measurement particle project plane galaxy uncertainty particle velocity denote galaxy hand usually implementation simulate define uncertainty measurement likelihood nothing suggest angular opt inference mass bt monotonically decrease numerically motivated community refer purpose mid hyperparameter prior experimentally probability density constant define integrate interval within generate metropolis hasting write joint next make along constraint perform metropolis maintain inference let scheme current let achieve variable zero mass radial variance choose empirical I ease proposal acceptance criterion metropolis discuss accept update line inference directly let current empirical th update element nf proposal interval inference rhs equation proposal density sample ratio posterior I propose vector accept tn lf numerical synthetic display explore trace posterior part learn synthetic black solid high credible represent bar line learn plot parameter dot line synthetic histogram size sample mark solid mass state space learn galaxy exercise relatively run isotropic achieve fix bin dimensional isotropic model isotropic result chain relaxed incorporation learn plot value recover modal learn bin middle parameter estimate chain bin long estimate state learn use isotropic black figure estimate I panel green focus measurable physics connect comprise training relationship either wavelet estimation bin write could missing project onto term domain projection identification prior perform galaxy galaxy kind particle density r dr function compute learn plot infer mass turning result absence prior isotropic state galaxy really mass galaxy mass distribute space crucially exercise possible space vector vector state remark mathematics associate research department statistic theoretical physics university department statistic university al physics unknown model parameter relationship function method modelling
construct contain follow connect besides vertex connect group distant object operator minimization eq sort matrix namely motivation intuitive graph similarity reduction method perform reduction kernel bring gain computational discrimination power reduction space empirically stay classification accuracy crucial classifier experiment conduct instead part classification serve relevant algorithm adopt new query transform basis residual discriminative show algorithm corruption design dictionary lagrangian formulate penalty representation vector lagrange multipli alm lagrange multiplier monotonically increase implement iteratively q shrinkage come transformed kernel project classification apply normalize regularize normalize unit mu mu identity locality use measurement term training sample fine small increasingly sample nn similarity near dictionary locality locality dictionary adopt straightforward way constrain locality reduce computational cost locality category biological criterion recognize speak locality constrain discriminative distance discriminative curse kernel idea locality mathematically experimentally atom example formulation mu first obtain via different atom fact locate near efficient enough database gd simply several perform measure perform distance conjunction flexible review subsection several pixel measure simple effective situation city pixel grid another assume make move move move count horizontal lot pixel distance present texture texture similarity main idea appearance characterize distance difference texture sum response histogram filter metric measure histogram texture initially surface category surface crucial tangent distance affine stroke descriptor relative descriptor iteratively match shape final match shape transformation shape descriptor define gray shape descriptor map sift descriptor descriptor method combine texture unified locality construct enforce locality via texture similarity three size set diagram atom dictionary combination similarity construct combination similarity e lc lc lc texture unify use suppose equal use unified similarity measure recommend raw use dictionary namely sample sample per generate set gd gd b slight classification gd stay global dictionary projection use consist face individual face experimental setting evaluate namely dictionary sample person mostly dictionary informative adaptive important dictionary degenerate gd become locality become well less enough performance dictionary sample gd size global note pyramid similarity database contain object background train rest use global sample comparison spatial pyramid feature use classification competitive especially classification improve high lack extra discriminative gd obviously feature category difficult categorization database experiment category constrain svd classification dictionary contain scene office select rest testing global dictionary setting lc superiority htb pt c c gd gd gd conduct run time evaluate public mnist experimental table use compare approach locality fast gd gd circumstance htb category pt gd gd gd mnist metric greatly enhance classification superiority discriminative compare database euclidean mail image fairly rate width setting detail local histogram divide face area histogram histogram database handwritten digit randomly size raw result pyramid experimental properly select enhance power distance metric bad extend mnist since work bias fortunately metric adopt framework show flexibility allow flexibility unified unified still close basically automatically select reason unify complementary complement diverse enough use complementary tangent believe could bring htb pt extend euclidean n technique smoothly combine discrimination reasonable additionally locality locality exploit enhance mathematically help database also reduce discrimination ability approach discriminative coarse strategy intuition appeal comprehensive superiority public database efficiency validate moreover construction experiment public database simulation discrimination ability idea discriminative greatly create toy experiment unified effective kernel predict become thm thm via collaborative locality propose collaborative representation similarity query atom locality dictionary addition measure unified superiority validate appeal aspect incorporate similarity theoretically perfectly classify conventional scalability state kernel regularize near neighbor locality constrain dictionary recent year great bring robustness regularizer ill condition represent basis basis call overcomplete atom call indeed impose sparsity return help recover despite fact nonconvex norm study et dictionary report representation promise approximate input atom predict select despite fact use still representation classification zhang far unnecessary enforce feature enough work representation cr cr improve reduce collaborative nature make poorly distribute function kernel principle component svm overcome sparse classification particular nonlinear infinite kernel group become besides major propose locality unify give gain cost gd cr database gd gd enable database classifier enforce locality serve locality appeal motivated finding list similarity important mathematically incorporate theoretically training obtain operate feature dictionary typically informative often bring gain globally consider concept link query atom play role support recommendation report gain fact atom yet well performance atom regular dictionary near categorization demonstrate classification fundamental tag advance since far able human infer abstract remain motivate discriminative efficient image outline discuss relate locality discuss follow remark technique al promising zhang al far mathematical conduct comprehensive evaluate overcome handling liu et address kernel representation incorporate model practical recognition discuss image however kernel formulate hyperspectral image method classification difference exist incorporate consider hyperspectral contrary strategy discriminant reduction generalize extend improve formulation detail reduction kernel notice idea locally dictionary pruning apply extension combine locality dictionary link conjunction bring scalability extend feature dimensionality propose briefly code whole essence class dictionary th denote I ji error formulate combination determine residual key reasonably regularize least signal prove collaborative important machine whose member task component classification raw explicitly specify similarity function via kernel easily kernel enable operate implicit computing often sequence reproduce rkh kernel include perceptron many transform non handle smoothly nonlinear mechanism map high via transformation mapping empirical mu mu transform
numerical nature superiority make skew viewpoint numerical outcome little want classical perform slightly numerical commonly triplet experiment figure would regard point examine comparison aspect opinion closure interpretability viewpoint plausible great possibility factorization independent lead preference skew statement lm subsequently adopt close skew variant prefer finite modelling limitation examine statement adequate correction avoid reader acknowledgement grateful discussion various aspect relate material theorem p skew continue popularity effective include formulation model literature property various claim well datum year dedicate class distribution special normal univariate elliptical study alternative elliptical impose symmetry simultaneously component closely meet symmetry elliptical skewness explain skew already quite stream skew symmetric elliptical similar recent account generate nearly compete alternative preferable specific considering examine far although parameterization early one distribution construction skew normal skew shall skew normal coincide dimension differ arise difference formulation viewpoint carry arise underlie parent normal wider elliptical call elliptical family examine classical skew counterpart skew question form skew reason great flexibility shape arise freedom traditional via substantial involve mix parametric th family role elliptical parsimonious possibly interpretable special distribution view quite inaccurate potentially impact community brevity shall lm paper appear adopt framework overall formal apply lee role elaborate skew distribution skew normal positive definite playing role parameter representation follow normal mean ct obtain detailed present chapter skew minor symbol retain parameterization write symmetric independent satisfied wise moment two analogue originally use skew match display scatter ht institute sn panel similar mass contour curve panel contour mode somewhat curve corner extensively illustrative literature become shall come qualitative include completeness parameter coincide family subset moment become factor handle burden rapidly purpose fitting common bayesian lm skew statement consider equivalent diagonal matrix allow independent skew component factor non vanish involve require classical incorporate logical frame describe subject selective expression skewness skewness excess numerical maximal appear coincide form case among thing hold remark formulation emphasis depend even wide limitation introduction skew possibly factorization consideration lack closure negligible skew advantage factor take interpretability remark relevance qualitative matter illustrate refer resemble regular student follow normal independent skew representation skew thesis range coefficient skewness marginally instead skew source point flexibility version appeal explore aspect lee early lm relate formal analogous concern defer section skew skew refer name incorrect constitute section lm skew view extension latent replace analogue stand multivariate skew near end place component add level highly restrict adopt terminology broad generality see early wide generality furthermore skew lm skewness sect study extension skew et limited form skewness expression skewness true classical skew marginally globally case coincide univariate skew expression distribution employ purpose potential additional variant tail prove considerable numerical diverse area employ context focus application lm place statement therein motivation present statement theoretical discuss lm contain claim superiority fm extra fm restrict besides general superiority formulation believe purpose systematic due selective reporting arise section note skew formulation lin lin lin amount skewed analysis herein skew herein conduct analysis correct rand ari perfect agreement zero classical software within package start lie skew classical skew component figure mixture performance give mixture ari h five datum package r base various focus size rw gender skew fit classical skew ari give poor produce well classification ari misclassifie present accordingly comprehensive wherein skew distribution triplet note consistent ari
overhead global gp modelling present derive perform guarantee achieve induce parameter communication embedding experimentally study suggest inference resource time power implementation demonstrate process show gp improve amount million test implement map multi architecture package derivation additional robustness failure drop software package extensively explanation recent propose variational train use mini batch successfully learn undesirable variational marginal variational induce target analytically derive additionally need analytic gradient induce advance strong correlation parameter heuristic work difficulty review variable aim function location input precision convenience analytically marginal prohibitive approximation aim term induce complexity input correspond infer posterior induce equation relationship induce link conditional overall make tractable optimisation modify fitting alternative greatly reduce fitting computational approximation distribute supplementary material latent aim mapping prior technique see derive finding next distribute model derivation inducing distribute allow easily model explanation derivation regression latent variable modelling identify marginal introduce induce expression multiply induce jensen low brevity integrate use obtain use derivation identical derivation break induce represent individual integrate triangular distribution calculus variation analytically plug q input obtain mi I sum communication supplementary using hyper induce calculate set global induce additionally px calculation supplementary material node central back global local posterior map central follow sum point material contain partial optimisation local range criterion well increase scaling compare scale distribute system square exponential ard automatically add conjugate inference give improvement running time transform linearly core space total run spend iteration show improvement close achieve core little sign interesting overhead inversion carry global core show assess increase resource available computation run per iteration effectively extra total take compare inference see sequential significantly computational resource inference spend step core research parametric sometimes state requirement practical equal load reduce computation worker maximum execution node core difference node suggest load describe series experiment demonstrate gaussian big task gps perform often regression point use present modelling perform imputation test digit far aware mnist na na k delay record distance datum test experiment setup different stationary nature performance dataset c c baseline depth estimator good different resource induce optimisation root rmse point core minute baseline take several gps big advantage uncertainty principle increase train large converge optimum bfgs mode optimisation likelihood mnist example large train use test take likelihood py e induce choose marginal converge get
notable feature build recorded ns performance sp stand quantization pyramid respectively projection convolution sp total highlight inference time pixel image matlab dual gb ram framework soft convolution pooling sp bottom manner adaptive map classifier table multiple additionally feed forward kernel table detailed comparison method one one sift descriptor extract sift descriptor generation adopt locality sort descriptor amenable parallelization gpu expect gender svm lc evaluate gender age net act baseline ar database code online include lc lc training testing code setup first project dimension matrix fed face recognition gender learn mid partition spatial neuron two comparison neuron display n compare furthermore neuron ns layer capture characteristic intuitively demonstrate propose ns work classification net include use appearance codebook spatial list demonstrate well year old deal perform slow resemble mid model reveal age face mae mae image nine learn original image upper panel image map sift acc infer lee zhang al l acc wang al lc popular database training testing resolution preserve aspect recent unsupervised learn sift feature psd method mid codebook code dimension model learn neuron category pyramid partition public categorization marginal reason foreground appearance sophisticated neuron alternatively sgd fix basis normalize pose negative newly adaptively class response n appropriate rewrite q sgd derivative calculate couple newly simple w hadamard initialization convergence decoder pre train merely pool activation specifically calculate pre initialization lack allocation category pre also initialize variable matrix purpose symmetry breaking inferior architecture ns layer random initialization work well descent initialize eq appendix due paper mid level soft convolution significant preserve orientation map illumination illustrate trivial filter illustrate two highlight full figure illumination soft convolution illumination require operation convolution soft thresholding put several feature produce convolutional map tensor normalize third scaling comparative third map threshold map produce normalization similar benefit local descriptor illumination invariance face individual challenge vary illumination illumination select image person original normalize soft map max maps illumination illumination evaluate framework database face face performance ever background soft convolution pool weak response remove soft operation keep three database randomly category face consist descriptor patch sift consist sift descriptor center pixel image generate display gray sift map sift mainly focus mid lemma mid feature greatly enhance automatically manner paper efficient mid level operation pooling quantization simple method need much time boost neuron ns layer mid ns neuron inference top extensive database achieve performance gender categorization recently code image sift convolutional network improve classification level far mid hierarchical neuron layer feature high fed panel descriptor window salient mid layer despite mainly nonlinearity pyramid apply sparse pooling nonlinearity focus nonlinearity sparse psd introduce absolute however point et al moreover complex simple carefully factor size density therefore instead design complicate mid efficient mid feature consist operation convolution quantization shown sift produce desirable might descriptor consider mid boost accord build neuron ns demonstrate neuron signal specific bottom inference ns improve notably summary mid feature give approach generate argue might ns mid classification ns support inference appeal achieve art describe level section mid mean build without structure via code sift despite promise accuracy extract low descriptor amount knowledge system autoencoder empirical confirm rule nonlinearity mid performance well include densely descriptor suitable difference simple study mid learn call mid sift adaptive mid generate filter decomposition map see rd tensor panel worth convolution adaptively feature several step convolution normalization advantage first convolutional behavior descriptor preserve information illumination filter thresholding sift pooling robustness map pair within macro size capture neighborhood pool nonlinearity map code feed forward descriptor result demonstrate splitting map map densely extract sift descriptor patch descriptor dictionary pooling code predefine partition different task detail pool code representation usually dimensionality reduction normalize mid involve oppose project discrimination reduce produce guarantee person illumination image display three filter map thresholded descriptor sift within unsupervised despite share similarity hard example eight capture subtle descriptor resembles derive forward pathway max build layer architecture produce complicate deep architecture comparison say use descriptor statistical incorporate along max also map convolutional map interpret consider contrary propose soft convolution negative interpretable mid manner build mid feature boost neuron principle neural signal category stay therefore call layer fed denote class build layer ns mathematically specific mid ns generate activation turn logistic activation encoder inference process activation pattern present produce structured activation top decoder activation successful field decoder weight decoder back neuron reflect appropriate level encoder bound generality consider decoder mid level length reflect neuron hence specific signal stay property structure impose eliminate besides activation denote minimize mean take column activation time activation activation h h h penalty several decay neuron automatically break separate part behavior rigorously three intra
circle specify etc loop generate diag library diagram option distance option useful want diagram persistence diagram bottleneck wasserstein package r bottleneck wasserstein two persistence diagram circle diag diag diagram bottleneck nd wasserstein diagrams code specify diagram loop diag diag summarize information persistence diagram briefly landscape persistence landscape sequence piecewise encode diagram landscape create obtain graph persistence landscape persistence landscape function max max middle diagram treat diagram persistent low persistence conversely persistent feature see value half life blue landscape right blue landscape function function persistence landscape specify interested st landscape length kk return band scenario landscape build compute prohibitive instead draw landscape subsample yield identically distribute approximation construct band valid illustrate sample circle xx subsample diagram subsample seq store store subsample diagram feature diagram kk construct landscape alpha plot code main landscape col col tb problem topological kde suggest describe work width band scale describe measure kde kind tradeoff maximize illustrate follow circle plus clutter specify limit evaluate xx xx xx limit among number bootstrap band progress bar alpha alpha display value criterion call kde f alpha parallel persistence maximize persistence kde example let density l dimensional subset maximal simultaneously tree tree tree branch tree particularly whose organization difficult cloud three well separate x xx x nearest knn alternatively use kde density algorithm connect density contain tree object middle lambda lambda knn access homology library package implementation persistent homology become comprehensive topological keep library new topological information analysis like thank ed discussion author develop package available algorithm package present package provide tool topological implementation given provide topological distance salient topological quantify persistent homology provide interface library include persistent homology persistent homology grid result diagram recently package include implementation allow visualize dendrogram persistent homology cluster recent advance actually take cloud topological lose collection topological persistent homology homology multiple simultaneously quantify nest persistent homology topology change diagram death devote presentation interface algorithm library topological information underlie estimator devote computation persistence diagram persistent homology set function grid diagram build cloud challenge persistent homology represent topological topological persistent homology exact persistence infeasible often confidence diagram allow topological compute draw x q prove validity kernel persistent homology implement package figure band kde grid grid show surface surface provide persistent homology reader basic concept persistent homology grid construct persistent homology arbitrary choose compute persistence library code compute persistent homology cloud object evaluate smoothing bar
exist partition core core regular degree particular remark hardness long dependence correlation hardness suffice proof literature express subgraph entry describe prove define constant leave universal fix black box give backward mapping model possible produce approximation trick probability mapping make call least take proposition get proposition marginal desire marginal know backward amount gradient optimization project reason project onto project marginal standard project np nevertheless projection address difficulty inside lemma consider operating amount lemma omit condition approximate project converge purpose completeness minimizer oracle error gx gx gx tx cs l gx translate approximate approximate equivalence convexity strong implication appendix x p difficulty black approximate mapping obvious closely polytope even onto np project onto goal optimization thresholding project project translate project simple per suppose consider section devoted procedure sx update tx sx p x I v prove subsection property keep iterate inside notation independent collection shorthand write lemma specifie want imply rearrange read fs fs fs fs simplify close polytope corresponding require show note b new check I contradict spirit next allow sketch give section supplement sx follow hyperplane hyperplane definition imply coordinate h h inactive require desire gradient h h argument require care prevent despite negative already complete paper address hardness backward within hardness even weak exist constant acknowledgment thank helpful discussion manuscript office award nf material amount node marginal subgraph claim obtain remove label induction base trivial formula summation inductive order approximation give approximation maximum independent choice suffice argue follow set map neighbor set removal lemma convex modification projection onto convex contraction prop inequality definition gx precede define divide convexity apply jensen give right show recall change triangle direct calculation value p p see chapter bind derive measure I I subsection follow restrict fs fs fs fs finally f sake argument contradict goal sx initial hyperplane except negativity write vector fact together zero observe h similarly critical active first inactive constraint require gradient constraint close x qx inactive consider fact critical bad rough coordinate increase sufficiently iterate argument might prevent projection start increase coordinate consist rearrange give follow prove ready prove coordinate know pp pp affect increment hence coordinate limited amount satisfy additionally eq together count move away hyperplane complete proposition corollary theorem david laboratory department school management mit edu specify undirected graphical uniquely mean parameter principle feasibility parameter learn canonical unless rp polynomial reduction approximate core know hard parameter reduction entail show polytope optimization procedure ellipsoid powerful high core intelligence variety finance communication biology undirected model rich applicability canonical consist wise marginal graphical computation marginal learn parameter polytope backward capture study subject interested task basic compute efficiently well approximate computational core simple pairwise define sharp compute core model tractable exhibit property exist despite hardness previously obtain undirected et show hardness show hardness require know section hardness mapping thus backward vector v physics literature activity eq serve normalize note play major role define eq core polytope equal hull vector polytope structure need large depend polytope condition notion entry next notion bound rp np difficult
slowly rescale treat saddle argue saddle instead become attractive newton minima entire justification quasi method e rapidly bottom less optimization previous report order rapidly saddle curvature fundamentally different method also outperform involve suggest minima provide come nature domain review particular derive point distribute negative critical attain plane critical concentrate monotonically imply error much exponentially saddle necessarily chance small r unless critical stand close one positive show eigenvalue shift global shift shift negative well eigenvalue typical slow yield perspective geometry surface saddle event pick become pick positive negative saddle qualitatively similar derive function error surface generic chosen minima exponentially saddle point negative result applicable analyze surface perceptron mlp linear layer surface show saddle indeed analyze saddle scale space mlp dynamic deep follow transition performance aspect soft explore network randomly choose teacher importantly unit within cause permutation among associated symmetric teacher interestingly curvature saddle curvature hessian measure relevant property surface develop generalize mlp matrix critical small mlp train sample version mnist newton fig setup detail test qualitatively critical point concentrate monotonically increase plane saddle seem accord leave critical give understand behave near us saddle point analyze appendix supplementary material detail new parameter step point sgd eigenvalue step along restriction drawback direction eigenvalue absolute saddle structure dot stand free rescale gradient eigenvalue approach newton descent move towards move along direction eigenvalue saddle point newton order curvature add rescale gradient modify eigenvalue every eigen direction increase drawback potentially eigen incur approach ignore regardless strategy bfgs saddle ignore curvature follow natural relie curvature behaviour take similar newton fisher matrix argue descent saddle effectively resolve arise however descent suffer curvature issue fisher gauss direction distant converge point matrix saddle exhibit negative mean landscape mean rescale descent use rescale vanish critical much near straightforward trust region taylor instead rely taylor trust constraint taylor region describe arise special saddle family near saddle sec simple eigen rescale newton preserve turn saddle hessian suggest example aware justification dimensionality saddle alg empirically theory suggest saddle validate saddle function network near order minimize approximation scale mnist direction cifar minibatch descent newton saddle free select coefficient small update saddle likely sgd near saddle point fig clearly small size saddle outperform large margin close behavior algorithm fig error see saddle get near saddle sgd epoch observe saddle newton rapidly fig shift toward shift suggest successfully error large recurrent saddle seven layer neural deep autoencoder benchmark descent newton rapidly sgd even confirm shift saddle newton follow free get art hessian free method well feedforward want saddle free method avoid saddle recurrent hide sgd saddle free newton method see trend feedforward sgd quickly suggest around soon drop find newton negative eigenvalue saddle method see sgd physics matrix theory neural saddle dimensional intuition minima exponentially dimension provide first neural surface test application confirm qualitative index critical positively generalize region saddle curvature fundamentally different define trust order saddle free method theoretically recurrent network saddle rapid descent newton trust show sensible saddle improve neural first beyond hessian second critical training neural far generally deep property surface guide design non impact engineering acknowledgment thank cifar research compute computational google fellowship thank vanish characterize symmetric number scenario eigenvalue minimum eigenvalue local eigenvalue zero critical saddle restrict span saddle maximum example saddle restrict direction move correspond pick sign point presence structure maximum saddle shape look along equal almost eigenvalue large structure structure circle shape max direction shape taylor neighbourhood critical reliable vanish eigenvalue span eq plot discover nearby saddle newton saddle seed select run among saddle free amplitude pick weight cube layer activation network different randomly train newton allow different critical point along trajectory absolute corresponding subspace slightly note find useful direction v beneficial subspace hessian w feedforward network strategy rate minibatch momentum sample maximize among protocol classical recurrent weight orthogonal rnn hyperparameter gradient
briefly introduce formally loss unified introduce misclassification denote label set penalize class hard soft formulate case hard little abuse predict rule margin combine obtain commonly hard classification margin approximate correctness non hard sign may margin direct non infeasible surrogate surrogate margin loss loss prevent fit commonly exist loss soft providing spectrum soft classification naturally formulate base describe option formulate form notation option pre specify express mention fit weighted specify class simplicity predict account commonly constrain interval generality weight margin surrogate consistency statistical surrogate classifier fundamental expectation theoretical g respectively margin function classification bayes weight calibrate theoretical loss monotone pc rejection option rejection option hard platform gap formally introduce underlying population shown p separate c soft show span entire soft formulate target unify margin insight order interval obtain splitting belong framework three dense correspond discuss later illustrate spectrum framework generalize collection classification bayes boundary simultaneously boundary formulate weight throughout loss class show theoretical usual multiplicative precisely figure function figure horizontal axis correspond giving except case smooth panel boundarie observation boundary incorporate step soft become loss indeed correspond learn task precisely theoretical loss hard option rule classification theoretical loss limit theoretical bayes correspond soft find describe section optimization use surrogate generalizing rejection option classification prediction common negative formulation theoretical differ along vertical f margin formulation surrogate first surrogate consistent class piecewise surrogate include propose empirical surrogate include piecewise surrogate consist segment boundary become dense tend denote convex measurable provide necessary sufficient surrogate consistent naturally surrogate loss condition exist possible intuition justify soft loss soft loss meet loss corollary next surrogate svm satisfy surrogate boundary contrast consistency surrogate surrogate build option circle theoretical loss panel appropriately first loss panel boundary consist non respectively consistent control segment surrogate loss segment surrogate loss observation denote intercept segment express intercept loss linear consistent piecewise surrogate piecewise specify b denote location along loss piecewise intercept slope decrease hinge eq segment order degenerate align hinge guarantee importantly next obtain satisfy logistic dot use dash piecewise tangent construct rejection state piecewise satisfy let loss construct line logistic piecewise surrogate satisfying figure dot surrogate vertical denote equal tangent logistic correspond differentiable highlight appear negligible notably piecewise suggest may additionally spectrum explore issue simulation surrogate loss show risk respect risk risk rate minimizer bayes optimal derive rejection far suggest separation function bound error piecewise class surrogate piecewise non differentiable svm quadratic qp also formulate qp complexity intensive moderately propose project similar rewrite surrogate define space rkh norm formulate intercept review kernel boundary write denote estimate iteration iteration let mi mb b b b iy iteration project illustrate illustrate achieve consistent piecewise loss piecewise constant derive boundary grid tune logistic loss time piecewise setting panel along black rotation variation respect space three piecewise bayes optimal setting linear loss boundary illustrate minimize tuning along decrease intuitive converge improvement classifier theoretical confirm classifier piecewise furthermore boundary black comparison piecewise logistic vary panel median loss standard replication minimal spaced sample rotation setting heavy consider asymmetric optimal boundary linear classifiers simulation piecewise great converge logistic nc institute drug private private md california effort many co date pc nc subject color estimate ad vertical consist feature nc ad process logistic validation determine principal pc subject subject interestingly appear ad subject include subject mild cognitive stable subject depend whether ad consider nc ad distribution ad correspond boundary vertical group appear peak appropriately divide subject disease discrete class several hard rejection option learn conditional framework hard soft classification provide perspective hard family unified margin surrogate previous behavior problem class problem part national health grant share u institute imaging association discovery foundation sciences company company company research company health provide clinical site private contribution foundation national health www organization institute education california imaging california research grant material option rejection option loss loss soft deriving limit q note bayes classification generally similarly recover let option rewrite equivalence option loss rejection option although traditionally limit eq minimize choose appropriately boundary boundary separately condition consistency necessary sufficient condition boundary define wish satisfied equivalently pair r r exclude boundary wish satisfied non class loss note equality satisfied derivation three convexity satisfied rule rule rule suppose express pg pg cf suffice pf pf g convexity eq rule margin theorem additionally inequality therefore combine choose letting fraction right surrogate boundary denote k k desire bind define f I exist bernstein tail h f I result since pf pf pf b pf rewrite eq p b similarly measurable uniformly class without f q net bernstein note necessary plus disease abstract learning covariate class literature distinction soft model extensively propose spectrum span reveal novel classifier convex surrogate descent disease keyword excess problem supervise similar regression describe generalization denote covariate probably covariate task correspond briefly target prediction hard soft classifier example include rule predict commonly hard probability classifier traditionally soft question classifier differ recently unify relationship classifier connect several discrimination far extend category base
ij topic cluster find mean partition compute th l high approximation dominant topic succeed probability find complexity notably document dominant document need coordinate note get long lemma essence proving identify partition call center correctly almost document prove theory thresholded document document able dominant topic assumption fraction topic topic help pure find complicated induce conditioning thresholded available author empirical synthetic real life coherence know advance test multiple give empirically well initialization project svd mean report dataset step standard select vocabulary term frequency remove less word paper vocabulary consist york dataset document dataset ng document vocabulary length sampling state burn iteration posterior show document corpus minimum topic fraction document pure also indicate justified fraction topic assign document high cluster table topic ij per column intuition local recall analyze synthetic corpus plot separately fix show plot monotonically hand unimodal c dominant pure topic mean topic topic ng semi corpus ensure corpus retain gibbs run final topic generate synthetic draw topic gibbs topic summary evaluated rigorously evaluation l reconstruction algorithm good dataset multiple real justify conclusion real corpora document present svd able thresholded svd svd massive corpora apart recovery thresholded broadly similar minute game team team sale book school book game game drug find patient drug medical music band company web www site computer software mail look room show look home house room look water house trade death car car com com author player team book author play character goal play team award million school teacher child plan plan million stock market percent team home room school shoot home company law company software company window million million company stock percent company million business com company com site quick product percent home com shot team team team political com www room million movie music movie character percent stock market percent price share wind weather water weather air article ball country percent right study student plane pilot company company medium company business customer million worker company pay shot shot game team team black primary blue blue lemma proposition exercise em height width microsoft topic latent word inference problem np strong give provable algorithm widely lda give provable vector aim develop intuitive svd provably solve lda co occur specific occur strictly topic individually major realistic corpora value svd step recover dominant sample empirical evidence corpus propose assume distribution word convex pick multinomial combination topic recover provably gibbs provably topic topic collection document sometimes topic pure word topic dirichlet development give provable corpus assume exist word topic single topic learn try occurrence keep group call occur topic frequency weak separability weight significantly motivate assume corpus paper document collection document dominant topic high topic every nearly purely contribution provably topic grow dictionary unlike grow semi synthetic corpus several topic document let k l give topic column pick document pick topic weighted topic I trial trial pick wise provable excellent provable start successful document recover collection document base primary indeed method numerical like satisfied anchor topic word polynomial note linearly dictionary every seem realistic ask like run assumption informally dominant document corpus reasonable assumption base propose inspire introduce individually occur topic much matrix subsequent mean dominant corpus negative dominant document dominant sl sense identifiability vector model unique group likely co occur assumption try word pick document could think pick pick multinomial dominant whereas technical weak sense justification generally word plot plot expect curve frequency reach unimodal empirically size close asymptotic think infinity large thought intuitively document intuitively need mainly need refer different svd
high fit spam response comparable spam lasso show estimate interval set spam estimate np loss convex j discuss loss penalty generalize descent solve choose eq special case omit brevity case logistic amount q form generalize penalty solution logistic piecewise figure set value set display expectation average replicate combination two model interpretable fit adaptive knot limit knot fit jump knot knot basis knot constant flexibility knot accommodate trend large fit suited trend previously future package make develop interactive demonstrating note fact u degree differentiable thus stein ts around block solve yield freedom add omit take p objective pz ji pz union n v u ip plug p n plugging inspection form begin lagrangian z note partial respect b v writing lagrangian obtain ball solve ex minus plus ex ex ex section lemma department variable observation interpretable desirable propose piecewise adaptively knot optimum provide show degree keyword predict response feature observation task offer interpretability piecewise pre knot flexibility instance non interpretability flexibility select feature include knot th element element reference contain review implement examine analysis set close section generalize restrict attention function might spline additive dimension propose extension spam induce estimate spam solve j td order datum fit capture complex relationship j td cubic predictor contain inner cubic spline also recently partially linearly basis choose adaptive interpretability many cubic spline clear change knot knot proposal knot adaptively begin seek piecewise constant knot parameter derivative adaptively choose jx permutation exclude solution value tend therefore optimization provide encouraging piecewise induce purpose much descent cycle repeatedly hold fix partial operation iteration leveraging solver solution directly optimum made initialize compute r p repeat fuse filtering show th trend filtering equivalent regression spline interpret variable locally adaptive spline illustrate portion spam however impractical v u u center triangular remove difference order equivalent solve lasso allow fit fitting modified ridge ensure convexity prediction zero compare predict one coefficient finite assume p provide smoothing package spam cubic knot space quantile ij four scenario display spam area area scenario scenario scenario scenario validation training freedom spline spam range degree freedom degree multiply freedom covariate intercept freedom estimate display mse versus three achieve low test mse spam outperform need preferred noise exception scenario scenario scenario scenario indicate replicate additionally summarize mse optimal j freedom except dimensional setting scenario achieve comparable explain performance dimensional cccc proportion mse freedom spam spam spam spam spam spam spam spam spam strength local constant qualitatively examine truly replicate fit constant spam spam impose encourage entire control flexibility variable fit fit varied greatly domain fit function purpose spam correspond simulated consider estimate country national country approximately country publicly unite
value namely reveal non step optimum decide round rather threshold truly semantic fold evaluate rank iteration top illustrate curve high compute mean deviation list optimal rank dataset gradually continue reduce explanation much overfitte likely htb reproduce previous program outperform sparsity degree table range b word truly effective semantic keep constant extent explain model extraction perspective criterion experiment feature exploit classification overcome significant improvement open relation extraction reconstruct item plan improve capable acknowledgment basic grant cb national science foundation china grant laboratory division innovation development laboratory technology university china china advance china com essence extraction label tackle sparsity noise problem completion factorize minimize classification complete label matrix underlie leveraging solve widely baseline knowledge text traditional hand label corpus achieve precision recall corpora satisfy increase demand large text distant supervision improve effectiveness paradigm intuition automatically text wikipedia york corpora heuristic account basic entity involve come occur appear text current diverse relation combine label relation paradigm corpus automatically come sparse kind nlp stanford extract variety name tag speech tag path unfortunately leading noisy instance relation explicitly noisy case extraction incomplete mention incomplete instance therefore distant incomplete corpora essence incomplete multi perspective use knowledge technique relation extraction supervision specifically sparse pair column relation relation classification transform complete unknown item item incomplete factorization de label contribute supervision extraction completion recover simultaneously logistic function influence incomplete suitable binary modify find global optimum widely discuss compare degree distant supervision firstly bioinformatic discover entity article maintain expert date web text build corpus without label adopt scale crowdsource knowledge online relation name wikipedia entities mention relation variety regression inspire relaxed replace sentence al entity multi approach label relation extraction jointly text basis address entity tag relevant pca collaborative distant supervision e perfectly promising apply area computer vision recommender controlling model classification robustness relation basic vector test matrix complete entry observable entry rank observable feature observable impractical entry thus label z optimization weight nuclear employ w w another call relation generate sigmoid entry derive function complete sigmoid calculate entity n relation np hard es suggest relaxation solving modify optima formulae modify solving contain infer follow gradually minima minimize singular value svd cut assign matrix parameter pt accelerate
input residual split suitable regression procedure apply lemma empirical taking arrive desire ready score either satisfy additive test non characteristic regression procedure additive population bivariate c bivariate satisfy test noise bayesian model input closely closely relate penalize likelihood minimize respect yield formula identity formula derive obvious penalize identical negative rewrite around value relationship score random standard sort white noise rule integral follow noise characteristic gp input add scenario without scenario scenario scenario gaussian simulated setting cause four scenario benchmark form competition benchmark consist cause pair consist pair statistically variable know cause task sample publicly available set select agreement truth pair consist weather obvious cause though due hide selection cause process relationship system generate whether intervention distribution perform practice original generating long available unfortunately available clearly cause effect set criterion cause relationship set cause follow subsection describe pair detail motivate decision scatter plot horizontal cause axis include overview benchmark dataset ground hour pair age diameter age hour consumption consumption weight consumption age stock age pair duration temperature pair age age age heart compressive compressive pair compressive pair water compressive compressive coarse compressive aggregate compressive strength compressive strength consumption consumption consumption pair consumption body pair age pressure concentration day temperature pressure pressure pressure relative day temperature pair concentration temperature concentration consumption acceleration dim dim life capital pair capital capital life capital life capital life capital life life capital pair water bank stock stock return stock return stock send http pair dim disease global temperature co energy life per pair growth temperature net pair temperature population protein pair temperature age merge weather weather datum six value year duration notation causal temperature temperature pair hour temperature ccc pair pair hour pair temperature elementary place tend level roughly think intervention temperature high hand perhaps location happen air let statistical south also place high lie south temperature empirically direct temperature dominate since occur air force rise air water due indirect via relation less influence temperature main direction wind relevant intervention allow pair temperature detect relation intervention west even unlikely east therefore duration positively high weather sometimes day sensor increase duration whereas causal influence dependence early temperature move change necessarily north south movement obvious somewhat weather west east expect relationship east west change adjust cloud uci repository concern nine height whole directly age year six cause relationship diameter pair age weight pair pair age diameter age height pair whole weight pair age weight height obvious intervention since possibility change observe consider intervention provide condition clearly whereas change change difficulty define agreement cause height age count change length natural good proxy age census census repository study age hour stock pair pair age per hour pair stock age hour instance per age already argue age difficult problematic intervention background life job experience year later old work job however sometimes long job experience intervention change hour intervention easy imagine would certainly age stock instance stock age vs hour intervention theoretically stock hand age influence stock thereby stock indirect age stock less hour uci city consumption car several attribute like weight acceleration come american association thereby cccc pair consumption weight consumption acceleration acceleration consumption air engine draw cycle engine consumption change weight change air consumption measure power engine consumption add engine car change consumption change consumption change consumption car powerful consumption causal consumption weight vice pair acceleration air design able certain maximum give car indeed engine big selection acceleration acceleration two combination variable multivariate three consider cause comprised consumption concentration chemical compound child year package language pair plot pair pair age concentration pair age concentration cause old set package contain subsequent old national usa consist collect pair scatter current duration interval interval duration repository patient use causal patient length h ccc scatter age pressure certain partly affect temperature storage daily air day air day time two drive scale weather causal day pair pressure pressure pressure surface mostly weather large scale weather pressure gradient hence pressure stem day pair pressure pair relative air day air movement place occur stay affect place reason scatter pair reason temperature surface pressure level pressure influence variable dataset website contain various national road connection around south scatter variable day indicate working day day cause introduce political amount traffic large number day certainly book pair temperature temperature strong impact adjust air little much heat deal relationship daily air produce presence surface give go detail complex chemical mention chemical instance apart air may influence temperature traffic weather occurrence path phenomenon lower high three temperature scatter plot temperature temperature concentration pair temperature temperature concentration daily concentration daily mean daily temperature daily place day concentration complex wind air global affect height formation influence contrast drive g place environmental wind temperature consist daily time pair temperature wind direction speed air wind air vertical source mix air different wind database united division cccc pair scatter pair life capital life pair life per pair capital life consist life year birth country capital various china correspond period respectively pair capital life pair life reasoning life water access percentage access water change people access clean water particularly disease feedback country development aid towards increase access clean water contain emission together sample mix world energy co amount across source change co use country term decrease change per life collect life birth country general rich thus care believe life human impact country vice versa collect live reasoning influence system determine minor disease per reverse causal yahoo database stock stock bank prop subsequently follow adjusted price yahoo finance base day use interpolation stock price calculate ccc pair scatter bank return stock stock prop stock bank return bank stock whereas pair stock stock reasoning stock prop prop major stock file http server institute internal website request send interval interval minute pair scatter internet internet website raise transfer create additional website access transfer website fact make inside outside datum room outside every minute day locate explain large fluctuation collect pair scatter outside inside outside causal expect outside temperature temperature heat capacity inside house reasoning pair take face face interpolation component face image answer scatter plot pair certainly intervention set repository order suffer disease choose decision temperature patient occurrence yes yes yes group six dimensional pair disease think disease create uci say create expert diagnosis disease represent patient consist temperature unit east conjunction centre office express mean anomaly description website count ten average ten small pair temperature phenomena surface cause entirely temperature anomaly correlation less proxy activity believe temperature organization un cover area period one consumption population describe consumption day c pair growth consumption growth consumption regard cause mainly people availability advance increase market economic short scale probably might consumption population could imagine fed reproduce response datum filter light response obtain consist three measure total net define release light intensity nm nm visible light time measure measure several forest available ccc pair scatter pair net direct temperature collect aggregated day site exchange set quality mean credible nan pair scatter temperature temperature exchange approximate release largely mostly consider truth pair site growth population nine file eight logarithm people eight scatter plot pair seem reasonable total versa people people believe rather employ might status child trial protein allocate mixed measurement drop stop value datum week discard drop week organize set see relationship protein protein produce c pair scatter pair cause vice website demand student interest size room month remove scatter plot pair daily daily originally concern historical daily bc temperature h pair scatter pair temperature tell cause effect whether dataset change consecutive age average take pair scatter relative age cause european union institute university statistics max institute institute kn institute systems circle fill black minimum pt circle black size black black thick discovery relationship purely observational science elementary discovery cause alternatively cause joint consider impossible causal different cause pair various evaluate performance bivariate causal discovery benchmark cause purely observational noise causal causal relationship rather association effect gold identify relationship control expensive impossible identify purely observational constitute influence cause influence common cause condition acquisition discovery attempt distinguish require condition call causal study purely copy task impossible could causal approach distinguish cause attract recently cause supervised shift variety able argue marginal distribution cause low factorization intuitively appeal precisely measure contribution extensive family bivariate discovery original benchmark collect year discovery definition causal review idea discovery review cause effect free appendix benchmark various joint observational distribution intervention consideration aspect particular perfect intervention explicitly force leave system notation intervention force lead consider label graph causal cause illustrate marginal infer inequality feedback relationship variable e present intervention costly still relationship I common effect implicitly feedback relationship infer decide upon causal say direction xshift node var z var z cm xshift var bend bend leave yshift node var xshift yshift causal relationship two observe variable latent cause causes explain condition hide variable explain dependence valid although except article review discovery exploit bivariate detail extensive literature assume effect field although linearity mathematically convenient generally np model possibly function cause latent effect cause cause intuitively reasonable model relationship nonlinear cause lebesgue assumption common upon feedback latent cause unobserved cumulative variable easy distribute construction direction another interpret q gaussian prevent draw introduce show noise allow distinguish recently variance lead high variable relationship identify causal influence precisely consider class consist function say interested introduce satisfy identifiable contour joint scatter sample distribution contour mean different model additive typically identifiable non gaussian distribution identifiable fall imply something might expect intuitively additive multivariate identifiability provide identifiability transformation call post know either additive I cause regard rigorous rather exactly quantify conclusion cause possibility happen satisfy identifiable would special unlikely discuss way helpful bivariate bivariate additive finite induce fx induce sample model residual estimate datum testing residual consider scenario split independent typically split big two identical datum different couple suggest come additive noise estimate residual one noise test parametric residual dependence care threshold ensure tight far choice lead way compare need algorithm scheme identify simply regression dependence decide additive I method measure estimate residual calculate measure principle subsection hilbert schmidt base alternatively statistic consistent differential entropy residual score originally finally briefly message score idea minimize possibility originally hilbert schmidt residual input definition propose score eq indicate independence possibility value option certain technical infer joint either splitting datum scenario kernel definition consistent additive dependency cause show weak vanishe method explicitly residual input entropy score use reproduce joint shannon proof application differential entropy identifiable additive exist e one order causal shannon advantage entropy marginal entropy mutual certainly disadvantage rely effect differential term identity identifiable noise show suitable assumption standard generative gaussian role calculate marginal evidence infer causal case dependence typically instead reason implicitly distinguish splitting use decide noise bayesian mml score function fit measure combination mml construct score trade infinity complexity mml direction identifiable mml refer mml like conditional mml identical mml base length mixture gaussian px optimization problem nonzero score difference former gaussian combine single bayesian score generally measure residual minimize respect respect challenge multiple local guarantee find addition strongly automatic prove consistency challenging residual section discovery cause effect cause base effect build mechanism formalize case toy strong assumption causal sufficiently provide introduce deterministic relation via translate contain intuitive case equality q interpret variable side distribution equality positively tends region contain illustration intuition behind cause correlated density high employ expression side zero section variable introduce rather define scope perspective base also priori range may choice let cause deterministic density exist interpret expression alone density special reference instead substitution side kullback therein amount infer cause density quite gaussians infer rescale specification essential implementation choice shift map preprocesse slope assume increase give entropy denote equivalence slope whether deterministic I normalization scale deviation discuss sense twice fine preliminary conceptually solution order remove occurrence original ignore repetition occur describe implementation criterion present section source code reproduce make available open platform library gp parallelization process gp implementation use exponential constant reduce spaced computation scale introduce error therefore try call order behave asymptotically remove way deal discretization comparison previous implementation entropy estimator toolbox entropy sp shannon sp shannon sp sp sp shannon shannon shannon shannon also make toolbox release near expansion detail toolbox gaussian take kernel asymptotically see also permutation gamma mean nan hypothesis estimate description benchmark set cause consist extension eight form competition publicly appendix cause justification ground scatter plot scatter plot cause effect collect benchmark ground decide ground straightforward method process ground simulate way realistic plot simulate look world idea structural want similarly standard normal causal process measurement default level expect finally gaussian approximately nonlinear cause expect scatter scatter scatter cause scatter scatter effect scenario standardize affine transformation empirical perturbation perturbation perturbation variable discretization discretization repeatedly cause merge add small add ideally robust perturbation estimate direction affect real effect weight pair come weight correlate age whole age curve whether accuracy increase evaluate force visually interpret significance experiment carry plot indicate weighted confidence interval confidence indicate evaluate bandwidth bandwidth datum splitting entropy reporting show simulate benchmark variant estimator different perturbation benchmark variant show obtain accuracy way additive measurement effect result turn perturbation additive misspecification originally size represent decision value direction use value suffer identical perturbation discretization slightly show performance base noise depend detail combine generally little differently consistent standard accuracy set right figure entropy six variant perform result nonparametric simulated exception varie discretization effect entropy treat occur multiple example occur lead chance level majority quite nonparametric well datum seem perturbation mml score score score chance probably typically violate accuracy satisfy scenario evident perform scenario employ measure setting practice perform scenario probably mml simple measure benchmark perturbation parametric well accuracy simulate behaviour understand match actual distribution datum simulation setting report variant show variant accuracy perturbation base accuracy accuracy perturbation bottom gaussian measure base accuracy around chance gaussian base low chance high chance require reference measure scenario
already process increment operational environment arrive fix half entire training feature human regression subset performance pearson coefficient comparable regardless train smoothed average model sample random replacement random well baseline change increase informative margin baseline small show percent fisher transform level test unimodal relatively quite interpretability percent calculate pearson mean anomalous achieve typically select select see htbp average perform prominent enough illustrate notable deviation great bad figure robust human well human validation production reach noticed entire perform optimal cause entire poorly also poor misspecification misspecification semantic human primarily avoid misspecification evaluate uninformative report generalize task promise score less answer answer employ tree forest model know use consequently row matrix row relatively active sampling agree subsequent supervise thus short human supervise forest train human maximized minimize perspective primarily human quantity enable safe solve large allow computer human effectively human score automate writing effective quickly adopt large scale context human requirement costly evaluate ensure consistently informative training maximize thereby reduce cost discussion integrate automated writing language statistical create automate scoring e english even wider automated evaluation recently length answer help platform massive contain automate writing train trait batch language input learn vector previously process require least requirement system human cause system allow yield perform adopt technology per paper barrier context example enable system choose human effectively integrate method literature choose sample old space optimal clinical study optimal design sample vector variance begin label unlabele paper active design choose prior learn good assumption match regression ask really active stick mostly terminology literature score automate assessment foundation set summarize ask write computer letter set read text evidence grow must tend experience important element target score range target predict set performance quite suggest student focus algorithm mechanic lexical style range b source automate student narrow determine inherent variable design number allow since feature eight ordinary square solution would suboptimal regularize ordinary square ordinary augment usually ridge penalty irrelevant towards since linear integer map back onto value score determine adjacent reasonable threshold use candidate wish give predictive score formally lack row predictive regression result consequence choice illustrate value great perform concept feature algorithm choice constitute maximally distant uniformly distant implementation implementation implementation exchange algorithm purpose design criterion optimality maximize set difference tend somewhat exposition limit optimality scoring randomly replace row row optimize algorithm optimum one maximally distant centroid see choose vector index q subsequent add initialization exist mahalanobis feature design distant distant another cluster extra final
c ic cost happen one select worker worker involve select set bid tc apply transformation post post transformation ex monotone allocation randomize post separate lemma constraint cost uniform probability round give number round round round round hence turn exploration e tt ucb difference ucb regret situation expect combinatorial solve optimization problem naive implementation ns lead computational complexity underlie combinatorial due able approximation algorithm monotonicity rule solve minimum problem np approximation problem ensure select general quality worker combinatorial eliminate worker eliminate elimination incorporate approximate return monotone give rule essential compatibility monotone allocation ic ic algorithm arm explore worker probability worker target decrease exploration select emphasize se identify worker worker cost require choose arbitrarily adopt ucb target check target ensure extra worker rest cost worker bind check run none violate average greedy true sample ns reduce iteration worker figure greedy number worker private worker aggregate select worker attain target novel setting develop constrain confidence bind post individually rational mechanism exploration depend inherently exist approximate monotone ir interesting research convergence attribute exploration separate solve optimization strategy example generalization possible require assumption soft form future thm identical labeling worker outcome aggregate problem challenge even develop accuracy mab constrain non upper algorithm worker cost ns adaptive call post allocation compatible individually give also select bind upper insight illustrative efficacy financial security company pool opinion aggregate majority probability increase company business provide minimum threshold set private report service threshold company learn assume learn give homogeneous abstraction homogeneous another incur design aggregate certain provide value level right answer call agent cost therefore sensitivity goal select subset give agent absence play know reduce learn worker minimize cost though worker worker significant cost thus face versus choose optimally learn natural multi mab work crowdsource challenge ensure unknown address try cost mechanism ensure theoretic learning cost simultaneously need game learn mechanism refer mab mechanism induce achieve require accuracy highlight solve problem select worker target optimal solve first version learn paper propose framework mab call ns make sure probability suboptimal worker set level true achieve match may modify ns separated confidence prove ex post cost post adopt technique ex mechanism separate exploit simulation knowledge learn agent information mechanism crowdsource reverse summary next stage discuss section mechanism extension incorporate worker avoid high cost exploration section variant mab address version crowdsource learn worker answer crowdsource provide natural review mechanism crowdsource quality satisfied hold worker certain meet micro aggregate answer worker assume crowd general number worker arm heterogeneous select go micro assign task task quality et predict analytic guarantee predict formulate oppose subset worker none address challenge also cost mechanism crowdsource literature crowdsourcing involve price online worker mab determine pricing crowdsource homogeneous know assume cost private information propose price mechanism mechanism maintain online consider consider heterogeneous et involve people market crowdsource winner worker period adopt mechanism preference task theory crowdsource either homogeneous assuming set worker learn mab rich body literature available mab problem far moreover satisfied oppose satisfied round probably pac closely pac subtle obtain approximately arm round provide approximation optimal set high satisfied respect moreover exploration round mab chen wang relevant pure combinatorial subset feasible learn mab discuss arm reward round exceed instead armed mechanism set combine area mab mechanism phase regret round arm slot develop adopt make click multi oppose traditional armed arm constraint consider forward mechanism procedure monotone rule randomize input allocation allocation rule exactly mab post transformation post monotone allocation rule reverse bandit setting propose translate accuracy worker preliminary appear ensure accuracy current improvement crowdsource worker work homogeneous crowdsource worker associate labeling worker quality task assume service quality worker incur target accuracy determine label l ex round confidence require worker aggregate worker right quality task ucb task bind task worker worker incur input depend rule aggregate abstract let capture select profile seek requirement monotonicity say profile I decrease bound smoothness satisfie increase continuous profile difference error profile continuous error probability next monotonicity majority aggregation monotonicity smoothness player task report label vector label label majority voting rule aggregation likely outcome lead worker mistake satisfy constraint respect give monotonicity focus assume verify smoothness simplify monotonicity describe recall fashion thus goal worker threshold make follow worker priori learn repeatedly also solve accuracy bandit mab typically measure achieve later way satisfied thus algorithm satisfied probability expected incur satisfy large would involve regret start important property profile follow separate sf framework suffer time worker algorithm reward function quality quality profile worker profile respect come profile change event markov eq worker profile worker get version strong law number fact suboptimal optimal arm number various mab ucb et mab arm maintain exploration number regret increase ucb reward arm work monotonicity satisfied reward confidence arm similar ucb high algorithm function know make algorithm error monotonicity bound smoothness assumption interesting assume label observe complete motivate trade company satisfy learn algorithm select complete enough worker worker assume publicly know learn since worker accord true satisfies smoothness algorithm black box aggregate aggregate label opinion aggregate aggregate voting aggregated ns ns ns worker confidence ucb select worker f explore worker observe true label k ts tt observe subroutine minimal return minimal present work ucb constraint input worker level predict decide worker set initially estimate report next observe assign similar maintain bound bound hoeffding prove lie worker since constant worker lie key worker effective meet add subset subroutine minimal accuracy meet even low error target round find upper confidence even use stop task minimal accuracy simply ns satisfie hoeffding worker I q monotonicity ns equation round assumption satisfied ns algorithm set set rest unique though easily return p round ns stop explore ns solve optimization eq aim non say round say round set optimal exploration algorithm set task bind overall regret round e get round f unknown algorithm require ns adaptive let h hoeffding n n select worker ns exploitation optimal whenever tu tu far exploration round total eq incur satisfied ns bind assume proper cost section call monotonicity theoretic worker agent allocate worker satisfy smoothness denote incur constraint equation eq linear every heavy incur wrong thus compatible say compatible report dominant worker rational worker always utility characterization mechanism provide player generic transformation take output compatible design mechanism set set monotonicity allocation
sampling motivation observation count sparse count discuss later applicable inference fast start dense maintain nonzero fractional term advantage construction linear procedure benefit contain mass one topic via synchronization scheme datum introduce literature fail handle huge inference mention big exceed ram share parallel inference correctness performance update heavily condition evident synchronization share variable statistical huge copy load place help memory worker share disjoint motivated addition specifically topic word outcome straightforward scalability allow share burden replacement flexibility inference rather model lda realize algorithm block block assign correspond worker worker token sample block worker another iteration worker block sample process synchronization share mainly asynchronous incorporate worker effort synchronization worker construct incorrect become even evident cluster cloud service parallelization proceed receive task request block carefully worker store server purpose distribute memory partitioning frequent background asynchronous simple hash implementation suffice dynamic partition strategy demand communication key store round key store task thereby global accelerate communication block block synchronization communication synchronization combine dynamic demand dependency frequently fast token fact method much complexity another count impossible term value denominator change final worker round highly receive vector key store worker aware change sense similar relax requirement major maintain show compare actual empirically negligible combine demand communication avoid parallelization protocol separable dependency magnitude time implement design illustrate partition worker generate assign coordinate worker maintain special key value store distribute synchronization requirement job worker scheduling constraint consumption acceptable since omit hardware equip specifically high network interface equip ghz gb ghz ram machine corpus wikipedia word token original word token phrase occurrence phrase vocabulary demonstrate scalability size topic extremely case surrogate measure lda sampler optima topic progress rise measure gibbs unlikely local reach might ask employ surrogate practitioner improper evaluate goodness model evaluate alternative generalize new system generalization issue learn introduce factor optima sampler optima control external measuring inference end parallel reach figure log trend similar dynamic partitioning suffer begin copy progress iteration almost synchronization worker leave round relax requirement minor huge count affect overall proxy copy worker round token local copy lie collect observe drop stay close procedure demonstrate parallelization error fast ability big table yahoo lda vocabulary copy long memory model big able perform indicate ability yahoo lda indicate parallelism big sized demonstrate effectiveness partitioning strategy c yahoo cluster memory ideal memory consumption observe parallel nearly ideal scalability start drop much indicate storage unnecessary yahoo usage parallel word machine machine big also machine different yahoo node traffic increase bandwidth contrast speedup closely inference effectively utilize resource significant overhead demand strategy parallel inference traffic guarantee ability inference handle end paper present parallelism implement parallelism efficiency parallelism process bring capability big metropolis speed already significantly block broad attempt parallelism investigation interested parallelism challenge dirichlet hdp regularize big cs application topic conceptual high become next especially grain online usually million conventional approach topic inefficient heavy centralized pose small ram address another parallelism namely parallelism enable integrate parallelism parallelism distribute element ability tackle collapse gibbs algorithm experimental computational ml advance technology big massive various ml resort parallelism task partition pose mild synchronization assumption valid huge file database suit parallelism due element share variable entity model clear persistent converging need convenient machine program model large basis topic basis couple normality negativity estimator must procedure treatment dependency parallel explore sub decompose strategy evident synchronization logical correctness cycle scalability fine grain variable early engine grain mechanism parallelization case prevent asynchronous update art lda yahoo lda advance server extent little correctness study theory support model update parallelization improve parallelization pose accommodate handle unlike convention application modeling instance online beyond extract topic visualization scale vocabulary topic readily available copy vocabulary require real feature augmentation word large conceptual raw model storage issue type parallelization parallelism parallelism parallelism update dependency among specifically make sampling share small computation find subset completely assumption block exactly result serial inference requirement handle modeling end model parallelism program parallelism parallelization
gradient algorithm smooth partly fulfil manifold identification light small bregman divergence linear book overview sharp extended prove term functional see literature state I condition lasso author prove though similar establish author nuclear norm note invertible restrict condition operator cover total fuse lasso generalize partly operator cover dimensional variation equal also performance family decomposable norm nuclear show high random operator noiseless ensure completion noisy signal level minimizer sensitivity non seek ensure usually hence assess stable sense stay notion partly smooth existence manifold partly behave identifiable move manifold behaviour smooth sum sufficiently locally partly first equality fact cone vanish u manifold class deduce p p derivative recalling manifold use constant manifold show together arrive accord lemma state normalization homogeneity sub simplicity slight abuse approach since continuity continuous applying lead contradict fact classical fourth moment finite scale partial relative enough partial smoothness partial partial particular smoothness mapping partial subdifferential continuity property smoothness solution particular define continuity subdifferential contradiction unify performance partly problem function popular regularizer use feature generalize guarantee acknowledgement author er european sigma vision least square partly convex class function solution notion force solution problem dim manifold make low generalized tuned level regularize tend regularize correct dimensional manifold generalize statistic operator science recover inverse impose solution consider value prior control amount follow canonical without stand pseudo inverse goal e understand close noise stability identifiability associate body literature include regularization turn special general theory partly smooth subspace hull set interior interior affine hull affine contain guarantee partly smooth originally hereafter contain continuity continuous say partly neighbourhood partly unique regularity proper discussion convex continuous subdifferential everywhere automatically verify continuity converge converge characterization popular smooth regularizer use imaging check partly smooth literature basis pursuit literature name capture sparsity overlap block group group typically therein detail partly value impose understand rectangular completion nuclear generally consider function partly smooth smooth proved matrix absolutely invariant define piecewise constant image sparsity enforce partly partly manifold start design impose sparsity use see partly partly force convexity state main introduce stability enough linearize pre deterministic show robustness perturbation design locally partly smooth constant deterministic vs typical one statistic machine consider regime row show hypothesis close sharp characterize solution
engineering examine split nonconvex direction multiplier alternate method multiplier sufficiently stationary nonconvex additional algebraic furthermore condition guarantee boundedness satisfy wide include identity gradient efficiently apply optimization problem twice value well element minimizer indicator engineering regularizer norm regularizer use nuclear therein proximal mapping stochastic linear block discuss map splitting apply proximal generate convergent globally choose modulus nonconvex cluster proximal apply one feasible alternate apply admm nonconvex particular author measure show square successive change iterate case nonconvex sum euclidean show iterate assumption motivate admm identity contribution characterize cluster generate admm replace admm approximation variant subproblem involve solve quadratic furthermore boundedness generate condition satisfy algebraic cluster actually semi verify recognize cover concrete admm show nonconvex statement preliminary material next devote proximal numerical concluding remark inner denote induce denote identity map adjoint product map semidefinite nonzero resp real value equal proper fx limit subdifferential immediately subdifferential subdifferential subdifferential z continuously subdifferential variable subdifferential resp respect general subdifferential enjoy base principle particular solution optimality always throughout continuously bregman continuously word union finitely strict inequality semi algebraic semi semi algebraic nice structural property proper continuous differentiable hold satisfying property proper close algebraic function study direction multiplier nonsmooth follow continuously termination criterion subproblem call subproblem proximal hence popular remark study suitable hx x say would chosen least continuity modulus pick interest update second subproblem pick hence iterate hence convergent subsequence stationary definition pass follow global conclusion point sequence admm produce quadratic indicator euclidean convergence admm proximal establish work idea admm note subsequently change primal like modification directly subproblem introduction establish lagrangian existence hand boundedness sequence general enough literature study scenario suitably initialize get strict improvement suitably initialize stationary stationary strict improvement initialize sequence decrease proximal admm stationary choose initialization approximate close obtain relaxation stationary point relaxed initialize observe take norm I obtain assumption make relation definition establish q operation preserve semidefinite point modulus minimizer summing eq subsequence put pass make use conclude desire minimizer see py py discussion precede stationary suppose initialized chosen end proceed consequently since must lagrangian combine recall conclusion inequality convergent modulus one take third ii choose picking suppose vector adapt apply algebraic suppose sequence admm converge stationary stationary point establish subdifferential py together constant decrease subsequence l notational minimizer relation continuity subsequence together lower imply combine existence claim furthermore decrease x z k kk meaning terminate finitely conclusion terminate finitely next function definition pick hard I dl fact next concavity eq divide take rearrange hold induction proof inequality consider q inequality monotonicity induction moreover hence claim relation complete comment inspection show theorem continue augment lagrangian property read z property case least interesting assume suggest experiment preliminary admm solve concrete convergence nonconvex set admm map problem reformulate admm let denote multiplier constraint iterate take ambiguity update nonconvex iterate initialization routine show cycle x k convergent successive change look nonsmooth mapping proximal backward update q hard stationary convergence flexible size exist continuously moreover indicate restriction small modulus allow continuity modulus definition subdifferential apply descent rearrange convergent subsequence along convergent x px px px px take limit along convergent subsequence conclusion concern hold example continuously function modulus concrete hold semidefinite open step lipschitz know lie clear lipschitz continuous modulus step choose h hx hx tw px n conclude cluster exist incorporate difference initial backtrack search perform numerical code matlab experiment bit intel cpu ghz ram matlab close present full count proximal admm admm guarantee latter solve admm obtain convex condition successive change hand generate instance relaxation report distance report cpu initialize origin allow violate always close obtain close initialization cccc ccc admm cpu e e e e e signal indicator slow heuristic generate initialize terminate occur benchmark solve use generate piecewise specifically matlab r consider computational cpu second cardinality error original always correct piece always original noiseless cm ccc ccc cpu next present visualize recover signal via
ray voxel row positive voxel neighborhood system twice continuously preserve edge choose class due quadratic linear standard come smoothed different determination ard originally ard likelihood diagonal treat evidence refer posterior interestingly many concentrated zero readily show concentrate concentrated mechanism specifically tailor likelihood type originally expectation em base step oppose mode ard variance ard computationally demand method mainly task extension ard approximation despite ill significant advantage ard avoid nuisance trade variance experimental design scope ard conjugacy ard poisson law conjugacy step equal hessian map result ct high inverse hyperparameter irrespective laplace principled lack objective conceptual difficulty reveal ard contribution extend determination ard convergence reveal ard preserve similar gain previous likelihood transmission adapt reduce considerably simplify parallel search pixel voxel analysis scalable guarantee global imply guarantee world ray rest ard poisson likelihood surrogate feasible sec analysis property algorithm connection method present ard model sec real parallel present sec transmission directly basis transform newly hyperparameter choose scalable discrete fourier wavelet transform ray ct explore fig usually row voxel average pixel e boundary neighbor outside domain boundary condition neighbor voxel agree many zero voxel piecewise smooth difficulty trying extend ard gaussian evidence q variational evidence low kullback kl divergence free maximize summarize solution zero kl minimizing kl nonnegative evidence increase likelihood e expression direct calculation accord integral prohibitive force posterior long evidence kl either decrease increase update evidence local except consider ard previous ard gaussian immediately clear extend ard poisson preserve gain ard surprisingly variational discuss still principle ard despite evidence bind call ard mention step perhaps common distribution mean variational conditional restrict addition restrict distribution univariate propose modify em two step repeat rewrite omit backward step reduce long distinguish similarly step provide estimate negativity add define form repeat reduce iteration increase importantly highly make sec drop forward operator deriving assume invertible square provide sec remove recall objective feasible practice ard evaluate prohibitive approximated straightforward verify fix jointly imply minimum jointly minima guarantee evidence understand somewhat estimation sparse explain kl transformation invertible exploit duality monotonically side obtain sharp envelope ard illustrate fig latter level kl shrink illustrate much mean concentrated posterior require responsible noise approach energy apply even expression evidence unknown gaussian noise force work al consider original ard backward level curve prior curve concentrate chosen kl student dash black form envelope mean prior posterior around fig domain lie simultaneous task compute scalability separable surrogate q zero th entry iff readily jensen jx j proceed definition simplify derivation define variance quantity associate th call type projection posterior type iteration back projection respectively jointly make separable denote set involve way modification requirement em instead equality obtain since need repeat separable lemma substitute estimate pixel voxel difference max deriving combine q surrogate step iteration step parallel describe initialize projection g mean forward projection require variance compute iteration line search component line search voxel variance parallelization provide mean share algorithm iteration algorithm number considerable advantage compute computer dedicate hardware gate array constrain obtain unconstraine problem sub solution measurement voxel complexity neighboring pixel voxel typically detail implementation search line give present convergence sec refer tucker kkt optimality solution give kkt kkt minimizer subject replace replace definition plug substitute remark update end iteration necessary kkt necessary kkt condition solution follow kkt plug solution let negativity solution introduce variance shall em necessary kkt substituting point reduce condition monotonically iteration par iteration divide done state assume ti term side sequence limit iterate zero imply par solution kkt iterate contain guess theorem positivity compact satisfied divergence kkt convenience proposition continuous map nonempty continuous existence follow line set several also constitute mapping therefore closed point composition close mapping close convergence present view sparsity minimum obtain formulation differential std domain note term without one global combination term serve avoid separable ard reweighte likelihood consists follow tune provide equation fidelity respect approximate difference variance serve pixel voxel remove substitute minima penalty easily avoid correct critical facilitate comparison reweighted sec correspond surrogate likelihood penalty reweighte consider iteration replace execute f vector estimation mle scope reduce mle surrogate close update surrogate decomposition execute parallel j shall huber huber pixel q th map q approximation posterior poisson b factorize emphasize principle ard alternative et inversion voxel feasible approximate since super super gaussian building sec extend sparse overcomplete square considerable update function take pixel voxel prohibitive problem objective rectangular objective long interpret due replace since independent minimizing minimizing kl interpret kl formulation although interpretation concatenation locate horizontal image horizontal pixel difference replace difference original neighbor pixel original correspond index respectively variance vertical axis neighbor ray ct medical ray avoid level intensity typical clinical par posteriori huber reweighte notation slightly par choice complete complete representation implement execute intel e cpu core run platform implementation search parallel core objective source intensity clinical angle full object detector measurement square map tuning reweighte one tuning run trial rmse comparable reconstruction table observe high much rmse properly choose important object rmse consume merely reweighte reweighted reweighted horizontal std realization negligible std reweighte reweighte estimator tune low rmse image visually map reweighte find fig occur tune significant fig reconstruct show observe region recall domain pixel std careful std posterior objective method image display include enhance frame mle use penalty vertical iteration recommend variance run pixel initialize reweighte initialize fit first objective iteration objective predict theory use assess practice reconstruction acquire view angle detector use setup section specification water inside air reconstruction different image reweighte show reconstruction produce reweighted repeat reconstruct correct pixel difference fig letter object mid air experimental letter horizontal one neighbor recommend figure objective monotonically water angular grid view reconstruction case fig include fig reconstruction filter reconstruction fourier reconstruction prominent view reconstruction fig display range enhance std fig reconstruction defer publication lastly fan geometry geometry search parallel line search lead whenever regime huber use trust appendix execute newton trust appendix newton work dominate region take converge minute trial reweighte take store transpose region memory back projection could take portion search lead comparable back iteration resource limit slow map reweighte twice trial low std view also determination ard call ard ard use transmission mean reveal mechanism establish previous ard important avoid tune good
important tuning likewise nuclear minimization enhance code original algorithm access code detail general tune noiseless result additionally emphasize superior clutter presentation limit variational limited completion publicly focus nonetheless conduct experiment thresholde inferior avoid presentation separately comparison fr begin reproduce hide uniformly trial percentage across varied capable limit beyond number freedom represent candidate pool tune author although parameterization entirely benefit strong theoretical always match art display motivate reproduce completion design superior generate varied evaluate reconstruction combination value reproduce challenging case superior reconstruction difficulty fr fr define blind rank result previously mean failure achieve next arbitrary constraint nuclear minimization type mapping ii latter condition operator display somewhat ill condition display include comparison case vary consistently able general explore range rarely always theoretical boundary explore actually np recovery failure certainly circumstance possible scenario probe carefully condition difficulty reduce measurement measurement even fix reduce exactly equal degree examine uncorrelated far examine failure singular case notice classified error state threshold almost correct hence nonetheless feasible theoretical maximally size feasible spectral indistinguishable importantly test fail much apparent motivate success account respect relative percentage trial whereby find denote trial new criterion failure become involve actual rank solution completion involve reduce fr break result perform besides metric achieve fr adopt limit reveal failure specifically dct coefficient process linear information figure replace purpose thing stand metric failure mostly rank secondly dct outperform report avg summarize demonstrate capable theoretical limit process even feasible nearly suggest failure failure tend near display test situation reveal nonetheless promising noise reproduce design observe observe although heuristic four adjusted value report exhibit superior class updating generalization performance limit special circumstance comparison true rank knowledge specifically algorithm test introduce priori nuclear rank match especially bad decaying phenomenon multiply th large decay drop finally regard complexity scale completion exploit difficult recovery increase though limit highly overcomplete show effect relatively versus world formulate consider collaborative recommender former latter observe basic idea order taylor approximation around estimate jacobian transformation accomplish project side feasible result original nuclear norm term simplify comparison result small successful significant transformation estimate collaborative technique recommender user entry item task collaborative knowledge estimation recommender system per appear strict validity assumption remain entirely unclear globally low observation computable necessarily lead fact report tend around almost imply provide discriminative type compare heuristic modification underlie algorithmic estimate completeness adopt offset weak image red transformation derive show comparison wide strict dataset million rating assess test ability rate item select user strong generalization three performance metric minimum result include algorithms gm generalization well course apparent fall narrow make optimally necessarily translate truly practical collaborative argue explore conceptually matrix affine capable broad nuclear norm break adopt principled justification entirely theoretical local empirical exponentially regard nuclear appendix brief lemma address compressive presentation basic aspect adaptation minimizer must infinity infinity span else objective drive infinity constraint constant ultimately maintain unbounded statement objective rank behind collection secondly ensure assume r scale minimize rank complete sketch column indistinguishable achieve theorem become note minimum positive infinity I drive construction involve first likewise take I kk great exceed consequently negative reduce except moreover consider shrink generality translation condition great preferred unless display display minima reveal counter iterative bound tailor symmetric likewise via practice sufficient obtain good application require recover minimal affine constraint notable special replace nuclear act convenient elegant theoretical replacement restrictive fail ambient high constraint poorly non alternative carefully tune locally failure like wide empirical theoretical measurement unknown rank surprisingly possible affine ill prove nonetheless condition whereby point locate optimum exist involve completion recovery subject involve mapping commonly apply collaborative problem low np rank non smooth consequently alternative denote concave nearly special retrieve surrogate prefer reduce norm quantify condition heavily matrix nuclear coincide minimal restrictive convex broad art operate constraint follow derivation technique adapt pca describe connection norm issue special whereby stationary discuss algorithmic performance contain efficacy image collaborative filter proof rule appendix proceeding take offer little substantial gain tailor modification probabilistic lead systematic analytical empirical insight justification consideration merely qualitative underlie convex avoid solution inspire require balance minimal truth include direct head design code original carefully tune even algorithm never demonstrate previously rank develop evaluate attempt locally derive maintain rank behave scale albeit well perform minimize homotopy merge replace function minima progress reasonably spurious avoid procedure pre reduce schedule specific solution ever derive apply function unlike suggest tune different class unclear choice substantially norm seem slightly quadratic less stage optimization trajectory minimization limit equal degree recover hence good boundary practice ill achievable nuclear truncate convex image via contain setting poorly somewhat non derive alternate low matrix solve approach require parameterize contrast emphasis regard experimental aware embed low rank typically feasible tune previous rank minimization affine constraint build summation element penalty apply iteration analysis affine apply similar model completion competitive state mention intrinsic focus challenge consequently general estimate refer particularly solve problem first close intuitively discuss desirable global minima conclude brief convergence minimize minimization reveal function nothing suppose apply term equivalently solve demonstrate optimal simplify nuclear arrive limit become conclude distinction nuclear intrinsic section convex substitution probabilistic function technical duality compressive please penalty approximately view still minimum constraint become possess attractive invariance mean rescale rescale optimum optimization much surrogate local defer small feasible block invertible block iff likewise correspondence rank theoretically restrictive require require likewise essentially guarantee possesse optimum rank limit indicator via argument certainly adopt function minimize rare largely process provably specialized quantify suppose diagonal restrict nonetheless case include generalized element instead block furthermore satisfy always global minima great imply cost function minima condition intersection merely rank simultaneously measurement still highly ill condition minimizer globally standard typical rank additionally unique one solution crucially minimal unlike true underlie importantly
emphasize complement either numerical quadrature mcmc number algorithm g riemannian sample correlation well mcmc posterior hx hx rx give rao estimator exploration update rank achieve effective argue involve introduce since replace reduction particularly space mcmc subspace offer additional advantage apply full low scheme full method root stochastic newton handle full much handle carlo product function either multiplicative sum expectation subspace numerical particularly useful example analytical reduced variance reduce reduce reduce estimate result constructing require much ensure capture variation choose solve markov mode project follow adaptively dimensional last state construction basis checking terminate evaluation fall return incremental distance inform heavily basis consist compute distance sample adaptively exploration might ignore direction inform update complement would construct numerical describe good construct posterior give essentially course algorithm sample mcmc pde inverse demonstrate construction mesh mean vary observational pressure pressure govern pose boundary boundary impose superposition width center corner bilinear endow log normal prior I prior length true pressure synthetic h pressure field collect black dot figure observation operator correspond mask operation deviation prescribe ratio show draw prior prior run order slow spectrum truncate frequency unless retain construction refinement mala simulate threshold dimensionality versus refinement carry coarse grid iteration diagnostic discretization h sample use black blue marker grid grid show show subspace h level discretization column order rapidly reflect log observe grid slightly grid effect discretization grid weight adjacent inform diagnostic order magnitude begin rate convergence diagnostic three level suggest local variation course explore refinement grid show figure subspace grid refinement refinement mode close hand reduction describe pde mala hessian full space hereafter result discretization langevin sde inverse sde empirical posterior mala dimension setup examine project onto vector result figure discard burn row benchmark mcmc produce decay autocorrelation use run mcmc iteration cost full difference second benchmark cpu immediately observe autocorrelation per cpu time reduce course construct roughly mcmc step cost h subspace result distinguish field central measurement sensor variance right region carry affect structure demonstrate likelihood sake computational mesh impulse system section evenly center place domain distribute evenly domain inter refine four pressure sensor algorithm spectrum note eigenvalue decay amount reflect impact lead direction area domain inform subspace however frequency might differ basis corresponding eigenvalue share similar eigenvalue different pattern carry basis realistic use stand year lose different intensity spectra inversion infer ill spectra minor totally inform small briefly theory setup reference transmission spectrum measure ray model height call cross section laboratory measurement discretize inversion resemble density assume within spherical height inverse layer fix layer approximate integral choose n geometry contain length line layer cb stack top know variance note linearize measurement synthetic solve profile discretize dimension simulated profile profile denote matrix value prior choose profile rough magnitude density know well totally construct diagnostic threshold compute hessian mala figure profile standard complement column show horizontal axis apply addition contribution cs entirely determined contribution low mean result avoid accurate illustrate plot six basis mainly inform expect mix space space mcmc compare mcmc test computational mcmc simulate subspace mcmc second cpu second algorithm iterations cpu cpu approach inverse approach divide subspace inform posterior distribution complement dominate explore project onto gaussian problem chain treat complement estimate expectation rao randomization greatly reduce particularly solution heavily dimension majority handle analytically approach show update generalize vary inform subspace global adaptive meet first pde flow parameter remain explore analytically treat produce via full computational dramatically problem infer chemical star dimension full specie appear offer exploit inference algorithm curse reduce order applicable acknowledge provide code sense support office advance scientific de sc sc example department usa intrinsic inverse affect relatively identify inform characterize influence support identification efficient bayesian chain monte sampling low dimension monte expectation variance pde sense monitoring carlo arise indirect parameter high moment quantile event parameter dependent prediction quantify posterior carlo mcmc affected dimension degradation increase higher posterior recent share scaling argue randomization estimate explain propose inverse identify subspace notion nonlinear approximation develop case reduction combine likelihood inform wherein independent datum particular approximate dimensional marginalization complement benefit evaluation expectation likelihood inform enable great efficiency step allow complement inform avoid analytically condition expectation previously way construct truncate expansion likelihood hessian log inverse problem nonlinear construct stochastic approximation mode either stochastic tradeoff hessian posterior proposal proceed low dimensional projection enable rao mcmc posterior amenable integration present mode choose orthonormal precision matrix preserve important seek form determine inform thus inform embed manifold aim global majority nonlinear inform forward linearization forward provide sensitivity observable inspire linearize newton approximation hx jx jx
describe descent practitioner decade inherent conceptual simplicity ease work code largely ignore recently report remarkable come understand cyclic greedy next coordinate increasingly clear complexity analysis link random describe subset coordinate select iteration capture smoothness useful function admit would allow would column identity denote property unit coordinate context descent focus extend reader inequality eq hadamard term hand side latter hessian importance descent perform design nontrivial deal design search influence complexity see table updating coordinate update lead perhaps resource whether understand study complexity dependence vector soon satisfy lead natural coordinate descent variant coordinate accelerate study study deal algorithmic aspect aspect mention employ serial sampling serial appear bound directly year nonsmooth primal n alpha method arbitrary apply serious problem function lipschitz regularizer nonsmooth arbitrary thing accelerate enjoy slow uniformity variety basic overlapping overlap nice serial parallel non serial sampling assign distinct necessarily lead bound speedup product associate intuitively speak sample linearly sample far almost inequality recover general give large submatrix briefly terminology matrix parameter inequality describe paper section review elementary satisfie however often entry eq functions require help assumption randomize coordinate descent method assume pick jx h jx inequality identity matrix element proposition matrix separable function coordinate gradient appear formulation th coordinate eq eq q form role design accelerate function appear dual apply set value value terminology shall never never coordinate descent key elementary elementary elementary associate intersect refer reader condition necessarily every sampling uniform additional uniformity property name doubly prove notable doubly pick uniformly give refer standard mini nice sampling arises distribute coordinate nice q nice sampling define nice pick subset uniformly basic combination intersection nonnegative scalar sum accord elementary doubly arise nice let sampling doubly nice statement definition intersection eq sampling necessarily eq collection constant add I via eq consider hadamard formalize independent ij ij restriction several alternative writing keep distribute sampling definition note belong partition nice sampling nice finally doubly uniform q nice note semidefinite diagonal denote normalize eigenvalue recall semidefinite resp resp quantity later useful compute I since elementary simple see consequence es cauchy matrix since add give sharp bind identity simplicity identity elementary whenever result combine upper study quantity statement maximal I bind elementary tight view normalize eigenvalue associate family rough sampling doubly mention upper apply proceed fix nonempty intersection eq apply q cauchy plug j obtain nice sampling let nice nice sampling apply calculation e give large doubly doubly develop hadamard product semidefinite study bound hadamard hadamard eq substitute expectation next direct consequence regard reasoning lemma sampling statement sufficient view hence pick similar class partially separable function arbitrary degree separability correspond see normalize consuming pass prohibitive next follow one issue avoid decompose different vector individually recall set coupling th j completeness equality finally theorem proposition conclude matrix large eigenvalue upper proposition see illustrate precede eigenvalue computable sampling way distribute sampling doubly sampling sampling serial direct vertex remark part improvement quality involve part compare small well admissible effort lead nice admissible dedicate formulae appear admissible computing size pass return approximate apply semidefinite number number notation pass n parameter strongly setup time epoch pass pass convexity formulae report big preprocessing compute formulae normalize use product partition method multiply value processing formula formula also magnitude take pass formula enough convexity
spectral previous scheme could boost spectral approximation show leverage let set leverage estimate follow iterative immediately enough allow sample matrix row cut recursive give clean argument prove version slightly theorem technique believe leverage score actually sufficient obtaining row powerful coherence row specifically coherence intuitively give describe exactly leverage row coherence spectral reweighte thus reweighte score uniformly sample never great sum score reweighte score row reweighte trivially leverage score leverage need obtain row score bound ensure spectral frequently lemma simple randomize numerous multiplication linear helpful survey runtime gain alternative algebra tool gain pattern require linear algebra process algebraic roughly multiplication approximate select projection challenge correct focus reduction step approximate randomized scheme combine require projection subspace go back recent progress significantly iterative specifically spectral incidence commonly primitive graph graph potentially row leverage spectral ensure reweighte vertex incidence matrix preserve row level accelerate edge incidence evaluate li fairly ultimately projection row preserve converge step r diagonal let singular spectral approximation imply preserve multiplication consequently singular leverage row also define leverage score leverage orthogonal row would change leverage row coherence removal affect composition row characterization help intuition optimal entry must constraint ix j furthermore score compute leverage pointing row remove could simplify approximation generalize score multiplicative leverage spectral lemma score fact exact score suffice I let denote otherwise completeness result argument diagonal valid leverage score indicator prove trivially formula fact bound break select always process select return score leverage score low fundamental proving prove study conjecture otherwise bound satisfy set exist total incoherence show reduce prove prove score evolve reweighte decrease leverage row leverage score rank update diagonal next claim row allows arise continuous ki place ready main require consider weight see decrease give lemma score non construction u ii increase leverage row remove row enough variety approximation however slight bound correctness sampling score intuitively coherence portion loose leverage score estimate score leverage leverage upper set bind bind come require sampling give probability statement match leverage leverage show hence choose accordingly clear gives start rate score cut leverage restrict leverage converge expect keep cut sum far correspond differ early g algorithm maintain row iteratively leverage enough row score summing cut row eliminate consider zero score obtain row c reduce sampling actually sample row rate computing return match introduction theorem yield extremely simple clarity initially present version first art solely preserve improve usage system recursive estimating row course compute leverage input matrix output spectral rescale make show set leverage quality leverage round obtain row output rescale reasonable fact solve time exponent emphasize trade rescale think fast primitive showing leverage efficiently compute generalize leverage rescale di obtain idea within multiply height score solver explicitly slight need example primitive analyze runtime simplicity spectral furthermore leverage constant row runtime refinement induction cut uniformly cut instead leverage increase generalized score increase another constant row runtime score termination terminate iteration factor decrease technique idea leverage use rough score respect row row finally leverage take come refine entry dd term hide tradeoff algorithm summarize use note matrix process recursive call output rescale note incur create result generic simply call sufficiently recursive modification head algorithm head recursive giving tail recursive w step error factor sample w give situation likely still give thank helpful discussion support nsf fellowship grant grant fa advanced project probability score random matrix independently concentration corollary semidefinite ip desire eq consider u trace cyclic semidefinite q equation directly ic lemma random concentration probability nonzero leverage change update diagonal eq
biased toward predictor ranking rank rank highly desirable find structural different ranking discover create aggregate link pair criterion whole consider ranking pair label fact output select ranking go name rank simultaneously ranking goal true value indicate contribution merge ranking predict pair link step exactly high true prediction come purpose link link window highest select throughout slide window one contain process iterate summarize cm x cm x cm x x step ranking gray window I counter prediction I l method step window represent gray initially pair exclude randomly join accord learn training practical test high already ranking namely consider rank step c phase pair predict give test learn top item etc tr n benefit ranking moreover store window array update complexity go ranking yield memory due ranking space complexity preliminary part aggregation numerically q first consider slightly aim q decreasing add recursion argument high eq I contradiction previous practical related ig life aggregation hypothesis classify rank increase condition nearly fulfil process mean problem get order process order efficiency first compare classic restrict near tree technique recall experiment explore involve view social crucial depend connection say link access phone order simulate divide three link phase define link link guess assign contain link link derive situation performance obtain predict link evolution vary one slowly reach maximum smoothly slowly aggregation improve precision rank exploit difference profile difficulty task recall impossible distant increase dramatically learn leave versus curve description discover merge link scale adapt learn ranking argue aggregate ranking information ranking redundant addition additional ignore performance plot curve obtain aggregate classifier apply well consequently well method explore aggregation area recall quantify performance quantity increase concern supervise comparable performance magnitude range poorly region minus average prediction retrieval benchmark leave prediction versus table comprehensive matter long ranking restrict example merge fluctuation rank source prediction suppose bring ignore process critical link consensus performance dramatically poor choice experimental encourage x x x x dependency indicate performance possibility phase intermediate small ranking c differ belong link link guess link guess method phase guess phase large miss ratio much average degree expensive costly consider large make ranking focus ranking ranking ranking experiment factor plot show begin closely follow curve performance aggregation mostly aggregated initially come rank soon allocation take good complementary aggregation always choose pair rank notice ranking poorly dramatically comparison supervise poorly limited highlight limit prediction largely learn result five link ratio area generate efficient unsupervised test unsupervised vanish grow stem dominate merge tend stick perform curve miss link ratio leave practical briefly magnitude experiment throughout cpu unsupervise ranking production index merge shorter framework combine ranking straightforward suited tuned need design prediction rank significantly improve purpose structural ranking node additional classifier option also consider user interact short connection theoretical mechanism link identify quality apply difficult detect application include security detect existence connection engineering combination active acknowledgement office scientific acknowledge european rgb rgb networks link relationship simple supervised rank aim various illustrate social improve performance rank also standard selection area relevance prediction link key mining practical go recommendation link view identification behind evolve example closure core link drive force link seminal snapshot network predict link attribute involve aim typical node irrelevant handle typically shall challenge computational indeed specific link role different type environment misclassification combine support biological scientific prediction desirable social network address establish accord scalar interaction predict top rank property example gender profile interaction last framework combine link method chain supervise retrieval document spam recommendation author approach pointwise feature fit undesirable rank link connect link popular account version direct spirit final adequate goal engine provide highly field measure discount however link whether quality work resp top quantity score effect imbalance
study population participant also requirement control design input p p compare decision trial requirement specify design sc power expect trial table file table report figure report create package center input panel main drop different show batch mode must results panel main panel output describe summary trial trial z z plot stage bar sake design h center default input top display power sample expect panel loading input save input contain trial automatically file parameter panel design bar web figure sake performance advanced parameter interactive change interactive mode automatically user batch top interactive advanced save click basic select save file input regardless load e already available real give match one must indicator contain treatment arm must binary outcome file header adjust setting give observe parameter basic proportion prior estimate design item successful control arm expect design outcome consider possible advanced per participant participant k stage include k participant alpha requirement design item alpha allocate use efficacy boundary applicable delta efficacy boundary simulation trial power trial duration time stop user reach cpu limit either time extend second item total item stage comparable efficacy boundary sc identical efficacy sc ss k constant ss final combine plot performance display performance three expect duration treatment treatment basic treatment advanced parameter specify metric plot metric page three metric table denote different ad reject h reject reject sc reject ss h effect participant ss duration necessarily total stop sample necessarily duration stop come plan trial goal iii trial aim stroke outcome plan iii phase trial little pressure monitor participant yield treatment effect ci think assess evidence efficacy scenario special treatment effect difference small participant treatment participant zero participant follow item furthermore control participant outcome participant p participant p true project design ad goal iii fully corresponding design goal achieve goal generally expect return goal achieve goal iii ss achieve recall treatment design default mean treatment scenario remain ad base design design goal achieve c f ad ad default achieve goal power power reject reject scenario reject scenario performance although specify alpha rate alpha package application criterion give explanation application output current limitation also outcome delay requirement acknowledgement acknowledgement research support institute ns drug comparative science contract institute environmental health sciences es publication solely color interactive designing trial criterion date design plan benefit trial goal specifically criteria duration user require programming experience application table sequential randomize trial occur strong evidence early trial benefit treatment study restrict example stop evidence benefit focus introduce combine feature design design define sequential design change trial except entire stop efficacy introduce package user type package densely application ideal user little experience core input allow standard several oppose automatically full use comparison report software planning phase treatment stroke new plus rt pa describe participant refer participant treatment participant baseline phase trial combine small determine small prior inefficient simultaneously answer combine focus trial throughout many formally design software currently available pe package computer locally web discuss interpretation demonstrate adaptive two refer certain likely trial example refer adaptive standard include rule stop trial early consist k stage assume newly participant correspond population pi participant stage entirely stop adaptive design restrict describe stage maximum end maximum cumulative participant end stage k k sample successful I stage treatment I outcome outcome available outcome c p average comparing versus give overview understand discussion efficacy hypothese nan treatment analogous simultaneous nan hypothesis h p compare hypothesis adaptive nan ad compare standard design standard sc design denote ss test stage stop early stage trial differ change switch participant discuss simultaneously implement standard area research global p p mean treatment cumulative statistic base participant participant standardize compare k population sum k right sc st st difference mean control combine stop analogous restrict formally z statistic follow subject k sum n I sum right left I I replace stop boundary statistic end study wide error strongly probability nan c asymptotically go ss control design ss nan single rule criterion standard efficacy statistic c stage efficacy design sc stop trial boundary stage stage complete k sc stop trial reject make simplification k efficacy boundary efficacy stage delta total range sc alpha cumulative sample sc sc boundary set delta efficacy sc z covariance sc c k delta stage nan define section ss ad bind boundary boundary ignore motivation prefer bind boundary control despite boundary trial duration assume sc user default negative stop although trial set equal efficacy boundary ensure efficacy boundary decision boundary ss analogously design except make simplification ss user efficacy boundary delta ensure alpha delta final efficacy boundary consider adaptive ad specify stage combine regardless stop end stage reduce turn option paragraph stage trial ad type two decision ad boundary cn delta ad ad ad nan hypothese alpha boundary control type rate level alpha ad c e ad certain ad boundary relative z delta ad delta k ad ad stage inf indicate reflect ad decide continue population trial stop entirely rule describe rule carry assess efficacy k reject c stop trial assess stop reject else stop future stage following iterate reject stage trial reject stop else continue participant ignore continue continue pi pi motivation evidence stop trial stop modification incorporate testing hypothesis consequence stage l remainder compute constant ad define efficacy k control alpha type suffice error global nan hypothesis follow size wide alpha interval type alpha initially allocate describe algorithm compute hypothesis define compute constant hypothesis nan hypothesis alpha ad leave k ad k kn right limit software comparable pe conversely pe pe tool implement many
angle might obtain simulated anneal cost anneal permutation convergent short annealing period physical six machine universit institute employ three tc enable forward enable enable ignore current outside force substitute part impact force passive coupling yield passive consist front front middle part connect enable around axis mainly inspire robot six light dependent sensor arrange front body sensor front body sensor pattern detect front light generate inside signal digital interface pc neural controller power sensor implement hz robot front wide experiment r period I right robot robot implement learn suitable period pass deviation period robot deviation period learn period show experimental diagram show experimental supplementary website directly transfer result mechanism flexible configuration robot possible c scenario r l work multiple complex behavior exclude three manual tuning robot introduction complete paper elsewhere implement control require precise calculate lot resource inspire already experiment control independent additionally modular independent biological perform loop sensor require benefit system loop instead implement robot carefully replace loop regardless robot four successfully present active generate pattern continuous self add selector artificial anomaly method realize change mechanism planning automatically adjust tolerance realize emphasize mechanism multiple generation adaptation orientation sensor converge easy controller control especially center maintain ground able body weight load result way configuration effective responsible robot body support robot robot overcome control balance investigate beyond work one two start exceed threshold learn algorithm period controller modular flexible six modularity offer possibility drive behavior obstacle module thereby focus neuron inspire generator demonstrate behavior self main advantage control precise mathematical robot multiple system frequency robot cause setup deal line learn mechanism simulate anneal technique automatically thereby considerably deviation original movement cause base learn converge acceptable getting combination period demonstrate effectiveness learn investigate might mechanism advance suitable furthermore cognitive goal addition employ controller reliably stable periodic unstable periodic controller pattern education grant bernstein g national foundation project natural science foundation project innovation foundation european fp specific communication agreement thank frank technical simulator implementation check language universit mc institute control implement generator pattern behavior controller deal present movement desire trajectory extend single simulated synchronization dynamic automatically resemble well first robot six result approach generation part independent multi robot pattern generator neural humans movement show level adapt create elegant vary report demonstrate achieve central pattern apply type kind control control review inspire way movement many loop control model robot integrate adjust several base sophisticated deal task control problem controller identical failure immediately adjust frequency individually change frequency independently also appear cat cat movement stable complicated happen proper contact procedure intensive traditional develop multiple deal find system controller output independently suitable automatically demonstrate propose real allow perform adapt rely multiple learn mechanism structure also state introduce platform effectiveness verify conclusion controller controller extend multiple also become indicate neuron generate input weight bias generate like pattern simultaneously add e signal period neuron detect control p period adjust control circuit sigmoid activation neuron weight w dynamic obtain period calculate every adaptively use period change pass post process module produce robot walk slow period slow wave fast wave blue area ground white one stop ground unstable pattern period generation useful neural circuit dynamic important principle mainly stable unstable period without dynamic stable period usually pass module neural diagram complete circuit since neural control study discuss briefly shape subsequently module feed forward phase shift simple feed network tc act capability increase tc even reverse tc turn finally neuron delay determined phase side delay see setup motivated perform period controller thereby lead drive single situation robot robot word robot contrast real control trajectory effective pattern inspire module processing output send line describe figure front neuron master neural gray module consist depict circle dark indicate module indicate synchronization mechanism delay output neuron signal neuron send last I mean period store store range increase time close much probable return combination period trial conversely probable period loop period straight robot automatically combination find parameter create tradeoff acceptance slow vice additionally observe learn often work empirically balance angle period simulate employ six controller controller frequency hz initially work master master period frequency achieve wave slow wave movement set consequence affect normally robot stay simulation orientation sensor angle orientation detect six lose synchronization angle subtract window deviation current period test different combination find robot maintain straight important period change period affect period keep illustrate column leave front six trial degree column decision period return one ignore initially robot right front robot turn balance body front randomly change angle period decrease right change deviation period trial return rather however keep combination period show advantage anneal provide change leave possible cope r l angle l body contact force environment figure fully phase air phase similar relate input example r body especially neighboring fix depict fig trial diagram start case state six one six scenario nine exclude counterpart situation depict r resulting
letter refer set blue mutation analyze detect tumor copy region proportion cell mutation mutation allele mapping contain genome panel histogram heterogeneous tumor sample large error dna even cell division mutation distribution central particular cluster tumor example mutation tumor sample attempt detect historical sample evolutionary tumor node currently tumor connect assign assign frequency infer frequency sum frequency child index object use cluster concentration dirichlet place assignment prior draw component distribution finite extend dp prior result unlike estimate thereby fix restaurant crp dirichlet assign object exclude generative infinite chinese restaurant associate object assign word unique chinese restaurant describe cluster produce rooted tree structure tree object proportional exclude child node exist read read matching allele variant position total read let allele population two copy recover frequency equation read count likelihood frequency infer assignment tree multiply identity equation complete pass asymmetric dirichlet burn acceptance discard burn gibbs allow result paper merge case infer natural ordering sum natural merge unlikely accept merge likely accept select tree merge parent child become parent child tree select select node e split leaf mix split node frequency uniformly population frequency parent decrease show leaf merge move construct either split merge move split merge move type gibbs simulation population population allele read drawn population read depth table quickly prior strategy mix fewer likely remain accuracy sample first divide actual take divide calculate package efficiency consistently imbalance population precision co simulation correction correspond difference suggest difference finally situation cpu great effort experiment heterogeneous run read runtime runtime ii variant adjust remain slow time ability dataset apply sequence patient five reconstruct sample simultaneously examine recover cluster together structure publication recover nearly difference frequency estimate direct parent child bottom substantial population occur cancer find patient middle bottom simulate runtime per per great accuracy five test amount cpu fold run furthermore decrease large number suit genome sequencing number expert dataset remain question decrease flexibility two acknowledgment national engineering award support science statistical increasingly popular infer chinese restaurant crp represent propose merge tailor improve time comparison stick prior superior sample cell
update solution trick also redundant identity admm iterate simplify simplify convergent split converge denote radius however really rate find split redundant convergent apply transition note determined penalty parameter split admm solve equivalent penalty transition surprisingly achieve asymptotic case lead small beginning proceed small section consider selection algorithm invertible high filter frequency band component band extremely huge algorithm case hence admm eq admm estimate method appear rate determine algorithm estimate sum two additional converge seem inner square minimization least square happen asymptotic solve still rate sometimes verify parameter discuss image show instance middle converge reconstruction penalty regularizer well resolution tradeoff note horizontal vertical solve efficiently instead inexact significantly thank curve error setting image setting solution convergence split immediately rate come inexact exhibit slow estimate admm might fastest split suffer paper regularize method convergent alternate multiplier admm regularizer nice admm lagrangian inexact method image reconstruction deep understanding admm analyze split admm method edge preserve regularizer insight tune region work interested convergence inexact inexact square proximal mapping complicated affect rate acknowledgement part intel split splitting image problem yield subproblem separable proof proof impose subproblem impractical many mr ray ct reconstruction inner least square usually shift image problem alternate admm inexact special admm augment concrete admm term measurement square smooth huber finite regularizer one propose lagrangian iterate least proximal efficiently soft thresholding method direction dual show convergence hold edge preserve assumption differently inner update impose convergent solve unfortunately inexact mr ray ct reconstruction lack equivalence method convergent inexact update inexact satisfie condition application ray ct pass necessarily shift difference laplacian high pass non vice versa nan ray reconstruction convergent admm inexact update ray inner iterate optimum absolutely look combine initialize iterate admm find
clearly stationary mab instance mab problem stationary formulate mab formulation reward budget grow time set reward evolve study mab formulate adversarial arm may study suggest finite regret regret change allow necessarily regret describe approach identify proof relative good action benchmark fully adversarial environment draw horizon batch batch perhaps accord arm expect change follow fix fix arm condition arm arm arm respective assume binary reward expectation k recall batch one addition j sum arm hold j sequence reward batch arm distribution one k tv last establish conclude batch single batch relative adversarial describe tune correspond associate regret j kk v kt dominate arm whole contradict e hold batch establish conclude proof sequence batch analyze action regret exp compare sequence compose part follow possibly parameter regret exp tuning difference policy throughout decision batch eq sum jx hold exp batch j take tuning parameter use exp subroutine increase one regret incur kt pc arm armed mab problem arm characterize reward arm play simultaneously exploitation due extensively assumption sharp characterization regret range reward maintain fully mab establish extent reward variation achievable analysis connection rather adversarial framework exploration exploitation stationary minimax presence feedback face collect trying optimize future fundamental variety internet web site seek recommendation priori price maximize select large preference customer select internet efficiently datum user know delay well instance decision daily future effectively acquisition future instantaneous base paradigm armed bandit originally context drug testing place general reward realization reward characterize maximize possibly discount reward receive mab modification extensively statistic economic dynamic clinical trial price innovation name reference programming formulation cover machine mab one typical benchmark instant select growth regret horizon converge oracle order characteristic sharp characterization regret traditional reward mab design growth domain several reward ignore mab formulation origin arm change rise term associate arm accord stochastic line lead various relaxation reference therein stationarity dominate mab raise fundamental uncertainty realization game see significant review adversarial reward realization single benchmark action reason static perform sequence time limitation adversarial regret static regard lie characterize regret mab problem non establish extent achievable bad four formulate stationary quite phenomenon remain mathematically constraint impose reward bound adversarial reward pick maximally policy within treatment mab focus reference adversarial treated evolve accord brownian explain second non bound reward policy sense variation arm sublinear adversarial stationary number time regret relative oracle treat broad temporal establish optimality regret horizon result order minimax regret range order grow linearly well achievable performance exploitation trade set compare stationary forget algorithms mab literature weight past stochastic associate time risk bias interesting draw adversarial mab stationary environment adversarial suitably calibrate optimally set establish introduce basic provide regret admissible lower contain brief discussion proof let decision epoch decision maker epoch maker arm random variable expect decision epoch reward addition assume beyond different collect capture second tradeoff stationary environment past reward reward old potentially relevant stem reward change reward old turn encourage enhanced exploration achievable build adapt set deferred idea admissible must regret order partition horizon batch except batch exactly good arm thus batch numerical realization reward batch realization expect correspond variation budget probable realization expect reward nature prevent batch since history sake simplicity discussion budget proof admissible identify arm batch epoch policy horizon select variation satisfy low develop variation budget policy number batch repeat initialization arm receive arm update begin allocation subroutine batch regret well adversarial regret dynamic oracle budget policy policy include weight chapter tend well numerically guarantee oracle study class algorithm regret order action adversarial subroutine two tradeoff versus exist capture subroutine good compare exploitation incur gain expect second tradeoff versus capture exp old discard characterize regret multiplicative logarithmic quantify impact extent environment achievable broad achievable upper experiment environment two arm arm epoch evolution select arm pointwise incur pointwise summing change reward approximate oracle regret incur instance display second depict third horizon fix tt describe change environment instance spend throughout whole spend horizon depict upper plot growth figure policy identify reward select policy reward selecting receive quickly arm keep trajectory reach policy tradeoff occur tradeoff subroutine exp exp explore epoch batch batch epoch exploration rate reward
seem latter new truncation contrast motivation aim loading rotation step loading rotation loading since pca loading approximate pca loading truncation tends produce vector equal less greedy globally block contribution devise truncation unify sparse pca together unified view find relation version drawback imbalance loading rest organize introduce present truncation performance give unified series relation new conclude c pca st decade ad hoc loading process interpretability rotation loading thresholding zero explicit objective maximize impose facilitate elastic net technique spc loading impose one suffer getting transform semidefinite guarantee unfortunately computational high expensive elimination order make large author greedy instance complexity lead solution solution cardinality range improve far full review generalizing recently power relate call propose augment lagrangian orthogonality correlation among loading global computational complexity summarize first give type behind understood loading constitute orthogonal span basis instead sparse approximate loading rotation solved alternatively optimize subproblem basis type soft thresholding percentage truncation sparsity linear diagonal express orthonormal thin mean loading obtain clearly eigen idea rotation sum hard rotation approximate loading confusion eq version approximation key sparsity penalty simultaneously fix subproblem close become form eq z tx z entry thresholding z far decompose entry otherwise express thresholding normalization compare add practically sparsity feasible evenly otherwise determine rotation orthonormal orthogonal basis basic table pca loading linearly loading truncation loading arrange wise pca initialize rotation truncation switch sort ji ix normalize discuss truncation type hard thresholding truncation sp truncation hard truncation operation nothing else devise truncation irrespective whose energy take percentage sort sp objective type correspond hoc st systematically seminal rotation simple discuss orthonormal four truncation study much orthogonality variance explain bound explicitly orthogonality angle define angle dimension span intuitively threshold sparsity truncate original bad side deviation sparse loading expect cumulative percentage variance maintain similar control appendix surface axis angle determine sum small orthogonality entry deviation generally well q hard usually guarantee moderately discrete truncate moderately en deviation direct orthogonality explain en preferable moderately en nice eq advantage lie want finally two tailor let svd loading close explain loading possible conversely much loading tends variance guarantee less loading loading another estimation diagonal sparse loading sparse loading pca loading explain loading energy loading guarantee en en sum project onto likely achieve proposition finally usually percentage subspace basis project span near proportional independently originally spc approximation variable alternate two subproblem substitute solution loading substitute two independent sparse net problem tx spc artificial original deal objective follow fundamental one appendix objective mirror exist penaltie ty unit combine original one unit loading via loading penalty constraint spc finally serve length small spc searching difference eigenvector rather key success drawback block ensure orthogonality also tend length length lead unbalanced among loading loading goal sparse mode loading less set weight appropriately align besides may loading truncation output loading normalize contain loading truncation loading arrange wise matrix compute svd major thresholding length type sp insensitive length improve algorithm block version matrix loading moderate comprehensive evaluation gene dimension random purpose speed test toolbox implement version similar code mainly five criterion sp loading std standard loading explain loading denote loading loading among loading instead std imbalance loading comparison direct sp let number loading original fall solution pca distinct set ensure avoid termination change loading exceed code common computer ghz load sp sp gp sp en sp en whether gp loading consider hidden word mutually correlation particular mainly common mainly dimension generate correlation latter feed accept datum reasonable share loading find support focus whether acceptable pattern detail nz loading std sp ta optimistic sp sparse accept artificial make test loading nz sp algorithm test criterion although suffer unbalanced std mainly lead maximal lead pca improvement tradeoff std sp focus bad rank patch vision pattern make comprehensive range gray patch dimension remove stability truncate energy loading variance explain evident simple provide section guarantee evaluation level absolute bound assume especially situation see upper cause dimension find specific well universal bind sparsity section comparable criterion plotted verify significant block orthogonality uniform unbalanced criterion get bad orthogonality orthogonality criterion perform generally good cost increase unstable sensitive consume increase sp en plot variance insensitive satisfactory across loading loading globally evident perfectly recover globally similar loading pca loading loading dataset motivate thousand gene determine show figure run involve mean report en fair sp grow much slowly already figure loading sparse pca call gp comparison exist conduct accord outperform trade sparsity explain orthogonality balance sensitive sp capable deal high sparse loading type work hard thresholding overall soft get good sp sparsity sparsity energy future effort objective en many prove bound achievable part deviation absolute value support overlap q soft thresholding operation combine z upper case pp element p k p p sx svd
five high follow one digit store visit calculate probability line store q k mean constant weight coefficient acceptable accuracy turn record path evenly mutually exclusive calculate validation initialize svm calculate eq classify number misclassification sample accuracy converge follow path mapping produce hybrid processor digital processors gate array parallel device time number one reliable node article use l visit global visit subset fx sign optimization convergence accuracy many genetic ga handwritten recognition particle optimization article graphic unit article improve algorithm large svm text bioinformatics slack determinant c algorithm optimization huge role simulate mix algorithm svm ability mean number sample svm means mean functional margin classify credible scale functional scaling b maximize slack basis
tensor index entire cross matlab standard three apply add operation multiply scalar operation use circular convolution circular convolution orient l di illustrate multiplication tensor size product develop fouri matlab notation tensor see proof transform rd fourier inverse rd scalars scalar refer form equip invertible module think generalization correspond scalar analog subspace rely property orient matrix set tensor size product framework multilinear form identity orient form module closed free multiply product illustrate generate linearly dimension htbp give transpose slice slice mathematically rigorous sum exception independence elsewhere product scalar algebraic definition see scalar orient article consider I independent zero union zero side orient orient come union derive section product write include help th use focus orient illustrative purpose position nonzero select matrix orient scalar shift original oriented matrix subspace contain permutation shift copy combination combination signal combination filter pattern argue matrix datum set adequate copy cluster believe suit many variation move subject camera shift usually collection application video pixel recognition provide useful framework capture natural circular family replace image step version algorithm algorithm arrange representation affinity n spectral clustering develop give analogous subspace clustering indicate ability traditional setting theorem strict must linearly merely special contain sum performance introduce notion cosine individually define describe algebraic proof find supplementary result speed sized synthetic datum lie mnist handwritten digit dataset synthetic image varied large database image total test display portion ssc entry penalize clear could perform tool face recognition know condition approximately subspace cluster basis face pose face via use successfully ssc segment person another training choice four nine ssc narrow succeed useful sign furthermore precede paragraph ssc parameter ssc unable face test original individual subject expression configuration w reduce pick first subject illustrate ssc ssc htbp experiment base image factor along normalize intensity lie take number pick person various pose different vs ssc ssc believe invariant preprocesse handwritten character digits instance digit curve show turn ssc competitive shift respect pixel good method preserve way aspect plan carry american sign video processor dataset take minute test effectively project fast ssc work state present computer engineering mathematics university combine recent field ssc come subspace respective subspace introduce multiplication ssc affinity build representation unlike ssc self flexibility ssc special array matrix whose element scalar take multiplication module retain property leverage vector preprocesse mnist handwritten digit database object assume embed option graph strong take approach come disjoint spectral theory final belong variation diverse array ssc employ resolve even reject outlier subspace two potentially exploit outside many reduction find approximate reference therein exploit multi present algebraic subspace column slice use tensor strategy incorporate cluster achieve less preprocessing necessary background summarize generative order call characterize performance add construct bad however generalization framework obvious key paper make present manuscript conduct face handwritten digits ssc
copula unique u dy eq margins cdf copula regression discrete explanatory copula copula univariate parametric th univariate margin margin response discuss simulate likelihood ce sequel since dependence longitudinal pmf response pmf rectangle dominant monte carlo reduction transform quasi efficient pass package advance probability high copula model discrete response simulate hereafter sl maximize univariate copula newton four place evaluation work poorly numerical log variable rectangle probability sl initially longitudinal reader copula hence numerical calculation simple approximate probability copula density dt dy univariate close q identity multidimensional copula asymptotic study asymptotic surrogate dt asymptotic sl estimate maximum sl copula exchangeable structure dimensional integral integral integral integral exchangeable pattern sl already take bernoulli binomial parametrization ease margin covariate distinct limit sl limit limit pass mle dimensional limit compute variety likelihood good limit comparison quickly vary effect finite truncation exceed probability ccccc limit copula margin bernoulli margin omit place sl lead regard surrogate asymptotic univariate latent correlation asymptotic individual count asymptotic count response univariate asymptotic decrease ccc ccc limit nb margins truncation point exceed sum dt discretization individual mean discrete decrease surrogate likelihood simple error se root divide margin place cccc se c dt error limit mle poisson truncation exceed small study estimation datum dependence concentrate model spatially spatial adjacency degree row label adjacent car copula construct inversion univariate variance car proper conditionally autoregressive margin consider nb parametrization binomial comprehensive chose covariate j count binary mean take poisson logit probit link poisson coordinate lattice size truly small per lattice car dt sl cc sl dt sl dt sd rmse sl dt sl sl bias sd rmse c sl dt sl sl dt sl individual dt decrease section spatially aggregated cancer second subsection incidence large section efficiency surrogate dt efficiency dt criteria aic bic aic include penalty aic nb spatial criteria dt sl parameter small aic fitting cancer incidence incidence period provide supplementary economic determined institute also interest cancer count assume poisson nb economic status fit ignore independence depict independence probability suggest likelihood reliable cc c size assume datum car copula perform via dt sl give estimate parameter along fit likelihood nb aic nb regression far car margin improvement one h cc nb sl dt sl dt sl estimate aic sl data copula margin horizontal line estimate profile profile significantly excess incidence status count response level gender observation whether different illustrate method analysis year well size discrete cancer count assume poisson nb preliminary ignore spatial dependence assume independence depict reveal individual probability suggest dt c cccc cccc regression dt est se est aic est est se dt est est se aic estimate se aic sl car perform dt sl parameters se sl obtain hessian usual theory bivariate margin test penalize copula nb margin copula poisson marginally nb cccc se se aic regression est est se value aic se value se aic standard se sl discretization car margin count surrogate application interesting glm reference dt poorly gender gender interaction gender significant conclusion latter analysis dt result true interaction good aic discussion paper study margin surrogate binomial car substantial estimate correlation parameter precision car discretization probability sl highly response although burden increase rapidly compute dt sl rectangle calculation replace simple however since cdf histogram reduce burden worth health interest field occurrence site occurrence threshold extreme area dt latent structure mat ern isotropic structure dt replace rectangle calculation hence discrete response structure pt sciences east uk pt distributional transform dt amongst computational multivariate normal copula model analysis normal univariate margin dt lead biased dimensional discrete simulated multidimensional randomized calculation maximum dimensional multivariate illustrate aggregated datum show via datum distributional generalize quantile rectangle spatially aggregate continuous straightforward distribution transform thorough categorical name hard limit marginal concerned generalize linear initially analysis correlate parametric family come continuous multivariate choice seem appropriate spatial flexible choice paper copula interval dependence modelling dependence copula study explore copulas experience discrete margin copula pmf copula cdf generally statement dimension negative pmf
section mixture experimentally acquire signal enable fouri conventional quantify chemical chemical place strong spin produce imagine bar bar field induce current frequency chemical mixture current call free local molecular permit conventional assume generate channel exactly perfectly model frequency intensity transform principle spike delta frequency relative magnitude spike adjust intensity relative due explicitly mixture conventional fourier general conventional procedure outline give intensity within frequency chemical group differ discrepancy know intensity calibration experiment measure slice imaging adjust theoretical intensity accordingly list experiment differ give composition mixture peak chemical section chemical show fourier transform dft fill take fouri transform peak dft width delta spike look like cauchy noise peak practice shift see figure together peak peak belong chemical index adjust conventional chemical q along peak chemical spectral conventional calculate concentration chemical credible ratio snr experiment definition eq peak snr peak simulation real conventional domain exponential fouri correction perform fourier spectrum national institute technology chemical frequency weight calibrate propose conventional ultimately wish induction two phase e signal white chemical specie mixture chemical intensity decay shift intensity frequency procedure section synthetic physical interpretation arise rf take sensitivity current stop period typical fourier delay correct fourier transformation raw signal delay relaxation homogeneity place effect amplitude ideal signal exponential perhaps range decay describe illustrate behaviour parameter generative model frequency intensity table panel dash line simulation multimodal phase explore surface frequency accurate optima gradient optimizer choose improve function every converge undesirable likewise popular metropolis hasting reject also optima frequency sensitive decay explore unimodal show performance compose component miss less initial decay information away thus increase surface natural decay decay decay find anneal simplex optimization reliable estimate chemical follow procedure uninformative analytically estimate nuisance parameter p variety operation channel chemical specie operation cholesky decomposition term motivate briefly selection provide review quadrature local shift taylor expansion gaussian however highly approximation multimodal break quadrature specification unclear test frequency metropolis hasting sample parameter quadrature time conventional mh explore multimodal surface become undesirable optima extremely unlikely find move gibbs involve parameter conditioning gibbs mix poorly gibb many focus frequencies monte carlo experimentally spectrum spectrum develop water similar coefficient water mcmc determine whether generally applicable quadrature focus snr spectrum description scheme construct domain modelling phase correct baseline correction leverage positivity requirement promise conventional discrete fourier dft promise statistical frequency quadrature develop heuristic quadrature robustness synthetic well transform focus aspect quadrature modelling develop novel chemical quantification quadrature solution thorough comparison conventional transform experimentally acquire behaviour section fourier ft decay signal automatically correction simulate signal intensity table presence use thus challenge ground truth synthetic resemble experimentally stress response ratio define uncertainty concentration gray bar credible interval standard datum show snr true concentration decrease snr figure snr around true identifiable std credible interval empirical percentage reconstruction marginal snr snr increase increasingly converge truth show snr bayesian credible true behaviour chemical credible bar confirm consistently accurate credible always conversely prediction ft approach vary interval broad bar ft unable concentration ft particularly sensitive become apparent study bayesian consistently accurate uncertainty investigate ft absence chemical chemical chemical hard snr whole ft red species concentrate specie peak concentration moreover duration acquisition frequency truncation peak make peak frequency window frequency nearby principled reduce difficulty absolute error spectrum indicate box peak peak truncation peak distinguished concentration bayesian show truncation concentration give ft systematically bias region bar experiment reliably distinguish concentration x axis experimentally ability level range increment create per snr ft figure expect mixture error bar horizontal sample ft consistently concentration bias ft scatter prediction always bind uncertainty conventional intensity peak peak almost quantify
classification carry linear range space input ik observation alone information side clear independently misclassification low regime feature condition overlap sufficient pass source upper misclassification extraction zero without unique misclassification diversity tend infinity signal condition information class distribution joint index ik ik ik ik ik ik ik j j ik ik otherwise ik condition classification expand fix obtain note particular ik specific value verify imply upper decay ik ji extract verify moreover achieve decay depend space span side intersect signal concatenation intersect span project intersect class interested reconstruct determine condition guarantee perfect generalize characterization also region simplify e r conditional moreover express sufficient guarantee regime approach condition upper observation alone e regime consider reconstruction directly perform diagonal property straightforward hand necessary decoder noisy noise vector stem coincide mean r side represent possible reliably r happen overall span project signal input dimension span moreover need span dimension alone sense projection capture characteristic share mean fig dot source possible conditional sufficient leveraging result provide regime correspond decoder operate decoder estimate associate classifier associate immediately gm immediately misclassification characterize noise transition draw condition distribution see show extract great large space component enough component information trivially consider side sufficient transition intersection region decoder reliably regime source obtain g signal correspond obtain formula inequality optimality input information derivation number model input condition satisfied phase input feature side reconstruction provide distribute union class characterization misclassification draw give misclassification expand report step consider expression integral expansion pair wise diversity leverage lemma asymptotic expansion bad indice diversity classification classifying whether associate reflect side entirely determined mean diversity possible classification leverage theorem determine misclassification hence accord condition ik ki ik misclassification classification ik ik j upper corollary number extract discrimination guarantee pair characterization noise expansion distribute nonzero appendix provide expansion case difference lie bind decay thus necessary proof expansion characterization source code counterpart entropy counterpart joint condition different immediately signal basis sparse innovation decoder result theorem gmm sufficient necessary class class condition briefly outline theorem transition associate derive draw use wiener filter bind carry leverage phase upper misclassification gaussian input accord class condition probability base estimator derive condition gaussian input meet condition scenario case size common innovation component mapped innovation additional reconstruct pattern report synthetic real cast aim theory approximate present diversity characterization behavior dimension respectively ik ik ik unit span class r ik ik ik ik ik different share sum space span signal different condition diversity yield simulation error fig misclassification well term transition diversity impact side representing value presence information obtain transition input linear span projection signal moreover increase feature increase analytically characterization base diversity align report need image correspond nm single snapshot accuracy reference image acquire hyperspectral image image perfectly measurement fig reconstruct hyperspectral information furthermore though still reconstruction correspondence block reconstruction six channel table notice fundamental classification art image extraction decoder carry classification also consider joint side correlation marginal likewise marginal misclassification construct asymptotic characterization quantity sharp sufficient misclassification necessary phase source numerical principled integrate reconstruction compressive hyperspectral imaging presence information direction one information source source possible model point appendix scenario design lead phase decoder side scenario side gain interest follow generalize translate signal modality recall misclassification give k k c lower simply follow derive induction upper differ multiplicative diversity integral also upper misclassification j ik ik noise diversity index expansion side computation diversity computation rank ik ik ik ik j characterization rank compact drop result ease notation straightforward space dimensional observe represent remain row tight proving hold consider span nan consider rank two subspace space conclude pick independent r n last observe force equal identity case leverage fact similar use r r finally lemma conclude pass generalization considerably complex misclassification transition occur semidefinite verify ik regardless assume ik separately follow r ik ik ik r simply leverage ik ik ik ik ik ik sufficient measurement ik ik r ik ik ik r ik r ik r r r ik ik r combine previous guarantee ik r ik r j ik ik r j ik ik r ik finally combine ik ik ik ik characterization expansion upper misclassification start expression leverage class otherwise expand expand index verify necessary verify define ik ik ik similar separately ik ik leverage error observation alone respectively ik ik ik ik ik ik ik ik r ik r ik r follow ik ik ik verify ik ik ik verify since eq previous expression condition ik ik ik ik state ik r ik necessary sufficient ik ik ik ik ik ik j order regime condition fact reconstruction observation consider bind incurred recover equivalently write regime approach zero introduce inversion moreover note immediately order leverage separately summarize state proof union necessary expectation hand simply complete positive semidefinite generalize positive probability consider apply recall rotation generalize substitute condition conclude necessity proof report ik p k c verify theorem ik prove leverage imply ik r respectively ik ik ik misclassification step proof misclassification gaussian class measurable converge ik separately state regime prove probability guarantee noise total argument bound c right use reflect gaussian provide input space denote ik ik ik j ik ik ik ik prove use write ik ik ik ik step proof theorem able follow cyclic independent identically division generalize interference interference interference multiple transform quadrature amplitude ratio identically input division additive cumulative tucker power profile quadrature shift compressive sensing matching pursuit side distribute reconstruction distribute pass isometry discriminant ratio code snapshot ci de da e mail fc university college email ac uk edu fundamental limit classification dimensional access linear interest feature information signal side assume signal interest draw correlate component specific gmm misclassification associate reconstruction associate interest transition quantity regime condition extract framework offer principled integrate high imaging art compressive image digital mmse misclassification concern extract salient signal method feature extraction dimensionality unsupervise dimensionality various dimensionality theoretic mutual information mutual information criterion linearly reduce lead art divergence express unified poisson channel enable signal reconstruction become acquisition paradigm offer simultaneously sense seeks extract low show reconstruct dimensional signal sparse projection greedy pursuit compressive paradigm processing compressive detection popular attempt aid dimensionality reduction prominent high include union wavelet tree manifold union lie collection leverage reconstruction conjunction root connect need reduce manifold linearly volume regularity decoder know beyond exhibit correlation concern high signal aspect attribute signal often live affine space side correlate connect feature dimensional relate compression classification distribute whereas side namely characterize two decoder side surprisingly rate compression associate joint compression discrete presence code compression encoding optimum compression distortion decoder contrast encoder suffer general information loss small case relate problem compressive sense side compressive sense compressive compressive entail desire signal leverage support decoder use decoder associate previous certain dynamic imaging minimization account distance image snapshot number image reliable recovery high mixed reconstruction compressive reconstruction necessary perfect innovation specific multi terminal spatially couple minimization matrix show ambient consider signal domain derive sufficient well algorithm multi compressive sensing description infer extract datum demonstrate various experimental datum signal unlike signal underlie represent adopt conjunction formalism represent counterpart signal signal lie affine translation rank prior approximate mild provide result dictionary classification video compression inversion noisy feature within moderate reliably extract relevance wireless communication metric gain measurement asymptotic carry decoder generalize joint condition low interest side also side use real resolution compressive hyperspectral imaging side traditional constitute work characterization compressive allow basis allow provide signal processing extract input classification presence remainder throughout section contain misclassification misclassification side notably sufficient classification proof notation case matrix low letter symbol identity drop dimension represent transpose rank respectively denote subspace expectation covariance denote symbol side feature associate desire projection extraction system decoder signal projection kernel side additive variance write eq draw distribution n rotation invariant special rotation entry fix cm cm draw east inner width mod sep west east inner pt height cm anchor west east west circle east sep height width anchor west east inner anchor west east near si anchor west index aim estimate input underlying purpose perfectly index component minimum probability classify classifier class reconstruction side objective decoder signal conditional mean observation minimize emphasize distinction previously multi task compressive recover compressive consider recovery addition objective jointly latter side information map aspect relate signal correlation adopt I ik k ik ik ik ik ik motivation fact accommodate joint pdf incorporate note c c ik p ik ip kp state reconstruction hyperspectral digit also common component generalize condition pair ik ik ik ik ik ik ik ik n ik ik ik ik factor subspace vector characterize share subspace dictionary signal respective perspective generalize previous scenario ik ik ik ik ik ik irrespective ik satisfy require prove contain leverage clear interaction connection ik ik obtain combination atom dictionary underlie phenomenon condition class hand ik ik statistically thus phenomena condition sensor describe innovation characterize formulation respect pick common exactly innovation basis range ik ik ik hand signal therefore express rank appear rank appear span gaussian span signal represent sum span input signal index correspond index span input side ik ik ik ik ik define represent span projection represent span projection draw index subspace span projection signal information component provide source
eq thus contradict complete ex approximation inequality desire contradiction ready put everything together complete proof corollary dropout effective setting sound theoretical need dropout exploration regularizer misclassification loss logistic minimize regularize remain regularization weight regularization last particularly surprising proxy contrast formalize compatible regularizer exhibit provably insight bias regularization provide result prominent dropout study inductive dropout dropout shape search strength co adaptation concern try function inductive dropout understand classifier remain popular dimensional third thorough understanding dropout inductive deep architecture preference learn decompose system node clean artificial classification dropout independently replace replace q note obviously dropout probability converge broad variety condition abstract dropout view stochastic view goal inductive variance case rather dropout may decompose eq negative marginal lead view style like rhs motivation proxy classifier assign weight preference strong get penalty show dropout inductive penalty surprising regularization monotonic absolute incur infinity prefer never reach extreme remain convex penalty matter detailed infinity constant convex shorthand drop play example remain dropout effect informally say dropout parameter dropout inductive bias source align inductive compare regularizer help illustrative handle use ensure control difference respective regularizers style useful tool study inductive work dropout family evaluate point case dropout regularizer regularizer discuss regularizers experimentally dropout view ensemble adapt require dropout variance study variant dropout compatible source preserved complement focus original dropout characterize unique minimizer regularizer separate dropout provide section feature optimizer drop keep paper introduction independently random analysis easier randomly one loss consequence optimizer well follow either attention feature r tie assume contradiction perfect tie xy decrease loss assumption minimizer remain therefore bind p r pr x e strictly minimum p summary vector weight dropout keep regularization additive dropout dropout dropout optimize vary criterion generalization regularization present specialize decrease exponentially prediction dropout second inaccurate experimentally evaluate dropout accord suggest penalty increase linearly proposition e w penalty new next show dropout regularizer regularization dropout regularization penalty remain single vector infinity figure dropout index substitution complete line use I function neither range expectation different character go remain dropout instead dropout vector initial remain go small consequence already penalty theorem dropout penalty zero surprisingly hold regularization proposition write q make penalty product prediction dropout identical us dropout monotonic fact dropout fix arbitrary locally behavior prediction like prove support dropout probability sign penalty infinity straightforwardly infinity penalty immediately lead nonzero support infinity together approximated dependence allow show whereas infinity sign show remain compatible alignment go infinity together complete multiple go infinity dropout discussion theorem proposition suggest dropout regularizer indicate grow suggest rare less weight frequent rare feature perceptron base approximation empirical dropout discriminative use suggest limit sign hand proposition indicate encourage help share weight correlate turn dropout pair strongly qualitatively prefer regularizer w separate dropout exist c family separate use consider weight classify perfectly regularize optimize criterion distribution vector classify encourage drop weight enough minimizer dropout criterion expect plot left dropout regularizer low dropout right green region mark compatible regularization recall contrast criterion p consider pressure pressure prevent grow correctly hand nearly wrong mean light proposition give advantage regularization multiple simultaneously go penalty remain bound weight increase base regularizers go infinity individual infinity characterize cause remain put sign suggest regularizer grow linearly dropout provide dropout work definition regularization regularizer regularization complicated dropout pair dropout particular dropout generalization natural well devoted prove separate pair dropout separate strong separation find criterion wish separation range dropout amenable exact difficulty separation dropout sign moderate analysis plot illustration dropout regularizer preference case single multi layer neural network deal open separation gain regularizer setting first show assume without generality multiply multiply change q continue sign numerator numerator since numerator locally note may slight modification nonzero sign negative move fix arbitrary support dropout sign regularization go go depend go discrete limit w w analyze inside third expectation I go infinity also goes drop drop non drop precise notation proof scale obtain regularize criterion simplify expression lemma repeatedly use closed separate start w decrease derivative increase proof continuous derivative term negative term positive complete correctly much assume contraction rhs negative contradiction suffice partial positive go infinity evaluate eq decrease increase negative desired show classify throughout proof subsection dropout criterion component independent bernoulli simplify minimize equation optimizer correctly one prove prove partial drop matter open negative proof multiply whenever proof misclassifie complete proof ex keep might classify prove misclassifie scale let simply implicitly minimize derivative eq rhs rhs show large contrary give bind prove lemma convexity either show complete scale criterion independent scale objective partial correctly suffice decrease even recall equation go infinity ex show term dominate assume fact give ex ready combining imply eq imply ready ex e complete piece succeed lemma classify complete ex perfect p minimize frequent partial majority vote notice existence node fill inner negative infinity positive hand increase infinity partial magnitude magnitude reach meet hold complete optimize dropout make difficult make jensen dropout convexity form last drop whenever become optimize fail want contradiction begin consequence assume contrary jensen inner w w w optimality w bn drop drop plus give bind bn n w dominate majority vote lemma ex loose large conjecture criterion fail produce bayes theorem minimizer let minimizer symmetry recalling equivalent value dominate weight circle blue red right
relaxed minimization nuclear sparsity typically incorporate norm trade alternate implement directional guarantee retrieval quasi polynomial sparse approach rely linearization constraint induce problem structure linearization sign original signal propose sparse constraint cone programming formulation extend noisy algorithmic benefit noiseless presence noise note system case term one phase clarity detail sect extend sect effect result discuss signal complex sect deal invariance measurement circular shift test matrix write bold letter bold letter matrix part transpose respectively j retrieval symmetry solution obtain interested solution condition method let ij j symmetric denote component phase rewrite jk nx objective reformulate nonlinear pose one recover relax estimate nonetheless positive group overlap aim surrogate diagonal cone j jj jj sign ji turn approach first sparse solution unique n solution least sparse contradict solution jj jj regard coherence coherence n recovery proof give theorem solution contradict imply solution multiplication unit thus z infer property invariance prove real linearization map x equal either nx I map complex jj jj jj jj square act positive number arbitrarily fix equation definition lemma jk jk jj ny jx problem rewrite nonlinear optimization relax substitute yield modulus show relaxation proxy sparse contradict unless jj therefore relaxation j r unique hold define index due introduce eq unique exist contradict definition imply perturb equation ie goal become eq sect convex form must omit except conclude detailed variant perturb noise via relaxation n j jj real addition path theorem adapt complex minimizer statement rewrite must satisfy replace due j j bound bind introduce notation relation I r j inequality eq let together constraint rewrite since positivity stability ensure consider dedicated sect particular allow order account appendix real typical subsection technique deal invariant circular shift problematic version circular shift linearize combine different pattern invariant version vector effective estimation retain define singleton make stand circular maximal shift linearize constraint ensure valid solving assume formulation convex shift shift additionally define x maximal apply name sum first first small shift magnitude finally give invariance circular shift invariance proposition x I statement invariant combination shift directly useful suppose check inequality involve square magnitude advantage shift reflect shift reflect feasible solution feasible kk show notation obtain I apply require map magnitude large compute satisfy implement shift unique correspond magnitude transform square issue shift determine jj jj therefore know dedicated measurement fouri restrict measurement shift fix zero j remove corresponding program lead final formulation modification approach greedy optimization implementation two method slight complex particular enhance support add normalize detect small estimate complex percentage carlo experiment distribution unit random nonzero
mix blockmodel allow membership flexibility formation analyst compare attribute post manner introduce blockmodel actor social actor form link whereby occur throughout aspect political method examine summary statistic occur frequently chance treat underlie structure example triangle could reasonably occur evidence whereby link actor link actor two represent blockmodel sbm membership two actor connectivity map actor onto form actor cluster extend actor determine sbm latent network difference model actor actor network conversely sbm whereby connect weakly mixed overlap actor membership actor interact sbm attribute explain occur school student likely gender play important formation belief reflect appear literature share interest status collective activity clustering additional hoc manner extend sbm incorporate link actor specific level covariate include gender age relate actor physical actor specification beyond link explicitly incorporate actor mix expert blockmodel terminology framework model covariate information adapt terminology incorporate membership model model thought actor characteristic network formation converse tie actor rest structure briefly review detail provide relational actor link present pair actor interaction actor represent link think share symmetric undirected otherwise say direct interaction refer school student interaction consider entry sbm assume membership group actor model interaction actor indicator follow multinomial prior ensure frequentist framework sbm variational collapse fully sbm substantial multiple group interact framework actor assign individual membership indicator actor interaction actor interaction model manner parameter distinguish interaction quite group exclude hyperparameter ensure mix beta treat nuisance parameter far covariate terminology literature refer covariate paper restrict actor incorporate individual mix hyper treat membership set beta multinomial decompose py z n np np np section sbm htbp b fashion employ bayes approximation previously useful membership setting intensity approximate concavity kullback true approximate distribution restrict factorize kullback multinomial beta introduce update extension ij gp ip py ij form make newton hessian follow np nr np nr np np experimental newton vary approach dirichlet wish estimate probability rather covariate prior probability intuitively think covariate parameter serve newton another encounter use separability model whereby pattern method suggest regression model prove force office entirely five actor location assumption hoc approximation making criterion criterion difficulty determining occur perform fold roughly drop fold straightforward value miss simply hold likelihood high uncertainty assess check total detail notable lack strong couple behave interesting environment formation link strong focus actor available previously incorporate conduct former office year impact position include covariate membership find evidence location affect effect network figure actor still form partly gender clear age facilitate covariate attribute rank low indicate associate gender school university office exclude fit value validate group somewhat satisfactory b b figure actor half actor display membership represented label accordingly overall weighted correspond indicate interaction include large font font half expect enyi whereby actor fit interaction occur check profile membership actor actor actor lowest seven actor involve respective exhibit membership actor belong actor exhibit three actor exception actor indicate full participant group membership actor appear highly social figure plot chart represent mixed chart mixed actor statistic popularity actor prominent green represent membership community red dark blue occur actor actor split actor purpose impact covariate facilitate obtain optimal obtain parameter inverse approximate fact firstly whereby twice create difficulty behaviour bootstrapping generate fit use record reliable outline worth reflect degree selecting appear impact bootstrap replication agreement estimate mainly significance covariate whereas status actor appear base quantile box parameter covariate perhaps obvious behaviour worth indicate group poorly explain four zero covariate term partly explain membership influential status year law school correlate parameter reflect inherently evenly nature covariate strongly positively skewed tendency retain term group consist establish actor actor actor strong group standardize mean deviation occur year age despite less age positively skew actor assign prior high explain actor probability within actor significant significant note upper parameter close particularly actor uncertainty would long exception school membership comparison seem quite actor appear impact range estimate quantile highlighted intercept status htbp dash occur examine fit predict link fit outline two observe link predict hold operate roc curve show appear well auc almost htb b checking
item well statistical pm option e difference exist carry post hoc pair result pm significantly improve half pm improvement wide item compare always statistical analysis right method seed select difference clique often sub clique seed high effectiveness attribute select seed criterion strict item smc semi superior heuristic list difference former item expand latter explain superiority come inference former chance weight heuristic treat equally c c run amazon expand experiment expand stopped ignore expand hence conduct intel ghz core g ram virtue ideal parallelism item large machine efficiently overlap call item present item devote efficiently quality seed range network seed method resort semi result advantage statistical demonstrate item run unweighted direct prediction also make item would like anonymous comment support grant china grant ac advanced chinese sciences china university chinese china expand scheme network seed non principled expand lead diverse propose new transform corpus treat corpus effective seed expand significantly improve performance complexity scalable large system elementary part link many network tend distinct subgraph community module occur computer reality active group friend algorithm discover belong community scheme step seed seed form seed select expand quality seed important detection lee clique seed select expand process method detect community improve give method em kind often seed select lead unstable due seed scalability algorithm may network node due pruning remove seed removal rare sequentially rank improper community seed select seed community easy end aggregate select seed rank network high rank seed minor diversity seed guarantee rank root rank drawback three kind mention globally expand decide community link community decide belong expand fast heuristic lack principle use lee fitness function appropriate datum drawback independently without highly community share post merge merge community merge difficult expand replace global optimization edge naturally virtue wide applicability propose theory em discover corpus edge community assign network belong extract effective drawback traditional expand treat seed set supervised edge item improve edge organization section skeleton item expand section experiment suggestion provide reproduce result publish notation g double subscript use prefer subscript edge clearly skeleton help reader rapidly ht cm terminology definition comprise network figure number single subscript double subscript nod v iv e e ice ice ice ice exploit mining community network propose key origin matrix motivate commonly similarity index gmm please table simply display follow list feature discard discard resemble preprocesse mining remove discard operation easy node similar document c p p p display item classify researcher semi supervise exploit na item seed use expand edge community expectation unlabele label community unlabele refine nb classifier item expand stop predefine match score middle color seed seed thin edge add two seed avoid drawback specificity local give information information compare filter select seed index filter second seed select fed representative seed candidate view lm figure bit select completely ignore specificity make measure link suitable seed specificity strong cause common edge suitable seed formal definition specificity common concept seed strength specificity get technology evaluate technology convert bit denote detail due information please evaluate number intensity specificity similarity neighbor specificity specificity mainly contribute locality comparing make edge appear seed split apart adjacent slow convergence process em filter select detail seed efficient selection method effectively representative term information lose merge virtual document document virtual apart try release seed create edge candidate e hence q apply make select candidate check include clearly scale th e network community narrow bold color final seed select get seed supervise add edge thin color edge color expand expand community lm select seed take clique classify community exploit topological potential include potential evaluate lastly community community neither adjacent include status generality include potential k ode j I figure triangle triangle give zero impose rigorous requirement order community treat I ensure ensure classifier belonging maximization divide kullback leibler expand resemble distance follow edge maximization edge evaluate weight occur term occur drawback color bias high specificity value respectively consist synthetic subsection subsection brief compare commonly algorithm clear benchmark control size govern distribution little community respectively network vary mix boundary edge community membership overlap fraction value generate figure average measure ground truth community normalize mutual use value facebook include year truth facebook lee detect value relate community accuracy indicate find well ht devote overlap prior 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nuisance construct value identity inference quantity alone let introduce predictive random valid predict valid center predict suppose iid unknown moment standardize joint minimal like latter satisfie nuisance eliminate ignore hand distribution namely central student centrality parameter student I q simple think box develop promise j generalize inferential share asymptotic base method study I differ truly property really school carry well base scientific effort focus develop fundamental building additional far read current still computation application believe development author thank associate suggestion research support national foundation dms dms statistical inference perhaps extension meet requirement free inferential I show promising generate valid probabilistic brief introduction principle principle I discuss principle I condition illustrate bivariate bayes belief inferential principle experience observe knowledge population essential part observable parameter mathematically distribution convert assertion concern plausibility plausibility meaningful probabilistic interpretation work make effort argument use change nothing well create confidence central frequentist great develop valuable live ignore yet fisher effort generalize inference reference argue mention free paradigm read reasoning seek free mainly principle namely principle upon principle basic marginal inference realistic knowledge belief plausibility stand belief represent support assertion true support assertion assertion plausibility function support also probability discussion total formally evidence available either assertion discriminate immediately satisfactory represent informative relative specify nevertheless bayesian probability proper improper constructing method close informative limitation latter discuss model take look precisely simple observe continue regard distribution distribution regard reasoning change status represent conventional goal view agree replace degenerate conditional must beyond inference formulation I framework know reach statement hypothesis interest summary observe support constraint general motivate principle next question measure bayesian approach meaningful scientific arise first encode informative prior sample view free probabilistic question difficulty fact basically bayes apply theory opinion correspond scale belief opinion essential success scientific belief desirable properly belief value need expand auxiliary design meet definition goal efficiency consideration though I explain prediction discuss principle quantity observable precise carry operation well valid random nested subset include serve satisfie simple example valid realization I make quantity eq valid predictive combine prediction belief plausibility need empty I set I valid stochastically assertion true meaningful scale plausibility belief value consequence test frequentist development efficiency principle validity efficient information marginal example difference I framework application future pose one marginalization handle unit size function auxiliary variable observe unobserved association build chi auxiliary freedom association argument attractive identity conditioning amount valid variable association rewrite term relate auxiliary normal partial equation give complement I technical produce exact multidimensional
mx mx mx dx mx mx dx v provide eq n next density eq xu convolution via function mx x xx z z aa z aa analogously turn right k k assume e turn p set p borel r complete evy triplet q eq q lebesgue obvious shown write hence positive constant q thm thm thm example thm remark cm evy embed estimator turn convergence consider problem evy support dynamic process laboratory method rf evy transform laplace change call embed solved formulate embed give stop integrable martingale se draw see item fact solution se currently statistical pose study call statistical consistently estimate closely multiplicative construct norm rate asymptotic normality brownian l evy turn use combination laplace construct rate basically coincide statistical already poisson let motion deconvolution variable near obtain example probability mainly independent probability random variable due problem inversion empirical fail operator support sequence tend convergence henceforth take simple principle decay fast throughout notation constant multiple arrive effort prove strong turn example density asymptotic let look therefore polynomially fast follow class assume choose eq closely relate context deconvolution n nn l rate factor infimum take sample q n h evy process follow independent evy situation since evy long evy triplet domain h follow let moreover bound eq real x hand straightforwardly derive replace empirical case need inverse transform define find evy two show logarithmic change brownian proposition moreover recover logarithmic fulfil exactly logarithmic theorem sense basically coincide logarithmic rate stress class know family beta rate mixture mean model mixture theory dependent heavy skewed coincide variable example density density statistical literature parametric treat nuisance zhang estimating mixing well knowledge first inference density generality minimax rate theorem variance mean evy change evy change evy suppose evy arrive follow exponent evy want time l necessarily property generalize evy frequency consider chen let evy high many therein al mixture normal distribution univariate member inverse function density density simulate bandwidth density estimate p tx procedure turn decomposition e km cauchy quite reduce compute integral nh realization xx run depict observe performance similar rate
precisely system throughout process generate instant subspace algebra take space mechanism unnecessary technical absolutely lebesgue theorem density surely constitute assume nevertheless state validity partially process define family partially markov markov state member family canonical constitute gaussian tx functional describe evolution state markov get equation z constitute process first hmm equation reduce arguably observable system encounter signal important application specific boundedness sequence reference involve sufficiently bound well continue observation lipschitz exist z lipschitz xx continue almost say modify replace lipschitz everything former everything consideration proceeding estimation later natural algebra variable mean square error ideally stochastic estimator interest one measurable nonlinear filtering frequently process concept stochastically hide system formulate extensively discrete filtering discover various markov purpose intuitive include direct far consider base responsible coupling measurable constitute tool measuring contain base serve channel result arise first ask assign another transformation second latter way behave constitute stochastic possible derive question key assertion induce process absolutely question conditional bayes density consider integrable base imply existence density resp support contain stochastic process ty actually coincide restriction denote collection rigorously absolutely exist characterize demand demand show derivative everywhere apply process comprise partially observe respect consider hide usual surely constitute alternative constitute constitute gaussian white noise identity top page additionally later first measure existence define measurable absolutely tx replace event restriction alternative belong fine since interested measure augment arbitrarily valid density equivalently statement demand density case adapt alternatively take iv use notion conditionally probability thus notion convergence consistently suit metric another limit borel converge equivalently random induced definition present rather present appropriately specialized later triplet space constitute nx constitute whose equivalently concept instance reader refer article let arbitrary limit possibly define base induce measure hold everywhere generic nonlinear asymptotically would like would yield practically present establish filter strong rhs constitute filter constitute mmse process side recursive realization focus rhs constitute process observation simplicity replace true employ employ thing argument classic filter way attention choice approximation coupling state filter stochastically make observation restrict filter change provide discover realization see treatment rhs resolution observe follow law course question operator member sense operator converge resolution appropriate formulate constitute definition ii constitute atomic equivalently pick either follow cx tx tt cn cn note begin filter operator fully devote facilitate subsection part useful scalar coincide absolute version consider norm present sake norm either constitute definition complete matrix multiplying yield temporal iterate proceed bind result preliminary towards state ii trivial member frobenius norm c norm eq trivially member must use product resp depend certainly correspond right yield may constitute significant throughout analysis simply constant affect constitute throughout effort reasonable limit relate member functional family euclidean proof square positive regard write next numerator express c tx constitutes remove equivalently therefore measure paper consider choose constitute conditionally admit pp existence supremum explicitly assumption rhs rhs however naive equivalently hold constitute mean matrix element make define complete leverage convergence connect weak stochastic tt member cx tx tt tt hypothese continuity set true x tt dominate would desire course member define bound lemma conditional must interest definition dominate prove result rather since constitute sufficient resemble situation equal unity strong condition next contrary convergence nice intuition respective sufficiently state intuitive closeness example base filter feed resolution parameter uniquely approximation additionally proof elementary omit natural circumstance ready establish subsection tx exist measurable subset measure eq concentrate determinant rhs rhs expression attention rhs know yield numerator bound member line rhs arrive equivalently eq rhs eq rhs expectation adapt measure statistically last uniform rhs ensure bound immediately exist subset either upper bounding comprise obviously define get either give since member also imply turn existence limit true adapt recall denominator q adapt least base trivially lemma put since nonzero tend complete condition measure filter compactly time occur nearly essentially framework nonlinear recursive grid perform appendix notational conditional existence
cone need separability programming et nmf instead scale algorithm e grant knowledge view preference go nmf algorithm reformulate detect exist constraint lp large data modification require extreme proximal algorithm lp addition entirely regime factorization extreme cause issue dd find elementary extreme cone dd computational advance help address issue dd organization brief nmf perspective explain propose algorithm reformulate lp paper conclude capital letter letter matlab transform argument non nmf nonnegative approximate solve optimization factorization negative column algebraic factorization refer geometrically generate depicted mind follow definition extreme convex combination exact generate unity entire vector index unknown equivalently constitute therefore efficiently context nmf lp belong incremental descent prominent exist large drastically constraint lp optimization multiplier dual I separability exist feasible selection dual imply program find dual use factorization identify factorization low zero let cost belong determine lowest readily obtain proximal algorithm pre column sure normalization rewrite lp set column stop update project constraint positive element switch experiment gb ram matlab version generate instance extreme create element column generate combination select carry nmf analyze topic allocate begin set effectiveness achieve list regime namely basically gray randomly image
sentence et achieve encourage likewise address memory sentence phrase base simply source whose model feedforward translation need close work show vocabulary almost simple lstm extent word sentence conclude great dependency much unable train rnn translation sentence although ability lstm translate initially long memory researcher performance yet dataset little difficulty work translation accuracy well xu google team google le deep excellent performance well sequence end make long short deep lstm english produce lstm achieve entire lstm additionally lstm difficulty use lstm aforementioned score task also sensible phrase word passive source sentence lstm dependency source optimization easier excellent powerful parallel surprising power ability sort neural conventional learn supervise backpropagation set result human solve rapidly backpropagation apply target dimensionality significant limitation whose length recognition likewise see word sequence therefore clear useful pose challenge input straightforward application long lstm sequence read fix vector extract output lstm essentially language lstm lag output neural map produce phrase system novel attention mechanism network elegant translation assume monotonic alignment ensemble far direct large lstm vocabulary score penalize translation cover relatively vocabulary architecture room outperform publicly baseline improves publish surprisingly recent experience well order sentence many dependency make sgd sentence source sentence lstm sentence variable tend sentence encourage lstm representation mean different qualitative aware word fairly active passive rnn feedforward neural standard output iterate h sequence input ahead length strategy target al rnn dependency short learn long dependency goal output sequence lstm obtain last lstm lstm lm whose initial softmax word vocabulary sentence end symbol outline compute representation actual use different output negligible lstm language second outperform choose lstm implement size lstm well lstm score lstm lstm solving discover learn much well sentence lstm drop score believe cause introduction short normally target sentence result minimal time lag target language unchanged source close target language minimal lag backpropagation communication overall early target confident prediction sentence sentence train see input lstm deep embedding input vocabulary thus deep significantly outperform reduce nearly state naive word result lstm pure recurrent lstm initialize lstm momentum learning learn half epoch batch vanish scaling sentence length sentence minibatch sentence short long sentence minibatch sentence minibatch speedup implementation lstm slow purpose different gpu activation gpu soon remain softmax gpu responsible multiplying english minibatch training take ten score quality score get report result initializations lstm pure phrase mt vocabulary list system lstm lstm ensemble ensemble et baseline lstm baseline well discover lstm show present du du il une des ann pour les collect es les du les du es est une des les une les dans air les un cr la les es
utilize cyclic add nucleotide template nucleotide dna signal know nucleotide nucleotide complementary template synthesis reaction subsection nucleotide nucleotide read dna template nucleotide depend nucleotide reaction condition determine incorporation study fix actually length cycle previous length natural previous previous yield investigate subsection old new model incomplete nucleotide incorporation eqs incomplete nucleotide incorporation eqs special complete nucleotide incorporation condition mention technology add four pre determine unnecessary specification name four kind permutation previously cycles cycle successive cycle nucleotide nucleotide template dna template dna nucleotide nucleotide incorporate certain kind detect incorporate detect nucleotide incorporate nucleotide ideally error nucleotide complementary template nucleotide add nucleotide complementary template possibility kind deal follow variation sequence sequencing incorporation sequencing sequence identical template sequencing individual template lose gradually decay sequence reaction synthesis incorporation nucleotide base incorporate basis nucleotide cycle template dna nucleotide cycle incorporate template nucleotide context flow order uniquely signal nucleotide incorporation strength signal incorporate nucleotide cycle use indicate cycle flow cycle example number sequence cycle first cycle utilize dna sequencing technology sequencing sequence instead template avoid advantage sequence reaction control adjust nucleotide incorporation sequence reaction incorporation basis nucleotide cycle region sequence technology try cycle incorporate incorporation rate utilize nucleotide nucleotide incorporate nucleotide incorporation delay next cycle nucleotide incorporate reaction flow incomplete nucleotide incorporation template signal combinatorial question complicate complete nucleotide incorporation tool solve solution complete incorporation incomplete incorporation united use template dna basis position sequence nucleotide flow nucleotide nucleotide incorporation flow fix length cycle incomplete nucleotide incorporation determine nucleotide incorporation generating length flow cycle flow cycle assumption nucleotide incorporation first value table arrange length cycle show nucleotide incorporation readily evident dp dl l cccc cycle base interesting cycle look table fix transform normalization factor find small negligible become big somewhat flow cycle sequence flow cycle create part cycle cycle long cycle flow cycle nucleotide sample example ba get get hence second length sequence nucleotide incorporation cycle finite subsection analytical enumeration previously check however analytical c program mention avoid unnecessary specification name four kind target sequence number avoid ambiguity add instead development nucleotide incorporation type delay number nucleotide cycle complementary template nucleotide current cycle incorporation special q situation extract coefficient power place detailed deferred flow permutation nucleotide nucleotide incorporation nucleotide cycle cycle nucleotide nucleotide incorporation conditional factor incomplete nucleotide incorporation flow cycle last nucleotide nucleotide cycle ix ix ip ix nucleotide incorporation recursive analytically transform exact cycle sequence incorporate nucleotide flow cycle nucleotide incorporation start nucleotide incorporation later nonzero evident probability flow cycle irrespective nucleotide part flow cycle must cycle nucleotide flow cycle sequence happen part reason first list table note factor table complete nucleotide incorporation factor list sum equation bivariate elementary flow similar generating function sequence fix distribution flow cycle nucleotide composition mean flow cycle close eq denominator factor small module series expansion formula close equal base difference exact nucleotide value flow approximate exact cc c exact exact calculate eqs nucleotide incorporation together normal variance nucleotide composition probability exact variance eqs normal slightly thick approximate number flow central cycle st depends last incorporate sum incomplete incorporation may incorporate give nucleotide flow incomplete incorporation term incomplete nucleotide incorporation nucleotide incorporation incorporation flow incorporation still nucleotide flow cycle fp nucleotide flow cycle nucleotide incorporate flow cycle irrespective nucleotide nucleotide cycle next nucleotide cycle incorporate cycle next incorporated cycle reason incorporation reduce factor eq
overlap bic tend group proper experiment report cccc group run improve hyperparameter set default sensitive hyperparameter show agreement actual point figure well find centre high dataset final respect probably separate distant galaxy lie understanding nevertheless affect configuration pruning also extend split latter classify observation position care leave extension interest know true acknowledgement author would thank comment early complete school sciences sc insight centre foundation grant research foundation grant ip proposition corollary statistical sciences centre school sciences college abstract criterion base clustering automatically effectively thereby include allocation selection practical use prior exact avoid one cluster observation algorithmic mixture model greedy search cluster crucial number reversible mixture alternative author efficient carry label similar approach context model rely throughout observed allocate categorical parameter approximate evidence maximize variant introduce base integrate complete differ complete use bic complete cluster framework assess bic latter include entropy group well prefer configuration good distributional solution appear indeed usually refer homogeneous point penalization discriminate overlap group become gaussian exact common thank conjugate distribution allow apart allocation block framework finite exact formula heuristic extend capable f return algorithm little answer straightforwardly allocation univariate situation explore section form framework routine drawback hyperparameter paper end remark ease framework univariate namely multivariate although applicability iid b focus context allocation call log define involve mixture dirichlet allocation iid variable every set hyperparameter model differ symmetric label wishart scale dimension modelling outline general exception assume shape crucial return complete already set collapse distribution group obtain factorization evidence take parameter formula final term hand centre determinant bic estimate hence every depend partition regard naturally take modelling among group fulfil exact yield advantage allocation indeed convenience depend suppose arise hyperparameter infer optimization simply restrict hypothesis value course overcome specify asymmetric similarly distributional possibility brevity leave task work result select cluster solution parametric include indeed paper serve mainly direct scale main global optimum second concern hyperparameter issue routine complete combinatorial greedy conditional mode mention rely configuration greedy employ block work number infer idea routine configuration informative clustering start random update complete loop yield stop configuration index change let number dramatically search usually reach convergence merge I completely end loop make soon group collapse get poor sensitivity rather assume split able mainly increase time obvious optima able leave must tackle issue author propose routine start configuration final merge avoid local optima merge current frequent finite mixture regard quantification wise state need less iteration merging cost interesting greedy routine objective algorithm calculate computational introduce greedy rather drawback local optima observation whereas change yield greedy gets indeed easily happen enough allow exploration therefore propose large leave well wide combine update multiple instead allocation belong nearest evaluation objective actually realization beta binomial trial group allocate need although objective trick reduce algorithm pool pick observation allocate increasingly number point combine thus dissimilarity storage proportional distance usually fine job proper transformation couple near hyperparameter version require simplify objective affect hyperparameter choose possess one interested specify informative standard extend mind limit interpret hyperparameter information symmetric make remove equal concern proportion realization value jeffreys whereas well chapter essentially small rise default hyperparameter choose observe datum constrain suppose narrow elliptical high default choice propose account many accord range choose diagonal diagonal describe cluster position yield combination shape identity omit univariate default wide case note concern parameter default hundred observation careful default however concentrated overlap may distribution routine update propose describe several clarity solution obtain greedy algorithm maximization model choose denote configuration expectation comparison create approach differently model procedure impose covariance simple maximization clear describe expectation bic suppose return contain thus compare maximization completely quantity proportional maximize obtain consequence update maximize one mean must suitable point exact version specify informative solution three mainly hyperparameter solution intend observation bivariate figure solution htbp right configuration correspond bic stress really agree configuration simulate represent value correspond show clear cccc
two treat extent relation continuous follow reduce write singular value decomposition closely principal approximate marker fix eq alternating step svd pre regression interpolation individual em summary project onto usually point marker vector scale find predict project q solve marker divide marker square length value label calculated average result like character individual marker generalize bilinear expect possible centre straight line mean prediction e project marker marker see obtain marker marker label reference practical view interesting point representation predict point axis explain observed trait separate maximize logistic column response column separate part row individual score package representation contain nominal category individual last use probability category present trait trait make identifiable log odd relative category b odd odd would interpretable probability category logistic relate relate component trait predict long straight make surface category logistic describe binary contain measure ordinal order column indicator categorical indicator expect individual value ij cumulative probability vector trait item variable define category intercept set response restriction probability obtain high dimension variable formally boundary scatter diagram establish search homogeneous characteristic multidimensional help search responsible difference individual htb logit binary categories eq jk contain odd ordinal cumulative share geometry geometry calculation category subtract cumulative htb equivalent fit proportional odd regressor response surface surface long particular category lie straight category item straight line project direction would segment except many segment separate contiguous equal represent point contiguous divide span region predict particular category boundary intersection contiguous must q cumulative probability hold calculate several category never rest category separate point calculate htb pt existence contiguous may example simple deduce combination equation intersection category solve solve equation j root negative intersect calculate transformation would calculate axis parallel prediction category order hide order category category high intersection back step start could equation calculate predict sequence precision step obtain previous search one regression ordinal regression individual estimate response paper regression interpolation change procedure quadrature integral procedure individual choose category individual part maximize maximize perform ordinal regression column procedure well quasi span explanatory see classify search responsible probable existence logistic see procedure remove problem choose simple maximize change affected way variable likelihood part could penalization posteriori distributional marginal gauss quadrature quadrature represent quadrature marginal posteriori score q quadrature individual dimension span among reference year organization department office european survey characteristic european members usa national institute focus effort carry availability resource science technology carry european union office european technology production resource technology survey try level group carry main international group quantify focus degree public private frame operation provide institute university thesis university database comprise belong international education award devote advanced study design region equal probability selection equal random select region assign rest measure module website http www process answer study attention module find aspect job code total algorithm two indicator loading table classification high challenge separation problem logistic interpretation high job security work almost condition job job status factor htb security job challenge degree social status benefit challenge independence status observe figure htb variable job security hide partially appear organization public job majority htb ordinal point representation angle benefit job security present first away aspect behavior slope although happen represent information challenge job security behavior category challenge aspect htb finally mind detail outside category possible challenge
choose computationally efficient sample query strong set strong possibly adaptively unbounded oracle simple differential theorem well differentially private threshold minimal notion privacy line connect technology bound private interactive analysis task appear hardness privacy digital introduce use code certain setting nearly establish introduction extensive code interactive give influential work interactive code name hardness false discovery give construction suboptimal code give code interactive guarantee intuitive first algorithm answer arbitrary adaptively query answer privacy black differentially answer many adaptively rich computationally inefficient accurate exponentially construct interactive code main ingredient code hardness non interactive code section read motivate definition interactive helpful review motivation code code movie piece company may copy company copy remove code ensure combine user create say trace construct key drawback single user prevent identify interactive company content copy distribute episode tv internet episode show combine stream stream company stream soon company identify continue copy another code consistency constraint say remove code robust robustness ready code game may user output vector empty user let c want consistent succeed recover convenience round execution notation notation user formally notation require user interactive say interactive robust error probability depend constraint call interactive code interactive adversary inconsistent may seem recover notice mean interactive establish interactive code every interactive failure bit traditional regime code code match code failure large logarithmic construction robustness give interactive interactive setting weak robustness completeness require high rather version set application false interactive code variant modify distribution support function pc nc pi ht issue receive j ii code figure addition precise set parameter help convenience intuitively call user correlation answer measure choice ever exceed mean answer thing never answer unknown answer must closely interactive random fix randomness unknown analogy one would round prevent like tail completeness gives specifically imply show interactive see answer score set ensure answer fraction round say force inconsistent eventually reduce answer prove equation simplify analysis issue fix force tailor concentration take equation choice follow suffice ensure order fourier always arbitrary fourier also density proportional handle round adversary round happen inconsistent answer normal round inconsistent answer round since round round mean small normal imply round conversely round concentration concentration user essentially form prevent instead proof verify desire bounding q ia add take ij answer draw likewise interestingly interactive still fail identify indicator event take adversary result code identify lower bind tail consistent establish good score adversary must constraint bias fourier relate round adversary adversary firstly lemma pp product fourier analysis bias expansion give expression effectively integrate calculus n nj jx accuracy n interested say oracle query circuit evaluate circuit attack triple take security randomize key message output ct message ct roughly security even access message security polynomial start simplify definition security need adversary pair ix interactive code robust fraction let let ct user attack comment structure attack oracle must true eq oracle oracle effectively adversary computationally oracle mean answer query arise oracle respect interactive th query false therefore enough require interactive theorem way adaptively sample start establish interactive however security enough user small security user whereas query user entry oracle formalize compare zero adversary attack without break security scheme let n interactive I otherwise sufficiently straightforwardly security depend adversary adversary efficient attack ideal high must hold event polynomial every definition defer claim easily eq answer consistent answer choose query every attack input query assumption query every eq second ct ic claim accurate error complete argue attack answer q deferred computationally adaptively choose every easily security answer polynomial put claim together main sake oracle theorem attack claim contradiction simplicity hardness prove theoretic datum level fact rely need security security message slightly discussion simply unbounded adaptively query adaptively private define notion appropriate respect change mind privacy game ht jx j qx n qx shorthand adversary q computationally accurate answer choose attack code user scheme let I ct query let start establish user small security interactive ideal attack parameter I column ct ct ii n ji every straightforwardly security query entry adversary access view adversary also fact argue attack attack probability must hold game security defer combine every claim answer recall number answer inconsistent adaptively polynomial claim terminate unless assumption sample argue attack answer ideal attack let computationally computationally efficient every definition security defer claim obtain adaptively every claim security thus oracle put together adaptively query contradiction reach claim terminate early contradiction theoretic hardness privacy discussion early interactive code attention claim claim security section claim claim modular claim section relate fashion omit brevity begin security via take key whereas security scheme choose adversary whether interact q claim claim let polynomial construct adversary attempt break security construct break security simulator new sx I ix ct ic occur claim efficient construction efficiency notice occur oracle return holds show query either unknown moreover message choose choose complete interactive interactive interactive answer construct interactive code parameter improve robustness interactive interactive interactive length user error specify answer c c continue terminate interactive code user error failure suppose non interactive interactive adversary adversary c receive c j sa interactive adversary round consider theorem lemma theorem claim corollary edu edu bind adaptively computationally efficient answer accurately unknown expectation statistical correct study al answer bound hardness assumption efficient give valid answer adaptively implication answer query choose call optimize hardness fouri analytic code simpler flexible construction query finite summary outcome unlikely occur chance discovery occur analyst incorrectly observation decade discovery highly influential control false scientific research typically discovery attribute possible inherently query interaction recently paper formalize give universe suitably provide answer generalize answer achieve analyst adaptive previous query answer adaptively choose arbitrary adaptive analyst answer answer query probability however situation turn query ask adaptively al computationally answer show oracle answer whether achieve oracle privacy assume answer importance discovery algorithm answer unfortunately assume function computationally query conceptually interactive bad oracle answer adaptively statistical private al query query sort restrict hardness whenever dimensionality requirement query
set estimating assess ij introduce estimator graphical neighborhood glasso precision entry correspond define family minimize ham stress standard calibration close performance besides lasso selection tuning estimate node correspond via similarly hamming highlight p c ij ji ji subsample fix invoke q ef pl km km estimate ij ji topology example confirm follow insight six graph topology vary range graph edge add edge e edge uniformly e edge uniformly number entire construct edge add selecting proportional current random consist set node set precision entry precision entry eigenvalue dimensional avoid new assess accurately strength neighborhood unknown outperform spectrum become pattern biology interaction expression protein sequence graphical represent underlie simultaneous gaussian know entry become particularly even precision global art solver graphical framework gaussian model parametric advanced calibration scheme approach practical introduce neighborhood particularly novel assess strength method neighborhood across wide scenario since unknown therefore promise
stimulus passive formula active environment active et implicitly consider explicit relevance handle active learning emphasize applicable handling miss case many convex machine literature class however approximation publish machine learn stochastic researcher expert common al objective important objective strictly example logistic semidefinite minimizer hold assumption satisfied important practitioner machine checking might require considerable assumption require literature conclusion obvious impossible typical mathematical statistic example state critical point desirable white theorem theorem risk al et type approximation review stochastic et variable metric stochastic converge white theorem risk wang variable bfgs random mass generative modern include mixture variable novel contribution situation passive assumption unique strictly stochastic descent proof theorem design interpretable theorem practitioner finally applicable five machine literature new minimize al design fundamentally development field ensure correctly apply specific also general handle involve minimizer exist necessarily sufficient condition continuous probability strong stochastic approximation partition let let number dimensional bound td bound satisfied subset value condition appropriate hessian encounter likelihood uniformly bound hessian layer perceptron evolve closed bound ensure solution practice empirically rather verify discuss analyze stochastic learning machine begin initial machine update guess refine iterate update assume mini batch identically mini value upon passive statistical function function attempt guess observation equivalently stimulus parameter give magnitude govern strictly search direction determine appropriate typically compute direction condition commonly ensure deterministic descent appropriately choose search relation mini increase tend law hold increase direction appropriate sufficient stochastic search type increase eventually decrease et specifie period stepsize period stepsize search direction size one call critical variable direction adaptive e et et al quasi likelihood cross realization estimation find v immediately use descent term relatively evaluate correlate whose computationally method evaluating expect correspond term multidimensional mini highly observation kk mini independent density mini initially increase integer environment update trial hidden consider variable characteristic architecture partition dependent realization visible likelihood rewrite derivative eq obtain substitution h imputation realization give expectation maximization define stochastic multiply integral mini trial sampling parameter estimate learn stochastic method analyze behavior positive integer case expectation algorithm use current probabilistic iterative machine environment characteristic deep representation learning machine implementation environment suppose episode episode independent identically addition episode episode episode machine passive specifie learner learning select action state machine mass environment characterize specify initial episode conditional state episode state episode episode j incur machine episode dependent allow possible adaptive machine formula derivative operator carlo approximation derivative gradient see episode since open system however episode influence next action methodology involve episode episode identically distribute interact environment episode sample overlap code dependent learn environment passive proof combination appendix et et review variation see appendix et al expand function theorem substitute identify condition bound bound exist iii piecewise continuous bound asymptotic function conditional respect
nature minimize complexity reason depth consider computational computation conference cut hyperplane line bivariate variety attempt take contour exploit circular suggest subset depth contour well depth cloud depth contour single complexity depth complexity line depth calculate depth successively update low coincide algorithm latter run sphere accord define region sequentially exploit depth updating continuously handle contour quantile univariate projection envelope region connection multivariate directional quantile correspond hyperplane depth contain direction hyperplane set algorithm hyperplane intersection form depth first search algorithm direction suggest depth special fast algorithm elaborate try save weak I small projection univariate depth explore generate line connect line normal hyperplane pairwise distinct hyperplane claim direction prove depth deviation real work try achieve precision exploit author one depth suggest framework computing affine tuple project orthogonal complement depth orthogonal tuple capable deal general tie lead theorem present issue non general notation hull complement w contain short depth eq depth compute integer remove add integer write subset position hull contain index denote contain lie hand contradict hyperplane whereas linearly linearly I every choose proposition therefore complete subset follow immediately precede projection orthogonal conclude simplification point map depth compute finally therefore independent fall orthogonal former category second computation precede space step arise section rise reduce dimension specialized algorithm bivariate choose result hyperplane hyperplane hyperplane general position never enter recursion hyperplane overall new min project point depth since subset complexity combinatorial combinatorial independently I new min yield outer loop hyperplane exception point map reduce algorithm recursive variate w variate variate stop case basis hyperplane j new min min section external library source code request easily implement orthogonal implement routine precede point calculated datum origin number store scale depth problem improve loop drop zero however experiment due independent operation algorithm base possess high parallelization exact execution depend step etc report table later remain performance differ intel core processor normal origin present execution cell middle line try try extremely vary increase exceed hour far outperform outperform former superior framework violate also call execution additionally heavily position unstable instead report effect dimensional design quick handling get large tie
orthonormal r dr jx u proposition function function perturbation assumption necessary perturbation identical alternative form functional taylor construction vanish segment positive third g separation eq need hellinger construction hellinger apply proposition hellinger divergence proposition far allow hellinger bound constant note n use contradiction guide framework desire conditional good grey sift project method hence hellinger construct similarity metric image denote sift image divergence clustering depict image class affinity exhibit hellinger patterns image nn achieve apply pixel intensity accuracy find perform divergence result imagine treat similarity metric classifier htbp p x f I iy nz I conditional divergence version rgb rgb pt false department computer science address address estimator distribution assumption statistic leave favorable theoretical apply derive popular quantity exist theoretic play statistic mathematical sciences addition analytical tool hypothesis functional algorithmic task develop influence range intrinsic several statistical recently gain object model mutual conditional divergence mutual information building graphical allow estimator functional post use von idea statistic study split setting function expand propose ds framework contribution however estimator perform analyse achieve parametric rate estimator normal sufficient approach estimator entropy functional list functional available despite generality image focus quantity relevance technique brief post hoc correction influence detailed approach functional knowledge paper post hoc follow paper integral extend functional density consider splitting build functional superior fundamental inspire split estimator functional analysis rely statistic look compact equipped lebesgue measure measure absolutely focus functional twice permit density ease exposition presentation definition distribution absolutely derivative belong development von distributional impose notion satisfie uniqueness domain consequently define term dirac delta sufficient assign term measure density control density functional satisfy form consequently write taylor expansion lemma appendix q expansion basis functional distribution argument influence satisfy estimating suggest construct influence right side expectation half expectation preliminary averaging confirm smooth datum trick commonly several work analysis cauchy schwarz inequality theory good stand decrease efficient except theoretically extend functional version cycle point however latter modification estimator entropy divergence mutual several hope good reference practitioner unconditional list expression software implement functional estimator property method several functional require integration estimation define r ds smoothness density kde bandwidth nh smoothing kde kde estimator form kde truncated kde use satisfied smooth functional table achieve square mse review note self contain directly bias follow schwarz attractive property agnostic correct rate estimator bound summation lead bad rate version bound correct couple use kde indicate limit challenge summation asymptotically rate estimator empirical asymptotic one theorem satisfy normal normality allow confidence theorem give valid interval use asymptotic differentiable technical zero order vanish rate behavior unclear degeneracy occur arise important two let h alone sufficient guarantee functional divergence assumption difficulty bound minimax functional define positive functional minimax optimality functional however gap ask improve rate regime functional taylor functional statistical gain believe estimator conceptually favorable functional assumption attention decade body work focus estimate shannon nice include enyi entropy paper method estimate kde near neighbor conceptually several drawback secondly kde obtaining plug require aware principled hyperparameter use optimal validation secondly method kde integration avoid functional see divergence estimate analyse rkhs straightforward clear convex problematic use weight divergence establish normality parametric strong work divergence method applicability include divergence compare software estimator estimator polynomial plug bandwidth perform plug make perform estimator inferior require software case poorly hyperparameter asymptotic dimensional hellinger divergence estimation repeat experiment compare asymptotic plot suggest hellinger whereas hellinger superiority focus density reduce expansion equal eq compact finally make stein analysis stein inequality intend estimator estimator originally von asymptotically begin proof lemma prove conditional bias preliminary condition consider derivative taylor normality addition begin around q step add chebyshev note step q ready kde achieve bias far bound bound schwarz n proof error bound kde integrate squared stein set sample stein shall note inside summation substitution lipschitz constants stein expectation first remove twice inside apply stein get complete analyse begin ip p bias bound bias condition first follow boundedness taylor hold asymptotic normality
conclude despite minimize good problem light consistency eigenvalue robustness certainly help behavior covariance build component soon choice quite within framework minimizer pr sr result lemma apply aim small critical threshold happen sense exploit everything polynomial package eigenvalue covariance course contrast converge estimator vary approximation consider autoregressive difficult construct estimator risk risk computation perform figure correspond red green stein good seem quite scenario ar bad frobenius little thing loss estimation frobenius restrict minimize risk invariant loss noise normal build robust good aspect assume work construction proof quite heavily outline quite depend behave wishart assumption restrictive high extension addition describe invertible absence risk mathematic method outline could consider might surprising present behavior estimator covariance zero since show risk equal contrast unfortunately wishart extensive behavior limit analogue frobenius allow unbiased adapt study tackle eigenvalue optimal recent look frobenius appealing account top eigenvector behavior perhaps know automatically condition covariance quite parameter interest appear reasonable although currently stein compute risk estimator identity implicitly unfortunately covariance satisfy compute pt conclude inside associate decompose pt eq get eigenvalue b notice generality would similarly iii loss possibility q iii joint define j pn pt polynomial happen l z dl proceed change pt respective dx consider first one l z l q integral converge long case b pt integral treat constant calculation old pt eq lemma collect define algebra part proceed weak regularity first smooth condition weak satisfie find variation stand calculus obtain pl notice estimator notice weak minimum simplify define p parameter carry argument notation p theorem weak rewrite eq twice satisfie therefore equal root positive yield obtain similar spirit therefore n mt weak conclude eq ii denote limit writing except cl almost surely divide eigenvalue therefore quantitie white f distribution q statement follow poisson substitution obtain variance contour transform form therefore go back bound lemma therefore use done obtain simplify decompose hellinger affinity density cauchy schwarz normals hellinger affinity supremum define note q contradict term conclude estimator definition nn l n k problem frobenius pca investigate use show use theory strongly consistent essentially asymptotically past year context study sparsity zero coordinate covariance distinct eigenvalue eigenvalue good high see also theory pca traditional retain rigorous require good estimation although covariance analogous context asymptotic regime strictly covariance frobenius essence really estimation level consider estimator finite rank finite serve principle propose restrict perform correction class unbiased invariant loss h refer noise calculus variation spirit idea directly risk truth turn depend dominant prove behave strongly estimator essentially minimax rate estimation construct worse worst remarkably show contrast estimation zero perfectly think structure generally accept unless robust encouraging construction simulation proof claim simplex pa norm notation stand wishart stack let distribute wishart construction convergent restriction eigenvalue estimator satisfy form expectation similarly weak regularity finite previous associate estimator similar dimension satisfy regularity regularity pt strong weak pt careful reveal spirit emphasize merely weak correction like performance common convenient move mention early loss thought frobenius stay unbiased risk rich shape part depend eq functional asymptotically sense explicit construct estimator
throughout especially towards illumination change sequence consist illumination frame sign illumination appear severe illumination change slight motion fail light accommodate robustness illumination whole book severe appearance change book book appear track tracking video drastically illumination pose purpose adaptive object appearance change argue model appearance apply template lead robustness record template novel manifold finally track inference propagate comparative video art illumination appearance unlike method motion formulate updating measure add bag resolve issue enhance introduce particle filter extension object track join acknowledgement department communication centre program figure track school technology appearance able pose variation approach base subspace several image accommodate use subspace represent object may propose approach euclidean inference filter quantitative evaluation challenge video approach obtain considerably track fundamental surveillance behaviour retrieval problem appearance design appearance intrinsic pose camera illumination rather rely object discrimination illumination variation achieve modelling subspace believe via adequate subspace origin translation meaning point shift small attractive purpose generally maintain location account generalise subspace affine subspace subspace track find affine subspace frame affine subspace novel distance affine via combination mahalanobi conceptual propose tb point subspace subspace measurement group consecutive object frame use frame third right region frame b object image dash wrong wrong location represent model addition frame distance result select candidate appearance affine object tracking approach tracking subspace update recent drift location object measure subspace subspace track update precede frames subspace distance measurement method subspace manifold wang online scheme contrast point subspace distance comprise block propose component take history previous create candidate object frame consecutive frame filter monte find probable candidate module encode appearance subspace achieve decision module bag object model encode affine module module history bag primarily drive attain variation desire appearance whole body encode object moreover person visible tracking appearance upon termination keeping model body whole tracking cope subspace scale frame blind inefficient since plausible monte space probable key idea become briefly algorithm track virtue probable candidate estimate I object recent bag frame object bag recent template affine track bag track challenge scenario rate use extraction slide template frame address particle appearance bag subspace module bag although euclidean distance minimum distance angular affine angular origin affine simplify address limitation manifold j mahalanobis orthonormal basis length geodesic geodesic q subject angle compute mahalanobis affine subspace distance distribution kl identify distance manifold define choice measure length mahalanobis result template tracking framework likelihood normalise subspace bag likelihood template opt sum frame update object state object frame state affine region calculate likelihood bag final likelihood eqn object framework generation geodesic operation compare affine bag operation tb box sake clarity demonstrate overall pose variation b appearance face various involve pose change analyse propose eight publicly project consist tracking task face tracking face face book frame video bit image sake efficiency affine candidate model template trade appearance precision affine performance model significantly idea affine performance affine center video subspace face assess method instance boost track collaborative publicly code propose box frames book
column work rank problem wide group analysis lrr solver focus follow compatible reformulate unconstrained low accelerate alternate sdp sized require continuous widely convergence divergence linearize suffer issue propose accelerate trick lrr drawback singular partial fast full problem reweighte relaxed smooth weight analysis converge indicate lead sparse rank iteratively reweighte work norm affine non robust completion proof norm thus application actually logarithm introduce relaxed iteratively reweighte least future regularize lrr theoretically general solve propose art avoid svd iteratively reweighte square lrr lrr low minimization lrr vision lrr reformulate smooth two I smooth call smoothed smoothed lrr lrr bring lrr smoothed smooth make easy convex globally solution easily convexity third order eigenvalue say furthermore solution say converge update solve separately break solve reformulate denote column svd qx z equation may matlab solve fix motivate algorithm separately treat matrix lrr problem lrr lrr compute art accelerate fast choice tune good accelerate worth though lrr structured group non overlap structured completion quite objective smooth popular norm variable main much guarantee logarithm induce norm derivative nuclear iw jj z g iteratively reweighte sum term square smooth proof show concave concave proof norm nonzero differentiable concave definite concave differentiable let get sequence limit globally convenience description implementation shall smoothed lrr lrr check worth nontrivial affine work unconstraine rank unconstraine minimization weight variable update weight usually easy update difficult updating also proof affine constraint square see handle square simultaneously also concavity synthetic real solve lrr behaviour converge appropriately decrease initialize norm experiment fix synthetic basis random corrupt gaussian show value lead inaccurate similar phenomenon lead accurate solution fast well denote type method svd matlab third party package default converge code lin set pc intel gb windows version time minimum time method datum emphasize lrr usually large solution rank plot computing see except linearize svd hence unstable sized high rank completely database face conduct subject face image pca normalize cut affinity lrr iteration linearize within iteration database draw sequence first onto subspace pca lrr project lrr fast c c subject acc acc inductive principal solve sum cause continuous smoothed solve iteratively tx face recognition learn training remove corruption two consist face acquire pose pose image subject svm recognition see accuracy different solver run much large figure plot recover obtain successfully remove face c iteratively reweighte solve minimization joint relaxed globally problem lasso overlap group lasso concave xx yy use solve dot together non eq minimum q imply sum eq globally converge j denote rewrite therefore mathematics university master technology china currently student
paper copula bivariate marginal square traditionally measure strength dependence index course order pose must impose require every reflect degree tail necessarily certainly interval property relate copula leave admissible admissible admissible motivated determine co risk admissible probability equivalently q independence representation diagonal serve tail path may maximize illustrate follow view index neither measure consider check admissible correspond right hand equality hold copula symmetric motivate definition give call maximal eq simple arise admissible path refer maximal employ variant classical index namely assume limit exist assume function illustrative concentrate conclude section copula example claim rely behaviour copula non rely behaviour copula explore model dependent inter amount et dependence w et zhang wu development topic excess economic pricing work research goal aim maximal role diagonal dependence copula investigate copula aspect reference let random pareto quantitie let copula et et al dependence newly suggest index define z z weight therein calculation noting dependence observe classical copula dependence formula next function unique tail maximal maximal copula symmetric copula whose diagonal copula give path k right panel none coincide low tail ordinary subsection symmetry imply path copula copula another example copula among illustrative example motivate present sometimes whether positively dependent specifically copula consideration maximal solve eq maximal path reach admissible fail cf end fr rewrite copula copula low tail maximally interpretation every admissible path generalize index copula obviously weakly maximally dependent compare copula get moment asymptotic formula c copula index interpret whenever course expression maximal demonstrate importantly close generalize eq closed path copula generator hold copula copula increase tail behaviour copula diagonal tail conservative dependence notion herein main assess copula paradigm modern management york much research environment research science engineering discount aggregate pareto distribution asymptotic capital expectation large claim york copulas tail dependence copula management ed pp mean copulas universit finance tail copula near independence extreme copulas density york nj characterization proof loss decrease achieve split decrease point maximal formula two index maximal conclude equation equation finding denominator xx solution derive form furthermore also lack close maximal expression arrive rx side tail proof tail li li order risk york mm section corollary theorem property mm path maximal department mathematics york sciences mm numerically measure extent extreme dependent risk paradigm management phenomenon hold asymmetric copula
et al identify situation simulation study set proximity gamma normal long central chi use circumstance brief overview issue involve population distribution like fisher justification approximate provide implement binomial test unknown estimation state satisfy demonstrating issue study several distribution simply asymptotic particular derive asymptotic variance large finite moment population density sample statistic nan distribution mean population mle remain rao justification asymptotic variance make accommodate total gamma variance measurable function translate know expectation easily obtained consider statistic nx expectation follow ne chi degrees prove theorem log calculate value simulate usage ratio respectively aforementione point incorrect exponential shape mean b location mean mean scale scale shape parameter way check fourth population population nan condition condition mean nd rd central ratio freedom like variance suppose differentiable nan apply moment population third fourth central population asymptotic normality smooth moment map neither equal otherwise write delta distribute asymptotically example view corollary poisson delta since nm delta n note theorem datum implement goodness kolmogorov suitable good fit implement context total incomplete gamma modeling purpose compute west implement basis low week total east implement series gamma nan associate china variance accept case adequate necessary result simulation study implementation nan however set statistic present accord freedom line nan expect demonstrate rejection nan actually equal weight mixture component mix mode variance specify rejection case nan lie demonstrate gamma illustrate false rejection acceptance nan obtain dataset http www observe goodness fit value gamma shape test statistic statistic true turn acceptance gamma fit figure rejection formal gain analytical strongly nan test pearson chi goodness test show conduct follow pearson chi goodness context followed however gamma largely shape test central chi square follow normal modelling chi distribution freedom aid article way check necessary applicability check whether ratio applicable well chi test class theoretically use fitting non kolmogorov chi goodness reference evaluation hazard journal pp square poisson binomial mm mm c daily series expectation method chi nj pp generate forest fisher statistical research worker company pp medical ed pp incomplete
risk difficult might label describe formulation seed one detector detector risk one frame negative seed generic detector obtained run binary aforementioned sample overlap ann car mean car incur classify regularizer calibrate enjoy optimization express equation minimization w x w impose task comprise model lose seed closely design streaming classifier past similar sense prevent fitting appearance individual object object appearance importantly justification category detector approach interpret category model average learning could difference datum benefit detector quickly seed thus category handle seed could alternative regularizer replace cost demand complex compatible regularizers regularization eq potential multi call average rate rule previous maintain update detector include detector one advantage suited rely practice mini potentially multiple pass datum share window also maintain depict simplicity issue unsupervised line hyper amongst mini appearance stream surveillance learn update scoring improve detection detector sequence annotate static correspond scene duration k allow equal self video detector domain evaluate video stream frame right frame ols detector detector generic detector pre online order detection detector detector experiment fisher reason art efficient category level highlight category appearance model learn able use application yield competitive result slide window ols ols seed algorithm wang detector without penalization detector self camera multi quantitative area detection report detector applicable set I alone video wang regularization compare unsupervised adaptation fix label video improve detector scenario confirm detector video tune object detector second video detector thousand substantial ol scenario illustrate multi tracking help well detector seed appearance trait useful handle intra small improvement discard generic enough video stream seed whereas must generic neighbor stand alone detector improve additional cost gray center amount label camera mobile drive paper detector efficiently stream sources ii contrary without manual intervention tuning objective recall detector exploit line instance level object detector approach world publicly mm learn identically distribute sample unobserve one explain collection particularly video typically collection focus unsupervise object continuously adapt rely related availability black classifier along differ adaptation easy inspire see discussion present mm approach along unseen video stream generate box score seed allow detect seed rest self adapt suffer transfer fit positive lack target address leverage spatio structure learn track automatically positive negative hard negative task yield gap instance learn propose applicable recent stochastic optimization novel parameter challenging world adapt detector domain approach annotate often pass annotate adapt impractical continuous streaming leverage unseen classified confidence classification minimize unlabele readily approach suited exploit spatio video collect sample learn detector rely set video move detector adapt video self learn fine target source use track inspire instance rely learn detector apply video stream track detection tracking detection zhang specific aim adapt also multi across category straightforwardly applicable category object detector parametrize classifier compute region category
algorithm generate posterior correspond full true particular weak scheme subsample interesting future development design likelihood hope researcher contribute area contribution fix collect complement decompose l term draw noise evaluation zero prior available likelihood contribution data point set covariance ard attractive choice error v mode hyperparameter optimize change compute multiplication evaluation optimally costly approximation spline spline surface radial approximate contribution thin spline coverage boundary predict likelihood contribution eq expand shrinkage analogously mcmc vb thin spline gp preferred refinement pair log contribution change drastically transform link logit categorical regression category I many category spline predictor dimension link datum connect tp k proceed since mcmc work computational dominate proof use om om ii straightforward taylor suppose prove r cauchy complete ii computed e part define follow q iid part conclude lemma assumption lm take I note l expectation follow similarly important ready prove stress notation make follow bound py pd iii py py part p py mp py py py mp part expression clearly iv therefore take theorem axiom conjecture mail liu se grant ce like discussion express author view markov many observation costly propose likelihood substantially scheme inclusion observation likelihood broad class approximation present subsample small adaptively choose apply bivariate half million observation survival data bayesian monte big proportional monte distribution early desirable full approximation furthermore selection etc costly evaluation pose however demand computing dataset become increasingly common computation abc vb stochastic great approach currently produce acceptable computationally demanding via augmentation hasting efficient show design crucial chain simple si order magnitude si mcmc draw time budget regular proposal introduce metropolis likelihood prove note even biased sample perturb exploit scheme subsample likelihood accurate application error focus effort advantage first adopt estimate sampling optimal us control improve mcmc budget organize subset section propose sample inclusion probability methodology application discuss implementation main vector fix augment marginal informally draw obtain move propose accept note arise regard choice imply intractable simulate h likelihood depend crucially however covariate observation log contribution log likelihood survey sampling problem total introduction likelihood unique piece example longitudinal problem subject individual series subsample indicator replacement easy estimation error selection determine motivate mcmc unknown true simplify hold verify kp iii motivate iv remainder assumption iii remainder check order taylor v hold subject bound highly posterior approximation chain observation noisy reduce per efficiency markov vice computing factor autocorrelation th lag limit factor mcmc obtain sampler produce vs efficient proportional perfect around fairly normal independent value high conservative obtaining assume approximately unbiased tune independently see tune variance likelihood iterate guess new size bring estimator particularly attractive sampling estimator guarantee crucially variance solve optimum depend computed tuning grid log contribution refer obtain surface mcmc approximation predict contribution approximation exact perform great care computational surface fit gaussian thin spline appendix surface fitting kernel likelihood multiplication appendix surface fit construct period year analyze set variable use bivariate hold control severe choose excess period bivariate probit multivariate probit augmentation illustration bivariate integral probit total draw discard first prior different sampler metropolis rwm rwm approximate log contribution probit link four response outcome separately put example vs proposal rwm rwm efficient draw rwm around plus thin spline expect around overhead thin fraction decrease negligible obtaining approximate rwm confirm decrease htbp also rwm maximum allow mean fraction rwm rwm proposal reached report efficient draw three rwm allow scale rwm show acceptance posterior mcmc nearly indistinguishable explore fractional theorem create subsample fractional error subsample choose average discrete survival illustrate period logarithm total sale logarithm age hazard probability tx ij ij obtain estimation parameter tp use rwm sample equation mode consecutive vast use effect give however illustrate dash panel rwm explore log panel logarithmic show draw vs rwm particularly size decrease
observe consistently classifier attain combine averaging provide classification achieve domain e acoustic representation combine approximately cross point db far g regime attain improvement classifier db snr conclusion apply front end speech operate acoustic address acoustic aggregation demonstrate outperform db perform gain level combination primarily focus term robustness implication extension speech straight forward approach combination classifier error code speech train alternatively propose phone hmm token passing speech former baseline hmm system first pass possible svm baseline though classifier construct solely due solely decision svms determine recognize algorithm subject confusion classifier snr confusion class confusion towards figure classifier attain confusion figure suffer high closure among group cumulative multiclass reduce use level follow instance condition figure meta improve gain outperform high attain classifier db complete hand level subset moreover classifier small adaptation ac uk automatic severe degradation presence additive human central behind art combine front end base compression modelling bring human effectiveness context model elementary gap human system human recognize isolated speech level chance snr db snr levels speech recognition exceed rate human error although conventional reach human attribute fundamental limitation front end extraction frequency cope environmental show take place front end severe degradation environment machine recognition speech front year variance normalization taylor front explicitly reduce effect front end distortion feature depend speech feature previous acoustic robustness front end end derive speech well draw motivation experiment linguistic message decision narrow reasoning accurate signal put consider speech beneficial exploitation inherently secondly sufficiently narrow approximated spectral additive aggregation selective de emphasis result improvement recognition band counterpart end resolution front extract acoustic retain speech potentially discrimination representation investigation propose front assess robustness filtering room cause spectra performance attribute primarily window conventional front end short room impulse response cause filtering speech conventional noise environment several speech literature scope channel filter robustness front compare improvement achieve speech framework hide hmms base architecture token speech classification term robustness linear classifier ratio classify classifier db improvement classifier section additive draw suggest future direction towards front continuous speech task machine increase speech use conjunction front fix front end speech hmms variable address fisher lie scope paper hence feature front length speech study possible extension front section future decision surface maximize two training training multipli bias predict simple k produce boundary linearly feature potentially therefore effectively classification polynomial sophisticated kernel acoustic describe follow svms combine code class complexity capture predict loss loss hamming hinge perform discrimination svms predefine code train code element predict loss classifier feasibility regard loss various ham hinge frequency maximally perfect bank decomposition decomposition cosine comparable somewhat inferior summary obtain different decomposition filter prototype filter bank representation coordinate bank primarily sampling avoid unnecessary limit burden believe redundant expansion speech signal filter could advantageous invariance speech construct capture identity even kernel sign speech polynomial kernel act vector baseline acoustic feature typically speech explicitly take feature evolution individual first divide energy form form dynamic correspond acoustic form component acoustic classifier obtain decision multiplier corresponding assign simple scheme score predict conventional majority voting map label alternative voting condition argument error probability voting contribution overall cardinality ideal decrease speech error majority voting may considerable improvement furthermore aggregation drawback specific use stack weight specific pair aggregation practical stack generalization hierarchical layer svm output svms aggregated meta level meta artificial real speech interpretation isotropic class domain find music fan nature etc sentence impulse impulse conference room spectrum impulse room substantial difference spectra mean view approximation reduce test proxy record location room convert dimensional vector second frame duration frame frames sec representation vector extensively art scheme estimate speech speech noisy relate speech clean feature gmm mixture spectra clean additionally fix variation training compute training consider classifier front end svm clean via perform via noisy use filter error unlikely case filter convolution particular svm exact knowledge offer expect scenario impractical nevertheless furthermore note acoustic ms centre decompose bank examine effect number bank accuracy speech effectively capture frequency relatively speech sufficiently narrow performance demonstrate increase bank reduce extract ms centre standardize within always perform meta weight scenario classifier meta classifier svm score clean meta vector contain clean db snr meta level svm clean meta classifier setup error stack yield voting achieve voting decomposition decomposition cosine bank achieve large composite therefore far list scheme classification white curve correspond combination stack scenario multi style stack development development consist clean noise corrupt presence linear filtering discuss figure frequency acoustic dash curve ensemble method vote stack meta level classifier section meta classifier stack train stack attain vote poorly composite acoustic stack quickly meta classifier assign robustness train base clean corrupt white explain show weight standard deviation assign stack binary classifier train style component style hold speech provide reliable amount speech observe stack classifier improvement match noise stack match performance acoustic classifier white result stack classifier match stack exhibit snr whereas composite db db snr stack range quite remarkably stack train test match db db snr train number weight small fraction style match db performance degradation noise stack figure attribute acoustic colored noise approximate narrow band white comparison report obtain variable condition stack classifier train match match match classifier db db suggest high
uncertain interpretable vice versa denote diameter fix statistic follow statement uncertain diameter dy interval interpretable interval axis indicate key proposition interpretable interval describe distance reach uncertain alternative may test determine definition percentile prove uncertain u u continuous member similarly continuity know imply contain uncertain infimum supremum answer since claim establish increase exist critical word equitability formal equitability interpretability term proof fix statistic fix worst interpretable proxy every base least equitability interpretability fundamentally able signal weak essence equitability interpretability power independence definition equitability extent iff exhibit interpretable confidence continuous base distinguish independence equitability interpretability power requirement interpretable detect independence ignore interesting relationship hope power address concern agree thing work greatly enhance art power think equitability base solely way dataset trivial dependence hundred thousand become manually examine define strength statistic rather power nan hypothesis independence alone paradigm sense reasonable ignore equitability statistic specifically broad equitability discuss detail concept relate implie property sample statistic limit definition estimator focus introduce prove discussion immediate section devote analyze state begin define sequel grid possibly row analogously let represent information population characteristic q refer mutual interpretable section characteristic name characteristic type relationship value let information rather abuse notation sample point order sample characteristic eq set define maximum edge analogously present variable statistic fact consistent consequence theorem statement equip projection eq continuous supremum realize maxima segment matrix consistent analogous corollary refer priori uniformly sample trivial therefore consistent consist normalize consistent true characteristic consistency abstract consideration necessarily suffice grow notice sample characteristic heart use grid quickly lemma build dependency grid consider master cell master sub grid grid seek bound grid variable consistency seek require entropy contain idea grid grid allow capture dependence distribute grid cell induce cell marginal argument analogously since sub grids master give grid mass two refer horizontal vertical line line add line line grid account eq triangle bound define concrete bias grid integer fix column away provide proceed show small allow inequality bind rest write I pair desire cn concave must sum observe horizontal involve observe give final grow fast lemma yield specify fix q sufficiently entry suffice difference n come hold least ensure desire ready infinite equip supremum norm pointwise sample show write n fm fr nm fm vanish pointwise adaptation continuous second map mean latter lemma tell probability consistent define essential actual quantity something intuitive rigorously previously choose investigation asymptotically different tradeoff easy relationship alternate alternate definition distribute normalization subject grid grid grid complex grid well show penalization also think necessary result prove relationship achieve statistical prove restriction grid ensure hold lastly simply consistent exist theory necessary equitability independence prove provide alternate jointly distribute supremum population characteristic interesting reason light characteristic normalization turn achieve mutual pdf version mutual variable abuse notation denote support observe consist pdf increase integrating set finite uniform distance pdfs function deterministic increase obtain uniform strategy characteristic resolution need issue possible entire characteristic continuous answer normalization move small cause however uniformity mass factor normalize matrix normalize information grid mass suffice decrease distribution distribution suppose move arrive write relate prove side lemma bound entropy complicate let leave cell notation obtain x x lemma appendix sum entropy combine line give resolution show continuity map complete form uniformly since consist give restrict ball around ball tell become map establish continuity see family finally within supremum norm give continuity corollary exist characteristic introduce continuity pdf normalization contain characteristic normalization normalization I distribute uniformly distribute loss grid place uniformly consider row mutual infinite supremum respectively view information supremum characteristic supremum population goal us foundation new introduce observation population let empty know k define boundary characteristic important boundary one partition dimensional grid former exactly proposition characteristic equal piece every partition column imply partition column characteristic rather matrix fix notice either case hand corollary supremum exceed follow introduce alternate characterization previous estimator approximate therefore efficiently compute wherein partition dynamic mutual maximize however rigorously justify give statistic consistent fact realize characteristic matrix easy compute consistently notation dimension column variable grid analogously define otherwise matrix supremum monotonically increase hold fine axis thus proposition non analogous order pair consider subset grid consider consistent estimator statistic consistent efficiently heuristic adaptation variance equitability stem expression maximization one grid type propose reason rigorously limit work estimate give datum formally follow theorem additive numerically distribution prove column dynamic programming row maximize row step bind show exist mutual achieve close use appendix row line probability lie cell contain cell line though grid contain row remove line fortunately though pair mutual merge leave obtain integration introduce choice numerical integration arbitrarily tradeoff corollary give compute additive q continuity give give rise density estimator function equip consistent sample variable continuous formalize develop equitability coefficient equitability specify interest reflect equitability statistic extent know denote show state power property independence equitability notion power equitability turn original quantity prove continuous equivalently open lead define new efficiently addition describe precision probability density analyze statistic density leave question valuable estimator variance difficult understand second statistic form canonical gaussian contribute understand capture notion theoretical equitability infinite limit highly desirable alternative characterization first open art size bias equitability go detail address idea author acknowledge constructive claim scale respectively analogously proceed proceed argument proceed term mean together fact inequality bind use e line give complete bind grid adjacent variable induce merge merge column since merge column identical merge expression ia complete non number equal equal probability function bind equal observe use lemma bind grid sub grid horizontal remove line mass cell union line column suppose lie leave mass right successive lemma binary apply next treat general case column l b lemma probability column contain vertical horizontal line entirely grid column analogously distribute upper hx hx hx hz hx z hx hx hx hx upper hz hx thus variable add mass lemma give variable q magnitude total mass mass go lemma b second please david relationships useful give similar equally equitability analyzing set formalize behind equitability equitability generalization independence us compute generalize mutual enable reason continuous canonical prove alternate estimator pair random variable hope provide rich theoretical foundation equitability extensive empirical discuss aspect equitability statistic suppose thousand association pair hundred million manually examine pairwise scatter plot context commonly take compute approach choose meaning score exactly question systematically relationship crowd potential list concern comprehensive trivial association care relationship detect sufficient power could reject excellent method allow exploration identify association sift relationship reject strength address introduce equitability dependence assign relationship type notion notably relationship cover similar note reasonable equitability statistic reflect coefficient determination regression possible characterize seem reasonable give perfect score relationship maximal behave desire original equitability much publish concern perhaps concern rich theoretical equitability main issue allow unified language use equitability limitation equitability formal equitability equitability equitability hypothesis whereas typical association yield relationship strength trivial relate concern power benefit consistency together easy trivially generalize estimator separate finite rigorously mutual information correspond orientation ask whether equitability power soon goal remainder probability relationship theory give begin analogous define call view supremum practically easy original heuristic previously proceed give rise sample density approach expect extensive analysis method thing equitability runtime introduce compare original statistic discuss exploration question compare size defer issue rich equitability equitability informally ability statistic similar equally noisy relationship notion rigorously equitability interested equitability variation adapt incorporate variable formally define equitability give formal overview brief benefit ask like tell setup set distinguished g quantify ask evaluate back finite want dependence criterion define equitability add particular strictly equitability equitability one equitability equivalent hypothesis though primarily various functional relationship change previously noiseless population one interested support manifold add perhaps simply mutual information goal make impossible dependence good equitability reason equitability functional motivate generic concept equitability unified explain equitability require matter motivation state perfectly make say equitability criterion equitability term perfect equitability informally equitability notion equitability equitability interpretability sufficiently equitability recent follow ed impossible nontrivial severe understand amount trivial proxy random arbitrarily crucially fact depend arbitrarily value may pointed issue allow relationship indeed quite translate thus apply address perfect primarily discuss section equitability give equitability equitability incorrect perfectly may impossible model equitability desirable remain provably empirically analogy science fact np want solution solution provable dependence comment publish discuss property interpretable define amenable efficient interpretability define separate incur performance cause choice equitability reason distinction need evaluate interpretability give definition analogy reliable interval interval whose value analogy depict figure table analogy interval would however detect good estimator subject care bias since rank p value value sample wide sample infinite relationship interpretable reliability reliability simply relax equivalent require concentrated area statistic x see look noisy correspond requirement exist functional relationship
crowd annotation sample annotation error problem mean cost structure collection formalize follow e image audio costly crowd worker sample nature procedure unbiased want difficult draw estimate unnecessary auxiliary arise statistical generally correction label shall se information hand machine insight simple correction independent construction annotate hybrid annotation auxiliary hybrid significantly utilize possible define introduce traditional collect annotate primary transfer design mining survey second moment denote per annotation annotation expensive characteristic primary precisely state design define number annotate sample population allow omit second primary unbiased third uncorrelated theorem prove sampling design unbiased unknown enough ensure p standard upper size annotate auxiliary annotated si expression derive si large annotate auxiliary first sampling auxiliary design auxiliary annotation traditional formalize p theorem collection certain cost general determine sampling begin eq trade figure less annotation achieve auxiliary annotation reduce pdf ad hybrid design indicate operate relative annotation operating point along curve relative follow hybrid eq annotation correspond annotation compare traditional sampling effective primary error variance hybrid primary annotation design annotation design mark black design cost additional design denote confusion value space derivation contain si annotate obtain confusion characterize misclassification binary matrix specificity unbiased recall annotation unbiased inverting abundance correct correction true value combine yield introduce abundance expression iy size give annotate primary transfer traditional transfer cost primary strong always rely hybrid commonly texture location annotate machine identify annotation estimate validation human estimate annotation pdf monitor community capture annotate automate annotation camera thousand image task daily abundance image annotate machine variance cross validation annotation annotation annotation task abundance day upper validate design carry survey collect annotate sample size determine hour draw replacement estimate investigate value plot cover image validation estimate survey estimate hour remarkably use sampling transfer sampling hybrid design manual effort equivalent upper calculation readily day manual annotation effort survey survey increasingly new automate approach occur coincide rapid automate annotation determine need error traditional still machine difficult subtle underlying density machine though many annotate traditional unbiased support hybrid mean confirm low sampling design b expect uncorrelated correlation verify experimentally correlation hybrid must take account simulation indicate validity assumption well option validation camera site visit satisfied would time affect appearance water vary due etc color third camera simulation estimate valid cover pp believe difficulty application differ estimate sample biased severe shift confusion valid estimation single g survey g sampling extend situation utilize separate machine matrix confusion unstable full rank modeling incorporate hybrid minimize procedure prefer hybrid hybrid unbiased maintain level base base survey audio survey population analysis survey corpus surveys medical solely design knowledge perspective future notably procedure like science california author acknowledge nsf national foundation division grant collection annotation survey wide digital form audio addition crowd utilize annotation collect population sample new novel hybrid utilize cost annotation key amount annotation need efficacy hybrid demonstrate application utilize hybrid reduce expert transfer auxiliary annotation
rich dynamic computationally tractable brief summary glm train latter section spatio temporal field number relate neuron stimulus glm relate precede stimulus former latter intensity glm spatio temporal field bold letter matrix point neuron poisson observe spike q expect spike train define spike brief resolution time spike train vary intensity entire log observe spike spike select calculate manuscript likelihood train solely intensity constant practice neuron fine effectively simplify form likelihood ignore optimization determine train stimulus spike glm intensity neuron field relate invertible infer stimulus quite provide neural response neuron glm predict current spike history stimulus vector count intensity stimulus stimulus neuron post spike history etc firing intensity decrease fire stimulus term select term simplify calculation interpretation individual precede time match increase gain likewise spike prior convolution occurrence spatial sample integration slow potentially exploit equation let simplification spatial component field algebra everywhere maxima exist particular lack concavity arise dependency model notable reduction replace derivative neuron neuron influence observation fire chance correlate stimulus reflect mechanism e electrical coupling etc naive implementation glm intensity fail distinction stimulus hold stimulus attribute stimulus correlate add post filter neuron fire rate conditional intensity subscript throughout sum activity internal neuron population subscript drop spike train clarity although likelihood implicit subscript spike coupling must generalize care spike conditional intensity preserve term pair appropriate intensity hessian neuron linearity intensity cross term neuron fit simplify neuron fit manuscript provide brief analyzing
wide composition library well phase presence phase signal library center diagram phase match ccc ex pt cpu integrate unsupervised challenging variety scientific discovery motivating contract sf suppose augment still subject admits augment constraint optimizing follow hard belong centroid non non assigning centroid close correspond initialized property proposition edu department science university van department university identify large noisy key material aim discover cell incorporation constraint addition solve outperform material precise enforce see growth many science combinatorial material discovery material property obtain hundred sample composition tool automatically analyze determine material composition important material activity cell evolution likely break material science role accelerate material accelerate discovery new material develop create library intuitively generate mix small promising library ray composition specifically x ray signal sample material material discovery call phase gradually composition term intensity phase underlie separation source basic ray source therefore factorization nmf formation map basis lattice recent enforce peak decomposition programming constraint peak respect nature lose become noise filter approach new integrate additional introduce decomposition allow specification negativity labeling general rich dependency example encode specify law propose technique call integer program constrain outperform scale world recover value compactly column ray pattern interested low input namely basic phase basic pattern point belong approximation datum mining denoise compression symbol singular decomposition produce approximation obtaining point interpret instance represent intensity ray intensity motivate compute svd undesirable example image superposition simple patch ray researcher nmf explicitly negativity coefficient negativity domain correspond upper boolean negativity hold science underlie structure chemical compositional variation lattice compound follow variation basis pattern define constraint negativity individually connectivity complex constraint motivate decomposition subject constraint minimize entry wise frobenius additional possibly require variable formalize value variable denote stack vector entry programming general encode combinatorial encode negativity supervise example suppose want explicitly formulate coefficient want entry column model application topic encode rewrite compactly appropriate semi include labeling belong q analysis basic grow framework supervise information incorporate typical enforcing point framework link constraint interactive scheme account feedback refinement merge supervise label alternatively desire temporal shift work literature single dimensional basic pattern incorporate logical negativity expressive class constraint specify among unfortunately nonconvex programming see progress counterpart either well even constraint rarely heuristic widely project apply call exploit advanced solve report procedure enhance sophisticated key quadratic operation literature leverage program integer solve improve warm search problem loop program presence inspire seminal coordinate descent upon number project quasi newton novel one take feasibility every iteration even fw fw w hold optimization problem feasible monotonically function monotonically non consistent depend scheme nevertheless evaluation play typically converge regardless initial provide experimental result capture label link complex logical describe high level knowledge motivate map nmf cluster range gene expression determine reflect basis obtain normalize entry belong assign consider information subset assume pair belong link uci repository truth generate various select point constrain first enforce non capture constraint approximately update supervision link constraint accuracy ac ic labeling match ground truth efficiently average knowledge intuitively properly take combinatorial sound take closure must link link implication deep reasoning computational overhead runtime couple second dataset g whether number class problem enforce negativity capture biological fact enforce kind prior knowledge totally technique ccccc avg std avg std accuracy approach average run show limited standard running second capture key law map identification temperature pressure phase occur chemical library involve pattern composition indeed compound lattice constant composition position ray pattern isotropic lattice peak shift signal vector free encode
beneficial close beyond limit besides decrease grow condition bind moment change turn resample unlike soon strong already note experiment common gain gain term term resample impact keep step impact marginally xlabel ylabel label anchor style axis cs south legend entry step legend style legend pos south west font opt table opt mark xlabel ylabel axis anchor south legend legend pos south west font opt mark table opt mark xlabel axis anchor north description cs south font step legend south width mark opt opt convergence xlabel ylabel description anchor north axis legend entry step legend pos south west font width mark small opt mark weight use optimize overall notice link descent update step early iteration behave like switch step crucial highlight try evidence unbalanced yahoo yahoo compose million triple ad click yahoo front page click rate row row click click compare incidence weight divide ten figure step perform well tend weight figure observe bias negligible obtain figure frequent graph mostly resample regime well divide yahoo divide potential expression consist infinite stream decomposition variance figure bias curve reach regime slope expect size difference iteration symmetry however dominate get become xlabel ylabel cs anchor style axis cs anchor bias legend pos south west style font width pos pos x variance south slope convergence synthetic xlabel iteration ylabel legend style legend south west description cs axis cs anchor south font table coordinate pos pos xlabel ylabel width south axis description description anchor south gamma opt sep mark table var regime opt txt triangle regime opt txt provide regard deduce sampling depend regime asymptotically gain however limit besides dataset difficult quickly happen slow dependency sampling allow take large extend simplicity focus least decomposition explicit interesting see logistic resample acknowledgement european project discussion hereafter otherwise thorough operator live proceed detail allowed derive need preliminary schwarz inequality one definite orthogonal eigenvalue eigenvector orthogonal definite form orthogonal form decompose g condition denote conclusion definite positive implie first long I also orthogonal restriction bilinear definite finally want direct let assume already tell fine denote large eigenvalue eigenvalue result compare true eigen divergence theorem matrix us eq matrix depend term look term always appear express operator recover soon never soon notation iterate exponentially bit bound exponential assertion proof variance indeed remove instance rest mostly bias term sup france sup paris consider average strongly case expansion exponentially decay new algorithm tight bind error may variance decay decay allow dominate dominate density lead gain term choose size significant improvement optimization together optimal scaling however convergence rate reach term come square split characterize fast show play strong practice behavior special sampling traditional bound potential gain lack asymptotic exponentially decay give tight variance decaying bias decay uniform choice dominate section dominate density gain dominate step size improvement denote optimization problem identically sample cover situation consider unseen select minimize often dedicate describe average gradient user step particular study center estimate recursion study cm gradient heavily mention convergence depend general convex logistic least sgd sgd general function partially without analysis decaying size dominant already view literature provide sampling optimize compute problem second large strict definite conjecture bind initial condition covariance bias give detailed know derive dependency term equal behavior decrease square also possible simple go assume usual rao also exact obtain decrease finally unlike bias cm expansion term tr tr behavior real often difficulty change several presence problem cost previous try optimize increase since wish objective
mean covariate eventually model model part covariate represent respectively zero base decision correspond coefficient zero first scenario important second variable rather qualitative covariate illustration interaction treatment correlation right panel choice coefficient interaction weak correlation opposite observational randomized assign patient treatment assign implement replication three aspect report discovery identify among tp identify treatment treatment estimate estimate outcome replicate selection table result estimation summarize follow first include lasso model ii small correlation average three get reasonable provide score small variance correlation variable much table treatment implement treatment especially approximately select important estimate treatment regime decision partly big treatment regime value among rule treatment lasso method iii method may treatment regime lead eight table magnitude nonzero treatment regime compare path trajectory variable increase path solution path allow ability setting number important large method include iii mis model method still lasso implement learn mis seem method method treatment select important moderate identify variable treatment competitive advantageous star conduct effectiveness treatment participant age initially participant level participant without satisfactory option acceptable level participant seven strategy switch augmentation participant cognitive ct available cognitive without satisfactory ct call level participant augmentation either li participant satisfactory randomized participant follow year base estimate treatment significantly bootstrap sample bootstrap value bootstrap confidence interval value article treatment important make select final outcome clinical trial important variable regime small value propose combine rank select decision rule combine comprehensive stopping tune maximize outcome treatment extended censor modeling expect treatment decision disease desirable stage h x different group treatment treatment correlation panel fit line triangle treatment dot treatment select nonzero score stand discovery stand standard setting size ii observational model ii treatment mean mean regime average replicate stand error treatment proportion individual value cs score model ii iii observational model iii x score identify variable tp positive correctly important value cs size tp randomize iii model model ii iii iii x opt treatment mean treatment regime outcome treatment correspond cs score iii iii observational three variable dash blue dot eight combination three choice black solid red dot dash three three choice solid blue dot h choice eight important combination baseline covariate red line dot dash lasso loss energy ability concentrate death patient friend patient complete medical private important able thing impact family status current status currently rate rating quick history diagnostic screening axis baseline axis ii hx hx hx abuse family abuse family intermediate medical patient rate percent sr frequency rate intensity score risk patient study daily care clinical optimal treatment clinical get attention focus ignore treatment qualitative interaction indicate characterize treatment individually article advantage method select qualitatively marginally versa optimal propose stop handle small perform practical setting clinical trial strategy heterogeneity characteristic clinical genetic paradigm duration type adjust tailor effectiveness traditional benefit interest regime clinical trial observational study optimal regime sequence stage latter treatment regime sequence tailor expect dynamic regime clinical trial study marginal treatment two popular backward induction deriving regime modeling value regime extend enjoy method outcome weight treatment regime experimental dynamic treatment regime star intervention disease study collect clinical clinical collect medical history effect collect clinical covariate decision interpretation treatment regime select could facilitate work alternative star star study clinical trial treatment strategy provide baseline patient medical intermediate medical decision next select useful decision expert regime area statistical selection technique focus technique medical make distinguished predictive qualitative effect variable vary method implement regime carry qualitative interaction design categorical covariate test conservative test least regime estimate variable propose penalize least correspond shrinkage penalty interaction select directly select interaction characterize qualitative patient treatment many covariate examine spurious advantage goal qualitative regime sequential advantage additional improvement treatment new advantage variable sequentially size another procedure include marginally jointly avoid redundant information propose stop criterion sequential decide introduce identify regime treatment decision demonstrate illustrate star clinical clinical trial observational exposure give subject interest option subject denote possible code accordance find treatment regime covariate subject response response summarize treatment mapping covariate find treatment regime treatment regime introduce potential outcome concept opt regime mapping assumption essential make identify regime treatment outcome influence straightforward show find expected response estimate need optimal current pay attention select information evaluate quantity characterize degree interaction score show possibility covariate eq tx eq note always covariate treatment covariate optimal covariate capture magnitude interaction subject treatment example treatment interaction treatment stand change knowledge reflect degree qualitative characterize help find regime limitation tend interaction nonzero covariate variable evaluate individually score variable crucial account base score sequentially covariate convenience arbitrary
gives project adapt increase achieve lm scheme slight overall improvement show effectiveness couple minute confirm effectiveness adaptation different pair translation score see hand baseline english language pair may complicated system day day lm adapt baseline n adapt adapt post addition investigate adaptation consecutive day day correction system correction new third quickly target list epoch day day create corpus day respectively remain randomly datum task incremental method yield english experiment five consecutive coming decide precede day day fourth proportion decrease combine various baseline day day day day importance adaptation domain available relative day vary project effectiveness impact adaptation various translation experiment provide score adapt sake clarity score system day adapt quality improvement day also score use three human translation provide european gain style adaptation beneficial improve task thorough technique adapt language want integrate correction statistical text concrete need propose speed work already perform document explore strategy lm weight network train combination sample original avoid lead fast language gpu experimental effectiveness translation english observe school science university le france fr fr fr neural outperform back like recognition day efficient adapt language instead mixture adaptation result cat environment fit lm important role natural back gram base embedding jointly language model popularity confirm many systematically gram significant last recurrent lstm translation system entirely network large corpus adapt new ability change property operational occur translate daily news article environment typical integration cat want correction finally want lack various involve neural lm representative overall perform correspond concrete need human human propose translation hypothesis post day already translation sentence propose translation perform translation day sentence adapt next task usually around next continuous space popular adapt lm corpus merge minimize domain development integrate specific linear system extract available lm turn lm cat environment investigate datum investigate neural community incremental presenting could perform adaptation namely recognition mention convolutional adaptation adaptation al explore output study corpora sentence three model gram rnn lm system variant adapt recurrent lm area speech early rnn speed adapt lm history automatically example want translate language align popular phrase base model translate short together translation lm gram log translation optimize idea speech build explore variant technique interesting nn work closely cat post provide human tool update phrase language european text resource summarize c en en l lm approach day day adapt system build procedure selection parallel extract representative development language translation method back lm base feature optimize score development source log optimize final translate human translate process day post hope rest translate procedure rather human translate approximately day percentage generic example per epoch none sec sec sec lm window experiment dimension layer neuron short short account token corpus converge epoch hour gpu adapt system analyze project document two adapt translation name limited could quite easily informative development ideally text loss generic adapt keep achieve always randomly
modify use context first introduce give exponent clearly away density exponent determine calculation consider boundary version describe bold width situation equal need ignore logarithmic rate rate typical rate assumption condition need impose exponent level smooth exponent usual begin data separation satisfy data set like yield provide n compare cluster exponent corollary extra corollary controlling sequence case estimation easy estimating illustrate estimation detail well estimate sufficiently small fast choose decay sequence corollary choose slowly decay decay motivate thus consequently exponent one hausdorff assess essentially however scope interesting whether true hausdorff achieve report last derive convergence case good rate sequence parameter little issue propose dependent recover without know recover present strategy user specify run family corresponding ensure apply end ensure elementary assertion sufficiently show n n yield establish observe right assertion theorem less b second smooth smooth exponent combine check end nn n sufficiently yield c sufficiently elementary calculation moreover always suitably guarantee apply n replace consider give proof use goal theorem hence sufficiently analogously g monotonically q estimate theorem proceed intermediate goal fix show use assume generality assertion write nn sufficiently restrict conclude n monotonicity q sufficiently c combine estimate obtain assertion case exist assumption analogously far sufficiently nn assertion finally prove recall already apply right side eq appear g n assertion proof suffice assertion sufficiently inequality consequently sufficiently third fix elementary yield inequality combine assertion theorem theorem issue propose generic small feed analysis show consistently small present strategy involve density class definition cluster propose I cluster set study reference unfortunately number cluster generally rule use couple create clustering dependent estimator useful tree avoid level focus identification structure level detail show linkage recover similar nn propose pruning remove recovery component level estimate set estimation article quality symmetric hausdorff two respect hausdorff metric clearly structure eventually fix contrast respect suitable set topological recent base use water modal flow suitable smoothness modal lebesgue single cluster one dimensional see none infimum usual avoid detail define make difficulty rule neighborhood mass topological connect zero address issue avoid underlie kernel compact distribution lebesgue recall functional remove whether cluster infimum component structure persistent persistence either implicitly g seems deal uncertainty namely seem opposite exactly connect compare consider look similar cluster modification discuss scope generic estimating level enjoy vertical horizontal estimation guarantee conduct level first derive consistency connected move main establishe well know describe mass boundary restrict boundary rate choose suitable therefore drive characteristic strongly build contribution consistency establish consider density know density new contribution add impose last least paper generalize feed description drive proof section proof auxiliary example paper refine notion definition cluster develop notation assumption throughout always borel fashion course interested lebesgue measure sphere possible interior closure boundary denote indicator set paper deal distribution density generality make density consequently notion somewhat canonical serve choice density long densitie distinct component become inconsistent neither two alternative readily available suitable relevant set horizontal thin line cut h issue fix bx define consequence call way whenever e level small closed satisfy modification say every continuous satisfied density density think range level absolutely help notion relate connectivity subsection make motivate definition generalize idea note partition informally speak break fine partition emphasize iff distinct iff relation case say empty set closed component well discrete connect partition call eq minimal figure illustration large horizontal order sense whenever aa dot indicate contour distance sup norm since component situation concept subsection cluster cluster normal three either cm condition see component two persistent gaussians together level coincide connect consider component open dimensional contour line line indicate thin line level level time uncertain vertical uncertainty cause subsection complement deal horizontal cause quantify horizontal denote operation closely relate operation base use estimate vertical sense ideally cl cm cl unfortunately absence cm cm cm cm cm cut connect connect component rest condition thin p rich lead indicate thin shape thin thick turn part summarize thick level follow statement hold add remove surprisingly meaning express even specify together effect significantly thick solid thin solid component persistent dotted line indicate within cm connect thin show dotted indicate within left thus behavior exclude help level summarize borel absolutely addition denote present analyze generic generic vertical horizontal relate component satisfied follow disjoint suitable detect identify identify c relate component step figure small stop soon ex thick plug solid line bold horizontal component vanish theorem bound component extend connect component satisfy dl feed estimate ensure partition geometrically well behave recall partition moreover example family partition partition lebesgue surface compact haar sufficiently manifold equip slight abuse dirac provide let c ph feed parameter satisfy conclusion estimator approach establish uniform unfortunately level become include approach open question differently address issue result construct density approximate grid bind cluster lead result modification c
relate university xu reviews american ann mi mathematics ann usa department computer usa global ny usa dedicate fuzzy combine diffusion map theory algorithm paper reduction achieve diffusion system representation description use fuzzy dimension reduction discuss describe nonlinear reference enough integer nn connect edge gaussian nonzero real moment property interpret scale result symmetric w interpret probability sum define increase local geometric datum integrate make possible broad path connect high path diffusion use aim find preserve optimally mapping distance preserve eigenvalue j choose element eigenvector point reduce dimensionality dominant correspondingly rate reason spectral range method correlation influence affinity metric interaction dominate affinity wide kernel kernel application sparsity handle isometry section describe second fuzzy real fa architecture consist node neuron field neuron activate pattern already correspondingly neuron encode layer weight pattern neuron neuron calculate define break prototype fuzzy subset input winner winning becomes activate determine match eq weight rate hand criterion meet back win competition project meet neuron represent art advantage stability plausibility advantage scalability speed parallelization art ability network complexity robustness map site origin tumor particularly important cancer diagnosis profile blue cell tumor publish diagnostic child widely extract information set hyperspectral band pixel hyperspectral contain band amount hyperspectral bring processing hyperspectral sample address hyperspectral identifying
highlight application easy programming library architecture share memory core cluster hardware graphic gate array integrated circuit distribute programming library various architecture inference underlie hardware automatically task system light upon problem memory medium low gpu low medium medium medium medium library machine core visible shared pass cpu core storing low meanwhile hardware reason write program towards drawback include capacity library support prevent resource atomic operation increment besides alternative framework multi multi queue library specific parallel pattern synchronization barrier green choosing device computer single gpu provide small compare processor easily easy maintain code dedicate device low power consumption pattern core ml limit cpu framework core core user specify gpu gpu move framework intel counterpart accelerate gpu parallelization population well smc develop hamiltonian demonstrate example accelerate collapse gpu framework particular gpu cpu use hardware investigate distribute user data synchronization process decide include receive barrier process besides framework synchronization handle message globally process invoke procedure process pass execution provide primitive library parallel review distribute framework execute task reader book large online computing read disk transformation disk machine key system pass key user generate hash intermediate pair machine parallel pass key user often often latent likelihood gradient aggregate ml ml open source collaborative dimensionality topic read disk overhead iterative well interactive distribute distribute disk automatically user store computation parallel parallel serial look iterative ml gb communication interact read key variational bayes require come compute computational show graph engine receive send last update vertex along gibbs easily conditional gps feature e compute asynchronous flexible scheduling pick queue pass vertex data vertex adjacent vertex finally adjacent vertex queue note long serial ml task matrix gibbs gibbs graph disk base version framework restrict communication worker communication allow pattern worker share count collapse lda asymptotic probabilistic derive simple scalable mixture go let progress dp mean extension dp mixture svms progress advance scale method derive back model moment always produce estimator algorithm matrix algebraic name moment perform extremely show computational markov wide domain include recent advance big subsampling distribute helpful exhaustive field learn database parallel system language considerable model widely become day science project google effort provide cloud service microsoft similar provide interface help language specific sophisticated automate process automatically interpret easy automatically report effort currently data acknowledgement support project cb cb nsf china project research program finish version put later section put concrete example independence base divide style asynchronous variational lda model get variational approximation posterior purpose collapse gibbs collapse conjugacy analytically k c z collapse collapse integrate ik quite optimize loop component count sparse reader reflect circle update change communication parameter however immediately train chain partly identifiability dependent style mh divide choice seem serial corpus computation worker count global fit server mention implementation detail ad lda counting aggregated update processor update incorporate parameter interesting look inference em step fix sample integrate well explain without account assignment processor divide infer lda aggregate local model optimize divide general share ad lda implement aggregate note ad lda lda drawback synchronization end sampling network yahoo lda issue replica machine processor avoid update put count finish document document vocabulary likely matrix yahoo lda overlap count local global yahoo lda synchronization policy yahoo background word w server server another direct instead want employ coordinate update eq document independence integrate ci natural reduce style mr phrase independently phrase simply aggregate triangle update algorithm integrate ci document wise variable ci integrate become style require algorithm core machine ad yahoo yes mr chen cn growth availability interest learn system bayesian scalable survey bayesian term include infer regularize flexibility algorithm subsample deal regularize inference live engineering massive stream become increasingly besides volume increase highly uncertain primary machine become field challenge big cover big problem big need deal learning dimension development spam explicit dimensional many scientific problem challenge regularization salient classifying image thousand million imagenet consist million concept average thousand wikipedia document category often category structure direct acyclic explore massive million become extract multi grain datum application speech model neural auto probabilistic generative model practitioner often slow factor conjugacy intractable integral several deal physical randomness incomplete principled combine evidence intuitively flexible offer characterize big elegant deal collect bayesian rule suitable deal big stream grow exponentially grow slow amount shannon increasingly leverage powerful computer deep capacity fast therefore serious effective become increasingly relevant big capacity address approximate learn term model learning must information change goal evolve must feed dynamic datum flexibility must flexible handling side structure digital sensor scalability scaling modify advantage grow article provide literature survey advance big hide variable specify core give datum likelihood marginal involve intractable integration th bayes sequentially useful formulation distribution make objective optimum identical fact add minimize equal interpretation significant aspect variational method make flexible incorporate soon bayesian posterior conventional distinguish bayesian post posterior bayes project density general bayes minimum bayesian ml range single variate regression semi scenario essence p p assumption assume ratio automatically model costly abc bayesian biased integral involve typically analytically category see method omit examine single integral variational history physics economic machine theory graphical reader seminal book nice overview cast feasible variational formulation posterior show objective maximize bind intractable target parametric e parameter solve descent often optimum integral approximate replace parameter variable assumption q em bayesian mc diverse repeat explore systematically method set common give density pointwise normalizing replace carlo converge number suffer severe limitation dimensional space book article detail monte powerful scale importantly advance later pointwise weight heavily proposal construct ergodic markov converge e hasting construct unnormalize prohibitive massive mcmc iteratively draw standard gibbs sampler sample draw convergence effort spend gradient mix rate langevin anneal sometimes handle mode gibbs sampler include sampler ordinary convergence replace ordinary conditional distribution marginal distribution question use method theoretical state property informative infinitely bag improper jeffreys prior use admit good frequentist contrast may since practical method hierarchical bayesian assume prior weak thus convenient put hyper empirical hierarchical dirac section progress make characterize empirical bayes empirical practice tradeoff computational conjugate inference belong dirichlet multinomial conjugate pair likelihood prior dirichlet z normalization factor posterior explore conjugacy allocation fig document topic vocabulary topic iw z lda popular application impose except provide impose correlation dimension topic model model infer correlation flexibility scalable prior integral practitioner including review much emphasis bayesian scalable review nonparametric bayesian infer improve flexibility leave section order challenge property learn requirement evolve feed time scenario objective handle rich input visual sensor scalability massive advance flexible scalable interpretable parametric pre specify matter may limitation especially priori may optimal datum change ideal figure latent factor structure factor abstraction level nonparametric bayesian elegant adaptation capacity define rich space dp ibp briefly review ibp reader article nice comprehensive dp first specifically concentration dp dp discrete surely atom independently base assign atom constructive stick fig unit break infinite segment stick remain segment beta break remainder dp insight develop variational algorithm chinese restaurant crp define partition integer crp derive restaurant customer restaurant customer subsequent customer table customer define property integer crp parameter dp crp step crp enjoy nice sampling dp mixture slice infinite sum condition assume single reduction analysis assumption weighted combination factor loading influence term usually ibp variant factor grow binary indicate latent ibp define process unbounded ibp derive crp infinite arrange customer choose first customer customer customer choose customer ibp role crp play unbounded latent dp de distribution crp mix distribution ibp admit stick length break dp stick length stick length development part machine extensive practical probabilistic function gps supervise model condition training matrix target value involve inner product feature bayesian simple example gaussian define gp characterize stochastic I examine requirement task conjugate review research process meet big develop sophisticated group spatial group different multiple work present ibp topological layer latent layer number hide unit recent extension concern dependency special hdp training framework review dependency nearby important dependent dirichlet dependent crp dependent ibp dependent ibp biological relational adopt ibp feature nonparametric membership discovery latent community advance extend scope equivalent variational build define nonnegative ordinary fig bayes question enforce constraint incorporate sparsity example max margin paradigm supervise lda additional design likelihood imbalance problem margin simplicity discriminant I ij average expect surrogate classifier classifier randomly make regularization adopt strategy relate upper bound lead imbalance issue relationship part choose likelihood second restrict regularization much flexible prior regularization bayes exist achievable bayes rule affect difficulty solve formulation duality theorem deal dd stochastic iteration fast convergence c approximate mh big scalable advance sampling multi inference optimally efficient unit computation model variational develop explore redundancy subsampling example reasoning create overview state variational descent sgd randomly update estimate gradient estimate appropriately infer update two global global lda assignment collapse illustration model inference consist draw mini parameter use search manifold probability tuning learn rate average stochastic propose use gradient trade variational model expectation method additional tight variational bound another monte variance auto bayes learn neural continuous ig gradient naive depend variational underlie effective technique need known parameterization representation parameterization cp exist minibatch estimate l maximize possess strength cp conversely hmc differentiable posterior dependency extend deep similar sophisticated applicable exist group three category sample mini gradient mix rate systematically various langevin example langevin dynamics mcmc produce noise isotropic p proposal prevent correction successfully monte gradient dynamic hamiltonian dynamic replace uniformly sample mh acceptance zero rate rigorously justify improve scoring fisher stochastic randomness subsample similarly riemannian manifold stochastic riemannian langevin simplex mh another category approximate subset eq linearly prohibitive set mh hypothesis testing allow reject fraction require mh theoretically compute easy frame replacement decide mean prescribed mh derive new stop bound bernstein adaptive mh target mh apply datum augmentation present method posterior novel augmentation indicate pz alternate update condition conventional likelihood random subset method stream mini know streaming go measurement update time tt role variational naturally stream rule perfectly suitable stream challenge evaluate posterior conjugate hide updating analytically kalman contrast complex bayesian model linearity close posterior intractable various develop bayesian p one streaming analytical form stream generalization passive result svms latent structure latent representation pa complex discover structure allow complexity latent hdp model resolve topics monte approximate smc resample large smc store expensive particle degeneracy processing bottleneck simple model derive kalman broad model density filtering develop extend basically approximate conjugate filter df df draw surrogate sufficient along df produce recent progress made distribute parametric family family solve tool
symbolic combine extraction symbolic computation symbolic context proof try often external select thousand theorem corpora ii internal reasoning evolve strategy corpora specialized work start complement first corpora main idea level library library attack library number approach number corpus contain million motivate lemma ai deal rigorous experimental evaluation corpora mining core scenario discuss conclude task automate large knowledge proof reasonably library construct isolate axiom previously prove theorem range thousand library parametrize proof number promise parallelization theorem measure loading library design fed find write theorem find already nothing current conjecture part far like include reasonable year experience library following indicate human name name considerably good library large weak various statement library library experiment ai library ai short turn hard name corollary many lemma omit prove depend focus variant complementary also necessary extent alternative ii far atomic inference library million hundred million efficient small orthogonal corpus name theorem corpora way quality reasoning system proof summarize tool initially art implement loop hundred lemma run index redundancy control age similarity lemma inference tool lemma produce successful run problem extract generalize thesis estimate dag acyclic inferences tool lemma lemma inference subgraph node axiom well ii stop minimal select lemma characteristic lemma characteristic include complexity implement idea experiment automate use lemma exist proof similar successfully mainly algebraic add large million early complete library library index limit style contribute importance graph try relative large popular appearance web network centrality implementation easily nod corpora core corpus corpus version core name algorithm intermediate name name initially theorem lemma put pass argument may common like alpha de represent keep operation table htb name name constant mb mb corpora intermediate style inference whole formula big neither disk trace obtain inference additionally intermediate obtain trace take hour cpu gb ram ghz consumption version intermediate call graph lemma trace table graph lr lr inferences inference trace edge lemma additionally symbol trace free type external together originally trace present trace theorem theorem explain trace alpha check alpha would obvious kernel keep trace lemma normalize variable lemma external produce version program replace alpha still keep dependency hardware core leave normalization process graph clear lost information lemma keep graph produce differently atomic construction intermediate drawback make mining big arise decision proof thousand core inference notably inference inference proof encounter step produce justification produce intermediate lemma justification level execute visible execute recursively step perform detail typical natural proof give trace small typical order magnitude order formal development decide look building gb million normalizing well distinct alpha normalization leave intermediate dependency whole post dependency edge trace format come trace theorems create goal proof user version proof interesting point view removal measurement version lemma clause operation like interact level record define perform efficiently define notion explore combine various dataset follow direct implementation modification way cut advantage tool necessary directly available define trace dependency iii use symbol general lemma name lemma axiom formally axiom stop axiom name lemma recursion define stop dependencie eq apart behind heuristic necessary hard need useful lemma conjunction lemma quality record protocol lemma express count main load hour take gb ram unfortunately experiment always integer quickly integer wrong extent chain inference q apart modification minor change need inference clause clause additionally create clause create artificial clause centrality graph dag neighboring node income node advantage minute ram disadvantage comparison previous take account already modification initial score advance perhaps could still keep overall reasonably another disadvantage weight quite counter base need important turn important reverse normalize combine sum try lemma come mathematic choose final dependency choosing theoretic cut cut library many cut cut cut htb graph library name mark gray give node name marked gray dependency newly dependency provable exactly assumption easy ai start name node dependency dependency choosing cut edge include edge edge make slow finding take previous subsection try limit lemma parametrize lemma already predefine naturally name name name name choice empty set name depend whether want lemma complement name theorem expensive second graph change mean lemma take cpu hour experiment limited core mine several scenario rigorous formal corpus quickly rank produce method look plausible scenario corpus compute corpus set newly theorem originally name theorem element way human ai parent ai direct produce train dependency precede new lot success first start metric precede equally use precede lemma consist lemma close lemma early newly parent whose take proof exist prove evaluation whole core remove resource hour take cpu hour evaluation core scenario lemma guess lemma still must provable lemma allow measure good new kernel trace strategy lemma trace hardware use server ghz ram mb cache lemma trace statement lemma store run part need feature lemma independently trace trace due intermediate implication take hour extract lemma lemma without hour take usual evaluation detect preserved version recursive theorem information library service theorem evaluate preserve name change preserved perform far choose evaluate ai method core union optimistic limited metric go theorem theorem scenario precede mining learn stack easy including solve together solve lemma show table old experiment cccc strategy theorem name combine evaluate trace theorem rate new alpha alpha normalize version come theorem seem table add big add million look whole next strategy support computation consider seem suggest focus either bad seem divide change real arithmetic big intermediate lemma success success create trace try translation hand semantic formula involve try name much structure theorem preserve preserve initialize translation formula suggest rather reverse opposite big come htb cccc reverse success formula resource intensive small core confirm method mean mining evaluation solve original inference solve divide evaluation middle good mining solve cccc unique theorem almost table various theorem comparison old
preserve view operation compute projection satisfied projection compare projection converge plain forest increase behaviour rademacher variant optimum improve random forest slow notable random forest theoretical sparse projection provide well tradeoff one algorithm popular dimension principal repeat projection generate decrease eigenvalue accord previously implementation package computing time mac perform mac os load memory execute condition output sub project projection respect time project output tree thus forest explore projection label study variant approach theoretically outperform term allow drastically size output time remarkably adjust jointly output forest improve adjust reach time tree prediction output similar multi classification sparse office sum variance pair eqn divide subsection adapt bias supervise algorithm carry assess effect projection obtain perturbation scheme e bootstrapping selection denote decompose l l rx bx l respectively residual variance decompose law first term randomization forest randomization decompose variance forest randomization eq ls ls fx ls ls ls ls ls e form I random different choose algorithm compute denote form projection like tree take l ls ls thus argument term decomposition ie l ls f l e l ensemble thus second put one ls rx b l rx xt l rx dataset biology scene domain music descriptor video domain classification study multi treat hierarchy dataset image infer feature entire drug drug interaction infer protein output characteristic htb dataset medical cv go go cv combine projection see brief description random lead forest ie superior dataset increase randomization projection improve forest bold last deviation bold highlight rf one deviation dataset grow decrease randomization output like drug robust however drug interaction really baseline forest tuning randomization may different dataset adapt projection output enhance complexity prediction different bias broad lead reduce burden supervise multi train label typical application determination topic address object category image many hundred hundred output pose address approach classification call relevance train independently classifier ie tree split sum score leaf label relevance build single account label dependency requirement storage compare addition output intrinsic irrelevant make attractive address multi problem complexity similar feature limit deal approach reduce label compress output original space decode compress explore compressed case linear learner times stage predict add decode error projection explore random forest label subspace forest score exploit ensemble decode leaf label empirically ensemble space reduce accuracy computational inherent tree ensemble different output theoretically idea good problem significant input randomization la output optimally large scale paper multi projection present property whereas discuss number sample nn supervise minimize output subscript vector input follow pre pruning split among feature selection subsample average leaf obtain aggregate output reach leaf output variance vector material q furthermore wise notice output multi statistic make thresholding tree build unseen predict aggregate learning generate among optimize split selection irrelevant simple price high high mind recall maps notation lemma matrix probability draw sparse rademacher obviously grow sparsity sense exploit random projection computational burden multi tree dimensional space idea subsection analysis point output single tree constitute bottleneck variance projection space modify denote project output generation projection project aspect empirically carry derivation supplementary material multi correspond random capture build bootstrapping error square bias decompose sum l fx ls ls ls algorithm supplementary material l l fx ls variable respectively parameter appendix l l rx xt rx result bad generate different problematic always prefer tree randomization randomization could nevertheless output term learn large variance dimension project subspace tradeoff randomization randomization low affect test ranking express label learn discard thus express htb curve represent value split ht label rank average display fold cross validation mean random forest standard deviation mm mm mm scene medical go cv drug behaviour feature cart converge around expense compression behave forest notice tree outperform output accordance inferior assess collect different dataset material reference range ten fold see compare learn learn subspace three value
illustrate consequence coverage confidence greatly bias thus coverage probability smoothed year nearly deviation fourth ps generally bias expect mr survival correctly extreme proper survival general give reasonable lastly censor scenario decision time point treatment assignment applicable bernoulli distribution probability baseline generate survival first function censor censor patient uniformly survival censor censor consider scenario easy year survival treatment time complicated function induction regime g x clear regime optimal ii treatment regime maximize year survival design finally base empirical survival search method st year treatment regime normalization scenario know simulation result summarize smoothed estimation nearly unbiased estimator treatment table survival regime se iii estimate regime nominal level iv smoothed survival largely coverage group randomize clinical trial four group plus large count treatment curve well survival give survival day treatment patient baseline clinical covariate historical found cd age may covariate goal survival notation cd come hazard counterpart study associate survival number intercept age treatment year estimate treatment early may treatment assign another patient treatment assign j nh nh identically zero process regime derive score specifically specify w establish consistency asymptotic respect n x e mean process delta ii theorem maximizer u iii theorem finally argument eq cumulative cumulative distribution function thus fr integration density process second taylor f fu combine p prove regime augment estimator survival model model numerator converge correctly specify denominator equal second term numerator equation term equal cs asymptotic survival u mean zero due expansion algebra note equal survival correctly I u weakly mean gaussian process iv establish regularity condition accordingly incorporate stage proof omit ps se cp mr f correctly specify ps ps cp mr f f ps correctly specify ps mean ps c c censoring rate indicate ht ci denote treatment approximation ps mr mr ps mr rate rate mr mr rate f ht ht low upper low ratio upper upper upper ratio ratio ht vs ht vs vs ht upper vs vs vs vs regime patient grow finding regime clinical outcome patient disease primary patient survival article estimator treatment regime treatment treatment regime index survival regime suitable conduct propose various clinical trial probability treatment survival disease cancer treatment favor heterogeneity study primary death plus plus treatment divide age plot treatment specific figure plus treatment treatment age plus probability plus old ht raise question assign clinical interest year survival probability regime decision patient complex disease addition disease may time treatment rule observe fast estimating treatment dynamic regime parametric call addition enjoy robustness estimating equation model value specify intend regime estimation learning method treatment weight machine regime treatment outcome clinical observational treatment regime maximize survival focus compare observational treatment assignment survival give treatment give doubly specific survival base observational patient predict level develop pre time recommend different accordingly optimal treatment maximize develop censor treatment finite bound policy learn proper time incorporate treatment covariate interaction effect method maximal year survival specifically survival regime regime regime maximize associate year treatment regime suffer numerical instability sample introduce value numerical survival investigate generalize estimate dynamic estimating single dataset clinical trial discussion proof option patient baseline covariate patient survival survival censor distribute risk treatment regime map simplicity index contrary potential counting risk survival time g maximize year survival year find treatment make uninformative censor survival censor estimator censor observe proper make causal unit treatment I assumption zhang et cast miss patient actually receive patient miss modify incorporate pa clinical need observational maximum specify derive censor clinical study censor e censor censor censor restrictive relaxed censor assignment treatment specific censor survival censor base censor censor inverse score estimator time numerator denominator censor certain treatment regime year survival rely specification improve proportional ph conditional cumulative hazard g I respectively two augment doubly property unbiased survival base fit censor base treatment regime year study smooth plot intercept curve maximization study conduct section survival treatment regime bias specifically g cumulative go bandwidth tc bandwidth ensure smoothed bandwidth red see curve regime estimation dynamic treatment regime incorporate point simplicity presentation decision patient covariate receive baseline beyond covariate ii treatment e coincide assignment consistent regime patient initial coincide censor survival patient treatment g commonly inference study dynamic outcome potential outcome actually receive consistent assign randomization treatment receive potential year survival inverse weighted survival regime patient censor take weight ia I ia observational say logistic smoothing h survival conceptually accommodate treatment decision may become reliable patient treatment regime asymptotic propose theorem regime weakly I
optimize log call encoder decoder kl bind ascent gradient straightforward gradient introduce trick variable univariate kl divergence integrate analytically refer appendix encoder set recurrent state vector sample encoding decode rnn hereafter update file binary know video game sample hz inspection become song optimizer make learn representation especially important optimizer inspire momentum bias create divide overlap instability learning decrease gradually final position song certain space modelling underlie also train time start learn generate music epoch show space order visualize ht latent decode train generate train overlap sequence yield representation dimension point music second encoding point randomly create call use possible rnn effective improvement dividing song possible point reverse step strongly time current approach lstm direct denoise unsupervised improve music addition complement supervise rnns com rnns variational auto generate facilitate rnns rnns exhibit suitable capture temporal music model development consist
study epoch act parameter universal sure risk sharp sample iterate towards understand multiple learning rely procedure massive deriving procedure generalization allow rather amount focus observation iterative data termination early regularization recently machine learn learn bound square updating process iteration large cost practical develop sequential aim develop keeping emphasize role complexity help avoid achieve achieve suitable original minimization restrict suitable alternative possibly way procedure process pick mention adaptive assumption strategy analyze number pass parameter trick example property heuristic online solid term towards excess iterate sharp matching possibly develop condition cover entropy dimension theoretical early stop incremental towards epoch rest defer material composition denote hilbert schmidt essentially paper hilbert rkhs consider develop functional reduce finite norm study distribution let define priori sized suitable increase fast failure pass averaging epoch recover recover gradient choose sure rely finite nonempty stopping help error excess since proof statement conceptually section great equation arise epoch optimize priori act suggest multiple pass beneficial rule cross adaptively low capacity sharp iterate sharp improve far therein non incremental incremental behave prove incremental capacity several square include different proof term build two quantity bound contribution prove due fact pass statistical iterate known expect iterate triangle q summarize error paper xt state main step equivalent step lemma derive follow recursive step initialize inequality assumption plug error obtain
specifically invert convolution convolution evaluate illustration uniquely show form time output multiplication matrix multiplication optimize multiplication less accordingly convolution multiplication mention early parameter mean approach care early however disadvantage forming involve implementation piece iteratively mini batch parallelism implementation lead multiplication effectively utilize gpu computational intensity write read memory traffic direct accordingly opt implementation explain another fast engineering neural use must input especially costly small compare often happen convolutional additionally early reduce subset nature follow step drawback although agree useful approach directly efficient specialized implementation handle many corner implicit implementation often optimize part convnet poorly batch optimize library maintain architecture something easy architecture routine fraction throughput size successively compute submatrix memory matrix multiplication take arithmetic although solution memory rather matrix routine require matrix routine mapping boundary convolution accordingly map load correct dynamically proceed convolution modularity modularity deep compositional schema backward engineering derivative flow device accord framework unify memory interface allocation raise framework make self contain descriptor function modularity framework preserve isolate layer definition purely additive development comprise implement layer protocol buffer schema include computation layer protocol layer scheme library descriptor setup backward call make respective layer implementation drop interface store device hold descriptor solely descriptor exploit reduce consumption group convolution filter backward pass gradient second speedup backward propagation testing illustrate gain integration schema unchanged layer implementation engine fall outside scope execution identical deep project integrate internal set convolution use domain besides processing speech language non square experience consumption multiplication mini batch integrate thank expand firstly convolution bring attain matrix multiplication hope gap secondly support useful speech video would like library multiple accelerate training library reliable provide require evaluate parallel architecture continue library provide library com berkeley berkeley present deep consume evolve make maintain issue long address library basic algebra analogous deep library implement processor implementation computational must parallel address optimize contain integrate exist framework convolutional reduce solve many processor computation arise network efficient implementation provide implementation explore significantly lead speech among neural implementation convolutional cnns deep neural network kernel differ traditional dense algebra deep framework implement operation activation execution deep community successful kernel architecture evolve must significant optimize kernel understand careful scheduling movement acceptable performance believe library computation several benefit deep kernel hardware secondly parallel evolve diverse diverse hardware separation concern allow library deep understanding architecture make framework take library flexible framework immediate rigorously maintain reliable processor architecture library minimum auxiliary case mini batch primary goal neural framework even provide abstraction low computational simplify integration primitive operation store keep low level support variant single arithmetic convolution pooling activation library allow indexing section image auxiliary tensor easy
vc e decision tree support smoothly parameterize relatively assumption dimension support datum vc principle estimate simple use rademacher bind allow remain piece absolute rl mapping complexity estimator equation use simplicity rl similarly I add additional policy provide note e indexing specify see fortunately example margin ordering structure rl consider consist combination may impose limit magnitude therefore may policy policy fix requirement form eq policy two maintain problem rl leave investigation automatically future maximization rl compute use therefore batch naturally grow follow demonstrate return mr learner variety domain world perform return evaluation comparison mr mr knowledge rl chosen approach discussion policy provide impose expressive allow mr maximize toy monitoring domain policy amount mr comprise radial function piece artificial trajectory episode toy understand reader world attempt presence dynamic evenly spaced radial impose performance solid mr figure mr fits illustrate red large mr select class fit place evenly impose figure toy fit early policy class amount datum grow policy monitoring world camera sensitive location camera observe locate sensitive additive camera dynamic take max radial basis limit red domain mr grow see relate reinforcement knowledge develop mapping represent rl structural rl additional theory bind performance extend policy aim prevent fitting amount frequentist lack true dynamic function deal grow see work require setting treat either class value indirect policy reinforcement appropriately sized policy provable extremely weak assumption rl allow theoretical previously bound allow structure maximize demonstrate mit reinforcement attempt choose policy return amount class size available principle structural statistical rademacher complexity identify maximize return policy class give unlike batch require system reinforcement decision reward agent straightforward batch rl data dynamic minimize error e rl prediction overcome limitation explicitly maximize datum explicitly estimate return poorly return estimate overcome principle instead return return control allow policy principled main contribution return weak standard batch rl result rl study single transfer bound return family policy review move rl return tie section provide bind policy discusse exist rl demonstrate build intuition reader discuss work paper completeness clear reader input class decision formally commonly solve give risk attempt principle analogy policy return rl hoeffding holds bound thought rademacher literature additional g rademacher complexity study dynamic constant mdp unable overcome interaction equation lie collect type policy episode use empirical n n n n hold episode policy call monte evaluation policy evaluate policy build section episode free carlo attempt artificial episode policy batch artificial episode episode start ns episode bind eq maximum see regard expectation begin move use equation least
remove data column project break equivalently projection estimating massive difference critical value large test deviation explore simplicity assume identity contain iid ignore follow pp km hold detecting differ coordinate relatively difference keep unchanged complete competitive next follow great consistency detecting search sufficient consistency compare asymptotic test first major compete numerator formula r sd z superiority diagonal sd test covariance p natural alternative coordinate equal non rescale alternative rescale two distribute choose dimension choose dimension describe section projection choice matrix choice theorem figure appear indicate invariance cutoff matrix power significance level nominal indicate level case monte first empirical test bs power two marginally large bs choice alternative b summary test choice bs matrix choice choice helps verify ccccc ccccc cp zero zero ccccc ccccc expression datum contain high minimal intensity gene derive pair filtering transform apply propose cutoff significance turn bootstrap p bs bs randomly choose repeat exercise median bs sd exercise repeat median respectively size compete paper mean population value indicate analysis illustrate practice compare compete asymptotic situation freedom centrality regularize incomplete beta beta project aa observe u nf n ks k ks nc property evaluate conditional identically variance central limit kn rr sr depend show kn sr inner parameter positive integer use depend turn imply hold integer v depend convergent abuse notation subsequence converge claim p n u pt dms usa university usa school engineering ny usa classical tend matrix overcome project multiplication est exact equality normal dimension often tumor normal multivariate generally derive either become poorly equality occur example limitation alternative work testing dimension work test equality independently respective testing sample pool use pool covariance become researcher extend bs establish normality statistic modify refer show asymptotic approach propose normality appropriate alternative test b early normality location transformation test base technique statistic transformation propose bootstrap likelihood moment degree p b test derive base high microarray hundred power upon absence structure clear exact preferred asymptotic preference reference projection space well know value upon nan distribution projection ignore enough tend infinity power past covariance incorporate test previously bs study biased situation power utilize projection previously organize test random section value test projection critical present apply conclude remark test solution project make arbitrarily matrix row project multiplication euclidean moment independent moment assumption pool covariance definite project randomized extension let numerator denominator freedom alternative converge b assumption randomize exact far randomize empirical randomly gaussian adopt standard conclusion projection value generate test recall matrix test projection projection satisfy project dimension
free illustrate singular svd allow unconstrained risk physical polynomial svd np ill nmf np also nmf ill pose illustrated fact decomposition therein uniqueness smoothness gain lee publish algorithm nmf multiple impose reduce freedom variation tv orthogonality development nmf provide issue machine one map nonlinear datum idea trick allow inner transform datum without map e reproduce hilbert infinite prominent machine support machine entropy analysis worth attractive underlie g classical employ recently attempt kernel nmf nonlinear nmf latter write map nonlinear transformation unfortunately first unknown space curse drawback reveal feature ill pose yield difficult deal tn assume propose nmf curse pre image oppose derive snapshot end explore investigate lie turn thank derive two additive descent kernel conventional propose tv approach hyperspectral paper introduce nmf nmf feature nmf several extension nmf incorporate illustrate hyperspectral conclude width old pre nonnegative notation f norm advantage iterative technique keep algorithms rule multiplicative column n scalar model n investigate derive illustrate nmf hyperspectral mean abundance decompose incorporate impose regularity overcome curse pre problem spectra mapping nonlinear transform product latter define nmf propose unfortunately machine feature lie drawback show side evaluate difficulty rearrange problem form simplify nmf problem element constraint drop tackle nmf relax semi nmf drawback give determined need call pre ill problem consist nonlinear ill pose base nmf pre image challenge attempt conduct homogeneous argue author map solve optimization problem moreover subsequently factorization base relevant curse kernel nmf model entry semi variant mean space oppose element expand expression take obtain nmf algorithm iterative additive solve gradient descent scheme accord similar update stepsize matrix obtain entry rule generally slow multiplicative derive multiplicative stepsize expression multiplicative stepsize multiplicative nature hand element trick decompose call gradient obvious equivalent input moreover smoothing n neighboring nmf similar express respect equal gradient easily multiplicative corresponding omit limitation penalty rule derive multiplicative expression term denominator physical often impose spatial influence study detail estimation namely norm spectrum tradeoff respect update rule multiplicative mt image technique variation tv penalty tv penalty framework worth derivation spatial regularization application extend direction image pixel th abundance abundance use four ni represent abundance pixel neighbor impose spatial pixel four spatial leave denote abundance spatial ratio leave particular abundance get cost respect expression update mt I nmf extension hyperspectral image band band dominate three material water water band remove yield band evaluate mean reconstruction state join extraction abundance estimate simultaneously spirit algorithms extraction abundance spectra since comparable abundance estimation abundance sum two additive nonlinearity yield recently generalize bilinear factorization require complete identify jointly dispersion dispersion nmf minimize impose convex basic nonnegative interpretation nmf kernel nmf least alternate constrained square curse explicitly oppose base nmf provide comparable nmf nonnegative embed provide comparable unconstraine nmf multiplicative denote since depend stepsize note experiment relate generalize bandwidth kernel stepsize wise involve lin lin linear lin gauss gauss abundance estimate algorithm image aforementione despite feature reflect inherent abundance whereas capable nmf counterpart analysis abundance map new kernel input explore nature curse pre multiplicative several incorporate regularity reduction bs china degree mathematic apply mathematics economics degree security france toward university technology interest hyperspectral receive degree engineering sc receive ph university france associate laboratory technology france interest analysis nonlinear wireless network process hyperspectral nonlinear co author award machine signal past publish review paper receive engineering spirit master degree control university master technology france security system technology france engineering research university diagnosis france email fr fr france email fr conference explanation e additional detail derivation update state hyperspectral image nonnegative factorization widely include blind separation hyperspectral sense deal nonlinear formulation framework suggest
htbp c c c applicability consider use researcher pl iii comparison criterion aic parameter statistic n x dataset wang appendix set appendix bic statistic system strength stress stress ii intuitive big equation stress strength invariance invariance property estimating sample mn eq interval generate minimum illustrate compare model e department statistic central ac central university generate offer limit investigate maximum applicability show stress reliability maximum attempt engineering literature clear distribution form additional attract researcher modelling originally exp detail al say advance point modelling generalize poisson distribution introduce geometric et convolution work reliability family survival give new variable survival family distribution cdf refer probability pdf eq follow cdf correspond random survival denote paper illustrate organize generating like show procedure performance assess simulation give method show shape pdf decrease rx f rx rx hazard proof straight hazard parameter quantile detail reader q branch assume take side get function real immediate check therefore property negative branch w substitute moment generate distribution give constant display mean central tendency also skewness c median median mode c c c c mode c c c mean c mode c derive distribution minimum shape parameter h rule follow al corollary constant physics entropy concept popular r expansion j reduce http com enyi moreover special derive q know value
generalization asymmetric know impulse vector play successful heuristic nuclear convex minimization nuclear norm reader compare nuclear version formulate reduction perfect comment weight appropriate weight constraint reformulate also outline regularization path call solve eventually far decide solve problem region optimal solution respectively keep fix evaluated decide point increase stop instead upper bound reach certain upper duality upper subdifferential solve particular inner product singular relax since side computed projection theorem bind become eq follow duality gap unknown integer solve solution solve record diagonal subsequent path solution solution I implement two choose relatively order order enough truncate impulse response negligible choose figure green line dense say black vertical line axis start gap exactly extent division far certainly truncation round interest approximated system exclude negligible user decide model order h promise show computationally approximate path approximate efficient selection g perform iteratively outline path explore cost possibly another input output turn subspace identification se dynamical matrix determine methodology solution path calculate base duality whole tolerance illustrate approach regularization minimization principle simple preferred engineering science translate intend preferred low advantage order include control implementation discrete dynamical impulse problem take balanced norm eq reduction alternative chapter problem problem find identification np relaxation explore minimization use nuclear define singular correspond nuclear attention aspect understand aspect heuristic work aspect concern often upper impact regularization issue
rsc condition largely inspire assumption rsc important rsc consider rsc separately rsc combine product could significantly ingredient apply bind satisfy regularization parameter bound dimension gaussian sample covariance sample choose combine verify conclude graphical use range financial recommender play central computationally unfortunately graphical gaussian latent marginally low regularize mild exist learn open possibility statistical distributional often ill problem particularly observation ambient arise recommender microarray financial inference dimensional structure distribute alternatively precision concentration non zero regime force impose statistically achieve complexity true approximated paradigm due enforce sparse dense suboptimal new extend model motivate many portion stock movie rating conditioning sparse marginalization observe regime graphical inference correlation conditionally marginal regularized previously utilize derive bound strong convexity incoherence precision precision rate zero effective latent general significantly offer structure section review relevant prior present letter frobenius nuclear norm learn covariance often glasso author selection consistency certain sparse latent tree inference tree conditionally consistency insight estimation error practice provide insight performance also derive fundamentally model model low salient video detect decompose low dense focus formulation importance formulate graphical regularize element edge connect include property respect ji assume generality property sparsity property statistical propagation portfolio financial precision unfortunately world capture global specifically construct precision knowledge variable covariance precision example structure variable remain matrix l marginal matrix write standard covariance conditional observe assume variable restriction dependency dense potentially property matrix recommender return motivate example effective consider tight frobenius improve upon effective assumption design similar regularize ml constant define optimization solver ml adopt decomposable prior regularizer encounter two derivation convexity incoherence low component fisher subsection necessary prior main subsection regularizer detail p decomposable function subspace u rank decomposable pt pe let perturbation nuclear decomposable respect eq norm small structural true sparse respectively pair shorthand later restrict direction interaction subspace loss also denote fisher evaluate define restrict fisher precision sparse structural error exist constant restrict sparsity assume eigenvalue information away identity property trivial denote depend tight sparse plus incoherence set ensure estimation incoherence interaction interaction inner motivate generalize fisher constant relate period true parameter constant discussion estimation fisher onto matrix subspace pair low detailed behavior project control page consistency contrast make explain bounding quantity frobenius estimation establish consistency algebraic precision marginal program estimate proven estimating superposition structure regularizers critical estimation condition log convexity rsc structural incoherence si rsc specify function hand si certain interaction element rsc si loss problem log previously establish si behavior taylor series approximate sum residual condition rsc cf lead hold detailed remark bind appropriately additive capture capture many additive derivation apply estimation sparse derivation largely estimation sparse pair approximate relaxed incoherence follow derivation incoherence assumption term vanish disadvantage overcome incoherence nature apply program regularization bind sample particular sample specify parameter derive hold high obtain inequality lead assumption constant regularization satisfy term bound estimation however requirement next disadvantage largely remove covariance matrix advance asymptotic obtain bind state assumption theorem give regularization sketch choice corollary end sharp spectral deviation covariance constant high significantly also requirement low simulation derive bound well effective hierarchical capture represent sparse whose global concentrated contribute eigenvalue magnitude characterization monte assumption latent dense submatrix sparse observe vary magnitude variable submatrix rank dominate ratio effective covariance local effect become observe effective effective simulate effective observe theorem hold precision matrix ii range regularize ml covariance predict pn b ac validate configuration rescale align rescale predict variable whose low likelihood extend grant nf author anonymous valuable along lee discussion use world example stock return motivate manually decomposition see whether
idea final component concentrate estimate state span grid require obtain span converge eigenvector span necessary irrespective wish true demonstrate information yield mixture additional average q separation necessary phenomenon happen variance span necessary phenomenon gaussian span though assume away nonempty mi coarse first single linkage group sample apart norm eigenvector project onto hierarchical contain close contain component ok perform exhaustive describe state start merge large eigenvector project linkage cluster g c w run result simplify run repeat chernoff precise gaussian tail give single linkage routine cluster linkage scheme point close specify single precisely concentrate respective separation hence linkage correctly identify within form apart divide perform accurate component separate cluster cluster single small weighted radius eigenvector give c within grid possibility obtain dense error k note calculate implementation linkage provide use decade estimate simple one mixture use span whereas estimation gaussian underlie therefore probability occur hence none none occur interval translate select candidate one close nx w triangle grid dp union bind nn bind mixture k state distribution identifying mean paper technique strong relate product bp parameter see hence relationship chi bernoulli coordinate normal bind gaussians approximate spherical distance provide gaussian extend spherical gaussian spherical gaussian around origin shift distribution shift separate last component p ji code distribution differ separate distribution mixture overlap least component distance distribution bind kl divergence convexity construction eq concentration distribute eq lemma equation minimum q similarly equation show equation equation proof component quantity hence equation relate cluster distance empirical sample average sample component use fact weight union bind triangle inequality hold component component eq hence inequality get inequality immediately follow sample component gaussian mixture mixture rewrite hence component error component mixture lemma term nj I I exist j j cn make subset prove equation error discard affect calculation cluster I loop exist c last inequality component probability eq inequality component apart non cluster cluster cluster concentration cluster differently irrespective sample total union show conclusion hold hold union satisfy ci projection eigenvector show probability lemma probability bound total exist choice w inequality immediately discard discard affect ignore lemma first hence would triangle enough eigenvector matrix prove g j eigenvector inequality fact single theorem theorem claim conjecture exercise theorem remark etc gray pt algorithm frequently much costly computation provide mixture mixture spherical use sample sample complexity previously know complexity near optimal derive simple contribution include meaningful band influence parameter document topic genomic consider source correspond document mixture distribution therefore central method mixture initially method maximization decade spherical gaussian consider mixture mixture component maximum sample polynomially dimension show use requirement complexity slightly relaxed notion sometimes approximate component instead derive give distance error letter seeks also often accept sampling distribute count topic follow mixture person gender presence various independent bernoulli coordinate mean observation topic hence population person gender gene independent population bernoulli product special coordinate variance extensively study separation provably reduce great document every human dna costly broadly recognize quite accurately factor approach modification modal concave monotone hazard rate unimodal compare mixture previous increase bridge gap near pac c bernoulli axis align mean spherical gaussians align algorithm divergence similar complexity kl divergence symmetry boundedness triangle inequality distance main pac dimensional near paper pac gaussian consider mixture product pac learnable factor eliminate probability show discrete pac learnable gaussian gaussians normalize deviation pac learnable divergence would similar also modification complexity spherical main contribution gaussian pac learnable theorem spherical learn sample ok contrast component mixture require sample addition one gaussian time provide estimator mixture gaussian basic one dimensional learn f estimator take independent construct underlie consider gaussian component coordinate show vector
read report management nlp extraction challenge focus much processing datum focus feed challenge relation two check global consistency present programming ip multiple temporal across algebra provide news illustration potential survey event graph annotate feed document diversity annotation feed diversity generally classifier improve rich reasoning expression composition set relation form event acyclic event graph must define closure define relation contain path begin head recent adopt temporal capture temporal arc event order event base mutually exclusive interval use variety machine inspire maximum entropy naturally one want performance score generic ensemble relation relation enforce algebra ip linear grid polynomial unless np turn practice implement ip solver significantly relaxation range semantic environment manually annotate participant classification dataset precision henceforth classifier notice ensemble whole set evaluation ensemble difficulty way procedure classifier c classifier score well individual table note quite fair procedure ensemble table detail recall notice score individual suggest ensemble often per ensemble throughout two procedure compose label use classifier enumeration notice u classifier albeit start outcome support assertion diversity recall c classifier c htb n c u u c u allow switch open source art solver intel processor ghz large large classifier htb cnn build demonstrate enforce consistency individual classifier improve precision overall practical direction research explore alternative mean classifier soft c f
result asymptotically aforementioned technique promising tool verification synthesis article far derive formal probabilistic tailor characterization dynamic multiplicative possibly discount cost work include priori additionally tight probabilistic avoid problem sample benchmark valid propose approach controller synthesis critical introduce theoretical characterization reach put forward approach discuss implementation error characterization reach avoid priori probabilistic posteriori bound general discrete markov comprise continuous space number measurable action denote py kernel admit xy dy policy horizon action process policy execution characterize evolve product endow trajectory condition state st instant obtain realization control borel measurable initialize cf consider horizon reach avoid safe horizon trajectory reach avoid property reach property avoid system fix markov policy reach formalize sample policy logical formula contain write indicator expectation trajectory else widely study theory verification simply obtain select dual define start reach within safe reach backward initialize probabilistic avoid property follow scheme prove function avoid analytical possibly discount focus synthesis seek markov maximize reach emphasize policy state policy characterize backward probabilistic avoid express eq avoid mapping programming reach write composition often reach time notice general solve point k exactly analytical expensive analytical reach want seek obtain take scheme curse markov exist k approach second learn particular algorithm consider suitable markov replace evaluation base evaluation adopt fit finite horizon achievable accuracy convergence bias conservative assign accuracy bound bound markov specific avoid priori notion possibly discount cost adapt reach avoid safe horizon base point base table let remark generation step backward mapping realization estimate eq fact base use independent distribute realization safe horizon set sample minimize condition initial computation rather single state adaptation sufficient synthesis synthesis policy estimate argument secondly classification use approximately say solution accuracy study error complete horizon bound model computable apply alternatively posteriori propose b well function weight eq optimal error space employ quantity inherent bellman cause point integral contribution transition dynamic follow subsection error introduce recursion inequality bound recall x ix individual state elaborate random quantity kx kx identically q error incur quantity extend express via empirical follow norm draw accord express probabilistic error uniformly reformulate empirical informally uniform standard related bound employ capacity concept rademacher number pseudo capacity complexity uniform parameterized generate appendix pseudo introduce inherent bellman iteration bound free define state space borel q approximation p assume solution quantity express global derive mapping single successive value precisely approximate th recall piece together accumulation iteration horizon reach problem safe table probability kernel initial equation scale influence error increase depend set transition distribution markov display density lead confidence less per event error large long value confidence dimensionality directly grid numerical space diameter grid furthermore memory usage add comment firstly inherent bellman dimension former directly low capable function good accuracy bias due bellman large minimize align reformulate polynomial lemma whole probability compute converge reach sample reach obtain dependent also sample sample manner k base collect step propagation composition term write weighting estimate bias ia bias employ propagation bias consider reach problem reach initial quantity accuracy size set accord two estimation k ar closed loop trace hoeffding triangle k x bound inherent bellman error less conservative iteration dimensionality approach temperature reach horizon affect study attain reach safe fix implement ghz intel core I gb temperature markov temperature possible configuration relate random characterized stochastic ambient heat room heat room room constant heat room process l multivariate give distribution transition denote determinant matrix mean reach obtain k x kk temperature within inside safe set solve less radial function uniform width toolbox matlab layer unit radial artificial require obtain state space probability characterize reach avoid contour approximation radial basis function safe reach approximate contour plot green action solution reach initial optimal action via employ correspond action temperature accurate flat blue far heat room high turn I shape stay room room interested performance compute last little fitting estimate namely fall interval easily later height ylabel xlabel blue marks solid forget plot crcr color green marks mark mark forget sep crcr accuracy w h ylabel xlabel log marks mark mark option solid forget crcr color mark mark option forget sep crcr iteration cause programming fitting consider grows exponentially cause accuracy good probability property specification reach avoid focus maximization horizon avoid approximate fit make neighborhood inherent bellman sample probabilistic error programming control assessment concrete approximate propagation lead tight optimize closely employ ensure exponentially deviation interest know hold sum inequality suppose support realization lemma provide base express operator event action follow point via realization kx kx jj jt hoeffde long sufficient chebyshev chebyshev alternatively bernstein bound whereas also variance range derive exploit function space analytical backward recursion notion fit hold characterize endowed pseudo finitely parameterize pseudo dimension instant sake substitute draw realization l l rewrite expect value cover metric function draw include finitely concept number cardinality evaluation l minimal deviation proposition draw trivial isometry covering number l independently tf xx dd cardinality therefore also pseudo natural sufficient pseudo let invariance allow invariant composition conclude real pseudo parameterize especially class empirical class cover average use option pseudo number overall gain option explore alternative inequality bound random bernstein tight improve inequality conservative sample complexity adapt fit minimize norm draw instant realize x sequence inequality simultaneously also inequality since union occurrence first always bind function follow w w w bind true hand w p observe define observe event define fourth depend backward
everything systematic resample ia nu ms nc leave invariant walk strategy particle form site exponential family cycle site cavity divergence global efficiently property write site gaussian equivalently ij ij multiply vector cavity must inverse latter compute equivalently cholesky multiplication cavity normalize ij property exponential family parallel compute cavity moment v cm marginal ep smc stem mode approximate slightly run ghz kb cache model c auc dna thm corollary section university paris scoring pac pac bayesian asymptotic spike make amenable tool gold expectation propagation approximate extend method essentially score label bipartite elegant way estimate threshold accord fine negative false receiver operate characteristic criterion scoring auc curve roc auc appeal auc score equal resp draw resp class auc base skewed positive much classifier small auc way instead bayesian consist pseudo exponentially auc risk bayesian establish bound part amenable powerful tool expectation propagation iid counter class denote notation hyperparameter respect lebesgue follow assumption hold bound density j I discussion less satisfied surely satisfy soon see proposition prove regard survey excess take optimal choice accommodate sparsity spike number non depend explicitly suggest lead performance prior one recover assign however pseudo mix dirac thus expectation depend way I latter appendix side respect hyperparameter commonly dependence recommend validation discuss practical implementation beyond brief mention difficult fix arbitrary make possible perform overhead smc start sample successive step weight proportional skewed particle resample replacement move kernel smc make adaptive numerically impose degeneracy always walk calibrate matrix product precise algorithmic ep implementation highlight site interpret dimensional gaussian experiment even drop depend global implement cross little extra adapt spike add product site un normalise site update straightforward advantage bernoulli dirac mass methodology non functional associate trick pseudo except apply straightforwardly smc sampler ep implementation simple possible implement site update ep match computed ep deal non identifiable auc hyperparameter maximize evidence ep site gp version exponential balance datum criterion unbalanced ep auc auc refer ep gaussian roc comparison covariate ep auc logit dna perform give approximation figure pac dna spike coefficient spike sparsity variance one decrease dna blue circle denote bayesian theory propagation fast unbalanced work rank multi consider hold probability density spherical bernstein bernstein upper hoeffding inequality prove permutation jensen lead sum version bernstein chapter k
unbounded price reality first arbitrarily algorithm proceed utility could price utility introduce yield price bundle price optimal substantially actually perfect extensive game pick simply algorithm first exponentially algorithm price specifie receive efficient utility except price price price gradually preferred note value bundle solution unchanged coefficient impossible learn preferred gradually switch necessary finally together price interact price set price utility face price vector quantity interest many mistake incorrect bundle analogy polynomial price mistake iterative polynomial polytope represent hypothesis mistake expectation function fix mistake mistake inequality get mistake directly reveal differ slightly query price learner et specify price budget main goal broadly reveal survey effort focus seminal explain monotone construction proportional generalize utility formalize learner pac distribution seek perform observation hypothesis restrict utility algorithm linearly concave utility preference choose without utility separable unable class function control correspond query wish price price adaptively arrive mistake result inspire classic finite majority remain discard mistake maintain hypothesis vote volume bound mistake learn utility specify power good represent normalize price good price preferred bundle subject bundle unique utility maximize bundle value utility maximize fix know budget attention vector assume discretized increment fractional capacity weight production bundle price minimize obtain maximum something utility algorithm select measure optimally round possibly set bundle predict get actually mistake sequence price begin consider first price maximize give utility efficiently maximize price combine yield pricing section optimal price perfect nash exponentially straightforwardly inefficient nevertheless observation family contain exponentially precise derivation efficiently let price operation ratio specify might production cost therefore bundle xu b x k actually therefore price attain nearly specify three optimal efficiently uniquely nk k kp compute price list establish pricing correspond bundle I note threshold whenever whenever bundle utility per least good otherwise also maximize solution lp straightforwardly characterize disjoint optimize set lp lp ip claim per good sufficient v v decrease price additional discretized bounded price consider fraction good cost order decreasing learn irrelevant always give price price output ratio price discretize increment specify bundle price make price particular price bundle information prefer good algorithm price good next learn ratio price item occur guarantee minimize binary search ratio ratio attempt originally learn already know initially set low adjust price point switch learn price v whereby gradually eventually reach preferred identify high must none good learn must quantity minimize increment linear search require identify manner preference search critical arise value always occurs set optimally unnecessary price price price vector good low still good become must price budget power running make complete algorithm remain approach achieve round achieve optimal generate optimal price approximately pricing learn could maximized bundle invariant scaling bundle ratio actual bundle optimally price receive approximately less opt query might regret possible regret model bundle motivate scenario force accord choice parent company day observe say mistake ever algorithm price call upper informally describe give algorithm maintain consistent see initially round constrain particular immediately solution optimal decrease polytope idea uniformly mistake eliminate probability exactly equal volume eliminate consistent mistake final volume however need way volume efficiently hence exactly coefficient dimension fix maintain consistent index among yet together new begin never go epoch fix epoch challenge I z ic ic z convex polytope round integer mistake mistake algorithm epoch final mistake bind time mistake per epoch epoch remain coordinate track volume set epoch stage epoch hypercube incorrect eliminate make mistake round fact mistake epoch find epoch mistake epoch linearity expectation therefore apply chernoff plugging allow leave whether discretized wish devise approximately finally different stochastic lemma california edu california institute edu research author university period bundle price observe seek adapt price perhaps follow online price purpose management work strong utility maximization theoretic reveal preference problem utility price sensitive period observe possibly fractional utility optimize utility mild objective price competition set maximization unit associate minus bundle maximize every round maximize bundle price budget optimally instead faces give price quickly learn knowledge instead
see distribute present new topology grid system exhaustive optimize name dynamic algorithm improve performance propose system distribute grid power device g unit advanced communication decentralize grid literature exist link affect optimize bandwidth power strategy exploit performance minimize mse associate dynamic poor select improved topology adaptation neighbor choose mse sparsity topology reweighte topology usually employ steady employ hasting link introduce combination neighbor topology adaptation neighbor specify estimate automatically performance poor mean topology distribute estimation rule coefficient incorporate algorithm organize describe letter inverse system instant system quantity measurement standard control pt focus linearize dc j branch angle therefore aim distribute ls name modify report measurement vector vary exist literature grid system communication cause neighbor performance link experience chance need dynamically topology estimation aim algorithmic give neighbor mse performance note describe combination rule rule cardinality satisfy distribute divide step combination step strategy first describe include combinatorial strategy strategy complete eq equation combination need low propose simplicity adaptive vary report system small result steady zero link poor performance follow adaptation combination norm task strategy combination excellent log shrinkage magnitude intensity stand minimum neighbor simplify describe neighbors eq include vector generate adaptation devise inspire combination change algorithm perform dynamic adjustment mse value
rejection bivariate criterion via nonparametric procedure control false sim multiple control false illustrate simultaneous testing become familiar field economic finance genome testing association typically hundred thousand image measurement voxel determine area cognitive false discovery widely massive interesting case meaningful structure microarray study gene structural usually valuable suggest likely false spatially hypothesis likely multiple attempt prior instance appear comprehensive find reference therein arise remove generate uninformative stage testing procedure pass filter quantifying test statistic weight affect choose filter weight loss validity recommend testing adjustment weight test hypothesis independence filter weight setting proportion incorporate information multiple testing bivariate hypothesis hypothesis primary unlike impose independence statistic scope filter wish explore region multivariate hypothesis project take interval single estimator comparison mild method substantially long component value correlate extensive validity rest paper review testing present control multiple evaluate section section end proof test corresponding outcome significance nan proportion incorrectly hypothese e rt introduce rt pt true frequentist group cumulative nan true probability equal threshold weakly dependent level hereafter control detail test l parametric nonparametric projection direction r p start intuitive rectangular rejection bivariate derive rejection region false rejection value b proportion nan part direction control discovery rate recall region rejection bivariate intuitively discovery rate bivariate rectangular notational simplicity bivariate define joint probability f bayesian rectangular rejection event infinite rejection region choose rejection power rectangular rejection region preliminary primary fdr insight multiple testing statistic compare find control filtering choice rectangular rejection high seek form optimal region let region f special restrict rectangular group discovery denote rejection fdr f optimal equivalently rate equivalent large propose constant traditional testing lem powerful hypothesis rate homogeneous version discovery distribution test correlate true nan bivariate nan true nan extension bivariate bivariate normality true transform bivariate bivariate test bivariate see general serve motivation develop rejection rejection bivariate normality take eq rejection formulate intuitive viewpoint researcher prefer reduce component aspect transform search eigenvector common covariance find direction project hypothesis act parameter define threshold call index bt comparison region stage addition procedure stage hypothesis filtering project bivariate index weight testing generate combine specific proportional parameter valid choice region testing form p multiple investigate follow utilize estimate direction sequence marginal distribution define rt ft rt hypothesis reject project role discovery theoretical true nan follow uniform nan hypothesis category uniform true far simplify correlation shrink project structure area employ image derive specific relax normality respect flexibility distribution example statistic appendix estimating relaxed cause achieve robustness normality hold true use eq parametric select come nan hypothesis close interval drop newly z follow distribution pi nan dynamically estimate unified dynamically dynamically nan dynamically choose sequence nan boundary procedure range hand verify hand condition pt consistency efficient f algorithm paragraph fix vary projection utilizing whereas inference generalize recall rejection value correspond shape value suppose pt differentiable equal constant particularly depend bivariate normally identical covariance bivariate figure vary slightly study confirm bivariate selection restriction impose identifiable formula equivalent ft plug direction rt parametric proposition p parametric control naturally project q reject comprise denote propose estimate substitute counterpart method control denote consist incorporate obtain control procedure investigate preliminary suppose bivariate calculate bivariate perturbation observe contaminate nan incorrectly question sensitive method carry wrong suppose primary side nan respectively p side hypothesis symmetric leave hypothesis indicate test statistic measure restrict situation true indeed method tail conservative asymmetric distribution satisfy fr x rp carry wrong justify power ft f quantify much figure setting control valid pt consistently higher alone remarkably contaminate procedure testing stage bivariate far illustrate compare multiple stage testing preliminary bivariate preliminary primary independent stage test significant procedure bivariate positively structure bivariate flexibility example except generate distribution degree freedom identical rest come hypothesis provide conservative panel ii conservative increase unlike large since freedom normality p normal independent assess bivariate set contaminated method conventional conventional procedure nonetheless information control case contaminate close illustrate stability appear method indicate advantage bivariate practically mixture comparison bivariate similar prior multiple conventional structure multiple testing two stage conventional three bivariate locate neighborhood bivariate cluster evaluate sided hypothesis test independently serial consist cluster mean conventional mean filter serve ii conventional information ii conventional alone neighborhood pt c microarray experiment detect differentially suppose gene obtain value chi degree freedom independent utilize bivariate true comprehensive comparison particularly case parameter copy parallel gene level contribute disease discuss study gene gene study analysis source interaction therein cancer comparative genome pure capture genomic datum gene population low calculate sided gene copy preliminary show scatter copy correlation motivate apply significance dna location show projection preliminary plot bivariate geometric location use significance scatter bivariate value geometric gene valid gene dna alternative dna find differentially express dna dna apply favor unbalanced copy gene serve scatter passing symmetry fortunately gene line come weight pass symmetry small gene dna level increase threshold gene perspective function simplicity threshold present scatter testing estimate comprehensive comparison reject three term associate organization go bind activity map mutation list mutation support gene notably top gene particularly gene identify integrate gene complex experimentally gene list perform gene map recognize gene david functional content term gene table report infer activate cancer propose multiple overall microarray imaging availability nan project quantify novel procedure establish projection mild operator index normality generalization random bivariate test thorough scope paper future spirit power maintain rigorously theoretically proportion uninformative reduce stage view independence aim test bivariate change hence increase change powerful proportion nan change beyond make sequence strong primary statistic publish handle multiple structure much strong ft rt analogously process f ft prove main involved go converge first regularity derivation exist g rational denote quantile continuity c condition continuity ft ft f c prove proposition normal h ft uniform directly condition consistency partitioning l give pointwise pick pt j j l km ft similar argument ft ft result nonparametric uniformly fdr derivation yield f fdr exist hand take side f f satisfie formula solution continuously g f differential intersection solution equation unique equal normality appear leave side true derive nan function nan nan use curve f
similar black individual heterogeneity mark median laboratory evaluate blue total black allow individual heterogeneity demand implement allow however properly perform approach worth accordingly impractical somewhat reduce produce history computing evaluating find implement somewhat slow likely correlate link movement account semi place likelihood somewhat many possible low probability instead draw movement improve draw correspond basis population e g record name match record heterogeneity integrate simple heterogeneity unify miss constitute step toward synthesis multiple auxiliary facilitate inference use multinomial unobserved probit provide sampler bayesian avoid need tune probit alternative capture logit desirable interpretation odd work potentially gibbs use logit capture abundance sensible heterogeneity survival describe model extend population formulation accomplish substitute desire relationship formally describe method analyse source arise dna method heterogeneity heterogeneity broad maintain reasonable sampling e g challenging need extension evolve mark examine allow individual acknowledgment discussion study finding paper author necessarily represent view service imply us public center usa aid department resource united service increase allow capture simultaneously behavioral individual heterogeneity parameter probit present metropolis hasting algorithm monte abundance model visit capture population usa find temporal behavioral variation estimate laboratory effectiveness technique commonly reliably estimate evidence mark broader explore properly account introduce detection convenient bayesian capture population passive become common largely less individually technique evolutionary hypothesis passive sampling study entirely match individual genetic observer design differential home heterogeneity detection abundance approach analysis capture individual identification occur contribution focus abundance temporal accommodate level detection survival develop simultaneously variation behavioral g individual heterogeneity bayesian probit augmentation technique classic capture sample three record encounter encounter first second history probability observe nuisance parameter case unique encounter inference never precede scenario may type identify denote encountered yield present whether one uniquely make assume close abundance allow generalize much accommodate behavioral marginal history frequency indicate record derive distribution describe applicable review could error make estimation adopt bayesian perspective use mcmc abundance survival encounter encounter history individual represents identify pt latent frequency denote column record history denote example record history record detection history population abundance encounter time record arise history rise record p p I I I I p p history history history encounter pr individual history record history kk implement method construct latent history indicate individual history record record table example correspond column simply replace row example row latent history record treat individual evaluate joint h deterministic rather consider propose accomplish utilize space nan solve vector heterogeneity explicit however allow heterogeneity explicitly population therefore abundance proportional mt b alpha illustration abundance temporal behavioral heterogeneity extension effect account modify adopt utilize augmentation formulate probit detection data treat binomial parameter indicator real individual individual individual remain proportional continue proceed j r rx respective j j never capture individual individual accept order conditional inclusion bi sampling page return repeat calculate iteration concerned heterogeneity detection quality artificial mark genetic material vary individual modify accommodate temporal effect heterogeneity specify probit individual intercept term individual identification latent u u u component modify model full pt tn mt mt u notable detect individual never detect analytical applied incorporate abundance usa occur hence addition provide error collection genetic visit dna capture close abundance allow behavioral heterogeneity motivation heterogeneity incorporate model rely meet capture fit probability method indicate behavioral heterogeneity allowing occur p mcmc algorithm accordingly reduce update distribution data sample correctly assign investigate sensitivity conduct uninformative specify programming pre post interface integer tune mh sampler divide acceptance basis accept million analysis analysis ghz intel core processor long movement rate movement result possible chain assess visual diagnostic package analyse diagnostic uninformative parameter prior credible interval find behavioral response suggest population report black dna low c individual heterogeneity cause abundance explain heterogeneity interval probability sample collect auxiliary informative analogous uninformative yielded record another way nevertheless informative little contrary take specify informative could sample degradation environmental could abundance prior conduct laboratory
despite simulation fair would reader expect perfectly fair introduce literature good reader situation important major aim implicit important software compete unify implementation give advantage incorporate initialization present soon package problem firstly fit plain namely treat improper density secondly decide exploratory likelihood way scope lemma di di universit di mail college uk mail c ac abstract introduction optimally comprehensive mixture density look treat idea benefit comparable fulfil apart evaluate standardized misclassification rate usual one one keyword cluster em improper likelihood robustness h optimally robust improper approximated distribution simulation currently comprehensive involve careful discussion issue assumption improper pseudo component define small capture outlier inspire show improper comparative study may cause problem use cluster certain maximizing give general multivariate shape want rely really generate method call reflect example use mixture fit way illustration use city discuss fitting plain outli collect robust dominate produce method recent overview level improper tuning introduce comparative involve introduce unified noise outlier prototype cluster set mention along song region issue additional dataset simulation study computation supplement property cite exist ix pp x ml mixture distribution mixture density parameter assign mixture estimate interpret proportion point covariance th cluster popular implement prove method assumption attempt deal ml mixture suggest outlier add mixture noise implement result uniform fix hull datum package include note proper maximize estimate point belong component v cluster drawback affect point reduce formal method replace density outlier noise far away area pre specify small central definite freedom consider freedom ml assign component result ml quantile degree freedom method cluster recent cluster amount index cluster model cluster triplet spurious outli cluster every point author maximize hand side outlier introduce method regularity propose consistency methodology partition proposal approach base find adapt estimation behaviour approach robust improper idea improper improper vector include improper used probability assign fixing define model region density distinction uniform convex hull cause problem require suitably discover easily extend well way prevent eigenvalue spherical relative scatter among study gaussian algorithm multivariate alternative see disadvantage affine would component transformation activate prevent still improper propose constraint quantity interpret point implement familiar half plain consistency topology extend result mle lack matter dependent choice quantity rather device enable good approximate cluster region look produce gaussian minimizer measure cluster prototype cluster th approximately good squared distance component indicate cdf kolmogorov optimally tune denote discuss good development version beyond paper however important precise normally bring see effect maximum enforce solution optima far compute recommendation fix initialization possibility assign observation time result pick comprehensive initialization consuming use initialization actually big outperform partitioning recommend spurious initial pr min attempt valid identify ml initial gaussian noise point partition program within initialization end happen enforce lead particularly set interval often contain large candidate discard later normally simulation assume true problematic consider ml gaussian method implicit way classify outli comparable assignment method triple permutation compute scatter compute expectation matrix study explore central student marginal marginal reference although case close non assume affine package mass supplement level idea decision p level play original freedom assume across datum covariance constraint freedom incorporate motivate consideration design avoid spurious ii design base extremely variability student decision mixture suggest initialization see notice allow allow initialization additional op op compare use cluster task estimate average standard code monte pair gray scaling emphasize difference high get precise misclassification online contain misclassification pair clear robust important perform gaussian work slightly bad time involve outlier suffer dimensionality u ml contain estimate generate rate generally though automatic always proportion p point compare suffer situation overlap completely dominate see separation student marginal perform particularly show good overall misclassification method comparison encourage see basically produce get case h disagreement substantial mainly sometimes merge cluster structure integrate investigate behavior independent compute interval add main produce whereas figure small constant constraint become active clear minimum minimum lie nice dimension core look become happen unless constraint latter section pattern mostly seem quite stable different h process produce estimate standard smoothly estimate noise proportion scale impact strong sampling consider rather discovery cluster change section example percentage classify involve central core distribution assign percentage point assign whereas noise really involve gaussian clearly separate separate need decide desire toward distributional dotted computed plot section situation affect dataset axis first separate therefore separate large clustering obviously analyze regard interpretation depend well treat noise outlier merge two cluster tune produce noise component artificial alternative real ground truth one class usually guarantee clustering reality prefer finding example give city originally competition conference fitting logarithm death birth balance divide divide number variable median figure death move figure moderate
moreover spirit dimension rather translate fast recently prove complexity still room third could complex value establish instrumental analysis describe approximate expectation function isometry existence around use cumulative couple use recovery control equality introduce limitation limited coherence level isometry control frobenius one dictionary coherence atom possibly sufficient desire quality would around potential algorithmic spirit alternate minimization limitation imply surely expression spirit improvement involve improvement convex recovery main consider triangle inequality assumption yield denote r r u previous rewrite w except handle thank convention j inequality u lemma denote p b pp j k take support draw since f I denote restriction j similarly f term f piece kp observe continuity apply I simple lemma term u proof far conclude lemma definition assumption consist signal select paradigm lead image audio argument sparse rely procedure analyze yet paper probabilistic sparse signal admit reference complete key quantity coherence combination field statistic learn line development framework algorithmic tool make design good prominent deal effort dedicate efficient wavelet notably many compression decomposition simply powerful classification formulate problem code bring play non convex success analysis establish generalization quantify signal reconstruction sample uniform focus aspect dictionary minima identify dictionary important interpretation call arrival modelling related visual accurately learn also obvious carry code denoise distortion denoise dictionary intractable heuristic e behaved characterizing help exist measure importantly early identifiability combinatorial condition involve criterion form basis identifiability noiseless outlier arise consider bernoulli model without analysis extend dictionary compose signal exist handle none absence straightforwardly take minima outlier regularize least cost truth relate consider minimization algorithmic demonstrating provably complexity initialization alternate open source implementation online approach available extensively exploit application htbp c noise exact admissible coefficient characteristic svd nb frames rademacher sparse response decay overcomplete dictionary presence characterize ground whether guarantee whether output algorithm resp characterize minimum exactly noise finitely upper provide sample complexity level allow brief description support select ii nonzero decay coefficient random recovery atom penalize penalty probabilistic signal plus noise loose coefficient closely nonzero cumulative see minimum generating dictionary prove blind separation understand reference tend variance hope nature algorithm sample rademacher average level cumulative coherence involve may level precise control admissible demonstrate relative amount robust material integer denote transpose norm frobenius f place exploit extract conversely b n zero index index denote complement denote linearly gram inverse orthogonal span column ba ba bb ba hold nm column dictionary learn sparse vector n ik reconstruct code definition denote typically dictionary signal q basis processing minimization regularization parameter control tradeoff unit image depend unit simplex characterize penalty generative noise contamination outlier state show necessarily frobenius future importantly generate signal discuss blind source separation invariant permutation hope specific transformation describe equivalence local invariance issue soon sufficiently generate noisy spurious support replacement available shorthand eq jensen entry magnitude small conversely marginal coefficient dynamic way measure boundedness complete handle neither indexed fact stem concentration traditional illustrated decaying sparsity norm relate cover control early field sparse specific express outlier train distinct property relate manner dictionary function represent training n number require namely complete minimum rely recovery signal almost surely control support solution regularize reasonably impose condition dictionary quantity unit term correlation exceed cumulative conduct assumption coherence coherence consider previous dictionary bind rely weak assumption context role define separately coherence would fully relax rip sparse code problem admit local neighborhood control regularization main building high dictionary sparsity consider eq denote deduce minimum provide local minimum problem consider minimize signal yet compatible small scenario hand large hand admissible regularization enough factor small coherence dictionary measure p least coherent quantity f resolution limit outlier discuss constant explicit aside consider assumption assumption constant draw f r robust addition provide right impose outlier refined argument frame I decay model randomly sign vector noiseless resolution infinitely slightly bad resolution noiseless indicate ensure minimum around fine choosing recover known optimization soon boundedness soon infinite sample quickly outlier control admissible energy e precision ratio f rr correspond minimized variation orthonormal perhaps orthonormal achievable impose constraint check limit precision local minimum radius around q noiseless resolution reach provide exceed resolution threshold base incoherent dictionary plain coherence eq hold soon incoherent union basis fulfil soon exceed consider satisfy maximally incoherent large read amplitude hand relax j j j existence coefficient satisfy soon much restrictive spherical ensemble spherical ensemble obtain independent dictionary satisfy soon classical continuous compact constraint radius sphere q reduce h consist ensure ball existence asymptotic analysis vector cost eq depend via covering show generative nc desire sample interesting negative target arbitrarily satisfactory resolution independent refined gain collection sometimes contaminate irrelevant consider sense share dominant property training consider resp matrix extract keep column associate resp clean contribution together n induce norm tx robustness context interesting regime arbitrarily robust learn outlier seem f technical arise implicitly define minimization leverage denote minimization always denote make pattern remark arbitrary moreover guess p linearly assumption light switch lemma signal uniformly addition suffice match conduct assume restrict ball reason motivate strong assumption involve coherence exact dictionary reader hence prove average rademacher average set usual eq rademacher probability eq z equation absolute real r respect probability bind conditioning
assign rna moreover annotation rna may reconstruct annotation seq reconstruction abundance estimation achieve interested reader refer comprehensive relevant computational many develop rna expression accounting method design group per sequencing mcmc second assess expression posterior employ likelihood approach expression testing gene multiple use differential gene expression usage usage refer total expression construct log standard usage sharing start differential usage square root jensen shannon first perform expression comparison negative group mean rna category relation construct form address uncertainty inherent two commonly accomplish g additive ab bb situation datum compare allele allele cancer patient situation rna statistical test although population limit test case happen seq population happen group hoc implementation specifically assumes differentially express conservative limited confirm study develop name seq aforementione accomplish method treat rna gene cluster overlap whether rna covariate interest would rna large gene burden multiple multiple major estimation expression separately problematic perform rna identify one differentially material know previously input file rna seq non adjacent rna seq usage adjust covariate htbp indicate penalize employ binomial regression rna seq binomial variation seq biological replicate differential expression testing adopt distribution assumption negative binomial rna seq replicate therefore binomial value adaptive lasso broad penalization categorical size annotation demonstrate satisfactory seq analyze seq reference genome situation part belong cluster far impose unlikely unlikely belong cluster cluster subset overlap size portion bp pair end assign I ia abundance ip th intuitively position sequence vary gene set contrast effective length nonzero include effective rna seq length seq supplementary material effective estimation binomial count effective covariate reasonable across configuration equation challenge matrix first candidate binomial regression candidate seq supplementary material database skip negligible length across candidate informative high effective length candidate important zero covariate often correlation among employ log penalty require interpret material c sample estimate expression read depth sample depth measurement rna seq rna seq first variation normalize n problem write impose penalty supplementary covariate snp focus linear additive aa ab bb b expression reduce binomial non impose solve material n iv iv uv g multiple covariate effect impose penalize supplementary material study express examine count across expect component differential describe respect set covariate alternative helpful understand special solve binomial chi model regression second category categorical binary g chi degree freedom penalize lr statistic log penalize nan follow asymptotic distribution penalize nan count step number lr procedure regardless rna seq small value study sample size vs valid population study calculate interest unchanged alternative calculate ratio repeat obtain statistic differential discussion include rna estimation testing rna rna variation expression situation expression gene low switch rna switch use refer relative rna usage seq rna usage replace discussion million rna seq read single simulator annotation simulate equivalently gene differentially term usage supplementary rna read genome next rna seq set rna seq pair read confirm effective supplementary candidate vast annotation restrict strong supplementary penalize cluster annotation supplementary abundance conclusion include seq htbp status ok abundance use annotation abundance next power testing usage file differential expression usage site usage majority file status ok ok file file status trust reason case combine replicate lead conservative implicitly case comparison gene favor power base proportion significant high figure attribute supplementary issue compare use roc supplementary estimate replicate replicate biological replicate due resample rna seq read fair recommend control simulation work challenge situation usage respect continuous covariate quantitative trait seq real european select minor frequency follow cluster version simulation annotate snp body rna seq across respectively select drawing assess differential usage nearby snps body multiple nearby permutation usage figure b simulation correctly detect differential usage respect treatment drug valid td rna seq vs supplementary list treat associated neuron dna protein knowledge software availability package expression usage intensive minute gene processor need implement material seq rna seq treatment discussion name assess rna seq categorical resample component distribution penalty resample first choice model seq data biological completeness binomial seq count lead severe lasso inaccurate supplementary apparent candidate consistent previous finding positive penalty well abundance rna seq distribute dna sequence affect abundance rna seq read bias likelihood rate limited impact testing rna read type error systematically account future assess usage pair individual pair multiple meta usage side near future plan include large diverse genetic collaborative sequencing read bp sequencing seq rna seq assign differential usage software development rna seq rna seq rna seq data count seq anti separately partially grant gm rna seq rna seq end sequence end discussion pair read read read impose upper th effective fr j shortest effective word rna seq effective seq summation weight ht discussion notation skip subscript length length fr consecutive whereas cover h r derive follow part r use observe even may sequence improve robustness determined count define end examine end identify break point read depth adjacent specifically gene overlap apply chi assess whether different length break parameter break default value cutoff default th break select construct consecutive gene among cumulative trial effective claim construct set set default change situation negative binomial dispersion discussion binomial extend situation j log penalize glm employ non intercept impose maximize iteratively update regression I I adaptive estimate likelihood implementation include loop combination carry loop square quadratic current square coefficient need remove improve efficiency initialize update n ij coefficient estimate little change crucial penalization select strong scale choose log large grid tune minimize rna often small de hypothesis bic log chen chen extend eq simulation restrict number optimal framework rely rna seq p suboptimal influence capture resample tuning conduct guide care laboratory laboratory resource national care university bl treat drug treat acquire bar maintain hour hour dark schedule room maintain cm laboratory water release day age collect tail drug day exposure steady concentration ng ml achieve drug bl nm age day drug treatment remove home brain product ok extract rna rna life rna verify library contain rna per library read bl use alignment experiment merge specific genetic alignment quality minimize cause genetic reference read segment bp per approximation snps bl genome variant report sequencing effort include release read mm coordinate update position string mm annotate pair pair allow pair read merged position tune thousand bootstrap parallel computing processing however particularly maximization numerically separate make dimensional log parameter factorial selection attractive seek penalize mm rely strategy concavity indicate support therefore update perfectly preserve iterate effect cause vice versa desire update increase parameter g five mm univariate omit htbp control case control control usage per cluster usage quantify total htbp htbp htbp htbp htbp outcome htbp mle glm nb htbp genetic map pass map bl j bb bl cg cg anchor anchor protein exchange factor exchange contain ca activation factor core domain contain protein domain box contain translation gamma domain alpha similarity member gm protein couple protein b associate protein cell like contain
causal worth note represent physical direction yield although certainly actual particle fail stand value represent choose measurement however categorization kind five outcome observation probabilistic incoming incoming link along causal quantum kind system classical correlation coincide usual one arise kind hide structure able general imagine every take concrete experiment need write outcome concrete tuple tuple term difficult obvious subset past collection distribution consider give causal define classical show actually quantum derive candidate sufficient consider recursively set disjoint get disjoint follow condition upon otherwise thereby subset causal write minimal familiar consider scenario figure generalize arm party source outcome party party party scenario equation conventional state independent assume scenario formalism outcome approach automatically treat realistic usually correlation probability setting actual distribution relevant processing application device independent expansion strong limitation need impose another device independent key even way highly part establish put disjoint causal past hold suppose thank suffice check maximal past w interpret party analogous know formalism box across party individually derivation carry induction party equation place arbitrary party show induction third worth note need event causal past introduction every carry variable conduct basic constitute representation physical modify system additional necessary reveal believe hypothesis situation biological try infer correlation behind action potential measure certain certain cloud formation possibility make hidden live edge determine outcome certain node operate take incoming turn variable outcome think hide live variable suppose physical one arbitrary conditional v particular different kind incoming sort hide live develop equivalent previous information processing fit standard causal behind thick thick blue node node approach probability variation allow reader non may restrict countable explanation order concrete hand v integrate result getting speak obvious ignore terminology suggest correlation actually node determine sufficient hide variable prove base case formula precisely assume big place likewise upon guarantee induction conditional v integral evaluates complete case evaluate keep already suffice u exactly party variable measurable conditioning make classical putting eq free use classical let copy argument quantum desirable result correlation classical conditional general causal good display special outcome node variable applicable inequality problematic aspect scenario polytope soon unconditional well question maximally hide space outcome scenario polytope general hide correlation able approximate classical correlation space variable whether finite space generalize definition infinite informally every realize finitely induction number node therefore put start realize hide variable node take together apply need careful way need subset finite replace conditional take subset finitely many possibility hence index measurable equipped distribution virtue coarse arise come equip tuple modify associated source word outcome new outcome retain distribution form outcome define induction correlation edge nothing except together hypothesis arbitrarily make second sum know assume variable structure also plausible randomness randomness parent additional one randomness generation back eventually end except parent precisely represent deterministic meaning v cover variable finitely reformulate v involved intuitive randomness parent vertex hide realize assumption nothing acyclic find start incoming edge reach node consist randomness inherent induction plausible technical quite demanding hide turn regard assign probability regard list say independently respective denote variable new u w f w stand tuple consist together construction incoming nothing manner define classical hidden modify u wu u randomness back rise overall result u coincide original upon component list drop make randomness along hide regard node classical induction situation property information root put processing complete virtue arbitrarily http net question measure measurable might achieve proof framework definition correlation form physical quantum probabilistic definition reason general rigorous quantum correlation piece structure previous correlation thing carry turn index individual success get preserve operation physical piece category gets label gets label assume strict play role carry summing outcome result normalize normalization preserve cone add neutral precisely module consequence mixture rescale simply multiply bilinear module bilinear distinguish normalize operation nothing nothing operation act nothing precisely scalar go real unit terminal every normalize thick circuit causality list requirement possible example next classical space determine correlation idea diagram object graphical calculus another reason believe live opposed thing overall composite specific node whole operation outcome e index operation marginalization normalize processing gate turn classical think operation realize exactly outcome every try label diagram indeed category formalize diagram idea appropriately label diagram product product factor generally unbiased category acyclic graph diagram graphical calculus deal per se rather appropriate piece seem develop everywhere difference causal direct indirect link causal causal comprise converse simulate indirect causal link link indirect ultimately link indirect causal correlation causal differ add prove claim correlation object extend assigning consider correlation conversely correlation turn correspond except assume come information pass subgraph imply end formalism correlation measurable whose operation space object operation normalize category notion coincide classical integral composition besides definition quantum correlation arise category operation seem work correlation contrast box every box theory box world sometimes one everything imply correlation try symmetric addition scalar come equip satisfy law requirement correlation arise object linearity imply object result linearity govern theory correlation consequence classical describe first option sense existence non broad existence locality fine tuning explain formalism space e hilbert speak correlation formulation quantum however operator work finite definition correlation prefer hilbert quantum state prove definition whether quantum symmetric denote infinite hilbert space trace understand quantum positive trace mind space form map operation operation preserve channel straightforward check correlation proposition definition concrete thank general obtain quantum proposition classical quantum know requirement assign hilbert space interested reason skip assign hilbert crucially measurable measurable obvious object finite hilbert assign eq straightforward assignment quantum operation operation operation map quantum would quantum advantage stochastic operation gets map quantum operation identity identity indeed classical correlation realize correlation sketch full go outline proposition situation turn stochastic operation indeed trick replace hide finite every v v corresponding outcome along diagonal canonical basis adequate party scenario free equation probability admit hilbert operator depend completeness implicitly assume since begin party detail label space incoming edge collection index outcome statement apply likewise similarly correspond simply relate tensor product state measurement multiplication encounter begin free guarantee hilbert since completeness I I behave rewrite trace hilbert jointly become eq dividing multiply hilbert hilbert basis index define straightforward operator quantum proof proposition translate back scenario formalism vast ordinary definition study verify comprise usual concrete quantum correlation scenario correlation party classical quantum explore like quantum hide extensively quantum hide stochastic recall variable index underlie causal precisely informally correlation causal must like definition like thick draw fill right right right circle fill blue l l leave informally speak every empty subset disjoint causal realization hide depict impose requirement hide machine biology become correlate hidden easy handle mathematically useful classical actually admits define take coincide output gate take incoming sequence case markov commonly illustration way variable hide come equip pair think result update operation hide finite direct acyclic correlation description distance thick anchor south east anchor north west circle fill lb aa b la lb aa aa ab al la lb graph generate measurable space finite joint distribution constitute network idea hide whose dependency arise bayesian actually represent word redundant start space old variable space basic classical correlation gate v assign hide outcome tuple old old gate coincide gate eq sensible left hand become need indeed original correlation onto tuple edge result start representation form one carry space gate follow income actually inclusion support end process eq simply far integral equal show use definition quantum measurable proper mathematic hide variable classical model like recall deal measure notion classical concept quantum set subset measure size satisfy axiom always mean actually measurable necessity measurable measurable theory hilbert quantum function pairwise want make allow case less letter interpretation measurable take form integral correspond paper understand notation integral symbolic since measure set regard integral refer text crucially deal produce exactly suppose analogue reproduce definition normalization line case space satisfy property fix fix measurable obtain abuse element operation measure quantum map trivial compose mean integrate trivial proof operation operation mean satisfy consider measurable normalized stochastic operation coincide product operation form product contain form need measurable operation definition see detail parallel analogous unique property assign leave note particular measure prove crucial refer additional terminology write difference algebra set measurable main induction nothing apply obtain arbitrary sp b desire measurable equip finite every measurable exist measurable q nx ps ns approximate well coarse measurable algebra b require many finite boolean algebra atom construction element hence product set spectrum unique atom tuple unique union suitable whole rgb thm thm thm conjecture thm thm remark section participant development innovation author foundation scenario small quantum theory quantum graph conceptual measurement role scenario formalism understand contribution latent thing markov influential interest theory lead recently development processing protocol security rely crucially realize illustrate non quantum correlation generalization include party party scenario additional subsequently comprise party party recently scenario add purpose present thick anchor south east north blue blue minimum size general comprise scenario technical source measurement measurement three pair point one event think event specific outside classical outcome way link think represent event interpretation equally mathematic thick scale anchor south anchor west space fill la lb lb interpretation connect typically random causal describe distribution become central causal assume represent outcome physical point outline sample sampling point reveal like uncertain complete imagine biological rarely also apply physics mechanic incomplete introduce degree represent particle event relevant party extension proposition formalism equivalent point explore plus minus skip node anchor east node anchor north west circle la lb la lb lc summary idea outline definition proof sometimes de force quite proof demonstrate trivial argument mark reader simple completely proof describe causal course turn know influence one direct indirect link equivalently order discuss quantum causal scale anchor south east time north west circle fill blue mm b z drawing look four event arrange structure indeed induce example spatially event happen suitably time continue causal outcome independent soon disjoint causal assume likewise potentially physical show reduce introduce variable paragraph pass causal process compatible unobserved propagate causal link give sense feature domain allow live necessary outline show fact characterize correlation
analogy setting unbiased blue estimator minimize estimator minimize variance strong request unbiased linear couple continuous couple ol respectively ols ols estimator mean replication unbiased proportional square residual develop hand side vector whose th element derivative prove imply q dt model hand process generalize tf ols unbiased proof appendix functional ol blue present work extension bayesian might curve might add weight instance distinct space generalization different world without loss generality assume delta symbol schmidt apply corresponding let kt w kt base couple last definition tf representative easily project ol projection projection linear crucial orthonormal straightforward take obtain blue knowledge common work setup choose condition support choice estimate estimator analogy functional design minimize design definition maximize herein study performance design motion location experimental location angle form right location central overhead panel available website compare motion seem adequate collect location trial support coordinate provide continuous optimum exact factorial batch trial experiment three trial perform thus exact optimum whole domain factorial different factorial factorial grid spaced factorial find design design goodness efficiency efficiency list low design accord separable usually majority differently proper space guess provide derivative roughly speak incorporate gauss functional despite complexity obtain elegant belong space experimental present literature instance rigorous model matter optimality criterion context theory set subset scalar ol map unbiased operator write vector couple vector tn representative introduce operator q thesis immediately tt prove linear equality five f nan f l pl h pt equality due couple ft ft dt ij matrix orthonormal complete representative kt replication relation eq assumption distribute depend neither given imply choice linear equation q tf equality operator lt lt set equation thesis side eq proposition definition theorem observation realization science economic field process procedure derivative reconstruct separately obtain hence gauss markov linear unbiased observation realization continuous science economic reason area book book parametric approach cover classification discrimination cover situation response functional multivariate repeat design remain focus functional regressor derivative exploratory high derivative reconstruct recent derivative usual reconstruct justification curve function course take consideration curve derivative reconstructed observe directly function reconstruct smoothing knowledge adopt analysis description work present consideration fundamental practical focus experimental design estimation summary final remark theorems regression response random linearly need experimental batch form trial repetition
second inequality due rearrange feasible objective feasible key strong optimal exist hold c xx nh prove em ready complete follow relationship feasible optimal last quantity auxiliary eq q proposition k f expectation side inequality consider sum together fact mf eqs lemma hold convergence analysis theorem interesting whether let satisfie obviously sphere satisfy assumption solution imply constrain illustrate convergence optimization easy certain counterpart specific q optimal solution focus set constrain extend building establish convergence constant verify w imply theorem even effectiveness constrain logistic label I review three multi sparse class positive remain negative conduct sensitivity study distribution step vary fix onto ball thus dominant computational implementation vs plot non much remark step demonstrate size robustness ht tf plot average run uniform parameter average comparison sdca sag unconstrained sdca adopt solve optimization accelerate sgd gradient suggest stochastic hybrid sgd pass switch kf plot set sgd sgd comparison behavior report objective gap plot initial outperform quickly gradually proceed phenomenon commonly hybrid report reduce stochastic solve constrain problem establish strong reduce analysis wide lipschitz continuity eq il eqs define obvious assumption exist know empty assume f lead hx f cx use optimization slow sub linear inherent computation many objective strongly project stochastic problem contribution convergence convexity variance reduce stochastic role big optimization solve prefer due stochastic gradient optimization standard draw compute stochastic slow full proximal rate recognize slow stochastic include sag stochastic dual sdca epoch mix gd prox linear practical square extensively even rate full address prox establish convergence convexity without although solution still convexity address establish relationship current linear solution objective bind establish objective function however paper suitable address adopt establish linear strong general mild satisfy present constrain follow paper eq continuously compact accord notice convex onto must moreover compact example objective dd logistic tx tp k np k let objective value achieve respect sampling remark eq similar prox term propose large choose outer compute single count gradient l proportion uniformly outer evaluation complexity specifically remark prox need evaluation obtain contrast obtain high eq section several fundamental key idea establish distance feasible rate recursive strong convexity lemma constrain optimization adapt corollary establishes gap solution function linear bind
interval use measurement include ahead instant flow therefore name lag smoothing may lag smoothing lag measurement arrive numerical investigate pos run choose interval conjunction temperature pressure accuracy flow however variance optimize outperform one manually variance accuracy suggest model happen pf step particle match observation compute intermediate k ip k sir obtain index sample set index sample associate particle eq previous involve need adjust end compute proceed previously rgb international gate mail international author flow sense problem previously module model certain temperature pressure flow leave well jump design capture adopt measurement physical process two approach approach sensor advanced field typically equip sensor control rate require flow expensive result measurement risk collect flow exercise soft soft decision purpose essential production tackle soft sensing base instance extend soft sense lift adopt recently enkf flow pressure soft filter example employ apart approximate bayesian soft iterative current work module framework include flow conventional steady state behaviour describe time reason dynamical jump capture rapid water operational underlie flow measurement one estimation g mention paragraph aforementione sense variance manually situation aim fill propose variance criterion problem proposes previously illustrate reader short introduction equation temperature pressure instant phase normally contain distribution simplicity estimate jump process flow follow probability predefine normal remark firstly dynamical model state operator noise term distribution unknown variance discuss optimize optimality assimilation flow rate gaussian gmm assimilation gmm pdf instance view density kde aspect convenient development later lead problem section form adopt non assimilation auxiliary rate sir mainly sir pf weight particle ahead sample particle brevity detail q j r k j jj nj weight generate rate tw formula manually trial final may automatic optimize sense later measurement collect interval type approach smoother optimal carry respect certain fix time ahead correspond smooth fix lag smoothing approach idea notational convenience instant contain jump minimize certain joint pdf solve line k large optimization process become need assimilation scheme often fix long minute study fix interval offline condition past instant particle dirac delta elsewhere substitute eqs obtain approximate influence optimization optimize lag smoothing use estimate measurement kl compare interval fix lag estimation arrive rather time instant current lag minute lag however beyond em algorithm construct function interpret cost maximize value initial one construct dependent condition maximize obtain optimal start construct optimal stop relative estimate j pre threshold estimate loss generality instant line measurement involve construct dirac delta function jump log joint pdf joint f calculate conditional pdf determine jump rate condition include instant situation approach filter approximated flow rate write ignore purpose maximization one need solve simplify discard irrelevant obtain analytical approximate integral carlo approximations instant associate multipli kronecker whose elsewhere essential thus drop q note multivariate draw variance likely relatively away case substitute eqs solve q obtain iteration square light eqs weight update estimate positive definite iterative weight need needs iteratively significantly reduce iteration f l easy verify hessian iteration formula maximize estimate example due behave slightly first flow run resolution md pressure temperature assimilation later pos run fine resolution synthetic course datum assimilation model run system apart case possibility assimilation produce purpose examine correction assimilation different number present respect seed configuration show vertical horizontal curvature degree vertical md label label md sensor collect pressure pressure k pa detail reader refer flow rate z pos measure pressure pos depict spin flow simulation reduce happen accordance fig pressure record place contaminate meaning pressure experiment deviation measurement pressure configuration jump specifically take set take true rate multiply result smoothing first variance flow jump particle sample input well pressure record temperature pressure adjust relative particle match observe pressure also profile true red blue dotted rate deviation pressure temperature red simulate flow blue flow simulated temperature pressure record flow flow flow appear true constant variance situation rapid flow true average minimize toolbox flow rate flow pressure panel use flow smoothing approach assimilation fig appear time rmse become conjunction rate variance interval spread implement em flow stop either norm iteration stop estimate variance leave variance cross order magnitude time consequence correspond fix interval variance flow variance lag equip lag pressure fig fig approach term pressure lag estimate less smoothing approach change true fig lag rmse become fix interval pressure temperature fine resolution study case uncertainty error purpose variance way pos uncertainty flow synthetic well flow assimilation synthetic resolution well md assimilation flow instead comparison well flow fine compare fig temperature uncertainty generate fine pos assimilation
inequality iff inequality statement eq put inequality subgradient q proof state technical use smooth denote get verify corollary schwarz plug obtain eq eq proceeding optimize inequality let first derivative compact domain lie enough need smoothness derivative denote put allow q eq inequality elementary q reasoning denote tb jensen need lemma reasoning put together divide everything take use jensen denote scalability selection ignore fact theoretical make example solution practical remain propose gradient perform selection tune cross build regularization rate prove standard fractional scalable optimize objective way come source moreover theoretically yield well yet rate depend use size reality wrong strategy attempt prior knowledge characteristic optimal cross validation slow partially exception keep epoch procedure number epoch decide set exactly batch regularization take role parallel study turn solve correspond stochastic nature different stochastic infinite dimensional plain gradient procedure achieve idea change dependent know algorithm provable rest sec sec adversarial set sec detail relate defer sec associate kernel implement inner satisfie reproduce property measure w consider l loss differentiable subgradient receive pick minimize difference statistical sample integrable respect binary infimum misclassification measurable kl kx k compact fractional indicate l big role analysis training use propose fast average immediate show result step risk average close moreover amount regularization choose infinity predictor infimum attain guarantee infimum vast examine infimum attain hold optimal suboptimal unfortunately able self tune indeed like kind perceptron mistake hinge similar step unfortunately mistake specific measure different loss algorithm equality return algorithm main difference past calculation computational let dependency outline moreover regret absolute gradient tighter smooth loss loss grow bad bound loss regret grow norm appendix kernel optimal rate coordinate truly misclassification risk iff exist relate misclassification expect special one agnostic regard low square interpret space condition discussion low term lipschitz unlikely bad case cover rate translate average establish novel approach stochastically improve suboptimal locally loss origin optimal obtain rate range consider without obtain excess lipschitz loss solution convergence specific dimension guarantee good misclassification risk perceptron weaker present classifier simplicity assume show hence risk batch obtain square loss infinity also tune achieve cross optimal core whose performance close batch first note propose regularizer bind prove optimal weight regularizer paper capacity independent regularizer exponent belong argue make svms indeed give parameter implicitly thank permutation theory tool convergence require data work also potential world hence concept method preliminary fold cross experiment three precise replicate experiment track contrary intuition sample probably finite dimensional seem bad fold time fast gain question open empirical w risk prove strong probably finally improve would result loss take gradient design analyze copy follow one sequence loss lipschitz algorithm hold importance difference dependency essentially second important assume knowledge
sufficiently separate extract decomposition note however scenario subsection rigorously atomic get dense definition atomic norm link correspondingly nd formulation generalize prove limit scenario grid dense correspond atomic grid sparse line base advantageous lasso practice automatically estimate variance phase common music independent minimize reference therein criterion understanding provide subsection note optimization covariance respect exist discretization frequency domain eliminate alternate name grid criterion solve discretization toeplitz sense represent see give retrieve use determine among choose always interest toeplitz mt sdp accordingly set sdp manner case sample adopt eq clean complete q give sum apply presence explicit atomic explicitly sparsity see term fit neither limitation determination motivated atomic explore connection atomic suffer two limitation neither noise variance reason inaccurate common method frequency accord datum lie range rank otherwise superposition equal split present component possibility numerical reason highly sdp set component bring frequency report grid I nearby latter issue certain contrast splitting cause version split grid without confirm detail phenomenon report previous lead frequency detect frequency splitting cause mean adjacent result bring challenge detection overcome propose line consist estimation covariance scheme framework carry solution main contribution covariance provide solution covariance exploit toeplitz choose window potentially resolution paper choose music frequency estimation study good sample model carry correctly estimate beyond provide frequency frequency effectiveness selection conventional length criterion aic well challenging require date available globally solve missing carry convex need initialization nd atomic norm q denoise correspondingly common call sr connection begin lemma q lemma h easily estimation formulate hereafter produce scale frequency variance omit brevity hold q noise variance problem convex formulate follow identify simplify noiseless atomic ensuring interpret different entrie fit sr whole reflect base version equivalence exist formally show equivalent implementation limit scenario problem obtain remark carry case result overfitte indicate necessity selection frequency process formulation section inherently amplitude derivation claim simulation utilize aforementione modify constant give elegant solver empirically problem meanwhile follow theorem edu sg method line concerned develop sparse limit approach dense datum atomic incomplete noise far prove atomic systematic simulation validate exist line atomic norm spectral signal process spectral estimation communication noisy index jj mf ks ji f available call incomplete miss practice cause failure weather physical frequency estimation line estimation paper mainly focus estimation incorporate many frequency classical music limitation difficulty worth approach nonconvex optimization require know easy selection frequency music theoretic eigenvalue predict eigen threshold outperform later compress sparse frequency decade discretize grid assume true practically close observation accomplish support prominent usually since cs finite dictionary discretization early dense almost complete adjacent vector intuitively reasonable dense since grid grid frequency observation naturally exist infinitely grid proceed worth drawback finite discretization coarse miss feasible multiplier admm optimization version analytically numerical notation number norm transpose semidefinite notational numerical distinguished rest organize preliminary extend case present extend systematic equivalence method present feasible conclude recover noisy knowledge dictionary sparse norm denoise recover play fidelity call denoise nd choice refer lasso sr lasso lasso sr loose easy concept atomic norm generalize nuclear convex compact contain origin function atomic norm dual atomic h write
onto exploit projective paper borel signal respective dimension typically represent vector filtering polynomially filter kalman recently hmms description hmm space stationary transition emission filtering involve follow act measure computable follow structure embed hmm exist exist markov evolve system ordinary death hx compute local sufficient special dimensional emission exist bayesian think label whereby admits jump gamma dirichlet statistical parameter k infinitely many type slight law refer mixture whereby conditionally conjugacy projective u process subspace purely atomic measure hence describe evolve review attention mutation generator x jump whereby jump cf project fisher reversible stationary kronecker delta act property dynamic property support alternatively diffusion function finitely many denote closure product reversible paper model stationary model exposition focus evolve countable emission prediction extend multivariate let coordinate show death jump diffusion generator virtue establish together lemma appendix assume conjugacy provide let lying consider generator datum close update operator operation sum let generator independently q tie law arbitrary class result cell k mm projection suffice merge denote turn marginalization multivariate single evolve mixture reduce stationarity distinct include distinct notation q dirichlet dirichlet measure iterate propagation provide lemma computable partially evaluate proposition consider family generator prediction theorem gamma measure think counterpart dirichlet gamma parameter denote representation conjugacy property random poisson intensity conditionally disjoint mass independent gamma distribution conjugacy finally know independent gamma process version branching measure space review interested branching mean per population process replace subspace compact contrary whose substitution describe branching account independent individual heuristic evolve constant operator reversible partition branching reversible distribution generator q act vanish process speed introduce independent generator proposition identify ease divide write I kx ik generator apply hand collecting note previous duality dimensional death conditionally jump deterministic driven type differential next propagation independence equal equal solve follow death jump occur jump occur iterate conclude jump distribution poisson observation hence version size process fix mm term leave denote respect multivariate follow argument step z mm respectively imply distinct mixing measure successive iteration filter family finite mixture generator application operator update operation sum eq derive filter conjugacy follow signal distribution law wise information integrate knowledge mean operator yield interval observation signal gradually approach ergodic signal acquire gradually weight sequential govern death deterministic jump continuously propagation gradually ergodic state dual thus able acquire information filtering concern prove due high generality derive propagation nonparametric corresponding parametric projection exploit support lemma provide probability death death start generator proposition propagation step generator q follow mm mm mm theorem mm mm mm evolve random indirect diffusion dirichlet point state obtain filter computable projective take mixture two mixture measure prior bayesian gamma secondary unobserved observation filter optimally entail extend key
definition q violate remark research study due nuclear norm dimension input linear namely show set nuclear rank projection onto dimensionality excess available constitute conventional complexity hope within reasonable cost difficulty might data identifying want analysis matrix accelerate calculation approximately matrix provide elementary scheme allow weak notion intrinsic stable later text dependence indicate decrease place nuclear c rank embedding operate dataset preserve euclidean random way adequate preserve construction high geometry hand leave space improvement main contribution manuscript multiplication notion rank rich reduction utilize wide cover highlight provide depend classic e lie etc theoretical special unit point strong et require preserve improvement nice exposition see improved bound matrix multiplication bound approximation exist well perspective preserve query describe polynomial time relaxation know development refer scalar span homogeneity satisfy suffice argument without loss I leave singular c triangle appear technical negative integer highlight lr analysis satisfied use triangle inequality define event probability positive accord l c useful proof provide fix tb conditioning l upper bind last equality sr obvious sr violate satisfied give appear union l rescale question construct heavily theorem propose randomize low follow specify follow practice distance row additive htp height n substituting write restrict search space standard become remove get eq complete novel provably multiplication drive
thick color color k prove theorem characteristic write always normalize respect prove implicitly al vector span e k hold proving every combination combination k eigenvector error almost show theorem fact column eq orthogonal moreover cf k eigenvector norm next eigenvector whose coordinate reasonably able characteristic eq ji orthogonal construct column eigenvector contradiction imply th somewhat us q due contradiction eigenvector approximate step function enough entire paradigm method cluster approximation translate suffice box manner point follow quick mean structure analyze spectral embedding give algorithm seek minimize squared center cost minimize close choose define mean normalize laplacian point embed group widely analyse result describe spectral show factor nice embed embed clustering necessarily kp embed point big concentration however point suffice take transpose proportional far another volume embed far embed misclassification big could large eq eq p hold finish imply case explanation mean partitioning partition return mean map trick bind one cluster vertex value technical reason optimal follow assumption approximation suffice permutation suppose index lemma ready contradiction contradict optimal achieve leave edge proof lemma contradiction approximate follow function assume index case distinction let eq prove I complete thick color red thick color thick approximate embed vertex tu probability distance heat kernel distance equivalent invoke analogous computation entry vector create additive give run obtain tu time conceptually framework step choose good different center choose result center additional embedding allow simple sampling motivate approximately show ensure close vertex removal step vertex form proceed grouping thank assign near center much simpler far vertex correct center moderate almost routine near framework combine however expensive become large however suffice mean space directly heat approximate heat distance approximate consider gap cost framework compute involve due lemma guarantee return throughout assume vertex proportional sample vertice routine approximate approximate eigenvector computation else approximate tc distance q averaging argument vertex core fu ir cauchy schwarz inequality inequality assume argument next lemma vertex vertice core one every probability vertex core z contain vertex cluster lemma use bind core come never core bind event core close core succeed triangle last condition give lemma fact hence exactly obtain vertex analysis belong assign vertex separation ask vertex embed reduce follow neighbor near neighbor grouping use proposition grouping analyze ratio return computed difference correspondence eq q early discussion graph circle matrix corollary eigenvalue easily undirected polynomially formally edge vertex define kernel unweighted rgb rgb theorem thm thm lemma thm corollary thm thm protocol email inf de visit institute berkeley inf de suitable admit present almost partition graph partition cluster cluster well connect outside key wide key cut eigenvector rigorous guarantee approximation heat embedding neighbor heat piece fundamental partitioning side cut formally undirected unweighted vertex edge hard time subset e partition partition eq clusters network capture notion subset domain computer vision procedure region split turn graph multiple subset key unique game despite various expansion eigenvalue lee high laplacian matrix informally order large partition bind informally close connection vector show variant rigorously analyze exploit heat locality hashing algorithm achieve cluster first eigenvalue span close achieve matrix characteristic statement k et prove think version motivate definition contain proof improve inequality application set span spectral theorem open overlap practical comprehensive extensive answer open question whether spectral mean algorithm rigorously analyze circumstance guarantee spectral algorithm let graph partition statement os ik algorithm euclidean different versus moderately e super run moreover embed obtain fast technique approximate via allow avoid eigenvector assumption hoc linear almost graph edge assume nk
reproduce rkh endow closure function predictor class classifier attain f boundedness satisfied efficiently prediction training excess surrogate g hinge exponential risk similarly respect convex surrogate define line research risk find crucial binary active theory last decade relate binary excess risk excess follow risk find quantitative excess excess establish convex transform closure bound convex distribution q extension view sufficient transform calibrate surrogate function loss iii insufficient surrogate affect account efficiency issue practitioner deal big unclear loss time family convex smoothing translation convex risk binary risk state efficiently function problem term smoothed advantage hinge transform popular surrogate hinge loss immediately excess affected parameter smoothness relationship obvious convexity surrogate statistical consequence follow smoothness step function follow smoothed hinge parameter transform complicated theorem smoothness bind theorem demonstrate infinity hinge hinge approach smooth smoothing correspond theorem smoothed hinge smoothed excess smoothed f indicate theorem smoothing approximation excess smoothing good tradeoff smoothing parameter risk analysis comprise bound excess bounding minimize f optimization generalization minimizer surrogate unify arise problem numerically nice hinge smoothness proceed follow immediately bound gradient rule understand generalization give generalization error recent optimistic rate yield easy lipschitz loss perform smooth constant problem error generalization bind smooth solution kn constant r understand smoothing iteration give follow assume characterize z z respect characterize converge see sensible perfectly classify point q satisfy perfectly classify percentage datum use characterize excess state eq complete proof failure constant otherwise excess e converge fast generalization limit achievable small investigate use surrogate excess excess risk provably previously relate make towards loss guarantee desirable property generalization excess excess risk result favorable smoothness achieve excess risk proof zero therefore eq solving verify eq compute define expression similarly e inequality q complete empirical convex risk bernstein plug replace universal noting remark thm surrogate loss prefer bring surrogate smoothness beneficial computationally optimal improve optimistic smoothness viewpoint affect excess contrast favor may binary excess risk motivate unified optimization generalization excess excess risk examine favorable condition convex excess instance product endow learner sample identically unseen come z label minimize excess study minimize n usually risk understand erm bound excess take function condition minimization consistent achieve erm convex efficient indicator loss function logit loss svm f adaboost learn empirical loss bayes necessary sufficient consistent differentiable origin convex
exclusive school evidence suggest cast label careful label type graph let infer leverage propagation label labeling relax node node possess quadratic configuration link negative find fix graph handle minimize eq full simplicity intuitively propagation friend whereas suggest reason school college city etc represent set respectively visible publicly type eq constant share type reason underlie r long sigmoid greater dependent threshold sigmoid enable control explanation edge allow extension choice explain really well maximize high require turn word control need force sure share one type match empirical suggest matching type enough explain eq thought type use model uncertainty type exhaustive reflect belief suggests indeed gain consider whose label early label propagation friend city city inference completely correctly infer city friend marginal extra benefit current city significant benefit city property try friend p enable type pair affect say city friend reflect match necessary eqs optimization spirit label define measure convex finding correspond friend restrict label convex friend form iterative direction back simplex f probability possibly improper could close eq label store type set label type project simplex converge distribution value require information message architecture generalization applicability reason formation mutually exclusive strictly school friend city black represent friend infer high school small maximize friend introduce part fold fold use fold various fold place difference variance inference rank label user rank top type rank actual present lift propagation iterative graph processing friend communication overhead retain top entry optimally b retain friend friend age high school college significantly improve run time significantly show friend decrease demonstrate importance limit increase friend enable well beyond point trend baseline facebook appear many friend friend become label friend add friend recall propagation benefit school college lift propagation city improvement school college easy latter figure propagation pick instead less common explain able infer difficult circumstance even label inclusion improve wide already careful type recall city college within group use college turn college membership extensive one addition membership impact label lift membership friend membership benefit recall lift label lift significant compare propagation merely scalability also careful make membership redundant priori impact membership regard type membership actually redundant type broadly sufficient true friend college correctly label share show top prediction type somewhat easy hard friend share explain lift plotted figure lift qualitatively offer within match matching need thus jointly mode particular user high college identify phenomenon create create reason necessary two carefully model method equivalent run basic label propagation facebook benefit model network primarily consider interested infer label type accuracy tackle infer label label label primary focus city connect network fail property label explicitly enable distribute pass architecture facebook label propagation infer label node predict low node say edu belong student node infer often partially fill label dimensional correlate ad search friend motivate problem friend contain city actual city call explain city one inference try friend essence common likely connect optimize category relationship address formation reason snapshot city know unknown friend completely know independently infer friend city common city friend city city among friend friend city happen type viewpoint likely share value type two user college beyond label item propagate graph inference try edge point reason friend college primarily infer label node knowledge link prediction task recommend college friend high school solution propagate cluster profile try deal incomplete allow believe readily contribution formulate one label belong whose property incorporate infer architecture scalability facebook profile real dataset recall improvement scalability usefulness survey relate follow generalization prove prior semi supervise relational model base view label quadratic penalty possible modify random interpretation label propagation handle large number assignment count none interaction hence fail formation typically predict alone relational use collective neighbor relaxation labeling well outperform focus good
ce drive specify actually ce varied range ce effective scale treat need energy experiment ce factor smaller effective restriction factor combine ce ce calibration exercise calibration cope check understand setting synthetic exploratory analysis necessary ce gain informative team collect prediction shape disk heavily field circular deterministic large input much physical denote calibration label new outline involve limitation framework couple deterministic computer plus discrepancy systematic disagreement model observation reality process review detail value via prior model computer output consume fit train simulation run recommend gp typical recommend joint observation coherent thing jointly give even lead mix substantial effort context couple simulation still previous work bayesian joint uncertainty bias justification coupling amount enhance joint despite simplify gp reason shall normal define stationarity simplify zero perform gp conditional integrate degree freedom component define correlation hyperparameter matrix posterior prediction require obtain point scheme typically restrictive ba schmidt modeling especially fast run computer fast big gp recent search estimation decomposition avoid sequentially carry library gp methodology develop provide gp inferential modular path focus rather input greedy fashion pair efficient update approximation local ultimately subset computational design neighbor extend correlation thereby calculation dense design size hour implementation provide package calculation parallelization yield challenge calibration way posteriori prior calibration parameter scheme involve cascade maximizing perform calibration pair vector computer input column design location model independently mx value great expense fit u measure well fitting train give prior field prefer therefore suggest maximize parameterization shorthand fit rather inner method discuss fidelity field x j j b f b f b fu inner max detail automate routine package loop predictive subsequent prediction via execution extremely example follow neighborhood implement model fit whereas sensible degenerate prior identity reduce work numerical stability term un inner evaluate search illustrate numerically approximate one implementation mesh interface successive try sequence weak regularity direct call popular many optimization derivative approximation numerical search small decomposition solver recommend posterior distribution give calibrate computer obtain predictive abuse let stand corresponding equation student equivalent one augment diagonal depend locally step design implement step save moment location combine de distribution I ideally full step lead student comprise normal necessary sum sample convolution prediction observe field still option might good consider identification retain desirable attribute evident concern primary agree alternative estimate uncertainty full counterpart build section adapt datum unit cube follow keep generation replicate broken regime unbiased designed explore efficacy propose approach scenario experiment motivate regime mc repetition use variation replicate realization model design begin four trial per value simulation mc initialize obtain solver boundary search generate comparison replicate variation example modular cube directly predictor entirely alternative error decompose level calibration generating estimating ht cm leave deviation panel estimate dash line triangle axis set arrange six panel group three truncate improve visualization arrange number replicate six six label prediction fourth lead nearly first cost u span alternative third leave replicate similar former indicate job surrogate replicate replicate imply replicate low thing quite panel recommend choose statistic min inter range ten pairwise number clearly variation variance stage random calibration great uncertainty come three trend omit clutter value along straight line go point densely near ridge combination confirm true far weak weak prior move uniform discrete nature smoothly vary value change change ultimately posterior motivating take average repetition intel ghz machine optimization span bias model quick right alternative option closely bottom leave even recover nonetheless expense estimating cm cm explanation matter fit stationarity approximate gp thus accommodate explain biased full joint exploit lack identifiability discrepancy parsimonious large good meanwhile summary mechanism second average correlation return motivate explore problem bias extent simulator substantial distinction synthetic experiment concern local unit cube response vary magnitude biased preprocesse preprocesse input experiment isotropic discrepancy restrictive field observation virtue version scale obtain subset computer specifically sample replication estimate energy thick ratio length limit observe pressure length diameter cope common suggest fast roughly short energy divide cube input square root small vector comprise scale perform monte hundred repetition conservative costly search field datum energy pressure column drop diameter provide variation report exploratory analysis aspect bias calibration insight difference variation aspect input substantial impact isotropic preprocessing regime average figure panel response average input preprocesse specification give influential input energy energy less sensitive input calibrate prediction rely model un report leave methodology gain turn bar subtract leave obviously mean pair fail difference predictive amongst credible visually predictor seem small understand however reject nan may due turn calibration exercise plot evaluation algorithm combine search indicate open estimate discuss minute machine take ordering would however fast version surface variation surface consensus value mean ce bias unbiased largely agree set however isotropic amongst attribute pre add close separable illustration discussion model synthetic profile log although highly informative negligible noise evident surface flexible weak biased yield much reveal right axis correspondingly dot bias surface stop early second uncertainty bootstrap review experiment estimate open circle figure heat plot right cluster estimate pair important bootstrap general surface large summary figure suggest representative amongst open colored converge biased noise suggest highly value make input provide team configuration nominal setting table design input nominal fill pressure ratio measure field exercise energy three provide variation experiment conservative accounting specification ask propagate uncertainty calibrate manner exercise propagation quantification uncertain input uncertain far bootstrap produce spread show explore calibrate exercise ht show plot indicate focus panel predictive four input roughly remarkably method despite choose estimating lead red degree nominal setting dash red providing spread skewed suggest prediction square preference allow show output nominal great agreement calibration methodology generally provide motivating approach calibration increasingly work processor core whereas increase although something salient feature essential process carefully extra package calibration leverage aspect calibration paper motivate poorly identify providing calibration output scheme method believe computation option amount high price great effort calibration yield excellent estimate flexibility nonparametric gps design calibration exploit idea model coupling gps discover calibration method decade
proof evaluate frequentist statistic inner p statistic corollary define large family extend translation lebesgue haar measure associate translation translation sum prove concern transformation typical invariant action mean modulus insight haar theory assume corollary sample replace statistic trick one dimensionality concern haar belong trick replace associate whole investigate testing different approach frequentist estimation especially invariance frequentist haar approach explicit central quite invariance bayesian invariant procedure haar sense right haar prior domain equality need first condition call stein common stein theorem satisfy rx rx assumption use mainly hold consequence invariant transformation group assume equivalent identity stein imply varie accord observe question domain underlie vs ie section modification presented namely propose illustrate see carry property belief ratio testing establishing pearson hypothesis test rely broad composite vs hypothesis data parametric merging test family vs composite hypothesis domain improper extension make bayesian type pearson indicate lr central vs define posterior integrate posterior perfectly allow define composite symmetry composite test sequel quantity interpretation far improper prior smooth side posterior less statistic general frame pearson extend interpretation measure seem sense mathematic interpretation issue need subset set joint replace role remark measure posterior proper evident world remain consider single weather daily describe daily five statistical whether prior enable simple hypothesis vary impact display explain improper simulation perform simulated reasonably alternative characterized likelihood frequentist estimation couple mcmc slice sampling implement simply order lr combination threshold evidence favor practice read great likelihood alternatively greater correctly consider cumulative sided whereas bf credible vs bf translate credible also natural credible inference hold inference generalize composite bf pearson bayesian version frequentist pearson bf pearson maximize frequentist classical unknown composite hypothesis credible bf relate hypothesis namely gaussian invariant frame stein theorem credible evaluate inference new example invariance likelihood conclude equivalence connect equivalence credible hypothesis frequentist invariance hausdorff continuous support invariant haar q invariant haar replace haar multiplicative right modulus invariant haar haar note haar haar occur haar definition haar modulus modulus invariant haar measure measure right haar right invariant haar haar statistics concept transformation family property action group say invariant connection transformation lead could differently define invariance family would long presentation group haar induce action frame haar turns actually note subset aa group finally define marginal density always finite probability even mean measure specify note action element element haar transformation lr rl measure posterior integral c modulus practice datum see theorem depend instead follow shall haar modulus imply haar haar relatively modulus since invariant g px notice simplicity transformation marginal way frequentist order side x px px corollaries corollary likelihood haar absolutely lebesgue theorem domain measure integration integration haar lebesgue modulus combine get p simplify sx sx sx sx sx sx condition see induce random particular threshold notice equal statistic frequentist threshold define pd h pd pd reformulate use appendix equation lr equation call integral b check inequality true multiply leave positive term integrate ix conclude implication eq b b reciprocal generalization distribution testing hypothesis test many threshold equal extend nuisance extend frequentist finite analogous stein credible domain confidence frequentist vs hypothesis measure nuisance two first improper soon second role lr discrepancy variable invariance pearson composite give observe dataset q distribution characterize alternative test great type make decision paradigm type lie fix invert test notation side bayes factor threshold straight bf jeffreys bf strong evidence favor probability bf simple hypothesis improper proper partial account prior proper partial bf bf initially vs composite mean gaussian powerful property x size issue see recent study consider unlike bf suffer idea prevent occur bf argue frequentist several classical unlike bf frequentist frequentist statistic ie integration contrary predictive value discrepancy domain choice discrepancy variable bit classical approach introduce derive simple vs test tool one mathematically propose compute q relative evidence belief evidence resolution large contradiction simple hypothesis posterior ratio unobserve variable namely bf ratio random evaluate cumulative compare pearson paradigm read likelihood fixing reject sensitive making threshold decision broad range grow typically clear computation display later propose generalize generalize add reference systematically deal case hypothesis necessarily list different analysis claim unlike bf improper posterior proper subject invariant transformation consequence function last example consist bf compare bf compare bf ie describe general alone show bf show support low general addition result example lr bf credible way bf thresholded seem invariant transformation indicate cumulative practice straightforwardly carlo markov chain algorithm almost histogram lr cumulative lr detection extra image dedicate large image available extra present dataset extra dark less star star degree classical potential study investigate play test frequentist prior hypothesis notice highlighted equivalence equivalence credible domain frequentist hypothesis discuss example discusse obtain frequentist credible extend definition frame yet composite hypothesis condition fixed parameter see although condition remain transition composite immediate hypothesis still generalization extension simply improper soon proper improper prior proper posterior probability bayesian pearson associate discrepancy lead conclude section essentially mathematical often think evidence expect equal happen bf highlight hypothesis raise question frequentist hypothesis agree interpretation could frequentist test unified consist analyze likelihood seem frequentist marginal hypothesis particular broad reference include modify p make exactly
indicator q refer whose entry easily see b cover nonempty precede argument full clique vector concatenation vector copy match denote clique marginal graphical correct graphical n correct independence central note part consequence require justification material tractable approximate inference covariance marginal fortunately leverage compute effectively covariance inference degenerate fix transformation reduce vector covariance invertible work propose minimal graphical full covariance project maximal maximal note subset linearly variable redundancy way choose long ss full b indicate representation conditional distribution degenerate observation clique example noiseless di c decompose count node edge retain count marginalization root node away marginalization edge technique u v u v tu row distribution v u u u conditioning formula v different way sparsity inverse statistic graphical factor density translate sparsity precision reasoning derive conditional asymptotic noisy throughout remainder tree edge notational simplicity noise unobserve represent drop factored denote observation determine entry reconstruct poisson long apply propagation laplace omit simplicity mean covariance ep density uv uv uv uv compute step onto laplace need mode variance approximate distribution solve optimize value form term since normal density bfgs work observation mode variance project infer inverse laplace take outer suppose ep message pass time maximization laplace infer count consume inversion obtain complexity model locate bottom leave corner make cell employ four feature cell cell fall lie toward encourage cell denote vector logistic domain count cell number move count generation generate vector count consider infer observation node count issue count introduce observation instead estimate burn million collect million iteration relative mcmc experiment run generate edge count introduce factorial show map much produces obtain task via compute table coefficient varied population approximation although significant make approximation map exhibit much ccccc inference experiment magnitude logistic extreme evaluate accurate although cccc ccc explore effect random vary size accordingly small method grows map rapidly ht ccccc vary true generate curve consistent population map map overfitte create match performance em iteration measure experiment measure cpu count count edge count compute edge require node count also much reveal computation consume relative match small map edge estimate good transition near conversely extreme map surprising much introduce collective limit distribution population size matrix maintain method inverting covariance develop efficient simulation show exhibit variance material base national science foundation grant usual writing replace show nh showing order sufficient clear inspection remain tree hard enforce consistency count sample equal integer count variable global consistency interesting corollary argument base detail converge replace mean property linear entry show recover invertible di dd definition equation valid indicator configuration trivial linearly collective individual collective statistic e individual intractable previous explore monte carlo approximations study maintain follow poisson accuracy mcmc map exceed set wish collective count might education census reason census education region concern anonymous location arise construct set individual copy population individual permit clique count individual joint answer condition observation clique individual setting clique clique model also difficult configuration take
dataset spend subproblem solver give gradient train crf gradient iterate solve inexact far experimentally tree much slow leaf strategy still benefit logistic learn ccccc linear boost zero mlp mlp boost mlp bound define lx h line zero mlp mlp boost mlp true mlp mlp mlp boost c boost mlp mlp boost mlp zero I boost c mlp boost mlp boost boost mlp proof successful write iterate observe addition entropy message non logistic structured exist minimize choose output major challenge solve high output np standard lp relaxation inference parameter alternate message pass descent mostly focus adjusted useful ensemble predict independently unary either hold adjust allow general relaxation problem alternate pass update major minimization loss logistic need function optimize experimentally test linear flexible predict fix sum function subset consider structured problem directly handle generalize energy reduce linear find logistic regression select loss concern standard choice slack rescale loss measure experiment ham eq maximum range general solve approximately motivation relaxation bind relaxation fx polytope agree region eq restrict value take q objective constraint lp practice preferable message passing exploit inference approximation pass guarantee use loss appendix result previously difference without smoothing configuration evaluate specifically eq contain maximization saddle inspire work message write alternate message ascent update message trivially fix message section regression initialize set multi ensemble solve albeit function minimize evaluate solve message pass optimize alternating optimization fix pass parameter thus concerned optimize conjugate message bias substitute simplify marginal inner maximization close thus equivalent eq term fact give result learning summarize alg depend situation local constrain function constrain image mapping energy would select experiment ccc boost mlp ccccc boost mlp linear mlp ccccc ij zero linear boost mlp boost boost mlp c mlp ij boost ccccc mlp mlp experiment respect bfgs layer perceptron descent momentum mini batch univariate tree stochastic boost loss fit one control loss multiply add classifier message iteration synthetic denoise visualization create generate sample feature add classifier combination nonlinear result low rate plot
spirit lead logistic label resp obvious treat follow connection deep log define converge pointwise uniform respect strong f randomness assume implicitly mle logistic well mild check function together divergence iid kn jt kk jt correspond odd ratio represent jt semi parametric practice semi extend iid logistic begin iid efficient completely parametric ignore severe make detailed replicate toy chain transition probability blue solid dot strong autocorrelation course constant use quite form pair replace k per either true reference log odd semi logistic effect effect add offset completeness practical variant intercept constant mean replace intercept another way constant simulate fix increase value pick value repetition perform inference enough ml perform show asymptotic bias project ideal possible nonparametric valid practice generalise model logistic parametric good parametric poorly line right constant convergent estimator thorough location link straight predict likely case country past spatial predictor availability net context movement predict visual reliably draw stimulus exhibit dependency tend move currently bottom corner take step go something rather motivate represent purely spatial interaction spatial dependency centrality bias preference take distance location tendency current axis horizontal therefore decompose sum angular function smoothing therefore extend straightforwardly explain turn r package component uniform example function variability replicate central location dominate although subject display include preference suitable format around minute poisson transform turn otherwise non generalise additive reduce display effect centrality bias subject gray blue way become parametric turn semi likelihood monte logistic challenge covariate dependency dimensional design non constant great logistic able leverage nonlinear reduction development challenge transform difficulty proof derivative ns nf f shorthand ns ne ns ne e ns shorthand ss obtaining intensity density confidence inversion equal aa ba bb ab aa nn exponential concave poisson concavity natural derivative simplify ss ts matrix derivative simplify certain rewrite odd since law almost surely surely sum prove function attain attain take establish previous interval assumption iii almost randomness absolute difference three term converge bind replace n third law I generalise book bound assumption independent may rewrite theorem supremum application spatial model involve estimate magnitude matter variability band smoothing spline smoothing infer report band smoothing fit band repetition thm thm theorem contrary specify likelihood popular lack infer problem poisson specify need infer another non iid include model dynamical binary turn show extend spatial poisson non parametric core tool modern deep vision datum appear function technical whenever estimate prevent direct many technique recent year difficulty non include model density intensity expand estimate iid optimisation turn semi arise extend iid markov description movement idea appear form learn transform family bregman divergence generalise kullback leibler divergence learn study place spatial far show converge uniformly technique time ignore indicate convergent framework parametric convergent section turn likelihood information call non iid still chain transition constant highly poisson q along line sum value mean previous space corollary turn model estimate corollary exist possibly ie function require need classical solve parametric part belong include penalty parametric uniquely suppose contain exists maximum optimisation
r estimator illustrate estimator affect error inconsistent appear analytic monitoring reality g light subject essential use calibration measurement inference bias trend functional propose measurement regressor response affect error aware reduce completely eliminate knowledge error deal measurement error include probably literature variable model van reference therein deal mostly parametric nonparametric method reference instrumental variable quantile ability interest describe recent treating model discover nuisance regressor kind measurement recommend region insensitive regressor invariance rank estimate nuisance parameter consistent every show estimator slope bias precisely even unbiased situation far present depend generate measurement distribution unknown regressor affect measurement observe ni ni ni identically error moreover thus variable observable predicting become ni u ni ni interested slope estimator ni ni e ni ni ni ni b statistic invert invert extension location analog error subgradient exist generally literature absence error ni ni normal presence asymptotically furthermore locally bias fix mean sequel take unless convergence shall need assumption underlie entity square skew symmetric score order statistic error absolutely tu ni v generally absolutely continuous density finite ni n ni ni regressor assume definite function finite give f normally prove subsequent asymptotic local magnitude response bias valid class demand location distribution method regressor non respective definite theorem observe instead measurement sequel confusion step follow ni ni nn either ni b v ni statistic contiguous sequence linearity case asymptotic ni n measurement response ni ni admit rank ni nr ni ij w nr ir nr nr ir px jx nr nr two absolutely hellinger array measurable measure van prove contiguous note residual b contiguous e ni case vector ni n nx nx fx dx tu u tu u du dt apply tu dt n yield fx ensures complete present case observe w ni nk ni ni ni ni du de n u x n du us contiguous ni ni n bound countable observe w ni expectation k n w ni together sake brevity ni w ni ni b ni use corollary arrive na ni na ni linear partial speak b ni convex function b convexity supremum take argument alternative equal asymptotically normally v replace complete illustrate measurement estimate empirical r bias square deviation compare deterministic regressor statistical software r
eq ccc mathematics department business usa il usa despite variance largely accommodate minor regression sparse variance post step employ penalty variance mean theoretical finding estimation high extracting regime extensively among prominent procedure lasso work low extensively address dimension square procedure guarantee mean correctly usual gaussian covariate unknown variance log explanatory variable positivity also capable vary order magnitude study optimizer assume procedure lasso perform update square procedure estimate estimate predictive aside provide estimate model variance provide estimate predictive covariate may scientific economic finance economic autoregressive rating dispersion generalize fall environmental recognize add activity extreme primarily distribution variance relation specific region asymptotically penalize optimizer establish property mean require examine complement finding scalar vector response respectively indicate boundedness sequence counterpart throughout index contain index submatrix extension notational simplicity small jointly indeed vice versa method approach simplify perform justification coordinate early suppose result close think initial enough pseudo perform procedure fix set work enforce result comprise stage solve estimate result pseudo residual appropriately choose finally compute reweighte differ penalty adjust mention stage think statistic minimizer differ pseudo act effect make pseudo likelihood close likelihood known closely differ choice oppose penalty satisfy sparsity condition property commonly lasso provide call minimal hope concave admit unbiased smoothly deviation scad minimax concave mc penalty behind form neighborhood act effect penalty value shrinkage reduce component generally q aforementioned concave satisfy likewise parameter balance value tuning choose minimize estimated degree aic bic property scad define scad develop substitute scad initial solution penalize maximize expansion penalty oracle property restrict scad proximal objective minimized sub decay definition statement decay result guarantee minimizer enjoy style penalty result type suffer possibility select minimizer converge minimum poor program set theoretical likelihood optimizer unknown unlikely derive maximum sparsity set minimizer examine elliptical contour list eigenvalue tensor large state oracle program require minimizer enjoy oracle simultaneous estimation limit theorem intrinsic low similar ease condition accommodate dimension generally necessary exponential either refer oracle property regularity low behavior regularity parameter variance fitting residual construct remove estimate observation accord mean likelihood minima address concern mild condition attain oracle possible mild course would mean knowledge unknown large minimizer assume similarly eq local minimizer enjoy notice allow substantial correspondence variance sharing correlation problematic minimizer recover satisfy concern solution satisfied design loading consider eigenvalue restrict eigenvalue specifically exist loading combine obtain enjoy penalize program stage reweighte variance precise addition access oracle mle fisher reasonable recover mild assumption consider convex stage assume minimizer enjoy normality property stronger prove specifically mle oracle mle mention rate penalty theorem would variance regard section conduct small study toy model jointly marginally remain covariate independent line simulation illustrate situation procedure iterate figure show precision coefficient toy aic st nd st nd st nd st aic st nd st nd bic st nd report second normal know support compare result simulation consistently scenario complex although furthermore demonstrate benefit first stage estimate nearly mse asymptotic hence theory analyze estimate variance quite show oracle estimate similarly guarantee prove assume log function fashion non section nonetheless interest sort guarantee could lasso family acknowledgement nsf dms complete resource university dimension quadratic let compact define objective accommodate norm radius construct minimizer objective estimator demonstrate refer oracle likelihood attain minimizer maximum stage result likelihood demonstrate regard minimizer pseudo maximum demonstrate mle attain sparsity set minimize hessian tensor lemma let unit sphere minimal net among cover constant apply assumption uniformly expand around value infinity curvature optima fix ball verify expansion remove prove strong employ similarly previous except perform proof proof oracle property knowledge minimizer precisely zero let value
vc improper learner small class exponential hypothesis instead sample complexity improve improve et show large proper proper private learn private learning concept class draw simple proper guarantee privacy fail accuracy al improper private infinite boolean return exactly privacy complexity prove sample disagreement hypothesis unlabele privacy requirement start relevant boost big error produce show boost accuracy private algorithm private show boost private boost mechanism al representation class improve consider probabilistic concept collection represent list learn select representation complexity private new privacy furthermore hardness assumption avoid size size characterize define domain size bound domain exist deterministic small deterministic class apply take solution quality maximize exponential database reasonable notion representation size database interestingly list protocol search bit privacy inspire record assignment satisfying least clause clause probabilistic representation assignment database individual maximize preference meet privacy another database database original record use size give reasonable query dimension vc lemma private c candidate separation negative cardinality element accord individual call neighbor one preserve differential nearby outcome differential privacy differentially database output take immediate concept label either pac algorithm accord target error pac concept distribution draw satisfy coin satisfying improper pac pac hypothesis predictor privacy private pac sample differentially use scenario choose approximately choice mass assign exponentially input pt leave define probability sf probability exponential private pn chernoff sum chernoff concept sufficient show must cardinality big empirical error private concept ready probabilistic representation behind sample hypothesis h dx representation concept private complexity cardinality hypothesis set boolean r concept choose place see characterize learner show implie private probabilistic complexity probabilistic arbitrary class every ca mechanism cc label choose sf first event return ensure mechanism hypothesis obey show happen probabilistic class chernoff hypothesis mechanism least learner class complexity prove learner learner assumption step set claim initially construct description short efficiently impossible properly construction inefficient learner defer probabilistic learner learn class exist represent c dx dc hc da h h h bind learner concept see learner size exist pair order refine class learn connection representation hypothesis construction deterministic representation claim boolean h extremely sized union representation contain straight forward boost claim use somewhat h non say fail representation one least bind hypothesis learner first size later sample h eq entry sample bit union event h dc two happen high bounding learner learner p pair dc exist b pair b concept necessary apply private broad problem scenario optimization universe choose refer function f choose solution reasonable database database notation necessary correspond big size private probabilistic private approximation big hard achieve optimization universe b bm bs bs publicly solution could qx probabilistic database f fs interested differential privacy universe record differentially predicate take query q cx et al define release mechanism differentially another predicate database scenario view database input quality eq every neighboring database element every representation bound universe database element every follow parameter mechanism eq exponential differentially fix bad bad event occur exponential mechanism least problem exist database record pair database element ratio fix b denote universe clause set clause assignment objective protocol et al notation represent represent database deterministic algorithm clause use deterministic representation necessary pick assignment satisfied clause clause pick assignment none clause randomly pick assignment database database differential privacy differential privacy requirement change proof valid private almost minor see apply must security representation element differential privacy sample whenever representation notation learner proper learner representation consider work boost use et show every identical lemma learn algorithm oppose multiplicative boost back replace factor boost capability ability let probabilistic first indeed exponential private fix step dc sf good event happen hypothesis hoeffding exponential learn represent class om representation q show every proper require still concept vc show separation point existence probabilistic representation description value class appropriate need process distribution return contain condition construct good probability contain choice conclude probabilistic hypothesis construct random draw description hypothesis goal care polynomial degree description pp p induce adjust inefficient improper learner necessary randomly every interpret e section definition remark corollary hard conjecture partially support foundation grant cs ac il private informally apply privacy al sample learner private combinatorial analogous know complexity private vc dimension representation concept probabilistic class private class exist sample similar
digits number double version black triangle law upper panel skewness panel panel panel g use maximal black panel determine various black line square panel c go independent realization occur complex often necessity distribution usually uniform limitation consequence arise tail moment sample size provide range handle library analysis finding numerical open search science observe quantity proportional say law density pdf analysis pareto much note frequency natural language rank frequency speak pareto retrieve store drastically last continuously researcher biology science economics finance social science law self organize critical multiscale collective intrinsic organization natural computer understand system power analysis power extract power law whereas consequence context statistic far none adopt severe pdf exponent equation power fundamental generation preserve restrictive suppose want extract whereas many concentrate inversion come inversion arbitrary motivated popularity extract variable interval variate invert certainly l previous write use generate power uniformly distribute typical solid computationally severe random role play period however precision number create finite bit satisfactory translate sample drastically synthetic bit precision simplicity close integer normalize number region green histogram presence correspond histogram circle histogram region blue region admissible exponent point visible region red area arise want accuracy space large h derivative pdfs valid random require consecutive admissible rise presence define overlap jump occur among consecutive discretization get bad wide law pdfs mean valid distribution network extract define fundamental epidemic fig numerically obtain confirm dy dash vertical stand predict color bit round method sum law random material moment tensor pareto due model sum law variable pay company individual pay distribute play role evy walk length simulation context behavior law pdf variance generalize stable deviation behavior exponent typical skewness excess decrease enough symmetric basis cutoff slowly normality small one dependent illustrate precision produce bit point machine precision power limitation approximately reach unit machine I able bit eqs limit truly instead numerical inversion suitable synthetic distribution principle variate computer store number principle computer algorithm paper explicitly continuous extend discrete limitation pdfs rather law distribution instance even strong discretization pdfs moment presence dependent cutoff law tail practical generation discretization tail outcome significance cutoff certainly analyse account factor concrete risk limitation numerical computer counter message increase distribution preserve sample
year cifar regularization maxout tree prior improve neural maxout product neural network image regularization use digit deep neural neural application application google speech speech hide hmms speech determine hmm fit short frame acoustic layer train outperform benchmark sometimes compression create low internal representation fall traditional image indirect application compression diagnosis vast medical structured design group specific high classification annotation medical image challenge deep neural effectively advance field need reliable grow crowd create big like removed manually crowd develop use vast amount unlabele develop huge resource make need neural understanding ever neural without entirely high representation key knowledge network neural area indicate far paper briefly describe history describe recent advance recently provide pooling dataset recognition field intelligence recognize classify image system accurate neural class benchmark dataset neural design image knowledge algorithm expert system create network effort engineering detector drive self prior approximate determine network posterior establish rule vast extensive make impossible review improvement deep neural architecture lead record break object recognition organization introduction brief history sub list commonly benchmark classification net network simple electrical circuit neuron input depend give computer simulate neural big scale theory switch circuit computer stanford perceptron patterns read stream phone predict neural world use develop research neural cat primary identify pooling algorithm separately decade artificial neural could mainly requirement lack train architecture bank read build convolutional network read machine reading motivate microsoft convolutional system include chinese character neural face record google face also vision mobile participant vision obstacle lot development improvement performance recognition image win image year win google accuracy comprise neuron stack deep face issue layer modification overcome come connect region neural net divide region map neuron connection connection feature advantage architecture instead connect layer low upper drastically cut weight entire connected layer weight back aim new patch sample use auto encoder back encourage maintain activation bias sigmoid boltzmann boltzmann rbm undirected graphical sparse rbms train divergence sparsity penalty autoencoder mapping encoder thus algorithm differ primarily around window pooling pooling activation pool additional activation window account pool non maximal activation utilize pooling pool output pooling region map split pooling contrast generate learnable region rich depend neuron determined acting linear sigmoid running function fast equivalent activation interpret approximation learn activation give input maxout implement large easily simplest square set predict increase forced become way consider output presentation hide unit omit modified mask element co neuron helpful several neuron helpful correct answer context dropout output mask equation hold dropout mask weight neuron turn inference average give instead massive twice essentially mathematically write justified use activation bernoulli calculate pass averaging present increase reduce overfitte improve generalization consist simple rotation field work object transformation augmentation augmentation horizontal extract extract increase size set prediction patch average prediction softmax patch augmentation illumination difficulty availability set image create rapidly meet demand microsoft common object contain spatial precise pixel distinction per aid contextual million store label abstract english list lexical note reliable
calculate original tensor cnn drop cnn record cpu base size rank cnn maxout softmax classify digit plus character pre similar layer constitute channel output channel filter result layer tensor growth rank accurately show accurate properly approximate error firstly make fine tuning one finally layer approximate significant drop accuracy fine tune derive fast accuracy drop time big speedup incur accuracy convolutional train noticed rank achieve table clearly fine secondly fine observation hypothesis large poor minima indeed cp minima bad minima random layer rank entire effect initialization cp fine c ft ft ft ft ft cp considerable comparison character classification cnn however determined bi cp layer firstly greatly secondly spatial variation tensor low decomposition improve support foundation grant consistently linear next tensor adding highlight slice check numerically tensor successfully pt yshift em convolution tensor fine layer compute cp tensor convolutional kernel replacement training process cnns obtain cpu drop class character cnn obtain speedup minor imagenet overall cnns vision computational notably mobile processor cpu processors layer cnn operation dominate cnn convolutional often expense consequently strong efficient implementation convolution major package work cnn tensor detail recall typical cnn tensor array dimension dimension output convolution convolution constitute dimension map output map exploit work speed within cnns apply filter investigate tensor cnn tensor algebra least square cp full ease decomposition cp tensor linear exist compute ease convolution tensor four cnn therefore package need tuning layer replace four convolutional kernel straight forward tune network back importantly cp fine tune previous method compare practically architecture ram storage valuable feed forward spatially kernel locally connect layer side confirm cnns modern serve facilitate minima cp cnns character discussion cc convolution box correspond tensor cnn side mapping demonstrate scalar box correspond spatial approximate composition mapping mapping approximate array either pixel spatial decomposition low decomposition accelerate convolution codebook bank filter combination share bank separable decomposable decomposition share decomposition suggest decomposition effectively approximate composition experiment demonstrate compute minimize network fine tuning inefficient well even cp suggest base cp decomposition tensor connect layer reduce rank cp decomposition part compute outer add tensor cp directly full convolution tensor replace cp square discuss fine network approximated conceptually convolutional layer decompose cp fine tune entire backpropagation necessary layer review cp core two tensor low lead separate many canonical cp cp minimal need singular finite canonical tensor error enough rank cp cp package choose non least square gauss newton capable much may cnns feed unit within tensor consume modern cnn use linear mapping tensor dimension third spatial
span consequently conditional maximize need redundancy code maximum instance family relative entropy distribution divide possibly sign integral statistic value match alternatively divide w possibly total likelihood combination see odd role maximize correspondingly arrange representation simple illustration negative require three maximum imply weight weight trial sufficient take alternative application algebra solve distinct linearly sign bayes however effect weight alternative quantile respective panel point right prior guarantee exact note divergence deal prior mixture kullback bad n early study show mixture mild subset model consideration finding sign bernoulli grid mm mm mm panel panel mass implication fold theoretical view provide bayesian demonstrate counterpart early bayes offer extract sample arithmetic explore family multinomial relationship support prior produce bernoulli match odd odd count neither proposition maximize likelihood provide minimax compression play essential minimum description show normalize though weight address marginal code universal prediction minimax regret bayes mass denote estimator characterize arbitrary range datum compression free minimax property compare parameterize exactly maximize trivial require work pointwise regret course achieve take begin compression achieve minimax infinite nevertheless continue distinguish particular comes think likelihood conditional sum conditioning bayes study regret study nature play role minimax consideration likelihood mixture simplification determination family bayes turn first recall arise redundancy compression giving difference kullback divergence formulation regret decision loss specify divergence procedure mixture kullback function limit bayes minimax minimax characterize redundancy call favorable achieve pointwise provide upper redundancy max also mixture see traditional role bayes asymptotic alphabet family role prior close jeffreys fisher asymptotically regret mixture minimax pointwise problematic bayes mixture sequence arise match information jeffreys fail problem bayes mixture slight family difficulty motivate consideration sign bayes simplification possibly sign code particular tool appear normalize require sum yet remain intractable mixture simplify equal conditionally multiply one ready computation ratio marginal sum predictive g case mixture marginalization marginalization paper whether provide maximized likelihood measure string px perform marginalization maximize get computational sign finitely iii numerical bernoulli trial divergence finally calculation observable observable outcome sign sign role string sign marginalization proper bayes sign integral produce valid emphasis
update update several need compute unconstraine project onto nonnegative advantage easy implement interesting solution properly drastically current recommended issue switch alternate method subproblem solve many method dedicate practice implement fast gradient converge stationary subproblem framework algorithm iteration expensive difficult implement initial guess rather refinement nmf algorithm mu solve use subproblem write form interact see describe thesis solve vector unconstraine though negativity advantageous optimize smaller unstable difficult see mild assumption guarantee converge stationary particularly otherwise initially recommend initially update gauss functions taylor display evolution describe section classic document datum set matlab matlab intel core ghz ghz ram display algorithm slowly data quite poorly classic initially solution objective perform poorly book reference therein e impose robust stable conclude criterion initialization usual scheme evolution optimality iterate termination discussion issue assess criterion similar criterion sensitive scale subproblem multiply dividing handle e scale bad type mu monotonically mu guarantee monotonically potential update order magnitude interval sophisticated strategy obtain ii come guarantee e nmf np general direction list centroid spherical scaling indicator therein use svd introduction one frobenius theorem denote factor al either scale initialize point initialization keep subset column generate goal nmf allow reconstruct column fact compute separability document topic transpose document word anchor assumption thesis hyperspectral section pixel pure sense spatial successfully develop far clear behave enforce suggest improve weight enforce sparse entry diagonal condition symmetry distinguish robust distinguish near matrix see use k drawback expensive class separable insight vertice hull column geometric find literature pure historical comprehensive focus effective successive moreover behind heart geometric nmf look vertex hull column work onto column implement use formula ht rank column let let prove correctness noiseless case induction necessary identify convex point moreover strict unless ht projection onto complement hence reason separable nonnegative w w improve post interesting purpose various schmidt column variable fact name come particular variant volume discussion behind volume hull processing noise combine make extracted span refined processing norm good column polynomial logarithmic closely relate greedy solve self dictionary reference exist many g vertex component norm strongly successive nonnegative include provably several connection mine nonnegative precisely mathematic science graph bc complete bipartite subgraph need check easily complete subgraph bc low polytope polytope exponentially find formulation importance turn polytope slack ia survey reference approximate related hence mixture nonnegative compute nonnegative rank amount independent variable therein know variant receive start communication equal combination input logarithm communication matrix closely rank relate find number nest geometry nmf easily interpretable dimensionality reduction bring researcher future publication lee author thank book kernel machines comment paper corollary question nmf automatically property nmf mining hyperspectral image nmf review nmf refer separable polynomial presence briefly describe mathematic via technique tool use visualization feature selection space span dimensional linear subspace vector rank column column assess quality depend frobenius popularity implicitly situation introduction compute efficiently value see reference therein svd principal mean center datum datum origin aspect pca assumption assume lead component nonnegative factorization nmf aim nonnegative wise nonnegative introduce article lee explain nmf aim relevant contribution rather nmf reason popular interpretable nmf hyperspectral imaging application air emission biology blind source separation single source separation cluster analysis collaborative filtering let gray level th nmf interpret image intensity image latter localize hence simultaneously several image dense face e nmf approximate correspond word appear document column word document sophisticated construction tf document associate document matrix factorization rank nmf generate q decomposition interpret word nonnegative original much basis number document interpret set document document nmf relate exist topic semantic indexing column spectral signature scene spectral signature light reflect therefore hyperspectral usually observe broad spectrum hyperspectral see illustration hyperspectral blind two fold material example surface pixel mixed material popular combination signature correspond road surface mix signature signature road signature basis weight rank hyperspectral illustrate six column signature contain abundance pixel note decomposition section hyperspectral survey see previous useful compute problem assume introduction nmf arguably matrix use kullback leibl music improve vision consideration survey nmf particular unfortunately unconstraine hence practice guarantee stationary heuristic application recently et nmf address et algorithmic exact nmf improve polynomial solve real high cost run operation usually nmf generate matrix practice satisfy generate interpretation classification reader therein uniqueness prior proper regularization popular usually contain plain poor blind spectral signature coherence pixel likely contain material usually neighboring value preserve image e therein algorithmic refined various nmf nmf projective nmf model different good look expert blind expert scene reference focus first issue standard section nmf design
joint term factor respectively score rao outside markov hierarchical observation variational parameter variational compute unfortunately require iterate every need noisy estimator sample mean low compute substitute gradient need iterate observation shape parameterization variance gamma posterior model variational require effort quickly develop explore present box many model optimization monte develop gradient maintain derivation correspond reach method demonstrate inference explore quickly evaluate model latent modern latent model infer inference summarize conclusion compute latent interesting computing practitioner resort widely approximate variational try find member simple close convenient exist ascent algorithm family close form expectation generic derive rapidly explore modeling assumption impractical practitioner variational apply method practitioner quickly derivation time adjust frame optimize adjust proxy gradient objective variational optimize sampling evaluate form stochastic variational perspective method evaluate calculation variational evaluating estimate library share reduce gradient essential develop control rao second emphasize goal box variational adaptive generic subsample gradient close coordinate natural way compare hasting require effort predictive quickly ease variational several method approximate kl include adaptive subsampling approach inversion impractical alternative estimation family set box variational inference approximate variable family variable goal variational q maximize divergence intuitively mass configuration explain reward maximize many configuration practitioner maximize expectation close ascent available conjugate latent variational analytic computation variational outside set overhead develop optimization maximize optimization noisy unbiased gradient update noisy distribution variational method gradient application maximize realization finally th stochastic widely found call gradient sample variational algorithm function variational build package variety make log joint reduce effort variety tb data field variational initialize maximize variance carlo large useful gradient require reduce rao exploit preserve black box rao replace respect expectation variable rao without specific rao replace conditional variable expectation place estimate variational govern characterize member seek gradient supplement say main step schedule second subsampling observation rate intuitively rate large vice versa issue let matrix iteration gradient per eq learning since element capture vary scale algorithm rate inference massive idea subsample gradient similar model variable place conditional noisy iterate noisy gradient variational use medical demonstrate likelihood longitudinal patients york disease patient k visit measure consist measurement take value amount particular come visit indicator health evaluate predictive likelihood visit carlo initialize randomly factorize normal real doubly size meet gamma normal series draw factor positively affect let process raw normal vector visit draw parameterization supplement black family tend limit parameterization allow visit visit finally emphasize ascent gibbs conditional metropolis hasting fail gamma instead compare hasting complete conditional predictive method hold test box metropolis hasting inside budget hour hold initialization model variational likelihood get hasting study estimator patient series versus iteration compare variance carlo rao rao drastically improve black box variational without fail make progress estimator rao variate rao estimator box variational consider factor health name gamma ts normal visit positive variable note expect factor gamma patient draw save factor expectation parameterization hard natural parameterization gamma draw factor visit allow propagate patient ts simple model
cover technique dedicated fusion trend introduce old work consist estimate high take correspondence impose fuse band band describe fusion notational follow band observe hyperspectral band denote observe datum resolution definition place hyperspectral model spatial assume circular account image assume uniform subsampling grid noise measurement matrix hyperspectral couple form highly correlate normally live column small translate relatively accurate estimation work consist linear building correspond discard former however see detail problem try solve ill therefore adequate regularizer use denote element product horizontal vertical respectively boundary condition variation impose gradient mean except detail regularizer detail align among vector regularizer hyperspectral sense regularizer def formulate term regularizer convex quadratic term direct split augment lagrangian shrinkage multiplier simple auxiliary optimization problem take lagrangian respect cyclic efficiently solve respect variable solution involve advance minimization operation approach employ primal dual unlike arbitrary problem experience literature fusion publish fusion hyperspectral publish real image hyperspectral university acquire ground truth bands band subsection follow horizontal hyperspectral band snr band former latter add db snr paris acquire observe hyperspectral resolution resolution fuse hyperspectral see ground literature angle base quality implementation consider work real life reference estimate image respectively implementation dataset truncate ten singular vector ten preserve original share note due average index choose situation yield index perform hyperspectral band remove svd project solve via work yield use gram schmidt gs intensity fusion technique transform bt box filtering result table comparative band show fig paris seen find seem spectral overlap hyperspectral channel compare zhang manner similar estimate spatial address level transform restriction implementation work part pixel leave run equip intel ghz ram memory take second see bt gs zhang flexible hyperspectral spatial fusion present hyperspectral large overlap high spatial instead one augment lagrangian shrinkage direction multiplier admm convenient variable exploit live
sampler stable choice prior dataset consist km galaxy survey currently cluster give choose new sampler ccccc trend time potentially exercise note stable report similar py gamma prior mode around cluster good condition consist profile profile within kind useful small explain encode recover multivariate normal unknown covariance mean wishart base measure currently assign denote variate mean denote wishart definite freedom wishart matrix diagonal dimension make weakly average probability prior class predictive mcmc report curve per curve per reflect grant agreement file multidimensional create plot file illustration txt file data multidimensional section mat size set mcmc array eq corresponding size chain burn average predictive probability dataset run observation probability leave table fold follow replacement point batch predictive probability exchangeable introduce variable induce eq plug evy exchangeable induced dp parameter evy gamma chain initial transition kernel proposition formula poisson homogeneous completely size pick atom q index easily measurable negative function measurable positive draw measure equality denominator induction hypothesis numerator use specifically theorem proposition bayesian mcmc generalize modeling important concern specification model infinite dimensional active possibly infinite exchangeable parameterize define infinite identify component formulation set introduce model apparent replace break notable introduce also alternative preserve almost normalize allocation heavily distinct component flexible mechanism several monte dirichlet process exploit tractable marginalization dirichlet overview development atom measure stick belong model stable poisson dirichlet process limit property stable dimensionality graphical rely task multidimensional recall mcmc present stable model contain conclude start completely reader let separable endow borel take space represent nonnegative random location characterize measure mean evy measure evy atomic homogeneity identically normalization homogeneous evy random variable surely homogeneous law govern l evy normalize evy comprehensive homogeneous index induce show exchangeable exchangeable partition law account induce biased permutation atom block increase block appearance atom biased atom tend appear early break construction stick break first length piece break etc form initial transition kernel supplementary denote random generative part mass pick mass conditional subsequent atom multiply probability next exchangeable denote order increase biased homogeneous surely absolutely dirac delta whose respect positive representation evy measure distribution throughout paper distribution homogeneous distribution govern paper popular j jt poisson evy poisson view distribution poisson specify ns py positive correspond positive see stable lt also show obtain direct supplementary material next stable exchangeable set one derive comprehensive exchangeable partition section sampler effectively apply evaluation hierarchical mixture extend joint mass mass distribution mcmc package integration numerically generator polynomially exponentially rejection proposal auxiliary augmentation describe difficulty computation value numerically address problem propose explicit rest stable nonparametric distribution smooth derive conditional maintain representation auxiliary final representation value graphical partition read conditional next cluster proportional exchangeable cluster variable study chinese restaurant describe partition condition auxiliary extension component lead chain involve update cluster maintain potential
empirical improvement small imply grow linearly alphabet prove theorem conference paper weak upper completeness tight report alphabet respectively range frequently last first sublinear sample use shannon alphabet language genetic shannon implicit fairly order imply furthermore suggest r continuous order order entropy may approximate could reason r enyi show infinity alphabet note achieve sample universal estimator know r enyi accuracy equivalent multiplicative furthermore accurate estimator sum multiplicative accuracy additive entropy range give unbiased exponent frequency use consider approximation early consider roughly part use polynomial appropriately table c determine enyi entropy illustrate integer show enyi multiple every bias empirical enyi table lead term approximation l k enyi approximate correctness eq infinity go infinity figure show performance draw illustrate empirical bias work empirical exponent attain exponent possible compare performance enyi entropies parameter significantly entropy imply fairly enyi constant theorem organize follow present power moment poisson integral enyi entropy establish enyi entropy randomness see follow inequality monotonicity every monotonicity upon inequality monotonicity old final follow upon rearrange poisson draw poisson simpler facilitate poisson moment start expect power expectation establish eq fact either equality multiply side expectation side q nonnegative schwarz review approximate interval set less require polynomial approximation minor polynomial bound variance approximation absolute follow serve degree performance estimator proof bound general analyze estimator separately symbol x nh sample corresponding estimate error estimate simplicity randomize describe q reduction q remain right bias n chebyshev reduce estimate repeatedly specifically hence hoeffding claim choose note half p must remainder sample appropriate estimator estimator derive estimator bias bind sum q triangle inequality independence jensen inequality variance satisfy bias subset inequality first concavity complete lemma reduce integer shannon estimator traditionally shannon analyze reliable shannon estimation similarly reduce correct provide choose remove bias alternatively bias usual sampling similar albeit complete form polynomial motivated paper consider approach arise symbol empirical nearly large interest estimate sum suffice similar power fact accuracy estimation power insufficient estimate enyi show attain brief random use symbol estimating sample polynomial da use unbiased namely combine rely bias bound complexity polynomial exist estimator least satisfy throughout satisfy bias q hold triangle lemma use claim low estimator sequence say estimate suffice consider sufficiently contradiction profile omit contradiction consequently follow upon correspond therefore entropy requirement difference sum construct require make ensure match need entropy sum positive respectively integer eq constructive converse lower integer q vector discussion verify follow must inequality hold theorem since alphabet yield sufficiently arbitrary lemma small positive root root newton identity polynomial depend power differ furthermore taylor small zero acknowledgement thank helpful suggestion apply estimate need additive technique show obtain sum namely accuracy enyi sum enyi complement inequality follow upper estimator sum accuracy comparable ok constant range multiplicative sample empirical version sample bias appropriately choose q since get proceed variance complete upon show note summation equal title theorem lemma conjecture remark remark shannon symbol grow replace sample need surprisingly develop
explicit cluster column know furthermore misclassification step conduct column assign augment multipli algorithm augment lagrange multipli rank structure nice multiplier rewrite indicator convex set aim minimization dominate fact subproblem optimization large hundred thousand svd scalability surrogate eigenvalue minimizer mean rank leave large scale future research minimize closed entry great lagrange convention multiplier j augment multipli derive summarize numerical political defer clearly implement ordinary clearly term guarantee procedure community explicit condition capable presence portion outli let semidefinite gap density denote eq guarantee solution belong group solely impose helpful work even good community adjacency sophisticated becomes allow second assume cluster community sbm literature node guarantee let density example modify condition relax detection plant within usually group density barrier plant clique therein involve section helpful behind optimization intersection cone boundary point hyperplane vertex programming discriminate clear observation clear affect result cluster specific moreover norm coordinate nonnegative apply obvious km b distinct majority misclassification j give example misclassification ball box box ball ball color assume box distinct colored misclassification prove misclassification control less rigorously speak easy mean r k j misclassifie previous base distance less hand community great comparison detect accurately without cluster whether approximation synthetic employ effectiveness community mean lagrange multipli simulation ghz intel processor algorithm fix illustrate realization know detail maximum formalize change diagonal n divided cluster community second accurately misclassification rate adjacency cluster comparable propose misclassification rate profile sbm modularity method misclassification indicate maximize criterion initialization misclassification modularity independent repetition classical eigenvector distinguish conservative political ordinary datum penalize apply nearly paper robust presence outlier propose strong mild condition density order grow adversarial consistent art feasible sbm extension current detection depend detection hold big capable must modify replace dependent diagonal adapt high correct sbm cl simulation establish guarantee choice much well open redundant procedure cluster usually group connection interesting write use coordinate nonnegative norm represent norm matrix whose correspondingly numerical constant algebra hermitian eigenvalue arrange cauchy eigenvalue arrange inequality xx consider symmetric whose zero nc sequel bind bernstein theorem total q zero pair zero determine leave article precisely sufficiently eq symmetric constant apply relax let particular outli node ordinary sbm art literature computationally feasible detection rigorously formalize follow lemma permutation give analyze several inequality one feasibility sufficiently utilize need construct show establish guarantee sufficient use lemma previously consequently three exist uniquely nonnegative diagonal furthermore eq aim must indicate suppose pt form intuition rigorous give cluster get small reason get choice intend constraint sure first projection concentration suffice guarantee existence moreover large numerical jk jk ii jk style thm section dms dms detection group undirected block allow adversarial outli node follow accurately misclassification set grow admit good outlier fast kind outlier community spectral adjacency fail retrieve major cluster portion political showing method exist feasible well fast outli node wide range engineer recent interest characterize approach reference network aim cluster node community observe undirected detection challenge spin perhaps sbm independent assume node assign label label adjacency graph respectively graph self loop symmetric bernoulli refer namely assume detection minimum tuple sbm detection study greedy see criterion likelihood stochastic pseudo gibbs monte belief gm cl lr convex accuracy fully modularity profile method prove consistent computationally hard justified theory community fast spectral cluster dense spectral cluster sbm generalization mixture connectivity different group latent membership easy detect apply graph single generalization sbm sbm first fit form arbitrary outlier important community detection arbitrary main question wish portion arbitrary rigorous proof begin portion node model cover range sbm suitable assume undirecte among sbm connect arbitrary node event matrix connectivity arbitrary restriction connectivity outlier loop equivalently entry arbitrary permutation capture connectivity correspond usual sbm bernoulli submatrix parameter define parameterize tuple necessarily randomness depend word allow depend generalization sbm connectivity stochastically pair applicable cover common name sbm assume node portion belong portion outlier ordinary connection specific outlier employ cluster weak difficult essential popular modularity outlier classify belong group connection refer object neutral political portion connection strong nod neutral even modify sbm combination neutral setting complex model overfitte sbm property political discuss sbm prefer significant cause lie outside cluster take community sbm model robustness portion result cluster robust good follow directional lagrangian focused consistency optimization inference group specify analysis proof contain additional technical proof material community detection computationally greedy justified maximum hard stochastic variational prove block fix going naturally likelihood propagation rigorous theoretical unlike aforementioned easy various section ordinary spectral laplacian datum type ordinary perfectly plot eigenvector absolute laplacian adjacency eigenvector combine capable cluster outlier suffice explain random independent independent adjacency eigenvector corresponding eigenvalue absolute graph adjacency matrix discriminate homogeneous behavior thereby unable distinguish laplacian discriminate two major simulation apply spectral laplacian percent certain penalize weak graph influence standard clustering detect kind outlier
rnn rnn multiplicative time rnn mn multiplicative rnn rnn rnn feedforward ff plain rnn regime rnn rnn see stochastic dropping test dataset initialization deviation sampling weight noise c c rnn rnn rnn rnn rnn rnn ff get spectral fix epoch try weight big lack store time delay see regularizer radius grow albeit give regularizer show logarithmic test fig trend indicate error income recurrent trend indicate increase exhaustive update base show rnns loss eigenvector limit low suffer store analytic presentation also explain regularizers rnns past believe gap sophisticated issue vanish term dependency recurrent loss rnn define pre activation income upon vector consider constant expect value compute hand deviation put form multiplicative pre activation equivalence bring form place eq analyse eq usual backpropagation follow post term deviation actual value analysis eight hyper range limit regularizer l regularizer e optimizer momentum h l rnn mn rnn ff regularizer dropout optimizer momentum size hide rnn rnn rnn rnn rnn ff regularizer optimizer l layer l x rnn rnn mn ff dropout optimizer step hide rnn rnn rnn ns rnn rnn ff initialization optimizer momentum batch layer parallel lead deep past rnns additional feedforward provide treat sequential rnn conventional relate dropout noisy rnns rnn improve capability aim empirically rnns model train norm performance initialize rnn advanced dropout work rnn recurrent network variation information language analysis financial domain mlp go technique train rnn perform time backpropagation suffer problem gradient gradient grow vanish gradient large fast purpose instability conceptually vanishing rely particularly property simple series need store vanish gradient become unstable behaviour rnn extensively sophisticated introduce feedforward rnns advance solution lstm task date study one regularization recurrent growth vanishing evaluation rnn derivative ability dependency discuss trend rnn like integration unit also momentum descent sgd powerful boltzmann gradient trick evaluation paper music dataset addition corpus consist denoise autoencoder step rich audio signal ed model key rnn typically rnn evaluate transition layer problem rnns variation conventional rnn term memory unit randomly delay increase extend apply complex corpus value improve dropout recurrent noisy mlp noise curve noisy descent feedforward rnns rnns rather solve delay widely long identification rnn really validate delay relevant however applicable temporal universal input rnn exploit technique fully connect hide layer therefore rnn rnn activation hide hide layer linear explain dynamic backpropagation rnns rnn calculate affected multiplication successively may lead grow vanish fast regime update gradient happen opposite eigenvalue recurrent much might vanish linearity might demonstrate recurrent simplified explanation perspective find come step become unnecessary dimension reach initialization represent represent matrix curvature illustrate thing routine surface face depend learn ground simple objective local optimize term quadratic normally expansion difference make involve inversion matrix make possible regularization particularly vanish systematically prove generation prediction dependency plain rnn modify lstm memory cell link conventional unit hide take flow hide activation layer vector determine activation fed unit sequence lstm time hard train gradient dropout neural every neuron incoming connection probability safe tend approximately incoming connection order dropout match plain validity dropout rnn hide single apply connection rnn regularizer act regularization dropout initialization rnn evaluate hidden output matrix rnn improve network gradient effect regime feedforward network tolerance unseen search coarse region rnn adjust gradient weight behaviour regularizer appendix demonstrate stochastic recurrent layer much feedforward multiplicative noise model evaluate recurrent preserve recurrent follow analyze restrict cumulative model backpropagation rnn train set increase noise space decrease weight noise time weight additive vector sample additive vector distribution recurrent noise update noise noise multiplicative multiplicative evaluate variant model additive multiplicative model weight matrix preserve even calculation backpropagation recurrent
sequential set adaptive compressed various determine location single adaptively identify compressed sensing design exploit structure claim near specifically tailor sense strategy compressive sequential basis information imaging literature sparse sequential sense various place example compressive sensing design minimize introduce sequential adaptive adaptive compressed sense mixture objective function sequentially rank gmm empirical measurement difficult devise compressive pick measurement amount previous measurement enable establish theoretical guarantee sense often resort query noise analyze measurement also develop numerical organize formalism greedy sense signal gmm finally conclude denote th let denote positive determinant mutual elsewhere quantile chi degree typical compressed sensing let measurement depend linearly subject sense vector set compress measured low model use non subspace gaussian lie union video signal general manifold exploit structure fewer early compressed regardless signal measurement sense eq either vary fix repeat integer power resource allocate extract maximize sequential outcome ax compress design row recursively view dynamic usually situation one usual approach operate core measurement probe maximize much formalize initialize return information natural useful noise x j either reach relate mutual determinant ellipsoid reach iteration avoid greedy discuss algorithm measurement outcome resource greedy sense nonnegative case correspond certain measurement sparse factor noiseless noise another finally greedy consist uniformly modify description characteristic location amplitude minimum signal n ambient replace remove noiseless exactly measurement n greedy algorithm lemma obtain measurement greedy possible mutual roughly measurement entry correspond sensor reporting count relax allow setup generalize sum recover non amplitude measurement vary repeat overhead lemma measurement incorporate gmm gaussian covariance consider measurement compress allocate allocate snr measurement measurement reduce communication q snr actual measurement proxy amount resource allocate colored resource sensing derive similarly snr greedy gaussian iterate adjust clearly signal decrease order establish white add accuracy satisfy measurement simplify hold see unit eigenvalue use eigenvalue informally measure power reduce sense gaussian gaussian eigenvalue w recover every outer drop black width color add x give measurement give various accuracy correctness white u eigenvalue u eigenvalue u e colored noise color add prior eigenvalue recover use noise model outcome affect gaussian advantage inaccurate bring demonstrate sequential new matrix measurement induction measure hence several thereby split signal implement sparse large eigenvalue method correlate change change sparse sparse therefore matrix covariance greedy gmm class gmm signal outline derivation respect linear transform mmse moreover colored determined instance demonstrate order formula color white colored greedy sensing error obtain error fall tolerance theoretically batch eigenvector matrix greedy outperform due method performance sense rank another entry zero greedy sense precision sense method perform identical use sense sense batch cc gmm assume gradient descent fig demonstrate cumulative mutual average monte trial descent heuristic fig greedy perform fairly compare gmm signal versus iteration associate measure cc average trial c batch descent greedy single measurement generate eigenvalue show entry greedy gmm sense mnist handwritten true label training gaussian digit handwritten digits picture digit measurement instance measurement gradient heuristic normalize ccc false recovery consumption california consumption year single year demonstrate reasonably good fit test sense year coarse well compressed collect consumption region automatically wireless sensor platform embed efficient monitoring consumption large power year versus number measurement framework compress sense maximize condition previous help signal bring robustness assume benefit moreover sense sparse demonstrate potential establish obtain error sequentially signal prescribe formally family I operation I single pair pick enough ensure invoke e uniquely variable algorithm greedy information sensing learn measurement measurement well colored full note write b w ba u maximum sensing interpretation color snr large eigenvector u w x u measurement density mmse condition measurement gmm close derive turn gmm weight q hence base close enable gradient draw computing carlo summarize whenever drop information condition entropy gmm overlap k disjoint covering yet start line measure keep intersect great subset determine coincide x k meaning size whole k measurement difference block reduce error every measurement intersect exact might happen l consist coordinate terminate n measurement end estimator coincide outside support x great r proof family signal consist signal measurement ax entropy apply size domain uniformly residual consider measurement measurement mutual measurement identical size reduce induction maximize use upper n query take explain eigenvector unchanged measurement apply several reduction combine power minimum integer direction measurement suffice ensure algorithm gaussian return signal distribution similar eigenvalue leave sum eigenvalue large similar difference canonical make switch orthonormal write basis measurement identity basis add measurement every I ensure mean return sketch first
relatively compare towards problem effective increment cumulative adaptation arise positive model constraint handle resample choose study consistent assume step constant adaptation mechanism es constant optimizing handle extend result chain algorithm attention distribution copula recently trend copula evolutionary algorithm next basic formally feasible section gives aforementione verify notation integer vector sure probability denote borel es linear sample es composition objective strictly hold handle invariant composition w g g call I j distribution stand sample index feasible sample candidate feasible update parent distribution step internal parameter adapt require es optimize define handle resample l respectively yield admit absolutely possess yield vector function wants deal es handling resample g k distribution copula regard g ff consist marginal distribution copula use size theory remainder recurrence divide left vector order apply rhs previous equation chain precisely geometrically j state es optimize handle resample matter homogeneous chain density chain ergodicity chain imply generalize proposition handle resample absolutely function lebesgue measure irreducible small set markov fulfil recurrent furthermore geometrically substitution denote l absolutely show function l infimum reach measure since irreducible finally take lebesgue strongly define v want drift imply compact want g condition integrable dominate integrable respect count dominate condition suppose compact chain number divergence es optimize side positive chain number hand side finite invariant lemma covariance constraint angle handle resample k kk n g j h e sign achieved state es handling resample absolutely distribution isotropic use proof equivalent isotropic chain use proof normal es handling resample multivariate hold nh integrable dominate theorem markov geometrically ergodic positive isotropic fulfil every obtain sufficient strictly advantageous pay copula e continuous decrease denote inverse reason copula invariant permutation permutation matrix hold permutation matrix first continuous continuous positive density continuous strictly moreover generator replace strict positivity respectively monotone copula q copula combine turn q sign opposite completely monotone define indeed copula express particular eq describe paper present evolution general distributional isotropic problem ergodicity accept ergodicity stationary monte opinion condition insight different play design evolutionary solid multidimensional attempt bring contribution applicability copula evolutionary copulas realistic actually
construct accurate marginal note road extremely largely abundance logistic additive additive smooth decision model kind decision case small scenario adapt high penalize regression drawback additive searching transformation available fail knot every additive reduce bias comparison increase variance cost moreover admit interpretation building besides reference dimensional classification discriminant organize dedicated study present describe variant original transform one give pair g jx penalty plug predict repeat probability increase stability transformation unstable density take kernel good mcp take final make split majority vote split splitting prototype marginal prediction procedure assignment usage make bootstrap misclassification reflect reader similar balanced switching feature run splitting perform transformation implement penalize employ level coordinate constant whole particularly repetition compute core implementation application multiple henceforth whole penalize leverage computer nonparametric classification computation various repetition column report argue contribution logistic transform odd marginal enter help separate reasonably compare additive model regularize road discriminant lda nb fair simulation setting sample five validation conduct need replication long f summarize standard use pseudo setting vector boundary due comparable pay use bad complex unnecessary decision surprisingly perform poorly bad especially common independent nearly independent nb ignore classification multivariate gaussian class boundary boundary nonlinear distribution oracle decision method fail classification report extremely fast similarly simulation computation cost demonstrate spam satisfactory road nb svm l svm road study spam demonstrate power attribute word character email length case letter letter rest split repeat summarize competitive training svm well dominate different training fail yield proportion due spline define penalize vector estimate identifiability margin condition parametric introduce relationship set specify notation continuously real taylor old continuously interior level condition assumption regularity estimator impose absolute assumption guarantee penalize un penalize oracle measurement compact j density f impose incur marginal strictly positive put bandwidth absolute condition similarly bandwidth exist n hold compatibility take specific uniform margin involve formal omit normalization p sample theorem excess risk control use next excess let sub pm version oracle assumption order density regularity condition component big density estimate strength need link density due regularize covariate bandwidth appropriately estimator excess explicit excess worth accomplish bridge regularize work change model note addition transform adapt establish omit propose new classification leveraging nonparametric estimator achieve feature penalization linearly transform original perform dimension flexible curse array demonstrate procedure misspecification well compete perform slightly standard additive standard ex fair rule insight size recommend impossible look abundance rough extension far investigation future beyond specific establishe estimator might near searching combination difficult task application ratio approximated snps could beneficial notable front independence sis marginally subsequently independence screen sis additive model contain technical proof f e p r since bind give rise play simplicity plug h mn mn mn bandwidth lemma denote result b b inequality n view follow simplicity order expansion pm pm pm c pm pm pm side q combine oracle corollary classification nonparametric feature augmentation know powerful univariate transform subsequently penalize newly transform augment train equip simplicity avoid create decision result feature generalize naive writing joint relate generalize model numerical domain real email spam gene expression implement parallel nonlinear boundary augmentation feature parallel identify category spam detection recognition high gene fisher discriminant lda near neural perform much many dimensionality microarray frequently thousand computational setup limited access conventional high new setting regularize refer classification augmentation selection introduce motivation suppose code pair variable binary dependent density ir py decision setting fisher addition rule structure among essential help abundance sample bioinformatics assumption correlation regularize road road plug directly classification advantage un regularize pool gaussian naive bayes naive bayes however among feature motivate ask question advantage precisely decision necessary thresholding good ratio transform future effort transforms build spirit sure independence sis use coefficient wish take feature
amount proportional similarly multiplicative distribution variance show distribution pair power recover want outline run output strategy allow large total conjunction terminal option either graphic terminal graphic macro ltb lt lt lt lt ltb lt lt lt lt lt bp r r r ltb ltb terminal either load package graphic need graphic macro ltb lt lt lt lt ltb lt lt lt lt lt r r r ltb ltb package color conjunction terminal option explanation load package package graphic terminal graphic macro ltb lt lt lt lt lt ltb lt lt lt lt lt r ltb rank recover claim quickly practical core intel memory netflix experiment collect radial five plot illustrate value illustrate varied value slow initial value seem netflix recover dataset reveal rectangular run singular algorithm singular eigenvector plot plot illustrate runtime eigenvector sequential allow good open version non descent low problem variety globally initialization rather account novel optimistic apply helpful thank advanced research fa fa science foundation nsf award fellowship author acknowledge contract air force domain indexing fa contract nf mid technique library throughput grant simulation software national nsf stanford program author acknowledge support finding conclusion express reflect nsf example happen stochastic descent descent update reasonable independently condition infinity iterate decrease inductive q therefore proof exponentially quickly stochastic enter choose start account converge optimum entry let try decompose rank e update eigenvector global problem mean global illustrate optimization give factorize constraint complete q relax entry dependence entry particular minimum boundary imply value imply np suggest analyze constraint case literature list sample iteration rank applicable assume factor omit scheme svd alternate minimization flow rigorous follow definition time space event encode monotonic call except call independently zero unless therefore recall q rational expand product bx cx b bx bx bc ab ab b divide side first symmetry uniformly normal purpose initialization component eq must independent convex inequality matrix jensen call matrix simplify evaluation rank rewrite recall substitution result q leave bind cauchy inequality since rank upper eigenvalue statement minimum prof apply expression event exist follow expand follow less function eigenfunction fouri x du apply expression desire next lemma case lemma make incoherence condition symmetric incoherent parameter symmetric matrix incoherent must therefore show show incoherent parameter let eigenvalue definition incoherence desire part uniformly choose evaluating desire lemma logic choose desire schwarz rectangular choose entry pick moment trace must derive behave uniformly unit unit radial symmetry denote moment chi square substituting sphere component sample schwarz apply definition desire part want suffice pick lemma evaluating apply desire subspace uniformly lemma project subspace span basis incoherent space note part subspace incoherent bind part evaluate apply rough rate hope improve linear assume error direction rate expand choose take inverse across symmetry produce q desire low approximation matrix want decompose angular phase recover vector analysis constrain appendix solve problem perform quadratic substitution case rip restrict linear rip definition norm rip matrix rip prove transform objective optimization rip parameter small directional derivative second direction fy cauchy schwarz optimum follow previous desire result theorem dependent standard convex method rate imagine follow angular phase algorithm state main body rank iteration coordinate precision scheme reason sgd scheme achieve monotonically linear approach coordinate additional electrical science stanford university stanford factorization matrix relaxation exhibit least square prove runtime analyze solve draw form eigenvalue transform application include tracking sample store operate factorization substitute problem drop store size people sgd standard globally manifold guarantee motivate converge globally establish rate optimum prior analysis previous noise problem previously observe select observe entry martingale space technique optimistic factorization semidefinite analyze rate sgd exhibit local solution optimize low approach correct riemannian back onto manifold order sgd point rate optimum algorithm involve study recovery minimization provide retrieval operation initialization sgd cover slow respective eigenvalue familiar adapt stochastic stochastic wide understanding paper analysis local stochastic iteration globally convergent provide stochastic analyze matrix completion retrieval tracking factorize function orthonormal eigenvector sample factorization introduce manifold stochastic manifold group riemannian size choose sgd size iterate think give intuition particular update q property care compute simple operate whole operate benefit rescale angular recover radial notice unlike independently independently r analyze low rank decomposition introduce family point globally whenever non fix sgd choose lyapunov show decrease time lyapunov function matter initialize rapidly regardless hold attack respect way angular close recover indistinguishable expect handle span eigenvector matrix angular success angular require member occur value satisfy satisfie noise analyze rank represent size slow define standard variable assume satisfy angular phase radial vary angular iteration rate complexity unlike prevent also recovery option explanation load package explanation terminal need graphic macro ltb lt lt lt lt lt lt ltb lt lt lt r ltb ltb l ltb proof document since use outline occur iterate unstable determinant intuition occur failure occur success
note reduction review oriented want misclassification projection albeit last part check whether statistic carry choice statistic misclassification respect vector datum space error dimension abc represent local predict summary eq besides rate upper bind bayes admit loss replace classifier eq q classifier support proposition minimize integrate know selection classifier classifier last perfect train provide tend size dataset numerical local difficulty classifier naturally local probability relative return substitute restrict mention set agree train moreover algorithm knn minimize beyond good misclassification knn calibrate produce reliable estimating must kind issue swap otherwise section core proposal method expect call already simulate estimate error hope whole additionally limit must reference table bandwidth estimator know table table independent database constitute reference validation misclassification rate call consider reference always abc simulate database display surface second help krige conclude resort krige comparable evaluation support reduce valuable point abc dimension statistic additional coordinate regard abc sort dimension summary suffice accomplished misclassification know attempt cost potential curse dimensionality knn ideal remain perspective tune assess proximity adaptive classifier validation reference initial adaptive classifier surface collection summary table reference qualitative trait replicate predict initial classifier collection compose qualitative trait agree return correct axis qualitative trait know classifier rate algorithm table accuracy independently adapt machine community discriminate material numerically structure neighborhood system undirected generality focus representative level difficulty index call site color model digital site lie undirected definition adjacent include parametrize adjacent site auto arise normalize call define edge realization directly grid color statistical literature drop height width height cm vb four define eight close field permit modeling encounter latent field index conditional py parametrize scalar hide graph face double intractable issue neither likelihood latent color continuous well fit give compose neighborhood system represent undirected simulation compose lattice except boundary lattice another integral beyond mention sum triple intractable neighborhood structure field obtain abc field discriminate neighborhood structure color proposal interval share color free preprocessing perform via color group color color namely summary become q color indeed propose appropriate number ccc rate axis respect horizontal axis line error statistic include dimension onto six axis section summary give beyond component geometric begin abc normalize table respect summary axis scale draw thank favorable via markovian prior hour cpu optimize time abc wang motivate cut sampler simulation extend calibration neighbor algorithm show evolution evaluate reference six three impact solid numerical good calibration prior really reduce obstacle large regard confirm reference training curse dimensionality size fig latter prior misclassification replace last independently three carry prior substantially classifier train new summary base help discriminate highly reference exploitation ccc fig display plan range plan full call validation plot part krige geometric summary highly table dramatically error classifier explain interestingly informative reference design limitation connect latent noise rely color indicate capture information geometric summary reference simulation diagnosis plot rate abc cccc b prior error vertical horizontal train solid dash error abc summary framework include noise diagnosis table difference carry connected component adaptive simple misclassification albeit positive classification paradigm provide error section derive classifier curse locally around trade dimension proposal dimension besides inequality complement avoid practically machine viewpoint give abc latent method construct statistic induce construct statistic approach intuitive isotropic averaged width length summary explain continuous quantization observe site analysis numerical demonstrate indicate limitation approach believe road add edge color group consequently reference table since field negligible misclassification summary able acknowledgment grateful feedback present like thank anonymous comment suggestion lead thank model hide abc new summarize procedure intractable aim like highlight ability evaluate local wide performance relevant relevant hide field gain statistic distinguish bring little information happens describe extra information aim precisely size might extension geometric paper extend grey subject theorem dependency hide markov challenge due answer computation paradigm sufficient statistic fall summary statistic cluster plausible abc evaluate via statistic approximate choice misclassification nearest gibbs field spatially correlate mapping genetic spatial random grey color undirected grid perform popularity major difficulty view choice intractable remark exception small however time deal explore answer try reversible follow important adapt context graph observe address question infer color algorithm extend approximate approximate computation abc address paradigm observe numerous simulation monte carlo probability difficulty highlight sufficient know consistency abc check article automatic scheme construct rarely review concrete accomplish abc apart reconstruct competition pilot abc also consume section present near neighbor base general hide paper analytically choice challenge abc model fit observe approximate review wide choice assume embed space space density respect lebesgue evidence define probability good fit perform maximum predict respect counting measure since invariant whereas drawback mode abc numerous approximate posterior simulate eventually good frequent decision posterior directly face curse lie indeed impossible ambient dimension abc perform dataset euclidean moreover regard computer keep track commonly summary iid replicate literature j serve compose bayesian distance term name
imputation discard incomplete discard incomplete remain proper deal miss imputation complete simulate miss purely stochastic analysis simulate pool well strongly structure video reconstruction video media contain frame reconstruct video security reconstruction refer incomplete empirically nystr nystr om allow approximate equation subsample replace quadrature subsample manually effectiveness nystr om formula projection segmentation extension eigenfunction outside geometric subsample consist record method gram entry semidefinite homogeneous uniformly gaussian positive necessary put method hilbert space nystr must consider lead observe side generate family maximally concentrate generalize wave cf f restriction sense formula recover basis elsewhere value characteristic column extension scheme stochastically precisely step restrict row construct note characteristic characteristic prevent introduce bias degree description attempt keep notation become heavy see account roll q add height generate evenly spaced dimensionality roll embed noise ensure lie spread rate distance height choose point avoid influence dataset assess visually diffusion introduce introduction excellent diffusion provide van toolbox reduction find visually original quite easy see computer denote track iterate connect plot plot stochastically version begin step final point provide move point original image standard dataset parameter case approximately slightly plot stochastically initialize plot plot graph test face excellent dataset comprise people pixel comprise people web run example show face pixel approximately nonetheless reconstruct accuracy people miss plot pixel reason apparent output display result method show reconstruct reconstruction make neighboring value infer entirely image word pixel adjacent effect image pixel increment report toy car object sparsity successive rotation sample every sample time continue drop point weather locate international record contain pressure wind velocity cloud cover day initialization stochastically draw determine exactly imputation note scale axis actually method difficult analytically represent denote single comprise eq remark appear whose extension maximally energy minimize update replace minimal incorporate contribution every restriction energy geometric linearly find bottom relative versus range percent trend less consistent matlab code write test mac ghz
relate latent image cifar similarity fold recognition performance absence modelling seem beneficial suffer svm gain compare b try linear svm similarity cell result increase believe boundary mention believe heuristic complexity case rand basis refer kernel basis om spectrum report rand trial sophisticated similar little deterministic difference normalization nystr om normalization tend case psd kernels h nystr om bad h believe reason lack basis alignment violate dominant capture useful eigenvalue basis eigen decomposition show ratio negative two column reflect entity nystr om normalization pearson linear positive normalization entity eigenvector spectrum flip square om generally provided slightly comparison spectrum c fortunately svm construction demonstrate fold augment similarity order similarity measure complementary resolution resemble process level fine svm fold support use approximately support resolution fold furthermore support increase compare red curve improvement psd two outperform compare curve model measure support sparsity roughly different complexity utilize try psd sophisticated competitive less cost define mkl consist perform alpha fold cross kernel contribute resolution result approximately use combine basis variance normalization empirical discrimination prior utilize regularizer consequence center correlation dimension help balance irrespective overall affect combination measure option combine prohibitive search combine similarity measure section normalization feature vector normalization scoring center scaling dimension z svm center center svm report validate basis sub sample similarity validate observe normalization work scoring suitable similarity measure svm score overall work single normalizing feature accord normalize marginally affect conclusion svm kernel combination augment resolution observe normalization much combine similarity measure motivate benefit least dataset remove validation similarity center properly solver unnormalized robust rbf measure analyze scalable scenario similarity measure competitive scaling datum name expand analyze extensively cifar intra expect play crucial role imagenet future scale object detection strategy major limitation psd many psd implicitly explicitly paper investigate approach framework show constitute suitable despite complexity experimental result cifar equip svm support svm classification problem success achieve operate become complex linear classifier propose non linearity see feature augment non scale mixture component high space maintain support grow approximately linearly time complexity scale support psd expressive enough various measure similarity introduce solver result convex unless explicitly psd eigen alternatively dependency learn mainly two drawback cost without aim address expansion show complexity model without removal eigenvalue require eigen decomposition similarity propose analyze investigate visual x learn learn xy positive semi psd k reproduce hilbert space rkhs vice versa namely linear svm objective ik kx kx learn evident problem product involve psd kernel closeness case frobenius negative gram point guarantee psd psd classifier impractical scenario reader svm different cost cost research dedicated test approximate restrict w method synthetic datum rank gram approximation complexity prohibitive large contrary low nystr approximate matrix eigenfunction expansion embed nystr om space method exist explicitly implicitly exploit test cost support subset measure matrix nystr om psd need psd flip eigen latter achieved find close close negative eigenvalue low rank approximation psd measure psd associate section negative maximal regularize value suggest metric distance clear work psd well expensive select svms mostly binary one v simple argue formulation conclude competitive complexity svm unnecessary overhead well class svm class advantage training svm summarize complexity computation k svm require evaluate measure evaluate svm max svm via map derive need non unnormalized consider condition I similar svm bx basis margin bx bx unnormalized square compare svm straightforward contribution contribution basis margin bx nystr method nystr svm kernels nystr feature therefore appeal similarity say assume basis basis result violate large basis covariance nystr basis superiority nystr practice nystr rbf follow b weighted normalization randomly basis accuracy number average scenario different spatial good exact measure non zero support sample need evaluate irrespective see construction mainly
account another discuss neighbourhood size control neighbourhood pc reasonable allow restrict independence test pc also pc computational complexity respect instance test indicate convenient quickly efficiently structure real scientific structure network repository benchmark network actual examine package structural previous focus seven scale create random interval generate variance number false positive increase path pc six particularly hc exclude model cross perform poorly roc reconstruction accuracy exception previously comparable rate high improvement exception pc confirm observe previously skeleton low sensitivity increase overall sensitivity calibrate properly comparison mcp former visible network comparable sensitivity positive obtain consistent low triangular b q expression equal positive uniqueness cholesky contradiction suppose regularity likelihood issue density needs check fisher dag definite suffice ij j orientation extra subgraph order compatible simultaneously equivalence order instead directly prove statement case technical ensure behave respect uniform law likelihood concave dag edge theorem pn pn imply pn check mn ij pn equivalent follow figure mcp hc mcp hc pc mcp text test use graph solid runtime runtime runtime respective mcp bar mcp mcp pc tp fp skeleton fdr fp dag skeleton fdr mcp tp fdr supplementary comparison sparsity recent accelerate restrict search thousand concave regularization generalize exist notably existence way matter comprehensive learning package algorithm compete high sensitivity rate generate estimate dag hour network penalization acyclic graph receive past decade application range expert system artificial intelligence graphical phenomenon certainly nothing new calculus develop estimate slow formulated distribution direct dag observational alone interested estimate purely experimental datum unfortunately difficult nonconvex scale exponentially realistic thousand thousand great new network critical likelihood structure observational property competitive traditional neither work however high whose thousand key challenge bayesian mind score space observational work effectively capable handle several various literature cover requirement none aware score development application regularization understand attractive computational allow penalty result concave scad mcp performance advance learn highlight regularization generalize theory class penalty conceptual deep recent development regularization vs long application insight algorithm hybrid constraint organization remainder review contribution establish approach discuss penalize necessary justify describe establish necessary material development theory complete outlined section empirical offer discussion future direction sparse learn rely interpretation bayesian term coefficient nontrivial address challenge dimensional translate family fast traditional estimating idea continuous penalty via show competitive enforce intervention proposed use observational explicit advantage convexity thousand minute propose rely traditionally network score base search optimize scoring description hill various network posteriori base test existence node dag structure long assumption satisfy tend constitute main drawback conversely approach fast spirit constraint skeleton remove edge possible step fast score search advantage traditional previously approach structure scale ever hybrid time assume exploit pc gain hill scale advantage distribute thousand category contrast present efficiently graph thousand knowledge purely rely constraint approach prune hybrid purely rely cholesky distribution equation reader recall completely acyclic maintain translation assume generate variate decomposition regard direct acyclic represent contain cycle slight abuse dag weight adjacency denote node distinction node clear simply thorough introduction graphical concept unless shall matrix denote column rewrite encode avoid unknown nonconvex play important vast assume penalize point much tailor logit generalization remain assume may reasonable see dag parameter structural determine instead rewrite dag normal connection normal equation dag shall call class different usual furthermore dag speaking entail consideration theory strictly speak encode refer ambiguity may pair explicitly adjacency give uniquely define inverse view define recent connection detail connect statistically identifiable stand difficult underlying covariance dag prove difficult exist method factor make order amongst cholesky important similarity markov field far justification fits establish via simulation compare cite discussion focus network leave future dag variable ordering denote existence edge sort general sort equivalent dag easy dag equivalently adjacency strictly nod match sort strictly purpose use dag permutation topological parent sort permutation convenient interpretation weight also technical represent diagonal rewrite compatible compatible easy check fact dag permutation dag dag permutation arise dag want presence absence dag statistically indistinguishable observational edge estimate sense motivate obvious connection give permutation cholesky nonzero entry exactly permutation oracle across permutation weight sort sort permutation dag dag variable sort highlight I reader dag obvious different ii dag primary amongst dag wish focus causal view generation finding alternatively could wish commonly adopt public structural relationship causal relationship observational alone causality identifiability framework question necessary discuss paper develop far approach estimator sparse dag sparsity avoid overfitte penalize nonnegative possibly additional penalty seek score posterior penalty recover differ aforementione two choice traditional approach optimization constrain include dag fix topological sort sort parent permutation neighbourhood determine project node establish one nonconvex minimization minimize loss interpret nonzero entry acyclic acyclic parametrization define parametrization program instance plug back nonconvex alternate parametrization unfortunately dag nonconvex behind allow exploit analytical gain shall gain indeed formal wish emphasize shall interpretation evident loss simply parameter ingredient penalty effect select bayesian whose coefficient rescale choice notable occur special aic complex traditionally computationally dag nonconvex lose attractive alternative introduce fundamental theory concave estimation three principle guide guarantee sparsity note guarantee parameter totally development theory perform detail penalty scad scad smooth mcp translate scad key mcp flat bias mcp mcp concavity mcp sequel mcp difference potential concave satisfy condition estimate motivate consistency strong require employ penalty assumption relax substantially generalize consistent estimation structure learn estimation support equivalence typically number general evaluate however attention satisfying justification truly parent similar screening commonly tend typically order thousand thousand happen expect parent practice constraint use penalty superior establish furthermore advantage allow speed confident admit justification novel insight simply guarantee mcp attain consistency structure high refer iii nonconvex careful discuss define minimizer furthermore complicated alone dag usual theory estimation identifiability true identifiable establish existence local true turn finitely minimizer dag equivalence class finite local minimizer candidate minimizer produce quantity distinguish properly control distinguish dag small minimizer local penalize minimizer hence dag empirical indeed representation another dag many remainder stay likelihood technical distinction begin state high scenario depend consider alone develop identifiability error development general gaussian easier single uniquely inverse equation simply matrix element maximize represent dag spirit maximizer maximizer reason refer say long curvature penalty tend maximizer converge additional include q local maximizer must careful imply consistent show right condition maximizer conclude local select dag grow super increase achieve theorem dags common topological sort choice parametrization dag topological dag isolate positive definite proof appendix theorems dag maximize amongst covariance theorem local maximizer pn na flat penalty ij act equally lie penalty define hope sufficiently continuity intuition may unique essentially answer dag approximate unlikely statement assumption impose combine parametrization give state theorem distinguish previous also denote maximizer give tt penalty function immediate adaptive result datum order identifiability observational notion oppose rule edge role penalty satisfy condition maximizer constrain penalize carefully show slope concavity concavity overcome local likelihood continue simultaneously guarantee parameter estimation vanish highlight trivially local satisfied penalty particular mcp interesting concern dominate penalize parameter true essential control growth grow dag long grow dominate estimate theorem remainder quantify penalty need maximum rate alone ij requirement require sufficient sufficiently hence course simultaneously mcp satisfy theorem factor parameter model condition extra redundant factor formula mcp may penalty mcp theory particular give scenario formal direction already remark equivalent dag reduce high course know advance optimal whole estimator advance show compatible consistency estimate edge dag penalty computation nonconvex guarantee minimizer guarantee minimizer remark estimator special concave penalization relaxation discrete light work compare good believe estimate permutation require careful type preliminary however remain work hybrid estimate initial direct partially search dag subgraph search undirected dag estimation restrict search idea constraint nonconvex perform minimization employ naive gradient employ cyclic checking properly perform enforce technical overview approach simple attention minimization holding hereafter repeat coordinate detail coordinate know minimization minimize consequence substantial check add estimate respect alternatively induce cycle neither edge induce simultaneously outline sequel function fact rescale cause computing hold ignore express implicitly overview must outline remainder implement concave coordinate tolerance block respect block minimize respect jk jk jk jk equation follow appropriately unit detail mcp show solution give threshold choice mcp q convert minimize hence mcp similarly ingredient reason choose mcp comparison function apply parameter strictly minimizer choice penalty concavity consider sequel regularization maximum estimate decrease scale guess nan minimizer model condition exist correctly work preliminary empirical provide section penalty discussion difficulty particular validation suboptimal avoid present several exploit greatly idea excellent implement use warm start active block parameter speed naive incur prohibitive bottleneck operation sum inner product hence compute cost operation million reasoning computation highlight require total cycle leverage become calculation number parent specify appropriate computing proceed calculating estimate justify know true practice specify sequel section update every active sequence concavity tolerance datum inner product I mm active final note estimate adjust order assess accuracy efficiency pc max hill equivalent standard greedy hill base pre mean intend exhaustive performance experimental thus consist method hc hybrid method brevity frequently pc method form regularization implementation mcp mcp gives offer gold dags dimension dag purpose hill slow move onto assessment advantage constraint discussion property outperform method fact accomplish search somewhat remarkable statistical pc hc package optimize implementation recently date publicly available perform ghz intel core processor gb ram mac os dag randomly accord os add independently equal array individual test test randomly generate sample structural choice sample dag generate dataset random dag instead exception hc regularization pc significance choose use start choice recommendation concern smaller long furthermore edge excess hour mcp concavity choose parameter represent fair also minimal estimate simply large algorithm iteration traditionally produce skeleton phase relation discuss case method dag ignore miss situation final orient skeleton undirecte ignore skeleton wrong direction distinction course regardless predict positive false positive estimate dag estimate log convert absolute far hamming distance perform dimensional concern accuracy metric imply false fdr edge sometimes recall estimate hc one pc use produce total average complexity processor time require define runtime detailed result choice order properly compare keep thing theoretical consistent choose accurate select seem artificial potential sensitivity nonetheless consistent compare discuss issue low mcp hc combination expect performance rely consistent setting test also increase mcp hc fp dag skeleton mcp hc skeleton fdr mcp hc pc tp skeleton fdr hc bad term structure far discovery simulated dag edge hc exhibit great hc middle pc edge represent hc consideration section difficulty exist overcome hc attribute overfitte though produce alone influence accuracy graph estimate algorithm evaluate estimate equivalence correctly may explain hc job oppose sparse constraint thus approximately mcp bic fact hc still perform well far bic estimate dag however hc omit term test fix mcp result test test show value test dimension increase remain across discovery comparable true positive indicate mcp mcp maintain discovery
different make plant refer every integral solution failure figure explore recover expect fractional relaxation popular objective relaxation thereby round step light relaxation recover regime distinguish plant cluster high enough quantify contrast sdp center plant sufficiently precise separation research direction come disjoint ball interest investigate certain point etc particularly interesting overlap accord truth observe median lp remain isotropic lp practical ground recovery phenomenon phenomenon context alignment angular refer relax relaxation another analysis median exact recovery guarantee relaxation powerful tool many domain ask various partitioning thank anonymous correction greatly improve work rw mathematical physics separation median lp property cluster rhs small ball around separation property see receive rhs enforce easy hold degree main unit respect neighborhood translation integral least proof proof show draw p x satisfy first restrict attain proof done use symmetric continuous thesis would respect lebesgue exist property random converge satisfy maximum attain step claim show number large enough center ball great zero probability nx jj hypothesis restrict alpha inside boundary demonstrate dc h circle dash circle intersect inside big position ball intersection let nothing rest partial respect z j absolute constant union probability sdp satisfy fix continuous center ball separate sdp integral intend go construct bad ball radius center away ball ball consider unit create copy pick center group center initially exist center center true example also initialization analyze cluster method choose center choose choose first ball event consider center lie assign center first ball get center datum center take newly form lie assignment center choose center therefore fail cluster disjoint cluster far fails assign initially copy assign center distinguish initialize incorrectly copy succeed configuration version prune instance initialization sample center center proportional remove center let say arrange apart group far away center fail recover cluster idea illustrate choose center prune unit ball distance center ball ball center select center select select center rest case normalization select exactly center arbitrarily introduction median primal yet play role variable amount pay thought amount pay dual median select element remove share remove contribute increase increase uniformly di si ji di di lp suggest primal point let rhs attain equal median argue set dual bx ax bm bx theorem assumption conjecture question relaxation focus median focus relaxation optimality tool relatively parameter free tailor distributional cluster center need relaxation integral relaxation tight separation lp separation yet psd center heuristic recover set cluster k factor recovery open suggest relaxation solve problem theoretical relax relaxation serve round serve true overall convex constraint convex computer science relaxation integer fractional relaxation underlie ground truth give model particular mathematical albeit approximate relaxation round relaxation round directly solve relaxed occurrence exact recovery phenomenon question motivate integral solution typical relaxation good believe examine yield strength phenomenon recovery understand say maximum minimum flow generally integer constraint totally vertex solution vertex relaxation study lp decode programming code recently signal community seminal paper compressive np case relaxation phenomena partition problem example include study partitioning clustering consider relaxation show recover optimal solution instance partition problem relaxation initial metric solve point disjoint commonly cluster point assign close alternatively distance cluster necessarily minimize partitioning point cluster read x mean lp approximation optimize problem objective via round effective although provable sdp relaxation mean previously relaxation geometric rounding appear recently lp median point admit distance also lp introduce ball position minimum center draw ball recover two point thresholding also contribute showing separation distance median mean objective standard programming relaxation integer relaxation programming relaxation relate relaxation relaxation intra separation center sufficiently separation decrease begin overlap transition relaxation begin fractional statement detail ball high recover relaxation separation relaxation recover separation variable point solution problem consist disjoint star graph show broad sense adjacency inverse vertice component theorem tight relaxation mean recover separation assumption theorem heuristic high arbitrarily separation fail high recovery section derive call separation see prove believe refined conjecture addition primal median clustering center appendix consist use scalar version construct sufficient relaxation probability satisfied separation mean lp exact relaxation study condition disadvantage heuristic good solution even heuristic combinatorial look relaxation heuristic solution property appeal iterative body stability tailor convex relaxation cluster ask used heuristic mean median heuristic even procedure like fail within regime relaxation guarantee probability recover correctly sbm communities community behavior enyi recover detection also know plant recently sharp threshold correctly moreover relaxation show exact threshold share fundamental objective point cloud obtain establish establish deterministic tie maximum graph moreover sbm graph random create distance technical difficulty study point though might comparable profile integer relaxation linear program indicate indicate whether unique motivating let ensure optimality integral existence feasible intend degenerate since zero observe solution indeed motivated observation enforce variable within easily identify median lp cluster optimal intend cluster lp eq point restrict rhs intend easy constraint trivially sufficient turn powerful separation possible exploit primal rhs get contribution distance way feasible rhs attain dp condition variable exist ball ball see point see statement assume remainder ask cluster ball neighborhood follow condition eq say satisfy provide big essentially say attain small center ball interval center interval q require recovery show median separate broad set cluster contain positive respect neighborhood center n independently median least find ball large hold drop expectation function attain close enough jk mean contrast relaxation median natural mean integral center exceed particular median lp natural lp use cluster see satisfy distance every cluster cluster solution two integral cluster complementary tell combine tight sense separation lp draw symmetric support sufficiently plant high plant imply plant point show p plant remain parallel feasible feasible solution know hold equivalently maximize maximize square unless trivial coincide ball since come symmetric relaxation semidefinite sdp relaxation cluster whose separated distance conjecture construct deterministic sdp plant condition high random matrix explain proof cluster total cluster point define matrix ease dual unit indicator index j ie dual otherwise cluster coordinate intend dual tell complementary tell diagonal shall switch length notation easier complementary specify dual intend entry ultimately pairwise know semi imply negativity essentially distance cluster finally average reasonable hold remain cluster greatly simplify z tx tx separation separation xx tx
rnns sequence recursive convolutional share whole see convolutional weight single mechanism learn sentence neural unit initialize project space nonlinearity activation think node recursion choice activation leave activation child convolution adaptively hard code network leave perspective sentence source actual sampling search corpus instance convolutional source sentence convolutional rnn rnn last sentence work encoder decoder source sentence length denote decoder generate sentence rank system approach provide phrase concentrate direct configuration rnn trivial determine target length unit encoder convolutional understand inductive encoder decoder measure encoder decoder english corpus combination corpus word respectively news news test set sentence parallel corpus english sentence word english rare word map token rnn decoder newly propose recursive decoder minibatch transition matrix spectral radius element wise neuron embedding case update update ht phrase phrase result encoder unit either step translation exclude select high scoring candidate reduce reach zero approximately rnn lstm search good use usual length prevent shorter e htp cm union phone number clarity china le et dans le une r la est de phone un une face est la position en la en sa en un de ce union et en une est union phone est de et une la l investigation complete end finding bank recommendation action reference la de du de la bank send ann conclusion send la bank te de ann e les pr send es le la direction des te ann pr send la bank des n des source question balance adequate within school reference ce des en comment le un acc dans une un de une cr questions pr se des dans les une de de fa il des pour acc des il la le acc de du une le still reference il vote il il les il il des source work reference es cr es cr ne es cr es cr source lot right il ce un large consensus il de le la il un consensus la et il pour se un en tr une pour instant translation specifically translation sentence look score translation change sentence sentence rapidly unknown challenge increase system although present along baseline phrase still translation word source significantly per model phrase recently improve translation handle obvious explanatory length sentence encode sequence neural phrase translation fig high sentence fact source reference respectively even train evaluate machine translation system test set choose one word list long short short word difference job especially short source sentence degradation machine ht le pr des est le des est le pr des le pr des des est pr des est le des est pr des est pr des pr du des generate additionally recursive learn united parsing encoder united correlate despite compare property automatically believe investigation paper translation purely focus evaluating encoder propose task sentence sentence translation possible encoder choose two differ newly english sentence sentence existence rare suffer significantly well future research direction machine translation purely find computation much source especially language deal secondly research prevent neural translation
box expand omit costly box datum output boundary box minimal cluster low boundary equation equation un normalize meaningful easily create performance experimentally area frequently receiver characteristic curve roc count positive form roc normalize divide cart forest hellinger distance tree among mining cart potentially use distance skew insensitive baseline rule induction box shape search iteratively maxima simultaneously list corner publicly breast datum uci repository obtain extraction evolutionary repository imbalance corner corner breast fast box algorithm imbalance rbf kernel width language separate fold fold test decision tree build prune research prediction control box expansion performance statistically significantly well use perform small choose good fast box often always bring several question question address question fast box perform scatter plot quality box imbalance ratio represents test horizontal negative fast intuition box expansion answer pose effect box example number might allow effect provide box around dataset datum restrict produce draw classifier generally seem cm datum good box box box perform cluster use fast box box interpretability set window process window window include fast box follow building window must form practitioner threshold box allow generalization box dimension count number place say place large place possible drawing classifier box box risk draw way namely boundary upper boundary box k matter divide tight box proper subset another set box draw classifier suffice draw classifier box experimental competitive challenge way one neighborhood warm close least dimension pass alternatively one approach cluster scan single pass keep center new designing setting box formulate mixed program act standard interpretable moderately sized box characterize discriminate naturally cluster failure benefit limitation hope insight box gold interpretable box interesting art cm p p cm cart boost rf box corner square corner breast breast mm vast majority world classification machine handle classifier benefit interpretable programming negative accuracy use specifically box separate method consider feature interest derive interpretable classification classification unbalanced domain range text prediction diagnosis trivial classifier use human expert form parallel call box drawing create classify box create box drawing exact mixed dataset gold substantial approximation performance approximation justify make box approach characterize class alone bring advantage fraction datum high imbalance create locally number involve computation though analytical may computation solution boundary discriminate local analytical calculation much simple decision choose namely classifier become imbalance approximate produce interpretable mixed advantage approach box draw interpretable generalization box drawing discuss approach make imbalance imbalance goal interpretable conjunction like introduce use liu et note experimentally work sensitive version decision sensitive seem interpretable rule patient rule induction partition like tend recover compose splitting criterion iteratively part describe patient slow neither box box though fast box approximation useful characterization approach start mixed programming act gold box box classifier positive correctly box majority classify give notation subsection notation definition parallel number index index boundary box box decision otherwise otherwise box classify correctly index example regularizer encourage box majority space box axis f negative box way gold compute say away resp upper boundary box definition rise box classify q give positive inequality formulation derive derive analogously make obtain degenerate boundary upper boundary number base computing environment full programming formulation size produce gold standard box drawing box determine drawing permit evaluate quality box operate much box use example cluster negative might cluster discriminate negative discrimination box around adjust locally power fast box seem box stage follow boundary set tight positive dividing stage partition negative boundary boundary expansion expand analytical stage cluster technique axis rectangle small parallel axis rectangle take minimum subscript part subscript subscript datum boundary th dimension th figure illustrate domain dash h cluster computation parallel discuss determine discriminate positive negative create exponential box
corpus alternatively set extract skip language literature max contexts context context share context entail impossible maximize switch underlie result embedding hold hundred thousand make tractable replace softmax elaborate direction present approach way derive word negative come corpus correspondingly probability come corpus observation q lead trivial mechanism prevent way random incorrect name negative stem become get almost al difference et corpus present construct specifically et time large equivalent draw normalization context text model relate fix learn representation context representation reduce regression model jointly make list software speak sentence context come around word window parameter size word window sample discard appear consider frequent define importantly word sub improve benchmark frequent another effectiveness window content away word make similarity really distributional clearly one
crowd worker ask look image example direct inefficient choose triplet collect grid format probe analogous triplet show grid image worker ask image grid allow collect probe yield triplet user change triplet amount effort crowd worker question grid collect triplet technique investigate effectiveness triplet acknowledge drawback triplet rely modification researcher multiply change crowdsource fundamental change lead embedding grid format collect several triplet triplet tight create reasonable low crowdsource budget collect trade user burden effectiveness triplet uniformly triplet quality decrease ingredient annotation triplet upon publication embedding useful search cluster find wish collect embedding author use triplet collect embedding work collect triplet collect triplet human triplet probe similarity use rather triplet pt collect grid strategy yield triplet good embed triplet collect separation half lie within area collect grid crowd active collect triplet triplet behind triplet triplet redundant crowd triplet embed space triplet rank object place geometric lie histogram occurrence answer triplet bottom histogram triplet individually triplet histogram answer triplet wide sampling triplet effect recognize grid triplet study human probe ask mark image human triplet per allow allow crowd worker load especially triplet crowd parallelism level human involve human right measure respect triplet take crowd worker pay complete author work quantify formalize hard answer author use acknowledge triplet histogram object answer object occur suggest certain recovered well random triplet roughly keep mind influence course collect triplet either one thick line batch color number triplet top appear triplet human triplet individually triplet error aim answer question run experiment show probe grid object choose probe baseline triplet collecting triplet embed embed embed effort worker task perfect proxy let validate approach conjunction paradigm handwritten digit contain digit generate comparison vector music similarity dataset point collect triplet present face dataset identity set extract triplet contain happen randomly triplet likely near triplet correct triplet uniformity create show selection small spread across music low synthetic select oppose image select close precise location compare neighbor worker perfect answer expect human reflect wide triplet publication validate human behave similarly proxy hour metric via synthetic embedding inconsistent triplet run image image contain roughly avoid shown allocate quantify publication collect triplet probe grid object probe varied three repetition experiment return grid triplet generalization triplet construct embed embed vary embed human grid spend near answer click bar percentile experiment triplet triplet view embed find choose grid cost triplet cost triplet cost triplet collect crowd collect cost result embed triplet uniformly embed unlikely much collect possible unique triplet generalize redundant evaluation reference experiment collect grid triplet collect time would outside budget strategy triplet grid sample unique triplet actual object give triplet grid task bold hour allow job generalize unseen constraint embedding low triplet human comparison grid triplet triplet come ahead view large grid even grid triplet sampling time separation size grid yield triplet answer triplet choose triplet triplet pick triplet ask gain outperform comprise answer collect fast complete large second vary fast question average grid per per able hour median even hour fast grid could hour trade important worker acceptable receive worker exploit hour complete worker pass give visible drawback take batch save recommendation collect researcher task identical trade issue researcher create quality create triplet researcher chance list collect consider grid yield triplet appropriate select yield kn work human continue investigate triplet adaptively grid converge random especially thank discussion support nsf fellowship award nsf google award similarity vision unfortunately embed collect task triplet technique effectiveness drawback triplet display collection task user explore collect analyze speed display create cost collect triplet
unity unity magnitude unity contribution decay x evident expansion constant depend eigenvalue unity circle imply number machine stack put number decay unity always strictly decay product exponentially mode mode exponential unless seek case decay act extreme mode whole numerically indistinguishable analyze broad range know perspective assign mode observe length report length derive magnitude understand cf eigenvalue amplitude contribution exercise ml stacking stack simple stack internal require pattern analytical statistic iid iid process generate string stack structure conversely correspond stacked us constraint stack thus symbol iid straightforward stack physical stacking recognize stack stack stack fashion must h namely respectively mix expand machine machine single expand directly expansion eqs state tm tm length et process much information especially apparent become sophisticated inverse upon expansion special subsection expression tm piece calculate effort find cyclic eq quick hold range become yield stacking absolutely stack stack sequence effectively assign coin tm repeat coin iid previous result repeat apply inverse tm identity matrix find eq final eq eqs show tm complex plane varied notice start cube root unity degenerate coin structure even allow transition underlie transition allow next ml ml probability see nothing completely expression previously however recursion relationship iid iid compute asymptotic although iid recently simultaneous c parameter hmm thorough process reader comprehensive convention adapt fig arc take transition state arc transition course illustration consider emission symbol uniquely terminology prefer represent nonetheless develop applicable hmms density mix number exist orientation panel al show list failure organization process failure result primarily sufficiently observe state give method applicable straightforward versus fig produce al appear iid fig clearly expect material generation specify challenging process use resolution transmission kind notation alternate block ab energy appear nine isolated instance sf stack external merging process process inspire machine recognize large process although state prevent occurrence inspire ask produce suggest show candidate must thorough dp reveal appropriate causal primarily give h et deviation domain self transition effect domain likewise increase domain prevent al interest maintain reasonably possibility predict sequence et propose al begin hmm machine fig case internal tm six state machine explicitly expand straightforward evolve blue root unity unity persistent four approach unity eigenvalue throughout eigenvalue upon eigenvalue tm involve root plot complex eigenvalue root unity transformation degenerate towards cube root unity eigenvalue tm solution part eigenvalue plot fig responsible decay constant decay quickly quickly slow cf indicate kind process fig implementation increase nontrivial cube root unity loop toward behavior suggest structure eigen structure apparent hmms calculation analytically mathematical assume hmm pdfs bring understanding material thought object importance contain representation directly demonstrate invert specify underlie hmm highly nontrivial considerable effort transform structural stand presentation wide stack possible stack contact previous applicability stacking rule position amenable restrictive sample stack method perhaps new hmms third dp machine hmms become describe structure present formalism framework identify mechanic information spirit theoretic sequel efficient analytical author thank external member laboratory office contract evolve vector stationary eigenvector normalize probability word convention denote object scalar concrete stack accord accomplish examine symbol symbol symbol symbol say symbol alphabet give internal tm gm h gm stationary probability probability long gm hmm fig circle arcs symbol upon probability machine spectral express stacking h represent stack term expand machine vary calculation physical h representation stack structure state ambiguity ml degeneracy representation size three transition distinguish among triplet label indicate transition label scheme store ml transition machine label stack transition take ml completely analogous advance gm distinguish fig state label choose give state satisfactory labeling scheme machine x h machine add three transition machine us self transition abc still induce transition advance stack transition apply transition able write stack gm expand alphabet six state b give tm completeness stack gm b state example connectivity presence self expand expansion yield sufficient component transition transition done determine calculate ssc define machine h previously ssc ssc call one ssc expand machine gm case ambiguity occur subscript symbol give ssc machine gm expand machine machine least ssc machine far distinction two effect calculate dp hide index integer p jx p transition abc language transition split label machine onto distinct indexing consistency machine machine transition map submatrix label mapping visually set statement submatrix machine state expansion gm express cyclic x cyclic among perform identity operation directly
unit suggest non trend mae rmse song year use mae quantify mae rmse song audio impact music aim evaluate collection consider single sequential complexity compression estimate aim descriptor string descriptor track audio descriptor descriptor similarity prediction song track popular web rating agreement perform control rating combine bag descriptor obtain performance gain song year descriptor scale benefit music content category concern quantifying receive field music digital web base music database novel music find retrieval manual annotation process latter infeasible amenable distinguish music audio identification track identify track similar collection track music similarity specificity similarity rating song particular temporal audio audio song descriptor temporal representation discard temporal sequential sequential complexity audio quantify string scalar summary statistic retain motivate involve rating song year descriptor audio relevant determining rating human rating year give chart assume chart entry song song determine thus might music recommendation song prediction might incorporated task year descriptor low specificity experimental account rating respectively finally conclusion fu track descriptor second knn estimate target li wavelet histogram knn classification spectral al construct decision tree classification et use purpose classification approach determine descriptor track pairwise track distance combination track centroid assume centroid thus close distribution contrast cross approximation centroid track pair apply identification histogram previously refer discard temporal yet stand contrast et representation mid feature widely version identification specificity approach involve intermediate aggregate locally estimate predict global window computing variance purpose local aggregation alternative al w alternative original result describing frequency whereas lee spectral al apply modelling feature generate compression lee semantic count likelihood recent attempt temporal representation bag feature al classifier classifier successive tag multiscale benefit evaluate aggregation base vary window size pyramid technique smooth structural change multiple modelling architecture aggregation tag resemble apply summary statistic since metric indexing retrieval computing resemble differ propose pairwise specificity prediction report shannon date music investigation audio sequence descriptor quantify compression require represent specify invariant measure sequence track audio compression constant original feature sequence frame compute due region fig obtain feature may encode efficiently conversely admit efficient invariant ordering observe apply specificity task consider dimensionality feature informative use alone task similarity rating distance descriptor track descriptor similarity predict rating pairwise distance descriptor vector denote th track available descriptor audio component descriptor track whose seek total similarity determine ps song year chart entry date track collection predict chart date descriptor specify method song year motivate multinomial straightforward determine coefficient use american popularity chart track annotate chart date chart entry extract audio version exception feature use th p feature component use select peak attack duration attack slope attack centroid moment spectral spread spectrum skewness skewness magnitude spectrum excess spectrum percentile magnitude percentile energy wiener magnitude magnitude spectrum amplitude successive component exclude coefficient spectral half wave centroid centroid peak predict centroid l low score track name stop I come time guess rise fall seven day I frank I country sign lee cat nothing gene track frame level descriptor descriptor compute describe vector principal component analysis fashion preliminary seek correlation coarse frequently choose string compression compression uncorrelated symbol string compression compression symbol observation uncorrelate distinct length unlikely frequently track alone average level obtain facilitate interpretation chart score report rank rank additionally low move piece contrast stand music strong power van track support exception track expectation specificity subsequently validity similarity capture beyond scope paper track analysis evaluate similarity annotation chart music pairwise similarity subject ask pairwise successive track point ordinal corresponding assume internal scale rating omit rating similarity five point similarity use absolute track rating rating quantify agreement addition music inherently similarity dependent internal widely verify quantify present select song apply chart restrict chart historical change production affect rating median rating per rating track display count rating less relative scale content might form recommendation track recommendation alone form recommendation interest track track evaluation five previously merge rating score discard similarity rating score perform evaluation result h count rating additional control subject subject assess training subject impose per collect rating rating subject control condition rating similarity coverage web rating control quantify control rating report five four agreement correlation sample sample correlation subsequently rate agreement coefficient rating aggregate analogously apply base condition balance classification interval predict rating distance descriptor vector use section additional baseline use account temporal audio feature sequence follow apply state space stack consecutive vector determine distance normalise compute distance similarity annotation square prediction quantify rating ordinal annotate rating term term pair term pair tie tie denominator geometric adjusted yield value difference pair versus accounting product moment coefficient separately unique rank tie rank tie contrast tie view compare ordinal proportion explain assign rank ba note contrast ba order ba rating annotation annotation annotation testing apply descriptor distance divide variance training across audio feature audio track thus distance vector descriptor among distance track combine obtain weight regression likelihood parameter norm penalty statistic rating accuracy hyper parameter rating incorporate p descriptor measure frame sequence error th c std combine descriptor five rating individual audio depict cross prediction yield similarly yield apply correction cross descriptor versus specific amongst observe descriptor feature describe estimate use ba consider performance rating outperform use combination employ alone incorporate descriptor gain ba respective four rating scale gain respective rating scale confusion annotate rating bootstrap turn post hoc reject hypothesis h c c c predict fig display feature descriptor perform rating magnitude across classifier one diverse multiple within song year rating use chart linear
direction many shall generate ad distribute sphere independent normal efficient generating direction moreover mention qualitative firstly large need geometry secondly trade conduct datum stem elliptical rather world need datum guide choose compare number calculation depth hand invariant separation ad set direction though direction produce depth precise keep increase calculation computation avoid direction calculation amount set generation univariate order univariate substantially see classifier stability classification phase direction heavy approximate depth minimize randomly direction depth direction direction much frequently computational classified depth calculate assign mahalanobi employ depth origin classify outside depth mahalanobi employ scatter within class alternatively mahalanobis depth mahalanobis depth avoid phase class center consider neighbor rule place classify hard validated computation sequel several well linear discriminant mahalanobis neighbor call spatial mahalanobi spatial use lda knn affine mahalanobis affine invariant appropriate version depth affine approximate td td zero class direction bad treatment quality depth treatment give separate hyperplane lda datum follow gaussian unimodal elliptical differ shift real approach assume different classified classifying class neighbor small see produce mahalanobis pool neighbor handle supplement rule restrict classify hyperplane correctly classify remain step determine parameter box constraint name whole panel together separate solid panel datum right quadratic symmetric k still kernel two class linearly separable hilbert correspond every determine margin figure solid plot discrimination small simplest separate select circle come decision exhibit right panel solid line indicate optimal left panel appear rule need besides select straightforward task say achieve tuning parameter computationally intensive determine correctly procedure take second several reach minute four classification phase point classified classify otherwise procedure calculation choose evaluate practical variety set methodology obtain partition diabete include ed take package constitute subsample diabetes multiclass split slightly process dropping object class description consider refer short description find page outlier see tie break treat knn simplify tie svm account tie approach three lda knn sect lda knn moment robust svm spatial equal depth classifier leave rate space mahalanobis depth direction cumulative task patient substantially attribute bold dominate patient classifier mahalanobis depth show table dominate also dominate diverse classifier classical different five aggregate measure task calculate lda knn mention negative value classifier relative small task value mention five table measure bold classifier none triplet satisfactory lda value visualize indicate well figure easily depth follow use estimate perform good classifier half depth part depth transformation moment mahalanobis similarly projection depth random outperform explain approximation direction treatment knn lda depth depth breast cancer vs vs cloud diabetes l r r treatment knn knn moment svm depth patient patient ed segmentation cancer vs vs r knn lda knn light nature treatment practical evidence number need see classified depth solution construct fraction rarely share amount base vanish version heuristic close separation come optimal separability separate rule propose cross svm separate really table frequency leave cross histogram bootstrap case two depth type plane find hyperplane small separation space separate cancer vs cloud diabetes heart patient vs patient segmentation vs vs procedure essentially feasible attribute datum transform distinguished depth vanish depth vanishing induce spurious symmetry non though produce region inefficient good shape exact version depth employ direction direction depth classify classify lie classified percentage substantially apply depth either near knn mahalanobis depth moment newly separate possible choice knn cover performance practically cross validation quick sort real lda knn none classifier depth dominate maximum moment mahalanobi outperform mahalanobis depth explain aggregate comparison greatly depth good mahalanobis depth vary goodness visualization experience problem tell stop span point separate investigate involve vanish hull application calculate expensive simple burden acknowledgement grateful master student university maintain package valuable comment anonymous greatly cm cm universit nonparametric fast procedure broad procedure first cube separate projective class alternative notion mahalanobis depth depth regard rate depth class dimension available alpha depth depth depth statistical generally class theoretical consideration procedure properly assumption real parametric classifying procedure due establish mainly evidence usually arise practical give field translate exist simulation really fitness demonstrate classification real procedure cube projective transformation reflect degree centrality carry depth function depth binary procedure depth separate linear plane contain origin apply alternatively study mahalanobis depth spatial depth everywhere vanish outside hull use depth finite outside hull practical question treat point represent portion classify depth supplementary classifier consider classifier compare mahalanobis spatial recall depth simulation study important broad experience usefulness far investigate space robustness apply substantial amount comparison indicator procedure number internet field evaluate standardized www package name three procedure classifier case include exclude neural network offer many architecture computationally expect datum adapt perform approach hand tuning set regard network classifier exclude svm sect describe classifier extended depth direction phase arise several classical mahalanobis knn introduce simplified svm sect task include sect extended depth property conclude problem phase transformation datum depth subsequent separation projective depth map coordinate transform reflect centrality subsequent ordering order depth depth definition current depth training first three version take value beyond hull vector variate observation r location scatter covariance denote univariate univariate minimal lie hyperplane distribution lie hyperplane obviously vanish hull mahalanobis projection illustrate mahalanobi
focus united plausible result test model disease context another internet trace evidence pose trace operational disease surveillance activity argue wikipedia internet source meet four challenge open reliable robust operational wikipedia thousand location need understand disease test disease model context suggest wikipedia access global monitoring outline plausible reliable sound operational disease surveillance forward scientific make reality mac discussion improve part gm technology biological number cb cb program program security department energy de public release file inter language article mapping input correlation file lead public health economic stability structure effort monitor risk focus monitor slow data media query effort promise challenge area disease forecast examine source linear language proxy systematic yet disease day test suggest location preliminary close overcome challenge monitor effective globally comprehensive impact united states surveillance health laboratory surveillance method internet yet scientific review disease forecast capability argue available aggregated online wikipedia statistical technique suggest datum forecasting model establish wikipedia outline reliable sound operational surveillance system gap internet technique disease extremely costly majority condition infection great united reduce economic effective surveillance detect quantify incidence save traditionally form laboratory follow report costly introduce lag observation surveillance upon internet medium datum mining technique health trace stream model truth health forecasting effective operational google trend challenge disease surveillance reliably integrate review improvement third broad applicability source generally available must data wikipedia knowledge disease surveillance resource usually simply incidence publish model flexibility test many context insufficient incidence train health contexts great new incidence dictionary census disease forecasting make complex limited context insufficient insufficient understanding biological internet stream forecast horizon evaluation approach yield fine one address approach wikipedia proxy hope available feasibility upon stream map daily access total context forecast successful context test day overcome challenge resource wikipedia keep date datum share adapt new context incidence article demonstrate several simple disease version suggest inter language article mapping readily translate forecasting short tight short argument first source wikipedia surveillance forecasting previously operational estimating disease incidence internet wikipedia access turn thorough set well challenge laboratory disease surveillance internet base disease surveillance traditional surveillance upon direct contact biological test rely surveillance datum include clinical call room example well surveillance report identifiable similarly resource surveillance essence department base department laboratory health exposure disease clinical surveillance disease example laboratory consist surveillance disease agent human mild severe clinical environmental public health school counter sale call early report alternative detection surveillance system notably visit publish wikipedia million language top website visit search engine roughly request engine wikipedia two key read change publish review publication vast article surprising seem manner abuse wikipedia effective deal wikipedia measurement popular dynamic wikipedia application order measure flow world popularity political economic attempt forecast sale stock application include forecast information prominent research assess wikipedia health e cancer drug information four health study wikipedia measure traffic relate wikipedia evaluate article disease relation news health issue find drug sale wikipedia traffic health article none article fourth recent united access broad use wikipedia interest wikipedia health purpose quantitative surveillance stage recently surveillance social stream large property complementary basic insight leave trace activity relate capture medium face web search health fact volume internet traditional surveillance effort exist internet metric exclude public evaluate party crowd source health disease surveillance root metric exclude metric counter drug proxy activity trace media message web server trace extract phrase metric occurrence create value train period internet true I future past case availability lag typically total vocabulary estimate metric accurate model produce correlation surveillance cite incidence variety stroke west simultaneous effort united wikipedia access study lasso year variation replicate article statistical key improvement disease proxy note briefly work detail reliable location forecasting briefly location result internet disease surveillance might systematic article article traffic traffic article proxy software open depend study statistical use available goal applicability surveillance operational public purpose surveillance base query google near google trends level internet surveillance well prior access query chinese engine google yahoo engine mostly english website payment index google trend google view level research scale effective model situation somewhat surveillance effort twitter certain outside company substantial share researcher media site consistent able find either extremely make wikipedia cite et highlight google algorithm little wide review improvement trend publish summary highlight well failure google resource trend else resource contexts google surveillance nearly effort small expand key propose base discover medical mention lda co occur list keyword health article build medical method discover coherent disease cancer topic correlate united drawback text require expert interpretation location knowledge measurement solely desire algorithm offer translate one context inter language link translation effort kind forecast even forecasting disease class forecast internet shift signal forecast lag al signal hand lag shift week forecast horizon indicator significantly xu et lag forecast et month query forecast appear potentially google trend forecast include sometimes disease important simple lag previously week daily independently day separate challenge surveillance challenge offer plausible wikipedia article disease incidence approximately location contexts acquisition processing use web datum variety include file contain hour time compressed request article request omit request differ human view automate request people read article factor commonly proxy analyze day data file hour miss gap hour treat minimal effect analysis normalize request yield article request hour fraction hour request language period request request file request retrieve daily daily china chinese united states united english china chinese total contexts list proxy resolution disease incidence goal evaluate broad disease across applicability mode transmission type similarly location develop test first reliable incidence specific disease frequently location disease health well health organization count information present wikipedia proxy certain language country english country united language need article disease enough evaluate generate traffic reasonable disease list disease incidence form file count infect b present latter plot translate diverse form format incidence map count set wikipedia scalar location article need concept english wikipedia link article select along link article biological article article low wikipedia article language article translate percent encode replace ne become language omit article merely point create article wikipedia http manually cause request target reliably map wikipedia leave traffic follow article goal selection article forecast disease incidence incidence sum day week month incidence disease frequency ignore wikipedia time temporal offset hour relatively scale ignore disease incidence series country article multiple article disease series incorporate multi article qualitative failure individual article supplementary repeat day increment forecast incidence day day count day match disease incidence day day statistical likely effective forecasting current day incidence incidence incidence day anti forecast anti model give still mechanism internet location predictive shift article yield version article yield article wikipedia correlation language two estimate model extend yield note evaluate model location test meta pearson score language compute disease language value opposite ignore article language sense favorable location apply illustrate e model fail subtle fail ratio snr wikipedia subtle exploration discover insufficient goodness complementary qualitative discuss fail evaluation failed forecasting omit forecasting brevity estimate axis traffic five wikipedia year period wikipedia series individually remain context file successful context evidence feasibility access success unite somewhat english proxy united high united couple former fail absence highlight noise source many noise article carry could carefully article rather china marginally successful successful capture baseline disease peak suggest peak model offset day dot anti forecast four successful context forecast anti forecasting context figure case significant forecast effective comprise forecast disease simply correctly vary reader interest due indirect news coverage find news coverage google trend failure cause medium disease short period day soon ill observe remove hypothesis forecasting
experience bad method par method rule full support indicate likely patient care service sign need maximum care top characterization allow quantify restrict spam breast nf svm cart rf display even severe restriction par substantially due benefit restrict careful c spam breast public reason rule intend decision nf rule select poor contain fact small restrict dataset analyze dataset p sec spam breast display run anneal step map interpretable potentially major benefit domain state national institute interpretable actually use well require medical trust decision help trust nsf classify ii kind list patient patient first traditional decision patient handle patient second receive decision decision maker disease first check serious disease etc paradigm naturally logic logic machine method produce design leave predictive directly align aim resolve problem clinical practice order rule type success list rule directly whereby classified second might patient heart disease high pressure high set stroke neither low stroke example decision construct part second result tumor tumor risk risk next margin age patient fit calibrate else age else age else risk else risk list serve dual purpose patient sort expensive sort naturally tree highest much currently use decision algorithmic manually assessment possibly score name practical decision course interpretability drive manual round gain popularity purely yield tree decision list collection logical popular inductive return example classified exhibit predictive inconsistent dedicated greedy decision monotonicity study lose enforce rather severe monotonicity often interpretable knowledge ic learn matter measure medical practice benefit risk list one care look top patient obey clause list start statistical build model help computation interpretable building discover modeling choose determine property monotonicity online goal binary represent risk rule h l mm elaborate desire user result b rule proportional rule user preference clause though letting independently distribute truncate permit monotonicity constraint diversity encourage large risk would space top list risk concentrate rule monte decision adopt l shorthand unnormalized equation note closely posterior optimize anneal namely objective subproblem find discrete space temperature simulate anneal discrete state search optimization order draw define neighbor new list length uniformly operation ex rule select uniformly draw cl rule remove mm optimize perform augmentation hasting describe variable schedule finally individual preserve enable experimental rule list medical practitioner place extremely restriction per predictive interpretability dependent substantial interpretability aim quantify performance specifically baseline publicly else else else risk else risk else else risk else risk goal predict day release binary outcome patient prior detailed aspect like status may require collect assess patient experiment choose condition list constant
meanwhile hessian expectation approximation sense reduction per measure euclidean formally strong gradient way motivate view descent negative usual one adapt descent information probability induce rise fisher fundamentally connect short quadratic euclidean make observe series divergence natural gradient space locally divergence kl divergence general argument p df obviously depend parameterization kl predictive parameterization smoothly psd metric riemannian distribution kl kl sense divergence objective discuss nice objective locally smoothly natural geodesic path riemannian towards jacobian gradient evaluate show matrix fisher give hessian even choice analytically expression neural hide unit sigmoid compute complex network hide make replace essentially gauss instead straightforward efficient one product linearize pass sufficiently finally multiply use pass fisher deep gauss newton matrix exactly least square precisely gauss gauss fisher hessian function equivalence hold fx matrix equal may actually natural important depend eqn normal normal familiar square familiar pay computation softmax layer part fed entropy softmax instead slightly close less computational linearize taylor correspond j z nice side make fisher generalize negative taking sometimes substitute standard within basic schedule rate choose heuristic nature guarantee ideally apply path distribution usually practice could negligible fundamental natural original argue taylor remaining word derivation natural appear along direction unit explanation might case natural optimally trade st vs divergence meanwhile change st approximation st st predict break fortunately equivalence know discussion serve reasonable proxy small tend curvature meanwhile update g equal natural factor important approximate accuracy approximation potential poor simply subtle gradient sensible conservative fisher develop discussion practical natural gradient eqn distribution use yield simple sometimes psd essentially already might way easy diagonal motivated free interestingly estimation despite various advantage theory exact use perhaps reason turn reasonable general section give expect due similarity eqn formula eqn turn useful approximation whereas moreover perform various concrete evidence fisher choice curvature convex rate comparable eq curvature experience learn recently form possibly modification maintain schedule rate also sophisticated combine diagonal quantity gauss newton correct fisher ultimately given accelerate improper serious due cg invariant overall compute automatically diagonal thank accurately diagonal gauss newton sgd sometimes incorrectly attribute actually eqn error network develop last couple addition diagonal base newton estimate history appear various work prove surprisingly old approach ng naturally parameter important affect characteristic eqn subtle gradient issue avoid phenomenon unlikely regret modify research eqn think conditioning curvature boundedness prevent optimizer severe quadratic k chapter add order stay radius zero approximation explanation single appropriate throughout course optimization local objective change adaptive adjustment one exponent cg justify curvature close around eqn important proving prove one scale step parameterized say whether parameterization smooth invertible examine parameterization closely elementary apply general kind rise invariance algorithm large step behave invariance parameterization direction note equal jacobian update curvature gradient w b g g inverting side give choice invertible analogous parameterization thus type curvature eqn case equivalent fisher fisher empirical except narrow case curvature hessian curvature sufficient j fx rearrange give relation hand situation occur affine practical step step update j smooth invariance automatically strong invariance whenever st words thus invariant affine newton method newton fail path fail order sufficiently invariance coincide quadratic sf direction towards riemannian optimal change taylor interpret square change add st giving interpretation expression large square negative interpretation improvement nd model scenario also argue computing tend report discuss aspect natural picture version appear offer insight contribution identification fisher gauss newton equivalence free actually natural method technique design break quadratic analyze parameterization step parameterization invariance characterization gradient possess feed forward circuit unit receive unit previous activation input denote output unit last call formally bias activity compute give monotonic document arbitrary call closely possible consist function disagreement guess use familiar encode predictive distribution could multinomial discrete gauss gauss newton
presence data occurrence contaminate proportion quickly large leading overfitte high contaminate contaminate give component covariance identifiability contaminate gaussian factor contaminate identifiability family identifiability establish package specie variable day upper height induce purpose respect evaluating cluster spurious refer bad approach model comprise bad rather illustration respect member real respect counterpart bivariate contaminate percent show contaminate analysis detect bad application factor consistent regardless perturbation section work facilitate contamination explore realize flexible paradigm contaminate elliptical automatic spurious herein start propose method reduction contaminate gaussian factor contaminate datum control latent outline variant expectation implementation illustration contaminate factor variate commonly focus elliptical widely theoretical problem tail distribution distribution elliptical represent contaminate component represent simple occurrence outlier point refer bad herein contaminate firstly ml expectation maximization stem contaminate scale illustrate proportion gaussian adopt contaminate factor elliptical error allow automatic bad contaminate mixture distribution improvement automatic bad contaminate observation parametrize parameter variant via g due singular estimate relative factor mixture contaminate factor contaminate sufficiently large relative sample cause potential estimate organize contaminate contaminate introduce mixture identifiability outline give graphic conclude robustness sake way w tail include location product focus mass contaminate contaminate typically good represent special bad via eq random cn expectation likelihood extension algorithm replace expectation maximization em ml characterization contaminate complete accordingly eq step calculation q note eq first calculation directly update perform choice algorithm maximize latter justify purpose could require proportion pre natural robust half good cm perform use package consider choice start constitute select different position strategy suggest contaminate form operational monotonicity observe great equal test assess contaminate acceleration use estimate estimate reach convergence whether acceleration give eq likelihood l l l cf converge k analysis relate financial index span daily scatter multivariate symmetry package reject commonly focus distribution contaminate gaussian statistically likelihood lr denote gaussian distribution nan bivariate result rejection graphical contour line ht ml arise explain variability variate random model variate pp analysis sensitive bad factor gaussian consider error factor analysis classical apply bad problem recall introduce contaminate contaminate contaminate w n cn term long contaminate factor value sake infinity satisfied ml estimate contaminate alternate maximization extension allow step partition cm step cycle step th contaminate accord cycle missing factor w n factor trace operator cycle q last row w ie k k
reflect meaningful occur disease nearby disease disease per similar majority reflect meaningful different significant algorithm robust choice estimate parameter also result scalable first patient efficiently vary b furthermore maintain characteristic challenge location optima guarantee scalability application hierarchy discovery patient patient nsf author nsf award author microsoft fellowship nsf award award recall categorical define notation dimensional rank therefore distance b b b second rank grouping divide recursive grouping sub sub keep track internal node neighbor merge firstly internal merged introduce path secondly merge recursive grouping merge method observable discover neighbor observation completely structure see internal sub note equal internal pair topological least node tree share define center pair path locally tree recursive grouping manner automatically surrogate structure relationship additive metric parent parent child child place single parent finally connect two internal local latent sub tree complete merging argue correctness exact work idea notion surrogate surrogate note know relate underlie latent tree show surrogate grouping procedure view consistent argue sub discover hide form maintain surrogate figure path path argue merge preserve path node merge structure correctness consistency careful hide latent tree identifiable satisfy variable neighbor node observation surrogate neighboring node surrogate along path surrogate connect observable latent surrogate correctness equivalence pair exist easy occur overlap path prove correctness latent subset small merge algorithm serial manner latent node visit induction iteration hide immediate consider implement visit contract surrogate neighborhood step merge know immediate therefore two triplet non node group three reference node parent group parent obtain alignment complete permutation merging merge align tree nod multiplication moment node zero hence parallelism method improve per tree local neighborhood process parallel size homogeneity edge tree triplet node product need triplet care triplet triplet merging consist latent lead worker parallelism require precise refer probability recovery tensor decomposition moment constant return satisfying eigenvalue moment eigenvector zero column drop svd scalar range one non entry entry probability find use computing embedding improve memory definition integrate tree follow divide learn model group iteratively operation span construction recursive grouping parameter decomposition guarantee correctly unknown low discrete implement parallel scale variable linearly variable experiment confirm health generate intuitive meaningful tree model popular hide variable markovian latent carry belief model computational expect hierarchical relationship tree object human estimation paper hierarchy disease health co disease patient patient disease identify latent consist tree exist hide observable long complexity scalable typically heuristic guarantee suffer optima easily integrate approach model simultaneously automatically learn method structure computational complexity via divide present divide applicable class discrete distribution mixture method moment tensor guarantee parallelism asynchronous aforementioned technical discover concept occurrence particular patient care task manual automate discovery clinical modal method neighborhood valuable correlation denote parent denote depicted variable variable neighborhood conditionally rest categorical use q j natural enforce parsimonious interact ia ik latent active carry triplet joint node active merge sub tree path final recovery depict divide start computation pairwise extent minimum span parallel done jointly divide merge group within group span operation algebraic alignment graphical finding underlie observed involve distance fit additive multivariate pairwise matrix dimension variable node multivariate heterogeneous latent expectation two distance along multivariate v cv cv compute svd moment distance span parallel carry independently group internal refer group sub hide grouping introduce use proceed proceed follow common expect distance l infer construct hide node multivariate discuss computation whiten huge computational latent structure sub tree iv iv accord equation node correction far learn tree neighborhood challenge combine globally consistent achieve span group possess local need align globally nature precise transition triplet find permutation transition guarantee final tree recall pair neighbor neighbor path moreover resolve neighboring node short union break node pseudo procedure reference list denote triplet parameter estimate triplet carry estimation triplet acquire issue triplet refer alignment issue two triplet triplet say node triplet group contain design triplet thus alignment correction challenging alignment solve become align recovered transition state node merge structure align tree degree parallelism identifiable tree dimension sample operate local recursive neighborhood neighborhood satisfy dl many natural tree bound hidden markov lead parallelism complexity parallelism e server red processor c couple multi capability version incorporate projection analysis discover occur patient record
future bid engine dependent behavior converge apply preliminary approach promise baseline introduce maximization call mechanism historical mechanism mathematically ad click ad engine receive bid th web user engine ad product quality bid ad common score compound predict probability ad example yahoo early ad place user pay j index utility pay I engine obtain pricing rule mechanism confusion engine direction bad symmetric nash equilibria public usually assume engine reality information ad ad period access rational response maximize reality diverse behavior highly capable place aforementioned recent year researcher try avoid assumption author relevance historical another kind statistical model learn training future either historical change framework work level optimize engine combine game call figure mathematically characterize rf g facilitate space base historical figure kind web click record historical click behavior period dependent historical bid mechanism predict future bid user click mechanism outer mechanism like emphasize fundamental characterize example cover optimal characterize equilibrium reasonable predict prediction period infinity way optimization detail introduction deal difficulty process change also propose find bid price hope rank high position receive click number click satisfactory per high bid indicate bid change depend bid bid denote bid finite bid search bid price time user issue ad click pay amount engine pricing begin period ad report change bid eq element change bid price I consider markov bid f b clear stream historical learn probability parametric switch bid eq learn bid price latter accurate former parametric paper behavior search engine compute mechanism bid discuss noticed bid change bid price period subsection genetic method optimize formally period stream engine probability proof prove stochastically positive u consider formulate f clear bid u f stream stochastically sample markov transition omit due restriction lemma theorem satisfy previous algorithm learn algorithm ease table query click parametric achieve give follow time achieve leverage learn predict period please initial bid profile sample piece ad mechanism click ad historical user sample bid accord learn period complex price formula employ method artificial intelligence handle linear introduce name improve bid mechanism may infinite mechanism predefine mechanism distance mechanism avoid greatly improve efficiency report behavior mechanism stream predefine predict stream use fitness genetic mechanism experimental generality quality task reduce impractical online response mechanism simulation widely generally collect click time click previous mechanism use mechanism remove bid click art engine keyword datum bid keyword ad mechanism day test mechanism different discount aggregate click behavior assume ad rank ad bid simulate bid three assume exactly basis take bid assume bid price strategy utility rarely assume bid uniformly multinomial fraction rest behave sbm implement baseline mechanism quality nash equilibria optimal mechanism learn historical gradient genetic individual mutation model set set show performance learn baseline mechanism axis avoid sensitive figure test stable performance well relative indicate approach theoretic furthermore well pass bad decrease statistically significant demonstrate impact effect experimental due response simply adopt classical mechanism deal optimize engine predict result effectiveness proposal plan consider factor plan comprehensive acknowledgment support edu cn microsoft com search
iteratively increasingly operator theoretical coarse grain hide coupling coarse grain spin couple system integrate physical minimize difference free physical system coarse grain preserve result map spin grain rbms rbms neuron couple visible describe restrict binary coupling leibl relative variational distribution hide like rbms less mapping thought compression expansion individual rbms stack output rbm next deep indeed rbms suggest implement extract feature organize begin variational context ise rbms deep stack rbms variational unsupervised dnn illustrate idea neighbor ise model discuss implication mapping physics block spin physical system coarse introduce describe block group term block create notice iteration physics one consider position spin spin configuration boltzmann hamiltonian partition function paper without generality typically hamiltonian spin q grain spin system end binary grain characteristic describe lattice spin picture figure two dimensional lattice block visible lattice interaction induce interaction statistical coarse grain grain grain interaction physics depend constructing depend encode interaction coarse grained coupling auxiliary visible grain entirely hamiltonian free coarse grain system usual ignore variational intuitively long physical invariant minimize grain notice transformation general move variational interpretation energy call restrict boltzmann rbms restrict rbms draw distribution binary image spin encodes ensemble handwritten digits mnist dataset rbms introduce visible unit interaction hidden parameter observe configuration visible visible reference rbm hamiltonian unit rbm unit rbm unsupervised learn rbm leibler furthermore visible datum minimize usually method rbm make dnn rbms stack layer rbm serve visible configuration visible via rbm treat activity visible ise ise tangent b transformation realize ise successive layer deep marked dot eventually attract stable point visible rbms analogous role energy object encode one scheme hamiltonian originally hamiltonian coarse grain freedom describe desire entirely language theory operator variational conditional exactly e hamiltonian language variational exactly approximation distribution work level energy literature usually make minimize divergence distinct gain detail examine carry numerically neighbor rbm dimensional describe along lattice lattice coupling neighbor lattice coupling perform calculation relationship flow coupling weak weak naturally layer dnn layer spin bottom dnn identical every spin spin hide hamiltonian coupling neighboring argument architecture implement interpret require calculation half visible couple deep neural sample critical visualization pixel depict material visualization effective field middle field move network consistent expect successive representative reconstruction numerically coarse ise lattice describe hamiltonian couple neighbor unlike ise occur set phase scale near grain mapping variational temperature periodic use rbm layer respectively see fig l layer rbm train divergence see method penalty serve encourage rbm prevent overfitte ensure interact rbm use explicitly spatial locality result dnn implement coarse scheme spin fig spin intuitive coarse critical hide fig fig reconstruction coarse grain dnn qualitatively reproduce despite compression deep successful recognition image raise question deep neural variational construct dnn examine ise self implement coarse procedure suggest implement scheme important physics quantum central dominate fix exhibit develop identify salient long interesting deep idea learn contrast often apply suggest develop idea entropy create amount open deep mapping space physical real problem give si material stack rbms variant com phase perform rbms divergence epoch momentum batch ise strength decay regularization ensure dnn reconstruction supplementary
maximize absence projection metric introduce empty link intersection common nothing inner adjacency complete arbitrary metric say geodesic subgraph single analogue one graph classical eigenvalue great one define maximize within observe link contain link sense informative cardinality principal analogously give q principal simple subgraph space principal principal ergodic sequence converge spherical principal direction informative symmetry whose fig link model project two component dot depend edge height axis sample project coincide proportion graph also year variability b variance close peak reflect tend together population depth random graph surely normality stationary adjacency graph minimize adjacency iff condition maximal iff subgraph version completely analogous proposition depth let stand start edge population therefore determine measure row eq unique finally initially provide elementary triangular arbitrary norm metric adjacency hold sake state distance distinct distance determinant odd negative eigenvalue apply setup matrix invertible determine characterization component stand adjacency one link family link link principal within otherwise objective reduce link find maximize ai link optimum proposition ergodic surely entail large enough surely entail principal eventually next analogous ergodic surely probability strong element strongly mix introduce prove instance strictly absolutely addition weakly observe center consequence mapping conclude es couple decade meanwhile focus statistical graph technique unsupervise principal well network last year behavior line stationary dynamic static growing threshold analysis characterize module develop particular technique community spectral among fit introduce dot mostly sequence parametric interesting discuss analyze network unique graph dominate label real connection financial market internet reason graph manuscript base nice measure depth analysis problem multivariate exploit determine question define several exhibit formulae calculate believe present extend important present graph link edge consider family describe consider diagonal sequence link connect path consider link transform inversion operator entry adjacency adjacency nothing correspond adjacency endowed study dynamic graph evolve discrete stand function link calculate connect graph belong expect median subset notion subset notion expect setup median usual expect network definition empirical precisely define network exist unique uniqueness central graph characterization maximum link call characterization central si j si sa la li subgraph empirical endowed graph law set element ergodic word central central homogeneity notion dispersion problem contain corresponding empirical scale finish present example enyi center empty graph complete graph intuitive maximum center introduce population version expect support notion maximize sample indeed easy hoeffding inequality enyi parameter thus arise normalize explicit present unique central easy empirical center coincide come section consist daily amplitude six location daily temperature range year month fig upper rank sensitivity extreme graph statistically correct comparison p share show construct finance graph month evolution year obtain http www central month central connect distant fig exhibit variability month temperature country correlate one analyze france united obtain depth graph robust know half depth mahalanobis different problem year metric particular define graph median precisely depth correspond population main maximize contrary require maximize fast monotonically month fig distance year exist four year graph exhibit b implication cm year month important definition measure distance setup allow statistical space write technique problem relationship first also
pattern dot percentage dot panel disjoint sub image plane panel st stand motivated introduce promise started adapt imaging modality ray would several article art attempt think framework extend work body electrical g department r department perform university security grant computer panel feature dual hierarchical use keyword patch frequency pattern keyword use sub permit discriminate suggest unsupervised topic useful wavelet transform tree topic year wavelet element wavelet powerful capture wavelet scale analyze extraction machine separate digital see image classifier seek provide north art five panel attribute di lack side combine dual wavelet patch supervise probabilistic meaningful style organize review analysis provide conclusion computer graphic color code color color often represent double double express coordinate wavelet wavelets resolution clustering transform provide characterize dependency tree complex wavelet transform decompose coefficient insensitive shift orientation local six basic orientation persistence wavelet intrinsic wavelet model mixture high smooth component successive scale narrow set divide patch patch image patch convert normalize domain complex j patch extract independently estimate iterative maximization method patch transition signature feature primitive building depend signature proportion proportion previous learn usage bag element word create text topic word weight bag proportion recognition five panel overlap image patch extraction independent style definition pt basic sub characterize keyword explain keyword style collection corpus kt generate keyword proceed patch assign quantization represent structure share dominant digit binary expansion pattern keyword style pattern sample collection collection dimensional dirichlet describe pattern proportion sub sample patch sub process choose pattern th pattern represent patch patch hide pattern assignment one need infer lda
crp respectively link ibp represent network call binary possesse attribute crp sf range logistic kernel rbf undirected meet use modelling phrase markov monte carlo inference crp number latent almost surely always store hyperparameter iteration latent variable case crp gibbs point difficulty add repeatedly sampling likelihood variable slice possible covariance covariance well assume extremely sensitive small batch fast due optima upon way interact extremely interact add batch rescale count count initialization step node interaction multiply constant factorization adjacency element either contain information modeling identify undirected edge miss network person know person could interact paper fine grain people people send message adjacency model way report log hold report test assign perfect ranking use assign independence sensible hold scheme hold representation consist count go contain count treat obviously scheme hold mean high p high probability node contain list enable extraction interaction employ publish reduce paper together count publish collect pair collect sample ensure hold interaction component crp gibbs crp node belong unseen monte integration corresponding entry gaussian dimensionality scheme hold might reason hold pair hold crp crp crp crp pair crp see crp model oppose gaussian towards closely little slightly different hold hold trace plot comprised noun word extract word identify interaction collect correlation crp hold hold negative significance hold computation seem favor gaussian model hold pair crp figure plot figure likelihood dataset hold type sample crp ht hold infer encodes pmf either chinese crp evaluate noun paper make main importantly computational continuous latent refinement enable streaming sampling scheme langevin gaussian descent might worse worth remove would put dimension dirichlet develop approach component extend phrase linear matrix also thank anonymous helpful comment concern modeling link usage indicator closely nod network model statistic exist chinese restaurant dimensionality apply social word solve observe input model predict interaction unobserve probability interact assign infer explicit describe weighted mean language word count count appear corpus unobserved word mean essential frequently bayesian mean meaning contribution since interaction expressive adjacency I contain node network interact prior representation specifically chinese crp finally refine exist crp gaussian latent noun extract corpus marginal fix put crp crp concentration parameter create unseen code class case multivariate variable covariance finally z matrix reason avoid slice put individual
operator explore execution solution specific type processor align memory boost weight input bit range rather approximation spirit weight element contrast approximation potentially could conjunction expensive operation relatively convolution multiplication field convolution layer importantly low accurately indicate parametrize exploit finally exploit tensor evaluation cnns character recognition develop provide evidence apply architecture way significantly approximation propagate great compression convolutional result addition technique weight tensor fully matrix representation permit storage describe construct good section tensor section describe convolutional maintain approximation efficient elementary linear assume direction improve keep efficient mahalanobi first propose seek coordinate whose less system let softmax image give know forward pass dirac center propagate mistake mahalanobis report use invert expensive use consider diagonal covariance approximate mahalanobis distance run tensor denote multiplication standard f convolutional efficiently approach need iterate linearly value ks diagonal singular orthogonal approximated keep ki I along convert compress even svd refer decomposition denote first svd alternatively approximate tensor decomposition operation square refer ht layer color onto color channel feature high convolution cluster sized cluster approximated tensor convolutional layer cnn dimensional particular project dimension filter combine find f xy xy basis vector constrain discuss u f approximation illustrate figure redundancy tensor approximate cluster cluster produce cluster original w contain easy gpu constrain count implement modify euclidean find sub either svd tensor could approximation gain cluster lc approximation h h many technique present degradation provide gain approximate tune train imagenet network convolutional cpu gpu k image present show gain overhead pass use imagenet number forward propagation spend convolutional layer supplementary layer approximation easily several cpu achieve cpu implement library intel mkl intel comparable matlab cpu speedup convolution baseline comparison code run find difficult gain base arithmetic material gain cnn different make specific however regardless implementation detail reduce often convolutional output channel approximate layer approximation span figure illustrate filter component point colored belong color show filter approximation project version st filter color channel performance begin point gain result cpu gpu corresponding colors cpu implementation relative baseline gpu color layer channel channel filter explore input outer gpu configurations cpu approximation gpu outer product drop gpu approximation procedure follow first layer tune convolutional weight tuning continue apply convolutional layer color decomposition pass keep convolutional layer manner overall great layer provide comprehensive storage central concern memory file neural product require majority describe use hyperparameter conv conv outer h nk nk bottleneck operation layer factor negligible reduce memory layer layer vast majority layer mobile convolutional technique evaluation quantization work hence use aid post projection learnable suggest generalization potential rank approximation approximate layer appear well table forward cnn architecture explore spend convolutional majority spend per fraction conv conv conv fc fc fc conv conv conv conv fc fc fc softmax theoretically achievable target
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rough laplacian see example embed cloud construct curvature serve background recall coarse space operator space operator intend laplace construct point serve give du du define curvature via laplace operator agree order classical du involve derivative agree diagonal analogy introduce curvature depend embed around operator life without take du thought laplacian eq parameter family du appropriately integrable function simplify eq fashion iterate du differ assume atomic call tt course du bilinear iterate e du scale notion coarse curvature empirical coarse scale application learn embed function volume metric induce adopt operator take ambient geometry priori knowledge geodesic approximation geodesic distance scale ambient limit tend sized geodesic analysis intrinsic geometry embed riemannian induce embed metric ambient geometry hypothesis curvature manifold unit volume suppose locally locally scale surely curvature simplify presentation state simple case relax distribution smooth everywhere space smooth closed theorem idea point recover sample force law choice datum size surely sized replace mention uniformly recover curvature smooth computation adapt q bilinear form rest l result leave give coarse curvature general measure converge curvature riemannian manifold curvature thought extension curvature smooth obtain empirical scale curvature assume fit recently fit sample another converse problem development point manifold even surface implicit surface use cauchy formula author like wu author express thank tool empirical process uniform law operator du iterate du operator iterate convergence large hoeffding lemma borel algebra borel involve trivial interaction hoeffding law uniform separable replace paper separable deal totally cover let totally particular finite class obtain choose hoeffding imply ambient lipschitz respect ambient function mean ambient almost weak function fix denote subsection function reduce number define ambient particular lipschitz fix class corollary fashion eq say relate ambient distance ambient smooth sufficiently smooth unique orthogonal projection embed function net ambient particular eq du sample inequality eq convenient normalization form q enough follow variable q use lead class positive notation function du letting class lemma bind simplify fix eq eq demonstrate decay rate quantity sure borel proof introduce measure embed suppose I let fix plugging expression thus borel give almost sure sure reason notation sequel function plug dominant exponential clear eq recall tf converge choose lemma eq introduce require almost lipschitz formula corollary corollary l eq increase view corollary corollary smooth space ambient uniformly take respect net grow bad polynomially q
label ref expert section generating compare bottom reduce good label ii perform top iii far risk notice decide good vote galaxy galaxy summary far describe galaxy perform ii pick label majority vote fitting compare minimize yield use use emphasize hence get model iii vertical line indicate solid bottom horizontal error classifier vertical attained dash accord respectively line indicate minimum attain solid estimate vertical indicate bottom bold number stand ccccc ii iii iii iii dealing build performance traditional sparsity avoid prediction error tune introduction surrogate variable big improvement compare perform former reason happen expectation sensible majority accurate provide reasonably many estimate lead well derive majority deal majority label derive latent perform procedure always use true majority base recover propose advantage latent naturally allow expert classify instance easily new expert tuning different achieve expense introduce introduce new logit dependency linear introduce useful help would show might use would find unknown loss function cost ann lee comment would member provide annotation partially support noted maximization first newton maximization rewrite regularize relate maximization measure performance first class finally assumption minimizer minimizer minimizing result find relate empirical arithmetic mean q iv cauchy inequality follow conclusion mean close minimizing vc v wish w qr v vc putting theorem get expensive assign highly expert undesirable situation train spam patient although expensive amazon sample unit classify many reasonably expert people medical problem desirable detect expert information adequate train crowdsourcing method predict though predict deal expert majority vote scheme label suboptimal trying find task train expert incorrect essentially algorithm amount method consist probabilistic unobserved usually root emphasis way usual observe label available literature term crowdsource tool try suffer crowdsource parameter sample increase substantially introduce parsimonious potentially bayesian shrinkage therefore hyperparameter valuable example new solution specify selection induce error label th unit feature attribute expert label contain summary minimize one mistake calculate error empirical difficulty closely depend generate score sample use validate compose value different contain vote input training obtain risk risk new misclassifie shorthand risk I give pick flip coin additional perform traditionally define em algorithm introduce improve prediction account role find introduce noisy optimum solve lead maximum posteriori correspond em iterate respect calculate plug solve regression map accord guarantee converge reason due identifiability consequently expert discuss em select agree next model tuning explore two completely response response calculate datum uci repository appropriate complement vote expert probabilitie misclassification misclassification follow misclassification vote describe experiment present expert response vote correct expert vote would probability correspond subset expert majority expert experiment compare em denote sparsity available comparison
car nine three learn part failure datum sensor datum service shape method follow sensor part associated actual part fail car go service fail car soon service record car occur service car use find number failure exception skip step rather failure form present assume network predict number failure first name scale shape learn failure due insufficient failure part fig approach part case learn result improvement accuracy inspire life similarly heart failure network present application cost part failure predict paper information like diagnostic historical future failure approach fusion diverse enhance estimate learn source rate predict future failure test failure service predict failure failure good scenario conclude failure sensor part failure plan enhance incorporate currently predict failure part next suggest optimize real world dataset com com multi attempt hard fail market lead company financial cost item failure failure failure play significant study learn rate parameter however available fused failure estimate failure cost estimate failure company service available fuse past part thereby improve data multiple problem occurrence service part failure rate sale change product conditional multiple source result estimate optimum claim inspire life give immediate next formal paper bayesian bayesian start learn later cost summarize brief discussion section overview predict code every service regular service routine failure observe service service part observe equip central offline build datum available order improve contrast past capture network dependency failure accuracy traditionally estimation component however divide sub level wise failure enhance occur take service occur failure observe service service failure part occurrence much early node node distribution define network dependency approach consider duration product index service source product cycle hour drive associated part fail part may part associate diagnostic associate record index fail first fail cycle fail contain cycle part contain contain cycle failure diagnostic service record interval present fail occur index part fail time contain cycle occur observe learn bayesian direct describe g encode variable rule decompose parent mcmc etc method e bayesian indicator part every dependency occur roughly failure actual failure follow goal learn combine theory present bayesian service record learn dependency cycle fail cycle variable capital small letter take parameter r kp take dependency mcmc hasting expect failure detail failure rate network follow shape failure part number failure car interval fail time affect failure analysis failure incorporate service bayesian network explain failure occurrence time observe time approach expect failure failure datum service service record failure three associate present failure analysis network variable predict failure failure record explain explain step explain capture dependency part bayesian consider failure part fail
gaussian problem link believe investigation ac uk cox covariate markov supervise case probability multiclass generalization supervise method classical nonparametric cut give supervise supervise despite simplicity specification still active explain part vast cox originally cox receive intelligence scalable perhaps covariate energy strong connection familiar boltzmann model prescribe field possible specify model avoid overfitte e mark cox spatial latent intensity model process relation near interesting often nonparametric intrinsic approximate close fairly parameter training validation could develop markov field already work build excellent contribution relate cox process supervise include time implement test mention bias logistic classification follow sentence fields cox relation perspective random min cut shall requirement field prescribe work classification despite discovery supervised method discuss additional connection commonly hard elaborate general link section propose semi min cut exploit supervise lie point mark pair constitute mark use parameterized intensity integrable bound borel set intensity borel borel borel cox random intensity point cox process process log cox exposition restriction paper measure surely condition valid commonly kernel zero square length cox show ht intensity cox square exponential unit describe cox density intuitive interpretation density kx x exist differ quantify occurrence quantify occurrence possibility elsewhere density product factorial moment lebesgue subject meet shall convenience density singular original although model model log cox mean superposition cox cox term superposition cox superposition cox cox population spatial whose speaking apply may superposition consider fix borel observe I observation product cox superposition density follow would limit construction equation proportional product product divide superposition interpretation independent contain point precisely borel partly result obtain cox example marginalization fulfil process knowledge affect desirable characteristic sense interference fulfil treat limit process standard interference meaningful supervise exist material new behaviour predict measure covariate distribution component softmax predictive datum new behaviour normalize function point give total wish produce similarity softmax choose distribution cross validation instance negative kernel kernel regime frequentist equally regime guarantee field briefly exposition augment generalization denote delta type fully model energy covariate superposition constant model differ process view simplify assumption ise ise model repeatedly apply graph min cut guarantee local optimum thesis energy posteriori cox semi problem condition q simply relevant know exclusive logical substitute verify condition problem subset theorem pairwise submodular solver generally fast solver former extra specific multiplying mean change increase likely category stationarity assumption category equation prior perform software matlab toolbox min paper comparison support machine svm harmonic supervise original demonstrate illustrative dataset generate two circle clearly centre perform use mean exponential see give sensible experiment min cut describe recover make class perform circle min semi six commonly diabetes come come brain interface single category movement either leave restrict mnist handwritten digit highly dataset number multiclass dimensional covariate diabetes aid paper literature randomly classifier length first randomly mnist set validation take intel ghz gb ram mnist use dataset classification variational inference logistic function optima random initialization take default software wide variety aware surprising significance result technical provide multiclass diabetes lc nn ff net rbf conv exclude diabetes free lag publish classical perhaps surprising training dominate term sum lie manifold relevance supervise covariate complex feed achieve neural achieve low require deal scope bias relax large leverage exist perform svm semi et al compare dataset synthetic et classification possible visualize geometric advantage semi supervise supervise give reasonable take digit set class call mnist first covariate consist measurement pass homogeneous flow two varied label randomized test existence trivial bipartite near euclidean harmonic length cross free validation harmonic ie length fold take second experiment counterpart supervise substantial majority perform similarly exception small double harmonic rate near separate surface show exploit advantage semi
last four fit gram fit indicate include aic law aic none differ demonstrate difference power law follow gram also lie ccccc gram gram gram gram gram gram gram gram suggest cutoff law assess superiority test suggest fit pl significance none power seem belong distribution dataset language fit well even mirror cutoff behavior raise question answer linguistic simply effect language whether answer make language profile repeat size profile figure suggest genetic profile sample compute sample curve except seem monotonically gram gram family trend result gram family unit classification power law law gram profile law follow power law cutoff gram find gram corpus internet specie law seem also claim law law word density normalize power law cx take member call point nlp cl predict small law language form family list language know language family tree classification indicate family language observe plot language family seem log deviation strict straight line goodness classification propose law language linguistic feature value response commonly package parameter package apart power law model author estimate significance superiority table recent apply linguistic color template meta series test show cutoff describe law ht name probability law pl exponential cutoff min testing law hypothesis law rank likelihood preference power significance absolute computed goodness low half world language investigate world language database word language word list least include experiment dataset represent language list name language family word include know mark word consist precede symbol combine click click stress gram profile family show quite language merge recall word list consecutive extract total gram profile gram profile family right gram power power fit gram gram gram gram value language type leave gram macro european na improve kind familiar word
section subsequently ability exploit improve domain four step represent hilbert reflect last thereby dynamic order handle real distribution work comprehensive algebra rkhs rkhs image I lose operation use term clear context take product z n empirical converge consist embedding kernel would require infinite method require embed evolve generalize situation output provide summary search lf f reproduce operator contain span operator product g g inner operator completeness regression order distribution variable solve follow functional st derivation consistent converge infinity step necessary component available simply learn result approximate combination observe distribution eq computed set mean sn potentially particular lie outside hull observe estimate guarantee lie available value form suitably original last rkhs map symbol compute pre act proxy inner rkhs embed drop replacement hand sample empirical similarly know consequently estimate fulfil small sample appear purpose margin one prefer weight variant construct sequence iterative greedy interpretation show embed drop depend concrete whether practice require pre second execute n sn sn size tn z treat desirable put emphasis achieve square problem belief early reliable contain ordinary square concrete expression remain structural modification curve learn sample body classical probabilistic technique aim form transition difference approach become particle learn learn mean covariance conditional infer nature subsequent minimal scenario car decade correspondence depict exist aim predict people situation trajectorie video possible separate probability individual report real dynamic highlight methodology additionally limit rkh embed gaussian train set interpret surprising combination allow show sample indicate autoregressive justified predict step distribution support location bottom illustrate nine new input mode predict quality indicate situation clear whether model gaussian call three step nine observe blue despite concentrated remove quantitative table correspond one leibler divergence besides last much observe distance set true suggest exactly residual compare pt baseline rkh see measure kl show segment real video sequence semantic source represent temporal interest video one long create segment sample detect interest video show per segment vary segment video vary segment movie measure actual segment table result choice z category baseline last segment merge segment global video close true video baseline tie sign multi correction except sequence vary application look drift step time choice tackle unlabeled target prediction set spam filter stop collect setup let sequence e kx joint learn adapt classifier look correctly available one map induce numerically surrogate therefore efficiently sample would goal classify right replace lead use weight ti ti cb play rest hinge correspond predict package demonstrate usefulness training classifier set consist car year car come source year decade give goal learn perform task order source source improve though affect visual
hypothesis say exactly nan hypothesis briefly suggest li specific explicit suggest minor rigorous detailed case augmentation modify high require argument order cx cx xx n k k event calculation joint partial exponential x cx xu I provide throughout integrate index density final approximation statistic modification statistic must implicit modify obtain modify integral subtract integral precede proof simplify appear complex index subscript dominate overall unable unable recently cite term four statistic ii iii statistic iv number repetition conservative derivation involve significance threshold decrease statistic hypothesis level application section become call question advantage hc hc hc hc hc hc diverse aspect mention approximation extreme attribute adapt original involve brownian empirical exhibit denote brownian bridge play derivation slowly enough rhs probability maxima long limit convenient independent b xt relationship wiener relationship define obtain classical slow apply seem reasonable correct modified inversion give suggest reasonably perform inversion frequently apply seem successfully four equivalent sense threshold datum shift nan mixture reduce suppose function value transformation notice directly transform nk globally argument concave besides conditionally nk k recursion level modify exactly term indice smaller high power poorly easy manuscript alternative numerical fast suitable hc hc hc hc hc hc hc hc hc discuss mixture offer estimating small magnitude suffice rarely side sided distribution statistic list threshold mixture parameter two sided repetition simulate number except hc statistic statistic modify power small significance level use statistic comparison hc recommend even consider give confidence choose suggest closely observe false discovery similar behave perhaps statistic subsection mixture version describe description b nan hypothesis omit suggest simulation compare bound calculate precede consider repeat configuration provide table low column true hc compare suggest far zhang method contain interpret contain li detect signal make method overall significance level test somewhat single expect frequency intermediate threshold give statistic correct exceed l approximation higher sufficiently practice comparison power statistic statistic goodness mix statistic mixing difference percent advantage significance level sensitive level intermediate power statistic definition capacity statistic value statistic relate goodness focused design detect excess small elegant large theory well receive focus center focus focus small cc cx give simulation behave third summation level appropriate threshold power interesting see whether context band suggest similar heuristic proof seem impose lower different somewhat statistic statistic remark mention modify slightly original modify although two simplify write proof interest exclude hypothesis similar proportional rejection unnecessary directly rejection region mean difficulty primary interest impose condition statistic unclear rejection correspond curves cx convex continuously increase n f nx xx desire converge go rhs expression event n c kf kf nc k n j n nc independence claim decrease enough suffice decrease inequality kn q uniformly achieve mf dy nf n dy dy b ki version rhs recall last k notation independent variable p r n rr mf n c first k k n n cx ny n nc rhs decay slowly n mf argument ny ny max c cx proposition upper nonnegative exist increase denote dy dy fm fm dy dy dy bm dy bm desire f ny k k om nc nc mf n om mf dy n dy dy proposition inequality nc n k ny k n remainder rhs equation rhs tend proposition high statistic n k binomial replace n thank associate reading suggestion improve style claim lemma e compare statistic approximations numerical show broad sample size higher detect sparse suggest confidence false whether nan high number consistency hypothesis statistic uniform hc modify term include recommend value reject global excess small value denote global hypothesis one false although confident study goodness function result adapt often cite poor value often alternative
face study control object time stimulus duration rest interval repetition meaningful face ds public format include voxel brain template without process head movement sensor etc study six subject include fmri brain area thresholde brain near roughly term fmri point voxel fmri acquisition smoothing order enhance voxel different spatial mm resolution mm voxel perform average version non version variant bss via ica section ica fmri realization generate specification describe realization additional verification slightly significant difference component dimension present ica blue represent eight true curve represent recover source figure ica spatially proper slice error visible ideal recover ideal leave corner external recover eight simulate fmri spatially brain slice leave box activation area pre external box illustrate reconstruct fmri versus ica complete ica component use reconstruction correspond sort component small rmse subsequently set start one increase add reconstruction plot reconstruction perfect ica fmri recover exactly number reconstruction versus exactly eight practically zero real fmri ds single employ full fmri voxel evolve brain raw background flat blue slice ds toolbox matlab illustrate ica course plot map experimental protocol employ dataset successfully recover external shaped see toolbox red actual human experimental employ recover run dataset converge fail component course evident third top match component leave match fmri noise ds ica component run respect reconstruction component illustrate rmse ica represent rmse final maximum low rmse slope connect signal eight match source component shaped course extremely experimental protocol brain construct brain dataset ds brain voxel slice inherent clear ica work satisfactory clear recover pre e real fmri never word perfect possible retrieve upper limit ds ds dataset smooth hence fine estimation realistic fmri consistency figure use intrinsic value dataset realization less signal original use sigmoid log clear selection linear fmri table estimation table mean range clear smoothed variant intrinsic span voxel furthermore expect consistent voxel enhance content activity area detail activation smoothing option carefully select specification sensitivity comment non smoothed variant space span voxel becomes apply plot use smoothed sm method reliable box reliable fmri mention focus level brain complex task minimum actual cpu entire perform fmri fmri dataset ds simple recognition expect distinct activate brain volume brain rather response inherently cpu core process simultaneously digital ica reconstruction plot human brain concern strict brain fmri retrieve marginal impact model short task correspond g correctly drive especially bss ica major combination parametric dimensionality recovery develop brain brain usa evaluation activity functional level voxel million several thousand art million parallelism human project purely fmri brain activity bss fmri try core cognitive task fmri smoothed variant real fmri complex brain task response theory equivalent brain cognitive structure require scale hence state feature real human brain assertion parallelism project acknowledgment functional analyze brain activity aspect human cognitive fmri human purely drive processing specifically fmri bss real fmri combination component brain process run visual although level process brain advanced efficient signal machine correspond total body yet neuron cm analyze especially cognitive relation simulate structure neuron digital infeasible functional fmri powerful analyze activity commonly bold contrast translate detect flow activate brain achieve exploit property order execute specialized brain properly detect activation constitute brain voxel constitute elementary spatial act resolution fmri voxel component temporal spatial potential intensive approach actual brain signal eeg cognitive task brain project brain effort project like million grid extremely power focus neural high cognitive practice hardware necessary fully simulate artificial equivalent turn machine still application artificial visual capability focus aspect human brain specifically cognitive task try cpu active brain volume perform fmri include fmri brain signal processing fmri study fmri define understand source proper reduction fmri briefly describe estimation fmri space briefly describe approach blind signal fmri study include experiment method dataset early fmri activation scheme paradigm subject specific color subject paradigm series input stimulus need primary area relevant task etc follow external setup previously activation essentially source fmri signal acquire fmri spatial temporal result voxel voxel x second time voxel brain snapshot practice actual brain brain area remain typical protocol involve subject exclude subject fmri create complex demand processing activate rich time low pass temporal fmri activate neuron flow series signal brain fmri spatially vary slightly subject traditional regression like model glm approximation since construct transformation shift derivative universal difficult locally due physical property various head fmri include task fmri isolate appear super spatial localization task relate fmri multiplication matrix spatial relate brain put brain fmri factorize map collect contain along spatial glm permutation shift regressor glm specific relate external instead component ica context blind bss glm ica dominate blind respectively fmri analysis volume voxel signal demand memory resource identification universal multiple experiment different subject multiply run case identify fmri activate specific property clear fmri identification lack specification background hence define standard glm make bss ica dictionary dl compressive cs increase interest ica drive fmri notably dl fmri analysis variation fmri deal complexity bss task reduce voxel consideration adjacent neuron correlate network scale spatio bold consider scan redundant voxel task accuracy identify inherent data fmri voxel sense consider however processing step conduct choose purely since currently study regard quality spatio statistical dependency essentially fmri dimensionality voxel retain voxel conduct information voxel inherent property retain external like recover bss signal formulate task assertion signal decay variance external reconstruction become separable bss recover e correspond fully reconstruct fmri brain activity drive fmri year quantitative randomness metric statistical various eeg etc financial time signal characterize texture analyze modality provide distinction e space algebraic apply dimension use evaluation useful regard redundancy dataset order intrinsic specifically quantitative linearity dimension mean dimension discriminative separately e mean parametric task analysis successfully previous prove valuable compare dataset extract feature qualitative clinical commonly calculate calculate closure cluster group various sample size approximate exponent way accommodate thousand however essentially thousand instead calculate grid equal sample cell occurrence count correlation ideally algorithm calculate intrinsic would allow totally uncorrelated discriminative specific set study apply set qualitative characteristic expert constructed employ sample available box order calculate slope plot sigmoid fitting parametric sigmoid identify axis identify scale specifically affect sigmoid linear central curvature fitness sigmoid assume uniform percentage bind low plot slope calculated reason factor fitness calculation parametric function axis range window term range completely triangular window rectangular fitness uniformly entire range triangular calculate fitness calculation window factor slope compute blind source bss fmri year consist different source activity bss temporal spatial exact bss ica identify gaussian source separate essentially reconstruct original signal combination discuss spatial brain activity corresponding time fmri ica dimension voxel produce either spatial conduct bss fmri spatial spatial accurate useful clinical fmri common ica ica widely fmri recent identify advantage brain additionally bss identify dl since ica dl assume specific source minimal ica fmri constraint maximum error ica error approach dl dictionary expect bss fmri fmri dataset ica intrinsic signal pre define fmri ica approximate quantitative track reconstruction change factorization verification estimate dataset differ track valuable tool analysis fmri datasets investigation fmri fmri fmri verify intrinsic activity task recognition fmri generator toolbox create fmri main source statistical underlie source component include super sub
expectation front logarithm may follow make may expectation entropy setting compute backward density emission distribution mixture maximization hmm multiple observation extension sequence consider two long sequence write simply dependent reasoning employ sum nd term way become reduction statistical arise hope new insight perhaps help discover hmms lagrange multiplier universal introduction basic hide stationary probability emission observation desirable produce maximization datum em find parameter eq rewrite term term
imagenet help improve cnn visualization generate varied synthetic cnn year feedforward networks cnns cnns investigate reason potential optimum large study generative modeling cnn define image category cnn final give image reference study differ non pre stochastic seek log approximate keep implicit image generative fundamentally discriminative share computational architecture usual drive discriminative criterion gradient require explain imagenet training help improve cnns generative explicit parametric white noise draw result distribution accomplish hamiltonian top deconvolution directly draw cnn extra meaningful varied synthetic deep imagenet energy product boltzmann algorithms deep relationship generative extensively cnns visualization image present deconvolution employ image define hmc category reference category score unknown yx w yx qx normalizing function flexible want reasonably easy generative underlie intercept train notational shall intercept already uniqueness mention set label maximize maximize log estimate class category perceptron top layer throughout accord expectation approximated importance set attempt treat weight gradient gradient yet difference provide drive easy adjust predict reproduce parametric generative especially pre current importance skewed effective updating lead toward importance skew start indicate discriminative generative appear expensive batch approximation specifically seek via calculation discriminative gradient induce specifically layer calculation replace sample form yx generative variable rule calculation exactly bring use f layer make ix yx imagenet care gaussian noise correspond hamiltonian specifically write physics context position function hamiltonian dynamic momentum denote physical hmc random evolve hamiltonian include computation deconvolution max un derivative visualization sequence three study discriminative gradient generative discriminative experiment identical experiment benchmark mnist handwritten digit imagenet natural study generative pre dataset utilize accuracy utilize deep distortion baseline set perform base weight max stage stop epoch discriminative start base rate table generative rate cm cm imagenet utilize testing train set stochastic descent batch decay momentum stage stop discriminative start base rate center corner training cm fast discriminative achieve mnist imagenet show pre discriminative improve update network accord toward generative gradient discriminative convolutional connect layer number par visualize train mnist visualize first visualize final drop avoid unnecessary visualization initialize hmc far visualize intermediate layer final fc visualize intermediate layer conv generative visualization
minimization least bellman direct function storage device possible keep due time final policy evaluate path discretize information record percent percent sample randomness state transition realization policy method implement policy use average percent optimal deviation budget simulate sequentially choose illustrate significantly direct percent least problem direct policy robust direct basis increase addition choose domain policy significant increase good parameter depict storage device price price storage htb reduce small post decision value three approximation three resource resource alone decision well overall although state advantageous simplify continuous problem available c discretization c number load wind wind ba ba full full instrumental dimension visualize figure policy price particular sample tend device high section promise scalable unknown htb resource level line variant bellman least bellman minimization bellman instrumental bellman bellman error instrumental establish bellman instrumental bellman method result strategy fundamentally approximate evaluate numerically control storage evaluate bellman instrumental appear basic optimal produce percent perform percent ideally suited call produce pure give relatively quadratic bit surprising exploration research direct challenge derivative limitation may need instrumental technique deal explanatory consistent estimate explanatory respectively probably widely technique since error easy least equation equation positively square inconsistent instrumental discuss instrumental follow noise ij unlike iv method equation e instrumental ij z indicate column il l il assumption trivially assumption across method instrumental variable estimator uniquely rank consistency instrumental give limit limit true covariance corollary definition recognize discrete curse know flat representation approximate learn hope problem function use allow approximate version algorithmic policy somewhat find success computer science community effort call state action pair value rigorous table representation scale problem science result establish architecture comparison perhaps approximate attract avoid action enjoy strong convergence although perfectly satisfied benchmark balance power load time purely suit realistic minor provide rigorous benchmark assessment produce algorithmic software benchmark http www edu focus architecture value receive attention literature steady powerful building temporal introduce instrumental variable overcome reveal modify bellman project architecture refer bellman chapter equivalent instrumental benchmark insight strategy choice importance example benefit storage wind policy gain assume load load price investigate discuss incorporate compressed air energy storage wind generation market wind storage horizon market address formulate market price wind study potential storage device thorough grow reader reference therein paper address close backward application create variation estimate bellman instrumental bellman paper hybrid approximate combine instrumental bellman instrumental bellman error hybrid instrumental bellman able time policy typically calculate basic instrumental variable search instrumental yet algorithmic fall direct policy outperform policy since bellman error overview base bellman minimization search gradient management stochastic explain performance dynamic policy investigate compare problem conclude rely bellman discount factor expectation change state transition throughout use convention index computing expect version bellman thorough discussion refer state pre randomness explain post bellman decision state inner maximization problem low dimension application variable multidimensional bellman field research widely column f post fix policy fix exist architecture weight post record determine decision record observation update algorithm post post decision cs get observation fix expect cs matrix bellman error minimize bellman bellman include bellman instrumental bellman discussion linear variant td least square temporal approximate tend td instrumental monte simulation cs estimation true value lie increase technical fulfil architecture approximate action geometric interpretation bellman equation least iteration q minimize bellman bellman span function bellman fix bellman bellman operator nice discussion address condition policy evaluation subproblem bellman function linear project norm state visit trajectory follow policy bias rarely visit another disadvantage policy important keep summarize algorithm bellman instrumental case subsection address bellman instrumental bellman subsection use section bellman bellman error instrumental alternative bellman find consider policy parameterize post function contrast value search vector policy solve follow policy challenging grow classic algorithm use sequentially policy simulate dimension optimize nonconvex easily computable derivative introduce fairly experiment observation policy treat objective combine quantifie much maximum get noisy value formally sigma updated condition normally value description implementation q maximize produce within converge asymptotically involve demand storage flow vector amount stand wind demand assume except refer storage grid htb demand demand wind device flow send storage store future storage wind upon capacity indicate capacity constant device lead maximum storage device must equation replace similarly ensure device allow demand implement storage fraction device full stationary capacity constraint note dimension dimension minus goal find policy ergodic horizon planning maximize accumulate discount absence load device price step uncertainty wind demand price subsection wind air square coefficient wind speed velocity wind square wind evolve ar model min wind suggest wind average price figure begin heavy tailed price adopt constant parameter deterministic periodic hour week month price model price jump jump interval addition jump size normal nonzero jump count time time jump occur divide jump jump direction magnitude jump consider jump parameter price outline demand demand highly upon figure peak pm greatly day week month forecast end incorporate customer make expert adopt indicate hour week month component hour week calculate load hour week load evolve degree load energy load htb programming variable post decision fraction storage device wind energy electrical grid td tp maximize discount policy may modify policy
recommend user compare one similarity dimension concept location social dimension sim dimension similarity dimension use least choose q idea rather situation experience curve evaluate user show memory mind document distance time memory environment weight user forget related situation reduce document risk explore situation exploitation optimal document multiply allow exploration situation metric fix sim case add compose preference sim situation current situation use algorithm recommend regard click recommend ap one day mobile priori horizontal axis month fa fa ts fa ts ts display click regard different fig fa ts effectively well term average click user improve explain consideration outperform document consider table significantly mean exp impact fa ts well precision without recommend propose document regard demonstrate significantly increase performance moreover conclusion recommendation recommendation follow researcher recently start user aware exploitation propose user bandit introduce name aware sampling fa ts document intensive evaluation result exploitation behaviour mobile make collection recommender identify recommend relate situation friend document recommend document recommend document short recommendation need balance actually great exp perform ts drawback strength amount experience recommend long document name aware thompson balance adaptively trade situation ts strategy explore situation paper organize follow related evaluation illustrate section conclude refer dimension dedicate armed bandit rs contextual recommendation document analyse ts contextual balance document recommendation author arm arm need continuously explore armed content author consider music recommendation long risk criterion take account additional standard deviation rs recommendation strategy recommender study recommendation aware thompson propose exploit consider curve recommendation situation home user office focus introduce enable human vector nc represent accord situation concept location user activity attribute preference click document failure spend document recommend situation correspond user recommender system propose user recommend exploitation orient important like use home situation add cv c cv cv cv document game choose reward thompson
detailed generalization perceptron online using provide theoretical accumulate loss major ndcg map perceptron different ranking measure ndcg map numerous seminal survey ndcg induce ndcg make subsequent write emphasize rank easy eq reader note receive learn whether induce induced loss eq also convex algorithm require r relevance document since predict rank score sort perceptron algorithm initialize predict receive def else provide measure perceptron euclidean unless state let control norm subgradient feature sec subgradient set main propose though perceptron receive exist meaningful relevance ndcg induce perceptron perceptron conclusion let linear parameterize document correctly margin holds accumulate ndcg instance list document definition max I max ndcg relevance batch set learn solving analyze learn minimize main generalization ranking take close look actually dimensional parameterization sec rank sort via scoring map parameterize form parameterization scoring actually map generalization bind class parameterization linear rank surrogate notation relevance surrogate generalization bind restriction sample sample mean duality w w b immediately w point w constant lipschitz continuity w comparable exist literature surrogate lipschitz r norm generalization inherently technique force thereby avoid price pay surrogate generalization surrogate rademacher surrogate general surrogate show family condition ndcg map family ndcg set though parameterization scoring correct parameterization invariance dimension property mean independent list formally score rank permutation invariance permutation permutation invariant dimension full parameterization permutation translate ac pp preserve create permutation column except column repeat w position hence column match column match first matrix rank check satisfy surrogate rank represent surrogate relevant calculation margin surrogate design relevance conjunction calculation show surrogate lipschitz lipschitz actually bound margin perceptron gradient gradient calculate make surrogate realization prediction surrogate theoretically relevance distinct space relevance space remove simple minimize design relevance gradient hence provide like algorithm guarantee induce measure ndcg map loss suitable provide surrogate analysis role generalization surrogate introduce modify perceptron cumulative ndcg loss generalization lipschitz surrogate third batch imply good possibly kernel tackle preliminary case scope subsequent acknowledge nsf pointing theorem unless understand cr formulate crcr g sort relevance relevance relevant list incur irrelevant document place document sort irrelevant document score among map irrelevant document highest irrelevant relevant map case need upper bind v r relevant document irrelevant place map bound r r likewise repeat logic equality calculation thus however j j ks iff document place take ir ir j l l jj I I li nd last thus document great minimum definition r x max st nd x tt tw z max tw proposition fix immediately tu originally take carefully derive definition minimizer eq theorem lipschitz rw rw er plug back get theorem thus relation x thus r constant eq index query relevance w sum pair query eq index index index correct query choose om surrogate score come normalize guarantee first truncate ndcg gr mr ii important property depend permutation thus document relevance document rank sort r gr ndcg positive weighted mean less relevance keep mind property come ndcg hence property directly note thm proposition ranking supervision relevance score contribution rank batch surrogate generalization independent object query propose rank surrogate surrogate obtain large margin surrogate structure cumulative ndcg induce also novel surrogate satisfy generalization supervise frequently rank number query relevance rank learn hope document respective relevance level rank list performance ndcg map performance measure convex reason exist loss optimize broadly three predict relevance document document take binary entire document associate take surrogate minimize major usually rank use exist base publicly moreover conjunction surrogate question algorithm provable guarantee remain large surrogate surrogate use surrogate supervise exist popular margin surrogate rank literature standard supervision supervision relevance map good lead define surrogate since ranking arbitrarily investigate develop classification large online perceptron surrogate special allow loss perceptron develop different perceptron algorithm extend set loss measure ndcg follow modify rank rank surrogate vector give ndcg unlike yield varied purpose sample unknown iw mr prove appendix definition understand similar induce relevant fail less relevant modification correspond thus loss document rather weight rank near must loss penalize much perfect penalize weight vector document entry weight thus even loss intuitive truly emphasize surrogate structure framework surrogate margin extension framework
memory requirement form prox multiple phase accelerate discuss due section projection multiplication step cost cost discuss role feature transform solver several mapping suggest literature sparse kernel memory treat box highly modular kernel sparse dense input kernel map vector appear solely block treat ij know scheme map split like additional entire transform generating construct independently monte result row primary map solver family shift mapping appropriately cosine entry transform group collect inside operation algebra store costly avoid implicit generator part need map fourier feature laplace mainly focus fouri experimental recognition digit comprise test comprise instance derive intensity classification comprise comprise report distribute environment core per cloud distribute resource hinge fourier see store dense run processor size strong notion parallelization come tradeoff time admit execution little roughly accommodate memory requirement long input model explicitly tb multiplication dominate run test solution communication cost gb plan ccccc avg percentage communication transform step step solver solver solver first solver solver first widely solver compute parallel gram incomplete cholesky use accelerate primal interior version binary create version divide class comparison core solve large speed make computationally though fast solver rank compute locally cccc testing time cccc time feature classification reduce pass dataset core solver hybrid parallel capability lose default running include transform solver attain accuracy comparison solver method solver memory demand contrast never form entire resolve scalability challenge conjunction optimization lead involve implicit dataset handle splitting approach performance term scalability implementation various modular stochastic update theorem lemma claim problem remark definition algorithm section google research research high optimization randomization propose kernel randomization variant multiplier carefully memory parallelism modern performance support enable loss regularization dense sparse library keyword method input training process loss truth convex prevent tradeoff control enable unseen test large big impose structural smoothness practitioner strong constraint theoretically big tends quickly consequence practitioner turn million estimate often carefully design loss regularizer trend recent success intersection numerical indeed scalable implementation play rapid massive well truly big effective constitute mathematically elegant dimensional linear parametric span series testing model central kernel define domain define turn procedure directly poorly train cubic parallelization poor pose barrier acquire scalable algorithmic environment algorithmic distribute describe estimate example unify approach block admm much design highly need proximal regularizer admm kernel environment well indicate scalable capable return library vector machine highly favorable acknowledge technical admm paper influential framework entirely empirical problem carefully necessary block splitting partition consumption extremely large example stress modify admm become quickly experimental scalable solver available maintain rest article organize follow various discuss transform widely machine learning speech brief brief reproduce hilbert equip act hypothesis stem expansion optimization solution plug solve linear learn rise model suitable kernel rich imply capability still price scalability dense incur randomize key algorithmic device dramatically train linearization linearization method ridge operation require dependence furthermore show distribute algebra efficient improve modern view attractive randomize ty l linear return regularizer reduce solving choice solve prove extend reach recognition state application number speech challenge optimization transform though original review direction multiplier admm informally take heuristic building big environment partition build model admm similar presence model variety admm operator admm split function involve tie admm rule gauss update cyclic coordinate augment iteration admm penalty cast admm eq add augment lagrangian n j proximity prox projection operator constraint regularizers efficient dual section solve row random towards setup distribute computing comprise ram distribute across node assumption scale semantic collect cluster aggregate distribute cluster memory simultaneously store disk restriction read block transform block produce I e generate process discard construction operator variant admm suit partition towards operator graph k computationally preferable linear set derive matrix partition partition r evenly pass interface I parallel computation relate set option multiple imbalance interpret admm semantic agree ij ij I j separable block j ml see rewritten view constrain turn average eliminate turn similarly also eliminate imply derive step unfortunately
substitute eq almost ignore handle conclusion permutation contain apply permutation vector bin store empty denote extra feature one hash nan assign value empty towards offset show assign red empty along offset bin circular bin empty bin offset value bin finally proper without offset would multiplication empty bin value ensure simultaneous empty match new bin number rotation ensure empty consideration newly near non empty bin circular final equal irrespective prove fact lsh indexing sublinear search generating processing traditional hashing hash testing hashing require exist simultaneously bin eq event theorem eq event interested compute convenience expansion linearity take term expectation remove dependency box bin bin simultaneous picked simultaneously occur bin occurring randomness bin simultaneously actual simultaneously bin empty bin simultaneously empty bin locate space simultaneously empty simultaneously empty bin pick perfectly empty bin equally e pick randomness selection directly close argument change right randomness leave provably improve procedure simultaneously close circular go add randomness bit circular empty new empty figure I bernoulli associate check circular empty use circular improve bin use non empty bin circular circular bit offset empty bin move bin offset final empty circular leave circular bin empty continue bin offset empty bin bin bin go circular empty bin value remain bernoulli hash bin complexity hash storage bit hundred thousand practically difficult satisfie lsh q unbiased simultaneously empty square mse respect hash variance estimator unbiased plot summarize theoretical improve well variance scheme experiment two scheme lsh neighbor publicly train query parameterize lsh generate meta function different realization hash parameterize lsh hash table store hash report union choose base recommendation show result recall please lsh point near neighbor standard retrieve point since result run retrieve summarize clear around improved need point query improve provide balance retrieve achieve point well clearly superiority indexing improve hash indexing retrieve lsh directly hash number point moment reasonable estimator unbiased regard three rhs behave fourth necessary empty bin dominate happen practice hashing reveal sub optimality add randomness provably especially dataset come evaluation hope improve scheme ph partially li fa nsf configuration empty empty ball exactly non bin empty bin likely involve combinatorial argument configuration simultaneously way term empty note nm n random empty remain bin likely randomly bin bin empty empty bin empty bin I replicate correspond nearest bin circular come eq desire simultaneous two case close simultaneous bin towards circular else empty
foundation integral domain ready exist continuous tell term conclude continuous finally martingale martingale since since conclude ni f prove complete proof carlo sample exponential insight sampling exploit convexity monte independent identically iid reduce q reduce monte difficult use priori manually finding improve simultaneously importance generate form past adaptive determine choose fully instance distribution define density serve finally exponential per sampling variance sampling speak efficiently find applicable evaluating insight estimate eq furthermore per family perform adaptive importance sampling suffer become minima importance establish important convenient variance infinite importance function old log side pass strong convexity soon differentiable integral interior x dx take euclidean onto appropriately sequence intuition towards step reduce call x nf tx nf mention unbiased adaptive importance third alternatively view second estimate course operation easily evaluate family result mathematically proof compact finite conditional sequence n nf x however lead geometry eq q indicator otherwise see compute choose bivariate variable speak density inner parameter word update ns nm nx improvement distribution initial ever sample gradually match price arithmetic option asset asset asset discount compute choose contain normal shift run end asset price importance fact prove count adaptive importance stochastic setup make sample theoretical variance surprising beyond convergence previous subproblem update especially case solution exploit establish subproblem storage requirement represent subproblem grow size could separate point generalization omit sake density lebesgue exposition adaptively divergence estimator special enyi divergence method enyi
transform manifold autoencoder mnist digit pixel dataset leave bottom interpolation line stack autoencoder image back text look suggest manifold near volume representation close easier separate possible manifold flat manifold estimate low transform moving correspond hide signal change unlikely configuration picture basically amount distinguish probability manifold answer question ht concentrate representation transform datum factorize space capture space elsewhere parameter model put lot mass outside get parametric density see put probability mass hence encoder one element case direct q call decoder decoder capture conditional distribution put probability role training sure preserve role training sure criterion achieve unless enough condition unnormalize start multiply first condition term vanish consider conditional maximum use contain satisfied mean decoder prof capacity increase become factorize net capacity neural may desire make unimodal strongly keep least fit estimate recover normalized option associate could importance knowledge basically add kind encoder optimize learn maximum proxy example dataset probability simple counting maximize neural factorize regular input challenge deal encoder optimize want output distribution want keep reconstruct although gradient direction optimize encoder cost similar eq encoder linearity apply discretize interested pseudo gradient back propagate straight pseudo gradient idea bind factorize binomial value ht encode feedforward decoder loss factorize decoder direction decoder compute pseudo back propagate pseudo inside encoder encoder transformation autoencoder factorial experimentally without anneal greedy pre previously network stack rbms deep autoencoder stack autoencoder consider function term zero loss autoencoder considerably trade sometimes descent perfectly prior map point use tradeoff schedule rapid growth forget schedule thus reconstruct usefulness trick also value tradeoff parameter difficulty stage weight fitting stage unity loss must perfect information otherwise never recover decoder special autoencoder version reweighte encoding give training fact two log attempt maximize encoder I encouraging much close encouraging contract noisy equally factorize provide evidence feasibility technique use mnist handwritten digits mnist validation split compose consider code mnist factorize binomial minibatch minibatch cost increase momentum hide layer sigmoid output input sample bit select probability change randomly autoencoder encoder decoder bias treat estimate unnormalize perfectly practice find function allow unnormalized distribution sample partition estimate importance took expect centroid layer train give reconstruction deep deep factorize qualitatively mixture incoherent mnist decoder unit necessary encoder match input binomial digit due autoencoder reconstruction digit factorize entropy factorize necessary encode hard factorize thus dimension characterize align practically happen constant sign weight make table contain bit flip column table sufficient entropy low autoencoder prior factorize encoder dimension dimension measure output rd column table remove factorize avg output datum avoid perfectly likelihood world go small ball
encourage causal already capability distinguish capability looking consider first preliminary result start detect improve like cause markov search pair interaction descriptor distribution theoretic distinction within letter markov distinction make mixture first cause respectively cause second third moment mixture identical third instance associate population I cause moment descriptor replace descriptor obtain population I k encourage though able quantity informative statistic g insight explicit rely make consideration light dependency secondly expect second layer eventually descriptor configuration c link create relationship member link use markov evident cause expect rank second previous asymmetric descriptor quantile approach associate rank strongly rank term member position associate absence population descriptor introduce j j I k j ik create descriptor distribution population denote return observational descriptor paradigm induce quantile term effect note would informative term cause major c improve appropriate selector like difference leave estimation easily approach approach consume dependent suppose perform user perspective node existence link step markov information filter mutual may take consideration considerably approach package base l penalization greedy hill potential parent restrict constraint hc hill incremental mb min hill hybrid dag pc use configuration reason si discovery score structure experimental synthetic setting dag example pair direct descriptor return denote link result training forest preliminary perform compare art package grow incremental mb constraint learn si learning algorithms hc hill base structure hill version training compare figure medium sigmoid series consideration make experimental variate obtain several improvement art move accuracy improve increase competitive package c assessment simulate datum use causal portion portion section second include portion goal never encounter training implement unlike return rank gs grow pruning phase phase technique return rank area curve assess different comparison train synthetic gs c c gs number also take availability causal interest table filter least inaccurate algorithm result return algorithm outperform respect drive belong think causality leave stochastic retrieve causality observational challenge preliminary confirm existence causality link descriptor research degree causal relationship indistinguishable configuration improve address assess classification extend exploit relation extend dataset bioinformatics de email ac statistical causality lie heart recent show infer indistinguishable thank propose machine infer causal link rely relation supervise successfully extract descriptor variate lie statement dependency causality many formal approach causality justify detect infer causal observational influential rely independence detect causal ic accurate reconstruct pattern restrict configuration causal define independence triplet conditional unconditional sound slow development indistinguishable pattern opinion mean aspect notably conditional unconditional independence distinguish configuration result prevent indistinguishable configuration evident appearance tackle cause pair additive information geometry common feature causal reduce uncertainty direction recent organization effect pair pair idea success random indistinguishable rank competition common bivariate causal supervise feature describe dependency link pair variable encourage cause another need multivariate rapidly information existence causal variable return dependency remain one evident dependency dependency lead learn link rely relation member markov create relationship asymmetric descriptor classifier link assess competitive effect hundred causal engineering physics control artificial cause effect mixed produce outcome challenge took rank eight rely transform classification stand link direct input bivariate particular association residual nu nu auc confirm relate copula redundancy filter random forest regressor posteriori improve final four subproblem invert inverting accordingly present approach exist variate direct estimate multivariate partial arise causal configuration value difficult distribution notion parametric aspect take since want cause quantitative distinguish characteristic causal relationship expect asymmetric quantify set asymmetric continuous define pearson coefficient nan conditionally independent structural term mb belong operator independent theoretic say satisfy effective propose selection algorithm notion relevant causality science notion life remarkable dependent variable density conditional theoretic dependency dependent
position nucleotide x example g number give unique equal small rewrite reduce eq item unique example nucleotide position nucleotide force add significant reduction apply single variance sequence library library language sufficient answer library numerical many handle well program link library package numerical calculation furthermore symbolic calculation make web server user library nucleotide mixture separate handle exclude although include zero automatically library user ratio format artificial example nucleotide randomize nucleotide position library set library effect nucleotide library average standard library figure nucleotide ratio mixture impact library ratio library member librarie library ratio include library library mixture ratio mixture need library unique skewed ability one accurately complexity nucleotide mention nucleotide thing degeneracy standard deviation behave differently sharp ratio deviation broad multimodal peak shift library peak deviation distinct distinct peak peak right peak number increase sequence peak equal peak deviation support school author thank protein engineering statistic library guide library unique library handle equal mutation site formula calculate unique library mixture nucleotide computer utilize library statistic library nucleotide effect skewed large library expect unique library protein biological property protocol library gene incorporation degenerate synthetic dna sequence usually equal mixture create region growth protein library protein stand equal mix mixture mix library ask formula library variance within formula calculation library formula huge way calculation keep big library library usually huge library nucleotide basis sequence possible associate either respective give
g patch shift skewed pooling require stay invariance nature transformation identity change exploit target label subspace complex cell assumption possible approach way autoencoder natural autoencoder parameterize decode function input producing training minimize manner autoencoder autoencoder reconstruct corrupted denoise autoencoder implicitly corruption denoise provide probabilistic representation connection autoencoder feedforward option autoencoder deep architecture stack train greedy decode simultaneously intermediate depict encoding start decode layer autoencoder odd discard goal autoencoder higher retain abstract decode recover discard three autoencoder basic denoise autoencoder variant connection definition one multiple autoencoder identity meaningful denoise element hadamard learnable bias sigmoid ensure stay bound stay element decode motivated denoise source autoencoder learn good connect mapping inside sigmoid motivated encoding decode connection abstract connection drop redundant three connect input additional small compare million batch low denoise implicit probabilistic fair layer model size model scale datum find autoencoder notice weight weight begin denoise improve affect design tie tie denoise feature easy visualize exist computer vision refer limit million mini mini batch rate adapt million update analyze representation material division try stack fine tuning phase update equal million use global stack beneficial layer reconstruction time initialization sample standard low autoencoder dash two layer ratio since autoencoder information ratio far lower effective low beneficial model mod connection benefit well cifar benefit add connection significantly mod tb represent denoise color mod scale horizontal significance negative blue side mod practically neuron studying find typically several qualitatively layer neuron leave kind example neuron selective three selective orientation selective orientation orientation detail supplementary material tb c depict column follow layer neuron identify well view procedure feature depict figure show invariance autoencoder increase towards layer multiplicative connection encoder decoder discard level well able manner direction discrimination color orientation summary early autoencoder layer ability autoencoder therefore combine autoencoder operation way explicit much deep dataset std cifar mod add cifar mod na mod continuously image put aside generalization last allow million overfitte problem preprocesse apply cifar dimensionality reduce match dimensionality reduction retain input use adapt normal activation bias nonlinearity center mapping way turn proportion understand well go question form frequency invariant interesting form pooling neuron look connection connection figure order invariance leave color significance variance generate neuron initially try weight neuron connection remove take neuron neuron receive depend neuron small proportion variance output income output name significance significance depict coordinate turn neuron strong negative weight color connection significance invariant tend since nonlinearity unit negative tie learn add concave like generating detector form convex concave weight truth invariance rotation invariance sample image image invariance figure translation figure impact neuron stay even l strength significance neuron good relu function relu activation b c w b operation tb significance cifar red phase link identify pool layer belong visualize layer neuron link neuron mark group color phase perform encoder allow layer autoencoder regular autoencoder connection encoder decoder pressure abstract translate reconstruction allow strength connection structure connection use world verify representation whose invariance fast layer formation denoise autoencoder tr error autoencoder build corrupted decoder map back autoencoder need store recently become dominant large available difficulty autoencoder learn autoencoder try retain supervised image activation away detail perspective clear unsupervised must semi supervise variational autoencoder raise autoencoder connection level focus abstract efficiently result back store close select relevant investigate connection extend early comparison autoencoder network denoise way represent change balance bottom heavy invariance level regular invariant feature detail connection guide pool qualitatively selective aspect input size right layer ratio typically irrelevant recognition source orientation recognize
denominator theorem differentiable global give side condition believe thing denominator show replace q since function proximal optimality schwarz paper continuous close domain nr e iteration due eq strong incorporate recursively expectation proof fix compare prox mini eq notation need second value n h figure close figure close proposition question united pa university united pa mini scheme improve composite number nonsmooth computation gradient objective start stochastic step repeat last iterate becoming start gradient show predefined mini implementation acceleration parallelization interest average close continuous gradient subdifferential parameter equal activity solve problem many fista impractical process coordinate stochastic paper technique reduction particular mini batch variant gd motivate typical stochastic sgd limitation inherently sequential parallelism combine analyze ms gd proximal mini enjoy apart parallel hence speedup attain specify formalize predict batch employ gd upper bind stepsize equivalently proximal gd prox old reference past unbiased eq gd prox point reference outer ensure ultimately extremely gd max stochastic per epoch minibatch ix analyse case get accuracy guarantee decrease epoch define stepsize inner loop computational minimizing reach gradient decrease attain fix target evaluate give present parallelism http www tw dataset compare ms gd circle mini parallelism green dash parallelism divide stepsize star show formalize threshold straight ideal speedup ms gd lipschitz euclidean define modification lipschitz strongly standard norm define convex collection n nice suppose k strong cauchy schwarz define monotonicity prove obtain subgradient write change apply
iteration exist sis marginal statistic sis iterate iterative applie broadly screen fashion save decrease obtain recommend decay schedule algorithm design attractive update time keep drop way computational load large scale constraint accord analogue screening follow fine situation conservative screening stage role reduce fine prohibitive contain interpretability consistency dimension concentrate problem counterpart attempt insight principal component column wise write without solve q dimension input take cf sparse formulation coincide reduce pca greatly inner pca simply r get sparse pca employ loading multi sparse pca spirit procedure concern pca body share similarity subspace estimation criterion convergence terminate work ad easily self cause ambiguity free submatrix extract index production obtain vary code high level twice observation recommend predictor explanatory assessment raw keep ability identify split whole subset tune sf median show error yield factor however careful pair observation design response value data outlier varied factor perhaps say correction cc number predictor split design median model interestingly identify anomaly value low dataset summarize adjust variance variance p seem property load r soft show various loading r respectively cardinality mild extend rr r r e r due substitute remain p jj aa j aa aa aa fall c aa j restrictive satisfy oracle matrix exist e example random sub r jj j rs onto p rs rs cs cs j universal term side handle lemma instance r follow norm p j p rs p aa l obtain achieve j j avoid mn p form follow c jj mn cc nc b aa closure thresholding thresholding regularity rarely group kronecker p r rt apply get penalty give loop k triangle solution rotation step justification hence time beyond running algorithm handle non mapping composition map r f characterize accumulation proceed similar line theorem minor modification accumulation point boundedness close ft fix globally line omit proof h rp h imply rs r line tf tf surrogate f thresholding induce cf continuity need j b dd x r obtain hence sub increment entropy e dd column motivated space contain must standard volume universal manifold denote norm universal q cauchy schwarz integral freedom detail theorem perform feature extraction unsupervised propose multiple explanatory guarantee sharp reveal predictive develop algorithm penalty theoretical achieve efficacy simultaneous reduction modern analysis offer projection subspace variable pca n predictor regularization obtain x explain variable pca typically life irrelevant loading prefer fail principal moderate sparsity individual loading still employ fashion guarantee sparsity explanatory guarantee variable toward extraction desire row even construct drive perhaps theoretical simultaneous main sample yield tight provide unify able inequality convex penalty show past suboptimal implementation ease meet challenge iii setting come universal predictive framework extraction perform rigorous tight regression problem signal information criterion scale free develop penalty theoretical local unsupervised analysis conclusion begin illustrate motivation component vector r pca special section plain drawback conventional attract lot attention reference strongly fashion meet sequentially pca certain optimality orthogonality hoc ii conduct burden dimension unnecessary remove loading guarantee get may employ unfortunately optimization scheme construction synthesis perspective variable projection addition j previous discussion parameter bring extraction sparsity facilitate favor form refer vanish flexible ideal enforce elastic mcp also develop nonconvex penalty furthermore doubly datum point new factor column refer decompose matrix efficiently decomposition either case difficult reveal joint inequality estimator multiplicative constant clarity constant necessarily assume enough inequality hold type penaltie universal apply c c penaltie p addition form cost spectral applicability also extent provide appropriate tune obtain show lasso gain low suboptimal error order j large first incoherence remove penalty purpose practically however universal choice two parameter rank aic none job novel perspective among principle avoid ratio assumption share concern may notational denote model sufficiently penalization offer fact emphasize coefficient cover reference assumption give degree second term characterize risk response model q roughly finer interestingly multivariate df information familiar unknown could sparse pca cf supervise however could challenge scale simplicity iid entry suppose parsimonious define sufficiently enough prediction c c sf constant value experience know well sf recommend address issue rank constraint nonconvex penalty interest light recent strength penalty relax thresholding rather consider real iii moreover r
dataset laboratory model standard lda word assume offer solution enforce constraint word unfortunately hundred enforce alone sufficient induce achieve interpretability specialize control structured token comprise vocabulary disease concept dag keyword mesh mesh retrieval organize pathway interaction summarize human thought deal effort structure necessarily property window expert interpret understand structure modeling equip control propose exploit dag structure interpretable summarize annotated article mesh hierarchy word inform guide topic expert sparse patient spectrum diagnosis concept annotate structured subject mesh lda meaningful mesh term find lda latent ibp compound allocation provide along manifold summarize form patient instance draw align font height minimum width parent path south pt pt style style pt path parent document bag representation model datum consist model generative comprise represent statistic lda build upon ibp compound dirichlet addition unbounded introduce three process preference describe topic relationship dag nearby respect graph associate tree drug treatment anti treat paper investigate treatment sub tree intuitively summarize word many child modeling nearby thought core model explain replace lda word hadamard product ibp document topic represent vector ibp concept mask represent relationship word distribution concept form use length observe sparsity dark view allow variation document describe vocabulary primary care hierarchy patient expert think describe could cover introduce layer allow explain observe word sparse generalization time procedure additional metropolis help sampler move matrix specifically mh prefer proposal knowledge mcmc use move encourage novel rely intermediate assignment tensor count assign topic topic nk k multinomial count topic slice give topic multinomial assignment know never way document concept count derive assign entry nk nk nk concept entry k q objective concept induce prior procedure fast mixing count reach unlikely sampler concept fast mix document introduce mh topic word ratio mh sparsity induce prefer equation get term dominate allow toward sparsity lda recover course layer sparse topic information point incorporate control vocabulary art interpretable tb occur patient receive diagnosis organize structure cm hierarchy disease recurrent mention diagnosis one independent run divide mean graph lda predictive however summarize clinical topic lda corresponding discover correspond use hierarchy probability rather word topic severe publish clinical tb library maintain control structured medical mesh search sr look summarize evidence clinical question consume reduce involved mesh helpful annotation systematic review researcher decide relevant term manually assign article inherent variability specificity make leverage difficult identifying concept nearby mesh interpretable provide retrieval tb p blind double blind channel drug dataset document annotate mesh systematic channel produce concept lda mesh rapidly report trial investigate use without topic comprise hundred mesh concept sample topic discover know article report control trial concept instance systematic anonymous confirm evidence retrieval topic wide popularity flexible corpus assume consider scenario idea coherent interpretable identify word among automate interpretability rate human develop topic work focus indicate link human work predictive summary kind describe disagreement result non probability interpretability nest chinese restaurant learn topic specific nonparametric learn also use kind topic interpretation complicated require human concept sparse encourage part use graph guide formation concept interpretability concept structure also simple interpretability expert define context word sense tuple incorporate hierarchical supervision improve come relationship content website summary jointly word show hierarchical exist forest enforce topic hierarchy label specifically treat label assignment document probit assign parent capture hierarchical structure contrast focus prediction graph use rather manner sparse generation much consider allow concept imagine nearby word nearby entirely nearby difference model neighborhood hierarchical prediction classification enough structured knowledge basis often scientific domain resource exploit achieve state interpretable graph control vocabulary structure induce interpretable bayesian nonparametric topic leverage interpretable maintain ability
gap use matching map use sparsity choose perform slice posterior normalization run special hasting update acknowledgement thank device award markov general tool practically apply prohibitive iteration present auxiliary mcmc query likelihood potentially small proposal approximate asymptotic fast method feasible bayesian probabilistic appeal output uncertainty make provide selection often model form distribution use inference monte persistent challenge coherent evaluate evaluate target update similarly typical variational bayesian procedure approximation intensive online procedure subset make procedure build optimization achieve result chain monte carlo consider data hasting mh recently mh move stationary condition effort exploit mcmc leave posterior introduce collection effectively turn parameter improvement structure issue evaluate discuss limitation conditionally target notational convenience term sampler hasting unnormalize evaluate seek auxiliary following eq distribution auxiliary wang hamiltonian joint remarkable given evaluate form minibatch subsample much computational evaluate family compute statistic need computed make likelihood run generate alternate update conditional emphasize partition part remainder bottom bernoulli ignore chain tend iteration iteration markov likelihood dark likelihood evolution detail proceeding section picture illustrate version toy implement likelihood whole likelihood bottleneck mostly structure chain convergence regular average computational summarize important determine iteration posterior chain number important tight put easy family summarize set consider either vector negligible n stage scale describe parameterize tight bound cost tight example choose bit front well tight place perform quick approximate explore resample take drawback visit resample work practice bottleneck usually simple overhead replacement hard efficiently choose line descent optimization long access chain still satisfy seem unchanged distribution z n metropolis accept efficiently datum matter whether accept geometric tune evaluation iteration point course markov leave something ht nz valid markov minimize assumption likelihood dominate step linearly constant choose storing operation scale store value need return th track cache store array dark index keep track thus dark assignment maintain record array position useful mcmc certainly regular evaluate per expect mix slowly favor iterate much slow answer question set depend give mcmc likelihood iteration accounting autocorrelation offer compare regular experiment classify mnist use principal use evaluate hold metropolis yield metropolis choose tuned optimization summarize mcmc autocorrelation per na I perform bad mcmc per much burn give map tune poorly reverse true rl c speedup
classified weight learner accord learner weight learner mistake expert consider total mistake base learner therefore weight least side obtain conclude mistake learner learner predict label total expect mistake algorithm aggregate expected mistake randomized condition expert expert learner classified mistake randomize majority expert convenience learner complete proof compare bind rwm mistake rwm mistake expect mistake rwm addition need incoming less expert suppose mistake similarly mistake consider fact well obviously also consider instance instance expert mention hypothesis expert region generality fact rewrite eq simplification q rewrite true mistake rwm complete weighted bag boost uci repository evaluate aspect instance effectiveness framework classifier update bag boost naive bayes balance breast diabetes letter rwm experiment rwm depend experiment dataset near ht bagging boost rwm breast cancer diabetes letter another dataset great method rwm dataset support result although increase rwm power difference arrange ambiguity table pair test indicate show difference arrange red cell black cell well table cell confirm bag vs boost rwm draw breast diabetes letter view time difference rwm bag exploit label justify great overhead create factor balance scale breast cancer diabetes value well among base green fp good value cell true group base breast diabetes mistake bind rwm mistake mistake calculate formula mistake experimental confirm accuracy mistake small mistake rwm large mistake mention c mistake mistake result diabetes mistake well rwm increase addition reveal rwm input show new online among superiority different learner class specify class correspond dynamic powerful clearly base classifier utilize cause great base affect whenever important use base learner volume exploit algorithms ensemble classifier prediction well online ensemble randomize majority rwm expert define converge region well resolve rwm propose novel prediction expert rwm result also well sufficiently expert randomize weight classified identifying belong label algorithm performance paradigm prediction good study extensively lot practice spam detection object bag know weak satisfactory performance online bag handle stream mention consist learner select learner input classification recent machine recent area lead predict goal predict close expert majority rwm present mistake bind expert exponential fundamentally rwm instead zero rwm exploit definition well good expert expert error expert necessarily negative true reveal separately base improvement rate call cascade expert theoretically expert tight exposure sufficient practically contribution rwm know rwm consider rwm apply data number output w trial mistake make majority mistake expert far large rwm tends decide accord opinion expert mistake compare discover sure algorithm mistake say try good datum instance one opinion numerous fp false rate low fp rate fp rate either look expert low look expert low fp rate lead three classifier learner rwm exploit factor every classifier predict weight learner prediction responsible
separate protein achieve cb dataset challenge secondary structure prediction great challenge computational accept predict protein understand protein drug protein determine structural state thus use algorithm protein extensively study protein close since early neural core component many successful significant leverage information development capture use recurrent neural probabilistic graphical graphical neural crf secondary commonly classify combine secondary coarse grain structure prediction achieve grain state secondary reveal address address protein introduce protein crucial improve performance secondary secondary formation depend secondary far apart protein still limited capture spatial knowledge various structure speech successful lack necessary mid work tackle challenge secondary broad supervise markov output avoid marginalization deep layer advantage crf field mrf versus generative classifier dependency important supervise structure structure protein enable hierarchical hundred introduce multiple convolutional allow high feature suit make informed ht utilize without generative training train computational reconstruct generalized difference input intermediate boltzmann avoid marginalization explicitly graphical learn directly enjoy feasible back prove autoencoder irreducible ergodic chain converge introduce latent denoise iteratively px converge example sharing seem leverage supervise supervise generative network supervise analogous corruption let denoise auto ergodic estimate provide procedure minimize reconstruction reconstruction generative corollary corruption process reconstruction train regularize triplet assume train h xy p flexible noise avoid marginalization hide benefit capturing task distribution ht contain convolutional layer layer computation layer therefore location making sized gradually simple convolutional consist input channel convolutional layer feature convolutional convolution thus connect visible unit filter map bias visible noisy activation pre z calculate straight layer label focus secondary challenging problem structural simultaneously predict structural sharing position secondary previous protein inclusion program package content sigmoid original encode binary input feature encode protein improve performance protein training commonly retrieve remove protein chain state secondary label infer structure discretize absolute result secondary label cutoff coverage majority protein chain short protein short aa zero contain protein perform cb far filter cb performance measure run consecutive reconstruction randomly add post obtained reconstruct consider reconstruct multinomial secondary binomial number network motivate arbitrary away prediction trick start activation sigmoid layer gradient comparison epoch implement library train gpu segment layer sc sc noise sc sc state sensitivity loop turn bend bridge description one protein dark color indicate strong versus achieve layer structure conv conv pool conv pool type convolutional denote pool window layer architecture use channel convolutional consecutive feature run get close experimentally incorrect later organization secondary achieve major less frequent state unbalanced label specifically unbalanced effort identify extremely rare improvement public benchmark cb validation train sequence homology ss wang discover success architecture vary try start gaussian convolutional original good layer performance dramatically prediction start learn reconstruction layer seem necessary reconstruction validation reconstruction error go vs representation generative sized secondary structure protein structure network structure stochastic capture datum convolutional level structure sensitive inform high distant architecture structure bioinformatics scene parse segmentation architecture hard code organization
center reveal embed lose take singular contain add row important capture scalar basis feature span justification theorem proper complete orthogonal interpolation theorem say decision function generic kernel orthogonal intrinsic g kernel pca alg interpolation generative learn degree linear notice b approximate basis generator generators projection address degree increase computing power exploit power generator project approximation sequentially approximate rank onto general fix hilbert approximate strategy project feature various alg lie degree generator ideal generative learning degree generator read notation projection matrix threshold apply add kernel compute interpolation basis row maximum threshold generative discriminative entry thresholded return singular generative time discriminative compute interpolation space alg analogous consistent estimator storing parameter repeat complex form evaluate maximum evaluation one vs width thresholde singular purely linear polynomial gauss overall misclassification second subsample vary runtime detail conclude competitive handwritten close discriminative implicitly generative discriminative discriminative scenario decision generative separate hyperplane uniquely generative learn manifold discriminative way manifold moreover dual discuss principal idea root statistic symbolic practical reading symbolic major kernel trick reproduce duality algebra duality rkh consider principal learning feature explain detail pca learn embed also algebraic topic overview directly applicable understand structured hand algebraic structure scenario theory build polynomial outline symbolic back al polynomial svd vanish basis basis variation symbolic vanish learn coordinate duality duality symbolic algebraic outline theory ideal duality simultaneously component learning stand open method conceptual lt european european grant fellowship ex symbolic algebraic inherently dual structure algebraic generality kernel main kernel illustrate propose simultaneous accuracy propose synthesis symbolic algebraic inherent duality method kernel fundamental machine method kernel trick g major drawback learn kernel principal learn symbolic algebraic inherently structural representation major directly interpretable seminal allow transform easy major numerically unstable scenario address attractive major issue symbolic applicability issue symbolic tool simultaneously generally argue discriminative world discriminative generative allow combine avoid considerable introduce relate duality polynomial object involve treat paper field usual convention divide change show object link duality vector polynomial space homogeneous polynomial polynomial nx dd space decision let k independence claim pass limit usual map elementary map identify explicit dual namely f canonical identification canonical scalar product compatible need scalar property k must description product fix exponent extend hold outer product orthogonality let less express duality algebraic reproduce hilbert additional evaluation symbolic polynomial equation could obtain identification section purely algebra next beyond usual rkh alone concept algebra proper object feature multiplication infinite says admit multiplicative generator generator need class space polynomial analogue manifold relate algebra geometry relate decision generative reveal duality introduce matrix resp kx concept ideal orthogonal scalar intuitively basis vanish algebra statement interpolation let generic let hold ds n nk ik ik x x nk remove iii equivalently row enough contain kernel degree yield statement instead grow whose size even claim ii claim vanish therefore carry consist feature algorithm introduce subsequent enable variety vice versa kernel estimate feature manifold generator conversely estimate pca relate obtain discriminative permit inspire model vanish sample hausdorff ideal partition noise act dependent task classification irreducible name irreducible sample principle ideal take manifold label part could give th entry compute output entry sufficiently large noiseless alg discriminative vary
follow result call spirit suppose strictly ty ty scale model dt ty ty drift brownian correct density px easy researcher often marginal dynamic residual nonlinear reference example uniformity check transform process residual box al du serial independence although insight misspecification sequentially correct moreover mistake kolmogorov type functional test iid use residual require examine simultaneously uniformity independence uniformity kolmogorov control nan show analytically design base incorporate lag product standard distribution alternative take account parametric implementation smoothing piece hinge integration simulation nonlinear simulation form avoid tool evaluation misspecification simultaneous uniformity correlation independence inconsistent li uniformity imply distinguish depend procedure impractical close spirit goodness hinge original detect misspecification rest test bootstrap justification provide daily stock index briefly deferred generalized joint uniform motivate incorporate product uniformity independence implication theory useful goodness test univariate empirical inconsistent discussion issue bivariate result illustration stock return risk define quantile tr e exceed unconditional ensure literature risk therein unconditional rarely metric aggregate von kolmogorov statistic avoid process wise check write interestingly hold q univariate account unconditional lag likewise information lag aggregate test account unconditional misspecification combination sum scheme possibly drive address lag care lag power sample pairwise five equal already moderate box provide dynamic correlation propose box draw across account one want nonnegative estimate asymptotically practice know monte simulation parameter statistic recursively statistic empirical distribution percentile much bootstrap utilize block directly since nan misspecification nan necessarily transform case block bootstrap desirable costly many parametric block model application take simulation take second thus speed bootstrap small reality univariate substantially lag enter dimension discuss detail fixing along omit relevant practical asymptotic impose conditional form cdf dynamic increase first weak element covariance bridge brownian bridge parametric cdf composite nan hypothesis effect differ ba mn mn following truncation r f g empirical still assumption generate abuse unconditional last choose additional average misspecification aforementione extreme misspecification elliptical report carlo control dynamic uniformity alternative projection exist ergodicity p consistent might distinguish alternative might case say solely consistent whole complement dependence serial might together assumption imply necessarily bootstrap critical bootstrap prove analog test setup rv assumption restrictive linear expansion state denote critical value critical come bootstrap critical approximated repetition assumption carlo propose test repetition critical save detail misspecification model simulation request technique stock exchange index maximum therefore require assumption ml fan mixing applies specifically design stick evident misspecification easy introduce usually transform residual test univariate test goal one hypothesis order make martingale transform modification leverage experiment examine finding application test see eq take thick triangle marker circle marker lag dash line circle marker plot sample panel student triangle marker lag circle marker consider dash circle marker panel change size panel test show marker plot fill monotonically close decrease misspecification capturing case also dependence lag instance aggregate lag wide alternative effect box include repeat experiment power obtained report save summarize nominal similar normal student equal span apply transform reject significance reject example test table line subsection generalize n jj von test show panel von kolmogorov panel jj von kolmogorov ar von kolmogorov c lag generalize ks suggest capture test reveal ar generalized reject significance statistic equal ar reject mean ar ar type equal ar statistics literature multivariate vector td f past parameterize conditional dynamic copula joint specification test follow multivariate transform univariate df joint formula probability chen dynamic copula transform constitute series iid apply statistic model important multivariate series bivariate two reject nan ar misspecification undesirable financial fill literature helpful process paper useful develop theory include grateful question suggestion anonymous comment possibility foundation school economic acknowledge financial ref cm economic school graphical tool quantile duration etc properly control dynamic smoothing test new integral transform establish alternative justify effect often ignore monte carlo finite sample property test check popular stock exchange datum
es n pr name ensemble list illustrate experimentally base sub find combine give surprising similar user combine weight fill achieve model name inferior simple plan present rank recommendation name occur name name name scalability issue future parent name thank valuable feedback european community university student thousand name choose pick parent inform decision recommender ranking produce collaborative algorithm list experiment world search explore name discover challenge intuitive consideration parent parent relative pick name actor rule mean name family belief role choosing mark name stand crowd avoid source email address address thousand name parent pick right present parent take collaborative filtering give name pool name study context offline phase main easy simple perform letter user name item recommender use bold bold scalar item name occur name occur name item occurrence aggregate bag order search name finally represent user action occurrence decrease order query explore name collection comprise dataset contain activity name figure name observe distribution concentrate per name show type interaction enter website order click name website link search page user request name available name category name correspond challenge given recommend name search recommender evaluate respect name user restrict enter activity user activity name display set name enter activity list detailed assessment recommendation left name position order list recommend name precision position might happen handle name clear activity type concentrate name name name user name name user median name name provide challenge user name recommendation representative user remain transaction name ignore impose g recommender depth al network recommendation name name name scale predictor collaborative filter collaborative cf web site online input interest recommend item collaborative approach compute collaborative filter amazon com match name combine list determine match name construct co occur name collection behind many name co name transaction thus process memory name recommendation list name occur name popular correlate name name recommendation name occur number recommendation max max max neighborhood approach collaborative describe recommendation capture name recommendation ensemble quality value combine estimate within reciprocal individual filtering recommendation boost model describe create name occur name test enter name bag randomly select name give user specify proportional user positive towards name user name consist list name sequence co occurrence name sort use choose name bag occurrence assume respective bag recommendation choose select pick frequency would recommendation iteration include list finally recommendation correspond follow occur prediction activity occurrence name name interaction
network topic friend site facebook twitter ad topic describe classified ad relate engine optimization content describe rich article likewise link building site add link email google email post market black server month spam corpus relate particularly interpretable topic email strategy spam describe way ad google web corpora beyond corpus insight hierarchical corpus tb type program ad category topic profile type try sum topic proportion job table show result nearly third com abuse majority appear involve insight analyze example com project wide topic concentration reflect proportion email opt list topic concentration focus stream explore generalization introduce finite dimensional technical use dirichlet main scale corpora result corpora real world nonparametric become complexity stick break construction scheme arbitrary version potential batch framework recently explore also worth facilitate theorem derivative k gain term convex flip essentially concavity establish tb concave concave fx application jensen yield right side establish upper crucial variational need maintain overall equip give expect inequality analytically statistic beta distribution eq tb show conclusion c q tight important conjunction bind tb tune alternate appeal upper recall obtain specifically occur expectation set upper equality x last four recover factorize well around tight sufficiently factorize lemma detailed procedure log respect study corpora document know hierarchy newly available corpora security deep network computer decade topic framework bag word explain frequent occurrence corpus dirichlet great deal topic intractable remain research devise approximation paper topic corpora document hierarchy structure corpus subject matter news article business sometimes e international idea topic model simplicity topic estimate say validation node corpus hierarchy leaf represent reflect ensure topic proportion category parent variation hierarchy special hierarchical devise new child believe broad demonstrate subject corpora field security seven year job crowdsource view corpus site corpus derive internet create break ground model number depth corpus moreover corpus depth variational approach demonstrate describe detail em parallelization additional probabilistic corpora inference model describe procedure evaluate corpora result section conclude inference derivation algorithm tb illustrate seek corpora child root top level parent individual corpus essentially extension lda model proportion weight topic document leave level proportion associate non proportion root corpus topic sample corpus parent tree figure dependence scalar concentration topic proportion category variance formally denote likewise assume document proportion denote namely conditionally topic proportion category corpora document begin recursively lda refer inform tb proportion dirichlet proportion draw topic proportion draw multinomial bit topic model however occur corpus hierarchy level view case flat corpus dirichlet show perform demonstrate advantage symmetric draw dirichlet serve nonparametric lda topic generative special base allow possess application inference seminal develop gibbs track conference sampler collapse spirit drawback fast gibbs develop framework collapse level deep later framework truncate possible dynamically vary successful variational corpora actual corpus involve variational inference complexity auxiliary stick construction scheme parametric already sometimes identify go wrong direction average preserve lemma concave lemma prove sketch concavity lemma obtain term likewise obtain apply inequality concave equip intractable result value dirichlet shorthand appendix emphasize naive jensen hand direction need log thus rigorous low shall surrogate inference detail respect parameter perform coordinate ascent corpus variational different category bottom corpus corpus update topic reader recursive call hierarchy recursive perform subtree start child category inference node corpus em factorize approximation updating variational coordinate newton corpus refer variational double naturally tb recursion variable prior proportion train assignment compute evaluate corpus manner node corpora uci repository vocabulary token vocabulary news vocabulary experiment batch article million token publicly batch implementation former hdp gibbs latter hdp prior initialize hdp topic hdp per corpus report hdp hdp summarize experimental bar fold fold hold testing range fold hdp corpus explore hdp slow hdp corpora corpus appear order explore hdp much corpus certainly scalability demonstrate exploit corpora corpora corpora paper category million token job token vocabulary crowdsource collection token vocabulary seven year job crowdsource site internet corpus lda attempt job hierarchy interior job leaf
al agreement reasonably insensitive choice al assess carlo label pool population primary namely label many monte draw aggregate calculate monte p c c al meet meet meet se h c meet meet meet meet se replicate average table illustrate six good aggregate classifier averaging group six good lda meet meet meet rs rs se rs met meet meet se se c na I bayes meet meet rs se se lr covariate denote pe py py appendix diverse choose two class prior use source uci provide wide term covariate variety sensitivity presence absence problem dim class name dim generate sampling gaussian mixture prior problem multipli multipli multipli multipli multipli red decision boundary classifier implementation detail describe lda standard implementation apply covariate scale I implementation package predictor assume ideal svm package radial calibration mle fitting computing optimisation define al systematically train label use raise stochastic optimality produce reduction central whose selection define reasoning abstraction theoretical heuristic make construct experimental classification seek select example example diagnosis review performance label select central motivating specify loss function describe suggest selection classifier great expect construction great thereby sense work present theoretical quality eq estimate motivate loss reveal whose maximum abstraction abstraction generate insight fully al method difficult analytically performance since source variation comparison make shannon entropy motivate development issue far experimental evaluate al literature explore source binary eq al al al behaviour label label behaviour raise robustness optimal label examine structured illustrated follow conclude background brief review somewhat notation categorical model covariate response denote bayes thereby give classifier classifier class class theoretic allocation denote misclassification allocate j index indexing dataset training division show division training subset discriminant analysis regard length fitting notation slightly extend non object node near produce performance classifier assess quantify disagreement prediction define allocate log empirical generalise expect denote loss log hereafter classifier label example typically discuss classifier discriminant nearest na I nn na I independence give standard classifier label abundance good algorithm human expert label systematic example improvement pool example may covariate relatively label consider common examine repeating generate curve amount label grow repeat application dependence iterate al may reason turn rs example rs label rs benchmark experiment al benchmark performance assessment much rs hence address rs rank performance number uci base entropy second seek metric comparison desire goal label need level often context certain analytically heuristic example choose classifier decision idea uncertain tuning uncertainty classifier allocate shannon define j justification search hypothesis efficiently rs loose search label prediction disagreement vote predict kullback version control sense search valuable insight motivate pool cluster suffice optimal optimistic illustrate gain theoretical error example author examine experimental spirit current classifier label section examine construct account fitting datum example error rate define way replace approximate efficiency approximate rate pool calculate total problematic classifier uncertainty question error uncertainty hard reduction form dependence critical notation intend base already train much single choose examine define label give label reduction actual denote actual goal great turning pool unknown take expectation label loss capture difference loss exist loss improvement since define section extend al reveal bad expectation show selection calculation curve smoothed nn al behaviour examine label condition primary take marginal vanishe denote marginal optimal behaviour target maxima reveal behaviour illustrate raise behaviour address motivate introduction generalise central marginal illustrated create shift illustrate fully specify covariate infinite pool assume allow explore q cx knowledge allow method popular shannon binary univariate balanced later target define section rate assume split class size hold full calculation consider j decision boundary denote calculate examine decision rule c equivalently boundary lr class wrong around second straightforward give denote cdf result appendix al oracle new denote update f directly equation cx cx figure great close figure case classifier fourth great improvement toy example complicate j solid green case great move boundary close analytically examine two se rs pool assume al rs contrast se problem functions maxima j j black se rs show blue red estimate dotted green improvement three rs se thereby selection select whereas case classifier se never optimal se never great se suboptimal turning rs se stochastic nature rs far se explore method label se rs start variant average draw label comparison rs improve location quite b c covariate black se rs green case multipli multipli multipli multiplier label sensitivity fix rank b always sample equation conditioning rarely make explicit examine motivate address dependence raises label alternatively sensitivity second draw low pool similar visually statistically similarity pool ranking great sensitivity problem classifier discriminant single pool grid imply rank pool ranking test correlation correction c closely relate draw similarity toy dependence dataset case turn near different section target theoretical present different estimate label dataset method compare benchmark rs describe equation include three primary estimating estimating raise interesting statistical estimation choice I implication dataset parameter na I disjoint partition figure scheme directly take term I ignore estimation term label problem train estimate optimistic I motivates term thereby estimate subset partitioning subset estimate classifier parameter arbitrary random perform time result study fold pool computational explores section focus al varied know classifier diversity
minibatch divide result sgd sgd ss much sgd two adopt minibatch rate significantly small sampling improve observe significantly strategy unbiased mnist epoch minibatch size divide row summarize objective algorithm row test four variance dataset term gradient demonstrate propose sampling descent provide rate strategy conduct extensive traditional uniform sampling promise validate effectiveness technique thm zhang university nj optimization task neural network sgd minibatch often sample minibatch lead high variance whole technique significantly improve encourage experimental confirm extensively community every method example random minibatch uniformly minibatch unbiased gradient estimator relatively propose dataset reduce key idea unbiased end relationship strategy minimize sum standard rest review work study minibatch descent empirical conclude finite rate linear rate researcher return average last fraction previously similar polynomial sgd extensively study unbiased explicitly new call finite moment still employ uniform importance sampling consider variance minibatch idea complementary reduction importance function useful throughout paper please lipschitz function gradient strongly co standard multiclass predict classifier label solution regularization describe rule calculation large popular modification draw equal rely derivative descent disadvantage randomness initialize ts r ti propose similarly compute v accord maximize reduction minibatch solve dynamically gradient cluster variance derivative require clearly impractical cluster relax k r correspond optimization calculate iteration calculate relaxed simplify follow algorithm cluster provide present notation begin analysis suppose satisfie nb expectation obtain conclude lemma smooth suppose nb verify satisfy sum inequality ht ht inequality convergence smooth function gradient well technical inequality h simplify p use convexity property give conclude another propose smooth p final use inequality iteratively fact
independent purpose plain represent straight confidence rate provably pac blue symbol provably theorem green symbol rate appropriate stopping time symbol approximately slope visualize guarantee see gain arm add mention probability pac illustrate huge gain keep mind time really prohibitive nature second term negligible implicitly know zero multiply pac probability error need average twice large deterministic sample budget relate pac design uniformly good across consistency requirement fix draw often preferable bad observation sequential difficulty impossible predict efficiency experiment consideration low give relate different aspect distribution denote round draw let increase right hand large q correct one possible satisfy right proof find exploration normal let independent ts suffice tend context involve theoretic appear gaussian bernoulli match tight budget confidence numerical significance exploration deal scenario suggest implication testing like match reasonable strategy stop rule context mostly give lead regard proper provably pac exploration use sequential stop assumption exist bandit density introduce log key follow classical whose relate stop event also quantity well let lemma jensen lead introduce successively log rewrite apply tn aa conclude together region state technical partly omit eq accord subgaussian super martingale eq let dt give proof quantity implication eq conclude lemma follow apply suffice ss ss ss ss x conclude arm rely follow lemma generally family proposition apply respective two often refer chernoff xx derivative use one show sum de universit option alternative dependent bound performance testing improve currently confidence equivalent alternative provide stop terminate budget alternative though practice criterion identification popular website empirically preferable user value page respective standard determine high user become user b either number fix pair determine schedule determine source algorithm take past display sequel benefit adaptive ignore term present armed consist unknown motivate arm expectation choose receive arm resp law resp belong identify expectation agent define arm past word respect satisfy triple determine sequel correspond strategy bandit two setting consider choice draw draw almost surely recommendation rule strategy compare identification resp budget follow bandit use error budget determine fully alternative law variance pair h among sequential ii expect require test minimize order probability gain randomize conclusion bandit introduction set aim maximize horizon equivalently introduce understand parametric proper leibl analysis include ucb ucb ucb goal determine arm try observation arm identification interest problem go armed good arm parameter recommend sequel advance bandit work study armed bandit arm empirical arm strategy derive pair model budget strategy wrong strategy arm ucb strategy pure exploration involve divergence lower easily armed set confidence obviously always suggest sub expression specific bandit see obtain c equal case match indeed reference q shorthand early relevance quantity arm analogy conjugate bandit define prove reach recommendation differ rule reduction draw arm time arm simpler introduce uniform sampling optimal strategy collecting pair arm algorithm sequential propose stop empirical belong introduce sample pac lower apply case elimination pac match rule interesting exhibit pac explicit iterate logarithm propose elimination pac case sample govern ensure function elimination exploration pac match appendix elimination exploration elimination round elimination feasibility variance class bernoulli bandit define arm parametrize kullback leibler either express static theorem budget strategy dependent directly closely algorithm use set
right class arise notion probabilistic state mark finite initial finite alphabet specify symbol recursively extend arbitrary follow impose state string nan unity specify however strongly remove ergodicity next formalize generator initial induce function countable imply space initial probability generator let probabilistic recursively nan index imply yield mark immediately space ergodic whereas marked representation mark transformation entry symbol fix initial sense stationary state note exist stre beginning stationary equivalence unique canonical induce canonical ij satisfie property canonical initial mark canonical representation independent canonical representation copy exist q begin define stay within mark extension ergodicity state mark representation initial mark latter may distribution connectivity induce remove state arbitrarily state fundamental importance entropy synchronization fix determine current state analogous rgb rgb rgb rgb distance scale font fill minimum text draw yshift draw xshift yshift edge aa south anchor xshift east auto distance font fill text text right edge bend edge xshift yshift xshift yshift edge draw bend bb south estimating rate generator translate synchronization problem symbol determine finite history probabilistic graph arc probability trivially thus stre state contradiction generality x string ij tx j consider contribution arise nevertheless qx complete theorem induce string existence string may order string alphabet construction scale string string computation string symbolic note state state symbolic string specify alphabet set symbolic count string overlap string count occurrence imply symbolic symbolic derivative symbolic distribution empirical since I read guarantee complete describe string sample theorem arise geometric vector construct different string generate derivative geometry let sl hull recall lemma claim vertice state stre convex hull string string derivative hull e denote allow n kullback stationary complete easily bit hx I always conclude stationary time symbolic directly employ possibility entropy generative entropy probability finite entropy q perturbation cause change turn maximize perturb entropy cl establish let claim perturbation differential perturbation entry perturb small form perturbation note attain imply within set admissible perturb perturbation attain admissible perturbation monotonic globally admissible difference establish note complete deviation symbol stream string extension unknown conclude establish claim write x stationary time correspond thus hx hx hx h note hx complete next modify symbolic derivative generate pr limiting x chernoff e pr pr denote rt dropping first big hence lead complete continuity entropy hx x pr hx h x complete string alphabet string hx e hx hx hx hx stationary alphabet bit rgb rgb rgb rgb rgb rgb anchor south title title yshift align legend style pos north east fill gray top dash gray width thick color xlabel length symbol xshift ylabel entropy letter ylabel style yshift xlabel style yshift xshift table x figures south anchor yshift title style align legend pos north east draw fill style axis grid style dash gray width height top color gray symbol style ylabel letter ylabel yshift yshift axis axis cs shannon figures table east anchor west title yshift legend align style pos east fill gray style axis grid gray height color gray xlabel symbol style xshift ylabel letter ylabel yshift xlabel style yshift format format sep axis theoretical south anchor north yshift title title legend cell align pos north east white gray style false gray true height grid xlabel style xshift ylabel bits ylabel yshift xlabel yshift scale false false format sep axis cs figure c east west title yshift legend align legend pos north white style style axis gray xlabel symbol xshift ylabel bits letter ylabel xlabel yshift scale format fix format sep axis cs value table x figure south anchor north yshift title yshift align legend legend pos north east fill gray fill text style grid style thick grid axis xlabel ylabel letter ylabel yshift xlabel yshift scale false style format axis table south yshift north english text yshift anchor south text black yshift anchor south east xshift south east auto black minimum width scale edge draw bend xshift yshift xshift yshift bend leave south xshift south east scale font edge bend node yshift xshift yshift edge bend english experiment subject put bit letter achieve entropy generate e symbol stream generate probabilistic alphabet lead significantly approach finite string string satisfy describe hx occurrence corollary hx e complete binary rhs bound alphabet relationship alphabet figure band capture data length rhs confidence band function step propose stream importantly string effort concern level contribution constraint lead uncertainty rare ignore string occur application english shannon experimental english letter alphabet size verify corpora letter collect letter example allow comparison shannon author estimate mention assumption converge trivial get lyapunov finite ergodic model directly probabilistic e look walk somewhat compare application insight alphabet stream generator history tell show quite converge contrast base error symbol tell precisely indeed symbol rate symbol generate stationary process correctness exploit probabilistic finite string importantly free converge confidence bound bad confidence requirement compete approach sequential datum stream naturally quantify tool recognize perturbation source carry characterization make practice fundamentally demonstrate converge fast effective range english text additionally algorithm bound connection input uncertainty require pre symbolic dynamic kolmogorov complexity kolmogorov estimate insight drive dynamic tool detect dynamical ergodicity stationarity relation symbol stream dependency long decrease entropy pre error know algorithm compression rate string distinct phrase sure optimally one end string length source entropy parse report instead occurrence contexts string occurrence count interested former answer possibility exist rate limit analytical guarantee unable finite convergence trivial report algorithm issue theoretical entropy follow distribution free characteristic consequence uncertainty specify bound finite length font auto distance scale circle draw dash align fill xshift yshift west xshift fill anchor west anchor north yshift b south align anchor text south align estimate north align state hide anchor north east font yshift xshift south west align stationary entropy stationarity report converge slowly rate imply
work whose eigenvector discard two j follow j equality due orthogonality j begin relationship j derive j theorem prove theorem suggest discard low error actually shrink turn connection sense view say truncate distinct say tend explore connection associate distance truncate similar transformation matter j pair associate say shrinkage point bound error simple reconstruction accord equation discard eigenvector truncate pca lead perform aim addition check classification pca far result publicly machine repository discard distance pair accuracy state pca discard eigenvector eigenvector row discard eigenvector cause eigenvector discard pair norm wise mean square discard wise distance addition check effect accuracy pca publicly uci repository eigenvalue distance neither accuracy denote write may q fundamental zero invertible write converse invertible fundamental algebra orthogonal therefore invertible transformation system homogeneous augment matrix discard algebra state homogeneous system infinitely many
offset improve sequential scan require huge similarly scan due alternate mcmc intuition conditionally independent mcmc proceed gibbs mean scan useful determine pseudo fit value amount value deviation mle display along generate mle indicate room mcmc mle scan gibbs indistinguishable confirm asymptotic poor foundation inference massive restrict significant improvement close close mle full estimate explore novel augment new theoretical concern original provide cd moment cd visit advantage pseudo show cd like conducted validate cd promise intractable propose cd provide cd perform apply family cd inference turn quite mix perform present challenge traditional quickly alternative cd relatively stochastically result become context foundation devote family generating offset e intractable except actual evaluate every also parameterize inference equivalent kl p unfortunately impossible evaluate normalizing problem cd introduce gibbs term likelihood deviation chain start objective function form observe nearly minimize clearly necessarily becomes imply maximally suppose represent result mcmc regularity though may time equilibrium implementation result belong term rhs use third problematic dropping suggest small divergence theoretical restrict boltzmann gradient maximum gradient infinity upon main development whether ignore principled approximation index b ib b iy step unconditional satisfies distribution unlikely step choose appropriately drastically reduce augment subset marginal let equality term discrepancy distribution reduce lower perfectly far justify cause result close implie augment minimized log within combine generate since obtain aggregate information indicate poor find good number gradient zero yield mcmc reasonable restrict occur approximately condition exponential moment analog become iy moment composite objective become pseudo form composite cd maximum computational alternatively mcmc equilibrium write composite either direct sampling sampler cd kernel arrive identical pseudo long additional gibbs likelihood objective wish index choose random conditionally mr function weight among let gibbs sampling back expectation derivative express complex long chain case use cd moment family approximation geometrically computation perspective though mcmc make require chain limited approximation yield quick expectation approximate expectation regularity upon condition convergence unlikely meet unless mcmc run similar gradient gain approximately within composite hessian newton eq hessian expectation approximate take similar effort algorithm quadratic convergence family interaction typically fit mcmc study inferential mle typically represent people connection
induce automatic efficient bayesian inference linearly characterize tuning infer multilinear factor provide predictive distribution miss entry extensive simulation synthetic intrinsic capability recover truth miss synthesis demonstrate outperform approach tensor tensor rank determination inference synthesis array structural affect involve video represent pixel person pose factorization capture multilinear among therefore theory study apply social video brain process popular tucker cp exist arise world attract great research year tensor multilinear partially factorization missing formulate least cp conjugate riemannian optimization tensor incorrectly severe performance another technique exploit formulate technique nuclear norm yield split nuclear also scheme tensor define straightforwardly weighted norm mode addition completion factor simultaneous completion technique combine tucker auxiliary strongly also nuclear affected emphasize cp tensor np bind tensor ill rank tensor investigate fact determine miss attract interest tensor factorization bayesian monte carlo mcmc variational inference extension robust include bayesian rank computationally inaccurate issue either slowly address issue tensor factorization multilinear noisy incomplete tensor miss rank automatically specify induce individual hyperparameter latent variable place due resort characterize approach effectively extensive illustrate determination robustness application synthesis preliminary multilinear cp specification inference mixture datum follow conclusion dimension tensor g order denote capital tensor denote letter ni I tensor sum element product vector size hadamard without hadamard kronecker rao reverse define q order entry tensor observe entry tensor noise assume shorthand term interpret rank tensor factor row wise vector cp factorize parameter nr n index essential factorization multilinear model general latent selection computational costly elegant automatic infer tensor overfitte hyperparameter minimum hyperparameter rank determination relevance determination ard ard weight principle analysis consider parameter sparsity place factor govern precision share latent matrix far factorize dimension point effective dimensionality bayesian prior yield write parameter posteriori extent squared impose impose develop eq hyperparameter analytically vb framework cp seek low occur assume factorize assumption form th factor family parent form parameter graphical inference mode message co parent parent term factorize see n denote subset associate column accord q whose mode need introduce random n sec appendix attempt compute multilinear quadratic length efficiently fix rao imply interact take account simplify multilinear finally posterior approximation update moment intuitive give follow update information tradeoff fitness prior posterior firstly coefficient similarity scale fitness note via posterior crucial automatic incorporate posterior see sec th mode hence posterior intuitive sum square lead turn perform message parent include incorporate however posterior straightforwardly vector leave inner matrix expectation outer th expectation quadratic n see entry intuitive relate number residual fitting square entry essentially take follow eq intuitive related factor also rest get solution initialization point initialize strategy draw singular rank upper value tb incomplete indicator update zero component computation entire summarize automatically update new prior affect hence posterior becomes force prior information component unchanged iteration use posterior approximate yield n n sec appendix input size tensor generally complexity w automatic reduce rapidly complexity polynomially optimal highly tensor parameter avoid procedure require predefined completion factor miss entry exist point estimation deterministic develop powerful satisfy cp account local assumption rewrite q probability adjacent define coefficient sum firstly keep unchanged change conduct synthetic world fully cp factorization completion base scheme aspect reconstruction performance entry world image intel memory synthetic procedure factor n entry uniformly mark video material tensor factor component tensor monotonically indicate effectiveness capability denoise estimation evaluate tensor e extensive vary condition size matrix svd result tensor group tensor size evaluation initialization incomplete tensor snr db evaluation initialization see incomplete tensor evaluation vary miss tensor snr db evaluation perform ratio rank miss miss rank initialization initialization term tensor complete detect snr db decrease deviation noise miss high missing achieve db ratio note achieve even snr true fail ratio determination primarily true level occur may helpful determination tensor generate rank db miss statistically consistent evaluate repetition fig ratio perform achieve miss precisely rank outperform miss extremely completion sparse conduct additional snr experiment see h image benchmark show method image tensor conduct four missing pixel condition snr pixel noise free c experiment text text miss entry image pixel miss mask compare describe ratio completion tuning ground pixel lr lr nf nf c c mp runtime runtime runtime ratio superiority additive obtain noise obtain smooth color recover predictive obtain clean remove effect appear method factorization significantly overfitte result observe mainly intrinsic low natural pixel recover image cause
reconstruct clean noise white representative voxel corrupted choose comparison learn contaminate tensor reconstruct table summarize reconstruction set enable regard image measure value slice fourier result stack choice radial illustrate fig regularizer read compare minimize consist square total utilize adaptively synthesis slice noiseless mention table fast suffer eight time volume b separable operator multidimensional separable nature learn reconstruction reduce operator larger design multilinear demonstrate corollary tu project team de nd conference publish representation certain research representation recently gain increase multidimensional thereby drawback inherent structure achieve enforce separable learn deal multidimensional operator code great popularity resolution compressive sense last combination atom synthesis counterpart sparsity transform domain nm encode e operator prominent analytically specific signal representative follow give short separable operator algorithm framework separable learn alternate step current approximation noisy regard row operator signal orthogonal requirement cost design include ensure incoherence sense unit structure matrix signal ultimately restrict limitation memory computational tackle enforce additional synthesis examine combine author analysis svd separable introduce separability maintain inherent imaging capital letter letter letter index accordingly necessary tensor section able deal multilinear call product transformation separate mode j nn mode eq result n n I offer understanding constraint rewritten vector kronecker product interpret operator additional extended certain operator regularizer demand label row operator maximal row trivially linearly condition realize enforce operator enforce point separate mode unit norm rank kronecker trivially neither property kronecker operator fulfil n iii incoherence regularizer barrier barrier within propose control able gradient base manifold many range medical one represent mean hyper third encodes scene encode information homogeneous kind extract flat synthetic parameter operator act local signal appropriate entire constitute fidelity serve optimization approach multidimensional operator minute c method
extension set trivial intractable support foundation project research contract college fellowship supplementary material bring autoencoder learn fig h separate thin black draw dynamical e case system dynamic challenge measurement mapping practical identification technique additionally identification inherently high identification mean deep demonstrate enable predictive system stream eeg sensor network dynamical desire design important role translation surveillance identification mathematical dynamical measurement functional relationship different measurement study standard exist expectation maximization method linear inherently difficult active last decade state survey smc dynamical hard un identifiability local overfitte high possesse dimensionality dimensional purpose automate art parsimonious dimensional deep architecture stack auto convolutional audio product google amazon feature auto generative mapping encoder mapping reconstruction vast number study latent kernel two mapping tb model use deep encoder network low nonlinear system dynamic learn illustration encoder map low map subsequently predict access encoder decoder motivate encoder contribution paper dimensional representation dimensional datum experimental embed latent performance compare separate consider image low compactly represent since insufficient capture dynamic feature g velocity map feature dynamical turn instant neuron none height execute x try neuron miss neuron miss neuron every neuron try neuron z try center leave dim align align center input leave x neuron text height execute try every neuron try neuron neuron fill z every neuron try fill every neuron align dim center align control state input feature encoder feature map decoder identify dynamical prediction extensively identification decade measurement ahead achieve predictor relate measurement difficult special state past nonlinear corresponding predictor long somewhat work autoregressive predict predictor nonlinear function network parameter function normally estimate back additional access value compute approximate detail final illustrate none height cm execute begin miss neuron try try count miss neuron neuron miss neuron neuron try neuron neuron try optimize encoder model propagation compute auto strong encoder pca exploit encoder encoder pair encoder component layer good auto encoder machine pca dynamics link arm horizontal plane velocity serve dynamic solely pixel nine consecutive ground truth frames instant ahead prediction joint bottom plane pixel speed dimension velocity model layer neural evaluate prediction prediction precisely iteratively feature instance identification illustrate assume image row prediction auto encoder sequentially model obtain predictive step ahead prediction train optimize encoder perfect job frame ahead reconstruct ahead prediction see model prediction training separate auto perform model reason bad predictive believe auto tb joint separate training image display red validation display space separate present separate training enable dimensional dynamical feature even place enable extraction model behavior compact manner datum one dimensional would insufficient period far along output tb xlabel ylabel pos north east width height plot plot subspace separate training compare display blue great image learn parameter sequentially predict correspond poor joint give naive predictive slightly horizon compare subspace linear model restriction capture embed sub optimal frame pixel increment direction learn autoencoder use network dynamic long term illustrate validation frame accuracy datum display encode dimensional four corner corner within correspond case separate exhibit material display achieve fit error minimize fairly control interested long horizon
bandit policy round case run algorithm policie abstraction ability rich notion optimization oracle also greedy suboptimal class thompson confidence bind efficiently maintain compare algorithm barrier restrict access via use key similar distribution policy convex oracle rather algorithm program sparse general base warm start total call oracle round certain arguably simple variant contextual bandit technique let context policy joint context contextual bandit vector observe action receive observable record result round ta ta set interaction record round rx instantaneous reward maximize regret round tx differ interaction also reward estimate h tx assign possibly greater detailed agent probability condition pick history policy maximize score one policy enumeration impractical general work learner reward r r return choose randomly recommend drawing x maintain necessarily policy pick like schedule epoch schedule epoch solve constraint require rescale version regret mass exploitation require place action exploration side control estimate accurate good greedy style section constructive feasible solve require round eq epoch schedule allow initial action reward history history q algorithm analysis potential progress analysis substantially iteration give iteration express theorem output weight unchanged loop start call loop loop start historical context vector involve support compute long identify identify eq action check tx td recall round dramatically reduce epoch schedule call schedule epoch algorithm present call practically seem epoch precede computation nothing intuitively expect warm call warm start different schedule epoch schedule warm start call complexity also example cost scaling observe start represent context round operation function need obtain involve oracle cost update rescale step store constant enter scan rescale run attractive call policy specifically schedule weight epoch similarly oracle ever weight sampling action construct substantially low formally schedule q compute put entry epoch schedule deferred bind call give mode defer appendix policy control variance hand side x discrepancy compare informally ok constraint q summing apply martingale regret optimization weight write simply distribution potential epoch unnormalized vector nonnegative might context ignore combination measure far regret proportional thus encourage action regret aim algorithm appear definition intuition calculus partial roughly constraint negative decrease increase weight constraint fully minimize corresponding derivative large negative analyze argue sum weight step must weight remains challenge show significant reducing argue respect nonnegative prove substantial execute suppose weight copy potential maximize use taylor fact give cause decrease least lemma number execute call approach combine round round mean oracle less fact warm epoch epoch potential end epoch start epoch early round large least potential write epoch epoch change advantage term change relate specifically expect reward optimal high along use intuitive detail defer potential increase decrease update round round require lemma explore first cover bag dim minibatch algorithm several baseline overall still plus total section online oracle classification take class action set call answer question thereby complexity track good sample full maintain fix upon suitable square amenable implementation file due public document dataset tf treat action reward evaluation take report well explore exploit explore powerful baseline bagging predictor example replacement predictor evaluation impractical run dimension hour alternative hour somewhat surprisingly occur decay sampling impose adequate large cover baseline simple modification report use default contextual use doubly supervise multiclass simple rate report effectively loss achieve algorithmic statistically general call round remarkable work believe scalable contextual bandit directly acknowledgement thank discussion part microsoft inverse score transformation action action tv epoch particular proof sketch union probabilistic reader probability distribution nc schedule epoch high epoch epoch contain round policy epoch epoch follow apply union imply union choice allow epoch schedule q increase observe event statement probability epoch round union weight epoch constraint define need require follow outline large much hold epoch achieve satisfie inequality inequalitie reward together variance induction base triangle inequality km epoch q round optimality exist epoch hypothesis rearrange give moreover display simplify yield optimality epoch inductive imply eq apply display simplify yield complete step optimization low epoch epoch trivial lemma straightforwardly translate involve break round epoch epoch third simplify step evaluated recall q epoch schedule whenever probability r ta mt inequality union bind hold least double lemma whenever epoch constant th definition moreover epoch imply last elsewhere assume achieve km bind execution must already equality exactly recall let handle jensen concave compute change define direct taylor choice since break piece specifically algebra k throughout vector produce algorithm mean depend proof note thus lemma combine eqs statement rewrite cumulative weighted reward eq nonnegative expression epoch expression note nonnegative tx ta
convenience monotonically collection monotonically modular modular propose objective induce ground feasible set focus start exchange pose function induce detail trick build dissimilarity number image category synthetic gain fig display field demonstrate challenge categorization due foreground attack select rf suppose foreground object solve rf candidate measure building nontrivial call pyramid error pairwise distance sum pyramid simple classification foreground improve categorization image foreground object sift image region match alignment sophisticated mid level adaptively pyramid layer pool salient structure enhance classification handle gain foreground encoding segmentation essentially cast call find field pyramid mid pool predefined region however mid pattern category fail handle prominent image handle propose framework discover field mainly category merely say location find rf category accuracy note highlight mean receive computer capture foreground construct rf submodular greedy guarantee rf nontrivial mid layer pool usually dimension quantization code pyramid image low sift preserve meaningful foreground object design nonparametric match image benchmark database essential elaborate pyramid nonparametric concluding keyword framework field multiple vision field learn aid image detection discover salient region salient something meaningful require van use salient scale detect object propose spatial pooling location location foreground mid learn detector location et pool predefine pattern notable translation foreground improve performance introduce address pair require wide submodular finite fa fa fa fa aa state add help add please therein similarity recently mid svm mid generate pool pyramid demonstrated learn convolutional neural pairwise mid reliably quantization mid level sift descriptor motivates propose collaborative weak fed category rf capture foreground object training query preserve meaningful foreground rf candidate vision formalism selective suppose category generality template rf candidate candidate candidate overlap grid foreground reliably correctly part many select desirable cover worth difference rf pool grid desire rf method pool image explicitly consider scale translation extract size object detection region proposal rf force fed object appearance contrast method preserve valuable allow rf capture object salient image intra inter prior similarity pairwise besides multiple category least capture principle inter exploit similarity graph similarity candidate large measurement nontrivial elaborate put index sum meanwhile index minima follow indexing maxima sufficient h follow submodular property exist monotonically submodular benefit hereafter explicitly enable extract balance help overfitte call balance image positive understand please gx I demonstrate add achieve prefer another result balance follow monotonically bias location specifically image mild center mean search center still capture fidelity intuitively position descriptor point rf furthermore fig pyramid arrive rf index within pyramid third rf analyze descriptor grid define similarity accordingly two parameter control transformation actually similarity gaussian rf dense connect uncorrelated candidate threshold large nn similarity costly computation adopt dense extraction scheme descriptor extract complexity construct extraction descriptor consistently extremely consume either fast approximate construction incorporate detection descriptor extraction sift sift descriptor resolution calculate rf candidate similarity dense extraction produce unnecessary meaningful sift descriptor instance reflect principle bias intra field classifier incorporate rf put grid denote descriptor feed image sift rf category exploit tree image qualitatively validate effectiveness discover public benchmark evaluate object categorization generate first object meanwhile away intersection expect point intersection please center bias prior set control parameter gain bottom gain point intersection gain inter image demonstrate effectiveness approach correlate foreground object contain category intra variability location variability scale resolution aspect benchmark class randomly exactly extract split r lc several one mid level represent image network lc base locality constrain linear feature learn mid pyramid kernel svm lc deep classifier dictionary mid level codebook sift encode codebook quantization sparse code feature pyramid pool image neighbor closely relate use sift comparison illustration select outperform sift detection remove noisy descriptor position sift descriptor fail notable change translation reason object find behind small similarity construct find performance visualization suffer guess objective small contribute moreover image merely ensure center constraint foreground image help construct essentially normalize divide transform control plot database meaningful outcome curve demonstrate work image foreground object category exploit model select vertex suffice guarantee propose pyramid pairwise merely preserve foreground object final try result research similarity feature propose even construction sophisticated representation consideration efficiency learn deep
entropy play role non mechanic find field hierarchical structure e et al zhang document classification community shannon address nature specie interaction specie individual system thorough interpretation diversity effective specie index replace relative maximum likelihood bias community rare specie unobserve perspective diversity hypothesis specie first shannon symmetric first technique analogous prior neural response dirichlet impose narrow shannon priors diversity exhaustive intensive general prior specie possibly infinite preliminary contribution variance distribution et able result moment prior parameter refer extremely modern tractable dirichlet relate bayesian et posterior discovery large diversity notice briefly class infinite tuple atom exhaustive account random partition infinite satisfy relation usual specie observe th new unobserved specie kp partition belong symmetric poisson family mix discrete distribution characterize formula dirichlet prior exponentially normalize generalized belong derivation result paper stress allow nevertheless explicitly two moment shannon entropy induce et poisson infinite dimensional prior belong recursion index easily shannon generalize already function r derivation third omit size specific prior stick break kind j second moment shannon adopt poisson dirichlet prior inverse breaking recently size atom difficult close entire allow gibbs weight dirichlet xx arise respectively follow et stick break biased atom follow easy eq shannon dirichlet numerically prior suitably formula comparison purpose standardize shannon prior guess index diversity effect prior guess concentration around behaviour shannon al poisson dirichlet combination ht prior standardized index dirichlet stick break biased concentration symmetry produce fisher prior family law random discrete gibbs mix simplex shift species parametrization bayesian application formula index shannon calculate partial derivative respect sake brevity omit explicit formula prior shannon fisher characterize prior put population appear flat uninformative suggest shannon entropy parameter theorem et sect moment provide inequality study posterior moment binomial specie observe relative moment generalize entropy general substantial theorem moment close extremely calculate application uncertainty derive high density interval simulate observe relative dirichlet index shannon n variance moment mind agree moment posterior prior arise formula variance may mind generality evaluation frequentist bayesian already nevertheless provide procedure conduct bayesian count collect site refer hill count formula realization size observe give specie interval estimate shannon interval likelihood ml correct fisher table nonparametric thompson estimator adjust coverage adjust specie preferable frequentist fraction parameter nonparametric derive sample thompson correct account miss greatly choose robust conclusion population independently stress propose account presence unseen prior place theoretically relative finitely update unseen fisher prior low suggest posterior moment posterior summarize shannon index kind use notion rank atom discover th least eq still almost surely th integer pn v determine integer moment recall h recalling eq limit theorem specie prior relative specie moment follow one particular allocation integer q integer sum partition function analogously integer suitably countable cs intensive bias indicator species pr partition nonparametric prior cm marginal gibbs sample diversity specie cm neutral species dirichlet break representation prior discovery formulae gibbs ann relation species cm shannon species finitely exchangeable partition bayes entropy description study site size shannon wiener generalize cm hierarchical model inverse nonparametric discover specie
invariant geometrically transformation could expect world set sized texture mix obtain patch descriptor gray value class absolute sized descriptor use length histogram histogram bin size vector texture retrieval representation patch texture patch descriptor enable fig texture represent color visualization pca texture encourage cluster enable quantitative assessment texture patch compact descriptor accuracy accuracy lead provide comparable magnitude descriptor effectively obtain accuracy apply consist sampling patch multiple texture represent use descriptor color texture compare transformation descriptor assign texture inferior texture significantly texture connect mathematical transform class particular pool thereby develop histogram learn great discriminative illustrative world texture definition matrix canonical basis combine standard invert determined write pair recurrence deduce combine similarly zero da da da da tb prove result deduce meanwhile property trace make definition rotation achieve vertical coordinate horizontal coordinate index tuple implie combine translation coordinate give cyclic translation writing meanwhile imply combine give patch give cyclic rotation partition eq equivalence coefficient class claim mm mm mm mm mm thompson edu thompson ac uk transform transform hadamard multiscale dyadic show appear different phase change effectiveness demonstrate invariant invariance coefficient basis algorithm thus autocorrelation covariance sign group permutation view discrete unitary great approach autocorrelation describe multiscale texture operator harmonic continuous study ambiguity important role code new connection signal describe hadamard autocorrelation powerful tool detect impose shift invariance variety code removal include face texture pattern recognition autocorrelation cyclic dyadic suit represent multiscale texture hadamard suited counterpart wiener dyadic autocorrelation aforementioned autocorrelation invariance transform patch follow color patch sample texture though exhibit obvious translation able texture display sample patch break patch transform absolute patch display transform patch section transform invariant dyadic illustrate theory example transform pooling invariance significant distinguish class transform summary theory transform result invariance multiscale transformation inner section hadamard transform show material suit application give tuple multiscale group detail example matrix ht isometry whenever unchanged sign permutation property representation detect illustration figure simple texture identify come texture transform demonstrate ability classify patch capture describe moreover multiscale ensure transform patch texture exhibit sign matrix function transform equally view familiar tool mean hadamard inner sign detail think multiscale rotation four symmetry multiplication set label vector positive example permutation dyadic multiscale term give coarse give fine first display change product second row pattern hadamard third fig example multiplication associate inner frobenius b da b q signal matrix expand connect autocorrelation convenient expand orthonormal isometry next bridge autocorrelation equally hadamard transform index autocorrelation band information invariance binary subset hadamard index index tuple characterize combination hadamard autocorrelation define appendix fundamental transform way exploit transform equivalence coefficient exploit transform coefficient significant distinguish class ht absolute coefficient invariant say allow absolute value cyclic absolute pooling build invariance transformation often pool provide principled partition within follow absolute coefficient give equivalence average
huge assume infeasible homogeneous kind audio stream divide duration feature extract frame model compute segment acoustic pass slide window similarity measurement segment identify coarse present news language air air access language follow describe section consist acoustic call acoustic change common detection divide audio segment small speech bic adjacent high segment acoustic source adjacent segment duration different news audio air ground segmentation obtain toolbox news bic audio segmentation audio bic literature face automatic segmentation inconsistent result identify acoustic next news audio news length read propose measurement propose two technique news audio anchor identify change stream duration second use point detect pass depict pass feature vector feature combine together group second group model calculate criterion extensively segmentation metric efficiency window co vector segment dissimilarity show audio audio segment see image represent acoustic audio evident technique reliably audio stream detect change pass proportional segment point obtain second perform second pass detect news actual center stream audio extract audio literature automatic segmentation pass pass coarse acoustic point acoustic self identify acoustic propose audio segmentation audio news music actual news automatic audio news time align build corpora address read map text exist sub solution segment acoustic reliably acoustic pass audio process audio audio stream audio pre index news movie etc segmentation corresponding audio music news addition speech automatic extraction news segment correspond speech
avoid avoid explicit covariance enkf kalman filter forecast observation ensemble filter seek specifically posterior ensemble inverse minimum operates sequentially apply forecast eq scale perturbation ensemble covariance square background speed carry variable ensemble k solution read perturbation represent analysis operator linearize consequently linear operator jacobian observation give multimodal carlo mcmc algorithms metropolis distribution complex density invariant mcmc generating proposal accept reject generally powerful may hybrid present filter assimilation hybrid carlo hmc know hamiltonian monte physics attempt drawback reduce explore sample hamiltonian operate phase total describe hamiltonian dynamic differential evolution map computation flow replace reversible integration method five st stage stage take abuse solution draw probability make analogy hamiltonian auxiliary momentum hamiltonian logarithm target probability auxiliary momentum mass matrix hamiltonian canonical equal show momentum variable hmc algorithm build initial summarize state issue numerical represent mass impact final affect diagonal efficient draw numerical current increment energy hamiltonian stage stage integration discard draw variable accept proposal continue many distinct draw filter enkf representative member even assimilation forecast linearize member draw estimate filter principle remove logarithm alternative enkf describe assimilation sampling stage k forecast member next result forecast pdf providing give ensemble follow chain forecast enkf acceptable choice stationary calculate ensemble base forecast frequently emphasize build full definite vary member warm entire sample covariance increase state necessary typically fix ensemble member ensemble matrix flow lead water set diagonal variable describe x index circular fashion experiment component range simulation algorithm synthetic create reference background system different observation operator complexity level linearity six cubic obtain trajectory magnitude state square component operator differentiable absolute et highly nonlinear scaling control nonlinearity model step size sampling sampling first test guarantee satisfactory take cost tune trial observation chosen number calculation filter numerical realization potential acceptance different metric analysis observation reference assimilation trajectory span reach perform burn noticed converge number burn stationarity work burn member generate decrease generate ensemble retain number usually inter chain parameter control requirement upper need order step consequently stable ergodicity markov chain length begin hamiltonian step benefit ensure result analysis system instance filter median instance central variance vertical central box length outlier plot outlier exception hilbert rmse enkf closely indicate inter instance box rmse central represents extend height number reason size enkf unit chain rmse filter plot represent median instance blue height central box consider plot show representative sample satisfactory rmse red failure outlier hilbert space enkf suffer outlier indicate panel inter rmse sampling filter plot median variance vertical line height plot red scheme rmse make divergence high continue define hilbert fail rmse tune give satisfactory indicate panel time indicate inter rmse sampling filter box times length plot cubic enkf due sensitivity level nonlinearity operator filter sampling converge stage satisfactory h indicate inter filter rmse across central blue vertical extend outlier plot red fine convergence see reduce size figure lead use indicate step rmse instance filter box plot red median rmse represent vertical extend height red observation operator jacobian operator sign experiment mostly filter hilbert forecast enkf analysis almost identically quadratic operator h rmse filter show rmse central represent times box consider outlier result obtain high bad rmse increase panel time indicate step red line instance blue height central box observation perform sampling filter fail observation uncertainty level linear ensemble small reasonable panel inter red line rmse values box vertical times height box plot size four linearity outlier give satisfactory even analysis select h use panel indicate inter chain step box rmse central variance line times height box outlier red jacobian difference alternative factor differentiable perturbation nonlinear enkf performance performance factor indicate inter rmse box across instance central represent variance vertical time height box outlier plot tuning notable improvement stage behave change panel inter step plot represent across central box variance time central plot red sensitive uncertainty degree nonlinearity test observe test performance performance level observation level state frequency frequency different indicate background observation htbp frequency state background observation standard different indicate observation deviation htbp panel deviation htbp frequency background operator parameter burn tune step refer step test optimal suggest realization performance integration outlier infer principle enhance inter indicate careful tuning step number step lead filter deviation error panel box plot line rmse instance blue represents extend height plot red controlling setting number use validate ensemble hilbert suffer outlier assimilation several exclude combine ensemble care density alternative future test capability challenge factor observation perturbation change correspond measurement enkf increase length well short hamiltonian four prove observation satisfactory large step size indicate indicate panel inter instance box median rmse vertical times height box red hilbert rmse relatively acceptable reasonably closely plot chain figure obtain good result hilbert low stage outlier chain summarize enkf observation operator show size setting name short version respectively sampling filter include deviation step filter water sphere water equation provide simplified model describe mechanism angular longitudinal height homogeneous wind discretization discretization longitudinal vector combine wind wind height integration adaptive reference trajectory synthetic add three observation wind magnitude wind create reference noise model deviation background magnitude reference condition background wind wind model background account create ensemble kalman hour ensemble create add condition covariance ensemble base covariance covariance method totally perform well algorithm make future propose filter burn stage step step enkf sampling order outperform enkf filter study moreover forecast covariance h hamiltonian number step c observation min std quadratic min std observation threshold min max observation operator min std min std h c cm std std std std std propose assimilation posterior sampling avoid need develop adjoint solution operator nonlinear variance offer analysis implementation require matrix attractive assimilation operational experiment carry linearity enkf outperform enkf continue satisfactory case enkf assimilation machine challenge failure subset terminate member run ensemble member considerably herein replace implementation enkf ensemble member enkf member posterior probability add new direction computational integration successive ensemble tuning context operational resolution perform comparison acknowledgment fa computational present five numerical position sensitive choice position lead test design state space subtract infinitely total system hamiltonian system time experiment whenever target numerical nonlinear operators equation time stability achieve step one three advance hamiltonian time advance
small approximate mechanism generate matlab simulation numerical device hold device memory expect use large whenever measurement interestingly bad moment execute communication author binary nonzero normalize eq although conduct result confirm e perfect measurement use general recall collect measurement I ratio consider procedure variable compute derive formula tight eq turn expectation derive let complexity require write choose appendix bad analysis interestingly attain nonzero suffice measurement note hold know know bad would still change binary e convenience error binary essentially least simulation study least poisson use replace require measurement suffice hope demonstrate poisson illustrate reader lemma along poisson confirm accurate unless low case tool closer range bottom curve low figure small panel fix closely interestingly useful symmetry attain confirm suffice choose compress compressed sense stable interesting nonnegative highly entry nonzero entry stable maximally skewed theoretical extremely away preferable complete proof similarly conclude note furthermore thus increase complete computer university statistic university nj usa department nj usa recovery often nonnegative develop adopt compressed design maximally skewed average fraction become stable summary dense
original description kind scenario occur search efficient code case describe analytically happen sampler ask produce program originally dataset text generative text list lambda assume assume assume apply observe flip bernoulli program flip predict program text program result novel argument program abc lambda std std assume poisson lambda exp begin expression expansion program univariate fed illustrate sampler account endow argument entire family box parameterize refer conditional sampler abc begin assumption truly improper program argument generalize abc time summary individual summary take value correspond statistic hypothesis hypothesis equivalent coin turn penalty accumulate line aside probabilistic programming language program expressive level environment signature express random type constant integer crp discrete process prior base real mixture uniform common primitive compound compound sample representation discrete dirichlet base generate compound procedure count compound production rule environment incorporate input name name type current compound avoid program return possible manually took write translate common sampler example require single production corpus sampling corpus production smoothed prior couple inference programming perspective result approach abc penalty source final preliminary probabilistic production employ probabilistic six histogram sample program randomly sampler domain variance mode repeatedly blue histogram exact leave plot infer inference converge code abc lambda safe par par lambda stack safe par par bottom infer program text assign use sampler program continuous feature infer two kolmogorov test vs analogously histogram program program express program salient characteristic text bayesian costly posterior particularly repeat posterior predictive inference representation option particularly order language program aim probabilistic interest encourage take beta metropolis give successful trial probabilistic probabilistic repeatedly produce statistically distribution infer probabilistic analytical include program text generative tb observe flip flip observe flip program abc safe safe beta safe safe predict beta binomial interest top salient probabilistic induce probabilistic program bottom exactly analytical posterior posterior sampler indeed close exact novel synthesis raise answer key really synthesis synthesis inference goal program search single generative program possess characteristic length etc program text intractable future basically available employ latter whose match know actual temperature schedule ergodic require result certainly surprising good job report open well particularly goal genetic way cumulative incremental program convenient text normal already learn subroutine inductive gain structure continue internal experience ability text match human seem something powerful piece intelligence inclusion generalise human reasoning suggestion comment van david college conclusion author necessarily reflect air agreement fa u reproduce view conclusion herein policy express air laboratory false frank department science united institute mathematics program encourage empirical suggest technique probabilistic language sufficiently powerful enable also might future program complete high probabilistic language simultaneously procedure procedure text programming description merely particularly degenerate probabilistic programming language possibility text generative high paper account effort directly sampler similar observational datum potential collection distribution box sampler automate discovery might perform leave bernoulli impose hierarchical sampling procedure fit human random variate somewhat mh probabilistic forward generation generate ideally program result program aim program evaluate generate generalize posterior expressive family suffer distribution valid marginal high programming probabilistic program effort step probabilistic program relate former treat text find exactly match latter generalize either introduction modern program specify equation use traditional force enumeration find inductive logic genetic constraint language support choice make logic lambda theoretical program inference unlike learn program sample observation input pair generalize main objective statistic field learn text say parametric program code structure sense manner observational
collapse implicitly dimensional come automatically quite approach number low way crucial computationally require runtime obtain train dr seek see dr nets model dr learning metric dr projection quite rely objective lda input space class scatter minimize scatter maximize latent among variation manifold space use dr fed disadvantage filter objective dr proxy show filter place corner closely dr equivalently project nearby latent apart achieve metric however metric solve semidefinite learn mahalanobis optimality long approach dr unify far generalized hinge svm extract closely dictionary learn always latent implicit new need contrast explicit mapping test operate linearly nest linearity svd would train mapping reduction regressor jointly nonlinear result iterate provable objective filter approach secondary dr mapping separability scatter little classification optimize place collapse maximally dr algorithm generalize specific dr combination extraction mapping net jointly optimize classifier acknowledgment part nsf award section lemma prop thm corollary definition conjecture conjecture electrical computer science california false dimensionality dr use preprocesse classification first learn obtain optimize classification jointly particularly dr method algorithm train rbf svm step svm usual close art runtime dr mapping latent class train jointly dr extreme tends manifold centroid linearly maximum dr preprocessing task low dimensional costly importantly learn particularly dataset avoid reason dr remove uncorrelated mostly away dr adequate freedom label dr inform call supervise dr input dl vector supervise learn train input supervise dr sense minimal intra scatter scatter supervise usually encourage separate input manifold make filter even pca proxy real objective minimize particularly dr filter considerably nonconvex particularly important question arise propose generic jointly optimize loss rbf armed dr classifier latent apply filter nonlinear svm achieve art short version appear conference paper describe svm pattern dy n want optimize usual separate hyperplane give term weight bias linear slack difficulty heavily simplified use introduce nested idea break functional follow prove seem trick optimize original target rbf next algorithm svm svm ordinary work exist scalable svm train independently regularize low generic net focus special include radial universal commonly unique memory mainly drive involve gram exactly reduce warm cholesky run include slack alternate scalar cost kkt lagrangian kkt pass reduce kkt express dual optimum achieve solution summarize step contour increase red set plot corner dot denote dot step several vs multiclass vs decision label determine svms objective function parameter slack svm consideration variable typically active solve matlab toolbox binary class high test involve step vs take vs svms jointly optimize classification becomes iterate subproblem rbf form update remarkably although optima initial hyperparameter margin reduce initial quadratic parameter increase time early iteration validation go improve fast iteration suffice hyperparameter usual mapping massive parallelization suitable latent versus class indeed processor center latent state art consider ideal infinitely l summary approximate ideal extent close maximally separable finding map trivial seek piecewise constant hard learn pca lda test split standard neighbor kernel svms inputs unsupervise pca kernel lda neighbor hyperparameter algorithm svms choose mac hardware remove word appearing reduce extract pca create split item validation hyperparameter try margin mean standard rate classification dimension bring cost fix dimension superior outperform consistently binary mnist include balanced validation set separable infer svm nonparametric nonlinear dr explore two pca validate pca lda use demonstrate incorporate regularization improve generalization neighbor bottom projection training algorithm perfectly use due power lda svm though remove noise reduce show pca training c error projection classify digit original evaluating require huge store solve center choose hyperparameter unnecessary hyperparameter careful kernel width parameter width width svms explore lda svm mnist experiment lda poorly perform pca initialization rate vector use kernel time basis obtain large speedup explore experiment lda poorly svm similar fig bottom different dimension error quickly much configuration different projection bottom visualize latent lie dimension overlap completely separate see view nearest pca gaussian gaussian processor speedup function early latent avoid subset digit visual comparison vs parallel toolbox scheme alternate scheme optimize rbf progress actual runtime mnist odd minimize svm quadratic dimensionality weight solve solver nest iteration spend alternate alternate iteration suffice find optimal small eliminate increase progress alternate slow nonlinear gain algorithm nonlinearity thank auxiliary coordinate
edge hypergraph incidence none algebraic constraint finitely individual become interested characterize incidence framework check independence variety compute gr adapt polynomial incidence hypergraph show subspace take system unique incidence incidence hypergraph learn give point datum conversely support tight hypergraph finitely quantify corollary pick word straightforward set pick random sphere denote index row minor column represent minor set deriving equation solve span incidence constraint k x sd ss underlie incidence equivalently independent set parameterize put space give subspace degree freedom leave potentially remove section incidence generally existence system exist finitely real complex solution system incidence become I framework characterize incidence check ideal generate variety computationally well linearize singular algebraic incidence independence maximal geometry property generic intuitively dense variety generic property framework hypergraph property framework hypergraph alone framework framework variety avoid framework relate restrictive ideally variety possible necessary explicitly easily give appropriate hypergraph area purely draw combinatorial I variety combinatorial go theorem follow combinatorial incidence hypergraph furthermore capture incidence existence jacobian algebraic matrix space trivial incidence take jacobian coordinate jacobian corresponding give lie let oriented form coordinate add v td notice volume coordinate ct kt q incidence j j three jacobian jacobian form incidence jacobian row row per framework realization way subspace chooses pick simplicity pattern show regular sketch regular state exist jacobian algebraic isolate could explicit component imply take jacobian less corresponding hypergraph theorem prove follow define matrix graph index index accord map graph one correspondence loss switch variable name determinant notice summation ready expand hypergraph union incidence determinant trivial trivial subgraph edge copy arrange pattern hypergraph group belong coordinate laplace rewrite determinant coordinate separately row observe zero expand hypergraph decompose map contain entry recall decomposition always correspond row zero copy particular pick row row non zero lemma minor non observe full lemma incidence expand hypergraph form inside decomposition expand hypergraph behave case non combinatorial characterization pure calculate open avoid pure span otherwise expand tight pure give notion dense generic expand give hence pure subspace position original jacobian solution converse since generic independence datum point sphere corollary dictionary quantify pure fail framework subset solution system pick sphere less construct dictionary pick major construct underlie hypergraph subspace ss stage expand construct minimal hypergraph small integer vertex constant least subgraph word structure verify vertex construct hypergraph conversely hypergraph put vertex remove vertex tail contain one move one slightly expand hypergraph add edge copy additionally count expand construct put add move one shift move inside take os entire hypergraph iterate therefore regard construction underlie hypergraph get incidence arbitrarily incidence arbitrarily coordinate full row maximally pure theorem algebraic jacobian fail pick find similarly entire take time complexity paper point incidence completely characterize hypergraph recover number corollary main pick algorithm additionally independent claim theorem wang sparse obtaining upon geometry span hypergraph characterize underlie specifically incidence isolate specify combinatorial systematic algorithm performance datum point know dictionary satisfy consideration interested relative dictionary arise context process machine vector recovery minimize represents lagrangian iterate start solve pursuit update convex overcomplete ill point difficult reduction exact problem dictionary np directly though np learning solve produce dictionary selection alternate mod iterative formalism mod posteriori recover iterative truncate singular take atom atom form orthonormal sparse stage relaxation minimization well theoretical true several algorithm strong constraint minimization find dictionary require basis al provable overcomplete via iterative svd dictionary provable however overlap dictionary frame dx xx x vector resp resp dd hypergraph fit point dictionary hypergraph machine complete characterization yield solution dictionary e isolated dictionary specify size related however highly pick dictionary sufficiently sufficiently provide systematic relate together approach require follow combinatorial character incidence system dictionary dictionary system another know characterize hypergraph give call pure dedicated specific framework instead subspace directly linearize jacobian uniform generalize non impose systematic increasingly constraint classify whole independently interesting learning find restriction dictionary dimensional subspace point dimensional union frame satisfy lie basis deal find subspace residual iterate point robust method pca algebraic union homogeneous fit determine obtain small span subspace set specify give intersection necessarily general closely relate intersection condition come dictionary union small span subspace outside pairwise intersection directly recursive small set dense problem follow step solve decomposition always apply iteratively high step problem follow span class subspace
nucleotide cycle positive identical number equivalently seq length original sequence express nucleotide probability constraint signal range infinity frequently expansion power series bivariate positive denote nucleotide flow cycle show nucleotide flow flow cccc specify complete nucleotide incorporation nucleotide flow recurrence seq cycle cycle seq nucleotide flow seq obviously recurrence seq nucleotide flow cycle nucleotide cycle reason solve close form solve initial condition extract appropriate symmetric nucleotide flow cycle table together nucleotide probability elementary focus symmetric value add need probability give second sum sum probability become seq seq row factor expansion nucleotide extra small practically ignore big contribution clarity value eq converge upon dominant eq quickly expansion q availability make include number cycle cycle variance positive cycles eqs calculated series eqs dominant q interesting show eq signal twice cycle probability linearly flow cycle linear growth size combinatorial system govern naturally distribution detailed variance plot nucleotide probability nucleotide eqs exact calculate eqs distribution respectively exact accurately normal distribution variance nucleotide probability reach positive nucleotide broad nucleotide probability fact reach eqs normal equal nucleotide cycle number use fix calculate point dominant keep term ignore depend nucleotide probability linearly perturbation theory nucleotide perturbation theory cycle nucleotide nucleotide eqs curve calculate nucleotide accurately equal calculate eqs curve normal calculate nucleotide respectively instead fix seq cycle process create infinite let flow cycle seq flow cycle seq nucleotide incorporation infinite nucleotide flow nucleotide nucleotide nucleotide within nucleotide cycle length seq flow seq first signal must happen signal flow base stop nucleotide base part ix nice property mean calculate coefficient differ long nucleotide nucleotide linear still limit gaussian linear check analytical develop program seed give close simulation website simulation derive cycle signal run variance average put relation example first eq put fact get e e acknowledgement clinical science rr national center resource health sequence distribution signal sequence cycle nucleotide derive cycle nucleotide software development next generation research next sequencing length sequence dna kind add nucleotide nucleotide complementary dna incorporate intensity complementary template dna template nucleotide flow axis nucleotide signal correspond reflect nucleotide incorporation activity nucleotide incorporation unnecessary name usual flow flow zero three third flow signal use nucleotide table q call consecutive object various field r sequence assumption actual follow similar signal study first proper function readily yield realistic instead seq fix
pattern accuracy classify datum label fp limitation test set model yield accuracy parameter reaction system reaction trajectory optimize model include synthesis slightly abuse terminology say imply maximize range maximize find greedy quantization expensive simulate quantitative checking particle fitness operate particular require fitness consider reaction diffusion system formula l design implement induced processor ghz simulating pattern ls pattern explain maximize induced pattern optimize parameter simulate set optimize optimize formula negative new newly ht l cm hx xt maximize produce formula system system optimize repeat process user terminate iteration similar one optimize simulate simulate terminate superposition whose interpret partition efficiently quantitative semantic combine develop supervise synthesis experiment version biology several direction moment version exploit semantic plan multiple branch third method locally interact expect experimental technique biology circuit section assumption propose dynamical central novel superposition logic semantic image logic perform integrate checking algorithm particle synthesis reaction form single everywhere nature formation origin biology self though diverse biology physics pattern recognition usually formulate characterize structural area pattern formal foundation formal semantic pattern locally interact synthesis rule interaction strategy draw checking follow locally interact dynamical system give pattern network state base superposition spatial superposition logic tree partition decision image descriptor infer example parameter desire optimization fitness function semantic logic formula propose logic encounter pattern principle locally interact reaction diffusion pattern produce automatically paper organize section semantic formula generation check optimization remark technique class goal main input represent several survey accurate descriptor choose depend related pattern descriptor concern intensity gradient appearance interest rotation feature descriptor contour verification pattern verification possibly behavioral pattern logic formula descriptor verification logical descriptor spatial characterize texture combine pattern descriptor use modal logical operator intuitive inspire author superposition logic existence representative representative logic captures pattern consider rather oppose logic fitness search produce quantitative semantic discount inspire notable difference metric main pattern recognition quantify produce desire pattern quantitative semantic pattern denote negative integer x spatially rectangular specie define n dynamic system diffusion specie j indices vector reaction diffusion jt interest specie observable q example protein infer analyze observation steady steady check x n tt system trajectory steady observation trajectory reaction diffusion reaction diffusion location depend concentration cell fitness inherent ability operate space fitness finally solve application step decide user reaction diffusion element small tuple represent concentration observable specie within sub select row indice tree vertex sub child vertex example represent tree direction sub child north west north south se htbp rv se ts ts vs lf l ts v b ts v vs vs fs lf ts ts space vertex equivalent observable species superposition region avoid observation inspire author aim would four child hold proof easily expand matrix equal tree htbp notion transition classical allow check ts ts sm se ls start tree labeling ts generate self give represent direction give label path example bs ex set eventually eventually globally operator logic resemble main operator resolution select spatial direction allow work operate qualitative semantic spatial formula pattern yes qualitative semantic write follow qualitative semantic check spatial violate satisfied may guide exploration generation reason quantitative measure spirit region discount transition take quantitative semantic define follow semantic q proof structural
researcher extensive community challenge free free problem employ quadratic problem develop experience rl constrain zhang robust scheme nonlinear dynamic programming base estimate derivative construction introduce control problem work rarely system system arrange present subsequently brief conclusion appropriate definite appropriate dimension function derivative compact x tx dx x eq x continuous closed loop asymptotically generalize horizon functional briefly convergence establish optimal start admissible admissible respect continuous noting notation side along optimal nonlinear equation observe system prevent approach control action name function rewrite note represent action control control expression pt initial solve equation control involve basic operator improvement current control policy implement learn optimal policy design system approach refer exploratory exploration insensitive behave collect method demonstrate control ix ix q yield improvement mathematical induction accord satisfie hold lyapunov derivative along admissible policy admissible u generate ix ix ix u define follow similar eq expression sequence consider monotone always part ix theorem substitution u prove demonstrate residual develop linearly function solution iterative q tx x estimate truncation yield residual ix l x ix force weighted integral residual lx lx name substitution notation integration competitive integration computing lx lx similarly substitution domain accordingly iterative set vector index else go rl involve framework policy iteration section model control let policy continue generate q ix u induction stable policy accord mean mean hold hold similarly theorem proof complete solve algorithm avoid repeat omit derivation update law lx lx jx law implementation procedure present compute positive stop else go back implementation convergent employ real admissible control necessity converge matrix block denote policy give solve free policy continue pt generate iteration iterative equation solve unknown rewrite collect system least scheme vector square similarly error expression function expression solve pt q unknown continue algorithm expression equation equivalently vector square implementation omit accuracy vector angle attack unit quadratic matlab care parameter update figure algorithm achieve show dot gain observe obtain algorithm model vector converge figure gain dot line gain simulation fast converse follow function policy iterative ix x k free system convergence vector converge iterative gain dot represent note good convergent show close loop employ figure
hence weak task obtain open problem context recovery natural suffice question negative guarantee exist property claim appendix ht ccc dot line fraction solid line vary entry fraction recovery spectral success phase transition r success much ht frobenius vs gap prediction plot norm vs comparison compare synthetic mention operator gap block basic cluster sample spectral gap spectral small cluster case depend fix vary give go spectral gap go demonstrate trend augment lagrangian alm nuclear success ratio figure plot spectral gap dot success line color gap trajectory gap increase color indicate sample positively correlate exhibit type success ratio conduct reduce end generate gaussian let frobenius gap noisy lead error output temperature day test matrix algorithm singular output spectral rank compare closure guarantee proof hence adversarial contrast work dual dual true matrix optimum optimum notation require proof note generalize span projection operator follow write orthogonal complement dual construction show operator stress proof dual later incoherent generate satisfie pt characterize incoherent incoherent eq satisfy incoherent e z lemma provide characterize dual let respectively unique satisfie ready construct recover scheme construction give c r lemma satisfy trivially inequality q r tw uv tc contraction inequality prove recovery spectral guarantee strong sample result matrix analyse strong incoherence index give spectral result recovery incoherent alone property information plan guarantee signal definition conjecture edu microsoft research microsoft com matrix lot new provide universal scheme come spectral exactly recover satisfy incoherence uniformly recover matrix require several recommendation system quantum recently provable solve incoherent significantly second relatively might desirable processing application rank reduce large bipartite similar result large require explicit vector later strictly weak coincide psd certain strong incoherence require nuclear gap graph suffice apply matrix incoherence spectral exact incoherence universal alone universal incoherence property really particular block show success irrespective capital letter letter th represent format frobenius represent unit context organization discuss define bipartite use require require storage individual sparse application result universal critical difference approach highlight algebraic analyze contrast minimization sampling generalization exact recover index bipartite bipartite hadamard recovery algorithm completion recovery regular bipartite graph g note regular eigenvalue adjacency definition term singular property graph property decrease bipartite study generate briefly couple
due inherent social graph individual create identify interest although label security exist technique anomaly limited graph unable reason identify easily internal ip external alarm common contact save unnecessary anomaly level give datum three probabilistic rely recent enable improve structure detect estimate see newly observe anomalous simulation streaming parameter detector new update define new detector detect anomaly important inaccurate use share copy distinguish evidence node subgraph accurate detector establish detect conference application naturally detector node team conference finally interactive visualization analysis enable easily focus critical change datum identify graph common problem neither anomalous part static since availability label problem transform g et disjoint union subgraph anomaly finding anomaly technique compression rely minimum detect subgraph anomaly search subgraph almost hypothesis broad work residual towards detect mat dynamic include connect detector design anomaly cause gaussian anomaly focus kronecker work change generative introduce anomaly change overall structure detector tool explore nature visualization multi tool explore inform tuned star pattern region integrate multi extend anomaly flow et al method observation new anomalous sufficiently value value new notice side anomalous streaming model light acceptable positive identify operational user utilize simultaneously anomaly community significant devoted develop model capture broad require accommodate stochastic introduce membership generate intra community os enyi er community membership flexible degree os determine world occur er intra follow degree size describe version original generalize level os r enyi degree partition subset community intra sample os enyi er formally ic specifically note edge define degree exceed cl calculate community recall set exactly let denote see different community edge original great internal er allow assume depend node expected occur probability expensive anomaly describe sequence label technique infer input assignment edge density model probabilistic anomaly detector vertex community scale apply graph construct occurrence weight suffice survey scalability grouping world edge density estimate model os enyi er seek within subgraph assume beta lastly estimate poisson yield ig mode posterior gamma define leverage anomaly graph subgraph define directly inherent multi subgraph node intuitive limitation multi second detector build define probability external subgraph member nod multi baseline detect fitting baseline anomalous discriminate anomaly subgraph section datum first detector anomaly given value detect anomaly graph subgraph subgraph hence allow anomaly particular anomaly upon poor mixed geodesic employ geodesic strong baseline natural application capability model anomalous graph anomaly perturb anomalous graph anomaly streaming anomaly anomaly graph value anomaly detector label anomalous fall similarly detector lastly detector include conduct use ten sized degree vary eight accord create anomaly experiment three community six anomalous node anomalous anomalous see hold constant change node decrease intra extra four community together anomalous anomalous node anomalous receiver roc curve area auc display contribute community level r pr roc precision see dominate category inferior winner superior expect cccc cccc team team acc acc st big pac pac east e big pac big big big east mac st pac st pac acc acc st pac acc st big acc sec positive entry false ccc ccc acc pac big pac east sec mac mac acc pac big east illustrate scale real world statistic represent graph division team game play detection fitting year detector newly observe update newly observe detector apply two ground conference schedule within conference change graph produce value expect scale discuss community detect markov weight appropriate markov clustering identically posteriori refer conference name discussion detector detector identify anomalous anomalous graph numerical attain statistic detector indicate sample probable graph addition identify anomalous conference graph anomaly rank anomalous detector conference membership detect maximally anomalous negative conference precision detector anomalous decrease membership notice anomalous short graph anomalous anomaly scale detector user focus attention community interactive visualization tool fine grain anomalous graph figure illustrate prototype visualization figure community visualization provide little insight anomalous section alternatively indicate community figure allow conference name display contextual domain conference display inter conference indicate interactive anomaly west conference graph immediately apparent conference figure outside conference change confirm conference detection readily interactive interest resolve hypothesis anomaly occur force discovery anomaly team conference post color address identify anomaly emphasis anomaly context identify anomaly hierarchical allow community hierarchical streaming anomaly base describe community detector produce accuracy truth additionally detector gaussian insight scale capability superior multi experiment accurately anomaly subgraph expectation visualization inform give sample enable discovery anomaly occur scalability address community agnostic mention community context secondly estimate require calculate space aid optimize gain
via twice complexity variant introduce regressor performance twice robust also proof hierarchical describe string node label emphasize string child refer child string say stre empty string string string l nod leaf node give regressor presentation start root entire space regressor small region observe child regressor incremental hierarchical regressor e hierarchical embed regressor linear output model expert efficient assign tree present output significantly reduce certain regularity piecewise incremental tree adaptive incremental beginning combination output expert suffer sequentially deterministic upper introduce twice universal e universal though fine appear increase parameter incremental start begin single root node instant find increment generate node divide region disjoint plane child regressor child accumulate regressor vector child child regressor evolution node dark light regressor depth correspond regressor divide find regressor incremental linear regressor accord regret region minimize issue section incremental structure regressor incremental represent linear observe piecewise whereas perform combine piecewise piecewise model incremental output set expert achieve model well differentiable piecewise increase exponentially sense optimization framework final w iw hence combination entire e force online practically considerably problem assign instead illustrate calculate entire new leaf assign performance represent regressor regressor inner definition construct datum require use universal maximize respect adaptive incremental correspond weight performance optimal piecewise next sequential achieving end demonstrate great conclude sequential structural update growth complete regressor compactly root node regressor letter reveal weight root calculate mean concave obtain regressor performance final estimate calculate eq performance incremental regret batch region regret follow arbitrary model incremental th piecewise regressor vx appropriate sized identity xx regressor calculate region nonlinear note organization piecewise algorithm batch piecewise regret prove bind conclude conclude letter find calculate weight estimation p w incremental leaf fig see fig partitioning method light therefore tree case length theoretically complexity life regressor stationary practical bound discuss order achieve require sufficient regressor evenly regressor remark algorithm piecewise partition algorithm regret indicate introduce asymptotically however intuitively justified regressor fall piecewise mention piecewise tree limitation accord divide disjoint force computation accumulate regressor since process remain asymptotically implementation provide evenly regressor neighborhood multiply total accumulate regressor node index fine node partition vector create node introduce partition regressor accumulate regressor advanced anomaly method straightforwardly incorporate framework begin regret prove higher define suboptimal affine taylor twice differentiable function affine apply lagrange remainder obtain conclude algorithm regressor regressor slide spline update square state knot lr basis represent window provide computational emphasize create tree overall due regressor regressor computational regressor straightforwardly nonlinear regressor use update regressor update computational original accord regressor illustrate performance synthetic match signal map circuit life various benchmark mean white circular represent hyperplane desire regressor fig normalize accumulate propose performance include figure experiment illustrate normalize accumulate algorithm performance batch observation fine region datum increase highly circular hyperplane algorithm highly nonlinear comparable sequence extremely fine early processing unlike algorithm introduce fine e hierarchical universal increase limit number expert hence observe fig produce discrete generate fig normalize accumulate emphasize nature observe uniform curve piecewise priori partition limited basis fig algorithm partitioning regressor partitioning underlie regressor see illustrate relationship rest inconsistent prediction generate circuit drop simplicity circuit accumulate propose nature algorithm omit achieve average gain algorithm accurately predict subsection namely involve realistic link arm target use involve realistic simulation arm angular acceleration arm link medium accumulate respectively regressor achieve accumulate reciprocal result first b hence achieve desirable structural assumption nonlinear regression signal incremental regressor partition independent regressor base sequentially increase nonlinear performance define incremental demonstrate superior algorithm series benchmark edu tr study nonlinear sequence result guarantee statistical assumption address regression hierarchical incremental present partition regressor drive gradually drive sequentially asymptotically optimal length provide description demonstrate significant incremental study sequential aim sequence x find exist assume possibly vary nonlinear life capture salient desire use either extremely filter spline different scenario hierarchical recursively regressor drive model structure piecewise prove achieve twice tune algorithmic upper modeling regression algorithm literature accurately represent differentiable perform particular sequentially space disjoint create amount partitioning regressor rely hoc strong extensively attractive tree hierarchical piecewise tree yield satisfactory regressor achieve accumulate loss region minimize regression depth tree computational compare particularly however model structure learn locally introduce modeling twice minimize finer necessary create piecewise model degradation partition keep fine aside nonlinearity nonlinearity modify author technique straightforwardly framework introduce doubly example regressor region correspond interval internal tree exist leaf node internal regressor leaf union child region regressor represent scenario partition depth tree construct depth decision modeling limitation increment incremental decision potentially length achieve power infinite piecewise twice model certain
respectively equation choice build multimodal update rule build markov target think metropolis hasting introduce count straightforward check slow density thus markov instead markov ergodic markov chain modify update jj ix ergodic easily ni n use family deterministic consist iterate accord kernel simulation implement metropolis hasting density need cover case view physics factor numerically investigate visit favor transition penalization formula assume converge expect ix update heuristic expect metropolis rigorous see wang algorithm update stepsize weight formula stepsize sequence goes vanish ni original complicated change related quasi analysis back set check explain see wang stepsize sequence choose adaptively build establish extend wang deterministic update density imply hasting wang update wang change rule linearize satisfy wang converge sequence ultimately address wang stepsize random dx x positive stepsize sequence suppose meta obtain practical say n function measurable function sure random stepsize increase wang assumption update sense discuss stepsize n wang ni check purpose one sequence decrease moreover theorem easily definition allow eq algorithm corollary stepsize sequence large wang stepsize theorem sequence convergence result see update eq recurrence relation sa point sa raise past control whole trajectory come randomness subset step prove sa recursion establish recurrence infinitely let crucial difficulty fundamental address dynamic follow proposition sign total variation present recurrence mean weight accord weight recurrence existence sequence converge recurrence prove give markov state performance wang similar wang lebesgue measure target read temperature normalization present potential locate construct isotropic move distribute accord hasting reversible lot leave right precisely locate around state temperature main leave leave enter conversely numerical quantification point typical leave thank adaptive wang see asymptotic metropolis differ sequence dynamic potential position saddle point let realization chain report wang visit much prove stepsize already almost bias limit corollary leave around perform value trajectory parameter leave well compute quadrature integral stepsize bottom exploration influence time concern multiply perform independent realization dynamic large wang independent realization start library week machine shortest check large confirm simple limit wang stepsize convergence visit allow state respect wang type view wang aim stepsize decrease mean decay still combine averaging view natural modify stepsize scale update rule positive deterministic call choose iterate compute draw wang algorithm stepsize particular stepsize relationship relate update notice notice rule addition consistent obtain behave wang accord care avoid fast stopping weight stop stop define evolve accord logarithm subtracting exist stepsize count index logarithmic normalize visit number computer behave metropolis probability accept propose move draw accept effective behave proposition modify increment proposal move hasting attain leave precisely fit law law wang simple system claim one consider agree well various parameter law line observe power ccc study convergence logarithmic empirical modify realization scale result prove plot fit confirm variance except around asymptotic regime attain long decrease decay decrease average version decrease plot decay behavior value iteration bias time fit measure fraction density q walk favor unbiased well follow reasoning section r compute ergodic respect eq follow reasoning section new enter modify update main discrete reaction well reaction paper make stochastic stepsize version visit fashion wang vanish stepsize less dynamic method penalization sample become speak parameter free version wang explain specific stepsize penalization time wang linear deterministic stepsize explain adapt wang sequence look show sequence converge prove possible couple geometric cumulative measurable second extend main fact necessarily lemma expression successively sufficient give convergence sa continuously v assumption verify assumption stepsize almost surely imply sequence compact let check hence n h introduce whose state assumption equation admit additive hold first check sequence nk km integrable conclusion square integrable martingale converge hx thank hx constant k detail omit combine easily unnormalized ni ns proposition deterministic remark use induction deterministic ensure let n decrease small possibility recall c decrease notational next ni bound go lemma sequence increase g g gx n ns sharp prove denote simplicity monotonicity convexity imply go proof show concavity logarithm monotonicity set one set c deterministic constant writing existence variable first right converge since choice ensure converge conclude universit paris est la self sample multimodal probability measure method variant wang adapt convergence wang modification exhibit similarity field molecular consist building dynamic ergodic langevin hastings average trajectory ergodic interest measure multimodal region probability ergodic dynamic high region average converge slowly ergodic difficulty modify target order enhance average respect recover biased devise importance sampling
residual small throughout hc ica brain subject fmri hc fmri signal subject matrix subject spatio term ica fmri explain probabilistic first hc ica source signal network assume subject observed fmri mix iv spatial across independent voxel stationarity prior ica pre whitening perform variability across voxel follow isotropic hc ica subject combination population covariate covariate group biological trait effect th ic voxel adjust assume level hc ica adjust primary treatment control important benefit neural many clinical gaussian population level desirable modeling fmri signal location brain activate area exhibit fluctuation well suit mixed pattern capture type signal tractable estimation fmri background negative positive fmri bold interpret background rest facilitate derivation involve latent variable state voxel follow z q hc likelihood v I likelihood voxel ml estimate parameter hc ica conditional log v web supplementary material marginal step probability finally distribution analytical conditional purpose main notation tractable need step update estimate formula update rule supplementary material summarize section material obtaining level signal variability base fmri thresholded ic maps activate voxel supplementary value ica software marginal three k v evaluate k major em exponentially exact evaluate sum space variational tc derivation depend heavily specification tractable require numerical cause convergence model whole latent small provide show r rp j z qp supplementary restrict fmri characteristic state specify background background fluctuation voxel activate sparsity fmri hc ica implication chance voxel activate overlapping activate support finding propose em subspace subspace z supplementary material subspace approximate measure lead simplification expectation q latent reduction step specifically update compare result use moment j j summarize algorithm start k p v expectation regard v counterpart base modification replace expensive hc ica huge secondly involve mix ica high challenge hc ica connection directly rewrite hc ica hierarchical level side iv iv mean major effect ic em exact ic exact fmri two manner computation subject signal noise signal hc ica regression simulate covariate use versus voxel specifically hc ica conduct effect post regression estimate ic type voxel level voxel result ica method hc ica inference tc h v type voxel power voxel cn cn l ica hc hc hc ica hc ica hc ica hc ica estimate difference demonstrate strong visual two area demonstrate strong connectivity subject examine temporal time experimental supplementary material correlation result suggest task significantly particularly strong coherent find signal become prominent demonstrate connectivity region suggest compare strong functional estimate whose calculated permutation visual identify little difference posterior fail reveal central node language hc ica powerful detecting effect brain compare dual adjust multiple across conduct fdr correction hc ica testing network voxel significant fdr thresholded fdr correct hc test effect potentially help understand clinical characteristic brain develop hc statistical covariate covariate exist ica hc ica help finding regard brain challenge model heavy efficient procedure hc fmri base em dramatically ica state theoretically method fmri result correspond support conduct simulation spatially source signal moderate method spatially covariate effect affect sparsity covariate obtain shrinkage dr voxel mix relate additional experiment supplementary material evaluate provide mainly focus across subject second trivial u canonical general g q nee u p apply z z moment analytical update update p moment ic prove lemma independent interpret activate versus ic lemma odd lemma tv q qp conditional vector approximate exact conditional evaluation moment estimate identify activate goal naturally probability indicate voxel v within specify subject hc show task brain compare another three scenario contaminate snr randomly initial change sign scenario necessary change correlation group average across result optima group map covariate add
noiseless sketch completeness noiseless walk noiseless implement call upper bound volume body q total call base ball walk noiseless lie misclassification incorrectly classify outside optimally intuitive statistical confidence adaptive full inside let denote standard variable make decision decision decision decision dictionary take stop decision either outside inside dictionary probability fx query ensure query fx decay constant big enough say arbitrary decay algorithm oracle combination walk illustrate noiseless establish inside algorithm oracle geometry level ball area inside body geometry convex need crucial draw position behave isotropic involve term constant body isotropic position current vertical inside call hand probability close give alternatively bad thus save call hence give band illustrate direction vertical impact vertical cube final epoch boundary analysis mass error area avoid probability part come band statistical testing way step big ensure statement sharp body cone error noiseless query total behave choose error query complexity body lemma conclude query example propose attain query less noisy noiseless obtain minimizer reveal model basic learn need quantify number call know information lipschitz yet comes obtain desire dimension seminal work optimization dependence author extend stochastic dependence leave polynomial noiseless extended author consequence upper averaging decay method progress classical category distinction attack yield second noisy yet leave low hope walk reason optimistic ideally randomness asset disadvantage informally start body form convex see walk spirit obtain continue fashion analyze ball restrict verify current inside outside resolve mention briefly work whereby noise may gradient another assumption literature additional constraint objective boundedness observation average mean affine transformation fy check convex let vertical lipschitz noisy call align epoch remain contain introduce modify property walk algorithm round start warm maintain provide cut region compute third step body affine transformation near body affine calculate near nearly body start since seed run mix n back guarantee take isotropic call near isotropic position
scenario crowdsource task whose reliability priori appear economic significant ix hold prediction arise error accuracy unsupervise ensemble important obtaining collect instance wish pick one second improve multiple simple majority voting perhaps define crowdsource expert system year reference yet address em maxima propose perfectly totally develop binary limitation actually specificity consistently accord balanced accuracy assume classifier balance ensemble suboptimal classifier significantly make follow focus simple sensitivity specificity classifier imbalance scalar imbalance joint tensor share eigenvalue extract sec devise imbalance restrict imbalance dimensional make consistent also unlabeled elegant expectation maximization multiclass devise probability confusion prove multiclass moment confusion classifier motivate example datum ensemble learner competitive even scenario classifier make crowdsource study case distribution observation building estimate confusion matrix center tensor closely notable center class need resolve matrix even decompose divide simple imbalance totally tensor covariance optimize second accuracy tensor finally consider let realization class imbalance let set th fully specificity future balanced totally predict classifier assume knowledge accuracy consider specificity readily tackle problem instance I marginal classifier conditionally pair well variant ambiguity fully class imbalance imbalance certain disease population predict presence individual genetic profile know noise em upon approach motivate unlabeled mean know contain imply balanced accuracy classifier show appear appendix sign ambiguity balanced accuracy imbalance b entry classifier balanced practice quantity eigenvector plug matrix classifier estimate r cast rank construct resolve inherent follow prove property iii assume imbalance computationally explicit multiclass discuss unsupervised make label predict label ix specificity convex former value assume classifier via taylor show plug eq spectral motivation consistently classifier plug directly improve linearization around inaccurate may guess maximize imbalance approach estimate estimate imbalance tensor second exploit computationally strong consistency derive conditionally triplet tensor follow tensor imbalance balanced accuracy correspond denote equal unlike ambiguity eqs depend class imbalance q invert determine imbalance practice though maxima imbalance classifier classifiers eq q converge maximizer b consequence consistency ml estimator note g concave find global maxima operation consequently grid computationally class imbalance instead specificity confusion confusion classifier probability k regard error employ consistently confusion confusion build upon develop section split empty disjoint subset classifier consider vs diagonal estimating confusion binary classifier pose estimate confusion high order dependencie three tensor beyond scope simple method multiclass imbalance uniformly iii balanced vector class imbalance generate standard deviation imbalance unlabeled imbalance improve instance mse scale show accordance accurate tensor total dataset repository due page detail dataset additional appear appendix distinguish background classifier randomly thus ii ml full label stability realization choose balance upon approximately vote vs realization improvement particular observe classifier result unsupervised ensemble learner denote situation independent exactly several work direction relax strict assumption classifier instance difficulty direction prediction realization follow vector half recall diagonal element correspond q relation give plug combine number error method give imbalance transformation output new classifier classifier rather classifier un eq latter equivalently plug collect note p replace incur hence delta square delta estimate delta quantity beyond study dependence classifier accuracy asymptotically eq v writing error deviation gain insight comparable accuracy balanced b mae base true curve nearly accordance eigenvector jointly covariance tensor beyond scope lagrange multiplier constant k possible expectation equal possible choice attain outline follow probability couple probability maximizer fortunately sufficient property continuously derivative corollary continuously differentiable eq differentiable hence also inside logarithm hence satisfy equal confusion shall classifier confusion matrix nonetheless lead subset end first confusion small confusion
discuss selective test time model event ignore irrelevant necessary selective respect dominate family fact draw family inference nuisance exponential nuisance parameter exponential family sufficient dimension nuisance conditional eq letting eliminate alternative test alternative alternative unbiased selective among satisfy unbiased confidence region invert selective test accurate unbiased definition thorough review family model simply law selective consider selective selective worth mind way choose example tailed test law equal tailed rejection region side way test imply selective level selection another selective splitting nearly always suppose wish abuse generality result convex completeness admissible apply exponential datum govern parameter response linear regression different coefficient splitting stage assume define cutoff large acceptance conditionally note depend neither test cutoff randomize technical unless copy independent independent copy could must occur event event l natural occur illustrate theorem bivariate selection splitting could interval available fisher matter selective increase expect interval tail interval long need together plot consistent event unlikely discard unnecessary stage splitting use information effect law leave concrete exponential section selective arise multivariate model consideration unknown generalize selective test coordinate ordinary ol convenient remainder adjusting denote onto statistic respectively distribute henceforth subscript ambiguity selective test sufficient transformation selective q conditionally independent distribution observe ty test law recommend serious construct selective case natural equivalent testing selective good linear accord functional point lead poor job select particular adopt avoid need several article selective selection work assume variance know obtain square select nuisance correspond rewrite eq event base base unfortunately line sphere equally hypothesis conditioning insufficient carry meaningful carry test nuisance write choose whether increase conditioning lead per selective case make mutually selective conditionally play role determine conditional great deal whereas condition contrast suppose matrix sparse highlight whereas conditioning set union realize especially important step model subtle deal conditioning plot value interval multi adjust likelihood ratio density choose inference conditional uniform thus window logistic linear glm glm represent difficulty control variable realization selective trivial like gender conditioning constrain promising approach may asymptotic though selective illustration selective matrix column snr magnitude choose splitting half yield instance split partition contain data procedure select select left inference inference lasso test lagrange sign test distinction procedure selection likely select superior model noise respect selection chance fdr aspect stage performance incorrectly power conditional screening well select quality goal outperform fdr intuition use information stage perform stage dominate improve drop seem successful stage slow surprisingly hold understand tradeoff work check draw student five rigorously nominal alternative appropriate scientific propose control q interval control closely selective countable define enjoy interval author address propose chance incorrectly fail construct confidence interval least square parameter regression ever consideration matter always singleton selective condition clear rate control converse relevant still question consider also scientific bag fdr proxy goal genome wide associate diabetes quantity vary interpretation gender control title job control gender question selective inference carry question ask frequency selective ask principle simple matter diverse still selective design price little improve challenge difficult take procedure reality procedure lead property ahead key challenge research balance choose realistic article represent repository file first website support foundation grant dms fellowship stanford genome fellowship taylor support foundation air office helpful select show expectation kolmogorov independent sequence z arise one wish region randomization order region implement rejection region test boundary randomization q correct sequence would monte algorithm property carry test define family approximate specifically cutoff cutoff acceptance possible n region nz pair right quick allow quickly upper confidence search nonempty dimension intersection unit selective sample hyperplane selective ball radius weight sphere outline selective conditioning draw fill text width em text center corner font taylor stanford model selection control selective recover property select analogous context closely intuitive justification exploit unbiased selective inference selective generalize test think consist analyst choose unknown analyst informally determine ask answer choice model use formally subsequent inference prior collect govern physical least partially example often exploratory decide predictor interaction include properly property file suppose significance level intuitively recognize still high among nominal conditional control presence simply value cutoff control selective valid ask simplicity example imagine scientific estimate ever demonstrate compound choose analyst decide roughly scientific question address result publish claim lead finding explanation effect extensively simultaneous several author adjust construct estimator genome wide pass fix gaussian large conditioning drug clinical trial adjust file effect meta selection adjustment interval construct false expect fraction non cover amount coverage interval see relate selective employ control propose region brain view classical exist analyst classical model usual rejection describe statistical check model leave open possibility argue selective type hypothesis practically base random scientific typically random classical control implicitly randomness eq viewpoint science split scientific aside selection depend nominal nominal selective meta selective nominal splitting practitioner identify separate popularity justification imagine though take ahead temporal actually solve control selective amount available selection furthermore always series rule part article directly selective splitting treat though reveal treat paragraph term denote informally everything know complete think one decide two discover knowledge stage everything reveal stage fair control conditional prevent appeal surprising reject unless sense conditioning carry hypothesis interest discard little carry formalize selective property selective control key conceptual question major selective us test even exponential selective briefly derive unbiased selective exponential model conditioning stage selective computing prescribed focus regression generalize recent proposal derive powerful selective require compare post selective selection initial second section compare selective conclude conditioning arguably observe function parsimonious identifiable researcher predictor probabilistic coefficient selective select conditioning many range aic minimization selection cf selective lasso solve term encourage eq notice correspond selective event lasso figure partition different screening imagine stage datum package remain fall test careful specify consistent interpretation coefficient adjust effect effect adjust education condition test follow otherwise concrete mind framework selective broadly allow random carefully inferential goal discuss selective development assume measurable analyst carry base tackle pair nan hypothesis mean guarantee necessarily beyond design loss test alternative countable question question abuse refer selective inference possible analyst select analyst shown select completely explicit correctly specify contain importantly candidate analyst poorly formal guarantee perform misspecification rule exception whether adaptively adaptively analyst probably wrong collected experiment issue take case discrete randomization necessary level adjust event ask question ask question ever select conditioning selective interested test selective selective countable selective design valid concentrate mutually selective selection denominator countable dependence test devise selective test concrete selective whole long run control nan wide operate scientific share countable question research apply iy I research probability grow long control frequentist independence generalization multiple discovery rate fdr rate even group fdr aggregate across convenient think contain set selective establishe selective inverting selective analogy duality selective event suppose selective selective confidence selective coverage cycle cycle cycle circle conditional variable even selective procedure view conditional used selection whose informally condition fine say control selective error rate give take baseline selective type selective confidence fine suggest refine fine monotonicity selective type w fine control type choice control variable extreme flip proposition fine computational reason additionally nonzero event convex another reason refine inferential guarantee meaningful coverage rate control splitting correspond split informally information mean quantify amount remain decompose expectation eq average conditioning quite consider selective gaussian conditioning highly contrast law practically lost conditioning confidence invert interval
energy residual iteration step candidate time much measurement sparsity iteration estimate identify k pt iteration stagewise omp orthogonal super etc identify candidate correlation measurement residual pick magnitude index candidate predefine find index correlation greedy adopting include match subspace sp thresholding htp propose recover rip satisfy rip vector isometry exactly obey isometry conventional ol ol slight ol fail algorithm converge iteration computational ol compare ii notation lemma section give study performance conclude remark vi notation useful rip order refer isometry constant consequence see begin interesting identification th sort element kl one correspond implementation computationally expensive require construct desirable effective substantially simplify iteration index equivalent q noting decompose k relate q simplification offer mention geometric interpretation project orthogonal study convenience state success select one clearly make index observe first correct contradict select previous iteration eq correct build condition ensure select convenience notation large k contain select appendix note monotonicity slight isometry g fail justify eigenvalue rip ols incorrect index eq first study differ find weak case definition identify index th holds respectively combine obtain adopt testing recovery reconstruction signal construct draw variance choose signal amplitude mention reconstruct particularly omp ol comparative approach simulation omp programming er gauss eps eps plot reconstruction sparsity sparsity call critical sparsity signal exact reconstruction algorithm fig critical even method exhibit high bp bp h eps gauss ols run measure matlab program core processor ram window reconstruction much accordingly much less ol call extend ols allow candidate list method fact ols identification reliable utilize energy analysis recovery iteration kk coincide omp ols addition empirical conventional improve empirical promise recover full value definition minimum diagonal replace reciprocal singular lie together singular partition inverse triangle respectively lk finally combine set k k k together lemma due lk hand eq check equivalently q definition corollary remark deal year recover signal call orthogonal extend least square ol choose index support much improve ols perform sparse restrict isometry rip isometry demonstrate compare art algorithm cs pursuit orthogonal isometry recent attract attention processing main system signal recover minimization q intractable combinatorial involve impractical realistic devote efficient algorithm recover rely search principle combinatorial computationally pursuit reveal bp
association data auxiliary follow could observe lead next thing accurately predict intuitively clear predict advantageous auxiliary much distinct reduce auxiliary example I straightforward estimator residual minimal point variable variable typical regression coefficient possible far reduce marginalization association rewrite ignore auxiliary scale dimensional replace inference convenient rewrite diagonal diagonal consider furthermore one I statistic value admissible predictive step uncertainty probability assertion agree assertion adjustment formula need plausibility partially agree I success I validity desirable property I assertion I assertion plausibility essentially condition set call hold nest either satisfy excellent reference function validity natural one optimality consideration first optimal simultaneous multiple application selection association observable space sampling association relation assertion exist make belief stochastically formal give association least predictive respect predictive name result set simple define impose assertion assertion section proposition show optimal random relative satisfy support satisfying example assertion simple see write event random give versus insufficient assertion generally might even assertion I consideration think simultaneously understand handle fundamental section extend development normal interested assertion disjoint simple predictive inefficient likewise predictive possible general assertion write reasonable strategy element intersection random two complex assertion predictive intersection appendix simplify complex assertion resolve choice set intersection sided interval optimal interval symmetric asymmetric optimal predictive corresponding intuition intersection individually justify way measure multiple assertion respect intersection efficiency disjoint predictive intersection previous section simplify set resolve assertion variable resolve ambiguity element make use transformation fundamentally inference impact support write problem assumption association mapping implicitly act auxiliary relaxed group act act directly notational act variable fit usual transformation assertion write concern assertion unchanged transform help assertion change sign affect take mapping move property immediately property transform solve transform random focus admissible admissible probability useful ax concern unchanged reasonable require display belief element aforementione invariance hold balance reasonable interesting practically beneficial balance check calculation however transformation model balance make belief stochastically assertion maximize element definition main notion element particular balanced association center I subject collection event predictive sub collection hyper theorem predictive assertion intersection half simultaneously intersection shape box towards transformation variable invariant coordinate irrelevant addition label multiplication product marginalization specify box optimal support shape box invariant box balanced sense definition besides property hyper cube axis balanced cube random plausibility henceforth drop plausibility away cube shaped contour plausibility plausibility plausibility region frequentist coverage plausibility length characterize quantile norm consequence validity across plausibility region new hyper cube plausibility nominal frequentist conclusion consideration context argue turn identical naive plausibility region sub message probabilistic automatically problem demonstrate I drive start correspond truly coefficient kind assertion still cover selection use cube stochastically lead calibration stochastically plausible fix claim I drive procedure control error equivalently certain validity validity tb apply implement plausibility plausibility look sort permutation rank magnitude p clear large rather plausibility plausibility formula include assign table get q left cancer analyze examine association clinical among receive include transform seminal response compute plausibility see accord lasso select I cccc plausibility simulation study autoregressive correlation e six scenario vary result I variable section hypercube display figure percentage parsimonious parsimonious procedure aic cross validation tuning match configuration range plot parsimonious include variable give result validity I curve except lasso panel bic adaptive true parsimonious panel variable consistency I fix select attribute consideration I come multiple theory apply valid uncertainty important drive I valid post naive plausibility region base develop addition I base notion pick plausible connection I calibrate control family rate simulation demonstrate I emphasis meaningful probabilistic summary development application I already problem involve multinomial genome wide multiple testing expect consideration problem improvement extend development paper interest principle develop complex initial step case step complete acknowledgement national science foundation dms dms suggest comprehensive treatment goal amount technical focus random support algebra subset close topology mapping measurable define forward separable stochastic distribution include default predictive member next function rich consequently I nest nest usually simplicity move terminology nest support demonstrate random simple distributional restriction set give subset equality form assertion auxiliary equip measurable subset contain closed key relevant ax relatively condition proposition ax x ax since belief attain collect intersection define new predictive support random satisfie condition state admissible intersection random direction easy clear either case ax ax st disjoint simple optimal predictive index intersection closure vx vx j candidate predictive assertion admissible theorem remain theorem show handle split st ax ax theorem ax measure predictive element define core balanced inequality element attain balanced balance resolve ambiguity shape make predictive get assertion imply random support sign magnitude symmetric one depend invariant sign understand differently decompose assertion random balanced beyond balance optimality normal balanced unbalanced unbalanced balanced sign take side small unbalanced predictive random belief large say balanced sense q uniformly admissible go assertion transformation pair maximize suggest connection optimality symmetry I book multiple assertion application want assertion balance disjoint sense respectively complex assertion make connection need symmetry property index predictive optimal complex maximize assertion elsewhere towards assertion disjoint decompose ax unique connection simplify connect condition generalize assertion continuous sense sense get helpful equivalent admissible random nest index write include class care make clear rigorous measure ax measure stochastic address fix investigate index symmetry condition respect definition write generic optimality admissible prove state maximize contradiction balance condition set increase union get example correction might need put contain r intersection inclusion write easy immediately former less latter claim define similarly leave involve non evaluate respectively generality q h key motivate construction leave right hand equal fourth hand q contradict predictive balance maximize complete plausibility plausibility fix size understand read plausible plausible addition plausible really prefer one want select overall variable pick complicated assertion procedure plausibility reject coefficient predictor plausibility remain seven enter zero model predictor still therefore continue remain plausibility additional predictor stop also table eliminate prove dimensional present though symmetry proof take non generality distribution centrality optimality consider assertion association equivalent assertion say side assertion predictive random plausibility u u sa plausibility function optimality want show predictive random u u admissible plausibility nest large complete parameter zero mean multiple really assertion expect plausibility believe marginalization ignore leave association propose plausibility behind expression like variate student see singleton follow plausibility plausibility I suppose sa l claim compute predictive obvious demand instead assertion specific hypercube predictive case complete nan apply incomplete stochastically large quantile efficiency note whereas empty assertion assertion prefer proceed explanation discussion tool argue
bound really easily lasso risk predictor opinion work property computationally risk good linear three different stage study property think useful type study weak discuss develop call employ variant half agnostic lasso net half distribution inference question predictive contribute risk predictor use interpret detail let interpretable produce interval thus exclude question infer linear deviation validity confidence essentially purely avoid asymptotic could define obtain square procedure summarize figure pt n split select subset forward lasso confidence similarity inference valid inferential statement dataset thank provide select ph percent interval percent figure percent parameter property explanation scope current datum predict gues augment residual permutation test easy interval validity depend desire device neither method residual prediction change variable remove assumption free interval free interested task infer attempt assumption regression indeed tendency three idea correct careful interpret article describe change hold change seem like pick change refer change put assign otherwise claim causal world alone eq change prediction change bring I exercise user likely interpret advance understanding inspire model easy low much hope author low assumption world I important acknowledgment thank helpful taylor force together array test automatically adaptive remarkable author make strong quite make advance understanding procedure paper error design weak form incoherence eigenvalue certainly place indeed think also highly exception exception design matrix random design matrix
set computational overhead herein approach determination finally iv end dissimilarity representation cost dissimilarity measure term generation respective euclidean second quantifie give cycle iteration derive induce cycle grouping compute putting depend derive good cluster determine membership derive overall bad main determination modularity system operate cost significant objective capability blind derive suffer simple model induce community one form identify especially maximum modularity force derivation additional read account measure performance combination final necessarily focus perform well recognition choice characterize modularity also code herein train scheme exploit derive remove edge likely induce form edge higher set fuzzy construct get increase remove derive consider group fuzzy substantial change scheme modularity additionally test see cost affect effective running start sec sec provide explanatory take uci repository lastly sec dataset contain pattern synthetic multi target non target specify train non target pattern system variant denote genetic mutation dissimilarity implement check fitness change vertex membership intra distance membership parameter intra cluster distance define perform base setting determine preliminary fine software implement library equip gb ram class area roc compute average target pattern rank evaluate confusion analyze precision measure measure define report average run execute seed significance class separate spherical test green actually belong blue target solve membership plot modularity demonstrate euclidean apply select target target nonetheless reliability achieve define taking target non indicate describe dissimilarity euclidean herein reference ref show auc uci performance variant seven seven achieve auc normalize usually affect exception worth note still sp dataset degradation three bad nonetheless worth point pdf observe severe initially degradation grow ab bc contrary demonstrate dissimilarity base adopt deviation breast diabetes pp ar normal breast cancer diagnostic bc bc sp gauss nn som gauss auto som dataset na nn nn auto encoder som na auto encoder letter letter show adapt target process graph mean distance solution solve greedy characterized parameter control importance substitution obtain knowledge result consider applicability herein result confirm hard p notably dataset auc classifier consider gap accuracy number solve e letter l letter p novel classification design make dissimilarity employ e classifier decision form concept modularity decision equip suitable boolean pattern validate two type benchmark base uci label comparison uci demonstrate several art result dataset prove effectiveness less pattern term dissimilarity representation allow accord suitable hand direction usual herein change graph map dissimilarity representation volume therefore future span optimization technique interest g membership devote usually pattern recognition pure viewpoint generalization system produce easily human expert insight since future goal viewpoint rgb name outli anomaly target pattern term recognize classification base primarily approach input euclidean derive effective region vertex dissimilarity optimize scheme consider design boolean decision test allow description contain either pattern effectiveness technique involve pattern orient character class deal involve real target instance determine device work properly correct device trivial instance model method take rooted decision adopt real video medical end develop dissimilarity although allow cover context adopt etc embed dissimilarity ds input edge normalize ds us ii define additionally represent ds relate embed span minimum far analyze induce concept modularity membership soft decision classification attribute experiment offer comparative benchmark follow overview provide clear introduce technical background material use successively detail evaluation conclusion provide direction iv reconstruction finally information theoretic generative parametric include describe operate suitable distance measure input technique category group neighbor approach drive base around vector model region surface optimize finally theoretic entropy mutual one important support training svm like particularly domain employ hyperplane like forest refer comprehensive survey state aforementione categorization herein intersection distance information theoretic notably exhibit sense substantially theoretic fuzzy graph partitioning concept moreover dissimilarity aspect however programming prototype cluster popular abstraction sound mathematical entropy model literature fuzzy establish one classifier subsection modularity discussion dissimilarity entropy finally modularity dissimilarity characterize dissimilarity rs determination span prototype strategy embed row dissimilarity fast way dissimilarity dissimilarity detail datum I realization nd gx ne ij enyi span literature connect enyi length e vertex edge edge relation normally value determine weighted degree ij weighted intend partition establish measure quantify cluster vertex compact intra cluster great inter modularity formally follow partition rewrite edge intra cluster modularity heuristic propose modularity assume normalize input domain requirement notably implement embed sec dissimilarity accordingly rs pattern distance suitable representation high determine rs prototype small informative euclidean see entropy descriptor also guide synthesis construct framework develop synthesis module I consider concept modularity sec vertex induce partition whose edge denote modularity contribution modularity therefore need take account boundary efficiently boundarie fuzzy decision synthesis optimize model objective optimization combination calculate two spread separation ds modularity group solution validation effectively instance contain considerably stage fig intuitive
time faces student linear subspace dimensionality speed access project reduce storage requirement quantify reduction projection ssc dimensionality order significant degradation engine behind quantify subspace change reduce challenge datum find low dimensional lie union extract formalize assume l subspace refer literature hybrid application inter unsupervised disease may dimensionality reduce speed acquisition even directly often desirable lead reduce reduction appear general cluster privacy widely dimensionality reduction linear euclidean distance property map popularity purpose characterize namely sparse ssc thresholding subspace subspace intersect quantify impact dimensionality cluster letter letter matrix denote ij identity refer stand sphere random subspace first point segmentation ssc segmentation approach perform high dimensional incur operating quantify ssc characterize specifically ssc apply reduce set quite provide contain impact explicit ssc even reveal affinity dimensionality reduction engine state project subspace space increase quantify impact briefly ssc adjacency construct cluster ssc lasso construct point spherical set subspace perform z z element spectral subspace discuss step j j z segmentation connect adjacency ce misclassifie ce inherently quantify work albeit sensible specifically absence impose reduce chance splitting increase select drive ssc connection virtue automatically I statistical connect ssc subspace estimate insight eigenvalue start throughout randomly point element j j direct say property inter c c subgaussian store high dependent propose matrix hadamard hadamard result moreover subgaussian isometry property rip sense conversely establish randomization column satisfy rip subspace k ks ks ks ts start main result ssc adjacency ssc adjacency apply cn state affinity ssc high small hence ssc impact quantify affinity reflect subspace reduce dimensionality orthonormal basis basis subspace suppose x j probability proof give ip reduce project orthonormal generalization basis basis dimensional randomly project satisfying formalize u result theorem imply I obtain case estimate obtain standard adjacency set n j reflect upper violate I start I high dimensional e j probability cause projection theorem accomplish perturbation concentration result violate false connection impact dimensionality ssc problem cluster image pixel acquire
evolutionary represent model higher imply long matrix mutation evolutionary therefore alignment type evolutionary abundance protein result usually rescale express specifically form choice find entry matrix score align sequence element integer similar sequence give could information match unknown large distance treat knowledge commonly think principle knowledge evolutionary naturally allow uncertainty evolutionary protein alignment evolutionary distance describe similar h influence volume long accept previously suggest evolutionary model match match mean respectively increase previously value h alignment account configuration must preserve new model widely penalty match also protein structure alignment model quite new small additionally view laplace marginal distribution uncertainty uncertainty numerical comparison body impose flexibility handle transformation challenging find sensible easily incorporate evolutionary protein inference distance cm sequence alignment ex school sciences mathematics university sciences technology school mathematic abstract know protein protein also influence bayesian align protein protein gap incorporate insight protein bioinformatics gap bioinformatics alignment structure protein aim determine sense primary protein position may sequence two complement alignment evolution protein position exist structure remain essentially unchanged well closely protein protein available bank reliable becoming developed ce design uncertainty uncertainty alignment allow quantify mathematically protein point point alpha configuration protein rotation configuration main interest shape protein alignment viewpoint essentially two integrate alternative consider model manner uncertainty correctly underlie flexible body demonstrate similarity transformation match application demonstrate range applicability alignment protein matching setting match every consistent match consider protein think therefore impose constraint prior incorporate fully section describe sequence illustrate describe bayesian constraint challenge previously literature measure evolutionary conclude pair element letter string protein alignment align type complement informally score type align give score entry express score score score overall providing nan observe g e score necessary gap sequence show alignment gap pair evolution instance mutation occur one position type largely region sequence alignment highly align region pair gap achieved interpret sequence figure alignment gap allow alignment matching penalty number gap alignment imply gap impose preserve configuration label body transform form configuration observe observe regard location spherical model therefore mapping point particular impose constraint configuration configuration row one match poisson point integrate support match normal align meaningful ordering preserve alignment well necessary gap sequence alignment match alignment extension illustrate gap consider sequence alignment match align indicate gap gap create figure first sequence length three gap count number gap would gap alignment configuration consist penalty start sequence formulation way decompose sum contribution pair indice form alignment extension e multiply q haar special unlike prior discuss sensitivity generating metropolis hasting alignment acceptance probability match suppose gap comprise form current alignment random n select propose match however currently match say switch proposal currently match configuration match match propose match acceptance say propose also match add accept currently match interval match switch current retain otherwise match reduction negative add match term q note perturbation alignment remove match switch number propose iteration mcmc posterior distribution programming analogous sequence generating computationally intensive converge method move deal joint integrate rather treat treat place q hasting proposal lead summation programming appendix extension similar alternative integrate adopt alignment recall total p ga h ga use mid quadrature grid stability rather costly evaluate scheme integrate new methodology analyze study align structural analyse pair analyse identification give sensible investigate abstract represent representative also tune great initially expect prior use suggest gap extension gap example case parameter parameter state mcmc burn translation take centroid set keep inference match typical match match probability probable least already appear match match align region uncertainty evident gap create diagnostic bioinformatic deviation median corresponding posterior report correspond modal use three different value specify match match describe principle obtain define cost match specify incur match point regarded note match incur relatively miss true assignment use give match distance guide obtain plausible uncertainty alignment converge chain run value value trace mixing value log manner good benchmark determine subsequent perform start angle draw nine run mode therefore discard remain top probable match st probable additionally run evidence converge plausible confident include match converge confident global dominant time prior expect value previously respectively posterior interval previous appear compatible distribution suggest sensible value match similar top posterior agree previous consider range evenly spaced range careful inspection investigate wide trace chain chain alignment pair match th probable assignment match give match reference match appear give report situation type number give context convergence value apart median log run diagnostic trace together convergence reach good range converge give probable find initial evidence reach plausible quantity sample posterior interval estimate match
eq find negativity triangle inequality evaluate dimension feature directly learn database z match want match pairwise must link modality modality want generate similarity form item logistic sigmoid function often hinge function indicate pair specifically stand deal unbalanced experiment divide norm formulation treat scale domain nuclear make desirable machine discover modality gradient find order regularize non constraint eqn accordingly eqn point entry product feature sign positive contain modal definition simplify remove term soft thresholding technique thresholding eigenvalue assume rank accordingly descent update way step objective omit algorithm iteratively search combination solution alg eqn low bilinear similarity popular modality image medium retrieval field pls microsoft generalize cca cca space different modal match pls another classical method cca semantic gap correlation cca pls cca important label learn discriminative great recognition cross orthogonal modality aim point unlike cca pls limit paired feature network cnn cnn code namely research layer output dimension pca method cca pls query cca pls wikipedia wikipedia article article build select associated word total document derive cnn feature pls dimensional thus experimental pca pca preserve cnn image feature dataset wikipedia wikipedia back number blue bold precision scope propose algorithm good database pca pca perform comparable outperform pca reason wikipedia may less category globally note fail work feature need reduction well without without lie reduction remove redundant pca may discard time pls algorithm pca table also supervise clearly outperform one pls partly reduce semantic semantic work find semantic fig precision curve curve text retrieval performance bit display pls database wikipedia high especially wikipedia similar retrieve experimental learn modality algorithm accelerated explore modality database document show objective consider suppose subgradient function subgradient nuclear svd part great denote turn bound institute chinese sciences ia cn medium receive year internet modal feature thus get heterogeneity match different hand metric explore learn heterogeneous heterogeneous modal penalization accelerate proximal gradient text medium database performance compare internet decade display audio requirement modal researcher media retrieval image several recently new author feature key heterogeneity modal common latent modal match classical solve aim modality mutually maximize similar cca pls semantic correlation semantic suggest combine correlation work helpful reduce image level beyond method analysis weakly analysis learn study near neighbor relevance component margin et aim gaussians traditional suffer difficulty modalitie metric similarity medium modality algorithm solution
cca literature involve objective really due replace problematic control concave monotone solution compress problem penalty eigenvalue basis pursuit lead iteratively reweighte selection problem surrogate function function surrogate fig three surrogate approximate original still continuous original section concentrate algorithm note exposition matrix pair matrix hermitian value approach complex ig px modulus applicable construct complex method minimization expectation em approach principle behind transform reader therein sequence objective start follow easy scheme decrease monotonically every I first inequality follow apply surrogate maximize produce iterate refer problem appropriate solve e iteration quadratic due differentiable concave surrogate ng px c compactly k follow equation htbp quadratic fact idea penalty quadratic propose solve iteratively weight problem reweighte sense quadratic function example coefficient tackle imply iteration minimizer fact potential function surrogate tackle propose incorporate systematic differentiable function follow aim function lead surrogate smooth huber penalty huber smoothed fast htbp smooth become smoothed smoothed surrogate quadratic irrelevant function view incorporate issue k I k I p x k p pp px x pe p apply smoothed smoothed answer define everywhere concave monotone smooth define low quite smoothed smoothed f ng ng solution smoothed original high smoothed say general global local advantage gradually large begin probably undesirable maxima success numerical ready differentiable smooth smoothed nc objective smoothed quadratic follow ignore eigenvector k tw iterative since base iteratively reweighte repeat eigenvector iteration generalize eigenvector since iteration drawback make attractive become ill condition suffer extremely slow convergence difficulty ascent free ill conditioning let maximize l l search scalar achieve go infinity follow lt l lt lt lt lt lt let xx lt rr direct accord easy thus ascent worth multiplication product efficient though slow become accelerate convergence large linear widely ascent introduce multiply ascent leading summarize practice particular positive direction ascent direction reader refer book ascent usually converge decrease minimize similarly scheme preserve ascent need objective ascent initialize generalized eigenvector ascent property guarantee choose repeat l l ascent assume special notice special sparse min admit q equation otherwise return smoothed use iterative algorithm generalize eigenvector iteration another exploiting algorithm close term suggest solve accord problem nonconvex let diag repeat proposition mm fact apply diag diag rewrite diag diag quadratic plane quadratic term form need back summarize require positive general diag easy require diag diag repeat absolute diag problem thus accord sequence generate algorithm compact thus guarantee algorithm sequence maximization smooth neither convex concave shall surrogate maximize prove prove function present useful later differentiable surrogate table show continuously everywhere except let satisfied continuously see continuously differentiable continuously f show smoothed maximize compact ng sense solution ng continuously differentiable lipschitz proposition ng p ng vice speak nonconvex hull attained supremum attain relax set optimal let referred sequence limit equivalent solution objective q n converge subsequence j easy necessary construction imply generalize exactly recall guarantee lead generalize compute stationary long guarantee exist experimental pc ghz ram subsection propose generalize complexity extract dc c maximization knowledge case problem solve dc solve iteration experiment set subsection dc algorithms ascent compute eigenvector matrix identically normalize size average trial see fast dc note implement attribute iteration evolution objective one figure much converge notice run versus average trial eigenvector diag sparse generalize generalize eigenvector successfully recover list result respectively initial regard three plot chance recovery parameter recovery versus versus chance achieve dc become exp stay decrease lp lot surrogate function two surrogate much make easily exp seem choice choose gradually probably smoothing fix inspire apply decrease smoothing solve less specifically step apply parameter decrease step choose random scheme decrease setting surrogate show decrease exact recovery htbp chance special eigenvalue pca receive attention covariance vast literature essentially generalize benchmark code website surrogate function call refer need eigenvalue cholesky decomposition matrix subsection mention entry smooth four different penalize propose fast four norm penalize may specialized algorithm deal penalty average subsection generate sparse achieve covariance diag orthonormal eigenvector randomly four eigenvector successfully recover chance successful wide range plot high chance achieve chance versus regularization dna allow possibility answer complex experiment usually make component potentially subsection breast set five scheme explain pca eigenvector scheme keep entry value explain increase cardinality explain high htbp trade cardinality allow generalized pair approximate nonconvex generalized turn regular problem point numerical propose outperform include special closed solution derive scheme show similar easy q maximizer know q p conclude complete notice define different case optimality left notice guarantee satisfy still satisfie satisfy arbitrary rewrite eq form integer get necessary integer satisfie need
variable originally propose extension single elimination originally omp ol aim yield homotopy algorithm regularization reconstruct homotopy procedure pd algorithm different minimize maintain maintain list recently approach optimization address together list handle gradually continuous hyperparameter opposite context involve hyperparameter solution warm value homotopy exploit affine track consecutive piece spirit interpret homotopy procedure solve pseudo convexity metric consider resolution warm reconstruction rarely gradually increasingly measure choice grid grid modify definition value rather adaptively similar homotopy pd propose nonconvex penalty difficult inverse additionally automatic cardinality rule usually approximation observation assume independent submatrix index rank hereafter notation forward selection introduce notation stand frequently resort slight abuse terminology line path may regularization take constraint penalty generic path statement greedy refer convert stand support indeed read minimizer minimizer minimizer penalize curve concave envelope affine contiguous case support ks set minimizer provide property set appendix path envelope may coincide state reverse inclusion penalize search notation deal output compose support pseudo index geometrically support segment fig lb lb lb lb lb lb sort order support associated extension start dedicate equivalently extension decrease adaptive dedicated three sn removal replacement eq terminate replacement function subset removal coincide ol generally replacement trial compute fast stable cholesky ss stand submatrix active unnecessary standard version implement ns vertical top bottom iteration select update support new dictionary support equal iteration none font lb lb lb lb lb lb lb lb lb lb lb lb replacement top refer selection four support de inspire homotopy minimizer continuously homotopy denote optimality first condition main homotopy solution minimizer terminate illustrate line black point separate eq atom support font lb lb lb lb lb lb lb lb lb lb decrease represent line cc lb lb lb lb lb lb lb lb lb lb lb lb ss axis without output meet I limit replace list support illustrated line lead first repeat call whole appropriate deal overcomplete dictionary early stop consider rule rule minimum step lead process plain repeat terminate compute concave might jump slope increase structural gradually candidate impose curve fig concave curve envelope domain font pt lb lb lb lb lb c font lb lb lb lb lb font lb lb lb lb lb lb font lb lb lb lb lb lb lb lb lb lb lb lb lb cm font lb lb lb lb font pt lb lb lb lb font lb lb lb lb cm font lb lb lb lb lb lb lb lb lb lb lb font pt lb lb lb lb initial configuration b line concave interval support edge update pd subset include concave decrease illustrated fig c envelope line remove new subset compute line concave empty support subset cardinality assign exploration removal keep similarly iteration attempt possible illustrate call explore support include pd concave reduce horizontal correspond extension iteration explore compute either support remove low cardinality explore list call sort decrease lead cm terminate support decrease replacement jj least cardinality candidate correspond alternative adopt call state empty detail omit brevity reason concave j firstly notice j prove sketch stress identify pd iteration concave next atom pd upper reason value improve early within computation hereafter height fig mm fig plain deconvolution spike db impulse dictionary sparse deconvolution problem jump db jump db kind involve ill condition pd analyze simple example detailed mm width sparse j line deconvolution impulse response gaussian convolution impulse thresholding toeplitz dimensions gaussian b respectively define jump atom code jump match height jump sparse generic either dictionary overcomplete dictionary deconvolution neighbor highly correlate condition dictionary may recover difficulty deconvolution width impulse overlap detection atom support pd result sparse first seven detect jump may model category ms cross derive expression cm cc pd data signal spike pd fig solution support cardinality pd rp rp small pd white curve almost coincide pd curve deconvolution cross spike contrary yield accurate short record moderately q number spike spike find spike db spike typical pd curve circle replacement return white pd coincide continuous low pd grey grey bar reach provide insight pd deconvolution horizontal axis single initial empty successive pd support include candidate effective increase cardinality fig improve solution decrease fig pd early subsection setting ratio cardinality width impulse deconvolution sparsity restrict f generate overcomplete gaussian impulse cm jump jump focus penalty estimate penalty algorithm simple ol reweighte ir cyclic ls cd smooth sl resort penalize least algorithm ls cd simple thresholding compressive sensing behave ill l cd efficient thresholding cyclic descent become popular although hereafter allow rough initial fast measure suggest choose less sl increasingly penaltie low relative nonzero sl implementation dedicate noisy work efficient inverse replace strategy solver crowd homotopy limit homotopy crowd mainly matlab implementation crowd author large problem compete grid grid vector randomly simulate specifically location nonzero I trial pd value support cccc time second width width fact snr mm snr width snr width mm k width mm mm snr ccc width fact snr snr snr snr first cpu pd viewpoint viewpoint propose solve either minimization apply output norm strongly ta average another represent separately pd cpu evaluate support negative st positive st average se tp order false positive negative fp tp analysis se tp algorithm likely perform use provide additional pd subsection reconstruction sl strict output se score run cyclic cd nonzero regard perform norm post processing interpret iteration cyclic l squared cd towards minimizer cm cccc cpu second width n snr width mm snr fact snr mm snr mm snr ccc fact snr width fact width snr mm snr width mm snr fact snr pd clearly group cd sl hand ols pd discriminate accuracy behave contrary outperform obvious advantage ir homotopy adaptively output relate whose ir solver tune stop pay burden fig two line pd horizontal trial computation start termination pd expensive however want reason draw pd viewpoint pd pt l pd ls cd ir se tp order true se tp ls ir se tp pd ls cd ir se tp true se time depend many implementation storage stop rule follow comparison two depend medium relaxed avoid huge stop pd medium pd l amplitude sl step last nonconvex dimension remain reasonable arbitrary rule comparison trade favor pd ir numerical ccc mm width snr snr mm fact width mm h snr width snr mm snr fact snr width mm snr h marker appear l sl performance noisy often specifically least always exceed discriminate positive localization non account wrong estimate cardinality subset order quite jump true support partially detect pd well correspond free I tp ls cd ir se se tp pd ir tp pd cd se tp tp provide propose algorithm competitive overcomplete detailed experiment space reason deconvolution overcomplete although qualitatively good pd se tp hard discriminate often consider spline generalizing jump detection piecewise jump think piecewise inspire regression jump shift version side overcomplete soon competitive carry pd noisy matlab choice desire ready suited induce highly correlate minimizer algorithms usefulness extension low range gradually greedy improve large enable classical selection criterion estimate include backward potentially pd refer removal dictionary atom remain testing simultaneously become carry replacement test omp ols spirit consideration propose path
new criterion explicitly account dispersion study performance traditional numerical regressor propose less costly joint proposal perform precision find computational cost greatly present dispersion fast two monte simulation model evidence conclude section random beta write eq dispersion beta independent differentiable function link function logit log complement cauchy link parametrization also carry inference find dispersion statistic final last fisher likelihood usual regularity thus upper nominal importance typically iii regression include adequate also link assess misspecification outline selection propose regression widely coefficient determination regressor add selection aic model introduce criterion good identify asymptotically model follow sense alternative measure relative location focus beta correlation denote maximum likelihood covariate pseudo take pseudo pseudo estimate log model measure goodness quantity eq measure goodness propose modified additionally recommend selection far goodness criterion define criterion minimize estimator dimension candidate say minimize aic instance asymptotically leibl accurate introduce unbiased leibl distance linear regression regression autoregressive aic regression bic include correction namely consistent autoregressive incorporate correction account dispersion inclusion extra covariate shall two inclusion goodness way account second selection dispersion penalize regressor approximate propose model dispersion eq numerical quite inaccurate moderate computationally dispersion beta mean regressor dispersion introduce sequentially outline dispersion select regressor adequate selection regressor selection regressor dispersion two entail estimation different figure time intensive approach ten covariate dispersion regression entail model computational efficiency suppose scheme run day minute combination criterion shall carlo dispersion regression parameter use implementation step file computer datum use different monte replication draw uniform throughout ht identifiable easily identifiable dispersion weakly identifiable identifiability approach influence mean intensity weakly identifiable dispersion easily identifiable weakly emphasize usual relate monte give respectively logit link generating model model case since regressor likewise take evaluation criterion use criterion able replication criterion select regressor correctly covariate dispersion dispersion variable assume regressor first four strategy frequency additionally discuss implement pilot simulation great weight one inclusion regressor heavily model model aic model aic joint accurate weakly identifiable correct small sample dispersion identifiable reliable selection weak identifiability small accurate good criterion nearly scenario well balanced display selection achieve criterion top dispersion identifiable recommend regressor notice finite performance selection heavily dependent identifiability table present correctly dispersion dispersion identifiable dispersion identifiable generating process good good winner figure correctly specify weakly identifiable perform reliable monte obtain dispersion constant focus select mean interesting correctly especially correct correct dispersion nearly dispersion far indicate regressor dispersion well come dispersion explain criterion regressor quite naturally selection scheme combination implementation propose ps monte present table nearly scenario model propose among implementation dispersion identifiable accurate emphasize specification dispersion vary dispersion dispersion dispersion identifiable identifiability perform equally even numerical practitioner recommend use misspecification school response read index student capital eight year transform original code dispersion regression inferential inaccurate dispersion dispersion vary regression logit consider covariate candidate hypothesis test mean include covariate logit reject nan constant nominal notice sample close evidence perform scheme arrive covariate dispersion namely diagnostic correctly aic arrive regression covariate dispersion covariate statistically nominal std constant considerably happen dispersion specification assume correctly
still ambiguity make identifiable estimate positively compare gradient bfgs newton implementation adopt bfgs describe bfgs glm software simply ignore r r compute subject extract r md normalize positively glm voxel take core intel ghz total four hour discuss section pure package validate bold presentation natural bold fmri descriptor publicly twice image per second rapid acquire comprise image within image time align run alignment manually additionally datum preprocesse detail perform window extract training original result beta map proceed encode handle image spatially smoothed pyramid modulus scale generalize learn bold original full necessary overfitting assess method otherwise coefficient activation yield activity present prediction bold highlight unseen prediction leave fold dataset task subject ask reject gamble chance gain independently varied level potential gain loss label challenge predict brain publicly available mixed gamble task slice correction segmentation normalization interface subject consist tr stimulus perform run across create correspondingly run predict gain correlation true metric well suited regression order label sensible occur always lie interval perfect ranking perfect disagreement fmri previously encode activation beta method standard glm glm design glm separate form dispersion size second form reference across region decode decode voxel true glm estimation task variant glm see display count identify chance identification algorithm map intermediate however expect correct directly translate estimation separate outperform classical range r glm worth score whether statistically test success recover probability recover method hypothesis probability equal alternate probability tail np p distribute performance glm identification subject count correctly total chance metric less voxel benefit range score basis subject bold average voxel outperform use identification subject glm subject study glm voxel voxel wise encoding score time voxel rank separate design element axis dispersion derivative axis trend suggest improvement basis peak reference around canonical score basis design give sign test leave one average voxel superiority design estimation separate generate gain basis axis time dispersion derivative abundance superiority color peak observe local model design matrix voxel score basis axis give sign leave confirm superiority design voxel acquisition slice estimation pearson correlation column voxel thresholde contour value show green area produce highlight visible r produce glm method bottom see perform voxel follow gray matter voxel wise acquisition slice plot voxel thresholded testing contour line top voxel matter relate shape canonical decode compute decode univariate selection parameter voxel cross consider assess superiority encoding difference score fold great performing sign report value together encode high basis basis size tb average consider well second dataset estimation outperform fix reference glm basis follow software sign order examine generalization task linear separate voxel constrain basis omit efficiency reason possible bayesian spatially adaptively learn subject work latter case level analysis consists reveal possible boost appropriately metric assess estimate metric identify encode predictive activation use novel activation compute benefit range method r glm voxel full observe increase homogeneous already glm good design basis provide find variability subject constrained decode classifier long basis ten derivative difference region involve observe correlation sensitivity incorrect used generate bold test signal bold natural correct estimation high impact decode evaluation accuracy stimulus type decode sensitive procedure encode activation glm spatially condition voxel previous cast efficient newton glm design quantify encoding decode glm outperform compete grant le de france france despite common canonical region subject datum lead power fmri constraint yet differ across voxel exploit model glm glm improve decode activity compare decode compete decode functional fmri machine response machine technique predict cognitive functional record study decade cox bold task stimulus although possible bold signal common consist extract beta bold analysis voxel base model activation coefficient bold addition third know quantify quantify activation mean linear glm wide suffer limitation glm commonly response activation know substantially suggest improve overcome aforementioned limitation finite impulse propose within glm estimating modeling e generalize general model characterization study primarily focus detect activation chapter freedom propose possibility combination three consist time derivative nonlinear desire space long overfitting approach require level share even choose inherently costly case regularize explore reference focus basis hyperparameter goal increase brain advantage fmri estimation voxel development voxel development voxel naturally translate robust voxel method simultaneous voxel previous smooth newton briefly conference experimental presentation glm separate design ten brain two encode drive decoding provide comprehensive glm glm improve decode computationally tractable open letter denote size kronecker concatenation notation k concatenation slice array first describe extract coefficient bold stimulus trial presentation stimulus signal voxel acquire glm response linear underlie matrix convolution stimulus software amplitude one voxel consist glm design temporal basis form respect work refer stimulus correspond taylor impulse ambient shape canonical stick duration basis generally give form stack element successively stack regressor element size kk instead long activation possibility trial single assume peak bold possibility coefficient peak amplitude glm reliably large perform poorly limited conditioning increase sufficiently similar robust spatially estimation unique obtain column event estimation glm method stem glm across translate rank amount enforce glm glm coefficient estimate bilinear nuisance regressor ambiguity positively sign cost feasible jointly convex practical formulation advantage contrast glm rank estimate coefficient factor parameter subsequent analysis rank equivalent latter beta method normalization average project matrix readily prediction non unseen link unseen occur encode trial part response formulate analogous discuss classical glm estimation correlate voxel regressor regressor glm estimate rapid design boost decode task extend predefine function predefine set basis construct concatenation
firstly inner integral calculate analytically principle break curse integrate analytically secondly two k page see useful factored reinforcement freedom function correspond therefore restriction inner unnecessary freedom unique constraint k ol due nonlinearity induce greedy algorithm basis kx k fit prior immediately enforce especially lack derivative uniform discuss limit infinite eventually example uniform correspond reflect frequently yield shall high formally generality cast project regularization diagonal element call derivative equation matrix page input factor everywhere uniform smoothness everywhere kf kf page basis initialize optimization improve cost equation calculate change add factored basis function factor x dx consume gradient precise equation proof similar basis state optimization solution statement empirically however improvement stay minimum optimize factored solve derivation inner influence randomly cache loop dimension fall inner loop change linear problem propose newly basis linearly test dimensional toy sparse traditional test artificial toy label variance plot factored basis randomize sample region prediction difference corner algorithm converge runtime influence strongly quality bad approximation converge efficiently data gp benchmark repository mm compressive dimension describe various concrete cycle describe value variable output dimension describe white red sample quality scale dimension value factor toy draw construct factor factor fitting taylor virtual rmse comparable set factor considerably compact seem face challenge function convex local control performance slow runtime shot cut preliminary experiment number adjust noise factor ideally principle dimensionality enforce paper poorly result function predict mistake difference sensible update algorithm area break experience regression sparse calculate product expense multiplication allow inference summary factor basis promise perform gaussian less space potential extension improve upon runtime benefit greatly algebraic author thank helpful science input dd k factored factored function integrable factor though factored product trick integral heart factor k j well choose kx k kx j discrete kronecker universal combination denote k marginalization basis analytically solve product wise elegant basis low pass filtering optimize randomly ex lf derivative algorithm fourier function product parameter ex ex lf k set unconstrained solution normalize g z z x z z kf twice g f ex kf kf derive unique diagonal absolutely continuous training fourier l covariance kx cache sample b improvement kf k lf h k eq function optimize basis outer new df tu face curse integral class regression factor structural property allow point product application reinforcement break curse speed computation derive greedy factored basis regression perform benchmark factor compact introduce competitive process yield factor like analytical wise product marginalization kernel suffer curse computing network application like belief kernel like support vector de mainly due sparse classifier call method gp average wang choose svm task restrict everywhere function require equally state exponentially lead bellman curse effectiveness function due though small take thousand construct propose basis directly support pose select former function factor factored solve analytically
news autoencoder tree yield autoencoder capture method map hide decoder reconstruct autoencoder perceptron neural implement idea stack autoencoder representation work autoencoder use decode soft tree soft split output leave section soft tree use simultaneously hidden decode pass encode layer external autoencoder autoencoder hierarchical decision decision leave internal decision give child traditionally implement soft probability decision consider soft internal child name left outcome univariate single split geometrically speak though split orthogonal split orientation make applicable region right child two classification implement logistic leave child xx equivalent supervise predict scalar sigmoid nonlinearity convert nonlinearity output want soft decision give response splitting hyperplane tree supervise square order backpropagation efficiently gradient parent decision structure soft train supervised well layer follow chain autoencoder back autoencoder tree encode hide representation decoder want initial e update additionally derivative decoder representation decoder layer encoder level slow layer autoencoder decoder tree epoch leaf continue depth increment update introduce allow split additive random mnist handwritten digit database set mnist handwritten image pixel output output denote matter category sort two one map nonlinearity map decoder stack perceptron autoencoder gain gain result extra level depth see especially dimension digit result representation autoencoder perceptron tendency nonlinearity sigmoid corner autoencoder observe multiple leave hide rather tree assign representation size fashion closeness important behavior autoencoder hierarchical soft small gain increase dimensionality layer distribute high encoder depth histogram since soft every count leaf leaf blue include digit learn locality region locality learn mnist decoder digit certain child phenomenon digit autoencoder reconstruction see tree bag representation word path omit clutter capture fine fine model map leave leaf distribution extension tree response modify rule modify response input autoencoder local linear projection degree locally partition assigning distribute digit dimension indicate effectively autoencoder tree representation move away intuition incorporate locality get reconstruct extension representation digit see class capture autoencoder tree provide smooth representation space move like addition locality small reconstruct figure soft encoder reduction comparable autoencoder autoencoder decode autoencoder dimensionality within opposed apply reconstruct centroid process hierarchical autoencoder
stand source thank parameterization equation accord operator hadamard vector parameter estimate updating turn update mix look term fidelity differentiable fortunately proximal interested calculus quadratic differentiable entail forward backward splitting disadvantage dramatically increase therefore subsequently rather resort simple derive subdifferential subdifferential follow subdifferential admit explicit would source independently expression proximal w important equivalent amount thresholde quadratic fidelity burden orthogonal consequence updating provide rough might prevent later assuming perform admit algorithm estimation problem nonconvex minima initialization diversity equivalently threshold value greatly improve minima algorithm towards point computed guess source improve update source choice discuss thresholding thresholding hard main drawback soft substantial improvement consequence update replace stand operator name stem well discriminant verify algorithm detail k therefore relaxation precisely nonconvex cost sparsity constrain blind source separation alternate converge critical firstly decrease threshold strategy minima minimization guarantee step threshold help prevent secondly update iteration prevent lastly spirit reweighte update motivated numerical tends display distortion ratio sde evolve hundred iteration subsequently transform frame assume orthogonal transform allows transform appropriately extra freedom help produce bss redundant transform yield improvement make redundant wavelet type wavelet translation frame practice dominant tight imaging amount allows transform tight case choice role concept diversity true amplitude discriminant sample source hard select threshold initialize maximum amplitude initial guess source decrease final threshold noise practice threshold stand th gaussian guarantee noise coarse fine property source estimate amplitude ii spirit simulate anneal minima entry choice rely weight low sparse source trade lead penalization provide sparse desirable separate source begin access value mis mis start test turn lead good carried bring evaluate bss rna newton blind separation general mixing source negative ii quasi disjoint support separation complex etc negativity performance bss p make monte various call spc introduce entry activation process entry source draw mean model gaussian law laplacian width half mixture resp right redundant translation invariant wavelet pick level actual spike combination follow interpretation ground decomposition need account part due noise stand neither global name distortion performance denoise effect method ability follow mix matrix criterion introduce inverse correct permutation performance monte carlo simulation unless tb visualize bss display source observation source turn active spike bss seem prominent spike exhibit certain poor visually source retrieve rna paragraph level correlation coherence varies distribute opposite source accord source correlate source performance sparse experiment channel db display leave bss rna quickly sensitive behave show much entry correlate higher behave consistently magnitude bss display bss behave quite source db algorithm source spike source tb tb performance bss precisely bss sources diversity significant entry source claim early high share dynamic correlate method number proportion entry leave resp amplitude standard bss behave db dramatically db bss tb dr recover generally grow limited source entry source limit next fix db panel source performance source recover less high performance decrease rapidly keep mind sample say amongst independently source discriminant increase simultaneously grow might source explain decay mix method seem improve behavior notice mix average scalar product matrix matrix involve source measure source precisely distortion source actual bss method scale impact weight algorithm estimate source source via precisely across turn rather perturb large amplitude weighting source weight proportional column favor entry affect presence prescribed level comparison demonstrate contamination whenever discriminant turn noise discriminant consequence impact proportion entry fix evolution reveal noise increase tend method surprisingly algorithm presence bss behavior matrix naturally step order reject eventually amplitude detect algorithms tb long recently modern analysis method separation play recent light crucial play accurate background multi observation component foreground decompose source contribution introduction entail component blind separation focus datum remove component prominent emission impact blind source separation study carry simulation observation ghz model simulation major center simplicity prominent area source locate region correlate figure translation wavelet signal bss default separation bss noise column source resp nominal noise db feature mild value snr exception emission spurious source residual belong emission seem correctly define estimate figure display error keep reveal difference error free tend emission evolution bss normalize varie actually bss seem db already view algorithms matrix suited separation figure confirm criterion regime value noise source detect likely origin partial correlation tb input scale tb scale ff residual scale residual tb rna ff residual scale rna rna rna residual tb ff scale residual tb ff code introduce article available blind separation partially correlate bss tackle article retrieve emphasize discriminant propose adaptively weight adaptive component bss experimental correlation source slightly entry source retrieve finally apply separation suit nature blind bss technique context retrieve purpose bss discrimination distinguish reveal rarely valid practice partially bss novel retrieve partially correlate source precisely correlation source technique field source source rapid development extract development dedicate survey blind suited analyze linear observation source quantify source term stand additive well row source blind separation infinite problem require information source purpose classical rely ica far context differ focus separation source harmonic analysis apply attract lot compressive reconstruction dictionary sparsity source nonzero coefficient source generally entry coefficient vector negligible representation example signal representation wavelet adaptively learn signal sparsity exploit blind study rna source orthonormal adapt retrieve source generalize source support active one source partially disjoint bss series signal complex especially exactly sample source vanish alone introduction diversity md concept source amplitude verify source salient source share coefficient building concept diversity show bss developments bss emphasize negative matrix focus set matrix necessarily non life actual neither statistical source may dependent source particular research development dedicate already correlate discuss tend source paper good result alternate algorithm iteratively admit close project square th known thresholding operator classical least partially correlate sparse blind clarity transform essence low limit problem find jointly possible precisely norm source bss tend mix matrix source radius notice sparse independently source
finitely f fix sentence exist finite finitely computable validity unary predicate binary e validity hard rational f language duality imply number unary predicate finite validity countable infinite unary predicate countable hard rational iff countable look favor possibility countable proof carry replace complete work alternatively problem countable requirement predicate predicate logic requirement logic seem play theorem unary predicate logic logic aspect property fully sentence partial join tree ask meet operation deep information regard fact reduction fundamentally require rational inter eliminate unary restriction crucially force equal reduction validity solve arithmetic operation rational language carry case fortunately example illustrate magnitude hold motivation logic see theory classical theory concept mention two benefit technique convert version probability expression bit canonical ordering counting etc analogue complexity abundance like indeed plausible one could description class logic game another develop complexity I cs equip final thank throughout away start work discover I logic make smooth finally importantly mathematical detailed mathematical world world able subsection thm corollary thm example thm theorem thm title logic paper first order logic equip interpret previously rational hard general language logic complete countable logic countable remain largely individual language unary predicate requirement language language finitely unary predicate countable artificial neural ann age big increasingly ever technique inductive logic attempt pac pac logic context investigate meaning hold measure logic logic inspire logic measure interpretation keep asymmetric logic learnable learn roughly error bind universe logic property seem carry turn reason much hard order logic fact arithmetic analytic decide language whether sentence table knowledge validity definition logic tuple language unary predicate binary predicate infinite number type empty tuple suffice denote valid coincide sentence logic language e validity also complete open logic unary predicate language logic valid iff counterpart validity countable model answer answering question vice versa assess calculus e c validity complete complete tuple notation denote language sentence coincide regard validity ordinary general mechanism finite rational relational language unary predicate function symbol equality e logic transform sized sized countable knowledge countable countable validity complete notation logic logic whose interpret analogue straight motivate research logic assumption edge artificial prove countable sentence development tool idea semantic e allow form sentence rigorously inter different rational equip powerful tackle completeness finite hardness utilize perturbation simplify value distribution last validity coincide countable model countable valid sentence valid validity countable finally mention possible letter underlie write countable signature sequence free variable formula shorthand shorthand sentence formula convention formula addition use place subset recursive hard every computable many hard resp resp complement resp resp complete please concept denote algebra define boolean additive algebra boolean say set least implicitly discuss e measure letter start etc context please concept theory triple variable linear satisfie equality two equation find nonempty iff concern feasibility program identify division feasibility arithmetic rational path program consist inequality concept regard formalize definition logic e logic dual logic abundance let countable signature universe algebra x logical example distinct iff treat interpret q universal variable general split x implication symbol boolean combination symbol reduce implication eq implicit make sure sense impose refer model ordinary language countable signature possibly language formula relation measurable include equality denote iff formula iff call model similarly concept analogue countable thing likewise make satisfie analogy normally record often countable possibly contain write interpret strictly formally every atomic q logical distinct variable treat eq iff interpret split atomic implication implication similarly logic q definition duality iff states table validity complete iff iff show model theoretic property exist f logic model counterpart theory automatically countable let formula complexity involve measure middle derive fact agree finitely additive countable iff finitely countable notice atom restriction reasoning must atom nan atom extend measure inconsistent measure countable countable contradiction everywhere treat make element measure main object e finitely call validity resp x countable validity countable likewise sentence finitely satisfy countable distinguish finitely formula finitely first would validity logic logic goal concept develop section possible motivate point mention later reader check apply respective exhibit example highlight logic let true equality iff singleton also logic nonempty true also countable let logic parameter q b common follow axiom sentence resolution analogue logic logic iff logic record ia x x I ib logic conjunction sentence rational number sentence finitely though expression graph vertice graph artificial example logic ease read logic context logic note sentence z boolean symbol define shorthand composition symbol rough picture implication follow identify indicator oracle randomly return accord pac accord concept return take note see language express assumption order universe everywhere define concept word concept assumption relational hold eq represent iff class collection assumption conjunction iff conjunction concept list triple decision procedure proceed returning represent proceed iff quantity mention illustrate dimension express formula hand logic express strictly straightforward generalization free part expression complicate boolean combination irrelevant label depend note along would concept apply establish kind question unary predicate also express size parameter sentence weight vertex language logic pair vertex loop moreover vertex vertex edge thing population express undirected complete unary express unique positive vertex vertex graph graph express existence carry vertex logic make statement element element undirecte complete apply subgraph valid iff claim countable model size universe measure interpret ik v kk reasoning fact modifying imply let universe subset finite place countable measure meaning realize arbitrary valid q universal formula equivalent valid certainly satisfy contain f universe number time obviously finite restrict appear appear desire tuple language valid yield language set finitely valid formula finitely coincide sentence unary predicate call countable f well essence predicate number indistinguishable part elementary unary predicate universe subset partition immediate interpretation measurable elementary equivalence strong sequence iff construction iff eq x imply converse apply start sentence node subset child child child call denote level write measurable additional simply uniform z x distinct valid e label q level measure level f reader choice node across apparent statement ready tackle relational language e countable particular suffice universe everywhere sentence free iff height finite devise suffice universe iff interpretations structure level program lp replace rational rational finite effectively logic proceed involve strict lemma vector exist iff rational feasibility elimination exactly whether affine plane index yield rational feasibility equivalent feasibility solve ready characterize relational logic roughly relational universe everywhere property check universe restrict everywhere interpretation condition verify universe probability everywhere iff existence strict duality language rational countable finite validity like express order string form eq variable form sentence weak sentence likewise sentence order sentence clearly every sentence term q differently tree generalization tree tree generalization analogue well pair level leaf node expression eq singleton tree x x order tree sentence verify definition concatenation eq thus property hold immediately simultaneously iff subsection rational would like countable symbol equality language unary predicate iff reduction work computable sentence sentence exist apply full preserve q finitely rational finitely repeat simultaneously lemma therefore logic case former half completeness countable unary predicate predicate rational reduce specifically show proof proof hardness reduction finite e rational initial respectively interested input assume represent symbol break section finite section describe first language section construct finitely indeed vocabulary vocabulary constant unary predicate state still intuition vocabulary formalize linearly order element roughly logic specify measure use around element avoid force equal become encoding encodes reduction iff finitely formula recursively free x nx consist conjunction interpret deal model positive sake clarity axiom atomic element measure respectively eq initially function shorthand conjunction iff great say shorthand implication shorthand expression shorthand p p transition construct machine head point cell head explicitly head character belong conjunction condition symbol conjunction exactly head cell symbol hold q x least conjunction write clarity set take element formula rhs assume shorthand probability say satisfy finally sentence parse therefore sentence strict segment purpose use conclude reduce satisfy universe natural iff iff position define iff symbol iff state since measure clause thus fix q conclude probability q e therefore assume sentence encode chain element axiom restriction relation element rx thus clause equation hold n mm satisfy ny
eigenvector score associate division operator take b work researcher log aware blockmodel spectral necessarily highly likely misspecification regular secondly previously cluster several cl concern robustness blockmodel bic blockmodel community choose community cl bic bic composite cl paradigm cl relaxation complexity relational complicated implement working estimator misspecification likelihood density statistical inference capture cl bic relational datum go would like misspecification joint consider stochastic blockmodel ng parametric impose specify full access univariate blockmodel family composite marginal first log unbiased usual regularity associate composite minimize blockmodel datum replicate common form individually regularity arise include argue much correlation among composite score retain good correlation consistency asymptotic normality scenario take context composite likelihood consideration k k result community eq different estimator distribute replicate asymptotically normally establish model universal though misspecification blockmodel replicate bic true community whenever correlation severe consistent asymptotically consistency form blockmodel composite bound community conjecture node increase community number leave work treat composite work assumption variable denote derivative q matrix u complexity specify cl bic reduce bic estimate cl bic naive vanish composite likelihood let adjacency ab ab ab la l n see multiply ab parallel partial ab cl bic number community simulation dataset blockmodel set independent label contaminate bring degree binary independent variable regular stochastic blockmodel thresholding gaussian correlation il w l correlate correct blockmodel carry matrix record criterion agree apart cl integrated likelihood bayes vb estimate community select bayes value true real datum set additionally incorrectly community median indicate correlate decay respectively correlation bernoulli cl bic cl cl bic prop proportion deviation deviation bernoulli w cl bic bic cl bic correlation common whether belong collect table il collect table c cl vb cl bic vb w cl bic vb cl vb cl bic cl bic bic bic allow topology connect form community size result network blockmodel identifiability replace unit q slight abuse expect table contaminate impose world cl bic correlation large instance community cl successful community generate purely blockmodel correlation cl bic vb select impose vb cl bic simulated vb yield median translate selection noisy estimate nevertheless consistently correct bic setting addition present stochastic simulation correlation simulation tend even grow community bic true vb measure quantify assignment community represent pair estimate agree assign second ratio within community imply detection cl bic grow blockmodel record score median ratio c c bic est md mr prop md md deviation goodness fit median ratio est scenario community label community community grow blockmodel cl bic across result aside fact decay scenario potentially indeed label ahead exhibit obtain evaluate cl whether increase number cl bic line international trade originally contain trade cl bic bic pair blockmodel focus year form weight country country finally show bic community correspond blue european medium south select south american split traditional yield divide community bic community little obtain longitudinal health http edu student school cl correct blockmodel figure show component result select community actual misclassifie score cl bic community still cl black cl criterion extremely even correctly specify fail black community student close among student goodness cl bic assignment slightly mr cl bic indicate superiority pair example bic penalize cl bic cl bic robustness issue due misspecification underlie capture amount recover structure especially literature study property blockmodel misspecification blockmodel blockmodel likelihood select simplicity robustness work spectral interesting explore cl dense real another whether correct blockmodel supplementary material file replicate simulation upon request read among community relational blockmodel inherently mixture raise selection community bic stochastic conditional assumption community different edge usually violate propose composite bic select blockmodel approach simulate material contain relevant code online community correct blockmodel blockmodel network analyze interest researcher study underlying structure world
stochastic gradient hessian product form replacement classification establish algorithm descent hessian vertical label objective dotted black mark cd outperform sgd objective h epoch epoch effect setting epoch memory helpful stage article manually economic market convert represent appearance word extremely sparse nonzero fourth market accordingly function numerical quasi aim curvature sgd method choose report iteration objective increase iteration fast initial eventually variant size figure mark axis epoch illustrate vary batch fix improve computational effectively hessian choice e g b parameter value blue spike occur term bfgs formula lead monitoring say indicator update skip small memory size consistently beyond improvement comparison reason deterministic bfgs observe poor unstable varied eq entire indicate therefore ratio computation display error norm batch exhibit non sense less norm gradient batch great show gradient decrease batch tendency inaccurate hessian stay accuracy need fw calculate form differ bfgs perform curvature calculate use gradient evaluation indicate sample compute algorithm numerical effort limited poor quality implement follow setting unnecessary rescale fw iii hessian updating average last recommend compare realistic observe batch h speech set stochastic entirely bfgs implementation latter use bfgs enforce uniformity resample datum evaluate present sgd diagonal rescaling quasi newton similar update rescale hessian noise stage online stage stage minimize use step take newton employ product compute derivative newton approach iteration minimize employ idea geometric presentation method seek asymptotically method concern address fisher context network interpret direction maximize outline implementation empirical additionally maintain argue contain hessian needed cope hessian improve al employ hessian curvature note nature regime paper operate stochastic convex quasi curvature interval gradient incorporate useful say make quasi product may essential goal require uniformity prevent task establish convex indicate quasi solution analyze applicable problem provide mechanism ensure satisfied thm conjecture l incorporate stochastic application quasi update lead curvature effect robustness propose quasi efficient scalable employ formula beneficial collect interval sub hessian product arising learn method machine massive impose batch feasible however scale suited computer sensor operate approximation limited bfgs produce regular hessian vector product uniformity avoid potentially gradient consideration convex arise simulation instance refer machine input typically take form collection refer objective empirical amount mini batch gradient yield operate stochastic substantially entire employ hessian hessian stable efficient manner scalable operation limit quasi optimization method solution hessian gradient gradient affect ill conditioning completely remove appropriate choice next present quasi employ limited bfgs correction recently define correction corrupt effect gradient numerical arise organize discuss experiment illustrate contribution paper term stochastic sa use programming standard machine sgd discussion quasi newton method fact curvature good enough bfgs minimize correction always strictly scale application scalable enjoy linear rate scale direction evaluation extend quasi newton stochastic update one quasi inherently average reflect hessian entire something stochastic achieve hessian calculation flexibility add new curvature emphasize curvature update schedule gradient curvature iteration eq define potential approximate taylor sample define training example emphasize code regardless give length gradient fw define average essential iterate bfg update correction mathematically describe update new matrix bfgs newton form employ formula loop step correction strength bfgs classical pair interval extra hessian iteration method sgd even early stage newton representative namely test give gradient hessian evaluation require multiplication evaluating batch approximately product assume total operation sgd appear report use choose logistic nevertheless newton method loop recursion bfgs employ limited bfgs updating outer product effective around compact main iteration since suffice curvature iterate lag product several select minibatch experimental environment well experiment cost namely cost range choice bfgs similar good value range continuously function strong slow employ iterate remain take convexity h fw entire nonnegative regularization assumption satisfied show hessian generate assumption satisfy analyze allow literature newton formula iii definite update zero denote boundedness set determinant show since eigenvalue use away therefore satisfied next establish beyond contain objective assume bound case follow assumption well hold iterate q fw
arbitrarily determine feasible separable present error induce separable roughly sample indeed contribution work compute contrast deal aspect contribution induce bayes consistent denote main formally state follow bayes consistent give unit diameter lipschitz refine enable generalization optimal property dependent expansion computable finite chapter rule terminology extract induced assume compression optimal generalize compression discover nn various heuristic leave reach nn label extension classify decay oppose compression space normalize diameter lipschitz small lipschitz denote metric cover ball radius finite agnostic whereby label example nh bayes classifier define infimum surely slight abuse distinction space partition positively subset margin minimum opposite label naturally induce h sx margin agree previous definition binary double style double double distance pt scale shape mm fill ps ms ms leave none dash p fill none ss minor terminology connection make member extension actually explicitly minimization paradigm minimize penalize explicitly term motivate analysis sequel nest inner n slight propose analyze sketch ff ss minimum cover routine matching computable ff bipartite graph compute opposite label minimizer total small u formally n penalize empirical risk ff f margin optimal r risk n fx pt technical sample break basic decompose excess decay surely proof proof appendix order connect rf concentration bind unknown would overcome introduce surrogate follow illustrate empirical common double deviation nf term decay almost surely index risk rf nf r rf rf nf n nf since nf nf n nf l make form penalty yield nf l nf inequality follow approximate margin particular hold r n margin surrogate loss every hence large r estimate analogously bind run various convergence compare actual take put x illustrate compare four classifier classifier rbf validation describe cover matching searching runtime nn satisfy contraction lemma essentially contain idea l writing bound term lemma write term since l take r recall define n diameter imply totally cover finitely ball diameter imply continuous set generality normalize pointwise lebesgue dominate imply hold choose sufficiently small prof claim rescale ng r decay lebesgue
plot atomic bandwidth figure repeatedly increase separate observe distance choose range number dx figure curve cluster sigmoid bundle bandwidth resolution shift identify atomic cluster dash line could potential outlier similar signature signature simplicity signature acceleration tangent signature imagine signature produce unimodal distribution acceleration acceleration signature signature good replicate original signature acceleration multimodal suppose different e signature signature output surrogate assign acceleration close allow signature cluster formal analogy simultaneous test local mode identify mean therefore effectiveness curve strength reduction choose priori cluster furthermore functional tune distance implicitly scalar functional shift properly select selection shift algorithm bandwidth drive propose optimally estimation bootstrap selector arguably bandwidth shift incomplete proposal strategy introduce selector propose maximization mode significance maximize hard come selection regard investigation datum bandwidth interesting research gate obtain explicit highlight correspond ascent analog publication r estimating mode functional bootstrap estimate methodology shift algorithm sort e shift original signature shift draw figure depict function distinct unimodal density intuitively shift axis direction ascent converge either leave leave local cluster idea generalize figure unknown empirical corresponding empirical density paper idea modal euclidean natural dominating find local surrogate estimate due surrogate require existence dominate open space approximation call fast local mode density repeatedly close use bandwidth update density local mode eq density activity apply signature mode unknown early several asymptotic normality mode derive risk obtain minimax dimension mode attain propose root seminal shift algorithm science segmentation mechanic shift arbitrary repeatedly bandwidth generate ascent close unknown precisely line repeat solve unknown also call discussion ascent ascent corresponding find justification density critical ascent see initial assumption distinct intersect gradient naturally form collection equivalence inferential viewpoint sound foundation definition mode correspond associate estimate connect component px estimate definition depend severe represent drawback furthermore pose often different persistent although euclidean challenge functional branch deal datum dimensional surface last decade nonparametric fully euclidean realization theory therein attention principled vast density attention difficulty define probability address define algorithm euclidean towards let x x shift gaussian euclidean local mean profile shift consist update simultaneously datum tend converge number algorithm perform cluster converge repeat update generalize shift clustering consider isotropic allow presence mean act opposed version apply simultaneously entire generate multi algorithm shift keep operation call equation depend position rewrite unitary q estimate density position gradient ascent large correspond position direction size visit repeat update candidate view ascent allow set trajectory equivalence class unknown define shift number imply ascent unknown recent consist suitable variable potentially dimensional measurable space induce generate pair difficult natural dominate replace radius induce function behave consider process decomposition value cumulative px bx fx fx op lebesgue dominate measure propose population surrogate profile distance two assumption henceforth shift exist worth surrogate space may define basis component small help mean numerator turn interpret population informative flexibility user pca force towards shift incorporate prior implicitly many save example uninformative semi avoid alignment fact systematic among curve show shift hilbert ascent gate profile notion ascent characterize element maximize gate derivative proceeding recall definition gate gate banach let gate differentiable gate differential gate differentiable gate linear gate gate estimate gate gate differential bandwidth possibly entire profile satisfy truncate instance gate derivative still surely fix long verify among purpose ascent say matter coincide gate functional rewrite unnormalized estimate surrogate operator analog equation eq ascent conceptually conceptually fix point update see mean ascent functional problem generic hilbert starting lipschitz gradient map therefore surrogate gradient flow estimate converge enough flow guarantee flow population surrogate regularity address address sequence shift sequence element approximate think associated flow fix adaptive bandwidth incorporate unnormalized analogy evident eq root bx hx bx closure open local maxima satisfying consider local bx maxima functional shift shift local develop hypothesis functional next impose profile profile ascent scheme profile differential ensure right table satisfy another useful shift automatically separate respect atomic p bandwidth note functional obtain iterate q simultaneously functional principal intrinsic coefficient th functional essentially projection mean produce clustering cluster know elliptical shape speak cluster cluster principal score elliptical shape yield similar however intrinsic dimensionality functional great evident space span principal oppose functional coefficient component top dash elliptical combine situation shift depict lie fail circular pattern seed suitably depict obtain cluster shift meaningful effectively modal previous show shift surrogate satisfied statistically functional mode whether point second gate derivative definite assume unknown surrogate twice gate gate analog hessian point surrogate negative pointwise constitute instead test return point similar adapt functional natural test statistic p give gate differential surrogate gate surrogate gate order optimality profile involve whenever element adaptive divide size shift subsample step subsample next subsample test statistic second subsample statistic unknown use construct take purpose
ph co analyse optimisation un mod un de est co en un pour des code des es le dans pour la la code la de la dans une base un les en pour la de pour est simulation tool study computer time consume run study sensitivity reliability molecular convenient code since suppose take less actual say design already widely study deal code numerical code code consideration call conditionally function code set type literature stochastic shape distribution normally author output approach variance author model generalized quantity process propose estimate one moment hypothesis output estimation moment carry computer consideration input return computer single support access probability input density I represent real choose non density bandwidth estimate output stochastic relevant express coefficient link apply analytical carry application consist choose classical regression problem consideration kernel estimator hellinger suggest I four dark light along black hellinger vector major drawback kernel curse quality dimension sample average poorly variable bandwidth kernel technique vary parameter several adaptive numerous use capture curve quality form function regression goal form former sample function datum lie provide build basis coefficient f I therefore integral way adaptation orthonormal experimental subsection negativity account adapt function property function experimental impose orthogonality ease prediction experimental coefficient among know build orthonormal onto maximized maximization problem spline propose apply discretized ensure sample decomposition ensure propose however approximation et al propose put negativity let free first basis force non function interpret section analysis build basis interpolation denote define function uniquely pick parameter g f specific hereafter mp interpolation point basis compute heat dependent channel algorithm therefore mp approximation interpolation lose basis greedy associate f coefficient solution quadratic
rate estimator sparsity favorable aspect simultaneously convex discussion section beyond bandwidth precisely aim beyond regard penalize group restriction generalization lasso nest structure group parameter zero hierarchical convex employ hierarchical penalty tailor semi extension penalty connection also write datum formula efficiently relate establish treat population recover minimal make employ show frobenius multiplicative logarithmic factor appropriately define population thorough demonstrate example use covariance discriminant describe estimator define desire sparsity pattern triangle form index notational useful express group example extreme may seem outside indexing triangle element panel depict index frobenius second express toeplitz w already bandwidth corollary matrix positive high hold moreover version guarantee course behavior give w estimator group lasso notice traditional act size penalty zero recover bandwidth estimation norm minimax logarithmic factor pattern hierarchical term employ triangle recall note principle weight consistent selection minimal weight refine norm particular fact include penalty penalize hierarchy within exhibit hierarchical factor appropriately covariance estimator diagonal pattern coordinate convex guarantee separable separability block correspond update involve ellipsoid explain ellipsoid remarkable proved weight start covariance large triangle pair thresholded triangle ever step bandwidth next show estimator dependent contrast estimator bandwidth toeplitz dependent begin adaptively estimator section property begin state assume marginally sub exponentially true nonzero enough demonstrate estimator recover need adaptive hold result high see proof assumption couple inspection need directly see avoid probability regard ij distributional marginal x continue hold possibly bandwidth prove mild next intuitive bandwidth able detect require measure root size noise high require sufficiently might weak showing exceed threshold previous may excess requirement however without bandwidth bandwidth small theorem apply scheme requirement positivity unable either minimax logarithmic population study begin state general optimal minimax immediate corollary bandwidth positive covariance deterministic theorem result weight achieve variance trade probability price constant multiplicative motivated general class henceforth trivially notice magnitude entry class vector assumption covariance would shorter tailor penalty adaptively minimax summarize suppose either logarithmic similarity base single realization discussion distinction penalty optimality weight suffice one must resort previously consider relate interestingly exactly force neighboring decay necessarily contrast type matrix neighbor decay immediately far enough specifically appropriately constant factor minimax respect frobenius adaptively approximately semi convergence frobenius require prior rate frobenius block statement class minimax covariance minimax sparse cite inspection state follow clarity estimator condition even illustrate theorem recall imply op estimator focus behavior matrix norm instance approximately immediate adaptively slightly suboptimal strength optimal rate class adaptively sure criterion bandwidth choice adaptively block minimax study positive quite circumstance prove assume assumption either provide reliably definite definite suggest choose conclude note light corollary property ten spaced value simulate move covariance frobenius operator quantity vary linearly frobenius side align closely pe simple weighting weighting scheme vary along equally space simulating suggest corollary f pn pn phenomenon phenomenon suggest section performance band band band nest lasso cholesky series estimator vary procedure element evaluate way frequently lead semidefinite require section definite problematic consideration suffer three description approximately take increase particular lasso outperform simulation scenario matrix parameter scenario likely nonzero find block still covariance nest lasso slow rest frobenius norm find band perform operator bandwidth essentially note poorly contrast poor two op increase frobenius require convex observe label quadratic discriminant assume assign estimate binary consist goal classifier automatic predictor intensity within class inspection size five parameter estimate rule regularize typically kx lda regime introduce kind covariance practical estimator problem admit result adaptive rate operator matrix multiplicative logarithmic matrix establish allow bandwidth grow estimator rate optimality truly contrast propose guarantee exactly version guarantee finite estimator fast linear indeed procedure require semi appropriate package name implement code method equivalent minimization primal duality dual function eq coordinate clearly lagrange multiplier equality make replace make rhs decrease simplify r h turn w numerical tuning range wish eigenvector op kn constraint may drop meaning update involve project onto semidefinite cone similar explained initialize positive part subroutine r return proved inspection directly place avoid ij distributional assumption marginal c q next lemma exist constant let n ij j jj constant find find lemma eq easy q proof equivalently hold k theorem wish recall separately theorem show w exceed follow requirement establish complete theorem q suppose recall amount net amount weight state prove instrumental first matrix newly contribution treat differently arbitrary second set denote vector penalty term write new sub b gradient monotone dual cosine simplicity focus constraint hold eq follow next equality eq finally let
high confidence propose reach prescribe hoeffding matlab part guarantee integration library variable quantile integral optima case ensure almost interval know confidence conservative bernoulli possible include failure process complex able iid simple analytically tolerance tolerance confidence publicly next automatic integration present review constructing rely theorem mention know get variance need desire width suggest construct confidence interval proportion add pseudo failure adjust since well interval carry suggest tail calculate exact uncertainty confidence interval however small follow inequality need algorithm prescribe cost computational confidence interval end discussion sample random variable review chebyshev mild chebyshev inequality random variable interval know letting mean width interval eq costly quantile gaussian variance bernoulli satisfy slow term chebyshev interval guarantee need provide conservative width confidence random use inequality inequality variable suitable chernoff special lie hoeffding inequality eq construct width uncertainty iid mean computational hoeffding algorithm release guarantee integration library library automatic pp replication confidence answer return show tolerance replication replication result guarantee replication ask exceed encourage answer case conservative least loose try construct guarantee uncertainty ratio reasonable price pay htbp recent practical carlo monte carlo numerical require input practitioner justify motivation construct bernoulli soon matlab toolbox
answer differ problem advantage towards challenge dataset latter learn mapping linguistic question answer task end enforce task learner benchmark base answer task exhaustive symbolic annotation answer question large answer task world real scope beyond tuple recognition dataset rich suitable retrieval challenge reflect represent content contain different question answer stanford consider question segmentation method v discriminate object category color stand refer thing color annotation heavily concept frame include language work number predicate word word error reliably common cutting cut front restrict sum common treat logical database assign similarity aforementioned requirement metric quantify architecture motivate accuracy via membership answer produce respectively bag author membership suffer aforementione whole question recent work direction improve score collect answer first run answer answer answer many answer answer agree answer average answer call extension potentially base coverage issue problematic concern abundance improvement consider similarity success accelerate believe consist learner limited hand build learner source additional vision resource contribution architecture identify exhibit answer challenge architecture task scenarios max institute inf language visual rapidly observe increase modality allow towards open hope achieve ai machine towards quantify increasingly ask open summarize challenge answer answer task base unique truth annotation carefully drive force benchmark provide output challenge progress machine understanding task progress inspire researcher architecture language sentence alignment question answer argue open answer attempt tool benchmark quantify carefully well generate grow benchmark evaluate increasingly ideally yet want metric evaluation assign domain challenge base limited coverage third aim issue bind reference frame answer task answer inconsistent obviously human inherent compete true truth answer true look consensus take multiple answer interpretation metric idea entirely build open aim demonstrate challenge exposition helpful building challenge open work modality either world machine aspect open answer deal challenge prominent order guide scalability natural reasoning serve human spatio architecture reproduce diversity thousand ambiguity category grow boundary become inherently instance sometimes difference reasonable architecture create human prototype concept limited category color learn noun g white white ambiguity reliably human depend context may observer frame moreover unclear predicate aforementione adapt symbolic reliability
contamination asymptotic standardized contamination replace collect require form influence robust base idea robustness corresponding estimating technique bivariate random marginal z ne bn k contamination point induce proceeding estimator approximate simplify influence imply corresponding hill clearly unbounded nature hill robustness correspond er assumption suitably follow derive influence similar non term detail index case distribution may transformation use approximate contamination sample also contamination simplicity come let regression tail dependence coefficient tune contaminate contamination bivariate could contamination assume contamination say contamination influence distribution influence estimator correspond approximation distribution boundedness influence supremum decrease great robustness extent contamination hence increase quite intuitive robust contamination contamination influence derive contamination point influence boundedness present illustration first robustness contamination consider bivariate coefficient stochastically independent bivariate distribution variance coefficient compare tail dependence standardize bivariate tail generate namely mse base performance brevity similar report far also quite similar p th er er er f er er er mse increase bias decrease near near give exist estimator hill extent increase mse slightly long propose estimator depend robustness simulation say observation er er p shift contamination robust tail copula close er er er er er er er th er er contaminate empirical bias contamination heavy contamination interestingly contamination case consideration contaminate model case mse increase contamination optimum become choice increase contamination bias slightly mse mse increase estimator quite contamination contamination structure performance contamination extend mse increase pattern even well exist finding clear perform value mse large remain near respect estimator hill small bias influence robustness near however dependence propose univariate generalize hill structure density achieve illustrate extensive estimator exist estimator contamination member outperform bias note performance parameter choice real estimator bivariate future consider view statistic moment estimator tail sensitive presence propose robust robustness classical estimator illustrate extensive bivariate extreme economic finance become day big loss predict case may help analyze grow trend value model model generally tail rare event heavy univariate characterize tail measure tail multivariate marginal characterize distribution effort dependence linear market return mostly tail function specify coefficient dependence parameter extreme see work al exist care estimator coefficient significant portion external ignore factor recommend bivariate proper outlier tail drastically objective big produce estimator tail univariate see lee receive attention bivariate dependence see well result illustrate bias mse consider kind structure presence outlier start brief dependence coefficient influence section examine though section finally end short unit fr pareto tail assumption obtain limit threshold depend bivariate extreme asymptotic assume probability vary slowly vary function txt transform pareto transform technique univariate property tail dependence index researcher hill estimator classical hill basically logarithmic likelihood regression alternatively univariate recent attempt exploit weight density robust lee hill achieve proposal recently assume exponential regression apply study bivariate tailed consider bivariate tailed unit hill lee fr transform marginal view tail tail index excess exponential distribution exist positive number identically distribution power divergence outli lee routine show estimating estimator estimator
predictive work dataset inter one machine strength predictive six find reliable predictive model efficient var k provide much wide band inefficient var ignore reliability conv conv var sometimes band interval cv conventional solution confidence interval attempt compare effective nine benchmark distinct superiority var precise conv conv value small envelope conv interval wide general trend regression square interval conv conv fix var hand quantile general trend estimator efficient quantile reason superiority occur variance global function square base size confidence inter response global introduce interval model proportion specify regression literature extend find properly side introduce statistical interval rate efficiency envelope prediction perform method rank provide figure compare art interval locally idea behind exploit prediction interval contain desire could bandwidth technique parametric interval bandwidth bandwidth differ conditional response variable find confidence interval inter quantile distribution conditional quantile response obtain estimate quantile propose predictive contain least desire proportion variable quantile suffer quantile occur conditional predictor conditional conditional estimator converge important note suffer base reliable member take local conventional asymptotic conditional remark limitation list reliable prediction square regression bias error address desire
procedure w nice error show initial f projection procedure nice get rotation simultaneously diagonal since norm bound know rotation invariant imply triangle project convex column initial within column thing prove multipli never initialization slice expansion enough eq tensor let run satisfy incoherence least norm probability least last lemma proof show consider incoherence correlation condition draw component argue bind combine calculation incorporate valid draw dominant good initialization propose lemma tensor tensor assumption denote capture correlation weight suppose relative define probability expand prove accord span first column row orthogonal operator similarly subspace svd ii therefore top leave right singular leave singular spectral norm second inequality equality b equality lemma therefore apply I I I inequality auxiliary entry last inequality taking next h maximize finally noise high tensor
fx hx agnostic learn hold restrict version agnostic hypothesis boolean p agnostic draw randomness pac agnostic say differentially neighbor satisfie definition change hold neighboring refer justification say fx differentially pac refer deterministic denote value characterization agnostic characterization learn let coin way produce communication bit randomize protocol input public coin one way algorithm algorithm output coin private protocol coin protocol public coin protocol compute require notion complexity protocol unchanged protocol
low near high gradually reduce noise conceptually overall step step anneal schedule target length sample improve anneal length anneal anneal fraction spend anneal run handwritten digits mnist train validation architecture hyperparameter agnostic deep layer train decay schedule epoch fig subset collect sampling baseline assess anneal configuration sampling empirically confirm burn run chain
error system examine whether able detect cascade large environment epidemic source truly close almost principle might seed rarely epidemic graph entire truly report truly report node epidemic source internet internet good ball infect report reporting finally truly report degree closely law network test algorithm report classifier uniform report classifier frequent turn increase achieve rate uniformly report truly reporting scale free positive carlo infect node whereas node report state often implement term environment finite optimal optimize preprocesse theoretical adequate section thm thm epidemic extremely human reliable phone infect people home readily secondary stay home stay home precise identify epidemic local require knowledge
observe generalize secondly observe consider estimation initially individual sample size deal problem incomplete maximization recent successfully maximum article branch organize describe subsequent study devote study obtain develop section conclude remark provide readily read devote theoretical control branching mathematically growth define follow variable identically dimensional common respectively denote control particle generation law control generation process individual remove presence etc type add population etc population probability surely verify development law control seem formally belong power I subset shall henceforth index regularity condition exponential family include many poisson binomial negative binomial belong distribution depend term
sense occurrence assign preference switch switching incorporate go already active base problem finally analyze problem classify sense comprise instant basic equipped solution include switching field divide discretization utilize gradient time programming select priori simplified determine end algorithm network switch initial cite initial select solution optimum trajectory start drawback especially base fact lead great potential horizon cost horizon bellman approximate state potential motivate author study solution interesting feature development condition
deviation c c c statistic carlo simulation sizes square corresponding deviation error theoretical previous confirm monte small size favorable regardless gap scenario preferable geometric transition relatively square well estimator seem impact establish reveal favorable case consecutive observation binomial rare remain consecutive somewhat model statistical queue queue time every hour transition consecutive observation assume iid maximum queue
input sampling step compactly store find rank provide run input sparsity time weak frobenius e replace spectral large low directly find first weighted alternate different sampling alternate algorithm need length though alternate run spectral application matrix small processor good burden distribute guarantee tight capital typically denote th denote unless denote denote operator denote frobenius norm denote principal angle distance span denote constant
expect cumulative regret gap dependent tight constant gap upper finally practical important minimum cost span optimally algorithm solve optimization formulate find modular work variant modular unknown episode episode agent observe weight zero receive product basis payoff maximize cumulative return equivalently expect cumulative formulate delay network initially unknown basis span delay span delay contribution bandit bandit successfully optimization extend combinatorial problem solved propose explore uncertainty episode gap bind also gap upper structural bound semi bandit problem synthetic diverse movie range problem write write
classify hierarchical classify class associate example train case misclassification design classification multi temperature conjunction object benefit advance area occur acoustic scene update category signal likely either car strategy employ acoustic environment environment multi acoustic database database environmental emphasis computational design perform introduce propose modular present challenge fail significantly outperform significantly comparable benchmark acoustic misclassifie correctly scope reach human environment produce anonymous read early manuscript substantially also technical audio frame spaced bin map band human meaning capable well frequency result logarithmic result process constitute capture spectral envelope periodic exhibit spectral peak frequency cosine encode include frame govern discrimination feature belong relative belong rather perform aim avoid variation accomplish subtract global extract global deviation vector normalise standard infer interpret generative distribution notation identify extract model generate operator class equation accomplish parameter rule indeed must sufficient component generate large tend spurious variation datum
recursion depend alpha add unless three coincide full method largely stem subproblem describe alpha comment key depth proof alpha subsequently alpha special simplified benefit reader form write implementation avoid alpha general minimization comment alpha establish alpha assumption first describe identity product space equip norm definite hadamard abuse section alpha solve domain vector block fact terminology relate broad proper uniform serial choose start generate block sampling enter nonzero proper move change
upon close map image result count perform capture assume window grid flexibility like actual observation counting explicitly often count histogram consequence live conceptual matlab grid page parametrization histogram representation task art extract generative improve reason apparent need concern outperform bag word computer consider latent community recent scene image originally capture count text image constrain way text building drop give way find possible constrain property location window capture count model grid count need model window feature count count demonstrate representation count combination accurately traditional come camera model variation represent bag due computational achieve ignore spatial patch bag arise variety extract low image cluster discrete assign descriptor codebook ideally categorie sufficiently discriminative count become exist histogram multinomial validate bag feature extract natural relate image consequence classification
rate convergence rate approximately less smooth result alternatively process rt dominate matching possible term large rl match third convergence rate excess risk b ab b b b ba bc bc bc bc derivations supplementary material x excess derive excess bind z obtain hold argument bound x mu bound counterpart e upper combine bound union bind simplify bound illustrative supplementary comparison justify experiment alternative probability measure endow sound ridge regression embedding rkh special prove regression kernel distribution old
supplement data indirect supervision pair collect website tag distinct label cause noisy hence consider token appear size end symbol either vocabulary concern question scoring approach label image replace question triple intuitively consist project treat gram hand triple share scoring eq absence map kb triple sparse absence relationship might seem however uninformative form relationship another entity besides entity bag word lead lexical counter answer add embed modeling entity appear hand triple sum triple embedding encode entity easily question triple triple kind rank hence consist question triple
discuss scalability graphic upon permutation marker computational burden become prohibitive randomly calculate statistic percentage desire extremely many simultaneously lead resolution permutation make moderately sized association partition marker represent e effect marker trait employ marker use novel simulation marker perform costly permutation half million marker recommend quadratic snps popular notable tree decision appeal assume little phenotype complex generality box little typically study black box little relative variable measure occur without permutation association study issue previously task significant bioinformatic direct deterministic posterior inference parameter walk hasting rw mh explore slowly predictor
remain hold try fit full misspecification benefit generative robust misspecification goal dropout dropout difficulty analyze dropout tell generalization interested term measure conditionally thus analogue section classifier poisson model dropout eq binomial generating whenever large average average node thick pi connect z connect connect box theorem provide appendix heuristic intuition fix test central roughly j distribution
neural current analogous connected architecture layer deep network note convolutional attribute architecture standard code code produce explicitly assumption process easy analyze activation composition exhibit connect layer kernel fix finally model tractable dropout give desirable suggest architecture acknowledgement van helpful appropriate architecture problem construct infinitely architecture capacity capture degree degree limit suffer covariance infinitely perform dropout architecture neural without training relate examine
case expect necessity error neither require q likelihood truth px logarithm ignore maximize x prove achieve completeness minimize regime node decay consideration observation question whether precisely recovery graph sufficiently least vertex admit recovery whereas intuition path noise extremely arbitrarily constant contain edge imagine adversary truth pick give edge consistent rest inconsistent guess mean distribution error respect choice thus interesting grid central machine poor pose interesting technical computationally matching maximize cut plus edge get node among possibility choose hamming observation compute label agreement observation agree majority precisely otherwise predict incorrectly note edge v graph reduction weight hard full
fa state action represent feature linear slightly abuse action fa bootstrapping difference resolve issue keep per slowly behavior gradient reformulate ensure td td maintain along regardless follow stream experience bottleneck reliably order run scalable et formalize parallel learn orient w reward single experience successful take angle exist literature power reward additional originally
insensitive presence however typical reference reference elliptical tail occurrence point contaminate mixture one represent typical observation prior mean represent consideration contaminate gaussian contaminate gaussian interestingly density commonly chapter distinguished failure vertical leverage regression distinction good bad bad leverage outlier fit improve indicate yes yes bad leverage datum probability classify category proposal regression distribute introduce mixture define distribute approach allow actually could stem square mahalanobis observation group leverage point belong class covariate
distance otherwise least update cluster member least cluster history whether every stream distance distance distance exceed historical deviation call begin slide advanced stream treat cluster examine old long slide removed refer old empty remove list aggregation match assign move close assign standard deviation historical historical keep begin present evaluate dataset experiment assess cluster purpose comprise monitoring appear twitter platform minute trend work extract contain public tweet day ground sharing cluster volume tweet per hour dataset tweet volume series twitter bin one hour algorithmic cluster assignment assume availability ground cluster operate produce cluster define whose correspond confusion
repeat objective nonconvex cd iteration different ordering test penalize one j warm calculate computation instead active criterion full ji generalized model fix however existence work estimate estimate penalize ji criterion purpose dag select induced maximizer difference ratio dags tuning parameter index thresholding accept substantial group assume rearrange individual component p hereafter pp k edge point matrix likelihood h likelihood group influence likelihood distinguish follow
w ds get therefore q get upper size boundary control claim term claim recall ty inequality partition length sdp easy triangle balanced cost exactly conceptually important main structural every graph uniformly every feasible solution main exist plant cross cut kolmogorov bit uniquely number bit store permutation encode string kolmogorov unlikely identify set whose I store size graph e w right constant subset belong belong edge hx hx binomial distribution therefore bx bx v v bx gx x gx xx x exceed space transform pick embed encode satisfy permutation restriction bit encode bit use xx bit eq bit encode extra finish low lemma nd nd encode bit encode kp r g kp kp kolmogorov complexity use structural immediately
maximizer step henceforth call lie form em log unique estimate estimate care estimate justify find un subspace original em find maximizer
learn researcher practitioner fewer publicly successful application machine speech recognition surprising kernel theoretically model highly connection note nonetheless impossible deep address sample barrier benefit tailor aim automatic innovation advance much propose fast hundred million hundred recognize thousand scalable multiple way representation multiplicative well additive validate extensive benchmark effectiveness counterpart accept provide kernel important light readily hyperparameter architecture valuable test comparative view method original improve either suggest two yet believe line theory scale organize relate account approach report extensive automatic speech future direction training example summarize implementation computation keep portion inside early cope
standard demonstrate variance standard obtained need em observe satisfied truncation criterion change less verification converge local checking maximization routine converge maximum likelihood choice estimation gp algorithm advantage program frame teacher include handle package complicated program build matrix contain although tailor code mix matrix iterative nonlinear mix nonlinear integral also autoregressive toeplitz serve mention update computational lack readily available nr solve initial far stability routine modify hessian hybrid reliable observation student year gp fail gp run much cp fail fail fail supplementary material mathematic standardize test score student large pre process remove student link observation teacher link student
satisfy cumulative risk start optimal deterministic convert second recursive union version fast rate analysis order rf reasoning moreover convexity rf q combine new proposition convexity end argument convexity list rate sum n tx converge thank loss nx fx partial sum negligible fast price study aggregation tune large similar list contrary price constant grows extend precede iid setting condition cumulative risk adaptive n c n order provide iid eq
memory split subproblem requirement reduce generic solver solve optimisation bregman proximal therein admm useful solving optimisation handle formulate reconstruction paper interpret reweighte lasso solve problem base admm optimisation apply reconstruction dynamical input equation n gaussian white noise positive covariance description cover nonlinear form satisfy assumption system
grain sum network coarse grain evolutionary dynamic matrix simulation discard detail would lose coarse grain presence multiple pathway summary artificial trait acquire order trait trait acquire simultaneously trait acquire trait inferential point determine sensitivity accurately determine trait acquire I I inference comprise phenotype summary component sample summary infer matrix encounter compatible intermediate trait miss biological record allow place unobserved trait compatible possibility infer trait biological intermediate prediction predict inequality deviation strict explore trait compatible single transition compatible several account possibility trait acquire acquisition trait trait acquire first tracking network acquisition give higher acquire vice versa rna www com use life rna rna spike measure abundance normalise describe dna bind green sigma usa species www parameter minute second follow
assumption embed identically error result achieve order mu selector discuss introduce well adapt beneficial difference fix particularly order take mu rate assumption sparse selector assumption follow bind lemma small yield generalize selector design modify selector admit follow
interaction core multiplication min distance cubic short efficient algorithm level already message short total supplementary material complexity definition em pattern alphabet index integer symbol arbitrary word empty always clear context empty word e position pattern equivalent condition let empty place cost least derivation pt end pattern belong construction direct check forest tree treat cost efficiently edge root leave go index supplementary minimize dynamic
projection onto algorithm f k stop appendix proof theorem cg convex curvature subproblem respect norm h triangle department electrical computer superior email lx support ci e grant recently good statistical property regularizer generalize call algorithm regression contain contribution derivation atomic derivation onto conditional cg also wolfe problem accelerate projected evidence project considerably atomic norm dual
interpolation bilinear coefficient scale paper transformation scale dominate increase please refer supplementary convolution modification propagation implement pooling transformation bilinear use please material derivation refer si convnet baseline carry architecture except convolution replace open code si dataset come variety scale invariant scale variation unfortunately category handwritten digit foreground pixel architecture convolutional map kernel fully regression architecture model art pre parameter note augmentation convolution convolution convnet dropout convnet convolution weight parameter mnist
input datum criterion reconstruction dissimilarity quantization pairwise misclassification method effective low dimensional code generate binary code huge projection overcome method high bilinear projection shape respectively bilinear variety work impose fourier extensively process propose enable embedding slow table dependent time frequency alternatively optimize domain extensive show fast
pac bayes data experimental indicate analyse algorithm select vector machine svms well bound early rely consider pac learner involve pac probably correct bayes reinforcement data rely signal encode knowledge increasingly enable signal big datum different
ridge shrinkage approach collection mixture prior collection model behavior predictor mixture general version partitioning predictor block submatrix subscript prior distinct differential distinct amount govern block reduce ordinary prior design x block orthogonal block allow concept analysis motivate measure construct indicator essential block affect block prior recommendation choice hyperparameter default specific view prior block criterion hyper limit predictor orthogonality asymptotic simple summary essential rather orthogonality encountered design successively condition prior use variant result elsewhere subject mean component posterior g block orthogonal block similar satisfy define result behavior avoid hyper prior satisfy condition theorem prove say coefficient display relatively block drive yy lower avoid main hyper block satisfy away block hyper
property projection detail order logistic observe log l ny solve subject write square subject old order apply logistic approximate algorithm subproblem extension model version application dynamic outcome generalize high dimensional notion monotonicity implement
description scope paper focus mainly aspect click count j dt n j j quantify r decay influence choice I example simple view connectivity parametrize social recently lot see emphasis theoretical lasso process application structure intensity allow direct intensity baseline self influence want produce achieve dt empirical assume ground intensity model easily lead
state product gaussian view prior depend approximate sequential monte carlo posterior particle represent particle approximate weight observation parent propagate particle approximate setup learn hyper represent particle filter state filtering particle one particle receive resolve regularize introduce work marginally markovian therefore markovian four main part auxiliary importance consider would represent second chain line represent forward
behind aic rewrite account maximum hessian continuous ni definite proof modification start x vanish equivalently generate q rearrange term represent conceptually new rest derivation plug normality first grow prove adapt choose call term write chi degree free multiply take side know affect second equality entry identity increase surely design surely definite equality requirement surely em corollary theorem theorem method combine prediction implicitly make input hard prediction substantially differ aic useful shift suggest substantially aic bic averaging function
feedforward complex map region lead compositional layer output give replicate computation input expand basic definition intuitive composition layer unit define weight l activation precede activation activation lf drop subscript activation maxout network refer unit arrange specify number width classify function structure choice region piecewise connect input linear full depend hyperplane come linearity n distinguished hyperplane hyperplane distinguish hyperplane hyperplane several complement point hyperplane towards characteristic well hyperplane region attain hyperplane identification neighborhood formally two neighborhood input say identify carry layer feedforward region
univariate linear predictor expression response non randomly snp treat response analysis build three control adjust shrinkage layer layer control specific loading third control selection loading column wise avoid shrinkage dense loading dense require genomic capture effect batch effect factor couple pair simplify ard priors column resemble induce wise loading recover substantial dense loading recover match genomic expression observation capture co gene sparse gene module annotation observation component exposure specific gene expression include data loading association perform association low projection high trait result interpretable affected modeling gene jointly covariate datum supervise guide space maximally wide status may include co vary uniquely scientific maximally application identification subset exclusive direction carefully identifiability recover recovered scale genome wide association gene expression level averaging maintain interpretability enable finally extension allow heterogeneous currently heterogeneous believe
cell location know step time worth spend connected component plot policy video method synthesis temporal logic develop control temporal logic focus motion planning possibly ad hoc method extension game also game challenge player game product synthesis synthesis synthesis memory besides logic minimize method underlie maintained gain past synthesis objective retain interest requirement base share state obtain transition notational step iv iv ix I q term
laplace unknown furthermore integrated nest laplace approximation nine demonstrate acceptable self similarity natural flexibility modelling variance auto gaussian mean homogeneous specified variance nest perform carlo need precise second minute marginal parsimonious largely handle call use report sequel code kind multiscale transform standard transform direction wavelet model discuss consider denoise concern test pixel represent pixel model wavelet
participant survey assign likelihood relie assumption normality violate risk implement statistical software package likelihood function function maximize estimate variable upon number complete determinant discrepancy calculate assume conditionally
scientific separability lead insight approach algorithm typically optimize keep seminal lee alternating base least architecture entire optimization disk fit main contrast separability big nmf goal apply reduction execute matrix new motivated section leverage algorithm near nmf deal generate cone extreme deal separable since column find extreme successive alternative base find residual index subsection devote find column cone extreme
disagreement failure probability rx without classifier draw replacement validate rx derive unconditional integrating combine union complete query part h precision classifier validate px unconditional section pac combine get state way extension match match score match extension rate matching field different network explain node verify sample actual section validate single application produce match match score indicate part place identify match simultaneous
around constraint arise construct outer set instance separation relatively optimize property get upper cover space constraint ellipsoid possible ellipsoid dimension convex constraint outer relaxation depend rademacher duality multiple constraint upper rademacher rademacher q rademacher tight geometric point infer sharp imply illustrate single region cover set circle make large intersection region value version recover offset attempt relate compressed involve various assume former context deal whereas intersection multiple ball aid compute subject survey fundamentally semi supervise supervised exploit distributional unlabele distributional unlabeled distributional manifold restrict empirical focus zhang rademacher ball researcher arise unlabeled knowledge base simple modification incorporate kind knowledge focus algorithmic constraint modify
repeatedly problem either gain one always distinction unique straightforward utilize step denote initially look complicated problem exact outcome dx possible substantially simply criterion lasso augment rank various implementation address arbitrary somewhat naive sequence problem denote require qr initial change solve take maintain qr improve naive strategy essentially order magnitude detail implementation qr section implementation filter lasso trend laplacian fuse lasso computation generic system least offer considerable boost various complexity proportional iteration across always point complexity nan unbiased degree freedom lasso fit mark formal solver sparse cholesky tight empirically linearly discuss general operation qr outline aside overhead implementation filter fuse straightforward dx long trend filter fuse problem fortunately specialized implementation filter fuse general note ever early termination complexity various note implementation concentrate track dual enter coordinate leave enter boundary set step greatly exceed exception signal never boundary sign throughout
multiply initially take place basic alphabet space vector letter inverse dimensional cosine angle sum information pair store utilize summation preserve unique easily sequence fix permutation store say c see cosine uncorrelated show represent appear vector show
define bandit measure mean linear notion regret motivate framework usual distribution one risk risk root variant mean variance present logarithmic continuous function face principle sublinear another use kl use usual good
divergence initial target quantity horizon choose lead run example iterate multiply second carry experiment conduct pc follow intel core tm ghz ram use parallelization example density fact strongly lipschitz continuous constant explore section sampling bernoulli gaussian z z c experiment table vector histogram v qualitative add histogram equal equal density direct dimension overall generating explain fact consideration logistic feature label conditional estimate logistic parameter rely covariance logistic last perspective seem geometric justified rule especially ensure introduction
setting publish paper configuration basis basis configuration patch show dense original sample class tree codebook representation scene lot large high computational image level auc label c iteration scene scene yet solution even image capable comparable rank base treat feedback report close converge result
learn order upper train gradient particle stop perturb take local optima reach use perturbation multiply perturbed noise go back temperature
challenging vision widely adaboost result adaboost htbp cccc toy eps width pos eps eps width toy width network procedure understand reasoning logic advanced field boost variation boost large weak classifier overfitte problem train perform classifier combination classifier may answer yes widely weak speed small discrimination comprehensive boost tree network correspond thresholded adaboost remainder display failure adopt boost author
iv cf part part second part verify additional eigenvalue eigenvector assumption clearly absolutely positive open neighborhood origin nan inspection element impossible establishe claim argue nb positive large eigenvalue dimensional span unique eigenvector large eigenvalue impossible nonnegative w f symmetric nonnegative assumption absolutely r open origin nan ii limit large eigenvalue limit large eigenvalue kb b x assume part proposition assumption b belong x eigenvalue coincide lemma know continuity square always find subsequence subsequence assumption observe random converge distribution precede subsequence limit claim immediate one consequence claim w z z nothing lebesgue surface cf observe b b establish claim every borel far gaussian show borel unit various place particular fact origin obtain bb b n nan ng square random freedom gaussian joint mapping non borel measurable establish part prove measurable part lemma see hx g x gx must hold assumption obtain together establish denote measurable replacing argument part almost root freedom obvious concerned distributional probability rich allow independent simultaneously almost everywhere choose theorem theorem claim ac behavior autocorrelation spatial study literature test circumstance build finding unfortunately portion serious build framework specialize indistinguishable keyword autocorrelation correlation hypothesis important test autocorrelation time regression ii autocorrelation spatial overview autocorrelation regression low alternative note early seem show limit autocorrelation one become follow autoregressive see either depend certain observable intercept span regressor intercept typically integrated context test general power covariance responsible
e skewed sentence regardless alignment hence eqn simple solution effectively uniform learn fine grain embedding frequent illustrate figure subsampling subsample english train extension source implementation online asynchronous gradient time simply individual per improved pre wikipedia training perform alignment train directly raw text file obtain standard preprocessing gaussian eqn naive advantage multinomial setup next sentence make update parameter compute due log
technique efficiency publication result big work quantum amongst thing circuit heart many come note boolean circuit study result decide concentrate supervise straightforward studying compare neural net deep net despite establish technique add hill focus finally detailed analysis color course part benchmark benchmark refine optimistic fact vary engineering believe preliminary justification paradigm section describe framework binary boolean circuit short vector classifier circuit could else encode classifier obtain
weight net hide detector useful neural molecular descriptor encourage net train baseline least list use multiple family aid inactive c group expression cell protein alpha identify specific identify specific channel interaction cell cell rna generate molecular descriptor descriptor exclude molecular descriptor descriptor score neural generate binary select threshold ensemble limited inactive formulate screening ultimately rank optimize performance relevant virtual hold leave learn baseline forest rf ensemble lr fold model report result validation datum well particular extent baseline tuning g performance network include optimization long stop train net neural
overview optimisation brief survey process gps optimisation offer belief behaviour process prior exponential opt mat ern automatic determination square scale hyper parameter completely characterize mat ern make exponential thus optimisation corrupted arbitrary marginally gaussian specified behaviour acquisition function acquisition exploration
employ low estimator compute propose square mmse start carry follow bar optimization fix ki k ki ki ki ki ki ki arrive ki ki ki ki ki ki reduce
enkf consider possibility formulae formulae divide framework introduce coupling system derive joint mathematically possibility divide counterpart whenever convenient joint conceptually still aspect may appear attractive term far organize enkf estimation framework multi divided framework condition extension divide finally discuss potential development enkf example illustration extension divide differ thus focus hereafter ease drop involve quantity couple sub covariance overcome technical describe unknown observation operator e affect convenience vector respectively assume different sub separable corresponding sub say depend certain system augment become separable noise observation uncorrelated scalar formulae assume I still derivation let I member background
illustration link shape piece connect fit entire appear dictionary although symbol write boolean algebra non diagram comparable graphic recent os ds ss ds cat additional show indicate connect noun part image take pt formalism dependency idea tie formalism take translate relatively difference linguistic choice algorithm compact converted phrase believe linguistic phenomenon natural entropy mutual word pair dependency work viterbi discussion formalism control thought piece valid syntactic order single piece sign indicate connect indicate discussion marked head become head dependent word lexical word lexical entry group lexical entry conversely allow lexical noun different lexical past lexical lexical rather fundamental single link fine grain object object indirect grain reasonably take serve rough rapid extract syntactic sentence assume capable dynamic criterion semantic guide parse parse static mechanism outside external realistic viterbi operational general viterbi plausible apply analysis limit oppose roughly human dependency limited include semantic class role partial relationship syntactic criterion parse inter relationship inherent parse assign source inherent strength relationship inherently long well possible formalism semantic semantic lexical absence entity different syntactic expression particularly special regard relationship predicate predicate atomic predicate semantic sentence predicate subgraph hypergraph clear implicit require entity action formalism structure topic linguistic appeal algorithmic computer text base transformation transformation thus short appendix formalism ultimately nature extend describe syntactic relationship rather point one relationship internal constraint specific relation rx graphical summarize software linguistic exercise summarize viewpoint linguistic capture piece reproduce concrete write denote current might write
imputation implement causal nonlinearity special package imputation package matching fail solution imputation predictive fit check visually lead estimated spline smoothness estimation repeat ten analysis benchmark estimate fit observational distribution estimate multiple case systematically demonstrate variability issue choose nonparametric successful effect show causal calculus multiple
penalty penalty j could penalty elastic net class penalty r generalize relate p differentiable lasso derivative elastic net mc penalty fail unbounded create path previous theorem say
difficulty give table core believe incorporation sufficient test text learner separable varied experiment paradigm quantitative ordering see difficulty see cause discuss order correlation actually human order slight prediction neither consider difficulty observe differ complexity boolean complexity moderately boolean find fairly combination low case two triple one identical match htbp cc cc prefer complexity bold depict coefficient determination boolean depict boolean give case note discuss prediction experiment predict set child categorization difficulty identification please see explanation subject implicitly explain categorization good category
feasible quickly demonstrate linear classification classifier specific p later exist demonstrate algebraic tie geometrically way interpret dual setting relate ball classical like insight perceptron von concept get unit length represent surface ball obviously equation angle boundary allow give interpretation since instance condition feasibility feasibility ever extremely popular literature summarize sec margin turn
inf mh monitoring convergence compare dominate iteration runtime high acceptance proposal approximate derive chain case take chain much chain indicate sampler modal assess report expectation generate add expectation would agree individually value differ choose high sampler material follow simple graphic scenario modal room light center room viewpoint camera room camera describe position orientation roll angle estimate multi room symmetric camera result position camera roll camera resolution single core ghz take gaussian deviation infer location angle inform histogram orient descriptor image feature cell mh compare overcome improve technique inf quickly experimental setup mode analyze sampler visit mode ever pairwise mode change mode way correspond random
measurable spatio space space concern closed ball satisfy many construct point reasoning give underlie temporal information mark explicit summary measure regard measure lebesgue usual process spatio temporal point space section irrespective mark measure borel discuss turn functional induce probability e consideration copy may choose specifically stochastic discussion close wiener measure e induce type definition count e distinguish whereby thing probability simplicity measure irrespective recall element g l consist point take tie g call enumeration vector geometrically spatial functional aspect note support later process however already stochastic path tm possibility mark turn explicit deal spatio simple classification thing begin q location occurrence mark functional mark connection apart non spatio enumeration assign occurrence support may write l far require constitute process simple part ground constitute say additionally stationarity intensity notation let completely uniquely bounded irrespective usual marked process invariance case stationarity say rotation rotation temporal stationarity stationary refer stochastic auxiliary mark support I furthermore g x I r
base method note level propagation reveal worse mae find visible emphasize long propagation accuracy strategy filter prediction almost good suggest believe propagation aggregation method information incorporation trust factorization improvement improvement achieve type affect incorporate memory base exclude rating rank relation devise aggregation method mae rmse rating include rmse mae rmse mae mae mae experiment start user new recommender huge many system handle challenge randomly start include training table clear affect negligible exploit trust reveal lack incorporate social trust propose datum trust sample trust perform rating l trust relations mae mae mae rmse mae rmse rmse mae rmse rmse mae mae rmse mae potential trust relation relation lack trust compare utilize setup subset trust gradually effect relation trust table number uniformly rating remain feed reveal trust enhance utilize trust trust relation exclude summary enhance rich source mention go triplet could due triplet overcome efficiency turn try subset triplet gradient stochastic derive learn test evaluate mae terminate mae start reach dimension gd mini batch sgd sizes gd min batch mini sgd gd triplet sgd computation updating rule first gd simple name use size simply exclude figure gd although gradient suffer slowly comparison time individual gd sgd need gradient iteration gd take less iteration iteration accuracy attractive gd sgd least gradient compute finally progress make beneficial propose matrix factorization incorporate trust relationship potential overcome traditional summary incorporating indicate
subspace optimize single shift standard share follow literature use define subspace empirically meaning projecting maximize minimize shift leibler minimize result convex need derivative iterative subgradient terminate iteration binary multiclass benefit classification score multiclass stage target basis differently exist representation learn directly classification generalize target batch converge calculate derivative I v validate approach adaptation follow experimental report detail
deal proceed either consider optimize new objective ascent equation clear hold constant concavity hold constant expression concave due constraint describe handle ascent concave eigenvalue minimize distribution dpp compactly simply lemma formula dpp e dpp update eigenvalue need derivative sized impractical lemma dpp dpp marginalization marginals derivation self normalize explicit unnecessary exactly practice keep slightly turn eigenvector respect sum simplify gradient simplification contain identity
proof example part section continuity assumption entail without continuity entail eq equivalence modulus continuity fix expansion r bound put prove crucially use uniform continuity show strict take pick sketch case know discuss importance part know convex game payoff maker require knowledge subtle program depend indeed singleton crucially actually adapt case yet prohibitive beginning constructive well prove definition lack continuity theorem regime ensure randomize pick finitely element actually calibrate simplex norm norm space dimension auxiliary calibrate q use definition inequality norm substitute well example toy perform simultaneously incur player opponent combination opponent index
leave reader constant completeness inequality consequence bound since assume first triangular concavity use triangular combine immediately r argument f f apply second independent conditional vanish contraction finally eq n imply claim proposition bernstein give
template fitting example patch employ regression forest vote haar like refine guess location guess initialization critical contrast deep take pixel importantly template build fit fitting detection pose differ task task apart pose attribute gender useful robust detector difference feature rather face use formulate face alignment coarse fine cnn cnn pre partition face part cnns output layer successive auto encoder coarse alignment pre network lead still achieve computational scenario embed addition task reduce overfitte model local place method extraction whole region simultaneously aim mutual old machine task since allow objective task prove difficultie rate across work pose regressor body part detector optimize
random case give f ds ds exceed finally ef e e frequently property large tr spherical lie hilbert eq embed feature map product clearly tr see large time matrix r tr gaussian decrease course decrease nevertheless seem suggest conventional replace scale covariance
nonparametric fit additional complexity however note fit range quantile addition readily interpretable calculate margin form discuss mean correspond study choice amongst al incorporate variance stable development year effect dominate different distributional feature take utilize year incorporate quadratic gb purely perspective sampler rejection hasting pp easy satisfy variance gb variance give al dynamic function variance follow gb good complexity gb similar fit besides clear trend development covariate believe largely suitable tailed run heavy tailed variance clearly choice pp analyse go gb h quantile gamma pp al standardize fitting looking display gb provide good standardized display predict gb compare prediction fit percentile percentile arrange losse al close gb h model gb report fit predict loss level present quantile level tail range increase fast across quantile figure figure percentile quantile plot percentile quantile line gamma moderate gb reasonably observe percentile quantile gradually quantile al
leave note leave represent leave article represent heterogeneity image representation lot investigate heterogeneity incorporation use covariate information develop longitudinal patient task remain direction event say homogeneous intensity variable coincide condition critical reduce tree condition condition setup mean attain unit attain correspond eq suppose unconditional map coordinate clearly continuous density permutation unchanged ease tree leave question question whether projective basis probability conditional ignore kernel density first need establish determine law moment scale normalize positive project correspond implication theorem statistical analyze goodness test random arise model process basis distinguish population generative easily interpretable simulate tree statistic thorough heterogeneity brain novel representation wherein develop test heterogeneity brownian tree goodness test wherein underlying euclidean increasingly encounter several datum tree structure hierarchical include database involve detection record structure rna human brain task protein treat tree observe observable atom tree acyclic distinguished represent ease topological tree
marginal latent realization less z ij ij observe response sample weight group trace latent trace plot trace mcmc loading parameter ordinal wish college numerous suggestion university grant grant grant health surveillance asset economic status population south survey ordinal nominal absence explore homogeneous group asset status variable ordinal item nominal survey item factor nature probit use underlie structure exploit combine hybrid provide mixed nominal mixture md survey cluster homogeneous group within paradigm monte algorithm economic within region surveillance continuously south early south since goal contribute status asset index way population study accounting survey landscape explore recent survey result datum contain binary nominal item concept literature cluster analysis explore
feature space common square euclidean leibler divergence objective desire structure activation whose control euclidean problem could minimize fix minimize invert overcomplete recovery greedy separation exist example activation spectral speech solve set classic use together code discriminative aware discriminative sparse propagate solution reconstruction ground depend dictionary typically would need
interval affect population close enough use reduction conditioning differ expect uv uv independent variance range accuracy conditioning actual rejection skewness central moment skewness x z z moment enough moment skewness large skewness hypothesis test z desire rejection probability skewness simulation close interval reasonably sided population theorem act correct correct procedure bootstrap order bootstrap percentile interval error skewness statistic test obtain equation interval skewness estimate bit non respectively small comparison percentile third discuss application except constraint size procedure produce sample combine conditioning come subtle way context rule sampling draw except suppose store basic bootstrapping bootstrap residual give bootstrappe row bootstrap include bootstrappe square left panel bootstrappe residual predict residual original prediction residual sample randomly bootstrapping correspond bootstrappe principle bootstrap formula standard se practice bootstrappe huge bootstrapping observation say resample level software high factor interaction combination sample combination bootstrappe residual bootstrapping rule estimate bootstrappe fit calculate helpful bootstrapping residual behave lack refer lack systematic random bootstrappe affected bootstrapping observation variability result slope large relatively small slope height value residual prediction bootstrappe linear resample residual condition two bootstrappe help line help student slope intercept either variable help interval narrow variability individual constant much effect confidence interval parametric scale model parameter bootstrappe introduce bias bootstrappe bootstrap reflect smoothed nonparametric population may empirical datum positive transform smooth smoothing common rarely bootstrap mean bootstrap practically continuous except one situation data procedure one draw bootstrap systematically remain add right amount exercise mathematical original bootstrap effectively correction factor bootstrap error create population copy bootstrap selecting
remain time seven
kx kx result kx leads trajectory vi compare side sum similar except trajectory composition finally state remain remain origin theoretical admissible guess challenging approximation straight boundedness evolve improve mathematical field engineering south school rapid city sd edu remark control vi theoretically aspect include stability limit effect error involve boundedness vi system evolve estimation initial within region remain reinforcement rl dynamic programming powerful obtaining solution mathematically tool attract researcher application need analysis convergence learn besides hdp vi investigate pi despite vi remain pi seem adapt pi initial drawback vi learn
complexity dissimilarity implementation straightforward penalty constraint vanish feasible solution write row rewrite lagrangian term iteration consist fix variable update multipli matrix implementation ds define q update multiplier respect shrinkage onto ball notice minimization optimization parallel resource simplex constraint done randomize notice program thus parallel resource reduce expect time similar column propose resource provide solver cubic generate dataset vary server cpu gb fact admm framework efficiently study ds propose program regularization change representative possible base dissimilarity ds find partition set important case implication put obtain select less increase put emphasis compare large certain dissimilarity follow proof theoretical material representative different dissimilarity dissimilarity
attribute description benefit visual category visual research recent discriminative typical assume class scene setting meet rather closed detector like effort essential cope tailed object dynamically define datum shoot train learner mid semantic human teacher semantic predict category amount define attribute signature etc interestingly support cognitive literature researcher explore human object natural category conceptual evolve cognitive effort category human associate predicate biological attribute would offer elegant novel perfect attribute accurately often abstract linguistic property diverse visual road attribute shoot proof alternate transfer propose account exist value
necessary regime regime ptc regime uniformity least uniform sufficient prove tt nc tc bound rely slightly count sample property kk size linearity expectation minimum number exceed uniformity run draw chebyshev far focus dominant fall directly implicitly uniformity matter choose output achievable repeatedly run failure majority vote uniformity vote draw failure probability failure uniformity bind prove pick sample case small low follow bind give give metric behind proof pick term fail characterize sample uniformity regime tight remain small upper require norm whenever prove follow uniformity logarithmic possible guarantee output uniform opposite bind uniformity necessary theorem version prove mention guarantee correct guarantee lower prove splitting case probably condition indistinguishable uniform adapt tight distance sample relatively conjecture regime small seem
underlie black grey process true branching dash solid line correspond median dot poisson significant estimation case observe negative exponential strong instance ratio median bias exponential efficiency exponential density step complete dataset standard e point estimation around bandwidth mass one mass interval solve outside interval another select nice unbiased error branching simulate density branching simulation intensity model realization false true homogeneous random follow allow percent convergence criterion cumulative absolute difference summary estimate across fig branching grey summarize consistent branching branching branch approximately inter process apparent estimate become short consequence dispersion cluster long attribute
pool region show mit coefficient cnn score part correspond illustrate clear clean filter come part even training become selective face localize c c test image cm cm cm cm illustrate score part image though consistently multiple sharing appear capture image belong conversely multiple capture concept part capture seem specifically respectively object object object composition part appear game reveal high weight identify low suggest part negative class rather surprising people examine image class image visible face cm cm part part detection slide window fashion purpose part filter weight drive hoc discriminative diverse part concept base perform previously cnn accuracy improve selection level
sensitive admm slightly slight freedom experience rarely always correct grid check axiom theorem condition criterion author large fdr automatically find test region elsewhere manner discovery power separation signal optimization augment lagrangian demonstrate fdr exhibit simulate fmri work plausible fdr control smoothing multiple concern nan simultaneously simple problem summary statistic nan ensure standard testing control successfully apply across notably analysis dna sources genomic exhibit microarray include fmri statistics brain allele population correspond physical fraction spike location lattice environmental network spatially localize method learn exploit fdr find spatially statistic discovery rate pre increase signal raw score distinct research incorporate popular advance composite review multiple group wide composite regularizer recommend
bx q assumption uniquely proof many g van lemma definition separate separation result section rely mle although separation property strong condition common support compact compact strongly separates consider continuous separable since compact fx fx n value banach number banach similarly complete assume two strong guarantee analogue however separation estimator strong solution property inconsistent counter chen wu let px px px unknown lie whereas always produce value likelihood subsection conduct great density q estimate I normal degree remove select repeat estimator well mean median inference unknown parameter parametric base observation use assumption describe mixed probability huber draw population eq q contamination simplify randomness take observation asymptotically equivalent
exact grow approximated issue arise continuous economic university science department multivariate time series circular rely project skewed cluster relax independence circular burden involve justify carry data scheme focus recover finally bivariate time hide frequently natural multivariate series longitudinal generation unobserve hide modelling tool education modelling multivariate series component type mixed mixture univariate notable conditionally application properly accommodate complex distribution moreover unnecessary number often reasonable price computational burden result circular
process observe converse true learn learn stay dictionary learn serve realistic algorithmic process filtering take consideration examine explore interaction example investigate upon heuristic examine relationship encode environment generate element dictionary activation norm end wish posteriori calculate subproblem q mod alternate locally
basic requirement learner stable cart base subsection bayesian aggregation simple canonical scenario component remain vary let minimax well setting shrinkage robustness empirical regression lack justification sample sample calculate root rmse iy mcmc iteration burn mh acceptance ridge ridge cross coefficient covariate moderate lasso comparable predictor nuisance la la lasso ridge dramatically appear second htp htp dot dot display burn although fluctuation negligible like fig whose magnitude typical la predictor however predictor suggest coefficient coefficient truth covariate affect impact predictor ns predictor response ns la ridge ns lasso comparable moderate non excellent comparable la ns
every selection combine corollary distributional spread conservative quantification precise term parameter distribution j cf mode laplace possesse attain automatically make standard object correspond speed therefore quantification know minimax euclidean loss nearly minimax sparsity regularity parameter choose next show choice lasso put ball substantially big intuitively explain prior good lasso due induce mode mean vector identifiability posterior base solve design inspire full pac paper modelling prior model combination choice prior coordinate heavy tailed general kullback measure shift leibler might kullback divergence heavy signal quite q constant pac technique constant pseudo take posterior address question achieve slight large show corollary slight prediction dependency natural corresponding subspace sx collection distinct subspace define
dash performance reconstruction together baseline essential directly quite expensive predict neighbourhood fig fig rotation cell array impose priori fs nearly equivalently neighbourhood cell sparsity preferable result feature paragraph relate measure practical learn experiment qualitative use technique indeed nearly identical evaluate discriminate cat face template evenly rotation classifier nearly classifier rapidly fail rotation conclude transformation encode visual effectively rotation oriented dash line representation cnn use composition group comprise relu three layer relu convolutional conv conv linear layer relu hard negativity method learn mapping fs neighbourhood sparsity sect orient report sect imagenet report validation
average speed first benefit em rely insight noise sometimes lead well apply detail apply sufficient benefit quadratic define speed em combine noise iteration fast reach stationary figure ht noise lead general tool extract algorithm iterative show benefit chapter backpropagation feedforward neural network backpropagation theorem proper noise feedforward neural backpropagation log detail backpropagation backpropagation backpropagation ball illustrate ht noise sphere change benefit boltzmann rbms depth pattern speech deep deep consequence pt black circle inner neuron fill red neuron text width cm dot layer network cd give noise benefit rbms deep rbms backpropagation bayesian effect function model statement likelihood domain pdf also approximate freedom quality approximation model fuzzy system tool uniform linguistic build approximate fuzzy discuss fuzzy address hierarchical iterative context contain minor result maximization mm mm algorithm mm extension mixture alternate algorithm end benefit medical incomplete automate speech image genome denoise disease track prominent even analysis researcher em ten step current use likelihood distribution chapter em mle motivate generalization formulate chapter end notable estimation method preference preference quantify well summarize likelihood fisher fisher formalize use evaluate estimate rv pdf low observe likelihood contain assertion often implicit statistic question observe simplify class parametric pdfs provably convenient rigorously give parametric describe likelihood estimate e joint pdfs optimize log transformation preserve point log ml incomplete fisher attention reformulate outside bayesian year state mmse method select criterion finite minimize mean square estimate suppose sampling representation identify moment mmse invariance fisher argue invariance hold change alternate maximize pdfs formalize measure probability foundation likelihood integrate marginalization em marginalization unobserve maximum estimate probability weak mle eq mle normal likelihood analytically numerical method root derivative newton nr series derivative score nr use experimental corruption group additive extension mle complicate ml observe datum basic treat augmentation complete fit address sequentially well guess likelihood equivalent imputation nonlinear good guess compatible statistical deal complexity coherent idea decade ad problem use truncation censor information family little standard behind hoc extend miss synthesis field datum formulation array include censor group truncate mixture censor algorithm schema family handle incomplete simplicity schema least old iterative ice backpropagation schema cause explore surface log fast em boost em enhance fast subsequent enhanced like bt log likelihood likelihood maximize apply incomplete instead corruption loss complete lose corruption likelihood optimize address derive surrogate replacement ascent lead perform ascent result step iteratively suitable output remain ascent surface stop successive give tolerance converge data likelihood complete likelihood specify random likelihood random crucially model careful analytic convergence algorithm identify corruption pdf pdf mixture sub population population illustrative exposition observation explicitly another transformation complete right censor gamma give censor analysis random subject medical procedure right experiment keep track unobserved exceed time main change generalization whose setup assume select admissible reduce delta function eq allow transformation transformation admissible space add flexibility admissible speed ascent em first proof statement condition observe q q leave unchanged maximize force term kullback leibler divergence negative ascent relative e ascent produce limit limit mean point saddle iterate point example converge convergent maximizer guarantee log present apply map point set
sufficient element span partition span dash span assign assign dash assign left side remain polytope thm pt minus pt pt plus pt minus pt er analyze reweighte compact ground propagation lead marginal
see perspective sparse previously efficient gradient update remarkably without ever factorize invertible matrix initialize update update change computationally manner maintain version representation propagation principle target square see direct naive way prohibitive rewrite
considerably higher obtain accuracy comparable statistically another advantage second use rest text run entropy na recent benchmark author identification task training text english author word long data section problem write latter distance document give language report correct solution well rank english set gram second
sensor grid diameter star fig cycle eq spectrum laplacian star graph eigenvalue communication star cycle hence eigenvalue therefore take complete use derive require iteration near star cycle rate diameter network signal present transmission via receiver channel end cause channel resolve channel I copy digit respectively receiver recognize digit accurate digit receiver digit observe state none solely digit formally simply however identifiability information generate connect matrix exist discover true word agent equivalent
fix every every average radius let point large infinite result preserve preserve original let say preserve error distance least preserve distance hold isometry preserve pairwise multiplicative interesting property application addition product additive say literature say map multiplicative error subgaussian space isotropic subgaussian briefly linearity isotropic subgaussian equip euclidean isotropic subgaussian set subgaussian map mean subgaussian particular due occur less storage
expect overfitte feasible htbp ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc range ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc size ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc remark minus em possibly great obstacle human simply machine intuition could validate decision primarily focus design complex spam vision turn create interpretable system artificial intelligence decade even interpretable likely accept numerous capable produce insight score national medical diagnosis scientific input predict response create understanding inherently highly affinity datum address process practitioner adjust cm cognitive entity refer constitute standard sparsity drawback interpretability think complicated limited estimating association medical scoring assessment tool enhance coefficient adopt human tend aware illustrate idea believe sign expert correlation predictive produce constraint relationship match view domain build interpretable primarily accuracy surrogate interpretability proxy heuristic fully accuracy interpretability practitioner training practitioner perform tune simple accuracy interpretability introduce interpretable predictive framework integer ip classification primarily help accurate flexibility practitioner optimize produce accurate completely achieve theoretic balance via meaningful regularization without sparsity monotonicity difficult produce exist design scalability mean ip solver polynomial time linear argument flexibility building
jx mx ip efficiently take validity easy imply know completeness decomposable node completeness sum completeness necessary ensure validity completeness density probabilistic reason appeal brevity completeness univariate identity say none child validity value equivalent decomposable modify product remove redundant require introduction polynomially many additional light consistency interesting know node univariate function normalize associate interpret normalize weight interpret top child correspond factorize distribution accomplish top like acyclic formalize paper old justify definition cite generalize take function choose integration generalize observe always decomposable separate additional useful come section tuple formal arithmetic term polynomial term polynomial whose associated collect say function sometimes zero domain distinct negative negativity circuit monotone arithmetic variable factor arithmetic scope scope polynomial relate multilinear polynomial multilinear polynomial multilinear include determinant matrix multilinear arithmetic circuit arithmetic circuit compute multilinear say child node arithmetic circuit multilinear circuit open whether arithmetic circuit multilinear without case formula multilinear multilinear syntactic identity scope become syntactic multilinear arithmetic circuit view decomposable somewhat less obvious useful later decomposable view multilinear circuit similar affine univariate monotone multilinear circuit node notation partitioning polynomial collection circuit consist scope e multilinear polynomial multilinear multilinear determinant non multilinear polynomial row multilinear circuit circuit multilinear circuit node child concept circuit relate hard dependency scope become concept multilinear circuit usefulness remainder connection motivate scope dependency member depend denote scope novel relationship completeness circuit review give quick behave differently sense expand tuple fan moreover decomposable completeness free one way affect expressive validity definition
different network success adaptation overlap da method meta observe source classifier correspondence adaptation adaptation successful low classifier pick difference layer layer label adaptation incorporate train blue source one correspond adaptation method experimental case domain considerable shift domain summarize mnist mnist experiment deal mnist digit patch color pixel channel patch invert pixel pixel digit become hard dataset digit still cnn train domain distinct long poorly lead successful adaptation consider quite difficulty adaptation task synthetic address common synthetic house synthetic
remark subset usually advantage compute line result tree interpretable last process forest full create tradeoff interpretability score predefine number decision three subsampling column leverage feature replacement test policy development geometric subsampling norm column
product paper mathematically equivalent distance discuss mmd experiment version independence latter indeed mmd sample look except pairwise statistic statistic match use informally definite formal iff suffice kernel property base proceed calculate independence keep dependence randomly observation like time accurate calculate reject prove fix alternative sense type go calculate fix draw independence two repeat number conduct independent section test dimension
set I support lemma put pack vertical case pack next near long random q must exist make vertical horizontal packing vertical finite act horizontal take packing pair sphere row uniformly intersection control first intersection bound hoeffding replacement reduce live random span precisely
correspondence parameter mis see correctly also large plot none perform positive change log involve un
recently machine learning regard conditional describe conditional condition kernel survey private author characterization dpp question arise naturally mind model
vast store explicitly store value library store cross heterogeneous hessian heterogeneou two level figure dot ram nine million unique add element linearly hessian figure hessian cost log graph group color one affect derivative variable still able recover hessian number conditionally independent need add figure four heterogeneous across group many reduce sparse describe appropriate group hessian substitution grouping classic software difference pattern estimate package interface large finite gradient explicitly user gradient store hessian densely attractive assume log twice unimodal use algorithm approximate curvature log posterior available need quasi newton format trust region exploit memory cost predict convergence log require extensive long optimum find fast specific application optimization difficulty find mode hessian mode generate cholesky solve
improvement potentially edu theorem gamma involving mcmc stick constructive correctness stick break use process random measure explicitly poisson truncation variational dirichlet variational bayesian structure likelihood finally nonnegative factorization task corpora beta prior pure increasingly popular bayesian exchangeable ranking prior infinite latent indicator matrix
local clustering maximally cluster arc helpful exist assign noisy problem convex hull generate reduce figure contain non realistic reduction region discard convex devise middle distance measure artificial variation hull w w si extreme far must avoid middle ready set distance via set realistic variation control cluster threshold convex basic step cluster complexity evenly compare pair comparison run ie apply adaptive clustering cluster
significantly train example training help ensemble svm ensemble scale aggregate implementation implementation framework primarily nonlinear novel ensemble software svms equation default
fc learn machine memory experiment include encode ten category near machine svm category retrieve base encode mesh multi voxel category aim predict type cognitive process pattern activation brain acquire fmri learn major fmri cognitive behavior human couple scale record brain interaction spatially distant brain connectivity describe process outcome element brain connectivity physical connectivity dependence brain connectivity effect activation path decode selection define neighbourhood neural analysis correlation partial causality functional partial elastic net combine correlation imbalance connectivity correlation cognitive construct increase subtract connectivity matrix pearson graph modeling process structural brain voxel intensity connectivity utilize within connectivity form neighbourhood local
improve interpretability technique increase applicability technique high figure omit htb artificial htb htb dataset sensitivity obtain sensitivity map space sensitive analogously input artificial visualize ability decompose knowledge acquire train classifier powerful methodology classifier literature ability reproduce base human supervision
xx symmetry obtains directly behave permutation obtain output quasi precisely posteriori function addition similar xx dissimilarity inverse minimum infinity chain analysis asymmetric case q hence show complete quasi diagonal operation matrix equality inverse limit dissimilarity u census year construct asymmetric dissimilarity function similarity dissimilarity scale proposition form quasi particular choice decrease dendrogram quasi analyze highlighted proximity preference blue cluster six cluster east green west plus us dendrogram pdf influence proximity cluster record interaction input economic economic interact focus use production north american production interpret direct closeness decrease input come combination input rather economic dissimilarity input production use production similarity dissimilarity though minimum height cm anchor thick method quasi algorithmic facilitate quasi ultrametric node quasi partition focus economic four code extraction pc rl mp service support service dendrogram capture influence singleton dependence except service leave mp mp resolution formation singleton sc explain diversity service economic engineering service production engineering pc financial
subsequent cutting plane post dual optimization calculate minimum obtain search method far upper repeat mode refer iterate mode upper flip portion local solution bind consider greedy depend highly initialization repeat several method cut refer refer relaxation approximately simplified difference impose list consider follow produce relaxation introduce triplet long cycle cp author website add triplet plus cycle add cp cp relaxation art interior binary submodular energy approximate energy trust tr principle evaluate b refer bb upper branch tr ed bb default initial iteration implement branch b mini limit quickly paper limit set program evaluate experiment perform fair implementation compare website toolbox test ghz memory gb evaluate cpu ghz procedure processing reduction processing write mainly improvement speed implement state branching cut cp sdp produce interior cutting give bound superiority unary strength connectivity dense sdp sdp lp class impose scalability cp lp
principal component additionally grid performance implication functional rgb quantum mechanic know prediction cause basic quantum mechanic instead greatly reduce dft exact theory prove chemical solid matter middle body onto interact reproduce exact density reason ks dft successful interact approximation body interaction much reliability dft scheme build sensitive approximation functional past four decade extensive improve empirical functional require well trial great success dft bad dft continue exist accurate approximation ks ground enable dft calculation dft fraction benchmark free calculation capable treat unnecessary dft ks standard approximations ks dft utilize however comparable total euler equation energy functional
sis elastic complexity cv majority vote insensitive predictor b outperform conclude rank predictor lead robust cv bt method selection tune challenge validate speed accuracy selection b bootstrappe proposition claim bootstrappe moreover argue set biological insight finally yield contribution challenge lasso valuable wide guarantee toolbox make thank
material analytical landscape often sec modeling tend topic sized topic sec detail thing asymmetric discuss hierarchical topic science sec english wikipedia sec technical usage computationally landscape affect performance generate accordingly measure consider examine generative section show topic well prior assume commonly asymmetric prior language likelihood generative model limit document even increase relatively estimate topic topic sake word topic actually deal language document word across topic generate language let document later stress log model language english language say merged topic word english divide two english former general word might fit overfitting improve english portion enough word vocabulary log document generative big regardless per vocabulary document precisely english document difference english topic pay language merge per compute log likelihood know language english fraction document pick symmetric asymmetric point treat next section pick make uniquely therefore limit equal
cell expression low separate add variation gene cluster gene practice normalize deviation spc result spc gene cluster shown address proper cluster regard include side cluster among cluster assign column cluster merge form gene remain gene gradually converge cluster c identify cluster path ht b compare spc good value produce inferior add category calculation section near neighbor result sensitive identify parameter search value resample result gene classify require run spc c cluster spc tight see table none misclassifie spc add cluster gene spc table random gene spc spc spc attribute arguably separate indeed big gene start confirm specific biological role small three chance specific dna spc expression possibility discover far examine even strong fail small big cluster leave ht p dna specific dna binding
consider subproblem hx rx x tt k k concavity encourage reflect force solution x expect equivalent concave general dc consist hx program new dc problem function dc dc q k solve consistency approximate problem bottleneck start apply algorithm section machine class generally respectively seek discriminate set separate use adopt notation introduce take nonnegative slack value lie side hyperplane hyperplane first objective remove select observe special q sparsity induce problem n strongly simply solve briefly let compute nx dc dc dc generate converge critical finitely nh sect local solve program case dc apply critical nh differentiable local iteration consist k nx solve convex solve compute nx program n dc generate finitely nh x tolerance sect approximation beyond threshold proposition general enough sect exactly
letter use n answer follow datum highlight concept regime adopt coherence property parameter identify second identify collect behavior could confirm exploration mixture structure go second keep coherence parameter cluster please thing reveal remarkable useful coherence behavior affect analogously coherence general coherence large answer highlight beginning illustrate large accordingly might assertion experiment e dropping additionally reflect
detection sensitivity parallel resource neural networks cnn achieve great advance imagenet study cnn imaging classifying cnns cost implement computer graphic hardware investigate feasibility effective reduction facilitate object classification preliminary detect candidate ct volume automatically compute voxel heart
user proposition type development adapt machine set statement set disadvantage theorem dependency automate avoid first globally constant dependency set allow etc dependency minus name proof heuristic statement theorem work development manually check behave obtain lc
image transform rt rt rt orientation texture essentially need rt discrete grid ft rotation
expression represent replacement accommodate window follow operator kernel kernel outside certain region experience typical follow operator introduction currently rigorously ability randomly equal training testing datum half square divide small box version perform outlier mkl despite introduce parametric language would shape inference fully inference location broad trend explain parametric form raw supplementary acc interpretability spectral sp bayesian mkl exponential
message pass terminal assign variable show node correlation distribution ia terminal interaction message argument message pass one run dense norm close message pass terminal message follow residual respectively abuse notation important mention field approximation pass whose remove correlation measurement estimate system size evolution se output equation threshold pseudo surely satisfy noise conditioning single terminal characterization add get square noiseless mmse mmse
classifier svm classifier make sample pt propose mkl rt nn dr perform rt mkl rt dr sample class mkl dr overfitte rt mkl c class nn rt nn c rt mkl rt svm c rt svm svm rt svm mkl rt mkl dr svm c c mkl rt show mkl rt split consider contribution see mkl rt clearly approach successfully dimensionality dr select kernel great lack
implementation resort multiplication require hereafter memory bandwidth fair way non selection regularization run setting manner cost value exceed training optimization therefore times range website source solver cd iterate vary zero less cover complete picture wide relevant summarize news benchmark experiment cc cc operation cd list factor zeros cd slow cd obtain solution indicate mark significant extensive svm baseline permutation addition shrink heuristic remove word solver frequency tie adaptation list range medium r exactly differ coincide significant compute comparable code report compactly include fold interior completeness list mm algorithm red circle cd parameter curve curve validation percent green configuration well choose cd logarithmic dimensionality highly imply optimal represent subset adaptation coordinate overhead coordinate adaptation trust cd method baseline set runtime font mark finish l solver cover type training runtime second small font finish
corpora language successfully directly representation autoencoder english train autoencoder dataset compose million sentence translate relevant language use english section corpus provide news english category create top hierarchy document raw form setup preprocesse classifier document embedding heavily
sparse precision past recent directly involve distance incorporate discuss denote f partially empirical observation remain clear lead reason matrix pair remain sparse effect three point conditionally autocorrelation cut ar complete matrix stationary empirically location magnitude diagonal decay sort element compare functional approximated covariance matrix c ij sort plot equal fix increase increase diagonal precision fix illustrate numerically precision generate domain scale diagonal certain j comparison asymptotic become close much estimate different density point fix nonzero truncate
psd uniformly without order eigenvector norm since directly discussion skip divide objective difference convex
experiment star assign experiment uci census example node kernel fix training objective also drug discovery challenge irrelevant value method decrease slow due irrelevant section admm popular compete method admm discuss data atom draw true zero norm try relaxation figure versus require tune admm edge properly seem perform communication cost practical conduct experiment compare admm well communication leave synthetic finding overhead irrelevant generally expensive solve gradient idea spend ghz
notice case recover know proximity operator exploit regularizer positive satisfie v leverage note show step obtain furthermore term permutation denote writing assume decrease
clear passive learner learner draw draw draw label learner set label property square logarithmic splitting several perform ol result useful appendix universal constant follow sample return explicit wise label gain execution instance guarantee might ol
hide layer well hide propagate consist user compute sigmoid predict click layer record sequential span make rnn applicable sequential represent correlate represent behavior historical testing carry crucial achieve prediction group consist ad display text time whether head diverse large setting fairly capability predict whether user information temporal dependency quantity big rnn sequential behavior ad display order averaged likelihood sample label click ad feedforward backpropagation basically
consist combine begin recall principle sketch functional excess even dimensional return dimension later follow yx x intuitively value unless
risk satisfy say alternatively scale shall theorem establish asymptotic elliptical hold remark fulfil e derive asymptotic shall restriction differentiable function von formulate assumption ultimately monotone
vector e addition feed conventional machine stage softmax already column hold logistic advantage unlabeled paragraph bag word powerful paris paragraph gram gram lot paragraph gram gram representation tends generalize consider concatenation next window input force paragraph reality iteration text window text paragraph bag paragraph version dm addition conceptually softmax oppose similar skip gram paragraph one learn paragraph dm one distribute dm alone usually try strongly recommend benchmark paragraph
try small learning momentum setting get level paper feed restriction improve efficiency around integrate restriction like belief propagation model inference mean approximate marginal update getting convert feed forward iteration tie layer enable extension paper tool discriminative preliminary mean
make learn mapping approximately preserve theoretically recover local validate employ image nonlinear encoder preserve auto encoder make generality generate learn layer deeply couple transform handle complex auto encoder couple cross local consistency discrimination stacking gradually share experiment task superior ac cn comparison heterogeneous sample extensively image simple couple autoencoder seeks couple every stack auto margin intra inter penalty makes simultaneously preserve consistency enhance capability
control kernel weight define learn less generator prevent overfitte nominal attribute sect preprocesse attribute nominal sect return normalization later instance kernel line store learn center weight activate gaussian illustration extract consideration attribute normalize normalization kernel weight store center activation estimate width std discriminate kernel learning instance activate one kernel overlap compete narrow near instance take dimension width present spread dimension return consist task nominal pseudo code preprocesse I preprocesse imputation encode attribute x j j x binary accept miss line several advanced classification accuracy importance base category attribute nominal nominal line integer problematic converted attribute category convert mean close problem parameter line nominal attribute parameter category attribute encode nominal equal category
match various age status euclidean distance result principle define entry adjacency course manifold see illustration cone manifold database wish age group site necessary group different average understand broad almost surely consistent half provide general state develop analysis network state population appropriately behave manner possess distribution normal center define expectation equal pre specify correspond reference connectivity large assume true moreover target laplacian know drop subscript nan whether sample asymptotic every sample subsample asymptotic chi evidence state population special evaluate whether eq pool knowledge traditional unstable definite estimator area research year regularization strategy frobenius solve generally choice term assume covariance also substantial closely inverse network interest roughly magnitude find covariance entry procedure assumption possess briefly independent identically thresholde thresholding follow denote ij nx il
change step numerically learn close type initialize paper differentiable simple perform empirical search proximal fig illustrate basic idea algorithm proximal another direction ideally direction gradient preserve geometrically line try towards optimal line search proximal solution
work fairly experiment projection asymmetric transform perhaps indeed perform slight advantage transformation universal tuning reason prefer simple option previous exploit hash asymmetric lsh normalization lsh beyond theoretical user interested give item hash symmetric consider asymmetric unfortunately free asymmetric extension call inner summarize previous hash lsh lsh definition similarity apply contradiction hx hope valid
capacity partition depict figure key propose interaction potential constrained key function sequential carlo also design target smc direct chain generally see smc undirecte smc estimate limited channel introduction sampler known theoretical result
acyclic dag vertex correspond vertex literature notation random probability parent arc vertex simply denote joint indicate indicate parent probability n parent goal maximize posterior probability equally likely simplify account prior certain modularity dirichlet hyperparameter draw e yield condition encode bit define pseudo boolean encodes anneal analogue classical simulated drive fluctuation interpolation ground state encode solution desire guarantee state formalism useful hamiltonian hamiltonian final monotonic real monotonic function vary slowly
previous practical third distinction endow case initialization report superiority unsupervise scheme initialization small aid potential determining channel layer focus community theoretical capture typical acknowledgment thank dedicated carry partly intel grant center reference need follow induce orthonormal hc ng psd I n vector follow stand transpose psd non hermitian readily circle eigenvalue definition get j small inequality define v eqn main mapping hilbert z u u u eqn obtain I cauchy vector real w dependent multiply apply vector equation fix z take logarithm outer z equality reduce di eqn fix stand contradiction x u u u z u arbitrary contradiction accordingly I I create vector indeed enough exist j z orthonormal constant inner lemma impossible contradiction show state theorem locality pooling reduce vector rather enforce locality
total consider parametric easy linearity expect alternatively tight expectation formal method testing happen statistical viewpoint go hypothesis occur background model word procedure instead detect eliminate need event eliminate vast clearly poor value critical inherently robust show false positive relax allow preference validation preference considerably eliminate keep elimination eliminate preference feed speed claim tight preference readily bi number keep cluster minimal consensus discard relax value bi discard meaningful fix bi belong bi next present experimental multiple build non object specify several example propose bi job recover linkage see j linkage tendency line circle compare b linkage assignment assignment residual find linkage obtain decide discard stable decay overlap limitation linkage object figure algorithm recover scene build reconstruction correctly detect detecting discard base element line segment detect distance equation pixel notice adapt would far refined cell leave ccc segment assignment thick green green thick model assignment
jj c consequently resample mechanism permutation carry child similarly proposal proposal access child c state I c tree structured step resample node child particle weight ratio recursion leave sub root child weight situation execute run via computing requirement stack child note usual internal procedure describe step merge via resample care well study two result justify appendix first normalize constant c unbiased exchangeable consequence second particle list q strategy building structure graphical appear although tree come come undirected graph salient present discussion give situation collect hierarchical structure collect school school integer assume correspond variable specific hierarchy leaf internal parent encode tp dimensional ise vision encourage nearby field see grid bivariate connect actually describe collection index example integer encode hierarchical integer grid formalize encode configuration least back unary add unary factor tree start illustration self exclude unary subgraph distinct consider factor graph subgraph fact copy subgraph formally family respect variable lattice least without pick tree decomposition recursively construct index tu indice sir
chernoff kullback leibler two namely observe get subset law q adjacency namely vertex line lemma similarly lemma thresholded exact bernoulli diagonal center invoke lemma covariance page lemma rip universal rip numerous instance rademacher satisfy follow let bernstein hold proposition article concern large
svd approximate spectral ice edge deterministic scale matrix product tolerance approximation expression term discard eigenvalue small low approximate note gain filter consequence choose prior spectrum retain eigenvector prior hessian inform low hessian several mesh dominant eigenvector filter independent since surface refine mesh invariance mesh refinement also dominant content dominant inform portion truncate compute efficiently free slide parameter field satisfy forward incremental stress incremental adjoint adjoint stress incremental incremental adjoint linearize version forward adjoint counterpart linearize operator amount adjoint linearize equation since hessian linearize solve typically order magnitude linearize solve inverse characterize construction rank observable construct linearize forward compute linearize similarly adjoint usually forward adjoint apply dominate linearize svd require product component scalability therefore product discretization situation typical pose problem neutral content ice figure uncertainty plus carry algebraic scalable covariance inner quantification ice freedom discretize incremental incremental adjoint uncertain slide core laplacian take parameter characterize velocity magnitude observational iteration residual outer problem solve inner solve decrease newton
child implementation q lr learning extract child signal refine another signal time
jointly model document modeling allow model individual approximate eq demand one need store evaluate issue sample retrieval illustrate achieve eq weight ignore effectively observe even without impose favorable simplifie considerably weight concept explore research retrieval usually query utilize concept posterior experiment rank ensure preserve rank translate satisfy binary constraint top top rather experiment therefore focus preserve pairwise preserve rank experiment reduce
level rely framework tune aforementioned gold comprise relate match last baseline rule extraction seed report entity extraction table tweet label entity treat wrong h distant distant crf boost entity distant supervision entity separately incorporate tweet entity entity extract question tweet word skip gram draw context embedding similar extract entity manually sensible tv movie book identify match human label like attribute extract network extract twitter follow learning reasoning logic weight logic expression team world predicate convert predicate node framework optimize denote logic rule iff predicate propose effective another sort logic truth logical conjunction way formula say distance far rule term distant weight inference calculate predicate compare framework efficiently distinguish feature use soft one extraction list attribute preference
dimensional version solve regularization address six fista investigate problem letter one briefly let everywhere proximity define recover matrix q seek data fidelity consist encourage
know matrix element sparse become evolution instrumental analyze uninformative completion informative noise lead informative point stable coincide bind noiseless completion matrix uninformative initialization time evolution equation give uninformative verify fig ht phase transition count completion treat low concern mse phase perfect recovery phase mmse position mmse function transition mmse fraction signal variance suggest case much set count threshold case g present pca evolution condition small small low completion uninformative evolve uninformative correspond observe ht situation rank case second phase mmse beyond count mark short presence fraction phase count smooth decay mmse respect case difference matrix intuitively easy transition coincide mmse largely explore mmse however seem performance g dictionary factor variance index index modification state expectation maximization scheme conjunction obtain line mean element matrix denote analytically solve specify state mmse dependence mmse analyze stability uninformative number uninformative initialization evolve expression coincide transition mmse consistently analysis uninformative unstable solid line transition point initialization correspond expression stable factor peak qualitatively phase analyze obtain measurement computational concern scenario independent bayes us dictionary big amp discuss present form section amp algorithm bethe bethe evaluate equation approximate log large bethe mmse bethe free section alternatively direct expression investigation rigorously approximate compressed sense derive factorization statistical mechanic particular cavity section replica derivation rigorous conjecture asymptotically include compressed main evolution simple equation mmse reach amp obviously concern mmse amp interesting factorization asymptotic diagram dictionary blind matrix calibration
optimal representation compute average decrease shown conclude outcome useful include survival outcome increase retrieve investigate separately show beneficial repetition try overfitte dataset dimensional train individual time subsequently predict mse mse increase kernel increase generation also corrupt alone also examine noise mse increase level value dimensional h generate datum infer investigate
noisy experiment gaussian digit experiment normalize randomly sample cross purpose repeat noisy trial learn mc fig error exception mc well know feature space denoise particular train infer performance comparable outperform small relatively digits produce miss fig entry miss lot extension term subspace mc ambient complete method high mc complete superiority mc ij ij ij possible sum v q processing edu modern rely axiom structure drive geometric put describe relate suit term mc two term mc generalize regard follow motivate mc mc second present mc outcome numerical superiority literature drive mean analysis subspace subspace subspace decade modern statistic
generate space phrase similarly visualize phrase word visualization clear rnn encoder decoder capture structure phrase phrase duration cluster plot phrase right neural able length sequence possibly different encoder decoder pair term sequence novel include gate gate adaptively translation rnn score pair linguistic rnn able propose phrase encoder improve translation find encoder decoder decoder net language capture linguistic suggest language application encoder decoder phrase let decoder phrase note language future application acknowledgment bm cg thank compute cifar partially
approximate function central component paper find sign proximity basis method chebyshev distribution somewhat flexible similar theory tool construct polynomial lipschitz polynomial additional component simple small marginal consequence know close learn close instance box localization algorithm start minimize yx label moreover
compare specification genetic framework genetic reject weighted package assess goodness manner function fit routine closure generate fit specification accordingly goodness fit model along reference reject mark fit family version much well model indicate goodness structural accept reject visual help regard come case absolute mean comparable goodness fit propose star construct serve addition process network tie fit h simulate node star tendency star tendency fit among red similar nan tie node major aic see
update svd except local site keep derivation atom dictionary respective computing singular dominant vector update respective reduce set computable site dominant eigenvector collaborative power classical assume eigenvalue eigenvalue span eigenvector paper interested variant eigenvector tn site end site initialize site site attempt site site dominant eigenvector precede iteration popular consensus average doubly design rely topology consensus average initialized consensus consensus carry consensus iteration communication neighbor iw system z nz ti denote one imply consensus obtain nz j within power consensus standard consensus iw consensus iteration estimate finally carry site successive eigenvector fall prescribe local doubly site solve x k z iw ref r k full collaborative term initialization cloud differ svd site reference
user transform become interest new real significant lastly quantify dynamic lead descriptor management database application term network online site share site friend user content post choose connect information type connection underlie social second dynamic flow network user well new examine study consumption body quantify user however less flow along particular user create content flow user drop piece content get find might decide connect original access content consider interaction sharing event change detect predict challenge establish require explicit trace traditionally share hard quantify fine grain effect diffusion rich mechanism content connection information user examine change old complete subgraph english speak twitter tweet million new million one twitter highly connection change month month overall slowly background get particular information
belong class maximize sparsity decomposition datum propose search form svd singular nonzero result form call heterogeneous approximate plus identify identify fail identify simulation exception exist differ variance svd identify presence arbitrary method identify contain primary misclassifie false false positive misclassifie significant vice versa iteration later propose r algorithm genome edu recommend available http www http impact impact pre easy accuracy care feature default setting force use ghz intel processor base variety simulate describe method identify three observation misclassification sequential simulation simulation previously correct sequence prediction specifically simulation identify percentage instead record entry simulation study
recursion generator movement stream anonymous file feature use prototype contain seed generator create seed start stream advanced initial generator sequel simple seed working environment represent share parallel processing frame omp h include omp omp long seed omp omp std std std header file processor respectively seed seed desire array object default design seed seed state default call seed array share memory random subsequent benchmark actual generation number parallel correspond unique execute simple master worker receive seed advance involve package
value posterior measure measure monte estimate speedup target study speedup full drop speedup dimensionality grow become computationally expensive thus gain limited algorithm advantageous value keep computational reduce low bias accurate small result depend full sample drive inverse construct adaptively construction simultaneous exploration full accelerate approximate together sample posterior attempt square carlo adaptively algorithm preserve ergodicity sample posterior accelerate expensive distribution approximate comparable build offline vector order magnitude reduce build solve drive preferable especially though concept build orient surrogate polynomial li helpful comment discussion united department mathematics er mathematics integrate capability inverse department institute usa de er sc inverse govern equation repeatedly pde monte drive technique tailor use distribution evaluation couple together
average record reach nmf algorithm utilize graph contribute exhaustive comparison report algorithm randomize achieve run number base uniformly news mnist k provide algorithm obtain f table sense magnitude take add benchmark graph benchmark community edge node community node share become increasingly hard datum author equal degree node algorithm vary construction mix c c l
justification divergence follow bregman convex n l require discuss pl obtain eq slight improvement pi useful potential hull k q follow know proof observe corollary dimensionality notice standard simplex difference get standard loss k radius one union plug bound corollary detail interpolation norm choice depend choice give q show plug achieve form norm interpolation norm additionally strongly function private extend show descent algorithm matrix loss draw privacy guarantee matrix refer hull rank matrix bound immediately get excess excess empirical guarantee purely mirror consideration noisy mean
recover example def sigmoid poisson def gamma gamma support inner product control gamma gamma need distribute put mass soft spike spike unsupervise discovery gamma spike def weight constrain positive probability mass move poisson gamma draw close soft spike figure visually demonstrate plot gamma gamma concept corpus present portion hierarchy sigmoid use bernoulli combination pass bernoulli derivative def identity sigmoid weight factorize share model interval family deep
simple meaningful form go share dimensionality reduction resemble net autoencoder scalable involve discover difficult justify hand scale theoretic nevertheless hope preserve challenge diverse source biology method perform unless encode complexity make difficult free successfully coarse grain representation fully dimensional seem diverse without label system able robust face redundant start several stand preliminary follow neural scale g side generalize non representation acknowledgment thank helpful w nf discover explanation
extra additional pass constitute singular randomized range indicate intuition relevant canonical space canonical question extent randomize range effectively
logistic gain albeit high gain click statistically logistic regardless click agree lr lr fig support click question improve world performance increment measure report seem small third ad put ad support often also identify change level improvement remain consistent hold auc testing technology production consistent line live review combine output probability share numerous strength approach consecutive day transaction manner reflect world show predictive use performance set particular involve
follow avoid correct give validate frequency module module delay window require implementation module perform fourier establish propose detailed module comprise act behave cascade depict complex novel nonlinear recurrent neural show note remain neuron layer increase computational scheme improve predict recurrent cascade scheme detail cascade provide process validation
triangle bound
exact help gain insight recent participant conjecture nonnegative nonnegative kronecker nonnegative rank result usual multi ms counter one fact rank slack nest outer slack decrease square fit hence large existence nmf nonnegative rank kronecker two relate rank question tool address conjecture explain introduction slack matrix regular polytope give extension circle allow approximate cone program program nn hybrid observe slack matrix match covering improve slack exact conjecture never able rank run display nmf initialization rank conjecture interesting increase slack illustrate rank slack due important nonnegative slack proportional increase
tail near satisfy theorem note gamma satisfie requirement assume value either fix link logit probit basis h old true lipschitz continuous n nf w p relation hold j taylor hellinger easy covariate respect spline identity expression simplify use ia ib poisson away zero infinity link monotonic lipschitz root poisson parameter absolute choice hold fact constant grow polynomially kullback leibler divergence fix lie compact n contraction contraction
turn state seem write opinion far pixel hence individual template look white favor black encode opinion symmetric part look object part half turn rule observe argue noisy part composition rule motivate drawing probability compose responsible generating responsible turn noisy template part unless opinion extreme tend focus aspect likelihood improve
discuss ise mrf ising inspection relax formulation mrf precisely cg review cg mixed mrf categorical cg model besides mixed mrfs potential application return motivate particularly useful binary mutation point binary snps form edge mrf graphical contrast permit value real domain consider linear say need crf mrf first crf eq conditional gaussian long previous mrf distribution x eq normalization crf mrf count value gaussian crf discuss trivial dependency value mixed mrf allow interaction term count value continuous log word poisson exist product continuous implication dependency possible form homogeneous ising specify conditional poisson poisson ise mrfs ise value distribution random x xy value mrf distribution permit node exponential interestingly class model value conditional positive real conditional specify exponential xy homogeneous pairwise build within could expand formulation poisson homogeneous pairwise heterogeneous pairwise may count node bernoulli conditional gaussian conditional
include maximally effect classifier histogram analytical calculation potentially improve risk proof assertion optimization risk formula need possibility estimation risk estimation statistical modeling practice empirical theoretically problem recommend rule pattern supervise space value algebra subset probabilistic
reliable recover procedure oppose theorem pair leave form triple arbitrary leave specie gene exp observe proceed conclude reliably recover bottom agglomerative dissimilarity tree less long gene dissimilarity agglomerative triple leave let triple look second condition leave gene ab ab equation upper hoeffding eq follow ab substituting error pick define ab prove tell dissimilarity leave show ac definition gene low
observation follow instance notice default case choose episode vanishe default value specify study cumulative regret roughly worth result bayes theorem problem bound gaussian distribution speak cumulative scale dimension indicate linearly hence suggest tight figure vary vary show robust choice perform wide range identify people accept subject representative census feasible person offer people offer age whether person education year construct dataset offer generalization offer report
grow affect map jump coincide p accord set minimize direction four winner minimize treat reference contain attempt detect right check come move direction interpolation cost new original
position equally discovery background classification remove anomalous discover systematically learn task job behind capable increase amount type phenomenon outline deep anomaly light term worth note anomaly big datum many outli unsupervise technique information comparison light curve anomaly subspace massive number object describe method apply outli area variability class unfortunately massive set explore outlier create deal point discovery challenge contrary example approach advantage anomalous certain unsupervised anomaly prior whole outli many find would technique supervise outli obtain meaningful anomaly illustrate green two space grey isolate outli method red middle outlier region point separable outli product probability adequate outli joint occur probability build advantage precisely vote training classifier confusion assign possibility feed classifier attempt classify object anomaly previously mechanism outlier
cnn seq region size neuron pool seq table exceed supervise effectiveness seq cnn predictive seq seq seq layer neuron entire bag gram multiplying nb performance lm third layer vector variable size vector gram learn seq cnn table except region seq layer improve seq learn effectiveness indicate sequence gram size might focus baseline sentiment reduce vocabulary gram lm practice improve gram categorization gram lm effectiveness dependent nn nb lm seq seq cnn seq error comparison bag sentiment classification document categorization k training gram indicate
rate count area typical application intensity poisson process bayesian form intensity mode spread posterior explore early gaussian smooth spline cover paper also potentially inaccurate finite approximation transform process gp multiplicative endow hierarchical satisfactory aim
else numerical far know despite attempt proposal high alternate method multiplier optimization trend problem particular parametrization leverage choice big admm parametrization computing trend filtering linear operator specialize implementation strength admm implementation reliably wide tune small size value situation admm however specialized produces visually perfectly cover setting achieve converge specialized admm display speak admm consider behave order implementation routine specialize quite considerably extend trend trend filter trend filter worth extension univariate readily block generalized additive reader well iterative much motivating illustrate inferior trend short heavily difference order specialize primal dual conditioning subtle ever
line form work focus set mean perform inference distribution moment play stein straightforwardly moment stein infinite work measure sample draw set gaussian contribution shrinkage theoretically improve however require relax require well estimator continuous reduce norm bound implication propose theoretically practice wherein shrinkage different cross validation shrinkage specific present consistent shrinkage refer section leave cross parameter open dependent difficulty answer complex construct empirically shrinkage scenario include window discriminative shrinkage estimator already appear extend provide justification contain estimator rest section present various include aa continuous hausdorff say vanish borel banach lebesgue f l df endow semi definite uniquely rkh
plot crcr nan nan green marks mark option forget nan nan color marks mark mark solid forget sep mark mark mark sep crcr unbounded scale xlabel ylabel title red mark mark mark option forget crcr nan nan nan mark forget plot crcr mark mark forget sep crcr nan color blue mark mark mark option forget plot sep crcr color mark mark crcr matlab height scale xlabel ylabel data red marks mark mark forget plot row sep crcr color green marks mark solid forget crcr mark mark solid forget sep crcr mark mark solid forget crcr mark solid plot
complete theorem normality divergence estimator root divergence asymptotic dimensional variance divergence slight modification lemma belong family asymptotic turn independence derive regularity interestingly minimum estimator expect additional along variate variance although estimator bandwidth converge value corollary simplify case correspond clearly relate condition explore detailed consider simulate minimum mse divergence estimator kernel normal reference bandwidth smoothed density density choose smoothed pure without contamination density report clearly mse slightly estimator increase quite application reason divergence option estimator influence function suggest study several
relate electrical load learn device user aid interaction test system subject robot trial human basis feedback significant compare feedback device contribute initial benefit expect acceptance active learn affect artificial device birth course adapt device job independence major current insufficient properly lack area insufficient need control channel device channel alternate location body acceptance clinical lose device lack especially prominent versus type despite potential challenge device learn interact motion play
observation carefully grow baseline make characterize value regularization classification expect advantage smoothness improve classification smooth respective infer smoothness balance modification may idea develop restrict assumption set margin behavior mass around hand complicate large closeness sake convenience satisfy assumption turn hypothesis mass assumption guarantee possess balanced sense reliable lipschitz refinement assumption pointwise discussion infinite dimension worth point continuity setting since check eq soon assumption obtain dimensional compact support possess regularity compact support set follow example satisfy difficult check covariance satisfy belong dt taylor still symmetric laplace small laplace belong standard cauchy distribution possess close setting compare analytic ensure assumption set although minimal base restrict neighbor minimax provide margin smoothness away suitable choice classifier compactly context near neighbor neighbor
count match comparison handle particular instance broader ranking preference suppose pairwise c define match sign sign count comparison reference item compare cc similarity easy player player player show comparison consistent identity assume item rank replace property pairwise permutation r rank tie definition diagonal hence glm outcome pair comparison item increase define observation j since mean item order decrease infinity glm enough get low remove absolute k n q similarly participant play several time spectral originally introduce vector symmetric nonnegative irreducible irreducible pre value respectively permutation decrease matrix next technical lemma first irreducible matrix monotonic use irreducible eigenvalue since subtract positivity apply index value
fourth gradient state satisfy smoothness limit perturbation form theorem hold recursion derivation show specific square order examine recursion dependence matrix I r expression shall provide hessian hessian value notation next explain steady approximation sufficiently network hence hold agent type remark manner theorems establish define variance depend whether agent group refer therefore one addition entry influence agent belong level subscript denote hessian weighting scalar hand belong agent combination sub group argument consider dramatically agent play observe belong belong hyper connected topology careful deal agent scalar effect evident fig able determine
come solution original problem subproblem fix subproblem fix second subproblem subproblem decomposition imply take subproblem th derive matrix one converge stop give matlab g g direct propose version year admm algorithm apply estimate cite
assume problem component appendix theorem exploit recall attain selector suboptimal simplicity show optimal logarithmic sphere motivated presence deterministic mean nuisance parameter risk one possible deal result measure suffice example detail gaussian summary assumption triplet jointly minimax lower exist constant denote infimum estimator denote selector benchmark selector know selector ignore error study use generating element
rate boundedness depend occur e analogous fashion derivative evaluated rearrange jj eigenvalue whereas mle true similarly expansion contract exactly proceed identically change one part ii state dispersion proposition ii ease drop conditioning log mode vanish point ensure model regular hence infinitely ignore set direct hessian hessian unless ensure converge regard term prior plus converge element h pn p pn pn pn pn whenever conclude recall modal hence approximate add across mode hence I pn pn pn drop although part condition make explicit orthogonality assumption proposition posterior mode condition guarantee regularity remain add proposition x probability mode element hessian obtain proposition mode value maximize expansion mode n nh
use datum consist audio model gibbs truncation sampler fairly binary mask multiple mask correspond heuristic component corresponding track signal standard evaluate sir ratio sampler yield separation performance
method respect estimate generalized criterion apply grow observation method comparable cv expect ol drastically condition rise tune comparable modeling hand outperform local
unit per dramatically time description examine toy showing section anneal case describe cross categorization million subsample generalize commonly consider separate operation subsample schedule operation use mixture motivate section wide latent combinatorial nonparametric assignment cluster detailed list set represent disjoint subset chinese restaurant crp base call component recursively follow assign cluster draw probability clustering crp index order exchangeability suggest simple gibbs sample posterior clustering
use detect community though measure modularity focus graph overlap nash equilibrium experimental result relate material article draw distinction approximate nash equilibrium apply detect partition direct community prove bipartite experiment author community reach modularity represent adjacency weight kronecker elsewhere herein interpret community modularity link numerator margin cell many satisfactory example tends merge bipartite formulation consensus nonetheless regardless graph remain type graph transform bipartite action formal bipartite belong belong bipartite margin margin transpose margin margin block square symmetric node distinguish community diagonal bipartite detect detect determine validity partitioning liu introduce block bipartite notice author take consequence modularity graph partition graph
balanced sbm attract tend block small interpret sbm nonparametric general way piecewise approximate reasonably popular estimator propose histogram appeal control robustness noise case inherently measure preserve map restrict ambiguity preserve map histogram sdp relaxation like organize introduce sbm relaxation compare pca consistency result brief discussion implement sdp devote network histogram conclude discussion ease cm vector inner cone natural matrix confusion act square act produce norm kernel nan space th sub indicator elsewhere vector index kp formally sbm simple adjacency edge belong exactly community belong community sbm form independently bernoulli variable write think operation pointwise write sbm symmetric derive two sbm plant pp determine identity pp plant
square least propose briefly algebra advanced read several conjugate modulus say root unit unit angle zero geometrically rotation vector pure unit propertie eq calculus general derivation recent elegant calculus comprise derivative derivative
exact approximately stay try formalize fuse tend follow go ever interpolation deterministic must stay boundary outside change go lead change issue fuse consecutive spurious section
policy reader actor mention programming set architecture policy technique instead discrete govern incur translate map action identifiable paper policy go discount sum receive start initial discount rl cost go however challenge instantaneous next govern hessian cost optimization minima go project iterate projection iterate hence scheme cost go adapt monte horizon discount step transition artificial could obtain carlo simulation discount cost trajectory use estimate build loop direction expression bias
international student anonymous helpful determinant simplify specify width demonstrate error recover main normal identical assume get find collect identity infinitely broad depend text condition fulfil q result correction inference department college laboratory road centre mathematics university department
fluctuation player requirement rarely wireless general presence payoff perturbation payoff specie fitness variant aggregate weather effect fluctuation jump incur dominate strategy eliminate strategy likewise strict nash equilibria stochastically mild aggregate presence surprisingly variant deterministic counterpart dominate strict nash equilibria stochastically stable irrespective perturbation broader stochastically reinforcement learn derive player become explicit strategy focus long stability stochastically lyapunov trajectory nash equilibria stochastically irrespective principle distribution play equilibrium matter vector dual real span denote slight abuse delta simplex shorthand tuple dependence write consist per player player denote profile player space otherwise x account mixed regard variable payoff stochastic dynamic process employ cumulative
mle never determine interior require polytope polytope claim empty degeneracy condition definition add vertex give add vertex necessary integer difficult simplex conv ne n r sequence omit polytope polytope hyperplane integer correspond lie boundary interior observation large mle exist check sufficient statistic sometimes graph positive indicate performance graph convenience produce
optimize optimal generator optimal conclude interpret maximize whether minimax game eq reformulate global virtual see subtract recognize previous expression shannon show global process enough algorithm allow consider convex supremum attain descent update converge conclude practice adversarial represent excellent guarantee adversarial
reach curve panel ratio number negative quantity curve show close unity estimation curve right panel addition red solid maximum roc curve terminal roc roc keep increase low parameter illustrate see value training well end compare result regularization coefficient indeed change
source give train distance simple powerful online iy n consist class riemannian essence trial trial classifier defines alternatively difference classification involve class eeg class output class trial riemannian mean discriminant indeed matrix structure signal covariance temporal knowledge signal pattern processing stem covariance matrix embed average response index trial belong target trial super covariance decompose covariance trial covariance however cross absence cross eq
law subsequently exploit several previously ergodicity overall indeed online state reveal rely forecast optimal control law provide sharp sensitivity law misspecification beginning require perturbation result advanced program contrast without inspire less type control law preliminary version appear conference publication limitation mdps leibler online offline setting addition perturbation state cost omit treatment demonstrate role strategy type compare paper report thorough evaluation tracking graph particular carlo simulation bar compare strategy policy choose pool randomly policy without cost strategy passive r good grow simulation strategy well randomly sample policy frequently formulation mdp theorem contain control include policy analyze contribution direction future work result index stochastic cone row variation leibl divergence sup f
pca direction researcher center information extract gram favor motivation application many thus provide sort non pca central signal dictionary towards keep component analysis motivation measure hyperspectral computer signal processing machine learn gene bioinformatic computer human recognition online handwritten character numerous nonnegative consequence lose center mild negativity unique large eigenvector positive component center issue center versus pearson versus coefficient pca matrix counterpart center product examine counterpart devise bound connect matrix large gram examine outer matrix relevant eigenvector non matrix way center base center beyond extend machine eigen gram multidimensional centering extension conventional centering discrimination
z ji kx sdca exact coordinate sdca brevity sdca originally output expand prox stay incremental exact sdca perform current option prox operate dual zhang conjugate conjugate operator primal proximal basic sdca conjugate completely eliminate sdca ensure trick interpret intersection algorithm primal sdca convex rather strongly sdca sdca variants sdca expand perform coordinate operation interpretation nf
design thank guarantee mechanism optimize know therefore exactly optimal guarantee mechanism bad mechanism equilibrium particular bad equilibrium sensitive algorithmic intractable running mechanism algorithmic preserve section introduce task standard definition example function could clear context omit subscript sometimes shorthand use shorthand except omit whenever notation estimator property readily available expert return point expert produce unless expert worker effort worker characterize effort estimate minimize effort unless otherwise worker payment amount randomness definition problem estimate access worker suppose access know mapping worker estimation test worker payment produce worker produce minimize mean square
series indicate state emission multivariate covariance element interpretable minimize extent fusion eq weights control contribution fusion weight eq learn absence motivate freedom informative state apply reversible irreducible mathematically eigenvector evolution complete hyperparameter subsequent independent recover independence modification enforce adaptive identical affect th update solve th identity approximation close approximation stability eq emission irreducible must satisfy detailed balance ik
variational fast netflix however research property suggest depend distribution feasible vb prior dataset introduce conjugate posteriori discuss simulation study strength prior dataset test netflix dataset denote column respectively matrix entry define decompose together observation suppose summarize rather
label task gradient propagation technique due vanish gradient addition limit rnns range dependency step relevant elegant lstm design special unit memory connection store temporal multiplicative unit memory controls activation memory control output flow lstm continuous stream memory forget gate internal cell adaptively cell internal precise conventional rnns sequence well rnn sensitive language
edu weakly label demand object cope supervision discriminative visual formulate benefit discover together weakly challenging image cope minimal amount supervision prominent availability image annotation consequently handle need annotation effectively ever grow annotate available addition robust noisy ill motivate explore rely supervision early idea successfully albeit learn data set intra appearance variation background clutter cope difficulty multiple retain one frequently positively
translate piece clutter clutter ram reach outperform ram ram convolutional attention deal clutter look learn policy include supplementary material intermediate reliably explore avoid clutter translate mnist piece clutter table similar improvement ram convolutional capacity amount computation change image convolutional appeal recurrent attention interactive input dynamic bandwidth play binary pixel agent aim ball pixel bottom gets begin mean capture know precise position location attention game action nothing softmax action
alternate pair parallelism work give compound topic count document denote dot subscript integrate show dataset lda mini batch strategy mini processing read process block mini x subset mini period global across mini batch update determine accord explicitly denote pass stream examine validate gibbs virtue parallelism throughput thank hardware accelerate must accelerate bottleneck early online
I resample filter theorem filter exist condition constant unnormalize x qx x hold necessarily wise cope combine tx tx qx identically distribute
initialize code initialize near indicator update fix rank fixing code n tf nd time propose explore popular representation important connection code neighborhood unify construct ranking regularization
gaussian n follow assumption omit deterministic apply quadratic objective convex suffice ensure hold system geometry concrete acknowledgement office grant national foundation grant dms support microsoft fellowship second calculation vector j embed complement thereby rademacher width claim consequence expectation empirical randomly remain contraction inclusion put piece begin rademacher j guarantee accordingly inspection definition claim next vector shorthand jensen inequality ij j g claim convex belong closure eq z observe put piece previously hull define xx frobenius nuclear unitary take generality let index write section
matter fulfil let arc nice matrix ix x candidate rank decomposition hold function point indeed analytic sense map derivative u nontrivial kernel satisfy derivative already point obvious notational modification exist rank obviously neighborhood neighborhood u decomposition hold consideration imply contain requirement apply corollary fx choose find rank analytic sequence possess due instance corollary part possible tangent assume g angle share nice abstract convergence slightly fx g iterate possess rate abstract theorem act validity case leave
fusion base signature compute feed determined briefly image verify person test architecture signature representation take dot exceed identity train depict feature video face considerably locality sensitive hashing less unsupervised image choice temporal window confirm main resolution clutter class test assess transformation table compare perform dataset align version really state appear model fusion create randomly scale temporal image
maximum framework yet optimization method capability deal follow introduce propose cauchy simulate conclude method non approach weight influence item replace sensitive large iteratively method eigenvector weight corrupt large noise alternatively remove ability integrate sophisticated probabilistic popular laplace formulation alternate solver component error magnitude noise subspace small recover claim choose pursuit also poor incorporate
record need interactive explain concept world web far content internet medium video student allow frame medium video change face education providing fit hundred specific varie similar format content question advance online resource assignment interaction attract online circuit student first least student pass pass score complete first complete perspective student take course year mit education year online people complete considerable illustrate scenario student drop make course course finish analyze hand understand student usage response online thereby feasibility analyze rate student student student may lack motivation leave reason rate exercise believe understand student student reason prediction increase provide motivation prevent certain student able predict accurately base week imply manner design accurate statistical accuracy student student wrong accurate predict student would due reason paper three tackle predict student persistence focus aforementioned course circuit student indicator present comprehensive produce consider yet tune ask accurately persistence first week course week course history many accurate week ahead organize describe model build machine belief decision summary play role
log sample number follow solution numerically fisher method asymptotic derive asymptotic confidence denote l n fisher definite mle arrive theorem straightforward value q covariance result invariance asymptotically normal transformation mean variance
site fire solve multiple learn square root rank highly scalable one data rank future also scale berkeley edu berkeley edu leave share root sketch data low retain sometimes even real efficiency arise typical many problem
satisfy odd dependent model response monotonic transformation marginal odd ratio category produce remain collapse
stationary point window write cardinality arbitrary window simplify unit canonical gibbs model ph core direct intensity convention h u estimation standard configuration set location intensity datum configuration keep indicator well logistic offset computationally hard
power replace simple convolutional output apply input maxout weight reproduce hinge index filter dramatically unit suffice thing output multiple architecture replace experiment perform use hyperparameter regularize scaled optimizer choose large value balance lead instability add
need dataset training datum expert wish scalability curve time second ylabel legend pos south east restrict title header false sep comma txt table header col sep txt core linear plot well early restrict advantage namely solver optimizer solution hand partial optimizer saddle show derive optimizer solver wide applicability experimental competitive art sgd optimization asynchronous version algorithm equivalent serial
tree serve discard tree dependency word dictionary word dependency matrix fix unsupervise et al bias nonlinearity validate sentence wish describe work description attribute context scene naturally motivate context particular convolutional neural imagenet detect location activation parameter cnn choose discard simplicity cnn architecture describe contain million resemble parameterized sentence similarity sentence turn compute intuitively match rise criterion design objective image sentence
operation incorporation extend maximum perform mutual discriminate candidate evidence discriminative power train allow spectra rigorously paradigm l begin protein mass mass round roughly represent spectrum identification sequence database responsible constrain denote mass spectrum identification construct comprise let shift relate offset mass atom offset water plus atom unity mass unlikely search doubly unique convenience benchmark algorithm heavily ms spectrum calculate p mass number score spectra benchmark variant implement ms begin spectrum vector equal length construct denote shift foreground e fit
develop aggregate loss loss shall case characterize mention density consistency several reconstruction reconstruction observe keep mind procedure whose respectively measure mae distance reconstruction histogram histogram notice density apply throughout cc true density mae mae rmse mae c cc norm l method probability losse interest functional total distribution reconstruction laplace real besides quite reasonable admit analyze analytical solution hard variety determine probability loss record pass quite error datum operational loss frequent situation amount account certainly besides potential method lie explore operational risk problem tail economic
feedback present gradient descent representing initial interesting deep method author organize formulate follow extension section follow conclude remark f state time integer dimension definite definite first argument assign scheduling may latter weight usage resource set continuous feedback policy asymptotically system minimize order study e simply assumed copy policy delay motivated operation bellman equation sometimes denote incur throughout remain consider recursion x bellman state simply give feedback eq eqs desire utilize parametric e approximate backward fashion use find detailed select interest use repeat process offline conduct correspond nn provide continuity continuity switching versus early simplify
finite mdp special terminal episode starts end reach terminal transition loop mdps th layer move consecutive episode reward end learner reach belong equivalent transition mdp denote policy follow
discuss issue square problem though generic solver fast obtain design dedicated algorithm leverage active benefit subset leverage subset strategy detail bt j q j open solver slow solver toolbox ii iteration quantity implicitly work update active computational algorithm linear active theory much practice set shrinkage experiment
complexity summarize note present cause slow coefficient obtain emphasize effect allocate size outperform complexity em energy market http com use repeatedly n compare complexity section design w w w outperform initial aspect respective counterpart exhibit performance size kernel allow representation approach outperform approach would effect accord experimental study behavior size subset realize section efficacy selective strategy complexity em propose orthogonal projection multiple include
mit mit expressive scalability gp leverage low full input result approximation guarantee close kullback leibler criterion subject considerably refined gp utilize low conditional trade order markov represent vary markov approximation gp two amenable parallelization core thereby scalability evaluation real dataset significantly scalable rank achieve rich provide measure uncertainty gp poor scalability limit scalability gp suit slowly vary correlation require high capture e correlation fidelity localize gps compactly support particularly rapidly however utilize
concave oracle rate target absolute constant agnostic number dc c disagreement bind address agnostic directly priori achieve roughly bind disagreement one provide disagreement lot progress decade amount annotation search hypothesis hypothesis achieve excess minimizing query study noise advantage inconsistent agnostic consistent agnostic disagreement analyze notion coefficient disagreement passive practical give noise label disagreement classifier sphere limitation apparent apply rate relatively development include recent learning address rate version space measure agreement extend zero error generally fact plug recover formal disagreement confidence
north e north south cl cl south north cl north cl north south south message schedule use consensus send consensus send contextual message desirable point message reduce consensus iteration lead accurate remainder unchanged manual highlight latter hierarchy make send variable immediately variable simple layer much less layer layer final regressor capacity regressor make perfect enough heuristic way complexity capable regressor datum important predictor layer wish model latent message income consensus message task parameterize regressor context consensus pair discuss data message run collection message pass consensus message message
execution execution sound costly copy program copy memory program rewrite use branch execution explore possible path path process space handle inter communication must store handle mutual object particularly useful counter barrier prevent proceeding barrier execution trace statement observe run program call constant appear choice exclude manner code sequential trace
cluster appearance gibbs become evident value group demonstrate topic corpus group th group document vocabulary unique dirichlet topic express construction topic j topic model also poisson count factorize n full conditional hand hand lead directly bayesian collapse gibbs form collapse sampler topic dirichlet topic nonparametric remove tune may hdp assignment sampler process globally share dirichlet distinct hdp lda additive combine ji jk
transform laplace zero every fourier remove logarithm measurable minimizer l confidence e f z f l give every xt inequality give density function bound prove help logarithm minimize denote function end error equivalence minimizer estimate depend c leave zero uniformly change variable de de lebesgue theorem find position know proposition
chi freedom rv support mutually consequently j b function stationary u u u follow constant inequality hold p establish generic stationary satisfy triangle chebyshev inequality constant u claim application two additional condition process recall independent constant large constant
measurement prediction possess gmm exact algorithm depend application simulate cross discriminative class unlike cluster due uncertainty instead dependent reject everything sure massive classified even rest naturally object hence never undesirable svm classifier hard uncertainty fuzzy bias uncertainty another discard surface figure boundary fuzzy together scatter random draw randomness formation account effect nature even uncertainty study two physical mass properly use temperature limited drawing
steady mixture outcome unobserve markov concern make idea appear think idea represent sense realization identify decomposition fair coin equip correspond coin notion appear satisfie reference terminology asymptotically identifiable implication identifiability every update agent learn state belief equip weak element necessarily agent outcome consider introduction dirac product suppose agent belief parameter agree belief accordance agent gain insight learn outcome make run observer ergodic typical infinite decision past observation
relative error confidence tight take per asynchronous mdp states action wherein action pair choose relative action asynchronous slow relative clear significantly asynchronous fast ball frame theory k contraction k b define deterministic affine contraction martingale difference matrix matrix mb decay third represent noise converge absence slowly middle desire counterpart tune well surprising iteration initially sure need scheme increase practical using enough new offline online discount
concerned quantile curve associate asymptotic scale deviation empirical process concentration inequality general factor main estimator proof defer brief limit constant hold k difficulty impose methodology necessary context set compact subset term expansion much stochastic center side give standardize approximated dominating term type process field symmetric q hold hand therefore statement xt xt consistent estimate xt xt convergence sup converge sup pn additive error detail definition lemma quantity quantile differ find approximation regression constant estimator section limit hold scale u defer explicit cc theorem xt xx
purpose ease attribute file format essence comma file attribute comment meta datum explore effect mostly set give meta predict class hyperparameter hyperparameter hyperparameter entry seed level meta datum instance provide bp predict meta act bp bp general fold set meta meta fold cross hyperparameter set set meta study use hyperparameter accuracy hyperparameter setting learning denote refer hyperparameter set bp set aid high meta
method nuclear minimization rank soft thresholding singular analogous iterative compressive case vector counterpart robust solve formulation avoid mc mc pg approximate soft mc warm accelerate pg accelerate alm augment mc descent factorization pmf mc mc alternate mc mc mc online match pursuit hard em vb bayes develop decade comprehensive solver low noise provide detailed comparison reader experiment present publicly reader modify code evaluate relative low stopping merely see closely estimate ground algorithm run show convergence average problem low binomial location outlier sample test alm solving vb test noiseless noisy addition test vb author matlab alm inexact adopted provide practically svd singular accelerate svd solver implement coordinate algorithm update update experience simple implementation package author default paper outli entry input simplify help tune guess rank os
point letting svd eq equal eq complete solve svd singular newly rate algorithm compute application tag gene website separate detailed htbp instance cardinality biology complexity parameter validation ml trace solve multi problem norm
effective relaxation numerical comparison subproblem acknowledgement give helpful comment closeness image cone suggestion subproblem discussion linear anonymous carefully read valuable corollary lemma exact liu abstract problem penalty induce propose decomposition minimization handle weighted subproblem develop partial proximal point subproblem limit bfgs bfgs newton finally penalty subproblem l bfgs newton cg type penalty decomposition propose term compute minimization vector product minimization feasible nonempty nonempty set globally wide application sparse noisy difficult addition feasible solution optimal bring solve favorable minimization regularize
resort trajectory completely underlie random measure illustrate mixture estimation key measure definition quite since evy simulate jump suitable sample conditional aspect address paper organize section provide explicit random survival framework gamma process dedicate moment section methodology survival use survival section two sake exposition simplicity full presence censor within multiplicative arise end recall random disjoint mutually random importantly reference existence say henceforth correspond identifie call evy distribute hazard censor
addition decomposition author report factorization function matrix factorization implement author pca implement default value require metric absence matter variable infer scale go decomposition scale limitation absence permutation matrix row correspond row quantifie implement first zero likely vertex unobserve require diagonal corresponding pearson find bipartite unobserved equal number unobserve allow correspond
discard hasting tune trial run conjugacy normal innovation sample truncation bottleneck execute ten mh draw worth point collapse straightforwardly original filter innovation let innovation ratio likelihood simulate system initialize innovation histogram true fall well credible region identify month disease activity subsequent year prediction sample running particle particle prediction disease activity month line kernel simulate trajectory enable particularly useful complex markovian two main sampler correct second movement drastically show empirically mix exploit pass compare conceptually require backward pass markovian effect error weight arise also worth point markovian model storage quantity need discussion storage future ergodicity ergodicity encourage particle find informative interesting work also different amount dimension ease
application defer chapter main low case identical distinguish draw randomness internal describe uniformity testing adaptive support uniformity outline turn principle deterministic carefully pick proof argue deterministic behave regard around sequence family due query us triangle inequality variation adaptive sample apply alone enable uniformly uniform pick uniformly argue distinguish case indistinguishable would indeed differ distance uniformity apart generality let loss write preliminary expectation subset query group intersection choose eq result roughly tail proceed contribution query recall partition say intersect instance sn j j
hide unit output give output eqs lipschitz q rs n ne I define similarly rs rs ne ne lb b combine completes consider hidden layer three dropout complexity three eqs lipschitz dropout lead complexity within drop improve
query categorization train correspond text linear detection task object class top convolution operation slide image window product model match semantic rely match propose product approximation composition code use compact compositional code database efficiently estimate query compositional approximation compositional dictionary compositional show dictionary motivates generalize yield learn source large compositional compact code length experimental sift searching interest similarity search near study research computational geometry computer vision paper neighbor inner study analyze product product
norm use slow applicable motivated propose scalable convex researcher value rank involve integer nu nu size tensor unfold nr j small size ni trace unfold norm formulate svd burden tucker rank usually sensitive liu al norm splitting technique auxiliary g nr parallel admm problem formulate follow sufficient matrix orthogonal solve analogous b problem orthonormal proof material order propose
tune vb accurate mean agree quite fact vb seem reconstruction suit reconstruct sharp density estimate example small map work produce smooth solution noise level smooth indicate tv prior produce preserve large reconstruction become smooth converge reasonably iteration need need large iterative gradient also notice amount like parameter converge infinity noisy converge test mostly good encountered converge either image illustrate might improper computation case gibbs vb prior parameter convergence ensure value avoid study prior make choose crucial
additional benefit alignment naturally source phrase length counter word nan propose long sentence perfectly accurately part sentence sentence admit patient medical centre un est un patient centre correctly translate source medical sentence status health worker sentence de translation preserve meaning sentence un patient un un centre kind series build translation type exp pour la de des les mean sentence word phrase quality translation mistake lack mark translate exp des efforts de cr des conjunction quantitative confirm reliable translation google reference input
follow use sample exponent thing concentrate obtain improve improve bernstein replacement oppose detailed eq follow assume rescale excess first page rescale excess ef fx ef ref ef f gx q consequently notice either proof theorem ef define notice convex jensen assumption eq rescale excess way achieve definition great use q otherwise certain combine probability recall put conclude proof variable repeat obtain sub root let f r rescale
week order accord appearance texture individual time texture name f f f f f stable f f concentrate f f f concentrate fit model effect occur trait htbp dim treat treat observation group consist mainly product mainly consist mainly consist product four mainly product missing find similar accurately consist mainly thin texture overall mainly colour among group average overall consist thin texture product texture observation fall therein response attribute
dataset conjugacy effectively show quickly accord becomes collapsed vb report find superior find occur illustrate level nature pass algorithm ability good merge good solution use run algorithm convergence also result merge trial often procedure hyper vb ten find periodic group capture periodic advance light dark effect model gene rbf hierarchical allow introduce far inspire model methodology incorporate share series series currently explore motion modification variational speed implementation merge collapse collapse wide implementation website http uk publication bayesian innovation enable structure wish intra group variability dp fast collapse
library online attractive reason single process distribute environment online consider small extreme avoid second require iteration produce classifier consume training classifier observe instance learn learn classifier notable pa linear passive correctly classify weight weight current training margin passive consistently outperform numerous algorithm task binary passive propose pass online typically pass training convergent set train size restrictive algorithm instance train
phrase maintain active list contiguous pattern frequent algorithm active index addition assess document considered phrase certain document guarantee pruning phrase early termination search phrase space ni p h algorithm p di c increase slide phrase obtain aggregate iteration fix phrase index counter algorithm candidate index refer implementation closure line indice prune search natural termination require occurrence minimum grow linearly minimum support frequent transaction pattern mining search exponential candidate phrase candidate phrase entire minimum document space prohibitive ensure separate segment phrase invariant allow effectively phrase closure mechanism serve far reduce runtime traditional phrase extraction reflect key keep rank phrase external basis nlp filter phrase phrase phrase implicitly document segmentation return construct phrase phrase intended contain number phrase mining enforce bottom merging decision agglomerative construction significance agglomerative phrase quality phrase upon document employ agglomerative phrase merging construct phrase phrase phrase induce valid implicitly pass chance aggregate obtain frequent phrase
manually annotate interpretation point class ground water bridge road construction concrete table ground rare class design experiment sample sample metric type random replacement straight second sampling try
connect convolutional mnist digits informative one image gain severe alternate next measure belong category computationally forward pass moreover sequential fraction benchmark encourage I work also suggestion gpu project accomplish google brain team google vision systems appearance computational pixel limit resolution image appearance constitute major recognition make address challenge object series system rather moreover potentially change handwritten dataset nature vision optimize
instance descent lipschitz function convergence bad smooth function sdca sag lipschitz dominate substantially smooth loss w respectively support many find good moreover question work improve well dependence condition size establish modulus work online importance average individual bad lipschitz explore reduce number contrast work technique technique use
b combine two claim combine say eq pa ba ba px b mu cx mu cb px px c px px c mu acyclic order eq node rewrite additional argument follow argument one possibly
underlie dynamic drive thank helpful acknowledge fa air force office scientific research project people represent provide snapshot analyzing evolve identify scale pattern interaction fundamentally change occur formalize network framework reliably combine hierarchical point occur resolution evolve social align know frequently framework use large highly interact functional approach non system interested understanding identify social change periodic case result response change network could stress introduce infer
start let eq rank rank notice target form prove proposition arrive line triangle algorithm never fair let incoherent approximated subsampling probability appropriately rescale n phase approximate distribution multiplicative f effect uniformly rather average norm weak coherent bound lemma dominate return substitute consider low completion sampling overcome uniformity focus former contain direction know absence uniformity art uniformity conceptually simple direction discuss adaptive low approximation understanding broadly adaptive unsupervised practically acknowledgement nsf grant award fellowship completeness apart concentration proof similar decomposition provide invertible follow control inequality result notation application bernstein coordinate absolute inequality plugging ensure bernstein proposition denote orthonormal probability q translate
effect discuss health big track rt walk fellowship university student health aid collect thank valuable comment ram tucker e medium track population determining develop novel detection individual develop accurate diagnosis twitter sample disease twitter develop combine text anomaly
vary section weak bounding annotation box define box subset annotation define bound z else union box label category include bound third column bounding box category map coefficient assign corresponding column depend box specific profile bound z z z make coefficient work really try linearly affect bound box label outside bounding box decomposable w r clique bound box generate intersect treat way modify label unclear penalize intersect equal bounding annotation infer category label
ty accelerate provide deal multi categorization problem option class th ti ty py iy conjugate categorical induce th entry set similar proposition class ic partition I ic reward opt challenge calculate efficient dirichlet reliability multi contextual crowd popular crowdsourcing require worker label crowd label worker reliability first incorporate worker reliability reliability true introduce worker reliability set model worker reliability multi utilize two static basic building several devoted allocation crowdsource particular assign instance worker accord although minimax rate new labeling repeat worker incorporate online dual assignment investigate guarantee error gold worker reliability mdp address decision crowd maker instance whether worker fix amount budget budget amount could knowledge characterize allocation mdp crowd labeling level budget crowd fundamentally noisy active difficulty instance crowd labeling model worker reliability require crowd label label learn budget allocation crowd many instance essentially horizon bandit horizon bayesian mab mab cost bayesian ucb policy different mab problem reward note stop optimal stop horizon must conduct study interesting opt soft lead adopt unless h h thus omit visualization first investigate worker
pc admit equivalence many fail generate causal structural modeling form ica optimum joint external enable identification causal identifiability often search contrast recent derive uniquely identifiability e variable bivariate regression advantage principle additive identifiability condition variable e identifiability identifiability unique identifiable order another propose regression causal additive noise introduce hilbert schmidt unique solution convexity issue discover assess cyclic modulus bivariate find mutually independent limited
uncertainty either seek optimize mean two objective solution select single offer genetic optimizer pareto minimize objective typically pareto front pareto another major depend pareto front multi sensitivity direction uncertainty difficulty multi objective strategy receive formula entire system engineering structure example cite use fidelity physics take minimize make reliability various source reliability formulate reliability practical structure main tail lie moment metric fail service behavior express multivariate limit
factor q eq estimator modify sure denote suited remark bandwidth operator experimentally bandwidth excellent estimator operator bandwidth thresholding tune sure modify covariance either compute mean square estimator
classifier regressor least square lexical convenience text technique kernel time support force linear reduce suggest significantly rbf task approximated et provide conservative empirical speed significantly number approximation applicable popular package derive svms applicable process kernel identify vector belong increase rbf kernel prune explicitly low therein
learn comprise filtering comprise step prediction filtering suppose px qx filter qx z pz px pz px dx pz respectively I nx mb operation datum I z algorithm summarize result case filter kalman kalman filter kalman particle restrict observation observation permit belief prior ni z experiment mb filter sect describe ground truth mb difference present filter synthetic nonlinear exist bayes filter incorporation application world vision validate mb concept sect mp ax gx ia analytical rkhs mb top misspecification scale bottom misspecification vs right combination mb I r mb mb mb probabilistic describe well learn mb show norm degree
key determination combine sec posterior update explain magnitude gaussian noise interpretation residual explain minimum explain information expensive represent increase incorporate show also factorize independent parameter zero versa small magnitude force element large enhance essentially observe datum incorporate message distribution sec appendix update compute straightforward need assume sec posterior expectation evaluate explicitly sec detail posterior residual measure square frobenius lower bind expectation entropy sec detail small leading automatically hyperparameter type achieve w guarantee get important initialization result initialize n initialize scheme n diagonal simply sparse tensor e efficiency manually procedure summarize bottom fig message start indicator evaluate
coverage size cp coverage cp unbalanced present compositional proportion team attack serve error team regression transform transform attack serve mean team effect game error vector variable distribution estimate proportion e relation proportion transformation inference covariate probability parameter n use
th sum substitute error derive follow symmetry summation set depend get note replace establishe encode message differ submatrix n x exchangeability px px px n x py result achieve derive exponent argument taylor series decompose use eq note lagrange taylor value derivative evaluate ix preliminary necessary due sparsity ix rewrite note incorporate consequently analysis herein channel code code candidate true set item candidate hold partition equation derivative px px n py py yx simplify second equal add subtract mutual independence necessity suppose element salient index error variable
image instance tend segment follow distribution paper propose law cut whose law fix achieve goal treat partition objective objective solve locally objective performing receive considerable attention computer vision science graph cut cut ratio one utilize clustering cutting accord theoretic objective cut problem cut circuit show approach despite spectral algorithm suffer
art building noisy house still recover manual outperform tolerance another drive effect large possess strong power collection conclusion encourage news match object match relaxation provably error free achieve remarkable efficiency greedy round strategy ability even turn perfect long partial severe occur situation ability nearly broad finding combinatorial integer programming perfectly semidefinite relaxation appendix algorithm simplicity matrix operator matrix lagrangian program operator convex negative whereas represent penalty optimize primal dual close number iterative update procedure q operator resp project reasonable return even dominant portion behave small component randomize procedure generate ni absolute constant since rearrange presentation diagonal perturbation include block convenient decompose comprise check minor come row column index eq quantify eigenvalue eigenvalue column universal constant positive eigenvalue
reduce computable effective class dimension vc denote subset computable vc converse obvious index effective note computable far concept computable statement condition dimension comment definition completeness finish vc complete index effective concept vc suffice property sequence function take set serve
backpropagation order visual word could path image top bottom image randomly implication effectively overfitte well visual outperform piece wise gradient input compute parameter efficiently parameter gradient descent ip target I I part image car appear lot exploit successfully seminal histogram substantial follow whereby visual appear specifically represent decompose r implication binary leaf visual computation grow deal modality annotation annotation note annotation annotation image annotation mixed bag annotation framework treat annotation word annotation word joint annotation relationship spatially annotation annotation give possible annotation path leaf annotation decrease order predict achieve model previous work capability base binary derive propose autoregressive
domain basically find ks fail reject significance nan solution verify random reject cdf b random line empirical cdf grey around discuss clarity empirical latter address next employ inspection introduce inspection unit comprise day inspection carry give inspection day inter arrival inspection reject associate plan classify infeasible plan
ise strength ise time achievable strongly model exhibit use strictly challenging e warm hard least assign vector goal give access h es every sufficient edge set actually note ml estimate add function independent set thereby likelihood follow hard many correctly complexity least observe
hash table store location hash asymmetric use preprocesse create phase query report hash table hashing sensitive ideally operate diverse rank neighbor ratio query hash effect good run hash report per recall recall independently fraction gold neighbor number retrieve query scan recall dataset scheme hash varie level see much compare hash scheme asymmetric outperform hash irrespective superiority indexing product sign perform traditional news worse plain mnist look mnist fact mnist plain close course negligible effect penalization ep news variation hence poorly variation always product difference plain asymmetric proposal require implementation adopt hash widely popular indexing practice originally
covered process assign asymmetric cost tune relationship attention discuss eqn fisher eqn like two conduct view mirror set negative difference criterion aspect epoch score hamming distance fisher criterion epoch cost view reflect matter contrary test drop gradually converge epoch gap obviously big binomial epoch distribution generate hamming rank performance figure ideal negative wide coincide fisher nearly different suitable affect solve leave configuration fuse connection function repeat rate list c c distinguish modify training speed improvement table outperform include
pz z set variable pz z recall pz conditionally
number topic next done learn moment complete full bernoulli gaussian entry span combination sparsity problem relax programming row explain output l sophisticated find vector subspace deterministic version require quasi problem setting assumption weight learn function ax score bernoulli deep activation layer assumption ax
h hz quantum exception infinite temperature success initial temperature often field state typically employ reduce fidelity coherent gibbs state fidelity least protocol use rotation apply success failure branch successfully fidelity state trick e h gradient state order estimate via optimize utilize quantum amplitude search repetition need distribution field efficiently true give state uniform ideally result final field state performance quantum sample boltzmann machine connect edge number operation visible use gradient see draw boltzmann expectation likelihood state quantum success average algorithm reduce repetition success call require compute configuration rotation visible rotation error energy thus claim contrast optimization scale boltzmann machine constant follow asymptotic advantage practice two objective train model optimize finite advantage quantum mean perform assume require arithmetic computer development quantum synthesis could use remove store consider access training require require bias regularization learning array contain bias bias em em k train train j estimate one access quantum oracle either efficient boltzmann memory quantum unitary procedure ability superposition seem powerful resource sophisticated need wish quantum utilize provide circumstance compute visible hide boltzmann scale scale q
general profile location scale multi overall point approach belong method explore information availability multi tend reveal accurate information global single matrix capture geometric eigenvector infimum eigenvalue scale neighborhood al support svm rule distinguish two domain surface image later classification scale necessary section feature homology review persistent homology good reference homology group make usually topological equip integer summarize appearance homology take closed persistence diagram appearance merging interval turn persistence diagram make properly
report superior max threshold multi base stop min classification change consecutive learn min satisfied criterion report stop operate stop min far applicable stop learn annotation label begin require counter al example prediction performance versa annotation sp set prediction occur stop type example encounter factor check quickly make stop perform stop stop select informative preliminary made performance
opposite opinion choose outlier whole foreground attention lead attention notice background lead dominant explore dominant agree removal point another less way select subset clean validation result highly competitive alternative rank exploit reliable sample national year match match regular pair comparison comparison head advantage home capture intercept team home support inspection regularization path reveal outlier outlier select lasso outlier team rank conference rank conference regard return see could lasso importantly large unbiased score big c team team c c name net intercept match least match winner tie eight outlier return
root localize anomalous could anomalous due rare attribute typical situation anomalous corrupted provide pattern consideration corruption anomalous contrary corruption child however corruption attribute localize right child pattern b realize corruption accept reject continue attribute child fig recursively alarm reject e anomaly search tree stop parent branch corruption corruption action correspond node continue correspond finally anomalous opt accept leaf favor well cost alarm illustration progress fig corrupt attribute locate attribute region partitioning improve complicated nonparametric situation accordance regard localization plausible density mmse averaging generate solution computer denoise instance smoothed mmse imputation cause fail satisfactory detection reason novel map imputation technique generate feasible likely approximate map size estimate original attribute training replace attribute statistically treat corrupted attribute attribute note sufficiently model accordance estimate condition localization tree attribute certainly detect cf introduce novel map corrupt attribute base rank implementation output corruption localization phase therefore computationally imputation phase require computation attribute initialize
value state ill many modern domain utilize extent often available covariance computable setting albeit practical pairwise environmental science field often correlation distant spatial modify bivariate covariance readily demonstrate formulation organize follow brief inverse methodology follow consider extend generalization numerical life section inverse portfolio portfolio provide appendix primal proximal belong subsection briefly coordinate block coordinate box qp al notice qp solve block coordinate descent correspond primal dual theoretical proximal newton author use nesterov algorithm converge provide enough optimal proximal global moreover operate outperform glasso alternate composite variable z z iteration
current modify recent work label read introduce classifier support correlation problem seek graph dependency limit markovian relate theoretically sp algorithm capable problem mod example sp mod assume correct whereas algorithm output state mod relation approximation learn relational relation mod correct traditional definition take account label correct dependency output part training contain
mean semantic semantic website associate query website associate query also compare sensible associate website e com dnn semantic website shot improve margin like hypothesis zero shoot compare svms shot supervise task identify semantic label label datum achieve label well low costly class h svm bag word dnn dnn embedding embedding semantic
possible state optimize contribution account map average significantly svms sample framework smoother finally unify special insight practitioner follow introduce unified algorithm model naturally extend numerous general perform hold extension structure svm application area human action recognition sentiment rely joint perform well framework discriminative graphical hide incorporate framework entropy minimize
suggestion start relate text removal provide classical backpropagation actor back respect e verification support human assessment name classification control make human decrease backpropagation implement feedforward train gradient neuron continue train discard learn classical overfitte show develop reinforcement line represent fed send mean modification represented modify adjust epoch represent delay module delay architecture test several collect past text extract
feature selection method call mr method select relevance mr base simple applicable use relevance input focus manner large nn lars lar criterion lar similarity output measure couple lar large feature get select regularization analysis implement practitioner benchmark propose compare exist lar dataset select feature focus select redundant manner scenario lar nystr om theoretically justify feature review feature n
k iterative understand yield mean high gaussian provide synthetic datum capacity know mean penalty short k unsupervised technique include cluster cluster spectral rapid automatic acquisition encounter problem huge attribute weight grouping portion responsible example responsible activity synthesis activate relevant make inefficient feature eliminate automatically importance feature approach dimension reduction principle nonnegative principal cluster detect log fitting dimension framework optimize particularly statistic zero weight feature work still keep word many final example seminal cluster relevant cluster relaxation improve put penalty cluster intractable overcome
successively tight bound high dimensional pseudo code accord dataset iteration update marginal calculate mini size stop iterate typically sec build construct label empirically quantify small total among stop explain criterion explanation similarly contribution factor quantify iy iy add factor beyond certain write ratio use distribution ratio construct implement accord domain factor binary specify special create recover correlate outperform task imagine independent bernoulli half define
negative kl distance kl represent concept encode imply soft gradient yx dot product draw two gaussian gaussian percent get choose principled bound chebyshev landscape regularization differently grow hard constraint keep reasonably sized hard lie constant diagonal involve keep hypercube energy score small dominate rest worth qualitative quantitative task asymmetric linguistic word diagonal note learn corpora token match aside leave publicly token appear less drop
stationary point discussion shall clearly consequently theorem applicable generate immediate nonempty closed algebraic algebraic algebraic practice design guarantee sense describe similar discussion initialize exceeds iterate huge finitely many boundedness clustering get stationary point guarantee solve feasibility next consider close algebraic certain exhibit lemma homogeneous proceed suppose letting eq second shows contradict conclusion constraint nonempty algebraic suppose generate addition limit cone assumption also proceed without z z conclusion z z consider precede follow
cause binary constrain baseline constraint extremely compare run objective net dropout layer layer momentum momentum choose size hamiltonian carlo mcmc take gradient rapid hmc careful tuning basic hmc include introduce parameter experiment optimize hmc spend measure correspond minimize sample absolute across worst large chain across variable package threshold base hmc integration also additional constraint datum repository normalize unit initialize deviation input compute
solver outperform year integer create program cut iteratively new optimum cutting solve simultaneously score build formulation unconstraine maximizer polynomial require cut technique clean call optimizer understanding approach prefer relaxation estimate stop computation solution cope even obtain solution minimum cubic large probably little situation observe report require large amount much deal domain devise approximate computable sample appeal whose take relatively empirically double tree variable effective optimal set conclude note background learn direct acyclic graph dag node categorical value
obtain manner site square gaussian burn previous convergence result shannon diversity illustrate independent horizontal axis represent shannon represent model highlight covariate red credible interval seem restrictive goodness sort remark help regard impose mentioned distribution stick important smoothing operate dependent clearly visible dependent smooth fit regardless make dependence diversity estimate proposition ht posterior diversity dot shannon dependent weight name construct thank stick breaking bring flexibility define define kernel whose allow dependence flexible counterpart advantage ready marginal scheme conduct mean compare sample arbitrary
ensemble neighborhood moreover optimize intrinsic induce reconstruction since geometry tangent principal obtain singular scale look reasonably expect correct scale approximate differential implement manifold sample manifold gaussian result pair heat approximately equal test order magnitude logarithmic point evaluate pointwise svd size cccc standard vs noise svd indicate interval could optimize distortion dimension metric matching interesting thing typically synthetic explanation near tangent select
score direct translation penalty phrase length section single translation reverse short phrase importance translation translation model incorporate short denominator score penalty model help see observe sentence length word similarly propose existence mean sentence avoid effectively deal htp early turn american year old increase health segmentation early modern turn american old department reference
query suitable query retrieval distinct multiple image pareto front manifold individually database dissimilarity query sift vision image dissimilarity computationally intensive dissimilarity sample query manifold need efficiently underlie traditional manifold image retrieval query next rank produce create pareto point dissimilarity sample pareto point compute pareto one pareto depth front dominate remain sample return pareto stanford scene dataset key front importance retrieval illustrative image pareto front forest accord within tail locate front query image necessarily image front contain desirable retrieve pareto pareto characterize convexity pareto database establish pareto theory connection front depth admit maximal
search ad user age gender search display query query user title appear title keyword keyword ad unique assign assign click number ad number click label six indicator age year old age third imply unknown three component unknown gender gender make next component display ad make use component display position likewise word appear element encode create bag bag word bag bag th bag indicate component bag component may nonzero much bag encode keyword encode title ad component encode keyword search title product keyword distinct component encode observe important cost regressor nonzero estimate regression contain describe section classifier want take ad ad observe write read click contain give mle model mle find minimizer likelihood loss see follow datum determine query user feature ad ad display say offline former use stochastic computing infeasible recall performance element online update whenever vector adapt preference process metric numerical select sample select consecutive
extreme statistical law identically distribute frequently extreme extreme write mean minimum extreme coincide useful distribution reliability survival log independent systematic component eq q unknown continuously possibly nonlinear finally monotonic respectively matrix respect analogously value substitute maximum extreme result adapt minimum index extreme represent scalar nuisance parameter similarly form drop column additionally symbol indicate sign likelihood
table use network good behave realistic imagenet example validation imagenet reference experiment validation convolution mlp relu use prevent overfitte parameter parameter fully connect take reduce imagenet ad ad mlp convolutional weight transformation mnist extract last pooling layer common convolutional imagenet layer connect investigate train mlp reference activation setup incorporate conclude imagenet activation nonetheless able comparable mlp adaptation
high array outer array th comprise vector outer product arguably tensor direct discussion tensor decomposition consider analysis social time straightforward survey decomposition reader refer tt tm compact decomposition term slice recall apparent represent constitute cf likewise write diag n diag n slice first dimension stands build matrix feasibility fundamentally assume couple entry well effective freedom rough idea obtain ensure imputation completion tm otherwise incomplete compactly see also generalize completion tucker argue frobenius outline rank completion provably encourage rank tensor decomposition note tune appropriately stream real setting incomplete slice acquire sequentially depict right leverage subspace devise tensor factor minimizer cf normalization coincides adopt upon obtain accordingly minimizer cost computationally nonconvex bilinear could instead instant namely update ii procedure necessarily recursively impossible aforementioned challenge big requirement sgd alternative
begin constraint z setting optimization eq subgradient subgradient compute r method three iteration optimization introduce refine initialize tc z constraint grow link constraint hence fall unsupervised membership de via coordinate mahalanobis many traditional fast near neighbor hashing require explicit address near advantage greatly reduce need near forest leaf seek begin point identify parent yield full hierarchy
considerable inherent biological camera dependence sort view assess assess different size place randomness variability region mm voxel mm averaged error strictly simulation deconvolution sp curve worse show completeness technique detailed deconvolution order negativity known perform show reduce estimate spectral spectral analysis pre process methodology consider interest study method structure assume useful something assess size parameter coincide compare mse indicate spectral five even though perform well rest surprising favor spatial reconstruct parametric competitive assume however involve level ccccc sp perform brain gamma pdf incorporate identical pdf region make real
super know present term ccc std model train half recover model figure see good due orthogonal ica point fail quality orthogonal one always two far mix order large mix anchor likely activate activate hide unit reach
allow median condition space follow step proof strong theorem exist aggregated subset imply follow lemma determine subset choose satisfie particular recall assume reasonable possess assume ols select square eq q notice select letting correct probability choose c k essentially result subset update
consequence eq minimizer existence determinant establish suffice expand eigenvector objective unbounde unbounded boundedness unbounded contradict unbounded unbounded frobenius eigenvalue great accord partitioning permutation principal submatrix irreducible simply permutation irreducible correspond conclude unbounded reasoning unbounded scalar e diagonal kk pattern entry finish remain note occur eigenvalue satisfy hold bregman q virtue bregman divergence duality opposite note condition hence parameter ar satisfied decrease order optimality immediate consider inside recall monotonically induction hypothesis verify check kkt optimality cf define feasibility remainder diagonal consequently concern converse stationarity remain follow note small eigenvalue matrix hard concern block particular block minimizer limit sequence existence contain establish reasoning proof general maximizer sample certain
able recognize place task instance recognition categorization place recognition component detection scale qualitative type place recognition work discriminate place configuration place analogy model analogy integration method formalism gram extensively aim structured modelling conclude base approach
generated implement software generate exhibit scale define result model time sample normalize diagonal test sequence weight sequence random test result roc reconstruction present superior dataset encourage particularly weight scheme use give inferior reconstruction force reconstruction original graph roc gene association leave submodular middle reweighted covariance gene experimental gene activation
partition file reduce introduce node indexing column trajectory name name really recover correspond name recover object efficient entity system indexing static create name string key key indexing accordance singleton pattern e name node name array index argument select name specify argument name two indexing index synchronization depict simplified node interface interface define abstract class implement related implement requirement quantitative intensity discrete class implement implement continuous node continuous e example create string string string add add state add add node state parent new double define interface classifier implement class implement time specialized process rely example indexing string nan state string state new state add add add false new definition double double double model classifier separate format format start bayesian node follow separated node define bn start list parent format format contrary yet exception specify class compose sign file distribution bn parent sign bn file format currently nevertheless format support
expand show kernel svm away two message quadratic enable discrimination object prior noise preserve able suggest image order principle accurate control learn predict performance attribute edge encode figure emphasis object boundary around svms visual preserve heart finding visual assumption combine perform well figure
equal factor reliably bind se find wide close possible improvement moreover c spirit problem latter agent repeatedly select arm observe reward ask high reward elimination find single gap well fix particular elimination algorithm section arm algorithm arm
number score calculate return quantity composite may resample outline far variable band light scatter correlation carlo resample perturbation
annotation pixel semantic label presence insight convnet pixel learn learn pixel model image cast bag pixel need annotation rarely segmentation
ask fix reach max eq factorize problem eq eq reach restrict bp write believe bp bethe multipliers eq q lagrangian enforce lagrange multipliers lagrangian read lagrangian derivative impose use way q bp try actually reach
k tensor nice f element assume j k j j kx assume predictor index belong order replace distribution follow basis active predictor spline stand prior suppose distribution harmonic rate isotropic logarithmic variate harmonic prominent obtain strictly obtain naive application smoothness condition co co interestingly automatically level ambient affect exponentially integer smoothness integer smoothness restrict case empirical n define hellinger distance space rate predictor density satisfy respect say predictor
speed apply tuning compress dense connect quantization convolutional layer facebook ai com cnn recognition repeatedly demonstrate result image detection year cnn many layer million storage extremely deep resource hardware embed device tackle investigate cnn particular term demand layer clear lead good balance recognition accuracy category task imagenet compression loss classification progress standard object almost cnn
x bx complete ii eq bx g bx lie follow clear equation g x p iv equation I p p therefore root therefore hence respect bx bx proof complete distribution eq I unique n h unique since h
say formalize entry constrain say constrain picture iff q explicitly conversely fewer free arbitrarily infinitely q nevertheless infinitely subspace free last dimensional essentially explain lead direction directly behind every contain equation contain else behind know determine clear entry disjoint set row edge look like bold correspondence title eq q label j vertex line title title title jj title title jj label label vertex vertex label j width title jj title jj j title title font line edge determine conclude entry respectively vertex width pt vertex title font pt edge edge come direct must despite highlight generalization start loose really lie span subspace course could ignore way generality trivial discard easier determine determine subspace satisfy subspace subspace fit obvious essential single lie use generalize behave dimensional subspace
h argument popular intend highlight admissible place l produce cluster embed consist point perturb simplex lose however generalize symbol work spectral connect q size connect isolated vertex let partition mi jx symmetric instance proposition yield v v orthonormal hence give point ray mutually orthogonal directional ti isolate accordance lemma main function demonstrate continuous accord local maxima g imply maxima outside identify simplex strictly give contain maxima suffice weight apply notice piece u ij bring equation maxima note mapping without system let sphere suffice maxima open strict convexity
sampling e region posterior sometimes indeed draw method via exhaustive interested detailed discussion bayesian hereafter prior pd information simple chain monte step start step increment new calculate accept repeat termination control begin burn discard prevent h
within belong eight global group choose least ols ol commonly model comparative base stable solid versus curve axis positive false positive show estimate base five repository alarm child adjacency supplement additive stable assign coefficient dataset sg weight perform correspond skew however relate infer since avoid sign estimate q convenience alarm comparative infer direct figure true negative percentage learn blue performance ols method result line difference varied away true ol decrease remain reliable infer positive similar figure show comparative
combinatorial signature log correspondence root mathematical dynamically generate move contingency statistic r f two cell thus move connect table add move contingency call walk start table contingency exist existence generators algebraic linear equip walk result irreducible move metropolis use adjust return exactly dynamically log linear restrict appear parametrization mention specifically useful decomposable divide apply statistic complex log table encode hypergraph vertex one normalize convenience log denote parameter set edge hypergraph instead usual list represent understand parameter vertex hypergraph collect map contingency hypergraph familiar two discrete see hypergraph hypergraph complete quasi independence probability hypergraph complete vertex partition remove hypergraph hypergraph complex
shift sa sa sa sa cycle sa sa sa sa sa sa shift sa sa sa sa sa cycle fill align quantization xx shift scale xshift inner sep size shift fill drop minimum size shift cm shift fill drop size pt thick draw east fill draw west sep font text width leave self check fill drop minimum thick fill green sep east green west inner font draw circle drop fill draw xshift east draw west sep align self check circuit shift auto node cm font b bend left node loop yshift draw loop draw bend node yshift align group inversion align stream solid cycle cm edge bend leave node node edge bend none none none align sequence align draw drop sep pt height grid gray axis anti stream xshift xshift yshift thick background layer yshift xshift sa sa sa sa sa sa shift auto font bend yshift yshift yshift bend leave yshift pt distance scale bend edge loop edge bend node distance scale edge loop none axis axis align center sep minimum rectangle fill north align thick scale scale style dash axis center draw height grid major dash thick axis align sequence thick align flat center flat white black xshift yshift background yshift xshift sa sa sa sa sa cycle symbolic causal assume existence probabilistic algorithms b sum purely circuit allow table stream conclude length circuit carry rgb rgb rgb rgb rgb rgb rgb rgb thick green drop operation algorithmic stream generate independent pt stream symbol move position go operation stream thick fill drop size sample path copy read symbol position pt thick generating source read symbol write move go fill rectangle drop deviation symbolic stream symbolic section hence evaluate choose correctness see symbolic derivative rigorous actual implementation compute deviation since stream anti quantify dissimilarity similarity stream classify efficiently stream near stream ultimately similaritie structure quantify mutual dependence compute generative mutual datum identify stream stochastic stream similarity clearly stream find reveal without knowledge reveal detail stream confidence minimal length reliable scalability learning task depend notion identify instant predict step series strictly superior certainly well claim outperform par yet set applicability system assumption notion dissimilarity individual measurement set possibility metric universal least nature process sequential observation discrete symbol quantization range symbol represent slice range slice quantization alphabet consist represent fine large stream thus symbolic alphabet symbol alphabet quantization fine expense increase scheme
incorrect incorrect nearly appear many purpose suit world application letter solve efficiency series stochastically embed operation fluctuation algorithm adjust often dynamic volume physics analog variety
manually initialize weight bias add weight mlp predict good decoding reach development finally include pre maxout dropout decode error development point error analyze importance decode narrow width reach decode width slightly replace put used decoder acoustic language important really learn align sequence rescale predefined rescaling failed converge low stage training address stage procedure affect rescaling
cardinality example cp generic mean structure provide output compatibility iii retrieve violate qp fundamental please output boolean word definition concept incur formulae f indicator boolean evaluate compatibility incur world contribute additive formulae carry contribution formulae total maximize opposite
simple vector make lsh lsh satisfying feature call neighbor search arbitrary extend lsh projection appropriate item database publication computer vision community relate build suffer drawback approximate construct become projection even use vector conceptually simple section new performance runtime accuracy crucially reveal boost particular improvement examine nystr om discuss subtle nystr om aforementioned demonstrate advantage nystr lsh example lsh lsh projection improvement draw two largely lsh schwarz fail
posterior first investigate update covariance matrix optimality lead likelihood class leibl distance propose prove optimality approximation offline manner costly approximation particularly require demonstrate theoretical ray heat example exploit indistinguishable inverse inference approximation approximation risk optimality approach treat endowed distribution encode model interest incorporate forward distribution bring structure encode kind among inversion parameter action operator feature may identify approximation bayes lead substantial approximation inverse prior define approximation characteristic storage requirement fast form negative covariance matrix class structure also arise kalman challenge update within suitable optimality symmetric metric broad relative update class along lead generalize eigenvector hessian assume posterior extend kullback leibler divergence hellinger low yield optimal optimality especially linear inverse regularize estimate matrix appropriately action hessian precision regularization effort concern exactly easily square precision million context bayes risk square exploit optimal become minimize bayes squared analytical square weighted direction low approximate posterior fall posterior
error construct give sparse factor first construction define statement prove reverse induction pi di refinement degree root I back affect multiplicative define claim linear factor find factor pn contain entry factor identity square multivariate normalize multiply side clear range
h p p p edge word select probable meaning word average close compositional deep compositional sentence phrase furthermore strategy word second measure sentence vector compositional reflect sentence towards purpose
version cox modify derivation new estimator adopt measure bias equivalently application conclude scenario tool system operate several returning capture characterize illumination require analyze image complex wishart successful equip parameter covariance look former hermitian mean statistically relate severe various exist approach method choice mainly estimator usually bias indeed bias size
high binary fast automatic discovery perhaps datum meet cumulative present handle variate variate boltzmann inference procedure people across competitive art collaborative filter public rbms extend diverse pattern recognition university ordinal feedback preference datum boltzmann machines rbms variate variate ordinal able opinion profile competitive state technique public extend recommendation review boltzmann attract due deep bipartite undirected enable sampling
detector still fourth plan use scope improvement category begin target domain assess discover multi belong classifier kn ny classification use basic writing svm projective variant flow require target domain include margin learn discriminative use target domain feature analogous c c c svm source domain combine overlap computer video terminology adaptation internet google come domain amazon image resolution resolution digital camera one pose illumination experimental follow source number per category vary category category source example role select per target domain testing without split source
ideally tailor simplify come information identical reverse causal need future causal predict future minimize expected distortion equivalent causal distortion minimize distortion forward minimize expected state measure prop square error distortion measure emphasize reverse causal another even infer maximally improve nearly distortion though treat reverse leverage retain future causal reverse statistic use statement appendix directly thm seem clearly good knowledge prop prop intractable tractable causal reverse practical information function finite countable simple allow shape though yield illustration substantial display elsewhere distortion entropy causal state single state provide distortion limit monitoring calculate feature ref self one solve maximize iterate iterate eqs explicit find maxima thus choose initial force analyze suboptimal solution sophisticated ref carefully state use equivalence sec distribution analytically produce entirely deterministic expand codebook necessary start codebook decrease allow usually maxima zero start increase key difference compressed condition calculate maximally describe phase curve describe function feature time describe symbolic dynamic retain length calculate sec process
formula low identity factorization compute combine recursive strategy interpolation radial basis dynamic mechanic hierarchical covariance solver process multivariate tensor efficient dense factorization large important several field others monte carlo dimensional variable obtain covariance normal symmetric symmetric definite factor computational deal conventional symmetric base cholesky rank represent symmetric factorization diagonal e decompose major versus cholesky long block
formulation disagreement loss extent discuss metric define full laplacian matrix well frame rotation nontrivial robust visual surveillance summary end disagreement loss view version correlation cca certain kind transformation rotation translation scale different visual desirable
gradient descent equation compute w implicitly proceed update ann accord old weight without parameter pair set pair u ij increment ij l u b via increment b update ann via ij l ij repeat step method extract inter variance fit graph embedding class within school extract projection
h c u u also convergence tn arrival consistent update may estimate time parameter vice term broad applicability algorithm conjugate compress linear response accord proceed ba center normalize diag draw associate yy f xy xx proceed successively conditional jt yy xy high autocorrelation face mix joint parameter space partition proceed observe yy tm inversion b close form expression update jt jt c jt ts j modify surrogate c l jt k draw df dramatically reduce requirement see shrinkage bl infeasible mcmc iteration suffer severe particle dimensional statistic particle store propagate impractical approximation n ga density j kl df numerous arrive sequentially predictor jk
task access device capable algorithmic task capacity recurrent neural additional keep time give constant challenge result evaluate error method different analysis time refer normalize square square normalized root variant cover attempt normalize distance output deviation produce percentage value throughout plot surface create accord equation point blue solid line depend optimize offline validation
correction second order third order write wiener weight wiener pass vanishe choice posterior minimize theorem integrate argument uniquely kernel arise part match major community basic method define wiener th choose meet conceptual third solver return main aspect wiener integration simultaneously add improper wiener wiener linear exist infinitely often coverage ode posterior context wiener prior
associate statistic record divergence arise substantial abc mean allow substantial speedup p p preliminary kl kl approximate yes abc mf mf comparison posterior approach preliminary trajectory trial residual proportion mf protocol inference statistic inference kl divergence kullback divergence use approximate abc protocol demonstrate infer experimentally study number copy quantification well mean copy infer ref behaviour induction per produce geometrically paper avoid cycle rna cell
demand take site compute life necessity reliable mobile device explore speed cnn approach speed cnns redundancy cnns consist convolutional bring performance boost exploit redundancy within keep accuracy approach redundancy conventional filter convolution channel vertical accuracy know cnns express visual interpretation similar rather redundancy unnecessary feedforward backpropagation cnns sparsity accelerate computation align neuron iterate successfully filter locate position due filter
converge functional instead prove dense easier deduce every exist converge deduce say functional hold increase natural bound previous necessary restrictive notion useful property minimizer approximate minimizer minimizer guarantee energy energy statement convergence say convergence functional identically compact minimizer functional index namely functional say every functional convergence functional interested functional probability functional f n dependence understand section consider remark namely riemannian structure metric remark precise space borel allow measure product space contain graph pd remark introduce use identification space note contain product opposite couple g equation hand great conclude space lebesgue imply sequence convergent converge true impossible convergent completion pd pd dirac delta dense pd dirac
analyse size approximate complexity affect rich space accurate hand move core issue appropriate count simultaneous low bound bind process ratio difficulty require boundedness generating function indexing satisfy truncate ratio prove meet requirement study devoted topic main rate tail bind multinomial make metric kolmogorov deal upper approximate metric finite approximate lemma devote hellinger distance partition partition j translation calculate metric unit nu nu dependent f accord construction euclidean nu nu order
learn variate setup require sample accurately extend obtain derivative use purpose fisher derivative tensor representation refer thus decompose svd case term present analyze hand unlike matrix tensor large regime less overcomplete lead rich obtain option perform discriminative prediction
limited therefore fluctuation state propose mining discover machine anomaly density anomaly many develop mining deviation detection density extend previously distinguish trace average state quantum include circuit mine key tool broad quantum broad necessary diagonal
log population two side n u generalize nuisance iii hypothesis
match perfect significant neuron understand relationship connectivity processing either graph chance conclusion consider iii inference information figure match extract statistically neuron connectivity classification neuron il neuron neuron vertex number four category employ four category proper match measure match second seed category seed amongst category match match vertex correct category indeed seed greatly match plan heuristic optimize seed summarize mc replicate graph contribution least
electrical engineering university respectively associate electrical computer currently electrical engineering company foundation frank distinguished fellowship award research interest high adaptive communication imaging edu acknowledge support nsf large low step adaptive sense theoretical show achievable summary original rank plus logarithmic experimentally robust collaborative task vision automate surveillance investigate sense pca paper address suppose admit decomposition nonzero ultimately interested identify location nonzero column particular focus may identify investigation identify array arise anomalous pattern component task increasingly leverage dimensionality idea line utilize reference therein compressed burden inference operate task surveillance application ideally region numerous give map aim map perform image linear wherein interpret overlap patch image equivalently matrix version patch previous effort natural subspace approach estimation salient region model common subspace efficacy visual recently rapid security surveillance imaging task entire successful salient
virtue reveal analyze apply orient laplacian even subgraph induce remove index remove connected component let eq lemma describe smoothed piecewise assign multiplication interpretation term electrical perspective graph freedom number estimate k xx number support reporting degree freedom graph filtering reveal mathematical estimate aa interpret electrical perspective graph describe go induced accumulation current network q say form number node solve current repeat current new induce current iterate even say node assign current potentially overall structure estimate informative tell rd order induce piecewise vector nonzero induce current piecewise extension propose filtering trend perform weighted incidence forward recursion could loss order datum say explore imputation investigate potentially penalty extension wherein graph convex principle moderately sized reliably variety order handle problem describe procedure take
start first moment uniqueness equation eq gaussians excess recover imply coefficient nonzero equation solution lemma gaussian add construct mixture moment gaussian differ contradiction repeatedly add take explanation bin setup alpha beta beta alpha gamma alpha alpha beta gamma alpha x alpha beta gamma beta alpha alpha alpha beta alpha x x x alpha factor alpha alpha goal alpha alpha alpha beta factor gamma eq b z alpha q gamma gamma c b divide unbounded exist per lead lead sufficiently polynomial normalize compact root lemma homogeneity polynomial coefficient magnitude regardless conclusion bound lemma I perturbation loss normalize combine root desire since max therefore max max c cr max suppose good approximation therefore conditional roughly would max x follow cc branch whenever take whenever remainder take condition branch setting clause
correlation carry hierarchical recommend linkage within correlation cut dendrogram cluster linkage minimum correlation cluster highly assign
interact economic connect indicator extremely mean number indicator indicator indicator theory prove purpose monitoring forecast proportional identifying vary order average averaged indicator equal pearson product indicator comprehensive dot country position node centrality economic edge colour code pointing regressor colour quantify relationship centrality connection high capability might strong connection span magnitude call financial tracking inherently couple fit possible account track average maximally error along aggregated regressor forecast exclude operator column besides forecasting indicator turn exclude forecast capability especially international trade flow lag
rating without b particular technique greedy achieve netflix dataset superiority previously work shot variant first variant active sequentially rating second variant one news recommendation arrive short require identify user tradeoff select complete present multi explore user review item offline setting work item start essentially mainly whereas tree solution start currently implement however optimal user potentially lead empirical theorem thm remark conjecture
weight finding problem algorithm stage initialize item add weight graph cycle edge edge may network span delay link initially unknown variant maximize modular address problem formalize th entry weight item stochastic item negative associate arm assume unknown generality stress equation episode episode goal basis minimize episode optimistic maximum weight statistic et et
robot trial reach number trial reach run map million evaluation map contain behavior extend behavior arrange shape heart shape curve place robot black draw position degree freedom angle theoretically achievable performance performance adaptation trial goal cm test save reach often specifically iteration second explain median except trial optimization robot continue reach scenario level never classic scenario challenge trial case substantially case median accuracy pre post behavior post able cope able behavioral descriptor choose simplicity sophisticated likely cope experiment show behavioral descriptor able position traditional extended fig value art policy search three map find behavior physical robot prediction note perform search prior alternative variant variant search evaluate search search randomly select test well one keep variant evaluate prediction performance map performance trial bayesian process randomly let choose evaluate bayesian classic obvious policy behavioral search directly dimensional dimensional dimensional variant map ahead time search directly algorithm automatic experimental variant gaussian initialize constant select experimental dimension automatic variant break modified simulator experiment section simulator involve remove create variant map simulation case replicate eight map six replicate variant time lead replicate per experiment roughly real robot simulate perturb multiplicative noise analyze fast speed variant number case trial trial trial experimental trial robot outperform demonstrate perform variant variant variant work policy search map contain result search evaluation nearly variant randomly policy design trial prediction introduce trial process learn trial low six
give sometimes give supervised dataset drug target network one network know disadvantage predict give recall ls ls ls ls laboratory molecular biology uk network largely partially framework train separate node train node systematically theoretically formalize problem pair bipartite discuss global approach extend later unseen drawing carry biological network highlight relationship biological entity gene protein micro disease network experiment practice entity researcher take interaction experimental prediction formulate inference consist pair adapt consider input feature second train predict direct neighbor exploit
algorithm inspire cluster cut different accord gap statistic candidate comparison follow refer introduce present simulation compare discuss complexity microarray conclusion general resemble size candidate spc rank consideration candidate determine combination gap statistic conduct high across global remainder detail cluster inspire cluster propose split hierarchical recursively stop gap certain produce label input dendrogram result hierarchical bootstrap gap default reference clusters remark examine dendrogram dendrogram assign root gap current
compact also return range learn affect generalization tool continuous hadamard matrix word could expect per concentration need continuity refer continuous constant normal invariant around mean showing lipschitz function cosine construct combine choice play condition inequality lipschitz union order magnitude exhibit significantly memory simplicity focus penalize able compute benchmark cifar achieve art expansion even begin investigate purpose uniformly see tb fidelity imply unless assess carry experiment uci e least instance investigate approximated whenever completeness method exact rbf kx albeit practically desirable due matrix recent achievable theoretically advantage function exponentially retain
take advantage consumption day day separately big markov hour day begin day remainder consist column vector time capacity take level take amount unit grid update dynamically interval poisson number average time estimate monitoring
reliability contribute bayesian integral predictive numerically expense ignore probable value expectation yet also several principal parsimonious suggest information high live subspace dimensionality less find practice combine involve possible integral analytically without simulated curse conclude model base discriminant implement discriminant applicable class
arguably inferior avoid fail put configuration sampling limit degenerate zero avoid respect mixing specifically marginal two quite little iteration method approximate randomly generate rapid mixing approximate compare chain calculate marginal exact tree project divergence avoid algorithm minimize lead exact useful belief propagation
drastically varied way ground example assumption deal metric dataset reconstruct query break neighbor one digit figure able recover density axis agreement axis globally query top behavior unweighted know short path near neighbor region ht life production fall place finally metric
cross parameter compute predict classification heart covariate set covariate observation hold ten see result h logit breast heart subspace something table state ccc post class dim ccc dim post present infer subspace sphere utility sphere angle avoid represent different dimension embed allow sampling well dimension great efficiency model acknowledgement would useful acknowledge grant biology gm grant dms mathematics institute nsf grant institute environmental health national health proposition corollary
towards goal issue address understand generalization appear functional room thus h notice come size rhs eliminate qr proof aspect question answer recommender estimating automatically training often reduce issue design effective rank functional correlate metric evaluation technique loss wise loss weight derivation piecewise loss theory markov field answer web retrieval answering recommender typically keyword return rank relevant potential recent survey approach automatically train relevance context profile object estimate query return object reflect query mathematically set goal sort list
semi nmf ill pose optimal nonnegative semi nonnegative rank initialization nonnegative factorization q frobenius nonnegative use focus nmf fact column input matrix decomposition column interpret cluster centroid reconstruct approximately column indicator discussion nmf motion super hyperspectral nonnegative also nmf compute semi rank usual nmf rank unconstraine r unconstraine necessarily singular abuse language rank sm procedure nmf decomposition svd exact nmf sm vector column sm nmf irreducible generalize belong class well often semi section semi already light nmf much computing nmf nmf ill pose exist
compatible ignore site copula raise influence typically site ignoring author impact alternative establish asymptotically joint instance comparison censor historical copula case study context g highly maxima development peak still active spatial modeling would allow joint advance give cause expect future development peak estimation semi parametric generalize pareto margin augmentation enable inclusion censor datum apply france historical objective historical quantile site model version ignore historical local strongly version illustrate extend historical implement estimation ignore historical quantile result likely complete
pathway conditional type model differently type protein dna bind co attribute data domain main obtain aim measure physical protein bind reduce function pathway count edge false positive aim measure factor dna bind peak seq bind event bind event mixture pathway bind whenever distribute nan hypothesis relate beta change gene variable existence direct gene gene mixture normal standard deviation expect
market index conclude research appendix paper analytical separate univariate structure correspondence multivariate copula construct flexible f normal possible term copula consist part multivariate dependence principle margin margin split distribution flexible asymmetric normal special split degree freedom skewness margin link covariate covariate margin margin split model link mixture split therefore split also margin multivariate involve quantify correlation variable usually methodology dependence extreme express bivariate principle dependency attain copula specify whole copula strong relatively fr hoeffding bound help leave
similarity fraction overlap chance feature calculate index health patient total model heart failure top list table discrimination respect auc hyperparameter auc hyperparameter control select
step variable lp double denote derive necessary sample w ij w j ratio competitive permutation ratio cr table average ratio report take matlab linear matlab toolbox ghz cpu gb ram problem bid match achieve ratio datum table suggest order achieve despite competitive r cr cr
identifiability model uniform mutual undirected graphical modern probability underlie capture aspect task partition computation graphical case markov variable ise long history start spin domain finance computer processing sampling significant relatively easy hence largely liu paper give markov node weight tree thus find liu mutual graph loop much reason neighbor indirect correlation distant discuss next subsection structure ise node search possible neighborhood whether imply conditional hold show greedy
study merge property propose dissimilarity agglomerative modeling aggregate agglomerative heuristic first one one new cluster denote merge cluster model less explanation accord factor infinity weight leibl bivariate brevity mainly rewrite sum kullback good choose merge dissimilarity infinity dissimilarity leibl difference property leibler impact unbalanced merging different code plus principle agglomerative cluster merge successively cluster dendrogram dissimilarity measure dendrogram euclidean criterion dissimilarity measure dendrogram trade merge similarly merge agglomerative dissimilarity partition sense less distinguished
kernel change frequently case large sign away mapping magnitude depend consume prediction produce rbf network network compact reduce still quantization approach idea constrain growth quantization quantization codebook size codebook vector close input quantization adopt codebook size allocate similar code add center size update convergence nonlinear regression give convergence successive converge noise zero sequence independent ai assumption two steady
rectangle rectangle right green rectangle dot red color width join central concept nash nash play equilibrium precise equilibrium perfect agent strategy ne discount solve formulation cause problem significant global via conference detail formal proof presentation sub ensure bellman particular agent game equilibrium henceforth refer sg sp game sp direction carefully choose descent point sg sp propose aforementioned ensure ne follow centralized base scheme assume known localize one running require action observe policy strategy irreducible rl good converge self play discount aforementioned desirable property follow economic learn play convergence nash meet unlike discount possesse guarantee style node cm policy actor bend east bend leave bend west employ actor conceptually nest operate fix value update minimum mention simultaneously albeit vary loop formal considerable ode previous paper usually show track ode reach solve hand adopt show
risk road road reason shorter narrow road scenario policy go expert make previous narrow road eventually fall share provide policy convenient deterministic fact coordinate frank collect explore distinction policy execution action interestingly go expert approach theoretical justification previously conceptually make policy expert go collect current detail alternate available addition general policy spirit dynamic programming proceed provide learn time step potentially time justify state ideally policy expert distribution policy determine exploration proceed follow current
reaction prescribe law proportional product reaction eq reaction reaction rate reaction bind quasi constant simulate use numerical integration step formal analog perceptron input similar early perceptron capable feed network model produce external internal analog yes analog control weight consist vs four implement weight rest species weight target input
generalize limitation accord theoretical spectral frobenius yield bound thank david nsf grant u advanced project social communication program agreement number view conclusion interpret represent great follow term great term choose third probability term less great formula third less probability eq permutation
information ground segmentation use combine mt trend accuracy stage stage division essential dataset image method mean roughly bound box compare method baseline obvious combine consuming performance well without trick feature study ignore svm vs training image explore vs one besides class one class image compare little unable supervision class imbalance improve increase capable stanford specie use
prune eight wise column wise eight wise call calculate full version operation ccc ccc ccc c shift add shift add shift dct definition dct l cc nz nz chen dct image employ set
negative approximated therefore performance necessary base population goal notion distance clustering well know notion population clustering recall method produce space deal clustering survey alternative apart proposal partition interest lie develop notion population clustering clustering moment introduce distance measure hausdorff set compact try proximity set set refer content identify although return hausdorff approach primarily practical seem cluster desirable assign portion correspond close clustering sensible distance cluster counterpart component minimization linear problem comprehensive assignment resemble adapt possibility add describe equally possibility
feature contain four attribute wavelet transform skewness wavelet transform apply method point cluster examine confusion purely apply uci machine repository white detailed ph score score blind range bad dataset rule selection pick sc plot figure notice fourth appear sc visualize involve rating scoring mode confusion cluster interpret like large center cluster good inside cluster score seed uci repository seed author mode mean type soft ray technique image image first normalize bandwidth pick confusion parameter seed
cluster ellipsoid hyperplane solve format particle search well programming particle constraint maximize publish simplex solve use simplex hard could ellipsoid algorithm simplex prove iteratively ellipsoid barrier interior one area way label store near learn optimal formulate technique solve svms bag neighbor model network approach nature minimize
effort level know maximize agent contract observable agent contract long minimum work ensure effort principal effort observable principal pt theorem complete microsoft nsf finding conclusion recommendation alone crowdsource market platform matching complete particular set quality dynamically adjust whereby effort observable pricing armed bandit represent contract cope propose new algorithm contract space treat single arm discretization adaptively region contract eventually discretize sublinear improve discretization compete several crowdsource market dynamic pricing subject descriptor analysis computation abstract behavioral crowdsource agent pricing armed bandit regret crowdsource intelligence complete computer alone crowdsource market microsoft universal relevance pay worker course human worker human english require less effort produce dedicated make sure complete properly encourage payment worker valuable view worker contract specify quality examine dynamically quality task evolve payment specifie payment output worker market whether task effort level observable observe worker worker contract contract worker formally expect make worker choose effort effort complete incur bring quality worker observe unknown worker decision treat dynamic contract mab problem arm represent contract structure algorithm region choose region treat promising area action promise general discretization appear discretization observable structure analyze propose width horizon round bandit illustrate via corollary special low theoretical worker choose accept task special dynamic pricing prior practically crowdsource market novel derive corollary specific
person voxel world resolution camera compose calibrate video moving employ voxel frame rate second voxel figure frame voxel voxel set quite sequence motion link nevertheless qualitative evidence able sequence truth could compare seed pass ensure perform time k time analyze eigenvector desire embed neighborhood resolution voxel finally topological assess assess segmentation produce analyze ground truth voxel link model voxel g figure work firstly body compose cut link belong unsupervised contain secondly body counting segment unsupervise label retain voxel obtain score take cluster segment score finally obtain segmentation store link voxel link histogram proportion cluster segment body p run consistently six comprise turn type boundary unsupervise lie correspondence score priori score sequence link rather point embed bend correspond consistent clustering dramatically bad segmentation segmentation appear relatively favor arm
summary say theoretic reduce sufficiently theoretic noise reduction formulate regularize solve computation illustrate propose future extension support aid scientific aid scientific research
low sake find aim work finite observe receive reward variable good suboptimal arm sample recommendation generic identify arm eliminate batch elimination depict integer elimination maintain remain arm within use functional sample define later arm functional proceed generalise sequential arm eliminate round k sequential eliminate remain round round arm
reject retain see result vary vary lr statistic confirm obtain parsimonious gain note report likelihood good model information concern note practitioner randomly close lr assessment adjust eigen orientation ratio version remain without reject motivated consideration extend test multivariate family eight gaussian extreme configuration estimate requirement beyond seven test likelihood ratio overlap
fig unseen semantic neighbor semantic class unseen connect class exploit unseen bipartite semantic bipartite weight store parameter shoot connect bipartite direct class unseen short graph image probability unseen fig classify unseen end unseen class start terminate class unseen class include bridge connect unseen unseen computed shot find label high linear neighbor unseen make unseen
c c c task visual system sequentially promise biological sequential step far propose internal image terminate return presence measure compatibility evidence region step set image region action three decide terminate four maximum termination learn sigmoid search select target image offer bottom effect evaluate region define evidence use normalize bounding box corner policy express execute repeat action detector bag maximal strong supervision find model maximize confidence
width width pt rgb line round line join pt color pt pattern color pattern node shift point note think intersect study translate relate distance whenever close ball center ap angle pair face along ap ap ap bound later point bind angle face statement hold replace corollary angle relate certain convenience though implicit characterization subspace idea connect angle eigenvalue orthonormal number
diversity gain pair decrease pair proportional rare analogously label cost label transition count coverage
single point different concave concave identical induce set tuple sort magnitude within break know boundary active subject either convex subgradient least know construction differ I
also large matrix center sparsity store plus low format ideal rank package present demand empirically consider estimate rearrange proportional deviation get eq say likewise iterate four residual zero fast slightly matrix practice subset omit center skip iterative guarantee center center likewise row center unbalanced way gauss certain degenerate guarantee formula simultaneously learn problem solution alternative solve row costly solve operate strong advantage time early make imputation alternate iteration hence acknowledgement thank author discussion lead center dms national science foundation begin concern regression eq follow observe bind clearly
knn strong often multi modal result use framework adjacency graph many contrast edge local neighborhood superior robust graph sl ne neighbors consistent requirement knn learn constraint four knn sl ne sl comparison list discriminate grain subtle grain category vs vs effort optimize sl embed high accuracy knn metric organize propose similarity sl ne discriminative similarity sl ne learn use relevant covariance matrix tie mixture localize fisher affinity note neighborhood dynamically update metric sl neighborhood
augmentation nearly experiment library reproduce acknowledgment upon national nsf research acknowledge code thank helpful discussion suggestion institute advanced technology il edu recognition number recent advance augmentation dropout improve unsupervised advance incorporate recent unsupervised analyze unsupervised augmentation cifar unsupervised supervised finding discover pre unsupervised surprisingly ratio unsupervised color training deep linear unit augmentation convolutional cnns
x x algorithm read strict lagrangian augment generate respect separately long augment lagrangian alternate lead direction back r ax ax r ax prox r ax r overview parameter consider admm e convergence averaged find program depend variety course program addition first admm modification inexact first handle parameter parameter vary quadratic augment lagrangian replace receive r ax ax r respectively dependency flexible monotone admm also consist find consider triple understand subdifferential maximal monotone vi problem solve new parameter admm vi ax ax equation sub problem study reference overview find want see rule subdifferential calculus find minimizer nice minimizer prox point rise prox r prox recent minimization prox prox g algorithm dual relation algorithm first hand equivalent rule add apply step prox iterate give definition general solution linear equation avoid modify step taylor method alternate relation straightforward equation x using rewrite prox several modification basic linearize admm multipli primal hybrid primal
approximation diagram region provide characteristic affect correlation standardized fluctuation feature divide two feature irrelevant feature note coefficient relevant ratio standardize always strongly regime order give general feature selection statement satisfied gap relevant correlated note indicate regime accurate sign much strong estimate absolute approximation square rmse probability function vertical variable red increase pearson coefficient health measuring special pool http edu percentage body regression body index upper arm purpose blue red pearson percentage body mass index red figure demonstrate
slow dl present supervise net direct early supervision constraint formulation performance exist justification stochastic gradient function loose pointing promise particularly worth wise minimize classification reduce error individual backpropagation unsupervise semi carry deep classifier instead standard softmax function framework softmax classifiers supervision softmax cnn softmax mnist cifar cifar technique maxout careful main style introduce classifier model layer early combine dl
object track use tracking collect structure represent location associate descriptor frame typically template calculate distance parameterize correspondence tracking frame template adjacent frame construct task adjacent frame learn collected frames w column concatenation part process template learn verification structure compatibility scoring method compatibility specific eq transform template spatial norm
write q censor augmentation manner censor ensure form nature conditional standard see make predictive want r rt us hierarchical generalise becomes hierarchical sample censor r location want use sample use subsection briefly describe ahead temporal ahead forecast two observe want temporal predictive calculate forecasting already forecast predictive daily fit discard mcmc chain converge quickly brevity mcmc table autoregressive autocorrelation successive autocorrelation reflect fact skewed simplicity spatial decay ci vary km spatial dependency
iid distribute n v generate loading noise performance z repetition loading fit range fit fit success fit ordinary correlation correspond effectiveness regression ridge regression surprising ridge look though bad focus essentially give naturally suit canonical cca center cca maximally correlate mathematically cca constrain close corresponding role cca tool find common
duration give effective mass barrier produce effective axis auto dynamic purpose delta intend dynamic sensitive convenient follow stationarity energy analytically solve constant write ref dynamic transform ref place motion reversible explicit force problem hamiltonian hmc table effective size force duration mc mc effective center table indicate propose hamiltonian method sensitive effective hamiltonian effective sample force evaluation c employ hz vx I
completion possible agree produce completion span clear nonzero discard residual zero validate em drop incoherence validate disjoint column state correctness completion whether agree ideal incoherence
generality classifier reduce rather volume reduce per classifier cross computational problem enable execute brief present brief overview classifier motivate test discuss classify reduce
hide observe use variational illustrated collapse parameter maximum support annotation lie bag positive bag negative necessarily level absolutely bag false practice instance bag svm constrain hyperplane maximize among bag use principle max sim softmax softmax sim yield max score super partial annotation different transform bag bag label logistic stick link instance maximization generate consider independently relationship solution pl instance address aforementione framework instance membership single instance deal miss amount generative outperform develop discriminative annotation finally like maintain bag bag dataset bag bag instance include exclude annotation bag bi bb b c class simplify notation annotation map graphical illustrate notation bag instance bag bp next label th feature vector weight th practice indicator take argument discriminative
attribute dataset attribute input root square error convergence part cross validation rmse ten example improve confirm theoretical exponent significantly quadratic growth rkh lead compare proceed grid vary rmse select ten fold grid likewise validation logarithmic spaced rmse std important validation take capable achieve good performance decrease practice range ccc rmse std
partial remainder roc analysis roc false various threshold curve unit variable two diagnostic population convention patient great therefore roc give give survival function auc summary index define conjunction carlo valid
distinguish input iy label generation term normally variance inference proceed impose specify typical non squared rule simplify process integrate term free function function variance cdf likelihood couple function adopt equivalence probit accept latent role slope cdf sample describe next section paradigm unseen instance exploit additional need
kernel fact less various depend assumption non partition hence method jump boundary effect cast multivariate represent expect case express information still nevertheless investigation generalization investigate statistical inference respect estimator consider profile profile square fan investigate whether limit hypothesis alternative x fan profile regularity nan q test chi square fan demonstrate testing nonparametric constant price introduce nuisance parameter statistic bootstrap j j j bb root component nonparametric component categorical efficient series example
focused task alone without example relationship norm bayesian inference via nuclear popular nonconvex minima nuclear norm application domain hull rank far generality finite dimensional seek optimize ex ex nuclear index unconstraine feature take approach illustrative mn mn kl mean constrain direct storage inversion become infeasible prior computationally feasible approach optimization cost independent gradient gradient compute use collect term convex unique identity function replace norm regularizer equivalent ex ex ex valid variate posterior representation theorem parametric function ex ex strongly rely kolmogorov extend use extend consistent covariance sample requirement kolmogorov extension training set covariance trivially representation optimization addition index
couple beta cause peak mass near yield tail express gamma gamma hierarchical layer tb member represent mention know half cauchy consider normal layer beta near zero cauchy compare distribution tail inference calculation intractable inference utilize expectation unknown subsection
heavy kl member associated member regret whether entropy simplex motivate look closely log sum conjugate property similar sum series go tool able express bregman notation analysis semi continuous suitably relative make bregman divergence subtract
propose consist straight tuple systematic position architecture involve tuple also investigate tuple move circle place piece piece k player perfect piece white black form player put piece color consist place piece diagonal piece piece piece pass game pass piece position reason become popular domain intelligence goal player evaluation player simple current player apply position desirable determine move play tie position piece counter
outperform classic tn tn step square svm paper acceleration proposition sequential direction abstract overview write style research introduce unconstrained subspace optimization usefulness signal compressive imaging machine explore combination separable method obtain art plain trust region easily invertible truncate nested improved merge accelerate lagrangian constrain science dimension use interior applicable problem exceed several become possess bad fista
utilize density functional probability induce shape maximizer excess probability induce geometric information shape gr class comment reconstruct probability avoid literature set assume priori disadvantage mode ms value analyze top vertical clear unimodal ms ms
label q ray stand usual significance range give complement interval proper stand observation cf first centre see divide residual well e presence high iid probability iid finite statement sequence iid finite infinitely many strong number condition remain combine fact bound contrary infinitely e imply
amount compare rnn actor actor task difficulty actor actor question object already statement question oppose rnn table conclusion similar outperform rnn support time c difficulty actor actor actor time conduct answer sentence answer rnn module rnn fed rnn effectively feed output module see difficulty object module feed module generation perform choose correct correct question answer incorrect answer incorrect believe correct indicate strongly rnns variant rnns c rnn lstm com reason component long jointly write context answer long effectively act dynamic toy task world latter reasoning support sentence question understand machine way read write large long memory combine modern
effective number test correction define eq controlling hence effective particular significance level label although iteration overcome drawback consider subgraph since ignore subgraph control permutation eliminate subgraph throughout recommended occurrence investigate gene study database subgraph large distribution study membership drawback threshold difficult graph knowledge subgraph use feature correspond informative subgraph subsequent view false positive detect positive domain correction prominent correction correction exact test correction expensive
name collection average sequence converge linear th convexity turn
seek output implicitly primarily interested approximate employ evidence dimensionality present common wide nonlinear vb density model parameter iterative identification information theoretic determine dimension demonstrate problem mechanic relevance call assess motivate mechanic describe equation calibration readily forward utilize engine response quantity measurement include material behavior coordinate configuration static case gradient lagrangian tensor primary consist body force second stress tensor material value point stress find aforementione condition encounter material far solve analytically vast majority resort numerical field prominent employ well interest reader book available term solution algebraic equation r r mesh shape discretization value frequently assume finite number discretization discretization aim infer variability discretize fine might
queue child advance resample particle child child simulate particle leave queue fully I queue particle create child release queue seek queue full instead particle particle single virtual keep track propagate particle multipli average account preserve reach complete particle reporting initialize particle initial constraint operating optimization demonstrate overall validity particle hmm associated emission second linear gaussian
brain graph hence graph degree normalize improve section colored brain region additional node plot brain spectral rsc spatial ba tuning value ba yield spatially cluster densely spatial spectral densely coherent regularize spectral two covariate cluster less node coherence increase uniformity size demonstrate spectral cluster brain regularize govern region brain brain brain graph treat brain covariate discovery highly brain break visible importantly approach align region connectivity ba rsc ba adjust ari quantify partition cluster brain partition sc covariate spatial spectral spatially brain different configuration ba ari brain similarity spectral partition alignment brain conduct
bilinear extract five store recurrent allow several appear record break neural demonstrate effectiveness store layer major two secondly store image rnn model word utilize capacity achieve recurrent strategy lead good evaluation tb start sign model layer word recurrent multimodal softmax deep cnn frame view recurrent type layer layer activation recurrent denote current representation representation vocabulary calculate vector element learn adaptive connect accordingly backpropagation propagate recurrent time multimodal recurrent rnn two recurrent multimodal softmax word embed one encode syntactic relevant word euclidean
phrase exist perhaps label review name movie sort infer sentiment word sort sentiment embed compare belong five subsequently choose highest predict output negative class ignore prediction test high negative use strategy achieve instance achieve precision neutral approach boundary sentence fall sentence multi learning obtain transfer learning require supervision obtain approach
propose probabilistic valid correctly therefore first low purpose examine bind obtain posterior vb leave one solution sampler leave manner inference normalize constant namely serve pseudo pseudo likelihood figure evident quantity converge iteration naive low increase proceed decrease unfortunately update one cost extra computational would technique inference monitor solution low technique emphasize technique applicable nd equally burn gradually decrease completion burn variational th variational burn ratio fall predefine final anneal concern evident take iteration variational eq automatically difference posterior bayesian clear whether true stationary average stationary want stress remain unknown true solution point convergence knowledge naive solution user enable speed one drawback vb dynamically cardinality computational sampler maintain heavy computational cost intrinsic simple ignore inference cluster accordance stick cluster avoid evaluate truly set intrinsic complexity proportional experiment implemented eliminate cost vb full variable eqs need expectation apply object remarkable relational almost beneficial
type tuning tuning selection challenge also domain interpretability assume involve arise biology super network may role fundamentally degree connectivity scale network fact exist free accommodate dense intend entirely almost order encourage effect enyi formulation accommodate idea estimate model graphical work graphical convergence degree whether graphical lasso network perspective publicly author acknowledge nsf dms fellowship nsf mf mf sl appendix recall scale augment lagrangian follow derive update separable update update depend address respect lagrange multipli bit algebra c permutation row
one achieve learn classifier favor body denoise autoencoder know stack denoise autoencoder able across allow domain explicitly learn identify origin neural include hide towards connection predict membership objective domain adversarial confirm experiment toy performance neural network performance representation improves rely robust label provide marginal classifier tackle adaptation approach method intuitively risk focus
description stable spline system property highlight alternative close factorization highlight computational associated computation stable corollary university united ac uk impulse process spline impulse surely stable entropy property stable spline independent
inverse statement true universal barrier describe time distribution define via technique simple computable barrier universal barrier make optimal briefly whose lebesgue family inverse mapping onto strictly elementary recover exponential family differential entropy barrier moment x easy exercise partly function satisfy function
word lda early vector use category clarity index subscript style comparative classifier grind end respectively consider date time different distance distance high distance subject matter similarity showed figure figure lie dimensional dissimilarity denote geodesic along short distance short ht similar ht mention similarity define need encode work define influence influence strong similarity mathematically define asymmetric distance link influence tendency measure alternatively think hausdorff measure link hausdorff unlike point point view context close capture take favor vary spectrum asymmetric hausdorff evaluate
overcomplete tensor optima characterize true efficient guarantee establish moment suffer spurious optima long overcomplete spurious optima especially grow compare dimensionality recently overcomplete random incoherent tensor component orthogonality correlation true model overcomplete hide time dimension improve analysis iteration initialization require mild overcomplete spherical naive bayes multiple view conditionally hide assume component convergence mixture mixture model spherical dimension vector sample correlation overcomplete model direction almost noise establish level gaussian low use spectral clustering impose tensor orthogonality separation additional work model satisfy noise provide cluster spherical mixture mean vector tensor turn tensor
solution example bound strictly semi online cost stream expect viewpoint reasonable rough dependence semi order operate set number guarantee learn powerful par neural example intuitively assign surprising therefore powerful importance recognize retrieval investigate argue require online yahoo user
degree freedom dynamically tolerance tolerance low tolerance negligible change step curve iteration current find value matlab find solve degree freedom I use contrast discrepancy mdp lc average regularization parameter final one ms indicate c mdp lc result mdp lc indicate entry lc invert contaminate regularization mdp respectively right either consistently poor except either respect appear noise
analyze rewrite proceed algebraic relate j
adjacency regular regular map perfectly regular never always distribution world eigenvector informative alternate time update link include hyper induce ignore number obtain tell permutation affect generate associate row vector look mathematical describe particular find motivate section encodes graph informative contain small perfectly eigenvector always fluctuation connection example equal compute fact eigenvector property polynomial expression recover polynomial equation form theorem value although may hard characterize many satisfy reasonably think sharp concentration adjacency random power eigenvalue obey capture different polynomial expect different link spectrum far know graph triangle
one number item default worker maintain simulation task worker b behave differently increase worker worker accurate boost know worker reliability htb variant specifically task bar iteration step figure accuracy fast take step run similar confirm change omit htb strictly worker simplicity compare worker violate toy suppose worker true error achieve rate misspecification misspecification extent publicly sample label independently vary see error clarity figure result usually among class worker web search first identify worker label potential e expert htb ccc visualization done conduct label independently vary proportion vote em label generally dominate run label maintain compare em bar impose label achieve algorithm misspecification worker set complementary worker example misspecification ccc percentage assignment increase task error impose bar language collect worker task e question worker binary second sentence label pair similar different assignment
study linear bundle formally utility generalization decrease separable utility call decrease concave piece denote segment specify rate agent derive good define applicable utility proportion utility next economic capture range substitution utility assume let h far behave utility derive zero regardless complementary utility set precision define rational size formal utility reveal preference introduce label measure multi say distribution informally amount low error statistical vector input bp output pair learner consider learn say reveal preference price utility bp reveal preference correspond demand sense actually notion query learner determine utility instance access query price vector bundle reveal
hilbert sensor sensor infer sensor formulate generalize design trace expect value character seek solve measure pde solve independent parameter facilitate construct posteriori randomize trace include characterize describe posterior vector configuration penalty newton adjoint pde function elaborate determine sensor configuration coefficient pde log medium flow pde require gradient essentially candidate location quasi newton exhibit invariance design trace q bayesian nonlinear inverse govern sensor experimental collect field minimize sense subproblem experimental challenge particular discretization map require underlie problem turn forward problem algorithm maximally exploit tractable large scale state datum include development concern inverse problem increase govern numerical nonlinear pose large frequentist objective finite inference amount condition address mathematical incorporate operator infer objective entail optimality e second order I scalable cost forward pde solve effort area include author sequential quadratic different criterion govern nonlinear system equation govern paper
assumption discount establish govern almost surely project bellman equation bellman projection detail transition component diagonal matrix diagonal policy eq aim asymptotic bound quantify convergence td well td asymptotic challenging reason fix bellman underlie mdp influence come mdp difficulty number mixing occur td start stationary rate amount underlie approximation order
find persistent diagram one persistent mobile sense environment distinguish assume motion agent subsection weak localization probabilistic movement characteristic group behavior sake behavior individual boundary specifically boundarie orientation characterize line change average velocity line segment length angular isotropic switch velocity follow leave central partition model second tangent boundary rw movement agent period time movement stop model memory
also normality expansion specifically formula taylor residual
immediately measure frobenius logarithmic rate bit minimize max matrix completion consider likelihood family instance logit belong bind achieve frobenius extent determine minimax integer infimum let observation bind logit probit theorem lower square low proportional multiplied sample
exploit word machine baseline correlation embedding correlate cross bag gram like possibility language amount language allow build level corpus explore use autoencoder language word representation align language rely simply bag align language quality since autoencoder word certain correlation maximize regularizer improvement empirically classification english approach state art percentage good english annotate processing language annotate sentiment already english language situation language dominate digital content elsewhere however
term fw sum lipschitz derivative explicit bad case pass oracle may access copy subset I
representative inherent challenging goal control capability recognize significantly hard difficulty complex linear noisy wrong expensive controller reinforcement orientation keep close episode location stationary absence wind episode end pre specify whose cyclic cyclic collective collective adjust pilot around axis design use action detail discretization process apply adjusted action discretization dimension equally sized interval spread process decrease agent sample transition batch transition representative representative state interval transition episode length decrease since step execute interval decrease episode clear significantly cut episode approximately argue nothing slow part consequence performance episode computational peak model begin considerably come warm start policy update section compatible computational task version address literature algorithm guide datum evaluating problem definition start discuss theoretical guarantee assume uniformly restrictive collection direct interaction environment strictly indeed relaxed assumption reward require policy collect choose section good discount case assumption regard kernel unfortunately usually empirically speak problem perturbation perturbation balance task problem like task representative result computational crucially ideally state would row hull contain state representative state problem state representative experiment cluster state center representative show section possible simple perhaps seem reasonably together average resort quantization approach update learn fit state prior know reasoning task policy vary space regardless representative add representative use representative resource strong task varied course good problem theory practice decrease number transition admissible proposition
ty ty conclude alternate prove rate convergence hessian descent order ie ie w w subproblem step lasso problem solve package coordinate p w ie ie k theorem proposition corollary stanford university department department stanford graphical penalize though scalable advance optimization address pseudo base propose minimization
vb robustness model drop expect overfitte increase cause free em variational overfitte integrate prediction performance vb tensor miss vb stay approach link attract date restrict probabilistic order deal variational bayesian matrix method derive variational form relate motivate variable conjugate et naturally tensor tensor well learn maintain distribution put
rather forward g field vector modality early situation also detail one want try image box annotation occur recognition challenge incorporate strong annotation form goal per image object categorization similar supervision distinguish available first tracking use time might capture auxiliary additional information image available interest different kind appear literature modality learn compute share alternatively image fill improve object categorization show training image author term gaussian noise approach specific applicable instead applicable framework individual framework formalize classification multiclass versus rest training vector vector encode computer vision image histogram
source representative subsequently process subspace filter process slice sample dynamical observation transform spherical across proxy uncertainty information available datum resolution segment associate use divergence bregman stress uncertainty
mean index represent label label vision derive mrf labeling energy correspond mrf decade technique computer vision cut belief propagation convexity method performance minimize extensively study study conclude visual labeling statistic solution benchmark pair suggest complicate sophisticated high high clique potential handle difficult visual labeling function exist optimize mrfs arbitrary pairwise potential boolean approximate extend submodular type enable automatic transform provide graph cut property scope
reduce multiclass newly category rank assignment rank category person express like music create tie assignment tie complete tie occur possible extremely order important partition choose possible model tie intractable resort pairwise comparison separately denote replace comparison ic il I translate original preference drawback relaxation guarantee hope enough preserve specify adapt comparison il occurrence lead estimate visible dimensionality extraction
segmentation reach image g crf crf crf traditionally field smooth segmentation typically contain couple node spatially qualitatively clean spurious classifier hand feature classifier produce score semantic prediction qualitatively illustrate homogeneous result detailed conjunction range thin require discrete connect crf employ label assignment unary label pairwise potential otherwise pairwise matter fully feature pixel adopt kernel pixel color intensity hyper scale crucially pass
numerical original cover originally miss complete observation htb uci uci heart uci yes yes yes uci uci uci yes uci uci burn yes uci uci yes uci misclassification scatter plot none besides see majority accuracy dominate perform fall time acceptable accuracy dataset good accuracy hard scatter glm penalty reflect score p p bias applicable variable omit comparison give may dataset table lp se se se se glm model se se se se glm table bar dominate find apply plot se bar expect although model model majority rule simple ht omit bias correction correction visual se provide prediction categorical bias improve se se work general se se overall correction algorithm necessarily section negligible bias htb se se se se se se se se se section verify great interpretation ability illustration employ analyze section census uci repository extract census database original current survey characteristic census split training machine learn leave
handle depend label label third assume depend previous labeling process crf toolbox brief description label noun noun head noun noun phrase task assign sentence test parse noun phrase phrase etc sentence phrase etc choose segmentation meaningful text sentence etc word segmentation chinese assign denote begin inside word test choose sentence consider word sentence input component alphabet component template crf package property summarize
map patch patch input image rgb characterize feedforward number subsample ultimately cnn convolutional kn patch shape map filter parameter k p l define activation z operation convolution non pixel pooling pooling recursively hold b sufficient purpose approximate approximation provide use replace finally grouping multiplication weight pixel principle anti limit result give product z replace substitution z z approximation z k regard
understand although stress redundant merely mean change correlation even growth static distribution constrain type type scenario v virtual likelihood approximation systematic analysis virtual less static forest datum constrain fix calibrate growth considerably substantially constrain calibration derive datum expert calibration supplement table virtual supplement mostly correlation correlation particularly supplement understand hierarchical divide growth diameter high grouping structure strong interpretation compete difference parameterization evident value high growth specie point difference mid think systematic estimation reliable manual fit fixing
recover consist visually entity dc air dc air chemical family balanced test class confusion importance partitioning discover control top multiply improve base ground significant
calculation expect another propose signature variance n var signature small variance similar index denote base distribution signature boundary use sort absolute signature signature follow
outlier effect equally include inducing show boundary potential induce boundary boundary cause true induce boundary misclassifie precisely differently effect backpropagation mlp g great effect classification large particularly stage mlp calculate propose method mlp simple gradually increase deep localize assume contain instance induce instance datum set instance determine instance exhibit examine misclassifie misclassifie b misclassifie weighting instance motivate misclassifie
usefulness use error differentially express patient spectrum available selection want modern throughput often detect control number approach unstable automate bootstrap penalize elastic selection widely use network genome association study stability combination stability component specify descent introduction introduce section common evaluation boost section examine patient boost conjunction aim phenotype measurement try pathway generalize outcome response predictor covariate fit aim fit example square linear obtain analogously
vc exist l rhs entropy logarithm number metric relate probability least thus equation eq entropy upper bind rademacher originally n lx lx thus upper bind solve suppose approximation give hold exist finite n whose na bernstein arithmetic mean least focus event auxiliary n n q xx maximizer action x second third x x likewise write get optimizer apply finally n n n n event least lx satisfied space add lemma l r c minimize property bernstein n e arithmetic l separate term l inequality upper imply convenience bernstein number random pn range functional fr ec minimizer however error prove l least estimating problem exponentially supremum policy evaluation extend similarity proof considerably constant x bernstein appendix show arithmetic focus inequality auxiliary
eigenfunction langevin case dramatically eigenfunction operator langevin indicate weighted outperform langevin index eigenfunction less align cs observe th produce rw slowly sampler plot chain provide proposal shorter langevin rw langevin rw sampler direction smooth eigenfunction langevin langevin snr significantly outperform langevin high eigenfunction langevin rw langevin langevin proposals comparable parameter surprising move weak weighted proposal well langevin lag autocorrelation b star symbol langevin symbol langevin lag langevin stay constant prior eigenfunction lag langevin eigenfunction low average langevin lag eigenfunction operator proposal brevity h useful quantify impact contrast weight proposal posterior
document represent high lda first network capture document however relation representation representation well visualize similarity measure topic represent capture similarity topic word kullback comparison triangle define base complement version kl divergence employ hellinger hellinger q topic network node represent discover topic similarity topic edge even collection place matrix format currently employ format store nevertheless parallelization implementation computation break parallelization core structure case cell column row word appear key key comprise tuple map key input key value topic appear operation complete expression hellinger every represent simply sum pair topic hellinger multiply subtract network
open bias fix train compare wikipedia token build occur corpus word technique
outside control kkt center combination multiplier boundary solely support allow hence ball triangle centroid radius f density cluster account method ball ball leave ball cover since cluster address inspire always mode shift centroid well formulate lagrange multiplier kkt lr recalling reformulate j programming towards name difference right model worth play role would enter actually reduce cluster discussion modify triangle encoding encoding kind encode pilot experiment conduct particularly atom randomly face encode learn encode encode atom corresponding feature last largely understand phenomenon plot treat patch atom one local appearance
normally direction model discuss calculation normalize existence section idea regard likelihood goal probability network degree consider node minimal parametrization parametrize q statistic arbitrarily remove sufficient information non normalize constant calculate directly normalizing correspond nan graph equation imply pg k n
definite one easy without identifiable matrix induce focus model simplify pair minimax known let minimax bound ml three setting pair bound combination technique main technical constructing packing semi induce pack cc main packing constant ensure pack detail impose constraint bound gradient negative log f cauchy recall induce put arrive effort convexity parameter et main focus walk case theoretic study analysis however discrepancy minimax us estimation error ignore specific see eigenvalue standardize
assignment identity matrix specify censor observation e survival person age interval specify great point tracking become standard rbms ordinal ordinal observation order observe threshold read offer alternative ordinal rbms categorical observation category associate category I large utility fix treat logit suppose categorical observation observation ask team person pick without z z translate rank category stagewise well category illustration particular tie impose rewrite z z mcmc inference alternate eqs inequality limit assignment remain unchanged due conditional independence
policy require induce inner actually hilbert allow function ms exclude however probability notational restriction practical future policy accord learn future value examine possible distribution mdp mdp episode time let make reinforcement sequence episode incur regret th episode q note regret expectation review thompson reinforcement later regret begin mdps episode
multivariate covariance allow size sum simplify kind kind index force pair pair nine index kind six pair remain last expectation vanish unless euclidean rank invariant orthogonal distinct way quadratic analyze sampler invertible identical choose loss put see carlo powers lemma term leave assumption expansion power calculate expansion enter combine substitute collect equation yield
set partition call subgraph consist undirected graph node chain line order fig b separation simplify induce prove provide type walk connect walk section choose suppose walk connect outside walk suppose keep non
eqs small ignore expression practically average increase flow cycle equal sequence variance cycle read length achieve probability great sequence nucleotide length base cycle nucleotide flow nucleotide probability right nucleotide eqs cycle previous distribution cycle two nucleotide perfect normal exact distribution long tail slightly shorter
trading literature short base temporal variation price stock without mainly limitation market individual portfolio optimization company level finance comprise mid portfolio financial decision company entire york stock new public manually handle company technique portfolio company stock
pass concave region expand bind since quadratic second ensure follow compute finally derivative prior range follow b c ccc ccc c ccc ccc c microsoft microsoft com draw discrete convert work sampling convert describe mathematical algorithm search maximum correctness evaluation closely rejection draw sampling predict work generic exact interest probabilistic build case desirable exist specialized rejection chain discrete return configuration perturb work sampler energy force resort observation perturbation infinitely many perturbation perturbation determine one irrelevant
correspond slow signal spike occur transform time series backward q small threshold yield indicator process clean spike objective step procedure network spike part fire observation make low additionally well activity context neuron fire activity signal output q spike case global activity processing simplify apply show figure neuron
subsample among versus across combine benefit production subsample census notable production fairly analysis suggest conclusion adjust month home researcher record across two benefit tend pool error benefit rely error population exhibit variable sd compare sd production age compare sd production number contribute substantial census examine communication demonstrate serve reliable imputation outperform modify incorporate dataset good experience rely potential default substantial sample appear misspecification default joint potential systematic future optimistic computationally time roughly likelihood large require much categorical clearly simulation dimension continuous likelihood mixture multivariate normal specialize exist leverage adapt extend context fully design fitting
dimensional except sign recovery observe sign sign prohibitive generalize computationally even check uniqueness application rotation specifically difficulty basis randomly select position rank dimensional decomposition principal problem name inverse efficient sharp inverse problem well study see elegant characterization theory modulus theory rely convexity functional estimation space convex present highly technique readily inverse early exhaustive lead statistical prohibitive problem solve prove case appear mathematics statistic generalize geometry notion low show width size study phase transition paper suggest geometry local sample ensure successful noiseless setting decomposable norm set focus detailed minimization erm loss excess risk subgaussian class proper localization radius convexity erm bound need
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dd trivially integral reduce consideration via zero multiple implicitly device determine theorem detail involve convergence k r obtain combination quadratic remain complex j employ deduce establishe state derivation conjunction r j n denote similarly estimator eq theorem quantify property estimator process demonstrate alternative frequency domain maximum likelihood converge representation provide theoretical estimator mis model demonstrate pseudo long distinction frequency domain domain accuracy replicate match overall double likelihood mis specification design domain long true primary secondary paper property domain sum square employ mis true generate long history series back dependent well
square goodness fit apparent goodness purely weight replica calculate inverse apply simultaneously figure equivalent give base accurate square systematically shift towards value potential review odd application discuss four area detection characterization scientific molecular characterization biology power deep understanding system scale range write dedicate j meet influence student clearly train responsible member scientific award university associate physics usa journal company experience entropy physical science search characterize receive start research biology biology max institute interest receive ph university usa laboratory california institute development temperature product interest include phenomenon development sense
various help evaluate vote necessarily adaptation optimization mean map training subset associate prediction function fusion majority vote measure function j j evaluating map rank mm positive prefer pairwise preference instance know performance map notion loss multiclass follow idea relaxation fusion pairwise rank pair want q force
character symbol language extend previous assume length message know advance code always know many last limited guess average string independently alphabet length string note string perhaps suppose random independently x simplex count vector count model form total count accordingly target mn poisson equivalent provide scheme string also assign string produce length use logarithm regret regret redundancy string q redundancy pointwise provide x become
k mn upper kernel operator l moment eigenvalue oppose th moment th moment x integral involve fourth dimensional learn grid agree dimensional diversity use weakly informative q extract color descriptor feature assign coordinate color sort axis produce histogram color process set sift descriptor descriptor commonly recognition image descriptor give sift sift feature histogram near cluster descriptor process feature scene dimension normalize norm combine dpp partial dpp dpp choose
consider input noiseless exactly evaluate expensive ir obtain initialization except nb portfolio ei poorly decision space ei es nb ei nb well treatment parameter novel theoretic optimization predictive maximize step since function approximate original entropy predictive evaluation produce entropy acquisition easily approximation es synthetic world es observe produce ei popular greedy decision tends result ei simple often get approximate path gp maxima derive approximation formally definite
screen respectively good dct table approximation structure assess coefficient transfer function relate could square expression select transform dct approximation thus comparison fig strong similarity spectral dct present actual
figure htb h htb construct time thresholded graph note screening via perform spectra series relation specify note threshold critical performing correctly discover series frequency diagonal covariance discover screening resolve synthetic pass gaussian band see complex screening correlation time chapter present time series focus identify thus statistical screening variable discover threshold positive negative significance quantify consider small number experimental validate screening partially fa circle corollary rgb rgb screening series chapter discuss series domain goal time time show time asymptotically statistically permit challenge series correlation screening accommodate fouri component degree regime specify threshold significance usefulness
nn sec sec v nn sec sec classification collect reference divided macro nn nn perform develop dissimilarity perform ds nn equipped euclidean differ third reference nn rule equip configuration label consistent additionally e genetic omit deviation sake classification gives expect inferior regardless system general variant affect significantly performance accuracy inferior r valuable absence operate search ds deduce l h dataset r consider configuration comparable reason yet require future deviation always small regardless demonstrate complexity cubic quadratic move effective computing calculate serial cpu show whole improvement exception especially speed properly ds reduce computing evaluation report specifically calculate demonstrate superiority cpu viewpoint variant operate synthesis employ process stream focus perform big operating especially I h involve
normalise give track candidate rotation determine track use retrieval pairwise report track track song accuracy precision ranking track interpret average precision track track query track compare accuracy distance across query adjust error comparison query precision determine song th measure transform distance distance apply outlier ensure rapidly track similar monotonicity rank combine average pool vary basis rank probability cover identity probability form utility versus straightforward yield baseline include correlation random examine distance measure relative codebook exception codebook size range improvement codebook size relative compression average result qualitative whereas loss relative appear advantageous compression whereas advantageous compression rely markov
e e p ce reduction design least favorable consider case j choose later b g ec j rip leibl combine low treatment omit p j lemma involve favor certain logical hierarchical focus hierarchy mean existence attract lot slow meet challenge big importantly study hierarchical difficulty sparsity type structural simultaneously reveal purpose strict iterate efficiency efficacy propose notice additive include effect adequate helpful sometimes behavioral science full n n hold hadamard pose plain variable
difference lstm exposure amount content gate hand control location input gate gate lstm unit new memory content control amount lstm forget gate information add tie gate difference alone type unit would well although preliminary translation apply motivate thorough lstm
profile concern depend profile candidate little benefit profile page short profile length candidate length correspond page something act short text acceptable accuracy candidate three profile word accuracy correspond text word case profile short text opinion piece couple page play claim accuracy profile correct rate achieve profile contain rate well statement evidence nr c text thousand rand c l text thousand rand l l texts rand profile length two length consider increment resolution text permit estimate short text attribute accurately experiment perform table last accuracy report end correspond page long middle act play column table correspond scene page novel article text text accuracy increase medium decrease mean decrease short text profile available achieve accuracy short text acceptable two four length
classifier kernel histogram overlap considerably reject scatter plot contour fit gaussian algorithm knn figure plot univariate algorithm post hoc univariate knn sort method low difference rf rf pairwise clique lda knn significant also post hoc separately lda order knn rf clique give information cc b b plot contour respect another type tp tn learn
incorporate common learn number one effort learn space number per metric weight information expensive prevent generalization near method learn attempt propagation unlike discriminative class code author website matlab code generate locally region select neighbor segment letter dataset uci reduce use normalize except letter split reduce overfitte training tune use basis dataset global global letter misclassification perform dimensionality high fast psd compare learn near neighbor c avg bold misclassification along
sampling carefully proposal low model operator standard probability rbms deep agree closely rbms compute agree full rbms one optimistic one encouraging suggest conjunction probability mrfs mrf boltzmann rbm mrf bipartite unit purpose exposition assume binary case distribution write q visible bias weight bias rbm v tr tr tr likelihood intractable exactly persistent rbm average v unnormalized intractable rbms
quantile measurable quantile function loss l use empirical interval figure follow accord along contain probability optimize conditional achieving contain variable intermediate interval intermediate prediction intermediate good approach aim prediction good training prediction class hinge use support set take thing empirical compare come mass around scalar trivially ideally know still conservative formalize equation capture let derive model residual good residual two member show know singleton contain construct define quantity depend capture section probabilistic solution depend able closely capture necessarily loose definition quantile outline quantile scalar eq robust equation conservative possible probabilistic guarantee sufficiently small empty conditional quantile function
improvement prediction u denote pdf evaluation point extract process function evaluation improvement consider permit underlie common define may variance process practical
dominate purpose follow respectively recall martingale generalization analogue uniform bernstein sequence uniformly probability therefore control martingale condition order iterate bound counterpart hoeffding inequality uniform constant p theorems identical appendix unchanged wiener far proof detail therefore defer super resp give low identical bound immediate martingale concentration without explicit basic
know margin penalization lagrangian problem primal reliable class assign eq associate positive class majority class solve tucker point rbf smoothness suit demonstrate much long rbf classifier difficult set reinforcement tuning method tune consume e rbf drastically classifier employ nest design supervise search determine close parameter
gaussian gp would derivation acquisition show six acquisition target surprisingly portfolio acquisition evaluation thompson mat ern form purpose sample minimizer need hyperparameter amplitude prior ei pi meanwhile keep thompson thompson namely report propose split equally among gp draw sample experiment evaluation optimum performance time performance repetition one standard commonly optimization three respectively
purpose text file consumption computational fairly long combine write library excellent convenience format file handle call currently file normally treat rest request call likelihood file discard alternate allele alternate retain alternate call miss variant alternate possible code support allow arbitrary name handle also improve expand mini manual search mini manual associate keyword keyword comment issue comment analysis trait respect quantitative transmission likelihood advantage speed increase meaningful procedure al robust certain permutation applicability law combine vast speedup decide rate table collect
improper obtained finally use walk rw acceptance likelihood mh gamma proposal moment instead prefer walk rw conduct specifically log recall ratio approximate mh metropolis gibbs fashion primary appendix update memory far distribution amenable gibbs particular use dd may inefficient less regardless refined method truncate little coefficient approximate use one variable sensible necessary value big context sampler backward observe length current reverse since sampler acceptance section primarily auto acceptance e mathematically convenient determined although assume short motivated specify precisely infer parametric bayesian modelling alternative primarily relate alternative model whereas approach allow computationally spend towards primary interest develop efficient properly reversible jump mcmc chain within marginal integration therein previously apply
svd attempt solve local optima lack relaxation problem notation nearby error notation linear constraint heterogeneity quantification know exactly e entry toeplitz element measure high metric mean error huber difficulty case structure exactly tractable formulation convex relaxation upon nuclear matrix singular nuclear vs norm choose large small nuclear robust problem dense theoretical next plain effective come weight describe relate
outlier amount event scale detect specifically separate separate tweet cache due lack group tweet take square square pm indeed semantic link event tweet quite text mostly term separate term square cache cluster mainly similarity graph cluster mainly construct two similarity similarity computational graph total filtering term popular frequent substantially affect reduce algorithm largely also filter scale event term share pair similarity cell compare keep computation due daily stream area york city experiment take second finish construction similarity minute code mid core due social medium online decade rise series research event user event use popular platform twitter attract significant due early approach detect specific rely stream keyword indicate wavelet developed well meaningful reduce noise medium analyze event tag associate propose temporal spatial tag author tweet measure detect keyword temporal hierarchical procedure twitter temporal similarity tweet co occurrence keyword
optimization guarantee find optima consistently estimator convex year advance toward zhang zhang show optima leave find optima fan et optimum wang establish guarantee lie substantially simplify work establishe agree well regularizer advance nonconvex problem agree establish recovery understanding point nonconvex objective objective technique variable tucker optimality primal state certain class nonsmooth theoretic smooth possibly nonconvex regularizer allow nonconvex regularizer suitable mild condition early additional minimum signal strong remarkably regularizer include mcp regularizers usual incoherence condition guarantee recovery provide nonconvex establish several nonconvex estimation weak nonconvex mcp develop theory absence incoherence regularize possess optima nonconvex however paper wang homotopy obtain homotopy oracle paper purely concern theoretical consistency stationary finally zhang show eigenvalue weak restrict certain nonconvex regularizers estimate provide approximate early however stop recover vector organize follow material estimator primal method concern corollary graphical case regularizer regularizer implication support contain contain illustrative simulation confirm theoretical universal write simultaneously write frobenius mh subgradient write radius also technique proof
use slice argue previously space fit memory gpu additionally divide step correspond implementation gpu cpu gpu pair master pair care computation master cpu communication node configuration must merge special configuration model update via node also dimensional index pattern novel implement flow node implicitly communication distribute implementation node configuration htb initial guess divide direction probability copy cpu execute gpu execute gpu direction execute
yield notice convergent subsequence boundedness take side hence hold kx q limit ii proof fact suitable termination accuracy parameter establish define proof q ready second minimization em pt define arbitrarily solution subproblem go go inner termination satisfy inner sequence accumulation stationary accumulation stationary moreover nonzero inductive argument imply p proof statement fact outer subproblem value arrange accumulation subsequence
background dominate residual capture proof appendix residual graph bernoulli graph subgraph include unit positive residual assume implication subset power vertex involve subgraph vertex concentrate foreground intuition subgraph vertex mean detect always relatively subgraph embed activity relatively remainder subgraph easier detect put language much easier detect less working communication enyi generate subgraph embed horizontal expression hold order maximum within subgraph closely scenario random subgraph tight subgraph provide good detection desirable detect small subgraph may stand eigenvector technique detect anomaly outline symmetry enable detection subgraph stand graph project principal rather entry enyi demonstrate two anomaly subgraph stand apart background compute detect presence anomaly chi contingency number point chi calculate favor radial symmetry anomalous spectral reliable anomalous behavior small subgraph identification complicated
direction class distribute sub develop allow uncertainty scenario traditional approach improve optimization method uncertain sub gradient know despite explicitly address aspect optimization uncertainty demonstrate time impact machine incur cost online regret sub multi study introduce decentralized sub interact path neighborhood online prove regret extension convergence regret aforementione corresponding algorithm used centralized effect favorable due rate failure inter sensor link uncertainty distribute fixed switching graph use
appendix offer insight student equation submatrix n marginalization sufficient kn k gives require k elliptical collection analytically theorem analytically solve n able analytically derivative hyperparameter carlo derivative eq wishart wishart generalization positive definite multivariate function q recursive generative q marginal distribution equivalent thm thm remark thm student nonparametric student integrate away wishart student derive inverse process overall student retain attractive gaussian
loss produce almost surely converge condition thus corollary proposition show uniformly bound uniformly conclude converge accord imply eq converge surely accord surely tends subgradient continuously continuously differentiable continuously j accord continuous hessian shall eq sample norm follow bound accord eq lipschitz finally taylor equal since vanishe prove proposition converge surrogate since vanish lipschitz derivative bound first taylor tend infinity fact since multiplying follow tend hold minimize implie tend investigate recovery expect online pca corruption justification whose pca draw norm change
vice versa citation separately identify original citation top corner exact match identify microarray laboratory generate laboratory responsible much laboratory affect retrieve laboratory correct drop effect laboratory six five paper act date six link cluster cell line cell
challenge spam semi supervise filter show supervised differ classified challenge collection email task filter cycle combine challenge challenge remarkable classifier train test perform filter semi filter support dynamic compression logistic spam spam train publicly test attempt whether semi report replace challenge challenge delay message six batch message message keep reproduce train task experimental outcome show version respective delay hand supervise perform
actually upper argue encoder provide close reconstruction reconstruction error generative elementary generative actually closely small check cost draw negative represent w r contribute evaluate generative quantity easy introduction bring inverse auto structure minimize depend feature reduce upper two f term minimize use compact representation reconstruction reconstruct auto another tight feature space involve integral dirac infinite encode value underlie network activation layer bernoulli distribution covariance matrix intuitively encode accuracy reconstruction
alpha plus collect high possible index high index index mean vector represent count number k u care look follow plus log logarithmic time summation running become hessian run mle precision initial run equation newton step newton put order bottleneck read run
program robust available rest notation diameter ix describe similarly primal rely guarantee target parameter infeasible ti mp u infeasible robust infeasible terminate call begin rule update apply directly instead require observed dual variable note primal variable together obtain condition variable z u ix lemma summing inequality prove return infeasible exist counterpart return infeasible recall
constraint completely analogously remark remain see input approximate note formal learning perhaps surprising provide justification tackle task inexact without manuscript aware hardness approximating case two unfortunately contain write case general f k problems dictionary relate potential dictionary admit approximate within factor problem preserve slightly weak
category property improper make health system benefit collect analyze increasingly find property correlate time may prefer member population receive benefit portfolio risk student college include code fitness predictive power category use machine naturally dependent everything place exclude physical fitness vocabulary success character trait power contaminate predictor underlie mechanism access problematic
sample n fp p ip evaluation turn predict input convert output dimensional project crucial speaking onto function nonlinear operation risk embed develop triple novel scale first kind massive furthermore risk assumption lastly improvement magnitude estimator well world datum set nonparametric perform hilbert noisy henceforth work give empirical
show unique extended driving discussion multidimensional rbm dedicate follow brownian motion deterministic simulate without definition rbm increase behave like brownian motion appear drift generic transformation difficulty acceptance multidimensional rbm dominate directly multidimensional rbm absolutely note rbm challenging arise one reflect core lie observation rbm us density contain accept key simply decide direct n x
use relevant spatio temporal classify spatio temporal behavior spatio temporal covariance avoid impose priori instant frame temporal denote grid perform regime exceed covariance order know undesirable poorly poor inverse
nh nn map eq h q claim x problem important integrate knowledge structural contribute incorporate algebraic algebraic independence trick demonstrate usefulness ica specific constrain underlie hand hand cause respective invariance transformation corpora transformation invariance need ambient noise speech robustness translation transformation handwritten digit recognition g popularity knowledge bioinformatics amount fundamental formalism trick feature ask kernel invariant sign mirror common complex phase factor rotation algebraic call semi suitable invariant trick twice namely invariance
filter element shorthand much degradation supervision incorporate idea discriminant compose ease enhanced supervision classify index patch image spirit intra pixel wise covariance likewise inter class lda eq q trace know pseudo full handle lda mat deeply build repeat propose variation verification digit texture discrimination face person pca b face illumination select subject expression pose pose manually corner image pixel corner gray together subject subject four remain neutral illumination classify variation test illumination pose illumination illumination pose illumination expression pose cross pose pose impact filter cross illumination one network set accuracy similar however observe randomness impact block robustness illumination set artificial observed percent translation direction plane rotation suggest various block may histogram aggregation
ba z n r
successfully apply topological count number ideal partition solution discard lie apply strategy number ideal vary odd number partition da di partition critical ml ml degree property hold ml degree ml generic cubic notice cubic surface equal suffice choose equation degree partition minor solution ml instead trying determine solution sign topological euler possible distinguished hyperplane degree distinguished degree intersection lying solution conclude root intersection correspond root f dx thin w fill black circle fill circle inner sep
discretized proposition loss consider tuple nonnegative integer since partial define since constant let assume side center z every pair discretization particular long precision threshold define combination shift center negative perturbation support exist vanish f always discretization put bound introduce know degree gaussian function bound proposition close look high tangent span top span output output simultaneous utilize weak specify convergence extended dimension size add adequate universe neighbor denote let dl sd ls orthonormal let kk dl sd
estimator obvious equal distribute strictly preserve protocol protocol feature v x differently distribute invertible information protocol map exist extend yet support intersect everywhere similarly imply differ b let integer protocol xx value identical definite thus invertible mp finite disjoint linearly independent v invertible position show periodic differ v argument yield determine analyst extend subset analyst select user repeatedly interact analyst feature p b estimator r v x able generate rating negligible determine preference restaurant rating movie readily item may private rating give modify protocol reporting rate reveal rx ensure subject respectively rate item profile information extract comprise privacy pair rate jx x jj gaussian item protocol summarize bias ratio item include reveal construct reveal rating subtract mp analyst feedback mp behind immediately privacy preserve rating since formal reveal rating establish among attain minimal protocol
handwritten incorporate target multi special handwritten example fall category group whole membership yield weight put membership weight non yield superior estimate treat old shrinkage available target shrinkage multiple constant finance one structure constitute choice base expert superior slow cross computational validate extend multiple section introduce quadratic program intensity optimum sample fulfil asymptotic observation limit structure theorem capabilitie letter denote symbol eq unbiased general index index matrix datum denote p follow omit obtain less setting est I give consider assume assumption increase dispersion eigenvalue increase dispersion assume behaviour imply
affine connect differentiable work directly pose challenge address manifold tangent rkh embed space considerably simplify preserve manifold burden extend euclidean rkh present random projection first hyperplane projecting present various vision task superior discriminative typical outline space discriminative completeness random hyperplane recognition person texture svm riemannian locality preserve relational bring power
estimate add preferred accepted compose hence add likely accept cluster wireless estimate factor solve determination cluster wireless security propose attack device communication wireless serve wireless framework develop deep nonparametric structure correspondingly factor estimation error
data foreground video entry sample performance fw fw prominent grow frame visually appeal fw video medium frame grow background illumination rotation weather fw accurate iteration significantly overall still fista illustrate plot increase fw c video cpu cpu square visually recover background capture foreground per iteration fw linearly advantageous take illumination superposition sparse matrix form stacking term capture smooth term represent cast full assume experiment summarize fw fista clearly fw scale r visually rank smoother condition scalable call fw wolfe fw norm combine frank
representation document wide variety modelling include classification model create sentence simultaneously explore representation exploit direction future rgb university united ac uk capture compositional word central challenge language retrieval represent embed low preserve crucial capture convolution document lexical concept model achieve compact advance vision present novel technique network text symbolic decade network model translation name entity research compositional space algebraic approach simplicity arguably network use great
pool magnitude velocity fashion distance maintain trajectory energy coordinate velocity equation move along along trajectory hamiltonian reverse together use move transition hmc understand operator act operator hmc cc b b involve hamiltonian monte carlo hmc base represent momentum momentum dynamic indicate dotted randomization momentum horizontal movement occur momentum vertical movement hamiltonian flip
dataset face individual capture condition expression size select half per person project zero row dl common sub include subset contain illumination sub round class satisfy great variation illumination viewpoint different sense adopt feature descriptor performance dataset leaf dataset use carefully flat clean background setting make easy application advantage focus shape specie art testing sub size dl descriptor relatively visual difficulty benchmark point fast image chain
heterogeneity also alternative pc assume necessarily principal discussion within covariance across major smooth common pc percentage care take dimensional slow lee et al insufficient complex covariance characterize partially approach surface although currently whether lead estimate eigenfunction regularize two step whereby curve compute pc compute eigenfunction pca smoothed cubic spline pc sparse involve wavelet pca unified pc advance allow pc grid model source variability lee surface quadrature score grid across develop sparse curve eigenfunction pc lee level di et present level multi functional method densely observe di al random al extremely et b al al pc score functional introduce encounter functional functional coefficient response many work introduce linear orthogonal discuss truncation penalty several reduce interpretability coefficient make across choice depend function basis purpose measurement involve non reduce measurement bias al accommodate predictor vary predictor general strategy regularization relate model flexible beyond present response inclusion fix extension nonlinear introduction highlight roughly develop basis sample common grid regression ridge result multiple orthogonal principal regularization necessarily first pc regularize pc et discuss smoothed regularization truncation basis remove error pc principal across curve estimate pc ml di subject predictor subject pc basis truncation truncation estimate eigenvalue still bayesian classify functional predictor transform time frequency construct logistic predictor function regularization structure incorporate estimation weight point inherent datum functional spline assume common spline basis regularization introduce use design use spline subsequently way additive scalar fix model multidimensional natural cubic spline basis represent regularization account
hardware gpu decomposition accommodate add combination nearly core alternative leverage library together compactly correlation small reduce pseudo input truncate expansion like parallel option extend magnitude illustrate big capability sum tree allow interface million size hundred core reach method whether surrogate show magnitude fast krige focus quickly kriging involve subset base give typical rapidly decay correlation simple fill response sensible fast accurate full high work choice scope sub spread optimal design criterion would search design building criterion lead prediction nn scheme design iteratively calculation regular exhaustive green roughly split comprise near one even relative early location exclusive iteration search choose variance much make design attribute aspect novel exploit trade
describe utilize uncertain information describe paragraph carry look credible improper combine b parameter reduce posterior base vector independent real improper attractive feature tail credible know b density easy marginal posterior credible union interval illustration tail short credible interval posterior unimodal credible program matlab scale offset contradiction follow scale offset
team play difficult college arrange highly rank contain mostly low rank percentage centrality specific short path node metric conference team closeness centrality also particularly graph team connection instead centrality contribution connection centrality centrality therefore rank great influence graph metric centrality assign win address limitation centrality win many high graph node adjacency centrality node eigenvector centrality neighbor notation identical equation place eigenvector represent calculate final numerous intuitive calculate highlight proportional
sensitive accelerate robust specifically attempt employ representation dominate robustness enhance speedup solver via alm derivation theoretically extensive ten verify outperform fast increasingly wide highly sample visualize huge text video greatly additionally outlier side outlier reduce subsequent significantly promise accuracy sample although efficient effective
face inherent extensive diverse accuracy label face face verification face verification face topic computer decade surveillance retrieval mobile device visual verification face dataset face large complex variation pose gender prove difficult automatic face verification work accuracy improve establish study close gap human verification human reason verification drop however scenario cross appearance collect training highly verification domain face verification modern face verification category extract low building classification exist face flexible deal level even projection center need specify similarly deep layer etc
form attention case practical lower entirely lie bound incorrect framework practice issue assign miss term although density remark miss miss beneficial assign miss discuss suitable improve chain model movement density miss miss prior density assign density offer aspect fact target infeasible introduction thus vector augment augment distribution follow distribution multinomial probability full conditional distribution I often mix denote
resemble linguistic simulation even able truly low unlikely give identify number demonstrate amenable combined tree furthermore retain proxy distributional look explicit moment high variability include language least mode language explanatory power linguistic apply positivity four positivity constraint explanatory identify false suggest identify tree amenable truly component amenable develop insight language separable hadamard product language dimension matrix indicate overall co projection particular standardized illustration second language remain language many contribute overall whereas distinguish component hz hz hz hz show range interpret hz range likely relate hz relate portion spectrum hz round frequency around hz likely human speech data frequency affect show interesting range difference particularly effective separate language effective numerically course require feature exploratory place especially distinguish variability acoustic evolutionary interest identify prominent feature effective
satisfy convert theoretic assignment seven relation table relational logic learn reproduce behavior show boolean atomic symbol logic output accurately bit demand present atomic symbol value interpretation formulae l randomly generate formulae contain logical operator compute relation discard seven relation formula partition operator formulae bin implement similar statement formulae six relation almost balance without basic task modeling pair statement six variable unseen structure logical yield short k example across bin
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vary rise marginal market market determine adjust five minute accommodate real constrain typically formulate lp determine incremental min index minimize achieve demand balance via flow cf bound bid solving determine lagrange multiplier associate lagrange multiplier define express eq price practice transmission loss calculate correction consist price flow line complementary imply loss ignore would either readily latter effect isolate entry argue subtract way collect
computational amp group test fig projection constraint amp additionally closely match transition performance amp algorithm match amp requirement practical projection case superior convergence amp burden r evolution thorough analytical future part union th grant triangle
among produce processor compute avoid processor processor function fortunately processor hypercube score processor keep take compute note processor processor compute collective execute compute processor processor evaluate processor high computing constant message hypercube processor number processor hypercube hypercube hypercube processor previous definition denote binary string denote string include partition hypercube lattice hypercube processor function hypercube parallel processor hypercube cluster string adjacent bit processor responsible transform subset bit string processor hypercube encode bit string subset processor processor processor dt kt subset encode processor hypercube encode string processor processor processor processor
relevance purpose unfortunately metric estimate nature interested click click may different art user click highly evaluation model always control randomly split statistically control baseline search engine group modify engine baseline component metric click system reach statistically prove successful allow engineering business decision manner nontrivial engineering resource consume effort need optimize click often guess like ndcg engine later control proxy determine modify indirect inefficient offline evaluate log
still spline lda cccc ccccc performance lda spectrum categorization speech consist frame dark frame dark water frame widely website contain measure nm interval nm low high spectra use face recognition task contain gray image individual normal dataset categorization contain category contain object view align pixel database probe randomly spectra remain spectra remain set contain recognition spectrum lda function radius recognition method baseline lda available deal image svm different linear radius basis
lead lsh desirable query dominate evaluation section examine cdf q figure apparent quickly keep intuitively undesirable code performance similarity sublinear time search especially basically minimize region figure plot region value normally pre
arbitrary compact solve apply classification case consider setup scalar array array non kernel compactly measure reproduce rkh closure one point trick index embed index space trick continuous compact hilbert define closed kernel define maximum margin u u solution project fundamental separation empty separate support integral characterize x write difference probability measure integral v k support hilbert project onto ray favor second label index set back kernel classifier coefficient convex coefficient find solve tucker infinite technique infinite control necessary kkt maximal principle subset equip hausdorff ct c ensure svm good classification ol krige unified ols gauss imply geometrically svm classifier geometrically margin suggest ols leave open article present approach parametric formalism deal infinite applicability theory measure vector ordinary ol conditioning uncorrelated covariance ols krige array arbitrary index support classifier hope deep connection extend result equip ol joint ol long version long version space banach
receiver look receiver post receive processing square mmse channel adaptation abstraction quantization feedback indicator feedback calculation map bit feedback map value feedback map index value index compute report delay henceforth delay delay lead problem alone feedback mechanism prediction scheme change reason interference gradually effect change active change different band example couple stop inactive band case net interference macro dynamically dynamic frame employ macro become dominant due fully resource load lead improve link perform channel employ treat filtering square treat effect partial loading algorithm transmission different user compute technique invertible furthermore unknown hence select rate wherein temporal build exploit technique come feedback ms sequence predict
ssc overcomplete sample theorem matrix union position point sufficient use sr use ssc algorithm cluster representation task assumption like face recognition segmentation see make use dataset dimensionality preserve dataset aforementione without precede allow fast compute result datum expensive cubic projection sample nature dimensionality tool projection become essential technique efficient evidence cosine projection
fidelity investigation different negativity row induce drive relie regularization positivity otherwise impose form lagrange penalty flexibility fact split subproblem lagrangian quadratic independent reduce minimization split q admit proximity proximity indicator projection positively extension
variability fit design principal pointwise pc pc solution pca towards pc target knowledge nan subspace toward pc checking opinion interested sampling component calculate design section specify also useful may interpretable calculate bootstrap explain span projected bootstrap procedure pc rotation towards eeg true scenario sample basis vector denote measurement simulate ik kk true set eeg dataset draw empirical univariate score pc random noise score imply proportion explain score variable coverage simulate simulated bootstrap sample comparison increase measurement eeg principal score variability fit population score basis variance score eigenvalue eigenvalue consider eigenvalue measurement conduct total hour simulation management job simultaneously job gb gb virtual depend scenario simulate pointwise coverage right coverage pc pointwise coverage close third consistently give coverage percentile give poor percentile interval skewness interval
metric rbf propose similarly justify small volume bounding mahalanobis idea preprocesse characteristic optimization researcher investigate fusion svms approach vector count centroid k completely reduce remove sample additional another related classification exploit independently split analyze classifier propose gain one rbf building htb allow dependence point transform generality calculate
learn adopt cnns imagenet much alone comparison newly train annotation classification moderately improve sentiment analysis far sentiment useful include business sentiment sentiment image much relevant propose sentiment noun vs svms consider base leverage concept mostly activity trying solve fine grain recognition organize try non concept deep study computer
degree correction line community separately event almost non partially overlap corresponding benchmark bipartite usa datum collect class interaction perfectly match literature partition al dash fig largely group modularity type community consensus slightly bad modularity list appendix find consensus minimal benchmark review ref majority identical empirical human system via var create constrain genetic giving structure type connect network somewhat document word cover var gene broad make corrected force fig correct recover community adjacency gene nearly gene overlap community degree community correct analyze find gene analogous finding bipartite maximum correspond classification broad heterogeneous degree movie actor directly edge exist actor actor movie show database movie study
simplify cost evaluate totally second difference easy rely flexible position capture position example expect movie encode prefer e therefore energy function un step potential resort distortion current permutation move pick keep order rest unchanged place new permutation move operation relative preference order randomly pick two item swap swap
recent result indicate bayesian coin outcome coin coin weighted tail coin use specify beta new coin give conjugate binomial nice posteriori equation give equal probability q correct bayes pm similar inform model inform prior ignore uncertainty uncertainty greatly affect characterize choice ideal draw use simulate eight analyze dataset prior directly simulation population divergence similar model bias time difficult dataset model method summary contain minimal correlate four default statistic contribute information time uniform summary prior limit five unit little divergence time assertion entail statistic coincide analyze draw vast yield cluster around text plot select avoid analysis much argue dispersion time plot essentially
preference try find human preference alternate robot determine action observe human together find human type cluster expectation maximization rank distance criterion integrate another cluster chain partition partition transition correctly sequence hybrid framework learn robot act robot vice versa uniform partition rather use bic ideal use cluster x human robot turn transition act robot action vice versa matrix must sequence parameterize denote otherwise repeat step assignment ix j converge algorithm call transition em nz calculate current high return maximum transition matrix randomly
explanation visual cause term operation macro target image assumption exclude target behavior black white relation unobserve discrete short possibly contribute omit simplicity noise incorporate behavior stand relation generative model image group observational observational clear observational observational associate label know observational image allow image take excellent predictor weather weather cause weather particular read whether visual cause ability visual pixel pt generate standard causal distinction detail target image space target behavior causal cell partition consider equivalent partition visual visual whose stand relation visual image know allow visual long relate observational observational observational among induce observational almost
epidemic likelihood paper observe set share lead leveraging solution one spirit leverage seem advance brief example continuous infer bernoulli bayesian prior posterior deterministic numerical integration detail example abc range traditional recent reader library quadratic discriminant matlab employ regularize polynomial library matlab interface regularization value classification implement nine chebyshev multidimensional project principal prior rescale one amount whiten multiply fold max try several give accuracy move average exclude pool computation iteration propose simulating base comparison user discrepancy implement way perform comparison use tie abc monte know carlo abc start sample weight implementation threshold quantile accept schedule pre schedule quantile quantile accepted slow take choose schedule epidemic transmission inside center uniform work assess expert rooted proportion infect
subject censor rmse forecasting table low apply motivate percentage force volume cf patient rate differ infection patient model establish association subsequent patient acquire patient index patient recent choose subject focus share variation hazard negative worth association logit link keep ease clinical hazard show smoother estimate penalize spline hazard flexibility hazard capture need assess hazard function end may forecasting
metric root rmse higher well quality rmse approximation include another pursuit fr extend match pursuit method give input initialize matrix set u k construct fr mp optimal weight similar svd matrix fr mp find know algorithm necessary complete netflix netflix dataset movie netflix customer dataset applicable characteristic propose increase iteration netflix limitation mp logarithmic number iteration verify convergence speed propose theorem theorem singular though bad f empirically different n mp er axis axis plot residual
vector anomalous instance assume generate different inferring latent dirichlet prior multi anomaly stochastic effectiveness demonstrate term anomaly view anomaly great interest information source wide variety naturally view page represent word occur page audio visual anomaly multi task horizontal anomaly view anomaly anomaly multi instance view detection anomaly anomaly figure multi anomaly view
nevertheless process meet challenge people conduct reduction advance robust high properly direction burden reduce valid robust appear dimensionality serious ideally mean direction presence back transform estimate coincide loading reduction contain much row corrupt tp true subspace orthogonal curse surprising finding connect outlier much severe subspace roc roc pca fashion significant loading vector inversion update contain drop batch batch size satisfy procedure roc significant intermediate kp pc direction column increase fast formula recommend assume unless speedup scheme tolerance computation batch share similarity problem svd accordingly cost affect roc pca subspace generate vary simulate denote tr denote use measure detection probability fraction label ideal estimation serious serious distortion matter simulation intuition choose combination small true
algorithm area input threshold classification specification algorithm likely recall kullback divergence random bind success probability minimization hypothesis sample indicator start last determine algorithm threshold binary misclassification training involve measure hypothesis family accord fix know convert error generalization part kl u sample make average learn algorithm h td difference triangle since low substitute low bind generalization complexity look reveal relative effect
asymptotic expansion iii assumption likelihood simplify consider expansion method error expansion likelihood unified replace expectation expectation asymptotic express high difference see include maximum connection lemma bayes method denote relation ii iii table type iii corollary summarize magnitude base leading term magnitude type
hmc hmc size tune acceptance step time time effective efficiency ess simulation min ess ess hyperparameter tb run ess ess min ess hmc hyperparameter ess ess min ess hmc lag hyperparameter low ess run mass matrix px semi step level hyperparameter burn iteration ess
k nj regime expert also hybrid vector direction multiplier follow interior phase adapt incorporation propose address challenge automatic high effectiveness compare synthetic real include natural language despite present subset particular observe good interpretability popular support svm devote order first memory optimization problem major disadvantage identify second memory requirement demand since store newton
claim exercise arguably american option exercise option round execute trade decision natural constraint turn exist american option continuous exercise dynamic american consider round adversary exercise option adversary also movement specify movement lie extend upper american ordinary gs reach gs gs gs gs gs gs gs gs unique round american pricing set option paragraph need iteration never exercise need elaborate little move round american option payoff controlling option need write round american option remark applicable recursive option price pricing option game binomial american type minimax payoff converge price uncertainty step st underlie asset decide nature lipschitz payoff bind price define motion volatility control sequence intuitively continuous option upper gaussian convex volatility surprising time counterpart start denote markov chain take lipschitz locally x hx xt adapt call condition control value namely value continuous process speak measurable martingale martingale put delta continuity compact discussion discretized variation continuity condition exist subsequence uniform match control consist dimensional wiener process control value p three first lie
flip occurs condition independent draw independent plug finally event distribute twice chernoff union substitute back pick early deterministic ensure randomize first exist utilize lemma constraint rewrite similar absolute simplified consider occur occur back third strongly attain strongly use fact maximum lower straightforward expression minimize show hold finally constraint low eq desired define sized block suppose target induce diagonal compose equal use block symmetric well notation write coefficient k exercise
mathematics nj semidefinite relaxation mle tight recovery problem noisy mle recover sdp regime
cnn availability multiscale convolutional cnn object extraction proceed multiscale version plane plane descriptor descriptor dimension convolutional conv conv maps image size result feature conv conv conv conv conv eight pixel incorporate local contour multiscale train test original two result contour fine tune pixel contour detection exclude two fully imagenet pre five layer top
turn biased estimate mmd mmd spirit take estimate square quasi sequence sampling sample q amplitude linear il basis efficient calculation mmd similar unbiased mmd eqn provide appendix aforementioned mmd describe equivalent appendix htbp shift denote mmd approximation set k I ia k ia l speedup bring gain mmd original aforementioned approximate time calculate complexity entire speed basis mmd mmd utilize mmd approximation consideration sample accurate thing calculation compute stream usefulness prove mmd
strongly proximal ascent prox le propose sag complexity sag variant et store gradient prohibitive reduce favorable machine zhang call method employ stage scheme gradient complexity avoid storage past analysis sag analysis extend prox solve prox incorporate weighted proportional lipschitz complexity upon one substantially prox uniform slow large much work explore
global rd ct important tune validation pre define consuming model adaptively resample candidate discard illustrate resample understand resample bootstrappe machine neural parallel use past inferential create process focus calculated datum effectiveness measure refer model fitness square rmse determination categorical outcome predict rate might create machine tuning data near neighbor model grow maximum fitting complexity pruning alternative pruning factor cf parameter determine depth partial pls
e simulating convnet clean good class likely label model ground imagenet clear behind superior training noisy information table error adversarial overall level table learn superior cm none none error true simulate outlier cover class chance cifar class training image outli know clean cifar amount outli significantly reduce without nevertheless outli reduce particularly hyper eqn right explore ability train noisy softmax
first pick pick gold standard side get consider gold standard result question gold decompose question include second first gold standard second gold satisfie induction hypothesis remainder include gold evaluate eq q question complete compatible mechanism free payment gold incorrect proceed standard gold first question incorrectly question evaluation proceed worker question free payment hypothesis gold induction since payment must non induction hypothesis furthermore permutation payment answer incorrect gold payment form algorithm payment gold gold incorrect answer let remain repeatedly apply answer wrong payment argument question uniqueness add payment payment mention proof desire sake brevity must property first proceed prove separately involve l induction rearrange get q rewrite expression simple desire l consider subtracting get q subtracting rearrange opposite sign consider know lemma recursively give payment answer evaluate ensure satisfied compatibility sn si sn level answer confidence employ expect payment expect payment prove allowed worker skip great skip level confidence follow payment worker report claim piece notation payment answer respect gold x worker payment compatibility form mechanism coincide define l payment worker answer evaluate worker select requirement compare vice versa
relate clear substitute kk bn self logistic completeness self self logistic derivative quantity achieve equivalently proof tail inequality random vector standard bernstein almost tail concern accuracy empirical moment bernstein random copy wish unbiased function aim sample absence minimizer commonly strategy desirable convergence erm minimize resource usage streaming regularity linear observe single super polynomially moreover quantify finite consider optimization euclidean minimizer sample sgd practice ease wish compute erm erm maximum certain regularity specification argument approximation
brief forest reader forest skip section forest represent direct tree internal leaf direct node hierarchy circle text em thick black minimum thick circle font draw black style level style cm parent tree b internal produce tree apply send repeat process node leaf make reach prediction component great sample phase construct greedy algorithmic construct arrive hand also one index optimize node split setting create splitting among child depth allow split leaf start training split make bootstrappe subsampling sample tree subsampling randomization reduce turn forest assign importance individual overview popular reason focus refer time entire tree value non relevant ranking threshold split researcher heuristic limitation extension record follow also underlie indeed frequency include feature theoretically provide informed however share drawback principled way determine beyond subsampling bootstrapping tree
tc point literature argument theory graphical lead close dc tc see dc kernel interesting happen dc entropy extend family stable spline stable spline exploit stability burden evaluation dc organize introduce gaussian via briefly review entropy completion property dc resp semidefinite denote order element invariant convenience white impulse response
vector manifold either suggest though bias make already improve recognition performance relu method triangle inference amount activation activation relu confirm cifar figure autoencoder hide unit train permutation cifar ie activation performance well invariant sigmoid relu detail light precede hide become small hide remove satisfy separate function active activation define
construction optimization path random natural function measure fx q algebra generate iteration write close numerically
plot benchmark associate euler univariate euler approximation euler posterior evaluate filter ng paper associate euler approximation reasonably inaccurate term location shape interesting consideration however dotted score key remarkably posterior replication rejection abc minute time case euler production still abc summary statistic euclidean reduction apply statistic produce parameter arguably provide exact despite slightly inaccurate reasonable estimate perform score produce reflect impose parameter provide quite poor marginal upon literature give typical mcmc note euler well exact effective use neither accurate linearity space confirm qualitative abc see produce rmse approximation indicate abc procedure statistic latter poor score abc accurate notably persistence parameter fp panel capture basic euclidean magnitude inaccurate ability consistent score euler dominate abc marginal rmse multiple score ss abc euclidean fp refer abc base marginal score euler benchmark top panel produce three run highlight
suffer global guarantee situation negligible remove project refer hard project non feasible projection efficiently low convex exception work demonstrate convex however able penalty scad mcp commonly use iterative thresholding htp pursuit sp however traditionally setting satisfy restrict isometry analysis sparse vector universal constant require rip wherein arbitrarily require perform number
complex reconstruct heuristic far classified genetic knowledge advance knowledge
latent continuous undirected model relationship compare four implicit sort model opposite figure discuss one prediction rating deal miss item certain take value user consider curve plot state pair predict rate roc plot class item rating rating rating play
trade computational estimator estimator statistical trade understanding phenomenon setting key come year section introduce notation paper ij u extract column let eigenvalue arrange decrease component length place eigenvalue eigenvector measurable unit angle loss change argument convenient bernstein directional condition turn particular size level whenever e u u every p pp n principal prove class distribution convenient level symmetric denote small element measurable sample bound subgaussian consider technique facilitate eigenvector minimax suppose restrictive mention introduction
window dependency field model rnn vanish gradient secondary prediction learn target cell gradient learn application secondary structure prediction feed neural concatenation
embedding unsupervise context paragraph achieve slightly paragraph version indicate add tuning well comprehensive favorable play role review positive exhibit significantly grain coarse grain svm movie review review sentence experimental protocol word document next compositional keeping obtain sentence representation recursive sentence sentence representation cross review baseline bag diagram quite much state
valid eigenvalue always indicate space expand determinant use verify association observe ii observe idea second use hypothesis follow positive infimum little extra capture complexity denote structure row resp mixture testing show calculation add see choose assumption estimation display imply complete section section remark grant dms dms gm part nsf grant dms award grant dms university nj ignore stability lead depend functional motivated difficulty functional correlation simple correlation rate exhibit phenomenon illustrate arise financial functional minimax matrix procedure component moreover characterize focus sparse
explore minimizer function purpose turn subtle structure observation formalize last section useful relatively include sdca accelerate ball characterization claim polynomial vice repeatedly apply produce chapter novel prove inversion sense carry low bind modulus upper root vast knowledge chapter develop tool root last lower precisely develop know strong requirement indeed chapter conjecture prove imply seek create accelerated descent root analytic theory polynomial use well gradient present believe future section notion important specification task sequence may randomly whose iterative increase denote possibly method random method draw previous mainly method explore property method nature sequence content elementary theory square pair denote root last spectrum likewise simple spectrum entry zero square note may size demonstrate matrix strictly exist sufficiently denote namely eigenvalue index plug yield convergence suffice norm zero derive aforementione
decay show generate propagate backward compute derivative near qualitative give intuitive task feed neural meaningful mean much architecture tree neural evidence dataset code website node evaluate code neighbor query sort symbol euclidean see seem meaningful reference break control moreover group table confirm cluster representation almost related mainly flow conjecture cluster group distribute vector representation symbol human program learn even though measure similarity program fail relationship different symbol metric e contrary aspect abstract benefit program compound union continue cast switch program interest feed representation base convolutional neural student student system run validity code code along cv cv
random univariate way avoid ill transformation table noisy create pdf pdf pdf pdf pdf pdf pdf configuration specify title vary axis bottom similarity euclidean noisy transformation truth noisy pairwise create column simulation row type transformation slope amount noise transformation information increase slow pdf pdf pdf pdf draw every run shape comprise
kernel som performance term seem section goal exception som aspect strategy som som represent give relational som dissimilarity som therefore equation part equation determination relational equation determination som equivalence variant equivalence relational som relational som som extend euclidean embed practice som som look relational kernel try address combinatorial generalize analyze differ use careful equation prototype neighborhood relational assignment rule value soft coefficient equation algorithm anneal implementation hilbert dissimilarity variant summarize variants som som relational give som htbp online som som batch som na som p annealing algorithm prototype batch som
lt lt lt lt bp ltb ltb ltb ltb ltb ltb ltb ltb l l l tw ltb ltb ltb traditional spin flip also evidence advantage increase arise result problem constant factor low spin flip gpu might generalise spin extra hard dynamically mention confident guess graph advantage considerably strongly subgraph would become restrict perform interesting describe markov field give motivate em p em use use value partial implement store slow would possible simple look table number instead store look arise slight arise small neighbourhood vertex convenient involve multiplication potential particular ideally able binary double bit processor
toeplitz realistic systematic naturally discussion section eventually procedure low suggest efficient parallelization whenever base probability relevant number include poorly informative provide clear contrast many subject expectation chose criterion stable setting intend retrieval namely preferred report protocol overall picture first give illustrate see error guarantee small covariate suggest error positive determine large bind false fp supplementary behaviour guarantee figure corollary coincide achieve value describe less figure positive positive determine notice false lasso latter include result finding positive factorial group toeplitz seem symmetry violate situation positive positive lie stability selection use loose probably room idea count outside scope vary disjoint false positive plot
show kl divergence expansion kl kl scale q continue bind multidimensional euclidean ball calculation ellipsoid claim two interpret eq mdp uncertain regret scaling demonstrate stationary control server queue mdp plan single queue customer queue bernoulli unknown mdp state action service resp service queue resp service service type action instant respectively hold queue whenever queue correspond total policy optimal policy policy monotonically range regard start estimate candidate q kl vector non degenerate mdp possess policy assumption theorem regret bandit policy arm completely mdp contrast huge compare summary state thus uncertain flat bandit large yield furthermore unable exploit state action exhibit scale expect return uncertainty scale recurrence cycle thompson completely flat force arm reward break cycle random
involve obtain density extreme surprisingly stein performance sample splitting difference I approximately microarray process accounting correlation effect size estimate extend correct bootstrap procedure purpose quite normally distribute future covariance additionally might explore connection correction discuss connection discovery provide fellowship grant dms solely author necessarily view manuscript extensive expect argument consider jensen b jk j left sign
state space comprise state assign value high process assignment denote want huge amount computing define separate module evaluate instead state denote concern future action propose actor carry deal huge likely problem turn actor algorithm add
study develop particularly since real like crucial comparison evaluate present taylor propose novel powerful covariate nan regression coefficient elegant thank simplicity comment issue condition
shift able leverage unsupervise selection robust mode wider evolve scale supplementary shift away calculation cluster cluster final converge right bandwidth isotropic respectively location plot correctly coherent mode water place smoothed fail detect mode conventional typical tendency segment boundary illustrative conventional level plot segment effectively local salient background exposition style kde px I np ic ik kde estimate set rise mm support satisfy regularity x normalize
protein odd make main collection protein present database contact graph available file refer graph contact consider atom connect protein attribute term attribute pattern literature aforementione physical label euclidean among characterization graph physical information regard protein protein remain note obvious ray physical analysis protein ds represent sequence thus subset ds consider effect viewpoint divide test obtain graph vertex explain real value vector aforementione denote graph algorithm ref accord matrix contain e contact edge atomic analyze eigenvector markovian walk however share eigenvector contact provide consistent protein principal walk far less split compatible ds protein ds ds sequence character usual dataset label describe protein vertex auxiliary ds direct ds g value hoc classification dissimilarity
score procedure budget differentially use score regularization combine parameter candidate parameter minimize denote h h l q r th entry generalize convex ensure perturbation differentially private perturbation noise noise noise procedure produce parameter estimate strong simple noise density xx need kb b
eq third adaboost specification span tree however structure natural tree maintain tree stand show chain bottom tree drastically interact adaboost mrf bp inference evaluate effectiveness hmm encode feature previous one differ original hmm extend partially state test whether improve datum level learn segmentation purpose macro average basis mrf vs limited newton suggest crf slow converge solution exact experiment forest cg per boost round meet inference stop message converge round final bp sensitive choice respectively appear stop converge adaboost mrf alternatives
future direction reliability cm work desirable important scalability combination expert independently learn expressive expert valid natural finally robust prediction theoretical combine obvious framework four I train
convnet great convnet compress kind remarkable large representation cnn compress marginally visually ranking compression pt dim dim storage I gb gb mb mb mb bin mb ms mb ms compression add encoding indicate gpu compression additional add time scenario exhibit compress hardware accelerate hamming feature work convnet cnn one amenable compression compression method ratio achieve instance test solely nonetheless representation retrieval dataset noisy across return image versus fix pool negative set result image datum scenario give pool query facilitate broad suggest diversity make convnet use correspond dense ii improve fisher encoding
positively neighbor scheme space make provide iteration eq positive determine adaptively alternative scheme maximum assumption survey fisher alternative score analytical naive gradient however name iterate derive artificial white noise component particle dynamic model ascent unstable carefully vector expectation popular applicable term argument characterize explicitly e path cost variance linearly computational use lag present path functional non vanish asymptotic improve asymptotic forward well filter smooth experimentally typically linearly result performance forward suggest admit mse dominate forward dominate understand mse sum particle forward confirm experimentally smooth limited path ml parameter accounting smoothing procedure approach applicable fast gradient prohibitive moreover run sequentially recursive variant ml justify ergodic ascent time increase upon ascent conditional new ascent except time form suitable evaluating time relie notation score filter use approximation fisher identity score property recursion e limit infinity study regularity algorithm assumption recursion possible originally propose space em
class replace bound rademacher say bind rank situation present run differentiable arbitrary subgradient property guarantee set batch arrive predictor side plug get optimistic terminology smoothness assume twice differentiable use express say usually
nmf role play nmf nonnegative create nonlinear norm matrix possibility must user nmf place rank singular svd two consequence naturally great factor nmf interpretation consequence processing document collection basis column nonnegative correspond term weight term assign document http www h familiar dataset nmf interpret instance heart basis similar factor topic sparse element strength document clearly nmf individual sparse I nmf nice interpretation individual structure svd create gain interpretability come perform equally reconstruct especially svd strength computation nmf unique nmf convex
decoder output style source source encoder pt encoder decoder transmission process square nx drop subscript distortion denote rate theorem encode mean infinity assume random independent minimize distortion store furthermore want possible bit communication incorporate variant coding encode r decoding call q write go define minimax base denote
minimize propose optimize order mn model complicate add gradient surrogate online see perceptron algorithms motivated exploit robust function surrogate cauchy confidence present presence function design traditional uncertainty guarantee classifier dataset conclude goal I many world consider corrupted classifier instead machine intuitive classifier decision risk df e fx minimizing
object voxel concern cardinality discretize coordinate grid reach point discretization ray discretization point represent concrete discretization correspond rotation angle rotation image discretize cube discretize rotation sense discretization value pixel way note discretized distortion phenomenon work element particular orthogonal checking assumption imply considerable coordinate ray sphere relationship hence span require subspace discretize sphere since permutation orthogonal employ grid transformation denote finite rx group coordinate view instance discuss power however impose strong grid graph behavior collect manifold build allow paper property fundamentally knowledge geometry familiar topic denote riemannian manifold embed bundle plane canonical ambient derivative laplace operator tangent connection spectrum l l eigen resp eigenfunction lf lf kf manifold inside f ambient lead indeed borel sigma absolutely volume associate abuse focus k case sense affinity connection mention decide dataset application context processing mahalanobis underlie state synchronization affinity
overlap exhibit variance systematically dispersion higher depict four result diagnostic plot color dark light symbol describe member class outli plot neither distinction influence correspond different majority space since outlier influence fit loading correctly identify outlier include four pilot laboratory production setting extract heterogeneous majority high ht depict spectra analysis valuable little fit presence many ht figure diagnostic scale enhance dark triangle examine use
affect change represent exposure people exposure change say reduction episode individual trait episode trait degree episode trait speak ratio causal exposure numerator denominator expectation however restriction difference expectation exposure paragraph matching subject additionally full match subject result match effect remain one match discard match variable control population discard discard parameter full incorporate individual effect individual fact control like conduct effect propose statistic statistic difference individual
sometimes form exhibit regret aggregation sequence possibly moment bound term bind term possibly remain bounded innovation slightly error I even quite work let start see recent extension assume predictor form large explain large page aggregation aggregation corollary moment aware introduce define denote denote interval older define extend differentiable rescaled sample locally stationary vary autoregressive observation represent unit variance say sample extend process time vary assume e see historical tend cope say sequence begin aggregation procedure situation result historical available allow infinitely past derive away vary condition ensure
factor model bayesian continue summary still open sufficient available statistic epidemic decrease abc material foundation grant ef work lee support office nf research utilize nsf grant grateful discussion datum lee consider biological environmental case stochastic candidate experience intractable dynamical suitable disease main hierarchical dynamical markov stochastic differential appear
adaptive use burn sampler benchmark iteration within show started fail adaptive part allow preliminary conclusion range general conclusion benchmark crucially start quite start maximum maximum pseudo start satisfactory yield variance start distant likelihood adaptive universal improve adaptive justified basic version
normalize vi consider arc six dataset model average method dimensional filtering ensure smc require smc neighborhood neighborhood intrinsic locate set vi smc illustrate superior compete robustness summary robust smc presence noise run propose tangent enable eliminate exhibit presence close smc appear dependence smc levels htb model ii appear contrary exhibit figure vi smc appear affinity spectral identify sufficient accuracy explanation outperform manifold computation neighborhood structure complexity scale datum affinity optimization principal eigenvector contribute operation perform fully since computation per neighborhood identify tangent entail calculation principal eigenvector ratio type manifold readily extra one smc ratio ambient vi unit sphere cost smc exceed ambient orthonormal unit sphere vi imaging methodology lie associate view orientation function nothing discretize pmf describe water pattern object image direction pmf image map map try modeling pixel segmentation modify similarity modification euclidean coordinate fit cubic cluster u randomly color cubic pass figure around spline cf region snr snr bar whose ten
mention pass state suggest camera pair static configuration ignore dimension parameter intrinsic validate simulated use basic experimental observe scene behave track multi tracking section strength space account use single advantageous amount newly object capability base two figure centre second translate along axis first around object locate axis distance model particle filter run particle ray move step ds pf ds pf ds pf estimator map map carlo result estimate truth cover particle notable dependent second space configuration camera similar one camera camera locate camera depth run algorithm approach cope linearity space limitation dealing depict propose propagate uncertainty move camera analyse distance camera camera angle successively update acquire initial solution
union vertex phenomenon fact understand fact optimization well return optimization continuous show easily f evaluate access oracle oracle oracle could construction prefer ise generally unless special vast traditional ascent approach statistic cover implement bottleneck quite np hard possible basic reconstruct
incoming activation unnormalized weight multiply w interpret model gradient compute derivative propagation plug numerator move continuous get value back propagate biased ignore fact bias relatively criterion hold usefulness choose serve importance sampling express dirac expansion normalize
propagation similar benchmark mrfs potential topology grid random edge present parameter field strength mixed interaction strength figure interaction bar interpret confidence interval fair otherwise parameter onto suggest large projection simple vertical horizontal chain randomly generate span cover use fix gibbs systematic scan variable maintain
estimating regression model influence early work misspecification extend notion linear predictor derive sparsity interpret even sparsity dimension maximize remainder organize background necessary divide agnostic inference result reader convenience j j submatrix row q order eigenvalue entry decomposition span gap pt eq column unique consider principal equivalent advance estimation require minimum well sense semidefinite equivalent row sparse assumption investigation gap principal correspond subspace sparsity basis assume
illustrate feasibility structured manner fully connect deep structured inference aim alternative encoding automatically determine auto aim explore efficacy scale image decomposition support science engineering innovation yu com interest connect graphical structured structured deep structured largely concept deal deep tractable structured structured connect field intermediate layer problem illustrate
assume small bound constant third select balance term order reasonable start small identity proof bind identity turn prove identity however stop time natural kt kt fs kt n kt kt obtain give state let definition index observe round period indexing round start sample hold fs v f furthermore proceed prove definition nonnegative notice stop stop kt turn hold eq calculation introduce hold appropriate last proof straightforward kt decay fast negligible handle lower bind introduce c keeping follow kullback sequence distribute independent distribute random clearly independent independent identically value even odd natural truncate parameter scale
far balance cross four work run default library svm model show obtain analogous outperform bad nine ten dataset support prevent worth even though svms build decision fold method outperform vice versa support claim different optimization computationally comparison svms perceptron h starting perceptron quite reasonable solution solution completely seem possible exploit already dimensional uci solved start random sample unit sphere hard valuable initialize approach last behave parameter control strength table score fit balanced h breast diabetes heart balanced result one superiority big interpretation error lagrange formula width part evaluation task chemical protein protein test compound ten ht fit
node preferable though propagation kernel proceed simply symmetric adjacency matrix kernel input green idea weighted transition obviously partially label graph marked accordingly learn involve graph attribute chemical annotate secondary measurement image inherently compose channel color way essentially similar advantageous per kernel attribute bin experiment normalize attribute set disadvantage ignore attribute graph database matrix metric initialization tw attribute combine attribute propagate attribute similarity attribute efficiently propagate continuous hash node associate attribute attribute graph edge challenge ensure compactly represent update matrix attribute gaussians center node share set calculate compactly spread attribute update derive attribute node attribute kernel initial equivalent correspond attribute distribution edge accord normalized transition associate attribute edge weight vector compare kernel ignore attribute vector exchangeable kernel reason associate space node exchangeability compact hash input performance accordingly blue green couple experiment iteration description
method often approximation perspective variational dynamical connect particle particle multiplier unnormalize approximation normalizing particle repeatedly iterate maximum variational free principle sophisticated mode coordinate practice maintain asymptotic importance converge combine advantage decrease kl correctness filter markov hmms hmms dependency filter history construct variable update markov select current set variable past product
primal decrease iteration line choice minimizer make parameter equally logarithmic decrease choice work cover theorem stop active motivated reach condition step yield eq q condition stop analogue problem community expect discrepancy always discrepancy satisfied solution resemble closely convergence separately proof strategy essentially monotonicity two evolution active elementary mutual upon let function identity imply denote characterize important monotonicity convergence arrive clearly
entry let fix model abuse result width yield state proof group similarity large size structured sparsity say zero encourage sparsity define penalty q element shorthand write norm group define structure constraint set form keep exposition work rest emphasize de emphasize almost need sake proving theorems literature know representation achieve objective relaxation constraint subsequently obtain observation number remark yield binary regularize group remain overlap group bind reduce ambient bind become combination structure regression structure know logistic group special efficient proximal recover elaborate detail bound sufficient ask interested constrain zero bind vector everything overlap logarithm number term pay price group recall penalty explore singleton sparsity lie group select
side probable correspond large quite monte carlo base dependent partial finite possible approach determine extensive law empirical density standard would theorem axiom claim theorem exercise proposition
value post literature propose correction usual controlling produce methodology computationally currently applicable correct invert kkt correct drawback interval restrict usually specific regularize estimator forward selection marginal screening follow marginal lasso orthogonal least primary model selection conditioning selection focus screen event limit marginal screening apply wide greedy
theorem show calibration without discrimination capability histogram method definition concentration transform histogram fraction positive calibrate probability notation define triangular obtain bin calibrate bin bins probability histogram calibration converge histogram proof proof state supplementary theorem show measure term histogram show base classifier measure definition classifier transform calibration calibration auc classifier auc third theorem bad due limitation theorem theorem calibration measure discrimination histogram classifier histogram mini
close request distribution skew trend toward balanced dataset could pa al pos aim aim pa pos corpus since label data pa process pa small step execute
yield superior learner base mean various scenario fairly dataset datum space difficulty generative method method apply try complex bp passive odd svm gmm naive scenario gmm naive bayes regression scenario naive regression gmm naive logistic gmm bayes svm gmm logistic regression naive scenario prominent combine batch illustrate comparative across multiple resource naive nature one present result winner b comparative unlike scenario employ arithmetic modify winner presence framework employ discriminative stream herein behaviour affect practical present scenario successful base criterion effort scenario stream address learn generate surveillance scenario
accuracy reduce kkt element vertice thresholded represent equivalently l l kkt block optimization precision fine thresholded admit statement thus induce component thresholded covariance resolution induce precision conclude equality labeling thresholded nest component partition thresholded nest prove give contain inside vertex precision l ij kn ij kp kk p lp stanford university rule discriminant naive bayes path
topic support addition hope gain deep understanding depend representation explanatory factor conventional nlp take word challenge one word representation semantic semantic index much representation deal semantic allocation lda word representation unfortunately quite train
measure draw positive drawing previously draw discrete useful cluster hdp hierarchical define group global concentration global vary control group control smoothing dp variable dp conditionally share extend multiple require hdp tf measure score query hdp topic document predefine e appropriate base hdp hdp base topic grow problem number topic reduce overfitte fix topic transform normal may glm linear capable relation covariate covariate generalise specify link relate response canonical link function family response dispersion dispersion generalise take q response choice binary trial canonical choice supervise extension topic learn control learnt act corpus regression coefficient response document generative document vocabulary topic proportion response draw ij iw ij response implement
request produce request active general request eq plug theorem agnostic theorem sometimes specifically compare maintain effectively replace replace state achieve algorithm mixture input budget request request produce er pf k j sign sign imply contain convergent subsequence j w kb b w kb sign I kb x x j surely sign x fx hx px gx px scenario recall vc get request constant since appropriate establish analogous axis align agnostic active discuss several specifically study thesis q b request q least return er constant request
mc receive nmf detector secondary set variate threshold true scatter set orthogonal diagonal scatter note estimate propose parameter detector employ distribute clutter sample clutter covariance shape towards zero gets tail nature estimate depict detector correspond curve remarkably huge desire especially clearly good maintain desire length well slightly would draw though performance underlie outlier recommend often heavy tailed shape whereas remain regularize secondary free clutter pd average detector set give study pd detector
fact transform algebraic link duality let tuple polynomial degree dimension polynomial polynomial ng nf ng express dx let let dx f f df consequence polynomial feature interpret decision kernel denote f analogue reproducing associate hilbert degree identify replace symbolic precede could also combine sec beyond
enable formulate formulate bayesian account available incorporate suitable mcmc asymptotically accord posterior turn approximate metropolis hasting admissible log computation inverse large image cf expansion adapt efficiently evaluate spectral effective small size assess representative construction cascade compound poisson cascade large pixel self construction outperform image value first patch pixel remainder main formalism framework underlying image result world benefit application analyze bound follow solution aim characterize call say belong small large process behave therefore fluctuation hausdorff dimension denote point take precise hausdorff increment wavelet formalism tailor orthonormal
suggest care correspond optimize first introduce component q moreover hessian x respectively since definite observe case convex propose gradient particularly suited introduce descent compute medium accuracy solution strategy problem close employ negative gradient definite iteration introduction scale choice describe respect norm induce definite positive step backtrack loop parameter fix parameter diagonal scaling projection backtrack fx go set accumulation sequence point iterate belong sequence admit assumption bound stepsize freedom choice exploit significantly practical main scaling stepsize behind stepsize approximate hessian objective minimization
processor keep vector processor continue standard condition asynchronous preserve convergence memory parallel coordinate descent randomization effective asynchronous decentralize communication failure optimization resource article dependent algorithmic synchronization tool continue discover ideally heterogeneity increase composite model map smooth big problem cope l obey certain say composite model whether fast yet work support european grant proof foundation grant schmidt support laboratory big review recent advance communication overview technique scalability survey parallel computation principle attain problem back area importance formulation dramatically last decade rise new successful vector machine wide process compressive sense medical imaging bioinformatic reason obvious
right core member university split optima division unsupervise learn overlap us division correct discuss elsewhere perform poisson community divide node contrast tailed degree network correct achieve overlap even emphasize analysis carry correct use interesting observe succeed value independent different parameter advance right learn known panel show move
van scheme group review computer education liu generative analysis online development social computer journal submodular maximization enable massive scale community conference tx framework online environment journal lee education plausible prediction bayesian computation membership journal research master university automatic coding act protocol international collaborative g support collaborative historical ed collapse journal multinomial mixture overlap community physical spatio stream journal rich scientific report pp mind university
sample expect perceptron recognize reward draw limit practical use exp probability explore previous algorithm advance adversary apply contextual propose estimate reward hypothesis separability reinforcement reward
k ci ic lie th row interpolation equivalently equation I hand c hold theorem sx subdifferential cost value assume w condition contradict conclude walk motion process brownian motion distribution bridge eq eq notice give fourth appear therefore q result say unfortunately opposite infinite come sub reasonably bernoulli cumulative cumulative sum zero nx absolute constant min max sum sub restrict last show minimum random attain
hash key sensitive along preprocesse call query hash uniformly satisfie note query transformation create let lsh transformation counter fact similarity failure lsh hash preprocesse sensitive explicit call without loss simplicity easily define concatenation qx q obtain thus suffice approximate neighbor transformation partially hash repository clarity
loading observe structured loading induce prior remove unnecessary factor load sparsity factor load iii neither gene disjoint may none iv possibly gene expression covariate unobserve systematically observation work induce jointly adapt loading zeros zero sparse dense use load dense mixture favorable gene number gene affect batch intractable enable possible subset sample search shrink zero markov mcmc expectation b gene entry response row generative loading select remain sim simulation ten dense loading factor component correspond dense vice simulation scenario residual simulation five method run set initialize warm final thresholded hoc tend recommend sim correct control
suppose si si si scheme obviously since verify numerical utilize establish imply compute compute partial take section corollary lemma g national foundation ed enyi institute mathematics email school institute technology national science ga email edu dr science foundation email nsf measure pair distance covariance correlation disadvantage compare fast distance computationally formula nice derive computing synthetic applicable much wide induction life aspect straightforwardly computational
express infinite moment follow lp orthonormal discrete variance admit jx representation direct fundamental random probability publish outline fact variance decomposition lp eq lp shape continuous derive scale interpret modify table numerical moment continuous first four moment quick marginal depict application normality tail lp orthonormal lp score obtain follow expansion moment decomposition k table list bivariate scientific question collect child aim chart help assess child normal comprehensive fisher child fisher properly tie recognize discrete tie surprising almost surely linear mid applicable discrete detailed investigation idea beyond scope elsewhere tool nonparametric
set imaging dimension vary million leverage space move towards high speed decay case exponent closely match leverage score set decay sharp decay present world would empirically usually decay law help theorem law decay prescribe algorithm orthonormal norm generation basis completion n kn k main km choose sort gaussian htb plot vertical correspond offer set row first equal perturbation avoid every
projection recover pose incoherent generate independently incoherence ensure mass spread fraction instance p pose serve warm exercise main bring analyze potential descent completion subset update iterate
sample practice make sgd solve expect network able fix exist observe rewrite follow bound bound
fdr c indicate procedure margin moderate proportion relative normal ise normal whereas increment set initial summarize figure similarly see incorporate testing improve dramatically voxel simulation lattice group mrfs simulation present multiple whereas procedure fdr automatically heterogeneity appropriately control indicate utilize dependency testing evident weak procedure slightly outperform globally procedure brain voxel median voxel goal identify voxel different rate nc nc procedure procedure distribution two cumulative procedure approximate three testing
recommendation mainly relation domain filtering relation comprehensive collaborative filtering information focus rich side information survey emphasize importance setting domain type auxiliary additional setting vertical focus study representative perspective setting interaction without overlap explicit overlap uncertain side tag two fm work recommendation scenario ii usually improve technique leverage auxiliary exploiting without difference iii goal recommendation efficiency aforementione survey research area recommendation worth exploration direction security
rewrite vector l stochastic angle block degree gradient add stochastic effect per difference quadratic recover directional curvature along stable technique descent rely gradient refer parameter expensive minibatch bias
consistent effectiveness ensemble far allow major classifier hand subspace reduce observed conjunction distance successfully three ensemble wide range base diversity technique mostly classifier robust variance subspace improve primarily decrease bias step technique ensemble employ subspace ensemble attribute important rsc investigate embed attribute promise preliminary rsc learner understand quick train ensemble accuracy comparable popular evaluate alternative classifier rsc rsc base nature rsc make ideal ensemble two ensemble tailor rsc classifier subspace rsc significantly ensemble bad demonstrate six subspace ensemble high attribute analyse source sphere rsc classifier rsc create classification
soon necessarily highlight exactly regard scenario ghz party numerically increase considerably increase reproduce quantum explore detailed dag input index parent equivalent demand q ac etc quantifie fulfil demand impose quadratic convex nevertheless minimization cast parameter illustration obtain projective identical inequality give correspondence inequality relaxation reproduce operational causal perspective independence locality quantum correlation causal model measurement quantum previously think consider scenario regard dependence finally quantify believe motivate importantly basic tool understand derive useful context randomness expansion possibility treatment characterization convex compatibility complex quantum support grant university research office w nf nf characterization verification support sake respectively norm causal model tool theoretically therein standard basis vector
sequence unstable bound fig box observe able unstable region highlight colored notice area correspond turn individual thus result salient motion similarly region sequence result global capture crowd motion robustness deal inconsistent subtle crowd synthetic potential region bridge note herein consistent interesting subtle motion discover employ similarity perform et sequence obvious region detect bottleneck able region addition
example likelihood approximate probability use iterative posterior bethe energy expansion rule general determine specify relationship model potential contain basis quantity scene labeling estimate objective vector interesting easily lead iterative log take hidden variable posterior variational inference belief nmf sequence simplicity ignore compute expect problem iteratively update step note step belief pass simple flexible descent iteration way consider algorithm unfold neural index node layer activation node tie different help fundamentally unfold allow course formulate derivative recursively sum intermediate derivative derivation give sigmoid obtain field markov mrfs level conventional unfold mrfs generalize change unfold mean field lead propagation deep architecture architecture power restrict mrfs mrfs high order factor easily mrfs create variable give formulation state
rule outcome target assign assign child c conjunction variable aggregate current split whether extract decision split tend happen top reduce extract metric rule rule popularity define incorrectly instance satisfy squared error value satisfy condition define frequency small preferred interpretable one accord combination pair rule tree include pair rule measure small error rule rule rule leave variable remove currently
eq hand bound cauchy schwarz inequality submatrix gram obtain remove column small eigenvalue investigate criterion write use correspond summation lower derive acceptance dictionary atom linear get low proof investigate summation eigenvalue distant dictionary lemma coherence dictionary coherence quadratic bound q q one hand second condition atom result acceptance criterion error approximate atom unit proof substitute term thank eigenvalue derive appendix conclude propose quadratic bind previously extend relevance onto subspace bind derive bound result
semi bandit online agent observe weight receive sum payoff close computationally combinatorial ucb like solve number tight factor tight choose subset ground item subject observe receive variant combinatorial combinatorial stochastic combinatorial practical application recommendation variant bandit access optimal cardinality set gap return suboptimal exist bandit variant call confidence bound chen recently contribution two bound significant improvement match factor consequence
important preserve linear block coordinate converge convergence also review choose choose q guarantee algorithm proximal operator ignore backtrack shift k I converge widely iterate another active pair decrease search algorithm start line paper algebra write notice observe union ensure finally coefficient asymptotic e note q u ni ty choose spam term regime error eq cccc report time step
var cast covariance available fundamental problem portfolio fmri study challenge grow natural ill medium approach obtain shrink specify autoregressive reduce view structural covariance attractive since provide covariance simulation show covariance proceed var integrate propose reduce fitting scenario var fit scenario ar var procedure reduce rank apply example concern stock return china derive reduced covariance estimator var modeling integrate reduce fitting var estimation apply dimensional latent latent independent replicate
rate difficulty control result problematic recursion simultaneously circular involve emphasize crucially keep penalty ensure would seek convexity strongly provide remove bind strongly loss strongly high put term extend case orthogonal genome streaming streaming regression average exploit theoretically rate try un exploit however add also streaming method goal competitive implement software experiment handle simulation create run linear instead tw c third method geometrically distance specifically random vector py w entry draw ccccc parameter regression linear time prediction measure aggregate realization slide window example addition online plot outperform correlate norm algorithm margin bad correlation rely strong perform difference particularly logistic incur achieve possible prediction recover optimal comparison treat streaming expect bring method around phenomenon fact desirable note term runtime fast run
size linear minimization option segment piecewise polynomial jump recursive recursion end provide parameterization calculate denote worst computation together complexity adapt present onto union subspace polynomial accelerate scale instead one ideally several update nearly natural maximal stopping course constant I estimate original modify impose continuity create binary program element change zero serve program continuity otherwise token thm corollary proposition thm thm cs il representation methodology variational strategy
estimator distribution perform enable divergence inference pair distribution experimentally theoretical use achievable nonparametric divergence nonparametric divergence consistency already mean divergence field generalize leibl enyi divergence rate probability distributional application information compression channel code mutual machine processing clustering entropy special distribution intrinsic estimation however beyond inference divergence detection hypothesis divergence e specify divergence establish
video forest choose million pixel require take tree train comparison forest forest test show accuracy forest huge cause high consuming maintain image forest colour f precision forest threshold quantitative analyse acceptable nonetheless able cope large complex tp noticed processing region return segmentation false face classification process region classify
tag provide estimate tag vary impulse third set quantile performance around evaluate accuracy sparse code matrix loss encode code example code dictionary dictionary negative typically solve fashion code update turn update fully frobenius may update perfectly loss penalty across consideration problem scheme requirement constraint specifically quadratic penalty recently broad statistical interpretation devise interior incorporate method code update make particularly model representation penalty discuss update scheme quadratic support reader exposition arbitrary nonempty matrix
environment simulation deviation guide provably construct prove impose guarantee principle sure respect denote action heuristic level deterministic path approximately reader manuscript result mention strong converge differential denote absolutely continuous strong deviation principle clear coefficient obtain recognize come deviation imply simplify qx bound almost unbiased via monte generate copy sample precise copy desire efficient jensen deviation q therefore actually opposite almost call importance notational convenience define wiener control sufficiently define
subsequent fully net remove interested identical allow big mnist version feature size initial decay momentum dropout reach fix comparison train class representation comparable artificial augmentation non augment conjecture
see article accurately relation word iii vary rating positive datum user examine recommendation movie show iii true movie unchanged new movie english american belong movie change get phenomenon observe number increase enough rule complexity representation user dimensionality computation update bias layer thus total matlab gpu acceleration second epoch dataset second epoch satisfactory large show scalable change pure significantly art jointly perform learn collaborative far first bridge state rs generalize propagation bag word representation alternative bayesian nature performance boost incorporate admit
consistent code word fairly reasoning therefore similarity number report kind insensitive sensitive sensitive insensitive publicly available embedding provide initialize denote good refer skip gram relation explain frequent initialization great job frequent bad showing skip gram greatly reasonable improve quality embedding context information note performance little embedding besides rare initialization word embedding embedding train recursive structure train less minute since update relation balance hierarchical knowledge rnn knowledge type actually knowledge basis leave word skip knowledge skip gram combination uninformative problematic inaccurate competitive noisy sample rare word embed embed cosine similarity five investigation accord combination task therefore skip combination skip gram relation besides four type process denote overview knowledge look actually job
identify extract salient demonstrate technique review scalable automatic extraction compare reference document create extract much preserve extraction convnet extraction extraction model acknowledgment would thank nlp early rgb united research present document computer vision extract scalable sentence avoid consume human symbolic researcher decade recent
effect mild learn outperform routine value reach reliably benchmark baseline analyze network insight challenge establish post becoming win correct development interpretable bayesian interaction interesting follow architecture domain uncertainty ideally offer balance real process benefit frequentist emphasis stay characterization conventional work experiment elaborate stationarity yield acknowledgement thank provide time google award amazon sciences institute advance
xu value lead conclusion additionally plot xu xx observe xu go generate xu xu relevant xu xu relevant compute xu xu xu xu hypothesis force x step collection suppose compute x x x I interpretation claim interpretation claim c claim unbiased report xu variability claim bias nature bias conservative compare deal support testing adjust numerically support multiple property microarray give gene white cell measurement basic h cccc cccc gene vs u w goal
center first center new analytic hereafter denote z z z global definite attempt close onto configuration fourth multiple argument global minimum challenging involve existence close local sufficient nonnegative definite encounter learn set empirical pc section c collect build rule classification train predict new select dimension sample whether discard considered selection space greatest z contain index group r modify statistic multivariate population statistic separable section choose principal inversion cause numerical exploit variance discard covariance accordingly permutation usefulness choose pc feature translate hypothesis test alternative calculate
parameter method mcmc source factor separately beta iteration report fm improvement fm dataset netflix dimension expressive rmse demonstrate fm topic use fm baseline experimental publicly implementation posterior training skip train netflix rmse baseline fm addition
split part surrogate f approximation moreover convex strongly also proximal gradient minimum often form f proximal proximal soft review proximal operator reader g appear dealing logarithm amount reweighte reference adapt logarithm replace consider real value function indeed convex note lead alternate proposition assumption surrogate useful instance regression huber huber smoothed loss represent associate problem formulate minimization x linear describe begin satisfied reweighte least huber inequality presentation smooth l l rate surrogate believe present jensen nevertheless instance procedure concept algorithm exploit concavity logarithm jensen
image scan disease assumption assumption positive novel standard assumption example instance belong suitable might problem describe bag classify anomalous advantageous classify cell classify decision influence generating application label face detector instance reasonable person oppose patch group bag recognition example image frame person image group annotation sense bag segment bag label belong background object song belong specie annotation annotate segment label costly weakly annotate foreground present bag fraction instance information output classifier get label name another spatially likely interest medical weakly annotate benefit bag patch
element wise restriction set simultaneously need adaptive measurement group restriction boolean gain cs obtaining therefore snr attempt evenly achieve least sublinear bit interesting measurement open question whether performance similar nsf corollary bind complexity sequentially low mutual information
corrupt conditionally conditionally mse attain limit ml noisy observation corrupt additive fc use estimator estimate sensor reach limit expect estimator therefore threshold htb relax conditionally observation conditionally sensor convenience fc observation derive optimality chen simplify system fc conditionally introduce variable chain hold conditionally independent n equivalence inference optimality optimality section derive manner pdf random positive early
generate feature episode twice represent link hand later extraction structure hierarchical preserve typically produce pool realize training aim linear risk parameterize weight loss follow convex order employ several ordinal loss negative likelihood outcome model lasso sparsity come instability theoretical intuition knowledge since independent clinical realize relation disease link ensure serve precision multivariate present transform temporal ed piece diagnostic exposition diagnosis version scheme applicable disease cover code letter digit digits head classify head medical contain represent clinical often episode episode visit end death health major contain event diagnosis admit home could come intervention assessment may list multiple problem historical transform sparse extraction technique precise instead exploit bank filter resemble filter sparse maximum history observation event discrete event diagnosis code duration parameterize th event convolution effect
correspond monotonicity positive value threshold put serve dag handle assign variable convention positive row argue monotonicity rather negative sample reflect monotonicity constraint strategy monotonicity penalize sample row large exponent specifically likelihood proportion row course include regularization combine define brevity asymptotically reconstruct correct correctly score monotonicity weight structure infinity play role overall stable choose develop score asymptotically especially enforce absence edge monotonic convergence size temporal see definition optimize
ordering count validate machine implementation count heuristic available package record count expensive none prohibitive apply heuristic give quantify output quantify single false might correct heuristic feature aspect express variable quite choose feature affect heuristic feature consider work h degree degree among degree occur polynomial proportion polynomial occurring occur place label heuristic could input defined polynomial feature feature across validation test section svms dimensional sigmoid
current psd practice project psd return mt task network common space mt treat follow enable transfer mt regularize learn qx rewrite incorporate constraint q ts ls q project psd q qx qx variable solve respect thing calculate partial subgradient metric update use triplet project psd hold regard dimensionality real world datum apply mt citation mt obtain article area wikipedia search also article article solely
em regard determined equation iterative mi tm substitute equation value sequence q limit since independent maximal correspond coefficient condition choose unique estimate close note solution penalty good initial quick super em tx procedure theoretical regularize large mx condition mild trivially standard chi trivially refer covariate consequence consistency condition j tend eq chi follow immediately chi recover mild oracle hold minimizer satisfy let tending theorem determine choose minimal adapt stability ss
hence fourth operation expensive decomposition typical identify dominate part right side approach exponential complexity assumption study uniqueness problem column affine independence factorization iff iff e affine figure uniqueness interpretation valid fail hold replace contain sequel play role solve extension negative factorization world factorization ask adapt solve column account negativity impose however return converse factorization principle feasible upper proposition bad vertex contain uniqueness uniqueness recently
provide first step understand brain network relaxed covariance independent normal whiten step preprocesse develop spatio possible instance study subject incorporate distance voxel assignment point choice voxel future research latter omit ignore irrelevant solve derivation equal derivative glasso like acknowledge national grant aa ai ns university research award institute brain pilot award university start em university american international publicly available brain attract fmri tool recover
tensor compute furthermore determine approximation np challenge rank become multilinear tucker unfold tensor multilinear rank completion multilinear multilinear problem open question class approach pass minimization though empirically achievable exist need substantially achievable formulation measurement focus find message pass need study tensor completion specific type oppose include specify show class pose state compute nonnegative pose study define risk function risk return question incoherence structure loss amenable loss partly justify interest approximation tensor compute loss approximation discuss soft attempt limitation move nonconvex combine hard soft issue design fractional factorial distinction incoherence differ factorial design write contingency combinatorial note combinatorial write many
bandit bernoulli bandit simulation allow illustrated robustness misspecification scale exploit replicate replicate regard tuning apply concentrated dynamically arm evolve understanding replicate favor work develop analytical acknowledgement work member facebook core team anonymous reference thompson bandit allocate arm thompson demand scale bandit dependent
minibatch minibatch method natural describe natural implement change convergence descent suggest improve rate poorly condition problem suggest method define outer job disk produce store iteration weight try work training datum number speed bfgs fisher average like weight stop aim optimize assume layer case would less optimize model average parameter duration find improve gaussians recognition use mixture gmm idea neural speech write mix dimension softmax number layer sum index class try class class group class evenly count class row old matrix plus term value modify normalize slightly result may truly effect rate describe note far conduct wu mixture regard mixture class multiple able improve result remove scale softmax proportional count average mention normalize zero mean affine transform accumulate multi discriminant class dimension fortunately lda actually space covariance desirable direction never type mention transform covariance singular decomposition singular motivation rarely encounter lead large transform rarely decide well establish improve get give improvement
monotone bound bound proximal possible difference prove solve low candidate exist solver broad nonconvex solver surrogate additional verify many surrogate table logarithm mcp laplace scad penalty bx bx gx kx identify give satisfy lie useful solver lie intersection bx supplementary intersection b satisfy xx bb denote
sparse component pursuit mention mild via involve nuclear norm recent many derive provable form proximal involve singular bilinear structure rbf low small trace error measurement robust robust account show call plus noisy component incomplete corrupt measurement calculate rbf scalable structure orthogonality convert scale problem linear constraint direction method solve linearization analyze remainder review background propose scalable develop efficient highly corrupt model denote subspace completion index result suppose incoherent r sign dimension haar measure probability measurement recover develop solve
synthetic dataset web collect amazon movie netflix diverse transaction click check etc ref ref predict preference behavior user pairwise build preference information novel rank generative comparison account essence approach ranking especially individual preference influence preference similar item iv comparison typically rank date aspect literature category optimally agree ref ref ref user preference category ranking population single ranking ranking ref ref heterogeneous preference preference behavior mixed membership capture multiple share ranking inconsistent preference mixture paper development efficiently consistently ranking topic corpus view probabilistic leverage recently topic modeling estimate approach run
artificial neuron reach human sense computer truly master visual e child consider core parallelization unfortunately despite year computer deep neural additionally fast ask high decade hardware evolve propose conditional deep positively correspondingly net
operator project one call amount compute perturbation state branching level insight proposition remark assumption infinite discount formalize markov variation policy conservative infinite recently policy per iteration error comparison particular attention highlight cost increase enjoy guarantee iteration contrary constant iteration problematic discount scheme confirm infinite decision mdp bound discount
extend dimensional extension dimensional chain integration dependent general formula explicitly q last sum denote set close eq I last unique point corner normal exist assumption b furthermore product therefore follow r dx last equality dx dx note plug standard ratio multiplying divide inside divide inside integral also function h h additional vanish old q therefore h know q draw assumption require perturbation eq r second
comment short test treat regard sdp methodology reconstruct incomplete noisy sdp lack bound take completely approach heavily distance semidefinite sdp lead convex nice importantly derive result social show treat work regard approach distance follow proof plan rank note j meanwhile since ta ta know directional jj j j thus jj conclude pt know desire show hold shall least completion r bad event interested shall x meanwhile rademacher know thus p finally exist apply bernstein bind old q x r proposition nm nm eq case know exist ex l thus random sub ex l ex l nc term dominate proof diag p continuously value
one want find marginalization direction encoder frobenius e generality regression minimize problem variance incorporate recover encoder replace x n unit supplementary material treatment achieved provide intuition equal join take choose poor trade prefer depend choice hidden solution highlight marginalization together capture principal axis non regularization cross optimal regularization hold trial record neuron form dimension marginalization hold repeat different train result clear yielded value argue axis encode decode axis try real material assigning variance argue toy axis stimulus direction vary correspondingly marginalization inside average arguably utility prefer projection full decoder therefore encoder essential equal figure marginalization matrix marginalization direction decompose stimulus decision bar plot figure decode necessarily variance work memory well average neuron explain simply axis explain decode axis stack row matrix standard sum eigenvalue covariance sum eigenvalue figure correction signal trial independent neuron random signal follow text assume variance therefore figure marginalization compute compute marginalization total angle star pair axis orthogonal sphere dot angle deviation quantify contribute activity
positive cone semidefinite cone simultaneously widely sparse signal variable group convex nuclear estimate lasso penalty encourage ensures interpret low aa throughout thing different statistical rather generalized especially sample take advantage see problem organization
perform via system implementation strongly assumption however physical addition ghz processors gb ram queue execution fix carry begin aim worker precisely worker perform computation complementary estimate kernel parameter dimension choice capability assess synthetic let mean consider addition range processor note experimental practical pass simulated dataset contain remain distribute acquire computing uniform bar
color correspond blue indicate green red triangle make individual individual use snp marker come train abc reference table subsample rate na I baye discriminant lda nn initial axis initial summary summarie axis axis neighbor initial summary weight neighbor regression neighbor implement I classifier number neighbors abc estimate minimize calibration error moderate I neighbor logistic regression minimize minimize use due calibration set heuristic summary use axis normalization use provide prior rate reference calibration table summary lda axes population albeit local bring expect optimal neighbor need quite large aspect local consume stress solely indicate forest calibration constitute I lda abc initial summary axis lda axes forest use summary summarie axis I bayes discriminant lda abc summarie lda axes local two axis random initial summary forest summary lda axis I discriminant standard lda lda axes initial summary summarie axis classification size reference solution average summary statistic lda axis contribute population meaningful discriminate important variable
often advance initial readily modify version dna fashion drawback error often propagate fine often employ extension network variant build stack rbms time top layer networks crf rao conditional random cubic standard inside algorithm quality respect early stage promise
se sp accuracy roughly specificity consequence misclassification kind make adopt science notation replace false positive false negative equally classify reverse formulation follow problem tractable false false negative use reason svm point section class feature traditional perform whereas measurement suffer would preferable new use fewer suitable combine feature preliminary version report training sample size repeat number choose formulation randomize vector threshold average zero originally determine use time assess classifier testing remainder performing value retain wherein mean statistically cpu comment comparable set half total application testing nonzero weight run another instead average run large adopt rank time randomized retain experience index retain iteration choose randomize right step training testing final specificity advantage many svm eliminate algorithm
weight defer carry believe situation evaluate compatibility factor evaluate restrict constant focus provide compatibility factor even correlated design belong set jump drastically result tv estimator ever lead importantly much piecewise vanish one bind first consider jump moreover work precede section fast bound two turn present suggest fast incorporate refine interest deduce minimax monotone example follow slow tuning lasso correlate classical logarithmic theorem rate set close span constant satisfie q euclidean span covariate fast effective number refine replace effective number correlation exhibit perfectly design belong span rate corollary differ fourth introduction finding comparison dependence corollary application
iii iv vi analyse vi conclude remark vector nf integer minimize convex positive definite denote fx policy within define policy usual asymptotically besides continuity lead characteristic tail convergent small consider contain origin exists control policy infinite exist admissible satisfie control bellman however bellman optimal curse look table neural term within approximated denote interest envelope valid state trajectory remain vi find vi initial guess iterate guess converge monotonically utilize converge reconstruct rarely problem parametric purpose function
kolmogorov cdf decrease practical reason rank weight calculate statistic monte carlo discretization observation expression sample level real low next point give gene inside observe vector test gene set sum return procedure k step function jump supremum test rely gene compute nan goodness fit theoretical cdf cdf
fact k correspond slide simplex condition course map k category tie sample mapping mapping crowd simple exact item favor expert vs crowd opinion entire scenario change g vector item category computation give assignment simplex basic affinity remain unlabeled satisfie harmonic unlabele respectively u linear affinity take train free eq clear laplacian laplacian effective I structure essentially set category follow principle harmonic lie maximum label lie constant thus assignment nonnegative sum predict assignment subject explicitly simplex simplify item belong category implement coding widely instead unnormalized latter variation square free intend extend replace divergence add term exactly meet play role propagate smooth item rely give
score preprocesse phrase provide author fold ten fold recently publish stanford sentiment include phrase sentence sentence sentiment label convert ordinal use structural phrase label partial sentence experiment employ ordinal multiclass setting nonlinearity softmax ordinal corpus experiments accuracy experiment randomly initialize word rand dimensional google b word reduce additionally initialize rand words acc word nb recurrent report bottom rnn
extract descriptor information response neighbor entire depend standard save hold dataset split aim adapt nn base cnn subset training error output cnn improve give algorithm accommodate centroid regular centroid solving solve test rate descriptor detail compare test training large large training curve intermediate compression ratio reduce nearly match nn marginally ratio superior baseline subsample cnn notable final
l noise omit clarity effectiveness cifar mnist image handwritten training experiment implement model multiplicative incoming weight mini rate factor epoch accelerate linearly momentum epoch reach performance result low validation train final good validation model random initialization outperform maxout achieve good
bfgs standardized estimate behaviour first correct regularizer bias structure meaningful rarely confident great research well calibrate equation collapse estimate show quantitative gray behaviour error relative estimate cg bfgs instantaneous linear leave projection px mb element q show blue stationary model figure stationary drastically figure show bfgs standardize prior outer albeit loose one course cg probabilistic interpretation exposition cg converge identify converge intuition one bfgs never explore block arbitrary elaborate course choose interesting way inversion like gauss direction text interpretation iterative solver derive posterior mean rank conjugate bfgs update rule cg apparent inference perspective bfgs well scale standardized sr lead correction form cg consistent definite cone bfgs cg possible rule
integer element intensity parameter density poisson likelihood evaluate give factorize close bm denote value marginal k observe define comprise initial treat mle nmf formulation want nonnegative one kl write q similarity generalise
low low excess differentially private strongly decomposable half decomposable minimizer terminology excess differentially eq give low every whose q minimizer case extra first universe choose entry change tight tight bound desire construction differentially private whose construction prove isotropic position reader deal general necessarily isotropic run distribution hypercube hypercube lipschitz sample property let j next walk reversible stationary walk fu mix walk distribution output step statement cell whose guarantee chain walk towards rapid chain space reversible markov set p observe eq lipschitz plug p completes bound output distribution close define standard trick weight attribute outside namely extension define function cube guarantee p position first namely variant
dictionary achieve fr truncate nn le de truncate le english en fr translation produce neural network system handle word highlight token novel achieve conceptually read translation appeal domain knowledge suit formulate network generalize phrase sentence store explicit phrase table conventional finally decoder unlike base despite advantage rare word vocabulary force vocabulary word sentence translate poorly sentence frequent word phrase base
dot represent series term dot represent dot pt repeat necessary diagram expansion suppose hamiltonian dominate deviation sufficiently series calibration response uncertainty show concept ref include uncertainty amount weak uncertainty reason clarity uncertainty ref taylor expand effective hamiltonian form strength contribution call new become justified correction diagram diagram identify pseudo accumulate uncertainty correction value e emphasize unity justification expansion sec small formal definition eq whereby formulate
bic n minimum cm white bic style mm cm thick white circle sep draw thick color bic black bic circle sep mm fill style sep white fill sep cm white fill color bic circle inner mm cm thick white color black style circle sep draw fill color black circle white style inner mm size draw thick fill text black bic style circle minimum cm thick text minimum cm thick black v v n v v v v v v v v v v v v v v bic bic bic bic bic bic bic bic bic v bic bic v v v v v v v v v v v v v v v v true include never choose htp n n v v v v v v v v v scale bic bic bic bic n bic bic n bic bic bic bic bic bic v v v v v v v v v v v v v v v v leave bic correspond high selection node never without htp v v v v v v v v v v v v bic bic
pre inspire predict automatically filter entity type object w evaluation table metric drop word modeling entity compositional phrase entity name promising greatly benefit train level relational scoring relational embed framework investigate bilinear also entity extra several interesting finding enable
lie lie strictly length least contradiction recursively construct definition real cube compact cube contain case subset assume diameter length recursion set vc existence construct property recursion say nice satisfie even odd addition every origin leave semi infinite semi final technical bring set help
consist define minimizer focus transformation sequence transformation representative algorithm consist minimization local whiten global structure view trade interested soon formally serve maximize introduce replace otherwise computation preserve iterative think modification include additive force cifar consist color partition contain comprise dataset
cl al deviation justify cl finitely elaborate cl considerably explain insight improve would true tree cl replace mutual estimate namely naturally small identifying however bad mutual information mutual theorem highly regime minimax worse require essential optimality latter cl intuition fix star independent give independent distribute entry probability normalization set overlap wrong edge set
select initial constant achieve early study fix boltzmann base axiom arm mean pick softmax select boltzmann mean randomness boltzmann act infinity pick uniformly decrease fix pursuit explain essentially pursuit maintain policy arm inform use version pursuit algorithm start arm actor problem pac form reinforcement pursuit maintain directly reward select increase decrease scheme design account case similar value maintain preference boltzmann play reward preference turn exist date family simpler elegant ucb planning prove go play program simplest maintain arm play pick fisher ucb bound ucb achieve multi armed tuned perform come without guarantee ucb variance arm pick maintain mean provide regret ucb ucb tune instance characterize aspect distribution affect relative performance surprisingly consider importance compare goal setup characteristic bandit affect arm type arm admit tune optimally setup learn curve
form structured selector dual suitable ds key aspect term original recently atomic estimation framework unlike consider norm atomic norm aspect norm selector primal homotopy linear immediately extend formulation alternate direction multiplier problem linearize prove ds inexact admm primal interestingly turn proximal update conjugate indicator decomposition suffice side interestingly set trivial operator efficiently support focus proximal setting provide error yield
analyze combinatorial sequentially construct count time new row previously unseen feature similarity argue flexibility ability count make suit wide variety world framework bayes random exist require predefine vocabulary share category beta binomial show outperform categorization need count arise text document term record many appear site record observe arise count relatively moreover major conceptual count row add sequentially row previously unseen feature word specie require count obvious row count unseen bayes classify predictive account feature ignore previously unseen issue investigate prior construct poisson gamma binomial lead count matrix time count exchangeable underlie arrive count take highlight certain evident rely novel
involve dimension cutoff subsample sensitive additionally possible merge easy user interpret cluster result potentially help overlap truly homogeneous cluster subsample sequential generally accelerate advanced method inferior convex cluster mechanism center iterative subsampling need number well noisy datum simplicity intuition magnitude present appendix n ari true estimated index respective indicate true respectively contingency ari base count c cccc sum use row row sum effect misclassification identify misclassifie noise estimate table count account number datum calculate year fellowship part nsf grant dms efficiency spc spc regularization distance cluster center capability recognize solution subsample order include mechanism ultimately tight cluster simulation ability handle dataset application gene class large sophisticated art
heterogeneous employ sparsity system minimize tr sn net aware introduce proportional heterogeneous region choose rest I network signal spatially input keep
inaccurate sample define event require additional since selection fs genetic sample genetic moreover
university usa group unique character certain people service become year study implement graph weight edge graph develop analyze characterize present dirichlet allocation lda lda predict generate representative group determine topic content author distribution topic preference website gibbs
nd x follow next lemma proof switch notation satisfied assume iii get calculus put contour picture g spectrum therefore order singular
distribution point nearby achieve parent marginalization tree tree label categorical normalize stable process parameter take discount variation prior ty p e consider modeling hence resort fast popular smoothing approximation straightforward instance refer reader detail use specialized general operation new split split update split exist child knowledge decision third unique tree follow walk input x j j proportional split leaf j stop leaf continue discuss version partition figure step h second third iteration partition though split gray rectangle new new lie extent
neuron item depend neuron let typical neuron neuron brain neighboring neuron fire action potential enough cause various neuron precede item strength neuron income time plausible firing fire threshold huge firing threshold small assume long consecutive step naturally comprise simultaneously fire direct neuron every discrete boolean whether fire another memory finitely keep consist another understand operation state influence potential fire status update q potential component neuron incoming notice function happen fire certain plausible description algorithm instead present state shall strength income next join operation perform exposition join operation operation basis least desire call fire nan state strength income total come fire strength strength incoming enter incoming
short backtracking variable short remove short negative even birth track integrate short confidence track style default vertex default default vertex default style vertex default style vertex vertex vertex default default style default default edge forward edge style edge draw forward style edge style forward forward style forward style forward style forward draw style blue forward blue draw blue forward draw edge red blue forward blue forward forward edge forward forward draw draw blue blue birth death flow red edge node pass successive difference dynamic programming shortest backward include pass dp array pass backward last frame frame original backward come one pass forward cyclic forward edge node short along variable iteration pass dp dp behave path pass go go track splitting track choose entirely track terminate flow quadratic eqn
nonnegative I directly neighbor comparison show without formally consider problem boundary outlier amenable theoretical analysis simplicity practice regression widely modification alternative regressor benefit include weight remain unchanged variance view much condition next interpretation prediction problem predictor predictor prediction preferable local develop locally conditional estimate cubic query consideration near neighbor recursively gain within predictive cart develop definition reduction heuristic split algorithm average split axis align split cart construct mode splitting leaf node intermediate align thorough base represent disjoint region correspond directly average point propose follow weight piecewise partition recommend decision constant problem recursive fully decision truly also cart interpretability interpretable tree interaction dataset simplest admit problem forest ensemble train component subsample forest one flexible tool ml thorough random extract forest propose weight combine idea aggregate cf forest weight extract tree ensemble practice rf flexible prediction algorithm perform predictive rf overall evidence rf efficacy absolute efficacy bias favor overfitte disjoint estimate black relative quantify overall universal scale determination absolute
tie tree propose locality allow multiple exponential quantitative set helpful left distribute preserve hierarchy activation ability subtree subtree induce tree feature
convolution xu wavelet section applicable hence general spatial rank determine throughout discuss n n easy var explicitly location apply decentralized filter posterior server server j central illustrate server central server move server computational moving plus central communication scalable massive matrix achieve assume compactly compactly parent case correlation j server treat contain know usually implement fashion extend vector dimension three parameter
evaluate perfect distance family appeal exist divergence chernoff divergence tight estimate come information theoretic bound kl divergence tv divergence inequality bind tv drawback inequality uninformative kl go tv refinement chernoff value minimize bc case bc motivation literature close form many chernoff measure bc beyond number exist author yield arbitrarily tight set bound empirically know bayes area david label new subset generalize author
k put iteration fix say clearly k x run b time fact well speaking possess dimensional tell solve influence good briefly discuss constitute suggest time find row partition less c taken solve leave heuristic ensure direction follow understanding say suggest heuristic idea dominant section example indeed parallel computing imply processor secondary device system computer equip storage device objective necessary additional resource solve usual row consideration readily preprocesse divide roughly equal feed parallel iteration main involve phase read memory modification already observe first phase multiplication operation concern across processor give processor secondary storage stream idea suggest parallelization secondary storage perhaps require hardware understand
achieve well denoise amp amp wavelet amp amp haar thresholding min haar choose well signal amp patch length search window parameter setting choose allow level amp original though domain capture effectively amp amp reconstruction algorithm amp bm bm amp filter omit result use illustrate correction considerably compressed test measurement matlab ghz processor wavelet optimally amp describe iteration amp run amp run iteration amp yield improvement code fail initial noiseless test iteration bm amp estimate final amp six processing house image present figure rescale restrict entire matrix store create amp store version signal begin recovery amp bm bm amp outperform vs dramatically outperform wavelet amp also comparison amp rmse bm clear amp bm amp majority denoise amp compete presence amp bm amp amp bm amp bm amp cs bm amp bm amp bm c amp bm amp bm amp amp cs amp amp bm amp c c cs amp amp bm amp realistic setting subject measurement sampling
show completion capable entry provide theoretical ad calibration increase incorporate property characteristic improve robustness accuracy extraction hoc organized pairwise distance coherence mathematical euclidean dedicate theoretical guarantee hoc array completion relate draw locate position circular room signal room locate room near base express distance represent sensor observe characteristic miss imply locality miss lead short distance miss underlie acoustic furthermore locate whose position euclidean transpose construct write hadamard square locate plane place circle rank exactly hence dependency low property pairwise distance introduce square distance kind miss distance structure miss noiseless recognize know square matrix noise matrix random
condition al orientation stand inverse inverse et al lda require cost release overhead right hand rank orientation present year reduction mining et et et
token member assume make restriction latent intersection label simplification model variable set disjoint restriction simplification length example shown illustrate labeling sum latent formulation crf crf see special employ calculate summing feature represent represent local consist parameter perform optimize objective term training reduce define analysis base
hypothesis parameter problem arise association set assume normality definite interest section restriction matrix shall composite density nan chi freedom coincide classical contiguous alternative n non central chi centrality contiguous centrality however contiguous htbp significance l examine test result nan statistic influence influence unbounde imply robustness corresponding statistic plot figure extend influence contamination point decrease increase test robustness type
method genomic bioinformatic brain kind neuron overall neuron differ chemical also type connectivity prominent processing movement make circuit challenge use e discovery aspect early property scale present understanding work repeat node represent repeat topology well commonly probabilistic connection rich cell neuron importance connection traditional arise genetic address challenge describe nonparametric automatically pattern location incorporate additional agree identification cell recently additionally compare discover human agreement future build probabilistic begin unobserved type cell connection cell nearby cell broad generative connect
generally something single science point general lie situation information exposure pre treatment situation unobserve exposure finally variable conclude simple information randomize study ann exposure potential individual randomization far suppose
quantity onto equivalently norm leave vector input outlier provide full truncate score well algorithmic date semidefinite interpolation rank leverage interpolation maximal ridge coherence also sum capacity provide notion statistical counterpart leverage well ready guarantee nystr sufficient number sample loss come sensitive ridge column construction find nystr om column kk matrix om induce provide scale effective
prove recognition acoustic dimension conventional dictionary dictionary new training svm classifier utilize voting class kernel version rbf bandwidth select via method exploit heterogeneous show modality employ main coefficient use sum coefficient vector final occur generalize aforementioned combination di category feature fusion fusion signal although sensor combine information sub optimality exist observation decision result misclassification aforementione verify test sensor different purpose nine sensor acoustic sensor combination process nine conduct six notice two acoustic sensor corrupt sensor sensor acoustic sensor utilize acoustic effectiveness sensor clean acoustic sensor nine sensor segment extract overlap segment dimension interference process sensor set coefficient minimal define six sensor validate efficiency method similar take accuracy single sensor set iv nine sensor interesting sensor three classification human h
obtain purely illustrate life worker collect additional care precision define integrate nest motivate present concluding remark mixed extend predictor consider response group distribute let known link ij ij unknown n q distribution say flat assumed parameter derive specification decide upon range mix interpretable
maximizer give start iteration facilitate chain run one could easily devise elaborate optimal rule open guess start use component final ingredient procedure quantification correction estimate intensity quantification empirical quantification straightforward example credible interpretation drive interval take regard frequentist property understand amount I bayes interval use bootstrap technique aim achieve sample standard arguably desirable actual physical quantifying technique frequentist approach result observe poisson regard one different resample empirical spline coefficient irrespective hyperparameter spline intensity result bootstrap procedure illustrate interval resample basic percentile interval clear superiority scheme usually difference basic interval percentile follow enable bootstrap probe large percentile poorly interval implicitly scheme bias conceptually percentile sufficient computational resource
cluster information exploit dynamically outside able high able successfully motivate examine already disjoint decompose separate seek combine denote combination random agent intermediate iterate iterate I k indexing estimate agent later small enough gaussian semi agent example difference matrix result infer indeed concentrate sufficiently agent hold probability observation agent determine member I I threshold agent test reach miss detection exponential therefore long agent successfully infer acquired dynamically adjust iteration accept hypothesis dynamically evolve neighborhood introduce diffusion iteration causality neighbor already summarize recursion update recursion key arrive theorem derive useful intermediate first recursion shall therefore examine evolution recursion influence study introduce error k use minimizer throughout network follow dt rewrite worth indexing definition combination possess diagonal
adopt recursively subset partition ask dyadic adaptive construct next question na dyadic dyadic policy simplify dyadic dyadic policy partition recall dyadic question f distribute take dyadic policy accord expect beneficial value dyadic actual term second converge martingale show sure asymptotic normality direct dyadic deterministic I random seed history include dyadic random binomial accord final cardinality dyadic policy nonempty product differential entropy denote term dyadic furthermore analyze term almost sure martingale convergence hold nz
efficiency optimal otherwise efficient separate inefficient fully always feasible model difference matrix exclude evaluation super inefficient never matter unchanged matter use super efficiency holds change diversity efficiency efficiency accord efficiency score indicate mi base redundancy perspective exist average label never accuracy finally reflect exclude negative express select link point natural handle redundancy label call separability follow feature always range classification form kullback leibler value assignment interpret program mi feature calculate label calculation process separability take weighted redundancy mi fig mm cc label cc cc ct cc cs mutual change calculation mi
bootstrap bootstrap shall selection regression consider maximize extension consider identically I estimate context selection shall criterion log correction try obtain bias adjustment use cross cv widely selection cv bootstrap base selection cv reduce introduce cv cv bias define follow thus cv correction direct let observed let estimate set replication define likelihood quasi cv distribution sure replication
select without low fdr good model lasso one unnecessary include plot whether look right lasso property oracle possible unnecessary oracle region inclusion offer validation would also phase tend validation specific figure plot cv sure poor nearly phase transition diagram replace make lasso probability cope independent screening consider cv carlo bic stein sure lasso sl
algorithm completion problem continuous choice list find recovery performance technique enhance dynamically decrease stop reach predefine initialize nonconvex function nonconvex experiment free rank r compare augment lagrange solve task alm evaluate recover relative regard recovery
computer store technology facilitate functional motivated various functional back extend detail extend relationship include functional logit functional polynomial nonparametric linear due popularity specific predictor predictor variable field functional density understand residual assumption cs symmetry residual density useful financial asset return hold wrong error density produce inaccurate asset unable risk estimate motivate response variable value continuous deal regressor power continuous scalar density kernel distance fit
odd behave predict conclude across message likelihood therefore connection turn question perhaps find way majority serious traditional word co carefully light finding help getting adopt responsible complicated think remain player careful consideration accounting wu economics business interest include hold management publish article international conference book candidate social
association correspond nm perspective devise mcmc two main mcmc design follow explore later move aim order new target mcmc nn nj mcmc move association algorithm explore extension da linear designing demand loop data step hmms alternative clutter hence convenient magnitude propose new exist modify link modify link observation variable four move move mcmc dimension birth extension move death reduction move move leave invariant self reversible pair essence move nz nj z z calculate move z move jump different birth death exist target sketch birth propose base change birth trajectory time proceed define logic behind candidate space observation locate randomly provide empty terminate
intuition behind view contain low monte mix carefully appropriately particle improve particle particle particle augmentation key particle process observation analytically particle allow less information particle believe augmentation particularly particle strong parameter process particle deal stochastic end marginal conditional function parameter process often depend approximate consist path full particle filter within
input autoencoder stack bottom top observing objective distribution arbitrary regularization ingredient lead autoencoder autoencoder objective autoencoder dirac since autoencoder joint mention autoencoder optimize reach objective reconstruct accurate intermediate aggregated loss goal furthermore representation vary continuously change focus reconstruct explore deep autoencoder nh layer enable regularizer single autoencoder autoencoder feed output feed layer reconstruct activation reconstruction input use layer gradient global activation autoencoder second autoencoder look optimize exactly unnecessary reconstruct back input parameter level input address drawback long
specify particularly work namely method correspondence way analyze parametric us framework transform regularize yield estimator shrinkage meanwhile substantially perturb lagrange q think make perturbation non yield desirable seek via feature closely connect aim network training dropout work maxout imagenet provide serve specify underlying estimator surprising isotropic gaussian equivalent shrinkage induce stable autoencoder solution autoencoder
suitable subsection update collect input agent intuitive reduce adopt ec explore toward visit everywhere central base update coverage simplify aspect agent movement particular markovian depend whole history function dynamic agent fully posteriori generic control phase coverage estimate control coverage review section variation movement another establish statistic vary past history
context content model inform information topic robust partially miss achieve cluster performance domain extensive scope topic counter part hdp word model cluster employ lda hdp extract proportion input lda hdp affinity human activity elegant discover first nonparametric atom achieve introduce base model document document corpus share throughout recently mix topic parametric fixing crucially attempt utilize context model brief account variant dp vast relate building constructive property stick break stick convention hereafter mix associate stick breaking use
real fast convex thus suggest robustness analyze special assumption recovery minimizer nevertheless important aware single outli magnitude relaxation fail normalizing center unit discussion compute subspace rigorous treatment guarantee result follow l l try minimize minimize whole iteratively least l rigorous explanation apply pca center ability attractive procedure outline first compute top scale point fast vector iterate
expectation numerator denominator value account auxiliary expansion relate f da upper easily verify spectral bound magnitude large recall contrast z fx recalling recalling equal last fact orthonormal plugging derive equal focus substitute write justify lower bound first least get derivation derive term plug back simplify upper place constant sufficiently recurrence epoch
digit choose aim extract linear database roughly worth point set challenge justify database individual subject near image pixel face database contain image pose expression subject image pose handwritten digit database contain handwritten digit use respectively image database normalize unit norm tb construction euclidean kernel configuration construct number set respectively line near construct essence toolbox lasso construct
proper convex true exist sequence convergence influence advantage fusion spatial response due cause scene mention response different variance response strict sensor response fusion relative spectral response use call response sensor try able approach estimate account regularizer discuss consideration possible reason hyperspectral normally full drawback subspace fusion onto subspace hyperspectral involve observation cyclic support column denote kernel denote j def bc def impose note constraint therefore even function minimize correspond noise remove neighboring horizontal difference adjust approximate without normalize dc unconstraine cover specify support estimate long actual concern deal overlap band constrain band denote hyperspectral band band band row contiguous hyperspectral band
ratio split reader depth discussion split significance assessment full raw online last ratio aforementione split assess significance employ sign sign test two repeat sample match order rank student use fold accuracy besides basis employ average accuracy common threshold significance additionally comparison apply post hoc method wise series furthermore optimize cross within combination grid specification give paper linearly spaced integer use spaced value linearly space possible distance low leave error keep test extra result measure outperform perfect accuracy e achieve good set perform fc perform ar count statistically thus
implementation streaming supplementary extension intend reflect detail across scale variant learning project learn frame spherical temporal layer technique abstraction occur towards stage pathway classification start file rate file overlap filter reduce environmental normalise root apply reduction useful median finding median band subtracting spectrum every spectral energy background cope simplicity across make onto learn spherical mean benefit model short variation derive order summary common alternatively reflect spectrum frame overall feature overall six six pool forest implementation issue datum pre release manually tune parameter label different decision label classification assume specie label match task potential potentially relevance classification forest full label situation difficult task large volume full situation model dataset comparison classifier multi relevance width fig contain long audio yet annotation time specie common format annotation annotation specifically automatic file make file decision
operate stream proposition simple defer distribution eq w w follow theorem stream adapt level exist active hyperplane bayes omit dependency show achieve previously without know label distribution characterize prove explicitly construct principle margin slight setting help adaptive provide sketch generalize concave neither differentiable minimize surrogate logistic loss surrogate passive learning achieve excess surrogate probability least nd minimizer generalization number I example theorem hinge c f excess working error many include hinge loss margin active parameter
ise ise use goodness biological organization interact ise arrange represent mathematical ise integer consist except associate edge adjacent associate quantity quantity count configuration represent configuration color white configuration read represent diagram tb ise configuration normalize sufficient sometimes consider temperature depend replace configuration get sufficient correspondence representation notation lattice nan configuration describe range homogeneity define distribution condition statistic computation give use chain mcmc ising grow lattice markov achieve way overcome markov chain formally basis limitation approach first note every uniquely nonnegative ise exist
slice give ica convert problem mixture exploit method precisely mixture polynomial certain non degeneracy incoherent dictionary minimization exact handle sparsity tensor program enough also handle level require signal expense complicated consider work overcomplete setting topic provide tucker tensor decomposition identifiable order observe decomposition decomposition technique empirical lead bad rademacher context spectral reduce bound tensor concentration trade rough cover dense require fine classification tight moreover general rip norm notice standard clarity asymptotic say real member vector may tensor I I pt convenience rest tensor instance mode refer column refer row fix index arrange tensor slice index slice rd multilinear tm mi u multilinear tensor similarly multilinear combination slice rd tensor rank vector generality said write closely multilinear form denote matrix since weight norm operator operator tensor rd section latent mixture analysis detailed decomposition efficient tensor decomposition latent variable provide tensor introduce exploit latent provide guarantee argue propose throughout simplicity high order view independent categorical kb define parameter cp decomposition rank hence third factor addition covariance denote noise
autoencoder learn instance word embedding goal place lexical offer strong performance easy word embed representation dominant unsupervise heuristic novel
control term vs account analyze recovery incoherence theoretical minimax regularization unfortunately practical problem tune additional cope practical tuning issue model cross easy give ideal could probabilistic problem efficiently rely heavily algorithmic formulation separation general rapidly evolve hence important highlight convex formulation constraint solution constraint also cover project accelerated gradient disadvantage formulation
consider th error bound approximation influence get function quantity get combine present follow influence propose quantity show influence satisfy parametric robustness propose statistic robustness get power substitute expression corollary whenever order approximation contiguous hypothesis contiguous contamination interpretation indicator derive put thus influence statistic contiguous subsection influence simplify univariate distribution thus contaminate contiguous chi square freedom centrality influence influence zero tc explore robustness stability contamination contiguous may ng restrict case make routine density nan
mid audio motivated improve speech convolutional network apply music spectral domain base feature mathematical stable via deep computing module invariance visual extend hierarchical additional invariant signal explicitly arbitrarily music sensitive small affect characteristic stable pooling smooth variability conventional wide band instability high keep pass component
intel ghz single improvement delay logistic regression simulate environment obtain main burden mixture jeffreys prior constitute set exactly logistic stem explain use benchmark paper indeed big mcmc researcher keep effective focus computing sampling classic scheme contribute attempt datum play computing control instance incorporate control proportional variant generic approach pick branch especially pick delay elliptical normal proposal runtime runtime stand delay hasting combination core delay acceptance version computational chain asymptotic delay acceptance classic metropolis colour sd mh highlight
twice almost surely poisson various characterization conditional model atomic measure sigma generalize formula classical kf functional fw w term apply denominator simplification q line completeness numerator yield undirected rest latent count rest hasting pl stepsize gradient momentum hamiltonian p exposure omit index simulate discretized q accept write pdf allow intractable hasting acceptance improper prior ij truncate efficient accept probability bipartite iterate distribution eq axiom theorem criterion exercise remark summary proof figure fill thick fill fc european intra european fellowship support fa fa modeling represent namely adjacency exchangeability apply necessarily empty rely certain choice underlie construction process degree distribution use derive representation network explore hamiltonian exploration range range facebook social circle political citation web include hundred thousand million class rapid availability importance drive behind attention build history os enyi fail world model recent conceptually
analogous detect relaxation statistic present computer simulation compare tractable experiment choose covariance assume nonzero choose varied entirely symmetric maximum canonical top eigenvalue mdp require power thing confirm maximum canonical eigenvalue well really top table eigenvalue competitive implement iid htbp vertical proportion though remain identity know oracle tight moment nan htbp cccc vertical leave interesting paper obtain rate whether computationally intensive even moment method analyze theoretical adaptation throughout shift hard unknown minimax low bound unknown hand easily lee way procedure accommodate scan procedure know canonical correction simultaneously variance nevertheless rely together concern moment mixture covariance paper population two alternative seem meaningful complex group however way population case identity alternative affine perturbation natural base top relate result investigate population population issue consideration researcher relaxation context detection covariance seem extend suboptimal test relax case prove bound line calculation versus hypothesis nan alternative sequel isotropic reduce hypothesis non zero implicit make problem versus bad case risk testing last risk lr versus low lr pearson lr simple cauchy goal lr reduce follow refer random red red
edge value conditional jump membership connection probability evolve propose membership rely extended try briefly think challenge heterogeneity method large recently cluster network handle possibly independence namely undirected group rely replace step low group small still room network second give social associate suffer obviously complex statistically valid challenge property model procedure asymptotic recently case theoretical inference definition corollary st fr present selective modeling heterogeneity extension development field application biology internet individual interact represent node individual interact relationship molecular interaction presence absence record huge graph reader e general appear method detect heterogeneity still cover quite past year present suppose complementary mention complementary focus review homogeneous vertex literature friend index
influence contain propose party gender year benefit likely associate influence relation adjacency common follow context however utilize challenge twitter probabilistic topic modeling extensive science apply lead analyze extremely big text statistical algorithmic see reference therein model apply tweet tweet tweet landscape medium assign topic entire preprocessing apply every use measure computing interval I th interest interval investigate capture active period limit concern user extend occur ten influential account financial time influential twitter media figure account mention
use relationship shape gain recent attention image digit image al human pose latter work train multiple part contrast powerful share mid feature part several analyze explain success suggest localization et convnet representation individual meaningful convnet sift comparison visual beyond correspondence perform architecture identical dataset publicly reference activation
clustering seek output shift trivial rise modal proportion modal modal jj ix spread modal manifold mixture density gaussian behave mixture like modal like regression mr variance depend datum base mode method term several component mode likelihood indeed np mode shift algorithm unlike algorithm np number method bandwidth mode mode kernel shift cluster mixture em modal simplicity eigenvector eigenvalue density span express modal set difference point modal align coincide ridge conditional mode locally ridge state saddle local condition axis modal
variance estimate identical hasting even dramatically ability analytically propagate easy influence graphical leverage applicability large experimentally long unobserved column specify belief yield approximate minimize factorization obey suppose natural family exponential write exponential log linear exponential equation derivative latter relationship eq vector
penalty lipschitz partial penalization keep constraint shall exact locally objective result penalty subsection eq nonempty assume lipschitz continuous suppose hold minimizer minimizer together relation fact immediately relation ii continuity yield minimizer local minimizer theorem omit continuous local subsection lipschitz derive cover lot minimizer lipschitz local exist minimizer indeed globally minimizer minimizer moreover minimizer corollary qx conclusion proof explicit modulus resp continuity modulus present penalty problem specific globally continuous modulus minimizer bridge locally
mid property steady diabetes l set testing training diabetes training htbp l diabetes diabetes include measure mid classification testing spend spend node feedforward extreme interesting cm plus ex ex plus generally lie selection algorithm select bias also bias avoid yield singular
inductive conclusion suitably inductive eq last rather exponential work subroutine r r prove iteration thus epoch drop ambiguity shorthand notice tangent angle obeys eq inductive hypothesis establish q along eq outer inductive conclusion base establish hold step run induction goal go deep algorithm noisy view special precisely also iterate analyze theorem follow trial similar requirement choice imply necessarily rely claim terminate final lemma noise add favorable conclusion g iteration noise requirement inductive inductive line lemma hypothesis hold proof inductive imply lemma particular requirement satisfy must suffice sufficiently large imply fix iteration union
bn consider facilitate comparison structure algorithm si reference discrete mutual bn sample bn file repository appropriate load alarm sim round skeleton dag backtrack false mi alpha skeleton sim backtrack hamming r hamming node whose unlikely algorithm power small hamming give dependence order backtrack note vary accuracy focus backtrack skeleton bn exception alarm ham great backtracking sample size hamming appear contrary increase get trend use backtrack range distance bn
add pixel confidence respectively fast derivative function everywhere sign adversary adversarial rotation perturbation process differentiable reaction adversary find perhaps adversarial well inconsistent adversarial yield apply layer activation unbounded activation usually original perturbation unbounded activation make comparison able maxout perturbation however additive adversarial training capacity adversarial universal apply layer sigmoid function apply final perturbation perturbation reason adversarial seem poor space live hundred another serve people think capacity different low exhibit rbf predict elsewhere default confidence layer get
graphic macro ltb lt lt lt lt ltb lt lt lt lt lt r r r r mm train propagation mlp neighbor na I rule forest produce fold induce use induce diverse hypothesis train training training diverse handle filter classifier vote train backpropagation forest uci repository attribute attribute pt c categorical mixed post breast breast anneal heart voting record heart tumor car evaluation census vs noise ten split randomly level sign test suggest classifier list classify nominal produce learn produce delta biased approach induce misclassifie weighting score instance induce
without constraint population extract nonnegative block dimensional localize task nmf notation q intuitive nmf vector product free multilinear multilinear widely exploit overcome discriminant later minimize element wise nonnegative define straightforwardly infimum optimization partial fixing optimize equivalent nonnegative square problem extensively accelerate proximal free extend gradient respectively equivalently base exist nmf develop gradient verify complexity space demand especially scale reduce consider decomposition perform reduce space less intuitive tensor simplify respect product I time matrix significantly reduce memory consumption
whole language multimodal language learn dense embed word semantic temporal recurrent image part deep multimodal connect language representation learn use detail parameter incorporate deep multimodal validate tc method significantly task image retrieval image extraction network potential sentence deep network field computer al margin recently al framework recurrent recognition learning describe retrieval
purpose first sufficiently hence stop width interval specify superior criterion propose since relative article sample property deviation modification high modification provide computer good knowledge attempt formally address long run stop rule hundred complicated fmri thousand diagnostic terminate deviation consider univariate weather collect united ii selection demonstrate deviation high illustrate rule automate provide confidence paper introduce relative modern illustrate hierarchical dataset discussion general target small restrict unfortunately setting analytically frequently basic construct ergodic invariant regularity
intensity function count dirichlet mixture gamma data technical instrumental n convenience follow dominate log likelihood denote kullback leibler th moment likelihood concentration aim usually first quantity stand number radius introduce kullback leibler eq posterior neighborhood define q prior express construct measurable fix sequence j enough complementary posterior concentration posterior bad n concentration converge global satisfy modify conclusion loss modification respect assumption posterior instance associate need construct transfer lie nu controlling become typically control density dominate dominating section dirichlet process dirichlet mixture use intensity context mixture conditionally cdf dirichlet stick break instance cover induce represent cumulative give radius study large exist ball test q eq
variety mobile pose behaviour promise overcome detection audio recognition usage mainly focus start insight user stress level mobile software state usage usage self longitudinal use phone mail collect user four day reach participant daily phone call discrimination mid term monitor student week period relate pattern social interaction phone call detect limitation comparison adequate daily people situation report recognition mobile sensor subject limit day major subject variety background usa china etc equip phone sense software collect mobile run manner phone device minute datum participant trait big report daily proximity minute enough resolution list phone participant ask daily
consensus show side show particularly encourage even gap eigenvalue cluster collection consensus cluster raw iterate consensus refine way element outside block iterate consensus heat consensus cluster dimension red pixel considerable block consensus heat consensus matrix refinement clustering diagonal show high iterate gap spurious relationship couple together h h demonstrate datum cluster consensus clustering step mi md remain stop agree stop solution matrix clustering determine repeat step ng common benchmark cluster web article evenly attribute
non default rf complexity parameter label expect nothing much would scope sd sd sd infer rate specific evaluate train test provide al gain simulated label study label quantify scope score normalise metric monte carlo varying statistically analyse careful refined methodology assessment iteratively grow method budget budget iteratively batches sense experimental resemble challenge realistic explore label experiment al plot experimental firstly sample pair
generality solution backward find point compute generalization matlab structure need matlab contain field return vector return way former matlab also matlab input return value matlab contain parameter describe fista method stand backward accelerate
useful network node connection come multiple example study membership association individual primarily facebook etc demonstrate machine algorithm quality heavily edge predict good source qualitatively subsequent graph difficult domain rigorous apparent big challenge connect aggregation underlie requirement locally aggregation incorporate good absence fashion inspire demonstrate community graph source community evaluate locally represent
right box recent deep require beyond random graph try individually definition hope mixture presence e change arise property suggest possibly seem quite plausible leave even though vision intend assume independent variable bernoulli relaxed concentration column denote denote ease exposition describe generalization nonnegative dictionary practical generality expect ax bipartite define magnitude j later effect every pairwise intersection among
solution significant constrain function problem parameter projection critical maximization correlation chain matrix critical scheme enough ensure implement copula estimate expression copula principle applicable elliptical compute gradient
hessian determination curvature bfgs low convergence argument complement characterization convergence sgd establish minimization function vary vary objective comparable sgd vary dimension problem exhibit degradation dimension vector svms points improvement numerical also compare non regularize bfgs fundamental bfgs observe separate hyperplane definition average function function convex strongly strongly convex descent motivate actual realization define give implement gradient require determination gradient average sense along intuitive formalize convergence hold control step problem small resort order evaluate suited newton whereby definite know select include bfgs e bfgs since practice approximated variation hessian tend
regardless sparsity contour specialized believe remarkable moreover recall almost line straight log observe function smooth novel primal problem primal sampling flexibility variant parallel variant sdca serial uniform serial importance nice speedup direct primal dual sdca sdca sdca accelerate leave sense involve pair mutually dual function convex conjugate follow pair concave belong interior last relaxed ex theorem theorem zhang two would acknowledge grant coordinate optimization penalized strongly propose primal analysis directly depend serial distribute bound match sdca predict speedup drive depend calculate speedup excellent efficient batch importance distribute distribute sdca pair optimization vary attract
identify exist identify regime measurement generic measurement generic regime perturbation measurement generic another irreducible k identifiability broad measurement identify theorem yield exactly apply infer theorem reverse implication hold prove another unitary n unitary identifying let single identify reconstruct phase invertible unitary identify generic proof analogous noting identifiability terminology determine motivate family irreducible denote small identify call small completely clear call write statement terminology allow retrieval state exclude formulation observable view element perturbation perturbation view identify signal perturbation generic perturbation except hausdorff hausdorff take furthermore statement hold replace measure subset closure subset closure already imply note statement contain generic make valid case analogue projection real vice threshold
noise variance accordance eqs compute hardware propagation across layer operation index sigmoid activation train sgd size objective momentum descent exponentially contain matrix vector initialize decay network sequence representation describe network corrupt control network precise entropy error train error show hardware suffer degradation compare control contrary slight presence incorrectly network prior shown improve neural network hardware virtue
generate try match slice slice quickly belief converge increase method entropy reliably explanation much present suggest develop strategy classical iterative additionally time latent prediction expense prediction might area rather suggest help investigate improvement investigate distinct describe useful optimize
index error develop fast solver proximal focus develop fast subject consider future analyze robustness whitening source try whiten good sdp fact whiten sdp whitening follow tight analysis whiten whitening extremely oppose pre whiten affect allow illustrate synthetic show whitening show well runtime analyze analyze reduce bound bound robustness since goal bind solution approximate w sufficiently interesting trivial unique column since see follow prove satisfie q optimal value
quantify method diversity dpp diverse without indicate significance sum gene sort radius indicate find diverse feature distinguish ii breast construct identify breast cancer comprehensive cancer pathway profile readily predict breast however poor feature gene pair protein protein network form gene belong similar community detection challenge avoid step collect genome breast gene protein top accord univariate regression respect tumor similaritie nature identify higher component specify gene approximately average impose dpp lead
visit remove repeat parent step leave parent child dag label hierarchy learn time million intel equip vocabulary major belong least major belong visualize hierarchy occurrence separate reasonably mesh vocabulary please well visualization rest subgraph care leaf representation predict child tend share
result sparse contribution computationally cluster dimension come complexity scale relevant inherent ambient spherical without notion feature handle organize formalize complexity high dimension generating point generate covariance cluster scope cluster e error
measure sign set sphere dictionary basis dictionary square expectation situation practical equally weight one measure random sequence component exist proof simple incorporate coefficient motivate requirement original dictionary signal probability base sparse product atom maximal freedom construct ascent need truly decay large study maximum near generate warm first incoherent noiseless draw unit exist c moreover sketch ideas svd include ensure consequently signal response response perturbation c still attain typical sign sequence permutation scale sign sequence original already optimum generating dictionary obvious special signal almost get ensure large arrive q find signal insight
evolve continuous neural define ideally equation dynamic eq frequently strictly mechanism behavior noise contamination equation recurrent indicate otherwise refer gene capture relationship node recurrent nonzero link interest time n u c I represent form dimensional time always point sde discrete observation challenge rarely analytical treatment approximate eq system depend separable x verify parameter minimize number limitation interest reality stock price influence stock market parsimonious interpretable statistically speak necessity penalty function
original future prior regular regularizer machine robust pca network share across resolve minima utilize induce sparsity newly outperform approximate optimisation mapping code extra one behind
later inner product obviously definite dense type core perhaps type expectation product definite figure accuracie kernel core perform kernel see core high lin lin bin mnist summarize see without tune compare abc mnist
reliability divide verification quantification verification concern assess quantification assess uncertainty model uncertainty mathematical reality predict decision verification difference mathematical concerned discrepancy verification sometimes practice largely technique control impact ensure compare source verification focus quantification uncertainty prediction unobserve validity model assess experimental comparison validity assessment closeness reliability uncertainty model assessment reliability unobserve discuss essence question justify validity really validity general produce valid quantity validity scientific strict e mechanic nonetheless valid sophisticated procedure validation engineering development validity make instance similar framework model representation framework observational insufficient validation predict observable strong decompose make discrepancy refinement quantify importance health economic introduce discrepancy within technique create discrepancy address prediction model highly calibration trained discrepancy unlike et al guide discrepancy systematic validation capability unique remain issue physics reliable physical couple reliable structure unique modeling entirely introduce discrepancy enter et unobserved validation address spatially vary elastic modulus model calibrate experimental consistent question compare always fit problem investigate match experimental account
interest q give meanwhile establish positive go term theorem asymptotically unbiased converge weakly correlate must hold establish uncorrelated maximize contain order complete prop conjecture prop prop prop definition prop comment prop remark grateful stanford fellowship forest prove area forest establish prediction forest paper forest asymptotically subsample number show asymptotic ensemble characterize treat black become popular box tool machine
independently use emphasis scalar scalar product ridge scenario experiment attribute decrease significant method lasso behaviour experiment popular mnist focus distinguish lasso scenario ridge considerably train check theory also number examine attribute offline utilize towards attribute htb perform slightly variability small examine attribute set predict forest dominant multi classification species address scenario data htb ridge similarly online examine attribute examine outperform attribute one appear perform well converge towards attribute small perform examine attribute grow budget ridge distribution excess art prove demonstrate even though quite partial direction algorithm expectation question arise
commonly multivariate multinomial distribution explore vector appropriate distribution random useful symmetric set poisson ip independently poisson dp prior suitably gamma conjugate cluster dp observation also dp restrict cache end temporal model temporal correlate count hmm dp incorporate hdp model case rise dp dp temporal come introduce define expensive dimension trace extend extend denote active symmetric otherwise emission across define rest dimension account case dp indicator denote natural distribution parameter bernoulli inactive case dp dimension hmm temporal instance hdp q hmm hmm describe hmm capture trace dependency exploit inference definition introduction dp
backward estimate obtain algorithm final look sampling bivariate full use quickly gibb low effective model gaussian variance particular exactly method much gibbs fact improve sampling widely hmc distribution space expand method dynamic hamiltonian update hamiltonian effective way posterior exact walk hmc invertible volume preserve similarly hamiltonian obtain stochastically likelihood bind
proper beta restriction give negative likelihood give atom weight atom atom location prior fix whose proper since assumption ordinary find form ordinary hyperparameter beta fact conjugate process posterior also posterior conjugacy approach bayesian nonparametric conjugacy still guess right conjugate likelihood automatically construct conjugate exponential give family family development bias representation marginal condition exponential measure lebesgue measure atomic mass conjugate eq natural bayes quantity eq belong family conjugate exponential start notion family location location atom unique location weight density density statistic share across atom ordinary component rate measure share atom unique automatic nonparametric prior automatic conjugacy accord exponential atom weight distribution fix accordance th atom sufficient atom ordinary weight rate measure conjugacy
minimize gradient useful ab q multiply side integrate eq divergence expectation density approximate average include analytically q e cross j size candidate sample except eq denote cardinality choose solution compute gradient descent euclidean give manifold ordinary tangent equip metric q denote
red black plane span principal vector subspace subspace let principal subspace preserve projection state state separate non remain help extend subspace multiple subspace ty separate margin projection vector respectively span use angle add dimension separate argument subspace preserve
explore soft margin use rbf range logarithmic rbf explore explore range produce tree bootstrappe varied boost fix tree range logarithmic ratio consider tree loss layer perceptron softmax minimize neuron value total dimensional space present generic different task crucial win either model ensemble run set good set winning frequency case
admm try fourth try calculate admm try admm adjust show admm completion effectiveness admm rank synthetic partial cosine operator compare rank take want illustrate noise compare admm show admm show fig run htbp propose noise namely approximately say robust corruption save illustrate effectiveness lr admm dct recovery illustrate advantage admm evaluate recovery compare reconstruction different sample fig addition compare generate run admm matrix lr easy increase achieve admm nuclear noise
match contain scale randomness operate lastly still hold partition approximate balanced cut energy property general multiclass unique balanced one sense balanced unique minimizer satisfie surface modify geometric substantially long theorem overall spirit role finally remark analogous remark convergence deterministic functional nevertheless random decide introduce space functional respect functional converge satisfying situation instead prove deduce precise converge enough deduce functional nonnegative functional boundedness functional property minimizer converge minimizer minimum make reason variational completeness benefit highlight work ultimately nonnegative functional statement cluster unique minimizer statement hold deduce arbitrary previous nx relatively know least inequality deduce imply prove dd kernel variation function rescale version sequence define surface weight define functional restrict functional characteristic function obtain convergence
order must strategy define stop determine base measurable control provide triple entirely setting strategy call natural draw advance almost choose recommendation fix budget strategy zero resp identification confidence resp fix setting follow confidence need average arm fix optimal require fix budget setting aim compare two complexity lower bind resp failure consistent sample resp failure lower present theoretic strategy round fashion sampling desirable armed bandit study regard hypothesis pair fix number budget rule determine sequential first confidence simple fully law know permutation pair hypothesis error small minimize sample type gain indeed gaussian armed bernoulli considerable interest introduction medical trial important perspective aim maximize reward equivalently expect complexity well understand bandit leibler
connection screen enable understand advantageous shot complex important hyperplane test among shot interesting study indicate significant sphere center spherical bind solve homotopy study wide range shot test select available successful application author constrain lasso concept screen lasso addition safe liu screening make variational sufficient screening compress select predictive outcome context apply screen music term codebook music xu et screening application dual give active change ai I nn lasso reject reject I e q w zero r yield reject nonempty hence diameter ds dr r r r lemma rt tt zero substitution ia substitution yield yield yield lemma empty nonempty reject partially nsf b electrical engineering university china receive ph electrical engineering currently
approach target latent bar intensity ground bar repetition selection gp rbf noise latter particularly adaptive trend datum book discussion model via already extract explicitly spike generate middle converge element hand converge element learn select inference gp cosine quickly ground truth provide quickly straight rbf ground select kernel
split coordinate child node inner facilitate formal version quantity tree begin expand otherwise height leave pt prediction predict feed observation predict h compute splitting x leave corresponding cm leaf bin storage size state depth store need store space order besides leaf need per tool sequence rectangle bb number initial jensen multiplicative want guarantee consider
text estimate extent political text note text text return probability extract hyper plane lie hyper anomalous encourage close meanwhile hyper towards text influence seem input alone gram alone yield suggest indicator author word perform poorly severe overfitting mention introduction ten agreement accept publish style death gram additional insight community author addition identify setup eight text membership identify close plane agree well preliminary seem p text non text correctly fact influence
symbol color three long represent red green choice make qualitatively wide typical give top lot similarity tend embed correspond dissimilarity identify rapid different score sequence whereas large sequence score bottom remove choice visually embedding embedding give correspond equal qualitatively htbp shrinkage multidimensional carry problem consider structure equip unique start extract exist form domain protein bank symbol contain coordinate coordinate case relatively compute dataset correspond evaluate
real movie sparse user rate bias towards rich user rate lot movie rating overall recover rating completeness present reader key study co algorithm well dataset recommend concern even study dataset limitation briefly user user continue well simple majority cluster subsection substantial recommend movie error rate rate htb show rate htb low c remark obtain obtain user ii movie conservative calculation rate netflix dataset movie movie select rating select user otherwise special computer svd ghz processor movie algorithms movie user error error rate htb evaluate rate among rate low particular error suggest robust independently cluster
strength nominal distribute alternative strength along nominal distribute evaluate compare test outperform test alternative strong right panel figure gain power become test propose test screening outperform agree screen substantially preferable whenever relatively simulation carry online supplementary material show particularly screen outperform covariance test maintain sample power sparse recommend biological know gene base focus extract biological insight increase identify associate biological state disease development research genome analysis produce name term category molecular mf gene group term statistically gene independent hypothesis model expression level gene gene mutation practice set set dimensionality set control wise procedure
appear good context mh algorithms reject single branch execution path evaluate core speedup core ignore circle font cm gray cm node state node state child state child child child child na I observe branch reject probable classic result scheduling proposal evaluate reject branch branch algorithm core useful reject proposal proposal root due core root computation speedup speedup core na I essentially advantage later consider special slow move evaluation likelihood slow
say different mirror image regard select score context globally discrimination optimize discrimination though probably give across especially behave indeed performance phone compare
differential object research rna seq relative assume come parameter dispersion individual shrink toward genome parametric relate dispersion notice estimation integrate frequentist paradigm place parameter thus able detection analysis typically start analysis gene de increase take cope investigate approach properly quantify uncertainty de
similarity yield much performance benchmark whole much well ensemble ill clustering add time base clustering method link graph link connect execution fig cost second pair method execution method two fast great partitioning ensemble crowd link crowd assess reliability individual exploit normalized crowd evaluate clustering unsupervise triple construct common reliability take consensus consensus term evidence accumulation partitioning conduct eight effectiveness method anonymous comment suggestion enhance science definition probably robust increase mainly aspect limitation exist firstly weight base ill clustering level ensemble fail integrate unify address limitation crowd agreement normalize crowd agreement quality
de en da fr en en es identify focus enable grain question similarity language determine similarity topology european language group triplets topology specify correct ordering fraction exclude subtree relationship language language similarity language nearby es ft triplet language remove ht cccc target es fr pt es es es fr fr choose four
denote homogeneous branching process root million old represent specie branch age branch simulate epoch give value binomial variation five lie temporal gap rejection simple apply require million simulator evaluation accelerate approach simulate due use generate estimate prior lead accept leave fit alternatively substitute less abc variance train noisy surface term approach successful result simulator evaluation rejection
node highlight central node centrality architecture network centrality connect scale centrality unity cluster centrality total independent band network free centrality dominate significantly method six condition exception recovery geodesic distance geodesic distance short path free geodesic scale network tend short feature small geodesic free lot band geodesic distribution band outperform control graph finally analyze synthetic band network connect component contain connect component band free size sample recover across figure nearly perfect graph inference construct module rare match assess accuracy infer interaction round final model threshold relevance infer node correct significance arrive gold independent test learn reference new network triangular distance vary interval ham repeatedly subsample disjoint repeating inference hamming edge disagreement bar disagreement highly visualization overlap color colored geometric relative american unify analyze reconstruct quantifying
identity lag walk forecast equation model parameterize possess hierarchical describe point decay range correlation reasonable lag lag hierarchical perturbation effect acquire kf enkf prediction walk dynamical thus solely term uncertainty quantification cost induce pixel inversion measure kf depend resolution resolution estimation error kf enkf produce ensemble realization accurate co day volume survey informative less improvement enkf represent simulate db snr instance day image plot function day co uncertainty quantification kf inside coincide ray coverage middle kf accurately enkf great kf plot three enkf solution
concentration model draw deviation table integrate ode htbp level range expression normalize normalization genetic example l error variance genetic l l pde forward pde boundary pde serve steady system head method function mesh endow gaussian squared length scale field eigenfunction integral endow prior parameter mode condition mode cover observational proposition proposition example mit monte carlo limited computational therein approximation hasting idea theory approximate inference typically sampler seek address characteristic local ergodicity distribution interest employ polynomial observation underlie regularity model evaluation greatly without carlo average multiple order forward ode pde inference computer experiment chain carlo computationally intensive markov monte scientific chemical often invoke constitute yield numerical cost forward quickly become prohibitive expensive cost recognize regularity interest characterize replace forward surrogate require forward evaluation chain reduce create global polynomial effort thus approximation accuracy expensive set approximate intractable significant improvement either induce conversely potential even delay eliminate need surrogate accept
svm aggregate train discriminate small instability sensitivity contamination unlabele bagging technique bag svm draw discriminate resample varied induce variability aggregation bootstrap bagging hold q positive unlabele choice bag classify instance high misclassification similar propose
self label euclidean fold fold describe angle translate seven rotation result c cm cm label seven source seven rotation angle high consider translation positive negative table remark change density robust provide performance necessity account disagreement label well result nn translation label nn labeling matching domain focus region remove match source close couple nice tackle target
potential first set item use translation map distribution setting space maxima vs plot original space hand map axis scale moreover matter map everything train correct increase cause map set spirit greatly simplify conjecture map consider appear similarity simple correct normalize item map length penalize correct rank straightforwardly implement put
response assess prediction unobserved response observe test response unobserve response sparfa range sparfa select control assess three metric receiver auc correspond percentage response predict py learner response classifier cf sparfa sparfa auc dataset auc trial sparfa achieve comparable performance cf sparfa outperform sparfa metric dataset emphasize sparfa
hence choice however require considerable htp htp solve investigate stepsize svm show evolution duality obtain stepsize mention previously stepsize theorem moreover computable comparable speed htp synthetic lasso generate average nonzero element maximal nonzero gb respect rate converge expensive compare angular
real multi modality datum label audio tune model baseline comparable modal dataset improve recognition video report result combination r v configuration range transfer deep fine audio semi bind achieve transfer bold chance performance video tune audio outperform audio datum show target specifically
patch patch patch difference final denoise result entire choice denoise patch case identify interpretation forward competition pass estimate scenario find balance generality cccc patch truth patch pool exhaustive patch adjust bm set show increase db increase db curve level propose mse recall patch clean distribution square define suggest optimal minimizer problem know never model prior patch difficult shape highly representative could draw center appropriately optimal diagonal patch measure respect improvement still linear mse define lemma patch eigen decomposition show noisy patch learn form datum eigen decomposition denoise assume subsection
half reliable systematically small magnitude real half predict correct qualitative sensitive contain shape frequency spurious predict obtain perfect impossible dot exact circle length dot denote green result representation denote systematically small coefficient magnitude ard result conclude learn v circle set exact dash denote length green learning learn time problematic green reconstruct give slice plus coefficient learn body b circle result dot red dash evolve set display look around half around applicable fraction representation set learn four fig dot dash learn
ccc ccc simulation lasso simulation c display b identical graphical correctly assume sparsity covariance overall little neighbourhood specialize intensive display adjacency matrix simulation c appear similar b simulation detect tune varied simulation panel panel correspond panel panel fraction set simulate datum examine
fact justification plug span td update parameter discount set instant td run length calculate td error value update follow td note bellman establish governed solution give vector rademacher component dimension unlike balanced one side primarily run come simulation side sf density integration fact chapter sf sf sf rademacher sf iterate define section necessary tuple saddle point sensitive around uniform rademacher eq similar eq expression operate finite sided gradient lagrangian recursion lagrangian enough sr discount actor sf sr objective rs sf recursion sr nominal sf algorithms sr optimization lagrange describe sr simulated sr lagrange trajectory parameter necessary sf td evaluate instant discount nominal perturb implementation would follow evident necessary base scheme simultaneous henceforth rs henceforth refer sf smoothed sf hessian confirm second inverting q update rademacher perturb update hessian estimate estimation technique chapter appendix expand taylor expansion observe vanish mean case update multipli hessian update ensure fast recursion see project symmetric projection ensure convergence hessian sequence matrix n eigen eigenvalue avoid incorporate newton update policy converge converge
nominal discuss great unfortunately share distribution high temperature rapidly start idea far intermediate traditional maintain high recent feature layer upper layer appear ever e ever play learn intermediate facilitate low hierarchy applicable energy equivalently rbms visible variable unit rbms much
model et al turn social approach analog grain infer latent influence variable validate compare several include movie model infer focus linguistic infer behavior combine model take linguistic possibility influence potential tool social brief description variant infer influence turn take et model pairwise action concentrate cluster influence scenario social nature relationship extremely valuable et al specify discussion correspond capture occur let pt calculate duration process
enhanced persistence characteristic illustrate validate correctness highlight visual persistent infinite hmms hyperparameter always informative often towards certain kind informative bias induce prior hmms remove indistinguishable hdp match explicit duration hmm poisson sample hmm construction hmm hdp hmm equivalently ensure persistence regime state encourage proceed monotonically encourage path skip duration emission emission duration emission give scale gamma poisson burn burn possible backward slice joint show score histogram explore duration posterior show duration fit datum distribute result hmm duration compare hdp hmm hdp state unit geometric normally implicit geometric fig poisson duration observation mean duration difference self observation difference ten dataset generate adjusted sequence state emission generate datum code audio utility duration correctness also
follow eq impulse response estimator estimator determine propose hyperparameter vector characterize impulse precisely jointly maximization likelihood integrate follow reason solution em method iterate converge global stop hyperparameter vector furthermore define
transmission measure image build transmission position across image replace detector array measure distribution angle frequency information smaller enable high resolution object need phase tractable redundant reconstruction often well convergence remain reader result see optical reproduce often practical refer convergence need global phase retrieval discussion phase recognize involve analyze projection descent metric focus iteratively enforce piece retrieval measure ap solution satisfy convex understand nonconvex still open question except quantify retrieval uniqueness proposition retrieval ap uniqueness show coincide factor result necessary sufficient condition ap unique solution factor ap algorithm become lead confusion throughout survey synchronization phase synchronization phase synchronization problem lead technique laplacian eigenvector initial guess accelerate speed large scale problem section show propose design convergence rate
curve figure learn adopt white notice region successful experiment coherence whenever show success sequence data matrix trajectory unobserve visual trajectory recovery corrupt recover art sim ssc std sim sim ssc ssc ssc algorithms trajectory correct dramatically ssc identity rate effectiveness dictionary environment sparse give might even typical structure multiple cluster go coherence overcome challenge arise coherent theoretically one capture structure produce suffer several practical simple lrr lrr furthermore utilize approximately still problem need domain document column new handle superior theory condition
solution hold instead b hold replace vx lemma relation fact rhs expectation high search expectation side show assumption u martingale martingale sequence lemma conclude convexity exponential markov view provide specific use stepsize policy corollary priori bound however need compact observe together part respectively definition conclude imply relation solution e point immediate exist stochastic b evaluation bound b knowledge time complexity establish slide generalize important cp specifically cp subsection nonsmooth give
toy class contain dictionary toy test joint ls group low eqs different toy region recover hyperspectral overall accuracy aa measure prior implement comparison whose fashion joint sparsity combine admm search among sort structured two prior prior row ht c c l
htb minimal quality decrease figure quite satisfactory active agree recommendation correct favorable stability time item recommend study item never failure account weak modification show offline evaluation factor recommendation recommendation user production offline contrary recommendation overcome
term limit two space correspond namely isometry relax quasi unity parameter dictionary atom say isometry entry preserve isometry constant isometry space isometry property dictionary investigate sparsity generalize atom kernel isometry property atom atom isometry dictionary gram large eigenvalue pair explore identify isometry measure besides expression theorem thank expression deal unit atom bit isometry dictionary r dictionary get bound divide yield isometry rescale atom proof theorem deal
dissimilarity tend prototype bag prototype bag reduce become consider rather bag concept instance originally approach ik radial however leave counterpart dissimilarity summarize representation dissimilarity information preserve per bag averaging dissimilarity preserve dissimilarity select relevant dissimilarity categorization problem bag image patch instance region include dissimilarity provide heavily redundant uninformative dissimilarity
intuition condition always strictly hold local neighborhood sufficiently concrete model equation function reasonable condition neighborhood confirm follow locally consequence sequence exhibit linear condition concavity side imply side bound combine original em repeatedly thereby give subset perform sample tolerance small tolerance require ball accordingly small scalar operator parameter ball belong enough iterate least size splitting figure illustration predict algorithm expect bind fix size suggest iteration particular focus least update increase choose remains fig conv part black ball logarithmic identical argument least union perform event suffice bind follow sum induction iteration whereas apply initialization bind argument complete gradient em separate addressing provide generate recursion analyze additional concavity assume pair intuition compare function update state strong concavity smoothness closeness requirement formalize condition verification gs positive regular hold oppose figs gradient whereas distant gradient condition radius triplet point gradient
character denote detail analytical conduct several efficacy algorithm draw note three choose index final lp lp lp problem analytical solution write explicitly objective inequality separately solution lp refer soft thresholding remark criterion speak whenever decrease slow term
worker disjoint summarize illustrate tradeoff along namely statistical efficiency epoch epoch ia ratio row wise zero cost writing read examine wise find execution depend hardware statistical efficiency row sparse read dense perform epoch combine linearly thus illustrate run converge error figure see number hardware efficiency change per epoch non subsample music detail ratio actual cost ratio row wise read wise method cost base optimizer execution read epoch svm row update zero scenario update estimate ratio read run long expensive reading cost access three similar traditional nothing architecture design machine difference model strategy exist simple leverage machine maintain version epoch share nothing art framework subtle worker epoch implement minibatch parallel calculate gradient implement dynamically requirement replica worker responsible update replica dominate minibatch implementation schedule way implementation overhead tolerance relative application paper implement implement force hardware deal coherence although reader converge single replica share core synchronization base epoch share approach consider fine sharing read
scheme accuracy small dictionary learn normalize vs binary classifier use regularization width validation dictionary match naturally sample previously svm train denote code comparative use supervise sample atom cluster close drop supervise previous rbf last texture task linear svm suggest classifier one accuracy well size moreover svm outperform test last rbf l coding coding last propose last compare different outperform code technique notable classification another mean soft consider task handwritten digits image training mnist compose image unit address class task use problem specifically separate classifier predict vs naturally feature different sgd dictionary nn code relu mnist describe previous sparse code addition building
similar predict protein protein know novel topological intrinsic develop second section annotation protein support machine spectral interior spirit make propagation obtain part semidefinite
detect sequence detect anomalous definition refer scale guarantee decay sample say exponentially consistent respect refer asymptotic regime go decay anomalous scaling develop anomalous I see anomalous sample draw anomalous capture level anomalous increase affect consistent mmd suppose distribution rkhs kernel refer mean hilbert reproduce clear map unique element associate distribution discrepancy mmd clear mmd equal embedding due base available paper scenario sequence start simple sequence case far anomalous detection anomalous case compute sequence anomalous constant naturally anomalous characterize anomaly generate apply kernel exponentially consistency desirable practice number small
complete target jump corollary definition em em height mcmc analyse move update particle filter application give estimate log proposal langevin investigate mala asymptotic dimension mala depends crucially accurately control sufficiently mala use proposal behave particle mala proposal compare walk furthermore acceptance particle roughly mala suggest monte well methodology year mcmc methodology tackle likelihood possible intractable monte replace estimate target work also particle refer mcmc particular metropolis hasting replace hasting sampler default herein walk rwm filter overhead proposal focus filter information region langevin
successful employ mainly probabilistic automate distance reflect mean language nlp ir use similarity semantic network base major classified base pointwise semantic term measure base semantic similarity direct graph arc partition dependency parent root one leave level third namely frequency define train consist path node believe explicitly scalable implementation improves implement outcome namely outcome probability v I px px fix identically outcome joint
drug chemical describe play compound represent discriminate document topic tailor domain similarity satisfy require instance learn typically problematic computationally expensive cubic likely overfitte especially rarely observe common dimensional pca reduce useful irrelevant similarity also discovery knowledge bilinear function dimensional mention sparsity parameterization frank wolfe learn incorporate pair providing ignore overfitte output
output identifie regard free recall mean simple numerical entry draw next randomly entry one plot trial range correspond estimation ix analyze reliable recovery output happen lasso strong convexity reflect change appropriately modify tail gaussian chi noise sub depend condition general recovery
increment denote cluster prototype assign empty centroid seed etc least one mean update empty cluster proceed point decrease decrease local converse suggest replace minima optima argue empty meet figure mean empty exception empty empty cluster create far copy however add seed decrease partial keep stage etc frequency
r combine get theorem execute quality privacy parameter hold differential call range therefore inequality computational quality suffice efficiently implement recursion operating easily call private recall show goal small error quasi concave algorithm label pt utility analysis let draw c j execute valid moreover execute execute quasi concave overall proper use axis align space think approximate private learner nc I align rectangle vc thus learn generic inefficient private learner give private learner efficiency however direct transformation begin sample draw component approximately likely fall proceed boundary use query probability positively place hypothesis th leave interval query private use transformation simultaneously could laplacian histogram straight approach learner overcome construct divide mass standard argument specifically interval class database b axis mention axis return database execute approximation roughly every execution theorem learner inefficient proper complexity fact learner must private concept class necessary database exist stand close database overcome tool approximate define domain approximately maximize optimization section database define function ii requirement could score increase neighbor moreover element private otherwise approximately gs gs choosing preserve fm output respectively hold two growth fact e proving
l e index transpose version call slow third runtime first evaluation call example sequential user parse restrict structure specialized solver inclusion decade solver solve alternate prox efficiently method like even problem spectral large sequential convex produce successful programming solution specialized branch name useful exceed care detect transform format subject extremely currently development parallelism capability leverage parallelism library simply implement software
concentrated verify variable mean event completely event claim turn convexity consideration contribute j q event difficult yield basic unsupervised family unknown parameter u goal natural useful subroutine various hand give recover maximize program relate limitation technique consider n reference coordinate feature first process sound wavelet useful primitive many state case recover nonzero generally entry approximation guarantee suppose arbitrary follow system satisfy equation solution degree satisfy moment cauchy schwarz sparse p motivate encode input subspace always close approximation ingredient subspace outline write subspace b gaussian
regression write regression k lk number serve effective facilitate selective noise component appropriately induce impose allow adapt ensure number interaction inclusion interaction scaling initialize se inclusion conjugate suitably section place component inclusion model inclusion c b inclusion default middle inclusion size vary fix inclusion prior inclusion depict configuration parametrize sample size large place alternate choice parameter play role interaction framework grouping within scaling realization parameter link crucially inclusion scale efficiency update discretization essential mcmc allow grid calculate enable vector grid ratio correspondingly length specify dirac delta standardize default sensitivity hyper though careful necessary allow nearby characterize inverse pattern surface scale characterize fine shape feature restrict smoothness grid range scale concentrate important variation response explain range ad pd active component across propose fast
assess rule choice perform extensive choice efficient method investigate seem yet criterion application real importance suggest numerical simple choice inefficient give unclear test consist end force second dimensional result experience two cg justification experience coefficient possibly scale condition cg cg however possibly conjugacy specific scheme furthermore proposal cg possibly efficiently aim effective tool context sequence conjugate context unconstraine detail study indeed choice might also
equip range expert convex corner gap cx corner robot find cc follow minimize therefore consist current linear velocity robot angular velocity expert always follow angular straight convex corner corner use controller decide version could analyze algorithm environment real characteristic environment length concave cx corner difficulty corner close front tb environment dim length cc cx home home office environment fig trace mark mark velocity environment grid map environment robot cm along environment robot real environment use obtain eqs universe rule fuzzy minimum implication universe variable min max distance velocity angular velocity three genetic fuzzy evolutionary fuzzy rule fuzzy generating genetic simplification genetic membership fuzzy base soft performance open software tool problem use perceptron hide bfgs number layer vary statistical software raw three group
base design operate pass thereby cluster find center select datum find center enable evolution technique divide task merge employ processor representative processor cluster cluster center cluster pairwise dissimilarity accurately cluster linearly capture develop som kernel full whose knowledge attempt scale reduce time memory mean reduce always produce matrix address challenge show cluster reproduce hilbert rkhs endowed follow cluster membership center domain matrix relax center find
follow p p f sec et al extend lemma directly adapt n hold km describe analogous e km km e proof operator k great second union partition union j p sp bound assumption corollary em address collective completion recover collection share partial noisy impose joint wherein across develop algebra represent collective collective tractable collective trivial
supplement exist hoeffding say almost nx imply directly theorems hand conclude corollary neighbor classifier theorem ni ok w ni ni corollary accord prior eq gain reduction relative gain logarithm improvement logarithm effective trading relatively research nsf award associate award dms stability concern conclusion population introduce measure instability capture variability plug classifier concrete classifier derive neighbor near trade stability possess minimax rate demonstrate near neighbor accuracy scientific scientific conclusion stability much many instability assess instability criterion dimensional selection stability tuning context purpose bag derive stability
item choose rmse yahoo provide rmse mf term criterion evaluation size also note give correspond default base user unable reasonable cs competitive less train version cs approach item expressive ability learn relevant item mf mf c yahoo na na quantitative cs present movie three last episode rating star episode seem rating movie half
run two reduction save nn classifier dataset computer package disease genetic heart apply technique dataset extremely snps space via multiplication dataset consider project table entire measure f area curve include process follow projection dimensional methodology testing describe projection dataset low projection course equivalent multiplication multiplication fix parallel roughly denote illustration projection ccc snps approach coordinate dimension projection realize take approximately month run comparative approach discrete contain snp divide testing see run observation select approximately day tree use test snps observation computational cccc total entire dataset observation snps testing accord training computation important snps random setting snps approach roc consider chapter genetic study projection follow forest conclude discussion apply projection nn dataset observation table result fold method reduce norm result value result roc result high value norm result area nn method validation meaning would two true area f area roc area roc feature forest observation apply entire regard result accuracy roc selection result table score three compare roc use snps area forest measure snps forest plot roc curve right consist experiment snps area roc snps area discuss table low nn successful predict disease dataset snps accuracy cross mention predictive bit genetic dataset hand snps observation able area curve curve snps previous score genetic
prior frequency run file partition classifying record record partition record one partition true partition measure proportion pair classify precision truly evaluate performance detection linkage big result present row correspond measure partition th percentile line refer recall gray solid average line show average average th depend amount identify contain file field naturally precision file general proportion false field file generally sensitive insensitive amount small poor mean truly detect file seven precision somewhat insensitive term precision panel easy truly prior indicate amount error potentially obtain specification performance trade recall prior wind prior indicate actually end simulation study file contain level intermediate list make publicly available thesis green utilize optical character technology transfer list list contain part current describe record database name name date death month article record file specify record believe field therefore truly confident name neighbor approach boundary two
basically estimator lasso isometry definition mse pack number application particular problem error projection contrast paper match error detail optimality estimate noise collection regression particular application among rank recommender system note write value motivate jx accordingly constrain square norm equivalently appendix packing pair see method match far sub optimality observe sketch vector abstract observe approach quadratic computational optimal accurate respect square solution set direction play important function unit norm important role sketch fashion sketch tolerance final sketch hold reference let matrix constant value estimation involve previously standard dimension sketch sketch
measure updating update must partial q distance assume topological space open question intelligence power otherwise distinction agent agent relationship mapping although satisfy exactly stage final minimized state indicate entropy mapping projection outcome total
candidate parameter ratio contaminate robust regularization grid parameter candidate need solve convex calculate present svm svm affect robust svm tends outperform svm choose straightforward grid search mean margin clear conduct another split test minimize prediction select computational svm necessarily classifier robust parameter carefully test outli svm train outli svm svm train outli svm svm svm present property compute robustness investigate show prediction classifier loss method another develop optimization although dc scalable deal massive
convex subset parameter convex entire show crowdsource function axiom axiom non convert aforementioned axiom applicable well exist inference crowdsource task axiom example two axiom section identify convexity would crowdsource satisfying axiom satisfy model easy objective answer non interval answer monotonicity worker ability exist decrease increase infer answer capturing
complexity capture behavior disagreement reciprocal survey summary behavior something maximize datum calculate combinatorial see section proof include kind star connection free growth literature large brevity may say star completeness nonempty star cardinality star equivalently data graph set node vc degree clear star guarantee exist bound gap generally infinite briefly calculation star classifier I least also contrast classifier I x align embedded star lie intermediate range x w h x hx x ready article upper low minimax abstract dependence logarithmic upper meaning reader logarithmic represent upper formal include mention comment regard theorem sketch underlying bound comparison passive aside refinement label mild case noise surprising primary prior root wide spread complexity active learning depend nearly passive complexity active know passive section thus hypothesis easy threshold classifier improvement passive classifier passive case literature label complexity exhibit spread literature star reflect passive problem passive class trend admit passive class passive come minimax complexity passive upper reveal sample passive improvement factor learn essentially logarithmic spread complexity long indeed vc dimension exhibit hypothesis classifier examine spread complexity increasingly complexity roughly increase strong improvement passive improvement dependence passive complexity learn naturally hypothesis complexity hypothesis class active complexity exponential though extent star roughly aside thus literature easy hypothesis passive reflect improvement consider regime hypothesis roughly aside make distinction easy hypothesis hypothesis always logarithmic label factor passive improvement nonetheless distinction easy begin hard dependence factor easy dependence dependence argue sometimes induce label gap low construct example class span gap instance sufficiently tight logarithmic suggest tight factor sx strong namely strong follow immediately fact refined loss generality introduce measure distinguish proof follow bound construction logarithmic sx tight bound embed variety machine maintain vc instance theorem tight another interesting implication separation classifier note result particular dependence reason separation interesting hx xy indicate achieve specific beyond discuss section complexity logarithmic refined linear unclear achieve separate generally hope bind improved match remain open extent restriction aside aside upper prove several bound
far corruption measurement additional signal smooth recovered signal interest graph assume require rank happen require sparse express bound formulate follow control frobenius quadratic nonzero recover use total reason computationally norm quadratic separate non graph form slight abuse reflect total recovered smooth low frequency rank force graph redundant minimize norm force magnitude coincide unfortunately hard replace sum sparsity minimization property practically alternate multiplier intend formulate augment iteratively update alternate leave summarize implementation measurement output signal stop criterion satisfy satisfied multiplier backtrack every element singular hermitian transpose stop consecutive cost completion review principal
look matrix summarize difference graph mainly pairwise random link topology central focus whose permit cast topology weight vertex respectively edge signal assign scalar unnormalized combinatorial graph laplacian graph particular column eigenvalue equal laplacian via generalization fouri graph extension tool detail smooth signal equip laplacian smoothness edge adjacent signal vertex weight consider strongly smooth supervise model statistical model try explain observation potentially unobserved q represents observe represent control signal adopt isotropic zero give classical key laplacian latent definition reflect link representation graph since many partition fourier
brevity htbp dictionary top dictionary element dictionary similar dependent parallel dictionary redundant layer propose dictionary convnet layer classification category infer send kernel cross model complicate layer imagenet sift training deep within bayesian map enjoy efficacy develop project high image accomplish mnist result near deep design jointly gpu scale deep novel probabilistic pooling operation integrate refinement
eq completes mention range consider random variable complete give method multiplier apply solve lagrange optimization monotonic far complete proof ready prove proof iterate achieve ready differentiable eq q conditions complete define follow kt iterate hoeffding satisfie meanwhile result kn arrive complete result deviation obtain kn k compose k n k kf kf z z difference ng similarly arrive least complete corollary engineering usa school science technology r china china
reach ergodic produce string bl note bl convex hull vertex produce string compute step derivative trace definition convex hull choose vertex generate associate string distribution state structure symbol find symbolic derivative define q terminate necessary connectivity output give asymptotic complexity essentially identical complexity input stream alphabet note corresponding take metric respectively stream generate probabilistic dependency process dependency learnable follow denote need result theorem coefficient dependence clear demonstrate directional process reveal possibly flow causality call stationary evolve need distinct process map imply encode inter dependency machine introduce notation infer machine string generate denote stream simplify coefficient give label strongly connect string run use establish stream run distribution current stream break j ik j dependence distribution stationary index equivalence minimal encoding strongly connect label g minimal converge stream state satisfie recall complete coefficient avoid composition correctness complexity inference complexity statement denominator appear ergodicity denominator surely infer bound surely refer assumption infer also see I cross establishes immediately establish small important bind symbol also imply relatively rare neuron network infer predict evolution evolve alphabet standard interested evolution notation accordance denote b
various hmms efficiency readily method seek active way possibly reduce try heuristic intuitive well practice exactly heuristic error largely theoretical result concern allow establish active strategy cost reduction maximum posteriori hmms bs brief active bs hmm study inference analytically essential hmms flexible tool bs insight demonstrate analytical map bs determine efficient scheme error remain unlabele allow examine active bs relate
ap fc yes baseline ap fc fc baseline fc fc sp yes baseline l method cnn acc kernel yes color bag yes ta yes pt table car car person person car person person person cat car map classifier activation local activation patch activation score train several confidence verify encode discriminative image patch discuss sec map utilize localization pyramid activation train several take characteristic activation consideration fisher uninformative however contribute invariance meet activation multiple reasonable equivalent filter enable pool multi
non distance close expression metric approximation indicate student test svm accuracy indicate score respective test c c dataset svm eq ten uci dataset learn former feature rkh respectively metric mahalanobis proximity learn distance mahalanobis cosine optimization computation finally compare classification multi multiclass svms psd computationally dataset low method set initialized identity initialize simplex satisfied default gaussian parameter parameter select
smoothed formulation variety weight need annotation latent annotation initialize approximately object manner effect initialization procedure correct feature box occur image short seek dense image find subgraph combinatorial encoding present signal old address description concept share information formalize flexible help box integrate cover positively versus generalize combination mode object appearance distribution image
emphasize solution denote optimal separate goal sample datum space q select unsupervised set full datum set main tool set score bss greedy name column v potential eigenvalue define iteration index include spectral rescale lemma theorem construct rescale top carefully e proportional norm singular select trial dominate time min rp
channel distinguish widely spaced artificial factorization gain source lee similar drawback stacking view instance spaced wide resource require resolution author point east west poor source issue vast majority extension audio allow structured decomposition dependency error etc paper could combine simplicity version nmf main
include several extensive datum represent network sequence denote adjacency direct general self e denote time respectively write quantity notation indicate node member node vector submatrix relation stacking index static consider snapshot time parameterize node relation node adjacency give random block dependent estimate rewrite number priori ratio estimation method sample switch spectral heuristic combinatorial possible class membership utilize adjacency membership temporal model call hidden extension conjugate initialize multidimensional approximation allow involve static namely blockmodel use extend kalman model decomposition
tackle limitation enhance software effort model present report rough extraction software project software software influence project effort fuzzy system enhance fuzzy enhance estimation artificial ann diagram help regression logic neural b boost effort promise comparative radial basis neural experiment carry dataset well regression genetic algorithm select optimize simultaneously improve effort al
convolution implement interpolation support input output transpose vertical zero input height width intuition summation reverse round operation weight combination element filter cycle right transpose filter store slice max channel patch operator sum detailed sect support convolutional equivalent extend datum zero boundary compute relu compute implement operator normalization independently location channel channel input dimension operator channel convolutional implementation implement batch somewhat whereas process image individually instance batch case array treat tensor additive map implement channel feature neighbourhood adjust must normalize section detail compute
work experiment recurrent ht module module neuron fully module module sequence modeling focus rnn long spirit rnn instead simplify introduce additional recurrent lag additional help bridge lag train difficult run slow term lstm architecture store error connect new gate network gate decay forget gate successful recently stack hierarchical hierarchy equip temporal
tell norm proceed induced atom norm follow immediately proportional know svd j sum symmetric non show psd span psd case would existence eigenvalue constant psd write psd matrix psd optimal decomposition positive might positive exist differently hull include cone nx ni generalize norm simple identity definition square replacing hand eq back finally give triangle bind denote sequence h fr fr ts inverse increase bound bound nonnegative scalar start prove tangent tangent closure inclusion prove k b duality corollary x x scale subdifferential deduce hull note tangent cone proof dimension part notation section appendix ab dimension normal cone subdifferential characterize cone introduce notation follow denote onto respectively form aa bb subdifferential write equal inclusion belong measurable characterization statistical give measurable provide cone fact freedom deduce simplify notation become belong sufficient g characterization prove inequality operator work know follow j I ij g j j v g g b u I derive
aic bic dimensional repetition examine weak pn interaction tn generate fit linear interaction typical example view ten form ten covariate oracle argue dimensionality prohibitive implement regularization fan candidate different oracle working benchmark select oracle criterion recover effect report portion portion simulation save positive negative report dimensionality tend criterion improve select small effect meanwhile interesting see correctly measure include supplementary several interesting performance specify except aic reasonably newly consistent selection misspecification interesting multiply oracle effect
trait control share evolutionary serious public health burden understand throughout evolution analyse human binary status assess trait trait pair correlation coefficient contain correlation genetic linkage trait present correlation trait correlation analysis reveal strong trait kb genomic among trait trait reveal history spend transition occur across specie binary trait pair definition correlation trait attract history question play investigate trait evolutionary trait population trait color orientation trait trait population evolutionary transition order latent include analysis use integrate strength traditionally alone draw trait matrix brownian
pt experimental rank normally sample zero normalize incoherence vary fix computational show scale increasingly dense fact hard intermediate intuition confirm show intermediate iterate frame time next foreground separation video form frame stack wise background static form foreground dynamic benchmark name restaurant dataset frame resolution extract several people near desirable
reward previously denote sect full portion maintain mean arm node directly observe q intuitively second account estimate value tight present identify along reach optimistic leaf node alg corresponding expand leaf arm maximum arm contain term add reward term point term become first mean uncertainty reward dominate reward amongst arm resolution need approach choose become occur sect supplement discussion expand two child lead accurate leaf create select big arm single episode optimistic outside optimistic remain unchanged node bound validity material begin need eq uncertainty resample internal optimistic fact second alg force optimistic become notice particularly critical choosing resample complexity theoretical feedback round upper iid iid find representative
snr sound model plane wave unknown arrival arrival isotropic spatial function sound component coherence aim estimate snr coherence sound field coherence first q estimate factor show coherent e coherent cdr signal bottom page omit cdr amount
length keep suffer challenge problem suggest length phase plan uncertainty margin appropriate regret solve provide explicit cost calculation paper control long arise classical parameterize sample start prior assumption positive semidefinite denote semidefinite main linearly parameterize dynamic choose kernel tw smoothly past know mx meet indeed latter markovian change example fit vector dynamic action select average loss e boundedness
feature large propose basic picking update instance feature norm adopt impose strong adjust primal convergence average follow let q tx execute factor make large robust unnormalized vector average describe method single coordinate update mini strongly lead omit technical considered optimization convex function develop dual convex appropriately obtain accelerate saddle form let batch update coordinate update update accelerate subtle primal auxiliary variable replace stay compare fact imply assumption batch equality bad become match case discussion order bad batch
combine feature exchange reversible implement name double reversible jump substantial gain double reversible jump enable model previously cost remain limit number propose direct efficient representation conditional elegant hasting bayes sampler cast bayes birth death set add remove associate birth event occur coincide provide substantial improvement status algorithm estimate connectivity
base localization improve initial notably degenerate class class remove point notable localization angular resolution increase manually position minute record minute propose h move scenario white move number red circle position correspond face video research world apply sbm method slide allow source direction fig frame number live manually select analysis center number high amount yield observed video speech activity detector adjust size segment sound source circle localization implementation face detector implementation cpu face detector annotation magnitude method standard histogram interestingly detect visible clearly may complementary visual face speak face even face ccccc wide field camera location location notice room impulse response circle find method face team research train system room room environment therefore likely capture room impulse response rely acoustic world testing position room impulse occur position moreover camera large right view room distance scenario online select improve use error average exclude large
english actually represent negative sentiment explore provide split involve classifier small sentiment explore effectiveness observe obtain fraction feature hope would guide future sentiment project number engineer university computer engineering engineering introduce sentiment date consist review rate star investigate property sentiment provide split validation testing rating unbalanced setting extend comprehensive classifier sentiment compound sentiment word explore available internet subject product book medium ever active sentiment among study classify piece opinion e either sentiment rating predict star
subject weight visual face multi modal refer comprehensive drawing trial faces filter raw hz hz ms stimulus trial channel classification decode regression show cross accuracy high leave pool trial use logistic penalization observe drop decode sg great table second trial
self normalize variance powerful alternative instance property consistent cite self subsample small necessary quantile estimator role study sensitive key ingredient finite product subsample asymptotically normalization nan directional statistic following distinguish brownian motion assumption pr dr multiple quantile depend process difficult implement nan simulated significance level percentile theorem local non p hold local alternative cross intermediate include reflect historical event ease rest dependency present lag quantile lag li k notice obtain analogue
take weight choose advance iteration store distribution spaced number grid value propose drive low computational trivial bayes approximated shape require implement efficiently perform hyperparameter stream derive update prior streaming update good covariance denote matrix wishart freedom parameter wishart joint lead expression conjugacy posterior assignment history
top identify identifie layer identify remain vertex cube specify vertex layer vertex identify contain identify identify main theorem review sect obvious always create maximum vc number dimension cube must use inverse move towards zero exist movement vc vertex shift cube anchor contain coordinate value notice preserve number fact neither must vertex easy number decrease offer subsequently sect existence projection sect vc embed vc iterate cube direction graph embed cube node face direction node contain correspond tb iterate every iterate reduction iterate color project coordinate correspond cube edge colour come cube complete cube iterate firstly maximum class reduction view time
term discrimination outperform focus development calibration probabilistic traditionally machine development improve discrimination improve prediction important make decision analysis probability outcome model method effective machine calibration modeling make may could lead addition affect calibration calibration parametric method parametric method model probability intend calibrate learn maximum likelihood distribution common non briefly associate estimate introduce method
power negative positive example argument language language trace rely positive large language language form language language build otherwise second family language language language language program intuition synthesis synthesis language language synthesis engine form language language synthesis technique access produce history recover easily program follow element far pick minimum see synthesis engine program return assumed iteratively discover eventually every positive singleton contain form large positive see previously candidate trace synthesis engine program language consider recognize language trace minimal observe z forget observe hence program class program fail program program synthesis
sa contain exactly solely learn call transaction factorization attribute attribute entity entity dimension attribute sa arrange tensor preference user item explicit preference rating typically sparse rating information entity occur cell negative preference assign real entity combination efficiently restrict training weight entity actual entity factorize entity dimension entity therefore actual although sufficiently generic weight leave exploration occurrence basic accurately predict entity one weight generalization square preference allow experiment usual framework vector consist hadamard product product linearity apply feature consist implicit method arise feedback possibility decompose computable part computation follow latter alternate usually accurate fast main square thus scale make conjugate cg square cg solve see linear generality base whether element reach column similarly simplify equation entity dimension weight sum difference part efficiently expand sum product argument feature vector rearrange feature scalar note change update solve conjugate high description
g curve give successful extraction source canonical correlation generalise noisy signal eliminate effectively blind implementation derive extract consider indirect preliminary research engineering university uk uk blind source establish
due induce high iteration good file despite fact file incur call observation accurate algorithm exploit popularity profile exploitation skewed popular file multiply also reflect period empirically content file cache cache replace file since cache applicable directly period file within period note history learn past period numerical mab section greedy terminal service average cache system otherwise cache capacity unit file user set file file size user uniformly skewness memory cache percentage refer content cache time algorithm switch greedy greedy algorithm mab plot lack theoretical practically steady switch figure mab mab switch period greedy counterpart greedy arm period reduction switch opposite ht mab inform popularity profile know advance cache horizon
integral case pathway variate density cm integral international journal journal pathway pathway h apply function pathway transform special pathway mathematic cm fractional integral perspective j york cm top bottom com mathematic university west cm mathematical science com outer united international cm
reward assume reward lie round apply martingale reward round decision holds obtain take wise optimal corollary bind regret end majority sr sr
differ proximal correspond hierarchical show proximal remarkably form write observation proximal solve exactly duality furthermore operation include special problem r method compete var modeling forecast dimensional fit square lag use aic per aic residual matrix lag selection follow square simple simply include var penalty lag lag lag lag pattern lag serve baseline unconditional sample ahead forecast form walk efficacy application evaluate method scenario component length describe
improve error medium sized modification produce strong test absolute unlikely grow bernstein sum variable empirical bernstein bound develop bernstein bound version variety bernstein small range variable random standard hoeffding bad increasingly likely binomial page bound advantage low q truncation inclusion refer depth depth multiply instead side bernstein binomial input randomly v
represented truncate normal tn distribution tn property mixture contamination scheme whereby amount proportion bad component g figure contaminate skew ignore write analogue respectively include fit contaminate schwarz good
estimation mse setting table display improvement significant mse improve primarily propose value similar unable abc currently bivariate analyze objective take recorded record refer ht ccccc represent customer propose bivariate beta binomial denote requirement joint beta distribute notation binomial introduce bivariate distribution gp b bf ep parameter determine valid beta form enable use bivariate propose kb beta beta furthermore
line line result performance recommender result plan investigate public mobile aware recommender model exploration exploitation exp current preference improve knowledge recommend user risk introduce name ucb user adaptively balance reveal exp feedback click near people become optimize mobile aware recommender learn may critical exploit appear frequently see select environment prevent maximize reward reward uncertain environment prevent formulate exploitation exp one solution exp arm hybrid combine confident ucb estimating interval reward algorithm document confidence essentially control difficult
completion completion compare well leverage perform repeat completion time attain weight completion procedure target output conduct effectiveness generate freedom covariance whose entry index observation noise unweighted collaborative collaborative filter datum unbalanced violate general experiment estimate score unweighted unless portion collaborative commonly unweighted compute solve repeat type report truth rank subsection test score coordinate synthetic descent vary sample set take row hinge loss coordinate respectively ht leverage result weight well intuition weight whether alternate weight procedure four ht round coherence perform procedure completion two set experiment
right dot node size right dot transform shape cm node right cm right b distance cm input node distance cm distance cm boltzmann hide bias hide denote input probability interaction visible differ bias share trivial regard wise normalize distribution namely yx px px approximate conditional represent conditional principle otherwise ambient polytope expect dimension triplet visible jacobian parametrization mild piece wise linearize version dimension rbm apply idea order denote cardinality whose small cardinality every hamming apart set implie joint know whenever conditional q proposition universal vanish imply conditional account follow divergence universal approximation analogue
shape three primitive record front fix isolated protocol consider illumination characteristic report recognition illumination canonical pm tensor standard pm characterize point product manifold high singular factorize represent modeling select approach perform poorly conjecture illumination hand tb use pm approach pm video traffic pattern light weather video record resolution range frame normalize version dataset normalization involve subtract mean normalizing intensity illumination traffic traffic fig select respectively method also compressive dynamical present approach obtain achieve worth et tb example traffic light heavy traffic video spatio compressive sc perform two simple property euclidean experiment realistic sample follow mapping back map create fix class variance problem medium hard give multiclass size map manifold propose sparse code approach task repeat ten tangent experiment sample turn recognition consider setup center projection sc fr characteristic prior approach
I word scope system collection use text specialize kp specialized specialized public public string double I I I else return j dp return use public count reverse public string return add dc else count return string solve string return improve collection use text dp vx xy long vx vx total total break vx scope write human understand human primary generative tailor base probabilistic extended variety baseline likelihood hold great deal human effort go develop develop tool development fast tool code problem outside machine public massive collect assignment thousand observe think code human human regularity combination work primarily
deep improvement unlabele minimal effort receive attention label train target improve performance deep bottom extend beyond structured output performance reconstruction objective prediction label bootstrappe prediction thereby structure may useful approach bootstrappe output proposal person agnostic region deep noisy multinomial softmax without noisy label addition log add encouraging multinomial feed forward posterior use softmax denote noise softmax follow learn
scene illustration independent human inference result mesh super impose image evaluate category internet significantly outperform dataset object internet superior current object inference transformation mesh challenge ask manually fit image mae score invariant surface mse much utilize approach attribute slight ground inherent object demonstrate b prior real naturally much part collect human person category internet comparison current art
rmse evaluation criterion order experiment result see vi design topology knowledge extract factor inclusion three model bad experiment layer eight verify offer design topology exploit properly knowledge stem good indicate nn network build fig omit modify magnitude negative large notice interpretation especially thick immediately influence neuron influence strongly neuron nd neurons layer affect moderately outcome quantify variable like valuable leave research try neural random forest achieve comparable necessary unsupervised incorporated try
situation variation miss simulation show simplified pooling yield short confidence mathematically trivial aware instance practical
weak tell efficient expect know perfectly predict expect size ensure tractable matter representation large etc statement fit hard possibility achieve unknown e agnostic edu demonstrate play central role multi feed forward argue analogy inductive induce sort encourage success expressive relative learning
extract pca percent improvement percent collect component ignore high precision optimization reduction localization position capability infeasible gps hardware gps service environment location location proximity localization measurement rely angle hybrid utilize various technique signal range technique provide movement location eliminate estimation surveillance object system divide define widely localization figure recognize location hardware manual reference transmission strength indicator reformulate discuss seminal summary column specify localization moderate aware moderate centralized localization centralize purpose soft localization moderate localization distribute localization distribute purpose localization svm localization fusion acoustic surveillance surveillance gp distribute spatial collective space som moderate localization som low centralize less distribute distribute less determination rl mobile localization correction method predict likelihood applicable system network thousand localization appeal investigate fu activity use phone music convenient way name ambient intelligence human home device automatic power management core classifier detect robust localization activity manually activitie limitation centralized system recommend investigate unsupervise automatic extraction localization scheme particular multi layer perceptron radial recurrent rbf resource requirement mlp likewise sensor node use anchor utilize system adopt node localization localization ability g alternative non probabilistic precision predict cost error device illustration mobile localization employ connectivity capability method movement movement detection localization design goal indicator though offer distribute effective outlier limit datum therefore idea sub start divide I predictor sub addition computational robustness preferred low tree develop target exact location target difference arrival tree also event
particular equation q langevin evolution fluctuation incorporate constant correlation root langevin equation convenience langevin easily arrive langevin concern markovian present mathematically markovian represent probability histogram ref concern moment average
x n f w encoder reconstruct conditional hold output describe symmetric model encoder decoder tie explore autoencoder denoise input autoencoder learn feature learn input train translate filter phase shift loss input entropy differentiable optimization momentum
one constant predict capture seem unbiased line metric popular regression tree package cart introduction forest longitudinal datum one subject computed entry cluster obvious accuracy cart forest tree forest subject cluster list close spline relation bayes forest besides distinction forest estimate test converge similar accuracy lastly focus area bic factor mechanism complicate necessity review attractive inclusion however
vpt def bl vpt fill vpt bl copy vpt arc fill bl copy vpt copy copy vpt arc fill vpt arc def bl copy copy vpt arc vpt arc bl copy vpt fill vpt arc def bl copy def bl vpt fill arc bl copy vpt arc fill vpt arc vpt arc def bl copy vpt arc fill copy vpt arc bl copy copy vpt arc fill vpt def bl copy vpt arc fill copy arc fill vpt arc bl copy vpt arc fill vpt arc c bl copy vpt arc vpt def roll exch def square vpt exch vpt vpt bl vpt bl def bl copy vpt square bl vpt exch vpt bl vpt exch vpt vpt def bl copy vpt sub vpt vpt square def bl copy vpt exch vpt sub exch vpt vpt fill def bl copy exch vpt vpt sub vpt vpt fill def bl vpt sub vpt vpt vpt copy vpt bl copy vpt sub vpt def bl vpt vpt vpt fill bl copy vpt sub fill copy exch vpt exch vpt square def bl copy vpt vpt fill copy exch vpt exch vpt fill bl copy vpt exch vpt vpt vpt def bl copy exch vpt vpt vpt vpt copy vpt bl copy vpt exch vpt vpt fill vpt exch vpt def bl copy fill def translate def stroke def translate stroke translate translate stroke def translate translate stroke translate stroke translate def translate stroke def translate stroke translate translate stroke translate stroke stroke vpt add vpt vpt vpt v def stroke exch vpt vpt vpt stroke def vpt mul mul vpt mul v stroke stroke vpt mul mul vpt stroke translate repeat stroke def arc vpt vpt vpt vpt vpt def exch exch vpt vpt stroke def stroke vpt mul vpt mul mul stroke def vpt mul sub vpt mul mul vpt stroke def stroke stroke arc stroke def exch exch exch exch add def def def def fill fill roll def get get get get translate mul mul def translate mul ne get add roll stroke ifelse true def def ifelse def def exch stroke stroke exch l fill exch def def l stroke exch stroke exch def def stroke pattern def pattern pattern landscape ifelse def landscape ifelse def landscape ifelse def landscape ifelse def ifelse def symbol length begin index def ifelse end begin def exch exch exch def roll exch def sub mul def mul sub sub def mod ifelse ifelse ifelse ifelse ifelse ifelse def constrain exch ifelse def add constrain roll mul exch mul constrain roll exch mul add constrain roll def rgb exch exch exch roll exch roll def copy mul add exch exch constrain roll copy mul exch mul exch mul roll mul mul exch add roll def ifelse ifelse ifelse true gidx gidx gidx gidx def loop def gidx sub def gidx gidx get mul def gidx gidx gidx mul def gidx get sub get gidx mul add gidx get le gidx get gidx def ifelse def def mul ifelse def pm gamma def stroke pm exch def stroke pm cf constrain cf constrain exch cf constrain ifelse pm pm ifelse ltb stroke ltb r ltb stroke ltb r stroke ltb stroke ltb stroke v ltb stroke ltb stroke ltb stroke v ltb stroke ltb stroke v v v ltb ltb ltb lt v v v v v v stroke lt v v v v v v v v v v v v lt v v stroke v v v v v ltb def exch exch mul roll exch mul mul def mul mul mod ifelse ifelse ifelse ifelse ifelse ifelse def constrain lt exch ifelse mul mul roll exch mul constrain roll mul exch mul roll exch sub roll exch roll def copy mul exch constrain mul exch mul roll mul add exch constrain roll def ifelse ifelse def gidx gidx gidx gidx add def def gidx get gidx gidx gidx gidx gidx mul def gidx gidx gidx sub mul add def gidx gidx get gidx mul add def gidx gidx def ifelse def def def pm ifelse pm def color stroke pm exch stroke pm constrain exch constrain exch constrain def ifelse stroke pm pm ifelse ltb stroke ltb stroke ltb stroke stroke ltb ltb stroke ltb stroke ltb stroke stroke ltb ltb v stroke ltb stroke v ltb v stroke v stroke v stroke ltb stroke ltb stroke ltb stroke stroke ltb stroke n stroke ltb ltb ltb v v v v stroke v v v v v v v v v v v v v v v v v v v stroke v v v v v v v v v v v v v v stroke v v v v v v v v stroke v v v v v v v stroke v v v v v v v v v v v v v v v v v v v stroke v v v v v v stroke v v v v v v v stroke v v v v v v v v v v v v v v v v v stroke v v v v v v v v v v v v v v v v v v stroke v v v v v v v v v v v v v stroke v v v v v v v v v v v v v v v v v v v v v v v v v v v
sensitive polytope convenient factor recommendation take actual one base decide implement strategy nontrivial perform experimental effectiveness impact design burden
corresponding argument observe round deterministic determine action round thus entropy relative equal relative recall determine pick adjacent go distribution coordinate node bernoulli l strategy plugging bind overall low least expression pick constitute round random neighborhood node need arbitrary expectation permutation selecting subset node term define arc consider j r randomness lemma outer remove conditioning conclude shorthand round outer expectation bind contribution overall quantity take bounding summing moreover action rearrange sum q I recall recall adaptively fall order overcome exp apply trick pseudo variant exp since guess run observe sum pay pay get take theoretic fairly analyze setting counterpart hold undirected direct let number induction start independence incoming arc obtain sequence small small sort node arc g state edge undirected refer arc recursively
x mnist original reconstruction largely sample heuristic multi heuristic well shrink shrink heuristic similar representative show note near optimal scale show efficacy heuristic pc elimination svm execute heuristic execution overall speedup comparison process original scale x speedup multi perform little execution approach spend reconstruction trend observe process dataset comparison set fair conclude pc extract shrink set shrink beneficial different set several hyperparameter fast elimination sample benefit shrink research shrink step shrink degradation shrink
might assign likewise test compare reward opponent pair test algorithm course reject ks rough ready merely mind interesting trend per opponent generator similar instance quite especially even block similarity show weak similar twice similar experimental metric study prominent us relationship converge consider equilibrium game relate payoff opponent finally tendency profile nash equilibria infinitely game difference could play static pure receive good static pure opponent play look regret record payoff respect significant thing ignore great reward learner always iteration approach infinity algorithm achieve lack regret achieve regret experiment algorithm positive achieve positive regret low gradient lowest achieve regret follow result practice come way see regret run negative slightly hand rarely run overall average regret converse attain take precisely wrong run involve around self outcome play claim break cycle improve per achieve low every distinct opponent well play converge pure opponent trend opponent regret dominate dominate gradient never dominate dominate opponent converge nash equilibrium self play nash equilibrium course play response hence dominate vice versa dominate dominate define another regret
factorize illustrate bilinear logistic bilinear logistic construct square become projection variational generate interpret mapping equivalent jk priors variational problem incorporate bilinear regression objective bilinear sparsity depend regularizers one explore future reason prior bilinear three bilinear ambiguity estimate estimate arbitrary rank unique ambiguity introduce logistic originally reveal spatial neural generating localize correspond reasonable sparsity third noise logarithmic concern regression intuition generalize show improve variational fix
lead quadrature quadrature intensity count component thorough study quadrature perform prove particularly realization localize section kernel estimation quadrature adapt inaccurate estimation mass spread study phenomenon financial issue frequency financial rather translate kernel second decay quickly small around much slow lag explain adapt clearly first step lag behavior precisely basically consist grid interpolation check approximation bring algorithm next last sensitive quadrature wiener adapt quadrature roughly kernel else precise estimation whole quadrature use point far able solve numerically consist point quadrature test simulate dimensional process simulation period second quadrature kernel kind multiscale kernel financial decade second
penalize need fix intercept control height dividing carry issue maxima challenging depend complexity consider quantification estimate perform latter avoid asymptotic number pointwise simultaneous bootstrap replication simultaneous confidence pointwise confidence contain fraction completely parametric hmms hold usually satisfied specific general condition zero guarantee domain sufficiently deal induce unobserved markov specific easier similarly distinct describe fit smoothing adequate datum drive leave infeasible generate
infection semi instance one could contact spline drawback involve cm school mathematical sciences ng rd uk mail uk occur methodology rely propose b illustrate show epidemic pick incorporate parametric disease prevent major high contingency planning past see mathematical disease understand disease nature disease fundamentally disease due ii little parametric assumption quantitie transmission infection period
draw black sep bl dot distance distance dot v circle width black inner sep cm fill bl dot dot h right h c distance visible layer cm rbm remainder comment present architecture extensively one feedforward hide unit understand composition unit tune input unit superposition linear unit see address accuracy feedforward maxout unit besides deterministic assignment give arbitrarily sufficiently stochastic see application understand development address impose subsequent connect restriction version approximation enough related minimal rbms nonetheless deep
method algorithm coupling e provide couple begin briefly present coupling diffusion suppose solve solution wiener univariate wiener finally orthogonal case need choice solution obtain stochastic indeed rotation however successful bridge close norm orthogonal onto vector geometric interpretation difficult imply wiener increment wiener increment increment minus increment increment increment direction increment wiener plane drive wiener meet treat define ensure really necessary bridge simulation rejection certainly ensure geometrically fast somewhat restrictive wiener initial path univariate wiener formulate review density function hence stationary diffusion solve stochastic differential condition ensure discuss lipschitz twice continuously transition partial
focus incorporate configuration detection within prescribed optimize auc positive trivial extend boost denote bold bold set letter entry tuple weak learner weak learner represent respectively predict output represent output learner training partial curve j learn learner score performance learner learner object practical successful slide main adaboost aggregated channel combine show outperform train bootstrapping train repeat bootstrappe classifier pool optimize roc efficient region proposal generation low descriptor spatial propose modification benchmark set pooling prove pooling pooling combine nearby statistic region preserve neighbourhood detail pooling learn lead state art object learn promise still time consume application low adopt simple operation enhance descriptor descriptor apply low provide relationship represent feature represent low derivative axis edge orientation orientation orientation feature redundant would select preliminary alone bad performance map image descriptor region store texture descriptor binary histogram version thresholde neighbourhood centre pixel
acc laplacian mode mode ex ex mode shift acc mode acc laplacian mode acc mode homotopy acc laplacian modes centroid interpretable centroid hyperparameter initialization centroid pattern surprisingly centroid object centroid branch manifold mode shape identity centroid overlap class expense class represent centroid receive centroid partition smoothness prevent centroid digit identity centroid move area homotopy assignment separate object achieve amount centroid mnist centroid valid image identity yield representative even mode produces shift nonconvex shift spectral find centroid pattern lie area individual
population baseline cognitive knowledge purpose eeg different baseline ec state subject room ask eeg data channel eeg signal subsequently apply anti pass restrict follow standard exclude retain subsequent analysis select available future condition ec eeg non epoch epoch consider observation state extract spectral assessment connectivity detail respectively eeg investigate purpose transform obvious physical eeg eeg signal extract epoch slide overlap psd psd ft characterize eeg namely cover eeg delta alpha
long remainder find negative baseline relative error indicate dataset hence claim include appendix average method performance large error extremely good run within dataset quite consider large see statistically efficient computational b appear computationally large implementation twice small primarily interested nonlinear expansion restrict dominate baseline dataset much essentially infeasible intuition adaptively effective sort try algorithm candidate rather parent inclusion also extremely base
change word change actual procedure call gets execute improvement thank pre central quantum phase artificial neural exploitation several investigate order perceptron method adjust weight though quantum equivalent classical quality purpose quantum perceptron onto compare perceptron able
case decision logistic sigmoid converse true return original predictive stock membership capable give stock membership incorporate far speak advantageous probabilistic sigmoid belong initial subsequent disadvantage poor relevant stock return address implement nn close neighbor within close neighbor may maximize quantity empirically speak case classifier single nn subset validation minimize large forecasting suffer bad intractable exist problem boost improve generate capable testing indicator evaluation use model ensemble behind return outcome subsequent classification prediction task meaningful future would application objective augmentation boost
classification validation cifar set consist colour validation size pixel colour channel represent divide validation dividing every get number choose epoch layer model pick da number training epoch da model time turn da combination parameter number step validation epoch da noise classifier pixel unit try outperform da error reduction achieve interestingly schedule detail supervise fine tuning yield low ever cifar error material test reconstruction error epoch hide unit worth
asymptotic present regard family statistic simulation carry comparison shall assume unknown estimating regard multipli lagrange multiplier n also test testing eq measure neighbourhood constant account fact complete assumption law consequently hold true restrict state rejection rule nan give alternative close term contiguous tend manner substitute n f obtain complete consider contiguous relax contiguous hypothesis variable obtain restriction able invariance empirical estimator inverse
bss task decentralize obtain multiplier denote derive auxiliary across bss w form lagrange multiplier three update lagrange multiplier quantity repeat bs neighbor via update decompose individual bss similarly plug simplify specifically become update perform locally iterate converge dynamic decentralize fashion centralize update keep essentially static mcp decentralize modification cut penalty cut bss copy bss cluster resource bss decentralized involve basic eigenvector subsequently execute implement decentralized fashion subsection bs initialize bb kt b start initial matrix decentralize equip th row localize let bs affinity b
hyperparameter eqn row order priori chi imply permutation variable propose modification convenience identifiable triangular loading distribution invariant diagonal entry truncate gibbs
find base hybrid outperform maintain start recommendation mf popular technique effectiveness netflix competition rating much matrix predict rating original previously mf represent activation mf value utilize phase weight simultaneously integrate description latent pure collaborative cf item involve separate weighted average combine depend technique content rating rating pass implementation address build description recommendation system input put backpropagation input weight backpropagation define sparse rating user matrix represent item
autoencoder variety mnist stochastic descent momentum training autoencoder manual search autoencoder decoder isotropic isotropic hide activation dropout additive multiplicative unless otherwise hide intel evaluate generalization autoencoder patch van image noise hide activation evaluate denoise equal average dropout vary activation hide effect learn autoencoder autoencoder structure learn digit
system reader reference article solver determinant contribution new direct solver semi separable matrix determinant separable determinant enable semi computational idea encounter computational autoregressive moving model sparse algebra
loss well equivalent w since agnostic ordering stream prune naive requirement arrive turn natural inclusion verify framework point loss l recover auc notion penalty function online interestingly exploit structural property penalty low framework follow style stress regularize batch problem let update require lipschitz continuous w loss decomposable function would would bound instead
feedback layer attention selective dynamically convolutional sequential allowing iteratively internal convolutional million dimensional space cifar cifar outperform deep convolutional cnns pooling recognition parse extensive stack feedforward bottom visual representation stage tend plausible detector detector object part train cnn evaluation discover feedforward pathway ask belong think answer imply feedforward perform
let get inequality new p c p r p experimentally verify minimization superior nuclear minimization establish observation sufficient condition quasi unique recovery quasi minimization theoretical
arm handle ucb arm confidence j optimistic reward express regret eq ucb comparison play strategy depend precise dominate influence ucb strategy explore arm arm good never stop optimal usually arm explore strength lie arm property exploration budget spread regret ucb play arm non ucb continuously avoid keep reasonable iii budget among promising arm automatically exploration grant ucb algorithm suffer greedy need knowledge reach regret contextual categorization profile core idea paper build
set stochastic sa briefly sa line fast pg sa aim analytical seminal employing govern solution cf sample plain propose try end episode shoot step size comparison policy recursion ode policy section likelihood ratio gradient ex markov single chain parameterize recurrent let sequence encounter optimize measure use simulate obtain sample gradient employ know estimate derivative
task connection establish market market period trading optimisation via market trading objective long market maker market multi period market pricing market maker aim optimisation trade maker quantity treat could apply translation invariance functional become eq pricing rule independent rhs substitute merge scheme converge lead despite behave agent contribute global goal trading sensible concern introduce requirement risk pricing objective optimisation problem risk pricing absolutely well functional essence pricing
input tail slight dispersion deep layer dnn observe inactive small unit input actually transform dnn depth dnn also code show trend activation subsequent hide true activation layer layer compare model model tend tb indicator unit behave hide focus unit number activation code layer tb length vary hidden information trend nearly deep observe unclear code capture enable versus redundancy hide unit length total trend depth reach trend evident parameter trend code length code deep task multi acoustic comprise address fundamental dnn design depth corpus improve overfitte utilize combine oppose regularization experiment suggest dnn architecture competitive specialized architecture architecture outperform architecture locally make meaningful property enable assumption specialized acoustic locally model work recognition task experience interference distortion train acoustic comprise evaluate date acoustic train optimization gain frame corpus reveal fairly dnn small begin encode information differently certainly layer fairly small certain point increase dnn depth yield gain dnn acoustic use optimization suggest dnn hide layer five reasonably strong modify acoustic procedure explore drive dnn acoustic stem acoustic train task dnn map acoustic input believe guide new dnn architecture train demonstrate fairly improve speech language understanding network component build acoustic design decision include offer investigation acoustic speech dnn final speech metric quantify factor task experiment benchmark hour compare network locally acoustic build systems corpus fisher corpora us performance
functional prove prune optimal extend pruning segment neighbourhood decompose pointwise cost condition pruning inequality generalise hold value pruning explain section pruning lead prune also functional pruning pruning demand pruning decide compare binary segmentation implement loss c assess well synthetic algorithm implement segmentation maximum search binary furthermore operation front segmentation point speed possible fast segmentation investigation occur database come different pac expect change profile run change execution algorithm benchmark array time
annotation assignment formalize assignment descriptor video annotation convenience element natural occur index action annotation background list actual illustrate interest regularizer f assigning regularizer scalar parameter jointly assignment follow assignment matrix one annotation put every obtain z indicator class classifier z z ta z ta assign row correspond em single replace skip replace descriptor notation want matrix assignment amount impose figure intuitive illustration column block contiguous occur
suppose bound norm tx similarity choose range wide close argument decrease large argument offset right plot curve big towards roughly big ideally small like big trend small hash maintain inner retrieve query conduct hash one reason popularity evaluate million increment rating netflix million movie movie rating integer form rating movie compute rating matrix appropriately rank call characteristic item row item product therefore rank method product outperform popular recommendation choice netflix proposal provable inner since hash
method tune bic elastic net smoothly bic scad discovery method simulation posterior concentrate posterior attain prior en bic fdr bic row correspond empirical bayes scad row empirical alpha beta pp pr contain pp pr fdr pp pr pr contain old simulation pp pr pr fdr sim pp exact sim pp pr pr contain fdr pp pr pp pr sim pp pp pr contain high dependent demonstrate posterior truth optimal minimax minimax via chain monte carlo simulation
show variance sgd enjoy propose reduce stochastic gradient dataset sgd indicate degenerate traditional prox sgd variance significantly variance gradient vary sdca summarize dual gap sdca uniform sdca converge standard sdca gap test sdca result sdca improve accelerate duality addition variance gradient sdca sdca enjoy sdca significantly might sdca paper study importance specifically sample stochastic sampling show depend norm loss relax smooth prox sdca importance show prox uniform sampling sampling improve rate optimization finally confirm appendix firstly bregman subdifferential inequality use cauchy hand result u divide conclude zhang department university
intensive experiment even practical often hold large outperform region provide valuable especially internet lead generation inherently dimensional data size long capacity dataset instance dimension reality web image word document power rarely document higher occur often presence absence binary representation locality hash lsh
close follow idea eq q keep magnitude drastically moreover also estimator optimal budget almost almost twice close pareto nest choose implementation law perfect hasting impact benchmark nested sampling estimation convergence ergodic dimensional vector stationary u reversible transition sequel suggest several final eventually sample keep theory practically speak approximately serve parameter need become generator explain combinatorial optimisation case reader compute walk simulation hence small implementation alternatively generate sequentially parallel recent sequential necessary
decentralize communication bit statistical instance theorem result binomial family communication linearly machine specify exhibit gap theoretic novel ingredient quantitative sharp upper recent subset current organize formalism distribute devote conclude variable abuse probability mass situation case density variable density divergence shorthand shorthand background minimax variant interactive protocol distribution consider quantile give expectation take risk define well bad via range paper minimax characterize problem sequel impose choice estimator minimax computer assume contain subset limited communication protocol machine local datum potentially past convenient model message central fusion denote collection message send protocol encode send correspond protocol communication variable length protocol distinguish two
irrelevant improve consider choose course pass possibility fold cross simulation illustrate grouping lasso use plus elastic publicly package fold elastic cv package noise correlation entry base average produce record htbp tp ms rmse rt elastic net rt elastic lasso rt elastic net rt lasso
context context element hypothesis correct context word pair lexical vector pair word call vector three use difference similarity reference similarity context vector word hypothesis similarity tendency learnable difference indicate lack similar reference similarity hand something colour thing label learn similarity affect lexical empirically algorithm three bad dataset lexical disagreement lexical issue examine present three detail present implication experiment limitation follow entail fulfil mean substitute occur sentence mean definition lexical present lexical entail able entail strong semantic entail fulfil typical semantic come semantic relation semantic relation outside semantic relation semantic relation relation typical relation agree whether definition lexical typical naturally suggest cat cat house cat word cat naturally mind cat sense cat house frequent sense lexical consider word sentence affect lexical decide lexical imagine definition lexical us relation connect condition relational determine entail entail cat semantic relation exclude correct cat cat imply lexical whether word implication depend semantic possible another implication follow relational lexical connection lexical relational case might cat relation cat house cat words house cat share cat house cat cat cat sense house cat agree cat cat relation say
efficient compute ordinal strength concept association discover association impose nuclear concept knowledge performance unobserve response collaborative interpretability relative tag ordinal sparfa pls provide feedback learner concept tag knowledge profile recommendation learner learner tag status benefit capability material support foundation grant air office scientific award science foundation grant pa f h k b r sec b n z cm sparfa com offer way wherein experience goal date factor sparfa novel framework base learner concept content estimate
distance node draw draw type rather construct tree tree count count action take local tree decentralize induce express flat shot difficult employ weight w otherwise guarantee develop priori widely machine however expect factor approximately information come factored mean show estimate converge plus policy dependent induce specify action global reward set cause value overlap component counting action value overlap expert recover action reasoning bound component assumption component overlap intersection profile profile expert return joint whose sufficiently optimize sequential ucb effective policy property bias interaction correlation
end include dyadic n l n note dyadic interval contain one outline provide theorem provide material find collect extension idea good since number interval help far q suppose anomalous interval efficient parametric basic dyadic introduce interval approximated property cardinality interval construct n pt p x successful interval outline proof supplementary material tp cause
ranking model user score decrease order transform collaborative ranking learn collaborative ranking correspond document observe train rank rank fu ndcg section extraction extraction propose tweet main feature part include rating user movie movie share rating twitter build tweet daily twitter original rating rating include extract feature feature tweet triple
filter individually focus orientation cause activation large try sensitive activate break scale small image small pick opposite column non scale object column work use fig map cifar size activation plot big response get activation filter drop gradually peak response peak size filter size compare activation weight however activate c htbp cifar drop cnn last drop
find cut cut frequency find node recover complementary note traditional reciprocal knowing automatically solution briefly refer graph non prove sampling normalize laplacian perfectly recover signal without identically belong lead clearly signal maximum bandwidth computing searching devise approximated minimizer numerically small eigen reduce complete control detail increase give hand due prove e subset signal converse question uniquely represent eigenvalue relax combinatorial need set hope reach cut understand relaxation define diagonal c hand side
object sequential learning also automatically order compare spirit attribute characterize different task apply variant allow capture task th eight annotation recognize specifie create task easy part create except part versus rest remain task image vs equal amount act negative different feature bag normalize act term repeat split across task mean value baseline semantic diversity marker vertical capture average rate background slice area reflect order purpose refer color number performance baseline treat prototype optimization
sparsity heart replace functional instance case reconstructed capture check regularizer smooth I vision image retrieval indeed compact signature anti propose quantification vector homotopy provide regularization quantization instance wireless optimization general complexity dictionary partly synthesis redundant directly j interpret regularize question solution problem give j minimal partly manifold manifold chain key equality know much complicated quite compact partly locally synthesis popular see comprehensive representation multiscale dictionary build wavelet isolate natural sparse piecewise dictionary make localized translate atom audio cope rich diverse dictionary research overview type regularizer terminology measure signal partly partly relative manifold popular example total semi piecewise image isotropic comprehensive review several generalized wavelet kind multi scale haar wavelet prior compose concatenation adjoint difference compose block signal define block write even block I e cope correlated type enforce unitary synthesis obviously overcomplete report compare distinction sparse theoretical development analysis regularization extension value low singular closed restrict smooth partly group build manifold absolutely nuclear j nuclear rank norm shown partly smooth around include low completion principal component retrieval instance collection relative manifold show also
involve clean eigenvector square associated eigenvector eigenvector eigenvector complete part eigenvector e degeneracy eigenvector column matrix leave equation complete proof distinct eigenvalue eigenvector last eigenvector back original know orthonormal normalize rewrite eigenvector arise dot j j last orthogonal kronecker delta term last state j equation two set arise equation release computationally explanatory examine version svd modern data analysis box sometimes poorly behind box focus build solid manuscript derive mathematic behind informally mathematic
recursion fourier pca interest thus maximum random tool try minimum gap maximum technique dependent process algorithmic learning mixture spherical ica gaussian extension detail improve complexity approach sample complexity tensor include I entry entry gap exploit approach tensor rhs copy recursive decompose projection random find pick eigenvector else ica jx kx jx lx maximum improve achieve apply fourier transform observe respect signal polynomial
efficient contour contour need contour computation simplify circular contour skewness respectively py py yy figure contour da b give f respect lot generalize skewness let dispersion scalar denote third stochastically matrix inverse denote density solve impose special special form write make canonical space depth contour depth family half univariate suggest contour elliptical approximate ellipsoid depth also illustrate contour panel vary
sequence quantify dna rna protein bind genome biology supplementary issue overview technology sequence review briefly double dna follow sequencing basis read end length basis end pair pair come double dna sequencing dna come plus read map genome position template read position read read minus plus read genome reference genome span read map length read pair derive minus read one read length every refer sequence detect copy read genome depend piece sample et simplify sequence read assume homogeneous intensity read genome call copy also content bias zhang process derive sequence jump number jump form symmetric walk increment jump indicate walk excess discuss except unnecessary since proximity ignore statistic effort make genomic reliable drop intensity interval reflect whether statistic involve maximization range perhaps obviously genome copy dna genome copy sometimes variation movement dna position genome site variant parameterize genome template read dna reference pair dna sequencing sometimes variant paired end figure region absence map span read pair apart span genome read overlap fail target genome distant reference read pair map
obviously solve score technique widely study accelerate base literature replacement score sufficient hold probability leverage score
establish asymptotic high setting parameter much assess develop paper aim mostly refer graphical lasso exist dimensional setting lie shall demonstrate carry covariance refer precision goal set candidate even constant exhibit impose consistent model impose restrict precision popular sparsity norm diagonal situation normally
mdps factor proceed policy episode produce proceed posterior factored choose confidence require graphical write return write return mdps remain treatment factored modification structure shorthand r obtain mdps accomplish impose artificial episode whenever factor strategy episode mdp simulation h encoding
come visual fourth discriminative cat cat visual appear visualize visual irrelevant look context among mi rs rs region localization dataset streaming method selection popular svm localization svm entire mi well visually region quantitative localization curve area correctness threshold detection arbitrary shape bounding mask yield localization pr value mi baseline also well linear usefulness
kind begin space type reflect spatial change feature reflect property dimension helpful strategy detect change system indicator eigenvalue slide line match report minimum indicate occur eigenvalue considerably significant occurring may detect subsequently overall instant day start recent weather environmental historical eigenvalue spatial baseline significance signal value cx b principal eigenvector principal c principal min p b mention type previous match setting criterion first fail contain effect fail recent solve typical usually perform construct inference environmental compare recent window current approach window ignore way baseline combination assume environmental setting environmental baseline tensor environmental frequent environmental main advantage fail deal
ii player iii maximal supremum player player inter hilbert open problem easily extend player game mean se determine player robust game determine mean payoff strong maximal supremum player could notation corollary play quantitative synthesis verification single average multidimensional vector boolean prove player finite strategy winner inter player game game partition vertex induce quantitative play specification graph objective multiple objective multidimensional outcome boolean formula atom well study payoff sequence infimum satisfy boolean
iteration one sample algorithm obtain consistently update irrelevant update objective number epoch amount especially possibly use gradient sample secondly training dimension different perform almost sg respect epoch outperform epoch value two ccc q underlie low standard figure recover low gaussian tensor middle slice recovered epoch sample solve shape slice along measurement thus objective figure objective see significantly outperform smooth local nonsmooth low cc tensor sample bilinear eeg apply format
maintain player player likewise payoff query record differ slightly public find private maintain private use instead argue privacy method minimal record versus trade since sample known hardness result release algorithm bad universe theoretically prior work guarantee practical benefit handle solver query data solver several heuristic improvement turn solver still good beyond maintain database let normalize largely routine later tune evaluation ingredient mechanism select output proportional sensitivity differentially privacy score round
unit anchor lda sure sparse show kalman filter separability weak instance column analyze uniqueness mild practice broadly number view j generates firstly draw draw view independently cluster cluster moment column linearity fall solve general matrix completion trick column sampling method perform multi randomly splitting obviously feature information rest capture assumption linearity actually general mixture hmm broadly use series depict chain state generate multi hold generalization emission consistency linearity convert hide current triple state triple build emission hull immediately recover give anchor extend
compute tractable challenge average variational maximum ise variational energy follow fig ik fc angular average energy derivative give element constrain momentum maximization nmf pair method choose cross performance assess
name bs anchor north anchor west delta bs c ex ex top plot rigorous surrogate error bind surrogate plot frequency plot report standard ex ex choose component dependent produce eight produce roughly rigorous perfect furthermore surrogate finally moderate around well surrogate rigorous surrogate e frequency fact compare detail surrogate recall figure surrogate overall converge surrogate characterize large interval compute align font legend column legend align style histogram anchor legend name title iii anchor surrogate gp kernel ex ex correspond curve depict report infer report frequency deviation variance gp accurately case align infer slightly realistic kernel optimistic interval predict correct discuss extremely confidence interval due legend style font legend cell align center label width center title anchor west legend name title ii mode bs north ii relevance infer affect measure surrogate method surrogate important recommend surrogate fidelity surrogate reduce basis decrease apply report figure plot order result surrogate legend legend legend align font anchor legend name title anchor north surrogate ex ex ex gp htp legend font
simulate realization surface observational observation difficulty latent non discuss appealing option spatial observational acknowledgement author university energy office provide valuable discussion approach especially provide thank university technology li special valuable htbp respectively set row show run htbp hyperparameter hyperparameter computationally property km variation model covariate observational grid scientific statistical wind follow spatial efficient mcmc sampler exploit construct vary quantile mcmc engineering sciences university mail result rare intensity public type
hypergraph mode briefly summarize color issue hypergraph count finding efficiently example contrast np optimization np exist appear generic interpret see algorithm polynomial number step need practically result example possibility sample exclude possibility polynomial good mcmc color care need study gibb c step metropolis distribution hasting mh way improve general structure simple bipartite blue edge red couple step exist propose eq add display move small allow move switch move mix choice property mh naive lead poor mix typically truncation edge weight certain zero achieve mixing without require factorization example w thus set follow differ balance improvement mix markov compare mh induce
regularizer regularizer involve connection regularizer rademacher denote mkl learn regularizer classifier I dependent rademacher upper complexity enhance flexibility optimize adopt updating utilize dual
notation permutation transpose inverse multiplying give row row effect multiply row permutation adjacency represent operate way characterize operator transform diffusion kernel follow similarity similarity correspond adjacency associate two graph adjacency matrix trace fortunately solution discount summation summation adjacency think series identify relate kind auto series model literature unfortunately consideration computing similarity expressive determinant kernel applicable lead dynamical
ordinary multiplication tensor tu w kk tensor establish return draw emphasize triplet expectation triplet equation take marginal distribution depend estimator moment estimator try function moment distribution moment result equation importance multi view ii ki ki phenomenon class moment multiple however structure mixture fitting circumstance rank know come allow difficulty try tensor store tensor option significant technical plausible individual argument orthogonal tensor explicitly online mixture interpretation potential
projection preserve randomly project row multiply matrix column suffice factor bound give communication algorithm improve reduction unconstraine application stream picture generalize approximation left row project row span small rank give strong strong line reduce cluster leverage specific specifically project dimension sketch sublinear offer interested know linear overview project top preserve sketch reduce projection result stream singular decomposition leave right singular kk frobenius remainder know multiplying scale imply multiply frobenius repeatedly dimension f semidefinite ni write multiply left project column rank amongst orthogonal projection formally set centroid mean cluster rely algebraic prior dimensionality assign ik construction disjoint rank rank possible projection find approximate indicator constrain either sketch certainly depend side tight constant preserve sketch preserve f f projection preserve final constant
benchmark identify test identify input show strong production weak production structure finding input application involve size experiment investigate size evident production efficient affected production exhibit somewhat except increase production improves increase influence experiment size second production method short execution execution table take c variation importance input production essential robustness result contribution output vary three production plot production process production process contribution inefficient input input
result vocabulary result construct dataset lstm rnns become architecture interpretation define could modify backpropagation validation set vanish forward gradient descent divide epoch moderately constrain achieve comparable lstm comparison forget hidden neuron favor neuron big contextual versus impose recurrent character affect dataset validation test cache lstm influence contextual corpus
pde pde require evaluation posterior amount pde occur optimization appear cost implicit accurate accurate without surrogate method significant neighborhood method around maximum sampling therefore efficient situation balance cost result surrogate near maximizer posterior maximizer maximum accuracy base moderate order scheme add use efficient mode cost
nine community database gap us air network contain network contain isolate isolate actually component large connectivity ns vi email email bind affinity electrical represent generator link network well ix energy www node locate cn table basic network divide treat probe move probe link probe consist link
basis sequence try incorporate cycle incorporate synthesis cycle flexibility incorporation rate utilize resolution consecutive zero incorporation case incorporate incorporate cycle sequence read analytical incorporation nucleotide unnecessary specification name kind permutation throughout sequence denote nucleotide incorporation kind template nucleotide incorporate cycle next next discuss special nucleotide incorporation eq
stochastic pairwise query coordinate optimization comparison algorithm convergence guarantee comparison tend regard descent consist determined search step along search therefore work contribution present block coordinate descent base comparison query show numerical explain devote proof
traditional singular fail bregman generalize dimensional occur logistic paper sequential property multivariate profile investigate receive attention wireless sensor huge amount streaming bring challenge storage overcome component find matrix explain portion pca contain naturally reliable largely bregman
configuration result mistake without adversarial mistake interpolation experiment span parameter parameter behave explore region maxout perhaps surprising induce run subspace span difficult cc cc explore coarse resolution structure run experiment dropout involve exponentially tend minima neural path two minima barrier random seed generator
sensitivity possible guarantee mechanism probability private cover guarantee mechanism find except sensitivity private unfold oracle eq probability eq provide include coverage cover guarantee unlike guarantee explicit solution output implicit describe pair approach interpret turn size database grow like continue solve feasibility private rescaled give approximate get kind multiplicative receive update aa normalize dense multiplicative maintain constraint two role solution useful define find least give ij oracle loss dual find may database vector pair think guarantee lp feasibility leave randomize scalar sensitivity slight offline building influential introduce express differentially private program throughout assume private neighboring database look oracle oracle private composition multiplicative private release exponential suppose score private sensitive output range oracle sensitivity private solver program mechanism scalar sensitivity find mechanism quality dual oracle guarantee universe constraint query privacy differ distinguish constraint differ constraint differ feasibility neighboring privacy want vector neighboring database think decrease
regard field conduct yahoo aim increase price work mechanism predictive intervention outcome effect causal estimation long effect work empirical mechanism desirable support analytical payoff payoff online adopt assume switch alternate design viewpoint make mechanism population mechanism agent mechanism agent receive round distribution select rule agent report mechanism deviation true type interest randomize experiment entire
elimination iteratively nested model pose magnitude regularization vanish regularization would regularization would dominate index gradient index manually challenge end adaptively fix advance fix relative procedure empirically performance encoder layer propagate threshold encoder train decoder reflect attain fix unit unit value retrieval demand code regularization stochastically minibatch correspond frobenius exploit structure allow long ability capacity decay across unit datum example tree balance completely representation however marginal representation bit encode construct code conduct retrieval bit b b correspond
however descriptor h lr adopt test blind transfer aggregate exponential class direct feature use additional attribute correlation give encourage restaurant restaurant scoring record overall multi task iii dimensional customer restaurant cover score set equally descriptor bit interpretation thus ignore outperform h go unified task core semantic descriptor variety categorical domain enable sharing novel shot
product stagewise keep unless reason stagewise zero unless reason keep stagewise lasso row distribution path stagewise middle stagewise path become stagewise ignore trend actually stagewise reveal behavior shrinkage factor stagewise algorithm yield effectively pattern capture path suffice stagewise scalability pure stagewise solution stagewise interesting study shrinkage algorithm say shrinkage applie shrinkage frank wolfe really strategy interpretation shrinkage stagewise history stagewise component implicitly place direction completion confirm stagewise tune even pure stagewise stagewise regularization parameter recursively inductive setup bind sake additional condition overcome inherent stagewise e exact path differentiable regularizer lipschitz constant stagewise limit number take stagewise reach parameter define effective lagrange exhibit weak constant stagewise satisfie need possibly nontrivial discuss remark technical hard interpret duality expansion difference fx explain stationarity z associate constrain lagrange along weak intuitively kind satisfied stagewise path exact display empirically replace decay condition furthermore decay topic htb stagewise figure ensure stagewise incremental stagewise constraint stagewise efficiently underlie exhibit stagewise estimate stagewise offer apart ability rigorous characterization stagewise future work throughout mark understanding stagewise capability work attempt explain stagewise teacher thank grateful frank wolfe lastly review constructive compare stagewise frank wolfe problem begin frank wolfe iteratively loss successively small minimizer frank wolfe gap could stop iterate duality sufficiently face frank wolfe stagewise similar former iterate whereas iteration substantial informative setting run frank wolfe run frank wolfe warm newly guess frank wolfe mind may compare something wolfe frank careful stagewise step frank wolfe actually quite make comparison direct wolfe regularization parameter step frank wolfe initial frank wolfe stagewise linearization around frank wolfe stagewise constraint point gx feasible point value frank wolfe new estimate frank wolfe opposed continue iterate repeatedly minimize stagewise consider construct logic finding maximally align frank wolfe stagewise strategy
benchmark model robust perturb input bag part distribution boost great deep leverage noise model relate classic model formalize pseudo ensemble collection child parent process define ensemble relationship pseudo ensemble ensemble create sec develop regularizer input state dropout reproduce fully supervise extend supervise produce state real world dataset recursive pseudo ensemble generate ensemble process parameter latent empirically type
generally examine improve benefit hyper optimization high maximize sense none validation provide perspective improvement result could analyze determine filter misclassifie ensemble search algorithm result high validation accuracy entire add hyper hyper give amount constraint show perform hyper optimization default classification increase filter increase filter demonstrate benefit motivation datum notation training instance equally machine learn concern input set generally
create process uncertain associate classification denote label recent numerous method management however none address collective possibility generate collective classification use final label major disadvantage collective label collective information algorithm collective account network encode link sometimes link successively link optimum contribution collective provide formal introduce labeling incorporate uncertainty edge enable across accuracy evaluate technique serve practitioner collective mining uncertain collective uncertain collective time present experimental conclusion classification especially context propagation refer class improve tend collective al propagation perform collective email speech al exploit et integrate leverage label collective
bellman bottom visit end algorithm algorithm fig respect instead solve path maintain close obeys weighted rule dags dag instance dag substitute add contribution show enhance contribution decay constant provide
data asymptotic journal american let hessian hessian ball expand maximizer interior x conclude hold mb hessian interior recall exact taylor modes bx kp assumption approximate case hx gs gx sx st x flow apply
represent pixel image utility indicate strength want ise self equation determine strength ahead compute common apply selection field notice recursive formula present selection design bs ds choose
super strength training explore master testing dataset take result one kind another relatively test standard dataset super list top decomposition subscript test super subscript indicate approach usual sense super target multiply successfully successfully domain view testing successfully approach likewise standard provide super train dataset success cross actual composition label show interpret composition estimate value value column prediction vary value show super composition column give may calculate match cd percent percent top percent percent candidate testing calculate cross domain super decomposition dataset need search consider matrix row considerably full full search super estimate able search pt column super column performance super candidate expect column decomposition expect use testing seem decomposition cb ce percent percent candidate source cc calculate search apply super decomposition super train interpret approach extend interpret b performance standard observe column exactly cb cd evaluation percent top percent percent percent source calculate super dataset unsupervise training tuning dataset unlikely much impact training improve
additional work unit integration minimal write short probability short running discuss mixture bayesian analogue mean methodology inference way isotropic covariance wishart cluster center speed finite chinese restaurant merge proposal two modification new testing implementation keep sufficient efficiently entirely standard term unit thorough
acceleration region ct scan figure image reconstruction reconstruct demonstrate os behave exceed image see algorithm light
stochastically dominate simultaneously write q fact equivalent numerator rhs satisfied one generative fact marginal orthogonal axis invariant first claim eq variable induction random base product iid unit independent square product joint induction assume joint sufficient joint j equal v new axis add term j k k k jk projection onto since j norm vector draw random define orthonormal row equality follow chi square freedom chi tail bind chi variable freedom
section focus click accurately usually compute observe click predict actual user click position alone differ click similarly predict click query unable user click therefore follow question well predict click click evaluation model correctly even make mistake actual sequence actual click rank term top click word click sequence click accurately ad predict click click predict reverse click intuitively high query easy low understand strength report across access query label human look rank score time post click relevance measure vs recall consistently particular able relevant document auc able achieve htbp recall xlabel ylabel ylabel legend pos north recall roc roc pm
although rd superior performance layer convolution benefit convolution term grow find performance always grow become increase size train either ratio per class suffer overfitte poor result sample class comparison around cycle train convolution layer choose comparison significant overfitte poor generate video dataset million respectively identical storage diversity produce recognition experiment examine approach video recognition entire video video frame video frame extract video video extract video result video aggregated video kernel compare learn nearly kernel indicates describe learnable initialization train pattern natural video intermediate simultaneously domain parameter update conv layer restrict convolution reduce learnable avoid improve depth initialization policy conv fc conv yahoo fc conv fc conv yahoo fc conv fc yahoo fc fc conv fc fc fc conv fc conv yahoo conv fc
r step everything know beneficial follow show alg algorithm purely term computable begin perfect momentum simplify c note matching noise place benefit inaccurate estimation example choice size dominate governed information alg assume allow arbitrary fisher per iteration update dominate gaussian covariance g diagonal large note parameter setting inaccurate decay mh likewise proposal increasingly close bias explore error supplementary momentum gradient sgd momentum formalize rule alg become sgd momentum reduce momentum setting sgd momentum guide e sophisticated involve use momentum scheduling elaborate select supplementary material without naive variant replace gradient use
cascade ensemble cascade procedure cascade yield place private incorporate cascade leave team competition separate challenge assessment reveal
understand effective convexity perspective characterize establish oracle lead class extend evolve seminal showing rate agnostic square loss function losse effectively pass risk minimizer bad way precisely heart minimizer bad learn empirical good bound unique minimizer sort converse problem addition stochastically class perspective fail look perspective separate minimizer agnostic bernstein include complexity first minimization necessarily excess good work complementary previous work come form set yet
smoothness goodness previously set gps se ard implement unsupervised method pca follow ard implement sample figure consider sigmoid x negative log predictive per function sigmoid capture rmse se ard identity gp ard intensity learn learn feature sigmoid presence frequency focus spectrum space compact compose layer neuron per performance outperform result confidence figure implicit appropriately
chance mass normalization reasonable note initialization cost reduce multinomial routine metropolis step proposal simulate true proposal term initialization long use sample hide coefficient lda superiority real root root node node node node grid thick thick rectangle thick thick innovation also collapse bayes hyper document word representation hyper vocabulary illustration gray correspond occurrence highlight ij associate realize vector update occurrence read store gradient algorithm completion lda update require modify update huge room parallelization exploit efficient asynchronous case lda change first dependency weak element summation negligible exist yahoo server server retrieve recent update certainly decentralized asynchronous
reader schema alignment schema encode formal language describe piece reality e diagram schema source schema entity call schema element schema relationship specify connection schema schema schema instance relational coincide table schema process often equivalently design find often call formal matching matching give semantic matching tuple schema confidence measure range state strength relationship iv specification name mean student point attempt schema match introduce schema level source schema schema data source body activity schema allow type infer semantic look source amount relate largely attract many researcher graphical overview idea main approach document dynamic distance idea coincide call transform former latter document direct document path root document leaf node associate base approach suggest distance vector real scenario along schema filter similarity among element degree instance adopt amount therefore computational hybrid map basis let kind find approach find group extract instance could element address could element address match return example system capable handle complex see explore schema form account domain schema model describe vocabulary variety linguistic neighbor develop extensively design deal language carry entity symbol character digit detect token article filter root exist rely basis example basis lexical university group english schema match system background knowledge element approach use knowledge wikipedia attract researcher investigate management level wikipedia mean wikipedia sense wikipedia couple yet contain million entity million fact whereas fact link entity fact coverage page fact entity therefore purpose provide rich consist thousand concept non create concept entity fact significantly quantity fact add individual specify semantic new schema handle fashion without promise aim explore
explicit construction intuition construction relation sp experimental sp bp allow max I message col color none color sum give sp combination bp message equation care valid contribution sp message factor product message assume false false however valid message basically message message cluster control contribution estimate sp refer contribute either former case sp sp correspond sp boolean fix options sp col sp clear advantage sp cluster choose difference reflect comparative success col see usually fix sp uniform particular assignment remain happen phase efficiently assignment sp switch local soon sp update need combination take minus update consider incoming exact substantially
see robust robustness see increase significant efficiency value recommend near fair know practice censor aspect complete driven choice might censor solve hope research present power censor robust applicability theoretical censor asymptotic scope censor pc pc conjecture random censor apply science medical planning etc non estimator censor covariate respect presence examine study censor covariate far censor robust power regression failure analyze science survival analysis observe model censor mathematically I observation censor sample observation portion censor event aim derive see limit maximum function presence
mlp exponential smoothing change associate player divided two evaluate price historical price predict forecast propose al ann south analogy fair week select bt component less exclude mae price forecasting hybrid ann hybrid demonstrate ann forecasting ann svm fair display ann svm paper problem forecasting system machine preprocesse technology use bt rf
suitable minimax low fix
scalability approach see field offer uncertain however learn human cluster reliable compare pairwise query additionally sake phase least cluster learn reliable option explore initially acquire may rare encountered framework cluster contrast active allow modification certain ground truth experiment specific translate selection query pairwise negative consist similarity similarity manner set constraint constraint indicate semantic certain set initialize select change indicate assume propose detail constraint empty raw reduction pairwise query choose assign create aside ability collect maximally one
consist update major analogously reference informally sum loop mod view sgd update less leave maximum begin whose randomness communication since characterize descent converge exponential mild bind variable define assumption load machine computation decrease invertible optimum bound var capture randomness update argue var monotonically decrease sgd close optima see variance much tuning term bound induce main weak sgd update step need directly rely threshold ever mail certain mail later day regularity reliable become ml drawback size carefully either knowledge sophisticated may
short walk code reduce size stochastic gradient descent line derivative propagation sgd begin decrease vertex far vs want say quickly reduce future walk vertex word long therefore affect nature asynchronous sparse scale show run may interest approach variant walk pass directly code instead learn worth application softmax tree leave still code decrease graph sequence page website stream walk graph sample also variant sentence view walk design language like streaming evolve
determine uniform dropout eventually optimize dropout dropout difference visible get close informative dropout regularization effect hyperparameter input unit require vast large likelihood mask always choose contain proportional discrepancy error map estimation use prior posterior geometry behavior singular increase necessary number explain big
neural baseline work project dimension net rbf svm std try rbf find critical differ order magnitude mmd
x x I I follow correctness dominate much way retain count generate use training instance random frequency neighbor nn weight rejection thus act choose balance variability nb localization separately emphasize preferred retain imbalance choose real number close search nn instance eventually classifier full multinomial model
gaussian express hessian identity derivative expression integral two back discussion rule back express main rule integral part line
accurate small component frequently expect regularize sparse quadratic computation previous iii occur solve optimization lipschitz term domain problem generally continuous penalty extend difficult optimize show algorithm also complexity nesterov require respect advance lipschitz regularization constant compute appear dominate change feasibility rescale independent care tackle arise always inside lipschitz moreover estimate dominate iii issue grow sample norm trace nuclear smooth
pass segment total validate list replicate replicate result successful replicate consistency replicate methodology region include marginal segment derive calculate posterior show sample accurate posterior way scale simulation method pass benchmark perform segment quantification focus change mean baseline specifies normal place segment bottleneck likelihood parameter parameter intensive
identical unified template follow package support importance constant carried easily rand rand definition co co kl knn co member member name
newton step majority computation spend al newton parallel library matrix operation gpu matlab gpu matlab worth interpretation datum elastic support vector feature select net equivalent recover special svm numerical treat call hard line algorithm p minus running spend result input advantageous implementation mode mode run dual well assume vector elastic feature keep
vector component acknowledge training image prefer use pixel sample unbalanced training pixel region eventually serious mapping handle base collect image graph homogeneous patch call region frequency detail color return accuracy image cost local close centroid within manually intensive adjust attribute region adjust representative training image reduce make strong necessary reasonable coverage select representative learn codebook descriptor training descriptor close codebook quantization bag descriptor furth histogram codebook histogram bins histogram initial collection accumulate histogram summation representative entropy evenly image vice versa encourage coverage feature essentially combinatorial seek add maximize cross entropy expand subset illustrate initialize propose suit highly nonlinear spatially rely content local sense operation could complex previous g design address challenge help contextual strong capability hundred report conduct technique learn apply material show visually numerically mit evaluate far conduct study positive throughout fix hide color confirm color transform reproduce color
equality follow conditional px I fourth equality rule definition n x eq third equality n follow formula mean eq exploration localization tn let obtain observe x ct tn tn access gp strength localization determine signal widely incorporate besides signal camera also bayes omit contribution gp filter resolve work claim obtain measurement give every call label datum pose estimating map offline obtain optimization localization bf filter highlight give trajectory divide slice slice slice calculate update mean
di contact test correction summary contact list contact remove gap reweighte count file estimate compute di symmetric entry score average exclude adjusted pair interaction predict rr protein rr pair rely efficiency introduce definition base existence unify concatenation interact protein encode protein estimate whereas family domain co decide interact horizontal concatenation ratio depend sequence come normalization intuitively score coherent assume rr sequence odd ratio conditional probability cf people action european fp grant agreement cb acknowledge european grant direct model quantify interact interact scene describe denote interact repeat analysis position correspond main interact p x
dnn formulation dnn character dependency dnn dnn output distribution enable temporal use weight hide distinction activation since depend activation work find nonlinearity select activation prevent set maximum act dnn like use backpropagation always gradient subgradient recurrent connection reflect perhaps powerful sequence maintain state
market circumstance able expect increase impact family see family tie make scoring score concerned scoring rule expectation infinite characterize also special characterization elegant rule score possibility seminal market sequentially score scoring rule space market introduce generalization continuous outcome market extend infinite outcome market well practice sound price agent neutral imply move belief extreme market information agent however fundamental mathematical finance portfolio notion convex measure indeed axiom draw finance market single financial market connection machine consist agent potential absolutely lebesgue outcome measure discrete throughout let denote interested variable device expectation score statistic scoring affine arbitrary value term implicitly affine transformation version strict avoid score density rather must large summary note place agent reporting expectation latter complete recover rule statistic exactly probability function point scoring case outcome generalize family generalize log scoring
k k combinatorial interpret matter extreme indicator complexity shape shape determine number space projection view require addition compatible multiplication minimize identity construction note enjoy definite keep randomly input k u input large choice application discuss solve exactly solver contain projection report across trial representative c angle norm minor major high minor c minor major major minor contain input give residual normalize angle mx solution normalize major minor path major
bound see mean p representation every functional exist df existence lemma f h ps h h l j j l j jx dx j allow integrable respect w equality dy jx dy dy e jx b third hermitian dx dy elsewhere last reproduce f expression proposition derive b w dy compute derivative simple computation notice h h obviously constant rest second hence integrate p equality operation fourth rw correspond symmetry partial derivative next summation product simply apply derivative combine theorem conjecture
intra inter vertex unify discriminative hypergraph partitioning formulate cluster hypergraph hypergraph hypergraph eq similarity perform solve optimization newton solution propose consider separability intra aim intra inter formulate denote vertex intra separable cluster partitioning maximize vertex membership concatenation membership p p xx conclusion diagonal form argument vector element matrix optimization sum sum ratio n reformulate trace optimization utilize newton three large well repeat step choose eigenvector singular solving obtain follow value hypergraph partitioning satisfy requirement may
though remain available moderately must meet since must bound word continuous continuous q old old cx consequence concrete cm cm continuity h also continuity metric continuous measurable compact guarantee schmidt k compact operator separability separability general section consider hilbert section due boundedness satisfied boundedness somewhat condition special output simplify boundedness eq map l kernel older suitable evaluate I k k nonlinear see natural exponential cauchy kernel continuous canonical see continuity continuity h present consistency embed theorem result concrete rate cope turn idea demand section
bc vs bc bc conduct cc nmf nmf vs relate domain em give vs em conduct performance recommendation user item combine different share propose novel filter probabilistic account knowledge across domain hand rating discriminative space rating indeed rate cross recommendation extension
gradient derivative iterative number condition example convergence matrix thought bring close would require solve cluster empty subset partition overlap cluster ie dataset cluster extensively field bayesian random partition chinese combinatorial define density description cumulative density moment generate program way alternatively program
collection grouping need proceeding shall set db rd r c k converge exponent suitably almost summation suitably adjust calculation exponent reverse order summation db value discrepancy original need sum term occur induce extra discrepancy geometrically complete universe number
nmf kullback leibler parametrize translate noise gaussian offer fit heavily coefficient update decrease update employ nmf solution proper update next heuristic scheme prove update turn e previous nonnegative preserve result per iteration involve nmf divergence purpose extend current step bind produce descent algorithm indeed I auxiliary rely concave decompose dx dx yy give dx c inequality lk kp lp kp lp use convexity
compute point point lemma e addition k therefore find appendix consequently subsection template c either terminate update update either rate discuss later alternate advantage nonempty norm dual iteration smooth analytical algorithm primal bregman smooth use use analytical achieve choice feasibility gap lead due know method describe unfortunately avoid expense section specify solving lipschitz lipschitz accordingly similarly f strongly rule bad analytical primal depend prox long absolute residual aim algorithm state variant limited technical assumption objective eigenvalue positive bregman cd pd pd c pg define satisfie pg td pd solution attain require fulfil strongly follow strongly concave result scheme corollary generate show certain problem priori augment lagrangian smoothing subproblem reasonable addition convex nesterov accelerate satisfy imply addition inexact inexact proximal operator start compute inexact whose analogously scheme omit initial bad sense primal feasibility sufficiently practically relatively reduce burden explicit gradient admm fast idea smoothing technique function augment lead nesterov smoothed dual either augment bregman
derive throughout focus sampling seem sample attribute natural p obtain technique pick mix strong direction zero coordinate divergence matrix eq budget iw analyze begin follow assume equal expectation eq let law eq divide observe eq side proof bandit dependence since low detailed idea sp loss
version consider nan enyi alternative increase difficulty resample among high enyi subgraph difficult choice plant dense make edge independently plant dense subgraph entire alternative arbitrary dense model first plant distribute hardness detection financial edge nan hypergraph big notation mean trial tv divergence base ask small detect plant dense subgraph question investigate statistical sharp regime treat regime sparse regime separately focus interested absolute asymptotic treat regime unified manner limit deterministic hypothesis result ok subgraph reliably statistic scan correspond whole subgraph counterpart minimax test test test q implication parametrization
update environment explain trajectory relationship relationship work plan planning learn tv human tv crucial planning show heat human center tv might move planning activity different function instance activity cost activity human object proximity around vary distance object angle certain angular activity spread wide fig human angular preference activity center angular preference parameter define human activity fig human project length prefer robot vision pass add center parameterized variance product von robot careful pass behind move back preference vary human cost capture activity along human user prefer robot cross whereas activity preference symmetric normalize human normalize relation object mostly commonly planning reason human human book display read bar
density pairwise intersect dp prescribe experimentally selection mixture lx n description length etc retrieve deviation complete dynamic seminal outline bellman paper contiguous element incorporate cluster size solve mean median center report tailor application contiguous bregman mean
grid plug back scale symmetric reason normality assumption plot furthermore individual contribution different calculation suggest play minor part compare violate implicitly phenomena world short tend model exponential homogeneity add flexible heterogeneity available otherwise
finite issue address ica subject degree restrictive permutation ica second rescale infer source assumption trick simultaneous goal scale covariance simplify remove prevent give challenge ica solution highlight benefit exist ica primary advantage fairly highlight ica identify basis maximize ica make apparent quite salient variance fail direction world lack orthogonal merely statement orthogonality factor recovered source uniquely source middle lie identify underlie source appropriately statement justify statement lead ica source permutation recover flip recover rescale recover dimensional ultimately heavily whiten begin examine assume factorial concrete arise invertible factorial ica optimally factorial independent distribution ica principal
manifold tangent fully alternative perform demonstrate easier able separate far good stack accurate previously generative among good mnist generative structure datum firstly demonstrate separation latent mnist latent nearby space correspond writing write right approach pass network generative class figure far house vary way feed significantly inefficient able perform
accomplish atomic similar require sparse overlap order ensure feature description report speedup parallelism cd svm cm cm avg cc couple asynchronous version basic problem consideration high interesting direction function obtain partly nsf comment expectation kronecker ready divide side statement I e l objective take expectation relationship substitute simplify theorem bind follow equation q take side give inequality fx step satisfy substitute require reader exposition analyze unconstrained similar manner recall iterate l fx fx x derivation schwarz inequality lipschitz continuous variable step inequality
behavior color estimate extract frame represent yield choose test average random depict accuracy baseline bit pl kernel reach hash outperform reach overall high tb b kernel pl body formulate atom depend compute dictionary atom well dictionary label comprise body static dataset video environment region represent video order yield point size b outperform baseline static maximum static art manifold tb accuracy sparse code introduce positive
compose unstable precisely map condition unstable stability absolute summarize p randomly hand symmetric dynamical equation write actually map essentially performance method sequential averaging momentum list asynchronous supplement scheme master unclear pseudo supplement center follow master center master move set dataset cifar imagenet convolutional deferred supplement gpu node gpu local gpu processor variable master store update denote color channel fully operator max denote operator softmax linearity inner experiment neural p bias initialize master break also mini
normal x position obtain adjacency embedding position identically multivariate position propose wherein incorporate adjacent vertex tendency present position instead dot gibbs position adjacency employ mixture corollary quantify suggest role bayes latent indicator identifiability constraint blockmodel eq embed gmm within present investigate utility blockmodel hierarchy name primary benchmark vector assume know gold prior position theoretical limit covariance distributional convention adjacency finally absence adjacency embed theory prior give rise spectral might increasingly see section corollary
give cdf length vary q cdf eq imply sphere fig moment formula htb ccc htb ccc sphere length mean contrary moment stand moment property
reduce ordinary singular I k symmetric problem coincide sign precisely identically note define symmetric form nk normalization q finally vector recover flip suggest constraint loss establish problem matrix coincide large turn result follow background exist eq evaluate large asymptotic digit indicate large tensor bind incur maximum likelihood proof real likelihood prove lemma expect sharp suitably constant appendix provide background random study physics spin particular rigorous replica spin theory confirm rigorously prediction maxima
application section panel first motivate time additive case intercept consumption price consumption individual min slope z pr intercept l type unobserved error visualization affect remain correspond extraction factor yield right htbp estimate panel individual existence check interpret effect testing describe less within estimator inconsistent least ph p consistent residual variance choose suppose nan hypothesis exclude true factor favorable violate iterate inconsistent recommend user follow argument specify argument test statistic message dim consumption dim level assumption fulfil unobserve true probably assume inconsistent alternative nan discuss way check whether classical sufficient reasonable decision additive section
analogously uniformly sphere thus interested variance projection operator must case assume careful detailed appendix say complement compressive tight qualitatively project subspace use per demonstrate figure predict quickly average formalize simulation plot thing simulation demonstrate
perfect promise possibility success practical audio speech recognition audio clean sound input situation
manuscript devise dynamic setting decision point observational collect decision disease extract medical record datum national survey assess health status children united states diabetes third wave control pressure manuscript simulate patient diabetes one challenge avoid result backward induction experimental observational start decision find option optimize decision point history regime nest key notion outcome regime outcome propose tool use regime time introduce subject censoring create
imply multivariate finding respective provide view main feasible solution match kkt become replace weighted benefit extra even vector condition design provide benefit scale restricted isometry property sparse condition restrict condition weak prediction cone group group cone restrict eigenvalue compatibility also introduce notion cone sign cone cone define sign follow cauchy eq eigenvalue imply scale
explicit mark x td txt thick color solid td txt thick mark black option plot xlabel ylabel ndcg pos north east color blue mark thick cd mark td txt thick color option table header td thick mark color minor title yahoo learning ylabel ndcg legend pos east color thick error cd mark color mark option header plot txt thick mark color black option solid minor title yahoo xlabel ylabel ndcg legend pos south east mark plot txt thick mark mark option solid index header txt black option solid header minor title xlabel ylabel legend pos bar explicit color red mark plot thick mark solid header true plot scale xlabel ylabel ndcg pos east color blue mark thick table txt mark mark solid header true header title xlabel ylabel ndcg legend pos south mark thick bar cd error x error txt mark color option solid index mark color mark header minor
list regularizer huber nonnegative unit regularizers advantage parallelism yet initial datum assign process column compute portion objective complete begin criterion add automatically function detect type appropriate implement method loss ignore frame fit function fit aspect day include value measurement miss encode pick encode use feature embed separate axis embed despite automatically course embedding regularizer embed embed set two code line across divided core restriction fit leave server possible hardware local many core avoid core process partition input copy column gradient subgradient signature implement square entry similarly implement proximal operation experiment several recommender netflix matrix dimension per row sampling location finally location uniform amazon ec master machine cpu core gb ram run matrix regularize capability depend merely illustrate c ambient table iteration useful iteration author grateful de sa art david price taylor van comment insight create support foundation fellowship stanford fellowship fellowship extend pca arbitrary categorical ordinal factorization pca mean maximum matrix heterogeneous miss simultaneously interpretation parallel generalize implementation send stanford mine large collection datum row represent column sometimes type record patient survey yes sometimes test question finance know asset class science record record customer difficult understand complex visualize example correlate feature identify anomalous enable low dimensional plot anomalous value well embed principal analysis pca find minimize sense extension handle miss extend pca square appropriate extension beyond pca add regularization factor impose encourage structure factor refer problem dimensional consist factor datum still familiar optimization formulation completion pca svd margin problem relaxation tight globally however involve
growth distribute challenging topic real world machine read order leverage hardware significant communication worker read popular version stochastic sdca propose distribute allow trade adapt diverse low core framework support objective model leverage avoid simultaneously employ step master perform data method round scheme naive communication theoretical reduction come moderate increase order sdca ascent optimizer smooth geometric platform propose variety show magnitude gain mini
hold mle belong function great mle study condition surface unimodal unique mode concave let symmetric concave real symmetric problem study impose characterize course seem asymptotically study concave uniquely characterize cdf stay integral empirical logarithm change slope modify cdf mass word nx nx n nf z dark circle knot concave mle mle nn main establish mle carefully density concave boundedness weak class compact start state reader b dt prove ft real sequence converge concave let class log concave class measure fix log hx mle mix g yield behind introduce avoid issue fact previous zero stay bound generalization result mle combine unimodal claim unimodal arbitrarily support approximate compact dropping towards quickly point convolution density center deviation choice problem exclude log concave log convolution importantly condition unable integral appropriate consistency proof two concave goal definition refine respectively location shift ensure stay uniform
regardless time additionally regular mean covariance matrix definite toeplitz solve fast current slice smooth interpolation ft tb pdf pdf pdf pdf finding laboratory exploring drive purely domain clinical stream prior intensity distinction intensity smoothing use discretized intensity rejection share intensity loss piecewise
problem work space call community problem issue flat sum need introduce basic construct volume probability partition disjoint subspace point uniform partition bin product estimate iterative generalized simulate markov bin reveal landscape mapping bin variant record belong probability among estimate suppose mcmc decrease schedule adaptively proposal move intuitively landscape improve method simulate annealing process sample visit bin equal step project identify local especially flat merge minima identify descent two minima barrier straight line minima energy dotted level descent
diagram reflect second diagram sample see experiment dominate variation persistence sufficiently instance near persistence thus allow distinguish class b differ persistence vision persistent homology valuable purpose outperform art problem would rather topological advantage shape benchmark consist shape synthetic human child pose human pose resolution classification synthetic real texture benchmark pixel benchmark predefine training image training shape retrieval ten choice piecewise mesh discuss compute persistence diagram texture classification descriptor report rotation operator
finally norm define norm shorthand notation pp ph turn start discussion problem square absolute favor model often take address overfitte measure take square absolute shrinkage employ sparse nonzero alternatively examine attempt language uncertainty set g via procedure procedure key rise interpretability n denote uncertainty norm fix regularization coincide extend setting begin identically triangle final follow definition dual homogeneity triangle completes result corollary likewise recover know equivalent follow conjugate observe square arise uncertain uncertainty induce
unified power within hierarchy vary nonlinearity generalize version complexity grow begin overfitte fluctuation aside decrease stop velocity connect vary consist release som infer dynamical supervision adaptive qualitatively produce elliptical monotonic trajectory color line show dash leave single variable use single recovered som biological hierarchy infer crucially distance exactly within system elliptical trajectory zero radius finite completely specify dynamic object som display object dynamic dynamical motion statistical fit trajectory would different understand automate adaptive inference sir software find perfectly biological phenomenon may interact able useful site yet site arrange affect site produce imagine setup evolution total site treat site due site measure corrupt scale decrease datum simple amount although expect likelihood dark
ex false em mu size th mu mu end align end align environment define style define array construct allow false tag true receive fusion fc via limited capacity cloud access communication receive
n stand determinant recursion condition natural initial satisfy rewrite nj ed arrive eq approximation argument finally stability derive state mse representation stack rewrite respectively kronecker stability
exponential hyper experiment schedule decay depth depth potentially small key aspect penalize learn decay get gets test standard depth deep rate important although study beyond gradient descent sgd minibatch gradient varied learning result roughly display value distinguish lower left except error combination nonlinearity result agreement affect accord walk initialization
solve multimodal get seem optimum issue former contain hard global one depend case cb seek quantify region common cavity powerful considerable supervise forward explain improvement datum show effect learn benefit play important expect annotation account training guide retrieval ec rely meaningful ignore ec also hierarchical ec number ranking support summarize rank obtain high indicate color higher unsupervised show correspond correspondence distinguish population within correspond similarity illustrate discrimination similar query overlapping supervise unlike match ec reason supervise value similarity relationship derive indirect cavity similarity division indirect show correspondence infer protein protein ranking supervise rank indirect expect indirect detecting entry noisy one indicate performance degree influence rank clear distinction roc
compute tv distinguish follow eq deduce imply distinguish case start two interest case imply mass apply schwarz norm assign bound translate case recall q take substitute distribution equation translate show plug claim square distance make constant choosing employ square distance distinguish show sampling sample distribute instead variance induce among number concentration sample denote symbol independently define
bias bias expense mse although use stop qualitatively case adjustment already adjust sample accurate adjustment ba produce size estimator part bootstrap increase bias adjustment bootstrap coverage close inferential estimator long integrate bootstrap filter preliminary parametric parameter simulation bias estimator via encourage suggest correction yield long encounter practice estimator result subtract triangular e less integral yield consider hz fact operation substitute sequentially recurrence h x neighbourhood origin small n extracting term ol regressor regression discussion nr j array weakly dependent e conclude ok ba analytically adjust
argument consequently omit proof extension easy leave note na follow inequality valid event large case let cart clearly rewrite argument thus collect argument property fix moreover k k supremum side grid k q finally hereafter tuple one level leave define fix inequality rest inequality enough confusion cut countable cut clearly result prove finite subset infimum modulus observe plug conclude prove fix exist exist satisfy p k fix conclude cut
projection node reduction project hyperplane span centrality well high component highly gender project central amount project variance allow decay classification accuracy heuristic extracting compare association provide association generation face fit avoid explicitly compute matrix association unweighte direct graph domain edge union set label subgraph note intersection edge ignore reason accuracy cost seem topological vector describe fast currently know carlo incremental reach get limit
ensemble detection unlike classifier propose detector preprocesse competition rank database publicly auc absence ensemble process dr serious disease diabetes common early prevent affected dr mass diabetes highly desire manual resource demand effort establish reliable computer system color promise automatic screening close clinical recognize fold first sign dr precise highly detection detector remarkable ensure reliability detector prove efficient aim value provide coordinate
collection positive location write density xy multivariate notation follow zero gaussian marginal recover process margin estimate bound rather mean problematic depend distribution asymmetric variance provide order input entry distribution cdf median median copula height x bottom axis line table toy txt mark densely mark black toy txt index toy dataset txt
principled research showing indicate positive element underlie link choose optimization problem relationship predict edge e real world co network ca test done relax solve link method svd train svd svd approximation omit link candidate computing false snapshot show accurately achieve large relax general overall link completion highly descent whereas eigenvector second semi supervise cluster inductive completion
slot execute eqn indicate action represent current learning know generality cr queue cr queue instant form represent total possibility value policy execute queue cr availability threshold capacity belong fig primary slot decrease queue hence slot figure demonstrate secondary queue secondary increase secondary service furthermore certain service share cr decide sense action
jensen mass false I exact false loose imply care worst bad absolute perfect suffice quickly like substantially convenience expectation become convenience order q
dash mark style fill red mark col sep comma fix utility col sep comma size xlabel utility legend style draw legend pos south grid style col comma expert thick black square comma utility thick mark table col comma utility ib xlabel ylabel none legend south major axis height sep comma expert smooth thick mark style red comma smooth thick mark sep comma size utility ib classifier measure reward contain discount property assess single return discount correspond traditional discount medical diagnosis category discount visit discount visit substantial risk apply discount accuracy score instance accord discount discount utility consist discount
fitting hyper penalization explore considerably markov mcmc lack investigation fitting prior hamiltonian monte short test critical scale real microarray confirm finding add tailed mcmc fully throughput genomic identify level disease classification diabetes response hereafter verify biological diagnosis disease collect relevant gene disease challenge analogy commonly univariate relationship ignore redundant gene include meanwhile weakly attempt take rather capture greater often maximize prior use tail propose prior hyper prior regression differently normal laplace laplace scad bayesian gamma generalize pareto unify review addition broad class commonly high bayesian popular mixture see express belief enough feature irrelevant totally sensitivity theoretically empirically jeffreys nan point markov mcmc lasso still lack probably difficulty likelihood hyper multimodal coefficient ordinary heavy tail cauchy
solve book numerical area medical imaging arise algebraic specific would mention sparse also topic network mostly spectral huge matrix algebraic seek complete scenario tractable size desirable scenario region resolution scenario recommender recommendation item recommendation theoretical practical kind optimally run many assumption work intrinsic ingredient dependencie circuit circuit kernel circuit restrict small circuit least costly ingredient inversion circuit circuit size circuit locality trading cost circuit capture algebraic
mixture analyze cluster laplacian operator compact weight belong interior probability simplex specify via weight mixture drawing formalize recover latent observe unlabeled course generally recovery become difficult overlap increase easy formalize overlap follow background normalize embed symmetric say kernel kx semidefinite throughout semidefinite function cluster relatively decay apart normalize embed associate laplacian rescaling kernel since symmetric construction large eigenvalue eigenvector typical cluster vector apply cluster embed reveal formalize normalize function conjunction suitable orthonormal result component section
operator generalize mild regularity regularity random joint function px lemma review first function regularity basically r integration ready vector respectively density parametric mild formal regularity stein parametric identity mainly differ elaborate see form closely know covariance parametric density identity function relation e since px score property stein property higher establish notation section formula product notation thus score function reason score enable stein order derivative yield differential statement mild regularity function operator variable formal description regularity polynomial satisfy orthogonality property polynomial know mostly involve thus order coincide polynomial instance convenient polynomial orthogonal w interpretation need provide order similar previous construct score exploit parametric stein identity parametric respectively parametric formal result regularity apply
approximate graph limited edge question spectral cluster assume distinct work obtain beyond beyond graph approximation approximation consider one important cut cut original graph machine etc approximation wish laplacian algorithms laplacian connection spectral analysis need cut focus behavior nd eigenvector laplacian crucially generic framework sampling costly full eigen experimentally appear range theoretical somewhat access g course edge sparse technique guarantee consider weighted graph graph
outperform td alpha bind iterative linear singular regularization notable recently temporal td keep varied estimate td error regard adaptive step alpha heuristic td error sign variation
notion utility diversity modular submodular differ entity maximal relevance order fs recommend item implicit penalize instance yu measure multiple objective consider tradeoff relevance diversity axiom distance implicit metric capture movie instance fs pe satisfy diverse propose objective recommendation offline document find case belong category diversity address et exploit user preference study diversity grow recommendation argue solely author similarity intra compute pairwise similarity item high diversity topic balance diversity zhang et problem find address maximization list recommend maintain item hybrid target weight require tradeoff consider idea marginal submodular approximation paper prior recommendation optimal target concern maximize diversity elaborate behind greedy online evaluation efficient contribute diversity motivating formal description user movie ccccc utility name x movie depend choose recommend movie match satisfied utility
analogously expensive smc use intermediate carlo stochastic approximation em replace simulated smoothing intermediate next approximation enjoy maximizer reasonably weak simulate joint smoothing structure rao start rao model smooth rao find apply mcmc base smooth gain jump separate infer jump compute
highly get additional change contrast x pure underlying solution implicit augment x square imagine panel ellipsoid
q measurement allow nonzero convenience bregman continuously differentiable assume inclusion subgradient norm addition piece compute linearize bregman govern exponentially bregman linearize bregman iteration image sense change variable iteration iteration augment linearize euler simplify wise motivated inclusion path always reach see convenience replace aside obvious piece wise though obey subdifferential lasso consider identity linearize let least solution oracle following evaluate hold estimator selection consistency also path refer existence select lasso provide consistency reach nonetheless bias
convex let compact define simplicity maximum possible optimality justify non estimator outperform r hausdorff hand considerably see enough necessary boundary ball interior meet intuitive sort limit see convexity et al compact convex reciprocal restriction converge desirable case account see prove proposition prove sufficient convexity convexity account pt pt general h compact relationship different geometric ready
significance p serious year du exact hypothesis possibly great tr et approximate normality sp normality sufficiently poorly demonstrate decrease test close comparative test central possess property propose
converge suffice p w go use inside supremum identically sm note possible c inside supremum converge far diagonal use bind sm sm sm sm e conditionally variance x cx x cx x sm find converge converge sufficiently sm x sm n x n sm c w sm cdf far obviously converge zero zero turn sequence inequality hand eq use lem k combine proposition corollary condition hence independent distribute distribute freedom convention possibly r r prop coverage corollary bind respectively instead use x satisfie maintain section far clearly denote orthogonal complement column triangle probability display p probability far display depend form square degree infinity finite turning term let denote matrix p converge definite converge x supremum precede prop lemma prop prop prop prop remark prop consider interval quantity minimal
search everything optimal r plug direction quasi bfgs update solve let maximal decrease overall repeat choose improve mc focus mc strategy fitting entire surface although mc penalty place certain subspace simple form produce notable scaling set expand coordinate standard descent step step converge simultaneously mathematically actually optimal appendix attractive non function appendix selective correspond put limit restrict search implement
affect macro optimal threshold theorem describe uninformative maximize base widely recognize particular prediction maximize expectation example prediction depend apply quantify optimal threshold dependence make difficult relate thresholded show perfect optimally thresholded calibrate get always macro argue actually weight rare label study consider article mesh control rate rare lose cause macro rare application desirable binary output label dimension probability column dc false negative negative
chapter ac uk sophisticated parallelization gpu acceleration parallelization naturally modular gpu million demonstrate dataset source integrate gps parametric reduction formulation responsible limitation output collect plus th
shape rotation fig contour contour plot large eigenvalue sign behavior kl change condition sign develop algorithm theory ml merge expectation divergence decide mixture merge elliptical gamma ml solution apply follow change helpful turn eq gamma em maximization step current weight k l l b ga step update maximize p I explain two modify case likelihood one sequentially step step maximize log
x filter x x x architecture use four layer eight feature map two model size allow translation invariance conv conv conv full x x pool also try three decrease factor validation stop momentum mini batch importantly turn layer normalization gpu training pixel preserve aspect computed randomly sample leave right global sample corner right visualize
cycle duration west mm fig south mm percentage west south mm transition west mm occurrence universit du financial existence learn learn perform unsupervised induce knowledge development dynamical theory focus broadly traditionally imply assessment post process long term continuous assessment behavioral
bound proof approximate coherence kernel since coherent side therefore complete strictly form root polynomial root diversity low rkh yield previously embed borel measure topological detail dp algorithmic one expression radial function radial function radial base kernel feature radial decrease namely decrease low distance diversity atom proof decrease building corollary bound entropy shannon generalize r measure measure generalize enyi entropy low window shannon entropy enyi order overcomplete provided examine linear dictionary examine show dictionary low
give intuition sequential allow follow since event observe achieve capture restriction first time final opposite versus intersection occurrence know presence access rough effect spin spin neighbor update time determine interval compare time start configuration output remainder towards runtime gives state runtime homogeneous let event none aside
low limit n discount figure bind number require amount grow require iteration simple impractical discount factor behaviour complex single mdp stepsize rule study special program extension consider value iteration derive formula l l n require quantity optimal minimum squared estimate constraint unconstraine objective produce satisfie manner e e observe e e minimize objective equal yield assume stepsize expectation decomposition write term expression complete positivity numerator complete numerator covariance stepsize explicitly account observation furthermore closed bias balance close showing behave correctly case collect simply add discount stepsize rule stepsize stepsize long easily induction reduce stochastic approximation algorithm convergent establish low proof requirement observation recent weak condition convergence paper condition common
x skew balance get x stationarity determine element eq want minimal equilibrium rate lead adjust skew ultimately db violate converge cover scale diffusion fig improve stationary apply transition make counter transition bias continue jump replica visit walk introduce site e copy auxiliary gb maintain markov chain site toward walk enter likely edge walk direction towards continue circle likely
train student thing teacher ambiguity mistake teacher make thought teacher help classify digit compare field create sphere gradient descent minibatch mnist difference sgd sgd stop sgd gd reach bottom gd mnist sgd noisy nature capable minima high gd
dependence keep notation cause neither confusion characteristic sampling lot accept accept stage overall operate characteristic stage operate thus control overall operating determine error equality want overall acceptance sampling give stage appropriately treat impose remain guarantee overall one impose obtain valid acceptance plan select give put hold wise risk procedure stage second put risk symmetric example yield acceptance plan measurement decision procedure recall additional usually different location identically distribute time instant inspection degradation effect relate determine degradation know work degradation yet enough knowledge degradation module justify degradation act power degradation practical
use plug generate subsample prediction simulation slightly limit sufficient necessary simulation find ensemble large approximately begin increasingly figure bootstrap limit estimate bootstrap build ensemble replacement estimate distribution normal move interval quantile form confidence limit estimate calculate prediction generate tree assess coverage true ensemble mean ensemble take ensemble close mean contain interval horizontal probability limit estimate high likely underlie external repeat internal remarkably external estimation ensemble build testing hypothesis assess feature training depend value additional feature sample uniformly interval independently look distribution point interested tree build notation run simulation consist estimate covariance estimate ensemble result figure hypercube predict feature total histogram statistic
eigenvalue zero diagonal quadratic since always small aspect stability regularization parameter network independent topology range go indeed one clearly step regularization topology low upper example become hand increase connect improve increase necessary distribute conclude step range small stability agent consider verify know diffusion stable indeed verify conclude example noise node consensus observe fully stable strategy al steady issue network leave corner al shorter increase fast vs good color auxiliary matrix select network approximation definite sufficiently simplify see assume follow assume positive definite connect surprising improve strategy act independently agent
correspond goal learn involve moment representation exact moment available carry throughout otherwise third depend score exist go support mixture input instead extend follow cross glm stein identity form moment separately message contain glm moment glm mixture glm section set appropriate continuous identity state satisfy mild regularity condition glm bias weight ix moment suffice let look moment expectation I iw subspace span however moreover bias vanish mirror trick span recover appropriately obtain cp
arm always arm fair mean value useful strategy least concentrate metric discount formal time skip relate safe exploration aspect sometimes lead outcome take account consequence htb parametric term discount return b average step forward per run concentration small mixture little arm multi bandit uncertain grow safe become slow prevent kind generalization policy skip environment justify explore cost translate generalization despite cost avoid ts suffer discount return intuition past value thus discover inform time success incorrectly pick positive arm exploration similar rely sampling yet challenge agent uninformative hyper infer data hyperparameter agent conservative explore robustness increase
bind bernstein v px iii expectation integral second combine inequality nc assertion rejection satisfy one instead define bind except obtain chi square freedom tail assertion small bound almost manner argument assertion apply old inequality recursively assertion assertion paper occur many collaborative filter spatio temporal convergence rate accuracy optimal rate cp tucker rank model high relation
probable viewpoint fc orientation train imagenet annotation ap r r c car train avg I aim angle instance far circle c c car leave right jointly classifier c c b database explore find yet stability issue conduct performance cnn baseline surprising error detection viewpoint baseline show effectively b detection detector cnn baseline score candidate square observe cnn really baseline investigate jointly optimize pose test variant probably lack improvement detection
special france paris en france universit e paris online aggregation prediction new regret version average loss instantaneous algorithm demonstrate expert interval use excess loss loss simple expert make choose nonnegative every incur k cumulative k k good improve possible possible advantage form expert loss unfortunately explain depend monotonic consequence tuning example trick issue sequential fashion suffer
indicate decay rate cross kronecker consider temporal q satisfy drop dependence belongs dependence memory let show weak decrease integer imply na thresholding dependence stationary process consideration together lyapunov apply n nn restrictive homogeneous decaying series factor result brain voxel discard brain grid place regular mm throughout voxel voxel center
segmentation coherence distributional count distributional advance contrast approach distributional drive supervision annotation parse task representation perform considerably generic propose recurrent parse idea argue introduction expressive generally distributional view recent literature distributional semantic entity take purely distributional logical representation try add amount formalism purely logical distributional obtain type relation logical semantic distributional compositional combination logical distributional semantic sentence generalize sentence argue distributional sentence element recent sentence text semantic word modeling factorization text nlp task sentiment political similar information call belief recursive bp
value penalty extend condition method ode norm method learn highlight classic ode realistic conclude loss classic rkhs scalar value kernel propose literature spline seminal kernel ridge choose positive definite smooth solve gram observe calculate scalar hilbert theory apply empty hilbert space adjoint
dpp np hard extract diverse run inference standard dpp achieve margin significant improvement performance evaluation metric f w broad applicability diverse frame open pixel historical summary independently validation frame trivially redundant low visual task public frames compute precision extract frame histogram sift extract intra compute standard quantile visual similarity neighbor compare select frame tradeoff precision multiple parameterization base flexibility point hamming advantage fine tradeoff adjust baseline statistically measure tradeoff valuable know want free summary recall preferable user take third party see video drop frame detailed analysis video material offer powerful status maximum avoid specification demonstrate dataset real conduct inference dpp call
theoretical rf rf computation estimation bias integrate addition choose r build tree use leave subsection toy bl situation fit scatter plot plot behavior toy rate section forest right rate indeed forest rf rate tree forest section particular forest tree rf result ht toy rf bl situation fit scatter act compare forest reach toy rf reach forest behavior hold model r denote scatter linear graph forest hold reach framework moreover suffer divide increase rf divide forest rf gain forest extra extra final risk improve risk plot scale rf slope fit regression five depend slightly bad forest decrease explain reason presence five well forest well obtain informative
constraint analogous r orthonormal basis ii theorems notation ai reduce aa coordinate basis principle distribution little definition use tail probability lasso simplify description dimensional write respectively estimate precisely value solve distribution expectation note trial distribution target direct routine weight trial calculate na weight sample sum routine estimate sample tune hypothesis coefficient consequently become distribution nonzero choose accordingly increase wide spread increase variance uniform procedure n n routine minimum trial dominate target value application give level expression factor potentially rejection trial routine lk lar algorithm direct sampler sampling lar comparable direct trial simulate effectiveness diagonal predictor vector lar along active turn design test aim calculate e routine choose dataset routine routine value estimation deviation estimate quantify estimate deviation across result probability coefficient
combine surface identification alternatively challenge relation stanford berkeley syntactic relation annotate argument sentence syntactic span branch multiclass identification entity distributional improvement show semantic
define formulation support zero reader number time parallelization year discard coefficient coefficient test rule discard screening rule overlap group propose screen discard screening nonzero use screen rule nonzero consider overlap perform screening test perform screening whereas contribution overlap lasso ol screening rule overlap
end monotone tv next say spread seed sketch coupling select edge live otherwise live live edge verify live graph subgraph live graph node must also could spread budget maximization influence spread maximized find influence greedy maximization mi v monte simulation estimate al exact spread marginal monotonicity seed estimation ps cascade mixture item use item topic mixture direct social influence spread exactly ic give influence probability topic aware influence seed e study topic aware include datum study due constraint section describe complete supplement dataset american movie discover movie learn movie direct rate movie rate movie later obtain topic mixture service search edge contain direct topic
equation compatible versus normalized distance dd give original find concavity turn reference boltzmann minimal nature binary perceptron classifying pattern weight force phenomena easy part potential vanish dp constant entropy curve fig dd dpp support convex tendency become constrain constraint explain simple local increase minimal grow rapidly consequence algorithm
iteration algorithm attract interest admit obtain rate optimization primal applicability high dimensional separately exploit explicit backward combine forward computation proximity operator computation backward manner deep justification terminology maximally monotone operator go rather view backward admm recursion converge fix mapping view extension perform respect variable solve saddle relaxation factor involve adjusted profile iterative thresholding rescale previous exist extension also symmetry primal dual obtain see encounter literature admm eq convergence guarantee condition g g converge primal generalization author primal dual hybrid accelerate convergence rate primal approach problem eq condition size restrictive convergence solution primal converge solution also propose specific primal interesting deal operator indeed differentiable proximity operator g gauss likelihood handle adjoint main inverse primal base forward primal proximal inspire seminal work extend convergence guarantee cm admit primal often enjoy advantage operator scale version addition satisfy condition respect slow dual algorithm rely tucker solution f terminate generate
subsample synthetic set respectively resample number like control negative alternative due incorporation selection structural precision discriminative fraction retrieve recall fraction relevant retrieve discriminative snapshot many fmri brain weight thresholded visualization purpose zero zero determine algorithm vision experience also threshold voxel voxel brain adopt scheme situation reach point prediction acceptable beyond validation kind fdr control main feature sensitivity specificity probably positive extra reliable least portion voxel truly achieve corresponding region accurate scheme permutation order positive finally voxel control voxel interest patient five voxel voxel element cluster three pattern I linear group cluster spatially cluster brain region voxel discriminative voxel last mean false figure demonstrate retrieve relevant also sensitivity relevant retrieve keeping control together randomized discover almost discriminative notice work well identify discriminative voxel different voxel right discover computationally visualize recall curve figure illustration specificity voxel aim chinese study history fmri subject kind
divide worker consider update column finally parallelization update independent fix versa mf h matrix frame schedule scheduling counter counter else return counter frame single empty list counter else col col frame counter k else x col counter mod support static take consider parallelization coefficient fail converge presence dependencie lasso challenge avoid simultaneous dimension highly contribute formally regularization loss standardize intercept loss coordinate cd soft thresholding schedule strategy pick dynamically observe j substantially
describe angle width b width width angle angle width set part snp available individual slope day measure datum al trait trait kernel linear use genetic define correlation phenotype set genome display depth region detail multiple building follow divide box multiple replication display correspondingly trait association previously trait width b datum chemical relate health property carry record environment year marker
perform due objective function perfect bind minimize trace consistent state need approach minimize corollary note centre france de langevin paris paris investigate simultaneously enforce structure
denote simply useful comprise gps coordinate respective q measurement nf assigning update move previous however reliability weight notion indicate user trajectory reason estimate expect visit assign measurement close proceeding follow pixel weight consider performance exclude pixel also provide novel analytical justification scheme choose estimation challenge briefly automatically detail scalar equivalently nj update dictionary end thus take measurement notational column similarly minimize cost scalar fit turn positive responsible second discard irrelevant datum dictionary change design provide feature help new hope scheme close optimization already measurement available hope track reason backward low function differentiable minimizer nonempty
mean tangent measure tangent work recently stein bregman matrix bregman divergence asymmetric undesirable jensen shannon divergence barrier cone obtain stein bregman stein divergence transformation computationally relate establish bind point matrix classify similarity vector aid stein hence convert similarity euclidean riemannian discrete dirac
speech low comparison return non evaluate system test file evaluate synthesis rnn hour train clean clean train noisy clean clean combine ai speech prediction system dataset clean number several inspire previous approach early replace effort classification loss rnns speech similarly simple activation rnn et multiple enhance scalability focus scalability simple lstm scalability dl instrumental dl reveal significant gain gpu hardware locally optimize library connection cluster inspire scalable utilize try potential train set large label set feed
previous result graph summarize acyclic decomposable hold reverse form add reverse add reverse operation neighborhood change change previous step denote node denote acyclic operation lead cycle add reverse remain cycle remain cyclic operation lead cycles acyclic add reverse otherwise cyclic form reverse operation follow cycle add lead lead otherwise acyclic lead previous need remain proposition search operation status score acyclic status assess implementation greatly graph sample second published replicate gb detail appendix bag aggregated prediction rule base version prediction build effective improve idea learn structure model dag highly drastically perturbation propose aggregation dag greatly dag bootstrap dag aggregate ensemble aggregation dag nontrivial straightforward
simplex axes de close plane hausdorff essentially orthogonal contain w kk w volume q ik multiplying side therefore return fact endow inner decomposable element entry odd note orthonormal square sum square inequality corresponding basis hand equal finite space easily cauchy branch easy substituting say infinitely jacobian either local direction imply say strictly indice jacobian matrix exist direction path w everywhere since prove isometry imply volume circle follow hausdorff euclidean cube isometry image jacobian r metric ready volume measurable exist possibly jacobian since constant prove singular
probability prove kn algorithm smooth k rewrite unconstrained apply summarize need compute operator soft thresholding operator sparse illustrate regularization edge experimental network world social transmission transmission transmission set cascade cascade infection infection illustrate consequence use cascade precision network present fraction infer network cascade successfully income polynomially neighborhood infer incoming hierarchical degree different super cascade pa summarize cascade neighborhood study predict value lead line regularization cascade large cascade may satisfy cascade hierarchical transmission transmission model outperform find competitive first score cascade contribute establish problem
interesting flexible spatio temporal separable acknowledgement award amazon research conditioning express give set posterior pf give predictive curve spaced grid predictive package mat ern determine refer distribution beta kernel epoch decay along place prior constant hyperparameter exceed bound name develop rapidly setting machine decide training start new previously machine
bayesian strong role much framework recent attempt indirect induce structure bayesian tool matrix good suffer attempt low introduce variance singular generalize sparse
almost attempt represent generative manifold theorem asymmetric kernel original limit work latter describe operator limit direct embed algorithm limit novel direct spectral apply directed graph density directional flow source separate directional geometry manifold hoc consequence principled model generative recover idea geometry result respect asymmetric kernel everything expression symmetric asymmetric kernel go elegant four able directional work direct graph manifold relate generative direct embed embed analog like undirected present asymmetric express
regression function rather accounting relate choose likelihood value observe differential variable independent distribute clustering assume multinomial additive function thus penalize maximized algorithm control complexity penalize establish closeness fit fit tend complex however discuss coefficient maximize robust curve training iteratively em cluster step penalize see initialization stop step compute log curve computation posterior maximize respect show maximization perform separate mix subject proportion obtain subject use lagrange multiplier eq role competitive discard update cluster proportion entropy stable enhance less finally penalization set describe competition enough discard cluster discard proportion sufficiently partition robust stand prevent large
rich label generate user tag human different vocabulary describe concept conceptual end multi net multi modal vector conditional imagenet dataset output image representation concatenation tag description skip gram appearing time
h nmf call tweet create conclude tweet nmf topic topic collection tweet prove fast provide tweet visualization datum nmf algorithm prove valuable since text well explore visualization aspect singular value consensus run reliable exploration would evolve national science foundation project nc mathematics
modern day record east united census want part country collect database record often single database database record corrupt quantity merge database return apply paradigm database corruption process provide uncertainty quantification generative unique record relationship database mcmc approximation
q distribution gaussian radial basis whose constrain dimensional interpret centroid manner model choose probabilistic gaussian pose gp euclidean I assumption straight line euclidean track recently pose prior application space metric space less efficient dimensional may reliable way deal geometry provide metric able smooth manifold gaussian model
eigenvalue alternatively employ initialization choose value pick initialize initialization initialization simulation stop value however progress likelihood stop determine basically acceleration decision make algorithm reach log value rand index true ari agreement rand ari perfect ari well chance alone although incorporate parsimonious family structure remain model
operator epoch initialize iteration ir I inexact sparse refer epoch epoch within ball radius center inexact employ efficiently level perform processor regularizer simplify form soft step update extremely carry average propose note convexity strong convexity curvature function relate variable continuously differentiable positive definite feasible regime matrix solution notion strong work f q bind hold eq q dual assumption epoch least last epoch improvement result depend bound propose problem multiple variable property bind constraint allow dimensional rate consist error gradient impose term convergence
attention distinction indirect cause application finance portfolio causality portfolio risk causality measure field information theory experiment perform well nonlinear causal structure well system set direction window economic identify range wide application frequency work separate briefly relevance causality measure largely ultimately researcher confidence possible relationship rather measure analyse causality still causality economic model gain even wide little nonlinear causality finance economic hand many could filter dealing parameter primal equation get dual weight depend parameter functional analysis space follow generalise modification metric fundamental concept space rest paper standard notational convention following prove continuous later operator use mean cross operate important functional hilbert functional trick explanation describe point future loss exist satisfy equality present definition hilbert schmidt criterion schmidt induce schmidt u rkh space define field joint expectation ensure schmidt schmidt schmidt norm hilbert separable tensor notational first denote application multiplication hilbert schmidt cross schmidt operator eq early element measure element rkhs respective bound denote rkhs strictly kernel k yx introduce early schmidt hilbert schmidt follow kernel copies eq cross eq normalise cross covariance operator normalise covariance gram equal u u uk
demonstrate feasibility base tight complexity result value worth utilize proportion bound cover infinity cover lemma class lipschitz cover w generation cover definition growth class dd vc lead hx hx pure bag correctly instance f probability least thus correctly bag draw distribution denote copy instance bag classify immediately z nr h bag bag select last instance bag treat tx I iy find two first trivially let constant see later find space three define equivalent ex x tt relation relation derive
drawback lack view derive reverse end start globally consistent first wish trajectory deterministic thus solution seek procedure point require class recursively assume define covariance iterate apply alternative future window window replace move retain computed implicit solver plot window xt
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public knowledge connection n I objective cell immediately protocol community detection maximize polynomial algorithm ease want infer protocol detect community execution protocol close limit protocol sequence objective accuracy maximization privacy max abstract protocol protocol symbol size challenge challenge successful accuracy privacy define denote exclude adversary get know
signature procedure yield first implement classical sampler virtual virtual order value rank become log quantity quantify goodness namely correspond value benchmark discriminate describe identify contain spectra involved optimally figure display histogram virtual log bold cancer group select detail optimize discrimination vs display virtual log value compute length bold vertical black true spectra binary two function clique restriction since value observation quantify two characterize fix precisely number determine equivalent pearson proposition symmetry concrete discrimination know independent consider parameter simple rule vector sign error precede situation matrix mean zero asymptotic variance supplementary material power decision affine length two parametrize fix two probability recall arbitrary pair material two parameterize configuration resp compute estimator correct resp number supplementary mass acquire patient patient moderate inferior spectra discrimination
low reduce original group classical lot attention approach discriminant propose centroid naive thresholding combine penalty discriminant vector scoring essentially reduce discriminant group direct avoid misclassification rate desirable unfortunately approach include one versus classification assignment usually multi canonical constraint undesirable viewpoint well canonical burden pose challenge guarantee bridge superior performance estimation group goal develop novel multi observation canonical form affect neither rule
bilinear model frame training extract try predict bilinear way transformation relational well relation transformation end frame frames bilinear infer transformation derivative temporal series way individually frame bottom subsequently tune parameter back may system equation stack account demonstrate surprisingly image multiplicative filter relation multiplicative recurrent neural work interaction gate state separate consequently work interaction transformation
associate transformation influence temperature etc measure variation computational transformation generally expert instance software decision evolution measurement instant record example pattern reproduce indicator normal behavior learn statistically model g measurement distribution failure signature describe indicator see expert generally specialize mainly diagnostic diagnostic take incoming come situation early sign failure could perfectly provide level trade false alarm general operator role sign failure recommendation identify failure monitor reach automate precise optimally visit
observe correlation fundamental range economic practical trial na correlation variable causal former latter seek likely case gender cause induce direct influence distinguish possibility leave trial carefully assignment drug common cause consequently correlation must causal fact common record correlation preferred ability complete solution variable cause strength precise causal whether correlation direct act possibility theory place restriction measure nonetheless show early constitute bipartite determine state device markovian implement type experimentally causal quantum signature causal word type correlation causal purely cause framework place common cause extension devise impossible scenario experimentally passive problem quantum causal quantum relate nonetheless causal form influence common cause causal mechanism simultaneously possibility depict acyclic specify causal parent specify circuit depict specify circuit maximally subsequently unknown quantum operation parameter swap pure pure swap common connection middle hybrid causal pdf aim discriminate causal relation
train block size block cifar block c decision tree much well svm perform function step art loss motivate employ support codebook example increase improve retrieval dramatically binary encoding sample outperform speed precision decision hash encode suit plot support outperform also high codebook feature codebook dimension time map train cifar employ two block report cg show spectral
call turn stationarity bethe consider bethe studied point hessian one somewhat call point expression bethe rescaled eigenvector multiplicative correspond transition ise start identifiable give cluster motivate bethe hessian like graph sbm analytically limit know spin graph community situation interestingly spin pass hidden take place informative remain fig bethe hessian analytically
cover call basis pursuit sparse recall criterion suited value non interval overlap consecutive active cf note group tu tu tight x random datum report average figure recovery criterion use include overall sparsity suggest try solution show error group envelope ball tu text envelope tu envelope note denote bipartite linearization trick employ reduce program j focus translate determinant submatrix contradict tu def tu surrogate proposition intersection surrogate text
eigenvector equally good machine weight covariance equation nevertheless extreme order flexibility section principal new encountered dr eigenvectors set constitute eigenvector allow straightforwardly substitute start typically towards eigenvector eigenvector encounter start iteration near within amplitude number drawback retrieve remove find component filter thank regard norm along function highly dependent commonly produce orthogonality manually check obviously case example classical require operation solve matrix build follow compute algorithmic various hereafter solution algorithm mainly multiplication refine eigenvector give one spend build
ball density function two pn dnn l dl assume h hilbert rkhs h fy detail integrable stand transform large introduce necessity measure borel algebra finite geometric median rather condition hilbert see x another univariate median metric define generality distance compute median ellipsoid important transform collection independent significantly strong collection constant collection concentration median true fast preserve estimator disjoint parameter contain arbitrary amongst affected family distance goal admit assume measure special case include f x f wasserstein infimum take simply write
norm stay rmse plot assimilation window norm reduction middle reduction cubic figure analysis tend background one assimilation norm approach corresponding residual norm consequence residual reduction correspond panel residual norm panel certain norm magnitude consequently ensemble residual norm converge optima slightly spike adopt conjunction cubic fig close simply instead adopt sophisticated parameter equip divergence end element I observation adaptive appear panel assimilation analysis norm appear assimilation nonlinearity exponential operator well stability comparison algorithm observation understand view type suggest start linearization remain roughly valid regard appear flexible make step small large guarantee focus examine experimental adopt g assimilation unless default experimental follow
quality bootstrapping enable notation denote mean assessment standard confidence analyze set highlight limitation overcome acquire online action datum decision probability ask action reveal account discard acquire quite allow acquire cost evaluate bandit close exist adapt recommend cite know output mind thing may data unbiased dynamic could indeed argue everything change item news news hour two systematically reason believe contextual bandit
cycle reach separately first go back switch affect likelihood bic section switch trial flip switch bic switch compute switch perform require likelihood v j md obviously constrain document zero document trial update compute datum likelihood note topic switch unlike compute evaluating update minima use select occur document initialize likelihood decision assign topic use document topic frequency count topic potentially however impose complexity involve iterative loop topic flip require scalar part experimentally update estimate proportion trial switch loop active experimentally time compare choose bic range order sensible use order way range order bottom fashion initialize specify predefine remove least plausible minimize bic
measurement spectral project organize mathematical problem formulation discuss various conclusion discuss vi system equation represent however analytical find explicitly unconstraine b ax whose analytical ta tb available wavelet expensive instead analytical solution prefer initial another previous project expect algorithm select row select version
chebyshev lemma choice lemma hence upper meanwhile piece plug give w bound user neighbor type rate via user rate exactly user neighbor crucially joint exploration choose item item u iy iy replacement item apply current scenario incoherence item item represent entire preference yield argument lemma hold user jointly item explore bound different write reproduce lem begin rate neighborhood user neighbor rate stochastically low good user bind every item suppose neighborhood hold neighbor condition km stochastically dominate bad variant bad rating good neighborhood inequality choice finally eq item union bounding provide must exist item time
kernel deterministic simplicity ei use informative present background area regret cumulative bound enable rapidly converge sub formally say sub laplace example zero uncertainty mutual play central role note
development belief network decade network default option machine learn success handwritten digit problem hard classification cnn notable good respective mnist report recent achieve learn algorithm superior cnn achieve significantly low complexity publication cnn whether architecture popular choice step result point obtain past machine feedforward conventional hide neuron neuron classification similarly sized begin classifier input distinct matrix mm contain test size activation class prediction vector unit test contain projection unit include b column dimension element always unity leave description
network approximately network indeed compare sparse easy practice point transition autoregressive mechanism bring dependence g rule condition fx ax n definite imply place penalty via cb follow joint exist care graphical computation union node screen dimension reduction facilitate remove unnecessary edge small brain connectivity reveal datum divide region economic divide decompose finally helps improve overall identification estimation accuracy base literature evolve order introduce undirected either represent give toy share structure cluster two node integrate picture network
operator relaxation f interest different namely simple guarantee search alternate direction multiplier admm
paper take proportional norm project onto system q alternatively interpret stepsize expect pick respect coordinate minimize let coordinate equal aforementione univariate stepsize inverse lipschitz gradient convergence detail section examine behavior algorithm bring opposite behavior algorithm represent system happen rank call
mutation population frequency depth interval confidence allow pseudo second associate affect describe use frequency copy broken population frequency equation allele frequency distinguish copy assign furthermore assume copy replace lie affect possibility relationship occur occur branch site relationship cell copy cell allele numerator copy cell denominator average number copy numerator numerator case still infinite affected lie branch cell average affect expect affect observe affect variant allele case affect possibility likelihood copy decompose possible situation distinguish example condition unable circumstance branch consider tumor come add contain read variant allele frequency half variant frequency copy genome position copy read allele method incorporate incorrectly incorporate alone tumor recover tumor recover
concept seminal building work give uniform approximation symmetric subsequent approximate degree agnostic approximated combination contrast strong limitation polynomial hypercube theory limitation hold arbitrarily high class rely type approximation generalization classical degree polynomial interval inequality function weight strong statement origin powerful jump must quality full multivariate generalization rich agnostic agnostic hardness weakly even boolean hypercube line give independent hardness assumption show pac threshold intersection imply e et al work prove unconditional
inference factor random respectively correspond unnormalized joint iff maximize map message formulate sum function represent part represent select entity mean entity match least entity match since entity equal without match similarity weight constraint entity entity connect constraint connect definition show variable total similarity constraint entity variable relate evaluate variable plane plane evaluate infinity serve enforce matching entity entity sum entity sum max weight maximize objective pass message factor pass define message reverse direction message pass stand source entity message update constraint number node factor node message calculate affinity propagation intuition pass either message sum really variable concrete matching follow c c message message except similarly since ia x ia except jk subtract formula equal update rule match pass number
representation coefficient dictionary polynomial update accord real behavior synthetic localize place performance collect experiment graph wavelet transform ii purely numerical dictionary svd treat iii graph kernel otherwise fix toolbox always orthogonal normalize apply thresholding would careful stepsize code synthetic unit set edge thresholded ensure experiment set synthetic training localize dictionary capture component signal random learn expect dictionary collect infeasible training phase lead impractical training signal allow flexibility polynomial fig fig able recover dictionary learn snr sm signal training testing set compare performance run test approximation polynomial well
rank ccc netflix collaborative interested unseen remove movie agreement compute movie entropy proportion movie movie randomly choose rest testing keep report result fig show netflix moderate pmf seem information state c c pmf mf pmf closely ai social focus preference preference express preference ai limit complete order set close normalise normalise previously rank involve former additional offer explore ease readily capacity similarity assignment
convolution iterate heat smoothing diffusion eigenvalue eigenfunction trivially eigenfunction eigenfunction literature eigenvalue eigenfunction laplace medical imaging vision eigenvalue shape al topological al shape detection et al second surface eigenfunction scope paper issue eigenfunction heat heat heat kernel smoothing take estimate signal truncation automatically determine heat avoid discretization wavelet wavelet complicated study offer simple unified consider wavelet generalize euclidean difficulty arise one try wavelet surface unclear translation try modify exist wavelet immediately grid surface diffusion wavelet bivariate wavelet scale eigenvalue eigenfunction heat translation bivariate heat rewrite truncation wavelet framework wavelet therefore diffusion heat surprisingly wavelet
direction allow divergence change motivation physics sm entropy rise partition solve efficiently formula sm practically multivariate interested form formula sm frank close sm analytic gradient closed similar produce adopt output kl regression minimize sm relative entropy analysis begin new point similarity vector vector apply rbf kernel form k xx exp k overfitte noise otherwise firstly kullback leibler marginal gp focus result pose analytical define eq index kernel use optimizer search gradient gradient complexity store gradient present gradient detail sm matrix algebra k xx k yy hence could rewrite definite xx x k eq worth introduce large calculus logarithm derivation follow eq computational cubic due solve system
randomization nonlinearity first approach match set approximation pilot show decrease piece affect large number show accuracy piece wise fit tolerance distribution sample marginal sec coincide give different stationary generating process input variance monotonic fig transform black line grey dash fisher confidence expect cubic transform display thick line spread spread expect correlation indicate fig monotonic gaussian gaussian correlation var determine transform turn close correlation differ decrease na I denote expect
cascade object star model filter part filter coefficient specify placing detector slide window output pyramid specify position level scale pyramid possible position tuple pyramid detector gradient anchor root part score write kx k perspective part labeling argue necessary correctly treat score label receive stop incorrectly formally score denote letter order response negative make assumption response star root true
variate selection x x variance distribution representation worth representation advantage parameterization reflect skewness parameterization implementation em generate far consider conditionally random truncate namely pdf aa pp limit satisfy rule group rule function rewrite discriminant vector normal discriminant rule e group region
conjunction learn representation reflect layer supervise contrast classification body finally optimal maximize activation neuron resemble human face visualize neuron heterogeneous two type pose aim predict window human g use body bound box contain human summarize location joint normalize bound box use truth part window body part train part separate detector appearance contextual corner annotate body window window body portion inside
theorem cm cm proposition mathematic university department mathematics mechanic state pr university school economics department mathematics mechanics university abstract family test characterization statistic family integral degenerate reasonably efficiency alternative efficiency simulated type recommend practitioner local favorable fully
fashion claim exist drop negative supremum clearly choice nh v bounded right denote sequential tree later except minor fact affect proof cauchy fix one tree well everything upper conclude end sequential cover close eq follow claim provide fix tree along recall hold cover put everything together lemma eq simplify far conclude since take case value particular choice definition value necessarily except sake root
discuss cross theory use domain base graph represent adjacency n eq subject avoid constraint formula rewrite minimization subject extra specify matrix equivalent maximize stability introduce
substitution movie describe interest consider dependency illustrate consist correspond learner template share base training identically share object classic ignore alternative discard clearly suboptimal sample providing achieve
dimensionality image mention vision concern variation recent work neural network reconstruct view face randomly generate face approach allow viewpoint relie find probable match viewpoint procedure sample model fully face show variability expect model applicable task cm cm first generate high description formally n camera transformation target mask parameter increase amount variation reduce overfitte augmentation cnn color variation apply artificial target mask plane change
manner fast exploit ensure vanish assume normalize inner monte carlo gaussian gaussian rbf combination admit expansion datum recursively recursion dense multiplication subsequent achieve reservoir place ensure permutation effectively uncorrelated iid independent draw distribution change kernel remain adjust rather without flexible adjust spectrum way range invariant accomplish since term frequency frequency moreover spectral frequency translation frequency mass basis undesirable optima enforce smoothness therefore spread expansion fast expressive parametrize section already efficient framework learn formalism introduction expansion clarity likelihood innovation process represent demonstrate extended process particularly suit expressive kernel choice objective instance list set gaussian finite process equivalently parametrize mixture integrate away express solely term matrix parametrize independent additive likelihood negative likelihood covariance close inspection expression simplify greatly perform storing require regardless efficiently computation approximation even memory use preferable gaussian
describe decomposition row wang zhang state take theorem column prove actually take author define new eq round integer I p follow argument replace point difference moreover weak upper make little difference purpose fact linearity equality variance e remainder strong weak whereas let pairwise hash need trial pick correspond overall back state omit deterministic operation apply result transpose switching n summarize decomposition bss version extension original bss spectral frobenius randomly subspace see ns nr I I I I construct small lemma simultaneously apply times column result version discuss version randomize use take respect randomness immediate markov careful implementation routine
outli impulse model vector give stable spline encode stability tailed density student identification procedure mixture gaussians variable propose posteriori make derive identification optimization propose advantage new compare worth datum popular huber contribution noise find year description describe approach derive impulse response kernel particular subproblem connect system admit computationally especially compare alternative propagation fully organization introduce map describe time causal transfer system drive output corrupt noise sake output measurement
linearity nonlinearity curve meta rank curve strict rank fig principal translation strict perform know size interpretability rank principal curve call monotone give cloud explicitly parameter rule however lack monotone fig requirement monotonicity new principal model ranking five meta formulate point b place particularly complex bring problem simple represent possible monotonic curve rank cubic meta rule nonlinear end determinant shape curve point scale translation facilitate calculation control derivative order exist cubic type strict monotonicity point nonlinearity cubic coordinate interior hypercube monotone parameterize cubic monotone group existence fail exist monotone eq appendix control optimal curve
ks table deviation percentile standardized version ks statistic adjust variance rv choose testing computation quality normal table asymptotic normality hold well stable rv sample inspection normalize show clear significant level precise eventually conservative small practice actually constant reference therein relevant
extract small eigenvalue consequence nc nc subsequently eigenvalue next nc nc subsequently eq hand q q inequality due therefore combine mm general though method convex globally almost theoretically analyse method experimentally promise back age theory hyperplane case apply outli theoretic principle learn hypothesis finite equivalence observe hypothesis label
special player attention th matrix notation matrix row discount expect reward initial eq denote stage markov property addition state state eq transition matrix eq write notation transition value express note introduce use relationship stochastic rational playing player seek minimax player change game value favor start minimax simultaneously action player receive determined minimax person exist player reward player strategy play player express reward receive value minimax nash equilibrium minimax discount satisfie game remainder playing
formal intuition behind robust update procedure short iterate explicit sgd iterate explicit widely sgd motivate implicit introduce still maintain quadratic proof present appendix formula variance sgd implicit unbiased rate derive close statistical comparative estimator stable implicit hessian iteration although quantity explicit implicit implicit update perform approximation na fisher general analytic equally define generalized implement sgd essentially univariate interval generally sgd evidence performance goal tangent analysis procedure principled purpose identify seem viewpoint score learning rate add insight issue explicit help identify loss sgd estimator whether one show low mild condition sgd information sgd information become apparent represent amount else learn receive minimal despite much jacobian curvature show I refer explicit sgd parameter e way sgd use order hessian jacobian effectively iterative procedure thus uniformity practical arise normalize aforementioned proceed sampling efficiency quantify theorem consider explicit update simple
show present appendix expectation follow behavior large follow mle appendix simple bias substitution integral mle v rewrite q obtain
approach recover lose performance uncertainty human st switching complexity annotation variation synthetic approach template question typically spatially relate frame argue evaluate threshold order predict accuracy architecture nd segmentation pair class whole test yield total automatic rd table switch human automatic class propose approach across baseline th human ask answer question collect answer question answer class single world binomial check preference
quantify illustrate method performance equation mathematically phenomena heat structural pde law law practical application real simplification tractable ignore physics certain condition pose difficulty refer pde knowledge pde accuracy predict compare pde model acceptable current pde define model alternative assimilation broadly speak assimilation incorporate reflect inherent mathematical assimilation share estimation assimilation calibrate assimilation define possible good involve parametrize reconstruct experimental obtaining assimilation physical square however mean quantify recent pose
classify strongly ground combine category classifiers cnn imagenet object close coarse confident coarse belong category category rank nd overall cnn place demonstrate need category classifier impact coarse train classifier coarse top error overlap execute trade classification large lead execute category enable hyperparameter merely minor speedup time build net parameter imagenet size hyperparameter compression factor compression decrease mb mb class net execute fine category error hyperparameter imagenet dataset building net independently obtain error building block coarse category method top top imagenet net build net layer imagenet layer total layer
perform class extract class dataset e relationship dataset predefine unseen class fine cat cat cat herein attribute label confusion though method predict achieve lead well contain drop situation apply share public achieve conventional art cifar conduct experiment low possibility similar order employ aware semantic visually likely similarity electrical device pick relate nonetheless work coarse class achieve property theorem manually
examine nature choice familiar law model algebra fine grain become refinement think initial triple degenerate triple singleton way account broken choice measure mapping implement value swap measure together form specify particular mathematical decomposition purely usage probability implicit somewhat outcome outcome choose outcome assume outcome belief type distinguish belief impose restriction make restriction probably highlight illustrative cascade mechanism sub experiment implement conclude affect odd pick know stage increase odd hence affect belief future mean causal aspect concerned scope potential stage stage part namely sense separately exclusive historical pick nothing pick give know execution sequential experiment situation fail example precisely text htbp clear belief choice respect boundary separate operation parallel namely represent right collection experiment sense operation transformation put choose illustrate figure htbp recall situation choose know first discussion amount stage measure subscript similarly table list table experiment swap colour black leave black white yes white yes black plausibility q choice provide stage draw right plausibility obtain intervention evidence support hypothesis past intervention deterministic moment intervention subject external recall
orientation orientation velocity connectivity presentation source fig histogram noise velocity connectivity stimulus make long range finally group reduce kind error subsection consider perform spatial visual predict stimulus trajectory group apparent motion move coherent trajectory one temporal offset one explanation specialized motion contour orient study suggest rule drive orientation role play specialized evidence trajectory drive connectivity come population response change direction show low angular provide sort spatio temporal interpolation mechanism velocity already motion movement extend stimulus present area asymmetric column orientation role play horizontal suggest biased motion recent show segment support two mechanism spatio temporal moving shape twice bar dataset depict create segment move frame curvature bar opposite bar orientation embed consist uniform path direction stimulus frame full spatio surface stimulus rise visual grouping spatial level identify visual grouping spatio recognize object movement reasonable two interaction point integration trajectory cluster spatio unit hand grouping law contour different implement structure experimentally confirm composition mechanism result summation argument cluster sum affinity construct matrix additional coordinate
bias improve much bad proportion miss disk indicate near contiguous composite distant conditioning determine design lattice low incomplete approach compete month spectral stationary observation region south illustration unknown mat ern function effect miss total posterior minus isotropic process mat ern q smoothness modify find bayesian mcmc non square lattice lattice maximum original pixel approach radius lattice choice lattice lead eigenvalue sampler discard constrained yield embedding describe run burn period three draw unobserve closely smooth draw region figure
case loo conditionally parameter evaluate importance separately correction correction produce also illustration finally dataset put scale interpretation firstly sometimes secondly calibrate interpret pseudo calibration error loo quadrature loo loo really completely fail loo useful la loo fix whole la tp l loo loo na na g g la l loo significantly ep loo loo tp ep loo loo na na na loo ep na ep quadrature loo significantly well satisfactory loo bias correction therefore force truncate loo loo cm add well bad tp loo loo loo la la table loo data loo quadrature latent ep result loo ep already good correction worse
area may community community community goal community detection citation author undirected author paper replace direct author b easy analyze research discuss separately rather split community large community dimension reduction interpret community design study htb citation identify score stanford bayes citation large parametric network network connectivity sophisticated identify community research propose score community undirected community follow datum respectively investigate cluster al seem largely section discussion b citation citation largely detection method community multiple testing spatial statistic present figure convenience road map section connect also section therein trend citation trend finding limit period paper proportion decrease distant figure become collaborative competitive degree bipartite phenomenon frequently obtain quality hard true quality resource google try portion challenge author overcome primarily home attention except recognize country bioinformatic primarily period statistical area period seem serve meaningful community collect long
counterpart mnist exception establish evaluation variational report softmax perform well score lda softmax competitive dataset hide record dim news training intractable direct sigmoid belief demonstrate emphasis possible model inference architecture considerable gain expressive continuous latent promise latent appropriate conditional latent would require make inference would make model make powerful direct latent acknowledgement thank comment provide gradient outline computation
walk proposal inefficient explore offer use improve mix hamiltonian monte proposal would full whereas optimization objective surface apply simulation feedback determine next location function idea gps abc cost abc institute world demand challenging computation standard tool handle likelihood problem two sampling algorithm hasting adaptively gps simulate challenging realistic biological illustrate potential algorithm biological birth star weather rely naturally observe hypothesis generation evolve critical match observation hypothesis cycle trivial phenomenon inefficient potentially interact program scientific play
improve change strategy alone equilibrium one play could strategy game payoff player check alone nash equilibrium games game incomplete intuition know opponent playing set consist player denote player though extension game transition
disadvantage problem movie rating average user rate movie rate practice uniformity far netflix user vary quality dramatically order suggest nuclear norm column marginal improvement unweighted nuclear pattern bound low rank column possess incorporate completion form regularization netflix assume movie rating dimensional weighted represent define column denote imply laplacian column column row term add frobenius nuclear outer non propose outer
model target try risk rare influential event concern notation parameter term roughly estimate period crucial system capable handling include year extreme heat loss minimal translate book opinion incorporate alternatively calculation identically distribution practice drawback tail put second degenerate maxima sequence exist degenerate member q shape correspond fr support understand I mean maxima increase would relax limit certain unit fr q helpful consider member transform unit fr margins unit margin transformation assume unknown may taken extreme transform location fr site generality one assume unit fr extend multivariate let maxima noting unless occurrence block coincide spatial maxima location degenerate sequence multivariate distribution one
transpose jacobian relate row jacobian stack hessian hessian equation follow form definition trajectory derivation hessian parametrize assumption hessian give parametrize policy trajectory hoeffding inequality section conduct domain linear water
correlation output plane require storage testing memory tolerance template take elsewhere filter correlation filter design formulate form expression vector notation equivalent channel signal formulation due circular circular replace filter introduce combine localization base svms traditionally cf design g template equality constraint relaxed constraint margin negative slack variable image wrong margin svm class negative typically express obtain add problem formulation signal determine dft template q inverse dft wish constrain find considerably computational numerically outer cm minimum width height cm fill blue thick width cm fill illustration require computational consideration discuss challenge arise filter computationally intensive memory overcome design typically great conventional counterpart usually train usually image kn kn reduce explore several way explore portion template observation portion nearly dft tail circular
stationary stationarity hold stationarity constraint derivative lagrangian multiplier constrain substitute set stationarity burden lie th order fact prove two center straightforward characterize denote two critical vector much strictly half coordinate similarly second half u claim q inequality x x kn quantity inside kn ki stationarity equation choose first claim k true low precisely p k reach stationarity u ki positivity dual since positivity always define express stationarity multiply side since choose contradiction follow item lagrangian apply kkt condition sufficient symbol represent abuse constant actual vary theorem convenience suppose nk k fix use concentration sampling replacement convex gaussian union put constant bounding hoeffding inequality term union bind eq apply q union second inequality use part third claim let overlap segment segment take union
run prescribed weight tree alternatively run approach demand homogeneity possible study cause difficult small branch length random gamma log branch trait describe time view possibility selection procedure search rate change laplace motion brownian given brownian drift gamma increment interval increment gamma define
layer kernel extract pooling layer use bandwidth time show net jointly achieve net net two layer contrast normalization max pooling test extract top net dimension rbf times median without center result drop phase gradually achieve net net imagenet million color image randomly random horizontal jointly neural net dimension net pooling net rbf set comparison max voting variation color neural net produce rate neural much speed test ridge table dataset representation convert base randomly break
go blind dr even though computer field create automatic system manual effort screening raise financial issue study dr specificity screen sensitivity specificity combine novel screening feature make close meet sensitivity automatic early recognition serve essential dr screening system dr investigate exclude direction dr recognition framework extend component automatic dr center novel assessment feature exclude image
probe degree structure I denote entity probe similarity transpose structure basis stand bipartite learn edge total bipartite utilize programming solve solution return illustrate come shot shot ambiguity cross word occurrence slice pixel pixel operation single locality appearance change ambiguity view codebook carry information visual large codebook map codebook preserve appearance look distribution associate spatial collection visual word embed use visual make spatially locally distribution account appearance change similarity way embed hilbert rkhs universal preserve inner similarity two two end image occurrence appearance visual word rkh entry generative discriminative view insight activation view location joint j p eq inequality rearrange efficiency simplicity idea handle specific bind eq spatial fig whole latent spatial appearance slice
able extract descriptor binary picture form dimensional canonical h hence binary accuracy linear svm feature semi side validate alone statistically respect pair svm scalable training autoencoder nonlinear reconstruct regressor mnist cifar compressed mnist cifar full anonymous comment discussion mark van la method principal component pca cca able reveal relationship datum nonlinear variant cca prohibitive large scale randomize
extensive separation important annealing nmf decrease entry high amplitude indeed descent decrease iteration refine final threshold independently estimate level solution threshold choose amount keep last also q aim minimization use descent minimize subproblem differentiable must additional indeed thank constraint hence fidelity term domain permit prove rule subproblem line proximal able continuous low q close wide interest function proximal operator update account negativity soft solve proximal operator constrain diversity greatly improve enforce make separation complex see aim explore extension tackle impose transform noticed contrast prior negativity challenge focus formulation sparse transform
restrict constraint dictionary minimize datum contamination objective atom recovery compare svd al condition sparsity atom dimension k apply denoise image corrupt additive slightly peak noise structural similarity indicate efficacy metric technique gain community survey
change objective soon main evaluating criterion form candidate optimize degradation see value fall threshold could coefficient discard atom atom removal successively update stay basis iterate basis coefficient implementation allow atom add spurious atom add later phenomenon atom form iterate would form iterate computational comparable approach require calculation vector ht iterate column reduce conclude discuss practical guarantee converge decrease termination property proof appear sublinear convex generate true optimum standard omit formal solution boundary atomic place
context replace span relaxation plant recover provide essentially exceed provably semidefinite programming sdp q possible say relaxation naturally intrinsic price pay similarity efficient theoretic simply guarantee sdp relaxation order recently introduce plant recover perhaps surprising runtime sample size interest raise practical provably
neither true source subtree pg pg ml proposition definition claim protocol usa anonymous medium share anonymous message initially sensitive consider observe snapshot spread advance exist protocol adversary introduce protocol call message fast perfect network message nearly experiment sample facebook effectively source cycle core aspect internet propagate message text video platform link message rely post brevity interface party communication communication anonymous enable message friend message author identity message exist service store company access solution store node know never party distribute monitor architecture simple anonymous protocol anonymous adversary recent advance building protocol truly anonymous platform contact strong adversarial condition consider contact graph identification fast connection internet principle receive equally likely key real scenario message spread share filter inherent happen message standard delay message spread time spread continuous anonymous popular decade anonymous communication receiver
fig contrast analogous hierarchical able detect significant specificity glm ols perform poorly even corner expect rapid column significantly thresholded map ol well perhaps detect sensitivity specificity fig replication simulation show portion glm ols approach give significant brain result able consistently detect truly activate avoid finding glm perform time canonical computing brain significantly canonical voxel correspond closely voxel fig believe diagnostic assess glm analysis ability recover peak left show hand column estimate replication width occur shift result single slice slice illustration secondary work involve activation think discriminant voxel level shape group shape also another response delay reach peak roughly apparent slice problematic canonical basis glm show
parameterization enable candidate induce convexity partition efficient proximal svms initialization generalization partition global candidate increase fitting indicate candidate achieve practice competitive art locally relate category assume specific interpretability assign flexibility indicate clearly relationship input model predictor probabilistic framework partitioning sp sp utilize region predictor highly advantage sp vc dimension classifier develop utilize predictor uniqueness specific account test formulation sparsity induce relate motivation test kernel
angle triangle retain discrepancy map integration triangle important graphic infinite carlo vanish integrable triangular van estimate merely integrable discussion conclude give volume cube ball simplex generate space sphere cube simplex simplex cube cube lie spherical triangle mapping identity equivalent input differ corner singular variation present cardinality point hybrid computation triangle point degenerate triangle via corner convenient equal quadrature right triangle borel
skip gram embedding gram starting suggest gram embedding retain retain compressed specific lexical acknowledgement centre france centre word particular lexical relation
proper bottleneck reason may redundant computational might lead bottleneck bottleneck rarely address bottleneck choose bottleneck bottleneck layer metric refer train autoencoder show bottleneck layer pattern hence determine critical bottleneck notice slope clearly within bottleneck bottleneck change critical percentage connect bottleneck curve line percentage difference give large enable automatically
q easily notice equation equality independent complete section thm partially support rf grant nsf grant nsf dms city wiener detect minimum observation couple strength differ adopt approach minimize mean alarm partial correlation sum stop mean without bind emphasis decentralize centralize arise engineering detection analytical consideration concern observation delay formulation time constant interesting variation assume depend treat yet formulation far either one stream regard post stream couple assume wiener different
project people project project average project segment project frequent project display frequency display project distribution feature skewed update goal mm project comment available project mm analyze behavior resort type type e project q counting project project project project hypothesis project less support one ready hypothesis project comment increase level dedicate web site matter extent previous pearson management activity correlation seem decide project depend receive dedicated site overall project depend goal goal frequent likely high project support project confirm recommender project goal
resolution obtain coherent take player position predict coherence respect process path coherent interpretability kernel simple coherent estimator require long substantially realistic convolution manual normalize branch exponentially state continuous previous homogeneous depth transition kernel convolution formulation reduce integration state simple play expectation taken provide useful summary unfortunately appropriate tradeoff homogeneous define average together irrelevant begin pass currently markov would estimate
wireless channel network adversary finite enable static adaptive receiver discrete asymptotic static pair propose novel even armed bandit work armed armed bandit regret sublinear sublinear reward adversarial measure minimize knowledge pair achieve linearly strategy bound strategy wireless detail minimal early rest adaptive learning conclude paper receiver receiver amplitude phase pass value symbol normalize unity pass channel signal represent see receiver assume receiver perfect synchronization match filtering sampling kt k signal level duration word signal level power send probability analysis receiver receive along offset
computationally kl model vocabulary word content may topic model might aspect different similarity multiple possibly source compare similarity similarity significance apply
supervised classifier good four supervise rate well unlabele good improvement classifier offer improvement picture different supervise optimize option among supervise provide far observation implicit constraint lot consistently likelihood achieve beyond deep insight issue cast light safe attain offer improvement test perform
likelihood average stochastic drawing uniformly definition involve missing involve pick layer flexible dark input fix value add add deep boltzmann structure mask vector indicate component multiplication simple illustration sigmoid index eq evolve reconstruction eq
concern argue reasonable geometry insight dimensional method infinite dimensional measure space tendency disjoint mcmc define straight forward proposal different note though still slow geometry incorporate also geometry ensure review langevin langevin metropolis hasting proposal though facilitate focus appropriate manifold chain manifold set rotation directional manifold problem appropriate need area geometric problem discuss geometric mala cope target distribution heavy tail dimension mala geometrically tail scenario geometric ergodicity fail mala acceptance scale related picture show metropolis target identically symmetry regularity shape tailor scenario change serve practitioner requirement acceptance rate rwm discuss focus langevin exposition riemannian geometry full derivation langevin diffusion manifold highlight question geometric hope field award suggestion field smooth
right h node target structure sampler empirical center generate gibbs sampler strongly identical simulate simulation moderate particle emphasize kernel respective good drop considerably sampler autocorrelation strongly however autocorrelation comes set hand form fully sampler magnitude computationally involve partially sampler autocorrelation suitable convention recursively simulate definition document remain index implicitly artificial particle particle assignment eq ratio unnormalize explicit set explicitly particle factor depend interest
translation testing dataset summary noisy dataset technique address wikipedia annotated software language technique statistical machine translation evaluation human annotate work supervise procedure generate wikipedia discuss statistical machine use nlp preprocessing wikipedia summarize table previous language specific preprocessing stage parallel language pose bottleneck language cover contrast rely agnostic compare sufficient replacement dependent entity token entity classify word problem task tag depend neighbor competitive sentence dependency
versus list bind especially efficiently threshold desire figure integration play role determine b subset expression correspond small theory lb theory c c simulate compare three numerical determine threshold estimate delay scenario delay threshold mix community mixture know mix inside b demonstrate
encounter practice value asymptotic bias would trial baseline trial randomize patient illustration outcome treatment baseline dl gm dl base achieve normality dispersion five covariate patient estimate logistic treatment successively log correlation table ap ap fit value correct reduce reduce far mark
computable hessian precise subsection stem structural strictly eventually lead barrier move along locally convergent scheme assume compute clear much progress path need fast convergence rate attain idea complexity need define function precisely large compare newton optimize together standard initialize subsection euclidean matrix obtain start definite newton idea behavior start close enough strict local fast convergence assume hessian fy x eq newton well follow formula formula obtains write hessian note insight newton look start show word affine share affine invariance sis inner would without third differential operator idea fix issue replace euclidean function observe one use always definition self barrier keep mind self describe region convergence newton self important eq without standard newton iterate barrier plan plan since q immediately initialize remain iterate obtain infinity need control fx uniformly fx fx fx safe penalization concave arrive barrier self logarithmic barrier difficult barrier universal key follow generally ensure central generalize identity close central formally basic self barrier small describe subsection x k scheme describe show iterate remain central indeed theorem obtain thus obtain finally yield explain central path trick one path small enough central path iterate equation start k analysis correspond obtain path roughly self barrier om computing newton direction compute invert thus barrier nice barrier barrier come
mix marginal usual effect spatial section penalty scad interpret obtain sparse induce around zero subdifferential net scad example notation net pattern convex subdifferential origin interpret score penalty scad practice penalize necessary determine pattern convex penalty optimum respect use algorithm sparsity penalty leave depend yield meaningful instance regression meaningful large perhaps simple estimating nuisance penalize acceptable context bias incur score thought test useful manuscript focus score effect modification section instead effect group appropriate straightforward
good sentence begin repeating song detect choose sentence certain presence document frequency size sentence weight c c raw binary parameter music context future include feature type gaussian help markov model leibler frequency relevance singular value term document frequency letter generic successfully text speech
link direct centrality proximity distance short centrality quantify node centrality another indicate way divide subgraph great within modular theory shannon contain entropy
belong inter level document label experiment connection inter inter decomposition control seem appropriate interpretable question whether use representation dense one representation space primarily regard interesting benchmark modification flexible applicable semi science computational intelligence system x entropy ie x p eq iterative alternating rule iterative rule instrumental analytical
response causality locality require superposition rise cause event background precede formally attribute background rate augment product poisson density causal iii cause impulse single parent process unweighted indicate direct edge node whose indicate reflect recently framework conceptually class graph exchangeable relate graph trivial endowed characteristic convert representation transform many construct nonparametric fall leverage formalism combine impulse eq binary negative respectively capture interaction parameterize support us structure interaction empty background process recover make event allow strength interaction suggest
experiment except node initial choose identity evolve leave tp plot standardized factor predict still membership estimate checking mean calculate state tail experimentally deviation size factor class encourage priori cause asymptotically true class filter assume student posteriori set figure comparison accuracy adjust rand index snapshot sbm surprising probability follow accurate
modify sharp bound additional show excess hold excess risk exp unknown instance regression respectively domain classifier aim generalize unseen interested problem function logistic loss classification portfolio management arbitrary e minimize exp concavity
multi task figure achieve small recovery short demonstrate particular observe outperform test noise exceed solution involve sparse test therefore choice real world I high collect name english letter alphabet twice group subset refer task correspond feature response letter experiment letter treat different training evaluate multi average square whose test commonly task
tangent axiom positivity symmetry triangle inequality simplicity geodesic derivative geodesic view distance follow end field call field characterize gradient field manifold hold exponential cut cut locally integral curve distance function pass note condition pde simple example second appendix distance fig field everywhere characterize distance distance v require red color indicate value blue indicate riemannian embed space pf ready heat flow schmidt learn set illustrate justification whole heat gradient obtain geodesic function analysis control approximation rely cut query cut distance fail since field cut would embed undirected graph neighbourhood near worth note
wish dynamical drive external parametric neuron goal insight analog second nonlinear state spike count bin poisson relate second smoothing lag particle smooth training figure sample addition prediction clearly frequency datum capture show cycle tb trajectory trajectory smoothing sample posterior trajectory simulate trajectory inside cycle attract tractable dynamical suited principled learn expressive
partially input denote miss refer observe union realistic g human nevertheless model within straightforwardly model jointly contrast input take account uncertainty location empirical experiment well q observation model fully partially location input define incorporate partially principle specific predictive train incorporate auto case confident prediction outlier due together bad gp equivalent miss gp semi supervise use traditionally encounter know approach accord model label incorporate specific follow achieve miss confident label build instance framework adapt tackle miss appear treat difference input associate uncertainty concern method self also train portion however hence uncertainty measure discard contribute make self semi gp simulate give input correspond world datum body formulate regression portion input miss extend train gp handle miss straightforwardly reconstruct fully input size motion plot mse compete method vary percentage miss plot seed supervise gp make portion miss converge gp miss identically gp latent automatically constitute generic dynamical modelling able capture complex rigorous bind range world give research extension become feasible propagation gaussian allow filtering application variational system linearly unobserved state non past latent propagation lower also promise research place application formalism process function five learn complex
exist q permutation many annotation rank rank empirical rank rank key aggregation statistic list copula rank list rank rank rank mid retain definition define informative domain call rank miss expert rank number expert rank object expert axiom e consistent notation axiom monotonic axiom furthermore call extension tie map strictly important rank imputation whereby list rather
researcher relaxation qp seminal evaluate various qp show dominate sdp relaxation aim obtain tight add high interaction marginalization group practical open decide attempt semidefinite primarily state semidefinite redundant linear develop proceeding notation throughout conjugate operator denote nonnegative frobenius state configuration discrete consider parameterized potential mrf maximize energy probable assignment estimation hard combinatorial problem cast mx estimation equivalent follow encode hardness arise aspect binary ii motivate relax semidefinite relaxation step
safe test appropriate dynamic screening reduce radius region sphere improve test effect test may approximately induce regularizer separability problem sense assume advance weight group induce soft thresholding eq dual n optima link small publish extension except contain thank please detail safe extend screen st define capacity screen readily thank concrete propose safe static screening implementation algorithm usual appear line line backtrack dedicated safe l safe state successive use screen test lemma screening compose line
vanish perform embed define rx proof exploit embed distance towards first sum assumption bernstein term sum bound bernstein resp resp fraction nice neighborhood coupling generalize random node everything else attribute parent attribute child label attribute equivalently label random accord everything attribute parent one attribute denote node tree couple g r attribute leaf independent consider eq lemma lemma give symmetry
microsoft early widely proper noun coincide w shift occur analyze shift amazon review content short book corpus table amazon review twitter dataset like acquire sense movie game application change acquire east usa shift meaning release book capability method book aware web request intend evaluate quantitative linguistic shift corpus copy wikipedia corpus wikipedia corpora speech tag introduce word shift finally mean word pair exclude functional word occurrence word occurrence illustrate two type perturbation frequent tag might car frequency method observe degree consistently word category perturbation quality perturbation annotation
report community detector label four unsupervised anomaly report traffic detector anomalous traffic technique multi resolution method report traffic distant use useful kullback leibler detect prominent traffic report anomaly tuple address traffic select alarm avoid miss build retrieve
square essential present satisfie instead restrictive arbitrary excess loss give straightforward taylor around end minimizer obtain identify level loss parameter z play thus result may role require identify cardinality proportional version find theorem assumption question risk rapidly decay rather concentration argument small thus heavy tailed heavy target scenario bound empirical minimization precise space almost close close loss distance average distribute goal least minimizer unlike select describe price cost like minimizer partial note problem deal predictive reflect good exposition estimation prediction select element minimize
limit posterior obtain process relative n support I distribution variance square jk ki hold w jk w formula matrix check posterior select impose n observation bb reliable value way eq always nonetheless tie population function tie become represent expression verify symbol use indicate go sum ease equivalent sum converge vanish follow procedure prove equivalent statistic consistent conversely instance different significance return powerful conversely alternative asymptotically experiment range facilitate test traditional never outcome call issue chance stress
propose ng jj ng shrinkage method assign get sparse elastic net covariate elastic tuning estimator reduce lasso select cross reduce package coordinate reduce comparison package
measurement imbalance momentum direction particle number event black angle imbalance momentum angle observe projection little discriminate production production capture derive sum observe p p tp tp angular mass mass visible simulation first low level variable benchmark text neural
effort design progress current benchmark upon descriptor decade show arguably claim traditional hand make different discriminative robustness numerous complicated stage optimize module pre encode transformation careful individual module intensive ensure whole module individually present unified easy framework face basis convolutional neural pyramid face pixel
fx cx giving obtain since exploit ei replace readily familiar g analytic ei monte predictive equation simply average f j observed generally error suffice arise explore ei surface whole numerically whereby outer loop yield yield ei search improvement al composite cause little deep consider constraint slight composite removing case ei new composite notation ei substitution integration otherwise rearrange shrink besides point analytically ei idea avoid boost efficiency ei drop lead way carlo search know feasible lie drop large region towards boundary restrictive consideration modeling extend much constraint surrogate summing extreme former using could monotone include active discuss motivate implementation provide material
something wrong something effect seem isolated appear datum table nan test quality jump step trend behavior practitioner desirable smooth jump one smoothing bagging adopt choose large cumulative size bootstrap estimate size pt value median regard mean seem value step total view
definition lemma statement denote portion lemma choice next problem integer follow claim suppose begin follow z argument proof replace follow fix arbitrarily every three follow repeat throughout arbitrarily nonempty hold satisfy exist prove repeat argument utility establish cyclic arise context logistic regression estimation cyclic minimization subject symmetric negative number observe variate definite variate inverse minimize concentration model gain popularity particularly develop tackle setting necessarily inverse covariance provide cyclic minimization objective function example convergence guarantee however selection version
computation volume page world page page extraction author analysis international conference page organization title supervise principal journal journal american volume year supervise title principal via advance system page year extension author zhang yu reduce title reduce journal page american title feature scale le chen title motion motion author journal pattern intelligence transaction year title author journal page year title comparison
reasonable convolutional neural work architecture architecture gpu expect particular architecture sort restrict connectivity layer current might connect activation effective parallelism layer column dependent varied scale modern convolutional short new descent sgd present perfectly sgd approximation perfectly nonetheless neural way train parallelism parallelism parallelism
metropolis take keep gibbs dirichlet topic prior ip variable parameter text token stanford speech extract gram tag simple pattern phrase obtain vocabulary phrase summarize htp case token r phrase token brief phrase brief manually neither lexical identify section template classifier introduction statement conclusion evaluate classifier split tune set range limit evaluation posterior great obtain classify support side precision topic phrase control plan plan plan company plan health right control rational r country vice member limitation act security rule security exchange act act communication act vote vote political political free speech
independent define index I zero rearrange since claim
general family term surrogate option calibrate surrogate sensitive classification interest extend surrogate multiclass early work zhang multiclass framework multiclass use surrogate surrogate lee particular multiclass lee al calibrate multiclass multiclass surrogate calibrate multiclass study consistency involve space prominent problem consistency calibration contain model rank document relevance query losse recent year zhang subset discount cumulative simple calibrate et normalize ndcg consider focused subset disagreement pd popular surrogate r loss et show surrogate calibrate pd precision reciprocal subset calibrate surrogate loss standardize show ndcg loss standardize standardize finally consider al use surrogate regret certain surrogate multiclass loss et calibrate surrogate note develop consistency multiclass calibrate multiclass introduce measure certain algebraic geometric result existence calibrate surrogate type subset surrogate main conference paper
linearity upper need precisely go follow absolute eq gr adapt argument go end equivalently rewrite standard extend odd define uniformly balanced algorithm query achieve overall wrong exactly chernoff target fail stability approximate title corollary lem theorem prop conjecture example observation definition acknowledgement acknowledgement university institute technology cl l grant li monotone monotone boolean circuit paper boolean circuit
median maximal point base point adversarial outli always fraction straightforward see outli base output estimate machine deviation ground truth estimate point low bind number overall fraction outlier outli adversarial distribution make appealing average consider machine outlier outli fraction break aggregation severe robustness base outlier preserve besides favorable alternative randomly permutation uniformly take face arrive respective perform central often outli long randomization division average communication follow section
line sbm see appendix claim set succeed throughout learn maximization loop boundary phase instability dot black line overlap retrieval parameter sbm additional component group degree sbm regime find spin phase see appendix ground vary appear determine classic modularity exceed principled method network agree perfectly find modularity overlap grind google network common novel find retrieval state world google truth statistically height c social book political google appear community look structure work group
observe difference consequence interest future ga ga li li li ga respect rank determined procedure become ga ga li li li precisely response bring canonical inverting
suggest upon literature number constant state space domain e potential maintain dependent location stem e galaxy basic physics galaxy mass geometry galaxy constant universal isotropic wang respect angle invariant also recall isotropic remark thus motion dynamic isotropic isotropic attempt use ensure freedom keep ease space addition per cross vector q isotropic vector include isotropic aim model help demand space use ready express mass embed within cast bin min equation unknown rhs mass density plug rhs compute discuss domain space e invoke functional relationship angular momentum e v circular distinguished circular speed mutually speed parallel simplify thing coordinate rotation previous v refer
type unify similar intuitively naive pick near else construct neighbor convert pairwise trick query inspire boost coarse fine human human perform coarse pruning recognize distance nearest fine distance query sample query convert trick utilize global query htb aspect residual obtain dimensionality identity dimensionality rp b database randomly select per person person totally database comparison global focus dimensionality recognition person totally average residual rp graph see five approach achieve similar indicate reconstruction residual table identity rp graph recognition discrimination ability difference high rate pt identity rp htb direction gd gd dimension interval gaussian zero vary gd perfectly classify poorly gd datum case unknown conventional subsection evaluate public database reliable experiment htb contain digit experimental setting
bivariate employ sn employ indicate limit density diagram identify sign classical sn systematic log link another inferential simple possible case diagonal factorization translate product likelihood sn hence stationarity implication diagonal involve project qualitative front qualitatively family discuss match skew replace one student examine skew dimensional student result skew density degree version function univariate freedom version dimensional qualitatively correspond sn tail via
van ac uk inference desirable property uncertainty estimate way tune hyper scalability dataset remain variational inference efficient exploiting give induce formulate inference preserve load scaling amount variable mnist gps big show complicated fitting variable dimensionality use model big dataset limited scalability dataset natural ignore amount datum
pool list image easy sophisticated sift descriptor importantly magnitude sift feature map see superiority feature map map show average c sift feature focus object extent soft convolution sparse intersection large codebook atom complicated et type dataset performance purely level level mechanism therefore base task drive specific parameter model include ns layer also study ar database face accuracy sensitive stable range reveal bring performance gain vs neuron gender sufficient gender peak neuron say ns serve efficient mid hand propose produce argue much propose layer neuron mid feed support result perform order achieve comparable effectiveness propose gain
noise section visualize organization cluster encode hierarchy connect dendrogram compute sample recover topological study code generate construct evaluate library expand grid measure smoothed detail near among point follow knn kde kde h h visualize graphic package code kde plot col h band estimator
box core marginal polytope minimize project gradient computationally hard approximate backward argument constrain polytope evaluate polytope hard nevertheless knowledge polytope normally ellipsoid barrier polytope inherent keep inside polytope enforce constraint consequence property state proposition reduction approximate partition tractable polytope surrogate gradient entropy polytope relate polytope reduction hardness obtain polytope hardness marginal polytope manuscript learn independently high level differ establish hardness marginal specific core collection
replace take justification question arise might argue modification thing critical thing arise trust region classical trust region curvature constraint pick suitable minimum trust trust approximation answer discrepancy second expansion impose partial accept take curvature square would difficult solve difficulty lemma matrix measure upper generalize replace inequality minimum always jump trust multiplier solution constrain step cifar algorithm newton curvature information fundamentally shape curvature method unlike newton saddle unlike rapid even requires approach dimensional subspace iteration span
arbitrary boundary rule machine excess loss condition excess margin correspond convergence convergence risk excess constant eq generalize margin da provide limit neither depend explicitly tight may achieve motivating intuition formalize soft appropriate hinge loss corollary result surrogate linear condition use theorem show suffice minimizer loss size separately lipschitz modulus logistic separately surrogate section lipschitz loss modulus satisfy close radius cover b iy I I following make theorem provide bind consistent modulus state combine follow satisfie generalize constant obtain non loss surrogate modulus condition surrogate iy satisfy constant obtain piecewise theorem theorem rate
dirichlet topic length guarantee assumption test corpora recover align bipartite column accordance matching topic recovery across il il report recover kl mean document length result ng small sample recover show error topic dataset green quantitative measure algorithm lda word exp dm real hold consist document document dataset comparable moderately length coherence semantic quality approximate experience quality top word coherence iw evaluate tc top recover topic topic topic top match pair topic gibb recover
uniqueness freedom j j unbiased freedom agree p knot small provide freedom total knot fit replicate set freedom spam spam j j spam spam propose freedom compare replicate data freedom calculate axis overlap colored degree spam axis axis plot solid vary spam black dotted indicate consider solution sparse completely pg completely never corollary
filter feature appear set fair also compare f nf dataset state htb top top concern art precision increment alignment distant supervision generate incomplete label discuss tackle noisy truly effective information extremely poor gradually imply approach principal recover low furthermore discuss feature program show consistently
heart rate pair height weight heart height know cause heart age measure person biological body uci learn repository concern systematic regard compressive function addition coarse water making incorporate material chemical show water influence concrete compressive age measure per concrete pair pair scatter compressive pair compressive strength pair compressive compressive pair compressive coarse compressive pair compressive pair water aggregate aggregate compressive strength compressive strength simultaneously component water decrease measure cubic nevertheless uci repository medical thought arise consumption instance constitute record daily consumption number gamma available h pair scatter consumption consumption consumption daily consumption cause individual abuse drug lead mean reverse chemical whose concentration test daily consumption exclude evidence consumption pair consumption volume average consumption cell consumption cell great amount disease level disease highly disease consumption primarily rarely condition uci repository collect national institute diabetes disease usa diabetes risk criterion year age select instance nonzero seem encode yield cccc pair pair plot body pair pressure age body define height obviously mass age cause hour measure standard tolerance exclude cause age bias pressure pressure causal direction alternative age measure cause could way may weather locate include temperature height datum evaluation summarie weather plot day temperature day year temperature temperature month consist variable day year year year drop pm count human twice useful heuristic day cause angular position around true infeasible change incidence pair relation temperature year area day surface temperature average daily process interval assume cause air easy artificial enough decrease air temperature long also environment play daily average national research laboratory website observation weather date national centers environmental national realistic grid four cell air pair pressure pressure pair relative day day across north across cccc pair plot pair temperature surface pressure level pressure daily area km km propagate
select illustrate effective region choose bad anomaly pearson correlation coefficient possible performance strongly favor select effective model simulation time train train hundred implication possible necessary implication design rapid train student student select initial scoring quickly start score student receive feedback receive refine collection pool subsequently maximally effort ensure
return solution return incorporate rule see worker round round certain monotonicity regret minimum optimization greedy eliminate worker monotonicity avoid exploration elimination worker follow give approximate denote follow loss index ms mb I k l l k k l pick return allocation rule monotone worker task task cost element similarly big call cost worker decrease worker remain appear worker nothing change worker worker already remain worker worker never big element cost worker element follow change bid big worker become big big suitable set thus exists discard perfectly equal contain run meet big true candidate therefore candidate hence drop c r rule worker eliminate elimination se confidence bind elimination se transform mechanism compare efficacy propose simulation solve greedy compare four ns solve explore
token map list representation turn rather token sample sequential token classic search index bag word worker record eliminate addition sparsity first reason great efficiency change within incremental document frequently word every word e maintain token fractional along make note dense fractional mass partition however state require disadvantage sampling parallelism salient parallel yahoo available notable similar token throughput yahoo lda roughly yahoo lda per compute core per medium machine sample throughput yahoo yahoo show throughput yahoo converge fast per careful synchronization fail
function minimizer point essentially manifold function manifold function perturbation partly indicator constrain refer constrain convex compact follow definite imply bound set equivalently state let accumulation k classical whose solution constrain partly smooth nearby soon enough fact understand operator imply enough partly smooth partly
hold need pick choose linear map strongly specifically hold pick next give sufficient admm existence boundedness proximal admm exist either sequence generate ii l x plug hold boundedness boundedness third next boundedness boundedness finally boundedness relation complete proximal admm choose theorem shall follow linear include hold sequence generate admm bound conclusion hold end recall comment condition shall fairly large twice differentiable twice lipschitz modulus reader inequality modulus differentiable whole generate admm proximal admm algebraic rely kl kl property analysis like
technology division contract term inequality decomposition extension apply derivation substitute side q eq apply eq difference involve line solution single e surrogate decrease adjust trust update newton trust newton consider within trust poor trust region leave axis decrease huber penalty approach dominate reweighte image ray specification ms filter strongly filter continuous create spectrum fill divide integrate computed accounting detector detector pixel angle resolution david convergent alternate image reconstruction transmission extend determination law object voxel additional latent voxel importantly image comparable penalize exploit parallel line search voxel considerably smooth penalty employ allow positivity inherent demonstrate algorithm ray compute minimization automatic determination transmission ray ct integral along ill pose sense solution consistent desirable measure represent inside well know example ray must increase physical transform filter back analytic formula incorporate mention prominent type iterative latter incorporation knowledge
angle begin interpret generalization pca ultimately nonetheless emphasize employ entry entirely underlying belief validity unlike exist turn theoretical condition whereby substitution nuclear nonetheless guarantee produce rank dimension nuclear limitation fold measurement force certainly importantly provable recovery place strong restriction singular ever hold realistic condition check theoretically measurement violate guarantee support empirically distributional adopt enforce next denote aggregate definite symmetric define accommodate case follow convenience abuse notation come negative wise denote necessarily per se primarily desirable give define operator expression clear overall require common bayesian regard likelihood respect definite transformation standard convolution integration minimize employ standard bound em bound simplicity experiment herein currently explore add independent progress even low obtain optimal closed form optimize dependent iteratively importantly
independent essentially discretization invariant posterior forward model mesh certain regularity discretization organized introduce informed reduction approximation present construct likelihood inform pde inverse efficiency property discretization invariance technique conclude overview random appear probability product describe commonly additive data section posterior prior advantageous evaluation expensive pose equivalently compact decay thus project onto covariance good involve balance decomposition define together yield optimal minimize positive covariance linear reader detailed range inverse basis maximize direction informative represent mode parameter space output insensitive particularly inform conversely smooth variance relatively inform case direction
state assumption long represent processor section encode development probability associate indicate otherwise write hadamard hadamard matrix notation identity elementary give one semidefinite identity particular matrix elementary element right hand side follow already rely play identity identity fundamental identity identity far illustration identity hadamard linear operation h follow hold set I deduce formulae restriction combination formalize probability scalar j matrix intersection equal
structural result low coherence fast solve massive problem rely building block narrow shorter examine avoid entirely state proportional unfortunately leverage score regression place simply random work could contain row remove preserve product drop include mix avoid storage runtime elegant sampling score need row mind straightforward sampling cm label sample small row repeat reduce small previously prove small come routine analyze ultimately require mix something avoid second possibly condition rely primitive possibly
important approach consist transform couple item transform implementation item relevant link strictly provide generally primarily aim rank stress relative state improve scalable link graph call ranking close prediction ranking supervise network section article social node kind interaction phone dedicated metric evaluate unsupervise consideration improve aggregate unsupervised implement selection parameter suit investigate record european phone service month interaction user phone filter social link activate interaction phone link call paper
design participant participant per assume per year combine population duration trial design rate equal reflect reality likely slow assume implie amount regardless take amount item per n sc participant stage stage ss participant item boundary constant calculate boundary boundary design calculate section design boundary sc item design ss boundaries h ss treatment range treatment set effectively basic great treatment set output panel right design section software link full design sc ss ad efficacy boundary participant early duration design input design design fourth together adaptive number participant stage ad break stage table efficacy k ad stop boundary k stage inf efficacy construction see research continue stop plot page display efficacy boundary stage design sc
master order frequency controller automatically synchronization structure fig structure add mechanism enable master master neuron master pass output master neuron inactive cutting connection master pass consequence equation detail frequency automatically combination frequency neuron satisfy activity synchronization neuron active inactive master note enable adapt period anneal feasible suitable discrete search robot record direction step angle last initial angle deviation fig green forward angle otherwise occur automatically synchronization stochastically process value combination record new period variation combination period depict calculated deviation rate indicate trial deviation select period assign compare particular period assign angle measure stop loop
throughput sequence tumor broadly simple consist copy variation widely throughput sequence technology generate read rarely variant partially tumor reconstruct evolutionary history tumor evolution tree assign node representation tree chinese restaurant assign leaf knowledge assign e pre specification apply stick breaking
convergent admm inexact fitting term method admm admm penalty select verify fit two admm suboptimal organize method inexact concrete mathematically show convergence two regularize discussion admm situation support inexact split admm multiplier al stack equality compactly
arm arbitrary reward let decrease purpose epoch q impose non maker evolution form expect reward continuously change variation correspond budget ht continuous spend first measurable abuse notation action mapping together class policy past history difference epoch oracle performance policy take worst guarantee policy quantity establish refer constant regret formulation reward adversarial manner budget lower achievable performance assume reward bernoulli reward stationary well least embed stochastic non mab stationary increase variation budget regret run optimality imply regret reward evolve brownian motion achievable relative stationary
discuss idea extend overcome drawback achieve tradeoff among orthogonality balance loading truncation research encounter signal process computer vision etc area pca approximate represent orthonormal loading represent component project product loading loading decrease lead maximal variance lie mainly vary direction loading enough obtain achieve physical financial biological specific asset dense principal make difficult loading zero principal combination principal component far physical loading dimension aim sparse basis interpretable represent tradeoff statistical fidelity interpretability past decade method consider tradeoff sparsity three yet orthogonality loading loading global advantage loading orthogonal pca satisfy orthogonality highly indicate mean loading loading orthogonality support
parameter pc window bit core processor set calculate classification gain popularity area open resemble basically
example convolution periodic reflect condition extend development assume list orient respective sum frobenius norm ff group horizontal slice norm like problem tend inside enough driven hope tensor size orient seek point ideally matrix efficient generating build tensor problem give algorithm whole translate give I chance intuition zero multiplication fourier equivalent solve multiplication next build norm affinity
dt bias rmse rmse sl sl sl dt sl bias rmse logistic nb nb regression spatially aggregated count cc cc sl dt sl sl dt bias sd rmse c sl dt sl sl dt sd cc cc sl sl dt sl dt sl dt bias rmse sl dt sl sl dt sl dt bias rmse simulation study lattice spatially aggregated approach bias standard sd mean scale sl car nb nb dt skew skewed calculate informative bias conclusion follow sl method bias dt almost good sl discretization univariate
summary frequency estimation little intervention chemical presence absence transform robustness low noise peak analytic acknowledgement author thank microsoft connection grant dark microsoft nuclear exploit atomic chemical environment model weight function free decay challenge estimate general model decay noise offer practical solution conventional using experimentally acquire robust low snr overlap peak enable snr conventional nuclear advanced understand molecular analytical nuclear behave spin axis generate later use share develop comprehensive lead study protein specie conventional lead ever
start consider innovation component unit thus kernel entry fig show reconstruction different observe verify phase transition signal particular section use mean gaussian reconstruction conditional ik ik ik ik ik ik ik ik ik notation reconstruction side without decoder notice match numerical ht section compressive sense resolution pixel signal vector patch picture represent patch extract divide portion section e test ik ik moreover extract patch notice introduce substantial distortion relative value side train covariance imply almost extract image reconstruct test image feature represent image input side image I reconstruction number linear also reconstruction phase transition obtain match mathematical phase occur regime namely develop significant approximately natural exactly matrix term account component way extraction full covariance mathematically equivalent low covariance noise zero entry zero unit noise variance correspond noise equal offer principled effectively task error compressive hyperspectral imaging example image subject presence measurement collect snapshot scene easily obtain expensive hyperspectral device rgb improve camera image code represent patch hyperspectral image system analyze vector hyperspectral image perfectly characterize run time close camera ht htbp side c region real camera projection implementation imaging belong side image compress single meaning
plot intuition dropout expect dropout plus weight mark minimizer dropout criterion generalize perfectly distribution strongly separate section whenever dropout separate define optimal classifier denote region green section figure regularization bayes strongly dropout regularization predictor dropout create predictor error number draw otherwise uniformly achieve despite first vote correlate ensemble discrimination feature put bayes placing accuracy find optimal dropout even conjecture source generate equally accurate optimal predict small require dropout make enough even p number distribution start theorem call strongly interpretation regularizer interesting dropout regularizer dropout minimize plus property dropout penalty verify dropout penalty dropout behave dropout loss plus convex weight penalty training weight vector go
row fourier autocorrelation fig sparse use autocorrelation slightly support plain versus phase approach linearize convex real case circular shift retrieval problem break derive invariance another greedy algorithm group valuable universit france electrical engineering sciences university california berkeley usa electrical engineering retrieval measurement nonzero relaxation norm result circular shift complex recover typical
hyper principled complicated conjugacy lose similarly certain actor logistic normal additional inferential unclear class sbm dataset potentially space may fundamentally assess hold employ possibility primary view equally know give general ij vice versa conditionally independent presence diagram since calculate ij ij ij I similarly multinomial since give parametric e ij log intractable low ij ig ng estimate newton step iteratively simpler note ip np np np w nr np nr little nr np j nr np np actor manner due political may
applicability second semi label nb theoretically decrease nb document presentation community plus add iteration ideal learning third trying minimize competition former determine fitness conduct facebook dataset community exclude finish combination network display training year bar network probe item perform item find community year attribute large true attribute reason see suitably year attribute suitable attribute also explain item network infer behavior item combine multiple good discover community worth community item k expand method hierarchical agglomerative agglomerative cluster partition explore collective edge algorithm merely cluster second community community reason partly poor
I valid produce valid singleton belief plausibility plausibility specifically plausibility simplify plausibility frequentist coverage consequence variation challenging framework singleton plausibility plot plausibility fairly true plausibility interval particularly plausibility seem plausibility I plausibility region I around magnitude I arguably give method probabilistic fundamentally experience scientific especially live failure probabilistic argument
arrive q two satisfie e dy mx combine properly choose fix due enough take eq straightforwardly study q integral decompose decomposition q arc proof natural hence desire fix denote n sequence k intervals v v constant z pn
heuristic approach large class variety model partially observable stochastic refer usually usually begin usually mean mmse state greatly measurement nonlinear filter possess recursive filter state gauss representation desire sense chain follow comprise almost compactly state measurement constitute vary known employ change appropriate summarize ii result converge define sense compact probability unity nevertheless deal almost continuous result time fact counterpart treat mode stochastic convergence compare regard approximation paper heuristic approximate recursive approach wide variety limit recursive sufficient condition operator highlight
contrast factorization furthermore motivated problem entirely topic numerous represent often meanwhile rank form able inherent help interpretation intensity amplitude count value nmf lee parts
extremely reverse lstm ask proximity fairly fact easy sgd output transformation greatly lstm method english mt translate input list translation visualize sentence representation english dataset train consist english choose specific subset public availability typical language frequent target every vocabulary special involve maximize complete translation search translation decoder maintain hypothesis discard soon remove add hypothesis decoder
close linear ref table first calculate approximation increase reach become negligible respective value cc cc exact curve normal mean variance eqs nucleotide incorporation fix cycle agreement exact skew left complete incorporation flow cycle nucleotide composition probability nucleotide delay synthesis incomplete incorporation sequence leave work school technology next sequencing sequence length flow cycle nucleotide probabilistic incomplete sequencing sequence generation sequencing generate huge amount
velocity discriminate galaxy datum several optimal obtain combined htb galaxy dataset dataset hyperparameter group test paper mixture complete respect rather depend hyperparameter sensitive choice sense use update however informative hyperparameter prior frequentist cluster configuration obtain little advantage categorical situation posterior optimization routine stem iterate extend present improve adapt context greedy property negligible mixture special consider avoid dependence hyperparameter hyperparameter differently possibility exact formula several framework real various simulate meaningful however strong solution indeed finite mixture pure force use informative
ordinal logistic job research innovation use useful datum international economic operation development institute statistic become datum simultaneous data matrix reduce dimension row individual column measure method principal factor fa combination regression essentially square algorithm normal exponential family describe never never alternate perform iterative extend sparse none logistic procedure matrix score calculate external procedure successfully nominal region nominal variable nominal adequate categorical principal item response ordinal row along
c bring picture consistent half inconsistent correlation function satisfy correlation fc measure either code able force inconsistent one interaction code good need show sum concentrate standard unable apply instead proof verify concentration moment generate prove concentration novel lemma dropping cauchy k k three random suppose obtain score large correlation arbitrary deterministic may able round power round receive apply firstly however prove score score nj ii thus bound imply completeness suppose sake n eq q contradiction thus rearrange equation give substitute eq q contradiction construction analysis interactive code interactive code robust fraction failure false pair construction analysis interactive interactive code interactive user fraction false answer adaptive working answer statistical query choose adaptively hold value answer iid answer moreover choose formally ht dx
decade view currently consistency graph model estimation meet heavily furthermore standard especially validation classical criterion aic bic stability different
observable active develop shaped stochastic understanding variety e continue et et publish parameter formula r specifie machine stimulus possibly internal external source stepsize machine additional call assume identically observe stimulus adaptive specifie optimal behavior learn value training stimulus x x absolutely notation sum integral stimulus discrete absolutely expect loss equivalently choose parameter characterize paper consider machine stimulus
cloud dimension algorithm depth proper variant tie prove keyword combinatorial complement suggest propose centrality cloud call depth location motivated notion decade depth depth cloud point assigns degree centrality later depth notion possess affine invariance monotonicity upper finite point lie
technique procedure guarantee normality condition write term brevity influence derive influence function py q derive alternatively definition construct density function sample denote without taylor precisely remainder segment first term cross g mention boundedness density give need condition formally normal finally construct analysis form estimate estimate estimate estimator proof corollary normality begin influence q jensen last boundedness density function boundedness fourth compare shannon follow fourth fig dimension dimensional comparison shannon entropy partitioning power representation hellinger rather wish problem category sample
serve naive normality necessary high grow definite grow function completely member covariance matrix covariance discuss asymptotic sometimes sufficiently ratio eigenvalue consider show asymptotically critical mostly goal turn attention parallel loss estimation frobenius parallel begin interpret tend clear part asymptotically dominate might convenience pick eigenvalue property specific alternatively result could correction look simultaneously asymptotically form thought correction discussion regularity strong statement appear
assume particle equal particle time identical relatively element brownian motion model diagonal sample speed accommodate object module appearance conceptual tuple origin basis extract consider top corner locate appearance affine track specifically result track time bag use appearance particle specifically obtain singular choose dominant affine object along able reflect appearance change accordingly propose keep model object track bag model use
member department electrical engineering national university vision research group dr areas vision hundred paper range topic google citation index associate transaction receive paper winner task winner winner mention associate award research award award award minimization involve smooth reweighted updating solve affine mixed minimization essential formulation concrete norm regularize theoretically previous one depend
multivariate mm department york mail current framework measure risk involve risk cf frequently model copula tool financial cf reference cf g two risk leave rise upper concerned phenomena unify copula survival index dependence survival copula duality dependence low tail al reference restrict consideration behaviour copula instead
report environmental service md pp mm weather small poisson b c york journal p stochastic day axiom theorem conclusion theorem conjecture example exercise summary section test study case fitting researcher generally central chi check validity asymptotic chi square nan variance goodness spirit example refer
basis window apply search optimize correct frame involve update order efficient practice computation code description stream detector seed detector frame initial detector start negative pool alg case rank approach publicly benchmark long video object wang independent unsupervised set scenario detector adapt stream compare wang art video also rely train detector confident positive wang consist ranking scoring
keep around one small subsample much replacement si usual si assign inclusion illustrate variability variance fraction impossible scheme explain si design represent equal si extremely contribution contribution much include replacement equation suppose sampling probability constant sampling weight approximately evident likelihood way replacement ps obtain deviation vs ps marginally efficient p many si example probability generally impossible construct model construct weight normal proxy goal time common computationally costly taylor aspect subsampling refer consume aspect numerically quadrature
svm weight specific express base lagrange assign support stack generalization assign performance correspond combine meta binary multiclass environmental narrow approximated similar band white noise noise precede front full band adaptation feature account vary contamination colored normalization method additive noise furthermore stack generalization tune classification similar distortion range instance meta train score score mixture clean stack limited amount require optimal meta stack offer individual problem useful major counterpart front si diverse sx compact sentence core test sentence consist sentence train test set meta stop certain group class fu lee classifiers multiclass initially decide use overhead impact degree slack training sentence energy add sentence hence level snr ratio
estimation conclude statistic perfectly formal mapping jointly statistically independent dependence extent know standard relationship note imagine scenario reason exposition back motivate pdf horizontal interpretable interval colored sampling reliable interval straightforward restrict attention ask much let close interval proxy proxy exists size attention guarantee produce illustration correspond differently closely distribution measure dependence close contain interpretable proxy interpretable know guess relationship noise omit reliable interpretable reliability interpretability resp diameter resp reliability resp interpretability call approximate distinguish perfect imagine way reliability dependence dependence interpretable resp interpretable resp reliable interpretable reliability resp imagine belief various interpretability reliability interpretability give terminology correlation perfectly proxy bivariate perfectly interpretable perfectly proxy simply restrict perfect second sufficient interpretability equitability amount measure suitably reflect particular might equitability measure case versus model state precisely term relationship call noisy write functional placed gaussian way arbitrary necessarily functional property equitability relationship noisy relationship measure resp average proxy equitability uniform instead evenly clarity opt produce analogously also distribute difficult say follow ideally equitability behave model somewhat narrow might add coordinate modification paper equitability interpretability differently formal
annotate however si design inferior abundance correct biased account variance size third hybrid design achieve error wide array si htbp detail cost hybrid cover abundance perform structure variance commonly annotation investigate mining determine hybrid second simulation assumption applicability quantify monitoring survey collect survey site year image hour annotate minute annotation sampling identify randomly annotation use percent
delta period worth note refer eq importantly likelihood concave maxima maximum sp spike spike ascent g stimulus fit roughly spike generalize intensity notational simplicity q stimulus though stimulus number
discovery cell material thereby address energy generalize extend prior specification combinatorial specify priori include traditional negativity dependency come chemical system factorization call program show outperform art material discovery scale large knowledge direction namely concern formalism combinatorial science nsf award grant grant nsf nsf energy department office office energy sciences award research energy nsf national national institute award joint center innovation office energy de stanford national laboratory support u office science office
error square dominate gain descent optimize gain quantify limit quantify asymptotically expansion let matrix multiplication notation pair drop implicitly eq distinction operate usual application sense symmetric large stable take frobenius look precisely left operator denote operator denote vector setup small eigenvalue exact moment bias would model lead contribution tr extra supplementary material specify actual matrix term generalization eigenvalue supremum supremum necessarily follow proof definite dimension
compare participant randomize switch decision covariate feature gender status history care feature intermediate medical degree burden consider covariate treatment decision quick rate final record covariate outcome switch randomly assign participant treat score select cut reason include good response reasonable categorical information response lot indicate potentially include moderate important covariate decision regime patient treatment among variable answer diagnostic screening sr score change level patient rate rest select covariate baseline relate patient give treatment decision comprehensive patient situation baseline examine optimal regime ig ia ia method get subject treat regime
tune stage function open gram precision translation mean estimate softmax deeply hide softmax figure detail extend purpose major layer softmax costly solution short list class adaptation scheme equally adaptation scheme back selection model practice mean adaptation process train several g generic usually take substantial hour combine translation adaptation train new method adaptation outline weight update
achieves general assume define return finally achieve begin guarantee assume cluster exponent exponent nd n similarly establish simple choice recover show candidate independent exponent furthermore fix constant addition constant finally recover rate exponent unfortunately simple selection correspond unclear strategy component present select proof fx nx integer hx fx measurable integrable lebesgue absolutely thus former lebesgue monotonicity show case persistence iv imply c part lemma hand connect prop db find return notation return become write v v ia w b find repeat assertion small let check remain conclude assumption check obviously exponent exponent
cluster research unify hardware cluster area innovation valuable capability capacity template question investigate school mathematic american center xu acknowledge technology additionally national foundation support
part gradient sufficient straightforward implement memory computer distribute key store hash worker support entry server store resolve access update suboptimal address max tradeoff made return immediately server fast worker get fast server propose reduce improvement include filter support add remove server learn topic lda million token yahoo lda server specifically bayesian fast software model yahoo correlated topic parameter partition machine computation copy globally size large challenge model address challenge partition store part e update part carry parallelism flexibility worker fast converge delayed parallelism storage automatically compute together user worker apply automatic primitive user partition vocabulary assign full cycle complete data lda use read fast parallelism replacement complement parallelism show parallelism consist several parallelism asynchronous distribute linear algebra library intel mkl could library matlab server basic algebra vector matrix add multiplication algebra package include equation factorization implementation intel mkl implementation aware cache aware also claim always find level library build upon efficient architecture algorithm effectively linear easily base support choose toolbox topic model hierarchical review include boost bag linear combination tree iteratively appropriate prior tailor recent advance massive control important factor parsimonious regularize estimate truly optimize averaged take inferring include spike shrinkage heavily tail prior e laplace reader selection aim expensive planning recommendation big become huge tuning prohibitive progress method select multi core stochastic method space
add table cut complementary available htb cccc lemma number add lemma improve come complementary proof lemma theory newly improve effect however since try high clear prediction category consequence vector fact add close theorem work lambda function eliminate try find corpus idea logical dependency atomic element conceptual organization ai topic semantic corpus serve automate reasoning core automate develop improvement due introduce evaluation proof exactly measure contribution ai whole corpus work make incremental quality method style detailed level inference graph intermediate prove hard run help ji many discussion extract par lead library consist million step give rise statement lemma analogously mathematical statement formal criterion usefulness furth criterion graph library add good automate mathematical library intelligence decade corpora mml
tree project leaf compute result exploit standard make reach leaf retrieve sample see appendix b material split score approximate project subsample split construction balance small generate significantly original grow may without predictive idea exploit context building project grow pair jx jx exploit projection kind diversity among share output projection output projection output combine randomization grow ensemble tree already extra discuss empirically look single small suggest variance subsection jx jx n ensemble computational difference output projection ensemble tree
patient treatment day day another treatment treatment use day patient base addition year survival estimate regime simple treatment estimate regime two confidence interval bootstrap base run bootstrap confidence stay compare year indicate optimal treatment regime improve survival regime iii interval approximation stay contain conclusion bootstrap confidence propose various estimator follow treatment smoothing regime regime year survival work treatment decision generalize decision define multiple year clinical treatment regime take du restrict survival regime treatment regime interesting topic investigation establish regularity recall work assume parameter proportional model survival tw nu specify satisfy
encoder encoder intermediate obtain rnn recurrent auto encoder auto mapping decoder latent stochastic introduce year similarity auto rnn partly
excess interested derive sure result exist surely almost surely fairly output satisfied omit clarity know integrable xx general depend well linearly quantify r w define g minimizer nonempty
exploit indexing multiplication fully leverage engine multiplication complete indexing require division modulus division modulus operation shift reduce indexing convolution auxiliary require convnet implementation layer configuration average throughput convnet importantly show well l layer layer illustrate architecture build architecture peak precision throughput build use architecture peak single throughput range peak provide gpu gpu deep
description vc depend vc dimension rademacher straightforwardly dataset bind error erm intend large small say generalization since erm limit collection radial create placing limit magnitude function treat control equation must solve equation choose turn bound policy performance decision tuple primarily done facilitate analysis equivalent rl maximum episode without everywhere straightforwardly v reward
cp cp bs bs nan ccccc ccccc cp sd bs sd bs ccccc ccccc cp sd bs sd nan choice ccccc ccccc cp cp sd bs sd bs compare sd table show sd large comparable sd worse compare sd test dispersion diagonal favorable sd show large turn project search large benchmark power describe power power
decomposition multiplicative literature machine investigate propose conventional special kernel wise division hadamard nmf worth identity half polynomial kernel balance impact high common polynomial update rule gaussian kernel multiplicative nonnegative since q division wise nmf include follow impose typically motivate different impose improve estimate turn constrain function interested solution variation less exploit smoothness minimize space term cost get reconstruction smoothness estimate yield additive method multiplicative turn unconstraine denominator nmf multiplicative rule unconstraine add denominator easy
l good fit htbp parameter aic statistic e pl customer service bank et data observe exponential good fit among consider result distribution figure htbp r pl stress play important
duality equal orthogonal support word imply bind approximation indeed solution duality gap notion value value space contain satisfy eq version unfortunately nuclear calculation subdifferential nuclear giving give yet
incoherence allow vanishing go infinity applicable likelihood define incoherence two structural si relation rsc restrict fisher structural si guarantee si marginal satisfied lemma rsc si prove remainder negative gs ks define next family partition derivative show author q coincide norm definition relation hold due rsc precision inequality omit derivation incoherence si condition remainder incoherence due verify definition curvature theorem statement error sample bind bind eq verify guarantee specifically require bound first wise choosing
eq state inequality matrix help c eq lemma perturb ab consider span orthogonal contradiction omit ik positive semidefinite ta b I run time reduce albeit pre modify complexity pair nz py pz tie arbitrarily reduce half pair replace run winner sketch df complexity complexity add happen close therefore run union generalize spherical covariance along kl divergence spherical distance mean consider consider subset code distribution subset differ coordinate show first
role event possible enforce ensemble take annotate use annotate plus none case q paper score classifier suggest weight explore arc perform composition triplet link
traffic management newly specification fit value derive error assess meaningful furthermore provide tight probabilistic reach avoid fit iteration approximation assess process reach avoid concern evolve endowed control investigate theory quantify dynamical known reach avoid specification formal verification algorithm advanced process endow past evolve deal theoretical add markov interest endow state work engineering life science controller synthesis selection order maximize figure go investigate system theory deal tracking focus specification know formal verification field verification deal simple chain development computationally deal task synthesis instead specification model continuous endowed avoid deal likelihood within horizon trajectory avoid equivalently property express goal state reach invariance specification core modal verification temporal computational controller becomes either allow reach path
particle ess tune I na j nm w bring two extra make sample quantity smc sampler use gold simulation slow turn ep framework derive target distribution constant pseudo rewrite ep generate case approximate un lead type exponential family see update site site
along preference ordering tell quickly word budget large allocation lp lp cp price established take maximize bundle modify maximize undesirable modify price increase price still good either thus price derive bundle despite xu c fractional instance time xu np xu cr ir n order budget uniquely remain unless budget know bundle remain minimize remain min wherein bundle first induce slightly show like bundle rather bundle order desire cause item prefer item price preferred price fully fractional fractional price
outperform outperform remain technique rate angle gap still fast fast dynamic adaptation reach angle gap dynamic topology distribute grid dynamic topology adaptation employ result mse
formula power derivation yield power explicit component bivariate role use information nan positively substantially biology economic finance thousand hypothesis investigate asymptotic theorem present parametric long bivariate normality maximum applicable theoretical lemma appendix prove almost surely reveal propose conservative incorporating drive asymptotically f ft pointwise sequence integer carry simulation study aspect control various bivariate unless otherwise simply proposition replication compare procedure dynamically select scheme determine preliminary multiple filter proportion hypothesis share spirit serve example come mean normal right sided testing versus hypothesis test bivariate nan overall notational convenience rt value nonparametric role value e respectively case control conventional alone affect power thus combine detection significantly compare conventional procedure scenario take value compare optimal estimator close except phenomenon bivariate choice provide control combination panel
individual never population time individual correctly desire discrete prior heterogeneous link formulate probit normal cumulative individual latent version share similarity recent probit detect specie level effect probit behavioral w denote capture give record h assign close population abundance modify effect remove effect binary basis propose sampling distribution enable gibbs include feasible readily available assume frequency correspond zero frequency create none history individual history full tn update one update conditional update bernoulli full hasting integer discrete distribution integer tuning frequency
without component consider thing namely find unimodal elliptical away core clustering estimation researcher mind core look implication outlier truly belong degree freedom core identification whether separate one interpret consider artificial dataset distinct point start cluster generate red generate region mass reconstruct membership region figure reference expect measure imply generate although non base robust define noise definition component assign cluster outlier mean assign classification cluster necessarily cluster really gaussian covariance determinant scatter existence wide measure correct achieve equal covariance proportion could assign set define th ellipsoid region define union membership g membership need outli outlier
european please project small project life easy boundedness signal local reference title signal j appendix r f manifold vice versa dictionary b establish consider parameter basic proposition f hold identify assumption imply kp kp show also lemma remain leverage yield kp kp use need h eq signal surely lipschitz frobenius metric bind appear equation column hence far consider condition program existence uniqueness invertible algebraic plug along span approximation exploit conclude unique lemma quantity exploit desire observe surely shorthand r fm apply apply triangle j j j appendix technical
gene responsible rna seq whole four bl j treat rna seq end bp rna seq read supplementary material experiment differential usage bl seq read genome fdr identify gene differential usage differential gene annotation respectively gain insight gene list category gene annotation significantly associate relevant projection supplementary figure imply rather bl list identify annotation functional imply read depth without annotation manual recommend sample one one evaluate comparison annotation functional category gene identify supplementary gene identify target study involve could effect disease potentially candidate treatment well rna two incorporate snps genome rna seq separately genetic variant annotate rna seq identify much less gene bl fdr gene gene identify six involve bind great bl follow behavioral phenotype bl bl might induce allele usage rna seq allele map allele seq genome wide allele seq supplementary allele specific rna seq assess usage cutoff differential treat
category comment later order iterate given order detail need translate definition acyclic graph income e formalize consist ordinary edge incoming set together map incoming acyclic acyclic definition direct acyclic graph equip type understood tensor assign graph diagram tensor way assign satisfy contain node equip union carry intuitively place diagram interpret value requirement refer composition order carry assign coincide half edge compose obvious composite carry armed label joint evaluate diagram number correlation generalize see regard claim circuit thick minimum leave quantum application partial start follow use contain edge lie boundary whose operation need marginalization operation take new induced apply result coincide claim simply likewise put guarantee induction gives depict show far follow linearity interpret hand side diagram object label leave side latter exactly element iw ix e go composite system figure regard composition part label coincide rule tell income rule diagram coincide take complete rectangle thick draw fill minimum mm thick black minimum e rectangle mm e e smooth first compute value empty binary correlation equation subgraph disjoint operation choose still independent choice modify change permutation nothing exchange neighboring node among among correlation original new respect original order swap thank
function separately function derivative derivative concern provide derivative derivative mean function integrate reconstruct addition even linear find gauss prove carry precisely ols reconstruct variance unbiased compute explicitly dimension nevertheless practitioner represent expansion truncate ol blue moreover ol practically projection consequence rigorous generalization present advantage
n constrain satisfy constraint ball j project generate project descent gradient burden per size stochastic project stochastic compute involve suitable f slow key improve project inspire reduce strong efficiently solve estimate estimation construct reduce refer algorithm lemma technical contribution require gradient choose rate initialize mi
observation able reasonably anonymous constructive suggestion improve acknowledge financial research model production provide procedure auxiliary k k problem interest pdf instant kp recursive probabilistic recursively conditional condition incorporate evolve obtain time instant consist pdf pdfs conditional pdfs eqs obtain criterion maximum lead derive filter largely integral eqs intractable approximation situation approximate posterior pdfs kf extension extended sigma filter ensemble enkf assimilation gmm bank nonlinear g nonlinear kalman current work adopt prior particle filter pf applicable assimilation problem th notational convenience start particle instant together filter incoming particle remain unchanged update
case twice average return hold sec sequence minimize risk online batch datum batch obtain bind become solution immediate result average hence rate zero infimum advantage ball always term exactly price pay know advance binary lemma also misclassification risk trick suboptimal reduce applicability show tune assume guarantee otherwise regime sample big see misclassification worth adversarial small deviation theoretical
grid gradually modify frequency iterative refinement maximization method typically carry guarantee global optimality dense sufficiently variation discrete atomic version dense noiseless frequency recover atomic norm stochastic presence noiseless study atomic exact prove atomic convex programming include incomplete atomic measurement encounter method line estimation demonstrate base note method incomplete exist atomic norm possibly theoretical result miss spectral framework estimation consist improved estimation explore atomic
development try build evolve process see upon stick hmms emission resort strategy model evolve family rise dimensional value hide simple infinitely evolve continuous infinite distribution give signal space evolve former whereas latter evolve support treat evolve doubly intensity specification process control correlation infinite model distribution intensity bayesian statistic application popularity bayesian tractable family consider filtering
letter column letter g entry utilize norm dr large singular nuclear moreover entry failure mt theorem recall well establish lack parameter argument sphere union
thick cm circle pt look generality c looking contribution follow heat nearly heat heat kernel laplacian heat heat formally heat define similar form relate geometric continuous walk refer heat heat embed vertex heat embed eq q embed point heat embed show range definition heat q proof heat approximate condition contribution coordinate give contribution remark heat weighted reason heat embed spectral eigenvector linear algorithm approximate nx matrix nonzero algorithm compute correspond output follow almost
smooth beneficial surrogate motivate develop binary take main show condition smooth smoothed hinge well use hinge convex r ff b excess loss f learn empirical source error bound excess since error emphasize surrogate complexity affect excess bounded excess error affect aim investigate small parameter result approximation hand small understanding
pick school friend explain pick school would remain school small prefer school explain school represent friend trade explanation little must characteristic network facebook suggest mutually exclusive setting user membership keyword page membership college college aid college easily create node create member infer label explain college influence college college membership feature next first membership age difference infer handle modify modification previously intuition support type label graph run consist user profile current city school college five friend lift respect increase recall
simultaneously relationship field make physical model physical system specify inferential bias gp simulator output nature pair inference allow source contribute gp predictor know desirable burden datum burden framework calibration consequently methodology moderate large number computer increasingly experiment accommodate also recognize exploit relative fitting simulator argue argue mesh acting quickly describe application goal detail section canonical calibration limitation include gp design meet goal project together modern demonstrate proof concept variation discuss limitation motivate equip conclude brief discussion section team interested study matter complex evolutionary arise phenomenon super temperature simulator novel setting experiment disk front end wave energy material b physical observation pre experimental input list range field column three specify disk fill pressure image geometry perform circular
slope expression affect way example consider conclusion display cumulative correctly clearly difference axis switch simulated dataset display datum reject surprising might consider involve discrepancy highlight lr interesting hypothesis test classical parameter space characterization frequentist way test ie decided reject pearson derive bayesian pearson lemma proposition symmetry break adopting paradigm reciprocal alarm detection probability measure exist proper conditioning eventually depend type way error add frequentist integral underlying maximize lr equation pd x x contrary frequentist hypothesis pearson deterministic stand vs composite compare interpretable couple compare unlike bf improper unlike crucial fundamental natural alternative bf many aspect likelihood whereas bf compare likelihood composite
gibbs sampling exhibit slow tractable although approximation exponentially apply introduce multinomial reasonable apply raise maintain dependency critical count question prove asymptotically show correctly able preserve expectation highlight property property original inference arise propagation ep allow speed evaluate define set random variable assume domain configuration second show exponential set clique marginal node marginal rely existence relevant henceforth add edge
solve fix update standard lagrangian duality term region locally normalize normalize constant message essentially variant depend experiment use message region et al greedy sufficient converge traditionally conditional map feature margin easy conditional generalize way rather margin take secondly vector learn aside
transform interpretation generalise divergence begin treat poisson outline countable subset integral may course interpret sum distribution equal domain follow state treat expand preserved likelihood iid observation write estimate introduce follow alternative additive likelihood intensity estimation equivalent remark along incur treat maxima interval confidence obtain poisson addition maxima family even preserve yet compute integral possibility approximate via noise divergence iid
laplace distribution c logistic laplace pareto mm consider instead score function regressor generate table empirical variance base laplace pareto van simulation consider regressor measurement quite similar previous confirm relatively finite fisher r square estimate heavy acknowledgement grant research nsf dms grant support section remark section recently regressor
minimizer pseudo likelihood concern definition consider familiar cover argument uniformly metric entropy hold identical identical throughout rf constant proportional eq concentration assume rf iv knowledge onto preliminary essential remain ss result stage eq possibly grow unit cube eq cover element clear diameter gradient theorem constant evaluate derivative gram deviation define bounding term term chain eq control notice large lie within arbitrary choice element lemma eq eq hence mean hence nj long
exist learn exist sample define note pac randomness great dc sm unlabeled times construct take chernoff unlabele differentially concept either least pa om show boost combinatorial allow private move boost back private private boost algorithmic representation arbitrary probabilistic pair c h class every end choose probabilistic class concept notation align denote h h denote show probabilistic randomly tc think leave fail good hypothesis tx tx cx tc tc probability think experiment h h tr cx dt tr tr h dr characterize private learning concept
discretization space tail deterministic fashion matter never distribution particular consequence discretization serious produce third threshold proceed direction severe appear extract law pdfs cutoff cutoff behave dy py kk suppose instead practical importance situation typical generation prescribe object construct node ingredient consists compose priori
centroid generate create vector scheme centroid centroid close remain however feature mapping centroid feature competition refer triangle gaussian data mixture distribution feature map membership learn em nonlinear scan input micro unit multi contain unit activation map image output feature detector remove irrelevant robustness clutter think pool apart window center location window act overlap window map difficult commonly pool
order reach accuracy far rank speed conventional refine obtain apply fail find large lead gradient reduce instability cp issue scope future show superior architecture character net scheme accuracy b second drop model dashed tune solid fine third layer fourth ratio number point greedy cp bad degradation cnn fine try layer
jeffreys motivated asymptotic prior nearly boundary space weight uniformly spaced mass weight without negativity long magnitude absolute become grow non representation mass point suffice degree freedom equation mixture straightforward algebra hold implement minimize point log ratio except avoid initialized prior
source separation foreground vision extraction separability nmf closely numerical therein pixel hyperspectral long problem see therein convex nmf root comprehensive overview separable scope provably new show efficiently noise provably noiseless extreme span normalize discard identify vertex hull combination combination presence near separable separable nx w nr near separable classified geometric noiseless case separable formulate reconstruct minimize column reconstruct separable particular denote constraint reformulate relaxation problem relate relaxation exact involve system incoherence nmf highly correlate hence extend sensing model relaxation correspond potentially remove successfully
rao expectation construct rao low variance work expectation control variate consider finite eq index whose taking set estimate covariance maintain control depend easily see expectation variate scaling scale follow control basis control reduce black compute gradient maximize memory modification compute small set computing average require sample computation tb field family initialize randomly variational n p ix lt less extend
dr zhang provide code dr dr provide datum science education project electrical engineering university hyperspectral high characterize complementary characteristic image formulate contain data edge preserve datum quadratic regularizer align reconstruct hyperspectral operator infer augment admm convenient splitting exploiting
use agglomerative context cluster natural think average mean thresholding cluster ignore cluster cluster plot belong fourth plot curve dendrogram large specifie total obtain addition stable evy poisson measure mathematically apparent stable simulation marginal conditional advantage memory requirement store well ess conditional general purpose bayesian prior add toolbox could block research ia thanks european european research european framework fp
linearly alphabet size factor learn generalization shannon eq seminal enyi shannon namely order shannon entropy requirement entropy symmetry normalization grouping restrict many replace shannon interest diverse adaptation enyi entropy generators determine bit physical randomness help read dna motivate estimator enyi entropy physics study enyi differ shannon vary depend alphabet size show need fall grow linearly interestingly order grow
condition mean moderate node percentile percentile eliminate remain outlier usually significantly synthetic detection far implement mean plot detection mean show graphical datum correspondingly spectral cluster adjacency average rate percent cost second apply cluster equivalent treat outlier group graph adjacency misclassification correspondingly consequently let sensitivity community plot unless close major cluster cluster possibly fact synthetic large form second two solution figure theorem community detection test modify convex analyze collect compose political division conservative ignore direction component totally skew variability political membership manually evaluate efficacy adjacency political degree derive sbm perform well instead degree correct sbm spectral back modification heterogeneity early choose mean variability
minimum big explore ground onto hill repeat entire slope allow sufficiently routine may momentum add advantage preserve persistent velocity update momentum value learn partial accelerate momentum formalize parameter update trick input inspire gaussian behind matrix income layer second close decrease possibility vanish delay transfer draw pre output train involve calculate pseudo hessian setting though instance clear vanish radius restrict small free train deep network random initialization deep previously due concern
gmm exhibit structural exploit include derivation study behavior mmse draw gmm linear whether mmse converge zero heuristic sense gmm heuristic pick component greedy greedy approach use greedy error utilize component ensure mix signal x p power noise snr approximated indicate measurement covariance ia ia vector covariance gmm contain vector far acquire c distribution correctness initialize c eigenvector solve update parameter signal estimate application find add constraint greedy allow integer outer outer cutting cut optimization problem solve feasible region g integer software cut problem original repeating sensing standard deviation covariance n xx zero tolerance dash
model optimization linearly vary scenario optimize separability exploit another complement realize normal sufficient success es multidimensional marginal apply new distributional copula copulas evolution es free suit black optimize context normality however study call evolution exploit separability multimodal several modelling generate normally distribute
perform nb sophisticated error almost unchanged size nb allow road nb fair svm newly expression cancer separate vector along standardized mean road nb fair fail road nb fair inequality density copy jx jx later q linear empirical feature eq excess base procedure expect fix development label theory carry penalize notation regard couple sample bandwidth marginal oracle p define technical condition consist transform jx nj compatibility compatibility exist compatibility jx compatibility restrict signal risk density penalize
hold quantity eq agree definition body failure event time either failure occur rational rational initial value initialize lemma rank lemma success finally use definition outline bound occur theorem prove success event occur markov angular radial instance quantity order produce low numerator denominator next condition logic eq apply desire neither occur result p verify choose substitute arrive substitute desire probability event occur success stopping failure occur neither success stop failure event solve desire stop time run condition failure occur logarithm jensen therefore stop eq eq solve substitute finally angular theorem first notice failure happen event neither failure union markov part since
switch pixel probability set transition occur precisely adjacency model likewise noise parametrize conditionally change site least favorable distribution draw color latent noise interval change tuned stay range characterize model distribution field experiment rule yy statistic capture discriminate compete fall depend front substantially joint lie outside family sufficient bring concrete geometric colored connect component undirecte coincide site induce site share color component geometry vertex connect vertex form partition rely summary worth note component help search empirical represent abc dimensionality simulate statistic sufficient edge induce graph trial success component connect fig picture notation number component experiment apply approach base two pixel
reconstruct geometric thesis cloud characteristic forecasting purpose paper define classification consider partially amount iterate use data value column incomplete miss transmission topological comprise sense focus geometric construct extension directly geometry analyst choose determine particular integral semidefinite care
assume psd eigen decomposition replace column nystr embed psd one basis expand normalization explicit subset feature eq nystr point svm solver combine similarity measure svm solve eq regularizer scale scenario reduce preferred regularizer straightforward svm evaluate basis share measure lack psd argue measure likely particularly vision consider fix instance therefore similarity kernel differ similarity computer vision kernel pairwise measure position object image define similarity measure potentially pd representation latent variable possible similar mrf position pick variable
main advantage come mcp fast term runtime mcp roughly material runtime unlike previous long estimate take respectively spectrum network reconstruct analyze measurement continuous cell underlie experiment biology hereafter assess infer observational experimental presented test apply logarithm produce close analyze discretized measurement transform nonnegative correspond magnitude preserve interesting reason represent misspecification develop nothing prevent three proceed clean much make use unable final leading formally modify metric count true skeleton successfully orient pc treat dataset likelihood able dag log score compute run behaviour split choose comparable possible mcp figure consistently many edge positive small across develop mind discrete mcp hc fp log mcp pc fp dag skeleton largely negligible take complete processor fluctuation low report observe improved performance consistently fast bayesian along computer design test limit approach accurately handle node compatible high accuracy regime applicability domain exploit nonconvex nature progress break barrier one since around edge poorly combinatorial nonconvex affect already indicate improve even change core algorithm coordinate descent sophisticated nonconvex rapidly develop merely technique network acknowledgement comment van de suggestion dms dms conceptually simple would dag topology coherent carefully outline dag eq define map matrix match confusion write mathematically dags ambient wish cholesky recall definite permutation cholesky triangular cholesky take compatible triangular take deduce dag compatible proof original similar topological sort permutation order
prove c j apply conclude c jx b n jx p hoeffding know z c since measure condition illustrate figure horizontal rt jx eq cluster total index represent square distance dx I I intend diagonal indicator intend want construct show complementary tell us vector q mean give dual primal ensure linear combination get uniqueness result define intend rewrite equivalently note subspace span expression simplify euclidean distance row norm coordinate semidefinite state satisfy sdp unique intend mean sdp coincide intend condition satisfied next statement precise probabilistic model cluster assume section let measure center continuous translation center consider disjoint euclidean ball center center separation search attain minimum attain desire rhs coordinate constitute distributional identically constant row recall span point row j quantitative spectra
rapidly convolutional translation purely neural recently translation deep neural consist encoder decoder encoder variable target translation model mb contrast system appeal unlike conventional every component maximize translation much vocabulary affect performance translation crucial approach strength neural machine integrate machine translation neural rnn decoder replace encoder rnn recursive convolutional english sentence vocabulary nonetheless model newly supervision kind syntactic neural able process recursive ht
example positive objective creates naturally tend box figure illustrate visualize expand near set production trivial find baseline always class varied fine mean meaningful order weighting produce possible consider interpretable cart produce trivial box split option imbalance imbalance split data cart c fast box corner corner breast breast box fast baseline handle box ghz gb cache core box minute optimality minute instance provable repeat able show exact box box font summarize
precise really something com software gain lot state embedding research paper paper presentation obvious neural language
leave identity consider euclidean dataset competitive write rather synthetic embed triplet embedding leave near neighbor category label neighbor within recover thing triplet triplet unseen leave reveal embed hidden impact embed experiment triplet acquire converge triplet triplet figure music dataset converge triplets triplet time triplet grid triplet several triplet bottom grid triplet converge far quality triplet wrong metric triplet wrong idea consider inferior triplet triplet triplet individually grid individual triplet human decrease
agnostic state b except along degenerate throughout algebraic moreover immediately characteristic mode cf remain characteristic turn term exchange decay manner symmetry organization fig figure line space line correspond tm qualitatively different predict note doubly sum sum associate unity unity eigenvector unity stationary throughout unity eigenvalue h question direct towards answer eigenvector normalization denote easily check via obtain draw useful break mention early critical phase transition along eq eigenvalue case operator projection remain note obtain operator addition term turn contribution nan specifically process make
outperform alone observe accuracy observe gain mae prediction accuracy observe mae analysis post hoc reject difference across mae rmse regression chart magnitude interpret across individual across versus examine relatively note derivative mae window monotonically across consider descriptor set combine observe performance relative consider determine audio similarity specificity content retrieval task rating song evaluate utility descriptor quantify descriptor descriptor descriptor confirm propose choose domain specificity content retrieval descriptor track moment descriptor potentially content complexity descriptor suggest base temporal advantageous employ representation yield small frame domain advantageous explanation abundance observation choose similarity base short characteristic structure future aim close feature towards track chart entry control audio production song audio production chart acknowledge audio respective outcome future suitable
dimension consider class r tie breast cancer vs diabetes r tie heart patient ed cancer vs data column diabetes identify cut mahalanobis region quantile take outli observe regard outlier firstly approach cube robust depth depth employ dd suffer vanish performance obviously portion portion lie outside class leave inside least table portion vary sect sample relate train classify finally tie knn classifier tie point pool determined remove pairwise construct depth moment approximated depth depth choose depth treat number sect sect polynomially space polynomial fold cross validation complexity characterize need sect classify knn traditional knn neighbor perform wide save satisfactory max mahalanobis depth calculate estimate simplify classifier slight modification traditional knn
great surveillance possibility surveillance translation wikipedia know happen forecasting often motivate model wikipedia forecasting test horizon day preliminary important likely work article current try manual process yield candidate article feasibility comprehensive evaluate thousand million plausible article also facilitate predictive disease incidence proxy inherently fine country level traffic foundation wikipedia relate project ip address aggregated datum public preserve privacy traffic sophisticated try single wikipedia traffic disease incidence non wikipedia variety wikipedia share internet highly traffic cause news cause difficulty preliminary use english article roughly article month repeat inter lead page title none author google translate notice title article traffic day new period article change exclude article maintain aware tried latter manually sum discard history article transformation full available comment article would facilitate change health article often map international simply model testing recognize inherent internet source strongly internet access technology age gender education role quantify survey sometimes study limitation bias wikipedia
move list patient hyperparameter list likelihood posterior list list parameterized space boolean clause give advance fed determine node risk associate plus default patient monotonicity give early prior list consist clause frequent boolean output presence diabetes presence boolean return diabetes clause decision monotonicity constraint posterior log real constrain great
newton usually drop inside r evaluate alternative newton simply linear center replace g gauss semi generalize newton method parameter compute hessian free approximately minimize compute explicitly typically generalization method newton backpropagation implement point generalize previously alternative classical gauss situation arbitrary substitute formally maintain unlike linearly approximate capture think linearity compute many simple happen section psd positive estimate arguably conservative reason accord reflect version obtain whole contrast
create th highlight black increment bic concern apart perturbation former select fix sort bic always factor observation observation c rr misclassifie correspond misclassification perturbed lead ht c bad obtain perturbation number perturb recall increase contaminate contribution involve gaussian mixture contaminate cf sense view generalization analysis observation
describe integrated neighborhood adjacency list call along spectral guarantee recover via approach expectation suffer local three leave triplet hidden learn denote x triplet joint provably recover learn parameter triplet tree triplet employ tensor procedure triplet next decomposition require label structure grouping estimation triplet integrate bring merging step sub parameter learn triplet tensor decomposition recursive recursive node recursively group parent continue carry decomposition triplet find close decomposition carry learn moment continue active v v v decompose divide parameter estimation compare popular optima stable guarantee since avoid quantity
empirical propose able produce baseline search important mean internet drive pricing ad affect engine research design attract attention area intelligence require bid ad search ad one correspond g ad price use ad bid bid price maintain current rank role engine group theoretic symmetric nash optimal require information hold involve situation limit therefore game theoretic analysis group conventional machine assumption
layer pm grateful grateful comment work support foundation award modeling system partially grant gm learn broad learn directly yield break difficult language despite success little successful feature compression show deep successful theoretical physics coarse extraction relevant operator physical examine introduce deep architecture boltzmann rbms illustrate near neighbor ise employ generalize learn central modern directly promise successful learn representation training datum utilize achieve record
sequence graph fix order element law eq common testing item graph general dynamic address natural corresponding notion view present statistical parametric model issue result occur natural definition involve implement network exhibit become standard finance application development diagnostic base network particular cognitive fmri eeg state conclude propose similar em canonical introduce setup find fix minimum distance part center foundation central graph stand adjacency
ask could quantify art panel panel institute similarity red especially st panel stand reason work area restrict panel faces check particular extent separate method thin parallel panel possibly interesting
accounting category representation activation ensure contain row matrix contain notation column compound draw multinomial integrate vector interaction count interaction exist gaussian represent well ibp ibp chinese restaurant crp discrete comparison crp prior crp assign case latent name chinese process stem follow chinese restaurant infinite customer choose assignment node act crp
outer cluster decrease color speedup distance metric fine tuned color fine imagenet training variety svd institute york university facebook ai edu design million make internet cluster problematic dominate layer exploit approximation significantly demonstrate convolutional layer keep accuracy within vision test problematic mobile compute life spectrum scale electrical neural take extensive effort devote relatively effort aim improve test
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long satisfie turn borel axiom conclusion definition exercise euclidean metric set scale coarse curvature almost coarse underlie laplacian cloud data laplace manifold paper together demonstrate curvature wu develop compute vector point compute form coarse take euclidean discuss choice uniform motivation stem space coarse speak deal infer manifold give cloud
either follow assumption vc vc fact appendix minimizer minimizer reasoning roughly next sparsity literature select follow notation conditionally x x ij nx exp exception interpret expert feature influence difficulty classify sample influence assumption feature expert simple many situation situation similar product likelihood feature joint response explanatory indicate model log
e due good scenario fig comparison expect failure year good actual failure failure case reduce use early replacement take roughly period decrease time directly cost period slow optimal skewed towards expensive part study integrate source leverage interaction refine joint use combine expression similarly protein protein protein type base present diagnostic service dependency data
partition superposition partition use exist learn literature take boltzmann unknown label obey pr pr multiplicative assignment obey convention et term q consistency function exploit classification exact polynomial long expansion optimum mention semi community use min derive cox exist cox place additional maintain necessary seem motivate mrf use map hamming discuss function improvement mrfs use consideration section pairwise mean possible kolmogorov minimization represented cut exact minimum work result kolmogorov value input expansion
aic good simplify assumption fit however law necessary computation scale removal low kolmogorov minimize gram language family attempt establish classification actually power law subsequently gram database depict international similarity list item list stability
practical usefulness highlight application access example build machine predict interval stay short goal fact reliably next nothing aim make give computer similar study situation observe capture vary embed vector dynamic task learn adapt situation already one create sample serve sample blue dot point curve thick mean dynamic observe thereby prediction unobserve blue desire thin dash dot
increase become distance tight comparison stable analogous decrease slowly varie unless difference important gap suggest concern c occur fraction dna li normally distribute constant know variance increment align exhibit speak interval various evidence positive error candidate conjunction multiple comparison involve length actual problem often hundred relatively simulation compare modify modify follow comparative similar contain change high repetition simulation theoretically calculate false hc hc though conservative inclusion continue procedure signal high
ica fmri smooth ica versus component represent smoothed ica reconstruction error represent limit ica conduct fmri realization specification verification validate artificial fmri difference log box count simulate fmri blue versus curve fit sigmoid recover slope curve central log parametric recover fmri box counting realization simulate result complete estimate ds instead slice fmri employ fmri e voxel grid brain volume evolve ds blue blue sigmoid slope accord plot point sigmoid ds smooth sm box count ds analyze subject present estimation ds estimation ds confidence significance plain exclude large value table instance ds ds estimation sm plain exclude small complete cell correspond estimation range sm sm ds smoothing sm small x range significance range sm sm ica model activate brain area standard cognitive activation brain basic fmri modality eeg moderate regard see nature inherent constraint source perfectly statistical satisfied minimized fmri ica reconstruct
lagrange hmm generalization expectation notational expression inside
nsf fa materials generative likelihood generative log likelihood different category whereas summation want assign score observation discriminative want weak constraint generative continue refine satisfy example node final fully connect wu california model convolutional contribution include construct cnns form reference cnns importance
bellman instrumental bellman equation regressor deterministic typically inconsistent instrumental regressor regressor e method instrumental previously context reinforcement provide instrumental instrumental comment idea project bellman minimization call bellman space span basis function minimize project right hand onto span euclidean mapping well respect close typically equation least project bellman consistent equation instrumental appendix instrumental variable instead easy estimator equation policy row I instrumental regressor full ij il l assumption necessary converge mean present equation follow assumption consistent xy show every mean observation explanatory next instrumental uncorrelated variable assumption trivially law see therefore proceed instrumental bellman square bellman instrumental project bellman minimization equivalent start recall
risk current situation compute aggregate set line risk situation empirically environment conduct randomly split five mobile application recommendation software get document recommend system impact exploration fa ts risk recommendation fa ts fix ts document fa ts consider recommend
surrogate prediction directly relevance score relevance score surrogate develop perceptron ndcg rank instance perceptron sufficient query ranking surrogate rank instance consist list vector relevant query relevance formally input list document vector supervision space relevance supervision multiple ranking rank query technique learn score sort score scoring quality learn query compare rank relevance normalize discount ndcg induce gr popular eq number indicate place position rd position rank actually intend maximize ndcg function ndcg performance surrogate loss lipschitz relevance incur irrelevant score document induce score sec write convex follow q scaling wise
focus gain overhead possible fix fix processor convergence irrespective splitting introduce algorithm optimization problem optimization rigorously argue optimization increase admm among expect strong scaling thus split accuracy strong experiment show summarize partition memory consumption gb vary gb minimum node speedup nearly speedup high node parallel node increase cause learn experiment number column speedup parallel apparent imply processor per toward scaling stay processor natural contrary accuracy resource figure example per node increase time improvement improvement gain show effect requirement graph
scheme exactly simplicity number empty greater extra complicated change final argument prove hold new change web reduce overall variance theorem precisely empty increase empty improve compare large hash experiment scheme compare near lsh empirically theoretical effect many practical extract similarity web vector presence word united review mean
v conclude I I side v third inequality take full side c summation cg cg definition divide side surprisingly limit theorem choice choice optimize work likewise seem dependence
average encoder much simple manifold small generating challenge back find find optimize gradually transform easily machine multiple representation abstraction together conceptual challenge deep good question unsupervised setup whereas constitute good obvious follow unfold stack autoencoder rbms appear yield manifold decode experimentally probable like close rise input immediately recognize simple addition
mb mb rest descriptor local estimate entropy term three linearly rf step cost remark interested predict two force entire typically constraint entire parallel assess experiment publish address infer causal datum size continuous dependency parent model relationship lx fx lx three medium number sample number dag structure dataset remove introduce unobserved descriptor return denote existence dag dags descriptor positive direct cause negative cause forest train balanced test independent simulate dags dag configuration configuration select negative pair
question number go sequence library treatment firstly equal ratio define useful ratio nucleotide difference statistic library secondly deviation library around mathematical software library calculate library library program server derive
color practically either training higher explain scale translate activation example contain translate type invariance autocorrelation measure g translate autocorrelation wide main supplementary invariance towards much mod cifar mod whereas supplementary invariance good dot neuron reflect neuron blue horizontal axis much model analyze material notable neuron mod whereas invariant vast neuron invariance neuron use test mod stay invariant layer increase material invariant neuron especially mod nonlinearity unit convex side tie concave
kt tt kt uniformly iy tx update estimate mini loop comment let x h max mb ms gd consider nontrivial bound prox mini batch strategy fix simplicity
face align subject illumination condition pose sample pixel subject control experiment focus subject top pca light leave informative region automatically pca adjusted r ccc ccc time pixel high dimensionality interested recall pixel another pixel total reduce similar adjust pixel conclusion cause simultaneous prove sharp rate pure reduction deal sophisticated reduce term vanish sf issue pca strongly nonconvex challenge situation initialize case strategy study terminate analysis property big universal occurrence denote orthogonal onto projection orthogonal complement scale write ambiguity satisfy lemma stochastic combine statistical
proposal core propose merge move propose thus prior influence move probability create k merge move adjust merge pp ap denote constraint row lie simplex solve proposal need conservative recurrence correspond likelihood automatically document propose topic give graph sparse lda interpretable world art nonparametric focus sparse propose iteration factor procedure hold first toy arrange tree uniformly concept include initialization problem document comprise test likelihood hold skewed somewhat importantly graph
slice result outperform regular mcmc carlo proposal operator exact variable dark choose requirement likelihood must recognize obtain problem produce inference monte benefit black believe example mh proposal global mcmc mix carlo much empirically slow mixing offset fast computational hope chain continue mix closely challenging variable bernoulli become unbiased
rest ensemble expert mistake evaluate ensemble conclusion consist first learner learner assign learner label exploit randomize know predict weather think expert expert yes question opinion weather weather predict opinion since nearly opinion guarantee bad consist stage receive correct answer majority predict majority whenever true factor wrong label among describe pseudo code majority correct label
stack usually map apply pool adjacent map output output average value reversible pass pooling effectively high computational convolutional wise boltzmann contain convolutional intuitively layer wise pooling provide intermediate store recurrent feed convolutional spatial alternate pass feature map position path pass architecture top connect convolutional depth grow convolutional protein figure channel channel supervise reconstruct compute activation layer correctly predict secondary
discriminative row far appear novel one discrimination class generative generative describe distinguish view generalize discriminative pick manifold distinguish alg point point vs class features basis discriminative stress alg alg sufficiently generative learn manifold yield take instead cluster correspond manifold know motivate discriminative circle ht circle set basis small noisy experiment radius center origin space proportional generator illustrative purpose important black handwritten digit describe minimizer
eq q continuous q first statistical second impose restriction verify proof difficult cdf uniformly mean segment r u r r u uniformly gradient uniformly vector eq hold stationary ergodic unconditional appear around reflect effect hence admit ml square minor allow ny function cr c nr follow limit suppose asymptotic continuous nn nf ty uniform iid close drift tf conditioning away elliptical condition family drift ar ny ty ny ny fix u true
user final respectively map reach please model dataset name capture preference bad item co variant inferior achieve ensemble cf user preference factor rate extensive art cf algorithm evaluate predictor popular name name occur name criterion also try optimize map rank suggest approach factor apply desire reach predictor
paper corpora hierarchical seek rich form additional successful perfectly regime hierarchical direction relax structure document topic nonparametric effort include chinese restaurant corpus provide tb describe posterior corpus category proportion topic generalize lda base parameterized approximation distribution latent variable fully factorize accurate approximation multinomial figure approximation leibler eq corpus concavity jensen compactly variational far factorize approximation proportion document category prior express rise intractable average use dirichlet dirichlet intractable average distribution although degenerate work practice first error novel average arise average note
rs c method meet meet meet se se rs na I cover three replicate classifier effectively literature hold advantage explicit theoretical target vote somewhat less approximate pool compare outperform except fact naive implication suggest form resample future quality approach al explore behaviour generate new optimal selection al method suboptimal choice dependence target label behaviour primarily theoretical address application experimental compare estimation algorithm across range motivate address practice ground aspect weight straightforward estimator section describe function calculation brief exploration calculation loss see loss analytic calculation density give class give x j distribution respectively sample nc expectation term z nz nf nz eq turn np f nz nf nz illustrate complicated analytic calculation give express
introduce equal p inequality typical I batch index replacement gd update reduction compute every sample sgd minibatch practice minibatch reduce employ instead uniform separate k ts gd expectation update provide minibatch
lead parametric evaluation complexity unnecessary rather expect indeed gap approximation spirit gaussian strategy confidence budget use non bandit gain uniform gaussian bandit variance improve pac stop follow two provide draw kullback key increment spirit iterate logarithm structure distribution low bound uniform bandit model include illustration performance algorithm practical budget element leibler divergence q bandit call identifiable identifiable class provide armed bandit consistent algorithm confidence proceed
hx get e pr complete initialize integer synchronization stre sx string occurrence synchronization string negligible uncertainty symbolic entropy iii estimation satisfy e font south alphabet title yshift legend align legend style xshift draw white gray axis thick corner height grid background top color gray xlabel length xshift ylabel ylabel style yshift xlabel yshift scale format format figure draw black inner sep axis cs begin yshift xshift plot xshift font south title title style yshift cell align legend xshift fill gray fill text style black thick corner grid dash gray width background style color xlabel length symbol style ylabel ylabel xlabel fix format sep table figure circle fill none black inner sep axis current yshift text font current confidence letter alphabet uncertainty
statement state theorem map pca map r l map must use complex triangle write j prove connection correlate square
generalize e cd moment provide fit match thus near moment may arbitrary fy derivative substitute cause yield note q minimize e take recognize moment combine close fall observe minimize gmm moment justify alternate tractable analytically cd moment equation intuition work
produce performance compare table reflect ability completion achieve demonstrate outperform especially lr n cp naturally incomplete tensor employ bayesian significant automatic determination moreover prevent advantage amount pixel validate discover ground extremely synthesis superiority due interest attractive receive engineering department science engineering university china laboratory brain interest learn zhang degree china computer science china research computational theory brain compute statistical publish paper international receive dr degrees electrical engineering team laboratory advanced brain processing science technical associate journal transaction figure proposition zhang cp powerful completion capture multilinear cp algorithm
concept visualization call appropriate along mode tensor mapping interested unfold unfold rewrite operator another unfold kronecker represent concatenation introduce operation able multidimensional record separable parameter serve weight entry thing second minimization constraint trivial role
miss try neuron input north count node count dim output align align center gm encoder deep encoder low layer decoder consist reverse reconstructed version set encoder train simultaneously autoencoder since provide dimensionality gp laplacian locally mapping way deep encoder free parameter decoder train cost sum reconstruction find normally feature processing model training function minimize minimize respect compute efficiently minimize consider q
make mass policy support random always obtain context policy word action context obtain reward zero observe take policy policy experiment evaluation initialize sensitive control uniform policy ta tr ta ia step minimization rather doubly reward function train contextual learner repeatedly observe choose access classification guarantee call round number policy policy obtain contextual amongst excellent performance online contextual reward action take round previous round feedback observe contextual clinical supervise necessary exploration action result achieve round probability set policy reward cumulative collect logarithmic dependence likely reward use
rf center specify constraint candidate comprise specify reciprocal distances around intra balance mild maximize respectively submodular induce collection greedy gain speed greedy construct marginal gain element gain one merely gain idea increase return property naive method element gain element lead practice investigate rf rf essentially nontrivial mid pool candidate generate due sparse code metric pyramid grid grid please pyramid instead totally different pool grid represent whole extracted sift rf measure rf grid rf l x p grid grid please also
diversity dirichlet bias general whole diversity index distribution predictive species discovery estimation posterior flexible limitation diversity shannon diversity diversity health classify group produce index specie specie belong population actual distribution species shannon far widely measure biological diversity identify measure mean value relative sensitivity rare rare specie difference specie easily recover shannon unique year c introduce physics generalization shannon
compact observe invariance enable significantly accuracy descriptor l vs intensity classify texture descriptor adopt supervise support standard purpose denote class column distinguish classification separate assess coefficient pick thus coordinate pick texture image fig randomly sized gray scale patch texture patch texture patch use descriptor near neighbor assign patch texture report coefficient accuracy comparable long feature obtain accuracy descriptor magnitude short descriptor different
specific segmentation segmentation generally identify discard room cross audio segmentation overview note literature often type datum contain news result audio explicit segment
hilbert advance hamiltonian position show careful assimilation model prior dynamical kalman enkf ensemble assimilation gain wide ease practical enkf algorithm restrictive applied filter posteriori assume posterior design observation operator degree nonlinearity handle propose assimilation name obtain monte carlo hmc non probability distribution carry level water linear highly nonlinear enkf assimilation variational filter monte datum assimilation information uncertainty system family ensemble filter application variational root costly development tangent model assimilation scheme root considerable development enkf formulation class stochastic member update perturb lead root filter transformation apply correct enkf observation enkf accommodate linearization spirit extend kalman alternative handle non linearity use operator instead linearize mathematical pose subspace minimize depend operator jacobian nonlinear addition inherently posteriori multimodal distribution advance feasible promise towards sequential monte particle density
design essentially provide turn recent compressed signal signal technique counting entry maximally skewed stable develop compress compressed compressed linear
par safe safe safe safe safe par var safe define lambda safe safe var par safe program lambda stack safe safe safe safe safe safe observational respectively histogram right program e code efficiently many dimensional conduct ability automatically discover procedure encourage text common perform abc space pre text learn representative histogram summary statistic figure discovery sampler program sampling mechanism exhaustive computation tb lambda stack par lambda par
perfect achieve lk achieve linearly linearly place centroid regular margin grow place centroid maximally however margin separability maximum experimentally dr uniformly keep projection fig separable first column apart center big achieve perfect basis suffice projection enough role dimension number geometric projection nonparametric fixing projection find perfect vs svms different improve drastically critical separate degree experiment lie simplex boundary increase ideal corner simplex centroid compare experimentally simplex quality random c ideal reduction dataset boundary black achieve boundary
jacobian remain submatrix make independent row column continuous identifie level framework incidence generalization framework framework molecular conjecture body constraint system frame affine introduce hypergraph property useful hypergraph hypergraph combinatorial literature concept vertex identify tail connect admit orientation exactly nash tight map compose edge disjoint tight ready theorem combinatorial characterization finitely subspace incidence frame outline replace copy row determinant identically apply tight hypergraph show matrix identically long avoid main call behaved notice modify hypergraph copy expand hypergraph let replace last incidence row expand expand hypergraph incidence framework underlie expand
seq shrink letter reverse infinite sequence seq sequence seq study nucleotide incorporation interested cccc cccc flow cycle nucleotide flow seq self contain mathematical algorithm software implication threshold basis intensity variation threshold base drawback sequence availability positive potentially direction bootstrapping threshold call basis iteratively threshold test independence target
concentration specie assume observable system diffusion observe specie case steady specie regularity form fp ss patterns b concentration steady large fine ss reaction diffusion finite initial I p desire observation guarantee perform two design develop search require pattern descriptor goal logic superposition treat problem superposition tree descriptor reduce find logic specify machine observation step synthesis correspond reaction diffusion formula end semantic assign superposition tree quantitative
omit obtain continuous time consider purpose update policy offline linear remark establish introduce policy optimal use converge residual reinforcement learn collect increase collect learn offline employ real effectiveness algorithm verify free policy reinforcement method weighted powerful reinforcement rl great optimally simplicity rather policy
main additional observation seminal completion incoherent recover nuclear later incoherent subsequently incoherent bernstein inequalitie significantly simple proof recover incoherent svd manifold restrictive entry uniformly sampling power law universal robust devise matrix scheme universal scheme dependent furthermore important perform know problem recovery bit mostly rip
vary label regard er model label occur probability labeling namely edge probability edge graph label regard mode consequently case multi view anomalous less empty consider within community edge graph unlikely multi regard possible yet model characterize inter intra community give capability detector inter intra community edge degree external er multi anomaly detector probability anomaly subgraph possible internal internal unbounded degree excess sufficiently negligible detector subgraph graph fit statistic value degree give order use adjacency node spectral modulus computing observe three lastly newly statistic anomalous normal base
spline satisfactory basis regressor generate unknown implementation match regressor introduce regressor differentiable conventional basis well basis regressor regressor strong without significantly performance linear define incremental decision regressor approach performance initial adaptively dependent incremental tree complexity regularity meet length paper detail section performance nearly piecewise incremental algorithm twice observe datum start simulation remark nonlinear regressor regressor I ti regressor past regressor vector transpose xx piecewise divide regressor region regressor linear regressor regressor vx assign independently method mean square regressor process internal parameter e limit regressor desire signal piecewise partitioning regressor regressor achieve good partitioning
algorithm behave like wang table quantitative study fix depend still see average square confirm validity wang rate seem value assessment number provide magnitude accordingly vary varied slope c slope conclude numerical wang look behave wang table value times one increase retrieve asymptotic observe confirm since time illustrate observe close bit temperature wang choice times parameter wang histogram visit indicate vertical well visit wang section corollary expect behave regime wang stepsize prediction previous consistently state behavior visit contain initial condition drastically change approximately know wang stepsize discussion advantage increase exponential rate one
distinction involve similarity hc ica variance empirical variance estimate em modify I covariate voxel calculate statistic estimator determine covariate voxel level map apply testing rate false fdr effect map specific sd sd hc dual hc ica n medium sd hc ica hc ica dual medium n c em em hc fmri ten fit hc ica exact em approximate em comparable exact temporal domain population major advantage exact em signal time exact ic em exact em time
body near position inside body intersection know surface ball cone cone volume graphical illustration vertical transformation bring isotropic position scalar keep keep consequence hyperplane construct case event least oracle call achieve closeness variation quickly
estimate scalar least estimate imbalance appear lemma error even rate plug estimator asymptotic error appear regard issue covariance matrix tensor present imbalance restrict independence classifier mixture two product address contrast totally tensor decomposition imbalance imbalance suitable maxima b attain imbalance fx denote vector instance likelihood imbalance expression average instance numerically unstable close true imbalance imbalance justify constructive b law delta method prove
sparse select base know law viewpoint carlo suggest discuss previous paper viewpoint case extent height still inside illustrated figure region union discussion interval stream hypothesis test construct rejection specialized require base independent reference side large empirical expectation manner cutoff small test bit principle solve efficiently describe computationally apply away several effective sample get small carlo possible truncate multivariate gaussian hamiltonian carlo efficient multivariate distribution constraint selective cite sampling propose conditioning well sign fit event consist single polytope sign variable exclude conditioning selection selective validity power put nearly price acceptable quantify tradeoff far work selective add quadratic sample instead sphere section gaussian setting selective binomial scan statistic generally selective simple selective clinical binomial experiment similar clinical trial heart patient correspond patient heart trial efficacy treatment th order construct odd tie select law fix remain selection heart attack heart attack margin conditioning give right conditioning conditionally extreme selective fisher aside otherwise family interval observe poisson constant intensity unknown maximize confidence
isometry appendix far condition success iteration combine get recovery sparse matrix rip isometry ensure index clearly guarantee condition justify former k summary index condition I actually k totally induce exceed recover l projection terminate recovery reduce conventional recover
implementation validity property claim propose I model add plausibility think ok agree validity property wise error familiar paper selection paper right select unknown seek desirable dependence structure design know efficient xu zhang framework valid worked propose inferential I special variable regression interest develop optimal set product I validity I sub select I approach valid post I drive base example arguably one widely tool scientific possible planning subset useful variation inclusion explanatory explanatory variable fundamental go selection central pursuit general solve arise comparison variable always suggest variable overcome limitation add kind variable include method schwarz severe allow candidate naturally etc despite certain property ranking inferential meaning aic bic conclude correct plausible model criterion meaningful measure correct variant elastic net summarize meaningful significance test lasso resolve concern parameter search specification posterior computation remain furthermore
project figure inference answer appropriately involve incorrect free nothing put splitting lead power ordinary try inference parametric invoke strong focus weak like completely sample predictive seem property
fig modularity trend simple increment monotonic worth note early entropy fitness subtle problem pattern narrow target good auc consider eight misclassifie represent report degree see auc high membership hard decision zero please data may still picture additionally yield insight recognition among optimize towards relevant property denote metric would find perform role entropy modularity situation pattern leave corner green pattern fact red fig assign trend modularity peak I trend blue pattern belong blue one correctly misclassifie respectively assign depict axis perform accordingly contain isolated herein uci miss retrieve comparison version uci normalize ensure unit uci usually principal ref
operation exceed ssc apply title width legend style east legend pos north mark none dash face none dot index time face mark none black none name anchor west title legend legend style anchor east pos north x mark solid face none dot black
evolutionary evolutionary process possible set normal function consider function default protein think accordingly posterior use suppose move say proposal value possible density model distance keep gap setting prior distance various first analyze sequence method identity align identity suggest sequence prior distribution top leave panel evolutionary modal acceptance proposal modal even approximately contain dominate modal protein possess structural similarity top show unimodal suggest long evolutionary two previous mode mean large move towards around expect bottom model evolutionary alignment consider pair multimodal posterior distance mode consider unimodal mode setting little sensitive parameter discuss influence keep give comparable match difference
one paired weakly pair training precision precision recall experiment one retrieve document rank document retrieval precision retrieve document denote retrieve score precision display scope curve visualization retrieve precision present user pca cca pls method text pls texts database wikipedia database dataset split multi contain select training word frequency
simplicity tool problem eigenvector especially dimensional ad hoc absolute take fit original approach desirable motivate research especially pca et sparse nesterov sdp exploit connection singular svd extract principal component pc approximation et optimization involve although differently except phase base conditional gradient variety efficient multiplication extremely large algorithm suit identity direct deal substitute difference require qp every intensive amenable restrict identity special multiplication need shown adopt approach develop eigenvalue unified method maximize objective function consider quadratic reweighted square turn eigenvalue sequence problem efficient ascent lead generalize spirit solving type often suffer get systematic inspire minimization nonsmooth
knowledge choose noise subset know specific case unconstraine consider quality sparsity viewpoint maximum relate bernoulli dedicate greedy explore iteration atom current subset gradually improve pursuit mp orthogonal omp square ols refer match iterative hard subspace pursuit category resolution series consecutive error increase lead extension forward subset support element error exceed square inclusion removal induce decrease design forward backward search either atom backward elimination omp single spirit extension ol early wrong backward ol omp low complexity search dedicate regularize result nonconvex relaxation convex pursuit lead homotopy whose omp homotopy closely connect angle lar simple forward lar importantly homotopy solve dedicated penalize ss font lb lb lb lb lb lb lb lb lb lb concave piece regularization compose support instance minimizer piecewise appendix understand consider concave envelope curve curve support solution vertex advantage suboptimal greedy value good replacement repeatedly minimize decrease descent complex maintain improve decrease approximation error
dependent variable influence regressor covariate variable commonly proportion interval common line transform long transform kind usual thus model datum practitioner standard specifically tailor regression underlie distribute flexible rate proportion shape parameter mean precision mean covariate function fashion formulation incorrectly take estimate density maximum likelihood estimate slope covariate vary simulation precision incorrectly properly model large dispersion identify source variability dispersion model beta relate
r glm default perform cross report assess statistically metric evaluate pearson correlation bold encoding signal cross sign obtain significantly irrespective across study relative averaged sort rank induce separate oppose free glm inherent glm model purpose design follow glm basis sign subject use figure voxel encode score first code peak difference level voxel encoding basis glm voxel metric pearson correspond exhibit score glm result investigate encode valuable peak width characterize mis reference voxel voxel exhibit method commonly software define www fail voxel score encode score plot peak code peak even single volume second coordinate encode score canonical axis rank black glm previously estimate voxel canonical glm significant use exhibit sufficiently suggest computational
I independent grow increasingly face curse translate scale hyper cross hyper support feasible mp train different fold pair sample sign star demonstrate solution reflect rmse toy curse dimension factor curse gp automatically factor basis rmse cc cc concrete red datum sparse rmse fold statistically indistinguishable except concrete compressive considerably standard require factored regression
autoencoder autoencoder stochastic gradient online minibatch epoch autoencoder decoder epoch reach weight leaf autoencoder epoch stochastic gradient mnist report scale word relative mnist attain error reduce autoencoder perceptron ten tree dimensionality tree might autoencoder news less architecture autoencoder autoencoder reduce ten
vanish obviously source introduce framework negativity source partially non negativity combination positive source generate seek simplex volume spike source sparse spike verify support illustration leave mathematical perspective characterize column mix translate spectra different spatial consequence mathematically emission emission etc physical density temperature make detail panel report take correlation wavelet highlight partial chance finite sample realization theoretically uncorrelated likely available negligible large htb limitation standard source negative generally negativity negativity source assumption approximately transform vanish entry present correlate source build blind source separation sparse correlate source noisy make assumption non negativity range separation problem especially image organized review limitation correlate introduce report performance standard finally illustrate diversity q first fidelity discussion choice penalty appealing make
consist per semantic differ weight often assign graph language relation resp represent total equal probability model everywhere axiom relation incidence incidence every element axiom axiom relation identification treat relation axiom retrieve domain function uniquely graph logic axiom edge go opposite axiom axiom graph edge complete edge connect collective weight equality fix analogous cycle weight size construction item statement subgraph working logic weak statement logic emphasis axiom universal sentence presence weight difficulty investigate implication logic artificial technique recognition ann direct graph edge weight connect neuron connection say write activation neuron neural neuron activate previous language binary ann introduce new predicate represent neuron activate satisfie threshold update ann neuron begin neuron activate accord digital activate states neuron neuron iff imagine property phrase logic answer validity q conjunction axiom axiom unlike weighted artificial theorem finite particular order language finite unary predicate use predicate ann almost finite predicate language issue e still expression limitation many case logic mention quite logic locality reader logic calculus model countable moreover subsection countable regardless shorthand shorthand n yy number depend free finitely
close back euler characteristic physical underlie decade considerable cut minimize normalize modularity cluster edge removal among progress researcher spectral majority reader comprehensive review stochastic respectively little previously community input specify quantity address select wise edge bic criterion variational approach highly researcher blockmodel undesirable restrictive bernoulli observation apply misspecification stochastic blockmodel variant motivation conditional restrict exchangeable graph composite bic community blockmodel exchangeable simulated show outperform background cluster methodology simulation discussion adjacency diagonal zero random
google frames assign frame correspond nf logistic unknown index correct sgd method axis epoch improve find decrease version almost batch illustrated show improves measure marginally right effective obtain curvature figure memory mark degradation great yield focus datum dot train remain error measure percent correctly classify latter account yield suggest occur set monotonically quasi newton fairly large sgd since set compare cost batch explore efficiency quasi report performance product order line vs dotted vs sgd significant margin cost dot scale crucial quasi allow acceptable cost
prove lemma definition ac il show modification bayes work proximity near grow appropriately regularize enjoy considerable bound principle speed encourage empirical nearest nn continue popular practitioner despite numerous continue yield paper near since amount effective analysis vote neighbor guarantee consistency
shift modal signature signature may signature mean partition evidence contain signature may signature signature shift signature signature verification competition signature curve signature dataset priori group signature sufficiently separate shift modal curve analysis represent unsupervised nonparametric signature evident signature signature polynomial get polynomial smoothing estimate acceleration signature normalize norm depict apply asymmetric gaussian smooth acceleration distance norm acceleration depict figure figure acceleration figure show algorithm mean shift signature fourth original signature original signature blue middle signature cluster display functional mean shift homogeneous summarize signature green signature adaptive ascent scalar counterpart dimensional shift design dimensional functional applicability mean shift correspond ascent practitioner establish
nf propose impose constraint introduce write diagonal account influence code introduction function black dash function get optimization bfgs et kernel regression consider locally constant weighted equivalence prove appear distance hellinger hellinger analytical calculation hellinger
fundamental object yet greatly covariance decade additional covariance key tractable lie toward order spectra constitute observe variable order assume variable distance series assumption allow depend depend assumption process likewise model use whose apart weakly correlate situation specify unknown depend matrix I n norm measure use estimator toeplitz form allow frobenius relative member matrix decay diagonal estimation establish frobenius minimax utilize practice estimator positive motivated estimator partition block zero adaptive class technique norm construction obtain heavily decay away far regard covariance diagonal arbitrarily situation decay estimator thresholde guarantee positive estimator cone would norm notably
natural extension relative partially support dms thank international conference great author express thank united national title guarantee mean bernoulli random widely success
mean answer question answer task never environment task quantify evaluate answer answer question quite understanding involve concept hide ideal manually every individually since infeasible ambiguity interested inherently bias frame grain categorization interpretation coverage automatic answer consideration member attempt issue similarity score lexical
tail large tail particular type note marginal fr xx tx yy ty tx ty ty cf bivariate characterize useful example copula see popular symmetric joint tail correspond bivariate coefficient freedom tail extreme tail asymmetric structure return structure chen exhibit tail exp exhibit estimation main technique univariate heavy independent index standardize marginal fr pareto observe let minimum fr
validate scheme provide predictive compare fix predictive interval model regression conventional interval conv purpose upon interval hard interpret mis interval measure big dataset compare precision width conventional observation contrast assumption error occur manner estimate increase able quantile one hypothesis hence provide table display explain conv k dataset var value find minimize fold compare display constraint sign fail sign put pass mis distinguish two consecutive annotate bold proper hyper illustrate row look proportion var conv conv reliability constraint situation stay var conv nine look almost everywhere large wide obtain band eight list sake compare mis mis chart chart display mis mis figure chart mis reliable mis find reliable envelope interval chart small mis ratio chart chart predictive fix look see mis conventional conv ls testing stay var conversely fail predictive interval nine conventional conv l wide envelope scenario var envelope error look figure fix decrease model note conventional small nan accept model observe interval neither reliable summarize display row summarize reliability quality band fourth display efficient ignore value normalize c efficiency var var var var var conv concrete fix svm l svm conv var k predictive goal detailed manner reliability purpose choose
surely norm surely expectation amount affect column k treat rewrite independent ia bernstein theory equal hadamard product hand hadamard independence q dt matrix bernstein know use normalize normalization happen algorithm part contraction power iteration tensor contraction argument argument update overall guarantee convergence similarly argue local convergence update define notice constant contraction perturbation suppose hold suppose h asymptotic include rate algorithm actually quadratic contraction quadratic beginning involve quadratic observe quadratic appropriate initialization rapidly sake clarity propose convergence convergence lemma explicit contraction observe denominator numerator contraction result tensor perturbation tensor w contraction define addition imply iteratively tensor power provide proof notation incoherence notation th column remove decompose unnormalized unnormalized c follow derivation repeatedly assumption use inequality exploit last exploiting term eq inequality exploit j distance expand notice argument provide incorporate inequality definition denominator use coordinate
tree subtree subtree leaf node uniquely define suffice mistake subtree subtree lem prove show tree bind one quantum communication express concept communication numerous low sample pac vc parallel dd aware counting restrict ball b define see already separation quantum way communication bound coin simple randomized public coin prove differential privacy differential privacy somewhat stronger bind upper since communication asymptotically learner zero privacy sample sample subset margin optimal achieve efficiently vc
order objective replace mask th mask multiplication variable concatenation copy mask function training criterion user corruption process sample dirac delta mask take corrupted version q autoencoder effectively mask capacity encoder copy px assign input logarithm essence maximize eqs every chain sampler although train agnostic sampling propose deep select generate select dimensionality hide unit layer use
report reporting apply might preprocesse simulate infected calculate neighboring infect internet inter consider free note infected node near neighbor infect infection value preprocesse report large theorem infected internet inter compose maximal relevant various epidemic occur report epidemic specificity epidemic available test standard plot carlo scenario body paper graph perform report ball radius ball radius standard show distinguish follow describe false positive fig infection size report positive type assume likelihood epidemic reporting plot obtain facebook level extremely infection setting succeed negative show succeed presence information contact zero hundred spread across epidemic begin experience epidemic process trend public across common
generation know k continuous parametrize term parameter intuitive also assumption interesting take follow binomial c check straightforwardly viewpoint mechanism individual birth next remove subsequent could population take follow condition check I either expect another process growth e z whenever depend unity classification likelihood shall entire z lk k lk li li n lk represent individual exactly intuitively accumulate generalize likelihood use parametrization shall worth mle knowledge thus address obtain ii intuitively rise parent also order investigate estimator necessary power series assume establish verify establish preliminary omit remark I iv
initial effectively controller training simulation switching preference mode next control initial investigate capability controller initial capability simulate initial plot initially change controller control different loop moderate robustness uniformly act depict system loop open mode lead instability system degradation steady demonstrate apply switching admit incorporation assign different investigate go solve cost go current develop burden many value south school technology city edu nonlinear investigate adjust switch feedback development ability switch switch lead deviation mode preference
contain cause identifiable soon period provide insufficient occur equivalently sparsity degree obviously condition necessary much hard eigenvector study suggest condition irreducible everywhere identifiability unable counter critical assume markov assume I introduce let arbitrary affine fashion function convenience denote small letter g p lie work operator q view preliminary obtain arguably estimator obtain q derive minimizer
detailed discussion comparison propose analyze frobenius sample see good bound various completion recover observe minimization uniform incoherent want compute rank popular frobenius norm consider scenario guarantee frobenius present element rank comment ingredient number leverage tm neither leverage exactly leverage account given optimize factored sample element routine matrix give objective function follow sub routine set span prevent heavy provide main rank show I constant specified completion
proved reason analysis thompson resemble sampling carry modular particular set bandit whether idea generalize maximize modular variant problem adversarial base solution key polytope polytope hyperplane combination prove first exist vertex exist therefore contradiction prove contradiction polytope express vertex greedy I ie contradiction ne e se least event hoeffde event claim get q last fact note conclude span solve optimally greedy work unknown learn interact repeatedly bandit setting formalize know computationally favorable dependency efficient massive drive greedy sequential learn combinatorial item number potential huge
structure histogram image derive finally trend scene intuitive method consideration complex acoustic forest poor second might valuable finally exploratory section audio baseline existence fail performance significantly suggest effect modelling describe one drive development normalise compression dissimilarity investigation design individual public avoid addition choose human testing phase evaluate experience experimental choice human rigorous comparison believe reflect employ qualitative capability interesting significance test human protocol cross clearly algorithmic human result figure achieve similar human suggest median benchmark secondly misclassifie acoustic aggregating take acoustic correctly misclassifie encounter music retrieval whereby always misclassifie moreover unlike human challenging acquire human experience comparison confusion present reveal misclassifie human find misclassification acoustic observation contain sound event even semantic environment universe mutually exclusive exhaustive word include application ensure category important suggest far first consider design learn acoustic give mobile service place resource intensive processing line need signal application real instead
serial consider nesterov special convergence rate prove set result obtain accelerate proper vector z fy kx z result algorithm satisfy specialized serial q unconstrained case serial nesterov prove accelerated utilize restrict indeed simplify improve uniquely suppose block lipschitz gradient serial q useful confirm block pick update often mention algorithm dimensional operation unless suitable implementation assumption gradient focus algorithm variable note present latter require address k therefore set block
location full location visit sequence instant previous image mixture sift histogram sake extract descriptor within analyze subset pixel fair spirit life grid combine image together investigate separate grid visual input approach image pixel comparison place bag map bag agree fair complexity counting grid g train window scene illustrative purpose label label realistic train image likely field view simulate generalize scene possible count scene close question neighbor comparison none fig approach bag feature simple bag particular example reason window top window bottom sometimes track train infer track existence train track furthermore proportion carry layer previously elsewhere organization retain reasoning iterate eqs count window scene take bag low right window compute count appropriate matching reconstruct
present analysis approach stage bag bag bag define basis take analytical typical gaussians exponential log method reproduce kernel reference therein divergence gaussians compute product straightforward divergence consistency number include construction overlap concentrate dispersion type negative performance remain also similarity enyi index prior address estimation assume response consider covariate nonparametric form assume regressor reproduce use classical regressor construct continuous meta generate handle dataset estimation learn bag finite algorithm ensemble convolution case similarity establish introduction rate bag
frank com building answer question intelligence promise progress recently achieve learn logical database cost either amount human label define tailor practitioner question answer schema without fine tuning supervision resource empirically demonstrate meaningful supervision similar label challenging answering answer topic bring huge building way development store huge organize database connect entity answer define entity give express simplify issue collect search e answer open answering remain challenge triple million difficulty machine language problem semantic convert logical subsequently scale hand schema parse negligible intervention might generic database schema broad english
rw model hamiltonian hmc favorable hmc favorable scaling guide posterior neural use hmc traditional feed back propagation though hmc explore manner rw mh expensive large drawback outline challenge outline variable influence assess relative background avoid testing model genetic examine possible combination hmc mcmc method rw greatly massive real assess network popular method advance technique clear use basic net parametric method classification sense smooth precision need relationship net appeal include phenotype complex capable classical natural consequence strength fit occur start represent include noise may method exist method another net black
figure goal versus movie review train bag word example exhibit behavior simulation run size result learn good suboptimal error relative generalization suffer bias show behave link favorable tradeoff dropout implicitly assumption label original dropout naive naive generative recent survey bag document dropout state art accuracy suggest discriminative strength generalization interesting assumption explain helpful topic make training dropout erm
determination deep refer reasonable activation even supervise kernel model model propose unbounde layer connection adjacent introduce draw amenable series fix encouraging layer analyze recurrent network back make gradient hide stable know systematic deep composition polynomial application construct kernel kernel correspond infinitely deep attractive study first show view neural architecture capacity decrease degree propose connect input examine obtained finally obtain gives define
assume prove assume bad regardless satisfie cd kp leave side go sufficiently contradict q claim easy fp fp p approximate cut see undirected exponential number subgraph cf property calculation possible sufficiently minimal generate input decay positive fact noise bad instead independent chernoff hence interestingly accommodate adversary cf nature node label adversary input arbitrary bad case bad inconsistent semi well purely effectively set contribute adversary maximize adversary grid attain consider addition marginal mention optimal recovery procedure generally per instead resort mode hard relaxation locally marginal lp relaxation tight field interest cycle relaxation edge noise another local significantly baseline latter likely map
amongst much become unstable relate proximity policy policy reliably realistic setup reason believe ideally suited framework turn aim learn task neural ensemble diverse error part rl diversity aspect experience diversity diversity diversity experience high assume multi diversity issue unless sound diversity mdps generalize train mdps aspect diversity discuss express think multiple heuristic situation
conclude discussion contaminate gaussian random degree contamination bad common maintain distribution advantage establish either otherwise robust contaminate distribution unfortunately relation covariate covariate contaminate lead covariate property contaminate replace contaminate contaminate contaminate contaminate distribution mixture conditional gaussian linear also convenient square concern distribution model regression marginal integrate contaminated contaminate depend mixture contaminate nest family contaminate mixture contaminate particular contaminate provide maximum two
social medium root expand aggregation variant meta share similarity system cluster exploit social algorithm unfortunately assume twitter streaming scenario expensive obtain popular user mention driven similarity social medium author e title tag date location propose cluster label combine similarity reveal group medium event twitter incorporate module temporal interactive location micro event benefit variation mean spatio dense topic term message assign close centroid work deal media stream twitter pre aim token tweet strategy processing particularly efficient suited streaming scenario aggregate computed pattern stream observe build cluster carry track least comprise twitter systematically baseline tweet assume knowledge social network system cluster paper simplicity
method handle network competitive compare relatively tb fp graph come domain penalize logit separately individual factor utilize encourage solve quadratic constraint impose network descent procedure free world latter real demonstrate pc procedure data dag candidate causal network demand size nature logit room cd moreover since nonconvex introduce global future consistency node grows investigate review topological sort topological topological sort acyclic every sort sort
standard ball grow edge boundary decrease cut edge budget upper contain get control remove boundary maximize solve connect capacity capacity every source lie cut easy check minimize give detail remove remove edge edge decrease control procedure size control statement I trivial budget remove increase budget total budget budget budget execution control remove edge possibly assume argue great hence remove control boundary decrease apply contradiction violate upper trivial bind equal piece solution nd sdp mapping sdp sdp coincide connect active second condition property sdp solution ball radius around ball graph vertex first sum option remove cut iteration go length cut cut step none cut let go know cut increase budget lie initially budget step change
theorem tell twice exactly compressive cs general reduce signal directly paper brief conventional compressive approach compress choose suffice reconstruct
send barrier barrier overcome develop successfully deep seminal propose hundred million conduct extensive recognition automatic recognition kernel deep net method appeal cost training reduce two competitive question deep application automatic arguably instrumental huge million adopt drop parallelism new trade add hundred excellent various consideration dependency effective sample early reflect extend million svms solve svms approximately matrix kernel million million sample time publication none compare reduce feature feature dimensional optimization approximate dimensional recognize promising observation product approximate spectral weight major random random learning training report speech recognition vision context automatic recognition example task
affect student measurement student teacher teacher time corresponding year teacher contain current teacher effect subsequent score student expect diagonal teacher intra student student year block student observation year refer gp indicate intra student great flexibility future year teacher effect student correlation assume independent analyze year year correlate moderately effect average teacher single future year teacher aspect persistence require scale measurement refer current teacher teacher contain student year alternative autoregressive specification persistence structure teacher effect correlation intercept implement alternative except teacher g define exception teacher diagonal set corresponding year new definition student express include teacher effect fit without student compound block student intercept formulation
eq result batch prove integrate case involve boundedness apply cumulative risk boundedness adapt recursive apply end application recursive argument discuss optimally observable develop parsimonious strategy aggregate dictionary complexity reduce generality e learner empirical iid interesting deterministic easy equation sense generality remarkable dependence risk natural optimal rate general stochastic ms restrict strongly iid observation iid copy assume jx preferable convert learner average jensen inequality give optimal regularity satisfy
transpose side obtain system linear subscript find store consider regression determine secondly dictionary due selection criterion however find solution relaxation np certain incoherence basically dictionary co linear typical viewpoint incoherence requirement bayesian treat concave add prior typically approximate yield eq
majority review diversity pathway generate key convergent trait benefit effort acquisition genetic specie expand molecular toolbox trait thank n conduct analysis experiment model test extract rna j figure paper study supervise discuss manuscript source correspondence publication view website material c seven specie characteristic record intermediate c majority abundance bundle bs study employ methodology comparable cross quantitative partition score cluster em assign g presence absence allow component two complete linkage agglomerative partitioning euclidean common em quantitative study specie abundance band abundance band qualitative presence score assign appear bs cell represent string string trait absence trait presence trait miss pca fundamental transition consist denote label binary phenotype meaning allow transition transition change towards string possibility involve simultaneous constitute order influence evolutionary dynamic
proposal combination namely error design independent regularize cast computed norm rate mu selector embed zero mean order advantage convergence propose analyze acknowledge gain rate additional regularization main constant property procedure auxiliary rates convergence stochastic notation integer cardinality say sub gaussian variance sub
sum parse parse short message step message definition message part technique message current form pattern pattern consistent cm material step compute step take turn vertical message passing imply short correctly show make nonnegative add connect incoming assign since acyclic source path take compute checked rescale r rescale short path initial rescale suggest vertex edge equal iteration thus bad complexity cubic practice small reach good
write permutation sign stems combine conclude differentiable frank wolfe problem recently interest improvement become particularly atomic ball compute euclidean proximity frank wolfe gradient regularizer regularizers fuse elastic net en glasso variant rely depend namely linear usually arbitrary regularizer permutation neither outperform consist pairwise sparsity equality estimate proximity efficiently accelerate proximal fista refer regularizer object order weight
map relu three third fully relu final send logistic stochastic momentum epoch tune decay network run file architecture source code scale convolutional neural science university college md david department college md show excellent visual exception carefully object align naturally feature get increase appearance recent imagenet discriminative use learn simple feature manner mnist build convolutional neural network achieve excellent handwritten digits sign category imagenet come learn pattern increasingly layer
projection take input simplicity give computation degradation l projection bilinear entry bernoulli binary eq kk section apply sign carry preprocesse drop embed fast fourier matrix clearly need datum efficiently compute operator convolution fourier transformation dft dft dimensional dft conjugate transpose transformation convolution original hadamard therefore bit bit dft efficiently
platform pac bound analyse performance classifier adopt classifier centre prior bound vector another ingredient logarithmic determinant inequality lie evaluate source view offset explore pac view consider promise applicability content video audio view feature improvement split
derive set properly pdfs directly block r p k j j dt dt I find intercept r dt dt r dt proof appendix material information drop subscript convenience know imply bf k r dt rhs rest nonzero bayes drive bf k k bf k exponent bf true entity denote index recall block index model result lemma slightly b denote j tn n bi ib ir r proof attain tt apply two satisfied block portion material bf k I maximizer lemma equivalently large bf om om q r b b bf om f positive structure limit rate first term go
anchor south west box west east box south box north east lag ar aic k time specific result ar goal lag series lag divide series second half estimate order lasso aic freedom fit apply adaptively estimate non freedom
shall denote ds ds denote integral entry proceed namely martingale properly lemma large eigenvalue random trace ie side notion calculus let whose integrable quadratic martingale quadratic r quadratic martingale identically r twice self adjoint matrix follow twice continuously application act importance result trace martingale td entail q x
next training repeat process repeat sequentially numerical optimization time need short initial result jump large poor prediction less approximate exponential particle show predictive provide c dataset log method compare rank task whether significant analyze comparison summarize across dataset differ performance level statistically superior addition predictive overfitte dominate sign p pairwise comparison give functional log return
purpose test bayesian prediction extra exist end section consequence split input output look determine model believe consider global choose commonly use covariate give satisfactory try aic supervised version bias situation see practical usefulness need experiment consider determine bias aic focus perform experiment promise derivation start equivalent aic hierarchical greatly applicability acknowledgement thank comment regularity item wish use explain explain select likelihood validation variant estimate vary assume correction see aic adapt input bayesian averaging case also
space maxout artificial hide call popular success maxout neural become I e hide layer product unit deep sum compute certain extend kind analysis discuss estimation artificial feedforward borel lead choose weight behave activation precede time overall look give compute activation precede consider activation hide set vector activation layer subset map onto subset map common recursively disjoint neighborhood whose function compute layer recursive formula count along branch root space linear region linear activation r r construct region input neighborhood distinct maxout discuss identify fold coincide absolute fold twice
gene test complete carefully elsewhere factor specific gene co expression uniquely loading b regularize control loading x estimate w calculate zero suggest gene gene precision matrix know gaussian network use observation construct connect great combine single keep edge run network co expression observation run short pass node project datum snp multi trait nucleotide snps snps represent copy frequent genetic variant total quantile although snp perform perform gene single snp across perform association snps biological meaning build snps snps conversely refer trait gene trait association interpretation snps minor frequency snps let gene gene gene apply repeat initialization estimate sparse active snp snp trait association demonstrate model table l one average represent zero snp dense individual individual among dense next analyze sparse snp observation observation active gene level gene level snp zero
processor gb ht marker respectively disk represent interested robot previous grey represent cell robot cell marker step mean action become know terminate satisfy specification horizon converge close second become policy second output optimal state three observe maximal error loose bind correctness apply robot planning north west correct cell robot arrive adjacent intend cell north ne cell fig robot maximize
rr depend omit follow mean eq easily convergence effectively whereby appendix situation density find noise wavelet integral due integral quadrature rule q know obtain whereby mm mathematical ex promise usual hide specify new dependence wavelet variance equal provide flexible inference become edge detection image auto laplace statistical dependency coefficient wavelet take value application hide determine wavelet
frobenius relative spectral miss cccc miss e miss health survey approximately low certain specific human data combination compressive sense offer currently first normal successful define classification ratio number trial rate contain
hyperspectral motivate particular gaussian entry take assume normalize sum pass column separable column probability index translate stream implementation section index nmf upper r r I factorization nonnegative factorization work singular show scalable serial residual e error residual choose datum near nmf extreme solve matrix fortunately complete remarkably pass suffice achieve use ever matrix thin
example label goal available training datum develop assume link involve network node accurately pac apply big technique verify match however available match organize expectation sample replacement section present validate match refer match node match validation extend match algorithm rate precision subsample combine conclude research later average estimate example use pair finite estimate replacement estimate similar
sample belong label addition unlabele observation class predict future minimization lipschitz hinge supervise act come represent transpose operation especially powerful arise setting must link default label ji c constrain j capture label lead j semi supervise survey link equivalence I j e index unlabele nonetheless proxy encourage instead partial show constraint j ta ellipsoid ball restrict unlabele label unlabele constraint necessarily quite yet nonetheless verify c upper bind energy x mahalanobis act estimate follow constraint variation smoothly encourage example neighbor predict difference operator ill square matrix like also encode label constraint diagonal ij e twice label node encourage
cite role one compute another penalty regardless apply call wide strategy describe strategy make order qr block zero take case skip cover triangular wide strategy minimizer qr decomposition minimizer change boundary empty square old initially differ column qr square detect begin compute qr special permutation orthogonal rotation special qr refer qr exploit qr boundary set differ qr appropriate operation operation meanwhile naive simply encounter magnitude quite favorable operation operation drastically near strategy naive naive primary naive one summary total qr give detail latter implementation complexity filter derivative operator define trend produce piecewise fit favorable argue trend quickly via sophisticated trend filtering actually key row make boundary tr problem two system polynomial implementation time coordinate list cholesky practical I least qr practice though yield necessarily importantly qr operate k reason preferable qr package use qr cholesky qr utilize maintain iteration successful sort efficiency
guess permutation time text create together implementation create gram retrieve contribution letter long v multiply calculate letter new letter complexity letter roughly language project plane implement create text corpora sentence language easily vector
choose risk risk arm good goal identify measure correspond pseudo regret bandit directly correspond stochastic try empirical similar even switch good arm undesirable usual rx focus respectively kf rich lot stochastic armed problem risk measure
langevin h drift satisfy discretization proof c position establish constant produce control two langevin langevin discretize choose precise assessment vanish step go thus goal step get lead reasonable trade computational error complement lemma satisfy second langevin diffusion path formulae achieve initial draw view convexity divergence get h function point total variation triangle hand call error time hand side error tv formula desire result satisfy level time output step th addition h mt applicability claim
codebook iterate start classifier rf codebook training decision bag codebook learn quantization codebook decision construct replacement specific node patch respectively recursively right compare gain class serve codebook learn element number image accordingly specific likelihood soft independently patch object class region background patch label patch produce label conventional cause codebook assign soft estimate model
chain step unbiased sample cd perform chain visible alternatively calculation persistent cd assume efficiently mrf produce unbiased learn perturb energy perturbation perturb
perform introduce logic adaboost solve propose operation respectively demonstrate layer greatly improvement significant case traditional dataset vision though decision usage algorithm complexity variety vision support nsf nsf efficient key problem cart operation feature weak classifier combine implement dataset repository convenience tree boost dataset thousand million cart remain mostly logic incorporate algorithm
observe aa one ii variable weak accumulation x n tight apply weakly bound every belong rejection nothing assume would continuity interior rejection continuous part proposition remain element otherwise b b clearly construction symmetric definite nonnegative square root c choice continuity remain invariance assumption yy belong conjunction precede equivalent cumulative distribution evaluate lemma turn equivalent impossible kb first assumption claim turn directly dt assumption claim obvious remain claim obviously invariant w additional condition rejection complement second claim bt obvious otherwise imply kb kb kb absolutely satisfy c iii equivalent obvious relation iv symmetric orthogonal complete definite must nd dd du immediately last xx xu z l spherical symmetry clearly prove proof analogous represent suppose inspection arrive would inspection equivalently furthermore multiply q I dimensional equal prove claim satisfie radius conclude arbitrary every obtain since coincide eigenvalue make eigenvalue q convergent necessarily equal inverse eigenvalue view invertible showing see since w maintain establish since I prop bind phenomenon cite intuition main serious part theorem incorrect proof part particular concentration effect present concentrate already observe somewhat development distributional include weak well allow absolutely theory invariance test convert precise advantage weak argument avoid tool class test treat iii much distributional theory build test autocorrelation characterization situation invariant correlation help phenomenon organize specialized test appropriate test boundary belong complement closure interior rejection region constitute proof
name label language label optimize objective simple optimize objective embedding separately alignment consist project embedding word onto embedding obtain approach learn extend target embedding canonical correlation conceptual relationship word accurate pair word ignore sense natural language two align representation align due train expensive apply objective everything jointly tp approach al formulation train less train resemble feature disadvantage slow stem objective
article hour run svm rbf svm convex l train test train cifar image choose category cifar cifar versus original pixel per hill easily less train train test cifar bit train cifar bit k c train cifar bit remark novel powerful propose circuit classifier enable efficient operation extremely obtain framework compare conventional circuit circuit present circuit vector note gate bits output gate gate look bit boolean
decide hidden layer neural adjust since major safe range ever bad dropout separate number net hide neural net allow hidden net hide net unit except constrain delay fraction iteration anneal parameterization though program round initial minibatch allow neural net hide task neural net select type annealing mode discrete learn straight annealing start final stop anneal anneal ensure stop anneal final believe precise long momentum weight cost except single net two hide unit either unit force deviation initial use weight subsequent control natural bottom adjustment scale weight bayesian optimization large matrix notion
becomes find lead remarkable automatic configuration cdf limitation optimisation base tree understand determine compare recursive feature threshold reduce uncertainty measure mean average could gps find uncertainty follow reduction uncertainty objective cart splitting convenient child gap gp cover unknown unknown variance place exactly
final rank propose algorithm recursive principal subspace reach much requirement algorithm distribute numerical simulation system zero rank signal generate ki ki ki ki ki ki ki variance
divide possibility combine idea enkf variant inspire current status assimilation thus point conduct enkf couple assimilation two extension trade efficiency consider assimilation aspect physics biology flow name assimilation couple state treat dynamical conventional assimilation enkf ensemble kalman update conventional consist mean I forecast observation construct q similarly decompose ti decompose joint filter step kalman construct lead center matrix element reader center formula front dimension one obtain assimilation divide express root kalman respect divide mathematically counterpart formulae divide estimation give formulae formulae framework diag h therefore transform leading eigenvalue correspond eigenvector square root formulae centering previously discuss accordingly I n system background
form noun roughly constrain syntactic require simply subject rather fine needed date expression include address ordinal proper name construction addition usual linguistic indicate clause comparable learn strict connect type considerably english fairly rigorously answer several linkage definition type apply corpus parse sentence broad enable parse word pressure apply much thing lexical entry word noun leave least hundred lexical learn lexical entry common lexical grouping word lexical splitting differently song group together lexical apart lexical observed sentence complexity distinction purely syntactic relation content syntactic applicability fine scan corpus lexical mistake place belong likewise merely subject pressure fine appear syntactic extent place heavily input corpus contain suggest book perhaps semantic appear generalize relationship semantic semantic subgraphs subgraph may syntactic order need phrase syntactic level different subgraph semantic subgraph may syntactic semantic relationship category early stage word become many x entity physical person physical category understand set include set phrase learn expression whose challenge complex construction simple content phrase ground none usually manually construct dictionary contain lexical construction essence construction treat single already accomplish automate phrase lin existence syntactic dependency rather author attempt rather pre existence attack sense markov take abstract distinct term markovian express internal category theory distribute entropy lagrange I np function maxima hill evenly algorithm slow commonly successful network assign naive bayes fast naive immediately independence word independently nearly impossible english viewpoint really count neither entirely satisfactory keeping fundamental outline relation relation view form constraint
differentiable smooth causal calculus probability calculate observational causal effect form observational reliably create challenge imputation nonparametric transform complete estimate demonstrate nonlinear effect conclusion give straightforward parametric causal model unknown calculate turn implement causal could utilize nonparametric causal simple structural causal operator intervention calculus
equation together equation x give result orthonormal unchanged calculation u v v convert orthonormal column thresholde ordinary solution solution result generalize design design select remove
metric large paradigm boolean task take form qualitative difficulty order observe human value predict human literature date b paradigm iii iv vi application task depict definition subsection discuss mention predict general aggregate indicate relationship type calculate classification aggregate indicate outcome participant task look rule neutral individual aggregate reflect type may subject focus fix focus fix learn agent focus rule heavily suggest like vary also display level neither alone capture general order sum quantitative fit metric paradigm block complexity metric find human ccc ccc ccc
ab g g pe business feasibility aim primal dual classifier understand condition determine difficulty aid establish generalization theorem use end von relevant deep margin convergence apart vast topic represent concerned zero specifically exist generalization theorem affine applicable one characterize feasibility problem inequality bind leave right side later theorem affine margin explicitly familiar quantity sec argue subtle especially margin
specialized choice significantly feature identify body something take computer pose prominent influential recovery plan similar pose body mesh inf mh ground mesh term mesh person retrieve visualize deviation edge higher beneficial experimental help viewpoint record mesh sampling predict body measurement application observation distribution height fortunately original training corpus per relate parameter many regression regularize regression good posterior show figure measurement dash line correspond value recover corresponding ground advantage generative ability miss perform depth code parametrization non account inf mh compute show reconstruction expect work propose incorporate discriminative enable diverse computer analyse behaviour baseline inform perform applicable frequently main inference many tailor
central tool process functional h first show l r standard statistic mark intensity inference framework cd continuous r pd pd locate locate outside additionally absolutely assume negative measurable l provide explicit construct section section spatio light univariate property mark stationary uniquely turn distribution mark impose eq independent mark marginal g e f satisfie mark independent retrieve independent weak obtain eq f measure p ff functional mark choice looking mark wiener latter idea indicate measure see martingale see poisson marks setup fairly may filter construction pre l connection reference measure create mark since diffusion setup choose wiener reference brownian motion wiener assign sample brownian q ask adequate obtain explicit density n conditionally diffusion process explicit expression f fm ic c apply apply underlying extension discuss l sometimes mathematically explicit purpose process auxiliary mark l fm b distribution although analogous conditionally thereby mark absolutely case may functional mark markov g mark mark filter ti nm pm ia pm density become turn reference locally appropriately probabilitie
user majority friend friend denote respectively denote stand alignment conduct experiment base rating accuracy user class comprehensive recall map trust trust value e list friend option friend ndcg friend alignment user relation rating user process similar increase rating explicit neighbor compare table align ndcg type explanation application example may specific inaccurate information implicit trust distinguish utilized conjecture metric side distinguish friend lead alignment implicit consistency social relation align opinion trust relation user trust conduct randomly social relation social friend majority social people justify correlation set include review product user user private alignment friend accuracy mae l c mae rmse rmse mae mae rmse mae rmse mae rmse investigate social recommendation experimentally incorporate enhance trust recommendation different pure factorization factorization matrix factorization propose dimension early amount create four different increasingly evaluate mae rmse mae rmse four sample set performance surprising algorithm exploit pure factorization algorithm case indicate incorporate recommendation indicate relationship social rich source recommendation need incorporate carefully nature totally huge relation relation remarkable likely influence friend make trust private nature quality trust deep investigation verify question investigation art end subsection rating second utilize neighborhood crucial make propagation direct neighbor consider propagation trust propagation long propagation level propagation far away user affect recommendation significantly result trust propagation trust neighborhood sense perform propagation constitute user user less neighbor recommendation I
classify exploit crucial specifically since code pls guide learn much figure consistently consider always na initially na behavior almost class visual dimensionality lower exploit lda simply apply pca localization domain adaptation target present split extract sa split art classification setting present result classification theoretical david integrate subspace target subspace optimize extensive experimental principled combination evaluation domain adaptation part work demonstrate convolutional
moment match show gain gain category concrete likely ten em model corresponding differ replace support wireless color video establish em runtime em discuss practice moderate average explore parametric novel algorithm recommendation task naive modeling item similarly advantage acknowledgment work support part grant give give briefly nature condition formally dpp assign regular dpp dpp constant dpp elementary compute term marginalization unconstraine hold conditional recalling express
lie choose auxiliary range payoff round sublinear worst need exist polynomially weight front instance exponentially tune advance block increase length version fix block regret action response discrepancy vector payoff block round payoff component propose length denote let bound range difference remark proof sequence payoff possibly denote integer payoff last block comment work original recent form aim quite demand calibration strategy grouping finitely payoff quantity involve round minimize consist represent loss cumulative expansion control impossible example control block prove ensure hold terminology computationally strategy indeed existence axiom choice constructive explain reference know translate case straightforward noting try quantity towards
rating well alphabet whereas multinomial insensitive summarize new penalize provide estimation bit completion mild value formulation directly grateful en l des big support discussion lemma control consequently control distinguish need leibl point control e I score diagonal eq
eqn proportional q likelihood last eqn discuss eqn model factorize attribute attribute imply probability pl product probability correlation detection auxiliary eqn variance th corresponding respect dimension reject highly product error difference measure decrease avoid eqn eqn obtain eqn contain six simplicity remove regularization simply ignore minimized parameter although jointly term highly nonlinear respect cnn stochastic search demonstrate reasonably decay filter second eqn regard negative logarithm third dynamic update fix current value last thus write loss decay term decay eqn combine least loss
trace norm give large iid mp b express complexity bound theorem exist sphere even likely margin order incur lipschitz eigenvalue lipschitz ambient logarithm potentially curse grant ep international high kernel width mail replace paragraph please hour height ne b skip h ne
variance pp assume development year year year effect satisfy follow estimation quantile formulation procedure model predictive distribution density specification section likelihood present along associate formulate need present al gb structure pp adopt relatively uninformative reflect magnitude instance skewness regression select shape gamma distribution discussion choice al gb prior combine result likelihood model ensure proper set constraint parameter restriction derivation mcmc posterior metropolis hasting rejection parameter value fail slowly mcmc replace allow simple design tune mcmc significant gb intractable et metropolis hasting popular mcmc technique reader suggest mcmc implement available request iteration discard initial burn convergence also carefully check autocorrelation plot posterior predict cell involve quantification predict mean al alternatively central measure adjustment base calculate require parameter integrate triangle include mode predictive interested loss random distribution convolution state feature sum loss low note low tail vary long precise value regular
hypothesis survival length tree significance survival fail detail test test permutation importantly change existence heterogeneity goodness effort article appropriately inferential structured data variant distributional connection develop asymptotic statistical one functional popular albeit see reference therein detail functional know functional tree appear promise weak convergence paradigm wherein ground development worth availability combinatorial tree prevent idea validity result simulation conduct contain branch length partly wherein branch hierarchical approximated normal partial sum lead current binary hierarchical ultrametric use connection exchangeable develop fit informally ultrametric arise sense leave tree choose vertex step tool euclidean space dependence tree structure rarely compare straightforward account attempt model approach use representation path explore tree regression tree structure branch length model investigate brain couple use detect record represent rooted tree secondary structure rna sequence order trees rna describe suitable flexibility method depend choice develop inferential straightforward determining interest support simulation tree dataset contrast probability tree consider wherein topological branch
report report status yes phone status yes mobile phone status yes car car yes status report yes cart report cart yes status report report report status yes material main nominal respectively asset status survey data difference cluster scientific interest aim appropriately asset care project survey economic valuable input serve study structure present mixed idea mixed modeling employ joint mix categorical categorical recently miss latent factor context analytic base gaussian copula cluster early variable none capability model binary ordinal present factor model nominal establish root expansion include extension partial approach model latent lie interval model trait response binary response multinomial probit fitting response treat nominal multidimensional continuous depend covariate item response variable advantageous
constraint namely negligible pyramid train train dnn directly second layer layer wavelet modulus haar dnn unit leave chance coefficient linearity report neural create alternative impose filter architecture relatively approximated softmax mse reconstruct evaluation discriminative setting gender generic per gender include adopt division contain testing compare mixed sir bss metric symmetric discriminative nmf frame overlap feature
explain interval recommend interval bootstrap error percentile interval natural derivation asymptotic interval formula come different sample time ultimately indeed say effectiveness also report side picture biased confidence interval I side test says call short interval trade coverage reduce length statistical interval intervals confidence side cover cover test confidence interval side sided nominal test percentile error skewness adjust statement regularity statistic e moment estimate detail many bootstrap interval early day develop accurate accurate handle skewness transformation sample accurate little thing thing effect percentile poor order order accurate rejection probability differ procedure skewness behave circumstance vertical top bootstrap distribution side top bottom interval bias vertical line normally sample middle correct middle range include truncate center bootstrap percentile interval coverage include right bootstrap scale interval happen bias thing simple center statistic bootstrap percentile coincide exactly side acceleration vertical line truncate middle distribution normal bias coverage text wrong unbiased skewness worse leave simple bias r positively bootstrap biased symmetric end copy bias enough percentile bad copy correct percentile interval would percentile side happen middle percentile bold error avoid miss bootstrap percentile even interval bad skewness percentile skewed skew percentile toward counter intuition intuition may much confidence must especially interval reach right conversely average error parameter roughly normal bias give bootstrap bias bootstrap percentile correct interval whether leave correct whether cause bias skewness second accurate bootstrap information implicitly percentile skewness interval asymmetric cause transformation confidence differ interval absence quick early percentile interval use
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replace side leave hand per repeat scalar summation lead removed compare remove vi state use vi sum upper decrease converge positive assume analysis trajectory enforce stage side become moreover analysis present regard I x generate process implement optimal control cost system establish approach perfect e function play derive result use nonlinear presence problematic phenomenon complete analyze vi open publish include good cost solely valid computer science matter discount infinite analysis easily compare example approximation denote possible write boundedness include value function besides stability rigorous consequence error great practitioner reason great tool control practice vi online control pi start
trial activitie activity etc show ds length ap deal dissimilarity compute similarity nearest ap sc relationships spectral graph partitioning time run make follow conclusion come trajectory choosing point try dissimilarity model challenge sc ap yet still come effective obtain due different target result representative datum ds dissimilarity dissimilarity class ds length model trial change value local activity contain good underlying activity pairwise dissimilarity source consider efficiently encode set propose row problem solve grouping find reveal set parameter representative implementation admm hence reduce two categorization representative model motion representative model currently statistic theoretical theoretical make prove order need notice
attribute perfect attribute classifier score add km km draw mean keep follow noise specific level plot axis test pick market chemical art school yu share conjecture principle shoot possible specify category generic attribute classifier category property possess even provide standard zero shot suffer novel image random forest explicitly account attribute obtain robust discriminative unseen class devise extension shoot bias mid attribute develop signature category operate characteristic idea association shoot number demonstrate three show clear advantage suggest valuable attribute play object inherent reliably shoot two stage approach give novel attribute predict attribute unseen object
require depend version bound prove statement uniformity tight bind finding reduce attempt tight factor regime bound instead interpret regime match strong statement prove associate game parameterize set support random I correctly otherwise sample failure convert oracle run algorithm obtain member game occur output correct answer triangle inequality fashion construct sphere pack code constructive merely win game intuitively probability win look like imply constant win p nn nn mention examine another al equality distribution distance uniformity metric extend utilize uniformity hold translate coordinate sample exactly number focus small prefer tell require learn versus uniformity use author knowledge order regime open uniformity chi arise closeness establish question may seem author match come biased coin distinguish coin bound perhaps give result convert
spline locally allow suggest fitting bandwidth insufficient smoothness fitting occur measure degree select smoothing monte explain sec thus stable datum heavy tailed mass however cause curvature heavy tailed worth em realization shape I exponential combination realization event simulation algorithm exploit branching process formulation estimate use em choose bad start procedure parameter uniform random branching ratio choose branching ratio fix point expect process bad another introduce overlap clear branching systematic study mle em overlap decrease synthetic realization c address process simulate nan model generate aic ratio level ks discuss option test type test alternative hypothesis
pca share filter part find violate half run bottleneck grow template contain candidate location bottleneck qp line qp solver ratio bottleneck second method discover seed train seed candidate top part well despite use heuristic slow comment correspondence long independent multiple experimental well pool part note take extract apply transformation see definition optimization visual group visual school science university uk representations useful base view collection informative discover use response part randomly select good subset namely intend correlated part discover look correlation computer make reason part invertible nuisance factor generation include break visible reason part variant object scene often object cat faces discover contribution unified framework
correction field isotropic task fmri analyze first tool temporal cutoff number algorithm reader deep area test statistic conclusion robust arbitrarily strong statistic fdr use differ conceptually regression recently propose paper relate local fdr smoothing rather covariate fdr differ fdr ordinary smoothing differ ordinary build testing assume statistic mix describe report appeal offer may discovery discovery recover know estimate parameter important call either control adapt unknown involve conceptually simple use spatially adaptive discovery defer fmri encode three change site site odd log odd let assume odd regularizer impose penalize define odd lasso differ mathematically differ conceptually size orient adjacency th
wang li al among establish depend two objective full section residual sum describe screen rule minimize separation exist p strongly separate large whereas large fairly separation imply screen n borel denote assumption hold separate n n nx n h nb nb nx nb nb h h ax complete assumption note restriction satisfy hold asymptotic literature de conduct simulation verify fitting well result multivariate whose vary angle fan independence screen sis well produce sis screening sis cr c sis solution optimization solution whose objective simple framework obvious effective theorem follow way separation basic include subsample robust statistic besides new subsample selection distance method well
state reversible jump crucial well want accommodate mixed support circular circular measurement circular modelling mix circular visit propose problematic particular known identifiability hyper addition mixture hyper density difficult introduce rely project normal distribution study behaviour association recover basis hide wind period km organize review discuss specification circular project provide large application real conclude circular direction circular challenge meaningful directional many circular book overview onto circle
express filter mechanism might odd goal also attractive algorithmic synthesis henceforth refer np ignore large penalty work mean lar package seek minimize approximate approximation penalty activation exponentially distribute mostly omp also problem simple feedforward hide pick encoding must update also nonnegative
propose serve tune double distribution linear updating equivalently aggregation add additional acceptance set propose calculate ratio corollary proposition remark section ensemble outperform linear approach motivated regression list adapt receive open truth convex linear tends concentrate good truth illustrate simulation dirichlet aggregation learn minimax misspecification many pick suitable model set problematic hence substantial practical aggregate combine obtain aggregated potentially single one towards attention search convex combination focus select combination aggregation function iid aggregation randomly primary interest adopt fix translate context basis orthonormal overcomplete weak learner special map average place updating
property contraction von distribution error typically derive robust misspecification slightly slow contraction assumption research lead european grant support grant bs e full high continuous compatibility contract sparse study credible quantification class regression possibly close selection coefficient priori prior usual reality select zero product coordinate crucial one finding weight decrease exponential performance theoretical paper identity see model share case must take account oppose factorization axis entropy refine prior laplace distributional insight nonzero coordinate contraction spike example routine dimension dimension result computation number develop cope consider recent various feasible value hundred thousand increase come clearly short dimension programming approach cope truly model present time surprisingly overcome see sharp
flip tf fs flip tf fs flip tf flip tf fs flip tf fs flip tf fs flip tf fs flip fs flip tf flip fs flip tf fs flip tf fs tf fs tf fs tf fs tf fs tf fs tf fs tf fs conv conv filter convolutional geometrically filter reconstruct output method sect horizontal flip flip rotation representation abstract mapping sect one look demonstrate representation layer manner fig transformation learn empirically sect importantly transformation sect new able property allow invariance build equivalence look whether information million optimisation may question whether apparent network equivalence obtain cnns sect sect
noise rbm term equivalent benefit bernoulli rbm benefit rbm positivity condition rbm put benefit condition benefit predict improvement deep learn careful contain cell need dna cell rna outside protein synthesis occur code gene cell gene cell adapt receive expression code dna control turn factor gene bind site locate basic bind specific complicated bind site fuzzy inexact give experimentally verify admissible variation bind specific single bind area miss exact dna model dna sequences discrete take learn sequence explicit em length mm generalization success human genome bind method success version supervise method strongly principled noise exponential em likelihood log reduce simplifie follow condition benefit replace art year idea emission advance advance gibbs hybrid frequency detail generalization accommodate maximization figure mr image segmentation mr brain voxel voxel grey matter effort automate difference water mr mr mr water unlike medical modality water mr within distribution current water segmentation post process automate annotation identify characteristic accurate water assess risk diabetes disease relate segmentation intensity mixture gmm pixel classification spatially aware hmms localize localize traditional algorithm causal involve show image segmentation also prior show apply mr image expensive speed apply noisy fast show improve classification examine fuzzy bayesian problem conjugacy rule rule bayesian idea inference convergence sampler pt chapter chapter school california partial requirement electrical grateful development work much growth year thank feedback closely grateful many friend support I like thank teacher united states control cause converge basic scheme neural fuzzy general derive theorem consist short theorem chapter broad em em variation algorithms review discuss expectation intuition derive give noise speed chapter derivation end enhance beneficial sparse artificial heavily mean algorithms backpropagation proof theorem bayesian chapter approach technique approach imply apply analyze effect approximation approximation bayesian uniform fuzzy approximation validate use fuzzy inference via chapter discuss extend show noise sample substantially maximization iterative corrupted speed benefit present derivation sufficient benefit demonstrate speed processing include backpropagation feedforward artificial analysis corruption general framework main uniform via multidimensional expectation maximization algorithm discussion chapter schema converge slowly dimensional incomplete problem expectation maximization chapter describe simple corollary cluster model benefit indeed theorem secondary expand function pdfs bayesian closed function function uniform lead maximization em lead map continuous property exactly replace equality define recursion solve condition general generate let closed convergent stop sequence infinite convergent subsequence monotone converge subsequence subsequence original advance derive convergent subsequence closure assumption contradict compact sequence apply interior current estimate element eq large compact generalization parameter general satisfying condition condition imply iterate point map map theory nash equilibria fix point give map apply always close map close incomplete base exponential pdfs application censor gamma condition partial derivative close map may restrict maxima instead non singleton flat sequence kind kind easy detect wise estimate sequence information carry approximately measure show information prefer measure standard fisher quantify complete complete datum miss datum observe expect fisher decomposition principle parameter estimate al em current em ratio neighborhood high slow number variation step forward variant root hard another multidimensional replace single conditional may perform multidimensional idea iteratively medical approach rate proportional amount embed iterative em may iteration e reconstruction space every pixel voxel liu liu
appearance correspond exchangeable log upper outer leave part show negative part show region exchangeable ce crucial exchange provide upper provide improve span optimize appearance exchangeable consistency constraint probabilistic work area demonstrate possibility highly connect arise frequently program language
forward propagation correspondingly problematic computation output output prohibitive since neural network output vocabulary become increasingly propose attempt address difficulty fall fall category impose define initial kind present actually gradient implicitly computationally manner actually compute
encode string report thing token pattern sort search store carry check contain dictionary extract match object size dictionary number pattern distance dictionary extract intersection dictionary concatenation drawback limited table totally token type constitute text contain plain english mean additional
expert importantly quantity expectation randomness signal conclude rule explicit form underlie intuition omit provide investigate exchange time reach connectivity play w knowledge markov invoke technical give connectivity w establish convergence rate govern characteristic generate true assumption strong agent lemma converge delta state let borel interested let proceed cost function round numerous expert adversarial belief bound identifiability strong connectivity q explicitly calculate apply primal dual bind centralized identity take right
minimization associated number completeness admissible triangle use observation hand find put estimate conclude subgaussian define assertion subspace subgaussian set recall main wish recover represent measurement terminology subgaussian probability measurement projective quantitative statement theorem entry assume dimension respect note statement subgaussian idea expect argument base approach riemannian induced notation terminology tangent associated tangent piecewise curve geodesic subgaussian map sufficient preserve
lt age shift lt college college sometimes le le sometimes heart abuse abuse degree include mml htbp mml htbp lift none size narrow lift lift class lift lift none rule lift color lift size narrow none none color color surface narrow narrow surface eq narrow eq leave error mml use fold cv experiment train long minute ip mn rule oppose ip thus mn rule involve compare result long classifier path plot mn train accuracy tailor tool diagnosis lastly numerical accurate interpretable many use ill literature interpretability particular predictive many relate comprehensive review popular assess interpretability distinguish visual popular linear address work table interpretability improve sparsity ensure monotonic relationship predict restrict popular neural support machine ensemble interpretability box mainly auxiliary extract prototype relationship predict useful practitioner accuracy interpretability practice interpretable require control multiple require address control multiple interpretability relate practitioner model monotonic exist way measure surrogate interpretability measure heuristic procedure round poor interpretability robust match non restrictive condition issue aside poor interpretability extensive adjust available parameter claim understand method design interpretable market produce interpretable domain study often predictive interpretable domain produce use highlight agreement outcome hinge discrete round ip classification discrete practitioner flexibility previously lee many train classification minimize early attempt minimization improve modify heuristic design specialize recent feature framework linear accuracy addition reproduce interpretable literature often accuracy pt control interpretability mention ability incorporate need mention address database address
discuss syntactic condition multilinear decomposable validity condition theorem degenerate strongly decomposable complete multilinear transformed degeneracy hypothese validity condition easy verify c section restrictive condition limit expressive ask efficiently criterion weak question prove provide extend np hardness property checking goal advance expressive ultimately focus well circuit efficiently logic circuit overhead polynomial easily constrain monotone arithmetic input circuit obviously simulation boolean logic circuit would monotone arithmetic circuit impossible circuit negative value monotone seem much insight general multilinear multilinear circuit moreover make arguably theoretical place conventional deep meanwhile ensure validity aware also condition validity seem arithmetic circuit constant kind structure kind perform something compute become impossible kind efficiently despite apparent efficiently example prevent basic operation work design formulae couple computational show simulate system close like vector multiply whole product give formally fix permutation constant input maintain crucially process limitation read function dimensional distribute possess limitation lie representation linear transformation positive entry simulate compute fact suppose one affine special multilinear branching polynomial require exponentially function computable polynomially section circuit decomposable univariate proposition convert decomposable size provide indeed polynomial require exponential prove fully characterize capability compute efficiently existence similarly even efficiently simulate work initialize state process fix stage transform real permutation sense state decode map g machine state size subject restriction input ahead allow large enough see compute would grow exponentially bit number bit combine ability combinatorial high capacity state establish size dimension proposition state unlike readily easy construct wish
assume shift goal x target domain indicate come source target example training time want feed predict decompose map vector feed label mapping finally map domain aim minimize annotated part optimize minimize ensure overall want want xt covariate shift assumption make prediction domain source measure dissimilarity change dissimilarity classifier provide train discriminate feature idea domain feature seek possible minimize label predictor multinomial domain evaluate saddle minus predictor prediction maximize
contrast behave fix difficult create incoming new overcome development systematic learner structure ensemble naive learner drastically ensemble component desire highlight increase sophisticated procedure construction practice subset allow acceptable accuracy generalization context sophisticated dimensionality play crucial role uniformly
rate quantity suppose estimator rate dimension demonstrate mix objective gaussian might hypothesis like attain bias seem population shall biased examine power alternative hypothesis increase might suggest power conversely power let emphasis gaussian dimension mean origin dimension need decide suggestion possibly unclear choice verify communication slowly affect author former power fig make justification
shorter noiseless fig determine unstable take prediction also instability instability sense analysis positive compressed sparsity show demonstrate gaussian matrix predict recovery dimension theorem conjecture theorem theorem question remarkably decade signal small synthesis measurement logarithmic factor signal signal use
column range true scenario see example value fit change interval aic tend change sublinear optimal
disjoint separate sufficient conditional practically check hard hand read kernel induce graphical important understanding thm thm corollary elegant graphical proposition
decade mention deep chain converge chain numerical correlation chain need posterior independently start collect chain take chain parallelization approach parallelization exploit example come target execute decomposition remain markovian technology generate single draw method gradient tool component function posterior model gibbs sampler implement small require strategy conditionally conjugate might possible augmentation transformation method action conditionally sale example hamiltonian hmc software posterior benchmark complicate assess method parameter think estimate reasonable illustrate inherent mcmc package example heterogeneous unit deviation place conditionally gibbs describe confidence indeed iteration get need proposal mac minute draw collect take minute collect sample processing median ten show population unnormalize tb example sale sale store week shape price volume week covariance inverse wishart conditionally
algorithm far variational similar nonparametric modeling variety nonparametric principled nontrivial literature use sampler sampling suffer poor scalability involve beta successful standard derive atomic random measure stick break yield construction integrate mean inference instance stick break seminal inference dirichlet stick break naturally lead model variational framework stick breaking process derivation measure
illumination illumination illumination cluster cluster matching compare balance hull near propose affine hull adapt show local distant cluster restrict middle variation enforce query constrain convex reduce convex treat constrain cluster adaptive adaptively distance reference take indicate distance fig conjunction comparison object method adaptively cluster base continue follow briefly affine hull technique describe evaluation comparison research represent affine hull distance near
base code link depend desire back programming facilitate particularly extensive aggregation scheme insufficient extensively broad overview development library mind major lower
mesh motivation voxel analyze voxel intensity neighbourhood instant possess illustrate intensity instant axis correspond separate cognitive discriminate voxel classify voxel slight squared intensity value voxel carry intensity instant vector voxel neighbourhood system high power voxel intensity mesh voxel employ neighbourhood fully activation brain spatially distant neuron neighbourhood model euclidean cognitive additionally spatially voxel strongly couple cognitive defining may redundant mesh arc improvement accomplished usage functional voxel neighbourhood feature vector whose consecutive connectivity voxel voxel example pair connectivity spatially distant inter analysis connectivity brain node correspond voxel connectivity study connectivity
diagnosis beneficial science voxel approach development may enhance interpretability even identify voxel biological etc classify human inspection motivated classifier attribute similarity superiority technique strongly suggest capture subject class dark knowledge within classifier study classifier effort make visualize
preference evident supplementary quasi restrict quasi ultrametric state include partition quasi dendrogram represent resolution singleton method formation asymmetric fig resolution form singleton cluster influence highly asymmetric resolution california every depict leave california explain fact california large area country california influence partial resolution permit california force minimum height leave thick bend leave thick bend bend edge bend bend edge bend bend thick bend right bend bend bend right importance resolution california important ordering form three precede happen already hierarchical quasi application asymmetric undesirable generalization asymmetric formalize notion introduce axiom framework rise new admissible axiom furthermore equivalence quasi invariance property application support nsf fa dms nsf dms map ultrametric dendrogram dendrogram quasi partition partition right continuity quasi ultrametric attain non attain negative clear identity imply boundary since conversely identity strong triangle inequality node ensures equivalence hierarchy property x partition definition eq q definition substitute consequently satisfy triangle ultrametric define converse result show
branch sdp cut plane evaluate experimentally suppose mrf characterize undirected node index unary potential mrf unary without generality lp polytope space q definition singleton node globally joint vertex extreme correspond problem difficulty optimize polytope outer l example vertex cycle fractional lie outside relaxation guarantee lp relaxation map cluster consistency constraint ed consider symmetric semidefinite discard rank convex outer generally boundarie polytope semi outer mutually neither dominate marginal polytope semidefinite outer work solve result general lp integer problem round lp sdp feasible round sdp relaxation base singleton pseudo would done label cut hyperplane round sdp relaxation expect objective paper simple rounding solver overall b subproblem branching present briefly optimum feasible relaxation feasible separate branch proposition branch global branch terminate lower sufficiently show selection branching branching subproblem selection strategy find investigate rule branching selection rely improve initialization bound queue subproblem
observe build fraction direction derivative direction effectiveness sampling manifold ref ml approximate quantum dft highly self energy principle dft test gaussian low cauchy cross test prediction hyperparameter generalization set construction grid calculation far explain origin search euler effectively project introduce pca constrain project work prototype yield highly constrain energy thank nsf kb correspondence li li rgb rgb rgb li li non interact functional explore highly energy accurate modify euler lagrange project derive effort research review dft functional reader dft typically fall derive principle tend work error treat certain semi
gene biological provide good usefulness rank set patient ac uk patient moderately severe patient severe measure intensity value indicator eventually protein serve discriminative outcome estimator though could favor framework filter step yield store patient iv demonstrate rank predictor version estimator fold lasso predictor predictor response average iterative sis
affect noisy happen far journal sharp observe topic journal decrease star system cell much vocabulary document document proportional edge likelihood document assignment fact use data compute mean never scale much see fig dataset example corpus computer minute minute english wikipedia decide link unique less million result decide total graph word roughly hour extremely take hour filter iteration take hour iteration run word interestingly change guess significantly lda synthetic likelihood explain topic merge small variation inference function topic heavily broad use fair comparison indeed high see degeneracy landscape provide well web topic area perform well lda splitting cell american economic comparable comparison likelihood model english sec english first group equivalent english fall binomial ingredient fit go back call english average q hold q word fall english entropy variable
spc performance scenario perform slightly tight initial categorization without include cluster clearly demonstrate sensitive pre specification noise recognize cluster group mean mean recognize resample show ari score well leave clustered might challenge become order reasonably correspond neighbor mention qualitative comparison lead previous demonstrate converged figure case take iteration solution table run fast spc tight calculate length include tight mean search prediction resample spc currently implement cluster write r spc good especially ht ccccc separate separate overlap noise performance spherical dimension group normal high correlation around one show true gene expression spc relatively size big green plot spc perform recognize small assign end normality data spc compete apply cluster dimension convexity ari generate present spc high score perform cluster usefulness spc high spherical comparison non comment path tuning parameter could
solution bx bx bx nonconvex approach consider nice dc decompositions dc program original sequence sx sides ii assume kk kx contradiction subsequence proving theorem replace zero approximation optimal tight approximate contain solve several context suppose concave define bound z carry resp concave bound resp general carry much result nonconvex motivate local minimizer describe problem I local optimum kx x implication obvious backward exist neighbourhood case case fx problem state equivalently solution problem dc dc program necessary local condition express equivalent fx able set eq q hx g gx fx ki b ik b kx k k prove mention approximations first concave context smoothly logarithmic apply concave increase verify particular dc general dc investigate
illustrate evaluate completion quantitative evaluate firstly incomplete complete error rate method subspace cluster sim ssc handle h preprocesse fail motion object produce miss error sim ssc segmentation miss possibly exist trajectory entry cluster improve dramatically advanced example verify realistic sim sim sim max std min std ssc algorithm ssc std pointed could call
deep feasible severe lack training medical imaging cnns generally number use image patch purpose effective amount apply classification improve view filter name max layer layer final layer recently publish call early dropout allow open source gpu
constructive repository proof algebra necessary clear distinction theorem proper dependency ultimately explain learn well need dependency among prediction theory corpus discuss future dependency extract development describe overview base formula encode logical essential bind environment
variety galaxy shape human shape code allow dimensional space shape statistical reduce
description involve additive predictive addition additive heavily penalize interpretability version accuracy predict divide small outperform interpretability construction great outperform mkl despite poorly dataset explain show outlier square linear regression occur centre supplementary material divide result supplementary material open detailed describe capture demonstrate discover describe pattern interpolation building technique non thank helpful part google code experiment scalar input
negligible simulation gaussian probability p denote gaussian terminal linearly joint notice depend value correlation expression mmse source noise denoise message pass correlate bernoulli measurement random see match se successfully recover run signal rate two contour rate distortion source dash line boundary case independent region contrary private obviously independence individual rate enough gradually result
preserve preserve embed canonical cca pls linear suffer input vector may bag similarity instead representation linear handle handle version end solve central choice make automatic combination application multiple initially receive significant attention excellent mkl extensively relatively explore mkl instance ratio optimization impose general lin extend cover ratio trace formulation refer reader trace ratio mkl result optimization iterative guarantee similar mkl lda trace equation lda cca pls optimum
active inactive treat free base early lin start pick index solve sub optimally newton express equivalently define onto hyperplane one convenience represent simplex denote interior simplex problem homogeneous collect elementary work optimum limit dirac peak insight actual regularity description scale consequence application amount operator scalar chain onto projective space line e equivalence projective identify sphere equivalence canonical step end denote distribution restrict union hyperplane superposition least progress rate eventually progress define nz define w tw tw trivial consequence invariance existence lift depend homogeneity homogeneity technical converge exclude cycle homogeneous coefficient progress capture define progress note coordinate everywhere set multiplicative ergodic corollary limit denote norm immediate consequence chain distribution obvious rate coordinate adaptation maximize function dependency problem aim towards cd converge coordinate arbitrarily boundary remain continuity maximizer simplex identification problem distribution goal adaptation could
vocabulary practice numerator normalize instead opt specifically probabilistic root node word associate leave branching path path branch observe bag use full decoder word bag thus decoder share internal decoder compute assignment leave well choose decoder eq
datum hypercube block contain partition block location next subsection convex solve precision matrix calculate distance block form always matrix assume magnitude diagonal attain relatively use multiplier proper close n similarly map efficiently auxiliary equivalently augment dual multipli linear constraint primal admm iterative primal tolerance condition experiment converge acceptable admm iterate guarantee constant minimizer penalty admm admm iterate linearly particular q c opt opt z rp opt return method block fit framework assumption hand selecting obtain
q assume strongly recover q minimized recovery sufficient mn mn definition since
distribute weak learner local base communication cost network graph depict star improved select parent receive child root back node work agree tree far operate neighborhood agree span communication construct simplex atom restrictive since proof design optimization amount communication atom node atom evenly two one instance select atom atom need show minimize f deterministic lead select atom bound precision communication algebraic approximately vector near base orthogonality near length precision near simplicity
break arbitrary form satisfie x lemma use u p v p positive homogeneity I vector increase norm homogeneity trivially convexity lemma combine
comparison passive risk default definition might achievable optimistic sequence regularity probability achieve challenge relatively extra active predictor therefore label predictor equality label budget ki kt list alg operate stage stage
rnn achieve capability especially long help accuracy click prediction unfold rnn algorithm unfold incorporate unfold necessary effect rnn unfold understand accord increase unfold good auc achieve unfolding drop check discover error vanish unfold explain unfold span several lead unfold dependency implicitly accumulate affected event explicit run history background intrinsic click click traditional click temporal recurrent structure art continue several aspect build user user pair even whole go
design set I usually lead excess bound independent distribution free excess parameter latter excess deal hypothesis large large factor close upper excess free
hold random euler gamma scaling sequel valid claim next mean et r proof q asymptotic vary suppose depending imply asymptotic j u
achieve relative improvement htb bt unlabeled bi svm center possible load hold vector gram news corpus word repeat experiment loading sentiment stanford rate grain grain task improve attention retrieval task length return engine content web page web page query test triplet result query three paragraph distance representation close paragraph paragraph paragraph call call report american paragraph call share information paragraph health patient pay pay triplet split
mf three continue rate test three layer usually quite usually gradient essentially sum layer try momentum lot mrfs edge energy sum potential eq crf make explicit simplicity apply conditional model mrfs image denoise optical flow field find factorial
use nonlinear transform reach stack reason degradation layer dimensionality layer improve greatly good pose view information view largely cca pt cca pls probe varied fix varied conduct view sketch view utilize benefit fold criterion gradually narrow gap two view cross image superiority method art world view endow capture reveal object image face modality condition pose computer
summarize statistic attribute ks test numerical attributes hellinger univariate define maximal numerical ks test size get picture hellinger discrete percentage nan hypothesis attribute average strict important similarity base rand set overlap u n u uv clustering base counting agree category contingency value indicate rand rand rand take value clustering clustering adjust rand contingency q ari ari ari instance instance datum cluster similarity near cluster get clustering instance union generate clustering base illustrate assign exist partitioning besides output use cluster distance instance attribute dissimilarity nominal attribute dissimilarity attribute cluster ari compute probably good substitute largely dependent classification classification performance indicator original outlier capture comparable generate show
open study examine exploit heavily topological effort precisely general fr metric satisfied verification increasingly geometry complicate various geometry challenge allow pose answer analysis propose remain additional fully establish capability limitation recently brain one tool order manner call leave determinant want ball continue first zero must thus symmetric determine block determinant block matrix entire perturbation preserve still conversely symmetric parametrize satisfy alone cone intersect open component positive zero conversely stay open prove manifold convexity statement graph rank show lie affine none hull point plus determine affine convex hull dimension convex choice since connect let euclidean segment neighborhood exist lie thus corollary proof condition basic manifold corner corner coordinate move top left corner take corner chart chart corner corner convexity affine follow nonzero determinant corner
search direction rapid construct intermediate direction variable general algorithm proximal proximal gradient proximity operator descent proximity proximal rapid involved gradient alg equivalent eq current efficient constructing follow guarantee comparison illustrate difference auxiliary variable rapid auxiliary
imply hadamard margin complexity triangular imply probability max combine select large enough include norm well use lsh fortunately require point generality database inside sphere indeed establish existence lsh refer pair query simple universal lsh need hashing database database two review threshold compare recently suggest hash quality movie parameter value follow pair mapping combine q
crucial step fully resample weight intuitively marginal assign specifically fully accord simulate optimal simulate point equally weight approximate monte carlo approximation primarily normalize partition turn provide estimator unbiased due channel
noting lastly encode direct cycle encode acyclic topological cycle admit topological represent zero encode positive well encode encode ensure cycle cycle encode cycle penalty represent direct graph inconsistent neither feature weight ji positive inner arc represent arcs network head spike arc bit variable fully arc bit hamiltonian row represent bit problem color logical set arc arcs bit total enforce slack bit parent bit hamiltonian indicate bit together hamiltonian encodes dag local term full
experimentally convnet whiten fig accuracy template template believe template improvement weight run convnet beyond gpu management issue similarity template carry weight template similarity template convnet achieve accuracy unweighted similarity hand similar convnet layer super linearly number word accuracy would template reach network template considerably accuracy template despite fact latter overall size worth performance unweighted convnet confirm hypothesis analyze shape decision region unweighte essentially hypothesis unweighted abstraction far defer accuracy problem network account variant eqn good code high absolute art observation whiten architecture whitening template use facilitate fair modify template svm sgd modify template reach high level author turn triangle improve accuracy phenomenon reveal local minima lead successful find report template code reproduce display template template template peak peak template almost fig importance sec template zero mean unit variance accordance patch
model solution tuning solution algorithm base occurrence leave bottom leave co occurrence might introduce employ algorithm specify linkage misclassifie bi f stand mutual standard real uci repository figure consensus four consensus solution contain breast hard optimize solution balance cluster cluster employ p lr lr lr cf c sc sc sc sc sc c c sc c ground b propose consensus propose mutual parameter involve dimensional ambient lie dimensional subspace aim subspace represent matrix case define parameter apply normalize node affinity wise cluster segmentation video object frame video object treat yet group instance propose bi consensus instance value consensus subspace sensitive wrong number analyze result range enable consensus low misclassification feature lr lr b b green computing cluster instance mark dotted line competitive misclassification consensus solution without video project pca base subspace misclassification consensus bi cluster cluster discard miss help understand need bi lr lr one size extract bi bi must row column cluster compare size bi return discard drop
success variable internal node put prior easily proper posterior make smc basic note conditionally value leave normal value propose node internal analytically pass therefore exactly two leave propose weight involve density compute message pass implementation perform qualitative agreement economic indicator five high child school distribution tp internal comparison method method within variance standard intermediate distribution forest post internal implementation implementation hamiltonian algorithm marginalization kalman inference limit baseline experimental setup measure efficiency effective sample ess convergence ess implement std non standard smc measure intel e processor ghz experiment ten begin ess run particle ess std perform mcmc concern ess min deviation inferior explain section sample impose ess smc mcmc investigate indeed challenge remain marginalization normal baseline std slight additional support normalize little k respectively reasonable assumption c outperform computational roughly std accuracy implication particle mcmc light recommend estimator particle deviation close particle case deviation std number experiment distribute environment idea particle description implementation implementation running particle second less single sampler
covariance matrix diagonal prevent issue power exact adjacency probability great property lemma nz I center gaussian take high bi unbalanced matter chernoff choose left degree may graph hence deduce draw let thresholded recover whose regularity bernoulli regular particularly lemma bernoulli q input important impact uncertainty aspect
amount approximately iteration conjugate gradient length present gradient via adjoint depend make recall direction free discretization I particular involve structure inexact newton references newton termination criterion cycle backtrack regularization warm section gradient hessian action cg express adjoint quantity action pt algorithmic scalability result component maintain core ingredient overall scalability exhibit algorithmic number increase indeed outer newton cg mesh hessian operator hessian exhibit mesh cg computation gradient fr lagrangian formulation calculus strong velocity pressure adjoint velocity pressure problem account pointwise adjoint stress fourth identity adjoint notable velocity nonlinear adjoint adjoint velocity pressure characterize linearize tensor act adjoint jacobian newton forward consequence self mesh solver problem newton step forward discretization field velocity pressure pair adjoint pair surface boundary hessian gradient evaluation expression form linearize solve hessian presentation expression defer hessian play role solution performance inversion section scale observational ice use newton slide decrease inexact characterize tie mesh slide velocity treat field make tractable inversion characterize linearize must core underlie objective vector product computation dominate cost remain newton algorithmic performance fine
choose back issue decide second notable go result mix explain filter denoise signal child heart
interested happen experiment experiment leave impose future annotation query match return excellent platform manual fit researcher find phenomenon take relation possible alone term take literature obvious limitation capturing demonstrate capture retrieval imply natural good additionally bring experiment miss feature computing model model include query however approach implicitly capable somewhat contradict retrieval experiment different small approach model
nonetheless hand great cccc model pre naive latent logic friend friend fan fashion fan fan friend live live device like friend devote automatic profile social major level public extraction study examine interest political year propose highly extraction education job base training distant supervision google treat use base supervision return predicate job education fundamental assume share attribute chance friend medium friend share adopt like attribute extraction extraction seed learn additional pattern distant supervision another methodology source supervision raw logic reasoning logic order logic representation day ai variety reasoning relational logic language capability type logic logic log linear incorporate relational dependency network combine bayes jointly consider reasoning logical reasoning language understanding web cluster propose
six fista algorithm fast support recover compressive consist proximity art array fast accurate algorithm recover sense array case ill pose address regularization seek eq form
identity replica bx ab yield expression average follow statistically independent finite q ab ab saddle expression operation lead restrict candidate dominant saddle point analytic restrict replica replica cavity utilizing provide analytic n df n account become lead particularly eqs dominant include complicated handle significantly simplified allow st replica compose z p ny ensures involve change saddle summarize equation evolution notably bethe usual replica factorization section evolution derive mmse matrix factorization bayes optimal datum evolution arise analysis derive section pca blind definition basically interest blind trivially take equation discussion present generate eqs obtain explicitly eq eqs channel additive eqs evolution greatly simplification equation evolution explicitly dictionary mmse evolution mmse find phase transition uninformative initialization limit correspond fix instability symmetry limit initialize plant constant entropy phase simplicity uncertainty completely unknown take learn blind calibration pca blind source separation problem eqs number channel distribution computation large obtain q point locally large bayes mmse remarkable implie set identifiable large give count bind next question whether mmse tractable answer uninformative behavior state expansion lead uninformative evolution mmse achievable g approximate message analysis see blind side let compress particular sense identifiable recover static phase amp match mmse sample take right transition finite value see phase dash calibration achieve amp surface plot mmse signal sharp replace smooth mmse link dictionary blind try possible plot negligible value less know matrix amp mse
shape satisfactory measure year fix previously repeat one converge upon hyperparameter eq likelihood numerically laplace expand allow partial brevity order hyperparameter require computationally arise search space direction occur invariance latent let unitary consequently invariant unitary transformation depend via unitary undesirable consequence firstly partial positive secondly representation infinite rotation latent eliminate
always noise error relative yield well representation next repeat paris city range pca art ssc one report result relative reconstruction thereby learn subspace capability mc report denoise handwritten digit demonstrate nonlinear denoise unlike mc belong compare kernel kernel eigenvector fix number te x te te noisy eigenvector oracle different subspace mc subspace equal mc training level te te te denote pre rely cope lie high always embed improve also help communication storage requirement geometry high dimensional utilize application denoise segmentation broadly fall two category learn latter drive geometric outperform lie near hilbert transform application nonlinear generalization linear remain computationally investigate decade popular generalization manifold also model mapping
performance phrase rnn encoder decoder contribution rnn encoder correlate expect try word neural network I shorthand opt penalty able achieve ht come analyze pair score encoder decoder translation phrase corpus score phrase sec expect without phrase rather linguistic occurrence corpus focus pair phrase phrase phrase target phrase translation rnn encoder decoder perform phrase list target phrase phrase either translation encoder phrase phrase encoder decoder short phrase pair rnn pair different fig arise rnn encoder rnn encoder learn simply frequency phrase pair
case case localization w return dp also desire runtime w claim w part tp hence cc large enough conclude lemma exist px let establish xx xx indeed dt te n know check polynomial proof
rgb new statistic fit well network observe take spectrum diverse algorithmic generative network statistic assess gap role inference dyadic covariate arise rule diversity research relatively develop assess goodness describe assess type community substantially gap leverage spectrum statistic percent make goodness provide observed exist assess roughly group simulate fit function assess fit structural actor orient know framework algorithmic subgraph approach fitting comparison simulate ask draw simulate network unlikely generate well specify readily example theoretical
ni consensus iteration distribute power satisfie mix mix u state exponential case tell estimate due iteration look block r k r help characterize cloud svd centralize perturbation recall definition sec sec express I mix along make arbitrarily long perform appendix rely usefulness cloud use demonstrate cloud svd representation deviation cloud centralize help simulation mnist cloud svd benefit distribute site enyi synthetic individual column distribute site site randomly follow noise carry average weight describe update power method cloud oppose canonical centralize canonical svd cloud svd see carlo trial centralize local cloud illustrate power cloud svd consensus consensus iteration denote eigenvector distribute power iteration monte trial power distribute fundamentally consensus highlight cloud centralized still first experiment experiment method iteration collaborative centralized error site monte
spread influence social influence medium compete information nature content recently diffusion work information diffusion diffusion property correlate instance event formation edge work focus arrival follow even behavior dynamic steady explore result like behavior user receive edge event cause information diffusion event lastly accuracy spike provide insight cause occurrence spike spike well examine aspect compatibility create content lead insight additionally tweet foundation grant yu fellowship microsoft like twitter give social consume content create underlie social individually study past much know user interact evolution sharing examine twitter post create connection structure rate cascade post create significantly discover cascade phenomenon way drop connection source refer create connection refer follow user increase cause discover interest diffusion connect exist similarity
rate feature sc na normality misclassification sc misclassification sc overlap entry valid layer layer entry sc mean variance layer layer na ht c identification misclassification sc misclassification spherical rate misclassification simulation primary identification datum identify sc var joint participant study participant show return contain patient individual follow exclude survival analysis section measurement pressure heart quantitative variable sensitivity experimental description contain third contain overlap select identify structure type exist computing case auto sc min var var layer df df df value membership risk subject exclude analysis third second neither sc fail ht c statistic df
seed force seed replicate mc windows program number single stream reproduce platform serial execution window explicitly create stream may key aspect latent trait response variable produce possible pattern variable parameter via monte analogously function justification focus example point quadrature grow error monte monte integration high dimension univariate integral evaluate integration remain dimension value general sense conceptual payoff mathematical simplification estimator typically suitable weight serial aspect primarily issue turn parallelization case code develop similarity solve two different include supplementary evaluate frame label double array
truncate capture retain separate norm posterior scale sensor error quantify average plot show reduce order model least rate reduce build prior reduce versus signal ratio amount order adjust examine concentration basis gradually increase record vector quantify posterior define marginal dimension noise amount reduce concentration complement ex ex ex ex ex ex ex ex ex ex high employ field point eq eigenvector preserve avoid inverse figure pressure head measurement sensor case simulate full discard burn mcmc approximate simulation mcmc burn evaluation reduce cpu ess speedup threshold mean standard field reference full approximate produce generate record provide maintain steady medium drive achieve classical approach order challenge modeling convert observational process play quantify framework model characterize chain carlo
universal method combine walk discuss main multiplication long reach start difficulty cluster refine cluster vertex vertex singleton mini accuracy news even trivial refinement achieve refinement fine scale slightly version initialize scale modification random walk fast start refinement proceed let hierarchy refinement perform level return seed termination denote seed initially level
respect quality excess excess q rd internal randomness risk mechanism preserve differential risk minimum achievable define min differentially private mechanism work fairly loss contain square give strong go beyond constraint paragraph privacy replace well quantity convex never large similarly bound assumption function bound analyze noisy mirror simple result precise tailor gaussian kn show interpolation norm view one assume satisfy appendix get perturbation width dependent bound width bound noisy mirror mirror descent often assume e entry lipschitz polynomial loose beneficial lipschitz issue bound differentially version wolfe constant precise
sketch algorithm present extensive report def layer variable gamma bernoulli two datum pt improvement collaborative corpus lead deeply sparse gamma overall corpora k baseline art observation generate def collapse def gamma sigmoid sigmoid sigmoid poisson poisson def def layer netflix netflix ndcg ndcg mf layer double def double double document hold conditional multinomial additionally percent document percent form document completion document fix correspond evaluate ten document differ latter difficult
write indicator overlap demand drop subscript effect solution must lagrangian self derivation remarkably number marginal marginal eq define follow sum value value constraint rough sec relax accordingly uniformly ix implement soft solution limit smooth choice sec tb py li calculate update iterate find optimum bound pseudo detail mini batch size estimate marginal representation consider use variable objective achieve mean set solution representation different tight bound ability begin synthetic
fit precision indicate time furthermore datum effect rather novel sufficiently regularization good inherent training objective dramatically worse sensitive
require e need loss report daily average performance performance entire daily prevent number entirely dominate instead assign b reporting performance slice allow circumstance improvement fig per total upper show performance click average daily apparent figure score click significant daily also evaluate score click test ignore entirely click use find period auc plot difference leave daily box circle mark outlier interval quantify order summarize relative day performance click set
parallel another picture evolve go represent dot iterate stop effort computational resource hence process describe greatly need network fig cascade architecture report predict confirm novel schema couple feed use separately predict similar cascade feed output stage connected nn cascade confirm effectiveness evaluation phase show nn architecture
svd approximation
rely nmf refinement heuristic tolerance assume soon relative error predefine stop conservative value numerical arrange display possible five initialization along paper local perform example really point numerical relative threshold choose high improve additional matlab precision open nonnegative g semidefinite see detail section matrix linear initializations refinement refinement idea behind obtain avoid initialization iteration algorithm pair small give state generate obtain fair different generator matlab execute outer execute far reduce time numerical stop soon check every ten display find run find exact ten display nmf find bold detail nmf algorithm use intel core
various sequence suppose assume sequence strictly nonnegative function constant verify definition verify sufficiently sufficiently control study balance contraction smooth collection spline wavelet fourier commonly hellinger w h p usual distance density prior sequence positive positive constant sufficiently contraction verify describe define way approximation imply rate polynomial series spline wavelet series wavelet j incorporate setup scale normal model suitable whose metric entropy control complement exponentially concrete theorem contraction crucial choose equation divergence square different choice
combine reduce part template chance turn resolution global attain bottom e value randomly initialize tendency yield situation max minus min composition parameter move opinion stage however gradient em beneficial initialization part care exist minus shift rotation part template version transformation allow transform template configuration probable procedure matching pursuit start add increase improvement material yield opinion
v intra similarly st inter pairwise solve regularize connection form generalize intra neighbor q obtain true block guarantee crf constant type intra inter satisfy eq hessian node universal constant solution recover neighborhood recover inter neighborhood node recover union extend selection evaluate numerical example ensure parameter meet discuss parameter simulate appendix sampler node conditional via gradient descent elementary sample operator roc recovery replicate top panel recovery instance variable panel panel decomposition x product specify bottom highlight advantage mix mrfs mrf crf crf poisson mrf represent mrf mrfs truncate variant mrf instead introduce simulation simulation selection recovery distribution homogeneous ordering sampler connect neighbor block curve recovery binary mrfs poisson mrfs mrfs able even extremely model set mrf recovery dag much treatment question beyond scope section brief demonstrate achievable via node neighborhood figure plot roc curves structure crf crf mrf good increase
implicitly bias value calculated get bias discrete minimize size describe sample take sample average risk mc bm misclassification additive constraint substitution plot curve figure thick grey us task suppose concave attain hence
ie draw therefore branch likelihood branch minimum accord exp convenience respectively work understand theoretical specie tree multiple focused tree reality tree molecular sequence recent understanding tree take fact gene generation associate assign move root change random leave length time site specie write mutation adjust branch xy disagreement character p xy goal learn character easy presentation realistic reversible molecular hold
aspect general focus modular focus maximization combinatorial triple item modular maximum short bipartite matching hard problem computationally similarly agent approximation randomize combinatorial recommender typically item value expect rating never perfectly refine repeatedly recommender typically write structure assume formalize bandit semi triple ground arm sum weight arm observe return combinatorial structure like stress expectation episode adaptively choose weight episode gain observe weight te learn semi goal maximize return randomization
layer gray convolutional consist size neuron pass layer project feed softmax produce bad leave tie except pool architecture network gray rgb neurons input choose well match cost network negative run implementation
human capability order become automatically variable variability obtain vote joint voting validity classify unknown light curve vote rare voting detect bayesian joint explore offline test algorithm million curve anomalous divide outlier air bad calibration instrumental remove outli list select object pass post stage match identify blue type outlier follow deep analysis important examine effect star nearby star star candidate visual discriminate remove systematically object run step repeat obtain apparent outlier execute purpose group interpretation finally match aim know belong followed analyze identity rf rf extensively use start rf construct bn purpose extract classification final outli method anomaly machine theory perform real training elimination analysis conclusion vast publish relation anomaly detection classify class unsupervise learner unlabele consequently training class turn partition anomaly method detect anomaly generality
cnn preserve internal input straightforward treat pixel cope word example vocabulary computation unit region rest image vector convolution learn call net convolution representation seq cnn seq keep potential seq cnn however rgb vocabulary size weight dimensional vector handle vocabulary desirable provide bag vector vector convert convolution word seq convolution size image fix sized output convolution also give layer pooling would variable sized output pass output require connect alternatively top receive sized fix dynamically pool datum cover overlap max pool datum whole
process develop tractable real work tune hyper automatically adapt degree poisson intensity gaussian advance term well nonparametric method carefully tune hyper reference therein challenge nonparametric devise procedure avoid overfitte intensity intensity location
well small eigenvalue criterion newton invariant conditioning issue filter discrete extremely ill reasonable trend filter gradient method perform primal dual descent admm simulation setup give appendix variant descent standard admm compute trend method drastically reach thousand latter technique admm suffer less issue regime accelerate accelerate form iteration specialize admm perfect descent filtering cover model specialized algorithm conclude describe specialized admm trend filtering algorithm differ admm require operation iteration lagrangian write implement cholesky update wise multiplication full admm time specialized begin augment yield update q small update trend fuse specialized solver
empirical kx dx f minimize upon achieve standard become get stein improve upon gain reduce incorporate adopt shrinkage estimator knowledge close provide knowledge interest investigate without require assumption dirac ensure case positive suppose non thereby yield characteristic function characteristic definite xx stein zero risk decompose part behave estimator easy great bias decrease hand become increase mention show shrinkage unfortunately fact know require oracle measurable bind uniformly virtue dirac kernel characteristic uniformity bind norm characteristic empirical upper transform ix proposition kx regard shrinkage dominate empirical estimator lebesgue
color green marks solid forget sep crcr blue mark option forget plot crcr blue mark mark option forget sep crcr symbol classification handwritten digits computation depict less solve toward set showing pair draw training equally test analogously misclassifie pair multilinear form infer relation equivalence relation misclassifie objective training height unbounde xlabel ylabel relative title test format cd fix cd white forget crcr nan nan nan nan nan red forget plot crcr nan green dash crcr nan dash forget sep crcr nan color forget plot sep crcr nan green solid forget sep color dash
justification contaminate gx minimize mse regularity condition asymptotic component pilot estimator recommend use pilot estimate general framework divergence different include divergence efficiency property explore estimator explore robustness performance analysis prove link robustness al family recent elsewhere beyond discrete distribution estimator family bandwidth smoothed density divergence equivalence practical implication well support real example anonymous suggestion lead version manuscript institute inference divergence technique divergence name family discrete develop continuous smoothing unlike avoid usefulness extensively et base quantification minimize include pearson chi pearson kullback leibler kullback hellinger bregman rao
pass slide communication device could human co operate power effectively lose control machine intelligence improve initial look loop artificial study investigate compare electrical load learn forecast load robot despite precision feedback increase purely feedback user control term lead great assimilation device predictive copy operate expectation remain verify life cm thank insight need individual one promising end interface supplement learner control sense live interaction part internal copy move cause desire
involve laplace cauchy boundedness even problematic dealing noise infinite far tailor situation reference section dedicate assume predict place take tail ensure set small notation assumption behavior satisfy tail investigate near family laplace distribution integration gamma satisfy immediate pareto regardless gaussian law tail univariate law satisfy inspection q spectrum justify influence consistency first minimal necessary obtain rule consistent discrimination classification assumption minimal mass non compactly density classifier discuss curse support real compactly support though show reference therein second mass compactly support proof mass major former family case excess risk strictly long neighbor occur neighborhood contribute near hold precisely assume hold sake convenience adapt tail note recover compactly density compact case compact
single outside leave unchanged observe together element exact remain r note remain even strict broken mean strict extend comparison rank index index j score get tie score observation comparison index shift row remain comparison intersection column corrupt generality get eq apply intersection row corrupt shift observe remain matrix strict except element row similar column tie break use mean strict matrix proposition corrupt induce score case pair adjacent break arbitrary corrupt exact rank comparison estimate corrupted maintain true corrupt set pair guarantee admissible pair random replacement denote pair sample admissible contribute pn notice provide small fraction
q serve pareto network specifically reasonable condition gradient theorem agent limit denote agent symbol gradient process past iterate agent verify whether agent dominate effect broadly consist flow illustrate particular agent equal limited sub sequel sub illustration figure strongly receive sub self loop indicate figure run diffusion mean sense third bottom network towards bottom third influence feed behavior third influence network still exhibit independent implication discover sub end sub limit sub regardless stand connected sub run independently network agent group use letter refer denote equal agent across would strongly truly
indicate denote product represent reflect simulation verify high robustness statistic decade extensively especially deal dataset fitting accelerate robust like name author fast inexact alm publish later outperform efficiency competitive robustness dense normal operation mathematical observation
selector inaccurate fail identify vector provide selector selector short mu selector subset discuss selector apply random mean finite admit drive estimator bernoulli value rewrite therefore model easily shown admit good available mu contain induce whose expectation vanish idea thank lead call selector entry constant choose estimator selector study design one interest selector enable error
advantage particularly efficient simulate pose perspective table clearly exercise draw burn auto mode middle simulation mean simulation setting leave zero perform simulation growing consider variance covariate variance pairwise absolute via simulate gibbs iteration burn death move probability posterior draw hyper prior posterior sum grow difference tend grow lasso ratio figure horizontal l ccc select predictor validation bit ghz processor ram assess experiment gene potentially show cancer patient predict datum gene predictor unit material assess scad method bp predictor moderately expect related observe prior remain parsimonious validate roughly remain small predictor drop observe predictor various finding effective detect parsimonious dimensional time order competitive implementation code follow storing likelihood memory focus mass burden tend
kt ft proxy equation enjoy conjugacy mask factorize separately correspond variational obtain initialize variational kt kt kt kt lt variational inference example check well separation bss comparison exact potentially complete conditional
convex modify ridge hold validation let fold tune relax kkt solve constraint fold avg intuition behind solution path curve relaxation compare use
assignment subsample mix schedule quickly empty pick assign remove assign little code effort implement linear subsample schedule initialize empty assign gradually latent schedule perform henceforth choose real consider iteration differ sampler subsample anneal hyperparameter hyperparameter inference per cycle early relatively inference subsample toy intuition asymptotic speedup subsample time simple two energy barrier way subsample anneal simulate anneal
membership partition community overlap overlap offer community measure vertex divide community attract relative benchmark event unstable suboptimal modularity yield modularity introduce new section define let community degree node modularity modularity value degree belong community e contribution w w simplify verify modularity calculation accordingly assume calculation perform vertex move difference step compute c z z md list individually compute r may infer follow c c simplify computation provide study semantic without discussion suffice say follow vertex apply equilibrium ne community assign achieve goal benefit player access strategy execute derive profile
different relaxation case feasible one relax remain since make enforce constraint easy think control surrogate cardinality enforce outlier li sdp model justify fairly chen xu belong sdp also generalize inequality advantage chen know number outlier natural restrict n balanced balanced relax note n kx xx x apart nonnegative remove generalization sdp pca relaxation lead generalize possibility leverage sdp c consistency sdp call sdp consistency mm note q block consistency differently eq plant kp qp follow eq show order let sdp sdp sdp sdp piece argument symmetric triangle establish sdp balanced plant rescale namely precisely similarly let c sdp consistent c sbm block point
jt q kt eq multiplying noting since real relate invertible also know optimization find proportional negative current scalar q gradient descent q denote increment step use jt kt update rule scalar cr calculus make derivative make derivative
mean namely similar also filter first detect change second datum detect use random analysis later publication scheme remove present example conclude trend derive iterate integrated trend
descent gradient choice vector call fill height width em coordinate thin distance cm right cm xshift block sim text ex sim update theoretical optimization albeit unlike noise component establish establish regard difficulty evaluation adapt carlo horizon discount set transition truncate trajectory approximation simulate want discount cost truncate artificial outline horizon set horizon truncate trajectory define trajectory limit induce discount current one il j c estimation algorithm provide algorithm free transition algorithm complexity respect
quantity similar matrix expand partial derivative valid two taylor independent variable independent width prior error reduce propagation variable use straightforwardly generalize generalise fisher fisher cast http www fisher toolbox cast grateful physics ann united states school mathematics university south sciences department
average recently player player strategy reinforcement response recently broad setting underlie consideration player payoff payoff specifically rank strategy score rigorous tune player response x assign strategy high multi value interior history theory refer discussion refer reader purpose seen x finally rate cumulative allow explore accordingly throughout reinforcement behind seminal bandit discrete set variant continuous systematic account two characteristic example reinforcement logit good response prototype unit simplex gibbs negative standard simply equation thorough therein consider penalty hx function project response support change game evolution play games reinforcement sort noise access cumulative assumption rarely
graph realize modal reduce distribution sample dotted sampling distribution capture go beyond initial concentrated estimate serve heuristic degree aspect seem property degree whereas thm thm condition interpretable structure statistic formalism core statistical consideration social specialize os construct serve estimation network concerned node record count familiar framework associate network degree comprehensive review base fail connectivity connected seem additional
language corpus involve discriminative usually rich primarily linear unit particularly behave deep impact difficulty leverage benefit piecewise linear propose adversarial net adversary learn whether team discriminative competition game article kind article pass perceptron adversarial net train use
ability difficulty function infer maximization likelihood sn set response expand student positive pair take share hamiltonian ise introduce several introduction ability exceed share extra adjacent could account reasonable room site situation demand contact range ability
offer auc really poor know subject performance significantly performance offer across notice subject calibration calibration quality point short minute acceptable game record generalization across dataset ii record realistic p p pz sample hz classifier subject increase calibration average subject ht trend outperform reference also despite subject cross performance show cross difficulty high result good adaptation propose necessary dataset iii record life live hold laboratory th
steady difference state steady unique invariant concentrate expectation much sum steady expect phase reveal phase expect cost strategy proceeding lemma throughout proof lemma heavily uniform continuity every unique invariant mx evident specify regret bind present reduction steady steady cumulative cost steady cost policy fix consider loss therefore cost steady complete phase e decompose nonnegative hypothesis strategy state decompose cost phase q steady get everything ahead last inequality due situation everything ahead result steady actually everything ahead q value repeat combine next bound quantity phase moreover matter routine calculation choice complete demonstrate tracking move undirecte probability dynamic topology neighbor simulation vertex passive random
free svm gram yield expression center propose center singular gram singular author consider optimize center centering model nest center either partially center comprehensive centering center beyond scope investigate investigate non update connect center gram perform matrix provide center machine subscript recognize center oppose center versus singular singular eigenvalue variance deal sake inner matrix associate outer machine present center set available inner essence focus gram non decomposition gram data decomposition realize financial economic coherent order non moment center center centroid decomposition rewrite mean contribute center center entry I n central rise
property term modification argument give apply rest lyapunov basically l convexity adaptive size round k unconditional drop convexity summation inside simplicity gradient element avoid illustrate correctly np np perform double lag lag double long amount lag lag amount dense double double double v double epoch long double store format double index double double np double track lag np
choose sure payment effort dominant eq dominant equilibrium proof course could arrive mechanism computationally overhead execute complexity ode statistic ode find certain criterion g unlike ode ode optimal ode solve approximate unique feasible minimization feasible preserve reduction minimization translate box manner special minimization regression restrict minimization behave separable minimization polynomial behave iff expectation regression bipartite ie convex function cost polynomial cost wants find minimized matching cost find min min optimal regression vector finding take well define hope
upon identify green blue respectively collapse ask clean fig manual piece together fig hmm difference uninformative freedom depict grey reveal critical axis motion loop display low axis show fig interpretation experimental reveal protein family role cycle md atomic effect protein biology gold provide know protein suit approach protein family whose lead role cell anti prominent member size complexity md task show water accurate inactive take adequate
consistency posterior assign precisely entry denote completion summarize survey reduce reasonable assume rank make consider e dirac mass large consider close lead variant consider iid conjugate among author give intuition seem computationally prohibitive
temporal classification layer backward recently acoustic deep rnn stack rnn predict phone show get art phone recognition database language gram phone lstm networks phone database result paper obtain art vocabulary recognition modify architecture lstm network address computational efficiency standard architecture lstm output layer lstm layer forget cell connect network parameter lstm
arguably category score htbp hard negative add positive region around foreground image overlap score intersection union foreground less precision improvement negative region automatically discover negative region green box discriminative part box fit configuration car cat tv svm negative neighboring negative discover negative supervise method frequent configuration discriminative visual show discover configuration provide coverage way useful hard negative together challenge berkeley due
fc layer scale ram ram scale ram ram ram fc layer fc layer ram scale ram scale ram test successful policy experiment patch enough experiment also test ability ram patch input also feedforward two unit test additional improve ram training full digit successfully combine information classify task translate mnist generate place mnist digit patch case effective size multiplied contain random translate several model translate ram network convolutional filter nonlinearity
require provide symbolic believe science division berkeley berkeley computer division berkeley berkeley lemma corollary remark em minus gibb several augmentation map gibbs allow expense slow parallelism excellent modern hardware graphical factor give throughput competitive fast symbolic fast report validate method applicable general distinguish represent miss generally want optimize value
filter right filter histogram model integration solution dense simulated particle filter particle visualization filter particle measure novel wise provide particle filter present filter
n substitute problem objective easily partial derivative regard ranking code complex use sparse code consider reduce solve sign rank code consider irrelevant turn sparse coding solve
claim lemma technique randomize tangent cone technique warm show obtain cover put piece z system prove theorem j width inequality calculation suggest maxima apply difference consider cover see appendix subsection result apply tangent cone guarantee u restrict piece q pair condition result state turn inclusion consequently apply embed result suffice least guarantee z rescaling throughout make convenient shorthand equality write conclude q ensure early bind claim dimension regularization nuclear contain analogy suppose optimum sketch eq must hold proof inclusion trivial q notation
exist definition reverse triangle n f continuous schwarz yield hold limit bind hand remove specify projection powerful corollary matter satisfie possess smooth hold consequently indicate validity cf remark since completeness sketch less elementary argument apply exist neighborhood orthogonal small limit taylor zero space mn euclidean equip inner norm save space prefer representation inner orthonormal writing onto subspace orthonormal basis convergence algebraic matrix see continuously fact assume analytic g dense relatively open hence real task tangent point tangent cone arise
nonparametric layer filter unlabeled task thresholded dot simplicity recognition natural computer solve face recognition task think require alignment able operate totally unconstraine face hierarchical architecture memory thus propose take advantage principle greatly simplify frame system architecture module abstraction cell input module considerably abstract represent input cell dot cell invariant concern architecture construct signature layer layer temporal association associate template training accomplish
laplace smoothness induce laplace model logistic highly resemble gaussian among cauchy advantage suitable second tail highly capable substituting scale framework cauchy q cauchy pca extend entry observe otherwise maximize introduce explain locate set estimator influence contamination estimation supremum take worst desirable estimator unbounded explain pca noise another neighboring stable possess cauchy estimator unbounde sensitive
figure reader dark gray background figure take legend title inside serve sequentially place graphic center figure wide figure place column environment format require input initialize ix center title title unless leave flexible breast meta table material figure material draw row table column place page please format regardless file include name name place otherwise separate author publication person blind please style leave subsequent reference give journal conference book author precede reference author year publication sure reference page strongly encourage publication version whenever camera ready copy reveal instead anonymous review acknowledgment acknowledgement version blind review camera acknowledgement thank give financial help student report student raw source involve crowd engineering extract student non model art examine model characteristic generally predict difficult attain massive open leverage advanced topic hundred develop time markov present make stack hmms finding contribution end raw advance methodology scalable interested fall extract combine student usage leverage collective brain crowd create temporal feature predictive comprehensive art technique
confidence coverage method work well asymptotic cat significance power cat point cat almost cat assertion test consistent sample consider home team call team time goal home team bivariate
uniqueness policy set bayesian hmm field health technology political business research patient com computer site percent company online analyst example instance lasso application speed approach optimization approximate sketch preserve computation lasso property solve root fast solving equivalent
recall bivariate partial odd model ml asymptotic penalize literature whereas patient
logistic describe requirement fast lead posterior technique often group g write usual glm notation datum form logarithm become extended derive posterior normalize logarithm inequality approximation equality apply logarithm tx tx tx bound proportional tx computed approximate
auc gaussian event background event baseline layer network whose hyperparameter optimize architecture top hide overfitte unit performance dropout method dnn relu relu relu unit effect benchmark activation complicate try initialize position activation keep
popular regularizer general xlabel ylabel avg legend style true legend north header x dataset server update processor main drawback update make guarantee framework although recent make coordinate observation gpu collapse sampler independently parallel good saddle parallelism contribution towards could epoch machine predict measure average epoch per machine value scale iteration ylabel font legend north title true serial txt
evaluation protocol keep sentence total group cnn semantic dependency tree rnn match global rank fair cnn multiple representation object cross evaluate prevent hyperparameter devise modification cnn discuss rnn global rank cccc search al devise model low good test cccc et al devise alignment sentence good quantitative table significantly rnn competitive objective suggest bring cost directly minimize global scene
frame convenience denote peak parent multinomial random peak subsequent th peak intensity value respectively center gaussians axis oppose peak consider rather currently consider gaussian bad far otherwise alignment enforce frame e peak score e alignment log random score peak likely somewhat simplified previously constraint variable utilize alignment subsequent current simplify come observe spectra resolution spectra assume state amongst return require among differently spectra comparable another essentially score score target let dd return score linearly calibrate order score percentile interpolation score largely target identifiable typically decrease
moment distribution section fractional procedure carry five full characterize parameter use robustness case case two density reconstruction criteria namely test start compound loss business density marginal diagram empirical use visual idea density cumulative whose figure observe difference fluctuation indicator good reliability diagram diagram measure probability beginning seem closeness norm distance mae detailed plausibility display mae rmse distance reasonably mae reconstruction correct consist transform distribute deviation uniformity poor display different power histogram seem robustness method perform reconstruction routine carry reconstruction
status physical consider ny memory bellman calculate respective scheduling conduct sensor side hence controller utilize transmission identical instant lose whether controller new aim controller side hence communication period end possibly measurement receive conduct k state variable update system physical system network reason dependency control controller new information derivation change sensor controller controller need use protocol fulfilled present simply communication delay extended delay loss utilize rl handle stochastic process delay losse bellman available besides mention behavior schedule capable handle moderate include delay losse final give weight backward dynamic concern lead unbounded control admissible control within
basic confidence round choose policy predict maximize implement construct set contain least confidence translate confidence transition select simultaneously optimization efficiently perform employ eq solve specify interval time ts confidence
almost choose ii aa significantly outperform iii unlike sophisticated convolutional aa helpful cost drive per aa sc visualization regard image internet video public methodology present example visualize request paris accord dense descriptor interestingly request paris city reduce influence outlier heuristic choose point even learn fisher point classical figure
linear gaussian analogy dictionary section compact cf exploit part enhance extend pruning dictionary strictly prune emphasize single approach contain multiple multiple compactly efficacy propose toy example red color color curve blue light color employ section linear adaptive filtering adopt basically filter kernel input sequence draw output corrupt trial mean average respectively complexity summarize curve reference include dictionary see outperform counterpart
b fourth last equality last definition expression fourth definition b u th machine core block band core compute respective block fig block main block band e follow block compute recursive step mr I block band nr r resolve transpose e recursive block ir u nm thus albeit black red likelihood curve jump area blue gps jump boundary yu chen department mit technology usa edu thereby perform region furthermore accurately scale locality
triple q check rate predictor guarantee x x rate distribution agnostic case algorithm logarithmic applying theorem number get agnostic isotropic log treat linear classifier complexity recall apply simplified computer science california la study generate main high requirement fairly major challenge achieve provide rate prediction rate study generate access large request learn pre label target aim active survey main challenge consistency maintain generalization binary search inconsistent agnostic algorithm agnostic minimizer maintain exist agnostic label sphere unclear major
error iteration pass predictor consensus message indicate red belief iteration message red significantly slope indicate bar also regressor capacity fig well comparable bottom message pass mm noise gaussian gate gate pr gate gate gate pc l south south circle pt red l circle south south consistently problem make low challenging devise inaccurate center circle square might wish reason colour foreground background make hard image perform sequential schedule marginal latent additionally place message lead pass fig highly inaccurate marginal center inference see quantitative fig predictor red consensus message message come pixel foreground internal design test two position
particle serial continue core theoretical additional computational overhead initialize run alternate trace new particle addition retain operating trace late branch arbitrarily achieve every process barrier compute weight terminate away branch loop new child process barrier albeit barrier execution program execution retain branch reach particle retain select signal retain branch process retain except retain trace retain branch
express c infinitely variable evy e find separate absolutely atom positive count dependent membership observe atom joint assignment size fix label atom joint likelihood r lead pmf order group membership size q lemma directly bayes rule augmentation marginalization laplace transform parameter analytic concentration use sample discrete distribution grid gibbs hdp thm corollary thm section integer stochastic employ count
equality ef lead appendix cover characterize bind number normalize n cover radius cover covering fx xx x variable complexity closely cover exist average uniformly f h lipschitz constant apply inequality I I sample h nh proposition eq h e vx u n yx yx j yx nx v vx h similarly e x yx
continuous theorem rv independent motion index pass coincide aforementioned let copy th skew covariance copy give stand gamma case coincide recent also excellent calculation extreme value normalize constant linear normalization supremum converge stationary limit weakly minimum process
maximize minimize bic bic parameter value component coverage model function bic bic number component configuration correspond diagonal dark wide wide classifier cluster simulation free measure observe uncertainty mass simulation randomly deviation scatter noisy decision thus draw contain training distribute uncertainty use classifier cluster inside repeating generate realization contain uncertainty draw make previous density cluster sample object statistically target gmm make straightforward method outlier target nature
learnable agent dark remarkable start track observe agent passive observer stationary condition occurrence situation observe finite let still admit decomposition process theorem obvious omit tail induce decomposition learnable stationary tail sketch ergodic theory partition atom refinement dynamical system seminal follow algebra every measurable q natural tuple stationary I I admit decomposition primary object interest outcome day history outcome previous relate prediction decision observer
q decrease accordingly proposition imply sure backward describe mdps action initial path merge common couple decrease path couple first chain time begin common state get couple chain couple couple overcome chain infinity simulate accord continue chain k define thus couple couple rhs picture henceforth q precede clearly proof k equation g provide bind developed subsection also early notable sure iterate however confirm simulation result
cc decay however suggest speed shrink construct confidence residual satisfy choose discussion stand introduce define bootstrap xx remark reason first develop suggest effective goal achieve coverage accuracy motivate lead representation also term bootstrap propose quantile regression iterative less effective term accuracy bootstrap analogue serve standardized section process approximate xx section ec regression replace limit ec asymptotic confidence bootstrap two coverage discuss bootstrap quantile affect quantile depend bandwidth hence location choose negligible slightly optimal order nonparametric eq validation gaussian study select rule package
redundant leave near neighbor decision gain decision split choose split attribute create accuracy return competitive meta decrease et identify instance correctly determine misclassifie instance difficult classify correctly hardness measure instance overlap instance instance measure cover produce feature subset tree percentage instance total kolmogorov instance depth tree provide instance belong calculated attribute feature skewness instance instance target
form good principal tool computation provable building recommender prediction customer base netflix entry rating customer movie dataset entry customer rate movie make assume rank customer movie highly correlate rating bound form customer turn recover often completion corrupt person identically sum loss least sensitive outlier classical example reconstruct would matrix presence become study call new representative basically approach minimize rank two factor rank call factorization rest review solve section factorization review finally conclude discussion recover rank minimize estimated rank combinatorial convex popular rank norm singular value two fold nuclear convex feasible global optima relaxed secondly prove mean operator analogy recovery norm model summarize nonconvex method discuss problem matrix element entry outside equality coincide exist early replace nuclear tractable work solve eq prove
describe label receive empirical label constraint propose rademacher global label analysis generalization global rademacher converge complexity set fast learn svd unitary value assume complexity consider orthogonality follow eq eq local rademacher erm label determine
reformulate exact penalty motivated structure zero problem consist problem propose partial deal method yield space numerical exact well quadratic past decade relaxation norm brief historical account signal reweighte due rank relaxation follow surrogate tend nonconvex study demonstrate reweighte nonconvex norm zero minimization mathematical constraint variational induce add e coincide penalty parameter though exact penalty solve special structure decomposition reweighte consist show favorable locally globally weight software subproblem bfgs newton section penalty method solve bfgs state code good solution feasibility
survival evaluate instant moment exist uniquely distribution moment density two entropy system polynomial turn convenient fast uniquely need survival moreover instability polynomial broad among densitie compact support contrary polynomial like prefer semi infinite transformation coincide approximate polynomial every recurrence relation moreover normalize uniquely decompose basis raw moment coincide sum give procedure make stress polynomial expansion might fail positive resort importance proportional normalizing require support support natural
interpretation intend since fa loadings fa explain loading fa factor result singular order convenience generate data sparsity form fa fa papers generative use tool interpretable combination variable consider perspective variable variable approach propose conceptually exist pca lar analyze arise variable affect strict subset experiment non elsewhere analyze scale accurately infer graph vertex define
consider discard burn compute running root compute modify smooth hence expect see five accurate filter simulator particle backward trajectory rmse show thick gray serious classical particle unknown already outperform particle gray line correspond smooth see sampler level order use system toolbox fix truncation nd system posterior discard rmse plot increase truncation level th consider year million severe hundred cause severe public ability predict disease activity sir environmental fluctuation specify discretized euler yield infected death rate r google infer epidemic proportion query relationship odd relative odd proportion infect count day month mh sampler sir model though observation base particle augment sir applying
whether constant achievable equivalence et almost question furthermore identity perform conditional demonstrate intrinsic interpret show fundamental distinction usual low support large al give doubly factor support adaptive support query prove low uniformity test conditional answering conjecture et al non probability conclude bind query estimation constant exist adaptive access zero perform query domain query indicate adaptive support lead low set construction carry tight query significantly provide low equivalence deal conceptual long quantify thing much restrict property namely invariant permutation show simple core distinguish work instance polynomially domain probability deterministic guess separate query
inequality n fact n rs complete lie complexity experiment decrease intuitively prevent encourage contribution feature phase try theoretical former leave introduce concentration real value eq write easy function regression network base n type neural random draw distribution random inner product
problem introduce source short compose vector short database superior group selection select dictionary term search efficiency compact experimental sift netflix demonstrate superiority area attract interest popularity compact code design inner inner product many scale cone design product fact exact naive scan observe similarity thus inner category code locality sensitive base hash iterative compression show superior performance little acceptable category specific adaptation quantization constrain research hyperplane hyperplane relate product address query hash maximum
u u tu n u u u u u r nu complete eq solution eq q complete rewrite u science chinese university chinese edu high tensor become scientific social mining challenge propose parallel trace formulate require factor mode optimization observe trace tensor core cast multiple trace alternate multiplier verify outlier term array
density kullback leibler respect optimal pdfs expectation unnecessary since normalise pdfs depend distribution vb expectation bar current variable necessary compute calculate h tx ty c wise hold verify write square linearity similarly horizontal covariance compute gamma independent derivation go tv case q case compute moment value update estimate one extract mode parameter density lr parameter gaussian isotropic briefly prior difference model next briefly distribution pdf bivariate distribution dimensional bivariate origin encourage periodic boundary
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formulate inductive empirical replacement various application fundamental replacement employ om technique concentration practical implementation replacement generalize bernstein function illustrative hold fx prove finite continuous technical x take let n get coordinate moment provide chernoff obtain trivial bind supremum calculus bind moment chernoff lemma sum obtain result derive presentation cube point thus uniform distribution define distribute thus thus concentration trivial quantity function modify assume step theorem
em guarantee need propose experiment carry mixture variable sample true mean mse true ex ex ex ex c c comparison approach ari ari standard ari small dimensional sample ari occur sample size good projection formulation widely vote house type vote yes party list htbp issue item water project share education right anti aid south binary code yes yes fit
infer assign thin observation solid line model infer panel gene infer subsequent function replicate cluster show supplementary material allow similar signal cluster slightly david tool differently cell whose arrive intrinsic show cell comparative purpose without hierarchical spirit maintain model vary microarray well variation gene activate temporal pattern vary correlate hierarchical cluster hierarchical dramatically hierarchical account significant model replicate variance discover supplementary version profile inference vb describe gradient find subsequently merge set sensible value optimization variational enable optimize fast vb biological time application capture consistency biological
unsupervise dynamic scaling affine operation correspond shift follow deviation compute standard training instance dataset time instance feature vector equation deviation define dynamic stream label vector feature th instance stream real feature e scale map range note irrespective original whereas relative enable appropriate range resemble typical affine deviation respectively frequently special belong follow write symbol overfitte
dirichlet collection assume document generative follow token phrase clique introduce lda assign clique token depict figure utilize encode distribution hyper simplicity conjugacy multinomial integrate lda assumption completely ignore infer great proximity phrase motivate strong graph relation bayesian association relation direct causal propose un direct dependence among near word latent topic un show clique clique phrase introduce express high clique define normalize joint collapse sampling variable ideally influence phrase clique topic state normalize get choose specific potential constrain merging significance clique possess develop efficient gibbs choice collapse assignment integrate configuration variable value take value pz w optimization lda experiment propose phrase visualization phrase label model attempt external wikipedia candidate parameterized allow user control provide phrase easily generalize probable phrase adopt clique guarantee topic phrase token iteration tf topic assign indicator instance document definition visualize sort relax assumption lda category phrase phrase incorporate use dirichlet gram latent variable word
go review four sampling explain usage popular cart automatically classification cover involve area improve difficult structure even offer array scan array characteristic raw return gps translate ground compute neighborhood systematic point
filter connect filter pattern match pixel predict resolution patch prediction fed image fit column location bottom add track digit move train connect image fully image experiment publish pixel patch resolution image give size search consider batch fully certain physical task vision decade application however primitive ability conjecture good human difference vision vision process around human integrate human currently instead extract location across different
limitation poor improve apply singular employ solve conjugate gradient ill unclear convergence regularize optimization whiten speed reveal propose closely whiten thank cluster computer overhead minimal computation employ compute present regularize loss introduce ii reduce condition boost convergence validate facilitate analysis henceforth loss respect gradient similarly lr scalar
path connect assumption connect function constant identifiability joint structural px px kx ix full connect function proposition none example prove density necessary intersection connect intersection call intersection direct causal additive assumption mu bx argument
interpretability way naturally dendrogram fit evolve combine change application perhaps social interpretability crucial interaction detect shift occur mean structure characterize norm distribution identify contiguous period relative allow subsequently process find detection change fact anomaly identify traditional point focus shift norm qualitative like parametric family slide one window represent nan hypothesis window conduct choose change past convert network scalar value traditional introduce novel define parametric graph compactly scale interpretable
proof various bernstein able build let concentration measure result coherence establish strong complexity completion use provide yield suppose tu complement identically statement note use coherence fact conclude notice ever fully live thus probability invertible th one thing either current happen ensure reconstruct live estimated strictly estimate event invertible latter via factor operator behave algorithm space fully invertible write invertible find q bound cost involve reconstruction projection multiplication ignore inversion take since run finally run gram schmidt take necessary argument specifically theoretic dominate minimax randomized say consider verify fact line inner minimize simplex deterministic suffice deterministic minimax guess
disease diagnosis develop medical mind question medical run available reconstruct diagnosis disease knowledge twitter could serious economic detection accuracy health expert surveillance traditionally report practitioner control surveillance costly slow public novel surveillance compare traditional system diagnosis automatic term self
method label image box exclude modification loss fig simultaneous term framework structural annotation specific semantic segmentation weak function level bounding box object seed annotation usage annotate annotation background give object bounding box segmentation quality annotation effort microsoft structure obtain try present instance supervision unified technique infer quantify annotation demonstrate effectiveness challenging annotation popular semantic segmentation compose generate pixel annotation bound label specify annotation dramatically improve compare classifier burden structured thousand scalar annotation human
crowd total cost crowd phase stage historical labeling result worker homogeneous definition worker worker provide label decision maker crowdsource service instead maker need infer aggregate size mutual overlap natural question ask allocation concrete example motivate c instance current toy worker noiseless aggregation vote put confident optimal lead policy value compute expect table assume table improvement intuition choose ask frequentist accurately limit accuracy instead optimally budget crowd discuss policy beta rich variety shape domain instead fix begin budget equally label procedure uniform distribution simulate section reasonably unless skewed term uninformative prior jeffreys adopt uninformative prior discuss fact bernoulli update th write current easy next choose current budget sequence formal collect historical labeling section allocation process phase phase since allocation phase aggregate aggregation process terminate stage maximize conditioning minimize inside binary write depend historical define q determined rule follow I proposition completeness place plug hand simplify estimate label confident
provide acyclic model definition say child causal divide class parent parent call discussion key regard principle identification causal fig cc ccc specify assignment x dx v give two variable unity accordingly reflect check conditional appendix accordingly determine imply way truth define identification structural causal advance contain finite unknown permutation identify boolean estimate enable conditional various boolean problem binary list compute list propose
uncertainty inherent underlie simulation uncertainty metric objective design uncertainty rise broadly design decision however probable approximation form value mean function taylor hypercube typically tail probability uncertain quadrature alternative single moment definition function uncertainty desire seek point match target density estimate pdf formulate suited software design improvement discretize pdfs histogram difficult optimization detail design pose elaborate essential heuristic efficacy seek pdf uncertainty target objective resemble target response physical
inequality show indeed rate proposition handle slightly submatrix q note divide block row block element word block diagonal strictly triangular figure form block estimator circle outside truncate know row
type product result applicable variety specialized high space decision involve form need decision leave interpretation completely computation aggregate vector also approximate architecture concept resemble speed assumption normalization derive constrain assumption vector normalize value positive none agnostic normalization provide conservative bind validity approximate allow algebra library demonstrate library significant
exactly laplace conjugate however mb mb model mixture distribution common rkh mean mixture x mb demonstrate note provide upper input pl w pl I j x j x fourth equality employ pl prove statement estimator iii pl consistency pn l consistency iii n function include upper sect denote mb sect mb f g ij derive equality mean q w mb estimator I I te j te f fourth equality explicit inner operation sect sect eq h q mb derivation thus omit present definition elliptical condition generate elliptical begin invariant rotation orthogonal matrix spherical characteristic scalar describe generator
form comprehensive comparison synthetic tensor promise tensor completion outli detection acknowledgment national china zhang national china grant china grant cb ph department computer science engineering china laboratory brain processing brain science institute institute technology interest tensor vision brain interface publish international ph china china currently laboratory advanced brain processing brain research ph china china current interest visual inference paper international receive ph dr degrees electrical engineering team laboratory brain translate chinese processing received work university brain institute institute international award award award proposition model outlier
description set first super website game round team attack serve gain opposite proportion point win attack point serve proportion opposite team remain part paper organize section definition compositional regression study methodology super remark transformation introduce
affect know coefficient importance recover probability sparse may correlation also error compare practical assume jointly observation correlate setup necessary snr note let variable mean iid zero conditionally analyze require correlation decay comparable limit fundamentally correlation f characterize achievable error set w bind recovery vs fig degenerate since possible necessity sufficient exact model along us recovery figure snr cutoff regardless measurement cutoff relation snr low theorem asymptotically identical bind incorporate show increase relative noise consider observe describe exhibit nonlinear noisy upper model mainly extra term denominator term reduce
break segment power whose fix exchangeable incorporate law objective power law simple iterative algorithm optimize datum compete baseline edu cut objective normalize cut cut become year widely simple computation cluster cut still suffer several require advance tend uniform case cluster importantly objective cut equal produce consider application normalize follow segment law frequently power cluster census fail cluster phenomenon intensity
collection globally ii maps ground truth map detailed consistency across map across truth relaxation tailor input investigate encode specifically encode binary I shall doubly block encode partial map I n nm diagonal notational input map encode constraint useful j component methodology novel sub start discuss ground map universe object element object truth shall connect point associate speak underlie universe clear clique positive psd explore collection obtain reliable universe motivate develop relaxation lift tight outli pruning turn constraint remarkably improve theoretical two matching procedure tracking spectrum bias effect represent candidate provide small degree exceed pre estimate h output decomposition estimate method output position guarantee maximize correspondence additionally map inherently natural add encourage intractable propose relax constraint semidefinite refer regularization compatibility sensitive default formulation
nonempty must contain computable use computable take example effective formula classical calculus expression effective examine finitely hyperplane computable half coefficient computable point boundary convex effective reference concept feature effective always behave concept class learnable effective david countable point behave
image percentage annotation annotation correctly predict annotation annotation train document representation rbf svm parameter rate weight rbf choose annotation word image base solely visual quantitative comparison feed visual word l c annotation separately modality figure exploit position usually beneficial exception grid slightly topic superior perform publicly separately accuracy less report highlight modality jointly modality treat separately pyramid adapt image classification representation vocabulary near knn prediction annotation prediction random split annotation figure confusion topic explore semantic associated class label unit visual annotation large connection visualize patch extract learn association window person word window deep large challenge multimodal art baseline multiple tag annotation among image class car etc
parametric test use assignment satisfy parametric assignment consistent fail parametric take general decision fail hypothesis significance classical model set model modelling statistical kolmogorov consider decision represent mean random identify refer determine less student hypothesis decision assignment reject identify hypothesis generate note equivalent enforce leave strategy direction variate
rather distribution take graphical consideration identifiable graph distribution find access access unbiased oracle access take access method statistical call fx mass definition statistical average dimension fix search pair consist set valid possibly solve call unbiased oracle probability unbiased graphical node degree computational cardinality assign accord soft let equal contribution choose
divide rank various hash perform lsh top rank hashing choose publicly ep news except mnist representation mnist consist handwritten mnist commonly generate two big partition create hash refer training statistic dataset dim std mnist news evaluation asymmetric section hash hashing describe symmetric hash base asymmetric lsh asymmetric lsh asymmetric hash hash measure task gold standard element hash indicator subscript distinguish generate sort function hash underlie follow compute gold rank list rank suppose gold increment count move recall balance obtain rank relevant average summarize clearly well hash hash confirm product hash different sign well lsh ht
image person report field well improve start author adapt negative pair domain experiment network illumination pose change highly nonlinear exact section architecture neural network output predict handwritten digit person identification subject generally style network apply include send probe figure compose convolutional cnn cnns connection exist share bias could share deal task person call mode view figure pooling connect channel convolutional pooling dimension pooling include channel filter filter neuron capture body train cnn person three train independently
variable derive subsequently related latent parent
exploit fisher consider score lead various score auto encoder efficiently estimate close tune score fit g autoencoder obtain estimation employ feedforward feed mild degeneracy assumption weight tractable output moment propagation procedure empirically improve dimension reduce term weight layer feedforward learn hide however incorporate label consider set learn correctly present
closely trend success field divergence increase understand scale function obey graph synthetic algorithm exact issue compute weight visible unit allow scale divergence rbms give scale arise deviation weight drop rapidly provide add decay success qualitatively necessary scale constant typical ml scale error obstacle apply natural generalization success state decrease case also adjust process risk produce address pose large wherein mass investigate superior result substantial improvement achieve perfect inferior fact need still later strategy divergence restrict boltzmann question whether substantial difference divergence find ml cd layer rbms visible hide cd training optima cd optima differ optima quite cd vector bias difference percent limit noise suggest optimization substantially difference tend unit model bernoulli tend distance arise quality find bernoulli determine relative error tend error machine trend significant location cd optima quantum closely current art reasonably suffice wherein cd ml base layer boltzmann machines quality optima two learn graph algorithm train field polynomially overlap main model exhibit superior train boltzmann slowly optima visible range consider important bfgs single visible four hide approximately machine unit unit although
homology vary work imagine side also cloud choice dash persistence diagram dot death indicate birth examine gradually neighbor note version point cloud union ball radius around cloud intersection increase infinity evolution homology persistence diagram persistence work space almost merge quickly merge grow merge birth death pair diagram persistence diagram three hausdorff z unit sphere thing restrict theorem close
provide user another sp behave sp stop tc name publicly past publicly available corpora corpus spam corpus perform split document category ten category tc tc dataset occur publicly surface feature word name word employ base model subsection select close example batch large great result method sp sc l spam fold avg cat avg avg dna fold avg cat avg avg avg avg fold avg macro avg
intensity neighborhood comparison package contrast take sparsity mark exhibit least exploit auc subsection top besides outlier furth outcome bias subset drop contaminate numerical experimental denote collect paired comparison live attractive property pair balanced live include video video database correspond round pair top vs blue circle c baseline rectangle baseline base rectangle pt line indicate line treat outli pair comparison pc open circle circle table base rectangle baseline base rectangle simplicity take reference illustrative video detect pair comparison pair comparison agree opposite occur video video arrange accord score calculate pick corner outlier preference order l addition corner confirm top pointing
region multiple interval ideally imputation corruption node varie opt clarity empirically digit response anomaly alarm depth leave pixel alarm vary detection alarm realization indicate statistical parent cf choose anomaly corruption alarm model show label local partitioning consider acyclic accurate mainly happen conditional explained distance pattern although label parent euclidean distance assign corruption nevertheless possible order reasonable euclidean contrary case rank euclidean rate detection detection corruption localization detection empirical frequency truly instance attribute alarm attribute localization corrupt experimentally localization small corruption widely spread localization corruption decrease since euclidean fraction attribute deviation algorithm local anomaly hence corruption stop lead distance emphasize alarm localization operate single phase alarm euclidean make determination reference imputation affect correlation imputation wise imputation imputation wise imputation test quality distortion corruption imputation define versus alarm large imputation even imputation case euclidean produce superiority unlike mmse replacement corruption imputation fair mmse imputation visually regard gradient naturally
incorporate additional pair specific bivariate incorporate constraint fourth problem superior recently undirected realization vector population relate feature specifically conditionally covariance inverse induce sparsity regularize x unbiased problem formulate covariance backward iterate always maintain dual estimate early dual use solve analytically close expression optimality rewrite soft thresholding apply entry observe yield x thresholding via identity convergence detail minimization sparse algorithm covariance tolerance backtrack step k x k covariance duality gap size bb step equation backtrack line conduct iterate satisfie k backtracking criterion safe take give write smooth
consider nominal feature mod task feature discretization either round real take allow accuracy many value oppose working mod metric despite issue frequency accuracy experiment synthetic uci value near portion create behave voting portion cause neighborhood correct output currently explore little value neighborhood give output use fold cross validation na I neural layer
stop dataset comprise evaluate performance system label tuned learn grid unit number use sample randomly dropout shot computer core gb ram want near neighbor share confirm network semantic h restaurant plane near display capture semantic well network special train deep layer dimension allow datum visualize network embed movie figure leave dnn comparison
like segmentation label may come maintain ambiguity propose encourage separate predict hide joint build map average interest optimize output significant improvement jointly example extraction semantic natural unfortunately graphical marginal hard structured efficient popular generally implement slow especially smooth general framework transform iteratively widely particularly compare consider output characterize unobserved suppose h
stop class text radial feedforward primary difference activation sigmoid statistically significant selection transfer kind activation probabilistic preserve neuron weight input pass input output threshold modify function arbitrary layer determine function capable g equivalent case create entire category capable radial basis variable act inference subspace topology fisher replace sigmoid activation function neural generates predict target mean score
machine learn categorization popular learning thus exist spectral mainly find method select successive regressor appropriate select output identically pair either categorical classification consider vector vector responsible predict lasso efficient feature optimize k regularizer lasso irrelevant lasso number sample dependency limitation wise linear introduce transform instance term instance use select kx nn
yu wang xu department university com com mail li old email cs edu aim dimensional redundant feature attract interest feature nevertheless serious framework neither intuitive framework lead guarantee consistency attempt drawback develop sparse cluster concept partition previously intuitively explain principle cluster realization extremely implement interpret exhibit well feature important high sample size problem example feature penalize framework analyze regularization analysis high report really rigorously feature comparable difficulty partition statistic set intuitive definition interpret property organize notion formalize framework k mean section develop brain
denote multivariate usual unit order construct information mutual first introduce correlation information multivariate mutual kl correspond terminology total correlation px extent explain correlation reduce argument mutual common exact explain also independence relation therefore construct model appear interpretation contribute successively tight constitute px conditional table representation quantify search maximally definition representation rbms auto describe generate grain hold negativity information representation thm next repeatedly invoke replace
manifold deep linguistic want require embedding relational schema hope tackle task similarity work center contract reproduce conclusion recommendation reflect work lexical representation map dimensional instead advantage capture uncertainty product cosine enable expressive boundary distribute embedding benchmark embedding asymmetric novel task retrieval relation extraction semantic model map goal low embed object map nearby embed approach prove single space embed represent estimate concept typically
neighborhood close point know kl algebraic case satisfy property continuous continuity modulus throughout compute find engineering application sparse take regularizer convex adaptation split termination criterion meet go optimality subdifferential convergent pass see point establish dr heavily dr motivate call envelope convex moreover alternatively relation elementary relation relation complete exist furthermore cluster rl moreover must simple middle comment split proximal mapping strongly modulus behavior splitting notice last next use minimizer relation equality
task difficulty acquisition address acquisition follow step task evaluate search criterion framework depend location extend solution seek reduce relative uninformative uniform intuitively maximize space differential affect entropy condition integrate gp hyperparameter full outcome select constraint hyperparameter several turn spirit discretization expect must approximate drawing find objective function constraint gps satisfied incorporate simply cost per become q illustrate minima disk indicate evaluation online topic
weight max set triple correspondence numerical specify assignment accord dag represent independence give bayesian structure challenge conditionally test stochastic alternatively structural pose relate avoid commonly include equivalent criterion dirichlet uniform depend local score function store base network learn seek dag parent e computation take unless retrieve constant despite optimization certainly introduce direct cycle say cycle undirected graph undirecte cycle four clique one undirected graph minimum undirected graph connect child drop network dag node order pairwise admit elimination elimination elimination elimination perfect elimination reason free
illustrate graphical represent circle model section propose priori across convenience generic build transform cumulative gaussian denote gaussian multidimensional methodology present care define field covariate distribution new apply marginal obtain beta idea include transform within stick break previous article copula process fully function covariance write normalize square rational quadratic choice apart shorter adopt deal gamma rescale bandwidth minimax contraction rate indicate correctly path smoothness function study behaviour covariate deal process multivariate gram write estimate
intrinsic dimension open supplement offer insight norm scope need challenge supervise asymptotically sufficient practically beneficial evaluate overall variable appropriately acknowledgment partially support content represent view national science foundation acknowledge idea height em bandwidth algorithm laplacian exploit connection manifold riemannian operator set preserve experiment tool semi model parameter neighborhood formally put common unsupervised obtaining optimize construct empirically geometry estimator manifold interest asymptotic
translation result significant translation effort machine phrase system recently entirely network promise system exist phrase decoder length encode sentence sentence sentence long long corpora sentence additionally nature sentence representation neural fail translate clause translate improve long translate segmentation encoder model encoder decoder independently consist rnns act encoder maintain hidden update decision
retrieval specifically query find problem query across search engine length utilize probability pareto log suggest density evidence indicate underlie typically concave pareto database automatically assign keyword annotate middle middle pareto transform label class label major annotation find organize discuss front property pareto pareto method result pareto visualize partially order retrieval decade image query provide user measure texture propose sift extraction vision research query query relevance feedback issue individually retrieve visual attention retrieval bm vector link manifold effective database rank anchor ranking assign pareto front wide pareto machine complex objective find pareto front use
number advantage regressor sgd accuracy conclude remark denote trace determinant semidefinite stand expectation sample sample convex strongly gradient descent size iteration identity reduce hessian quasi whereby attempt matrix method bfgs work since bfgs curvature finite define select tend completely resolve bfgs possible hessian differential close permit update hessian implementation avoid formula stay definite arithmetic operation step newton cost total computational bfgs reduce scale likewise bfgs storage motivate memory objective limit describe curvature curvature proceed previous curvature information idea restrict use early current iterate expect precise pick positive definite form curvature perform refine approximation cf plus inverse yield complete update recursion although expect recursive product information reduce storage operation yield iteration context compute gradient gradient gradient realization gradient curvature descent function past gradient gradient simplify discussion gradient associate subsequent cf purpose natural stochastic see need stochastic version identity quantity eq update matrix recursive except hessian approximation
model extreme regression special dimensional vector diag tr write exclude aforementioned approximation hereafter linearity long require likelihood ratio find parameterization parameterization interest note systematic specification dispersion sub sign obtain diag diag diag diag diag diag diag diag diag gamma sign replace diag wu adjust sign statistic q q exclude
collect approximate inner element wise approximate derive commonly kernel square distribution fact random kernel limit shift invariant instance relu invariant cosine angle compose alternate follow connected layer top perceptron much connected layer account often extremely redundant greatly layer replace new perceptron illustration despite simplicity feature storage reduce require name reduce constructing introduce eq diagonal
structure parsimonious rank decomposition see comprehensive application subspace track history representative recently term subspace observation incremental subspace converge rate incremental second seminal handle miss minima see lack unstable behavior especially amount accordingly data paper imputation offer provable performance stationary flexible accommodate model leverage subspace estimator exponentially similar put anomaly focus incomplete measurement upon separable nuclear amenable subspace track complementary strength convergence simplify technical assumption propose algorithm provably attain nuclear optimality complement claim present algorithm decomposition accurately simply reconstruct cube model assumption leverage stochastic case entail fitting criterion frobenius decomposition propose online offer solve large tensor massive main simulated internet traffic traffic anomaly superior efficacy conclusion draw bold letter denote letters hadamard cardinality magnitude denote n additive signal stand index correspond sampling entry
construct generate binary supervise encode semantic thus metric merge forest powerful semi random construction two contribution previous nonlinear point collect relax every available node focus address tree situation incoherent thereby data algorithm robust additionally propose tractable neighbor major limitation tree remainder paper describe detail hierarchy forest metric max learning section near near neighbor complexity compare forest element forest tree independently segment distance conceptually represent rather distinction
covariate principal involve reconstruct deconvolution impulse non somewhat similar bias traditional make deconvolution approach deconvolution dynamic voxel apply deconvolution substantial million spatial voxel temporal actual moreover focus study integral impulse voxel biological assumption particular assume moderately inspire deconvolution multiply assess moderately realistic image slice homogeneous take provide quantification discuss voxel eq input impulse response voxel reality contaminate version decay voxel ij j largely decay nature estimate deconvolution truncate could situation difficulty function reconstruct handle curve voxel voxel spatial spatially voxel voxel three signal dimensional smooth may available voxel feasible k
ica ht similar calculate mixing coefficient compare order consider total activate reach representation activate hide complete case prominent far complete frequency orientation filter see achieved use spatial orientation difficult propose success training patch train preprocesse datum especially
message pass indicator central essentially pass computer pass coefficient average negligible htb estimate median justification wide number coefficient c criterion q consistency term model note almost sufficient sign relaxed concern condition mean square aggregated choose theorem boost rate heavy tail addition preserve almost known significantly load demonstrate
replication per size star dense setting already indicate section difficult star next analysis ask score association biological concern behaviour range perfect datum strength center goal graphical representation estimation diagonal interpretable impose constraints negative discuss estimation lead negative entry graphical lasso impose usefulness describe initial quantify validate analogously regard ten fold value except report loss sign level come rather feature induce association specie link evolutionary model flexible alternative keep mind validate liu tree support latter extra relative validate impose sign concern spirit one consideration base help answer concern attribute answer clearly amazon simplicity treat minor modification reduce five fold cross different tuning entry square block contour present section modeling contain face dataset employ comprise collect cf panel
oppose mapping object localization recognition place categorization environment another therefore large database vast place recognition develop visual distinguish use global feature computed model feature cluster visual quantization discretized lose generally speak
identical present solve proximal optimisation operator regularizer complicated structure point use write make symmetric dual algorithm h constant initialize round dual main row feasible essentially decompose original solve operation complete sort ordering minimize plus differentiable subdifferential one subgradient hence actually ascent since fix sufficiently convergence use
parameter dedicate learn test section vs enable model one model force generation model ten discriminate related specialized high vs assign example assign trajectory posterior assign great specifie specify format allow file value file file load sep specify file load weather specify column file load default file incremental number column trajectory possible trajectory file allow specify file specify column except consider specify name model validation partitioning file section txt txt class probability txt true test predict txt predict txt predict file identify specify load txt txt txt test txt txt test txt txt file cv allow remove name file reason txt ex txt test txt txt txt ex txt ex automatically require definition file reduce trajectory percentage datum line force trajectory original default specify trajectory trajectory time transformation specify file file file file file force soon datum training section file file file file test file specifie store
length note red marked amount variation popular vision circle fashion trivial perform example instead insight pixel classifier setup expression make remain aligned translation example hard dataset perform second interaction pixel perform prior normalization add capacity locality go like error classifier fall improve attribute solely
dependency get draw testing distinguish suggest ratio test helpful two observe problem always since hardness stay roughly problem lead difference correlation random variable convention chi degree freedom deviation strength correlation estimator maximal risk use c universal
circumstance resample though dominate coefficient take account via calculate uncertainty uncertainty suggested discuss monte accounting uncertainty return coefficient
label counter simultaneously learn representation confident infer multi annotation make inference image competition infer class investigate idea carry preliminary end jointly fully instance
end factorize eliminate elimination operation marginal gm similarly priori would gm group gm linear achievable grouping might simple tree correctly represent grouping clique resort gm become exist gm remain far possess maximal clique vertex clique contain form connect arise illustration implication take complete three
require year asymptotic contraction rate regression et al stick obtain high van categorical consider allow exponentially bayesian finite spline construct show smoothness agree logarithmic use smoothness density conditional allow devise typically reversible vary give group dirichlet coefficient conjugacy structure utilize base direct univariate computing method close posterior discuss tensor spline contraction section uniquely closure indicator denote inequality pack symbol stand generic
state art mb mb interesting implication cnn times efficient embed device onto embed trend mobile bandwidth allow send typically applicability platform order storage requirement consider cnn eight layer three parameter art widely heavily parameter explore interested parameter reduce dense layer run convolutional layer network early work cnn publish explore method test researcher show
good distribution parameter respective ad aic p statistic approximately chi ad von give fit distribution plot cdf survival function fig distribution comparison ratio lr likelihood ratio freedom df reject significance significance significance conclude h freedom exp u ax b therefore bx
shorthand position define vector intersection hyperplane illustration characterize independent row algebra state clear linearly row move discuss position zero entry use arbitrary contradict one simple extremely zero non contradiction form zero contradiction discussion determine mind dependent identify give characterization notice represent theorem document iff row say play crucial lemma iff proper subset fact verify trivial would nonzero entry linearly first accordingly generality zero write immediately statement suffice bb linearly independent contradiction reduce q know nevertheless last contradiction row position position row linearly assume loss row loss generality may scale construction otherwise accordingly linearly particular algebra since transpose entry back common polynomial zero f entry theory algebraic zero ready course still subspace example contain vector vector incomplete
segmentation theoretical option relatively measure color let location color distinct radius enforce sparse greatly computation decay small use spectral hand label label number cluster human median clustering complete place specification appendix theorem hold idea sphere give basis system unit accordance p right origin maximum optima fully optima relative sphere body satisfy strictly constant mh local specify hx x hx e ii theorem ht h xu optima outside consequence necessity demonstrate necessary maxima valid assume twice differentiable twice maximum strictly g g conclusion enough convex convexity convexity
tailor speed difference root able generate candidate metropolis satisfie exist help sufficient location mode location divide distinct modal density design assign region determine likelihood know evidence candidate note pick requirement region decide candidate current locate normalise take center
htb dot curve cccc word symbol represent heavy tail convenient method fractional detail sphere characteristic stable graphical sg acyclic bayesian acyclic ordering order fact triangular diagonal determinant triangular equal jacobian determinant transformation z j hence bayesian sg multivariate definition sg imply concentrated number unit sphere sg represent lemma every use order unique use index base true assumption dimensional stable multivariate form forward variable verify bayesian criterion bic minimum score select acyclic major special represent gaussian
dynamically observe network observable represent table observable hypergraph edge characterize hypergraph basis move move edge summarize term adopt r negativity table move applicable ensures move task suffice goodness fit purpose large applicable discusse construct applicable metropolis walk output nf probability construct fu tu k symmetric periodic hasting ideally structure hypergraph employ crucial usual rely basis markov basis basis due rejection usual metropolis draw full markov impact contingency entry entry independence replace produce outside interesting similarly suppose independence hypergraph walk primitive closed correspondence primitive primitive walk correspondence perform edge say walk degree move depict hypergraph represent edge highlight table highlight seminal dyadic relational summarize
detect technique principle real access exceed accuracy specialize nearly discovery rely ability automatic discover system compare record classify odd rapid process quantitative knowing look search specify distinguish principle remove traditional key include correctness dependencie digits symbol acoustic intensity traffic road anti stream opposite stream produce inverting sequence appear digit common anti stream vice anti stream statistical relate stream character symbol simplest map symbol generate string select stream generate anti stream anti stream stream flat stream mine heavily prescribe encode stream result stream leading cascade application involve decision rgb rgb rgb rgb rgb distance font none fill text white none major axis axis none width height gray thick axis none axis width dash gray none none width major style dash gray line axis none auto distance font draw bend leave yshift b loop yshift bend auto font bend yshift b bend leave node yshift bend yshift c bend bend yshift xshift bend yshift xshift yshift align font hide xshift yshift center gray yshift align font space text black font align align center distance generator capture statistical symbolic stream model construct require either exact alphabet admit wherein trivial usual operation context key group observe alone synchronization reference generator imply sum unique hence generator make stream anti unique zero characterize arbitrary alphabet size minimal symbol realization variable uniform symbol alphabet entropy among sequence generate identical distance thus exploit observe alone without require give stream intuitively observe symbol alphabet deviation historical ensure contribution address occurrence system identify eeg hz font font thick style gray bottom font false axis scale style format fix cs cs axis cs none inner axis cs fill white sep none fill sep axis cs ia ia nature hide fail heart ex file sample letter font font background gray bottom font width scale x font format scale ts axis none inner sep none sep axis c ia art art ex series letter font font thick top bottom font axis axis false scale false style font cs none fill fill text inner draw sep ia accuracy achieve specific hand optimize classification ia eeg visually potential subject variate trial quantization letter style font font name axis top color gray style axis width height scale false format txt cs axis cs none axis none fill white inner sep cs knn ia achieve consideration database series letter font font thick axis style gray style height scale style font number format fix axis cs none
design investigate let select play take time play play refer machine play machine early central distribution new carry follow h playing describe function qx upper quantifie
cascade maxout last layer feed directional rnn hybrid architecture combine ability maxout transform nearby precede gate account term train observe condition add decay score phone basic treat error recognize convert sa testing model evaluate auxiliary development gmm dnn gram network use similarly result testing validation adapt gmm build frame feature per gmm force align library
translation impractical mechanism comparison summation applicable solely logical limitation formal logical reason complex language group arithmetic contain logical predicate mix rational constraint boolean value problem logical formulae researcher maximize formulae maximize amount formulae strictly solver
indexing search many retrieval key practical development locality hashing lsh rely projection retrieve near grow database long employ derive section work visual near neighbor asymmetric excellent performance hash kernel classical hashing variant nystr om able angle show angle provide theoretical issue around lsh pca demonstrate tradeoff classic bias variance tradeoff lastly importantly potential boost validate technique retrieval recall query implicit reproduce interested find database query lsh function result hamming note
advanced scientific technical result statement make previous section auxiliary approximation important semi matrix seek perturbation identity rank frobenius column identity end write symmetric gradient functional gradient iff condition equivalent iw nonzero eigenvalue solution critical diagonal either monotonicity subsequence prove distinct satisfie importance next negative look prior matrix root covariance update semidefinite update rank converse identity semidefinite update main approximation covariance minimize generalized belong also specify finding minimize invariance w lead identity unique th corresponding leibler divergence rank eq belong approximation follow argument hellinger hellinger dimension minimize turn q belong h proof turn optimality posterior root possibly root equation furthermore lead result svd use root square root w norm entry back yield ig theorem optimal give pseudo reveal q v g dd di dd di I expression recognize optimal mit edu inverse problem characterize approximation lead large posterior chain
compute power square polynomial let fix exist right equation p proof equation approximate factor polynomial extend parallel inverse eq fact approximate factorization polynomial apply task one construct sparse matrix om depth number simply maintain non achieve present preserve p refinement behavior analogous
model representation largely representation semantic phrase evident early attempt distribute multiplication representation representation model relational order contraction adopt mean composition observe compositional approach sentence recursively network vary semantics share take token
ph interest statistical sm sc engineering work research member electrical engineering digital communication mathematic paper maximum equivalent look parameter meaningful interpretation context synthetic coherent illumination call interference ratio describe scale moment quantity negligible substantial moderate thus correct bias used image propose correction ml estimator profile
ordinal input covariate cumulative know ratio ordinal good choose review development attract attention past decade add margin statistical multivariate ordinal study decade latent generate rbms interaction advance massive develop specification well procedure ordinal reach rbms wide generate response recommender review indicator current rbm hand ignore nature treat drawback category interpretable ordinal modelling ordinal adapt ordinal utility along
datum since typically expensive besides handle basically relation nlp base domain adaptation attention domain adaptation aim target consider domain adaptation validate assume underlie unknown domain fact difficult manually divide discovery discover domain discover testing benefit include object vision multi recognition detection part former vision latter rarely address viewpoint domain however introduce interesting challenge deal unbalanced vs background illustrate assume target traditionally target pool domain tree apply information capture datum regularization fact adaptation svm give rise concept sect adaptation sect develop multiple sect source objective adaptation target
dynamic certainly machine generate infer bottleneck machine future develop rather extend graphical compressed arbitrarily redundant despite restriction prop distortion suggest distortion property optimize predictive process symbolic dynamic dynamical distortion analysis provide tool identify stochastic phenomena key feature principle approximately train ref science relate kind organization encourage complicate quite fully automate author helpful member material upon support part laboratory office contract nf sm foundation fellowship berkeley fellowship proof elementary markov highlight statement prove repeat formal solution ref come anneal anneal objective find equivalence distortion straightforwardly consider process associate codebook sample unnecessary distortion codebook denote unnecessary distortion implicitly specify via instance codebook codebook realization codebook codebook codebook process distortion measure eq quantify probability distribution shannon jensen divergence q codebook consider codebook contradiction codebook random expect distortion codebook code incorrect code desire use early complete sketch next consider class state realization code information distinguish new rate thing proposition main prop relationship
three article fast symmetric symmetric factorization symmetric factorization identity hierarchical structure scale validate scaling encounter rank diagonal particle scale within group nest matrix hierarchical extension fast symmetric definite hierarchical factorization factorization scale allow generation factorization algorithm depend first arise dense arise dense fill radial density kalman filter efficiently recursively tree matrix represent
video record view record paper multi metric video combine advantage margin minimization ability find good datum meanwhile learn systematic video knowledge multi view video viewpoint metric effectiveness usually highly kind encounter processing
project layer denote transfer define sign derivative denote n f z ii training backpropagation make playing play put proceed one ann rule show clear index variable lf define l w l computing let play normally implication analogy usually backpropagation w ij l base lead computing ann b ann bias
j metropolis former latter draw df conjugate admit ss admits demonstrate df replace obtain approximate online distribution stationary ij l conditional j conditional close expression say base income suggest identify interested conditional inference nuisance df partition l j j update g approximate example c df choice partition df simple serve intuition parameterize sufficient computational benefit main assess sophisticated consider probit df augmentation probit df model update variational admit online derive approximate df conditional see df conditional distribution admit surrogate metropolis applicable conditional approximate c df measure define large plot observe time appropriate arrive sequentially horizon approximate accuracy addition square replication error appear table plot show representative parameter simple predictor proceed
try training well expect consequence easier generalize dl compare identical except time dl task complexity pattern able generalize pattern reservoir dynamical computation dl memory computationally capacity create dl place reservoir mechanism study reservoir usually place requirement reservoir understand effect reservoir architecture task varied requirement nonlinearity greatly outperform generalize novel increase reservoir dl result overfitte lead generalization
machine problem infer ode infer ode classic purpose well regression attract return instead solver way particularly candidate probabilistic solver deal uncertain definition gp ode share construct ode family first third yield combine strength gp ode solver interpretation classic question fit fit gaussian linear member family fix statistical ode find ordinary differential hold solution simple treat base extensive
dramatically resource variance first employ analytic expect quantity parameter stochastic expensive rely discrepancy mean measurement exceed threshold trajectory exceed threshold involve reasonable number deterministic behave related trajectory trajectory use eqn fall outside abc immediately reject determine check whether calculate exclude protocol implement rejection threshold perturbation obtain new meet describe theoretical birth death process replicate rate illustrate birth process trajectory record randomly time
performance graphic accelerate pass cnn extensive implementation cnn computation processing accelerate separability network filter difficulty optimization penalty consecutive recent speed cnns convolutional rank scale train minimized accuracy demonstrate speedup convolutional keep cnn connection layer structure state art cnns fully connection training output connection cnn also connectivity investigate open far apply structural conventional feedforward acceleration cnns constrain path difficulty optimization successfully learn
nonnegative since collection dx grid case author grateful valuable manuscript valuable point author grateful know elegant book valuable grateful dms support center nonlinear grant dms support fact lipschitz sequence argument conclude enough theorem find sequence bound constant bound sequence du exist x u nx dx axiom claim conclusion theorem conjecture criterion definition lemma remark pt measure euclidean domain cloud weight based develop available learn good address question scale connect hold metric allow suitably functional goal develop mathematical rigorously problem go infinity application establish algorithm task largely determine minimizer converge increase functional set setup cloud define sufficiently weight computational task represent cloud minimize graph functional cut balanced cut variation range variation term total dirichlet
dimensional topology induce hellinger distance expand haar volume support wavelet derive rate upper density easier successively introduce notation support rectangle edge power assume support edge fact support equality plug normalize support require support front support rectangle tensor haar rectangle number solving reach f application variable advance haar achievable traditional achievable convergence tuple nonnegative integer differential define differentiable density achievable major density affect decrease quickly cite straightforwardly advantage fact quickly restrict attention
new alternative discriminative efficient spectral matrix weight fast theoretically discriminative extract feature feed quickly classification moment label raw e guarantee contrast score feature show score provable semi set unlabeled handle processing framework operation also scenario input source instance crowdsource application different feature input
considerable information state detect background intrinsic physics express brevity remain experimental match normal true model rarely know applicable adopt k deviation measure trace discrepancy drawback error matrix method ideal ideal deviation
regard pa several test investigate mean et al exact difference log
effectiveness illustrative simulation two inherent graph also boundary fraction vertex match via increase fraction increase vertex utilize dissimilarity plot dissimilarity figure figure dissimilarity run begin simultaneously serve highlight strength cut edge pair new perturbation follow probability probability graph independent level increment increment dissimilarity dissimilarity dissimilarity datum drive though point future dissimilarity flip parameter seed increase indeed performance os enyi highly
set begin choice inequality communication engineering apply mathematic hold institute school science currently ph electrical supervision image process computer b ph vision framework employ dimensionality base exploit fact enable geometry formalize operate potentially use column algorithmic utilize separate convex denote nuclear column identification subspace span low component inference overall depict correct low rank component transform overall identification sense carefully column know facilitate compressive matrix column bernoulli set second approach reduction identify outlier nonzero subspace span low rank simplify significantly high ability identify outli detection column sample bernoulli regularization orthogonal sense acquire exploit devise reconstruct object sequential numerous recent area cs summary article therein subsample inherent strategy parallelization utilize one partition effort utilize formalize rank plus effort early robust seek sparse extension entire otherwise low analysis direct seek
mostly perhaps significant problem event city edge section time interest pm road count graph block day week eight measurement count note tune substantial observe description news st traffic could ground truth trend laplacian two panel compare laplacian term smoothing sparsity impose already spike able laplacian far localize event display throughout distant decrease flexibility increase display node truth asymptotic trend focus broadly throughout concrete statement graph defer similar argument basic inequality square q denote connect quite general yield sharp trend filter trend trend essentially simple inequality large k k small loose graph control bound converge nd stronger bound trend recall chain operator fact operator link trend tight univariate trend filter minimax optimal prove connection trend locally adaptive spline sharp rate adaptive spline
isotropic change deviation away approximation isotropic tight logarithmic perturbation let suppose q show group moment desire want sum p kx I pm tx call block block rescale give lemma relate gaussians gaussian eigenvalue project note symmetry triangle suffice variation dimension rotation invariance scaling assume term symmetric term term case option two eigenvector single dimension empirical transformation I getting let arbitrary probability concentrate univariate degree lead factor get eq proceed induction formal induction condition univariate get result reliable verify symbolic multiplying expression standard code explanation formally variable mu sigma sigma beta mu moment sigma sigma sigma sigma sigma mu mu sigma mu sigma sigma convert mixture mu sigma sigma sigma p sigma mu sigma sigma sigma claim excess moment alpha alpha gamma alpha alpha alpha gamma alpha alpha alpha beta gamma gamma alpha alpha alpha beta alpha beta beta match alpha mu beta mu
either hyper bf eq incorporate redundancy flexibility illustrate call redundancy scenario predictor association essentially specification receive
description region methodology offer computational economic couple generate concept financial volume factor five nominal past year form international web trade investigation economic growth flow economic readily project economic make cross country trade flow clearly united china macro point picture near assume weight node united states china within capability unite year china steady score global trade measure might node flow flow contain one element zero I act gate working show stream enter go stream flow letter one evaluate
movie arbitrarily movie select prediction rmse separately movie run evident qualitatively omit baseline arbitrarily movie baseline denote subset least since outperform base estimator user rmse metric baseline ensure optimal design base outperform baseline account identically distribute practically indistinguishable utilize rating conjecture clustering seem perform space result reader train state percent start try tackle item answer give rating collaborative filter considerable rating
novel gap free low match bind episode choose th pair suboptimal optimal difficult discriminate item item index cardinality motivated basis result ta ia ta item pair indicator function function event episode inequality gap basis episode regret episode indicator instead item episode kt kt follow every suboptimal match remarkable aspect regret decomposition rest gap expect cumulative regret regret episode choose
perform choose solution axis robot alternatively wish see vs height expensive behavior store difficult visualize refer define space behavioral characteristic behavioral dimension variation height descriptor controller genome location create behavior simulate space produce controller dimension behavioral robot weight height map beneficial performing solution behavioral search solution many way perform solution extended search perform search design effective efficient modify robot design begin generate solution evaluate record g space record robot performance current map behavior type keep beneficial reason newly candidate location initialization evolutionary improve generation pick solution mean chance randomly change behavioral determined keep parent meet g time stop map iteration number behavioral adaptation accomplish via k search unknown measuring model rigorous create use select update next acquire prior choice particularly cost nee several f f normal therefore variate covariance relate nearby value correlate via distant influence distribution correlate put differently distant almost nearby correlate kernel noise user specify optimization maximum select next improve explore part predict acquisition section classic likely concept expect behavior real objective incorporated process mean function equation previous replace prediction start code supplementary well store performance map pf function controller physical estimation test nearby test fig square mat ern kernel variant curve mat ern square become parameter fig function mat ern stein interpolation error select extensive acquisition finding optimization reality exact
ts ls l l drug protein drug channel drug protein drug drug protein interaction base chemical unified cv trees svm classifiers particular global approach relate specific choice paper base ensemble drug interaction besides local global include among completion method base base systematically investigate exploitation supervise biological formalize pair homogeneous global local interaction unseen node discuss highlight term interpretability unsupervise bi experiment predictive result family competitive advantage approach structured pair protocol local relate several biological conclude discusse find generality connect
yield least compare candidate rank dendrogram poor subset feature fail enough candidate subset follow average value imply subset may likely contain among candidate discard high rank subset increase rank increase incorporate large mean prefer smaller easily apply default cluster dendrogram previous candidate candidate unknown subset different idea candidate high average subset prefer detail idea list g candidate pt argument default size global candidate every size
element draw close iid influence hence row ensure row adjust suitably relate rbf thus radial normal analyze moreover consequently rescale entry gaussian sign retain checking albeit feature rbf hold considerably block retain change radial straightforward approximately length rescaling direction normalization radial part operator kernel key conventional requirement well need rather discuss determine first kind fourier function ball fourier transform iid use scale appear entirely also surprising rbf dimensional concentrate fix characteristic latter dot draw correspond direction draw operation address gaussian initialization implicitly dt equality homogeneous polynomial second reason stability access
similarity formulate paper good implementing outperform trivial one range learn meaningful conclude perform nevertheless closely probably receive increase online reinforcement customer simulation implement confirm reinforcement use middle plug hybrid home extra electrical deviation
discriminant model training observation class class conditional observation multivariate fit optimal set know covariance structure allow set call gmm matrix structure initial class probability optimal conditional probability bic gmm high solution maximum estimate calculation apply test high insufficient overfitting solution former risk fit information
propagation base graph field tractable quickly converge form slightly informally strength neighbor ise attempt model basic take mixing project different divergence show project iteratively decomposition secondly experiment project use avoid drop kl experimentally perform divergence evaluate respect approximate gibbs divergence approximation expand fully factorize mix extensive presentation wish draw obtain
condition criterion drift absolutely stationary suffice rescaling empirical converge weak combine lemma substantially define knowledge isotropic place integral scaling graph entirely underlying scale near upon generalization truncate connectivity decrease exponentially connect limit corollary isotropic graph vanish vanish bound tail ensure
thus change decay prior direct integral decay exponentially datum illustrate utility classic gaussian second classification gaussian data uci machine repository subspace normal plane comprise subspace plane challenge infer contrast mixture normal level gaussian subspace follow five truncate parameter conditional ten plane table range assignment ten perform poorly decrease temperature acceptance achieve affine precision normal uci repository breast heart breast objective classify tumor ten three multinomial logit mixture infer period theorem remark
geometric designing perform evaluation criterion task answer retrieval feature cost handle tie weight grow establish technique thus picture aim rather traditional setting probe functional application answer question site web retrieval another quality answer greatly question answer question rich contain linguistic often link page relevance search difference suggest object modality represent separately likewise rich content structure document retrieval relevance indicator well decade pair functional capture specification functional rich object first usefulness quadratic overfitte reflect particular place list metric
aforementioned effectiveness synthetic np hard moreover show nmf ill pose exist nmf case rank nonnegative constrain variant semi semi nmf sense discussion detail initialization thank mm nonnegative factorization semi look nonnegative approximation property nmf heuristic error nmf unconstraine singular svd approximation algorithm svd certain initialization extremely well optimal semi nmf decomposition situation np hardness notice paper available treat semi theorem contribution go prove orient algorithmic algorithm provably matrix prove hardness ill nmf imply usual nmf lem let eq geometrically affine triangle big sm tight matrix
mean impose seem return odd finding reduce one return interval involve large point plot return recent level estimate access multivariate quantity mean angular four variate bivariate version angular simplex horizontal dependent angular threshold six pair location excess another location tail outside location fr transform comprise dependence extreme independent explicit expression incomplete beta vary one strong dependence I six plot datum display easy marginal threshold condition threshold return period take excess value point theory increase horizontal estimate dirichlet limit
simulate recall simulate achieve strong integrate way importantly contribute overall independently contribute fraction pathway hypothesis support addition particular indicate location possible fix priori validity issue address future node domain diverse evidence perhaps category prior might evident result domain posterior notice perturb presence pathway however hope divide pathway gene dividing pathway various pathway heterogeneous include indicate
multivariate copula multivariate dependence base copulas quantile quantile work model drive force beta employ rewrite contour positive relationship negative axis remain copulas degree respectively copula possible extend property financial correlation require extra copula copula upper tail copula copula class copula follow new inequality hold numerically careful discover illustrative eq harmonic special dependence copula efficient proposal copula represent copula tail dependence describe parameter copula parameter give relate copula substitute attractive invariant monotonic correlation dependence correlation
assign cox proportional cox regression constant relationship ny py patient period right censor observation maximize partial driving weak towards zero cause instability instability choose highly correlate
markov combine get eq line get discuss subgradient point recall x n assign online example transform choose online lp differently explicitly appear follow dual variable rewrite ty kp mb x ip tx tn tt online toward subgradient size normalization examine four
applicability doubly run dependence question estimation influence zero generalize complicated require work unobserved know help generalization likely generalize alphabet complicate uniformly setting modify graph unbounded degree acknowledgement grateful david graphical topic year I thank comment proposition theorem reconstruct ise physics direct towards various model model neighborhood greedy allow assumption necessary allow ise maximum notation doubly exponentially dependence exponentially doubly probably suboptimal implication learn ise learn ise neighbor
bayesian behave fine grain easy processing aim successively cluster agglomerative dissimilarity new cluster merge artificial highlight interesting power consumption home year highlight multimodal balanced dendrogram concave pareto chart emphasize focus difference able work plan method distribution consider cluster universit paris paris functional sample evaluation pair arise weather hardware quantity sampling exploratory large consumption monitoring reduce small set segment part cognitive ask complexity representation unfortunately induce additionally adjust parametric make sometimes implicit density
know smoothing definite rkh smooth kernel size affect initial step online dependent kernel cv free plug plug computationally intensive suitable widely although multimodal distribution also topic relate goal kernel base normally avoid complexity mention online adaptive mode sample specify priori final practically real work treat propose paradigm adaptation filter size sequentially square incorporated quantization compact rest kernel section
agent pick free agent pick draw join round minimum cm rectangle em green policy update converge limit system simplify ode asymptotically ode sg sp result stochastic quite crucial stable give ne discount single game pure never show ne quick per review section use problem solve game sub present sg sp condition equilibrium counterpart single game stick conclude remark approach nash general discount game minimax sum meaningful learn convergence nash restrict equilibria solving equilibrium traditional nash equilibria propose reference model free prove converge self play however state game objective strategy definition play game infinite sum singleton mdps numerically equilibrium share aforementione offline complete game iteration converge ne design program involve find equilibrium solve equilibria nash strict homotopy tractable infeasible game model free cardinality nash equilibria rational agent combine ne setting
bayes regressor expert predict iteration execute policy regret compare additionally action rl I rl n n tn square go iteration sample collect loss ij ij estimate j ij ij ij km jk bound hoeffding nm empirical regressor rl nm nm nm obtain nm reduction alternate present denote prove lemma eq iteration policy time define guarantee online sensitive j policy I I
exceed provide relation weight simplicity bias processing consider branch integration concentration simulation interval plot want surface change neither close effectively perturbation concentration output surface genetic responsible determining species specie solid line stand specie row previous adaptation output incorrect analog replace adaptation weight sample vector define integration implement precisely rather qualitatively provide time
kk fact lead increase distinguish slow increase ht provide empirical spurious secondly high topic great topic utilize show see solve validate experiment average experiment keep fix norm large match theoretical serve upper red go structure vary keep parameter fix vary fix
top since low stage low stand eigen particularly matrix multiplication accomplish helpful large combination alg show picture stage epoch active triplet sgd recover dimensional extract activation convolutional network imagenet fully feature learn predict conventional reference cluster compute query distance center assign distance performance conventional smoothed number compute explicitly base use refer empirically original recommend fully exploit
less negligible complexity obtain pruning possesse comparable demonstrate relevance acknowledgment partially usa pt cosine require introduce assess compression show implementation nm dct degradation dct play signal image video stage dct dct stage
exp exp persistent homology topology relatively attract great deal allow object persistent resolution evolve identification cluster discover type interest notion explain depend adopt ideal modal yet background new risk similarity clustering applicable identify modal cluster estimator necessary type object formulation clustering comprise partition population measurable content define permutation differ probability detail distinguish concept essential r clustering procedure induce henceforth immediately assign get theoretical focus different clustering ideal population cluster well establish induce precise minimize denote usual assign group ideal corresponding mixture clustering
present combine multidimensional stress least square try preserve pairwise mapping put paper note solution eigenvector diag visualization constrain stage mode point mode mode scale factor stage individually work th jx j visualize overlap remove visualization treat mode improve visualization accounting connectivity measure connectivity connect straight connectivity adjust panel much summarize smoothing size rest parameter free bandwidth cluster bandwidth merge large reduction j experiment parameter bandwidth conduct merge visualize step pairwise high kernel
error ht cv cv error cv error cv cv rbf variant svm superior neural dataset achieve svm evolutionary quadratic evaluating employ carry future multiple hyperplane place kernel evaluate performance hyperplane may reduce particle method technique deal determine hyperplane boundary problem weight use programming constraint minimize neural svm quadratic
check contract observable effort ask worker level pay constraint payment increase back maximization problem effort show contract contract contract contract conceptual contribution adaptive contract design conclusion area version contract round basic agent set prior worker contract specify agent set effort outcome payment cost contract outcome contract maximize minus payment employ contract heavily influence work build line lipschitz continuity discretization require immediately natural shape arm reconstruct approach virtual width similarity information result mab pricing crowdsource market essentially round either reject offer call worker directly thorough problem describe capture extension static multiple worker pricing simplify classic bandit occur complete worker specify worker complete contract derive complete normalization minus payment worker observe contract effort set result complete choose derive contract offer emphasize agent effort randomize line crowdsource effort worker make error worker utility task payment minus cost contract worker effort expect utility crucially effort observable choice task model effort value nan level mapping call mapping effort call worker completely production traditional agent worker observable extension round worker worker unknown contract type effort outcome worker choose effort utility production observe specify contract worker reveal adjust contract round total utility round assumption relax
surface recover transformation shape treat transformation limitation model mesh interact exploit reconstruction enforce consistency surface volume hard surface sensitivity paper representation propose automatic spectral segmentation move coherent infer body action voxel rely body compute accounting technique g rely enforce handle enforce space propagate technique favor desirable segmentation virtue isometry reduce constraint relate body shape affect initialization issue technique pose multiple volume sequence body recovery pose method recover intrinsic shape embed able automatically term consistently spectral see stick recognize perform successful favor extract instant instant explicitly motion graphical hide extract technique reconstruction availability approach promise body consistently propose easily exploit preprocesse stage example segment feed hmms identify understand coherent physical interpretation segmentation initially propose extensive quantitative evaluation real comparison compete move voxel domain embed
hz original coefficient I fit curve original coefficient coefficient whose absolute show equation fig figure spline almost base fit well fit original curve curve
improve error batch elimination eq limitation analysis rely investigate might always functional particular function refine theorem corollary functional three specific functional decision mean risk problem formulation application finance reinforcement learn give unbiased unbiased good sample eliminate round imply mean case upper
accord adopt estimate initialize regard seven procedure em hierarchical initialization hard second hierarchy initialize high probability log high acceleration use estimate decide reach close asymptotic li respectively asymptotic log likelihood q cf em converge q adopt replication ic also definition ic definition aic aic mm definition illustration attention focus accord
see test visual similarity semantic representative semantic engine indirect semantic base advantageous direct verify early base semantic inter develop shot show bipartite computational class set dataset label take first support provide belong see element row test although relationship see class relationship employ word representation vector regardless unseen correspond connect find construct semantic node see I
bounding box aware publicly suited annotation reason many image people action target annotate evaluation action detector extent many often make localization action database class class slightly specialize keep trivial contain perform action annotate maximize set region define either space consist segment order box grind produce map represent human table furthermore show human match computational spatial learn ground truth bound annotation alternative annotation consist
use fact span without right similarly value vertex edge define complement sum unnormalize let laplacian denote nonempty subset index lemma show ap initialize like subspace origin ap nonzero result satisfy generate sequence ap attain attain imply ap nonzero write equal change large q eigenvalue suffice examine derivation therefore da e red theoretical low sequence plot blue successive ap initialization
equip eq monotone function submodular function shift apply greedy specific construction broad envelope diversity greedy
result notation inspire invertible penalty parameter suppose correspond slight abuse treat active variable element subset element assign know argue induction induce active select inductive
start convex convex operate rank large decide decrease value value operate goal operating take six hour gb relative package big gain per leave netflix fit implement right give early option reach package available implement number missing correspondingly center make prediction fit package compute large column center find package author upon factored split many machine transpose split row across current hold zero cpu core restriction memory machine copy cpu copy though core act core copy ram core exploit low signature method four distribute consist row full spread across method distribute multiply similarly
metric versus svm sl ne due formulation sl sl ne scalability computation class linear sl ne constant quadratic sl follow sl ne sl de sized middle large sl examine optimization sl ne sampling strategy selective lot initial attain similar retrieval sl ne shown see sl image class sl ensemble range sl de project single two ensemble observe learn sl sl subspace dimension performance validate several datum
encoder relu activation superior previous thorough analyze regularization augmentation observe indeed pre provide gain unlabele label art performance enough training label form pre unsupervised work help deep unsupervise zero encoding train unsupervised take improvement convolutional show randomly cnns boost technique popular supervise cifar unsupervise additional pre training unsupervise
fidelity non overview mainly latter mention algorithm originally variant applicability since fidelity appropriate split perturb variation technique field cf many apply image decade follow reconstruction receive ray ct encounter ct limited view image cf edge material inherently split receive strong ability regularizer efficiently classical ray ct ct image mainly invert case fast protocol cf fouri fourier reconstruction sense find exploit application velocity encode accurately operator emission nuclear classical imaging poisson available half life water suffer inherently splitting emission see modern allow imaging reach suffer live imaging imaging scale deconvolution address appropriate poisson quite beneficial apply achieve broad imaging highly perturb probably proper modeling essential g weak choice since marker decay detect datum model process ray cf ray transform coincide transform model describe property acquisition algorithm discuss split highlight strength carry image total variation tv algorithm backward metric expectation step emission tv adopt modify efficient weighted square problem within proximal problem stop
rx square define accurate equation restriction indicator k lead pearson indicator plug rearrange drop external selection subset challenge difficult intensive exceed often genomic bayesian penalize regularize computing ise show rapidly compute present result accuracy applicability regression gene broadly frequently hundred year challenge genome study potential feature regression statistic classical square turn quite difficulty feature especially general phenotype body seek nucleotide snps trait height mass trait represent absence snps highly explanatory trait thousand thousand million presence absence snps tend linkage
good expect rapidly region expect gradient vanish concern pursuit overall ultimately output experimental architecture one additional feedback think analogous produce local main act kind although error train result training focus raw input notational sample consider independently goal net convolutional neural layer error output weight weight filter layer layer map layer refer filter response map classifier softmax
record complicated background clutter object object rotation illumination variation motion find material extraction brief descriptor task task throughout consider estimate tracking criterion score I frame regard frame average success successfully frame sequence track accumulate accumulate detect frame compare approach boost approach
variance economic financial market management heavy tail problem kk solve constraint determinant setup model fortunately identification solve fix proposition modify var justification find approximate decay centre distribution apply addition involve choose threshold discard significant modelling distribution distribution credible interval ci return period extreme large estimate hierarchical framework section model discuss briefly illustrate temporal result prediction section finally discussion west reliable day region insufficient surface water rare duration important public planning propose weather resource management region
fortunately avoid exception convexity instead penaltie simplicity penalty add convert compete generate model n act reflect correspond difference another nan run repeat loading repetition predictor penalty repetition pick hard achieve start find result much coefficient seem much perfect bad first add job recover unclear exactly applicability run biological
space hmc satisfy stationarity balance sec hmc non must joint express hamiltonian invariant ode system invariant continuity term eqs reversible obtain apply simplify need aim ode explore rapidly reversible large size ref split split energy idea express integrate reversible discretization splitting product contain dirac delta law splitting law eliminate freedom exact value field f follow determinant facilitate sec
immediate generalization nonzero column I sort otherwise state infinitely subspace match give hold row form state column condition condition prohibitive state pattern assume notice typical compatible
extraction tool gain lastly use image grey image image refer isotropic voxel result volume voxel provide image measure scan lag slice right normalizing slice grey
rewrite bag write hence affect instance log set maximum estimation maximize log form efficient therefore approach maximization hide propose logistic lr hide framework surrogate function r specifically step pt p decrease observe begin derivation log rule likelihood logarithm surrogate expectation instance probability py kb px em surrogate equivalent force keep q obtain substitute b b bi bn b use summation instance bag overcome prior computational probability introduce programming approach efficient challenge bag associate instance convert figure allow grouping complexity chain dynamically compute finally figure algorithm b p bn equation incremental illustrate e probability exclude computed recursion recursion bag compute bag p py b py py replace obtain store store store illustrate bag label box currently b p b
function strictly root thus since root notion behavior successfully power element algorithm error stability uniformly realization copy x z follow kx hold reproduce hypothesis hold stability theorem concern proof involve generalize newton sum eq symmetry symmetric hold let generalize implie pose conduct address explicitly section conduct experiment world dataset uci repository attribute attribute instance
htp discrimination illustrate newly publicly cancer publish newly roc accordingly cancer patient cancer control observe poor normality figure closeness monotonic outperform roc auc obtain htp htp htp outline methodology approach utilize
ep prefer approximate namely n ep approximate non joint un normalise bi variate q ep belong product division show variate normal ep try fix factor probability convergence n z ng leibler divergence previous kl expectation derivative close mean constant find particular gradient prior investigate approach aim correspond
calculate value nan alternative population problem model right estimate curve surprising relatively power pick quickly birth biological indicator birth affect classic study large state associate birth weight factor population medical black white status yes history history yes yes birth birth gram set relationship birth show ols black vs vs history history positively birth perhaps surprisingly age check predictor age standardize residual adequate indicate relationship age birth could expand specify ol age fit value age
hierarchy release mesh link mapping disease produce disease th imbalance gene disease gp gp pmf identity gene generalize disease full disease disease gp pmf pmf gene gene difficulty reflect table extreme find approach trace baseline pmf outperform baseline model rank constraint fig highlight propose top unable require store size implementation reduce initial indicate inferior disease correspondingly achieve dataset gp baseline pmf perform disease level recall result disease disease describe association gp pmf gp utility compare approach gp suggest helpful essential recommender information service matrix effective recommender recommender covariance may rating pmf side measure metric recommender typically concern
framework laplace draw zero spherical ensemble times error take mse analysis experiment aim reconstruct regard amplitude vary zero show amplitude error error tail b laplace fail amplitude reasonable amplitude range fast performance measure reconstruction
aggregate update however fully aggregate distance serve kl divergence bregman core traditional generalise generalised aggregate use prediction expert update distribution aggregate via log loss kl divergence bregman divergence notion apply update
place tuple weight assign use utility empty location opponent player hand al present possible include network position capable position function line player standard heuristic exception position randomization move player long play correlated ability play player consist game play aggregate scalar count average score constitute paper eight time advantage symmetric eight symmetric expansion tuple return perform recognition game al advantage tuple network
detail svm physics randomly problem tn still solve condition life recently call constitute towards begin method gradient cg remarkable property descent rely linear quadratic depend hessian index lipschitz gradient present relate expand cg minimize span direction propose expand minimize function propagation gradient cg method increase gradient version cg neighborhood extend direction smooth convex equivalent
quality process detect false observation tolerance change belong good reconstruct quality control ba propose nh nh denote probability alarm specifically q propose new smoothing dimensional bandwidth asymptotic given plug assumption
modify smoothed negligible standard assumption asymptotically close unconditional validity whether assume make object relax distribute main result iid gaussian imply observation iid assumption relaxed interval surely upper end quantile moment prove discuss symmetric denote covariance e hand significantly significant obtain rather rely iid tail generate illustrate line line asymptotic
stream supervision question statement statement segmentation know label segment slightly know statement statement statement contain e office case write feature directly model directly add triple loop keep win winner value whether old old old support memory compute old old old support memory feature find first thing memory train write eqs loss match term final word remain rnn eqs triple term argument side every rather compute task answer neural train predict read stream encode accurately fact past compressed dense rnn perform task sequence term speech incoming signal term movie answer modularity level
subgraph mining investigation eight popular dataset propose always return art magnitude exploit dependence test increase power frequent subgraph mining correction object massive chemical pathway structure web mining mining graph database two distinct discover subgraph statistically drug discovery activity similar fashion protein require event subgraph candidate subgraph extremely check subgraph exponentially cause huge significant represent number subgraph mistake science subgraph far severe resource significance level control positive subgraph common highly conservative test
kernel x x taylor correctness mention lemma repeat element define stand
readily calibration availability problem solution associate well proper probabilistic setting density enable computation static challenge sophisticated content raw image ignore process emphasis mechanic denote infer datum also example outlier calibration generally observe attribute model term datum conditional bayesian framework specify discussion dimensionality aspect employ improper prior naturally available priori mention primary work low could forward specification fluctuation turned assign nonlinear map different amount different would mle equation extreme variation direction even would cause huge drop variation whereas dominate identify effort drive argument approximation algebraic objective bayesian employ motivation behind intuitive resemble component pca vector along span decomposition attention field signal image audio attempt encode possible
require keep bound resample particle branching particle branch event mutually observation order particle order keep arrive online relative particle posterior allow online asynchronous particle weight particle process average particle process far number child weight design particle need careful appropriately informally weight alternatively approach scheme could poisson integer particle number particle start particle particle recursively would
advantage spectral utilize structure notational convenience term element equal covariate categorical beneficial center simplify section form robust node edge also brain compute use tuning procedure describe next alternative classical canonical analysis algorithm inherently reduction run spectral specific tuning block information practice contain tuning parameter balance value approximately interestingly lead continuous stable continuous range consider eigenvalue within minimize find define appear covariate vice static static remain slightly perturb transition value occur square th tune cluster estimate propose model estimator assume covariate bernoulli variable membership addition support depend
layer denote multimodal tangent force lead rnn rnn softmax probability next adopt sentence sentence length denote sentence generating layer likelihood sentence calculate sentence generate differentiable train rnn generation retrieval image retrieval sentence image sign start word pick model sign calculate generating treat affinity image retrieval query rank retrieve rank equivalent retrieval sentence retrieval consist appear problem sentence condition across training normalize original image sample ignore lead much performance use task architecture word language initialize layer pre imagenet detection combine treat experiment well
friend child parent movie dataset originally sentiment total movie review remain label negative train label set break map sentence train sentence experiment logistic eq sgd mini document iteration total epoch different minute evaluation particularly contain identify correctly part sentence sentiment rate extract sentiment entire
sample proportion crp cluster assignment construction k analogy stick break stick portion break stick remainder proportion first break stick infinite proportion cluster place relation dp place set namely cluster first correspond specific domain correspond list item entry item record l j crp prior indicate strength typical assignment object relation matrix object index discover case binary domain describe j eqs domain object point explain imagine user mutually act user reach first domain user single reach domain applicable node limited applicability domain whole discussion old one drawback cluster assignment membership blockmodel multiple assignment change edge ibp handle attribute flexible many view model model superior interpretation somewhat counter intuitive simple advanced probabilistic attract pure limited nest chinese restaurant membership representation base possible connection three connect link separate combine triangular representation simple generative scalable limit cardinality triangle binary network consist million node consider cross collapse
free network htp color free network extend accommodate node extend accommodate optimization ai onto per complexity two penalize package penalize adjacency describe finally standardize standard deviation display simulate correctly fine obtain varied obtain figure display compare use appendix htp colored line proposal figure surprising explicitly estimate ise refer network refer discussion graphical model density ensure sum one network penalize log likelihood due difficulty several neighborhood solve separately penalize pseudo q proposal involve maximize sparse network admm solve step
sentiment want distinguish label review product movie book domain achieve transfer unlabeled review book hypothesis example hypothesis true suggest even generally classifier approximate example example sample train approximate equation example eq subset show empirical combine result similar vc tell risk tell find small fix minimize risk divergence strategy domain indistinguishable source risk target present
closed form know factorization spline exploit burden computational scheme spline spline recently become mainly due extension expression order spline factorization diagonal interestingly share closed term spline highlight spline error method model adequate
yy local ds ds show exist increase decrease section lemma going result lemma deduce correctness log concave convex characterize dirac exist measure functional clear dirac hold constraint iii either give eq q integrating useful continuously theorem try point
annotation combinatorial harmonic compute image reach database sample al pose question toward acquire strong supervision multidimensional pair landscape still landscape grey factor element semantic representation people ordinary texture proper color texture highly color analysis investigate great et reason make meaningful feature fully automatic method edge base arrange create van style help distinguish conduct van collection small state rather van test statistical et al ordering embed one tried apply unsupervise laplacian feature fast unfold capturing contain database name last name title style image detail primarily
overcomplete observe spherical iteration learn significantly large signal noise previous efficient desire optima overcomplete model mixture achieve gain speech computer unsupervise challenge task label extensive topic gaussian decomposition order usual tensor tensor method blind detection current decomposition degenerate complexity requirement exceed often paper power make progress repeat tensor main intuition characterize dependency component randomness argument idea inspire algorithm analysis aspect pass typically infinity handle factor amp polynomially idea vector initialize core state follow gaussian initial universal tensor multilinear form w correlation j statement dynamics rd mixture show initialization condition initialize initialization combine overcomplete
need enable parallelization streaming later build adaptive sampling idea branch technique back center focus whole cluster encourage understanding result provable presence base family study book survey aspect area suggest elegant primal area problem often present semi number simpler tight nevertheless online
g problem assess efficacy presentation illustrative use namely toolbox locate conditioning represent along straight ray rectangular situation sample problem insufficient domain take full lead right side take represent hand sample l curve know rate lead problem solution deviation copy level obtain matlab problem apply preserve find tolerance white variance approximately matrix white color error detailed table level order randomly illustrative rate noise
prove return enjoy normalize generate subset
network field researcher experimental specialized specialized compare possible classify contain high physics matter physics physics cover scientific author author undirected author network extract along physics physics matter physics individual neighbor undirecte summarize belong task belong physics matter physics medium differ physics cm classify physics classify matter vs cm classify domain know network behave network nod formation random graph real discriminate social closure degree twitter twitter make information adjacency undirected graph generate random node number add edge twitter binary consist similarity run classification base five
change vote translation score later potential label class formally identify true predict invariant gap aggregated score serve score item invariant quantity gap aggregate across item labeling meanwhile bind rate relate assign labeling besides interest introduce notation score gap aggregated error change amongst aggregated score translation label high bound error kullback divergence high model worker notation define require condition aggregate basically predict item predict label bind bounding solve solve analytically thus bound serve condition present notation entropy bernoulli variable notation formulate theorem practice might evaluate conduct take estimator thus general setting compose generally dominant inside min rate recall tighter behave tight proof defer apply various labeling correspondingly apply binary majority voting majority voting scenario cover case multiclass labeling weight voting since hyperplane rule later posteriori vote weighted majority
bundle bundle admissible bundle vector outline argument h see reveal accord need utility bundle map kkt bundle linear follow sort run allocate namely admissible bundle admissible cost allocate bundle bundle p bundle admissible design admissible bundle j j j admissible obtain bundle amount amount increase crucially good well bundle exist clearly good else case transfer bundle bundle latter amount increase contrary suppose allocate get otherwise case admissible contradiction well admissible bundle check next help lemma clearly bundle second partially bundle case increase come check namely second need mapping utility h time mapping sample preference particular class know segment see
prior sensor parameter many let paragraph note let bound boundary positive self adjoint trace respective unbounded know note choose assertion proposition summarize derivation employ lagrangian pde lagrange multipli function ik ik ik n ik ik ik space counterpart derivative state adjoint vanish weight basis vector derivative gradient provide appropriate adjoint adjoint vanish variable define require vanish notice adjoint rearrange system ik hand side reveal follow thus adjoint eliminate right side element adjoint letter discretize describe computation gradient q note rely gaussian trace justification procedure compute I adjoint variable point accomplish follow differential appear discretization pde block eliminate solve ik right expression discretization eq discretize email edu edu email edu email optimal differential seek infer parameter field govern small moderate aim inverse moderate parameter author forward compute design gain
rate constant case td derive require mix handle assumption td handle asymptotic broad reasonable mdp second inherent iterate reason carefully variance iterate average large succeed necessity mix stochastic optimization variant td use iterate contribution summarize bound probability td variant incorporate center easily approximate iteration scheme show finite obtain expectation eigenvalue
requirement tune extra information precise topological without frame distance stop agent move status rw meet stop correspond occur observation mode set metric cloud dimension cloud persistence agent make improvement metric incorporate static rw status nod percentage stop node event I configuration agent cloud space proper persistence diagram hybrid static existence environment persistent distant outlier world importance cause appearance
similarly obtain existence moment drop throughout calculation
request unitary multinomial purpose classical square contrary logit recover however estimate cdf logistic significantly low rating avoid summarize outperform h kullback leibler divergence remain split rating range binomial rating rating l suffice prove tensor get f control apply ensure plug eq polynomial relation
send en en tr cr cr could rely successful coarse sentence even able outperform ever language language language obvious act unlabeled datum prove useful nlp representation syntactic unlabele usually small datum exploit unlabeled generalization allow vocabulary thereby partly sparsity concentrated start look document representation align language align language develop english document train english learn represent document resource language project annotated datum another projection resource translation
fx f f I actual constant big large section nesterov prevent without fall bind uniform actually
evaluate element sense principle converge optimal available generalize space application third define convex crucially remainder respective near state desire precise representative fix order lemma appendix let assume basically representative partition relative correspond sum adjust lemma result guarantee regardless specific respective representative proposition small kernel fix impose possible flip side statement appropriately rather guide sound limit illustrate simple world capable contain move difficult task classical double balance drug schedule domain action discount algorithm derive decision evaluate task appendix capable two reach goal avoid along except discount evaluate state collect policy action random case define group mean place varied report pick maximum return use convention instance coincide effect representative improve surprising magnitude indicate contain depend reduction consumption resource replace approximately operation policy figure fix performance improve however linearly explain huge time evaluate compare reinforcement task iteration popularity build transition good balancing long unstable problem apply cart track cart balance hard compare single implement reader carry transition cluster version fair varied policy iteration figure call hard variant decision policy probably achieve rate balancing figure contrast decision able balance attempt interval balance difficult reinforcement result opposite conservative estimate run computer use policy minute compare evaluation step involve require contrast operation fit iteration fit conceptually also build transition adopt choose
x ever transform lagrangian lx ta respect wise ty ty substitute write n ty ta ty ty equivalently ty constant composite ty ty ty hessian boundary pseudo parallel extensive literature crucial gap proximal gradient method fista regularize yield payoff partial outside scalable rigorously demonstrate derive
scale try hyperparameter prediction show comparison three setting term shape hyperparameter factorization variational vb vb prediction receiver operate auc measure prediction extreme imbalance less motivate auc since auc view imbalance link independent term variational cp tucker factorization model randomly unobserve incomplete factorize cp extract factor full missing performance
positive fewer certain accuracy level might come slack formulation margin slack actual object interest classify hyperplane slack training datum proxy slack th slack scalar dimension grow number optimization qp medium hundred solver suffice qp suitable algorithm tackle general qp package fast specialized qp propose call solve solver adaptation setup example look example check example easy classify hyperplane subsequently know classify learn improve train result classify hyperplane easy classify even indicate hard classify train svm enforce margin small influence compare inequality value correctly possible weak optimization overfitte original tolerance
nature impose inter dissimilarity latent dissimilarity isolate need central spline priori dynamical evolution occur euclidean curvature sensor may geometry space euclidean euclidean measure vector incorrect simplification bregman measure bregman positivity bregman
study recommendation situation experimental result validate energy mrf image minimization play role computer vision early vision follow pixel define pairwise dependency graph minimization accuracy accuracy thorough comparative study dense three closely hardness figure hardness situation perform validate combination exist reasonable unary computer vision combinatorial graph minimization decade hard exist general problem characterization mrfs state art mrf include cut solve graph partial labeling
baseline projection multimodal clearly numerical require majority handle cluster original I replace present flexible capture content unit randomly country remove project posterior people china interesting north east south south task fill answer response answer ignore create randomly portion answer remain answer generative answer predict person answer multimodal collaborative filtering subset ccccc categorical continuous category six representative problem country country origin size country iv
attribute invariance desirable vision low level pose abstraction detail label spatial invariance signal incur combination max layer employ originally dense substantially relate invariance inherently boost fine employ fully crf semantic score interaction pixel increase crf efficient ability fine largely improve performance boost pixel classifier state couple pixel system virtue fully ii challenge good margin system compose cascade fairly technique make system potential
success logistic give interpretable ht grow tree predictor predictor role categorical include candidate categorical predictor fitting effect capture split split among regressor candidate logistic ordinary determine effect response grow wider able estimate logistic logistic logistic numerical regressor criterion regressor node terminal node stop fit degree minus turn terminal node otherwise select tree chi ordinary satisfactory divide interval construct table pearson chi result chi dependence also select predictor splitting interpretation cm variable take consideration depend fit split simple node good regressor conduct chi split fit candidate regard count failure expect instance success observation category cell thus chi square cell proportional count hence degree freedom numerical numerous contingency distinct count pearson number cell theory violate fit find additional contingency logistic numerical predictor chi nonlinear adjust compare group divide fit sample cutoff point table chi way pearson chi roughly approximated chi variable otherwise two lack summarize run logistic model construct contingency category cutoff
parameter intractable inference technique variational fast problem handle multi problem evidence estimate parameter inference technique approximate tractable approximate variational kullback kl maximize non act logarithm evidence apply low estimate maximize slow recent advance approach gps use variational variational write since diagonal diagonal diag variational variational bind log integral jensen bind write maximization positive covariance diagonal diag diag concave respect inverse determinant concave ascent parameter
map z z patch patch square point possibly locally shift parameter cm cm rectangle rectangle aa node aa node every c right leave yshift style rectangle node cc middle cm every grid yshift every aa ab ad ec ed aa aa ab ac ac ad ad ec ec ed ed rectangle xshift mm yshift patch right style bad ba bb bc
forest type know datum allow test manual calibration visual assessment parameterization adjust pattern typical create virtual output rate option model know priori sufficient virtual practical understanding process interaction posterior width posterior create synthetic supplement median panel visualize refer display low fit black correlation model tree specie seven detail availability distribution virtual aggregation output concentrate v brief detailed density parameterization table represent assign parameter retrieve order plot mean display correspond uncertainty substantially subsection represent sample along respect
train add layer connect exist stack head subset label cluster train augment pre minus hold pre make capacity rather adapt new tune perform fine method
minimize variation transform transformation form signature signature transformation transform signature denote robustness cluster art test parameter mean rely information cluster spherical split
whether extend paradigm instance methodology measure instance identify examine set learn weight filter especially useful affect backpropagation result induce generalizing however beneficial induce instance even datum label correctly model misclassifie remove training solution beneficial multiple misclassification repeat benefit beneficial significant preferable generally remainder organize review relate handling motivate weighting conclude c package conjunction terminal explanation use load package graphic terminal
red circle positive variable toeplitz assumption replicate circle positive influential predictor scatter plot smooth region definition fs fs fs height depth pt mid pt fs fs keyword modern set observation variable selection important flexible variable call resample procedure add control boost simulation insight minimize observation descent boost intrinsic one begin compute residual call base learner good percentage iterate final fit learner learner spline square boost iteration iteration early
follow illustrate potential policy iteration policy approximate generate extra exactly except induce represent decision represent incorporate action gap policy tree partitioning define extra obtain build fix represent corresponding decision vary tb describe space day high load bar impact performance compute well restrict efficient expect trend policy influence difficulty number adjust complexity specific greedy policy restriction benefit term general truncation policy tight vs vc gap property easy report order magnitude tree show conservative may thus performance performance algorithm stop monotonicity still upper actually distribution policy iteration converge actor evaluation implementation ac algorithms actor use gradient actor policy space would ac tailor continuous share form actor algorithm problem vs finite remark allow space slower reason complexity supremum advance localize rademacher modulus empirical hand relaxed gap regularity goal decide efficiently informative sample
discretization refine posterior along inform maintain mh proposal inform subspace dominate prior prior dominate adapt proposal adapt without target proposal proposal replace symmetric regularize sample independence absolute continuity condition comparison produce size direction influence autocorrelation mh improvement operator proposal several limitation art first representative structure non problem also broadly local target form great part limitation present describe simplify mala stochastic newton langevin sde method hessian simplify mala use derive proposal inverse hessian mala newton related function expect newton data operation expensive mcmc sample mala generally local riemannian metric insight direction curvature insight distribution particular
include major education conference number color topic actual topic uk actor computer computer operate system system label record music record label company production company company cc cc gold natural world ten discover assign characterize basic foundation consider study describe research year execute corpus number iteration similarity experimentally approximately node set candidate significance topic nsf grant ten see label two evaluate label show six mostly topic inter lda topic grant label rank return present comprehensive visualization reveal scientific subtle effort education conference display towards indicate color discover characterize
observation occurrence context pair may explicitly co occurrence local bias relevant context weight
able surprisingly benchmark deep architecture boltzmann auto encoder size adopt map otherwise map size really performance curve dataset besides aforementioned method local atom randomly fine tune sift increase consistent encoding encoding dictionary variation image opinion atom relatively advantage unsupervise mixture learn increase redundancy decrease efficiency hence desirable combine sift base performance table sift dictionary increase begin slightly k however c effect conduct original image extraction block map pool yield generally layer mean size improve translation add sift pooling effectively detailed fig representation size field pool representation well representation complementary individual stage encode sift representation voting
reach regardless add sum entail ultimately estimate discuss asymptotically case r enyi model hold pn n n enyi know nk nk nk nk nk na vanish instead claim consider parameter notice non degree os belong easy denote k proposition nb n
q one evaluate q aggregating observe across lemma construct packing every satisfie packing bounding desire issue boundedness final relate likelihood hessian q making allow must define minimize n must virtue coordinate sub substituting get everything final set guarantee binary code hamming code e ed vector pair satisfy j third vector l j step make value decomposition u pt minus minus r star empty name display tag display display tag macro name cr cr cr cr
constrain domain inequality distribution close impose categorical logit distribution rank supplement p z z read k tractable approximations box field employ q method applicable persistent per explore explore entire fit idea cd create discard e collaborative filtering maintain moderate parallel helpful boundary collect mcmc eq describe three namely handwritten digit collaborative survey supplement difficulty end progress reason map nature underlie norm rescale hide unit deal another posterior add evidence leibl posterior posterior give simple integration constrain
learnable supervise may arbitrarily long dimension space linearly c give well define actually calculate take early proof class tight dimension let dimension generalize number follow standard optimistic dimension confidence set function square l tf growth empirical c distinguished lead function set scheme argument essentially statement value derivation deviation
acknowledgment acknowledge helpful lin thank berkeley national laboratory energy office office advance compute research contract ac berkeley national laboratory york lin support national grant dms dms work office office advanced mathematic contract ac foundation dms lin sampler variate assimilation laplace asymptotic expansion implicit sampler regime improve confirm use assimilation direct sampler give variance weight determine independently literature introduce noise smooth density
distinct graph order graph node hence theoretical definition loop multiple simple assignment node contain edge walk repeat path cycle node cycle graph edge path edge path edge say node list
reaction nucleotide template determine light intensity detect axis determine nucleotide flow light nucleotide flow excess nucleotide ideally incorporate reality light incorporation identical pose inherent difficulty accurately specification kind permutation target denote nucleotide distinguish nucleotide cycle cycle show sequence particular sequence nucleotide need flow cycle sequence
stock portfolio specific portfolio need stock portfolio correspondingly examine investigate financial fast transform filter technique publicly financial rapidly hardware enable eventually stock automate machine finance frequency focus company reduce easy stock combine capability support portfolio suggest
likelihood constant linearly however condition von posterior concentrate around reduce see plot remainder linear multiple need maximum yield construct cauchy guarantee refer take outside inside linear line exact sample challenge address first mean way stochastic define generalize perturbation work require infinitely draw perturbation break dirichlet insight rejection lead likelihood behaviour illustrative proportional independent unnormalized perturb location basic ib consequence choice axiom moreover independent generalize continuous maximizing perturb analogous max adapt sigma finite disjoint consistency requirement requirement ensure consistent ensure requirement restrict
improve various parameter reduce prediction provide experiment dataset e neuron describe rank pair neurons association rank ground participant recall curve show sensible especially density small truth manually association roc pr curves network case partial correlation filtering function pca case increase really improve various minor last show tune good performance
usual hour work indicator term three child status mi width confidence property challenge high nonetheless offer coverage age figure regression improve method tend coverage particularly interaction toward effect expect imputation coverage drop coefficient hour work lack appear effect effective similar age ci examine quality estimating cell plausible display proportion child home perform rate drop coverage never drop every good coverage arise misspecification tends somewhat extent probably due large cell standard error evaluate field construct production brevity subset panel subset wave subsample size census researcher estimate production record however variable
high linear regression observe estimation regularize selector therein biased lasso estimator normality example base measurement wide recent attention observe measurement recover nuclear minimization noiseless sign recovery compare estimation theoretical confidence increase bias correction de wise constrain quadratic procedure high inverse structure pose proper relaxation atom structure solution efficiently geometric intrinsic difficulty tangent play role develop general induce answer inferential question local rademacher complexity studied capture difficulty pack volume paper quantity computationally difficulty main summarize unified cone formal ratio define establish normality linear compare consistency remark beyond intuitively present fold unified estimation feasible program provide collection provide feasibility guarantee
scale nuclear relaxation one
mis obvious ij convergent moreover I j var almost surely eq argument c nz c nz remainder decompose integral equal follow conclude precede analysis respect substitute nj cauchy n thus complete establish show deterministic chen q gradient examine recognition ever approximate fit rise response advance mis significant issue mis move memory chen incorrectly integrate model study demonstrate mis obtain inferential specific dependent precise degree mis mis specify regularity true pseudo pseudo limit proceed behaviour incorrectly dependent equal specific integrated move average distance extent mis several firstly derive commonly estimator namely
water essentially table use boost boost boost boost circular simple increase move emission boost variation emission boost variation illustrate uncertainty observe well star circular variation marginally include imply calculate turn well example biology us protein atomic datum determine mechanic force field calculation force field determination typically contribution force clear realistic field comprise weighted force field freedom atom angle inferential determination determination appropriate weight field good force normalize assess observed prediction field incorporate boltzmann partition prior inverse temperature force explain realistic temperature identical unknown force van link drop atom van force attractive van interaction force attractive
give remark take fusion derive fusion diversity classifier could investigate additionally interesting study performance video retrieval assess trade rank diversity retrieval perspective take classifier indeed pac obtain majority vote vote belief kullback vote problem part european fp agreement author corollary st de st france universit france past
author would like thank grateful family department introduce convenient strategy predict distribute random size implementation optimal moment condition string moreover string alphabet poisson universal law alphabet compression understanding outcome individual quickly shape small example chinese character chinese character frequent choose redundancy count whole n n minimax familiar universal pointwise p cs nm instead slightly suboptimal independent indeed preferable size advance count conditional sum count close relative unconditional subject expectation
I I lm ij lx lm lm lm lx ij lm lm lm previous example lp qr tp walk metropolis hasting mh slice alg tb dimension proposal tb tr pseudo handling scale slice reject reject lower decision slice accept need new inside slice whether interval slice upper bind outside low low slice make tb prior tb z tr l let inequality nonnegative gm inequality symmetric trivially use
hyperparameter broad uninformative gamma accuracy approximate entropy compare es compare rejection rs objective process fix data hyperparameter know ground rejection scheme work discretize grid mean variance evaluation entropy use rs measurement collect display x particular measurement display leave figure approximation ground truth es see discrepancy rs discretization draw ground truth rejection method plot objective similar rs ground performance optimum box reproduce comparison optimize dimensional unit first gp
video code good energy closely decade devise dct prominent work dct et different nevertheless multiplication year see advance new dct exception arithmetic cosine background recently yet scenario
complex correspond respectively phase formula critical fig transition threshold match leave gaussian random column predict obtain formula I case predict predict independence property theorem temporal perform simulation window monte trial observable htb analyze propose complex value correlation synthetic stationary mean white series band pass add ik ik impulse band pass stationary series stationary impulse band band ideal band finite impulse chebyshev part see sample show magnitude finance focus detect series correlation detection lead reduce computational cost wireless sensor network useful remove study consistently finance financial correlation become e scalar naive sample dimensional random completely correlation consider instant time
hand show pseudo herein operation compression technique us prototype element ds compressed prototype dm compression compress prototype set order execute retrieve prototype interval effectively proof narrow size extend evolves show tr set iteration herein expansion replace discriminate prototype perform threshold graph maximally prototype analysis outline evaluate dm two execution rs initialization size relate single fitness variant initialization detail fitness generation dm rs initialization compression operation distance characterize operation cardinality operation entropy cost cost relate embed dm cost expand deduce version sec implement operation notably rs synthesis depend tune compression seek ms procedure consider element different heterogeneous mode ms filter characterize neighborhood define neighborhood size weight characterize reason initialization synthesis systematically value neighborhood cascade compression computational fitness involve algorithm
markov since correlation limitation normalise compression define correlation string string short string common long string lag string string motivate consider inner cross involve lag string rely determine bit encode perform illustrate observe alphabet may interpret require optimal thus quantify bit require represent accounting dependency finite define give base increasingly term cross entropy quantify expect source code estimate iterate describe term source quantity denominator serve analogous denominator entropy observation prediction quantity b b mx nx prediction evaluating vector precede r parameter predict base determine embed space illustrate depict far future motivate
multiplier kkt satisfie let l c rl j j j sum cauchy schwarz oracle choose say moreover plug dd j p j coefficient integrate logical among g two main hierarchy interaction associate e jk turn achieve see benefit involve parsimonious inherently hierarchy traditional hierarchical turn multi ad shrinkage purpose vanish guarantee optimality slow large vanish main enforce magnitude coefficient desire apply introduce part constraint drop study decade
gradient short dependency dominant researcher try devise simple gradient descent use growth derivative guarantee interested sophisticated follow element wise nonlinearity unit attempt result recurrent rnn recurrent task capture long task speech et al evaluate recurrent lstm empirical describe
author far least text must author estimator adjacency text text text text proximity word sequence index word symbol symbol require direct proximity discount sentence direct proximity word word word word primarily include exclude gender type text sentence illustrate common symbol window worth load worth index accuracy unknown text correctly attribute denote write accuracy classifier construct direct network word edge represent edge text compose word measure pair select node discuss calculate sentence way discount factor apart similarity sum find position apart direct word eq similarity proximity word every amount text undesirable want normalize author positively node divide word normalization matrix inherent serve know text interpret mc probability word every
reject reject sample vector approach reduce post hoc difference due dimension eigenvector eigenvalue importance hoc significant clique reject disjoint post hoc find clique retain level post hoc correct experiment show univariate show multivariate seven discriminant discriminant knn neighbor random tree svm svm bioinformatics ec fold multivariate normality normality precision normality compare use algorithm
write derivation unfortunately proof essentially instance whose metric derive formulation svm segment letter tune validation misclassification average along standard across competitive kernel svm remark derive global algorithm advantage exist way learn analyze approach generalization evaluate superiority scalability measuring instance distance euclidean lead grow past year survey mahalanobis constraint close typically standard unable multimodal boundary overcome limitation work metric per locally simple metric perform integrate burden severe overfitte method
block belief layer distribution rbm logistic unnormalize low recognition p efficiently intractable case concern markov mrf rbm denote potential f latent p pair want compute test place mathematically outline closely applicable addition unlikely reason mrf situation appear denominator log partition translate log problematic inaccurate dramatically lead researcher direct performance fact sigmoid network cite
desire hold l sf nf ix bound thus uniform statement yy unseen belong specify unseen q ms mi give feasibility sf eq apply statement probabilistic statement inequality least contraction inequality bind yy side yy nr c yy nr c lemma yy nr yy n nr imply low bound yy yy statement unseen realization low simultaneous unseen realization problem sm j equation feasible every change appropriate eq add third value replace max term hand due concentration quantify get probability around maximum change follow statement right sl events ne ne sl n event happen e
architecture fundamental propose score datum fit great difference tuple circumstance know nature find poor local initialization multiple begin kind stop experiment step maximum
explicit hoeffding concentration inequality extend argument upper bound union bind long technique naturally case contain detail lead suboptimal optimality proof tight proof generalize appendix detail hoeffding type martingale like except bound explore emphasis work prove slightly stop martingale theory exploit favorable convergence nonnegative nonnegative possibly argument begin construction hoeffding
sensitivity specificity g svm evaluate benchmark uci rna graph library framework matlab evaluate implementation objective certainly implementation implementation normalize zero demonstrate data sn accuracy letter clean eeg proportional weighting study experimental imbalance ratio detail ann run deviation small bold acc sn letter rna acc sn mean
acquisition restrict minimization recommendation portfolio compute entropy detail rest visualization objective dotted plot acquisition ei blue pi ts red select candidate histogram blue dash draw blue green triangle vector bin minimizer depict utility first monte carlo integration e step sample quasi carlo entropy analytic simple impractical intractable compute replace desirable propose expect close target
software author read final manuscript acknowledgement helpful continue user additional program useful suggestion plot frequency leave variant chi table small summation use impact impact accuracy rapidly frequencie table consider skip improve problem increase plot space rs phase e likelihood total historical extreme value finish due monotonicity dot horizontal mac mac mac mac snp mac mac mac mac calculation dataset mac mac mac mac snp mac mac basic cluster mac mac mac k mac mac mac k snp mac mac k k
moreover change posterior bias varied plot appear value situation although bias appear variance precise dependence scale appear marginal standard via square instead concern long plot fitting fitting exactly hold reasonably consistency likelihood base estimator proportional impact improper twice repeat change difference well practically improper likelihood consider present table rr std ci exact process via approximate run via exact use red respective correlate interval slight shift exact imply less excellent frequentist level exhibit behaviour check method entire analyze result residual present rr std slightly close large small vary analyse similar vary posterior deviation result posterior plot log ccc three proportional reliable conclude posterior uninformative default ability explore
nuclear term step k alm reweighte norm like e lagrangian descent optimize fix alm rather update inexact alm projection soft threshold singular minimum follow toeplitz determinant heuristic weighted need solve nuclear subproblem analytic add definition constraint lagrangian augment lagrangian follow alm strategy weight nuclear completion minimization take gradient formulation update look reweighte derivative equation arise lyapunov alm reweighte nuclear section
initially negative parameter increase threshold true threshold event comparison stable partly term filter addition stable peak wider well noisy scenario time sect curve performance drop peak cc close parameter setting investigate temporal tweet sect b spatial tweet fig decrease hence especially temporal interval tweet generally noise finally influence relevant tweet increase tweet observe gain match intuition high signal detect event concentrate correct temporal spatial threshold comparison employ filter synthetic generally suggest detect different absence simultaneous spatial also lead event two highlight multiscale dedicated detection scalability public tweets york bounding box leave gps pair stream public tweet twitter stream request retrieval tweet predefine bounding box tweet tweet contain check implement daily detect day vector implement toolbox remove provide toolbox http minute eq term appear tweet evaluate standardized term valid minute number scale method post
ex ex school department berkeley pa berkeley dual establish selection even nonconvex guarantee rigorous nonconvex nonconvex vanish recovery may require incoherence condition corollary wide method composite loss square modify conclude study prediction body relaxation nonconvex arise see broad identify among candidate encode entry vector optimization hard much slightly problem constraint norm well condition relaxation estimate parameter therein encourage sparsity aspect norm value linear regularize sample setting nonconvex smoothly absolute concave regularizers become regularizer cause numerous empirical variable nonconvex paper penalty assume continuous allow apply one exist open may take extra meaning minor abuse notation appear side univariate act readily regularizer restrict homogeneous estimate function goal consistent consequently feasible study say symmetric differentiable amenable vi regularizer vi amenable notion past regularizers vi goal conclusion vi everywhere differentiable furthermore appendix useful amenable regularizer many popular regularizer amenable amenable example illustrate take form zhang take form eq amenable finally consider example amenable mention penalty amenable
potential structural biology determination interpretable protein enough particle particle intensity collect ray sample fact copy collect treat particle ray orientation pattern modulus particle pattern different slice complete recover pattern compression experimentally artificial coherent light source hz correspond hour european become operational hz alignment demand high volume achieve balance object light parallel keep experience heterogeneous computer fully implementation scale cluster hundred
one unconstraine bound singular approach find nuclear convex relaxation extremely find recently since problem indeed demonstrate extensive recover vector g reweighte minimization variant ji problem arise network localization al reweighte solve addition extend problem square produce method either processing solution apply expect reweighte capable yield aid solve variant study stationary minimization minimizer first order minimizer
small anomaly detect expensive enyi bottom outperforms gap stanford analysis project http stanford edu co record amazon represent product frequently represent internet take undirected link eigenvector large residual subgraph align amazon com modularity residual shown leave frequent co isolated set small large subgraph possible edge subgraph internal neither income compare take million comparable average internal none among internal great however primarily outside span anomalous consider graph represent norm get large index highlight deviation eigenvector align subgraph align vertex primarily external degree subgraph take sample size vertex share vertex dense external vertex dense eigenvector extremely external degree anomalous background residual principal algorithm simulation utility identification detection network subgraph show anomalous background demonstrate anomalous processing recent focus extend attribute particular dense subtract edge rather add
sensor switch dual past decade range environmental air increase decentralized scheme reduce rate ensure presence failure relevant centralized literature distribute adaptation interest communication range dynamically sub share contribution dual average distribute method allow adaptively diffusion network uncertainty third weight uncertainty reliable statistic derive regret highlight link organization background graph review optimization network weight topology distribute application operating environment online subsequently online demonstrate conclude future utilize system
somewhat behaviour mode move mode phenomenon high gp tp consistent close elliptical tp task show gps computational home message tp gps modelling flexibility extra suggest useful gps almost application tp orthogonal expressive kernel proof lemma corollary paper student analytic marginal enhance explicitly depend verify student situation change structure covariance good come additional rich bayesian non simple exact
minute nearly minute figure safe greatly superior pca develop algorithm problem norm convert constrain online stationary moreover empirically propose recently online nuclear suggest hard corrupt relaxation stochastic bound exist exist bound solution q examine uniform note uniformly prove constant eigenvalue equivalent basis produce norm upper consider uniform trivial feasible upper uniformly bound make solution boundedness surrogate lipschitz uniform bind uniform tf subgradient theorem measurable subset borel measurable nf suffice hypothesis corollary particular set uniformly proposition family pf ff p
ks score quantify lead text unsupervise term component quantify gene activity sometimes component characteristic set likely use collapse gibbs lda lda hyperparameter collapse gibbs yield em laplace select predictive repeat well fold prefer expressive fold separately optimal
fit label semi supervise good principle semi datum generate make like process unlabeled text text empirically task good well cluster unsupervised model many care serious b average datum explore supervise natural focus study paper book explore possibility semi supervise thorough carry explain limitation language large intelligence scope text organization present overview semi technique follow semi consideration conclusion label train little label generate label model label working principle value
reconstruction error depend vary component component error log balance bad relevant digits quantization numerically might want possible change change relative scaling theoretic homogeneous intuitively compression viewpoint divide error move bit per gain may actual error usual problem might prefer conclusion minimize reconstruction provide term certain regularization add reconstruct backpropagation noise namely provide noise differently translate feature auto criterion neural need include variance parameter well modify reconstruction involve square quantization still
want distribution maximize newton read entire propose modification require dataset substantially decrease analyze theoretically open dirichlet distribution long prior analytically bayesian maximum data recommendation engine rating user rate inaccurate prevent estimate opinion week represent week start update originally similar dirichlet biological core level
decision process markov robust amenable decision maker scale dynamic programming use bad leave adapt black available problem seem relaxed theorem corollary claim ie il ie ac il il ac il framework belong min solution realization type case uncertainty lead semidefinite paper solve stage program solve number call robust fundamental perform parameter convex concave optimization
otherwise size within fact clearly go well state sc construction special polynomially relate hold ready first transform find see every induce cover covered equality would conversely induce norm cover put contain exact cover conversely cover sensor problem choice sp solution p set
public invoke review role causality reasoning broadly concept fitness begin causal difficult persistence right context cause competitive training physical fitness account failed meet bar contract demand role causal origin lead mechanism connect character fitness might concern role student character access causal pathway argue fitness use decision origin fact
large correspond consider eq choose set minimize square think consist nonparametric tail effectively use number thus consist output mapping use linear mapping rbf dd project respectively note I I set q nf tn output rp compute multiplication vector include metric every input space complexity need store basis contrast space lastly evaluate
sequence rbm plan introduce multidimensional rbm drive motion right plan simulate free motion conditioning satisfy previously another namely continuous uniform simulation technique tolerance piecewise least complicated wavelet localization tracking jointly maxima minima dyadic ultimately dyadic reason preserve easy piecewise input gradient since computable lipschitz combine approximation fact strictly eventually construction indicate rise
regime covariance compare gender cover body g oppose spatial spatio temporal appearance highly gender especially heavily low result maintain low e avoid curse dimensionality paper organize dc feature
act na image unitary rotation act feature without cost iv canonical fundamental property readily apply implement sign simulation obtain sign separation ease applicability trick invariance induce transformation possible focused sign sign finally experimental result come class form one sign cluster sign namely obtain eigenvector norm thus interpret learn representation increase compare ambient try projection g scalar heuristic lead fluctuation scenario compress different combine kernel learn
experiment dataset randomly class filter filter testing error average show table outperform improvement follow classifier good method histogram object cifar image test cifar significantly scale object explore simple database face digit roughly align accommodate database spirit verify q mat k spatial pyramid pooling aim database help object experiment face digit train cifar filter overlap region bin pool pyramid pool histogram pool reduce learn descriptor paragraph combine dimension descriptor table gain degradation art method augmentation extremely encouraging triangle convnet convnet conv maxout combine arguably simplest unsupervised convolutional deep pca binary block convnet network layer train extremely efficient involve build comprise follow output alternative yet could facilitate couple extension image classification hand write digit experimental show outperform par convnet comparable hand feature task face recognition robustness
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task need ideal operation reduce first point equation minor coordinate lie generic form certainly avoid difference ideal symmetry restrict affine chart x assume normalize vanish looking apply yield intersect equal n symmetric easily compute determinant express determinant vanish statement lie lemma following hold proof apply describe u obtain replace th sign u jx h h combination polynomial comparison strategy ml precisely action symmetric group look ideal polynomial equivalent requirement
accurate answer mechanism exist super smoothness mechanism accuracy much inherent private mechanism answer direction exist algorithm synthetic database accurate smooth query specify immediate question output database preserve natural contain proof section query notation proof discretize dataset please step similarly laplace noise finally notation mechanism preserve privacy straightforward output private immediate specify later good approximation also follow equality correspond error error round error respectively equation five type separately derivative discretization eq let depend satisfies calculation yield chernoff q eq q calculation run arithmetic operation
privacy mutual information perfect protocol minimal vs theoretic privacy preserve mining investigate namely analyst privacy trade study consider analyst multiple clear analyst perform statistical operation operation publicly party statistical operation study set include social recommendation analyst enable user perform aggregate privacy private linear constraint amount analyze privacy recommender system recommendation focus make output application private recommendation differentially private privacy consider recommendation rating differentially recommendation user might link party adversary rating manner private none work study privacy perfect independence work privacy necessary amongst main theoretic develop asymptotic datum grow arbitrarily asymptotic private problem impossible practice privacy approach distortion privacy mechanism assume general private develop quantify privacy privacy design mapping inference information information private hard involve paper private datum privacy simple also limit statistical keep mutual review highlight challenge factorization address prediction analyst possible news article etc user item pair rating rating rating attempt comprise rating particular term da kb ij profile typically minimize
theory hold assume neither symbol meaning estimate unbiased unbiased mean observation symbol symbol stand data moment fourth observation shrinkage shrinkage shrinkage quality target contain target limit behaviour integer overview shrinkage combination estimator optimize manuscript generalize optimize turn linear arbitrarily rescale quadratic program equivalent optimize eq govern quantify target estimator unbiased estimator adjust target element contain target correlation entry variance estimator element target optimal shrinkage quadratic follow estimator relate limit consistency sequence index kl assumption estimator behaviour behaviour none target identical relative go limit property combination
performance texture highlight dataset create step datum construct discard particular class c sample exclude combination average hyperplane exclude datum train exclude still face well result euclidean sr locality sr code locality preserve relational sr synthetic datum texture recognition necessary improve texture sr code locality preserve seq seq plus accumulation locality projection divergence
evaluate matrix multiplication generative obtain px generate computed rest inference probability propose factor performance modify metropolis hasting infer hide dimension factor integer within draw modify choice positive integer run iteration expectation plot influence initialization great infer value
form wolfe frank wolfe reformulate value term slowly issue fw essentially wolfe cost problem feasible apply frank wolfe exploit step idea auxiliary summarize g however frank wolfe optimal convex set obtain one solution objective value imply bound feasible modification apply wolfe obtain hence produce straightforward sparse slow diameter frank method well performance frank wolfe generate iterate l expression gradient solve linearize subproblem subproblem homogeneous l easily verify fairly wolfe direct lemma algorithm k l v
star introduce neural able mean low preserve sentence semantic build document representation compositional embedding sentence sentence task sentiment nlp network entirely feed manner recurrent notable exception convolutional neural convnet short fundamental building network nlp fundamentally symbolic operate necessary create symbolic text word embedding receive several excellent mapping neural embedding relationship serve excellent neural document compositional word embedding sentence combine embedding inspire convolution great model
net state update step accept reject transition go change equation however incoming transition make transition modification transition state precede repeat transition etc two time flip accept step update fashion eq transition transition state exceed direction exceed rate direction alternate physic generalize poor behavior detailed balance remain assign momentum flip fashion take hmc rough rough hmc
various effectiveness extend dictionary propose approach sparse collaborative name dl aware kind extensive experiment superior art term robustness background present state conclusion propose version handle superiority suppose discrimination without well job lead result agree superiority dimensionality correspondence dimensionality discrimination zhang arbitrarily quite dimensionality dimension still effectiveness besides drawback zhang experiment limited face though variation collaborative another third party dictionary shot version extend coefficient face valuable party bring
past year effect sort choice wavelet single choosing basis function assess user assess publicly method contain computational implement al et al al software software start package software method response predictor general represent effect predictor position goal curve deviation frequently assume iid whose structure covariance curve deviation error primary curve ia covariance imply st enter model accounting potential later regularization interpretability potentially estimation interpolation curve predictor truncation penalty choose inclusion within correlation induce affect account weight lin wu et proper analysis correlate idea far affected function especially joint inference lee exist back method scope covariance aspect focus inferential capability fine common approach correlation handle highlight sample moderate sized although design sized iid classic represent rao pcs represent case determine pc mix fit form mean curve curve deviation frequently orthogonal estimate curve ar bandwidth could autocorrelation demonstrate mean penalty parameterize decomposition involve regularization truncation spline separate penalty deviation spline truncation inherent pc decomposition spline white parametric effect model spline basis curve perform curve mean curve spline location represent curve model matrix pc decomposition penalty curve deviation reversible jump bayesian average number wu spline knot deviation truncation inherent zhang polynomial represent mean effect smoothed subtract white level function pc decomposition adaptively free truncate spline sparsity spline diagonal fit thompson level spline candidate wishart function process partition sum weight kernel white al specific gaussian bernoulli spline determine four scalar green wavelet space regularize induce wavelet adjust like et fine grid flexible represent introduce mean curve parametric shape
nearby trade operational consideration search initialize restrict grid subsection c sequence supplementary separate solid dot image show criterion circle current supplementary high red undesirable greedy row choose circle span column focus first four eventually possibly third big show design choice impact choice red close corner third behavior surface create fourth column eventually fan behavior first column fourth search initialize use nearly zero follow arc first arc creating near final row start empirical local design robust parameter number orientation work regularity accommodate select along suggest strategy leveraging discover different structure partition place along exhaustive amongst quickly numerical optimizer
var appendix scale purpose vector suppose also experience expert opinion background word prior let transform obvious way uncertain prior uncertain p base statistic component appendix posterior follow f posterior integral box gamma completion square gm define thus density proportional g q g thus proportional eq f e
team team player team player team effect effect expression probability team mean team equation maximize accurately situation team team team comprise team team probability team margin solely eigenvector centrality maximize historical game occur maximize historical formally historical occur year algebra edge network calculus algebra three genetic search global team implement would maximize occur initialize player vector genetic run carry fit certain genetic take converge therefore decide genetic attempt optimization iterate randomly vector simplex
outperform accelerate speed method reduce accelerate solver achieve acceleration observation acceleration author like suggestion besides support project natural foundation china important proposition institute chinese ia ac ia ac cn massive increasingly various challenge generally speed selection subspace specifically rank subset suboptimal property find globally optimum category assume center center select
verification accuracy thank exploit performance face verification adapt approximation anchor graph introduce speed take addition application need dimension memory time speak far gpu large inversion issue online intuitive representation acknowledgement chen work partially vision project lc ie edu remain illumination single source insufficient variation propose principled process variable name comparison additional multiple verification unknown target adapt automatically therefore must determine since due use end propose name face verification source advantage improve constraint asymmetric task constraint maximize multiple gaussian gps parametric also
introduce define infection remove equally infected individual improper initial infection augment likelihood respect measure st j denote parameter period formulation common model although also densitie datum introduce allocation yield simple gibbs st st dt nm infection infected say infection sample may accept sir period datum shape rarely bayes epidemic scenario b respectively table summary respectively factor behave model scenario
determine set tree amenable projection diagnostic exploratory distinguish characteristic evolutionary brief overview concept tree acoustic study investigate evolutionary evolutionary tree traditionally object specie field formally graph graph network thus analyse assumption process oppose tree constraint unique connect interior node white circle leaf think language interior version attention interior node adjacent leave express three connected leave tree fundamental circle fill sep circle draw sep relationship case expand recently attract considerable example advance author geometry space model see constraint implicit binary understanding beyond active semi algebraic assess whether tree language acoustic set model applicable random specific order constraint yet exploit purpose investigate tree precision latent matrix univariate relate covariance result result covariance calculate necessary margin positivity strictly triple positivity impose moment derivation considerably emphasis give apparent focus solely fundamental tree constraint linguistic
language semantic acknowledgment citation page stanford computer science neural network mean open support demand logical evaluating whether plain neural learn contradiction representation first artificial relational recursive structure quantification natural simulate logical natural structured neural successful array sophisticated language sentiment detection encourage ability representation question whether learn achieve fidelity algebraic whether match ability core semantic phenomenon like quantification contradiction algebraic yield framework language representation yield effective representation typical reasoning
lp qp qp lp qp lead dual sparse svm solver separable p implicitly middle panel figure define cost k k k primal soft comparison qp rest derivation standard result qp equivalent standard svm solve solver r svm learn function quickly let kernel ranking function become learned summarize svm qp tune kernel tune hold cc pair pair compare pair inequality accurately test two
readily close minimizer soft thresholding multiplier update price ti summarize topology unchanged available period reality power grid frequently expect thousand condition cope challenge recovery enforce square show nh value absolute cost cope batch price online vector suitable tailor processing algorithm aim form regularizer leverage grid recovery setup problem whereas equivalent regularizer online coincide regard attain sublinear building introduce copy soon admm update upon complete reformulate linearly
individually fraction item pooling include multiple linear allow item detection first operator extremely due commonly amp since property match recent use amp optimal one fact however bp situation efficiency involve nan
exceed note technique usage dp adjacent dp show memory need disk store disk scalable large usage scheme recursively problem subproblem subproblem split space order space although scheme practical limited increase ai ability solve intensive serial much several already develop bn mention author pairwise scheme parallelization processor solve compare run parallel scheme subproblem variable parallelization overhead become parallelization present take subproblem control complexity algorithm redundant calculation dp solve day processor processor processor decrease processor processor pose barrier large parallelization dp efficiency dp lattice equivalent powerful topology computer exchange adjacent
validate offline live engine use evaluation subroutine offline organize describe capture interactive search describe offline discuss solution issue search engine section discuss finally section contextual bandit formalism classic armed information interaction useful important present content recommendation formally contextual learner contextual reward reveal learner contextual optimize action interaction environment convenience q run action round reward almost increase search engine include vertical news vertical web click variant vertical optimize give reward another
improper begin fisher mean multivariate follow discrete discrete dimensional sample white denote vector slightly abuse notation let bayesian assign conjugate covariance hierarchical mean covariance normal wishart distribution version introduce remain hierarchical bayesian conventional wishart hyper whose posterior belong simplify form gamma ax e bx hierarchical constant normalization incorporate prior state posteriori follow order derive calculus formulae ax linear I xx information thus require initial b iy ty I I converge surely
rank list large relatively dataset essentially plot confirm previous essentially well offset need ii target recall well relatively recall g iv retrieve top optimal fraction retrieve panel retrieve value code plot retrieve code panel retrieve target recall retrieve code panel retrieve value
restriction continuous exploit ol natural krige positive definite gauss krige mean banach space continuity ensure kernel ols krige restriction strictly krige maximal ols krige unbiased predictor diagram eq banach exactly ol infinite banach ol generalize krige hilbert certain array manifold krige function smooth prediction effect compactly consist co functional array array functional array compactly support value b ci evaluation array integral scalar array scalar space compactly support measure array co let co array functional measurable mean arbitrary quantity ai product co array assume throughout co array support automatically array satisfy e mi array array dirac mass weight dirac act evaluation embed explicitly structure integral operator ki ci operator closure existence justify diagram summarize necessary integral operator machine represent hilbert important continuous ci ci ct ci maximally almost continuity assume correspond array almost discover certain satisfied proof arbitrary compact cover entropy index
adapt system predictor outperform predictor electrical engineering centre ac electrical engineering centre wireless technology technology ac wireless try serve reliably use selection feedback feedback change set rate wherein build like prediction partial frequency prediction problem complexity provide upper information theoretic simulation loss throughput abstract evolution adaptation exploit variation wireless rate bit per suited channel condition support rate would optimally sequence joint user propose source encoding symbol study encode practically algorithm modification source encoding namely build converge depth asymptotically practical may stationary long period short furthermore difficult requirement use frequency depth tree implication discuss analyse sequence use extensive upper tree depth optimally pick reflect propose use order
core security system modality define structure datum project template maintain classification paper desire property preserve random good formal structure far main subspace say exist subspace formally state independent tell definition margin subspace separate q geometrically say dot subspace dot subspace angle subspace sample linear structure preserve
hyperspectral contain snr pure pixel run observation abundance different able rmse favorable additive snr converge equal reason mean row abundance matrix large value converge db db db repeat time large predefine threshold estimate detecting
bootstrap amenable parallelization store project high space access file minimal memory storage access requirement compute furthermore even necessary summary bootstrap distribution translate summary quickly bootstrap calculate bootstrap dimensional confidence region pc construct solely similar complexity available arithmetic pc e statistic calculate bootstrap procedure eeg procedure later pc pc purpose functional pc easily demonstrate feasibility health study design analyze relationship metric health patient entire use place use stage primary pattern eeg among quantify uncertainty variability reflect subsample history eeg hour eeg subject eeg eeg consist two measurement raw eeg subject windows proportion hz recorded proportion preprocesse raw eeg smooth subject result hour hour window panel example function across subject figure first primary way course four correspond pattern fairly smooth pca eigenfunction course five pc pc explain variation pc material
completeness adjoint matrix main difficulty side coincide hand rewrite analogously mean adjoint denote mahalanobis dm put implie since put direct consequence fact feature concept generalize mention approach reduce calculation rbf usage invertible invertible replace consequently retain geometry preserve rbf one restriction class lead restrict possible adjoint
sometimes fix huge amount annotation subset namely car face social form manually annotate positive evaluate precision rank dataset new indicate region spee berkeley vision deep clean architecture rapid file switch gpu gpu architecture convolutional neural visual sentiment concept mostly eight main conv fc five convolutional fully fed softmax class correct multinomial
place bipartite partition sbm perfectly odd vertex also bipartite whenever identical partition illustrate sbm force break symmetry find partition provide prefer demonstrate moderate find well fast solve small likelihood sbm explain quantity evaluate correct thus find vertex separate search search search community roughly fact large space degree correct sbm gene sec iteration well rarely find pure eight replicate find converge sbm take second replicate eqs difference sbm random initializations mixed solution exhibit qualitatively optima additional flexibility suggest sbm optima sbm infer replicate easy correct sbm perform small perform reliably partition sbm fail projection modularity moderately inconsistent corrected sbm nearby section partition
filter within ranking user item community collaborative rank paper pairwise simultaneous item less candidate far rather rank differ million preference item c important focusing assume user successive wise matter namely effectively distribution way introduce parameter community belong enable rich modelling community ranking generation process parameter unseen second approach stage model
predict computation correct implement event thus fundamental sensitivity demonstrated present herein prior table fig insufficient explanation support random dimension average strength incorporate parameter narrow prior parameter expect uniform exclude value likelihood similar impose prior often exclude ideal recommend look impose mixture low limit look divergence time uniform place uncertainty suggest divergence constraint divergence must fall state allow place majority believe still detect divergence reduce support cluster implement less cause prior unlikely exclude true prior likelihood rather causes exclude divergence show narrow weighted support model less space carefully bayesian strongly probability unlikely region support wrong evolutionary history inference historical event analysis treat cluster fortunately uncertainty avoid region improve inference
validation subject cluster two safe place immediately affect reinforcement function pass input value safe observable configuration op place action encode implicitly three demonstrate robot action probability use belief human subject execute actual robot predefine execution assume preference subject demonstrate task assumption later response post experimental human participant robot perform action phase reveal evaluate act human human indicate although exactly human enough validate leave cross validation demonstrate weighted assignment likelihood cluster compare manual label code type expert safe expert safe type robot place person alternate finish algorithm perform partition
visual nf training test feature error observational entirely spurious predict nevertheless may accurate observational completely inaccurate recall causal observational train learn hypothesis could section pt net mp pick representative class tell observational principle ignore issue observational observational causal class question represent consist laboratory causal observational input class causal acquire observational step predict learn mean metric experiment metric induced metric space induce distance propose way approximate requirement output random observational choose causal loop search image neural net general set output desire causal mean agree accurate want minimal variant break desire perform feature observational confirm learn
random posterior classifier expert solution marker p generation visualization fourth plus marker concentrate abc statistic let covariate classification expert allow drive insufficient specify classifier expert namely proportion individual infect posterior marker comparison expert summary marker affect suboptimal choice expert select expert additional consequently posterior curve triangle p expert pdfs reduce affected posterior marker abc cause triangle red uniform convergence three condition condition equation classification accuracy classification classification rule validation denote set validation set disjoint point datum disjoint splitting fold possible equation validation rule classification probable measure unknown rule obtain q unlike sum make weak stationarity belong expectation occur classification bayes step frequency law rule meaningful analogy sum consider sum logarithm opposite unnormalized probable
cox enable mention cox close baseline estimate flexibility incorporate recurrent event form censor time assign slot correspond binary takes happen censor observation period recurrent period start recurrent refer full event censoring express let value cox cox dependent logarithm transformation ease baseline hazard assume smooth process write intercept shift relative baseline hazard order accommodate
vector weight ki k output economic pursuit mp storage mp mp pursuit prove useful accord eq x x show number rank residual decrease observe hold uniquely stop nonzero hold singular r conclusion eq contradict consecutive residual back complete ready definition property property view extend low f completion study recall map define express rewrite ad mn mn mn mn ready tackle apply singular least square kk converge procedure present detail hold approximate set able rank recall
movie low anomaly score movie movie rate rate user group action star series specific lr lr title full vi mr generative multi view anomaly find inconsistent confirm detect view anomaly anomaly several relax assumption would application cca annotation indicate cca wide nonparametric probabilistic anomaly inconsistent view cluster view cluster view view
limit relatively minimize fitting design noisy huber loss usually original deal effectively motivation roc space outlier exist consist load tr pc low r pc q nature term matter robust satisfy vanish regardless loss outlier estimation outlier motivate mean throughout give project onto decompose part stand term describe entry finally sub problem challenge increase dimensionality subspace outlier outlier regularization refer observation sparsity project necessarily roc pca estimate pc identify rank reduction roc way enforce e take convex p suffer biased inconsistent fusion penalty scad hard ridge also apply row type roc row arise contamination reader roc conventional modify frobenius estimator subsection estimator build roc generalized estimator begin definition n pm tm estimator derivative
tm use q super search classification describe range allow framework admit fitness evaluation focus algorithm search drastically algorithm assume error iteration independent consist spike consist incorporate algorithm super greatly never design hand case find random guess polynomial necessary notice learnable necessary search computation algorithm study evolutionary domain domain analyze last datum whether acknowledgement national foundation china evolutionary purpose inspire phenomena pa subroutine
determine include parameter one definite redundant parameter well unseen position order accuracy task divergence generate give task prediction practical situation give make formulate latent observable gray estimation target figure observable estimation latent circle latent target estimation estimation panel show probability target
toy use challenge power hierarchical benchmark infeasible dimension usually within strategy useful much computational however gibbs sampler slowly scheme limit hmc novel separability hmc semi hierarchical vector hamiltonian example result hmc large restriction enable restrict importantly restrict g
k method take algorithmic prior requirement demand feature select possibly solve need easily incorporate exist meet requirement individually enforce sparse regression hand regularization presence correlate essentially correlation successfully regression incorporate knowledge process demonstrate alternate admm net quick solution converge slowly interior svm able svm qp solve quickly hybrid simultaneously combine novel knowledge propose able exploit knowledge feature
f r strict inequality characterization american european binomial suitable adversarial upper american black price calculation characterization always exercise merely payoff payoff calculation resort multinomial recursive tx gx discretized uncertainty length analyze approximation american give arguably american nature round execute trade decision discuss address connect adversary bag one price either decide underlie factor figure movement pick total payoff movement option high option worth option asset case free gain go back asset price replace movement binomial low rise strategy conclusion uncertainty game binomial bind price also equilibrium bound binomial movement movement neutral particular upper option price risk neutral neutral measure use pricing risk neutral measure analytical correspondence play keep mind price backward programming suppose entail movement write solve iteratively low match illustrate price option apply maximal next instead final grow make device pricing c bag bag adversary standard model price european uncertainty round payoff option black start analyze version write r formulation follow definition minimax represent upper
likely block presentation auxiliary definition column index instance return straightforward exercise finally never query result require quantify insensitive deterministic formal technical intuition kernel expectation choose uniformly point quantifie hold uniform sampling block sub query block member value member quantity query discover query use case k note belong discover neither occur term account one previous contribute discover contradict logic least eq query bind query pair r j tr
rank recovery apply mathematics nj department nj department recovery mle convex relaxation likelihood
improve fine significantly complementary property expect conclusion per fine tuning raise performance pixel cost improve support pr c ap token fields se stream convolutional contribute third mid detection provide boost feature stream contour performance contour task fusion contour performance sketch token information structure stream use convolutional layer achieve contour fine tuning
circular eqn circular q substituting eqn eqn eq demonstrate mmd circular discrepancy contour distribution fourier transform kernel generate sample unit circle calculate circular discrepancy mmd circular discrepancy construct implicitly furthermore note norm norm investigation begin mmd sampling spherical ratio large eigenvalue match mmd result fig bias mmd quantify absolute exhibit see increase quickly mmd unbiased indistinguishable method mmd prefer superiority mmd synthesis rectangle locate within circular mmd kernel family compose isotropic bandwidth multiplicative exact
arrive apply inequality present property prox eq regularization depend publicly dimension source list process choice machine benchmark illustrate characteristic prox splitting uniform function choose prox number effective pass effective pass evaluate component full pass appear curve prox big remark iterate dataset vary varying period leave right objective gradient
adaptive generalize simulation train median sequentially median hour process adaptive resampling would hour occur resample rgb rgb manuscript resample scheme value tuning parameter efficacy procedure sub parallel gain procedure explore possibility produce identical tune maximum depth prune another spline prediction identical fitness procedure never unnecessary resample risk use remove identical choice need performance might conservative could confidence discard setting adaptive procedure rgb end rgb rgb tune parameter efficacy
assumption empirically also package effect illustrate confusion large make illustrate left training therefore start decay away start base accurate however case clean parameter fix parameter cost increase want prediction identity belong additional outlier enable apply describe noise base sample outli across unfortunately exact add outli make become eqn outlier principle experimentally sensitive network without first label clean dataset flip
aim payment compatible generalized axiom eq length payment axiom payment worker expect payment assumption present section utility case describe experiment amazon crowdsource platform would emphasize mechanism research amount typical crowdsourcing task expect worker understand mechanism act instance expect significantly upon modification amount compatibility propose mechanism standard platform amount high worker researcher design game compatibility prevent check experiment reduction comment worker suggest work theory amazon crowdsource platform call worker exchange pre payment payment part task nine task range annotation speech answer skip mechanism requirement worker amount total task mechanism constraint attempt worker complete least task respective certain worker execute depicts worker baseline skip mechanism nine question incorrectly average payment iv break answer fraction plot gold question prevent result choice time time put payment mechanism self mechanism receive comment worker receive increase complete job complete thank interface task obtain solution website author gate bridge three fix gold compare baseline gold skip confidence worker number worker baseline correct gold standard base mechanism result worker convert upper remove answer error match true worker identify ten depict gold baseline gold skip confidence classify depicted amount
rule q twice apply cauchy schwarz arbitrary desire calculation inductive complete inductive relate bound use complete substitute use shall eq statement recursion second step ps ps ps induction hypothesis conclude necessarily aim seek algorithm regularity quickly usage certain generalize smoothed huber estimation algorithm achieve erm every consider quantify implement space algorithm decrease standard super polynomial trivially see handling importantly quantify obtain comparable erm trivial guarantee size number large divide order address question low erm rgb initial
nature subsampling bootstrapping formulate subsampling explicitly subsampling first subsampling feature formulate bootstrappe forest subsampling subsample explicitly apply scheme use various scheme forest scheme slightly popular accordingly make applie subsample focus approximate detail first mathematical framework key node equation randomness build mathematical support construct space naturally subsample us feature let selection situation use easy index feature determine word focus feature probability get equally likely number possibility hypothesis regardless stem limitation infinite detect label feature choice purely realistic many dimension case reality spurious relationship label feature probability select usual tackle spurious relationship bag bootstrapping solve problem get still useful relevant feature important make correlation say imagine nonetheless complex correlation derive assumption tool determine threshold odd feature surprising statement somewhat sensitive strength insensitive study affect change commonly
correlate tc spline ss problem pattern undirecte edge cycle great connecting specify band along complete fundamental result discuss assign become definite large possible view particular band notation convex extension admit entry definite call let gaussian independent word sample covariance equivalence
preprocesse whiten whiten patch expect performance patch suffer get activation suffer mlp relu initialize train hide logistic yield supervised fine increase performance slightly dropout regularization yield permutation cifar slightly whiten extra none video transform dot whose random image subsequent transformation video frame intrinsic frame frame model interpretation
generally good satisfactory measure put emphasis expense measure propose opposite endow g lebesgue remark
parameter produce overall sn sn ratio dot line correspond described box sn marginal horizontal dotted correspond box pdf sn ratio horizontal dotted decrease sn qualitative result sn impact sn summary statistic certainly tendency closeness sn volatility asset pricing portfolio management segment three decade demonstrate volatility motion asset price inconsistent volatility empirical see option pricing notably volatility also view evidence price geometric motion black price review response many time volatility stochastic augment jump prominent option price become option pricing mcmc filtering assess adopt simple square root logarithmic asset period day wiener restriction positivity previous steady gamma density eq kind function view discretized version diffusion return observe retain diffusion discretization step abc occur exactly composition match return daily realize namely stock reference inclusion volatility would necessity financial abc invoke euler approximation detailed outline sigma deterministic specification define respectively filter particular step sigma
limitation infinitely none analysis square recent attempt made extend differentiable objective continue restrict constant remain style method hold large satisfy rsc particular statistical result work show relax optimal rsc respectively offer employ greedy gaussian family satisfy rsc convergence solve hard thresholding prox provide projection crucial unified descent confirm prediction
distribution additional advance unfold histogram treat histogram distribution come demonstrate regularization similar reconstruct check
say rating ignore observe adjust bias perform movie rbm gradient eq use divergence cd much produce log minimize kullback divergence minimize usually learn use new unit activate add rbm could extra conditional unit distribution become hide affected activate
efficiency factor minimax write algorithm input attractive adaptive sparsity level adaptive sparse zero inequality provable ask computationally statistically tight occurrence indeed otherwise adapt plant formal definition discussion price efficiency pay estimator study essentially rate computable order pair countable distinct clique clique plant problem suggest variant classical associate word pick vertex plant plant input graph sample random nature plant clique possibly algorithm know plant clique clique point asymptotically vertex plant clique
go go forward backward normalize layer stack lstm introduce feed responsible network single softmax secondly expand feed forward recurrent recurrent similar idea explore show
zero task minor classification gold part parameterize dataset example optimize gradient efficiently scalar possibility project lie parent illustration sentence far continuous standard backpropagation understand standard implement representation sentence experiment binary label sentence constitute resource supervision setting fair mini batch fold implement embedding treat parameter implement
draw draw divergence constant testing see imply infimum class covariance matrix complete observe prescribed lemma imply desire lower trivially observe treat separately pt combine case define integer yield ii simply invertible invariance eq q schwarz yield observe result list illustrative comprehensive detection however literature focus various performance functional covariance covariance attention originally gaussian white density paper study estimation regularity estimation nonparametric become phenomenon regularity decade high sparsity unknown parameter parameter interpretability simply sub particular covariance zero uncorrelate
lot describe dynamic algorithms analytic polynomial framework analyze optimization algorithm formulate popular utility iteration principle able term iterative dynamic process spectrum able generate sequel sake brevity quadratic order define typically prescribe constant function cast chapter one way generalize sdca nature randomly reason inversion become mapping mx clear convention hold inversion matrix execution algorithm induce test assume sequence close minimizer initialization interest algorithm take matrix satisfy identical matrix method add matrix treatment examine otherwise choose dependence true see readily naturally ask range characterize speak consideration technique instead answer specification l gradient method see accelerate rewrite stochastic gradient sgd straightforward extension go x sgd satisfy appropriately choose stochastic coordinate repeatedly minimize coordinate denote th row matter sag like sdca sag closely sag method derivation result slightly order sag framework straight implication optimization strictly come q apply
choose report optimization neural network exploit field program exhibit rbms autoencoder speech like nlp word datum like gray value become accordingly pixel word meaningful unlike nlp feed index equation representation representation etc anonymous value reflect degree satisfied word fed work word modeling probability linguistic word conditional introduce energy maximize word calculate researcher realize normalization essential merely feature word consideration local pattern word recurrent neural rnn dependency rnn would either vanish propagation nature rnn treat language language rich structural solve problem detail representation node abstract formalize learn objective subsection answer fundamental previous section vector representation real symbol character token level character
observe transform transform globally consistency denoise manner close apply correspondence main contribution present consistency pairwise transformation programming permutation globally special matrix present invertible similarity euclidean transformation interest shape multi view transformation introduce perfect information straightforward extension handle noisy finally type transformation zero invertible transformation
generalize median dissimilarity generate difficult solution optimal som principle median som iterate well determine update generalized notice prototype variant solve unit neighborhood neighborhood dissimilarity natural know som problem batch som request unfortunately som algorithm rather cost careful cost actual per avoid introduce arguably drawback median som intrinsic restrict massive som sub need determination propose subtle restriction unit empty apart prototype prototype usefulness visual representation matrix avoid point form generic lift prototype restriction som intrinsic som prototype representation dissimilarity euclidean q som mean define solve eq
appendix tw strategy gs review respectively measure handle tw monte move respectively spin immediate carlo section big roughly term powerful search state big approximately equivalent configuration repeatedly subgraph subgraph update threshold back go step subgraph update run collection state way try long find energy find ground oracle know solver terminate target successively value look something let repeat em terminate run result interval generator energy obtain occurrence discard minimum serve purpose estimate accurate statistically also second reasonably confident energy energy landscape behave ground reasonable energy energy ensure chance course rigorous
ensure multiplicative front large choose simplify integer choose expect false decay choose large forget subsample close large probability close rank base contain maintain relevant set increase reduction quantify discrimination occur available trade extensively see theoretical intend goal goal truly strict positive provide relevance false improve recommend indirect assessment method scalability recommend computing capability take covariate selection probability effect subsample section depend iterative base see limit procedure result generally strongly previous already highlight subsampling procedure follow covariate signal covariate case expect amplitude noise subsample omit procedure output randomize return similarly consider uninformative score denote procedure pd covariate deterministic analyse quantity circumstance concern determined estimation true word grow
lemma henceforth consequence event posterior epoch boundary stop vector put epoch choose define convert precisely represent eliminate suboptimal time epoch distinction eliminate eliminate break resolve eq particle modification action eliminate early lc cc lc suboptimal define bound value arbitrarily assume region positive quantity contrary eliminate parameter epoch sample eliminate display follow nonnegative suboptimal policy choose let epoch step post respective elimination write calculation write iid bernoulli assertion large enough thank coupling iid bernstein n estimate fashion probability application give enough sum give complete lemma conclusion occur remain suboptimal th standard schwarz c hence ct boundary stop write regard state lemma epoch kl whenever write obtain thank infimum set finitely expression probability assumption observe complete epoch see factor
oracle oracle htp negative stein oracle outperform estimator unbiased instead statistic rmse generate equal note perform normally small large test statistic average replication table c freedom stein oracle model unclear version right wrong incorrectly also method show empirically model preferable wrong inaccurate group feature standardized control accord standard belong observation group belong wrong share pool covariance wrong pool covariance rmse small
remarkably scientific people different rational revealed preference expand upon despite process interaction agent loose make algorithms process viewpoint problem far namely process evolutionary cognitive decision brain neural information cognitive make novel take
focus range underlie derive computable high dimension misspecification covariate define covariance net contribution covariate mean variance ingredient inclusion covariate alignment parameter account contribution covariate
partition boundary depend bandwidth roughly capture take scale size near mode mode reasonable converge mode one reflect local bandwidth utilize usage shift restrictive utilize cover fix shift heuristic density smoothing fashion isotropic track evolve primarily aim result somewhat offline supervise process neighboring stability estimate recently automatic bandwidth gradient offline focus gradient error bandwidth computation domain shift create partition multiple domain mutually segmentation utilize
allow interpretation high loading high loading turn loading allow specie column structural analogous follow base solution consider classification compute three specie case obtained refer svm look sound substitution space mathematically translate correlation physical maximize combination magnitude heterogeneous inside inter field correlation statistically significant first canonical variate give explain table table among substitution canonical correlation transfer physical drive act second variate link minor discrimination scale significantly variate matrix report expense substitution imply change relative substitution significant canonical correlation substitution discrimination logic root move e substituting opposite moreover move negative loading aggregation move different move svm basis explain svm svm result cost majority component collapse general express without lack relevant aggregation point relation physical canonical report
snp various performing screening concentrate second differentially private screening paper procedure elastic regularization differentially private step accurately end differentially regression interaction elastic penalty extent penalty exclude absolute experiment high score snps convexity impose sensitivity privacy show often recover interaction term factor middle bar
also thus new domain h update learner requirement meet set tree guarantee carry incomplete loss move functional functional may optimisation author point decrease subject h translate select large boost proceed likely accumulate boost add ensemble control average mrf boost learner fail correctly instance mrf training particularly desirable formal tree selection direction condition search direction small scalar continuity differentiable constant find experiment satisfy mrf linearly take whole network example grid g successfully tree purpose tree subsection belong
capture multi explore gaussian depend rather valid filter individual expert overall parallelization property combine power allow subsequent property combine poor product expert achieve scalability result comparison mixture probability expert generally
entire dimensionality nonetheless common category retrieve pre consume intensive grow differently generalise category runtime attempt address however attempt limit produce necessary small stage complete inherent nothing relatively compare ac ac uk overcome world computer vision carefully imagenet category action video offer develop scale purpose category retrieval operate millions second system reach product amazon current system typically bootstrappe search source learn third video rank retrieve contain category aim stage happen matter second retrieval performance trade high severe penalty severe high costly training rank excellent compression quantization page entire aside gpu gpu
time typically iterate comparable experimentally discuss translate property provide use kalman grid black example regard iterate gradient em smooth exposition far fix either computation benchmark kalman shape scale subsequent example run particle normalize kalman bottom average true bottom variance n mcmc put model consideration problem numerical program page author first long statistic n nx ny result omit top panel show mean evolve ratio kalman exhibit bottom scale computed filter grid estimate decrease support approximate provide method portion confirm mcmc fail space false security scale run method experimentally estimate instead linearly mix suggest increase remain degeneracy problem particle convergence rigorously red versus truth one short know estimate ny single gibbs addition figure posterior independent method seem consistently inaccurate negligible increase methodology performance run additionally compare result estimate particle display improve particle mcmc increase computational particle gibbs display parameter prior favorable particle black dotted versus
query relevance get detail exist map document convert row abstraction consist iid document often vector rank sort scoring contain bind norm accordingly similarly natural counterpart ingredient function set negative vector score differentiable smoothness depend twice norm
h approximate replace alternate constrained define scalar scalar scalar penalty assign measure user enforce easy hoc negativity practical nmf dense use another solve e element solve thereby execute far solve fast available truncate comparative nmf factor incorporate nmf believe call reason well nearly multiplicative become iterative element remain mean improve toward continue provide flexibility path towards poor prove saddle converge local ad hoc drastically saddle suffer nmf guarantee prove otherwise superior alternative hoc linear whereby
carefully follow decomposition loss characterize codebook loss stein estimator simulation result show empirical compare estimator randomly draw rate comparison stein see estimator towards shrinkage rate stein choose signal ratio observation code method compute repeat combination recorded error theoretic low error convergence sharp bind estimator nonparametric euclidean similar ellipsoid principal ellipsoid great
generalization since df df r f consistency conditional presence asymmetric rate provide conditional accurately trade surrogate next estimate x show dx asymmetric x p rate estimate probability sample low universal guarantee n nx r kx separately guarantee mapping universal induced reproduce kernel hilbert slow preferable density hard density reduce curse accurately variable therefore distribution
power logarithm power turn level method perform mostly equation clean image relate manner replace sample natural kind practice analyst potentially indeed invariance general modification modify free noise datum result see put large rest spectrum eigenvalue proposition notation call I similarly since hold asymptotic assume assumption conclude come unit modify work instead diagonal matrix q large asymptotic eq clear spectrum spectrum try regime hence neighbor near robust avoid near intuitive context systematic future impact kernel started compare level much partly result focus laplacian map create difficulty rotation impact additive broadly proposition collection dissimilarity consider regime call noisy asymptotic though way change version clean version suppose consequence furthermore gaussian like robust recognize essential move situation elliptical scale model largely noise I modify instance situation approximate approximately spectral free suggest original noise past practical theoretical refer interested reader aforementione paper detail
method default yield evaluation criterion elliptical distribution covariance stand stand contaminate contaminate model fix rank contain equivalently outlier separate observation spatial configuration rotation vector greatly reduce scenario need contamination belong complement concentrate simulation subspace span eigenvector whereas setting online resource belong robustness contaminate contaminate simulation data facilitate parametrization drop measure indicator observation come location find separate outli configuration frequently
quadratic denote deviation assignment th odd receive unit odd match affect equation vector unobserve lower since p nan indicate reach value reject hypothesis quantify sharp nan note statistic simplify I use eq sharp nan hypothesis test sensitivity refer refer odd equation outcome change inference furthermore q let follow third moment chapter pg normal theorem page combine fact effect lead finally normal case strength identical look quantity
grid theorem face bind estimation loss random estimator thus require nontrivial rely intermediate leibl divergence construct certain spread pseudo consider define hypercube eq fractional definition expect check choose need appendix proof depend certain sufficient choice rely depend lemma apply constant simulated picking randomly smoothly bound obtain realization innovation sequence center display online predictor explain corresponding specification overall iterate aggregate aggregated equally shift loss right line predictor one predictor accordance theorem aggregate first outperform predictor achieve original aggregated loss original td ta derive predictor improve quality maximal
model description transmission role deterministic typically ad hoc preference systematic approach considerable environmental aware work illustrate example consider deterministic base show calculate abc smc popular choose dynamical modification develop option factor previously dynamical become approach keyword selection ordinary stochastic differential dynamical choice dynamical expert knowledge hoc function objective
infer need deduce conclusion square root two use synthetic gibbs subroutine newton mc log nr approximating follow algorithm fix instrumental produce basic version propose examine assumption ty produce stand sampler gs importance step likelihood subsection computation use nr
lemma imply q affinity matrix connectivity solely gx rx covering ball graph riemannian determination adaptation riemannian use geodesic cx neighborhood point cx imply tangent proof projection onto easily difficulty arise logarithm ball arbitrarily uniquely among minimizer inequality follow restrict chart two subspace chart mt rt follow sample technical bx htb rest proof follow appendix see equality combination h tc eigenvalue h remark inequality derive simplify strong requirement q immediately follow graph function constant imply bind conclude exist constant estimate connect origin proof geometric let respect chart projection thus combine furthermore fact minimizer appendix exist depend manifold consequently conclude claim carefully constant riemannian manifold letting immediately respectively threshold satisfy gx word proposition point conclude noiseless geodesic due replace parameter satisfy requirement requirement I next explain satisfy requirement sufficiently satisfy rh conclude portion fact sample multi analysis multi geodesic tangent without noise difference level precise trivial claim robustness noise experiment study space solve propose
represent actual objective though low show explain linearity make axis actually extend camera estimating assume object detect view discard fitting represent green straightforwardly camera space use statistically track object false method camera throughout move accord velocity augment respectively object often size additionally population construct motivate remainder reader within object object object object might probability either detect detect dl spatially distribute image plane object know association empty restriction function association denote density object conditional update costly density density possible propagate multi density cardinality set first density set gm object previous framework mixture require survival assumption propagate gaussian assumption strong consider might significantly differ necessary relax detail adopt birth detailed scene restriction
efficiently p f definition sx joint entropy last let emphasize appendix control additional bind approximation bound intermediate bound outline theorem intractable sample q empty soon even approximate adapt technical aspect mention achieve ratio state fact check return recall
properly could give capacity especially face outperform deterministic able reconstruction c c c bias na na na hybrid na noise weight interpret probability away particle show estimate final substantial challenge reasonably reasonably separate p like promise address issue latter explain neuron feedforward
specify sub thus principle turn expensive smoothness equality add extra quadratic arbitrarily closeness yield eq like subgradient slow general course different euclidean projection operator descent minimization place result convergence ij ij jj l ii indicator otherwise constraint solve like gradient solve quadratic analytically h
utilize elastic net concave interesting unique prove play role basic denote basic eigenvector principal maximizer j uniquely proof satisfy sample cover selection easier relevant regression analogous order q technical limited contain condition special pca imply relevant select thresholding thresholding intuition illustrate difference toy select relevant sufficient condition assume block negative control sample provide satisfie satisfy unique addition part positive part additional individually negative exact part
water variation deep table noise visually fig start able truth unary gmm feasibility structure computationally purpose structure structure connect sparse encoding layer inference interactive perform manner tractable manner solve field structure state structure wide application natural language use mrf limitation utilize unary potential neighborhood smoothed boundary range state boundary
sampler cost cost report reflect particularly cost k allocation training perhaps achieve indicate line estimator underlie allocation combine weighting still considerably variance especially observe consequently allocation thompson outperform static practice tune automate adaptation benefit simply new strategy monte pool estimator produce cost map correspond problem bandit take area finitely infinitely estimator sampler bandit study variance family thorough investigation acknowledgement microsoft greatly decision subsequence chapter iii need second tt return denote begin round recall round cumulative likewise definition estimate mean may give history n kn I sequence share k identity cf kn get mse prove easily
threshold monotonically move close center classifier sphere maximize generalization margin separate margin method truly aim search minimize show svm deal cauchy schwarz divergence projection prove constrain reduce instability apply maximization project big circle point express procedure rule window width ascent gradient formula omit gradient computationally issue practice maxima start sphere run also possible start solution model perceptron reasonable short investigate section operation construct tx tx qx qx k qx projection classification point decrease search iteration build however access expensive change parameterization width window width factor replace equation analogously evaluation beneficial tend tradeoff big lead simple
label propagate along edge appropriate accommodate tend induce induced propagation achieve competitive avoid pair exploit bin node new feed common scheme construct preserve show hash hellinger use classical chemical sized cloud propagation originally addition expand kernel introduce propagation numerous central huge graph represent video care propagation kernel easily kernel scene domain kernel begin introduce propagation walk propagation kernel section example information scheme propagation well respect choice graph bioinformatics real application image classification object kernel develop mining establish relational graph kernel class walk size subgraph kernel subtree base exist often slow attribute slow kernel subgraph subtree introduce compute signature set label compression feature fed base kernel iteration although kernel usually competitive runtime design label label propagation mark unique symbol however propagate run label observation motivate avoid termination similarity neighborhood recently become popular
entire variable truncate replacement induce choice closely probable variable assignment particle approximation connect compare first sometimes particle fail three dirichlet hmm state differ particle filter total particle ess sequence filter run illustration sequence particle filter conditionally degeneracy without time conditionally unlikely particle resample size fall ess ess place
htp accurate primal penalize square arise compressive sense technique incoherence property sense extensive competitive state recovery without exact sparsity dms partially national science foundation china section remark primal dual frequently compressed strategy identify dual variable square define update certain sense incoherence property restrict isometry noise global extensive present illustrate strategy global convergence ten sense amongst broad application formulate follow sense vector denote component vector
raw unlike generalize discover group pattern suggest voxel subject individual individual raw align across reduce validation relative glasso find align expert carry train fold voxel shift voxel dimension misclassification show small successfully cross subject location allow fitting note pre perform report train expert pre select assess select priori involved picture classification voxel significantly lasso overlap elastic net overlap return cognitive elastic net make interest force across individual mean subject drawback interest succeed voxel task glasso make figure overlap group lasso ill overlap force force undesirable select elastic treat independently across voxel also pick voxel interpret lasso elastic leverage inter subject similarity solution voxel allow task correlate voxel lead cross indicate inferior predictor like elastic
depend contribute specific coincide global method illustration plot component versus histogram figure eps know indeed bayes density em method classify show entropy
develop without distribution constrain affine b characterize truncate deriving conditioning follow q picture give independent decompose dimensional clear condition univariate fall q truncate lie plot truncate note width quantity cdf cdf f b v yy gaussian agreement summarize confidence interval
calibration capability base classifier assumption loss measure plus rank counter develop processing separable dataset calibration base learner discrimination auc acc perform rmse retain discrimination obtained prior post processing improve histogram size experiment fix testing method capture calibration size experiment see steady datum real uci repository information people decide letter cost whether person among include build two predict person letter people expect return letter concerned choice
develop svms cost corpus imbalance imbalance discuss experimental setup present evaluate hyperplane svms close positive pos recall suffer present pl many time tune training overhead subject max part quantify reduce
descriptor processing hence feature explanatory depict stream ccccc diversity appearance scenario identify literature experiment order study behaviour option choose popular member design classifier constitute quite situation unary strategy completely gmm use unary least batch slot combine slot little effort accuracy label batch refer misclassified batch formulate period label batch available effort knowledge stream non stream three baseline approach passive batch label obtain train stream online learning need complete second expect odd batch odd keep buffer time train buffer classify batch partly set passive however odd original
email spam relative occur mark choose split spam long tail community community fold yield number community correspond lr spam positively direct conference internet business email mail receive report address quadratic discriminant path graphical naive interaction single linkage cluster community procedure applicable diagonal sub problem show community sparse encourage matrix sparse produce interpretable
dim dim dim dim dim dim cause entail benchmark research word embedding microsoft research relate word describe select build potential scale recently rapid learning researcher start model amount text nlp notion relationship deep approach
model vocabulary integrate iteration chinese restaurant collapse gibbs posterior section describe general conjugate auxiliary alg word allocate hdp term conditional document denote allocate exclude word give word allocate topic empty topic weight new new atom range topic indicator remove unsupervised replace document sample assignment difference topic update allocate gaussian response though response rewrite cluster residual empirical finite count prior parameter remove document average use calculate regression glm sigmoid close coefficient method converge number instead sample common laplace unnormalized memory bfgs map estimate parameter sample document model simplicity also consider estimate find benefit sample word allocate topic randomly document old km parameter binomial value use l dataset classification financial direction price stock
reasoning law jx ix j reasoning event union event k ij hx x hx ij x hx k ba jx ij jx jx j j b j j prove differently reasoning implication combine implication label disagreement combine yield bind improve publish reduce marginal product space axis result simplify complexity achievable active request great maximum classifier value budget exist marginal active binary classification establish low p b k k thus aforementione establish second hx prove target allow request great return classifier ix xx h xx jx
b appendix shrinkage estimator scale interest covariance matrix scatter regularize estimator I condition automatically sample estimator recursive converge convergence mean convergence penalize respective shape regularize estimator shrinkage mse due general estimating problem scatter scale estimating question estimator shape select e derive estimator shrinkage estimator alternatively scatter close scale copy w hereafter seek minimizer mse q oracle oracle plug verify employ value matrix component uncorrelated toeplitz note identity rank measure
sample vector alone kernel duality kernel cut pca basis manifold necessity subsample ht compute cut component matrix dx kk compute truncate reduction principal agree pca right component onto datum ht feature like feature like entry coordinate manifold entry orthogonal apply right vector compute vector
j n c short connect parametrize positive numerical reason conditioning covariance limited dependence covariance log formulation block form work wavelet function convenience stack use independence q must admissible quantify admissible practitioner application additional minimum mmse bayesian complicated posterior determinant parameter situation common markov chain mcmc accord turn sampler successively accord conditional associate sample instrumental propertie reader tc p c r instrumental c r tt burn define n mmse subsection require inversion computationally prohibitive numerically due grow replace exact assumption realization regular lattice additive
tc stepsize matrix variable great equal problem multiple experimentally fraction instance discuss stop iterate improvement tolerance maximum impose scaling gradient scale gp whole scale scale method cl backtrack differentiable limit degenerate local experiment plot iterate fast decrease function observe impulse impulse obtain relative scale method scale importance versus performance ht ccc result test number nf average second whole run server dual intel processor mb cache gb ram parameter matlab interior trust region ip tr table option suit
variant computer software framework language communication datum provide tolerance communication central communication bottleneck method principle communication substantial develop version simply parallel term processor need coordinate need change serial furthermore classical convergent require step serial variant recent coordinate surprisingly requirement decomposable minor modification modify fact communication degradation decomposable compute contrast robust happen asynchronous show highlight model system globally ability prove useful formulation useful mathematical place pressure accommodate increasingly dimension internet text long progress utility interior go discuss locally algebra cholesky multiplication take prohibitive solution big inexact describe big review non term
transition recover plant hard hard transition occur transition transition bethe energy cross exhaustive search regime lie transition plane meet order arise introduction bp plant along height reach critical possible ground label separate realistic use maximization minimize discuss parameter namely political label
relatively proportion people people another final project sub group south massive infer potential explore influence enable practitioner connection meet global online finally note flexibility specification robust generative capable represent label validate benchmark make candidate towards learner group low future like run service course student learn feature reveal massive new factorization computationally efficient cluster topic local explore underlie user sub level statistically significant
art contextual vector sequence context reward advance adversary round reward regret hypothesis purpose contextual minimize cumulative one hide layer learn action know many
dominated exponent ultimately determine rate change change presence phenomenon force close therefore observe distinguish piece assume notice likelihood follow order piece character inspire estimator drop subscript kkt coincide formally establish follow originally prove fused coincide applie segmentation covariance scope huber instead convex filtering piece wise approach piece arise consecutive decrease however double constraint hence integrate walk z analyze detection fuse true piece
theoretical runtime note lsh scheme approximate threshold vary norm scenario approximate query change parameter also costly preprocessing query correlation norm keep inner product upper norm word transformation sign thing work norm transformation know precision recall well l parameter recommend correlation cosine note
skewed type configuration loading ss sample loading factor dense loading dd capture gene co er differential network batch ds gene variation dd variation affect variation gene population calculated percentage ss sd ds mean order run orthogonality component explain quantification assume normalize across wise calculated sum total gene contain loading component sd ds dd category ss ds dd number component total select fit identify differentially across identify er across er component er differential er component panel dd component ss ds component er er er panel component er er sd panels range minimum across precision matrix subset test
algorithm utilize second sort adopt sort via algorithm argue dyadic define close interval q evident time cite numerous disadvantage adopt correlation idea idea root idea utilize computing extend suitable also detail effectiveness evident size become availability detail publish study rest paper covariance correlation relevant property unbiased additional capability finally conclude remark make proof correlation co several pearson pearson dependent
shape peak flat shape end excellent compare finding fig substantially computationally intensive discrete datum lp framework discrete free goodness discrete develop continuous make test whether procedure em compute goodness test statistic justify h equality goodness chi square probable compare chi fit equal square measure interpret lp drive statistic interpret chi statistic powerful utilize universal orthonormal construct polynomial recurrence relation often complicated example investigate parametric fit select shape base model skew lp skew exponential estimate u plausible heavy tail performance statistic alternative detect lack contamination component traditional ad er von level nan cutoff level set
right singular vector top singular running appear appear completeness description construct factorization column k lemma construct k rank replace consider step arithmetic output modify imply bind hold orthonormal accord conclude improvement spectral lemma spectral aware deterministic randomized factorization method construct matrix k take randomized sampling modify let k imply square eq expectation respect proof mention analog decay leverage
observe fraction recover version pose machine task analysis several practical np hardness show problem broad constraint trace sum memory time prohibitive iterative call optimize hold
index choose g sgd architecture try break mistake classifier run iteration word iteration loss monotonically
fdr procedure develop three image field dependency generalization replace markov markov spatial image human brain assume grid follow two parameter ise voxel hypothesis normal extension hmc major difficulty normalize likelihood product tractable thompson may field mechanic intractable turn sensitive zhang algorithm et maximum despite incorporate name se restrict belong exponential family search mle prevent mle average adapt backtracking increase lead generalize et fdr motivate image subject patient ad subject voxel cubic voxel line posterior ac line wang
incorporate auxiliary design designs integrate feedback auxiliary item raw rating rating rating user movie measure actor extract wikipedia engineering restrict case fm description rich interaction fm capture recommendation integrate auxiliary rule transfer constrain vector tag inform incorporate obtain introduce basic knowledge near neighbor feature trust introduce different friend include user social knowledge friend feature similar term friend term one tv
noting assign current mechanism detect tends adapt equation grain reduction eq reduction technique cover batch advantage tune trade thresholding bound thresholded weight bias weight normalization layer convergence sgd condition size guarantee
v vs image rsc vs vs vs rsc vs reduction comparison due increase similar experimental bias use bootstrappe eight datum bootstrap replacement bias set show attribute four observe increase biased determined case reduce comparison single unbiased forward reduction complex structure pruning systematically reduce bias average rsc make result low rsc commonly net lie ensemble bagging rsc overcome instance rsc attribute new reduction rsc compression compression general keep large number fall knn explore rsc examine cardinality cover probabilistic sphere degradation possible heavily prune strong candidate create rsc control retain instance
observe efficiently derive analytical normalization drop without loss generality precisely combination solve evaluating denote standard vertex type q iii condition use similarly third vanish correspond vertex arrive c c take point write operator point condition instance inequality bind simulate illustrate box maximally achieve full distribution numerically impose plus normalization see arrive distribution achieve quite simulated moreover result another nice much bi full minimize value almost display fig decompose measure influence particular simulating drastically move region finitely analogously feasible polytope define compatibility region show main xy pa violate state plane measure angle obtain exceed compare even causal
capture velocity consecutive capture particle position velocity cubic velocity jacobian separation spatially instance jacobian stability root offset satisfie query point unstable vice stability field result time scale vice versa frame transform comprise act unstable point scene denote height frame collective crowd simple grouping velocity similarity group respect change crowd phase shift
unfold recognize sigmoid structure please layer frame weight relax constraint conventional sigmoid network note feed sigmoid start layer perform forward update structure schedule illustrate general mrf feed sigmoid interpret either interpret mf mrf may interpret deep unfold compact structure either one inference variant neural model belief propagation probability apply loop bethe approximated well investigate explore unfold without parameter bp bp mf update marginal posterior know mf update message contrast mf belief normalize belief output mf mf versus incoming message prevent ensure incorporate belief yield marginal mrfs cycle long prevent feedback true problem message pass field formalize two unfold layer implement update accomplish message optimize message schedule bp instead message mf schedule
add already calculate ordinary forest global score regularize gain concept particularly represent base maximum control long thus addition add importance follow normalize rf control importance also select score one rf calculate frequent tree association association association transaction transaction association rule transaction contain rule condition side measure follow define proportion transaction transaction outcome transaction contain item contain length consist use side
essence criterion approximation regression process principal operate current discard belonging include dictionary residual obtain project span coefficient cost current function eq removal budget concept discard nonetheless removal process issue motivate complexity dictionary computational expensive complexity scale dictionary operation moreover condition express multiplication instant computation affine reduce linearly dictionary instant error atom atom moreover discard combination atom possess duality discard bridge criterion purpose derive approximate atom obtain coherence criterion
gap small event eq suboptimal depend argument event happen time event remain choose monotonicity satisfy moreover q bind numerically upper bind event item specific associate q item observe sufficiently often event item gap far bound definition bound decompose regret part gap proof gap inequality conclude proof follow definition fact green green definition near whenever offline
linear allow feature response discussion support nsf grant finding conclusion recommendation express necessarily reflect view descent section exploit property optimization approach completeness proximal use solution new extension proximal framework subproblem block compute step size avoid proximal solve efficiently coordinate project backtrack search summarize coordinate cycle proper backtracking exploit quadratic basis matrix coordinate subproblem formalize problem minimizer eq operator one thus completely track search qr block gram
autoregressive pa model many science science finance aspect var explore var estimate covariance coefficient compute var forecasting estimator end residual condition underlie uk u show ensure identifiability variable first dependence account variability individual structure encode separate dependence pattern motivation variable relate unobserved provide motivation similar see relation assume identifiability orthogonal equivalent model model latent variable covariance latent variable isotropic identifiability become estimator observation ignore proposition exist analytical reduce reduce reduce link widely
batch loose respect minimax rate much modify slightly yield discard extra obtain generalization error besides direct estimation dealing feature batch modify replace replace regularizer q correspondingly large new new attention allow latter prove generalization online verify batch lipschitz idea section generalization present error decay early minimax long problem extremely bind loose end probably problematic weight streaming weak good analysis guarantee penalization sparse result run probability output direct expect condition regression streaming setting achieve prediction contribution iterate convergence obtaining convex still similarly work generate isometry latter regime restrict isometry enough usage imagine finally design orient increasingly important streaming analyze procedure computational property favorable use believe combination examine proof check obtain emphasize random also regularizer mirror sequence
one use sense denoise focus mainly extension simple believe scheme believe art segmentation denoise processing lead work texture optical thank provide code author helpful constructive greatly jump representation additive polynomial segment approximate parameterization polynomial temporal jump jump residual approximate equivalence method find close jump subsection criterion algorithm constant option optimally solve minimization index sub recently sparse optical flow denoise vary solve traditional framework recover piecewise
realization case hold certain eq statement condition exponent side argument neither one hold covariance become om decay however fact pr l om term third apply eqs om lemma w standard ann mi k k divergence machine information statistic particular normality propose estimator example divergence plug histogram scheme assume none
rate variety quality level illustrate detect face detect conclusion draw qualitatively sample fail image environment high comprise human secondly robustness illumination image camera characteristic colour qualitatively notice dataset illumination approach discrimination stage effective opposed column combination notice quantitative analyse dataset result comparable colour high true prove human visual colour
scalar equivalent newton bfgs line converge iteration huber rapidly convergent scheme problem figure fast huber quantile behaved b b loss study demonstrate importance adopt framework sophisticated emphasis briefly discuss possible benefit paper form corruption simplicity corruption additive happen sense transmission scenario ignore image improve loss function discover corrupted randomly occur image kind noise therefore huber penalty thereby loss infer tends spread dictionary image extract patch stack specific randomly replace percentage
assess monte multiscale environments investigation simulation associate issue partially science dms condition infimum condition generality hx assume lipschitz lose due notational convenience notational omit use control rearrange lipschitz continuity definition jensen sup old eq omit distinguish denote explicitly mention put sufficiently inequality conjecture question proposition proposition proposition corollary proposition conjecture proposition proposition stochastic equation scale equation ergodic provably sample rare event expectation functional perform simulation presence randomness construct measure prove asymptotically numerical
feature extract deep despite importance often side task notable variant representation distinguish example label representation par include also sensitive calibration person person object provide person
social behavior hand recommendation tb topic image internet community facebook friend recommendation contextual planning article search ranking annotation via topic online internet community share friend tie query tag search tag feedback event wikipedia indicator indirect article group formation social membership growth measurement crowd social tie recommendation collective social capital self site longitudinal increase human computer interaction topic flow cloud code lattice galaxy mobile wireless traffic fluctuation centrality heat article user top flow cloud boundary galaxy galaxy dynamical trust mobile wireless monitoring article galaxy formation
learn word embedding also use rare distribution word tail frequency compose word furthermore word appear thus make capability new know top close calculate representation score ht type compute definition public split implement tool test base type build specifically four set vote balance word use suppose vocabulary frequency reach feed similar frequency approximately total word coefficient similarity rare discussion balance knowledge character wikipedia totally million text process digit replace word replace occur discard result ignore compare baseline baseline use feature skip input vector length projection baseline skip gram update word design baseline coherent human cognitive skip gram employ type update note fix share fix context window
less outperform several example create see review description brief sentence since express opinion movie review ignore sentence consistently convnet documents architecture structure technique network language domain language model translation answer sentiment language task entity language us encode semantic language geometry idea nearly network
search already complete consist example handwritten demonstrate search speed search dataset consist example train show transfer see optimization task benefit search logistic get minimum able consistently evaluation averaging configuration toward high conversely low agree less roughly agree offset epoch minibatch increase model mnist solid line indicate mnist line develop process especially well suited approach
decision depend mid example reject mid retain decision retain verify mid retain conservative turn mid example valuable information formalize motivate propose hybrid abstract could whenever could lead decision occur choice opt mid affect testing simultaneously likely microarray reject utilize usual mid value natural mid section reject study section describe operate characteristic proceed abstract method computationally adjust abstract randomized motivate new adjust mid mid mathematically randomized remark method define relevant relationship value test countable variable specifie goal reject retain
web supplementary material set feature selection york supplementary include optimal section scale factor web section web table matlab file optimal classifier classification model introduce propose section hierarchical write classification k simplicity operator summation relate correspond work determine majority vote individual vote optimal lda poor performance simulate majority voting strategy suggest enable rule predict look angle case class use concentrated point theoretical observation classification thought building space remain question good high assess support motivation thousand observation investigation thus observation emphasize let ip rip pc pc empirical space sequence
history view word latent item get name fm model obtained skip skip maximize pi item feature fm item train order user item relevance handle fm model evaluate movie rating collaborative filter movie
may unconstraine minimizer replace introduce n tn nf inspire convergence sdca duality gap proceed sequence use e represent n e g use ng add twice eq come immediately e obtain rate use inequality prove become inequality replace sdca low play role sdca rate sag apply conservative heuristic section use size present update relate sdca strongly result composite one well surrogate even though suggest rely surrogate equal inspire part l randomize available warm surrogate composite minimizer solve address small value empirically warm could would proximal surrogate convexity even lipschitz
si bag weak proportion si mi bag contain test object independently underlie class learning general introduction refer train instance train classify individually classical dependency field mrfs popular originally describe labeling assign part speech tag account sentence label sentence achieve si task mi mi task si si observe label exist advantageous bag provide bag derive train correlate although convert si si mi instance correlation instance call trained feature label instance relational repeat encode learning mrfs simultaneously label bag object bag name scenario bag directly distance kernel bag space bag
compare necessary achievable relaxed imply bind similarly measurement achievable therefore contrast bit cs might room measurement sublinear performance improvement individually power information sequence maximize observation restriction measurement formula group testing bit testing bit cs
result low observation low resource correlation affect case distribution coefficient fisher contain observation go approach limit correlation sensor bit sensor sensor observation also make decentralized htb certain present optimal estimation independent noise noise local sensor locally process fc fc design previous section one sensor fc observation stochastic employ possible rao fisher conditionally probability minimize maximize optimization
assessment patient one assessment successive within week patient least event assessment patient collect gender country birth status status among consider age assessment broadly classify health refer moderate class risk assign look diagnosis convention event period risk code event open would typically complete rare month period attempt moderate r horizon day filter bank sec event primitive code map high level correspond severe episode hierarchy list reduce rare code diagnosis disease health diagnosis table alternative view health likewise frequent economic robustness rare event risk give good specificity intervention detailed convolution bank compound filter min medical risk suggest max similarly item suggest serious surveillance assessment moderate list max ii max item time item mean rating item rating l order network ease interpretation divide non distant encode belief old
edge recover place detailed figure realistic material area significantly bic relation bad trend relatively opposed finally randomly run average incorrect find non low mean almost twice result supplementary material demonstrate efficacy filter filter approximately possible hypothesis considerably less mistake drop score bic across level realistic training variable great surface leave bottom box demonstrate performance knowledge learn ten bad random measure positive true negative prominent fusion event persistent theory
variable output heuristic al know study alternative likely heuristic thesis upon problem relationship heuristic artificial intelligence infer make machine variable elimination quantify average heuristic heuristic first machine formulation algebra follow recent background learn section describe conclusion work free formula problem produce free formulae reduce prove algebraic formulae polynomial method elementary doubly complexity survey remain work produce polynomial eliminate polynomial decomposition space form real root
st observation suggest boundary mean test social circle statistic circle overlap ratio less compare exploit get inferior mt operate learn metric task final preserve structure network mt article citation circle well reasonable analysis importance lie task perform exploit mt convergence source author support award fa multi task learn number jointly network priori structure provide network come attribute associate variation relational entity often source performance work
parameter maximal mse choose bias em indicate mse well mse result correlation structure suggest large select right induce mse deal high structure report ccc r sf mse sf ss deviation sf mse square compare margin correlation among feature identify true feature mixed mse addition indicate tend independent regression htb panel increase go zero extremely parameter force estimate irrelevant go feature specific simulation three selection regularize without consume especially pick aic sf
distinguish specify advance typically factorization moreover column choice contaminate additive distribute achieve reduction reduce discard arise case identify one instead solve factorization row candidate subsequently pick alternatively pool backward successively apart k k alternate coordinate descent propose g along feasible perform exhaustive search possibility impractical stand alone initialize latter block step help dataset entry probability simplex size setup report average follow hamming denote box five
hundred alternate dataset fmri treat fmri bold connectivity bold spatial fmri insufficient thus connection simply treat direct hypothesis conjecture nearby voxel explain range group close brain nearby grouping provide group interesting biological develop precision assignment difficult network dimensionality fmri hundred thousand challenge applicability hierarchical fmri brain interact introduction interpretable enable infer hundred thousand develop update compute simulate go model tool relationship
estimate characteristic rank denote nuclear th estimation exploit method consistency thresholding np notational drop bound question ask whether could assumption result case keep sufficiently large contain large check extension technical continuously multiplicative specify measurement value change alternative focus iid one relax assumption unbounded direction appeal interesting sub gamma consideration insight simplicity boundedness unbounde numerical rescale condition iid measure alternatively tensor notation multidimensional array call clear low rank specifically upper substantially encode structure reason well low tensor ill pose might bound tensor converge fortunately
dependent observation person motivate computational need thompson become large require feasible get extremely good real problem online hard complexity scale straightforward computation core different bootstrap replicate later replicate probability empirical simulation bootstrap heuristic solve bandit thompson replace thompson
define expand substitute gpu equation convenience mathematically diagonal later due small assume expand never diagonal eq never instead directly note seem like factor term gpu gpu gpu scaling multiply gpu notice invariance lose proper fix great symmetric computed gpu cpu element nothing cholesky compute extremely rarely bad already disk minibatch initialize top top eigenvector let ii quantity compute fact square enforce per minibatch use inner product gpu factorization trace follow expand update need multiply quantity enforce towards gpu nature computation bottleneck try fail apply inverse fisher fisher expect unit normally experiment nonlinearity maxout like typical matrix something rank input matrix matrix sort greatest scale input value leave unchanged last like take scaling set unlikely sgd learning rate direction track change tune minibatch specify interpret minibatch reason believe speed update every update sgd summarize matrix fit picture ignore explanation instance correspond separate copy describe typical configuration
solver c l singular singular proximal separately apply proximal use subroutine nonconvex experimental show outperform previous small nonconvex minimization extend nonconvex affine alm acknowledgement national international centre office lin china china grant cb collaborative fellowship department electrical engineering school technology science technology laboratory university edu sg com com edu development differentiable bx give denote
svd bilinear factorization rbf product desire uv observe missing solution complement incomplete corrupt product lemma impose much exist solution exist note case entry observe alternate multiplier admm solver non algorithm produce different estimation guarantee theorem compare convex common impractical problem rbf complexity scale understand rbf relate method special desire qr update modern parallel architecture regard section algorithm run fast accurate bilinear hadamard product miss propose alternate multiplier admm extend analysis exploit admm assume simply zeros uv uv sd uv
isotropic assume distribution step inequality element wise conclude direction p ip j suffice union bound ij similarly result require proposition improvement eq could since isotropic lead overall projection let consistently recover permutation direction spherical normalize hull state discuss post computation propose preference develop generative account population user inconsistent natural ranking statistically model leverage advance latent ranking provable consistency computational complexity empirically art approach preference provably go comparison demonstrate competitive performance collaborative metric demonstrate effectively variability real world estimation partial extensively last decade various prominent user ranking homogeneous center truth
observing exploit unit bit bit give rate consist dimensional input parametrize bias single propose layer parameter computation consist parametrize basic bit weight indicator input may choose possibility predefine make sample
repeatedly compute able call apply value white amplitude project onto span quadratic state action mapping discount receive well know fix bellman similarly optimality point policy iff equivalently implement greedy input practice achieve call direct suggest approach hand shall policy scheme section comparative performance go describe error
without subtract supplementary large sequel use descent gradient theorem guarantee furthermore convergence sample var estimation reduction sampling mc sampling suitably modify unbiased incorporate fairly material provide detail significantly well problem detail proceed realization variable distribution apply obtain I smooth facilitate iterate common enough optima projection rarely occur return keep consider denote equilibrium suitable technical surely application hold differentiable surely discussion slow requirement algorithm rl involve decision
artificial wave ml example try pose light high input pose importance connect method successfully result simplicity nonzero furthermore percentage interesting observe treat three equally eigenvalue model well efficiency digits mnist set digit pixel two major two represent capture capture percentage variance consider image mnist mixed top obtain indicate projection two capture major image successfully represent digits face image gray pixel represent dimension representative face randomly eigenvalue return eigenvector matlab experiment mac os bit intel core ghz cpu gb memory numerical experiments matlab software package semidefinite sdp efficient sdp solve sdp network termination tolerance train inexact proximal employ terminate primal meet respectively tolerance choose train report eigenvalue h cccc ccc cccc problem e e e digit term
exclude mistake error correct additional rr frequency lead work spatial task correct stimulus arrange exclude trial position discrimination name distinct six one mixture two proportion uniquely water group right figure exclude trial categorization previous exclude either qualitatively exclude reward trial decision protocol combination trial cutoff neuron exclude neuron fire neuron fire fire rate hz exclude neuron minor transformation square transformation qualitatively neuron work neuron work discrimination neuron categorization preprocesse spike train filter kernel ms smoothed stimulus histogram neuron work direction fire directional show neuron prefer sort separately neuron prefer preferred dataset trial trial fire separately trial interest figure illustration water align four reward take trial water across discrimination categorization trial describe alignment error along time piecewise linear align introduce cut trial similar pooling datum reveal qualitatively marginalization neuron stimulus trial trial filter train vary stimulus trial fire rate thought condition dimensional collect column mean single neuron decompose part vary stimulus decision averaging
population estimation sequel constant play role result decomposition eigenvector eigenvalue f obtain j eq desire commonly give regard rate assumption small apply choose sufficiently suppose n assumption reduce minimax rate attain determinant barrier zero barrier
operate resource necessity tolerance continue sensor wireless obtain information environment growing acquire sequentially efficiently avoid communication fusion sequential decentralized communication central unnecessary designing analyze amount mathematical challenge question ask elaborate asynchronous computation develop deterministic stochastic asynchronous online property entity processor hereafter perform estimate meanwhile processor receive update form computation regression goal integrable assess classical offline setup batch prototype
nan distance fm across distance proportion two nm proportion ap population variance diversity allele var across mean diversity two allele two index dm coefficient population universit de france france france universit paris uk range purely motivation likelihood propose selection forest complex cover algorithmic output indicate probability correspond forest recommendation sparse forest severe table performance computation methodology illustrate approximate near bagging subsampling introduction approximate abc ever increase cover calibration always still critical implementation partly explain widely accept specifically major quantify vector practice finite lead summary produce summary crucial role provide inconsistent answer exact bayesian factor probability summary pool summary statistic simply avoid select random forest therein probabilitie approximation well posteriori try build lead compare dimension predict solely probability toward loss estimator assess reliability select posterior select wrong simulating tool random perform suggest tailor implementation support argument favor rely forest forest expense production forest value direct possibly large collection summary statistic
run divergence htb cc belief networks boltzmann share use stack simple block capture dependency correlation building field restrict machine dna building inherently dna distribute rich carry drawback inference rbm stacking field representation rao difference dna modelling purpose specifically design
say sparse sparse norm whenever disjoint subset g adapt argument norm sl decomposable notion say constant constant definition prove respectively ny ax decomposable q replace interpret state minimize decomposable near ideal problem group generality constitute future objective discriminant present context merely find discriminant relatively discriminant whenever third great choose discriminant devote amongst give label linearly weight discriminant separable weight r linearly situation dot dark dot clear circle circle circle circle circle pt circle pt determine feasibility lie hyperplane linearly way instance power behind high hyperplane separate many hyperplane support separate hyperplane point hyperplane formulation hyperplane illustrate concept separate hyperplane circle dash circle pt pt pt circle circle pt circle
become replace argument denote lead precision contain compatibility factor c well permit comparison decay invoke usual isometry restrict slow design consequence substantially compatibility concept evaluate application new weighted compatibility may apply one hand example poorly prediction moderately covariate develop optimally gram matrix define penalization lasso selector recommendation explore consequence result nonparametric least proportional th derivative study lead notation denote proof tucker infer subtract relation may appear somewhat c equivalently write display divide side accord eq replace introduce vector display combine jj hand negative n classical subset identical left reader I
hence may continuity approximation difference approximation former use iteration continuity function hx u directional equal select belong boundary follow continuity forming follow continuity contradiction side contain away margin contradict close reason versus lead function arbitrarily disjoint hence finally continuity sequence vi value function upper recursive relation initialize ix ix bound bind lemma mathematical induction x x ix comparing ix ix ix mathematical induction part lead tight see dynamic consider condition converge
carlo primary genomic treatment review set typically experiment may consist correlation noise repository gene cancer etc list molecular signature significantly weight size genome identify choose appear smoothly contiguous assume include gene practice subset interval say nan replacement gene fact set identify uniformly distribute tuple identically basic object inside
assignment exist author paper write regression given expect assignment intuition give solution derive conference paper category soft assignment category suitably define typically pair category category negative dissimilarity allow want optimally one nm qp row column column likewise index determine case assignment item category type encourage term encourage item although could fix mean dissimilarity laplacian separable problem couple certain solution extreme program lp separate k category tell category correspond assignment category quadratic mn e tell differ generic unique close n k nk similarity assignment large laplacian dominate follow assignment category similarity sparse category similarity point similarity qp minima since semidefinite multiple characterize give sufficient minimum assume
make prediction spatio temporal set type compute transformation input layer formally type memory output nonlinearity wise sigmoid nonlinearity softmax layer among unit connect intermediate score sentence discard output length sequence incur supervision whenever step since dense weighted transform mean interact learn interaction layer task sentiment detection consider come might model multiplicative multiplicative recurrent sentiment retain rnn
nn training discard find drastically storage requirement facilitate stochastic neighborhood visualization compression compression devise efficient cholesky change variable carefully world baseline error set lead order nn case dimensional vector input compute descriptor matrix via metric euclidean mahalanobis approximation along matrix affine geodesic definite rank matrix f accurately describe intractable moderately sized roughly burden distance similar call jensen q near neighbor classification nearly asymptotically
improve network see capacity early stop decay training involve proportional objective weight incoming addition effective equivalent regularization denoise autoencoder additive signal reconstruct denoise criterion permit overcomplete reconstruction regularizer idea fitness species co likewise dropout performance complex co feed procedure function define l
focus inverse update rule ph point direct rule exchange ph section inverse update bfgs cg arise ph equation note estimate computation projection however goal sub explicitly singular n nu matrix mt designing solver step identify span ideally regularity equivalence cg offer elegant way additional cost recall covariance span semidefinite property eq normalise positive matrix span covariance bfgs scaling prefer use standardized prior prior cg hybrid construct problem conjugate run store conjugate gradient multiplication cg estimate cost minor construct gradient I problem yy e estimate rule propose cost standardize sr corollary quite sr use cg use give overhead posterior algorithm alone multiplication cg store mean require relatively problem crucially external distribute eigenvalue external
compute sequence choice guess sequentially receive online rule subscript law compute sequentially use detail step execute add estimator discuss implement exactly implementation general implementation recall initial density since close suppose approximation dirac
convexity set dr hence excess condition brevity sign last inequality follow fact p tt exp give construction output arbitrarily close multiplicative interested pure differentially efficient multiplicative fact interested total could however achieve sample distribution well explicitly work mainly highlight construction construction private isotropic position eq choose write since bound term measure distance denote derivative point membership efficiently suffice oracle polynomial run highly isotropic isotropic particular however convex efficient place isotropic position apply transformation take fit inside finally transformation isotropic diameter put op norm lipschitz walk building done define walk input cube output close respect argue
nlp group helpful early lastly valuable feedback stanford google v le google google google york university comparable approach significant conventional correctly translate rare tend symbol vocabulary implement system later utilize post processing step translate every use hand suffer extent phrase allow extremely word strength phrase address rare problem corpus explicit enable corresponding sentence utilize translate use experiment english translation winner task translation map sentence target
include influence shape order reduce diagram flow boundary positive arise signal gain turn effect calibration sec numerical example reconstruction color structure calibration calibration respectively panel low exhibit mcmc panel mcmc explicit absolute measurement local tensor pseudo instance principle field optimally calibration start calibration basic idea infer vice versa fix reach estimator hamiltonian equation must iterate base mark new infer determine usage hessian
draw thick fill color bic circle minimum thick bic circle mm size color text bic style circle sep mm size cm white circle inner sep text rgb rgb rgb rgb rgb rgb style circle thick black style cm thick color circle sep minimum cm draw fill style circle sep mm thick black fill black style circle inner mm thick fill style circle inner white style inner sep draw thick black circle sep black color circle inner sep draw fill black minimum draw black fill style sep mm minimum black inner thick black fill style sep size thick style sep mm black minimum thick style mm color style sep mm minimum black style sep mm thick fill color black style cm color n style sep thick style mm cm thick fill text circle sep minimum draw white circle mm size draw thick white fill text draw white sep mm cm fill text black style sep thick fill style circle sep mm draw white color black size draw white white text inner thick fill color circle inner size cm color circle mm size draw white color black rgb rgb rgb rgb rgb style cm black fill black circle sep mm minimum draw thick
bilinear slice layer bilinear bilinear diag special bilinear matrix c linear bilinear bilinear diag result observe worst overfitte publish achieve discrepancy sgd vs bilinear consistently comparable much require parametrization relation expressive simple bilinear bilinear diag baseline comparable bilinear note bilinear
since odd every origin contain accomplished interval semi origin replace replace left interval infinite lie ball vc consideration family obtain let scale arise primarily represent accord collection vc recall final subset hull rest connect hull vc hull rest pair convex denote lie convex hull lie
connect filtering spectral insight particular stop filter introduce whitening demonstrate fast standard collect sample processing aim classification transform transformation unsupervise simplicity scalability three processing unsupervise encoding abundance image interact crucial ensure efficient paper study contribution filtering benefit stop performance
location model mle stein show towards uniformly low underlying prove statistical development shrinkage compressed shrinkage estimate dimensional small variance mle bias risk wherein dominate exist analogous x mle mle stein example good prefer uniform estimate dominating recall family bind mle perform parameter mle design handle save generally address solution idea shrinkage reduce methodology
strength excellent rapidly arm valuable choose specific problem empirical guide heuristic bandit consider clinical trial vary effectiveness unknown identify successfully heuristic assign patient extreme variability done subset bandit characteristic affect arm reward implication work regret reward et consider moment impact performance precisely identify accurately evaluate apparent tune bandit example evaluate easy bandit strategy could effort towards turn algorithm clinical trial whether bandit suit clinical answering question implement dropout identify good treatment confidence bandit trial offer trial term patient simulate clinical strategy randomization criterion successfully treat patient clinical trial conduct treat patient early treatment particularly suited context initially patient study patient receive day test patient treat provide patient individual assign treatment patient achieve success patient condition course test patient result mark strength indicate indicate publicly unclear
inexact simple inspire simple subproblem proximal proper define correspond c admm optimal initialize k work need efficiently proximal stem subroutine one function convex pt mm enable negligible flexibility algorithm proximal efficiently simply admm term spectral radius accord theorem error measure gaussian error span far result entry probability choice notion gaussian gaussian norm standard ds eq
conditioning thus truncate distribution involve kind fortunately likelihood augmentation negative generalize replace binomial share binomial dispersion row exploit binomial beta binomial jk express absolutely evy laplace express eq laplace transform express sum independent compound pmf pmf binomial pmf may truncate q laplace augmentation concentration step gibbs unstable calculate number rapidly allow precision machine numerically thm proposition probability matrix potentially unbounde three derive negative binomial binomial binomial lead natural count although wise distribute fact use derive explicit drawing column map random order certain derive random random count framework construct naive text require predefined account unseen analyze completely suggest propose poisson multinomial laplace
go compare low result spherical gene usually cluster splitting big cluster price discover tight cluster fortunately split detect merge reasonably number decrease apply cluster four preserve pattern four amount create splitting extract cluster biological small small dataset demonstrate spc iterative subsampling cutoff discover interesting pattern approach relatively trade large reasonable either preprocessing could remove valuable dataset filter coefficient cutoff remove carry profile function lose without knowledge number able decade rapid rich ever problem size computational intensity effort far create consist essential exploratory simple model cluster dataset numerous modify mean computational modification include random subsampling sophisticated review summarize abundance outlier
sparsity aware rest assumption give box aware doubly coefficient sparsity aware diag I fact simply express towards ii I I possible prove
lastly environmental gibbs control additional explore snps rest snps effect snps study
find author department engineering create account account receive research process lda appropriate format thus list similarity list likely package word word filter count tf word list project properly part five section generative detail describe estimation
present special interesting construction state spectral adjoint eigenvalue eigenvalue lipschitz equal numerical hand unlikely give satisfy metric space say property ball r lipschitz maintain
bad rf implementation optimize implementation test five exclude simple scale take range subtract common forest pass rf hyper default initialization rf vs training rf batch version evaluate split version candidate split split every recommend training increase realistic streaming setup store multiple pass vs time marker pass training batch rf mini mf streaming setup new significantly fast batch version tree balance mf rf remarkable label competitive similar test comparable rf world irrelevant method independent rf well mf attribute amongst attribute
identify total operational strength conclude four simplify join miss create cm operational operational state operational double strength operation elaborate join input join soon one firing firing come parent henceforth item fire fire accomplish parent state henceforth predict go back operational operational item neuron fire show fig diagram create operate algorithm transformation assume join implement fold fourth former subset consist strength argue operate diagram easy transition operational state loop operate join self loop parent come item transition event fire consecutive shall briefly actually implement operate algorithm require need besides fire neuron fire necessary plausibility fire soon primitive plausible pattern mean parameter visual one possibility special pattern item sensor remain unchanged period presentation presentation
maximal carry series round find round track unary include effect interaction activate track overlap commonly occur element direct acyclic short start path absence term correspond dp approximation solution quadratic track go could otherwise accelerate interestingly greedy update pass dp quadratic penalty pass find pass dp learn tracking potential feature depend extract video spatio relation candidate track parameterization sign convention maximization represent appearance template represent pairwise track temporal transition represent birth feature location make detector appearance consist detector allow motion connect later time window window overlap window lower flow transition vary track appear move thus single birth death geometric object spatial context bins window window location additional intersection box area set corresponding feature ratio video tw vector encodes geometric object w object way intra
turn internet movie database rt rt website review media user rt show meta datum rating vote actor review sale b entity extract content tv original release rating user tv actor rate pg rt score rt aggregate rating rt voting item american office provide scatter plot attribute week user rate title sale title report vote public engine office reasonably informative box office access american european quantify understand turn google trend volume google search give search shoot relative establish query volume region search volume neither else volume report would relative search world european country country community normalize query interpret fraction google search week search approximately common scale engine panel scale sale release search engine attention look close region search north correspondingly sale release panel engine attention sale engine advance particularly successful search engine sale sale suggest search engine local sale volume instance predictive consist cumulative new backward release release indicator vector country vector identity location search engine title release
fold cross discriminant tree tune rate pair sum report node tree distribute note almost leaf node percentage term four remain
water system ground gps operational environmental integrated system feature error vary server operational water national environmental service send national service movement water condition operational create lead boundary goal spatio low coherent uncertainty without transfer processor h c period hour sensor unite top sensor product filtering inference spatio parameterize isotropic mat ern intercept walk neither cost elaborate parameterization parameter also walk initial
confirm design evaluating task rate divergence divergence estimator divergence extensively process involve segmentation separation function divergence perhaps widely signal process family distance kullback leibler generally indirect class divergence measure useful dimensional two class assume restriction prove useful application knowledge underlie divergence measure parametric parametric introduce also estimate measure investigate unlike divergence fisher utility focus classification section iv come provide vi contain discussion future divergence without fr
match partition block come algorithm update step convergence rate strongly ht partition use block p remark method store row update translate run update method run choose strategy pick ensure row block store separate secondary storage convenience mention crucially difficult rate hence exist arbitrary effort line viewpoint fix strategy round easy evaluate alone find pick block iteration randomize pick row per pick randomized pick proportional use block approach effort correspond worth one mention selection algorithm focus greedy adaptive deterministic new estimate block hence greedy pick amongst candidate choice appropriate strategy end time briefly greedy projection block possible pick fact refer bs pick iteration emphasize b
correction carlo efficacy amp framework evaluate imaging amp recovery briefly review amp later demonstrate within reconstruct denoise denoise amp behavior simulation kernel implement name suggest apply filter whose gaussian noisy image gaussian pixel violate image convolution implement efficiently remove noise filter regression pixel neighbor close compute addition spatial proximity window noise dark prove amount extend average pixel neighborhood take neighboring neighborhood pixel however edge opposite side edge usually neighborhood wavelet transform basis coefficient transform hence inverse sort thresholde soft thresholding performance unfortunately image wavelet thresholding least overcomplete wavelet coefficient square neighborhood coefficient prior expect noiseless bayesian square remove noise coefficient bm filtering begin grouping perform transform dct haar transform bi haar amount group perform transform pixel thresholding second wiener filtering estimate outperform compete additionally author bm optimize complicate quite combine bm patch filtering help group patch mean dct haar bm retain performance bm unfortunately provide among filter non maximize closely signal use denoise rescale automatically
correct equivalently bound square aa respectively assume circular therefore hence small independent lipschitz base next rank orthogonal show denote therefore incoherence incoherence property enough prove j define correct bound incoherent right n array thorough evaluation conduct theoretical insight hoc classic mathematical matrix projection completion achieve array calibration position semi program stress discuss reader detail low space preserve matrix relative position first center distance position real scenario map compute shortest consideration sum minimize approximate miss distance short classical programming show reliability space semidefinite increase gradient method calibration topology hoc optimize reliability control state importance distance incorporation simple classical assume distance
triangular orthonormal identity c al theorem al nan eqs notice give characterization suppose contradiction nonzero q orthogonal x full
latent jj ix I I position respectively labeling definition follow inference analysis base extend construct valid clique length e setting score path contain horizontal lattice layer valid non node valid least pass correspond path node valid layer layer pass position connect position valid become remain
previous independent set section contaminate density hypothesis give distribution define g gd call information section type show belong g hypothesis statistic g follow nan type linear combination nan context chi unity unity useful robustness asymptotic test exactly illustrate consider fix contaminate proportion origin contamination influence function boundedness quantity towards explicit index statistic theorem
anti correlate ai electrical engineering university california berkeley department institute department department university produce massive new discover biological discover neuron crucial neuron type traditionally connectivity manner show neuron enable reveal structural enable automatically derive connectivity massive far circuit computational method impact high throughput sequence connectivity fig cell profile cell probabilistic cell belong type connectivity vary typical profile historical classify base give start stochastic block cluster connection salient logistic link additionally body validate perform simulation accurately cell simulate compare estimated job recover correct fig extent exist infinite assume
introduce potential potential value potential intervention sequel particular response pair ab rs rs none causal mechanism mathematically mutual independence ann experimental randomize property serve validity observe
n rgb definition main statistical sketch previous algorithmic result time number depend kernel adapt kernel lastly capture fine difficulty semidefinite emphasis obtain guarantee primarily work area focus issue guarantee inferential low result recent excellent combine obtain fast kernel improve relative state recall importance perspective input bad decomposition work leverage randomize positive rank parameter projection approximation recent qualitatively bad
solve multi sensor simplify method optimization constraint common tackle splitting multiplier splitting break close minimization together introduce efficiently optimize splitting rely burden dictionary yet augment smoothness fidelity variable keep updating present weight converge update involve subproblem intermediate subproblem update element separable mean operation equal summation constitute simplify solve svd determine soft thresholding q subproblem unfortunately close difficulty come regularization row group operation dictionary multiple modality alone restrict sparse regularization arrive sparse modeling normally require iteration achieve converge tackle exact third function taylor expansion achieve expansion last line I separable property simplify separately solve utilize approximation utility yet use whose sketch j penalty combination domain nonlinear empirically multi fact extensively validate become linearly onto
specify assess influence unit company expect vary political regard eight include potentially logarithmic width size medium random effect slope covariate beta logit beta complete specify remark flat improper component precision parameter indexing random wishart assume reduce define intercept covariate add related intercept large effect categorical
tail order account normalization mean decay rate mass invariant datum discretize bin width narrow central correspond pair event reference order dataset draw bin histogram marginally mutually independent belong unfold belong cb estimation cb invariant mass maximum indicate high cross check find agreement carry bin event outside extended side place knot result unknown sampler condition number hyperparameter initialize iteration hasting burn observation size extend square confidence interval replication core confirm converge iteration little variation roughly autocorrelation whole plot mcmc verify bias pointwise bootstrap compare unfold reasonably overall shape figure histogram count divide unfold reconstruct peak smoothness boundary intensity reasonably tail intensity invariant mass sample figure na I confidence figure
sufficiently concentrate around r k k ahead examine analysis eigen decomposition orthonormal far follow substitute element obvious independent hypothesis weight random square zero gamma distribution complicate pdf instead rely construction sequel error test type alarm rejection right tail ii correspond tail ii bound chernoff pdf origin skew long tail need subset chi degree chernoff central euler number I exponentially mutually function k degree centrality substitute c k h rhs gaussian last step chernoff enough exponentially pdf form pdf chi freedom central degrees centrality pdf pdf scale chi f plot pdf red curve mass around close curve red shape agree make evaluate mis rhs demonstrate assess error alarm type incorrect decision become proceed operate steady recursion cluster decision
discrete able modify policy set description policy make ability ask simultaneously good figure substantial benchmark dyadic benchmark contrast dyadic question nearly benchmark well dyadic object location dyadic policy dyadic question remarkable compare little lose hard compute policy dyadic much go dyadic computed quickly pre compute ask question setting compute dyadic compute provide least dyadic sometimes dyadic right dyadic low turn dyadic policy dyadic explicit analysis dyadic conclude lower expect first notation
consider assignment contain total estimate proposition straightforward similarly two sorting rank sort merge complexity apply conduct pair already process sort cs grow subset computational cs straightforward thus assignment order therefore case estimation cs refer programming interior polynomial bi thus f interior necessarily ten choose benchmark result discuss summarize uci dataset domain range within characteristic discretize prior selection lr kp discrete dna discrete discrete continuous evaluation uci namely support c near neighbor na I influential mining algorithm classification empirically three rank representative
model dispersion regressor specify selection criterion regressor dispersion present highlight propose criterion yield easily identifiable stand version parametric scenario compare performance criterion size performance become parametric weak criterion evident competition outperform criterion least relative regressor jointly weakly identifiable also weak clearly instance aic replication whereas regressor mean dispersion interest must mean identifiable display criterion outperform
select kind predict measure risk relevant covariate part discuss select coefficient measure technique criterion prominent one high denominator fdr lead achieve low fdr compressed noiseless thresholding identify thank situation estimate regularization good estimation compatibility order relevance strong bounding inequality key role screening find asymptotic true rate stein unbiased concerned discuss prediction selection
nonsmooth base reweighte nuclear solve compute proximal operator nonnegative decrease monotonically point guarantee note nonconvex stationary last nonconvex synthetic algorithm nonsmooth need extension nonsmooth nonsmooth concave differentiable nonsmooth inequality call denote versa concave versa subgradient useful explore monotone
infinite aic optimally gives forecast particularly attractive ease explore typical credible account variability plug bandwidth cauchy inf rgb rgb bandwidth nonparametric error density lin nonparametric value recent error density admit bayesian estimate simulation apply nonparametric type regressor chain recent advance bandwidth residual establish rate estimator framework recently propose bayesian simultaneously kernel error bayesian mixed regressor nonparametric regressor essential idea functional scalar
medium discuss implication google differently social key evident finding google controlling message length day exhibit incidence large indicate increase lead message increase decrease due sample result allow turn across picture discuss effect variability effect seven variability follow differ measure also clearly find indicate message successfully effect finding would contribute confirm
measurement observer observation assume mean covariance absence example set make uninformative posterior direction get k k posterior calculate method matlab processor linear evolves need mcmc marginal calculate continuous length performance da observation clutter loop move update jointly window loop move contain move alone da da particle line slow especially report time mcmc da cost resp particle include cost particle mcmc association respectively efficiency propose move particle da da region include inside target plot connected star connect measurement target clutter good non replace kalman kalman target particle per window parameter density z compare joint nz nz converge log evaluated truth apparent gap ground truth alg track iteration
single particle number calculate weight nj ip py ij extend stop particle particle iteration propagate particle particle first ii simulate particle proportional likelihood output particle choose particle proportional iteration monte estimate product unbiased implementation output arguably simple particle filter liu implement calculate depend latent
confirm would three autoencoder joint epoch autoencoder representation test model performance table report test classification computation layer table joint achieve error extraction unsupervised contradict performance regard good model regard helpful good generative necessarily translate good discriminative p l cccc j mnist rand h detail cccc cccc rand apart perform initialization train train joint beneficial perform another weight autoencoder train autoencoder table use performance improvement joint correspond irrespective initialization joint addition clear scheme scheme suggest use role autoencoder
eigenvalue recover linear autoencoder generic model computationally demand shrinkage note equal isotropic noise autoencoder discuss previous difficult heuristic selection problem iterate scheme low high unconstrained solution procedure describe update encode update show small cone ordering reason scheme subspace show eigenvector cutoff find isotropic case algorithm admit close give particular never isotropic iterative
centralized find also non plan provide distribute accounting phase trade coverage final certain proposition ensure probability relax belong kernel hilbert induce covariance see key question phase location markovian process adaptation happen infinitely converge se answer rkhs sup location location instant relationship estimation field rkhs q contain location think borel probability stochastic mechanism
distribution yet structure formalism effect hdp strength across group share make precisely cluster precise cluster share introduce dp modification definition group number exchangeable collection observation group level dp cluster pair product base dp draw base measure realization repeatedly within specifically base observation conjugate respectively stick break stick countable atom eq stick stick break form ij stick collapsed step refer chinese crp second conjugacy l ji z v integrate dirichlet conjugacy exclude ji ji z pl ji
plot ambient variance ambient varied runtime randomly generate mirror descent large require ambient ambient dimension gb ram intel cpu demonstrate also require synthetic accuracy draw outlier draw change variance draw plot scaling change dataset perfect scale offer complexity lead well demonstrating release digital survey database spectral resample gap correction use find center subtract spectra value pca reduce
generalize option tw tw tw iw iterate small therefore use iterative algorithm option recover singular via recover recover j l extend need form disadvantage straightforward runtime epoch implementation maintain scalar scalar appear sec eigenvalue sec sufficiently epoch practically relevant regime success algorithm succeed high
neighborhood characteristic informative researcher enforce graph sr derive representation select graph explore powerful discriminative sr noise base sr recent representation sample capture global drawback corrupt clean sr base capture structure liu representation graph graph jointly constraint capture global subspace mild lrr correctly preserve membership sample belong dense undesirable lrr interpretation negativity visual performance nn draw ks direct basis coefficient many
correspond bottom corner illustration perform matlab implementation intel cpu gb ram work pixel pixel pixel sort ground one similar due color band nine fig dataset term fusion obtain resolution problem closely relate challenge large normally image normally spectral fusion program solve split augment multiplier estimation sensor formulate intrinsic dimensionality hyperspectral space image define adequate splitting effective publish simulated life detail optimization form k k k q computation fouri include inverse advance involve q separate minimization solve th convention dominate perform multiplier acknowledge dr providing acknowledge zhang providing provide gs pca bt process corner fill blue style thick width minimum height text style gray es universit
capability early classification economic communication time author accuracy euclidean distance wang claim near experiment believe carefully assess extensive broader consider exploit world distance raw fourier transform dft notice parameter dft coefficient notice equivalent raw time usually take close sometimes effect rapidly relevant accuracy accelerated reduce extremely literature aforementioned acceleration capability et al financial use reasoning al use medical wavelet coefficient instance system diagnosis provide advantage generally series base piece wise aggregate coefficient polynomial computing similarity extract employ model computing use chi cluster coefficient stationary series chain discover
fold strict mm mm heuristic measure disk write store take setting reduction determine fig code save load state disk process classify disk time extraction take two pooling learn high dimensionality automatic recognition volume across potential specie realistic variation demonstrate case unsupervise learn operate knowledge training find large volume benefit apparent indicate lack feature create feature lead order magnitude order specie audio datum substantial attain raw transformation preserve implicit perform input often hold back spectra common low result use audio volume availability annotation crucial small label item increase annotation boost due single individual audio cause problem dataset come directly annotation format portion auc classifier suggest label yet automatic outcome importance annotate intend public collection highly specie automate audio width mm plot nr indicate scale width
label differ construction adopt threshold learner suppose every bound bounding standard robust introduce underlie coefficient unit pass origin disagreement coefficient result improvement term rate also considerably simple low bound close q say present unknown connection multi convex strongly tie together propose adaptive base datum furthermore parameter appropriate conjecture existence algorithms adapt notion convexity introduce mention direction unknown query target instead handle possible algorithm improvement convergence learn prove corollary convergence rate surrogate view
use basis move would alone hasting randomly preserve chain lead inaccurate value markov guarantee intractable following modify irreducible computing consist move expand sufficient prove chain form canonical lattice pair paper configuration simple swap construction configuration simple propose uniformly remain unfortunately irreducible unable leave swap connect sufficient htb ise configuration connect project result result reversible holding present maximize adjacent exposition subgraph maximize connect prove configuration ise unique configuration connect unique singleton configuration singleton configuration component unique max singleton exist q reversible
show number either rip claim detail partitioning concentration fine partition several similar concentration hold exploit trick spherical modify tensor exploit draw show median ica tensor modify observe dense case subgaussian nonzero empirical bound ica set ica ica bernoulli random variable subgaussian sample remark detail counter intuitive dependency actually expect require idea net argument addition ica subgaussian subgaussian see argue claim ica exploit assumption incorporate conclude section state organization propose section exploit learn model alternate asymmetric update update alternate mode view alternate least multilinear mode intuition tensor orthogonal tensor suppose correct tensor expand point power update incoherent power initialization successful recover algorithm moment learn provide initialization supervise exploit technique power multilinear l cluster member cluster tuple alternate output center tuple tensor draw compute vector initialize tensor perform many need expensive carry procedure lead model decompose moment tensor view hadamard thus matrix operation initialization bound distance distance ambiguity issue recover tensor unchanged one sign two section tensor k factor asymmetric always adjust appropriately simplicity dimensional overcomplete precisely assume highly organization section appropriate tensor concentration employ latent model mixture sparse semi supervised information
approach reduction time machine slightly although representation domain seek auxiliary prediction approach denoise provide exploit feature nlp processing
pursuit decompose signal sparse develop variational accelerate practical superposition surprisingly look different principal decompose sparse principal seek image scene mathematically state ex solve specifically relationship cite remark fast ahead greatly cross restrict adapt generalize smooth penalty frobenius include huber penalty besides proceed section cast general product regularization enable use discuss computationally accelerate project quasi formulation section demonstrate efficacy new
solely get actual unstable hand large value empirical stable almost region power satisfactory value size nominal pure even size generally finding indicate preferable would empirically preferable nominal level power test maintain usually actual unstable use worth note concept simple composite testing would scope proposal paper significance consider section proper coincide nan idea divergence restricted subspace restrict divergence base testing coincide robustness property easily minor change impose nan decide consider extension excellent robustness already empirically remain
set signature g k projection onto template implement histogram signature impractical require version property gx instead transform version neuron store transform version template template allow signature transformation visual template visual adjacent correspond memory audio observe similar template transformation computation
dominate relevant perspective fit metropolis nx ratio prior overall acceptance metropolis hasting beta posterior product bernoulli acceptance acceptance centre sequence application variate illustrate likelihood prior step often converse apply costly prior ratio eliminate unlikely normal confirm histogram result term individual evaluation potential rejection costly decomposition bring execution binomial replace sequence adequate sequence fall acceptance integrable therefore improper original note irrelevant sense try far successful value b high though requirement ergodicity
borel let nm n old inequality let q square integrable theorem surely kronecker value almost surely almost surely without evy monotone relate evy intensity tail evy intensity eq slowly vary satisfy part monotonic monotone infinity additionally infinity chebyshev inequality variable hierarchical node edge graph total function decrease work concave low markov eq q chebyshev type obtain conditionally poisson poisson go invoke gamma direct graph income count notion exchangeability represent variable indicate restrict generic space infinite jointly exchangeable infinite exchangeability representation involve transformation uniform constructive array study represent exchangeable either terminology theorem conclusion exchangeability sense sparse real network exchangeability obtain alternatively rescale network size lead sparse finitely rescale distribution node auto right node right node node benefit law aside array structure adjacency different notion exchangeability network link otherwise exchangeability exchangeable small unlikely fall lead intuition implication exchangeability
cauchy schwarz inequality moment cauchy schwarz triangle schwarz use schwarz moment limit supremum reasoning n combine eq q rely straightforward provide deviation necessarily quadratic random deviation four deviation obtain q back taylor k imply standard variable x ax power decompose product ax k fact nonnegative office la bs calibration rgb rgb theorem gaussian concentrate detection procedure top projection able keyword detection mixture sparse eigenvalue test fundamental aspect number number literature proposal convex relaxation greedy multiple use dimensional accord appropriate choose selection hoc performing preprocesse necessarily useful propose penalize method suggest crucial estimate propose method two psd stand semidefinite specifically sparse nonzero integer belong sequel parameter arbitrary note identity dimension read q exposition mild approach setting treat would burden vs otherwise limit adopt quantify test bad problem manuscript minimax way versus nr nr noise pseudo nr alternative hypothesis bad versus error maximize introduction alternative hypothesis maximum far enough infimum test formally speak hypothesis index correspondingly also limit well powerful separation rate satisfy
bayesian presence covariate generalize poisson q variational heterogeneity inference residual review case integration actor application consider sbm likelihood force actor sbm wise block rectangular height state sbm sub interval fall receive attention direct suffer intrinsic identifiability problem preserve paper cite interpretability result show subgraph characterize evolutionary network value vary consist typically time keep connect sbm suppose aim detect share connection stress node cluster latter former group never p fill white circle present particular performance review focus sbm constitute broad network popular social account social induce heterogeneity identification unobserve simplicity decide discuss model review general cover section assume cluster latent controlling
account account baseline general baseline large topic media stop discuss account account account model reflect activity covariate l require member without follow obtain estimate treat nuisance full form hazard pl get leverage measure total hazard change bring hazard algebra obtain hazard change bring value express scale next newton logarithm q smoothness employ use hessian given initialize optimum maximize update constant newton analogously entry proper book keep obtain entire matrix hence complexity iteration
replace diversity layer locality column pick cosine similarity instead average center entire neighbor color center near image together expand average conjecture pool mid hoc fashion use convnet style sift retrieve neighbor coarse dense impose smoothness image alignment neighbor alignment convnet location convnet response formulate grid feature image target source image edge
subtle determine membership total entirely trivial principle perfectly every determine clearly thus output shift simple determine output manifold modal differentiable derivative clear context manifold curve cover support vc fourth derivative derivative make sure represent manifold critical point mode saddle assume usual variance k condition regression integrate error proof pointwise kde kernel nx pointwise hausdorff q fix curvature density nx nx z conditional derivative uniform error estimating mode closely see uniform q pointwise rate additional pay square nx analogously mode yy yy yy nx confidence
section define equation fix inspire boltzmann propagation normal perturb often derive interpretation derivation j define stack substitute evaluating covariance impractical variational two example constitute often estimate go likelihood denote univariate mean precision th wish precede generative experiment approach covariance hasting mh
consequently iii also assumption suitably assumption bound accumulation point ii accumulation subsequence take side theorem arbitrarily f k go end virtue sequence establish penalty method accumulation feasible subsequence f penalty method fact accumulation subsequence point problem denote vector form sufficiently together sufficiently unbounded closeness finite follow contradiction pass convergent subsequence finitely q kkt ax kt ax contradiction unbounde contrary generality divide side contradict assumption subsequence take limit side along subsequence finitely see last relation together loss generality
n em w em I n n b n dx surely computation generalize invertible matrix dx dx ii denote sample algorithm train choose propose extra spend bias view part simulation ghz intel cpu fx create distribute normalize range long obvious quality learn
condition completion frank wolfe convergence typically require accurately use entry denote throughout change resp resp asymptotic span denote onto index proof involve integer mark gap spectrum decompose decompose j eventually stop concatenation matrix column point index definition bound quantify whether end q present f satisfy gap tb nb tu l kx tx matrix call result imagine thing obtain distribution independent split precisely explicitly present least procedure analyze subroutine subroutine control intermediate arise matrix tight coherence description iteration way top vector arise use reason ap see randomly main return good condition n element return guarantee
variable marginally conditionally rest choice local multinomial univariate variable frequently combination interest variable parent configuration parent advantage possible task regardless set separate uniquely parent child redundant inference bn implement consist encode algorithm seminal graphical ic bn test graphical information student correlation score optimisation technique candidate goodness algorithm attempt arise independence dag latter pc gs incremental hill search min parent semi pc implement across several package extension
nearly approach computer vision adversarial positive primarily linearity space extreme need yield rbf presence phenomenon sample fed assign probability maxout softmax mistake change top drop confidence mistake cifar sample convolutional maxout experiment suggest need imagenet image encode search uniquely able focused softmax confidence mistake rbf behave find error confidence mistake hard problem belong cifar skewed class maxout network none classify likewise network classify classify introduce propose succeed randomized runtime cifar success step ten step class image gradient method member dataset activation
lt bp r package conjunction terminal explanation load package package graphic explanation graphic macro ltb lt lt lt lt lt lt ltb lt lt lt lt ltb ltb r ltb ltb ltb ltb ltb ltb ltb ltb ltb handle significant high determine handle add effectiveness artificial figure graph reduction handle percent reduction percentage handling compare handling handling handling axis handling handle decrease handle increase exception classification noise broad application beneficial task weight filter algorithm diverse suggest produce result filter weighting great estimate handle examine forest generally high inherent instance
gaussian snr population nmf positive uniqueness uniquely end nmf rank unique describe case tensor nonnegative factor know uniquely recovered idea reduction propose order perform rd tensor obtain follow product rd tensor unfold original uniqueness close nmf sparsity moreover directly reflect zero focus core tensor break curse improve uniqueness ideal nonnegative essentially unique mild sparse suggest quite sparsity core tensor impose sparsity matrix pp db able approach improve q approach
extract sentence dependency recursive feature feature feature sentence generate generally two method category assume sentence attribute field kind sentence multimodal input e sentence probability sentence give affinity retrieval fall close build bilinear whereas store architecture allow arbitrary htb word image architecture illustration rnn share frame simple recurrent neural network widely language task speech type time
specify terminate simulation validity validity ensure terminate interval coverage rule difficult high dimensional magnitude component far unlikely suitable idea terminate simulation ess sufficiently bit variance asymptotic variance note general easy estimator standard terminate interval magnitude simulation terminate poor reasonable default minimum specify reflect analytical condition asymptotic validity relative stop three theorem strongly setting direct practitioner automate criterion applicable first one need relative knowledge magnitude suffice excellent however adjust balance effort exceed criterion
b theorem dependent step assess case super smooth case consist finitely fouri transform xt integrate tail relate exponent analytic law vanish case prove satisfying condition monotonicity condition respectively finite support strong bayes posterior condition hold satisfy mixing via case mixture gaussian derive rate wasserstein assess empirical deconvolution ordinary recall borel probability moment wasserstein metric recover investigate correspond symbol scale b mixture selection deconvolution smooth error optimal measure herein consider rate error dirichlet hyper mix super recently investigate frequentist rate deconvolution model density measurement convolution p yy transform density density iid mixing derive rate either ordinary smooth super ordinary super smooth minimax super corollary inversion inequality relate density density yield deconvolution follow
iii iv phone call incoming call receive diversity incoming unique call send entropy ratio behavior percent call call percent regularity average inter event inter event consist incoming total follow call call receive percentage ii percentage iv percentage text text receive user hour text characterization diversity apply online measure variance inter call inter quantity measure evenly distribute address three feature unique total entropy explain bin function fx np apply deal bias entropy filter case filter proximity feature table general proximity accounting see time diversity interaction regularity two event variance inter time extract general iii correction calculation formulate
explicitly beneficial great consensus superior add paper consensus herein matrix treat input reach initialize randomly consensus final consensus consensus run mean consensus via consensus final membership method variety mean ensemble pair unclear potential datum essentially round collect herein popular consideration refer resource spherical mean low objective matrix cut accord normalize cut ng choice membership algorithm mistake assume rarely algorithm mistake introduce consensus matrix proportion must agree keep vote initial consensus form
maximum compare pool largely available score rise degree autocorrelation contrast expect experimentally autocorrelation score contrast compare performance al al common autocorrelation trajectory al small would lead method trajectory dominate rs budget would present picture al much al performance expect comparison show would score resolve score autocorrelation show show examine base way compare benchmark rs stage first seek
five start structure represent former matlab take input strictly matlab contain field list help default default solver minimize gradient determine toolbox find contain cm cm prox proximal solve prox proximal proximal operator prox
goal offer represent observed network inspire boost handle noisy overlap signal noise plan thorough framework cluster potential corollary observation pt significant relational give challenge general boost inspire weak entity similarity graph suitable community detection real demonstrating measurement consideration local structure community real absence ground quality proxy learn measure contribution aggregation framework learn application heuristic measure operate incorporate make
indeed expand signature rough individually inverse polynomially correlate set empirical define simplify main explain correlate expand find signature rough solution get polynomially signature imply dictionary entry whether ability individually k j assumption large entry ns using argument shall dictionary formally write unknown expand large ki negative outline quantity exactly additionally deviation signature require additional standard deviation general signature
distribution show present strong package show thank valuable suggestion correction estimation student copulas I use univariate compute estimate matrix initial copula current covariance ii derivative
represent differential entropy matrix differential semidefinite close minimize differential among semidefinite condition strongly variation explicitly see positive definite conclude stay definite subsequent still eigenvalue zero mean arbitrarily bfgs implementation challenge introduce enforce matrix proximity exceed show restrictive solve regularize correct explicitly regularize bfgs replacement variation correct variation term bfgs regularize curvature iterate think gradient iterate stand hessian compute bfgs batch determine add positive differ account relative regularize regularize differ gradient observe add curvature necessary variation discuss hessian stochastic gradient satisfie explicitly eq guarantee variable cf cf approximation core comprise require
dataset ip importance convergence ip show almost sdca ip practical sdca u somewhat prox sdca explanation primal update q conservative indeed satisfy large primal variable confirm test change result display clear primal convergence prox sdca prox sdca less hence close difference rule prox sdca speedup predictor speedup focus hinge factor specialize nice dataset several value practical theoretical prediction large roughly speedup reach lot year smooth primal machine incur predictor regularizer especially interested big million much big let n nm mi separable list mind name know simultaneously solve primal however apart block choose like method dual perform primal primal update
mutually exclusive exhaustive mutual algebraic elementary logic equivalence triple triple follow signal identifiable identifiable converse identifiable equivalence assertion proposition identifiable open equivalence establish third triple generic measurement generic identify measurement identify generic measurement regime identify generic identify furthermore dense open either three formulation mutually exclusive exhaustive analogous retrieval treat verify generic rank enable signal reconstruction measurement signal recent furthermore algebraic algebraic term non intensity measurement image ray magnitude sample phase optical task reconstruct finitely rank algorithm involve projection fit nonconvex optimization semidefinite programming success grow algebraic formula derive require measurement scale dimension jointly algebraic semidefinite successfully reconstruct generic identifiability signal signal open enable phase retrieval signal phase rank investigate mention identifiability
expensive computation hardware corrupt descent true stochastic hardware shown avoid additional present proportional differ descent gradient descent proposition proof directly applicable enforce monotonically satisfy preferred fitting relaxed understand hardware induce successive discover hardware computational analysis lipschitz iterate eq batch gradient bit sequence converge variable expectation allow decay I hardware accurate
maximize bayesian design minimize leibler eq random additionally mix ct slice select entropy progress accuracy overall posteriori fig measure truth gp depend subset dimension learn seven possibility truth correspond fig significantly method uncertainty design perform poorly
continue standard generative hyperspectral full dirichlet without loss generality n go particular parameter characterize pixel simplify symmetric follow phenomenon uniformly decrease concentrate around vertex concentrated contain pixel separability implicitly assumption ii exist statistical assumption thereby implication pre formulate whitening q large small ba deduce whitening assumption n tw whiten aforementioned bind conditioning plug corollary provable spread play get consider fix natural concentration interesting specifically
dpp space space say mass specifically subset determinant contain column semidefinite matrix speak representation ensemble similarity dpp subset square tend volume co machine appeal marginalization model dpp preference six except ensemble group green rank condition dpp sample interaction probability proportional rest tb dpp toy except select although big block block standard model often make binary random specifie include assume bernoulli covariance spike
exploit hierarchical learn representation co occurrence qualitatively among hierarchy occurrence highlight facilitate reasoning space experimentally illustrate dependency multi classification area machine contrast instance goal model structure space occurrence attempt structure configuration label classifier handle classifier efficiently effort maintain update generate ground
result supplement assume constant setting probability depend let proposition satisfy define zero proposition establish result v result contrast provably identify method computationally primarily dimension however goal mixture two spherical
argument behaves simplify special noiseless signal order maximum within dictionary conversely expect dictionary incoherent dictionary take account need least incoherent imply reach small choosing translate mean maxima conduct input lagrange multiplier n arrive scaling ensure giving generating bad atom signal sign iteration determine multiplication comparison thresholde instead omp procedure algorithm local refinement furthermore version new old signal process learn accord algorithm conduct experiment htb generating signal far give decay factor uniformly vector sphere choose permutation compare error basis number perturbation correspond approximately noiseless signal
ridge thresholding network ij j fitting square j construct jk perform hard ridge j cf threshold medium large link must maintain solve single solution latter ridge impose application allow medium th exclude maintain link show implementation th integer decrease costly iteration simple directly network topology relatively fine either dynamical possibly nonlinear approach reason observation perturbation dynamical exponentially fast important investigate lyapunov representation degradation
transfer code much also decision result problem come gradient optimize code train create iterate create initial robust coding produce latent present evaluate motivation robust pca
demonstrate effectiveness core core basically q achieve mention package modification pre substantial store moderate issue summarize expensive kernel half total pairwise store realistic pc merely example computational demand another serious issue issue
use purpose predictive support source validation assessment phase inform methodology illustrative involve validation bayesian advance hardware advance fidelity enable physical phenomenon system complexity capability heavily nearly complex nuclear use area inaccurate inform decision could response critical reliability systematically characterize reliability science reliability computational necessarily computational thereby regard computational compete design meet operational decide evolve informed prediction quantitie important feature use observational scenario available would course engineer currently assess unobserved address propose call predictive broad define specify enable test predictive physical reliable theory law whose highly must less reliable reliable embed various modeling approximation empirical interpolation mechanic embed might molecular fidelity fidelity embed composite build highly enabling reliable though specifically require reliable restriction composite specific ingredient representation representation observational model provide uncertainty development model predictive approach discrepancy model unobserved advance accomplish embed physical directly uncertainty bayesian model condition observational uncertainty unobserved composite provide physics subject validation integrate assess reliability prediction describe calibration predictive regard uncertainty infer
formula appear subset etc uncorrelated follow result follow lemma desire combined hypothesis suffice definition way limit suffice infinity never condition hold replacement limit distribution forest example consistently forest prediction remain largely answer subsample training randomize predictor prediction forest averaging explain formally resample random forest main base prediction provide consistently thus bring forest bootstrap forest remarkably estimator surprising property
calculate size lemma k proof appendix enough know always bind dm however prior knowledge prove analyze second know able algorithm calculate prove eq phase examine expect bind expect bind km improvement attribute reason highly dependent require hard attribute eq efficient scenario call efficient present develop estimate operation update update result project ball yield build unbiased gradient modification p r tp ti slightly let idea unbiased estimator therefore use analysis full proof present notation author algorithm ridge scenario analyze tell
mt media file mt user file mt file mt monitor mt project file mt file trace cache simulator build cache service request maintain cache request simulator cache block cache record else record add cache cache limited cache trace cache simulator cache exist policy keep track previous cache sparse hmm baseline trace trace htp r r trace sparse hmm without dp mt mt mt mt mt mt portion dependency train operational mark real world set require vary periodic require explore version setting htbp minus pt pt pt though
useful specification additional mcmc adjust achieve even quite well solution small hamiltonian dynamic effectively result improve posterior step hamiltonian black handwritten digit handwritten digit benchmark consist spherical gaussian parameterize connect output distribution pixel consist datum necessity individual map jointly unbiased construct replace chain hamiltonian describe vary number auxiliary choose deterministic
demonstrate biased nonparametric prior odd bernoulli discover gamma process couple heavily description work particularly value atom interpret trait atom proof essentially fix ordinary translate bayes full moreover entirely process reasonable carry tuple need real believe broad acknowledgement project university berkeley fellowship automatic prior far generate bernoulli location atom accordance bernoulli rewrite conjugate fix location atom beta hyperparameter normalization ordinary component proper hyperparameter ensure measure must must represent improper beta either distribution integral hyperparameter restriction recover bernoulli parameterization condition q pick biased beta point beta condition pair particular construction location atom weight atom location recover posterior nonparametric prior sequence finite represent full bayesian via finite motivated prior exponential likelihood construction
none b comparable uci benchmark training measure choose summarize describe select significantly stock tend bold c reduction kde auto forest energy stock kde kde auto red forest concrete energy evaluate propose simulator body part roll roll roll value angle velocity
zero forward see finally real world even slight independence tackle corruption heavy reconstruct belong subspace coefficient reconstruct corrupted assign unconstrained quadratic program run loop gradient conjugate straight projection total matrix identity program technique scalable synthetic detail empirical face individual illumination face
successful describe agnostic agnostic successfully construct ensemble predictor constrain grid generalize agnostic attempt directly infer accord contrast bayesian implicitly data likely member irrespective ensemble approach risk hyperparameter loss rule posterior estimate repeatedly sample could vector predictor low rule repeatedly predictor obtain stand obtain sample replacement empirical risk
completion recovery conclusion accord aim paper effectiveness explanation experiment window core ghz gb conduct representative completion partial dct effectiveness admm admm admm former value performance quality admm nuclear norm recovery lr conduct visual situation dct conduct admm lr situation dct conduct compare admm rank dimensional dct zero deviation matlab generate randomly completion experiment randomly x terminate admm terminate criterion f empirically test parameter admm matlab code use peak ratio
construction lipschitz boundary disjoint eq disjoint collection satisfy together modification variation concrete visualization minimizer cut eq appropriate take datum distribution two rectangular eq easily domain appropriately characteristic analogously partition base cut problem optimal cut n nk k red line indicate cut utilize nearest eq near descent cut initialize ground truth algorithm three consecutive graph partition return quantify partition simply misclassifie I sequence satisfy last inequality piecewise way convergence context nn measure graph ratio maximal average realization degree compute become become increasingly relate perform exhaustive experiment three domain consider correspond distinct fall surely connect also rise rather structural increasingly graph see figure scale connectivity geometric graph vertex leave connectivity alone responsible balanced scaling serve benchmark context provide balanced cut test fail outline pose balance consistency practical difficulty may
abstraction prove useful statistic whereas well introduce generic notion consider confidence low theoretic specific armed bandit derive refined bound confidence particular fix set familiar behavior alternative addition improve sequential time proof result deviation lemma good arm exploration divergence sequential testing paper find arm armed arm arm option receive expectation agent goal identify index arm expectation tuple expectation sort decrease depend several analysis include kl ucb thompson without try observation study name identification advance consider confidence introduce identification successively discard example bandit arm sampling model subgaussian propose algorithm subgaussian upper imply rather comparable gap recent bound sample dependency term gap go remain exhibit pac algorithms exist improve upper work bandit follow relaxation consider literature tolerance optimal compare
iteration recommend pass execute break terminate loop early suitable input determine solve shot screen external set additional examine algorithm table generate rand mnist write digits sampling randomly subject face pick word uci repository represent occurrence document remove leave first solver solver give percentage reject speedup divide sum solve reduce lasso speedup use value select problem r feature dim rand mnist average shoot st dt spherical conservative shot screen dataset sphere dictionary also oracle sphere center provide bound shot test fig salient default shot indicate potential spherical gap indicate worth default spherical except value default dt x dt effect mnist speedup plot test dataset iteration confirm low sequential screening scheme salient mnist robust yield rejection speedup compare one shot giving screening use successfully solve
function posterior via regression function simple spike behave accordance expectation obtain performance improve performance since great ridge via development regard stochastic inference acceleration part terminal cell cell acknowledge foundation fast number alternate relevant state whereas computationally infeasible entirely approach problem selection variety inference challenge case hierarchical
unknown start sequence rate predict form number start weight forecaster start exponential wise loss strategy perform step sequence negative algorithm rule rate forecaster defer article expert advance final explain let increase sequence rate regret lemma lemma dividing grow last main derive stochastic assume formally learner ask knowledge observation strategy consequence show
character demonstrate recognize detect close admissible evaluate length admissible occur character character contain model bernoulli hoeffde document decrease exponentially acceptable proportion necessary achieve desire probabilistic right side proportion unlikely exceed become impose allow percent opt evenly throughout maximally construction character document criterion meet case order among application quantitative produce programming principle implements object oriented programming implement represent analysis let seek represent
thank addition amount pairwise distance geometrically shrinkage project euclidean encourage effect analytically give principle leave entry minimal eigenvalue principle q eigenvalue decomposition light minimum dimension equivalent assume eigenvalue apply amount shrinkage embed circle circle right circle circle pt embed characterization shrinkage estimator characterization clear convenience projection derive oracle stand close euclidean frobenius norm large tuning parameter embed explicit error general true distance eq light
word rich rate average assume across item item receive rating receive item item heterogeneous rich word item receive receive assume rich item model rich assume user item rich otherwise analysis relaxed comment modification proof condition form satisfied fashion matrix furthermore noisy channel two likely upper size size constant require recover rating develop scale asymptotically occur rate rating e function require recover cluster satisfy separability condition hold exist completion observe rating fewer accurately proof present sufficiently hold rating least item recover rating find present recover cluster key modification information rich clustering compare select user say prove rich algorithm select accord user majority
hereafter hereafter test via monte simulation propose value size test therefore empirical critical reasonably maintain weak signal loss take alternative take magnitude take follow identically distribute k block identically dependence e range possess robustness propose autoregressive distribute generate distribute variate move beta consider p covariance simulation test sparse sampling take significance compute simulation hc summarize figure test empirical test reasonably nominal maintain fail long range structure hc procedure fail maintain strong ex ex hc ex hc hc nominal diagonal autoregressive move consider level signal covariance signal strength matrix empirical took propose maintain nominal test screen
variate determined iteration time examine schedule expect speedup true quantity tree predictor predictor predictor immediate transition always choose available acceptance improve increase worker move sequence affect scheduling branch promise branch ultimately step chain compute completion distribution apply monte posterior unnormalized modeling correspond model decompose independent logarithm term mh eq form normal separately variance together expand perturbation concrete subsampling subsample construct term subsample empirically deviation multiply estimate finite correction eq form
work show side metric dnn phone embedding extend multi gold derive term feedforward neural input stack contain hide layer sigmoid activation embedding dimension type
technology quantify expression throughput possibility rna seq yet appropriate statistical assess statistical bayesian treatment empirical approach parameter quasi account tackle lead highly couple probable differential expression rna sequence uncertainty theoretically parameter gene yet small across log fold ratio continuous exhibit difference cause random variability
cl sl ensemble accumulation similarity connect triple time consume fig infeasible dataset method perform benchmark agglomerative cl sl baseline cl sl cl sl cl sl performance baseline dataset consensus dataset method stable pair good dataset overall method ill clustering heavily clustering pool ill pool clustering pool clustering clustering partition mean randomly cluster select heavily clustering ratio clustering experiment base clustering ill clustering link pair wise clustering clustering accordance common reliability collect clustering consensus accumulation partition gp respectively eight result effectiveness robustness propose ensemble consensus accumulation partitioning link fundamental purpose partition unlabele homogeneous
project axis distribution axis consistent discard might size powerful regardless compare asymptotic efficiency relative great size tend equivalent value relative efficiency relative follow theorem small equation case scale imply error conservative xy dependent numerator denominator test high test test random rotation assume size kernel associate uniquely respective reproduce hilbert xy
gps gps wave unlikely poor sequential size importantly gp achieve interest wave improve gp wide advance number wave sufficiently detailed whether gp become increasingly problem support model wave cope suitably say accurately predict posterior metropolis hasting proposal acceptance gps parameter likelihood mh decide simulator onto prior wave report volume dimension act
du univariate correlation generate marginal binomial via generate margin topology original element monotone slight normal sample poisson close great appendix practice package cdf several pre abundance zero account sequence normalize sequence multiply desirable depth preferable filter sequencing depth normalize serve count fit use target zero binomial good account count newton candidate superior normality assumption pattern adjacency undirecte graph topology generate step undirected adjacency ii convert assign diagonal convert focus representative structure degree recovery thus topologie maximum band model disjoint association across scale biology network serve comprise specie many specie connect sparsity control start adjacency type network chain connect neighbor edge fill cluster comprise divide approximately randomly assign scale network law specie connection adjacency matrix add adjacency adjacency precision
procedure kf use approximate mean kf q explicitly enkf ensemble bottleneck enkf describe converge small size degree require version enkf adjustment ensemble transform kalman subsection kalman take couple initial product second cross reduce kf gray operation state covariance cost nm tn essence kf store adjacent cost form covariance prior cross already kalman equation product equation walk fix variation derive formulation assumption store update assimilation step therefore kf covariance initial assume available model covariance spatial section kernel dense oppose provide sparse engineering application
ergodic converge rate gives point select log likelihood happen exactly quadratic match proposal move discussion three point mt mt mt always situation substantially detail setup genetic circuit response concentration chemical figure vertical correspond algebraic switch z datum six scale around nominal endowed uniform nominal observe low average without posterior remain seem useful towards quite add difficulty heuristic heuristic rigorous give expansion designing grid respect induce convergent density inefficient whenever illustrate overall approach provable convergence surrogate attempt resolve propose hasting local process use sequential experimental exploration design reflect quality local random refinement local quality practical permit wherein enable quickly smooth simpler asymptotically exact walk couple local approximation demonstrate posterior refined walk metropolis broadly adapt metropolis hasting time propagate broadly theoretical complement experimental several order involve ordinary equation partial equation approximation remainder organize describe exact defer several emphasize present representative therefore discuss several future metropolis infinitely refine problem forward
justify bag contaminate resample contaminate set contamination explain section contamination enable create diverse base resample bag exploit resample tradeoff place variability increase stability model influence svm quantify distinguished instance ii iii incorrectly classify margin bound
property return set every share notation associate firstly risk classifier subsample self hence close minimize risk deviation disagreement rewrite confidence majority keeping performance confirm label close source risk apply optimize disagreement risk label however decrease guarantee gibbs decrease avoid design tune question concern da reverse circular reverse perform intuition source label seen previously validate analysis make sample show
list originally english word obtain return translation one devise training distance negative margin negative hold descent tune task improve least propose cross modal mapping vice versa contain representation wikipedia gram mode image represent train train labeling set task label entire distinct return cat chance observe differently case cosine domain improve standard setting chance
logit lead inverse probit sparfa next give response response minimize observed response constraint use practice nuclear rank one constraint via validation emphasize sparfa regular sparfa negative logit fista start iteration perform aim follow projection make give size inverse logit link bin boundary measurement
htp red solid accelerate accelerate suffer slow amount overcome issue one strategy fy k evolution dash iteration dot one clearly already well solution help htp htp lasso htp physical average coordinate htp
modality normalize temporal original frame derivative compute temporal delta element vector calculate way delta static audio video dimension whitening frame result audio build network structure two letter deep machine final rbm unit unit letter size space embed although preliminary one embed
angle face second put clean patch image database adaptive denoise apply localize new exist bm bm bm new patch procedure determine text face demonstrate superior amount available generation give eq drop orthogonality take later lagrangian multipli setting pair must correspond observe trivial satisfied take constraint denoise column become sum lagrange take eigenvector orthonormal q recall simplify first difficult note substitute eq hold therefore standard exist give remark fact procedure propose denoise image denoise patch noisy generic new patch database denoise filter problem denoise filter solve sparsity generalize denoise offer systematic enhance second determine denoise localize bayesian localize intensive computation
approach generate expansion abstract approximate term give subscript representation example calculate cost multiplier introduce assume component separately separately column contain length specific kernel identity calculation assume multipli component may equation need obtain exponential radius process avoid square cholesky intel core ghz point define fraction ground way solution call remain apart training test accurate result use predict detail use sec interpolation shift parameter prediction example ii either literature general study quality assess mean predict predict different ard
human exist infeasible screening gaussian possesse screen gain popularity context possess go approach generalize generalized screen partial screening response select screen recover call motivated fact precision obtain feature onto neighbourhood sure show exceed threshold grows establish surprising exist procedure unsupervise conceptually implement estimate
away unstable equilibrium future discount reward actor convergence fix lagrange td td lagrange multipli constant recursion set equilibria ode ensure evolution ode stay recursion convergence limit policy recursion unique eq bellman transition state similar manner stable govern almost let unbiased vanish vanish martingale cf see actor recursion asymptotically track point ode eq bias converge uniformly follow discount show recursion saddle ode define surely proof manner discount claim use use convergence rs g ode satisfie context application delay motivation behind variation road infinite setting traffic formulate briefly recall queue time road since turned belong road factor traffic discount implement green simulator order neutral parameter follow td unlike rs neutral multipli neutral use smoothed sf technique neutral n j q rest symbol rs sf neutral counterpart sf hessian actor update form actor rs g attempt accord sf variant sf hessian update sf sf counterpart rs consider rs underlie boltzmann approximation road approximately order increase experiment phase nominal iteration policy simulation converge average snapshot road simulator traffic frequency specify traffic proportion horizontal fig set weight
proposal neighboring constitute single cause mix eventually low point mcmc move manner configuration accurately rbms typically train stack greedy wise deep belief layer latent feature another rbm mix well configuration draw top level gibb lead mixed well result appear consider layer model even phenomenon receive attention learn texture
make consequently cause al shape kernel learn non parametric fashion control person person decay person characterize decay enforce value instantaneous reflect material new dynamic bayesian specifie occur people person interval token word type token upon multivariate process token person categorical vector draw person specific base discrete vector characterize inherent usage person person self enforce make token type person end time person characterize decay parameter
duration delta duration counter always state hdp possible transition hdp restrict ps equal ps draw dependent hdp bayesian hmm setup hdp hmm replace auxiliary dp instead hdp create hmm reversible hmms cardinality death operator incorporate duration might would complex finite hmms emission slice sampling emission mixture responsible observation subset mixture relate identity state describe could arrive inference perform dynamic exactly compute map implicitly cardinality kind guarantee auxiliary duration emission sampling hdp jointly employ filter backward infinite duration hmms duration associate time log likelihood quickly backward slice sampler move trajectory infinite backward hmms state time backward difficulty overcome use slice auxiliary result sampling context auxiliary variable introduce variable
bayes relate optimization efficiently novel iterative maximization performance problem drive access identification specific namely interested wide engineering reconstruction science communication hundred method impossible thorough literature generally pose structure impossible retrieve partially intrinsic
ray foundation rna protein solution nevertheless phase mechanism year experimental enable potentially resolution chemical specificity ray enable combine high precision resolution imaging advance detector world develop hundred help ever material device study life macro molecular machine picture recover atomic detector small focus numerical exhibit encouraging conclusion organize introduce setup show sufficient ap section relationship ap objective ap synchronization propose accurate also design achieve field set non negative phase would set symbol form scalar hilbert lk ii f pi li I wise entry resp resp notation aa ba ij ij form hadamard matrix express format stack column embed j I record wise fourier magnitude relationship coherent pattern discretize camera
select detail bernoulli variable one cardinality presentation assume sign nonzero sign ij globally place elimination scheme prove distribution noisy follow tw like column equal fairly hold denote difference first coherence finish follow ie ie u ie ie ie ie norm norm sense u ie ie give ie imply u next sign entry bernoulli sup entry distribute u long ac u ie u ie see sum u ie randomness easy ie ie ie f vary ij u
enhanced also replace subproblem easier approach q iteration bound point subgradient function application problem gradient method latter new namely slide gs evaluation approximately solve subproblem accelerate method gs need compute pair use place output show subgradient subroutine nk ps prox slide procedure let parameter given observe slide compute approximate clearly problem since affine skip gradient differ remark gs firstly occur increment gs algorithm update secondly solve consist solution relatively notational convenience procedure conceptual yet
joint force pixel matrix pixel atom group equation sign inherent since compose several e dictionary classify pixel measure reasonable enforce atom accomplish encourage group active inactive group dominate classification inherently mixed optimization represent weight collaborative collaborative regularizer define refer coefficient lasso joint
vice versa item modification weighting scheme weight weighted item leverage one allow time limit bias induce equation thus optimisation amount dissimilarity reduce influence rare important calculation gradient system limit modification precisely subset large calculation descent notice point costly associate statistical
sense consider coherence function coherence initially propose cosine angle become j construct coherent candidate coherence cosine functions control nan computationally atom perspective coherence deal unit explore analogy norm gram eq atom exceed follow include j view extension coherence approximation extension dictionary eigenvalue span sparse dictionary sparsity low low bound investigate condition cf proceed bring mind gram dictionary unit norm atom every lie provide gram sparse dictionary
bag classifier evaluation undesirable attractive base strength ensemble classifier two introduce prototype bag form dissimilarity prototype bag stand correspond size stand offer default alternative well alternative dissimilarity set split data dissimilarity change expect choice affect well single dissimilarity ccc problem drug image dataset scene formulate concerned categorization song dataset audio feature zero
satisfy concavity operator previously subset size sequence use sample size contraction coefficient initial sample bind q probability guarantee dimensional instead analyze discussion state theorem algorithm term addition initialization iteration estimate illustrate iteration size figs em eps decay rate iteration number em fast cc fig eps fig eps regression em decay geometrically decay geometrically devote linear covariate vector miss ratio give choice define guarantee miss missing covariate miss satisfy ball previous corollary somewhat seem counter intuitive minimax covariate show gradient algorithm formalize amount information confirm show eventually decrease grow fig plot mixture algorithm optima splitting em em operator usual full subset contraction em iterate probability perform large constant logarithmic sample achieve analyze particular iteration sample give stochastic gradient eq appendix fig mis eps decay ccc fig em mis eps figs mis eps missing covariate
e achieve oracle property section equal follow reason variance advance accordingly even know consequence fact variance fact subsequent analysis notation l analyze lp discuss capability recover result l l definition lemma previous l ne
space tradeoff show access use allow well lp qp choose factor expected distribute overhead however svm outperform wise advantage incur node request contrast lr l primarily tradeoff descent number converge fast factor speed overhead break scheduling second use second scheduling inside remove epoch hold implementation tradeoff space difference qp row wise access wise fast primarily issue support claim interesting choose throughput compare throughput different system simple sum model core throughput figure high throughput system incur cache write single copy copy node cache single fast overhead dynamically schedule maintain overhead language scheduling computation fast impact modern architecture row wise mean ec validate access dominate strategy row wise force correspond report measure take achieve side segment epoch reach wise access least phenomenon column access slow simply read preferable coordinate descent wise amazon google lr qp lp dominate write capture describe impact factor wise strategy figure ratio row column amazon machine increase ratio wise become slow wise hardware
adopt paper logistic loss training keeps prevent simplify exposition straightforwardly vector learn nonlinearity jointly significantly learn classifier however mapping raise generality set rest fixing norm essentially permit adapt advantage set general focus go class general original cast dc program last classifier rewrite optimization rewrite eqs hold restrict component vector hyperplane remove small component positive class knowledge atom desire half encode prior balanced estimate assume proportion atom sample solid smooth qx parameter approximation become soft play important training easier soft latter handle essentially relaxed optimal variable still nonconvex type write dc solution dc function
local ratio cut define one choose cut correspond weighted think vertex divide unlabele wish assign assign represent vertex similarity vertex cluster give order
anomalous unknown priori test apply consistent stack sequence case anomalous exponentially comment scenario lack knowledge scenario study anomalous sample characterize sparsity anomalous consistent sp substitute theoretical mmd apply reference choose I choose anomalous laplace respectively change normalize horizontal axis clear converge theoretical furthermore curve drop mmd run number anomalous plot converge consistent threshold increase anomalous priori choose distribution mixture distribution figure converge confirm theorem test scenario reference sequence gaussian sequence see reference probability agree drop increase importantly error zero theorem comment variance variance different
proposition similar mala mala know mala rwm study asymptotic regime rwm mala asymptotic mala rwm natural whether mala rwm extend within mala log property particle mala mala depend crucially behaviour accuracy log decay mala behave mala asymptotic rwm mala rwm scale furthermore explicit posterior particle mala mala though implementation particle mala mcmc theory extend beyond model introduction sequential carlo filter filter guide introduce mala result asymptotic size improvement rwm discussion measurable density represent arbitrary sequence assume directly
would child scalable graphical hierarchical expressive implementation massive massive hierarchical namely bioinformatics graphical merge concept graph structure age extract noisy make network scalability massive datum massive divide arrange datum influence immediate represent bn representation handle scenario massive represent random represent represent disease city connect disease city assume disease bn outcome compose node probability entry represent massive efficient multi classification probabilistic hierarchical apply domain throughput
usually actually relevant similarity fast amenable usefulness propose feature irrelevant feature good knowledge reduce projection furthermore order provide original present experimental section metric attract lot ten effort towards comprehensive exist focus mahalanobis symmetric semi undesirable dimensional early resort dimension loss interpretability satisfactory limitation restrictive weighting r
coefficient method include bic criterion aic cv minimize stein unbiased sure setting variable location empirically correct demand know reliable popular high demand paper select computationally agnostic regression threshold become variable reliably lasso jointly estimate result benefit sparse property lasso research seek throughout
score merge discrete variant cluster iff well score initialization number operation costly consume solution split suited generalize follow center index centroid heuristic straightforwardly decrease round assignment stage stage neighbor center c lp tc j tc illustrate toy strictly decrease score repeat finite diagram check pdf means mean k centroid pdf k pdf center
quasi give learning construct private proper learner discrete algorithm exhibit private work separate pure approximate case agnostic class easily exploit bound sample predicate private database achieve tight work prove concept examine private differential label therein privacy label private term dimension completely characterize learner constant kind research direction work learner try construct private learner complex private hyperplane would interesting another research try understand learner construction know improve generic pure private learner characterize complexity learner early work another prove separation pure differential privacy differential demonstrate noise pure privacy gap private currently unknown show pure term negligible database shorthand use instead mx mx dx j privacy aim database say preserve record learn datum differentially database omit preserve differential pure case function access differentially mechanism differential concatenation moreover adaptively differentially private permit interaction preserve ensure privacy bind privacy oppose follow permit preserve privacy access ensure label domain predicate mapping example sample unknown successful say hypothesis concept I random coin proper improper empirical q class characterize pac learner behavior cardinality hence pac class theorem give sample dimension concept mc mx dy output concept agree learner pac use
specify format depend require solver interface sdp present transform form trivial say usually cone template function return return empty appear template point point lie use template order form expression every represent atom return otherwise child top apply optimization constraint objective side constraint add sense form argument construct problem variable dual respectively transformation primal intuition note expression
comparison meta moment expand hilbert survey exact hand way arithmetic inequality question whether polynomial ask square considerably polynomial polynomial polynomial polynomial conclusion equation oppose purpose transform inequality easier follow reduce clearly degree polynomial write polynomial see degree write represent write hypercube meaningful real value multilinear polynomial assume loss polynomial verify use gr basis put thing several informally state sketch view equation polynomial need incorporate see system form variable interested system small ideally even parameterize call operate exist system mx fine discretization number carry actual corresponding degree
move add remove single away neighbor reversible sampler bias move probability metropolis ensure balance markov chain maintain desire vector potentially serious scalability issue inclusion proposal evaluate neighbor hence inclusion back fitting scale although score evaluate parallel scalable inclusion interactive framework strategy identify quickly occur good inclusion well current many fit trade neighborhood disjoint configuration pair reversible neighborhood sampler remove remove swap reverse pair neighborhood quick dramatically predictor change proposal allow vary encourage component removal size set decrease unimodal p pair setting remain prohibitive computer component proceed move forward neighborhood reverse move construct reverse accept sampler satisfie detailed mcmc desire inclusion define neighborhood likelihood component intuitively seem sufficiently particular pair add configuration remove neighbor pair opposite direction require hence pair add need factor negligible limit random away appear good pair move markov chain resolve issue
table step scale cg stop else compute k ap h r h h ccccc h r h r r q r r diag h r h diag p n diag p k j kp solve z ta r r k tp observe coefficient table introduction cd cd cd cd kk cd kk kk k preferable cg investigate cg allow cg within generate mutually
dark grey indicate place robot behavior show ability learn avoid try robot fail avoid paper embed quantify fuzzy proposition linguistic multiple extensively environment environment compare mobile significance perform statistically grant support education national plan ap ram part european project cn education notice author version accept publication apply soft correction mechanism reflect document work publication j modification necessary adapt example production linguistic rule example different default use new population l le jj correctly j belong e c classified rule belong incorrectly classified rule take definition accuracy represent individual match mutation generalization individual cover change parameter regression default straight learn rule proposition rule interpretable rule also proposition number c tb proposition confusion learn performance measure show tb straight convex concave concave design stage transform variable second machine high actually
center ik bc expression lemma find membership small constructing problem outline view method approximation offer advantage top eigenvector secondly cluster instead early cc ccc mean cluster sample randomly center denote mean update number sample number initialize column matrix algorithm normalize hypergraph partition consensus maximize mutual partition membership represent assign label partition partition lie perfect maximize consensus represent partition hypergraph regular edge meta partition balanced meta cluster meta association meta
matrix partially fraction vary unnormalized theoretical analysis decay small normalize htb adjoint operator constant v v j c ta follow q bernstein make differential recall particular set sign p p dual theoretical collective requirement meet requirement derive optimum factor requirement program affinity relationship movie explicit represent completion task miss entry potentially core gene protein
x iv tx desirable integer inequality decrease ok ok conclude lagrangian denote derivative yield k ni w ni ni next therefore plug supplementary ni ok report good context contribution rigorously general characterize yield prediction call instability throughout classifier adapt elaborate neighbor one popular literature extensive theoretically justify risk comprehensive reader weight near risk attention find control regret nearest specifically trade regret new methodology minimax rate offer comprehensive near neighbor
consider answer map idea naturally lead item provide rating yield latent space translation rate translation rating rating rating item two rating interesting rating represented nan rate modify qualitative result write item rating describe consist combinatorial combinatorial optimization weight item clarity representation depend user item restrict rewrite loss optimize propose consist apply weight weight particularly fit
mass provide illustration mass interpret support finite mass provide project coordinate distance bar histogram together sample coordinate reduction observation prediction reduce space concrete ccccc observation c c c empirical distance distance selection ccccc mass computer first thesis brief introduction justify justify simply domain lipschitz theory generally study dual classifying optimally use hyperplane banach construct lipschitz lipschitz exist lipschitz margin lipschitz margin denote label equally mass follow criterion distance finitely supremum lipschitz support result tx define inside conversely metric mass respect satisfy measure g q margin draw without regard area consequently summary relate margin imply soft classifier soft margin mass use acceptable embed observation banach life mass extremely compute certain assigning individually keep classifier prediction chapter base new mass chapter chapter explain trained predict label life learn training evaluation set time iterate evaluation parameter receiver operate roc explain technique determine classifier collection confusion count occurrence comparative cm positive negative negative call negative confusion confusion subsection measure confusion quick well classifier predict skew trivial predict even power evaluate e confusion matrix observation actual proportion predict label precision use science especially retrieval disadvantage f account quick example confusion cc actual predict accuracy receiver operate characteristic roc prediction explicitly area measure classifier predict reference regard roc classification
field ci comparison assumption ci ignore fashion ci eq j ci l membership comparison make fully approach disagreement level represent agreement believe field increase mass move take fairly one set take general truncate comparison record disagreement large capture disagreement disagreement disagreement relatively concentrated value close especially amount reasoning specify remain l disagreement construct priori notice believe exclude inclusion level probability record depend use comparison nominal frequent take disagreement could supplementary material gibbs joint subsection brief na present illustrate record believe field specify year month inspire name compose correspond family practice record pairwise record record agree except month day month record notice record day pairwise decision take decision pair record fairly decide record table record could record miss name name whether record believe deal situation record datum report occur year month day date scenario report error name date place form information self
simple radius choose convergence update measure term rate increase long proposition bind suggest guarantee projection great cc factor plot expect blue run algorithm independently consistent bar naive sketch sketch roughly twice verify predict range choice predict blue bar height average marked instance iteration corollary imply high solution confirm bar show naive square roughly apply simplex g portfolio problem prove many matrix completion control user define radius observation observe link nuclear dimension scale exhibit qualitatively compare unconstrained square sketch poor solution exhibit near optimal classifier goal collection correspond individual x since classifier conjunction detail database expression neutral pose example image
improvement integrate element observe efficacy preprocesse layer beneficial concept small large scale visualize dimension time second although proper epoch result contain four field pseudo randomly interval trial last speed accuracy cluster time second example consider virtual agent
boundedness treat ensure tight result agree study cc theorems equal bias show figure obtain unbounded svm especially label contaminate unknown world thus estimate contaminate outli reach contaminate indeed non contaminate give satisfy worst admissible classifier acceptable search validation experiment let property calculate case classifier properly term advance however statistical prove property give sort negative margin statistic linear order property estimator investigate mathematical reference therein detail population eq denote nothing accord express three random omit suppose zero
crowdsource crowdsource design suitable mechanism expand scope model human crowdsource system low appendix proof employ contradiction base axiom q upon axiom eq yield desire contradiction property include choose property decrease lx w lx x axiom towards increase function satisfie axiom investigate convexity hyperplane mean proposition corollary crowdsourcing inference truth optimize function maximize
interest size complexitie theorem bn compare improvement aside logarithmic factor passive active conclude bn bn bn constant passive aside factor suffice active low passive learning aside hand bound ignore satisfie bernstein class gap bound unclear necessary sufficient improvement bound improvement reveal improvement interesting active learning since indicate quite passive logarithmic passive aside factor upper bind passive learn roughly roughly know model finally agnostic gap unclear range passive aside passive replace disagreement section base relate star logarithmic include bound know disagreement split x h substantial literature label complexity various ask bad maximize respective result last improvement express study family active certain quantity quite diverse pair inequality relate attempt relate literature isolate give plug bad behavior relevant star maximize collection definition entirely complexity additionally role proof diverse literature establish devote together represent summary know relevant present discussion measure star minimax complexity implicit literature case compare additionally complexity passive learn loose argue maximize low implication star relation measure star summarize literature thesis disagreement define let let er xy er hx xy bind label complexity learn effective variant reader thorough disagreement b thesis well work characterize disagreement various survey thorough survey disagreement bad disagreement coefficient survey ex ex general date disagreement survey survey survey detailed description well know logarithmic ex ex bn ex ex ex ex general literature advance understanding capability learn section ex ex ex find quantity connection play proof ex ex ex upper offer ex ex simplicity logarithmic ex bn replace ex offer refinement case ex I case replace theorem appendix refine analogous ex desirable bound case near intuition behind achieve rate approximated objective reduce consistent classifier observe far classifier obtain fine grain pair eliminate request worse eliminate classifier inconsistent guarantee eliminate separate arrive space return apply remain allow separate eliminate lead eliminate g hx gx x xy find request capture label request unlabele abundance unlabele potentially increase whether though intuitively however comparison section ignore logarithmic
define define project feasible backtracking project detail also implementation completion variation minimization backtrack search signal formulate unconstraine unconstrained generalize difference descent feasible differentiable solve backtrack implementation graph signal variation signal initialize stop backtrack search cost decomposition matrix discuss general focus whose corresponding minimize nuclear norm study connection f tr tr r cyclic total word rank naturally f frobenius equivalence variation relate nuclear norm nuclear potentially minimize rewrite shift insufficient cause return quantity vector smooth also quantity signal belong subspace span coefficient follow spectral signal belong recovery graph shift cyclic graph shift identity robust shift index matrix anomaly
learn visually especially ba er graph ba evaluate recover position testing average ten three graph competitive rbf ba graph c c ex measure ex precision well understand behavior number gaussian rbf behind behavior uniform entry decrease opposite increase trace tend zero decrease implicitly laplacian b lead ratio dominate fidelity ratio fig learn evaluate edge increase graph intersect reach peak keep drop edge similar graph match combination random rbf graph investigate present respectively different number signal initially deviation signal deviation random temperature measuring show period I month
dictionary middle threshold impose refined digit efficacy model half layer digits layer row furthermore refinement much excellent interpolation result upper digit reconstruction htbp htbp analyze face category deep dictionary infer discuss map fig third layer similar mnist interpolation dictionary max pooling part accurately face detail second layer dictionary deep convolutional testing via project layer deconvolution
provide rate give hoeffding rate regardless empirical risk classical frequently use theory generalization rademacher complexity inequality value domain difference eq show condition generalize domain domain condition furthermore similarly inequality coincide one match inequality manner generalization number vc dimension distribution k tn q show eq characteristic denote recall evident q q another clear result hoeffding range proof ready prove theorem law iterate z respect analyze one representative measure two meanwhile domain adaptation hoeffding
dependence recursive specification symbol note procedure sufficiently long elsewhere include sake seek symbolic sense infinity within string derivative turn reach general state merging process simultaneously symbolic derivative fail already encounter create state find match merge two string crucial seek stre right extension carry split ensure find error call lf observe consist trace set consist string large depth infer identify convex hull mapping qx recursively symbolic q qx terminate necessary connectivity initial run direct symbol read move arc generate normalization recursive synchronization step infer short large approximation upper step approximate arc normalization traversal traversal counting assume row state separate multiple however state algorithm modify two identical discuss carry efficiently use space complexity assume computation involve encode string inspection string identification traversal count normalization arc sum bound symbolic distinguished complete refer probably approximately pac said target output metric language class language efficiently pac learnable establish probabilistic appropriate strongly symbolic denote g define therefore string sequence satisfy triangular define symbolic derivative combination correspond class learnable pac finite long generate estimate runtime initial identify extension occurrence visit right
interesting cognitive posteriori hmm estimate develop problem relate reduction solution determine active reduction examine uncertainty reduction solution inference hide base improve information obtain costly beneficial inference notion active active optimize model principle specifically assess analytical simulation excellent within rest active respectively analytical finding inference follow discuss relationship inference direction realization noisy alphabet write give general overlap average overlap request
pyramid l cnn acc fc ap yes ap fc baseline yes baseline fc fc fc yes pool yes fc yes baseline acc part fc yes fc yes fc yes yes fc yes mit pt l description ft bb cnn fc yes baseline fc yes baseline fc baseline fc yes fc yes fc sp fc yes fc fc yes baseline bb cnn map yes yes ta fc mlp yes fc yes fc art ft plane car cat
produce similarity kernel function look angular project angular nonlinearity instance space similarity representation robustness induce similarity I instance vary discrete normalize anchor representation variation robustness objective function raw distance towards thus dominate objective kind fisher differ compare induce first specify proximity statistical generalize system metric coordinate induce differential map equation differential metric differential metric width difference
encourage logistic technique dc programming concave pointwise compute non zero entry complexity leverage simplex control sparsity heuristic justify perform reduce vector show word whenever enough operation use bounding box positive window box construct window htbp dataset bias htbp c c car cat cover cover multiple design bag
frame music iii iv chemical services environmental school bss use matrix remove word letter default try score full select dataset bss leverage score leverage bss list occurring word supervise bss score five five table five category belong bss document music bss select closely good error selection dataset result fig bss leverage score selection full
explore material describe model nmf nonnegative complexity technique magnitude audio signal simplify take seek approximately decompose factored form index prototype spectra combine activation equivalent maximize eq unobserved observe fill white thick f thick f edge z pt right
dot mathematically tractable dot draw attribute enyi latent process inference author count include operate dynamic temporal stochastic et al temporal extension sbm memberships sbm main author change specifie combination gibbs sampling algorithm demand base dynamic inference procedure hyperparameter estimation investigate approximation priori estimate discuss iid follow scale distribution gaussian observation variance dynamic available noisy evolution linear dynamic transition apply previous gaussian entry process matrix noise unlike construction evolve correlate manner state either estimate system observation define graphical linearity kalman
stage diagram explain e step classify simple actor interact complex actor person interact web page weight assign actor type average secondary scenario transaction activity either entirely count counting case interface interface entity user interface complexity htb complex actor count many actor degree add product multiply weighting add adjust value factor table assign project essential
derivative output obtain complex composition form dag variable argument implicit argument dag dependency variable bipartite acyclic evaluated iteratively sort depend come scalar network interested derivative output dag modify remove make evaluate dag likewise respect recursively accumulate feature layer easy integrate variant visual pre easily implementation cpu gpu version contain require gpu available cnn implementations capable cnn language google convnet project architecture file convnet somewhat individual block matlab gpu convenient toolbox implement cnn seem general purpose framework toolbox several computation library dependency thank
time inferior lstm cope lag transform rnn output sequence via linear analytically train therefore suffer lstm synthetic complex fail dependency optimize attempt hessian adapt rnn allow rnn solve term lag performance number great network training multiplicative rnn multiplicative unit factor way reduce training day graphic filter lag strictly temporal stack
trace penalization determine pair start op ij ij satisfied include solve singular unfortunately particular replace efficient algorithm pca significant relaxation heuristic truncate power psd provide accurate solution rip algorithm case step truncation denote truncation consist tolerance ab ab ab b b ti new active find small termination algorithm algorithm instance add type method maximum suboptimal component reduce thereby select solve matrix small computational bottleneck condition hold evaluation thin rip hold multiplication total cost reach complexity would warm run keep track fast reliable solver discussion go beyond report low rank theoretical assess several pca numerically add mean error dimension numerically take measure square constrain form norm projection noise fast constrained atom confirm summarize dimension trace depend increase confirm capture certainly set correctly solution z kk second set theoretical atom overlap roughly dimension regularizer sum atom bottom right shape curve support atom overlap case consistently outperform suggest linear support factored covariance form add k right make psd form method compare basic presence problem
small entail neighborhood dd state two proceed random norm hold sub lemma exponential gaussian norm I tt jk ik jk jk jk deduce constant obtain e hence last component wise cauchy schwarz derivation yield e argument lipschitz set assume implicitly yet misspecification application investigate leibl principle dimensional asymptotic expansion principle misspecification generalize dimension suggest logarithmic dimensionality complexity establish general misspecification kullback principle rapid advance modern technology throughput set genetic fmri functional datum economic finance frequently make contribute
trait outcome trait parameter estimate rapidly trait brownian along tree trait assess uncertainty space development different along branch thus naturally development area lead model dependent evolve variable aspect explicitly tailor assess trait history assess combine threshold trait unobserve brownian trait spend current interpretation outcome underlie represent effect number genetic factor evolution usually model brownian diffusion trait assess diffusion threshold create continuous development trait evolution trait presence possible importantly trait correlation control trait share evolutionary trait analogy inference association molecular pi less development unobserved tree degree degree independence refer
matrix level remove representation alternate projection see user heavily speed recovery complexity rank e number stage drastically number iteration require convex denote approximation thresholding initial km ts remain result correctness incoherent row respectively row unique sparse e ensure recover recovery result tight constant sparsity sparsity exceed sparsity
near optimality class problem translate alternative cover every around near optimality dimension fact another interesting entire drastically arm many application notably complex position portfolio management switch might formally node bound lem number thm lead exist arm context policy process notice extension observable mdps ergodicity mdp tuple p mapping pair action find long formally receive parameterize policy induce mt relate transition policy initial sequence ergodic reward goal find policy coincide mdp cover sect notably correspond search space mdp special context sect evolve time determine reward state history translate sect sect mdps sect simply advantage work policy finite mdps thm sub regret average transform accumulate readily continuous restrictive size extend comparison use strong reward globally finite
signal dominant font font legend style font system challenge extract analysis conventional perform maximum mutual information hmm clean read news corpus clean multi condition hmm technique know well alignment hmm implementation maxout accordance extract static
construction asymptotically optimal furthermore discount mdps problem keep denote matrix last action dynamic subroutine take action inexact nature near tv ta x state result meet special case assumption rate mean rate reduce study variance question directly ucb concern equation regular solution cost optimality action exist abuse call sometimes necessary guarantee condition continuity state boundedness
imply recursive formula look obtain calculate resort iii treating case ii closed form inequality hold calculate resort case treat proposition microsoft lin minimization concave saddle propose dual dual primal perform accelerate parallel extension weight theoretically arise often regularize minimization convex loss regularization predictor solve problem associate label linear vector machine hinge loss regularize regression obtain linear problem lasso regularize erm book especially interest develop algorithm case evaluating incremental operate component extensive incremental method gradient batch iteration complexity precision quantify complexity
functional independence graph interest definite inner product operator precision element correspond absence indicate convenience abuse notation edge present dependency prior matrix scaling evaluate hold distribution multivariate wishart wishart coincide importantly doubly intractable return implication wish perform gaussian around outline approximate wishart update either clique substantial making
automate start linear association follow learn use observe direction source spectra vary invariant feature spectral variation hundred thousand tf point source aggregate observation formulation lead problem problematic firstly case prohibitive secondly clear regression natural speech common source content frequency source long period devise recently use compose association noise ref inverse extend tf posterior direction gaussian covariance show perform localization separation consume runtime head mechanism collect association position keep position experiment head onto theoretical introduce elegant associated truth marker manually hold front camera place horizontal image head camera setup locate accurate direction quantify propose organized section sound localization mapping onto sound extend input section obtain localization source draw direction head setup record series direction sound sound center frame angle capture single sound static source setup static sound source sound subset possibly sound source general much set acoustic embed
simple classifier possibility classifier classifier use nlp consider sentiment grow small manually label occurrence compound sentiment table test outperform specific purpose specific book see opinion mining review e actor camera lot target challenge sentiment different language specific sentiment analysis english reason english visit internet internet http en wikipedia language complexity country even country widely develop sentiment lack sentiment dataset sentiment end book rating star comprehensive experiment expand set also extract domain sentiment training help space reproduce
accord trial belong sx sx classification discriminate belong source domain experience decode subject subject variability across situation share multiple train underlying idea try capture try instability diversity combine prediction ensemble learn decode set subject partition training one represent diversity train
issue extensive quantile return examine use sample period median box version throughout directional stock stock quantile stock bootstrap confidence replicate quantile lag low lag mean likely positive next half box significant quantile stock median return return lag forecast forecast quantile stock stock return note result first lag cross lag lag less likely half figure box stock return stock previous trend absolute imply quantile increase loss year significant quantile gain next year figure box lag quantile
fact follow hold follow innovation associate latter product appendix value satisfy integer kullback distribution form n duality recursively note nh also bias coin flip probability since measurable conditionally history repeatedly unconditional n concentration size lr kl condition thm fact assume imply rule gain
vc coordinate coordinate coordinate see flip immediately close desire build contain union complete inside contain collection queue comprise collection projection queue collection jump onto swap produce empty complement usual proceed iteratively canonical class consider maximum eventually within check whether project onto contain project retain recall collection height connect sect filter soon clear cube cube construct maximum filter subsequent early vc binary cube vc maximum vc yield simple picture embed compression trivial serve vc class table nd project maximum vc cube argument straightforward complement vc cube union maximum possibility two verify component component must apart diameter check vertex verify symmetry cube
acc rmse acc auc acc acc acc auc acc rmse auc acc rmse pt department university university calibrate calibrate prediction particularly machine present parametric algorithm learn apply step learn make wide select scoring measure second advantage exist calibration convert discriminative probabilistic posterior svm calibration calibration prediction I assumption method nb reduce remainder
minimal candidate program already trace minimal intermediate candidate minimal p either element language return trace result last last p last since element set empty verification engine language element stop yet minimal single progress parse infinitely case long progress intermediate nm mp fp p terminate program synthesis engine verification engine return history synthesis history program amenable long synthesis history synthesis arbitrary history powerful language program language second language language candidate program language form consist
field context aware interaction assume preference interaction interaction way include argue model train recommendation preference modeling perspective item preference focus interact interact additional justification compatibility interaction classic influence interact item dimension user role rank interaction additional context interact context interaction help well context follow scheme early indicate text green layer either preference dimension thing dimension one novel novel context preference divide user reweighte context dependent weight include simultaneously preference dependent interaction strongly affect interaction solely minor item relation context recommendation bias certain bias composite reduce interaction treat reduced way model context omit dimension interaction set reach scope nonetheless cm user bias item bias interaction six five perform traditional case interaction good case intuitively sound perform assumption preference model second interestingly difference heavily noise member much reduce pairwise pairwise
result pass randomly normalised check sure respectively linear give predictor normalise source configuration already prove correlation mixing normalise make stepsize fig define statistic generalize accommodate white bss mixture assume
identically past expect accumulate exploration new arm arm arm know optimal slot high difference accumulate reward good arm profile decide instant accumulate reward regret balance tradeoff show arm accumulate accumulate loss unbounde logarithm ucb mab several arm ucb prove file cache cost algorithm order prove order special number play content place content cache content popularity advance file cache user content cache memory cache user readily cache want rate request cache otherwise directly carry user example background user request either cache observe request store cache cache memory unit file file file divide instantaneous demand file request file period normalise maximum serve demand bound support popularity request cache file consider reward unit file gain user
cm certain transform reaction reaction diffusion j measure cm fractional cm j reaction journal cm calculus fractional calculus transform instability multi medium study apply mathematics growth journal fractional build output mechanism reaction reaction production physical technique describe tail law stochastic
access regard past observe policy execute round distribution nature choose th policy round round maker dm select accord nature dm issue mdp accurate value expect
propose new var call hierarchical selection regularizer provide shift obtain generally nonzero lag motivate goal interpretable model flexible computationally method lag lag procedure early attempt square criterion develop grow lag hold size work information criterion tend whereas tend despite tool lag practitioner approach typically lag reduce economic justification dynamic component forecast false specification add accord predictive inherently offer var specification utilize aic lag selection approach component specification
appropriate suppose allow grow also recall close neighbor index recall close multiple rate want validate rhs rewrite sign close pr pr inclusion rhs q rewrite use substitute expression rhs rhs eq empirical validation bind empirical term multiply term convert linearity apply know length range many term partition subset hoeffding place probability draw use
criterion contaminate outli estimate laplace skew approach noisy spurious cluster contaminate approach asymmetric aforementioned robust cluster model cluster use help point spurious noise discuss contaminate preferable herein contaminate
prior plausible accept parameter abc accept bayes estimator sufficient statistic practice statistic determine rely sufficient low problematic continuous compare summary set p sx generate set accept discard accept I I tolerance evident simulation study implement hasting mh target approximate distribution highly draw upon mh proposal incorporate smc technique propose partial control utilize rejection mutation incorporation mutation improve abc ar
exploration paper aware recommender validate series name compute accord situation recommend desire read interesting current case orient avoid ucb exp ucb mobile paper review involve illustrate conclude point bandit algorithm consider recommendation work dedicated propose base experience preference user preference period association concept profile experience combine describe three complement highly book social among user combine spatio quality recommendation device explicit rating rating proximity natural mobile recommender people read behaviour predict combined preference content provide mobile author
weight available matlab author home denote entry th standard entry remain singular row leverage score leverage replace coherence coherence sampling attain whose th equal row q leverage first two prove row rank leverage weight therefore leverage standard minimization complexity norm nuclear problem nuclear approximate study literature uniform later k th probability exact coherence norm model completion complete whose accord align jj devise paper tb partially matrix initialize perturb let matrix max factorization model alternate poor coordinate
section conditional hidden output face total entry py positive arise reinforcement observable namely state process bellman manifold conditional come feedforward arbitrarily approximate arbitrarily feedforward output sigmoid model arbitrarily feedforward network hide unit feedforward linear threshold example feedforward input output follow boolean function f k km class deterministic distribution arbitrarily hide unit k km paper give description capability conditional boltzmann machine relate restrict boltzmann theoretical rbms trivial study unit imply finitely choice bias quality input unit kullback form universal tight depend attain complexity star improvement worth open analytic integral upper bound expectation draw
lc depict manifold may solve sum one locality coding albeit intrinsic purpose embed manifold aside purpose embed exploit codebook learning explain dictionary analytic locality code represent point surface rank four square b locality code contribute neighbor get manifold initialization dictionary I ip ni lc lc f f dictionary label dictionary tie generate fed base like classification generate code code atom jx dirac efficient utilize residual residual eq class could like preliminary high compare aforementioned alternative solve mean combination represent project close point follow principle completeness closely adjacent intrinsic atom show formulation code solve scope coding efficient learning manifold n elaborate usually employ obtain common depict learn write convexity inspire set break update atom eq minimize r project manifold detail pseudo atom dominant visually informative
choose optimize divide program run incremental hold exception optimize rate code gradient backward optimize couple day smoothed model avoid knowledge internal nonetheless smooth I broad tuple token token distribution number child independently smooth support token manner low log scope test word length replace j word length word replace j j author microsoft research amenable statistical possibility learn become task level potential first code rely primarily integrate offer suggestion massive source public indeed show even improve idea suggest basis code enforce programming language code representation purpose visualization hope learn program programming program task develop tool improve
predict situation category within invariance preserve consistency loss state hard q bootstrap variant term allow predict object unlabele handwritten face train consistency refer describe consistency soft bootstrap reconstruction detailed section mnist handwritten digits degree figure vary label training model mini sgd architecture linear work soft consistency network layer initialize could quickly bootstrappe phase bootstrapping provide significant bootstrap nearly soft
scene affine transformation human mesh smooth generator representation baseline object lot attribute minor lack c illustration run image buffer mesh image close complicated program combine scene generator scene affine scene generator scene configuration induce mid simulator engine formulate interpretation inverting simulator observation widely source numerous graphic simulator drive put rich world contour recent mid
transformation neuron neuron neuron neuron neuron proceed achieve proportion explain compare regression performance rest neural backpropagation compare linear realistic significantly high one compare explain look plot model fig residual fit term besides normal two big curve end plot partial residual effect rest assumption explanatory establish log transformation improve quality still violate verify novel dataset b backpropagation begin analysis dataset find nine attribute plot value substantial drop indicate big
simplify imputation average estimate complete datum variance variation simplify create member
efficiently satisfy require perfectly predict unit intersection homogeneous normal hide incoming weight since input hence intersection hypothesis feed hide single hypothesis normal pac learnable show feed minimize class predictor different weight weight many hence feed capture behavior capture network expressive indeed know network minimize even unit
historical sensor network task without important reason dynamic change rapidly desirable operate environment collect water exploratory initially operate system newly acquire environment researcher accurate meanwhile task prescribe learn low system sensor unable correlation ensure connectivity requirement fulfil sensor hardware resource discover use communication internet introduce motivation support make learn ai limit human intervention drawback consider use technique sensor framework considerable hypothesis consensus trade specifically requirement system employ centralized resource unit speak intend capability bound full control process past decade advanced machine survey machine discuss machine wireless ad hoc network application tree communication specialized survey learn development outli proper action take meanwhile discuss intelligence challenge fusion scheduling intelligence branch focus inspire fuzzy survey decide instead variety strength provide comprehensive roughly unsupervised distinction survey way learn work discuss learn challenge encourage lastly survey classify compare effort provide researcher interested explore research introduce section review effort address localization medium essential determine enhance behavior requirement security specialize difficulty section comparative guide introduction wireless sensor sensor characterize collection create expert recognize rich pattern understand beneficial machine numerous flexibility benefit provide concept adopt context exist intend structure algorithm reinforcement supervise learn learn classify group cluster investigate third include reinforcement agent interact environment online machine characteristic supervise learn hybrid term supervise aim strength category adopt section omit please therein thorough discussion predefine input output represent parameter fact extensively medium security call output
forecast contribute monitoring protocol wind future next author discussion environment project generation wind author analyze introduce analyze load wind physics properly adapt multivariate comprise velocity power fluctuation produce wind langevin drive wind
unimodal operator limit issue adopt alternative generative neural estimator transition alternate main landscape although restrict boltzmann softmax fine specific autoencoder begin present overview include autoencoder autoencoder generative autoencoder autoencoder training autoencoder feed neural aim minimize input
way partition possible modal cluster scenario adopt use wu generate partition either u u last partition identify choice consistent parameter importantly cluster observation close true simple bayesian cart need one ht u show identify interesting deviation mean result count generation randomness reflect quite realistic heterogeneous datum form scheme overlap partition u u u
v v v stroke v v v v v v v v v v v v v v v v v v v v v v v v v v stroke v v v v v v v v v stroke v v v v v v v v v v v stroke v v v stroke v v v v stroke v v v v v v v v v v v v v v v v v v v v v v v stroke v v v v v v v v v v v v v stroke v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v stroke v v v v v v v stroke v v v v v v v v v v v v v v v v v v v stroke v v v v v v v v stroke v v v v v v v stroke v v v v stroke v v v v v v v v v stroke v v v v v v v v v v v v v v v v v v v v v v stroke v v v v v v v v v v v v v v v v v stroke v v v v v v v v v v v v v v v v v v v v v v v v stroke v v v v v v v v v v stroke v v v v stroke v v v v v v v v v v v stroke v v v v v v v v v v v v v v v v v v v v v v v stroke v v v v v v v v v v v v v v v stroke v v v v v v v v v v v v stroke v v v v stroke v stroke lt v v v v v v v v v v v v v v v v v v v stroke v v v v v v v stroke v v v v v v v v v v v v v v v v v v v v v v v v v v v v v stroke v v v v v v stroke v v v v v v v v v v v v v v v v v v v stroke v v v v v v v v v v v v v v v v v v stroke v v stroke v v v v v v v v v v v v v v v stroke v v v v v v v v v v v stroke v v v v v v v v v v v v stroke v v v v v v stroke v v v v v v v stroke v v v v v v v v v stroke v v stroke ltb def exch exch exch def roll exch def exch mul mul sub mul mod ifelse ifelse ifelse ifelse ifelse def constrain lt exch ifelse def copy exch add constrain roll mul exch constrain roll mul exch add constrain roll def exch roll exch roll exch def rgb exch mul exch exch constrain roll mul mul add exch mul roll exch add exch mul exch roll ifelse ifelse def def gidx gidx gidx gidx loop def gidx get sub gidx gidx get def gidx gidx gidx sub add def gidx gidx sub gidx get sub mul gidx gidx get gidx sub mul def gidx sub le gidx gidx ifelse def def def pm mul ifelse mul def def pm exch def stroke constrain exch cf constrain exch def ifelse pm pm exp def ifelse ltb ltb stroke ltb stroke ltb stroke ltb ltb r v stroke ltb stroke ltb stroke stroke ltb stroke ltb v ltb stroke ltb ltb ltb lt v lt v v v v v stroke lt v stroke v v v v v v stroke exch exch exch def mul roll def sub mul mul mul mod def ifelse ifelse ifelse ifelse ifelse def constrain lt exch ifelse def copy mul exch mul constrain roll copy mul mul roll mul exch constrain roll rgb exch roll exch roll exch exch mul exch exch constrain roll copy mul exch mul add exch mul constrain roll mul mul mul roll def eq ifelse ifelse ifelse def gidx gidx gidx def gidx get gidx gidx gidx gidx gidx gidx get gidx sub def gidx gidx get gidx mul add def gidx gidx gidx gidx def ifelse def ifelse pm gamma stroke exch def cf constrain exch cf exch cf constrain ifelse stroke ifelse ltb ltb stroke stroke ltb v stroke ltb v stroke ltb ltb stroke stroke ltb r stroke ltb stroke ltb stroke stroke ltb stroke v ltb r stroke ltb v stroke ltb v stroke ltb v stroke stroke ltb stroke stroke ltb stroke v ltb stroke ltb ltb ltb lt v v v v v v v v stroke v v v v stroke v v v v v v v v v v v v v v v v v v stroke v v v v v v
arguably happen solve spurious fw employ small time consume large sampling convenient cardinality across analyze minimization linear gap
inform learner select exp exploit factor induce feedback direct probability term dominate current computing dominating inform dominating dominate minimal logarithmic yet another proving operate feedback another call internal randomization ifelse b dominate tw tp ti ti ti see algorithm inform run time compute direct induce index since adaptively dominate choice feedback variable cause tuning reason exp slight describe appendix trick direct satisfie trick importantly graph whenever dominate system step exp dominating set regret exp combine bind expect inform set moreover quick comparison reveal feedback advantage inform symmetric factor derive informed direct theorem inform also bound unable bound exp exponentially weight programming action fact program name regret bound utilize e instead use weight action oppose decrease merely affect ifelse generate loss generate simplex w tw w ti p ti I ti ti tw sufficiently notation
reduce step communication ignore last globally ignore involve scalar al lin al demonstrate elimination shrink technique heuristic condition eliminate eliminate multiplier keep optimality eliminate primary behind shrink contribute working variable sample eliminate present condition previously lin shrink overhead condition possible previously eliminate eventually definition conservative approach decide likely essence difficult execute lin shrink reasoning value execute shrink discussion heuristic shrink reconstruction ensure previously eliminate positive step update gradient reconstruction shrinking correspond multiple eliminate gradient sample communication negligible heuristic
scope take largely literature generators generator interpretable different drawing generator last induced generator game payoff dispersion player arm instance match iteration take cpu year upon conduct reward conclusion sample mean confident conclusion way check experiment run experiment confident percent interval fact serve sufficient evidence repeat verify summary statistic experiment take day computer time bootstrappe datum experiment point subsample create form confidence distribution subsampling would diversity moderate distribution would explain bootstrappe summary also check distribution gaussian mean distribution ks vertical vertical analysis unless significant monotonic pair game relationship variable interested study convergence issue appear converge frequency exhibit whether early empirical profile mixed establish nash exact checking multinomial unfortunately situation expensive fail large balanced typically close calibrate incomplete ground truth positive vary roc concept distribution metric quality dominate quantile high quantile maximize dominate value attain probabilistic check probabilistic perform curve everywhere dominate latter higher shift discuss begin fundamental action mix formally record I average reward detail begin accord generator explore examine game across equilibria action consider different raw iteration game reward
kl k ff inequality give summing give knn cauchy note logistic accord convergence depend rate e constant c since k k hence n ns ns argument complete class ahead indicate point significant speed set follow large intuition taylor origin run benefit compare randomly datum generate dimension keep vary median iteration algorithm quite per almost linearly size degree speedup regularization dominate second regularization
book sake occur change book bid ask count resp mid resp order bid move price limit order resp bid move count bid move stress price move correspond occurrence market bid place bid ask distinguish handle consider characterize intensity kernel code intensity bid mid unchanged move intensity estimate book impose bid ask symmetry p estimation estimation scale method retrieve indeed unstable allow retrieve symmetry provide http www book keep year file list limit well ask book micro second precision thus compute ask bid market limit limit mid move shall book depend size asset large small asset asset summarize c simulation
counterpart regression ignore high persistence state encounter scenario simulation ms ms lin viterbi relate ms lin predict regime ms result exploit strength flexible ms strength inferential particular ease interactive formulation selection explore detail along validate aic formulation local class less price outperform need need regime apparent plot regime address fail short day variation interesting explore regime persistent autoregressive capturing
consider infection incorporate period parametric fit able infection epidemic individual remain arguably know epidemic call infect individual state contact individual infect remove infection proportional individual multiply give action individual recover take despite g disease majority assignment parameter advantage manifold avoid result fit often simplicity parametric often tend homogeneity mix well specify non allow
transform shape fill bl dot cm dot left l cm l l cycle leave right node cm l node distance h scale style shape circle line pt inner fill bl dots end dot style shape circle line width cm bl dot h cm r cm h node p rs cm probability distribution bottom top composition originally represent layer lk n lp define conditional feedforward output feedforward bias clearly conditional
bridge target conditionally unbiased diffusion bridge bridge algorithm bridge one simulate intersect simulate bridge go diffusion soon equal independent approximate mcmc nearly approach right calculate expectation produce diffusion mcmc sample simulate euler discretization avoid simulate multi version transform diffusion bridge simulation simulation exact even computational approximate coupling simulate approximate mix bridge simulation paper dimensional generalize bridge method process give wiener bridge simulate diffusion term see standard decide coupling process follow discuss meet manner meet simplify assumption diffusion ignore influence sufficiently assume diffusion dimensional wiener wiener brownian start brownian onto plane orthogonal pass process coupling occur q find alternative brownian motion plane plane
thus condition learner consider instance j learner index training equal tree last equation learner vice versa flip well exponential cut plane cutting constraint solution original start add qp continue program cut major bottleneck lie combinatorial combinatorial solve briefly optimization associate speed instances break maximization restrict order instance optimization problem compute search reader analysis decrease decrease q material initialize learn optimize range else cache access meet weak structured svm cut weight cascade adjust node partial auc cut plane mn problem svm violate learner cut plane weak consist step learn decision fast bin cost total sample call structural cutting cost scale linearly algorithm cost cut plane complexity spend train learner table compare complexity adaboost discuss adaboost next discussion adaboost detector identical adaboost adaboost play adaboost calculate u indicator minimal point subset instance tight instance position
wide cluster smoothness kde make construction powerful algorithm mode dataset laplacian mode mnist acc pt n dataset mnist handwritten digit object vary angle topic semantic category appear keep statistic collect pixel randomly special laplacian search spectral implementation nmf cluster walk mode normalize several graph laplacian graph build let select run respective hyperparameter acc run clear algorithm smooth superior demonstrate importance cluster perform poorly laplacian mode close good criterion find exist range hyperparameter homotopy decrease c
previous retained carry separately central activity pattern among select accord span area eeg signal ec investigate separately outcome perform test feature provide framework preprocesse describe section psd previously psd measure single region extract mahalanobis distance detail carry test condition preliminary report false maps channel psd ec feature j eeg head code axis map psd feature ec already eeg study alpha evidence close brain reflect genetic ec well
knowledge rate convergence strong analysis fail course limit guarantee final bind study believe intuition source improve standard thought weight non paragraph average result average without strong assumption algorithm utilize grow oppose unweighted set nice standard effectively control favorable one problem work deviation interaction algorithm predictor index potential author differ algorithm plausible weight weight justify weight single sparse feature weight magnitude easy sub strictly
gate quantum fix complexity serious advantage perceptron process superposition superposition information extract via quantum quantum discuss introduce variation quantum perceptron weight read consistent digit binary w u x control control parameter additional sign shift iteratively adjust
motivate improve termination beneficial decrease accomplish project vary learner relevance explanatory rely automate constitute methodology specification capability dependencie advantageous explanatory neighbor explanatory accord nearby probability weight intuitively value class different write increase positivity relevance n w experiment strength form weakly stock price indicate substantial furthermore boost procedure ensemble section identify important predictor consideration project demonstrate explanatory purpose visualization particular low toward indicate explanatory high predictive variable vary efficacy stock return summary explanatory b b b great period consideration
neural network without precisely get large boost supervise fine tuning conclusion put optimisation da train low yet classification minima autoencoder manifold though loss autoencoder way primarily stack would improvement diverse representation composite autoencoder proceed local explore dropout comment supplementary material sampling change batch try variant value setup unit first method much da experiment evidence denoise sequence success supervise tuning completeness try encoder layer size
observation spend require size respectively correspond analyze author interest mean unique note enyi I n htbp confidence level obtain interval summarize statistic interval wide confidence day presence see function despite significance ci ci ci ci ci first eq day fact notice day possible day dropping outlier approximate new outlier day comparison n detect lack study examine cc pdf fig
hermitian henceforth bs decode bss overlap sense sample b nonempty bss bs bss bs collect bss full practice bss whose sample helpful intend traffic simplest bs traffic diagonal sec bss overlap bss belong jointly user network architecture proximity site formation may gain pattern costly intra traffic hierarchy mode mcp simplicity exposition sec traffic proportional nonzero mcp favor position inspire advance mmse entry absolute discussion seek f control incur nonconvex norm absolute bs amount traffic site bs user incur even entry decoding
keep triangular nonnegative distribution low triangular loading concern continue prior property row inferential setting comprise factor row associate factor comprise nonnegative
initialize small train phase call train algorithm know item description pass along return weight line perceptron element initialize normal mean perceptron learning rate store yet initialize line train detect fail convergence detect fall specifie improvement else rate differ perceptron instead single held line perform phase perceptron refine together order ic ic backward f ic completeness train stochastic gradient backpropagation individually present conditionally line item
autoencoder validate effectiveness autoencoder useful supervise autoencoder dropout tuning noise large autoencoder purely gaussian noise mnist interaction unsupervise supervised suggest optimize stack type relative autoencoder one layer train input accomplish network object recognition autoencoder map hide encoder yield autoencoder typically square autoencoder autoencoder train reconstruct clean corrupt
support innovation global department contract er office program fa motivate whose b introduce form note equation bandwidth sparsity semi semi equation would variable extend separable rank variable surprisingly length
cut plane cp project subgradient training compare cutting method surrogate involve breast task dataset uci repository training average test use buffer length buffer perform pass c letter c cut plane task vary buffer gradient method accuracy within second even cut plane well trend measure second cut plane could achieve accuracy epoch
dynamic stay step policy content gate last directly properly expect gate regression result respectively conclude neural feedback connection reinforcement selective internal certain rapid shot stack feedforward filter feedback certain filter guess internal maxout selective internal art consider acknowledge include relate convolutional cnn neither try human strategy cnns
n sign lem fx lem immediate result moreover lem let p p proof easily need confirm consequence integrate virtue matrix solution recover furthermore let probably except result keep bound detail quasi
dataset truth actually small overcome ground truth generate type item user type truth rating user netflix yahoo music issue real dataset truth evaluation issue usually solve use reject known restrict choice time rating per meaningful comparison miss rating obviously live people full experiment mind allow update recommender policy need decomposition precisely item current unknown rating evaluation would methodology interest currently evaluate bandit bandit uniform strategy dataset provide yahoo available able rating netflix policy think methodology contribution paper
employ four mini batch sampling batch obtain ratio since tail speed incorporate several explore handle state context portfolio management finance business pt study constrain shortest consider conditional locally employ four importance incorporate procedure along utilize cost objective necessary first second mini spirit monte relate rare importance variance constrain attract lot recently learn unlike previous mostly return measure conditional
flexible analyse market goal market reason market understand mechanism objective market objective market find market aim motivate would analyse whole unlike agent measure market result paper establish strength mathematical giving aim market trading sequential optimisation connection market learn prediction market associate specifically payment unit security pay state require vector subset linearly security space discrete contain th continuous practice trade agent share security portfolio payment essence call
dnn much dnn much select hidden choose size slight frame trend frame suggest network objective theory deep counterpart act difficulty deep deep understanding compare system final component decompose hmm dnn layer dnn dnn largely substitution rate constant small component overall decrease substitution fairly quality confident matching audio component link system vocabulary dictionary capture variation tb analyze group understanding act root leave analyze hide analysis height bar bar break classification base error phone network root three base phone base phone combine set phone classification non dnn correctness spread substantial base base base error fairly base occurrence generally exhibit pattern dnn acoustic nature observe phone find model gradually across large improve expense tb performance level hmm speech decode yet address question architecture achieve describe algorithmic necessary prediction setting offer analysis aim first pattern compute forward dnn non compute fraction example plot sort sense activation help code code researcher learn hide active size unit layer equally share code perfect dispersion would flat sparsity representation axis size deep dnn layer every dnn case activation decrease deeply dnn per deep layer suggest transform compressed dispersion particular dnn fairly flat percentage typically use recognition result relatively dnn previous speech maximum training experiment dnn serve discriminative task dnn generally neural dnn acoustic drive unsupervised yield speech benchmark appear modern dnn neural approach
storing segment calculate high store update primarily still detect section heavily whether examine pruning partitioning apply prune neighbourhood call functional pruning segment neighbourhood prune functional pruning partition version functional pruning efficiency show slightly condition c part depend define segment thus q recursively return last vary firstly problem store update store empty define cost ft update candidate change least leave hand
close z since constant moreover since round therefore programming detail supervise annotation secondly solution annotate sense label annotation correspond every descriptor incorporate modify discriminative explain domain label assignment set subject link center quadratic b constant admissible due avoid class fraction ideally hard proportion problem everywhere except column make operation sec intractable dynamic programming modify constraint minimal avoid trivial objective desire incorporate multipli vector still heavily unbalanced towards deal unbalanced dataset square weight
neighbor popular underlie lsh lsh functions property domain function high formal hash mapping locality sensitive family sensitive satisfy task search query lsh provide mechanism create idea hash function meta lsh lsh need independent meta processing assign hash retrieve element whose query lsh element probability family function one query lsh query preprocesse existence lsh translate sublinear nn note lsh lsh near search dimensionality lsh widely popular present lsh scheme lsh family generate hash eq dx cumulative euclidean distance lsh part lsh
generate question sense properly ss follow condition ns argument model say assign mass strategy posterior vanish assume lemma simplify eq partial convergent since vanish ask posterior concentrate important answer suppose theorem event accord hold former consequently two get let entry min condition say give unnecessary mass together converge bayes start joint fractional joint due conjugacy integrate leave marginal model penalty complexity favor adequate provide rao
sdca minimize sdca importance sampling n option version square loss prox sgd sdca sdca sdca adopt subsection adopt algebra several world aspect dataset dataset dataset fair comparison algorithm adopt experiment prox sgd parameter estimate bound ratio sample sgd verify empirical sampling accelerate duality conduct fix seed learn measure objective gap examine generalization ability finally also gradient importance sdca uniform sgd sdca respectively test variance gradient learn dataset sgd summarize sgd sampling sgd fast rate two adopt propose importance error rate last two indicate sampling effective generalization right proximal prox sgd coordinate
existence cauchy rational adopt locality sensitive lsh lsh question clear answer literature study theoretical similarity similarity basis comparison retrieval lsh inequality framework indexing lsh popular lsh lsh work abundance binary question lsh prefer attempt various aspect example show
I c g r ht pdf monte carlo nest stop criterion ns contribution simulator chain analytical monte visible nest little accurate stop always see meaning estimation heavy bayesian originally bias introduce theoretical framework derive estimator allow law bring modification ideal infinite sum carlo way implement practically totally non parametric conditional draw hasting overcome sample markov pareto carlo heavy tail substantial budget approximately optimal variable far thank university paris energie alternative suggestion comment improve manuscript
protocol communication budget take jointly independent satisfy budget message protocol minimax understand definition instead infimum interactive protocol consequence metric entropy confirm bit problem bind tight problem refine substantially begin low geometric space capture define pack claim family distribution interactive bound q proposition interactive although exploit structure problem mean receive packing entropy distribute minimax yield factor achieve simple machine compute minimax machine result set message serve pre interactive protocol low bound receive unknown low communication budget universal constant section proof centralize sample low machine individually decentralize match ignore factor achievable machine average fusion center average technique family
function great exceed sparse starting point analysis let represent whenever restrictive ty demonstrate whose grouping constant penalize distance converge norm demonstrate sparse signal property recovery also bring solid theoretical justification let value noise identically candidate use plain loss newly behind follow possibility propose produce introduce objective
broad word include formal logic entail task lexical relation recognize pair important lexical semantic word semantic particular semantic relation classification work lexical lexical perhaps lexical concatenation concatenation six vector ease hypothesis tendency learnable context context occur context tend imply cat condition choice would make well possibility weighting choose non weighting word apply lexical semantic relation semantic might seem lexical result past past relational operational definition understand definition attempt lexical intend case exclude non lexical argue agreement lexical decision fit believe inter agreement trade definition lexical section word relation item attribute attribute fulfil item attribute fulfil fulfilled limitation lexical one speech entail noun substitute speech entail situation involve something lexical relational capture act reasonable say act entail noun address limitation one cope part speech noun something pattern could speech limitation definition relation lexical phrase entail phrase corpus successfully lexical lexical useful section preliminary inspection semantic relation systematically label entail instance relation nine lexical asymmetric symmetry apply classify word pair classify symmetric lexical relation capable symmetric asymmetric relation lexical behave behave case car car car see detail researcher apply classification lexical argue
see g rely question dependency necessarily dependencie solely data response theory use propose share analysis however rely estimate solely interpretable factor response ordinal determine several consider learner value response ordinal factor miss ordinal interpretability matrix negativity sparsity ii tag estimation ii tag question oracle concept association concept sparfa jointly association knowledge difficulty extend sparfa response correct incorrect exploitation concept ordinal sparfa estimate concept demonstrate real ordinal sparfa outperform sparfa collaborative predict learner response education interest
factor big generalization planning take way parallel randomize planning generator direction whether plan effectively factor e simulate en expert policy perform factor corrupt establish factor write neighborhood overlap agent constant sample bias pair eq action action expert estimate receive take expert expert e bias turn cause correlation count component payoff policy bias case bias action overlap contribute introduce mixture recover action q joint affect affect maximize joint performance case overlap intersection action profile lie solution joint cause wrong bind explain two omit
anomalous event denote type ii anomalous occur say hypothesis compound nature interval anomalous length small anomalous infinity detection length anomalous become anomalous anomalous successful change asymptotically successful bound distribution bound characterize candidate anomalous scale detector successfully gaussian anomalous distribution differ hypothesis correspondingly interval happen even hypothesis nonparametric distribution unknown arbitrary capture distribution
far tb evaluation consist tweet rank remain tb per evaluation task tweet triple high ir discount gain cut ndcg ndcg performance recommendation tweet predict ideal tweet evaluate formally discount cumulative measure maximum ideal give triple normalize ndcg ndcg user correspond ndcg obtain split baseline rank baseline describe factorization machine fm provide monte deviation
unique infinite number matrix row infinite solution choose similar decay fit flat generalization easy solution see angle original scaling implementation interpolation transform filter much consider horizontal filter capture symmetric invertible integrate tie filter equation transformation mean tie transformation back convenient corresponding filter training propagation distribute transform aggregated filter canonical filter experimental begin achieve follow baseline
exposition maximization expression part problem dirichlet seek maximize interpretation expand objective dirichlet guarantee strictly negligible due set select add maximize q essentially infimum fractional actual simple try unlabele weakly connect label end involve numerous high cost involve account expect sense underlie manifold comment compute eigen iterative specifically method write computation break product typically eigen pair computation iteration iterative filter complexity major atomic operation need store vector moreover structure aforementione suited package eq eigenvectors less
leibler variational change expectation expectation prior compute hold depend rewrite take term obtain nm probability task choose order combine obtain set hold eq nm use weight learner expect vote predictor multiply hand obtain vote predictor variance differ vector distribution learner use return vector computing obtain loss z dt gauss describe obtain next task transfer manner sequential information set q nm order group inside group
recovery deal measurement instance guarantee non noiseless sparse chapter volume author technique much wide ensemble finite value locally partly smooth locally solution concern stability additional partial provide guarantee correct unique guarantee source perfectly location norm matrix govern existence degenerate stable need norm minimal remarkable proposition concerned checking closed form pre define linearize definition note hypothesis set non uniquely similarly hypothesis involve empty affine uniquely affine terminology actually amount solve x appear compute close regularizers instance read svd find exhibit check sharp sufficient condition model case jx measurement rate assumption imply strong uniqueness identification theorem analyze clearly see minimal distinction theorem plain enough fact quite case minimal dominate amplitude entry achieve zero identification manifold conclusion strong guarantee fidelity account loss fidelity quadratic strictly argument prove show theorem sharp almost characterize unique obey correctly cover neither stand affine hull conclude either illustrate finite discretization operator operator make estimate require handle operator row sensitivity perturbation concrete row vector suppose model see g assumption index associate whose generalize previous machine jx state
road turn right road analogy road arrange address dimensional degree reduce must prove select direction suggest analysis order dependency exist remove second insufficient structure solution order representation term another direction require along statistically class algorithm term ica experience help underlie technique I I write section prove transpose matrix tt bi directional require start matrix hope address aspect well principal pca tool graphic parametric extract relevant minimal effort lower reveal simplify intuitive
desire answer probability set representative pick distribute multiply pick occur great claim clear gap monotonically true pick threshold large eq setting thus en inside right dominate order failure slight modification argument let degree set polynomial form iid straightforward estimate roughly speak large magnitude subspace accordingly recurrence theorem let similarly use number canonical
base want able whether importance plausibility depth e scenario calculate r plausible sense depth reverse stress relate set skew generalise family family skew distribution financial close exactly near behaviour structured multivariate briefly half introduce generalize univariate examine computation canonical skewness prove contour skew cauchy elliptical simplification construction investigate angular deviation angular measure explain quality approximation show misclassification interpretable approximation difficult give half hyperplane euclidean affine
benefit also insight approximation scan field even scan section relate power structural formulate approximation power real set describe section paper statistical genomic short read sequence refer successfully fail call alignment allow read often read soft also read produce map close expect detect structural end region genome read pair minus apart expect boundary produce minus read boundary plus read produce end entire produce read close dna dna sequencing read end dna call seq mapping read template detect genome read bind bind shape peak site roughly triangular et jump locate read plausible match kernel scale parameter width equal unknown one maximize statistic ds coverage equivalently process see effect tt ds ratio map length sequence pair map opposite orientation map read length map read variant mean deviation cause cause number variant unknown although segment target map detect consider toy field intensity read alternatively think marked mark starting read plus logic q likelihood index scan genome detect scan statistic simple product function relatively involve compound
approximately score probability contain initialize add matrix row proper size thin singular value singular order orthogonal svd row score fortunately
lemma claim multiplying element inverse q jj mn assumption together prove final note expression positive put sparsity graphical lasso oracle maximum eq q observation give sparsity equivalently suitably zero tail function put theorem estimator subsequently analyze novel minimize penalize determinant bregman
dynamic improve future poorly might reward exist knowledge efficient exploration exploitation cumulative vast majority efficient upon knowledge environment beyond design attain sample state strong difference agent cumulative controller kind regret curse level many practical prior beyond state time step machine direct reduce easily exploit factor
bag instance region region difficult say label generally determine instance formulation ki svm bag different svm instance positive combination linear simplex gradient optimize object take image bag derivative formulation combine respect calculation formulation sparse gradient compare art positive half extract sift densely select sift descriptor visual fig split extract sift codebook main purpose
dimension alone circumstance condition default optimum eigenvector eigenvector without average delay early take exclude term result reveal temporal detection false indicator track event study key movement movement constant spatial subsequently compare three scenario baseline environmental historical baseline strategy scenario compare consider environmental week data environmental day set day search environmental baseline tensor match set environmental setting compare different figure illustrate versus scenario versus result dynamic reason approach unseen environmental setting robust provide reference surveillance alarm art benchmark performance maintain alarm rate considerable extent feature introduce novel decomposition overall eigenvalue change dimension helpful
counter might purpose affect play become notion make sure simulation crucially supremum infimum objective consider objective temporal trivially quantitative vector atomic closure complement multiplication objective quantitative three game expression player determining satisfy condition checking objective extension player contain conjunction atom bound play provably fundamental long quantitative specification boolean union intersection complement strategy single game prove payoff payoff exploit different limit infimum knowledge consider next formal multidimensional payoff reduction counter counter intuition correctness formal
convex assumption convexity equivalently q continuity second strong side result note proof rate convergence generally vanish size discuss finite assume choose large since use induction complete algorithm remark let section restrictive nonsmooth start lemma find nonnegative scalar continuity follow eq summing note ik jensen eq follow boundedness instead generate subsequence large summing inequality sum side eq bound hence subsequence ki take inequality ik satisfy asymptotically condition note immediately
involve attribute often find solution attribute parameter update private setting particular predict depend heuristic come large attribute good attribute like get run range truly know principled handle private area g new mechanism handle order magnitude large continue answer minute dimensional remarkable free accuracy previous approach possible maintain distribution art private set specify subset query record record take like wise answering query answer wise record query answer like marginal query handle
apply table besides r another primary contribution divide find anchor merely low low hyperplane easily handle solver geometry partially probability solver problem plane compute thus learn hundred improve randomization rise multiple subroutine apply hull five learn latent allocation lda nmf subspace comparable generalize separability minimum hull model minimum present divide extremely original convex hull point vertex hull separable hull replace definition cone separability geometrically cone empty generator integer separability assumption cover finitely point cone generator algebraic form x ki give model also negativity allow contain rule hull extreme ray separability uniqueness constitute use actual
maximum ise variational computationally job mean isolate approach towards allow explain variational illustrated ise letter sample letter experimental valuable statistical mechanic drive provide insight behavior organization evolution protein mechanic construction
fidelity order approximate similarly residual approximation residual operator approximation would avoid high dimensional matrix construction computable computable model general residual indicator accuracy reduce order solution change basis increase surrogate dual attribute error even indicator generally frequency depend variable distribute validate surrogate enable perfectly compute inverse stochastic gaussian process generate prediction point denote infer transform reduce error contain joint distribution gp construct via training begin analytically kernel noise covariance kernel geometrically assume generate treat arise prediction variable n derivation expression q likelihood component account gp play crucial account I represent incur employ indicator therefore interpret quantify decrease increase interpret correction due high employ hand reduce employ indicator albeit employ include due lack I pa discretization k kk basis constitute polynomial dependent center radial basis domain employ approach vector hyperparameter affect compute maximum hyperparameter identify remove indicator kernel q graphic axis axis cs cs axis axis axis cs node cs cs axis axis axis block nine input basis reduce section introduce experiment
computationally furthermore modular likelihood choose need spatial choice pareto distribution present advantage choice modeling paper see lower low another choice marginal base extreme likelihood present appeal suited realization extreme year unobserved site observe implement within framework main spatial quantile planning extent simulate km km regular spatial predict method account covariate scientific statistical recent year various model extreme extreme distribution suit quantify physical argue model modeling explore pareto vast framework appear level
half cluster belong belong expect involve four report model requirement mostly force triple appear big might heterogeneity low region indicate possible direction see section region intensity estimate posterior background express reduce considerable contextual sensitivity detail test value much sensitive implement alternative represent association event observe cluster distribute figure value denote number pairwise value association suggest current reduce specie many measure literature interpretable influence section step configuration assess section time run intel processor design random model behaviour project strong prior estimate meaningful context accordance contextual see posterior design cluster complementary applicable context stanford biological context dissimilarity different result specie carefully consider computational aspect consider problem mh
node sum mkl I consist along path call path sum leaf layer consist combination embed atomic path color atomic layer
informative behind many associate permutation describe represent think subspace whole decide fit distribution covariance power associate independent functional gaussian close capture notational drop give graph compute run matrix permutation solve big open graph summary power discriminate effective graph encounter call distribution never think distribution analogy object intuition adjacency gaussians design kernel ensure positive previously satisfy property study semidefinite overall adjacency summarize
ascent rewrite kind objective function note building characterization degenerate completeness selective vector expect occur let lyapunov point lyapunov q negative quick iff negative assume degeneracy probability semi occur selective objective briefly like tensor rule power rule slide threshold neuron slide neuron rewrite define lagrange expansion orthogonal lagrange expansion eigenvector objective attempt degeneracy maxima eigenvector look successively subtract repeat orthogonal local optima tensor correspond optima stage need ensure ascent
sort suffice paper overview prior work really approximate furthermore write suppose dimension sketch column goal work attack splitting implicitly compare adopt make comparison explicit analyze separately section additionally generalize split r kk preserve sketch side constrain suffice side dependent note linearity trace step property trace lemma I bind equation immediately low requirement k f lemma sketch broad error next rank preserve sketch side k k specifically rank handle term starting analogous long eigenvector f f step trace semidefinite schwarz symmetric combine equation recall framework analyze dimensionality start sketch simply onto top rotation truncate vector claim suffice low enough appendix take robust computation often pass limit gain substantial let condition semidefinite q final small
therefore irrelevant relevant htbp evaluating production respective input parameter production column separate pc contribution description generate correlate independently varied input contribution correlate varied independently case independently generate variable specification remain candidate refer set efficiency obtain include input production concern adopt interval simulation try time trial note train full report result htbp parameter follow recommendation al significance comparative use begin full efficiency significance without generality include production
long context propose layer specifically design dependency nonlinearity gradient vanish connected time recurrent pattern suffer rapidly time removing keep constant allow period precisely efficiently recurrent would never vanish gradient variation type memory overview diagonal connection unit differ recurrent achieve retain topic large besides argue learn recurrent cache
error order element finite error accurate accurate inaccurate error table low estimating location generate operate compatible location center project precisely away boundary five away boundary step away mc truncate become increasingly implicit implicit sampling importance generate mode via sample algebraic implicit solving error define norm parameter
individual future mathematical prediction system reliable system may major yet lack indeed obstacle besides collective behavior challenge complex attract lot attention prediction estimate link help biological network protein fact recommendation spurious sound design recommendation irrelevant reliability identify noisy progress largely field accordingly link researcher apply field ref
cycle nucleotide normalization calculate length expand respectively formula separately variance cycle sequence q follow mean cycle increase linearly sequence incorporation extra evaluated value nucleotide incorporation complete cycle derivative vanish cycle nucleotide nucleotide incorporation nucleotide incorporation numerical integer expansion computer continuous calculate evident nucleotide incorporation accurately normal variance
optimization solution I algorithm accuracy output efficiently cope dimensional refer pc algorithm recover set approximate let solution lead solution imply x ta denote modify method reduce hessian method solution line
university california berkeley center education pt berkeley edu interest due streaming energy end use profile principal pca classical however introduce family generalization kl mahalanobis bregman come interested generalize view extend theory property discuss end
cc sgd variety relatively find high dimensional nearly neural perform extensive problem sgd easily likely aim problem easy sgd slow advanced enable could like review thank experiment hyperparameter directly literature specify maxout describe modify maxout network maxout publicly relu dropout intended nearly reproduce relu standard provide relu network dropout precede file
composition private program mechanism oracle find private everything unfold definition suffice previous note tight row amount approach final sensitivity depend private simple throughout weight q instance neighbor objective concrete part add laplace lp exactly private optimal draw perturb q sensitivity objective solve perturb accuracy least laplace happen event perturb lp add bounded perturbed find solution sensitivity trivial exception solve relaxed sense reconstruction attack privacy reconstruct database due completeness restrict entry round round input entry uniformly identical eq uniformly privacy private lp convert database lp solver attack lp likewise say optimal solution lp change bit change right mechanism least feasible lp note guarantee reconstruct impossible differential privacy zero lp zero change coefficient similar lp coefficient amount mechanism private additive lp solution probability solution place index observe private change small constraint want solution constraint mechanism sensitivity private feasible satisfie public probability lp
first realize realize outcome repeat estimate initialize tt give critical underlie imputation realize realize collective mechanism expect typical agent causal issue proceed make behavior prior outcome round consequence good mechanism agent mechanism offer utility strategy behavior agent expect benefit agent adopt game agent vector action softmax behavior agent nash mild condition agent strong high experimental function
lead without additional weak global show exactly eigenvalue exact semi nest dropout autoencoder principal component procedure intrinsic speed quality gain offer procedure na I hamming database semantic within memory associate retrieval complexity grow computationally prohibitive bit address code likely many query locality sensitive seek preserve projection inefficient code impose retrieval decay datum capture coarse allow logarithmic adaptively code large code hundred exist example retrieval dataset entry average per fast order also use continuous degradation compression give rise quality combination small
go tc audio affect notably live device versus combinatorial conventional domain adaptation shot adaptation address calibrate acoustic environment type music speech track categorical variable live lr sr iv live degradation toolbox audio split training overall hold good exploit semantic descriptor descriptor regular recognition environment environment demonstrates effectively covariate see data r origin lr sr avg lr lr tc attribute
special spline use outcome display array stagewise regression noisy curve draw equally knot stagewise figure show stagewise along dotted stagewise computationally update solve exact computed stagewise produce visually reasonable observation thick run stagewise step curve stagewise curve smooth visually reasonable difficulty stagewise set set easy previous iterative invariance around stagewise generalize give term since lasso special case covers choose wherein encourage order piecewise choice dimensional fuse statistic literature incidence correspond encourage component piecewise constant respect framework use trend filtering update linear perspective form switch justify two argument write q conjugate satisfy stagewise path convert primal stagewise differentiable stagewise aside reduce multiplication one fuse trend filter sparse make strategy estimate stagewise update outline compute stationarity th view iterate point simplify considerably eq interpret convention think stagewise strategy compute primal note stagewise estimate generalize lasso stagewise encounter far increase stagewise begin towards opposite usual direction note loss primal initialization update express eq stagewise begin trivial fit along form towards row vector concrete fuse incidence evaluate difference nonzero towards build shrinkage across difference move constant amount towards graph grid graph appendix general various computational three major far present find room comparison tune grid parameter compute recognize fair near computationally superior stagewise competitive capable produce overview path implement group apply accelerate gradient idea complicate backtrack line step refer section description stagewise step coefficient define uncorrelated population independently block predictor correlation group stagewise fit warm run stagewise top row uncorrelated stagewise fit underlie quite competitive fit plot stagewise fit exact top took stagewise fit meanwhile group uncorrelated stagewise take second surprisingly middle computer uncorrelated case stagewise stagewise estimate within criterion frank wolfe mean frank lastly bottom lasso stagewise component path one draw path uncorrelated setup correlate bottom display
relationship pseudo ensemble perturbation space notion robustness focus perturbation novel regularizer make behavior pseudo ensemble generate regularizer match unlike dropout naturally art tensor ensemble world original conclude approximate pseudo child provide pseudo dropout sample activity common minimize consist sampling mask extract use fairly form impose tractable allow variety ensemble formalize follow parent perturbation pseudo approach come broad
quality improve training induce benefit hyper empirically demonstrate improve great parameter optimization supervise generalize hypothesis induce algorithm data set noisy learning associate parameter induce datum hyper improve weighting noisy significantly hyper improve algorithm validation characterize validation induce parameter characterize g optimize training impact induced instance induce even instance induce outlier beneficial instance case instance induce b line represent classification dash line induce boundary
example link uncertainty associate network mining link change uncertain nature analysis need link failure human interaction recently mining investigate uncertain propose algorithm find database support uncertain propose et semantic represent move trajectory configuration edge region connect neighboring main originally cope directly despite number enumeration become intractable overcome simplify leverage simplicity tuple attribute uncertainty uncertainty tuple model database may edge underlying instance consider additional generation rule correlation study combine mine extensively management neighbor context mine frequent mining mining uncertain scenario though entire graph node label collective
end end output dag root guarantee process similarly tree version dag traversal distant sense child child responsible positive choice choice meaningful priori select maximize score available percentile example experimentally choose
part nsf modal mixed cluster datum journal american association shift mode cluster transaction intelligence toward transaction machine intelligence bandwidth selection unsupervise unify self coverage journal recognition research sure screen journal cluster american association journal selection journal american optimality et rkhs
log expression empirical moreover entry give standardized write identify learn especially computationally modern exceed number inference calculate model result take universal infinite limit derive expression selection logistic
multiplication power weight vector lexical noun represent component noun lexical like net tune take give experimental second calculate solution candidate super candidate single pass rank candidate top candidate last super task remove irrelevant candidate tends improve mean impact fortunately affect adjustment greatly evaluation mean candidate top percent candidate candidate super noun super model median composition similarity believe well probability kind see table pseudo context believe pseudo context vector want really learn section suggest red thing separate noun train target super noun noun noun never describe super dataset build table possible rank rank marked candidate build list super rank unfortunately candidate super guess super good member red specie specie table summarize super target evaluation metric top super work composition super super composition mean candidate percent percent top percent c c decomposition baseline vector take significantly super percent fisher test significantly super percent top confidence level candidate median candidate percent top percent top percent percent c compare baseline decompose super decompose although restrict super performance percent versus top versus versus test confidence level near
forward help help chain mix coupling implement mixture gaussian would many sample forward gibbs test code sigma replace subtle never subtle become top soon major drawback unfortunately therefore test pass test general drive development pass outline code method code modular
great q side q q absolutely absolutely inexact kk kk show update inexact saddle
inequality unit vector also inequalitie subspace column draw dimensional correct know pick among label oracle pick maximum success return failure original pick incorrect neighbor pick incorrect near pick among pick incorrect close oracle failure incorrect close point correct fails therefore analyze independent form dependent analyze stochastically dominate random onto isotropic subspace stochastically dominate bind condition least give simultaneously k j inequality since independent fix union choose therefore
online generalize click ad happen currently develop pricing model effort towards click strong assume query click iid let click indicator event position count click capture click click click beta update quantity involve click proposition remark conjecture theorem axiom claim wang click tool leverage feedback click aspect share sequentially click engine search key information important key argue click user experience incorporate search display click relevance search evaluate extensive engine system user query list rank link ad display call search click ad interact search particular ad click attribute ad never search surface user query understand document compare approach crowd employ human
either semantic concept various semantic concept popular show dataset manually annotate large class tag tag depend tag internet verification label noisy contain intra image benchmark recognition high semantic imagenet use imagenet visual recognition k internal imagenet guarantee image consider entire comprise million manually million report recognition manually class tag ground without verification therefore supervision noisy characterize intra class image visually similarity object video content manually video semantic category belong category video size set evaluation protocol binary multi semantic mean map activity publish video sharing release video release development training video learn video first frame phase score network architecture convolution layer convolution architecture compose layer convolution layer fully layer last classification layer resolution image resolution convolution pixel convolution layer max perform layer connect
noise come ignore three decay minibatch fisher gradient size keep govern control see size big big minibatch gradient cause especially close minibatch small per setting digit contain indicate model mnist parameter gamma precision result insensitive generate parameter sampling carry alternate follow step sample weight pa minibatch gamma conditional sgd try give good result use sgd momentum try setting sgd momentum find good performance factorization use movie simplify consider prior however benefit inference sample minibatch rating hyper experiment select try sgd momentum try sgd definition lemma hamiltonian mechanism distant metropolis enable efficient
cascade implementation base round overfitte cascade round team compute count run validation occur iteration additional randomized cascade cascade
statistical exponentially weight forecaster case recently learn property characterize special function space develop rich condition loss understand connection bernstein stochastic fast previous secondly establish slow notion stochastic effective convexity stochastic toward get rate involve er chernoff moment excess moment application er chernoff bind finite vc function describe extend fast notion call weak conclude connection topic theory discussion fx operate compose comprise loss compose frequently throughout
main gps integrate connect explicitly unlike gp corresponding discover employ one convexity lead multiple optima feature optima severe alternative bfgs hessian additionally beneficial extreme easy easy per rmse ard sigmoid number mapping covariance exceed notice undesirable slightly offset actual location ideally describe would intractable gps uncertain input kind although exist strongly appropriate designing covariance g
evaluate serial include follow sampling denote lda conduct yahoo approach wise sampling recently fair strategy three comparison result lda lda three document follow figure speed performance document wise lda indicate discuss document order document fast document compare justify use core distribute parallelization lead new lda huge compare yahoo implementation scale large yahoo lda yahoo server therefore outperform server yahoo disk base implementation assume associated token disk fair run yahoo normal disk yahoo run storage yahoo disk code disk yahoo yahoo parallel machine conduct dataset amazon present figure lda outperform disk yahoo well desire time fast yahoo next
great schema derivation could diagram piece reality expect diagram encode versa throughout deal library student university library offer book article scientific conference publication kind stock book diagram diagram entity author right author author book auto swap translate entity book author relationship book resp child tree schema correspond whereas represent resp option resp option third element attribute book author reflect practical good avoid b author book observe diagram relational incur point diagram orient organization diagram order specification child datum impact match consider map onto list specify concept element phone name phone phone convert negative drawback explore refined level meaningful reduction negative examine highlight semantic properly finding improve schema process user nice schema page extract subsequently exploit refine query page contribution discuss structural link instance schema match source exploit user search web experiment life car site semantic precision web far provide schema primitive build type instance offer integer boolean compatibility type whether violate instance constraint convert soft constraint constraint detect positive schema conclude semantic state reasoning involve primitive solve compatibility table compatibility string integer short second whereas report compatibility coefficient high compatibility type define usually associate management human type simple furthermore allow create type start exist implicitly induce base extend mean new construct intend analogous fashion derive assume type implement relationship implement concept specify sub construct adopt sub appear construct specify semantic exploit management compatibility type principle decide repeatedly simple already complex element match match cardinality specify occurrence element occurrence denote zero indicate occurrence allow analogously compatibility check two schema discard
picture maps assignment assignment constraint partition function equal solution notice abuse col j satisfied represent bipartite include two node denote neighbor factor summation true evaluate clause false circle graph versa refer factor summation ix message typically initialize distribution update estimate loop summarize bp take opposite bp true true true bp influential loop rather instance may converge time bp update message message substitute variable substitution repeatedly message reach point become individual update operator employ sp refer heuristic bias biased fix convergence additional assume iteration sp summarize schedule h return otherwise
complicate pick wang censor covariate usefulness need verify censor datum need lot censor robust slight section simple technique work optimality scenario due apply life efficiency crucially tune choose carefully censor proportion give censor enjoy many property inference censor chen f wang et many assume availability associate target e g medical diagnostic pressure covariate give instead infer response alone interested conditional covariate often study act nuisance component th parametric calculate eq censor distribution also adjust tie parametric censor prove several consistency functional life science know survival censor experience similar disease
hide constrain produce output reality allow criterion htbp hour ahead forecasting present ann ff ann sa forecast svm mae period ff ann sa mae mae rmse ff perform input wind wind previous hour time day presence feed greatly forecast htbp value convert algorithm obtain value train machine mlp neural
long admissible instead combinatorial hold trivially
e te cx z jx jk tw estimate influence derivative q note operate certain redundancy shift constraint component ambiguity scale sample ambiguity lead therefore estimate ambiguity ambiguity cluster similarity value sample neighbor nn efficiently entropy compute ambiguity scalable representation semi supervise spectral operation mm current mix cluster gmm ambiguity entropy approximately evaluate uncertainty qualitative shown notice appearance boundary increasingly model set complexity nonparametric parametric complexity active adopt large scale sample compute scale method
fact throughout expand expand term stationarity last thus sgd var var var ti ti optima capturing randomness tv similar algebra lead desire assumption ml size turn satisfy cluster considerable code framework specialized implementation recent server ps allowing distribute high throughput allow insufficient really ml algorithm output many theoretically computational throughput guarantee convergent exist ps ps communication mechanism implement ps enable ml volume internet activity pressure ml scale beyond single size ml single partition machine machine practitioner turn server ps server
network traditional improve distance representation influence choice label outperform method create label online deep tool analyze build representation evaluate representation task increase sparsity improvement micro outperform even training demonstrate scalability moreover build arrange section formulation classification relate outline relate work conclusion member member edge partially social attribute map label utilize dependence embed achieve superior literature traditional relational inference markov iterative topology label unsupervise structural task
close enable approximate optimize certain distribution good maximize bind overall algorithm initial randomly mask independently update update parameter decrease properly increment go termination r respect replace accord descent assumption intractable summation binary evaluation practice use carlo method last sample well two two think amount
focused datum instead large approach bias try give discrepancy sample feature trick inner equation feature mmd db kb
distribution interest inspection datum rank set arrange arrange arrange arrange constant middle rank rank large rank seven ten set small probability observe extreme discover pattern similar gradually optimal switching minimize mean surprising value choice nine pair nine dot degradation respectively bad w significantly marginally bad significantly accept large
rule student logistic class generalise example transformation distribution list exp control univariate baseline typically reduce poorly limitation consider weakly correlate
write take feasible outside x proper satisfying q subdifferential denote proper hilbert space fr continuous frequently lemma ball lipschitz nonempty briefly apply formulation cover broad input generalization frank programming gradient apply know computational geometry elsewhere appear various signal name boost greedy method orthogonal recent descent discuss reason conditional evident example proximal reference iteration low whereas gradient exhibit rate side step operator section
report encourage positive segment posterior bin interval obtain calibration seem calibrate present simulate detailed use call choose analyse platform replicate datum run pass replicate segment large would distribution uniform namely elsewhere justify simulate cc ccc start ccc length ex pass ideally infer replicate datum happen consistent across replicate infer replicate
extension dependence jensen jensen jensen offer extension control complex value dynamic combination beyond quick estimator cover analytical gram iii modularity modularity computation
gpu set axis run yahoo ranking response goal song audio dataset figure even gpu completely dominate computation massive speedup gpu marker leave fail complete solution recognition datum version match exactly parallelization improve scalability highly besides code burden reduce spend establish parallelization square hinge solve entirely large
hide dnn neuron random forest foreground dnn spatially color transform three regressor choose dnn forest forest scalable dnn forest color dnn fair adapt table foreground obtain slightly dataset forest dnn obtain foreground obtain dnn inspection color forest spatially enhanced image random forest retrieve node retrieve dnn color visual th pt p c method run method mit learn level mapping use experimental work testing mit hence c mit hence train column table capable prediction term predefine mostly concentrated show see enhance close ground could near neighbor slow search percentage contain pixel hand powerful thus method capability exploit near inconsistent color remain testing image th enhance expert enhance enhance effectiveness algorithm naive selection sensor interestingly achieve well gaussian primarily nonlinear deep network rich compare entropy method achieve especially select number mobile filter different color warm rise filter light choose mit enhance training half verify
vary densely achieve error gp I multiple field smallest tend low field usually field step gp e exploit temperature light measure denote colored end vertical horizontal show localization gp field gp small light field small low low field field geodesic e path true road segment road topology fig localization road segment average run cc mobile trajectory light b location achieve error average gp gp clearly scalable I offline incur paper localization whose constant memory filtering theoretically analyze outperform localization algorithm scalability robustness
form alignment gap alphabet denote gray help track protein family position alphabet go cf transform alignment gap one position zero otherwise denote length I cx entry measure protein similarly protein approach approximate value former enforce rely fact input consequence position e site anti work suitable anti correlation model bayesian inference behind density determinant block independence protein sequence read attain constitute inference proper parameter bayesian need introduction require computed accounting prior inverse prior p meaning inverse constant euler wishart integrable px eqs formulae posterior covariance abuse notation shall provide estimate differ attempt protein contact
unnecessary lexical stress keep discard rely cause corpus without lexical evaluation development subset vocabulary decode vocabulary convert window frame form final vector preprocessing alphabet token total recurrent accelerate maximum pass rate pass train gpu implementation decode cross set bt lm lm sort fairly mistake character word
correspond begin discussion scoring distribution framework scoring outcome attention mechanism relationship market semantic implication market behavior vary depend agent market agent accord characterize market reach potential argument market proportional eq though work density turn draw well duality specifically program care specific outcome outcome entropy tend towards always constrain entropy whereas objective let negative solution take form multipli integrate ensure family almost interest consider regular family regular family convex differentiable lie follow property onto inverse statistic statistic log log family relate invertible expect statistic family depend underlie proper scoring consider logarithmic density parametrize proper belief let strict converse score characterization proper strictly px show intuition bregman divergence strictly equality know relate bregman state full lead expectation maximum agent report interpretation principal according agent usual
set extreme point polytope polytope express k solution vi inequality simultaneously polytope reduction formalize denote set put arbitrary vector polytope exist inequality arbitrary inequality dimensional construct qx eq side product satisfie use point derive admit full rank whereby orthogonal eq ks induce orthogonal follow hold cauchy consequently follow unitary similarly since cf likewise definition complete cf theorem obey orthonormal probability depend else implicit
unlike optimize small become high one plausible greedy fast achieve good enough provide approximation gram check number greedy adaptive iteration adaptive subsection ridge ridge near summarize ccccc greedy greedy census ground truth adaptive error sampling since result report two much size case big case adaptive central bounding uniformity phenomenon large art sometimes dnn close reveal accuracy speech match dnn clear improve efficiency kernel high exploit integration technique context include analyze gram author would thank sequence wiener anonymous point helpful program project air laboratory contract fa detailed make term notational lemma claim improve
image hypergraph construction sequentially introduce clique star expansion propose probabilistic hypergraph assign accord centroid pairwise vertex similarity individual pairwise similarity represent designing share context corresponding similarity contextual take account neighborhood vertex corrupt still information corruption context aware hypergraph similarity measure type hypergraph nn hypergraph hypergraph pairwise hypergraph affinity vertex hypergraph hypergraph capture manifold structure modeling contextual combine aware hypergraph intrinsic robustness corruption contribution fold spectral order property build hypergraph encode affinity type end type information hypergraph similarity vertex
detail tackle distribution bag bag bag possible algorithm basis similarity formula finite gaussian heuristic relate parametric gaussians kernel hilbert reference appeal gaussians close product gaussian divergence lack theoretically approach statistical divergence metric non number construction metric kernel overlap concentrated value dispersion type hilbert define guarantee certain domain estimate kernel open plug algorithm similarity distribution index paradigm bag treat instance solve ray set label bag example fit configuration handle shape patch region document web identify link customer characterize record
domain recommendation rating user dataset train remain bc bc em give bc em conduct vs vs model outperform show latent domain common rating aggregate propose art cross domain enhance recommendation dataset give nmf bc nmf vs setting bc vs bc vs
space kernel propose q ensure cluster matrix mn multiplicative overhead greatly require case random partition kernel directly interpret partition sensible machine mean exhaustive example random cluster algorithm couple generate randomness algorithm exist randomized initialization bagging feature return
set separate system system primal build presentation shall familiar na I extend constant purpose like upper unlikely behave shall intersection detail construct element fix set prove system vc verify bound book discrepancy edge lemma vertex drop total drop
control fitting variational paragraph range matching term correspond independent half assign prior find model lp match lp estimate guarantee particular dependent plausible evaluation satisfactory use four model spectra extract image nonlinear pixel accord feature component interaction pixel p b fan bilinear fm generalize bilinear adjust bilinear interaction th pixel stand hadamard admissible set robustness pixel impose cutoff abundance remove drawn version namely two appear
example produce low sometimes hinge loss introduce slack apply result problem select accuracy ht cpu problem c cpu second result solver give relatively matlab get active almost regularization addition parallelization implementation immediately look number plot figure ht technique construct analyze thank smoothing theoretically rate surprisingly analysis enable inexact augment lagrangian expect deep smoothing help adaptive strategy connect european future foundation grant lemma feasibility induction notice assumption nonempty dual saddle inequality lead tt follow outline lemma lemma k definition follow write k obtain also h get subtract inequality definition equality substitute rule third ff substitute final third hand substitute since first inequality lipschitz continuity respect give refinement express kf line easily get prove q g second inequality refine since right hand find proof lipschitz inequality c update c estimate ga update k augment lead obtain combine q prove set obtain inequality imply
orthogonal basis learn distribution pose vector advance determine coin uniformly accord relevant define family underlie prove bind choose key ingredient relation learn coin relation first adversarial ki j identify fraction adversarial specify gb picking assume orthonormal hand attain attain reason observation th coin assumption previously informally coin formalize coin problem hypothesis use mechanism
n constant lin k scan union bind bernstein scan multiplicative chernoff eq several condition suppose fix measure fix suffice positivity q thus tv q inequality variation product measurable subset independent copy variation distance p p eq cauchy prove ball index bin index ball bin fix condition index index need follow association fix independent copy condition vector index vector moreover distinct statement expectation decrease q view therefore q denote bipartite vertex vertex denote distribution match edge proceed clique let clique define conditional
reach activity c room interact activity trajectory model pr robot purpose human environment pr presence available learn preference activity preference discrete preference expert rich encode learn pr robot video thm minus user preference trajectory environment human challenge trajectory interaction environment preferred trajectory new cost system motion segment neutral use preference express environment extensive preference plan preferred trajectory environment preference environment rich object human challenge define good trajectory environment trajectory trajectory preference expert learn video robot segment good neutral parameter preference run environment validate claim planning environment good define vary environment paper object lie type feedback expect user training trajectory currently system argue co active preference feedback non intuitive nevertheless match rate algorithm task
weight define px px property family divergence turn problem yield bregman bregman connect thus family property dp return family density point support include laplacian family densitie family contiguous property intensity grey small distinct grey
update metropolis hasting proposal accept necessity compute normalize acceptance current algorithm keep reasonable acceptance function symmetric current truncate zero proposal draw step package sampler assumption exchange tend auxiliary
solution matrix stack root via identify eigenvector interpret scale direction indicate transformation ica far operation familiar term whiten whitening remove second along term datum whitening demonstrate first eigenvector perform rotation order dependency mathematically transform expectation covariance dimension operation familiar principal pca eigenvector remove reduction remove low figure ensure preferred direction symmetric much sphere whitening simplify ica rotation simplification observe reduce simplified provide additional structure recover highlight likewise consistent whitening recover mixed decomposition whiten rotation statistic variable whiten remove correlation last dependency require correlation ica special term factorial search rotation order instead remove correlation rotation achievable therefore term order estimate
repeat optimisation output discriminative network generative model represent likelihood training use label label bind extension latent bind handle unobserved q entire unseen label contribute relate undesirable classifier ideally variational add also learn label purely discriminative motivating model also variational instead categorical symmetric unified objective optimisation generative optimisation jointly resort
show understand interest discuss detail mention one assume specifically denote let j j linear time follow auxiliary slack solve algorithm convex unconstraine separable separable encounter across rewrite splitting considerable constraint dual eq conjugate fit asynchronous manner describe include least agent allocation therein empirical examine coordinate descent analyze pt experiment convergence rate establish laplacian graph demonstrate topology star run follow decomposable constraint choose evaluate h clique topology topology acceptable long topology sparsity communication require portion essential pair compare star diameter diameter
hyperplane class origin kernel svms introduce derive note lin standard construction derive kernel principle nonetheless euclidean leave new base locality sensitive unless kernel except binomial recognition video comprise individual capture video represent recognition use video capture normal create subspace individual set remain new derive pl outperform new outperform polynomial achieve overall bar outperform performance contain pose image pyramid descriptor acquire pose result use compute
obtain integrate variable term equality equation htp move rate vertical processor plot dark red corner processor code deep color mse green color red color violate achieve significant reduction increase go iff iff next consequence let corollary processor grow mse rate average double equation converge weakly notation covariance interest double averaging convenience non vanish variance asymptotic variance lemma extend dimensional symmetric technique matrix generalize multidimensional go look notation move keep parameter symmetric precise contain scale become block
symbol membership gmm represent level green compete increase explain formulate assume membership although position parameter distribution true covariance figure gmm sign pair monte replicate indeed aside flat prior directly indistinguishable position utilize multivariate gmm method see case position vector distribute sbm note illustrative sbm specifically obtain mean rate see slightly empirical sbm demonstrate robustness sbm final sbm parameterize position case approximately error approximately pair gmm competitive bayes pair analysis
cumulative straightforwardly follow basic additionally unitary length end lie challenge still partially researcher number type regular cube generic examine model
op mat op suggest shall adopt unless drop subscript derive directly perturbation perturbation op singular op op op let top follow k op claim technical analyzing lie non leverage upon theory sharp critical loss signal tensor q large mat mat unfold almost limit subsequence invariance distribution limit surely eq normal since weaker sufficient odd iterate map find result use conjunction available literature amp qualitative asymptotic amp recursion follow establish iteration generic standard satisfied next two initialization ground
abc ic ed gr please integer factor feedback give high high f dt rely definite follow deterministic process mean propose ol ff ff iw start iteratively know feasible dimension spurious parameterization order jointly propose integrate penalty optimize define form follow minimize eigenvector minimize respect respect allow select alternate inner optimize outer update simulation finite estimation notable globally global optimum start follow eigenvalue unobserve span orthogonal possible regressor factor heavily specify exist recommend factor slope know z ik jt correlation time exist panel proportional center lead statistic overcome diagonal exist affect asymptotic term
contrast compressive measurement case justification exploit across become phenomenon matrix achieve concrete projection hope irrespective leverage approximation large usually suffice require achieve observe compared bind highlight average careful similar spirit entry column
guarantee method factorization utilize sound source type mix complex sound autoencoder separation autoencoder successfully speech
control backward induction residual dynamic treatment regime decide treatment assign effect cost attractive follow outcome depend also intervention population equal decision pseudo translate self method optimal setting design regime optimize utility fix manuscript infinite datum collect develop regime treatment datum trajectory remainder manuscript organize follow explain develop gradient descent minimize section conclude remark time trajectory th trajectory decision maximal take summary capital letter variable subscript potential effect treatment
precisely define quantity estimator base weighted mixed weight suitable strong sparsity give strong group assumption review support work work suppose satisfy bind scale corollary work assumption exhibit although dominate interpretation order initial coefficient projection nearly pseudo inverse propose naturally control tucker kkt automatically tx tx j tx closely relate constrain lasso guarantee tn whenever
south east table index header txt txt index txt index header txt table index true txt index header table index header plot index header txt index header minor title xlabel k ylabel ndcg pos east header plot header txt header header plot txt plots txt header header txt index scale minor title xlabel ndcg legend pos east header txt txt header plot index header header plot txt header xlabel ylabel ndcg header txt index header txt table header x plot txt index header plot txt table x header true txt index header plot txt header txt x index true plot minor title xlabel ylabel ndcg index header txt header true plot header txt table index header txt index txt table header txt table index header plot txt index table header scale title xlabel ylabel ndcg header txt
coordinate nearly plot good rank increase rank substitute width xlabel ylabel coordinate coordinate error finally depend find insensitive fit improve htb xlabel ylabel coordinate coordinate coordinate error add wish row continue set estimate new row computation alternate representation new global minimum new new three rank serial share memory implementation fit implementation subproblem implementation date encourage reader package implementation special subproblem program implementation find encourage interested date collection specify store array correspond specify miss correspond object characterize stop alternate procedure fit miss loss store penalty miss list fit automatically add offset scale py correspond iterate solve problem criterion meet regularizer support py quadratic huber hinge ordinal quadratic implement regularizer code model implementation aspect usage date encourage reader specify datum like tuple list loss rx rank code fit loss loss rx x regularize loss rx fit history mark convention regularizers may nonconvex simple globally useful factorization view unified parametrize modeling view pca loss model loss ica thesis hinge divergence regularizer nuclear max norm thesis regularizer factorization constraint literature change regularizer may pca pca svd indexing nonnegative divergence al community induce penalize decomposition pca review focus tool integrate heterogeneous canonical understand eigenvector structure de low data nominal ordinal et label label image text dimensional space recently language processing document computationally generalize np compute weight completion result distinguish way optimization refer matrix factorization present method alternate newton gradient relaxation semidefinite program iteratively entry solve observe method conjunction exploit gram semidefinite intractable optimality semidefinite lead semidefinite factorization
define claim bind enough lem worker belong part use rescale last inequality k k ta rgb op title title proposition definition conjecture definition berkeley berkeley berkeley berkeley significant performance dramatically necessary communication strong well mini batch converge quality quickly batch gain fast theoretically justify sgd approach distinguish communication efficiency optimization assumption cover case case mini though update locally process mini dramatically
near whereas near leibler contain case less ccccc lr tr tr lr tr lr tr tr tr symmetric first mixture number replication ccccc tr lr tr tr tr tr log density value replication respectively beta replication assign label fit two mixture assign label base posterior large wherein next fit component concave record density list size run local run different allow iteration case wherein likelihood flat would em estimate setting flat expect performance outperform two half component dramatically even cutoff point define move usually low difference affect label somewhat totally find log see detail concave percentage datum however uncertainty differ two particularly useful clustering assign less cutoff asymmetric g screen cancer want misclassification case near center great divide center component normality log concavity section new three estimation old national many old analyze explanation people day fit symmetric concave fit variance around bin plot histogram symmetric log mixture et fitting estimate component assume force
use minute matlab clinical intensity function asynchronous medical investigate around investigate make method intensity event increase flexibility infer abstraction event purpose transform raw form standard clinical intensity intensity contact system usually increase increase condition instability probably generate contact severe frequency medical contact similar value
eq partition small mass size capacity energy landscape bins space bin energy bin estimate size energy landscape contain barrier leaf node volume repeatedly smooth landscape volume merge measure desirable difficulty figure landscape look difficulty learn record length color bar leaf true node ii minimum minima curve error vertical like roc operator pattern recognition sliding threshold curve characterize difficulty auc area ii task close impossible problem correspond difficulty measure difficulty experiment move proposal convex involve use first
essential capture topology diagram moreover persistence diagram persistence diagram come persistence diagram create connect maxima merge birth persistence diagram way indeed parametrize persistence diagram fig rectangular particular pixel pixel threshold piecewise triangular mesh heat signature yet commonly crucial aspect persistence diagram respect perturbation infer persistence diagram consider map require diagram natural metric associate persistence diagram speak diagram two diagram persistence diagram infinite distance range persistence
norm complexity dictionary least except contrast yield still heuristic give amenable technique precede write version set possibly uncertainty uncertain corollary uncertain give fix dictionary occur independently signal contrast learning arise previous primarily uncertain implication equivalence empty eq equality homogeneity triangle homogeneity triangle satisfy desire complete attain pair regime course regularization upper problem recover long equivalent unless one discrepancy computable well see satisfie long equality dimensional long strict almost sense gap
circle gray measure order connectivity sequentially maximum c connectivity solid synthetic correlation measure range initial measurement ref hide carefully informative many som nonetheless comparable sir additionally give data specie confident specie infer interaction limit limited number specie output conceptually hard since observe dynamic chemical specie vary case select adaptive single range infer ability condition measurement select able predict chosen range range range fig equation dimensional roughly time measured chemical specie crucially computational complexity sir even hide adaptive sir model exponentially model impossible system many guarantee traditional infer require predict dynamic sir fall process chemical utility say mechanism response unseen qualitatively one necessity feedback importantly analogy model
limitation code hour height width ne u ne ne ne chapter head chapter head subsection head head compatibility corollary conjecture em fc maximize detection fc jointly use investigate adopt theoretic fc theoretic quantization rate address coordinate demonstrate joint approach
update severe systematic limitation hyperplane along filter onto instantaneous dictionary achieve project counterpart algorithm steady yet treat grow paper natural I input present ascent subspace steady mean
error among uncorrelated want make analytic average realization back present show excellent propagation specific task average matrix become eq determine critical determined chosen drop I definition nonlinearity effectively approximately apply row nonlinearity rely analytic need assumption product property vector layer calculation square precisely expand logarithm taylor optimal slope expression low indicate reasonably
cavity similarity indicate methodology near neighbor class annotate phase compare ec annotate ground truth I truth count successive match ec thus ec ec two ec belong conversely similarity ec relevant formally similarity digit ec figure give six correspond ec infer annotated encounter among rank condition query entire ec number annotate ground truth information perform call rank denote couple query database implicit couple conventional optimize together vector minimize set denote truth similarity query contain outer pairwise difference rank loss minimize dual q kronecker feature information feature individual kronecker easily kronecker dual universal indicate access training protein approximation probably kernel yield restriction fp similarity
fall restriction distribution uniformity optimal obtain work restrict broadly fall property decide efficiently property classic problem obvious candidate poisson truly know quite test allow vs contiguous place approach give exploit tolerance identity support effective accuracy barrier complexity answer independent might deduce low however easy member establish dependence tight related distribution
numerical table bootstrap technique substantial gain adjust bias adjust analytically bias ba iteration ba least importantly ba ba reduction expense mse systematically adjust adjusted detailed rule terminate iterative two middle table bias column fall bias record make mse mse conclusion record panel case analytical adjustment record third panel table column fall comparable realization improvement overall summarize confidence size nominal length nominal qualitatively bootstrap ba ba adjust ba ba bootstrap adjust record coverage poor ba produce accurate key point narrow expense inaccurate coverage secondly coverage accuracy yield expense precision negligible interval contrast adjustment analytical adjustment
satisfy latter forest work x notation easily therefore consequently reveal finally eq tend zero assumption satisfied cut recall prove let multivariate univariate know additive implie let tend zero tend assume decrease exist b x x k x since contradiction almost surely necessarily rest x deduce cut k inequality cut perform root case cut direction e cut perform along word calculation besides calculation tail deduce union eq exist q event occur remainder quantity illustration illustration dimension cell case
generate domain graph edge training vs part comparison class need become costly mean use classifying xlabel dimension projection ylabel accuracy plot illustration method art accuracy show generic enough quite author classification validation base feature include gender factor pos patterns report maximum seem acceptable tool gram split reference text vector cosine similarity improve repeatedly set similar accuracy note classified contain token hard reduction discover structure
dr database al match sensitivity specificity dr patients dr report et al automatic must absence clinical detector serve detector prove select modular preprocessing method candidate evaluate value competitive promise component approach support science project develop system office technology contract om om om p inf phone digital issue medical candidate use ensemble would pixel extract
table x index accuracy mention work hard apply task carry krige copula inversion cubic approach inspire informally speak perform primary individually reduce independent multi help yield multiply py x py approximation learn two speedup provide easily numerator advantageous distribution introduce analytical solution precisely predictive copula would obtain reduce gaussian
bind mn first ce ij ij first accurately deterministic propose dependent square recover biased matrix thresholding operator recommend technique provably recover convenience first define rx mn error weight weighted case show constant expect error relate minimizer thresholded order inductive standard inductive want matrix completion special inductive correspond also disease prediction theoretically analysis motivate world example
cr mathematically immediate belong constant rational cr sure avoid channel ps se decode primary queue full corresponding queue empty queue inactive indicate service throughput cr consecutive action maximize secondary service cr queue finite dynamical cr user cr channel cr cr choose feasible action receive immediate take place repeat reinforcement naive implementation discount idea value two
greedy orthogonal omp decode computationally expensive omp collect random random projection sense small sparse context compress sense recovery pursuit nice well page decode count achieve similar accuracy count sketch need
col sep comma sep comma ylabel style west height width axis font axis sep comma expert col comma xlabel ylabel post legend style west major height axis style axis sep comma col comma ylabel post inclusion legend west grid major every font axis sep comma expert table comma inclusion covariate gap size increase eventually low knowledge monotonic averaging repetition yield smoother comprise beyond significant different assess instance datum namely classified range close skewed accuracy instance site classifier return absence information auc area receiver operate insensitive auc auc perfect predictor xlabel ylabel auc legend none legend east grid major height
sampler keep move gradient fairly langevin metropolis reach distant start reverse back phase look origin large move ie change monotonically work well hmc compute minus log ad hoc close reciprocal nd estimate nd independent dominate hmc partial respect coefficient concentrate mode reduce loss fairly update time reduce substantially sum fixed want justify trick important coefficient coefficient update hmc ability hmc random may hard update need update iteration need square gibbs sampling phase sample fix hmc adjustment nd derivative macro pt pt bayesian hyper li high areas sciences example genomic research gene class thousand wish fit date method tail li knowledge fully bayesian appear literature laplace sake attribute restrict dimension laplace hyperparameter recently paper report hyper lasso mcmc
therefore center finite denote integrable kernel linear map gaussians since contain fx function since vanish center expectation xx european european carry support fellowship ex combinatorial locally evaluation method system allow heart allow replace inversion provide blue unbiased noise modern wide medical imaging signal big g recommender system whole property minimize explicit depend explicit compute locality yield tradeoff characterization include explanation circuit combinatorial finding optimality rather natural compressed sense dual sequel field
couple density normalized see smoothed shift bound region define solve poorly component example similarity decrease offset tell semidefinite solve simply couple modification trick show albeit loose bind capture exponential decay gaussian display offset line display straightforward location define solve conclude counter kernel kx calculation compare normalize span top eigenfunction map encode label important level spectral section involve intuition imagine density imagine mathematical bandwidth separate principal vanishe discuss goal quantify equivalently root density measure distance
fast discriminative feed quickly discriminative new task contrast previous learn raw theoretical guarantee contrast construct cross feature mining information provable framework supervise handle pre process score datum framework latent framework scenario input crowdsource different group approach construct score average leverage source challenge scenario section elaborate end supervised framework purely however datum challenge distributional likelihood convex especially involve incorporate generative extract input classification setting consider regression set compute r access understand label vary change locally learn form moment yield high function call order score learn use unlabeled feature apply tensor variate particular setup yield high derivative accurately use feature parametric framework incorporate previous obtain derivative find spectral find notation symmetric decompose hand
weight minimal inside maximal cut edge sample biased increase observation expect look cut inequality cut institute institute spectral computing costly limited notion approximation generalize cluster propose new algorithmic sampling improve considerably learn obtain costly example may metric biology case initially alternative approximation l graph laplacian graph easily value spectral see discrete spectral vs imply use min relevant include nd eigenvector simple example cut theoretical
stable stability implicit td show td td consider assume instability clutter let write assume iterate stay guarantee iterate td performance explanation start argument argument eigenvalue
assume function cover movie show return ranking value table list appeal high list cover diverse list seem cover movie depend need priori hard list diversity section list list property rank item item family x c list item action family maximize item diversity follow solution efficiently behind recommend control consider diversity preference item popularity score predict rating entry utility objective diversity increase typically g address balance increase diversity maintain utility consider primary aim recommend item maximize choice utility recommendation increase diversity diversity weight problem formulate order gain gain choose real instance movie recommend dissimilarity product recommend website diversity diversity item add diversity empty generality particular satisfy general cast maximum argue code work sort decrease utility sg contribute finally recommendation second movie movie list movie place
ip f ip ip ns jj previous markov leave invariant rao together step smooth particle intermediate put present smooth model enable simultaneous smoothing initialize compute notational convenience variable foundation enjoy property certain ergodicity maximizer particle internal together detail empirically
use nlp lasso equivalent equivalent model penalize mm mm decade derivative analysis coefficient popular selector iterate show deal
sx research support nsf grant n research support china cb grant grant dms dms grant w nf name corollary stanford e sparse noisy differential exist correspond sign path discretized setting path piece discretization lead fast sign minimax setting stop rule identify rely development differential time image euler discretization recover signal noisy eq oracle unbiased vanish meet bias remove see multiplying satisfie path time sign thus q motivate differential indeed existence uniqueness good path argue covariate uncorrelated proper early stop sign linearize bregman
convergence selector go zero deterministic convergence rate distance proposal numerical aspect detail incorrectly reject go reject fix ensure cycle must select initial uniformity reject assume enough select uniformity reject close cycle uniformity accept estimator initial nan otherwise hull et al nan uniformity replace definition
asymptotic size many dimensional high situation situation explain use local weight weight asymptotically addition comparable lowest asymptotic among introduce address power addition give asymptotic investigate conclusion summarize proof section average statistic
probability c cccc cccc independent c k bic aic bic exchangeable aic lasso aic bic coverage summarize target procedure set coverage probability independent especially naive interval selector however configuration line lem state vanish selection confidence constant valid moderately probability nominal aic use fail minimal probability selector drastically aic confidence nominal failure interval especially interval reason observation selector stochastic probability small aic bic regressor study regressor make cover former target word situation hold validation implementation lasso course thus depend additional repeat find h df df df f denote last view def establishes follow imply expression like empty beta beta p g prove far l far inspection every every pair satisfy generate construct subsequent abuse shall argument special influence suppose maintain hold satisfy irrespective predictor selection interval model standard target inference post prove
particular discard turn inconsistent example proceed exist consistently first minimize substitute single piecewise equation check small member absolute define interval mean linear zero leave change sign solve q see inconsistent discard accordingly minimize search reflect multiply general technique improve alternate conduct another meaningful mf recommender system descent penalize real mf pr mf construct datum saddle
positive negative class award micro precision fraction positive fraction actual predict harmonic express negative harmonic f translate hold concave accuracy offer discuss actual positive gold actual negative sum fixing linear mention mean assign gold score assign complementary gold see compare negative costly classification domain positive negative considerable cost information mention micro treat prediction score macro f similar accuracy sum monotonically retrieval researcher extensively
implement compare method domain big acknowledgment acknowledge project fellowship fig rgb rgb sections box box intuition appendix chapter integrate couple gps result approximation even cost big
factor radial sphere radial distribute density eq radial density parameter analytically propose reference parameter procedure next likelihood split depend whereby strictly concave whereby long hence attain remarkably possess enough still end maximize optimality definite solution thus rescale constant uniqueness upon existence positive appendix rewrite introduce transformed vector turn solution recover naturally two equivalently ii equivalently positivity preserve also sufficient optimality concave limit column ic prove
position resemble maxout yet filter pool convolution max pool convolution map product follow convolution offset extract patch primitive filter dictionary implement carry library stack convolution layer activation unit subsequent similarly final two imagenet layer supervise architecture experimental c mini cb mini cc pooling normalization mini normalization propose pixel retain response neighborhood pixel apart
c cluster c analogy cluster analogy h mm percentage cycle duration h south mm role theoretical view call learner dependent possibility highlight exploratory behavior simulator qualitative mainly behavior major interest pathway namely possible learn dynamical analysis
enyi entropy generalize gaussian convolution sum describe diversity l remove show entropy low summation core coherent furthermore nn follow distant dictionary equivalent normalize connection approximate become bound increase number grow decrease coherence measure measure online coherence less atom thus diversity within high threshold shannon p p j present kernel write norm span function r n n worth conduct satisfy approximate straightforward theorem lower distant bit turn r enyi entropy case former list include also small
write runtime primary exponent propose complexity algorithm either nature seminal liu make assume generalization decay distinguish require exception family despite informally graphical decay asymptotically know pairwise temperature ise lattice survey article possible efficiently correlation paper convex incoherence explicitly careful provably ise base isometry likely algorithm happen base markov mix closely mix thus class
proposition since proof recursive update stepsize long approximation infinity subsequence solve systematically period horizon horizon move horizon stage programming equation solve identically derivation l j use logic proof update recursively diagonal covariance gradually expand repeat horizon original store potential benefit optimally vary stepsize adapt outlined action problem insight sensitivity consider normally reward discount policy initial approximation furthermore use stepsize secondary stepsize stepsize minimize error scalar secondary stepsize bias stepsize behave early quickly converge limit point behave stepsize tends happen used issue tuning view slightly stepsize rule give yield parameter harmonic stepsize expect convergence million reinforcement issue logarithm stepsize stepsize yield also omit rule harmonic stepsize properly eventually exhibit improvement single performance rule contrast
bad alternative bad precisely statistical dynamical space go equilibrium review dynamical monte utilize method large point correlation partition function evolution auxiliary speed article modify accelerate stem mix equilibrium mathematic chain markov adopt section introduce mathematical definition hasting follow possible configuration markov chain specify chain switching path get independence row length evolution th px evolve ty assume power multiply also define scope
since obtain indeed landscape tell ground though hard away value lie actual growth still point design idea teacher student comparison mnist first half train train network hide layer relu last teacher mistake sgd teacher label second replace tag probability teacher h c error st student student teacher
randomly spread production critical see randomly arrange eq represent dimensional element batch worth discuss modification case lead explicit plan determine lead numerically appear plan study minimization n minimizer put fix minimize numerically denote turn search pass optimizer stage round control simulation conduct insight property procedure design distributional relevance see accurately second mind estimate sample affect operate characteristic first corresponding cf sample production line numerically invert standardize first four employ biased bandwidth gps indirect repetition second stage plan comparable stage gps selector plan situation indirect comparable represent symmetric small whose notable minima
provide prediction note covariance infeasible moderately sized obvious improvement achieve building appropriately scale uniformly replacement possibility detail build great depth produce incorporate estimator become statistic case incomplete situation x I go asymptotic present central without consistency guarantee incomplete order nx nh k n n base approximately root full maintain relatively subsample many computationally equivalent traditional bagging full bootstrap final theorem k though easily tree great assume response bound bound algorithm asymptotically provide load build tree subsample predict prediction final precisely bag use build full estimator may carry distributional tree prediction minimal regularity condition build tree ensemble build method building forest occur additional determine random statistic u
strongly distribute lagrangian connect responsible outline dual primal generally agent propagate c algorithm nc nc highlights sequel covariance diffusion strategy able verify theorem difference relation close desire range step size therefore stable also even dual utilize consensus positive algorithm study solution act enhance dual define always require definite converge prove consensus strategy become unstable agent note unstable growth consensus equation implementation remain show attain term range steady consensus detailed algorithm optimizer since error error vector evolve accord know optimize conclude create resort transformation ignore redundant dual singular decomposition partition diagonal singular non term furthermore
complexity analysis idea label value high q high variance dominant dominant perturbation translate claim rip need proceed thus complexity incoherent whitening since access slice addition perturbation whiten perturbation term whiten whitening differ sense analyze slice moment therefore empirical slice make perturbation new refer whitening term slice different depend detail claim perturbation translate concentration classifier approach except error rip bernstein tensor loose derive tensor z bernstein assume score function convergence perturbation vector suffice otherwise
start environmental dynamic receive environment update posterior bayes action intend maximize expect return criterion tt uncertain world balance exploitation discount play crucial role relative reward opposite illustration exploration exploitation policy index belief decision material find intractable state either potentially unbounded require trivial multinomial rich inference monte fit planning side planning call thompson capable handling involve plan adaptive plan directly albeit cost unclear integrate development approximate scheme perform base sophisticated sampling planning select action solve action current perspective bayes ts computationally proven empirically reach theoretical regret armed bandit handle mcmc generate optimistic
rhs eqs result establish assertion combine bound c c n k go eq rhs q hold unnormalized prior rhs evaluate third term fourth upper bound yield measurable eq mean square sample predictive expect error source time higher two require recommendation comprise rank analogous define singular enable decompose high several naive require
architecture section target choice viewpoint object space correspond classification contribution definition b possibility distance distance similar localization pose require optimize use variant loss network probability test belong prediction pose evaluated extend annotation challenge dataset annotation object annotation introduce joint pose provide accuracy viewpoint standard precision ap computing consider discrete viewpoint associate size threshold set orientation annotation network train imagenet evaluation mark truncate evaluate provide
conclude impossible distinguished expert exponentially tune interesting drawback come uniformity hard one instead bound like get expert regret key quantity instantaneous eq hold develop variant extend propose tuning rate additional factor secondly bring essentially expert report first confidence optimally return excess improvement begin introduction loss rate k cumulative optimize theorem respect desire bind work trick suboptimal quantity new
assume corollary subsection replace theorem corollary simplify treat fix value item give reduce magnitude dropping item however item show continuous simplify monotone enough validation step sized block validation remain drop randomly h nj overlap aside far exclude select candidate matrix base decay power gap separate validation fold validation variability restriction
semantic recent focused building evaluate separate relation focus evaluation closely possible section development parameter section composition surface employ multiclass four first level hyperparameter separately see optimize development follow instance positive published previously publish recent model inform tag predict relation show distributional obtaining score present publish metric expansion attain work vocabulary collect token token vocabulary experimental vs detection distributional semantic many lexical center shift identification relation key parsing task identify seminal task lexical word tag raw syntactic include identify subsequent explored aggregate multiclass relation
r principle many wish analytical available penalty ensures estimate use vector zero goal reduce autoregressive induce eq lasso set coefficient take impose exhibit interesting behave define smooth differentiable nevertheless eq algorithms problem show smooth proximal term purpose follow supremum gradient ss ns accelerate proximal gradient lipschitz
dpp empirical study beyond particularly diverse subset selection compact include quality item coverage supplementary material item sample matrix spherical exhaustive possible noise add drop repeat sample another diverse pair yield training ground precision quantity hamming mle dpp validation testing state various setting dpp thing item among conduct experiment impact fig know ground use outperform number increase method generally improve close oracle specification vary fairly mi quickly subset whereas mle specify parameterization include ground mis specification effectiveness similarity performance ground truth similarity nonetheless mle encouraging parameterization avoid specification outperform track margin similarity panel performance mle plot red color approach blue color along horizontal axis evaluate
step balanced forest draw take split partitioning recursive put choose choose put compare terminal uniform tree forest minimax rate paper performance risk general interpret focus mostly risk tree forest infinity rate tend decrease fast infinite forest estimator provide forest forest contrary empirical infinite bias theoretical illustrate simulation section three model section close rf notation appear line line decomposition purely forest rest build let let independent possibly estimator several every partition convention deal belong partition tree call forest define consider finite finite risk among function q decompose partition integer q proposition average every quadratic risk term rewrite name wise case tree conditionally asymptotically quantity integrate
ij j bb ab aa diagonal square identity let minimizer tucker kkt subgradient evaluate hereafter mean clear review uniqueness fit lemma proof uniqueness kkt unique regard assume column vector augment augment rewrite kkt condition unique uniqueness fix linear determine normal algorithm article basic detail sake understand low go technical utility couple joint subgradient assume mcmc joint sampler simulate error scatter confirm accordingly subgradient autocorrelation surprisingly complicated sampling section approach resample base method tail tail probability challenge accurately weight simulate sample absolutely impossible bootstrappe lasso simulate directly b diverse autocorrelation set writing examine respective space p vector equivalently vice versa equivalent immediately kkt condition rewrite essentially define unique helpful I p solve collection give easy extend nature illustrate map four
network rnn parent child tangent compositional bottom ultimately example sentence little certainly fail correct relation role play entity sentence address representation semantic play entity rather information logical parsing
overlap screening adopt polytope group experiment demonstrate rule dataset efficient rule discard determine represent different hence overlap lasso screening rule dpp dpp form tucker use kkt screen discard solution dpp include rule dpp screening group overlap group screening rule dpp range group derive screening rule dual vector variable dual project
propagation probability small mixture demonstrate topic table individual mixture mixture item htbp topic probability topic topic retrieval item idea raw topic mixture dataset provide nonzero probability column percentile percentile percentile show mostly topic topic propagate mean would investigate topic give iv ij ij j mean fairly separate topic coefficient threshold topic coefficient edge table overlap node network researcher number researcher overlap different pair overlap explain nature movie rating interested category movie friend interesting though overlap nature influence behavior among represent overlap overlap entry row overlap cccc summarize edge overlap coefficient statistic topic fairly area topic dependent topic mostly separate
close observe numerical parameter therefore hand get z z derivation fact limit base slope dependent term become collect supervise extensive find task relation weight hardness analytically binary equilibrium solution explore weight reveal organization isolate explain previously behavior provide analytic simple part physics physics spin tool constraint
definition keep integral dual schema condition basically note set feasibility constraint trivially satisfied element whose naturally termination feasible relaxed set next core high decompose small subproblem mrf computer great success flow estimation object set vertex every master mrf resource adjust maximize relaxation decomposition relaxation affect speed convergence decomposition mrf define consist per subproblem relaxation subgraphs hand lead large subgraph message interestingly consist subgraphs integer replace constraint negativity lp optimal mrf solution exist type subgraph efficiently belief message pass exchange message graph various besides decomposition small mrf even relaxation cut mrf mrfs know mrfs potential mrfs write eq source edge express one mrf see relaxation solve lp furthermore subgradient admm relaxation difference even subgradient long requirement converge alternative smoothed accelerate scheme apply primal area problem long successfully miss pose recover satisfactory need introduce fidelity possibly penalization justify determination posteriori regularization penalty employ impose lead feasibility hybrid
false positive one voxel discovery level active voxel false positive positive hypothesis logistic svm randomize voxel vary select voxel multivariate voxel voxel different restrict probability voxel absolute change change regularization force close show selection show top voxel randomize mostly introduce positive constrain block subsampling bring show work slice brain voxel whole generate show cluster represent index represent cluster spatially I index cluster feature spatially simulate voxel slice multivariate pattern bb eps bb eps bb voxel imaging fmri brain complete probably voxel improve interpretation discover potential voxel space training difficulty attention amount fail use rate probably discriminative voxel randomize superior negative stability selection voxel structural stability randomize block subsampling fmri feature selection recognition brain read kind pattern lead subject brain diagnosis person feature classifier receive attention application diagnosis select voxel candidate feature discover manner accurately mainly construct classifier ignore redundant informative ignore inclusion informative may
memory per partition primitive schedule update schedule decide variable simple possible schedule shall later schedule dynamically fast converging avoid already converge parallel dependency incorrect primitive worker compute write g fraction worker result iterate primitive worker mf application partition key standard array build primitive ensure date variable automatically variety store synchronization parallel asynchronous present worker ap usually risk algorithmic error error guarantee alternative like ap work shall parallelism cover ml
hypothese useful marker marker possibly nest principled technique model fisher kernel propose simple proportional correlation kernel fail give satisfactory setting kernel due setting genetic multiple q jk jj mutually incorporate contribution g jj jk input kernel marker mutually kernel marker effect mainly account sample incorporate via principal marker group denote write j
recovery ensure secondly way deal practical simulation perfectly even constraint early termination affect bound whether perfect recovery lastly approach often retrieval compressive analyze trade adjust aforementioned enable evaluate provide phase retrieval
precision choose namely snr illustrate decay mse algorithm calculate process behave mse fast measurement perform measurement point therefore seems outperform scenario reason behavior change easier narrow need convergence speed multi advantage adapt without manual estimation also complexity iterative weighting exploit figure indicate weight outperform online learn however compare art include mse highlight krige offline scheme entirely krige technique certain measurement despite figure serious computational number fact calculation new measurement apply gaussian krige certain measurement significantly observe base accuracy fashion iteration parametric sample simulation amount point iteration art learn reconstruct addition path measurement mobile smoothly robust low impose operate time predict location exchange point
comparative two public class vs person probe ii vs person available identification pls histogram drive accumulation feature pls first give image rich edge dimensionality space reduce employ person person people group rectangular ratio descriptor rectangular block ratio descriptor foreground construct consider person
hour speech dataset hour corpus model hour speech testing corpus able hour distribute linearly spaced log term window ms evaluate feature frequency critical system particularly hour normalize per feature decode choose hold hide deep speech rnn neuron hour input dnn gmm dnn dnn deep speech baseline al dnn fisher hour corpus system deep speech ht gmm hmm et dnn et et dnn dnn et cnn hmm mlp n n speech testing speech evaluation tv restaurant car drive text
table complex false positive demonstrate effectiveness aggregation false underlie complex figure table quite across distributional score normality instead residual linear df df node score comparison compare simulation log score equivalent posterior package extend dimensional p pa score default non false positive although result total factor least positive perform former considerably still couple mainly bootstrap algorithm drawback practice alternative aforementioned log likelihood operation average average improvement select building good result table couple dag hill dag learn propose structural hamming distance investigate aggregation able greatly compare procedure couple dag remark definition graphical field biology social linguistic direct dag bootstrap
plot expect intercept volume outperform mean versus respectively though respectively inspection criterion seem well addition approximate volume criterion cross euclidean cube behaviour large natural induce topological natural decomposition boundary tb map face blue greatly blue region greatly shrink red wise blue space ideal boundary natural space kind ideal sphere infinity hyperplane spherical rt sf relatively polytope linear inequalities union lastly serve sf ss decreasing make arbitrarily term approximate xx way end section raise possibility volume might generic qr qx integral allow bound neighbourhood sr nc z
j k kk h give contact infect initially adopt direct entail transmission draw transmission transmission time differently edge case infect neighbor continue infect remain infect entire multiple illustrate cascade cascade dimensional infected infected window never infect cascade trivially show likelihood cascade st ft hazard survival infect node cascade hazard infect cascade cascade cascade give st k instance contact associate ji nn cascades cascade direct cascade cast estimation infer edge correspond node subproblem per infer incoming generality problem subproblem cardinality nod node
generally explore space various hyperparameter promise epoch frequently epoch observe hyperparameter tuning drastically model fitting machine partial training curve tend follow decay stationary exponentially decay basis order accurately forecast variety decision dynamically create rapidly find hyperparameter potentially uncertainty exponential problem develop definite develop temporal theoretic framework process experiment several machine
trivial formulation follow efficiency variational technique rank matrix white gaussian unknown completion scenario choose goal measurement reconstruct study convex nuclear norm set regularization advance
alternate diagonal limit fact component tangent account perfectly compute diagonal numerically unstable discrete derivative sensitive grid illustrative figure recover quite straight form field recover component recover alone next two field tangent plane figure show clarity remarkably cccc show top bottom consider sample direction tangent map three dimension longitudinal survey united data consist token identify job pair occur take span node walk number job job diagonal step motivated essence nature phenomenon model largely job notice change affect
predict log class cluster miss unknown assess partition show log aa aa spline regression mixture five spline notice correspond mixture misclassification robust polynomial cc clustering cubic spline knot cluster em seven knot regression spline model variation number well value cluster decrease rapidly regression spline regression discard gradually converge curve objective spline mixture behave similar become cluster value middle experiment dataset cell cycle use effectiveness time course use standardized construct level point analysis cycle curve five cycle figure spline spline provide partition cluster bottom merge middle figure clusters rand index polynomial
train work even second conditioning multi modal boltzmann author generate sentence adversarial net way consist adversarial capture discriminative probability training perceptron generator generator output train adjust adjust min
use non negative factorization square text term topic document sort word word topic pick tweet tweet close consensus tweet connect cluster cluster frequently close tweet cluster tweet way obvious nmf color tweet relate topic tweet slightly create short right see figure topic find
million allow scalable derive qualitatively size entity merging database analysis process many modern database bayes realistic method elaborate merge pose refer merge share unique database unique entity include record identification entity resolution link record computationally
hessian appear show example write gp reconstruct image path result fig poor reconstruction straight go region little mean result reconstruction section synthetic digit motion datum image improve image camera background acquire give total per attain straight reconstruct along clear geodesic avoid region reconstruct image measure distance fig error straight poorly away point geodesic
completely unconstrained bic agreement pick lead group account dependency well performance set covariate respectively directly model well cluster clear numerically however prevent size minimum framework discriminant currently individually advantage response fashion tailed datum distribution multivariate restrictive incorporate mixed acknowledgement work support science author build prove hold almost integrate side g
model describe apply locally lipschitz care possible contraction admm end level epoch epoch cr ir establish epoch establish obtain result contraction break part batch estimate th obtain analysis stochastic I constant require strong impose enable tight strong convexity relate dual note lipschitz prove lemma weak lipschitz length significantly follow short overview establish guarantee build proof since need convenient merge sl couple together two separable part convenient epoch error obtain result play epoch e next add error care proof
estimator suffer connect problem high help overview possible popular access calculate technique joint marginal bias aspect lag involve additional cost slow dimensional naive histogram slow histogram high near would stationarity suffer degree causality measure asymptotic stationary standard framework correspond causality integrate source toolbox causality limited set stationarity augment test schmidt toolbox manual short economic adjustment article stationarity transfer entropy affect highly reliability density bias stationarity slow two result result parameter choice lag lag additionally smoothing understand kernel kx exp correspond infinite express expansion rkh q cross order covariance data parameter choose clearly drawback focus density histogram choice lag choice lag causality big range behave observe causality lag time perform single lag transfer entropy spurious causality case complex observe important general influence smoothness control large conversely overfitte consequence particularly popular convenient size ridge cross expensive determination lag method could increase lag pairwise increase lag degree decrease small lag apart suggest autocorrelation causality testing describe causal instantaneous causality instantaneous coupling simplicity several repeat acceptance approach acceptance loss decrease calculate test reasonable big decide window allow choose window discussion
proportion curve apply conditionally iy additional summarize b bag growth prediction f essentially instances build bag relevant real application political survey bag space instance formally generation let determine pick bag size variable let bag drawing bag p instance bag well cover instance bag bag draw bag hypothesis distribution choose sequence easily translate addition small subset section easily bag size bag hold immediately number bag addition hold bag algorithm version interest useful privacy learn preliminary align implicitly utilize one
implicit gp equation straightforward would thank helpful input differential suggest explicit analogous iteration previously suggest yield standard ode xt differentiable order embed order vast introduction several family implicit speed ode potentially represent value ideal problematic classical solution approximation correctly
ratio observation
tackle discovery formulate privacy preserve decentralize detection multi protocol detection walk preserve discuss future network people recommend people interest topology discovery operate social must interest people may raw towards routine necessary involve current service facebook play role party discovery accomplish node profile social return recommendation complex one level datum successfully
numerically evaluation positively correlate discrimination performance correlation strong value cancer z range cancer implement spectra detailed generate list reference z ratio indicate process spectrum code task outline section vs systematically explore belong three cancer discrimination vs vs cancer restriction force priori number signature signature length inferior computing force signature inferior mrf discrimination spectra outline yielded associate correct decision achieve derive procedure maximize optimize signature reference vs signature retain clique section classify sign evaluate absence see optimize spectra dataset map affine separate cancer display good affine explicit signature discrimination first signature fairly length cancer discrimination task length obviously advantage ratio benchmark discrimination signature take account mrf discrimination reach discrimination h cancer vs vs vs cancer vs vs vs vs cancer performance reach simulated signature discovery performance margin vs vs discrimination vs
consider canonical discriminant literature canonical vector fashion separate estimate come guarantee set express close simulation datum keyword block discriminant feature selection wide application finance biology challenge popular visualization seek maximize variability respect variability pattern novel estimation selection group discriminant goal addition group set propose efficiently generation point suggest natural grid grid carefully choose mostly computational simplify consistent user rest follow discuss insight
back pa ga ti step yield improvement bi prediction accuracy propose primarily way restrict abstract series fine tuning generative learn represent frame across frame video frame common bi machine model whose formally bi image angle pixel space frame hide unit subspace transformation hide encode image rather content
independently detect amount illustrate numerous shift anomaly scenario involve delay describe explicitly specific sign provide detailed setting aggregation maximize anomaly detector knowledge shift quantity early sign normally natural test test coverage case shift compare statistic two possible population length window change case rough maximize several recommendation scale simple ease indeed raw generally interpret easier still therefore indicator correspond rejection give case shift finally point increase specificity detector quantity compare
nonetheless achieve expect operator derive unnormalized semidefinite trace operator ref two semidefinite define triangular vary convenient parametrization q semidefinite vector define triangular take transpose number partial tr b additional relate trace unnormalized operator term cast norm enforce absolute lagrange multiplier enforce infer relation relevant wide give correlation whether influence unless intervention causal generalize conventional complete causal analogue classical alone provide optical optical randomization extract relation quantum correlation follow explain variable causal variable cause act ambiguity system exhibit quantum find surprisingly pair order system top cause common cause acyclic graph direct conventional quantum circuit quantum system box state operation discard circuit gate either top swap circuit dash scheme output state setting outcome passive scheme output measurement projective fixing setup gate notation optical
denote training similarity affinity pairwise relation hash preserve hash closely hamming calculate code mh similar hamming pair multiplication prevent task employ tree hash hash decision tree output hash compare method technique th datum bit auxiliary decompose problem code datum binary way complicated decision tree become task binary bit conditioning th bit equivalently quadratic code bit bit bit
edge argue particular stochastic able even limit entirely satisfactory dimensional number edge term secondly problem partially operator exist extend strong limitation spectral backtracking operator actually bethe hessian physics field show operator directly non backtracking perform stochastic sense community soon standard world benchmark give bethe detail property connection backtrack ise spin spectrum
incorporate tu penalty tu envelope worth positively homogeneous give cf q one zero x direct section ccc limit encode graph tu edge incidence bipartite encoding pairwise function fact envelope ball linearization trick reduce tu cc show objective structural minimizing envelope regularize regularization unit section produce spike value dimension compressive random datum produce datum relative compressive regularize formulation bp tu budget describe lie include constraint point
observation algorithm extreme quite relevant see distribute among fig significant general fail median median regard moderate huge attribute amount miss within result frame q would able unseen two test dr spectra ls rest frame miss spectra definition spectra miss component discuss may missing explain highlighted coverage range illustrate regard absence datum eigenvector show good algorithm ls ls noticed width emission miss consistent come large notice l subset fail converge large correctly reproduce emission
reader question necessity efficient hilbert directly wasserstein latter preferable nonparametric setting establish weak concentration around dirac wasserstein rather posterior respect hellinger universal detail outline yield I assume condition hold eq part independent imply typically slightly bad contraction difference handle bind throughout consider simplicity generalization conclusion obtain condition see address situation statement problematic hellinger wasserstein hard estimate practice next geometric metric distribution hilbert discuss define separable reproduce property therefore l define give hence imply convergence wasserstein hellinger metric translate constant hellinger two distribution q
cycle convenience scalar iteration previous update work task local linearization involve introduce accordance former linearization multiple assimilation justify situation conventional iterative enkf sub optimal instance nonlinearity challenge filter statistically show later consideration account iteration local evaluation jacobian discuss give formula formula root precisely transform observation linear error localization eq addition generalize introduce scalar front optimization algorithm special intuitively approximation estimate tends inverse implementation conventional use assimilation difference regularize method focus residual specify residual residual criterion change inverse useful iteration rule scalar gradually reduce aim prevent drop analytically residual norm sequence satisfied may desirable residual norm prevent either state reach b process outline eqs
aggregation estimation proof need theorem existence asymptotic expansion natural variable probability click valid bandit produce go concentrate static actually issue rather tend make evaluate policy algorithm produce long nothing due chernoff classical especially randomized example argue cost complicated smoothness non justify true introduce experiment compare model linear news news display kind news universal interesting ii news consist contextual approach contextual bandit recommend news evaluating fast
specific identify understood fig topic document topic document generation share topic iteratively current specific word non proportion document likewise fix topic estimate follow computationally jointly perform minimize widely accept comparison logarithm bic balance goodness limit type goodness derive bic modeling dimensionality huge bic aspect type share model topic validation solely corpus minimum topic bic estimation subject introduce prior skewness topic asymmetric prevent word dominate however parsimonious proportion specific fashion spike sparsity use share provide relevant topic sparsity word probability use model small proportion cross combination background argue generalize keyword introduce huge every topic topic proportion probabilistic give proportion topic specify topic lda exhibit salient fashion rest universal
step support common update index row measurement update individual projection w combine refer sc belong weight equation test correct also term notation express matrix form stack training linear assume go solution residual residual residual recognition task single available realistic assumption sequence comprise video frame
exhibit also movie via cluster factorization real rating algorithm simulate movie recommendation recommendation algorithm amongst friend star dense still miss entry rate item rating simulation receive reward mark netflix rating nonzero entry netflix result user u user recommend make movie recommendation movie completion item recommendation item feature rest furthermore reveal dm thresholded rating thresholded rating dm estimate training give compete outperform result netflix similar propose online justify collaborative work key type al neighbor collaborative filter predict unseen study popularity friend examine ability predict rating understand move limit user type type recommendation
grow literature optimisation software package hyper unknown hyper focus deterministic new hence base assume set assumption trivial suppose z combine get convenience know write know f f f c simplify
accuracy hierarchical hold water equivalent cnns feedforward one quick intuitive rigorous necessary van inspire become default report mnist handwritten digit straightforwardly extreme rapid training error implementation individually combination indistinguishable innovation ensure operate patch weight percent hardware iteration backpropagation number unit require suggest challenging dataset type network problem activation occur vector notation convert activation plus value utility combination describe introduce denote vector relevant class mathematically entrie entry class vector activation training vector ideally eq although exact potentially usually solution standard solution express equivalent solution
hard thresholding infinitely penalty exact group divide variable group entry belong gradient detail lb ji g ij matrix matrix cover scad many proper form especially nonconvex penalty enforce parameter easy prohibitive rather q control diagonal loose solve quantile j p successful base matrix set thresholding threshold element zero see thresholde minimum put upper belief network specific usually guarantee expression network seem formula cone difficulty size propose simple asynchronous type denote search subsection convergent satisfied basic line restrict along direction condition satisfied accept carry otherwise try iteration initialize decrease
conjecture thm remark proximal admm collection solver cholesky subproblems intuitively try binary mapping matrix rectangular subproblem briefly
far make preferable gain style algorithm know avoid invert storing form p expect row pick space positive feature ridge regression see main approximation matrix aim style never form update update cost per ridge cost per iteration parameterization latter track via
fig look right look copy make provide reconstruction tumor innovation previous method impact allele frequency contain incorrect inference frequency affect allele integrate datum seminal observation provide automated method reconstruction overlap alone describe infer branching contain lead read base accurately tumor clear extent automate reconstruction describe furthermore demonstrate correction change perform specifically much recover highly copy number state fail reconstruction rely sequencing assume copy occur copy development decompose restricted region copy branch preprocessing population completely detect population detect rely error reconstruction branch equally alternative reconstruction seq even highly achieve reconstruction limited region benchmark stage manuscript new combine reconstruction unlike allele reconstruction
program appeal strong threshold admit expectation polynomial idea polynomial exist make reduce theorem upper jump threshold impossible low formal infinite finite inequality weight finite essentially learn log normal laplace capture everything law main motivating relax concavity end result agnostic technique start concept learnable concave distribution absolutely logarithm multivariate logarithm concave concave laplace natural contain heavy tailed distribution smooth distribution hypercube sphere degree l prove interested
match truth two source calculate statistic tp fp x produce produce carefully entity source true feature feature score entity score randomly entity pair correspond add true entity start value hard vary create level entity source entity evaluate source entity entity pair correct precision harmonic recall employ unconstrained er denote give source score allow entity match entity datum help resolution pair world million scalable operate primarily answer multi match bipartite source statistical total matching movie effectiveness matching add bipartite matching source bipartite obtain movie source six record source result dot result pass plus record color pr threshold specifically regard entity similarity never match entity resolution new range threshold message pass greedy add constraint movie one new source go addition
appear unlike necessarily sometimes see fig subsection update dictionary step simultaneously traffic traffic representation implementation first depend polynomial total parameter practice easy store dictionary learn svd mod forward adjoint case ty vector compute cost fast way exploit ty efficiently compute numerous processing advantage structure dictionary representation omp soft thresholding main chebyshev detail third benefit costly sensor structure learn computed structure make signal fashion parameterize dictionary polynomial give belonging translate vertex polynomial kernel localize fact translate area lead comparable dictionary approximation dictionary robust available size training
latent stochastic update every learn progress markov chain chain close data acceptance rate decrease notable effect dual operator split big previous comparison method score preference analysis mf factored model use ascent loop also run provide mf mf randomly matrix rank mf accord ground rating performance normalise discount cumulative gain truncate ndcg reciprocal sure metric put measure ndcg rank evaluate collaborative netflix challenge apply movie user increment divide
ct ct varied voxel mm ct convert software package ks structure thresholding acquisition partial medical ct ct intensity density interior topological appear thin head cavity bridge extraction segmentation surface surface mesh remove equivalent sphere various topology topological already extract surface topological simplification directly correction surface surface intersect check characteristic automatically equivalent solid definition object image convert slice ct axis topology slice result correct surface topological operation mainly relatively implementation neighboring voxel image concave operation operation instead operation may sequentially binary volume operation operation direction operation operation cavity operation easily disk correction reduce slice unfortunately
could equation py px py compare p xt dt x x indicate optimization whether maximize directly maximize theoretically relationship sm derivation start agreement agreement close maximize maximizing maximize intuition subsection follow subsection clear motivation associate detailed get big bias towards indicate hence py indicate prediction px px py px hence sm predict output x gp maximize sum htb toy example task pose schmidt kernel pose dataset evaluation beyond estimation inverse behind distribution therefore choose start specification toy subsection setting select argument task value uniformly star regression suffer effect indicate space example use input prediction example ht challenge see error total toy
transform autocorrelation autocorrelation obtain order result double bivariate gaussian variance double integral enhance numerical use use eq piece piece wise transform comprised polynomial segment q piece partition domain doubly doubly truncate gaussian piece affect transform therefore estimate interval rather use continuity monotonicity maintain comparative bootstrap large linear accurate estimation adopt multivariate variable pair addition correlation sufficiently consider power frequency difficult achieve case match well spectrum realization decompose four
return alg require nevertheless practice execute likelihood part false p vs p vs p p k p policy alg select sequence pyramid treat alg summarize active htb pyramid likelihood pyramid compute break break compute location uninformative line round part store policy terminate foreground obtain final discriminative label background filter final compare score operation location search possible score alg total turn incorrectly
multivariate skew mm discrimination multivariate skew elliptical mm scalar skew graphical model skew j leibler asymmetric heavy tailed b enyi relate family pre mm e leibl la skew error application profile mm n incomplete j mm ng w skew elliptical beyond volume van discriminant van skew mm univariate york mm population performance discriminant
also layer prevent dropout dropout select select back propagation activation multiply normalization network refer detail heterogeneous collect pose label whole sa collect body body stick left body part head leave sa illustrate definition annotate bound dataset randomly box extract human mirror double augment factor body remove extended pose predict position window equal find initial start dataset initial
file efficient essential various analysis etc testing important book review property gradually new base characterization follow negative identically distribute f equally exponential able validity nearest proved observation continuous f composite family unknown f empirical f observation
equip denote net sup aggregate element give upper net fail metric slow one obtain trivially bound phenomenon concept mention empirical finite even upper bind illustration bound minimax rate couple example linear predictor combination cover bound metric space entropy banach rademacher yield control minimax rademacher enjoy rate specifically regime minimax depend rate depth say tree large tree denote cover follow
define domain make low respectively cross scatter clearly pc look pc rescale unweighted cca pc pc decrease rapidly datum treat input agree relate wrong distance situation bad
dependency template state rating person interest movie production movie classic logic template semantic also concentration inequality example increase interest structure datum citation chemical interaction important field assumption extract many
brevity term mask mask exploiting trick require white background figure public specifically provide angle angle include near color model around keep aspect subtract build cnn value pass whole epoch learn matrix recommend energy never start decrease expect network last use except viewpoint interpolation section reduce time task largely remain restriction interesting supplementary material successfully force row another column transform even transformation quality image basically office fine row
invariance preprocesse mm computation learn individually optima practice free free improve performance freedom need case retain invariance rotation adjust spectrum require instance rbf rbf suffer concentrate fix relatively design e part fourier transform analytic compute inverse fouri readily expansion parametrization provide expression function mode parametrize piecewise moreover piecewise draw occur probability invert quadratic note considerable choice location cumulative location particle yet basis I b feature rescale relevance determination piecewise approximate radial piecewise generate require basis partly gaussian matrix expressive obtain optimize scale property sample chi adjust radial marginal combine procedure input generalize additionally wide class hadamard preserve modify keep multiplication fouri unchanged repository flexible applicable million process beyond secondly hyperparameter involve order e ghz gb ram marginal ard gm grouping rbf ard dataset every pick test report result rmse partitions rbf form kx ard kernel kx ard
fa david decomposition column together row km via randomized algebra approximation input nr r factorization k desirable attractive algorithmic construct optimization k progress span numerical algebra science remain sparsity running question ask question stream topic second run polynomial error column row indicate column row construct section section fast dense contribution subsection subsection idea summary rank later algorithm explain implement etc summarize put context summarize present contribution answer section sparsity select optimal column theorem exist algorithm column form column row column match low ever relative select additionally lower progress accurate approximation non although include matrix input address although deterministic construct
density student distribution pdf density parameterize discuss student identification use variance residual probabilistic review identification set kernel due gaussian realization define scalar impulse response assume joint vector value represent covariance instrumental derive impulse minimum square bayesian jointly efficient marginal obtain integrate adopt paper description briefly previous apply deal represent normal adopt pdf compare thought generate two draw eq model due vector value
assume minimize thus infimum support part li lemma challenge grind applicable attribute overcome challenge propose essential unsupervised meta rule regard unsupervised attribute model cubic curve control hypercube meta rule rank able manifold interpret dataset comparison state art approach list unsupervise multi monotonicity smoothness skeleton htbp viewpoint machine hierarchical able evaluate ground seem challenge difficult issue viewpoint type rank divide multi data representative approach rank variant candidate link attribute attribute object summation scalar assignment give rank first learn object determine propose attack mapping preserve requirement neither vector quantization score attribute nonlinear
statistic rv analogous long move however von freedom sharp note neighborhood p detail repeat monte material sequence covariance apply transformation sequence nk sd base ep rv nk sd ep rv statistic case times ep construct rv standardize
exactly v result conclude root denote extract triangle definite third inequality due unconstrained optimization em qp definite know volume since space challenge instead focus hypothesis measure principal axis hypothese long axis near axis large principal preferable close volume loss fit binary setting naturally include adopt link ellipsoid volume unlabele
player though game rich attempt score depict b stay move take action turn north east west player player player ball toward opponent opponent act simultaneously period attempt opponent reach ball b attempt reach assign randomly correspond ball aim maximize goal subject discount party observer play try play statistically calculate rate h worth would offer help decide b reward brief decide reward opponent conventional opponent part respect specification prior later subsection quality reward negative state ball
n weight lead estimation sgd verify instance parameterization parameterization canonical generalize formal sgd parameterization family log idea carry extensive experiment fisher scoring estimator sgd computational fisher score r versus package implicit sgd package set analyze national air specifically numerical incur small efficiency cost quantify explicit euler approximate differential explicit sgd straightforward attract method large problem sgd datum find diverse area scale online implicit sgd procedure poorly understand arguably least filter implicit robust form usual likelihood normal model attract method mirror implicit proximal function minimize proximal operator establish proven problem present quantity know predictor affect variance appropriate parameterization appear furthermore convenient known function rest glm space hold n variance property devise sgd initially estimate reasonable every covariate vector distribution accord glm observe application definition stochastic generalize iteratively step vector implicit sgd definition concrete omit gradient see factored definition correspond baseline
apply sense bayesian context property estimator absolute loss instead brownian bridge study derive mle investigation interval base performance investigation similar credible
attention database exhibit human tend prominent object occurrence answer time answer capture produce wrong propose inspire measure boundary become fuzzy seek answer naive answer n answer set membership equality whenever variant wu word empirically require answer weight magnitude figure refer weighting correspond plain benchmark architecture require answer score highlight challenge unknown true form advantage world distinguish
next invoke eigenvalue covariance expression operator computationally avoid explicitly list equation nothing bayesian eigenfunction eigenvalue product trick j mathematical physical account mathematical rise pde functional functional determine pde physical distinguish incorporate procedure give form third parametric provide systematic optimally covariance observation posterior saddle admit variational permit formulation flexibility quantify prediction popular inference estimate estimate determined therefore inference powerful quantify prediction assimilation problem incorporate state gaussian
table build another fine precede initialize net minor share layer dramatically mb testing hyperparameter affect number large categorization enable obtain substantial minor much fine cnns mb mb baseline similar cifar convolutional layer cifar double cnn fundamentally average average classify category difference initializations cnn classify cnn average independently train prediction cnn average cnn performance category consistency tune multinomial fine tune coarse consistency method cifar imagenet generality cnn nets cnn significantly test block net ht truth net imagenet top make top fine public building
device box maker cd computer disk head pilot phone machine table gate light house pyramid medium air school daily item play top mask saddle music background unfortunately concept class manually class new newly class group minor fluctuation compare cifar employ interestingly relate similarity visual computer computer computer electrical device codebook help boost codebook might due complexity shot still object miss stage
know precisely characteristic rarely theoretic elegant definition express rich dependency one define tree measure set experience game name player action move causal experiment crucially player express special ability see assumption strictly knowledge player move miss state game indistinguishable player move game analogy people yet go great maintain people draw thing thing novel responsible maintain symbolic rule syntactic perhaps finding analysis appendix firstly interactive order logical state refer appendix consist represent weather pressure assume elementary physics know weather dag subject plausibility compete h depict crucially separate induction causal arc variable two single diagram add diagram dag importantly specify wu inferences htbp contrast situation tree allow causal dependency dynamically weather semantic self indicate probability importantly encode determined path indicate leaf multiply htbp belief encode node finally root causal w consequently distinguish purely decide player consist collection nest causal enough algebra instance suitable modelling state label htbp action observation course game subject keep track probabilistic move depend private subject interaction happen possible move namely experiment transition compatible transition
orientation cluster keep focus difference group connectivity kernel modify affinity geometrically adapt dataset positively influence capability already simulation relevance orientation circular clearly succeed distinguish remain partition plot diffusion constant correctly contour distinguish semi circle partition case affinity case coefficient stimulus wide nan half condition observe affinity object straight retrieve object two element circular contour get unit prevent contour recognize object embed orient segment kernel contour grouping affinity time dataset intrinsic causality reflect carried affect efficiency define cope modify still normalize affinity transition matrix balance reversible eigenvalue obtain modulus perform thresholding far cut approximately transition discuss minimal cut cluster different direct set mark difference symmetric hermitian understand n equivalent eq n rewrite natural double copy sufficient grouping broken speed orthogonal movement segment find maximum local speed make indeed random speed could background lead brain group together rely orientation coherent velocity field smoothly study determinant purpose contour trajectory primary temporal optimally measure stimulus velocity stimulus direction movement
disk bayesian jeffreys proper range ensure choose long tail uninformative shape uniform embed evaluate multiply conditional determinant improve block gamma form pointwise accept accept unchanged choose probability spatio provide huge large conditioning processor gb ram mac os system fourier package mcmc disk shape burn indicate clear unimodal histogram draw minus summarize spatial posterior mean panel draw agree observation converge unconditional unconditional center panel illustrate posterior field plot domain
sampling loo use used obtain laplace toolbox use monte provide result ep approximation hyperparameter weight point grid point usually usually small monte well loo remain loo towards integration quadrature loo loo quadrature quadrature accurate e true quadrature exclude however posterior unstable approximation accuracy tail loo contribution influential correlation strong la simplify problem approximation fail loo case removal datum failure loo la ep several severe failure la ep fact failure marginal ep la ep loo l loo alternative insight improvement write global cavity ep correction increase evaluation small point interpolation sometimes instability produce global result
cite sophisticated meaningful publication evolve centrality centrality fan fan cite author paper table among paper selection primary paper collect author wish influential important area need effort area author community area contain author interpret presentation name group author paper finally network real name beyond community organize centrality discuss contain discuss limitation pattern trend frequently author paper centrality measure centrality centrality closeness centrality reciprocal extent locate centrality conceptually centrality centrality author citation citation paper centrality degree paper table present author identify different centrality centrality largely fan author htb closeness fan fan fan identify centrality measure cite paper selector paper know penalization lasso many wave research devote penalization htb l ccc closeness fan variable fan al li bayes multiple variable li variable selection include covariance empirical www edu list file account number citation count list cite paper adaptive hand important lot broad community google cite
result dim c rbm cd rbm compare algorithm close applicability rate considerably rate along table deviation compare counterpart range layer interestingly autoregressive train model perform well train outperform expressive model rbm cd cd considerably sample compute handle posterior ability autoregressive within dependency considerably one factorial layer autoregressive connection r dim test factorial apply document train generative represent know bag train corpus similar model
kp vertical discussion promise perform improve abc call simulator naturally pseudo several technical gps modeling likely exception simulator noise noise limitation gp assumption covariance major gp abc role phenomena value expensive setting interested insight importantly challenge likelihood partly abc free describe section simulate proxy provide framework progress model inefficient mind simulation maintain surrogate parameter main able posterior accept expensive procedure expensive course model gradually time reject uncertainty insight determine proceed mh simulation
player know probability chain game base hide quite approach fact nash ad prove play nash equilibrium least equilibrium game show equilibrium type nash pure mixed nash equilibrium bayesian games player payoff profile tuple game player strategy
vary depend matrix benefit regularizer smoothness uniform pattern follow uniform user movie independently sparse identically epoch distribution scheme rating set surprisingly suffer uniformity fig value nuclear medium graph uniform report real result aforementioned rating star increment two purpose firstly arbitrary submatrix secondly construction detailed non uniformity sort movie th percentile observe correspond rating movie user movie discard quality important detail
important aspect identification summary model summary kernel statistic poorly information necessity abc summary probably principle identify develop score interest couple choice computational abc classic example imply development efficiency abc convert sophisticated importance generation early likely reject develop computation research state development tractable closed likelihood usage decade could write max stable close population use abc tractable far remain toolbox availability abc nature construction likelihood computationally prohibitive problematic purpose focus framework bayesian computation fit idea behind demonstrate interest modelling multivariate hold mind location due differ non unit use distribution transform multivariate fr generality fr margin preserve monotone eq simplex angular spectral measure determine marginal pz
account consider axis objective reward sufficiently one force pareto front occur return get wide tend expand opposite trend devote metric learn algorithm learn start indicator converge parametrization
search entire average training transform localization response sometimes filter provide localization thereby result localization ht localization inter mean across run bar design without account ccccc leave evaluated design consider window e shape dataset consist category consist decomposition cf filter precision precision object single template learn conclusion paper fundamental formulate frequency explicitly multiplication circular result exist intend function remove circular effect cf show derivation filter design address challenge introduce design eliminate author like acknowledge air force research laboratory resource quickly present bs electrical engineer ms electrical ph university currently future technical gold award student engineering interest filter efficiency processing receive engineering institute ms degree electrical engineering program currently institute research machine institute technology receive conference research air force laboratory sensor interest recognition computer
term derivative formulation replace constraint positivity ik ki convexity constraint verify jk compactly notation ik jk n kx center therefore eq alternative formulation define ki optimization likewise solution define jk k stage output residual dc formulate estimation subject screen consistency variable ac set part first sufficient sparsity stochastic argue probability change line ac dc output use optimization boundedness practice construct relevant satisfie kkt define show optimality index lie contain set th th large entry solution solution program hold regardless impose boundedness set additive convex set guarantee relevant analyze positive restrict analysis standard analogous nonparametric detail probabilistic treat setup positive response support f unique minimizer k independent subset twice differentiable f signal excess risk incur support strictly risk uniqueness play coefficient
length gamma variance notice parametrization gamma scale scale multiply shape limit arrive challenge series process main consideration approximate numerical treatment type search intel ghz implement alg homogeneous laplace allow predefined branch regime free specify branch share trait environmental see biological jj f gamma
sdca execute machine ghz memory gives save compare net compare nonlinear apply top select update resource utilize rate alternative synthetic dataset datum rbf bandwidth choose time median batch justify convergence blue dotted guide iteration could converge seven ridge demonstrate clearly full cost sdca nystr sc vs solve dataset sc sc kernel obtain trick regularization parameter feature go rate achieve sdca illustrate flat region unbalanced rbf bandwidth obtain median trick regularization block sc stop good rbf kernel
publicly specificity disease disease process dr diabetes frequent cause case work population people diabetes far million people develop suffer form people dr quality fm dr screen partially clinical protocol dr reliability medical ensemble applicability breast use classifier ensemble classifier improvement processing system fusion fusion detector efficient consider processing image regard dr
identifying loading cpu run kb ms contexts shoot shot formulate core entity match bipartite graph structure huge appearance variation ambiguity efficiency basis co statistic dataset demonstrate scenario demand storage computational apply real question consider computational calculate wise latent modify learn separable appearance accelerate would optimal affect behavior word occurrence interesting camera add match entity enforce acknowledgment material department science university award ed view represent express imply zhang person identification person identification surveillance maintain entity individual location camera challenge appearance individual illumination maintain entity camera jointly account weight graph base co occurrence occurrence base appearance infer unlikely occurrence appear shoot multi shot computationally person match visual co occurrence shoot surveillance camera deal maintain location overlap camera
canonical correlation minute minute c minute minute canonical correlation variant ghz amount run drastically I store order multiplication allow model cca implicitly randomness random fit available budget contrary art ten parameter autoencoder regularizer view validate grid desire several truncate learner exclusive feature teacher build construct help reveal little achieve leverage randomness key multivariate spectral demonstrate experiment compare art
toolbox frame adequate domain orthonormal synthesis accelerate describe level pre tune keep reconstruction bss al decompose source sum interpretation target word account stand invariant ratio sir result design latter criterion evaluate since account reconstruction provide section train decompose combination negative atom problem negativity w use version laplacian compose performance role level trend smooth enforce method type adapt cope peak suit impose orthonormal expect wavelet enhance separation source retrieve model mixture base tune tuned reconstruction source source particular information reliable later smooth nmf able recover adapt well
art technique therein seminal dictionary learning make k et mod svd denoise behind capable patch two clean image noisy suitable overhead svd fidelity perspective denoise image super resolution suboptimal moreover minimization learning solution
implement forward description dual essentially amount weight span norm solve implement variant achieve well difference likewise set fw description optimally convexity inequality note deduce use bind subtract define q decrease always choice iteration obtain subsequent choice eq substitute eq denominator sequence inductive step assume hand increase quantity upper substitute complete example remark aim atom atomic norm regularizer solve well atomic norm describe call produce reconstruction generally norm truncation exploit nature reduce via experiment signal forward outperform tailor
discover recover portion subspace hypothesis plant sparse embed nonzero entry recover comparison special drive round complexity scalable moderate regime algorithm well perform dictionary vector break barrier literature optimistic applicable dictionary nonconvex also global basis point formulation nonconvex optimize relax necessary force live give
virtual pass token perfectly connect virtual infer great message underlie contact g social anonymous communication dc vast topic accommodate unbounded fully network message model detect origin epidemic recent analogy receive infected act infection adversary contact network infect certain pose centrality centrality measure random increase multiple source source infect allowed recover sir centrality number infect node message platform overcome design protocol fast source focus anonymous underlie contact phone facebook friend system introduce delay delay user message discrete system message immediately protocol delay message inference message difficult pass model adversarial know contact observe snapshot state node receive message strong whole contact state adversary aware receive adversary device device adversary learn snapshot infection give observe adversarial also capture state continuously anonymous protocol call strong adversarial start spread contact edge assume delay user via deterministic
programming express entry package square estimation two way bold impose integrate feature constrain square voxel j like pilot nuisance projection form separable least admit closed form solve respect little singular solve iterative notation remainder j equation q initially pilot generalize least program update use lagrange multiplier compute singular find monotone omit give voxel induce voxel assumption sufficiently j multivariate distribution uncertainty vice result allow glm correspond across brain degree justification rely voxel activate unconstrained easily test pilot define asymptotic inference voxel sensible test activation activate voxel inside identically sized dimension bold fmri stimulus varied systematically duration vary nine create stimulus canonical nonlinear net
vector combination weight vector art locally svms vector local predictive model neighborhood process near neighbor model initialization locally anchor produce construct weight manifold value cluster svms learn svms simultaneously al kernel intermediate respect tree region change weight svms maintain ability art propose exhibit specific predictor meaningful case denote basic notation c index partition input weight th linear partitioning region figure concept dash specify weight whether apply point figure specific partition wise weight
triangular follow corresponding get digit within triangle zero beyond corner triangle converge convention center triangle yet arbitrary follow corner infinite integer rule triangular desirable balanced center symmetric final reasonable third first triangular van discrepancy triangle work unit discrepancy calculation side standard parallel panel decomposition centroid horizontal pointing purpose triangle integer degenerate numerous discrepancy attain hold sign discrepancy maximal exclude van point signed attain discrepancy contain point
relation set relation close range behavior relation lexical embedding linguistic observe rank penalty embedding model certain refine show task embedding benefit quality lexical
sentence translation automatic test sentence term removal refer keep character long appear sentence vocabulary dataset present qualitative reconstruction help aforementioned denote denote pre tune operate understand amount distortion network update much calculate softmax case comparable preserve reconstruction similarity descent backpropagation well assess understand notice half cosine report representation explain
prove time cholesky dimensional brownian motion hold tuple stream detect change independent propagate sensor unknown sensor consider markovian propagation different propagate configuration point max approach consider criterion delay path point objective stop stream observation couple correlate correlation correlation assume work decentralize sensor sensor cumulative binary asynchronous center well centralized distinct alarm alarm actual run uniquely appropriate parameter would delay optimality prove delay recent include sensor couple drift
project project category rank opt previously return outcome project project sort twitter recommend score content recommendation table static feature percentile mm initially project view nothing return list seek financial reward support cause contribute small traditionally fs friend site act act different project project largely project technology game frequent project rely friend family perhaps employ social site reach technology project look active frequent news identify frequent might recommend pool expand pool beneficial twitter project predict potential project twitter activity
predict beyond future history dynamic pass attempt shoot transition player transition influence transition pass include pass pressure shot outcome pass shoot informative yield combine contribution full resolution view full rewrite eq substitute reasonably briefly alternative proof immediately omit chain resolution illustrate conditioning markov resolution short state call transition pass shoot attempt mark shift form basis carry among process well pass
allow action power ratio learn mixed mab define triplet scheme indicate product ucb arm discretize learn discretized value discretization old continuity figure another advantage know horizon horizon stop end every round trick mab ucb completeness arm maximize tu duration play obtain algorithm choose I strategy choose algorithm maximize objective time scheme increase performance get converge decrease indicate separate euclidean cumulative increase learn learn fast converge strategy hence dependent wireless etc wide wireless case knowledge wireless channel bad confidence bound confidence regret arm define set arm optimality
support technology innovation publication reflect ex calculate similarity modality challenge many modal compare similarity realization similarity object link service face infinite
perform performance label object therefore propose eq unlabeled objective supervise lda additional assignment partial assignment class problem form particular labeling toy benchmark dataset train supervise website illustrate consider toy normal center bottom gaussian top decision boundary actually correspond happens draw force fall
fig boltzmann backpropagation deep boltzmann work except likelihood deep boltzmann probabilistic similar would somewhat flexible sigmoid mixture expert one ask happen would line eq ensemble hand expert criterion resemble handwritten digit see middle minibatch rate stop
analytically computational method propose hasting geometric rwm space proposal rwm proposal manifold take qx x g x simplified mala proposal markov chain et conjunction marginal mcmc article simulation highlight geometric mcmc inverse surface ground vary fraction scatter reflect later observation return return times challenge infer medium wave manifold mala figure density incorporate particularly biological nonlinear dynamical comprise six cell system nonlinear eight complicated observation rwm geometric mala report inference size per consider infer jump chemical reaction consider chemical seven rwm simplify mala report accord simplified conceptually simple correlation make al method clinical author highlight difficulty sparsity identifiability discuss describe activity likelihood base approximate spectral evaluate take less five
alternate left figure target elementary conditionally von dispersion sampler accomplish aforementione normalizing estimate partition function classical four node right add left display add node node circle circle draw right right edge edge edge cm main circle edge edge edge topic comparable square mrf relation variable graphical simulate seek distribution four propose one sampler sampler particle sampler arbitrary non gaussian discrete tree tree sampler partially gibbs top ordering give
size stochastic stochastic good step english wang english train uniformly equal understand model english wikipedia embedding english belong create name entity wikipedia second cope annotation extend matching maintain wikipedia article wikipedia language topic category entity region organization organization person map internal entity bias label overcome bias example section language language english identify entity article unfortunately english annotation mention page leave entity phrase english
statistic nonlinearity substitute mixture avoid es probabilistic mixture exploit fact motivated detect align dna knowledge connection two node inside community bernoulli inside community otherwise guess belong community one mixture scenario figure correspond community take advantage enforce community mix community interaction figure
consequently asymptotic probit probit probit analogue factor evaluate fix expectation conditional expectation coincide correction assume small derivation covariate correction probit conditional measure form logistic necessary fitting perform fit covariate omit model covariate normal correlation case probability take skew normal conservative response
nonetheless gradient convergence oracle go proof discuss interpretation relation read fy fx smoothness quadratic fy fy fy assumption precise framework duality clearly always project step size extract lipschitz come back project subgradient obtain multiply yield claim condition improve notation convex smooth descent satisfie obtain x g x conclude one factor complexity step gain reasoning lemma convex convexity smooth prove thus get give claim smoothness prove convex smoothness one x fx x simplify presentation black mapping simplify section make black box span n lipschitz procedure convex strongly oracle ask ask subgradient span necessarily remain compute value minimal prove take low proceed eigenvalue hessian small eq symmetric convex must imply prove remain minimizer span define immediately conclude simplify limit precisely chapter valid choose work clarity exposition let infinite already thank give infinite equation conclusion straightforward computation gap prove respectively gap accelerate published restrict attention case everything constrain describe nesterov accelerate descent context smooth strongly oracle reduce gradient descent improvement quite describe order numerical strong convexity case quite illustration token right token token label token token right token label let strongly strongly induction intuitively fine prove use strong equation understand far answer devote combine fy fx x induction true get fy fy fx fy fx
select section show diabetes datum section range residual htbp penalize score value alone show transform regression display respectively score widely piece wise jump phenomenon detail vertical tc zero multiple qualitatively linear regression notable strongly regression result suggest feature associate htbp black correspond normal diabetes calculate conservative formula value decision pattern correspond display thick black become instance threshold age coefficient well sample thresholding far limit canonical super efficient page lc observe penalize score lasso two variable exact conservative
stand duration overlap classification contiguous segment constitute baseline middle section dataset average successful accuracy frame frame overlap table although yield well yield yield yield well baseline dataset notice see baseline combination good work paper music evaluate contiguous summarize publish mainly music produce song summary way image short version song
possibility flexibility algorithm high work propose genomic gene sequence present genomic measurement complex genomic entropy entropy entropy generate observe respective genomic sequence way network
reflect inter decomposition representation plot tight representation inter good predictor importantly control single alone logarithmic growth break tail result inter high case strong latent classification nn summarize report peak micro macro accuracy yield correspond initialization achieved report show micro accuracy average initialization range dimension peak notice remark worth care
gibbs infer parent parent conjugate constant rate gamma minor elsewhere kronecker delta parameterization k conditional graph gamma weight color theoretically dot indicate show recurrent process feedback lead infinite perturbation mean deterministic rank one equal vector random large come uniformly achieve high weight asymptotic unclear various theoretical indicate dot figure variance example grow large eigenvalue increase greater unstable additional adjacency matrix feedback loop perform link process identity goal event hold link probability interaction compute roc complete set interaction similarly infer
scale property final respect satisfy sbm property node affect definition specify sequence adjacency aside requirement linear unlike stack column top system commonly process equation procedure dynamic sbm utilize central density scale sum bernoulli dependency identically condition asymptotically gaussian q time scale occur occur edge mean increase scale scale identically long apply
concave passive exp concave function several regression portfolio management batch sharp risk space result manner properly individual exp concavity goal help statistical investigate know combinatorial online theory trend characteristic convex condition concavity generalization recent smoothness
represent false error small see correctly nonzero component exactly true quite short algorithm insensitive attractive capacity demonstrate well propose compete multi multi denote th sample weight sample uniform vector sample I response compute recover figure test number advantage plain note estimation rapidly insensitive average estimation error default heuristic formula
solve system nearest search compute pca dense base cost column empirically manifold ambient cost could nearby simultaneously vector field vector field discover geometry however key firstly solve try preserve dimensionality try directly manifold note diffusion geodesic method approximate parallel adopt scalar heat propose learn heat field field gradient obtain try directly heat field note scalar field order direction initial vector manifold manifold idea another parallel tangent ambient another direction employ tangent pca tangent pca noisy method develop heat well partial empirically effectiveness geodesic algorithm representative le unfold mr euclidean distance
gradient stochastic introduce carlo issue mcmc alternative conventional sequential appealing variational gp predictive derivation prediction new induce approximate lp approach induce low expensive compute hyperparameter q summation unbiased segment sequence locally around segment justify time smoothing measurement future variational gp implementation interest
differentiable periodic use exhibit periodic dynamic incorporate prior nonparametric sophisticated modify compound periodic dynamic whereas periodic path draw compound covariance control variance intermediate shift experiment compound level bias dataset apply performance intuitive checking agree knowledge flow consist flow apply variational gp dimension ard initialize initialize neutral dimensional reveal figure visualization keep remarkably visualization visualization sparse run use induce variable variable gp initialize would reduce quantify dominant plotted axis visualization near variational temporal variational follow consider take motion frame time pose jointly paragraph walk learn investigate reconstruct test test test dynamical latent variable additionally test dynamical explore sub model model correlation discover reflect latent assess cumulative per joint scale space model initialize nine mat ern function prior infer dynamical mat retain quadratic dimension ard body ern smooth body model infinitely mat ern one easy motion significantly approach train successfully furthermore dynamical discover prove robust indeed table show gp outperform neighbor gp map worth intuition investigate encode smooth circular shape regime encoding explain force smooth interact supplementary video mat ern use quadratic correspond taylor space cl cb sc sc sc gp
object estimate extreme rank show unbiased parametric learn expert aggregation aggregation base aggregation score available work combine task refer input rank list problem list feature deal difficulty aggregation list geometric normalise rank spirit result build heavily theory copula find available rank describe distance ranking review copula function hypercube unit review bivariate multivariate next definition continuous variable copula distribution univariate effect ordinary extend
optimizer drop negativity constraint feasibility give rise semidefinite careful reader enforce relaxation marginalization widely lp relaxation map q turn redundant feasible solution feasibility intuitively propertie diagonal sdp approach problem employ similar state material despite hard difficulty advantage algorithm advantage property pose solver despite superior accuracy primal regular pc scalable solver propose solve problem negativity produce numerous dual solver restrictive solve pc exceed develop alternate direction multiplier
investigate paired screening deal regularization prevent toolbox lastly show naturally loss generalization able well allow screen dedicate case one corollary base concept extend group group inactive optimum construct contain prevent closed solution center sphere test sphere screening sphere require inequality obtain close rt give dual compute feasible screening test feasible dual feasible safe argument feasible dual dual problem conclude proof construct recall define contain point ellipsoid ellipsoid tangent lemma new sphere q check use convert
expansion fourier entail appear construct law decay gx decay exponent provide first distribution knowledge theorem improve good sbm block sufficiently spectral therein mixed block order additional spectral vanish community exceed universal show consistently condition similar attribute attribute recently show empirical probability apply probability low algorithm correct focus regime e degree edge linearly fundamentally infer distribution result endow label similar enyi isolated neighbor isolate label information
error independent define correspond calculate shift detecting shift shift series line estimate observe bootstrappe figure calculate bootstrap statistic significance observe detect magnitude score extend change user typically twitter decade review amazon book book different book micro google books corpus enable analysis social linguistic extract book language gram gram restrict gram focus span snapshot pos distribution google syntactic amazon review amazon include million start review rating plain text show amazon twitter span period month domain book micro dataset tweet tweet tweet location tweet text change usage time
enhance rest paper organize follow present idea discuss convert raw labeling section systematic conclusion come work lasso well formally training denote output n j j lasso accurate interpretable independently risk regularization
absolute independent complexity parameter well may replace indexing unit ball obvious parameter index result version average latter little heavy tail situation parameter tail typical trying probability fact extend loss explain throughout denote mean empirical minimizer procedure select empirical risk minimization erm context problem natural success erm erm close start know deal distance produce erm sign q exist target heavily target restrictive exclude like arguably independent corrupt since gaussian choice tailed term heavy tail
depend base therefore accumulation always asymptotically consistent prior zero bb test case study test bb percentage instance arrive almost regard surprising important consequence try effect medical well data situation bb always well turn coin guess equal test case rich analyst analyst information collect matlab code available probability let borel borel equip field measure element dirichlet process measure finite measurable dirichlet pa beta beta letter expectation dp reflect prior scale stand normalize define dirichlet atomic probability dirichlet satisfy property sense unnormalized easily posterior hereafter useful dp sequel
result htp result turn selective ol mcp turning kind ng turn turn well method respectively nonzero display htp scad mcp good tendency
deep fact deep alone train competition learning decay mostly visible particle contain feature million heavily regularize arbitrary quantity datum unnecessary neural network lead significantly well manner decide well prior train level variable perform deep suggest derive variable discovery accumulate network classifier
heavily representation huge face computation extreme addition pyramid multi scale face increase discriminative sharing network validate learn introduce face benchmark collect realistic representation present pyramid representation fast compact new social validate hand sift various heuristic descriptor aim design trial also semantic meaning
group many thank lee science foundation dms support apply mathematics activity office advance scientific computing de ac discovery grant david le lee constrain approach come address nontrivial scope handle noisy analysis narrow gap propose response surface expect lagrangian optimization hybrid allow think augment act bottleneck effort yet orient approach surrogate heuristic constrain minor modification motivated word nonparametric design mathematical algorithm provable algorithm optimize handle run computer little constraint modern guarantee statistical approach potential surrogate property objective evaluation expensive monte optima conventional usually local method statistical handle explore perspective consider scalar ba include assume feasible cx nonempty clear distinction difficult numerical allow
guide meaningful analysis may may aware modeling seem reasonable thing prevent try activity research take yet empirical suffer reflect piece finding order likely publication file fact light publication fact among expert thorough also one create ever activity currently account background valid selection selection
ball contain conclude contain fix arbitrarily let coordinate along follow valid along every fix arbitrarily suppose note argument need establish argument part r I follow hence every sequence iterate clearly distinct set still q various step case b generality let b I eq give result play convergence empty subset belong choice distinct follow empty hence set close follow solution identical set linear I lemma I x e r
filter learner advance embed advantage stand advance kernel rely formulation field see spin reconstruct shape bayesian discuss title probabilistic mit artificial title intelligence introduction title author journal volume page gene title gene regression base aic author journal mathematics author journal machine volume page year title bayesian paradigm journal volume page dimension interpretable cpu memory dimensional dataset subset
convolutional weight amongst worker many require synchronization convolutional worker worker particularly near convolutional net inefficient parallelization net dimension extend particular connected neural introduce asynchronous efficacy whenever part neuron activity output worker worker parallelism worker gradient ensure consistent exploit parallelism neither degree scheme informed
validation compare likelihood consider vs f component learn utility visualize specific death indicate high odd vote favorable compute ip support compute align issue capital issue conservative favor toward magnitude actual follow analysis use illustration ct power care topic support topic respectively figure switch notably support average concern support tend focus put concern individual argue fall uncertain analysis predict vote circumstance utility fig result support favor support confident vote favor except interestingly side success favor crucial turn conservative vote favor situation viewpoint
seen form rank due however rank
notion multiclass necessary condition calibration respect later introduce calibration dimension derive calibration dimension subset shorter include main maintain text closely proof completeness calculation multiclass surrogate calibration multiclass loss surrogate throughout denote integer integer e hull iii multiclass describe finite generality training instance goal many space e option notion common predict loss multiclass consider prediction away heavily satisfy ordinal evaluating system predict star hamming label bit bit representation frequently sequence hamming loss prediction reject loss assign incorrect prediction application costly diagnosis uncertain may well request intervention n goal prediction expect loss example draw randomly yx py ideally e difficult risk cd u vector risk achieve model optimal surrogate surrogate
apply unbiased learn membership contradict relation essentially derive hypothesis get optimal latter denote go hamming vertex hamming weight fraction lie middle level matter edge middle ratio go weight amongst setting imply level let study monotonicity giving circuit contain new circuit extend employ fouri analytic tool hardness circuit I boolean boolean study many decade depth survey many
important robust receive real response among datum par principal effect outlier example similar parameter affect close learning optimize require loading time method linearly intrinsic ambient robust linear rapidly computation obstacle big easily recent year rapid fusion machine order negligible communication cost implement distribute communication generic yet preserve reduce evenly machine aggregation operation compatible enhance efficiency factor potentially ability cope big yet machine reduce usage computation communication negligible assign machine step
modularity truth landscape grind truth gibbs phase energy modularity give low free marginal truth bp converge fix find approximate propose phase physics retrieval marginal group energy marginal generative model network contrast group free markov marginal need obtain thus cavity independence exact tree regime block marginal quite free message estimate send probability update message denote neighbor act update derivation bethe energy edge edge network moreover easily converge appear although grow least solution converge chance modularity
cm determine fluctuation level maintain compare versus elliptical variable co fluctuation discussion support order equation solution term confidence maximal interval individual q actual elliptical minimal saddle point
advanced term embed vector inside sub state onto term state likelihood convolution prior compute perform posterior illustrate synthetic real mm keyword model dark galaxy mcmc science understanding aspect behaviour system relevant select aim arise indeed though onto projection know arise science relationship functional situation unknown ill pose condition quantification uncertainty entirely scientific scientific namely datum comprise therefore refer whole learn measurement datum learn set however characteristic system physical biological system simulation consider basic behind learn system simulation useful way appear need add via thus datum achieve quality approach estimate bayesian learn measurement methodology remains learn model process however train formulation training possible form continuous popular covariance length impose smoothness learn
design location origin various transform separable importantly datum avoid however make perform dimensionality mapping feature mu mu j universal much dimension formulate eq hard dimensionality construct utilize related vector also j mu I find transformation introduce reduction particularly substitute equivalent add another specific complexity regularize problem handle corruption robustness term regularize rewrite eq augment lagrange multipli alm subsection review pseudo reduction present preserve utilize pseudo work pseudo transformation easily singular svd method stand within class scatter dimensionality performed view structured counterpart critical preserve less one elsewhere reduction perform especially small compare dimension dictionary size identity matrix pseudo graph one atom neighborhood neighbor among among
skew distribution triplet visible preference classical skew rw examine early point corner capability largely variable gender already appear herein reason extensively illustrative purpose literature skew symmetric proceed consider triplet ari gender reference classify variable analogous fashion clearly distribution comparative one argue package implement restricted skew terminology supplementary result figure dispersion around exhibit many repeat package circle triplet extra identity package attribute skew package
bad optima processing training amount prove art entire take minute model scale modelling present distribute dataset million experiment demonstrate gp amount show inference resource load gps perform big datum open contain extensively derivation equation present supplementary explanation google european fellowship lemma ex pt ex mark computation distribute implementation requirement evenly
overfitte neuron share neuron discrimination optimize descent please propose ns reflect see inference predict sparse decoder sigmoid sparse code force combine oppose include ns classification performance study highlight efficiency framework object categorization carry classification finally comparison subset ar recognition gender classification database consist capture illumination gender testing age database span age leave set finally categorization experiment toolbox level soft pooling different demonstrate face gender categorization absolute mae superiority ns compare control self st mid face database thousand alignment internet vary dictionary basis face record neuron basis unlabele reliable dictionary st amount basis ns level
diag kde library diagram diagram class plot band tb option produce persistence band kernel previous interpret band validity persistence diagrams kde detail option draw place diagram option figure tb c band legend col pt complex consist diameter simplex complex gradually radius create compute persistence diagram user persistence diagram library generate circle
know efficient say algorithm mapping also pairwise entail produce canonical determine whether np show element polytope discuss backward box variational characterization log project substantial iterate without enforce onto section give necessary background conjugate hardness computing partition family specialize duality statement subsection relationship conjugate conjugate partition finite furthermore supremum uniquely entropy express q forward mapping map unique canonical q useful describe hardness
momentum coefficient search like feedforward saddle improvement gradient saddle exceed since hessian plot universit universit universit de stanford universit cifar central challenge many minimize continuous high quasi newton almost often local method minima high global physics theory network evidence point minima interest saddle dramatically local minimum motivated saddle rapidly high saddle descent quasi newton apply recurrent neural provide superior geometric experience geometry surface choose error maxima probability typical random function dimension increasingly saddle point indeed saddle minima minima saddle curvature gradient away saddle curvature
approach efficiency high classifier soft connect soft consider natural connect hard soft rather specific soft classifier novel partial estimation number separate group clinical patient whether disease great setting emphasis individual estimation probability three boundary classifier approach provide assumption entire underlie conditional weight two briefly rejection option introduce third option reject label notably directly prediction probability belong exceed specified require classification view intermediate classification application option include certain medical weighted extend classification accounting difference problem soft remainder paper problem introduce generalize misclassification class surrogate surrogate hinge binary classification theoretical performance rule sub solve piecewise surrogate behavior datum disease database discussion framework
procedure raw word fail word significantly topic topic success correctly topic identifying frequency spread svd toy frequency say ideally rest threshold job extend toy different threshold word also ideally would either achieve condition split part partition thresholde clustered call project namely dimensional svd projection recently also yield start cluster center classic produce cluster dominant approximately identify frequently document nearly approximately document occur pure svd thresholding threshold let high ij b
report publicly indicator prior compare identity link package tuning fold set increase find association country journal association fit repeatedly split data gray bar predictor logistic order gene gene child available measurement cancer available measurement tumor normal available performance spam split training gene set cv percent classifier spam performance spam may successfully model display six
initialize convergence decrease criterion difference projection enforce matrix step pt step svd max z cm c c tuple tuple label relation label adjust four kind public two widely automatically use lexical syntactic name regard feature attribute whereas htb trade size contribution feature formulae assign z
turn plot function despite early surprisingly seem depend characteristic implementation detail often turn correlated direction cause happen causal chance level case discrepancy actual cause assume look accuracy chance benchmark accuracy around explain fact choose appropriately perturbation uniform base drop back add base accuracy slightly well chance become improved slope use estimator bit discretization still chance level amount benchmark accuracy depend entropy estimator due generally behavior method estimator suffer discretization affect perturbation perturbation however base level estimator perform base base make interestingly estimator distribution turn base obviously ratio support explain performance measure benchmark cause know ground benchmark several set bivariate causal discovery base additive geometric inference conclusion robust perturbation include implementation perturbation characteristic causal former conclusion report surprising report time method cut test one correct conservative correction would study method outperform benchmark conservative many result dependent across quantitative nontrivial believe chance continuous discretized digit record find use differential entropy coarse discretization cause entropy entropy robust perturbation include observation contribution extend straightforward distinguishing exploit certain pattern current enable closely originally finally fix bandwidth heuristic modification lead hilbert schmidt definition hilbert schmidt notation original biased propose joint define distribute choice justification stem characteristic special originally intuitively hilbert rkhs rkhs embed marginal give variable detail notion bias tuple center clear typically unbiased bound everywhere negative write kernel follow continuity show lipschitz novel consistency kernel q q start inequality take briefly main expect converge increase regression precisely consist q weakly moment many weakly might go training vanish asymptotically actually result seem kernel assumption bring even distinguish scenario half independence splitting training call use call mean residual vanish asymptotically take split suitable simply reduce scenario weakly consistent suitable state residual use estimate
centroid determine choose observation close uniformly centroid cluster close uniformity display real feature distribute smoothly centroid final less important sequential value perfectly persistence measure start two maximally distant feature add deterministic perfectly persistent avoid get optima appear however suggest optima criterion mean low persistence centroid htbp effectively request simulate much sampling entire replacement choose human
rule rule randomness involve learn seed worker affect success realization weak notion compatibility monotonicity allocation depend realization know ex compatible every irrespective ex rational success realization correspond allocation realization say post monotone mab call see previous need compatibility ns step add agent satisfy cost lead new algorithm monotonicity ensure modify ns select worker satisfy respect algorithm ex monotone compatible ex go analysis stochastically ex post allocation transformation obtain apply allocation stochastically ex ex individually rational success realization parameter allocation induce payment mechanism desirable game theoretic property label task ucb worker initially worker select k n ts tt exploit observe allocation natural apply payment absence worker determine stop compute compatible ex post individual rational post monotone achieve post monotonicity ex post mechanism algorithm post monotone monotonicity fix success brevity worker irrespective
make independent task variational lda synchronization scheduling thing become challenge due small gpu see parallelism subgraphs recent study variational pass scenario outline begin lda collapse big picture lda technical distribute finally section dirichlet allocation bayesian topic low dimensional capture semantic underlie widely amount conceptual overview inference collapse lda vocabulary word document token bayesian lda k intractable approximation gibbs sampling integrate dirichlet yield collapse gibbs algorithm current token token assign document assign topic token exclude th token collapse leave room improvement sub
hold critical case case cover proposition since noise example illustrate situation variation derivative operator turn design vector assume joint fourth canonical obtain say estimator consistent estimator split property implement j proximity forward backward identifie theorem hold close inspection apply generally variable forward assume
piece noise recover figure cm concave minimization whose projection proximal illustrate flexible stepsize affect proximal objective compact see cluster terminate successive instance random standard instance quantity upper usual proximal iteration less observe affect easy solution depend cc e e admm algorithm solve cluster point generate merely choice point generate admm actually convergent condition boundedness proximal admm future direction adapt convex problem case author discussion author also thank anonymous suggestion manuscript cm definition corollary remark partially grant minimize sum nonsmooth latter composition mapping
reweighte property par par image par comparable par trial previous ard objective par practically feasible assess provide convergence par establish addition compute posterior variance variance indicate variance therefore relative reconstruction variance suggest variance boundary subject future use os par split process lead considerable report literature os os thank provide access laboratory experimental acquire like thank mr dr trust newton mr david manuscript provide add penalty positivity consist detect ray ct therefore measurement noise ray medical community class measurement model accord inherent linearity transmission class statistical inspire automatic relevance determination ard allow incorporation positivity poisson underlie apart fidelity avoid parameter determine knowledge ard likelihood mainly ray ct medical method present transmission transmission denote exponential represent ray voxel pi py line source would th detector object standard independent calibration construct entry line source th voxel
minima nonetheless carry rank great explore visualize region nonetheless numerically feasible trivial convenient rank scalar penalty varied feasible minimum characteristic zero canonical finally element varied display nuclear norm produce rank cost display incorrect global appear success depend heavily carefully decay hope initial towards problematic additional spurious explore underlie regard guarantee unchanged iteration function unless satisfied show must bound cluster guarantee converge homotopy regularizer require carefully choose minimum factor perform poorly practice necessity prove formal practice zero promising remain important conventional minimization penalty whether fall nuclear respect consequently cost globally nonetheless distinction lead minima circumstance either construction lead nonetheless improvement covariance column overall kronecker sum q proceed fashion modification alternate upper accommodate use list compare art affine noiseless effectively
introduce change prior construction inform gauss function gauss newton give truncation threshold correspond eigenvalue impact likelihood balance retain informed direction typically threshold extend criterion inform direction global consider use posterior hessian feasible store scale construction global inform suppose sample truncate construct global hessian orthogonal access newton one approximate either construct dominant explore directional landscape project global inform subspace onto self adjoint induce complement factorization key correspond compute orthonormal choose form complete naturally decompose cs transformation illustrate hold prior reformulate r define rewrite reduced complement expectation approximate prior sample posterior importance weight far
worth formula c descent instance accelerate coordinate specialized nice sampling sometimes minimize restrict serial picking coordinate uniquely coordinate never coordinate serial form complexity eq equal time systematic inequality descent method normalize hadamard define already establish different potentially outside randomized c elementary intersection restriction vector define output diagonal hadamard vector index eigenvalue shorthand notation obtain pr day design complexity randomize particular variant refer involve capture way established develop systematic one eigenvalue
fairly relate score lem continuous ki comparing q scaling lemma imply contribution triangle follow trivially k title corollary mit solve massive improve processing time row sample unfortunately leverage score difficult compute row eliminate critical row information instance look sampling examine sampling fraction weak enough observation approximation preserve addition spectral sufficient leverage turn enough preserve key understanding preserve nonetheless increasingly leverage score iterative uniformly row estimate score leverage score reduce aim shoot estimate cut sum estimating
skewness link mobile phone many ratio much fall aim range visualize performance score base metric past example propose elaborate take classic follow number share correspond weighted class calculate whole distant compute weighted link start probability return steady process active connect exact metric expensive approximation use sum keep term predict imbalance distant pair likely pair class imbalance dramatically large distance expensive especially ranking present merging method system rank rank
limitation user priori stage even though large information contrast select design criterion may stage stage implement design pe implement test platform pe window package user generate design use phase phase iii package adaptive adaptive design multiple term outcome available stage stage though stage stage package platform open web little interaction knowledge language interactive web application computation manual application default www www package locally google view option free quick slow encourage heavy locally web run currently com window www window internet line third progress package library package internet library mm calculation directly compute design far application input table table plot see interface generally interface run locally panel output figure design panel describe
period actual need estimate one trial take robot hz try period result control trial minute finish learn limited verify robot controller master controller frequency hz synchronization mechanism controller trial start period e individually fig angle trial sd front front every scenario result combination period robot indicate generic learning anneal determine learn e possible period acceptable permutation acceptance combination period discard previous accept bad condition anneal reduce greedy bar represent permutation green bar greedy annealing situation value depict upon bar figure permutation increase large bad scenario perform period l situation increase combination discard however suitable annealing search bad
expansion recent genetic analysis tumor analysis insight understanding predict response heterogeneity population chinese reconstruction associate merge sampling overview reconstruction remainder consist description series variant illustrative tumor evolution visualization evolution grey expand tumor circle refer share distinguish parent parent mutation highlight
common simplify ray ct application often convergence show term convergent inexact method although term open admm admm convergent admm appropriately concrete mathematically regularize denote measurement image degradation simplify far I clearly algorithm quadratic
right trajectory spend third horizon generate along log plot average length plot natural logarithm horizon plot detail linear dependence natural logarithm linear value dependence measure average increase budget value imply imply depict trajectory regret plot figure average trajectory slope illustrate regret since hold affect particular size top structure describe linear log already depict bottom leave slope estimate bottom figure illustrate support minimax level traditional tight mab quantify price stationarity mathematically capture reward stationary together characterize minimax separate arbitrarily growth quantify price non stationary reward one allow
subscript remain loading remove follow previous estimate hold x give fix solution ty ty ty ty substituting problem maximize convex see split length column become solve back case sufficiently small ty w substitute ty ty back finally solution partly program cb national china education china theorem lemma component aim interpretability dense pca meanwhile exist problem add penalty various objective pca paper whose motivation rotation basis basis approximate three loading bound scale physical unified load loading loading loading lead loading dense loading less important indeed helpful property concern globally global otherwise property type loading exist compute loading one loading scheme former
associate lagrange solve problem minimize problem traditional rectangular significance digit varied digit five digit assume
recent cluster matrix whose relatively zero subspace express member subspace express combination variant ssc ssc bound representation lrr norm penalty regularization guarantee paper object maintain two form linear linear combination propose far page object tensor entry object write oriented weight scalar employ multiplication product product construct introduce scalar font bold matlab indexing treat
alternatively randomize quasi base asymptotic sample simulate different simulate ce random variable discrete base ce longitudinal count propose copula distributional transform early appearance simulate advantage copula continuous margin ce likelihood cl modelling aggregate simulated ce dt cl efficiency calculation ce inefficient lead univariate multivariate cl joint lead study base dt recommend dt previously dependence modelling dt study copula discrete simulate proceed brief overview provide theoretical efficiency discuss estimation two turn surrogate dt could background copula multivariate cdf margin variate cdf margin
credible fourier approach intensity phase distribution approach conventional transform acquire free induction signal quantify approach outperform uncertainty ratio peak induction powerful signal impractical chemical alternative technique chemical relatively alternate approach decay inference principle latent variable conventional transform decay frequency promise typically conventional research tool delay detail chemical quantification robust overlap peak fouri transform large apply resolution less brief conventional fourier transform introduce parameter resolve statistical prior number jointly phase multimodal surface
geometry fundamental reconstruction presence characterization misclassification tractable misclassification bound k k union asymptotic characterization identify absence misclassification perfect absence misclassification lead characterization within boundary bind bound boundary metric offer refined behavior upper decay regime name diversity slope characterize one characterize quantity term classification gaussian distribution ik observation feature pair affect key diversity order side information expansion misclassification draw condition give ik misclassification j ik ik ik ik ik see provide characterization feature description diversity diversity pair classification diversity pairwise dimension linear index dimension side provide subspace possible discriminate among correlation rank conditionally class kp ik r r ik ik ik diversity correspond span decoder discriminate class dimension space span diversity classification consequence characterization misclassification probability access upper zero upper misclassification feature misclassification zero accord ki ik exhibit ik misclassification zero j ik ik r ik ik r ik ik ik cycle node start dot cycle cycle dot ik j ik ik r ik ik r characterization necessary depend whether ik distinct r r r depict tradeoff case bind misclassification correspond index space matrix determine effect correlation transition transition free classification ik importantly ik side ik r ik
derivative derivative increase negative turn complete ex w q define dropout symmetry contribution affect express condition label satisfy constraint minimize define show output divide eq must complete sufficient make informally order term show pay margin error probability adapt analysis provide term actually something somewhat complete proof normal random complete drop complete lemma eq complete ex since show need proof convexity since complete use regularize specialized contribution case affect furthermore leave since fix symmetry identical recall q prove need rough proof inequality constraint x ex moment inequality relate binomial moment directly imply fact constant change enough ex work less rough minimize range give take among
location mean approach sufficiently perform recovery convex measurement greedy however measurement already exactly recover versus left trial figure report accurately estimate recovery noiseless robust regard recovery support plain dash sect start set measurement square magnitude fourier ii compressive signal measurement sparsity reduce seminal phase retrieval support technique signal idea apply linearize measurement minimize
fit main fit model property well observe geodesic distance plot figure summary line reasonably low degree statistic fail actor network high degree geodesic correctly predict actor connect number actor connect variability cause section htbp b recently introduce extend provide analyse relational mixed beyond interact majority actor assign partial membership group interaction interpret accounting heterogeneity model latent incorporate demonstrate variational computational still take single model outline validation burden likelihood variational effective literature group identification care appear treat nuisance observable blockmodel latent actor form
adjacent detect tried compare propose code platform good survey statistic local network filter clique lastly clique one clique another clique process stop clique share cover clique process maximal seed maximize function fc c I c ic ic label discover overlap receive variable evaluate measure extent compare superior superiority exist expand expand process network mix build expand clear smc identical smc identical except expand method smc expand method run smc parameter provide clique default community threshold value network iterate pick performance membership parameter subscript rd figure nd eight rd column first diversity
parameter connect auxiliary interest efficiency present contrast suggest start association variance nuisance correlation fisher regard auxiliary statistic perspective argument easy check assigning inference admissible auxiliary control expression accuracy predict result logical reasoning argument usefulness infer reduced q nuisance auxiliary variable association involve associate allow auxiliary depend nuisance nuisance
simple inequality dt obtain analog next suppose follow following hold optimal minimax us auxiliary function transform function function moreover mx dx furthermore x dx uv e v second statement lemma density density v eq distance mx p dy dy mx identity derive
constitute half boundedness although contribute belong correspond constitute edu optimal approximation remark ex cm asymptotically nonlinear filter stochastically convergent process approximation approximate mmse nonlinear filter property stochastically convergent approximation observable stochastic comprise stochastic state conditionally assumption time stochastically well compactly unity present basis prevent design nonlinear even practical design fashion apply constitute novel mobile research author elsewhere go deal mild technical definition prove later devoted result partially observable relate essential background theoretic measure mode stochastic convergence development
generator extreme negative apply identification show generator author acknowledge non deal separability introduce reduce constraint nmf datum nmf nmf point topic generate
cr il sentiment un cr une un un les phase du attractive clearly show insensitive replacement projection neural translation simple language feedforward list strong mt reliably improve researcher include source combine improve follow incorporated decoder mt system decoder useful improvement baseline sentence although sentence order word work rnn sentence back focus et direct network
incomplete incorporation base complete incorporation relation function principle insight q incorporation probability nucleotide incorporation agree count q elementary nucleotide incorporation reduce incorporation lead result compare check cycle pn f nucleotide incorporation distribution nucleotide incorporation nucleotide use nucleotide incorporation case non nucleotide incorporation complete nucleotide incorporation close expression distribution small module expansion speed technology give distribution associate synthesis practical technology development testing sequencing contain mathematical traditional termination currently available
combine hyperparameter note relevant solution configuration role determine analysis datum combine choose change describe exception cluster obtain way performance level overlap simulated true generation regard value configuration configuration centre setting hyperparameter sensitivity estimate group complete result ccc group hyperparameter default number repetition always recover separate method outcome obtain hyperparameter evaluate suggest rate want combine guarantee obtain well combine solution compare choice hyperparameter overlap final considerably hyperparameter meaningful reasonably return experiment bivariate essentially divide cluster make highlight
business attractive behavior la la htb pt contain author allow simultaneous continuous ordinal nominal logistic adequate theory ordinal use extend result term ordinal logistic ordinal logistic response dimension column produce span row odd obtain multidimensional item geometry computational calculation prediction direction logistic response surface project score odd multidimensional item prediction representation apply study knowledge innovation extract information international jointly carry operation development institute brief discussion concern nominal
accurately dimensionality hardness nearly dimensionality large computationally unbounded adaptively build use hardness result work identify hardness slightly weak nice code code achieve thus hardness environment interactive combinatorial object hardness hardness give interactive interactive code intuitive hardness able code privacy structure analyst know draw analyst answer reconstruct adversary representative adversary object query originally digital interactive code interactive adversary follow game pick specifie response adversary adversary refer interactive contrast code interactive interactive code identify suboptimal achieve technical result boost non interactive code recover code technique interactive independent copy interactive code shorter interactive give suitable extend interactive well interactive achieve still interactive except negligible false security detail section discovery roughly reconstruct answer private reveal hold notion privacy seminal attack use establish interactive combined framework every computationally accurate adaptively
remainder specify framework section dimensional definite summarize sample call precision precision matrix symmetric gaussian undirected node denote maximum
density eq relation step et al hold converge proof vector converge contradict conclusion subsequence equation subsequence subsequence converge since subsequence probability one year formal machine approximation typically possess new state assumption divergence representation generalization review joint large directly important ii learning presence iii active adaptive training statistical environment specify estimate density use paper white
execution decrease try second second average try variant present median reported present combination lie span orthogonal position perturb hardware assumption negligible time stable choose outperform speed correspond implementation hardware extend nature set complement datum q already hull proposition depth hull increase
normality suppose theorem regularity f pn begin expansion around assumption second use regularity second q therefore weakly class kde bound pn pm give bias via conditioning stein estimate respectively recall boundedness argument inside via jensen third precisely schwarz us denote obtain instance occur argument last term take bit difference stein complete alternative separation adapt hellinger divergence index follow satisfied q straightforward modification theorem tf np vs vs eq make complete proof index perturbation use result hellinger zero satisfie exist ball
think independent fix weak risk plausible depend derivative calculus variation yield minimizer however dominant let minimum context existence quite approach optimality dominant estimating make expression converge proof underlie one although successful intuition prove minimizer aspect behavior converge bound explicit result perhaps high start stand rate classic say shrink sense minimax shrink neighborhood say neighborhood achieve rate shrink neighborhood strongly consistent least shrink neighborhood sense
sequence notably compete second compete location performance indicate book average indicate second well precision video book precision heavy object resolve handle especially frame good mainly handle propose extent face object pose variation fail object follow object fail partly variation obtains successfully contrast severe illumination rotation plane rotation track method able track begin fail frame track person
electrical computer engineering interest vision recognition winner microsoft fellowship lin receive ph university key laboratory computer normal university microsoft compute technology chinese interest vision graphic learn associate intelligence international computer vision area extensive iteratively reweighte recent lead broad recognition collaborative filtering popular guarantee practice sparse either frobenius nuclear norm
utilize several index suggest conservative quantitative risk management able distinct fail dependence compare generally behaviour tail slowly correspond motivate search small possible organize idea include admissible maximal example demonstrate index tail extent extreme family copula somewhat discuss section general copula conclude calculation copula idea present
mm water requirement manual university publication pp frequency day consecutive journal engineering apply science mm chi square size rao poisson series modify statistic goodness test theoretical independently assume asymptotic statistic chi square freedom less et concerned potential conduct fit al et
first return label automatic target mostly correct correspond sample recall track seed inspire associated seed maintain detector pool candidate location region close flow object new location negative spatio temporal smoothness object instance negative target easy self successful object conduct detector seed positive negative transfer generic detector oracle seed box approach background applicable stationary move negative difficulty
increase bad variance bring adapt draw rwm proposal carry acceptance rwm estimator variance increase sampling fraction datum long period produce compare posterior rwm data obtain accurate posterior explore extremely explain accurate figure fractional subsample size work regular integration likelihood note lead even large relative gain note numerical integration slowly increase size numerical problem subsample tune dramatically time consume use efficient scheme bind scheme si work many order assign contribution likelihood sampling satisfied conservative evident application adaptive fraction tight avoid compute probit time effect
representation compare additive well present stack generalization noise style stack involve meta score clean db corrupt previous style classifier stack hence superior performance classifier match average improvement multi test difficult practical scenario classification result expect lie match white stack scenario classifier scenario filter speech agnostic rely second employ environment degree match employ accurate filter classifier square star stack exhibit trend generally improvement condition classifier yield well classifier db db similar conclusion comparative performance regime attain improvement across development input meta vector approach flexibility adaptation environment moderate correspond whereas combination difference confusion show suggest error independent classifier individually value combination parameter value empirically previous explicitly decision performance combination regime
fall sample suitably interpretability translate definition look key generally become due interpretable converge interpretable let small b require converge try construct measure exactly rather act distribution essential equitability measure affect hand fall definition equitability statistic measure intend independence rather idea approximate equitability attempt robustness measure utility two rather strict explicitly allow possibility preserve act measure distribution capture truly seek done unfortunately become however statistic reasonably define second explore fully worth requirement converge requirement merely largely subset equitability ideally instance equitability noisy functional relationship relationship equitability hypothesis testing measure equitability define equitability think nan equitability generalization power behind equitability equitability respect concrete noisy consider tailed statistic nan must yield tail hypothesis relationship relationship relationship reduce independence case nan hypothese non property tail tail distinguish particular subject constraint subject result ready equitability call significance statistical give tail distinguish zero value bivariate distribution ordinary distinguish alternative nan case relationship equal power tail nevertheless power besides contain right interpretability analogous tailed test hypothesis uniquely nature reliability need agnostic detecting aspect power reflect uncertain x
transfer knowledge auxiliary misclassification discuss extensively design collection traditionally advance technology digital storage information rapid collection format image audio document popularity attribute record automate expert reveal desire annotation slow development advance computer vision document annotation rely expert annotate utilize automated cost annotation problematic probability differ crowdsource alternative expert annotation annotation annotation annotation carefully design crowd worker collect
neural generalize glm glm flexible activity phenomena activity technique glm network visual devise quantify spike train student ph thesis pt increasingly tractable analyze predict neuron increasingly statistical framework flexible exhibit
ratio maximum peak basis observe region lattice smoothly datum point sample let subgraph enforce connect encoding define flow belong phase connectivity therefore variation triple straight composition phase enforce enforce obtain actually phase peak regular nmf range hour nmf sparsity violate previous soft assignment reflect fact multiple define c reflect amount violate constraint future report visual incorporate constraint exhibit well quickly analyze realistic hundred could hour dataset phase pattern phase chemical identification
notice point view restrict absolutely continuous respect absolutely key use least theorem indeed ignore moment resample order appear instance change trivial sampling section value motivation resample classification highly asymmetric click weight frequent reference towards multiply importance frequent likely impact case easy take gradient favor less performance ultimately drive case schwarz surely q optimal observe gain term large initially negligible optimize stay clear experiment tend
full ix define opt advantage base current select k I linear advantage improve redundant treatment advantage stop incremental advantage selection decision make step let assign quantity outcome randomly selection corresponding covariate advantage q update regime update sequential increment variable stopping criterion iterate sequence incremental advantage characterize variable point advantage newly explain explain currently mean make decision variable selection sequential selection propose select method way score score fit linear optimal penalty penalty simulation cross r study observational study relevant make treatment patient column consider three generate simulation interaction form baseline ty covariate entry
adaptation account train amount detailed adaptation prevent c activation architecture function adapt day l adaptation bold architecture hide layer bold respective shorthand tangent activation percentage corpora line accord topology layer initialize matrix layer backpropagation record select integrate system table lowest keep topology part incremental decrease minor use result configuration corpora clarity tendency end updating incremental compare last improvement part
satisfied strictly sufficiently also use assumption suitably histogram check cluster thick order thick level restrict distribution density recall dimension illustrate nonetheless exist continuous density use easily generalize establish usual assumption satisfied exponent exponent exact exponent fast component separation exponent monotone separation exponent describe illustrate ensure cluster exact separation exponent polynomial somewhat density asymptotically identical assume derivative vanish analogously class density exponent behave saddle separation influence begin let satisfied additionally exponent pick receive exponent replace theorem double bound cluster separation exponent receive separation exponent sufficiently large exponent discuss thus exponent exponent converges consider hence exponent establish
sample go extend framework high simultaneous integrate selection extensively throughput rna restrictive existence uncorrelated gene cluster gene offers automatically integrate generally address dimensional current related computational
disadvantage solver explore hierarchy encode rich second suppose split overlap often posterior posterior posterior explore variational exponential posterior approximately parametric reasonable right mc independent sampler simply aggregate intractable choice group chain single mcmc parallelism straightforward chain strategy adopt slow massive datum category compute processor gpu computation across combine local likelihood unit responsible update space extensive communication specific compute unit explore independence ci hierarchical ci chain posterior communication key combine sample posterior attract attention consensus monte directly combine valid implicit approximate follow build kde represent posterior embed posterior hilbert parallel mcmc posterior combination inaccurate aware efficiently eventually approximate use process multiple likelihood ahead hasting represent binary stochastically reject evaluate core respect core execution ignore communication efficient branch delay acceptance testing method naturally iterate early parallel separate processor extreme parallelism work fast parallelization maintain target correctness gibbs parallel sampler parallel various distribute sampler augmentation logistic normal mixture ci extremely partition stochastic exclusive often lead efficient solution optimization extensively particular sgd share component scheme particularly langevin specialized hardware use accelerate graphic graphic unit self contain device conventional computer distribute core processor graphic easily maintain code dedicated device consumption smc bayesian work demonstrate accelerate collapse lda likelihood eq gpu parallelization smc sampler gpu fast hamiltonian add acceleration e gate literature extensive progress beyond demonstrate word though
combination method solve performance good improvement show cccc theorem sec sec lemma total combine comparison several related report old old theorem preserve old new version additionally performance well well far subset big whole decrease yield new third learn human dependency quite inferior minimize fourth explore many learner slice old htb success lemma lemma name k nn type lemma type k lemma human nn full new mining conservative probably core plausible library much author cut produce small graph predictor add lemma edge produce dependency produce dependency correspond success possibly cut far account small translate allow automate ai immediately formal knowledge computer interactive library however library contain advanced form proof whole development pre exist symbolic reasoning amount library extraction later exhaustive
compute budget learn except fold observe lead precision forest superior value column remain baseline reach drug interaction go attain randomization optimal maximize dependent task relatively remain comparable consider base show trend randomization input build forest drug interaction feature label control forest randomization control accuracy decrease strength low case learn subspace performance behaviour completely phenomenon curve htb reader study appendix material show different time specific combine reduction trend matrix matrix replacement hadamard rademacher subsample space compare rademacher space sub
proportional hazard specify u weakly gaussian score specify weakly p consistently baseline patient independently uniformly covariate survival take extreme correspond odd censor choose achieve censor obvious treatment class regime contain treatment impose x need survival hazard distribution correctly follow compare well smoothed combination assume model term censor replication genetic implement package save presentation report logistic omit table survival follow treatment year plug establish empirical coverage survival probability cp true year misclassification compare regime standard treatment regime logistic treatment regime match relatively bias close one simulated treatment year survival year survival relatively bias estimate survival
present sequence variational sensible state state gradient likely make possibly gradient enable independently develop train expensive
random mean plot epoch act stop beneficial theoretical finding result suggest epoch appendix repository dataset incremental iterative kernel ridge report method inspection approximately remark definition laboratory usa learning
tensor language gpu analogously safe convolutional forward descriptor attribute filter descriptor specify along dimension tensor primitive convolutional neural special convolution list section forward form backpropagation closely relate mini input width output vertical height zero input convolution convolutional filter mini map per image feature map filter output tensor previously image height computing specify convolution mode set matlab
return estimate combine equation rl equation formalize rl learn policy class classification easily rl vc rademacher equation define use classification see go move equation encode reward dynamic analogous crucial relationship rl rl eq n return observable use calculate describe rademacher describe vc unfortunately vc linear indicator function
upon cutoff empirically simulate conditionally identically set burden compute approximate use become recommend large matrix satisfy theorem building random apply projection random expect accordance far small adjust correlation correlation entirely maximize appendix regularize emphasize power upon challenge minimize test simulation draw e haar matrix manner one sketch generality break evenly use project
curse nmf suffer investigate input scheme rule descent scheme multiplicative lee method problem hyperspectral art nonnegative reproduce become prominent interpretable data scope extraction compression recognition processing lee nmf successfully face recognition biology gene nmf consist approximate constraint physical thank part idea issue hyperspectral illustrate hyperspectral reflect typically acquisition reflect cube characteristic contiguous band resolution vary resolution superposition spectra underlie material hyperspectral extract spectra pure material area obvious spectra nonnegative nmf physical interpretation nmf
equal log function expression easily numerical regularity confidence interval confidence significance denote normal sample size base present simulation triplet initial element distribution independent estimator bias n htbp
interested suggest specific nuclear idea gap inside tolerance information arise differ structured duality possible approximate theory example section respectively dynamical system transfer truncate impulse negligible impulse create choose
extract low effect precision decompose regularize ml high rate rating completeness movie three covariance proxy element actual validate intuition span top residual capturing effect heat thresholded expect rank capture precision much stock stock similar low rank heat show global effective much number stock ac lead global sparsity remain conditional effect prove decomposable learning suffice verify condition elaborate taylor loss true perturbation rsc specify tolerance hold rsc sufficient direction convexity restrict imply rsc rsc precision
construct correlation column span approach column actual use top eigenvector matrix approximate span entire weak use span mean still corner separately modify algorithm single linkage take algorithm rigorous time near approach apply mixture spherical pac small close concentrate pac recent additional spherical align gaussians linear exponent spherical gaussians notation result bind give appendix product mix estimate empirical mixture collection output obtain contain class distribution work albeit constant estimate bind gaussians learn spherical gaussian mixture spherical near
develop university ie expression text sentence fail consistency present temporal expression whole text enforce publish challenge extraction text extraction analysis event increasingly challenge natural process nlp application
base h h p see apply bind chain introduce bound ax dx ax kx k dx n tw n kx x error x quantity lemma mapping chain xx follow expression correspond general error express apply lead approximation hold pick n single define I estimate give rewrite yx follow secondary variable statement accuracy union bound first long side obtain rewrite density rewrite scale deal process endow affect synthesis maximize associate property reach avoid reach deal assess horizon trajectory goal avoid express state reach formal relevant air maximize toward avoid model evolve continuous domain discrete reach specification lead rich unbounded focus specification numerically figure probabilistic study scheme quantify error bound
simple end mm measurable support supremum reach hand deduce write bernstein gibbs hypothesis extend note true rest equation bernstein case note infimum two latter specialized measure specification gaussian spike put consequence q obtain previous lift step define similar previously upper quantity soon put
stream price price parent company despite lack price nevertheless like able go period chain adversary utility efficient word bad function make prediction fall framework contextual web budget ad exchange amazon price power poor self category g store crowdsource management reward daily worker select task view price view purpose optimize fall large bad case online target try value bundle additionally despite linearity utility reward entire budget round repeatedly classic preference set restriction utility particular discretized differ amount induce price upper without set infinity
lead combine minimum mse minimum remain satisfied traditional performance angle gap mse comparison use accuracy angle gap rate angle corrupt white average fig
must supremum f ft surely combine left side decompose ft ft subsequence subsequence surely step complete violate k l almost surely contradict k l justify lemma I f condition yield suppose hold rt n simultaneously observe rt rt rt term converge almost surely suffice lemma suffice triangle imply rt ft rt ft rt ft part rt lem simultaneously conservative imply show readily imply imply condition consistency complete follow argument j j j j rt rt rt testing side involve joint x x give calculation yield bivariate section derivation plug explicitly consider plug p dp p dp p f associate co greatly real theorem support grant dms dms context large certain present sim maintain false availability bivariate preliminary analysis
blue ridge mark usa pool mark randomly mark filter hereafter blue lead hereafter assign randomized identification use light difference observer correctly influence example individual mark mark laboratory know individual variation analyze single error match satisfy history mark recorded history record history observer light would analyse blue light examine accurately temporal modify accordingly light million iteration pilot tuning burn start require long run due movement mh sampler diagnostic somewhat blue effective sample blue find median credible table hence reliably light probability
side fit plain outli basically dominate odd meaningful cluster produce side figure cluster high balance birth death cluster medium rather collect low rather despite produce observation look turn substantial disagreement even fix compare classified rate somewhat consist song true class compute discard either compatible discard correlation carry identical list mean mean std clarity feature standardized inspection reveal systematically although overlap extent coincide rand ari value clustering maximum agreement ari interpret third default ari ari ari suggest ari ari particularly high ari classify ari suggest core region run date compare robust shape cluster cluster method comparable formally
gaussian process ic reasoning function expectation function set frobenius norm g eq upper standard thus c bind decomposition metric regard denote kronecker deviation r overall l briefly outline guarantee asymptotic refine presence naive robustness reasoning naive factor completeness straightforwardly show locally dictionary lipschitz gain lipschitz show denote coefficient minimum yield reader notice eq well hence c eventually k eq cp r notice provide frame low frame thus lemma assumption corollary I imply refined result reason finite minima coding noise thus focus noiseless signal combine almost sure around truth energy predict similarly completeness develop realistic spike second dictionary structure dictionary blind calibration improve
contain member repeat macro member double bind heavy contain express protein protein protein pr psd sec contain protein type protein protein protein non domain alpha domain ig tm short member nuclear envelope repeat contain member specific protein sort member htbp bid interact domain death contain b member alpha bind class homology bind interact b theorem corollary rna usage continuous covariate sensitivity specificity method name usage rna seq correspond gene accomplish covariate latter situation compare allele one tumor specificity effect gene rna seq expression differential usage high dna gene often include separate several rna rna multiple may stage produce rna cell great understand functional change genomic disease traditionally microarray provide gene rna rna sequencing rna seq much purpose rna seq rna end sequencing end pair sequence end rna seq rna seq reference genome rna overlap expression th gene measure adjust depth th gene major challenge rna observe rna specifically rna seq compatible
scenario causal approximate rely auxiliary towards income income hide formalism theoretical satisfy physical satisfying exist notion correlation explain correlation structure partially order direct thick anchor east anchor north west space circle thick fill blue minimum mm lb build consideration quantum conjecture classical quantum challenge partial conventional scenario discuss classical correlation analogy network classical model hide notation collect notation usually omit acyclic think mostly rather similarly edge think link propagate exception hide network v v u edge put section keep distinguish carry notation somewhat since clear present intend rather constitute approach generalize arbitrary structure definition achieve adequate thing certain constitute scenario consider problematic approach observe situation statistic repeat assumption trial spirit justify neuron condition encourage conduct system causal structure compatible need causal structure causal structure may whenever event event regard obtain finite relax albeit technical challenge node write collection disjoint event node causal mathematical interpretation confident biology science situation influence causal formalism nevertheless causal whether compatible observation causal formalism come possibly standard commonly link proceed finite sequence shorthand direct absence conventional formalism network propagate indirect intermediate potential indirect happen depend structure long constitute merely environment expect occurrence causal member causal loop g graph sensible length zero figure acyclic graph mean direct
denote curve th observation th dimensional vector know function zero mean process th account correlation repetition directly smoothing procedure derivative obtaining reconstruct ft ft ij dt identically distribute independent ft dt reconstruct right equation case simple derivative inner product global covariance context functional blue
strongly convex make suitable convexity adopt note optimization besides extend constrain please relation component convexity ii I np prove em solution objective assumption k algorithm take condition next equality k convex unbiased estimate appear optimal l lipschitz gradient together fourth due
observation error matrix error mis show report simulate manually variance similarly present fix approach optimize match temperature case pos accordingly rate exhibit large manually optimize smoothing estimation manually variance perform particular lag smoothing panel variance panel variance flow rate flow lag panel brevity experiment term covariance manually variance lag smoothing lag appear low mean rmse pos cf fig assimilation may attribute eqs optimal circumstance one adopt assimilation precise beyond assimilation manual interval lag mean rmse time rmse soft sense salient flow jump ability dynamic auxiliary toward temperature pressure physical sensor flow rate jump implement utilize available temperature pressure
unknown tune standard error assume q assumption theorem see sake smoothed hinge hinge compose gaussian svm tune validation range test classification class amount gaussian dimension norm training close extract elementary bind key idea function idea derive q maximize use bound b concavity logarithm prove differently use vary often proof max w achieve expression elementary
compactly mf atomic compute via programming toeplitz entry presence propose atomic noiseless sdp frequency decomposition computational sort sampling practically since always ff specify definition exactly atomic atomic late analysis observe contaminate noiseless write sdp signal subset noiseless signal per square noise variance eq optimizer part derive appendix generalize complete sample guarantee noiseless show frequency recovery frequency
filter largely parametric signal dimensional infinitely chain hierarchy evolve dirichlet measure speak upon background filtering review hmms evolve mixture parameter projective process finite cox relate conjugacy static mixture exploit project admit computable result exchange filter valid depict former projective respectively obtain duality propagation cox process rectangle x n projection
low embedding additive approximation question lie substitution rank development assumption nuclear would approximate multiplication research framework fp agreement concentration direct concentration dt case use eqn thus become matrix
os k cluster primarily embed closely resemble practical partly heat traditionally within believe wide partitioning setting search dimensional partitioning factor cut lee study higher partition subset partition expansion formulate outer almost linear gap et center combinatorial contrast result bound perspective heat heat use partition current efficient cut matrix multiplicative use embed guarantee present cluster unweighte u ss extensively algebraic adjacency give eq give matrix v fs formal cluster eigenvector normalized characteristic thick
excess risk transform surrogate could binary risk smooth convex surrogate reduce optimization specifically achieve loss optimistic surrogate excess risk question reveal excess examine smooth account excess favorable appropriate smooth excess briefly discuss classification calibrate surrogate relie derive transform elaborate excess derive smoothness binary section omit n I draw dy
front typically focus label relational network network general gain variable emphasis make combination user wish simultaneously word model text permit incorporate attribute label million explain stanford several type high college public desire depend application five label school college city label primary situation people meet become friend school college situation mostly mutually exclusive may share school simplifying explain necessary friend indicate co locate building meet imply individual friend result type formation mutually
input computer simulate j adapt computer calibration couple simulate conventional call thousand replace ideal modern idea calibration local mesh methodology necessity recent trend simulation synthetic variation motivate limitation generally word nonparametric adaptive rapid increase computational power system field work physical phenomenon study induce bias must simulator extent account interested model calibration require describe physics center system develop field evolution input address mathematical small diameter conduct circular configuration aspect ratio describe shape circular field experiment aspect ratio circular exercise shape c design ce ce diameter length ratio ce ce energy input require detail separately super computer exercise third reveal range computer ce ce explore input circular region vary diameter geometry derive disk hold ce computer simulator energy unknown involve heat mesh output insensitive scale explain
hypothesis condition upon non decision domain bf test procedure frequentist relate statistic try bf frequentist hypothesis test propose test normal frequentist fundamental asymptotically regular smooth tend nuisance still lr x previous variance nuisance next end derive relaxed difficult within statistical frame general equality classical invariance arise haar topological notion necessary integral integration accord context give frequentist assumption parameter transformation call family densitie lebesgue group action measure absolutely measure marginal define integral frame avoid technical consequence relaxed random parametrize condition upon whose family sufficient statistic whose sample replace statistic frequentist measure call respectively haar measure induce absolutely lebesgue call marginal posterior frequentist sx
maximal clique clear limitation high property review detail number appear n ai setting particular count also clique mass nearly see proposition differ write term original instead enforce statistic clique adjacent implicitly prior expression expression work supplementary material analogous set vector factor count consistency approximation go derive distribution allow pass multivariate follow directly indicator fix I I
pixel constant vector pixel output figure benefit classifier convergence observe energy current possibility energy software fit rest problem impact future could overall systematically bethe propagation energy alternate parameter smooth algorithm update stochastic lp iterate perform optimization energy making pass slow
write lagrangian exist irrelevant equivalent lemma correspondingly fix application far advance transform constrain already start optimisation stop constant hold issue poisson give alternative involve integral langevin approximate likelihood iid turn classification f integral hand general likelihood availability estimator gradient counterpart langevin apply straightforward adjust see use logistic logistic appear place general call divergence divergence follow odd replace
library optimize generator seed result bias correspond bias slightly bias line measure observe parameter base parameter choose compare size asymptotic regressor fix regressor table uniform deterministic regressor bias deterministic regressor random cause slow r mm estimator score demonstrate admit test generating
q cauchy eq combined union consider q hence uniformly focus union value provide cauchy know cover entropy center cover probability cover conclusion demonstrate furthermore demonstrate consider rather negative introduce bound term perturbation argument consider second trying argue small let pair consider objective mean theorem assume thus able cover ball also cover radius entropy bound fix eq set dominating eq eq elliptical subsection eq r curvature control lemma imply oracle property
characterization large release motivated analysis collection probably learner accord unknown concept generalize example label take requirement preserve differential mean affect particular sample rigorous private survey determine complexity proportional complexity learner exhibit logarithmic class low possibility complexity learn analogy complexity private combinatorial sample learner towards characterization introduce notion concept vc learner computationally sample simplify exposition ignore dependency privacy follow dependency parameter later section private proportional informally concept ignore union argument use exponential mechanism learner show et et function learnable query learn efficiently positive natural computational et al study examine evaluate point al properly imply
impose forced cutoff define reduce support future must serious arbitrarily number bit power believe approach generative dynamical property power material mt code run machine dependence desire number show compare default run implementation mt triangle green circle run numerical square black triangle run square bit equal
offer use object semantic labeling subset derive label cifar organized word average concept human annotated imagenet label million image million around annotate imagenet class apart unlabele unsupervised belong resolution preprocesse see digit digit image significantly hard digits scene house image resolution mnist handwritten digit center intend object contain image generic human object degree pair good algorithm
shape composition tensor substitute perform grouping give sequence kernel intermediate network essentially wise compute flat utilize convolution backpropagation momentum layer fine layer gradient either convolution operation element require number rank comparable take convolution absence bi multiplication theoretical problematic bi tensor hand require rank considerably assume architecture character bigger train devoted layer make
negative distribution consecutive conditioning string conditioning string formally sign iid mention reflect possibly sign marginal without horizon characterize exponential horizon also horizon independent maximize horizon independent finite statistic may regard bernoulli distribution present sign maximize cardinality measure support tt
inexact alternatively factor keep precisely nonnegative square solve decompose independent since nmf descent differ symmetric input nonnegative factorization nmf generate see section update update update step section describe several nmf important tool order component satisfy condition formal explanation nmf naturally either multiplicative modify follow division develop nmf mu everywhere current decrease monotonically mu interpret rescale another intuitive entry derivative decrease derivative partial entry zero modify occur entry zero partial satisfy mu converge several way rescale descent see modify low bind update become scale iii research mu converge relatively slowly theoretical note
variational techniques variational provide scale normal gamma likelihood gamma correlation medical explore non approximate c predictive ts gamma gamma ts drastically analytic burden essential success variance gradient black work improvement library family simply gradient second dynamically carefully distribution g carlo significantly write expectation start derivative integral q simplify supplement rao integrate full family distribution field rao recall
publish real dataset process relative spectral acknowledge acknowledge generally effective illustrate hyperspectral imaging fusion vector variation nonsmooth alternating multiplier admm term spectrum field hyperspectral spectral visible near band narrow offer resolution range spectrum interest source image fusion
place draw converse potential new cluster every potential cluster draw conditional assignment simply conjugate replace cluster rule assign q straightforward density read state generic implementation detail prior heavy tailed distribution recommend transform j nc ic slice tw rest value break exact need author proceed number auxiliary lead slow memory requirement quantity point correlate slow quantitative ess
polynomial estimation estimator achieve time number distribution enyi entropy measure randomness discrete shannon measure diversity quantify activity anomaly estimate shannon give extension near complexity shannon precisely definition enyi mapping sample entropy minimum require estimating q additive great definition obtain confidence typically interested dependence size alphabet regime essential growth notation sufficiently k k eq namely grow linearly enyi order shannon result completeness
theoretical shot major among small portion section programming sdp follow feasible various solving theoretically sdp detect propose new inspire sdp consistently formal deferred cl first viewpoint viewpoint originally going consider ordinary sbm definition sbm event observe adjacency symmetric function rank ordinary give log function choose appropriate maximize entry let constraint check must entry infeasible relax semidefinite relatively requirement become ij hence nuclear penalization contrary impose cone outlier convenience theoretical objective equivalent great introduction penalization sdp recover use solve reveal penalization natural section optimization improve choose definite must ordinary mild detecting among pursuit
feedforward connection feedforward backpropagation gradient formulae eq step feedforward layer use co multiple drop improve dependency preserve tendency stochastic reduction recurrent apply way possibly drop dropout set drop step dropout search find dropout connection evaluate technique use music source classical music divide training testing approximately make necessity make network step long split dataset sized number unit tangent linearity sigmoid rnn entropy step denote note report ce error norm
sequentially correspond power consumption perform maximize measurement eq c k normalization ensure update component whose high error measure apply see else primary secondary om lm theorem observation protocol theorem insight em claim framework sense maximize family signal sense sequential query algorithm model sparse propose signal bring robustness application monitoring sensor call measurement hence gain guide sense big exploit measurement small ambient dimension work shoot measurement much combination entry seminal assumption help sequential adaptive compressed min algorithm restrict performance recognize adaptive sensing offer benefit metric large gain achieve recover signal know sense consider
concern adaptation evolution tailor careful evolution grant display color display display color color green display green display display display color plus paris france institute sciences investigate random size normal well among child I fitness become parent next increase relax normality movement general linear
classifier transform alternative logistic maximum likelihood regression capture decision boundary serve build know marginal transform feature boundary nonlinear transform feature interpretable combine model different marginal contribution boost view original marginal estimate run logistic penalization paradigm step feature augmentation implementation decision boundary separate stand naive wise mixture I pi tw error bar repetition suggest nb boost nb view compare road eventually well large surprising biased nevertheless road model small finally oracle example newly well reasonably suggest
denote occur denote either stop discrete whether martingale denote time failure subtract finally success union markov substitute isolate produce application average understand sampling arise importantly occur independently specifically draw type incoherent eigenvector incoherence problem incoherent condition neither rectangular recovery something convert constructing eigenvector dominant sampling satisfy entry bind rectangular parameter singular vary big time hold common uniformly problem handle flow sample uniformly parameter vary size convergence time let uniformly spherical distribution furthermore compute analysis sampling arise subspace projection consist column give select independent problem motivate satisfie parameter separate entry case additive independent increase
deal focused decrease chain carlo common condition reveal viewpoint abc nearest neighbor knn transform calibration accord replicate frequent highlight via knn mild knn nonparametric infinity return greatly differ provide necessary consistency include tend address shift approximation return solely knn classifier model obvious whose already minimization sake clarity validation simulate reference evaluate misclassification difference knn abc validation knn evidence indeed know purpose analyse irrelevant classifier prior indicator know whose apply predict give datum misclassification pair distribution loss fy fm suggest misclassification valuable posterior nevertheless simulate solely summary explain practically limited know function subtle basic actually train subset reference axis propose abc
intel gb ram os v depend retrieve record dataset negligible specify linearly time datum short value geometric scheme test natural dimensionality suggest converge minute imputation collection datum take suggestion test iteration remarkably iterate require weather conventional statistical cope invoke
definition consequently directly base boundary towards basis boundary cifar class play exist work regard dissimilarity measure fix set kernel base feature fast comprise image fold class fold fold th fold experiment neighboring cell properly effect paper consistency normalization possible ignore feature center normalization kernel primal objective square hinge loss k number fold show vector fold except pixel folds benefit general consider similarity cell measure give z rx hx rx allow q z cell cell cell pixel significant improvement probably
average fp skeleton fdr mcp fp dag skeleton fdr mcp fp dag skeleton fdr comparison sensitivity closely concave highlight conclusion literature surprising tie analysis method robust small dimension roughly mcp observe base reliable fact hc omit dimensional confirm expectation regularization dimension concave comparison provide display supplementary material fast approach particularly score hc single estimate respectively compare mcp method fast roughly magnitude pc translate runtime compute furthermore mcp runtime dimension limit large test fast mcp algorithm require versus pc term difference pc pc significantly fast material dataset claim dimensional subsection efficiently loss previous assessment number mcp purpose increase dependent relationship sense perform realistic scenario run result depend crucially take dags edge threshold six second per second node tp fp skeleton comparable notice fdr due increase sample combine confirm efficiently complete improve total runtime five minute purely report successfully runtime minute day internal internal cache standard vector stand optimize yield fast imagine incremental order edge penalize equally false one sensitivity efficiently pc bic note empirical suboptimal graphical behaviour develop tune mcp confirm edge suffer reason already performance pc use significance level run bic also select appear perform bad relative report section run restrict candidate pc material edge qualitative already provide briefly level provide detailed assessment algorithm increase decrease show behaviour observe section material speak dimension mcp improve graph remain estimation nonetheless
essentially point average distance simplify average equivalent formulation index hence euclidean cluster sdp optimal satisfied large put together isotropic radius distance problem occur currently center assign near center ii new assign terminate fail optimum separated isotropic median lp sdp bad center away ball ball create copy group cluster copy copy pick center cluster center initially group center configuration center away two nearby easy never cluster fail high space even separation center section report regard median mean sdp input consist disjoint center ball I ball use successful optimization separate ball respective repeat plot empirical success cccc relaxation color probability ccc high show range range lp achieve fact median seem comparable lp require plant test necessarily interesting diagram failure coincide clustering cluster plant disjoint support
lastly architecture decoder despite encoder suffer curse investigation require addition general translation system propose recursive neural find sentence syntactic acknowledgment author acknowledge cifar universit de university universit machine purely decoder encoder decoder correct paper analyze property neural machine rnn decoder convolutional network perform recurrent apply sigmoid variable distribution matrix result activation new activation logistic gate previous unit reader b activation ht besides
minimize x c example th cluster tend expand use parameter note give point opposite box away corner explain loss operate distance decision boundary boundary boundary namely analogously low analogous calculation boundary begin issue deal exponential numerical multiplying term divide term factor add least avoid multiply omit proposition solution close outside box set boundary effectively description classifier make start r box compute interpretation set boundary ensure box away nearest give subscript
share many also intuition crowd bit attempt distribute representation phrase chen point model context probability
simple crowdsource lead quality dataset crowd worker interest collect human triplet collect triplet human researcher variety set author learn triplet alone annotation specifically create embedding create dimensional axis create embedding triplet collect crowd worker become intractable difficulty collect triplet datum human relationship choose triplet intelligence provide affect collect triplet embed primary concern object way collect comparison crowd traditionally triplet collect triplet grid ask collect triplet whereas triplet traditionally collect triplet
product particular sequence way precise relation separate denote use tm formalism discuss become column rewrite reduce express already obvious analogy completeness concept explicitly stationary found physical machine constrain result apply spectral decomposition burden power expression power scalar also closed reveal type ever stack familiar make eigen understand behavior decompose operator moreover analytic find tm operate z contour integration complex unit eigenvalue large contour contour plane depend tm lie unit guarantee index index must algebraic eq strongly stack finite hmm besides form useful inverse stacking circle
display histogram chart song test title subset apply effect consist track respectively descriptor deviation beyond replace separately set descriptor descriptor subtract rating audio descriptor without intermediate estimate audio perform coefficient denote behave assign incorporate fall year individual track group track consistently combine group potential track slide window descriptor track window track window difference chart entry testing window descriptor datum section average vector centre annotate chart entry absolute error mae root error rmse display exploratory song year descriptor slide window restrict analysis spread year plot examine chart entry trend descriptor fine non day slide window chart entry mean descriptor examine autocorrelation lag correlation yet decay non stationarity
projection mahalanobi yield asymmetric bottom well extract elliptical contain namely quasi shape depth come need extra determinant depth satisfy represent projection depth univariate projection exact construct hyperplane attribute additional coordinate mention relevant final hyperplane element respectively extend select involve respective separate straight origin separate form minimal choose onto straight orthogonal separate one illustrate diabetes www ac tr seven pressure diabetes age represent unit square class direction depth depth space extend depth space small two reduce extended subspace separate straight line choose axis separate point initial similarly hypercube step iterate extend decrease extend stop step variate depth direction project depth depth implement depth question computationally
purpose study medium internet china case increase period long mechanism context suggest difficulty pattern china fairly noise context contain significant incidence change evident context flat capture trend traditional surveillance progress year meaningful could capture internet incidence identify plausible three pattern improvement linear could lead suggest tune illustrate fail wikipedia wikipedia actual observation infection cause become internet trace poor sub disease observe pattern activity united good internet connectivity incidence peak differ day internet even major news signal forecast china failure united states success united snr china success location list success forecast offset offset forecasting summarize model location context test produce disease line successful technique approach sufficiently promise explore related popularity wikipedia relationship article total level social internet key capture disease incidence distinguish trace health modeling broad article article query pair c china united united china china score test pair disease meaningful indicate list pair disease concern establish feasibility
temperature simplicity performance compare svm radial ridge denote cart forest denote package logistic hyperparameter std nf rf svm cart rule inductive form rule nf rule input nf note rule necessarily decrease monotonically rule increase use rule list dataset fold list fold roc cart unclear perform svm validate poor cart poorly relative
curvature arise w gauss instead negative curvature intuitively hessian quadratic offset since curvature arguably long distance curvature curvature manner seem regard method statistical enjoy rich apply approximation section establish ultimately define training correspond density divergence learn distribution learn eq substitute efficiently empirical substituting minimize standard maximum note extend agree kind fit conditional composition output mapping proportion fisher first psd trivially psd product negative w directly difficult come actually compute situation version give
cm step yield cycle iteration inversion kp diagonal matrix gaussian factor implement obtain estimate acceleration illustrate measurement make x estimate percent day temperature capital city area matrix dimensionality factor variability contaminate former package run criterion good bic contaminate analysis prefer quantify choice perform reject usual factor contaminate gaussian
patient store disease disease international code patient dataset dimensionality dataset history diagnostic event disease vary million million event disease patient disease consider disease dimension contain diagnostic testing qualitatively evaluate knowledge code high code code goal font w observe w vary establish scalability regime qualitative hierarchy hierarchy truth rf tree tree baseline agglomerative baseline advantage increase support md team e htbp portion represent code dataset together group disease group know disease clinical disease group latent reflect status disease common capture node dimension per variable figure portion disease
channel search select ad user determine piece optimize study combine learning game learn framework historical response mechanism mechanism learn bid next mechanism maximization predict bid period infinity genetic bid future datum know call adjust response mechanism mechanism learn historical overcome drawback novel combine strong handle second effect first historical mechanism predict instead mainly engine click average short
file visualize visible denote layer visible layer encode spin hide hide spin influence visible break labeling language network inspire statistical consist unit layer success unclear architecture possess architecture understand structured explanation dnn view coarse high level learn increasingly abstract layer feed dnn combine successively apply extraction simultaneously learn supervised set concerned solely training compression make successive coarse tackle physic describe phenomenon short
call let identify graph belong apart distance precisely group assign identify center case depth e look center valid cover look depth double peak pair triangle triplet circle triplet display graph fig large triangle triplet graph circle belong nonparametric near look near among sample assign label majority neighbor setup principal variable project equivalently iteratively element maximal
panel size extraction divide patch pattern pattern inference pattern sub image pattern pattern sparse binary concentrated keyword panel normalize heavily term panel sub weight pattern heavily panel differently visualize project
node unseen u pseudo count approximate main method cf dependency variable break statistical independence variational approximate crp however allow fit fix break model component prior comparable infinite predict node notably likelihood well however even evaluation often interaction link complete dataset c close difference use ibp
unlikely level operation comparison color negligible theoretically achievable speedup color increase achievable cnns vision widely result network parameterize redundancy non convex redundancy exploit technique appropriate tune performance decomposition base decomposition cluster similarity learn contribution generic redundancy inherent deep cnns imagenet show layer connect factor convolution tensor denote dimension value convolutional spatial location
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thus bilinear form eq choice without highly machine theory pattern geometry cloud laplace rough operator connection uniformly cloud laplace operator approximation heat rough laplacian cloud identically almost surely rough almost surely spectral assume normalize recover rough generally recover whole fm smooth tensor iterate approximate iterate wu develop
parameter variable logistic majority time logit normal label label ref expert probabilitie vote figure selection horizontal possible iii plain satisfactory hence become iii detect em behavior agree return bad feature expert ref expert expert probability generate increase scenario minimize risk graph case probably bottom table set red content quality unit appropriate label great
failure use failure know learn take failure step shape q set learn step learn previous step predict failure interval approach failure approach also consider along dependency variable consider known product another incur company part decrease period fail replacement minimize cost gradient specify comparison failure section service discuss section present discuss module learn
give semi supervise task demonstrate dataset establish give pairwise submodular necessary cut novel connection classical promise incorporate exploit markov literature randomized map largely acknowledge ep google award comment presentation clarity appendix cox cox comparison literature theoretically experimentally literature insight cox function product local lebesgue borel eigenvalue eigenfunction lebesgue measure use predictive density density contrast gaussian family cox process superposition appeal model unbounded chinese imagine cox tractable alpha hard compute approximate two seem approximate cycle compare interest
gram study open display power law size gram track linguistic phenomenon language behavior pl language relate originally database word list show test pl gram gram gram gram gram gram aic standard column correspond aic value
shift plan effect reverse top bottom reverse detail task evolution vary sequence dynamic hilbert give insight methodology show possible time vary learn restrict space flexible part european european fp grant token problem future interesting predict step set machine embed reproduce operator illustrate principle formally set set distribution possible distribute approximated main smoothly evaluate
band give operation goal approximation situation small ii power basic differently application paper scientific problem statistic normally price power nan become severe found particularly appealing suggest high side version henceforth design motivation ratio statistic hypothesis alternative change poisson generalize statistic order event th size behave likelihood maximize tail ratio statistic become consideration tail get argument compare completely ratio seem excellent desirable goodness theory include size apply discuss briefly behave organization expression approximation monte demonstrate comparative broad discuss confidence
super source relate drift activate time mixed fmri fmri input eight fmri eight map slice course linearize concatenation row voxel eight spatio map mix linearly eight fmri formulation final simulate fmri point voxel linearize algorithm slice voxel volume datum task comprise relate trial inter red green box appear participant leave presentation box red appear box appear red left trial per nan acquire dedicated image contiguous brain tr te ms flip angle slice voxel high acquire te ti flip angle slice dataset public include brain template head movement datum nine subject fmri thresholded area zero voxel roughly fmri voxel filter specifically fmri acquisition practice voxel enhance voxel mm since voxel resolution neighboring voxel average voxel version smoothed version section detail visual dataset comprise collect task reflect change six subject general ms thick te resolution gray face pattern category select comparison
take involve thus first minimize set compute
conv visualize filter assign convolutional final layer sample intermediate produce varied layer hold collection node fully connect convolutional recent cnns generative aspect show cnn cnn also visualize turn generative learn approximated sampling fundamentally yet architecture latter generative visualization cnn sampling draw give
section assess variant direct discretized benchmark storage management benchmark problem consider relative benchmark run continuous continuous discount discount storage period space horizon optimal policy typically cpu primarily discount consider storage operation management table refer demand ba consider trading storage price variation discretize benchmark true discrete process state discretize average wind divide storage capacity load hour device max storage device device within hour price fit real price load fit load mid transition wind wind resource wind price e demand computationally simplify interval day resource state c wind wind storage full c full ba ba ba ba reasonable instrumental bellman test illustrate policy instrumental bellman
ts randomize idea bandit reward initially observe failure select algorithm update generate select adapt fa aware propose situation ts fa ts let user situation past situation compare situation method document recommend observe user choose situation improve strategy retrieve
classification prediction supervision ranking compatible map ranking simply take full query competitive prediction surrogate less suitable set supervision rank full ranking sec vector play subsequent surrogate truly explain weight relevant induce measure crucial perceptron minimize measured surrogate hinge member end proportional vector member upper measure rank however technique different provide ndcg document sort relevance level bound relevance vector relevance document vector permutation sort score follow hold say dominate relevance map relation thus sense choice upper induced loss surrogate consequence ndcg vector ndcg determine permutation sort however still condition generalization
parallelism naive ij substantial hardware node example node gb memory amount presence parallelism fortunately note access block contribute appear ij use update much still call depth logarithmic dropping iteration clutter cholesky first fast cache memory use less node node associate also part regularizer initialize r j ij ij note enter admm proximal proximal admm implementation loss multinomial logistic hinge proximal operator hinge solution total requirement count amount term start cache requirement term undesirable however column splitting reduce memory
bin close consider simultaneous empty bin color simultaneous empty bin bin total position bin way empty add extra choose space bin bin bin q desire expression bin argue total position two empty simultaneously bin improve bin bin position empty bin bin bin empty bin
try use method gradient descent optimization well certain different add decrease variance advance weight weight spirit interesting understand thank art discussion suggestion research support
novel probabilistic interpretation autoencoder generative autoencoder low decoder furthermore keep information optimal unimodal neural network factorial show yield sample stack autoencoder long low gradually yield interesting reveal picture idea unfold transformation regular factorize help understand respective role reconstruction criterion regularize auto acknowledge cifar first rise autoencoder encoder encode activation autoencoder plausible mean near map autoencoder training example
calculus introduction denote variable extension disease cause cause disease notation intuition causal provide acyclic relationship dag prove tool causality variable dependency read graph asymmetric nature causality suggest dependency find descriptor causality suppose causal link order pair dependency correlation define descriptor call descriptor asymmetric property causality causality asymmetric descriptor define descriptor markov direct make connection two ii asymmetric create cause condition separate asymmetric q I I j j mutual term asymmetric descriptor
sequence library average know I joint probability obtain put piece average sequence calculate eqs calculation position along possibility nucleotide mutation position state tackle directly however unique probability nucleotide
object illumination classify information need need irrelevant simple naive invariance would list realization object vast amount realization generalization useful invariance represent invariant transformation kind represent set generalize list realization whether possible realization discard operation pooling implement way variable also along relevant often simultaneous operation extension detector sensitive type increase number expansion pooling stage visual cat feature implement competition call simple generating alternate various specialized pooling version pool convolutional review hand transformation
therefore put divide side tu sa co let cauchy consider convexity q h summing multiply kk right convexity hence equivalent max h convergence desire
choose explicitly later coordinate thresholding rule nonnegative assumption give start tf penalty theorem provide careful essential infimum thresholding penalty convergence nonconvex mcp bridge penalty p define p infinitely may frobenius nontrivial lead p type regularization screening replace satisfy supervised application replace let r hybrid perform inner enough k algorithm complexity initial may construct initialization multi frequently challenge variable screening group screen oracle predictor factor loose small type problem ps ps p define similarly break multivariate p rt modify inner
combination power efficient allow interpretability focus control domain could text corpus eliminate pre relatively straight implement factor relationship grateful child provide gs topic model powerful tool latent structure finance goal even modern discover interpret interpretability hierarchical topic encode word lda interpretable summary real matching performance probabilistic originally develop discover corpora framework domain latent allocation lda vocabulary contain observation scientific article image action modeling vary topic summarie document situation topic understand topic meanwhile discover publicly
query relative per regression tune datum cifar update map slice generalize classification softmax datum belong softmax log match softmax tight chapter cifar discover autoencoder prior langevin mala choose softmax experiment gave qualitatively tune dramatically outperform mcmc fold likelihood likelihood tail call robust perform robust dataset molecular property consist million
predict negative label predict third scenario multi dataset learner number weight rest follow two multi correct label instance learner produce learner rwm mechanism since rwm learner learner expert parameter output label nx mistake probably occur mistake rwm rwm mistake mistake subsequently show mistake rwm mistake enough mistake mistake far fraction total weight wrong answer trial mistake learner
protein structure form model architecture change connectivity unfold auto improve quality representation acknowledgment thank discussion acknowledge computer center university resource support nsf health gm national institute medical sciences center gm institute predict secondary protein new supervise network predict secondary hierarchical representation deep train deep generative extension learn chain apply protein scale sized protein layer architecture focus inform representation learn label state
leave almost neighborhood describe alg efficient way symbolic say alg statement noisy beneficial want remark classical consistency applicable lead sample limit argue proper notion manifold say intuitively correctness noise alg complement consistent thresholded span imply pass alg seem svd therefore work enable look span look reveal important identify notice mathematically free example take suppose feature vary principal generative case matrix row classic
series methodology goodness conditional distribution precisely let univariate real let sigma family subscript abuse distribution apply finance relevant moment necessary check jointly specify conditional feature quantile know essential likelihood fan conditional link hazard autoregressive duration knowledge assess finance description management serve risk finance specify location distribution unconditional student model conditional two moment cumulative ty dynamic scale process commonly interest examine test nonlinear obtain dynamic scale series necessary location distribution model estimation rely
include recommendation co occur search user goal capture towards correspond recommendation list bag consider enter name exclude bag co specific avoid name exception popular name build occurrence collaborative feedback metric neighborhood similarity metric compute co occur name recommendation biased fill list name user implement language library model table individual
corpus examine spam multi document topic modeling within corpora list document remain branch job corpus evaluate internal corpus adopt except learn concentration parameter note comparative result multi corpora sampler corpus result evaluation word fold corpora five test fold corpus fold exhibit regardless picture range model corpora outperform support topic share respectively overfitte tb ad building post fan post word page day comment write facebook origin follow topic hold likelihood corpus consist job table show job com
capability I classifier appendix explore appendix divide study error score h omit covariate location classifier label vector explore source variation al problem study evaluate many estimate se algorithms seq two four logistic discriminative describe random non diverse recommend understand experimental study density omit primary density weighting defer al provide study continue pool label experiment classification classifier seed seed affect pool seed experimental experimental evaluate method iterate al loss label illustrate b base provide metric performance label complexity suggest substantial agreement ranking reason employ several calculate ranking yield five metric primary metric al assess classifier performance single single base metric metric seven al seven tie
combine two r letter fair algorithm setup perform logistic optimization tackle classification multiclass step th minibatch sizes sgd ss sgd ss use cluster also via degradation fix seed measure training effectively epoch
little gain sampling match satisfie bernoulli useful far strategy bandit check fact together strong introduce appear fact illustrate fix resp budget empirical match strategy approximate universal bandit model arm crucial stop arm strategy analyze paper stop approximation exponent drawback relate coincide close stopping criterion kl exactly particular approach pac exploration goal result bandit model fix budget star circle specify running scale average
slowly value distinct strictly string aspect subtle search context yield future symbol stream lead irrespective initial course know priori least approximate always identifiable string parametric statistic error entropy alphabet px px occurrence base generally entropy express random variable entropy alphabet usual manner px px rule entropy calculation definition I x rate stationary still hold h define entropy stationary distinct literature formalism find include brief sake completeness denote string string string denote string cardinality strictly ergodic stochastic ergodic calculate sufficiently stationary moment formalize assume realization fix use ergodicity long string induce assume stationarity construct assume ergodicity extend via countable sum notational relation string string language consideration extension equivalence induce equivalence string
infinitely least zero consequently physical feature denote two nature pass truncate end side feature distinguish proceed
none undirecte national study health health health behavior record one school student th student friend student four edge indicator connection present student important social tendency statistic tendency display robust geometrically share specification statistic thorough justification curve parameter goodness friend
employ auxiliary show image difficult addition presence pixel sensitive carefully specific condition crucial drawback completion challenge truth extensively randomly pixel low approximation prior relate use rank mp performance image runtime mp improve performance recovery high mp completion significantly obtain comparable tuned image visually demonstrate mixture prior advantage recognition surveillance videos classifier pose illumination arise question condition highly model image introduce synthesis contain image people pose miss either intractable order apply ratio initial completion tune ground truth fig synthesis rank determination tensor rank exist account address cp treatment incorporate
preserve multidimensional tensor denoise flexibility wide prominent inverse imaging image fouri measurement scan desire clinical throughput convenience sense far exploit sparsity measure domain wavelet variation enhance quality learning adapt accurately measure structural similarity slice volume denoise aim
investigate ahead prediction xlabel horizon ylabel legend pos north width plot coordinate coordinate p computing error space dimensional accord penalty term order everything eventually less intuitive suitable initialization section network choose especially example true preferable system time combine identification auto encoder high move horizontal good dynamical possible denoise autoencoder noisy real b convolutional c exploit controller carlo investigate reinforcement prediction reinforcement decision
x ta therefore least appear eq h high piece get finally piece main warm largely apart bit objective initially might clearly regret constraint meet call initial recall decrease objective oracle call complete proof variance epoch variance policy every policy q lemma abstraction reward pair oracle create epoch time per suboptimal algorithm explore achieve logarithmic run trick adapt call oracle previous optimal however prohibitive scale logarithmic factor contextual require coordinate policy epoch policy update epoch trade concentrate burden round call round oracle call round stress total develop variant conduct show baseline
exploit matching matching exploit weak worth explore accurate region rotation sophisticated discuss motivate discriminative patch among work benefit problem grain recognition deal lemma rewrite lemma exist proper simply lead tune scalar derive auxiliary monotonically increase submodular split definition denote v increase derivation q h v h vx eq q auxiliary monotonically submodular monotonically increase proposition gx x end proof appendix monotonically monotonically without simple mean integer increase otherwise equality image turn therefore overall prove proof
conditional structure specie information theoretic train physical review cm inf measurement york cm partially exchangeable development game page institute mathematical cm volume page institute mathematical california cm process mathematics ann communication system technical boltzmann statistics journal physics mechanics york mutual document physical review completeness regression univariate generalize quantify presence cm key word entropy diversity estimation shannon entropy non overlap independent classification measure deviation
extremely group geometrically transformation image take rotation cyclic transformation length one rotation correspond cyclic translation mapping ht red k accuracy approximately descriptor descriptor descriptor transformation sub cluster descriptor texture transformation analogous cyclic horizontal following index tuple rotation cyclic equivalence partitioning class detail six coefficient member discard addition six class give autocorrelation also discard coefficient since member average thereby
become automatic alignment segmentation news speech corpora audio difficulty solution segmentation propose novel technique acoustic stream self similarity apply music retrieval audio audio audio segmentation duration news
speak high favorable stability range one area help control dimensional particle develop assimilation accommodate gaussian efficiently operational sampling posterior mcmc strategy generate specifically hybrid incorporate system new operator overview assimilation widely give hmc algorithm experiment filter enkf conclusion brief overview assimilation da highlight behind research assimilation information knowledge numerical physical background uncertainty model initial begin evolution perturbation state evolve tangent linearize time observation observation assume distribution observation assimilation combine background assimilation gain popularity ensemble belong latter two exist kalman likelihood filter review kf assimilation moment sequential assimilation algorithm proceed forecast equation produce quantify forecast ensemble kalman enkf take monte carlo member forecast ensemble simulate reality error covariance estimate background time due localization product forecast formula observation version kalman add assimilation kalman make linearized observation k kalman gain use version formulation
maximally skew skewness dense design fraction simply nonzero matrix analyze require need follow measurement essentially use measurement slightly use small
well inference greedy model ultimately stochastic capable high modelling language substantially particle carlo probabilistic inference approximate bayesian abc summary statistic compute generating vector f repeatedly interpret generate repeatedly time sample unnormalized compatibility text lambda real program text time variance noise skewness observe level probabilistic write generalization employ establish correspondence variable describe particular implicitly line implicitly return vector skewness draw normal mean text output skew square code highlight
class direct nonlinearity hard mnist parallel svms within toolbox replace point matlab processor run machine processor limit speedup right may model parallel toolbox quite inefficient proxy function classifier optimize use secondly without minimum view little yet obtain true minimum want ideal consist centroid locate corner simplex really filter nearly efficiently simply find optima classification allow role dr ideally dr variation domain nearly belong choice ideal theoretically sufficient limited separation nonlinear dr train work help move data space boundary simpler hold view good manifold dr dr informative something
hypergraph set xx xx h dictionary partition satisfy regard minimize little restriction question position learn conversely indicate section rise follow pure give size additionally nontrivial combinatorial give minimum geometric characteristic p dictionary subspace dictionary fit et satisfie frame certain pure hypergraph incoherent existence locally give minimum guarantee span unknown position question svd etc span span learn unknown illustrative p b section find hypergraph specify combinatorial characterization uniqueness magnitude projective notation mean fit subspace incidence convenience incidence subspace incidence index lie span follow combinatorial input incidence constrain solve variety characterize generic give x
cycle come twice flow nucleotide probability target nucleotide signal signal approach various limit distribution derive paper asymptotically organize seq formula cycle obtain analytical carefully simulation summarize far r length variance distribution incorporated flow nucleotide incorporation represent usual case letter throughout independent row nucleotide successive four cycle number
induction operational semantic basis technique pattern generation worth whether satisfy resort qualitative semantic appear deeply high capture pattern generate disjoint set accuracy rule translate characterize obtain eq formula predicate label write formula path translate interpret else pattern simulate reaction system generate contain negative choose observation steady state reason trajectory reach steady state discard separate step intel processor ghz gb memory first rule root explain translate follow formula
simulation norm converge figure control iteration dot value control approach gain convergent control evaluate machine theoretical incremental name td rl general trace two propose mainly mdp optimal thus promise introduce address optimal control usually bellman expression extensive digital sensor availability information past attract
construction assumption incoherence exist strong incoherence property necessity regular hence serve least square satisfy psd psd singular large requirement give standard psd stress os enyi fairly straightforward schwarz argument inequality second exist approach moreover sample necessarily element observe vector rip strong connection incoherence assume
change easier may increased lastly adapt admit gain parameter often discard update summary promising scalability cm engineering division ridge national laboratory tn com computer science north nc edu recommendation express publication reflect view novel graph address anomaly label stream introduce detection describe coarse build aggregate fine closely relate hierarchical simultaneously deviation technique insight internal detect event user narrow focus anomalous particular subgraph anomaly evaluation anomaly detector base detector baseline label accurately anomalie synthetic subgraph graph level possible anomaly interactive visualization tool identify precision social playing increasingly role yet insight
regularity seek represent regressor guarantee represent observe entire processing accumulate introduce twice regressor implies sequentially asymptotically regressor piecewise regression piecewise region accumulate asymptotically performance regression provide algorithmic detail let regressor fig denote node represent tree leave incremental asymptotically sequentially piecewise particular piecewise leave achieve compare introduce introduce piecewise whose competition base regression regressor unknown length piecewise performance huge high fine region show regressor competition significant upper incremental intermediate introduce present twice differentiable arbitrary regressor twice differentiable fig datum complexity introduce state twice differentiable note
build introduction let consider target state lebesgue partition choice discuss later multimodal word vary lot remove consider probability bias typically dynamic implement principle learn eventually adaptive adaptive daily practitioner computation physics context partition call give measure choice partition reaction coordinate course problem context weight free reaction focus method wang practical viewpoint main wang tune efficiency prove convergence wang spirit estimate technique wang adaptive dynamic efficiency wang numerical term set follow correctness wang argument wang draw conclusion convergence present probability biased density bias observe
change lemma mh mh grant complex network separate region characterize component goal fmri clinical ica covariate ica decomposition inaccurate inefficient hc ica testing ica hc em analytically tractable step develop subspace approximate em computation high test voxel expensive covariance advantage hc ica fmri connectivity relevant brain introduce hc ica preprocessing hc ica algorithms ica center whitening facilitate subsequent ica preprocessing hc fmri subject fmri acquire across voxel vi paradigm ica dimension whiten fmri eigenvector
vector uniform near isotropic position factor volume shrinkage epoch shrinkage cut body drop high third weak later improve affine q find isotropic least affect big notation bring isotropic mix ball independent sense algorithm call oracle upper
change indice entry confusion modify precisely two increase overall hold directly unchanged suffice end six unchanged proof value randomly k near trees forest tree sequential bayes test uci repository mnist digit short description dataset ensemble appear p classify background spam classify spam classify mnist binary mnist york ny usa various raise priori classifier accuracy unsupervise solve independence assumption fact classifier competitive artificial contrast supervise unlabeled label occur consideration train label
coincide nominal bias adjustment contrast severe selection reflect confidence long nominal significantly less essentially potentially severe selection address conceptual allow base reader may randomness carry second analyst control thing analyst carry include concern select select appropriate rule scientific ever use model split special reason choose require model clinical trial hypothesis selective writing state formal mathematical great mistake inherently rather give analyst must many know particular choice sensible analyst specify analyst course misspecification procedure splitting model leave bad design property selective selective constrain want behave give reasonable since condition selective setting converge truncate much bad threshold truncate become reasonable take provide extended regularization regression respectively present theoretical generalizing optimality generalize work view selective recent distinct work notably fix regard estimate apply selection statistic require work entire typically marginal py py selection prior reflect credible interval article present adjust goal inference make control multiple perform enough threshold simultaneous less
second category approach greedy signal construct principle greedy gain include orthogonal pursuit omp sparse signal index main difference omp ol greedy update support omp find strongly signal seek maximally omp computational call multiple orthogonal square multiple optimal ol reliable hence utilize
ranking inferential ranking bic interpret I plausibility validity interpret assertion specific hypercube random uncertainty plausibility ranking find validity sa plausible strategy select single base datum minimal select sa plausibility seem cf become reliable identify beyond classical frequentist bayesian challenge example objective bayes dependent inferential I feature assertion meaningful summary datum assertion summary calibrate easy interpretation summary guarantee frequentist error use random set predict auxiliary variable statement two development case multiple I complex identify shape involve consideration shape assertion simultaneous balance discuss distinguish optimal general hyper set I model validity I present I drive selection procedure base I present outperform several work meaningful summary must technical deferred I development design instead evidence summary property I focus specify efficient valid summary I model eq q predictor fix non singular without column center ignore intercept I refer
incoherent none try motivate ask early without model model raise issue tend ignore risk care worth point nice sound justify assumption sense risk I bl
generate obtained successively construct derive hard fuzzy dr scalar intra first dissimilarity vector dissimilarity w define soft value quantify target class membership function please close general adopt provide exploit extent dr perform might classify consider get membership target cast optimization well maximize entropy modularity ds maximize datum modularity partitioning solution community cluster suitably optimize separate modularity whose find approximation modularity follow calculate enyi sec edge separate component pruning repeat time partition use however modularity terminate greedy edge weight community modularity modularity return code herein describe vertex modularity increase remove partitioning consider
illumination condition cluster come insight given illumination condition terminology correspond would contain image individual minimization multiplier admm matlab reproduce http www collect individual specifically choose row dft time corresponding ssc apply average instance subset realization show theorems ssc ssc outperform run increase
regard whereas suggest contain fix include inference enable subtle detect integrate trace give great empirical increase accordingly extra add generally apart less match empirical high increase thereby barrier associate inter empty poisson increase imply equal account two way observe point inter distance match option match behave ccc alignment protein alphabet letter letter one sequence align measure develop scoring assign pair alignment evolutionary evolutionary would shorter evolutionary derive say substitution give chain evolution substitution mutation entire protein period place substitution via derivation use markov period evolutionary one substitution period denote substitution probability long evolutionary
literature function scale triplet metric modality realize wu modality apart similar pair information ignore algorithm two distance modality space al information graph accelerate gradient nuclear formulation modality learn medium retrieval database give brief firstly distance point feature besides binary call link contain pair learn metric follow
continuously concave monotone know increase increase continuity show lipschitz continuous p g thm penalize aim extract generalize pair involve tackle surrogate develop iteratively surrogate separable regular ascent eigenvector provide systematic arise closed derive experiment eigenvalue eigenvector extremely useful numerous analysis machine tool component pca cca instance eigenvalue interest eigenvector large although surrogate smoothed equivalent maximize stationary admit every iteration remain organize generalized surrogate use give brief review systematic issue arise convergence section numerical conclusion real field strictly positive low case scalar transpose transpose denote element denote diagonal main interest regularize include
subset theorem informative envelope subset affine ss would intersect contradict adapt replace interval obvious straightforward ss paragraph k ptc france sciences france mail fr david fr france mail ec fr china e mail mail edu cn document unknown know impose receive replace efficient solver homotopy minimize inspire homotopy minimization heuristic search backward previously inspire homotopy respect empirically problem involve dictionary usual least square homotopy homotopy least square sparse traditionally address constrain nonzero entry quadratic fidelity quality formulation adapt select contrary although objective indeed deduce square bi axis namely objective integer thus pareto kk classify former envelope point fig nonconvex well support reach objective support solution method answer depend size nonconvex area pareto font lb lb lb lb lb lb lb lb square pareto support notice objective pareto minimizer tt sum formulation constrain define constrain problem many able minimizer motivate strategy
select tend dispersion regression recommend beta whose present table differ precision function use regressor include covariate consider covariate diagnostic measure considerably bootstrap base testing inference must model dispersion beta regression model criterion dispersion must enter dispersion less costly viewpoint carlo criteria finite approach argue pseudo dispersion beta regression former sensitive dispersion application acknowledgement acknowledge anonymous suggestion regression criterion monte dispersion parameter include precision
I description possible condition vector interest optimization voxel brain volume voxel voxel great importance detail base quasi glm voxel one respect kronecker identity rewrite minimize update procedure high cost iteration problem optimize concatenation cast solver numerical gradient whose easily kronecker identity parametric equivalent partial derivative avoid product kronecker identity compute compute model case define concatenation f read play crucial role iterative non prevent glm glm significantly matrix case significantly thin euclidean orthogonal matrix triangular objective function reduce liu limited constraint constraint problem include convex box support solver arbitrarily degenerate case smooth cost since bias force direction search length wolfe optimization converge equality constraint
efficient product marginalization universal variability densely basis competitive possibility potential area real find predict unseen approximate notice conceptually noise source noisy noisy important observable distribution infinite sample least approximate training linear linear linear define factor factored basis I linear factor factor base ideally approximate basis restrict powerful fourier infinitely approximation factor gaussian limit reproduce kernel hilbert hilbert
imply efficiently combination multiple leaf allow thing instance representation though internal become increasingly class combination value root boundary locality different future direction promise would tree tree adaptively autoencoder tree internal node multivariate encoder tree generate leave original autoencoder handwritten
high large motivate procedure source nonzero source exception dot share two scatter plane estimate ball dot match would dotted ball along source correctly toy highlight significantly source entail low norm display black direction crucial source separation generally simultaneously amplitude locate amplitude source sample non discriminant mix likely one complement diversity hold discriminant relevant separation way restrict bss trivially substitute unfortunately extra reasonably directly extra flexible belong problem equivalent sample zero adaptive procedure mix estimate manner equation discriminant diagonal highlight previously bss conversely discriminant take amplitude source suggest discriminant weight th scalar exist vanish generally source choice particularly account sparsity amplitude entail source several high amplitude bss source novel blind adaptive analysis source build weight ability discriminate estimating source transform domain
ny imply ny mn must chain element maximal allow encode iff iff iff translation initially head clause head update correctly clause cell symbol state complete rational rational whenever rational uniquely ks ms immediately q rational gaussian elimination clear less rational rational value linearly subset desire rational rational positive may subset differ set particular alternative justification proof thm enumeration sequence element test signature finite turn model j proposition value integer integer outer loop show e problem rational f identically countable unary relation relation rational f validity finitely resp finitely resp sentence inter reduction result last finite countable order language finitely finitely valid obviously finitely finitely sentence finitely e valid work countable exponential immediately finitely finitely resp binary finitely valid sentence order relation finitely valid case first language valid complete computable validity inter reduction countable first language symbol add infinite unary predicate normally valid via computable normally generally work computable sentence carry restrict attention normally valid validity finitely validity model give finite preserve consider duality sentence sentence follow
benefit scientific field physics especially large internet variety arise network detection community detection definition often regard success depend consequently community unknown yield z j variable approach literature include blockmodel accounting log blockmodel maximum ab ab ab within community often tackle effect z satisfy community derivation allow self edge thus assume edge community assignment likelihood observe adjacency constraint give elaborate bernoulli reason approximate blockmodel computational burden parameter reader comprehensive review find form
fw h fw fw fw fw fw k fw take give relate bind construct inequality follow fw fw v start eq imply establish bound iterate hessian approximation satisfie stochastic sgd method stochastic newton arise supervised learning discuss method well amongst method choose set heuristic make neighborhood choose constant give experiment try decrease increment inferior us clutter introduce elaborate training size computation control frequency limited bfg update every curvature remove limited update sgd stochastic datum bfgs form repeat every guarantee equally form stochastic
dominate appropriately bayes present store answering query term storage memory come rigorous compression ill pose evaluation neighbor cite quantify tradeoff front generalization argue nn fall advantage seem nn consistent work recently say classifier induce opposite label obviously separable
stage validity confidence simultaneous test figure stage sample level shift candidate replacement element confidence let illustrate mean group xt ct involve distinct signal vertical perturbation basis datum modal distinct modal clutter curve spurious shift algorithm curve grey red mean identify modal asymmetric top panel percentile subsample mode bootstrap summarize curve line clutter highlight candidate local shift replication entirely lie leave therefore correspond mode mean shift cluster curve display behavioral experiment thank provide datum reach target virtual environment curve standardized activity potential specific array spike sort distinct curve tend characteristic spike analysis curve cluster curve summarize first curve fit truncate heuristic
program rgb rgb rgb rgb study phenomenon consume study asset management analysis molecular run code paper code contrary code stochastic paper build code output base density propose yet apply two les code es pour
q conclude follow proof immediately definition first follow immediately weight focus q r follow hence easy q recalling follow proof th k knn hamming k slight abuse zero enough eq theorem op satisfy q constant sufficiently suffice note therefore follow high variable order equivalently sample toeplitz optimality estimator convex minimax adaptive norm factor recover exactly convex admit empirical demonstrate effectiveness illustrate practical improve classify sound hierarchical high cholesky weight result sparse may large gap insight contribution high problem convex amount dependent allow logarithmic population proved logarithmic moreover member newly estimator also
tolerance p algorithm expect sample attempt computational although tolerance successful computational mention introduction cost solve failure variable lie article explore prescribed error tolerance
object cut option probably question like teacher mixture language understand detect person person understand gender person detect teacher part cut learn thing co together stand knowledge instance box never window help architecture cut utilize limit search common sense location front
tuning estimator hill approach robust regression bivariate tail univariate observation scale univariate exponential regression erm identically approximation likelihood lack robustness identically develop observation univariate approach homogeneous erm fr pareto marginal estimate estimate equation marginal pareto coincide obtain produce help behavior contamination variate distribution want robust empirical corresponding note distribution belong consider contaminate degenerate contamination space define thus measure
analyze guarantee run efficient vertex selection technique include parallelization efficiency social various scale accelerate server distribute machine hundred inference formulation partitioning also present brief overview structure instance solve measure fitting regularizer correlate exhibit sparsity many word attribute ad pattern graphical joint summation clique graph inference undirecte interact clique inference set represent vertex vertex neighbor computationally dependencie problem vertex illustrate risk consist sample consist zero working dependency natural edge clique add edge clique example challenge inference store machine divide exploit decompose solve simplify develop efficient algorithms variant execute server framework computational node divide worker globally server worker server two way parameter recent parameter dependency block loss generality consider server server aggregate without bipartite graph part assign server assign illustrate want worker server goal implement graph balance load computational load incur roughly small parameter keep ram cost memory communication minimize goal cost worker communication show server never maintain cost communication server worker request server minimize machine partitioning attract interest scientific scaling database social previous cut recently vertex cut different work partitioning give partition art several large quality objective note np complete rather task partition intuitively assign balance minimize minimize tb partition iteration improvement probability fail evenly complete several solve key important algorithm proceed follow add increment fu fu I detail quality algorithm partitioning generate partitioning refer iteration assume maximize fu u note monotonicity fu fu fu finally balance happen chernoff terminate increment hence finally fu server neighbor iv program totally constraint solve optimization index server maintain record whether integer exploit constraint consequence integral relax convex optimization single find locally optimal one variable due optima neighbor perform complexity could infeasible first server describe neighbor efficiently tb initial neighbor partition pick assign remove remove tb expensive problem strategy vertex find vertex load balance assign balancing improve naive calculate find fraction undesirable remain impractical graph vertex accelerate cost compute store obtain new adding add change vertex vertice sparsity cost small build datum store cost illustrate th th entry doubly link array increase low vertex update doubly list preserve portion cost small equal degree store small array head location cost jump accelerate list cover input partition union vertex update address sort integer bound degree update evaluate vertex doubly link list access vertex due sequential storing array find vertex list link keep cost bad head average fast partition balance assign additional address balance optimal remain computation randomly construct subgraph neighbor subgraph sequentially denote subgraphs feed union subgraph begin strategy subgraph advantage link efficiency place parallelization soon keep current partition graph size large memory quality efficiency extreme consuming though reduce remove quality single parallelization cpu time subgraph node server node issue partition progress maintain request worker partition subgraph worker read file subgraph subgraph use set vertex well potentially assign assignment improve result use subgraph subgraph old vertex old cost parallel partitioning start subgraph result initialization way want already partition old use initialization use neighbor set partitioning worth impose maximal delay
reformulate reject sample reject rejection quality leave rejection vary reject fraction initial reject fraction operating region accuracy respectively apply classification rejection data performance classifier supervise arise enforce accord improvement achieve rejection rejection rejection ht cccc derive minimization classification highly truth rejection bottom row color derive split learn design sample classifier learn classification parameter accuracy rejection misclassifie classify parameter vary hardness degree overlap gaussian ability cope hard table class accuracy evaluate classification comparison rejection measure reject assess usefulness electrical computer superior university center pa edu electrical engineering lx electrical computer engineering pa systems application three system rejection measure ability correctly classify sample ability measure relative incorrectly classified compare classify applicability rejection loss function automate classification application consequence critical introduce great advantageous classify diagnosis retrieval furthermore cope set noise classifier rejection sample reject fraction nontrivial classification analyze error trade allow determination incorporate reject option embed achieve risk minimize one embed option design framework assess rejection variant base rejection conceptual point feasible classifier reject combine prohibitive rejection world system characteristic roc surface obtain false positive belong positive outlier belong volume roc suffer ratio accuracy gain small measure evaluate performance regard reject evolution rejection relate accuracy accuracy insight rejection unbounded rejection classification allow assessment reject allow three embed reject option label reject correspond derive classifier rejection potential classifier measure show performance completely describe application conclude system see couple classification map dimensional nr represent whether rgb rgb cycle cycle cycle rgb rgb rgb rgb rgb rgb rgb rgb rejection derive system reconstruct probabilistic general rejection element reject small element thus reject incorrect classification classification function separation induce problem index reject pose separation define reject represent support norm subscript present approach lose correct classification element incorrectly incorrectly classify classified ratio incorrectly correctly classify evaluate compare concentration concentration note objective zero classify reject sample classify everything reject rejection fast option add variation express reject length variation allow estimate measure fundamental obtain give support define rejected rejected reject accuracy regardless separates reject correctly newly reject classified sample since express combine give eq quality decision absolute difference rejection q incorrectly since consider classification rejection couple classification obtain reject ideally incorrectly sample reject quality become rejection become rate fraction ratio classify sample triplet relate triplet confusion knowledge confusion binary pair mean triplet reconstruction confusion system sufficient describe behavior classifier q denote classified fraction incorrectly classify reject classified reject incorrectly classify reject give binary classification confusion rejection computation quality knowledge reject mutually exclusive reject unable reformulate computation measure rejection accuracy reject fraction quality accuracy reject accuracy entire quality comparison among classification measure
genetic marker specific mutation incorporate marker poisson allele profile develop mcmc origin genome come proportion population allele frequency highlight existence rare european highlight future development relax incorporate population article devote infer method automatically must careful setup generally sensible correspond common structure update remain manuscript manuscript iteration population restaurant update new link segment chinese restaurant customer ik customer simple parametric observation marker beta al iteration I auxiliary b r walk around parametric hence leave form snp section carlo nonparametric infer extend linkage allow infer infer population origin region assume mutation present mcmc simulation genetic consider variation frequency lead rare modelling snp data mcmc refer presence systematic genetic marker allele population population central history gene map useful gene event nucleotide snp year marker assess population investigate relationship focus particular occur isolated result population american broadly detection sample iv population population population proportion vi identify genetic segment individual propose two analysis estimation pca population decade low individual reflect genetic note principal capture may reflect linkage aim historical event explicitly association assign possibly membership influential early structure assume allele jointly simple extended structure determine genome comes model independently profile markov monte extension address dna mixture dna group pass along day contain original population shorter segment reflect long event occur segment recent linkage necessary thin link correlation quality et improve contiguous share proportion grain process allele model introduce profile segmentation profile rich dependence population important statistical concern model determination use selection probability bayesian though specification population population nonparametric offer unbounded size apply use nonparametric counterpart model linkage model design scalable bayesian dirichlet hdp uncertain costly simplify event identity scale nonparametric slice truncation describe gene section allele individual denote proportion individual individual work take population account employ split contiguous genetic denote follow split segment et complete specify l nucleotide snp distribution use proportion assume equal asymmetric population genetic material thompson control asymmetric allow nonparametric population limit lead mathematically symmetric hierarchical stick break representation diversity population large uniform proportion dirichlet extend specifically constructive follow assume infinite number population particular random number population use population subsection detail complete dirichlet allele frequency case beta allele population concentration independent reason specify fairly denote marker genetic available proxy physical measure interpret issue population range truncation allow resource propose subsection discuss hierarchical prior break population impose population higher undesirable modelling induce order artificial undesirable switch inference address abstract formalism base construction hierarchical dirichlet dirichlet accord dirichlet parameter consist positive concentration parameter base measure constructive process review nonparametric generalize dirichlet surely independent base atom stick break modelling correspond population allele random introduce variable next condition population fall select state individual update finitely achieve simulate stick breaking fall threshold forward backward slice backward variable slice dependency latent filtering instead ignore cause slice algorithm metropolis hasting proposal proportion slice threshold population index proposal slice population indicator forward dynamic eq starts proposal q denote way expression account effect conditioning proportion current backward population computationally mcmc since technology straightforwardly extend assume individual individual chain allele frequency follow straightforward correlate allele allele frequency another likely due specify allele population moment extra physical e sample individual source modify require specification measure hdp perspective employ generalization dp stick break chinese restaurant cluster size decay possible recover population simulation bi genetic marker mutation per population history particular iii population recover number follow population I specification number sequence marker increase spurious dirichlet nevertheless cover I prior specification difficult agreement value aim capture arguably implement parametric linkage model describe allele al value likelihood latter harmonic example seem log software drawback criterion interpret result give suggestion improvement demonstrate human wide study american uniquely advantageous history extensive occur year hard recent american branch separate drift shape genetic landscape cause drift east eventually become allele common snp rs allele european alternative allele snp human straight shape rs snps include strength absence snp rs informative snp act specify carry variation range effect mean selection individual snps available population cell panel publish american population west south european population run sampler iteration every iteration precision hdp mean beta observe frequency prior evidence mcmc minimize alternative major I occurrence use panel cardinality european origin cluster respectively confirm look frequent assign major verify firstly
concave conjugate machine reduce solve result saddle widely saddle exhibit say likewise keep notation learn convex concave saddle explicit form separable convex gradient smooth subgradient smooth strong convexity convex minimization erm eq convex function predictor convex regularization fall regularize erm formulation regularize detail erm employ eq lead sep compare general consider composite solve alternate multiplier admm problem particularly erm zhang propose dual descent descent primal dual dual stepsize unnormalized primal solving carefully exploit subproblem propose adaptive stepsize summarize primal elaborate theoretical comparison erm superiority conclude first primal could linear achieve method stochastic dual coordinate iteration intensive variant batch method handle however conservative constant stepsize primal convergence reality observation exploit propose adaptive solve configuration optimize alternatively update dual principled separable iteration variable follow coordinate block exploit erm couple block norm small block primal couple large help variable update proximal also incremental variable intermediate convergence adaptively contrary tb block initialize pick coordinate block configuration update update use whole procedure sep notable characteristic compare size whole bring independent parallel modern help use available possible convergence assume strongly n technical coordinate e practical implementation measure performance term w r pass firstly problem zhang point set diagonal optimization ridge employ conjugate ridge sep ridge regression dual closed objective measure entire ill conditioning observe substantially condition stepsize time sampling dataset number ccc protein ccc dataset protein compare real set benchmark list table collection obtain dataset take term aim minimize regularize method hinge smoothing conjugate hinge ridge necessity whose dual convex initial newton sag stepsize sag try stepsize result theoretical stepsize sag loss algorithm smooth hinge compare method task early epoch result epoch cause stepsize choice primal iteration explicitly couple block primal theoretically superiority erm immediate size theoretically optimization also acknowledgment support china thank discussion proof present firstly characterize semi partition consider configuration matrix definite firstly q cauchy schwarz fact view inequality obvious exist far simplify consider zero expand positive firstly variable th let I strongly minimize saddle happen round consequently mn dual strongly obtain side
similar classic correlation propose software package usa confirm solid thin parameterization compression wavelet evident wavelet match signal address wavelet signal suggest methodology wavelet version wavelet science engineering research development pt cr tag tag university electrical engineering I electrical role diagnosis low cost clinical conduct although extract information utilize recently advanced wavelet I wavelet offer frequency ii perform characterize global design wavelet match detection wavelet match signal systematically numerous issue generally regard represented wavelet wavelet often select research wavelet compression detect minimize compressed wavelet compress signal minimal wavelet consider original signal analysis wavelet aim six seven nine medical usa place projection eight channel internal present process internal visual verify normal diagram depict cc perform signal low analog use frequency analog solution bc configuration hour four body index sd one utilize university raw contaminate potential iv signal appear channel visually g identify discard practice recommend obtain reliable subsequent subject interval hand recursive iterate procedure iterate coefficient observe perform similar recursive analysis ii level compression discard criterion base large absolute coefficient retain eq original compressed analysis introduce reconstruction evaluation effect compression commonly percent root square utilize study compression signal subject cycle human scale period magnitude fourier consequently minimize difference decomposition wavelet wavelet wavelet human haar set optimize choice wavelet abundance wavelet prohibitive result wavelet introduce know wavelet finite impulse filter length great wavelet find haar wavelet one six utilize wavelet wavelet wavelet parameterization define every scheme generate parameterization minima surface wavelet
iteration rbf kernel distance neighbor far fine classification rbf rbf svm classification three baseline original kernel rbf figure fourier dimensionality grow accuracy rbf fact approximate use section mse show dataset achieve compact kernel compare fourier figure comparable marginally well lead well classification kernel map advantage train scalable nonlinear computational complexity advantage achieve observation transform dft denote element dft dft dft efficiently store element dft kernel nonlinear feature although fourier procedure element trick convolution nonlinear performance storage tend performance feature recent allow large map shift invariant substantially performance attribute mostly simultaneous along parameter neural nonlinearity bridge stream high dimensional impose capture deep compact li david discussion kernel via feature machine challenge method approximation pick nonlinear map prediction propose achieve kernel joint achieve map function definite induce could fortunately one utilize rich despite popularity high complexity vary prohibitive million tend growth svms application since space utilize method expressive reasoning nonlinear become popular machine formally find nonlinear even approximate design approximate kernel result work propose optimize directly definite shift learn approximate achieve predefine map compare baseline method propose structured projection computational number nonlinear input seminal map mapping form rbf additive skewed multiplicative monte build random fourier besides base matrix significant effort good kernel mkl optimize directly optimize joint kernel type machine decomposition limit vector scale truly datum alternative apply machine partitioning relate begin approximate shift positive fourier characterization definite borel transform interpret expectation carlo fouri use type kernel include despite popularity feature notable issue matter influence kernel approximate approximation technique try lead kernel work fourier lead feature function kernel shift invariance show map positive shift multi training optimize map dependent hinge initialize output challenge number optimize problem nonconvex propose find sgd traditional eq sgd mini point write set step optimize initialize random fourier sgd optimize algorithm classification also kernel mse note classification former compact
patch follow patch whole pose patch learn sum input output project forest suitable multiclass classification parametric tune parallel robust processing patch spatio forest tree tune depth laplace calculate pose construct drawback pose pose real image back follow training recover might choose sample test viewpoint regard labeling write appear pose pose assume view trend reduce range multiply smoothed step turn trend turn clean rf rf patch rf promise clean background real background conditional automatically rf clean set truth indicate global although achieve drop learn patch importance verify patch hence overfitte show test test pose misclassifie pose like square front often misclassifie estimation learn image experiment verify effectiveness future foreground background utilize image assumption patch generalize height title date em project aim pose crucial know pose object besides also drive good pose estimation driving system pose benefit field pose implement dataset multi camera however create extremely consume limited dataset research specific poor work utilize power shape specific balanced precisely million thousand image easily pose base predict category tight although choose probability vector pose give grid score combine whole set pose feature diverse project focus transmission method iteratively classical problem research line rely diagnostic object view category view invariant group part feature svm classifier pose latent discrete convolutional propose base simultaneous gain structural object like model base pose estimation represent obtain rough object annotate viewpoint motion limitation utilize solve collect model annotate thousand object category vision utilize evenly accordingly image first pixel divide patch patch preprocesse extract whole evaluate build increase test
dimensional significantly curse canonical design set variable instance complexity certain cca view regression cluster etc theoretically interpret cca probabilistic foundation practical cca view utilize application extract kind texture popular visual base video typical cca view correlation view drawback information obtained ignore tackle develop cca generalize view yet natural particular analyze view approximate covariance tensor investigate alternate square adopt measure covariance view represent feature whereas tensor illustrative subspace feature dimension useful extensive experiment challenge task internet cca approach confirm summarize work introduction extension extension view extensive experiment paper dimension find dimensional feature former subset variable original transform datum reduction e laplacian e lda amount label view attract multi view graphic multi view family weight focus remove irrelevant reduce multiple leverage dependency coherence view conditionally independent thus shared exploiting correlation exploit method multi dimension among view consensus share adaptively view pairwise constraint dimension incorporate use margin latent subspace al formulation share conditional constraint correlation analysis originally find basis variable project maximally pattern recognition svm view prove rademacher svms cca correlation conditionally simple subspace base view show weak condition addition cca concentrate tensor vector cca without extension multilinear cca consider study volume tensor flexibility obtained term cca difference multiple vector view closely work concerned cca cca cca least cca perform combination canonical representation costly svd train drawback reformulate cca couple ls seek pair cca efficient adaptively disadvantage cca ls namely pairwise exploit correlation view ignore framework introduce correlation generalization several column cca find call canonical correlation canonical maximize problem matrix stack cca constraint cca cca view cca canonical variable avoid solution adaptive reformulate impose l cca cca pairwise tensor multi reduction high view diagram kind color histogram wavelet texture feature extract feature intuitive illustration generality different data covariance subsequently sum r feature low projected briefly concept multilinear tensor denote I denote mapping associate multiplication store express series kronecker c p finally frobenius tensor give q instance view two variable tp appendix far add become identity nonnegative trade base instance include web annotation near neighbor feature nn view long cca formulation view accord web term average voting term cca view base ls unsupervised dimension induce cca empirically evaluate supervise dataset secondary sequence instance randomly unlabeled instance performance three cca method cca cca ls amount unlabele follow find evaluated set since directly learn datum large needs solve thus position divide attribute context attribute base current position middle position view attribute dimension c cat cca cca l compare relation dimension table concatenation comparable strategy single cat baseline dataset cca view utilize former accuracy cca increase avg l cca good dimension cca cca l significantly cca ls cca reason seek canonical factor decomposition tend uniformly factor discover explore kind explore information ad internet uci repository whether dataset instance instance utilize unlabeled sample use attribute represent absence l feature view view term site view feature cat cca cca avg ls show compare observation concatenation strategy cat cat high label much steady underlying correlation compare unlabele utilize order correlation reason effectiveness natural image conduct subset cat image distinguish similar cat label utilize unlabeled color auto wavelet texture represent annotation improve number l cca avg peak cca cca avg l accuracy avg cca ls decrease satisfactory even method label cat cca avg cca web task non linear problem dimension infinite linear visual histogram achieve nn averaging use formulation view setup tensor parameter optimize unlabele utilize separability trick strategy comparable slightly well c avg empirically computational conduct matlab computer ram cost result high decomposition adopt paper could satisfactory accuracy efficient demonstrate superiority unsupervised cca deal multi ignore correlation feature resolve tensor cca discover analyze view application conclude subspace view feature especially high examining may utilize outperform high cca traditional main disadvantage could utilize accelerate tensor accord element denote th additionally accord
label show dual kernel rank fraction maintain representation align manifold sort unfold form image synthesis remarkably distortion manifold unit author address issue machine proposition property laboratory de alignment match correspond property generalize align complexity unfold plus alignment cope multimodal align dimensionality robust transfer address issue reduce computational principle exhibit synthetic example recognition constitute interest pathway characteristic depend availability family attempt adaptation canonical cca source meet seek target discrepancy mmd geodesic datum suppose feature idea geodesic distance linear subspace intermediate flow subspace path dimensionality pca cca gm information projection discriminative label source find feature partial pls another know domain coherence plan source target try align feature move decision hyperplane know matching generally use appealing method specify amount pair semi alignment project belong become geometry preserve perform deal multiple cope nonlinear introduce generalization property allow dimensional property cope extract rotation manifold structure unfold simultaneous domain domain align manifold kernel numerical inversion permit alignment meaningful propose summarize align manifold kernel nonlinear cca align pair may useful align lead pair demand counterpart must stress laplacian take effort compute efficiently solve resort reduce remainder review section introduce formulation practical present linear method toy real visual conclude section classification belong close apart entity lead component class otherwise class similarity represent radial basis rbf near compute three entity joint contain column serve common project inverting domain adaptation synthesis hilbert replace reproduce kernel dimension ii block row hilbert thank sum hilbert theory resort representation theorem express linear map replace dot product contain become instead dual advantageous operate numerical computational current fisher deep latent space map define eq feature eigenvalue confirm use benchmark among candidate kernel tight graph artificial distortion mis alignment visual database recognition first series toy domain column fig scaling experiment hold rotation line third dim illustrate classification source align basically rotation manifold allow sort unfold experiment even alignment projection resolve provide picture green linearly projection trend experiment provide ccc cc cc space exp domain exp exp linear extract feature second right accuracy rbf kernel project randomly label resort operation simultaneously unfold full actually h c adapt svm domain show inversion direct linear kernel shown label keep fig reconstruction capable invert accurate basically significantly domain rbf achieve target rotation exp class domain domain class c exp plot average run consider amazon four domain amazon feature extract normalize histogram visual word obtained subset amazon dataset feature adaptation proposal supervise adaptation geodesic eigenvector pls use pls eigenvector source domain constrain source project correct decision source domain method label pixel domain sample alignment ordinary nn classifier label pls use target sensible kernel problem histogram intersection k j du nn top bottom report outperform compete improve supervised method provide art handling domain dimensionality classifier align
partially ac ac may satisfy acc ax ac different subset unconditional association set bc ac x cf ac z always hold pairwise ac z ac represent separate give separate reduce model explain separation criterion imply away association hold ac bc ac bc acc thus association connect graphical etc conditioning association figure ac notice appendix z condition bc one six ac bc ac bc also hold bc hold may six distinct rule supplement hand four draw dag consequently bc exclude ac ac ac ii bit conditioning weak square seem ac bc illustrate vc acc ac bc theorem corollary relationship ac vc acc example argue ac bc dag relation replace covariance comparison ac bc x ac ac z ac ac ac ac qualitatively ac iff theorems implication vertex compare square parent behaviour nature path illustrative theorem move drop move path illustrate drop ac z factor numerator denominator thus find theorem possibly whenever graphical association tree vertex definition connect vertex say subset give separate separate separate hand unique give relation set describe separation criterion satisfy independence formal operation triple define satisfy ac ac ac satisfy ac ac figure ie ac ac ac substitute direct skeleton tree path connect connect vertex ie path k ac ac ac ac ba z ac bc ac ac b require iff independent none find present none direction water point consideration measurement water level bx ie water neither however also ac ac none point ad stream ie ie ie section relationship square qualitatively compare well qualitative condition relationship reduce counter unless ie conditional graph law clearly keep plot compare relaxed z necessity figure edge respectively dash regression set plot title plot coefficient coefficient unconditional ac ac solid edge ie qualitatively ac ac ac z violate concerned square partial correlation condition violate condition square qualitatively compare either theorem ac z figure satisfy violate qualitatively correlation square qualitatively relaxed qualitative comparison sufficient involve mutual information matrix square comparison qualitatively rule sign comparison correlation regression develop definite e trivial constant k either zero denominator sign numerator algebraic positivity c correlation replace unless assume abuse still correlation ax cx ax cx ac ac ax cx ac denote inequality inequality ac ac ax cx ac denote bb ac ac bm ac ac ac ac enough ii hold ie ac bc bb b cx notice eq bb substitution proof ie notice zero none bb bb z ac follow ac result condition denote simplification ac give connect clearly connect connect clearly connect convenience square need ac ac result separate imply ac z I q show consider subgraph vertex c ig ix ib I ac I ib triple ig ac ac n conditioning suppose separate ac z ac z ac ac ac z follow assumption decomposition ac intersect g q v vertex arrange l k x result represent consideration ac z acc ac furthermore show prove direct acyclic vertex exist path non connection possibly connect say set empty separate separate dag whenever separate relevant vertex ac z ba ba clearly converse path however ac possible vertex imply least cv ac would ac thus separation cc give get ac path ac ac ac ac ac z ac b ac ac give ac ac fact three supplement closely refer treatment mixed graph contain three terminology describe relation graph define say vertex pt mixed condition sp separation mixed vertex path ie path vertex graph connect set every connect separate pair disjoint set say relation represent disjoint
box object view get fan pathway two objective fine tuning label multimodal initialize bfgs mnist pathway likelihood optimize fine tune pathway deep evaluate joint test joint analyzing image digit mnist randomly draw digit noisy learn dnn mnist digits default model dnn classifier test test clean testing noisy digit db train triplet clean digit test multi view svm indicate helpful multimodal svm fan multiple boost well achieve goal want output source boost matter objective fan deep multimodal powerful lr error multimodal fan share representation multimodal powerful branch deep function different step initialization tuning stage cd maximize joint fine stage define label object recognition answer leverage source boost experimental fan structure fan easily fan extend branch share node branch lstm joint multiclass object recognition jointly prediction experimental demonstrate baseline multimodal year approach propose multimodal joint rbm gaussian unit et al low image additional svm demonstrate various object recognition multimodal deep attract attention use autoencoder speech composition unimodal undirected pathway pathway large unlabeled potentially multimodal allow naturally effectively handle specifically paper unified fan learn compose lstm generalizing model initialize cd fine parameter optimize leverage multiple handle different visual recognition visual denoise autoencoder extend corrupt boltzmann introduce recognition denoise shape indicate ignore handle effectiveness regular structural multi propose joint method detection boost competitive baseline multimodal svms share handle modality clarity deep fan explain boltzmann machines rbms branch boltzmann fine deep pathway part rbms introduce rbm hide unit parametric form likelihood calculate cd ignore bias observation layer update bias restrict boltzmann stack rbms high activity hide layer energy configuration represent visible rbms real likelihood wise basically rbms cd rbm treat next rbm stack multiple deep kind couple stochastic binary unit hierarchical via rbms clarity way suppose input layer unit clarity modeling layer visible rbms bias visible clarity analogously get likelihood formula fan additional top right multi modal layer ignore input cd effectively tune parameter initialization joint representation modality discriminative multimodal update via multimodal deep basically stack rbms layer learn rbm treat next stack v x manner mention model approximate expectation mcmc procedure expect sufficient statistic learn latent replace update log h entropy approximate posterior naive factorize x field dependent item sample mcmc parameter cd eq fine determine visual labeling deep different visual labeling corrupt way clarity loss specify indicate share triplet l sigmoid case branch simplicity clarity note branch structure keep class object assume input object object belong purpose answer view image specify ignore clarity initialization cd tune w r via backpropagation bfgs visual label two even multimodal cd view recognition leverage joint learn joint output input latter leverage source bi feed consist encoder recover share feed multiple model adjust visual cd instead recover autoencoder multimodal unimodal undirected pathway completely unsupervised fan multimodal fan lstm cnn multimodal multimodal discriminative modal feed forward neural share handle learn together representation multiple
create density image plane translation add snr synthetic optimization perform seven traditional sgd accelerated optimization particle rescale deviation parameter radius minibatch size use except iteration minibatch minibatch trade small result size slow base tune training accord base rate momentum cm online hessian epoch online increase log remain initialize density result versus gradient online minibatch projection match density cross correlation search reconstructed iterate similar publicly approach run method evaluation five fail converge particularly approach approximately formulate cast result stochastic array seven range sgd quasi synthetic method little exist make epoch minima notably require per hessian free simply take iteration method problem challenge highlight exist manual amount manual every would new need automatically department determine structure biological key biology heavily paper structure latent model seven method recover less epoch initialization find slowly simple method converge optima method fast protein traditional ray limitation target grow impossible biological determination raise attempt thin pass measure produce image visible relate capture interference image nature biological keep lead extremely around imaging image illustrate particle image ct ct projection direction corruption density coarse shape visible fine presence practically explore em density introduce image formation em posteriori estimation stochastic optimization speed gain ability compute estimate stochastic less initialization previous bad initialization able quickly initialization compare em mixed result suggest serve optimization algorithm paper real benchmark aim refinement initial estimate project image consider orientation state slice origin projection direction particle fundamentally ideal circumstance impossible error refine poor initialization structure clearly wrong appears result publication incorrectly crucially use method orientation image orientation parameter analytically intractable numerically far perform marginalization originally image alignment marginalization originally perform find individual particle marginalization raw stochastic progress make quickly optimization recently fundamental sgd popular surprisingly momentum method sgd iteration natural manifold fisher accounting attempt approximate curvature hyper attempt operating gradient still limited strong scope interested reader compare evaluate performance good converge optimal simple complex costly formulate observe unknown seek projection density direction corrupt primary interference model frequency spectrum typical zero information zero vary setting condition every refer reader noise arise exposure noise model iid formalize density denote cubic grid orientation represent transformation project image denote image plane center represent denote consideration evaluate make slice eq fourier transform image fouri diagonal interpolation operator fourier transform speed frequency fourier specify provide shift however unknown cope shift double analytically resort quadrature quadrature direction account rotation quadrature shift quadrature weight quadrature scheme consequently frequency image combination exponential encourage density specifically prior possible promise directly marginalization
fundamental tucker tucker tensor q matrix tensor tucker unchanged size remove rotation encode abstract space eq invariance minima isolated isolate consequently systematic decomposition unconstraine interpret search endow hessian order riemannian metric hessian costly simplified index candidate riemannian metric simplicity consider element note convex convex individually td reasonable novel riemannian space early square propose tangent line abstract object total run total riemannian riemannian key riemannian conceptually transform unconstrained briefly development cost first equivalence blue color manifold tangent induce equivalence realize vertical space tangent equivalence sense metric horizontal abstract abstract tangent riemannian g x along equivalently riemannian well pose horizontal x tangent vector transformation straightforward endow riemannian principle concrete abstract g start dimension need tangent operation extract ambient normal matrix dd matrix characterization eq q efficiently matlab routine vertical vertical characterization dd horizontal projection lyapunov skew couple lyapunov routine combine horizontal local search manifold I depend representation horizontal abstract tangent manifold smooth mapping tangent tangent generalize concept manifold lift xt depend riemannian completion choice development use conjugate smooth complete conjugate remain cost end riemannian guess concrete formula total n n r n r dd x lyapunov order r f auxiliary euclidean partial derivative respect due partial scaling riemannian lift subsequently lyapunov routine numerical cost derivative initial guess square cost degree direction f algorithm state tucker decomposition nuclear intel machine gb ram matlab handle select entry dimension os create stop either mse iteration exceed five deviation comparison os asymmetric os euclidean benefit conventional symmetry natural randomly os simplicity compare descent backtracking give conventional consider os b fast error scale tensor size rank os outperform influence instance os complete superior decrease ill conditioning instance case additionally impose exponentially decay number cn influence evaluate property noise noise train os asymmetric rank along different case consider tensor size consider rank hyperspectral hyperspectral five random pixel os rank r adopt randomly split validation algorithm stop correspond ratio propose corresponding day wide bin size completion dataset reveal validation partition iteration rank second l os mse problem stem riemannian exploit fundamental uniqueness tucker decomposition riemannian enable riemannian concrete expression work superior benchmark future research direction look update rank tensor tensor tucker acknowledgment thank van present dynamical science office research national height tensor supplementary mm width concrete r tucker transformation riemannian product tangent manifold tangent x inner extract tangent characterization hold tangent space characterization skew matrix generality n characterize horizontal tangent remove along vertical vertical linearization dd linearization orthogonal vertical characterization space orthogonal relationship vertical characterization trace trace trace trace mode unfolding since skew extract component ambient definition tangent satisfy equation solve matlab routine tangent h result dd note tensor r plugging equivalent r matrix skew lyapunov equation routine combined gauss auxiliary unfold specific scale metric derivative scale consequently relationship lift requirement tangent space mode dd lyapunov equation square lyapunov solve routine representative comparison numerical span synthetic instance tensor os figure show different manuscript mean figure os figure fast consistently competitive especially ratio tensor rank os algorithm show algorithm instance result sample complete rank convergence os propose five dimension figure superior propose outperform additionally
fail hold interpretation differential natural tradeoff privacy release database achieve easy goal aggregate interest conversely complexity privacy complexity differential setting less recently show optimal privacy guarantee meaningful complete privacy give private choice smoothly approximate differential algorithm compute extremely fundamental database marginal row say sense md meaning e md differentially private marginal average privacy combine result must multiplicative answer family query additive necessary laplace differential low mechanism average case error guarantee guarantee mechanism matching surprisingly degradation answer marginal sample laplace mechanism factor widely technique query bad error efficient differentially private guarantee laplace sample differential cc c lower n code originally digital recent bound differential al code construct demonstrate mechanism accurately marginal private give database construct answer attack privacy accurately break private release specifically state algorithm differentially private private meaningful privacy guarantee size reduction start row uses differentially apply attack value low quantify pure sample proportional md correspond add squared turn well tail bound add noise differential privacy different union chernoff show namely ensure error marginal trick marginal rather row row binary database differ single denote replace another privacy differentially private adjacent database know differential privacy generalize smoothly say database differential privacy differentially private adjacent marginal privacy exist differentially private exist differentially private every accuracy bound namely differentially private main differentially private sufficiently large introduction rearrange convenient introduction code tailor originally dd code subsequent adapted existence code bind differentially choose create copy entry output privacy differentially private mechanism nk contradiction sound code completeness differentially contradiction ensures require k contradiction seem natural one way case error rather average bound mechanism distribution sample distribution function formally measurable firstly clearly verify shift distribution give bind obtain infinity surface precisely gamma q inequality eq require verify first gamma r I give circuit use privacy powerful take input database query differentially assume simply differentially define mechanism private differentially private differentially private require q guarantee hold q failure failure dominate work differentially simple private suppose special hoeffding hoeffding copy let give e dd row eq q rearrange require combine sample uniformly sample entry sample independently n let verify construction
latent latent version complete coordinate descent minimization constraint hence duality strong factorize case slack variable latent model product linearity expectation fact qx ix kl primal strong duality maximize constraint minimize kl minimal value fully variable assignment q fall slack relaxed remainder slack form slack equivalent rgb rgb rgb rgb rgb rgb rgb rgb rgb cs berkeley edu
vector real hyperplane side sum along direction consider shift hyperplane decision determine anomalous shift hyperplane store anomaly network design single dimensional weight regression hyperplane directly eqs hyperplane anomalous non anomalous distance fall derive threshold point form center correlate center class hyperplane anomalous identify anomalous vector activation identify anomalous point cm cm theorem corollary height em expect value anomaly give identification class anomaly identify neural describe decision separate offline recall class measure datum throughout probability measure weighted lee anomaly detection traffic case measure identifie measure accurately entropy responsible formation particular anomalous identify attribute sample assume subset bound area define partitioning size maintain partition maintain graph base anomaly composite score scoring structure asymptotically optimal false sample increase normal produce distribute maintain disjoint score anomaly center respectively hyperplane measure center hyperplane anomaly datum point regression greater bind anomalous likewise within hyperplane great anomalous shift show activation neural obtaining point value hypercube unit area dataset unit square plane give minimum sample region region radial circle q circle decrease probability contain within one contiguous otherwise order exceed maintain radial size less disjoint thm lemma circle radius partition copy contiguous classification remain new know member class member candidate outside new linear respective hyperplane question center whereby hyperplane linear hyperplane suppose regression give otherwise micro anomaly detection anomalous radius center hyperplane anomalous anomalous class triangle definition anomalous section use obtain dx dl anomalous anomalous dx show exist anomalous
flat fix representation second qualitatively show able automatically proper conduct extensive empirical superiority previous fed ensemble length word column distribute vocabulary vector pool vector insensitive order word length sentence recurrent network connection hide form nature recurrent sequential generation task model machine step time recurrent bias non recurrent summarize past composition recurrent htb build parse initialize word non child composition recursive network hide parent parse child connection recurrent sentence tree recursive parse recursive neural neural composition recurrent neural network chain recursive composition pyramid locally combine reach top pyramid global sentence refer build differ neural try form decide illustrate acyclic graph show input sequence pyramid let level define j tt th j view phrase tt unit consecutive phrases original rd two level whole sentence enter pyramid apply word phrase rich pyramid matrix embed u v tailor factorization help parameter composition pyramid range recurrent way length linear representation fundamental behind encode semantic phrase composition two phrase express hand hope phrase variation decide parametrize decide non forward future composition mechanism adopt mlp output system multinomial pyramid define apply pyramid worth note composition implicitly form layer along phrase recurrent net recursive net htb pyramid build phrase original sentence illustrate rd pyramid straightforward short phrase high focus part sentence interested review goal phrase opinion set question mr cr cv cv list class measure split train fold list autoencoder word gradually phrase along tree sentence apply generalize pool paragraph unsupervise public top paragraph continuous pool phrase sentence recurrent recurrent neural network network convolutional pyramid use pyramid difficulty train recurrent largely due vanishing discuss dag vanish still recurrent show application propagation problem frobenius norm recurrent act since computationally exact value direct formulate coefficient typical minibatch optimize word train wikipedia embedding performance composition relate softmax implement try mlp implement improve htb mr cr nb rnn compare consistently rnn margin comparable number code phrase dependency hard help phrase nearby word consistently also outperform set well fail length encoder machine quite task think due encode characteristic surprising datum average limit window mr cr rnn rnn also v rnn table time hyper initialization report run consistently outperform set htb study consensus pre visualize belief score vary sentence set train adaptively appropriate hierarchy give give concrete mr prediction level score fig row show ty incorrectly high representation first belief correct multiscale combined allow correspond belief automatically component st row implicitly property explicitly objective achieve sequence explore new direction sequence instead length continuous effectiveness short represent acknowledgment author technology helpful support china national cb conjecture com ability accurately word phrase self sentence hierarchy phrase composition adjacent segment mechanism network particular task effectively vanish persistent qualitative quantitative analysis automatically suitable task yield task cast semantic parsing describe sentence representation representation quite translation matching perhaps simple continuous bag word sentence max pooling word word unsupervised fashion effective
result compare perfect seem show affect show accurate design work consider several perfect assumption suggest type compare theoretical result rank long cdf order th rank resample denote result sample result bootstrap compare base literature liu functions empirical methodology powerful design comprehensive likelihood balanced assumption liu al likelihood method data cycle liu et al use overcome balanced figure plot algorithm liu third htb contain birth month weight research university well rank quantile month weight intensive active nature activity birth treat record month perfect seven month month weight process result weight seven month time summary histogram seven month birth seven seven month birth observe median month weight birth different design pt bootstrap et df show satisfactory perfect propose nonparametric cumulative et estimator exhibit order property balanced applicable result application et df size acknowledge constructive comment associate support national institute health grant gm department statistics usa mb usa nonparametric df base set et use unbalanced rank bootstrap estimator df asymptotically exhibit finite scheme use estimator estimator keyword rank powerful collection technique collect representative small fairly accurately order take actual costly unit find application environmental sciences chen rank balanced unbalanced unbalanced rank rank order statistic quantify size unit rank quantification unit order pre suppose measurement th underlie population denote n identically df reduce balanced show chen easily see equal df underlie property balanced chen use estimate practitioner standard statistic make inference characteristic interesting develop efficient technique exact obtain chen al estimator use resample underlie desirable outline consider validity method problem describe finite sampling parametric resample population mean consider five design propose liu testing problem real consist seven month provide conclude remark likelihood powerful nonparametric use subject et estimation spatial estimation calibration among nx px minimize dp I multipli testing p lead nonparametric distance q size estimator element draw retain generate bootstrap repeat bootstrap easily bootstrappe purpose unknown parameter balanced modification sample underlie desirable similar et bootstrap sample mx underlie simplicity write test incorporate estimator introduce lagrange coefficient exp nan testing bootstrap mx function estimator proceed select continue retain repeat step bootstrap interest resample perform use b experiment also summarize example design per denote design distribution proceed proposition df interest row represent population converge distribution normal diag f df approximately refer rest work consider approximation degree freedom bootstrap use nan resample calculate proportion estimate testing sample normal logistic perform see unknown let x calculate bootstrap nominal
feature approximation diameter grow preserve derivative analyze uniform convergence propose motivated feature map enable task derivative consider kx ph tu ph tu ph tu j impose constant square expense slightly q case p p analogue obtain assumption boundedness impose certain moment expense bernstein net union extend differentiable z dd conv h continuity z supplement continuously clear p kx qx theorem comparison handle unbounded function detailed feature context improve machine approximation compact analyze approximation reduce equality change lemma bind rademacher bind combine x follow jx ix I theorem conditioning bind imply covering guarantee center combine bind arbitrary point quantity center lipschitz uniform optimize p cover center net eq cauchy schwarz get imply l z linearity get net center use propagate center thereby condition notice bernstein give union substitute jensen dc matching prove boundedness imply let random suppose yield bernstein satisfy theorem college house ar uk kernel powerful tool tackle capability relation good intensive scalability require operation limitation construction literature fouri construction kernel popularity theoretically provide norm derivative quality success several fundamental range extraction estimation discovery hypothesis capability relation possibly heart product kernel kx flexibility operate raise serious dealing order resolve numerous solution design additive construct shift paper yet efficient rely appeal kernel low dimensional map empirically x explicit low dimensional feature primal fast solver thereby enable scale algorithm degradation approach low rank approximation approximate online applicability area privacy causal surprisingly literature theoretical insight pr number fourier feature involve empirically statistic systematic entire characteristic decay diameter guarantee asymptotic nature finite sample optimal optimal almost sure convergence apart kx ml traditionally involve kernel involve kernel gram consist derivative address numerous task supervise multi infinite derivative elegant method mention quantify quality summary derivative term rate preserve match growth various along brief section material map definition continuous f finite borel denote banach integrable lebesgue measure rl df ix da r gx p nb nr gamma dr dr continuous translation invariant kernel kx fourier e symmetric kx tx replace measure write inner algorithm primal thereby well complexity solve mr precise dd differentiable tail compact optimal order convergence excess order significance grow analogue rate dependence section introduce practical theoretically understand theoretical optimality section see improve logarithm discuss consequence next suppose kx interest hoeffde inequality refine apply provide show consistent topology convergence convergence sure lemma instead discrete absolutely lebesgue density interesting study diameter therefore mention rate write
office building day usage business twice investigation business warm day probably figure present usage pm business day type system day turn middle day day special spike energy consumption spike model understand exploit differ treat day replicate modelling usage arising give usage give usage observe directly infer form general arise unobserved function main latent single switching among process sequence estimate derive curve replicate replicate switching among author single realization realization paper discuss detail paper contain replicate principle unlike work focus generalize change control parameter maximum penalize realization consider replicate replicate replicate simplicity across replicate hide unobserve consider identically follow fix replicate nk nz jx jx intercept ik depend identity matrix variance variable govern distribute length transition estimate standard measure replicate user depend since expectation maximization maximize em generate arise section overview propose intercept cubic entry find criterion propose cv smoothing simplicity application simulation choose r joint complete application replicate write p r entry eq hide state application calculate supplementary function fix smoothing guarantee depend dependence via summarize f maximize parameter obtain maximize obtain present update show form k ii pz ik see dx intercept probability diagonal step still change however easy except algorithm ik eq py nk ik obtain propose hold maximize fix eq initial work test datum discard treat set follow replicate cross maximizer pf convergence computationally intensive fortunately replicate estimate present use process restrict ease explanation use derive matrix plug formula yield second ik process obtain simultaneously supplementary section q r find supplementary datum replicate usage give give power assume covariance intercept smoothing describe detail figure fit nk fit curve give assume parameter estimate pz ik report fit give usage assume smoother estimate follow k z nk also estimate curve replicate transition transition actually happen gradually incorrectly replicate come get fit switching nonparametric regression observation within replicate correlate describe analysis assume follow table present obtain observe agree table show figure power usage usage consider upper seem building intercept variance therefore current follow markov generate replicate replicate locate type type simulate datum markov study vector plotted figure space true simulation obtained generate distribution repeat step time replicate repeat step obtain calculate simulation assume markov figure study replicate value try converge take long choose fit curve simulate simulation quality via produce deviation supplementary plot simulation improvement could way adjust freedom estimate parameter process error desire coverage interval form quantile coverage close supplementary box plot observe estimate close true follow maximizer maximizer rewrite positive
detector operate reliable regime adjacency size community characterize adjacency matrix entry bernoulli overall stochastic two connect parameterize probability matrix adjacency corrupt adjacency graph connection community restrict increase plant clique detection restrict block study critical size network study external edge community dimensional one matrix noiseless graph laplacian community adjacency two obtain eq leave multiply since entry let th singular value large rectangular prove concentration mean n I prove square inner almost surely almost recall show transition exceed certain case grow sign two term psd surely eq transition whereas asymptotic low substitute asymptotic value stochastic matrix random prove c c c p limit community detection independent generate node correctly transition empirically almost perfect low derive opposite fig threshold empirically empirically reliability denote network community detection classify three category network network intermediate region american political books amazon estimate political book book frequently determine perform separate book book neutral label investigate sensitivity detection detection mostly reliable community mostly indicate fact may detect extremely phase transition network corrupt perfectly phase estimator reliability community say empirical network model transition theory community spectral corollary remark department electrical university usa community detection subject random transition community connection edge two arbitrarily connect community external specifically almost inter community edge connection critical low transition transition threshold noisy processing node consider regular disjoint community external edge total number topology characterize otherwise community graph partitioning correctly separate community cluster specifie cut dimensional define ni semidefinite small know connectivity algebraic laplacian vector entry
unbiased frequency component experimentally sample random convergence strategy enyi star simulation analysis briefly discrete graph foundation adjacency graph shift undirected underlie th dependency filter comparison normalize shift write norm normalize basis graph signal inverse may orthonormal stability constant review class sample work previous graph coefficient node signal measure smoothness graph shift graph signal lead graph graph graph frequency k note smoothness recovery graph theory requirement restrictive requirement real world approximately control contribution frequency speed decaying frequency component ellipsoid frequency flexibility subset follow sample signal denote index call noiseless recover either mean index clear subset experimentally propose experimentally mainly concentrate frequency concentrated frequency perfectly frequency provide unbiased consider follow node estimate frequency component reconstruct fouri transform bandwidth result appealing graph disadvantage energy first component recover stability frobenius due compare rate two roughly evenly element example graph discrete space unweighte theorem rate bound set bias upper see experimentally design nk k bandwidth bandwidth achieve type bias term experimentally exhibit much random sampling f definition asymptotic real graph bandwidth definition recover globally recovery focus domain optimum close frequency signal experimentally noisy signal biased frequency signal band project onto component sense recover low frequency need reliable ccc nearest neighbor enyi graph star blue mse compare graph enyi star generate frequency component nearest near eigenvector discrete evenly definition base connect near neighbor graph enyi since eigenvector enyi graph simulation connect simulation star expect algorithm star performance test represent red represent algorithm black dot linear lemma enyi corollary class two base sampling frequency design graph star simulation give bandwidth mm signal versus experimentally sample study recovery graph sample smooth experimentally design sampling use
branch metric vs sentence corpus nine much sentence approach propose annotated training rely sense sentence correct incorporate posterior complete deterministic transform text corpus wikipedia estimate principle small number hand linguistic process widely parse generate candidate pick successfully parse parse approach close parse e sentence result train parse manually annotate good train iteration unsupervise dependency search variant move represent hypothesis expectation whole complexity complexity increase rapidly reduce sampling difficulty search base orient concrete parsing unsupervised parse automatically set raw sentence search among objective measure search indeed greedy structure iterate firstly result generate list candidate good candidate number bootstrapping employ bring benefit allow expressive generative propose similar machine learning increase call dependency parse intuitive sentence head difficult long syntactic category would much easy length look usually name adopt iterate framework model set sentence traditional start iterate phase continue use start exception supervise employ extremely sparsity outside employ generative intuitively root generate generate root dependent generative continue current rooted generate respectively plus child token generate le context generate contain context allow traditional impractical thus estimate neural topology make rnns inner content phrase cover representation context allow flow rnn make tool estimate parse generative model first inside depict head approximation plausible dominate inner word vector initially pos tag dependent conditioning context context g generate clearly context represent context full head inner representation indeed le q equation generative ir third party dependency parse party generate list phase corpus moderate report provide merge sentence evaluate english dataset portion corpus corpus phase section worth note phase iterate search define list shape thus fitness far away proportional fitness fitness exist diversity set create constraint control point determine diversity want search distant area maximal inner expressive feed network universal capable perfectly avoid early system contain sentence length whereas phase force phase run parse sentence corpus training testing unsupervise parse sub appear remain tree head rule gold pos tag accuracy take occur label every digit dim word embedding word embedding learn wikipedia train rate recent system length analyse aspect examine lexical start contribute phase phase contain sentence e contain sentence length within phase explore area reason short rich difference wrong system less long role semantics role lexical semantic embedding sentence without word embedding use accuracy drop lexical semantic performance system generative phase vs suggest phase capable dependency conjecture capture claim examine experiment employ extension framework harmonic harmonic training sentence performance iterate start phase remarkable iterate phase find parse start experiment certainly order lexical semantic exploit iterate result end phase head improve except correspond increase attribute improvement context capture correct pos tag ir pos tag g jj ir conjunction cc modal md modal show expressive order generative context share avoid rare system distant node informative conditioning addition free recursive neural net able unseen map close exploit lexical semantic learn turn exploit lexical limited tag distinguish word mean similarity mean close table annotated ir thus relate ir framework option use parse explore experimental iterate disadvantage compare system harmonic innovation capable exist expressive external expect
solve solves write equivalently relax positive constraint solve parametrize constrain simplified framework going arise signal exploit structure part q weighted application appears signal diag diag therefore generality assume every linearly formulate elliptical complex elliptical apply go efficient function surrogate achieve element inequality equality know achieve hand find eq new require single tp j toeplitz toeplitz structure constrain apply idea embed toeplitz toeplitz size parametrize row semidefinite false restrict feasible fourier toeplitz eq take j bar stand wise satisfied toeplitz base noisy augment toeplitz structure construct l impose toeplitz toeplitz stationary satisfie correlation increase embed subsection formulate compactly define real relate toeplitz toeplitz structure positive w w algorithm q apply assumption limit generate conclusion continuity assumption arbitrarily apply adapt solve section handle go structure tractable applying refer differ essentially principle application follow inner reduce propose estimator mean eq normalized expect carlo follow four estimator namely constrain toeplitz parameter indicate small addition embed toeplitz toeplitz sequential estimation error toeplitz second estimator bandwidth choose semidefinite consumption run scale computational increase reflect different toeplitz second constrain simulation toeplitz fast decay slow decay impose toeplitz structure vary estimator fig either bandwidth covariance regularize impose toeplitz structure bandwidth examine robustness direction arrival additive ideal element mean direction uncorrelated e signal direction distribute sensor music locate interval stepsize arrival correctly angle denote denote fig constrain constrain accurately music constrain plot see fast unlike arrival music estimation estimator generate spike varied set obtain measure estimator structure error plot smoothly gauss latter double former loop fig kronecker toeplitz impose help reduce kronecker kronecker structural constraint kronecker toeplitz problem information log constraint minimization community although thm consider elliptical mean covariance structure account beneficial propose incorporate cost structural convex finding base tailor special wide range process relate sum toeplitz toeplitz structure addition kronecker derive mm show estimator low cost constraint minimization arise wireless communication financial engineering notice possesse special exploiting imply beneficial improve various type toeplitz structure application model sparsity inverse matrix covariance closely problem structure previously application follow attempt covariance realize normally poorly cause set heavy lead way address aforementioned find matrix perform seek worst precise belong uncertainty asymptotic estimator uncertainty contamination structure constrain distribution give independent elliptical estimator minimax distribution sense obtain investigate focus group symmetry cost symmetry global study generalize type numerical semidefinite prove suffer drawback either grow formulate generalize consider large convexity focus convex mm case application exploit structure load discuss end convex turn kronecker theoretically prove converge structure unique initialization convex structure consider efficiency tractable present vi consider number elliptical proportional characterize estimator random angular gaussian fitting word normal independent symmetric notice possess structure take account motivated idea focus include improve accuracy formulate characterize cone proceed minimizer throughout assumption exclude algorithm hereafter assumption continuous boundary sufficient assumption constraint tackle possess convexity instead try find appear reason point capable mm derive briefly completeness set successively simple point th eq surrogate function satisfy stand directional point problem assume level x stand partition mn update x h ordinary block update derive form surrogate detailed characterization fine move minimax corrupted belong kolmogorov class estimator define elliptical completely differ develop constraint semidefinite matrix convex minimizer
persistent generate improve adopt consuming since normally linear constant cd unlike cd adopted consider document novel heuristic develop uniform cd treat partition parameter computable ml setting model proxy document document assume sample gradient sgd per document distribution result indicate might satisfactory alternative part dd remain computation simply word advance derive readily remain assume length towards incomplete load fortunately also store fast computation cd implement length ratio accommodate probability ability necessarily integer choose value weight weight define thus derive cp combining testing instance label well sentiment sentiment consist movie divide positive select set advance count slightly improve initialize learn partition separately nearly surprisingly constant without enhanced force value implement cd gpu fair cd share learn dictionary size range computation cd variant htbp minibatch time slow cd reasonable evaluate performance learn distinguishing like feature generative fair reason evaluated model indirect precision curve htbp retrieve recall p mean map evaluation check testing regularize logistic train label rate sentiment several baseline dataset compare cd naturally without posterior unit use train cd show size greatly cd retrieval slightly find retrieval count document greatly ccccc cd dataset htb model full cd sentiment dataset sentiment clear learn cd outperform gram input also arguably reach extend consider syntactic dependency relationship achieve well outperform full achieve well however extremely explain drop dependent large nonetheless systematic strategy explore work undirecte efficiently allow size estimator adapt length learn softmax powerful extract representation novel speed length benchmark high powerful analyze document categorization mainly like vast development great typical boltzmann machine rbms learn rbms cd really rbms softmax great inside especially vocabulary require hundred thousand thus limit typical undirected bag word
come attack group student student act two introduce student exercise score score observe might bias supervise scenario score true depend whether actual use compare score induce ranking agreement inversion ranking equality set estimate median assumption exercise inherent certain intuition tendency give strict whereas account normally prior fit observed purpose give result elaborate procedure confirm find true score artificial tb unsupervise course learn learn reliability vary work ordinal bt suggest estimating might purpose assume rank two score induce ranking use truth follow naive baseline exercise ta bias compute ta solution set accord task another ta put ta automatically towards student ta tb actually raw able performance rescale exercise lie rescale lead get interesting surprisingly self tend look shift version ta histogram contrary shape distribute first obvious skewed exercise many student mark score multi modal big solve part tendency generally ta evident receive moderate look reveal mistake mark describe meet runtime template solution unlikely full lower tb simulate abstract reveal error due suggest probabilistic serious problematic get whether really justify solution mistake get score realistic abundance analyze simply optimize ta histogram shift example unsupervised model ta score might bias shift measure control strength regularization little accuracy reliability student reliability artificial datum sensitive choice actually em equal one believe improvement fitting explain ta different ground none outperform study standard ta result considerably outperform mean step look actual ta consistency record performance compare student seen leave amongst bias amongst student next histogram figure little amongst histogram looks consistently believe use ta major probabilistic mid bias student report quite half student least roughly gain value student mean would improvement correct confirm dominate strict serious lack understanding material come structure less used report whether student find scenario perform mean job bias room improvement improvement compare totally wrong student none job student try effort much however look reveal portion student figure picture change compare l error calculate value student place supervision seem effect evaluate structure positive optimistic ta size reasonably inform picture reason use competitive elaborate model helpful understand acceptable student complicate model final machine test mean enough much well job variance among rather lack solution student mention successful generate publicly material contain couple report constraint ask ordinal conduct currently release platform group course acknowledgment partly education pl universit universit author responsible content publication lemma exercise theorem evaluate literature aggregate come solution ordinal ranking none improve student work co scoring become increasingly student crowd hope student perfect fair accurate aggregate many challenge good aggregation take suggestion literature bottom science algorithm structure ad course student exercise apply see publicly aggregate contrary researcher none satisfactory baseline reason course department computer student active participant traditional two student exercise
consistency classifier projection margin sample set maximize margin margin perfectly project maximum small classifier particular many whether classifier able arbitrary model characteristic call consistency class minimum going search maximize cross estimate project analogy approach take minimization start notation accuracy eq unbalanced might make important despite lead balanced weight ignore distribution small measure accuracy linear tf bad whole obviously make classify minimum greatest average follow section density approximate grow infinity consequence result true limit case use classifier give begin simple perfect distinguish regularize consistent linearly separable perfectly linear opt integrate function zero integral result non regularize attain radial regularize consistent radial gaussians variance projection normal projection give maximize al minimize maximize regularize unfortunately neither consistent linear start integral integral function integrable schwarz prove main paper minimal linearly separable projection narrow q connect minimize potential also optimize log density hold integrable another dot denote solution confirm claim simple numerical life evaluation analyze possible compare hinge case non classifier behave radial gaussian distribution place dataset embed positive dataset notice hinge optima fourth optimum core local however term locate near sufficient problem loss convex distant underlie comparable away fourth dataset gauss gauss mixed mnist loss non regularize classifier truly simple class hinge loss truly lead nearly grow many machine minimization additive loss perceptron machine meaning sense grow infinity answer classic error directly translate function define risk even overcome classifier
vary level volume spherical middle ccc separation middle compare parametric select data hyperparameter hyperparameter great eigenvalue table approximate marginal likelihood volume poorly equal general good mixture provide respectively volume mixture separation separation separate c c marginal likelihood separate mixture c estimate poorly mixture ht c c log obtained propose well separate mixture ht log mixture situation situation table see select provide separate parsimonious misclassification error thing observe select cluster term misclassification model one situation volume obtain propose situation show evidence majority select competitive table bad mixture volume evidence evidence go almost substantial evidence especially cluster volume model volume highlight ht vs vs vs competitive situation respectively vs vs bayes select denote situation table estimate across mixture equal volume good spherical also figure structure mixture ccc see spherical diagonal different cluster equal actual cluster hyperparameter mixture estimation two parsimonious sample component gaussian follow covariance volume figure partition two datum respect hyperparameter hyperparameter degree equal variate control control consider four situation four situation model diagonal log value see situation correct cluster c propose model competitive accord bayes propose confirm stability variation partition cccc partition confirm available old diabetes summarize model conduct lack four datum ht l diabetes description comprise old national usa comprise minute long alternative vary report propose c c log marginal old parsimonious except number vary parsimonious model spherical stable marginal orientation notice term strong compare competitive likely high ccc old partition posterior component comprise observation describe length colour collect classified vary report value see good provide actual see provide estimate cluster hand cluster addition cluster four provide bayes set evidence value performance confirm rand index misclassification parsimonious one high rand lowest rand index show partition precise posterior close first principal axis actual leave optimal middle diabetes consist curve area area steady group chemical diabetes diabetes alternative report likelihood obtain correctly parsimonious k c c diabetes datum diabete compare term competitive k rand index parsimonious model k indicate error figure optimal quite rate ccc diabetes set area partition empirical posterior component set cover three specie feature propose propose high value partition solution approach four marginal datum width middle component propose rand evidence four significant show log report accord table diabetes latter three make old diabetes vs vs parsimonious base infinite eigenvalue matrix chinese restaurant derive flexible avoid encounter maximum automatically simulate highlight represent finite partition cluster use bayesian parsimonious potential future may concern parsimonious desirable extend simultaneously formulation figure plot sensitivity figure parsimonious show cluster likelihood parsimonious represent nonparametric parsimonious gaussian parsimonious parsimonious technique posteriori map framework bayes factor parsimonious mixture obtain parsimonious alternative parsimonious model mixture dirichlet bayesian selection parametric model weighted multivariate gmm focus mixture datum parsimonious gmm exploit gmm wide flexible clustering demonstrate cluster analysis likelihood framework maximization em gaussian mle first normal mixture fail bayesian estimation intensive dealing encounter allow replace estimator achieve introduce assume uniform mle perform parsimonious mixture mixture carlo mixture decomposition parsimonious gibbs usually mixture fold pre establish approach penalize bayesian likelihood via compute posterior factor parsimonious mixture indeed natural criterion etc represent extension analyse component death refer fully bayesian mixture jointly bayesian one parametric unknown model go take mixture capability advance namely method dirichlet chinese restaurant process crp represent principled mixture cluster offer principle jointly infer parameter rather form grow represent formulation mixture group flexibility approach parsimonious covariance structure dirichlet parsimonious parsimonious dirichlet overcome issue automatically infer structure simple one complex maximum posteriori model factor simultaneously parsimonious flexible modeling cluster infer organize discuss based evaluate devoted discussion conclude remark label point possibly unknown base base mix proportion mean vector matrix gmm generative process mixture mixture multinomial give proportion component correspond parameter mle maximize em extension mle mixture fail avoid describe estimation gmm conjugate proportion inverse wishart normal view datum gmm mix conjugate mean component multivariate inverse wishart summarize follow step wishart gmm wishart mixture maximize posterior em namely prior prior gmm gibbs gmm cluster extend parsimonious exploiting eigenvalue matrix provide range flexible parsimonious decompose determinant term previously unseen assigning possibly unseen cluster observe chinese restaurant crp customer table customer social th proportional customer previously new proportional positive real number crp crp customer correspond crp mixture complete cluster gmm one inverse customer choose crp crp density crp covariance common wishart dp parsimonious crp parsimonious matrix flexible volume crp table decomposition mixture cluster parametric clustering investigate eight parsimonious covering family family parsimonious spherical full summarize gaussian parameter ht l l type diagonal diag diagonal diag structure denote inverse mix proportion multivariate wishart parsimonious dirichlet parsimonious parsimonious mixture model mixture perform miss dirichlet label posterior mt complete prior mixture sampler generate posterior multivariate normal wishart gamma depend markov stationary therefore n approximately distribute couple give hyperparameter z z detail material also component hyperparameter dirichlet sample arbitrary consist sampling prior introduce conditionally cluster beta n hyperparameter pseudo ht nk nz I cluster get distribution output retain solution correspond frequently sampling strategy number parsimonious bayes general modeling
year extensively computer movement fusion modality dynamic movement device computer interface hardware like computer active mobile sensor much device form environment device active continuously study four representative modality usage behavior device modality due relatively consumption remainder four modality error measure false far false reject collect author subject mobile device day kind literature duration absence restriction usage factor mobile device participant good representation close world organization device organization decision modality serial strength level additional classifier add without classifier system multimodal characterize fuse decision fuse aspect particular portfolio behavioral mobile device extent temporal multimodal mobile device four behavioral analyze modality fuse system user datum get motivate recent multimodal verification human computer interaction approach min combination classifier classifier fuse initialization utilize multimodal system knowledge subject characterize overall significantly achieve fusion combination available already design allow contribute drastically decision active mobile device year rich ultimately one contribute fusion verification achieve subject incorporate modality behavioral achieve user knowledge subject duration day portfolio modality linguistic verification thorough domain along traditionally machine prove rate impractical mobile device come modality context block text location extensively aware web study purpose understanding web source identification far computer mit reality portfolio usage location usage position low gps trace utilize work behavioral subject collection carry author requirement study user subject version table device device track device period track device modality visit location gps characteristic duration participant area place usage device os long duration study could modality tracking power include front tracking face recognition gps frequency device text web location refer character soft refer new device location show modality aggregate instance modality single soft website take rather gps table order three remove period device consider minute minute minute date time change divide dataset reflect event compression period cross utilize window active duration hour active order htbp home facebook phone message home sense home gps united world characteristic location city vary day day human location modality analysis soft visit physical gps modality extract raw modality produce train describe detail classifier take event tx tt current classifier score fig mobile device varied activity majority facebook event therefore final record associate record actual principle well representation linguistic style capture analyze classifier construct count user domain number time training example com domain www com refer domain entity across user entity visit user quantity normalize frequency value visit user valid user maximum entity machine radial rbf kernel function score form regression svm score additional validation learn library htbp box fire event associate modality multiple decision modality fuse sensor describe parallel architecture describe comprise local detector fusion center detector make favor local detector order decide favor favor detector assume observation detector use set couple equation word impractical detector condition fusion center performance decision suboptimal since detector scalable design parallel fusion scheme observe make decision decision form fusion center combine optimum priori represent optimum eq practice determine false alarm interaction mobile device divide fold three fold fold phase purpose individually fold test individual fusion three phase characterization relate fold fold characterization fold fold characterization fold fold characterization fold fold fold characterization fold characterization common fusion measure datum fold characterization testing far phase classify window system decrease increase show window bar compute characterization result use fusion center indicate activity produce metric thought decision window size older cover consideration window add text even average hour show gps correctly verify four modality hour mark fire character small list h modality different text gps web show within window distribution period window decrease long asynchronous modality event fusion utilize operate characteristic roc far decision fusion drop window window utilize modality contribution global decision paper verification evaluating error rate portfolio system relative contribution window minute contribute contribute window explanation significantly short
odd expert remain measure might suboptimal regret note yet play role algorithm divergence begin vector call sequence adaptively could special recover eq lemma convention banach function respect step optimistic mirror yield z tight learner advance align learner regret tuning lipschitz lipschitz bregman case divergence uniform stay case constant develop optimistic mirror incorporate step trick monotonically trick monotone sense necessary variation throughout game gradually horizon take adopt strategy worst describe algorithm banach optimistic descent statement automatically cast regularity far round satisfied accumulate variation regularity note importantly non monotone initialize check tx tm td c optimistic mirror descent epoch number instance epoch technical shall become lemma theorem assume enjoy follow regret summarize v follow gradient regime allow divide batch play smooth batch gradient recognize mirror prox period far regret mapping dynamic correspond history mapping index prescribed strategy payoff player lk player arbitrary switch convergence constant static sequence propose vary dynamic regret fully adaptive environment derive dynamic regret capture sequence sequence interestingly consider rate minimax acknowledgement acknowledge decentralized online nsf grant dms proof primal dual pair cd entail combine return eq simple bind summing q complete define choice sequence belong last entail divide batch use single batch horizon let r ib tu tt b otherwise precede upper complete sake clarity presentation stick epoch recall bar refer quantity tune pair identity fails suffer simple function sum fact batch use precede prescribed correspond optimistic strongly correspondingly get I tt notice choice guarantee specify splitting like payoff get denote appear last player denote regularity combine appear account identity bind would like player regardless adopt player drop negative well entail statement pay average recent develop adaptive take observation retain guarantee direction method perform benchmark direction benchmark regret guarantee scale notably measure apply zero player game action play nature action nature reveal learner aim regret numerous static minimax algorithm direction non benchmark hard guarantee advantage sequence non adversarial quality move distinct investigation develop regret show respect benchmark dynamic generally dynamic regret bad obtain dynamic regret furthermore propose dynamical potential forward idea generic sequence learner achieve variation type adversarial capture intuitive sequence version reveal online get regret regret valid begin full setting receive gradient trivially obtain simply play round online study knowledge dynamic full online regret variant optimistic mirror noiseless priori automatically adapt length term second technical derive apply monotone investigate provide player play zero game player play drift guarantee sequence vary slowly generalization action
support weather phenomenon click decision history web theoretical formally world record imply rather degenerate single could entirely score probabilistic typically state world pose problem generalization value observation possible correspond model outcome like low expectation eq since hold square however immediately truth evaluate however algebra true assign outcome kl divergence shannon entropy divergence g difference sample want mean pair sign test test mean option trick theory score thorough develop algebra logarithmic scoring get entropy term prediction theoretically comparison hold outcome pair determine score indicate pairwise mean score rule less unobserved distribution model rule logarithmic distance kl former interest quick outcome report wrong often free add parameter report undesirable popularity connection proper use scoring cosine similarity correspond rule general bregman compare probabilistic proper rule score reward report effect nonetheless theoretically preferable un comparison score accuracy probabilistic
overfitte hour provide dataset roughly year period price load load load second load unit measurement variable additional source competition total load load skew decide natural log variable much target calculate variable date extract day week integer day year integer day month week integer hour month month integer early hour early hour difference z z strong hour show select shift autocorrelation window hour autocorrelation price hour shift day see autocorrelation value much hour mean shift surprisingly hour boost apart find relatively importance hour suggest within lr hour month fairly result use validation research competition mark par decide window day train substantial drawback leave day month period testing leave day total assess forecast compare provide mae root mean rmse boost mae price day table statistic daily day rmse median low average day filter represent error confirm difference two mae mae rmse count mean competition forecast probabilistic worth investigate energy effort focus develop forecast help fairly approach capable achieve competition price surprisingly conventional forecasting observe wider cover future competition filter greatly improve forecast hard task series variable gradient competition year forecast experiment reveal perform auto correlation box real forecasting price task participant inside outside market ahead forecast unique inform short term forecast beneficial business ahead market forecast help operator ahead methodology current last team position approach forecasting competition establish competition put forecasting forecasting track forecasting hypothesis series candidate consider linear follow denote combination auto around zero parameter element weight usually acceptable error suitable minimize error boost responsible
bayes always surely imply martingale one tp tp h let integer decrease h tp infinity tp consider generate lemma desire trivial exact rest devote one stop bt monotone convergence use recurrence inequality markov process thompson decompose tp tp tp recurrence previous choice tp p tb tp tp kp h finally small h get notation let value later decompose use use lemma lebesgue monotone convergence recurrence regret thompson tp tp tp tp fact p establish recurrence inequality tp notation see decompose devoted bound lebesgue monotone establish desire recurrence inequality lemma process thompson follow tp tp ta q two b rearrange newly get p tp desire recurrence observe q use p axiom microsoft successful thompson bandit property encode exploitation prior bad dependence fully dependence yet case result true generating low discovery thompson appear well know tradeoff repeatedly action agent receive action reward potentially randomness make reward generate countable underlie reward random draw know underlying yield highest measure incur always select optimal action precisely frequentist selection reward take randomness impose bayes regret discretize abstract bandit arm coefficient reward reward useful result still unbounded probably paper take thompson action accord equivalently thompson I thompson thompson gain lot largely furthermore often analyse strategy classic arm comparable obtain bandit prove provide insight assume informative prior thompson regret bound unfortunately thompson bind another prior thompson always dependent prior thompson thompson unclear good perspective bayes regret extreme trivially knowledge work frequentist thompson informative thompson sensitivity prior important question useful implication thompson thompson important characterize bad yet meaningful highly nontrivial main summarize statement section case mild loss generality let thompson bound furthermore factor exist instance respectively low bound low bound countable model exist instance thompson show true p bound remove logarithmic far small open thompson exponentially expert specify assign expert sake simplicity thus exp thompson partly bad advantage efficiently impose prior remove core difficulty analyze state upper rely proposition proof proposition differ limitation sketch proof proposition proof proposition thompson precede problem specific smoothness thompson fairly problem notation obviously simplify notation tp decrease notation function tp complete therefore word time lebesgue lemma regret thompson tp tp decrease recurrence assumption q hold sketch reward define upper lemma stop dominate lemma next thompson tp tp use decrease get recurrence rearrange newly inequality lemma frequentist thompson consider thompson problem recall moreover lebesgue dominate monotone aspect thompson bandit focus representative case fully bad prior good matching upper extend quantify inherent sensitivity poor bandit version true strong frequentist still sensitivity insight thompson bad thompson range underlie negative functional process tp carry computation p kl p p step follow p hand jensen definition absolutely measure eq complete count
sample conditional cluster shown likelihood introduce component hyperparameter equip form dataset finite exhaustive enumeration clustering practice implement greedy search space successive application operator chain considerable model heuristic advance follow simplified speed cluster specify advance first exhaustive enumeration special correspond heuristic success simplify largely database find condition equation recent year generation theoretic possess desirable strong mathematical foundation similarity conduct conclude preferred purpose denote clustering co occur marginal derivation joint occur cluster cc ns describe item formulate clustering reduces also measure sense model cell truth retrieve value typically give general truth subset one include exclude disease respectively measure take successively retrieve experiment number cluster initially select top low gene gene experiment emphasize stage find minor simplified suggest search heuristic range end al et ultimately specific initially choose six gene complete linkage cl euclidean pearson correlation correlation cosine cosine fix gene result performance show clustering result well proceed evaluate approach previously model approach clustering dataset maximum posteriori find cluster restrict restriction trivially retrieval evaluating distance evaluate marginal query recently marginal average keep comparable discussion third differentially conduct specific suggest differential expression profile obtain condition potential achieve retrieval performance hand assume preprocesse suggest default gene use correlation instead two result expression retrieval model expression formulate comparison retrieval show retrieval scheme clearly outperform result far indicate retrieval quite robust allow vary assume fix mean conclude proxy conclusion generalize beyond scope b approach surprisingly gene essence predict learn therefore instead query may necessarily aspect experiment task relevance aspect current setup ground truth cell type disease match require modelling evaluate ground truth require match class ground truth require relevance increase retrieval capturing agreement figure comparable figure ground retrieve query match b b note phenotype g age disease retrieval idea next model retrieval retrieve experiment multiply element clustering experiment match type combination result type retrieve assume retrieval ranking paper general probabilistic model al likelihood query learn query compare argue reduce nuisance characteristic relevant retain comparative simulation inferior encounter real scenario see outperform counterpart contrary approach family seem somewhat restrictive potential individual store arise result model fairly standardized repository assumption belong gene experiment cluster turn good purpose preprocesse prior experiment yield retrieval combine keyword would like thank useful centre coin public relevant improve search annotation retrieval retrieve express profile retrieve query criterion well retrieval general model separately expression model induce cluster gene pattern empirically fast mean suggest clustering scalable construct purpose use distance package molecular continue ever amount biological store retrieve experimental relevance make experiment current rely search
survey answer question sake privacy survey recommender miss entry recommendation netflix service facebook prediction problem miss entry demand clean lead also noisy datum component focus sparse miss completion element account know imagine miss sparse stage propose unified yield performance low sparsity datum optimization simultaneously learn underlying take alternate learn ambient performance compete conclude learn regressor miss prediction particularly sample see formalize notion index n entry index label regressor unseen predict label parameter statistical motivated even though ambient large close regressor predict miss exploit incomplete regressor regression inherent regressor relevant network sensor would entry sensor exploit increase miss definition mp mi pls assumption handle missing introduce dictionary dimensionality dimensionality projection central independence dependency fully practical investigate investigate maximum work different regressor ambient explicitly miss exploit sparse relate task pose problem form concatenation author approach exploit statistical test finally procedure perform pca dataset follow regression exploit limitation stream ensure algorithm thus track variation track quickly algorithm inherently unsupervised directly approach exploit coefficient aim incomplete product entry identification alternate minimization alternate case problem perform learn regressor sparse two problem utilize structure linear model algorithm label information inherently inefficient exploit reflect potential basis subspace know rotation well solve hard necessary purpose row correspond zero matter inefficient particularly deal subspace ct slice procedure propose joint simultaneously learn relevant solve optimization formulation measure regressor low third penalty encourage type formulation prediction unlabeled datum first project onto span column projection output depend shall detail jointly minimization alternatively call pass order update six detail initialize matrix miss singular vector value matrix completion literature project subspace initialize initialize update responsible replace quadratic system equation step mp routine estimating problem current mp solve optimization start implement recursive mp routine available rectangular perform solve orthogonal subject construction subspace perform gradient require form regressor achieves necessarily regressor modified term skip update linear take parallel row take time orthogonal numerical miss rough span attempt span need amount computational past etc similar guarantee sgd base drive use alternate minimization describe convergence far r x perform regression surely replacement n fp lf fp r tv j p j j connect rademacher lemma calculation technical convenience allow state possible trade classical rich class small error generalization error term think class imply richer bind potentially wish surface empirical predict many value still error easy hence small column onto dimension know discussion complexity fraction training sufficiently yet regressor regressor ambient generate orthonormal dataset entry standard separate validation generating entry resp simulate retain ease choose analysis behave similarly report five different generate choose hold search parameter mse sensitive hence coarse perform stochastic lasso allow decay try good rate noiseless figure dataset red break figure mse impact noiseless study impact gradually noiseless substantially small examine zero dataset increase increase plot seem noise increase like increase comparison discussion provide slice task prediction ct scan ct cancer paper clear noisy advantage gain utilize bad ct slice ct
ie sdp dimensional instead linear determine rearrange identity eq rearrange yield formula since feasible sdp goal sdp event theorem characterize acceptable fail motivate constraint dual start characterization determine may write formulate task clustering determine determine symmetric find analyze eigenvalue eigenvector lie span write finally eigenvector column correspond eigenvector want pick thereby impose strong remain impose satisfy since choice satisfie immediately satisfie also necessarily implicitly require division desire check suffice eq imply summarize cluster let sdp relaxation bind take cluster whose column center first triangle separately combine first whose entry eq triangle inequality equality verify point recall stochastic prove ball surely lift hoeffding notice sum hoeffding stochastic expand result sum explain definition schwarz inequalities eq finish argument apply subtract ball add expand get q triangle give occur cauchy schwarz last cauchy probability union triangle iid whose outcome determine hoeffding equality linearity q probability combine leverage assumption probability conclude combine combine result column eq pass frobenius term may second union q suffice rearrange q acknowledgment nsf dms reflect united air conjecture problem sdp mean problem model ball radius draw common probability prove recover two explicit ball cluster task machine k dissimilarity usually choose mind common clustering criterion set centroid c I solve calculate centroid may objective furthermore output far slow preferable convex relaxation attack hard combinatorial know round paradigm region set seek approximation guarantee framework relate relaxation particular rounding happen find feasible original phenomenon know optimality thank focus problem geometric question appear provide work separation main sdp recover cluster strategy sdp ball sdp relaxation vertical exact derive prof show high hard many solve entry observe give sdp relaxation ax give dual semidefinite need interpret remain exploit express
cover class interest amazon characterize south locate forest constitute background post consist collection type user construct portion begin change configuration partition series sample pixel year profile datum miss four batch minimum propose method threshold correspond metric recall predictive value precision accuracy classify truth configuration q denominator accuracy mention threshold furthermore achieve across substantial amount overall stem tendency belief proceed change region apply pixel locate amazon several natural include substantial area convert production area build ice class bar change aim expect change pixel quality call monitor amazon produce national pixel particular segmentation note regard year panel panel ground truth higher estimate capture lower infer estimate year procedure pixel panel probability change j pattern bottom panel agreement reference middle highlight panel across concentrated cluster cc change characterize top illustrate result pixel region panel probability change w dark gray bar truth gray background represent plot outline infer year dash represent see profile datum reasonably project profile close bottom cover probability change spatially study region see change gray highlight change v right region pixel colored class infer case panel contain pattern panel propose inferential routine art mainly band combine statistic finally capture perform world study interestingly seem follow spatial pattern going operate spatial pattern high whole region pixel localize transition background cluster large pixel major ground percentage year pixel profile profile forest background forest reasonable profile summarize year year sub pixel leave figure respectively top plot correspond classified localize cover isolate correspond degradation area region detect cover crucial use resource management modeling effort sense rich inference change broad unfortunately dataset hierarchical account miss site extensive characterize dimensional forest analyze pixel detect distributional dataset point posterior use informally change change series flexible change suitably define methodology propose recovery characterize essential sense hyper carefully region general em successfully cover change accommodate situation characterize surface estimate post detect kind change pixel effectiveness method art ground change infer localization overall consider change serie contain formally change define devise computationally efficient g g section filter pixel change derive update identify depend change update need jointly pixel value miss entry evaluate vx jk vx situation involve datum thorough history miss exploitation series remain image literature period experience change estimation dimensionality change amazon maintain insight change occur forest enhance monitoring must maintain area affect human depend extent modern resource management observation surface enable especially human activity decade advanced series name moderate imaging balance moderate high capability observation measurement error contamination geometry challenge develop bi temporal forest series grow exploitation e however nature optical cloud contamination pre great address structure change proceed neighbor interpolation polynomial interpolation imputation thorough handle statistical analysis handle imputation change statistical assess tailor series large characterize reasonably homogeneous background cover classify detect understand nature well assess forest convert cover specify change aim consider apply issue product due view geometry spectral band visible pixel seven band exclude proportion treat year unit effect subset keep enough year profile avoid band miss one band discard year since happen make hard change evident year hard attribute minor plot year plot highlight possibility profile seem background change class carefully establish international percent cover green never green percent almost green without green dominate height consist period forest dominate tree percent consist community leaf mix dominate relate spatial resolution step temporal band dimension partitioning variation spectral distinguish class code cover equivalently q datum employ kronecker temporal note cover covariance cover class dimensionality profile temporal variability nature dimensionality large transformation approximate k x opt pre jointly end approach simplify burden simplify notation remainder parametric require miss pursuit point pixel exist devise account miss model procedure allow segment post change segment background lack change represent configuration pixel segment segment follow multivariate mean flexibility pixel set prior pixel segment affect smoothed spirit interest parameter em pixel recovery pixel weakly informative probability change occur specify change equally one recovery two give
accurately like two costly intensive formation explicit retain generative capability original world effectively capture several implementation parameter runtime number mixed number specify base preference validation may number give hold choice another reduction form signature guide signature merge consideration use cluster suitable control take iterative begin size increase significant portion cluster keep proportion portion assign remain runtime assign center affect cluster determine total cluster result fast convex shape become normalize axis suffer slow density estimator retain choose subsample compute density result computationally pa usa unsupervised spatio without conceptually spatio process property regular discrete seek joint spatio temporal process causal system propagate concept formally eq cone set future event affect light joint product likelihood pdf given seek equivalence light similarity discover give introduce method predictive state reconstruction follow model extension cone forecast mixture predictive require light predictive considerably large introduce reconstruction spatio temporal scalable consist reduction instance require maximization density density differ cluster proof describe spatio lastly discuss result principle mix reconstruct soft light state light cone unlike retain benefit soft likelihood forecast appendix parameter arise algorithm htbp light successive reduce object light final cluster output predictive state nonparametric step density assume consequence neighborhood point assign effectively density coverage avoid formation cluster g cluster use density density affect remain point signature family state signature final predictive assign predictive form nonparametric decompose spatio light tuple density compute condition merge reduce final htbp simplify step light cluster consequence space difference state minimal new map step spatio temporal mean become user scalar measurement since resemble video prediction frame pixel light exclude cone extract result simple compare take value use prediction current consistency low error light directly future cone regularize implement learn package change remainder near regressor light cone near light cone output learn default setting experiment light density estimation well original method gaussian density parametric unable delta set error mse ground truth distributional per avg negative truth distributional test well compact apply likelihood show three respectively frame model remain percentage maximum range actual prediction predict frame pixel qualitatively predict capture much frame give smoothed extreme htbp like knn linear material regression low confidence pearson lastly one proof highest low overall one relatively spatio three accurately forecast
publicly achieve speedup merely center view imagenet pre excellent accuracy challenge object segmentation exploit accelerate fast r speedup degradation benchmark manuscript conference manuscript extend initial version acceleration deep among deep investigate important imagenet evidence share inferior discovery architecture acceleration cnn I decomposition reduce former attention directly address essential decomposition equally fine good several investigate nonlinearity neuron influential accuracy present whole present decomposition separate filter dimension scheme reconstruction filter minimize conjugate solve filter reconstruction demonstrate character imagenet evaluate unclear adopt decomposition imagenet single acceleration report fail sgd tuning suggest nontrivial optimize layer imagenet preliminary acceleration open research layer stream improve testing particularly hand also thin besides reduce run stream decomposition close svd nonlinear simplicity asymmetric reconstruction deep response pixel lie low decomposition find rank minimize response convolutional filter channel filter volume denote volume entry number filter rank response expand filter complexity eqn eqn complexity illustrate layer filter correspond convolutional spatial randomly note arbitrary dd approximate solve svd actually good response convolutional imagenet layer covariance plot large eigenvalue substantial portion eigenvector conv contribute energy original filter rank work adopt low input local volume investigate unit nonlinear focus relu drive eqn reconstruction nonlinear r nonlinear due nonlinearity feasible relax auxiliary variable penalty alternate solver involve similar eqn form svd reduce rank regression belong broad category problem let dd decomposition z problem follow consider ij applicable dimensional nonlinear least problem warm solver run gradually infinity find iterative solver increase run find matlab much fast approximated accumulate deep propose asymmetric layer deep layer map previous current layer term layer incorporate c channel complexity pool conv pool conv conv conv conv layer follow relu conv spatial pyramid totally bin fed layer fc follow another fc softmax column response sparsity proper used uniform layer solution approximation classification accuracy energy empirically reduce energy degradation classification linear pca reduce roughly product pca th layer approximate approximated whole optimize layer speedup ratio maximize accumulate greedy initialize lc small large iterate achieve channel operate channel firstly easily control rank enable svd secondly optimize exactly decompose close solution subset operate use speedup might combine spatial thank asymmetric reconstruction effectively accumulate architecture decompose conv rank filter output channel speedup ratio original speedup contribute decompose determined reconstruction adopt optimize layer reconstruction eqn layer without spatial reconstruction important accumulate asymmetric speedup ratio complexity may tune imagenet datum asymmetric version speedup layer conv conv conv conv filter rate compare asymmetric approximate case approximate conv approximate conv comparison multi layer involve deep asymmetric previous approximated version try symmetric asymmetric asymmetric layer approximate speedup effectively layer approximate simultaneously rate drastically result acceleration solver layer rank selection conv selection consistently outperform counterpart rank advantage observe often choose rank rank selection conv assign conv explain conv less concentrated high rank prominent diversity art acceleration rarely address ratio whole rate top view cascade focus evaluate single network imagenet filter reconstruction speedup increase report conv speedup evaluate another pt speedup ft discuss backpropagation reconstruction find nontrivial work imagenet observe carefully independently start converge initialization range imagenet optima report single imagenet filter investigate deep involve whole speedup speedup sequentially conv speedup conv decompose speedup speedup conv conv accelerate large ratio ratio asymmetric speedup speedup increase speedup ratio get version decomposition strategy small speedup solver extensively easy speedup completeness asymmetric drop accuracy speedup also imagenet dataset underlie architecture acceleration train much deep model layer adopt initialization method otherwise common comparison train worse accelerate effectively model redundancy increase c filter complexity conv conv conv conv pool conv conv pool conv conv pool conv conv conv conv fc complexity show relative number total convolutional portion c cc speedup c c c speedup ratio convolutional accelerate column filter rate compare table comparison performance also fast accelerate gpu ignore view one report top accelerate top fine accelerate per intel ghz cpu version actual speedup ratio speedup ratio overhead come fc speedup accelerate easy parallelism gpu actual recognition image believe practical significance accelerate view speedup ft top pt speedup view ft value single accelerate speedup speedup ratio firstly important without rank selection rank selection increase repeatedly evenly feature size besides conv increase rank table conv conv filter trade compactly maintain evaluate imagenet evaluate challenge conv layer absolute somewhat fine fine speedup previous suffer greatly accumulate fine tuning increase view error speedup degradation deep model redundant increase cpu speedup report tuning suggest fine whole decomposition current object method exploit model evaluate accelerate default publicly cnn evaluate average imagenet task approximate asymmetric unlike layer dominate cnn detection conv speedup conv result model detection speedup degradation believe speed advance fast feature extraction considerable conv speedup detector fast acceleration speedup ratio accumulate nonlinear asymmetric demonstrate complex imagenet c microsoft com aim accelerate convolutional neural cnns cnns substantially vision unlike approximate
formally strictly speak write limit write define regularity condition mean combine turning intuition datum set generate outcome fig mean increase partition two cluster high context appear yield region interior within square diameter represent diameter seek find clustering say reverse split cluster splitting give tend split consequence decrease assume item follow split write enough q repeat proposition establish part next linkage sl sl merge sl question remain two first give set point distance take approximately order program percentile linkage dissimilarity also give study well give perform unable suggest result give worse well suppose closure disjoint disjoint large strictly suffice regular two less ensure one perfectly sl point together point subset component metric close proved theorem dendrogram sl separate perfectly enough perfectly sl use limit convexity regard perfect representation key force component thin closure satisfy always find work well seem satisfy regard sufficient examine technique find outperform interpretable merely enough th much sensitive value cluster influential point valid point affect representative influence cluster merge versus presence preferred indeed extreme linkage random start randomness cluster point regard ensemble represent pool clustering analog play linkage group clustering result final algorithm dendrogram linkage desire case dendrogram must back grow dendrogram little may split remove outlier small result therefore course merge however merge use final representative place root design advantage linkage merge become remain justify see dendrogram define tendency usually put near leave dendrogram reasonable place well separate homogeneous dissimilarity small refinement separation among remove k table place place early despite poor average suggest spectral cluster fig good seem broadly size technique assessment ct compare seven half depict show clustering panel little theoretic well separate randomness include panel merely panel fourth panel cluster respectively overall inference h ct consider nonconvex run generally use preferred htp six indicate ct show six clustering density whether ct give poor merge half top see low three nonzero ambiguity portion half bottom ambiguity versus recognize ambiguity indicate panel ct fu dna website set method well none fail completely greatly contrast outlier effect ct put cluster indicate low htp six ct generate evaluate successful strategy projection onto however systematically difficulty dimension present table separate less likely dimension mean perform well observation give replication obviously ct two provide ct work data expression seven class expect seven missing profile present ct red seven find find white attribute case could said omit see poorly example example well method h datum example ct rarely perform aggregation never likewise never perform really normal design difference occur outperform lead challenge something know datum mathematically estimate cluster instance statistic implement package addition cluster different sub deviation take integer though identify indeed worse bad percentile good sd seem guide poor sd well aggregation differ similarity single linkage usual euclidean percentile establish formal geometric percentile sometimes euclidean performance test variety qualitatively clustering simulated clustering clustering component separate suggest lead clustering satisfactory course complicated shape little separation lead equal outperform hybrid eight form always yield robust acknowledge nsf nr example department statistics usa usa propose non convex three stage first stage use produce series clustering select cluster linkage stage dendrogram stage dendrogram variant argument justify step stage involve real keyword hybrid mean linkage unsupervised technique dataset cluster wide list reference recent centroid variant come agglomerative come scope limitation strength map precisely combine centroid agglomerative careful treatment influential convexity quantity dissimilarity cluster principle correct give rarely particular outperform clustering convex formal clustering ensure corollary method case give result ensure condition simple knowledge except establish iid variable need effectively assume draw variety size create linkage sl clustering size clustering clustering pool zero one ham sl choose grow dendrogram cut dendrogram similarity similarity cluster merge ignore cluster possibly merge merge small cutoff sl usual short one effective generate disjoint closure find distance far describe distinct close pool point minus linkage final evidence accumulation pool clustering way range hence membership hamming ensemble co third use grow prune dendrogram tuning seem technique conceptually hereafter separate centroid ct key use place ct linkage average linkage technique stage fourth grow unlike unclear ct technique conceptually pass first partitioning divide small agglomerative theoretic first simply closeness cluster boundary fourth look ahead contrast cluster mean enable technique propose hybrid clustering way estimate combine modification follow sec cluster unable provide interpretation present concluding remark sec begin generation five input large reasonable hybrid technique number serve cutoff cluster work reasonably find require right give start initial agglomerative merge clusters bm bm construct similarity ix I I I sd sl vertical dendrogram leave correspond namely maximum branch dendrogram length line dendrogram cluster final cluster write clustering exist adjust cluster sl submatrix use sl give final size brevity refer use dissimilarity
crowd recommendation preference collection aim order good preference grow recover order inconsistent preference acquire challenge explore existence ground parametric item assign preference restaurant etc compare preference number repeat comparison snr reveal item recover ground rank ideally rank partial well aforementioned adopt accordance popular paradigm arguably mle inherent convexity comparison efficient another within produce estimate nearly minimax square centrality finding parametric consider therein rank square reliable realistic scenario receive rank fall ensure top item accurate identification term question minimum comparison affect preference score address question algorithmic minimax optimal contribution two begin characterize fundamental three number comparison preference perspective emphasize separation quantify preference minimal evaluation reflect separation rate propose nearly identification soon exceed limit constant careful score sense rank iteratively pointwise comparison design primarily estimate minimal optimal accuracy furthermore numerical mle centrality ranking receive considerable item draw distribution underlie one observe identification model adaptive term rank multi value preference preference scheme exploration tradeoff perfect characterize complexity sampling relative provide item admit dimensional embedding explore basically approximately ranking accurate ordering approach accommodate query noisy item motivate generalize often top assume preference collect manner apart centrality mle variety aggregation guarantee convergence sample centrality mle nevertheless total ordering selection justification derive work principle existence permutation generalize involve ranking ranking another work distance rank broadly scope remainder key top main fundamental nearly linear summarize present detail spectral treatment direction proof rank mle e defer appendix respectively provide brief notation denote norm respectively independently besides mean constant formalize present performance metric item understand rank outcome numerous existence item depend item without throughout otherwise pair outcome denote indicate throughout ease presentation statistic acquire item snr comparison comparison observe item assume throughout dynamic score irrespective positive away regime grows readily translate separate vanish e voting like pairwise identifiable index denote characterize reliable ranking rank perspective scheme bad challenging aggregation distinguishing near decision acquire finite measure see play determine identification employ comparison absolutely preference would main finding tight condition identifiability state exactly assume throughout fed identifiable identification plausible detail control entry report identifiable behave choose worst compatible impose comparison necessary reliable boundary around sample complexity preference separation another barrier away remark dominant fraction prior focus latent score infimum almost identical potential pointwise achieve minimax pointwise present bottleneck top two base control estimation identifiable region separation item arise specify fine coarse readily minimax separation case fine item specifically minimax separation consecutive many hardness ordering item item necessarily easily unless impose fairly snr requirement comparison snr snr could increasingly passive requirement reliable eq employ preference upper q challenge dominant consecutive suggest rank outperform ranking separate active pair evaluation rank initialization around ground truth sense via spectral pointwise manner consist coordinate wise operate splitting within describe detail throughout preference particularly discover preference incur enable desirable fortunately method serve ideal initial guess seed average outcome large comparison I ji j distribution centrality word centrality proceed markov chain return distribution lead transition completeness centrality reasonably probability analysis mle e mle I rather graph model utilize mle iterate log coordinate method mle far apart contraction pointwise accordance formal summarize analytical slightly splitting recommend leave justification spectral recent arrive rapidly attract towards optimum restrict loss kind gap mle minimax optimal successive characterize interval role outli guess sense entry ground outlier mle computational initialization step rank centrality accuracy instance likelihood sum term find one program refinement stage accomplish mle provide np basically succeed separate high object long additionally cycle require stage spectral achieve linear sequel like interpretation estimate heuristic since present calculation suggest around control locally existence low leibler calculation precise truth rely surrogate result plug fortunately incur employ sense make I iw iw I truth dominate surrogate pointwise loss exceed procedure solution low guess converge rapid heuristic suppose constant simultaneously appropriate wise expect q replacement choose outlier one refinement stage another obeys recognize sequence recurrence point f bf b specialized obey sufficiently apart truth careful reader spectral centrality indeed analyse lead initialization estimate reasonably spectral mle converge one naturally seed refinement stage order mle desire via configuration establish analytical bottleneck bias tradeoff accounting randomness random general independent randomness go avoid nevertheless case tradeoff acquire comparison applicability important report calculate trial pair comparison centrality illustrate tradeoff repeat comparison sparsity spectral mle outperform centrality resolution comparison e next identification varies impose rank accuracy centrality situation interestingly mle relative apparent seem capable achieve randomized simulate future aggregation develop aware perspective return item accordance combine wise identifiability preference separation come develop mle investigate remain characterize choice pair comparison draw model collaborative rank pool user ranking guarantee mle ranking subject theorem heavily reverse coordinate likelihood resp vector resp np np coordinate lemma later concern separate fix eq derive call cover within evident one produce cover cardinality n l occur n recognize lipschitz result pick sure cardinality cover sufficiently large putting suggest truth use constant simplicity wise assume generate clearly obey calculate taking reveal kl use p say coordinate sense mle ground truth fortunately surrogate likelihood true coordinate likelihood brevity depend consequence gap iw gap obey two substitution give notably j develop
monitor spectrum mobile suit behavior consist link topology measurement one neighbor typically perform adaptation stage lead communication intermediate aspect network way essential role network diffusion classical optimally different combination rule laplacian neither account operate snr result performance snr vary across scheme adjust optimize performance circumstance regime mechanism switch consider compose homogeneous filter circumstance nod advantageous tracking capability previous approach adjust alternative deal case alternative drawback firstly local neighbor feed back present separation adaptation implement rule suit track scenario addition asynchronous extend combination weight mean illustrative propose analyze diffusion derive close steady square deviation new adjust similar introduce include stationary tracking theoretically analyze performance l adapt network derive close present main conclusion possibility vector letter transpose context length equal one summarize notation cardinality exclude index belong index unknown node node local combination weight assign node neighbor exclude connected topology depict share information neighbor instant minimize mse instant measurement length realization unknown length linear denote measurement power across possibly parameter c c kn k diffusion solve estimation manner iterate phase however differently diffusion purely combine combined neighboring straightforwardly write local vector typical projection characterize keep combine constrain coefficient explain section impose negativity constraint kn update recursively dash highlight adaptation phase combination phase adaptation even affect clear accommodate compose adaptation influence selection weight suboptimal adaptation phase affect deal adaptation independent node neighbor delay slow detail strategy steady present full consider compose adaptation equation consider diffusion algorithm prevent division fig steady steady deviation mean network convergence slope conclusion extract steady network heterogeneous optimal convergence reach optimal consequently select section state environment thank energy directly steady several difficulty encounter assume static although straightforwardly adaptation introduce variation independent distribute autocorrelation tn kn kn kn commonly adaptive throughout stationary input regressor u tn assumption widely analysis diffusion realistic many application assume white spatially n kn nk condition sufficiently independence similar behavior stand filter diffusion strategy local estimate notational convenience product across represent obtain stack entry ergodicity regressor filter series govern radius eq choose impose negativity limit scheme converge radius simulation converge much fast solution steady often diffusion scheme iterate limit analytical steady network replace matrix equal triangular arrive theoretical steady coefficient u kn u tn un equivalent apply average principle tn form delay line factor assumption adaptive filter analytical approximation q steady use iii crucial node operate instance favor fast favor steady strategy learn suitable kn kn tn kn kn stand optimize square bind emphasize note application order negativity constraint guarantee field remain subsection stochastic mse stack define kn tn kn collect rewrite newton control autocorrelation avoid division identity matrix average regularize kn kn kn kn l attractive implementation node invoke inversion inversion constitute adapt subsection replace cost temporal represent combine time obtain equation read symmetric th neighbor could condition filter interpret autocorrelation see correlation window weight factor towards rectangular convergence steady instability affine window rectangular window efficient term outperform state simulation rule stationary tracking simulate fig employ adaptation step size input multidimensional unless observation node variance length vector uniformly equation set aim validate compare state art carry node scheme stationary scenario expect objective subsection predict steady well although metropolis rule check would analysis plot variance scenario c last part study situation fig plot steady match good particular slow medium db stationary fig learning rule trade steady influence parameter couple factor correctly steady dramatically instability degradation art tu also table simulation b l provide combination consequently seem adaptive homogeneous network noise explain gap regard gain scheme surprising term clear network tr analyze track steady state keep rate conclude steady fast tracking diffusion rule conjunction rectangular low complexity number compare vector cost reduce regard equivalent paper novel scheme meaningful
train binary notice first subsequently draw mnist example fig illustrate draw sequence attention depict fig whereas attention attention like person motivation large look nonetheless image draw able capture colour composition architecture demonstrate highly house mnist generation dimensional attention embed beneficial generation paper ba draw image mechanism sequential auto encoding substantially improve model mnist view house distinguish ask fashion modification rough precise picture generation aim generative typically single possibility iterative fundamentally architecture represent towards create independently successively successive stage generation area precision width recurrent real combine family recently deep variational lead significant advance differ rather generate single iteratively construct accumulation modification part scene ignore result year capture sequential attention reinforcement policy gradient backpropagation sense resemble read neural machine present selective attention read modify mnist house generate conclude lastly like direct reader read generate variational determine salient information input network receive key decoder encoder decoder previous decoder secondly decoder successively ultimately oppose dynamically restrict encoder decoder well feedforward encoder pass feedforward decoder passed encoder produce compute px pass decoder step result pass rnn rnns encoder encoder encoder output may implement use architecture record handle paper notation encoder receive decoder form operation latent experiment diagonal bernoulli gaussians latent great propagate gradient pass decoder operation ultimately reconstruct specify advance h bias compute concatenation single logistic latent show fig pass omit binary natural define kullback leibler draw latent simple standard total expectation reconstruction loss interpret sample reconstruct total compression decoder generate draw iteratively pick latent decoder repetition generate operation eqs one selective one simple draw image operation create modify vector provide selective attention crucial generation draw attention selective without benefit training mechanism machine aforementione array filter smoothly vary configuration resemble computer base autoencoder image centre location indicate patch green indicate boundary patch digit middle patch whole image patch illustrate filter specify centre filter patch large attention patch grid filter column fully specify intensity filter attention dynamically determine ensure positivity ensure initial decoder horizontal vertical define attention patch constant ensure extract reconstruct filter gaussian filter display leave last bottom patch attention intensity concatenation error error write operation colour input read write rgb triple reading writing channel realistic three visual house cifar network always indistinguishable cifar image natural preliminary exercise module mnist classification bernoulli reconstruction cifar green colour emission intensity model approach work training cost image optimisation algorithm example sequence video efficacy attention aid image performance translate mnist like digit
every consider dnn well supervise dnn performance shape pre dnn cdf value see dnn compare non dnn confirm training deep many much explain enough second good training layer difficult final vanishing may auto error favor final training stack auto supervise criterion supervise learned intermediate straightforward output pre perform capability structural dependency face shape output space output space output difficulty input pre layer link dnn c experiment exhibit dependency consist dataset pre shape fig face model configuration learn shape difficulty dnn large opposite present variation pose well cdf curve dnn dnn emphasize test dnn configuration truth htbp htbp article generic incorporate neural base pre deal two initialization well hide deal structured train pre validate test two challenging outperform demonstrate output structured capability perform detection raw give application trick show supervise future plan help supervise partly support project cl france pre deep architecture vanish issue paper characterize internal dependency pixel label problem generally model fully architecture learn strongly evaluate building system generalize single mapping focus constitute sequence string tree graph discover unknown statistical language application parse output iii part tag bioinformatic model tree speech processing speech speech structure category discriminative category discover latter dependency output unconditional distribution kde functions space add reconstruction output back approach output task classification scheme support learn space inverse need graphical structured capability capture hmm output suppose many world relation conditional fields crf thank output widely deal crf propose random provide crf signals diagrams crf signal segmentation hmm crf cost graphical model structure generic unified incorporate regression task deep dnn training make dependency output apply structure real world detection deal structure add constraint output structure output structure structure discover output complexity help result generic incorporate output dependency final output input dependency framework formulate learn space apply input dependency part datum firstly cost function unsupervise pre dependency layer layer supervise back allow part input layer consist mlp new describe refine mapping input input stack stack mapping eq function input replace reconstruction x keep initialize output reconstruction optimize initialize link link respectively supervise dnn describe experimental implementation present version library role recognition study year task remain complex pose expression point image application face dependent task face widely capture constrain face shape shape image many face also match carefully convolutional field whereas consist define propose face discover consist training unconstrained variation illumination dataset dataset truth divide sample similar resolution collect box truth dataset image normalize face deep hide
intuition plain paper existence framework facilitate degree explore problem rough coefficient fast possible follow pde discretization mesh example symmetric fast problem type affect lack progress development robust method rough hierarchical resolution wavelet method wavelet application wavelet arbitrarily preserve classical wavelet away property rigorously prove sum harmonic provable rough coefficient pre operation reformulate quantification solve automate paper possible pre support fast direct orthogonal nearly bound condition surprising achieve wave operator essential playing miss find possibly strategy play game fast completely find estimator decision sample analogous theory generalization quantification model require analogous formulation guide discovery identity difficulty generalize concept rough lie priori accurate adapt idea concept harmonic coordinate essential rough coefficient assumption find ergodicity fine sequence lack robustness rough basis must provably localize identification element numerical replace try give rough spline spline discover numerical instance information optimal method optimal strategy min game choose finite incomplete player minimize remarkable von strategie deterministic randomized strategy although information decision theory sufficient compact game purely deterministic priori connection player place distribution candidate strategy b place candidate prior although employ player player distribution bayesian prior appear due min determine player employ prior linearity calculation restrict prior linearity investigation algebraic framework linear systems b optimal accuracy apply discovery elementary gamble characterize exponential enable localize high section game must nest measurement coarse fine resolution approximation form martingale conditioning martingale hierarchy interpolation orthogonal system number elementary gamble nested orthogonal scalar product norm solution compute condition enable computation complexity identify equivalent algebraic set recover approximate linear equation let element purpose eq measurement know vector definite matrix example instance recover choose b lead purpose result select accordingly preserve linearity efficiency restrict b step mixed player q matrix follow whose minimum define eq simple calculation nest define write write subspace write projection scalar rectangular nan follow problem unique norm respect product matrix note zero conversely belong dimension conclude particular z z belong complement observe k solution v k right calculation equation control entry power radial basis krige assume observe energy norm significance quantify simplify energy estimate choice ax z approximation approximation interesting knowledge corollary remark motivate problem good measurement use quantify recovery write write orthonormal form eigenvector measurement correspond span small eigenvalue associate observe minimal matrix nearly randomization sense accuracy multiplicative factor log law entry application derive see indeed conclude observe p measurement value difficulty associate small one problematic randomization play game modify game player decision player randomization strategy game computationally measurement positive symmetric solution ab symmetric constant measurement identify positive error factor pde discretization laplace pde piecewise mesh norm resp solution resp resp span row resp span right side generalize continue analysis design interpolation interpolation formulation choose approximate linearly express q interpolation condition acting subsection continuous covariance function pde algebra interpolation condition measurement since test kronecker delta formula expansion discussion solution admit formula formula variational property conditioning intuition precision inverse constraint optimal recovery admit unique minimizer define unique subject furthermore obtain proof define measurement dy follow respect extend green observe fy dy imply fy dy fy dx dy h directly see allow solution partition closure sufficient write simplify modify equal one elsewhere exist construction additional equal use localization subsection require solution convexity constant via present similarly use construction clarity degree via constrain problem energy local energy rough show property beyond span lie exponentially support localize diameter element write center exponential decay basis eq aa euler let contain closure domain contain closure define q exist jx dy monotonicity green jx dx dy dy x start ball center contain diameter follow j dx dy sum inequality constant simplification integration eq follow let naturally outside localization hold need unit center diameter piece direct piecewise apply let union union restriction zero w lead h combine decay deduce combine conclude proof simplification solution note localize I furthermore v square contrast need slightly change avoid technical without generality scale u r u lemma preserve localization localize numerical section building level resolution hierarchical nest decomposition game say resolution diameter diameter least constant depth tuple j ks say regularity resolution regularity h j hierarchical nest measurement mix mixed player value measurement iy space nest hold measurement element hierarchy computed question formulation player express replace mixed dy follow measurement e investigate game hierarchical manner coarse game value theorem realization martingale increment vx form martingale increment gaussian measurement martingale martingale property martingale time cost towards martingale l condition gaussian covariance q direct martingale increment operator complement within write direct element hold belong orthogonal u u k nest restriction transpose interpolation follow restriction interpolation player mixed strategy player information l k restriction true equality observe I imply coefficient observe k identity onto restriction onto vx symmetric matrix invertible admit symmetric positive restriction operator define representation definition imply j j consequence formula provide restriction interpolation nest follow transpose furthermore r k nested imply take theorem lead orthogonal decomposition q sense basis see wavelet orthogonal rather adapt space pde subsection induce decomposition condition uniformly hold q furthermore l analogously h simple consequence k scalar direct consequence lead good integral player rectangle rectangle level basis involve subsection basis unconstraine element k theoretic interpretation figure observe let k k ic I quadratic q direct therefore inversion effect decomposition system uniformly dimensional define furthermore eq consequence u discuss subsection system solution mesh regularity conjugate cg guess yield approximation arithmetic writing prove size k extra underlie localize remain error fine localize check define definite element fine give allow localize element unconstraine localize ai k k z control reverse induction hold constraint writing hold eq equality l domain I proposition constant aa imply finish proof solution b constant aa ax need symmetric let b aa j bound number k imply aa eq low imply conclusion replace theorem aa solution follow
parent eq derivative directly calculate transpose derivative gate accordingly omit check approximated structure complicated consider simply overall solve semantic piece understand attempt sentiment phrase within stanford bank sentiment early factorize consider small component phrase phrase recent start principled formation semantic enhance composition node lstm stanford sentiment bank evaluate benchmark work datum annotation tree comprehensive lstm stanford sentiment bank review discuss stanford phrase manually annotate sentiment split training predict sentiment root sentence phrase sentence sentiment sentence phrase sentiment phrase classification mention minimize setting phrase regularize th element multinomial iterate regularization tune data split structure conduct conduct accuracy stanford bank result sentence column root machine correspond confusion merge vector interaction nb svm lstm table lstm batch size hyper fine weight leave word unit word initialization word depict converge root phrase fast phrase task start minute efficiently lstm experimental first keep lstm depict name stand gold sentence circumstance phrase node phrase however comment sentiment bank bank enable study change keep annotation tree sentiment available cover vocabulary concern sentiment setting lstm margin lstm obtain compare label lstm improvements internal learn parameter hand ability model lstm root leaf lstm performance figure length unbalanced trend show advantage deep semantic lstm effort attempt utilize structure prefer chain recurrent neural implicitly linear compare first word lstm short right lstm read right left phrase correspond sentiment bank annotation version experiment root include annotation lr lstm lstm root lstm leaf lstm leaf lstm root parsing help improve label leave recursive lstm recursive lstm inferior use gold gap sentiment dictionary structure gap conventional structured short propose reflect memory provide principled structure learn sentiment text replace enhance lstm show useful lstm research community contain line attempt representation utilize believe recurrent actually structure implicitly give empirical toward answer input high root semantic structure short lstm wide speech recognition translation tree reflect recursive principled consider interaction language understand mean text art recursive model layer lstm helpful achieve well year long lstm demonstrate translation code hierarchical modality semantic language merely concatenation instead sentence yield art performance task scene segmentation lstm learn reflect multiple call neural lstm potential avoid hence interaction tree deep lstm together lstm instead structure lstm mean piece text representation text understand human language stanford tree bank determine sentiment favorable benchmark much annotation enable explore experimentally lstm art recursive composition lstm memory structure consider recursion modality recent year demonstrate achieve performance semantic analysis segmentation recursive tree recursive leaf node combine backpropagation effort neural include amongst leverage syntactic parse recursive neural subject vanish result difficulty compare claim simply result performance recognition previous lstm model utilize achieve ignore priori lstm structures child hence multiple cell hierarchical show blue figure line indicate often sigmoid later lstm principle respectively soft compose lstm specific gate gate forget child datum forget rather denote hadamard sign hide child right gate resource child forget gate forget gate weight combine formula regular child forget
factorize perspective arise application trick objective interpretation ignore dependency activation may retain dropout form row multiply dropout capture multiplicative noise univariate meet training uniform prior weight format store p interpretation put kl dropout prior possibility respect approximate discuss detail analytically approximate eq use dropout rate low learn separate dropout per layer neuron separate although specification beneficial set maximize variance rate optima objective cause variational bias introduce good recurrent estimator variable trick unbiased stochastic variational inference focus variable extensive parameter report application long history use infer probability type show dirac posterior variation dropout similar focus approximations monte compare binary dropout type pre name correspond name include type noise introduce type weight write choose fully neural hidden follow recommendation rate early method epoch empirically different estimator describe epoch epoch variational dropout independent gradient full gradient rather advance encourage result format number format type value number exponent closely format receiver approximately double common specification kl uniform equal transform interpretation control digits dropout column vector result previous input expect variance rate dependency dropout q gaussian dropout parameterize treat weight variational optimize w marginal wish optimize kl divergence p prior approximation straight bernoulli sign approximate consist kl divergence kl divergence conditional scale kl bit evaluate weight divergence part transform uniform log putting involve cdf entropy define q divide family gaussian determinant variable rewrite term depend kl posterior log prior consistent additional term numerically use rd dropout alternatively improper prior allow indistinguishable correspond rate approximated term vanish claim draw stochastic minibatch draw random weight hand q decompose identical trick vanish uniquely compare model variational noise dropout correlate generally regularization htb top bottom stochastic gradient estimator epoch epoch epoch sample var university california research drastically efficiency rely parameter minibatch variance minibatch drastically upon uncertainty global minibatch local trivially variance minibatch fast convergence dropout dropout parameterize posterior specifically propose parameterize posterior generalization million minibatch gradient due high neural capacity wide diversity nonlinear pattern lead spurious happen train various controlling overfitte currently popular effective binary dropout gaussian approximation call identical regularization much marginal extend exploit greatly direct generalization parameterize bayesian posterior neural network overfitte computationally design markov mcmc inference asymptotic network alternative framework modern variant variational infer neural show modern deep neural much dropout variational datum simple regularization trick drastically gradient uncertainty minibatch flexible popular relationship dataset tuple standard observe belief posterior rule pp p involve integral approximation necessary optimize parameterize leibl practice likelihood plus w maximize minimize variational exist minibatch base especially basic trick new minibatch likelihood minibatch random draw gradient correct proceed perform stochastic tell asymptotically local weight crucially depend fail make objective monte calculate approximated indicator minibatch shorthand rewrite give inequality arise nn variance minibatch however random entire minibatch variance moderately variance intermediate weight translate form independent global translate yield computationally statistically trick generally applicable explain contain consist neuron receive feature layer multiply nonlinearity specify factorize minibatch need million number layer neural would hard originally perform simple turn importantly device optimize basic algebra happen architecture library deal activation directly factorize eq rather gaussian result activation activation j million
decomposition copula allow gradient copula introduce scalable augment copula easy later I mean augment factorization maximize fix copula field special running employ learn outline alternate set iterative procedure alternate share objective function well fix px describe copula automatically learn structure copula family selection among family supplement preliminary copula change requirement variational mean arbitrary copula add everywhere thus augmentation naturally address augment repeatedly inverse cdf individual copula efficiently bad dimension distribution follow copula calculation conditional cdf tree copula condition loop copula copula separate log therefore field change mean augment space use analogous gradient pre copula gradient set marginal gradient simplify copula contain gradient copula gradient copula sum pair conditioning condition copula copula require copula gradient arbitrary family convenience augmentation easily incorporate efficacy datum use model feasibility case dependency arbitrarily framework mini variational combine idea adopt nesterov accelerate velocity iteration momentum look update allow change quickly adopt al rao replace expectation lagrangian meet sample normal specifie assign copula augment factorization pair compare response variational bayes technique perturbation argument estimate mf truth set display simulation effective indicate display diagonal mf variance well also mf copula optima mf parameter posterior hand mf use set handwritten digits membership example report mean model latent draw bernoulli independent factor mean field copula hold field minibatch take average take minute take minute mf fit copula perform mf require minute already inversion well fast apply inversion outperform drastically upon convergence runtime hamiltonian five variational mean iteration comparison field either field copula iteration already field upon copula preserve dependency propose principled perform field copula alternate scalable manner stochastic mean mean easily add bias form approximation capability variational achieving acknowledgement foundation discussion assume structure copula family black box inference tree copula family synthetic calculate possibility intractable grow exist sequentially dependency sequential subset fix require copula preliminary find outline copula family certain conditional bivariate copula among family maximize sequential package easy also family experiment frank close family correlation include version copulas frank copula tail theorem general family structure copula copula allow dependency variational distribution approximation divergence augment stochastic straightforwardly original mean reduce sensitivity local hyperparameter help characterize interpret dependency latent keyword bayesian inference copulas network efficient approach approximate distribution applicability complex tractable make either field original variational order preserve mean monotonically kl budget demonstrate fit bivariate addition copula field structure approach knowledge fall dependency augment variational calculate easily place copula example mixture consistently parameter reduce optima implication feasibility restrict inference make write variational preserve variational inference study solution class model differ inference explicit denote cdf bivariate pearson copula copula specify tractable much focus two dimensional copula student frank copula multivariate lack flexibility accurately model dependency successful bivariate specifie factorization copula conditional bivariate copula also copula specify
answer representation together cnn outperform image need answer multimodal convolution propose image question question cnn multimodal concatenation prediction concatenation complicate relationship multimodal input multimodal interaction answer multimodal convolution word lstm answer reason meet word exploit lstm without modal convolution question compose reliable high semantic pool reach cnn possess language high representation examine whether reliable randomly question significantly compare language language natural representation content answer question drop generate representation greatly demonstrate moreover lstm question learn lstm question well introduce lstm performance future representation paper propose cnn neural cnn model architecture representation answer public dataset demonstrate outperform com li com propose question end composition inter modal generation answer image cnn cnn question layer multimodal joint candidate efficacy recently answer substantially multimodal language specifically rapid sentence retrieval far explore complete answer answer image content produce building block vision language processing however understanding pose challenge regard automatic image ai multimodal learn image produce image condition related human computer question understand instead example question successful well represent pay multimodal input question employ convolutional cnn image triplet consist question answer cnn learn like question content cnn learn answer question extensive result dataset state architecture encode conditioning recently sentence automatic image multimodal require make use widely recognition cnn successfully language multimodal relation sentence retrieval term lstm sentence image representation generate answer question et al binary image answer al question answer visual parse question answer neural research formulate conditioning question compare image solely lstm concatenation sentence cnn however question question answer tend denote color neural inspire rnn name visual semantic embed image question multimodal correct lstm treat question learn treat individual word exploit handle drawback cnn employ learn inter relationship prediction construct cnn well interaction make answer cnn related input figure cnn one cnn high semantic convolution layer representation softmax generate answer multimodal answer train reliable answer question firstly multimodal exploit sentence individually multimodal input many cnn representation achieve image recognition work encode content activation sigmoid relu take softmax layer relu cnn mapping dimension provide benefit firstly meaningful meaningful composition pool composition convolution max summarize component convolution max pooling question scale whole layer max generate max last pooling representation representation sentence multimodal input generate answer multimodal cnn treat semantic consecutive semantic question interaction multimodal input multimodal convolution multimodal similarly paper treat image representation treat generation word far exploit vanish time step begin perform manner interact closely question question well exploit question firstly semantic question interaction demonstrate feed show softmax layer answer question introduce cnn train evaluation measurement employ convolution cnn accommodate length question length choose embedding obtain gram cnn top softmax relu map new eq dimension image accommodate convolution multimodal cnn sentence joint softmax input cnn train sgd tune image multimodal softmax prevent dropout construct however publicly cnn public database testing answer image image type specifically type object color comprise training testing constrain generation answer dataset large color answer evaluation correct testing question besides wu base subsequence require use respectively image multi employ semantic answer develop compare specifically neural cnn interaction world human answer word language multiple multiple single word guess lstm performance
vary reinforcement admissible decay reinforcement configuration mdps hierarchical reinforcement basis markov explore temporal aspect task subproblem agent mean observe agent action move rewrite give reward function respectively one framework option way call create idea hierarchy option option option call mdp define choose environment option terminate probability notice semi policy I next option entire terminate continue accord rewrite q probability terminate update eq propose human expert still expert manually define behavior specify behavior tool compact maintain scale expressive power rl agent situation adapt change configuration reinforcement together reinforcement modular composite action example different kind position case could robot configuration arm reward return value must represent child child return reward unique child rest behavior would child health could one formal exploit option approach use policy option execute past state reward begin option generate probability argue model option reinforcement option reinforcement primitive type trial room next room moment room lose agent room figure branch highest low represent learn use receive action node receive show activation behavior expect activation behavior expert behavior use behavior baseline save use room action intensity give room intensity specify perform take complete notice wrong room show agent learn one learn node receive wrong save room root child try leave room show behavior difference learn effective behavior create character game quick receive development prove tree hybrid dynamical work root acyclic graph dag implementation level modeling level prefer extension core description horizontal due sort et propose look reason tree reinforcement behavior option similarity reinforcement abstract general hierarchical reinforcement area manual behavior problem part manual behavior view expert reinforcement agent physical agent constraint tree composite local node work hierarchical ensure nest intra option validate expert knowledge confirm nest node action expand use tree expect capability us agent entity scientific inf use behavior human agent execute desirable adapt human environment discard due framework node behavior address capability agent option ensure show affect execution must right moment goal minimum ideally zero chance consider human robot operate carefully neither human bt plan make video appeal human expert control execution action agent reliability agent behavior build maintain behavior also good behavior differently event bt entirely manual create behavior variation behavior agent agent human agent adapt environment interact bring rl agent adapt environment real online discard agent bring problem stability agent bring robot character human generalize poorly converge add capability reinforcement behavior rl minimize risk behavior reinforcement overall tree agent option framework reinforcement remainder present overview tool reinforcement reinforcement learn validate empirically control relation discuss behavior representation make agent create game character character control hierarchical state machine code rule effort tolerance system notation formalize behavior tree controller relation dynamical coherent behavior tree current minor please behavior provide transition especially collaborative bt define explicit box design independently another add modify remove necessary piece model bt decompose graphical bt node project although another parallelization possible easy worker contain parallel behavior tree root direct incoming child child leave subtree single leaf node propagate branch reach return call immediately root sometimes back type stop reach tree category present category constraint category commonly refer propagate composite return computation necessary symbol change behavior return state child repeat execution child represent action node leave propagate instead environment internal robot involve play sound turn spatial transformation play etc action instead signal meet commonly criterion condition agent visible low represent reason could return still never handle differently return receive core action five composite node node controller specific sequentially child return return receive sequence child return node selector child sequentially child stop
batch mini size ms gd compare good performance ms gd mini effective pass count effective mini batch gd comparable well gd parallelism ms gd parallelism ideal speedup parallelism achievable could evaluate efficiently ex descent give effective pass give thresholding experiment version average analyze well instead gradient descent apply give good gd safe ignore give demonstrate superiority gd algorithms ms gd propose batch variance nonsmooth unconstrained processing ms enjoy former admit parallelism comparison parallelism potential q proximal collection vector apply iy k q q convenience iterate x v ty x divide side inequality notation analysis put decrease obtain eq available eq summing multiply side right combine strong convexity combine h define statement relation recursively expectation eq value desire strong also choice verify equivalent side positive trivial thing need verify denominator need apply operator give update apply follow proximal operator calculate separable mini coordinate list let k k g k eq conclude summarize liu text bind evaluation need predefined accuracy iteration speedup mini simplify n essentially always indeed resource outer translate case reach speedup assume condition prove fix set outer minibatch ms gd perhaps explain behaviour need simplify assume big long processor approximate b convexity also convergence epoch accuracy epoch desire expect epoch enforce rest choose small highlight limitation efficient speedup processor probably gpu architectures parallelism loop follow note evaluation remark exercise question plot liu mathematics university unite department university usa gd incorporate improve complexity semi stochastic gd sum nonsmooth convex perform computation follow become introduction mini compute gradient base benefit effect reach predefine mini scheme admit parallel parallelization method empirical risk descent reduction separable nonsmooth number smooth function convex lipschitz constant e q parameter allow bound combine develop ms gd mini proximal gd enjoy parallel speedup environment attain mini formalize predict speedup mini batch loop one gd intensive past year accelerate nesterov fista scale big iteration issue randomize closely mini variant gd motivated accelerate proximal batch acc acc prox largely mini stochastic update limitation sgd inherently parallelism mini gradient via assumption whenever procedure equivalently write follow scale proximal stochastic eq estimate gd old gradient already prox point reference outer index new counter instead notation outer loop square variance ultimately extremely semi complexity complexity clear semi outperform regime fista achieve momentum gd max stochastic stepsize start mini batch store ix kt size estimate ij k iy h tx loop index epoch counter loop epoch start compute statement predefine particular speedup target follow demonstrate base logarithm e gd stepsize gd translate speedup mini threshold prove outer minibatch ms gd related mini acc prox acc incorporate mini nesterov acceleration claim define pn acceleration batch acc prox gd theoretical acc prox ms gd numerically total component evaluation compare condition theory mini gd advantageous acc prox acceleration acc prox illustrate ms gd ms ms gd ms gd set regularizer eq conduct q set training perform mini batch parallelism regression lipschitz result evaluate ill give four sparsity proportion nonzero element constant h sparsity might ask ms gd sparse sgd nonzero test operation ms gd fully
patient diagnosis diagnosis along vector recover separate selector various grateful anonymous helpful comment author wang matlab approximate selector use direction theorem example iterative selector two stage formulation selector stage construct selector direction simulation alternate numerical fast selector operator consider linear predictor parametric among square great deal variable selector selector sparsity involve selector receive amount attention selector fit minimization technique selector strictly guarantee ensure uniqueness selector right censor outcome importance selector demonstrate work cast program interior interior large scale problem cast solve alternate direction find selector show usually cpu problem rewrite iterative iterate subproblem successively subproblem solution gradient subproblem linearize selector world datum proximity optimization via solve primal one implement achieve comparable consume outline follow section numerical efficiency propose propose first simulated patient rest let absolute th otherwise give vector hadamard denote denote whose th natural product df fx fx develop proximity solve optimization exist alternate direction method linearize propose reformulate augment lagrangian lagrange multipli penalty optimization elementary equivalently subproblem gradient scheme k subproblem complete square constant form subproblem efficiently iterate selector base matrix appear rewrite thank norm characterize proximity operator review definition map solution vector conversely satisfy solution proof straightforwardly chain aa previous problem comment proximity appear soft operator cube length far b check find amount couple iterative upon equation dual dual exactly scheme apply word introduce iterative scheme sequence iterative initial seed particular limit sequence selector often nonzero correct bias assume generate iterative step step submatrix extract coordinate eq set parameter parameter compute I terminate reach stationary terminate stop meet change successive tolerance stationary successive iteration fix stage complexity comparison loop relate subproblem approximate subproblem terminate stage complexity next indicate short situation require iteration follow proximity approach method present selector matlab center advantage typical utilize equipped intel core cpu ghz gb matlab pc intel ghz processor gb windows simulation generate recovered combination algorithm stop priori noise well approximated exist method event speed affect accuracy suffer gaussian support size select uniformly identically sample normal collection selector collection zero standard simulation experiment selector measure square ideal estimator approximate noise simulation standard ii illustrated selector magnitude nonzero component estimate nonzero curve represent vertical away
appropriately optimal order iterate learn main goal compare implicit update suppose approximately term limit dominant stability work implicit define explicit loss low initial prefer discount explicit term discount reflect misspecification cause instability initial decay convergence implicit rate explicit conduct standard benchmark simulate real dataset display behavior scale challenge alpha xx gradient descent xx xx stochastic iterate replace proximal reduction proximal gradient sag establish minimizer mle stochastic scaling adaptively second sensitive default version descent aware importance equivalent interpretation optimality normal regression separate normal n ny xx plot full xx optimality follow xx follow xx let step proximal method pass single pass achieve indicate pass converge perform varying affect parameter perform increasingly bad stable achieve classify cover type leave validation datum use use logistic set regularization method exclude rate specify setting paper hyperparameter misclassification pass hinge figure intensive hyperparameter easy way investigate change iteration ht digit indicate misclassification pass respectively xx supplement hinge ht accuracy term proximal implicit update size average iterate er rao performance significantly par explicit theoretically aforementioned robust convexity effectively learn stability come model simplicity comparison proximal tuning calculation dataset storage information understand strongly convex objective grain analysis average corollary university university university become unstable statistically inefficient information term combine proximal implicit iterate er non stability robustness learn respect function demonstrate state averaging method utilize update simple consider wish expectation typically loss wide mean cast approximate approximation seminal maximum posteriori learn mle usually incremental procedure realization stream subgradient respect realize combine idea implicit iterate implicit update hand operator point generalize splitting comprehensive implicit derive efficiently implicit proximal method geometry proximal idea replace stochastic gradient update average average accord aforementioned finite thus rate estimator well difference regime incremental simplifie employ keep average periodic several whereas storage averaging iterate analyze optimality application convergent average work analyze update superiority substitute convexity simple certain aspect f expect asymptotic iterate imply non averaging experiment several task confirm suggest combine method definite theoretical eq differentiable surely differentiable decompose yx dy convex sequence random surely hessian almost surely twice convexity convergence assume believe implicit vector see probabilistic implicit efficient restrictive include variety logistic linear time series constraint notable exception form well regularization confirm either subject use average study proof
identity testing closeness address mild logarithmic low indeed logarithmic section proof follow sp section motivate special highlight two aspect test distinguish distinguish testing later subroutine simple case testing determine belong suffice thus chernoff suffice set next element pick determine probability increase experiment several uniformity illustrate difficult find element distinguish distinguish observe differ argument method near proceed testing argument chi capture dependence complexity example consider distinguish suffice capture test exist chi two n else sample correctly hence give find select however consider way find element quantify idea pick element definition generality heavy pair symbol heavy distinguish easily auxiliary heavy property algorithm achieve good trade tuple useful achieve near independent tuple tuple return heavy lemma element apply pi find distance furthermore increase find element heavy yet belong higher precise trade complexity recall find distinguish near distinguish easy meta subroutine want use yield help complexity return tuple run previous tuple output recall distinguish distinguish element distinguish candidate combination element know set form ideally scenario arise constitute partition ss combination possible scenario find tuple case randomly entire test yet scenario output probability otherwise step propose identity testing test calculate complexity tuple g h g h test return tuple closeness unknown identity closeness testing identity part find distinguish distinguish algorithm closeness testing extend identity closeness distinguish order probability decrease distinguish closeness find organize identify distinguish formalize distinguish element one frequency distinguish closeness require additional testing order subset distinguish closeness test outline finding serve symmetric compare simplicity rest come find eq indicator variable precise argument remove ps ps pi efficiently show state distinguish convert calculation analysis expectation seem analyze success event take fair amount conditional generate probability show use need element pi pg heavy although probability tuple ir use sample know multiplicative factor later furthermore I I threshold ensure guess value output I search fast recall problem approximately know guess instead find assume closeness least remove pruning element probability step tuple tuple return main binary set underlie result n sn nj ks ss ni else return obtain set heavy whenever discuss sufficient closeness firstly remove element big number indistinguishable probability probability concentration concentrate many time address pick ensure none element consider find perform closeness use distinguish run run distinguish main repeat majority success close none probability complexity run otherwise tuple I p return thank cl distribute align universal suppose poisson variance term chebyshev inequality chernoff p qp bn chebyshev eq chernoff bind argument hence n tuple since count prove thus proof pi upon expand furthermore similarly hence lemma return th overall chi square numerical simplification lemma error use theorem result output probability return element let correspond induced output probability pg h pg x x g x g pg ab pg pg py py proof chi substitute minimize pick condition event discuss probability fall three complexity use sum tuple yield complexity tuple p g sample summing use define section interested element end notion element show property reduce index go sum suffice eq inequality side sign different sign sign find distribution draw sample output tuple py generality technique algorithm element large far element get pi pi good element satisfy heavy exist element appear j first state follow independently distribution find run underlie good tuple tuple else find remove heavy two small tuple run initialize set element p r set sn nj obtain else remove remove element chernoff less n remove remove call identity let element sample th heavy eq suffice call proof remove element q time substitute rhs simplify follow hence iteration ii remove element heavy pruning pruning rhs inside ir simplify union lemma since time union bind divide proof return notational return call first showing heavy never remove remove element two part event I p conclusion lemma show r time clearly probability bind contain j rs rs chi eq return event happen set pg pg happen output union element prune unchanged prune pruning convexity fact hence q r eq
surrogate problem bound derive use surrogate minimization address issue approximate projection outer experiment synthetic efficiency machine svm well machine hinge loss surrogate classical majority class poorly biological constitute diagnosis pearson risk consist type type ii risk minimize dealing framework empirical risk propose optimization empirical subject empirical constraint present deal propose guarantee solve pearson problem section deal art pearson classification finally experiment synthetic rna seq cancer henceforth underlie classifier mapping sign predict occur classifier e must shall nonconvex convex fig surrogate problem x x I classical surrogate al quantitative relationship risk use show excess calibrate risk type eq pearson introduce let denote mx unbalanced unbalanced alternative sensitive lagrangian svm svm lagrangian cross propose pearson calibrate classification empirical type risk property classifier classifier mm eq suppose surrogate ii restrict attention calibrate surrogate loss convex everywhere lipschitz twice calibrate posterior directly value connection assumption differentiable vision normalize high loss calibrate hinge boost satisfy solve pearson algorithm backward splitting method introduction processing offer guarantee reliability suppose satisfie reasonably note lipschitz split separately proximal forward backward algorithm implementation projection operator assume rewrite derive convergence iterate let except computation lower set onto fortunately compute perform sufficient efficient design level algorithm proceed successive outer subgradient projection dd eq terminate principle compute half space serve outer outer onto express explicitly describe onto onto magnitude perform q fp fp fp kp kp fp k p iteration contain low fp fp kp projection half need sequence onto dp k fp kp dp fp fp dp dp fp n k computable inner take risk clinical set work dna empirical recent review selection detail alternate pearson classification experimental surrogate half figure risk typical throughput sequencing rna seq microarray preprocesse library jx total number read library sequencing expression propose transformation optimum eq transformation last negligible count patient patient gene sequence generate variable modelling measurement randomly change value decrease impact binomial transformation whether artificial patient class unbalanced sample
drop ill unbounded irrespective geometrically r measure noiseless analog constrain least long allow could relate much class asymptotic wide pointed square logarithmic achieve regularize estimator nuclear regularize achieve error arrive example upper imply appear sufficient slow vector slow exist definite ball scaling small number simple satisfied interesting design matrix appropriate tail moment know concentrate hence weak broad finite specialized statement indicate constant eq begin quantity square hence feasible condition later high henceforth eigenvalue orthogonal z indicate noiseless case regression model yet quantity appearance bind possible improve place dependent nuclear quantity square instance standard much indeed fail trivial entail albeit convex problem block practical reach numerically find apart regime without result particular face stock different employ rip boundedness involve problem fix standard random wishart behave form probability moment order scale replication phase specifically turn quantile associate triple summary may mask observation identify quantile drop model concern reasonable give rise noiseless yield solid line curves fit exceed empirically relative wishart expect behaviour design replication wishart I parameter oracle error negligible see color grid validation minimized note ensure pick nuclear minimized nuclear minimization chen et specific choice regularize assess impose constraint add constraint yield parameter grid specify worse report conclude case differ regularization oracle seem present ir z eq factor random motivate connection explain popularity model straightforward available constrain square estimator take consist image covariance turn obtain price stock technology begin year total retrieve correlation precede wishart difficult observation point replacement range replication replication approximate measurement perturb reasonable albeit extreme million achieve reason stop picture use full hand fast ex f c ex paper investigate trace symmetric semidefinite excellent employ nuclear side usefulness finding recover li partially nsf dms fa invariance suffice consider orthonormal canonical basis operator minimizer coincide projection law zero follow cone z r optimization symmetric proposition contain dimension conclude problem lagrangian dual proposition follow remainder establish dual obtain kkt optimal pair obey take inner complementary substitute feasible choosing ingredient give write last eq equip inequality lemma h maximum case consequently obtain follow easily see substitute finish recall bind back v yield desire concentration extreme eigenvalue let eq expand theorem obtain two assertion sequel respectively minimizer satisfie contain sphere consequently conversely satisfied otherwise divide proof analog suppose inside lower expand back yield collect obtain ex theorem section theorem proposition definition lemma sketch l trace positive computer department science nj usa past year receive considerable completion quantum estimation notably nuclear great popularity argue long positive condition situation approach entail knowledge estimation come trace interest estimate measurement matrix attract focused set sense phase retrieval work nuclear amenable modern technique lasso arise regularization less clear incorporate present semidefinite denote cone interest gram method rank kernel estimation measurement employ nuclear norm regularization proper parameter interesting practice choose finding negative dimensional paper certain design achieve regularize generalizing noiseless paper good paper compress sense goal complement finding summarize terminology throughout matrix inner dm dm usual number b linear adjoint consider error convenience cover tail symmetric projection orthogonal complement sequel estimation refer
denote information contour say frame mass induce independent evidence use form contour contour function contour pl pl algorithm maximum missing cause measure estimation density express become know uncertain information belief contour uncertain observe likelihood contour respect eq regard contour observation therefore eq em e observe small threshold special incomplete datum censor denote de two observe censor let datum mix knowledge experience partial simulate label draw randomly change probability uncertain label em censor em run estimation bias commonly equal exact estimation c degradation noisy soft suffer label supervise learn estimation unsupervise learn uncertain traditional indicate follow experiment censor censor class label appear moreover maximum e censor datum e algorithm estimation available belief improve project failure device stage continue regard evaluation perform priori frank com frank er evaluate em compute maximum estimation especially uncertain datum e due censor uncertain derive base knowledge integrate life censor right censor term censor drawback removal terminal censor possesse become popular year censor kind reliability evaluation uncertainty ever tendency take uncertain account last decade uncertainty latter lack information restriction cause compose expect datum carry uncertainty censor e uncertain use analysis censor consider special simultaneously censor merge uncertain unlabeled value hide occur early believe therefore belief prior pseudo prior method maximize show label algorithm censoring attract recent year flexibility removal terminal theory belief first later give brief ii censor pc describe identical
neutral frequent remove stop distribution distribute top frequent neutral feature pool unbalanced positive select pool axis unbalanced dataset remove unbalanced knowledge classify obtain balanced movie unbalanced remove document unbalanced dataset maximum improvement obvious neutral entropy since assume neutral distribute suggest gradually bias label incorporate kl well balanced unbalanced compare feature balanced label feature pool conduct movie positive unbalanced randomly positive balanced little reason label unbalanced maximum neutral bad approach pool axis unbalanced construct remove unbalanced manually movie dataset distribution positive original balanced movie balance unbalance pool unbalanced result show unbalanced become remarkable neutral decrease significantly remarkably divergence guide label unbalanced kl robust model al highly unsupervise assignment al constraint instances newly propose several dataset unlabele cross constraint self et objective al framework project distribution al explore distribution reference propose incorporate explore monotonicity chen try leverage incorporate class distribution objective discuss perspective address paper try prior et instance feature et propose active learning problem model propose regularization term expectation experimental considerably improve comparative knowledge work leverage may present detailed discussion incorporate neutral feature simple require modification common neutral unbalanced entropy regularization controlling extra nothing corpus assumption violate unbalanced reason kl utilize assumption fact suggest additional kl knowledge sometimes fortunately insensitive rough possibility perform reality distribution domain china cn many approach knowledge robust justify experimental propose remarkable improvement robust baseline language processing task text categorization indicator sentiment leverage guide nlp previous study address problem line leverage encode commonly see knowledge variable dependency last knowledge latent variable crucial knowledge fan provide word less heavy handle undesirable investigate aim reveal factor practical regularization formalize output easy neutral namely indicator reveal neutral boost remarkably make manual annotation neutral regularization maximum class regularization simply use neutral neutral uniformly contribution regularization term neutral distribution kl outperform briefly justify survey robustness method label knowledge indicator manually example label sentiment criterion provide preferred guide parameter preference expectation give express et label label parameter indicate word otherwise number softmax bfgs framework term constraint indicator correspond elsewhere load annotation well number label often neutral feature feature frequent preference neutral uniform neutral prevent dominate neutral class term manual neutral take neutral work successfully way prevent desire unlabele take maximum principle predict x p x number label entropy empirical objective already distribution roughly labeling kl predict objective preference follow
reveal ensemble anneal inverse temperature heat peak indicate temperature dash critical swap rate schedule heat gray curve anneal literature possibility anneal near path histogram temperature anneal construction schedule energy histogram successive overlap schedule logarithmic grow phase construct schedule e approach temperature automatically avoid worker anneal decade algorithm find temperature schedule entropy production temperature root heat rule change proportional heat capacity entropy increment inverse temperature increment generate schedule annealing show proportional anneal computation put system anneal seed right panel exchange anneal drop close heat peak section multiple second order relative term vanish view definite define parameterized special temperature information therein switch state therefore optimal geodesic manifold equip metric present relative follow anneal successful relative entropy accumulate approximate resource ensemble canonical ensemble temperature temperature anneal intermediate ensemble multiple parameter accumulate discrete geodesic generally reliably entire connect generate geodesic dash hand curve energy peak dash true line major powerful simulate canonical ensemble algorithms bridge motivation due tail overlap exchange difference intermediate boltzmann confirm ise model correct canonical whereas correspond canonical however advantage reason proportional solve temperature canonical peak temperature temperature ensemble multiple run fashion control set ise peak around peak anneal systematic produce schedule phase separation less consequently boltzmann also energy difference large produce boltzmann anneal algorithmic nest inference bayesian normalization essentially inference nest view special annealing zero temperature relative entropy ensemble constant cumulative contour reduce nest anneal ensemble result ensemble implement build nest principle guide nest utilize truncate truncation fact therefore volume energy energy among result nest ensemble achieve nn next energy also anneal produce ise also estimate example run speed three order magnitude anneal anneal energy instantaneous nested ensemble anneal histogram contour histogram slowly protein computation nest anneal ise geometrically canonical compression rewrite anneal canonical ensemble fast anneal ensemble ise fig canonical sampling reason maximum nest many bridge anneal carlo step ensemble generate along also density manner maintain constant entropy variety canonical family close fact annealing aim implement compression reliable nest tie whereas anneal hybrid monte carlo canonical ensemble worker anneal rich find ensemble simulate difficult mean contribute ensemble ensemble anneal grant configuration also family protocol intermediate canonical configuration factor temperature obviously method parallel boltzmann ensemble prior function cutoff temperature ensemble configuration energy great nested configuration system potential plus state denote delta function energy choose intermediate bridge successive ensemble kullback equality kullback leibler divergence distance ensemble contrast distance broader contain support quantify member integral energy th relative energy article relative distance ensemble overlap ensemble might useful exchange ensemble jensen shannon control mainly ensemble article explore measure ensemble infinitely many annealing reach ensemble intermediate fix entropy amount speed ensemble need relative average difference problem outline normalization constant evaluate state integral reduce integral anneal histogram interact whereas anneal energy visit configuration entire simulation update ensemble relation histogram free energy histogram update partition start previous set state histogram estimate ensemble reliably configuration previously ensemble ensemble anneal iteration schedule infinitely slow annealing accord entropy lead schedule shift direction depend relative next ensemble impose close canonical anneal illustrative sufficient monte consist draw uniform relative configuration criterion temperature annealing localize accurately anneal decrease represent boltzmann could decrease number thereby resource explore apply anneal simulation lattice particle relative monte randomly lattice try flip spin initial start spin cover proceed continue energy range accurate produce accuracy
make implementation next short delay big sampling factor nature front regime stage subsample use front end guarantee frequency sparse bi successfully singleton high stage subsample front architecture brief reader regime use r front end period achieve operate less regime decrease function follow ok f front noiseless delay bin contrary additive need path role structure front architecture detail observation b w b refer bin measurement bin sequel vector generally observation vector bin stage architecture per per chain j bin sampling period delay circular shift r sub front sample dft singleton singleton signal singleton try bin somewhat similar compressed sense incoherence restrict isometry rip widely recovery sparse although sufficient incoherence rip bin measurement stable propose style q property measurement restrict positive rip characterize preserve capability matrix operate good vector since bound away point discussion reliability stable circular front mutual incoherence bin least easy offline incoherence unit mutual incoherence eq circular delay chain front column bin frequency circular shift front uniformly shift matrix make slow consist detect reliable circular shift front combination randomness structure enable consist delay delay circular shift shift measurement exploit structure intra singleton e end decoder recover big dft output front routine singleton singleton exploit bin bin determine dft connect explain operation focus bin square observation estimator observation estimate correspond potential bin estimate justify bin observation satisfactory appropriately choose bin cluster measurement mmse coefficient output boolean singleton singleton dft coefficient dft coefficient pt singleton false set I bt I process colored estimate thus estimate successively refine continuous slight abuse multiply two frequency length fold result increase h r architecture circular shift stage sample frequency red fig reconstruct fully frequency fig acquisition propose manner reconstruct brain image acquire reconstruct image fourier reconstruct differential vertical operation dft operation creates approximately reconstruct access fouri sample brain reconstruct chain stage architecture r differential brain image inversion fully frequency reconstruct brain operation divide dft center fourier non center total fouri reconstruct show column bin lemma circular shift term I support hoeffding inequality apply summation thus choice circular shift least consider circle eigen eigen absolute entry matrix equal provide absolute diagonal consist part first show r dft second decoder denote fail reconstruct dft bin fails correctly classify bin singleton bin bin identify dft coefficient event entire decode bin wrong decision put piece bin process entire perform noiseless front construction vector bin number iteration dft get reliability role event threshold arbitrary complex noise corrupt value cn please use bin cn bin bin multi bin bin possibility consider bin identify zero bin zero dft bin let bin bin zero inequality use singleton inequality carefully dft coefficient discussion bin let process classified bin bin cn bin bin dft compute compute dft binomial need end pre b l k c l constant r front end constant stage bin dft sample presence moreover decoder bin constant proposition bin process cn get pr pr corrupt periodic interval observation singleton bin proposition frequency estimator true singleton bin sample overall singleton assertion theorem conjecture edu fast fourier transform fourier dft arbitrary length dft signal question fast transform induce code chinese theorem exploit devise iterative compute dft computation applicable whenever attractive adapt corrupt particular compute dft particularly implementation feasible randomized measurement permit flexibility choose variant r present dft corrupt signal dft white like mr spectrum dft coefficient fast way compute dft arbitrary complexity signal signal fourier assumption noise spectrum dft compute length dft arithmetic computation million million relevant big gain highlight conceptual framework underlie paper adapt noiseless robustness modify detail front stage stage call delay chain identical shift shift shift precisely asymptotically delay shift random choice circular shift measurement good mutual incoherence property enable motivating show domain minimize fourier reconstruct image brain acquire mr fourier sample elaborate art technique promise direction demonstrate dft practice uniform typical however compute dft corrupt hardware desirable even sample process set dft noise variant present useful application flexibility randomize measurement system elaborate dft emphasize assume secondly sub noise rest provide signal overview literature key provide review result propose recovery generic description validate complex transform signal corrupt cn signal dft zero dft remark dft signal ratio front dft operation linear sample flexibility dft computation r algorithm dft property computational recover dft please dft perfectly corrupted assume reconstruction dft long zero dft coefficient arbitrary value dft applicability complex dft successful recover perfectly also successfully transform rich processing compressive estimation study decomposition music approach tool theory theory many issue manner sense literature random pursuit standard isometry rip characterize matrix unitary sparse like exhibit rip scale practice dft good knowledge characterization rip consist dft sub scale contrast bad fourier sampling innovation level though key difference dft indeed compressive sensing inspire work compute dft high sub require bound noise dft compute dft applicable signal sub regime domain process end sub front architecture decoder later front signal dft furth dft length input form sub front decrease path period delay stage input signal delay chain shift illustrative processing corrupt stage sampling stage furth signal delay output front end far grouping bin decoder big dft short bin bin obtain node bi dft node bin sequel vector edge connect dft contribute bin e bin e observation dft identity let bin bi dft coefficient represent bin connect dft coefficient contribute bin sub dft contribute bin stage bin stage computing dft transform decode support bi I decode bi bin contribution dft signal dft coefficient noise vector contribution non dft bin stage verify stage non dft g bin corrupt observation sample front bin stage height estimate dft denote decoder threshold choose appendix stage bin singleton singleton v multi r decoder function singleton exploit high addition determine dft select right graph remove neighboring contribution check decode successful remove decoder successful decode coefficient decoder bipartite sub
study sophisticated estimation fast pdf common pdf belong poisson parameter advantage often back environmental estimate nice actual lie occur potentially priori pdf estimation option maximize grow add penalty unique maxima compute typically lose combination scale kernel fall decade choose appropriate process additionally slow parametric square approach search discuss wherein maximum likelihood estimator lipschitz pdfs property efficiently computable form present ml computable pdf pdf band bl bl think however propose preserve property estimator consistency efficiency bl infinite pdfs case test grid outperform bl ij x j support si c solution locate local value global maximum maxima however exhaustive entail solution compare programming theory additional knowledge know bl strictly positive estimate theorem test remarkable test prove construct plug result divergence si consider fourier transform frequency follow cutoff data j multidimensional dimensional result si next step select solves computationally state si lie bl bl strictly estimator simplicity computational kde estimate equation solve bl quick briefly bins pmf pmf bandwidth pmf make true pdf small small bin bin reduce f lx c lie np toolbox improve nearby hamming equal neighbor nearby computationally surrogate pdfs several surrogate pdfs bl panels use surrogate strictly pdfs panel plot size pdfs theory whereas marginally computationally expensive remainder pdf strictly strictly pdf cut assume calculate compare estimator fast kde nd order kde plot adaptive bl non bl pdf respectively bl band pdf bl respectively alternate hz position spike sort accomplish manual mit care activity grid cell peak fire grid spike histogram position generate spike dot trajectory inside blue neuron factor x x allow nonparametric spike estimate kde smoother capture sharp spike kde cut frequency combination frequency fit low ks circular glm spike fit rescale kolmogorov ks compute cdf estimate quantify ks plot close ks statistic ks estimator figure estimate ks remain inside ks ks glm glm glm nd activity glm structure glm certainly neuron covariate glm glm neurons know kde show mark co position spike ks kde nd glm glm bl develop three quick presented generate estimate strictly remarkably estimator even parametric apply mechanic development si wave mechanic absolute density mechanic wave momentum wave wave position bl versa occurrence single observe think experiment box wave finite momentum wave bl pdfs bl macro phenomenon bl double set bl bl convolution pdfs bl phenomenon level phenomenon bl macro phenomenon bl pdf macro observe pdfs process almost bl cutoff lie finite impossible distinguish bl pdf logarithm exist method select estimate fit ml pdf band converge pdf band increase frequency cut frequency infinity e sophisticated likelihood cut frequency figure n ij x increase infer complete analysis leave present approach numerical technique incorporate idea study prove estimate clear normal study normality first
location useful model could applied stock transformation model median improve modelling log transform residual allow require model raw stock comparison enable come recommendation assess course assess question interest ask availability system available cost comparison several criterion know need fitting applying may add required model validation good know p yield model performing regard specific comparing since context study carry advanced approach covariate krige consistently improve increase one involve multiple tool study predictor turn bring additional instance rank stock drive factor recommendation recommendation france diversity make recommendation information contain relationship stock flexible prove map national care dataset model national systematic scheme check spatial autocorrelation residual simple fail national residual provide accurate stock prediction highlight thereby guide research model acknowledgement analyse scientific environment management institute research institute european author thank involve address technical bank handle publication result correction mechanism reflect document publication volume page play major global source improve stock national monitor study recent first consider several increasingly tree convenient multiple perform network procedure limited predictor prediction significantly improve modelling prediction adequate care allow contain behaviour source increase temporal pool distribution international reliability suitable content density comprehensive may parameter dynamic interestingly regard use validity scale precision extent mapping rule relate stock use model adapt tree regression study study extent km predictor extent study approach national mapping france decade specific stock location reach stock unbiased bias ensure stock national potentially room especially way improve recently relate environmental method design outlier nevertheless currently qualitative variable nonlinear automate share robustness problem autocorrelation stock consider aim stock france quality network use model useful modelling advantage stock france france stock site monitor base km locate grid cell possible site km center site site locate locate position cell site individual take cm within individual composite composite south measurement stock horizon layer density mass account reach stock variable variable depict available mapping national scale site management biological forest forest median ph per national testing estimate database adapt content european european link possibly type surface unit rank also include instance occur available application production combine matter concentration month potential month temperature km average observational variable estimate site spatial join grid map primary site content possible site correspond lastly moderate resolution image primary get site development mapping mapping tool software stock present area south west part belong model onto apply base learner final combination weight learner boost algorithm algorithm stagewise base learner specialized regression gradient boost algorithm rely base produce draw replacement besides determine base minimum terminal available stop internal avoiding thank stochastic aim minimize risk overfitte predictive handle linear among predictor qualitative variable assess contribution predictor partial assess predictor thorough guide use fit predict stock refer model represent lot know site stock additionally content predict complex occurrence site month mm month addition matter leave value recommendation investigate represent et different class spatially three method stock contain covariate observation transform due skewness stock vector residual prediction transform transform response assume spatial model observation identify extreme reduce effect spurious exclude predict close observation confirm stock dataset represent log valid left inequality observe exceed possible measurement depend verify check leave validation valid property square standardized error note cross median procedure validity aim estimate spatial counterpart variation model model spatially variation effect model use fit due anomaly ordinary krige consist ordinary krige transforming predict stock prediction lagrange multiplier krige variance lagrange multiplier unbiased ordinary krige six refer counterpart validate cross procedure involve stock commonly suggest mean root square error coefficient determination strength value distance inter whole well model error root calculate hereafter name provide picture skew use monte fold enable external prefer preferred leave dataset model dataset krige fit krige step spatial spatial validate present section check perform validation procedure provide one prediction counterpart indicate performance metric external model validation fitting krige distinguished krige fitting fitting result distribution performance use adjustment name repetition validation sp residual result dependence residual dependence indicate simple variability deterministic spatial model value obtain site stock location extreme appear evenly distribute yield valid spatial validation repetition f spatial counterpart model express result improve bias important appear root square error fig skewed prediction median bias skewness result compare median spatial resulted improvement significant variance spatial express km smoothness three plot horizontal bar cross repetition line diagram f map instance give minu indicate size dot absolute error c whole dataset improvement model add term reveal south west areas central area area bias part west france site error area prediction prediction strongly note strongly model model yield significant degradation spatial south west improvement limit site improve difference indicate spatial model yield spatial ht residual decrease control factor content include range lie km give correlation controlling factor stock look km content decrease residual around km handle ph parent material however include control spatial km many functional spatially component another explanation high derive difficult draw conclusion study deal density stock stock might great estimate accounting associate error wrong bias
distance shorter hold shorter predict standard great hold shorter long supplementary iv vi qualitatively remain predictor supplement predictor run finding confirm validity find ii offer parsimonious science major power component find iii novel record reason suppose governed law empirical correctness extent individual determine explanation broken power record law record explain component distinct individual hold record unchanged gender age number summary vi universal three predictive validation imply finding number law iv second describe great negative third middle distance vs separate fall cluster middle distance provide capture social attempt influence statement cause conjecture datum clinical leverage uk predictor middle improvement rank version low middle maximum duration check verify level oppose relative predict world km unlikely provide complexity performance long short may implication finding length primarily vs conjecture methodology difference whether capability age cross standard decrease shift et al possible longitudinal bias older amenable quantitative validation number attempt notably cluster long resp shorter strong high example half good performance yet produce even also quantification em plan especially event ability accurately difference achieve drop prediction summary description immediate assessment plan potential scoring population prediction precise fair run resp capable conjecture validate power collective acknowledgment provide code norm use thank regard implementation local high rank helpful comment early manuscript computation carry fellowship author prediction acquire jointly methodology working algorithm rank concrete performance jointly analyse carry jointly paper responsible raw process matlab upon request analysis obtain link www achieve event event country event obtain database automate www ten track road track road exclude reason attempt country datum consist table field birth contain record attempt event field date second set available request database error removal record range old see gender database record date birth miss due record eight record road road half record record record leave recorded attempt set preprocessing table index event contain index column index entry date store mode yield ff proceed find percentile year good event period miss mode table ff event table select mode ensure close fitness performance fitness high attempt per influence chance use depend fitness narrow specific summary attempt performance wise event percentile miss preferred geometric distance preferred corresponding event mostly train characteristic depend reference affect removal matrix obtain high score sub gender certain amount per sub text table ff row gender age well event discard refer retain score take measure squared rmse mae rmse sub sample obtain validation entry entry scenario question performance event predict predict causality preserve lie predict third iii make extensive task due bias old attempt many recent attempt argue absence technical ii iii performance matter opinion contrary influence history due occur scientific statistical viewpoint outcome present pass variant column column complete natural logarithm second complete event column indicate mae order base measure unless otherwise learn experiment level repetition intersect summary good relative rmse report fair describe fair bar event event fair short event typically certain distance case whereby predict lie curve vs distance fair repeat sampling point standard analysis series experiment term mae result mae qualitatively similar measure prediction rmse mae log time c predictor high great middle unit event rank perform predict w predict event performance prediction prediction run fast model proxy real singular recover appropriate group singular unchanged prefer optimal notable prefer shorter old prefer transition right panel distance anti correlation distance short three positively long mae table report method mae compare rmse mae indicate presence qualitatively report mae bias towards long qualitatively log mathematically rmse mae rmse mae accuracy rank predict time top year improvement rank tend short distance accordance iv indicate individual good descriptor stability learn predict learn influence prediction log learn calibration goodness time good error bootstrap significantly prediction time normalize overall prediction preferable stability effect predictor year compute close point formula show predict distance predict year stability improve power predictor five choose point main prediction experiment display table different residual case incorporation lead aggregation sign demonstrate event may datum formula performance affect attempt predict performance make table report year performance random prediction qualitatively use prediction time comparison matrix completion single generate describe repeat nuclear pre cross display figure nuclear matrix available attempt rank ii synthetic assumption generative synthetic performance synthetic datum number three summary independently gaussian summary estimate component top event model distance miss uniformly miss per entry non nuclear norm repeat result display figure robust norm affected rmse approach minimization compare plausible model pattern identical top plausible entry singular svd real datum confidence display component missing entry rank identical observe component recover almost exactly slightly sub methodology component old year percentile range consider respective display estimate law unchanged component reduction explain consider display compare iii summary prefer summary display individual exponent score score correlation correlation approximation prefer number vs vs vs score optimal distance predictor performance distance vs top event compute achieve determine percentile refer prefer attempt event shorter old long explain phenomenon train shorter regardless old c iv phase transition closely phenomenon discuss consider set sequentially predict predict km triple consecutive distance exclude study perform triple rank performance distance way compare correspond red term distance perturb perturb calculate perturb prediction distance short km slow predict predictor find distance great km shorter long middle axis equivalent performance triple line middle decrease increase keep evaluation event mean individual law power nuclear em log time log cccc cccc cc nuclear em speed normalize log good cccc cccc cc event data nn individual nuclear log well good cccc k nuclear mean error cccc cccc c event type individual cccc rank leave equally cc cccc law law corollary conjecture scientific level individual performance low dominate individual law evaluate individual quantitative scheme break law basis prediction contribution modeling implication focus collective uk individual law explain law record individual training performance prediction accurate date minute mae table performance leverage insight record improve human prediction briefly law model performance dependence know describe extensively run law practitioner formula fix performance performance parsimonious international association distance comparable present performance performance may score forecasting performance science speed max direct prediction clinical appeal none prediction parsimonious power accurate scoring interpretable may one clinical measurement interpretable usually predict explain present desirable b avoid parsimonious empirically database uk yield descriptor basis prediction explanatory database step parsimonious performance discover individual explain range summary relate number great interpret exponent hold remarkably average describe linear correction law correction record three allow assessment completion record distance distinct approximate law power display performance world record exact straight align closely straight line notable record straight line break world individual variation optimally break law record individual explain variation scenario base top green predict green blue account predict green blue account event red distance blue green whose remain supplementary phenomenon present throughout quantitative simplest display panel predict green pattern performance exactly red event explanation green mathematically demand sum red blue I mathematically blue red green blue vanish equation multiple pattern triple average minimize model optimal prediction scheme instance apply performance event completion matrix predict behaviour recommender system predict cope finding supplement see appendix detail performance parsimonious explain consideration translate power demand depend since coefficient remarkably find due miss data analysis analyse online www contain removal individual range old comprise km km contain th percentile performance main analysis supplementary request full link acceptance manuscript state practice validate subset square absolute mae performance correct merely report data validation setup supplementary finding uk accuracy evaluate include include near representative art quantitative formula exponent power predictor exponent exponent point completion expectation maximization b minimization completion method performance event performance
variance ni use simple fact assumption denote asymptotic eigenvector motivate eigenvector natural insight normality regime relationship understand notation denote entry generic investigate behavior denote th normality distribution theorem strong weak asymptotically unbiased estimate conclusion equal eigenvalue matrix reveal control divide part element jk furthermore normality eigenvector noise radius addition jk j k factor dimensional small covariance start fact eigenvector spike understand important high remark datum note create distribute project eigenvector detail factor secondly strong generalize distribution part invariant general regime component compare eigenvalue limit eigenvalue normality except bias cause high estimation motivate sparse insight show eigenvector consistently estimate eigenvector name shrinkage principal thresholding time result management portfolio false discovery proportion finance observe stock e expression latent factor loading uncorrelated index repeat simplicity respectively load matrix condition covariance away covariance exhibit particular nonzero element decompose matrix spike separately correspond equivalent entry threshold thresholde determined section measure relative error device weak meaningful scale discussion drawback empirical inconsistent significantly dominate non part see recent eigenvalue drawback assumption p addition bound specific impose avoid sophisticated condition need drawback shrinkage inspire shrinkage first simply thresholding constant three common factor loading ready abuse svd decomposition obviously thus separately notice error eigenvalue reflect control estimating rate apply relative norm norm different incoherence space come care error space separately eigenvector correct dominate dimensional setting see study shrinkage estimator reason recommend practice subsection assume factor achievable perform eigenvalue risk management finance volatility risk portfolio allocation covariance underlie portfolio although curse dimensionality basic bound risk early similar exposure portfolio mathematically make error large mostly drive regardless converge portfolio make management model limit exposure position application nd st rd nd rd st nd eigenvector rd short match theoretical asymptotic effectiveness I eigenvalue much generate model simulated multivariate row standard multivariate matter shrinkage necessary maintain comparable even decrease however increase order result spike serve benchmark compare get b right degree indicate accuracy scatter plot true basically meaning signal perform benchmark proof asymptotic weighted wishart variance classical hold bound weight omit treat lemma column assumption independent note row normalize factor therefore independent easily spike pn contribute limiting eigenvalue b bn sub therefore involve I idea technical invariance prove normality q sub identity eigenvalue eigenvector j b n c leave multiply equation employ define derive right show element complete proof r mn b r distribute unit orthogonal remain r j easy since r c p conclude r distribute distribute derivation prove norm need random sphere pc e n pn pc could notice inequality therefore claim prove hence indeed say proof lemma second fact pc n pn pc bound asymptotically n suffice vector element characteristic expansion easily hand side mean clearly pn element must lie element empirical eigenvector p establish list e u mt tc u jt introduce mix let coefficient independence u error lemma sequence pa adaptive estimator apply get p assumption together fact pc pc lemma claim pt tp one last rate max q simplify schwarz ii theorem ready build prove write nc eigenvector section imply right iii ii p pp already thus term pc q denominator bound risk rate come follow eigenvalue corollary assumption support nsf grant dms dms grant gm gm wang eigenvectors unify spike dimensionality play principal analysis device low set new insight reveal bias eigenvalue covariance shrinkage principal estimation risk discovery proportion statistic study principal component factor relative management widely reduction visualization eigenvalue regime substantial amount effort understand empirical eigen structure early establish normality sample substantial eigen recent behavior relate development weak asymptotic bound grow factor grow dimensionality consistently question asymptotic eigenvalue question arise convergence empirical eigenvalue dimensional eigenvalue around theoretical counterpart covariance quantity role determine asymptotic eigen eigenvalue theoretical pca three angle contribute whereas serve low component pursuit incoherence rate sparse assume eigenvalue correspondingly reduce possibility accumulation effort spike size covariance almost scale lie scale threshold regular investigate part eigenvector corresponds normally scale random regime study principal literature regime allow spike size lead lead perspective regime bound eigenvalue bound ratio offset assume signal factor
image represent category instance image training generate category label mahalanobis mahalanobis entire dataset select fold training run use different initialization select log approach perform cholesky frobenius log covariance treat element space triangular cholesky bregman affine distance make k highly distance riemannian geodesic cluster log frobenius learn original cholesky decomposition compare logarithm domain riemannian metric geodesic distance show geodesic learn mahalanobis face supervised categorization clearly learn geodesic well c cholesky decomposition frobenius gain gain frobenius reference x n riemannian riemannian manifold f descriptor detection dictionary v log calculus diffusion tensor learn label face recognition environment analyze contour object categorization similarity search divergence wang invariant tensor value segmentation covariance pt pt considerable propose compare matrix affine invariant induce riemannian focus riemannian geometry propose drive approach riemannian metric distance learn face denote product pt pt frobenius triangular cholesky exp logarithm vision involve underlie metric feature shape rotation matrix etc lie riemannian need develop inference technique structure manifold considerable vision diffusion tensor medical imaging diffusion tensor water tensor characterize optical motion often employ encode descriptor texture track recognition correlate feature represent cross sample filter feature histogram covariance rotation invariance different various measure literature comparison log log euclidean reason metric euclidean riemannian euclidean riemannian ns equip metric space geodesic distance remarkable community learn log riemannian metric correspond geodesic distance distance like near neighbor distance explore idea metric geodesic mahalanobis brief overview literature technique euclidean riemannian log geodesic distance experimental conclude section distance statistical consideration linearity riemannian property frobenius one information similarity dissimilarity constraint let point point point aim learn mahalanobis parametrize similar pair give threshold mahalanobi denote index component similarity dissimilarity capture control bregman publicly result n ns right geodesic distance map identity equal usual space denote vector form operation inner uniquely characterize inner product give define unique product simply extend lie multiplication euclidean geodesic give directly say mahalanobis vector geodesic correspond riemannian uniquely mahalanobis hence riemannian metric geodesic datum mahalanobis geodesic mahalanobis learn cm mahalanobis mahalanobis section propose riemannian application face label face ii dataset experiment pair image correspond person dataset design face dataset face development pair pair test image pair image person consist image pair pair randomly development subset final image subset convert coordinate represent standard experimental protocol development set pair perform split train split parameter face matching learn compare performance
flow q round need polynomial state introduction equilibrium game function canonical game thick color text blue right bend bend bend node edge bend bend right let exclude equilibrium consider equilibrium hard player edge experience half player total edge plus see observe matter player path edge plus edge plus go whereas go every total exactly thus since eq lagrangian play define game zero sum approximate equilibrium follow fix maximization game good response game write equality approximately hence find minimax equilibrium lagrangian pair strategy action p equilibrium concave know argument use round regret select particular good response round average approximate equilibrium equilibrium lagrangian game induced gradient observation well game lagrangian description algorithm present descent close convex gradient regret dynamic lagrangian gradient bound q let average approximate equilibrium plug approximate base guarantee plug recover call therefore observation q plug constant extend version theorem theorem edu play play payoff game capture choose maximize goal price production know utility price bundle efficiently computation complexity utility reveal preference access maximization natural choose utility observe bundle would like price road unknown determine minimize round quite share important feature choose function understand unknown maximize minimized concave unfortunately pose pose maximize resp minimize concave resp maximization instance non concave price price unit unfortunately concave generic high dimension efficiently maximize also convexity generally play utility objective minimize traditionally game assume know utility utility several natural equilibrium efficiently preference function clarity detail maximization reveal preference optimally class unknown apply general include mention version technical reveal preference simply game rd problem main challenge class many write price setting face price bundle induce concave continue arbitrary cost meaningful well family thus bandit maximize unfortunately get bundle bundle set bundle price reduce simple maximize price suffice give access bundle set next ingredient efficiently find approximately function strongly feasible specifically bundle price query precede induce strongly convex induce target interest subproblem procedure flow detailed optimize follow strongly action objective concave objective action finally third demonstrate tolerance simple production stochastic procedure similar variable solution variable survey substantially certain game np ignore computational efficiency assume knowledge utility game learn query pure continuous result problem learn security learn optimal strategy polynomial game pure strategy algorithm np neither despite give polynomial maximization recent pricing reveal special quite extend game also work find atomic function ellipsoid form degree induce target game game increase ellipsoid subroutine approximately induce induce flow exactly strictly comparable motivation recent line design game direct denote key ingredient ability minimize concave function use descent radius projection onto subgradient descent start algorithm descent q alternatively within essential every strongly extremely let concave xx useful noisy say df ff noisy optimize use specified membership concave maximize reveal problem want bundle assume allow price good bundle price maximize price bundle maximize utility period choose price induce bundle would design nearly mild convex closed contain unit least empty bundle lastly cost differentiable decrease important technical assumption satisfied class concave bundle concave price bundle iteratively price price price allow simulate query use query feedback iteratively quickly bundle carry demonstrate function bundle bundle might induce set induce bundle observe vector price characterize feasible vector bundle maximize vx since maximizer thus concave know ascent direction contradict desire function close vx cx bundle concave problem meaningful pose rx vx pose reveal preference without specify bundle sensible often formalize homogeneous homogeneous unit scale preferred bundle concavity differentiable rx vx prove claim invoke euler euler homogeneous continuous homogeneous conclude interest strongly differentiable concavity follow induce bundle price vector bundle actually x p concavity actual bundle close quantitative lipschitz norm vx cx x vx vx bundle accuracy initialize restrict update descent bundle analyze define whose bundle solution vx constraint price price induce bundle subgradient restrict price unchanged even primal program primal compact let strategy function guarantee later restrict play minimax strategy state sum minimax equilibrium observe fix response write eq equality choice pair minimax induce reduce equilibrium lagrangian game pair equilibrium induce satisfie definition fix concave function play descent good dynamic define action player average play equilibrium simulate dynamic observe induced gradient lagrangian compute recall lagrangian sum mean end average description restrict update action regret lagrangian average form equilibrium plug induce action ready utility write must play allow utility p quite often achieve reveal preference operate interior trivially satisfied whenever induce game application evaluation function value induce action approximation guarantee find optimizer iteration present initialize td fx tx px observation q approximate satisfie end guarantee plug guarantee require total value bind constant introduction find induce approximately flow atomic unknown graph represent specify interested agent infinitely agent aggregate decision induce flow ff flow equilibrium game lemma state whenever decrease associated equilibrium compute whenever call social equilibrium power edge induce flow approximately game player flow player l tool problem begin assumption function match induce flow implement cost flow induce flow potential flow condition guarantee potential variable imply assumption implement flow require sufficient guarantee vector flow fix game satisfy flow need without last solve induce use form additive manner edge ellipsoid achieve rate induce processing step maximal polytope span transform body round transform need behavior exactly run maximization contract produce produce quality stochastically map work effort agent dimensional problem agent dimension effort agent know stochastically map abstract away effort loss generality contribution strongly produce contribution contract however stochastically principal want optimize response realize agent optimize minus agent contract attempt contribute utility utility ap cx expect principal contribution realize price utility agent utility merely version version adapt assumption feasible contribution hypercube attempt contribute unit dimension nothing lastly agent learn approximately observe hold generality principal lagrangian dual induce contribution satisfie optimize contribution perturbation response subgradient price principal observe realize contribution unbiased subgradient gradient realize contract price descent satisfie target price realize agent theorem w consider induced satisfie agent contribution zero agent form approximate minimax equilibrium induce contribution simulate regret run gradient realize price take player dynamic recall form descent contribution need
multiplication multiplication require multiplicative reach theoretical complexity transform layer layer cope column combine order addition multiplication finally one multiplication besides two multiplication post multiplication minimum multiplication multiplication another finite combination depend field transform order second pre addition layer cope respectively combine column layer pre multiplication q multiplication multiplication multiplication multiplication column pre fast algorithm fast split example dft multiplicative regard roughly hadamard decomposition short low popular ft attractive implement high dedicated acknowledgment support proposition tag cr tag email de mail de finite transform code interesting field inverse promise transform concern digital access system spread transform existence transform ft transform operation short decomposition hadamard decomposition discrete hadamard transform multiplicative dft implement scheme processor high speed dedicated hardware compute transform gaussian comment minimal multiplicative ft additive implementation plus introduce pair assume computed observe combine hadamard reduce first column make q therefore procedure addition q multiplication let
correlate several numerical approximation still expectation propagation unstable specific reason guarantee need principle convex set objective multivariate truncate integration truncation inequality provide absolutely continuous respect entropy vb disadvantage vb general come disadvantage include base convenient subspace vb bind original integral idea able truncate denote negative energy univariate normal truncate zero variational q bind could iteratively solve round find parameter c vb ep vs ep vs property integration integration dimension consider ground pseudo main validity various truncate provide integral vb minimize bfgs matlab precision draw varied vary compare accuracy moment moment compute euclidean ep give value integral consistently seem interesting vb correlation unable use correlation family compose truncate moment give result also become respect vb ep correlation handle generalization old inequality binary recover method minimize multiple variational approximation minimization great practical advantage vb practitioner integration could express special unique approximation maintain heavy tail far behave posterior optimize variety speedup vb ep decade focus type integral common problem glm prior potentially discrete design dedicated graphical upper know old conjecture express approach span symbol measurable scalar assume inequality expand right pair old exponent result prove equation lead symmetry proof proposition q college old vb involve maximization respect minimization bind problem literature integrate ep art integration many involve integral approximation technique explore involve likelihood criterion slow require empirical boltzmann simple ml estimator function partition soon model include distribution graphical non need set design decade including performance field approximation algorithm provable guarantee guarantee hard one variational expectation ep reweighte classical scheme base basically information apply bind tend variance example interestingly inequality suffer avoid vb lack guarantee connection introduce old tractable possibly previous work focus product potential parameter tool compare ep contribution show well approximation highlight effectively bayes variational properly variational unconstrained minimization optimize find intractable computed fashion approximate amount seminal minimized exponent discrete far bind continuous make old illustrate previous define univariate lebesgue assume want common k regression sparse univariate number observation integrate remain
assumption improvement burden iterative framework report concern screen efficiency screening method projection new assumption focus computation concern design compute article lie aspect screen consistency unified method strong insight assess screening method relate sufficient sign another arbitrarily flexibility hold even ic carefully choose equivalence screen consistency comment relationship commonly illustrate study screening measure signal evaluate compare design assume ic article follow basic screening provide condition relationship condition design probability task learn coefficient impose portion coordinate split phase recover phase point dimensionality regularization raise computationally suggest find eq define hope comparable step usually involve screening exist comprehensive put definition article estimator strong eq definition much usual study weak see relaxation sign reduce screen choice article property take sure screening projection screening computationally theoretically ordinary least square estimator although sufficient consistent fix define contain large coordinate terminology help establish theory dominant symmetric restrict dominant notice dominant screening noiseless consistent dominant dominant hold prove sign notice necessity material noiseless good point intuitively preserve coefficient need dominant diagonal dominate combination theorem need change accommodate certain estimator screen tailor term addition current tight condition necessary consistency rule example satisfy sure sis condition ic standardize zero letting give ic represent matrix ic verify screening matrix dominant explicit ic illustrate dominant satisfie ic restrict dominant demonstrate ic however requirement ic impose make violate predictor contrast ic impose flexibility equivalently ic satisfie correlate weak sis screening matrix make ic follow dominant ic screen consistency sis imply illustrate necessary guarantee lasso avoid give advantage computational screening consistency commonly screen common high strong sub row contrast next estimator satisfy row tail screen random broad distribution elliptical focus gaussian screening essential magnitude chi square states matrix sis screen sis asymptotically screen lemma indicate necessary usually condition inspire sis I sis dominant plug solve inequality notice correlation rely heavily material sketch leave material define angular central bound projection decompose choose central gaussian conditional te te desire distribution te h screening illustrate lemma diagonal satisfy exist provide supplementary material combine assume satisfie restrict imply c union matrix entry satisfied provide precise condition observation expression suggest screening evaluate closely screen question study establish necessary condition verify design relationship ic sis arbitrarily predictor see compressed technique marginal selection proof theorem section dominant similarly consistency coefficient notice screen fix screening sign j argument choice notice consistency proof prove cover complete proof row except ic without assume become sign sign either q sign check value first argument restrict dominant choose q sis proof part provide degree q eq iid proposition diagonal prove lemma dominant plug solve notice section section proposition need establish result reader reference orthogonal manifold manifold mathematically call haar orthogonal left probability suppose positive definite orientation angular right become angular unit sphere let invariant decompose distribute property entry probability essentially small eigenvalue eq diagonal belong quantity interested coordinate divide second part take care diagonal q distribute distribute begin evaluate part vector denote although still due decomposition term decompose haar point possess identity determinant easy result definition simplify angular relate quantity
leverage theoretical connection field different toy datum follow imply express derivative gaussian combination quickly acquire potentially locate acquisition new datum e tuple toy region within ten correct approximate evidence region improve full quickly perform intractable simulation several magnitude toy optimization two major difficulty free discrepancy parameter small former give discrepancy region science generating likelihood form inference infer yield datum difficulty choice discrepancy difficulty discrepancy tackle difficulty play role statistic joint see interest realization stage sample implicitly define via datum compute computationally well possibility simulate likelihood indirect economic bayesian process aforementione highlighted measurement simulate discriminate similar generate parameter solve free inference identify chance sample normal curve simulate mean green density discriminant lda yield green dash curve example curve drop curve
relu respectively layer layer c acc acc layers acc acc ar c present show representation learn measure component ar show high compare layer ar contract layer due reason activation hide compute set ar acc acc lc acc acc lc lc extend select image category ar select change select per training rest six setting unit change maintain unit system improve utilize abstraction table classification accuracy improve deep good image construct stacking module module low layer representation spatial pyramid classification experimental public database partially support national science distinguish chinese education cb fundamental research central program team sparse code classification however computationally expensive though signal discriminative dictionary sparse simplified module avoid inference module module module stack stacking evaluate four database extend ar outperform reach practical learn code promising digit sparse code sparse sparse expensive researcher use signal learn expensive algorithm train dictionary avoid expensive fortunately simplify module train dictionary fast linearly map layer sigmoid function map ability output label infer calculate multiplication stack module far stack stack network increase speech classification retrieval additionally batch offer potential problem amount despite classification limitation conventional sigmoid nonlinear layer literature slow solution suffer sigmoid recent relu train play key role image noise high representation lead promise image classification evidence reasonable representation module technique generally zero unit unit dependency unit unit observe module connection dependency divide group capture local dependency among hidden dependency module exploit stack image stacking module relu regularization hide modular advantage code dictionary lead compare extract retain ar experiment get particular scene originally yu deep stack layer module perceptron mathematically describe follow I ji ci ji di connect layer connect hide form eq square derive gradient module element one convex basic module stack deep low feature module replace hide representation bilinear stacking paradigm module expand module modular different sigmoid relu sparse penalty add unit upper hidden follow activation relu activation unit group g objective upper hide unit impose sparse norm enforce sparsity representation neural advantageous representation dependency dependency divide within force hidden unit group norm regularization conduct modular architecture matrix belong solve layer fix gradient square objective wise division relu activation non optimization unit tb parameter epoch initialize random repeat fast find deterministic plug square e simplify derive gradient eq define process outline tb describe layer architecture module output label decompose three module square generate module module iterate construct summarize implement hide unit activation module advantage parallelism four database extend database scene database face original image normalize pixel ar database database color people person take image variation include illumination standard
b conversely prefer commonly kronecker kernel anti use symmetric learn preference successful theoretical learning eigenvalue eigenfunction obtain learn depend introduction several recent therein determination another tool analysis universal intuitively enforce experimental previous thus far rigorous theoretical enforce property literature anti pairwise kernel kronecker universal result approximate arbitrarily anti function anti expressive concern provide guarantee anti pairwise symmetric anti regularization symmetric anti symmetric set feature literature conversely write type mapping simplify consideration couple input bound function generate write equivalence unique known kernel kx kk reproduce mapping often define rkh hand change norm composition diagram hilbert schmidt indicate require consideration operator suppose compact adjoint operator consist notation advantage eigen basis negative eigenfunction eigenfunction yield self adjoint eigen system adjoint next concept cone monotonically hilbert infinite space doubly doubly denote banach operator operation operator hilbert hilbert hilbert iff exist doubly kernel write space kernel type inner product joint pair define pairwise immediate forward permutation permutation invariant invariant equal permutation give projection define know kernel anti symmetric kernel symmetric pairwise q analogously anti anti projection connection symmetric anti permutation moreover anti form projection invariant obtain definition express sum anti form invariant anti integral operator arbitrary anti integral permutation q projection anti look consider divide preference projection eigenvalue function zero anti symmetric risk consideration eq anti error counterpart hypothesis anti whether restriction aim anti discrepancy literature split error part cause bias cause draw input caused briefly consider follow subsection discuss roughly kernel value eq show eigenvalue operator prove know infinite length non segment end dimension inequality straightforwardly draw affect effective limited error guarantee enough approximate may universal regression anti restrict design equivalence omit due lack formalize concept definition space close universal rkhs universal approximate property rkh universal rkhs real accordingly kernel arbitrarily rkh armed definition next characterize anti kernel arbitrary q anti symmetric determined section theorem generalization anti kronecker anti analyze pairwise rank correspond interpret anti second thus formalize rkh approximate see completeness somewhat square eq k proof regularization symmetric anti symmetric knowledge regression regularization may type symmetric decrease bias bias v anti cause kernel moreover cause operator anti observe eq integral anti symmetric integral analogously mapping stack operator check kernel reproduce kernel pi k p together rv rv pairs function belong rkhs prove claim anti kernel start form operator bias product function inequality due regularization anti
ascent dual separate last analyze analysis minimize nonsmooth obtain free asynchronous parallel sg describe sublinear asynchronous successfully server deep randomly training sample index k star shape master serve master exchange master simultaneously basically step select master compute master master basically aggregate master source collect predefine master perform atomic network especially server asynchronous sg serial parallel sg update compute instead serial sg asynchronous parallel overhead delay value gradient vanish asymptotically section parallel implementation parameter master example value denote delay evaluate th th asynchronous sg shape summarize short read mean inconsistent read worth star structure asynchronous cyclic delayed architecture delay architecture independence delay worker might important pointed asynchronous implementation old intuitively age idea assume commonly asynchronous note roughly worker q ergodic convergence iterate take optimization nonsmooth ergodic totally think roughly theorem properly assume ergodic corollary basically stochastic gradient worker speedup serial sg optimization sg nonconvex observation long roughly compare serial sg speedup achievable compare analysis consistent result upper worker ensure speedup consider widely asynchronous cut completion always involve machine platform sharing randomly select index randomly platform exactly software asynchronous implementation sg preferred basically consider implementation share worker read modify simultaneously read read share compute training stochastic share worker share cause inconsistent read read share memory share memory eq close look properly hold iteration exceed consistent result argue numerator count factor comparison rate essentially read big difference result sg consider analysis consider inconsistent assume impractical consistent read absolutely speedup achievable maximal worker speedup long worker result strictly gradient k convergence sparsity reader validate property computer following validate speedup interested speedup speedup speedup exactly speedup count level achieve hardware running time speedup speedup affect hardware generally bad speedup deep evaluate package convolution max find website network mnist dataset cifar initialize server worker server worker handle gradient worker core default tune serial sg use chosen default well draw report table observe iteration speedup sense problem table speedup cifar drop bandwidth parameter full require communication threshold dramatically htp l speedup speedup image minibatch conv fc cifar full x rgb intel core datum totally total number gb generate core mini batch choose choose base sg draw speedup report speedup computer delay usually htp machine leave c c speedup study popular asynchronous system sublinear prove consistent result achievable improve early share proof proof assumption expectation q unbiased stochastic next equality inequality next use equality last inequality side substitute full optimization complete globally apply theorem equality complete theorem q expectation side due take take upper come complete lower bind far relax show satisfie bind acquire substitute thm counter thm counter thm counter counter com asynchronous implementation stochastic broadly neural practice explain speedup mainly asynchronous mechanism gap provide support implementation sg memory establish prove speedup achievable generalize asynchronous learn asynchronous parallelism largely overhead parallelism asynchronous parallelism worker synchronization asynchronous parallelism speedup many art stochastic coordinate ascent randomize asynchronous parallel optimization mainly research effort speedup people exist explain excellent practice due deep asynchronous mechanism people nonconvex use widely share memory system fill paper try first nonconvex smooth necessarily specification consider asynchronous originally memory architecture diversity key computer naturally efficiently read share system unable usually ensure read write coordinate implementation sg gradient consistent inconsistent asynchronous share memory platform theoretical establish rate size minibatch achievable number speedup property highlight theoretical many knowledge first offer support early particularly maximal worker ensure speedup accurate free strictly dominate apply scenario solution th natural take paper make item
title empty empty format empty title title ed ed format emphasize emphasize volume empty ed ed swap connect series emphasize volume number check empty mid sentence connect empty function multi page global format page page page multi page page connect page format journal empty number empty format page page page empty chapter chapter connect page format ed emphasize format emphasize format format thesis empty technical format article format name et format format volume nan author series format format format name format format ff jj format format name format name author author empty function empty key author key organization key key organization label empty organization author label empty key name empty author label manual author organization type label author organization organization full label swap label output write function output format check format output check title title check format format format page output note format annotation format format either skip year format check format new format new sentence format function format author format output year check format title new block entry author format author author check skip year format title format format chapter page check new format series format chapter page chapter page format book note format format author check author format title title check format ed output format format chapter page sentence format output format annotation author author format output year block format title miss ed format format series format page output organization organization format output page annotation conference manual organization organization organization format author format new format title output organization block address organization organization check block entry format format check format title new master thesis thesis school school check block entry format annotation write format format key year title format block entry format annotation author author format format title ph thesis format thesis school school address entry annotation organization organization format format output check format title format output output format annotation check author block format title title format tr check block entry format format year check block title title output format annotation default macro macro macro macro macro sep macro macro macro survey macro macro intelligence macro communication macro journal macro journal macro transaction engineering macro transaction computer macro transaction computer integrate circuit macro letter macro journal macro journal computer macro science macro journal macro transaction computer macro transaction system macro graphic macro software macro transaction office read integer sort format name name skip jj format name sort format title word author empty sort need sort author sort author empty sort sort name sort author key sort author key author sort format organization sort empty sort organization key format sort manual organization sort sort sort label title format max sort iterate string label extra extra last extra label year extra year nan max last label reverse extra skip year label sort secondly condition computationally infeasible methodology jump employ strategy accelerate rejection condition jump light develop enable framework establish adaptive simulating condition extend simulate condition framework represent mean simulate dimensional principle comprise definition principle outline skeleton diffusion approximation mean computation determine path constrain path process partition path simulate conditional proposal rv skeleton simulate impose coefficient sufficiently regular ensure existence unique continuity drift coefficient sufficient allow volatility time interval univariate transform let apply jump f v induce transform point measure induce condition jump diffusion constrain end jump diffusion volatility compound jump coefficient jump distribute compound poisson exist compact set order tx tx density condition form paper directly boundedness bound set suppose lx outline simulate diffusion transform represent sde simulate rao less section direct extension term serve consider bridge path algorithm sampler operate diffusion introduce simulate measure brownian sample proceeding sampling draw accept exact rhs diffusion path infinite unbiased construct entire path simulate alg inf skeleton compose else return principle simulate skeleton require path employ construct simulate disjoint layer proposal belong aid precisely ax approach make computationally efficient strategy reject conditional reject acceptance simulate additional event alternate graph critical acceptance formally implement rx exact algorithm ii letting reject return simulate skeleton per accept return computational link graph naturally want choose graph alg perform simulate exponential set order interval refine upper accelerate rejection essence find conduct remainder simulation suit simulate condition long infeasible path efficient simulate extent interval mid mid simulate equal acceptance three associate consider evaluation next computation conditional accept accelerate rejection begin evaluate computationally respect sub new sample path tighter coincide iterate notation comprise evaluate acceptance estimate comprise regard time interval bound note h simulate l layer information return return satisfy principle augmentation illustrative accept path two path trajectory skeleton sample trajectory skeleton methodology represent sde denote construct algorithm develop upon skeleton collection principle employ rejection computational rejection employ condition jump proposal measure key contribution alternate construct point compound simulated ensure bridge condition component consider superposition compound sample path start end induce sde measure proceed follow exact simply draw accept x I consider acceptance simulate dimensional evaluate without simulate sample leave skeleton construct denote law acceptance decompose acceptance accept rejection strategy acceptance compound jump simulate leave form jump path brownian methodology recall compute require acceptance condition jump exact acceptance let simulate finite suggest omit incorporate idea illustrative accept skeleton simulate compound poisson per x reject reject return simulate l layer information l reject return set skeleton x acknowledgment mp thank work k author base intractable new application along interest lie methodology smc email ac uk ty empty ty empty e option without option argument option without option def def def def def def def def def def def sp large rest j cause page inconsistent pt height ne em sp plain plain pt height stream stream stream stream stream start file percent
z lattice g lattice obtained reconstruct self energy three estimate solely polynomial learn fig typical weak different use example worst predict large study rigorous learning material detail median example different show size around predictive interesting choose scenario lattice function chemical ml reconstruct lattice g use result shift prediction problematic ml prediction function ml good job database fig ard ard global finally analyse totally database homogeneous database choose actual previous overfitte influence approach well predict database choose full loose predictive machine really use body physic show predict solution function output function apply change solver cluster theory self way ml material accuracy largely enough problem important might cost account approach adapt department er numerous discussion implement j department university national solve equation dynamical dimensional technical issue map distinguish validity machine full particle indicate development attractive computational efficient option prediction system body contain entity decade difficulty long monte generic exponentially property law approximate phenomenon investigate leverage exist solution generic quantum physics essence use matter physics context molecular density functional context weakly formation energy material physics particle quantum ml energy scalar equilibrium body problem solve many issue arise application include infer quantum body relate build involve function formalism capable solve question relation position exchange correlation approximate interact quantum quantum body physics local green consistency band material question parametrize self consistency member test machine neural forests accuracy problem critical transition outperform decide kernel ridge detail follow determine process detail explain later text first implement database condition span range possibility database site strength range density ed discuss implement descriptor output datum naturally formulate lp green energy section material function parameter denote scalar interaction strength chemical ml concerned database energy different
heterogeneity represent effect represent estimation class dyadic longitudinal dyadic illustrate package analysis keyword dyadic latent factor mcmc social individual measure dyadic relational particularly variable dyadic population individual node direction quantification person international trade log country country ir include package specifically analyze trade rgb ir ir ir na na na dyadic exhibit dependency row heterogeneity mean heterogeneity popularity evaluate heterogeneity around overall additive row heterogeneity mean normal heterogeneity equal normal r response df sum pr heterogeneity row comparison estimate column na usa close square maximum straightforward implement classical fundamental characteristic dyadic relation refer node additive effect evaluate correlate evaluate popular additionally pair node outcome correlation effect dyadic model volume highly volume country analyze node describe variability row follow conditional row specific heterogeneity heterogeneity column summarize describe equivalently row mean variability beyond capture effect covariance element covariance evaluation tool progress display via sequence plot store vc intercept include default intercept store style fit beta vb complete estimate give goodness summary deviation row deviation within dependence histogram represent histogram generally speak histogram model lack datum respect discrepancy surprising dependency variance often dyadic dyadic model vector characteristic receiver dyadic covariate characteristic refer sometimes student success popularity like fit dyadic log number membership ir ir ir ir dyadic store array dyadic covariate value psd col col variance psd vb divide mean posterior deviation standard normal exceed calculation appear association population share positively country assume I residual package model column dyadic false fit psd row row col col col variance psd vb deviation almost fit explanation precision via exhibit dyadic regard fail dependence plot similarity individual associate relationship suppose node indicator organization indicator co organization may anti measure dyadic covariate multiplication see dyadic cluster people prefer form tie lot triple triple link one explanation link occur must would tie node multiple link triangle visual network person relate person characteristic set covariate product element dyadic length regression account describe type network pattern equivalence group way related equivalence estimate multiplicative trade share dyadic fit effect rgb predictive statistic include show raise exist attribute multiplicative case characteristic unobserve factor characteristic describe mean depend extent vector magnitude type network dyadic essentially dyadic aspect model package option letter stand factor fit model provide adequate dependence statistic estimate share latent psd col col shared variance psd way factor blue magnitude indicate trade regression additive row effect example identify trade volume dyadic outcome case trade datum transform binary transformation binary friend friend neutral discrete event amount people phone pair population accommodate dyadic follow ordinal meaningful include discrete outcome binary indicator count order outcome medium high ordinal ordinal probit simplest ordinal dyadic variable binary whether dyadic indicate interaction include social tie member dyadic several display office location status office age practice fit without explanatory describe observe latent specify model probit contain simple goodness compare fail fit fail term common description positively estimate large illustrate fitting age age coefficient psd age age age col col result positive effect age older dominate dyadic effect dominate summary intercept intercept identifiable parameter estimate intercept term part transformation nuisance human social fix people friend national health ask school student five member friend five friend scheme ordinal friend ordinal also censor complicated ask five friend five people person people five person censor absence modeling develop dyadic treat outcome continuous letting coding indicate rank rank positive correspond relation people could consider person censor person rank implement fit base use study ask rank variety rgb na specify rgb psd intercept psd goodness plot heterogeneity simulate satisfy impose amount dyadic design ask participant friend dyadic censor way datum survey person less approach analyze censor option dyadic outcome ordinal case treat heterogeneity tie across rank outcome ordinal dyadic impose unobserved option dyadic happen cost pair node partially dyadic datum distinguished pair dyadic missing mcmc iteratively simulate along value value way approximate speak miss procedure specifically study popular dyadic node randomly sample ask tie tie friend participant friend friend friend illustrate analysis college describe record record code ai already design share na na na na na na na na na na na na na na na na na na na na na na na na na na na na na na na na na na datum bin fit ess beta intercept col ess intercept row col estimate similar second output fitting predict imputation dyadic dataset example relation obtain dataset na goodness respectively indicate reasonable obtain variability sample probit illustrate histogram illustrative exercise decrease sample concentration dyadic dyadic base accommodate dyadic model allow dyadic point word seem dyadic allow possibility certain parameter longitudinal relation college student person graph seven period surprisingly graph include student program member effect model indicator model include indicator product dyadic regressor dyadic program include dyadic possibility function array dimension dyadic array array datum analysis dim n dim dim using previously summarize fit bin psd intercept col psd vb indicate correlation status program evidence bit note whereas consider effect regressor might vary depend lag measurement possibility term regressor vary interval create dyadic covariate binary measurement regressor dim dim w vb ar vb psd intercept row col w col col
distribution concern three unknown benchmark useful procedure known analyzing rare detecting phenotype large rw although mh abc obtain locate finally proper allow implement assume sequel type explain analyze aim mutation occur dna site mutation mutation summary generate independent mean mean n ns exp w ns marginal poisson employ report result pilot run top calculate jacobian bottom simulate top leave jacobian grid pilot bottom credible posterior mean choose summary statistic site may every course statistic abc method approximate jacobian nearly relation calculate relative figure respect rather discrepancy prior posterior quantile parametric abc variance sample deviation normal n parameter linear illustrate residual output former posterior center contour posterior vertical line constant report scale variance move variance differ quantile stochastic representation available hence difficult focus stochastic z skewness abc quantiles range transformation describe pilot show var report residual variance abc rw mcmc four density posterior posterior chain assess dataset expect abc error mse automatic abc fp abc compatible observable summary one application dna phenotype dna single nucleotide observe million fast need question snps mainly disease usually collect open open nonetheless certain genetic snps disease come I human collect genetic world method develop genetic population compose family snp snp white affected red snp configuration phenotype snp observed phenotype level usual independent instance fisher test association snp former individual affect snp status treat highly genetic variant transmission relate phenotype configuration snps constitute relate logistic also volume disease genetic inside determine individual later observe similar disease snps level inside separately overall analyse snps logistic log odd snp transmission snp configuration transmission assume usual law segregation individual configuration summary odd affect among individual configuration specifically number occurrence individual numerator denominator constitute limitation perform pilot point grid pilot depend observe phenotype effort pilot see hypothesis genetic illustrate logarithm respect algorithm conditional conditional simulate snp term make distance tend match configuration obtain acceptance rw mcmc figure chain rw b dot factor along snp exhibit large posterior posterior snp skew center around also reflect logarithm bayes snp risk snp analysis snp rs bayes factor first precise signal summary seem vary grid suitable abc proposal rw fashion rw constant definition quasi likelihood proposal analogously scalar summary moreover happen part happen mutation properly discuss another lie argument perform quite notion rw mh focus constant assumption find lead discuss proposal reflect costly likelihood use abc di di computation abc bayesian practice basic may inefficient presence discrepancy elaborate monte abc difficult automatic proposal likelihood model value statistic sampling pilot establish conditional construct extended variance value many application biology involve computationally rapidly literature lead set leading later become area sample index prior aim n observable g quantile etc author suggest say accept assumption moreover sufficient agreement improve abc drawback inefficient easy issue monte carlo method abc analyze lee monte carlo smc attempt require analyst major literature concern suggestion current propose posterior mean pilot specific composite focus mcmc method building proposal proposal model account adopt approach function indirect distribution arise tractable context kernel transformation scalar typically pilot run regardless sample thus serve routine analysis appeal application genome fact end available asymptotically target follow throughout propose abc formally discuss illustrate propose conclusion remark conclusion profile receive wang summary whose observe assume convenience summary lie real line suitable specific replace pilot run state sequel abc purpose input density convergence rp smooth monotone differentiable spline possible convenience analyst diagram goodness estimate depend resource achieve make wide increasing curse large observe gain precision monotonicity abc relation g monotonicity automatically recognize proposal essentially simulate fix quantile include approximate jacobian reduce interpolation spline mcmc jacobian estimation effort desire proposal regular taylor around mode monotone possibly conditional covariance
section want exploit sake present respect argument suitable change expansion denote decrease non eigenfunction moreover pc admit j j decay exponentially tend decay decay super decay exponentially hyper hilbert pca basis carry orthonormal basis provide pca present decay variance devote define functional result subsection illustrate deal discriminant define I maximization approach identify parameter mixture highlight carry information orient tool detect latent proposal group surrogate large surface assign group consistently proximity look pc assign mean nn modal q empirical version discuss estimate projection span eigenfunction semi attain case b times bound integrable support follow thus justification univariate smooth scale whose sense specify concern spurious select mode jj use estimate play coefficient identification mode graphic visualization system software prototype estimate pc belong upper datum proposition avoid curse dimensionality estimate non explain large practice solution external criterion index depend combination accordingly validation criterion choice lead criterion find reference therein index pre extent class datum cluster proportion proportion calculate cluster clearly range obtain choose trace estimate matrix select discriminant differently presence establish model aim new incoming group structure typical one assign class correspond equivalently known simplifie follow argument apply straightforwardly setting consider classification assign eventually tend hard thank whenever pc group asymptotic parameter possibly different dimension simplify mixture decay start represent straight min operator order eigenvalue least eigenvalue much concentrate concentrate decay exponentially simplify similarly surrogate th group tend equivalently one density parametric introduce interesting parallelism conditional density assume mixture theoretically finite subspace slowly ensure projective discrimination full probability context gd eigenfunction occur balanced concern span eigenfunction pool discussion coefficient bandwidth particular behave converge tend subsection dedicate control cluster dataset dedicate simulation quantitative comparison goodness detect measure misclassification error exercise cluster noise point detect keep mind simulation exercise expansion element control mean shape coefficient beta scale coordinate spherical g limited semi unitary whose center choose un easily identifiable concern choose avoid noise direction pc replicate set proposition equally exponentially correspond great suggest generate sake depict plot algorithm htb middle right set mode space monte return misclassification mean mixture first pc code summary misclassification error quantile gm combine bic misclassification whenever configuration correctly recognize misclassification equal gm whenever bic cluster ccc st km dataset aim bring phenomenon domain analysis consumption estimate composition light decade widely explore kind cluster near nm grid water chemical original avoid calibration presence shift since represent way chemical composition chemical available chemical correlation linear particular water equal content protein positive water chemical composition first explain kernel pc figure reduce look local minima whose present three modal well pca concentrate first pc explain variability use reach internal external criterion computing couple accord summarize couple possibility reproduce distribution chemical measure curve heat centralized location entire system scheduling generate line demand flow mainly demand load aspect weather forecasting load demand application cluster heat consumption west centre produce generation previously regression consumption year privacy figure display behaviour demand intra daily due demand aggregate behaviour differently demand difference appear dataset way period daily load discretized mesh figure display procedure perform functional spectrum pc sufficient limit provide contribution exploit mean curve plus suitable see weather highlight difference demand heat less counter pose three systematically day cluster algorithm use choice reflect daily level demand peak moderate peak effect element cluster represent load daily modal curve plot grey moreover box plot daily label multi modal external cluster exercise pattern mid forecasting performance predict activity condition central acquire essential contribution activity procedure neuron sort detect thought correspond single neuron experiment reach target virtual detailed description neural activity channel versus discretized analysis also perform spectrum explain variability observe appear good build index lead admissible correspond cluster combine produce maximal level procedure curves briefly simulate consist misclassification fold validation evaluate estimate remain glm basis functional nonparametric discrimination classic see computation setting training set balanced translation variability around two small medium high cc cccc ccc glm nn discrimination exercise pc obtain misclassification summary deviation error comparable due spherical data glm result performance dataset belong domain dataset come website curve discretization relate berkeley growth data fit discretized datum aim discriminate curve base gender detail
impact output specifically address three type consist likelihood estimate characterize resort expectation attain iterate experiment reconstruct impulse compare base scheme account initial regression long history tool reduce mean regressor compare square novel method identification get estimate impulse class recently kernel spline estimation decay kernel identification see instance spline two estimate effective rely bayes argument exploit interpretation regularization impulse response model estimate marginal output impulse among impulse compute situation preferable record five reason standard variance mse g time time ignore rest experiment discard depend preferable initial condition context first incorporate unknown assume autoregressive average stationary estimate initial minimum initial marginal exploit problem iterative method technique method blind identification close involve grid search organize follow review estimation relate system algorithm discuss error figure output impulse convenience delay capture dynamic corrupt zero mean interested impulse q contain problem instance discard collect however considerable rest improve present interpretation impulse hyperparameter structure determine velocity impulse response introduce follow write follow posterior estimator sense hyperparameter quantity need datum consist compute hyperparameter quantity estimate computing variance residual identification approach ml integrate hyperparameter effectiveness bayes approach serve new aim initial straightforward quantity impulse maximization condition nonconvex possibly dimensional impulse response devise maximization method iterate suppose calculate impulse well impulse response introduce impulse toeplitz relation toeplitz impulse response definition iterative solve hyperparameter guess initial convergence sequence global marginal impulse limit use information available miss rational spectrum realization white noise namely probability initial write joint probabilistic available size hyperparameter amount solve problem unknown hyperparameter estimator guess update estimate impulse exploit miss less joint statistical propose mixed map highlight act put agree case find consider start guess initial hyperparameter incorporate information covariance general conditional case estimator conversely set obtain conditional yield degenerate iteration rely monte number carlo system radius plane impulse filter unit order filter filter carlo noise noiseless hold correspond priori avoid initial discard conditional estimator present base hyperparameter score impulse response test ccccc kb ic zeros kb kb ic kb kb kb ic oracle percent impulse response information discard kb kb zero suffer effect wrong system estimate impulse response record performance kb ic mean estimator kb ic perform oracle initial
measure code ab days total ab million ab constraint support constraint ab patient contain medical prior ab ab instance lift great non cause square ignore medical record ab record ab association risk ab reaction drug record time month never month association rule generate rule patient minimum confidence constraint patient contain medical record one association b association lift cause square value risk adjust failure severe occur month month rarely record instance refine adjusted show refine pair pair evaluation may show refinement able read unlikely able refinement c failure due read code rule may generate rarely record read code limitation c probably reason ab record prescribed commonly database medical ab consequence interestingly rule identify cause lead support refine consider work short medical history prior require item newly unlikely item record medical medical consider read prescribe thing read record frequently age patient year birth record rule set support minimum confidence tune base common rare whereas increase rule tend suggest poor apply rule rare common small contain lift account restrict rule three mining help refine improve adjusted filter signal occur still amount concept refinement record history instance cause drug efficiently refine signal require tuning confidence suggestion efficiently implement distribute enable contain investigate age signal cause go thank side prescribe common occurrence effect automatically leave likely essential refined correctly majority correspond paper filter patient medical require parameter patient aim improve health unfortunately majority induce drug clinical view positive often researcher identifying generate drug refinement severe death thorough generation database present additionally occur decrease take conclusion recently refinement thin database uk million record thin birth gender decide record novel medical remain use method develop subset patient thin million patient thin event via structure read code medical specificity medical diagnosis laboratory read consist alphabet dot level read define level medical parent read direct parent parent child read medical parent read code correspond child correspond code cause drug calculate implement thin issue still main issue move home within thin database medical new incorrect patient medical event find record date start prevent newly patient drug exclude identify drug record month thin database exclude identify drug prevent reporting mining frequently transaction read frequently occurrence event occur association rule similar manner patient medical record thin medical rule rule outcome pair correspond adjusted identify formally hypothesis method refinement find database prescribe two summary work base exposure occurrence ignore adjust remove patient period drug determine outcome gender refinement patient day patient patient date record date date drug ab measure record day measure quick numerous require present ccccc code ab ratio ab death unknown secondary rule thin contain version read code consequence read gender record date occur patient association record tell chi lift identify insight occurrence patient compose maximum lift maximum chi item patient record indicate value cause consider cause rule lift great lift occur patient population lift calculate base number instance lift maximum chi square calculated rule
svm compute hence binary vote vote opposite vote majority total vote weight win weight give vote stochastic call moment draw margin al tight account light vote inspire name whose state multiclass weighted majority design complex output classification generalize multiclass label section recall output give pac risk majority vote margin define margin let accord et function chebyshev inequality z counterpart justified thank elegant pac simple generalize important pac stand multiclass space recall look majority vote multiclass q vote realize version multiclass margin notion multiclass let margin vote regard strength multiclass draw vote classification lastly call example output margin majority sign come vote binary majority make mistake margin multiclass leave side hand verify vote necessarily correct weight mention margin differ multiclass margin definition decision multiclass strength true class combination versus base multiclass multiclass multiclass q binary multiclass vote equation multiclass margin relation vote term margin minimize much every inequality sum drawback minimize finally multiclass result able al multiclass margin label majority number output among label otherwise label majority label low square euclidean cumulative margin margin bind label vote classifier developing depend derivation minimize generalize margin multi distribution margin coordinate second calculation hyperplane
mistake row mistake efficient mistake dimension leave question mistake precision bound dimension mistake motivate instead constraint sample polytope objective unknown constraint change mistake ellipsoid coefficient change objective implement separate hyperplane mistake reveal preference relate multi problem observe show continuous finite give polynomial pac reveal preference connection learn prediction efficient compression efficient complexity meaningful reveal mistake price maximize also learn strongly leverage et adversarial objective know change optimizer constraint change similar distinct preference optimization vary think say problem trying predict lp partially choose coefficient learner goal predict learn mistake never use mistake program polytope take adaptively output first study give polytope learner polytope change change give finite mistake learner problem study polytope change refer study receive lp partial known polytope know observe define mistake partial example total sequence learn bound mistake bind mb put definition learner coefficient denote coefficient region change way polytope know loss generality polytope boundedness name uniqueness probably remove name write finite precision polytope tell encoding encode typical constraint program define need finite differently converse precision polytope without bound vertex polytope write precision necessary uniform mistake next mild polytope rank degeneracy organize present learn bind assumption necessary precision specify adversary force learn mistake like round rather finite precision part still move avoid complexity arbitrary observe polytope know day avoid inspire et ellipsoid mistake result ellipsoid assume represent day ellipsoid terminate coefficient objective denote denote make feasible write imply define polytope coordinate lie region write infinitely polytope informally arbitrary solution rate precision specify solution feasible polytope fact program vertex write vertex polytope satisfy adversary constrain run copy ellipsoid feasibility constraint define maintain ellipsoid candidate mistake constraint solve nonempty true lie ellipsoid polynomial mistake ellipsoid volume ellipsoid use n whenever mistake hyperplane ellipsoid current separate hyperplane ellipsoid tw formal formalize leave solve simplify lp prediction vertex remark lp solver exact vertex polytope mistake make ellipsoid find bind ellipsoid intersection constraint bit access solution precision separate ellipsoid empty number ready mistake lemma instead observe mean region choose objective value solution contradict hence rule round e ti predict outside along feasible prove initially mistake round adversary pick bold point matter learner predict return different guess learner adversary pick process learn formalize high procedure adversary matter adversary ensure interaction adversary output polytope actions adversary algorithm precision polytope day nr tr tr ta mistake mid middle top small procedure take learner mistake produce choose adversary infeasible optimal learner adversary pick point computed constraint uniquely feasible region round r r qr polytope polytope subroutine make every learner mistake round polytope present round return adversary lp subject feasible remain show new always bind check intersection polytope newly hyperplane high second equation accord hyperplane polytope intersect adversary polytope intersect hyperplane intersect unless happen never hyperplane proof modify furthermore modify newly add effect hence fall hull round polytope linear constraints eq denote hull randomize unknown problem mistake dimension constraint mistake polytope unknown algorithm randomize algorithm completeness open problem mistake achieve hypothesis form multiple generality denote consistent day polytope day consistent update consistent optimization wrong mention I change optimize algorithm maintain instance see polytope set round solve name select real round round repeat number randomization might adversary mistake round mistake round eliminate round product round mistake mistake rearrange first describe new input two line return otherwise empty j j j j subroutine identify vertex underlie polytope infeasible coordinate tv e exactly contain vertex exist separate disjoint interval note therefore subroutine eliminate polytope contain several claim interior interior take line must intersect denote place gap interior combination interior ready theorem know interior otherwise claim segment polytope three generality write belong hyperplane
pp valid replace theorem furthermore constant orthonormal satisfy satisfied sup proof theorem pp complete assume space notation preliminary dimension localize eq exist let localize constant straightforward spirit exist take basis sup n ks ks p ks l q dd dimensional explicitly k notice deduce find sup respect vector norm hence admit eq axiom theorem claim example theorem exercise proof investigate selection regard regression endow haar optimality calibration procedure call penalization slope recent penalization perform existence behavior slope heuristic thus method successfully wide applicability slope justification indeed study framework theoretically optimality calibration show validity validate heuristic framework selection histogram extend slope density slope heuristic histogram density previous optimality heuristic general model endow orthonormal basis sup element number intersection element assumption analytical bound risk treat context haar expansion noise model ideal penalty resample candidate mild penalization describe framework slope heuristic validate hold penalization proof upon conditionally assume variance independent sample follow dimension paper introduce detail p x pf pf square eq regression consider possibly empirical image estimator excess excess least give loss excess quantity depend penalization dependent aim analytic model selection provide exist constant note localize basis orthonormal intersection support orthonormal orthonormal localize state let deduce satisfie give moreover straightforwardly satisfied finite interval large notice assumption localize proof totally trivial argument theory orthogonal polynomial basis notation convenient existence strongly localize exist partition orthonormal p dimension constant I noise assumption give relation specify quantity strongly localize latter ensure uniformity along constant define localize collection model polynomially complexity want choose estimate concentration deviation uniformly collection put extra inside depend reasonably depend suffice low upper ensure assumption property term assumption especially inequality supremum empirical used inequality matter include extension unbounde state derive context need concern heuristic piecewise lead slope heuristic penalization def decrease power hold n penalty twice estimator satisfy oracle select dimension remove remainder ensure assumption regular partition theorem opt pp identify empirical excess generalizing endow localize check unknown level slope heuristic issue slope shape prove calibration linear optimal procedure situation remain ideal thank slope heuristic devote penalization ideal index propose ks
limitation especially perform multiple interact allow challenge mostly focus memory doubly link dimensional memory head head move nearby current head move move list fix position b recurrent sgd propagation pattern controller learn hard sgd task supplementary material seem robust well operator introduce partially rnn stack stack round sequence generate goal learn rule understand scope pattern model stack list rnns backpropagation prevent learn use baseline rnn unit hyper baseline validation short long sequence unsupervise sequence pattern evaluate rule produce sequence training epoch predict sequence correctly bold recurrent mechanism stack list c c action stack stack b b clarity stack stack first stack empty interact deterministic bold pattern count table report sequence rnn stack rnn either list operation use table rnn unable able generalize sequence count unit parameter round require obtain stack show discretization rnn stack element input second stack start empty stack keep track stack unit rnn use random repeat multiple stack rnn rnn lstm seem generalize unstable stack frequently explain choose versus stack discretize stack read sequence addition supervise token addition ask reverse order train choose equal less digit stack average rnns generalizing run bar previous example addition moderately rnn stack keep track sequence e read read write stack interestingly capture stack store stack stack result finally stack care carry state explicitly say cache lstm stack validation test stack corpus recurrent stack rnn lstm corpus capture similar bag stack well stack decay bag memory efficiently learn rnn rnn rnn algorithmic attempt problem motivate algorithmic involve discrete algorithmic stack rnn input output format flexible allow loop access pattern algorithm possess automatically hard recognition solve recurrent memory stack operate complex currently memory fix learn would like rest facebook team comment facebook ai research york deep approach limitation complexity simple recurrent model capacity show sequential recurrent memory perform various task major source recent world research explore neural successful task lead vision recognition commonly attribute hierarchical recurrent theoretical instead represent learn current art approach past well linear one layer method describe demonstrate layer guarantee deep architecture layer non currently deep pattern model interestingly deep capability recurrent net allow memory structure stack matrix net multiplicative mechanism learnable memory simple read stack among work aware neural research done study sequence generator nc nc short building predictive mostly discrete pattern memory precisely sequence denote symbol algorithmic algorithmic example simplicity focus unary binary represent design free short length controller clear external memory module recurrent stack stack prior problem supervision problem sequential token character design symbol stream recurrent network rnn rnn layer recurrent delay recurrent sequence token rnn encode token predict probability base token follow sigmoid activation coordinate token recurrent weight hide network token number token architecture learn pattern one capture gram rnn
pose variation part face part probe method enable promise probe pose variation face image subject outperform degree pose encounter practical access scenario subject section review alignment probe combine method face model mixture handle intensive propose illustration part constrain sub ii transform sub appearance iv pose expression probe learn locally wise neutral expression neutral optical mrf local patch pose pose give probe face image localization alignment pixel accuracy wise correspondence pose exist one across pose expression rely relevant pose variation model detection pose objective score detector star shape constrain base method pixel image rather appearance evidence similarity align constrain part tree constraint complex property beneficial also cope illumination subject method follow collection surface piece expression change furthermore prevent face canonical template template part vary arrange define template template database region canonical alignment align template alignment realize transformation transformation yu yu md affine group similarity satisfie little simultaneously compose assume subject could term corruption intra leverage extend base give encourage mmd part dictionary whose align image denote reconstruction unfortunately unstable often initialization flat face indeed face contain structure overcome incorporate structure individual motivated shape constrain transformation different term determine structured shape parameter tuple tree consideration structure alignment weight address shown propose dictionary face produce correspondence pose face intra subject variation match fit disk center white line link display vertex set edge compose transformation root use face canonical consider simplified order associate node edge parent child use distribution dp I thus gaussian assume independence tree structure network joint logarithm joint regularization structured term localization ignore associated pair equal constrain shape model also enable later probabilistic prevent degenerate jointly tree integrate strongly supervise joint block degree tree form structured shape globally directly shape constrain call keep tree shape main difficulty convexity constraint domain alternate fix update efficient apply note relatively learn effective issue moment gauss taylor e iy linearization lead follow optimize e repeatedly linearly converge show convex lagrange multipli alm augment lagrange multipli denote frobenius matrix alm search saddle I alm directly alternate manner turn form let q operator sequentially eq auxiliary convenience basis equation form td ig summation hence large part simultaneously describe expand converge x inexact alm update group alm alm inexact alm summary solve linearize solve alm optimize technique section alm linearize paper propose without perform aforementioned specifically iteration keep unchanged indicate denote fix propose jointly efficiently solve outer section initialization face detector bound location bound available structured shape initialization act template probe detector fortunately alignment bad time handle face subject image face align form part face similar face subject perform subject wise optimize variable alignment residual subject sort subject wise residual subject small residual dictionary together part align transform transform subject part recognition perform aggregate decision basic adapt aggregate scheme nonetheless basic choice illustrative module note detail inconsistent different combine substitute high part small line adjust average transform subject dictionary align I prune subject sort ns j recognition exist recognize iy label section associate present face compose part align form dictionary propose part dictionary relational different part shape dictionary part couple present algorithmic first align part constraint serve alternate dictionary face stack face part ni part base contain part correspond appearance region ideally due inter illumination word rank aforementioned model frame sequence face leverage align directly apply alignment alignment often converge meaningful solution show similar apply probe instead constrain part simplify notation part dictionary surrogate function reciprocal root row denote solve alternate strategy face illumination conduct illumination expression outer corner center inner corner r corner corner r remark conventional region window outer corner pixel dictionary design whose neutral consist part part availability parameter model evaluate face pose illumination alternative face face face face demonstrate robustness state face section initialize corner assume pose expression probe conduct automatic face pose face detector across illumination use multiple probe face label corner manual face recognition manual automatically probe alignment recognition term alternative similar recognition comparison consist subject third probe face use pruning base face strategy define accordance name pruning face pose fig face neutral large publicly contain subject span accord illumination illumination appear structured model subject subject specification pose neutral angle face image neutral viewpoint otherwise mention use section across pose different subject probe illumination neutral lr lr lr align c manual lr lr manual report recognition alternative degree compare manual improve automatic alignment consequently face pose even tell baseline probe viewpoint neutral give high knowledge recognition across mostly due appearance period illumination ideally vary nevertheless piece alignment face recognition extent overcome report view image expression alternative recognition pose outperform practical face illumination people near neutral like minor strength subject lr lr align cb cd ce manual consistent perform degree alignment weak strong final individual investigate discriminative exist representative face show efficacy pruning dataset pose illumination part align individually report align individual part recognition within fairly recognition individual efficacy part part alignment recognition align realize face ns discriminate lda local recognition pruning also stage align experimental previously ns shot face subject also shoot illumination summarize illumination section probe di lr lr align di di si report table tell ns propose part alignment superior face alignment confirm effectively individual pruning investigate algorithm simply prune scheme bottom table list recognition rate together difference original pose drop prune remove impact pruning non pose time recognition error recognition opposite precede standard correct occur opposite pruning correct introduce rr pruning pruning report partial probe image position varied experiment setting recognition fig portion face drop rate pose pose good require face subject equip different feature fairly lda method experiment thank run model advance probe face illumination largely except face probe subject subject multi pruning neutral pose automatic manual initialization know pose report case semi use pose pose perform degree similarity individual training face image shape necessary knowledge able cope face pose encountered practical control reliable pose detector coarse face pose pose probe face either advance al pose illumination normalize illumination table report recognition row automatic change probe work coarse fine sequentially alignment part alignment experimental coarse fine strategy work well reasonable pose drop consequently failure tell equally pose probe either automatically confirm propose face probe illumination cc ccccc initial pose manual auto auto auto detector follow alignment pose pose recognition alignment appearance part structured shape part probe constraint formulate regularize optimization experiment efficacy handle illumination pose change integrate illumination change research adapt computer thank produce report table equation array parent similarity seek minimum keeping update horizontal translation similarity u c f unconstraine equivalence find eq link holds link q solution solve step gauss update constraint nj k linearization problem repeatedly solve converge solve adapt alm lagrange q lagrange multiplier alm I ii directly instead single variable notation use inexact scheme alm update alm n specifically present appendix might manual fairly improve reduce alternate pdf wishart wishart conjugate distribution different give estimation wishart nh distribution additional weight algorithm practical face system vision face alignment key achieve face piece wise surface surface develop structure shape probe face appearance consideration tree shape integrate classifier part recognition face par robust face across illumination develop system past decade control variation cause illumination pose illumination illumination use multiple carefully choose image vary probe face subject illumination face illumination generate face image
predefine dictionary wavelet central adaptive dictionary directions mod dictionary representation two often entire central increase feasible practical database entire access compressive data compressive version certain inference problem within compressed study line spectral compressive compressive domain performance towards compressive compressive general random propose computation onto bernoulli apply scale compressive tracking setting improve share several attempt compressive roughly three algorithm compressive measurement inspire aim compressive minor take overall none aim compressive maximally large scale moreover none work give efficient extend compressive scheme efficient key wide general compressive dictionary projection review similar signal dictionary member minimize coefficient matrix pseudo nonzero unit intractable solution via measurement product column drop compressed attempt solve follow learn minimize code strict optimization distinct measurement variety omp find atom hold penalty compressive tc minimizer ki preserve performance svd attention special guarantee closeness generate increase close increase gaussian intuition random dense mp generate sparse see get fix increase increase theoretical analysis give insight number tradeoff also accuracy distinct accuracy random reduce matrix divide pl control nonzero average interested cost collect datum representation coefficient penalty term code except omp omp dictionary penalty compressive quadratic first block sum square coefficient relate atom block l l represent li k k synthetic plot successful vs svd compression observe computation examine algorithm propose method implementation svd entire dictionary atom draw normalize atom gaussian corrupted draw compressive measurement factor evaluate magnitude inner successful recovery fig trial practice may indistinguishable see reach tradeoff memory vs factor reach eventually level grow access dominant support science foundation science foundation award university university university center shannon electrical engineering university
immediately iteration converge embed reliable worse suggest large dataset use projection projection run time error drop monotonically transform monotonicity decrease important mention figure need randomize embed dimension must conditioning yield ccccc e ccc projection dimension method fix completeness run set trial ccc embed method rest method quantity relative objective quantity versus independent perform median report machine considerable electrical scientific research originally use randomization resource algorithm several remarkable computation ram importance sketch solution subproblem sketch construct review highlight modification problem scale parallel increasingly importantly though scalability come communications improvement chebyshev query advantage avoid profile nontrivial expensive projection strong perform scientific looking would helpful like acknowledge research office advanced project energy provide large hardware storage massive storage store currently datum later create daily implementation simple processing method algebra traditional environment load disk easily dominate greatly increase develop implement randomized scale environment randomize problem deal implementation projection algorithm randomization relate solve exactly pass empirical highlight importance various quality versus etc medium exist sized data lie heart many signal processing compute convenient structure arise broad range value object use adjacency graph represent region band similarly dna nucleotide microarray represent snp condition individual internet automate set record task environment typical machine scan hard disk make cost application low solution paper overview recent numerical environment apply resource regression presentation rectangular sized least robust rectangular hold formulate result ram parallel environment relatively straightforward manner aspect algorithm aspect parallel environment random projection approximate digits precision medium digit precision user return principle develop high interested medium precision traditional subproblem understand principle relate principle important high development number subject freedom sensor number sensor word gram document stock extend restrict attention rectangular arise learn apply list health million snps genome subject disease determination target rectangular one collect internet spatial discretization partial differential freedom grow exponentially increase reach discretization cubic time dependent stay depend discretization spatial especially number sensor wireless hour back nlp gram grow geometrically document frequency trading stock well example daily file size great gb restrict rectangular develop method traditionally extension approximation svd qr decomposition connection qr decomposition rectangular similarly programming special problem class development environment thing term scientific researcher versus database researcher difference achieve parallelism share memory pass alternatively massive datum describe framework computation require want evolve evolve evolve interested reader quick parallel scale go tends share core core core memory memory core parallel linear generally provide computation analyze highlight design algebra survey algebra focus develop algorithm iterative expect interesting development idea distribute traditional performing distribute basic idea special case regression ram general traditional rounding embed method review implement sized provide discussion general interested reader addition review overview couple overview central theory ram important principle extend scale environment describe l two precision randomization conclude approach give follow ls interest meta take matrix estimate exact corresponding element sample row subproblem subproblem return meta term draw zero indicate trial equal choose meaning algorithm construct solve sample problem quality uniform subsample simple implement easy perform tune leverage score easily probability error mass crucial meta ls generalize rectangular leverage two type understand randomization designing practice ram well environment l problem think low precision inverse span diagonal element row alternatively express qr thin row coherence measure well vector basis well fit compute leverage key need sampling na manner leverage lead informally euclidean matrix axis provide key must leverage think small nonzero singular value norm compute solve precision ill speed iterative quickly informally aspect quadratic iterate precision precision key quantity condition guarantee provide meta solution incorrect relate randomized provide type vary application preferable discuss notion score generalize long generalization important environment meta time former depend flexibility provide score exact approximate I qr thin obtain na ram original ls problem practical implementation meta run dependence roughly scale prohibitive moderately value three meta fast ram meta run hadamard perform basically random quickly compute statistical leverage algorithm run approximate distribution use construct extremely input sparsity input order term implementation ram run ram construct iterative hold coarse obtain precision practically iterative construct high approximation l subproblem iteration could draw extra iteration could still failure convenient application might answer undesirable monte scientific moderate solution algorithm might expect run bad meta represent qualitative bad ls go elimination meta margin bad asymptotic matrix several remarkable ram open large environment principle want want parallel environment meta algorithm principle importantly situation must algorithmic principle must section review paper section traditional solver list letter letter scalar g etc use frobenius norm norm letter span except g solution e give problem important special square l regression absolute error former solution optimal particularly make theory algorithm formulation homogeneous elsewhere unconstrained problem scale l rank minimizer length minimizer unique min length computing define define regression problem linear system number rank let large underlying notion characterize simplicity conditioning let minimum condition matrix always notion definition factor factor matter formulation matter condition establish nm application call instance solve algorithm iterative number problem take form system consistent unique ls min min right follow hold certainly respectively leave problem ls regression arbitrarily well condition qr decomposition optimal solution solve well reduce ask orthogonal qr exist consequence round matrix full column lemma provide subspace preserve sampling build square classic problem linear algebra detailed survey certainly beyond know direct reader min svd singular singular calculate aa unique factorization factorization complete orthogonal factorization usually compute qr make triangular qr determining solve least correctness replace svd factorization either normal expensive mention especially chapter van sparse square problem column storage method refer iterative minimizer solution cg min preferable cg arithmetic numerically chebyshev semi l problem rate affect condition state z l thus cg like remains estimate iterative lack regression program programming formulate solve convex solver come example easy due therefore solver gradient interior cutting solve regression discuss solver reader smooth specify use simplex solver subgradient feasible initial search speaking conditioning regression make step solve square problem choose smoothed divide zero theory certain assumption rate hard relate low distortion technical regression emphasis environment full lemma condition qr linear particular section practical matrix speed pass condition quality algorithm family round roughly speed ram one practical level consideration consideration round data matrix property geometric property point preserve building projection linear problem dependent time storage storage input matrix construction embed embed important typical implementation algorithm round round subspace vector preserve subproblem round complementary reason latter introduce overview lp low solver precision solver lp lp round round low round datum aware datum aware right align center align center node align center fast leverage round start round n dimensional respect find round convex graphic connect round round round ellipsoid volume ellipsoid contain lead hardness ellipsoid round round call see polynomial special convex hull algorithmic work round call oracle er er pass er focused ellipsoid rounding method slight fast call find slightly round separation rounding describe ellipsoid origin separation oracle subgradient initial immediately improve algorithm conditioning take computing immediately present give subspace aa optimal important observe embed scan portion guarantee observe take norm summarize well discuss distortion subspace qr distortion use see table detail run run depend embed method time ccc mn score sampling within algorithm approximate leverage score accuracy embed subspace embed different embed introduce result embed embed name run ct l transform exist trade distortion linear embedding provide distortion embed dimension obtain application deal first distortion distortion cauchy ct nc scale least construct sum half cauchy gaussian random variable large always well dense propose construction sample combination fail independently diagonal comprise ti assume power integer informally effect spread entry comprise cauchy small finally quality fast cauchy transform construction constant ct lead dense distortion dense somewhat bad distortion matrix matrice ms choose standard vector independently cauchy theoretical subspace transform describe summary embedding several aware preserve embed previous subsection algorithmic advantage embed even embed hard embedding probability random whether aware embedding could conditioning yes algorithm preserve sampling leverage matrix l result give desire several completeness regard subspace preserve obvious compute leverage involve form normally undesirable application score done perform pseudo score specific leverage embedding idea point estimation satisfie sampling accord constant say require gain theory suggest preserve give ms ms n ms n n use asymptotically quality application qr base summary subspace aware embedding aware embedding idea aware distortion subspace regression subspace embed depend reciprocal ms coordinate orthonormal nice basis subspace preserve dimensional nr n na eq least choice lead typically condition row compute exactly central multiply idea theorem preserve sampling mn ms eq subspace preserve ms constant exist compute condition norm norm speed sampling affect theoretical formulation complexity still worth norm explicitly obtain small trade implement preserve table round subspace describe subsection problem tool introduce subsection solve subproblem construct ellipsoid round able relative regression interested precision medium solution principle condition basis elaborate solve summary several representative ram section distribute environment example embed subproblem svd stable high qr pc pc normal equation iterative pc c reference subproblem low er order pc er accelerate descent subspace preserve sized obtain step construct preserve embed box fix mean matter distortion reasoning indeed solution include state distortion optimal sized approximate deal meta many require subproblem subspace therein simply aware subproblem use algorithm run projection alternatively hadamard projection solve subproblem reference therein aware hadamard combine input sparsity asymptotic practical ram parallel distribute particular still matter refined implementation preserve use original original somewhat detail approach invoke system run moderately embed low precision high solver solver completeness solver depend first apply solver system condition small constant iteratively nm author randomize use chebyshev detail solver use various approach run conditioning quality trade size embed trade computing solver work nesterov employ combination round method technique generally also solver medium several implementation environment regression appropriate environment among need result precision subsection implement environment comprehensive completeness describe implementation illustrate several implement computational subsection implementation solver design design high random provably chebyshev cs iterative step prefer formal system size qr svd length aspect transform cs couple nontrivial way start among choice conditioning depend certain system q parameter algorithm condition probability slow transform several large environment transform fast environment easy implement partition along big lastly properly random nontrivial dominant na projection generate number use cpu understand preferable communication cost account precisely fail slowly control expect cs inner require conjugate cs iteration multiplication need synchronization strong conditioning expensive advantage environment consider single beta frame beta alpha describe detail low precision precision solver scale environment implement standard massive regression step well importance base norm sample subproblem problem subproblem key thing note job extract basis three ellipsoid round er qr low qr qr er summary conditioning conditioning implement cauchy transform ct dimension implement manner consist denote associate collect qr complete subproblem several approximate solution original exploit pass say marginally expensive pass almost effort provide node cluster query take second took come almost free basic solution desire return approximation original ip collect row aside increase evaluate use several ct etc solve precision summarize although interior cut plane need pass dimension resource computation tradeoff medium precision pass subgradient large environment design distribute computation similar except high region ball solution describe first ellipsoid construct multiple precision solution pass query iteration query point per multiple use convexity query subgradient serve separation contain return perform iteration solve determine embed crucial part aware subproblem iterative low medium datum choose challenging matrix stress variant describe range uniformity condition four dataset uniform uniform bad leverage score condition score matrix score generate list matrix b random control leverage exactly bad good single determine ram first replicate na stack alternatively ng manner call stacking stacking summarize yield stack solution possibility
kronecker htb normalize magnitude background number show compare calibrate incoherent previous capability gain pass single rmse normalize ratio background outperform htb htb paper propose rejection multiple detect stationary clutter signal rank factor clutter result clutter detection gain corrupt training analysis experimentally confirm multi change detection gain achieve rank element via let right positive semidefinite hermitian positive semidefinite hermitian iterating complete hermitian iterate hermitian hermitian matrix project estimate clutter form filter hence improve temporal channel clutter covariance since h h temporal lr thus prove apply proposition target image stationary target well move due use processing spatio clutter covariance stationary clutter enhance target note clutter naturally low clutter kronecker provide corruption due move target theoretical property experiment challenge advantage exist track move scene activity cause synthetic particularly task surveillance area regardless work move object move move perform frequency frequency detect shift significant include low small imaging view detect limitation move lose resolution use multiple moving cause stationary especially grow integration detect clutter either scenario velocity filter clutter otherwise target search massive velocity acceleration often complexity compressive approach intensive design channel potential benefit potential move configuration include spatially separate multiple pass combination create collect fact pass long delay issue exist move target background track center detect target applicable scenario detect phase clutter threshold amplitude well dim clutter parametric channel generalize advanced develop adaptive use spatio training across channel clutter classical low large guarantee due dimensionality spatio rich freedom exceed severe overfitte coherent receive delay design angle across etc clutter paper due computational efficiency excellent clutter regularization problem none exploit spatio contribution exploit spatio structure significantly corrupt return th th bin adjacent freedom greatly exceed bin covariance particularly bin standard show introduce reduce note clutter portion clutter significantly training reduce significantly involve addition structural via subspace corruption addition none spatio temporal kronecker kronecker product two temporal rank pass covariance include recommendation rich covariance kronecker l method many prove significant kronecker spatial clutter calibration model hence kronecker l observe theoretical project spatial clutter effectively project thereby clutter filter approach improve robustness allow remain clutter covariance factor sum different rank product base filter algorithm temporal clutter highly result demonstrate complexity extension organize discuss extension kronecker pass move change detection give return decomposition noise return clutter scalar distribute return bin isotropic nature phase target filter clutter preserve specific motion target clutter ideal noiseless return return single consideration ideally locate space clutter linearly spatial clutter turn depend characteristic clutter interest exactly rank significant principal time return target bin j platform shift target lie outside clutter long time move stationary clutter correspondingly b f clutter filter make orthogonal project clutter exist orthogonality spatio clutter projection basis subspace algorithm kronecker additional available moving lie call spatial kronecker clutter datum note temporal relative entire move lie outside temporal clutter mse project filter small outside primary allow kronecker reduce kronecker enjoy benefit arise clutter covariance motivate thresholde problematic kronecker kronecker combine specifically kronecker svd b I separate hence necessarily structure covariance addition subspace training include create signal additive superior clutter rejection pca energy clutter product clutter share target surveillance interest determine change reference later appearance platform generally reference image change detect change scene move mask change scene advantageous target thus arise clutter subsequent change detection spatial form history involve calibration clutter subspace project away pass pass filter output suppose clutter lie spatial clutter lr component temporal assumption lr spatial asymptotic regime loss lr lr typical become kronecker naive spatial equivalent except kronecker trivial temporal reduce via spatial become stage project clutter achievable case clutter gain turn analog apply temporal first instead stage occur spatial specifically show loss give spatial fix target integration follow singular giving move target ideal sample regime clutter temporal small spatial still use release dataset circular move formation divide integration truth currently dataset complicated real resort roc gain experiment reference target target section plus bin kronecker covariance synthetic clutter learn spatio spatial lr mean training filter clutter slow mse go added testing clutter instead value potential potential range contain clutter move target filter response vector statistic
polytope percentage cover decision attribute capable discrete provide vector categorization output requirement formalize definition dimensional mmd attribute partially great space polytope identify coordinate class counting instance value constraint datum decision datum structure tree kind convert space besides classifier train range pass minimum range small find consistency sake range shape svm shape assign rectangle convert define axis family impose intuitively along rule figure space use attribute strictly pattern actual bind consider pattern specify attribute pattern attribute range denote find attribute denote follow exactly coordinate specify attribute mm xx xx xx rule convert element one rule conjunction pattern element space pattern low op include space op lower k low bind upper mind mind upper pattern bind exclude space space element fundamental merge span value refer space merge use principle merge principle subspace intersect prediction vector explain least algorithm merge notation way element intersect attribute cover space cover element word element cover property establish decision component intersection element denote first principle handle third resolve two create assign specify cover weighted average space intermediate introduction contribute mx space normalize indicate specialized base small metric pure meta learning use repeat overhead metric distribution attribute attribute formulae specific require differently ensure formulae merge merge algorithmic merge add contain xx xx xx new associate value value resolve create initially share add decision never handle else update previous handle solve space element return intersection cover space cover definition remainder remove remainder merge element merge principle principle partially avoid already hash map cache hash map update intersection intersection remainder add figure range range range age grey pattern intersect convert intersect contiguous age range degree range percentage percentage attribute show decision contain rectangular element rectangular rectangular remainder element use algebra element computational reason merge operator merge consistent another depend identity space merge neither theorem merge satisfie algebraic merge geometric empty intersection union intersection resolution algebraic therefore combine appropriately operator consider could keep result change show keep result add create consider unchanged relate intersection space involve element intersection intersection place intersection track remainder updating create remainder add respect empty space first element loop execute show identity second cover divide line different two strict particularly element discard lead incorrect add element vx mx vx mx mx xx yy z hz line unique space algebraic structure operator merge get remainder value result merge remainder merge depend cover intersection thus result operation operator result depend merged scheme sequence merge specify decision merge merged representation merging specify decision merge represent tree leaf decision application represent final two merge decision merge pairwise merge describe x x introduce decision space merge bias space merge merge third describe merge decision merging impact impact space reduce decision space large merging desirable merging show homogeneous partition among unit final merging scheme time distribution example stream time may trend situation soon receive merge everything stream want model receive specifie operation place develop prevent result number furthermore power decay decay generalize overcome limitation subsection decision space binary operator combination handle computation merging introduce within discard otherwise intersection assume corollary merge concentrate merge merging space cover proportional operator path leaf merging scheme node leaf subtree account root internal scheme account account fraction number contribute fraction value root merging final path show scheme value merging x n space account scheme mx explain early space merge exponentially storage provide impact merge merging value vx formula value bias merge one merge order use value vx mx vx mx mx mm mx x x vx mx vx mx mx prove merge vx mx vx mx mx vx mx mx vx mx mx vx vx mx vx mx jj mx vx vx mx mx mx mx vx mx mx vx mx vx mx decrease new create carry information technique time could consider space vary attribute evolve broad restriction introduce operator apply algebraic merging space intersect mm space intersect element line unchanged algebraic derive property every restrict element decision space element element matter take continuous range unbounded identity space space create line space matter merge compose variety merge create create operator restrict element significant two consensus weight restrict exactly operator definition reduce intersection merge intersect attribute element lie intersection measure identity sensor classifier instance classifier merge global merging classifier create classifier examine operation classifier merge operation classifier merge property merge operator show merge behaviour apply soon trend bias bias also homogeneous achieve bias storage bias operator use restriction combination decision rely element shape intersection large attribute space shape shape become hard direction shape complexity suggest complexity type choose many scenario distinguish type uncertainty concept element predict consider dimension measure since fuzzy become fuzzy value predict class yes detect element make contribution mechanism transform measure represent transformation element value vector combine solely belong confidence generate supplement raw classifier interpret reliability type classifier classifier decision support general performance large specific combine transformation uncertainty element consider type take beyond sample area essential system framework address idea algebra datum highlight decision tag result union intersection table merging element guarantee element value uncertain element combination answer advantage lie mathematically express helpful discussion research observation classification observe environment among integrate merging furthermore classifier merge setting ad hoc framework possibly classifier merge desirable discuss main mining setting decision stationary impact decay database partition learner ensure model storage develop also operators meta entity entity record database tuple change view classifier prediction often might global resource storage entity arise illustrate international classifier classifier way group key distinction p classifier specialize restrict instance classifier design typically classifier classify say instead operation several firstly classifier differ prediction heuristic g meta train undesirable cost transmission consider case meta potentially type meta describe previously validate researcher suggest question understand classifier combination mechanism formal algebraic investigate behaviour propagate local combination support kernel rely observation heuristic merge merge
reference signal controller illustration image display feature correspond experience previous value value ahead optimize planning penalty greedy conduct trial auto display relatively prediction advantageous compactly figure controller start current trial trajectory experiment stage display case gradually number trial case framework policy task use collect transition auto find good material detail policy computing nonparametric overhead day run overall policy exploit dynamical learn fairly datum rl learn action space deep dynamical controller need crucial learn feature mapping jointly term art rl learn quickly scale learn pixel acknowledgment foundation dynamical contract number college research draw thin em efficient remain develop fully instance challenge must loop learn ingredient use deep auto learn dimensional also datum crucial long lie predictive control strategy art action scale toward fully influence many decade devise process image summarize behavior environment uncertain adaptation rely aspect scene camera robot configuration agent environment want action space natural promise toward pixel reinforcement rl principled mathematical deal prohibitive working use efficiently keep experiment dynamical internal rl dimensional use pixel possess thousand dimensional dimensional network stack auto art parsimonious high successfully google amazon facebook rl deep game purely learn slot raw employ deep architecture find representation however neither either discretization low limit applicability exploit transition internal purpose employ low feed forward dynamical closed loop practically information however exception neural data direct access principle along suggest solve rl need scale approach propose directly unlike exploit classical horizon objective minimize cost tf tu control face additional challenge trial set practically agent robot video robot fully jointly deep auto learn forward control input property compactly dimensional predict paper measurement encode measurement neuron draw none execute begin node cm count neuron fill neuron try input count miss try neuron count neuron neuron try neuron try fill neuron try output count north node dim hide output reconstruct align count align tm decoder map high observation none scale cm miss try neuron try neuron try every try neuron try neuron try every neuron try neuron try neuron fill try align high dim align leave z node control iy tu h color represent dimensional image encoder decoder deep compute activation layer e image g k number encoder ty encoder negligible compact auto dynamical encoder allow step history input feed forward nonlinear model performance control section exploit ready piece prediction encoder predict image prediction image decoder auto encoder minimize reconstruction learn auto feature transition material gradient ahead multi crucial pca exploit auto auto compute high recursively auto restrict low exploit use policy optimal sequence minimize small sequence control signal dynamic determine control sequence trajectory cost associate trajectory control determine control observe entire loop exploit predict requirement reference frame encode reference function online work exploit predictive image good turn together learn states input inherently prediction quality dynamical learn controller fashion gradually without system collection close loop divide multiple sequential follow strategy collect use record simply apply controller close converge reference imply collect include suggest imply strategy latter choose greedy exploration feedback select initialize greedy happen collect propose methodology framework image input gray angle angular velocity deal
asymptotic member exponential family distribute random define diagonal invertible coefficient column th nearest euclidean summary value correlation undirecte represent magnitude correlation I vertex event equivalent hence relate poisson summarize theorem row reduce diagonal positive variate dispersion valid differentiable everywhere except continuous component member family unknown follow q note parameter detection sequel correlation density reduce subscript plot various consistent arise purely summary use detect sequence summary pre dispersion sparse summary depict theorem diagonal pre dispersion diagonal problem unknown post dispersion assertion stop away form member parameter change summary model due rule optimal fix supremum achieve ensure leibl density asymptotically use variation suggest dispersion row pre variance post wishart delay log alarm change value test alarm post alarm approximately kullback large delay value parameter divergence delay alarm fig quite accurate similar simulate size detect htb predict ij mining discovery parametric sense among detection statistic future experiment real random partially verification technology department energy nuclear na remark claim problem base change point consider row belong density parametric correlation stop asymptotically sequential detection sequence post change distribution belong finance failure change stochastic finance stock detect interaction dynamic sequence random time experiment case coefficient successive well separate change series reflect finance stock change day week consider problem dispersion sequence paper precise datum know detection neighborhood statistic big summary parameter treat screen mine specifically discovery decision time decision stop maker subject alarm change overview problem minimize suitable metric delay subject false alarm correspond user time false alarm pre maker sr family sr optimality formulation post change parametric strong asymptotic
organized smc simulation genomic integration discuss application reason result give supplement tool develop supplement block notation rest row index indice matlab integer represent alone entire stand nb iv diag decrease value b iv case frobenius onto rank gap tail provide accuracy detail relatively analysis perfectly recover show svd fail observe supplement block fails recover high addition include constrain relaxation k k smc row well small method would unfortunately rank assumption unstable failure recovery approximately rest sequel u k obviously obtain estimate subsequently recover use know locate principal h u nu u typically serious near mis specify overcome difficulty introduce know heat htbp first move front row clearly orthogonal unclear back row remove helpful r r rt recursively finally propose h v v equation singular break algorithm construct column nearly singular large non investigate theoretical property section lower together certain class approximately choice tune discuss corollary helpful explain intuitively dominant block necessary dominate theoretical theorem significant gap properly lead accurate recovery exist constant q thresholding besides involve u singular quantify difficulty hard class rank theorem yield principled choice generally depend setting rank least give randomly column decompose replacement respectively satisfy column thresholding break number row algorithm uniformly select replacement necessarily column break satisfie amplitude row amplitude row probability mean hard case generate orthonormal column sample haar uniform break parallel orthonormal haar measure column column break examine numerical matrix setting singular exist investigate setting smoothly long sensitive compare smc finally replication supplement generate diagonal accordingly different setting haar measure specifically I significant singular q major loss spectral loss perform small get large gap adjacent work singular fast htbp singular singular demonstrate continue well set threshold affect report thresholding column thresholde similar norm frobenius norm decrease high decay fast across optimal choice thus fix htbp cc versus column decay original generate randomly choice j penalize recover solve supplement set singular propose method substantially outperform frobenius next decay decay study regression simulation randomly fix table relative loss small substantially penalize htbp ccccc frobenius lr lr smc procedure integrate genomic cancer cancer relatively heterogeneous substantially year survival cancer majority iii iv disease year heterogeneity part underlie lack successful treatment strategy motivate genomic identify molecular signature help optimize example cancer genomic sample gene highly survival include cluster expression interesting survival compare alone limit validate construct denote facilitate alone measurement imputation purpose imputation previously gene signature cancer unique gene information imputation imputation gene expression remove platform batch indexing subject indexing thresholding suggest penalize imputation yield leave observed variable smc imputation recover projection predict survival select marker marginally survival nominal lead principal component marker survival imputation integrate study substantially assess survival fitting cox integrated cox observe individual hazard integrate magnitude base substantially se size reasonably study compare combine study meta improvement statistic suggest compare smc smc c summary suggest procedure accurately lead add imputation significantly improve confirm method genomic smc correlate pc outperform conventional method completion analysis adopt smc poorly number reasonable cancer signature highly pattern gene expression signature gene imputation promise signature expect imputation sequence particularly gap dataset analyze smoothly significant gap fast theoretical smc implement major decision thresholding row row thresholding thresholding close randomly close consequently row provide recovery implementation multiplication sd possible accelerate computation space work acknowledgment associate constructive comment supplement matrix completion genomic supplement additional theorem key main consider effect generate haar measure thresholde different loss similar row thresholding interested vary set see increase accurate keep decrease converge increase collect technical first singular perturb suppose n standard divided block submatrix suppose orthonormal submatrix haar exist know ba nr proof end theory q convenience extend p equality achieve minimizer p two symmetric replacement eq know matrix clearly need prove must measure beta unit construct net v imply n finish set corollary row row besides submatrix namely linear p perturbation yx u ax u combine n stand supplement finally lb work orthonormal basis denote exactly proof due assumption supplementary z nu besides denote span span space want try actually besides perturbation inequality v derive follow characterize supplement finally separately b proof fact break show break adopt hence orthonormal eq prove know prove construct differ fix number real small b svd specify use construct svd eq small q q column know hence theorem part besides take transpose similarly eq corollary haar find happen q minimization comparison base integer denote
gibbs sampler parameter denote discard validity check assess good ss require adopt estimate replacing ls square estimator employ parameter aic bic ss ml exploit practice square access ls estimator score compute impulse compute test noiseless ht identification evident despite cause quite scenario difference impulse response ht monte compare oracle ss ml l method ss standard non ss ml moreover introduce identification subject particular gibbs exploit conditional highlighted numerical experiment affect hyperparameter theorem assumption example theorem conjecture remark se introduce identification impulse response process kernel spline information start identification framework employ markov monte provide gibbs sampler converge substantial art identification system area communication bioinformatics identification constitute dynamic system cause standard identification square performance technique system propose paper tailor measurement fashion exploit performance specific handling quantization recently identify exploit identification similarly impulse system gaussian mean system identification purpose permit flexibility identification e hyperparameter maximize marginal integrate impulse mean bayes g system admit think solution end system obtain non hyperparameter unknown noise impulse response contribution show distribution criterion sample quickly target base popularity identification organize introduce dynamic system system propose performance conclusion end eq impulse characterize unknown fed time corrupt non whereas shall consider exposition paper measurable version particular guarantee system determine formulate system impulse response independent th row entry hold truncate know marginal think follow hyperparameter posterior improper prior support variance equal case assume improper situation mild measure sufficiently reliable obtain replace estimate recall unfortunately bayesian still hyperparameter deterministic discuss section bayesian model previous system identification drop impulse computing section use carlo function close solve special namely sampler e idea stationary state markov sample conditional depend density factor becomes gamma distribute carry redundant discard density expression mean covariance matrix density position propose ht input initialization initial draw sample clearly large guarantee number discard conditional draw iteration get
switch switch matrix stochastic focus analysis rule signal uninformative epoch time step agent exponentially fast inefficient costly demonstrate achieve rule communication proof concern behavior agent switch almost log assumption imply switch agent follow invoke graph existence real number connected node leave technical switching dramatically communication future focus would turn threshold strategy proof agent recalling since equivalent number signal assumption apply view lemma tv sure neighboring agent eventually since switch almost recall write preserve limit derive surely guarantee since identifiability take convergence per exponentially fast asymptotic vanish denominator routine routine nf attempt observe private condition true sense distinguish benefit side observation rely protocol propose efficient switching bayesian regime exchange informative regime efficiently communication preserve cost verify distribute attract decade application range sensor economic scenario spread adequate truth result instance agent observation achieve belief social benefit private learn unknown exist learning focus mostly individual particular seminal observational agent perform use linear belief opinion inspire rely time protocol rule et effective switching topology assumption agent belief change drastically private agent private bayes private use average refine opinion neighbor observe signal give private criterion switch bayesian vice versa regime agent use update non agent rule due evolve time mild able switching regime efficiently provide discus remainder organize describe characterization switching subsection follow illustrate concluding remark future direction proof denote number capital letter letter vector denote transpose interact doubly assign edge agent j agent decide refer prior agent agent triple assign consistently I identically epoch initially kl kullback divergence strict induce agent false agent detect false globally identifiable paper marginal true path every content signal guarantee likelihood make uniquely identifiable aggregate guarantee end node exist direct time instant mass represent agent convergence sure asymptotic sure formalize agent learn true surely realize agent initial opinion initial observe give calculate vary update detail evolve observe realize opinion alternatively bayes signal information elaborate update incorporate belief particular likelihood neighborhood refined opinion view repeat update contiguous interval neighbor interval communication successive private verify choose protocol recover case characterize bayesian protocol switching offer informative influence agent opinion private private b threshold case binary agent
consider even behavior hoc center lead meaningful asymptotic one less bootstrap usually performance trial independence sequel via cm trial draw count seem natural avoid diagonal I particular square behind independence drawing index h c c permutation permutation uniformly permutation index use avoid pick twice train sum however bootstrap q bootstrap resample thing randomness precisely satisfy unconditional c ts trial fire hz window line conditional approximate simulating times randomness trial approximate carlo obtain thank line pick twice trial full bootstrap even observation realization unconditional conditional visual unconditional unconditional bootstrap testing quantile conditional quantile value reasonable nh happen conditional surprisingly none three may eventually gr develop computation carlo exact line figure set approximated distribution basic paradigm nevertheless widely spread aim explanation one explanation center prove mathematical center particular explain mean sufficient replace empirical bootstrap denote bootstrap conditional fit see construct impossible quantity observable nan check u couple j nn correct similar computation show directly h black ts ts ts ts u first line simulate obtain quality either simulate accordance wasserstein prove accurate statistic explain computation trial correct finally intuition work exactly figure indeed account approximation finally extra simplification permutation may surprising rewrite action permutation quantile conditional conditional distribution close c work base work phenomenon permutation full investigate purely one critical approximated monte method pass acceptance conditional usually realization value despite keep test present statistic value quantile way may permutation equal indeed test slightly version note correction five trial u version couple e firing rate hz window length firing rate hz trial f exactly equivalent f small gray represent center window detect horizontal correspond vertical plain dependence dash negative vertical separate left dependence dependence cm accept c fdr fdr discovery discovery rate leave adapt test fdr run fdr homogeneous firing rate hz window trial carlo approximation windows correspond theoretical dependence ability publish experimental old delayed point front vertical panel sensitive light six place degree hold delay ms ps six green delay ms ms turn red pointing first signal ms pass occurrence change reaction rt release movement mt record seven hz filter hz window spike neuron isolate along behavioral event signal store pc trial rs consider expect confirm pair previous consider already presented detect permutation window detect behavioral may false pair neuron unitary count sign correspond behavioral black vertical bar ps vertical bar es bar response rs describe delay count focus recorded train count count naive distribution suffer sharp trial approximation turn bootstrap namely method independence test real simultaneously record spike train message method center approximation second phenomenon combine observation classical think center apply center statistic approach first line still figure correspond thank make run gr exact algorithm count elegant really possible long monte simulation use work bootstrap value trial bootstrap contrary adequate quantity precise behavior small two right decided delay count much apply independence several individual test use treat window name well discovery classical trial test take table datum exist likely thank conclude article permutation unitary event delay free suffer count despite prescribe control fdr trial compute sensitive reasonable definition delay count recently notion still question acknowledgment access centre de universit nice partly la bs dependence graph region cm event base delay count fr fr fr fr france nice france france keyword unitary testing investigate several principle unitary permutation delay test prescribe testing simulation single fdr negative area nearby electrical occurrence potential spike neuron one record spike train dependent detect among popular gr un apply decade vast amount therein main popularity precise period degree substantial develop rough level low hundred may induce idea keep level despite define shift ms want analyze value poisson bernoulli commonly validate accept spike train thorough surrogate datum particular trial assess trial available bootstrap paradigm assumption underlie method always whose parallel practice main intensity fire poisson practice replaced take work unitary event include delay suffer detection process propose delay multiple preprocesse permutation propose delay trial finally similar method share drawback sequel two point spike train neuron couple independent observation I copy notation couple x correspond expectation neuron distribution stand event stand denote delay neuron potential spike train due temporal spike informally spike neuron delay less order several count process delay eq informally count bin contain one spike count sequence count recent delay count introduce shift define discretize necessarily point delay count point informally spike one delay delay given sequence order length two point assign assign govern interval length precisely step point segment namely homogeneous poisson intensity require parameter hz delay linear advantage exploit train new corresponding trial statistic either practitioner would choice observable compute denote several paradigm reject reject quantity critical
tree illustrate element formula publish two empirical literature highly mining piece extension use social participant hard innovation participant numerous introduction refine variance design estimator propose effect eight large variance bootstrap perfect converge slow result network tree relate string variability chain sampling index primarily motivated markov chain indexing node text reference markov mix apply sampling apply visualization store retrieval various bias sampling mechanism mining seek edge conclusion extend entire graph highly rather theory mathematical participant friend piece literature refer transition simple walk friend usually piece tree child draw subsection give notation associate people adjacency graph element friend node matrix define pp less absolute spectral extensively one social bottleneck satisfy markov simple walk term estimate many friend population estimator thompson spectral calculation lemma chapter piece let reversible orthonormal lead step write play fundamental lead eigenvector process community east west walk east west partition sign look sign partition east west make concept spectral let rooted graph cycle vertex unless fix seed node parent step close denote property chain call state initialize stationary contain index participant sample example node person individual network represent height number denote characteristic e wish estimate subscript sample thompson normalization properly thus estimator estimator transformation estimator study independent negligible second eigenvector second eigenvector social graph west potential correlation contribute bottleneck irrelevant mean constant functional select node uniformly tree graph generating function function eigenvalue one absolute define eigenvector correspond closed variance number grow eigenvalue remain unchanged function change piece two transition q summing motivate grow tree imply term dominate satisfy exceed variance outcome interest bottleneck enough var obtain design effect necessary depend otherwise design give bound interpret subsection refer exactly future refer iid select distance seed node diameter define slow correct dependence design effect converge fast converge rate subsection political blockmodel study network indicate support feature node six feature negligible portion whether population white worker drug follow use large component political contain average display political colored lead emphasize eigenvector figure individual title give create bottleneck political bottleneck lead likely error display sample provide decay especially htbp display five highly display network eigenvalue great instead figure cumulative covariate composition population display panel composition panel leave center right panel average increase political standard covariate line jump indicate variable panel correspond make line covariate five translate correspondingly computable tree previous across size effect legend drug city tree study tree line strong tree present horizontal strength potential bottleneck legend tree line sensitive illustrate practice reason practitioner obtain ensure large tree much present effect bottleneck preferred sample achieve variance realize illustrate sensitive insensitive bottleneck critical identify drive network node theorem close form construct drive combine follow eigenfunction eigenfunction tree rate rate converge give critical ignore ignore match understand balanced require illustrate analytic simulate empirical political relate political quantity subsection design synthetic empirically tree sensitivity large distribution herein disadvantage fundamental obtaining error avoid eigenvector require define reversible node v x yx yx ease subscript yx yx xt constant yx yu xt yu p yu yu yu yu yx proof completeness proof inequality loss generality variable pairwise cauchy nb eq nb procedure repeat function towards zero sequence upper bind jensen next use notice fact fact upper bind di di k bit growth term idea apart two di k di j j di k c h di k growth rate hz di z contribute q fraction proof fourth balanced distribute denote moreover single generation I drop correspond iid correspond wish random borel exist variable balanced chebyshev constant eq borel theorem w conclude convert section proposition helpful thank liu helpful course web construct index tree correspond observation indexing chain sample unit popular estimator effect network critical social eigenvalue large bottleneck finite design effect grow long converge slow introduction drastically statistical take classical population individual sampling frame available cover response become typical survey difficulty require frame interest network reach people use reach go name sample united provide researcher reach population friend context population public drive technique population e people worker conventional technique international quantify population include center control health organization united serve result herein model walk person exactly model index individual indexing
shown model bm sample gradually outperform rbm adding rand cv trivial connection add complexitie rand estimating rand increase model sample size increase cv gradually rand cv principle preserve confident real could benefit way kl divergence sample real simultaneous rand complexity second third column rand cv divergence sufficient cv rand tend select visible rbm difference rand rand fourth worse small gradually outperform balance complexity could simultaneously useful reduction problem theoretical side principle maximally preserve confident confident theoretically general work orient interpretation bm building block deep boltzmann architecture sufficient abstraction describe flow transformation illustrate maximally preserve confident achieve tradeoff preserve layer architecture maximally preserve indicate fisher confident specific adapt series density plan incorporate modify confident could respectively partial derivation partition part cc verify coordinate imply fisher diagonal er unique asymptotically fisher involve parameter recall block ball surface kullback sample fisher fisher fisher exist diagonal matrix apply decomposition become rotation tailor index tailor mixed fundamental property ellipsoid monotonicity base hyper ellipsoid ball surface indeed surface ellipsoid eigenvalue surface main eigenvalue ellipsoid parameterize term axis eigenvalue ellipsoid spherical coordinate prove diag diag ellipsoid surface integral definite prove integral partition finite let sum limitation sum multiplication arrange order monotonicity ellipsoid coordinate extend ellipsoid hold monotonicity preserve dimensional standard ellipsoid monotonic maximize preserve top eigenvalue block element bound upper operation affect integral give maximum complete proof element bound x obviously diagonal diagonal great complement bottom equal hence complete vector l h I realize prove mixed projection mixed coordinate tailor stationary learn thus denote datum since define solution gives hence preserve complete proof complete expectation respectively equality expect divergence vanish completes prove uniqueness projection bm thus unique divergence fast treat bm learn gradient choice coordinate exactly ml bm distribution manifold bm treat unit visible projection qx theorem corollary em plus em minus height width depth orient focus feature situation limitation method scale datum feature aim consider oriented dimensionality space propose call confident preserve confident less confident parameter assess fisher manifold neighbourhood boltzmann bm perspective visible bm general formalize essential bm aim bm discard theoretical sample sample study series boltzmann belief stack auto encoder deep attention application vision language despite principle search difficult introduce region parameter could capture data distribution thus aims learn meaningful representation parameter describe since block deep architecture formal essential part density bm parameter underlie respect small occur trend moreover overfitte adjust complexity selection could criterion I confident universal probabilistic model system phenomena parameter reduce parameter adopt various criterion rao formalize theoretical general dimensionality manifold smoothed free restrict major difficulty choice keep geometric preserve project distribution perturb true surface assumption without well belong e define problem maximally preserve rao distance unique close fisher distance information assign free parameter appropriate free neutral zero use section advance principle close turn maximally estimator estimation normal covariance asymptotically er rao ml exponent opposite square respectively maximally preserve projection maximally e maximally effectively class class beneficial sample among reduce noisy maximally preserve capture dominant discrimination class principle fisher decompose two orthogonal parameter contribution former distinguish true confident minor reliable preserve confident confident optimal equation parametric reduction fundamentally reduction extraction focus offer deal scale contribution intrinsic datum rise maximally preserve confident confident one binary analytically parametric boltzmann bm visible bm bm unit certain propose scheme experiment develop manifold simplex foundation ig family differentiable manifold parametric coordinate multivariate coordinate coordinate exclusive index regard nan index indicate zero coordinate respect denote ij solve subscript order parameter index convention coordinate equation meet identity coordinate define great row regularity partial derivative measure carry inverse fisher tight consider coordinate call vanish influence uncorrelated rewrite j g otherwise generally develop proposition generalization information q probability three calculate g g p j give share n manifold could target reduction reduction construct confident distinguished confident one confident confident neutral confidence parameter assess contribution distance coordinate usage infeasible coordinate since orthogonality hold show mixed coordinate bm neural visible hide stochastically depend visible interaction interaction visible interaction visible self hide connection express boltzmann joint normalization factor boltzmann realize bm actually see role coordinate bm general bm unit rbm rbm sample ml use ascent bm maximize equation likelihood phase sample positive sample stationary second call negative estimation adjust gibbs phase avoid difficulty gradient follow markov run denote cd expectation end number dimensionality endow design direction coordinate define maximally preserve fisher rao learn stationary exactly stationary ml uniquely hide preserve unit bm distribution bm bm visible leave bm activation bm due bm manifold projection framework rule learn bm theoretically algorithm reach fix iterative property iterative guarantee hold projection proposition minimum mixed coordinate unique one proof ip investigate confident neutral equation part learn bm hide exactly confident empirically density boltzmann machine adaptively bm bm modify confident connection give hide restrict rbm connection visible unit emphasis confident among analysis use confident expect denote meet incorporate adaptively follow edge confident graphical comprise assess could follow infeasible tackle connection orthogonal decompose independent coordinate respectively fisher fisher note equality hypothesis nan alternative chi investigate rand cv perform connection fold validation topology adaptive artificial jeffreys dimensional learning cross kl fit various size generate distribution focus divergence give offer trivial result qualitatively variable number reported average could n relatively sample effect gradually cv rand cv performance connection column complexity rand term theoretical insight confident cv gradually rand cv explain preserve confident sample increase benefit rand column
show category image non property play role superior sparse solve reflect sparsity loss max numerical school sciences technology china china wang mathematical technology china china technology china china ny usa mail wang com recently successful capability incorporate sparsity representation code pyramid matching transform descriptor non character experiment show improve use sparse iterative part research vision machines pyramid matching model extraction treat document keyword text match document frequency keyword apply method image processing turn image histogram capturing shape object discard pyramid match successful model correspondence discriminative codebook generative partition segment histogram promising image illustrate sift extraction firstly descriptor sift descriptor extract image codebook descriptor code layer pool average sub image task chi nonlinear complexity svm impractical pyramid achieve art image categorization year like locality code image representation code widely pool mainly representation image however max sign coefficient condition consider representation sum coding section describe propose present two basic image quantization trade balance fidelity term trivial solution unit typically normally overcomplete e consist code phase restrictive constraint sc achieve much code image patch help salient model popular due sparse work image difficulty behavior hard structural incorporate follow satisfy component max pooling bring moreover propose convex non code iterative truncate nonnegative component correspond remove plain practice able value coefficient regularization remove truncate kind practice correspondingly short reliable solution correspondingly alternate repeatedly take convex iteration truncate detection name describe give extract codebook initialize stop universal l magnitude estimate pooling pooling define th row report comparison convex implementation especially window matlab gb descriptor widely use
u h remains show inner need g g follow h g h j nh g unitary nr f nd r fx ix n dx ix ix fx nr product uniquely consequence algebra department engineering university ann mi statistical problem measure mixture model parametric instead measure uniquely provide moreover latent identifiability hilbert base number measure realization random drawing mixture primary question concern mixture model identifiable explain identifiability consider iid impose group component call group sample mixture random element identifiable simple paper show identifiable per improve regardless mathematically measure probability arise naturally see concerned extraction sort structure popular assume question latent represent determine statistical utilize interested different perhaps regressor another relate dataset directly measure anomalous similar paragraph probability sort consistency require make unnecessary consider penalty fixing clearly theoretical couple important implication firstly practice twitter keep analysis lose result seem suggest technique significantly identify past couple decade application identifiable sample identifiable cdf mixture closely measure domain group show different rely theoretic basically collection measure technique tensor tensor proof totally algebraic previous tensor treat measurable dirac measure fold contain denote probability dirac unique ambient mixture probability follow probability want law mathematically bit construct integral principle must exist representation index measure mixture permutation henceforth summation summation minimal mixture component measure dirac define minimal integral algebra sign v minimal law derive v definition central object interest mixture measure measure modelling collection identifiable mixture complex practically random confusion describe literature illustrative map tensor separable demonstrate give dy dy g beyond tensor product hilbert unitary u h u hilbert decide associate span product introduce rest modify purpose connect product tensor product unitary u follow unitary unitary n u nh space hilbert proceed induction clearly finish without hilbert schmidt hilbert schmidt hilbert space unitary operator linearly inductive previous define time continue measure finally need technical derivative product nonnegative n proceed exist measure l l assume simply side normalize pair mixture share let lemma v derivative derivative equal multiple example p l therefore everywhere generalize first suppose evaluate yield p contradiction pair satisfy cf ff nonnegative without p ii since unitary lemma unitary r dimension h h I linearly conversely remove follow exist generality thus p I mr right combination let k know exist orthogonal z p apply p distinct measure whole
evaluate biology preliminary material conference paper preliminary analysis derivation comparative genomic biological make serve benchmark show outperform particular refine task multiple structure view cover formulation formulation box kernel another tailor kernel toolbox empirical artificial well dataset task comprise us insight special model regularization supervise functional term measure regularizer control trade easily task interested past discrepancy st st similarity develop multi allow similarity learn equip weighting instead priori automatically part comprise line include line also jointly make loss besides connection importantly novel special label independently measurable training point mt encoding view multi learn mx mt tw mt vector concentrate encode loss task r ms tensor space direct sum hilbert us hilbert regularizer insight paper representation primal problem base theory space review remove dependency need adjoint I definition adjoint identity prop example way retrieve index task alternatively index use adjoint map conjugate write prop mr mr furthermore q supremum optimum w duality dual problem partially primal maximization present formulation completely exploit first note affine exchange c c dual completely solve optimality assumption differentiable requirement analog kkt stationarity lagrangian note rewrite definition previous rewrite optimal introduce kernel generalize multi loss function several multi dual hinge increase start single task standard towards novel many multi conjugate hinge loss verify calculus q c hinge briefly may single onto thing first believe facilitate reader familiar task formulation yield non sparse multiple equation greatly simplify svm give mkl obtain special case case mkl corollary restricting task first case section definition similarity appeal fix assign pair example task publication domain two task fix idea base similarity task read express idea center following group within respective term regularizer cluster assignment regularization center assign least primal formulation may regularize constitute date regularizer framework adjacency encode task view regularize task laplacian invertible regularizer eigenvalue relation capture relationship task task dual involve weighting bioinformatics form furthermore consider sum kernel corner kernel dual interesting within consider also constitutes actually choice discussion kernel relevance work ahead novel importantly mkl engine mt mkl tree similarity mt mkl exist similarity combination graph adjacency laplacian extension readily task multiple kernel access suited laplacian weighting give prediction accuracy couple equation coupling advance formulation introduce task weighting decompose arrive term tree graph assume relation computational biology different task expect evolutionary history beneficial share terminal terminal node task discuss regard mt mkl require give square length scale use mt mkl weighting length scale hierarchy trade task transform scale algorithms matrix mkl implementation tailor large allow datum demonstrate set employ efficiently computable certain string toolbox convenient way mkl completely completely mkl along without step mx constraint eq whenever update need objective decrease epoch need keep date change result algorithm computation feature map cf iterate line stop define line kernel primal last objective alternate h task matrix precision initialize inverse satisfied accord store primal primal decrease hypothesis mt task kernel implement toolbox implementation classification furthermore optimization mkl solver use analytic cut plane novel computable conventional svm integrate perform require modification currently module lastly scheme describe module implementation truly large mt string interface module mild exist strictly contrast concern direct rd rr descent q set continuously minimizer cluster minimize data domain similarity iterate cluster corollary unconstrained put indicator shorthand problem sequence thus order recall initialize f q cn cn cn eq function finally trivially fulfil employ paper string fulfil infinite kernel exist representation map framework range control toy experiment diverse review experimental work closely one case investigate genomic computational biological early h task independently pool evolutionary power biology illustrate case multiple successful application computational biology joint multiple problem two experiment describe sequel beyond investigate framework generality evaluate hierarchical artificial generate vector inspire evolution hierarchical accord leave root node subsequent hierarchy tree leaf carry dot product pair figure clearly valuable leave couple mt mkl create node mt mkl mt union combine treat task report roc accord detail comparison mt roc mkl good perform margin suggest beneficial next simple considerably improve performance observe improve non mt mkl mt mkl accurately identify genomic sequence genomic whereby rna copy genome sequence genomic resource bring annotation nine take annotate select jointly learn treat initial similarity extract similarity genomic hamming rna genomic change evolution different class similarity refine mt mkl create task similarity mt mkl exponential task different collect include positive label consist mt mkl scheme split split validation ten set nine svm pool task improvement union individual indicate similar discussion improve least marginally individual seven nine individual attribute matrix speak propose mkl eight bad improve mkl achieve task similarity able learn mt mkl beneficial several believe biology potentially application multiple computer security present regularization refine kernel hermitian numerous primal formulation hinge integration toolbox framework could norm mkl primal efficiently solver special software machine toolbox term predictive intersection analyze outperform baseline theoretical good computational great instance international drug breast early helpful foundation well european support research foundation grant kl support duality real present reading refer introduction duality machine presentation conjugate hilbert g g supremum affine duality indicate conjugate semi appendix definition adjoint real hilbert euclidean appendix known duality hilbert theory helpful computation proper hilbert g hilbert space conjugate loss conjugate note unbounded supremum translate show rgb remark corollary center york york ny usa computer university cancer york usa computational center york ny multi task similarity refine multiple mkl general
via tune r training convnet layer select inference module integrate principled efficient extensive demonstrate efficacy convolutional novel pooling yield demonstrate capability statistical model ultimately map plane map multinomial block next set impose activation pool bottom initially sequentially bottom layer refinement learn jointly readily variational image hadamard product indicate assume layer top layer stage view plane pool map partition contiguous pooling pixel pixel location block pixel stochastically pixel equal amplitude associate block max learning part fig constitute excellent initialization refinement top generative process constitute w
analogy especially well suited sense lack make also token improvement al focus describe understand paper neural language extend traditional gram represent token indicator jointly train replace likelihood character model network gram incorrect replace normalization probability distribution consider extremely computationally log linear neural language training context window language learn neural embedding nlp dependency parse sentiment document name recognition parse among less word construct multiple representation extend document multiple dense embedding word sense cluster context expensive token learning mapping type embedding skip gram learn predict sentence gram model word vocabulary embed probability eq context sequence noisy consists sample noisy context w window maximum window noisy context noisy randomly sample tw skip sense induce cluster embedding token context word vector type maintain sense token predict predict token embedding perform jointly predict use current tw tw w close associate context vector let v global avoid complexity predict formally mean context belong similarity context sense observe embedding noisy skip predict predict word global sample noisy update randomly tw tc ts tw tc tv tc tc k learn number np vary relate non type proportional near vector initially vector cluster create new vector create online word r vector order skip gram skip microsoft pc l gram abc tv skip gram net try offset loop pre ball tv roll np run run run run operate operate np walk run run walk operate go limited run present neighbor associate various embedding near neighbor word compare similarity embedding parametric skip p skip ms sg seed sg seed comprise drug storage power physical agent np two related contextual similarity evaluate vector rate include contextual make overcome issue stanford word pair context noun noun noun noun evaluate embedding corpus since per contexts sense measure embedding k compute embedding address metric pair fit metric ignore select word independently probability cosine center report similarity human dataset r tf gram skip gram tf al np outperform measure bold face skip np np np tf representation face dimensional show well network dimensional perform slightly skip task give achieve art achieve measure metric since well improve word analogy introduce np compare state skip gram np show np model embedding fair mainly frequent vocabulary multiple top even embedding frequent art show present extension gram embedding word perform word sense type state word task token embedding nlp acknowledgment support center reproduce finding recommendation material author necessarily reflect computer science edu interest space embedding nlp single ignore thus usefulness skip learn embedding recent jointly word discrimination embed art machine corpus token represent dense value commonly help curse improve generalization semantic syntactic dramatically processing benefit arise volume considerable gram log high wikipedia day much common input name extraction parsing substantially continuous name extraction continuous skip supervision similarly dependency parse skip embedding recently apply notable prior string embed approximately contextual relate biology moderately space close triangle word example without triangle discover embedding multiple
order conduct synthetic remark preliminary experimental nevertheless nearly histogram sort add overhead need sort match quantity conduct use intel core mb cache ram mac os operating report error illustrative sorting point std sort take distribution gaussian gamma support size consider differ style width axis title north inner sep grid xlabel number ylabel shift ylabel label font style width list blue mark mark legend style anchor west format legend style anchor west style format anchor north east title gaussian density gmm txt cycle title beta table plot beta txt title histogram pdf give sort algorithm histogram constant piece record time exclude achieve result scale three nearly size constant note three sort sample run essentially desirable show achievable piece histogram shape hard gamma mixture beneficial rich piecewise next algorithm decay regime dominate htb exponent minus plus data histogram gmm avg histogram txt avg minus error error histogram gamma txt table avg time time high avg error beta txt x avg time high histogram gamma format sep std gmm avg std beta x n avg std table avg std gmm txt avg std beta txt std dominates demonstrate attention interesting contrast linear somewhat simple polynomial encode instead potential programming lp solver run far utilize small account repeat histogram bind give relatively root negativity root correspond find nevertheless certain regime leverage negativity proceed remark utilize arguably elementary build part ii unit satisfying polynomial root necessarily root root require sect still achieve remove divide degree act return formal notation formal representation fact give def root root run quantity simple compute scan eq right side expression root lie root root z pz lemma dominate runtime root let ps polynomial evaluate point return ok running claim correctness clearly always return ok choice since algorithm correctly negativity assume lemma since approximation definition either return satisfie theorem throughout focused metric naturally goal norm subsection algorithm discrete set minor piecewise distribution sample degree pi th hypothesis c algorithm distribution albeit slight modification set difference definition total formally well highlight modification move continuous notion vc discrete particular maximum disjoint interval property inequality guarantee projection computation efficient interval dimensional program appropriately generality polynomial use interpolation formula polynomial representation bound analogue feasible robust efficient separation feasible recall negativity fast polynomial non find root precision evaluate distance exist root kp ji notice integer argument section quantity return discrete separate current c mit li mit schmidt mit distribution piecewise polynomial piece degree draw yield structured family domain nearly consequence complexity meta experimentally demonstrate practice level piece piece fit interval iii finding efficient density claim density observe distribution fundamental history extensive estimating estimator estimator technique decade large body belong approximated number distribution precise define output density learn computationally whose time nearly linear additional agnostic misspecification family merely univariate family several decade yet understand surprisingly distribution mixture gaussian use wide variety three agnostic nearly polynomial employ context family polynomial existence approximations ingredient learn prior unfortunately polynomial exponent quite turn yield nearly wide range family domain nearly broad single stress number idea describe overview consider univariate function finite loss generality focus standard notion natural analogue boolean notion minimax access cf cc algorithmic result interval follow give tolerance h work slow approach principle efficient high run necessary level ideally show indeed good exponent substantially improve running remove information theoretic nearly nearly linear time natural family modular distribution order extend polynomial discrete design linear poisson nearly optimal basic structured family prove appropriate value target approximation minimize would like product theory family example family lead algorithm complexity concave piecewise linear piecewise polynomial show theoretic concave density nearly agnostic piecewise theorem nearly linear mixture concave matching theoretic show approximate piecewise agnostic time natural mixture family note several previously study cover unimodal monotone hazard concave yield nearly linear family unify way range structure modal hazard family provide aim cover rather power method crucial component know agnostic univariate distribution equivalent proper hypothesis search find efficiently roughly speak find mixture provide overview technique supremum gx probabilistic tool inequality let pdf theorem follow piecewise hypothesis quantity time involve main interval learn sub remark appropriately dynamic programming roughly formulate interval discover interval theoretically polynomial slow application particular hence fast lp implement run overall follow idea iteratively merge interval become subtle vc speak consecutive ensure run improvement roughly speak exploit inherent problem solve convex separation optimization ellipsoid efficient long research family structure sort pdf reader book early dimensional past decade start monotonicity concavity focus restriction monotonicity reader refer book mixture structure attention theoretical common address shape mle variant mle quite mixture piecewise polynomial spline extensively inference density moreover spline work mathematical non linear wavelet smoothness remark work unknown recently give seem run require sort logarithmic run iterative merging current show efficient constant emphasize easy distance significantly paper algorithmic subroutine find simple subroutine indeed exponentially many constraint section application univariate finite interval denote family set disjoint generally define metric norm measurable norm supremum take different vc bound pdf empirical pdf elementary fact finite set sign change k f ii piecewise top merging close approximation merge broad class hypothesis satisfy intersection hypothesis efficiently learn present constant capture many proceed merge merge algorithmic challenge projection coefficient polynomial approximately empirical give oracle allow solve feasibility checking whether k convert feasibility variant polynomial feasible semidefinite program simplify suffice set interval replace supremum finite set distance suggest solve black box application sdp solver run constraint inequality increase contrast achieve dependence sdp additional structure projection importantly separate desire polynomial equivalently interestingly significantly lp cut plane distance coefficient polynomial root large necessarily separation oracle outline algorithm proceed point assume reduce jointly distance nearly current variant biology maximum scoring segment exploit give variant nearly number point convert separate polynomial polynomial guarantee start generalization piece arbitrary integer histogram denote density g start provide follow single point sample merge notion crucial j th partition merge form interval iterate quantity eq error interval track error perform arrive respect formal trade piece output histogram n j te characterize establish run exponentially th merge imply construction loop substituting show generality proportional sample algorithm proceed learn return terminate rest partition boundary interval jump event throughout final jump j create jump jump create interval eq use ta suffice prove set jump eq indeed hx interval constant use analyze interval interval definition interval contain may jump singleton include jump interval contain assign sequence finally merge iteration jump triangle jump change bind distance prove lemma complete b use complete proving merge except error merge large suppose merge candidate merge iteration li te lt ti summing condition give plug since create merging interval recall complete ready general merge version algorithm introduce histogram histogram hypothesis intersection hypothesis find efficiently note class sample algorithmic generalize piecewise hypothesis variant I intersection ii good fit function distance efficiently distribution histogram merging hypothesis piece definition aforementione mild family subset whose domain hypothesis ti h ii main intersect formalize interval contain example histogram piecewise framework set interval piece note histogram degree polynomial throughout sometimes denote piece piecewise negative two intersect easy ready fix integer sign restrict pdf want constant iterative take arbitrary output hypothesis agnostic assume call computation oracle sign pi pr respectively abuse support take ready projection n n remainder explanation merging analysis formally proceed interval interval histogram oracle well eq keep merge interval partition formal explicitly merge defer oracle fix interval run time sample interval algorithm conjunction projection oracle turn main subroutine general merge unknown hypothesis minimize merging depend underlie present non polynomial degree polynomial interval formulate construct approximate combine exist achieve k convert feasibility empirical convert scaling pass subroutine transform empirical distribution feasibility also empirical set feasible polynomial polynomial consider original want find slightly relaxed version find negativity additive truly negative small ei collection write polynomial negativity constraint space lead could establish intersection comment intersection negativity let negativity encoding fix expand constraint cx ci ei ei ii ci ic worth note feasible negativity restrict polynomial simplify location replace supremum show suggest black sdp encoding constraint polynomially super run black box lp solver importantly separate allow interestingly lp k efficiently separation see utilize structure separation show notion separation return yes separate yy cc perform basic hence resort accept approximate separation hyperplane yes cx return hyperplane hyperplane note definition separate hyperplane membership hyperplane employ several separation oracle exist still use approximate oracle contain moreover radius return yes return ellipsoid cut plane oracle technical order suffice separation separation oracle e cx bind separation initialize need ball bound polynomial let polynomial coefficient px px lemma upper radius radius ac kp define length separation reduce volume feasible region reach volume feasible separation find conclude achieve infeasible radius feasible also easy polynomial change stay hypercube next constraint distance relate feasibility optimization binary carefully order canonical separation separation polynomial l small empirical achievable return coefficient cx maintain clearly begin trivially cx loop approximately preserve loop ball must identify loop consider inequality feasible thus oracle empty radius return c concrete cut plane method run lt multiplication separation complexity running time dominate iteration operation oracle ball combine run oracle run ss projection separation oracle ball define oracle along part c whenever polynomial separate hyperplane hyperplane hyperplane run none happen return theorem run claim formally negativity approximate negativity satisfy polynomial ok return polynomial approximate negativity eq prove negativity simply hence correctness correctness run runtime claim oracle describe v describe subroutine ki I yes v q left hyperplane argue guarantee hyperplane inequality indeed hyperplane entire claim define number number interval maximize q collection interval suitably length support consider maximize let interval maximum rhs q let otherwise exclude ei claim direction denote disjoint achieve maximum support interval follow set consecutive pi put way interval negative smallest contain associate sign support small reasoning pi I namely pi j transformation claim transformation pass solving consider compute present analyze description alternate otherwise consecutive number ki merge q compute efficiently store weight collection interval amongst formal definition algorithm let weight nontrivial return solve attain boundary collection interval every collection atomic maximal say maximal either respect suffice long atomic maximal stop iterate either atomic maximal let contain piece atomic contain subset atomic maximal since maximize attain subset every contain interval interval subset ever end sign modify property maintain resp left sign
fortunately recursion normally call max convolution compute recursive operation pattern total pattern normally still moderately discrete max analogous allow nevertheless convolution operation though first recall convolution piecewise max convolution belong moreover f slope fig symbol write iy ci x thus retrieve ic parameter c ic ic max concave let concave interpolation imply domain piecewise discrete sketch max piecewise concave sort piece slope clearly pass cavity convolution omit simply remove convolution step trivial affine mapping easily generalize trivially concave ensure concavity computation binary simplify tw w tw w tw eqs correspondingly eqs simplify order result drop different presence impose arbitrary single special configuration ti flip respect configuration partition sort index order correspond subtract order index turn index optimal sign f index define variable clearly get picture except shift shift turn shift quantity expression computing provide report update cavity sum e bp behaviour reasonably suggest vertical reinforcement value solution perceptron log bar fit show estimate critical perceptron layer concave iw zero e qualitatively similar around close connected layer capacity single layer still capacity unit test store thus demonstrate great increase clear due permutation unit replica effect tend intermediate state mix different solution make still help achieve constitute improvement bp limit extremely approximated ms approximation naive normally slow purpose show extremely max valid advantage break thank hoc additionally max contrast equation extensive weight algorithm achieve theoretical cavity full detail note text cavity analogous cavity change turn point change express global choose convention obtain consider cavity concave maximum fact expression cavity field consider simplify I I I I plug expression cavity algebraic I I I turn region jt jt jt f last kronecker jt jt jt I go cavity eq efficient np complete call algorithm independent update put par scheme bp interest perform perceptron inherent naturally break ms feed neural etc problem obtain assignment device test give example discretize sigmoid scalar vector input weight unit operate unit reach popular successful even usual drawback gradient presence minima slow circumstance problem even simple version become hard complete storage capacity device bad robustness application theoretical long storage rather network hard potential practical biological light upon origin hardness physics isolate energy landscape minima tend poor cavity instance show belief propagation correctly output association single simulation simplify version simple work complexity measure performance thank approach deal tree structure fully use least straightforwardly arise address ms reinforcement analogous reinforcement use temperature limit bp approach zero temperature ms addition error add field go small field layer ms bp use shall binary computation ms bp storage capacity time reach polynomially well rest organize present solve thank convolution complete detail implementation binary layer throughout unit binary transfer I evenly spaced unit convention single omit layer unit layer receive device kind also consensus machine like input would share overlap tree architecture generally capabilitie storage fully connect situation also possible fully machine symmetry since second throughout context desire association correspond desired extract random input pattern still extract usually teacher device rule low architecture student device student teacher permutation pattern
classifier algorithm applicable form use produce write discuss approximate equation simple average observe instance previously chapter prohibitive storage evaluation another optimal misclassification v surrogate regret bind surrogate usual work function linear throughout shorthand minimum classifying equivalent classifying include completeness straight cauchy schwarz quantity self similarity ex verify distribution pac include label add label flip sigma noise robust negative feature mmd see restrict divergence q commonly yy ex negative equally mmd solve take cauchy objective theorem classifier regularize optimize margin suggest high feature normalization idea kernel kernel density instance use optimal rule ensure feature kernel function class hull calculation feature linearity loss word pick usual multiple general pick generalization collection draw dependence correct definition application frank z begin point average choose run obtain term line available view minimize approximate concentrated originally produce super standard ex ex time use rate rate fast rate search cd approximation appear statistic closely frank wolfe sparse approximation material split disjoint use approximate sub separately tolerance margin therefore assess compression motivated mean mean compression classification n produce classifier let compression probability eq way tolerance maximum sample mean theorem suggest stop optimize justified theorem tight contribution highlight set keep classify mnist comprise kernel bandwidth classifier parallel split range step blue entire baseline test training obtain rapidly obtain roughly perform mean entire margin assess validity loss place show maximum discrepancy regularize machine surrogate speed evaluation margin relate degradation incur instance ie q x pac prior draw couple term refer furthermore linear identify posterior calculation kl divergence assume prior posterior identity restriction see weight vector begin theorem posterior standard moment normal quantity decrease bind minimize map define union posterior normal ip furthermore fix previous posterior normal hand final moment line cauchy schwarz eq recover take distribution minimize hand bayes theorem present assess generalization useful hoeffding draw feature least hoeffding yield distribution collection draw use hoeffding union mean result let say ball define grain assess decay hilbert exist diameter know equation
bethe bound begin experiment use mle analyze various matrix diagonal partition gradient regularize run entropy upper run display upper high regime estimator rw reweighte interestingly objective regime produce value moreover inequality rw regularize bethe quantity use fw compute already learn red incorrectly probability low blue nontrivial red force pick pick l c c computer vision sequence follow setup datum consist frames house separate frame toy angle label point frame divide validation split learn loss parameter house gap little difference synthetic tune indicate make probability match albeit permit bp approach unstable bp fail converge step mle go year student return student year return student preference obtain room major year train student live remain student node gender entire datum student create many feature indicator feature matching mle table coordinate ordinal ordinal value see agreement appendix perform multiclass hamming model classifier plot roc curve demonstrate false perceptron structured svm decoder obtain test bad pt hour take long image segmentation formulate keep image result approximately try naive subgradient descent slow partition loop run bfgs another objective optimize optimize parameterization mle fastest mle evaluate error describe curve average substantially smooth raw curve quickly attain low test error curve run inner loop finite run algorithm confirm convergence fw early low attain iteratively move value initially inaccurate value dimensional result prediction nevertheless move compute expense minute portion objective hamming estimate local classify dark region classify already run make essentially correct texture algorithm hamming get internal criterion exponential free energy free energy concave add enable maximization leave efficiently frank wolfe scale dataset coordinate wolfe rapidly achieve map practitioner either employ double svms competitive margin error fast compete mle simple work combinatorial regularization technique employ part setup part fw bethe energy derive full dual search fw appendix explain far low likelihood two sided evaluate evaluating display likelihood compute fw procedure likelihood small dedicated hyper ghz intel gb physical ram run run matlab combinatorial author matlab extension year period assignment addition student survey ask preference feature level student question create several student matching learn describe unit assume ordinal qualitative relatively indicate perhaps second predictor successful least structured employ publicly available code author try lambda configuration achieved ham significantly mle profile go usual hour study audio entropy local polytope show iterate fw decay subproblem wolfe curvature quantify linearization twice upper heavily influence piece depend unbounde fw curvature require part entropy depend look function long inside fw always guarantee produce inside box typically issue appropriate propose bp add combinatorial matching pairwise binary mrfs become gibbs assignment ever well define iterate boundary reasonable reweighted rough high bethe energy necessarily graph bethe proven argument argue polytope follow concavity bipartite perfect free polytope graph bethe written entropy concave entropy concave formulate describe fw match column ki k contain separate graph conditional arbitrary maximum matching denote absence item x coefficient single replace add regularizer bethe write bethe likelihood program linear constraint begin eq derivation mn sequel pairwise matrix additionally stack thus rewrite justify product later minimization objective maximization compact domain unconstraine function attain stationary simplifying iterate fw plug product fw bipartite feature able occur perfect link perfect discover matching neighbor polytope infinite match feasible overcomplete parameterization matrix treat learn fu e exchangeable replace parameterize likelihood bethe bethe convex mrfs span frobenius penalty reweighte approximate entropy n nh singleton entropy marginalization use identity stationary objective frobenius rearrange outer eliminate similarly reveal gram stack stack entry n mu v flip simplify sign later write h nm nr gradient w products bethe coordinate frank wolfe computing denote step contain row row row n md e w md add value wolfe fw perfect use code marginal marginal force call match use bipartite call affect aggregated bethe approximation provide fw substantially fw converge accuracy differently h fw bipartite perfect speed trade complete edge lda run sampler average algorithm termination accuracy bethe specify present ratio fw fw slow within affect advantageous include rand rand lda david many formulate predict structured output framework support vector structure discriminative apply posteriori decode likelihood probabilistic structured partition paper bethe approximation mle remarkably connection bethe frank wolfe fw partition single efficient double approximate maximum estimation outperform exist segmentation vision learn markov mrf conditional crf parameter learn regularize mle maximum include regularization principle ascent repeatedly gradient log partition surrogate likelihood perceptron map solver quite approximate map black box user abstraction mle goal practitioner superior offer interpretability time marginal approximate access solver bethe energy frank fw method naive fw mle perform marginal call experiment gradient solver accurate answer achieve avoid costly double first generic reweighte technique bethe style surrogate model dual approximate problem minimize subproblem formulate separate accelerate fw use fw test interact allow variety map max pairwise ise match apply pairwise binary bipartite mle problem mle method student sample statistic hypergraph ease well linear regularizer central compute work learn bethe recent mle convex free energy fw convex iterate define search fix step minimize independent parallel depend application combinatorial reweighte lp solver project onto despite parallel prohibitive large problem space perform fw convergence know fw include fw algorithm input uniformly computing useful
record distinct extract news follow experimental dataset yahoo criteria bandit algorithm dynamically ucb single single ucb algorithms ucb ucb algorithm assign context user actually quantitie subsample suggest apply user proper tuning maintain fair dataset run maximize suitable range payoff dataset make plot version test average variance observe small summarize yahoo retain far dataset record discard ratio cumulative payoff aim three consideration web yet point dataset fairly way consequence dataset provide huge imagine population yahoo moreover collaborative strong article yahoo item high chance user preference collaborative effect fact clearly win comparison suit start accord unknown c logarithmic denote w big big clear easy j achieve partition upper become relevant distinct partition influence yet worth repeat role whole kind partition ahead simply contextual bandit grow factor r shall exploit reveal replace term n become bad scenario result extreme single many operating scenario group similarity universe item group similarity provide operate encouraging operates simplify content universe conduct reliable annotation yet potentially research lack infer factorization technique subsequently clearly combine co far see adaptively item computationally amenable advantage stage get stage somewhat similar meta investigate content recommendation exploitation user item possibly clustering take advantage preference collaborative filter world show scalability increase bandit regret within web collect preference service enable interaction content recommendation recommendation web service core business web universe popularity service adaptation preference algorithmic interaction group similar e static specific type user community content clustering user dependent music cluster music change item could group tend preferred notion side group user base similarity item side user see suggest recommendation scenario movie recommendation computationally double simplified version technique double recommendation contextual associate exploitation work user tend item recommend cluster need different universe user clustering induce compare perform context real scalable exhibit bandit algorithm also hold stochastically prominent bandit content main information fact embed preference relationship click exploit filter technique typically collaborative user often impossible adequate aim exploit collaborative effect co technique batch one recommender g whereby lack suboptimal recommendation approach behavior clustering specific consideration partition dm user belong cluster like lie cluster significantly behavior u common thing context assumption threshold cluster set e unknown upon user value conditionally bound variance observe setting break sequence step receive user recommendation pick ti ta goal learner comparative theoretical interested bound cumulative regret learner extent good exceed aim section kind contrast content universe universe p partition cluster user induce induce I e possess common resort content make user method whereby preference relation rating collaborative group item item pair e grouping recommendation social beyond bandit specific piece bandit paper try assume combine seem large scale analyze bandit cluster completely like result technique lead rely user side emphasis recommendation see spirit author author none author specific effort dependent present rely tradeoff exploration exploitation call associated base feedback operate linear algorithm available subject initialize identity bind counterpart need regret repeat holding high practice order define compound turn compound exploitation item put emphasis compute receive algorithm perform user neighborhood compound di ni brevity aggregate ta tm drawback clustering base item fact dependent item make store clustering maintain similarity behavior universe affect aggregate user belong round lack reason despite drawback reasonably stream ii approximation generally exhibit good priori maintain clustering bandit description contain maintain clustering item clustering represent connect undirected index graph cluster cluster clustering exploration user item dm di ng u ni determine current clustering w brevity quantity ta update tn te tn ti te item single correspondingly cluster make unique item depict therein candidate eliminate depict elimination algorithm compute neighborhood accord compare split item new cluster user clustering overall main point square clustering unique side point item item side change item item edge user item get imply naive would allocation maintain prohibitive moderately usage approach start complete create randomly graph la
noiseless size test setting equitability across sample intend achieve equitability budget size setting independence insensitive level analyze lead size fast yield achievable compute translate runtime minute variable analysis snapshot currently improvement way algorithmic allow computation equitability power several dependence new exposition equitability independence community rigorous side exist state finding equitability noise achieve superior equitability equitability variable poor maximal examine statistic share independence perform independence albeit differ equitability independence testing much examine substantially high correlation power independence equitability estimator equitability characterization equitability give lead high expense weak relationship equitability expense runtime trivially runtime fast large result suggest rank strength sound imagine keep equitability examine enjoy alternative relationship type broad simultaneously appear preferable choice measure dependence demand equitability goal low possibility power evaluated set finally understand comparative analysis exhaustive date dimension equitability relationship statistic analysis hope enable precise trade one setting equitability functional explicitly possibility besides intuitive equitability interest method perform much bad equitability attempt scope relationship equitability add result theoretically equitability understand strength method direction measure variety superior different appropriate setting understand insight inherent trade allow landscape effectively ultimately understand acknowledge k constructive table use equitability analysis cosine cubic cubic shape x xx xx x u exclude due poor across portion drastically graph h quadratic xx statistical equitability power perform require ability result case expression wherein solve use population equitability worst worst interpretable shorter interpretable colored red interval white equitability robust analogous present supplementary setting size value present computationally expensive equitability interpretable length short interval color interval white length equitability factor distribution setting include material expensive analyze equitability curve relationship take quantify poor equitability model square low mid parametrize whose parameter equitability size use maximize equitability across noise test equitability material correspond equitability vs mi infinite equitability mi noise compute newly interpretable interval equitability mutual square correlation large setting composite curve noise analyze relationship strength quantify mutual mid powerful parametrize equitability equitability test sample table equitability material power curve relationship aggregate power power comprise area curve statistic parameter area vertical average relationship turn poor testing equitability suited equitability dependence aggregate relationship power relationship test determine relationship type relationship dot average case list line represent default set use equitability independence parameter equitability see equitability noise runtime series correspond indicate equitability equitability equitability present value test maximize equitability examine value test include constraint supplement test equitability default rbf kernel sample test pairwise computationally test median pair pair median analysis examine area range test statistic c pair c runtime method setting default information parameter independently test median present parameter equitability fast equitability interpolation point set example relationship low computationally exploratory analysis association accomplish compute possible examine assign type equitability formalize addition equitability assess independence runtime lead dependence include statistic newly introduce equitability primary power regard find relationship mutual estimation prove setting test trade runtime fast compute trivial achieve equitability relationship appropriate tool guide statistic equitability hundred thousand association within analyze pairwise association search common compute low scoring list depend statistic statistic zero statistic contain many relationship relationship fact though trivial systematically score relationship relationship crowd relationship list manually relatively small relationship strong detect power trivial relationship weak allow relationship exploration set goal many association task utilize equitability dependence equally formalize power hypothesis relationship strength nan hypothesis zero equitability intuitive functional relationship reflect determination respect possible equitability difficult measure equitability mind relationship efficiently computable relate coefficient essentially compute translate benefit extensive equitability power runtime correlation mutual hilbert schmidt framework rigorously equitability yield main conclusion regard estimation setting four art art also achieve outperform outperform alternative examine mean poorly detect power independence competitive albeit parameter equitability characterize equitability free consideration final conclusion concern find fast method run cluster near equitability take compute together conjunction achieve mix filter result equitability explore together first equitability coefficient introduce dependence measure wide setting power independence focus primarily performance comparison provide hope paper expand review equitability equitability analyze characterize tradeoff independence equitability analyze extensively definition relate informed reader coefficient maximal score quantify strength see goal coefficient statistical grid plane point analogously discrete variable denote empty finite g maximal achieve type population define quantity object characteristic population supremum jointly matrix shape type different relationship maximal maximal corresponding jointly variable population alternate characteristic estimator population original maximal estimate characteristic order maximum size grid resolution order let define prove consistent dynamic introduce prove consistent estimator contrast although whose characteristic matrix turn compute distribute define set grid analogously pair define programming include control remain consistent statistic maximal coefficient aim relationship absence independence maximal behind coefficient characteristic tend grow imagine bias equal power well bias independence goal noise maximal useful signal information coefficient avoid total pair consistent tailed control test ht p quantifying value equitability dependence formalize exploration equitability review equitability way equitability roughly equally noisy relationship viewpoint dependence equitability test distinguish relationship different amount reject nan strength test nan relationship highly power relationship formalize rigorously equitability specify strength relationship diverse relationship add determination respect return build intuition define equitability broad equitability interpretability equitability specifically reflect notion concept even statistic let standard relationship satisfy distinguish equitability question call equitability discuss property statistic equitability independence equitability statistical statistic independence extreme extreme say identify relationship weak would equitability differ analyse independence statistical hypothesis equitability equitability contain distinct relationship analyze b tailed test base hypothesis show alternative power instead heat instead consider set hypothesis plot result color correspond size right distinguishing surface attain row define equitability take intuition assign equally type invoke estimate exposition term equitability bad equitability reliability statistic reliable close interval reliable diameter v pdf pdf c relationship amount since interval interval analogous plot relationship range blue red interval interpretable interpretable reliable interval shorter interpretable interpretable bad case interpretable interval solid line interpretable interval red representative relationship reliable interval hull interval reliable reliable noisy functional relationship consequently reliable central b type distribution three sampling small union central acceptance define interpretable interval small close interval interpretable diameter show interval noisy different type pearson interpretability reliability statistic one time way basic one measure resp interpretable resp resp dependence reliable resp interpretable reliability resp worst oppose prove interpretability gain intuition interpretability interpretability interval case bad interpretability interpretable interpretability perfectly interpretable arise let interpretability functional depicted figure interpretable correlation worst shorter interpretable turn equitability assumption state statistic respect interpretable confidence interpret way strength relationship measure vice versa relationship strength statistical independence equitability news news one equitability relationship equitability require large equitability far conceptual equitability relationship mean write variable equitability amount relationship dependence bad resp abuse terminology equitability equitability respect set relationship detail review analyze equitability interpretable interval pearson coefficient trivial analyze standard relationship noisy equitability th reliable reliable enable interpretable interval equitability reciprocal length interpretable interpretable many large value sample relationship type different different amount interpretable thus poor contrast notion interval course equitability question relationship trivially perfectly correlation normal cc pdf v respect pearson central distribution interpretable interval relationship different interpretable indicate red case possible illustration monotonically use proxy relationship receive equal review equitability equitability dependence begin quantify equitability interpretable interval conventional connection set analyze representative coefficient total view exploring grid draw assign aggregate normalize mutual aggregation supremum explore grid except summation method use pearson statistic explore two grid differ test stable variation model good equitability equitability model outside noise affect outperform mutual problem inspire equitability mutual suffer superior equitability model substantial estimation equitability setting equitability test occur population effect equitability information insight demonstrate minimal superior bad average equitability noise add mutual perfectly examine surprising broad test examine equitability specifically mutual analysis show sample contrary picture equitability estimator mutual technical comment publish author theoretical see figure table demonstrate correlation poor equitability equitability equitability high equitability equitability correlation measure dependence examine demonstrate examine equitability perform interpretable interval present material composite relationship quantify heat map tail distinguish hypothesis composite come well method average set hypothesis list maximum achievable information parametrize statistic affect present across test assess equitability statistical confirm conclusion quantification equitability interval distinguish examine small equitability estimator task size test even variable see maximal test degree equitability noisy relationship property method traditionally detect deviation yield high nan statistical independence due fact hypothesis method highly simultaneously detect deviation display may detect relationship bad relationship course nan allow hard nan composite correspondingly suffer equitability lead vary type setting sample mutual outperform equitability poor equitability come parameter setting good power equitability demonstrate inherent statistic equitability establish case interestingly estimator maximal alternate expectation correlation hand lack equitability return result bind computable achievable extent relationship world relationship large claim understand crucial determine extent important development grow hope insight together enable investigation equitability ranking rank measure dependence assess dependence examine notably upon hypothesis alternative allow aggregate across gain view power perform determine last analyze depth use achievable performance large analyze examine equitability analysis perform relationship choose analysis similar material manner equitability analysis add eight noise level evenly range noise noise substantial sample level understand affect power dependence automatically need way eight type power compute power relationship integrating amount add area amount result power statistic noise choose uniformly power noise drop set threshold supplement power curve score type plot list choose parameter red substantial attain threshold material contain quantitative ranking dependence value method parameter several across relationship quantification quantification large outli accordingly supplement circle analysis supplement threshold use generally threshold material aspect ranking statistic closely coefficient aggregation summation mutual grid score grid via characteristic fundamentally promise dependence task demonstrate translate note power discrepancy due analysis intend equitability show use equitability trade final perhaps result method small method gene show observation true statistic relate detect rank relationship pearson correlation coefficient rank result believe set relationship independence power examine strength detect statistic question threshold systematically score equitability provably low threshold converse true test require achieve relationship minimal examine threshold threshold besides supplementary material grain analysis statistic strength independence type grain picture relationship ability power curve directly axis independence examine figure result choice base previously report substantially set test indeed strength dependence test grid base base summation choose result method find provide relationship examine maximal entry across relationship one promise exploratory discard burden power relationship use whereas time since power appear important eliminate power independence base yield test close art differ suggest trade additionally find test vary considerably across different whereas sensitivity substantially finally bivariate many appear last observation magnitude case answer wish high datum exploration measure already reliably thousand able relationship additional relationship strength scientific uncertain exploration maximize relationship think statistic inspire framework pose numerous challenge establish test parameter optimal equitability power
l est une pour es de mahalanobis des es les pour par pour la q eq iv e est e analyse en de eeg dans est en une se la dans la les les et es la est en n stimulus une li du stimulus les une et est ce un des dans la es par exp e par pour correspond une portion du si dans les observation figure des des et les par les es et une la projection dans des class es des des le rare se la de analyse une de des des svd une analyse des les la en dans la de pour analyse spatio des et es dans eeg signal optimize et journal statistic et mapping k variate lda spatio feature eeg eeg localize analysis sim r international cm eeg universit universit universit fr l analyse lin dans de lin des des plus par composition la des les des et des la de mahalanobis des dans des une la la cl es analyse lin es eeg abstract focus discriminant variability average row dimensional discriminant multi relevance separable eeg analyse lin es es est dans en es eeg une de les la les tr et est le est sup et la tr de de il une se la des en des es par un mod le de la covariance et de des les es se dans dans des eeg de ce est de une lin les en de structure des es se de la en la des les des des de mahalanobis les les dans en des de des es dans des une des dans les eeg dans un par dans pour eeg dans un interface machine la est notations pr se en de et extraction pour des de eeg des est de la est pour matrix l op trace la b bc aa de kronecker l mod dans le de dans des les pour r es r une de de les de covariance des q p
subsequence hold subsequence ne ne subsequence subsequence becomes restrict converge large v rewrite subsequence boundedness converge subsequence sufficiently necessarily nf large inequality countable going establish main boundedness q n conclude simply show hold virtue argument consider straightforwardly several need q know boundedness show acquire outline divide show triangle expectation conditionally n define lead second inequality rank one perturbation old obtain eq moment matrix write ensure n positive solution definition prove unique conclude hold define uniform derivation use take deterministic provide generic result purpose et entry zero I nc ne nz transform eigenvalue next describe successive moment generalize valid successive kk step theorem corollary large covariance comprise sample arbitrary upon advance robust estimator scatter sample infinity introduction behave asymptotically different enhance outlier mostly outlier thus robust estimator bring benefit estimator sample within class huber favorable risk estimation momentum big advance large dimension complexity fix e detection started adequate instance toeplitz structure particular scatter come regime huber classical unlike classical nature know robust successively majority gaussian arbitrary give aforementioned scatter take form implicit application limitation regime large regime several estimator scatter behave explicit fully nonetheless independent zero elliptical salient work elliptical henceforth normalize apparent versus fundamentally estimator outli comprise amount outlier focus scatter aforementioned robust behave easily find scatter impact outlier normalize demonstrate robust scatter outli control appropriate huber substantially remainder rigorous statement proof defer attention analytically case outlier either I conclude remark provide stand hermitian transpose transpose dirac unit denote stand part support denote order hermitian x diag stand matrix compose almost sure stand weak ni ni deterministic hermitian moment deterministic vector consider assume last column outlier merely request moment technical n refer merely consist discard outlier lack know immediate alternative estimate norm robust normalize diag arbitrarily bias detect significantly differ majority robust estimator scatter huber study precisely outli identification scatter vector shall scatter continuous increase equivalent accounting multivariate call show shall however estimator later implicitly assumption hold relation take theorem entail allow property implicit matrix matrix interesting outer product scale along outer expect set emphasis weight merely ensure outli impact especially small immediate eigenvalue follow empirical define via n nz equation importantly imply support determine respective k df n df deterministic implicit transform successive formula precisely formula albeit behavior quite weight relate implicit deterministic get insight property successively specific scenario simplify assume maintain assumption perturbation find remark eq shall denote arbitrarily individual involve specific one read simplify side left hand increasing depend come isolated calculus asymptotically choice scatter tend strong mostly outlier norm essentially thus gain dimensional coordinate font style yshift near white anchor north east font xlabel ylabel near bar width mark plot n ij xt come subscript let independent th n surely real function density obtain monte carlo average soon numerically figure suggest somewhat confirm observe approximation decay factor versus font style densely yshift near west anchor north east font xlabel ylabel major xlabel r j xt interesting arise majority become difficult general n impact enhance sample version n behave asymptotically neither contrary capable reduce outlier n outlier previously outlier depict n f optimally discard thus highly confirm show close tail contrary main matching yshift anchor near anchor west fill white anchor north east font xlabel near ylabel width grid major xlabel plot f n f f nu various estimator affected outlier interesting investigate moment case initial expect induce bias fair scale normalize moment successive moment relative scenario moment hold rather important moment ccc p ij estimator large lead conclusion address scatter elliptical reveal behave similar normalize conclusion suggest conclude behavior versus unlike normalized scatter compare value even might probability induce lead tune close rejection property measure asymptotically oracle estimator interest come isolated finitely suitably isolated bring important performance gain finance heavily isolate relative outlier one moment close oracle improve account observe successively regime estimator study frobenius nothing suggest alone quite sensitive outlier study essentially say noticed proportion versus towards let truly provide well behavior scatter finding aspect relevance rejection risk inherent nonetheless implication plug detection isolate global information
nice odd popularity logit link come simple maximum equation fast theory vast reference fraction extensive characteristic generalize answer ever logistic probit link second function probit binary bayesian reference point probit mining deal logit attribute experience regression link strongly support tendency give place rest word appropriate practically suppose represent complete chi thesis population development height weight logistic law even theoretically subject laboratory normal play distinguish role theoretical support observational material matter obvious logit surprising apparent due fair recognize definition probit logit yield two essentially determined solely mathematical convenience theoretically equivalent univariate characterization tend context throughout information view ability good work perform dimension life reveal link univariate sharp begin dimension increase organize general namely equivalence structural real section clearly describe demonstrate claim moderate goodness probit logit provide probit logit reveal input section spam conclusion along insight compare goodness goodness perspective ask prediction probit differ logit often logit function yield difference prediction mining force focus predictive equivalence binary classifier define cdf binary majority rule take component logit probit classifier give q one misclassification hx practice close form empirical classifier randomly tr te te te te te te te te split test error yield classifier dimensional shall say disagreement classifier say perfectly classifier space shall e version cdf almost perfectly standard cdf probit logit perfectly write thank straightforward link dimensional equivalent nonzero pm sufficient logit probit thanks deep logit relate parameter conjecture linearly intercept replication copy compute probit logit probit tendency replicate logit probit relevant sample generate intercept ir determination equal indicate logit entirely help determine probit coefficient probit logit slope neighborhood probit replication determination replication diabetes logit probit logit patient diabetes obtain use probit logit probit logit ratio probit around appear pattern variable theoretical univariate intercept benchmark logit probit logit give characteristic display cl probit ratio probit logit still around important justification simplify univariate intercept complete multivariate relationship neighborhood regardless consideration support confirm conjecture relationship probit know know confirm already noticed express pp proof mention equivalence logit without loss denote variable probability logit probit function equivalence show probit logit coefficient slope task approximate confirm consider probit probit logit logit link taylor approximate log logit ignore high expansion probit function derivation ignore order get c c straightforward inexact confirm derivation reveal probit link equivalence concern similar logit relationship logit verification replication link function consider function data error classifier corresponding probit logit replication realization four calculation replication skewness etc assess similarity difference similar replication bic verification cauchy namely eq replication test error suggest link almost indistinguishable equally htbp logit sd skewness min verification equivalence diabetes diabetes diabetes arguably use statistic diabetes link equivalent logit mean sd skewness min four bic aic slight link function goodness yet evidence claim logit simulate link percentage
penalty derive several variable put moment condition heavy general theorem residual enable imply nan x lemma provide indicate nan statistic also technical suppose cumulative achieve x f part indicate region theoretical imply exist achieve three fp positive entry nonzero false nonzero penalize estimator replication vary cccc lasso adaptive scad scad fp penalize fp increase fp size get k eq trace n complete term q third entry indice group step sum combine lead j pa thus complete limit follow remark rgb measuring dependence statistic distance adjust correspond nan latter error load root asymptotic distribution test domain strict asymptotic significance efficiently generic building result superiority word dependency various signal bioinformatic whether conditionally mean ij precision rich normality always real tail skewed propose gaussian network flexible still transform restrictive find graphical would propose natural way conditional give remain node hypothesis decide present economic nonparametric test function calculate test lead value dimensional hellinger conditional characteristic likelihood motivate respectively n I nk f independent I rest node proposal dependence rely heavily pearson satisfy normality non measure robust deviation nan hypothesis statistical measure relate include method consideration benefit distance independence true second correlation dependence test exceed distance measure propose test covariate estimate mild error main contribution conditional independence dimension covariate relaxation gaussian organize independence construct conditional theoretical type study demonstrate set short section technical introduce notation norm th matrix frobenius function characteristic tool covariance review section gamma nonnegative joint covariance show nan hypothesis corollary property surely corollary constant dependent test seem covariance true observe result step ordinary least ol statistic level nan one replace ol penalize conditional justification dependency
result performance positive lipschitz ac transformation minimizer adjoint let initialization terminate generate terminate stop show c lemma singular inconsistent subset minimize unconstrained initialization iterate current soft arise inconsistent soft multi constrain cut appear noise experiment turn satisfied illustrate increase satisfied cut error observe satisfied enforce always constraint choose option integrate set partitioning problem generate bi among constrained always link pose difficulty multi link procedure satisfy cyclic constraints recursive bi early split issue binary split specify derive class assume add constraint one vertex link constraint although cut derivation different sign encoding introduce towards classify vote point measure supervise give table spam uci mnist illustrate unbalanced problem digit versus generate show plot enforce cut value initialization even cut suit normalize cluster consistently case unbalanced spam vs rest significantly outperform cut moreover hard encoding multi leave versus middle normalize cut violate constraint ccc versus middle cut constraint right violate dataset breast heart mnist ccc middle constraint versus middle fraction violate definition technique relaxation cut constrain always moreover soft trade consistently cluster grouping similarity alone give item domain give al encode available ml short cl incorporate constraint performance constrain research base originally normalize spectral relaxation laplacian relaxation quite loose recently rewrite combinatorial problem nonlinear graph laplacian cut far balanced cut tight continuous approach integrate spectral idea modify order enforce another idea embed graph laplacian close original start encode link constraint inconsistent normalized relaxation continuous spectral fulfil present inconsistent constraint thus extend handle optimize trade cut violate scale problem non convex satisfie stop omit find supplementary notation correspond paper normalize cut vertex formally graph vertex vertex volume partition ratio weight graph element specify suggest allow follow theoretical statement inconsistent relax framework function q link show constrain normalize correspond relation violate quantify lemma partition connected assume minimize lead contradiction constructive lemma consistent constraint equal constrain normalize cut problem combinatorial problem combinatorial optimization elsewhere minimum non c diagonal equivalent particular indicator partition functional thresholding yield small second denote cut convex positively homogeneous convex homogeneous symmetric convex last change limit integration indicator shift follow choice normalize right statement show theorem hence immediately solve normalize constraint integrate correspond vertex derive cut link must constraint merge edge note many must integrate link merging preserve cut constraint prove merge reduce merging vertex vc partition reduce must cut
use different easy check everything induction repeat previous q accounting use coefficient conclude lemma writing couple jensen uniform choice I q since absolutely return complete contain recover straightforward calculation binomial difference replacement improve sample replacement set element hold course straightforward calculation probability eq theorem proof difference union inequality appendix improve discrepancy eq invariant improve de observe label final giving introduce study inequality prove tight process also relation popular complexity rademacher complexity argue finally combine concentration provide risk rademacher complexity widely empirical measure inductive thank attempt apply notion observe label goal correct many mining recommender object generate training realize cardinality source gain probably due fact imply risk bind essence second deviation risk compute follow emphasize learn standard inductive trick translate inductive sample inductive complexity make difference replacement rademacher application author depend contraction know loose dependence concentration without base measure include interesting continue empirical index function replacement nature limitation nonetheless illustrative expect empirical replacement constant remarkably new hold additive additive order low upper order suggest measure learn measure provide achievable low bound relate rademacher achievable also improve comparison show apply obtain computable theorem discuss advantage symbol finite set input function loss defer arguably statistical rademacher conditional fix subset quantity sign take simply rademacher important role learn complexity mainly replacement rademacher fix un remarkable additive I low boundedness necessary I conditional cardinality eq conceptually rademacher sign permutation contain minus idea sect multiplicative much also significantly improve relate class rademacher result even class map absolutely sect b multiplicative bind improve fix un sign bernoulli complexity notation mf sect risk set fix receive consist element remain learner predictor fixed measure nx hx h mh risk sect least satisfy sect conclude concentration expect appear meanwhile computable contraction dependence loose complexity label write note identically low slack significantly small latter cause show appear two time comparison also tight rademacher extra argument one simplify expect appear theorem eq marginal appear equivalently uniformly sample without replacement replacement jensen take distribution rademacher sign random plus minus
review music specific follow review section centrality describe experiment perform multiclass result conclude remark work music however algorithm summary automatic clarity coherence generic segment represent part song segmentation frame detect repeat along song self segment contain rank segment cluster output song structure boundary cluster middle produce strategy segment similar segment song summary extract measure modification ensure extract piece start calculate compute similarity aggregate similarity song pick summary another method filter filter segment lag frame filter lag classify pure feature frame selection duration human since summary naturally people generic song since human differ create meaningful segment specific allow variability aim human consumption instead specific research effort centrality social focus improve input weight corresponding sentence encode incorporate outer stationary accord convert sentence iteratively select accord number rank far item arrange list follow correspond item fundamental give start centrality rely cosine represent summary central sentence centrality rank graph build edge create sentence pairwise threshold use weight unweighted edge eq convergence total vertex high sentence reach sentence sentence sentence high sentence score text reduce dimensionality original building term sentence element number sentence translate composed sentence apply singular singular matrix relevant sentence select index singular sentence include sentence sentence value never never singular fall value sentence selection diversity low redundancy speech take centroid sentence different sentence previously sentence base diversity sentence represent centrality text centrality sentence sentence certain q set select sentence set recommend eq previously unweighted centrality degree sentence recommend allow sentence oppose sentence recommend corresponding sentence set centrality relate sentence threshold explored set specifically heuristic cluster distance first cluster initialize sentence document heuristic set several metric distance test multiclass music consist classify base scheme song address task hold comprise compare standardized step song summarize signal select process step classification feature feature per song first texture song centroid skewness ms frame solely compose task feature candidate extract consist music several music although differ similar deep solely usually usually influence post country music characterize repeat phrase builds represent several wide vs vs validation classification experiment segment begin middle end song baseline classification summary multiclass dataset yield dataset c library extraction operation summary algorithm music algorithm operate concept extraction sentence song vocabulary frame use assess obtain cluster centroid centroid frame vocabulary frame represent discrete nature song size since sentence discrete represent occurrence frequency exact representation sentence cosine size final sentence cover value size second size word instead weight effort music signal spectral try choose impact classify vs orient nature classify result htb second send thing vs task job distinguish drop full beginning section drop lose full heuristic beginning second htb c c send accuracy binary binary binary set task poor job describe distinguish perform worse vs vice versa segment classification equally reach improve performance baseline classification classification task calculate set analyze music go beyond classification look confusion obtain carefully case understand confusion identically sort ideal diagonal confusion diagonal show individual result classify confusion group sense present achieve accuracy share confusion explain music sharing characteristic virtual produce perform classify accuracy track strong important much information track remove instance second incomplete song blind summarize music extract interpret heuristic extract segment music second whole song time ccccc h classifying begin full classification using drop second low energy contain relatively part whole take begin ccccc htb ccccc classify middle drop accuracy get track way middle segment section though low track part vs nature human would probably unable distinguish classification drop mean segment ccccc f ccccc classifying second end section compare misclassifie mainly mostly share confusion second song good htb ccccc htb middle beginning still average feature feature perform well signal second may happen song sufficient diversity segment accurately represent whole song distinguish structural example accurately distinguish need part song summary classify detect relevance diversity inform important fit summary table claim classify summary vocabulary sentence weight accuracy middle lose section individual since middle perform distinguish summarie accuracy mostly increase summarie diversity several structural remarkably job classify original full data ccccc ccccc confusion section respectively second vocabulary weighting compare section namely select diverse part include able interesting individually increase htb ccccc ccccc confusion correspond difference middle combination second vocabulary word frequency weighting apply performance sentence tend get summary document even appear undesirable bad job describe song aspect frame cluster presence frequency representation section increase explain summary improvement ccccc c confusion show middle word word sentence weight algorithm use overall improvement section improvement explain produce htb ccccc f ccccc confusion classify summary specific vocabulary sentence weight create cosine similarity section lose summary confirm include middle remarkably improvement namely ccccc b htb c ccccc run sign rank confusion scenario second middle full drop full term accuracy e statistically speak accuracy previously summarie diverse amount second audio allow piece automatic consumption could
independence suffer type systematically assign relationship avoid power threshold across relationship equitability define threshold equitability prove equitability converse criterion equitability ask standard relationship low detection tail sample equitability low straightforward interpretable confidence assume say equitability make need imply equitability therefore minimal criterion equitability equitability equitability achieve sort robustness power independence relationship miss intuitive heart equitability low pre fine grain relationship precisely preliminary question argue perform equitability equitability commonly correlation mutual property size eq detection equitability prove serve primary purpose provide equitability context central power language formulate achieve connection current future exploratory independence measure dependence currently relationship whether one relationship however relationship equitability dependence independence understand equitability achievable besides relationship characterization set equitability behavior question equitability certainly develop varied new interesting case particular assessment formulate bivariate equitability change exploration ask author acknowledge r constructive useful legend type color equitability legend equitability correlation information use distance correlation require overall mutual parameter equitability specific david measure increase exploration trivial relationship kind tool nan variable attempt trivial relationship matter relationship meaningful impossible need set characterize equitability measure aim challenge statistic assign g idea call relationship interpretable draw testing moderate distinguish kind relationship regardless strength relationship kind strength equitability thought power interesting equitability evaluate like interest minimal dependence candidate pair variable evaluate follow size grow dimensionality meaningful gene analyze reliably thousand significant relationship percent pair manual usually characterize impractical challenge test examine small poor depend thereby list toward systematically assign relationship pair cause set crowd relationship would examine rank guarantee datum pose identify kind identify number kind equitability previous equitability informally one measure assign equally relationship regardless type theory object interpretable interval type strength interpretable act good belong estimator narrow narrow interpretable explain measure connection equitability interval hypothesis test typical dependence analyze distinguish trivial association independence assumption statistic strength regardless question equitability independence ask detect deviation also distinguish regardless connection equitability detection fix sample minimal relationship minimal relationship kind threshold strictly high equitability equitability converse equitability ask relationship strength reasonable goal example formalism relate equitability indeed analyse analysis equitability popular introduce aim good equitability functional power statistical equitability power dependence understand equitability discussion around equitability equitability accommodate variant allow theoretical equitability allow explain limitation equitability additionally connection independence question concern maximal information defer paper use equitability analysis hope paper equitability method achieve equitability goal setting equitability informally formalism equitability rigorously statistic dependence relationship idea way ideally quantify inference interval invert certain interval statistic narrow constructing use equitability use notion equitability appear discuss equitability practice equitability generally generic accommodate exist potential motivating often functional relationship coefficient determination correspond functional form statistic previously standard relationship interpretable ask statistic reliable sample reliable diameter illustration reliable hypothesis central reliable relevant interval question interval small reliable acceptance level interval interval interpretable interpretability statistic value interpretable interval correspondence coverage interpretable value sample counterpart sample interpretability small interval denote small interval region relationship value black reliable interpretable replace interval reliable interpretable imply respect summarize reliability interpretability interpretable resp say resp interpretability reliability average resp reliable interpretable imagine grain summarize interpretability accord equitability equitability case interpretability equitability interpretability equitability measure often use interpretability mean bad interpretability equitability interpretability reliability sample interval respect uniquely value reliability interpretability perfectly reliable interpretable build give example perfectly interpretable limit perfectly normal deterministic correlation perfectly interpretable reliable set bivariate normal perfect interpretability bivariate normal framework ideal contain guarantee interpretability perfect applie equitability requirement exchange equitability view tradeoff tell give vocabulary concrete equitability variable trivial variable denote functional equitability respect equitability functional functional observe function encounter enable empirical make equitability realistic distribution leave relationship past examine gaussian depend lack description noise easily besides allow might importance modification formalism design equitability functional relationship impossible introduce equitability represent variable conditionally coordinate otherwise interpretable serious first limitation point technical comment extremely arbitrarily noiseless prove model second limitation result address perfect equitability approximate primarily dependence part say notion equitability also mathematical equitability latter define perfect equitability informally equitability approximate notion equitability mutual scheme empirically conclude perfectly another perfectly perfectly impossible equitability property question remain measure analogy science np want approximate solution search appear practice formalism equitability relationship expect broad fact dependence population value trivial widely score equitability equitability evaluate equitability generate uniformly spaced maximal percentile minimal percentile reliable interval reliable interpretable equitability reciprocal large expect contain function assign type pair figure depict relationship depict way analysis interpretable equitability pearson relationship give interval indicate illustrate show relationship large interpretable line achieve limit relationship monotonically proxy relationship correspond equal see notion reliability interpretable intervals interpretable interval estimate define bad equitability size interval speak equitability natural interval arise equitability rich relationship monotonicity sufficient equitability obstacle equitability bivariate gaussians many asymptotically differ motivate exploration scenario relationship different relationship value relationship equitability specify population due effect depict population identical instance might correspond different operate interpretable interpretable sample interpretability picture though fundamentally data measure robustness make large functional parametric asymptotically perfectly undesirable exploration lack e circular leave next thing whose approximate equitability section though largely noisy application application functional relationship decide mutual deterministic possible impossible make equitability property equitability term interpretable via inversion natural ask equitability test alternative answer question equitability equivalently nan hypothese strength equitability power equitability analysis fix statistic reliable state equitability sided interval invert interpretable sense interpretability consider level reliable tail base nan interpretability direction maximal element provide reliable interpretable interval state describe equitability term power definition power uncertain interpretable versa statistic let interest right test hypothesis alternative hypothesis level composite complete parametrization independence give distinguish main result information tail nan view interpretable interpretability precise uncertain x set interpretable vice alternate equitability term two short lemma minimal interpretable give statistic know connection reliable interest level small test supremum ready prove characterization equitability statistical function statistic bad right tailed test power proposition fix equal closure closure illustration show pdf indicate denote interpretable power axis indicate interpretable uncertain equal statement prove claim show simply observe empty construction equal show already tell tail note option give equitability interpretable distinguish illustrated figure pair small relationship dash critical tailed nan curve power define instead heat nan nan critical heat result
period convention active hyperparameter perturbation bayesian optimization et al author knowledge depend reduce sampling hypercube literature avoid room optimize classical stationary poor slightly fewer clearly improve require counterpart find parameter many sample mcmc hybrid monte carlo monte carlo sampler slice recommend benchmark result remove statistically passive simplify surrogate among benchmark gradient low respect advantage perfectly figure apart regression bayesian fast less comparable fraction optimization intend per minute hour like bayesian local minima surprisingly computational cost case heterogeneity continuous categorical present section order reduce cost loop evaluation local fail converge fast real deep dataset row row second combine improve speed achieve art fraction idea surrogate efficiently gradient heterogeneity input optimization automatic reinforcement design popular rely form many rely stationarity able state computer economic function nice numerical point sometimes smooth g however many trial consume furthermore function multimodal although classic become popular method system adapt human machine automatic reinforcement kind box target available trial criterion decision condition improve optimum generality value associate function observation easily summarize generality remainder hyperparameter acquisition representation hyperparameter acquisition upper bind among n paper improve combination achieve reach evaluation improve also exploration drive gain simultaneous estimate variability comment bayesian deal space application regression base stationary many isotropic isotropic gps quite se represent smoothness variability function capture property behavior idea exploration achieve characterize mat ern instead everywhere might unnecessary may variability smoothness different scale nan reward actually attempt function process popular gp project stationary recently idea base input region combination gps gp even local gps region weight decrease second hyperparameter optimization hyperparameter hand bayesian root idea hyperparameter become problem learn kernel hyperparameter explicitly explore area gain hyperparameter surrogate reduce bayesian improvement confidence etc exploration predict principle expectation proposition hyperparameter add purely exploratory criterion need combine annealing focus several iteration gain pdf represent importance expect early result hyperparameter shape find highest relate certain numerically considerably accuracy however use require simultaneous perturbation classic space approximate perturbation although perfectly perturbation bernoulli I algorithm simultaneous perturbation also compute theoretically burden bayesian negligible respect update perturbation anneal many optimize discrete categorical etc model space acquisition optimize available implementation bayesian combine space reduce result
network filter much asynchronous compare weight initialization solid line deviation see line use demonstrate streaming evolve track subspace suboptimal contribution past track evolve forget old discount eq tracking derive follow keep output get eq outer drop last arrive discount descent neural asynchronous update rewrite discount activity local except discount suitably stay sufficiently two discount cost presentation numerical asynchronous modification section eigenvector row eigenvectors subspace initially subspace reach low rate allow fine filter jump rapidly neural filter project principal fall jump interesting lead slow decay extend filter error reach high small fine filter increase adjust neural filter error fall change lead slow extended memory past track asynchronous statistic describe initialization db definition text subspace paper make mathematically dimensionality satisfy possible single network show find neural input subspace show learn algorithm principal principal minimize projection component feedforward yet form anti update initialize stay mention exist filter orthonormal yet maximize input norm component however multiply mutual functional form feedforward connectivity architecture interesting converge filter filter feedforward numerical simulation subspace advantage principled activity inversion inversion local implement inversion iterate similar scheme guarantee guarantee criterion still suffer plausible network yet available stability start dynamic reach perturbation around stationary ever fast available generalization approximation recently bind dependence regret convergence speed streaming limitation neural use formulate exact pair stream reveal dissimilarity problem euclidean distance general dissimilarity distance derivation streaming version dissimilarity current past datum reveal approach datum implementation biological plausibility necessity much impose biology could conjugate method truly see generalization tracking finally similarity representation computation acknowledgment grateful discussion iteratively covariance hand strictly upper triangular triangular substituting iterate convergence symmetric gauss explore value recommend matrix multiply upper move component implement represent feedforward connection neuron plausible require algorithm rewrite result main text calculate evolve e introduce notation weight denote perturb network perturb w express definition relation eq side note stationary f next calculate go relation finally proceed far simplify get final skew matrix row row eigenvalue row remain e future linear treat separately start stability ease express convenience change eigenvalue could calculate eigenvalue stability multiply orthogonality filter would f contradiction eigenvector eigenvector eigenvalue share eigenvalue prove derive plug perturbation neural orthonormal decay perturbation make orthonormal perturbation perturbation restrict calculate perturbation therefore extract subspace neural cm title anti neural streaming medical york college tx keyword multidimensional typically dimensionality streaming adjust principled time rather principled bridge derive plausible subspace stream principled cost multidimensional plausible rule project principal principal subspace algorithmic theory process high input information million cell subspace project subspace simplify plausible dimensionality offer plausible reduction seminal figure point response furthermore transpose neuron eigenvector covariance importantly mean update neuron plausible derive square view compute numerous network example algorithm coupling previous able derive principled single rely feedforward neuron activity network update neuron anti derive perhaps come separately include contribution neuron numerous reduction biological plausibility requirement perhaps originally streaming collection reference minimize wide notation row span matrix rank point learning rule use neuron appeal anti rule b principal streaming datum traditionally dimensionality cost scaling connection neural implementation streaming organize minimize online single asynchronous analytically neuron numerically generate generalization subspace neuron compute subspace feedforward weight follow weight follow anti output similarity center vector space capture product datum product gram gram matrix center identity find minimize discover low cost pca projection rather eigenvector rotation stay evident cost streaming minimize stochastic compute batch output cost keep equality keep large simplifie far grow linearly dominate drop arrive term cost sufficiently admit close inversion plausible algorithm map onto asynchronous minimized coordinate optimal component converge mild assumption produce output meet well paper asynchronous nature subspace input set take principal analytical stochastic algorithm input project reduction rewrite eq matrix filter term perturbation stationary stability principal eigenvector filter cost optimal correspondence filter stability analyse pca network ode stability ode method time network stationary set zero input covariance perform weight update average average rest drop dynamical due neural activity assume algorithm sufficiently algorithm calculate stationary matrix collect relation weight summarize stationary state kronecker stationarity rule immediately stationarity rule equality hence ij e prove filter first equality ik f stationarity state filter partially question filter span eigenvector share unity rest value decomposition eigenvector span combination span eigenvector high eigenvalue need orthonormal row orthonormal row orthogonal skew h g arbitrary imply decompose row row skew rotation keep
recurrent network language explore phenomenon find translation new recurrent implement continuously traditional superior recurrent rnn input language nlp view translation rnn ability encode string elegant rnn task require store string imply short string inspire synthetic task rnn long far neural stack double stack suited processing structure language neural middle ground rnn implement powerful access read write unbounded constant result propose particular able range deep rnn lstm cell learn test string sequential reproduce perfectly input encounter training nlp translation symbolic base free rnns attractive symbolic algorithm expressive beyond capacity model neural network idea operation recurrent stack queue fully control obtain rather parallel explore powerful random operation whereas efficient believe task closely relate work continuous stack unlike operation mix across stack limit symbolic memory stack queue queue describe stack modify queue act stack traditionally operation letting intuitively interpret controller onto stack top stack formally stack form upon controller recurrence receive state stack pair dynamic represent represent maintain change strength first remove object lowest next remain quantity stop next strength current equation dynamic read quantity copy strength quantity preserve read strength scalar output read row traversal stack illustrate third step remove stack ignore stack cast differentiable arbitrarily case supplementary material neural queue stack exception operation read strength high read front queue top operation operate end call end write bottom follow read dynamic stack queue direction equation equation decompose update notational clarity neural except eventually affect read versa three module recurrent state input fully differentiable training controller exchange offer logical size controller controller enhance neural queue give neural stack wish replicate interface layer dot line take recurrent transform setup tuple state stack exception randomly overall read pass controller controller state scalar pass stack well projection projection stack pass stack next read stack tuple controller serve recurrent adapt queue stack module configuration enhance rnn controller controller linguistic non terminal begin root tune generative relatively training terminal balance terminal terminal generate terminal generate vocabulary word class reproduce level syntactic divergence english phenomena challenge unbounded recursive space include purpose subject rp gender free article gender translation english sentence noun infinite conjunction noun neutral article sample sample change observe measure well capability beyond length train sequence separately unlikely ever set read symbol begin next symbol take maximally likely softmax symbol produce correspond corresponding sequence generate formally end batch correctly predict th task benchmark evaluate stack queue enhance run size stack embed hide architecture stack queue enhance two deep come extra memory module regardless logical htb c stack lstm queue lstm minibatch across used gradient run seed number generator training calculate batch accuracy batch overfitte train coarse fine grain accuracy architecture deep benchmark well automatically lstm benchmark similarly across random try stack enhanced lstm yield sophisticated c lstm queue lstm pt layer lstm stack lstm pt lstm stack lstm stack queue outperform benchmark stack queue enhance lstm partially enhance whereas drop experiment gender task enhance order enhance htb sequence serve controller learn nonetheless regular benchmark expressive power ability perfectly notable finally queue solve stack solve simple controller operate solve flip sequence half reach token success deep lstm benchmark exploit short local dominate dependency overall rapid manner sequence enhance enhance unbounded memory capable act stack queue task task benchmark accuracy enhance accuracy require considerably
path explain insight imply bn add dag allow factorization effectively avoid tree observation help first bn alg htb bn terminal bn build build bn create create variable terminal alg creates hide build observable appear line create alg point observable variable root htb add add retrieve cache root terminal root create th find cache root alg build add observable variable variable hide alg build obtain find add build observable basically recursive scope terminal node child scope create variable associate child respectively current product node recursively child equivalently alg contract obtain alg bn add fig x alg indicate show add alg verify parent alg iff appear root terminal consistent parent contain correspond def bn construct encode induction height height bn alg consider root child root say iv ir iv ir ir contradict construct bn forest root height hypothesis separation rule note component follow child scope present variable take edge except add fact appear refer eq hypothesis complete thm bn observable scope include otherwise node increase construct hence consider terminal node node node variable root one edge variable node edge add graph size add alg alg construct bn consider child visit line create hide connect root observable alg alg alg alg thm transform know bn table bn tree tree size exponential bn representation generate power compactly alternatively bn add convert ac root leaf add pre ordering extent restrict add add normally refer detail add alg sub contraction node node pre order add add symbolic x x r r r r r x x htb symbolic symbolic appear weight replace common intermediate process allow symbolic operation internal add add view symbolic add symbolic add alg return symbolic simplify topological elimination use help symbolic add elimination detailed symbolic add alg recursively simultaneously root root node line root label variable line variable recursively simply create symbolic share rooted long share root alg occur contradiction share variable add rooted hence scope occur hand appear scope apply share root node appear multiply child line add rooted child correspond sum branch appear node two scope root link node symbolic create process alg line simplify symbolic merging encode suppose connected node symbolic add remove connect encode unchanged sum alg replace symbolic label elimination alg alg alg respectively htb add ordering add x hide variable ordering multiply alg alg keep symbolic add give bn return multiplication symbolic effectively redundant hence alg represent distribution illustrate bn one htb alg sum apply alg alg apply alg otherwise global ordering sum add topological alg implement preserve alg also view apply result bn add alg alg build bn add alg multiplication importantly branch product share scope view product alg end hide leave may add begin multiplication hide operation alg bound thm powerful informally exist polynomial represent key recognize proposition add record bn actually store sum product online quickly field bipartite relate depth low width bn consider height node sum path accordingly bn hence bn clique size minus tree reach bn author exist family much I substantially internal give convert exponential exist convert bn width high high inference precisely enable add compactly represent distribution width machine undirected network exploit practitioner resort require bipartite diagnostic quick medical causal independence insufficient present establish precise provide constructive consistency analyze tree also learn add correlation directly correlate causal relationship explore since bn convert example establish connection product network network key insight algebraic diagram add compactly bn independence generate bn direct elimination bn history inference process help paper depth recently tractable deep inference distinguish network task need approximated inference broadly clear introduction seminal understand expressive joint clear convert occur small bn encode belief lie context belief ignore due correct direction representation bn done translate bn compact prove adopt algebraic diagram add bn space generate bn structure bn generate bn add time constructive add give understand semantic bn recover relationship third normal bridge direction produce convert reasoning model counting count I evaluate boolean weight sum truth assignment important stream exact weight relate style exhaustive decision diagram decomposable form circuit broad field divide answer knowledge language target key shift common phase offline phase study recently convert mutually exponential closely relate decomposable formula dag enable formula count reader discussion class probabilistic discriminative propose apply classification later structure directly field promise activity modeling modeling investigate quite provide examine introduce capital random bold capital letter bold vector omit subscript variable letter denote subgraph arc refer need material reader familiar skip subsection whose characterize support characterize dag edge bn bn variable parent bn encode among topological bn bn variable bn conditionally independent probability admit reason comprehensive definition diagram domain algebraic diagram algebraic graphical function root dag terminal whose internal edge allow boolean discrete internal label take represent set henceforth add local constructive representation representation local branch internal node alternate node node bn generate decomposable boolean distribution allow add allow consistent transform network derive proof note refer complete let terminal complete decomposable sum every terminal distribution scope scope terminal nod complete decomposable f complete consistent decomposable let topological terminal node topological order pair iv iv network root expand polynomial law sum example iv apply iv jx x mf expansion otherwise generality expansion I eq root principle intersection scope due removal ensure local respect affect polynomial transformation alg complete decomposable v iv v root contract transform consistent informally iv union appear kind inside line induce root use nonempty intersection child build multiply link keep unchanged create node decomposable remove unchanged network affect line example transformation htb construct decomposable sub highlight highlight red outside dash
set connect balance even without membership constraint balance function hold partition function sf simplex equivalently continuous solution c cf note round trivially yield balanced relaxation exact e partition construct f k obviously simplex constraint problem membership imply exactly element I sf c lc indicator partition constraint enforce belong avoid column proof long indicator round constraint round solution continuous balancing round yield theorem control idea suggest image variation piecewise follow cut sufficiently propagate neighbor k sf c approach usage constraint symmetric balance asymmetric balancing employ prove result desire trivial balancing cut optimal construct non negative asymmetric feasible achieve low partitioning relaxation f l f feasible homogeneity indeed continuous relaxation obtain argument asymmetric set matter balance zero show bad asymmetric graph c c ff k homogeneity theorem introduce relaxation balance asymmetric balancing also relaxation indicate dominate illustrate toy desire relaxation enforce converge continuous solution round converge degenerate dominate generate yield partition solution round converge color point apart relaxation key paper problem reduce like guarantee moreover work negative balance monotonic eliminate introduce sf k sf hence proceed iterate approximation iterate result feasible monotonic ratio submodular convex element iterate lc j l definition subgradient sf sf amount change ratio decompose sf l rewrite break constraint relax I f I cluster unlike vertex automatically discuss use minimizing solve follow q feasible l sf l sf terminate theorem state predefine monotonic sf kf tt terminate inner strict terminate inner eliminate smoothness introduce additional lp lp material edge new equality ij l optimal constraint decrease finally rewrite yy order propose class saddle vector huge appear saddle introduce lagrange optimal value indicator negative elsewhere b saddle point dual q primal matrix introduce practical diagonal row completeness explicit dual iterate l k lagrange multipliers simplex ji primal iterate otherwise diagonal matrix adjacent vertex dual iterate give q per reformulate lp integrate label overall solve constraint sequentially via round see repeat double compute l b large belong natural fix top vertex membership minimal increase another lie center cluster membership depend well constraint degenerate stop see htb initialization sf li c kf sf kf c rank j membership f f f diverse method algorithm superior publicly code default initialization cluster similar initialization addition dataset variety uci repository nn weight news experiment term balanced balancing table report fraction construct per cut cluster font balance outperform across result show recursive bi affect cut directly asymmetric ratio cut significantly qualitative degenerate solution pt ccccc well strictly strictly good good cut balance method cut case integrate truth help cluster performance one fix percentage label truth cut initialization strategy work cut unlabeled method fail completely class news minimize cut continuous relaxation specific balancing splitting approach method enable integration link monotonic difficult ratio contribution acknowledge grant lemma sketch base cluster exist computation balanced heuristic weak original relaxation cut recently relaxation loose relaxation new minimization achieve outperform approach ratio cut pairwise formulate balanced cut cut
I state exist transition count occur state merely count without previous state take avoid characterized parameter prior tendency tendency markov chain depict explore short long coincide correspond stem perspective derivation section dirichlet reason gibb elaborate pt recall interested posterior gibbs turn moreover full sampler turn variable emphasize dynamic conditioning merely count leave characteristic chain sample concrete transition restriction unchanged probability suppose force take value precede regime join regime regime exist concern series alternatively think assign nothing regime suppose take table take innovation count different htbp py py currently sample normalizing mention sampler simultaneously transition prior draw take place conditional section study shift state change inherently estimate burn pass markov hence theoretically begin convergent initialize rather allow grow suppose reasonably change state chain normal shift change variance gamma derivation full conditional specify specifically occur realization see overlap range htbp change inverse gamma hyperparameter variance conduct burn sample period reduce dependence figure show intersection line demonstrate break true number specification replication detect change correctly point number case point estimating model gamma subsequent approach maximum posteriori estimate obtain newton together parameter result slightly burn great indicate explore besides model metropolis hasting walk value previous standard normal incorporate show table conclusion explore correctly change simulation bayesian parameter modal therefore empirical model first apply model mining poisson plot assume case prior prior perform sample metropolis hasting sampler burn sample reduce sampler draw sampler estimate probability indicator point intersection line location interestingly produce exactly one identify occur posterior deviation use deviation standard match literature robustness prior robustness replication collect replication detect one point find without number point number structural let index change frequentist autoregressive structural subject prior gamma hyperparameter sampler sample draw regime exist year change second mean deviation consistent estimate replicate whole detect change number suggest replication detect point model sense misspecification need change hyperparameter three discrete poisson result empirical dirichlet true change location appendix give conditional sampler q conditional gibbs obtain unknown discuss department finance business economics university economic economics finance chinese international economics university china edu quantitative management hide markov specification number specification model need sample state around normal mining united provided detect change growth model assume probability chain derive depend propose change change recent include change introduce random indicate regime specifically unknown period remain vector assume parametric parameter indicator take model transition regime point hide markov specify alternative vs multiple introduce hide right dynamic impose restriction appeal specify determine method state utilize dirichlet mixture provide introduction process study section discuss provide dirichlet technique derive dirichlet
lastly treat inversion still process work relatively obviously rely requirement set actually rest reasonably range accurate observe use hyperparameter recommend ht component component posterior component component last posterior ht component role propose complete spaced treat play another role univariate quick evaluation fitness evaluation would auxiliary although seem trivial truncation dimension sampler avoid approximate truncation stick break event break transform multinomial event model probit binary outcome certain difference finding full fortunately make easy simply l child functional analysis nonparametric spatial functional analysis provide nice interpretation regard sophisticate simple put novel assume latent utilize spectral enable miss transformation datum surface air spatial additive separable functional rapid interpretable spectral non nonparametric dimensional easily probably commonly spatial equivalent add base kernel therefore smooth light wide computer inversion fit solve many classified category reduction learn nystr om approximate top another approach spatial function thereby simplify refer utilize location reduce bottleneck multiplication one concerned whose determination resolution cost interpretation become commonly example correlation decay strategy issue focus spectral discover nice eigenvalue density time lattice school advantage firstly operation inversion completely avoid multiplication carry fourier complexity secondly exponential ern lattice rely property finite non effort limitation bayesian procedure treat lattice one operation stationarity local stationary process integration address call reduce approach issue solve bayesian example demonstrate capability dataset section briefly review study regard likelihood dimensional study lastly utilize consist north american dimensional gaussian z fourier transform covariance complex number angular frequency symmetric conversely represent fourier transform approximate review obvious lattice regular distance location increment location diagonal follow verify c conjugate straightforward eigenvalue inversion determinant express omit weakly stationary regardless assume always secondly frequency real backward dft dft either evaluate need dft operation multiplication operation involve operation dft lastly commonly increment location appear g one undesirable fortunately easily augmentation nuisance section substantial computational advantage see application two restriction sufficiently high lattice initially datum partially lattice lattice location beneficial noisy realization miss update complete smooth process degenerate form spectral scale range vector interest miss realization around use eigenvector transform imply note likelihood notion low covariance generalize value lack within element regularization definition covariance help one traditionally parameterization affect formulate widely function form ern close fourier covariance improper originally tr xx spectral domain xx quite efficiently range interpretation consist covariance utilize miss independent copy equivalent I distribution inversion sampling efficiently dft dd involve conditional variance index way would need create bottleneck avoid simply normal parameter update inverse gamma parameter metropolis hasting one focus however reality phenomenon edge appear corner observation instead rather dft dft represent distance cx kn knn cn remain proceed reverse fortunately exhibit edge know small value since manner point nuisance edge obvious double prevent edge appear practice repeat close zero one adaptively number negligible effect height issue might rely accuracy requirement solve lattice rest reasonably range quite even find recommend functional spectral approach counterpart bayesian inversion approach maximum estimation implement software cost lie subsequent failure cause correlation process either drive neither aforementione package square exponential ratio correlate reasonable possibly stationarity convolution location location small parameter assumption mixture process collection process define stick confusion frequency unless state context flexible stationarity capable address gaussian distribution worth correlate vector dependent consideration mean vector smoothness take vector stick break realization address flexible formation functional probit apply element range functional efficient functional process condition set update inside different besides functional gaussian essential probit correlate univariate multivariate distribution independent secondly assign component th multinomial sampling illustrate multimodal height ex n dots datum frequency frequency difference appear cluster stationary within subtle correlated curve switch middle process pattern become indistinguishable correctly estimate weight carry suggest interpretation transition suggest smooth suggest gaussian distribution process north american assessment surface temperature collect weather forecasting exclude year induce year intercept second second assume gaussian kx sx smoothing call filter choice spatial keep spatial additive last forecast correlation conditional space affect filter performance modification spatial multiply spatial accommodate benefit smoothing represent therefore drawback still assume limited additional example quite flexible describe model reduce separable ideal filtering create result impossible computer however easily solve ns functional mean process weight distinct exhibit change fouri three datum evaluate forecasting posterior mean location forecasting run take minute sophisticated hour converge converge smoothness parameter change subtle range autocorrelation weight allocate domain help nd rest change mean thank evolve change fix l rmse rmse ns fit l rmse rmse list rmse forecast complex forecast suitable filtering purpose interaction separable slight illustration plot temperature height height ns ns decay location independent contour covariance whereas latter parameterization application surprisingly sometimes unless close spatial association exclude alone ratio term density criterion cause curvature time change framework spectral new back despite ease estimate signal processing often locate characteristic prevent fouri dft study framework assume observational partial noisy realization surface miss underlie requirement integration benefit dramatically reduce allow feasible rather recently gradient compare major underlie enable sampling provide cause comparison study
g qx n minimize estimation slice challenge difficult challenging aim set framework estimation recall sir linearity reduction follow condition easy necessary sufficient version linearity let obviously identity hand side e therefore sir interestingly description linearity allow unable equivalence describe model careful consideration focus generalize concept classical new find flexibility dimension properly extend data theorem equation inverse regression reduction pt li chen tu institute statistical email tw south email abstract inverse base rigorous possible reduction formulation unify mahalanobis sir discriminant naturally reduction space measure separable phrase dimension functional linearity traditionally space column satisfy em variable problem inverse sir lead eigenvector sir construct linearity condition require x method angle normalize covariate new dimension reduction enable simplification permit reference therein model univariate response vector product covariate think functional straightforward may g inner perform case form dimension correspondence covariance consider vector operator space correspond careful rigorous restrictive message relaxed investigation formulate reduction problem subsequent mathematical kernel summarize outline either understanding motivate wherein dimension function fall require model model together sir standardized restrict equip inner bivariate linear operator say operator v ds dt ds induce positive semi definite definite equality definite strictly ball define map say ball bivariate induce xt xt easy verify continuous function integrable hence continuity chapter definition semi take expand convergent series eigenvalue eigenfunction sometimes write strictly orthonormal strictly still complete regression study semi integral integrable independent random obviously well solve problem domain hand require indeed sufficient classify group categorical feature reveal response categorical phenomenon categorical normalize within covariance form conclusion domain purpose classify response xt example regardless slice present main link reduction relaxed reproduce hilbert induce define range role equip interestingly product mahalanobis covariance study relaxed eq boundedness induce norm f bound il new compose three well rigorously extend stochastic xt st proposition exchange order double integral jensen apply eq large flexibility dimension define surely e reveal belong analysis dimensional finite component functional ensure integrate arbitrary variation variation degenerate sufficiently finite much subspace subspace ensure inner obviously I equivalent linearity condition quantity verify identity sir main problem lead requirement subsequent etc functional operate restriction lead relaxation flexible useful st define proposition therefore
decoder constant rip q combine column observation kk dt c singular good nonzero entry small quantity satisfy v union low denote exist inverse constant yy thus unique minimizer analysis c hold triangle unique minimizer sufficient exact minimization minimization relative hold minimization rewrite reformulate eq thus polytope positive vector hull x minimizer minimization c q c kt express vector q easy identity write equation rip additionally right side side I inequality solution second triangle set condition number q following hold q depend generality c integer lemma lie polytope vector q right eq want
previous state wavelet put probability ball satisfie consider wavelet continuity conclusion coherent investigate nuisance sophisticated regression model handle metric prior investigate slightly result handle adapt impossible mainly come function wavelet measure dr value retain nonparametric consider case design ahead covariate fix neighborhood design n n n sequel z ahead compact uniformly spread sense partition lebesgue covariate identically assume bound lebesgue consistency theorem kernel coherent wavelet whereas theorem soon satisfy may quite explain statement correspond coherent coherent state dp c dx true know let wavelet dp dx pd statistical assuming give let denote joint coherent necessarily complex window put positive function consider real wavelet wavelet situation get value kernel fix mixture begin situation make nature kernel function concern mixture coherent wavelet independently identically distribution denote variance stand product positivity neighborhood n existence exist function f c model dx computation let finish dense wavelet coherent converge lebesgue measurable proved exist converge previous follow write inequality nk existence test miss recall exist p f f f also increase eq f enough metric entropy desire concern ii finish existence test need correct complete metric equip locally bound closure series representation series intermediate compactly ensure continuity computation without generality moreover check bind eq choose small entropy desire remain prior probability decay notice nu nf replace consistency gamma consistency theorem verify weak satisfie condition gibbs mixture kernel duality dirichlet dirichlet apply I dp reversible thresholded approach rely measure chain set allocate dp inspire p tf almost surely sampler sequel particle unique iteration successively pn l stand allocate evaluation require knowledge hasting I metropolis scale parameter wavelet cn rmse wave spike corner cn rmse rmse wave angle spike corner run simulation find dispersion criterion run htb wavelet seem offer take carefully change dramatically estimator already crucial estimator opinion mixing prior fact wave coherent state wavelet theory choice carefully physical consideration geometry guide discuss wavelet candidate perform situation computation technique credible band credible good sample band efficiency done develop new necessarily kernel generalization done believe gaussian result measure weak consistency slightly different arise use outside measure properly prior whole care propose easy consistency allow translate fact dirichlet appealing decay jump dominate lead sparse reasonable bit conclude result desirable occurrence coherent quantum physics front square integrable group consideration could eventually guide right example namely coherent possibility entirely totally minor modification acknowledgment grateful helpful throughout article list laboratory france regression problem underlie probabilistic analogous density induce mix contrast function topological fact class way jump treatment locally separable variable definition poisson surely mean characteristic convenient interpret characteristic disjoint surely atomic let ia ia ia variable mean hence proposition mean purely almost surely number almost surely atom atom recall value theorem mean linear surely completely functional product jump jump either context random assign borel l consider functional spirit strong condition random say l consequence almost purely surely atom surely jump coincide atomic analogy case moreover measure homogeneity first z dx reveal context throughout gamma homogeneous letting turn onto characteristic aa two gamma gamma characteristic appear simply distribution dirichlet study bayesian literature complex computation derive dr assume support define borel gamma completely measure classical worth note satisfie strong convenient construction formalize next proposition variable v pi vx surely notice derive happen gamma normalization value scale variable integrate carry paper situation could obviously shall kernel overcomplete hilbert banach desirable function basis expansion coherent wavelet kernel promise leave haar space preserve representation irreducible strongly square integrable eq continuous subspace unitary irreducible say invertible carry simply identity normalize integrable mod carry bound onto closed sequel slight abuse notation stand inner complex product linear df real group endow canonical euclidean compact hausdorff haar unitary irreducible strongly however borel localization rewrite define stand mapping connect affine know endow topology topological hausdorff group leave haar irreducible well space subspace fortunately unitary irreducible strongly integrable unitary irreducible continuous allow irreducible call wavelet wavelet follow condition basis see previous wavelet basis wavelet dictionary b deal supplementary restriction kernel coherent topological homogeneous topological locally hausdorff interesting let locally q follow devote let u automatically verify dx u recall u bound rhs rhs g satisfied wavelet admissible b remark compactly take arbitrary recalling lipschitz rhs last rhs term exact thing let follow merge abstract measurable prior follow investigate property go
nature experiment estimate collective perform unsupervise move supervise formulation demonstrate base yield std mr lr see regularizer method oracle note though mr successful baseline lr svm show loo difficult illustrate actual accuracy compute nest loo loo whereas select value hyperparameter enforce result many bank loo colour influential weight introduction avoid also character letter shorthand else tweet truncation weight consistent sign suggest avg whether output assign inside train ard good spectral relate medium news reporting broadly relate central high mr benefit additional supervision mr drop bank explain effect case bank tweet positive positive negative classifier learn non stationarity collective unsupervise supervise domain find loo achieve return inspection learn reveal automatically data access annotate learn difficult allow minor difference stationarity community yield result baseline inspection correlation quite enable justify apart future explore feature contribute exhibit social ac media community determine belief collective ground formulate collective annotate reflect characteristic share annotate several thousand tweet seven successfully increase interpret act upon social medium especially circumstance spread twitter active turn attack child social medium text reject really trust setting access exhibit characteristic linguistic little annotated outperform multi collective great topic reaction broad tweet deal thousand annotate training build sophisticated novel classify collective adaptation show successful multi spread media flow group source tweet flow rank proxy information flow manually explore twitter along user former website match facebook false receive comment website none automatically automatic medium detect political twitter task define piece cascade classify cascade credible difference entire necessarily carry whereby classify classified support train identity pool tweet transfer classify unseen hard classify tweet set tweet temporal community r bank false twitter dataset annotate manually study medium medium support include tweet corpus tweet class refer corpora various corpora label modelling automatic without supervision five finding true rare primarily interested community nevertheless strong ratio ratio support vs tweet prove true third ratio come prove break many nevertheless overlap confirm event tweet first loo training transfer set leave formulation half training leave set may tweet classifier logistic word addition rbf first use naive nb component log treat mr setting jointly towards training tweet feature tweet gold bias regularizer learn present training use vector exclude underlie signal useful gaussian well nlp art widely use process explicitly handle imbalance process generalization mr central latent function specify degree function radial rbf lk latent map probit interpret q j integral intractable technique various calculate posterior propagation refine kullback true conduct parameter evidence ep gaussian experimental handle nlp compare rbf rectangular specify relate control way fact simple empirically frequentist lr use set unsupervised adaptation loo hyperparameter selection via nested loo hyperparameter
sequence get dd consistency let I write follow fail convergent basis focus multivariate transform consider obtain may row project orthonormal basis calculation obtain eigen structure generality independent ta ta rewrite obtain coincide easily obtain note basis estimate immediately functional discuss property concern prove cauchy sequence previous variance subspace add follow ty ny tt acknowledgement cover thm department universit di mathematics di via year concern explanatory curve curve instance chemical variable predict environmental weather pattern temperature variation year like spectra classification classical technique functional estimation contact range deconvolution reference base see reference term eigenvalue eigenfunction basis something properly justify proper give procedure automatically neither functional real link compact call identifiable space estimation pose related moreover reason come discuss influence orthogonal firstly parameter inferential large dimensional size introduce auxiliary analysis carry consider functional model deterministic collect correspondence treat among notation inner norm assume realization neither distribution quantity main focus describe functional paper small sub call coincide random support small space coincide function orthogonal odd present center straightforwardly verify analysis well straightforwardly generalized context entirely functional estimator least square finite classical particular mild pose compute reconstruction provide minimum situation avoid choose formally lie sub reconstruction consider uniqueness restrict reconstruct identifiable search sub thing perfectly projection component datum orthogonal vanishe searching suggest guarantee determine finite sub typically reconstruct imagine actually answer discuss central rewrite slightly operator I orthonormal explore relation describe orthonormal basis analytic since identifiable dimensional projection element square minimizing obtain behavior mention wide behavior sake define replace follow q estimate dotted belong eliminate priori choice observe irrelevant related quantity pointwise differ component hence sum influence explanation phenomena lie divide simulation simulation appendix mainly consider sub space situation simply fact influence analysis contribution take reconstruct among figure among pointwise mention due fact correlate put related close well contribution observable compose eigenfunction basis construct solid dotted coincide subsection bias concern introduce trade tp operator relation obtain eigen denote eigenvalue respectively eigenvalue one hence projection direction I total distinguish bias choice sub identifiable minimize suggest estimator situation estimate variance estimate discrete naturally chance minimize variance choose space ease notation variance coincide component priori different space principal pc minimize bias describe panel panel pc pointwise discrete prefer chance minimize choose discuss arbitrarily compute infinite closure countable I construct investigate asymptotic dd arbitrarily make consideration mention identifiable curve greater implicitly arbitrarily present consider analogous space define
estimate examine prediction n prediction error match theorem slope thereby scale theorem triple block zero normalization prove appear suffice sake shorthand scalar loss generality bi column make hold swap swap side moreover every minimizer lower complete convenient event condition may two zero independently piece show conceptually similar detail proceed derive contradiction finitely convergent unbounded claim let sequence beginning write definite unbounded guarantee must claim imply boundedness proceed monotonically guarantee minimum point ball eq contradict specification prove claim always minimum iterate shorthand center ball disjoint volume within air force office research fa office convenient priori let sub design minimum hence bound bind u minimizer interior say interior remain prove low ii since ta notice combine yield non global definition hold define triangle inequality find yield proof convenient omit shorthand representation construction matrix update statement suffice event ib claim final fact thus iteration assume integer integer suppose minimize inside ball function turn eq fu fu minimum contradict f prove argument induction well claim definition prove argument minimum belong derivative namely prove contradiction mean calculate fact assumption establish claim second equivalent hold chi square inequality ccccc berkeley edu department engineering california berkeley high estimator absence restrictive condition slow intrinsic broad estimator estimator square function together local optimum associate bind apply optima popular nonconvex regularizer scad penalty mcp addition regularizer optima broad local minimization typically bad minimax possible problem order feasible implement polynomial study computationally estimator question coincide differ fundamental classical counterpart explore gap classical risk high regression minimax broad coordinate separable regularizer include regularizer linear throughout entry matrix variant integer sparse parameterize triple ambient dimension use denote measurable take quality response obtain estimator denote constant deviation compute manner subset motivated heuristic basis pursuit selector replace constrained covering reference focus design satisfie certain restrict property know design matrix computationally efficient matrix conservative prediction error lasso question establishes result complexity condition polynomial achieve hardness possibility dense polynomial show slow namely base square wise decomposable consider choice least regularizer popular pursuit square nonconvex regularizer mcp nonconvex local argue bad optimum statistical concern focus descent isotropic result way paper gap dimensional closed application broad conjunction add fundamental gap remainder definition estimator illustrative prediction achieve statement provide lemma defer discussion previously link linear predictor prediction analysis regularization separable regularizer example analysis provide bound estimator risk adaptively ordinary instance counter slow achieve rate nonetheless design adaptively error square root minimizing criterion different root relative advantage require purpose lagrangian lasso minimizer square varied root apply square root due form nonconvex penalty due fan li mcp family scad penalty similarly mcp mcp regularizer illustration scad regularizer turn precise good matrix lasso achieve prediction information theoretic avoid include restrict good easy recover intrinsic et strongly column prediction absence constraint say simplify notation follow consider normalization integrate tail follow range could main theorem discussion consequence local minima local open define object triplet typical descent method converge applicable local sparsity separable coordinate eq statement see low estimator result adaptively choose nature allow choose minimum execution convex provide minima match bind separable regularizer establish existence bad adversary power minima concern typical optimization function local general g iteration also algorithm unify begin observe iteratively give stepsize approximate minimizer ball radius neighborhood thus iterative current algorithm choose close tie randomization terminate minimizer belong interior loss define global nonconvex ball powerful observation impose regularizers origin jx iii
operates score binary word minimize error train principle layer activation batch word target lexical translation lexical word take multiply sentence translation restrict book avoid difficulty sentence calculation word calculate sentence naive pre probability slow calculation source sentence comprise pre calculate target phrase word whole corpus phrase translation configuration add component corpus log utilize house phrase decoder nc corpora train correspond corpora plus news package big corpora domain describe conduct reduce sparsity describe house phrase decoder search hypothesis error weight optimize consist contain k consist investigate impact describe model corpora mainly corpus quite speed process testing except validation table cc avg send avg include linearity consume boost unknown value keep hidden gain strong baseline source context whereas use source show impact extract source appear l fr sc contexts include help case case good improvement cc system fr sc sc source vocabulary size indicate context architecture architecture consist train hour hour whole translation time decrease use architecture deep stick architecture big use big corpus quality domain similar model broad domain report corpus conduct experiment english order baseline would effect language long build similar integrate translation table cc system sc translation increment baseline come frequent word context baseline frequent source context notably sc phrase enable abstract representation dependency decoder help baseline linguistic resource preprocesse process linguistic resource translation language future try integrate linguistic feature might fp edu contexts discriminative deep neural network leverage dependency linearity model reduce approach art pair translation task attempt approach draw community huge address depend context standard phrase translation translation phrase pair extract corpus phrase segment context phrase leverage wide context phrase discriminative exploit predict wide feature discriminative hence translation dependency source perform abstraction input linearity lexical translation rarely share could model well address issue discriminative lexical train neural share word source linearity exploit semantic dependency among organize lexical translation translation model neural network provide experimental result lexical lexical decision present phrase base employ translation lexical advance propose either enhance model context approach directly target sentence gram phrase mt thereby boundarie phrase gram basically build principle joint linear relationship le gram translation phrase long context joint word source aforementione work essentially translation inherent exploit sequence word global context motivate et two belong another lexical approach name predict word model employ classifier perform lexical word phrase extract rich source sentence input opt network architecture employ global original describe finish decode determine individual train train word give sentence sentence represent sentence bag indicator represent vocabulary example target neural next sentence
analysis proceed summarize pca performance area diverse expression implementation package comparison robust pursuit discriminant pp projection consider datum remove along find pursuit pp projection dimensional reveal detail interesting surface etc extract analyze widely blind source pp projection extract seminal like pp variety pp outlier pp high space combine definition dimensional four seven later though technique fail intrinsic dimensionality existence intrinsic ensemble often data optimal typically unstable tend get contamination contain cause aggregate bagging offer one variance aggregation aggregation combination form equally weight th base regression bag outlier class come bb bag vote frequent replication namely bag learner bag b paper bagging help achieve robust classification space bag datum contaminate estimator subspace attribute bag proceed bag add crucial consist select building randomly draw replacement index build ingredient estimation bag tree make ensemble base replacement sample drop dimensional build base learner base b rf max max pair training size generate replacement denote turn base deduce outlier replication base aggregation bootstrapping mechanism learner performance life thing apparent training true error replication define throughout randomly assign consider none tune replication six microarray gene clarity completeness present performance explore analyze diabetes deal obviously dataset reveal difference performance method small available package ii please dataset qualitatively explore microarray gene expression iii data microarray subset dataset classify recurrent recurrent tumor observation cancer cancer dataset vi contain subject classify fail pp huber pp rf diabetes microarray stability strong aim robustness notice unstable producing despite initial finding general thorough study determination well generate density contamination capture contamination scatter furthermore study pattern parameterize identity dimensional one classification throughout paper use consider namely contamination mild contamination contamination performance e simulation look reveal pp appear pp pp huber pp categorical pp performance discuss much pp typically pp least low pp notice carry rate contamination clearly show contamination among rf whereas pp fail pp fail ideal namely binary large correlation consider technique contaminate regime number class investigate space dimension technique fact favorable regardless contamination htbp fail notice rf well strongly regime look combination class effect htbp htbp see figure contamination categorical contamination well take rf predictive rf realistic contamination thorough predictive performance robust interesting large rarely yield technique one remark pursuit seem matter fact pursuit fail regardless aspect two predictive suited lie indeed overall random forest yield explain early mechanism forest estimator subset crucially mechanism leave proportion certainly indicate early forest inherent forest robustness subsampling attribute fact selection inherently address dimensionality elimination outlier subsample sense connection however loose yield determinant bag forest relationship forest currently way acknowledgement express infinite ever especially flow powerful proposition conjecture corresponding institute university mathematical sciences institute ny usa pattern recognition subject research year explore performance special concentration variety forest ensemble inherently specifically robustness focus classification dataset precisely large class may consider building throughout shall th estimator build portion split recognition create solve precisely amongst discriminant classification tree forest tree relevance name datum instance state namely call small since common day especially microarray gene disease cancer matter consider six contain various namely brain traditional discriminant near neighbor mainly lead method even solution severe curse loose ill several achieve discriminant analysis microarray expression regularize regularize logistic dedicated typical contamination get ever outlier relatively extremely high ill outlier literature discriminant scatter apart discrimination fact reveal covariance estimation context approach explore stage technique get life datum reveal test context simulate contamination scatter matrix correlation value contamination location rest paper two present brief emphasis use limitation method life five dimensionality contamination correlation various choice prediction replication section six brief introduction work dedicate arguably group estimator discriminant object whose discriminant give gaussian presence discriminant lda share sample group membership discriminant give ik observe membership finally pool turn make robustness performance outlier explanatory cause discriminant bias need method scatter much deal version regularize application extensively area need subsequent contaminated dimensionality combination robust pursuit presence author aim know minimum determinant robust discriminant recently explore extension compare extension explore also performance one extend mention package immediate scenario contamination technique
ensure symmetry upon inversion back spatial subsection propagate gradient fourier transform layer apply inverse dft input dft remainder dft propagation gradient dft introduce symmetry distinct choice technique function project onto idea pooling stem observation input discuss technical detail advantage h map output pool cm n implement output map dc shift maintain central submatrix denote approximation take dft list conjugate symmetry special case subsection break truncation value treat individually effect various procedure intuitive supplementary material address dft dft layer apart gradient appropriate note convolutional employ pooling implement additional dft regardless spectral spectral significantly retain pooling representation information resolution precisely linear low pass uniformity respect mass frequency elimination reconstruction suffer dimensionality pool specifically freedom input specify arbitrary gradually pass small frequency outside central maintain output dft resolution stochastically truncation truncation axis outside nest e uniform high resolution filter cnn directly frequency domain advantage empirically layer seek frequency domain attain inverse dft cnn input mini batch back dft identical outline discuss change cnn explore epoch filter frequency transform domain representation pattern considerably considerably update cnn filter across domain weight capture degree freedom hand provide appealing basis characteristic often localize spectral representation observation filter tend narrow qualitatively speedup descent linearity exactly regardless whether frequency transformation space parametrization meaningful relevant modern rescale able leverage axis small optimizer element exist number promise future elaborate discussion lrr pooling maxout imagenet fraction parameter keep measure error achievable pooling b augmentation optimal pooling network effectiveness run optimize hyperparameter evaluation validation imagenet result pooling observe subsection spectral pooling permit number pooling control severe quantization bind preserve horizontal permit produce smooth choice pool filter size spectral layer output layer filter convolution relu nonlinearity height r think parametrization dropout layer decay weight decay epoch dropout augmentation hyperparameter want success perhaps hyperparameter assign map randomize constant setting attain cifar cifar competitive employ architecture deep generic sp remainder past match mark light achieve speedup architecture negligible parametrization cnn optimization architecture different pooling layer size architecture attain competitive classification deep equation increase difficulty optimization reflect considerable horizontal dropout spectral filter spatial spectral fourier transform attain variant optimizer find figure convergence speedup table surprisingly negligible speedup expect much room exploit spatial work rich spectrum spectral pooling allow pooling dimensionality significantly addition parametrization fast frequency employ fourier transform convolutional ed multiplication ed desirable transformation forward sensible nonlinearity switch significant dft difficulty impulse involve sum hence locality fouri locality spatial locality employ wavelet provide wavelet throughout machine great learn cnns center research support mathematic office science advance compute research department contract ac resources national thank school university school engineering science discrete transform speedup computation deep efficient computation cnn employ cnn perform dimensionality considerably per parameter flexibility pool output representation modification competitive dropout effectiveness complex convolutional filter observe configuration cnns result across science video cnns expense train convolutional key ingredient cnn deep natural demonstrate convolution filter computational arise convolution wise domain offer provide cnn study spectral representation cnns fourier filter representation mapping correspond unitary transformation however argue spectral representation show filter tend spectral thereby reduce optimizer align representation dc domain domain conjugate gray symmetry fourier transform dft decompose signal introduction dft input
random scheme develop minimax promise currently code scheme greedy sparse apply acknowledgement support nsf thank valuable comment theorems minimax clear non finitely prior integrate optimal residual r ci show schwarz show achieve suppose theorem asymptotic minimax risk normal ellipsoid argue problem reformulate last support provide parameterize optimal bind suffice attain jj j feasible show l solve feasible plug calculation particular base correspond favorable sufficient base follow choice lead complete write expectation show argument lemma algebra note eq equivalently universal apply schwarz need lemma inequality denote brevity inequality easy minimum small non increase simplicity go observe since hence achieve must switch thus justify fact complete suppose degree centrality central chi poisson weight follow recurrence derive prove replace prove thm thm remark carry storage placing limit bit encode estimator excess risk quantization level pareto tradeoff quantization space technique analysis minimax storage size place datum construct understand error let resource fall within minimax risk region pareto versus vary case storage procedure formulation problem naturally motivate instance analyze estimate send communication cost become lose risk quantization scenario cloud environment process store limit storage dominate scenario quantization tradeoff processor limit precision arithmetic computation precision motivation storage nonparametric estimation lie classical distortion theory minimax theory idea refine fundamental connection code wiener deviation periodic constrain minimax q bit total identify regime bit r pt bit quantization classical minimax regime however certain pt number bit insufficient preserve minimax quantization dominate minimax insufficient regime show regime error insufficient achievable quantization threshold surprising classical optimal risk achieve keep estimate smoothing level minimax compute least favorable source base mutual combine quantization establish achieve quantization classical distortion generation codebook determine source allocation bit use bit adaptively smoothness quantization estimation adaptive coding storage operating correspond regime establish minimax quantization regime outline proof procedure stein quantization detailed supplementary material obtain estimate lie subscript mind nonparametric subscript maintain minimax infimum estimator respect abuse minimax risk bayesian align whose support minimize integrated favorable prior lead provable scenario constraint bit formulate follow codebook estimate need store index simply encoder decoder denote storage constraint decompose expectation equality form decomposition due express abuse infimum definition distortion mutual denote minimax since low q goal least favorable turn simple mean hypercube show low risk estimation quantization large scale low mle close take constant ball scale tight bind show mean euclidean nb show asymptotic euclidean quantization construct bind budget euclidean ball case relevant estimation sequence stein suffice new code strategy clear sequel allocation determination detail bad series variational asymptotic minimax tight demonstrate quantization asymptotically space white diffusion equation wiener observe diffusion goal choose arbitrarily recall white familiar carry encoder section composition encoder decoder call risk asymptotic model specifically coefficient ellipsoid hold actually long expand orthonormal convert reformulate collection follow establish explicit distinct quantization regime play interpretation apparent q achievable asymptotic budget turn threshold value surprising classical first keep basis regime bit great convergence rate bind directly proof possible regime bit suffer regime optimal quantity term sequel constant constant dd regime bound choice parameter large favorable prior insufficient regime point third regime communication insufficient optimal long quantization dominate risk decay matter fast analogue simple informally term quantization regime outline defer gaussian term section concentrate prior q mean optimization infimum define classical distortion give analyze term quantization follow moreover close insufficient regime regime begin solution decrease decrease correspondingly optimal reverse water scheme third exist integer series three reformulate writing bandwidth function bandwidth choice omit argument solution express summarize achievable code scheme quantization together block system cardinality infinite index block ready scheme distortion codebook codebook adaptive block budget separately code magnitude detailed generate codebook kn encoder decoder optimization dependent codebook codebook keep normalizing row illustration codebook encode thin color gray dotted rectangle node color dot thin fill thin dotted thin gray gray kt store store index codebook get base reconstruct codebook theorem establish asymptotic constant respect remark estimation bit asymptotically budget namely size codebook suppose low radius codebook parameter function bit fact adaptive grow formally modification numerator expectation denominator achievable grow need code bound accord expect codebook similarly codebook inside imply achieve ellipsoid generate codebook universal risk ellipsoid code detail outline term
adjoint operator expand numerically take discretization skip minimize live functional fix initialize convergence result classical gauss example smooth popular proximal thresholding fista employ fista great backward see proximal contrary forward backward limitation fista convergence prove far know property fista rely step soft operation practice evaluate iteration procedure backtrack choose l z reduce closed simulation request motion smooth brownian motion bandwidth denote medical challenge know language framework instantaneous record varie fast study signal vary instantaneous patient instantaneous heart evaluate r peak peak non instantaneous heart cubic spline visually instantaneous sampling analysis fourier transform window please see extract instantaneous fail addition factor process signal take window window deviation clear dynamical hand instantaneous frequency blue add noise perform model noise snr noise signal frequency curve curve mode function harmonic function vary instantaneous find representation refer encourage several thing fista still usage take minute finish analyze length find sensitive choice choose behavior moment robust numerical theoretically study interesting finding compose intrinsic varying depend weakly domain window profile window signal show window ideal nevertheless nothing window support wide window standard signal note start indicate achieve window make effort determine kind energy window extract topic wu research support eq c part determine positivity amplitude condition eq q change globally change integer hence q sign inside calculus know n n loss exist last part thus know thus sign inside sign inside finish take without calculus hold claim come taylor cosine q contradict claim amplitude generality calculus claim contradict sign bind immediately control qualitative dt nf lf lt lt zero taylor second expansion component taylor recall short transform window need universal fix argument immediately q first claim particular v ft g define enough distribution hand result g lt v kt smooth bound right hand simplify clearly conclude finish eq g claim lt lt equality obtain proceed finish define also lt claim enough generality point choose condition argument lead assumption claim lastly phase function na lt kt dt na lt lt contradiction theorem definition thm motivate limitation signal compose vary instantaneous frequency harmonic signal property representation fista shrinkage thresholde numerical coin confirm analysis fista instantaneous frequency proper feature step toward signal inside traditionally commonly however compose harmonic perform behavior capture lot decade time frequency attract example short time transform ensemble decomposition solution analysis however mathematical foundation several limitation name attract issue introduce coefficient phase technique class share precise preserve technique trick wave mechanic finance energy physics signal slowly instantaneous want study signal instantaneous song wave kind signal frequency instantaneous requirement auxiliary function quantify fast instantaneous minimizer guarantee apply shrinkage thresholding fista way adaptive harmonic model provide fista vary frequency mode consist adaptive harmonic fix constant consider functional function kt c satisfy c c instantaneous control proceeding component reconstruct th component integrate eq realize spectrum ideal function restrict compact connected numerically function ft reconstruction property visualization achieve take equality hold due one h ft kt frequency kt kt k lt argument hold q ft observation variational discuss follow well carry come term might te special case know thus follow cf capture consider function plane restrict representation
substitute exclude stable solution correct among stable say negative number satisfy strict give set much prove stable cx cx cx cx I prove condition already stable otherwise process say right consistent merge otherwise consistent merge otherwise consistent otherwise continue process construct note stop kind stop prove equality thus main conclusion reduce problem sequence maximized solution follow v ty b prove ct v construction start leave consistent right combine get go show contradict correct mention main consistent obviously among solution prove solution solution cb cb cb cb cb thus note equivalent wrong say always text sub suitable consistent solution pt cm pt supplementary material dynamic specific degree accept document text mention text claim prove loss move around actually finally stop component exclude sort
repeat arrange half I define simple vector repeat span precede unique transform update possibly repeat also uniqueness express cholesky practice square scalar iterative practice precision svd machine p extend application image applie tr u tr sparsity iteration matrix l yy compute see minimize p tend study behavior code equivalently establish transform update corresponding solution orthonormal result coincide solution employ alternate minimization specifically eq update denote svd optimal solution unique zero proposition replace constraint additional proposition provide appendix orthonormal synthesis dictionary denote dictionary immediately alternate update orthonormal alternate transform computational cost matrix begin onto ball employ sort hard equation update computation multiply update exclude pre scale step cost coding scale overcomplete synthesis synthesis scheme also computational analysis converge iteration svd typically translate e constraint use barrier violate otherwise w unconstrained objective exactly equivalent objective minimizer whenever two objective unconstrained minimum constrain formulation unconstraine formulation error control avoid interested know alternate minimizer result alternate minimization g accumulation denote magnitude wise ij k x k iterate sequence decrease say accumulation point minimizer define arbitrary start large choosing magnitude local accumulation local optimum accumulation particular equivalent equally minima objective minimizer irrespective empirical transform insensitive initialization conjecture potentially provide illustration corollary convergent refer globally convergent set minimizer learn requirement distinction convex extra g tight isometry convergence control arbitrarily perturbation half accumulation half perturbation perturbation outside local maintain sparse could choice perturbation small directly adaptive blind denoise compressed sense formulation highly see solve transform denoise guarantee converge trivially p obtain similar objective monotone value moreover iterate sequence accumulation iterate minimizer accumulation proof result corollary corollary demonstrate transform usefulness alternate learn initialization sensitive initialization study provide understanding compare patch representation usefulness implementation code version intel cpu memory bit window operating system image representation patch mean stack patch display removal adopt compression denoise work experiment metric learn code lose fitting domain learn transform learn transform ratio transform patch serve simple surrogate information overlap learn transform transform scenario initialization initialize algorithm iterate code initialization dct kronecker dct second initialization invert learn instead row matrix zero show initializations b converge error decrease require importantly nearly initialization indicate algorithm reasonably insensitive initialization initialization dct frobenius norm initialization condition dct learn display call transform atom texture initialization equivalent appear somewhat permutation transform learn initialization provide differ experiment learn overlap patch propose closed compare version value determinant p fix dct analytical extensively compression near integer simplicity execute update figure plot learn transform patch dct fig c close via dct normalize close recovery analytical dct patch size transform size cf reasoning learn experiment number gap patch learn transform conditioning restrictive normalize error algorithm identical algorithm involve fast actual general another analysis blind source separation similar work ica representation learn ica signal correspond ica independence matlab code learn recovery replace sparse ica seen propose transform recovery approach superiority compression ica fig slow ica finally transform synthesis heavy burden scheme compression classical involve dct wavelet learn adapt specific transform adapt variety transform compression goal recover represent vector corrupt whose variance present denoise use transform patch patch representation unknown propose estimate iterate propose algorithm step average respective brief employ image couple fig different image compare denoise overcomplete synthesis denoise scheme matlab available website use setting maximally patch result transform denoise usually transform increase degradation overcomplete setting various denoise number denoise work pt algorithm denoise iteration step patch transform brain couple denoising obtained obtain well image average svd db transform algorithm denoise involve close fast c four average transform denoise denoise compute svd denoise svd method speedup transform mainly cost base multiplication see sparsity method synthesis sparsity since sparsity svd threshold threshold svd speedup transform higher point actual speedup information synthesis typically fraction speedup denoise svd noise increase increase patch image explore transform denoise involve close overcomplete adaptive transform denoising become well choice parameter use experiment simplicity bm bm adapt transform formulation square efficient become transform orthonormal synthesis provide establish convergent minimizer guarantee rely learn transform obtained provide dct provide well k svd fast denoise learn involve discuss extension overcomplete elsewhere easy eq subscript element optimal whenever consider corresponding term objective either hard fix discuss remain depend square root specific us svd index converge singular hand aforementioned limit behavior formula degenerate accumulation hand scale constant clear coincide solution alternate transform coding aforementioned simplify problem orthonormal update problem value latter provide proposition denote every accumulation r q tm subsequence obviously limit inner product sequence orthonormal diagonal matrix entry maintain decrease diagonal svd precede indicate accumulation similar difference barrier replace penalty brevity sketch use operation optimal ball minimizer follow vector prove property existence accumulation point accumulation iterate every accumulation every sense monotone converge transform global minimizer objective decrease thus explicit therefore bound sequence value generate least accumulation point convergent subsequence bound accumulation boundedness simplicity inequality denote k n unbounded c inequality follow triangle since section boundedness previously lemma accumulation accumulation iterate satisfie subsequence converge accumulation objective monotone converge accumulation accumulation sense subsequence converge accumulation define subsequence accumulation point barrier continuity converge accumulation iterate accumulation algorithm converge accumulation linearity r accumulation matrix subsequence limit equality fix sparse accumulation accumulation subsequence iterate sequence converge accumulation imply deal uniqueness satisfy similarly ii imply feed stay accumulation finally fix iterate accumulation equally minimizer g ng produce stationary hessian definite trivial sufficiently furthermore obvious code thus q sparse code give fix point perturbation suffice otherwise barrier trivially rest preserve preserving expand norm term trace inner x simplify q I aforementioned positivity assume neighborhood become lemma define simple x sparsity great column column thus follow zero outside thus proof discuss bound converge singleton sufficiently combine sequence singleton tie support coincide support one converge point case belong theorem unconstrained barrier function perturbation I ix repeat theorem signal certain transform dictionary synthesis denoise medical instead square transform alternate code transform step provide globally convergent minimizer transform practice insensitive present promising transform sparse convex dictionary widely year study notably specifically transform sense unlike synthesis measure residual transform capability model scalable synthesis briefly sparse reference difference find representation transform subject w contrast synthesis noisy np synthesis analysis code especially involve various synthesis computationally adaptation sparse adaptation synthesis dictionary signal
deep acoustic deep net neural net really really train improved discriminate highly yet acknowledgment model imagenet helpful discussion google google google machine train prediction unfortunately expensive especially net possible develop use compression surprising improve acoustic heavily single new full distinguish fine grain model expert train energy completely optimize requirement large machine typically training stage requirement like highly operate real user requirement easy extract model regularizer dropout train transfer already rich conceptual may investigation knowledge make keep abstract view learn learn average incorrect answer lot model tend generalize mistake probable accept reflect true closely optimize generalize generalize normally large however generalize train transfer class produce soft target transfer transfer simple geometric individual predictive soft target entropy much mnist correct high much ratio soft target may way information define rich say look little stage softmax softmax produce call temperature softmax suitably target temperature match soft target show model transfer consist entirely unlabeled original training find especially encourage target match target provide typically match soft target turn neural typically produce use softmax logit compute temperature normally higher produce class knowledge target case produce temperature softmax temperature train significantly modify soft target soft softmax generate target exactly softmax temperature magnitude gradient target important multiply soft target ensure remain roughly unchanged change contribute cross logit soft target gradient approximate transfer simplify high provide case attention advantageous almost unconstrained cost training could noisy acquire dominate strongly suggest ignore unit case net weight share pixel net small net add task match temperature soft target deal generalize translate transfer two give unit work try perspective digit error cause much increase optimize right get test training model error approach try relative hard indicate extract hard single improvement frame similar improvement preliminary mnist word frame recently relate acoustic train temperature unlabeled gap hard advantage usual ensemble ensemble though easy give total computation fine grained overfitte target dataset million label deep convolutional large two parallelism net core mini batch replica mini gradient server back parameter gradient server last send replica replica core put neuron parallelism two lot several train option fast number make ensemble contain many highly type low weight slightly coming sample remainder training correct bias logit class derive decide full even though confusion cluster party bridge cluster line algorithm obtain produce result investigate happen ensemble deal step according call use step take special empty intersection note empty class minimize class plus class arithmetic optimize carry c c accuracy baseline extremely completely show absolute combine model test overall report accuracy belong class restrict r delta accuracy disjoint cover class table show improvement break trend claim soft target hard target lot target could hard demonstrate fit speech early baseline hard target lead severe reach train target information remarkable early stop converge soft target effective train another training train frame test baseline target collapse
two correction copula study fix keep unchanged gradually get tendency simulated investigate work iid chose show histogram ten burn reduction uninformative towards correction seem binomial particularly among difficult cox datum one everywhere else section study correction future making set less extreme simulate appear copula correction general always circumstance intuition underlie strong case problematic correction limit applicability maintain normal cdf jacobian jacobian ix qx n I collect term substitute eq correction account thank provide r code relate study grateful hold datum thank comment correction mix integrate laplace low count somewhat particularly problematic new implement part package evaluation simulate indicate bayesian copula generalize nest part accurate extremely software rich generalize prior somewhat inaccurate replication new ten million red curve correction scene minimal let iid prior prior intercept precision histogram red show run quite experience difficult fix reasonably problem usual ensure asymptotic validity see effect parameter realistic amount spline smoothing lack parameter little add panel observation predictor bottom derivative near frequentist attention recent computationally laplace propagation type costly use speed requirement correction proceed present correction section show method work brief conclude remark hyperparameter observe want laplace approximation find curvature approximation skew approximation detail notice marginal approximation second improve approximation structure fix solution additionally correction soft define increase correction determine since zero close moderate large increasingly find let job investigate retain skew cdf complicate derive find simple case preferable skew simple version try correction significant show approximation inaccurate case correction iid per give sample correspond page difficult error see empirically laplace use prior correction chain million burn million posterior close reasonably mcmc parameter see show simulation difference mcmc deviation effect lower improve major effect value correct c correct correct discuss
infinitely density activation converge surely activate infinitely often activation bandit structure bandit activate well surely without strong law finitely infinite optimal activate time write trivially eq define space relation yield logarithm similarly large rearrange positive eq note relationship take unbounded eq sufficiently relationship follow eq take proof bandit population random slowly increase regularity almost surely additionally remainder construction herein effectively control exploration tradeoff contribution derivation upper regret policy establish ii term assumption law iterative logarithm action arm sequential sequentially population population receive time sample restrict though ix purpose establish important distributional population alone discuss bandit let indicate event population I interested outcome controller complete would bandit measure regret notational form explore pseudo satisfying give inherent controller reasonably gain lose activation reasonably value sum assumption law number modify sequence force play winner able case bernoulli bandit collection consist density respect scalar set vx leibler mild therein require policy slow population subsequently collection type population poisson bandit due set show regularity therein population satisfy far I sample population dependent asymptotically I simplification requirement policy together alternative way policy derive asymptotically reference optimal strict construct regret therein big paper assumption law type policy mild index almost asymptotically bandit large particularly sure growth motivation know utilize minimal considerably expectation might reasonably interested currently explore minimize path outcome offer sense intuitive policy essentially set explore experiment available high explore slowly slow grow explore enough bandit past implementation policy guarantee asymptotic behavior bandit guarantee bandit broad additionally index policy individually capture many policy behavior perhaps emphasize asymptotic basis essentially asymptotic policy good definition good satisfy mild condition bound order growth remainder attempt change dynamic remark roughly equally single bandit rarely growth pseudo way function bandit essentially good trivially e concave differentiable linear example policy unbounde positive concave let define way blue arc width bandit sample mean broken exploration bandit view variant sample good sequence convenient define sub activate state policy policy good statement theorem arc follow every almost surely exist considerably non fluctuation true go almost bandit activate result e linear occur brief whenever integer point raise activation equality fact discrepancy occur increasingly rarely hypothesis specify scheme break explicitly sufficiently leave tie statement make appendix ucb define blue width briefly mean policy large mean increase exploration traditional upper refer policy ucb necessary exploration simple simplify give proposition proof q optimal bandit application give regret hold limit leave produce prop verification minimal bandit activate hold bandit activate must activate activation hard indicate phase bandit roughly policy single bandit optimal bandit rarely offer potential circumstance optimal activate coefficient activation govern comparison fluctuation term assumption activate linearly assumption grant law iterate seem index optimal bandit greater dominant dominate reduce optimal roughly often bandit index policy essentially play winner fix bandit period phase dominate property problem side phase variance iterate bandit bandit activate infinitely iterate great alone theorem arc width short index eq restrictive since traditionally logarithmic trivially proof sub surely leave extend pseudo surely policy one thing improve pick sense certainly optimal always fix however bandit constant order present control theorem bounding fluctuation around indeed bound however sub optimal index sense bandit equally regardless quality sub rarely optimum boost bandit contribute bandit fairly matter discovery unknown though use measurement play proof distributional property surely law iterate utilize bound remainder independent never necessary bandit regard arbitrary multidimensional process remove way proof somewhat pseudo reasonable finite horizon regret complete policy bandit horizon bandit suffice result bandit implement hereafter two alternate phase mode play winner activation activate bandit least activation winner activation great activate phase govern winner mode activate bandit bandit least activation period winner bandit minimum activation activate follow hence repeat base sub reach bandit activate policy period winner loose sub bandit period winner bandit point relation time bandit activation activation additional activation activation activate
hull projection algorithm constant rank outer matrix unfortunately parameter trial find latter operation operation iteration case I force bad base cumulative current key perturbation regret distribute lead perturbation parameter achieve find perturbation instead random gaussian consist recent consider connection calculation computational dominate component perturb bad case optimum respectively algorithm independent bernoulli flip add closely grow distribute linearly analysis eigenvalue independently pose pca version operate deterministic top eigenvector online wolfe base perturbation except inferior method dense benefit pca finding goal good reconstruction guarantee study suboptimal due fact adapt perturbation clear use perturbation independently zero vector trial tune perturbation matrix skip instance matrix trial predict perturb coin half trial bernoulli quickly extension perturbation unfortunately force suboptimal regret counter dense counter regret force conjecture achieve regret w gain good gain share instance vector expert gain follow algorithm dropout provable share essentially bound perturbation rotation pca nk reveal receive call positive eigenvalue gain oppose protocol choose respect randomization algorithm matrix specify generate gaussian entry grows tune single choose accord rule sum sum picture trial variable perturb relate give sparse sum separately start eq eigenvalue generate orthogonal sum bound bind p nn proportional similarly symmetric furthermore function change integration convergence replace integration w last instance dense opposite old instance summation rule q sum second dense instance instance compare minimax regret instance case instance present suboptimal factor informally bound general soon since large noise eigenvalue par eigenvalue trial regret determined condition claim length suffer suffer little suboptimal bound minimax sense action follow expert small perturb cumulative analogy algorithm unfortunately perturbation would basis act differently perturbation would need eigenvalue try pca dropout perturbation coin flip record predict record natural noise conjecture achieve asymptotically conjecture standard set expert dropout loss gain often provable case word seem online perturb perturbation avoid top eigenvector worst regret close online generalization version support st summing trial since jensen eq plugging put pl technology university california machine vector parameter inefficient address learn obtain parameter trial per trial ideally decomposition per time line predictor minus key analysis achieve optimum algorithm per small paradigm
discriminative tune discriminative regard generative datum goal study coupling scale model empirical tradeoff neural mnist weakly mnist approach svm likewise rbm weakly perform classification label mnist deep currently good performing variant architecture protocol usually involve unsupervised supervised phase combine algorithm classifier highlight tradeoff system include usually additional validation set come set consequence net amount increase argue favor label account comparison net naturally approach deep learn architecture total label model label operate perform reduce number system likely number require fully hand outperform compete back finally unlabele direct integrate application involve inference observe regard evidence potential unsupervise direct favor integrate unsupervised supervise favor far appendix execution black graphic structure matrix multiplication ideally parallelization graphic cpu concept mini learn update gpu mini gpu usage cause negligible mini batch parallelization parameter choose optimize else gpu rate size test likelihood converge ccc mini std log std hide unit turn complete validation reason number available prominent account find start validation indistinguishable normalization softmax winner closely able broad learning set express proportional hidden unit iteration label value keep learning might stay learn hide yield step optima overall trade encounter gradually paper keep choose unit dataset validation decrease recurrent feed forward possible overfitte point hide normalization experiment rate early optima general long validation training choose experiment deep n em directly maximize free denote entropy depend parameter em optimize iterate posterior posterior expectation maximize locally maximize detail behavior sum respectively dynamic apply non result assume weight learn equation learn add repeatedly learn step form expression give product denominator regard growth give product approximation inspection approximation hierarchical step small approximated cd use supplement detail sum approximate sum supplement last furthermore apply find analogously convergence recover converge individually experiment verify come balance preprocesse applicable small explicitly free refer motivate unlabele parameter tuning construction require sparsity anchor near dimensionality reduction write mnist label test probably dimensional manifold classify support vector result number manifold investigate manifold dimensionality nearest take part testing state test label test low dimensional manifold combine result use measure tuned model combine model deep decay hyper use deep neural architecture paper validation publish contain deduce protocol combine represent system component boltzmann typical consist layer decode hide unit bottom discriminative comparison mnist validation sup test unlabeled point include summary protocol tune diverse report also protocol difficult exhaustive believe eqs sec eqs eqs eqs eqs eqs eqs eqs eqs eqs com de mm institute advanced university von study supervise often complex heterogeneous parameter work integration highlight define learn mixture classification network scale learn quantitative benchmark tighter competitive amongst label demonstrate applicability competitive operate favor strong focus simplify capability deep rapidly progress modeling belief convolutional network machine autoencoder hybrid unsupervised make benchmark difficulty mnist prominent example complicated architecture unit data initialization parameter define number sample initialization systems vi anneal parameter parameter drop early integrate require theoretically network architecture hyper g importantly hierarchical unlabeled seek optimize principle combine derive network thereby probabilistic minimal number allow classification model hand class within write generative model top abstract comprise layer occur digit final observe poisson distribution I observe note normalize weight dimensionality large order maximum seek come summation maximizing expectation em iteratively low log setting update update aforementioned transform happen scale derive neural circuit proper derive show plausible formulate make neural mode neural analog neural neural obtain unnormalized datum activity normalize weight assume assume satisfy th activity become limit rate derivation taylor expansion technical normalization activation long identify convergence sep sep black color black inner fill gray cm cm cm cm cm pos cm width dash edge yshift pt amplitude mirror xshift yshift black yshift pt amplitude mirror xshift yshift pt black yshift mirror xshift pt yshift cm cm cm pt yshift width center corner minimum em draw thick font mid label mid distance blind label em label unlabele em unlabeled block validation label em text width generalization height general right em keep half tuning training protocol label per set risk optima layer compute denote uniform range overcomplete far label separation problem use information second dataset belong topic comprise raw occur tf stop investigate total per fully set mean error value total initialization procedure describe good knowledge report task break binary merge topic work experimental setup discriminative rbms perform weakly optimize supervised validation compare l cr experiment r achieve error ff test decrease ff label early optima reach fully tune base set additional training free could potentially setting performance handwritten digit mnist mnist test gray handwritten mass minor font legend font xlabel style ylabel font scale false xlabel xlabel near pos ylabel ylabel near pos axis thick mark none black txt axis axis cs access height axis ylabel ylabel near pos line right thick mark black txt overfitte layer feed forward influence layer therefore recurrent unit substantially likelihood criterion early monitoring soon overfitte first influence datum start stop overfitte compare
spectrum exposition practice rarely strictly stationary usually stationary slowly change spectra range suitable determine integer extra average kind value fairly reasonably terminology localize sequence range integer please hand follow average facilitate enable driven approach ideal number index fact multiple relevant say wide frequency variation know window scale multiscale result low frequency channel recursively low frequency every simplifie range nonnegative via processing convolution later stage deconvolution subsequent convolution stage stable later window wider precede recursively two one nonzero transform recursive consider sequence doubly provide smoothed alternatively multiscale yield fourier wavelet sift et constitute choice whole recursive high channel serve desirable member gradient calculus calculate spectra classification regression integer integer locally infinite integer implementation computer obviously finite sequence furthermore locally stochastic filter regularity easily filter offset integer construction enable stochastic process regularity distributional versus poisson parametric convnet relevant nonlinearity stationarity local translation convolution correspond subsampling correspond convolutional convolution follow entry subsampling entry correspond absolute entry convolutional essentially convolutional filter precede subsampling acknowledgement remark convnet implement follow operation convolution complex absolute result average real multiscale spectra multiscale spectra drive configuration calculate multiscale convnet filter complex exact correspondence wavelet whereas analogy drive multiscale certain modeling series natural include patterns multiscale spectra nonlinear connect concept information original value theory leave far exposition nothing basic treat limit consideration doubly nonnegative range integer range latter sequence range terminology refer fact
face phase enough dm optimal arbitrary accurately dms dm good th adjustment approximate wish find achieve dm q algorithm provable stochastic discount weakly acyclic well weak strict dm deterministic e dm call differ exactly dm dm strict discount weakly acyclic strict strict path start joint equilibrium policy good path acyclic games acyclic straightforward adjustment introduce dm strict well would clearly converge acyclic next enable dms adjust strict path adjustment dm algorithm initialize iterate mm cl u ik ix ix ix ix ix u ix ix k ix ix cl game acyclic strict game hand define acyclic dm policy assumption q exist finite eq dm learn policy baseline experimental dm switch strict well dm switch dm algorithm cycle switch strict flexibility come baseline policy flexibility lead convergent behavior drawback phase inconsistent spirit synchronization hold th dm denote acyclic value learn exploration consistently equilibrium predict even set path policy converge equilibrium tx assign initial condition respectively non convergent infinitely w contradiction show initial w hence know dm dm random assign eq exploration phase dms finitely joint policy follow appendix combination policy either use uniform ia j finite number uniform dms deterministic v ix optimality introduce upper bind due acyclic strict minimum length exist q recursive part pick satisfy existence integer eq assume due borel borel lemma contraction lipschitz th dm random assign exploration finite dms finitely hence desire convex form dm use uniform ia joint policy exist uniform deterministic policy ix tolerance rate upper event since acyclic well exist satisfy rest recursive reasoning proof iii proof sample process show learn obtain q lead recursive part completely analogous proof part assumption mathematic university algorithm dynamic game stationary maker minimal also cost decentralize dynamic game weakly acyclic algorithm surely stochastic game desirable game specifically acyclic game oppose static game future applicable broad application literature stochastic dynamic game learn gain popularity interest generalize reinforcement dynamic lead set primarily reference tend real agent find play equilibrium objective serious agent maintain action compute use value action objective player learn enable learn optimally environment application information agent burden agent make analytical regard property game also attempt extend algorithm present computationally prohibitive quickly agent state action claim convergent equilibrium single game fp stationary past fp convergent even simple state agent move zero game strictly adversarial resemble centralized learning relate include actor equilibria stochastic game higher store give state specific rigorous viewpoint agent short localize reasonable system agent organize devoted paradigm standard first introduce present generalization follow paper remark proof technical system illustrate generate controller generally state coupling lead dynamic discount dynamic game follow dm dm states decision dm dm determine decision game induce denote initial dm make joint decision together well distribution select dm appropriate dm policy dm decision solely policy dm interpretation dm policy randomly set policy dm primarily policy dm find dms possibly cost may dms adopt represent ease policy dm joint perfect discount stochastic possess equilibrium define since may many game possess particular game arise application dms benefit possess primarily denote deterministic class game dms identical team although team useful model system game arise system dms necessarily team deterministic joint policy dm least joint low dms generalizing notion game weakly acyclic set well dm characterize factor satisfying denote call respect policy position say dm strict discount weakly acyclic strict dms constant learn factor visit often proper rate environment exploration randomly small however decision expect persistent team weakly acyclic games repeat singleton
apply obtain place paired basis pair next derive multivariate bound condition section exploit result derive definition independence bss derive bss satisfying invariance discrimination minima derive section derive ip define reference ip derive expression report empirical verification derive independence section article end conclusion f say hull say finite support volume surface constraint condition differentiable gd mathematical integrate side equation integrate respect bring let without let accordingly none derivative cx derivative get get prof combine case mathematical differentiable gd g induction fx integrating integrate bring integrate bring sake hold generalization f gd none zero loss f function equation integrate equation dx dx gx x x x equation base inductive step induction differentiable gd g hold prove loss integrate case side bring integration real number integrate respect coefficient ij ij tc zero number none bring coefficient bring prove equation odd base sake let obvious without f gd none loss generality pf integrate get dx integrate equation integrate equation dx g dx dx gx f gx g n x x integrating prove inductive step induction gd n bound converse differentiable gd principle implication value prove generalize derivative available differ constant derivative would decide prove bring equality strength prior conversely equal derivative line derivative everywhere lebesgue unity bound seem restrict say restrict natural think extend independence look similarity score develop matching dimensional product marginal pdfs independence variable independence find define difference pdfs independence joint gradient product difference vector analogy gd satisfy definition add pdfs order support result obtain hessian product hessian pdfs hessian hessian hessian component prove pdf pdfs bound independence bring interpretation independence vector goal quantity nonnegative quantify derive independence quantify integrable pdfs norm add invariance detail appendix pdfs invariant respect translation variance affect reason pdfs shape desire invariance invariance apply pdfs pdfs vector integrable joint marginal pdfs deviation independence vector differentiable marginal pdfs correspond standard deviation differentiable condition apply prove necessarily support pdfs order differentiable pdf prove independence blind bss generation instantaneous variable mt component problem bss mix matrix unique bss accordingly bss mixing obtain separation source contrast bss criteria correspond formal bss let transformation contain identity variable mapping trivial leave unchanged follow property source maxima discrimination spurious achieve maxima need scale whole permutation scale operation state widely divergence invariance accommodate without bss define invariance quantify property q transformation satisfied justify invariance predefine act per argument measure satisfying scale equivalence solution bss precisely bss solution mathematically whole contrast easily convert demonstrate whether contrast bss invariance scale filter diagonal prove normalization scale invariant norm normalize density invariance property definition verify bss contrast linear bss verify invariance trivial simplify start prove neither invariant relative invariant though apply density invariance permutation discrimination property normalization bss p bss source verify invariance contrast diagonal scaling filter simplify start scale normalization bss independence already mean invariance way prove discrimination advantage minima minima still respect also contain optima correspond optima analysis bound differential pdfs optimal thought imply spurious maxima eq indicate c independence zero indicate maxima spurious optima prove prove information prove gradient differential mutual mutual minima desire investigate obtained try small pdfs convert simply integrate prove serve integrated difference similar proved reason effort prove actually result proof spurious optima maxima maxima focus measure avoid estimation sample marginal pdf estimation pdf entropies bss conventional pdf marginal gradient pdfs derivative achieve base histogram estimation fast pdfs pdf estimation pdfs indirect estimation suppose though theoretically say pdfs derivative require derive light square select place sample require basis select place basis pair un pair location computation estimation note order gauss similar direct estimation difference potential reference extend concept estimator realization unknown pdf bandwidth usually definite single square q thus way avoid continuous theory information ip independence information appendix interpretation describe done bring field field zero potential reference analyze absolute potential reference local reference field potential neutral system express quantity reference potential example electrical circuit reference assume ip concept far derive field ip concept first question define rip quantity initially set place select basis rip analogous approach bring quantity place act potential linear select potential information rip rip integral pdf gaussian definition rip bring cross potential pdf reference newly concept location basis location sample space vice analysis scalar intermediate information interaction interaction due potential force interaction size similarly reference potential gram potential interaction potential already scalar x nb target close expression concept stage approach least direct estimation square mutual worth contrast euclidean quadratic square generalized estimate potential actual require correct assume select number basis parameter depend rip identity obtain r rip use rx ij dy bx rx rx ij symbol operation rip multiplication term replace hadamard addition entry exponent obtain multiplication number require available complexity measure term multiplication multiplication time reduce correspond directly number specifically place estimation multiplicative cost gx kx accordingly matrix nx l n nz way dimension explain previously number multiplication estimate equation q require equation complexity multiplicative pair expect accurate dimension mention article elsewhere calculation way intuitively convolution article correct think delta pdf justify popular square rx kernel multivariate multivariate major equation estimation basis replace rx let square least square contrast combination correct obtain kernel basis place term purpose rip rip estimation bandwidth multiplicative pair pair derive basis computation kx xy product estimation discuss write require solve derive verification ability dependent testing bss kde cross demand choice balance select uniformly distribute sample let generate signal dependent entry trial ht definition opposite uncorrelated component mean imply orthogonality rotation transform component bss give contrast dependence theoretic bss local design existence spurious landscape derive contrast bss keep test selection bandwidth selection selection bandwidth bandwidth bandwidth validation parameter distribution first type r suggest type type skewness skew skewness skew add experiment report justified bandwidth bss previous application set new every consideration demanding assume expect property vary ideally bandwidth parameter gaussian assume bring bring need rule consideration distribution solution derive assumption near gaussian estimation bss show ideally multiply comparative minima spurious minima unimodal landscape minima everywhere hessian everywhere accordingly measure ica contrast discrimination spurious multiplicative basis place pair computation verification bss quality vary vary require restrict future function negativity symmetry triangle inequality four condition contain concept distance geometry generalization space define norm
scale data factorization utilize randomize user already indicate category represent multinomial category corresponding category popularity transaction recent restrict transaction day recommendation pick product interest category brevity lift term acc different observe lift bias matter lift partly mf poor bad factorization binary factor construction incorporate c ccc mf email add recommendation early adopt base recommendation percentage customer split one item motivation temporal score attempt temporal g highly likely recommender keep recommendation hand time item recent feedback capture trend algorithm rank base acc map ndcg guarantee step reasonable offline boost base recommendation capture learning independently encourage practitioner add standard module recommender couple remain open though computational bias nature difficult power distribute need improve recommendation transaction recommendation encourage tail understand balance relevance popularity top recommendation shift dependent recommendation engine engine capture trend shift observe resolve criterion user online always boost recommendation encourage practitioner recommender temporal dynamic technology database recommender extensively domain movie music news recommendation etc recommendation content neighborhood base model recommender body work evaluate item user collect paper take real world website recommend observe product week week day user demand external demonstrate rank sale product week line figure indicate transaction moment among product website huge figure rd rank product temporal challenge recommendation ignore bad experience instance recommend na I deal discrepancy restrict model transaction occur day transaction severe majority customer across transaction wise choice fewer couple shorter small may appear window plot model window baseline train transaction imply collect training possible exploit transaction sparsity capture trend recommendation capture discard item bias capture recommendation recommendation model optimize bias carefully examine recommendation dynamic interaction proceed use generality throughout total user index user position recommender item typically email item recommend email template practice yield score denote user yield item default rank top scope find th item evaluation item relevance item rating collaborative filtering application item relevance particular user irrelevant presentation rank acc ndcg since top individual user many care item enter item rank essentially ap average position denominator relevant item rank map discount ranking ndcg relevant rank individual user eq item presentation convenience mention suffer transaction dynamic attribute kind huge discount external likely capture external factor temporal challenge collect day determine bias evaluation maximize state information user item score user find bias accord performance maximize bias recent transaction capture change analytically shall iteratively guarantee terminate step optimal total item recommend user index position score prediction relevance top bias aim rewrite difficult resolve calculate adopt score bias optimize next describe find item contribution one solely item simple assume user recommendation accordingly discard item four consider relevance eq item share irrelevant accord accuracy associate particular discard item item discard top recommendation item recommendation bias add enter change determine maximal utility convenience utility recommendation serve utility kb k f inf via subroutine present respect bias utility compute bias prefer utility possible enter recommendation infinity constant include recommendation lead negative exclude recommendation subroutine bias bias lead utility sort time single previous subsection update detailed b cb top necessary loop cycle item outer loop stop utility line item straightforward save record recommendation top recommendation user immediately decide whether update item recommendation bias need care similar recommendation user speed computation recommendation cycle utility terminate couple iteration state consideration recommendation utility change top recommendation always increase utility item top recommendation essentially remove consideration recommendation suggest remove positive transaction remove propose parallel size item couple heuristic might consider adopt save medium score memory keep top recommendation recommendation multipli instance recommendation remain item observe matter recommendation complexity type item bias frequently raw score tend past popular item appear frequently one negative item candidate without dramatically reach local optimal stop cycle yet criteria percentage threshold percentage bias cycle reduce time yet find optimize bias optimize recommendation ndcg item top play item rank rank utility change bias th nevertheless position example average recommendation gradually score rank position compute utility item currently item unchanged change item relevance utility item term utility move utility change original e top replace ndcg hence rather item find optimal bias item suppose potential change reach single item fine reasonably criterion candidate set u compute difference b subroutine setup include benchmark recommendation transaction construct benchmark split date user activity date week evaluate corresponding measure acc lift improvement lift behavior different quantity efficacy setting recommendation model chain current action another estimate consideration item instance however consider pattern action pick high transaction validate experiment recommendation well learn day transaction fine bias without consider change utilize long history long window user baseline transaction method train month item compute item bias metric item month specific bias probabilistic recommendation engine popular another incorporate temporal decay date decay work recommendation factorization music rating root stochastic application response acceptable conduct series experiment construct sensitivity base metric various table lift bold winner lift observable thank number look mind year thousand improve netflix competition improvement suggest recent trend propose weighted decay help bias individual preference improve recommendation especially huge acc map ndcg map ndcg cc popular bias macro tend shift item towards ground item reverse order test prediction respectively check overlap trend add bias recently
statistically mis specification indicator superiority initialize since initial allow reach likelihood vertical concatenation lag empirical block next asymptotic unconditional purely q corpus output filter whitening whitening fit perform scale extremely matrix vocabulary describe novel leverage scalable indicator corpus structure shifted use generative fit structure assess usefulness variety high lag approximate covariance corpus denote count corpus minus matrix operate second matrix strictly speak size co sublinear ignore rare unfortunately every word vector one orthogonal degenerate direction structure instead perform projection need fortunately lie embedding train level embedding train maximize might capture syntactic semantic release code algebra library consist modification public svd randomize approximate repeat multiplication low handle individually reasonable employ spherical poorly property diagonal modeling anti correlation would capture pass multinomial function prevent kalman filter conjugacy practitioner upper multinomial hard estimator poorly exploit minus low infeasible multiplication formula efficiently evaluate training formula usage use technique particular indicator multivariate maintain minus rank minus low em linearly transform post hand affect whitening minimize coordinate whitening obtain high whiten similar canonical correlation context successfully word embedding identify language appendix provide whiten ensure return psd manual correction unnecessary psd smoothing equation vector smoothing identical inference first alternative appendix inversion avoid token distance variable coordinate sphere highlight similarity commonly kalman rnn explore sec softmax sigmoid initialize steady kalman online kalman parameter nonlinear rnn sigmoid regime behave identity capture coordinate anti evolution rnn pass rnn separate word mapping dynamic evolution explore dynamic state study namely word likely reflect interpretable strict transition invariant salient community share express recommend get unsupervise pos response pos token predict pos embed include token embed set token kalman smoothing use universal pos tag original embedding york token maintain text type terminate broad maximize local pos network outperform word tune ignore instead classified tag hour train since heavily fast sublinear corpus tag v v table compare em initialize feature token embedding embedding local initialization find maximize force token embedding statistically significance level use vs initialization column universal tag contribute statistically significant tag expect context independent capture consider token baseline expect outperform pattern match long v highlight rnn application permit language lm context unit could restrict leave softmax architecture work rnn whether improve final optimization obtain hide tuned rnn comparison train substantially corpus popular rnn schedule hold decrease tuned value crucial jump start plot number minute rnn core far em iteration prevent rnn epochs initialization allow cpu table set find train specifically initial rate ignore require small set train embedding unclear baseline assign token syntactic single tuning rate rnn architecture leverage posterior work complete microsoft helpful comment empirical also various detail section scaling simplify coordinate orthogonal rather consistent procedure semidefinite psd singular span psd critical psd kalman structure datum fix unit vector ignore find factorize next expression kalman solve property use directly still solve form mc effectively ignore invert employ term inverse plus formula therefore intermediate filter equation worse possible indicator subscript denote along ignore zero therefore posterior term invoke come formula consider posterior smoothing depend model eq inverse inverse low useful algebra large example place trivial recursively invertible identity quantity storage compute inverse inversion compute contrast progress randomly nearly identical fact post hoc looking course em iteration quickly begin take find pos nlp ignore natural success word word posterior dynamical efficient kalman filtering learn extremely scalable operating employ task see recurrent neural recurrent neural reduce training nlp datum text nlp unsupervise independent context oppose context per token bank financial would dimensional obtain token embedding variable specifically moment perform maximum amount operate aggregate co discrete encoding token mis generative draw vector multivariate system desirable scalability filter simple learning moment svd co simple form multinomial second problem input employ token embedding pos name recognition pos apply local embedding token pos gain baseline embedding salient name kalman filter update rnn rnn however pass token left parallelism aggregate matrix distribute fully characterize consider given provide every invert substantially filter update key update actual
connectivity population set cover spread brain structure recover display obtain weight cutoff dark dot start involve different level structure notice generate even randomly cause optimization early tend dag constraint rich medical list produce edge connectivity happen temporal play role connectivity ad related increase corner figure temporal study mention connectivity increase ad cognitive patient significant change important one income connectivity communication increase communication dominate please influence factor limit datum disease imaging study worth exploration paper whole nc whole edge initial learn class axis quite weight leave fig adjustment edge increase mm kl mm paper variable discriminative achieve optimize classification demonstrate utilize bridge classifier margin problem analyze advantage discuss propose guarantee validation prediction improvement discriminative associate direct node exist constraint equivalent topological page eqn topological dag meet prove condition node similarly contradict impossible directly link path node compose another path node compose eqn eqn eqn far explain eqn hold topological come link dag topological order node come combine eqn dag iteratively alternate dag dag parameter column alternate dag x pa eqn I mean monotonically f second prove therefore optimization problem alternate contradiction ji ji ji ji order constraint regardless ji pa pa contradict sufficiently dag proof alternate eqn dag sufficiently edu liu university north hill usa semantic bayesian bn discover exploratory brain variable bn necessarily subtle critical investigation population propose power bn bring framework gaussian employ bridge discriminative svms convert fisher framework upon explicitly advantage framework paper ensure learn max probabilistic wide diagnosis bioinformatic effective infer dependency bn acyclic dag bn absence bn focus bn bn score use independence testing consistent include recent score approach define scoring candidate optimize strategy length vary heuristic dag consequently restrict exist tc tc stage identification candidate parent stage prune certain computational drawback never stage stage prefer gaussian lasso stage approximately demonstrate improve bn traditional bn task diagnostic purpose straightforward usage train bn high kind train bn optimize discrimination reflect difference bn stage parent bn face exploratory unclear explore discrimination interpretation interpretable finding mathematical term requirement come demand understand domain sufficient identify purpose accuracy also generative bn amenable generative necessarily discriminative share critical group unfortunately happen clinical consequently inferior discriminative exploratory aim gain discriminative bayesian discriminative variable brain moreover bn bn individual discriminative base gaussian kl employ svms learn learn maximize svms contribution framework include induce fisher discriminative svms bridge learn optimize iii advantage kernel induced simplify discrimination svms second learning term optimize upon motivation optimize svms framework jointly class discriminative learning framework contribution dag ensure validity simplify optimization consequently parameter block coordinate dag parent discriminative propose application early diagnosis disease newly image brain network attempt study interaction region image brain semantic bn connectivity connection pathways region affects find many disease novel disease diagnosis framework test demonstrate capacity conference publish extension aspect used constraint theoretically verify benchmark set structure learn method framework constraint fourth difference discriminative framework comprehensive framework vary organize review brain framework propose topological ordering notation occur summarize ht random represent represent pa realization bn bn background brain please generalize beyond example define bn direct closed path property pa pa gaussian parent I bn graph parent bn traditional consist stage parent initially criteria drawback true recover stage sparse implicitly regression optimization sufficient dag require ji ij ji eqn difficult solve method whole iteration update search node simultaneously obtain conventional network recovery modality sensitive traditional cognitive assessment diagnosis subtle brain suggest necessity brain complex mathematically align affine transformation interest voxel brain region brain connectivity brain technique dynamic causal bn causality bn lag network fmri bn note region employ handle relatively bn construction identify parent bn bn underlie context represent ad train separately likelihood may ignore subtle critical distinguish argue learn jointly essential discrimination integrate discriminative important divide incorporate train generative discriminative train influence paper propose kind discriminative framework induce discriminative kl performance base discriminative optimize subject capacity framework fisher discriminative kl model one map represent vector feed error adjust improve convert kernel feature learn way sample intuition object similar compute describe change direction fisher weight identity categorization induce feature gmm visual vocabulary despite success application mainly publish discriminative confirm separability sample fisher develop criterion log factorize pa learn kernel require possess property firstly separate secondly learn maintain reasonable capacity strategy separate hyper plane svms fisher leave minimized good svms secondly control model minimize parent enforce dag ensure validity develop margin optimize svms meet discrimination exploratory network different conventional classification bn learn reflect hence drawback regard discriminative framework discriminative framework classifier sample likelihood accord train specific couple well discriminative svm root relationship classification simultaneously explicitly optimizes implicitly optimize implicit svm direct outperform kl advantage bn optimization problematic increase mm fit two svm solve package mm optimization increase medium could hard solve kernel kl edge introduce vector word convert study body option encounter node handle keep whole compute option fisher pearson high correlation simple avoid dag remarkable selection discrimination kl mm selection step candidate tractable would could switch target gmm indeed simplify favorable property gmm propose constraint simplify discriminative recall h utilize implicit eqn optimization search could high optimization solve change induce obtain interpretability dag change experimentally solve dag solution satisfy optimization propose dag topological variable separable dag use direct node predefine bayesian direct dag constraint eqn topological order link remove compute ji ij ij whole avoid order worth note provide number could reduce long eqn satisfy topological dag however continuous noting therefore drawback point dag could single column absolute way alternate process strategy eqn dag warm strategy increase eqn tb datum initialize eqn fix optimize solution let enforce bias improve pa equal bias dag dag yet challenge noted eqn problem conceptually link framework analyze require deep investigation significant work aspect learn process connectivity brain network set follow h constraint ability recovery iii discriminative iv learn connectivity conduct publicly disease two imaging modality include weight mr cognitive patient control removal gray matter gm region normalization gm image node include disease gray mr patients nc belong partition partition image image disease use typical brain spread partition test class learn optimize column dag whole whether whole column implement code website whole attempt objective implement square provide warm applicable whole dag dag compare solution time arrange initial ordering average whole almost identical feature order feature order contrast variation change fig give quantitative averaged present solution affect permutation solve dag constraint regard order quantitative permutation average correlation whole ability whole since truth available due unknown conduct nine set come repository literature nine arc arc arcs arc chain arcs arc arc arc water arc whole eight bn mb gs tc tc repeat parent eight predefine nine bring behave mb show total number mis
term short term load balancing learn present balance instantaneous load context select short consider instantaneous instantaneous exceed transmission bs propose accord consider aware scheduling bs criterion velocity introduce rank candidate velocity many fair avoid enable fair resource cell base scheduling incorporate interface macro instant average mm historical associate past define capability balance reinforcement mm perform load balancing represent player I player optimize perform scheduling short step utility update select bs allocate base velocity mab maximize overall analogy machine maximize collect iterative select trade mab mm approach macro bss db due partially simulation basis scheduling utility mab part time maximize whereby mean action select iteration drop radius bs drop bss number large time due simulation velocity direction movement macro macro simulation baseline perform average filter entry cell baseline proportional scheduling exchange macro refer throughput mm mab improvement respectively hence mm throughput center throughput throughput base approach term update achieve approach hand aim maximize cell throughput gain also reflect throughput mab improvement classical approach compare fig versus bss bss yield per ms ms cases classical propose approach classical htb besides gain propose depict fig modify simulation setting classical higher obtain depict pp decrease high velocity half pp mm show behind load balance cell tries extend extend coverage history scheduling approach system maximization second aim cell compare method gain throughput base approach heterogeneous network essential provide user throughput cell however due macro base bss introduce additional requirement challenge dense base aware management mm cell use macro jointly long traffic schedule term throughput throughput probability approach expansion management reinforcement scheduling evolution heterogeneous approach meet ever capacity entail management across management essential ensure connectivity mobile user maintain service project macro dedicate key interference plus ratio failure probability unnecessary pp unbalanced load cell entail resource efficiency experience dynamically optimize traffic essential bias margin management investigate literature main cell speed simulation interference resource velocity co interference approach consider broad moreover adaptation reinforcement robustness selection formulate proportional problem load cell management jointly balance scheduling scheduling reinforcement base individually optimize schedule limited macro optimize traffic long term process history velocity scheduling bandit mab base improve overall reduce pp difference basic classical approach exchange among traffic cell individually optimize among mab aim satisfy capacity macro optimize result load balance short carry scheduling velocity average rate effort propose long solution traffic balance scenario load balance mab base short aware consider throughput enable fair enhance association paper organize system conclude transmission model bss drop
stand inequality element wise stand wise spectral hyperspectral ni slide adjacent column q abundance column abundance zero noise physical abundance coefficient namely relax say abundance efficacy lie characteristic I low justify abundance consideration correlate compose material although suggest abundance recently powerful g mainly nuclear norm impose another present spatial area mark safe abundance use robust already successfully contrary hypothesis low impose report sparsity low property knowledge sparse optimization q rank term fidelity term parameterize become flexible enough impose result accordingly estimate advantage low rank later surrogate attempt robustness consistency enhance weighting utilize knowledge simultaneously solve differentiable form regularizer optimization tackle alternate direction multiplier admm smooth incremental proximal exploit splitting function indicator function w differentiable require minimize incremental proximal consider recall proximal function soft thresholding perform manner ij ij diagonal matrix belong mind singular q thresholding correspond next projection nonnegative number utilize operator nuclear singular value computation projection proximity operator differentiable fidelity follow gradient approximately operator cyclic convergence select abundance norm constant analogue summarize table concern complexity complex svd take dictionary ill condition usual hyperspectral high signature convergence framework slow convergence robustness admm type develop alternate direction solve abundance consider lagrangian matrix lagrange case lagrange multiplier augment sequentially remain variable value lagrange multiplier alternate minimization cycle elaborate far admm remain auxiliary involve norm indicator regard minimization experiment highlight follow simultaneous incorporation abundance initially counterpart algorithm ignore modify low accounting solely say drop prior next simultaneously abundance entry illustrate fig eq db fig specifically exploitation sparsity counterpart see visual inspection recover fig rmse c early sparsity respectively abundance herein optimal rmse abundance abundance matrix snr db show rmse minimized region grid form subject rank abundance proper mean concern abundance aim end way depend range db realization rmse calculate white rmse obtain compete compare admm snr additionally slightly db method outperform characterize abundance colored actually hyperspectral structured assess method realistic linearly illustrate algorithm therein compete whole snr robustness c rmse rmse experiment highlights estimate abundance mixing mention simulated hyperspectral image consist block abundance solely row produce sparse abundance pixel abundance level rank description mixed pixel corrupt snr pointing row ccccc cm ccccc cm illustrate algorithm apply hyperspectral scene capture usa pixel spectral band nominal water low band examine algorithm term minimum dataset assume case pixel characteristic prove compare pixel reflect lie lie build dictionary signature display abundance map worth point evaluation careful inspection map algorithm produce similar pattern nevertheless algorithm reasonable meaningful abundance material hyperspectral hyperspectral sparsity spatial comprise proximity component norm abundance treat sparse rank two solve incremental admm call simulation synthetic art derivation computationally efficient scheme svd investigation relevant research abundance impose could topic interest work deal compressive processing interest know structured form suitably impose sparsity enhance paper spectral hyperspectral specifically two novel lie homogeneous region hyperspectral nuclear abundance determine slide window conventional impose sparsity abundance cost incremental proximal alternate method admm illustrate conduct hyperspectral alternate multiplier abundance attract recent year process identify spectral signature material whose generate mixed deriving correspond formation constitute call abundance vector two rise put literature address extraction research spectral signature available abundance fall make fundamental mix signature pixel latter adopt therein pixel signature predefine dictionary abundance henceforth treat due consideration impose various abundance regression attempt well abundance result idea incorporation knowledge justify dictionary pixel signature accept dictionary one abundance zero practically norm bayesian scheme abundance constraint incorporate region region among signature pixel hence correlation abundance lead development whereby abundance estimation spirit term dictionary large assume translate abundance sharing present abundance pixel abundance structure impose abundance regularize direction multiplier fashion propose image abundance central take signature
technique know originally stability bottleneck overcome develop condition recursion control iterate inclusion measure explain motivation stem property dependence vary reinforcement iterate stable pose reinforcement paper word markov finally application temporal weak literature guarantee convergence use involve show iterate outline specifically set reinforcement notation al et inclusion di guarantee absolutely reader km lx dx say chain compact great sequence dx dx x call dy control eq jointly compact lipschitz without integrable martingale contribute related assumption call value markov process state let iterate detail limit x limit set sufficient constitute lyapunov explain fixed choose dy sc h assume n dy paragraph hx h us c n explain early definition convex compact hx hx functional say hx sake convenience ax proceed ax convergent nk nk nk nk nk nk nk nk nk nk compact convergent subsequence notational contradict f nk nk fu nk tm tm n tt k control martingale noise surely follow rescale recursion q unfold expectation get mn claim surely initial almost follow stability iterate lemma limit c rl rl tm tm generality far know h kl tm tm tm tm n tm prove kl xt xt compact hold tm tm lemma sense claim consider manner p n hold x l mn nn ml hence implication see tendency fall unit jump inside unit outside jump outside trajectory fall since force jump jump jump use get contradiction present impose recursion couple stable measure list control process take compact space borel respect measure ergodic prescribe show proof reader dy tn tn bs almost remain hx iterate stable converge invoke converge idea several vector recall recall value control detailed exposition van impose condition convergence say van regular usual analysis range l component trivially n nx nx c auxiliary
number approximately serve compare early rounding mode layer format observe point near begin epoch appear upon procedure stability bit fractional precision baseline proceed sgd stop reduce precision tend despite round small gradient suffer result achievable error reversible increase bit bit precision format rapid improvement reveal promise network train stochastic round epoch high precision computation employ explore arithmetic fact processor throughput double precision concept conjunction stochastic extend share orthogonal use conventional point arithmetic neural network hardware mini batch descent dominate feed error propagation calculation computational throughput translate training processor high memory hardware hardware enable hardware cost secondly modern early potentially gain choice prototype multiply mb block ram gb typical dimension storing store memory perform throughput measure second multiplier parallel sa multiplier great subsection block ram cache fraction matrix store read request income l cache back compute external sa column array controller ip interface operation x integer capacity constraint continue multiply employ allow show logical array implement every cycle ram either column cycle element early accumulate accumulation partial datum round implement store external memory throughput operate example neighboring delay improve operate frequency cycle multiplication accumulate product accumulate multiplication throughput array feed incoming cycle multiplication output array length typically bit input typically less element round truncation bit operation unit feedback shift bit add bit capability determine excess bit detect max complement rounding result vs number follow examine result remove enable compact external memory logic unit overhead hardware array implement synthesis frequency consumption throughput efficiency w compare table resource throughput benefit series operate frequency flip network influence low unit specifically substitution arithmetic circuit come gain energy throughput precision fix computation conventional round fail adopt bit computation implement throughput efficient multiplication round little overhead software scale inexact deterministic across stack right hardware large deep precision context network train use round degradation energy hardware implement point arithmetic round deep hardware ability perform capability rapid architecture thorough search space hyperparameter come recent large computing effort include distribute compute thousand core graphic processor time natural apart computation context unnecessary moreover improve neural exception asynchronous reduce purpose hardware design need traditional often unnecessary overhead computational resource possible tolerance relax hardware software energy possibly computation hardware error ingredient develop need introduce preserve application towards cross co low precision arithmetic rounding fix point motivation arithmetic computation firstly point resource circuit area secondly reduce enable fit give memory capacity dramatically improve parallelism key finding exploration neural train fix arithmetic validity approach neural network mnist cifar deep network train round achieve throughput use arithmetic unit round module determine critical design analog digital surprisingly literature aim performance majority study focus implement train high study processor employ hardware throughput implement propagation present bit arithmetic understand precision network back propagation learn probabilistic rounding update precision requirement technique neural variant perceptron contain unit result neural million trivial precision deep dataset procedure recent hardware neural employ find achieve number round computation knowledge round neural low arithmetic implementation network correspond integer fractional number plus bit paper notation representation correspond length integer fractional small format limit precision set element result number mode round number fractional length weight layer format present investigation dnn convolutional network dataset comparison reduction error set bit arithmetic constrain intermediate propagation layer output update bias format start parameter keep unchanged one word length bit allocate fractional fairly restrictive choice implication perspective parameter update store loss information update significantly fix point format consequence progress worse near scheme oppose stochastic rounding maintain zero probability secondly offer limited relu hardware bit length carry simplify connect network relu activation train recognize handwritten digit comprise pixel contain digit lie augmentation perform weight minibatch minibatch size cross baseline achieve bit show two rounding mode bit leave bit integer portion seem network degradation either bit begin impact primarily reduce fractional precision zero rounding preserve learn precision round unable prevent ht rounding
noisy ica consider unit sphere turn slightly notion iteration complex hessian expand uk real derivative derivative kk iteration perform candidate inner ta orthogonality inner invertible largely issue noise free aa form aa inner define zero tc contain proper particular aa pseudo ia eq basis equation issue column tb choose vector couple remark modulus constant guarantee modulus convergence cubic limitation omit proof near make progress predefine k return result orthogonal modulus minimizer g e derivative let g minima maxima let recovery demand simultaneous row recover complement recover column achieve use aa kk kk kk kk c tb b access allow use ica mix orthogonality column td kk latent know find properly cite gaussian paper transpose space lot ica mixed sign joint sp paper trick paper pca rely sir highlight paper direction arrival problem thm thm thm ica blind ica assume signal linearly ica inverse datum recover latent however optimal measure interference plus even reconstruction propose model source signal improve ica provable practical original quasi step lead typical ica dimensional kk latent entry may throughout achieve center ica eeg removal signal recover column direction application processing recovery recover recovery impossible uniquely generally natural question inherent ambiguity part noise preserve optimality interference optimal b good signal remarkably different optimal noisy orthogonal whitening noisy even noise datum provably summarize contribution demonstrate optimal setting noise upon ica ica arbitrary complicated preprocessing necessarily experimentally ica algorithm noisy moreover noisy matrix infinite ica wise conjugate transpose transpose proceeding important somewhat arise ica nonzero contribution equivalently scale convention modulus factor addition ambiguity permutation indistinguishable say recover choice unit modulus permutation source define invert obtain align random indistinguishable particular ill define length mean even obtain source ill despite possible additional assumption portion source simple form inherent probably blind number generate vast reader book broad overview allow many recover ica mix noisy ica largely split higher another manner somewhat noise ica directional line advantage e latent signal sign algorithms tensor redundant also high complexity deal overhead directional derivative characteristic rely heavily upon continue start ica address practical issue ica originally sign ica source preprocesse unnecessary modify experimentally generalize source work attempt ica note noisy ica ica outperform first recovery assume recover precisely permutation zero constant ica recovery give kb noise define divided interference contribution access b estimate ica column white maximizes generalize spherical non proceed supplementary use even fourth nice algebraic ica random homogeneity vanish section unit sphere f property derivative ica necessarily unit f generalize iteration direction hide orthogonal multiplied treating update iterative converge rapidly one transform make orthogonal fourth positive may fourth sign preprocesse variant get issue distinguish ica one ica ica step notion inner candidate orthogonality issue inner invertible ignore issue aa noisy set let entry tc particular ia unseen obtain column unknown scaling choose unit fc fc fc worth couple remark notion exist guarantee choose exist converge cubic due omit proof ica test significant progress k convergence check meet recover however column demand idea simultaneous instead column using simultaneously find orthogonal complement within product k ta kk kk kk k fc fc fc c k zero let k us orthogonal gradient idea recovery algorithm step orthogonality column show td kk k signal tb tb inner product ica ica ica fourth ica design implementation free set gaussian opt reverse singular decomposition random singular singular choose datum ica aa noise variance variance ica sample power reporting error bar interval distribution ica degree freedom uniform unit create compare optimal ica ica use ica result formula apparent sufficient additive keep algorithm compare ica
type analogy detailed analogy question give pair implicit apply wind pressure answer temperature connect pressure analogy also model identify list form example two one analogy pair chapter book act answer book several act difficult analogy question combination candidate answer classification question also study classification ability majority odd one iii relaxed sound mean pick close word identify correct typical word close iii iv lose question close question require pick word list opposite question test ability identify select word typical ii answer type solver divide good framework basis question three component framework classifier identify different type short use build representation svm portion label include analogy component representation word answering requirement multiple word focus rare relation multi relation second sense challenge capability aim text employ label relational embedding combine sense relational word embed many text mining limitation single multiple study representation capture word pre cluster discrimination knowledge dictionary discrimination skip slide window input text stream slide window try word stream skip q word stream probability input output e vector window occurrence use skip pre learn embed weight regard context window calculate frequency window denote window calculate average embed context spherical representation cluster number cluster sense window include cluster denote vector online description word represent sense word representation vector close I match pair strategy recursively match pair find word sense use embedding different occurrence correspond sense pair word sense pair integrate relational skip entity operating entity function relational knowledge extract dictionary etc form triplet knowledge word sense word make relationship sense away word unified knowledge denote distance embed simplicity corrupt triplet construct replace pair select trivially norm word representation constraint norm commonly trick enforce representation adopt relation relation within combine derive relational sense pair calculation combine question online final provide section manually test type analogy analogy ii classifier set correct book effectiveness framework test set corresponding statistic list record standardize united student advance framework public analogy age analogy analogy overall education analogy analogy school master candidate experiment new baseline guess straightforward guess run design intelligence quite human answer question human amazon internet allow intelligence five question people collect take several high restriction worker american demonstrate accuracy intelligence worker iii age education continue believe sense latent distributional word allocation word gram sg word train skip gram skip window embed count multi ms word respectively publish multi word embed author frequent adopt propose method learn embed e triple adopt public relation build solver www computer question solution question wrong answer solver empty cause e analogy classification solver answer ccccc analogy school degree master candidate degree sg ms ms human amazon human question statistic participant participant age table find participant report distribution education background test participant hold school education education normal intelligence demonstrate mention accuracy empirically superior gram sg sg sg aspect performance ms ms sg multi sense word bring much rare contextual training linked train rare empirically superior sense demonstrate effectiveness adopt building quite indicate knowledge level intelligence answer various compare sg question incorporate word improve table accuracy type answer question human tend achieve accuracy consistent reach competitive involve amazon worker certain intelligence accuracy answer question different education people education degree tend common reach involve amazon master question potential human intelligence sg people school well overall assume school master candidate overall sg sg although concern due deep relation framework type ii representation relation design dedicate solver relation address experimental framework exist even exceed amazon involve experiment work early solve ai leverage intelligence plan enhance sense embedding bin liu intelligence standardize question human intelligence question measure understanding test solve automatically artificial intelligence apply simply apply word mainly relation among tackle challenge analogy word leverage novel word consider nature word dictionary solver outperform exist question exceed amazon might far intelligence deep embed intelligence deal experience test intelligence design individual adapt intelligence year measure test education successful life people still tag intelligence human like intelligence software activate attract ai design agent maximize success artificial intelligence interesting like robot answer far know limited develop solve test human develop agent question near question recent progress natural language nlp word advanced ability ai mean relation leverage however attempt application could satisfactory performance occurrence multiple capability embed aforementioned component first recognize type question usually sub type analogy question kind therefore effective question leverage novel consider knowledge word word retrieve dictionary tag word sense addition embed relation incorporate relational word effectively dictionary third specific solver representation relation representation find answer word question word answer question calculate offset answer conduct use experimental outperform question human amazon interestingly propose appropriate intelligence intelligence intelligence usually contain complete limited score calculate answer several age test behavior perform normalize less contain category picture question several detail algebra geometry logic reason identify element sequence effort develop method www computer program solve test score average number solver target level picture question recently approach
predict thereby reduce burden foundation practical multi domain demonstrate margin several far bn structure learn concern future parametric suited control false graph gene expression pc track dependency previously redundant calculation pc code maintain cache store research extension modification search procedure promise optimize multi several thousand label multi underlie joint composition need acknowledgment thank grant joint integrate lp j lp lp fact contraction lp label definition lp lp weak union completes consider satisfie composition lp contradiction lp lp composition contradict minimal parent form lem mlp suppose exist condition intermediate undirected minimal recursion I parent mb boundary lem boundary union property induction second markov define lp lp lp lp lp lp lp lp lp lp lp contraction statement lp thm lem gray gray network structure pc skeleton bayesian greedy hill edge series experimental hill bayesian structure use eight benchmark various assess return algorithm outperform goodness structure second solve characterize identify minimal irreducible factor condition multi classification multi encode term ten cover conclusion structural pc form motivate empirically effective suited source code boundary bayesian bn probabilistic structure bn acyclic distribution associate dag encode attract dag useful observational ideally coincide dependence identify structure terminology classical model basically cb systematically independence orient bn search ss evaluate hybrid attempt skeleton cb constrain consider ss present novel call pc skeleton scoring hill search constraint target variable think extra arise expense positive false among modification al later modify pe na fdr edge scale thousand series comparison pc hill powerful algorithm bn bn benchmark assess new dependence pc investigate ability joint challenge video annotation web page categorization large number category recent research focus exploitation dependency combination bn offer elegant powerful establish markov markov boundary theorem offer characterize decomposition ii predict thereby input burden solve datum assess comparative ability pc encode dependency structure mean indirect indicator make rest review bn evaluate show several involve artificial world learn empirical methodology theoretical raise issue future section upper letter set letter bn tuple direct acyclic node represent satisfy subset parent denote easy factored allow parsimonious enable determine huge determine relatively bn structure entail denote path parent parent path bn converse necessarily remain variable conditionally none parent drop separate adjacent iff find iff encode conditional via criterion dag show establishe undirected arc undirected edge class uniquely represent partially dag dag link dag orient dag equivalence structure learning sound dag database ideal test decide let denote four disjoint subset probability four z property z composition z x super probable bn bad need resort able cb ss cb relatively quick stop rely significance test unstable cause many error graph ss incorporate user database ss method deal prevent find optimal bn currently available like computational variable burden impose parent ability restrict cb method construct local around target bn scalability balancing accuracy hybrid min hill conduct show fast outperform reconstruction dependency equivalence greedy hill parameter heuristic powerful state art bn capable enhance algorithm bn e constraint super dag ss optimal vertex structural feasibility bn super structure scale hamming long prohibitive interest hybrid capable accuracy cb ss contain skeleton super small control discovery false attract false introducing behind cb induction bn handle cb method identification neighborhood scalability cb systematically check relationship target either datum discrete acceptance discrete global distribution multinomial latter independence test discrete table frequency c shorthand classic independence mutual mutual log ratio test differ asymptotic degree detail limit problematic small contingency implicitly conditional heuristic literature heuristic perform assume power count user heuristic reduce contingency freedom adjustment heuristic hybrid hybrid skeleton perform bayesian hill search cb subroutine call parent discuss view learner attempt produce learner pc false discovery fdr fdr node hybrid benefit incremental start extract severe size order increase reliability pc obtain pc think negative extra true particularly dimensional domain criterion say less severe original weak learner fdr aim proportion among fdr extension pe na modification incremental association control false positive estimate simply remove condition five indicator density equivalent think support suggest bic goodness new new measure generalize hamming distance learn learn dependence match undirected remove orientation sample learn pc compute indicator pair set generalize reconstruction two posteriori probability encounter practice score sensitive equivalent typically score rely gold instead employ network also skeleton phase value average depicted indicator express false distance clarity mention interpret regard advantage consistently negative expect come little false precision recall benchmark independence cb test worth capable reduce sample result much cc c alarm cm link dag obtain size bic training clearly dominate generalize network dependence datum heat find h rapidly plot sample average involved somewhat linearly h time slow mainly expense incur obtain pc challenge assign label throughout vector find function amount way suppose capture relationship straightforward meta binary method opposite sense independently lp consider label question capture label exactly multi attract purpose review fundamental wish involve feature subset r multi recently encounter minimal proper subset label behind partition j wise generalize simply mode seek minimal factor facilitate also characterization boundary depict dag dag minimal either ii parent address question markov answer markov boundary dag boundary problem decompose multi combination whole label summarize label respective boundary minimal lp boundary aggregate probable assess separately label markov boundary scenario without denote classifier exploit serve comparison selection label boundary evaluate effectiveness feature label dag denote minimal label dag forest classifier achieve good handle continuous rf rf default summarize come biology music repository come dim example label dl label continue bn learn datum biology image medical multi label assess focus decomposable score global term accuracy exact match label predict set hamming inherently label conditional fold table report report node dag figure illustration purpose conclusion inspection label several dags densely dag display figure conditional sep use many inspection dag bottom dag advantage information label beyond scope deep h pc significantly degree label degree resp nice pc consistently reduce negative discovery reveal mlp approach extract mlp confirm h interesting looking size mlp part lp see label decomposition ignore dependency expect mlp dominate dag pc compare mlp mb pc h sign statistically improvement pc approach without mlp mb pc identical hence column reach conclusion mlp average actual difference accuracy significant sign pair surprisingly feature increase due restricted feature though interestingly densely connect selection reduce input effectiveness balance increase burden drastically relevance loss boundary
roc lower u contingency table least curve pr contingency generate roc pr curve space dominate pr mapping pr true pr roc roc dominate pr curve skew balance unlabeled large pr loose demonstrate effect via roc table property confidence cdf statistic via strong law number empirical cdf pointwise cdf area curve amount positive ranking depend ci upper bootstrap number positive practice fraction unlabele violate within understand effect positive correspond ranking cutoff well correspond rank cutoff bad baseline none black corner none text corner use interval cdf standard aspects pr show estimate cdf cdf practice outside interval true roc pr curve close roc along curve strict plot treat approach pdf curve wide roc space assume yield poor estimate note estimation pr curve rough available pdf contingency upper base use efficacy approach curve positive cdf rank positive estimate sensitivity space recommend use roc unless good estimate roc curve thorough assessment set address project circuit logic grant research fellowship medical e van cancer cancer plan bm grant research grant assess performance metric label unfortunately many furthermore unlabeled example available able unlabeled bound empirically able roc curve quality critical rigorously evaluation often report summary metric area operator characteristic roc curve visually operate roc curve construct contingency table confusion contingency relate ground depict contingency predict predict positive false positive negative false negative compute contingency label label costly impossible special research cope labeling oppose partially empirical evaluation superiority mean assess partially label alternatively unlabeled certain circumstance describe show compute example contingency rank compute contingency table overall theoretically give low upper bind false empirically efficacy pr curve review partial classify learn logistic naive value belong typically belong order list instance sort value confident characterize value rank top treat complement prediction predict compute fraction label incorrectly contingency positive negative instance within instance cdf r eqs rank overall interpret label top rank x receiver curve rate axis vary axis cutoff point curve give positive draw straight line consecutive rank extensively assessment operate commonly range random perfect precision curve axis function recall positive positive skew set know positive negative unlabele instance unlabele negative focus available include negative contingency curve rank positive directly tackle theoretically positive knowledge disease either roc pr however positive available rough pr treat unlabele inherently cutoff positive treat small curve positive relationship include unlabeled latent positive positive implication disjoint subset overall r numerator minus r positive rank positive distribution rt three positive within equation equal rt lemma relationship rank contingency overall rank table require account unlabeled proportion positive example label greatest bind rank l r surrogate positive define u l greatest great via due contingency greatest bind contingency discuss contingency cutoff rank contingency create positive construct distribution contingency table rank describe construct cutoff decompose separately partial contingency table via contingency surrogate positive great via contingency positive translate require ensure may assign surrogate contingency contingency bind replace equation rr available rank upper vice versa obtain positive similar eqs deviation independent l know positive l equation violate biased occur application via cdf since perfect estimate cdf interval ci cdf ci cdf
directly fit reference therein copula posterior density type transform datum capture mixture model variational hierarchical vector variational posterior representation applicable require conjugacy within family assume jx prior vc mean field vb special independence differential c hold px parameter diag univariate concern simplicity copula margin include analytical copula univariate histogram shorthand cdf copula copula construct copula marginal pf offer variational framework representation j c j copula allow margin I c achievable kl optimize free margin c copula function accurate term posterior although form margin copula function approximation univariate margin j p copula still discrepancy margin compare upon update parameter lead calculus convenience incorporate exploit gaussian copula non decrease directly optimize one diagonal apply variational bayes j diag translate jacobian cholesky triangular detail deterministic stochastic introduce desire posterior normal auxiliary form illustrative show capture form normal gamma consider single observation conditional p sampling available fix margin jx nx cc deterministic special algorithm baseline gibbs appendix mixture automatically desirable polynomial uniform nonparametric simplicity add construct cdf parameter cdf exponential help chain rule derivative q term analytic derivation due expectation analytically tractable non nature locally solution contrast likelihood respective derivative derivation accord respect diag jj alternative way express gradient expectation already j derivative define polynomial low b limit prefer degree issue variational bernstein much simple proposal assume integration hold j gradient gradient turn enable stochastic constraint simplex projection introduce prior optimization summarize improved adapting seem kernel variational continuous margin involve cdf need entropy mixture pt j demonstrate able preserve margin constrain certain parametric form relax nonparametric construction refinement polynomial k gamma accord margin hierarchical avoid specify margin nonparametric base flexible univariate margin copula limited interpretability via pairwise dependence discrete margin parametric copulas rich dependency future copula automate posterior efficiency metropolis sampler mcmc proposal derivation model hierarchical variational combine copula log margin part width conjecture copula constitute unified construct posterior divergence posterior propose copula bernstein polynomial characterize posterior inference distribution represent intractable integral kullback bind tractable bayes field readily gaussian incorporation correlation continuous unconstrained skewed heavy tail
sample statistics nominal ess code summary ess st rd significant rate well size coefficient quadratic nearly improve mix lead much low strategy partition space beta mh diag dimensionality state partitioning burn interval sampling sd ess ess min max sizes trace confirm select part stochastic newton twice differentiable pdfs taylor series hasting dimension empirical assign dimension preliminary sample subset well call mean within subset potential go dimensionality tradeoff complex evaluation algorithm generate mcmc less correlated univariate metropolis proposal center cost generate hessian show next technique e software per sampler extend utilize computational cost gradient evaluation seem implement newton sampler algorithm multivariate local expansion newton advantage geometry capture hessian sampler independent resemble gaussian require hessian negative valid property easily glm initial take augment fast problem lower apply improve convergence visualization capability predictive distribution world pdf multidimensional univariate sampler slice rejection pdfs univariate sampler however less correlated development efficient black box importance machine recent development sampler carlo hmc view implement stochastic newton metropolis hasting proposal locally multivariate current equivalently everywhere satisfied much coefficient appear variation square glm metropolis close glm hessian hasting identical dimensional provide adjust langevin rate software write package extension mix argument diagnostic method review overview hasting package offer provide summary future research begin metropolis density mh property ensure transition ergodicity mh density locally density hessian respectively globally negative fit think counterpart nr nr mh reject proposal identical case acceptance describe begin overview responsible implement follow log function hessian multivariate prop prop hessian old old prop old old r prop new prop prop new old via call argument control choice newton nr control partitioning argument generate via fed via unnecessary proposal collect diagnostic probability series relative series diagnostic use illustrate method diagnostic visualization capability mode multivariate pdf lead small overlap nr mode accept next pdf mode argument nr guarantee nr maximum fit gaussian different derivative observation bad strategy partitioning subset subset gibbs argument list space belong convenience respectively rich applicable chain mh specialized adopt create specialized calculate summary include quantile effective plot proposal mh test log pdf iteration nr exhibit mix valid ensure end full arbitrary across expect value proper important mathematically expect apply incorrect interval correct require glm nr x primary ml rather switch remain beta diag examine trace plot pattern non peak pdf chain area plot select offer plot besides diagnostic sampler space iteration nr burn deviation sample sd ess min st rd max half burn argument counterpart function package observe rate contrary pdf secondly relative non pdf still lead good next want predict poisson explanatory distinguish predict predictive illustrate
pt mm mit mit edu microsoft microsoft unconstrained nesterov accelerate geometric ellipsoid method nesterov one verify descent view combination descent optimum search optimal rate convergence acceleration practice behind nesterov accelerate descent new interpretation reason possibility acceleration problem acceleration strongly convex gradient ball contain optimum maintain ball intersection radius shrink accelerate iteration contain intersection section convexity imply q recall smoothness allow factor obtain style pos right pos node pos node node pos pos diagram intersection one geometric accelerate optimum combining obtain already center ball ball formally easy calculation eq q fy fx fx fy eq radius see acceleration shrink hold ball observe ensure geometric well insight center square radius ball formula r induction case display fx fx fx fx square fx fx assume point intersection two consideration intersection segment reveal satisfie ball half ball ball remain g straightforward recall instance correctness iteration guarantee property believe integration new suited radius radius h initial fx kx fx fx bc bx bx descent accelerate accelerated method bfgs updating bfgs ii
input measure eq absence concept capture notion sample visual combination q cluster q cluster label visual number representative cluster max cluster list refine cluster base label share concept least compute label exceed label sample split break differ neighbor static neighbor seed sample create centroid centroid unlabeled compute diversity angular distance choose centroid gain speed diversity unlabele unlabele label diversity approach differ notably disagreement cluster entropy refine batch label decide batch newly beyond batch sample entropy cluster sample mean conduct divide training test video vocabulary divide part initially label test concept algorithm batch reveal label algorithm annotation score test start corner skeleton annotate set arm exhaustive annotate level frame computation angle maximum whole annotation dataset available research testing rest select datum positive size sample select seed method propose validate second baseline annotation round weight work assign top compare discuss since htbp baseline well annotation capture inter train concept base jointly pattern case al select informative first train early monotonic decrease ap annotation ap sample get correct concept dataset round baseline average run round indicate help concept model well believe one like propose combine novel uncertainty sample approach baseline retrieval reveal active learn recommendation author reflect author like member ari suggestion feedback operation base finding recommendation reflect additionally thank member dr suggestion feedback conventional normalize label good determine annotated learning combine measure novel diversity integrate refined level agreement corpus annotation retrieval video disk daily effective tool annotation tool treat automatic annotation whereby explore correlation rank sample relevance user query exhaustive normalize like annotate effort cost annotation raise way annotation require address issue active unlabele query provide system output label annotation typical engine annotation retrieval engine responsible labeling unlabeled system video retrieval combine measure refinement density art relevance generative annotation technique define label concept word word et al suggest annotation pick high task query pick video high estimate video fully annotate integrate selection annotation combine uncertain engine label use svm
constrain collect mini calculate monitoring statistic specify environment decision specify make svm precision exist present rate unlike approach detect presence label outperform detection drift arrive batch concept stream limitation performance extremely magnitude hence limitation propose identify fact imply define tn fp denote underlie percentage tp negative classifier characteristic value predict stable joint concept worth step pair refer influence note concept suggest even classification current stream use easily adapt efficacy detect drift concept detection estimator concept stable reject level statistic user level type point stable false every fix accordingly cost simultaneous inference correction run alternative na I na I four drift I four use rate r essentially previous decay weight instance class imbalance detector test characteristic test detailed significance level detect significance level concept c tr r c tr cd decay significance level decay current framework practice detection rate application control move fair learn geometrically sum variable historical take weight stationarity overcome reliable geometrically bernoulli random simulation decay carlo surrogate generate upper significance serve bound surrogate detection corresponding obtain bound four framework instance classifier class bound likely reach detection drift store since store new one sufficient cross correspond fail reach new monitoring cycle underlie decaying factor significance significance level bernoulli lb investigate bernoulli variable stable concept I change equation stable concept set distribute realize time imply thereby statistic compete empirically dominate achieve lag power exponential short lag limit alternative variance htbp km kk maximal lag rate regard detect drift kp lag occurrence drift detection observer adjust cost incorrect prediction immediately concerned occurrence drift stream provide statistic convergence long guide r synthetic public consider imbalance public dataset across various baseline poorly utilize bootstrappe generate algorithm classifier public create stream stream fed visualize detection point concept avoid redundancy present histogram remain one compare identify drift false numerous cover concept drift mechanism algorithm identify discuss typical three classifier drop cp cc cp cp count top bar bin mechanism consider concept imbalance balance imbalance cp cc class occurrence imbalance classifier unable alarm drift trivial false increment respectively imbalance ratio unchanged cp cp decrease decrease select cp cp public generality choose machine misclassification majority concept svm classifier adapt concept example hyperplane hyper dataset drift detection create dataset stream collection difference magnitude drift user topic table imbalance c balance balance balance imbalance imbalance imbalance balance balance imbalance imbalance imbalance concept early false delay rotation dataset drift drift experiment dominate drift superiority detect decay minimal early poor missing delay count drift count false detection simulate dataset table dataset concept span concept concept histogram predictive drift period bin span summarize particularly dominate performance small benchmark regard detect produce false delay drift concept belong allow work stream use user parameter detection term low require however change relationship observe concept new concept concept drift apply stream response underlie specify compare benchmark public span type benchmark term detection concept mining stream stream always income concept stream business intelligence data stream paper focus detect binary classification drift say predictor intuitively concept drift underlie detect concept widely use detection streaming employ hence detect drift false
useful ps call whenever agent correspond exist value categorization common place layer action layer particular operate introduction act traffic drive traffic among continue drive stop car category four combination color randomly choose action setup describe would compose four shown try associate rich connect associate action elaborate auto circle minimum sep font edge node edge enhance ps illustration green create step green encounter place newly create edge line dash line similarity third create place second left cause share red establish modify auto inner pt thick font node cm style inner sep dash bend right bend node auto thick font dash bend edge dash edge dash bend edge bend edge bend edge bend left cm sep thick font bend bend bend edge dash dash bend edge bend right edge bend dash node node node dash bend node dash bend dash bend bend node dash dash dash edge bend right node edge bend node mechanism categorization categorization ability recognize achieve come think red relate stimulus fulfil connect zero categorization realize flexible fulfil environmental adapt scenario begin drive green light irrespective opposite stop green driving phase drive stop leave color reward choose irrespective signal phase ignore action clarity explain later demonstrate connect correct correct build update present flexibility configuration network adapt fast fig display practice strong edge address configuration rate thick style minimum edge pt circle sep thick font node node node bend bend cm node inner font bend edge bend right cm size sep thick font bend bend bend x south north west efficiency ps agent function agent choose adapt phase impose action asymptotic action extent chance learn stimulus trying find similarity consider analyze environment different background different agent direction whenever follow irrespective call color auto thick inner thick right right dashed node dash node dash dash dash edge dash node dash dash edge dash edge dash dash dash dash edge edge dash node dash edge dash edge scenario ps two time correct never later step symbol twice thus infinitely many ps ps behavior color twice thick style circle sep thick font right edge bend dash dash bend edge dash bend bend dash dash dash dash dash bend bend edge dash edge node dash bend bend dash bend dash bend dash bend bend node bend bend dash dash bend edge node dash edge bend node edge edge bend bend edge edge edge bend dash dash bend edge dash edge bend bend bend bend dash dash bend left difference enhance ps successful enhanced look strong correct tend imply probability unity colored efficiency ps prefer reach enhanced ps precise auto cm thick sep edge dash dash dash dash edge bend bend bend cm bend cm edge leave dash dash dash cm dash dash edge leave dash right dash enhance solid curve efficiency completely reach asymptotic eq figure hundred analytical efficiency ease analytical curve qualitative difference x east dash blue dotted qualitative advantage quantitative fig indicate nonetheless network action majority voting occur agent whenever gap random ps follow simplify approximation actual agent efficiency look take argument lead show configuration conceptually action eventually edge correct present finally fact attempt approximation time plot simulated curve red cause whereas reduce nonetheless certain ps start analytic lead high asymptotic efficiency agent certain reach express cumulative thought express series reflect thereby converge configuration action give namely irrespective agent unnecessary would affect h auto distance style sep thick font style regular side thick font style size inner sep thick font dash dash bend bend dashed edge dash bend edge dash bend leave dash edge bend right bend node cm edge bend cm cm bend bend bend left edge bend bend node dash node dash bend dash dash bend dash dash edge edge dash node node cm edge dash dash dash node bend dash dash bend bend left node bend answer look efficiency lead action hand correct accordingly lead asymptotic e contain reduce irrelevant agent categorie irrelevant generalization make generalization efficiency also enhance ps tends show fig h c image east image west category expect action irrespective asymptotic efficiency eventually connect action enhance environment stimulus chance unless common recognition categorization classify new meaningful dynamical enable ps generalization generalization property mechanism ps arbitrary ps learn extreme introduce ps practical handle also conceptual enhance ps rather enhance dynamic walk give sophisticated complicated overall preserve inherent enable model rely existence induce generalization successful potentially create exponential intermediate agent efficiency agent performance color relevant present recognition particular predefine circumstance associate blue together even similarly generalization single category color single possible require far dark divide systematic take order addition structure whose predefine ideally even completely priori generalization thereby speed subject acknowledgment discussion project universit universit universit er universit cope simply several enable projective projective novel artificial intelligence recently perform standard simple reinforcement problem complicate canonical world car projective principle allow performance generalize ability extreme environment otherwise impossible ability act upon new experience extensively life need recognize light correctly traffic may color neither shape reaction aspect common generalization learn red signal signal action relevant whenever mechanism illustrate generalization handle traffic
berkeley edu microsoft com microsoft com grow make crowdsource important expert knowledge option address issue introduce voting utilize worker couple strictly proper mechanism guarantee together conduct empirical study amazon validate big ever deep demand grow collection crowdsourcing web service like amazon crowdsource worker perform exchange obtain crowdsource worker lack interface express several improve via interface mechanism increase labeling crowdsource label task question option category option worker require option formally involve single crowdsourcing extensively empirically vote worker wherein allow figure example voting system worker experience test confident alternative remain one individual vote interface worker mathematically problem belief worker set crowdsource multiple allow worker valuable question identification source item vote worker second worker worker worker answer worker second flexibility partial voting crowdsource partial knowledge crowdsource worker option problem couple compatible payment worker receive crowdsource payment strictly proper scoring want possible select option result prove compatible section coarse rely people belief payment requirement mechanism coarse select option belief enough also conclude report preliminary basic discussion voting candidate preference among interface crowdsource candidate option question crowdsource voting focus fundamentally irrespective rule design payment payment event payment strictly event strictly maximize belief however mechanism specify mechanism present score support belief question crowdsource worker response propose mechanism suitably worker free mechanism interface turn satisfy axiom adapt set presence answer design operate gold typically additional predict worker mechanism design therein weak absence option option assume contain question mechanism know answer worker individual worker option belief respective option belief distribution worker across shorthand goal worker worker formally worker option value worker select precisely option worker response gold question question question select correct answer select gold question none correct worker question option response payment worker note restrict payment crowdsource solely perfect worker answer amount mechanism quantity expectation choice question among belief question formalize belief select option sum option consequently option worker payment summation gold question worker choice assume extend general compatibility mechanism expect payment worker option correct theorem main result prove mechanism requirement mechanism algorithm compatible compatible mechanism amount worker attempt optimality uniqueness claim mechanism contradiction argument specific belief worker act belief coarse belief option know sure option incorrect among option thing payment make remainder devote jump without continuity first suppose generality worker belief option maximize worker payment select support option payment worker option payment mechanism inequality payment different support strictly payment upper strictly case worker equal maximize maximized component correspond set early expect payment maximize act outer respect question expectation take worker belief condition gold argument prove inner maximized payment worker act subsequent proof equality mechanism compatible lower thereby complete coarse belief worker belief compatible support belief worker worker belief mechanism belief turn break something desirable option worker belief contribute verify early theorem suppose question q eq follow result coarse select derivation mechanism belief axiom say worker question question select wrong option select option question gold wrong payment axiom notion qualitative goal interestingly notion value belief option present evaluation mechanism amazon crowdsource platform goal experimental perform basic check practice evaluate voting whether worker interface reason design quite respect worker understand unlike exist benchmark position primarily theoretical detailed publication conduct separate worker language display text identify display worker mechanism execute payment gold question payment mechanism start interface start fraction question zero incorrect voting interface payment interface payment entire author evaluation response response comprise selection option figure depict turn figure depict response select plot reveal voting approach successful fraction incorrect answer depict per worker increase
relevant arise becoming choose sample wu zhang discriminant ratio determine relevant one dominant range image ratio lie image give relevant lie dominant weight give choice distance successful literature combine instance retrieval distance consist query useful euclidean move space cluster relevant thus respectively set define apart set account relevant image inspire relevant database image define euclidean define score interval initially consist score mean high membership alone able reflect completely may relevant far centre relevant outli modify relevance score cluster evaluating define calculate database average database average scope provide retrieve lead increase relevance scenario image early retrieve follow change time relevance feedback one adapt precision evaluate total image retrieve scope equal scope involve return image relevant retrieval therefore aim retrieve new set contain retrieve sense retrieve number relevant consideration retrieve expect expect type increase value scope basic increase clearly establish really behaviour become counter hence evaluation remain irrespective rf search database image retrieval segmentation visually object indicator contiguous segmentation contiguous adjacent merge recursively contiguous homogeneous hand sub one contiguous segmentation terminate isolated image mean cluster view preprocesse computed rise mean segment equal cluster discard illustrate show pixel group detect correspond segment row image base segment motivate image segment database rank th th obtain segment segment denote segment segment th amount segment close denote rank similarity propose segmentation define image retrieve st relevance refer retrieval reflected perform henceforth shorthand respectively conventional involve propose scheme section implement relevance combination cluster density rw propose assign distance update every scheme refer brevity experiment whose table propose rw feedback difference lie initial retrieve rf rf additional apply rf retrieve image retrieve specify retrieve feedback improved propose seven rf iteration carry retrieval apply six retrieve rf five time retrieval retrieve set six rf cardinality retrieve image image integer six five rf rf accounting image feature perform image take five new assess retrieval rf iteration converge iteration rf discovery relevant retrieve retrieve retrieval possible crucial aspect standard describe feature texture colour occurrence colour value colour proportion pixel colour co occur colour position diagonal give colour image shape colour pixel diagonal single represent column indice researcher quantization specify co co pixel horizontal direction occurrence need demonstrate approach briefly name db db database subset db illustrative image database image per category database db db single idea diversity improve thereby justification mm database retrieval db db rf compare rf iteration c propose rw rw rw db db db db database evident table mark change rw scheme table table shorthand name identify initialize improvement evident case higher encourage discovery achieve high relative clearly lose c database db db iteration database present illustrate recall precision database db db content retrieval conventional retrieve query image segment match conventional approach segmentation instance employ scheme improve acknowledgment author put record ray school technology university discussion former student crucial insight ph thesis contribution dr university north university division institute road north hill mail ac ac com retrieval essentially extract database similar content image bridge texture human system typically use rf mechanism iteratively incorporate user give relevance retrieve approach vary work alone implicit incorporation image scheme relevance time suggest limitation approach keyword retrieval segmentation relevance direct consequence rapid imaging technology million generate source surveillance device security purpose scientific imaging home availability digital device internet maintain enable retrieve require indexing keyword file name index book result indexing picture carry represent school lead second content retrieval objective colour shape texture role attribute associate image scheme information segmentation propose retrieve rf efficacy approach establish six representation color describe couple feedback overview
right point take irrespective initialization challenge stand implement idea exist dimension finite vs ahead section describe trust address f f e carry section n invoke trick plane capture composition verify contain open instead avoid trivial minimizer locate study three eq region figure typical style describe look orthogonal plus perturbation argue perturbation affect ip behave u sound write magnitude simple aspect know ahead time knowledge appear spurious minima order information saddle riemannian trust region sphere unconstraine successively order approximation iterate order geometry movement trust induce subproblem trust region make progress typically radius dynamically accord text generalize manifold natural tangent iterate k riemannian riemannian usual riemannian subproblem ball trust trust region take basis equivalent objective trust ball admit numerical minimizer solve issue tangent additive resort tangent see movement iterate sphere geometric w small show decrease curvature trust sequence eventually move stay strongly converge minimizer quadratic sequence far include development dl application book survey summarie development recognition technique relevant current dl start algorithmic correctly extract generating require exponentially many subsequent study polynomially many ensure local however efficient algorithm dr show recover solving per provably overcomplete incoherent initialization refinement include recovery run super provably recover complete dictionary aside theoretical dl result cast show sublinear vector improve recovery nonconvex sequentially core blind separation nonconvex optimization nonconvex structure include recovery tensor pursuit signal refinement exploit refinement dl also guarantee optima complete geometry initialization algorithmic design may prove valuable nonconvex trust sphere build effort generalize algorithm riemannian reader survey development conjugate spherical trust riemannian manifold adopt either global critical force piece together entry similarly b u function dl pca ica multi nets architecture provide provable subject go recent deep ica matrix study assume row perfectly fourth overlap considerably help improve stability component connection help ask play role dr help landscape section even independence familiar figure broken sparsity level become believe generalized correlate case hand landscape grow minimizer adequate knowledge unclear objective distribution rotation exclude course bold capital scalar several specific work sphere ball integer confusion entirely vector th operator norm max norm distribute identically n bernoulli compactly frequently state prove unless otherwise note notation organize geometry riemannian trust sphere corresponding discuss main present main discuss direction major algorithmic deferred cover result proof reproduce figure find characterize landscape n n landscape orthogonal minimizer claim hold np n constant failure probability obtain sphere riemannian part stick capture x np n map minimizer sign local ic theorem local f x w q argument sign minimizer symmetric section cover sphere calculation sign local contain uniqueness isolate minimizer equally sense produce close row characterize high landscape hold simultaneously probability w w claim fail hold np p numerical np particular minimizer sign ic constant identical e quantity detail nontrivial landscape successively nonzero directional negative curvature depict constant every r c r h w page every hold gr page w qualitatively concentrate nice extend next every page w see next provide desire w w l fix x lipschitz l lipschitz integrate thing dictionary landscape look qualitatively dictionary condition define landscape u enough canonical comparison constant np see hessian cause section detail small large result local recover row one saddle motivate riemannian descent trust sequence minimizer asymptotically produce algorithm target one minimizer exposition try technical possible seek initialization iterate repeatedly minimize approximation iterate sphere instead approximate tangent space sphere obviously expect eq q next solution choose movement iterate read understand interpret particular f riemannian notion subproblem orthonormal recover classic trust region subproblem solve method numerically suffer approach e trust practical follow recommendation chapter backtracking base whole algorithmic procedure algorithm n r trust subproblem k general chapter produce probabilistic assumption guarantee iteration problem algorithm efficiently produce close summarize dictionary orthogonal dictionary dictionary orthogonal p riemannian trust minimizer set constant positive complete dictionary suppose condition positive p c riemannian trust region size solution near minimizer constant convergence target polynomially estimate relatively adaptive size produce relatively goal state provide analysis riemannian find nonconvex entirely nonconvex ultimately minimizer show start take consecutive unconstrained unconstrained iterate stay ensure stay gradient point minimizer continuity magnitude show iterate stay fact unconstraine mapping lemma magnitude gradient see section c r compare magnitude unconstraine crucial bind magnitude iterate trust region f p trust subproblem iterate make upper combine condition give also unconstraine detailed trust cc claim carry unconstrained k provide next forward calculation subproblem interior inactive q q constant unconstraine note claim simplification complete iterate actually nearby minimizer serve good proxy distance f relate magnitude point lemma component lemma make characterization locate sign obvious symmetric unconstrained iterate integer lemma local k line apply condition thus combine estimate moreover region thus proof ready piece together proposition enough lemma proposition satisfy minimizer contain interior connect respective sign component closure r minimizer proposition objective fix call unconstrained minimizer iteration consecutive step consecutive phase never drop continuity whole whereby minimizer either case drop many within unconstraine three drop stay connect component iterate finitely take step unconstrained decrease future must unconstraine unconstrained certain point minimizer iteration claim away target riemannian produce solution away minimizer simple linear programming round procedure optimizer sequentially recover row w sparse dictionary summarize relevant dictionary though recover dictionary resp start appropriately find target recover accuracy q show resp recover riemannian conservative produce row recover round input repeat procedure recover dictionary cn failure numerical constant towards correctness round desire provide already orthogonal round small induction procedure mind solution first vector correspondingly dictionary objective matrix learn keep setting reduce everything original effective lemma recover h small overall simple simplification tail inverse work might scale affect recover dictionary dictionary number algorithmic setting recover dictionary failure need round round round page argument compare orthogonal case part necessarily u v small perturbation perturbation matrix orthogonal bound see come instance dictionary prove solely rely c u theorem provably perturb q next identity suggest w obtain combine choice word return small constant lp round upon sketch plus round union tail inverse simulated recover row coefficient rotation l np entry bernoulli reasonably produce step size vary primarily vary pair repeat simulation sign e return reconstruction trial near likely recover lp round implement obvious dependency complete dictionary lp program approach recently improve matching obviously complexity intrinsic work suggest necessity efficient complexity proxy adopt amenable affect algorithm work directly sphere treat complete bound treat transform change transform current geometric structure plug play deal orthogonal nontrivial relatively forward extend recovery tight frame though elimination marginalization overcomplete marginalization behave coefficient likely physical ica characterization modify theoretical insight heuristic plus refinement mostly whereby initialization analytic highly coherent problem framework experiment believe nice spurious minima either surprising point predict modern reasonable generic initialization maxima saddle iterate saddle continuous counterpart theorem complete result state section deal convenient random subset moreover last line composition similar argument establish twice claim composite eq particular result integral form taylor w q result let q approximation f optimality substituting establish lemma certain work basically continuity need operator operator respective tangent place translation play lemma translation moreover e complete establish property riemannian fact lemma substituting claim f direct state work section canonical w riemannian differential justify enough scalar part apply tail monotonically decrease similarly tail summarize n ready result together radius define net np q e hc obtain take eq complete given consider get complete lemma local property manifold version k translation isometry theorem lipschitz complete taylor theorem q parallel define isometry isometry combine bound obtain need lemma dictionary cn least homogeneity consider random bound moreover integer simplify fix n numerical z n q cn overall lower suppose riemannian return nearest round take form vector enough return relaxation obvious objective thus solution necessary r therefore imply orthogonal subspace solution argument exactly basis complete define invertible lp eq imply v complete r e bind see iii upper large principal angle eq column rank projection onto span nontrivial pt conjecture width electrical engineering york recover dictionary modern machine give efficient provably recover contrast guarantee algorithmic certain spherical manifold provide riemannian arbitrary initialization presence saddle light structure spherical trust region landscape second geometry approximation c thank private foundation partially support grant thank mu mp possible model seek input signal signal role compression useful acquisition traditionally heavily explicit analytic constructive successfully numerous advance ever range classic modern multidimensional basis wavelet practitioner adapt describe signal challenge couple rare put effect codebook naturally occur datum challenge question ever empirical mathematical science discovered encode atom early process discovery application dl past decade process recognition deep architecture review development particularly learn great variety text genome series derive optimal manner hand dictionary increasingly promise armed sufficiently allow successful contrast theoretical application dictionary free desirable guarantee correctly important algorithm broad understand arise consider formulation attempt dictionary fidelity coefficient quality impose desire admissible orthogonal set optima unchanged example hull effective symmetry break tend play together relaxation come bilinear conceptual diagonal diagonal denote transpose isolated amenable relaxation unclear relaxation g problem sense relaxation provably hope dl seem always produce towards basis
without backtrack next time select multiply drastically reduce evaluation variety crf character recognition dataset name entity speech task journal pos letters image sentence syntactic sentence token tag correspond etc pos tag entity classify quantity gram pos case character token pos task assign syntactic tag token sentence data follow division development gram shape task competitive five variant average basic averaged py meta initialize dynamically classic power work comparison deterministic bfgs include online heuristic author sg non test sag give initialize sag size figure pass time gradient small different extra forward line evaluation backward computing use across strongly unique running indicate accurate plot exclude interpretable poorly include test plot plot dotted gradient size outperform move away early meaning early outperform view recursion outperform hybrid early relative rank change choose perform less effective initialization obtain competitive runtime outperform substantial good error far always reach speed pass reach e x x positivity lyapunov term lemma followed expect lyapunov add round q third ignore obtain get part proposition section strongly sequence probability except use choice uniformly maintain useful summing minimize side next x nf l nf l nf k expand l similarly b define lyapunov straightforward expand apply k simplify term k combine term row attempt constant require relations eq zero look place n eq expectation constant get error plot plot appear despite value exceed sag optimally tune gradient lack tuning could method salient utilize crucial performance perform poorly well obtain part label figure remove permutation perform bad perform nearly sag previously perform although performance set value lead runtime plot body pass independent except implementation tie result hardware thus little runtime hardware set runtime general runtime number several perform slightly runtime seem extra root implement runtime fast high relative bfgs bad runtime proposition rgb average use crf gradient improve uniform reveal training objective well well optimally tune stochastic conditional language label extraction parse name processing vision mrf dependency advantage discriminative building mrf disadvantage slow train crf due cost crf single training example advantageous deterministic single example deterministic method fast deterministic method require might optimization community consider vector objective objective like regularize gradient contrast deterministic low cost sag combine training example iteration reach accuracy fast convergence rate sag traditionally implementation sag step show sag binary sag tracking marginal crf drastically reduce sag uniform adaptively frequently datum sag particular fast compete task part speech parse optical character indicate sag outperform term require perform optimally error pair comprise standard minimize likelihood summation chain backward compute gradient solve grow thus like pass prove inferior strategy newton evaluate training traditional attempt improve deterministic example process reach dual online sag achieve fast classic sg cost quasi hybrid deterministic method slowly decrease method state fast algorithm bfgs accelerate stochastic gradient sag objective constant bound gradient primal sag stochastic iteration write slope example sag use instead keep iteration sag randomized gradient aspect classic fast major requirement nice use sag select regularizer crf likelihood gradient start track see normalize update sag nd py py I f would gradient fortunately dense scalar constant gradient small quantity change gradient unfortunately crf typically approximate standard backtracking since use value step monotonically slowly decrease g g py since stop see full zero difficult decide continue often memory crf marginals respect feature example feature write term take algorithm difference old parameter thing similarly pair store pairwise marginal
speech search infer probability stop variable stop factor affect model random capture parametric overfitte sensitive two fourth previous query query query hard parameter increase need result nice nlp use armed multi article improve click weather user want place search repeat treat online generalize thompson build hash term term associate query lda treat topic get collect predict adopt generalize thompson treat model original ten behind vote weight average vote rank feedback whether logarithmic ir ir wrong adjust engine date base user generalize thompson quickly identify cluster current sampling payoff rank search think high put pdf step update behind click observe rate position instead ndcg feature describe feature huge statistic one user term htbp short relevance level short relevance term time relevance history history level term relevance history time long term show history user user history cross user user number number time cross user history number time algorithm compare exploit default search engine ndcg default ucb article high yahoo thompson term thompson propose cluster thompson corresponding ranking htbp achieve default high thompson htbp generalize thompson train model model payoff explore suffer insufficient htbp see high htbp propose improve term short improve long term arm short use thompson thompson linear experiment efficiently improve rank popular armed bandit engineering extract lot style train rank novel way long term behavior use algorithm new default rank arm project start begin web topic I semi short user period turn query several keep query show armed learn quickly multi ad recommendation none armed web decide web web hard challenge user totally project project focus armed news user web search propose behavior site engine index engine treat query search really want movie read review infer interest previous query user want search engine return ignore interest display user basically technique context interest user past short user type contextual different user relate request search query refinement refine refine refine generalization refine mutual home home refine user profile build history current try interest example click technology article engine company try model behavior user generate query try interact user cluster current infer term behavior build interest profile adjust interact remainder solving describe short long term web search work short user good variation query benefit issue thing query express need return benefit popular people adopt click measure variation experimental show web significant click query click bad effectiveness argue query handle user history profile user profile represent history interest capture feature noun phrase tf tf bm treat inner profile user query train probabilistic look behavior construct history consist search model weight history query em estimate lot interest category category training interest distance user profile cover return web metric word semantic classify category learn history adopt sensitive behavior aware query query map tree effectiveness include click post category base occur prior query rank cosine search train behavior latent former n represent vector perform identify web page rank include decompose temporal segment discover temporal long short term search feature bandit news web news set article one click baseline try expectation click model article traditional regression predict estimate always bind thompson old heuristic lot posterior normally function choose however true function implementation simple
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paper propose deep auto encoder acoustic feature speech stochastic conventional analysis synthesis speech speech include auto simple encoder synthesis wu school computer science national institute propose speech synthesis feature spectral linear conventional synthesis text speech confirm speech dnn auto encoder current synthesis use probability apply language offer advantage know however speech model still artificial speech partly remove speech room improve spectral speech reconstruct fine lose synthetic experiment dnn indicate synthesis suffer due average due feature bring representation coefficient intermediate representation acoustic result speech answer propose deep technique linear synthesis extraction discrete cosine dnn denoise auto encoder condition finding use speech information dnn apply acoustic modelling synthesis dnn restrict boltzmann modelling hmm recurrent neural network modelling modelling work use auto encoder speech encoder base bottleneck group deep auto encoder technique paper binary deep et try discriminant discriminant encoder continuous calculated spectrum synthetic speech auto encoder encoder decoder encoder vector eq dimensionality frequently sigmoid relu map decoder mapping perform linear alone employ bias mapping encoding layer stack fine tune reconstruct maximize typical error mse e denoise encoder auto encoder auto extract encoder auto first corrupt pre auto encoder encoding locally layer layer desire decode layer deep corrupt train denoise auto encoder auto architecture back mse encoder training derivative represent neuron I il function l gradient parameter fine tune report encoder independently spectra evaluated encoder method synthesis condition synthetic use art synthesis dnn consist long extract spectra straight dim spectrum autoencoder sample extract encoder acoustic f energie reconstruction error auto spectrum hide layer decrease encoder bottleneck encoder dim acoustic input dim global randomly value produce good experiment dimension l dim encoder c auto report synthesis build auto criterion show spectra encoder denoise auto encoder reconstruct frequency part spectral distortion original spectra reconstruct spectra test observe auto encoder distortion compare distortion preference number ask auto da ask auto encoder encoder speech
map inner inner component tangent fx depend perform develop algorithm together crucial line condition ensure algorithm k equal inner product gradient algorithm strong condition search euclidean line find point wolfe find interval point condition euclidean detail reader behind satisfying initial previous gradient follow turn find minimize along geodesic point combine length numerous experiment effectiveness method good extract report cg notably fast except cg without manifold cg cg cg without iteration lrr rr riemannian counterpart model enable potential match accordingly yield quite suggest optimization may potential open algorithmic work effort extension rich class wishart prior leave simple actually fit broad enable incorporation penalty avoid easily easy moreover manifold optimization topic explore table pt thank author title title author corollary definition mit laboratory institute technology new mixture em box optimization fail slow intuition convexity consequence manifold match em highly encourage record outperform variability strength optimization tune method prove exist tool hope encourage wide widely variety process quick search maximization numerical conjugate gradient newton inferior programming covariance euclidean space especially boundary iterative affect cholesky also point program nonconvex stationary formulate convex resort sophisticated slower statistical high viewpoint nonlinear optimization em believe em remarkable substantially motivated observation positive numerical quasi difficulty implicitly simplicity em may justify manifold turn inferior discard refined outline idea intuitively whereas make manifold turn remarkable consequence ultimately enable contribution pt show key development key procedure help beyond outperform usual cg independent real show comparison manifold perform across em run encourage new mixture ensure service release implementation publish huge summary impossible line examine several counter claim think inferior purpose nonlinear programming method value amenable order convergence refined paper convergence gradient positive constraint suggest decomposition covariance single nonconvex add spurious report dimensional spherical near spherical matrix gmm branch nonlinear classic reference toolbox substantial study sometimes limited spherical gaussian focus highly algorithm background material serve establish notation quantity gaussian sample estimate q time seek compute manifold em canonical programming em especially stem cost incur enforce covariance avoid motivate take view manifold manifold applicable box optimization dramatically geometric intuition see manifold resemble convenient riemannian smooth equip inner product tangent manifold possess usual nonlinear rely locally join along short generalize euclidean convexity say geodesic within convex symmetric definite tangent entire riemannian metric nonconvex remain globally geodesic convexity coverage recent emphasize geodesic play convergence expect method much intuition geodesic close ultimately em subtle handling apply single geometrically suited invoke far reach impact transform prove cg manifold omit reason theorem maximize decompose eq must eliminate gaussian leave impact conjugate cg riemannian gmm likelihood minimum log theorem
forward backward capacity lead redundancy independence high strong may ignore many capacity highly pairwise dependent also contribute largely discrimination subset salient since dependence redundancy develop precise correlation effectively try redundancy introduce adjust redundancy organize review theoretic metric include experimental conclude study decade kind general aim class redundancy selection know correspond discriminate class label class distance may since weight feature remove extension incomplete still unable feature redundant accuracy speed method feature base inter obtain relevance separately measure class relevance feature score select fast correlation feature yu another method separately utilize would redundant mention redundancy identify index redundant ignore complementary discuss remain feature tackle former wang information mutual intensity relevance redundancy implicitly complementary complementary criterion identify redundancy complementary criterion mutual complementary salient identify salient although mention recognize measure among complementary select correlate select approximation turn essential describe fundamental unit assignment convenience hereafter logarithm quantify mi q consider amount two note mi extension mi q third mi entropy mi solve mi imply potentially cite justification mi write simply take top number decide stop assumption feature make mutually far widely recognize salient redundant redundancy variant generally relevance redundancy redundancy discrimination enough effectively redundancy word redundant may weak individual particularly microarray complementary modification identify complementary mi explain great significantly weak give word redundant conversely relevance complementary simultaneously redundancy complementary magnitude criterion sake q measure redundancy complementary relationship feature correlation among strategy well still suboptimal although pairwise handle relevance mutual independence word mutually identify able inter feature thus redundant I hereafter term select salient hereafter specifically give candidate feature candidate fp interference give status influential recall candidate interference fig green proportion pairwise distance complementary correlation ht scenario candidate show distant redundant complementary redundant feature candidate likely complementary value complementary correlation candidate reliability candidate distant candidate candidate complementary likely salient distant reliability make interference reveal interference dispersion likely complementary negative would redundant vice versa apply dispersion interference standard instability eq less interference salient less redundancy less dispersion interference end adjust e value negative account candidate piecewise use rather analysis relevance redundancy dispersion search candidate ht code th dispersion f algorithm repeat loop loop predefine one end additional newly add feature could record summation summation take fast ss f ff algorithm algorithm method representative review five selection describe algorithm relevance correlation criterion take fs fs redundancy mi select greedy manner measure relevance f note selection order redundant redundant feature eliminate method suggest mutual maximization base mutual information concern also select field feature ranking search instance throughout platform use dataset select implement conduct ghz cpu ram computer window validate method ten frequently six kp dna datasets tumor microarray mixed supervise feature r lr name class kp tumor I bayesian nn adopt nn kernel show check independently thus classification feature check feature result fold validation fold conduct classifier classifier collect classifier significant dataset range report classifier feature ten approximate fig fold rate different type classifier knn effectiveness consecutive select average fig superiority seven kp dna tumor breast cancer begin several tumor fig probably consider measure pairwise redundancy ignore redundancy feature higher select well never dispersion redundancy influential evaluation feature e respectively find inferior exist dispersion ht ht ht record feature indicate mm mm mm dna tumor cancer breast cancer avg rate c ten fold cross conduct evaluate sample correspond record fold cross significance notation test correspond feature high bold good average ten last see value rate show outperform dna error rate ten give diagram apply visualize box box represent red worse indicate l pt degradation level improvement level use ten feature less feature row among significant bold value classification selection na I correspond performance
estimate gradient ascent update optimisation intuitively mode behaviour beneficial probability alternative hessian result newton optimisation however hessian difficult abc prohibitive evaluate proposal problem construct local hessian newton make use limited bfgs set currently propose markov user crucial memory correct requirement fulfil restrict accept hessian psd correct standard remove hessian approximation denote eigenvalue smc discuss fix smc rely nonlinear consistent new variable model perturbation manner require intractable instead need distribution approximation determine formulation practical balancing requirement return study impact smc abc perturb particle tolerance level est carry I iw u case z I compose particle trajectory particle generate step particle deterministic smc unbiased carry variance high difference detail proposal particle together quantity close discuss evaluated gradient particle quickly lag reject step algorithm markov chain detail evaluate first evaluate serve comparison intractable appendix mix markov autocorrelation lag burn discard indicate uncorrelated imply index investigate smooth gradient dot indicate obtained see error log gradient minimize grow estimate result suffer parameter optimal alg acc median abc abc median computed monte abc pilot require abc probably match exactly model log use prominent financial presence jump return denote symmetric stable previously simulate appendix similar posterior indicate estimate volatility obtain smooth abc upper left obtain pooling dot grey density posterior prior quasi proposal enjoy require size provide hessian length experience simple derivative density prohibitive proposal hamiltonian algorithm likelihood resource provide national lag smc map accordingly use discard burn iteration discard hessian pre parameter gaussian denote distribution transformation simulate v transformation mail united mail division control mail hasting parameter proposal pilot inspire quasi newton hessian proposal inference likelihood application benefit new modelling return bayesian parameter inference intractable latent eq q intractable lie exactly far evaluate wise prohibitive evaluate effort intractable correct together inference intractable smc random walk evaluate view intractable also
rr rr c test e letter e e optical description rr rr c letter optical l rr rr rr test optical label unlabeled addition calculate lda hoc unlabele close repeat use follow criterion average likelihood denote test likelihood unlabele percentage time semi strictly log read supervise calculate training improvement supervise optimal l calculate determine average ad hoc supervise column comparison supervise pair likelihood show retain hypothesis average though reject equality optical statistically significant difference test likelihood reference number respective result strict sense carry readily train set turn indicate probably supervision estimate might substantially condition ad hoc concerned provide well bad look ad optical reason unclear seven nine set lda classifier immediate improvement supervision discriminant kind lda many classifier family make equivalent bernoulli study context class consist mixture concave outside supervise seem still appropriate need carry seem though try combine gradient ascent finally try rely interpretation consider generalized concept much broad merely make supervised principle generally certain concavity worked illustration lda acknowledgment de tu david tu I eventually simplification I thank give great decade like anonymous critical input laboratory image technology mail http improvement currently supervise estimate never bad argue upon example prove supervise well counterpart concept estimation principle refer objective estimate supervise explicitly improvement latter experiment improvement contrast discriminant parameter estimation widely far field diverse image quantum communication tool ml develop modern field satisfactory supervise however develop general supervise additional typically unlabele come improvement supervision contrary study theory improvement rather deal improve supervise essentially close estimate obtain supervise close estimate label supervision instance former strict relation classifier readily datum semi resort generally order applicable principle supervise objective account explicit refer treat behave bad supervise benefit conservative possible unlabeled encounter principle main theory core principle supervise section work theory semi lda really regular one employ tackle optimization problem section compare regular semi supervised put somewhat broad raise conclude begin put estimation lda principal early log take contain sample label pair class model consideration ml semi focus attention supervise lda broad review essence ml already consider rao exploit eq labeling estimation supervise parameter unobserve label unlabele maximize fairly sample procedure refer though classifier available label train give newly classifier unlabele iterate convergence initially unlabele remains probably suggest computationally tractable alternative procedure couple decade instance propose learn classifier subsequent self year possibly know semi likelihood treat label nuisance parameter come model likelihood maximization rely classical hard rather posterior thought assignment consider already formulation apply modern overview find seem different way soft assignment datum assign self make major aforementione suffer increase unlabeled behavior cause misspecification e class properly note supervise capable typically display supervised treat label datum might change go idea marginal subsequently density exploit weight author asymptotically semi procedure counterpart asymptotically behavior supervise learner depend set may problem previous choose marginal could performance improvement reflect year different take conceptual put relationship label exploit class prior supervise set benefit arrival adjust dependent well lda improve involve notably total refer semi supervise significant improvement classifier aforementione impose hoc constrained likelihood give maximize original reference reason applicability currently broad suggest find allow solution include label version unlabele instance well suggest solve ingredient augment unlabeled label second ingredient proper original parameter classifier need unlabele rao intractable overcome introduce possibility fractional result well behave classifier put appropriate supervise task priori constraint correct learner benefit may lead motivated method motivate fix soon make work concern technique take write function note ml lda supervise rao treat univariate semi set lda contribution lda finally remark contribution employ decision early widely self inferior contribute scheme currently generally applicable come make devise strict consider similar contain true unlabele upon information result obtain mean seem helpful supervise trivially fulfil take proceed strict improvement argue q likelihood label supervision lda prove construct learner difference incur first introduce give refer precisely vector simplex provide posterior likelihood dependence explicitly indicate side soft hard equation semi supervise express supervise datum enable extent possible deal semi supervised go assume soft labeling product ready general ml estimate maximize maximize lead nature objective take minimizer consequence never look even worst hard labeling expect semi estimate obtain supervision different soft adversarial bad case consider quantitative often happen imagine expect instance ill label nothing gain extreme exact copy semi outperform regular proposal semi explicit specifically briefly subsection demonstrate canonical eq fix fix concave compact invoke minimax allow maximization saddle unique definite follow definite pose parametrization canonical triangular square well invertible cf come back cl leave fix essential merely offset maximizer maximizer lda every average datum weight well supervise solution unique simultaneously e distribution unless empty equal upon prove semi lda vector continuously eq inequality label unlabeled probability pose eq expectation mean always subsection provide know look saddle optimizer know semi maximize equation calculate guarantee linear opposite experiment decrease maximum number addition reach every care quantity determinant covariance high fairly result obtain determine take semi start log raise firstly unseen compare concern remark
introduce abc simulate model intractable abc abc abc let abc model parameter generate decided accept py design possibly multivariate statistic empirical moment approximation operate rather convolution impose rejection use delta form j posterior use introduce simulate summary measure actual rely likelihood sl abc perform mcmc accept acceptance distribution quadratic true vector fit summary likelihood mle fit observe zero search close observe positive conditional embed operator statistic kernel need firstly I canonical map q implicitly transform non thereby nevertheless eliminate mmd abc element rkhs associate definite whenever bound moment w unbounde discrepancy mmd simply distance capture particularly mmd example use kernel rbf laplacian expectation mmd x n embedding measure f learn distribution insight essential measure nonparametric distance mmd distance yy almost summary biased estimate population simulate histogram mean mean shape notably differ true range term euclidean comparable abc rejection abc refer abc b j coarse estimate measure euclidean estimate good second algorithm match insufficient give inaccurate row sufficient histogram sample posterior abc sl posterior black bar posterior obtain close sl difference drastically mean red example inference system population model equation observation denote determine put broad draw vary drastically challenge b sl abc adopt mean peak abc gaussian three ran sa simulate use matlab sa split test set run abc value coarse error fig euclidean vector summary simulate estimate sl operate summary statistic abc attempt summary affect mmd discrepancy measure abc embedding take sufficient simulate world posterior statistic mmd abc widely rkhs capture domain datum adopt ard include adaptively gps save sample worth total abc approximate mmd university college ac uk contribution situation observe intractable simulate base choose summary rule summary incorrect partial paradigm manually mmd observe datum reproduce hilbert statistic scenario effectiveness abc bayesian likelihood abc applications bioinformatics abc posterior distribution interpretable natural phenomenon intractable evaluate integrate likelihood marginal computationally issue readily mcmc evaluate observation actual abc actual observation summary statistic abc partial rather poor difficult quantify summary summary transformation statistic minimum regression least square boost method summary focus may suffice still heavily inspire indirect auxiliary thorough review advantage control aic complexity principle way one exponential family cf sufficient dataset light
underlie opt latter mahalanobis simulate statistic compare projection ks test word multidimensional taking difference function cdf mahalanobis combine mahalanobis constrain root galaxy correlation sigma sigma root component explore keep permutation particle reduce percentile calculate good magnitude radius light two component target early linearly wise distribution calculation intel bridge ghz hour line consistent approximate behave standard deviation posterior fourth refine towards value behavior threshold iteration fast decrease r target sigma sigma sigma model blue line denote parameter parameter configuration approximate forward likelihood intractable combination perform pool particle solution iteratively improve gradually threshold automatic impact particle toy software prediction agreement threshold calibration application distance measure mahalanobis goodness infer configuration find abc reliable input map correlation simulation implementation promise modeling problem acknowledgement thank statistical thank discussion software grant national package website package package development commonly parameter observe various high parameter space hasting hamiltonian affine ensemble rely situation function direct make modeling include observe body semi galaxy weak simulation measurement example highly measurement process year bayesian abc gain attention need systematically explore space distance metric metric threshold data calculation efficiently monte advance sequential monte carlo abc problems instance carlo galaxies ia variant constrain disk formation simultaneously software base abc principle implementation calibration wide simulation software fast make mahalanobis represent refinement carlo control loop calibration method weak difference simulation reference property test error numerous couple purpose simulation crucial discuss principle abc consideration compare gaussian toy simulation conclusion release implementation abc algorithm find appendix set bayes probability derive rely quantify difference simulate sample accept retain approximated specify small abc q expensive available useful summary statistic amount refer sis explore discard inefficient rejection particle gain attention advanced population sequential monte smc sampling construct converge intermediate distribution besides abc likelihood framework therein adopt small abc represent position refer pool typically candidate pool assign iteration threshold pool update approximate pool choice threshold apply perturbation appropriate balance slow expect hand slowly fast often select ratio poor choice preferable sort particle distance typically ratio final approximated poor kullback distance desire proposal maximize improve parameter space particle current constant correct discrepancy population current explanation able increase good consist iid drawn distribution deviation seek mean abc py normalise data simulate standard flat analytic cumulative cdf express expand distribution variance increase abc acceptance precision parameter cost improper draw normal distribution standard define mean deviation percentile pool particle describe eq estimation abc panel depict display sample posterior analytical small beyond change approximated variance line image generator field generate use simulation consistent dark loop rather pixel analyse package produce identify
classification latter allow small somewhat feature condition p fan ab closely study mean normal index generate reduce vector generate replication furthermore mean j nm new sample feature selection threshold classify vector unique choose suggest variance ratio may generate somewhat analyze lead feature proportion feature combination mention single essentially reduce pure guess mention impact weak small combination misclassification see error tend large misclassification failure feature significant see comment guess increase significant improve precision exhibit observe increase grow contribute successful complex misclassification always become enough offset ht indicate class conclusion observe weak feature contribute class independence ignore preferable high acknowledgment foundation partially science foundation nsf grant dms dms start lemma q k l calculus since generally chi calculus logarithm generate calculus show obtain complete x truly obviously independent recall let consider central chi rhs corollary department department mathematic class significant distance class successful interesting counter intuitive classification accurate selection coarse nevertheless strong illustrate misclassification multi context interest classify imagenet http www represent discussion challenge handle dimensional datum name exceed solve require rigorous fan fan case normal selection pure although literature obtain classification therein comprehensive survey high fan ab generalization design statistical mention adjust pairwise hill adjustment support technique class lee lin expand well class affect classification number class hand distinguish thousand human reason class dimensional easier grow even feature sufficiently coarse classification small nevertheless impact phenomenon first attempt investigate rigorously impact accuracy selection class truly really separation carry observe close euclidean select condition require successful finite size finding indicate define without become misclassification error assume euclidean center satisfy follow classification misclassification class bind classification confirm evaluate assumption th infimum rule consider realistic irrelevant zero obvious chi centrality threshold th q theorem minimal feature correctly identify unknown feature j truly per increase weak become contribute effect coarse fine feature one truncate vector rhs result assume condition eq theorems minimal different order asymptotic consider classification fan contrast class define grow exposition separation distance
factorization discuss solve low solution challenge fail problem find negativity etc issue several explore th matrix variable norm appear name include projective atomic norm generalize regularization negativity norm norm many norm choice generally factorize still work several explore factorize q still factorize show idea factorize subject minima significantly unfortunately aside norm result force replace closely problem fact equal search converse minima formulation typically useful optimality local minima regularization application tensor form neural context network neural network single hide induce train globally optimal unit hide produce analyze idea additionally framework sufficient globally purely recall simplify use capital dimension dimension letter dd n r nr give element tensor slice along I slice along matching concatenation dot space image multiple portion gradient number integer general use tensor equal function direction differentiable return motivate family framework sum specifically tensor define slice factor requirement impose positively positively homogeneous k place restriction mapping must positively capture form several mapping q positively homogeneous column slightly cp outer multilinear apply wise training td relu linearity connection utilize complicate consider broad neural architecture hide x x neural relu architecture layer define fed layer positively map neural architecture compatible homogeneity note max positively homogeneous fall imagenet series convolutional pooling layer normalization layer layer take define transformation removal positively note however rely potentially change applicability architecture cast wide framework mapping positively degree requirement non follow positively homogeneous degree essential match homogeneity idea factorization function norm allow regularization place input slice factorize tensor requirement positively semidefinite formally semidefinite positively homogeneous framework variety semidefinite positively commonly compose multiplication power homogeneity combine positively homogeneous pseudo norm function positively popular interest semidefinite constraint also typically positively homogeneous transformation positively equal homogeneity formulation positively homogeneous degree arbitrary mapping additionally element compatible specifically give positively homogeneous x k xu uv scale convex infimum finitely sized dd size infimum u suffer issue early namely due complicated allow purely tool tractable factorize formulation build analysis typically eq factorize example intercept model possibly minimum note impractical optimize even solution would need desire merely next minima slice factorize initialization global purely local begin lemma relevant first verify positively positively degree recall concatenation tensor positively degree x infimum norm fact semidefinite infimum degenerate property infimum complete note homogeneity x x xx trivially factorization yy result property property gx hull equivalent w r dx I combine homogeneity pz pz factorization characterization subgradient dual give conversely associate concept norm solve still limit derivation subgradient characterize w regularization function factorization r gx x gx iw trivially gx I I also gx w x gx gx produce contradiction present local homogeneity local take p rearrange eq take qp rx z combine rearrange result preliminary ready corollary optimization problem begin factorization equality infimum satisfy minimum function subgradient condition condition gradient minimum satisfied optimality leave turn minimum f rx rx homogeneity rearrange take note side definition directional direction get complete result regardless give global local purely function global clearly minimum reach local theorem increase path minimizer must exist r zero generality scale homogeneity construction r recall x k gx theorem exist iteratively reach complete also meta outline global factorization bad case grow bind choice maximum require tb x I minimum define factorize terminate begin challenge alternate guarantee critical minima emphasize minimizer descent strategy guarantee entire space balance homogeneity mapping map conjecture result likely mapping save positive homogeneity match show factorization guarantee regard phenomenon positively assume homogeneous provide counter demonstrate minimum pg gx additionally exist neighborhood sufficiently happen always minimum origin always take arguably situation oppose positively homogeneous mapping depend decrease grow factorization uv decrease scaling factorization degree dependent choice column decrease neural note show neural positively mapping outline partial explanation replace traditional positively homogeneous output allow purely local conclusion sufficiently assumption initialization objective traditional regularization training form tend practice regard critical balancing degree homogeneity regularizer prediction ensure homogeneity significant deep noting limitation current framework state art must previously well however apply optimality reduce function entire limitation possibility future implement architecture advantageous experimental imagenet parallelization operate largely gpu focus mapping believe mapping principle future present wide variety problem analyze tool particular guarantee global minimum factorize tensor factorization range field common vast majority disadvantage associate typically multilinear idea
constructive algorithm analog theorem use line theorem begin analog complete low follow bring eq prove low case upper constructive smoothness role constructive greedy chapter q lemma get allow case argument case notation lemma eq prove choose note correspond low univariate kernel er nonnegative frequency enough imply need adjusted case old eq continue smoothness let case factor low case prove theorem relation class obvious opposite wide well theorem effect small characteristic smoothness prove kolmogorov class differently approximation detail study abuse ball lemma make constructive constructive proof bound typically dyadic depend let coincide mean wider pointed providing constructive small achieve traditionally research go paper approximation progress constructive still progress smoothness result present arbitrarily order right approximation detailed discussion banach constructive provide order respect system paper recent main mixed derivative smoothness constructive interesting history brief history system multivariate class smoothness definition periodic advantage univariate polynomial improve relation uniform constructive method result term respect interesting phenomenon discover establish decay fast bound constructive theorem smoothness time interesting late technique version use reference therein obtain order constructive powerful probabilistic suggest constructive approximation greedy banach chebyshev constructive inequality constructive chebyshev greedy give polynomial integer denote nonnegative exist constructive term polynomial formulate term typical constructive let upper constructive class point denote tn tn r q define proceed define embed control control logarithmic logarithmic scale smoothness recent book eq complete bound begin upper q establish take eq eq complete low proof give proof analog lemma weight one monotonicity prove case corollary combine bound prove follow univariate introduction convenience upper constructive wide build term approximation remainder function later q equivalent q index cardinality large include nonzero sum q continue q error proceed choose frequencie yield prove
form triangular window window adopt decrease incorporate understanding come loss minimize loss arise saddle rather minima gradient allow saddle reason optimal often htb implementation modification else triangular start policy cycle stepsize stepsize lr cycle describe start learn boundary exist example implementation difference number minimum policy cifar create curve rate cycle return start iteration accuracy peak decrease iteration well though policy policy cycle cycle rate drop quickly learn vary linearly minimum maximum vary boundary cosine investigated report briefly triangular follow drop fix learn policy reduce less limited number run investigation run policy need file either schedule factor half iteration number solver reader ask regard subsection question epoch epoch divide file cifar experiment cifar well simplify drop stop cycle train drop resource run addition iteration work stepsize convergence rule reasonable boundary epoch boundary epoch vary linearly reasonably experience idea approximate one repeat exercise twice set epoch maximum plot rate start fall choice first use two show cifar dataset start converge right reasonable rate get rough eventually begin reasonable exercise offer set policy wants slowly show reasonable reduce range iteration epoch maximize slowly use epoch applicable effectiveness method subsection policy cifar imagenet subsection policy policy valid use cifar run gpu gb memory architecture k memory htb htb website assume fairly architecture setting file website website fairly baseline recommendation optimally train policy run max start full file learn example cifar train lr max momentum decay snapshot snapshot triangular solver show policy four cycle rate cycle obtain classification policy setting might benefit policy derive reduce learn accuracy implement value linearly reduce iteration show table benefit compare file policy go substantially obtain method compare use rate dramatically tb cifar triangular cifar cifar exp cifar cifar fixed triangular exp exp triangular exp htb many website architecture parameter file architecture file website baseline avoid difference initialization number imagenet architecture file next minimum boundary figure converge set reasonable fair baseline necessary else apparent accuracy small rate rough drop learn reasonable policy accuracy file net test exp lr start display snapshot snapshot mode versus policy policy list table peak accuracy rate quite show table accuracy stepsize compare run architecture since accuracy around indeed accuracy policy around accuracy policy go finally comparison policy accuracy exponentially counterpart win entry imagenet file fortunately website architecture file use hyper file situation one cnn rather optimal architecture use hyper c max start max next run epoch increase result run cause short divergence accuracy file net display lr base max lr stepsize start weight snapshot snapshot solver gpu running limitation stage fully case peak cycle produce accuracy policy policy good guess versus architecture produce policy present benefit cyclic epoch rate vary near easy additional expense report present several cyclic test give cyclic drop cycle factor reduce learn tool train convolutional explore full plan learn rate perform furthermore recurrent analysis improve understanding acknowledgment author david david suggestion helpful laboratory name rate need find schedule instead rate report near accuracy tune often many describe reasonable epoch addition demonstrate training imagenet train network cnn give face speech car technology train global update loss book recommendation deep architecture say optimize optimize hyper one use worth tune know rate converge slowly rate tb monotonically demonstrate surprising phenomenon fix eliminate need tune near additional benefit see figure reach follow order magnitude drop adaptive
network recurrent expensive well rnn strength allow suffer less vanish hide wise w hyperparameter subsample network module correspond marker module return word notational convenience future module refer subsample embedding vector module basis module question use fact consist weight module input question reason answer module answer importantly fact focus attention fact pass episode summarize memory module fact important later allow pass retrieve fact ask reason iteration retrieve facebook place weight sentence intuitive module episode mechanism fact module episode mechanism question gate initialize practice vector help scalar episode sequence fact endow episode finally episode memory state attention highlight modularity mark fact important facebook end pass fact choose gate supervision pass module end straightforward sequence modeling wish one representation episode final modify replace pass final module go answer module send answer module module initial output softmax end token training cast error sequence gate supervision cross module differentiable deep network backpropagation application several task nlp parse sentiment analysis answer logical lack memory module sentence another chain network kind recurrent successfully speech recognition sentence relevant translation al extremely deep sentence sentence memory map sequence directly memory learn work recent network add natural language answering module cognitive human whose existence store module memory human spatial might argue form relationship spatial responsible module specific relationship module inference module human behavior answer speech sentiment preliminary train use development hyper early backpropagation employ word dropout facebook synthetic ability retrieve test th support fact conjunction support fact compound basic list path agent pass task pass begin training switch subsample module end token gate supervision modify episode gate result pick sentence list table bad long sequence recurrent model input suffer view module significantly require iteratively retrieve fact store slowly incorporate information sequence use sequence position part traditionally every classify speech journal iii split word produce th ccccc al acc state art reach accuracy model gets achieve stanford sentiment classification level fine grain label train result grain negative neutral neutral sentence classify positive train grain label neutral phrase label cnn mc ct lstm grain key et le cnn mc ct al sentiment gate function task grain list well incorporate experiment result lstm sequence special case input small english news use seq seq lstm seq seq lstm list nlp end albeit complex idea model multimodal input acknowledgement discussion style circle fill color color fill draw pos pos draw thick circle color color minimum minimum height natural cast answer language introduce memory input form semantic relevant answer iteration recurrent answer end facebook modeling speech classification sentiment stanford sentiment rely representation require manually answer text ability fact task cast answer like translation task name entity recognition problem sentiment like memory network fashion answer triplet question answer generally answer task require reasoning answer text semantic search reason retrieve fact answer pos tag take jj r est overview full detail generate answer module process raw input video signal nlp input long news article
select relevant show without conversely pls specificity false low avoid positive approximately whereas number relevant sensitivity pls selection process standard compression impact approach situation datum breast cancer level breast cancer work breast gene occur contain one restrict gene differentially condition expression express express center effect pls log observation split resample tune fold validation linearly space perform even prediction method simulation variance assessment h different important regularize usual convergence log severe converge tuning parameter cross validation pls hyper repetition c percentage coordinate component observation score axis compression technique would separate pls tune first method appear component produce discriminate procedure previously correspond pls bit easily combine differently log compression component sufficient separate indicate discriminate properly lead efficient evaluate use component depend method principle estimating size probability coefficient estimate method one expect positive expectation number positive fp relevant determine false positive penalization fp maximum false stable probability subset grid empirical positive stability gene log discover positive positive relevant gene approach compression positive support previously compression suitable dimensional small perform compression ridge iterative least pls logistic particularly throughput sequence consideration logistic properly glm framework appropriate ridge ensure confirm sparse pls pseudo pls moreover combine prediction turn technique pls stability validation eventually provide website f france france curse context number far observation thus partial pls perform combined logistic particular pls improve simulation ensure concern tune classification pls expression thousand concern patient breast implement dimension dimensionality challenge genomic record like gene classical classification method spurious unique call development statistical compression technique project information square pls correlate construct variable base mean contribute penalty constraint variable sparse pls combine introduce step component combination pls reveal advantage high lasso among one pls predictor correlate occur elastic net pls response adaptation sparse pls preliminary discriminant analysis classical pls solution logistic regression classification method achieve via reweighted least especially high difficulty rely step logistic intuitive handle propose pls step idea within generate make classical pls coefficient develop sparse pls compression inspired selection art especially increase hyper pls propose update soon adaptive discuss finish comparative eventually prediction breast year tp use predictor rely newton explicit observation observation interpret pseudo successive weighted issue give completely identifiability concern exist infinite ridge optimization regression replace ridge unique still exist produce ridge predictor suitable pls metric covariance center order intercept pls compression suitable particularly square covariance new continuous denote respective exclude inherent variable pls framework construct sparse weight response weight zero lasso difficult overcome sum concave penalty easily separate instead one stay metric case matrix product argument separate function covariance adjust constraint could lead compression penalization lasso yet classical constraint adapt penalty account predictor high weight successive square pls reduce base issue remain explanation iteration problem contrary optimize another achieve classification issue propose pls constructing choose generally high pls treat one add pls totally sparse summarize follow n pls weighting product replace identity precede pls discriminant pls classifier pls pls denote nevertheless concern pls separation compare art perform eventually reference perform solve maximization norm know elastic computation come pls pls da da package purpose evaluate compression aim crucial performance inspire redundancy within degree relevance predictor redundancy predictor block noise block block choose block response purpose selection tune validation ridge linearly space sparse linearly range especially crucial pls point analysis essential ensure proceed criterion follow low ridge systematically ensure sparse pls redundancy example contrary resp iteration cyclic confirm sparse contrary confirm seem procedure c consider suppose return value become uncertain hyper parameter return validation repetition variability adaptive cross validation ridge parameter fig contrary validation method return one consideration instability cross hand select influence validation precision deviation repetition another cross ridge variable determine
fashion protein similar protein diffusion vote reciprocal protein vote method diffusion capture long range topological local prediction figure advance diffusion capture association protein accurately neighbor majority vote level strength topological integration protein explain capture network effect specific vector fine topological visible single approach canonical vector majority vote function top string functional relation explain diffusion demonstrate plug protein multi problem toolbox representation radial nest five within parameter solely topology score observe improve supplementary diffusion novel biological exploit extend heterogeneous perform jointly optimize feature demonstrate exploit topology molecular network predict molecular demonstrate substantial diffusion accurately encode local topological predict gene vector informative describe protein term topology readily exist future plan improvement example simply straightforward ideal overfitte specie numerous annotate believe hierarchy challenge prediction evolutionary include identify functional module analysis discover term molecular network computer artificial intelligence mit usa mathematics mit computer il execute rna develop throughput htp two hybrid molecular interaction genetic interaction study often incomplete interpret thus functional annotation perfect interact likely type diffusion extensively context biological effectively propagate indirect gene relation gene researcher select distribution method mainly ability topological neighborhood still partially incomplete nature throughput high false principle pca effectively high dimensional reduce linearly project variance predictive machine application effective overfitte little spirit improve improve dimensionality design capture nature diffusion novel framework reduction topological facilitate protein idea topological run molecular node distribution multinomial parametrize node minimizing leibl divergence relative parameterized pca reveal internal explain variance compute extend heterogeneous perform apply predict substantial next capacity heterogeneous genomic resource string train machine node metric gene prediction svms test string annotation remarkably feature machine useful certain node characterize similar association interact rise molecular phenotype find node either select diffusion setting topological lie beyond neighborhood advance diffusion method provide extract information encode compact representation walk network vector describe topological minimize diffusion logistic walk analyze network probability take consideration identify adjacency molecular interaction protein entry eq control influence global topological place great emphasis entry visit current correspond iteration start probability state position suggest protein capture topological association instead simply achieve approach part quality dimensionality original spurious network fact biological incomplete greatly reduction logistic latent node dimensional assign context close direction inner frequently random walk fine topological retrieve function vector allow extend next model optimization take diffusion input find dimensional representation good approximate kl divergence probability guide entropy express objective low dimensional diffusion optimize respect objective use standard newton bfgs almost solution framework novel interaction variety network integration identify gene take weight manner integrate analyze topological confidence genomic bayes confidence network network specific integrate network mix extend integrate perform diffusion node distribution logistic assign encode intrinsic newton bfgs assess representation obtained consider protein network protein interact protein annotation characterize protein predict function topological proximity protein capture similarity protein close protein rank make topological protein similarity two protein representation protein cosine follow distance ten protein assign unlabeled protein unless majority vote cosine protein readily feature machine various exist formulate task machine svm functional annotation train assign protein annotation method annotation protein available via dimensionality integration network support machine able five validation string network annotation remarkably art diffusion string database variety throughput database exclude text prevent protein edge
syntactic implication far provide whereas quite complex perspective briefly full fail confidence occurrence consider provable observation jump nontrivial follow statement wrong explicit proper besides generalize implication partial implication proof construction value attempt generalize rapidly reach difficulty lack property identify turn find connection state implication confidence use connection partial enough use case implication boolean algebraic set partial implication explain partial condition linear programming situation surprising lp merely characterization decision seem follow program big alone attribute discuss section receive semantic simply terminology close analysis call sometimes dataset transaction thus attribute transaction simply subset attribute think attribute transaction subset attribute transaction cover datum set transaction transaction alternative write transaction cover implication set union fully partial implication else specify conditional partial note much symbol logic expression entail implication proper subset without properly number confidence lp follow real program feasible unbounded feasible arbitrarily objective call primal duality exactly infeasible unbounded infeasible feasible optimal describe partial implication start comparison notion implication consider consider start develop discuss otherwise everything strictly implication implication case confidence threshold nothing else say algebraic set implication differ equally well entail confidence occurrence everything course partial solve true interval intuition combine implication discuss incorrect appropriately cover implication implication iy seven simultaneously tight suggest zero state vector involve although elementary build hoc seven inclusion fail value generalization case somewhat subtle point seven proper clause begin move comment seven inclusion far intuitive discover right generalization turn getting discuss section discuss duality interestingly statement generalize merely useful characterize set characterization variant standard tailor apply consequence make dual play want intuition terminology usage implication e transaction cover cover intuitively extent weight give transaction implication mind read follow whenever weight non negative non negative combination say classical necessary sufficient confidence formally component q useful lemma weight implication transaction number time appear complete transaction w x yx yx union mean z z parallel resort duality lead natural lemma check correspond leave min w z ix programming feasibility really characterization prove equivalent prove certainly feasible solution w continuous rational component preserve feasibility objective natural transaction copy feasibility ii direct feasible reference transaction let number alone transaction dual feasibility positivity read non w follow call whenever characterization deal relationship partial implication x denote every smoothly prove v z u z would removed validity obviously contradiction read case fact know x iy I item j wrong low follow state partial implication k iy argue implication I l l I l l characterization theorem look say li easy solve theorem close get say implication homogeneity hold enforce homogeneity either cover implication homogeneity empty implication homogeneity nice homogeneity requirement show exist implication x statement cover cover read iw I put together get every characterize counterpart satisfie classical implication implication homogeneity homogeneity homogeneity else implication homogeneity quite subset fails note decide formula k ready implication equivalent l x k homogeneity iy clear l say index statement unless empty properly inequality fact proper inclusion inclusion lemma straightforward implication follow nothing since trivial assume empty suffice accord inclusion exist fail cover homogeneity rest non assumption prove case course consider violate lx lx iy ensure z also hold hold enforce homogeneity turn key result quite implication trivially nice partial implication nice bit exactly implication homogeneity happen symmetry conversely implication homogeneity recurrent concern implication lemma every implication homogeneity direct application rest hoc implication define confidence implication v z convention take occur negative make numerator point convention function inside comment ensure turn obvious well la un si partial set side reference notation first sort notation main theorem confidence x homogeneity iy iy theorem negative expression follow argue therefore also cover case empty max I enough point direct would state early max definition expression write turn right side hold additionally particular hand inequality hold assume definition max I w side theorem go one critical partial certainly theorem sound case find ab contradiction side big claim expression solution case check theorem say among implication however question fully
noise general lie union video extend via low popular approximate gmm recently develop density information able technique support award grant song h school consider setup measurement form suppose copy signal sketch sketch noisy result gaussian matrix equal copy average introduce view q far expand unbiased bx measurement may let eigenvalue mx ax measurement absolute tr power measurement definite apply let otherwise thus signal condition relate follow therefore nonzero positive semi chebyshev subsequently update sequentially q follow hadamard inequality conditional recall direction correspond eigenvalue eliminate eigenvalue lemma perturbation eigenvalue suppose stop step ideal q note exactly therefore ns else om lm proposition sense greedy sensing model may estimate setup matrix gap recovery sequential compressed sensing acquisition image large sequential nature problem either due fact streaming process compressed sense develop classify graphical exploit distributional signal seminal compressive gmm work general sensing refer consider greedy optimality greedy aim design subsequent mutual condition measurement mutual capture result orthonormal decrease eigenvalue consequence almost always estimate quantify performance sense proxy estimate theoretical include relate entropy measurement characterize covariance establish additional present initialization numerical example good greedy spectral matrix vector quantile chi degree semi definite sequential unknown sequentially goal assume choose goal measurement precision sense eigenvector eigenvalue measurement covariance illustrate dominate performance greedy sense note calculate iteration necessarily reach power k constant trace surrogate calculation matrix easy eigenvector power update take form decomposition update trace become bind signal amount hand hence characterize amount reduction number measurement versus roughly occur upper simplified suppose eigenvalue characterize versus covariance allocation prescribe reach precision establish extra reach precision give recovery level require expression require measurement one full covariance sample
scatter mention voxel lasso voxel probably voxel early region select voxel fold brain although apply diagnosis apply point nonnegative fista thresholding provide natural foundation china grant corollary analysis involve thousand million number regard lasso select diagnosis usually feature stability perturbation explore fuse lasso feature diagnosis incorporate spatial voxel optimize novel variation network nonnegative explore compare analysis sparse gain great statistic absence major performance apply diagnosis disease brain image interpretable instability mean perturbation include bootstrap unstable lasso dimensional result undesirable diagnosis issue selection less study induce structural imaging image brain voxel induce disease correlation prior brain exist label ad positively cognitive disease gray cognitive accordingly enforce name model non fuse fuse lasso enable select measure feature demonstrate model stable worth g although solve diagnosis ad make fused solver feasible regard propose solve generalize fista constrain prove use element post tv solve duality formulation include apply fast precision high sparsity people leverage underlie introduce strong voxel grouping voxel coincide various topology consequently overlap ideal problem adapt fact select necessary fuse successful induce nonnegative select stable positive partially support neither provide minimum tune loss variable assume variable suppose correspond many brain directional feature penalty tend sparse I spatially coherent also nonnegative select unconstrained systematically greatly encourage disease relate apply thousand mainly framework table propose modification scalable explore lagrange deal iterative follow constant kf convex file solve fuse show utilize separability term define denote element solution nonnegative tucker kkt necessary sufficient condition sub objective derivative lagrange complementary condition q variation perspective natural novel flow easily parametric flow inequality duality I proper cone file since rewrite generalize primal equal tv convex define lagrange multipliers derivative dual write k please supplementary file dual omit change illustration flow highlight illustrate flow take minimize cost quadratic flow minimum via include limit minimum via flow recall ij exist decomposition moreover possible devise flow tv efficiently prop prop transform duality flow compare solver node sample cpu ghz efficient diagnosis issue ad normal nc mild cognitive voxel nc classification use q voxel spatial adjacency
outlier significantly different non involve already outlier frequently version crucial recover pca contaminate pursuit construction universal pca assumption moreover convexity np hard pca extent relaxation paper aim reconstruction adopt classical outli observation row outlier introduce square regression ix I small error low default see estimator datum corrupt free minimization lead formulation separate method lead quite large error component state direct minimization desirable annealing approach regularization trivial section minimization convex manifold material identity nm concave employ technique linearization fix r u I pointwise minimum sum linearization concave tm u nm set eq I u ss u I coordinate concave matrix orthonormal symmetric semidefinite factor using svd singular uv uv tolerance output robust center pcs nu km I point objective reconstruction finally monotonic k u u terminate compute immediately small tie break minimizer k I one u I objective smooth neither construct data use handwritten digits digits mix proportion handwritten digits equal mix fig exclude value ground default simultaneously robust influence additional one supplementary material cc cc ccccc person visible end reconstruction person robust robust interesting pca sequence slight change whereas foreground model outlier pca background frame foreground performance water surface move background dataset feasible background algorithm optimize hand bottom crucial reconstruction cf foreground background mistake supplementary background water set present fast initialization sufficient runtime parameter r present reconstruction efficient default setting perform similar solve h partially partially extraction system cccc material background water surface object person frame person outlier cccc cccc water surface set fig change foreground rescale maximum note cccc frame water surface similar red frame person leave scene high reduce dataset figure please outlier compare scene foreground background foreground segment separately center main experiment method cccc cccc cccc reconstruction perfectly oppose large frame know principal affect put directly avoid often pursuit fast probably tool exploratory analysis reduction g see datum fact strongly influence outlier indeed one pc drastically robust pc receive lot attention recently pca base one pc perform however positive affine informally arbitrarily still upper inverse projection deviation lead non smooth search disadvantage compute technique lead poor pc vision
precise error condition motivate minimize u resolve label fraction coarse close need analyze lemma suppose regularity third minimize pose sdp clear aim minimize I u v constraints argument prove lemma equivalent theorem begin illustrative example negative matrix depend result tell label minimize illustrate matrix question assumption final happen high whether sophisticated acknowledgement order modification omit generalize suppose iid smoothness smooth follow sense point probability regularity interior invertible namely relate fisher regularity first l satisfy q tell wise combination write satisfied lemma hold follow u l side eq pt exercise remark example well rigorously characterize pac agnostic pac shift provide widely popular linear class conditional upper match sufficient estimation sample pac agnostic goal learn belong shift attention estimation mle active satisfied widely model multiclass field active mle statistic class asymptotic statistic label machine learn special involve consistency full generality goal minimize log estimate sample query unlike towards except low perhaps round likelihood either high implie observe classifier sufficient optimal classifier class pac generalization search style inconsistent case disagreement confidence active passive agnostic gain requirement agnostic active set classification kind random previous active regression algorithm fit increasingly refine partition refine complexity result exponential apply general provide selective decide whether label mostly estimation variate suggest select sample regression variate notion fisher information trace directly optimize consistency regression base work provide promise consistency algorithm apply guarantee moreover unlike single interaction single sufficient order begin pool example draw belong label example give py generate also abuse notation define goal label time matrix negative learning number brief generalized use hessian label solve sdp refer behind well respect essentially I minimize finer condition respect formally step case unnecessary skip directly step regularity quantify standard study regularity interior ix exist neighborhood lx extra essentially vast model lemma mild regularity estimate rate right main proof support l estimate follow
estimate grid mc correctly identify mc trial music considerably low snr db music fail figure note rate music localization source whereas guess rd set difference low music definition generalize huber sparse sparse unknown signal careful characterization huber devise simultaneous conventional refer heavy yet negligible usefulness localization application sensor array single measurement I unobserve row unknown signal q reduce row non non ij lead computational reconstruction accuracy recovery matrix level eeg arrival source process cs algorithm guarantee suitable noise condition meet heavy generalize huber huber value regression generalize huber estimate scale robust require huber matrix error simultaneously particularly obtain estimate challenge ill elementary huber devise problem offer outline background robust recovery section huber huber localization cardinality resp index column hermitian transpose row f function argument equip usual hermitian trace entry denote nonzero row equivalent statement operator possess set refer version index unchanged row suppose circular distribution density scale reasonable square l scale factor minimization l small residual imply even influence least use huber require difficult involve compute special start proper loss symmetric circular fix jointly huber elegant devise huber generalize huber scaling preserve property bound real derivative calculation minimizer minimum fisher huber behind sensible choice residual simulation aim pursuit due objective huber loss greedy recall offer estimate onto huber criterion iteration update signal matrix stepsize refer pseudo huber function divide build stepsize compute stepsize stepsize tune simply criterion n stepsize need adaptively control minimizer ascent simple minimizer solution fp huber huber case fp huber loss minimizer find fp fp iterate previous word consist receive wave point time array weight linear kt mt distinct represent known localization parametrize source cast recovery overcomplete source location measurement
block belief partly efficient train g cd present step incorporate deep propagation bp message two drawback cs continuous distribution necessity iterate message amp amp cs especially forward observation unknown corrupt zero subscript notation refer matrix order amp moment mmse contrast utilize inverse give estimate factorize posterior read calculate p ix refer seen know interested regularizer solution inverse within probabilistic amp convexity induce utilize gauss bernoulli gb ip bernoulli draw accord expression gb gb split normal control informative measurement infer course truly account signal truly sparse merely support instead specific site support identically refer reflect partition sub term support write mean variance gb nice mode coefficient amp amp support e coefficient existence dependency natural model support binary rbm rbm rbm train boltzmann restrict boltzmann bipartite physics joint visible layer rbm rbm coefficient hide sequel connection simplification rbm field first second order give energy hamiltonian free minimized simple I field within energy rbm equally visible hide factorization rbm visible site give fix variable eq equation line assume nmf find site activation literature use free tool nmf approach correlation situation make popular one way refer sake statistical recognize expansion constant show bethe densely proceed visible repeat eqs mean eqs action tool use rbm approximated solution equilibrium eqs eqs iterate possible arrive field balance enter reduce assume magnitude scope amp perform inference accord factor depict utilize rbm coefficient classical form amount bias visible variable amp rbm energy effect rbm amp influence hide visible thus influence visible respect within sigmoid nmf eq right represent observational text direct amp influence construct rbm prior term dependencie rbm unit successively via specify attempt persistent throughout amp force undesirable minima alg rbm factorization post value rbm factorization current smooth rbm sufficient post play role enter state minima observation side prior side rbm carry act show efficacy prior amp handwritten digits digit value binary support handwritten value rbm mnist set divergence cd epoch additionally impose decay draw distribution linear projection subsequently utilize digit comparison mnist reconstruction digits percentage successfully measurement visual reconstruction digits row correspond reconstruction amp gb propose nmf rbm factorization approach factorization column represent approach last digits top bottom amp amp empirically pixel expect least gb properly correspond rbm zero amp rbm amp rbm approach version nmf post test strict requirement amp desire number rbm epoch epoch use unit fig percentage recover successful reconstruction easily prior version amp gb amp gb upon rbm support support rbm gb demonstrate correlation amp cs factorization provide factorization matrix iteration rbm achievable oracle percentage amp rbm rbm exactly rbm order rbm attain necessary increase show rbms stack improve upon scalability inclusion rbm factorization burden reconstruction proportion require computationally amp rbm rbm support properly provide cs reconstruction superior empirical assumption
unfortunately guarantee rank critical point vector theorem exist since follow truncate expansion second assume eq rank nu u entry indeed feasible critical critical rest without increase assumption proposition correspond kkt partial cover strong critical critical similarly order bring point resp point critical orthogonal span dimension iff fx xx xx interior order critical second face restrict put latter result linear equality characterize result either globally rank dimension suboptimal strongly concave fx order critical point kkt convex unique optimizer exclude comment optima face global optima optima optima proper mild set refine leverage hessian assume order relative interior rank rank eigenvalue theorem critical mu np tangent semidefinite hand orthonormal space span vector span w follow denote likewise cauchy determine new add admit version nonzero raise q combine meaningful linear intuition critical optimal produce latter summarize fix global optimizer uniformly face almost answer face theorem motivation upper face attain inequality cover impose denote select independent pick iff linearly always indice select row slice index row see define py e ij ki subset constraint linear least constraint count use matrix expand term slice c c te furthermore expand namely k q attain make many combination confirm act conclude argue tight indeed repeat slice contribute yy pp pd combine ensure point kkt critical kkt point critical point apply conclude cut sdp second critical certainly solve practice empirically sufficient bad theorem provide require increase rank proceed move operator xx definition eigenvector bring cost compute simple kkt critical warm start iterate way yield kkt grow call list hope take integer I iy assume availability procedure inside zero saddle able make eigenvector saddle eigenvalue nonnegative return kkt kkt allow take discussion unlikely event terminate critical point optimize proposition compute critical kkt otherwise decrease iterate need cost decrease rank never exceed terminate admit kkt procedure toolbox descent toward iterate convergence unlikely impossible happen unclear ideally modify global order polynomial number step knowledge riemannian set light recent modification trust method polynomial critical region seem reasonable expect interesting drop return exactly point numerically exceed say future kkt I bound eigenvalue project algorithm thin retain slice return optimizer hope help bad remark remarkably low sdp follow sdp admissible strong condition solve sdp illustrative compete sdp orthogonal synchronization matrix relative transformation benchmark random rotation level independent solution equivalent admit phenomenon partly take solver run guess return force version top reveal empirically weak node node node problem solution return numerically close grow merely dominant synchronization gaussian naturally measurement square alternative minimize error orthogonal minimizer ij similar spirit relaxation subspace round program regime non noiseless tool nonsmooth prove typically higher formalize restrictive x ij ij kkt otherwise ij contradiction kkt block positive semidefinite show w ij contradiction apply particular thus convex small case even though two differ constant difference speak cost nonsmooth concave minimize smoothed note coincide affine function large explicitly appear consider norm minimizing appear kkt guarantee point strong concavity kkt reveal kkt empirically excellent quality kkt increase warm start rather convex pay lack purpose global orthogonal synchronization fact permutation modify synchronization notably arise computer vision permutation size perfect achieve perfect recovery phenomenon exactly assumption even cost node huber permutation vary horizontal axis square vertical close perfect get remarkably huber cost accommodate outlier still fast unfortunately grind global concave hand come guarantee solve start warm previous start identify appear optimizer optimize cost rapidly robust synchronization solver cost reweighted least successful preliminary work call compute relaxation orthonormal reduce riemannian effectively increase involved investigation riemannian portion usefulness bound smooth manifold cover address sufficiently kkt offer nonconvex kkt kkt give critical could compute second possibly spurious admit see sufficient perhaps investigate via regard cost admit second point acknowledgment author thank conduct research support research paris et synchronization cycle measurement convention condition exhibit formula attention rotation orthogonal reveal explicit formula check carry let dm semidefinite perfectly set inconsistent guess criterion spread evenly th root build part recurrence q ij j semidefinite unitary diagonal unitary operate use unitary proof general measurement none condition surely almost face minimal part er face select yy statistically vector vector surely recurrence almost statistically x ignore hold proper yy px q nontrivial combination scale density mass zero rgb line def grid step def pt anchor east proposition corollary example remark propose solve optimization form constraint block involve phase rotation orthonormal combinatorial cut exploit fact admit optimization convex characterize reveal kkt semidefinite magnitude fast well code consider identity far focus available application product indicate sign allow product correspond orthogonal relative rotation stack matrix correspond product define stack concern twice continuously symmetric negative datum encode induce invariance action group orthogonal cover max semidefinite block rank factor optimize cut linear constraint motivation relaxation linear dual may project initial discussion projection pay dimensional solution sdp name iterate search form matrix quickly operation seem admit intersection semidefinite cone affine subspace geometry solver phenomenon apply lagrangian sdp powerful bring great insight want matter guarantee nonlinear penalization solution like nonlinear build upon observe certain elegant put algorithmic true smooth replace nonlinear algorithm riemannian address invariance optimize equivalence full rank advantage geometry become difficult well justified increase practice often target quite theory nontrivial lift describe riemannian geometry frame toolbox critical riemannian gradient vanish critical critical unstable orthogonal determinant relax nice furthermore riemannian iterate iterate numerical tucker kkt condition convex sufficient kkt dimensional second order reveal point warm computation allow reveal terminate formalize rest improve avoid geometry tie reference cover grow view theoretical investigation call inspection particular face effect describe always tight essentially face generalize description concave critical reveal kkt sufficient mild critical point result convex paper efficiency solve synchronization rotation permutation note simplify exposition easily accommodate size development go complex relaxation numerous application consist estimating group measurement ratio model seminal class maximize proportional laplacian application cut interaction effect final matrix structure sphere appear notably fundamental solve determinant exclude pairwise rotation measurement model come camera sensor localization rotation separately determinant pick connected relaxation cost orthogonal matrix notably nonsmooth propose robust geometry equality count tangent sense affect tangent restriction riemannian embed thus problem euclidean tangent riemannian hessian directional use expression riemannian away next extra require achieve slice thin slice close consequently cost define compact matrix hope recover block proposition show consideration equivalent hull rank handle optimizing solve probably hull submatrix block semidefinite singular orthogonal conversely multiply notice orthogonal onto span remain span orthogonal may decompose dimension face face line relative face relative form interior call exist linear exist unique maximizer compact hull extreme point notably arise concave attain minimum construction proof admit optimizer value soon np hard figure admissible matrix lying segment admissible extreme construction full extreme support many meaning rank extreme singular proposition consider rank schwarz equality attain max admit extreme latter let nonnegative x though prescribe norm critical kkt critical saddle point kkt explicit critical point kkt bring ingredient proof kkt
denote visible unit respectively energy rbm specify visible bias statistical visible rbm visible boltzmann physics rbm rbm interpret free visible unit refer important rbm compute visible unit rbm bipartite however calculating computationally scale unit train rbm ascent carlo rbm computational nevertheless draw derivative sampling cd present physics inspire rbm free refer reader review apply rbm start possess binary boltzmann usually base multiply boltzmann inverse temperature apply write energy newly introduce external field recover transform maximize auxiliary variable inverse dd average configuration boltzmann free transform minimize allow temperature expansion carefully temperature arbitrary accounting obtain correspond entropy interact term I field order order systematic correction return rbm remainder theoretical denote hidden recover lastly visible sigmoid well stationarity condition utilize term consistency relation couple define satisfying field recall obtain weak coupling expansion couple system equation spirit propagation iterative rigorously demonstrate spin remain relation index careful minimizing provide running eqs converge present iteration second term estimation log rbm note utilize cd gradient ascent derivatives eqs obtain procedure visible hide bias merely unit respectively weak expansion poorly rbm recover calculate take greatly third easy include fact rbm admit triangle sum pair triplet exclude bipartite rbm couple fourth require utilize rbm number separate cd dataset handwritten digit comprise mnist rescale pixel construct consist foreground scene background represent image rbm visible study rbms unit adopt mini batch learn point mnist test present implementation mf great complexity investigate consistency converge instead iterate similar cd persistent maintain persistent epoch epoch therefore converge persistent iteration self mf persistent algorithm use comparison lastly evaluate rbm rescale raw design training stack rbm unit rbm operating term comparison train rbm cd rather training free neither momentum implementation however decay necessity rely compare free training average independent standard likelihood divergence log demonstrate advantage computational auto em mf generate rbm handwritten perhaps preferable certainly preferable mf logistic epoch comparison perform black raw refer training code pseudo log epochs implementation cd procedure find show panel fig fig ascent pseudo log epoch interestingly persistent epoch contradict common approximation gradient represent explain persistent epoch resolve informative examine rbm fig choose epoch p display rbm digit yet generate particle particle identifiable digits qualitative improvement digit visually particle possible rbm log confirm cd persistent iteration preferable fast moreover demonstrate surprising consist rbm training perspective rbm deterministic binary visible unit hide unit learn usefulness task unit map toolbox order place emphasis quality rbm training rbms training cd training yield cd persistent although decrease likelihood train datum raw marginally test imply real successful consequently treat interestingly persistent observe deterministic cd present rbms field I design rbms show I bring practical deterministic rbm cd algorithms file implementation online rbm monitoring throughout real show gauss rbms rbms field stack rbms jointly separately boltzmann deep generalize boltzmann hide difficulty expansion start
vector match sequentially accept word historical rnn reach last word natural information rnn rnn long rnn perform click manually label insufficient feedback limited click provide similarity exploit objective lstm query respectively learn sentence lstm rnn activation different input process learn lstm rnn effectively keyword lstm indeed keyword lstm read activation word encode entire reason application web query document cosine web task method word sentence sentiment performance design capture fine grain sentence recurrent develop retrieval also treat explicitly gram window pooling nlp capture dependency belong develop speech treat word encodes differ leave encode semantic embed vector reach view feature representation english convert lstm rnn lstm rnn sentence maximize probability predict sentence word summation bi pair plus linearity lstm letter gram keep encoder decoder jointly align translate rnns concept attention decoder discuss study train model sentence embed description ahead another rnn robustness example web ranking come equally salient limited memory ii word query robustness lstm rnn lstm serious use size lead retrieval advantage keyword require capture contextual recurrent embed rnn lstm rnn networks temporal rnn dense sequentially word sentence representation figure th code hashing letter gram recurrent word semantic bag representation whole use slide capture layer feature rnn neither sized pooling recurrence sentence express recurrent word architecture traditional convert letter supervision sentence detail although sentence principled manner dependency vanish neuron originally lstm forget gate connection fig gate forget gate gate vector connection connection connection vector consider forward pass lstm rnn model hadamard element good representation input challenge practice since collect manually semantic nevertheless widely web massive candidate usually record binary semantic query supervision signal embed achieve click engine complete please section adopt cosine similarity vector sentence length click sentence document subscript sentence lstm take corresponding want document rnn include query corpus document denote sample expression logistic accuracy measure cosine large help train rate momentum scheduling equation please refer nesterov momentum step rnn necessary derivation accelerate mini batch train incremental update back method accelerate convergence nesterov find rnn yet rnn model update update train lstm present scheduling gradient maximum length model document set number lstm rnn minibatch compute compute l l understand lstm perform visualization tool analyze question dependency critical embed semantic iii lstm rnn question lstm web search engine description query document pair supervision lstm relevance follow query sample try around query query collect web query popular retrieval finally pair query text retain comprehensive train lstm rnn visualize behavior output lstm reveal lstm keyword detect simplicity query keyword extract list interestingly cell keyword specific cell cell mainly keyword c cell cell cell cell cell cell cell al much make community infection infection infection health pressure cell cell cell cell cell hash replacement replacement infection infection health high pressure high pressure embed retrieval engine specifically rnn query vector similarity query document show standard mean discount ndcg performance rnn lstm rate train baseline evaluate performance fair lstm train include retrieval ir sake bm state document base match baseline ir latent semantic side give report base dirichlet smoothing lstm rnn significantly exceed good baseline ndcg statistically point sec come vector hide rnn lstm parameter number c ndcg ndcg ndcg bm rnn rnn comparison train rnn fig conclude rnn optimize please number epoch address short information sentence semantic evolve input detect due limitation human label signal user click engine performing show allocate finding example concrete sentence important web show outperform significantly work early deep model effectiveness extend include develop directional version propose embed processing sentence answer information structure cost present derivation gradient supplementary material recurrent subscript th subscript model please eq rnn cost nesterov rnn architecture omit simplicity parameter cell lstm rnn subsection subsequent lstm rnn connection subscript input eq gate eq q connection eq forget gate forget gate bias value connection bias q back material gate visible gate word word document value color title document good match reveal rnn sentence embed rnn train dataset document pair relevance generate relevance bad excellent rate assign assign lstm please score mean score assign generate neuron cell rnn rnn active neuron result query interesting sometimes assign assign neuron rnn c lstm rnn number neuron example rate human assign rnn lstm generate lstm neurons rnn rnn lstm rnn assign neuron neuron neuron table assign lstm rnn assign rnn c c c cell lstm derivation rnn lstm rnn subscript divide rule rule step backpropagation back use procedure follow rnn architecture eq diag entry entry substitute derivation gate gate solution therefore resolve bias equation forget gate derivation gate forget gate connection update forget gate substitute q gradient equation connection example lstm dataset document activation gate gate cell cell document fig lstm rnn three semantic cell state evolve gate cell word document lstm blue color try similarity query gate state valuable context information store
local minima dot act signature add represent pair local minima lastly adjust width edge strength cell describe paragraph evaluation minima discretized denote near neighbor ascent method fit multidimensional scaling mode minima denote plot connect proportional approximation consistency stability function nonparametric density like panel useful density estimate density work surrogate rather first estimator estimate square third difference define satisfie omit signature converge apply derivative rate also derive signature interested estimator summarize signature density region may unbounded kde majority cell pre drive case minima distribution minimum except move boundary minima unknown moreover statistical establish essentially minimize look good piecewise cell within cell good linear predictor cell except covariate finite moment assume use sufficiently critical counterpart regression difference follow show consistent estimate smoothed smoothing estimate cell assume hold eq regression via focus region covariate occur frequently put weight term version aim look good seek many similarity pilot find mode original lot effort come clear population quantity converge pilot construct pilot estimate theorem quantity test difference signature within cell density want simultaneously interested estimate strong reject quantile deviation control type control signature visualize approach provide use knowledge cell visualize algorithm compute region cell ratio every pair cell multidimensional center chart j adjust bootstrappe quantile visualization affect preserve visualize chart provide significantly cell connect introduce datum splitting describe cell visualize visualization slightly half certain direction cell alternative conduct energy iid useful goodness fit name recommend two sample version energy numerically use value quickly introduce testing consist split e kde cells energy cell energy reject correction provide use along twice test cell since density randomly split find second mse consider flow start whenever away close flow able bind flow move infinitely value eigenvalue gradient minimal absolute eigenvalue evaluate change whenever must pick sufficiently constraint minimal follow near remainder use bound flow lemma solid mode box minimum empty dot line gradient flow region equal flow within flow line close stop local complete eq eq region around fact critical boundary possible region note lemma link mf derivative constant condition basically extend bx bx theorem condition replace constant lemma x project gradient notation theorem derivative let near dx turn proof rate hausdorff critical distance constant depend actually small whenever point vice versa j small intersect obvious intersect bm f distance attain hausdorff project use connect segment normal space slightly distance hausdorff contradiction x sufficiently hold bind rand index mode specify small thus mode estimator x ab mode cluster boundary pair definition rand rand index volume probability vc vc process theory vc theory mode local generality sufficiently thus whenever equation note finite estimate local mode field far nx set note dx source different second level theorem part theorem assumption absolute eigenvalue prove second assertion assertion theorem version connectivity connectivity mode recall local eq convenience within region thus ji ij ji show convergence recall point define q within least majority set uniformly consistent transformation q q compactly within around denominator comes put equation occur take eq prove consistently extend prove define consider gain consistency far analytic proved split part agree second remain put matrix nonparametric regression solution one mode cell conclude prove convergent part region estimation part region estimation translate complete assertion regression first assertion prove assertion equation lastly k corollary give nonparametric high complex problem smooth define complex consist call intersection maxima minima generalization function roughly speak piecewise monotonic shift certain show visualize representation wish visualization circle cell large density latter dataset density visualize difference green circle neighborhood denote red chart ratio region complex estimate theory develop three two smooth boundary maxima regularity hausdorff satisfy mode let kernel boundaries rand kde use sufficiently fx visualization show function see test two information test application include vision topological previous stability pointwise sufficient prove regression visualization code use paper definition start simple maximum maxima plot intersection call definition derivative critical sign critical partition distinct minima call call degenerate critical function k c c ascent flow move along individual correspond point ascent converge flow element theory manifold manifold flow point descent precisely gradient start flow share flow limit function share dimension consist manifold thick curve thin thick thick curve blue manifold intersect see assume g collection subset function call manifold consist cell cell give manifold manifold use manifold fit individually cell empty intersection manifold measure boundary define boundary derivative eq measure difference function set hausdorff denote manifold show sufficiently derive collection manifold embed every span space simplicity column depend abuse notation eq space manifold gradient flow eigenvalue matrix scalar vx require flow move mode flow move derivative imply along behave behave flow move flow boundary thus minima manifold let f bound boundary manifold sufficiently close two st make manifold ascent manifold define analogous quantity consider vx result manifold imply nonparametric g manifold manifold density estimate regularity manifold plug regression regression condition consistency mode mode cluster mean shift mode use manifold point region describe denote boundary boundary boundary symmetric kernel exist l kde widely essentially norm function kde boundaries mode sufficiently small simply combine kde rate decompose nh estimating sense mode density way quantify rand partitions kde namely rand index versus rand rand adjust rand rand mode basically incorrectly mode nh asymptotically application plug cluster kde version level set level distance infinite cluster level distance design connectivity cluster mode clustering plug estimate kde estimate kde equation
simplify computed approach powerful kullback leibler measure demonstrate modeling conceptually link addition reduce unbiased information information aic widely applicable unfortunately contain nearly fisher greatly complexity failure analysis frequentist criterion account depend analogy aic applicable change frequentist test already canonical point interesting relation newly derive find approach fundamentally understand level bic suitable segmentation datum model write role partition place along axis red line represent minimum datum dot red respective change bottom point dash signal observation temporal shall q true cross entropy entropy expectation understand distribution change index mode fundamentally shall regular typically consider model zero consequence couple n define determination index change index transition binary study e g initialize sequentially greedy step say nest successive exist previous always binary process show panel statistically state mle differ binary explicitly selection entropy eqn estimator estimator bias phenomenon add reduce reduced parameter estimator cross q regular criterion aic generally approach complexity criterion frequentist use distribution frequentist parameterization small approximate direct eqn tractable complexity compute observe denote consider new parameter new essentially harmonic procedure aic model compute evaluate difference series complexity remain question first complexity compute state break information information per information constant kronecker delta analogy taylor expand parameterization fisher refer th parameterization eqn expansion variable information rescale coordinate interpret unbiased mle rhs forward intuitive mean dimension complexity complexity complexity binary eqn convention point segmentation ar overlap partition mle nest partition eqn since convenient brownian bridge walk brownian algebra bridge brownian step place relation disagreement illustrate bic bic justify observation bic always due clear fig constitute much large produce complexity complexity problem result twice complexity since circumstance intuitive pick change segment must reflect add must observe background discuss significance state occur significance complexity eqn identifiable change predict change aic predict complexity include slope exactly predict significantly aic fails terminate bic initially match simulate generate four family nest model fit plot corner eight true change minimize information red entropy plot split entropy green addition increase cross entropy green red continue compute panel complexity observation aic bic estimate aic nan hypothesis partition parameter test partition divide test statistic statistic expand analogy eqn approximation derivation frequentist statistic accept alternative equal interpret frequentist statistic determine also critical statistic base optimize large approach extensively apply four cell molecular paper information change point use criterion parameterization advantage frequentist analysis observation unnecessary develop hoc approach confidence level automatically information prior like bridge analysis widely author design perform global segmentation min information implicitly segmentation min plus iii terminate implicitly segmentation correspond positive acceptance plot false interval dimension analogous false rate false entire partition eqn relative bic complexity use change slowly normally iterate see
datum set noise drive analytic sketch mse depend moreover approximate diffusion step verify eq euler ask complexity euler eq constitutes compare euler reduce algorithm expect approximate reasonable q ensure turn satisfy regression model assume model fixed put subsequently rwm run iteration effective inferior drop mathematical expansion power size identify unbiased construct match euler asymptotic bias decay rate complete extensive toy analytic moment quantification error result batch ergodic function concentrate posterior mse express derive recurrence equation recurrent plug result repeat sum explicitly geometric sum q equation recurrence derive express term order elementary newton identity term compute e x newton identity similarly nn ix n nn jx lx knn nn p nn nn sketch derivation mse take analytic expression illustrate calculate e e n n nn deriving require mse es toy similarly explicit expression file cm proposition remark section large inform proposal usually whole propose gradient langevin subset accept reject sequence decrease mathematical central decrease zero bias size stochastic modify remove obtain obtain toy study euler datum markov langevin dynamics big fix many molecular expectation differential equation sde run scheme approximate dynamic one average finite use proposal algorithm machine modelling informed proposal require sde standard brownian appropriate dynamic ergodic arise approach would evaluation langevin dynamic generating subset q subset algorithm appear front unbiased limit behave investigate satisfie decay asymptotically achieve asymptotic size gradient hamiltonian monte decrease point analyse particular dynamic ergodic numerical measure apart invariant remain question arise lie respect tackle paper particular generalization power explicitly contribution modification asymptotically euler modification bias bound finite rhs extend main relate establish existence nice poisson decrease rate finding confirm numerical study toy bias explicit connection confirm equation match analytic expression toy importantly allow significant study mean square average second comparison euler quantify mse quantify mse useful proceed study regression observe behaviour weak apply finite long time behave euler discuss toy model section term acknowledge thank mi section preliminary backward section global expansion ergodic average invariant condition dirac adjoint generator moreover backward kolmogorov equation taylor rigorous taylor integer follow assume bound derivative fact assumption enough derivative growth regularity bound numerical deterministic assumption satisfy reasonable derivative order assume deterministic initial taylor series form smoothly depend assume coincide weak immediately integer remainder study invariant error weak time time numerical sequel ergodic assumption imply ergodic compact question though condition ergodic answer case relate drift behaviour euler solution ergodic theorem derive euler infer particularly equation generate step equation follow satisfy one ergodic average smooth ergodic deterministic similar difference euler explicit expression invariant key long say average discrepancy true time respect measure choose reveal formula hand case process study light cost calculate euler method likelihood imply lead corresponding presentation illustrate main calculation dimensional calculation q notation expand expectation respect variable obtain see expression leading contain error give theorem euler precisely make term see section introduce extra appear relate average like imply error front time way achieve time amount datum available since amount introduce modification calculation imply ergodic euler calculation appendix form simple equation contribution analytic expression subsampling variance extension average methodology adapt poisson pde transpose poisson equation sde average ds rhs control derive ergodic elaborate reader therein euler taylor expansion express summing divide remainder rise state derive presentation compact argument sde read unbiased diffusion result sufficient diffusion theorem believe scope notation instead drift write taylor yield sx yield h derivative drift suppose fix advantageous step mse agree decrease equation confirm toy poisson term assumption infinity twice satisfie v lyapunov sde e exponent subsample however even strong toy model simple posterior eq equation numerical read generate replacement toy illustrate clear theory obtain analytic moreover confirm toy mse derive section correct computational effort toy analytic match amenable expectation limit capture investigate limit law combine eq fact subset limit compare error coincide expression particular compare euler replace fact toy model simple integration section calculation discussion modify obtain euler point n rh superior outperform step euler evaluate observation indicate dependence bias asymptotic take step term subsampling form choice stay contrast consideration subsampling also regime small recursion letting subset see rhs restriction desirable need increase sde section bias becomes euler additional regime efficient computational efficiency accuracy far investigate variance study calculate
beneficial bootstrapping scenario report measure top reliably use overall predict miss kb recover entity type extend kb completion kb completion treat unobserve kb negative unobserved kb negative example positive largely method unobserve object systematically negative sampling snapshot kb kb kb type construct two set entity set precisely number negative entity type pair scoring type fact margin control experiment sampling entity fix global relationship exist map entity represent entity feature entity embed propose algorithm use scoring se te optimize eq modify handle vector refer ensure optimum certain adopt give update vector te te te te te initialize randomly ei te se entities projection se l column describe detail embed expressive model objective relation extraction entity type cast extraction entity ne objective entity model motivate make neither foundation previously method type min positive perform manual evaluation method fact data type statistic add recent snapshot effect entity example datum challenging feature description type description w w ne global objective ne ne nt model g automatic evaluation aspect system empirically perform final boolean wikipedia text map try yield improvement ne objective use show achieve score classifier among ne ne perform type expensive result show expect type despite popularity embed nlp verify correctness evaluation perform manual show evaluation kb incomplete result indicate automatic manual manually miss effectiveness highly type performs compare frequent bootstrapping technique supervision method would find prediction use linguistic entity word music frequently l g entity type focus entity type level develop kb kb new york article method sentence use entity wikipedia article perform kb completion relation extraction majority infer within kb kb train different suit comprise metric evaluate miss verify automatic publicly experimental objective produce baseline method future plan information link human get conduct microsoft computer science edu microsoft microsoft usa microsoft com knowledge kb completion relation extraction entity kb fundamental task kb little automatic completion information kb external wikipedia individual train consider consistently prediction baseline manual evaluation base correctness automatic evaluation miss write basis community contain fact drawback incomplete fact usefulness answer lead complete completion subproblem completion infer entity kb completion entity nlp extraction entity parse question answer add entity relation importance little publicly dataset infer entity kb dataset use snapshot kb could potentially evaluate ability already two kb predict add snapshot enable realistic challenging prediction compute ideal treat type equally consider measure prediction measure within type example entity side perform well global metric entity global combines type side low dimensional produce ranking negative side reliably confident entity candidate base summarize develop comprise metric kb miss relation evaluate fact snapshot miss entry unknown potentially use evaluation evaluate ability predict fact kb drawback snapshot early treatment snapshot later newly snapshot fact miss kb snapshot fact test snapshot subtle fact contain predict newly add fact hard enable realistic evaluation kb snapshot relation automatically kb fact kb automatic characteristic snapshot advantageous snapshot snapshot wikipedia fact snapshot treat unobserved instance unobserve instance snapshot example training fact newly add snapshot
kp long analytically multiple method stochastic deterministic counterpart inside partition sep fig protocol different approximate kf kp computation almost copy like uncertainty cavity distribution kf compute cavity f f cavity q kf sep keep computation sep consumption sep propose hidden factor hide variable eq sep next approximate need maintain point cavity n n involve retain also globally share piece sep accordingly assume k memory consumption memory sep year sep test gb ram huge test sep sampling probit probit unit sigmoid make moment analytically subset sep almost ep indistinguishable reduce nm mb mb mb protein mb year extension vi relationship introduction power alpha alpha q ep alpha change alpha projection return alpha alpha difficult motivate practical alternative power ep converge however equivalence apply sense stochastic spirit keep current discuss connection variational natural natural n another approximate compute form evidence zero recover current n reader local update global parameter typical return well imply factor power alpha divergence recover change notice dataset q interpret impractical like inner move n use technique outer average answer optimisation inner local optimisation alpha divergence inconsistent vi condition nature point average ep q fix
value enumeration slow exact permutation fix margin uniformly compute value exceed margin although per interestingly permutation test rather equivalent section use accord occur occur binomial large perform enumeration compute test enumeration accumulate large compute perform enumeration high need value goal since typically collection except dataset approach sample collection hasting mcmc proportion exclusive set collection reason mutually exclusive suboptimal may gene summarize collection use denote undirected weighted vertex connect weight exclusive identify exclusive first section graph set choose topology pathway two cut observe collection score chance evaluate significance score sum column permutation collection satisfy collection important cancer pathway exclusive primarily mutually exclusive biological pathway advance one analyze unfortunately reduce identify address one new contain surprisingly exclusive row one note run exclusive set contain cancer run summarize collection mcmc frequency collection marginal collection exclusive permutation gene high use test use ease exposition define contingency reasoning behind coverage gene exclusive high coverage contingency table freedom one gene contingency table contingency margin could result interactive visualization see website module marginal graph user dynamically module module module view sort search collection module mutation dataset publish mutually exclusive gene section result simulate dataset function tie e gene pathway show bar comparison adjust rand collection compare cancer sequencing pathway exclusive highly whose exclusive rate highly gene appear cancer exclusive section pathway vary ran set average rank pathway alternate rate dataset pathway able reproduce possible average pathway figure coverage rank pathway rank even extremely approach comparison respectively increase across much fast superiority score pathway simultaneously collection exclusive overlap non overlapping set important collection pathway exclusive gene pathway third total also pathway default consensus ari well pathway ari measure agreement partition ari partition ari partition maximally table fraction pathway ari furthermore ari dataset value varied multi range demonstrate parameter choice multi even fairly statistic gene cancer census version dataset sample besides dataset recurrent identify collection score remove least significance cutoff run match run identify collection significant accord census collection identify pi pathway surprisingly exclusive range less surprising dataset identify exclusive overlap report association cancer include spurious exclusive set handling cancer l without filter indicate difference filter frequency argue less biased gene mutation mutation gene breast run mutation cancer array datum repository availability cancer remove highly find without table supplementary multi include background collection high collection largely demonstrate frequency dominate coverage come gene gene gene real cancer run cancer analyze nucleotide copy gene detail supplementary section detail gene genomic circle indicate edge collection value output four mutually exclusive module fusion publication mutual increase number mutually exclusive module six mutually exclusive gene six together module target protein involve set clear notably show gene set largely fact pattern collection module include member pathway role cancer function significance unclear module include cell cycle pathway pathway module four include gene moreover cancer suggest role cancer perform breast cancer merge run introduce breast traditionally classify analyze include classification number module first association report module module relationship associate pathway module contain gene highlight high consistent associated form contain mutation pi pathway gene annotate part pi pathway study circle module p report breast role cancer breast cancer factor involve breast report tumor breast cancer note multi report tumor explain occur suggest breast neighbor tumor breast cancer report occur suggest gene pathway domain possible role exclusive interaction pathway due allow refined interpretation mutually exclusive collection mutually exclusive knowledge exclusive frequency rare large contingency genome datum identify collection exclusive superior real large mutation cancer identify significantly exclusive overlap pathway illustrate subtle pathway protein hypothesis analysis variety site annotation cancer cancer project cancer analyze exclusive analysis finally novel tail enumeration broad examine dataset biological adapt type supplementary approach exclusive ta wu correspondence define whose collection collection ergodic ergodic chain converge want proportion use exclusive high hasting algorithm chain modify ergodic desire stationary distribution hasting collection state bipartite apply exclusive number occur check weight examine sample significant relatively exclusive distribution initialization supplementary gene assign gene collection uniformly else gene note unchanged p algorithm initialization converge similar fail converge create perform initialization initialization initialization multi initialization generate precisely gene union gene union chain gene variation total million iteration initialization examine calculate initialization process stop otherwise iteration final example mutation variation million figure obtain clique marginal corner precisely log choose start horizontal show dramatically find first form corner slope line negative move run million initialization run equal define million initialization supplementary large meaningful practice large long mcmc since space collection large alternative approach g clique clique suggest collection detail contain array patient genome use annotation together expert protein protein patient copy genome include include identify significantly single nucleotide variant file copy output gene sample nucleotide variant cancer include copy remove value four instability level number unstable call contain copy number patient cancer genome approach necessarily exclusive pathway fraction proportion sample gene gene gene probability introduce noise datum match mutation frequency empirically breast mutation remove occur result run million initial start h cancer l cancer cancer page non pathway overlap non overlap overlap overlap overlap overlap non overlap overlap consist pathway rand choice use run value sample exclusive color dot occur note value co occurrence fast tail enumeration green dataset tie score pathway pathway runtime bar gene first edge slope subgraph h dataset whether occur grey indicate colored exclusive colored blue rgb combination mutually exclusive ta wu correspondence cancer heterogeneous disease combination drive cancer pathway pathway incomplete identify recurrence f pathway mutual expect pathway several important distinguish analyze mutual include exact sensitive detect combination rare simultaneous mutually exclusive simultaneous ensemble collection mutually exclusive multiple pathway cancer outperform real application hundred cancer reveal mutually exclusive overlap pathway cancer cancer international cancer genome identify genetic project genome sequence thousand role cancer cancer directly sequence datum challenge heterogeneity collection interact perturb heterogeneity cancer motivate development examine pathway review pathway database network lack specificity accurately examine require biological combination mutually exclusive observation observation relatively tumor pathway gene mutually exclusive de recurrent mutually exclusive module identify gene mutually exclusive protein protein identify coverage mutation approximate mutation combine equal minus coverage overlap co occur find chain carlo set identify exclusive set program weight cover exactly mutation gene identify however show sized require contain mutually exclusive frequency bar exclusive weight like common genome gene tail use exclusive occur test highlight exclusive cell limitation combinatorial subsequent algorithm gene mutation high dominate towards identify majority gene include version study model ratio identify mutually exclusive sensitive gene limit applicability identify gene feature prove overlap overlap option cancer show multiple pathway mutual signal introduce limitation outline towards discovery combination low tail enumeration approximation simultaneously identify collection mutually exclusive collection summarize result marginal pair enable include overlap without knowledge simultaneously discovery mutually exclusive avoid identification mutually exclusive approach cancer datum breast cancer type pathway novel breast cancer simultaneously identify mutual pathway first transform value collection two collection whose exceed summarize significant indicate corresponding pair algorithm mutation nucleotide mutation variety change binary surprisingly exclusive mutual motivate score four exclusive sample weight later least mutation mutation give mutual single frequency three low like common cancer recurrent spurious therein score test contingency b entry equal occur margin sample occur statistical reduce score product score nan set collection grow exponentially typically compute use markov mcmc collection pair connect pair exclusive remove call collection specify tuple measure observe rate generally unknown mutation nucleotide vary uniformly
theoretical applicable temporal scheduling markov stay integer demand service census decade common center indicate van care stay ultimately cause cost al study induce census variability show effect job census excess resource material frequent instance expensive patient census estimation term typically patient census optimal resource allocation body address census integrate less developed work integrate significantly quality et van appropriate characteristic account rs assume type trajectory employ analytical technique patient patient stay express call outcome lack suited capture patient sufficient depth interaction issue develop patient historical patient classifying optimally form cluster impact scalability patient heterogeneity many traditional classify patient diagnosis service work large patient insufficient patient two historical factor gender determine trajectory patient trajectory patient entire location must patient different another patient visit skewed census forecast classifying requires perform significantly second patient classification solely identify patient cluster gender define shape patient associate location statistically validate finally cluster phenomenon capture patient force group assign type closely patient estimate manner hoc gender patient statistically classify patient identify classification seek develop method cluster patient type interaction heterogeneity begin cluster trajectory patient census serve rs module resource schedule traditional novel patient literature cluster patient trajectory group patient module output cm van optimal resource schedule system enable maintain internal traditional hold review new detail follow validate apply study historical finally exist focus rs integrate rs least aforementioned characteristic develop integration rs account heterogeneity interaction various rs green patient arrival optimize resource scheduling either develop flexible consider interaction pathway al al al rs isolate feed forward ignore interaction address issue propose scheduling patient pathway interaction portion consider properly patient heterogeneity moreover patient census estimate arrival rate arrival patient census forecasting review daily largely hence week ahead capable forecasting patient flow multinomial prediction feed flow implement present challenge even variety source burden scale patient al scalability exponential distribution reality patient characterize patient phase combine network include inter relate variable represent causality type begin phase enter fail propose complicated interaction build mix e account scalability interaction heterogeneity htb patient trajectory heterogeneity patient van partition diagnosis diagnosis et planning provide cart etc attribute diagnosis find service necessarily patient share age diagnosis trajectory figure diagnosis trajectory literature high level methodology flow admit record patient location cluster combine arrival stochastic capture census level network census product accurate census programming scheduling patient initially patient stay another serve patient service follow cluster conventional applicable problem patient trajectory mix patient develop semi mixture model patient trajectory predefine validate important generality scalability patient patient type trajectory patient patient cluster estimation patient patient type patient trajectory number semi equal mixture henceforth different markov estimate sample state patient indicate patient stay death initial patient enter u length stay subscript sequence capture behavior restriction visit component initial array hold mass spend patient les belong trajectory transition probability transition time patient amount spend function sample trajectory likelihood observe zero issue conjugate pdf denote q patient cluster give membership unnormalize maximize iteratively regularity satisfied guarantee ultimately optimum distribution iteration membership plug kp kp kp kp kp kp yx kp kp number sequence kp kp make sequence convergence increase redundant test control type chi develop compare transition compare hold merge cluster detect cluster patient visit number patient u one input scheduling transition recall patient compute period day consequently express sum eq semi estimate stay compute research objective purpose integrate process method arrival create stochastic census scheduling patient throughput volume purpose brief detail focus patient cluster integrate optimization approach herein two develop census set section cluster trajectory trajectory one arrival semi patient type section create census demand arrival process arrival combine trajectory poisson demand deterministic assumption approximation reality widely literature control theoretically possible close beneficial patient flow toward deterministic management deviation incorporate approximate particularly arrival arrival patient take account arrival homogeneous poisson vary day week combine poisson markov process arrival census fix distribution define demand rs integrate census objective focus metric minimize maintain throughput perspective objective allow increase consequently patient present van formulation similarly schedule optimality apply approximation metric formulation patient attribute week unit day reward reward patient trajectory type day demand day patient day decision day patient plan horizon system day week throughput patient planning horizon column flexibility allow type patient project patient day set subsequently distribution patient leave side level management add formulation model constraint approximate limit census census patient able proper mix patient type week ensure resource frequently conclude type trajectory estimation integration scheduling develop accuracy patient trajectory flow validate scheduling simulation state sequence patient four semi different cover transition gray high mixture additionally probability generate output would general htb c c model plot drop indicate cluster weight find pairwise hypothesis estimate cluster generate representation estimate true probability show figure chi square test equality estimate parameter table equality hypothesis mapping demonstrate patient flow scheduling model rs patient type effect scheduling schedule trajectory patient use obtain schedule trajectory compare resource optimal utilize naive patient age gender diagnosis patient cluster patient frequency perform attribute datum generation attribute patient formulation show percentage improvement setup naive metric patient result optimum naive patient significantly benefit patient close optimum attribute lead understand resource indicate accurately estimate yield near outperform apply propose resource schedule historical patient complex system year include stay age diagnosis patient leave death
set algorithm table table iw column name dp compute exact method posterior assumption modular dp list dp iw show iw clearly use short computation significance dp meanwhile mcmc stability similarly much short case exception several thing term method large phase obtain dp occur letter child mcmc dp mcmc unable reduce dp please iw run dp dp method dp dp step dp dp factor hidden difference iw mcmc iw run data iw corresponding time phase less run dp section effectiveness strategy range across show cumulative mass iw note formula p five tumor letter five iw small dag tend local large number iw include structure node inclusion improve show small iw great datum sound iw increment case letter correspondingly figure increase increase running figure clear achieve case second dag mean interval respect increase actually decrease benefit strategy iw vary letter child iw run iw sample direct edge totally mcmc iteration discard mcmc iteration burn dag sample run sample experimental setting case run run iw demonstrate soon letter run program memory expensive iw child outcome run get experimental result show performance method edge bar deviation iw dp iw run run smaller compare iw iw real sample iw significantly show figure term mark compare iw advantage iw combine figure dp iw shorter iw second phase dp second dp compare good iw shorter iw letter experimental examine material experimental modular method modular force enumeration dag approximate tool estimation posterior edge mcmc comparison one modular showing fundamental structural direct complicated perform experiment uci well heart modular five modular influence length length intermediate variable feature eventually influence eventually influence combine path eventually influence path influence x x z jj iw set iw significantly compete mcmc iw compete modular support b iw try run iw times standard direct modular feature run iteration discard burn dag run mean deviation run bar dp iw correspondingly three modular iw dp iw method modular dp edge significantly two modular iw well real iw significantly p one consistently modular iw good value kind mcmc edge investigate mean iw significantly corresponding variance result iw good sample iw iw compete five investigate modular iv letter hypothesis guarantee case letter serve performance requirement dag come performance inequality posterior easily event bernoulli independently repeat success figure see much mark bar edge even trial side hypothesis reject nan hold next demand requirement dag sample show come guarantee hoeffding logic repeat estimator even histogram even hypothesis reject hoeffding table figure edge even correspond among sided conclude less clearly even sided testing case increment hoeffding hoeffding decrease figure edge run algorithm max clearly maximum furthermore side reject nan conclude hoeffding hold efficiently bayesian network dag sample dag feature interest show empirically estimator considerably outperform art without modular capable estimate arbitrary modular modular prior iw structure modular modular modular time modular modular bottleneck dp application able implementation use cluster totally memory proposition along eq prove probability proposition pmf assume modular derivation dag relate sample occur order consistent step get total formula pg sample sample p pg prove pmf bernoulli pmf ii almost since law converge random bernoulli thus central theorem eq denote mapping theorem theorem hold equivalent conclusion imply straightforward q define equal choose every dag dag exclude dag proper p p p f f I pg p op g order modular dp notational convenience essentially set lagrange q neither eq limit eq constant done prove lead whole done converge g positive super exponential thus algebra define g g probability g ng g ng ng nn g whole surely imply proof proof quantity proof essentially proposition direct dag dag accord feature theoretically estimator outperform art learning sampling widely various task network bn bayesian network dag whose represent direct dependency encode node parent semantic compact joint answering query semantic insight effect decade network unknown learn motivation learn make use predict another closely semantic discover domain context dependence different often interest semantic node directly influence interpret eventually influence furthermore node node cause learn interesting gene turn examine path structure kind structural goal discovery define dag find equivalent dag dag use single posteriori lead conclusion average challenging super special develop probability structural space capable handle bayesian moderate due big posterior via node limit path path compute path assumption modular soon dp handle mainly modular problem dp algorithm far feature independence hold generally user upon feature typically need limitation compute posterior one solution modular approximate model develop draw dags monte hasting dag develop procedure operate show mcmc outperform develop proposal mcmc mcmc converge fast mcmc structure superior mcmc nearly mcmc method operate order appropriate common drawback finite show competitive hybrid several work approximate order mcmc special form computational convenience please begin modular assumption modular prior desirable prior bias dag topological assign modular inferior please develop modular strategy dag order dag accord considerably considerably mcmc moderate applicable moreover desirable unlike algorithm estimator control dag address limitation dp whose restrict modular posterior various arbitrarily exact dp develop iw bias modular theoretically prove base empirically superior modular iw address dp usage modular posterior additionally efficiently prediction application avoid modular prior briefly bayesian network dp iw demonstrate advantage finally appendix proposition theorem node dag represent convenience parent dag dag represent probability dag assume global local local likelihood score local close score efficiently eq modular modular marked distinction dag often structural indicator otherwise average issue describe dag infeasible problem approach posterior sample dp algorithm space rather dag linear order vector dag order denote subset linear notation nonnegative define return example modular setting accordingly dp step step compute degree truncate technique extend standard algorithm compute define follow forward contribution show start take compute whole dp algorithm modular include dp step introduction backward q recursively dp contribution structure limitation posterior modular use dp compute posterior idea sample invariant able compute result constant eq modular proposition state definition choice parent order order non modular draw draw sample dag draw parent dag prove subsection show compute efficiently conditional respectively follow q ni appendix u sample order element element order sampling order sample pmf posterior modular result guide view dp answer oracle aid counting generator reach situation count information compute dp perform parent describe algorithm term order accord dag one dag sample parent correctness dag iid posterior algorithm take overall time complexity efficiency fairly dag tend maximum biological assumption widely experiment dag dominate moderate reach several thousand develop dag step dag exact construct dag take need sampling modular edge take take note order dag memory structural algorithm ii converge q limit iv hoeffding state order least require property obtain guarantee iw weighted dag correction sample dags dag check dag time hash time iw iw joint store construct construct iw structural dag method experiment property iw converge convergence represent cumulative mass sound sound interval sound dag possible tractable desire please note express equivalent give correction strategy iw solve desirable dag strategy keep sample sum dag problem instance dag maximum possible negligible majority dags dag cost dag sample expect time requirement hard disk store dag take dags dags dags dag sample reach order cost per avoid meanwhile strategy therefore gets sample times dag implicitly serve strategy guarantee dag large spend huge dag sample dag dag sample dag keep dag none chance get guarantee dag state actually ensure good art applicable moderate mcmc first compete hybrid mcmc second eventually hybrid correct come dag posterior shown converge fast structure order performance mcmc mcmc moderate applicable mcmc iw infeasible large limitation iv hybrid mcmc compete apply collection dag score use dag constitute advantage algorithm specify iw complexity spend good search increase cost one iw ordinary computation severe usually iw interval wider set memory please language demonstrate capability set include ten real uci machine letter tumor synthetic gold synthetic contain letter child vary instance tool pc intel processor memory extra specification maximum partial order art order modular partial mcmc implement language estimate state readily dag consistent c tumor letter variable moderate able use language dp posterior every modular prior therefore absolute posterior essentially j j modular indicate discovery note mean since base value one datum tumor child fair set cause run iteration mcmc sample partial p sample information order consistent great method inclusion efficiently modular edge compute order performance performance approximation come dag avoid mainly decrease dag modular feature table list cost list correspondingly total run precisely speak total running method report run relatively computing include dp six run step due factor percentage performance
require block voxel matrix human matter acquisition normally diffusion direction brain contain gb format format five memory requirement memory system research dramatically storage requirement life life via tucker decomposition std introduce tucker tucker specialize propose approach multidimensional factorization array tucker rd approximate decomposition decomposition guarantee array array e tucker powerful compression store tucker explain compression obtain tucker tucker store instead require store compression mean classical tucker allow represent dense core tucker array dense tucker achieve core array tucker decomposition array tucker core array entry entry store require storing coefficient location hereafter refer tucker std compare tucker model std compression ratio low sparse array show life explain diffusion voxel dictionary signal extend decomposition matter voxel orientation white evaluate orientation hereafter calculation represent arbitrary dictionary prediction signal predefine resolution diffusion contribution q atom fig discretization atom spherical sample spherical construction diffusion spherical coordinate orientation diffusion dictionary different diffusion measure direction signal voxel discretize life signal organize approximated combine diffusion atom orientation zero entry indicate fig sparse atom voxel give voxel store individual contribution voxel store index atom avoid signal std life matter prediction life life comprise sparse v direction voxel array use tucker equation rewrite array n correspond white voxel eq life tucker n vs along main diagonal combine sparse tucker life also decomposition introduce optimize std life build white brain std orientation entry voxel atom version life sparse tucker std life evaluate brain variety negative least problem hereafter exploit std life life core life std version store describe use report implement w result matrix std write life q kronecker write follow stand stack vector avoid multiply allow maintain usage plot memory requirement std std measure diffusion std function compare usage life storage requirement std analytically nod per store proportional storage conversely store std non core std storage n without straightforwardly fig b memory direction node storage direction std fig memory number storage grows linearly grow much sum substantial reduction memory consumption std become direction std approximation life discretization fig std std predict original mean validate discretization std original std original range show discretization varied difference r whole matter volume comparison relative std life fit std discretization spherical relative w weight std respectively test tucker original storage show std major original validate predict diffusion deterministic std replicate finding demonstrate life implementation life software std matlab tensor toolbox file mac bit software test computer gb ram brain ghz intel gb scatter plot error predict probabilistic computed single probabilistic base std discretization grow time human grow require collaborative community focus modern big application imaging measure white brain major put improve white macro micro level digital availability develop compute computational trajectory white matter node first compare connection identify compare validation establish accurate core matter global global plausibility estimate alternative matter computationally intensive routine brain population recently linearize method separate similarly family linearize modern new linearize road investigation matter linearize impact decomposition road population human increasingly resolution modern clinical report pseudo decompose life option parallel decomposition component atom voxel acknowledgement chen comment version support computer support c code tuning project block mm ia la brain sciences program cognitive science usa edu publicly grow increase availability resolution require modern linearize decompose matter achieve comparable approximation brain big datum mapping brain network tensor major matter diffusion probabilistic deterministic transforming start share brain collect share potential group analytic tool attempt extend replicate scientific fundamental modern separate paradigm landscape imaging motivate storage sharing hereafter show development processing challenge resolution signal algebra object dimensional array array tensor vector number dimension primary compare algebra multidimensional turn address convenient substantial reduction briefly array diffusion combination track measure property white measure health disease estimate white matter routine b life brain candidate white generate candidate generate white matter volume error predict use comparison method model fig predict diffusion signal linear within voxel predict diffusion voxel model direction voxel red voxel combination predict green voxel measure matter voxel organize long full volume assign use optimization notation voxel life life panel measure double fundamental task evaluate individual one substantial white matter spatial directional resolution modern size model linear primary describe directional primarily matter signal directional relate combination brain measure signal signal strength denote strength voxel candidate matter voxel non apparent diffusion f definite signal example exponent tensor semi ellipsoid radial tensor diffusion array array array array multidimensional basic discuss array notation definition array letter call th order array generalization denote capital letter e slice slice use array along array slice let index vary array slice hold index unfolding array array address dimension mean
big appealing scale accordingly hence interesting concern finite subtle analysis fail connection cluster conclusion view open question mini batch policy minimal inf two proof distribute mdp correspondingly assumption satisfy show transition matrix correspondingly trajectory cluster least bind trajectory obtain h h unbounded maximal distance must reasonable trajectory order misclassification occur quantity leave corollary problem static strategy accumulate context interact website gender age device etc website customer interaction focus horizon know context provable context optimize naive implementation extension process mdps commonly dynamic health management trajectory observe trajectory answer transition standard estimation method affect temporal diabetes influence measurement variable incorporate create mdp reduce generalizing incorporate static form transition zero instead sized world website website activity suggest ad relevance ad age gender device determine website mechanism http mechanism insufficient website gender observe website word trajectory page user profile take type exist device want child parent suggest interaction fashion user elaborate want ad user time spend website interaction optimize line solution valuable case underlie contribution present general provable horizon contextual empirical discuss case infinitely context reinforcement rl reader mind preliminary present derive trade bar research setup markovian consider hmms hmms mdp essence context arm bandit mab present user similar setup reward selection mdps architecture task domain gate mix different area could model learn source fit representation deal state interact environment differently mdps perspective relate model user type conclude short comparison know affect complex generalize hidden parameter constant context reward compose suitable state rectangular singular determining formally setup problem analyze extension setup mdp setup tuple py rx py rx interaction episode accord learner action accord environment learner q distribution mdp finding maximize cumulative mdp rl definition establish action correspondingly map cx model action generalizing model observable arise number problem behavioral pattern customer side gender aggregate available independent greatly simplify writing finally adopt setup episode episode adversarial fashion state distribution generate mdp maximize reward increase optimize trade exploration exploitation unlike rl setup reward cumulative reward agent cumulative discount true therefore converge similarly know apply correct regret respect notice though new loss different define value obtain agent explore classify trajectory mini batch mini batch distinct new act possibly way issue next length recognize correct exploratory part addition closely problem mdps formulation mdps transition uncertainty rectangular related trajectory impossible consider unless subsequently pose confident trajectory embed improve notion separability trajectory separable policy fulfil policy logical maximize distinction optimal consequence one lead area reward action distinguish still problematic underlie exploration open solve mdp require sample need identification infinitely model substitute simplicity trajectory analysis modularity follow state transition set score trajectory scheme inefficient exhaustive cluster adjusted policy step mention decide confidence could combine strategy choose set wrong apply estimate sophisticated would rl whose scenario constant satisfactory necessary assumption result trajectory visit separable enough sample trajectory model policy realization algorithm assumption full available supplementary material assumption mini relate proper scale require enough third correspond trajectory extension previous consider complicated scenario evaluate consequently regret precise context context application agent trajectory step trajectory latent length trajectory naive employ rl algorithm approach share context trajectory scheme choose classify choose exploit length exploitation decrease approach trajectory short trajectory regret contexts trade exist cluster part draw length sampling action correct context thus worst use bar deviation examine follow long trajectory cluster experiment plot trajectory present trajectory part bottom trajectory length episode phase threshold follow adjustment period succeed certainly trajectory fail episode sufficiently cluster quality simulate episode portion trajectory dedicate identify exploitation exclude
symmetry finally v repeatedly pick vertex reference try require high vertex less likely good vertex give lowest well around spectral obtain directly grow community g logarithmic extend sized community degree overlap sized overlap motivation vertex vertex path multiple short use would community attempt get relate connect vertex integer cardinality intersection vertex community around assign subset edge even generate edge probability approximately eigenvalue operator hold ie terminology ie ie get estimate start leave r determinant follow e r ij q linearity one product contribute determinant product involve dominate follow find graph nonzero list approximation set r e se use solve repeat estimate approximation input distinct approximation create take return median eigenvalue product vertice integer already course flip get away away equality sufficiently follow vertex input decision proceed follow I iv iv j solely compare vertex fairly vertice subtract inequality strict representative community give vertex without classification vertex number suppose minimize community run fairly algorithm follow plan select vertex one anchor vertex attempt community actually community approximation bad accurate output follow randomly assign g v comparison community otherwise fail bad vertex time classification half classification little two contain none could bad average classification together agnostic start use pick run combine classification explain require sphere comparison integer list community small rational numerator q small hold classification small classification disagreement classification define disagreement classification minimum disagreement remain classification assume community great overlap return combine satisfied run comparison symmetric row equal reciprocal rational run community graph draw without beyond multiply corollary row agnostic comparison average degree row positive l agnostic corollaries connectivity matrix grow almost exact recovery agnostic proving require terminology distinct eigenvalue order also addition sum vertex well key behind short graph vertex short refer whether good q otherwise combination rv I follow vertex symmetric gp vertex compute count return count plan rigorously choose randomly edge fact never change unless double something furthermore whether whether proportional deviation complete symmetric eigenvector determinant lemma draw independently ij r g convention probability bad vector I u e mi mi j mi li bound multiple r u mi mn mr mi u h ji upper constant multiple choice bad expression link determinant rr p either bad determinant desire establishe prove order entry exist draw eigenvalue eigenvector large eigenvalue case eigenvector fix within furthermore standard mean distribution sum poisson c randomly select vertex j e give randomly member hold c om n algorithm classify r hold good within sufficiently select least vertex r h ij allow pick member community probability long initial vertex run I j h om member community e c r om vertex combine result reliable tn e agnostic comparison rational small classification half classification discard classification classification disagreement remain classification community claim correspond great overlap exist reciprocal rational probability tn run tn tn vertex improve assuming find satisfy large reciprocal graph success half classification give classification least classification classification less ensure classification community minimize vertex work correctly eigenvalue run om tn computing degree agreement classification find classification take whole run om tn proof desire positive stochastic block community matrix assignment call assignment recall exact community partition assign contain community recover q hellinger maximization far often vector instead e community recovery recovery solve precisely recall th p solve exact exact recovery failure rate need algorithm agnostic use step run agnostic sphere slowly corollary agnostic second making whether node another step solve problem profile sbm mis exponent agnostic splitting output community intend large subset subset agnostic sphere edge node likely belong profile computed classification size claim density degree profile compute preliminary belong notion repeat profile graph vertex community community relative independent ba enough regime particular le give rely side approximate draw sbm plant reveal profile vertex classify vertex resolve observation poisson hypothesis take value distribution mean goal minimize likely condition e posteriori resolve allow eliminate decoding give p time hence h p multivariate regime probability ix jx high profile divergence word find hellinger although exponent previous recover however infer recover narrow hypothesis composite poisson determine true disjoint subset hypothesis belong belong note disjoint wrong hypothesis realization test minima sum therefore control ji belong community belong group group error profile draw profile summary prove let subset subset degree correct q information j ii ax previous testing classify arise misclassifie let drawn community density corresponding size within expect vertex apparent change apparent follow let disjoint draw assigning subset profile wrong vertex graph less hold vertex eq let neighbor least chance community report community report degree profile profile conclusion follow possibility result converse know hence let recover partition behind technical step handle graph step agnostic vanish sbm parameter unknown hold time independent negligible impact classification error rate adjacent multiply rate plus plus pt plus recent block sbm linear three regime develop require community communities ii regime degree enhance agnostic overall regime agnostic parameter achieve limit quasi sbm study block slowly regime provide background detect community graph science apply variety network try community people social recommendation classify detect sub tumor decade understanding appear sbm sbm canonical community connect recently back center attention practical allow massive due phenomenon latter sbm communities community identical intra consistency regime community completely recover positively sharp recovery recovery achieve introduce sharp showing detection conjecture besides detection recovery property ask community vanish misclassifie two almost recovery generalize discuss conclude achieve threshold shannon capacity constraint development field g survey likewise identify threshold community guide reasonable regime fail particularly question whether sbm answer question work community sometimes exception provide sbm sbm sbm community recovery sbm concern regime connectivity symmetric separate solve exponential snr solve recovery symmetric equal sphere comparison community cp snr must exact recovery consequence previous sphere comparison almost entry sufficiently graph draw sbm sharp characterize divergence distance provide operational divergence analog channel solve show computational gap community characterize community definition exact recovery model partition ii partition recover exact recovery close sbm size community community logarithmic degree sdp agnostic estimate cycle count walk community namely aware private regime obtain sbm result allow rely major open without agnostic degree achieve node solve know community ensemble plant connected independently specify draw cluster label reveal observe focus community either grow relaxed motivated average degree fact phenomenon regime partial community agree agreement recovery recovery take element community community recover theoretically solves algorithm exact extract merge community merge vertex high connectivity therefore sbm entry nonzero sufficiently agnostic sphere graph community decrease refined word entry eigenvalue whose eigenvalue k recall information partition next show without know parameter recall partition ensure agnostic run theoretically proof rely fraction main procedure clean degree recovery already use technique difficulty know vertex short short path probably value require vertex attempt relate connect compute unfortunately whether cause differ get assign hence define vertex independently equality ie simplify terminology ie p ie dominate get term note expression estimate r e linearity determinant column determinant dominate pick several
slice method focus recover efficiently regularization follow tensor view computational explore involve rank tucker analyze heuristic tensor completion tensor extend version date sample resolve provide provably enjoy order contrast provable appear tensor symmetric factorization contrast employ programming base hierarchy even moderate sized numerically impractical semidefinite program grow rapidly guarantee recovery alternate relie underlie symmetric optimally dimension careful initialization method solve tensor nuclear regularizer norm guarantee optimally rank finally base tensor recently optimally aforementioned conceptually brief complexity third setting well key approach ccc tucker unfold tucker unfold rank decomposable tensor nuclear norm computationally intractable completion random projection separable rest introduce setup approach set also tensor upon projection tensor case validate section outline direction vector matrix case character e bold character two tensor define inner frobenius work third array mode refer keep third mode slice similarly mode tensor decomposition tensor r r generality tensor analogous result tensor tensor indeed challenge tensor r tensor mild assumption algorithm somewhat I r set vector precede measurement separable formally separable third operator q definition extend separability mode separability decompose action linear act mode application involve mechanism indeed sense study compress sense tensor argue separability desirable precisely recovery example outer slice slice typical separable form ensemble unit sphere suitably measurement case projection process play development compressive section separable share rank tensor completion entry reveal tensor separable depend nature mode slice n trivial extension slice slice separable completion tensor completion due contextual recommendation separable mechanism separable mechanism gain interest appearance retrieval rise similar spirit notion problem recover measurement interested recover spirit sketch slice recover natural tensor sense sense rank slice also think separable complexity exact aforementioned completion extend natural separable sense mechanism separable measurement measurement third recall weight vector require mode distinct separable weight vector introduce formal measurement refer vector potentially concatenation vector concatenation subsequent diverse suitably efficiently unknown tensor ingredient bridge inverse problem thereby allow contraction matrix slice tensor contraction primarily role recovery slice completion x v w verify enjoy decomposition contraction singular sense trivial particular form degeneracy e slice slice lose contraction situation extend terminology tensor tensor slice extend degeneracy trivially almost contraction suitably situation non degeneracy tensor choose suitable ensemble pick orthogonal vector concern w ix sample sphere case pairwise completion matrix r k I I abuse tensor slice see ensemble appear underlying degenerate generic suppose n matrix suppose degenerate eigenvector eigenvector determine ambiguity one arrange common eigenvalue entry decomposition next build algorithm inverse bound algorithm essentially turn tensor degenerate compute lemma turn finally obtain invert exact directly pose preserve tensor recover recover input degeneracy pairwise compute eigen denote eigenvector arrange solve w u last solve linear obtain whereby compute minor modal factor repeat perform eigenvector matrix modal properly align rearrange column sign observation drive separability separability word act measurement act principled tractable informally define nuclear succeed exactly recover recover along furthermore non pairwise generic exactly recover follow meta low measurement pairwise succeed exactly unknown complexity separable completion degeneracy naturally start describe main recovery assume separable advance eq efficient computational study problem matrix form input reconstruct pair normalize last linear input separable optimal eigen eigenvector sorted arrange respectively compute eigen let matrix sort common rank arrange simultaneously column q output tensor v move situation separable also provable complexity bound random I solve recovery measurement detailed thought measurement identical us slice establishing random matrix establish projection problem prove second identical manner lemma rank nuclear heuristic succeed recover high sub event event take bound event also constant recover complexity bind let describe succeed recover lemma tensor modal generic recover determine high simply take failure optimal tensor become unnecessary directly step solve factor nevertheless problem fast exist problem minimize tucker rank consider problem matrix far note sense seem compressed need store matrix store operator perhaps sense tensor require sense uniformly truly dependent recovery instance respect completion reveal slice slice precisely reveal slice require distinct slice say eq slice measurement mode precise key slice index slice without replacement context tensor completion slice contraction slice use input incoherence rp p canonical basis vector incoherence condition slice slice slice thin incoherence slice slice max slice mode slice slice condition slice condition literature instance point restrictive incoherence inequality away complexity multiplicative incoherence slice suitable ensemble bound wise decomposition direct contraction incoherent incoherent slice satisfy incoherence slice row space slice incoherence detail tensor give tensor slice definition slice k unique theorem correctly slice incoherence corollary sample complexity requirement recovery sub event occur change degenerate pairwise algorithm succeed recover decomposition probability allow slice I slice scale step nearly order comment projection necessarily slice result sample replacement uniform furthermore completion rely nuclear norm alternate minimization completion slice remove slice remove slice sample order straightforward way remain notation omit third order kn interested slice pick may slice tensor define mode deal contiguous slice collection mode element tensor refer notation contraction tensor matrix third case either coordinate analogue lx straightforward rearrange get eigenvalue along line notion pairwise degenerate ratio degeneracy tensor appropriate ensemble describe input tensor degenerate generic eigen decomposition column zero sort order common perform simultaneous obtain call recover r l tensor n k third assume separable separability norm via two mode algorithm tensor provide successfully recover nuclear non ht k eigen decomposition column sort let match result eigenvector determine linear recover k k k natural specific operator expression k randomly euclidean distribute identity b equality ii separable recover solve nuclear sub lemma solution suppose solution theorem concern outline succeed exactly recover rank omit sake complexity constitute complexity freedom e freedom whereas achieve sample e third tractable operation routine efficiently enable straightforward tensor order entrie along mode distinguished slice distinct slice slice reveal straightforward exercise argue obtained slice correspond separable recover slice optimization recover correspond slice incoherence degeneracy slice high slice uniformly sample unique solution recover compute factor recover exactly summarize km mode slice furthermore incoherent slice condition slice sample degeneracy pairwise succeed entire support conduct involve suitably tensor projection completion transition plot unfold transition plot propose sdp tensor even sized section run via tensor whose vary look recover tensor repeat less number tensor recovery convert varied tensor slice slice figure method recover trial far scalable unfold method compete along compare execution sized magnitude
step backward collection hyperplane b input precision subproblem essential correctness hand vector long horizon modulus occur lead precision tool algorithm convergence tolerance guide precision stop gap give solver condition feasibility feasibility could increase long hyperplane value bring error acceptable interior active solver often interior employ relative condition modulus hand solver terminate infeasible negligible appropriately brief tucker problem interior method barrier infeasible stop interior find condition system desire precision address concern allow difficulty tolerance convergence failure avoid manually adjust solver readily feasibility maintain throughout course interior result feasible computational resource size conduct optimize level storage device write storage drop price meaningful wide storage device conduct network storage device high focused water distinguish storage natural minute price price update storage device hour divide time quite algorithmic technology stochastic investigate effect regularization storage device determine post decision quality aggregated version grid transmission line produce transmission generator include generators generators nuclear generator wind wind discretization period run generator forecast wind wind make use storage however optimize generators device grid stochastic wind place device wind high demand energy efficiency storage device challenging factor device transmission daily variation demand take store generation avoid peak period balance store challenge environment source information wind wind wind historical wind month period consider wind weather regime information characterize stagewise former appropriate stagewise equally scenario assume stagewise period assume period weather percent nine regime visit percent anchor north simulation grid coordinate align date zero xlabel ylabel cells sim true col comma sim col comma cells data table cell sim col comma sim col comma sim col comma sim sep comma cell sim col sep comma cell sim implement solver quadratic optimization problem tolerance storage term examine device resource stagewise leave uncertainty show convergence bound especially consistently problem height title stagewise major xlabel ylabel value baseline south grid title xlabel iteration coordinate height stagewise xlabel ylabel coordinate coordinate major title uncertainty xlabel coordinate h anchor south height title stagewise xlabel iteration ylabel coordinate anchor height major title markov xlabel coordinate south title stagewise xlabel ylabel south width grid major title xlabel iteration table second storage device stagewise uncertainty iteration application offline example day wind cut time day policy c c l iteration c c large long numerous world energy finance demand often trade ever grow practitioner framework cut hyperplane quality development approach consider energy storage grid transmission operator unite fast counterpart great gain several would involve regularization possible path exploration involve time coherent additionally like obtain insight also interest pdf definition pt plus pt develop period stage hundred resource finite assumption storage transmission propose exhibit counterpart gain problem keyword nest scientific development finance create practitioner meet ever speed reliability problem issue significant gain important understand implement integrated work program satisfy horizon potentially hundred period stage realization stage dual receive considerable attention real framework widely energy account area popular approach medium planning scheduling al portfolio formulation method plan transmission market traditional planning technique show planning operation system wind introduction fast time optimize grid level storage subject weather planning demand stochastic dynamic asset end horizon account transaction model consistent employ popularity exhibit resource introduce markov surprisingly grow improve issue technique method however framework growth require propose reduce around distant cut hyperplane contribution label stochastic identifying exploit post decision regularization optimal sample regularization avoid growth scenario exist computationally tractable involve exhibit fast classical useful resource work relevant wide practical work use set device transmission realistic setting allow state day increment produce period linearly horizon paper review relevant problem formulation traditional programming dynamic overview dynamic main contain quadratic method describe algorithmic tuning issue behavior dimensionality resource state normally handle originally eventually approximation nest scenario stochastic dynamic practitioner upper guarantee many version despite curse essence cutting class exhibit slow convergence curse computational period cut plane n problem enhance improvement utilize chen point randomness dual identical realization multiplier small use approximate solution correspond realization explore cut evolve stochastically go beyond side warm tree cut prune cut et al hyperplane large furthermore dual programming adopt scenario strategy scenario selection successfully problem low proper stochastic develop version develop utilize tree entire history applicability period sigma algebra throughout convention measurable surprisingly standard solution programming realization nest partition enough computationally information history employ dynamic state sufficient decision cost transition represent pre resource please necessary formally evolution difficulty among equation place method solution result since approximation high address concern follow semi definite matrix resource substitute converge solution detail present study kt tr subproblem pass value original problem include suppose approximation let forward pass k basic solution pass every algorithm converge forward tr tr tr since dual basic similarly function post collect forward cardinality complete know subproblem backward pass therefore vector know iteration obtain without formally solution current solve suboptimal basic moreover hyperplane update pass r occurrence converse borel event draw realization pass one large number draw update perform finitely exist realization theorem every vanish coefficient however rely might properly relate way overcome impose assumption finite iteration assumption vanish therefore complete proof drawback stagewise generally problem often involve temporal approach adopt address case occur vector autoregressive stagewise autoregressive period additional accommodate realization model autoregressive advantage stagewise independence dimensionality history dependent convergence rate setup increase information rather resource discrete probability current realization stagewise autoregressive constitute realization historical weather indicate distinct explain around forecast properly weather dynamic different weather regime inherently multiple approximation optimization problem
break tie perform obvious edge ensure cut find step stop label label subroutine mid short short subroutine subroutine proceed mid point step compute repeat describe sub procedure label label min label propagation entire sake completeness run theoretical subroutine subroutine connect label return learn labeling conceptually think operating vertex find vertex opposite search phase pick short set apart end keep short among connect vertex cut pick vertex label shortest shortest connect thick short short path subsequently find edge short boundary finally query main reason firstly keeps presentation extra incorporate know criterion label stop achieve secondly recover boundary handle quantification challenge let label edge graph pose underlie making labeling partition collection component induce partition cut boundary sc illustration correspond colored cut vertex correspond interested clustered component cut might easier cluster cut argue complexity instance term boundary first natural induce label large conversely label query linearly label labeling vertex n see knowledge unlikely vertex might expect easy see key graph sp g path fy fy sp g cut imagine pair discover lie pair search path label edge quantity exhaustive drastically cut edge rise third near connect component correspond component cluster turning observe cluster path resp define cluster path dot prove binary induced cut remark g strictly well integer version quantity term phase component balanced situation ideal consideration assumption various line ignore error outli component allow readily generalize case labeling argue parametrization show graph query property near optimal discover direction show query optimal begin observation label vertex label path cut b phase observe know entire search cut imply set lemma graph subset random least proof combinatorial phase attention algorithm search recall remove edge shortest short connect short label phase short even step phase length short current short length drop thing get nearly step span step end step boundary might query phase end connect label vertex end new greedy cover connect discover sense discovered cut cut component per run cut component found never discover bind put characterize rate active characterize boundary assume curve analyze cover difficult apply flexible result practice gap adapt boundary minimax less good knowledge class binary box counting boundary generalize piecewise smoothness connect realistic detail classification predict cell intersect boundary box counting set follow bayes connected component hausdorff label generalize away zero away check distribution satisfy bf regular lattice center indicate ensure lattice also label active lattice partition request noisy correspond near optimality excess active learn minimax logarithmic significant nearly achieve classification problem achieve enough require everywhere use regularity complexity arrive excess allow estimate excess function h experiment red grid preliminary digit originally smoothing gray pixel separate task vs might simpler randomly choose digits nearest vice versa thus node undirecte unweighted nonetheless edge drastically vs vote voting record uci repository create graph retain boundary grid positive core square algorithms b need query completely guarantee query sensible experimental analogue query figure experiment clearly quite surprising perform mind believe try national foundation health ai grant grant ai prop access noiseless oracle request label vote bound union fact query voting statement label cut long small union integer effect l hand drop conclude recall feature cell oracle access intersect boundary however fail correctly vertex cell detail affect oracle query take majority noisy oracle sequel assume free label vertex lattice connect randomly vertex discover component observe connect cut know add query component number cut cut cut query cell fix cut hypercube contain furthermore vertex cut easy order edge cut component since complete repeat mistake denote excess observe everywhere intersect bayes probability number eq observe large depend c w derive main near trivially true cut fine restriction expense clarity induce cut component component notice low complexity cut cluster value result tell sake evenly odd evenly compare manuscript disjoint vertex copy clique show copy obtain subgraphs replacement way chosen pick edge path way subgraph total leave label determined cut follow hold cut number give result particularly relevant active grid bottom vertex count connect labeling notice query set size way component top right cut box contain cut contain cut furthermore cut cut locate contiguous box box contain cut box bind walk observe walk must box box symmetry walk end box valid walk go restrict walk right notice walk towards end observe walk terminate walk make entirely move count attention walk contain move move walk walk start box end suppose walk block walk move move therefore reason conclude walk one leave bottom walk start end correspond cut two edge valid cut walk r observe cut sum size conclude discover like walk observe tell query argument interesting black learning application nonparametric classification label label select structure label short pair need demonstrate data performance theoretical guarantee demonstrate implication theory show minimax excess important problem keyword find classification suppose vertex unknown
reflect transformation naturally unique polynomial surrogate account us department office office advanced award de address field uncertainty reduction traditionally use hyper parameter despite coordinate infer infer seek parameter use inference end dependence hyper expansion acceleration bayesian coordinate enable avoid expand explicitly solution uncertain hyper feasibility propose diffusion spatially log datum infer profile profile markov monte inverse arise many information computational view challenge inverse ill multiple continuously affect error infer challenge motivated ability posterior carlo hundred thousand method prohibitive large acceleration surrogate much forward mcmc step instead pc pc extensively setting flow model inverse involve number unknown computational challenge arise suffer curse dimensionality hard achieve expansion endow process surrogate evaluate parameter quantification incomplete bayesian inference calibration hyper propose effect scale hyper accounting variance attempt prior uncertain hyper work methodology expansion term parameter expand pc surrogate hyper hence estimate extension explore base eigen eigen eigen form hyper utilize basis kl hyper distinction pc uncertain expansion hyper apply transformation expansion expansion hyper avoid complex pc even development start provide account uncertain hyper change describe role pc acceleration section toy conclude statistical include model briefly formula prior relate model rule express additive assume become issue discretization need uncorrelated problem reformulate coordinate lead q p kl expansion generally improve hyper generalize bayes covariance hyper equation model kl sample multidimensional suitable carlo rely metropolis hasting flow chart evaluation sample dominant realization value fed solver model demand computation prediction particularly differential motivate substitution polynomial surrogate compare resolution complete surrogate offline subsequently likelihood approximate use construction detailed section introduce polynomial together domain equip inner product belong covariance function semi eigen equation kind value countable eigen continuous constitute orthonormal eigen value decrease retain stochastic uncorrelated meaning expansion mean square kl converge employ inverse infer directly infer structure motivate introduction parametrize family section gaussian completely covariance function aspect covariance stationarity easily confirm large parametrize symmetric definite know uncertain covariance interpret hyp great importance trying understand traditionally infer end set noisy location hyper kl expansion uncertainty expansion model address uncertainty develop enable infer hyper kl formulation generality center uncertain hyper become eigen scale eigen ensure function orientation eigen covariance representative deterministic random order eigen orthonormal continuity coefficient computational purpose generality allow translate expansion eigen variable specifically observe correlate cast shall invertible every determinant brief illustration error center hyper parameter correlation affect shape eigen assume hyper important note kl mode small small mode case decomposition numerically mode space leave plot right plot respective eigen use decay expect average uncertainty note l average right truncation approximate subsequent reference estimate realization equation corresponding realization observe average basis comment error spectra report decay precision mode suitable numerical eigen eigen accuracy indicate remain achievable double precision continue average yet range decay reference low eigen represent vary htb indicate error hyper plot reference error increase truncation involve behavior report increase precision cl far highlight robustness coordinate even prevent determination eigen yield error small exhibit denote subspace length speak eigen represent process range provide less select accelerate inference process pc numerical section exploit surrogate efficiently handle uncertain previously provide brief error input let formally express uncertain parametrize probability restrict second functional dependent complete expansion form eq expansion mode functional lead polynomial practical truncate truncated total term pc exponentially expansion number expansion denote series series converge determination pc distinguish realization pseudo method reduce computational equation level project pc procedure result couple efficiently couple operator reference return problem want pc prediction coordinate hyper notation previous expand dependence advantage discuss change mode map accurate varie reference diag construct approximate mapping field variable reference propose relate numerical avoid trivial note non trivial reference pc consider complexity simplicity change order gaussian independent gaussian case using continue uncertain condition simpler high consist equation deterministic boundary log uncertain function away ensure set addition setting classical element piecewise time investigate approximate r error incorporate approximation approximation truncation pc expansion discretization inherent counterpart provide section mesh discretization parameter ensure dominate effect process mesh deterministic diffusion project subspace kl translate approximation period counterpart depict covariance indicate pc truncation dominant low use reference scale minimum competition effect increase cause translate low marginal standard depict global kl mode increase pc show covariance previous see dominant reference appear superior choice monotonic l function present plot local combine approximate report plot pc truncation depend scale close achieve error ensure remain hyper local monotonically contrary local error strongly surprising expect pc order account local error minimize base independent standard pc aim hyper value induce transformation quality depend well insight measure plot eigen high thresholding see reference exponentially denote maximal reach impact clearly much moderate pc direction low insensitive decay contribution short scale filter coordinate robustness minimal yield maximum yield pick minimal quickly decay confirm covariance ccc indicate average solution stress first readily alternative consider technique polynomial instead proceed pc basis truncation involve certain focused uncertainty mode contrast uncertainty variance additional dimension propose amount uncertain numerical demonstrate variance average covariance remainder pc expansion expansion problem solution construct expansion prediction addition measurement direct discrepancy could advantageous illustrate interest field parametrize inference serve accelerate comparison purpose value consider inference covariance advantage field use diffusion correspond test field base profile cl inference perform measurement profile time uniformly solution measurement noise avoid diffusion equation solution observation significantly fine log profile unless depict location observation x solution observation times gp parameter choose inverse enough scale contrast diffusion problematic orthogonal gamma moment uninformative specify determination give instead chart require new finding reference average dominant mode solve obtain problem unless discretization involve approximate extract component constitute sampler pc determine surrogate transformation remain enough present multiply distribution acceleration distinguish offline adaptive posterior bayesian assign infer properly explore chain kl show kernel density kde coordinate posterior gaussian distribution quantify divergence quantifie indicate quantifie find kl plot posteriori map finding inference brevity ccc hyper covariance divergence kl plot quality infer median quantile posterior quantile large attribute pre hyper everywhere infer inference use gaussian function median posterior profile repeat previous hyper mcmc necessary pre chain hyper illustrate
laplace operator generator sampling section grant spectral operator regular basis orthogonal satisfy respect additionally need follow generator reflect brownian motion closure basis drift volatility satisfy strong regularity obtain weak namely bernstein fulfil applie fix correspondence denote ergodicity ergodic eq define projection introduce wavelet scalar since diffusion verify unbiased action basis gram v g ergodicity high argument prove j scalar schwarz inequality arithmetic vx vx easy eigenvalue function j invertible estimate laplace measure distance I structure show uniformly strictly invertible view formula plug diffusion need generalizing eq via continuous unstable boundary choose n view density ergodicity due estimating laplace neighborhood latter ingredient find control achieve magnitude generalize use posteriori bound g bound tend markov difficulty theorem eigenvalue conclude volatility ill pose drift estimator rate achieve convergence infimum estimator bound sample strategy observation alternative admit measure operator accomplish use schmidt explicit generator present numerical volatility estimation throughout mean drift volatility two use euler scheme compare four sampling shape observation symmetric beta rescale interval mean together distance depict scheme tp sample process construct oracle minimize present root square volatility interval obtain carlo surprisingly n exception decrease across allow small beta distribution error surprising beta yield uniform integrate estimation carlo volatility estimator problem see beta volatility trajectory four randomness observation ignore estimator design time throughout generator normalize increase generator uniformly proof gap equal due tv dt e tv tv tc ts similarly tc mc establish general integral follow variance nk kf l ks analogously cauchy inequality kf l ks easily geometric exceed upper equal cauchy cauchy series first assumption furthermore choose event inequality I I enough constant bernstein order solution eigenvalue q take suffice inequality adjoint definite u u consequently obtain claim since j l operator restriction operator hence generalize invertible generalize eigenfunction expect posteriori error control norm operator analyze norm rather refer finally prove argument invertible j v hence operator schmidt consequently chebyshev norm r eigenvalue nontrivial set hold j subset j restrict finish bind j event choose eigenvalue start continuous first estimation yield determine taylor intermediate denominator event cn supremum process relate yx envelope inequality exist proposition proof derivative exist let proposition especially high event v xu jx xu xu jx xu xx xu bm b triangle volatility since xu separate zero cauchy schwarz grant xu uniformly hence furthermore consequently drift obtain bind drift bernstein proposition obtain j error b yield easily since laplace transform parametric suffice use distribution lebesgue alternative compactly wavelet vanish jk k generators converge uniformly alternative therefore yield eq leibler uniformly leibl account times tx bx dy drift know volatility assumption ensure define obtain generalization assume nx nx lebesgue ensure dominate n lemma kullback leibler divergence r r l schmidt generator contrast generator laplace functional calculus map furthermore suppose decompose term take maximum estimate generator prove compact adjoint order eigenvalue eigenvalue eq project x variational eigenvalue v I consequently project operator gap projection eigenvalue cauchy z operator see z remain eigenvector x v want sketch posteriori technique extensively chapter error v problem symmetric generalize matrix q furthermore eigenvalue normalize nx I generalize order eigenvalue furthermore orthogonal definite purpose posteriori matrix cholesky normalize eq condition disadvantage relate method suppose provide exact intend compare theorem hermitian definite reader convenience n exact b theorem thm thm thm thm assumption theorem volatility coefficient scalar diffusion random construct ann estimation minimax sense adaptive sampling operator performance illustrate numerical secondary diffusion convergence continuous instance price scheme discrete distinguish remain argue realistic parametric naturally arise bad observation study nonparametric drift diffusion generalize reflect scalar one hand technical difficulty investigate economic reflect within reflect option individual affect act force f reflect brownian
use choice slice receiver slice summarize actor step observe infer likely year aggregate tensor cover aggregate tensor interaction omit receiver country actor activity receiver actor tensor usa china actors usa tensor million six million oppose exhibit show insensitive report error mae error mae nz hamming zero z correspond portion reconstruction incorrectly across display dispersion leave portion l consistently low mae low mae nz score unobserved portion l score simply many zero opposite observed complement predict portion low mae mae nz score magnitude suggest non portion consistent unstable section compare explain sparsity issue focus interpretability factor index date range tensor infer span date year infer component infer correspond decade anomalous tensor summarize relation take interpret cut tie correspondence five year difference date range illustrate correspondence also sparsity receiver depict ten receiver actor central possess activity factor component component receiver exhibit top actor dominate relation international exploratory tool explore localize anomalous interaction reflect activity span two year date range ever become zero alternate poisson away estimate latent factor practice solve instability mle efficiency empirically performance verify equation specifically expression arithmetic ik except point factor replace clear inference lee implicit correction suffer bayesian contribution geometric expectation define geometric construct variational arithmetic inference implicitly reconstruction expectation difference geometric arithmetic illustrative arithmetic mode parameter mode probability monotonically depict arithmetic low upper arithmetic grow yield much obtain arithmetic geometric approximately figure geometric probable arithmetic expectation close factor geometric use expectation r mae mae top top e political dyadic event inherently phenomena green et analysis dyadic biased independence continue dyadic instead event opposite viewpoint dyadic conduct analyse phenomena researcher begin dependency thereby effect analysis line research viewpoint latent factor exploratory datum specifically identify characterize international dyadic event exploratory reveal interpretable capture persistent anomaly analytically variational tensor instability issue negative tensor factorization count provide empirical demonstrating arithmetic recommend subsequent inference tensor david helpful discussion microsoft york city part part nsf finding conclusion recommendation reflect david tensor structure dynamic interaction decade political record country country know international develop bayesian poisson interpretable salient predictive performance tensor variational counterpart use bayesian exploratory tool political infer international behavioral dyadic international social process pairwise actor actor researcher facebook however explicitly time dynamic infer process task international decade collect record country country e traditionally help international group study less scale create several set automatically extract encode dyadic event news modern differ previously behavior document micro behavior day although potentially picture international effectively new latent relation event component span china aim international concern north top top beginning component infer span lead attack month occur attack paper poisson factorization infer dyadic scalable variational exploratory international relation party localize activity attack dyadic aggregate element count country toward country tensor salient latent tensor derive dyadic rarely interact another non interact traditional unstable count tensor tensor gamma avoid validate outperform highly algorithm traditional relationship explain construct latent researcher geometric use geometric increase infer factor use involve bayesian matrix bayesian tensor factorization exploratory tool political demonstrate infer interpretable international year researcher dyadic extract news database daily parallel started collect dyadic event forecast political instability publicly integrate comparison code dyadic piece type target actor code country information hierarchy action sentiment action study international code datum origin actor country actor recognize dyadic aggregated country aggregate event element country toward country date entire set step tensor tensor million element zero must dispersion decompose factor representation salient pattern tensor common tucker canonical cp generalization singular decomposition tucker decomposition way count tensor cp treat count know tensor aggregate ik kn four example vector receiver factor length infer perspective reconstruction poisson e perform via maximum count often yield estimate matrix draw assume poisson rather impose prior full bayesian inference pmf image music topic recommendation bayesian pmf bayesian pmf impose gamma thus mass zero maintain tail induce prior define impose four prior latent factor gamma mean throughout encourage sparsity interpretability factor invert posterior hyperparameter approximate typically facilitate product gamma ik pmf variational parameter fit exact kl eq factorize achieve coordinate ascent iteratively update hold parameter auxiliary pmf omit update arithmetic expectation since expectation g q sufficient efficiency efficient even optimize via empirical iteratively variational inference intend support arbitrary tensor addition tensor describe use complement portion slice percentage non element dispersion portion report type absolute error zero mae nz hamming loss achieve slice four portion kl r mae mae nz mae mae nz nz z top top top e validate
pr py source termination message form backward combine reader familiar bayes refer probability source solely forward global focus maximization translate example apply kkt initialize n f usually numerically discuss block forward message distribution architecture follow building latent show finite alphabet architecture code label alphabet come heterogeneous powerful whole increase bottom belong box upper connection bottom stochastic pointing assume source child architecture categorical alphabet represent independent visualize draw index nm mx px propagate collect usual flow simultaneously follow rule incoming combine produce imagine act combine connect branch handle architecture backward message come towards branch message latent feed forward build generally collection spatially distribute architecture red blue architecture represent layer bottom patch patch latent message among subset build across scale system inference generative latent message forward cone reveal associate cluster image layer code role propagation delta root reveal specific system impulse response encode bottom variable backward delta diameter hide contribution message soft code pattern completion message patch patch patch example patch termination source iterative outline deep deep network large mode check check generalization specifically follow architecture pixel randomly select patch layer build connect variable another pixel bottom layer backward learn b block connect block pixel backward message learn block cover take car filter diffusion filter filter obtain car alphabet filter patch extract show fig previous use matrix learn mode generative distribution delta gray pixel symbol complete respectively forward quite forward represent orientation early visual distribution large scale pattern patch extract bottom considerable amount patch delta backward resolve quite uncertainty experiment pattern pattern never obviously succeed quite complete learn generalization easily determine space result paradigm successfully deep architecture retain contain internal layer layer choose extract paradigm salient structure object character different constitute proposal deep literature flexibility modularity network introduce type scale add let take care adaptation mix supervise unsupervised architecture complexity issue paradigm embed
version inequality variable could matrix choose distribution computing surely handle n rp z follow complete product subgaussian tail lemma go bernstein inequality prove bernstein concentration prove jx xx z ok therefore complete use ok ok ok claim bernstein state technical lemma control variance follow z ax si repeatedly incoherent recall subgaussian fix complete next ok sx ia ta si term separately column column si ok ok expectation ok ok ok ok ta ok ok ok complete bernstein ok ok nz claim truncate log bernstein spectral norm x xy ty xy bind calculation bind term eq matrix bernstein complete order bernstein form claim subgaussian hold tu entry subgaussian yy v yy apply bound notice yy yy ok variance apply bernstein number field implement field figure simple layer neuron output decode respect middle layer weight middle moreover top layer equip layer bottom outside remark update attention accomplish spike update weight execution new bottom layer neighbor decode layer repeat implement appropriately sketch implement neuron network perform product neuron even receive sequentially generalization principle top singular also inner onto required life accomplish stochastic fs include basis enable optimization minimization work result provable somewhat outperform heuristic give understanding minimization provable architecture motivation introduce algorithm work limit incoherent dictionary previous parameter improve upon application setting function think minimize allow leverage variant common methodology field enable datum convex heuristic alternate provable somewhat surprisingly outperform alternate minimization heuristic alternate design provable seem architecture field give code almost limit finally believe framework code patch number many wide turn appropriately representation improve efficiency brain sparsity constraint fitting tool feature important segmentation retrieval super building involve notion formalize dataset vector matrix whose encourage subsequent choose large overcomplete adapt give coding call recovery gave try give evidence matrix image portion author place familiar setting whereby datum assume appropriate lead surprisingly seem unnecessary practice energy hard constraint work mod svd fact method rapid progress design polynomial code provable guarantee paper discuss generative viewpoint success largely far new simplex ellipsoid behavior successful new time code simple intuitive important beyond efficiency architecture roughly weight entry potential also general analyze algorithm relative field see rigorous bound sparse case dramatically recent add literature heuristic solve approach representation alternate solve problem update appropriately rich heuristic neuron remarkable e plausible basis optimization problem many heuristic offer computational explanation possible rigorous sparse broadly computation instead rely solely code give solution give column work overcomplete give overcomplete incoherent next former work give depend alternate minimization close give square sparsity run time however exponential give error decrease geometrically work sparsity remark empirically alternate minimization appear much generative similar rigorous paper polynomial assume generate dictionary normalize pairwise subgaussian support noise motivation representation requirement expense restrict sparsity iid little like singular vector long stay incoherent include wavelet whose column incoherent relax permutation sign distance allow high sparsity framework instead try update rule provide good gradient heuristic plausible converge geometric complexity step implement algorithm sparse provable general technique use alternate rule whose initialize near error polynomial modification carefully project component along complement analyze variant however analysis initialize algorithm near high setting help return sample complexity algorithm admit implementation appendix currently logarithmic incoherent algorithm exponential implement operation framework sparse code simple context suggest new way analyze algorithm update opposite appropriately challenge identify progress lyapunov improve probability progress difference subsection identify inspire try convex strictly analysis turn propose though movement decode procedure functional specifically true gradient really movement flexibility different need quite either compare bit next bit please movement setup move expectation contrast expectation true gradient bias bias algorithm approximate error connect relate sa abstraction general look desire essentially sake relate correlate fairly flexibility decode turn variant take useful et em require gradient close plug allow decode step step check movement conceptually solution step guess natural sufficient correlate convex correspond strongly geometrically systematic optimization solve recurrence z second part theorem constrain iteration euclidean z far lie update systematic derive functional various rule move approximately direction certain course know update direction unknown nevertheless whereby converge update rule get optimum enable analyze sign rule approximate project descent error project along column currently go get early rule analyze denote decode main give useful expression generative order correlate lemma across take simplified essential suppose iteration geometrically defer show ng I correlate k ok mn proof omit notation assumption pointing direction extra triangle g I corollary apply I I ok mn lemma last notational I note close distance term right get simple I complete near previous crucial rearrange g q write convenient matrix side first straightforward use term introduce auxiliary matrix I claim iv moreover frobenius spectral norm ok ok last inequality put piece complete induction induction hypothesis inductive say inductive invoke prove large norm induction theory term initialize usual analyse give novel succeed work support vector correspond recent throughout execution suppose moreover incoherent k define theorem main invoke time order support share element show sample let q norm expectation v v sx ta ta j independent expectation whose rearrange next useful set recall incoherent subgaussian diagonal invoke eq q term q complete inequality invoke ok one symmetry collect frobenius om claim main conclusion go beyond idea minimization break alternate minimization random heuristic analyse minimization method correspond incoherent dictionary incoherent I ci ic concentration intuitively strong randomness independent subgaussian disk therefore subgaussian union complete even long correct randomness strong randomness elementary wise I I j choice j ok nr kk however expect hold bound proof mx jj r high subgaussian variable ci complete really require expectation maintain estimate proof repeat maintain give analogue suppose remark lemma although condition indicator invoke event happen let event happen omit eq strategy next linear lead contribute term ia mn bound gm norm term complete proof ready except invoke notice project constraint projection subgradient method nesterov seminal book detail update rule preserve property summarize follow claim second part complete fact increase analyze particular converge globally project ensure correlate frobenius norm theorem suppose choose proof substitute recall write q recall contribution early rewrite lemma imply near third project onto rule maintain avoid intensive difference repeat calculation sketch denote also hence term term verify remain note
gaussian place universal place fix say admissible moreover notice fix mixture admissible remark somewhat arbitrary choosing obtain fraction minimal fix recall statement obvious half also place half interval intersect split contribution first term interval since let p b bounded follow xx interval rhs intuitively close unbounded far since gmm concern interact need suitably parametrization linearly transform domain gets map fix write rescaled variance denote give say rescale property rescale follow variable know furthermore set follow suffice return algorithm inequality find back polynomial line know negative component valid k rescale round weight simplex check p lemma argument complete proof show univariate parametrize set fact zero algebraic plausible candidate satisfie attempt believe natural interesting proper agnostic mixture algorithm linear brief give agnostic mixture laplace addition mixture gaussian run class mention arguably difficult mixture condition parametrization laplace trivially care demonstrate parameter however issue introduce polynomial program degree taylor expansion laplace mixture thus complexity nearly constant schmidt gmm fundamental theory learn gaussian norm give univariate k ok logarithmic achieve good k replace moderately dominate run achieve recently state researcher progress approach offer agnostic gmm gaussian agnostic closely agnostic recently question proper fit solve entirely deterministic carefully polynomial encoding inequality apply learn class besides popular practice sample gmm encounter normal study seminal include biology gmm sample rigorous notion gmm community year relate algorithmic decrease hardness gmm mean variance mix weight variance proper equivalently total variation hypothesis unknown small htb ccccc learn cm ok estimation low bound arguably guarantee recover instance physical wish cost recent parameter show necessary mixture univariate tight give scale exponentially univariate quickly prohibitive mixture reasonable improper small work possible gmm tight logarithmic factor produce gmm disadvantage interpretability attractive gmm unknown somewhat proper many gmm require produce accuracy estimate increase gmm allow expression quantity learn expression density return produce gmm output peak smooth make interpret identifiability gmm ideally proper learn interpretability density recent factor complexity understand estimation nearly run become scale much density resemble exponential bind show avoid nearly run time nearly worth proper moreover arbitrary explain properly parameter offer generate assume gmm produce give sample truly gmm gmm rarely think phenomenon distribution far gaussians set pdfs achieve incur onto gmm universal pdf agnostic agnostic approximation hence agnostic produce hand agnostic gaussian agnostic open question see progress increase agnostic outline contribution paper restrict attention many understand main notation pdf let run optimize front instead learn gaussian closely make question imply time special simplifie bind moreover complexity density factor compare algorithm run moment algorithm offer agnostic guarantee hope worth agnostic significantly agnostic hence understand tractable agnostic guarantee gaussian mild offer particular applie appear polynomially polynomially convert purely learn algorithm mixture distribution learn available differ essentially proper gaussian core mixture invoke complexity piece sample obtain density fit gaussian entirely deterministic fitting reduction gmm solve carefully design polynomial solve system reduction polynomial inequality main technical directly fit challenging pdf gaussian gaussians e piecewise consist piece restrict polynomial must pdfs inequality easy convert restrict back proper polynomial good polynomial encode guarantee proper approximation require intersection piecewise polynomial integral difference instead minimize minimize vc theory exactly intersection norm norm norm polynomial inequality norm would exponential polynomial polynomial inequality give gmm gmm doubly work real ram different specify sufficient rescaling parametrization inequality length interval piecewise fit near serve simple identifying combine idea outline mixture gaussian carefully design crucial inequality lead polynomial inequality impossible summarize body work attention provable corresponding notion outline subsection picture current reference therein seminal work start long community g reader give tight parameter mixture univariate gaussians necessary price give complexity offer weak many attractive become tractable dimension smoothed knowledge take mixture unfortunately polynomial dependence underlie note necessary complexity component paper properly candidate construct hypothesis none conceptually gaussians component gmm component return moreover improve worse hypothesis polynomial advantage section utilize assumption introduce shape distance proxy adaptively restrict polynomial properly inequality approximation denote fx x measurable one paper pdf precision pdf iw iw k component k kk result work algorithm density estimation subroutine gmm estimate carefully system polynomial formally inequality either boolean formula predicate relation predicate inequality satisfie result let find solution case run gaussian pdfs estimate directly polynomial piecewise order piece flat tail piece taylor around expansion always exist shape polynomial inequality restrict q triangle encode estimate distance density denote family interval kf q property gaussian zero proposition gaussian pdfs distinct fact corollary variance distinct imply claim gaussian behave use polynomial live scale solve polynomial inequality work density obtain suffice solve follow use restrict proxy part polynomial norm approach per component weight precision quantification quantify boolean polynomial finally solve behave know polynomially good approximation parameter yield first run subroutine occur system precision could extremely approximate take arbitrary shift scale shift since solve apply rescale estimate interval note mixture large accurately clarity presentation ignore control appropriately point show overcome limitation formally behave estimate rescale support behave gmm behave parameter learn detail behave case illustrate difficulty proxy shape require zero polynomial zero auxiliary simply introduce run since polynomial lead become goal avoid dependence system inequality mixture gaussian contribute significantly q estimation succeed gaussian triangle connection norm attention computational perspective polynomial regardless encode convert solve feasibility encode norm section general particular wish optimize polynomially adapt algorithm well section setup let polynomial piecewise algebraic moreover assume membership formula predicate degree moreover satisfying polynomial maximum polynomial polynomial indeed suitable piecewise inequality hence suffice set constraint combine constraint piece encode know become polynomial encode integral piece set ii appear iv boolean constraint encode set permutation represent piecewise piece integrate inequality encode integrate expression trivially integrate integral individual piece valid depend fix moreover check tool encode constraint interval satisfie follow simply count addition optimize boolean predicate require encode quantification variable variable precision system system
yield grained experimentally show explanation augmentation extension architecture yield competitive mnist fraction unit deep turn input response linearity dropout multiply mask probability half unit input drop magnitude active hide formalize work help set learn function input need datum mean training region learn mapping cover portion help boost coverage make noise regularization increase hide view hide less clear globally transformation possible activity transform project noise linear input dimensional noisy bernoulli adversarial square equation generalize define generalize unlikely find minimize detailed subsection fortunately hide layer representation th layer sophisticated augmentation respect raise question answer trivial inside network without noise make hypothesis dropout adaptation projection differ always version layer large reasonable dropout project noise back view argument experimental hide section dropout layer representation roughly corruption present manifold span entire manifold along linear manifold non let represent dimensional map representation representation could overlap small case representation overlap insufficient representation least overlap representation overlap view data arc belong arcs arcs class separate arcs class create arc length arc arcs class total train single contain randomly run representation presentation number training mainly arc single layer mlp arc dropout result corruption view create space sample work corruption report noise anneal help supervise assume imply noise layer rich augmentation schedule propose alternative avoid schedule rather slowly change corruption sample training sample uniform show unit actual test non linearity function half run relu unit train noise give mnist cifar experiment dropout give mnist cifar relu latter seem mnist cifar inference turn binary result mnist cifar rely magnitude consistent hypothesis sparse sub region sub region run experiment mlp architecture mnist cifar study effect scheme mlp relu three mlp architecture hide mnist split epoch training mean percentage run permutation consist pca whitening preprocesse reporting evaluate fix noise noise increment experiment hide increment dropout noise scheme layer level increment experiment cifar dropout dropout input dropout train cifar mnist layer zero noise corruption plot layer scheme sparsity representation sparsity limited experiment corruption unit zero deviation varied level effect back input second input dropout back generate approximation solely epoch epoch generate noisy input hide bernoulli mask cifar dropout input suggest project neural generate drop cifar idea dropout augmentation suggest network plot layer dropout corruption cifar representation term class lot new work suggest augmentation give evidence view require understand augmentation evolve architecture augmentation space rich augmentation test work support education project research award grateful towards find property composition rewrite exist expand compose apply modification p ip percentage probability hyper layer sparsity start fall cifar probably cifar dropout
kl heavily auc semantic smoothing wasserstein favor auc achievable deal value combine loss two test illustrate relevant despite overlap irrelevant error example music car band water water water wasserstein wasserstein costly describe regularization connection index wasserstein encourage respect output tag baseline interesting work may wasserstein encourage equation relax objective w diagonal eq risk minimizer bind bound wasserstein depend empirical rademacher value call random control risk stability constant empirical let lem hold probability jensen role conclusion finish est kk space wasserstein lipschitz lemma wasserstein define probability valid around function softmax softmax proposition wasserstein distance wasserstein follow continue gradient theorem immediately conclusion follow trivially zero uniform softmax lipschitz singleton identity sample calculate theorem thm lem vector specifically mean th column zero equal h feasible plan wasserstein prediction proposition case become diagonal assign arbitrarily long meet q scale wasserstein define wasserstein distribution overlap mass mass cost q conversely give without generality follow alphabet integer pixel metric plan euclidean ground feasible direct support match respectively denote q notice obviously assume lem derive relation upper inequality prop fall illustrate multiclass label correspond lattice semantic similarity flip neighboring category figure level repeat multiclass classifier standard wasserstein average performance function tag yahoo filter well dominant lexical noun location noun proper vocabulary remainder occur select tag frequently occur tag art nature family tree water architecture car live cat sign road build new cloud tag flat master drift wasserstein loss wasserstein loss use enforce mini momentum optimize function redundancy semantic tag tag tag tag image pick run country build road house family education weather people b center machines institute brain institute mit international e predict output challenge output improve wasserstein distance optimize wasserstein costly efficiently compute efficient regularization wasserstein unnormalize statistical connection index wasserstein loss encourage prediction output tag yahoo dataset achieve superior baseline eliminate comment problem use scenario multiclass class imagenet challenge acoustic speech segmentation consist model metric semantic adjacency capture euclidean distance pixel classification task organize hierarchical label space also call presence metric measure severe intuitively incorporate metric favor completely r multi learning incorporate measure cost plan move match apply include propagation represent use performance multiclass indistinguishable category human label make class plane switch neighboring loss wasserstein loss prediction euclidean incorporate wasserstein ground truth noise section describe experiment cm contribution learn wasserstein divergence loss wasserstein loss base justify erm wasserstein draw connection synthetic real world annotation demonstrate incorporate decomposable popular value lead algorithm nonlinear post distribution distribution wasserstein simplex graph minimize dirichlet wasserstein formulate compare retrieval contour problem discriminative estimator cost space measure set shift recommend choose tangent normalized segmentation example shape represent truth simplex generalize unnormalized propose unnormalized measure penalty divergence result z division optimize satisfie hx u represent element multiplication unconstraine gradient hx smoothed enough smoothed normalize output relaxed wasserstein nearly b unnormalized learn wasserstein two full let risk q composition function softmax lie simplex probability constant commonly decay train achievable important minimize risk wasserstein multiclass multiclass risk note semantic prediction point capture wasserstein probability wasserstein intersection region metric space wasserstein plane ground wasserstein grind w pd incorrectly far
unbiased unbiased selective state selective selective hypothesis away selective scenario root simple hold stage root detail turn main inferential also law selective likelihood lead root inferential tool multiscale derive lasso shorthand select kkt condition shorthand simply subgradient note square root square root usual rewrite kkt calculation kkt write strictly speak calculation define quantity event question computing yield take closely usual imply one test selection explicitly design deriving eq easily computable inactive event inequality recall question interested q provide simplification show independent though use formally every dropping law fix specifically convenient convex otherwise goal q without like test nan q relevant law freedom computable truncation law truncate precise choose convenient parameterization inequality inequality solve explicitly one intersection convenience happen event event restriction residual fit inequality define event model observe inside usual p law score selective orthogonal consider separately pseudo setting rather estimate plug quantity law statistic statistic estimate consider perhaps selective direct inspection equivalent truncate law truncate root truncate pl estimate vary limit likelihood ol obtain estimator root call pseudo likelihood degree freedom estimate think improper density proportional recover estimator event lasso compare literature validation generate absolute sign lasso root discover ratio selective variable screen close usual ols root include comparison pseudo variable coefficient result ht coverage consider example various anti point ht mutation naive selective v p e drug marker predict quantitative square tc square generally impossible multiscale problem piecewise prior get noise level formalize problem part assume discretization recover inference balance goodness impose penalization motivation solve scan statistic get active knowledge level analogous subgradient selective l repeat q connect multiscale work author multiscale column easily highly screen kkt kkt repeat residual get solve residual copy genetic variation human genome dna link development disease provide useful simulation gm ratio ratio well selective splitting produce example demonstrate splitting dominate procedure second datum even yield fact gaussian formal justification consider square root description interesting partition group lasso sign identity select two affine affine stack partition set course setup law result reduce inference parameter projection subject affine conditional notation jx jx j drop set distinguish two either function law linear functions eq law active drop inactive event interested row law ti es ti es constraint possible insufficient otherwise estimate order must natural projection sphere ti e affine truncate scheme gaussian stage rather pool unbiased ht interval short interval use inference stage mutation datum l p p p hold tc table selective scheme tc name proposition inference level selection square lasso selective selective test inclusion select root level perform well root hold datum inference observe drug selective consider model natural canonical continue fair recent work assume though crucial selection procedure value focus suggest angle could carry unknown approach distinction tuning post equivalent square root modification eq lasso know lasso hypothesis equal parameter roughly view would enter lasso square root convex differently literature analogous use convex objective view procedure model square sign notation investigate kkt tucker basic selective describe broad post notably canonical begin specify formalize determine ask selective attempt selective critical vs selective error determine take distribution
connection trivial integer real usual lemma concept dct consider average impose dct concept previously describe formulae dct describe must generality signal nan assumption affect dct spectrum inverse admit derivation arithmetic dct average dct tailor average next specific derivation mathematical average exposition order average th act average algebraic one thus perform suitable recognize always consider impose act issue precisely find inverse say separate useful notice lead unitary sequence situation address inverse analytically dirichlet eq digital multiplying number bit possess regardless expression direct inversion tell unitary alternate unitary constitute simple dirichlet however formulation nevertheless indeed definition establishes connection formulae follow probabilistic integer zero assume signal value convert nan subtract consequently separately address problem fractional sample interpolation way let integer dct formulae construction take direct formula invert act weight sample fractional combination uniformly fact stem orthogonality transformation kernel investigate suggested function inherently act usual derivation summation expand follow run obtain return double block act dct approximation argument illustrate act example short dct widely adopt code subject extensive act identification interpolation fractional depict diagram act interpolation procedure fractional employ act average block act averages implement act combine function cf approximate dct spectra randomly signal distribute uniform result mse due figure comparable approximation dct cosine transform construction matrix v matrix accord nk matrix need construction matrix thus singular determinant unity inverse inversion element stems additionally express spectrum act spectrum dct due combination ii function term dct dct spectrum decompose part dct extend interpolation act thus invert inner weighting depend independent separate define weighting average eq row weight proposition indicate across application part q effectively relate function nan mean always dct signal establish dct consider algorithm nan signal contribution arithmetic transform ii arithmetic interpolation issue introduce arithmetic cosine devote dct computation therefore application dct differently formula framework arithmetic adequate interpolation exact dct length particularly exist error tend consider evaluation issue could adequate solution act order collect fashion act need sort dct concentrate computational perspective transform filter motivation arithmetic cosine paradigm existence act fundamental building arithmetic transform direct could attractive exact interpolation signal open topic author provide scientific partially dirichlet inverse examine dirichlet series result q find put product dirichlet accordingly final already format simply return write denote convolution conclude odd q admit form integer multiplicative multiplicative sequence maintain q dirichlet convolution possibly conjecture example cr cr tag de work electrical ab mail act act design efficient implementation new mathematical tool demonstrate interpolation provide application transform arithmetic fourier tool spectrum main nature part additional improvement expand dft fully arithmetic lack impose sample sensitive incoming sample necessary essentially two signal hand drawback sampling discard option sample method obtain arithmetic way order interpolation consider although interpolation attain acceptable block length consider length imply enough totally meaningful way spectral component domain testing build hardware consideration direct enhance dct describe dct obtain mean dft spectrum map dct spectrum goal dct cosine act method examine arithmetic average exist arithmetic fourier transform average tailor dct act exact
summation know growth know pick decentralize bipartite illustrate similar bipartite n exploitation il il player event il il event hoeffde bind combine unknown similar omit scenario chain arm irreducible represent I stationary markov since ergodic arm x max xx x max I ip irreducible reversible let gap obtain jt denote contain j j ji mutually player reward player successful play without player frame time player pick max xx jx j x x ts prove reward lemma occur tx jt jt jt jt n bipartite representation decentralize markovian ii de choose slowly increase arbitrarily decrease regret exploitation clearly similar equation ct il il theorem problem bandit bandit pick arm distribution design policy maximize reward player separate communication costly yield expect grow low centralized case algorithm work fundamentally address question incur decentralize solve relevance domain decentralize system cognitive system model understand choose bias repeatedly instant instance coin reward know bias coin help discover coin bias exploration versus exploitation must well option current widely planning sharing etc say formally policy could employ show grow slow subsequently generalize play pick multiple bound reward seminal simple policy logarithmic policy see interest well bind deterministic sequencing phase also appear approach like probability exploration policy bandit general finitely arm policy single player pac bandit grow bandit motivate network wireless spectrum try channel channel look user statistic wherein maximize channel dedicated channel expense imagine setting wherein user arm rise question decentralize inherent second index regret bandit decentralize regret logarithmic achievable reduce ranking solve decentralize appear setting make become decentralized quick propose mechanism player answer two question decentralize insight exact decentralized policy regret however partially policy stand spaced exploitation policy regret grow near assumption factor exploration exploitation pre long exploration phase policy answer fundamental question inherent cost cost policy introduce policy e chain reader extension markovian set simulation conduct evaluate compare decentralize previously know formulation prior work new study performance player instance much appear present armed instant identically process support distribution unknown choose history reward action arm objective choose reward mean solve play reward notion compare policy cumulative obtain formally player minimize policy eq arm take great bandit sensor useful computation unit index computed refinement compute arm armed bandit refer channel dedicated player however play arm signal signal regret hence communication choose player pick regard player player unknown bound playing time player want horizon know bipartite unique player computational cost communication cost distribute match exchange incur occurs expect minimize q define feature exist single single player bandit capture class policy al seminal index regret work play large mean arm play exploration exploitation incur study context quite arm gets play become unlike omit dependent numerically empirically outperform overcome frequently difficulty extend regret regret computation q worth sub actually logarithmic growth major encounter bandit policy channel player natural expect q frame length difference difference reward precision matching slight unknown policy player bandit generalization player exploration exploitation epoch try player choose maximum index policy exploitation player epoch initialization play arm compute lt l phase play arm play arm perform else largely arm exploitation value concentration reader hoeffding inequality range well bad e make arbitrarily extend bandit effort section generalization single exploitation communication exploration phase explore arm round phase represent turn end exploration phase player protocol aside bit contribute regret bipartite match line yield k k player process early bipartite stick distribute phase phase successive initialization exploration player play arm match player initialization set exploration arm number play reward trial success else obtain play player price prefer player player price spend consider term end add cost precision index bipartite matching precision run round precision two index close happen player bit specify cost communication communication constant phase phase player preference bid index player receive arrive channel optimal exploitation denote context bipartite index know choose de de ts note choose sequence thus slowly increase arbitrarily slowly sequence make close near policy cubic extensive performance propose policy respective scenario simulated mean ranking algorithm player policy compare known consequently logarithmic performance slightly account scenario see next computation retain logarithmic grow linearly present policy generate independently horizon million tolerance performance average unit distribute bipartite
relationship function copula hx continuous uniquely conversely copula joint margin continuous margin function copula generate transformation extend representation joint induction generator approximate joint start f x x conditional x px result formulate joint let convergence x unit hypercube theorem tensor n equivalent give belong copula joint minimal euclidean respect copula density function cx x j relation instantaneous process represent htbp approximate present result sde secondly unitary lastly martingale continuous dynamic close corresponding purpose n fx infinity converge function pp detail norm approximated would imply approximated assumption sequence chain generator function statement process converge generator let follow uniformly chapter path use generator follow generator convergence discretization unit locally write taylor obtain ax x x ax ax ax ax ax ax uniform ax x I ax ax ax precise see process infinity n fs n x generality theorem distribution consider integral sufficiently small second integral h like limit get eq way order approximate marginal term calculation n cn equivalent cn cn cn h h mix derivative h exactly term result diffusion belong martingale problem assumption smoothness sde extension martingale central martingale existence sde refer property sde per db continuous f martingale pose sample matrix definite ns ns ns r r local martingale martingale particular valid x relatively stop convergent subsequence x kt stop process bx lemma generator well uniqueness stop hence imply dependence attribute describe unique multidimensional interpret functional specification involve generalized process develop demonstrate obtain multidimensional start multidimensional decomposition define general generator multidimensional generalized demonstrate copula provide equation convergence multidimensional sense microsoft property max gray light gray department university college martingale decomposition generator multidimensional diffusion sde symmetric nonnegative matrix value sde denote c smooth compact aim address propose weak generator emphasis derivative decomposition approximation scheme generator conditional generator copula propose develop characterization furthermore support constitute little look multidimensional investigation focus aspect decomposition martingale continuous chain approximate dependence structure throughout volatility form joint coupling structure drive framework specification motivated deal representation process markov room far chain fact always treat algebra model multi whose construct generator characterize construct among apply discrete generalized diffusion approach associate multidimensional markov process approach theory diffusion decomposition literature manuscript multi process construct instant drift process interpret terminal extend functional take functional equation markovian framework process conditional diffusion coefficient project manuscript generators multidimensional generalize define copula specification paper organize introduce correlate characterize generator convergence approximation place zero measurable countable probability sx ty denote operator axiom relate law start assume contraction generator continuous property entirely fact solution compactly belong generator equation sde bs cs dynamic law drift operator specify solution rigorous formulation martingale straightforward sde martingale martingale operator representation uniqueness result define mention develop sde adapt condition growth theoretical construct specification process sde approximation involve characterization cross tensor generator algebra make correlated generator emphasis mix decomposition generator first illustrate sde approximation block approximate construct element n discretization boundary consist possibly complement construct building equation ax ax ax ax ax ax bx uniform alternative discretization generator markov process particular approximated generator process furthermore process discrete impose condition state law previous instantaneous moment coincide x z instantaneous local moment introduce approximated generator notation process generator multidimensional process matrix instantaneous moment describe generator markov jt jt two jt jt x rewrite independent markov operator dependence way namely dimension operator action state space approximate orthogonal introduction x k process approximate generator markov act local x cn cn mn z pn pn pn n furthermore differently conditional n x z correlation finite partial x j f note apply decompose j x observe operator act joint product discretize discretized act along difference pn intensity x l l mn pn pn pn generator bivariate correlate act hilbert generality cn z mn z pn z let generator generalize assume locally jt generator operator dimensional consideration therefore let generator whose matching x ax x ax ax ax ax jx generator instantaneous intensity report instantaneous calculate states probability rewrite py j py x px I nx jx entry identical impose large two straightforward independence multidimensional
rademacher complexity borel lemma university ann microsoft research microsoft research bb v prove minimization unique unlike first sequence predictor risk source hold minimization technical free space np even approximate computationally attractive utilize amongst predictor limit error effectiveness specifically address predictor minimize sequence converge concrete object minimum rather minimize high dimensional infinity straightforward predictor minimize draw say unclear highlight thus boundedness question minimization main theorem function set linear consist vector function countable formally instance arbitrary collect differentiable loss defer study loss exponential population size lastly define excess risk probability logistic sigmoid early limit probability hypothesis sequence existence existence sequence compact metric construct duality duality carry consequence utilize approach handle secondly conditional probability convergence property property resolution eq satisfie natural apply introduce exhibit make essential sequence encounter strictly generalization particular coordinate logistic put rectangular red region define separate negative convex attain norm grow lead sequence regression give unique resolution point analog uniform apply margin dependence achieve sequence behavior concrete g gx rr might relationship exclude desire occur true along metric choice equality proposition I risk classification risk example generalization classification error problematic behave provide consistency function see dense complementary small sided close collect symbol precede subsection continue view whereby drop integration subset symbol ambiguity sometimes risk excess risk paper meaning satisfie class denote continuously differentiable restrictive classification loss existence conditional challenge hypothesis space begin study hypothesis finite uniform problem exponential recall infimum exist strictly separate example achieve risk study minimizer convex minimization linear entropy loss different kind unnormalize dual unnormalized unnormalized entropy uncorrelated make reweighte note unnormalized entropy slope zero always unlike primal minimum attain differentiable optimality rewrite define q absence label q example value infinite space infinite construct finite minimize dual unnormalized return formally give rise negativity objective question intuition construct qx qx qx large absolutely include slightly objective finite one candidate space banach measurable allow function work space call detail construction begin convexity serve role th define ball ball q outside measurable choice ready introduce taylor space conjugate contain finite iii constraint via adjoint transpose adjoint map constraint require write fact constraint apart result dual well optimum look technical primal optimum exist use qx qx yx qx nan obtain follow differentiable optimum optimal appear prove slope loss informally dual avoid unless force fundamentally see call difficult build addition alternate positive negative predictor return predictor equal predictor margin receive always still weight prediction wrong origin risk along case orthogonality slightly perturb risk along pattern minimizer exist example minimize perfect achieve call separate non nan measure receive margin difficult hypothesis optimum call risk prove highlight implication difficult reason hypothesis optimum arbitrary measurable difficult optimal dual optimum attain maxima moreover dual optimum difficult main restrictive class twice classification loss lipschitz every finite recall easy take becoming give loss measurable let r r r control constraint difficult positive margin incorrect prediction increase minimizer reasoning risk must fortunately hold e difference split difficult four either turn control require range low derivative mass control technical include bound risk key need separately optimum scalar apply piece handle proof treat easy risk predictor actually view half finite generalization remainder focus eventually span optimizer enable application complexity generalization span take hypothesis correspond complement kernel subspace span interesting perspective risk orthogonal complement risk convexity negativity rearrange rademacher generalization consequence class difficult loss optimization force large finite hypothesis finite correspond canonical difficult set base imply control piece scalar subgradient canonical give db heavily influence point risk moreover instrumental care behavior lastly notable prove multiclass classification find extensively boost analyze convex without adjustment space allow treatment non point topology adapt hand appendix cover banach measurable finite p analog inner banach banach space bilinear describe versa banach endow topology imply topology topology topology pair topology bilinear construction compatible begin rest pair banach endow compatible topology give banach everywhere graph close banach close theorem conjugate fu statement equivalent optimality measure assume give banach bilinear measurable function establish conjugacy adapt say banach decomposable proposition page decomposable close decomposable pointwise optimality equivalent complete proof contradiction I segment connect pointwise differ banach map finish state duality adapt strong space close proper function operator maximizer discuss appropriate banach appendix banach space space begin around zero norm ball clear space definition summarize part function identically hold also banach space norm measurable must rademacher rademacher random variable rv rademacher absolute summation essential thank task control deviation approximate collection scalar rv z alternate iii iv consequently l census eeg letter horizontal quantity bfgs scale quantity relevant appear rademacher function please wide little observation primarily motivation depict conduct uci repository split logistic yield set logistic bfgs point split bfgs code relaxed termination order provide early avoid please roughly capture norm predictor error whereby behave whenever satisfie imply grant since first grant positivity precede property grant concave hold note increase statement continuity large eq concavity manually check loss let e consequently jensen normalize measure grant maximize positive expansion eq convenience start entail entail secondly entail entail entail everything bound already taylor lemma establish inequality tight derivation consequently mind agree along combine see go suffice measure note optimum evaluating turn moreover odd lastly statement follow h follow side provide equal particular imply norm dual taylor obtain banach space topology weak finite function mean definition yield decrease also therefore eq e law via duality banach space topological argue invoke denote yield remain continuous map finite decomposable imply show continuity finite optima since may condition start qx qx q adjustment q feasibility construction far attain consequently consider adjustment furthermore primal r inequality actually differentiable imply strict qx qx ir ix rx rx yield differentiable strictly whereby ii optimizer technical useful proof optimum satisfie everywhere strict nothing entail derivative coincide suffice scalar piece piece along every univariate neighborhood lipschitz continuous lipschitz obtain eq definition subgradient grant finish otherwise piece appendix part course attain attain imply consequently eq imply also primal coincide prove establish proof inequality finite hypothesis set dividing give prove apply control increase structural optima use whenever extra purpose suffice predictor lead hypothesis give difficult also optimum let already suppose define adjusted meaning must remainder continuity u u u u sign set r rv rv way q presence incorrect prediction precise control briefly logistic loss derivation tie pair handle thank old cf controlling behave advantage region rearrange suppose scalar construction imply interior image optimality taylor inequality final integrable side make rearrange taylor expansion q convert iv grant q jensen bind immediately imply convergence depend two function definition mean split subsequently control simplified apply bound definition equal apply construction depend desire show suffice contain expand case grow go text develop degeneracy problem scenario learn since equivalently mind relate boundedness hypothesis definition whereby hypothesis every whereby end result follow continuous infimum compact give
eq wise batch nonlinearity replace normalization need dropout style add target give denoise traditional encoder decoder calculate connection vertical connection connection unit vertical project wise connection drop sigmoid nonlinearity parametrization use layer low denoise multimodal multimodal ratio distribution decoder connection analyze path training combination determine much parameter encoder encoder ten impact auxiliary training evaluating model seed supervise epoch minibatch weight adjust accord schedule learn linearly last starting choose hyperparameter tune report validation multipli auxiliary task beneficial test hyperparameter choose worst misclassifie significantly lower report comparison boltzmann train include target target input connection connection input go layer initialization activation inference feedforward classifier train maxout autoencoder connection compatible supervise denoise training achieve margin conjecture due supervise unsupervised supervised find propose feedforward back quick implement function connection without many currently study impact cost work extend dataset autoencoder connection unsupervise simultaneously unsupervised cost layer wise improve significantly permutation combine auxiliary help hide generalize auxiliary autoencoder perform noise denoise autoencoder unsupervise show connection denoise change way permutation classification perceptron bold fully
bit h use instead bit hadamard normalization bipartite access pass immediately let integer give time ns position b bt kx desire suffice accord notation explain theorem algorithm form stack approximate accord entry index ta bb scale generality entry output sure integral ingredient prove kx ensure exponential progress corollary suffice deduce formal run vector position value precision entirely except along since polynomial time see computed assume represent trivially length operation take loop take time part light fact computation procedure number time loop present aim output satisfy goal arbitrarily round near integer integer range round entry near integer satisfy nothing round integer therefore note averaging position different round cause position coordinate add cause proposition focus achieve attain place achieve ingredient graph edge edge unbalanced intuitively pick magnitude tie break pick set unbalanced denote q equality regular argument note union regular denote goal iteration precise lemma suppose defer satisfy particular q triangle add subtract inside rip use use increase imply every constant pick sparse deduce plug triangle add subtract definition enough exponential guarantee attain end finally discuss proposition magnitude coefficient represent note slight abuse notation index position integer one rest pick hash output induce thus respect hash fall recover formally set coefficient later intuitively dominate correct implie must would produce bit bit note recall whenever equivalently correctly equality agree outside outside pick adjacent namely collect decompose hand claim suffice recover guarantee bin note intersect recall ready since h ti h ti ti ti ti si zero last lemma plug optimal randomized perform finally recall compute follow combine theorem randomize adaptive query access coin algorithm perform arithmetic one bit use ok proof modification algorithm absolute coordinate round vector suppose h hx leave randomness bipartite right contain contain right union vertex determine choose random seed shall observe result underlying respect assume conclusion deduce case note since depend coin determining hold analogue least stage condition conclusion lemma coin bad happen proposition independently choice particular rip particular support long expansion applying support take union thus follow tool analogue proof instead randomize random coin stage however conclusion coin union bind happen throughout error condition bad happen union final proposition least os write sketch sketch oracle queries sketch throughout execution optimize sparse take arithmetic operation naive sort nonzero entry add indexing update entry logarithmic would indexing observe run since corresponding copy additional per arithmetic procedure loop instead iteration plug value run proof alphabet q seed length irreducible polynomial interpret give shorthand follow construction provide choice theorem fix arbitrary explicit rd r construction apply mostly et require integer regard expand vector upper seed q desire upper task obviously nu nu n efficient algorithm integer polynomial integer imply claim hash normally technique construct seed universal family turn proof random basic universal hash family contain extension field map mapping universal element random nonzero uniformly imply universal ready hash tx variable independent min argument support hx z vector two express assign zero write rewrite probability universal family schwarz domain bind distribution set contain support california berkeley mit thm thm rough thm construction htb compute hadamard transform dimensional satisfying use algorithm start important technical tool use construction optimal deterministic general algorithm improve explicit construction would improve lead reconstruction exponent allow run improve discrete hadamard hadamard coordinate index entry variation dft hypercube notation hadamard well time fourier dft scenario hope signal improve rely process design efficient dft sparse last decade development efficient recent mostly discrete fourier transform dft cyclic focused hadamard term approximation dft development survey aforementione randomize desirable deterministic although subject transform one sparse dft signal run time run recognize considerable interest run reader think regime dimension like exponent query access deterministic give parameter compute constant approximation value exact goal formulate sparse recovery think resp objective comparison follow capture constant let bit output long constant fix appear solely state unbalanced use technique unbalanced immediately potentially dependence hope running theorem absolute simplify optimize exponent regard say exponent however randomness run improve use exist family graph construction rather large exponent asymptotic article hadamard transform adapt substantially fast randomness hash randomize access bit long internal coin absolute constant bad ok arithmetic sparse mapping coefficient bin locate position key need typically introduce parallel aggregated identify coefficient eliminate proceed fourier cyclic run h show guarantee aforementione implement e bin bin furthermore number different result reduce select still deterministic however relax mapping formal expansion mapping lead near need simulate black induce mapping implement paper observation explicit number mapping find construction yield extra query identify procedure query analysis considerably thank call isometry rip norm immediately hadamard linearity discuss reduce compressed notion unbalanced focus amount compressed sense reduction section main sublinear algorithm main deterministic improvement comes add let index coordinate agree elsewhere equivalent hadamard approximate k role hadamard equivalent hadamard computing equation compress sparse hadamard formulate thus goal present adaptive linear requirement isometry combinatorial need rip order eq rip approximation satisfying use normalization equivalent unbalanced formally unbalanced mention characterization matrix unbalanced rip absolute unbalanced recall xx regard hx distribution min entropy computable achieve equivalence unbalanced nb bt tx hx vertex choice bipartite associate unbalanced focus algorithmic hadamard hadamard sense focus discuss h rr bipartite query suffice namely real rip order constant give compute combine arrive absolute constant follow deterministic running adaptively query kx adjacency bipartite unbalanced satisfie rip constant assume sufficiently constant suffice product efficiently well explicit use technique list construction make result prove p parameter deterministic run ok absolute constant set derive proposition linear simply hadamard algebraic q v
covariate result efficiency elaborate efficiency depend bias thus strategy future focus recommendation cm de la paris france universit paris de france universit paris centre de paris france recommendation integrate online user influence interact system consequence recommendation via application offline reduce filtering frequently aim suppose interest system internet online social adapt obviously evaluate monitoring achieved click historical offline several real offline etc profile item rest profile various influence historical specific product etc limit strategy recommendation protocol principle weighting propose practical relevance constant recommendation recommendation user collection phase build public business organize describe weight scheme demonstrate relevance denote item historical recommendation build instant associated possibility item recommendation instant profile user scheme user business favor evolve classic offline calculate instant joint moment item select item certain lead selection process similar evaluation protocol moreover bias indeed recommendation discount major constraint business thus classical static value lead weighted leibl asymmetric useful reference reduce influence time offline production tend evaluation unbiased production offline evaluation seem class apply collaborative filtering consider recommendation suggest user quality recommendation estimate simulate repeatedly item select user collaborative user present collaborative filter cosine proportion associate item algorithm recommendation high method different collaborative one present h value day day date feature important notice score
volatility abc smooth estimate filter cumulative denote compute procedure return exception volatility vary return heavy confirm filter accord approximately copula residual drawback directly suffer accuracy tail cdf generalise nonparametric q denote location right low tail model marginal tail asset type kernel tail filter residual parameter make likelihood filter residual quasi newton present low tail cdf residual dynamical contract simulation perform resource provide national like thank discussion lee make calculate gp covariance log acquisition hyperparameter gp th algorithm sample obtain hypercube lee optimisation solve use direct implementation abc space use follow make fix lag smooth lag smc discard hybrid particle real world price contract obtain use particle additional setting adjust hessian smc abc tails newton solver department electrical computer university technology university nonlinear intractable likelihood sequential monte approximate smc costly novel smc laplace intractable abc volatility conclude enjoy comparable significant reduction optimisation portfolio use copula margin optimisation dynamical modelling datum extensively volatility stock financial volatility important risk management copula risk var portfolio bayesian nonlinear limit model aim bayesian nonlinear say wise analytical recursively computationally prohibitive denote denote abuse py evaluate intractable possible problem likelihood computation result apply standard estimate perturb perturb quantify challenge problem abc smc abc estimate suffer problem require mind optimisation extract posterior iterative operate constructing computationally evaluate approximation contribution introduce result pass computationally area alternative work combine smc maximum biology parameter usefulness inference volatility model useful investigate impact likelihood standard consider financial portfolio smc abc inference margin copula result considerable speed compare method use related alternative log estimate ii robustness approximate alternative bayesian optimisation latter simultaneous stochastic drawback problem practical parameter gradient ascent smoother wise continue motivation overview abc continue discuss smc abc intractable log construct exploration exploitation numerical highlight aim method construct complete square second log express hessian observation see around bernstein von concentrate laplace require obtain see laplace ii result newton gradient many suffer slow log posterior use optimisation gradient even costly obtain suffer variance estimate analogue log furthermore problematic main laplace approximation contribution difficulty create laplace surrogate surface aim create smc abc evaluate resemble around make optimisation operates sequentially surrogate use sample predictive process briefly account globally evaluate noisy log posterior model abc already hence predictive could determine heuristic phase method optimisation return gp acquisition therefore could abc alternative laplace proposal challenge write intractable except instead propose smc particle sequentially predictive quantity obtain dirac denote normalise dirac particle particle evaluate consequently filter model require algorithm abc variable express assume augment denote density non refer bandwidth density fundamental abc select particle first perturb transformation later assume correspond easily use transformation random perturbation simulation denote particle require wise use perturb reformulate tolerance est operation carry resample propagate extend trajectory particle particle iterative particle particle three iii carry randomly multiply discard small part implementation usually recommend practical application propagate simulate system third weight present denote particle implementation know alternative useful perturb estimator biased particle variance variance estimate use simulation repeat likelihood estimator unbiased view bias decrease likelihood von motivate laplace quantify abc construct surrogate posterior reference gps see infinite variable distribute gp see value see surrogate priori log accord gp infinite value specifie correlation encode gaussian estimate later posterior theorem calculate notation available parameter hence construct function posterior use obtain however application calculation handle computer demand filter gp decrease cost memory grow properly explore discuss issue vary space flat arise five ard function posterior calculate necessary experience major paper recommend plot mean mat ern ard function type equivalent mat ern posterior smoothness laplace weak smoothness property assume replace mat ern covariance square mat ern function choice regard design mean gp hyperparameter bayesian approach use e marginal get optima next parameter aim heuristic next available point log user optimisation evaluate test research field paper make recommend derive ei rule denote exploitation exploration maximum assume large peak expectation predictive posterior gaussian often add exploration area increase parameter covariance matrix similar exploration optimisation therefore two local optimisation gradient section put together abc outline combine log surrogate function suitable hyperparameter prior may fail converge properly hyperparameter log ii quasi iii hypercube sampling execute update hyperparameter pre interval hyperparameter costly recommend algorithm parameter covariance parameter est hyperparameter k direct compute direct finite execute suitable ei predictive extract map carry optimisation newton wise laplace approximation require log hessian hessian function choice covariance note solve hessian adaptive obtain accuracy smc smoother costly negative previously discuss optimisation noisy prior ei achieve convergence mat ern section usefulness accuracy rate impact tolerance second illustration return illustration management illustration abc algorithm competitive practical collect consider stochastic q parameter assume random increment synthetic use evaluate smc abc tolerance parameter posterior gold standard laplace smc estimate observe
scenario due improvement proposal achieve remarkable conjecture whenever know cause negligible organize include complex signal exploit study kernel kernel apply calculus value derivative illustrate conclusion transpose hermitian trace determinant sample real throughout gaussian distribution regression regressor relation follow model nonlinear linear square wiener gps solve mmse filter input input value herein immediate dealing output wish dependent resort multiple relate complementary joint construct additive follow circular complex training function factorize multidimensional proper complementary covariance jointly number positive definite marginal later equality opt choose tune well obtain previous output remarkable good nonlinear channel mse increase steady second try hyperparameter capability first train fig procedure indicate step criterion algorithm prove valuable learn similarity well underlie tb cm output widely study resort process hand end endow gps output availability optimization derive reproduce conclude complex cross output nan must part need prove exhibit end fully remarkable independent process produce output white iv integral value use value many fundamental often value proper uncorrelated complex simplify value paper develop value reproduce complex value input convolutional covariance pay preserve input maximize use derivative besides scenario solve challenge scenario deal signal literature benchmark remarkable proper kernel vast engineering complex availability processing value processing interest processing widely case complex uncorrelated conjugate useful simplify solution admit complex version improper nonlinear complex address network recently reproduce rkh regression mainly component regard author complex discuss involve besides neither may suffer improve previous isotropic value fed real physics drawback solution approach develop kernel although bring study know technique successfully gps interpret advantage provide hyperparameter likelihood benefit novel provide output reproduce output proper signal provide covariance complex kernel output part design part addition skew definite covariance pay
implement continuous specie one belong share describe semantic behavior allow previously unseen computational reduce time cluster collective repository solve first collect large software piece experience program hardware believe process practically infinite space collaborative one currently many optimization software cm repository find meaningful architecture mainly benchmark htb cluster software across several architecture experimental result update public cm repository mind pool distinct cover software piece set benchmark benchmark achieve high speedup optimization number benchmark distinct benchmark benchmark across piece help substitute manually practically improve approximation decide associate unseen unique optimization base software decide purpose generate repository software specie semantic hardware collect monitor software piece share resource os pass share one past classifier full htb demonstrate high software support similar I much share dramatically drop close behavior collective mind model simple cluster relatively program pool cluster remove prediction leave semantic ft count basic ft exist relevant hardware understand improve since feature counter correlation simple try manually code highlight add feature effect perform transformation speedup show problem machine realize large though highlight repository usually community practice share publish improve machine specie public benchmark various researcher switch share possible continuous share balance reduce complexity usage neural good help hardware problem htb methodology improve lack record behavior automatically customer software surveillance require distinct execution intel core image set cluster classify detect explain software specie gradually specie add statement additional branch rarely additional cycle dependency solve notice effectively convert customer self minimize energy tuning effort market plan software therefore identification extraction frequently consume software piece real program plan use extend interactive interface connect framework simplify integration care automatically dependency add repository methodology physics community benefit powerful source gradually furthermore immediately validate novel optimization across realistic system big repository ai physics characterize interaction share software piece fine specie repository possible like algorithmic specie continuously optimize hope hardware boost innovation implement idea regression include run particularly mobile device avoid subset representative interestingly eventually large enable repository support software engineering present support foundation fp thank ed feedback collective mind technology feedback considerable improvement version technology share graph wide user flexibility repository force possibly possibility crowdsource exploration dimensional apply service user mb web service plain line slow slow indexing option decide original open repository format fp project collective fourth technology publicly core collective fast orient analysis service separate share service conceptually collective collective mind kb mainly furthermore package automatically dependencie believe considerably technology collaborative sharing experimental interactive report term effort share interactive graph gradually past format focus solve program run time collective predictive available repository gpu scheduling video processing mp gpu exploration local frame processor apply active build refine share could cpu improve frame second believe vision towards collaborative systematic engineering framework brain quick idea knowledge innovation get consume issue experimental wikipedia extend fix error improve miss many entry unique collective module dedicated page allow researcher evaluation lead along useful tool arm international uk develop know hardware eventually run mobile cloud service unfortunately optimize keep ever ever change system code collaborative publicly solution help software gradually available connect collective repository characteristic exist hardware configuration environment collective mind consume mobile mobile science continuously track win solution hardware skeleton parameter minimize execution spend failure specie pareto mind furthermore continuously classify redundant one various software input set hardware similar technique create public realistic evolve benchmark knowledge gradually improve depend usage scenario continuously grow collective become hardware self management collaborative collaborative knowledge crowdsource active learning hardware validation pareto code interactive ever diverse computer power consumption reliability hardware availability specialized hardware working memory storage often run heterogeneous multiple mobile device cloud force software rely exist hope fast efficient scalable hardware complexity ever change hardware hundred fail produce code energy include decade evolve possible software empirically search combination implementation scheduling among choice execute program cost thousand per considerably performance life mobile device promise advance report discussion show miss software fundamental already grow exhaustive find good recent acknowledge continue efficient hardware development usage htb pdf hardware fraction whole hoc heuristic vast hardware furthermore specific run time mechanism costly practically framework execution vast practical engineering tune behavior conceptually hardware propose background ai help start gradually real software range whole line piece depend hardware possible requirement recent collective mind repository share open piece together meta gradually extend easily format describe build piece dependency hardware software include operate run library piece randomly execute window device device share resource mobile cloud service gradually cover configuration community continuously software nature biological treat piece continuously track behavior versus configuration environment record win solution minimize execution consumption size failure memory storage time software piece pareto repository project optimize specie continuously reduce cost hardware aware software hardware conceptually help create first knowledge public large diverse evolve continuously optimize benchmark knowledge gradually software apply physics community learn optimize large share piece whole library function consume loop versus global coarse gradually move fine include versus internal decision interactive already methodology adaptation include begin usually focus execution consumption deep combine ad hoc architecture large benchmark allow methodology sciences biology ai big datum predictive cm continuously win specie distinct community unified background extraction continuously add specie thus practically software automatically environment cm continue species execution predictive find feature complex intensive technique neural decide manual option exist predictive manually simple fast decision explain predict specie web link share software hardware wikipedia engine success depend active try use weight share resource specie software manual gradually work together describe partially share specie together gb storage moment share I benchmark validate help tune production customer consumption specie combination cloud mobile derive distinct class cover share specie manually semantic dynamic feature predictive analyze predictive end correlation isolate share counter code wrong classification substitute ad architecture verification derive optimization dramatically eventually involve software engineering development improvement hardware continuously grow repository unify service practical hardware design decrease development cost importantly side help international engineering present life motivate engineering example encounter briefly introduce collective mind repository collaborative associate across provide demonstrate continuously big datum specie predict realistic representative section demonstrate improve demonstrate miss specie improve optimization tune computational conclude development direction various brain inspire brain function layer popular choice modeling pattern image include implement fairly regular neuron receive weight input output process activation sigmoid many implementation filter activation switch neuron neural determine processing capacity correct prediction failure heavily total neuron speed resource include specialized co involve careful balancing versus associate cost consumption development price usage improve evolve modeling vary minimize execution hardware center cloud service minimize cost include consumption network surveillance mobile internet thing strict place hardware memory system year software engineering neural relatively straightforward simply select hardware tune achieve nearly peak figure configuration arrival hardware hardware would double software consumption dramatically htb mind contrast software execute market frequency core cache hardware gpu neural power consumption access parallel home popularity cloud services amazon google microsoft others experiment mainly service time improvement operate together numerous free software development tool may expect advance practically generate efficient software piece hardware nevertheless eventually decide validate sake start collect c system multidimensional choice whenever real experiment execution include code usage decide see room improvement fast many projection multidimensional characteristic cost track win solution minimize physics pareto filter quickly necessarily fast efficient require move improvement execution degradation achieve improvement execution old furthermore internal parallelization scale old linear scaling parallelization htb mobile architecture execution dramatically power consumption drop try specialized hardware execution hundred considerable development cost performance encounter problem cache scaling core static fundamental lack run try move language achieve similar software considerable ad hoc good program cost summarize software often aware improve usage cost cost balance execution accuracy device execution balance gain care therefore believe current performance blind engineering change improve innovation science technology start search could software relevant job account gradually software try connect keep track within production several severe difficulty evolve difficulty reproduce machine web service collective mind biology repository optimize tool whole considerably usage cost briefly help decompose software piece currently support major language community gradually optimization feature dependency software hardware characteristic cost unify format allowed formalize almost exist finding function piece run computer hardware software system software
rely result half give two classifier improvement change pac resp marginal next give hypothesis share source support upper inequality chi could unsupervised way appropriately two distribution hope control transfer definition theorem st france universit st france provide base pac bayesian theory improvement previous et way bind tighter easy algorithmic adaptation generate source generate analysis adaptation belong domain generalization express average pac bayesian new pac improve moreover appear design able term paper introduce obtain pac offer majority vote dimension priori quality identically accord pac weight low precisely vote related consider da al difference da target information source label sp majority vote domain recall usual pac generalization gibbs al risk notion r disagreement marginals eq h g reflect well favorable situation achieve source derive promising minimize domain
obtain let great lead real I value case derive eq therefore hand equivalent noise probability lemma imply desire test multiplier see phase transition sharp comparison plot phase generally cs achieve illustrate empirically grid reconstruction signal separation guarantee edu xu explore superposition projection arise application biology imaging rank subject sample show long incoherence separation interest superposition measurement require experiment practical model superposition choose acceleration medical digital imaging nuclear biology signal superposition superposition consider linear superposition satisfying word superposition superposition less ambient reconstruct assume order enhance matrix notation shown directly reconstruct recovery tend matrix nuclear I norm q theoretical via theory measurement nuclear minimization contribution exceed scale reconstruction toeplitz superposition complex signal signal accelerate superposition analog digital possible compressed uniform grid fall domain usually exactly fall basis discretization conventional compressed recover grid complex atomic minimization prove reconstruction separation frequency enhance matrix chen et play similar complex apply aforementione exist frequency achieve comparable accelerate frequently molecular monitoring chemical application etc et low signal result explain give apply incoherence uncertain diverse chemical sample require organize extend main rank toeplitz numerical base observation whose consist lead rank enhanced constraint row specifically let column propose eq rank correspondingly problem hard solve possible recover nuclear norm minimization likely guarantee require robust much degree freedom special interest extend complex guarantee incoherence available realistic diverse chemical biological limit applicability recovery theoretical ensure scale result incoherence arbitrary number small obviously get accord order choice error easy form orthonormal standard linear adjoint identity onto simplify introduce diagonal letting q case gaussian dominant recovery minimum nonzero gain minimum respectively let minimum condition concept atomic norm give set unit gaussian map corollary part gaussian unit sphere convert real matrix complex letter value therefore gaussian mean get desire follow eq gaussian instead give cone accord cone hull respect whose part I variance need r calculation satisfy singular decomposition define linear q adjoint singular decomposition subdifferential estimation width check choose get convert vector second q definition orthogonal check satisfie imply line similarly together prove give tight gaussian mean part constant proof relatively introduce idea lemma even integer kk k jensen inequality easy see ki utilize index specifically node I edge equivalent class class index accord traversal play bound reader concept two equivalent
forward recursion complexity array support increase exponentially recursion notably array problematic deal deal describe two name row share composite maximize em longitudinal satisfactory name composite row composite likelihood analogous construct column dependency due latent row composite array r request finite simulation simulation cover array small give chance quantify due composite likelihood another relevant aspect tackle particular support variable cross extend finite implement independence devise cross cell array index version estimate training illustrate utilize leverage inter intra comparison type throughput genomic rate mb genome process publicly genomic window thing aspect dna insight feasibility utilize methodology array genomic contiguous mb window along article assumption section section outline likelihood genomic offer conclude row basic conditionally give identically distribute column chain initial probability coincide parsimonious chain parametrize complete pair natural depend requirement mixture application section transform normal mixture suitable parsimonious incorporate impose mean observable comprise covariate fix obvious replace normal family array variable parametrization covariate full feasible array switch methodology mixture initial distribution implementation account covariate denote denote configuration latent eq denote trivial involve relatively strategy joint column compute becomes indicate use introduce indicator reference decomposition comprise latent comprise column matrix comprise step compute indicator sum extend v v step value update mass probability constraint also number way array time increase however latent variable prohibitive infeasible application propose composite row great potentially efficiency datum underlie simplified version variable row composite base importantly readily computed treat useful log satisfying assumption chain need target otherwise express composite cb z ij e definition term approximate model ij one e maximization finally mean composite column datum py uv composite row composite log regard latent meaning section former separately indicator express w cb compute update update perform simulation study assess likelihood another estimation datum array design typical application full likelihood fix benchmark suitably scenario scenario underlie row fix apart apart respectively parameter bias root likelihood likelihood likelihood table median median deviation likelihood c bias rmse rmse rmse rmse rmse c c rmse rmse bias rmse c c rmse rmse rmse rmse bias bias rmse rmse rmse row mass probability estimate approximation comparable available either composite approximation comparable approximation fact even sophisticated rely independent approximation former estimation composite approximation fast benchmark design importantly pass order run second hour size simulate still effect see average remain instead general row approximation e appear fast approximation iteration even consume large array composite row literature suggest information bayesian see select free maximum computing penalization complicate hessian rely regard deal independent composite cross straightforwardly extra cross splitting treat either half repeat base maximize cell remove result quantity average log consider quantity pair either maximize close maximum course derive consideration illustration study university human base four comparison contiguous mb overlap window try relate landscape dna molecular along genome author publicly window produce segmentation mutation simultaneously characterize utilize segmentation array comprise measure contiguous mb overlap cover table capture dna composition dna e g nuclear associate I site site bind dna level code standardize normal prior ht line nlp h rna ii average es dna n l l lines ac structure form site code ht critical number latent cluster genomic number distinct strategy composite compute correspond denote report log c result high composite likelihood one well obtain alternative compare similar quantification compute high model c towards base form genomic distinct table report latent convention modality order decrease modality table stationary ht figure color code way feature segmentation e latent posteriori predict horizontal represent contiguous bar top reporting code vertical dimension genomic reporting code black red range report green green characterize genomic vertical bar mark horizontal bar concern note comprising estimate small mass detail cluster proxy proxy include proxy region activity concern segmentation cover approximately estimate estimate e least characterize cluster feature cluster state feature cluster whose profile former feature cluster feature see represent alternate approximately window towards figure cluster strongly level strongly cover window strongly cover article array contiguous segment composite approximation optimization specialized show methodology methodology demand clear composite row
covariate hold eigenfunction uniformly bound constant depend cover square minimax follow eigenfunction probability least square also contraction rademacher find book van processes immediate result connect literature assume ball estimating assume underlying regularity zhang similar mixed norm yu highlight rkhs ensure hence question strategy estimator ball surprisingly answer challenge occur smooth minimax optimal rate rate choose optimal rate attain unit ball multiplicative obviously suboptimal attain minimax main theorem brevity shall assume argument natural whose specify nonzero jk generate clear set step ns ns brevity jk kx jk jk jk q jk dx jk jk jk follow specifically infimum measurable ny nx fix g cm l kullback leibler conditional ns eq yield constant q constant complete immediately q write lower together derive inequality separately value n h l inequality fact hand l exist constant probability event hold l inequality write light l contraction concentration h g gx gx n n g bind uniformly argument jx u conditional constant conditional hold combine c e exist constant event constant first step proposition corollary establish reveal component smooth sufficiently rate identical dimensional smooth rate curse dimensionality transition reproduce advance technology finance devote understanding challenge dimensionality development methodology counter fan li selector progress understand extent regression reliably van reference therein model restrictive alternative attract much attention past several lin zhang van yu couple amount certain kernel fix idea follow support product compact subset component rkhs clear model identify h obviously view trivial take collection another canonical unit interval g note additive representation define quasi h minimize rkhs number function interest optimal rate th interval imply side dominate always eq pay rate nonparametric closely learn aggregation machine combine kernel single achieve study david expect understand organized concept reproduce present basic use reader rkhs symmetric semi integrable hilbert completion product shall rkhs assume paper cauchy recall shall repeatedly later spectral theorem admit eigenfunction marginal delta
participant slightly ensure camera overhead video capture angle place face fig rectangular person meta datum player starting video videos pt videos rgb videos depth encode rgb videos distortion correct frame camera pose participant face participant whenever available object bounding box position track focus annotate divide absence segment video develop annotation schema draw concept social literature schema series question annotation schema social predicate focus object language similarly involve movement rate six student video introduction survey ask video video annotate ensure student accurately reliably detailed annotation h h conduct skeleton generation partial entire visible layer raw player average generation relatively case except similarity visible demonstrating effectiveness also across indicate relatively stable possibility show detect task skeleton actor novel essential predicate attention collection new audio visual dyadic research social discriminative conditional boltzmann generate datum purely decomposition powerful offer possibility substantially advance mid predicate layer beyond generation multi understanding semantic extend multimodal stream include audio full behavior make multimodal capture rule interaction automatic efficacy interaction interaction behavior virtual reality environment thus well robot interaction foundation establish systematically scientific acknowledgment nf view conclusion view imply david novel predicate sound social methodology collect game consist dyadic expect provide new research computational social restrict boltzmann combine combination accurate predicate exploit capability actual training mean frame decompose behavior purely computational determination human processing scene research bring problem leverage computer vision social interaction aid worker country course interact well worker success general would enable smoothly interact extremely useful identify detecting predicate facilitate irrespective interest aspect reduce trust predicate attention orientation social sense exist infer internal instead body social emphasis cognitive action interaction approach apart social action jointly insight involve reciprocal act joint behavior nest demand behavior participant interactive social interaction behavioral movement pattern social interaction establish essential predicate mind focus computational social multimodal deep model recognize discover past decade machine advance furthermore complex full body pose many activity multimodal multimodal dyadic social model maximize solely often unable incorporate hybrid address combine discriminative level discriminative model allow recognize propose answer question approach attempt detect social qualitative multimodal human variety must span everything lexical pt modeling essential predicate multimodal temporal model social interaction predicate multimodal annotate publicly interaction discriminative conditional introduce enable learn advantage result detect behavior sec sec specify explain sec quantitative sec conclude interaction implication science focus theory largely much show infer participant interaction sequence social dynamically participant behavior social require realistic human interaction detect overall state person external speech multimodal also model activity involve deal physical rich participant consist learn representation low input tend solely joint method single generative energy iterative consist discriminative train separately learn generative project rich unsupervised powerful boltzmann rbms building block train cd algorithm demonstrate ability deep representation rbms deeply boltzmann capture complex recently deep capable rich include rbms temporal rbms motion human pose phone recognition parse music generation describe similar prior define parameter rbm generative hybrid rbm define distribution visible ensure bias architecture h factorial case real binary layer code prove energy slightly rbms visible distribution hide function term history previous gibbs time model autoregressive visible equal history rbm vector phenomenon factor restrict boltzmann boltzmann however complicated involve factor layer layer similar equivalent visible visible label generation bottom type deal miss fig miss visible goal label cd obtain update combine respect expectation reconstruct reconstruction visible generate visible finally label activity recognition dataset annotation interactive behavior demonstrate contain relatively action involve person interaction collective activity dataset lack rich dyadic dataset dyadic interaction child dataset collect structured format child interact pre narrow social behavior child another focus study social analyze human format human different aforementioned activity limited coverage diversity activity class lack narrow behavior g issue game
enkf enkf describe enkf filter localization equivalently member enkf member assumption enkf enkf denominator gaussian read easy correlation successful enkf become calculation successful estimation enkf require enkf sampling proxy assimilation illustrate observation enkf successfully sample enkf condition success use importance success particle filter see appendix particle two regime success enkf observation moderate noise thus particle filter enkf perhaps surprising particle particle additional thus optimal sequential avoid observation past broad enkf filter uncorrelated illustrate show particle filter noise noise enkf sequential enkf enkf particle enkf think variance improve enkf improve surprising update enkf marginal I go limit go thus posterior enkf become inefficient enkf assimilation success filter fix observation hand filter also successful fix keep quality improve keep fix optimal behavior importance importance filter inefficient enkf posterior derive enkf gaussian localize enkf draw marginal posterior density localization draw posterior target since enkf moreover enkf assimilation enkf ensemble properly produce marginal various enkf attempt importance enkf carlo know marginal numerator evaluate write integral integral density know multiplicative constant exact enkf integral carlo become ensemble size infinity ensemble define approximated analysis ensemble however ensemble carlo enkf analyze weight compare report difficulty assume observation assume available difficult enkf finding impractical frequently enkf posterior brownian discretized order forward euler brownian motion step collect independent noise deviation enkf perturb localization ensemble necessary comparison filter coincide implicit consider enkf numerical synthetic perturb assimilation compute weight store equal collapse weight enkf member enkf maximum panel particle resample replace weight weight enkf assimilation plot panel particle filter however joint frequent enkf explain synthetic assimilation weight enkf member marginal enkf require connection enkf review feasible enkf may slower assimilation scale enkf require dimension problem may numerical confirm enkf localization system ensemble size small enkf adaptive multiplicative also implement enkf directly appropriate noise compute times deviation mse dimension section far plot numerical figure constant left panel dimension panel show find equation predict system moreover localize enkf direct sample confirm statistic enkf agree compute localize enkf also localization without localization localize enkf mse insensitive sensitive tune localization e satisfied vary change reduce state refine assume show enkf localize enkf confirm assumption confirm feasible scale linearly sample dimension also effect uncorrelated violate attempt posterior bias localization work enkf produce show typical tuning adjust adjust posterior mse size enkf enkf moderate relevant infeasible enkf dimension rarely behavior reason application assimilation case mid dominate far mesh region rich discover mesh refined decrease mesh refined assimilation mid process dominate would observe behavior enkf resolution mse observe practice mesh mid effect resolution assimilation summary mse connection feasibility theory find assimilation may sample operational assimilation summarize density assimilation current filter enkf expression enkf broad enkf suboptimal particle filter broadly enkf joint imply enkf sampling enkf suggest enkf explain usefulness enkf require ensemble size enkf suggest ensemble dimension bound connection feasibility assimilation enkf assimilation enkf mse rather insensitive material base science office advance apply mathematics program contract foundation grant dms office research pe would thank interesting comment feasibility discussion provide wish gaussian problem become note independent expression simplify component simplify eq denote covariance independent thus add assimilation also origin component denote one component find filter give substitute expression eq assume enough reach success filter constant rearrange tb tb tb tb berkeley laboratory mail berkeley california ensemble kalman enkf widely condition evolution condition marginal condition filter imply enkf marginal posterior localize enkf useful explain applicability enkf tune enkf mse sampling huge moderate model density careful distinction two model condition describe condition filter determine variance one develop enkf enkf joint enkf broad one assimilation cycle filter unless past observation enkf imply posterior insensitive error make assimilation derive weight enkf localize enkf importance question assimilation weather forecast forecasting assimilation cycle irrelevant enkf optimality enkf posterior enkf explain applicability enkf dimension connection enkf datum assimilation true ensemble investigate localization mse enkf marginal variance error mean localization mse insensitive huge successful moderate consider assimilation vector smooth dimensional discrete random identically iid assume covariance assimilation perfect deterministic explain careful elsewhere assimilation consider describe assume specify entire trajectory joint posterior history variable dimension posterior large small assimilation large record weather forecast frequently posterior hand kalman covariance require impractical case enkf use enkf make monte forecast requirement kalman filtering ensemble member member obtain sample perhaps localize perturb approximate marginal implementation enkf enkf particle use empirical e recursion approximate weighted function posterior need collect extend recursion account lead filter particle unnormalize else particle condition variance normalization obtain close mean contrast normalization small nearly posterior desire carefully recently logarithm must infinity summary condition success wish particle filter filter particle go infinity achieve cost bias apply assimilation assimilation question assimilation implement concern numerical explain variance assimilation datum particular forecast need intuitive feasibility assimilation deviation e extend high assimilation feasible covariance norm data assimilation uncertainty kalman kalman gain mild steady algebraic steady state kalman gain kalman assimilation rather tool derive asymptotic steady state assimilation steady small suitable frobenius define covariance connect frobenius correlation g correlate red large uncorrelated random frobenius norm imply project onto span reflect empirically fewer easy assimilation motivate effective assimilation noise actual large non dealing need frobenius norm precise requirement resource interested assimilation behavior dimension finite limit numerical weather assimilation problem pde connection limitation particle reason mesh discrete mesh mesh refine small imply feasible fine computationally tractable feasibility may hard assimilation large reasonable positive scalar varied system attract literature particle filter posterior scalar well steady covariance compute assimilation balance represented set sufficient qualitative feasible assimilation generally constant believe datum assimilation feasible dependent white hold grey feasibility assimilation else confirm assimilation infeasible counter intuitive scalar feasible theory label infeasible frobenius norm invariant rotation coordinate scalar sub structure
quantity generic tail adopt particular approximation posterior rather base theory incorrect situation valuable tool model marginal spirit interpret represent variate cdf variate mf jx account among component cdf copula assumption marginal copula paper distribution parametric copula limit particular approach empirical survey produce nonparametric likelihood particularly readily available completely replicate vector express functional sort profile moment result whereas obvious independent third towards quantity computation likelihood expensive repeatedly consider achieve posterior avoid crucial abc relatively easy new simple propose pseudo randomly distribution actual represent highly inefficient strategy available example concentrate abc expensive likelihood family importance sampling sir generate draw propose bc partially quantity set meaningful although statistical nuisance produce might robust inference parameter way important lack physical meaning estimate dramatically circumstance reasonable specify adopt frequentist simulate estimation copula paper spirit propose goal specify model assume distribution sample representation copula use copula skew parametric investigate mainly partially popular literature obvious imply robustness method essential aspect dependence disadvantage parametric demand posterior though density require huge iteration might algorithms avoid computational burden modify run among pseudo lie modification ht marginal distribution j ms draw row eq f store b datum know counterpart say nothing write know coincide actual rank original evaluate p general multipli produce quantity size comparative frequentist describe confidence interval construct frequentist three low limit nominal limit box posterior take computation procedure precise estimate median behave length length tail credible simulation short copula raw value histogram mass entirely close perspective follow describe estimate give many properly treat il il marginal represent correct variability work notice slight towards large incorrect report contain monte di bank log return available model student innovation may via return bank distribution package simulate parameter follow simulate simulation consider row return di posterior acknowledgement provide universit universit universit di simple make functional multivariate carlo algorithm functional particularly costly evaluate work
point reader field log relaxation space dominate relative propose sampling key collect estimate understanding likely future etc effort generate possible pt run adversarial news b rule marginal deep nlp e rule negative example conduct incremental system program last competition quality sometimes assess blind assess program development technique development speed news adversarial name refer domain figure six template four category focus system build base relation corpus million news page generation extraction supervision span spectrum collect sentence journal article precise text relationship text news write relationship e g belong pt learn incremental component write core ghz ram adversarial build program technique show speed development quality incremental sample collect collect combination similar system focus news competition six sequentially score cumulative execution take significantly time win indeed hour take minute fact extract end task fact issue differ incremental compare evaluating given understand total execution part extraction classical incremental news speedup free key speedup news across high update original acceptance execution extraction supervision speedup rule attribute contain contribute speed produce incremental low execution application interesting cause fact introduce large case need get factor therefore factor hour spend sample single conduct verify leave report evaluate impact sampling variational news sampling approach variational slow acceptance distribution change extraction rule sample use group supervision variational slow rate pt baseline switch group experience build quality challenge accelerate build incremental component statistical approximate improve face order keep aid acknowledgment advanced program fa program national office national national institute image research fellowship foundation american finding conclusion recommendation necessarily view find namely sampling provide proof describe rate summarize result markov chain represent difference assign event metric comparison sampler couple statement follow argument low lb voting voting variation unary exponential eq meanwhile flip similarly event could happen event state q event bound less least step heuristic behind variable specify iteration set rule dependency change want active call inactive next active inactive create new factor inactive conditionally appropriately group decompose presence inactive variable collection inactive inactive partitioning inactive line conditioning independent inactive separately variable pair impact avoid grouping inference grouping make simplification specify inference group concern hardness g cg allow reduce contain inactive heuristic accord active inactive variable I independently height pt connect remove variable set condition j j decompose strategy find compare actually fast less determine extraction rule news show change follow experiment hour common run execute phase variational finish many sample collect hour hour happen document analyst would efficient interesting distant supervision able create human intervention perform online method model last start experimental adapt standard outperform compare approach namely descent without new new training proxy stochastic separately pick grid hour pick fast loss epoch percentage within learning epoch loss achieve reach within sgd fast descent sgd converge drift stream keep resolve concept machine adapt change consider update model forget cause impact drift incremental learning impact solely focus target significantly drift require difference target amount change concept drift approach second learn target section work concept follow dataset email spam use testing train drift converge drift allow low loss term iteration use almost active change original concept due inference small current design component motivated aim improve design justify find able incremental decade incremental rely classic incremental technique operation individual like iterative segmentation database graph algorithm reference remark stanford stanford edu database extraction recent deal dark datum system combine learn idea help observe develop technique inference optimizer five showing speed task two order impact structured community profile effort processing learn community place emphasis extraction community common technique mix datum quality structure information database entity complex relationship assess complementary claim tuple tuple many actually massive far document count effort shoot algorithmic arguably question good use rapidly number language axiom rapid move quickly construction execution language perspective perspective logic language semantic execution go phase evaluate describe tuple gibbs output tuple google expect e subroutine loop computationally tb ram machine iterative use technology field drug recently compare system provide ten database precision entity win entry iterative arrive concept lead contribution inference incremental incremental feature extraction series make specify change systematically system due phase change datum compute problem work database new new change simultaneously use inspire probabilistic database technique apply incremental clear experimental diverse program find approach largely orthogonal axis change performance highlight neither choose optimizer experimental highlight improvement describe program development run incremental incremental snapshot approach fact throughout technique incremental outline rest paper depth development presentation system incremental exploration tradeoff description optimizer present study decade base system machine study improve quality build formalize ease accelerate hope feature google knowledge incremental pt language goal construction relational classic study maintain inference focus related work focus incremental specific structured low degree factor graph much examine modification graph aware single end build language first definition semantic heterogeneous collection system contain put schema may extraction integration illustrate system news article incomplete kb person linguistic pattern roughly indicate terminology four object system seek input entity person thing entity person another individual entity entity relationship mention span text refer entity mention entity phrase connect process entity detail end engine walk phase manually store database default sentence nlp pre speech linguistic parse loading type mapping entity relation candidate candidate mapping mention sentence mapping query must candidate chance extract feature markov user phrase phrase two whether people think say influence phrase indeed indicate return two receive explain detail arbitrary operate tuple allow example bag aware nlp feature dictionary specify ability specify rich entity rule helpful integration supervision markov logic particular schema relation mention label distant supervision illustrate system kb entity pair q incomplete world entity entity incorrect generate sentence people technique redundancy cope phrase generate largely distant supervision relation furthermore integrate unified logic semantic run phase inference obtain final confident repeat understand feature mistake facilitate find three aspect believe enable program program sense probabilistic provide algorithm allow extraction language familiar stack visualize datum user construct end system pay traditional time spend extraction evaluate pay go inform logic language logic implement weight rule across feature every rule single couple writing easy user user specify extraction allow bring optimization implication semantic way noise semantic default semantic give semantic evidence define weight semantic logical tuple user schema predicate variable supervision part specific class evidence assignment negative ease exposition domain boolean predicate variable rule substitution variable replace conjunction fact rule three q real add weight indicate world less likely motivated semantic boolean world allow framework compactly specify distribution database illustrate web could extract one vote think variable relation indicate resp vote size consider vote close semantic depend vote semantic give logical ignore voting level semantic raw count ratio raw semantic ratio semantic suitable want semantic theoretically even logical semantic less write resp expand way symbol create create weight allow model g indicate tie logistic formal weight return value explicitly construct learn triple node correspond identify possible tuple correspond datum outside database database rule ground semantic probability follow factor statistical value compute return system tuple efficient run incremental phase evaluate program delta modify factor modify relational operation incremental incremental inference change advantage decade input query schema output modify view technique schema additional tuple represent update delta relation tuple update delta delta execute generate modified variable overhead gain load present incremental produce incremental two phase access entire attempt information store call phase precede variable factor respect change study tradeoff infeasible use explicitly store fidelity store infeasible moderate sized world perform speed arise factor store update store hasting scheme store
adapt target design away prediction expensive another measure symmetry upper mse mae lift bound hypothesis cm la paris france universit paris paris france universit paris centre en de paris consequence percentage regression find weight mae universal mae study paper goal quality traditional application quality q risk choose accord opposed mae mse practical well order minimize mse rather determine theoretical see obtain adapt complexity independently copy set addition problem indeed fix weight therefore support weight use notice verify car mae mse simple stop car goal change result summarize expect loss relate practice challenge mse mae view uniform function introduce give notation supremum cover controlling supremum cover function unless assume mae cover number easy q get class classical mae bounding assumption case bind need indeed eq expression role hand might hypothesis covering role play mae interestingly replace mae vc dim indeed yx yx vc dim equation erm
algorithm converge fixed point increase avoid cycle expect normalize message follow minimum would stop reach threshold propagation propose uniquely dna resolve difficulty continue throughput short read sequence counting accurately repeat exceed example version population np copy contain protocol randomly assign one letter probability binary template copy template introduce per starting read generate determine template assign regime expect read template hamming read template bit introduce error rate per bit obtain computer template various determination count even high determining template bit generate read rate perform accuracy mis algorithm average simulation hamming distance incorrect red hamming distance correctly edge regime property propagation balance impose question dl grant work grant foundation award prior cluster real equation become section give x h ik h spin degree however pairwise spin hamiltonian transition critical critical vast spin aware alternatively configuration constraint run partition blue uniform partition recurrence principle use compute intuitively effect favor quantify average behavior blue fraction see phase sort balance consideration weight configuration function order estimate entropy cluster limit balance phase impose partition cluster cluster configuration recurrence configuration definite order prior knowledge recommend uniform choice simplification eliminate since message eliminate message give simplification dependent moreover shall difference write explicitly contribute contribution effectively eliminate arrive involve ahead ij become author cluster pairwise belong triple assign implement sequence random word noisy channel cluster cluster paper wherein govern describe describe assign zero configuration likelihood pair assignment different sufficient ensuring constraint acting triple determine affinity propagation graph message pass configuration complexity usage rapid force net force require calculate distribution illustrate trivial future constraint e pi pi edge assign edge hypothesis matrix cc belong blue edge count point triple triple figure choice consequence hypothesis calculate denominator solution maximize result drop effect result operation ij interpretation energy minimum term force pair minimize solution applicable datum separate decision edge condition energy section represent configuration maximize equation graph variable neighbor every represent square every variable graph depict
transition different wide series nest well drawback complex predictive proper selection procedure appropriate natural unfortunately fail regularity nan hypothesis aic bic introduce estimate ar general inaccurate criterion specifically gap identify describe add new improve goodness base non goodness fit maximized criterion process smoothly root whose polynomial evolve behind observe knowledge remainder organize follow gap state section specific model state assume markov emphasize parametric consider propose multi new effectively impact bad initialization maximization present approach point curve new distance turn ar fix approach ar outline selection elaborate subsection ar filter subsection distribution symbol bold face generate single filter instance x eq subsection selection criterion add goodness ar predictor filter ar stable within distance simplify curve criterion root determine filter intuitively root characteristic small filter h ar model algorithm compute root ar denote filter uniformly curve fig ar h element filter characterize center w w examine filter close update subsection explicit formula distance assume generate stable filter use integral conjugate assume mean curve reference general reason ar reference measure ar filter explicitly filter generate notation power reciprocal multiplication let give determinant associate identity b determinant mention statistic require calculate filter sample generation obtain I extent reveal equivalently summarize lemma procedure long generate recursion k z z z z behavior model hmm hmm px method ar transition series function auto assume brevity unobserve indicator nm denote multinomial word tractable maxima em step predefine criterion brevity unknown expectation side take value old omit brevity cause away optimum initialization choose time consume new initialization reliable technique series economic probability state close hence adopt initialization retrieve em ar mean update achieve normally around style elsewhere et al filter cluster curve curve maximize synthetic scenario scenario scenario ar generate transition mm algorithm ar aic represent gap filter root inside unit independent gap statistic gap filter root inside
strategy finite strategy allow maximally include win exist proposition sufficient existence win characterization extend necessary precisely maximally strategy property environment characterize precisely property check satisfy exist strategy general win sketch relevant briefly take n formula game win construction additionally offer check win condition property translate construct tree commonly property get state accept tree never visit state visit check checking solve answer win proposition maximally existence game special proof limited winning formula maximally solve number software extract strategy gr condition maximally extraction application greatly simplify enable performance strategy non thus game remove game counterpart induce vice versa win win induce counterpart run result win acquire correct respect specification move strategy respect priori instantaneous reinforcement reinforcement discount work form optimal win reinforcement discount reward case concern strategy implement word act game equivalently environment system strategy loss algorithm markov game choose well learn interact condition discuss ready maximally strategy win divide formula win system maximize maximally game ss reinforcement maximally strategie compute game include win win win preserve win strategy subsequent correctness requirement reinforcement algorithm win strategy summarize proposition maximally expect proper win many include win intuitively would bad demonstrate robot motion plan different win game first maximally compute strategy strategy robot square turn know cell robot go adjacent cell go adjacent stay current cell robot avoid environment always observable pos pos pos pos pos pos I leave stay ta change atomic proposition l j k requirement I maximally reward robot ahead numerical encourage robot reach environment possible available robot advance reveal instantaneous take cm spend extract action tuple ghz greedy simulation result adversarial environment system robot position environment strategy environment reach position step show converge coincide optimal strategy iteration example construct game win robot system robot visit corner infinitely often low cell instantaneous remain maximally move win ht example trade win system visit cell say infinitely gr game way add counter control move visit cell maximum satisfy extract maximally force visit strategy extract maximally maximally strategy increase game allow counter discount maximum discount system counter trade learn counter discount reward n study respect logic criterion unknown infer idea synthesis reinforcement provide specification need planning fact corollary synthesis priori interact specification subproblem maximally way satisfy specification quantify priori reward use establish correctness logic specification respect unknown technique specification specification correctness preserve sub overall demonstrate requirement motion plan adversarial logic specification criterion seem effective supplement description hand concern rule specification temporal quantitative criterion help encode subtle application system requirement e jump light criterion specification design human synthesis specification reinforcement unknown solve synthesis focus static know dynamic crucial nearby adversarial environment strategy criterion objective logic specification deterministic environment environment objective guarantee adversarial quantitative payoff crucially rely quantitative gain experience direct interaction coincide multiple reward use application process specification expect modify case paper optimize priori temporal specification decomposition subproblem part encode adversarial satisfy specification quantify apply reinforcement operating envelope synthesis guarantee satisfy specification win concept rest model care system external interaction control play critical correctness specification discuss tuple control action finite action win action correspondingly state control set action available game state exist assume logic evaluate take specify otherwise state infinite finite memory strategy win initial formula win qualitative win model reward maximize choose evaluate system nonnegative consider accumulation instantaneous run game add instantaneous acquire instantaneous reward reward acquire weight example payoff function instantaneous independent define strategy give used run environment strategy interaction experience environment another use game win condition describe formula player game win maximize give instantaneous necessarily exist win strategy maximize win specification w fig r cm auto white edge loop leave loop optimal winning
density estimator density memory requirement small requirement entire present mixture logistic gps lda competitive root bayes kullback sequence bayes differentiable appendix strongly divergence compactly sampling normalization integral meanwhile numerator perspective scale inference leverage advance optimization resort particular mirror descent gradient unbiased stochastic bregman mirror descent minimize draw stochastic mirror iterate prox density divergence prox prox resemble rule furthermore pass arrive appear prox stepsize imply mirror scan dataset iteration issue tractable may mirror prox provable convergent particle later map eq update reduce usual stochastic mirror regard mirror inexact prox mapping step give recurrence sub q sa scheme average prove experimental suggest behavior last iterate prox essentially involve intermediate introduce particle already guess deal case initialization interestingly two yet good guess cover sample intermediate particle q form prox mapping ignore unnormalized version working incur approximate integrable latent variable lda incorporate guess could summation several difficult way lead particle inaccurate develop estimator leverage mirror suppose approximate smoothing derive mapping location solution version location associate normalize weight function working show possess formal old kernel kernel bandwidth om kde achieve rate stand lipschitz far linearly kde dimension weight solve prox section mirror weight estimation appropriate particle location divergence ht pt input density px present mirror descent incorporate posterior iteration maintain exploit benefit prox either weight therefore connect carlo particles integral return stepsize algorithm monte smc importance smc gradient smc utilize visit also share algorithm approximate density density product efficient make scale assumption rate sublinear sequence sublinear posterior term convergence return algorithm give difference commonly monte langevin may proposal integrable stepsize particle mirror consist integration optimization particle overall convergence integral true density generate later kernel function old almost surely boundedness depend assumption especially automatically validate bound convergence inexact prox mirror descent main state stepsize kernel tc b om apply b solve recursion sake sample convergence ratio number grow divergence average decay decay first directly achieve conduct bayesian process dirichlet multiple mode deal conjugate sequential langevin dynamics sgd langevin variational variant inference lda detail please observation p tie make mode pt ccc pass langevin ccc smc bandwidth theorem batch burn langevin repeat times recover method fit fail multimodal density kind quantitative way visit understand behavior perform begin utilize stop noticed smc start particle langevin dynamic bad one fit mode sgd langevin contours langevin find mode simultaneously sgd optimize inaccurate totally dependent logistic regression conjugate handwritten digits classification mnist function identity first period langevin initialize prior distribution pass whole time repeat obviously gibbs need scale well search notice achieve comparable performance nonparametric datum carefully flexible bottleneck sgd optimize ccc sparse gp wikipedia year conduct gps smc approximation represent randomly select year map standardized induce hyperparameter sparse fix bandwidth median distance point particle demonstrate advantage initialize initialize report pass baseline synthetic demonstrate sparse conduct generate rbf times batch illustrate evolve datum see gp well mean sparse cccc iteration ccc lda wikipedia dataset document vocabulary estimate separate document since follow particle smc save set fix solely pass stepsize default burn provide search
contrast fine domain unseen domain table contrast pos dependency f bt l feature tb l pos google gram word syntactic syntactic pos embed syntactic embed syntactic pos cnn word dependency bilinear embed publish compare previously publish compositional ne tag tag help may make expressive introduce distinguish embedding fine report task traditional cnn less next feature template head surprisingly show binary template remove degradation remove entity need md attain nothing strong move engineering hand capture linguistic word strength relation back tag semantic back relation move lexical word embedding word dimensional space back designing easily incorporate incorporation dense value sentence certainly challenge mention extraction consider limited hand property contribute utilize compositional deriving sentence level word embedding compare compositional arbitrary type alone state art extraction obtain art approximation embedding open domain sentence contribute component parse application gold entity prior instance entity labeling pair instance gold train type ignore relation token report domain report additional task entity semantic entity entity report comparable super tag entity paper use tag table entity type greatly remove contain entity baseline drop gold entity unknown play role improvement become context head dominate make entity type unknown baseline tag resource encourage entity predict gold bc pm pm pm baseline pm c author md intelligence technology china compositional linguistic rich compositional extraction expressive domain idea learn embedding able tackle difficulty meet compositional embedding handle arbitrary sentence annotation global propose relation extraction compositional traditional rich relation tp sentence drive united people drive depend word generalize g operate lexical insufficient extraction lexical appear sentence entity significance fine information nlp feature word extraction lexical word linguistic help capture lexical embedding embed task parse semantic extraction capture lexical insufficient linguistic context compositional model linguistic embedding extraction contextual feature construction generalize composition linguistic begin construction compositional use feature capture compositional treat word embedding rise annotation utilize associate annotation stage annotate sentence annotation combine word sum compose softmax layer output dataset relation goal identify pair construct sequence tag name entity direct relation type towards embedding compositional nlp relation still e relation sentence extraction name entity boundary type available assume two entity relation standard within sentence entity complementary illustrative example develop representation capture word entity incorporate work use entity feature insensitive show word propose framework sentence annotation focus relation extraction benefit lexical annotated embedding embedding special subsection annotate sentence first vary sentence hand vector matrix herein annotation annotate sentence distinguish annotation task utilize extraction sentence embedding input case relation extraction annotate sentence many nlp highlight annotate embedding specific form polynomial specifically literature suggest powerful directly log model form annotate embedding previous annotate sentence produce entire formulate dot product matrix normalize matrix model fix bilinear feature drive indicate appear label embedding drive dependency label dependency path weight generalize across word share product smooth lexical nlp lexical lexical property lexical word outer recover word form therefore view lexical keep expressive cnns rnns zero sentence cnns rnns optimize word embedding gradient application equal gradient bilinear deep embedding training easy incorporate separate feature use feature vector path head refer entity name head two conjunction indicate entity whether entity feature entity result help embedding predict introduce lexical embedding embed linguistic annotation pos stanford gold entity entity tag embedding train portion corpus l set head h
code region optimality sequence convergence convergent minimizer well minimizer respect irrespective code outer large enable insensitive even convergence result theorem correspond except theorems corollary theorems blind compress sense cs involve patch level tune contrast involve set mr berkeley simulate include encode reconstruct step hard ball well p slightly transform learn well unitary transform application compare specific work formulation algorithm behavior reconstruction variation transform overcomplete recent use redundant geometric early method software implementation respective build wavelet image use image solve size suggest four fold overcomplete learn iteration overlap find empirically execute iteration per employ sparse code dictionary threshold patch vary linearly set patch patch dct initial fourier code matlab version implementation optimize execute computation perform intel core cpu ghz gb operating quantify mr reconstruction signal express db peak image root relative reference reconstruct fully k quantify symmetric whose deviation pixel filter reconstruct reference magnitude learn complex display reconstruct raw weighted brain raw fast spin sequence te fold peak reference execute converge quickly show e successive iterate converge theorem metric quickly initial measurement scenario db hand final reconstruction enhance db db problem identifiability noiseless scenario learn transform part frequency image experiment execute lead variable density normalize reference correspond zero reconstruction reconstruction various scheme well mark see various significant improvement db transform partially finally quality fold overcomplete rich show reconstruction difference magnitude reconstruct cccc reconstruction magnitude update since arbitrary establish lemma successive iterate converge subsequence index iterate converge accumulation convergent say argument respect convergent subsequence monotonic decrease lemma right together set patch always minimization unique minimizer combine work convergent subsequence finally subsequence accumulation coincide accumulation converge proof consider x convergent subsequence thereby inferior superior bound convergent subsequence subsequence convergent sequence convergent subsequence next accumulation set accumulation denote state subsequence iterate converge accumulation partial accumulation point simple sequence every accumulation svd accumulation iterate algorithm involve code every outer outer update use consider index accumulation full x due limit column l h l b subsequence aforementione square root svd subsequence get immediately establish accumulation iterate critical sub matrix accumulation algorithm subsequence iterate converge accumulation lemma singular next g finally eq easily thus accumulation establish accumulation point accumulation iterate small region subsequence index iterate sequence accumulation lemma accumulation perturbation perturbation preserve utilize equation side replace hermitian operation involve long whenever easy preserve support zero b therefore accumulation sequence region subsequence accumulation lemma consider perturbation column denote order perturbation b trivially energy preserving follow expand drop argument equation unique argument simplify scenario preserve perturbation b minor barrier set replace operator theorem differ local perturbation particular accumulation extend proof lemma finally theorem negative barrier unitary keeps otherwise national foundation nsf grant correction email signal know property heavily exploit medical imaging sense exploit domain synthesis reconstruct measurement blind compressed reconstruct well transform descent involve closed form importantly although blind sense formulation nonconvex converge objective define formulation usefulness image reconstruction highly measurement involve model provide medical compress sense convergence extremely popular application exploit sparsity patch reconstruct investigate subject aim scenario good image transform briefly review topic model blind compressed contribution base image various synthesis dictionary give disadvantage synthesis sparse deterministic hard various tend large transform approximately transform residual error rather image analytical cosine transform wavelet difference advantage synthesis compressed version fouri fourier recovery acquisition incoherent sense sense expensive linear compressed sense typically representation stacking measurement sense measurement typically choose orthonormal satisfying transform domain equation represent true rewrite substitution hermitian transpose synthesis np quasi replace convex problem reconstruct image cs fidelity depend physics recently image technique ct emission imaging demonstrate quality reconstruction compressive advantageous reduce scan time clinical throughput well reconstruct subset pixel compress sense compressed technique utilize wavelet reconstruct instead focus blind compressed compressed simultaneously enable datum drive dictionary transform advantageous application adaptation synthesis study transform prior successfully demonstrate compressed sensing image overlap patch typically learn compressive measurement much compressed method surprising method specific adaptation one primarily focus synthesis transform model involve transform simultaneously highly measurement propose transform importantly converge critical highly minimizer compressed compressed discussion work measurement sense denoise deconvolution imaging technique mechanism excellent visualization space acquire drawback affect clinical especially dynamic imaging application relatively slow imaging technique advance hardware mr limited mr physics constraint energy compress either aforementione enable reconstruction mr reconstruction transform reconstruction involve transform importantly synthesis reconstruct speedup consider mr amenable clinical describe transform blind compressed formulation efficient coordinate problem present novel scheme proposal compress reconstruction view particular constrained regularize reconstruction adaptive transform suffer high blind compressed directly object overlap patch assume patch along fidelity patch synthesis number sparse column learn additionally avoid scale ambiguity consider strong flexible patch code sparsity practice appropriate coding sense measurement standard level estimate synthesis dictionary reconstruction adaptive convex hard g synthesis code repeatedly convergence overcome aforementioned drawback synthesis transform transform effective learn constraint within arrive transform patch patch denote term number notice enable level patch p enforce range transform patch denote inverse trivially transform prevent degenerate repeat help ambiguity sparse representation penalty cause penalty together additionally control scaling learn regularizer condition I easy transform tend unitary learn via even transform previously well strictly orthonormal scenario representation denoise patch range well learn unitary transform regularizer follow formulation unitary transform problem p unitary sense jx jx simple consider error unitary w global p solve jx possible low notice satisfy triplet feasible case minimizer patch constraint unitary guarantee solve lack image minimizer propose admit code minimizer another minimizer permutation alternative problem replace problem weight p use absence space code objective finite constraint feasible potential unbounded iterate version constraint propose extend propose transform formulation scenario single extend image frame slice jointly reconstruct transform p objective video formulation extend compressed block formulation alternate code transform variable alternate step describe detail sparse involve follow transform patch notation frobenius set choice large choose unconstrained solution case index column page definition choose solution formulation variable keep p analytical term decomposition definite give eq solution invariant factor cholesky factorization eigen form nevertheless scalar assume convergence one standard accuracy aforementione give global minimizer non singular problem least alternatively solve exactly lagrange multiplier corresponding lagrangian formulation lagrange satisfie transpose real matrix unique algorithm patch direct sized gradient lagrange multiplier optimally j tw hx optimal multipli monotone guarantee rate solution practice loose solution minimizer obtain cg additional computation avoid way find repeatedly tune employ cg various matrix exploit enable show assumption efficient application structure toeplitz efficient case patch formulation overlap overlap opposite distance location clear patch around proposition establish block fourier encoding include column matrix except tw entry overlap image apply shift version correspondingly shift operator circular convolution sum standard state typically diagonal computed first factor case unitary assume matrix arrange first entry column patch operation correspond corner zero elsewhere extremely negligible aforementioned finding dft impulse efficiently fourier modality obtain obtain subsample grid multipli computed computation assume around image include matrix multiply diagonal q location lagrangian space location need newton obtain update fig would specific dct measurement sparsity iteration adapt transform code overlap patch repeat full svd hx l l set column generic p h z image ts w code transform iteration computation square root operation code therefore scale notably projection ball sort hard cost transform operation fraction zero application I etc cost patch pixel cost dominate hand various operation multipli latter take typically newton argument total
index column tolerance sample sample row random index block k ic general psd finish need prove guarantee form fully show compute linearly guarantee enable calculation computing discuss psd alternate proof prove linearly remark linearly hold terminate early early termination choose column nystr om I x x low right exactly rank vs random sampling b terminate trial exact guarantee recovery choose redundant accurate variety column contrast choose redundant rank gram nystr om precisely combinatorial nystr om indeed guarantee exactly recover column expressive although recovery step center psd rank column rank iterative compute column terminate machine theoretical guarantee guarantee selection redundant separate trial sampling sampling frequently select span step limiting equation enable must considerably computation result score computation dense nystr om mean form full result useful nystr om method gram finding slow uniform speed compute way regime complexity p slow regime make uniform competitive form many substantially expensive adaptively parallel appear column selection third adaptive advantageous run see low complexity accurate low runtime see nystr om approximation nystr om form impractical third problem class consider contain sum tune om random ii leverage nystr om iii repeat experiment uniform intractable store uniform sampling describe matrix fit memory sampling method tractable generate calculate consider dataset use matlab processor large competitive accurate convergence vs sample show vs second rate column second curve fair see consist point arrange kernel point characteristic age without dimension around cube vertex distribute cube impractical explicitly rather sample frobenius discrepancy score full representation compare follow matlab ghz processor gb show implicit mean mnist problem handwritten benchmark mnist datum contain image pixel similarity point hyperspectral band classify area assign class ground represent assign class light field dataset intensity plane patch stanford camera array angular point split uniform long frobenius discrepancy randomly entry eigen node core ghz processor core table size available increase among point sample two tolerance random capability subset consist size reference store binary color channel focus determine kernel maximum become approximation trial image kernel achieve uniform sampling leverage score full addition nystr om addition deterministic enable adaptive scheme know priori column leverage one must guess appropriate certain primary scheme accuracy example uniform random sampling well continue add figure second efficiency dimensionality reduction kernel nystr om clusters compute centroid compute kernel mean overcome observe flat leverage entire grow large primary runtime accurate give mean cluster gain nystr fast run multiple time nystr second run take furthermore sample selection invertible sampling calculate step gb infeasible column intensive straightforward appear higher random fast column take regardless selection compute computing data form less take residual sparse extremely matrix store novel om approximation requirement random exact demonstrate efficacy matrix processor size regime numerical competitive scheme able approximation cluster remark conjecture proposition g adaptive g gram approach reduction form intractable incoherence without guarantee recover numerical achieve adaptive cost complexity low psd machine machine framework require formation contain similarity nonlinear storing grow matrix increasingly store extend extremely nystr om rank keep capture majority ambient lie approximation information nystr compute broadly selection apply application range segmentation factorization success identically underlie sampling draw practice sampling far efficient uniform advantage entry adaptive computational burden reason entire form store require even store zero reason apply extremely collection possible kernel form candidate kernel matrix small sample un column principled predict informative matrix form operate select explicitly submatrix dimension make order runtime recover preserve sample column enable efficiency computation fill provide tractable adaptive apply accuracy comparable dramatically usefulness long entirely work processor efficiently column incur overhead node message pass interface om regime addition exactly recover greedy guarantee linearly independent inefficient organize introduce nystr survey important motivation behind column incoherence sis accelerate sis call theory determining exactly demonstrate efficacy approximate common machine nystr om describe write respectively denote product row sum describe matlab indexing index widely classification cluster lift linearly separable help measure frobenius accurate combinatorial complexity adaptive build nystr select residual large update accurate residual calculate sufficient obtain make point consist cluster centroid centroid describe zhang compute centroid exist mean np hard cholesky sampling parameter reveal low use example self expressive seed select point match pursuit seed properly dictionary use adaptive adaptive subset complexity address om approximation sequentially select nystr om lie column already add ideally significant impact span column incoherent already criterion find incoherent psd recall contain x ix select expand base upon denote second comprise index contain approximation entire apart sequential incoherent follow collect index element maximize value terminate
eps eps comment procedure kernel unbounded correlation difficult problem provide issue dimension eps eps wherein design inside section kernel suppose center optimize second figure contour length optimal move correlation center information length decrease become away reasoning show place area average eps eps eps besides another way length cost several experiment perform try another objective experiment lie exist point information length lie cost exist optimizer good phenomenon demonstrate minimizer near minimum eps eps eps numerical show word influence geometry evaluation adaptive closed loop procedure design experiment evaluate update hyperparameter g log function maximize gradient base sequential programming couple design describe adapt batch batch change rapidly batch leave batch size overall sense greedy design context maximum proceed batch greedy function hypercube endow weight exponential kx il l impose adaptive batch hyperparameter inherent error entropy adapt design batch hyperparameter outperform however marginally achieve error decay batch hyperparameter optimization experimental domain describe hyperparameter domain ill place new close size initially large subsequent design feedback large counter robust situation hyperparameter unbounded gp another understand approximation ever gp quadrature appropriate comprise advantage onto subspace basis sense grow exact quadrature seek inner q still external necessarily span rule subsequent express orthogonal version basis polynomial quadrature rule already decay interpret modeling impact regression space approximation rule gp kernel kernel function eigenfunction connection comprise difference gp identical immediately corollary case eigenfunction infinite consist eigenfunction quadrature rule clearly among generality quadrature posterior kx difference approximation source eigenfunction span contain gp case equivalent space gp simply due yield approximation example kernel construct fully polynomial eigenfunction quadrature rule bind depend condition sufficiently situation poorly condition difference approximation imagine invertible quadrature whereas proceed manner approximation experimental posterior eigenfunction quadrature splitting eigenfunction second term expansion contribution extra eigenfunction integrate comment special property eigenfunction thus integrate explicitly enter gp rewrite eigenfunction eigenfunction expect design eigenfunction error incur numerical eigenfunction eigenfunction order design ensure integrate capture precede underlie practical implicitly assume projection otherwise indeed vast regression properly expansion rkh accurate readily interact integrated variance ensure span eigenfunction correspondingly force measure uncertainty retain contribution subspace difference approximation extra tradeoff generate perform regression gauss quadrature polynomial degree kx ix normalize polynomial decay surrogate experiment closely surrogate approach evaluation approximate infinite eigenfunction emphasis project external figure error approximation gp improvement associate figure onto reference right panel show energy basis magnitude projection compute extremely high quadrature onto even projection basis begin decay error index function begin decay somewhat slowly basis lie red line error gp design spectrum error grey panel quadrature trend compare evenly design error spread broadly ht surrogate gauss quadrature gauss quadrature magnitude space similar experimental contain rank gp eigenfunction numerical like gp gp regression principle gp wide eigenvalue kernel endow standard eigenfunction polynomial version I eigenfunction present section adaptively greedy sum approximation regular infeasible grid dimension grid provide benchmark mit quantification framework domain endow gauss quadrature additive separability favorable use batch loop noise result trace relative f mx carlo h eps right hyperparameter trace design panel trace design adaptive evaluation begin error precision quadrature hand gradually additional add gp already trace fairly fairly indicate fast decay converge eigenfunction converge eigenvalue quadratic hand order order observe decay roughly dimension matter next note expect see well example product eps eps begin roughly evaluation error approach several gp reach relative reach attribute require interaction interested repeat hyperparameter adaptation length randomness choice start figure show square exponential typically hypercube domain approximation unbounded weight significantly challenging tail error gp using show well benefit include eigenfunction calculation batch obtain experiment future adapt iteration fall basis require gp family truncate kernel eigenfunction finite kernel adapt criterion surrogate design design criterion present integrated process continuous spaced avoid undesirable interpolation design substantial benefit strategy nonetheless greedy standard minimizing demonstrate adaptive update gaussian approximation simplicity gp use eigenfunction difference eigenfunction projection average quadrature quadrature node performance adaptive gp easily approximation couple domain current approach eigenfunction integral eigenfunction couple present work experimental way interpolation node radial basis compare quadrature great broadly rigorous reasonable ahead design adaptation posterior may effective author acknowledge research notation associate separate denote rest orthogonality rule mx ax define square orthogonality orthogonality property simplify expression take eq equality equality substitute cauchy matrix multiplicative j ax cauchy schwarz orthogonality begin trick replace come orthogonality equality arise arise come split mit edu experimental design procedure explore use gp minimize posterior integrated gp treat domain point good interpolation entropy mutual second perspective gp identify regression coincide polynomial orthogonality eigenfunction approximation set adaptive approximation function favorable experiment interpolation quantification computational essential design quantification system require large prohibitive expense surrogate relevant analysis computational simulation surrogate experimental view attempt input relationship obtain suitable choose contain parameter convert simulation include span radial experiment designs design mutual quadrature linear independent procedure preserve orthogonality relevant design surrogate space use approximation analyze minimize design process criterion criterion broad surrogate difference space finite call minimize involve add experiment candidate criteria mi alm criterion maximize alm criterion consider effect well design expensive mi sequentially maximize gain mi obtain domain complex direction computationally expensive deal combinatorial optimization explore ad hoc optimization sometimes design input help domain create barrier benefit complexity minimize undesirable cluster radial alm well approximation performance alm mi finally perform continuous good lebesgue raise quadrature use eigenfunction kernel eigenfunction assess design eigenfunction eigenfunction entirely integrate illustrate qualitatively quadrature select experiment outperform experimental test outperform approach paper organize part part review section section describe second background describe comparison gp process approximation posterior begin covariance semidefinite suppose simulation parameter evaluation result notation component covariance interpolation however ill condition result eigenfunction recall countable endow borel suppose eq operator countable eigenfunction form eigenfunction eigenvalue reproduce rkh kernel represent convergent let first countable locally compact strictly series converge absolutely compact subset regression later use kernel equivalently q polynomial viewpoint ill practically singular precision decay integrate point integrate become prior always integrate prior experimental integrated gaussian process choice motivate inferential consideration oppose procedure quadrature large add experiment location commonly call alm change h f mutual maximize greedy fashion candidate experiment greatest entropy mi perform greedy inversion contain set simulation set candidate invert effort aim reduce cost specialize kernel mi depend crucially candidate base differ approach avoid challenge follow optimization descent multiple experiment simultaneously design advantageous interaction account design take beyond early help design domain proximity location also address problem correlation kernel sufficiently expensive find design procedure one effort require experimental summarize computational experimental objective consider scenario carlo sufficient mi criterion candidate evaluation typically alm must location inversion evaluation design carlo pt mutual pt alm option minimization continuous candidate discrete design mi remain tractable possibility choose design play important role numerical objective write q become increasingly must similarly th data informative mode
word top near topic eight typically respectively understand difference like cancer topic return medical care share semantic fashion topic topic distinguish result lda qualitatively lda firstly explore structure inherent topic simultaneously acknowledgment cn mining document play language however relationship occurrence corpus good word essential representation lda representation topic space alternative show interesting model text document language nlp ir enable benefit similarity relevance past decade solution bag tf semantic probabilistic latent semantic know allocation relationship word document distribution word occurrence document lda high probability probability choose representative drug technology probabilistic language represent word document dimensional nlp word propose could syntactic relationship like paris state sentiment analysis paper answer semantic idea propose propose topic incorporate topic topic semantic relevance use cosine representation aspect list topic result achieve allocation lda latent vocabulary lda generate n word topic sample topic document infer latent meanwhile inspire skip word sequence maximize base skip gram skip window calculate representation w incorporate representation skip situation predict word skip predict word document infer maximize likelihood gram pz skip topic approximately maximize probability softmax without softmax stochastic sgd model dataset word english learn fundamental extract document choose document besides word word training contain million word
compatible adopt different problem investigate various user requirement wants meet practically reach statistical cluster respect generate first base technique result complexity investigate obtain standard setup supervise representation implicit cluster quantify define pac learn introduction notion capacity complexity combinatorial notion version mapping losse true embedding minimization erm successfully particular mapping paper organize section define investigate erm algorithm uniform sufficient present conclude provide subset denote cardinality respectively induce accordingly difference partition fx centers center minimize cost mapping mapping mapping formally mean let every pick mapping generating readily cluster output pac stand probably pac mapping representation size least regard formal pac problem good mapping class intuitively rich mapping cluster introduce appropriate address bind representation need erm learner sample implicitly access go minimize note studying formalize representative mapping respect cluster representative uniform convergence mapping therefore upper representative solution find mapping clustering learner interpretable mapping way include mapping solution concrete define notion introduce slight uniqueness say k mean cluster satisfie satisfy eq degenerate solution uniqueness require cluster arbitrarily small totally uniqueness mapping mapping uniqueness argue subset useful therefore rest paper next prove class representative algorithm feed make sure actually representative formalize mapping sample select paper devote provide mapping exist number small specify interested note mapping real prove complexity mapping beneficial capacity help easy analysis sample complexity class vc output function number pseudo generalize notion permutation binary union formally set able investigate mapping representative I value provide vc however provide dim sense class next introduce prove purpose uniform argue care class section uniqueness lemma previous let cover k basically cover mapping uniform result mapping constant cover hypothesis precisely e theorem number mapping notion mapping define let real define let l q value pseudo reason cover cover cover rewrite uniqueness show sufficient sample follow combine pac mapping logarithmic proof mahalanobis metric mapping result pseudo value dimensionality case value scale factor statistical representation mapping target cluster cluster finding mean pac notion erm technical uniform result complexity mapping notion mapping complexity uniqueness reasonable learner mapping unique otherwise interpretable uniqueness pac notion result define make challenging analyze open question acknowledgment proof uniqueness property note due eq term prove center ready first small cost follow lemma second david david school knowledge propose protocol relatively come datum align formal analyzing paradigm capacity spirit vc representation learn induced embedding task divide set coherent subset serve apply dataset likely dramatically answer critical
ergodic reversible spectral gap provide stationary random define version q spectral form spectral stationary eq length within multiplicative require average expectation average form markov chernoff bound result significantly bad requirement instead adapt tail iid directly article yu ergodicity state eigenvalue use frequency eigenvalue fact reversible difficulty invoke page whose exponential th root avoid return question fully notice directly suitable arbitrarily possibility width increase effect chain slow mix small confident empirical subject achievable sample path compute visit smoothed empirical bound sensitivity confidence stationary idea dependence interval include probability path markov frequency state visit lead slow rate strong inequality estimate form plug form mixing indeed estimate exploit confidence smoothed frequency call accuracy ergodic chain smooth sensitivity plug form spectral relate relate perturbation entirely observable g estimate computationally note reduce matrix implementation state concern input path ergodic reversible chain confidence spectral unique stationary q moreover obstacle encounter avoid establish observable martingale tail comparison validate bound remain part simultaneously hence interval empirical determine lower plug bound center confidence detail new interval u v furthermore surely apply generally I asymptotic random deviation help bernstein tail due divide contiguous block size resp let dd position inequality contribution block note ahead observe copy apply copy follow sequence let suffice surely block q norm sum cauchy schwarz gm simplify expectation third place last step bind result yu sx h product form marginal joint yu imply event denote recognize process chain integrate give time proof q conclude eq last step follow return combine obtain least combine probabilistic condition least probability finish replace observe recall trivial violate chain chain must markov swap construction indistinguishable eventually reach path length c chains eq ergodic reversible chain j di regardless must visit distinguish chain stochastically dominate generalized number random geometrically least visit smoothing positive ergodic unique stochastic martingale measurable union view constraint unobserve event hold avoid spirit quadratic conclude claim stationary unique eq matter start role central capture ergodic transition role j define analogously perturbation comparison p ji j establishes interval bound quantity q p q entry bound norm validity interval li improvement continuity proof term state sensitivity let compute bind fully yield derive complete claim proposition key true lemma definition bb c university article provide fully dependent prescribed stand previous either require additional interval zero length path sample require constant multiplicative low restriction place chain procedure achieve direction research work challenge fully markov irreducible finite stationary time assign trivial nx x scientific interest arise involve mix effective estimation quality knowledge develop construct non trivial empirical confidence reversible ergodic suffice main summarize guarantee multiplicative visit provide estimation estimator multiplicative notation feasibility interval unknown quantity chain task explain turning fail theory avoid interval valid vast literature markov instance converge surely central distribution deviation zero asymptotic nf result help limit behavior mean chain little behavior need evaluation numerous chernoff provide probabilistic corresponding bound identically due temporal dependence intuitively draw bound chain effort unknown context provide estimation asymptotic hence path another theory sufficiently yu mr providing case however estimate derive mix coefficient limit possible eliminate difficulty flexible sampling oracle generate independent device mix hand circuit transition probability exponentially large diagnostic integer expression exact elsewhere symbol
establish polynomial np clique show clique instance give input vx opt opt prove minus top write eigenvalue equation solve quadratic root expression decrease removal stronger exclude constant natural version clique subgraph subgraph maximum ds admit algorithm vertex every algorithm one determine polynomial determine clique distinguish clique graph construct let adjacency run solution clique contain clique opt clique radius graph edge distinguish clique every efficiently clique degree ds constant one clique subgraph vertex polynomial plant approximation plant clique definition axiom mm mm give clique establish exclude np weak exclude classic tool challenge interpretation component combination original significance application desirable obtain goal hardness clique traditional cardinality maximization sparse exist I radius eigenvalue resp zero resp exist clique fairly clique need order clique
code symbol symbol decoder distortion prescribe fidelity specifically ny alphabet fidelity letter operational indirect distortion block indirect shannon infimum mapping nd source code operational indirect rate direct observable indirect distortion fidelity criterion shannon source theorem imply q indirect indirect direct distortion computation alphabet perform transition probability letter introduce constraint lagrange dual give transition rhs non satisfy necessary achieve maximum number satisfy satisfied equality hold context auto source node noise plus channel hamming fidelity receiver correspond give source reconstruction view remainder infer fig decrease hamming distortion correspond case define distortion measure distortion intuitive interpretation make list step solution outline proposition equality imply maximize rhs derivative non decrease single conclude rh equality rhs special substitute q symmetric compare observe correspond domain decrease therefore increase reduce correspond vertical dash slope determine return unit system confirm increment bit describe less effective distortion intensity rate coding need provide tight convexity closure illustrate theorem summarize rh indirect hamming distortion give noisy channel indirect distortion investigate bit although conceptually model important distortion level error balance existence treat lead write derivative domain monotonically whenever since behavior root substituting remark department electrical stanford department electrical national height circle fill blue cm bernoulli compress manner consider noisy symmetric control classic distortion distortion function source rate distortion term indirect distortion close expression return increase bit rate distortion channel distortion allow average extension scenario source directly obtain noisy observation environmental source process statistically know motivation indirect source centralize restriction computational consumption local infeasible distortion fidelity criterion square describe description allow quadratic general possible paper binary hamming distortion introduce shannon provide source indirect indirect implicitly equivalence indirect problem fidelity fidelity identify compute indirect source
statistic reproduce hilbert mapping optimally learn pair scoring turn asymptotic see anomalous rank advantage false alarm nominal level preference sort x sort anomalous standard setup characterize high sec step rkhs kx cross learn adopt weight pairwise disagreement loss quantization complexity high training complexity closely ranking nn distance noisy datum raise fairly sec insufficient false alarm demonstrate next connection svm lie build connection anomaly detection natural ordering create apply detector unseen produce approximate unseen object justify linkage fall quantile toy example approximate consider quantization impact level set show appropriately quantization show alarm rate reject toy example demonstrate nominal plot appear reasonably algorithm preference pair vary surprisingly approximate notice generate level quantization preserve order discussion turn quantization alarm htb x approximate oracle density curve appear vary curve approximate peak step pair sort training stage wise algorithm stage evaluate binary svm bp point note input pair much worth distance adopt reduce stage analysis mention nn distance guarantee asymptotically reliable slightly purpose worth paper quantization conclusion assume none quantization tool consistency quantization arise score distribute probability nn cause confusion eq wish solution consistent fix rkhs rbf see claim concentration measure relate make except rank sample learn compute value test evaluate prescribed alarm region r nx integer give newly draw nominal lie denote contain fix permutation nk remark precisely draw score percentile bound asymptotically approach irrespective give emphasize context sort increase small low correspond extreme carry anomaly experiment sub isolate forest svm code configuration configuration fix bp use step calculate rank accord level whenever adapt routine version implementation statistic calculate rank vice rank average nn experiment nominal value performance quantization nn validation use disagreement vary ability bp comparison significantly bp surprising statistically appear marginally careful reasoning significance anomalous consequently extension marginally bp statistically resample see sec validation cv use anomalous argument smooth well account level unlike nn present discriminative ranking learn nominal asymptotically high region allow alarm rate approach state art grant st view interpret necessarily policy express ease divide point provide near among omit st near proof involve converge empirical concentrate hoeffding inequality step properly q last discarding follow check replace easily verify diameter inequality variable rewrite divide part relatively close show exponential converge x fu min fu min fu fu df df combine converge min rx bx mu lx rx l bound lx cd lx diameter upper relate notice loose c min r mr smoothness density bound bx r q min lx ax ax line combine input set variable main body small assume preference pair let ad svm rbf associate cover radius appropriately let appropriately kn us argument author minimize borel q derive consistency enough due concentration inequality inequality inclusion compact compact cover radius inequality covering finite cover attention disk radius also quantity enough schwarz tf tf n cover radius center covering rhs bound verify difference change ready result complete minimize asymptotically recover preference relationship give surrogate differentiable density j fx surrogate non differentiable correctly hinge svm preference sample proof line l pg pg gx gx I dr gx dr j gx gx requirement function increase six sum pg gx g pg g l I gx gx dr dr dr k gx dr however requirement complete theorem prove lemma set system measure event remain portion corollary thm parametric score nn accordingly train limit false alarm percentile result anomaly detector alarm decision superiority exist nn anomaly detection detection statistically deviation behavior area detection security surveillance parametric characterize class anomalous parameter provide likely view find least unknown test set estimation include estimate suffer high statistically unstable volume volume test avoid compute paper improve computational however stage runtime ambient test different db method reference therein possibly db issue specify work anomaly anomaly generalize auc anomaly scheme false alarm characterize leverage db identify outlier anomaly well estimate addition poor produce
immediate neighbor refer treatment interactive perspective reinforcement correlate assumption arguably way social norm take adaptive connect regret reinforcement algorithm isolate game past decision regret matching degenerate pass cyclic path visit among rather adjacent implement diffusion strategy make play human social interact environment sensor beyond physical sensor preference particular physical social sensor interact sensor reveal preference reveal micro economics parametric agent influence utility maximization preference single agent measure discrimination analyze relationship price internet service position page search google interact social test typically interact require utility detection player game extensively maximization agent utility resource social scheduling demand scheme introduce social refer tendency various individual similar relationship share motivate communication elaborate prominent comprise entity make sensor mobile device etc available decision action establish agent k k agent exclude bound reward cost outcome payoff function reflect privacy maintain link reflect consumption production content production sharing capacity restrict agent situation model economic literature formally form non action space e speak identity transform payoff market model cost far share communication among agent capture graph respectively agent except neighbor network aware agent outside social neighbor payoff decision exact however even know utility straightforward generalize form social present cluster network modification simplicity continue use paper part define correlate equilibrium show page pick equilibria several motivate adopt correlate simple nash equilibrium among equilibrium lead higher require realistic naturally agent future correlate equilibrium interactive decision private recommendation action agent recommendation draw action recommendation recommendation decide follow result neither wants provide recommendation agent device trust reinforcement attract much attention processing accord r ni average loss select action play action see bottom page strategy agent utility jump denote true update rely realize pt imply gain reinforcement assign positive fact name inspire enforce neighbor via regret regret agent belong utility decision uncertainty decentralize adaptive real diffusion game learn social share matrix connectivity agent immediate global identity denote represent one respectively agent combine realize rescale individual fusion neighboring belief approximate far use ordinary differential ode experience enable algorithm summarize protocol human individual ordinal place set update member group local summarize term part valid term action space force action play speak statistical large lead behavior correlate introduce evolution agent successively old vanish strategy ordinal action order exception equally see pick decision make play bad cycle equilibria assumption game conditionally agent chain feed sec derive agent individually time action profile space profile adaptation exponential profile enable evolution old decision repeatedly action agent straightforwardly evaluate global via recursion reveal global game matrix experience average characterize average behavior dynamical work work accordingly abuse notation vector rather represent process far represent euclidean characterize local global algorithm real exist agent follow distance usual distance global equilibria sense theorem play game exploration state collective equilibrium distribute fashion behavior equilibria polytope theoretic view rational agent sophisticated rational argument weak determined theorem differential inclusion appendix differential generalization path derive analytical illustrate c cm agent exhibit characteristic isolate agent arise social agent aim coordinate decision group place receive connectivity belief strategy reinforcement however replace algorithm view polytope quantify evident correlated equilibrium reinforcement stage share monotonically reveal preference equilibrium play setup address yes agent learn network fundamentally different compute minimum preference approach wish interaction reveal seek agent maximizer subject budget constraint micro economics google mx mi response external agent dot line aim nash equilibrium game reveal preference utility maximization mx select function non rule utility optimize amount social influence consumption agent external influence resource impact therefore resource impact provide maximizer maximizer exist concave scalar axiom reveal namely point remarkable feature theorem trivial concave monotonic utility way continuity concavity monotonicity demand comprise alternatively ordinal transformation function also ordinal geometrically utility finite hyperplane theorem social potential wireless network influence social agent p mx play potential fig external action formally agent utility denote action game individual satisfy nash satisfy nash equilibrium strong nash play ix nash maximization budget budget total resource nash concave game differentiable agent give statement consistent nash scalar follow potential axiom reveal single monotone several option produce preference potential provide necessary nash statement intuition connect statement provide concave potential response strategy nash equilibrium parametric nash involve determine feasible solve linear constraint polynomial short flow parameter fail nash result action noise statistical detect nash action consist feasibility clean nash denote clean nash hypothesis satisfy nash vs source eq give agent action nash equilibrium play game significance ii solution characterize appendix consider influence recall inequality allow feasibility straightforwardly guarantee less detection nash definition enhance adaptively optimize external reduce achieve dynamically external external influence density satisfy nash definition gradient simultaneous utilize estimate corrupt reach probe use indicator construct generate realization parameter accuracy probe estimate gradient compute per iteration exposition point detection apply nash real aggregate consumption energy market social network comprise agent agent section consumption price consumption system operator website economic rational self rational utility nash true associate construct management control consumption power consumption price set management respective price utilize aggregate consumption maximization nash detect demand power consumption model construct potential external agent price several behaviour weather define denote day action aggregate consumption respective budget unit aggregate consumption satisfy utility result consumption agent limit suggest aggregate follow variance term satisfy utility provide fig h consumption day start west east maximization stochastic pass consumption independently maximize grid however concave surprising result distribute demand management consistent nash test power nash construct agent prefer marginal construct potential suggest prefer agent give consumption agent player agent theorem agent preference improve program attempt site facebook twitter agent comprise depict tx detect social known detect tweet request tend behavior human twitter tend tweet far human tend connectivity network friend political consider depict fig agent design detect eliminate account network denote response agent account action agent agent static budget agent total total resource agent query friend capture agent decrease resource limited friend capture attempt respective agent preference fashion agent maximum concave probe distribution agent represent magnitude iterate fig via gradient decrease allow reject optimal external influence type error test satisfie optimize allow distinguished error statistical agent agent equilibrium game agent reinforcement equilibrium correlate wherein network topology rely follow agent attract set focused parse construct equilibrium probability example example detect property consider ordinal nature human behavior step proof omit adequate reference characterize system represent differential inclusion differential provide inclusion nonempty convex make form markov least characterize invariant take light protocol successive affect strategy regret use stochastic piecewise derive limit simplex joint profile kronecker product process define
x kx kernel scalable gradient idea inspire recently maxout random maxout feature locally boundary component interesting linear interpretable map linear bandwidth kernel involve product linear map feature main introduce analyze maxout advantage training utilize avoid take maxout scale classification unsupervise maxout follow pca data reduction visualization maxout locally maxout maxout relate approach set corpora maxout maxout maxout unit give precise description maxout maxout unit maxout study shall consider dot z independence expectation hx I maxout maxout random x polynomial proof material maxout unit linear estimation piecewise linear q locality value particular case coincide oppose understand locality non linear radius locality pool look effect gaussian vanish hash qualitatively locality qualitatively far apart radius closeness size pool radius hence locally linear solve reproduce kernel I derive linear radius locality maxout random feature map definition dot therefore I get sufficiently next incur translate convergence rkh locality hash cx jx w c qx j z binary hamming sensitive hashing show maxout space allow linear classification locally hilbert functions maxout approximate dense belong unit loss intrinsic ball cover radius dimension use intrinsic ball moreover suitably iid give gaussian dense approximate replace practice maxout feature q numerical choice training example loss bound relate quantity dimension maxout risk achieve nonlinear class locally precision maxout statistical function suggest nd space live g dx q f f ce e c e q projection assumption us study maxout first element value locally interest locally dimensionality reduction empirically efficacy
possible change combine sparsity encoder lemma discussion paragraph inside previous separate reason discussion interesting interesting show happen constructing h batch encoder achieve iterative feature though iterative flexible able iterative encoder separately bind sparsity parameter simplicity slowly additional encoder compute encoder vector encoder vector h loss encoder example message iterative encoder sparsity encoder reconstruction encoder encoder vector getting encoder encoder sparsity tradeoff iterative reconstruction trade algorithm non row increase batch encoder every row encoder zero non row first encoder batch batch encoder k define quantity encoder eq q claim summation handle modification use method apply n h auto matlab implementation implementation ghz intel processor gb ram measure expression psd matrix hence try explain information versus achieve sensitive goal preserve possible across come choose optimize figure highlight also give batch comparable despite bad comment run exist run ten notice considerably concern want error exist finally mention optimize run empirically fast version accurate version considerably fast call generic specialized future cardinality optimize reflect preserve learn dimension preserve much asymptotically relative yes encoder connection clique pca rgb axiom york ny enforce generalization improve interpretability auto give asymptotically tradeoff give algorithm feature pca auto encoder transform encode bottleneck close encoder preserve auto reduction encoder encoder important construct low auto auto auto encoder decoder encoder map perhaps linear auto encoder pca linear auto encoder enforce encoding map encoder formally auto linear encoder decoder encode feature reconstruct reconstruct minimum reconstruction formula perhaps encoder information approximation k chapter encoder opt opt kk k early visualization feature extraction simplify component application desirable dimension direct significance gene financial application seek tradeoff ability reconstruct introduce column zero encoding original interpretable know formally rr seek let usually clear singular view matrix k k v I ir frobenius ij encoder first encoder optimal set approximation pca result encoder encoder simultaneously selection loss pca low bind information pca construct simultaneously first must orthogonal batch extract provably guarantee theorem factor point provably iterative construct guarantee approximately iterative experimental performance benchmark standard predict produce auto optimality nonlinear auto prominent auto address auto encoder lot recognize factor encourage use scaling attempt rotation thresholding iteratively residual projection get principal pca straightforward max clique see problem maximize generalize know via reduction hard pca maximize historical symmetric var decomposition loss auto encoder explain maximize explain view capture symmetric lead historical approach capture explain sum information variance unconstraine minimize information encourage objective solution reason information general intrinsic unsupervised secondary encoder constraint symmetric translate suboptimal encoder place loss convert approximation inequality relative explain immediately give careful orthonormal diagonal factor constraint property typically one compute quality completely satisfactory sequentially interested produce aware exhaustive require perturbation exhaustive heuristic direct convex refined tractable simple greedy backward develop greedy branch run backward principal quite theoretical aware polynomial guarantee optimality additional negativity give construct sparse guarantee trivial rapidly decay guarantee apply pca clear extend apply guarantee top good approximation optimal pca encoder polynomial pca box encoding column selection construct provably auto finally modify main prove iterative f column vector range span sample e r rr non zero jj span whose kk reproduce kk k mention quickly projection main guarantee rank rank modify rr algorithm invertible modify factor actually strong column locate compute compute additional svd asymptotic running multiplication affect produce encoder e satisfie say rank sparse encoder decoder hence conclude find give approximation simplify way main black box give run r follow notation initial approximate svd ensure reconstruction part expense time reduce still reconstruction give expectation least factor run right via svd approximation also recent deterministic detail achieve apply constant approximation de randomize step appear result trivial increase entire time sparse auto auto encoder ok encoding identify decoder rough sketch doubly construct column ok ok ok ok combination getting doubly encode decoder reconstruction column doubly sparse encoder identify reconstruct entire black encoder obtain provable rank column sparse pca choice construct result deterministic expense guarantee kk dense show pca error require guarantee encoder optimal encoder optimal encoder approximation much challenging approximation sparse encoder combine single loading factor algorithm produce sparsity linear achieve compare require constructing
eq martingale construct respect sequence eq give respect apply suffice I obtain relation cp give q follow step relation pp op facilitate account mean ki h h n cauchy n h h h take account obtain c k calculation term take independent e h e k n e h absolute term hand cauchy term way notation obtain n lagrange behaviour nan hypothesis relation involve relation neighbourhood write similarly last term notation n I pn pn obtain relation follow give n norm n n n I nan hold op right hand side independence q q hand side q relation inequality eps fill stroke cm cm cm remark proposition corollary coefficient variable increase increase test nan behaviour ratio test easy practice test nan law fix find asymptotic confidence phase carlo statistic technology numerical refer explanatory infinity traditional cm type principle precisely type penalty parameter penalty penalize number explanatory consider lasso penalty concern reader review automatically dependent model accurate type devote high refer paper interest explanatory converge cm technique fairly problem approach traffic hypothesis brownian main maintain change presence change adaptive lasso estimator paper choose number criterion propose yet sample make behaviour empirical likelihood first change point vector explanatory vector coincide observation variable suppose explanatory depend depend take enough last use confidence nan assume change first present two notation define need theoretical study behaviour test statistic analyse accuracy confirm improve coverage result cm notation assumption likelihood ni identically distribute thus view introduction variable obviously empirical likelihood likelihood nk r nk n lagrange optimal probability I lagrange empirical take account respect j imply pp pp n jk tn two k I remark restrict study instead statistic particular empirical lagrange brief notation convenience matrix bold square throughout denote generic line even formula whose begin notation simplify notation phase follow matrix also nr particular kronecker explanatory assumption need keep property high dimensional change n np nc q k p op op op bound probability assume point expansion assume asymptotic statistic hypothesis build phase also model change obtain nonlinear degree different normal intermediate study asymptotic behaviour emphasize break without nh first concern convergence rate valid lemma lagrange multipli accordingly follow give lemma satisfied n proposition satisfy statistic approximation proof suppose hypothesis lemma establish asymptotic normality statistic explanatory appendix presence essential way proposition ii nk I n n immediate nan confidence asymptotic theorem build quantile normal simulation calculate firstly monte replication coverage cr divided explanatory ls quantile convergent ti iy relation calculate give less change hypothesis accept know iy absolute compare statistic point fix phase test phase system apart fact consider theoretical multiplier easy unknown estimate convergent conduct term statistic theorem approximate ie relation monte x monte model cr error distribution mean subsection throughout monte replication study behaviour power cr summarize give corollary trend accordance without power error cr cr precise false carlo replication quantile respectively nk exp calculate large hand approach table give power jj even remain unchanged cc cc asymptotic involve region parameter coverage nominal coverage level phase improve critical value generally coefficient decrease approach divide proof proposition theorem lemma proof one n probability relation equality use notation follow pn use pn pn pn q n p hand
define wavelet wavelet wavelet wavelet coefficient moment wavelet wavelet hilbert transform real wavelet hilbert belong hold condition coefficient stay energy coefficient vanish vanish moment nan frequency response generate wavelet additionally shift phase response generate addition operator wavelet wavelet toolbox signal illustrate wavelet wavelet translation scalar preliminary investigation analytic analysis analysis assess wavelet figure frequency change well low modulus analytic wavelet high view scale high wavelet wavelet low signal wavelet modulus fouri wavelet individually occur continuous anti versa pair wavelet fourier wavelet derive fourier write illustrate wavelet transform symmetry additionally fourier wavelet potential proposition corollary example cr tag cr tag mail de mail wavelet wavelet kind symmetry wavelet like wavelet wavelet transform like wavelet analytic wavelet wavelet wavelet idea introduce wavelet function symmetry odd present name analysis simultaneously odd wavelet represent account scale one derive scale compare play wavelet coefficient harmonic component fourier scale hence associate odd vice naturally quadrature xt look kernel analogy transform replace hilbert order allow fourier wavelet review result wavelets transform eq variable hilbert transform fouri transform operator define hilbert impose interesting hilbert function odd vice versa wavelet anti symmetric anti verify wavelet wavelet explore proposition view transform wavelet fouri hilbert coefficient generate wavelet belong straightforward conclude moreover energy coefficient vanish vanish moment vanish moment proposition wavelet number moment wavelet define look analyze asymmetric kernel write jt naive observation wavelet impose coefficient wavelet also energy wavelet transform energy ft moment vanish moment vanishing follow th eq moment fourier like nan multiply typical signal fourier wavelet design real framework wavelet
learn order albeit connect base marginal dependence fs label conditional mutual link co occurrence g generally graph intuitive benefit together parent vary near parent possible loop bring one ignore assumption make problem segmentation valid problem th highly correlate discover burden specification structure involve discover impose priori order label label fix pattern parent label label frequency information sensible try maximize parent label dependency dependence follow outlined matrix mutual placing leave corner vertex discover seed parent label direct parent case construct classifier output l w computational calculation easily thousand limit search label building pattern acyclic cycle consume employ classifier construct refer monte interpret provide undirected dependency classifier py compare comparable term connection refer argue undirected typically easy relation construct undirected slow stage notice ensemble improve build seed label majority voting follow concern exclude classifier py ct discard description classifier majority discover sec direct sec alg sec alg chain discover like achieve present improve scalable list represent dimension ensemble experimental summarize collection dataset vary familiar community th sort represent feature number instance train complexity time address problem metric return true logical label relevant audio image biology text text text localization localization confirm hill beneficial cross display increase relevant scene confirm hill performance sensitivity list classifier fit logistic probabilistic logistic e obtain well svms accuracy recommend tune classifier wish focus use problem within framework svm sgd implementations hill hc cv hc confirm run use hamming run superior run namely dependency exact measure orient whole achieve competitive full method consider possible initialization improve hill strategy initialization regard scalability note roughly instance run approximately conclusion large compare instance running show require run wise finish within hour gb scene local music scene medical local avg match scene medical avg test test rank table place bar span average rank critical bar method statistically considerably strong match particularly well hamming score propagation behind localization table present second finish gb memory dataset music medical local avg music medical avg rank base bar overlap slow I desirable application prediction segmentation sensor real arrange detect person segmentation synthetic top room light sensor arrange window thin target target come light light source target corner h localization sensor arrange around thick line horizontal axis divide th pixel active sensor inside detection color simplicity specific avoid super let position triangle corner triangle indicator pixel pixel sensor low much see sensor height create corner light room light sensor source initialize k j check ji consider also interested study directly received consider trial success probability prior map robustness algorithm address beyond achieve remarkable emphasize property exploit knowledge sensor result sensor precision correspond obtain ct table detailed section increase fine explain see measure ct label consistently complex contrary label model experiment particularly dominant ability respect significantly method surprisingly structure base versus difficult justify modelling dependence improve dataset scalability crucial ct place order computable information match surprisingly much begin overall statistically significant able excellent number time test prove experiment structure degree project c vs deep learning feature vector dimensional acyclic one parent uniformly probabilistic generate qx normal consequently equal dependency show generate term truth discover visually discover appear improvement confirm random difficulty range hard number leave medium medium accord novel describe monte carlo procedure scheme I order py py py bayesian fully similar sample independently py graphical exact chain monte technique target adequate undirected configuration repeat py py conditioning neighbor py ty certain burn task scheme chain converge produce mathematic signal communication de circuit multi become increasingly year popular multi label classifier particular constitute ct method scalable important competitive multi problem scale thousand even thousand keyword chain multi structured classification multi instance rather label correlate allow expense increase computational label classification label binary attract deal interest development author recent recent show vast learning relationship gene tag category news may relevance add month gender month simply irrelevant receive however treat multi paper family integer binary vice versa focus effectively large feasible many tend label present scalability show powerful suited deal cascade label possible chain complexity label main contribution novel highly ct impose ultimately chain ct capture essential among efficiently underlie label sequentially probabilistic measure ct namely thousand label ct run naive ct seed outperform ct organize formalize various strategy label dependence augment theory early carry two ct secondly competitive typical output prediction localization segmentation conclusion help complexity dependency carlo require notation traditional take instance example many possible finding np find around excellent discussion complex measure co label latter author measure incorporate edge network occurrence mutual hereafter moderately dataset final end rather involved graph inherently demand input course strongly particularly try label label facilitate train label build dependency learn following treat label find conditional dependency small sized suit pc
overall overall sum magnitude square system generalize group unit tight size capacity induce convexity apply regularization equivalent novel base regularizer network capacity per unit regularization capacity norm magnitude system depth far aware capacity feed forward unit reference begin scope significantly characterization perhaps natural use analyze feed forward often depth network light difficulty optimize compute direct acyclic graph income special output node add rely input internal output propagation wu graph refer network depth direct feedforward vertex partition layer layer maximal connect layer per layer also shorthand h parametrize mostly relu relu relu several convenient exploit share function pt hinge share property hard threshold activation without change network f g relu realize unit point simplify calculation layer weight gives realize norm weight go network group type parametrize q context group regularizer impose constrain incoming regularizer e magnitude decay layer multiplication summation activation homogeneity relu activation different change q two family class effect rademacher excess minimization complete treatment exact particular rademacher typically effective capacity class rademacher norm prove induction show neuron supremum attain output highest rademach complexity layer top rademacher magnitude capture node layer width magnitude happen whenever omit width countable allow complexity investigate width subset vertex hypercube first layer connect desire sign layer recursively copy unit add rd layer copy network h p h repeat process understand dependence width depth scale group avoid construction factor logarithmic indeed width magnitude control capacity theorem offset use indeed sufficient condition convex independent specific weight never weight instead alone result complexity may respect combination pf convex show place side side interaction come come complete proof homogeneous hypercube input weight value match vector connect pm qx hidden unit connect construct case weight except ph focus constrain incoming separately unit show two network relu consider ability suggest unit look input product path control motivate node weight going really regularizer look aggregated weight regularizer finite emphasize edge go unit magnitude edge regularization imbalance think regularizer refine consider dag path notion proof f combine allow bind non depth width independent bind limit network make problem kernel allow two network e infinite unit bound output unit beyond control width immediately width might hope might make easy however point require relu indeed apply hardness intersection two layer conclude efficiently pac even increase efficiently pac even margin moreover short learn even version corollary time though might input output feedforward prove homogeneity clear incoming vice versa homogeneity triangle function complexity input output vertex since observation per graph depth internal subgraph tree per unit think fully network mean unit low stream unit intuition power deep network feature encourage learning task multiple per impose namely share vertex pick edge copy vertex edge incoming vertex incoming nod vertex regularization overall weight system generalization guarantee input lead capacity class even increase depth condition unit confirm convexity leave interesting characterization show net convex neural net network consist unit class finite unit support think impose similar see relu equivalence overall constrain unit bottom geometric equality balance homogeneity hope might make intersection conclude learn polynomial subject assumption margin short discuss rely allow infinite still depth derivation depth avoid already depth unbounded increase depth arbitrarily graph toward equivalently arbitrarily sensible dag sensible depth generalization unfortunately width bind dependence avoid anti lipschitz per avoid anti lipschitz proof base inductive argument relu inductive argument rademacher hull class inductive complexity depth need complexity example negative cone control feed analyze control depend parameter control controlling size control guarantee perhaps regularization still necessarily depth although multiplication behave differently experience relationship novel regularization precise tight though require go beyond bound real class regard independent control another expressive going provide expressive monotonically increase continue related decade circuit might resolve correction rademacher rademacher class represent width activation bound induction neural rademacher upper establish lemma find cardinality rademacher complexity linear bound norm rademacher p member hypothesis inequality output technical lemma contraction lemma lipschitz class maximization rademacher independent defer obtain rademacher apply vector row equality prove less fx side reduce need show since need rewrite inequality hold inequality true proof neuron contraction lemma absolute absolute anti lipschitz anti contraction lemma rademacher proof df satisfy homogeneity homogeneous triangular df df iw activation moreover norm establish df homogeneity homogeneous
ice member ensemble ice help rigorously characterize uncertainty even systematic error also environmental projection rise ice linear model process principal component mass ice make substantial level rise ice pose substantial people ice sufficient ice west ice might future rise east ice might increase surface air ice may next rapid ice significant population people level rise future ice characterize ice importance project future challenge ice advance ice many representation important ice towards investigate confirm knowledge uncertainty contribution characterize key produce ice projection ice challenge expensive effort ice model significant advance ice calibration limitation sigma rule choose error set ensure robustness assumption run ice thousand setting applicable three ice ice height area cover approach even ice analysis base aggregated quantity spatially approach standardized approach probabilistic calibration approach unable utilize source ice ice output observation ice core last period whereas rigorous dataset aggregate ice profile make suitable aggregate uncertainty would properly ice propose calibration appropriate ice pattern ice binary absence extend calibration linear framework considerable pose logistic avoid latent variable computational burden ice ice extent ice presence absence ice west ice large rise ice include rapidly ice interior west ice terminate ice ice thin contact ice elsewhere line narrow separate ice ice know ice ice ice core smoothly ice equation describe simulate ice computational description remainder organize output use introduce explain computational spatial formulate approach describe pc application discuss implication west ice conclude boundary technique capture challenge enough basically evolution change flow slide surface ice manuscript nest span west km nest grid boundary store year appropriately present bp modern improve proportional record year ice prescribe ice modern reach day year basic linearly year temperature longitudinal correspond observe ice decade future ice west realistic model ice poorly effect uncertain ice model uncertain vary hypercube comparison stein reasonably range relatively r confirm design parameter generally recognize important ice configuration see tune previous design input cube infer ten day geometry modern map ice ice modern original cover entire grid binary outcome presence ice figure observational observational dimensional pattern challenge pose structure development computer calibration new calibration binary spatial computer calibration stage computer calibration infer observational calibration make easy identifiability model binary specify framework construct discrepancy possibility parametric uncertainty fast spatial datum example uncertainty identifiability issue considerable challenge feasible stage description framework calibration spatial detail observational computational inferential challenge provide approach challenge notation output spatial cover pn ni ice calibration outline calibration composite subsection reader reasonably fit computer represent trend matrix definite output setting estimate calibration observational model approximate run dimensional spatially zero parameter observation choose infer remainder model bernoulli write eq output conditionally model approximate natural parameter input spatial e function vector maximize ill pose respect fit predict calibration consider systematic discrepancy observational observation value parameter match covariance process standard logistic q discrepancy define posterior density carry face computational challenge spatial describe point also straightforward pose parameter flexibility result process natural pose computational numerically infeasible function cholesky decomposition covariance translate correspond hundred thousand compute ghz moreover store double calibration step challenge discrepancy integrate cholesky matrix scale ghz challenge previous build principal inferential dealing help approach remove integrate process cholesky decomposition covariance subsection logistic detail closely q rewrite q th th minimize iteration iii minimize find hand side plug ij imply mt I log function th result local maxima algorithm need ice study ice input make projection numerical real ice calibration lack identifiability design hence provide calibration ensemble cover entire lattice cover field supplement overall grey grey area percent principal rate less construct synthetic observational choose run truth mean considerably experiment realization discrepancy process supplement realistic synthetic observational approach work ice ice problem complicate input member potentially select truth reality operate actual ice ice ice pattern construct ice spatial follow realistic observation run observational patterns ice root rmse spatial location observational run select common ice pattern ice pattern synthetic observational ice ice step observational ice ice recover discrepancy well supplement panel component validation describe percent cross validation confirm precisely error translate prediction binary informative prior input range metropolis hasting integrate look standard size accounting plot density plot change system intel pairwise sub ice slide display reason density peak around recover value dispersion compare informative probable well limited modern ice load effect ice ice relax towards modern past position estimate likely able constrain mcmc projection build ice ice convert mcmc projection show ice projection comparison likely ice synthetic truth year cover true ice change projection match ice volume ice due term ice discussion ice dataset focus ice uncertain though behavior interior ice relatively ice accumulation slide ice advanced maxima ice ice warm ice affect ice rate main cause ice increase ice thin interior ice line advance regime effect ice extent ice variation little consensus highly ice et al decrease extensive interior slide coefficient slide ice pressure drive stress slide mainly hard slow slide soft fast slide relate stress velocity effect ice profile affect coefficient modern ice unconstraine modern advanced last area cover generation ice advance wide range slide slide modern ice evolution year response ice ice year lag deep ice represent elastic sophisticated model suggest short west know ice fill narrow nearby ice ice coarse size sufficiently lack plot mid close somewhat small find correspond nominal primarily affect ice little modern ice favor less nominal weak well slide quite valuable validation modern direct effect realistic ice affect significant influence ice profile recent relatively consistent ice modern relatively thin control ice projection leave volume seem counter impose day drastically early advance discuss produce modern line despite run time tail difference year value unconstraine well past long future level occur reasonably behavior consistent recent study ice calibration framework discrepancy identifiability issue calibration specify discrepancy pixel exhibit persistent see hoc detail discrepancy discrepancy section ice paper ice application discrepancy true create scientific subject run present projection base simplified representative pathway model modern day position expect lead well constrain mention future direction formulate calibration enable information geometry ignore ice ice ice incorporate information zero multinomial beyond calibration challenge comprehensive formulate calibration approach computer describe ice result calibration uncertainty ice principal analysis
university california california bayesian big year attempt efficient scalable monte monte hmc probabilistic construct collective geometric whole along unit optimize scalable lead substantially statistic principle powerful several decade underlie mechanism bayesian uncertainty reveal landscape global method computationally intensive inference require simulate intractable simple algorithm explore inefficient successive state autocorrelation movement effective tend quite low convergence slow hamiltonian hmc walk hamiltonian state distant nevertheless hmc explore metropolis fully space dynamic method hmc geometry space improve hmc automatically practical big analysis scalable technique bottleneck big evaluation involve provide balance accuracy cost common base contain retrieve criterion strategy effective subsampling surrogate substitute usefulness moderate approximation effective whole randomly unit criterion implicit subsampling result framework geometrically manifold follow overview explain detail present finally devote discussion energy always analytically statistical increase however simple method metropolis become especially dependency geometric target dynamic auxiliary momentum explore jointly quadratic correspond log multivariate hamiltonian system stepsize accept probability simulate hamiltonian hmc generate separate q l u distant proposal autocorrelation acceptance hamiltonian allow efficient exploration although explore fully model since flat use geometrically substantially call riemannian geometry target hmc automatically adapt identity commonly hmc fisher hmc hmc shorthand partial th element p dynamic mechanism time reversible dynamic use deterministic reversible however require intensive hmc contain energy included walk geometric quantity evaluation example potential hmc mass inverse extremely expensive involve remark piecewise use develop method difficult extend due grid commonly learn limited computation inverting space train neural surrogate function approximate accuracy network incorporate criterion subsample easily complexity layer layer number hide provide neural network feedforward network activation scalar define hide weight unit hide unit bias q hmc network standard algorithm base alternative key full subproblem run hmc collect state accept explore region train collect need hmc surrogate function hmc function construct parameter extend version pp p l partial pde hmc hmc ess mcmc ess monotone ess size step hmc stable effective acceptance discard proposal iteration unit logistic improve counterpart logistic regression observation design parameter sample x I independently energy everywhere hessian surrogate step hmc substantially almost compare ess marginal hmc present figure experiment speed lr bank hmc lr hmc pde hmc hmc next bank dataset feature make uci machine repository bank datum summarize hmc counterpart intensive pde inverse coefficient pde flow medium coefficient pressure forward govern pde coordinate exponential length eigenfunction operator define kernel endow target particular expansion add uniform grid hmc function local hessian directly diagonal two positive part surrogate posterior hmc hmc hmc improvement efficient hmc hmc substantially hmc metric hessian remark usual bottleneck another example complicated simple pde evaluation use neural surrogate huge advantage experiment high improvement amount datum increase scalable explore space neural surrogate explore completion problematic demand involve apply chain dynamically history drive force hamiltonian well function sampling well distribute dense well gradient h state journal chemical physics relaxation intelligence physics letter dynamic markov b langevin methods journal series pp mathematical via langevin dynamics international page online
histogram brownian learn capture stock price correctly clearly learn question hard edu collaborative kalman filter collaborative filtering relate collaborative filter evolution brownian whose parameterized dot product relevant brownian moment filter multiple interact dynamically evolve drift handle posterior via preserve calculation manner similar quantitative evaluation million netflix dataset qualitative stock return make historical learn interaction rating example netflix movie star rating amazon allow user video user filter address preference recommendation make user interest predict collaborative filtering include outcome dyadic want team b game wish predict stock price exchange effective factorization prediction dot prove powerful relate wherein sample recover otherwise treat handle static meaning latent learn batch assume movie team stationary good information dynamic collaborative evolve exchange tracking student competition collaborative location fix multidimensional brownian motion since motivate real problem ideally event update preserve calculation evolve probability location location filter call stock parameter volatility market geometric brownian motion approximation nevertheless basic factorization collaborative filtering evolve modeling collaborative kalman filter review collaborative filter dynamic basis generally speak factorization incomplete location latent location approach collaborative distribution depend product logistic probit could probit value univariate unseen interpret collaborative consider let rate eq rate location influence way collaborative filtering variation development drawback temporal goal easily brownian motion briefly review mention many filter technique temporal filter naturally arise develop kalman filter relevant kalman filter vector linear vector additive state evolve markov current previous write inference observation state control dynamically observe initial require allow analytical calculation extension require continuous kalman filter variance drift difference make extension brownian address drift present present highlight contribution smoothing store datum handle inference evolve brownian learning ability significant probit ordinal observation movie rating model gaussians discrete motivation behind collaborative kalman extend section kalman framework object brownian function dot focus rating etc kalman prediction probit problem real partition region denote star rating class fall relation rating rating binary unlike collaborative times rating may problem stock modeling arrive interested inference discuss multidimensional brownian duration event tu tw j kalman filter multivariate tw tt u tu iw dynamically index assume posterior continuous extension u tw posterior interpret design equation also kalman drift modeling delta drift move make information concentrated impact allow dynamically volatility stock notational specific straightforward motion define ta brownian state ta ta tc play brownian motion model purpose volatility geometric brownian motion modify integration imply integral treat posterior inference evolve model control vector point break presentation inference part deal overview inference present stochastic update brownian motion eq g subscript perspective multidimensional motion generative draw brownian drift unobserved become inference learn parameter time update note modify note skip directly change first posterior unlike kalman analytically tractable typically nonlinear kalman filter employ evolution state instead field tractable hide kalman calculations approximate factorize approximate variable significant advantage define divergence true posterior equivalently variational convex variational involve add entropy require hold ignore notation calculate pz z ij truncate ignore result ascent updating appear mean ty depend fall normal cdf evaluate u symmetry update j tu iw mean covariance therefore update derive infer brownian individual u process since depend posterior eigenvalue second expansion give respect remove expansion modification make necessary integral time interested tw j unbiased expectation fast approximation collaborative netflix contain movie rating movie rate roughly movie movie rating rating half star across movie user stock datum measure time total stock million require set dimension standard probit parameter geometric brownian motion netflix try share movie shift overall user movie landscape probit link equal width set netflix account star online vb case vb use batch map version probabilistic pmf probit estimate pmf probit mf mixed membership factorization drift sequential big variational algorithm iterate user single exploit expect limit movie instant drift figure histogram dynamic movie netflix vb map rmse algorithm hold value small omit discuss online modeling preference evident difference introduction space vector improvement batch vb pmf em variational variable probit rating find treat generate probit important well compare pmf pmf rmse static calculate rmse rate realistic interested rating calculate rmse clearly bad look user movie test batch netflix
item presence ib entry entry independent order item general comparison origin meaning show arbitrarily informally arise form shift shift one identifiability throughout denote score vector case special case form q concave see hence comparison purpose entry remain eq error pair induce edge determine time sequel central play laplacian ordinal related measurement dx ti jk nj k laplacian semidefinite eigenvalue eigenvector jk j jk emphasize comparison quality identifiable semi norm sequel norm natural metric estimation semi generalized model arise naturally discuss norm risk square semi pairwise statement etc numerical sample parameter subsequently b semi model consequently risk factor follow carefully construct difficulty construct semi norm laplacian appendix proof minimax euclidean norm present minimax norm ordinal instance ordinal strong concavity maximum estimator next topology topology complete depict simulation draw follow plot average reduce predict predict normalized curve conclude pair analogue pair eq conjecture risk ordinal topology identical pair suppose pairwise ask worker option assume theorem risk pairwise via understand exponent depend every item underlie quality score sample selection item large unit identity visualize choice item hyper contain observation represent shift observation invariance concavity also propose ordinal hope true ensure hyper connect every comparison hyper also assume illustrated model concern score make concavity little give cauchy yield scalar concavity goal capture scale minimax subset well understand human storage recommend restrict selection depend noise set incorporate define wise comparison comparison topology call comparison reduce laplacian early comparison square bound respect take dependence occur multiplicative pre one standard square dnn euclidean always evenly across nevertheless analysis require understand precise number pairwise comparison application one compare demonstrate play laplacian comparison design comparison application good topology pairwise comparison setup popular topology evenly distribute sample unweighte unweighted evenly concentration inequality see let laplacian q context square norm interest eigen scale laplacian small conversely scale matrix eigen minimax claim see condition claim strict canonical euclidean ordinal spectra regular graph various text graph edge complete spectrum scale risk condition regular scale equal respect give minimax optimality bipartite partition set comprise say node edge eigenvalue regular laplacian star scale minimax condition star bipartite scale discuss minimax risk associate edge pair every regular xx class strictly cycle spectrum regular exactly dd turn apply result minimax establish arrange product second laplacian angle risk low lattice hypercube pair hamming path scale laplacian bind know hypercube practical prefer conjecture minimax scale optimality establish topology path one uniformly replacement line worst predicts topology give topology uniformly random chosen determine pair comparison generate score estimator model employ average employ respective uniform draw set packing choose eigen generation construction low packing procedure graph packing identical use packing star procedure laplacian star shift bb various topology star consistently star graph phenomenon plot vary bad simulation multiplicative factor describe experiment conduct amazon henceforth crowdsource choice platform individual put exchange payment along worker complete worker answer worker allow question involve american require worker ordinal choice question worker circle worker area ask topology involve aggregation follow via validation employ pool worker experiment estimation topology consider relative consistent graph well bad misspecification case outcome effect misspecification consideration address approach comparison ordinal enter answer figure ordinal compare approach fine ordinal binary argue ordinal convert ordinal processing estimator ordinal set conduct seven investigate response obtain subtract conclude collect subtract amount ordinal task select several important knowledge conduct worker paragraph audio sound worker audio tag clarity relevance image worker relevance independent collect distance audio relevance std std ordinal std experiment worker worker ordinal ordinal answer access ground truth answer remain ground truth compute ordinal ordinal important measure interface see worker first website ordinal one directly ordinal response unlikely set table show sometimes significantly ordinal explain inherent human human first evaluation typically ordinal particular introduce assume discuss ordinal contain ordinal encountered vary bias give evaluation also recognize human allow evaluation cost clarity versus ordinal form reliable pair comparison noise comparative preferred measurement answer help determine response evenly accordance assume priori ordinal gaussian capture ordinal deviation retain set order bring specialized gb b place minimax setting risk risk coordinate measure number reduce model ordinal treatment ordinal reasonably error scale consequence result allow base ordinal well minimax exact sake completeness ordinal setting three normalize three execute iteration procedure select ordinal practical system pool worker ordinal pool convert task audio coefficient result rest error observe per close mistake reflect table estimator incur ordinal experiment mistake whereas outcome go remain need constant order address present broad preference demonstrate utility choice ordinal potential improve mechanism effort noisy characterize finally useful variety future semi pairwise acknowledgment office national foundation grant dms support microsoft fellowship ordinal low argument see minimax estimating index refer packing construct pack risk construct packing require auxiliary q result bind lemma dimensional vector symmetric semidefinite nonnegative denote collection jj eq fact choice construct packing prove claim give z calculation case prove kl b bb bf claim ordinal squared purpose subsequent prove minimize differentiable mle semi mean perturbation convenient semi convexity piece claim inequality lemma mle ordinal need verify strong hessian simply ft di parameter lemma concrete remain control component n square reduce fluctuation moreover mean straightforward v I gaussian imply algebra universal tail low respectively minimax euclidean norm ordinal part semi norm describe section substitute upper comparison semidefinite nonnegative scalar specify later vector boolean keep first z comprise coordinate remain choice trace turn employ z b k packing integer scalar subset boolean hypercube length j entry j apply pack large claim consider packing claim turn paired likelihood w quadratic convexity hessian lemma quadratic application lemma tail quadratic form see appendix since integrate yield semi bind state packing laplacian packing remainder identical except requirement packing square form q n claim respect suitably prior vector bayes lead distribute n apply iterate subsequently proof laplacian underlie bound lemma rescale take verify dual log follow consequently hold likelihood n eigenvalue one shift invariance imply write every pair evaluate I well recall define since chain k employ k z il b note r final scalar whose let packing appendix satisfy j yield pack b applying prove packing make k prove bind follow theorem minimax b show scalar whose later construct set j vector yield general element pack packing claim three construct packing calculation pair j k n state exist hyper suppose hyper edge correspond schwarz v span I desire eigenvector eigenvector similar argument matrix invariant collect useful form valid variable jensen complete laplacian graph semidefinite decomposition diagonal
binomial distribution l stack bit estimate window code show comparable ccc replace isotropic noise reverse trajectory probabilistic sampling rest image long region delta know train train technique model reference utilize available roll use radial basis generate roll successfully train simple occur use multi layer reverse trajectory figure perfect convolutional architecture illustrate direct previous mnist variety result mnist likelihood comparison window image consist circle capture complexity therefore probabilistic texture straightforward evaluate figure novel toy real test algorithm stay often extremely algorithm diffusion noise make distribution sample evaluate straightforward posterior extremely helpful sharing window office office foundation conditional f f u distribution occurring occur finally perturb trial b roll initialize normalize radial single hide x step share top pass sigmoid restrict roll successfully bin step binomial perceptron sigmoid reverse independently step share pass sigmoid restrict successfully bit nearly convolutional pixel sized convolution per dependent consist identical combine variance perturbation apply wish goal make use multi output vector every map convolution step mean multiple power perform convolution resolution pointwise nonlinear relu operation resemble multiscale multiscale dense act image act pixel follow nonlinearity experiment pass pass dense pixel generate image cifar dense pathway convolutional pathway dense pathway scale pathway learn family computationally essential equilibrium physics systematically iterative learn learn probability generative thousand probability model additionally reference probabilistic model suffer unable describe rich flexible normalization generally flexible expensive variety tradeoff expansion variational kl contraction score propagation parametric novel pt flexibility structure multiplication model log convert physics chain rather chain define probabilistic evaluate perturbation tractable explicitly full distribution analytically target utility roll sequence handwritten digit mnist cifar cccc identity slice reverse gradually term process generative decade develop develop directly paper early motivation advantage idea physics static bayesian method multiply learn training inference challenge inference objective inference restrict generative layer layer production time em summarize develop flexible trajectory reweighte improve generative train match equilibrium autoregressive recurrent deep extension tractable train attempt distinguish mapping marginally factorial density learn mixture mixture causal neighborhood additionally datum network generative compare experimentally adversarial idea equality learn markov slowly constant trajectory langevin realization diffusion use stochastic kolmogorov forward kolmogorov correspond kolmogorov knowing cccc cc b example b forward diffusion datum distribution learn finite process figure first generative probability derive distribution multiply denoise label gradually behave rate perform q binomial diffusion binomial diffusion kernel binomial distribution reverse binomial continuous step forward trajectory covariance bit flip binomial f reverse transition provide cost time would applicable include assign sample reverse trajectory evaluate trajectory reverse identical sample require substitution correspond quasi static physics amount provide jensen entropy divergence derivation trajectory correspond equation become equality reverse diffusion probability perform regression gaussian flip performance right schedule greatly improve estimate schedule move diffusion learn schedule overfitte dependence additional hold partial binomial gradient ascent forward diffusion schedule per diffusion image identical sample note display multiscale object produce especially high task signal multiplication produce x distribution difficult variational autoencoder distribution treat either perturbation multiply demonstrate diffusion multiply multiply
address initial answer question initialization b c estimate perform comparative among state study different compare real text identify method initialization candidate background sec discuss along propose sec finally conclusion sec mixture optimize objective mostly consist step maximization likelihood use multinomial use accuracy em solely focus significant value function initialization detail investigate em empirically discuss order automatically generate set select optimal need model consider focus model task novel exploit hierarchical cluster hierarchical agglomerative generate mixture difference employ require among inefficient dimensional simplification use approach propose statistical selection method list criterion select mm likelihood recently mm aim comparative generate investigate range datum complete cluster order assume multinomial distribution multinomial mixture k estimate maximization maximize number expectation q sample update certain criterion meet parameter examine value start short initialize short stage assign component probability step conditional candidate model generate cluster explicitly generate model employ em begin continue within em mml detail propose multinomial agglomerative permit three dissimilarity select merge merged cluster symmetric measure dissimilarity criterion besides use linkage criterion merged issue mixture base function employ minimize criterion augment model criterion complete likelihood add classified minimum message mml probability select minimum criterion graph consider idea point minimize root automatically generate method experiment simulate adjust rand ari stability ari discrete count type type verify generate sample dirichlet determine text collection cluster reader construction dataset initialization list consistently discuss sec selection strategy trial trial trial initial em dataset follow good competitive provide second term real emphasize accuracy experiment real generate discusse among implicitly method initialize initialization maximum threshold fig illustrate method stability bad real outperform perform performance fast prefer however prefer accuracy b compute sample
evaluate applicability see two possible dataset tractable approximate acceptance ratio illustrative simple dimensional normal distribution illustrate misspecification assign flat mh isotropic stepsize set reach acceptance applicable posterior coincide reference decrease bernstein von gaussian center true minus simple subsampling lot help pdf chain pdf basic pdf histogram fit deviation mh bernstein baselines mh run depict green green later tackle run combine equality importance batch note assumption propose artificial batch multiply justified posterior gaussian go infinity accurate property estimator batch multiply kernel estimator additional simplify chain assume independent target final bandwidth importantly batch grow propose disjoint poor approximation approximation propose transform kernel transform interpret extended copy conditionally first approximate speak propagation cavity cavity prior term batch iterate simulate cavity distribution fit cavity computationally experimentally posterior convergence another research avoid introduce wasserstein median propose wasserstein develop mean robustness median drawback circumstance valuable contain batch technique appear datum issue multiplicative number often rely unnormalized version useful several potential scale mcmc describe mh present access surely unnormalized pseudo mh realization replace mh considerable practical application particle mh paradigm worth investigate problem large mh qualitative mh preserve invariant mh might largely difficult pseudo involve mcmc estimate fix variance log likelihood ideal mh access generate quasi large autocorrelation ideal implemented recommend keeping ensure actually incur mh describe marginal mh unnormalized without replacement unbiased log denote unbiased interesting whether almost surely nonnegative use recently possible generalizing unfortunately shall result typically relative result poor define estimator replacement integer value whose decrease ease computation geometric correspond tail finally let mention logarithm difficult proxy appendix order impractical geometric truncate reason section expect mh rely none yield satisfactory exploit methodology datum methodology posterior expectation unbiased biased mcmc mcmc suggest algorithm expectation unclear whether alternative simplicity experiment also make heavy follow admit marginal pointwise evaluation must lower log replace target bernoulli lower exploit previously see e mh sampler equilibrium related evaluation explicitly specify pseudo elaborate extend unbiased unnormalize precisely pz ib accordingly obvious unnormalized give note choose integral integral similarly evaluation speak although pseudo propose disadvantage require integral tractable advantage first sampling require compute bottleneck avoid explicitly resample easy marginal hence particular ergodic ideal explain variance let notation tight variance big tight bound meet fix fraction outli log figure taylor taylor cases evaluation per initialize try small evaluation joint gaussian number evaluation replace algorithm expect behave chain towards slowly pdf resample propose rely pseudo langevin iterative iteration update sequence score langevin mala proposal mh mh compute decrease zero analyze target central slow carlo unclear compare subsample need reach practice stepsize recommendation choice subsample iteration mh subsampling draw away variance get big lead length pdf k pdf length p length figure subsampling monte carlo hmc hmc inspire stepsize explore approach suffer heuristic demonstrate rely violate ratio example likelihood result accept average proportion use acceptance often easy benefit inherently affect mh acceptance simple unbiased unbiased likelihood approach review attempt could try proportion newly target suitably assume new draw likelihood pe pe variable I r index contribute nontrivial rescale simplify denote roughly decrease binomial probability unlikely subsample roughly broadly contribution likelihood end likelihood much unlikely contribute subsampling also absence naive subsampling nontrivial additionally variance log keep mh entail gain evaluation term likelihood naive subsampling allow tackle mh controlling ratio author theorem justify likelihood equal mh correct target estimate inexact inexact chain approximate ratio tail tail decision result mh likelihood introduce correspond pseudo proportional pe ideally want variance log subsample point importance weight obviously purpose subsampling e spline total subsample likelihood control probit use full suffer unclear good proxy fit train obtain likelihood subsample take acceptance note level confidence aware anneal annealing mh application receive attention apply mh mh like incorporate rely average use mcmc approximation aforementioned initial slightly effect seem remarkable inaccurate choose subsample explain posterior fit gaussian include tail dataset subsample tight choose log ratio heavy tail go von run require illustrate use approximation size pdf pdf pdf plot overall mh little inaccurate also assume amount care recommend sure realistic note way obtain subsampling weak approach impractical illustrative potentially subsample illustration soon theoretical independently replacement inherent acceptance exploration accept move develop heuristic anneal presence formalize symmetric assume make third px I px apply px yield find probability accept move check whether acceptance target concluding distance control positive ergodicity walk temperature bound proportion even case control moment subsample inherent acceptance ratio one upper refined log likelihood ratio eventually sure inequality lipschitz know inequality yield bound take bandit subsample size acceptance take probability improvement ideal mh regression mh ergodicity mh speed ideal show uniform ergodicity extend result geometrically scenario sampler require equilibrium subsampling rely lead result sampler access estimate take subsample likelihood even basically pdf basic basic pdf basic inequality worst theoretical come locally gaussian bernstein von good gain example current sampler target empirical confidence sampler tool proxy likelihood act variate taylor center obtain stochastic evaluation expansion sampler proxy outperform represent log require less likelihood evaluation combine iteration mh figure big give detail basic basic taylor proxy likelihood act start confidence replacement generic empirical concentration one think deviation likelihood emphasize concentration concentration refer reader correctness confidence implementation ht px n keep track already replacement bt bt bc tt geometrically cb accept introduce performance confidence sampler w likelihood w control variate lower proxy ratio acceptance equivalent checking replace q confidence ergodicity underlie mh sampler mh confidence exist proof straightforward algorithm concentration confidence lead turn availability proxy assumption although basically identically possess third typically expansion section expand around obtain choice defer section bound lagrange proxy mass concentrate estimator taylor proxy section concentrate inaccurate insufficient subsampling gain mode close chain agree whole iteration I dataset reference nd taylor expansion example gradient need aim proportion ideal mh easily generalize number section evaluation subsample mh fundamentally first contribution budget flat stop rule loop confidence meet eq bind grow strictly slow often dominate lead bind logistic asymptotic proposal loop consider taylor correspond standard order time dominate asymptotic good set posterior drop matrix assume say approximately growth third case independent still factor avoid load proxy relate quantity build disk database http www define taylor expansion within gaussian consider start single report number fraction likelihood evaluation roughly evaluation break barrier reach http www edu tw dimension without require complex sampler mh cauchy solid chain vs dash run iteration drop every explain stop
plot show direction median learn infer round mp direction ty u gp candidate localize physical direction conversely model allow draw condition normal lead pc via pc condition eq pcs form candidate directional average sum conditional gp px mp choice zero absence localization average choosing absence inaccurate generalization fortunately infinite target substitute label reduce eq allow approximated train band tend negligible low sub band derive consider minimum round gp eq whose represent improvement minimum expect loss analytic weighted expect model denote cumulative low improvement explore property fast empirical human gp mean efficiently little query criterion direction gp covariance gp subject round necessary improve query selection close localize horizontal plane towards head near localize confusion angular initial localize median plane plane consist report alternatively play independently min mp trial proceed round direction trial conduct case large occur percentage gp error human report direction exhibit variance test may future measure human localization via variance ht sound nn accurate randomize full develop query offline model localization product improper formulation ability model parametric receive direction output summarize spectral response base achieve training sound localization subject test base find direction infer error interface localization localize close intend head regression sound localization possess remarkable sound localization ability subtle receive human arise acoustic wave head reach center head absence head transfer wave direction allow accurate audio function base sound localization acoustic event receiver sound solely receiver actual robot receiver transfer map place direction envelope near neighbor nn spherical coordinate learn self map horizontal median coordinate frequency posteriori transfer closely relate match leave right sound filter space frequency source computing measurement belong maximally pair sound feature measurement label filtering address number report ordinary ol regressor perform poorly inaccurate linearity predictor include arise parametric fitting datum linearity gp place weak joint observation observation represent predictor spatial smoothly model gp allow function realization select goodness belong hyperparameter learn without cross intrinsic high dimensional space feature uncertainty variance tractable thus linear inference observation training set vector make linear evaluation observation general gps method encode select prior observation selection inference large dataset active research fortunately quality overcome previous work dataset base interpolation gps model relate refer vector standard collection aforementione quantity sound gp specify train belong ht selection run cost randomize trade cost density linear accuracy cubic respective localization show gp small informative spherical randomized simple selection sequentially iteration subset add subset generalize small spatial audio resolve front back directional indirect measure query report new pool graphical interface move towards target develop query localization rank candidate along unknown propose give candidate determine would within recommender select query refer cosine unlikely round cost reasonable adapt predictor round base query predict trade require model prediction fortunately suit probabilistic query gp sound invariant transform representation e mx mx alternative common gps specify share gps physical topology thus inference three gps front back report posterior realization depend differentiable r mat ern covariance dimensional specify gp product mat identical product function time length bandwidth hyperparameter remain large length smooth hyperparameter maximize derivation realization likelihood sampling sampling large represent goodness mean covariance assumption different compare without domain order size dimensional sample conditioning operation compute store exact intractable practitioner cost expense analysis evaluate demonstrate localization despite gram dominant one minimum mapping contain space dimensionality kernel analysis gram derivation eigenvalue capture dimensional score feature apply allow reformulate problem contribution gp pd mp belong subject hyperparameter iteration domain use angular gp train infinitely model direction approximate sound pressure continuous well fitting sound source record spectra suggest relative intensity pd compute dataset total energy specify pd mp lead eigenvector suggest feature train angular separation predict direction method train parametric across mp low across ol log ratio predictor pd feature insufficient pd ols gp gp gp randomize select input risk efficient good approximate determine optimal exhaustive prohibitive evaluation greedy rank risk minimizer consideration future see gp full condition input point input risk empty r rr specify must gp covariance approximation remain evaluate require class point update define gp posterior eqs evaluation kx
theoretically hinge converge option dataset difference option converge machine precision option clear conclusion smooth pick check output let equality follow arbitrary q everything independent treat proof quadratic reasoning hold lemma claim theorem exercise question com ed ed ac introduce adaptive ascent sdca empirical minimization method adaptively probability dual variable throughout complexity distribution theoretical propose risk minimization supervise identify throughout differentiable lipschitz problem considerable attention recent year usage supervise erm analyze include sag gd gd consider coordinate sdca operate erm problem primal sdca select dual attract considerable attention year alpha advance mini variant review naturally randomize individual coordinate exist work assume descent subset coordinate allow optimize individual variable smooth primal explicitly typically quantitie development field literature probability aware piece resort heuristic theory method sometimes sometimes observe progress call propose selection construct base quantity summarize theoretical algorithm describe provide introduce element conclude technical numerical find highlight propose stochastic dual provide rate analysis enjoy rate sdca effort issue algorithm heuristic variant computational heuristic make sdca computational summarize first method variant numerical experiment appendix dual follow dual dual update coherence alternatively coherent primal ti mention solve exactly deferred appendix lipschitz let constraint obtaining hold defer appendix special case lemma hand proposition know q therefore plug bind primal dual letting optimal serial sdca convergence sdca next suggest sdca probability program theory sdca direct quadratic corollary sdca know propose variant avoid issue set importance v n tw divide epoch begin epoch one option update every intuition coordinate set p cost gradient probability epoch affect epoch result probability begin could sample even correspond work choice begin choose option close option serial one sdca differ sdca iteratively yield fast sdca uniform probability change take change epoch epoch calculate operation need
convergence property figure employ reach consensus state assume note mention show algorithm fix consensus reach weighted average node node address issue exploit devise distribute parameter identify weight estimate assume identity identity I close maximize w w close centralized extend centralized centralized weight problem pdf evaluate node attack figure roc curve learning see detection learn weighted detection scheme average effect performance conventional consensus attack show certain fraction exist consensus robust consensus technique operating fusion update effect attack interesting remain future vary topology analytical certainly attack topology question topology incur fast conjecture k paper fully exchange information absence fusion characterize steady detection detection address problem perspective robust attack node consensus devise datum statistical distribution node base distribute consensus datum attack literature traditional detection framework comprise phenomenon fusion fc decision many scenario centralize fc fc become bottleneck failure absence hardware wireless medium alternate decision one local exchange consensus algorithm consensus explore approach update summary fusion combination network global author consensus detection small fast consensus scheme however consensus author distribute weighted fusion show consensus detection performance gain consensus detection attack attack attack originally attack attack address centralized attempt make address security recent work attack exclude neighbor significantly different attack detection large mean step state node distribute spectrum sensing threshold adaptively eventually isolate exclude perspective early centralized way identify potential outperform design fusion decentralize attack arise node recognize identification operating parameter complexity update operating approach differ base exclude consensus contribution follow steady consensus specifically statistic probability investigate degradation detection conventional weighted expression weight attack likelihood operate enable adaptive fusion paper organize system attack study consensus scheme section mechanism effect attack link link define six consensus figure laplacian base sense summary summary state consensus steady state statistic phase decision phase energy detection node instant instant detection product square gaussian chi square freedom chi snr represent snr detector give algorithm step node continue information consensus nothing fusion update come neighbor node asymptotically reach information initial final consensus state I iy special average consensus whole network reach consensus node decision predefine threshold beyond reach consensus paper refer final test discuss attack scheme degradation average consensus base attack network node update suppose strategy prescribe attack w jk ik initialization allow completely appropriate adversary perform attack consensus consensus attack attack investigate past literature see theoretic mechanism capable consensus attack attack introduce initial devise influence attack try iy reach interpret assign node detection performance always statistic dominate statistic analyze detection performance analyze fusion weight value degradation cause attack attack detection characterize strong global h coefficient receiver operate general monotonically fraction need statistic denote free topology whole consensus process reached analyze consensus detection presence attack conventional consensus detection characterize steady alarm degradation loss generality index rest define n tw condition coefficient n local coefficient seek iy computed substitute condition appropriately attack make coefficient zero insight plot coefficient global statistic consider node give figure consider channel gain coefficient zero six attack less blind fusion scheme analytical detection alarm consensus come function notational denote clarity derive two later generalize arbitrary j j summation pdfs normally variance result eq due nature behave strategy mean node behave index notation arbitrary node ji performance scheme alarm network detection alarm iteration represent expression false alarm node combination cardinality represent compactly w tw wise w j ta alarm gain result figure node attack assume consensus detection alarm consensus consensus figure attack probability alarm consensus alarm consensus discussion detection approach sense weight assign issue datum assign high weight final dominate detect type choose conventional use e concern propose robust average issue propose consensus weight node want statistic make every majority otherwise treat consensus attack detect satisfy
mf predictive predictive tree forest mf implementation default use simplify propagation ep incoming message uncertain message numerically crucial dataset dimensional message message predictive uncertainty distribution show red unobserved region predictive rf prediction confident mf quite region mf region smoothly red predictive uncertainty mf blue expect predictive less smooth compare mf experiment compare forest variant process predict delay eight namely age year need arrival week month month dataset state art predictive train create use split train mf rf comparable rf significantly split stationarity burden gp well exhibit result gps phenomenon forest achieve rmse significantly compare useful decision forest achieve large believe rmse failure forest gp forest complex regressor forest split batch rmse rmse rmse rf novel scalable methodology regression sensible application planning scale delay demonstrate framework provide art rmse uncertainty forest acknowledgment share helpful bl foundation research fellowship college newton international ep fp grant agreement appendix hyper use hyper optimize integrate noise variance unknown fraction could actual belief propagation leave optimize increase tree tree depth assume fs fs fs post pt fs mid fs alg college department sciences forest efficient application quantification demand measure uncertainty measure poorly variant tool application area goal uncertainty model perform bayesian within tree light finite carry propagation design typical probabilistic demonstrate far quantification exist forest little cost performance turn probabilistic delay forest gps little gp due ability accurate also prediction application optimization balance exploration exploitation unfortunately gps cubic computationally non parametric scenario dimension progress decade big recent randomized popular achieve popular forest variant rf computation attractive regression task forest yield classification forest gps decision forest distribution move away smoothly uncertainty uncertainty smooth exhibit desirable combine gps probabilistic decision forest forest usual forest probabilistic specifically prior compute randomization property demonstrate forest well online set focus organize forest forest regression discuss forest uncertainty rf ii forest outperform gps term negative probability large regression uncertainty introduce forest task begin adapt completeness review decision describe n task label point uncertainty triple root strictly binary partition root represent parent denote denote location b rectangular put point dimension precisely restriction split forest I justification recall sigmoid encode child parent depth hierarchy align hierarchy marginalization intermediate change test branch mf hierarchy smoothing goal note hence message analytically hierarchy structure scale hence efficient compare message tree study chapter inference pass bottom pass parent message two child top pass first compute child message message root node parent fashion reach hyperparameter detail appendix optimize ideally optimize computed descent individual marginal close solution assume estimate reasoning increase depth tree assume tree stop split identical splitting label stop homogeneous strategy sample forest tree except gaussian rather denote branching reach branching away branch create instead prediction I prior bias predictive mean gaussians w think
class score pattern build appropriate great class develop quantum pattern class generalization introduce formalism purpose discrimination pattern specify quantum state quantization dimensionality also quantum form quantum extend present mixed pure state enable superposition quantum quantum interested representative classify order quantum pattern represent normalize superposition calculated product first list quantum require feature dimension quantization symbol onto quantum quantization crucial view correlation define quantization obtains classify unknown execute average quantum classify pattern accord great value class superposition quantum state quantum quantization pattern recognition perform quantum mapping calculate pattern vote however case utilize superposition base distance represent numerical start vector quantization subspace appropriate context transform represent interval order fix state space require encode quantization also require fix state encode element nearest sometimes function pure procedure implement system ns join ns join ns ns quantization representation represent unnormalized superposition flat score consider number normalize superposition represent eq representation amplitude note extension example encode pattern counterpart tuple quantum represent vector quantum representation obtain space introduce scheme dirac notation interpret quantization representation quantum dimension one main classical mechanic describe space describe quantum case system describe use kronecker product quantum classical expect tensor structure encoding correlation scheme separable quantum simple separable representation consist eq map feature dimension implement ns ns ns k plus norm v scheme execute real separable execute specify overlap quantum state represent th separable allow pattern produce describe class separable quantum assess positive classify class appropriate ht fig encoding symbol present introduce enable world quantization dimensionality quantum state require execute framework conduct possible separable representation mechanic work quantum beneficial security view party quantization quantization apply framework formalism quantization advantage approach possibility measure use classical information use measurement acknowledgment support project quantum theory imaging author would thank via suitable introduce computation resource quantum illustrate difference example discrimination machine merging processing engineering pattern recognition research quantum problem quantum discrimination connect development quantum mathematical formalism suitable recognition main method quantum investigate separable flow processing execute standard computer introduction dimensionality utilize notation initial consideration quantum purpose classification propose encode datum formalism quantum describe basic datum formalism
pattern form order sup sup sup b sup b sup sup sup sup sup c database change turn order construct worth develop length general sub pattern pt sup sup c sup sup sup sup sup sup b sup sup sup sup b sup sup c sup sup sup sup sup sup sup c sup sup sup sup c sup sup sup sup sup sup sup sup pattern contain binary sequential binary partition sequential sequential important support partition consideration length search pruning mechanism explore space exactly top interesting mining regard depth branch exploit anti space lower sub depth advantage node extensive maintain open store give explore locality variant explore disadvantage larger explore search evaluation exponentially burden explore adapt sequential ensure moreover sure space explore differ sequential specialized end note hereafter ensure late search queue item pattern queue queue repetition adapt expect detail sequential regard queue queue top sequential empty space database detail leverage extension directly bind exploit regardless use expect depend pruning dataset maintain set sequential explore prune high scoring first propose bootstrap quick limited bootstrap depend domain es ms add sequential high leverage make availability sequential application sequential sequence website visit represent transaction location series visit customer market action course patient dataset extremely rarely make scientific addition assess discovery discovery sequence discover scalability public select book many extract sequential database topic largely motivate mining aim demonstrate extraction use dataset token token flat generate accordingly shift make average occurrence average deviation embed sequentially associate embed select extraction extraction allow would return datum embed pattern use support leverage ever extract sequential frequent item explain introduction pattern compose frequently chance support difficulty extract highlight point single embed embed pattern rank pattern base approach large embed embed recall decrease three factor keep extract increase consider pattern interesting mean counterpart sufficient mining whereby overlap independently though actually within top dataset illustrate table match adopt embed pattern represent extract depict pt pt depict pattern actual next extract pattern base informative pattern actually chance pattern difference explain token relatively frequent exactly happen pattern token appear frequently see pattern show conversely leverage embed already appear observe thus effect accordingly hand introduction sequence accordingly depict pattern two although attempt extract without filter discovery sequential mining mining book domain build dataset book consider abstract sequence highlight regard vocabulary sequence characteristic avg max pt modification build book sentence processing reduce stem say make word linguistic meaning use sometimes extraction report critical one make extraction correspond repetition frequent word support example case seem extraction book similar claim contrast observe approach book retrieve give focus surprising know classification typical english frequent surprising ed vs numerous statement activity mostly decide frequent book past contrary rare observe early sentence person place english indicate compose say mr mr way almost sentence finish top pt pt sentence cc pattern pt cc cc pt sentence cc pt top pt top paper cc pt finish showing report surprisingly order extraction fast encounter top bootstrapping leverage relatively make extraction high little twice base extraction slow efficiency high interest turn pruning book book high execution long support extract top pt support pt pt pt introduce definition sequential specifically efficient exact exploration contribution introduce constitute high leverage validate consistency question issue heuristic integrate background knowledge common measure highlighted table assess pattern pattern joint differ investigate pattern joint framework china research research air office scientific research office contract fa rgb university edu framework exact pattern combine expected concept measure top carry confirm consistency approach introduce efficiently interesting measure core paper early pattern appear something happen body pattern frequent even event completely independent large database create frequent even problematic frequent pattern interaction within novel insight database issue even repeat extremely frequent relatively table five frequent pattern rest frequent sequential observe permutation frequency discovery high individual surprising contain occurrence enough believe probability occur length event demonstrate frequency sequential item determine pattern frequent capture argue pattern capture see frequency possible close question pattern database contradict sequential standard involve support formal motivation specifically tackle order high leverage note extract top expect research discovery sequential section conduct pattern sequence pattern occur build different sequential derive score strict episode difference independence consider overlap partition important reason expect activity sequentially parallel nan frequently paradigm paradigm relate
galaxy treat apply algorithm version splitting result panel bandwidth panel smooth panel panel right third coverage panel leave display smoothing smoothing see optimal smoothing method coverage integrate risk bootstrap estimator bandwidth selector work coverage limit instead learn risk risk select selection tangent sense projection eq hausdorff smooth curve parametrize end q property rr eq convert bound law contribution small define think du du small bind simple lemma property angle imply angle projection onto tangent bound projection since similar hausdorff du side combine conclude note derivation work constitute asymptotic nx bootstrap consistency easy nu nb proof replace lemma let uniform distribution q second assertion nx similarly need pick restriction department university density ridge coverage generalize two risk select tuning parameter estimate dataset density thm thm definition density dimensional characterize high vision imaging density universe detect propose shift algorithm modification usual algorithm adapt geometry unlike mesh mode move project gradient nearby kde act bandwidth despite coverage generalization integrate expect smoothed selection choose parameter propose dataset follow independently identically order whose dimensional curve dimensional manifold nh play role kde detect smoothing smoothing bottom introduce coverage risk geometric concept let hausdorff length area projection uniformly ridge random b dx bound distribution eq cover omit lemma link hausdorff call project apply set nice outlier risk hausdorff outlier risk kde datum manifold half du du rule minimize select principal curve different coverage count self monotonic bandwidth derivative high coverage risk pick estimate concept dimensional cdf contain coverage region cdf link diagram manifold coverage diagram use similarity coverage diagram serve nice cdf mesh consider curve coverage diagram green curve difference expectation take risk analyze prove particular risk since jensen bound define orientation collection associate specifically usually eigenvalue require r require direction common condition kernel kernel assume compact derive coverage eq density thank jensen convergence risk square theorem require decay come bound derivative converge give hausdorff density density appear agree hausdorff nonparametric converge fast
review weight adjacency length input shift operation adjacency filter polynomial q input signal depend filter polynomial describe method note similarity present section selection consider learn describe typical identically desire lead optimization sample useful dynamic evolution value power correspond undirected structure autoregressive coefficient conditionally accord mrf assume give matrix evolution process time ij ij use graph challenge single adjacency unweighted property process define single weighted model describe non graph new relate discretization partial differential signal framework series index sample represent graph sample graph discrete follow form collect affect influence limited provide continuous signal index describe first approximate discretization grouping arrive see causal naturally fit model big true graph product sum jointly basis jointly eigenvector term q adjacency cyclic shift represent direct dft temporal arrive estimate time series wish adjacency first follow represent represent ensure adjacency first nonconvex find locally near instead separate polynomial must mutually ip q still hold naturally coordinate sub formulate regularize incorporate especially new running except ic linear autoregressive drive white follow correspond maximum posteriori framework extended process nonlinear value loss convex prior polynomial estimation reduce complexity find term polynomial polynomial adjacency matrix polynomial function generalize outline adjacency filter behave initialize ti denote number maximum count estimate direct minima initialization summarize discuss basic estimating problem norm nonconvex like ensure coordinate global mild objective level norm block descent extend section converge compact produce appropriately choose optimum nonconvex example vary temperature sensor year location unite least square projection reconstruction mrf matrix proximal descent estimate create element draw make thresholding scale direct os enyi topology edge ensure stability arbitrarily result stable form unit additive simulated graph generate sample structure basic initialize graph direct see individual direct qualitatively sparse sparse close thus see extend square error extend across different carlo mse decrease propose plot produce mse compute decrease suggest total error interest temperature average temperature united pass filter cutoff mrf distance neighborhood city estimate consist index odd training testing different prediction since experiment see corresponding task compression prediction entire train leave proportion well mrf sparsity mrf perform training error mrf truly capture dynamic process temperature mrf estimate mention previously axis correspond produce mrf produce magnitude pick west east wind country multiple south chain knowledge experiment estimate describe believe model underlie temperature sensor process tractable vary
cubic interpolation framework global interpolation interpolation perspective naturally enable toeplitz structure gain scalability target grid ht conventional induce form perform w x perspective induce approach well popular rbf expressive induce interpolation nature go global cubic ability greatly induce function versus interpolation greatly scalability gps interpolation kronecker gp though write cubic evaluate model particularly approach superior similar efficiency recommend understand comparison recent storage le locate input consider ghz ram accurate gp predictive likelihood evaluate induce induce sort randomly use cubic figure indistinguishable absolute generally great ht cubic interpolation c average reconstruct entry cubic interpolation form weight black show vary point interpolation cubic grid blue input distance linear interpolation mean upon induce input allow precise interpolation less long coverage cubic interpolation regular weight global kernel interpolation red correspond strategy global kernel interpolation cubic interpolation reconstruction qualitatively cubic reach find combine cubic region ultimately runtime linear cubic large general go interpolation interpolation cubic interpolation great boost without runtime give importantly cubic alternative test perform covariance matrix moreover yet toeplitz accelerate gaussian dataset large provide discover rich statistical representation improve process show typically expressive suited great first place arise inducing require computational necessary suffer efficiency gps gp attempt product operate figure distribution even sample sophisticated intensive taking instance enable underlie likelihood gaussian process py kernel wish equip learn induce grid induce figure reconstruction reconstruction provide whereas unable reconstruct kronecker induce second hour induce point sample exploit method effectively exploit kronecker limit well exploit multidimensional scalability series exploit structure computational placing induce grid create toeplitz exploit ht automatic speech deep sound series consider context datum use large contiguous region grid locate therefore direct inducing point toeplitz scalability show induce mean log scale empirical value correspond runtime hundred runtime confirm loss cubic interpolation gain induce less runtime generally infer curvature function unable induce tend add scalable gp require computation overall induce induce form cubic scalable combine kronecker gains toeplitz arbitrarily locate input show ability induce expressive kernel learn improve improved order magnitude simplicity major strength explore interpolation create entirely new scalable strategy could remarkably perspective interpolation well induce point induce need combine model orthogonal benefit recent process kronecker toeplitz provide motivation toeplitz hope improve understand dark medium gps kernel framework quality induce interpolation covariance mechanism scalable kernel choose kernel scalable point alternative enable toeplitz substantial additional scalability require fast expressive storage gp sound process gps exactly flexible capable learn expressive requirement gps contain empirical thus induce method large computation storage inducing induce purpose require require number expressive learning exploit toeplitz advantage induce exist highly accurate scalable kernel dataset lattice product make kronecker locate input likewise costly similarly restrictive require grid induce kronecker toeplitz scalability induce critical covariance induce induce scalability fast combine exploit show interpret underlie help accuracy induce point interpolation interpolation strategie cubic interpolation create kronecker toeplitz induce point toeplitz gp computation kronecker gp require input view toeplitz locate input gp induce order magnitude popular extension toolbox simplicity generality make easy scalable gaussian section structure interpolation reconstruction sound conclude process vector yx gp collection fx kf gaussian nk jx kx fx smoothness kernel example rbf hyperparameter target additive yx covariance gp evaluate depend obtain condition separate calibrate fit complexity hyperparameter integrate iterative covariance form complete theorem prove eigenvalue asymptotically bound complete eigenvalue approximation pca determinant marginal approximation expressive input space general setting toeplitz complementary toeplitz stationary kx kx spaced toeplitz along kx toeplitz product fast g gradient storage grid gps computation flexibility scalability box require point suffer major predictive accuracy perform expressive valuable large structure gain requirement grid place induce method kronecker toeplitz algebra complexity computation computation product cross covariance induce wish bx u ix weight extremely interpolation per weight grid regular use distance weight expression approximate gp essentially induce make directly exploit toeplitz vector cost storage
health institute image big http www fellowship foundation google finding conclusion recommendation author necessarily reflect view nsf tradeoff space empirically cubic notation secondary one observation type resp resp resp large matrix involve size input relative two analytical differently dominate expect output channel type versa approach magnitude strategy channel demonstrate ratio output channel dimension fix channel vice versa optimally cnn narrow major validate heuristic scheduling device contribute follow protocol vary ratio show fraction scheduling essence gpu also use peak scheduling tried estimate device speedup optimal edu present compatible end examine characteristic purpose convolutional neural architecture employ cpu throughput improvement cnn directly cpu hybrid cnns networks cnns research application recognition cnns perspective database concern contrast cnn technology key choice cnn modern offer grid slow microsoft project cost effective generation intel cpu parallelism likely continue user center issue amazon ec neither google compute study architecture conduct open cnn call version output bottleneck convolutional execution technique focus tradeoff layer batch one standard multiplication multiplication compatible library intel art pick optimal depend ratio usually dominate optimizer pick space automatically network contribute execution system fast simple achieve convolutional cpu device create cpu gpu typically gpu reach almost use argue hybrid layer ec gpu instance core cpu throughput become effective homogeneous open question amazon ec end gpu describe definition convolution operation popular operation convolutional element index kernel operation problem highly multiplication convolution layer take channel transform tensor multiply phase multiply multiply create back strategy correspond sum let indexing array slice describe e submatrix dimension expensive expensive balance create multiply trade start index let eq multiply expensive balanced spectrum either phase expense call q two approach multiplication intermediate appendix conceptually experiment report number fusion discuss optimization discuss partitioning partition partition convolution partition indicate default process process single parallel split partition partition show indicate partition currently parallelism layer model share decision simple heuristic input device gpu conduct compare library cnn system neural architecture use cpu version gpu imagenet diverse ec machine illustrate per tolerance concentrate throughput find thus remain compare ec instance iteration image b ec cpu instance cpu physical speedup cpu use fast increase conv appendix probably comparison cpu gpu gpu core run gpu cpu expensive gpu fact gpu however far magnitude associate cpu suggest cloud services gpu microsoft google train deep cpu validate accelerate purely cpu gpu training running convolution operation gpu hybrid batch image run gpu run ec gpu bridge cpu core report figure group significant gpu batch cpu cpu gpu core available hybrid gpu cpu figure ec gpu execution per give speedup number parallelism fully layer briefly study focus improve although decade specifically multiplication due year library framework
entropy bind proceed lemma always eq combine lemma assumption study randomize reduction reduce projection result randomize reduction hinge large address randomized regularizer reduce dual mild dual study present randomized method communication scale datum continue application bioinformatic finance vision medical critical solve big great dimensionality reduce also computation efficient reduction lot e hashing latter refer randomize examine reduction generalization model strong rank assumption separate weight assumption regularize randomized leverage solution scale dimensional support compare example dual implication randomize hashing compare mild assumption subsequent interpretation exploiting classify people genomic important designing addition performance exist feature propose feature dual recovery reduce rely realistic ii analyze smooth iii method projection set analysis application distribute learn combine benefit address optimization receive problem especially dimensionality reduce employ develop ascent observe communication let solve parameter primal randomize randomized reduce reduce dual previous reduce random construct matrix recovery recovery error one dual reduction corruption original strong plan limitation different relax assumption contribution trick sparse solve regularization whose reveal later understand dual regularizer svm hinge non smooth hinge q change give reduce margin smaller reduce reduce square hinge loss primal eq regularizer emphasize dual error trivial assume set complement set remark dual proportional x bind assumption contrast require x dimensionality affect recovery result hadamard hash om recovery low connection signal theoretical smooth recovery restrict eigen restrict another restrict restrict eigen condition recovery function bind sn recovery nearly smooth quantify except top satisfie distribution dimensionality reduction lemma reveal subgaussian universal hadamard projection dd ii hadamard typically compute choice possible provide sample randomized hadamard sampling universal projection apply random mp dp dt md satisfie please speed hashing dimensionality analysis rigorous recently hash hashing algorithm denote rademacher random equal write type hashing remark compare remove extra discuss another one consist reduction random aforementioned randomized sampling implicit explicit scale method recovery transform introduce randomize smooth bind om idea relaxation show conv conv supplement entropy arrive remark result amount e sn section conduct contain split norm report hash randomized function hinge aim motivation affect vary among randomized reduction trial square hinge loss indicate magnitude variable support consistent dual decrease increase large certain making threshold recovery exhibit threshold consistent trend recovery much square loss sufficiently solve sometimes experiment randomize multiple distribute distribute associate among total communication reduce distribute original demonstrate effectiveness stochastic ascent reduce high use reduce problem record running step optimization ii recover communication method stop run improve
salient query guarantee discover left size example query fact three many query various fail query completely recover satisfie recover query good conversely good hence salient return recover video sign image tie video initially unlabele goal triple crowdsource platform amazon type crowdsource worker ask specify feature task worker ask feature assign non triple pick triple triple except figure run datum gender early choose face gender learn feature compare adaptive baseline adaptive pair worker show tag ask return feature complementary discover many distinct relevant two fraction hamming say differ redundant sign face product triple triple feature terminate done replicate seed triple discover dataset adaptive triple sign face triple obvious choose distinguished order learn feature poorly distinguished algorithm run query guarantee feature hierarchy triple efficiently feature partition feature discover induce belong agree average indistinguishable discovered perfectly distinguish benchmark function query scatter compare triple require indistinguishable triple triple adaptive pair rapid decrease discover triple discover introduce framework query inefficient feature experiment three set theoretical prediction unlike detect redundant human process avoid redundant feature place outperform feature salient face language product direction would investigate type challenge crowd direction work attempt salient imagine aggregate addition different feature salient diversity crowd triplet low common low also otherwise recall adaptive double feature rest one let triple low triple triple feature triple write child child triple set triple nice property triple query feature map combine therefore triple subtree make triple begin observe query step feature see query leaf leaf query moreover feature queue currently explore induce subtree root underlie subtree subtree triple query star subroutine stop query algorithm draw double query p yx follow similar lemma randomly triple triple query query discover feature discover maximize discover triple course minimize triple choose argue triple triple random pt theorem remark ex ex ex microsoft research ca microsoft approach discover crowdsource crowd member common two display provide binary discover triple adaptively label hierarchical simple similarity recover less finding discover statistic crowdsource discover merely label hand address crowd diverse salient name datum example face salient gender numerous binary feature think mapping feature refer string describe value crowd worker exploratory machine feature learn significantly compact exponentially grey crowdsource ask people multiple word phrase example fail ask crowd worker tag sign american gray tag tag equally none could discriminate sign inspire prior familiar belong name present crowd triple three example common feature meaningful datum choose triple salient triple distinguish sign size salient people often triple address feature crowd worker accord label necessary eventually require annotated discover query triple salient adaptive say gender assume thereby avoid equivalent face illustrate hierarchical orthogonal applie across large state query adaptive response e car sophisticated independent random find use expectation since incur moreover worker second feature batch label image feature learn ask query one seem could green similarity theory arbitrarily discuss hierarchical section triple adaptive triple query feature analyze expert approach crowdsource name automatic representation e summarize inspire worker right task times query vision order direction crowdsource show worker positive example feature reduce adaptation order grain terminology incorporate finally receive every triple distinguishing return already discover imply common triple definition common two child child child distinguish never observe triple particular internal node example leave parent leaf leaf distinguish triple figure triple would uniquely distinguished distinguish feature close point say specify advance triple answer could anonymous think purpose least proper non least query figure discover choose specific example child leave internal order discover triple must triple triple fail answer leave set difficult triple triple triple triple triple insufficient motivate triple moreover proposition query feature tree pair determine identical maintain queue explore discover initialize default feature initialize f jx query query label consecutive go feature triple efficiently internal child internal branching root tree rule feature terminate triple star triple use limitation defer appendix example face thus feature abuse much even triple among crowd triple always
analyse classifier kl moreover majority vote case classifier x point kernel express linear precise pac domain idea decompose risk expect disagreement disagreement label domain weight latter domain divergence limit worth note recover well domain relate enyi specific measure domain estimate sample hypothesis last old p bayesian adaptation divergence contrary measure domain oppose hyperparameter disagreement successfully need make adaptation negligible pac bayesian attractive consider different necessary sound domain marginal xy unsupervise interestingly correct capture pac bayesian justify generalization present precisely simplify h equation function eq shorthand jensen markov extend pac bound quantity appear especially interested possibility control help parameter disagreement source consider obtain similarly reason thank optimize domain real h disagreement respectively separately combine suggest minimize former disagreement choose ignore result hyperparameter tune hyperparameter trade optimize linear prior center e x figure adaptation linear achieve equation descent starting give trick weight augment rewrite fix rbf positive resp firstly decision toy label label domain clearly succeed adapt misclassified disagreement target secondly use amazon com benchmark review product book attribute originally rate simplified rate product appear ten domain process tf weight perform adaptation book train co train amazon com thank reverse search parameter logarithm report algorithm label unlabeled report evaluate separate imply method reasonable algorithm overall six increase confirm novel improve risk bold l derive context majority analysis domain joint source crucial divergence domain trade give major domain tackle name adaptation future aim extend divergence like covariate issue suppose two domain actually disagreement give weighting bound depend enyi second unlabele rgb universit universit france issue express majority upper trade source disagreement target easily rely pac generalization propose machine learn corpus e unknown wants study computer vision etc simple spam adapt another significantly different often importance divergence bound pac focus vote tackle domain fashion share seminal bound trade marginal ability adapt paper adaptation trade upper half disagreement bind relation source distribution weight target domain pac analyse closely gibbs predict draw return call expectation risk pac bayesian literature deterministic classifier study disagreement seminal analysis
obviously approximation allow sa intensity cardinality target target previous birth particle array acoustic sensor frequency sa measurement cell standard formula sa give factorial moment predict cardinality filter describe update distribution filter efficient tracking proposal equation design sensor sa filter accurate cardinality filter exploit approximate contain I distribution label particle label process label birth update time rewrite birth extract mean gaussian alternatively density kde follow cluster sake exposition survival target mass newly constrain probability constraint impose since track posterior subsection density proposal survival target birth survival birth proposal construct survival eq birth summary target exist newly target code multi use proposal grouping particle death newly generate target I k ip q k lead particle sa cardinality match cardinality prediction important designing proposal snr snr multi sufficiently estimate general become detection multi filter seek proposal match cardinality individual compute specify component cardinality account individual density straightforwardly choose preserve label mass cluster construct resample much thus implement time weight probability track elementary construction sampling cardinality newly propose particle sample ix code implementation k q evaluate transition bx k ix I transition due proposal technique computational load sum problem particle transition guarantee k target sample target proposal closely spaced target k k coordinate modulus amplitude velocity walk fluctuation complex q tracking measurement usually refer measurement consist power return cell mean white symmetric spread cell cell coordinate cell centroid I complex denote cell write statistically random reduce define I centrality distribution cell modify hence form template e consider incoming highlight grid notice target share cross consequently snapshot domain report initialization mainly propose incoming particle velocity towards position birth particle symbol acceleration initial state birth birth target assignment fig fig increase target enter scene thank convergence difficult target due fact retain target confirm behaviour target reduce phenomenon surveillance confirm challenge spaced multi filter enable filter result confirm spaced target use snr approach problem devise target target interact satisfactory load separate spaced target cardinality target mean measurement process number measurement process edu surveillance area call model finite estimate lead version bayes filter however straightforward smc feasible sampling space multi sa recently multi bernoulli applicability application closely target label filtering problem arrival array wireless amplitude tracking merge multi track spaced target contribution estimation sensor dynamic base typically collect measurement facilitate efficient tracking important target trajectory real describe target assume target generate belong target preprocesse raw set usually application space standard approach may adequate case make information necessary require advanced measurement time superposition multi approach smc method passive application target indistinguishable approximation target inspire propose target sensor track framework label enable track direct problem arise propose particular call sa design particle latter require spaced definition label sensor target particle label density numerical present section present description sensor report random euclidean practical tool notion integration theory standard system model specify notion integration multi contain measurement target denote target target track history track simulate technique tractable omit history multi recursion prior equation start target h convention density index satisfy interpret object term exponential family form give set existence probability track need filter principled bayes tracking capable tracking target describe general fortunately target greatly transition drastically subsection present particle complexity review ensure order integer target target kk k completeness multi recursion posterior integral computational requirement filtering technique approximate approximate integral converted density particular dominate measurement integral interest construct generalise context vary particle w k k value posterior z p description filter
high word ability direction work second accommodate arbitrary apply quadratic variational appeal structured penalty lasso group penalization systematically track promise exploit screen quantify response penalization rr rr rr em ar ex simulation design adaptive original ols intermediate calibrate control rr rr r rr r rr design em ols ex sparsity stage purpose possibly estimation coefficient estimate penalty procedure benefit transfer stage operate distinct crucial calibrate discovery setup gain range art discovery explanatory attract attention decade development penalty selector show relevant various recovery apply differ coefficient accuracy numerous stage predict stage relevant operate candidate accuracy assess perform ol lar relaxed modify reasonable marginal bridge statistical perform regression method rely relevance select region hypothesis testing bootstrapping differ rely stage summarize estimate support variable focus first coefficient improve genomic regression quantitative use penalty variational optimization hierarchical detailed benefit inference empirically ensure coefficient family false discovery criterion reliable input alternative numerous perform fairly nan hypothesis expect one usually error genomic false appealing context attempt follow path first consist permutation computationally theoretical approach credible define projection function semi propose whose build later extend aggregation response unknown dimensional vector rely lasso tackle define hyper fit throughout discuss regression numerical acceleration apply address approach view adaptive viewpoint mostly formalize formulation ridge coefficient return proof instrumental define dependent implicitly determine stage process primary retain estimated stage small large variational bayesian covariance matrix assume model stage two approach second state support recovery reveal regularize condition incoherence state truly variables retrieve provide irrelevant covariate correlate rate slow noise estimator procedure produce small restrictive experimentally design ol variant adaptive consistency since believe low interest estimator relevant predictor ols decay lasso ol respectively mean experimental validation thereby theoretically prediction performance actual penalization cross commonly penalty follow ols regression arbitrarily ridge adaptive choose apply serial overfitte sense result optimistic conclusion test dependency denominator randomization set exact exchangeability exact multiple take variable estimate test approximate block screening prescribed design lrr ex c permutation test ex satisfactory level either calibrate conservative prescribe false level explanatory problem propose among correct calibrate show establish clean procedure partly fact relevant establish describe remain validation mention independent rely summarize modify procedure devote stage gain overall illustrate set regression screening stage design cross importantly support limit support implement respectively criterion datum truly truth analyze present application genome association infection variable variance know numerous magnitude relevant varied predictor mean unit variance dependent mean belong position relevant randomly distribute block belong enable noise discuss design report three medium medium drastically compare variant mean conclusion measure coefficient display snr snr ex group lasso high medium ridge still slight step improvement benefit ols ridge adaptive par option jointly optimize stage respect penalization improvement compare study mainly focus setting beneficial addition design considerably penalty within cross post lasso screen ols ridge optimization penalization serial se though viewpoint also square discuss representative setting feasible respect control discovery significance true false negative control procedure univariate variable selection experiment noise level calibrate stage calibrate procedure ridge original ols ridge rr r simulation ar ex lasso numerous determined model poorly lead high statistic stage block group ridge line ridge dash ols dot rank dot line mark observe consider far design group approach high level extremely systematically dominate weight regularization ridge base plain improve ol ridge bring thus table procedure threshold level always clean design dramatically gain ridge variability regression follow ridge ridge bring stability c lasso ol r ol r ols dark red lasso clean significantly high sensitivity dark red box properly selection procedure statistic affect less origin design compare ol univariate testing screening perform lasso calibrate rr rr ex em ex ar ridge ols correction univariate design dramatically original gain effect beneficial transfer ridge difference ar original ol univariate calibrate rr rr rr rr simulation ex ridge ridge ar original testing calibrate rr rr rr rr ex ex ols ex variable wide infection study identify genomic influence rna level infection
enough sentence order vector need close additional run record fast computer error lattice learning problem great relevance well variant adversarial succeed time due run time grow learn question simultaneously number draw linearly consistent open whether make ia current elimination noiseless two confidence mistake strong pac function consequence statement mistake comparison relate comparable improve thm remain gaussian make explicitly keep track general observation seem explicitly make give learn use conceptual carry complexity run instance run immediately get confidence complexity consideration runtime bad exponent result thm pac possible enough ok example instead randomly mistake hide specifie lying take element cover factor every example vector learn think intersection gaussian increase collection sufficiently size sample order contribution reduction noiseless learn exhaustive search drawing devise well presence non amount noise rate open attribute hamming weight associate work mistake round round boolean must predict mistake learn mistake example unknown learn pac complexity example e consider pac fix half mistake pac learn concept mistake mistake run convert pac learn introduce correspond pac source rate every mistake bind mistake per round kn red pac statement pac mistake mistake mistake mistake well see slack divide roughly divide run round improvement improvement pac model prove thm basis large constant later arbitrary part let ensure km k apply conclusion rest reproduce span n kn tt nm kn tf happen efficiently elimination ready describe begin space ia ia constant individual enough thm fix lem maintain obviously terminate hidden mistake make notice begin whenever make mistake predict space reduce factor lem learner mistake learner store vector treat subspace elimination lem ease calculation round simplify ignore term eq gain learner mistake calculus long improvement mistake mistake noise rate half
examine theoretical cv formula fact predictive firstly cv second equivalent optimizer secondly lastly mathematical relation also minimize average statistic design regularization depend foundation enable call hyperparameter design hyperparameter optimize likelihood rational procedure minimize paper average generalization validation validation important property study applicable leave cross validation cv hamiltonian turn view criteria cv criteria estimator generalization value equal equal whether cv average prove regular cv make whereas maximization second loss sample take hyperparameter minimize random loss generalization problem criterion employ determine hyperparameter region cv criteria heavy enable candidate variance cv small new formula eight section section learning devote proof theorem result discuss conclude future definition bayesian statistical learning real euclidean probability train nonnegative call study improper normalizing predictive generalization even asymptotically validation leave calculation markov chain monte method define widely show paper posterior cv cv pn however asymptotically equivalent statistical regularity minimization method marginal minus marginal likelihood eq method integration improper hyperparameter minimize minimize several notation condition definition mathematical arbitrary candidate proper denote loss posteriori estimator depend equal log minimize define simple power adopt convention suffice mean automatic summation order singular set infinitely time parameter unique minimize convergence regularity matrix invertible almost inverse well k k finite let q equation regularity say definite satisfied hold singular machine expectation mathematical reason definite saddle order outside integration zero empirical mathematical relation define k w w note relation average relation manner mm calculate eq k k k j k empirical mathematical relation hand mathematical derive prior hyperparameter find relation asymptotically asymptotically unbounded phenomenon discuss asymptotically relation map replace average neither generalization prove regularity exist theorem true learn mathematical self average estimate hyperparameter widely regular directly point nontrivial example note improper prior prior therefore relation minimize exact cv nx criterion definition cv n nx z px numerical ten thousand collect prior firstly average average deviation give std std hyperparameter candidate hyper compare choose minimization interval proper average std generalization whose improper whereas energy hyperparameter h predictive design c std prior immediately derive hyperparameter cv calculate xx yx criterion nx n z free conduct dimensional identity candidate hyperparameter variance rigorously hyperparameter phenomenon contain std lemma arbitrary px training leave test log leave validation generalization proof px w I px px px n w w j half half px n px px px half obtain mm function map v need mm integrate region taylor expansion respectively give fw l u derivative define use notation remark matrix sum pair combination symmetry odd integrated follow sum put complete leave hence odd eq condition moreover n map proof equal parameter note map therefore apply therefore lemma eq term complete five mathematical average generalization prove sufficient prove define let odd even number q prove prove lemma eq q subsection minimize
computational limitation limitation model approximate evaluate summarize principle meta expansion krige model new krige formulation pc krige krige pc krige spc spc krige employ least determine orthonormal polynomial polynomial krige meta krige spc krige iteratively universal iteration case polynomial trend select loo model find krige krige spc krige krige error analytical function result pc good distinct experimental design krige preferable spc kriging reduce spc krige research realistic reliability idea design input zero level reliability instead everywhere initial interest add preliminary investigate carry france research fill blue width text center height text height text center width minimum height rectangle draw fill minimum height draw fill text blue rectangle text rectangle fill text text height width pt fill inner sep input vector two less researcher paper modeling polynomial krige regression set polynomial pc krige validate benchmark reference krige perform large limited asset demand computational model keyword model meta pc modern make order ever complexity structural new behavior acceptable basically input parameter understand eventually constraint similar common dedicated aim physical possible fidelity modern high fidelity typically mean run may day material apply exhibit simulate know refer probabilistic model variable prescribe joint assess uncertainty consequently onto system uncertainty call input million compute architecture surrogate surrogate capable predict input realization analysis among option construct meta focus non black input realization provide additional knowledge build expansion krige machine extensively investigate decade polynomial expansion krige expansion expansion polynomial variable traditionally expansion partial differential expansion obtain among method specific though treat especially code hand call projection review development spectral compressive polynomial paper reliability optimization find meta technique interest krige known gaussian krige meta interpret gaussian find field structural reliability krige technique toolbox r toolbox attempt krige bridge aim distinct powerful call krige sequel accurate flexible detailed organize krige approach combination distinct meta section benchmark analytical equip algebra variable denote capital letter realization low letter capital low letter system behavior introduce uncertainty pdf joint pdfs note variable model input value output cast expansion orthonormal vector index input dimension independent margin orthonormal construct candidate variable summarize orthonormal classical pdfs ht orthonormal polynomial orthonormal uniform beta bx multivariate polynomial compose multiply polynomial handle series truncation response accurately truncation consist bound total polynomial maximal denote polynomial polynomially truncation thus tractable highly input vector low interaction polynomial thus interaction degree tuning maximal decrease small interactive univariate retain vary part solid line represent index define candidate next consider repeatedly evaluate model realization decade pc namely square minimization minimize evaluate empirical denote error derive read function polynomial able output thus type regression expansion thus quantify l residual respect pdf error auxiliary validation rarely purpose model analytical eq normalizing variance output number polynomial sample experimental design tend phenomenon predictor vanish loo general build n loo theory computational loo would determination special analytically build proof build model technique krige assume response realization p value trend variance unit variance auto various mat ern autocorrelation generalization mat scale call shape euler modify kind simplify apart correlation part krige namely krige assume trend trend unknown unknown formulation trend sum pre krige krige discuss define krige auto hyper calibration parameter eq response development approach preferable autocorrelation assume family cv shall discuss multi minimization algorithm cast distinct algorithm quasi newton genetic differential evolution algorithm spc krige behind procedure polynomial trend universal krige procedure orthonormal trend universal kriging base error krige figure box blue box represent spc krige realization node auto distance cm lar autocorrelation block sequential pc krige distance cm prediction line lar lar prediction line line lar pc krige spc krige krige krige orthonormal algorithm spc yet krige consist iterative add auto calibrate pc krige meta model meta minimal leave error krige universal krige trend polynomial ranking box universal krige loo mark cm cm autocorrelation node block node krige auto auto loo loo line right cm auto loo loo loo right right distance distance loo loo line krige loo I cm prediction krige krige krige view krige meta model trend part valid loo pc krige meta krige comparison krige illustrate analytical evaluate verify new krige approach uniformly distribute input two gaussian original benchmark analytical independent input use method sensitivity analytically behave smoothly point space eq consider x function last function pc multivariate model hypercube meta model ordinary krige spc krige krige krige combination gradient order compare output value krige meta follow variable uncertainty carry th percentile boundary dimensional approximation krige krige meta minima maxima analytical component high global characteristic component whereas ordinary frequency lead input visual krige meta design choose yield second result ordinary show spc krige krige ht modeling meta sample ordinary krige perform box spc krige krige bad overfitte pc ordinary krige due polynomial krige krige spc krige ordinary krige krige perform value krige well krige accurate spc krige range ht relative generalization error purely model design sample krige relative generalization model need error behavior among krige krige approach sample krige significantly require properly surrogate krige resemble case capable analytical various function fig estimate highly input previously fig qualitative pc krige quantitative combine krige visible krige traditional approach follow krige low error whole generalization meta pc kriging resemble ordinary krige experimental design krige slightly spc krige krige experimental accuracy input krige approach like pc krige krige whereas pc kriging resemble whether krige provide accurate pc krige combination krige accuracy high computational krige spc krige intermediate
specify reconstruction move filter might significantly variation step modification expression noiseless next convenient perform majority side sparsity varie perfectly idea entire successful k independently probability reconstruction length albeit number close condition get insight q suppose signal tell large significant ignore remark inherent difficulty establish namely quality continuous reconstruct quality arbitrarily reconstruct far stochastic compressive video compressive signal introduce resolution instant collect decompose foreground typically background respectively take measurement tell foreground still measurement give compress mean suffice perfectly foreground frame motion image laplacian perfectly integrate nothing show compressive background translate diagram frame pursuit depict motion construct past rather foreground former texture yield prediction foreground operation model take reconstruct obtain output input optimization input foreground proceed current subtract module foreground frame z l dot dot dot op op op op op op op op op op op dot op dot op op highlighted technique generate decoder motion motion reconstruct consider require motion metric spatially smoothed vector coherence outlier motion far field linearly belong overlap block pixel predictor motion neighbor pixel pixel scan frame correspond pixel white white green white green black red black white solid red dash line dot line illustrative mostly background reconstruct reconstruct foreground k visualization oracle gray oracle figure gray reconstruction sequence sequence frames people office top panel fig show background frame several background foreground frame fig frame reconstruct foreground dark setup sequence true small value initialization adapt frame camera e isolate sequence motion block mention frame improve reconstruction solve remain frame benchmark cs oracle fig measurement estimate foreground frame quantitie fig fluctuation clearly advantage foreground standard oracle cs recall cs line oracle though fact small less resp frames frame reconstructed show quickly fig relative error around solver varied reconstruction error frame foreground ill condition frame closely figure figure noisy oracle figure gray gray cs oracle noisy noisy gray gray cs truncate measurement vertical e compute yet curve shape noiseless reconstruction order online reconstruct dynamical minimization perfectly explore background real image video sequence notice perfect reconstruction increase time perfect hence function independence recall definition assumption consist contribute probability contribute conclude event c event condition sparsity step prove inequality state third value expect note component contribute cf event independent equal simply value condition use corollary conjecture rgb rgb department electrical engineering college uk mail n ac electrical engineering university mail laboratory e mail reconstruct sequence signal support evolve nonlinear recursive compute compressive problem image background image foreground reduction respect background compressive reconstruct sparse signal signal evolve otherwise describe nonzero reconstruct measurement online reconstruct measurement measurement formalize online measurement acquire possibly sparse invertible transform f sequence signal track wireless application background detect video example surveillance traffic image g compressive expensive uv compressive video access frame conventional video frame notice foreground pixel frame sense minimization reconstruct foreground frame mention exception compressive compressive frame succeed cost unnecessary measurement address problem online use reconstruct foreground area foreground extra foreground frame give frame fail adaptive sparse reconstruct optimization measurement know reconstruct say generalize pursuit show draw measurement require reconstruct small pursuit furthermore choice irrespective free address reconstruct use estimate require compressive side contribution summarize contribution adaptive reconstruct satisfy compressive motion next acquire word know characterize compressive background incorporation number measurement another make fundamentally prior reconstruct signal sparsity slowly contrast operate theory slowly quality e side extent pattern sparsity nonzero use establish compressive background performance conclude proof reconstruct signal limited overview solution control terminology filter kalman filter e across filter knowledge state incorporate procedure kalman filter signal time instant nonzero signal compress sensing assume vary slowly number measurement assume along relate work include knowledge kalman measurement name briefly overview probably reconstruction scheme euclidean problem version replace measurement problem characterize computing assume measurement signal review
error interest solve problem regime present sequential scenario pass prohibitive minimal thought contradiction default sequential fashion assess enough evidence nan stream enough even far statistically easy conclude confidence look fraction dataset discard expensive test subsampling know hard subset suffice could resource address rest subsampling entirely stop would suffice devise formally mean q think parallel stream resort memory keep track single vs process main issue apparent multiple testing observe rejection conduct reject chance correction conservative produce walk move intuition law alternatively algorithm batch automatically stop difficulty near nature type follow imagine fair coin assigning tail keep coin basically remain envelope early walk behave play role new fair flip walk around envelope outside envelope hypothesis practically examine asymptotic version depend empirical tool independent non contribution control non uniformly control uniform desire stop propose intuition detail automatically concentration always absolute observe binary coin may bias detect test alternative size involve deviation hoeffding nan fail reject reject fail reject sequential arrive time size sequential define threshold rough argument statement introduction sketch formal corresponding result test threshold control type also type concern asymptotically iterate non sequential insight p n treat small powerful sample sequential early motivation use reject statistically distant alternative work coin full would q examine therefore low bound binomial test stop definition hoeffde first infinite geometric fact stop soon could almost good formalize precise non reasoning sequential seminal line testing implement upon result together heuristic e clinical trial perform loose version bind though scope current nan stop apply may additional cause practical implement rigorously test template refer batch importantly simplicity denote h v u nh h sum ensure large estimate base specifically whenever least consistent test test specify interpret unnormalized statistic assume moment subgaussian special suffice schwarz difference however analogously batch tight control computable priori concentrate around run simultaneously combine uniform inequality deviation analyze test bernstein walk exist theorem calculate time test generic control type basically constant result basically favorable high prove tend argument identical hold walk finally bound version walk capable extend outside scope stream I word discrepancy mmd mmd ball correspond population theorem mmd iff differ alternative sample kx kx limited testing sequential test note batch similarity testing get type error factor also early stop independence involve test population quantity remarkably characteristic joint conclude process b I calculate scalar assigning expectation batch n design previous section control hand present sequential nonparametric testing alternative analysis term desirable property empirical type compare importantly essentially early present simple extension setting theory next form upon v concentration inequality p second goal moment prove bernstein however version convenient follow take define
occur need recognize occurrence encoding speed stream meet number discrete transform dft apply spectra decision tree firstly fourier spectrum aggregate spectra spectrum memory reduce fourier redundancy advantage arise new combination recognize stability stream secondly devise thresholding compression obtain concept optimize dft remove potentially vast literature drift recurrent exist fall broad category store meta learning mechanism match drift method past belong category concept design unlabeled datum issue explicit recurrence overhead module whenever difference estimate concept store repository show outperform global dynamically classifier individual meet accuracy acceptance train ensemble function stream conceptual together build set version et approach consist classifier classifier concept concept drift state incoming classifier instance threshold validity also design delay similar accurately express base classifier newly receive confidence learn al recurrent concept concept accuracy observing design dft tree highly future dft improve classification maintain spectra parallel dominate fourier spectra match current fourier transform dft turn dft apply et distribute capture decision algebraic representation preserve represent dft fourier give jx coefficient fouri approximated computing storage overhead fourier consist thus mechanism capture classification fourier inverse dft expression value thus avoid tree fourier symbolic classified calculate maintain forest hoeffding tree drift divide tree fouri pool maintain encode hoeffding tree forest drift fouri spectrum spectrum fourier maintain pool ensemble ensemble pool ep carry reference structural similarity describe discuss special place h ep structural root datum set randomly select hoeffding forest pool read classifier forest pool classify embed detector window classifier correct else drift identify perform fouri dft threshold pool aggregation pool ensemble pool pool hoeffding attribute create good empty pool income forest pool drift signal drift instance good term thresholde help change tree high accuracy repository whenever concept spectra repository nature identify subsequent good pool classify subsequent point classifier prior dft reduce redundancy pool firstly check whether good spectrum pool threshold step succeed dft produce make pool pass integrate spectrum separate spectrum spectrum great structural similarity currently evaluate disagreement decision disagreement exist ensemble pool update ep single remove alternative aggregate spectra define fourier state call ep et show sensitive high energy great thresholding energy obtain inherent tree iterate spectrum energy order drawback proportion energy fortunately equal coefficient energy single spectral denote order compute exist extension illustrate character begin vector without validity cardinality straightforward optimization increase process speed calculation character absence present hoeffding computation generic domain schema optimize inner product character schema else exactly combination illustrate character occur beginning position validity attribute dimensionality value save multiplication case scan vector overhead multiplication coefficient calculation large derivation fouri aggregate spectra represent produce spectra produce different point spectrum spectra set aggregation stream advance still use express q spectrum produce drift accuracy comprise inefficient bottleneck spectrum major advantage overhead effort initial spectrum extend attribute transformation define split tree integrate integrate account attribute expand incorporate schema expand spectrum add attribute expansion remain index position unchanged position integrate spectra produce localize attribute essentially implement mention focus assess effectiveness ep consumption assess ep pool impact spectra significance generator recurrence stream know ep recognize generate span occur stream order challenge add noise inverting instance spam spam spam informative attribute datum price move price dataset simulator file four scenario record every instance velocity feature maintain stability take velocity directional moving average window instance spectrum reveal approach employ storing concept repository advantage reader refer comparative spectrum mind design practical dynamic stream accuracy take minus sized stream entire fig individual across strategy dataset contrast ep follow show clearly dft fouri spectra store environment memory limit large ep pool spectra ensemble examine usage time claim aggregation ep recurrence counterpart segment concept occurrence span concept represent recurrence ep concept aspect consumption assess consumption influence great store repository accuracy consumption use exclude memory relatively experiment pool ep spam present memory consumption pool forest repository distinguish focus repository exception ep hoeffde together spectra produce chosen candidate aggregate result provide memory high ep ep achieve consumption figure benefit apply aggregation fourier dft application stream variety maintain relatively spectra ep aggregation reduce dft computational term process speed r dataset hyperplane spam fast even though simple suffer counterpart ep current winner tree fourier pool ep aggregation strategy effort stability drift demonstrate expensive operation aggregation yield work environment ability minor dft application mention coefficient minor capture inherent provide dft therefore experiment aim non decrease level clear interesting tolerance ep counterpart similarly interval metric superior performance explain generalize
sample line median network network average prediction bayes nearby oppose confident cast bandit set label receive receives reward receive thus receive reward measure difference reward achievable receive take sum agent various bandit task agent scalar represent action output selection keep tuple buffer size bandit heuristic trading exploration exploitation greedy policy good figure compare bayes agent greedy purely greedy agent enough greedy explore agent nothing approximately explore begin ignore almost perfect learn good classify mnist digits performance bayes comparable demonstrate non problem allow reasonable unseen bandit bayes automatically exploitation readily scale optimisation asynchronous furthermore readily gpu david comment backpropagation compatible compression principled comparable dropout weight plain feedforward neural network reinforcement uncertainty train confident correct shall use call introduce rich representation contextual overfitte decay principled build upon backpropagation predictive little systematic exploration greedy task network rather learn computation exhibit learn thus instead learn train unbiased gradient inference network form neural integration upon gradient prior attain dropout relate deep modelling variational apply unit unit might thousand network magnitude make optimisation large hide allow complementary averaging problem thompson weight great uncertainty network naturally decision make deterministic understood neural learn network classification conclude brief view probabilistic give weight categorical softmax pass normalise gaussian square input map onto several layer transformation learn likelihood backpropagation placing upon weight find map eq laplace answer expectation possible configuration accord label test item expectation ensemble neural practical learning parameter sum shall trade satisfy readily interpretation length prohibitive various certain express density probability transform posterior work proposition monte backpropagation algorithm inference bayes unbiased gradient learn trick operate great apply unit fewer unlike complexity require close combination variational family cost posterior term weight draw technique common number use part cost posterior gradient find kl compute bad cost variational posterior gaussian weight obtain gaussian deviation posterior posterior optimisation pt pp calculate calculate parameter deviation backpropagation remarkably learn deviation prior posterior simple diagonal minibatch partition kl cost epoch partition scheme heavily influence largely useful influential contextual bandit persistent context choice different yield expect present context agent build reward action use pick importantly receive difficulty absence train upon exploratory action perform model neural weight observation high reward explore sometimes exploitation leave future investigation thompson popular pick exploitation pick pick suboptimal thompson bayesian treatment thompson sampling pick probable often fast thompson new pick sample thompson adapt neural pt sample variational posterior receive receive go mention decrease variance trade reduce monte pick posterior action pick begin converge selection focus discover estimate lead exploration mnist contextual bandit task classification ensemble dropout sgd sgd sgd mixture mnist digit training image label nine mnist generative etc shall improve ordinary feedforward softmax label exclude augmentation dropout attain around sized digits set used hyperparameter learn rate protocol descent mle size average bayes sgd unit initially overfitte dropout converge bayes expensive dropout slow eventually bayes dropout posterior
efficiency rwm hmc hmc riemannian langevin ess mcmc kk ess normalize overall measure computer illustration start bivariate gaussian box limit unit rectangle directly panel density panel rwm hmc hmc truncate seed method overall reasonably ht c truth repeat set covariance obtain discard overall hmc efficient rwm hmc rwm propose state reject constraint efficient slow high spherical hmc handle suit ccccc rwm hmc e rwm analysis tend group magnitude estimate optimization alternative method call penalty term replace represent full sampler spherical augmentation handle particular distribution hmc algorithm propose box constraint evaluate diabetes coefficient sampler spherical hmc ordinary ol choose correspond shrinkage vary spherical fix comparable compare impose tight low shrinkage spherical hmc substantially hmc discuss section fact family call residual sum square constraint magnitude allow force exactly model bridge flexible effect shrink limited bridge constrain domain unit ball apply estimate bridge diabetes spherical hmc tight fast shrinkage reconstruct take value translation form function gaussian type box sided spherical hmc form low upper unit form component side absolute discuss summarize rwm hmc hmc effective sample implement spherical hmc normalize ess interestingly hmc ht ccccc rwm e hmc hmc popular model draw mix proportion document assume draw assign probability semi collapse index factorize method stochastic riemannian langevin dynamics sg langevin use mini batch hasting langevin regard step refer sg sg modify sg follow metric logarithm volume adjustment sampling assignment conditional exclude decrease predictive method assign training test document calculate train document wikipedia vocabulary project text evaluate hold mini sg sg sg compare show list sg early stage number increase sg absence fisher sg sg sg introduce sampling sphere explore map mathematically framework augmentation original slack augment energy geometry volume adjustment total take advantage split efficiency split lagrangian velocity momentum avoid requirement embed could spherical geometry introduce directly start informative spherical geometry augment could might able add benefit future explore possibility spherical augmentation elliptical sampler slice sampler general infinite involve infinite f drop quickly increase geometry sphere r map introduce induced define metric metric dot euclidean call canonical lead fact foundation hmc invariant regardless right side ball view system jacobian way term canonical yield form canonical metric way c invariance analytic determinant determinant determinant canonical metric matrix lemma inverse change measure functional compactly coordinate chart cd euclidean geodesic q symbol cg ij cg rewrite augment multiplying obtain sd td x spherical coordinate element volume change measure jacobian matrix jacobian determinant need jacobian determinant weight follow note jacobian splitting hamiltonian usefulness hmc dynamic discuss split start hamiltonian eq lagrangian solve solve first dt dt g u tt discretization difference locally expand equation prove eq k ft iterating provide hmc error right determine monitor intersection coordinate constraint approach norm constraint leave state hmc boundary solve find element find consequently determine instead find intersection sort order intersection sign point k constraint constrain domain adjust velocity velocity u point v h v conjecture pdf figure distribution lasso probit domain challenge commonly novel augmentation constraint domain sphere sphere generate remain back spherical augmentation computationally sample state hamiltonian use example process lda modeling constrain geodesic lagrangian commonly statistical bayesian regression probit many copula intractable simulate estimation improve quite zhang due target deal mcmc typically proposal ensure boundary impose quite inefficient especially domain alternatively remove inefficient explore involve norm map augment explore way implicitly remain within focus hamiltonian carlo hmc discuss modify sampler boundary go create boundary approach henceforth inefficient domain follow propose hmc handle interesting type applicable present brief overview hmc variant underlie idea norm type spherical augmentation hmc constrain evaluate method section devote discussion direction hmc improve upon walk rwm propose distant current accept distant proposal numerically simulate hamiltonian denote denote common symmetric definite often convenience hamiltonian define density sum hamiltonian evolve equation available need time sake numerical usually solve system metropolis reject metropolis acceptance although hmc explore rwm geometric hmc riemannian use position explore sphere hamiltonian riemannian endow momentum hamiltonian unfortunately manifold hamiltonian become product consume g follow g reversible volume change jacobian determinant satisfy condition throughout term handling type restrict augmentation manifold ball hyper way target change sphere recognize chart collect sphere discard affect apply transformation adjust change corollary respectively transformation result implicitly handle impose original illustrate sampler move translate boundary original hyper constraint hyper constrain ball thus ball type rectangle side spherical augmentation use carlo particular hamiltonian however generic vector domain vector variable formulae present dd spherical like change transformation energy coordinate system hmc sphere coordinate could convert later besides handle hmc technique efficiency hmc endowed v sample define hamiltonian change chart minus derivative adjustment contribute extremely adjustment adjust integration v hamiltonian recognize standard hamiltonian augment due invariance appendix velocity c z hamiltonian g equivalently symbol preserve hamiltonian approach euclidean avoid assumption hamiltonian split lagrangian follow appendix tangent energy riemannian circle sphere analytical detail define simulated discretization size improve computational efficiency rotation proposal show henceforth h accept dt hmc metric sphere g start dt hamiltonian change potential energy round spherical adjustment minus leave volume adjust estimation integration x xt recognize hamiltonian explain invariance see
generate output input store across machine could represent structured consider communication alone work training support inspire constrain alternate multiplier admm primal theoretical show convergence solver method training model organize briefly introduce implementation discuss present follow input together minimize loss limitation focus well applicable still valid constraint view set write one respective solution satisfy fix method iteratively work well iteration work work current work distribute store call instance store index inference call optimize inference consider round single cut plane inference small work working dual inference multi core parallel challenge iteration use compute use communication form box problem apply distribute box constrain decide communication eq tune decide decompose sub locally solve exponentially machine coordinate machine inference two scalar feasibility directly base admm reformulate unconstrained mention split minimization maintain update require problem intuitively solve problem regularizer close equivalent penalty function minimization solution environment summation require add average consensus consensus outer update parallel weight equality augment lagrangian multiplier add objective duality saddle substitute next consensus sub problem convergence require convex converge consensus convex satisfy admm converge parallel many rely select sub problem work solver communication machine depend usually grow binary depend decide solver see feature index inconsistent across part speech pos machine observe order machine round incur huge communication overhead tackle issue hash strategy use unique function feature weight vector task hash use string integer environment note hash structured task distribute quadratic admm study extensively use primal rely several fortunately leverage framework modification non method dual outer affect costly many consume essential low structured perceptron method major optimizing perceptron simply guarantee converge single also effectively inference require modify minibatch update machine share unclear parallel part inference setting assume communication overhead whereas communication substantial environment core significant speedup expensive two pos parse dp pos label aim tag sentence tag tag speech pos viterbi word tag dp sentence structure syntactic parse formulation high liu evaluate parse score word correctly portion test machine result reasonable quickly machine perceptron train separate final eight conduct experiment implement set split eight partition fast method inference round communication improve realistic time parameter inference communication identical communication round admm ten speed tune affect fine set use solve
membership membership membership component update respective minimize log likelihood due accounting partial membership minimize maximum update seminal minimum message length mml message mml model mml scheme encode generic summarize encoding encoding message length encode cumulative fisher summation ji j encode message goodness negative mml minimize message membership update respective mml estimate converge model need estimation consequently quality mixture thus reliable tradeoff balancing aspect determination mixture evaluate mixture function balance quantify bic integrate complete mml parameter far criterion mml incomplete address tradeoff mixture criterion mixture vary component score convergence effort get optimum issue arise play role split merging component enable optima notable amongst mixture select two split leave component candidate depend simplify mml bic facilitate mixture component chance like scoring lack mixture fit explain address limitation propose conjunction comprehensive mml approximation infer establish outperform widely mml adapt modelling idea perturbation merge improve assume mixture child locally optimize mixture message update separately split merged operation merge great message length consider possible operation give component mixture good chance none perturbation explain split critical lead desirable splitting mean reasonably unchanged subsequently perturbation result improved component mean direction work directional three spherical describe moment estimation minor axis submatrix dispersion root equation hence maximum note minor axis parent align onto major minor axis plane split align op standard let sphere co second child angle mean parent start child component locally child serve parameter mixture component membership adjust remain component estimate em choice merging determine closeness identify close component kullback consideration membership component weight membership respectively em merge analytical form two kl ac respective normalization constant analytical b spherical find leverage state splitting merging attempt optimality mechanic suitable method refer reader example angular axis green mixture heat figure visualization spherical co transformation elliptical contour surface shape pattern begin split component child initialize mean explain child optimize two message bit ht merge figure optimize child subsequently optimize result component use algorithm illustrate splitting splitting mixture note split different show em case em state mixture improve amongst perturbation split splitting component integration black unchanged colour merge third carry figure depict split merging component splitting initial mean child separate optimize message length improve mixture merge appropriate close accordingly g perturb step ht explain message mixture component increase mixture overhead mixture weight mainly message correspond optimal associate examine variation beyond start mixture total message length reason decrease increase curve marginally increase thus encode parameter affect minimal gain log increase effectiveness ability mml sample procedure moment ml mml version form parameterization correspond parameterization version show inconsistent dependent mml expression likelihood estimate optimize require metric evaluate kullback divergence various compare use statistical mse decompose significance value figure ht considerable compare mml great map mml mml kl divergence compare moment compare bias mml mse mml tradeoff lead bias mml also proportion map mml traditional considerable especially size bias correction mml reduce moment mml base estimate bias far mml estimate competitive parameterization inconsistent therefore avoid mml regard parameterization applicability mixture directional chain atom position protein atom sphere radius atom co determined atom set form directional protein protein consider publicly comprise pair directional describe mixture distribution previously explore concentration use mml taylor mml truncate comparison work equation estimate modelling employ determine optimal employ mean current reasonably apart good chance form terminate iteration involve split merge model terminate iteration iteration mixture merge appropriate mixture terminate perturbation result search begin component split merge observe curve component mixture behaviour component increase method infer rise perturbation final characterized behaviour iteration search steady increase message length series characterized component message decrease search terminate increase dominate increase mixture mixture effective visualization component include contour sample heat concentrate characterize typical component observe component compare mixture model model mixture protein directional modelling reflect explanatory power mixture parameter complex show encode part bit message hence gain cost length lower serve well explain component mixture great difference gain bit low message length address mml htb c length bit million bit circular contour shape contour contour onto contour shape depend define explanatory enhanced compression descriptor directional protein previous descriptor base uniform sphere due mixture nan description provide model use offer oppose angle equation surface successive corresponding mixture encode accounting distance atom total obtain descriptor infer translate enhanced compression mixture bit compression follow application protein directional demonstrate mixture model nan mixture component million bit mixture mml computing mixture separately encode mixture mml use two introduce constant parameter negative aic bic modelling criterion follow heuristic mml perturbation mixture determine criterion bic involve estimate mixture em algorithm note mml em section mml aic em ml ml estimate approximation mml mixture mixture bic mixture moment resemble mml estimation mixture bic employ change evaluation c mixture maximum bit bit traditional aic bic minimum reach value behaviour initially decrease likelihood difficult appropriate trend distinguish moment log huge amount part message mixture encode length apparent moment ml mixture length ml difference encode mixture obtain unlike aic bic mml criterion also component mml htb htb aim project traditional aic evaluation mml conjunction search amount empirical vary range fixing change infer mixture base mml see mml trial agreement complete aic result great distribution unit sphere theoretic square traditionally moment maximum mml transformation estimate mml traditionally use conjunction demonstrate model protein spatial modelling mixture describe descriptor modelling task biology minimum message length von fisher modelling statistic grow science biology directional directional type surface compact von density comprise distribution unit sphere respect model directional generalize fisher form characterize value parameter scalar compare distribution factor additional directional statistic pose difficulty also lack order achieve balance suggest alternative relatively interpretation equation orthogonal vector axis spherical analogue serve surface argue unimodal gaussian spherical surface value scalar entity infinite importance angle mixture joint task bioinformatics complex mathematical estimate approximation considerable effect mml reliable parameter ml mml unlike ml mml map mml invariant state estimation use parameter mml take account parameter state determine mml framework inference part encoding encoding parameter select parameter base mml mml wherein demonstrate mml outperform one directional datum demonstrate reliable mml base traditional mml model mml well traditionally also invariance mml compare model protein distribution serve mixture model directional organize mml framework highlight explain associated construction likelihood parameterize mml implementation constant derivative mml base describe section experimental discuss respect selection paradigm criterion overview mml develop per probability per event code event require shannon comprise bit result give odd hypothesis rigorous compete message vary explain may state come message goodness fit generalized set reasonable derivative likelihood mml free lattice quantization comprise mml difference ml mml ml effect consider minimize map estimation self state precision incorporate determine volume region center multiplied probability use compute message encode precision comprise directional parameter parameterization relatively along axis rotation matrix transform orientation axis support co determine show bring plane operation transform axis subsequent rotation angle major minor axis plane orthogonality preserve orientation angle axis plane matrix scalar concentration contour spherical visualize relate allow correspondence elliptical understand vary spherical contour contour moderately major figure maximum ml require density widely estimate formulate estimate derive moment moment alternative adopt n normalizing moment step align co transform axis frame angle variance axis rotation define direction require low submatrix decomposition subsequently dispersion rotation orthogonal transformation transform axis standard coordinate axis transform axis moment estimate moment eigenvalue quantity simultaneous equation conjunction estimate limit approximation accurately optimization library obtain estimate minimize solution numerical routine root start previously moment typically explore compare distribute n non space drawback formulate give space density derive prior angular density uniquely define direction mean spherical surface angle determine orientation major minor angular definition range conditional joint density reason consider parameterization map invariant space characteristic inconsistent parameterization non linear prior j parameterization f likelihood across different estimate various dataset show random size parameter library conjunction derivative optimization maximize observed counterpart obtain ideally maximize
massive long term twitter explore via stream reasonable power adopt twitter share facebook friend activity find twitter stream without content devise reconstruct tweet study individual response suggest average exposure production tweet occur average exposure positive relationship response whose week experiment highlight occur highly differently different highly equally likely adopt interaction evidence yet avoid consequence experimental reliable forecast circumstance sentiment text tweet positive sentiment score provide text twitter linguistic rule correction medium tweet positive range neutral single sentiment express tweet define positive sentiment tweet range sentiment tweet tweet neutral bar neutral neutral tweet negative tweet author tweet baseline previously neutral tweet baseline proportion bar sentiment tweet collect consist least tweet provide week twitter collect user tweet produce week produce hour precede tweet tweet english contain medium video finally tweet target annotate justify english medium attribute sentiment tweet choice limit last limitation tweet week discount reconstruct user sample medium study tweet within description user finally separate tweet neutral focus intensity facilitate overall piece intensity hypothesis medium various pass via online typical interaction essential ingredient reconstruct tweet allow whether correlate response subsequently study purely observational control differently sentiment neutral tweet exposure tweet less one fig positive tweet tweet neutral positive amount exposure tweet model notably sentiment tweet neutral one perfectly neutral observed positive test significant neutral far illustrate narrow negative positive negative response response positive seem neutral particular another call tweet sentiment tweet formula fraction tweet large since tweet produce precede hour publication allow represent calculate tweet stimulus associate tweet value fig response neutral value stimulus illustrate show stimulus response stimulus stimulus response suggest strongly stimulus follow positive stimulus neutral neutral individual tweet cumulative tweet affect user previous measure tweet stimulus focus tweet user tweet produce prior tweet proportion neutral proportion sentiment tweet user tweet proportion determine euclidean distance sentiment determine nature stimulus exposure neutral less equally likely adopt adopt great tweet similarly distance tweet vice versa tweet fraction tweet fig tweet exhibit content presence select fraction tweet positively affect positive versa adopt high one perform twitter facebook control exposure design nan highlight user history response week number insight exposure negative exposure response suggest divide significantly adopt suggest observational possible separate entirely dominate observation user vice real mixture experiment need social medium channel million individual day produce daily medium micro communication therein facebook spread typical person massive content unknown consequence use content
obtain single source distance improve cv computation distance computation simple source computation source distance storage inherent query graph coefficient node sampling probability usually sample choose adaptively space distance size metric running dominate computation size base cv suffice node need computed randomization introduce precisely randomization ensure obtain probability relative exceed polynomially node identify initialize node dominate computation node graph short path uv query member compute apply computation query show include either cv cv relative polynomially section establish base regardless lemma close algorithm distance l since substitute eq z substituting contribute v z k consider situation uniform expect suppose uniform node close position sort uniform particular iff choose accord z uniformly z conclude per precise definition close distance accordingly space median within median close close node within mean distance probability proportional distance substantially large show universal contain node show partition node remain definition u z v nz corollary lemma grow concentration conclude apply chernoff node exactly contribution variable range chernoff expectation node equivalently high apply estimate relative polynomially query metric space would mention way step involve source ii alternatively polynomially small placing source computation contain uniformly select polynomially error mean polynomially probability polynomially establish identify polynomially distance provide important property relax high computation base total small time sample query relax definition claim property quantile distance distance close v well verify yield universal polynomially distance computation point sample quantile quantile candidate half nc polynomially error establish claim useful single computation apply coefficient apply return ensure cv correlate must amount computation polynomially polynomially single source computation base sampling obtain sample base probability single source computation treat apply space start overcome explicit calculation first nearly little satisfy p c replacement compute unbiased cv definition therefore direct hoeffding inequality size obtain probability relative exceed small efficiently like express probability v uv relaxed desire guarantee detail use pair condition within polynomially next subsection efficiently implementation fairly completeness independent replacement obtain independently replacement arbitrarily order associate interval randomly draw sort pass sort completeness describe sort set operation draw independent iii exponential transform sum hence precision identify scan point randomly point take jump sort scan array point size randomly bad per improved tree total contain clearly practical per cost search universal computed set claim computation upper say quantile get eq side side far thus wu c nu eq use rough accordingly definition apply weighted estimate query guarantee algorithm uniformly node factor size vertex decay lemma pair use ideal sort high bad adaptively another computation compute node sort exact consecutive mention completeness provide apply metric compute iteration fraction current exclude increase point linear arbitrary probabilistic create copy times max absolute weight g copy weight ab bc ac complete truly algorithm compute short show detection undirected triangle negative number triangle detection computing instance triangle construct problem union copy correspond complete negative claim path cycle see direction contain v v correspond regardless shortest path correspond shorter path get u triangle centrality distribution average point upper value low satisfy inequality z half node spread centrality consider node point whereas isolated network contain separate centrality comment cv point diameter restrict single still weight entry obtain respect nonnegative symmetric observe close row member therefore universal realize triangle absolute embed universal intuitively size distance inner product stem something whereas large like centrality reflect consider weight particular sampling extend node equal respect point perform bad median exact median separation fewer adaptively determine approach node maximum apply skewed hence easy sample well median z median working simplify necessary estimation increase way smoothly minimum tracking stop correctness note estimate summary weight weight even skewed cv factor single probability term space sample depend surprising linear section thm conjecture thm thm thm thm thm corollary thm com il cs ac il conference volume query fundamental classic centrality popular measure study social novel insight relation via fundamental centrality computation metric preprocessing estimate use computation error ensure error exceed polynomially structural centrality study measure classic closeness centrality term closeness centrality closeness centrality centrality reflect ability fundamental cluster distance centroid datum consequently relevance relate distance distribution cluster advantage parametric similarly near knn distance knn target outlier carry incorporate classifier demonstrate uci repository accurate knn notion centrality extensively aim provide facilitate metric correspond length path input specify graph round distance node input mention difference application relative imply distance centrality list sum give centrality metric seek computation node worst seek computation computation perform nearly unweighted pair suffice sum also distance node suffice relative scalable easy path tree contribution weight estimate statistical suffice ensure relative polynomially query probability least compute probability sample expect sample
every hx h part turn part depend set may combine guarantee error aggregation lead begin depend probability satisfied turn invoke fact every clearly nontrivial class g dictionary assume type equivalence useful ball member fortunately concerned inequality contraction principle bound establish side involve contraction high symmetric multipli concentrate exponential obvious contraction base totally definition theorem corollary theorem theorem mathematical institute national act fast happen always attain procedure procedure attain rate norm span square integrable space let predict effective cost measure squared predict behaviour minimize risk take distribution mind sake minimizer exist choice predicting present extend loss function sample independently hope produce random minimizer integer learn set potential class respect endow find feature study rate scenario bound alternatively rapidly decay tail subgaussian however explain fast function unfortunately mean often common fast scale rate imply reality straightforward construct simply matter size correct happen mid procedure rate suitable see precise statement thus reasonable fast rate must reflect highly ambiguity fast optimistic roughly location accurate intuitive optimistic ignore mid time procedure attain optimistic location select belong framework function survey broad restrict goal aggregation attain optimistic minimal aggregation require denote avoid confusion optimistic risk erm outline follow description analysis specific fx minimizer minimizer q one look note every optimistic straightforward important close closed product target independent cm satisfie rather low class small purpose norm class process index star f f x I rich end belong convex star shape around symmetric belong measure indexing statistical view indexing form detailed explanation role definition define reason consider happen coincide cm parameter multipli excess square multipli component satisfie term erm perform optimistic indeed base straightforward verify convexity interested satisfie exist least optimistic rate optimistic ball constant specify cm happen upon select extend erm optimistic assumption include structural assumption optimistic learn sense good let wish occur achieve achieve optimistic thus may occur stress attain optimistic procedure may depend let assume every notably side central median class member norm result find consist dependent way behave apply erm element contain observation follow always f lemma whose excess relative small type perform somewhat one construct know oracle independent case q convexity apply recall identify way class empirical estimating mi nx hold right nonempty matter ready aggregation eq w one follow right constant event cm devote event original dictionary unlike aggregation carry excess eq claim apply n f therefore every every thus q thus verify complete proof focus show provide properly constant main low f ball obtain rather every follow proof thus approximate truncation probability absolute next claim observation similar constant note set class abuse write minimal norm th minimal every minimal observe large n fx union equivalence absolute constant end fix observe contraction argument g q fix name function f f mf mt j u ni v star shape standard verify sl k note provided therefore specify eq rd lemma least every f star sl homogeneous star ball suffice ensure distance member behave tail certainly mean distance unless obstacle use median functional show little need small estimate independent copy application independent copy applying depend eq
attribute object multimodal merge build softmax layer multimodal predict word share rnn model sign sentence stage image start sign pick word softmax generate weight two layer word embed use compute wise one note calculate multiplication multimodal softmax softmax activation role encode back softmax operation equation dense dense word reduce parameter decompose multimodal activation intermediate strategy accordingly element wise scale tangent well view intermediate multimodal softmax sharing layer rnn model without increase concept connect softmax concept suppose meet sentence annotation consume unnecessary whole access fine model cause concept decrease originally concept concept learn learn specifically word cat associate cat fix sub tendency word think similar associate word new datum tend overfitte new concept enough intermediate change baseline activation layer original bias fix call strategy sentence word embed lstm layer part speech image example description play cat contain sentence playing model see word cat vision imagenet category useful classification vision combine vision effectively new annotation image novel learning release image annotation around description nc cat cat accord instance annotation check whether description cat validate validate concept tb nc cat nc three dataset learn concept ms construct contains create derive contain activity concept sentence label sentence describe annotation ms sentence annotation sentence firstly imagenet train secondly rich description compare cat concept nc annotation figure randomly separate table issue dataset base add test pick cat denote testing add image organization nc three publicly available publication encourage future description concept b evaluate overall generated previously conduct comprehensive evaluation concept cat cat dictionary follow sentence sentence represent testing nc set always indicate balanced measurement show concept tb share original layer model layer well performance scope validate task cat word cat term mean layer activation affect deep deep deep represent tb ccccc c nc b cat deep cat deep datum add sample base image concept stand set image image concept base set deep achieve improvement novel concept test reach demonstrate word deep successfully exist sufficient similar helpful easy whether shot randomly image cat randomness training limitation consistent metric full indicate blue red nc dash line line shoot scenario draw show performance train concept nontrivial metric b deep deep l deep deep ct base deep model nc firstly counterpart secondly rarely imagenet pre train vision deep cnn describe concept requirement annotated nc ms effect decrease cat table nc learn semantic show successfully new word sentence low concept concept ms cat word embed novel vector generate visual learning sentence task description image describe method allow image novel large concept particularly share validate university com task description linguistic new module improvement share scheme suitable task prevent overfitte novel concept dataset construct task experiment effectively learn visual without learn concept observe description parent slow accumulate enough concept children quick rough meaning sentence previous word describe vision field handle new sometimes novel learn category add novel however concentrate mapping novel computer task concept concept child seem call visual sentence novel address large amount allow dictionary extensive concept concept validate rnn base large object provide method multimodal e retrieval method recent multimodal rnn performance task sentence model three use cnn semantic language
inequality get lipschitz use introduce probability expectation really multiplication somewhat proof attribute significance coordinate property apply definition consequently since enough clearly belong write therefore assume know estimate q trivial fx overall b k second get assume say k get careful aggregation contribution version totally symmetric submodular f li corollary si jk polynomial several notion value boolean function influence total finally apply generalization influence degree degree corollary lemma far learner pac independent agnostic value eq learn access draw remark respect use range pac agnostic learn rely agnostic degree excess hold arbitrary uniform model pac submodular function two influential function fit example simply variable degree plug uniform example output time use find influential ensure variable therefore analogous degree coefficient substantially simple proceed exist set set define outside namely equivalent need bind number degree fouri coefficient submodular relevant submodular random example least satisfying run use exist close distance triangle suffice prove j index simply estimate q within lem time application chernoff obtain desire follow least run influential completeness corollary agnostic submodular submodular exist learn bound submodular self exist monotone stick variable fouri coefficient bind careful low concentration stick closely relate majority know see kx otherwise bind sec ks w j k dt use condition pac submodular use result submodular variable learn class bit particular random boolean small every boolean monotone submodular function time random hx ok construct middle whenever notice see example give hx fm tx tx convenient switch notation indicator set verify j fs j fs fs fs fs fs monotone function algorithm pac submodular boolean access translate overhead use approximate overhead choose statement function mapping formula give verify corollary low reverse sensitivity spectral noise sensitivity sensitivity noise satisfie follow result w exist obtain low every exist linearity coefficient differ uniform error theorem slightly weak learning argument imply prove pac function constant small pac reduce use achieve imply example low spectral concentration submodular function self function construction code briefly concatenation string bound self point therefore exist differ fact property hamming point q avoid calculation hamming code via bound algorithm learn query reduce variable bound uniformly otherwise function example could obtain use bounding acknowledgement thank useful discussion convergence generalization unknown value rademacher rademacher variable rademacher view complexity self bound rademacher show small class monotone self remove definition since affect membership naturally ia fact fractional equality proof also study geometry place rademacher use definition clause negative lipschitz monotone self f bounding claim submodular equivalently play role combinatorial algorithmic theory polynomial norm fourier concept approximate norm function exist degree improve well previous monotone submodular technique reveal nearly learn fouri hypercube notable object research devote boolean hypercube attract algorithmic game analytic apply arise rich real focus property fundamental value analog play combinatorial game game submodular find function return algorithmic characterization rademacher play role bounding class well function polynomial define standard polynomial degree degree concentration low well analysis number algorithm rely crucially low polynomial application privacy approximate norm degree analysis noise establish degree subsequently value decision also give bind submodular suffice addition show submodular self bound learning class function notably motivate polynomial approximate know approximation approximate low meaningful variable learn work investigate denote bound submodular bound via suggest self bound low picture substantially rich class complexity bound summarize submodular prove submodular k k f c show general class constant total namely bound match upper small approximated polynomial free degree constant bind approximate within improve lower prove even total fx upper spectral total influence discrete measure pair particularly set drop due presence submodular function constant careful degree self fx monotone random submodular behave individual threshold norm result everywhere show namely rely boost prove submodular partial warm submodular substantially simple totally symmetric function factor exponent quite actually concentration require totally additive applie error new translate use brevity describe model improvement query ask low learning algorithm exist output far use run use complement upper nearly tight value see statement algorithm proof bind learn testing pseudo boolean unknown function submodular pac value query use value define submodular combinatorial optimization lot algorithmic expressive self monotone submodular natural necessarily many especially approximation shift invariant submodular range way equivalent nonnegative monotone monotone appendix full rademacher vector well tool learn equivalent monotone bound broad broad generalized function bound bit lipschitz condition normalize value bound possibly monotone submodular function equal coordinate function monotone non decrease assume bound learn consider error relative negative submodular bounding scale additive scale within rely value combination expansion exactly df w iff df fx approximation function e particular lead seek degree define follow f k maximum define clause arbitrarily fix achieve maximum since monotone fx w w fx fx fx j although bind actually tight prove choose coordinate order fx fx fx ci ci ci ci ci
feasibility derive utility homogeneity example unclear explain explicitly inter variation demonstrate size allow though article believe regularize recent advance collect adequate sample ratio necessarily estimate zero often covariance still heavily biased size achieve unbiased trade therefore focus aggregate achieve allow stable toolbox sophisticated simulation experiment explore serious helpful much herein marginalization ease drop subscript eq recognize unnormalized inverse n reciprocal normalize simplify concave analysis concavity two proposition consider mixed concave sum since gamma well log multivariate log characterization log proposition lemma see irrelevant assume hold data generality continuity due monotonically increase positive definite partition possible infinity since combination go definite log hessian log conclusion directly reference page compute derivative q elsewhere derivative straight structure I hadamard vector space expression definite need hold point z stationary zero multiplication substitute ax inversion lemma whereby need ax yx px wishart compute straight maximized recognize model derivative zero yield imply I number estimate scatter rgb rgb lemma fp science technology innovation jensen university covariance class meta meta applicable intermediate basic compare homogeneity fundamental correct mle poorly become ill condition approach central utilize covariance standard analysis pca linear discriminant quadratic discriminant example gene fmri many expand list publicly sample become effectively ridge precision still dimensional scope total exceed exceed major cancer genomic laboratory accounting assess motivated interaction contain gene method limit genomic ordinary various inter treatment effect effect meta inter analysis study model observation multivariate hierarchical ip probability generic notation pdf respectively inverse freedom pdf generalization inverse wishart exist wishart control inter homogeneity correspond homogeneity versa around inter variation homogeneity preferable much especially near therefore parameterization remainder x pi study wishart wishart evaluate cf arrive eq expect state likelihood concave concavity proof defer fix definite lemma state therein moment pool observation know pool scheme maximization derive compute conjugacy inverse wishart expectation wishart likelihood scale precision appendix current update inverse repeat iteration maximum derivation defer keep iteration subsequently numerical define estimator pool estimate utilize log implementation yield disadvantage identify maxima saddle likelihood class heterogeneity construction large homogeneity estimate homogeneity homogeneity hypothesis equivalent wishart become test leave hypothesis however nan simply number acceptance region likewise addition denominator add approximately intra intra class know meta well determine ratio variation abuse let interested quantity variable proportion study law variation equality agree ij need fourth observation wishart cf imply continue term substitute expression naturally straight plug estimator gene though variance exist precision stability datum variability contribution homogeneity low suggest high covariance color identify module color key edge dendrogram color color weight plot outline simplicity employ analysis study identify correlation agglomerative hierarchical cluster linkage minus arbitrarily height produces name color heat hierarchical module show graph hierarchical top gene cluster gene yield repeatedly suggest homogeneity select sd fit next module relevance use module base go term significance module highly c cn cn cn cn chi col cd cr cd col col col il col check identify overall os r treat module cox os module module matrix module represent module gene interesting module arise result survival analysis mark identify patient outcome manual screen module l degradation suggest activate degradation possibly express activate activity system il also link pathway associate poor outcome degradation central study link poor disease enhanced cancer thought role interaction manual screening go module meaningful covariance application discriminant utilize discriminant suppose variable suppose lda e assumption yy intermediate forward implement derivation analogous determinant simplify generalize becomes consider cf procedure correlate design perform similarly bad belong multinomial class round observation
high one regard period period long due change number period equally long totally uninformative one big figure seem uninformative prefer focus divide depend multidimensional example volatility depend price associate period period six frequently forecast day use viterbi evolve visit evaluate reliability predict come mention method idea define day one assume look sometimes wrong period divide vertical dotted line day make consideration day general secondly stable look graph day day really general say due change lead moreover limit instability publish researcher similar past day average average day plot influence strongly quantity give formulae book forward variation consecutive wider predict continuity keep amplitude variation error small analyze type work noisy right behavior recognize temporal time three forecast daily unstable mathematically forecasting problem value inaccurate respect price viterbi try mixture appendix want construct show divide depend hide viterbi belong state histogram gaussian try histogram htbp model see associate seem collapse mean around htbp model leave right mixture histogram clearly leave totally uninformative base mixture portion htbp figure forecast use ergodic show one closure vertical line testing period htbp htbp figure forecast consecutive normalization use model notice state price price lowest associate consecutive complete htbp mm cm remark hmms forecast hmms financial depend subsequently analyse previous literature put primary b phrase market big market change easily access market online call trading rapidly spread trading consist software decision market lead interest intelligence finance neural network hide hmms focus paper hmms forecasting publish md study paper forecasting analyse critical daily create set value year build test behavior impossible predict organization description markov firstly choice issue relate lastly understand hmms explain thorough performance datum conclusion mention finally hmms present relate initial chain process matrix markov q describe visit emission density distribution hide observation continuous emission gmm constraint let denote hidden emission gaussian observe process price daily determine dataset structure way turn state ergodic look put financial context price probable adaptation standard us ergodic right implementation step divide divide consist group assign gaussian worth note use transition matrix choose uniform group zero transition transition upper triangular leave right prove wu maxima three researcher great scientific advance wu performance combine start idea hide actually state idea prototype abstraction build method gap optimality try
prove slightly weak uniform ergodicity base kind lyapunov perturbation error lyapunov take supremum take obtain setting lyapunov lyapunov provide transition lyapunov uniform ergodicity constant requirement constant ergodic measurable lyapunov q constant vx p suffice assertion state requirement perturbation play first ergodicity quantify second appear norm measurable finite restrictive classical perturbation chain see restrictive perturbation ergodicity imply ergodicity ergodicity relax requirement lyapunov p ergodicity imply ergodicity limitation separate role uniformly lyapunov uniformly ergodic measurable lyapunov eq vx know fix real consider lead q finally yield proof complete distribution lyapunov far comment kernel calculation interpret family perturbation state converge bound begin study consider quantitative perturbation prominent namely hasting langevin let autoregressive q variable I say moment easily transition exist metric e assume wasserstein distance l obtain imply give inequality distance emphasize estimate w obtain eq assume thus eq bind autoregressive also ergodic imply analyze value norm difference measure inequality perturbation metropolis hasting analyze either hasting wasserstein ergodicity ergodic kernel uniformly ergodic interested realization serve proposal metropolis define probability step form work sample define accept else reject proposal unable evaluate force behind hasting algorithm random variable algorithmic work independently else acceptance still hold acceptance wasserstein transition transition form acceptance random independently uniform arbitrary fix variable analogously dy dy within eq wasserstein perturbation bind acceptance satisfy lyapunov p essentially acceptance probability numerator separate part integral remain suffice make subsample moreover choose arbitrarily obtain combine eq norm geometrically sufficiently main geometrically measurable denote probability transition ergodic measurable lyapunov number e possess stationary set use easily corollary lyapunov function assertion corollary wasserstein last statement follow corollary instead many corollary ergodicity imply satisfie drift argument proof satisfy sufficiently ergodicity metropolis hasting algorithm langevin implementation overcome approximate langevin mainly base noisy langevin gibbs field define lebesgue langevin euler discretization sde langevin diffusion value sequence variable diffusion stationary stationary say depend random distribute normalize carlo substitute define markov langevin algorithmic draw call independent fact langevin lyapunov l stationary kernel argument irreducible lebesgue weak thus set stationary ergodicity q statement consequence right side assertion fact obtain perturbation langevin number independent determine ergodic number remark state result lemma thm theory address question markov reflect difference flexible two markov satisfy wasserstein ergodicity condition monte lyapunov estimate geometrically autoregressive bound show quantitative estimate approximate version prominent metropolis hasting stochastic langevin carlo mcmc computational respect unnormalized method application available demanding see small two expensive likelihood contribute evaluate approximation rely moderately random subsample value naturally cut metropolis hasting budget bias bias discuss understand behavior bias bound ergodic chain implicitly appear ergodic restrictive markov chain provide perturbation wasserstein lead mcmc wasserstein distance turn consequence wasserstein geometrically ergodic chain ergodicity extensively use exist generalizing finding noisy gibbs integer space mapping equip measurable another markov ideal like simulate perturb actually transition wasserstein distance marginals wasserstein transition kernel wasserstein ergodicity condition assumption curvature lyapunov perturb transition number eq constant lyapunov wasserstein parameter weight supremum lyapunov wasserstein always satisfy suitable q denote corollary autoregressive former turn perturbation chain kernel geometrically ergodic ergodic suitable drift ergodicity wasserstein ergodicity moreover pair measure wasserstein observation carry wasserstein perturbation bind ergodic hasting particular geometrically ergodic markov geometrically ergodic distance quantify replace weak easy control theorem perturbation lyapunov distance perturb langevin refer concentrated measure eq next equip property linear homogeneity obvious fact form nothing transition define notation measurable whenever exist interpret introduce curvature convenience reader finite interpret property argument geometrically ergodic transition finite reverse complete linear know stationary additional assumption eq estimate trivial metric indicator set dx apply triangle variation obtain consideration exist ergodicity impose metric contain ergodicity exist wasserstein ergodicity eq resemble impose wasserstein ergodicity wasserstein distance high transition stationary approximate use chain namely initial markov chain call whereas perturb bound difference suitably step wasserstein perturbation measurable lyapunov p vx induction w allow existence lyapunov weak uniform ergodicity
successive behave allocation like example recall budget must limit arm budget concerned identify arm recall merely arm however impossible final converge theorem successive allocation set successive recall uniform allocation factor stress merely fall back ensure bad uniform successive strategy practice observe experimental compose minimize x xy mp output algorithm train validate hyperparameter I mf I selection minimization necessarily generalize give output iteration compute eq assume necessarily issue loss sigmoid put hyperparameter namely arm develop section hyperparameter uniformly log scale within region valid range sufficient cover grid grid source package bandit arm evaluate leverage nature machine robust fashion review relate context show hyperparameter optimal without explicit function reject hyperparameter assumption select per song prediction sample dataset year zero variance small error fast term plot successive successive top exp validation require allocation successive svm rbf train hyperparameter uniformly trial hyperparameter allocate calculate use store magnitude fast successive respect successive iteration plot ht consider bi objective objective hence hyperparameter rank choose scale result arm trial observe two particular successive successive theoretical present direction analogous arm variance switch cost intermediate memory resource various balancing bayesian considerable notational ease infinitely limit read single element consider singleton successive algorithm identify arm consider subset one complete showing contradict prove state run loss last envelope involve arm almost integer effect favor simple rearrange interpretable uniform follow arbitrary bandit literature objective good identification stochastic framework know set leverage nature algorithm cast hyperparameter stochastic good identification resource promise hyperparameter setting accuracy method become widely simplify accurate hyperparameter learn search optimize since many nature working scale evaluate intermediate partially example show hyperparameter via stochastic ask hyperparameter vast black box fully make attempt intermediate work form simple build work multi armed arm hyperparameter intermediate loss widely applicable bandit solution remarkably exist bandit fail exist fail two main loss monotonicity non obtain costly case compute drastically bandit identify set confirm theory relative standard baseline applicable good identification behave source subset paper set survey work suit set baseline experimental bandit identify average versus try maximize latter analyze arm objective stochastic suit cumulative necessarily suit good arm h observe recommendation receive continue arm armed stochastic multi bandit choose loss get versus good arm future loss arm play loss adversary generality start game propose arm probability adversary stochastic loss something behave minimum return arm bound decrease novel I hoeffding stochastic increase tell nothing even decay consequence reject possibility arm arm good attain despite challenge measure idea stochastic activity decade major branch set budget arm total algorithm attribute amenable propose successive non successive fix input minimize many decide suboptimal long discard successive elimination elimination implicitly e ucb generally exhibit undesirable behavior observe c observe loss elimination ucb exp total arm cost partially validation market probe case assume horizon budget total observe arm include popular minimize cumulative practice successive particular attractive along observe loss budget successive originally propose novel arm predefine bad repeat arm initialize k k budget attempt progress effectively parameter notable worst ever place budget total sample eq next identify without assume wise addition order final relative reasoning heart theorem tb arm return merely arm
turn refined partition local modularity sophisticated heuristic modularity approach biased tackle correct take degree mix usually euclidean space edge pair draw maximum markov position euclidean network similar latent practitioner extend representation vertex mixture approach bayesian inference look block look sbm latent distribution parameter ik ik node extra sbm heterogeneous account generate scheme well posterior adjacency parameter dependency derive tackle variational alternatively gibbs sample even unfortunately tractable either criterion aic ic derive criterion case sampler development sbm deal take assume subgraph look cluster belong vertex cluster relation vertex involve multiple multinomial distribution sbm replace bernoulli allow belong last order dynamic evolve time temporal hmm linear usually focus choose social affect future unobserved structure like highlight approach homogeneous poisson continuous removal occur approach graph build large literature machine target case numerous graph point least instance undirected biology neighborhood use early appear tendency become decade single quite common main idea extract whole graph supervise two adapt graph distance couple technique numerous adapt specific graph neural neural process vertex leverage structure maintain already process neuron solution consist building dissimilarity graph base relational difficulty detect far expect class belong complete bad complexity determine subgraph np nevertheless numerous solve problem computation introduce substitution series individual cost least costly transform graph kernel reproduce generalize numerous walk choose relational machine mean surface vast know extract efficient topology lead supervise exploratory analysis label concern vertex vector via challenge consist vice description issue two source medium temporal graph ignore issue propagation lot task actor another generally massive spread decade object complex etc etc ignore numerous cm universit paris f paris france commonly object straightforward formalism many scientific computer give introduction rely include supervise algorithm usually cluster focus topology static graph edge evolve deal evolve infer numerical characteristic balance source challenge especially locally context object interest give full possibly complete task produce cluster graph rather associate model scientific pathway science tie actor web network powerful tool extract complete survey refer highlight share goal cluster finally technique take often indicate short generally unsupervised vertex share connection general vertex define topology top vertex community appear densely connected group effect discover maximize score
hundred set human activity segmentation comparison dissimilarity poisson process measurement vary detect occur task show contain allow performance several point theoretically block regard dpp block semi psd assume pi psd conditional proof trivially one hold definition proposition pt minus plus plus pt exist map need present great application successfully difficulty detection new name preliminary point candidate study candidate dpp dpp conduct estimate effectiveness demonstrate five process elegant model selection quality diversity dpp subset q write quality item angle diversity feature measure assign quality diverse problem attract note inference greedy submodular decoding minimize exist computation take time become become nevertheless almost block fig inference replace inference kernel size item similar different away manner mainly aim detect period refer segment candidate change candidate quality diverse state g preferred point dpp purpose meanwhile almost block g far dpp become decade broadly classify frequentist change location include run improvement e advanced monte posterior big world frequentist core testing general statistic past window move metric value threshold ratio kullback leibler divergence explore threshold determining study heuristic dominant peak discard peak close require threshold dpp metric create preliminary candidate much treat point dpp conduct dpp obtain final diversity contribution name rest organize brief give theoretical real dataset interested dpp ensemble almost diagonal zero bottom leave namely diagonal contain dpp sub matrix sparse sub index correspondingly square definite consider motivate element see dpp partition correspondingly c c tell sub largely dpp map far dpp inference mc invertible rewrite map represent area zero block recursion objective optimize depth essence ic jj I subset item perform inference c almost diagonal series inference sub kernel comment optimization sub conduct depend wise greedy achieve sub dpp dpp lemma third apply dpp unique whole block leave study partition partitioning dpp block every size bottom overlap adjacent diagonal area note partition way value obtain balance small achievable partition smaller small adjacent sub inference empirical illustration fig greedy inference realization randomly sub area next vector separately step entry kernel generate partition greedy map original use baseline run much drop within map plug play inference ask connection map dpp relation dpp successively diagonal kernel eq dpp correspond conditional belief conditional fed selection result form allow map entire kernel latter small information incorporate generate kernel conditional map average error bar input ii let dimensional interval segment explicitly denote interval new build dissimilarity arbitrary dpp popular decomposition kernel magnitude view angle diversity allow construct utilize metric candidate create move adjacent window likely value peak value select
second unconditional guarantee tolerance rely describe state suppose batch monotonicity complexity monotone success absolute combine distance estimator monotonicity apply modal sample failure monotonicity modal modal weakly monotonicity modal describe work complexity original distribution sample straightforward knowledge though sample preprocesse step sample able make many correct question without approximate whole answer completely require learn restriction strong query access original ability make query distribution outperform query per first monotone achieve latter constant query exist sampling sampling decomposition let monotone fact without piecewise following find j l index turn constant complexity already strong indistinguishable close unless take already strong guarantee namely access well detailed level idea decomposition approximate histogram certain weight get quantity carefully monotonicity gets correct ever actually occur state interval constant let I monotone time argue significantly total otherwise closeness monotonicity jump mixture start two consecutive observe loss generality one monotone close assume eq I sum satisfy distribution put monotonicity suggest immediately output sample normalize whose non increase non monotonicity monotone claim bind sample adapt consecutive c proof previous otherwise closeness consecutive interval parameter describe completely fashion failure nan sampling monotonicity idea since monotone monotone apply scheme monotone inequality derive triangle latter feature query sampling imply query dual follow fully access expectation possible cumulative usual draw monotonicity cumulative query approach group interval optimally group every coarse fine correct correct lie inside boundary inside overall monotone kind global average inside monotone boundary last cumulative access problem e contain allow keep correct implicit query coarse monotonicity optimally linear ensure weight issue go correction budget entirely used process remain effectively end match exactly correction essentially weight include weight extra allow correction whenever way first correction query every point inside inside boundary weight water employ context order algorithm use budget sure never remain portion whole distribution sampling correspond know access cumulative access close close reason hereafter entirely cdf query quantity average subsequent water budget correction back average remain increase spread uniquely explicitly cdf fourth core subroutine perform correction range average throughout element great move stay middle whose maximum px ie say spread water allocate amount would pour total amount weight move weight full I happen might yet reach list portion w distribution cdf two monotonicity sample ignore extra budget try partly implicit coarse inside monotone even process e remain regard monotonicity proceed correct monotonicity execution water procedure allocate budget pour use beginning ensure use first domain support exactly ensures pick initially weight modify either pick draw event observation process determine uniquely define query query stage identically distribute bernoulli take select begin fact monotone fact length weight also detail therefore increase moreover construction monotone within explicitly sample change water act stop prevent remain monotonicity violate consecutive guarantee monotone consecutive budget guarantee indeed budget pour uniform weight correct would pour ib distribution constant jj hard g process total variation close satisfy monotonicity separately jj negative minimize clearly moreover ps px px allow conclude process budget allocate put differently amount budget write j element execution monotone monotonicity disjoint boundary consecutive j monotone add correspond minimum weight bring total put finally main theorem taking arbitrary j distribution hereafter denote original monotone access call approach original try detect miss sample perform shall follow appear utilize subroutine miss error batch proof next describe approach influence remove weight miss model interval monotone occur interval length element monotone monotone get monotone monotone weight e I violate interval weight end domain move add distance p monotone claim draw detect exist hold monotone inspire equal take hereafter partitioning care big monotone must potentially big element monotonicity xx claim observe miss modal monotonicity far conditioning either fall close suppose accept reject soon interval observe rejection guarantee failure explain I n n obtain probability kolmogorov derive close monotone conclude finish proof apply distribution bound encounter part weight quantile estimate correct stage hereafter convenience phase pass return either monotone second observe thus support monotonicity case denote expression fact define eq claim close monotone monotone distance n interested uniformity domain allow amount task arbitrarily naturally slightly bad query hereafter construction correction fidelity closeness complexity construction term uniformity domain extend level idea first von closeness drawback lie operation argue sum exponentially uniform however get close guarantee distance precisely distance possible close get essentially enable distribution combine idea use generate coin whether bootstrappe describe arbitrarily uniform von uniformity failure cc bias coin applying retrieve truly repeat time precisely variation variable parameter bit failure take bit hereafter view group distribution close key definition exist uniformity complexity extend draw ks bind k hand side far might achieve even n exist uniformity n support p get query yet bootstrappe exist uniformity guarantee bootstrappe recursively recursive resp j applying get recurrence recurrence give upper conclude randomness improved fix determined call easy observe generalize result unknown unknown sample could conditioning rejection argue great generator let cyclic moreover non imply sake hereafter generality absolute overhead possible discussion uniformity dd generator k h correctness fact independent uniformly random kk x break find union event amongst h event h relatively ideal therefore adapt rejection uniformity uniformity uniformity cyclic query uniformity query complexity rejection identify try draw randomness provide source source random bit uniformly distribute distribution property goal similar sampling uniformity extra setting difference randomness bind min input variation sampling since weak additional sampling use extra bit unlike sample original distribution tight result particular randomness low apply extra bit conversely uniformity min also vary uniform even sample could even subroutine truly sample correct learn monotone access monotone element sample take start approximate cdf additive define fm element effectively purpose k k check quantity also access leverage output direction interest example property agnostic efficient consider use sample correct follow work set testing query either get domain query cumulative provide one sampling original black succeed precisely batch get test vote well test straightforwardly generalize fully finite denote property I unique element variation g furthermore tight inspire question one nn l nk range obtain thus notion distribution testing fix distribution access call output least relaxation item fix output concerned setting namely call dual access oracle behave cdf put algorithm precise distribution exact variant paper contain omit sake arbitrary recall obtain monotone part lem claim theorem prop conjecture claim acknowledgement acknowledgement rgb support nsf author microsoft nsf grant grant situation address act end connection utilize expand applicability property algorithm improve algorithm proper analogous cumulative obtain sampling monotonicity stronger monotone namely addition restrict miss significantly learn whether additional bit require correction process distribution bit sample methodology work gaussian natural define methodology correct much principled include question inherent studying challenge science basis draw modeling address fashion propose methodology property property within distribution hand correct defer describe state informally measure guarantee distance number need correction naive learning find accord inefficient agnostic sampling monotone get output guaranteed necessarily learn return distance reader truly need sample quite although simulate truly random tradeoff furthermore parsimonious extra bit correction factor reason track separately main complexity agnostic approach throughout arguably illustrative challenging insight sampling non probability body cover decade e detailed list reference monotonicity wide concavity hazard risk evidence monotone direct implication correction shape constrain begin implication existence imply class dependency learn exist sample agnostic family monotone poisson binomial sum next algorithm efficient agnostic agnostic hard third estimate imply latter decide rigorous general get bound low specific application achieve improve various turn monotone cumulative well monotone distribution approximate histogram small carefully decrease monotone also access cumulative cdf monotonicity complexity level combine approach correct coarse fine correction within monotone mostly concerned distribution provide sample concept cumulative summary comparison access query access justify sort query implement overhead unless otherwise formal mention although definition present total analogous definition give draw internal notion allow close desire convert access access sampling evaluation property oracle term simulate query improve oracle sampling query maintain ensure
present detection convolutional cnn cast object suitable weak pointing converge boundary network object object accurate ap architecture recent cnn object vision human limited application object thus visual recognition move towards rich image understand pixel level many still box existence far region score far level detection mapping bounding cnn think must room cnn regressor straightforward detection integrate object cnn bound aggregating combine name bound box pointing leave bottom corner recursively direction fed network bounding box object slide cope state single everything performance window single include direction mis box object verify strength model primarily detection contribution fold suggest box aggregating involve bound regression art performance class task decade handle part flexible compose object severe another demonstrate competitive numerous variation activate vote location recent development detection advance cnn imagenet detection region cnn represent activation cnn proceed probably object object proposal feed pre cnn proposal mid cnn activation convolutional svms object proposal merge fed regressor mis despite limitation quality fail procedure reason agnostic proposal improve quality reduce cnn individual component feature box r cnn another cnn detection train cnn map rectangular mask object approach directly estimate method proposal leave bounding method also unify verify component detection operate extension summarize fig cnn feed directional corner bottom input possible involve right go image f let prediction feed corner corner return f corner end instance corner end detect project bound box image detect box activation several benefit portion proposal window object proposal carefully maximally proposal guarantee object reach terminal obvious corner fig previous detection cnn weak direction short length strong bound convolution connect layer filter layer layer compare adopt please refer detection prefer max force bound direction force maximally bound enable return prediction final conv conv decisions size conv relu max make operate devise quite image form test decision possible pair evenly case process original multiple augment generate satisfy positive top fig area vary complex scenario scenario multiple always narrow instance among instance follow rule generating iterative stage way bound region ratio scale region feed fig train cnn select portion region portion remain average loss extend instance verify effectiveness proposal direct coupling cnn separate use feature meet discriminative study maximally activate proposal ht l proposal svm score toy human reasonable pre cnn svms evaluate svm cnn svms classification compare much weak correlation neuron body start target reach maximally activate discriminative face human b initialize merge follow intersection box feed finally merge detect single include proposal separate merge several bounding box procedure us region feed entire body combination instance instance logic sure instance include boost region slide paradigm feed slide window successfully require input layer layer compose regular fed spatial activation significantly slide window method cnn diverse ratio scale also slide window feed scale image obtain thousand slide window feed scale aspect box instance region proposal produce fed image box merge decrease box minimum average employ bound regression bound detection employ box refinement box b window initialize fed chance reject false fine localization bound merge final verify primarily detection human wide object beyond human center stage decade nonetheless human image still severe rigorous verification human primary class compose web diverse pose variation set ap rigorously evaluation ap value one relatively training weight optimize pre conv conv initialize conv conv whole regardless parameter follow length feed prevent divergence scale scale aspect ratio accord box set merging refinement ce method without refinement ce achieve refinement score extra extra ap refine
recall lemma need cyclic permutation proof abuse notation rather since satisfie thus observe respectively simply exercise claim novel stochastic via order suggest attain sgd guarantee assume deep experimentally form every loss categorization I later generalize regime stochastic sgd basic index random gradient perform mind much distance control stay keep simplify around q subgradient two use notation j instances specify distance frobenius regularizer easy sgd rewrite measure give conditioning reason become conditioning consideration converge equation possible would naturally sgd algorithm rely pick minimize assume lipschitz matrix equation optimality condition know norm eq optimality typical lead order correlate nc scenario relatively sgd get back issue update overhead sgd time column invertible sgd intuitively capture deal show enjoy speedup advantage become runtime runtime sgd magnitude follow survey discuss variant preliminary showing find technique choose twice differentiable method dynamically coordinate hessian utilize hence compute convex example preferable newton method case base direction batch approach operator applicable obvious see adaptation bfgs approach estimation hessian low aforementioned order sgd always hessian approximation tackle rely gauss newton discussion come guarantee approach name algorithm adapt algorithm along form several convergence bound discuss gap storage time apply update rely diagonal replace iteration previously start update form positive denote apply update given equation inner inner provide lemma family proceed inverse informally combination identity subsection rank choice denote eq straightforward leading eigenvalue diagonal recall previous require decomposition fast calculate behind depict blue vector random coincide multiply right probability formalize set outline feedforward neural composition layer predefine training network amount function fully connect perform transformation usually variant gradient backpropagation affine calculate calculate unlike layer iteration note process cause main gradient descent technique convolutional weight convolutional col besides convex conditioning particular batch nesterov describe initialize choose random accord uniform mnist view house input function output channel pool kernel convolutional size layer channel relu affine channel prediction architecture conv conv affine relu affine training test multiclass axis height log legend legend align draw black color blue sep crcr color mark option solid crcr b width height axis legend legend align align leave white black table sep red solid crcr width height legend legend align leave draw white color crcr mark mark option row crcr e e style legend align leave align table crcr solid sep crcr much terminology architecture conv x conv relu summarize width axis true align align white black row sep crcr color red option solid sep crcr height legend style legend cell align align white blue sep crcr color solid sep crcr width height style legend cell align align white black sep crcr color option style legend align align leave
penalty conjunction convexity fidelity estimator efficiently far improve estimate matrix equal estimate column perform sparsity lasso selector observation location entry coincide node nonzero column need diagonal coincide entry construct dependency course estimate estimator straightforwardly partial infer clear mean absolute b concern describe dimension without panel variance relax maximum ingredient graph feature present study term residual variance rv likelihood estimator column wise sparse linear briefly handle next section present figure accuracy estimate detail introduce present material unknown note equal integer integer complement singleton matrix every diag transpose square note element row resp whose give resp element th row column resp pseudo element wise norm sample unknown consider present estimator diagonal development always center variable two coefficient residual j j estimating consist vector solve sparse estimator use explore empirically different perhaps offer trade complexity lasso matrix fit matrix element small separately computable even appeal preferable column lasso establish investigate estimation prefer root confusion scale aim estimation therefore gain insight natural consider matrix lasso expression orthogonal subspace vector multiplication intercept reduce follow residual estimator regression residual coincide maximum likelihood variance use distribution therefore complete handle similarly result observe usual estimator quadratic independent maximize likelihood lead eq recall view decomposable maximum clear bit rv truncation sufficient satisfie j jj provide discussion entry combine proposition variance contradiction estimator explanation really certainly parameter ignore allow explain proposition denote b jj b kk independent view variance jj jj fourth truncate right hand probability inequality p suboptimal lack constrain proposition error loss generality consequently therefore relation entail entry estimated risk th apply write maximize vector check condition precision necessary semidefinite b positive maximum jj cost last jj jj aforementione differentiable point derivative vanishe provide jj b note trace jk j give differently quite entail description set component vertex cardinality denote indicate class equivalence connect distinct vertex matrix h h readily introduce entry path connect reproduce argument estimator belong connect compare proposition estimator outperform symmetry belong ideal ideal systematically outperform variance estimating need original th mention early quantity always provide convenient keep connect estimate correlation ij somewhat replace necessarily connect loop depend choose try span short combine algorithm short path span threshold short tree span root tree increment behind span tree favor contain partial correlation aim minimum spanning ht follow ols rv rv enforce symmetry raise issue relate penalty measure intermediate penalize responsible trade constraint enforce symmetry extreme coincide play role feasible equal assumption objective feasible set feasible lipschitz algorithm indeed descent upper hessian unfortunately value loose resort descent f experimental explanation symmetry penalize likelihood precision square lasso rv follow rv without rv comprehensive experimental diagonal precision order many situation possible six several precise matrix use matrix equal diag diag six entry ij diagonal entry row introduce q result experimental compare follow section rv residual correspond symmetry maximum estimator diagonal conduct experiment matrix column root penalization commonly universal lead fairly scenario square entry root aforementioned value scenario correspond know scenario include experimental empirical precision configuration configuration estimator rv replication expect r table along error conduct square lasso cone root rv square lasso follow c error rv root rv root follow ols rv ideal empirical reflect comparison preferable residual variance symmetry refinement vast majority slightly bad size quality use happen step quality estimation mostly estimator thank nearly variance rv variable graphic vector central ol root lasso convergence speed fix rv root rv c rv ht c root rv follow rv rv explain term reduce suggest fact entry include suggest look contain short path weight among span tree short give tree path among bad complexity construction tree graph operation connect node short short degree complexity try related weighted short node component overall choose root large variant well rgb rectangle rectangle circle circle circle circle rgb rgb cycle cycle cycle cycle rectangle rectangle rgb rectangle circle circle circle circle rgb cycle cycle rgb cycle rgb circle circle circle circle cycle cycle early descent scale one coordinate descent perform size opposite increase constant mathematically speak operation gradient thank guarantee start iterate limit tuning parameter cross validation choose geometric range result plot nearly choose introduce entry precision copy precision commonly residual significantly symmetry mle numerical entry noisy small realistic root ordinary conduct accuracy residual mention introduction novel observation partial
nn per jj nn market jj nn relation concept trading evaluate ground especially inherent relation cluster discover algorithm model per ranking vocabulary multi noun phrase relation cluster strongly strength association sample relation coherent relation reasonably summarize position though reality toward second restrict basically pick related organization entity topic resort city near france entity exclude seq notably tendency form entity rather entity issue relation clusters persistent relation look topic cause allocation topic absence special set share illustrate incorporation syntactic syntactic pos tag basis concept like core flow rather notably requirement word explain absence broader share explain relation relation behave much like topic word semantic content likewise although syntactic abstraction away word syntactic one syntactic overlap relation first relation group level resolve believe modification lead significantly inference minibatch sample correspond chain corpus iteration plot achieve level contrary explanation inherent good redundant corpus although discover drastically alternatively need simply variance yield computational drastically reduce likewise minibatch tuning base simply far redundancy appear case would gain gibbs sample regime hundred million document also advantageous statistically efficient corpus amount store minibatch structure promise extraction modeling assumption coherent relational moreover acknowledgement author like improvement model explain mathematic alternative setup fail however fortunately much approach analytical variational collapse goal hold fixed sample use document document mean batch document gibbs unbiased natural little bit work ensure iteration follow minibatch document without burn n overall update learning update note update track raw code actually global manually remove limitation extend hyperparameter begin variational objective depend give use fisher indeed identity fortunately know calculate analogously gradient thus obtain second centre author amazon amazon access web scale automatic base large corpus unsupervise machine text recently tool scalability obstacle rely scale sublinear inference qualitative extraction web corpora gradually automatic resource relation extraction inherently encounter unbounded encountered need success probabilistic unfortunately prohibitive time impossible incremental training model lda apply extraction process support stream plain variational able qualitative fraction reduce include unsupervised show major prior group cluster sentence document use separate entity word syntactic paper feature vocabulary size feature document notation set scalar discrete associate sentence draw assume access pos name entity name entity simply kullback posterior small impose factor entire minibatch document carry iteration parameter relation variational times carlo supplement minibatch origin supplement likewise natural scheme consist article new york time entity leave sentence nn pp vb seq pp vb seq exclude
maximize size respect minimize depend like several evaluate empirically dependence initial weak nearly problem another way aggregate partition experiment construct walk computation completely cost aim minimize channel good compression motivate procedure completely differ section contain benchmark useful general ground element see political political truth graph ground truth partially manual process introduce error ground see partition political particularly fit sbm ground also easily check ground ground problem sbm sbm partition degree community real parameter mix community edge go community lead disjoint separable graph boundary become community community normalize shannon entropy mutual partition coincide take otherwise overlap overlap community propose refer somewhat subsequently set community paper really communitie varied generate respectively correspond standard deviation bar experiment precisely material run algorithm identify reconstruct good performance evaluate result overlapping observe operate subset vertex recall random start note set start set community overlap community via introduce community algorithm overlap benchmark run value obtain benchmark alg inf less detection situation note generate detection community long overlap know clique average sbm define consider model assume follow partition iteration sbm step somewhat bias c p linearization whether versa amount third number two path p proof material note length essential argument never require adjacency consider path target algorithm result initialization initialization algorithm observe behaviour suppose truth cluster original part find precisely usually use construct co q regard cluster often initialize set choose repeat empty precision course also imply standard finite indeed rearrange j j jj non negativity equality iff recover iteration proceed plan state chernoff theorem binomial lambda use sbm initializations random sbm node graph next denote generality expectation count component degree degree obtain least union hold node union fluctuation sum necessarily independent consideration often omit degree total concern write ds ds lambda equal union upper probability assume intersection conclusion quantity expectation shorthand relevant assume partition statement individually probability claim union inequality two large thereby prove expression similarly c thus finally length path path concrete path let path path type expect hence consideration concentration bound neighbourhood set argument lemma obtain concentration one lambda order conclusion carry precisely full make shall either without note deviation guarantee deviation hold satisfying assumption randomness partition satisfied discuss claim symmetry reverse inequality obtain examine write denominator quantity summarize q proof plug use similarly proceed enable expectation fluctuation incorporate satisfied specify degree size case set overlap specify multiple specifie number community software overlap specify section strategy instance use threshold figure spectral package run final euclidean k improve different return somewhat different benchmark discuss sense heavy different sense cluster overlap overlap cluster post community non try processing comment regard structure discussion restrict setting community heavy remain belong community hand node belong community community community almost intersection property start chance return much measure chance common explain graph section thm claim thm proposition thm definition community walk therefore easily benchmark community benchmark performance previously prove stochastic community cluster problem subset connectivity subset connectivity rest happen community application may instead individual node application communication traffic design biological meaningful arise survey euclidean transform weighted survey community depend whether question weight direct another allow overlapping notion several adopt benchmark produce computation develop recovery graph variant diffusion entropy non space measure short walk certain detailed evaluate benchmark find alternative insight reconstruction evaluate random benchmark significantly similar performance performance spectral modularity clique benchmark introduce modification enable detect overlap community overlap benchmark perform evaluate stock sp correlation return right stock overlap community community node belong community fact community algorithm benchmark largely stochastic block reconstruction mean suffice first purely probabilistic motivation shall analytically analyse name spectrum behave probabilistic generally dense constant reconstruction dense fix size approach differ spectrum equally behaviour study graph sbm high concentration inequality conclude remark sbm mention view euclidean vertex diagonal walk community algorithm walk notion sampling vertex randomly start
sequence hx penalize erm expect penalty candidate overfitting reveal penalize summation solely nearly distribution inequality fulfil straightforwardly extend minimum prove bound excess reach erm summation attain rank minimizer summation solely value take measurable denote independent x x base pair sample degree rx rx yx q equip notation statistic key noise hoeffde true section goal cardinality rule candidate finite optimal rule simplify version nr fulfil constant minimizer whether occur excess fulfilled least soon reach ranking minimize cluster subsection sampling expectation realization replacement approximate computational scheme bernoulli explain result interest subset cardinality power general belong draw second inclusion probability inclusion sample scheme bernoulli given observe plan inclusion plan addition bs survey role wide survey scheme view refer account seminal conditional eq equal situation sample advance choose among inclusion probability else sample result obtain thompson estimate collection symmetric assumption incomplete statistic without replacement eq highlight perspective replacement advantageous replacement stochastic loop precede section ignore technique provide machine sgd investigate gradient estimate incomplete statistic draw space machine eq rate compute gradient unbiased estimate statistic construct draw symbol refer k ki alternative sampling statistic build function smooth one show small estimate get variance strategy risk variance average potentially form combination summarize convergence use form strategy report literature semidefinite function gaussians mean share overlap proportional respectively handwritten digit class consist image extensively benchmark reduce retain unit merely involve subsample erm gradient erm scheme risk statistic pick statistic empirical pick project random testing risk pick reach average accord modify reach compare strategy reach quality dataset though quickly scheme reduction erm compare analyze complete subsample gradient sgd incomplete paper incomplete statistic sgd complete experiment mnist project batch namely batch sgd risk mini sgd sgd complete batch size comment large strategy support though rate sgd incomplete similar strategy sgd incomplete lastly expect gap implementation small mini batch size hundred wide learning estimate risk seek optimize increase functional involve sum rapidly become implement counterpart pick randomly replacement refer novel deviation learn preserve certain situation occur extended scheme bernoulli show purpose base beyond experiment technical proof independent jensen use argument rademacher rademacher symmetric sign variable inequality shall eq convenience sequence k ki equip notation observe almost surely virtue vc major dimension q next independent I assertion formulate turn straightforwardly first proposition assertion direct assertion combine h observe triangular yield whose reader follow integrable assertion h start direct apply nx x decompose expect pick complexity incomplete statistic criterion successively notice prove follow assumption theorem fulfil probability straightforward assertion namely bind observe solve partly slight fulfil q assumption virtue eq union proposition focus sample degree argument easily express variance hoeffding see subsection equip orthogonal hoeffding decomposition eq center nh nh nn pointed subsection sampling replacement asymptotic rate big metric estimate average highly expensive moderate term procedure functional feasible empirical risk replace drastically base refer rate erm result describe approximate incomplete version sampling technique stochastic erm numerical display provide evidence largely empirical sample erm develop erm essentially rely study maximal deviation average expectation adequate assumption purpose inequality wide deal range recognition view pairwise empirical error rule statistic hoeffding ingredient studying allow establish maximal deviation erm minimization df statistic law pool moment establish mean representation functional completeness investigate average sum symmetric integer number observe eq representation refer illustration copy couple find pair close bound function hinge estimator one two minimizer framework algorithmic robustness without contrast naive subsample seminal contribution calculation summation index solely simple index incomplete replacement stress build replacement involve summation depict incomplete statistic sampling pair base population overcome issue depend summation
improve bf vs notice correct quite inaccurate since lead student interpretation accordance standard approximation likely indeed look bivariate marginal shape elliptical nevertheless laplace pressure write choose student error bf integrate whereas assume variate student take jeffreys improve laplace comparison comparison dimension integral reasonable amount correct period draw use consider draw df equal inverse modal c laplace improved laplace correct student bf vs approximation normalize bf offer model laplace approximation laplace due shape approximate fix generally write give integrate density consist three involved group r five group constraint experiment result indicate successful model successful pair across define type normal separate follow approximate mle approximate laplace modify approximation effect poor observation specie involve integral random compare laplace approximation approximate marginal improved laplace take gb ram iteration mle method laplace laplace approximation laplace compute derivative beyond second order joint computation three method available effect particular specie laplace start minute converge laplace mc gibbs sampling quasi laplace obtained improve mc quasi approximation approximation improve mc mc intervention laplace widely frequentist second inaccurate improve show superiority laplace three method well gold demand scalar integral parallel accurately integral burden enhance analytical strategy example analytical automatic evaluation david unimodal necessarily tail asymmetric heavy always regular unimodal laplace quadrature quadrature accurate indeed package effect perform quadrature large quadrature point seem context package plot consider reproduce plot use file reproduce skew reproduce file reproduce analysis code file reproduce datum notice package require package corollary multidimensional integral frequentist shape laplace inaccurate asymptotically standard formula also dimension frequentist superiority comparable keyword expansion integral likelihood numerical integration involve smooth depend quantity density approximated quadrature typically accurate curse feasible low especially sense reach value alternatively increasingly dominant expansion laplace formula computationally convenient expansion analytical tuning unlike monte carlo moderate may monte carlo focus laplace marginal frequentist widely bayesian approximate density bayes bayes framework effect likelihood markov nuisance interest draw posterior invert laplace lastly laplace context joint survival longitudinal improved laplace integral achieve order setting behind density normalise easily approximate point laplace spirit optimisation demand requirement method detail issue rest section background improve final remark ease twice differentiable minimum laplace sample line interval panel laplace achieve laplace order propose give multivariate identity skew marginal skewed degree df control ordinary student identity df normalize skew density multivariate dimension figure standard higher worse obvious laplace work skewness versa quality laplace contrary improved laplace answer scenario similar laplace however substantially less laplace approximation variate modify illustrate method application focus aim know maximum mle analogous compete package available material
numerical references langevin energy boltzmann detailed study langevin dynamics balance formulation actually demonstrate beneficial improvement acceleration relaxation reduction correlation steady state interact intractable etc direct monte master promise convert initially method relaxation efficacy accelerate report relax initially realization simulate multiplication step realization transition transition approach detailed balance bc reveal superior confirm mean solution present provide validation study langevin evolution alternative fast convergence langevin write equation equation instantaneous function steady divergence analogous bc context master steady correspond force bc confirm different reach present analogous bc steady review mcmc langevin equation find present several artificial force convergence analyze briefly review simulation fundamental evolution master equation degree define master cp satisfy transition numerically perform update master steady state define balance bc restrict aim generate ss x ff temperature nontrivial bc confirm satisfy bc cp nontrivial bc lead construct matrix follow system former propose diagonal configuration accept summation element diagonal summation cp element bc cp cp symmetry read find rejection symmetry probabilistic flow efficacy theoretical eliminate accelerate relaxation reference conclude efficacy come rejection decrease element sense efficacy method yet understand skewed replica transition master equation swap transition give manner balanced impose steady share satisfy bc simultaneously bc bc transition instantaneous master equation bc master symmetry rescale transition matrix present author prove rescale acceleration drive rescale emphasize rescaled transition dynamic master langevin force wiener freedom give equilibrium purpose hereafter force fast achieve langevin flow define steady vanishe therefore equality steady satisfy steady master hereafter thing divergence bc spirit divergence steady vanish divergence change ss steady particular equilibrium exist confirm formulation transition force solution indeed demand detailed free must fact denote element probabilistic element unity force careful instability term temperature instability force divergence analogous may permutation reach permutation scope nontrivial bc composite analogy denote flow define system divergence system steady state system ss nontrivial solution consider force steady confirm flow immediately exchange monte carlo extremely performance equilibrium naturally existence analogous mention study publish subsection nontrivial confirm interval obtain location transition omit argument right side derivative backward confirm probabilistic flow first heat regard excess heat signal coefficient degree case ratio probabilitie excess heat also confirm heat artificial implement artificial flow confirm artificial force numerical restrict force minima locate equilibrium n x tn minimum potential relatively equilibrium force accelerate system evaluate ordinary plane case trace exist additional force describe area steady state however steady without addition observe steady integrate steady lose avoid show confirm integrated reach limit increase decrease efficacy beneficial addition method spin temperature region method remove xy steady mathematical additional force steady red plot stand location tb origin acceleration steady force mathematical understanding acceleration relaxation steady define operator eq q one use transpose vector hermitian boltzmann nontrivial force effect additional force nontrivial express reach p anti hermitian right side anti hermitian steady operator hand quantity anti hermitian part free nontrivial additional rewrite x x relation trivial one k vanish reach force satisfy free anti hermitian system anti hermitian accelerate relaxation master equation transition matrix master characterize large operator identify gap force eigenvalue asymmetric operator restrict
user depend namely bit htb histogram protocol privacy parameter user else set compute private server server jj privacy privacy privacy basic utility unit contribute complete error suppose modified reflect round round simplify proof occur least recover item round algorithm whether counterpart item I unbiased triangle get eq claim ratio eq numerator hand depend pp condition decode implicitly assume perfectly tail conclude pp private histogram error protocol discuss protocol heavy lack interference user heavy heavy factor hold simulate choose repeat protocol times interference free channel hence eventually item list separate channel protocol protocol result frequency frequency random protocol description server access randomness integer run result string construction problem protocol section protocol namely construction modular protocol moreover aforementioned object theorem idea condition separately heavy hash item holding report user channel user choose protocol get interference construction channel eventually item high list know result oracle list frequencie suffice pairwise independent member construction hash length family hash server assume access input generate random server string user protocol histogram user input privacy initialize heavy ki v v pp modify item add user frequency hard construction cost sub algorithm frequency estimate protocol verify discuss expense public seed public need privacy protocol given differentially observe protocol channel separate hash channel user seed channel user item differ item channel privacy note privacy report ratio channel protocol privacy argument paragraph v list frequency mention frequency item implicitly change expression item v list implicitly item seed independent get hash report random execute component basic work string sequence kt step sign sign otherwise pp pp protocol protocol htb ii k frequency histogram pure pure meaningful trivial algorithm clearly private differential privacy upper histogram section construction show construction construction ms construction inefficient construct statistical first bad use channel uniform item whereas scenario bind second channel proceed derive mutual item output prove scale scenario channel mutual information mutual information together let denote simplex corner probability assume differentially report arbitrary randomness estimating estimation randomness randomness error hoeffding prove technical histogram I fashion respect define case example probability turn hand application hoeffding mechanism sake contradiction item follow hoeffding contradict notion channel randomize define uniform variable user item independent copy output replace let eq independently follow input consider scenario iy equality show complete follow scenario uniform item user apply fed output report output wrong minimax reach show n nd claim denote discrete dm bad bad information variable originally pure differential corollary therein next iv differentially private similarly finally imply complete channel denote differentially private put together iv complete histogram low advantage inspire multinomial pure make modular differentially private protocol bind state discuss scenario draw independently bad show frequency notion otherwise channel mapping every compare channel local whereas scenario user item low result would show channel scenario derive mutual item local namely composition algorithm imply channel mutual information inequality acknowledgment nsf university award grateful frank point transformation compression pt lem lem lem lem lem fact lem lem edu give protocol match differential privacy individual user differentially private server protocol heavy along user come universe protocol run time regardless one protocol either low adapt result server protocol transformation software web management want want collect datum provide store private datum call signal server figure provide public visible randomness local private remain protocol equivalently v protocol estimation local privacy local differentially private web task mention protocol coin server value universe enable analyst summary definition together item histogram implicitly estimate frequency frequency item implicitly measure frequency structure aim never item estimate frequency may list error ignore histogram oracle analyst oracle query oracle item retain universe home page summary histogram useful protocol storage user protocol satisfied protocol frequency protocol histogram heuristic none accuracy differentially histogram protocol regardless public coin also code construction inefficient take rather query oracle determine execute frequency construction sublinear protocol recover either server idea error encoding server receive decode protocol low sense g universe likely heavy unique privacy essentially copy protocol add private protocol protocol private computation frequency long efficient protocol give item appear random instance implication equally distribution protocol universe item differentially error universe assume proof simplify framework develop privacy modular low differentially private protocol possibly state mutual theoretic unless modification compression coin model common string dp protocol user bit server efficiently protocol literature protocol heavy public utility privacy transform server apply expand short send generator transformation rejection fix player whether kept ignore local privacy rejection procedure little quick far frequency paper protocol task add relevant every function exponential technique recently class giving protocol low protocol algorithm sense protocol hash heavy appear context approximation arguably root evidence see g add introduce construction use tool ensure generate hypercube symbol pick choose bit j construction encode serve purpose construction item basic string jx uniform bit choice depend randomness output represent hold matter long input randomness important utility outside send server public server receive use private provide inefficient private heavy oppose provide private I enjoy depend finally review histogram clear later binary mapping constraint decode element fraction mm encoding decode several construction example thesis describe basic construction ensure differentially private bit string special vector pick string later construction bite special symbol special construction item bit string uniformly eq uniform bit index privacy om output come privacy hold randomness help ensure come outside send server public situation server receive bit private projection line construction differ oppose estimate heavy local use copy add pure differential carry private efficient denote theorem parameter affect guarantee protocol randomness generate public shared server note construction generate much less independent construction protocol frequency protocol input privacy ni server compute length user note need bit part fix item give htb privacy guarantee frequency privacy oracle private set error randomness output input rely behavior inner formalize independent taking also hoeffde triangle inequality least probability hence term upper give efficient private protocol provide construction difference randomization user noise differential guarantee oppose difference carry oracle construction compose binary user report frequency give use oracle inner encoding aggregate protocol generate bit generate take construct differentially user bind theorem rely aggregate encode detail efficient histogram frequency subsection construction simple call heavy general heavy hold item special symbol represent item construction constant protocol differentially private input output protocol encode user code result server redundancy item require length constant say relative error asymptotic behavior several construction thesis example encoder part know convenience hypercube encode report code server round aggregate near argue combination round sufficiently close describe code note report report htb pp encodes else user server server compute jj construction differentially private directly differential privacy rounding code since unit contribute hamming complete common least frequency error reflect round alg simplify part heavy occur show suffice round would item unbiased part triangle apply distribution claim use bind fact assume depend pp pp correct perfectly remain tail property provide private histogram protocol sub protocol lack interference idea separately heavy cost channel item choose large protocol heavy assign interference construction channel item heavy item hash item separate parallel frequency protocol like frequency item pairwise hash whose input seed protocol description user server access randomness generate string see random string histogram protocol protocol namely private oracle private problem modular internal protocol object show lack interference user heavy create computational channel user hold channel item sufficiently large repeating time heavy free one channel eventually contain hash overcome separate protocol result frequency oracle frequency output item purpose suffice distinct uniformly member efficient construction seed instance family efficient family hash return server assume random server result hash family htb efficient protocol confidence heavy empty ki n pp pp pp pp modify set unique frequency hard protocol frequency sub protocol compute protocol overall user bit expense public construction rely public string seed protocol differentially private protocol differentially channel protocol channel seed hash report assign channel user seed hash user channel separate channel privacy ratio put paragraph frequency satisfie mention implicitly zero theorem mention estimate user item run channel every occur without interference inequality tv channel whose channel heavy run channel possibly channel event guarantee frequency item algorithm actual item great frequency complete give protocol private distribute report add bit original mention transformation technique private protocol server protocol iv public report string server server report randomness output give bit generic generate string ni ib server server obtain desire protocol cost transformation probability transformation computational privacy bit protocol bit item right side iv construction iy iv note take randomness two fact protocol thus protocol affect sampling formalize statement necessarily protocol point negative randomness bound least randomness respect transformation discuss protocol protocol efficiently local describe hash channel remainder user report uniformly random execute see public string kt dependent user public string compare bit step sign sign desire otherwise step per pp nm ii privacy th group I encode construction oracle bit protocol differentially private protocol bit protocol follow item least pick lead transform private histogram report bit expense add randomness bit protocol mention introduction transformation general compression let server I randomness output simplicity string statistic server server report report randomness output algorithm report htb generic bit protocol privacy independent string ni iv ip server server collect report transformation done preserve protocol bit output item right hand privacy public string user p iv iv e public string upon user server view report take randomness protocol sampling sampling transformation transformation efficient protocol protocol histogram bit error key statement histogram differentially protocol protocol argue compute efficiently parallel channel seed hash independent component user get hash remainder uniformly execute basic string kt kt corresponding step item sign step pp bad protocol protocol full seed construction oracle local low namely pure meaningful yet low upper histogram section construction yield construction construction ms construction histogram inspire expect pure model item user show low error estimating frequency obtain low channel noise input output output first scenario channel local generate scenario normal scenario user directly argue error lower next scenario derive bound channel item privacy namely first scenario mutual information item denote denote item differentially local generate report arbitrary fix randomness identically distribute observation error minimax estimator maximum distribution eq argue hoeffding asymptotic result minimax proof give first private sample directly section turning case application hoeffding sake contradiction item use contradict channel mapping user apply report proof input consider replace iy complete scenario user apply independent copy fed local let output wrong hypothesis incur minimax nd establish claim probability mass density simply continuous bad bad inequality iv pure local like iv inequality differentially similarly complete channel hence write fact differentially private claim v n complete lemma frequency advantage algorithm technique bound estimation modular show prove differentially protocol formally sample refer reader version draw independently bad right distribution maximum item bind notion item uniform channel scenario channel output second scenario suffice true channel lower proceed mutual item channel namely composition channel information conclude r university grateful helpful frank lem lem lem lem lem lem claim protocol bound local differential individual report datum histogram frequent item along implicitly user whose item universe protocol necessary regardless computational efficiency protocol run adapt al public need server know protocol preserve computational people software anonymous may share datum want collect raw subject collect provide store private also server summary public visible randomness extensively private protocol v describe bound local private protocol web basis empirical mention show protocol public coin set bit server value universe label wish enable analyst frequency look summary definition algorithm computing item histogram produce least list estimate oracle measure distance list never contain frequency may price ignore oracle retain frequency universe home page financial histogram useful protocol communication protocol protocol frequency bad protocol histogram bad protocol match bind provide polynomial differentially private protocol protocol coin server length time code construction inefficient state oracle query determine protocol execute construction previous error construction sublinear piece first protocol recover player server value server decode protocol input idea low compressive specifically use hash universe item run unique privacy essentially copy sense private oracle protocol protocol differentially instead show regardless communication protocol efficient protocol instances rise item frequency remain instance bound minimax bad protocol universe local item much contrast differentially private protocol achieve universe assume frequency simplify statistical privacy modular private protocol possibly state mutual information one theoretic bound unless compression et yield coin server player string dp bit server transformation probability protocol literature protocol heavy randomness public affect transform protocol particular expand seed send transformation rejection public keep server bit server quick reference recent work protocol specific task paragraph algorithms utility measure show statistical query show related technique communication give error guarantee basis basic protocol theoretic protocol idea large stream compressive protocol hash heavy fourier arguably root provide close g context subsection introduce construction describe basic construction user private one basic input either string vertex hypercube special represent pick bit string bit bit clear construction bit input unique special symbol serve purpose special situation construction user basic bit string uniformly e output pair bit every depend randomness om note output represent bit output bit come privacy matter independent
score main expert semantic form really indicator choose indicator function qx qx select indicator principle g take binary indicator generate indicator highly redundant imply consequence naive naive bayes assumption estimate use motivation classical property performance motivation successfully expert probability indicator basis allow discriminant box kind grey long performance mention previous realistic selection range backward method chapter redundant binary remain note recommend text book see give feature former fortunately backward strategy efficient subset provide already subtract time evaluate backward backward apply arbitrary move magnitude expensive g simple mi search cost cost classifier incremental base incremental generally addition take firstly lot performance mostly reliably classification however additive identical enough feature estimate uncertainty decision note conditional poor scheme indicator obtain section identical example example belong switch standard white trend anomaly trend trend amplitude step point central area score slide window explain variance successive window detail value train class good subset choose among feature report mutual mi rank classification search good step set backward vice improve one last backward summarize outperform redundancy indicator construction mi tend redundant indicator obtain mi error backward search forward backward always allow accurate ordering infer procedure alone move search select performance reduce filter avoid guide greedy comparable performance search good recommendation mi cm cm universit paris paris france france method datum study indicator reasonable context body available simple
figure fold plot scale change figure demonstrate explain protein fold strongly pair instance change fold across time show vs fraction fold fold change vary across constant three example analysis serve proxy variability protein combination figure pair protein pair result observe due focus distinguishing investigate gene protein fold gene may predict reliably fold cumulative error indicate protein change gene fold level specific post functional gene lack protein noise conversely ratio explore across ratio gene reflect biological specific define ratio divide median gene evaluate significance variability type gene pool across use quantify statistical result go protein small trend account fundamental type far set fact highly mode increase tf decrease indicate variability across reflect noise quantifying demonstrate level noise far take account rna seq mass large systematic protein ratio dna sequencing bias quantification systematic minimize intensity dna sequence quantification quantification variability start reliability reliability simply reliability proportional strength estimate replica rna seq protein derive overlap quantified estimate two figure b take account reliability measurement protein variability reliability reliability reliability variability protein level across type leave accounting explain level remain mostly post reliability lower large measure noise reliability explain post likely major determinant variability highly level role full distinct poorly role protein degradation protein level estimate protein level contribute post bias influential protein systematic bias variability within post error alone correlation protein figure reliability would level account estimate thus accurately quantify mechanism strong post noise figure indicate post substantially level significantly dynamical response variability increase level must across cell bind protein synthesis specialize degradation substantial big much post perturbation fold less due substantial contrast cell acknowledgment thank constructive grant institute grant health gm research fellowship contain rna seq level human similarly corresponding normalize protein multiplicative factor raw additive normalize choose baseline normalize measurement set conduct normalization specific scale level raw median represent protein protein protein noise level way error via correction measure datum decompose signal signal decomposition correction variance observation estimate reliability define fraction measure variance simply protein estimate protein level de protein explain c region identify restrict attention group go quantify relative protein exclude large protein likewise noise specific eliminate variability ratio conduct vector comprise index index kolmogorov ks difference systematic fdr fdr group due mit edu abstract post type specific contribution factor determine protein factor variability protein type variability find level protein contrast dominate protein fold highlight type specific introduction ease protein level protein proxy level conversely set protein independently case classical division understand trade principle assess contribution protein mostly protein level mostly post view quantify correlation absolute mix many protein variability protein across condition biological interpretation implication variability protein protein gene protein widely refer source protein protein order across principal protein orthogonal distinct different counter intuitive conclusion illustrate context gene scale measure gene trend counter intuitive correlation large conceptual datum demonstrate
expansion empirical eq uniform drop eigenfunction hold theorem expansion correspond drop also one theorem also obtain lemma let theorem expansion among iid moment exist emphasize validity er hand expansion expression contain expansion asymptotic small resp enough valid expansion actual approximation even zero hold addition possibly fix hilbert denote correspond indicate readily compute fix h equivalence long limit completely discuss optimality draw heavily suppose c speak rank infer computation order cauchy schwarz arbitrarily moment eigenvalue optimal expansion index still converge result eigenvalue eigenfunction weak remainder write measurable allow flexibility j physical dependence concept popular etc consider k p require sequel follow resp sharp reference relate result weak introduce well h structure ht follow hold mild decay assumption provide gaussian explicit condition structure impose assumption assume addition condition simultaneous important relevant stop rule develop require point corollary also covariance uncorrelated run serial perspective appropriate condition general exist weak take correlated sense approximation appropriate optimality already substantial reference therein datum basic plug lead choice result remainder section necessary bias optimal decay note bias negligible regularity quite decomposition eigenfunction natural term decompose degenerate setup result convenient include assumption present b discuss condition quite condition essentially common degeneracy encounter let analogy eq general transfer remain substitute correspond place version bound provide pointwise uniformly whether assumption everything express general optimal hand drawback interested mention mainly therefore desirable precisely uniform turn occur model imply valid everywhere apply section dependence precisely routine reveal analogue depend mainly convenient decomposition eq recall j nc n convexity require geometric condition process note mix surprising relevant general simply impose already lead simple polynomial boundary variety though boundary usual degeneracy already mention analogy expansion assume assumption eigenfunction proposition little explicitly assume may theorem normalize j operator mention test test serial stationarity many canonical augment increase since account view rigorous minimax theory estimate cf amount see mention necessity control field learn base highlight usefulness theorem reformulate framework convexity lead condition particular result range allow principal typically reflect degeneracy usually necessary discussion hilbert relate fundamental detailed discussion impossible theoretic perspective current cf mutually x assume independent involve truncation motivate eq sharp general necessity control actual deriving expansion goodness confidence reformulate change underlie hilbert condition validity discussion validity possess decomposition eigenfunction sequel let operator eigenfunction decomposition candidate operator eigenvalue eigenfunction distributional q certain see elementary schwarz validity relation hence complete shown schwarz invoke bound cf note assumption cauchy schwarz follow since rearrange schwarz similarly markov inequality combine inequality assume assumption recall cauchy schwarz claim follow sake ready readily treat note triangle inequality eq triangle cauchy schwarz lemma give hence first treat proceeding claim proceed iterate lemma claim follow result valid proof find manner function polynomial concept univariate polynomial invoke method partial moreover one denote suffice consider lemma fix eq since may slightly weak sequel note space want compare prevent necessity coordinate key ingredient variable q sequence grant alternatively may bernoulli theorem major tool subsequently reduce iid resort simplify sequence iid h l role finally n sequel suppose lemma grant p q constant u let q clearly monotone put j pm bm conclude set balancing arbitrarily grant theorem covariance claim iv follow elementary computation establish schwarz grant note obtain remark combine gaussian require denote gaussian follow adaptation result lemma slightly adapt mean covariance covariance proof employ obtain next verify deduce remark verify readily lemma gives hold next remark theorem first need assumption observe moreover schwarz eq verify proceed inequality far bound hence verify bn b routine calculation reveal claim turn briefly elaborate difficulty main objective course everything imply well sufficient guarantee validity order work truncation detailed long equality eigenvalue eigenfunction observation heavily reference eigenvalue theorem step verify deal introduce observe schmidt analogue put obviously hence remain proof convention absolute line write large version schmidt eigenvalue eigenfunction next tool summarize result sequel notion hold b j j n proof throughout proof analogue see claim argument computation routine calculation establish due representation cauchy apply establish proceeding get claim establish follow hand get ba schwarz application readily proceed grant construction desire lemma three lemma validity j note uniformly end separately schwarz times fu elementary calculations j sufficiently large manner obtain I virtue establishes show way difference assumption triangle inequality convexity q establish large b lemma conclude routine calculation q complete ready proceed grant sufficiently large j j proof cauchy schwarz triangle select virtue negligible precisely due get thus proceed lemma suffice replace bound uniformly frequent sequel consider establish preliminary yield relate via triangle give proof exclude complete omit analogue computation proof theorem bernoulli shift e inequality readily derive contradiction assume converse side arrive kolmogorov also get conjecture assumption mu expansion eigenfunction lag eigenvalue spectral gap underlie memory process study deviation among show extreme rise construction also asymptotic transfer covariance operator latter functional become comprehensive overview assume usage introduce convention eigenfunction functional principal empirical eigenfunction define lag j fundamental eigenfunction eigenfunction result become important well bound cf simplicity unfortunately perspective covariance operator expansion prove name correspond heavy structural spectral assumption iid also section presence serial generalization dependent general avoid previously mention derive expansion optimal dependence allow short memory weak memory strong condition optimal application eigenvalue precise mild high dimensional interest particularly cf section functional outline key assumption introduce notion weak discuss expansion context additional emphasis linear memory eigen section long prove devoted proof involve multiplicative give denote complement variable sequel convenient assume j consider variable part sequel depend
observation x tx probability natural likely paper correlate follow concentration finite obtain conclusion suppose condition lf lf measurement keep cascade number require cascade quantity interpretation node time simply infect cascade cascade necessary constant goal necessary recover cascade obtain sparse inverse cascade cascade output true follow sketch kronecker kronecker validate empirically assumption probability state art extra benchmark approximate evaluate algorithm real social network graph edge recently kronecker report cascade ic obtain commonly link compressed paper signal research direction solve adaptively cascade david grateful feedback suggestion proof section show lemma give q martingale assumption lf surely apply union proof contradiction positive get mostly rely show concentrate around cs h thus h mn cs guarantee exist w distribution set support follow draw quantity together graph cascade figure algorithm benchmark fix cascade fast algorithm cause time cascade increase linear expect slope large overhead fact seek recover cascade approach sparse cascade include graph recover parameter context validate empirically graph extensively diffusion take place presence infection graph precisely design cascade diffusion goal understand decompose focus single discuss identify among graph cascade influence cascade recover influence cascade non state step literature cascade sufficient contribution cascade sparse notably cascade cascade able efficiently robust recover prove guarantee tight survey cascade conclude edge active decade approximate cascade later network discrete cascade obtain algorithm one analysis decay limit cascade need suggest obtain recover weight cascade prove close wherein consider standard recovery strong exploit kkt cascade analyze model orthogonal describe node condition previous event state mutually graph eventually reach probability word cascade describe diffusion influence become node uniformly source verify cascade overlap cascade start infected treat cascade context cascade extend cascade direction consider transition probability cascade problem linear fact diffusion true special draw linear cascade indicator cascade state become time step condition cascade provide node sample interpret link cascade cascade either remain infection mutually succeed success stay cascade terminate rewrite cascade inverse cascade fortunately independently color cascade stops fix blue node time cascade cascade discretize independent temporal discretization whose infection interval infected indicator infect interval contrary cascade remain random property discretize induce cascade cascade link intuitively infect node get cascade inference become inverse cascade central present work cascade influence parameter note cascade mle prevent overfitte control cascade decomposable write equally measurement step reach horizon cascade node deterministic contrary cascade condition concave function iff cascade regularity lf hold cascade lf soon data regularity cascade cascade equal explicitly case constraint obtain verify need exist add regularity assumption program recover parameter cascade estimate sufficiently recover provide network support exactly relax decomposable henceforth focus omit notation analyze edge influence parent standard symmetric define q cascade binary vector lf let convergence rate different cascade
computation denote instance correctly classify check f h quickly always sign discuss situation incremental completely optimization incremental completely framework suboptimal solution compute gradient optimization gradient current gradient obtain gradient proceed gap stop incremental sensitivity analysis optimization sign low upper become sensitivity task describe summarize experiment repository small lr compare incremental part use approach conduct core ghz gb use rna million task compete problem regularization parameter nonlinear case rbf task speed also trick lower bind select well meaning error stop trick conduct operation increase observation mis classified compete table computational compete tight novel provide actually instance particularly relatively instance three plan propose stream prove function theorem complete square noting compactly obtain lagrange multipli multiplier rewrite constraint strictly active let optimal write obtain convexity inequality rewrite multipli technology technology introduce framework problem use remove quickly update classifier incremental although large completely incremental might expensive novel update without actually quantity cost property advantageous instance update demonstrate applicable sensitivity bound provide sufficiently tight incremental sensitivity leave training logistic support simple except acceptable thing care datum particularly design instance remove instance add solution linear efficiently framework original helpful reduce incremental computational incremental learning expensive except incremental mean complexity great intractable completely every nice actually unless want interest update suppose could change minor modification order propose quickly compute depend unknown classification low bound linear specifically denote linear obtain upper score compute property advantageous number update bound useful sensitivity test interest make positive mean label instance available study sensitivity closely relate design example literature check classify update exactly fit svms exist build idea sense bound obtained propose exist bind inspire safe screening coefficient use lagrange multiplier optimization actually contribution bring sensitivity develop framework cost depend rest organize describe task present low upper update addition discuss direction present first conventional incremental propose study train use conventional incremental update hereafter denote label remove instance index instance one want modify classifier predict class label give consider represent classifier represent empirical control differentiable example include logistic case number instance entire difference small use incremental algorithm incremental work incremental learn conventional novel make inference framework compute low upper eq computing depend update quite advantageous base propose framework classification problem sensitivity propose might element new j e toy bound bound beneficial making decision practical task old blue bar us sensitivity test follow tight sign update relatively entire size would demonstrate empirically many case toy blue bar indicate unknown rd sign instance update validation leave regard bound use correctly leave classified mis classified bind
hold case outcome occur frequently research recent finite maximizer although rigorously uniqueness maximizer stochastic unique maximizer satisfy imply diagonal maximizer maximizer entry maximizer nonnegative column proof correspond outcome maximizer proof unique maximizer element maximizer first maximizer follow form entry row third third row uniquely maximizer sum discuss construct maximizer let observe result marginal subsequently conclusion sample power dominate dominate fix increase marginal power decrease increase weak increase power methodology power specify instance guarantee summary randomization depend marginal difference outcome power marginal fix difference conclusion confirm easy sharp marginal easier reject sharp potential table contingency limited amount become negligible asymptotically paper sequence hypothesis systematic quantify hypothesis retrieve discussion distribution example university recognize several researcher construct large systematic ordinal outcome power randomization datum treatment extension contain contingency nan introduce randomization refer permutation test test randomization test randomization ability clinical trial describe status improvement ordinal limit assess power test ordinal assess randomization utilize assessment power randomization construct nan hypothesis experimental unit identify sharp hypothesis literature assess power randomization super finite population potential outcome something refer indeed ordinal outcome quantify nan develop alternative hypothesis close vary study power randomization proceed review focus ordinal outcome randomization sharp introduce sharp nan discuss test systematic hypothesis report demonstrate assess randomization completely randomized experiment unit ordinal category bad category value potential treatment science unit potential summarize view later reduce science play hypothesis p marginal outcome treatment take unit otherwise unit assign element science early observe miss treatment assign cccc summarize outcome manner similar term represent unit outcome count sum express sharp nan experimental unit choose suitable specific randomization randomization involve outcome hypothesis potential outcome assignment mechanism assignment simple realization randomization nan note significant ordinal question sharp issue examine power create sharp wish joint increase hypothesis infinite way make create intractable make tractable impose follow interpretation term distributional effect alternative sharp scope help hypothesis introduce quantify hellinger quantifie sharp nan hellinger intuitively distance sharp nan however need hellinger sharp complete picture commonly categorical hellinger solely rely joint subsequently hellinger converse hellinger interested power randomization test hellinger increase construct maximize constraint minimizer maximizer construct minimization somewhat associated potential outcome minimize minimum problem however expression maximizer stochastic triangular proof first satisfying note element define upper low p j p entry corresponding column
model discover concept human expert assign direct exercise exercise correctly recover pre knowledge dataset non simulate case hyper variation additionally simply get exercise virtual student virtual concept response generate two answer student knowledge exercise single difficulty student get exercise difficulty student model classic theory e guess student time affine exercise understand incorporate simply exercise exercise student concept sample student usage core complete across contain working govern agreement design privacy accordance particularly learn interact site student often self topic public student school latent direct set influence equation concept occur far node exercise nd ask dependency graph concept tag pair remainder product obvious student answer correct baseline generate relationship education expert require intervention coherent rnn education particularly annotation pattern input disadvantage simple hide require amount suited education small rnn future take explore dropout pose literature space model student input track especially student task develop program able include material efficacy propose acknowledgment many thank support cp appendix exercise exercise probability display less element capture get exercise google stanford edu stanford edu machine interact education effectively student knowledge task inherent challenge utility rnns student rnn advantage encode human domain capture substantial improvement range suggest promise knowledge rnns education open access grow cost building trace student future student hard delay already tune content show gain machine could inherently complexity human brain use education rely deep along allow representation student knowledge code main recurrent neural auc knowledge benchmark model annotation exercise generation formalize interaction take student next interaction tuple combine exercise exercise tag exercise whether show root problem correctly single intercept incorrect intercept prediction exercise next interaction visualization show prediction exercise type previous exercise tag assign exercise leverage expert annotation absence modelling predict inform diverse education cognitive social influence complex macro motivation challenge micro human complex process knowledge pose heavily aforementione nevertheless concept answer incorrectly formulation assume knowledge learner difficulty extension suffer mapping onto concept exercise several refine concept exercise mapping gold cognitive analysis domain learner process kind observable behavior although present require exponentially implementation restrict discrete hard code latent make overcome limitation analysis predictive combine combination adaboost forest feed ensemble limitation requirement work model kalman promise expensive recurrent neural neuron hide evolve input activation education rnns notable rnn ability point lstm instance translation amount training result suggest successful student formulate govern diverse property presentation individual rely principle attribute model rnns sigmoid student response upon recurrent neural rnns map illustration compute state successive sigmoid parameterize input recurrent state short term lstm variant rnns powerful forget gate thus retain make easy interaction complicated transformation equation rnn interaction necessary convert encode student interaction tuple combination exercise exercise space encode assign dimensional compressed recover constant sparse tuple exactly encode vector deal extend complex student interaction
structural illustrate triangle convolution represent vector feature evaluate subtree follow eq parameter fix layer design allow short propagation position tree feature effective tree base tree number vary feature fix rest explain detail subsection subsection deal third problem training illustrate leaf leaf g noun phrase stanford leaf node embedding rnn representation convolution process subtree window node child associate child indicate leaf child straightforwardly exponential node cnn add amount dependency dependency representation lead child node cause weight parameter window traditional c convolution g position believe much reflect relationship generic convolution parameter parent child frequently occur sentiment offer education bring year obvious slowly question place base topology size technique deal heuristic generic criterion pooling include pool slot neighboring slot pool approximately aggregate intuition global pool heuristic include slot preserve slot pooling b pool slot pool low position obvious aforementione slot pooling improvement address slot pooling dependency tree word order order slot position sentence extract pool efficacy method along pooling task penalty detail evaluate sentiment question conduct widely discriminative stanford review setting prediction fine grain strongly neutral negative coarse versus sentence neutral prediction simple class neutral class discard take stability place difficult list control comparison consistently outperform rnns extent flat cnn show important sentence tree information rnn integrate far evaluate model sentence plus split target entity location svm rule cnn cnn c convolutional propagate model various art traditional utilize code human engineering knowledge classification reduce extent architecture qualitatively light mechanism pool fair reasonable tune hyperparameter consume largely hyperparameter sensible hyperparameter report protocol different initialization summarize complicated sensitive pooling mainly serve necessity deal experiment slot pooling literature pool efficient rnn achieve sentiment typically epoch ccc slot sentence group smoothing sentence length comparison rnn achieve overall slightly bad think fair sensible contain confirm analysis sentence rnns rnn difficulty explore convolution propagate especially long sentence mechanism neural process pooling ultimately supervise back come layer pool pooling tend vice process global sensible slot windows sentiment convolution pooling know mostly sentiment result tree convolution window window root act child see window sentiment window neutral sentiment window sum window novel discriminative sentence parse build denote variant achieve high sentiment slightly outperform state art task convolution sentence effectively useful li xu cn com software institute china convolutional modeling model leverage either dependency sentence structural aggregated max underlie detector effective feature extraction state dedicated rule effort visualize convolution modeling aim capture sentence g various attract attention nlp community feature engineering dependency subtree dedicated one svm specify sentence advance neural bring language considerable propose unsupervised learn word real sentence neural automatic learning sentence cnn recursive rnns cnns neighboring capture inherent structure parse rnn composition parse long cnns rnns variant recurrent review combine advantage cnns rnns whether rnn propagation cnn propose neural call parse tree tree variant convolution subtree detector slide entire parse sentence extract dimension architecture feature propagation path learn sentiment outperform understand visualize code result website present architecture discriminative convolutional cnns processing language depict classic convolution detector sentence window word convolution position detector parameter concatenation convolution pool fix size convolution effectively interact though convolutional local
role rewrite less demanding finitely finitely section polynomial exercise chapter know exercise follow regular sequence finitely dimensional whether polynomial polynomial lemma large arbitrarily fact span example swap show suffice hold identity block nonzero polynomial involve proper subset polynomial contain univariate polynomial evaluated observe since common root elimination chapter conclude partition polynomial uniquely contradiction exist choice position prove satisfy finitely dimensional independent basis imply indeed case unique system equation finitely mention permutation row large say uniquely useful relation hold column r assume arbitrary hold matrix column follow satisfied per form denote independently term sum may rewrite last follow ready fails first fail characterization set finitely unique uniform lemma remark circle font deterministic pattern low completion university completion wide previous provide completion miss finitely observe matrix guarantee contribution set derive sampling unique high column observe attract attention range recommender filtering entail study know approach require coherence gap theory loose condition incomplete finitely agree infinitely additional guarantee depend entry characterize question main characterization pattern complete condition finally organization formally leave statement additional nonzero location binary entry dimensional span entry place may column subspace subspace consistent entry consistent entry different paper result sufficient guarantee constraint turn introduce express column block constraint determine redundant indicate statement hold almost zero statement one main necessary sufficient hold every subset satisfie follow uniquely follow satisfied form form column satisfie hold satisfie define little additional uniquely pattern prohibitive especially pattern satisfy probability scheme column independently least vector compatible I observe formalize coordinate degenerate minor determinant determinant measure almost every subspace subspace exist infinitely hence system become impose subspace characterize finitely many expand equivalent eq recall may subspace nontrivial infinitely infinitely associate subspace observe
behave presence trend structural var variable value causality topic phenomenon use short recall focus particularly autoregressive move detail reference therein indicate total observation lag period similarly denote j j ty variance simple draw identically variable average respect resp ty concern relate time past completeness process ar forecast forecast square ols regression choose line closeness make hence give slope represent error nature later iterate take process time condition eq rewrite lag act follow k te variance constant covariances ar generalize ar random zero operator rewrite framework four e regressor forecast become forecast biased inference perfect regressor perfect introduce predict quantity forecast forecast coefficient ol estimator forecast p forecast refer estimate actual forecast forecast mistake make forecast actually occur measure forecast arise estimate estimate useful forecasting regressor claim lag check causality coefficient optimally choose lag aic bic minimize choice bic residual contrast time aic main large choosing length estimator proof relevant due trend trend persistent movement variable trend trend time trend focus simple stochastic trend forecast u forecast drift time stationarity trend bring issue coefficient series ol autoregressive toward cause trend interval valid example namely statistic hypothesis mistake hypothesis reject true hypothesis statistical probability call sampling nan hypothesis actually significance reject hypothesis approximate standard limit see g moreover call e model trend detail side hypothesis u standard test one hypothesis hypothesis ol alternative deterministic must analysis forecast section model forecasting var future autoregressive var period appear process impact variable model regressor lag case say var consist form normally var ar var ols reduce vector u compactly pl pl pe stationary lag polynomial stationarity root say exploit follow l u bl bl z remain move accordingly vice versa lag length determine var rule past typically include lag full cycle lag carry lag six lag capture year residual component usually decide lag lag use limitation amount forecast lag let ol residual aic compute modify replace set lag minimize iterate forecast compute var forecast use main forecast forecast forecast forecast period period apply compute ahead forecast ahead previous stop forecast ahead var ol coefficient question multiple series variable predict another answer determination causality g concept causality alone coefficient past nonzero similarly cause formal causal nan causality x p tx ty ty degree freedom statistic critical significance reject effect cause drift root equal difference follow random accordingly random walk trend say integrated trend say two say integration autoregressive stationary test unit autoregressive stochastic trend reveal series process always trend integrated coefficient integrate say decide exploit expert qualitative graphical checking common perform statistical initially ol see augment test concept extend say need see line regressor number ol past ols two relationship regressor along lag multivariate influential focus introduction regression correlation check goodness devoted application able example augment statistic exploit causality test information approach constitute core second aforementioned acknowledgement author acknowledge excellent dr give constitute core autoregressive time series work project deal depth study effective make forecast concrete framework major mainly consider causality trend useful constitute core present whereas project present data concrete area causality bic criterion trend
dimensional space physics geometry physics neighboring matrix geometry play important determine neighborhood grid become mrf provide preprocesse involve random enable pdf statistical article briefly definition terminology computationally explicit interpolation interpolation four potential current connection present conclusion research n euclidean boundary hull n tt transpose interpolation observe field regular grid estimate spatial field denote probable give accord integrate statistical analysis distribution mathematical property group spatial prediction grid whereas bandwidth support notation bandwidth indicator triangular neighbor choose local bandwidth accord euclidean space determine compactly infinitely avoid zero bandwidth sampling depend purely compactly support imply kernel near neighbor fail bandwidth represent point value field normalize preserve unity configuration energy lead explicitly local maximize joint functional following h average fluctuation euclidean dimension two curvature influence motivate positive parameter contribution curvature control bandwidth functional partition condition metric intuitively justification average positive multiply increase multiply sign space express symmetric square identity otherwise kernel index curvature term h follow row vanish eq diagonal expression non I maximum likelihood leave calculation operation datum bottleneck optimization computationally efficient requirement store use use follow base reduce apply interpolation involve vector exclude negative respect validation matlab function initial parameter use assume lead optima value optimum optimum use global point unknown value functional give mode network concern interaction term neighborhood interaction sampling case bandwidth control illustrate distinguish difference weight compactly kernel imply pair contribution denominator point weight combination q energy functional precision ph bandwidth determine neighborhood illustration root side root diagram represent whereas represent term eq minimization mode modify vanish transfer element analogous krige predictor validation obtain compactly support neighbor distance ii k well dominant term computational time investigate approximate double denominator analytically evaluate double minimum globally estimate respective mae rmse measure mae mae reflect optimum slightly optimum quite covariance truncate dark online correspond area online value scatter respective daily automatic monitoring spatial exercise study thus comparison investigate measurement release corner simulate dispersion value magnitude rate measure hour h set gaussian minimum optimal illustrate isolate low sampling along boundary convex hull imply validation recent mean spline exclude cross measure range I mae map generate interpolation cause bilinear interpolation bilinear interpolation matlab b interpolation text measure predictor I error bias mae rmse I mae determine use leave point show whereas spike plot vertical range variation latter determine ensemble believe partially slow function ern slow variation parameter degeneracy may recently involve exponential quadratic last table result include optimal close counterpart high variance cross validation exhibit variation precision non imply sparsity matrix rely estimate thus reliably pearson correlation low extreme interval mae rmse neural look mae rmse close validation describe I rmse near neighbor knn employ knn determine estimate locally imply vary locally neighbor algorithm type improve use location input value output variable long plain version complexity fix neighbor alternatively locally efficiency estimate mean leave dependent version involve regression neural similarly surface case correlation abstract suitable imply curvature coefficient characteristic length correspondence establish rectangular available datum case parameter reasonable initial support neighborhood contain least arbitrary exploratory run help global approach use cross validation functional use involve measure square investigate search investigate include herein local optimization lead method performance improvement set bridge machine covariance implement neighboring algorithmic missing point cross validation efficiently rectangular grid calculate without function store large big spatial investigate extension model case web address laboratory manuscript space operational education european machine framework model scale require application present employ combine idea physics computational define involve mean sparse matrix expression interpolation number avoid big expect abundance similar scientific engineering field design
intersection contradict step therefore yet time customer change increase bandit mean despite foundation fundamentally dependent pattern account implement greedy exploitation period period understand correct present insight would want exploit excellent exploitation yahoo heavily series substantially contextual management regret classic pricing maximize price mix explore pricing exploitation yahoo front goal ad g classic static trend customer peak trend google query com would ad trend ignore classic armed bandit analysis arm pattern many form case effect beneficial customer certain day yahoo front article short much well explore article insight key exploration period likely make optimal suboptimal simple reward multiply know multipli high explore reward approach general micro I playing period round price dynamically result reward armed become explicitly care arm exactly suboptimal multipli classic bandit logarithmic inspire exploitation phase mining competition recommendation yahoo article article would short period available click article keep alone almost percent allow setup work differ work dependent multi bandit arm exhibit multipli accordance work know analysis discount ucb slide ucb change stay extreme exhibit brownian motion differently rather one work extremely micro occur length multiplier certain threshold stop information round propose consider available form dependent trend reward assume advance periodic google etc draw time arm mean let round let play jj otherwise play update provide regret reward last round initialization expect low playing make finally arm good arm high reward period usual rate harmonic want compare usual greedy suited set multipli discount divide greedy present play suboptimal arm reflect moreover usual less easily give jt jt jt positive mean reward arm lower greedy hard exploitation arm generally usual output jj mt uniformly play arm appendix logarithmic hold term regret bind give initialization play arm good probability decay jt grow remark appendix jt jt arm quantity remark appendix grow arm change w md w order interpret happen multiplier grow brevity abuse still little probability choose arm qualitative choose bad decay greedy direct consequence slow typical much well greedy introduce ucb multipli reward exploit good round reward call order let high total round high define short play regret achieve way sequence decrease amenable type ucb bound wrong arm choose proof leave calculate weak regret ucb regret high collapsed term round threshold round ucb soft ucb gradually contrast threshold reward exploitation enhance effect multipli exploitation let define follow q correct decision chernoff hoeffding second jj mt grow regret soft ucb bound identical reward modify multipli three type multipli wave want find far peak arm game figure wave normally arm choose make play essentially good multipli reward multiply estimation period website people peak multipli click ad show obtain comparison version reward discount round divide algorithm usual present multipli figure figure bernoulli success advantage reward ucb use h initialization mt output loop initialization jj mt play motivation scoring exploitation well recommender yahoo news recommendation yahoo unbiased evaluation challenging trend click arm news article access handle lead key insight although feature feature substantially turn insight gain time arm appear show arm drop global trend need modify stop exploration click rate stop explore old stay old
matrix ignore argument dynamic world normalize stationary toward equation stationary generate exploratory plain global system rule stop decay generate motion system lead signal avoid activity give plain describe sect novel propose body stay perturbation body external vary change initial act individual external influence quantitative learning sect control base position harmonic example xx u x factor average sum replace integral stationary dimension carry jacobian pair complex observe intrinsic generate sensor complexity physical deterministic inform exploration notion gain tool tool paper proportion weight human body degree arm robot use position perturbation position sensor force appropriate internal inverse sensor realization behavior reach exploration predefine work produce motion never self quality robust phenomenon circumstance understand broken symmetry individual symmetry break nature phenomenon root dynamic sequence situation author acknowledge discussion comment gm grant receive european union grant agreement institute system fundamental behavioral development provide answer unit require high level construct learn specify level intelligence surprising specific modification break brain plausible target investigation argue evolution brain small progress mechanism signal transmission enable understand organization however gap bind leave open question interact self organization gap circuit acquire make local together together early rule however like scenario learn drive generate perspective determined action self organization past contact relate behavioral feature together bootstrappe behavioral plausible naturally raise question whether aim encourage yet decade paradigm ai ai role environment close hypothesis argument present generate phenomenon self behavioral dynamic without construct motivation support material overview generic realistic link joint angle measure like implementation drive force simplicity complex external controller code neuron transform sensor feed forward activation type simplicity translate network particular intelligence plan brain require organize internal never mode dynamical body environment feed forward pattern rule plain simple robot controller law rate would proportional multiply concrete fix suggest time activity derivative focus demonstrate sect behavior main reason behavioral involve trivially lift entirely outside physical control let assume basic sensor approach realize approximately relate cause time lag r never exact reconstruct copy formulate write scale dynamic fig decay principle relate controller step together physical explore time idea neuron experiment threshold dynamic periodic inverse difficult different lead unfold motion pattern separate basic sensor capability development behavior matrix relate robot force constraint classical method alternatively order self organization experiment make whole exploratory behavior normalization latter feedback strength perturbation active maintain see sect edge argue life development finding see sect system strategy long spend behavioral reflect spatio pattern simple local rule eqs understand behavior generic call biased central obeys geometric also physical system transformation sign angle expansion situation learn symmetry breaking preference breaking picture break symmetry event rich scalable symmetry break newly propose specifically study due lead specific behavioral cause signal model simple calculate inverse back neuron neuron shift correspond classical pointing way little neuron latter term additional effect fire unchanged accomplish enhance intra series experiment demonstrate self organization interestingly behavior seem system turn task orientation always neuron start first study stage common understand cause abundance assessment robot neuron choice motion primitive duration realization storing alternatively normalization gain threshold another behavior formulate contrary dynamic nevertheless mention converge create large highly coherent depend combination body couple definite threshold include organized ground additionally behavior robot slightly fall arm first contact ground force create shaped environment interaction either different behavioral mode video meta stable perturbation mainly external observer look robot explore body change behavior generate inspire degree joint delay row step two record e interaction environment dash line indicate preference forward back specific body break forward backward backward break happen break desire way element additional facilitate circular delay sensor implement inverse organization connection delay forward backward posterior link direction anti two room behavioral first anti result see connection leave pattern transition front additionally fig second resemble observe subsequent delay sensor subsequent opposite side anti smoothly decrease delay sensor type small delay call loop cluster supplementary switch robot video sequence video notably motion pattern physical body robot interact dynamical massive robot massive force robot angular velocity action recent immediate learn process lead robot eventually end meta periodic crucially mass must large feedback external observer say channel search physical position robot force role couple axis force robot immediately rotation otherwise opposite video robot find influence variant experiment interestingly robot need much two upper body robot indirect sensor actually correlation occur force guide robot specific paragraph interact coupling let exchange extend drive force induce perturbation sensor lead eventually video outside level effect local plausible enable self show self determined without level orientation understand exchange organization artificial neither task specific system discrepancy artificial dynamical reality self multiplicative behavior evidence adequate behavior interaction argument edge allow break dimension human nature apparent goal search open nature dynamic
increase dt sn field variant name particle filtering interact theoretical extensively algorithm problem arise state framework occur recently class address space generalize smc support term smc sampler static regard smc particular refer target arbitrary distribution weight sequentially process mutation particle distribution aim weight n n n give filter interact particle sensible would construct asymptotically integrable denote example one follow modification smc develop final kernel define particle backward correct framework distribution smc stage whereby via mutation particle reweighte via incremental importance weight measure particle diversity commonly reduce detail weight particle particle use mutation kernel new w incremental importance weight sampler specification subsection basic sampler explicitly utilize expression need perfectly acceptable sequence normalization constant correct lack importance normalize obtain population event smc insensitive depend target follow cell let eq particle construct sequence population probability smc sampler posterior weight n n abc tolerance draw l reverse describe mutation particle abc incremental bias n sequential carlo joint mutation weight incremental induce identical smc marginal joint sampler sampler theoretically use suboptimal approximate approximate backward calculation denominator abc posterior smc develop abc specific target call anneal abc consider abc emphasis increase abc construction may consider smc procedure discrepancy stop online discrepancy tolerance fitness particle indicate take perform particle tolerance equation specify level cdf distribution quantile stage abc way tolerance schedule adapt local approximation efficient tolerance calculate strictly mutation smc sampler specialized mutation genetic search mutation particle mutation mutation smc formally mutation operator genetic distributional provide mutation follow select mutation say stock figure asset month spread ease financial asset chi operate book option book meet relate consist visible book process exchange matching engine range size limit sufficient construct much picture previous consider aggregate volume level volume per spaced construct precision fitting auxiliary describe volume subsample price volume interest auxiliary purpose compare indirect term make basic estimate term agent reasonably produce realistic intra daily reduction dimensional summary volatility dynamic inside spread instantaneous volume bid summary employ characterization capture specifically mid return suitable interval aspect parameterize volume bid side volume distance algorithm specify tolerance force iteration specifically procedure schedule employ force specify result test tolerance estimation configuration low value quantile tolerance show forced schedule run mutation compose mutation component degeneracy practice efficient simplify mutation mutation eliminate weight numerator denominator produce issue dimension due nature cross particle exclude possibility particle section abc trial axis distance intra day volatility price intra smc obtain estimate agent replicate feature relate price dynamic clearly value particle sampler standard optimisation evolutionary present discussion pareto intra high weighted figure intra repetition particle volume market ht result abc respectively indirect multi estimation abc easily introduce distance metric ii procedure consider multi ii vector framework parameter vector use mutation kernel solution abc return comparison highest return former non simulation financial market particle try fair comparison mutation operator highlight difference smc present firstly procedure suffer particle degeneracy mutation mutation secondly default iteration particle degeneracy additional front chapter stochastic demand asset exchange real perform via inference adopt result estimation perform adaptive compare indirect class quantitative finance attribute book primary intra every exchange place one fundamental dynamic important attribute asset recently asset explain real range truly pointwise trivial addition amount big day asset one calibration calibration indirect inference adopt algorithm widely search mutation sequential sampler smc sampler abc framework finish european secondary chi x recently structure financial market financial market provide drive market participant trade matching mechanism stock together pure york stock exchange stock exchange hybrid operate less trading activity chapter market participant allow place order price execute price order execute immediately order opposite book trading specify interest book display snapshot book particular stock chi market share share share remain share execute immediately enter limit book second share simulation model allow day trading process behaviour market participant financial limit market order depend model instantaneous mid high bid ask price modelling intra dynamic stock volume well reproduce address hand intelligence consist trading order possibly consideration impose action reproduce feature market tail later realistic agent financial asset necessarily price output reproduce one prominent power law intensity limit level unlikely modern presence strategy dependence clear present formulation representation model rich propose section abc purposes section book intra present stochastic representative reformulate computational day every bid ask side dynamically evolve respect equal price low price interval price interval bid book away ask book away expect bid remain price period course bid price order away subscript passive order order order understand reference price start level bid assume activity uncorrelated top unlikely impact price volume volume model unchanged interaction level inside band model dynamically evolve observe feature modern present detail include agent order model level random book side order ask order activity class model activity interval passive agent order vast majority typically execute particular level order generally consider small assume responsible market ask market participant execution take market execute bid ask side trade multiple market asymmetric information market size include order bid ask vector order furthermore order level conditionally stochastic order arrive bid property multivariate cox l intensity monotonic distribute accord skew kind skew copula importantly scale furthermore order independent si market maker activity limit order place evolve trading day place bid allow capture structure level bid analysis frequency dependence skew exchangeability intensity bid tail occur intra market produce bid depend book equally would occur skew copula process skew intensity bid therefore order specify stochastic model exception number order level order level c construct latent matrix k cox distribution order high order queue assume remove critical market ability adjust activity execute maker removal activity market evolve intra reflect nature trading limit order ask bid ask skew copula non exchangeability stochastic bid ask book feature mean market adjust volume order create new occur trading addition principle limit order volume book instant representative specification intra daily book market participant market participant type throughout day often market order limit chapter activity model dynamically evolve place market order compose component stochastic structure k order furthermore stochastic opposite order l bn accord cox k transform intensity strictly monotonic mapping characterize n k ki real dataset volume level ask stochastic agent model specification describe demand type intra synthetic state time denote obtain base transformation map activity incoming event limit process section trading activity simulate trading day algorithm model
run dataset gradient crf learn report average learn amount practice sampling computing require partition function structure prediction pair test significance legend anchor log xlabel ylabel bandit collect performance collect explore exponential thin tail hold detail performance affect experiment train increase improve trade probably prediction often produce hard legend xlabel height coordinate improve train varied quality crf hyper map keep condition policy affect hold multipli derive variant generate change deterministic predictor derive long still upon stochastic recover achieve ht legend style xlabel ylabel coordinate coordinate become within trend across remain unchanged include supervised capacity serve robust batch feedback risk bind hypothesis away dependent robust principle prediction optimize rich family learn massive classical generally learn non loss function extension relax assumption handle etc research rao anonymous constructive principle batch bandit feedback g ad query bandit feedback user click ad nature problem score generalization constructive minimization policy optimizer prediction decomposition objective enable efficient substantially datum record recommender ad little interaction contain record e feature describe g recommend news article feedback article read feedback provide partial feedback fundamentally correct prediction g news article provide feedback bandit feedback interactive control offline cross perform etc batch system control estimator develop interaction bandit algorithm evaluate perform sufficient unbiased policy reason pick conservative generalization error structural minimization family constructive nature principle bandit derive optimizer structured prediction decompose variance linearization effective classification problem verify support detailed risk principle parameter structure output optimization discussion approach fall give incomplete feedback estimate prediction supervise feedback generalize sophisticated cost offset allow perform output sized batch bandit candidate exhaustive approach gradient family equally expressive build develop optimal doubly additionally focus inverse doubly estimator tight concentrate historical stochastic like exploration bootstrapping allow strategy pick bind successful problem like supervised risk armed bandit regret contextual bandit beyond approach implication several application feedback warm armed bandit pre retrieval evaluation bandit ht supervise hx p xx mh could indicate assume fix unknown hypothesis note hypothesis notational convenience assign interactive system observe sample indicate risk maximize user wish interaction batch assume historical collect system typically supervise ideally need full explain set candidate test candidate collect comparison hypothesis principle supervise outline fundamentally feedback class equation finally constructive motivate learn bandit optimize well serve analogy risk minimization call pick true additive translation multiplicative risk crucially equation degenerate degeneracy arise objective conservative selecting via biased estimate similarly hypothesis avoid unbiased h un particular tail sampling really explore region favorable algorithm classic g structured machine use weight multi news label could simply concatenation bag assign efficient think multipli induce deterministic gradient classification interactive predict system document feedback correct loss outline analogous adapt yield equation prior sound batch bfgs box optima implement code need variance obstacle develop spirit multi svms converge local optimum training objective decompose differentiable optimize overall taylor concave approximate w tw du iw iterate proceed input popular problem structure svms simple crf essentially independently outline experiment collect repository range l supervise bandit policy collect principle arbitrary stochastic policy crf amenable loss supervise hamming incorrectly label false negative I I h feedback g learn supervise report loss rw hamming agnostic handle parametric
elimination wise appearance length representation per frame popular space align feed train spatio train cnn attention simply activation last combine mechanism frame cnn cnn element cccc cnn global video description incorporate lstm outline incorporate feature add mechanism incorporate otherwise whether performance video generation estimate maximize description description backpropagation word maximize probability validation stopped update sec cnn cnn confirm temporal convolutional report automatic first use four line comparison beneficial exploit benefit attention exploit temporal free across exploit temporal structure fourth consistently design reflect description intuitively table reflect description present video corresponding description model fourth video description correspond video dataset perform evident second mostly panel frame type cnn correctly oppose simply work video capture temporal structure frame wise fine grain motion consecutive frame order global propose learn frame decoder generator empirically validate approach indicate model furthermore result text use addition preliminary gain possible leverage generation direction thank acknowledge cifar li universit universit e universit universit universit recent recurrent neural rnns motivated video static video require dynamic description temporal video description incorporate spatial convolutional short temporal cnn tune human motion automatically select temporal current dataset large challenging pair video description open activity challenge vision hour automatic video help index online video conjunction synthesis video description visually description consider challenging automatic carry contain moreover video description sentence characterize typically frame interaction actor evolve amount vast collapse fused video description generator exploit underlie argue video structure grain motion information characterize action answer action localize time consecutive temporal video refer action video video summarize elaborate description focus salient salient video framework video appearance frame input train neural network frame collapse via simple entire ignore two collapse paper introduce exploit encode structure derive spatio base recently context translation generate attention small frame generator activity see start descriptor abstract action emphasize video generator effectiveness temporal domain call video description per video much track movie video use encoder decoder capture spatio promise static frame description method suggest temporal generate mechanism experiment static throughout video video exploit cnn describe approach purely description decoder generation decoder neural encoder decoder encoder encode representation sized architecture choice rnn length symbol input convolutional neural cnn generate encoder encoder decoder choose type rnn decoder step internal symbol rnn run recursively symbol detail choice decoder automatic video cnns successful recognition beyond recognition task cnns variety task localization vision representation cnn feature activation layer video temporal spatio temporal convolutional neural cnn recently capture cnn build representation preserve local motion descriptor frame sequence divide spatio histogram orient orient flow motion sure extract cnn convolutional activation relu max pooling convolution relu layer temporal video feature temporal feature spatial width height frame within finally combine concatenation image frame position train cnn train recognition activation relu pooling feed classifier compute aggregate dataset video video complete architecture cnn use convolutional ultimately temporal feature cnn duration action complete video averaging exploit frame action entire event allow video duration short duration attention xu exploit structure exploit temporal video averaging vector decoder weight reflect relevance temporal feature video input decoder summarize previously temporal return together normalize attention unnormalize score normalize attention mechanism allow subset frame temporal selective inclusion decoder exploit temporal empirically fig graphical illustration attention generation study work domain video embed video tend translation video generation low video representation intensity sec sense closely recently static approach apply et adaptation generation limit recent annotate observation description accurate description visual content well activity base cnns
hardware gmm apply l nearest mainly cross illustrate impact fig big lead accuracy table report art part cnn train layer imagenet deep table already outperform respectively stage annotation utilize conclude bag superior pooling bounding box demonstrate gain gain effectiveness view particular representation method fusion pre train benefit detection classification paper instance solve problem exist work level ability boost annotation framework extensively art experimental ability possibility scalability possibly improve establish criterion noisy proposal secondly suitable candidate pool directly select proposal pool without aid bounding box framework edu sg yu zhang advanced digital sciences yu com bin laboratory technology china edu wu laboratory novel china edu university edu sg great cnn enhance discriminative feature framework framework transform multi object proposal view representation proposal exploit label utilize ability boost category art method power combine deep art dataset availability cnn great task powerful problem generalize category application propose address label recognition level label indicate truth box label utilize tune train cnn representation diversity image car force hundred car multiple fall instance feature region densely fail poorly inspire utilize weak supervision label fortunately many strong supervision box solve understand ambiguity obtain box bound box annotation boost unseen category label two multi tackle task separately view utilize box form view spatial configuration weak label tune cnn view encode view global representation view train classifier balance abstraction local similarity thus enhance importantly utilize encode distribution proposal feature partial bounding box overview area vision machine cnn multi cnn adopt label recognition cnn cnn global dataset different image imagenet different global method help improvement weakly max pooling score employ multiple scale achieve multi view deal learn combine view supervise employ separate contain propose several study combination computer vision mainly vary utilize near vote linear analysis tailor neighborhood learn local satisfy train metric metric vary region space formulate label recognition object proposal detection become e target proposal background object classify multi complexity scale single need object cnn detection selective good hundred enough proposal cover category every instance particular employ selective object selective extra ground truth box proposal selective search find traditionally optimize alternate typical ability limit applicability efficient originally image assume parameter k mixture mean whose soft assignment probability map proposal abuse representation proposal proposal feature proposal subsequently utilize good accurate local intra class focus spatial view enhance effectively encode configuration candidate local determine candidate relevant particular ground label could pool ground object study local study metric mahalanobi satisfy eq certain symmetric definite decompose distance equivalent map transform cnn metric nonlinear minimize loss encode discriminative distance instance margin encode neighbor positive neighbor class distance employ near belong learn discriminative specifically replace neighbor utilize label abstraction build pool extract cnns convolutional conv three convolutional specifie apply pooling row full conv conv conv conv conv full full st st drop soft pool pool encode information near neighbor structural although linearly issue extraction inspire incorporate neighborhood encode extract proposal feature pool label label label annotate label utilize bounding box ability unseen category exploit visually close category box annotation cat cat train exact annotation visually object exist strong supervision boost validate feature view label representation view cnn convolutional fully connect fine full cnn dataset train network process subtract image also relevant currently involve stage cnn network logistic label vector iy fine seven last connect large use view execute tuning tuning process object fine logistic final margin neighbor fine accelerate seven layer connect view instance label evaluate challenge dataset dataset ap test cnn pt em train cnn cnn svm extract layer train representation imagenet visual limited imagenet first layer cnn layer replace adaptation train target extract object image cnn level tuning label tune final
auto encoder attempt contraction denoise chain perturbation decode deep direct generative neural carry likelihood stochastic descent secondary network variance gradient finally feed forward feed directly use transformation space learn cumulative density mapping sample exactly train minimize mmd minimax use adversarial specifically generative layer top indicator long enough draw pass map vector multiple layer architecture relu layer output cc jointly define sample prior pass neural net network model indistinguishable generative carefully involve mmd mmd kernels generative complicated auto arguably capture reliably encoder create visual exist manifold mmd reliable literature refer operate auto produce code auto encoder generate code proceed greedy encoder fine encoder code mmd final encoding add encoding layer term manifold code motivation denoise play crucial determine mmd optimally open heuristic advanced approximation use kernel span range mixture sufficient obtain weight kernel far equally well result drive difference behave write especially factor maintain gradient issue mmd usage kernel overcome mmd minibatch update draw minibatch mmd minibatch mmd minibatch pf minibatch generating pass train minibatch throughout straight protocol density model scale gaussians search auto dropout encoder unit tune likelihood mnist stack deep adversarial net stochastic net dataset competitive significantly outperform despite mmd effective decoder powerful produce visually appealing reflect window likely perturbation transformation along decoder noise learn merely visualize distance merely tb mnist highlight red box interpolation go interpolation highlight highlight aspect explore uniform space correspond datum highlight right bottom projection realistic attribute change gender simple framework call moment network approach maximum discrepancy indistinguishable mmd trick explicitly compute moment use minibatch descent training combine model fed original auto generative simple combine mmd readily mnist achieves superior demonstrate discover implicit manifold direction mmd alternative mmd criterion possibility time develop literature possibility expansion inner explore mmd long grow minibatch original advantage statistic entire another like treat encourage well generation latent explore posterior straightforward way predict recognition auto mmd fair representation attribute mmd could statistic change sensitive variable utilize encoder create complex possible convolutional create color acknowledgement david helpful regard provide deep generative feedforward recently generative adversarial adversarial technique hypothesis mmd simple statistic sample model backpropagation generative encoder mmd generate produce combination compare baseline mnist area deep neural memory task recognition translation supervise feature recognize promise inherent good offer generalization qualitatively assess generative matching begin easy draw network output quickly oppose mcmc necessary boltzmann recently adversarial unlike difficult backpropagation behind maximum discrepancy minimize discrepancy moment trick mmd minibatch contribution encoder behind encoder code encoder rich encoder model original decoder auto encoder yet effective produce generative mnist face dataset comparable baseline include available set ask answer question sample statistic similar distribution formally follow mmd match high moment write product
ising make early bayesian core neighborhood distribution base pseudo follow candidate conditional differ define equal joint equation conditional marginal support equal support marginal everywhere describe clique neighbor clique regular manual point square clique array array determine derivation normalize array define involve summation correspond summation neighbourhood j array array array produce exact array neighbor array exponential term summation require show compatible conditional distribution conditional conversely conditional involve q exercise essentially exercise array neighbourhood neighbor whole simulate sum pixel odd version wang exercise simple case improvement wang sampler obvious book therefore odd powerful property gibbs color exercise find exact involve sum though reduce overall normalize joint associate conditional resolution exercise use arbitrary joint compatible conditional obvious probability proposal proportional step propose multinomial neighbor purely especially value wang exercise simulation account sub grid sampler piecewise explicit directly chapter appropriate correction boundary loss therefore every give mean posterior mode differently error basically look first look risk equal pick sum allocate reach configuration produce necessarily minimum experimental way check different compare deduce slowly run symbol integral call neighbor pair full conditional virtue exercise ise modality ise distribution invert pixel ise eq modify model simulation color image perfect site modify cx col col col k x cn book increase manual histogram grid detection cutoff empirical simulation replication opposite decrease manual histogram return manual grid empirical ise posterior statistic exact could tolerance albeit high computational cost gray chapter marginal q correspond mode locate expectation n jeffreys give associated bayes identification standard proportional gamma distribution proportional binomial proportional distribution proportional power transform conjugate exponential observe observe conjugate proportional choose integrable jacobian unknown part make impossible separate exercise student since jacobian integrating lead proper inverse conditional get conjugate student normalize student q scale prior jeffreys generic location py pz z determinant constant jeffreys provide change parameter jacobian negative value density interpret conditional product eq cauchy integration discretization compute regular summing product density inverse fix dimension operate conjugate prior arbitrarily family unbounde limited hence appeal un normalise post sd post constant integrate proper level seq seq cover alpha low simulate z go vary probability take go ratio go varie ratio less analytically maximal denote sigma manual bt evolution factor denote example denominator integrate evolution xy exercise numerator numerator normal simulating follow bf seq le bf bf setup express normalizing exercise denominator happen support evaluate integral normal density degree freedom finite denote student exercise integral show discuss student density simulation experiment lead three infinite alternative symmetric integrable proposal integral since exercise integrable integrable student infinite integrable density behave importance weight cauchy tail importance face tail evaluate use nu df df col col gold code variability huge jump series manual bt importance solution therefore take text exercise integrable run r alpha alpha alpha show considerable improvement evaluation manual gamma bt evolution three importance evaluation gold cauchy sample harmonic simply harmonic mean inverse pi repeat true invertible transpose produce use word column invertible nx linearly solve solve equation check lm call intercept x q conjugate give gamma identity integrate q virtue q n n exercise sense matrix unobserve predictive q covariance matrix deduce mn gamma produce prior nc xx associate joint show g orthogonal obvious generate eigenvector subspace dimension determinant jeffreys predictive predictive jeffreys prior therefore limit gibbs irreducible sampler work rotation axis respective center radius connect manual mean thus similarly next sampler depend whether start disk bt gibbs sampler conditioning union jump probability disk distribution irrelevant role derivation symmetric consider correspond full full conditional compare gibbs conditional gibbs col sample true marginal namely col bt gibbs output posterior associate ga full sampler associate conditional full eq q code conditional gibbs sd remove grid le seq log post post col add sample fit bt gibbs superposition conditioning give corresponding family associate eq conjugate n canonical function case poisson poisson link generate pdf q property table margin margin cell inverse parameter write independence sampler gamma proposal metropolis al shape mcmc else col col col illustrated manual show fit target histogram proper behaviour example acceptance gamma histogram maxima histogram output mcmc distribution iid cauchy estimate normal cauchy check graph function random metropolis code x df df df checking length mcmc n mcmc mcmc rt valid mcmc comparison manual cauchy noise convergence noise number iteration curve observation induce acceptable comparison gold mean metropolis sampler al mcmc mcmc al rate mcmc sampler repeat call output three matrix take range something manual valid range l remove loop replication running simulation manual sampler metropolis hasting may converge problem approximation iid hastings walks compare code book sigma exp I version n quite improvement hasting target mixture top histogram target probit flat prior q inner volume infinity probit exercise additional power nonetheless traditional control limiting include intercept probit bank intercept need add code function lag autocorrelation produce bank intercept factor manual book intercept posterior perspective take magnitude last covariate book inclusion intercept bank probit coefficient intercept flat iteration right last latent variable probit introduce respectively mu mu application bank output manual unobserve py iy I symmetric obvious mu inverse call library conjunction complete irrelevant regression x introduction gibbs variance truncate respectively base call library mod x probit coefficient sigma inf mean sd inf sd sigma beta mu sigma function represent figure manual manual converge induce move transition compare bank code iteration respectively bank probit intercept histogram iteration auto bank probit k n approximation simulation suggest book adaptation x c mean log bf complete get individual tag loose capture obviously summation sign acceptable keep sum obtain model book constraint r c joint distribution conditional exercise q capture indicator derive necessary life history consider approach schwarz begin likelihood case accounting constraint accept reject replace marginal still know normalize eq uniform matter normalise examine associate reject even thus output fall within accept reject algorithm trial accepted parameter conclude trial probability uniform fall surface fall repeat accept bounding performance exercise constant log ta x nt pi prop prop sd degenerate rapidly moderately uniform accept reject section gibbs conditional conditional special gibbs slice exercise since distribution density high conditional sampler slice inverting take case exercise rate reject algorithm exercise normalise simulated algorithm since since rate lead eq reproduce exercise exercise deal simulation modify return n test consist monitor rate return test q I show beta distribution get easily exercise law number coverage vary coverage vary schwarz group location devise block joint distribution special partly hide individual constitute capture relevance conditioning time past location conditioning location unobserve block addition bring block extend respectively mixture bernoulli consider mixture define identifiable restriction solution part classical usual solution box trick box identical box allocation ball ball equivalently box count remove extreme value partition sample partition empty indeed partition mixture unknown eq q latent variable prior propose imply normal allocate value factor sum sum exercise em apply test influence representation imply get possible implementation tt sd sd sd em em em sd em em sd manual manual increase start start convergence independent sum clear sampler parameterization unchanged since x gibbs therefore loop allocation mu mu mu repeatedly highly illustrate manual seem start sampler iteration show exchangeable weight necessarily exchangeable imply apply proximity map annealing also simulation integer share get neighbourhood behind anneal first concentrated necessary simulate whole convergent slowly iteration consider application mean modification simulate since normal hyperparameter modify version gibbs z move guarantee walk behavior proposal ratio logit walk show simultaneous power normalize simultaneously importance difficulty normalize denominator run experiment influence observation share global get increase global define neighbourhood anneal concentrated mode large necessary whole convergent map increase slowly iteration much requirement book immediate simulate simulate p prior conjugate trick mean gibbs iteration I compute completion mean difficulty around sample close may many simulation mixture unbounded likelihood sample procedure image sum partition partition allocate single term unbounde grid mu seq length seq surface like manual exhibit illustration unbounde random depend bt tw bt bt h suppose show stationary moreover sufficient ar autoregressive integrate jacobian expand eq main close predictive distribution acceptance also reduce write hasting current indicate iid double impossible stationary ar point satisfied p eq associated ar define proportional posterior integrable integrable coefficient derive recurrence expand q j recurrence derive root test degree denote reciprocal polynomial
transform author investigate transform fundamental symbol coincide et transform know although investigate admit relu challenge transform admit relu approximate b transform space denote sphere respectively write class refer lebesgue dual polynomial smooth compactly function distribution compactly smooth decrease function consist vanish space space identify treat list range may associate fourier eq hilbert principal satisfie denote design fouri fundamental proof eq fourier leave right dimensional every respect old transform dual way say admissible provide belong class admissible q radius use symbol parametrize b absolutely immediately transform justified follow one integral integral iterate integral immediate right always exist dominate write remarkable product color depict absolutely convergent wavelet absolutely distribution coincide ordinary locally integrable contribution balance theorem converge pl get versa composite verify restrict use l f balanced assign table fact approach direct instance et admissible reproduce always instance laplacian accord convergent non stem thus dual transform fix classical uniquely mf uniqueness condition formula fourier domain domain key constructive derive constructive fourier slice transform theory say zero admissible exclude unique distribution product modification example problem origin equivalent change condition justify admissible pair construction admissible admissible k computation satisfy almost addition convergent fouri convergence theorem fourier transform another wavelet reconstruction formula q three p p j v identity j limit fundamental identity j pa formula proof control constant control compatibility fourier admissible admissible converse correspond side lead lead lebesgue relation relation identity k reconstruction topology uniquely universal recall checking table list activation belong subspace admissible approximation activation truncate z tangent radial dirac rbf derivative dirac proposition slowly function contain relu belong tangent obviously accord since bound proposition z derivative recurrence recurrence rbf derivative dirac derivative theorem admissible z k z calculate moment admissible even straightforward admissible odd obviously kk z e k perform experiment reconstruct image theoretical list theorem admissible necessary admissible necessary activation sigmoid sigmoid dirac relu activation derivative relu unit step dirac addition examine activation admissible fourier polynomial origin fourier domain numerically integrate radial grid potentially bias cccc fa truncate power function line draw reconstruction draw figure theoretical reconstruct rather completely fail necessarily work reconstruction dirac fail reflect difficulty dirac origin fail relu lack reconstruction relu work note step relu reconstruct sigmoid reconstruction fail consistent polynomial transform origin draw dotted cccc real dotted h cccc derivative gaussian dirac step relu h signal define reconstruction formula b list result fairly bottom reconstruct dim understand cause pass z unbounded activation universal property neural coincide construct respect wide activation traditional rbf truncate relu dirac precise admissible expression transform distribution existence distribution suggest convolution converge balanced coincide dual long admissible transform unbounded admissible polynomial exclude condition direct consequence sufficient investigate reconstruction slice approximation identity reduce inversion latter construct filter
frame similarity expect representation extract accomplish temporal temporal formally node divide induce diversity divide constant long weight ask happen initialize initialize matrix zero extract meaningful representation limit despite initialization might get reach image temporal frame experiment change result negative weight weight encourage matrix exclude sign constant absolute initialization accuracy express architecture detail convnet imagenet yield good imagenet choose study drop dropout fc transform convolutional pooling convolutional start imagenet network large evaluation split spatial convnet negative convnet fc lstm composite lstm convnet initialize ia spatio temporal convnet look whereas convnet look label edge detector spatio temporal convnet combine rgb optical flow spatio convnet single optical optical flow give slow fusion slow give softmax score optical slow fusion lstm fusion slow flow early fusion lstm give result slow additional average ex stream convolutional fc convnet multi stacking spatio convolutional net create spatio incorporate spatial convnet despite limited label video outperform scale label video dataset spatio convnet initialization performance label challenging incorporate spatial train spatio temporal initialization spatio convnet convolutional temporal video method video contain shot frame video represent continuous live temporal information video recognize getting get training spatio facebook consume require nearly spatio severe convnet imagenet tune video solve overfitte give model representation tackle propose several way convolutional temporal representation video convolutional dramatically overfitte video dataset learn improvement convolutional layer otherwise spatio convnet spatial video appropriately composite lstm example nearly match classification action mainly drive extended deal video traditional spatio temporal video orient wang propose trajectory feature fix fisher vector lastly classifier distinguish state dataset availability deep motivated scale label video deep spatio temporal originally propose et convnet fine tuned frame extract dataset show deep early spatio convnet dense flow extract video
band dynamical benchmark offer scenario multiclass classifier vc vc capacity generalize svms support paper describe dynamical system analogue circuit use ordinary ode solver uci repository improve accuracy reduction vc minimal system svms widely machine employ application cut edge utility demonstrate svm formulation quadratic programming admissible margin suitable constraint identify involve system algorithm learn vc value undesirable svms vc imply give equation recently show term minimal exact vc solution programming generalize benchmark dataset svms term use far support instance well variant dataset paper solution vc dynamical converge minimum vc classifier analogue system complexity modelling domain provide hardware implementation base dynamical attract significant attention last decade real realization circuit recurrent system interesting biological solve problem symbolic memory among integrate programming propose network surface breast diagnosis dynamical system barrier present neural programming extend originally recurrent programming bound wang wu neural include dynamical mathematical assignment liu demonstrate neural constraint utilize introduce minimal describe system discusse conclude remark sample dimension interest separates show vc imply close capacity minimize minimize attempt small make misclassification error fractional transformation lead transformation machine capacity formulation determine hyperplane map solve equation primal resp dual denote evolve couple moment formulation equilibrium element equation represent occur resp order differential determine coefficient matrix use make positive positive asymptotically imply trajectory equilibrium matrix redundant converge assume hence equation find vc classifier aim augment row matrix find notation represent multiplication represents matrix show equation also term matrix write equilibrium visualize equation system namely dimensional belonging draw normal variable horizontal axis fig
every edge locate within community graph spectral assignment assume dynamic divide label permutation group random bp vertical recover threshold phase increase occur choice agreement network deviation numerically overlap plane bp along notably large indicate structure weak algorithm transition zero transition agree past bp large especially away transition bp spectral threshold instance draw derive mathematically limit threshold assume structure previously develop community edge give latent structure bp dense consist edge handle step extend evolve continuous hard regime factorize describe fix regime theoretically possible believe group grow far include handle network independently annotation thank helpful financial research science grant fa u air force office advanced project foundation author pz us asymptotically consider detect well labeling latent sharp chance recover latent difference internal sbm phase membership I call take add temporal adjacent imply locally become temporal along copy community label respectively multiply distribution eigenvalue node temporal two branching edge give temporal adjacent version temporal give rise temporal time spatial describe expected child multiply population ref large correlation recover fix make arbitrarily dense imply community threshold fall two pass edge correspond noisy leave propagate rigorously static community conjecture theoretically community run exponential graph control bethe propagation bp cavity carry asymptotically backtrack perform way infer node time belief propagation tree locally nod marginal temporal neighbor pass along edge temporal fig illustrate scheme message node asymptotically partition linearize community verify average reflect permutation symmetry system unstable due perturbation parameter bp equation system bp factorize correlate latent spin simplify equation study message factorize solution linearize version static sbm linearization equivalent backtrack rewrite message deviation linearize neighbor derivative amount eigenvector jacobian derivative bp relatively backtrack convert temporal ta uv use eigenvector absolute compose node cluster vector community group complex circle give panel area region static point dynamic principle linearization eigenvalue non backtrack bp equation I real value unity
use construct flat test level prior encode object class predictive hierarchical prior prior flat prior hierarchical model degradation model characteristic hierarchical model improve decrease large visual appearance frequent large abstract appearance category formalize class share statistical strength sharing understand prefer hierarchical one avoid sharing perhaps prior suited place great category appearance maximize primarily lead hierarchical might political science census growth answer batch online batch relevant image whether batch prediction formulated training category sort reconstruct assumption learner one predict output suffer observe attractive circumstance bayesian think employ robustness speed allowing give guarantee future unseen generate particular simple bayesian follow explore extend certain glm logistic proof defer answer question important derive gaussian parameter lead great robustness next hierarchical gaussian complement learning theory prior model finally prior parameter small online must observe make glm py py choose modeling analyze world scientific application py density observe observe learner predict py py cumulative aim fix glm log rely loss bound glm py throughout remainder first understand likelihood require scale versa py py logistic respectively proposition learner distribution write spectral uncorrelated repeatedly choose attractive deriving regret directly posterior analytically logistic regression generalize originally glm specifically glm py matrix derivative hessian mean p multi label class belong combined likelihood make desirable bound develop pac relevant receive probability generality pac consider predict bind distribution pac risk regret fix attractive rely pac require calculate remain risk erm proof see l choice tight almost circumstance satisfy poor state erm bayesian predictor generalize small batch bayesian bound prior robustness share statistical strength selection relate choice hyperparameter hierarchical parameter place freedom one multivariate heavy tail decrease exponential cauchy recover place prior yield regret bind behave would heavy thought switch logarithmic behavior concave possibility priori might moderate value guarantee logarithmic large robustness specific choice bind bayes nr bayes cn c prior prior allow statistical strength provide answer condition strength hierarchical preferable sharing strength formalize framework number carry begin problem manner number example idea pac learn hierarchical bayesian pac task theoretic typically goal learn alternatively advantage use online receive task observation image source source accord learner observe place one level similar discuss qualitatively prior significantly nz hierarchical shot learn source make make large task cs cs hierarchical n shot provide new source case consider investigate bind two prior prior let appearance would constant furthermore vector explain poor performance class appearance parameter vector object class investigate space dimension example prior bind l regime g meaningful interest introduction dimensional desirable feature selection dimension achieve performance increase induce bayesian lasso convert prior place laplace seem dimension put lasso though unable match common induce place exactly density spike keep regret increase component ensure logarithmic maintaining generally n choose exactly zero choice purely reason set theoretic benefit hierarchical main three specific prior likelihood widely often substantial generate mechanism offer analyze variety employ represent hyperparameter group create complicated simple one result important hierarchical use result insight batch statistical risk applicability acknowledgment gr comment anonymous constructive presentation u fa air office scientific engineering fellowship proposition cumulative taylor f assume z combine yield theorem q q define occur reason nz py ct observing factorial odd
large document technical least intuitively say topic something reasonable assumption appear context large arbitrarily probability must q dirichlet namely draw probability reasonable range imply large well essentially enforce topic view notion separability sense topic document topic document appear mass word let non topic call document sense topic specify number initialize support topic matrix topic proportion multiplicative document correctly identify support large topic topic actually initialization within document effect multiplicative improve technique one c something large come existence local anchor anchor furthermore show decode quantity evolve system evolve iteration quantity quite topic support portion roughly devise document try common false positive never say might topic topic look initialization user document show work firstly anchor assume word topic proportion similar small document appear recent discriminative word large topic difficult roughly range topic proportion reasoning follow document kl proportion carry first long topic anchor progress anchor word show long dominating phase keep drop reach show dominate identify step finally improvement round begin positively topic large proportion try whenever quantity kl f kl minimization respect anchor namely use simple maintain estimate divergence estimate able anchor never analogous manner round argue reduce initialization section remark proof dominant topic weight dominate document anchor namely satisfy proportion sufficient proportion anchor word word explore relationship anchor bad initialization anchor word low ground truth even fairly stage variational challenging significantly new leave important acknowledgement helpful discussion use equivalence say want study specify initialize kl incomplete modify need sure topic maximal dominant multiplicative topic property multiplicative step logarithmic document effect cause multiplicative word incorporate correct support also fairly mean version among rest support put way imply value variable everything kl variant thresholde namely unique large constant number document kl problem kkt topic satisfie appear support topic next topic document support former easy look would one finite happen document estimate appropriately variable iterative correct side split notational proceed crucially topic support topic document topic appear word certainly document summation vanish however update document topic call analyze topic correlation assumption independent belong topic claim establish throughout together topic keep improve support without certainly statement lemma proceed prior document document denote ensure equivalent look ever upper step constant support look kkt word certainly j I enough corollary nt c conclude take consideration value correct iteration achieve portion technique keep quantity iteration evolution message pass evolution average track track tc c iteration quantitie cause update alternate update constant iterate well precisely tc kkt part non previously ic claim contradiction little translate c upper imply certainly monotonically decrease interval absolute case upper c monotonically hence tc tc proof proceed lemma tackle upper fc get want yet relationship get right function fc put suppose tc kl minimization respect iteration iteration satisfy tc lemma immediately proportion entry within multiplicative section remark iterative proof show iterative incomplete work whenever topic lemma happen kkt separate ic argument furthermore prove guarantee completeness fairly recall goal support I present document support topic speak devise try determine topic positive common ensure topic pair test say yes intersect support find roughly document word formally f min j min correspond neither contain inside remove remove list analyze proceed describe determine topic min topic contain let contain topic belong property topic denote belong word belong least say intersect say pair intersection show back round many dominating topic difficult track discuss formal state anchor document document dominate minimize quantity say something min j variable start anchor min kl kl respect variable f kl upper f dominate c r identify distribution p outline anchor identify word topic assume throughout topic prove claim outline virtue initialization enable iteration basically value lemma suppose j variable eq constant get kl minimization anchor document minimization optimum b kkt condition denote j rearrange need place intuitively view combination unless belong topic multiply something say reasonably large somewhat let anchor topic split update q partition three contain vanish word topic upper b ok dominating contain inclusion document dominate third reason actually preserve show initialization word j prove induction cover topic topic seed document claim low hence claim almost q certainly want min establish previous logarithmic number factor cause support discriminative word correctly topic crucially mass outline word discriminative word topic cause decay topic belong dominate discriminative whenever zero maintain multiplicative anchor word correctly whenever j claim I j c expression eq get claim maintain document I certainly focus furthermore bit upper inductive show equivalent together belong drop let topic belong hold constant ic b namely topic least multiply want bound term eq weak property di dominating belong must document since topic correlation claim want I complete claim correctly dominant maintain discriminative large point follow anchor discriminative kl simple since increase simple plugging exactly state combine anchor word discriminative c l l bc achieve dominant l bc l proportion want bc roughly argue round support discriminative word multiplicative correct support discriminative kl whenever kkt summation however contradict want correctly identify namely analogue basically back support thing deal quite anchor word iteration eq brief look nothing upon read prior happen specifically weak dirichlet dirichlet topic negligible prove elsewhere include concerned cc ic proportion document relate coordinate large prove small rest topic proportion correlation correspond ss claim individually since hand since inequality dx eq dominant topic namely topic whenever big show topic probability mathematically prove jx dx write bind ratio dx b dx way analyze divide integral portion much difficult last come pick part imply claim let rewrite little proceeding c course large c finally portion dominate portion portion x dx first expression evaluate numerator inequality portion expression bound separately low dx simple independent inclusion still within topic modify handle handle filter want variational handle previous section argue common argue common progress argument fail scenario study small certainly b mass dominant loose really proportion dominate require clear topic include break calculation generalize sketch outline variable prove multiply inequality fraction word document want correct discriminative whenever support furthermore multiplying get c indeed equivalent easily deal alternate lack anchor way document behave like contribution intuitively show correct current tc tc max claim contradiction translate side upper certainly monotonically interval difficult calculation lemma give fc lower get next lemma suppose ic c dominating sequence q upper simple eq bind side prove side hence bound analyze expression let bx indeed derivative fairly gm sufficient true proceeding expression bx bx bx bx bx bx j j ic c c c c finally estimate word dominate dominate document high topic word total union thm proposition section inference efficient infeasible despite popular current theoretical understanding effectiveness inference base optimum topic show inference provably learn satisfy topic expansion introduce anchor topic prior dirichlet prior introduce modeling role initialization fairly lda variational insight might nature force combination estimate eventually reach document word document span heuristic phenomena machine variational like practitioner alternate relaxation easily theoretical know guarantee quality direction algorithm relate convergence em algorithm setting another set appropriate initialization alternate minimization ground prove address provide assumption initialization strategy converge number difficulty somewhat closed second variational operate introduce stress identify rather understand behaviour method method variable generate q term achieve back step compute set scenario common relax e min pz e step none family approximation guarantee ensure one optimum explore optimum model model prior topic pick topic pick result document topic type commonly assume satisfied one popular originally vast theoretical relevant context paper work work word anchor topic topic partially topic work certain say support topic topic almost word one sequence long prior
triplet triplet multiple work sketch generative parameter mlp exponential focus diagonal depend triplet triplet multiple dependency potential integrate variable exception variational come crowd maintain flexibility cost goal maximize likelihood capture triplet involve latent class model equation resort employ doubly stochastic highly provably maximize evidence standard bayesian speed benefits monte inference define approximate resort mlp parametrize act predict latent input write evidence low act need infer inference variable resort trick variational unbiased expectation predict variational inference unbiased sample shape learn drawing triplet time becomes learn triplet carry carry combination oracle upon inspection component form variational autoencoder triplet triplet happen formulation force share information triplet implicit teacher turn generative model act fine grain run belief network determine performance absence basically belief network information hold datum hold triplet datum crowd inform act proxy loss distribution momentum optimizer crowd triplet run graphic processing hour logistic house natural comprise train counterpart digit perform density unsupervise unsupervised observation autoencoder oracle crowd identify digits triplet picking report visual inform model clear benefit desire triplet triplet task prediction dataset comprise condition image take source varied way depict change appearance apart variability identity depict person unsupervise use architecture series crowd ask simulation question upon present triplet enforce question term light similar typical answering accurately require understand variation concern physics tackle detail crowd triplet question match produce answer resort distance angle angle triplet influence question architecture unsupervise fair triplet inspection figure representation hold triplet triplet hold crowd drastically infer representation inform face identity inform pure inform flexibility predict unseen triplet unseen suggest inform crowd learn image content purely triplet slight space physic beneficial label knowledge automatically improve triplet triplet space strong well result unsupervise generative triplet contribution probabilistic triplet knowledge crowd rich result improved ability predict reasoning system medical theoretic distance unlike commonly approach triplet framework conjunction infinite partition spatially topological vision segmentation shape oracle work regard crowd multiple promise conjunction amazon finally wish mention bias study usa program institute york york usa systems label effectively parametric convolutional building representation compressive criterion inherent lose semantic always label crowdsource implicit feature take advantage come algorithm standard demonstrate image drastically triplet crowd supplement label shape vision development system use image video crowdsource practical representation crowd decision typically employ subtle automate reasoning system deal crowdsource frequently noisy expert alternatively similarity understand initially come apply object question oracle paper learn flexible observation learn human observation train interpretable exceed purely case representation robustness base instance crowdsource community crowd similarity constraint crowd embed probability rather fix density attempt learn assume similarity ask weak similarity probabilistic introduce adaptive crowd without mind triplet feature crowd flexible model employ learn work multiply usage crowd performance generic fine grain categorization density estimation proceed probabilistic crowd triplet principled combine crowd triplet graphical transfer triplet knowledge explicit remainder typically give object visual infer order group object observe crowd evaluate report close candidate repeat procedure triplet treat oracle internal structure function latent approximate oracle mapping conditional arbitrary uncertainty crowd triplet capture replace define triplet motivation heavily raw statistic provide constraint explicitly quantify
name cyclic low adapt impulse periodic incorporate simple state minimize call monotonic integrated algorithm minimization admit simple solution prove implement operation due double slow especially propose acceleration minimizing organize formulation present review section review convergence iv acceleration spectral vi present conclusion case letter matrix low scalar denote field denote phase complex consist element diagonal form diagonal stacked minimize periodic periodic metric express frequency rewrite expand square constant theorem e ignore simplify tackle objective nonconvex modulus follow step subsection simplify look good periodic zero exist length sequence impulse name low directly simple alternate eq summarize ease nk convergence author point two exactly minimize local even original minimization although show long periodic sequence view adaptation periodic show initialization although original metric acceptable problem metric general briefly scheme adapt periodic mm method refer difficult simple problem reference therein include suppose mm optimize function produce accord update generate say construct meet easy decrease eq second follow mm algorithm easy purpose first construct hermitian hermitian easy check problem define tr p eq n know maximum eigenvalue ignore constant sdp relaxation yield sdp rank scope complexity sdp solve amenable apply compactly k element absolute clearly choose max h p diag rewrite closed solution although minimization twice minimization minimizer integrate derivation carry periodic apply periodic initialize max k diag kx nk monotonic maximization minimization scheme sequence guarantee converge convergence algorithm point introduce minimize smooth tangent condition refer exploit property consideration real optimal obvious problem versa facilitate analysis minimization follow ready property algorithm limit respectively denote n fu accord assume converge subsequence also ignore easy I denote equivalence problem multiplication compose similarly product computationally sequence say describe principle speed noticed convergence may double scheme carry derivation acceleration accelerate adapt accelerate originally accelerate cauchy combine em update implement mm update wise accelerated summarize algorithm general nonlinear constraint project descent property backtracking adopt repeatedly e maintain original practice backtracking need loose original accelerate approximation iteration preserve require rather mind kl fu choose worth note take know guarantee backtracking choose satisfied choose result monotonic call backtracking require initialize repeat n max diag h ki k u k backtrack cognitive designing correlation satisfy like band low sequence challenge algorithm spectral constraint band denote hereafter express denote minimize constraint transform q follow derivation rewrite step list acceleration readily elaborate initialize repeat diag n spectral pc ghz cpu gb numerical propose design matlab code website book measure mf sequence algorithms criterion n uniformly fig different backtracking accelerate mf computational complexity among three length accelerate mf versus sequence curve versus sequence average sequence accelerate perform property accelerate initialize accelerate accelerated evolution shown initialize accelerate local minimum contrast increase probably also tell initialized accelerate thus frank probably htbp sequence impulse periodic exist periodic correlation h periodic sequence periodic correlation db
capture positive accuracy tweet score range neutral sentiment tweet one positive sentiment define range indicate extremely tweet vice tweet label negative sentiment tweet tweet neutral score sentiment calculate tweet neutral respectively adopt contain public tweet twitter sample social stream store tweet english contain medium month capture sentiment content processing content external etc dataset comprise tweet facebook produce distinct user tweet process sentiment score calculate sentiment neutral neutral account capture around neutral tweet overall distribution skewed observe tweet example tweet thousand opposite panel second pass tweet number tweet sample represent concerned sentiment diffusion sentiment dynamic popularity tweet function express fig b original tweet speed diffusion reflect second original fraction tweet never fig tweet report tweet comprise tweet tweet times tweet bias twitter reason average fig capture well know broad content popularity medium mean toward large spread broad neutral collect spread neutral tendency favor neutral far accordingly negative spread albeit neutral tweet negative generally twice cascade less popular consequence popularity sentiment employ discover axis represent proportion tweet occur color dot event discussion black number bic well star mixture investigate affect popularity exclusive occur twitter topic tweet roughly public tweet exclusive never appear study aim many type characterize three proportion tweet peak tweet proportion tweet produce peak popularity simply day exhibit tweet expectation maximization gaussian determine three gmm spherical fold validation bic quality vary bic different mixture bic process determine optimal four agreement optimal gmm four represent tweet occur axis peak popularity discussion blue dot exhibit activity quickly complementary discussion dot reach reaction discussion quickly decay discussion dot balance peak discussion black event attention accord exclusive observe obtain b example dynamical length day center peak day fig event capture game nature generate peak namely release exclusive immediately discussion rise band car cause death activity discussion perfectly reflect discussion day peak four day stay lastly event namely ed away four tweet tweet proportion four class symmetric discussion bar tweet evolution sentiment class twitter useful neutral discussion positive peak sentiment constant notably amount much average average content event intuitively carry stay dataset popular discussion dataset discussion toward complementary happen grow shift toward short length discussion exhibit high level around average ability scale tweet role medium diffusion find light sentiment affect spread neutral pass publication original post positive tweet might interpret reader chance neutral analyze tweet individual prefer tweet neutral amount neutral clear popular quick reaction negative neutral reason online environment aim diffusion ensure reach sentiment discussion signature discover sentiment yet event brief medium like twitter sound suggest relate etc vast characterize future political match release positively represent room etc characteristic exploit finding practical consequence relevant dynamic sentiment crucial want policy effectively policy management recent highlighted diffusion information yet spread short piece social strategy social medium
cm cm cm rmse versus data purpose set rmse multi sharing across task gap suffer lack interpretability compete lr rbf task approach electrical treat period model hour e g burden unfortunately discover consumption interpret framework electrical hour load forecasting challenge year temperature response moreover weather day year daily every day forecast response series set half lrr rmse lr additive specifically hour lr compare comparison signal rmse roughly interpretation fig temperature display b hour per day give index temperature covariate hour model similar shape tf lead load prediction intuitively air effect hour activate tf air occur tf hand tf active tf represent transfer function transfer tf tf temperature day learn additive fitting conduct recovery distinguish coherence wide range realistic correctly underlie compete term predictive demand multi competitive learn extract correspond customer improve scalability involve million task j j recover sufficient condition bc omp recover correct residual pp bind upper eq l l atom select therefore inequality term assumption conclude bc omp select correct correct recover orthogonal onto span relation dimensional flexibility key benefit transfer identify output loss interpretability cause interpretable forecasting key exploration corpus establish sparse new fitting transfer step pursuit whose result literature latter experiment world baseline method yield interpretable structure datum extend benefit additive task additive dictionary learn widely extensively theoretically successfully machine ingredient covariate additive manner flexible allow good additive understand face task additive independently several firstly domain expert visually transfer essence interpretability human view independently task model overfitte overcome introduce novel output task sum task interest variable obey law neighborhood use consumption pattern novel univariate transfer function covariate represent covariate specificity task scale use constraint recent field fitting update transfer scale function update transfer constrain matching pursuit bc omp extend pursuit coherence accurate extend theory transfer function learn analogy update experimental world demand accurately learn addition propose method outperform baseline candidate interestingly correspond small fraction independent benefit experiment maintain small decade independently task impose close weight live linear group context family enforce involve covariate significantly common across transfer covariate dependent expert candidate transfer transfer even thousand moreover covariate relevant hence input covariate small involve paper review formulate model explain sec sec real datum load forecast multi task performance comparable baseline interpretable customer vector refer element product n denote pseudo moreover moreover count operation pseudo inverse moreover column entry wise negativity constraint additive diagram briefly intercept represent covariate transfer transfer continuous covariate commonly model spline spline spline spline use estimate intercept consider centering constraint convert optimization center basis specifically ij equivalent unconstrained order commonly add simple easily quadratic regularization solution additive share denote zero new combination set function covariate constraint prevent transfer capture new candidate non function task offer wide keep non negativity interpretation transfer could represent opposite effect lead high demand task lead demand illustrate additive diagram covariate set task task model transfer set diagram arbitrary similarly done sec transfer spline denote spline basis rewrite j jx nj iid find p square residual avoid overfitte prevent note transfer regularization spline strength inherently find code minimize desire exposition analogy us scenario equal define lie former constraint code entry constrain efficiently approximate challenge linear solve successively spline mod sparse code dictionary coefficient global recovery bc provide recovery right superposition belong simplicity atom nonzero develop correct atom bc omp omp omp algorithms search select dictionary bc impose constraint available atom previously bc omp recovery bc omp omp conditions omp recover every superposition whenever q call similarity leave omp omp recover correct close j behavior sec inner j jj seen estimate population transfer coherence definition bc omp bc omp recover coefficient detailed condition coherence recovery particularly interesting application strong function sufficiently statistically bc still succeed recovery recovery satisfied bc omp valid omp sparse representation reader relate recovery author build incoherent coherent condition correct show word exact atom drop weak correct atom signal appear rather spread throughout omp differ whereas assume occur section implementation sec sec result section load problem background additive demand forecasting example transfer dot dash line represent lead transfer transfer correspond transfer regressor independently parameter task regressor implementation additive task package step centroid prediction centroid aspect simplicity likely give moreover world application parameter manually domain interpretability goal parameter employ bic know experiment evaluate affect problem empty transfer per large checking currently approximation eq candidate transfer simplicity task transfer randomly number iid distribution assess transfer synthetic set treat show estimation accurate coincide highly noisy prediction rmse significantly synthetic despite limited task improve rmse method experiment perform task fourth model complexity task basis simplicity theoretical numerical lr half customer come survey economic indicator number business time customer consumption total customer stochastic aggregate signal measurement hour day week covariate test transfer method outperform perform slight
backward convolution bank channel filter minibatch multi demonstrate ability fast convnet gpu spatial gpu domain speed shape execution time throughput measurement efficiency device peak gpu kernel evidence create great reach suggest create gray include create code deep minibatch channel filter channel convolution offset parameter assume scalar algorithmic multiply count wide shape implement convolution inner batch dimension calculation single coordinate multiply contiguous input column offset extra integer multiply operate contiguous block datum convnet employ programming load index arithmetic load convnet gpu half memory gpu extra share source create gray scheduling allocation include implementation project interesting also implementation advanced texture load increase texture indexing calculation memory share share iteration store output share bit point share load limit per demonstrate power parallelism project perform load combine convnet convolution multiply patch filter block block basically adjust indexing calculation indexing map patch channel offset offset add offset compute replace loop channel patch offset location map boundary graphics gm gpu convnet architecture bad efficiency ratio actual throughput peak throughput gm processors core device peak throughput count single execute processor independent speed efficiency unnecessary strictly multiple number mini size modify efficiency actually imagenet minibatch computational efficiency range bad patch loop compare section reach due intensity respect memory cache device additional minibatch device limited experiment determine efficiency significantly efficiency network efficient
account joint interaction next score drop include dropout give abuse notation removal remove sort also sequential marginal view measure contribution look removal measure look include challenge involve anomaly many benchmark real evaluate ground truth evaluation address issue work construct number benchmark second supervise learn construct analyst evaluate expand recent describe methodology systematically create anomaly give huge world benchmark huge anomaly benchmark benchmark create anomaly frequency anomalous main class represent rise anomaly union anomaly detection benchmark anomaly point benchmark experiment anomaly detection benchmark benchmark anomaly analyst distribution normal formally analyst specify describe obtain analyst uci curve use computed anomaly lead analyst anomaly metric number reveal detect anomaly evaluate analyst detection normality analyst analyst drop value threshold experiment uniform result consistent choice remain analyst anomaly benchmark construct set normal analyst learn include generative practice arbitrary require widely discriminative possible subset evaluating subset learn subset cache need learn anomaly al commonly detection experiment anomaly detector describe et al range benchmark ensemble learn replicate threshold varied ensemble mixture retain approach address poor model bad local optima number component gain advantage straightforward form density gmm model marginal easily dimensionality analysis obtain analyst regularize analyst primary reason accuracie competitive train must fold worth evaluation framework potentially choice analyst bias beyond scope replicate qualitatively type benchmark point multiclass concrete anomaly benchmark uci benchmark benchmark total anomaly benchmark serve train dimensionality number analyst publicly across optimal analyst subset minimize threshold constrain produce compare method represent simulated analyst benchmark anomaly rank compute six choice actual rank use analyst threshold value across anomaly derive confidence first note focus anomaly top indeed also percentage point anomaly observation qualitatively choice suboptimal detector outli anomaly constrain room worth note set correct reasonable expect version compute computation difference nearly performance exception gamma statistically advantage interaction critical anomaly explanation able recall evaluate make appear dropout normality score see overall dropout close weak signal early decision difference produce feature often marginal make robust early decision achieve score dropout prior investigate decision detector quality detector method detector analyst compute minimize sequentially constrain show result detector detector reflect sequential news motivation reduce analyst analyst rather arguably less desirable perspective often sometimes contrast anomaly observation indicate reasoning anomaly leave open question anomaly benchmark gap benchmark marginal generally decision method uci point http contain anomaly represent detector infeasible analyst thus evaluation overall approximately train domain quite rank rank evaluate domain figure clear marginal well particular indicate combination analyst weak decision number significantly outperform bad dropout detector little difference independent detector outperform suggest outperform recommend computing introduce require correctly detect quantitative evaluation explanation overall prefer introduce usa edu anomaly detection present anomalous analyst instance truly security unfortunately anomaly anomalous leave analyst evaluate anomaly detector analyst order analyst confident analyst effort investigation explanation number reveal attain one contribution novel quantitative evaluation analyst benchmark artificial simulated explanation anomaly benchmark several insight identify anomaly anomaly generate distinct point correspond meaningful anomaly security statistically activity reality true analyst decide anomaly analyst face analyze challenge especially interaction feature case even anomaly detector pass analyst recognize key due effect overall anomaly analyst improve reduce analyst anomaly intend side effect analyst reduce analyst detection effort anomalous analyst effort focus contribution intuitive score present analyst analyst acquire information anomaly security analyst roughly feature reveal minimize analyst identify anomaly contribution quantitative comparison analyst benchmark ground anomaly analyst evaluate reveal reach methodology anomaly contribution detector operation support finally fourth contribution empirical method anomaly real provide investigation recommend insight paper organize review anomaly anomaly concept section describe method compute introduce quantitative method framework unsupervised detection relate supervised aim classifier instance propose produce relevance score score classifier point anomaly anomaly detector anomaly detect analyst may consider chance detect anomaly analyst effort toward anomaly propose analyst attempt efficiently indice feature appear order consider notation feature onto analyst present analyst able make base feature add give analyst analyst continue analyst able process terminate early normal incremental analyst point anomaly reasonable expect analyst anomaly consider amount reduce amount analyst effort monotonically measure reveal analyst anomaly analyst detect anomaly quantitative require access analyst term section issue wide anomaly detector computing detector either compute particular detector avoid former attempt indicate
name survey exist try attribute string measure heuristic name etc high f around collection citation attribute along medical incorporate previous predefine similarity specifically obtain specific accuracy specific dnn feedforward many initialization deep rbm sparse normalize method automatic al big dnn recognition convolution neural train back propagation outperform method mnist complicated process yu feedforward recognition ability robust variant state feature normalization feature dnn author name explore system author combination many compute dnn learn exploit type neural layer parameter initialization scheme activation perceptron network stack upon connect correspond layer correspond dnn interpret direct approximate posterior dnn layer layer layer give denote express vector hide unit output represent output basic output become input next sophisticated compute multinomial dnn vanish traditional activation propagation algorithm issue adaptive dnn ability affect layer layer layer achieve however slow capable yield overfitte hyperparameter base experiment fold change hide hide unit create network size dnn learn must proper representative datum publication author attribute accord survey string match automatic author name author digital library present build acquire library name instance name author name van etc author name dataset create author understand label person research totally detail number tu le pair publication publication record original fold tune hyperparameter overfitte split percentage particular pick cross network layer unit hyperparameter layer hide many five size hide unit high validation predefine apply nn forest bayes respectively suitable predefine implement architecture implementation term separate test predefine set achieve use predefine feature relatively predefine set method predefine k forest c learn dnn evaluation benefit moreover automatic feature expert knowledge dnn learn feature successfully capability complex research overfitte hyperparameter tune early stop architecture dnn vanish train sigmoid sigmoid thank smooth activation dnn label train dnn improve hand automatic dnn ability complex recognition object raw rational create pixel dnn deep network feature automatically author name ambiguity additionally general system architecture author author dataset significantly predefine feature achieve prediction predefine extended solve open author benchmark unsupervised encode research university technology artificial intelligence digital deep deep neural frank com p edu name ambiguity decrease reliability retrieve digital library automatically ambiguity architecture name dataset contain author name significantly outperform use predefine method achieve relatively compare use predefine ambiguity publication author name appear name distinct reliability digital library author task digital library author
relationship case discovery structural proved distinguish cause depend functional attempt field salient speaking notion property one separate several purpose concept relevant context originally inference likelihood idea suppose conditional parameterized marginal alone say variation three strong weak subject restriction view imply posteriori exploit parameterization applicable argue parameterization causal pass fail compare distinguish cause without structural see algorithmic condition kolmogorov paper explain adapt weak statistical statistical nonetheless exploit direction two random draw density one set adjust setup say admissible take depend clause ii defining say operate cut computed concept generate contain conditional model concept also cut operate bayesian independent sufficient sufficient generating cut marginal model bayesian counterpart classical cut equivalent cut theory regard relative operate cut generate note requirement respectively otherwise ii trivially meet take trivial situation iii violate posteriori htp sep fill circle fill circle scale width sep circle node fill circle width circle draw node circle circle scale width sep node circle circle draw circle draw depend sufficient modeling unlike super regime relative observational causal common cause relative indicate determine separate argue direction turn direction admit causal infer sufficient accordingly indicate causal direction familiar identifiable cut cut give identifiable identifiable situation non gaussian follow mixture gaussian involve independent alternatively priori priori see direction direction identifiable illustration shape parametric constitute bayesian cut nonparametric bootstrap coupling examine cut htp bootstrap density estimate literature assess parameter e g clarity bootstrap pair bootstrap drawing replacement calculate estimate statistical test give point evaluate evenly length mapping mapping setting imagine effective case observable previous argument bayesian way assess determine causal input involve module input three possibility non direction replacement sec otherwise simplify estimate covariate shift exploit avoid flexible verify effective bootstrap estimation density method inference base find e generality otherwise instead nonparametric take optimizes produce posterior likelihood functional estimate conditional point pe test experiment similarly center size row nan hypothesis actually schmidt independence reasonable dependence width row evaluate bootstrap method ground truth data two examine causal geometric causal assume effect plus additive cause way reason use replication htp multiplicative super additive various inspire pass sign control range purely purely multiplicative effect control produce situation different datum see able causal replication although try replication speak improve replication include structure influence exponent change performance four bar correspond tend significant ht situation reduce pair direction seem scatter
derivative contribution literature langevin mala mala special mh discretization langevin drift diffusion proposal depend mala way sampler monte far current proposal numerically integrate often necessary hundred time propose achieve reasonable mala hmc rely user scale reasonable efficiency propose mala hmc manifold associate bayesian model admit take namely fisher associate negative choice demonstrate highly suit langevin hamiltonian rarely mcmc sampler derive simplify manifold negative hessian explore note paper degree software definite application overview full hessian cholesky produce hessian whereas present potential exploit generality handle cholesky region zero primary adaptive leave stationary hmc length mala adaptive step regime standard implementation methodology pilot illustrate methodology realistic model methodology mcmc provide admit log di take simplify mala mala hasting distribution symmetric matrix sensible explain modify cholesky adaptation aim cholesky decomposition low q diagonal negative definite particular cholesky positive definite see optimization ingredient newton optimization basis simplicity sparse factorization upon request hessian cholesky metric reduce q iteration metric chain rarely near tend chain approximately typically take small proposal mechanism local scaling region derivative observe transition occur region chain proposal variance phenomenon target degree see rarely region point tend cholesky computed modify effect generality derivative fisher undesirable effect either work purpose duality mala hmc mala integration hmc length single mass hmc absolute provide integration comparable across area pilot trial relative trial step evaluation note energy probability hmc mass target update individually mechanism updating depend manner position specific momentum start proposal equal l typically tune produce acceptance acceptance candidate equation reason approximate integrate computationally integral computed numerically obvious condition consist mh accept markov distribution gibbs block trivial former mh g make metric see problematic around forward simulation figure dot adaptive pd figure present see length selection step actual energy criterion discriminate behavior point reasonable backtracking application necessary backtrack iterative scheme inspire search length fulfil factor great length inform observe include slow may tune refined long dominate computing need considerably costly hmc tune translate hmc average burn however tend acceptable moderate selection typically contain scale act step substantially interesting overall rather q translate thus distribution walk consequence primarily primary section consider step modify mala highlight behavior mala posterior non part parameter space response regression allow follow estimating ess monotone ess second spend computation carry ghz intel core gb ess ess ess ess ess mala apply burn write time ess time comparable remain run mala value panel latter visual panel trace forward depict panel show density current first realistic illustration return take default prior specifically truncate p exchange return daily also include constitute log observation mala hmc hmc take attain rate around addition one th th mala cholesky potential problem prior table provide cpu ess per repeat sample iteration cpu burn package partially remain mala produce per matlab mala produce indicate little add modify cholesky depict various diagnostic mala highly ill informed illustrate different mala stationary regime mala shorter approximately show base enable mala progress keep setup indicate stationary regime visual indicate small zero minimum eigenvalue confirm iteration indicate useful take short little curvature direction acceptance mala minimum ess ess ess ess mala mala probit probit adaptive logit mala logit probit mala probit probit logit mala logit probit mala probit probit logit mala logit mala logit mala logit probit mala probit effective cpu times regression logit correspond link probit correspond fisher adaptive adaptive length selection inference response model include compare propose methodology specifically consider response inverse correspond probit close logit fisher whereas hessian still definite probit collection datum reference logit mala work mainly numerical occurring burn time simplify logit full mala minimum produce per logit mala interpret length version produce iteration approximately reference mala par hmc coincide slightly effective sample favor mala consumption cpu roughly logit find mala denote improved feature hessian rather relevant paper make cholesky produce hessian simplify mala metric near curvature length also develop implement intractable even admit factorization example propose perform mcmc dimensional hmc need soft enable
achieve performance augment mark extra table validate effectiveness learning cnns train much large k table method perform good category r training crf extra extra extra extra crf extra crf extra crf rnn extra crf rnn extra message crf reveal new gradient message potential cnn network conventional learning segmentation demonstrate effectiveness usefulness deep message readily acknowledgement arc future arc fellowship gpu appendix show segmentation propose cm van university centre output great segmentation scheme cnns message potential calculation significant perform crf cnn potential class contrast output cnn functions exponential cnn message large good method cnn message method deep attract deep convolutional cnns combines cnns feature complex relations cnns cnn objective cnn crf joint segmentation cnns sgd optimizing require calculate typically require repeat slow avoid repeat final prediction potential goal final prediction potential optimize learn conventional crf prediction pass cnns calculation efficient cast cnn potential potential pairwise etc scalable iteration message inference learn procedure message crf message message pass apply method semantic achieve intersection union report image cnns several resort joint crf first train apply dense crf post processing extend jointly rnn crf cnns dense implement cnns unary train unary potential capture jointly explore estimation explore field predict mention cnns crf incorporate pre train potential learn estimator rather potential machine propose relevant idea estimator learn potential train traditional regressor cnns deep cnns end style feature factor thus formulate goal crf sec learn review crf cnn approach mask function crf factor potential index function unary unary segmentation example predict image joint also calculate marginal exclude crf np hard type apply belief propagation reweighted message pass mean crf cnn joint cnn optimize crf construct cnns add minimize log crf segmentation mask stochastic sgd function easily compute apply bring direct calculation infeasible iteration repeat marginal extremely expensive cnn training usually huge sgd thousand approach infeasible conventional crf message potential calculate conventional crf approach follow message marginal discuss involve message estimator function calculation message depend message variable input learn estimator depend message estimator message estimator pass asynchronous message pass simple asynchronous simplify pass pass propose learn variable estimator interaction neighboring denote encode nod connect estimator become hence incorporate however general exposition describe inference dependent message clearly message message iteration inference become formulate formulate cnn output denote network indicator correspond implement cnns analogous design convolutional feature one crf likewise feature node connect exclude pairwise factor pf fc feature vector input alone final generate final connect please refer potential potential class dimension significantly increase computation tend training estimator need output write ideal variable marginal truth ideal marginal problem message network aim learn formulate neighboring node factor neighbor fix neighboring generate classification thus traditional regressor contrast train cnns learning procedure define sequence message aim quality contrast potential function message fast might preferable neighborhood effectiveness inference c c cat person train tv crf crf rnn message publicly background category set
outlier contrast return r outlier represent example character ten different template aim ten image centroid template average appear preserve reason font template information template may adjust provide centroid template fig fitting risk update covariance affect eigenvalue thereby bias another soft svm slack employ sample reliability hyperplane hyperplane outlier maintain separable kernel trick replace dot effect map inner pca projection eigen covariance procedure trick dimensional transform high dimensional possibly hilbert equip definite kernel trick typically dc laplacian flow show formulate instance general slack regularize far derive lagrangian dual applying trick dual simple trick normalization become maximizer coincide simple minimize variable globally efficiently necessary practically iterative either grow nearly minimum object construct outlier regularization prevent overfitte recently connection introduce slack object insensitive slack sensitivity parameter slack serve lead problem geometric optimization regularization labeling understand lagrange dual analyze n v n w lagrange multiplier three lagrangian infimum rewrite minimized z dual quadratic qp vector tc comparison elaborate characteristic template use dual therein four case lagrangian multiplier feasible constraint must iv become objective effect primal increase force deduce coincide lagrange lagrange dual dual except upper bind occur frequently simplex nonlinearity space handle space costly impossible product mapping immediately trick allow propose template applicability experimental original dot dual mnist handwritten image digit respectively carry sample take transform two centroid visual eventually centroid b test centroid region necessary aggregate confirm put template centroid close carry consist reflected reflect implement centroid comparison regression neural exist kernel test validation determine auc template nn template centroid template auc r produce svm regularization shape term follow distribution sake visualization discard membership depict color code result incorrect kernel correctly separate data membership method compare base original iterative carry one object vary fix window equip intel cpu ghz mb gb ram digit additional consequently regularize long show vary present microarray variation gene expression patient add runtime dual form demand take less time well change runtime design toolbox exist summary formulation form advantageous otherwise form desirable form complexity number primal visualize template vector imagine image stand step class word vector aggregate set correlation increase case maximum average slack mean effective outlier reveal result gain consider vector slack r mention fig representative picture extend fig plot correspond expectation fig fig fig analysis fig become close last fix experiment correlation template vector tendency know kernel certain approach originally context classification problem optical automate aggregate template application misclassification neighbor setup nonetheless limitation demand wide drawback instance present result successfully limitation original outperform classification adjust flexible addition benefit formulation regularization make appeal range analyze code become challenge example font automate neighbor bad procedure despite optimality drawback limit wide practice handle dataset suffer limitation improve regularize approach programming problem incorporate slack derive trick handle make propose accurate grow r run substantially solve primal neighbor qp trick neighbor parametric classify largely retrieval video indexing tracking location nn computationally intensive neighbor break dimensional space approach complexity adaptively neighbor template match pre representative use near template create template template classification aggregate template template minimize risk originally propose font optical character automate protein cluster despite theoretical limitation wider well propose limitation original centroid maximum member outlier add return angle represent appropriately r find robust represent centroid aggregate template ten image
line copy different note slight abuse translate condition denote chernoff note projection span result km singular soon condition second statement condition combine line k smc step find span share span extract u bm b rate find vector find extract split store whole easy span thank column span km km first first technical arrive belong span signal estimating span span desire compute process count zero entry zero law process orthogonal negligible km noise span thank split sampling simply span reason see multiplied lemma signal thing result km span use schmidt become q compute ready analyze smc first check contradict obtain high analyze required pseudo store sparse sampling bit finally thus circle q equality stem condition satisfy replacement nu matrix proposition sum sample replacement q dimension k w equation recover remain part satisfy eq lemma find w k upper entry independent last equality stem markov high v n eq randomly u km b k b k stem km bs k u u b high stem k n os km inequality inequality conclude k km mn projection linear span mp k mp u mp mp stem high independent markov x x ji ji j observe therefore thus k km k basis linear span orthonormal basis row ns chebyshev lemma indicate probability row coincide since markov inequality schwarz stem sample column streaming completion interest store column matrix sequentially miss memory propose streaming estimate original vanish linearly ambient store output number stream computational reconstruct noisy constitute collaborative attract recent motivated system amazon netflix google propose user rating naturally translate completion row resp user rating rank inherent similarity extremely become store rapidly matrix designing constraint system collect reveal request store particularly recommendation system actually understand machine technique approximation consider constraint detail exist work transpose singular svd unitary contribution let truth wish noisy observation assume ii large tend corrupted observe entry operator wish observe tend infinity completion streaming column among smc streaming completion complexity matrix construct assumption accurate square exist denote small satisfy iii main result paper smc asymptotically accurate pass soon rate smc estimate soon ambient approximated output smc store treat observe instead singular column find keep step right extract vector right singular matrix top easy schmidt process orthonormal span top vector realize follow subspace span row define operator number survey recent rank could build block section follow rank approximation efficient show absence rank propose program spectral reconstruct asymptotically depend adapted presence improve memory guarantee streaming approximation pass column matrix appropriate memory think accurately asymptotically mn f mn mn easily mn algorithm asymptotically recall stream reconstruct arrive identify pass right propose frequent direction sketch problem kk kn efficient use stream pca apply author randomly I low randomization way reduce algorithm survey devise rank mn sparse mn operation explain asymptotically completion initialization qr deal column information batch arrive column algorithm address singular value dominant vanish entry become consider interested provide estimate singular spectral simple observation constitute basic theory signal order qr issue avoid entry decomposition diagonal whereas diagonal remove dominant spectral present analysis decomposition suggest random theory argument loose ensure separation spectrum need span nearly analyze store store value store memory apply
place plant community whereas easy increasingly high measure plant partition mutual information partition decrease resemble plant er modularity clearly range mixing indicate performance inequality mix er modularity display benchmark four clearly much modularity modularity former benchmark graph community community grow threshold go structure modularity worse large consistent prevent er cm modularity community indeed community achieve complex develop accurate kullback text internal dominant observe exactly ignore use read ignore q divergence interpret move community density approximately straightforward logarithmic odd positive dense density significance expect significance alternatively leibl significance average internal arrive hence significance generally general entirely significance size dense community write dense partition keep mind low significance general repeatedly refer community group usually recently form optimize analytically clear discriminative modularity suit detect early method fail describe simple form compose node reduce establish protein interact connect scientific pattern heterogeneity node triangle diameter another real densely connect know share community field numerous detect structure ultimately regard partition either implicitly community modularity extensively belong wide spin quality give energy mechanic unable capture fact modularity community measure probability systematically modularity different framework describe divergence enable analytic compare modularity significance nan model behave addition formulation develop good community especially graph apart community quantify kl difference believe improve link overlap consist contain overview relevant provide bt number density edge community mm internal total edge internal draw edge way internal observe population derive difficulty implement mainly may assume consider elegant appendix measure metric case probability lie community whenever otherwise look dense generally original base binomial approach former link replacement fraction internal edge would eq development dominant binomial binomial become replacement negligible partition benchmark meta solution develop straightforward fashion initially move node long aggregate repeat contraction within keep edge keep initially upon total number community essential ensure aggregated graph move aggregate internal notice formulation aggregate improvement move internal possible edge total self loop internal complicated summarize graph weighted version graph internal unchanged open flexible suitable review compare still comprehensive modularity extremely method detection want partition original modularity assume nan internal total community modularity compare value sum difference model node refer modularity modularity propose nan assume every enyi er plug modularity version eq er modularity kullback leibler unlikely eq sometimes tight make likely modularity sense modularity consider circular neighbor exclude group consecutive node er modularity reach indeed modularity partition many whereas go er modularity go later er modularity simply unable partition show axis present approach describe internal look measure immediately significance contain community unlikely many community elsewhere random exact statement asymptotically significance eq graph leibler great benchmark scale resolution base divergence internal compare difference significance affect actual moving decrease significance intuition confirm convexity kl divergence kl appendix
architecture directly result retain neural restrictive consequence restriction plausible nonetheless influential development recently accuracy imagenet benchmark relate indeed act affine transformation abstract class neural another cluster activation abstraction composition sparse high really quantify similarity goal focus architecture suit large amount nuisance variation two derive bring nuisance enable build well representation task address limitation principled google face architecture art face benchmark use architecture crucially classification learn good basic behind objective light connect objective perspective triplet nearly global explanation dominate equivalently thus deterministic indeed implicitly criterion employ irrelevant nuisance transformation multiple abstraction interpret iterative recent understanding deep upon physics construct boltzmann rbm block physics configuration configuration variable long correlation range fluctuation show analogy goes create exact real indeed irrelevant multiple level strong literature summarize explicitly variation level abstraction factorize dual purpose exact product iii regularizer overfitte iv configuration justify low setting process deterministic vision invert powerful powerful currently employ limitation room refer follow broad back impossible top unlabele task moreover affine transformation capture phenomena figure clutter brain feed back dynamic unable unlabeled amount overcome design extend new rule summarize explore design realistic assumption cause include translation rotation perspective graphic geometry order computer graphic enforce initially affine transformation weight rotation nuisance transformation interest motion away camera template google directly scene representation example encode depth useful thereby benefit image rich representation inference equation geometry limitation correspond million order nuisance result learn contrast accelerate acceleration change implicitly probable global however potentially max message optimal wide temperature smoothly well pass variant approximate bayes em knowledge top feedback task low feature suboptimal higher implement top message properly outline convert top network implement pass inference max top scene understand segmentation clutter bottom feature principled define back propagation often algorithm inefficient implementation approximate em whose probabilistic contrary much exploit e bottom top fast computation sufficient sample covariance efficiency substantial fast tradeoff moreover although incorporate visual static video static input cause change little supervision would external supervision supplement enable focus nuisance factor difficulty purely discriminative technique thus armed enable benefit principled manner dramatically power encouraging factor hybrid generative relaxation perform naturally parametrize likelihood natural label achieve world classifier sensitive mis specification principled training span spectrum discriminative stochastic descent back em thank thank detail instrumental maxout consist template operation also claim proof eq limit hypothesis independent channel abstract feature assume matrix dependent template nuisance application matching template c maxout template reduce convolution operation activation unit us max pooling relu activation relaxation single convolutional consist convolution operation relu random nuisance switch independent evidence nuisance invoke noise free datum specialized structure derive mathematically nuisance line calculate log pa cg use write expression operator invoke v v modify w constant outside drop operation single traditional origin relu log likelihood measurement relevant important miss hypothesis relu activation relaxation maxout nuisance distribution write generalize exponential simplify greatly play familiar partition importantly counter maxout robust discriminative mixture maxout let nuisance exponential family maxout except generalize describe definition serve generalize quadratic depending amount label many world scheme still avoid data dropout consist neuron output activation neuron rely every piece evidence force prevent adaptation detector perspective show answer yes dropout generative completely miss noise free miss dropout em train strategy training utilize discriminative generative miss iff pixel intensity nuisance variable g since soft yield sum soft yield characteristic step generative however end derivation distributional generative flexible mathematically q pd partition allow pd label feature discriminative random subset sharing since intractable sum monte yield theorem definition mm edu pt computational outperform task visual speech numerous nuisance variable unknown speed recently layer large massive success system capability fundamental question analyze architecture probabilistic deep generative learn model lead system convolutional decision forest human expert wide array complicated object image despite orientation challenge nuisance nuisance simple inference car lie object challenge decade vast approach nuisance super capability learn linear computationally massive amount training architecture convolutional see recognition localization recognition part speech forest image system fundamental success focus raw amount coherent understand architecture framework insight learn principled design improvement latent nuisance combine class nuisance extend transformation abstraction enable pass via optimize lead nuisance additive directly insight answer suggest pathway improvement wide classification etc task nuisance focus paper follow map onto insight provide operation proceed derive suggest promise include generality generative model operation extend define layer represent inference feedforward propagation enable message probabilistic pixel multi intensity channel seek infer prior chinese crp classify map image likelihood bayes rule high amount formation nuisance transformation endow generative model image subject nuisance problem graphic engine c add practically poisson identically distribute iid function nuisance categorical probabilistic mix natural statistic gaussian image q matrix generalize I nuisance label gmm direct figure depict gmm fig image location pixel patch pixel patch nuisance variable pixel image nuisance omit quite maps nuisance template speech recognition might represent speed amplitude alternatively part represent nuisance approach one sp nuisance choose likely inner product definition compute approach conventional ms ms mode nuisance image global configuration target nuisance image justify setting approach commonly sum max amount inner product template nonlinearity nuisance isotropic diagonal simplicity treatment generalize manner non quadratic depend please ms impose type assumption template pixel configuration see accord operation generally pool see fig explore model inference implement sum message b several recent normalization know come later batch principled implication unclear probabilistic assumption old arise image filter template template value template fully object template layer object pixel position likely ground air connect equally present invariance global template relative occur filter overcomplete filter convolution convolution factor pass nuisance image nuisance must two parent might undesirable force active formalize via image patch whereas patch switch inactive strong correlation neighbor must prevent realistic measurement sparsity extensively spike coding variable think nuisance q bias soft unit relu modern last line assume drop world abstraction hierarchical accelerate abstraction concept category inductive bias exposure informally light natural giving summarize abstraction order concept image face detail level abstraction specify identity face specify fine fine location pose e close without fine shape continue fine scale pixel intensity channel image refer become increasingly abstract detail preserve high level essential overall back conversely move concrete formalize process deep start abstraction object overall pose follow level intermediate detailed fully nuisance level abstract nuisance transformation fine abstract concrete bias clarity factor location hierarchy incremental form intermediate path bias covariance kind face location whereas direction generate abstraction function composition product factorize exponential parameter number enable hierarchical deep difference model detail example classify image deep iterate fine coarse bottom abstraction infer high section explanation clutter top essential hypothesis fine pass class top make use classifier message fine coarse abstraction infer coarse fine abstraction infer intermediate mention fine since template eq image diagonal affine classify omit clarity fourth max index channel intermediate care infer coarse pass suffice since relevant integrate eqs eq simplify come feedforward th layer yet softmax regression layer network miss level fact label probable configuration interpretation become clear need input feature model label discriminative train know since discriminate classifier learn define procedure generative discriminative employ relaxation obtain discriminative mathematically likelihood extra introduce old generative equality differ generation pass family conventional ensure line relax learn optimize classifier step e arrive discriminative comprise probability relaxed brain world transformation graphical introduce discriminative generating label instead imagine via classifier graphical parameter represent relaxation discuss invariance fine abstraction switching variable layer relu activation eqs discriminative conjugate training day train mini instead dataset light development sgd generative classifiers discriminative inference configuration softmax layer end activation detail interpret back discriminative batch principled motivation pass generative help misspecification issue slow require datum light connection new insight work answer question importantly fail see explore insight graph interpretation convolutional pooling layer operation apply generally architecture type neither hoc entirely two neural table knowledge generate template via affine thus examine exercise mathematically notably study computational also strong representation visual suggest brain factorize appearance search image maximize cat train million result image store maximize mathematically seek score class ic g ii activity individual activity patch pose activity patch fig nuisance pose ig key factor image formulation classifier factor nuisance architecture learn prediction classify use systematically variation location much information latent nuisance possess variation nuisance require evidence traditional distinction supervise unsupervised ill bad formulation capture nuisance parameter fig show believe notice early probably nuisance max serve purpose nuisance forest machine seem arise model vast array explanation understand quite successful segmentation task prominent use pose track microsoft medical wherein distinguish quite expert section assumption regard nuisance switch additive mutation nuisance category evolution heart derivation possess cast sum message pass series determine branch tree node question repeat reach leaf leave class posterior decision classify input send average evolutionary category start root template randomly template template mutation parent mutation parent child evolutionary template assume add course early distribution exponential family mutation irrelevant nuisance evolution individual abstract template sequence local nuisance many additive incremental intermediate evolutionary path mutation add template evolutionary evolutionary path start end leaf leaf specie template q additive last sum iteration layer go fine deep decision difference evolutionary underlie mapping single decision label histogram miss sec treat understand interpret configuration wherein world infer leaf inference critical gaussian finish need function internal analogous thus discriminative counterpart relate forest
surface subject optimal respectively cross subject via evaluate examine hold actual response hold voxel false discovery small panel correlation panel know interest separate examine canonical bold cca canonical identify weight accounting delay second voxel surface canonical subject movie frame low voxel module canonical correlation module integrate complex hyperparameter easily datum fmri response movie movie stimulus component voxel surface blue four voxel voxel colored variance overview initialize hyperparameter range cca two hold finally variance component fit movie response surface voxel colored interest outline mapping response accurately predict histogram correct bar component subject response movie stimulus canonical plot voxel negative indicate blue indicate four negative positive variance component colored rgb rgb false false false frame title cca valuable covariance cca cca similarity across fmri subject resample template introduce module implement regularization gaussian cca functional response across movie demonstrate set discover cca subject response analyse similarity covariance analysis method relationship decade variety field cca conjunction ica apply find subject mapping activity network subject technique call functional instead voxel alignment space equal dimensionality relate correlation input quantify package implement cca matlab cca toolbox analysis package either minimal seem active package option kernel cca cross validation cca fmri activity multidimensional cca onto basis theoretic mutual transformation zero maximize q pair subsequent analogously uncorrelate precede canonical less equal constraint reduce generalize q canonical extend multiple additional cca trick cca relationship cca project x k transformation invertible accomplish norm analogously partial least become cca ill cca adjust lagrangian mathematically cca pls maximization pls reformulate cca thought cca machine package include variety mining develop many cca cross pls module orthogonal exceed kernel cca module develop seem development access use keep library minimal module experience module organize class file main common implement object implementation visual representation create dataset cca dataset break first cca dataset procedure overfitte explore example include np var random np signal component var var var var var cca component cca train cca cca object cca initialize component retain use regularization canonical retain validate employ cca return canonical quantify correlate complete mapping dataset well correlation prediction actual canonical disk load disk use library save object share module object object initialize regularization point default retain integer default kernel cca boolean type regularization cutoff compute canonical weight hold prediction number boolean object parent cca fit dimensionality argument held dataset dataset dataset feature format compute cca mapping method accept file object load newly array default array canonical retain test integer default analysis get cutoff canonical weight hold default proportion select boolean value object include object differ validation range cross validation fold split hyperparameter large base repeat hyperparameter aggregate fit rest proceed analogously demonstrate software run fmri natural movie find canonical similarity fmri response cca mapping subject hold bold subject movie fmri collect day
likelihood estimate use denominator jacobian dimensionality strength sx classifier job ratio loose point make particle community towards concern value face nuisance associate generate concrete training balanced datum quadratic py px monotonic various surrogate lead discriminative monotonic function ratio must parametrize parameter nan alternate could desirable convenient training target input specify via classifier roughly flow generate expected loss influence g histogram approximate point point run generative demand particular context impractical instance human intervention classifier pre parametrized compute value thus concrete realization probability language imagine estimating depend dimensionality cost generative leave optimization problem future presence event describe uncertainty physics response associate search test generalize use discovery pseudo generate correspond correspond signal alternate hypothesis approximate one alternate advantage help classifier capacity relevant region perturbation I continuously signal htbp feed event evaluate presence nuisance thought typical usage machine systematic parametrize nuisance propagate uncertainty statistical improve parametrize correspond ratio outline distinction classifier work stochastic try uncertainty offer explicitly event intensive approach element intensive integral response handle even detector minute single conceptual detector offer evaluate initial practice measurement physical describe measurement mass particle cascade decay involve parameter nuisance coefficient detector learning associate pearson work access goal pearson generalize perhaps formal general validate classification metric parameter classifier different use discriminative make purpose calibration importance estimate decision naive various approach calibrate individual event calibration non provide calibrate take machine algorithm scalar achieve dimensional approximate generative available dimensionality distinction come classifier real know correct go outline perform repeat work correctly identify classifier approach eliminate approach parametrize jacobian factor parametrize evaluation likelihood avoid occur ideal technique high multiple univariate score parametrize simulator offer approximate bayesian free frequentist formalism strength separate quality calibration involve estimate difficulty calibration perform calibration depend dimensionality complexity result run simulator thank challenge early work project grant grateful science national york university e particle physics statistical evaluate likelihood function parametric central summarize information experiment need key area science result test interval simulator generative describe process measurement often impractical prior construct ratio datum calibration search particle simulator detector processing confidence interval likelihood test discovery single event feature process label typically simulator interpolation parametrize improve hundred search utilize supervise high physics library multi decision tree progress classification lead test long true composite parameter suboptimal high event event demonstrate intensive optimization motivation extend usage classifier nuisance scope result expand offer require parameter frequentist separate target discuss map aid statistical value density interested alternate lemma state powerful evaluate able density increasingly act instance high physics forward generalize prove ratio test form discriminative discriminative classifier generate extend result generalized ratio parametrize original e make accept reformulate per event assume vast exist generative predict bayes contrast classifier model threshold model explicitly learn familiar lead confusion current confusion traditional simulator motivated domain source confusion classification term ratio include lastly frequentist generative produce data mix monotonic desire target ratio test well think map point
sequence apply gram gram embed datum gram skip try maximize training constraint assume training center number representation softmax softmax sampling softmax negative utilize train size thus biological size qualitatively gram embedding dimensional van volume property protein well small calculate gram g metric score gram strength task protein total distinct family represent summation gram sequence select form negative machine classifier protein prediction sequence distinguish structured protein bank short set length protein neighbor distinguish typical binary positive aforementioned protein protein gram average protein short sequence maintain distinguish characteristic use release order sequence protein protein offer implication visualize different criterion mass van gram qualitative analyse neighbor diagram color accord gram group protein prove useful classification reveal bundle module mcp terminal protein tm tm ab tm tm tm type tm tm tm tm protein protein protein bioinformatic ability character protein sequence structure protein fraction set structured protein protein bank order visualize reduce histogram overlap gram occur structured protein exhibit histogram binary protein sequence gram protein shorter trivial case maintain obtain classified protein specificity present order region analysis size comparable c see visualization sequence reveal column comparison experimentally additionally versus order region classify accuracy respectively dense biological call sequence diverse meaningful physical chemical dense representation family show protein visualization providing interpretation discriminate sequence protein furthermore http edu visualization matlab request advantage embedding train encode biological deep application bioinformatics representation phase future aid wide problem prediction extraction identification block department california berkeley usa division berkeley national berkeley ca berkeley edu berkeley biological sequences method bioinformatics visualization protein identification protein protein use evaluate classifying protein family average obtain method addition predict protein database database region rich classifier distinguish protein bank protein embed diverse information regard pre train deep bioinformatics berkeley visualization use certain language biological sequence dna protein language language information conceptual analogy natural process nlp deep discover encode biological protein bioinformatics visualization protein structure interaction biological embed dimensional skip explain embed work database subsequently qualitatively evaluate classification protein average obtain family mapping sequence database database classify bank protein region advantage embed train sequence base tool datum available language processing http berkeley edu visualization become machine year main approach store establish inspire item fashion store partial description item store close generalize attribute nlp semantic syntactic embed define characterize neighboring word context consider amount via way architecture gram vector using skip gram vector representation show degree seek unique biological sequence interpretation skip gram train sequence sequence chemical purpose protein sequence bioinformatics prediction domain illustrate tackle protein protein protein relate protein similar structure gap motivated identification sequence classify study classify protein classifier structure sequence perform van volume secondary protein perform protein frequent exhibit result balance study biased protein secondary abundance role cell biology protein protein study characterization category region experimentally order release protein present nuclear mostly computationally identify interpretation work protein confirm furthermore
lemma result suboptimal arm suboptimal suboptimal arm gap corollary main difficulty dependency random gap layer arm bind number arm depend empirical gap allow suboptimal prove small fraction near conclude bernstein empirical probability reader define event arm strictly easy mean problem since number chain rule get arm select arm know simplify enough add small conclude assumption follow distribution arm reservoir budget denote arm union know large write define notice assumption follow constant depend large v imply implie cm rgb many chance try certain number design cumulative learner simple infinitely bandit arm either multiplicative extension several near small case horizon sequential decision number action make classical action choose among call choose round respect cumulative simple bandit small number decision extension infinitely many action impossible try practically face extremely large first already past whose sample challenge infinitely armed bandit respect bandit good arm sample arm order reasonably difficulty ask link selection principle subsample arm want chance sample good least tradeoff infinitely armed bandit problem reservoir learner work give optimal constant reservoir characterize specifically reservoir reservoir achieve limit factor arm rate arm sample reservoir bernoulli spirit distribution arm refine even factor prove low even sub case reservoir come tradeoff likely output therefore arm selection cumulative regret optimize problem want optimal select minimize finitely attract develop aim find arm arm ultimately infinitely bandit number arm available relevant biology even infinitely achieve regret infinitely armed extension concern replace hoeffding bernstein extension treat precision implement algorithm implementation simulation arm optimize cumulative multi armed infinitely many bandit many infinite class example bandit setting consider assume infinitely arm classic assume reward arm combine reward restrictive contrary arm armed call arm set arm already always learner time output assess simple regret p right arm reservoir arm bound distribution domain distribution hold also relax different imply arm assume distribution classical standard first infinitely bandit present bound corollary crucially depend depend algorithm problem regime result one characterized rate characterize regime selection tradeoff correspond reservoir put mass close good reservoir come parametric regime output close arm reservoir many arm arm exist regime rate one interesting difference cumulative e valid value simple fact exploitation everything explore reach cumulative practical implication examine base arm reservoir arm eq maximize ucb lead order divide logarithmic classic almost confidence regret infinitely bandit target cumulative regret number arm sample regret whereas interesting ucb specific specific number arm select ucb indeed decrease gap infinite infinitely arm useful refine update seem hand constant present easy state main characterize regret constant enough short sketch result control control mean arm high approximately time eq suboptimal arm suboptimal arm high since suboptimal arm time factor except consider specific result conjecture relevant either practical reason particular horizon exist corollary concern highlight present include theoretic general state near small e imply particular limitation simple modify bernstein display note general modify variance sample armed term refined variance thereby term proof conditioning immediately corollary arm bernstein minimax problem minimax sense proof immediately theorem rate therefore bernoulli discover valid particular distribution bernstein decay arm rate limitation general discuss limit never yet regret enough good interesting vary therefore index extreme exist tail hill estimate slightly prove arm directly estimate arm q concentration assumption bc assumption ht reservoir run knowledge note enough imply situation minimax round ucb use learnable reservoir go loose another question quickly trick double size away ucb air modifications straightforward algorithm simple modify air regret regime performance regularity different
get reduce approximate sequential monte carlo techniques employ expect gain use summation available filter b use update update particle particle particle particle markovian achieve draw accept proposal usual hasting probability material smc describe rely heavily fast accurate posterior filter might achieve development kind simulation nature space integration need repeat calculation subsection way infer third leibler approximation dimensionality simulation show supplementary material distribution accuracy numerical calculation extend technique detailed calculation find supplementary material hence mass poisson distribution analytical unimodal feature study depth equation filter available type integration likelihood subsection alternative normal rewrite normal sn u un multivariate laplace transform employ proof derive sharp laplace normal unique stay laplace equation scheme limit kullback calculation particle well approximated interval unimodal decay quantile normal derive pair focus poisson tt unconditional distribution state tail tail eq standard expect upper quantile tt likewise moderate true far low tail equation simulation practically quantile tt help calculation example template assess template compare methodology frequentist inference community sequential status example discuss work ucb sequential totally chance integrated intensity b gs template primarily illustration affect clarity presentation perform large template template methodology easily adapt use template k l calculate ten filter available gs smc bias gs gs apply case gs case motivation use template filter scenario model gp scenario prior correctly template matter gp emphasize template gp approximation instead integrated intensity differ intensity away truth compare prior region marginally outperform posterior adaptation lead template truth prove potential practical gaussian process choose optimize alone review work bayesian sequential experimental fundamentally optimum criterion seek aggregate bayesian design global bayesian nonparametric seek integrated mean paper sequential know paper aim understand target study literature intersection differ lot scientific formulation inferential even though literature measurement gp focus active conjugacy analytical approach capable deal conjugate poisson generalize nonparametric even also sequential neither criterion study inference portion none demonstrate nonparametric ed conjugacy nonparametric observational monte overcome computation log cox process exploit log emphasize mainly study ed employ sequential carlo easily nonparametric work topic design application theorem third gain equation p b b b b b p sn sn te te n sn ts sn multivariate normal sn un calculate fine j b log ie iw xy I w w k k w I I k w j w suppose know update e k k already w w j k k k w gauss quadrature km c g c e illustrative simulation fundamental approximate histogram path job approximated particle bind observation filter weight markovian move ny l ci p ig break axiom criterion lemma notation remark proof em energy galaxy able object average perspective employ accommodate combination know study conjugacy monte efficient tool volume distribution inference process normal simulation usefulness literature inferential technique general spectral log cox fitting branch characterize rely strong relative ease fit exist template object classify source geometrically weighted template template however cost observation necessary observational strategy template specification filter equivalent specify filter aforementione scientific specification datum role preferred oppose advantage motivate firstly good secondly find np hard armed sequential smc efficient calculation sequential difficulty come combination make gaussian conjugacy several make knowledge ed accordingly induce uncertainty goodness posterior serve primary proxy filter play finally address iii convex template nonparametric motivate study design ed statistical well ed
control cf depend show nf minor detail hellinger formulation proof inequality notice follow corollary lastly inequality conclude let vector result brevity j j conclude proof first bind argument arbitrary low proposition identity matrix q similar algebraic direct consequence property toeplitz strictly triangular imply regard simple integration combination conclude one algebra n q ultimately grant additionally cs partially nsf grant edu definition example lemma section claim macro format cd fix set domain arise likelihood minimax salient exploit capture covariance plug asymptotically contrast standard theory applicable class plug dense covariance change point observe detect shift temporal audio eeg health sciences advance algorithm asymptotic theory context exist work shift temporal statistically less structure modelling detect change collect base change normalize series california despite consideration researcher seek exploit temporal detection single contribution lead detection possible algorithm simplify process point observe sample moreover nb infinity one fundamentally framework arise distinguished distance gaussian process study asymptotic domain asymptotic suitable bound observation three dimension also adopt speech finance cast involve smooth pls pls current pls asymptotic procedure account procedure aforementione obviously ultimately suitable treatment first aware establish domain set second vast increase domain focus carry detection ignore suboptimal fix domain serve spatial spaced asymptotic notation perhaps work author gaussian ratio bound probability hold test statistic work direction wish chapter test variance sum random test continue admit ignore dependence level regard phenomenon x covariance negligible accounting test domain contribution paper suboptimal fix domain moreover detection new statistic increase domain encounter fix considerably challenging one need way point process increase domain show account analyze base upon ratio draw know dependence asymptotically increase domain setting hold structure class term determine spectral density decay fix confirm confirm paragraph covariance know address scenario plug method covariance covariance consistency regardless estimation plug situation plug inconsistent case contribution integrate exploit property absolutely toeplitz matrix classical norm either several beneficial detection focus one gaussian hypothesis spectral adapt plug subsection focus shift mean devote plug estimate serve optimality discuss numerical experiment assess section contain direction proof present auxiliary appendix inverse toeplitz useful stand minimum maximum operator length one matrix ij usual inner pm ij mf p fu fu fourier denote absolutely toeplitz nf na nb nb nb stand lastly gamma present datum account underlie assumption kk subsection mean symmetric denote toeplitz integrable definite exist depend satisfie hold bound origin closely link origin section turn essential statistic fairly explicit spectral regardless density class function close assumption rational real unit lead note since favorable easily assumption set mean zero endow toeplitz set datum observe symmetric toeplitz view accordingly denote fact symmetric f real universal scalar regard infimum necessary infinite polynomial state condition fourier common analysis toeplitz proceed change composite domain satisfie either fix setting restrict spectral admit increase domain moreover hypothesis alternative notation occurrence time nb denote change state tt nt union composite generalize process essential threshold depend false alarm also substantially exposition propose form unknown present unlike function explicitly account specifically asymptotic estimate propose approximate matrix indicate later let plug strictly consider domain consider subsequent section detection false detection take choice become seek result admit polynomially gaussian polynomially decay admit exist depend constant several comment role various choose apply appear shift tail notion standard observation implicitly challenge sample rate one hand appear closely nn possibility small shift detection parameter variance size analogous difference process chapter small lead shift smooth jump satisfy assumption label parameter jump contrary exponential class previously since great algebraic effort easily process regard super actually detect small jump quantify assertion smooth turning describe satisfy admit scalar comment exactly fixed set structure play factor gaussian arise encode r priori take plug approximate method definition way assess focus dependence parameter space base mle detection affect namely component fix domain grow detail zhang show consistently behind existence mutually absolutely model realization generic refer induced measure word algorithm equivalence grow shall exhibit performance fully whenever dense dense contrast estimate approximate proceed state non sequence small sup must remain speak weak condition sup norm estimation result condition imply denote horizontal covariance scenario base upon compact wu likelihood determine strictly speak verify imply correspondence space weakly consistent validity assumption sequence regard satisfying scalar depend c n q brevity drop vanish namely enough previously clear appear detection finite situation equivalence detection aspect rate plug plug disadvantage handling difficulty introduce isotropic explicit formula class parametrize example f associate rich efficient inverting significantly cost compare accelerate generalize approximate flexibility rest matrix eq broad condition decay give show satisfy du et show label proposition ready plug satisfy universal give theorem appear theorem universal covariance asymptotically plug plug big plug almost surely apart usage test detection early highlight accounting fix nan alternative hypothesis fix control mild hold covariance paper integrable jump increase least guarantee although detection error vanish verify early vanish jump vanish qualitative hypothesis shift mean deviation careful look reveal small standard bn nu test distinguish shift remark gap critical drastically change favor universal exhibit optimality study use compare especially thus increase preferred jump theorem plug theorem nearly density demonstrate minimax optimality domain spectral consider restrictive k generally speak contain spectral density condition qualitative class satisfy stand rational spectral admit salient qp although spectral consider due add density kn kn n result near optimality density study although establish near optimality plug spectral satisfy conjecture broad turning domain size shift unlike distinction assumption mention term universal test detection control simulation goal assess fix fix regime sample area receiver operate characteristic roc refer auc standard assess alarm curve roc curve pure line origin auc realistic compute auc repeat repeat correspond accord covariance shift shift curve roc curve k assess literature dense due estimate apply figure detection smoother impractical slightly auc figure plug polynomially decay compare detection panel rapid remarkably jump establish section study detect recall figure apparent plug covariance show choice furthermore detection robust estimate recalling method analogous rate panel decay exponentially decay panel function polynomially decay r diagonal satisfied case exhibit slightly gap auc curve polynomially decay presence regime panel display jump three dash exhibit auc structure plug nu right panel estimate panel display three curve green plug nu introduction detection plausible subject thorough probabilistic extend extended rigorously minimax detecting recently specifically scalar intensity detect observation optimal dimensional domain study behaviour paper valuable intuition minimax aside set detection proof main establish toeplitz proof interest algebraic derivation save follow unknown triangular addition brevity nu n z alarm non centrality lemma demonstrate eq suffice universal lower bind identity nd alternative show indeed algebra get arbitrarily note inequality obviously hold large identity verify basic yield inequality conclude proof tight w upper imply cf g obtained get universal universal n kn proceed manner precede cholesky refer gaussian observe n tend get
trajectory crowd small label least crowd try give reasonable small make positive example trajectory expert expert bad match auto encoder layer occur layer layer try build auto encoder corrupt corruption max tune negative stochastic mini prevent datum percentage neural network find infinitely trajectory across sec pre make fix cloud cloud grids size represent remove normalize trajectory preserve status trajectory g rotation status propose trajectory dynamic length match cumulative match strength maintain contribute cumulative length match index di ic local later complete programming di contribute cumulative match precede match encoding order rotation object normalize contribute e path give form tb crowd platform platform crowd tb cloud trajectory via object vary quality likely task successfully leave dash unlikely successfully line collect crowd crowd web platform see virtual web without expert presence user unseen manual web user cloud manual object start demonstrate fig select object use work like video bar show gray user click bar trajectory full expert similar object experience show cloud move try modify orientation pr position extra gray click minus remove occur bar update status open user click broad build point example crowd platform trajectory expert amazon completes ask complete object manual follow take raw microsoft fusion cloud object dataset model robot never see trajectory execute pr able turn right two trajectory pr model dataset test show dynamic mt first column manual consist mt mt intuitive percentage mt value survey find mt trajectory robot reasonable test fig trajectory axis sec locate orient differently successful dc slow successful difference turn switch object last transfer affect method find coordinate sensor allow tb learn node last cloud picture visualization cloud trajectory show select point cloud high activate input execute object fraction robot never correct planning trajectory correctly execute trajectory visual exactly object expert collect purpose compare crowd see believe crowd well crowd amazon vision art handle cloud expert pre object cloud even still extremely multiclass svm accuracy trajectory large outperform art give give access test lead time difficult modality design axis language act turn node act deep modality give noisy label handle give show crowd learn modality pr robot front language cloud trajectory cloud language robot trajectory controller successful pr project website introduce never see formulate structured output completely modalitie cloud trajectory deal crowd platform non expert deep baseline dataset crowd share learn engine acknowledgment building prototype website useful discussion microsoft award office point part highlight pt pt object human program planning object formulate plan structure handle three modality collect large test language far show robot never person visually read manual possible human vast experience differently shape generalize water figure build key object share similarly among completely even robot never machine able previously similarly fashion carry name rather shape completely robot cloud find appropriate trajectory experience consist distinct object part trajectory object part cloud color use object label center operate unobserved successfully trajectory previously cloud noisy black robot machine could execute environment variety object object alone rather rely understand cloud plan key modality cloud language crowd deep impact language architecture modality collect expert expensive presence robot work web platform outperform train expert previous entire oppose sequential state validate via web platform present key planning via object incorporation planning modality crowd dataset activity image use part however object activity reliably object many need direct focus detect part pour predict motion sequence perfect pour instance interaction track vision daily significant demonstrate trajectory via transfer sub sensor sensor trajectory consist status translation rotation origin r pr rotation instead euler angle trajectory trajectory translation linearly spherical trajectory transfer conceptually necessarily inconsistent make compatible new modifying rely object object challenge depend variation object degree commonly angle configuration need trajectory small orientation error execute modification robot position orientation object modification via cloud many object base individual part translation limit object lastly even different shape cloud even angle intrinsic shape part trajectory compatible trajectory space configuration principal frame rather task position orientation
capture decrease solve recurrence epoch resemble recurrence gradient incremental study sgd scenario massive grow dataset increasingly impractical discussion empirical necessity version competitive practice benefit type algorithm understand instead focus effect use face consist example pc generate normal spectrum finally point spectrum theorem vary rate particular slope log convergence graph ref thm exactly occur dotted line guide higher explain target fix rate dependence due spectrum plot second considerably despite pc variance good argument initialization result exponential remain incremental scheme result analogue calculation q extra lemma martingale n nm n version instance pick therefore draw chi freedom draw independently square one freedom characterization specifically second function finish unable reference suppose concavity apply deviation q use establish inequality length value eq q repeatedly shrink shrink n lemma reasoning careful finish pick summing yield lemma lemma hand expression expectation claim epoch yield bind theorem eq write q b recursively recurrence q finish definite thm thm thm draw wish eigenvector fashion maintain new give finite sample principal dimensionality reduction project top prohibitive update estimate eigenvector study elegant closely proportional point covariance paper estimator eigenvector eigenvalue treat q achieved point identical term recently lot descent convex non end system lie analyze initialize unit time receive next perform gradient adopt show progress behave time forward initialization average sensible fail coordinate far p pp ne x likewise initialize orthogonal remain avoid problem intrinsic random update ignore merely interested progress potential random initial normalization update likewise state final knowledge eigenvalue sigma outcome include x pick surface sphere initial rate epoch drop argue times epoch use argument careful specification denote space moreover consist include build martingale argument conditional step size time nest subset also proof yield line perspective generative matrix batch computationally intensive incremental bad case iteration purpose recent attempt inherent present pca mild detail sigma nu improvement measurable follow appendix identical henceforth quantity additive term monotonically decrease arbitrarily close bound away recall advance skip unit little bit much establish surface sphere start recurrence generating nonnegative ty show define derive
bias arbitrarily number ensemble sequence bias process sequence denote infinite process produce coin single come particular trial vanish however statistically mean another almost fluctuation process bias future bias information mn random variable mechanism drive random relationship excess quantifie correlation second quantifie past due fix ergodic mutual closely parameter estimation theoretic identity continuous differential entropy choose finite come divergence divergence cover ref parameter realization consist noisy fire spike train transition generate probability entropy x x hmms internal entropy generate hmms divergence h trial one might suggest multimodal normality posterior carry essentially rely log behave bandit process normality calculate normalization normality posterior generalize statement arm normality attention limit capture error distribution normality correction infinity long decrease entropy know recall ergodic immediately recover likelihood excess spin lattice finite divergence ergodic period alternative divergence language text empirical sec analyze asymptotic aim asymptotic normality recover aforementioned power analysis utilize ref recover storage require amount accurate inherently observer note arbitrarily excess statistical infinite still excess reach predict necessarily accurate agree ref proxy introduction choose within least choose important process store memory highly vanishing generalize bandit ergodic even trivially transition variable thus agree criterion look spin challenge yet theoretic ergodic look forward structural biology semantic human operate separate transition lead signature thank member upon u office nf nf sm student fellowship berkeley fellowship intuitive suggest truly complex condition familiar truly complex purely interaction allow spin lattice critical power law spin autocorrelation asymptotically configuration surprisingly dynamic spin ise lattice evolve lattice configuration spin stochastically past future imply finite spin familiar concrete near otherwise take possible give bound directly standard ise lattice bit excess entropy continuous global spin utility order likely maximize temporal phenomenon spin purely truly first contradict ise lattice interaction coupling strength iteration concatenation bandit observe logarithmic fig node cm bend auto every style draw loop style leave style style loop loop style right loop leave loop loop loop algorithm color green infinite build process familiar known multi bandit theoretic understanding highlight distinct divergence ergodic draw consequence resource divergence structural hierarchy truly many biological phenomenon measurement neural language failure transmission grid apparent particularly challenging reflect resource store parameter memory require resource analog mechanic suggest divergence since resource divergence sensitive inherent organization uniquely indicator date tractable construction relationship construction class repeat vary stochastically trial trial stationary memory decay many trial past future bandit answer remarkably process memory mechanism select insight derivation structural universal property unique present estimation divergence past divergence divergence bandit process structural principle learn view truly simplified introduce ergodic process theoretic approach review alternative construct structural statistical approach highlight discussion nest organization relationship hand difficulty sample predict attempt express persistent consequence closely resource specifically minimal reconstruct series randomness description complementary resource logical length series irreducible randomness though model fortunately
bootstrap avoid analytically use highly favorable analytically example scalability complexity bootstrap bag little fast distribute computation advance digital technology lead phone health inferential crucial correctness hypothesis traditional inferential storage massive architecture massive methodology massive expensive addition assign computationally conventional assign uncertainty deviation commonly apply two obvious computationally impractical volume point process advanced computing massive even variant subsample problematic output bag make massive datum massive subsample module bag massive store moreover subsample module process compute construct assign weight massive bootstrap yet number bootstrap sample expensive commonly modern estimator demand numerically primitive ls original statistically bootstrappe massive introduce low robust bootstrap possess scalability significantly low subset robust avoid point bootstrap possess conventional scalable system consistently accurate preliminary present conference review bootstrap implementation consistency new section big idea process store estimate g confidence interval etc great confidence often informative plain estimate little scalable disjoint bb resample replacement assigning subsample computed population within subsample module module produce effort computation subsample nevertheless thousand impractical even complexity modern maximum likelihood estimator primitive statistically robust l face even sufficient outlier fp equation dependent value bootstrap conventional accurately reflect correction need n statistically compatible distribute system massive many pca combine desirable method smooth fp bag little scalability computational complexity burden estimate drastically replication bag let subsample randomly replacement equivalently random weight b distinct subsample bag compute distinct low allow fast computation confidence modern robust draw form module bb subsample generate form assign weight initial solve compute uncertainty disjoint datum set subsample module estimate uncertainty average subsample order statistically robust random robust continuously widely subscript different estimator iterative iteration obtain turn need present fp scalable step obtain modify follow let subsample b replication statistical bag observe outcome denote dirac n subsample bootstrap b side class pf fp appendix quantile word pr q robustness break bag b upper minimum proportion subsample sufficient proof n general estimate accord reliable sample quantile one bag former latter b explanatory quantile close quantile sample draw quantile efficiency significantly estimator important setting purpose e variate variance scheme efficiency original bootstrap simulation compare side setting right side draw subsample performing step e distribution well bad average along uncertainty see element n r performance assess estimator ls setup subsample maximum module start add subsample module bootstrap sample illustrate contaminate robustness setting accord theorem sufficient bind choose multiply resemble world lack accord setting upper estimate multiply still proportion severe lack robustness face contaminate make comparison deviation compute cumulative report relative error time remarkably
bound entail old regularity know entail regret optimal optimal nonparametric run average net note regret indeed regret entail well explain technique build perform simultaneous cm scale aggregation exponentially forecaster competitive instance extension competitive increment increment scale gradient core lie use already present argument scale unclear besides spirit square online contrary build use discretization g linearization aggregation consideration suboptimal exponentially forecaster address lipschitz slow multi crucial linearization type section design efficient old knowledge concrete proof endow sup small net subroutine extension algorithm extension minimize loss simplex simultaneous convexity function jointly k algorithm derivative scalar variable map sup vector value assume upper partial tune bounded cm sake prove bind consist main aggregation level scale cm proper net kk cm definition follow predictor tu define exactly vector output forecaster apply j weighted forecaster define initialization j n n n predict ty low k high new weight vector exponentially forecaster average type forecaster tune introduction pp exponentially forecaster tune I address associate corollary sparse high dimensional spirit see yield could slightly regret slightly spirit omit know advance forecaster adaptive prediction modification regret multiplicative constant know advance adaptation tune without advance even forecaster useful small regret also assume forecast round measure exp convex loss bound rr depend loss quantile regression aggregation replace proof level cm explain tuple forecaster assumption cm differentiable convex norm k ki value ix f ff inequality jx b inequality substituting entail b claim apply cm infimum dirac monotonicity substitute tx jx tt convex intermediate weighted forecaster tune since intermediate prediction square concave infimum g set f f eq obtain expand fx conclude exponentially forecaster exploit net complexity exponentially actually class sufficiently enable adapt technique quasi net easier exploit algorithmic viewpoint old class introduction forecaster net viewpoint quasi regret see fix role approximate piecewise constant follow quite ai construct net net dyadic discretization note every partition mf cm restrict cm see continuously fortunately combination n two dot line function maintain exponentially dyadic parallel round combination aggregation simultaneously u u cm nj aggregate tune e every make ax assume dyadic dyadic tractable fall round overall factor complexity tractable round polynomial tailor future process reader g g subgaussian I z lemma close technique maxima formally approximation k f provide adaptive horizon advance basically vary rate positive integer k k kt jointly convex loss multi jointly variable bound eq k correspond exponentially forecaster page expert therefore well e note lie need upper conclude proof appendix sequel exist norm continuous denote introduction forecaster modify net viewpoint lead section special function consist net discretization let play fact discretization fix constant final define set define ai constant cf net net net discretization partition note partition refine dyadic set center consecutive level denote replace definition play refine partition take look piecewise polynomial value replace let f tractable precisely algorithm parallel round fall multi aggregation jj simultaneously instance loss define convex competitive j exponentially average forecaster q prediction ax nest h logarithmic factor partition nest partition nest follow proof defer appendix assumption theorem nest h complexity omit paper classical linearization splitting gradient eq last vertex side sequence correspond exactly weight vector output weight page g eq last conclude net lemma x c c fx ai aa f fx argue fx x ai induction lipschitz figure illustration indeed ax ni na f ai ax derivative width setting choice conclude p q mx fx mx f cp fx nx I ni x b triangle choice n explain incur small cumulative regret inside previous fix time new fall multi perform low thus follow two aggregation start one apply forecaster theorem appendix instead norm gradient mx
make threshold analogous centralized condition thresholde nk n nk nk pn nk nk nk nk centralized nk nk shot procedure average study dual regularize form evaluate estimator cost central l server average back average form store form round remain machine solve scale row jx estimator l k ty l l j j nk ty show subgaussian converge usual subgaussian show match centralize subgaussian plug subtracting take similar together grow grow machine average estimator comparable exceed threshold grow machine exceed term sparse dimensional setting first estimator machine communication bound communication risk regression impose sparsity mean bit need bit among algorithm amount far establish communication average machine generalize coherence subgaussian thus bernstein ns obtain desire union component conclusion lemma express norm subgaussian subgaussian recognize nx variable constant simplify union subgaussian subgaussian subgaussian recognize j proposition simplify union lemma remark lee equally college devise one shoot key dataset machine modern dataset distribute arise work fit multiple machine computational bottleneck processor shoot highly round pose challenge design shot popular datum locally master estimator produce average multinomial non average centralized machine stochastic sgd subset dataset thing centralize recently study erm mse erm erm match centralize erm optimality setting number set average erm order centralize erm erm suboptimal centralize erm work generalize risk minimization regularize minimization rely beta min strength centralize rely aforementioned min ensure recover square restrict machine desire contrast average work correlate make study divide hypothesis devise average centralized idea estimator thresholding average nk nk nk total machine aforementione minimax optimal factor lasso shoot algorithm centralize average subgaussian subgaussian seek regression norm regularize signal processing develop say lasso nearly solution support bias proportion order gain nothing average lasso formal bias estimator lasso ordinary ol coefficient correct incur shrinkage term incur previously refer term depend play suggest form solution proportional variance refer generalize let q keep feasible subgaussian feasible subgaussian subgaussian occur see lasso decay suitable require condition positive direction relate call replace right side gaussian zero constant extended subgaussian design covariate subgaussian subgaussian q occur probability consistent suitable intuitively large dominate empirical part typically subgaussian event occur probability practice lasso estimator condition lasso give convergence rate bias small constant incoherence occur plug subtracting set old generalize occur piece decay fast decay comparison subgaussian variable variance nm
universe output approximate private release natural problem complexity minimum number differentially pac give privacy sample pac al suffice class possibility private private pure function universe vc properly privacy improper class pac differential sample complexity approximate differential privacy et show properly I threshold learner complexity proper pac privacy properly datum universe extend threshold threshold complexity concept vc totally properly privacy interesting characterize proper private learner learner differential leave possibility improper pac privacy sample question present improper point pure privacy infinite domain et point privacy grow give mechanism domain mechanism inherent countable improper pac learner function pure privacy release analyze sample totally universe every totally order universe four privacy problem release interior query release threshold kolmogorov distance proper pac thus bound release proper prove interior interior universe privacy h p database I simply output point hence universe size hard database reduction interior differentially mechanism hope construct reason failure always output distribution similar mechanism extremely generally feasible evidence computational hardness al class polynomial hard al proper universe concept need private continuous show class learner pure characterization private sample necessary sufficient subsequently equivalence representation way equivalence pure learner private al show boost query guarantee answer query showed transform private learner error learning minimization denote order domain differential early show release noise answer random variable say function laplace let sensitivity add preserve differential algorithm access differentially private mechanism overall privacy guarantee differentially private differentially private second argument close query solve complexity size every interior least monotonically construct differentially sample element set differentially private agree private private inductive depend fix define positive differentially private appendix combinatorial call interior code recently differential induction claim database differentially private mechanism claim construct follow ny sn ny bn sake contradiction differentially private interior solve dd sn dy iy agree private pair adjacent database eq argue great agree first digits interior point succeed interior agree digit except database fix randomness st everything else fix except union give desire contradiction introduce q induction get upper point bind mechanism low set proof guarantee solve totally order interior interior idea paper full solving provide goal pair element length recursion number several exclude least one pair agree agree random agree scene element agree element follow element agree two one good use technique recursive finite totally differentially private probability construct database small universe every element construction present tool primitive denote finite database universe define domain finite database maximize mechanism solve specifically exponential differentially sensitivity tn differentially private building differentially define domain database sensitivity growth without function choosing mechanism differentially private solution approximately instead set gs lemma utility mechanism slightly result bound function choose mechanism execute growth quality contain present privacy kn choose l start execution database recursive database observation recursive call input recursive call database motivate value agree element common pair stability randomly bad distance leave side place element approximation database recursion least guarantee w help identify induction call e mechanism output suffice recursive call database element choose pair close except continue database inductive pair agree least agree element thus mechanism exist mechanism satisfie case agree condition good output hence analyze hence big fail appropriate execute call differentially induction recursive call denote differentially perform call database call consist denote database similar database probability recursion private preserve argument formal first one database exist permutation composition desire whenever database execute differentially mechanism composition get close gap roughly interior recursive limited affect privacy preserve assume preserve privacy grow exponentially result low hand recursion pair new ensure change limited input affect element element pair pair twice database carefully dependency think pt pt common twice every database acceptable still begin input ensure close agree identify randomly element pair change every rapidly interior learn kolmogorov proper pac translate bound version write row query release private approximate answer count simultaneously query query universe answer output database interested release counting interested relate release universe pmf qx qx count totally cdf totally domain f distribution kolmogorov totally order cdf release collection counting closely release empirical distribution theory query sufficiently approximately agree answer privacy consideration privacy incur actually improve privacy computation large offset result mechanism differentially private database row let replacement row answer qx return accurate appear differentially differentially private algorithm operating database replacement row run result private adjacent database index sample index subset consist since observe privacy e n nn n concept take concept example unknown probability accord unknown output precisely respect target hx cx error pac pac class target distribution draw coin otherwise improper hypothesis statistical necessary concept suffice properly agree recall totally threshold sample differentially complexity differentially private require database database include correspond concept distribution threshold measure later analogously similar learner cx cx take without error every database case learner privacy every concept every differentially bound complexity next complexity size fix denote contain add concept exist marginal distribution consistent must hence consistent privacy show differential privacy multiplicative complexity properly error totally domain differentially solve interior point error private accurate pac learner pac learner differentially problem complexity solve interior every argument learn totally order domain differentially private solve interior differentially private properly threshold differentially private differentially private solve private interior threshold privacy hence apply differential reverse direction size change hence preserve proper totally yx suffice chernoff probability output consistent concept generic free uniform differentially pac learner concept differentially private proper result empirical database draw sample subsample replacement lemma complexity private learner extend general release private item bit dependence much even negligible separation sample complexity private private vc dimensional threshold vc obtain class vc totally concept differentially private learner hardness proper concept different concept datum universe consist universe element database example fail hypothesis differentially proper complexity note element concept evaluate justify necessity iff complexity point evaluate use sample identical show require complexity element class proof toward contradiction differentially proper use essentially output hypothesis consistent example contain embed axis axis axis element evaluate ni observe private limited entry differentially consider execution correctly generate observe bad random axis axis axis axis output contradict hardness find threshold universe algorithm fail algorithm query release private approximated differentially private answer prove query release predicate differentially private differentially private pac sample differentially private argument incorporate contradiction query complexity contradiction construct apply database answer every cd nc cd prove reduction database specifically totally order domain maximum every differentially mechanism relaxation interior require technique bound note ask work domain construct distribution every private mechanism solve generality totally domain contain infinite take mechanism increase bound unlikely note problem mechanism problem fix sake contradiction bound solve interval universe tt bind idea learn point evaluate packing privacy concept whether countable length resolve impossible even infinite privacy countable countable collection hypothesis differentially private pac point finite proper however consequence clarity loss suppose sake learner countable subset hypothesis establish sequence packing constraint infinitely disjoint construct wish hypothesis anonymous helpful suggestion guide point reference privacy mechanism neighboring database r ss two every fact imply hold fact output r must follow output gs k complete analysis choosing mechanism define fail solution event choose f gs get mechanism code address digital content piece digital content copy content user produce copy hide copy uniquely informally produce still provide certain traditionally assumption require show code roughly speak work nontrivial accuracy algorithm satisfie mean back differentially solve prove object traditional lie order interior follow order randomized codebook symbol coin adversary subset say security completeness completeness error probability take could code use interior produce copy existence interior lower solve completeness differentially algorithm solve database codebook replace differentially private eq construct interior idea allow interior every interior domain user domain completeness perfect suppose behavior user nx sn nx sn digit every codebook codebook maximum digits agree agree digits agree codebook security check completeness consider code produce output index agree prove perfect prove let lemma somewhat reduction show interior enable accurately release idea reduction differentially private quantile input release strategy tree generate leave node leaf sample path sort database block value quantile final differentially formally describe differentially mechanism interior succeed database include empty sort power n dd rd differential release let noise partition partition accord differ block cf moreover noise vector sample produce succeed suffice execution succeed give section theorem remark fellowship support technology information security fellowship computer differentially private threshold evaluate otherwise first differential privacy impossible require grow technique apply properly pac learning threshold differential bind separation concept differential properly extend direction small construction bound differential privacy pac threshold differential aim analysis privacy sensitive individual privacy differentially private introduction effect individual differentially private nevertheless rich many compatible privacy individual infinity still asymptotic vanish get
deep boltzmann loop train architecture stack let construct factorize distribution belong parametrized distribution evaluate form normalization dy fy dy maximize reach maximum implicitly find decompose divergence pz obtain low analogous seem important conceptual divergence give training quantify train bind remove normalization problem combinatorial bind follow direction see subject argue beneficial light formulate model let train close bind bp original prefer maximize minimized decomposition bp approach optimal intuitively qx complexity k wide range wide concentrate sigmoid bx ba sigmoid important equation evaluate eq estimator update reweighte derive gradient supplement p concern fully observe final determined computing normalize weighted individual basically optimize contain random proposal sample use proposal algorithm proposal candidate resample accord resample resampling evaluate relative end procedure resample l k l kp k draw sample distribution proposal chance cover weight mixture include equip distribution straightforward initialize p gibbs multiple chain converge bottom fix approximately reconstruction provide map give estimate conservative probably might likelihood experimental would normalization equation select experimental result various discuss competitive description initialize implementation available http uci repository mnist dataset translate robust distribution describe method gmm importance general repository summarize offset conservative generative evaluate c connect dna web auto ar gmm lower assume unknown accord train otherwise converge epoch importance report unknown reasonable digits digit sharp biased one obvious correctly variability assign probability seem one source variability gray short stroke gray pixel variability within one would propose detailed failure learn relatively ability highlight digits reconstruct partially layer proposal goal formulate stay weight whenever approximate inference weight occur quality roughly symmetric control plot p mechanism totally proposal visually indistinguishable show supplement show sample generate database pixel gray bernoulli proceed rapidly epoch mostly learn face epoch sample variable hide hyperparameter generative automatically inference derive likelihood deep generative multiple layer force model different approach solely train something future serious attempt normalization know would enable tight bound test report differ mnist attempt could certainly make apply involve nature generative always direct might make suitable task observe least wide range choice parametrize assume eventually suit training acknowledgment uci experiment hide letter web sgd research university unsupervise challenge problem dimensional auxiliary help fit start run
il operator backtrack implicit fix yielding implement use minimization matlab implementation bfgs mx dynamic consistent prediction figure bethe root bethe hessian construction cluster available direction replace pass type computation theoretically transition lead receive european research european fp agreement department physics sup paris france universit paris paris paris task application aspect problem reliably performance reconstruction propose completion bethe hessian negative eigenvalue bethe discrepancy estimate matrix reveal analyze random statistical mechanic neural efficiently matrix depict root square empirically compare exist infer entry motivate collaborative observe widely question complete assume address reveal question motivated generic detect reasonable expect existence impossible achievable root square estimating unknown provide rmse call rank eigenvalue bethe completion rmse exist contribution construction via spectral method use part spin phase transition fraction element call observe reveal reconstruct difficulty reveal entry per shall rank iid algorithm parametric analysis completion algorithmic associate programming low considerable completion entry rmse empirically achieve regime proceed observed reveal reveal entry resp singular decomposition keep ratio consecutive minimum discrepancy initial first improve replace different minimization detect community spectral traditional spurious singular show backtrack bethe spectral reliable inference performance completion analyze spin mechanic transition rank unable completion see optimization careful adjacency refer bethe hessian q parameter neighbor assume center numerically solve build bethe resp rank function alternatively negative bethe possible backtrack weighted spectrum bethe backtracking next motivate wise infer illustrate bethe justify compose remain positive belief size bar convenience eigenvalue bethe bethe free increase shift eventually merge increase plot uninformative region motivate graphical perspective generalize bipartite bethe problem minimize call bethe read degree model study decade shall well bethe energy energy initial expect correct critical mark appearance spurious minima bethe approach bethe detect retrieval look hessian bethe hessian involve vanish involve derivative remain bethe hessian picture motivate backtrack equation bethe bethe mathematically simple handle use statistical mechanic rigorous argument investigate phase transition method mechanic bethe hessian repeat computation cavity interested mechanic phase vector sensitivity random perturbation existence condition exist spurious condition meet bethe critical implicitly population remarkably population compute suggest matrix become simplify stress regime decay simple expression limit
variant hilbert embed markov hmm predictive exploit future reformulate stage instrumental hmm instrumental identify use dynamical reduce dynamical auto determine method similarity encode distribution next three learn approach distinguish work limit observable uniquely state handle choose window instrumental noise window estimate system rather multiple whereas regression establish consistency regression perfectly hand convergence section main theoretical instrumental true regardless regressor triplet input instrumental equally well successive convergence rate possibly estimate ensure invertible future mean future closely think main quantify regression independent regression satisfy q proposition hilbert schmidt hold test result proof go theorem generic main regression estimate operator estimate exact show use vanish middle replace eigenvalue completeness example bind estimation g rate regression generalize next address functional embed regularization error accommodate dynamical define assume dynamical stable get perform well onto x sense since prediction demonstrate learn specifically limited feature pick history window reduce consistency attempt student interactive computer question student learn student correctly answer question incorrectly transition summarize observation solid horizontal maintain represent answer student publicly call generate geometry student knowledge typical attempt correct iff student answer try student constitute discard sequence length begin handle history window regression important sample observation training regression incorrect restrict binary correct incorrect reasonable state observable predictive indicator denote statistic conditioning simple result hmm fact hmm incorporate knowledge intuition unlikely event aggregate observation indicator optimal must learn exponentially parameter predictor easy window length train logistic advantage paragraph need predictor near approach logistic close variant two split train split error split depict accuracy turn outperform expansion increase non work supervise propose stage stage history future estimate successful system history identify latent exponential increase scenario would like extend framework dynamical instrumental framework I estimate start restrictive invertible zero least equation learn specifie easily observable eq constant move realistic dynamic span hmm rp rp p replace regression sake detail rank decomposition rr operator use replace estimate b b tb bb possible history column represent possible give eq extend observation follow enough singular invertible match instrumental steady case gain specify marginalization instrumental regression regression action variant reproduce space operator possible implement non parametric smooth depend produce arbitrary conditional weight expectation action application kernel regression produce combination training operator shift future condition stable state regression estimation regularization bernstein state surely q eq q recall test error perturb version covariance operator effect regularization characterize effect text define basically capture addition sample regression important observable quantity depend respectively quantity effect covariance positive assume surely let union set suffice remain suitable similarly argument term case test regression define eq unit invertible operator triangular error assume ab v I rest triangular within x x cd geometric harmonic eq cox allow simply union instance projection proposition remark edu cs edu cs substantial interest state dynamical system algorithm tradeoff speed despite predictive sometimes practice contrast literature belief predictive restrict linear view instrumental dynamical learner simply effectiveness propose non linear regression outperform correctness substantial belief observable could invert often intractable sometimes invertible replace th seek moment discover tool expand consider remove difficult hmm expand observable brevity call offer computational statistical hard state prior structure dynamical remove directly derive analyze implement require difficult discover average track algorithm well fact interpret instrumental dynamical supervise additional ability arbitrary regression problem
hierarchical exactly model dispersion positive relation independent bound straightforward unweighted weighted semi scale size bind unweighted semi insensitive subscript inversion identity employ lemma method estimate quantity proposition group weight positive definite define invertible proceed statement contradict nonzero full hold positive semidefinite far one show suffice converse exist nonzero norm force moment force invertible imply additionally together follow force support size show markov imply final assumption q imply probability set choice weight sense equivalence efficiency effect assumption force imply exist neighbourhood asymptotic heuristic neighbourhood q write let identity series lemma three establish concentration bound near estimate asymptotic infinity certain weighted estimator consistent share estimate bounding constant bound assumption unweighted away infinity bounding constant asymptotic result define eq similarly asymptotic go typically condition go unweighted weight show thus necessary away infinity proposition establish consistent force force immediately proposition let show efficient choice bound assumption set sequence condition two result estimate multivariate random identity er device show quantity converge follow lyapunov ensure converge normal cauchy schwarz thus standard normal proposition force zero normal moment estimator counterpart study hierarchical regression describe logistic simulation similar behavior group replicate replicate draw wishart freedom splitting evenly replicate draw exponential point draw multinomial proportional equivalent drawing give rise group effect zero draw variate variety population empirical base estimation moment two programming language procedure laplace implement splitting estimation split combine estimate average implement quasi likelihood package iteratively fitting procedure maximize detail intensive loop outer serial tuning evaluate replicate indicate error visible moderate still appear panel method likelihood factor without include validation substantial improvement computational utility estimator recommender specifically user movie moment hierarchical minute serial hour section preference user rating recommender user population meaningful coefficient specific available rating datum rating movie star rating star movie rating relate rating use effect let specific plus indicate star set encodes movie rate movie reduce list covariate assign score score category list category zero predictor motivated intuition recommender popularity movie rating measure whether capture user overall predictor depend past regression rating order treat vector ordinary likelihood specific action child rate robust logit popularity rate movie movie recent review movie movie review rating rating assume moment rating compute approximate come normal marginal marginal approximately elliptical evenly spaced normally part bivariate look coefficient look association estimate follow affinity intercept tendency user action child movie user like movie action tend prefer movie movie prefer popular movie tend preference allow diversity encode regression also primary system rating compete ability strength obtain generalize vector linear likelihood model fit moment maximum randomly review set test fit test user aggregate average indicate error global flexibility model outperform estimator general hierarchical model unlike proposal predictor appeal large rely assumption distributional likelihood estimator procedure mild asymptotically fix effect vector linear hierarchical group size theoretical apply theoretical condition ask handle hierarchical hierarchy feasible implement derive guarantee care obvious proceed crucially conditionally coefficient likely impossible application datum item popularity perfect solution fall within simple predictor context normally datum volume continue advantageous computational primary concern demonstrate keep gain improvement speed author anonymous reference suggest heuristic minimax consideration show weight use eq make unbiased minimize minimize let denote square must lagrangian multipli eq like measure generality eq ideally minimize practice find instead risk attempt find gradient constraint solve situation computationally expensive hierarchical simplification weight motivated correspondence practical use semi consideration replace invertible define first case compact value identity force sum write matrix cauchy schwarz event least standard markov event triangle bind define replace imply identity force note force assumption lemma define follow identity along inequality give definite semidefinite exist case positive last q let eq markov follow define imply fixed pointwise similarly cauchy lemma right triangle put lemma markov normal write fix maximize depend gradient randomly th rate value appropriately recommendation effect effect simulation th well iteration perform simulation linear regression logistic gain reduce statistical method competitive perform four loss simulate sample choice replicate replicate generate effect covariance effect wishart degree freedom draw predictor I compare moment weight choose implement maximization programming language free likelihood package splitting split compute separate combine implement intensive c serial time random effect loss replicate base slightly method consistent method method range size panel computation simulation likelihood procedure procedure clearly fast follow exact consideration gain reduce term statistical efficiency competitive exact heterogeneous strength across likelihood hierarchical computational propose hierarchical consistent recommender application compare standard method hour minute multiple sub exhibit city period member ignore independent account observation specific social reference book describe detail explicitly hold accounting variability hierarchical second well prediction latter relate recommender system user item user preference specific recommender mixed item specific collaborative base preference similar user many recommender system iterative high letting denote effect likelihood maximization profile likelihood computation cost computational proportional substantial sparse exploit structure impose constraint estimate situation processor estimate cost cost processor reduce likelihood criterion descent series lower fit report propose extend population parameter exist alternative locality dominant procedure across computation factor demonstrate moment amount likelihood stochastic gradient descent propose first mix implement example cost dominant time validation time notably split running moment procedure improvement regime trade introduce detail exist procedure asymptotic normality simulation method additional support lemma generate individually jointly goal specifically associate predictor vector dimension let predictor dimension effect expectation population far within group give lastly identically distribute assume region random exploit root typically generalize non relation hierarchical linear response relation q variance dispersion get group get hierarchical inferential formally effect effect vector response computationally fitting restrict denote density maximize likelihood maximization newton iteration optimization use technique describe expectation quasi profile software maximize likelihood negligible additional effort construction unbiased introduce first value denote notational convention matrix multiplication invertible subspace symmetric relation concatenation matrix matrix due negligible practice handle cone semidefinite employ similar modification continuous continuous consistent exist allow rank estimator approach estimate consistency minimize weight discuss practical alternative option call unweighted correspond second call option call two semi set repeat prefer unweighted variance group unweighte big specific much conditional weighted light prefer show effect moment base estimation nonlinear moment exactly relation relative moment theoretical approximation
velocity buffer buffer buffer could also arithmetic buffer save store compare store bit integer point save fraction miss backward much net hyperparameter optimize rapidly train run allow hyperparameter advantage show several scheme would previously impractical net employ heuristic hyperparameter validation schedule choice intuition argument objective directly jointly rate neural separately sgd mean schedule initialization seed seed initialize mini batch enforce stop rather optimize schedule optimize deep neural network choose layer choose several optimization include sgd minibatch conjugate section momentum meta size cc elementary meta meta elementary demonstrate schedule average seed iteration optimize bias initialization scale bias initialization line total subsequent interestingly first say penalty neural role hyperparameter network improve individual neural simple see neural network layer mean might relatively hyperparameter scheme automatic determination view gradient transformation gradient objective augmentation procedure show example optimize label respectively label light class difficult remain sgd treat vector distinction rate exactly reduce elementary optimize come adapt domain recurrent neural net net think architecture weight hard architecture illustrate pixel character character alphabet character learn separate alphabet net generic like filter maintain specific alphabet weight absence quadratic weight weight infeasible implicitly build structure penalty net alphabet three diagonal matrix correspond diagonal level similarity character character distinguish character constitute learn correlation low partially equally share interestingly share input weight separate tie pdf pdf pdf pdf low force weight hyperparameter automatic ad software package development provide access internal automatic containing loop code back later engineering difficulty address practical explore issue learn dependency depend hyperparameter thing elementary depend network thus issue sometimes make elementary learning induce gradient uninformative term relate gradient illustrate phenomenon layer learn high gradient become uninformative maintain rate approach minimum problem learning rate relatively stop meta magnitude meta gradient optimize limitation overfitte objective I rough guide hyperparameter hyperparameter affect regularization discrete manner choose closely derive l bfgs update hyperparameter crf I available exactly contrast exact gradient hyperparameter learn converge svm loss weight tight optimize selection likelihood gradient hamiltonian several reversible memory dramatically approach could base optimization could incorporate computation require much elementary gradient chain mostly small update parameter dynamic memory elementary evaluation long trick paper derive procedure compute momentum approximate drastically reduce requirement hundred gradient validation something infeasible allow automatic tuning training tuning detailed regularization neural acknowledgment thank helpful discussion device advanced institute technology reverse individual derivative input output reverse mode differ work reverse opposite final nest scalar scenario reverse clear imagine intermediate mode scalar multiply vector jacobian general yes case multiplication sparse jacobian reverse intermediate maintain drastically reduce requirement reverse gradient exact validation gradient thousand include momentum initialization parameterize exactly descent momentum machine system penalty specify size initialization choose crucial hyperparameter selection gradient demonstrate automatic performance optimize gradient mode allow cost elementary hyperparameter gradient compute inner loop elementary I reverse describe technical gradient eliminate high elementary parameterization give flexibility explore hyperparameter back elementary hyperparameter momentum exactly gradient descent momentum continuous momentum reduce gradient allow thousand hyperparameter optimize fine initialization preprocessing insight optimize asset backpropagation allow compute backward evaluating loss obtain either mode force difference would make entirely infeasible hyperparameter however I approach sized maintain reverse division multiplication bit concern carry reverse require repeat multiplication learn procedure end usually unfortunately reversible set ideally initialization hope inversion another move analogy dynamic analogous generate
visual inspection designing recurrence analyze system build adjacency recurrence interpret network extend recurrence paradigm classification use compression distance another way build adjacency extract different unclear topological relate representation image angular transition apply convolutional neural classify previous inspire rescale auto da imputation mse test learn cnns da explain introduce framework encode angular coordinate actually cosine summation angle duality complex framework bin encode real rescale series value angular radius factor span angular span water encoding map monotonic time coordinate inverse second opposed coordinate preserve future rescale angular cosine angular discuss angular angular rescaled accurate transform angular interval angular angular follow transform time define inner type angular preserve dependency increase position top interval angular reconstruct level feature deep trend illustrate extend markov sequentially given identify quantile bin construct count quantile chain frequency quantile quantile dependency step demonstrate get loss overcome divide magnitude quantile bin bins temporal denote axis matrix position quantile quantile actually encode probability interval illustrate special capture probability quantile overlap aggregate subsequence encode nn fast raw control n trace cnns classify signal processing pre facilitate compare classification publish compete window bag space classifier recurrence pattern symbolic svm bag convolutional multiple map parameterize shared producing cnn learn overcomplete sake please details cnns illustrated bins window time bin enable construct small discretize quantile construct size kernel size quantile cnn soft cross image cnn penalty factor finally classify test use prefer help selection provide without cnns compound sake generally overfitte generally high note rescale time series map image mapping image mapping show later rescale come hand image variation signal dynamic recognition pixel encodes static depict channel image e channel static embed classification tune compound classifier competitive state time series previously mention function series uncertainty among come ambiguity precisely predict missing series break manually noise randomly transform data auto encoder note add break help last break train model apply back series broken help recover extract series imputation raw input da run change hidden type imputation shape remain da four totally imputation initialization descent repeat report mse mean complete sequence imputation unknown interestingly da perform sequence gap full imputation mse da raw well predict always mse imputation mse imputation raw stable performance raw raw trick augmentation dimensionality datum information image da utilize temporal spatial subsequence stable full mse mse contrast cnn neither interpretation edge angle cnn work illustrate reconstruction six map cnns eqn cnn patch essentially move nonlinear integration consider dependency benefit preserve temporal observe layer cnns dependencie convolution image preserve address feature cnn orthogonality advantage
bit put differently player bit capture payment player analyst select player uniformly report bayes nash positive maximize affine player bit player payment informally payment induce payment agree bit note would equal payment rule induce report regression unbiased nevertheless preferable desirable consider vector r follow condition close decrease significantly variance invertible depend large although expect ever follow notion provide small spectral theorem whose restrict attention ball simplicity useful differentially nr r jointly short lemma imply payment observable differentially jointly mechanism jointly differentially output player sensitivity player follow sensitivity arbitrary neighboring database say r r neighboring bound change estimator satisfies show differentially database differ denote compute output choice computed noise difference fix arbitrary plug p joint differential computed subset theorem differential use player uniformly theorem report partition differentially nash lemma player expect run different dataset player differ lemma noise add preserve privacy symmetric profile player fraction player player cost player player report I fix database player q player sequence database input player run database ridge differ database report exactly player maximize inequality add equilibrium group accord compute let report strategy compute ease estimator nash player receive individually player privacy unbounde maintain privacy player compute group report ridge follow algorithm player ease bound payment low bound payment player receive input input payment payment payment payment utility mechanism negative next require analyst budget payment bound receive payment thus q continue since infimum expression feasible region finally cd jointly differentially private bayes nash fraction report estimate individually rational fraction mechanism analyst cd fraction union bind private guarantee report approximate achieve among thing n due dominate nash fraction estimator dominate term third rational player bb bb final always entire simplify bind suffice individually rational fraction mechanism analyst n characterize draw must identity suffice calculation q minimum n contradiction eigenvalue case n contradiction mechanism far upper depend report privacy define expect expectation player report mechanism I privacy bound log particular composition setting differentially private report player sketch inequality specification privacy inequality utility ic g I mx I fc interpretation loss plus convexity eigenvalue function prove convex identity strong convexity reduce requirement quadratic strongly strongly notational ease denote loss denote th coordinate identity thus positive vector sum also psd corollary formal hold individual privacy mechanism guarantee participant individual loss immediately poses differentially model exist mechanism privacy sensitive well challenge computation fitting perhaps fundamental experimental many model learn hold analyst task must medical trial census survey behavioral currently massive hold interested enough wish influence outcome either benefit directly concern necessary design mechanism proper tradeoff budget participant concern privacy easily handle clarity privacy hold analyst linear analyst player computation minimize analyst cost establish player analyst differentially private pose differentially bias individual payment issue mechanism appropriate mild technical square receive positive individual assumption experience loss accuracy attain establish provide vast decrease effect technology agent cost cost would interact series paper acquisition agent concern vast operate agent lie private cost privacy explore notion bring technology prediction report presence concern report privacy player report report agent player simplest sophisticated accurate deal private ability regression different context analyst consensus come agent loss show minimization albeit establish agent datum without receive approach consider agent privacy body risk outcome perturbation instead even choose perturbation mechanism characterization preserve set technical preliminary review prediction privacy player eq analyst infer player player properly analyst specifically measure analyst physical extract medical record list player preference lie either payment analyst privacy design mechanism take perturb response negative informally mechanism accurate budget accurate mechanism part player guarantee privacy detail player rational formally throughout analysis independently ball discussion generalize uniform response conditionally support boundedness sensitivity finite natural response finite support imply response estimate eq minimizer unique estimator regression classic differentially database quantify privacy intuitively differential privacy output payment player insensitive like sense shared neither publicly player publicly payment mechanism publicly comprise exclude payment consider portion mechanism outside jointly private player observable q privacy also require payment differentially private player deviation payment roughly intuitively report emphasize learn privacy mechanism differential notion differential feature treat privacy certainly attribute medical medical response response paper privacy privacy modeling player characterize sensitivity analyst privacy describe cost incur differentially private payment utility assume arbitrary bound increase privacy player intuitive imply privacy player assumption quadratic hold cost formal reveal formally conditionally conditional illustrate idea report player concern simply player formally present spirit player payment depend agree produce report bayes nash x li l report nash equilibrium player payment reporting well uniquely report x x report nash scoring affine argument importantly estimator case appendix replace response restrict report characterize trivially absence privacy cost make arbitrarily analyst example budget possible devise report player agree report depend analyst privacy reveal publicly analyst payment differentially estimator result differentially private add differentially private ridge estimator construct technique ridge class differentially private though perturbation construct accurate mechanism replace ridge respect approach score player variable report report quantity concentration ensure grow linearly long grow slowly ensure choice approximate mechanism remain differentially formalize intuition prove mechanism regression indeed attain approximate jointly private private incorporate perturbation add noise mechanism output joint differential draw accord laplace ab jj dp r r formal version algorithm differentially approximate nash equilibrium player accurate individually
likelihood order bf analytically rejection testing wide used number figure give figure well computational estimation importance intermediate rather multiple result point box plot infer sl outperform outlier figure bf slope example work quite bias bf conclusion would affect slope see bias trade evaluate ensure might sl work bf bias limitation bf use highlight sl introduce bias assess abc bias impossible assess sl simple implement wide limited avoid use possibility use form proxy exact find consist point abc highlight sl gaussian assumption appear variable note sl much inference dependence sophisticated possibly sl sl simplicity find obtain exact reason internal place sampler smc take reciprocal tend inexact bayes particularly implementation see theoretical current datum consists ise ise via evidence truth take weight introduce distribution bridge estimator high variance tail expect bridge external since cost total bayes compare stage firstly exchange estimate smc move target employ effective ess fall normal variance run exchange space method bias improve may move take account statistical efficiency regardless whether alternative approach link style sl alternative estimate much attention appropriate avoid abc sl sl unable estimate use essential sl must disadvantage method figure accuracy advantage variance sampler problem inexact smc sampler avoid simulate monte require exponentially practically dimension simulate doubly like focus estimate evidence sampler mcmc inherently carlo beneficial examine estimating dimension section introduce alternative sampler intractable distribution marginal smc sampler smc sampler sequential particle normalise target pf ty z particle reweighte weight represent choose differently alternative mean normalise give resample main smc simply take intermediate distribution idea explore gibbs random offer method approximation weight marginal negligible compare correspond consider avoid calculation namely proposal within smc still denominator try pf p tu py rw weight reversible mcmc choose invariant incremental incremental presence ratio mcmc involve precisely weight update place direct spirit unbiased although precisely appendix incorporate smc target find useful consist point exactly ease add increment smc mcmc standard pz pz spaces eq f move provide become method sensible choice know early aside time may find point every update would add target describe sl approximation likelihood may smc target idea explore abc also provide smc sl explore previously context sequence target even obviously examine smc sampler evidence relative consider precision observation evidence analytically ease cholesky decomposition element wishart simulate use space thus motivate suited smc sampler particle target target pf tt one consist particle choose fm internal sampling systematic resample perform effective ess evidence smc run sampler median summary c st rd evidence example indicate advantageous within weight somewhat analogous exploration analyse biased weight bias u admit deterministic variable generally practice flexible formal setting one section expect weight law estimate compare eq square enough sufficiently assume would bias suggest qualitatively bias bias small increasingly could argument trade bias section investigation importance bias weight effect inexact produce sampler way motivation assumption error principle small level particle monte understood field approximation allow denote time auxiliary random variable space transition combine smc need assume iteration assumption proposal weight naturally finite employ formalism relative exist slightly control error employ inexact weight dynamic forget condition suffice stability correspond approximate iteration appendix demonstrate accumulation error intend qualitatively accumulation weighting potential ergodicity framework somewhat broadly strong allow establish simply justify scheme introduce sufficient together present suggest investigation effect bias smc simulate single estimate alternative smc sampler add target smc internal bridge estimator specifically q v smc run time particle examine bias inexact sampler compare sampler perfect evidence exact inexact smc observe inexact exercise sampler weight sampler present effect improve smc may simulation particle mixing bias inexact inexact observe decrease clearly biased weight useful doubly biased sampler sampler square square evidence inexact sampler compare experience suggest sampler use estimate good theoretical investigation idea worth situation involve likelihood result mix accumulation situation useful intermediate biased decide resample particle choose describe smc estimating outperform previously paper also generally context biased weight accumulate bias commonly accept science network however bayesian use due likelihood develop describe weight monte intractable investigation much interest intractable situation pointwise application occur pointwise example case big consist constant random give model overlap consider previous work introduce simulate challenge methodology whereas depend consider presence simulation specifically approximation us smc complete flexibility counterpart analogue concern acceptance mcmc applicable base sl consider success usually briefly problem outline outline bayesian discuss method metropolis mh simulating look section mcmc avoid evaluation lie normalise arbitrary estimator unbiased appropriately mh interest automatically extension instead q variance strongly appropriate ideally tail suggest reasonable likelihood choose particularly lie motivated importance suggest name make auxiliary low alternative place seem improve present main ratio usually reliably tractable knowledge publish evidence abc approach large
adopt transform inverse transformation image reconstruct image measure degradation structural index spectral residual base sim distinction consistent peak signal noise ratio figure capability image fidelity measure image measurement qualitative compressed visually indistinguishable objective assess transform code embed employ software library video stream employ dct bit bit encode video frame public video library simulation control video step ii agreement metric include logic block flip ff count delay static dynamic final transformation pixel percentage dynamic consumption area resource propose ff ns complexity propose low tool compression compression adapt computational suggest capability transform asymmetric encode several device propose decoder context power bandwidth capability alternative low accord meaningful quality hardware exact consumption decrease field approximation quantization acknowledgment usa cr cr tag pe j des sciences france mail free number image hardware consumption literature image video tool recent several measurement blind verification blind medical image compression transform dct bit encoder quality dct however arithmetic operation device demand consumption exact dct general approximate element possess arithmetic mean bit shift prominent sign dct dct code capable code hardware associate analysis perform transform image video compression scale final remark polynomial matrix aa factorial synthesis x nk entry integer derive require arithmetic consider transform generally less dct matlab language exact dct matrix propose approach discrete seven time large dct less evenly rescale accord formalism detail parametric family aim result arithmetic analytically tractable find satisfied obtain complexity q obtain transformation expression return therefore synthesis equation transform matrix scale may compression diagonal embed quantization explicit transform coefficient
likelihood although use composite various highlight calibrate composite approximated posterior substantially low figure rely eq include summation set lattice overall constant lattice generalise computing simplification arise un likelihood dependent lattice except last drop n represent straightforward exact minimum lag lattice first neighbourhood occur lattice additional straightforward compute summation conditioning true misspecification approximate posterior aim identity gibbs identity gradient moment namely identity express adjustment simply substitution map approximate address note concave however log semi unimodal example optimisation map difficult time find calculated evaluation provide bfgs algorithm carlo bfgs point bfgs algorithm estimating algorithm little despite bfgs algorithm modify remark curvature scalar directly link hessian observation choose block weight everything write identity deal solve lattice compute exactly serve composite computation carry gb computing normalizing take cpu bfgs take second map one approximately minute situation require wang use dramatically simulation simulate bfgs stop monte whereas covariance place integration mcmc burn interaction critical ise exhibit spatial around parameter value right sum evidence constant expression turn plot example clear un posterior denote respectively posterior adjustment provide correction posterior ise variance square average distribution h use law moment estimator figure case adjustment allow option variance correction seem carry approximation magnitude ise kl structure abundance parameter interaction induce figure misspecification evident magnitude adjustment yield see curvature adjustment correction illustrate conditional statistical analysis field likelihood typically concentrated contribution replace composite extend number acknowledgment grateful anonymous insight centre science grant foundation grant play due normalizing likelihood principle approximation paper result illustrate play important distribution lattice exponential arguably popular social include biology physics popularity field parameter trivially replace full generalization refine purpose consider composite inference focus collection variable influence approximate posterior use mis description gibbs composite likelihood especially formulate likelihood issue composite bayesian illustrate various remark finite undirected define adjacency definition directly major due normalize constant depend parameter summation possible trivially pose serious difficulty parameter composite likelihood outline binary lattice lattice normalize write dependency henceforth point index bottom column column order neighbourhood interior along lattice exclude lattice abundance aggregation allow variation
task include metric auc predict order relevance rank hypothesis available belong reproduce product space definite k reproduce work batch scheme rkh formulate q rank metric kernel furth subsection concept robustness despite potential capability deal dataset recently analysis pairwise space establish guarantee almost surely polynomially require iterate restrict strongly associate unconstrained novel mainly operator probability hilbert schmidt paper organize introduce example specific discuss relate present technical lemma pairwise loss measurable difference see study follow learn usually prescribe ball f implement unconstrained generate algorithm sequential access training hypothesis upon reveal iterate obtain iterate functional rkh x mm functional approximation theorem surely implication recall universal universal kernel kx pairwise fractional kl select subsection pairwise f indeed specific pairwise let x positive definite pairwise characterize rkh induce statement gx g g gx use assumption kernel apply equip statement associate define univariate similar proof corollary remain remove firstly using see discussion example author generalization risk estimator sample eq rkhs inner loss definition formulation formulation kx gx gx g pairwise batch bound establish case follow ensemble hypothesis function z fair let hypothesis technique average online project regret rademacher average together f f term side l rate iterate f tx difficulty analyze novel enable overcome characterization mainly prove necessary notation j define derive well theory error j depend subsection term turn attention sample side end establish lemma inspire similar gradient certainly prove equality f ts f k complete denote j kf operator imply lemma hold remark easy inequality variable hilbert difference hilbert surely k fact schmidt see let hilbert hilbert schmidt operator inner usually denote ready accord term j l j first term side k put estimation back observe combine l recursive equality sample hold j apply yield basic notion quantity statement functional b k part prove follow side apply start technical proof appendix ready establish theorem consequently back desire definition prove well g exist equivalent easy q consequently put turn univariate us proposition pairwise n span span section gx gx g gx gx gx complete proposition part online theorem b learn unconstraine rkhs non strongly perform unconstrained aware pairwise algorithm surely rate polynomially decay size discuss form f save improve implementation square particularly result acknowledgement support grant
intersection small correspond assign point outlier let core bind every need cover conclude se se still mistake ingredient conclude definition reliably shape cluster unlike art cluster contrast technique provable skeleton continuously see local skeleton set geometry cluster outlier infinite stream provide theoretical quality cluster massive stream become medium finance throughput community evolutionary web activity email service challenge throughput scenario real typically provable generative force retrieve convex exist assumption heuristic lack need stream use finite stream assume technique difficulty handle effectively time massive stream propose skeleton online address challenge basic cluster skeleton capture geometry skeleton maintain skeleton density skeleton automatically shape skeleton skeleton update procedure outlier recover strategy skeleton allow adapt drift merge split guarantee quality huge offline k mean median cluster radius poorly cluster shape survey variant density combine perform suitable stream another variant technique center continuously point belong rich complex main nonparametric cluster guarantee tune cluster encode split variant center keep purely outlier share exist random arbitrary shape however offline agglomerative size slow aim encode idea intensive store skeleton variant provable inherently offline often pass time clustering shape mainly also assume several iterative propose know online set essentially initial laplacian infeasible shape constitutes shape likely belong datum neighborhood point complicated idea graph utilize via correspond skeleton belong number call clear skeleton correspond weight encode around skeleton skeleton belong skeleton set update skeleton mention skeleton cluster stand skeleton vary initialize skeleton take translate strict provable regard quality maintain skeleton skeleton relatively entirely skeleton skeleton cluster grow skeleton never cluster overall number skeleton generality skeleton variant merge merging splitting cluster cluster extra turn keep undirected element associate skeleton encode denote skeleton cluster assignment iv iw I bx rs algorithm turn skeleton stream create skeleton weight skeleton belong cluster skeleton merge skeleton empty singleton start split turn I bx bx un un merge j vs merge merge v un merge merge min v min w min min merge merge merge assign multiple basically act merge unify scenario cluster initially one combine true skeleton skeleton merge important subroutine skeleton size newly add skeleton independently skeleton seed initialize randomly alternatively skeleton conceptually cell latter correlate far away skeleton number skeleton next consider skeleton skeleton merge merge skeleton relatively merged cluster skeleton set consideration already denote minimum skeleton skeleton merge merge skeleton skeleton point newly skeleton copy point merge point newly contribute weight skeleton close skeleton find increase contribute total skeleton replace skeleton cluster merge skeleton skeleton merge small skeleton point encoding pool entirely conduct skeleton exclude skeleton singleton correspond accord add skeleton algorithm skeleton singleton cluster skeleton aim cover cluster sample form skeleton sequence generate triple complete skeleton initialize newly cluster skeleton create skeleton singleton create cluster update singleton accord new splitting variant handle breaking skeleton weight skeleton point determine skeleton connect mean split component cluster responsible graph newly cluster replace skeleton skeleton merge graph combine newly skeleton close vertex skeleton radius cc subroutine skeleton skeleton x bx I g merge subroutine regard describe introduce dimensional disjoint call core great I arbitrary give rise core probability formally give outside important core due presence word cluster good quality core separability recover nontrivial offline connect component online bring algorithmic computational challenge definition cover ball arbitrary ball express denote ip p kp fp outlier outlier cluster core keep skeleton cluster point core also error online lack enough phase reach reach ready core core see phase least algorithm merge contain upper skeleton cluster practice theorem error come theorem skeleton point word rate produce rely intersection outlier mark green phase theoretical whenever phase skeleton equivalently treat skeleton formulate follow lemma contain skeleton skeleton outlier lemma accord take core merge skeleton skeleton thus four synthetic set contain randomly draw shape outlier shape letter shape shape deviation affect fig cluster quality guarantee art nonparametric dominant comparative produce use cluster dataset handle work fail fail dataset
related rademacher index set sub quantum e rademacher directly independently distance measure calculate invoke depend exactly determine quantum since real relax criterion accuracy cover identify element entropy quantum quantum element speed optimally distinguish discrimination zhang goal given belong concept separable exist separate exist difficult quantum discrimination classify great ask cardinality show dimension quantity function respect generality assume choose denote convex argument level cardinality complete quantum n independently sample hypothesis input ball class collection aim quantum quantum parallel measurement duality discuss relationship quantum code framework linear functional element derive quantum state theorem sphere since embed rademacher however learn rademacher bind series deterministic hermitian rademacher rademacher variance hold concentration inequality eq realization n complete repeat space rademacher assume rademacher q due duality complete proof entropy input proportional intuition unit ball large ball radius perspective fact evident lebesgue measure banach calculate effective exponentially word demonstrate state quantum definition code map bit exist tr th set upper level om ps upper success however relate dimension dimension order consequently om inequality recover integer show directly space coincide previous section constructive way implement quantum ml framework material derivation appendix affine pseudo entropy rademacher paper analyse problem measurement state outcome quantum dimension also show learn quantum entirely tool classical proof also derive summarize unknown reasonable finally sphere show learn ml learn measurement provide viewpoint quantum connection field existence code state quantum area integer linear operator conjugate transpose schmidt inner product stand conventional trace operator trace identity standard norm reduce norm element output hypothesis independently ensemble set dimension cover metric call big constant introduce deviation express sample boolean pac vc provide boolean constant finite provide bound analogous finite absolute n bounded entropy closely et al q therefore sample therefore provide another wherein classical traditional sphere direct gain dot euclidean q state act convex consequently functional operator associate useful algebraic property operator space projection orthogonal mutually rank associate simplex angle kp convention centroid face intuitively interpret kp quantum tr relationship associate input e functional act affine consist bound functional however easily convert behind quantum need reader ref detail basic network simple call scenario sign satisfy condition boundary perceptron adequate parameter misclassifie example compute add termination reach output integer procedure adjust dimensional input datum activation bias perceptron infer perceptron htb kk height title author subject green issue date journal proposition section quantum engineering information university technology edu tw com tw receive significant attention promise progress make quantum theoretic training predict arguably theory setting complexity paper unknown dimension hilbert result solely complexity explicitly able connect quantum science quantum discrimination lastly representation learn quantum mathematically apply artificial intelligence aim devise systematically typically ml unsupervise machine hide clustering supervise characteristic learning machine time determine query hypothesis approximately generalization closely require target complexity trend big feature balance complexity set well without overfitte active capability system recent large integer search feature improvement superposition contrary classical bit combination consequence mechanic wave superposition store give device quantum phenomenon quantum resource result shannon theory feature area broad promising application advance subject classical quantum machine attract substantial classical task totally precisely accelerate computational transform quantum hand fundamental quantum state underlie system statistical certain accommodate value subscript refer set learner accord pure learner access classical ml belong class current quantum procedure consider membership quantum extension quantum classical polynomially process problem memory store method procedure unsupervise additionally quantum execute big verification computation method quadratic optimization problem microsoft approach quantum neighbor algorithm surprisingly number depend rather wang phase hamiltonian ml fitting learn quantum pattern computer important task interested reader comparison quantum state accord fidelity cluster statistical study quantum operation hide quantum statistical model quantum width corner draw center corner mat sep mm block block c wang south north south pos north pos north pos north south pos south c work quantum quantum device quantum randomly state quantum need learn machine decide optimal measurement measurement statistical theory proper quantification propose outcome learn quantum exploiting banach measure rademacher complexity derive require quantum measurement proportional hilbert quantum formalize employ tool solely theorem cover three proportional quantum sphere hence ml apply may relate quantum state physics exponentially point serve quantum surprisingly dimension quantum hope quantum quantum ensemble mutually perfectly discrimination minimize dimension guarantee quantum bad reasoning hypothesis stand bit receiver successful provide quantum alternatively coincide work discussion background theory supervise describe relate function derive quantum addition interpretation cover rademacher quantum code formulate sphere representation network implement paper hilbert orthonormal adjoint inner subscript omit norm trace norm define norm operator likewise finite class operator operator quantum serve yes measurement constant parameter change notation table start mathematical formalism examine error complexity speaking supervise ml observe agnostic pac comprehensive introduction refer reader e supervise aim approximate train performance take absolute convenience easily generalise quantity lipschitz constant complexity measure therefore homogeneity assume square derive sample problem since risk minimum I infimum take possible measurable deterministic almost surely identify collection call hypothesis assign eq effectiveness almost agnostic def hypothesis agnostic pac learnable sample train empirical erm principle assign way evaluate relate risk reasoning algorithm output uniform uniformity respect member measure quantity confidence require class class agnostic criterion interested reader therein train call domain agnostic erm algorithm agnostic result complexity criterion underlie learn agnostic pac fundamental agnostic pac determine rate theoretic hypothesis next complexity introduce size hypothesis far interested agnostic model resource class introduce dimension let domain every generalise vc quantify complexity introduce real n nb bf bx bf bx bf main theorem function dim subset dim domain bf bx bf bx bf bb underlie constant dimension value side pseudo even addition combinatorial quantity measure concept back kolmogorov area mathematic cover number cover cardinality metric endow support fx I fx pl entropy q significance loose technique concentration measure capture sharp bound rademacher bound function variable rademacher associate complexity use ref convenient far measure combinatorial hypothesis eqs sample functional justify quantum practical situation physical aim three measurement state measurement experiment quantum count functional correspondence measurement furthermore unique e proposition every measurement identify subsequently either linear outcome eq space linear effect hilbert determine probability correspondence quantum coincide state matrix quantum banach furthermore value target banach space hypothesis represent functional nonempty interior body symmetric unit ball dual schmidt product duality formalism banach follow set linear functional pac learnable duality operator fix hilbert map trace class operator conversely investigate dimension set functional banach every ball banach restrict core calculate banach rademacher series value formula helpful duality formula upper via rademacher series remain banach measure proceed practical may yes outcome outcome perfect
solution sufficient also say establish existence find solution exist definite matrix proximal operator value g expression observation make metric proximal framework proximal newton proximal quasi choice algorithmic approach starting update size search direction size subproblem n selection prove moreover bad second hand decrease easy verify selection decrease regard backtracking describe ht e use proximal gradient k combine follow equivalent subproblem kf k explain theory backtrack assume inspection maximum many principled way quasi backtrack oppose check far backtracking mild find ff moreover convergence impose evidence need union impose twice restriction probabilistic cf restrict number small hessian become explicit backtrack attractive certain count step newton newton proximal newton newton proximal start q moreover size step whose proof appendix I solver collection problem first denote sp ratio use profile profile self lipschitz fast applicable figure prox operation medium accelerate bfgs update proximal good prox operation solve proximal newton good prox operation via problem bfgs omit exhibit linear fast show variant nesterov dual exhibit perform total respectively surprisingly consistent indicate convergence satisfied practice proximal calculate gradient algorithm record ht number exhibit meet believe form vector form full especially like definition profile proximal bfgs performance good prox operation prox convex modify programming random show mf proximal rather test outperform plot prox prox operation achieve fast variable method minimize applicable usual assumption smooth convergence highlight backtrack new basic local free proximal practical former assumption fast plan version frank com composite minimization function laboratory self like concept minimize analytic property numerical test real composite eq possibly nonsmooth composite naturally arise application science image science loss trade optimally exist lipschitz gradient cf definition development sublinear smooth regularity within quasi newton convergence algorithm exploit principled bfgs bfgs instead focus develop convex self composite minimization function multinomial logistic first metric backtracking operation guarantee proximal adapt convexity convergence great life lipschitz exhibit sublinear help solution form geometric contribution self function descent second pay particular subproblem variant locally variable metric hessian locally
co threshold neuron undirecte present see quick te cause cause method improve incorporate add vice versa instantaneous causal instance insufficient therefore solely pairwise indirect network strategy eliminate indirect strategy derive strength neuron deviation asymmetric standardize complexity neuron symmetric unsupervised eliminate indirect normalize pairwise indirect integrate network capable method average rank sort know refer formulate description supplementary material subsection since indeed conservative correlation discriminate effect experimentally correlation statistically informative plain quantile correlation recover correlation hundred sample capture quantile quantile pair neuron feature complementary spike feature square normalize center signal particular rely quantile extract square influence measurement nearby neuron dataset dataset provide neuron neuron network see either assess receiver operate characteristic roc recall pr propose metric roc auc roc optimistic encounter network recall curve compare method big discard link curve apply good result value small font show observe network cc cc normal normal statistically highlighted use original use direct auc similar conclusion table auc material see additionally unknown area roc rank advantage cpu compute different intel mean minute instantaneous compute proposal improve unsupervised improve art network rely among compare experimentally namely dataset communication technical university bioinformatics biology universit du present neural fast connectivity neuron activity art entropy process remarkably simulate time challenge competition reconstruction elimination understand capability brain cause treatment network neural connect neuron perform circuit responsible well easily group neuron record neural one study population simultaneously neural recover brain neuron
introduce shorthand continuously cauchy schwarz turn hx u hx hx v x z kt x kt hz kt hz kt x z kt kt order derivative eqn directional v u hx lemma guarantee integer hx hx hx v hx hx hx hx hx hz z v hx directional exist directional hx hx v hx hx v hx hz op directional hessian lipschitz uniform solve stein notation limit pointwise hx hx p py dy py dy hz kp compute stein graph stein gx jx b x gx gx gx result graph stein discrepancy stein discrepancy equivalence stein jx stein compatibility imply establish second j g k j g z g g exist z stein inequality fix j b b fix compatibility invoke inequality boundary compatibility property turn establish g b j j z b deduce z acknowledgment share implementation triple early manuscript support fellowship foundation fellowship theorem em em em theorem turning procedure sound rapid create challenge quality stein expectation biased assessment quantify bias target often turn chain hz hx qx target recent researcher asymptotic mcmc procedure asymptotic correctness rapid variance estimate bias add flexibility challenge sampler parameter pool address quality bias sequence stein program design section illustrate application assessment quantification work often gx jk density integration intractable aid point encode mass probability hx sample approximate quantify converge iii computationally starting consider eqn hx large measure converges weakly term many include generate hx distance generate adopt measure computation infeasible generic integration intractable could focus know many class function know track ii practice question distributional return third characterize value act function generator op px gx stein operator derivative computable even normalizing boundary boundary gx gx x smooth boundary suitable domain p study classical stein converge wasserstein set analogous distribution analyze extend reach literature classical stein discrepancy determine wasserstein large include prior stein strongly densitie p stein multivariate sufficient stein stein pz implication proposition px pz analysis paper readily accommodate uniform stein eqn stein gx nx exploit flexibility bound relation stein discrepancy wasserstein well free stein stein stein discrepancy equivalence q efficiently stein discrepancy property discrepancy restrict unconstrained section present domain evaluate function stein discrepancy qx px gx gx gx x stein primary difficulty classical stein stem constraint impose stein way difficulty impose classical smoothness collection gx gx gx classical taylor compatibility remarkably stein strong stein program v j l represent function value amenable prohibitive unconstrained extend coordinate program boundary compatibility b b appendix stein discrepancy strongly stein summarize recommend stein unnecessary complete stein q g sorting point enforce stein computation solve th program bound n qx n x I turn evaluation program simple stein diagnostic scale degree complete stein discrepancy decay student stein stein well notably student function exhibit relatively middle stein stein appendix stein discrepancy compare wasserstein target provably uniform target generator seed point I non classical wasserstein distance apparent graph stein discrepancy classical track wasserstein magnitude separation wasserstein fact stein discrepancy langevin bias mcmc design scalable inference approximate langevin metropolis hasting use grow meanwhile explore space stein diagnostic adopt minibatch sequence select high quality ess diagnostic autocorrelation discrepancy across stein discrepancy select ess greatly greatly slow stein diagnostic resemble posterior ess maximize stein minimize ess stein mh biased control mh qualitatively fewer rapid rapid reduction mh stein discrepancy trade cancer patient whether spread node bayesian batch discard evaluation thin remain stein computational langevin length surrogate normalize jx tw quantification discrepancy sequence appendix stein sequence deterministic dx sampler slope good square metric bind suitable compare bias infinite class functional finite collection functional distinguish discrepancy mmd distributional kernel approximate mmd access ground truth target ex boundedness langevin generator second function solve stein equation equation hx hz hx hz hz hz op hz op hz hz hz op hz op equivalence stein imply wasserstein wasserstein inequality standard follow inclusion smoothed function close integrable h hx hx bound fix lipschitz admit gradient representation tx hx hx tx hx z v hx relation yield h tx b td chosen stein discrepancy pg l z l l l l z objective q stein factor lemma hx hx x langevin useful proof langevin concave langevin diffusion generator lyapunov vx cauchy schwarz arithmetic together q x constant continuously differentiable result growth proximity ultimately smoothness hx op hx hz hz hz z op op op w w op z op x establish taylor hx schwarz yield hx op x difference mean remainder hx hz hx hz hz z hz hz hz hz hx hz h w h eq invoke difference apply cauchy schwarz definition operator norm hz op x cauchy schwarz z z hz op eq hz hx op norm w x op w four proof serve follow term inequality strongly langevin diffusion differential eqn sde p v x kt kt kt x k second order eqn v x z kt v b coupling h op x z ks apply continuously differentiable pz yield ks ks second concavity may desire kt ks ds conclusion order difference v v ds invoke kt produce kt ds x couple together third bind h h u z ks ks ks presentation pz op pz op op x v ks x concavity reproduce conclusion kt ks ds ds langevin diffusion wiener
surrogate three label distribution prediction pt matrix prediction different conditional induced classifier loss respectively represent l ty bayes goal learn close algorithm w optimal distribution function training discrete computationally approximately algorithm approximately e argument optimization seek predictor hold continuous predictor excess immediately consistent w derive vs surrogate excess relate j one loss threshold break favor result proof point previously zhang surrogate hinge predictor however dominant learn predictor instance class manner instance dominant vs hinge surrogate like surrogate conditional domain minimize surrogate real convex surrogate call risk relate loss calibrate dimension assume one onto define evaluate clear partition predictor result excess surrogate suggest good choice surrogate intuitively prediction sense close predict low noise predict make choose via section minimize surrogate set simplicity label norm em ex r md I I block ascent algorithm fix type problem db j projection ball excess bound surrogate loss modification surrogate n em u b generalization proof along omit surrogate hinge hinge extension generalize hinge dataset vs surrogate space class incur bayes randomly prototype vector mean prototype vector generate pick surrogate reproduce hilbert regularization parameter search cs incur risk excess threshold surrogate support cs imply result algorithm less cs error poorly optimize show much ccc ccc cs cs repository class regularizer choose split train simplicity cs algorithms level choose comparable time algorithms run fast option reason speedup function problem speedup cs reject powerful abstraction capture control classifier diagnosis formalize give excess relate surrogate operate small metric direction break excess vector u n prove satisfied surrogate position everywhere else theorem simply linearity expectation rhs equation last equation u equation u u yu also u u n crucial straightforward inequality else f f linearity rhs hence trivial u u u rhs n u also u follow surrogate f follow linearity rhs become j observation j equation thm thm conjecture thm class prediction say reject thereby extend generalize surrogate yield consistent design also consistent operate dimensional generalize surrogate consistent would well take predict make wrong problem medical diagnosis test convex logistic loss adaboost consistent problem hinge require consistent double hinge segment flat segment multiclass double multiclass reject reject seek reject option incur denote call svm binary minimize piecewise arguably widely surrogate like dual
suffer drawback mlp therefore city sentence learn extract structure match drawback architecture sentence desirable sentence meet representation basically slide sentence one e segment clearly convolution segment layer max non overlap illustrated layer perform window eq could pool analogous architecture cc level two sentence high encode dimensional segment among pooling resemble dynamic similarity architecture rich r c convolution pooling preserve information although may segment retain triple consistently gain find correct usual happen choose turn separate mlp actually map devote devoted pair naturally divide filter denote slide rank essentially pool pooling preserve convolution abstraction fully preserve offer capability individual internal abstraction interaction sentence sentence fusion hence object rich structure intuition verify give rank include layer architecture propagation section adopt turn descent batch size easily machine core regularization architecture early medium training less early dropout deal overfitte word embed english learn wikipedia chinese learn experiment tune cope sentence word optimal performance eight three perform layer convolution relu convolution mlp comparable like propose match different nature compare namely sentence tweet match identification natural three task language write prove applicability matching text calculate mlp two document matching layer layer use unfold autoencoder get dimensional sentence mlp sentence sentence sequentially mlp score coherence performance layer correspondence clause basically clause comma match heterogeneous relation lexical hard clause similar original million test positive pair negative negative sentence show convolutional perform fairly well run behind sentence surprising come last cause word embedding split sentence parse sentence original tweet collect major chinese service writing style positive ten million triple translate english tweet select tweet match original negative report task slightly purely negative model loose margin determine sentence language acc object benchmark contain instance early stop state require instance achieve hand significantly design relatively superiority deep structure matching rely raise match whether something convolutional perform indicate importance utilize sequential sentence interestingly experiment train negative sentence surprising auxiliary correctness word response enhance match notice reasonably act composition meaning segment word rarely focus match score product building text largely representation aside nature section fairly embed text mostly convolutional network work pool relatively tailor deep architecture language sentence outperform b chen national foundation china li part china project cb department science school central importance successful matching need internal step convolutional adapt convolutional capture rich level generic nature different language empirical variety demonstrate efficacy superiority object central similarity g retrieval correspondence linguistic level translation english language sentence therefore internal rich towards propose convolutional prove successful natural representing devise hierarchical sentence comprehensive convolutional architecture require tree put part understand contribution summarize first convolutional architecture sentence pool abstraction characteristic architecture sentence modeling architecture detailed architecture section report propose convolutional architecture illustrate take embed backward cc convolutional combine compositional recursive autoencoder example illustration purpose turning element focus segment word offer code layer choose
boost multiply convolution instead stage stage classification cifar scale run bottleneck runtime necessary pc ram peak gb hour cifar percent far convolutional three follow apply pooling stage variety whereas use whereas mark fellowship project acknowledge david dr university dr discussion resource acknowledge present design rapid frequent filter inspire value stage regression stage unit efficacy art classification database times network google house competitive database achieve neural rely convolutional feature extraction rotation memory computing hour gpu periodic able desirable therefore seek rapid even potentially expense image achieve network neural architecture surprisingly entirely select although application filter relatively train show produce classification database set computer apply still result dataset optimisation parameter gain recent review context convexity fact optimisation solve unit pixel train classifier excellent performance yet convolutional architecture together ensure namely classification least square feed output classification database cifar google view house network state near present benefit clearly require competitive hard cifar imagenet result cifar core attribute filter generalize aim remainder organize description obtain classify generic apply describe obtain learn achieve remain specify weight text pixel classifier stage three pooling layer fourth output conceptually divide stage combine first convolutional stage largely exist approach classifier largely stage algorithm image filter describe size filter transformation represent input kk cc ci channel k algorithm suggest image sequentially applicable feature operator construct multiply matrix hence total introduce size concatenation convolution diagonal matrix copy flow mathematically argument matrix toeplitz instead entire pool intuitive explanation receive sum simple response operation often help also nonlinearity nonlinearity classification root projection layer pool instead q effectively l operation follow description whether raw treat feature introduce numerically represent label value stage unit logistic g activation vector employ pooling pooling use describe default randomly non training lead train iteratively backpropagation determine pseudo solution solve overcomplete often follow ridge qr equation mention eqn follow runtime bottleneck multiplication image contain w filter k output inverse constitute close output activation classification decision image value method list database comprise channel namely channel raw pixel convert lp scaling affect preprocesse mnist reason previous convert add rgb contrast cifar convert raw whitening image objective layer dimension filter consider corner filter centre filter imagenet patch obtain randomly class filter database channel channel convert filter implement consequently filter channel dimension filter normalise summing filter convolution using obtain valid convolution remain feature exist tradeoff filter factor point enable filter comparable hyper hyper choose previously superior weight nonlinearity choice sigmoid sufficient nonlinearity good classification strongly presence see choice remain classifier examine vary optimize validation generic parameter layer image convert image fourth kind filter filter marginally cifar
participant final ht compound ccc auc auc mt nr nr nr nr nr er nr nr sr sr sr sr multiple within water actually structure together automatically compound clean run clean routine chemical coded consistently encode calculate describe layer ta layer ta layer ta ta layer ta ta define optimize goal set label eight task validation set add training b avoid sample sample within chemical participant world allow receive well scoring team avg nr sr er average stress panel challenge name save deep performance participant never place total challenge sr nr panel average challenge winner nuclear stress display network show deep highly chemical could previously decade expert field reason also lie representation application set confirm lead research ability greatly environmental health drug become support european union author acknowledge true might exist biological neither performance method chemical state build chemical descriptor decade learn never clearly outperform paper net establish challenge set standard people variety chemical many exposure day drug candidate clinical health environmental future activity effort drug goal develop scale demand testing rely throughput screening investigate whether chemical compound concentration exhibit certain different vary chemical compound reliably determine compound activate pathway interact time intensive typically test several whole many time compound multi effort project thousand highly exist approach compound applicable interact infeasible compound predict effort score density machine feed network challenge chemical compound win multi biological activity protein inspire activity involve whole biological specifically focus measure measure compound compound cause death affect seem well abstraction chemical structure include architecture concept idea depict chemical layer center higher ideally compound investigation several show help chemical boost task integrate task utilize representation latter may fail effective representation task boost task furthermore train one system take descriptor compound input tries predict type act pathway task solve present chemical compound want whether compound property compound property predict behavior compound compound binary later weight entropy binary multiply relu one output different scale try standard deviation nonlinearity bring descriptor scheme filter bring try combine chemical descriptor well amount additionally early cross validation contain hyperparameter consider molecular descriptor similarity hide number yes dropout predefine type determine molecular descriptor feature use layer backpropagation decay weight crucial application different storage format chemical representation connectivity currently perform compound drug presence chemical column produce approximately informative compound ie report literature compound chemical often additionally descriptor compound descriptor group around property count molecular feature extract chemical include van atomic involve quantum area calculate descriptor software median deal hyperparameter deal hidden parameter single gpu gb ram batch since store dense format tb disk sparse storage mini batch gpu convert multiplication validate challenge program challenge collect framework research criterion hard public database compose different challenge different sub challenge split seven nr pathway remain five sr pathways nuclear component control development play stream measure modify gene challenge include
approach second order difficult particular challenge learn dependency propose recurrent network properly entirely possible enable learn initialize recurrent eigenvalue name trick transfer information little initialization gradient backward rather effectively experiment dependency effectively difficult number number handle ability dependency enable classify minimal processing achieving state network temporal connection initialize rise recurrent turn recurrent something convolutional neural hide bias positive rewrite expand convolutional matter keep linear learn recent success backward sigmoid expand temporal gradient identity convolutional balance tendency experiment standard prevent hidden activation equation traditional rnn short gradient search good result forget gate give long set forget gate bias lstm gate add rnns random mask signal mask problem mnist read leave corner image corner ask predict category mnist h rnns fail fully neural convolutional network succeed term treat succeed impact problem input repeat test recurrent language model public benchmark modelling tend per hide confirm par sometimes language need gram give neuron help project term important wide variety ai rnn end page people actor question year birth answer simulate end answer retrieval top retrieval finding keyword document return document wikipedia entity robustness retrieval certain retrieval read top parallel combine unit last answer setup answer token birth softmax year birth token page page train page sure birth set rnns rnns bag softmax birth bag page bag rnns gaussian sigmoid rnns identity rnn dependency network vanish potential overcome identity temporal trick initialization dependency extent short challenge rnn ai problem language dependency relationship event sequence dependency across
find expression since usually call gene filter non biological responsible phenotype distinction effort search small dimension phenotype classification discriminate show informative linearly discriminate couple discriminative exhaustive perfect subset classification weight et propose gene preprocesse highly variability subset likely sensitive microarray limit challenge similarity paper run selection centroid use predictor predictor construct less biological result gene gene group co generate gene year yu similar employ univariate independence phenotype biological reflect exploit gene since account joint cluster classification complex phenotype cancer gene rather genetic emphasize exploration among cluster selection optimize secondly generate yu inactive whose centroid relevant phenotype remain non phenotype classification set discrimination experimental test importantly centroid cluster proximity global advantage researcher decide cluster implement matlab pc windows operate website http ac svm toolbox detail binary evenly phenotype sample evenly divide contribute independently centroid label position depend n centroid expression value performance generate keeps go test jump go truly active cluster truly discriminative gene terminate figure recall gene cluster since train simulate find centroid discriminative model reflect calculated euclidean centroid cluster close centroid distance trend base meanwhile cluster dataset logarithmic transformation reduce gene validate three fold validation randomly third testing phenotype dataset rescaling variance value gene gene restrict discriminative start input set correlate sample metric summarize centroid discriminate decrease generalization sample e f active perfect perfectly voting result perfect roughly use set generate gene highly probably sensitivity select experiment gene appear input size contrast centroid run step reflect stability rather individual discriminative biological active focus biological go hierarchy gene active biological geometric convert score cluster average process associate active refinement cluster third gene four monotonically suggest meaningful one response convergence closely hold biological process activity bottom biological besides dataset follow phenotype study dataset cell breast gene phenotype er er positive contain tumor normal level gene tumor b expression ratio consist expression gene blue cell consist divide class bl dataset phenotype derive phenotype rule nb bl filtering logarithm sample phenotype classification split way last subsection summarize value exhaustive globally cluster apply cluster last l l e breast e bl nb e e among phenotype active increase slightly start execution highlight three indicate decrease ability cluster training separation sample improve set gene appear cluster reflect step range closeness different reflect limited absolute generation value centroid keep keep iteration distinct general show microarray closeness know optimal refinement previous work performance classification table sample version cross direct performance comparison validate validate cross pt dataset yu multi three validation slightly breast however inherently range none sample comparable breast et good match yu also multiple class version method yu seem high nature unstable subset guarantee individual gene subset propose backward gene gene provide phenotype class future study implication regard phenotype performance gene cluster generally exist centroid gene consume seek number gene centroid drawback cluster centroid optima optimum clustering completely require cluster centroid avoid optima overlapping gene reflect gene contribute pathway discriminative certain gene statistical conduct inactive pattern appear test consider improve exploit well provide various computing module way sample phenotype integrate multiple phenotype classification occurrence gene multiple different experimental high occurrence frequency confidence truly subsection execute generate discriminative module well sample phenotype cluster number stage stage involve evaluation discriminative validate use cluster gene average distance gene width cluster width fall width discriminative active adopt find decision hyperplane maximize two class hyperplane vector linear indicate gene base discard single svm systematic elimination reflect biological inactive gene gene consideration partition group microarray active discriminate start gene select gene pair iteratively large euclidean distance near take account iteration group form cluster inactive factor remove add cluster determine linear like eliminate discriminative centroid sufficiently representative expression pattern measure width eliminate gene whose pattern little cluster iteration factor objective multiple adopt popular test split problem remain class construct use active centroid cluster multi cluster accuracy generate gene cluster accuracy correctly however sample average accuracy fold biased estimation distinction cluster closeness cluster euclidean cluster close cluster construction precise recursively l l l significance average distance gene dataset calculate distribution randomly dataset recent microarray technology significantly identification disease phenotype cause individual gene effect insight disease pathway understand disease pathway microarray deriving module gene pathway expression microarray pathway pathway serve fact pathway manual condition activate integrate microarray infer pathway disease activate identify pathway may complex amount microarray data series microarray measuring expression indicate gene cancer dataset cancer module co topology conceptual width eps co similarity previously define low dataset high similarity suggest cluster similarity link activate similar module interestingly order analysis module analysis gene pair share divide within connect component likely hierarchical gene hierarchical modularity scale module module similarity select module infer module activate module apply related microarray cancer relate second cluster network activate cancer identify breast specific module tumor importantly gene play tumor module microarray total disease disease cancer non correlation percentage bend normalize bend correlation gene pair gene gene co network differential co expression scale distribution co topology act highly gene cancer phenotype gene play cancer fall core division organization interaction cell neighbor interact interact ip division ip likely characterization member long serve involve neighbor cancer supplement differential top degree together account degree differential fall main core gene dynamic cell gene movement cell dna l cell degree gene degree node play cancer behave gene tumor differential co summary frequently however tend inactive type cancer expression large connected differential contain connectivity break coherent type dynamic connectivity cancer network module second cluster utilize dynamic connect co calculate microarray term correlation profile distance expression profile unlike commonly pair gene base profile frequently occur relationship dataset likely link co expression module turn phenotype gene connect suggest pair component module cluster fall similarity within connect number test distinction select module show within pathway addition activate certain phenotype retrieve component connect diameter scaling logarithm especially cluster top score remove overlap result comprise module edge module statistically biological annotation cycle division stress mechanism width supplement linear diameter discover cancer module activate module activate cancer involved division genetic represent signature cancer figure module solid gene module involve cell solid tumor next module activate breast width eps cancer solid module network dataset module division genetic stability another module activate consist module locate correlation module tends activate solid tumor expression edge dataset co estimation tumor dataset result order module breast tumor dataset rest dataset correlation module breast tumor breast tumor dataset module protein gene breast tumor module express breast cell increase activity express breast rich play crucial tumor breast cancer induce rich involve survival gene allele c breast cancer gene involve cell suggest tumor module width cancer eps cancer tumor module main module tumor high degree module gene precise encode aa significant indeed cancer sample locate suggest breast cancer tumor gene recently depth decrease breast module find bind growth cell induce similarity recently terminal interestingly pathway direct member bind significantly positively production example breast tumor specific col col gene tumor function adopt tumor identify network module involve cancer tumor member arrange cd line individually breast increase associate advanced breast breast death breast module tumor cd induce production cell factor bind predict bind report expression module reveal breast tumor beyond expression elaborate reader assess pearson correlation connect within activity relevance cancer b module play different process cell division respectively division order underlie member module distinct average correlation module breast module role breast tumor development member gene module exhibit pattern module weak similarity across seven cancer module within second second rise connect module module co cancer pearson module within gene pearson correlation module active bend module pair cross module pairwise pearson percentage bend correlation gene highly module discuss previously module pearson dataset module rapid microarray molecular mechanism disease study utilize microarray derive list specific cancer little characterize et predefine meaningful biological pathway activate module wide pre module limit association study tumor functional growth cancer derive simultaneously cancer study topology characterize module identify module module activate molecular mechanism importantly discover potential tumor network particularly breast tumor commonly advantage simultaneously condition cancer activate thus provide insight complex mechanism characterize cancer approach identify densely module solely incorporate co expression diverse type module regardless network biological necessarily densely module pathway apply molecular beyond available framework module manual intervention systematic way put robustness base observation relationship suggest year aspect network essence exponential integrating provide well accurate module explore future depend sampling different phenotype cancer pair co expression currently impractical potentially imbalance strategy correlation determine non cancer stanford microarray database gene convert gene bend percentage effect calculation number calculate estimate dataset gene expression reduce sample size effect r calculate standard may reality enforce distribution correlation invert gene differentially express cancer positively set correlation I gene pair estimation valid estimation correlation cancer cluster differentially correlation cluster program complete euclidean simple existence correlation miss correspond second cluster gene case size differential process hc pair process separately meaningful module normally edges thus cluster keep module gene go biological process go hierarchy gene directly gene consider homogeneity model significance gene kb scan bind cut sum negative keep per sort predict factor module bind section first like support place want ph thank li zhang dr song manuscript thank go biology stay great friend frank dr zhang dr tu dr li dr xu dr zhang especially wu dedicate modern phenotype approach inherent phenotype make throughput phenotype propose method automate gene similarity robust different aspect technique phenotype phenotype complicate consequently gene subset sensitive phenotype tend novel increasingly gene simulate result perturbation sample phenotype performance phenotype cancer gene network module multiple gene cancer module module coordinate detect breast cancer tumor module gene important module throughput accumulate nucleotide number alternative phenotype association genome project post genome vast still remain largely question gene gene product interact identify interaction various cell gene change biology use poor question address advanced expression transform vast possible functional refer wide make information divide gene dna rna template production gene expression indicate time certain monitoring gene protein throughput several develop understand behavior particularly use magnitude many dna label finally store image process quantify intensity factor variability probe process undesirable effect intensity comparable comparable mean gene reflect introduction microarray principle phenotype observable phenotype activity environmental phenotype sometimes categorical phenotype unify medical language recent year language control science mapping structure give researcher ability translate terminology comprehensive concept network category relationship classify relate processing support tool system accumulation gene rapid large contain genome gene phenotype expression measure value phenotype manually record develop method combine type well phenotype indicate propose automate bridge search provide discrimination phenotype gene primarily phenotype involve study aspect phenotype focus phenotype mechanism difference phenotype difficult quantify throughput fashion lack comprehensive phenotype prevent chapter enable phenotype principle phenotype microarray microarray method cover confirm profile highly true description unique capability phenotype design disease profile factor inference direct comparison different method entire body platform microarray produce phenotype profile facilitate quality phenotype accumulation microarray datum gene phenotype gene differentially phenotype hand since microarray contain gene desirable identify phenotype couple unfortunately sensitive selection training sample gene usually tend overfitte supervise unsupervised increasingly gene combination iterative exist produce combination phenotype backward highly phenotype classification prove stable combination study phenotype integrate consist kind phenotype cluster truly discriminative enhanced technology differentially disease disease well phenotype individual thus important variation chapter cancer develop multiple microarray discover disease module propose dynamic topological module phenotype activate de activate many module consistent annotation module activate breast cancer tumor individual module associate never adopt perspective gene important tumor base provide insight complex characterize cancer cancer incorporate co expression dynamic predict phenotype pair value validation requirement match phenotype phenotype moderately phenotype agree among quantify phenotype measure description share description denote cosine angle map term identify pair threshold phenotype phenotype description highlight effectiveness exploit phenotype select dataset calculate original repeat removal dataset phenotype pp removal demonstrate derive average assign dataset generate prediction two describe various correlate correlation dataset platform predict highly correlate focus remarkably job separate separate capture traditional classification case describe phenotype collect phenotype uncorrelated figure order multi dim eps derive phenotype profile phenotype dataset individual pp train correlate recently merge excess share similar clinical biological regarded distinct disease derive pp help lead improved treatment set design gene phenotype profile patient would treatment serve method novel phenotype dataset phenotype confirm examine whose significantly profile number concept particularly interesting patient mutation profile significantly correlate know affect response profile demonstrate specificity phenotype utilize large microarray due phenotype many microarray thus inference
keyword incorporate kind specific superior offer elegant way keyword knowledge keyword framework keyword signal keyword dependency probability mass express model estimate maximum maximization active keyword joint keyword detect respective student mixture model variable people processing enable aid human home environment wherein multiple machine understand machine able speech recognize speech stream effective solution environmental large etc target hard challenge identify speech recognition speech scenario challenge different recognize target scenario wherein sentence record recognize letter digit signal wherein white although restrict author identify note performance interference sir speech complex recognize segment respective noted vocabulary suitable task drive home environment formulate keyword denote relate home environment create student pair pair contribution sec ii estimation em newly database home environment contain subset give signal passive assume keyword let introduce boolean keyword iff active frame iff p probability active frame jj two assume keyword active combinatorial jk k th jk denote collection k homogeneous one keyword ml jk u jk likelihood data data collection denote one repetition choose vocabulary distinct distinct moderately word inside record directional audio specification hz sample separate system build order albeit overlap sir relevant characterize mixture signal fig available propose category keyword eight frequency acceleration obtain shift keyword assess average percentage pair leave one validation explore clean ml use mixture phrase phrase detect correctly detect correctly detect correctly detect correctly c phrase phrase phrase detect correctly detect correctly detect correctly detect overall phrase phrase phrase detect detect detect correctly flat comparable detecting least correctly keyword pair henceforth experiment sample scalar value yield inactive active mm pt mixture recognition accuracy performance perfect one intuitively content model task detect error task small like confusion outside part scenario keyword answer detect initial expect keyword least
intermediate medical health either drive help indicate almost neither category motivation improve model predict low risk patient know status resource allocation clinical accordingly advanced compare approach patient goal classifier whether linear adaptive regression actual adaptive lr improve application application cutting case justify development effort practice improvement precision specificity general perspective result knowledge combine behavioral operational inference history effectiveness public satisfactory classical equally rely basic high outcome fact outside scope patient practice rely tool massive format medical leave datum contain create serious produce measure simultaneously miss maximization miss iterate public health produce characterize ideal risk assessment machine tool relevant clinical operational comprehensive store database consideration nature medical database continuously data acquisition effort cost predictive progress adapt clinical scalability rarely raw medical problematic entry inherently sparse clinical widely patient outcome motivate severe extent medical datum problematic perspective definition ever short use advanced overcome multiple integrate work project division clinical routine medium scope project combination patient clinical medical health plan aggregate metric patient risk treatment first motivate example feasibility merging claim clinical diagnostic risk addition thus make patient claim base use standard near neighbor empirically weight preliminary investigation clinical patient patient likely predictive logistic pose question would advanced imbalance set feature denote vector solve margin map misclassifie slack penalization soft svm usually transform problem implement tool reliable cope weighting build decision relative penalization class tucker radial rbf performance function bandwidth rbf achieve time consume classification scalable algorithm belong whose system multiple scale final solution combine information efficiently large construct approximated graph ann phase method coverage lead well ann suggest coarse support create number update optimize level refinement project coarse level easily adopt classifier issue prevent create small even majority method classification poor situation prior method domain occur datum completely feature datum occur instance dependent desirable many frequently imputation miss either directly purpose imputation imputation knn imputation imputation regression fit local em miss explore incomplete missing demonstrate achieve storage implement eq em relationship select miss imputation value optimize distribution complete control represented preserve evaluate performance confusion tp class negative tn acc sensitivity sn specificity namely uci rna real life refinement graph framework nearest neighbor typical fold cross validation setup create discard select datum selection base due superiority
entropy distribution typical fluctuation around centre mass replica framework replica breaking fluctuation separate yet learn probability target probability unknown increase move small treating otherwise orthogonality similar low imply must condition originally pure block part get lead positivity stability rs stability mm de de sup universit et paris france sup universit paris france target probability expectation compatible expectation bias give boost increase entropy inverse smoothly measure version space pointwise concentrate replica version corresponding qualitative target vary mean close compatible multi system beyond modelling define number degree possibility interested proceed consistent datum available yet inference parametrize distribution configuration distribution amenable computation largely I I informally possible compatible I propose alternative foundation mechanic illustration boltzmann back I knowledge I information theory reflect property alternative somewhat formulation operator compatible word possible I enjoy valuable sensitivity error I come biology literature history work carry consider take value straightforward entirely sum unity therefore pick hereafter polynomial prescribe admissible set admissible distribution contain distribution I flat admissible objective volume target I inter precise mathematically tractable statistical ensemble precisely uncorrelated realistic indeed reflect space compare adequate hypothesis consider average may reconstruct non negative scalar product typically informative scalar reconstruct correct small configuration distribution boost entropy continuously space infinity amount select I distribution depend bias I entropy apply within replica infinity rigorous check replica replica symmetry breaking fluctuation call mode expect large design sample small remarkably good agreement organize reader sec necessary sec calculation result numerical conclusion consist ise configuration hereafter target purpose lead term convenient introduce entropy curve system denote lie line tangent entropy per bound characterize dominant contribution illustration spin htbp curve mm let system label simply observable admissible hereafter define space logarithm I distribution eq lagrange multiplier enforce distribution consider spin spin I average multiplier coincide field pairwise act realistic perfectly affect tolerance hereafter introduce dirac delta sure admissible vector constraint term refer define probability measure purpose study I shannon entropy introduce define I hereafter spin quite contrast statistical analysis existence entropy since multiplicative eq expect calculate value replica hereafter outcome replica sec picture interest I mass distribution angular version distance center hereafter square distance represent fluctuation mass root square side rectangular lie observable observable mean variance quantify observable fluctuation square fluctuation average scaling due symmetry pure exponential average give dirac saddle saddle locate give agrees behaviour volume calculation replica method presence exponential scaling bias entropy impose non bias report sec sec sec major replica configuration probability marginal close depend later see configuration value dominant later infer boundary hereafter edge separate learn one remain observable tolerance change value ratio transition process htbp learn replica calculation configuration qr switch dominant sl phase separate label value denote phase turn get insight plot continuously reach agreement fluctuation volume largely manner ratio negligible tolerance phase fig grow negative expect become tolerance phase call agreement range take place interpret measurement fluctuation fluctuation performance entropy I dependence quantity importance negligible replica calculation show consequence I close pick distribution version fluctuation measure replica two place observe though transition contrary remain mention phase phase show right panel illustration cm phase tolerance tolerance fluctuation large take large become case negligible behaviour become irrelevant hence limit irrespective I enough compare fluctuation govern large irrelevant replica volume constraint introduce notation integral square replica moment perform make normalization irrelevant discard hereafter rewrite identity approximate saddle assume symmetry replica delta obtain integration assume appear vanish need find consistently analysis outcome meet case eq direct one treat put introduce natural sharp difference argument complementary dominate complementary sign get logarithmic formula expand saddle exponent saddle determine dominant contribution positive imply saddle feasibility compare width q nf saddle justify configuration dominant domain integration expand identical calculation expression indeed approximately decay function equation parameter get eqs match dominant remain meaning contribution probability negligible sec large configuration lead see long configuration accord accordingly ignore case consistently determine solve equation critical separate phase I critical associate iii check distinguish solution large phase case validity rate determine eqs get write hold iii iii stability rs integral call exponent write term change peak saddle approximation apply irrespective saddle mean edge configuration substitution expand configuration saddle dominant sec require normalization saddle edge configuration dominate imply determined correction valid yield trivial derivative eq omit saddle quantify around saddle consideration dominant correspond easy satisfied validity correct compare example p p accord relation apply value imply region solution coincide peak order expression satisfy saddle decay exponential respect large height rapidly peak consequence rapidly converge pure case eqs critical sec replica symmetric rs fluctuation detailed calculation sec stability fluctuation use eqs relation obtain interpretation stability rs become unstable probability configuration eqs irrespective phase marginally stable iii lead order iii tolerance regime sec rs phase marginally simple distribution space linear combination normalize version version instability appearance far place growth confirm calculation insight finite effect restrict tolerance throughout mc method update step random vector satisfy orthogonality fulfil initial condition orthogonality restrict move direction move positivity constraint intermediate may calculate accept reject unchanged along dash see convenient instead wave configuration component mode mode orthonormal basis note wave configuration choose observable random limit choose uniformly fouri mode fouri imply orthogonality soon fouri normalization easily exclude mode carlo markov detailed balance procedure quantity spin correlation target consider ten different plotted figure bar deviation carlo move irrespective show carlo step indicate equilibrium see
propose multi instance histogram wireless detect include medical video situation system discriminate experiment propose construct video belong dataset image detection classify end patch patch instance bag framework learn extract patch texture texture feature instance pool quantization histogram represent vector machine type conduct entire dataset overlapping fold fold set remain fold histogram instance framework histogram image train basic histogram histogram vector classifier please notice parameter turn exclude svm positive measure recall receiver roc value traditional histogram recall figure performance improvement original histogram feature employ function datum improvement increase classification sc sc validate appropriate roc class auc shown clearly method challenging highlight histogram search geometrically protein bind protein drug protein bind site histogram multi protein bind evaluate histogram bind site bind site protein bind site belong site vary select dataset dataset class could site htb query bind site bind protein site rank site site belong rank return bind bind site bag prototype bag feature bind site sparse code rank curve auc curve also measure rank roc histogram roc base discover auc given figure bind site system sc value compare sc histogram htb type function compose histogram reconstruction pool regularization code programming algorithm sc outperform previous future apply security powerful representation method attempt reconstruct sparse sc instance histogram histogram use reconstruction novel performance histogram instance image wireless video bind site retrieval encourage histogram code sc effective representation method try reconstruct new end minimize norm usually impose code vector norm regularization version multi learning traditional could many instance sample histogram sc histogram fact function leibl divergence function quantify error histogram metric paper problem loss replace propose especially formal programming discuss give instance learn vector instance th quantization histogram histogram nd dx ni sparse histogram histogram md dm mi traditional sparse code penalty could impose zero keep however smooth combine avoid norm notice go zero error sum objective trade basic please obtain regard substitute slack release slack directly turn slack slack sparse code optimize optimization iteration optimize fix fixing optimize vector vector code variable turn please variable long
network capture complex relation ibp give accuracy case affect ibp training size good improvement make augmentation improve even ibp aim solve ibp datum much see least ibp efficient dataset possible also plot plot demonstrate curve cifar curve achieve low error clear sign ibp dropout increase confirm add cifar cifar classifier small slope pure dropout give add cifar ibp advantage standard theoretically ibp ibp bp mnist cifar cifar see procedure batch augmentation experimentally confirm cifar shift scale rotation use implement augmentation simplify design estimate fully use cifar summarize far improvement tangent less ibp require structure obtain result unfortunately could significant improvement dataset ibp bp bp tangent bp ibp mnist cifar invariant backpropagation ibp backpropagation learning algorithm require pass derivative calculation derivative around might useful believe useful usage network area lemma vast transformation vector translation rotation variation vector backpropagation incorporate noise extension consist backward apply confirm theoretically establish demonstrate backpropagation network learn relation class also suffer overfitte technique crucially good number preserve label usually variation location image rotation area result speech knowledge process invariance robustness variation vector extension backpropagation combination call simply regularization time act network train training implement increase claim decay reduce learn early implicit regularization employ convolutional layer widely locality fully number give flexibility pure layer tune locality reduce next propose similar nature transformation invariance case cnn convolutional layer follow scale translation invariance variation way transformation sample convenient analytical moreover object also part obtain exist sharing cnns autoencoder variation architecture show object representation able deal rotation sift attempt transformation overview article qualitative comparison present invariant part practical implication part accordingly ibp pseudo code number layer activation map pixel multiplication non empty activation forward pass backpropagation specify layer prediction prediction achieve softmax transformation derivative simply derivative denote derivative differentiable computation activation whole reduce update specify usually time input space move direction classification prediction backward direction loss prediction length specify versa specify pass transformation invariance would need jacobian matrix autoencoder minimization change move class invariant aim good need look derivative output propagate later derivative activation backward pass also quite initialize perform three specifie step minimization find trade crucial algorithm bias notice backward pass propagate third compute bias pass implement contain write sigmoid ii linear iii except cause backward iy filter also immediately iy w dy rd pass layer fully map transformation propagation backward passes loss forward pass derivative activation activation backward linear max pooling propagate position derivative bias rule attempt use achieve author make neighbourhood small rotation tangent initialize network propagate output set vector towards direction vector derivative linearize add propagation appear propose end derivative invariant vector case derivative output rotation loss make robust variation opposite lead huge transformation express combination basic transformation rotation scale list tangent always incomplete general exist implementation tangent suggest require smoothness basic simple operator transformation require vector training repeat tangent tangent rotation perform training experiment backpropagation ibp want determine baseline consider dropout always value within evaluate modification mnist experiment batch exponential decrease function layer follow region within except normalize px maximum scale dimension cifar database epoch initial permutation initial rate benchmark describe mnist contain
pool subsample insensitive rotation global global pixel movement totally subsample metric even mahalanobi belong class face belong class e face people mahalanobis metric semi mahalanobis negative margin metric learning psd maximize close multiply distance scalar normalizing normally minimize constrain convenient later program slow state psd solver semidefinite cone perform solve manner relax psd constraint look solver map function property quadratic use auxiliary everything objective rewrite svm bias prove convert standard form solver solution semidefinite svm objective psd order solve run solver kernel thus avoid look benefit solver vector need store computation far rank many matrix algorithm solver effect include locally vanish rewrite forget finish equality optimal form iv equality first separable easily need add slack second add preprocesse previous run svm solver semidefinite quadratic solver case mnist result size currently quadratic use solve take program exclude semidefinite solver perform slow solver svm shift mnist digit comprise mahalanobis distance negative perform local example compare lead svm invariance add shift metric linear learn negative applicable mahalanobis svm shift intuition shift well variability divide subset one subset training image use align cosine svm shift image shift mahalanobis svm see improve albeit degree show mahalanobis metric query natural insensitive recognition insensitive implication important many tolerance primarily vector efficient mahalanobis metric learn metric algorithms neighbor find appropriate improve successfully face identification metric global mahalanobis matrix psd make distinction psd optimal limitation overcome approach nonetheless svm suffice behind object class mahalanobis try weakly supervise case positive query image bank train face image person unlike metric method need person metric twice look equality minimum taylor value interest mahalanobis mahalanobis costly decomposition dimensional regular semidefinite solver second
must linearly linearly guarantee linearly induction linearly unfortunately algorithm terminate construct matrix occur early termination succeed succeed linearly note choose initialization full co chance select select perfectly variable random variable hold subset exact gram select column column independent precise correspond recovery recall gram equal rank g equal state assumption exact recovery satisfied thm simply combine lemma lemma return terminate return column index column ssc rely dataset provide subspace make subspace reference point matrix union recovery reference subspace theory decomposition combine seed sampling face mnist leverage detection describe evaluation face pixel illumination hyperspectral spatial scene signature class dimension mnist handwritten fire neuron collect perform position target synthetic subspace overlap coordinate add create gaussian vector size point evaluation mnist fast omp fast representation experiment processor dataset run evaluation matlab approximation matrix utilize selection obtain matrix contrast base decay evaluate seed approximation iii iv leverage five achieve equal recovery linearly wide interestingly error seed roughly sample ii behind quickly iii leverage suggest contain structure well make significantly less seed representation signal self ssc consensus see sec dataset subspace sparse seed rectangular spectral think represent bi use method cluster introduce provide elegant relaxation bi co approach eventually find eigenvector matrix lead visualization embed union subspace red dot star feasible efficient seed capable separate seed neighbor weight motivate fig hyperspectral observe improvement sample seed noisy hyperspectral highly seed tolerance omp apply simple subset point point image compare performance seed randomly cluster error mean denoise seed obtain component raw ratio face image cut vs six face illumination face illumination condition full respectively single low dimensional subspace seed behind outli try subspace require contrast try collection signal exploit rank signal lie subspace whether outli self expressive diagonal alg ssc provide tolerance determine column upon threshold column dense case threshold segment bi multi however difficult threshold display ground datum via fig seed dataset corrupt outlier sparse seed subspace omp constrain outlier outlier seed column base determine appropriate rank structure outlier challenge column sparsity modal set explicitly threshold segment matrix seed provable omp solve range approximation outli seed recovery thm real world e obtain column recovery guarantee explore classification cluster contribution show expressive computer column selection approach expressive basis amenable independent exact thm imply select linearly problem prove strong ssc rank overlap direction future implementation sampling provide alg na I sec inefficient step addition calculate result previous formula rank alg vector criterion index denote block note invertible complement non zero case w k b b k eqn fast incoherent alg like thank discussion lee collect nsf nsf mh distinguish fellowship rgb k g aware reduction matrix aim find learn express self alternative introduce scalable computing expressive decomposition seed greedy incoherent form basis seed develop seed low subspace range denoise numerous result seed attractive complexity factorization expressive clustering subset recovery method combination basis simple provide extremely basis often mix geometric express element expression successfully classification provable discover idea expressive approach lrr represent contain zero along interpret expressive principled segmentation apply big challenge ssc construction affinity greedy ssc data affinity development dataset discover upon datum express approach self expressive seed select sequentially incoherent uncorrelated use call incoherence operate subset thus column entire gram second use vector first step compute omp seed detail sec alg demonstrate thm return capture datum linearly ii estimate remain stop aid solve include cluster denoise sec performance seed dataset neural result demonstrate seed scalable ssc fraction organize selection subspace introduce seed motivate study seed four application ii iii iv conclude appendix column concatenation let span say collection th ik dim subspace selection index approximate sense collection signal invertible recovery np force body study propose sample target ii leverage compute svd column approximation computing leverage approximated sample select column pi highly costly matrix computing operate column self expressive q represent reveal representative collection aid hyperspectral group norm within aim form consist method orthogonal pursuit omp subtract contribution atom repeat satisfied target reach residual alg termination either approximation coefficient vector residual signal vi learn ssc follow user expressive underlie ssc coefficient ssc matrix strength edge live affinity graph affinity subspace motivation underlie ssc consideration subspace ssc lead provable least span provide reference subspace sec lie union seed reference set subspace alg complexity seed select termination termination error sparse normalize solve omp stack seed column step form accelerate sequential incoherence column incoherent uncorrelated range machine learn use way find gram motivating image face illumination fig return varied illumination incoherent iteration return highly redundant illumination appendix alg implementation motivate suppose already index loss generality column nystr guarantee select linearly justify one via wide illumination whereas redundant new approximation write q span scalar measure poorly column computing span instead influence current greedy decide approximation correspond denote entry diagonal calculate change nystr approximation select user
remark summary recommender actually highly attack since since far small extract design profile statistical diverse general boost adaboost effectively improve conduct compare technique attack boost recommender variety suggest news book product recommender successful carefully attack term attack lead construct method crucial profile important attack attack formulate user attack traditional knn attack handle kind issue fail effectively aspect firstly statistical attack profile significantly source rating item rating extract possible classification become attack profile profile sophisticated make much easily classify general adaboost feature experimentally classical type specify appropriately comparable type adaboost technique hard adaboost employ weight gradually emphasis concern operator conjunction type learner improve attack experiment conduct alarm rate knn effectively give brief attack attack model analyze focused attack attack profile utility attack knn detect attack et try user detect attack also detection introduce rough theory attack user profile attribute overall attack wu et hybrid attack metric technique zhang capability base nevertheless attack addition et svm profile attack svm create profile decrease speak attack leave desire introduce design distinction attack profile attack secondly conventional particularly attack apply variant adaboost gradually emphasis concerned attack attack bias attack classify way attack attack attack rate low attack high target item attack profile form detail item singleton multiple call target attack attack rating rate minimum value rating function random rating utilize attack attack profile involve attack profile list model attack attack c c c c rating c nan mean nan around item randomly segment item reverse choose nan rating l power item rating item power rating normal c nr item nan l copy profile rating profile nan nr copy user profile nan attack popular item si max mini contain item reverse attack popular entire attack item aggregate score user rate attack centrality item high neighborhood item apply rate discard neighbor item similarity score item top nr item high attack aggregate select require least rate significance power neighborhood every significance discard score select nr select base number rating user overall introduction approach aspect feature extraction attack approach phase phase phase classifier via phase test construct attack profile attack attack attack etc generate mixed profile increase attack construct aim extent imbalance detail attack size attack size form feature extraction employ subsection extract profile characterize set composite retrieve generate result phase testing characterize user feature generally generic basic attempt discriminate attack profile profile type specific detect attack signature attack employ besides employ base additional specific maximum rating minimum rating item attack profile high deviation generic specific difference profile profile item rate score entire item rate user item rate rate rating rate rating rate rate minimum rate otherwise size rate entire rate raw profile transform attack know attack detection formulate conventional e knn handle kind issue prove boost weak learner fit gradually increase emphasis model weak learner model precisely interpret adaboost dictionary misclassifie normalization g adaboost loss boost numerical optimisation add weak learner gradient boost prove show slow lie slow hereafter lin xu gradient boost aforementioned controlling step variant like boost truncated novel improve numerical consequently capability boost good regard certain extent may possess learn boost idea adaboost initialization weight iteration find sum misclassifie tf g introduce experimental metric environment secondly impact compare svm knn attack attack mean set recommender collect researcher field collaborative attack recommender system movie rating mean movie derive website rating approximate besides rate attack profile attack table rate low attack attack analogous attack attack attack ensure item randomly attack attack select movie movie rate attack movie reverse randomly choose movie movie one user whole profile random nr attack diverse size attack dataset mixed training training attack profile exploit attack size size profile profile include attack attack effectiveness detection classification detection alarm divide detect divided alarm attack profile divide numerical ghz ram gb conduct list several diverse utilize maximization create generic specific aforementioned relationship extract feature reverse attack false alarm diagram attack fix illustrate design svm knn test validate follow default profile classify unseen knn knn algorithm fold pearson correlation utilize build week learner fold equally spaced boost fig surface aforementione nr attack example different attack size surface knn present rise attack svm effectively attack attack fairly classification attack also accuracy almost produce little concerned profile detect naturally knn essentially svm attack low much detection false alarm rate knn fail within attack nr attack attack attack nr knn attack knn indicate attack profile profile attack profile knn boost improve iteratively correction adaboost enhance detection false alarm
otherwise general feedforward adjacent unit layer weight activation network label probable minimize posteriori map online arrive sequentially update q intractable store field marginal appendix like marginal likelihood contains generally intractable summation summation approximation assume fan fan normalize neural quite distribution end p ij directly calculate taylor around used expansion pass backpropagation pass pass initialize follow value input layer bayes initialize refer q learn output define alternatively eq value parameterize way process parametrization backward substitute iteration show step weight configuration I l ji h ij evaluate task binary examine multiclass deep fan layer hide convert mnist handwritten digits method method spatial configuration similar cnn unit receive input layer unit connect input therefore cnn map connection implementation unit element whole fan map neighborhood report handwritten digits contain set sequentially identify classifier training recommend backpropagation neuron treat image vector neuron hide add architecture hide neuron layer configuration architecture filter method filter layer unit output select configuration large well hide network configuration configuration lead fan unit size layer second layer layer vector employ architecture dropout network efficiently dropout demonstrate neural backpropagation gradient method investigate hidden unit dropout net presentation presentation deterministic sect sect train epoch performance table mnist observe good layer outperform p outperform grow slightly well although hide one increase unit standard optimize hidden fan fan fan network compare structure table result show dropout configuration hide layer dropout increase besides b reasonable validate dropout prevent overfitte c c unit algorithm performance binary unit bad input contrary performance dropout table input output layer method light extension network block connect convolutional b finding algorithm task performance real fan hundred well improve mnist report backpropagation evaluated dropout spatial weight study classification validate image dataset cifar neural use explore initialization acknowledgment visit
nearly define concave lemma log concave simulate annealing proceed epoch sampler annealing arise convex fy ii ki x j run transformation warm guarantee anneal warm distribution successive epoch round guarantee away warm next epoch concave density support next account impact warm let approximately concave suppose algorithm step prescribe epoch anneal walk round near epoch follow theorem convex prove hold improvement almost instead concentration heavy tail isotropic see bernstein inequalities log concave measure way achieve goal invoke random row isotropic almost surely since isotropic translate log concave isotropic final temperature annealing need oracle informally per epoch corollary approximately epoch unknown lipschitz aim sub chernoff bind decrease repeatedly average observation concentrate randomized optimization value upon visit well precise three return twice query equivalent version optimization problem take box net return information net query parametrize random gaussian call give oracle denote q lipschitz random second affect oracle dependence optimize per oracle optimization union exponential somewhat artificial draw spatial alternatively could union visit function convexity decrease minimum decrease break optimization stage optimize region guarantee convexity aware problem surprising provable consider decrease complexity decrease formalize discussion convex non radius mind property fall order simple target additionally deal constraint handle smooth generic forecaster repeat suppose forecaster observe observe forecaster define mapping vast majority write follow relaxation relaxation call sequential involve gradient might approximately evaluate saddle point proof function view want point function loop maintain either length interval satisfy case essentially two case argue remove still enough hard point support convex fact similarly terminate log view consider interval current continue gx l x l claim terminate soon interval imply gx terminate concave associated level moreover either imply e stationary distribution scheme accord restrict classic reject page error variation restrict quantify tail event sample define shorthand eq bound proof main theorem variation walk start u produce see accord along let truncate distance elliptical tv hold eq compose truncate fold iterate run satisfie induction eq follow closely argument proof device random define tb yx inside suppose assume define q imply follow since although sx hx next contain ball first page lemma q take induction end identify epoch prove claim arbitrary give sake completeness relate arbitrary convex increase linear complete claim hard add effect eq proof theorem time round however close guarantee need oracle complexity term logarithmic distribution log concave distribution implementation algorithm anneal first essentially motivate optimization method produce induce decay minimizer program approximate dynamic particular factor require amount convexity optimum computation access may function oracle value return subgaussian probability oracle know detail motivation study problem derivative essential sketch area readily private programming algorithm present invoke run anneal approximately fast approximately log line sampling concave already early walk discretization motivate central theorem lead walk strategy walk long step motivate somewhat hard pyramid evaluation hence walk present reduce achieve optimal include simple evaluation mention boundary denote negative denote concave log gap long high dimension establish less section section induce run generate step initialization choose pass act sphere line line address cause deviation include sampler search yield build optimization problem line segment function segment produce function e gx method concave initialization show gx gx c gx x fx stop concave lipschitz convex find step initialization find desire distribution furthermore query approximate concave particular concave accord analysis round mix spectral gap markov turn spectral relate call
interpretation mutual show comparable modify optimize log somewhat word hyperparameter embed conjecture corpus eliminate occurrence word less text context count function approximate transform differ form fw dot see try fw information order occur optimize pair corpus monotonically rank advantage rather instead model order fit analogy word find word problem occurrence noisy due word co phenomenon document corpus diverse help effect token use word context occur context word inner convenience embed c product large want rank parametrize sort specific context write follow machine replace popular hinge loss enable upper certainly desirable list minimize quantify monotonically goodness model later assign versa ranking consider monotonically concave function monotonicity concavity imply loss sensitive sensitive list interested context co bottom make either due four alternative interpretation relate reference plug replace sgd sampling around linearize concavity bind tight motivate parameter due also minimize optimize unlike admit sgd c unbiased use put everything detail another line corpus hand complexity two update computationally involve multiplication via step c w converge converge remarkable update another triplet perform key optimization multiple lack include code indicate proportional c monotonically consequently context list give view parameter view modify ranking problem fashion appendix give word introduction work perspective rank score retrieval rank indicator rank observe discount special one increase concave efficiently apply across processor work distribute stochastic impact rank small dataset closely follow combine token consist wikipedia tokens token benchmark token around tokens wikipedia article already plain format processing sentence corpus stanford character discard short token long token token token small corpus extract token must window follow finding symmetric decrease word apart contribute count specifically corpus corpora large window corpus corpus vocabulary k dataset embed boost combination ensemble later interpretation cosine add add experiment word analogy token word substantially employ word analogy six word rare word together human evaluate similarity task google dataset question syntactic question contain five capital people syntactic question nine opposite comparative question correctly select word thus mistake namely score using indicate give vocabulary reporting vocabulary question zero make across experiment argue present function report million dataset table note trend analogy set comparison include unweighted attribute performance implementation text token extension text token test original update google analogy function perform similarity analogy large dataset scale good configuration weighting comparison input occurrence embed dimension train skip art nlp default suggest produce well occurrence corpus small three follow large corpus eventually word analogy consistent finding indicate token case word similarity optimize rank token score somewhat close report token discrepancy processing corpora similarity train word dataset google analogy syntactic result list table google htb token task google semantic syntactic word analogy token achieve appendix result performance analogy achieve loss analogy therefore important emphasize expensive setting extra robust ranking supplementary material worker mutually exclusive exhaustive approximately sized w q begin outer partition context induce context update
passive ep coefficient passive ep illustrate ms ms time ms ms ms ms error passive time vary computational list recovery highlight fast corrupt list linear algorithm suggest ep use improvement bit recover regard hinge many perform robust bit improve recovery bit hinge linear solve ep generalizes propose suitable ep trade hinge improve advance relate quantization implement operate bit compressive attractive model function function hinge bit classification measurement sign hinge binary bit experiment hinge cs observation bridge bit hard thresholde binary hard suitable sparsity true elastic net machine passive propose suitable ascent propose solve prove trade hinge loss improve exist bit dual digital quantization extreme measurement need benefit hardware low signal analog sign number bit find sign q measurement system measurement direction magnitude measurement e lose quantization make assumption meaning bit recovery explain partition hyperplane number hyperplane feasible recovery subset additional assumption unique tell measurement rarely consider application advantage attract cs try sparse sign small measurement cs cs fundamental bit cs count convex algorithm approximately variant hold real accurately bit noise component transmission sign magnitude true analog sign transmission among mind deal sign true empty utilize hinge sign change hinge attempt minimize hinge loss robust algorithm review ii cs attractive condition bit nonlinear heavy bit sign recover regard binary problem hinge widely svm hinge calibration task linear loss rarely enjoy yet linear quite hinge hinge cs closely definition characterize provide bridge loss hinge regular task cs hence expect trade bit norm unit non constraint consider name effectively ep ascent show optimum exist cs section iii introduce conclusion end vi bit introduce attract hard minimize nonsmooth hull bit sphere sphere constraint constraint model reformulate programming become satisfied however solution project sphere project sphere mention binary change application bit cs infeasible feasible feasible classifier sign loss bit signal approximately loss understand replace one sided side iterative one robustness sign improve robustness measure deal change estimation approximately convex propose equivalently put come play model loss rare task rule yet norm hinge observation task motivate special cs hinge bit analog near hinge distance correctly classify correctly contribute optimal measurement sign contribute optimal solution many idea incorrectly classify datum classify example encourage large well influence sign replace transmission detecting measurement sign performance sign counter performance two different number binary well algorithm performance algorithm able detect mention able detect detect analog noise snr quantization flip quantization green dot dash dot solid respectively sign change sign four performance bad main happen mainly sign fig confirm change differently sign main cs leave modification advanced loss naturally replace sided establish robust illustrate solve guarantee global derive specifically convex term model experience task expect suitably may performance great analytic ascent indicator primal problem follow u passive primal get sphere imply small separable apply ascent efficiently subproblem subproblem separable parallel via subproblem let increase u previous discussion give coordinate ascent end analytical passive coordinate ascent ep optimal optimality optimality condition coordinate j coordinate j u j j optimal theorem svm passive matter happen choose maximum number choose stop small though passive analytical problem use ep similar way experiment flip evaluated choose experiment average recovery plot htbp c experiment performance measurement performance contour
htb kl message length analytical calculate derive appendix compute parameter minimum message encode previously table list metric mml across combination mml notice extra compression make mml song mml song mml e e e e maximum mml base reduce mml estimate always mml htb l song mml song mml e e e e e goodness behaviour generic derive score degree compute test exceed critical nan conduct know infer table consequently imply nan reject use incorrect distribution hypothesis htb song mml song mml e e e e e e e test mml behaviour mml size plot demonstrate low variation behaviour irrespective continue value produce low mml actually perform bad song due mml identical approximation accurate limit f depict increase note produce highlight mml fundamentally show note extremely rate limit mml coincide compare mml behaviour dimensionality amount ml mml one htb empirical propose method mixture gradually increase simulation mixture direction true component parameter mixture mix proportion component align axis illustrate variation infer component concentration angular separation become datum correctly concentration concentration angular infer search even different easier whose concentration mixture component present average number component concentration chemical directional protein atom motivate structural modelling generate protein three protein alignment secondary pattern encode protein offer serve varied redundant experimentally protein publicly version coordinate transform directional characterize co measure associated consider precede comprise transform translate origin lie x axis xy yield orientation previous coordinate use coordinate direction repeat transformation successive protein result total structure database protein component infer model comprise protein search mml mixture model structure correspond mml base mixture mml result empirical directional belong frequency protein major mode correspond secondary structural middle direction chain exclude due truncate model point would code protein possible rare empirical mode mml allow compete recently nan nan atom use precede highly compression gain orientation atom surface sphere equal area cell surface sphere uniquely encode encode description state length orientation angle sphere directional protein build mixture descriptor encode angle length orientation correspond unit describe surface equation two compete directional structure htb mml bit uniform divide message length statistic encode potentially various protein modelling introduce robust infer mixture ii von minimum length provide tradeoff model message length search effectiveness directional well current would thank protein provide author like acknowledge l technology involve inference component discuss unsupervise use mml criterion demonstrate effectiveness search parameter key handle fundamentally different type mixture modelling euclidean multivariate address model directional unit contribution addition methodology mml expression von fisher derivation mml test simulated mixture experimentally determine bioinformatic demonstrate search encode state model mixture model offer explain field biology engineering economic amongst compose model kind model aid hide sound probabilistic model extensively machine mixture component respectively sum determine number mixture mixture difficult balance objective determine data parameter fit strategy use control assess mixture ability message length bayesian method prove effective concern model widely poisson von comprehensive summary research partly motivated mixture von sound justification compare decide component traditionally estimation work mml principle unlike invariant unlike mml mml mml estimate scheme cost state parameter continuous precision encoding state map mml process precision message thus length mml mixture challenge model limitation mml incomplete drawback propose mml formulation select formulation demonstrate effectiveness estimation use mixture extend relevant directional use von fisher establish component estimating rely boundary attempt mml simplify matrix diagonal mixture far different number component select good search iteratively begin redundant error search component mml result sensible component component child merge start number component iteration avoid unless require mml expression length estimate discuss model directional von fisher mine significant scenario science physics directional several surface sphere ellipsoid fundamental von symmetric unit mean direction random dimensional kind mathematical use many perform inaccurate demonstrate experiment conduct size evident mml section mml reliable art mml use von circular von demonstrate protein angle text analysis cosine text evidence compare statistical bernoulli widely package parameter mml mml candidate mml mixture version effectiveness implement modelling allow mml framework optimal effectiveness mml protein mml estimate multivariate mml select mixture component depict competitive conduct mml support application section function matrix traditional give maximum likelihood solving equation estimate covariance unbiased give likelihood involve conjugate literature unbiased von fisher obtain equation estimate difficulty analytically solve equation improvement respective improvement provide easy demonstrate well start equation utilize fix conjunction interpolation point heuristic provide perform newton result order taylor series accurate estimate demonstrate iterative truncate iterate propose likelihood govern concentration considerable counter minimum base result unbiased several compete mml mml extend work mml estimator generic exist develop rigorous compete datum mml information theory whose probability length event shannon insight length result compete odd compete encode comprise two take bit bit complexity explain state fit mml paradigm infer several gaussian vector give datum involve choose evaluating length mml lattice quantization mml framework compose part encoding encode mml key mml estimation ignore measure parameter finite precision mml incorporate determine region multiply message encode mml formulate derive von mml precision prior dimension wishart density q computation fisher fisher approximate fisher analytical element correspond c describe due multiply derive message encode substitute mml need minimize mml mml need compute mml observe mml preference traditional parameter explore mml estimator dimensional mml inference formulate generic reasonable support evidence parameter suggest uniform fisher general dimensional equation net message mml equation estimate minimize mml linear discuss comment approximation experimental guess iterate mml appendix mml newton mml mml observe distribution mixture observe data mixture log eq j traditionally standard em standard maximize equation closed gradient employ achieve step fractional membership partial conditional belonging assume expectation log membership maximum likelihood iteration step estimate sum ir update parameter describe methodology involve mml function em discussion describe behind model mml encoding message mml require encode parameter cost encode datum decompose encoding encode message require absence like model belief decrease bit uniform within prior minimal magnitude treat expression encoding encoding state precision precision dimensional encode message mixture note lattice quantization equation parameter update constraint derivation mml update appendix use membership component intermediate mml update ir mml obtain solve successive less firstly optimize correspond include correspond state secondly ml whereas update mml component reasonable initially strategy well amongst converge covariance singular member number numerous attempt infinitely mixture method aim cost associate end penalty certain explain simple add variant aic suggest penalty aic multiple free coincide aic serve model fit formulation adopt type statement associate free cost characterize proper criterion mml wherein part message multiply formulation mml formulation argue task potentially grow proportion mml bic determine mixture aic bic mixture variate mixture complexity mml formulation incorporate discuss mml scoring author gaussians fail method rigorous one length scoring derive tradeoff fit likelihood hyperparameter pre define compute use hessian equivalent empirical identifying variate prior prior assume element assume consider joint p h previously discuss propose scoring density covariance ignore fisher dependent mml gaussians hessian diagonal eigen classic mml component run em within optimal cl likelihood equation arise mean entropy cl parameter score model good amongst choose mml criterion formulate score two message encoding free component weight assume component jeffreys describe density encode mml scheme encoding parameter use parameter detail make follow encoding assume length vector treat independent jeffreys prior jeffreys ignore jeffreys note formulation difficulty compute length greatly notice weight number bic discuss highly mml entire essence goodness accurately em point component consequently component search mml however remark search allocate component ignore subsequent assign hence bind reduce assign consider scenario mixture equal mix proportion infer wrong relevant component decrease optimum update low increase subsequent attempt robust place overhead handle component achieve true rigorously mml adopt simplify approximation mml formulation message formulation fisher distribution section trial elegant comparative superior method outperform base regard alternate heuristic infer limitation infinitely mml mixture overall see mixture namely mixture parameter propose mml unsupervised operation htb current split component well merge current search mixture suboptimal smaller perturb new new retain split great line first two optimize algorithm component integrate optimize reason rather already similarly component mixture record merge match divergence mixture evaluate mixture merge initially start child optimize split well retain splitting component perturbation improve current repeat improve provide observed membership execution membership adjust subsequently achieve optimum membership operation membership component index current generate amongst component fractional membership carry assume membership remain component carry sub state subsequent update goal identify distinct within provide maximum variance parent component side parent ensure reasonably apart serve good optimize membership membership child computed component component maximum characteristic mixture adopt distribute randomly membership mixture initialize update membership belong child child describe effective membership eq child substitute update give equation eq mixture parameter equation update equation since mixture multiply influence child integrate component usually member exploit start upon integration ij new follow initialize membership start estimate maximization component perturbation component result new goal remove current mixture check whether mixture explain component weight membership good r ij mml update ensure see initialize membership possible complete membership membership perturbation result new explanatory improvement join component improve merging runtime another merging identify compute kullback kl runtime constant close match identify merge component involve membership optimize component merge let form merging pair integrate mr merged component merge ir merge component merge membership message length perturbation merging result current mixture propose consider gaussian component simulate point strategy search various detail begin infer split step explanation fig depict dotted mean dot initialize use child length update mean black dot denote mixture post em phase bit iteratively merge show splitting first component show merging first identify close kl merge operation improvement operation initialization operation update fig message length discard merge perturbation total message terminate htb green denote mixture post bit htb blue component merge merge along parameter optimize bit stage intermediate mixture employ consider option possibility getting optimum propose balance tradeoff useful prior knowledge nature way appendix evolution infer explore overlap terminate demonstrate optimum number infer length show length algorithm converge vary length curve drastically initially start length gradually clearly suggest encode parameter complex decrease length mml evaluation criterion message comprise statement state complexity mixture parameter part increase increase illustrate message length axis fitting decrease message mml behaviour consistent sharp error datum overfitte dominate message optimum number metric monotonically component mml illustrate methodology cite information integrate discuss far section experimental true time mixture mixture mml allow length encode message odd mixture give mixture explain methodology score mml scoring infer score mixture search optimize score expect mml mml mml method however scoring demonstrate superior lead define prof use kl metric give two distribution metric relate simulation kl expression experiment mix proportion bivariate infer experiment separation gradually increase percentage simulation determine two htb correctly data b illustrate separation number correct message divergence infer therefore experimental performance roughly apparent similar close separation mean infer table depict fig many overfitte correctness mixture infer comparison difference message mixture propose infer simulation mml mml function result optimize mml mml function bivariate across close divergence kl denote red zero compare vary value kl divergence zero variation kl suggest mixture employ mml scoring htb along line conduct proportion distance plot htb component show infer search low correctly infer quality mixture message kl infer low message fig message length suggest search sub optimal mixture fail analyze kl bivariate fig infer kl value kl component however htb gradually increase average infer component simulation fig show infer average
mp mp generalize lipschitz high compute stop feasible compare trial weight computing consuming consider linear current trial atom svd efficient low storage suppose tensor combination rank store store help break curse mp completion iteratively redundant completion prove generalize generalize select atom nonconvex successive sr sr trial see sr strategie sr either intractable need alternate allow closely wolfe constrain recent cg cg atom trial lie convex significant cg constraint cg constrain counterpart nuclear norm exactly approximation computational derive approximation power type normalize singular inexact pair correspond lead th tensor recursion subroutine recursion although complexity lead singular svd go acceptable compute normalize pair x establish tensor recursion normalize step subroutine also unfold induction v induction k inequality relationship frobenius subroutine subroutine get x obtain di x course subroutine apply subroutine trade computational cost time discuss subroutine subroutine subroutine together subroutine method base space svd require high whose odd always tensor key tensor unfold organized subsection extend tensor convergence l l uniformly value tensor w subroutine l l q l clarity k k k naturally term k r w relationship frobenius multilinear letting form stacking row task represented unfold write convergence positive uniformly large eigenvalue generate subroutine l ls denote r w l l k f naturally hold f km recall numerator f k however every positive definite size follow add rewrite square symmetric tc f assume ls ls subsection conduct function end characterize denote finite assumption restrictive variety huber fair define huber parameter huber nonconvex loss mention still begin analysis completion satisfy assumption say tensor derive hadamard e define loss deviation cost completion completion w w w k k follow boundedness uniformly hand tell k consequence q diagonal diagonal ii unfold q act remove deviation x apply multilinear assume bound eigenvalue mp ls w q k w f recalling recall f last complete make modification discuss x th column linear multilinear learning assumption satisfy define inequality improve synthetic well focus tensor completion numerical conduct intel support loss prior fp solve value convex relaxation tensor completion norm fp tensor factorization use sect criterion generate ten tensor randomly mr report particularly mr perform mr value table show efficient mr optimize use matlab three two three mr fp htbp r mr e e e e e e e e e e e e e e choose hyperspectral brain image hyperspectral tensor rd fp consume due case less particularly hyperspectral mr relative size take performance fp mr might consume slow speedup compare fp useful miss intuitively performance method miss fp mr fp art nuclear latent norm scale nonconvex nuclear nonconvex regularization treat stop tune validation specifically follow available education student student school indicate bias task index school year index therefore jointly learn mse var performance restaurant dataset rating restaurant rating aspect restaurant space index aspect tensor mse compare school restaurant dataset school well accordance method still bad eventually slightly view bar mse efficiency around iteration give desirable tensor scale school restaurant examine effectiveness completion cauchy loss employ robust completion robust criterion previous randomly mr vary entry contaminate outlier fig mr mr increase rapidly experiment method give tensor treat directly consider setting miss image recover g short paragraph seem remove recovered tensor cp tensor th image I row recover entry store recover store speak storage ccccc examine axis iteration stand fig ls l red ls fig plot plot robust curve confirm derive sect tensor completion logarithm axis confirm art storage cost sharing convergence dataset receive european research european fp author contain grant grant project grant project medical science policy office definition subroutine multilinear propose pursuit cost matching type large problem store tensor storage help break curse dimensionality various circumstance provide approximately compute tensor provable help analyze experimental synthetic effectiveness key pursuit learn nonconvex tensor appear generalization vector make represent problem tensor goal rank tensor provide allow pursuit several task represent lie information high rank learn low tensor encourage rank widely mode nuclear encourage tensor low tucker scale many exist main rely singular decomposition category several factor minimization avoid tensor algorithm solve scalable scale motivate efficient mp tensor propose pursuit mp update trial combination namely cp tucker define
equation markov expectation development demonstrate easy simulation adopt bayesian expectation section contains demonstrate conduct future difficulty deal expensive feasibility issue transition mcmc langevin hamiltonian employ finite batch long induce convergence practical approach difficult mixing use hypothesis might confident accept reject decision substantially construction quantification one towards induce original explicitly respectively orthogonal expectation contrast construction require elegant exploit additional computationally due size subsequent asymptotically availability appropriate tight likelihood precisely bayesian investigate promise generality applicability mcmc mcmc complexity mix mean evaluation iteration require chain propose provably sub number unbiased low likelihood several expectation available likelihood pseudo section coherent big exploit device develop approach unbiased expectation intractable infinite expansion attack arise contribution complement monte carlo procedure static target partial target px tn pl subscript construct increase batch whole exposition geometric batch set possible small consider assume integer use experiment present device component transform expectation posterior lemma provide way construct unbiased estimator converge evaluate correct bias different present recently hilbert space convention moreover applicable finite regime truncation variable replace sum estimator require intuitively tail match expectation simple setup along truncation fashion replicate procedure reduce variance copy scheme repeat empirical illustrate discrete corresponding number replication desire tolerance truncation variable tn explain key methodology datum I I compute stop posterior dot dot line connect procedure dot dot list advantage something think generate estimator mcmc chain batch evaluation result l cost reflect amount single core partial posterior expectation order te stay large bound expectation e speed fit tune complexity namely budget result replication work figure see correct asymptotic finite markov overall corrupted practice careful burn run reduce bias way address asymptotically give sequence unbiased expectation unfortunately expression unbiased line partial expectation noise augmentation acceptable sharp exist scheme decade mathematical engineering effort methodology software package expectation mini size decrease posterior often structure independently expectation true replication batch size roughly computational resource speed practice mcmc accurately clear produce sample burn estimate approach fair comparison likelihood evaluation mcmc notable posteriori inferior challenge standard optimisation evaluation example bfgs commonly benchmark iteration reasonable somewhat map estimate one sufficient avoid challenge extremely base indicator variable point mcmc dark obstacle dark point point need suffer compare fair replication take median likelihood sum replication already converge ground stress extremely conservative appropriate computational mean reduce bar additional outli convergence replication behave similarly bar line ground plot replication correspond evaluation extension compare experiment inference large unbiased expectation require eq simply access sharp computable form prohibitive typical gp eq inversion mcmc suffice look posterior cost ap still infeasible practice apply combination induce perform fix mini batch toy map feature choose mapping eq covariate spread add observation predictive mean explore partial repeatedly observation top show mse get zero almost unlike functional corresponding average posterior top prediction multiple mini batch reveal knowledge none sub scheme therefore resort compare inference feature subsample bar note mse eventually vanish compare replication mean cost slice plot prediction size mini batch size give zero approach inference gp combine gps descent huge streaming allow cut induce combination fourier predict delay record involve consist label covariance via fourier sake apply include centre preprocesse essential however adapt covariance match induce gp match computational roughly replication iteration remarkably rooted rmse conclude tune experimental protocol etc instead make achieve highly leave variational convergence mini batch random report hyper average reproduce rmse chosen obtain rmse constant batch bar repetition formalism streaming scenario expectation unable force batch discard still possible fix budget large process hardware restriction replace truncation still stream fully constant mean estimation gaussian biased bar batch make replication conceptual address unbiased need assign truncation result expectation runtime also result due resource limit arise covariance matrix side partial posterior exceed memory allow truncation develop solution computation outperform connection truncation functional dataset aim weight note dataset take figure top partial tend figure behave partial truncation relatively happen around sampling yet reach acceptable regression look around perspective many serious arise employ transition chain simulation statistic partial posterior implement exploit exist parameter furthermore conduct accurately compete simulation methodology likelihood carry experimental stochastic area future explore tradeoff detail deal use iii thorough formal show increase batch variable variance ratio express evaluation truncation tn truncation normalize constant z addition depend convergence partial full partial point estimate almost sequence posterior suffice grow undesirable remarkably slow partial expectation moment remain variance remains bound large rate quickly converge independently fit practice investigate partial comparison among posterior stress small variance figure determine derivation select key bayesian expectation monte consistently expectation average sample feasibility goal scalable full straightforward leading
one see approach semidefinite relaxation maximization semidefinite q anchor estimator give approximation submatrix sensor sensor relax constraint rank via last equivalent formulation introduce enyi uniformly increase performance essentially anchor sdp formulation show superior sdp xy message pass practice would choose formulation pass expensive hide offset outside plant average experiment classic rank inconsistent numerous vision rotation usefulness demonstrate numerous recent graphic sensor localization biology semidefinite sdp round numerical college game propose ranking guarantee synchronization amount ground synchronization aggregation approach art rank truth propose sdp synchronization technique subgraph locally ranking identify small player pairwise comparison noisy identify relate open question angular synchronization semidefinite programming ranking least angular rank eigenvector sdp synchronization aggregation section hide ranking player ordinal pairwise comparison application information especially noisy fraction inconsistent ordering total consistent instance meet circumstance outcome cycle seek rank player rank comparison even incomplete considerably distribute affect procedure noise recover partial consistent investigate possibility clique dense subgraph graph scenario dynamic structural exploit aware citation modern efficient sort main especially modern internet application engine google feedback amazon crowdsource individual human popular netflix college economic system item worth exchange reciprocal matrix reciprocal pair aggregation asset pricing market universal perhaps noisy offset note economic triangular traditional theory prove datum ordinal numerical true system somewhat preference movie movie rather application outcome often via team line reflect intensity propose preference microsoft assign online outcome engine update level underlying inherent assume early player pairwise order perhaps popular date google rank web relevance structure web note website spirit identify user assign page high page weight page high another output order adaptively assume available reveal somewhat like event strength compete ranking mle datum idea explore social pl utility numerous pricing much relate focus refer liu comprehensive tool community another boost preference base boost machine et propose parameter break parameter moment break ranking pairwise comparison synchronization author break together parameter propose adaptively certain recover choose computer science literature every meet encode preference np pair player meet seek rank come huber row et seek rank angular aside work angular synchronization explore yu embed compare embed later traditional formulation come impose additional satisfactory result image utilize iterative rank estimating encodes outcome aggregation rating compare address permutation inference comparison available less realistic recent summarize make similarity item assume decrease along furthermore demonstrate impose contribution summarize explicit connection rank angular spectral relaxation robust ranking numerical rank literature variety graph numerical pairwise addition compare outcome game english microsoft game college game two recently state art aside traditional simple singular independent interest currently investigate art player prescribe adjust approach rank aggregation inconsistent pairwise produce finally ranking extract make angular synchronization sdp relaxation robust ranking semi supervise player also integrate information provide incomplete inconsistent pairwise player single numerical exist across comparison player algorithm aside traditional furthermore decomposition theoretically separate art method graph also outcome english microsoft computer college identify ranking extract advantage ranking stem simple may come comparison exist group translate noise allow almost recovery underlie truth point angle angular condition perfect ranking ordering angle preserve paper serial algorithm angular synchronization literature describe rank eigenvector sdp method aggregation sdp relaxation angular synchronization problem problem sdp synchronization section variation open relate propose extract comparison serial english briefly serial perform compare method summarize centrality et aggregation discuss rating make amenable ordinal singular decomposition svd popular square l player pairwise player player construct rely recover ranking another related ordering goal recover similarity summarize additional rank svd stem observation noiseless one column random perturbation ranking choose singular sign order naive approach comparable recent serial centrality explore relational develop interactive refinement aware svd research svd rank random note enyi method amenable analysis word skew decompose skew zero decomposable rank skew investigation decomposition noise similar entry matrix long limit dependency comparison whenever enough e ranking setting common highly master much investigate additional amenable light random literature relax dependent dependency nearby ranking square vertex incidence vector contain least solution rank et show square graph far reach various area theory graph laplacian systems topology et item encode outcome aggregation pairwise comparison player set member goal rank often across match player player never play word chance frequency reflect rank distribution associate interpretation science centrality design structure within application web dynamic propose temporal human imaging markov adjust applicable rating system equal player player otherwise player assume w associate start player player long next transition result denote node make sure stationary top entry score sort applicable rating system ordinal alternative win inherently win transition stationary average result winning hybrid approach centrality serial matching comparison third reference intuition behind player final solution ij two close proxy difference win player unlikely play match one two player player mind spirit design win meet balance favor assign win keeping proxy eq version centrality rating system well centrality measurement win incorporate score intuition behind win large possible winning large small possible mind win next play find ratio synchronization group estimate matrix ratio represent confidence noisy relaxation solve instance angular synchronization rotation angle eq difficulty amount subset available element ratio realize edge vertex correspond available edge measurement probably use random complete relaxation angular eigenvector hermitian soon eigenvector successfully recover angle measurement available enyi phenomenon encounter soon build hermitian element preserve angle perfectly e assignment exploit frobenius rely non follow replace individual magnitude weak maximization form eigenvector hermitian matrix orthonormal v relaxation large estimate hermitian v angle rotation angle additive eigenvector angular synchronization normalization eigenvector consider previous formalize normalize bottom unknown rounding via semidefinite programming attempt preserve angle good may follow maximization hermitian give diagonal exception angular synchronization remark size distribute alternate multiplier large problem statistic point cut finding difference fact optimize matrix estimator cholesky rank sdp ranking phenomenon noisy favorable sdp program able explain sdp relaxation bring impose magnitude constraint enforce via angular ranking suggest similar denoise angular embedding observe robustness measurement recover rotation motivating compute circular ranking minimize initially pairwise player lie comparison noiseless ordinal angular embedding rank player circle say angle correspond angle imagine player around angle angular angular synchronization around circle play role choose map circle would cause ambiguity end highly rank map unit ideal synchronization player upper circle anti direction however solution angular synchronization problem plot figure post processing step accurately underlie ordering match end circular ranking minimize illustrate step instance plot ranking induce angle recover synchronization show ranking angle angular synchronization sort label position denote associate outer vector hadamard edge offset cyclic shift repeatedly circular shift tuple circular shift tuple circular norm take account sign enyi measurement initially angle angular synchronization angle respectively recover truth rank simple angular synchronization around say offset eigenvector cyclic example wrong figure ranking adjust permutation rank synchronization rely sdp relaxation matrix pairwise comparison noisy comparison transformation build hermitian h e angular synchronization sdp recover angle increase order good circular permutation comparison output induce ranking different l glm enyi measurement comparison outlier level satisfactory able accurate solution outperform respect glm sdp comparison ordinal undesirable scoring accurate rank pair order serial algorithm base offset preserve want reflect offset one think player rank favorable player offset though could also perhaps adjustment proxy small number strongly synchronization denote sup yield initial synchronization set apply twice method rather base investigation somewhat ordinal winner game similarly across measurement record winner frequent e synchronization superiority give second order glm run ordinal measurement record winner rely difference case rate inconsistent ranking item player perhaps usa match question rank player consistent possible setup inconsistent partial comparison nonzero correspond pair item measurement inconsistent individual b prefer vote rich sciences literature date majority become option b aside appear social recommendation movie possibly pairwise comparison rating mathematically aggregation formulate pairwise illustrate slice set however usually approach parallel produce rating eq final player one ranking sort decrease induce rank par par glm par sdp aggregation run average ranking ranking albeit naive would available circle figure svd avg avg avg avg glm avg sdp far result simple bottom matrix sdp naive rank counterpart hermitian I denote graph connect system aggregation formulate q synchronization mind angular correspond whose constraint eigenvector solve constraint correspond long feasible synchronization unfortunately long cast eigenvector simply encode constraint sdp semidefinite one rank eigen perfectly sdp solution practice eigenvalue necessarily piecewise whose support induce accounting circular minimize aggregation illustrate confirm sdp significantly accurate aggregate rating ordinal comparison measurement naive serial f lot redundant subset correspond essence sdp row inner product aggregation eigenvector implicitly write involve sized sdp solve rank eigenvector sdp relaxation aggregation ordinal rank centrality plot measurement version centrality adjust rating simply win compare uniform yield accurate result naive aggregation avg par method glm come good sdp ordinal four sdp par sdp avg yield accurate especially enyi measurement illustrate sdp follow closely par par complete come next performance rather avg glm avg almost especially gap two one avg glm order point centrality l plot l plot glm plot p plot par avg par avg par avg p avg glm avg plot avg experiment suppose readily subset rank player obey impose constraint propose relaxation angular synchronization post able case rank small synchronization constraint dimensional graph application biology latter realization euclidean one subgraph know biology node spirit refer non player shall know priori mathematically synchronization formulate measurement element compose sensor noisy element offset node previously enforce hard sdp relaxation see spectral relaxation section angular synchronization semidefinite programming light sdp angular synchronization real matrix diagonal anchor hard relaxation angular synchronization noiseless sdp return one top eigenvector recover angle ranking rotation circular anchor player search circular permutation anchor induce right illustration truth place half magnitude anchor free player mind propose prescribe keep anchor player apply cyclic player ranking rank rank anchor player stay cyclic pairwise associate ss outer choose circular total denote stay fix solution preserve possible relative player recover alternative apply cyclic total permutation position rank word guarantee happen return program rank hence projection several model ordinal note measurement significantly accurate future direction concern possibility player rank sdp post constraint alternatively may spectral synchronization ordinal encodes pairwise synchronization constraint parameter available pairwise angular could also generalize highlight future believe interesting improve question may denote hermitian pairwise angular similarly hermitian offset user enforce angular one sdp synchronization case offset soft pairwise thus wish maximize form condition constraint absolute number kkt condition remark constrain weight subject sparse respectively cluster work hermitian one matrix recent principled constrain eigenvalue recent laplacian solver explore whether soft allow enforce enforce player instance sdp furthermore sdp pairwise comparison alternatively explore synchronization structural biology interesting variation could concern enforce certain I one game lose word reconstruct ranking order arise genome sequencing overlap read reader relaxation sequence svd remarkably global information h r concern outli setup multiple voting pairwise rank aggregation provide incomplete inconsistent player natural whether derive estimating consider recent van introduce class vote yu embed angular synchronization angular investigation angular synchronization use orthogonality often regime relaxation well computational direction find perhaps oppose consider point player rank south extraction collect longitudinal player roughly speak player rank relative ordering establish consider sphere parallel nearby perhaps also investigate extent extra overall robustness find circular eliminate freedom search rotation sphere result agree robustness remark exist guarantee synchronization trivially noise exact ground perfect angle synchronization necessary perfect enough angle could synchronization svd rank minimum give cost incomplete angular synchronization relaxation exist provable guarantee computationally spectral rely extensive english match microsoft college pairwise relaxation player prescribe consider aggregation rating system inconsistent pairwise produce global aside traditional amenable tool constitutes plant partial ranking player pairwise trivial extract partial fact unlike synchronization preserve order player plant partial acknowledgement author thank institute fellowship support stay berkeley fall carry grateful suggest year possibility apply angular would grant fa serial discussion literature beta give microsoft aside real life one subset uniformly throughout pairwise lot noisy rest network raise whether partial ranking locally consistent empirically preserve partial addition ranking rely spirit exist plant clique subgraph approach extract ranking rely recent multi partitioning inequality partial ranking necessarily residual seek whose union final set expansion plant locally consistent rank player pairwise one enyi consistent ranking start estimate rank result hadamard product product adjacency measurement method instance ensemble position ease visualization consist corner whenever offset induced measurement conversely offset initial measurement expect magnitude ranking player red highlight perfectly method fail note title plot ranking ranking contain correspond sub long identify subset average inter residual know science unweighted concern maximum clique np restrict bipartite graph become problem seek maximize either seminal enyi place clique efficient spectral adjacency
evaluate harmonic reference accord author label model reference gold standard another reference either make accuracy trivial well performance class division zero negative influential reference one influential show ten fold validation baseline reference predict citation macro measure table add feature use four model cf table sign statistically two tail pair level greedy high achieve model semantic add bad hypothesis task remove high title cite core section base influential model count useful combine calculate individually reference svm reference influential reference model high influential reference influential svm svm describe logistic distribution binary maximize gold label weight classify instance value predict influential set logistic indicate matlab conduct logistic assign value select important correspond likely influential regression classify reference experimental svm logistic vs threshold result reference axis curve also table curve percentage equal peak axis table figure test gap perform citation count make count cite design ignore count citation conventional citation counting count type ranking influence citation count citation count direct loop rarely period slightly citation label graph direct edge cite direct edge might influential way citation count citation citation drop paper cite citation count citation add count great try threshold reference e pair network building network cite paper paper citation mention body relative list sort weighting reference citation count cite paper body apply convert citation count function might edge higher citation direct citation paper cite cite twice add cite paper count cite two conventional citation formally let exclude citation stand large least author cite citation author conventional except least author cite cite mention influence receive paper large q stand influence index network citation paper publish paper close publish citation network impact researcher community desirable try identify statistic paper count regular expression precision manual regular line multiple author another automatically distinguish main reference may increment count numerical g use paper manually citation citation count section paper citation count reason network dataset contextual precede differently highly influence cite work citation raw unnormalized process automatically manually manual scale paper versus count citation count influence two ranking paper paper group paper rank count correlation coefficient influence paper highly cite paper accord citation paper increment count paper vector citation citation two highly cite paper count influence citation drop agree rank paper less cite group author calculate correlation index row table trend go count count different index correlation identify association identify select predict future paper paper top index precision divide show range top rank rank find two mark seven equal index negligible show commonly list engine list document case range equal otherwise author otherwise score whereas conventional precision encourage evidence weight identification author paper author paper several annotation could quantify delay paper interesting machine approach human purpose text available indexing restriction limitation extend database another without feature cite title feature overlap could cite cite similar work author moreover view adversarial attempt game exploit count survey manuscript normal review majority report influential reference approach art precision influential circumstance occurrence citation resolution convention ever mention citation number influential fine reference influence bring avoid present two influential reference hard task occurrence metric simple citation influence refinement address citation distinguish acknowledge type research paper cite influential paper investigate issue contribution significantly influential one counting cite influential reference confirm citation fairly scientific reference list alternatively count section cite assess research robust track research reference superior benefit combine grateful identify science grant thank comment importance article cite treat equal variety central variety citation label automatic good evaluate number feature conventional citation determine impact approximation count cite refine yet score article threshold publish cite treat citation hence equally error page number time likely citation reading cite illustration report cite citation create paper cite read count reference raise serious quality citation count write cite hundred reference find reference redundant attribute fraction tool determine reference inspire core contrast journal many make influential reader often citation effort linguistic near citation body pt citation text cite frequency article cite literature reference cite effectiveness identify influential reference four particularly useful cite core secondary purpose citation future author researcher reflect citation frequency well measure account cite give weight ability determine cite substantially influence attempt identify researcher solely publication compare unweighted measure well say degree precise influence researcher know write evolutionary likely influence influence good paper good decide label label influence give influential influential acknowledge wrong whether actually plausible say influence author might might admit nevertheless reliable determine influential author label prediction influence motivation paper citation rather reference alone influential one paper influential potential application citation potential follow long reference could reference familiar field nature read material prove author index citation count sensitivity could contribution survey highly cite far recommend review perhaps thing author rigorous writing methodology influential filter influence impact survey methodology put author journal less sensitive citation organization impact count also impact citation benefit influence track science interested idea noisy way track reason spread people may network link web page link research citation could could improve web page need read filter might help recommender work count field phrase back early day citation indexing reason way reason previous work article physics distinguish class conceptual operational evolutionary reference cite wrong negligible citation indexing similarity document first automate select category machine distinguish category identify via linguistic classify weak neutral positive neutral classifier rely phrase citation self citation citation text manually acquire annotate access text build machine svm na I bayes library rank superior methodology differ significant author identify influential reference use believe analyst seem classification knowledge report moderately inter agreement concern propose measure citation count benefit good assessing arise annotation thus citation provide rich machine characterization relation identify example generalize broad one acknowledge one author journal weight citation publish less paper appear journal intrinsic citation journal cite publish propose importance citation get paper frequency article original contribution closely cite relate reference times least ten common give classify concerned research paper reference create take author paper cite influential create author gold approach generate vector reference manually reference testing standard vector contribution wide pair count feature position however feature intuitively attractive extent follow subsection feature count frequently body likely influential reference five count pt count count introduction core section include exclude already exclude conclusion future section feature reference feature appear even lp similarly applicable useful influential influence originally suggest inspire subsequent expect preliminary report old early update extend greatly drastically weak create category three mean three bad strong represent factor six end end citation label influential occur citation automatically extend label cover word feature citation ten reference cite body increase citation context citation sentiment human annotation association word annotation whether eight basic trust number eight basic sentiment citation indicate citation influential sentiment ten occurrence citation citation body cite influential intuitively important seem base location citation sentence pt pt pt binary indicate citation appear appear cite time begin sentence feature next base location citation pt pt sentence reference include mean total position range sophisticated location reference influential appear solely fit arbitrarily put together pt pt citation paper receive literature occurrence metric estimating evaluate organization journal cite highly cite likely collect raw citation reference accordance convention self citation refer phenomenon know citation citation among old cite influential publication year calculate publication result non negative length reference range paper predict influential normalize raw range normalize kind normalization mining improve time reference cite cite ten reference would cite score cite normalization contain normalization let reference reference pair distinct nf ij f r f correlate achieve gold reference author help create reference direct fill online paper essential reference highly influential influence experimental choice research reference merely believe expert assess reference essential reference know need give reference without different online table usa researcher lr country france uk usa mathematics physics gold standard dataset us benchmark give paper indicate reference convert plain extract influential influential total boundary parse reference document scientific run hand code expression citation occurrence detect paper annotate name standardize publication paper body second manually correct google citation include item manually correct citation explicitly reference mention precede preprocesse annotation paper pair paper datum occurrence reference reference text reference speak influential influential research determine influence reference pearson various influence simple correlation base
ie ie ie ie w j te ie j te ie tw design bernstein w ex ex ex ex ex substituting rhs eqs bind z k k ex inequality recall random interpretation independent triple overlap triple schedule match three triple index consist triple prove twice round triple exist prove order cover triple triple triple contradiction order triple whole triple triple copy triple copy triple proof threshold since f use theoretic low follow packing simplify notation suppose draw uniformly randomly item good generalized denote leibl divergence partial ranking sd since process drop alternative marginal respective jensen inequality kl ranking draw exponential alternative rank choose remainder packing integer pack set bound ex ex last ex hold last last inequality consider note nk independent change exist subset k j b b item upper technique independently fact supremum sum supremum follow inequality inequality ex bernstein inequality ex ex follow j j ex ex ex ex appendix long account take winner directly appropriate row one sampling respectively uniformly always count occurrence remainder rely pack integer positive packing entry lemma imply ex last hold maximize hand side desire claim succeed produce desire prove generality orthogonal random notice ex ex uv ex uv ex uv ex ex uv term bound probability find term exist ex follow satisfy entry event hoeffding th variable draw sphere theorem prove concentration fix xu application recommendation management preference predict logit hide preference low rank reveal preference various form relaxation approach context interest collaborative choice convex relaxation upper many recommendation preference predict assumption success collaborative model learn ordinal collaborative preference subset obtain item tracking activity spent page rate make want user similar unseen item predict prefer discrete choice model describe typical else particular connect significant optimize offer accurately history type capture interact category predict choice item choice multinomial logit describe ranking rank preference provide rank preference item represent low rank represent item match correspond first item preferred user pool user preference whole population noisy true preference category norm minimization ordinal data two context choice provide result finite minimax information theoretic factor interpretation upper interest analyze context collaborative rank pool exist work wise propose relaxation matrix bound statistically optimal generalization similar comparison match result general sense analyze comparison refer rating ordinal guarantee remainder collaborative ranking provide collaborative analyze similar relaxation ex ex ex ex iv inner indicator event integer model collaborative preference widely similarity preference logit capture capture item small rank item simplify number analysis might differ choice give ranking ranking v underlie preferred likely nature capture reveal model describe decision alternative dimensional decision maker rank utility draw intuitively rational proving appendix notable pl special pl widely machine mle centrality quite beyond pl overcome restriction pl rank algorithm provable apply recent advance pl clustering approach heavily mixture additional guarantee mle polynomial time provable solve relaxation observe preference rank negative accord I convex surrogate optimization search maximize nuclear minimization provable extend identify convexity satisfied convex performance guarantee notice equivalent give rank list estimate class ij item way characterize quadratic norm incoherence draw replacement independently draw far treat item I apply technique analysis rank assume provide show term potentially rank hypothesis theorem solve regularization matrix need achieve arbitrarily degree directly match logarithmic range sub scale dependence although linearly sub dependence also simple special paper range advance illustrate choice wide another underlie realistic approximately formalize ball decay relatively optimize get result matrix suppose hypothesis least strict recover factor low exponent panel scale line rescale choice dimension mean scale square plot analyze illustrate actual insensitive broad rmse rmse leave plot versus rescale rescaled rmse broad convex next fundamental limit counting indicate scale degree construct packing accurately estimate true probability generalize constructive argument minimax establish sharp logarithmic prove nuclear universal numerical infimum list theorem provide interest ignore regime comparable regime theorem upper bound factor another scenario interest category second category denote present present fixed simplify notation set alternative user independent accord equivalent class sum alternative relaxation observe compare rank correspond user person subset preferred alternative alternative category draw respectively precisely draw necessary analysis appendix corollary matrix optimization corollary show sample need scale order factor degree fundamental bind ball rank theorem since identical omit suppose sample universal infimum measurable observed term comparable establish theorem factor research still slow want method provable initialization simple model pl general analytical notation conditional position p ji v v ii hessian follow ex constant least interested number hessian restrict nuclear convexity restrict strong collaborative section divided case ex ex least assumption prove desire ex singular decomposition orthogonal respectively projection onto tm rv rv topic form row concavity cauchy inequality ex ex ex ex ex notice u rv tu u rv tr r ex ex ex e absolute ex e
definition analogy vertex fig hypergraph incidence vertex denote diagonal form matrix hypergraph model hypergraph attribute correspond hypergraph correspond share attribute attribute matrix strong break penalty hypergraph provide correlation regard heat clique certainly scheme apply hypergraph utilize model hypergraph regularize classifier call attribute predictor hypergraph cut cut whose hypergraph cut keep attribute relation label normalize relation hypergraph cut denote row attribute vertex sign identical reformulate normalize hypergraph supplementary material besides measure attribute attribute obtain euclidean shift label attribute lead hypergraph introduce hypergraph preserve th attribute relation hypergraph cut hypergraph row actually hypergraph shift attribute space attribute cut find feature align space mapping whose predictor attribute correspond hypergraph substitute avoid overfitte equation positive positive matrix regularize square solve derivative zero solution give prediction project sample span vector encode predict attribute th span attribute specific introduce meaningful enhance attribute section example information enhance exploitation enhance always share attribute approach incorporate first hypergraph hypergraph relation subsection hypergraph hypergraph second construct pairwise encode attribute connect belong hypergraph heat adopt finally laplacian hypergraph equation correspond laplacian hypergraph predictor hypergraph graph denote short capture intra manifold preserve pair shifted align empirical rkh attribute representation embed evaluation associate number equal attribute assign support scenario shoot shot sample predict sigmoid normalize scale probability class label calculate posterior class sample label maximum regard annotate attribute attribute template probability classify sample attribute template class class attribute activity attribute contain annotated image test roughly attribute facilitate rest test video video around video class disjoint class testing report dimensional baseline dataset already feature represent database database already provide histogram rgb histogram sift histogram shot database database attribute prediction accuracy report know attribute approach indirect run surprising class label limit comparison semantic compare discover attribute performance provide feature notice propose outperform gain accuracy comparison approach gap attempt sample preserve manifold use unseen structure unseen enhance attribute attribute complementary attribute ccc accuracy shoot conduct interestingly significantly outperform confirm capture incorporate grouping together quality intra class structure performance dataset one testing take consider common categorization define employ equation near classifier sign accuracy cauchy apply accuracy accuracy kernel algorithm database accuracy attribute dataset originally report supplementary involve complexity consume quite take dataset time second matlab configuration cpu ghz ram attribute hypergraph attribute predictor collection hypergraph hypergraph cut hypergraph exploiting attribute incorporate hypergraph mapping attribute also extensive attribute effectiveness shot shot categorization integrate shoot propose perform shoot paper science china grant program team university grant author relation derivation hypergraph vertex instance element incidence ed respectively hypergraph return prediction attribute vertex normalize hypergraph hypergraph derivation control attribute avoid overfitte adopt control loss aim pay attribute categorization shoot learn shot categorization system pay parameter decide separately choice cm influence tune one tune procedure tune fix value parameter selection learn correspond initial replace accurately value performance database database relationship sensitive conclude value influence shoot need employ obtain aforementioned start accuracy database database big demonstrate accuracy figure similar peak optimal number also database respectively report performance however really conclude uniform database performance indicate get performance university usa edu cn edu hypergraph attribute predictor hypergraph attribute regularize hypergraph projection hypergraph align directly act attribute linear consider incorporate class shot achieve intermediate encode share across play role semantic communication show supervised attribute object description category encode task problem unseen training lot approach attribute fundamental start pay attention learn vs exploit attribute attribute categorization shot category label ignore prediction independent attribute exploit correlation exploit attribute power attribute competition framework suitable describe retrieval annotation exploit attribute preserve natural shot attribute break semantic point water attribute correlate classifier learn separation attribute water cluster although attribute utility subsequent categorization overfitte besides cluster preserve hypergraph general classifier flexible information attribute prediction computational activity attribute categorization consistently effectiveness rest organize work propose experimental evaluation classifier suggest attribute domain svm categorization show
randomize define prove last assume argument conclude conclude initial end definition strategy swap supremum previous upper bound jensen equal let rademacher rest precede expression relaxation lemma bind pick node sort step random irrespective forecaster pick expect forecaster distribute modification proposition shall simple modification adversary pick random adversary construction pick analogous relaxed simple tc condition hold k constraint hold recursive condition randomize recursive conclude q definition split v k statement theorem corollary remark define combinatorial aspect find nature combinatorial burden interestingly compute semidefinite program however enter regret rademacher benchmark trade motivate let prediction evolve social round user network observable type covariate may gender age education system predict user outcome conduct unseen stand type prediction behavior person devise aspect consideration second covariate leverage global computationally feasible edge strength dissimilarity system make binary reveal develop instance roughly mostly encode class side label class side nevertheless information measure therein similarity dissimilarity minimize weight negative laplacian latter propose obtain feasible expense start solely model shall item involve formally real assignment computer combinatorial cut unique conjecture relaxation ratio goal problem online learn allow reveal forecaster sequentially moreover identitie little except evolution forecaster particular constraint graph know ahead take network prediction constraint forecaster distribution identity side stochastic mind situation local global coherence label model appear involve yield tractable improper develop slight guarantee towards develop presentation provable future constraint arise classical rademacher complexity piece value conditional rademacher combinatorial relaxation framework suffer semidefinite prediction prediction two distinct upper minimax obtaining increase solution sense online relaxation relaxation distinction semidefinite relaxation compute relax round improper still effectively quantify increase large rounding procedure crucially multiplicative increase gap regret constant front opt relaxation relaxation statement modularity soon one small gap regret employ tighter base prove weak note solve situation offline extend additional prediction tradeoff remark spirit third tensor level tight relaxed involve framework individual stream fashion individual manner know like allow describe formalism relaxation state random guarantee admissible relax gap alternative lagrangian several low near shorthand observe set forecaster make prediction vary forecaster shorthand constraint assignment unweighted labeling cut rise item induce represent example hyperplane classify margin way indicator forecaster respect forecaster locally recognize forecaster face set information forecaster able conditional constraint edge connect reveal accord draw fix constraint accord property employ average rather nice unlabele play pool example variant generative model formation though much upper horizon concern easy incorporate let mention literature prediction graph node precisely notation static expert node class strategy draw next compute water argument outline regret course many solve might computational use pay bad rademacher topic next previous section randomize employ semidefinite program efficiently constraint depend information good formulation henceforth reason hierarchy interested labeling write ready sdp sdp relaxation vector correspond constraint first constraint program standard label whenever assignment common similar hierarchy detailed treatment semidefinite perform efficiently give maximization obtain think solution hierarchy sdp describe set sdp level regret randomize round strategy solution two program purpose analysis serve bound end define solution level expect main theorem provide convenience guarantee hierarchy via forecaster level hierarchy sdp term value correspond sdp suffice go observe solution feasible cm km use solution second since remark really refer draw improved gap stress gap prediction require round strategy round rademacher already mention benchmark optimal benchmark computationally hard improper nature around clearly thus lp gap weak immediate implication minimizing violate alternatively think constraint round problem consist sdp step sdp level relaxation put constraint us level hierarchy sdp notice penalize provide let solution optimization gap sdp context clear cost hierarchy constraint sdp prediction far randomize relaxation rademacher vector variable end sdp view regret rademacher sdp multiple satisfie inequality example second version problem introduction weight information l generic rewrite sdp integer level hierarchy opt since rademacher conclude small normalize graph well behave like near analyze set game similar problem quadratic constraint randomization incur metric labeling problem aim assign item combinatorial part cost graph cost assign space multiply encouraging item pay map singleton type separation edge otherwise polynomial provable penalize relaxation
respect bind prove supplementary let condition due achieve accelerate incorporate mini like direction mini setting stage cm algorithm k iy z ki x sgd step introduce multi key method give insight fx fy fy fx fy additional require assumption condition mirror fx k fy I fy fy k respect history get fx fy fy complete option moreover lm b b fx bind total solution respect k k analyze method boundedness compact change modification without material modify objective every assumption achieve accurate consider objective minimum satisfying k pn fw fw fx stage monotonicity notation pn fw small outer accurate generate bregman parameter condition acc boundedness complexity sag acc prox logarithmic calculation harmonic outperform c acc sag acc propose heuristic first sufficiently complexity upper estimate easily drawback adaptive technique perform inspire start third idea exceed run description website use perform well outperform mnist r quickly work tendency evaluation mini batch far accelerate gpu ccc propose incorporate acceleration convergence incorporate accelerate descent achieve support strongly strongly minimization optimization type empirical minimization smooth term follow smooth latter assume strongly strongly obvious paper propose effective sag sdca gd acc prox sdca prox acc prox gd reduce linear computational efficiency deterministic sag problem strongly add increase difficulty recently propose accelerate prove insight specific converge complexity difficult decide section heuristic determine present show norm distance generating function
instance input column vector dimension wish series convert segment scalar constitute concatenation segment mask divide trace duration time series delay couple combined recurrent neural recurrent connect state delay role network analogy connection interaction node period low filter self connection interaction pass significant performance affect interaction happen use practice infinite signal always constraint limit bandwidth signal consist fully number convenient convert depict immediately pass filter operation delay trajectory project picture rnn show grey infinitely many dynamical differential compute gradient involve sequential trace costly especially need multiple piecewise approximation combine full adopt replace duration derive combine eliminate lead relatively quick simulate measure datum highly good correspondence simulate one challenge experimentally fit exactly turn slight long period hour consequence turn challenge apply directly apply output next input physical useful feature system train systematic difference setup physical part limited serve input term offset control able cover start roughly accounting simulation argument map back range subtract due delay chance argument fall occurrence rare could benchmark task first mnist classify use essentially input segment time change classify dataset period depend initial practice use digit nesterov coefficient learn duration regularization shift digit shift include output present original example training simulate cross digits nesterov momentum momentum test directly compare noticed figure indeed result signal field group together confirm feature single period new mask ordering also employ scale column difference cause notable optimize internal offer comparison add art mnist comprehensive mnist mention website experimental datum simulate pre process dimensional energy enhance call delta delta delta common arrive wish potential demonstrate physical computer extend art performance addition perform wise informative determine randomly sequence frame nesterov momentum duration far simply meta depicted mask process strongly rescale input emphasis delta delta channel repeat scenario previously optimize simulation random present right optimize result random bad optimize comparison mention though state work value even mention please quite improvement detail dynamic less crucial mnist need suggest reason dataset second provide value act indeed therefore already secondly may pose obstacle recurrence task provide current time way mixing occur physical delay optical physical setup input system fully optimize usage dynamical system input inefficient input relatively digit optimize boost encode mnist directly utilize inherent system resource reservoir give task greatly scale effective good achieve reservoir setup step reality two problem measurement part keep mask physical hundred potential advantage optical would research future current hinge ability mathematically unclear usefulness direction improvement apparent could process could phase current backpropagation simulation end without next direction machine current physical backpropagation reservoir research argue recurrent state reservoir always use dynamical remain restrictive possibility optimize system system recurrent connection relevant dynamical accommodate appear backpropagation currently recurrence delay desirable optimize internal accommodate alternatively loop rich recurrent connectivity backpropagation abstract use design analog hardware perform signal scope realization neural result good processing capability consumption acknowledge office european agreement brain project f acknowledge les european acknowledge les grant acknowledge l develop interesting fast great parallelism digital far employ processing paradigm applicability descent backpropagation optimize encoding system demonstrate obtain work reality system common reservoir may inspire analog computer influence architecture availability computation require magnitude development allow researcher dramatically turn lead major limited effect recurrent rnn processing series account arbitrarily long implication system depend context present feedforward dependency scaling carry relevant update practice recurrent suffer important drawback first feedforward fully benefit architecture recurrent inherently nature rnns acceleration number operation rnn slow learn recent solve hessian promising attempt heuristic idea rnn grow branch employ initialize term series create still possibility physical address find remarkably complicate encode physical paper beyond work experimental strategy physical dimensional validation input
interesting bit odd difference vanish I last differ bit use min confirm format hashing bound bit simply finally reader report keep bit bit clearly good min bias bias vanish conduct svms hashing hash discard keep number generate practice typically often store obtain store bit store effective experimental figure present variety panel dash blue bottom linear curve bottom dataset test accuracy min bit interesting bit bit dashed view feature allow practitioner generate approximate scalable online equivalently nonlinear pay price linear datum might dimension use random resort model intensive interestingly develop way approximate max consist three firstly conduct extensive nonlinear svms answer min max linear kernel secondly surprisingly implementation validate via extensive real finally demonstrate svms min generalization design paper extensive min min use massive linearized hashing practitioner min max svm remarkable work consistent form record hash unbounded building large scale simple discard extensive bit essential approximate validate publicly expect work interest among practitioner like utilize nonnegative nonnegative entry dataset show build via hashing define popular term write kernel soon clear reader vision min max intersection extensively intersection interestingly outperform min max existence conceptually intersection design show affect marginally min max apply hashing hash concerned kernel example combine fashion e multiplying type convenience enforce normalization recommend max literature widely binary hashing logistic issue max mining kernel table public compare linear min kernel intersection min kernel kernel hash max remarkable min max form theoretically effectively implement need provide surprisingly completely discard validate set hash bit min max extensive present kernel machine kernel pre summarize classification kernel svm report accuracie fine figure individually high result figure confirm min typically kernel justify max application min max kernel color min max kernel dash dot linear max kernel expect boost combine multiple kernel core multiply chi kernel projection difficult fortunately consistent good min max cost paper classification figure effectiveness min accuracy mining hashing technique relatively algorithms practice truly scale application click prediction hash min vector alg procedure time clarity vector basically matrix projection say probability conceptually positive basic building approximately kernel clear briefly achieved bound unbounded alg note sample bit space mining bit hashing ignore alg encode information rigorous turn scope try observation call bit call since bit h f min r air job united states list english tailed dramatically application sense bit proposal challenging
become underlie widely perform importance connect define kronecker factor kn along fold asymptotic behavior rectangular unfold nontrivial signal without square unfold tensor conjecture us mode wise truncate unknown rank tensor tensor take form unfold k sufficiently contribution unfold phase fast decay error decay subspace directly tractable base estimator become trace propose increase threshold also previously empirically demonstrate tb recursive unfolding ordinary unfold bind norm theorem prove tb c p p cm recursive unfold ordinary subspace ideal notation summarize number denote square unfold unfold partition part index rectangular unfold denote inner tensor norm norm tensor recover plus direct classic perturbation rectangular desire result ratio view insight decompose lead add wishart gaussian correspond wishart noise speed phase exist figure perturb happen predict tb regime observe decrease unit n slightly recover rank propose trace nuclear norm nuclear unfolding define achieve trace subsequently guarantee ratio scale propose square side translate become recursive unfold k pn square unfold unfold see recover information rectangular unfolding next consider tensor contaminate exist rectangular unfold odd general kn behind show estimate two horizontal product tensor dimension tb k product go around tensor multilinear span leave singular vector exactly recover inspire mixture tensor eq kronecker dimensionality define prove km k norm semi eq old inequality last arise choose kn kronecker rao k kk k noise could tensor mode kn regularize associated input rank f solve direction multiplier n tensor residual primal introduce write ball update step finally solution primal see lagrangian multiplier write singular soft tb notable subspace dominate multiplier let tensor q estimator matrix kk restriction incoherent matrix mode denoise norm q incoherence kk k norm kronecker construction regularization scale projection p scale dimension sum rank square case conduct tensor denoise observation choose spaced decomposition latent approach initialization assume cp know true norm top constructing space norm admm measure initialization select leave measure line show line theoretically confirm sharp increase around place see see grow relative panel small choice parameter addition place plot optimistic tractable cp clearly error optimistic grow reach critical subspace reach subspace subspace hard optimization dataset semi commonly benchmark modeling five sample contain amount emission measure tensor standard spaced fed cp cp initializations space optimistic scaling put number context compare similar synthetic cp behave near cp large regularization overlap norm latent norm observe tensor noisy tensor normal intractable case linear optimistic ignore minimizer feasible nuclear eq spectral reduce cp rank tucker denote orthogonal r r k orthogonal tail second eigenvalue eigenvector derive condition consider first second theorem construct exist universal n inequality inequality last follow bind union
accuracy accuracy sign stable random svms panel panel present accuracy svm sign regularize svms tuning projection high good panel accuracy sign stable projection svms panel panel also present accuracy curve mark accuracy sign projection regularize I addition present accuracy svm mark experiment bit select clarity sign stable projection demonstrate bit mind bit nonnegative row bottom stable consistent curve mark result linear four solid label represent result correspond bit higher conduct require accuracy sign projection weight sampling curve label stable projection dash bit achieve accuracy compare sign sampling curve solid svm four curve label respectively different curve correspond higher conduct achieve accuracy compare random solid marked value curve bit high bit require much achieve bit dataset straightforward min kernel explicitly outside strategy expensive prove correctness easy except would able kernel example random accuracy confirm bit compare sign stable consistent weighted panel mark respectively sign correspond bit provide extensive large application present task mind neighbor search interesting sign projection empirical need line research stable projection work department university usa data tool neighbor sign process thus provide approximate linear nonlinear kernel arc arc provide two stable projection ii bit except practitioner sign ready scale application parameter literature effectiveness sign projection variety dataset sample comparison comparison large sign exceed typically sign projection number bit regardless favor bit consistent nonnegative advantage stable projection type problem bit sign projection core focus sign projection application matrix multiply projection context stream scan bit parameterization stable available definition eq make sense I e dense limit issue largely max variant normalize max similarity efficiently min also sample use call consistent traditionally consist unbounded make much convenient scale machine although sign bit stable accuracy important individually dataset report kernel
matrix class draw obtain example covariate variable class covariance element equal elastic net elastic net package regression respectively net adaptive elastic two performance despite increase elastic elastic net dominate elastic produce elastic net penalty relatively mistake penalty similar example ht ic ic ic hinge make lar algorithm square error time logistic article propose high efficient compute competitive valuable toolbox high classification minimize th power inverse usual second programming problem find readily try tried drive opt leave report paper associate suggestion appendix suffice check derivative e first lead directly setup support penalize art cone dimensional order overcome fine regularization publicly classification svm vector svm widely use modern classification consist seek maximize define th slack margin tune trick produce boundary separate hyperplane extend reader refer detailed explanation notice lie hyperplane boundary phenomenon generalization method discrimination find separate inverse margin point inverse margin replace generalize thereby improve whereas svm svm novel geometric cone programming solve primal interior dimensional covariate much size dimensional affect variable discard one use cause accumulation estimate classifier classifier generally classification svm produce svms svm scad net work consider penalize dimensional solve cone challenge cope associated penalty dimensionality derive combine implement package give quick observation gene panel depict take code elastic observe several sparse formulation standard svm often quadratic equivalent hinge therein poor svm replace norm eq elastic penalty lasso elastic show elastic important grid cross validation refinement elastic penalty replace adaptive elastic enjoy net adaptive penalize far consider adaptive adaptive computed elastic net trivial handle penalty propose strict net elastic elastic net focus sake presentation standardize u coordinate close solve principle form part intercept algorithm summarize cyclic descent iy ix update intercept u iy r warm strong increase implementation path warm stable grid small point warm solution warm start sufficiently kkt scale strong likely inactive set correctly check whether incorrectly discard solution incorrectly discard add survival eq update incorrectly discard back survival boost apply another cycle investigate set finish change active margin update default show strict descent use elaborate
include minimal performance like review routine retain transform retain block method dct assess image common size dct sufficient adopt retain dct q permutation performance dct trade additionally separate measure quantify compression arithmetic complexity arithmetic elementary multiplication bit shift transformation focus attention context video quantization diagonal contribute dct resort assess table display complexity also dct calculate dct employ modify approximation dct low arithmetic shift definition exact dct matrix dct matrix iii measure compression first code efficiency snr next subsection description dct dct total quantify transfer matrix angular frequency per expression quantify energy dct quadrature signal unit satisfy mathematically minimize maintain approximation dct code compression tool mathematical code gain transform uncorrelated efficiency markovian analysis image compression compress assess image degradation original version transform mathematically transformation divide sub block block particular retain employ reconstruction range reconstruct subsequently recover literature regard assessment take consideration similarity employ support adopt quality collection image instead particular robust procedure public bank absolute compression ratio outperform I dct coefficient discard ratio table show measure measure ratio approximate transform method proximity measure compare dct good transform arithmetic complexity modify compression indicate qualitative comparison show dct approximate dct propose instance although could well computationally demand code transform improve section offer comprehensive several figure proximity dct dct implement implementation intermediate result wise transform dct adopt transform block buffer circuit order dct circuit format buffer block transform signal indicate wise transform digital dct hardware flow diagram architecture cb fig modify cb section bold box realize base rapid hardware co architecture digital matlab synthesis option auto transfer hardware description architecture ff device architecture realize increase fast order delay fine realization measure hardware loop verification digital resource varied range adopt word test within matlab physical hardware device time logic flip ff delay maximum operate synthesis tool run flow estimate evident modify cb fast consumption hardware ff dct cb architecture tool environment circuit cycle software architecture convert digital hardware design generator tool lead physical implementation architecture technology lead extensive hardware hardware verification hardware language design contain transfer library verify environment mapping technology guarantee could design environment follow behavioral adopt library behavioral source cell fix consumption logic adopt figure synthesis area path delay ns consumption area display complexity adequate area throughput hand real drive force logic design clear technology area algorithm cb low power dct approximation hardware transform prominent possess complexity compression modify propose good dct speed among approximate examine implementation dct approximation tool nm technology operation much architecture optimize digital library realization way post test dct discrete require support usa engineering research processing energy consumption development approximation dct due remarkable approximate transform offer circuit digital hardware lead consumption conventional integer transform multiplication possess peak dct candidate video several dct digital architecture realize digital prototype circuit technology map nm dct compression consumption year significant system digital video device video internet protocol prominent area requirement field traffic surveillance network hardware throughput well context cosine dct video dct energy image first dct substitute two several imaging h h scheme video employ integer transform operate capability achieve performance demonstrated especially possess operation computationally dct approximation video include literature dct operation consumption operation issue approximate transform introduce dct possess complexity optimization minimize computational cost hardware implementation dct approximate dct transform dct image compression cb dct round modify dct vi dct architecture base successive call take advantage separability kernel dct algorithms video propose possibility video rapid hardware realization dct describe associate fast transform discuss dct quantify assess dct digital architecture hardware field gate array nm circuit conclusion current dct calculation dct approximation meaningful low dct totally requirement arithmetic prominent address transformation nan require dct provide low power design transform characteristic approximation exact dct hardware multiplier dct video provide operation availability fast digital valuable asset consumption driving factor quality reasonably low important system picture device video device demand extended life library algorithm dct engine dct offer master device device switch low dct storage certain alternatively dct picture quality snr video metric dct video intra frame dct information measure metric certain demand picture foreground frame say switch dct intra basis account vary picture clarity digital dct mathematical select dct matrix format contain number positive number diagonal require image quantization approximation bound power nan multiplicative
stop average stop achieve divergence eq optimum dependent pac martingale use union bind carefully choose technical recognize problem achieve follow presentation closely rest stop version favorable property nonnegative infinite whose concentration show exponential would every proof fully e statement kl give desire stop nonzero expectation pe f pd consequently outcome control thm main previous continue write w early expectation condition event go need stop analogous exactly precisely stop time involve calculation e simplification conclude thm thm definition give pac time inequality simplify state flexible usage pattern stop data consideration know determine occur frequently practice descent sgd empirically highly concentrated choose manner fundamental limit law iterate logarithm lose concern instead result uniformly time manuscript focus issue general present large hoeffding bernstein view version martingale induce fair coin repeatedly write rademacher variable discover random walk law iterate logarithm rademacher walk rademacher generalize half rademacher true upper bind capture regime interest regime encounter sense concerned failure examine dominate tt discussion focus rate follow result statement occur occur absolute definition martingale positive martingale bernstein sequence bound iterate logarithm explicit mention allow mixture evolve index dependent martingale find process tailor posterior manuscript method prove though complicate paper none sufficiently low compare inferior uniformly time iterate
strong appear upper rest main outline proof theorem ensure algorithmic approximately concentration gram correct interest canonical basis hull cardinality sphere matrix ic write absolute constant namely subgaussian analog observational subgaussian random subgaussian subgaussian assumption subgaussian identifiable particular next state theorem consequence interested eigenvalue wise approach row restrict eigenvalue state entry minimal understand integer still lasso condition ensure suppose hold matrix vector satisfy give outline denote imply general condition case bound statement hold require increase correspondingly dominate paper large suppose model independent entry satisfy programming admit f absolute outline condition obtain somewhat show restrictive subgaussian regression bind lemma lemma condition modify gram theorem suppose theorem lemma pass lemma essential state immediately reveal regard hide precisely enough fm k low denote curvature smoothness mf assume condition define lemma appear sensitivity jk requirement enough satisfie combine lemma give yield leave choose f x ty prove belong feasible event proof lemma goal section see corollary immediately condition definite let define probability c show norm isometry property estimating gram x f state stay positive probability rank let ai fm b constant depend state fm proof corollary appear eigenvalue correct row across subgaussian newly derive framework work follow definite moreover effect investigation model future extend method measure multiplicative additive study moreover current measurement acknowledgement mark support fa part nsf dms corollary corollary section rest present variation concentration stochastic cone vector let location absolute k condition fact part fact k choice gram negative general bad eigenvalue stay tx state auxiliary may independent interest independent satisfy x hold random copy symmetric copy z lemma f r f define lemma large I c c c state immediately corollary check condition theorem pt pt ann mi parsimonious fitting dependency dependency matrix wise dependency set representation variate kronecker covariance n generalize x w subgaussian analyze restrictive eigenvalue able recover model single observation response vector variate social science become increasingly popular biology process communication graphical structure recent kronecker equivalent stacking call contain column mm covariance stack covariance column row see relate kronecker encode high error decomposition matrix kronecker sum identity measure practice exception work deal match pursuit omp recover case subgaussian entry bound dependency compose subgaussian vector
alignment kernel dna protein kernel graph validate method apply challenge real datum set cancer clinical record cancer patient year brain cancer group together clinical important background knowledge essential alternatively disjoint subset series permutation object lattice partition gauss integer mode distribution sequence forget induce integrate one well define k chinese restaurant crp see instance partition covariance kronecker process invariant permutation reduce b ij covariance eq block define assign joint read one viewpoint partition possibly wishart directly dot suffice conditional assume zero n crucial calculate analytically impose severe problem derive move similarity similarity pairwise transformation assume without access rotation information replication access plausible strategy empirical row dot procedure since mean probably requirement wishart matrix correct replication subtract row vector operate center transformation relate row replication subtract x xx certain might rotation use principal coordinate kernel decomposition project axis direction principal axis estimate highly lead fix rotation contradict column normalization pairwise dissimilarity even solution might avoid invariance constant move vector similarity pairwise depict move information rotation lose translation whole matrix red reconstruction directly dirichlet cluster cluster observe suitably pre process euclidean characterized mean equivalent absence distribution generate wishart jj ij transformation general notation distributional generalize wishart observation transformation kernel follow within covariance generalize copy square eigenvalue argument conditional probability read serve fact infer wishart parametrize care influence encoding versus row conduct present novel evolve dirichlet exchangeable differ different cluster evolve exist cluster static able structure completely identity object notation section cf block size size th chain first markov notation manuscript consideration want infer adjacent adjacent expect result cluster independently cluster evolve assumption observation arrange static x describe left evolve flexible allow distance cluster cluster couple rich obtain cluster object belong element per cluster imply different right cluster centroid general iff ij j prior dirichlet partition prior partition generative sense idea forget partition denote generative point static generate label introduce integrate dirichlet partition point note define invariant permutation wishart freedom wishart distribution change size differ possible cluster case reduce matrix need obtain draw way detail k degree freedom wishart generative model te fig generative te correspond inside matrix tw tw tw cf apply mcmc assignment conjugate sampling algorithm infinite model exist epoch epoch totally object belong prior object belong table c neither e cluster probability variance one whole denote metropolis hasting choose lead old include cluster degree shape applicable scenario crp view process label switch crucial initialize block rate influence decay estimate belong distribution weak effect conditional estimate pre parameter contribution assign eqs metropolis te define complete consume part characterize row remain probability new partition determinant regard work investigate track example grouping article topic topic become popular invariance longitudinal additionally track course pairwise distance object construct definite decomposition project axis axis underlie structure hence cluster need grouping datum point already assign track preprocesse identifiability compute sampling routine require computing routine slow way point generate point per large dimension way sample draw create distance matrix draw point new sample sample draw store pca per correspond illustrative separate te burn algorithm analyze trace block trace plot usual perform take sampler ground rand te te compare evolve te model well linkage single linkage separately static te burn phase repeat tree cut nonparametric compute separately scenario static well evolve expect group single point run cluster te separated cluster te comparison pool pairwise distance single point pairwise distance across repetition cluster object point belong compute rand pool fig explain datum new shift object group together cluster cluster group true te combine pool evolve te linkage linkage separate except pool second experiment overlap computer roughly hour performance translation invariant evolve static state te gauss overlap probabilistic linkage linkage fail te dynamic model te gauss demonstrating yield directly statistical te model h te synthetic simulate highly color significantly outperform baseline generate performance te independent demonstrate repeat highly period point cluster consecutive multinomial sample gaussian large result overlap randomly compute move synthetic te significantly baseline method show fig comparison dynamic model te gauss clinical brain brain patient highly variable depend age gender average first total health record group vocabulary treat binary vocabulary use rank comparison sentence cluster use obtain patient rank patient document window year available compute patient represent patient entry correspond patient specify cluster find ten year vanish year year remain patient cluster decrease patient patient year patient suitable cope kind datum patient differ patient time course death leave document year patient appear occur flexible suited model change change every patient computer take time switch tumor status year analyze cluster detail analyze cluster would scope death rate see sentence treat combination explain ex five cluster cluster describe patient addition brain cancer death consist patient speech vision
lemma open counter pair event word position weight let random know probability p therefore
parameter constraint xy dependence nb follow regression methodology poisson nb carry initialization scheme concern em strategy large explore poisson involve sort count assign observation nb generating model simulate order integer observation come group try poisson score regression right ccc aic score minimize fail produce demonstrate methodology tn summarize ht type education line logarithm median show fm fit proportion significantly count component significant indicator belong indicator respectively count great proportion school low decrease also indicator black proportion indicate proportion white predictor inconsistent conclusion ga numerous however show tell count reason education good size education education investigate public propose novel framework response count systematically arise dataset contribution poisson able select determine count suggest count disease disease number city count finite involve come framework carry criterion demonstrate variable responsible interesting trait datum people treat count probability event give event regression linear variable covariate via define effect exposure incidence subject exposure responsible determine define mean count violate say binomial poisson binomial come treatment observe count count come mean fm modeling within past unsupervised task principle mixture treat concept useful heterogeneous appear model extent largely covariate covariate heterogeneity context throughput sequence online develop literature criterion aic mixing give force proceed information binomial hypothesis statistic assumption confidence level aic prefer choose
root leaf subtree root child compute pos tag th child head word embedding representation embedding concatenation operation relative sentence relative distance map dependency tree pos tag embedding composition matrix pos tag capture syntactic example noun noun activation dynamic child fix information pooling subtree root node figure tree parent model phrase nlp task parameter nlp interaction head child semantic plausibility subtree dependency parse node child simple correctness final terminal parse head correctness subtree pos tag goodness tree sum tree rank output base criterion dependency score count eq set final objective minimize plus score incorrect decrease gradient direction subgradient use diagonal step rate subgradient use discriminative parsing dependency parse third generative ranking list forest dependency parse substantial parsing give sentence combination hyperparameter base sentence train discriminative way base tree approach dataset chinese use score english split development tag development automatic pos tag way pp bar pt symbolic md ylabel limit true style font legend anchor north style sep black pos result perform slightly good base limitation add line initial discount also experiment final previous oracle achieve minimal engineering rank affected base overfitte although result also work large think large increase greatly large need multiply output experiment show achieve significant improvement add list dependency parse neural parse dependency parse tag arc make action transition parse nlp propose compositional rnn image difference node compose subtree parent pooling position parent convolutional vector recursive probability utilize treat tree regard recurrent unlike discriminative base difference compute besides also sentence dependency address level phrase dependency capture syntactic compositional phrase architecture parse tree therefore nlp effort engineering dependency paper regard semantic sequence length fix nlp text research limitation investigate thank anonymous valuable comment national science foundation china program technology laboratory processing school science road china edu cn problem phrase dependency dense convolutional syntactic compositional phrase dependency convolution pooling layer model informative discriminative list parse tree effective improve art dependency english chinese dataset discriminative much dependency parse million feature ability complicated distribute semantic extensively language nlp semantic phrase help address generalization representation parse dense representation complementary focus keep unchanged parse optimize task important unseen phrase vector parse parse recursive neural binary parse parent child node tree phrase dense propose convolutional architecture compositional phrase architecture parse tree dependency parse give dependency first unit interaction child recursively output input parent output length illustrate phrase red contribution paper summarize architecture phrase sentence dependency regard sequence length jointly nlp classification complicated child pool rank parse parsing decision experiment model briefly neural architecture language nlp phrase sentence sentence every context classical layer whole sentence binary structure length leave word recursively length whole multiple tensor product figure illustrate rnn branching triplet triplet either word node give p ap bc p bold font letter compositional syntactic compositional compute plausible syntactic parent high scoring standard parse apply recursive phrase rnn enough
eigenvalue number way eq unique function maximum integrable therefore martingale reduce proof mind proof correspond estimator form change h equation scalar product rewrite auxiliary integral unique get eq integrate get du du outline correspond however case calculus integral fractional fractional integration q constant st kind transition equation help operation get equivalent right rewrite rewrite equivalent h v arrive part formula rewrite eq thank helpful discussion theorem maximum two independent integral equation fractional brownian index require case develop tool mixed process demonstrate likelihood regime fact stochastic calculus support center possibly interval space worth mention also form integral verify step extend define square integrable h beta map isometry hilbert wiener process put well call fundamental martingale square martingale reduce existence uniqueness concern existence st prove fact unique tp fractional integral simplicity formulate let h x brownian motion square observation process wiener drift x combination consider proceed probability topology probability measurable set measurable functional e x give wiener independent also use center measurable q still projection alternatively arrive exist constant change line apply inverse
variance closely call vary model explicit view pose used remain network see test ability generalize pixel make flat rotation represent dc achieve image include previously unseen transition intermediate pose seem sort angle angle unseen flat train deep graphic interpretable graphic static image utilize convolution train use force face encoder latent decoder network never dc component arm see example less handle complex deeply handle large object architecture spatio utilize motion visual move handle recurrent network also replace decoder hope motivate interpretable representation variant access fellowship helpful discussion science intelligence laboratory mit brain cognitive mit microsoft research uk edu mit edu microsoft edu paper present deep convolution graphic rotation convolution convolution train use encourage pose image pose qualitative model engine remarkable automatically hierarchical cnns boltzmann generative successfully relatively little characterize et al consider propose theoretical irreducible come open question work theory representation work representation abstraction represent happen world graphic go compact description graphic typically fine transformation pose compactly identical al graphic code align recent work graphic probabilistic latent et beyond stage encoder domain decoder produce interpretable graphic reproduce interpretable complex transformation rotation variation hybrid encoder transformation object plane rotation direct graphical convolution variational bayes encourage representation train mini batch active inactive transformation value learn function texture inactive automatically create quantitative efficacy convolutional inverse graphic dc encoder decoder autoencoder consist decoder neighbor training produce consist pose texture shape gradient back force dc show mini batch inactive transformation g face light pass etc graphic graphic model propose representation unlike relatively recently et al feed forward neural encoder serve handle grain geometry face relatively extend apply jointly train utilize convolution de convolution encoder respectively convolution massive increase recently use cnns object specific supervise truth directly image task amongst encoder decoder comparison proposal intermediate variational encoder spirit assume representation piece spike comparison encoder interpretable graphic graphic rely work work use graphic depict attempt face camera source target mini batch scene angle source might occur generate batch scene hold face consist many different face pose property mini intrinsic stochastically sample batch reflect identity desire unchanged batch hold neuron force variance batch full change neuron receive reconstruction close likewise neuron proceed representation make gradient figure correspond angle source intrinsic mini batch minibatch representation calculate entire output gradient pass continue backpropagation encoder representation dimensional intrinsic work work encoder decoder neuron force batch change neuron value gradient encoder put variation qualitative capability dc latent smoothly leave smoothly leave unchanged strong face encoder transformation encourage neuron wish transformation mini dc train inactive act encoder point invariance close care care face matter way scale small reconstruction qualitative capability learn dc original light neuron change dc train batch generate face shape texture pose meta momentum decay also perform varied
learn potentially overlap neighborhood denote bias feature produce encoder g w induce pool learn although recover sufficiently reconstruct correspond possible reconstruct reconstruct group sparse activation additionally penalty activation include nonlinearity critical inference convolutional dictionary show architecture convolutional pooling conceptually identical connect described york ny new york ny edu classification rely coherent video datum feature unlabele adjacent exploit train encoder establish connection coherent neighbor likely space example video likely adjacent frame assumption feature introduce temporal coherence feature slowly discrete adjacent frame degenerate degenerate mapping informative input discriminative criterion pairwise geometrically weak high propose term prevent constant act preserve priori like preserve possible optimal extract slow
note last equality z schwarz use invoke b inequality proof use mp precede small sufficiently conclude eq equality next invoke ai invoke entry moment due satisfy e adaptation precede difference satisfy precede martingale satisfying constant subgaussian increase positive constant every j uniform l inequality last inequality concave cb independence across invoke constant precede array triangular measurable plain definition inequality dynamic dynamic regressor fix show one uniformly valid asymptotic allow conditional important time allow band contract dynamic widely economic social extremely differ unobserved repeatedly dynamic however inference model presence lag effect regression seek explain economic growth determine factor panel big explanatory control result explanatory arise control form economic panel access reason decide investigate subset control one propose inferential procedure dimensional progress decade popular lasso research however recently possess estimation independent plain linear treat establish oracle dimensional datum study property penalize gmm high arise panel datum coefficient shall involve panel consecutive individual dependence panel reason assumption panel case correspond approach static panel consider effect nuisance parameter time straightforwardly difference model error correlate stand effect intrinsic interest hypothesis simultaneously involve side explanatory truly zero classical severe impose vector effect reason sparsity instead total magnitude effect control variation expect albeit dependent interpret percentage change fix remain percentage variation control vast covariate sparsity actually variable deal assume structure group dimensional sparsity inferential regression lasso invertible gram context regression inverse covariate suffice inverse weakly entry need contribute group behave differently joint asymptotically three type parameter increase consistent robust conditional panel consider error asymptotically uniform subset show band uniformly rate type size organize introduce next robust seek sample construct contract parameter section carlo defer denote norm unit column entry dimension maximal cardinality index kronecker product fix exist constant maximal minimal give rewrite p np confusion however argument tend often assume observation source lag next write compactly compactly something linear difference heavily propertie block properly gram impose sparsity oracle inequality technique fact get expression instead characterize equation variable hand side may reasonable heterogeneity high dimensional effect logarithm fix effect unobserve factor motivate sparse weak instead infinity weak sparsity strict exceed work handle define equal start point panel differently observation solve weight probabilistic scale different must break step turn need inferential procedure impose data expectation martingale respect error martingale considerable high furthermore need distribute individual rule term conditionally terminology lag introduce scale matrix singular conduct suffice compatibility type tailor define integer eq restrict make effect sense write diagonal really submatrix bound away assumption trivially moment impose compatibility standard literature various version investigate subgaussian plain static common covariate assumption dynamic panel generate completely subgaussian property behave wide defining inequality inequality least q valid well use end inference novel inequality allow grow even upper go correspond oracle panel technique quadratic equation analogy inequality linear inequality finally independence concentration sharp mixing restrict increase fast conduct observe convex belong subdifferential multiply leave q would invert inverse shall oppose term add back define sparse lasso limit consistent presence interested asymptotic basis interested show discussion work regression high importantly row regression properly j z jj j kkt subdifferential shall rigorously kkt write eq z eq inequality require argument needs understand construct diagonal removing row submatrix row remove column th remove except multiplicative row sparse generally weak sparsity sensible population coefficient shall define write replace row respectively reasonable assume I section bound subgaussian imply zero translate sparsity impose row entry equivalent justify dynamic panel reasonable mostly conditionally adjacent conclusion important sparsity part part impose term regression define large establishing asymptotically induce however parameter limit uniformity result limit estimator reduce low corner follow tn motivated following establish uniformly bound need order allow note allow sample hypothesis interested simplify hypothesis correspond assumption necessarily necessarily moreover cardinality provide stress total much allow consistent hypothesis involve inference extension relax inverse exactly furthermore vary dynamic depend relate inference static panel classical setup interested inference gaussian equal exist illustration variance equal hypothesis similar reasoning hypothesis involve asymptotically convergence accordance straightforward usual asymptotically inference restriction differentiable usual even impossible involve weak expense show band contract precise satisfied pi nj nz percentile standard let coincide convergence consequence reveal band uniform band important z guarantee irrespective guarantee coverage value achieve desire clearly confidence optimal particular narrow contract contraction fast contraction show band base band one contract worth non inference investigate calculation carry formula naive monte replication estimator square rmse procedure monte replication construct lag regressor interval involve parameter construct evaluate carry level confidence nominal coverage regard plain lasso report dynamic burn generate datum generating root lag disk toeplitz th entry dramatically covariance conservative report precise form I lx turn calculation reveal construct unconditional finding drive plain change unconditional carry non entry test power follow consider replace zero entry theorem variation consider baseline far experiment eight replace freedom rmse lr lr lr dl b l dl l dl h square variable encourage base see due wide confidence assessment uncertainty size superior least fix effect fix assume actually band superior coverage rate procedure interestingly affect result less expect towards lr lr ls dl l dl l dl turn estimation nominal price band wider believe band due accurate uncertainty narrow band experiment assumption accurate oracle one rmse lr lr lr size power b l dl ls increase compare oracle band nominal rate band test oracle procedure may add rmse lr lr lr power b l dl l dl surprising go furthermore band base confidence band belong left hand variable become narrow fix method similar exact relaxed experiment well band become wide experiment rmse lr lr power l dl ls dl l final add tailed covariate high set table error increase band increase sparsity fix roughly unchanged procedure addition conclusion consider dynamic panel increase test hypothesis simultaneously towards valid matrix inverse next band contract contraction simulation assumption extend subgaussian covariate error allow rule panel uniformly subgaussian assume furthermore subgaussian root outside monotone exercise page f tu p assumption make proceed theorem shall event event valid minimize lasso yield use trivially hold positive bound event compatibility equivalent estimator satisfie constraint introduce compatibility valid hence upon combine tx yx quadratic right root second right minimize desired formula root namely oracle n arrive uniformity enter oracle deterministic oracle norm random usual norm provide let positive entry th l xx lag lag ns l nz l ny kt tt ti conditioning zero martingale argument martingale arbitrarily conclude assumption calculation natural precede positive satisfying definition hand satisfying define thus assumption block I
observation observe shall natural bias serious validate dataset optimistic estimate validate define predictive performance complexity penalty rigorously perspective estimator criterion ic historical information require code although promise problem unknown approximation originally aic encode aic information true green green dot truncate add due content encode aic aic lead consequence aic lead information limit continuous freedom appendix derivation detailed meaning exactly approximation subscript understand result aic expression write play role illustrate encode mle goodness dimension increase encode optimum aic aic plot red powerful difference model parameterization small great complexity simulate black aic estimate initially accurate rather cross aic aic selection clear immediately large analyze computed evaluating eqn invoke correctly predict clearly significantly figure aic dotted model dramatically true complexity clear key assumption aic mle normally context aic fail sensible fail often propose base selection frequentist criterion frequentist analogy aic predictive dataset draw unknown compute approximation parameter complexity index analogous hypothesis estimate complexity construct call frequentist aic evaluate minimize small true expect good asymptotic approximation rather decrease generic feature nest parameter space associate see complexity parameter identifiable figure motivation describe analytic complexity complexity increment specify th write slope figure show identifiable justify identifiable justify clearly justify add predict plot cross entropy parameter entropy understand identifiable regime entropy ambiguity cross parameter complexity analytic equal ambiguity appear nest un nest complexity expectation chi freedom harmonic degree harmonic derivation give cumulative write model nest nest un nest write distribution relate model clear eqn slope slope closely q interpretation use inference large true exactly regular absence approximation accurate aic inference include statistic clear interpret test advantage traditional frequentist intuitively clear assign hypothesis hoc frequentist base specify automatically ad hoc confidence suggest bring acceptable example complexity compute simple generally necessary could result explicit tractable realization resolution beyond note probability offer promising model context exist na seem intuitive strictly decrease therefore probability model expectation dataset cross rule observe rewrite probability replace sum naturally measure cross appeal principal maximize equivalent leibl measure expectation identically think metric since distribution drop rhs equation compute relative minimize information maximize minimize mathematically great insight might significant approximately gaussian mle matrix expression normally fisher approximation independent term eqn distinction information estimator unit modify one derivation analogy derivation goal execute start distinction degenerate model true since minimum degenerate cross entropy understand follow strategy aic taylor expand information order perturbation follow definition nest un model true entropy remain perturbation acknowledge term term drop perturbation replace fisher simplification eliminate cross eq expansion minimize result write dependence explicitly clarity analogy aic harmonic normally precision equal perturbation chi parameter keep chi square correlation adjacent minimize entropy therefore choice chi need chi square entropy complexity define note chi square specify square must compute eqn eqn perturbation product line relate eqn eqn heuristic criterion aic apply frequentist analytically many context aic understand unbiased therefore limit exhibit aic like instance observation unlike bayesian hoc information inference theory justify narrow predict finite observation parameterization degradation mechanism degradation intuitive accurate fit reduce parameter ii high respect qualitative goal realize quantitative minimization maximize upon independent succeed context failure criterion frequentist criterion aic broadly divide description establish connection frequentist approach reference reference application application approximate model probability write absolutely distribution model
agent turn policy heuristic trajectory exploration tree agent accommodate policy rl horizon balance avoid confusion refer primary distinguishing encode exploration denote choose one policy uniform success agent execution evaluation next evaluate large control policy convert episode high reward although vb vb special dp close well dot solid line flexibility explain paragraph cr unknown r time time c box c advantage variable episode accumulate reward average test episode collect episode collect agent follow agent procedure approach use report exact controller node number perform suffer maxima issue see bar level employ uncertainty policy infer controller robustness low value em traffic control reward leave controller algorithmic iteration several monte policy rl approach exploitation reward iteration summarize category optima algorithms policy allow flexibility fix indicate trade time periodic generating model allow trajectory policy outperform produce low near solution produce solution previous rl worth discuss rl number traffic agent control traffic intersection agent locate except code compare traffic direction heuristic generate fair addition examine exploitation initial behavior exploration exploitation produce high solution scalable framework decentralize exploration exploitation reinforcement experimental benefit infer scalable domain problem size allow quality policy acknowledgment support us office research award nsf award variational standard divergence minimize low n decentralize vb update equation follow vb inference maximize joint vb n w r solve give reward allocate step vb k weight compute construct vb gamma issue non maximized way operation derivative form difficult search stick break connection characteristic useful detailed reader stick measure disjoint associate weight stand generalized dirichlet truncation density use keep equivalent indicator argument zero backward ki pz n ki k backward message compute recursively impact sequential batch exploitation five batch update body thm thm definition remark thm question rgb maximization decentralize size converge maxima far consider construct stick prior lead controller variational available demonstrate several showing algorithm art decentralize decision sequential numerous exploration control control make decision stream observation action dynamic decision generally belief decision making make planning horizon optimally scalable infinite continue infinite agent represent scalable em show learn trajectory without know work affect unable sub yield sub bayesian nonparametric controller previous assume centralized execution decentralize accomplish offline decentralize controller stick break prior controller posterior trajectory recognize prior contribution algorithm directly operates shift simple bayes net framework moreover vb agent episode vb agent problem size scalable domain simulator realistic adopt learn reinforcement knowledge policy able propose method describe relate tuple action agent joint action receive world state rs receive discount agent agent observe local observation maintain observation policy local policy infinite horizon belief objective bs maximize tuple n denote controller action policy notational cardinality joint agent obvious thus take history agent controller choose problem transform introduce binary reward pr pr net optimize policy representation vb problem bayesian stick break used specify structure formally decentralize stick representation index notational simplicity define dirichlet stick breaking I specifically base rl nonparametric decentralize domain world plan previous method employ hide hmm controller conjugacy hdp impose therefore storage employ gamma hyperparameter bias among node reader note process encourage compare dp sparse transition allow correlation break always episode agent vb low lb lb use hyper return policy controller govern dirichlet multinomial variational accommodate unbounded node apply prior agent stick break construct prior hyper application increase replace normalize w ki equation theorem vb prior reward reweighte improved step improve
tx x algorithm track particle keep track come past history use appear leave kind kernel particle extension localization guarantee consistent go forward sign yield un p dx semi growth map py infimum expansion tw go direction worst could grow basically particle contraction pf fix frank wolfe translate rate error kernel wolfe obtain particle explicit particle standard filter depend rather distribution rate would translate tb start investigate mixture give different gaussian fw though rate empirically significantly increase method remain application kernel filtering use frank wolfe quadrature monte quasi system detail experimental dimension switch govern series mutually order difficulty filter kalman filter run kalman albeit nonlinear closed density run pf particle reference batch system allow exact filtering fw compare pf resample carlo particles discuss assess compute rmse filter filter along quantile run nonlinear benchmark improvement somewhat difference see upon bootstrap pf mmd section propose algorithm six pose consecutive frame motion estimate pose model comprise velocity acceleration orientation bias position currently number rao filtering extend kalman filter remain evaluation modularity approach simply simulation setup fw run fw pf pf use run method fw comparison run time reference pf average second give assume know zero natural available unstable basically keep fw give improvement bootstrap pf particle error particle role focus investigate gain implementation evaluation update online scale particle pf spend step particle bottleneck ghz overhead fw fw fw experiment practice fw pf particle fw particle still pf filter particle performance quasi monte modular particle filtering filter particle filter future future work include convergence theory acknowledgement centre european project supplementary improve pf rao use rao tractable pf assume system comprise conditionally transform standard normal argue sort discrete mixture accord transformation naive gaussians store nice sensitivity implementation detail synthetic matlab control toolbox observable pair observable pair correspond observable main text report figure plot obtain stay carefully current effective optimize mmd many particle filtering error kernel matrix nonlinear mmd numerical precision ask increase translate reduction filtering error particle fw seem suffer big tb tb py px nuclear q py x f dx dx hx hx dx f hx dx upper expansion gaussian thus big decrease rhs thus take quite norm px conjecture result n ny py nx compute fourier transform representation integrate dx xt xt w dx may w w change variable w w n c w e dx w b w w w perform b w w n continuity small small less un thus linearity related norm scalar multiplication mmd term frank wolfe z finally control normalize repeat argument convention back quantity fw rewrite normalization constant go worst grow back would interested note explicit example really say c hand whether f without disadvantage presence quantity explicit though close upper repeat working quantity similarly get tw tt extra preferred tight remove really unit augment hilbert rkhs frank wolfe analyze analog instead fw mmd vertex extend step wolfe size gaussian experiment give step objective mmd fw vertex r g fw ball radius center eq fw g crucial interior yield giving imply become rhs convex maximize get induction thus strict translate back appendix say fw subproblem wolfe mm proposition radius unlike conclude seem might infinite thus worse previously around arise bad reference definition definition frank procedure integral reproduce rkh potentially convergence monte integration special replace particle filter frank wolfe quasi monte emission additional localization improvement quasi explore idea approximation constitute involve eq space set beyond sequential monte carlo inherently challenge computationally common relate robot vision synthetic evaluate image pose filtering solution reason bottleneck arise improve complicated allow filter leave standard inefficient option contrary nearly acceleration develop toward class filter bottleneck computation upon particle avoid arise simple monte bootstrap build appear frank fw quadrature particular gaussians past convergence preliminary give accuracy particle filter particle integral belong reproduce kernel pointwise reproduce property feature rkh briefly integral associate empirical mass independent use unbiased variance estimator analyze cauchy bounding approximate central quantity quadrature rule act standard rbf fact refer regularity object lie closure hull finite insight frank wolfe quadrature wolfe algorithm iterative algorithm optimize general banach iterate obtain vertex g k iterate suitable high decade old survey hull vertex run frank wolfe yield g k I optimization reduce negative frank wolfe quadrature rkhs g k maintain px include normalize tx propagate normalization constant frank wolfe rule shorthand adaptive quadrature rule pair fortunately insight central quadrature call fw vertex search non optimization exhaustive fw call sample wolfe fw vertex search show material add bad return quadrature choose search hereafter fw refer always weight alternative previously visit vertex wolfe hereafter refer k quadratic simplex min active reader frank wolfe quadrature fw guarantee summarize follow infinite
later construct give inclusion construction paragraph trajectory follow ball surely subsequence rescale choose fix inside example could linearly trajectory force jump recursion conclude tn tn tn q trajectory sequence ball around origin tm mn tt tt rescale noise converge surely sake completeness show independent unfold recursion expectation norm use eq letting rescale convergent consequence assumption claim assumption need begin show rescale trajectory sup l mn rearrange inequality get e kt dependent purpose fix solution clearly lemma inspire follow limit coincide limit sup thus compact let n xt x since map observation early deduce weakly lk nm lk xt xt dt xt lk dt nk xt nk bound sake convenience h nk nk xt xt contradiction subsequence k xt sample ready stability stable nothing hand xt tt assumption ml explain xt origin globally asymptotically stable loss generality assume martingale sequence loss assume work bm bm surely converge possibly compact invariant recursion early drift begin prove map follow lipschitz set xy yy hx nn ny nz n hx ny hx bm satisfy approximate satisfie map cx cx cx h cx let xt lyapunov origin proof n yy h x h nx h corollary approximate almost addition close connect since true even require special happen give previous consistent find list solution value initial refer proving outline prove length point prove lemma proof work assumption bound subsequence non empty every hard dc cx dy containing neighborhood h x h cx upper xy yy xy h exist linear functional convenience claim ax claim true proceed pick ax n contain n nk nk nk nk c nk nk nk nk h nk ax nk n nk fy contradiction ready modified retain assumption let h cx update h stochastic inclusion statement cx early stability iterate prove identical manner invoke iterate converge set explain immediate stability follow question sufficient stability lipschitz x h xt subset define cardinality show recursion limit exist satisfy iterate connect generalize assumption recursive chain nx c c nx h c section show version convergent omit proof iterate stable converge xt relax fall exponentially prove iterate follow corollary set origin similarly nonempty change make statement differential inclusion remain theorem aforementione explain deduce trajectory fall around origin rest extension mean value set recursive inclusion immediate drift problem corollary exist average stability two sufficient stability one natural theorem recursion lipschitz function recursion refer limit asymptotic detailed exposition subject show recursive specifically continuously iterate compact sometimes refer iterate word iterate guarantee stability several discuss show dynamical system extend iterate map reader martingale refer see special cardinality stochastic recursive overlap respectively stability case accumulation present unified take care aforementioned extension relaxed notation connect set prove stability stochastic outline work boundedness map differential inclusion let iterate sufficient iterate
consist modify present dataset patch extract task versus separate two disjoint piece take patch patch patch transform norm binary build subset dataset class image position multiclass image handwritten set image image norm class cifar parameter unique technique low burden threshold quantization run reader understand one one test use build methodology substantially accuracy considerably reduce bit multiplication show b c bit precision point throughput present differently classification accuracy ht cifar error report cause random gd portion mini sample train affect half range highly gpu cifar substantial half bit ht feasibility bit shift instead multiplication slight consumption expensive application technique technique enable single operation precision almost computational throughput worth note nevertheless accuracy dictionary set perturbation training increase generalization unseen investigation learn adjust integer multiplication bit shift dictionary thresholde entry run introduce substantially reduce load cost tested increase technique power consumption partially improvement high education allow access nsf mr mr code last detail von discussion classifier learn soft threshold shift method modify energy dictionary apply soft dataset indicate solely sum shift integer valuable implementation throughput decrease energy consumption cost enable instead bit almost double throughput resource trend feature feature overcomplete dictionary learn dataset classification drawback application computational resource drawback learn approach multiplication soft realize due parallel multiplication hardware much work consumption explore derive reduce four group image raw replace costly operation integer operation hardware accuracy dictionary classifier near multiplication bit shift slight dynamic image reduce quantization range value train vector multiplication slight decrease technique algorithm name soft reduce substantially substantially simulation test technique last sufficiently general different use multiplication extract extreme dnn paper approach learn valuable embedded consumption necessity operation architecture image briefly representation signal identically sparse pure good learn soft differently map simplicity process jointly hyperplane n loss prevent overfitte extract sparse follow classification class return reader deeply understand simplify finding technique purpose operation approximate relative power md show behave begin train atom build multiply open evaluated set figure ht classifier representing include eq penalization solve constrain problem iterate generate compute iterative method computing require dual constraint lagrangian problem lagrangian lagrange solve gd method upon gradient evaluate n establish modification comparing
sample strongly completeness simulate classifier result data mapping predict object old object ground label classical machine vision bioinformatics micro formalize decade work produce many e discriminant analysis lda artificial vector tree attribute instance justify image object g face variety approximate affine camera coordinate point move object lie subspace rise study subspace important branch compression know subspace contain adopt class instance obtain subspace discrimination variant version way show subspace face model handwritten speech classification explore theoretical justification interest justification know variable independently functional whose minimal rule predict minimal actual classifier small spirit fact function converge word large misclassification matter knn boost consistent certain condition consistency linear comparable consideration complete experimental classifier rest description suffice integer restrict classifier find subspace dd form singular class n desirable property rule say strongly consistent rest obtain follow theorem classifier variable center reveal prediction condition weak result svm boost simple linearity important easy analyze order good interpretation limited foundation generalize therefore long point bayes rule partition optimal practice form form observation rule classifier nx q p main proof basis consider plug difference fix therefore show due estimation mle proof evaluate lda svm demonstrate result show serve complementary perspective reason note significance study classification simulate brief find sample experiment subspace disk angle class recognition chemical region determine find machine repository dna find recognize dna sequence retain dna represent three class neither database digit service recognize digits image signal acoustic original evenly impose recognize collect experiment dimension na na na news real repository collection news principal explain reduce subspace carry matlab default toolbox multiclass realize author split subset split record pt htbp lda dna c dna news result know comparable lda meanwhile computation roughly lda lda require covariance positive reason ambient news restriction review simple model subspace prove prediction show result especially
datum fuzzy flexibility binary fuzzy value hyperplane optimize fuzzy hyperplane apply propose artificial world obtain fuzzy machine fuzzy hyperplane idea introduce classification success character outperform precisely bioinformatic version square svm task accuracy suffer drawback fully assign strictly assign class many consider main concern assign importance degree moreover classifier ability approach cope fuzzy fuzzy analyze fuzzy concept operation introduce offer capture inexact fuzzy unlike phase propose treat datum point importance apply fuzzy membership point fuzzy fuzzy bias fuzzy rest include model fuzzy quick review main behind training point follow solve geometric interpretation depict toy group hyperplane hyperplane weight sample geometrically toy hyperplane close class indicate class hyperplane slack one desirable penalty employ equality preserve accuracy constraint theoretically four standard hyperplane hyperplane explain fuzzy improve notation sample sample represent slack equation transpose application belong uncertainty elegant cope fuzzy final degree influence influence fuzzy assign application induce discriminate class vector symmetric triangular fuzzy fuzzy fuzzy component fuzzy define fuzzy fuzzy equation inexact eq slack rewrite rewrite equation would appear rewrite substitute hyperplane hyperplane equation q fuzzy find hyperplane hyperplane definition find point hyperplane fuzzy data fuzzy hyperplane n nx fuzzy distance fuzzy hyperplane membership determine fuzzy hyperplane fuzzy hyperplane hyperplane accuracy experiment environment pc intel ghz gb ram false false negative cross methodology focus first svm record record determine circle circle show result algorithm paper svm accuracies hyperplane responsible line section exactly amount high responsible classify two line fuzzy nature line discriminate data b dataset uci machine repository heart cancer dataset represent detail four accuracy note version two algorithm version propose non version meaningful lc lose heart cancer
fix shape sufficient transpose see univariate univariate identically distribute iid exponential iid standard model family extensively sequential hypothesis widely literature reconstruct example family matrix column matrix row statistic define ty cumulative call statistic short denote give iid belief vector assumption belief reconstruct natural statistic mapping k component third definition fourth involve family family rule k completing suggest belief sufficient statistic determined dimension hypothese interpretation essence prior period dimensional embed statistic space dimension sufficient belief statistic bayes namely assumption reformulate minimal statistic space belief truncation explicitly oppose become compare contrast two acceptance yes apply statistic review however scalable testing simple assume variance concern prior belief one cumulative optimality equation illustrative describe belief belief lie fourth panel clearly path remain next describe sufficient statistic acceptance interval low action decision independent acceptance interval implement desirable increase heuristic draw sharp multiple hypothesis period period natural state dynamic programming conjugate conjugate fact arbitrary flexibility solve belief flexibility collect figure acceptance interval figure increase long require go high dimension chart clear gradually increase observation square suppose natural difference identical still different matrix statistic ty x obtain correspond mean give prior dimension example figure horizontal axis cumulative vertical square improvement provide policy development asymptotically sequential observation identification zero case hypothesis distribution zero series posterior study compare cost combination simulation less average cost percentage table error optimality consistent optimality policy hypothesis difficult another response easy satisfactory range matlab intel core prohibitive application adaptively multiple mode diagnostic power hypothesis decision hypothesis decision maker one hypothesis terminate choose mode sampling true hypothesis generate observation know decision prior family fy k sequence sequence clearly follows denote rank full rank sequence sequence brevity belief sufficient statistic control beneficial use many action systematic observation matrix low sufficient mode account acceptance region discuss yy p dy variable test multiple maker take multiple stop sample period observe average x mf global let become unless take generalization widely discuss structure policy difficult implement devise efficient scalable without assume conjugate sufficient statistic dynamic natural standard belief approach desirable quick natural use variable learning problem often sufficient chapter need sufficient solution become commonly table observe dimension sufficient especially situation policy increase curse draw intrinsic dimension exceed family commonly moderate computation illustrative suggest also extended hypothesis test sequential alternative hypothesis observation maker one accuracy goal quickly translate minimize expect incur incorrect involve trade arise vast include security monitor clinical target recognition study sequential hypothesis test maker identically iid statistical one strongly stop accept observe policy implement hypothesis policy hand numerous asymptotically study policy note review view multi observable identity generalizing stem curse dimensionality dynamic hypothesis scalar one vector belief increase make come exponential family central theory dimensional hypothesis belief many binomial reformulate find moderate even hypothesis solution rise region opposite standard belief experiment substantially suboptimal delay parallel involve mean contradict specific method scalable hypothesis grow reconstruct belief technique observable limitation tie suffer curse see come exponential iid observation function distribution hypothesis observe alternative accept make new stop multiple hope identify desirable quickly respect historical acceptance
bn e converge model sample structural every dag ss subgraph super vertex average degree super around feasibility bn vertice cluster search improvement hamming benchmark report time prohibitive great hybrid cb ss thousand vertex skeleton contain true network extra extra bn attracted recently control e false compare hybrid ss allow fair difference ss max min parent child learn variable subroutine parent child combine incremental divide cb candidate empirical conduct pc min hill various formally bn tuple direct acyclic dag bn parent statement extract graphical note exhaustive intractable separated separate bn converse denote parent common unique bn handle cb identification neighborhood scalability cb systematically independence independence fisher test decide dependence upon rejection acceptance nan independence discrete multinomial represent independence discrete datum function frequency c l configuration shorthand classic mutual define factor degree particularly sample contingency nan increase heuristic perform sample user power structural zero contingency degree brief overview reference combine weak learner attempt pc learner inter computation may think false weak perform extra receives node hybrid combine benefit extract conditioning increase pc run pc learner decentralize search candidate pc true working domain effective restriction neighborhood less severe decentralized significant enable neighborhood correct parent tx add true positive tt positive change xt set dag xx hill try inter incremental receive return rough de pc omit brevity conditioning size de significantly increase reliability two hundred thousand variable restrict relevant relationship ss phase discussion idea efficiency appear candidate unconstraine greedy idea candidate hybrid identify parent hill begin edge direction score score continue similar recursively search adding discover list list keep last good local change list attempt occur score ever encounter search terminate score heuristic enter pc ss range c c c experimental comparison datum benchmark learn algorithm claim possible compare bn benchmark repository investigate parametric reference pc implement integrated package develop pc publicly type pc cpu ghz go ram run window bit investigate quality skeleton return pc cb phase false ratio number output true positive euclidean assess dag report five dirichlet equivalent single sample support learn goodness fit generalize network generalize distance quality dependence require match undirecte add orientation edge network structure performance new correspond posteriori distribution degree hold encounter experiment prior learn rely gold employ several reason report skeleton ss phase benchmark benchmark depict table increase gold improvement bad clarity mention regard quality observe false improve benchmark cb maintain reduce quality dag obtain ss bic improvement goodness clearly dominate ability generalize regard dependence pc significantly rapidly tendency less average overall increase factor grow somewhat linearly nonetheless worth package employ pc efficiency compare code currently allow fair consistently generalization cause maintain couple child alarm conclusion pc promise construct bn performance possibility hybrid keep low focus study heuristic combine dependence cb independence type size applicable large sample structural time slow behavior independence test permutation pearson permutation mutual test shrinkage single bn permutation structure goodness output graph one test test outperform parametric test network structure fit picture open reduce super super structure sound improve learn sound rather skeleton expect sophisticated strategy hc sound super easier learn many extra edge thereby result type burden involve hybrid gain accuracy rate miss independently keep track found previously lead redundant design infer target optimize version super get computational maintain cache store reduce computational author hybrid bn hybrid extensive experiment outperform goodness generalize overhead time currently structure structure find outperform margin experimental edge crucial super like
dimension overcome avoid employ filter dictionary convolution receive attention globally decomposition provably activation draw model work extend decomposition invariance convolution operator denote element tensor denote similarly matrix cyclic convolution vi j convolution operation cyclic cyclic convolution twice cyclic convolution cyclic shift discrete entry important use improve computational extensively stack matrix q concatenation stack additive noise incorporate sample activation active coordinate encourage small extend limit ica simplicity estimate decode criterion map map focus develop estimation paper rd tensor extend tensor I stack slice v v ab column ab cm order multivariate tensor use moment third tensor bx ax ax convolution show third nice form denote column third univariate third activation order activation fourth method manner cp usual decomposition unfold order filter minimize frobenius enforce rest devote throughout paper matrix block non relaxation mode perform compute product efficiently implicitly present computational utilize various rao carry incorporate stack reduce completely stack partially column matrix consist column stack introduce stacked identity thing stack filter appeal notation denote nj block stack diagonal simply inversion need result fix iteration note unfold compute inverting processor take degree parallelism serial computation parallelism time degree parallelism combine discussion decomposition framework activation convolutional recovery error alternate fact spurious increase experiment report order minimization alternate scale sample linearly question investigation plan task scalable long learn extend dimensional signal replace block generally framework lie algebra advantage tensor invariant expect embedding block inverse multi know order entry know shift eq tensor format q therefore rao therefore decompose propose via inversion inversion stack partition r invertible invert matrix inversion invert multiplication indicate inverting matrix inverting processor simultaneous block therefore multiplication processor processor lemma corollary fact paradigm learn component learn cp tensor convolutional projection onto operation fourier multiplication compare alternate minimization map decomposition dictionary learning convolutional ica deconvolution convolutional model generate unknown dictionary unknown activation activation speech sentence spike train activation language processing usually loss employ filter add sparsity alternate activation vice versa alternate expensive modern run optima reading np convolutional compare dictionary shift unchanged fundamentally ill pose impose design solution huge dataset paper answer framework convolutional moment via decomposition convolutional map convolutional ica whose component stack invariance popular act sample operate average moment closed use operation fourier multiplication degree parallelism estimate length
diabetes heart cancer screen moderate size repository book short intercept laplace laplace concentrate require logit challenge ep deal ep contrary probit site update moderate accuracy error versus accuracy accuracy laplace panel fig plot scheme component fig dataset supplement box plot accuracy across plot supplement sake scale ep ep left panel ep cpu intensive standard improved expect course cpu hardware note pass laplace suppose replace student supplement result scenario addition represent ep panel laplace breast complete sampling method dataset produce nearly instant along essentially gold standard nice discuss gaussian laplace proposal gaussian probit particularly favorable next factor cpu cc ef mt mse mse breast heart importance dataset probit ef cpu time spee intel hyper core cpu gain mt efficiency mean parallelization speed implement core virtual amazon ec virtual factor run time report median mse improvement mse dataset gain parallelization evaluation chance shall section sake completeness see supplement scenario order relative criterion eq resp square obtain resp cpu sampling importance sampling sampling term posterior observe fig median across reader difference use ill discrete proposal construct nest regression correspond intercept successive regression mcmc sampler binary intractable approach hand see ep sampling smc thesis smc pp importance laplace valid pseudo argument compare approach sampler cpu minute figure smc sampler estimate also consistent reversible jump sampler estimate covariate box pass hyper generate selection interesting extent type posterior deal prior ep important end routine right near regression concerned recommendation always fast implement author ep logit drawback ep lack theoretical learn however manuscript ep well assess second exact particle reduce single even run alternatively perform well calibrate main message title leave alone serve benchmark elaborate distinguish algorithm gibbs sampler covariate matter even offer generic metropolis hmc amount practice propose novel compare gibbs sampler small dataset numerical properly computation scenario binary possibly covariate seem critical complexity therefore computation perhaps strong approximation ep family gaussians alternatively active covariate finding generalise assess statement hand study recommend account approximation former regard latter binary certainly fast whether laplace smc relative performance use acknowledgement thank common regression moderate discuss extent sound review fast laplace extensive might hard day markov inspire hmc smc nest sampling approximation variational book even approximation methodology thing approach binary regression g probit logit benchmark remark bayesian optimisation practically regression question title suggest finding lead current gibbs dataset well diabetes basic toy large seem remain competitive computation algorithm obvious criterion discuss posterior whether easy change link complete extent require manual tuning obtain performance important fact easier free manually tune discuss tuning computer hour cpu pay require much manual may serve review believe already develop relate criterion parallel method perform core architecture parallel architecture although phrase already certainly get big big dataset fair really big datum away kind encounter bayesian paper cover deterministic offer method discuss part contain discuss selection discuss end computation generic expression datum consist cdf transform linear form probability cdf probit take logistic cdf e accommodate outlier predictor preliminary deviation mean range intercept weakly assign outside reasonable default centre predictor henceforth independent deviation scale henceforth jeffreys prior determine method one cauchy difficult tail explain quickly evidence quantity later particular tune derivative compute two concave probit regression stick gaussian map point posterior iterate iteration approximation log newton work concave variant pass estimator infinite variance mle properly complete hyperplane separate outcome extra inference occur variant newton adapt automatically g determine line replace iterate reweighte interpretation seem roughly stick newton cauchy prior shall cauchy ols ordinary happen section come include ep e laplace ep vb vb field probit may marginal component expectation directly mind fast approximation preliminary method describe laplace approximation expansion minus refer particular laplace phrase approximation discussion drawback marginal simply obtain laplace mode q vector stand laplace deduce see pass connection scheme apply posterior hyper grid improve empty describe cauchy recommend student prior log guarantee prevent converge work reasonably briefly describe em em student implement deduce aim single newton approximate one newton iteration laplace refer reader detail laplace laplace consensus well match posterior ep compute iteratively parametric density natural exponential q natural gaussian family could gaussian natural ep consist update equivalently keep match z gaussian site update turn achieve implement must compute hybrid probit compute supplement link logistic dimensional quadrature simply never course simply ep support result sense posterior ep work many variant determine intensive complexity ep site observation laplace perform ep expensive laplace remark modify ep end may parallel factor processor inversion improve laplace marginal describe perform basic laplace vanish point remain implement result adapt different choice simply write evaluate correspond adaptation ep fair model shall see laplace sampling method small form calibration prior since previous laplace ep compute preliminary generic posterior q marginal estimation restrict ep recommend student ensure variance bit call stress however iid assess auto normalise also advantage importance amenable quasi carlo integration explain offer marginal error cpu trade know suffer curse variance grow exponentially large meaning get negligible hand moderate see smc automatically perform something elaborate assess compute roughly approximate target require simulation compute instead ratio elaborate technique carlo estimator express vector inverse cdf replace sequence vector spread evenly g condition monte background construct possibility conjunction importance sampling mention however often often ability quasi monte way marginally average become unbiased assess repeat assess variance repeat reasonable approach chain carlo markov leave invariant drawback g specify start determine burn period assess chain invariant fair regard b start draw cover assess visually consider augmentation formulation vector variable probit model sampling iterate sample b thank conjugacy stacking hence gibbs drawback particularly switch student scale cauchy well cauchy prior b yet require strategy turn thing derive sampler first augmentation mixture finite second paper logistic infinite discuss conditional since implementation main justification great generic investigate numerical hasting consists iterate describe metropolis generic take approximation practice usually importance input hasting metropolis critical move slowly move rarely choice lead fast exploration close ep validate bad news move tends cite motivation elaborate strategy hmc cover hamiltonian monte mcmc perform determine accept hmc make jump metropolis excellent un normalise physical hmc position velocity energy mass trajectory constant practice proceed new velocity keep practice step perform third accept reject see summary rely volume preserve jacobian probability output output momentum perform hmc mass stepsize approximation obtain rescaling incur correlation difficulty drawback hmc tuning seem currently popular vanish adaptation acceptance optimal hmc take acceptance iteration fix much exhibit behaviour large may long distance come spread already take interesting hmc correspond hmc locally geometry main drawback derivative expensive take account well exploration might relate adaptation hmc instead aim trajectory
envelope computation available successive segment th envelope whole envelope calculation persistence landscape alternatively one intersection line segment bad also different landscape intersection persistence landscape calculate persistence bit paper subsequent end landscape clearly computational construct number point landscape bad algorithm number death empty intersection interval calculate persistence encode persistence landscape number interpolation persistence special n average persistence kt give k kf see summary k n consider death constructing may calculate repeat landscape improved min case death evenly spaced combination simply combination two persistence q calculation complexity distance formula sum consecutive write integral dx ap start diagram since calculate persistence death lie evenly spaced calculation persistence nontrivial diagram birth death lie grid distance persistence combine previous calculate combination birth death pair important distance persistence persistence construct calculate interval grid present experiment dimensional persistent homology uniform dimensional normalization difference higher project cloud homology cycle subsequent rescaled range new back scale persistence distance average explain combine compute equal proportion great difference various believe ccc dim dim dim dim dim dim dim dim scale one dim dim dim dim dim dim dim dim scale dim dim dim dim dim dim dim dim ccc average persistence dimension normalize degree persistence landscape implementation bottleneck distance wasserstein current death uniform birth persistence diagram bottleneck landscape bottleneck wasserstein distance currently present w collection independently aim landscape birth landscape distance calculate persistence landscape birth death calculate persistence implementation procedure library maintain user tool user familiar programming program library plan add program describe program illustrate toy program window os available result program form file birth encode also file contain persistence landscape form diagram sequence critical follow file persistence landscape diagram degree file combination persistence name contain birth pair persistence file consist persistence program file death union circle radius measure uniform tb persistent homology use time result name circle file encode degree file homology particular degree list file circle dim txt dim txt file dim txt list file persistence diagram circle persistence persistence txt point example persistence persistence circle circle txt persistence txt persistence txt example persistence txt circle persistence circle circle persistence txt txt describe input file file persistence diagram landscape calculate dim contain landscape produce file obtain file generate plot software build engine instead create plot program name file persistence diagram persistence landscape persistence remain plot use h ccc circle circle circle file file persistence program compute persistence combination persistence file contain persistence diagram persistence degree circle file file contain persistence diagram combination persistence integer text example file txt diagram persistence output matrix file persistence name file persistence diagram persistence file suppose contain class file suppose indicate try permutation indicate distance please lot time user program file circle expect section implementation near classifier persistence topological distance matrix implement persistence diagram class sequence option return program vector coordinate usage program determine program landscape locate training name file file program file later classification classify create use option integer indicate name file indicate supremum value persistence diagram landscape distance average calculate parameter average one run indicate many name file file file name classify indicate file txt order classifier persistence diagram calculate class half diagram classified file file name circle classification element make interpretation easy work except get wrong reason turn interval correspond circle sort well follow outli classifier persistence persistence landscape birth death construct landscape n persistence landscape sort point next k kb db kb kb kb acknowledgment author thank valuable suggestion topological multiscale geometry quantitative persistence topological give persistence distance average procedure intend facilitate statistic topological topological persistence landscape module average summary calculate topological summary tool provide summary useful method prohibitive implementation tool publicly topology machine purpose convenient summary summary call standard simplest encode filter homology field obtain space turn module nonzero homology homology give basis consider pair consider generalize correspond increase sequence summary perturbation lead perturbation choice successful breast signal tracking landscape birth death birth death th large exist persistence landscape extend persistence kt kt distance persistence persistence persistence landscape various library death pair persistence diagram wasserstein distance statistic necessarily adjust metric procedure testing persistence diagram brain persistence diagram apply obtain confidence band landscape average persistence landscape persistence landscape bind persistence landscape kernel kernel persistence diagram reader also persistence landscape birth death persistence death naive persistence landscape min clarity coordinate element clearly linearly total number value persistent landscape array large equal persistence vector persistence piecewise encode persistence landscape map persistence linear construct landscape list death variation appendix computational complexity persistence landscape evenly grid k landscape list sort accord initialize add kb
solution quadratic grid part million cell simplification criterion analysis simplification grid structure iteratively degradation dissimilarity cluster impact merge dissimilarity grid merge minimize degradation minimal grid w r distinction agglomerative stop choose analysis follow cell hierarchical agglomerative categorical cluster increase representative cluster model evaluate average representative numerical categorical result frequency visualize frequency select interest contribution provide visual mutual select part mutual partition observe excess interaction locate contribution interaction expect visualization highlight valuable part bring complementary add mobile day computation confirm cluster traffic study time categorical apply inactive area hour call record cluster nearly amount datum million indeed distinguished plot ratio cluster hierarchical interestingly cluster study satisfy number partition stay rest dot strong country cluster country four due city phone traffic cluster place locate use cluster dot typical locate main already area influence country cluster city city big city less typical less region recent growth country locate area intensive city part central central business locate city cover neighborhood mainly area previous one separate north localize match party area last two group locate area similar locate city differ dark grey introduce mutual visualize traffic call inter traffic visualize fine segment draw position segment proportional contribution information proportional call map highlight call visualization country capital big capital recent city activity explain excess traffic phenomenon phone west country around area densely flow note track traffic time period introduce section categorical call high call study call interpretation treatment ten segment miss indeed period time segment miss group call color call green locate localize miss short localize activate provide understanding year three describe week hour simultaneously partition week discretization hour keep obtain cluster day simplify fix segment acceptable four cluster display day column segment red blue contribution mutual partition discretization business call pm office hour phone traffic business cluster area pm day cut pm last segmentation interpretation period pm follow pm user map traffic period color partition discretization connect locate east pm day connect pm cluster user people area part area experience pm pm business le neighborhood west city economic sum live area work week localize area aim extract different mathematically make first country network country call difference mobile user live profile usage discuss interpretation besides level country confirm impact economic identify branch em mobile service still grow mobile phone first case call available network answer question quality pricing discount depend valuable information spatio pricing receive much attention benefit system public mobile may g mobile challenge etc process phone million daily spatio equip activity suggest activity health monitor al approach mutual minimize obtain locally cluster matrix significant progress way analysis mining sequence forecast combine free categorical dimension therefore network suggest base provide explore exploit component applicability user datum data model aim joint type categorical numerical take cluster category categorical interval numerical variable multidimensional grid whose cell partition partition grid posteriori minimize bayesian implement trade robustness follow analytic cost combinatorial notice mean data categorical univariate categorical categorical variable grid stand priori constitute categorical numerical stand model close prefer get priori grid high cost nan priori probability likelihood cost value grid indicate simplicity logarithm minimum code length grid value categorical equal bn n obviously bottom strategy pseudo code grain make partition interval evaluate merge merge iterate grid consider case e categorical numerical point implementation greedy grid grid resp grain advanced mainly exploit grain pre sparse cell cell empty contribution cell grid stem hierarchy model part interval cell interval merge perform instead grain concern grid dedicate heuristic locally solution post alternatively partition value across move interval time optimum search meta principle consist consider round allow detail optimization method mix available name several real study cl record consist call reveal valuable human show g paper suggest methodology contain well explore original relevance activity massive mainly service basis mobile generate date duration call exclude initially purpose social interaction activity derive unite sum recent valuable information development purpose leverage country rise application improvement economic indicator population city planning management mobile prove mining technique source temporal sequence source generic methodology retain model technique simultaneously variable discretized categorical group see constitute data result brief lead data principle exploit result experimental relate work conclude come set communication million mobile
unit first mask matrix impose property must sure unit connect input unit rule define intuitive applie mask connectivity last layer sample great minimum connectivity layer conditional model conditional beneficial approach order stochastic minibatch miss partially invoke unobserved secondly ensemble construct exploit conditional slightly easily vector original conditional random order first hide agnostic randomly advantage agnostic train exploit create ensemble order unit train lk uniform l w b choose order hide connectivity imaging training agnostic instead minibatch assume whose connectivity input training absence indistinguishable situation inform additional learnable apply strategy weight also parametrization sometimes useful treat every cycle list connectivity agnostic illustrate layer along value lot work feed autoencoder generative behind research test intractable partition design neural autoregressive feed architecture extension state make unfortunately deep code reproduce tag likelihood respective sgd batch early binary uci evaluation put university california repository letter stanford overview dataset name input valid connect dna letter run hide update varied cycle sample chance validation hyperparameter layer activation relu make competitive otherwise clutter deviation deviation supplementary connect rbm mask reveal winner however mask help negligible letter mnist mnist digit update help single mask second hide make gpu model gpu building uci relu conditioning varied hyperparameter value hide rate report table result make network well forward deep yielded layer add pattern case illustrate layer train vary mini batch rbm cd intractable tractable order mask mask compare near set figure ensure simple use mask modification autoencoder distribution direct autoencoder evaluate high probably maintain acknowledgment compute mask mask dna letter make mask make mask make designing modification autoencoder autoencoder autoregressive constraint reconstruct input constrain autoencoder output interpret full multiple framework architecture implementation fast competitive art autoregressive definition general formulate learn scenario miss imputation synthesis many make challenge essence curse impact grow good fortunately recent progress task great scaling focus attention operation explore simple adapt network make estimator alternative mask autoencoder output autoregressive solely precede preserve implementation gpu straightforward autoregressive explore simultaneously multiple observation connectivity binary description basic build upon clearly example concentrate observation motivation hide representation input reveal statistical structure distribution autoencoder attempt learn feed forward close matrix activation function input connection autoencoder specify cross loss treat take negative autoencoder usually descent paradigm autoencoder layer input disadvantage trivial copy reconstruct perfectly consequence equation since perfect x could autoencoder output valid probability specifically able properly correct autoencoder product imply always decompose conditional thus become negative q autoencoder particular form unit sequentially modify autoencoder satisfy autoregressive computational matrix multiply binary mask autoencoder impose assign integer give hide depend conditional exclude create encode overall encode connect thus connect hide connect output notice rule unit mask construction autoencoder autoregressive connectivity
output hide stochastic sigmoid number advance hide unit sufficient work compact representation feedforward markov organize definition deterministic present tuning bias layer keep section bias offer number string binary negative strictly source polytope form consider sigmoid compute scalar activation affine output probability feedforward layer unit output nc feedforward kernel represent feedforward kernel give feedforward network shape every circle sep dot right cm dot minimum size dot node transform cm dot node node dot l probability eq parametrize feedforward tuple distribution approximate arbitrarily closure euclidean topology property k mr free minimal unit k feedforward weight multiply feedforward network threshold extensively classify deterministic hidden function unit term marginal activity independently hide second bias bias tight tight approximate reveal theorem first well depend idea illustrate arbitrarily row copy finally hull model individually hidden piece compact lemma trick input us unit trick produce flexible layer simply shape style shape sep dot dot shape inner cm distance right dot b transform minimum dot end dot draw sep dots node cm dot node style draw size cm dot l node swap l swap follow bias dimensional cube face cube support hyperplane plug previous approximate kernel arbitrarily divide successive pair vector unit except unit unit hyperplane dimensional n kernel let l l deterministic arbitrarily indicate zero vector lemma weight b n precisely strictly entry entry wise map consider n fig pz eq arbitrary choose appropriate refinement arbitrarily certain mutually th indicator assign make relative sum next input bias entry map consider sufficiently large gradually th continuity nz li claim arbitrarily lemma entry p proof make irrespective make pz pz p z arbitrarily transition continuously value transition one upper stochastic feedforward sigmoid probability free hide suffice suffice kernel boltzmann machine show suffice unit suffice bound feedforward
pz model scientific date historical introduction thus integral evaluate generative least precision solver equation transition say give triplet numerical solver general know perform computation aside differential probability density evy drive stochastic previous deterministic function transition analytically integral exactly markov pz denote integral hide outside yield estimate desirable produce numerical exact tuning produce go infinity go consider exact contrary extend kalman filter systematic return uniformly already arrival new available advantage sequential illustrate introduce uniformly rule case importance draw prior per yield go infinity however problem next approximate satisfy implicit pz model free perhaps plug abc posterior markov give go understand true retain go instance summary abc quantify abc estimator exact abc filter plug method wise particle deal smoothing task since publication introduce integral draw k require weight propagate resample weight without significant describe scheme consist draw independently resample approximate filter approximated manner go filter limit bias remarkable since make particle sense filter tool filter kalman dimension particle filter estimator typically improvement study filter filter constitute product form update particle linearly observation prove particle per observation infinity guarantee particle particle step define define k k go infinity however variance well path degeneracy step population distinct time particle replicate precisely element path particle resolve degeneracy issue smoothing value particle lag degeneracy consequence particle method model initial represent dirac measurement obtain approximation obtain recognize degeneracy indeed transition delta fewer recognize early function random walk introduce monte move move leave posterior well high correlation move early method move degeneracy attempt state year advance filter iterate filtering rely filter model particle particle filter efficient recall particle call particle metropolis hasting pseudo particle path trajectory pseudo compute distribution infinite perfect filter estimator yield perfect thus metropolis hasting proposal remarkable algorithm x particle mcmc method study particle variance observation although informative optimistic assumption independently would overall filter particle one path tx compute eq base constitute practical consistent approximation general iterate batch upon arrival begin design proposal number introduce address take play model step estimator section light particle mcmc go infinity ideal smc markov incremental model resample invariant resample n w structure equip particle difference obtain instead incremental move instead complete simply mention smc distribution number fall particle define consistently go infinity design turn computational us mention evidence retrieve algorithm consistently compute integral ideal smc fortunately slow occur typically assimilation perform happen occur operation overall memory involve thus keep available memory cost error equally across motivate algorithm smc reasoning run step overall smc sequential online upon arrival piece computational effort uniform smc automatically along enough make adjustment stable performance currently exist pose challenge series term scale algorithm amenable architecture algorithm computation year pz use generate algorithm distribution particle resample particle systematic resample behaviour smc run ess ess decrease slow time step whereas ten occur half move end run precisely plot transition particle call reach call indicate call trend cost overall year daily estimate set four since call pz differential minute use occurrence incur collect minute algorithmic five run represent pairwise contour indicate dot explain instantaneous population posterior locate recover sequential ability investigate grey quantile marginal use go parameter accord asymptotic fisher information imagine observation reach figure predictive infer particle successive grey plotted circle triangle predictive region observation expect fall region focus time step predictive circle outside indicate triangle grey estimator introduce another pz pz except term use uniform pz odd py approximate algorithmic bayes factor dash support pz pz observation pz bayes pz keep generate pz bayes show simpler available enough confidence accord bayes criterion particle confirm initially simple pz prefer datum strongly support light review parameter filtering integral approximate online filter exact parameter smc estimator update arrival complete applicable run reasonable hardware thousand number change open area development one difficulty cost filter yield evaluation linear unclear whether likelihood could super moderate prediction dimension state variance particle filter typically scale particle dimension another memory store particle involve whenever memory usage storing path study method reduce hardware adapt play compatible measurement particle filter markovian requirement meet markovian particle setting recent markovian instance motivate direction article hide recent transition put gaussian particle identifiability case infinite particle markov recently space distribution markov constitute current methodology ep thanks taylor useful comment dedicate discussion bayesian time series arise various treat arbitrarily complex smc markov fix review development allow currently object interest illustrate toy open challenge scalability review long state space flexible les mod markov les des de smc des pour du les observation une en des en le les de la angle er pour analyse mod en population de pr plus des plus des mod plus constitute series value countable collection time observation arise latent markov chain specify initial successive state call transition specify current distribution parametrize integer explicitly write collection resp represent daily water omit indicate candidate volatility financial series phenomenon study phenomenon choice synthetic dataset give trial one intuitively hope future observation reliable inference ad hoc procedure connection section transform integral introduce desire computing section review compatible implicit meet desire requirement smc monte method smc mention section methodology open challenge goal hide refer available article refer normalize filtering path trajectory prediction refer give current depend use prediction product denote smoothing refer state realistic lie interest observation prior evaluate marginal bayes normalize useful uncertainty account filtering smoothing account next several compare chapter comparison evidence evidence normalize normalizing introducing account smoothing refer average task assign prior odd
runtime run line theorem use randomness space distribution taking banach exist absolute need bernstein random satisfie find still cauchy schwarz cauchy schwarz inequality degree proof induction schwarz eq equation result singular proposition definition example application tensor th rd tensor decompose decomposition polynomial decomposition relationship matrix concentration tensor represent involve th array outer product th entry tensor application tensor define sum agree correspond tensor hard problem behave survey unlike matrix decomposition tensor specialize decomposition algorithmic idea latent gaussians model dirichlet allocation previous inspire many work limitation although attempt decompose tensor overcomplete tensor machine order tensor base rd preferable interested overcomplete decomposition overcomplete rd tensor understand explicit rd nontrivial circuit rank explicit rd order matrix component also case quasi sdp see recent survey difficulty overcomplete rd tensor unfold tensor matrix unfold rd unbalanced intuitively base order moment allow particular component closely give find subspace closely relate many dictionary tensor almost randomly third close close decomposition close close tensor distribution recover high uniform spherical accurate dependency however refine find refine close prove component multilinear tx k tx x key sum random tensor tensor far unfold corollary unfold rest relate tensor give polynomial rd tensor key tool section quasi algorithm decompose norm spectral norm sum norm tensor decomposition inner matrix block notation dependency throughout high tensor array paper simplicity rd tensor depend goal homogeneous polynomial corresponding sphere hard concept reader section system inequality polynomial define constraint satisfy form easily generalize variable constraint schwarz prove square useful matrix random matrix turn argument expectation think expectation distribution distinguish expectation polynomial degree pseudo polynomial x pseudo obtain pseudo constraint satisfie expectation tx ta concentration random tx net vector close later give observation suffice vector idea pseudo expectation pseudo pseudo though maximize tx pseudo p yes otherwise norm case suffice concentration randomness follow careful noiseless norm hold polynomial rd tensor tensor high unfold algorithm success degree nc basic find find moreover intuitively remain constraint valid pseudo pseudo e kk formalize I use time pseudo satisfy I kk vector intuition old claim average argument detailed prove suppose vector like algorithm follow sdp high satisfie previously find unit follow expectation constant must appendix proof decompose overcomplete rd order rank almost match unfold concentration technique useful tensor decomposition machine although initialization algorithm idea help solve david discussion randomness schwarz claim bind property fy diagonal entry bb apply cauchy schwarz spectral lemma lemma j tm direct consequence simple allow simplicity let ia incoherence sum bernstein sum thm basically say concentration concentration independent exist absolute uniform proceed q n tt ia tt bernstein bernstein spectral matrix spectral individual incoherence variance bernstein randomness two claim bernstein individual probability claim product psd suffice decompose write eq ready return yes randomness guarantee unfold prove return pseudo expectation know tx show return show expectation tensor unfolding unfold term high pseudo exist repeat bound pseudo expectation take expectation assumption imply cauchy
orthogonal significantly outperform individual patch trade capture considerably intensity centroid voxel redundancy independence rotation global complicated post process observe centroid redundant hand patch raw intensity centroid require preprocesse centroid future improve example coefficient plain negative cost future research sophisticated function class imbalance dataset region account account carry volume region try voxel per good training huge train fairly well unseen reason relatively variability brain explain voxel training capture increase unfortunately expensive consider artificial one transformation rotation create artificial distance centroids legend legend legend column college uk ac image assign voxel mr brain capture voxel patch capture context centroid spatial consistency contrary commonly segmentation technique mr model manually brain tackle brain quantitative brain quantitative often brain volume mr image essential segmentation brain require protocol manually consume full enable systematic image acquire benefit dominate patch consist assign voxel neighbourhood intensity deep neural prove art computer vision imagenet contrary traditional feature engineering crucial learn raw input development deep automate briefly review deep architecture dataset segmentation classify voxel protocol segmentation brain segmentation implicitly manually brain consist mr manual widely new query consist query query finally combine strategy heavily non usually perform critical enough accurately map introduce change region boundary identifiable intensity intensive responsible whereby give voxel correspond region particular recent advance learn extract beneficial learning concern hierarchical one imaging segmentation approach neuron mostly patch computation graphical despite increase medical yet brain slice input comparison whole pathway hand side feature merge convolutional layer follow colour share patch scale select layer max windows activate training architecture design voxel corresponding vector network particularly mostly spatial precision first patch local detail patch voxel add capture slightly broad context around patch small amount memory dense patch allow big patch ht input design preserve global segmentation region arbitrarily consistently preserve position subject obvious would simply patch span distant input require add instead figure operation reduce average intensity window full patch size terminology operation pooling patch patch voxel intensity coordinate informative absolute require perform initial generally consume one additional centroid image mass voxel voxel belong voxel region centroid voxel belong absolute invariant rotation distance invariant scale centroid coordinate average centroid voxel precisely voxel increase learn field considerably thus overfitte layer convolutional also impose neuron value layer decompose weight field mean detect connectivity sharing constraint model convolution operation neuron operation feature convolution size field map map bias convolutional reduce merge precisely pool layer neuron discard map field lose windows layer make reduce overfitte connect layer layer apart layer activation neuron neuron unit relu contrary traditional sigmoid function vanish layer activation neuron input output label voxel output reason share orthogonal patch patch lowest learn patch orientation layer experimentally find vector zero one position evaluate dot product space represents carry update weight error beneficial long narrow average momentum scalar rate momentum respectively centroid new directly centroid propose iterative network centroid pathway train mr enable centroid voxel approximated centroid centroid refined refine centroid two time change observe already really poor segmentation slightly long illustrate initial improve even though centroid region lie distance centroid sure network distance distance particular approximated approach competition pixel team require segmentation quality assess mean coefficient image manually translation region win obtain overall coefficient performance learn require computation run gpu gb therefore trade dimension decide randomly brain approximately voxel voxel purpose voxel dataset voxel patch voxel intensity three intensity voxel intensity centroid patch validation early error epoch early stable
cccc panel depict action indicate b various update deviation policy depict move bold cost think boolean hamming distance specification use e sophisticated variant descent pass detail pos use sequence search dependency parse sensitive multiclass search set request gray width text inner corollary edu cs edu microsoft com microsoft com edu microsoft com microsoft demonstrate suboptimal learning learn poor learning compare optimality guarantee unlike enable prediction application structure learner joint variable observe parse output achieve commonly require neighboring solve structured feature capture policy policy structure exist step policy implicitly reference attain typically word pos tag reason constraint say contextual page high website item position item font plausible display page user feedback reference namely web page learn something keep full feedback learn reference goal improve upon optimal core locally operate fashion achieve regret reference sub operate past secondary good variety include whether reference policy reference past algorithm dramatically hand superior poorly hill confirm policy real bandit level distance right bag child bag child bag child right child loss choose new kind regret modification contextual bandit output nf minimize expect search induce consist initial transition pair end structured convenience define clear express input action approach search use agent choose terminal specify policy action start repeatedly length trajectory trajectory reach follow policy accumulation cost state generate state training well internal randomness choose action lead decomposable trajectory generate optimal state trajectory twice grey algorithm deviation bottom reach collect learn top middle decrease I roll initialize loop generate reference ta ic ta assume sensitive predict receive loss perform update online sensitive learner define give online cost sensitive om binary operate algorithm algorithm policy sensitive optimal proceed online fashion along roll roll roll generate trajectory roll decision round reach multiclass multiclass correspond assign difference taking reach roll multiclass learner default roll roll policy batch final across round pick section answer question throughout obtain act begin discuss roll roll summarize use roll roll obvious choice roll roll blind reinforcement hard l roll roll inconsistent learn rl generate marked reinforcement much hard cm node child label edge parent child edge parent end node child child node parent node child right node parent grow node label child edge parent c child child end parent reach policy go bold branch learn policy pick randomly policy choose action please roll roll cause never learn mistake poorly test discussion sensitive whose action use since uniquely reference finally though state take deviation result state learner action achieve cost job unfortunately actually run perform expressive policy pick branch state sensitive generate sensitive example complete roll crucially cost take reach roll policy zero cost regret despite robust mode figure pick roll generate sensitive similarly sensitive learn roll roll make blind hold term local policy change sensitive om arbitrarily suppose one structure depend roll observe policy cost choose depend however well learned instead motivate generate policy deviation reach decision reach notational simplicity take use state act expect action complete roll regret algorithm reference capture regret roll let mixing parameter appear combine scale appendix comparable assume classification formalized arise solely restrict asymptotic gap vanish obtain state correspond average correspond avoid reference individually rather exponentially research result theorem consequence guarantee reference reference policy regret incur reference policy alone reference combination factor evaluate term case irrespective suboptimal term guarantee stay term learn improve poor overall either competitive locally demonstrate reach optimum exponentially reflect equip local optimality establish search policy trajectory feature depth state policy policy index bit trajectory step level deviation bit string distance consider powerful algorithmic give cost deviation learn access cost algorithm powerful reach step deviation class reach policy show compete reasonable step algorithmic policy class loss start construction apply hamming variant contextual bandit structure set round learner suffer search emphasize reference mixture initialize policy step follow end output common partial choose whether recommendation probability perform update policy base round step step average regret early definition algorithm q setup able evidence application multiclass multiclass cost search label recursively label half subset search root leaf end reference result use bad action detection training roll roll reference roll roll reference cm highlight roll roll cm reference reference suboptimal bad learn reference highlight part pos leave prediction loss trivial train suboptimal hamming roll immediately dependency parsing learn generate tree describe syntactic dependency system sentence deterministic lead end policy reference suboptimal apply otherwise arbitrarily choose suboptimal policy prior work deterministic journal
effort difficulty develop rnn easily lstm perfectly learn however general introduce multiplicative connection backpropagation demand long time implementation parallelization difficult due dependency internal feedback loop stream employ parallelism huge memory serious bottleneck implementation parallelization rnn rnns lstm perfectly parallelization parallelization intra parallelism increase mini conventional parallelization stream parallelism result parallelism paper generalize propose derive intra rnn explore inter stream parallelism experimental concluding remark parallelization various rnn introduce rnns basic generalization cover advance lstm forget connection base rnn every propose basically direct consist node layer delay delay amount delay signal connection output weight activation source value delay connection connection state activation gate lstm multiplicative layer multiplication input subscript represent generality introduce function additive wise multiplicative nonlinearity gate connection direct edge introduce multiplicative regard normal structure lstm rnn error convenience derivative q back backward pass layer initialize accord error criterion activation layer minimum softmax output layer indice derivative acquire wise multiplication layer become multiplicative multiplication connection error gradient parallelization rnn dependencies frames rnn determine parallelization intra stream parallelism stream parallelism separate part special direct mini batch feed neural internal recurrent group subgraph inside subgraph find inside remain otherwise node group recurrent ready parallel lstm one find strongly connect sort useful feedforward dag operation frame dependency different isolated node node sequentially step pass delayed connection exclude delayed connection dag compute topological order recurrent quite bottleneck employ multi parallelization parallelism stream mode stream context independent multi parallelism overall execution train gpu mode connect order sequence sequence long apply efficient truncate denote network however pass output error stream output gradient weight pass throughout equivalent feedforward neural increase mini slow easily modify speed simplicity sufficiently lag style font xlabel stream ylabel speed xlabel shift pt ylabel shift legend font label font legend style north west pos align major lm seq txt lm par txt lm seq txt lm par width style font xlabel stream ylabel xlabel ylabel legend font style north west minor minor log pos align inside style major lstm par txt lstm par lstm txt lstm par txt txt gpu experiment since mathematically mean square parallelization language model stream mode rnn architecture network batch amount gpu stream error step comparison forget connection note self connection compare number stream parallelism employ intra stream parallelism stream stream nice advantage stream rnn learn mini batch gpu forget operation
tm tm measure classification contingency versus stop definition true example stop stop set unlabeled nonetheless exist use truly positive truly contingency versus example stop label contingency table versus true label cccc total contingency stop truth cccc contingency stop truth table see example classify truly positive truly count contingency truth convenience show contingency table contingency table truth cccc total iteration truth contingency count learn versus suppose implie meet turn c notational convenience pick notational convenience ad c aa b ba u observe inequality hold used classification assume minimize measure maximize limitation loose tight perhaps expect bad make additional tight note c case practically substantially theorem prove tight utilize insight statement contingency perfect precision stop stop set classify therefore state general prove helpful contingency contingency follow theorem theorem case handle theorem contingency proof classify example stop show theorem scale implicitly precision theorem generalize scale precision factor place theorem difference learn issue connect stop stream generate issue would proportion count simply count stop unbiased select infinity stop probability approach date stopping criterion dominate heuristic dataset widely stop method remain inexact forward mechanic level achieve effective analysis reveal central sp success stop transfer unseen test proof agreement consecutive agreement precision conjunction assume setup serve stop switching region agreement relationship work publish conference learn association proposition em height em em break md nlp bottleneck nlp theoretical stop sp reveal element success agreement successively model impose performance stop result example successive proof relationship agreement difference consecutive exceed bound conjunction active query selective reduce cost create train considerable interest g nlp widely bottleneck new nlp system main effort require learn develop focus annotation effort al knowing annotation process challenge stop early useful wind model generalization lose recently stop al although coarse mechanic therefore crucial achieve effective terminology conservative conduct publish stop behave prediction publish empirical test agreement al exceed three consecutive iteration al heuristic well present stop prediction help deep work class perhaps important enabling stop active paper useful work switching strategy switch strategy similar case proportion instance sp particular stop size relationship agreement sp stop contingency classification learn iteration classification model model place category population place agreement indicate probability agreement chance classification independently agreement chance give cccc contingency probability learn true resort use table classification model frequency proportion expect classification agreement cccc contingency table count learn iteration delta describe estimator variance accord stop stop although work task sp variance stop dna fold cv cv fold course fold fold cv
passive predict whether tweet problem news tweet recommendation formulate content temporal message user social go break comment news et dataset sort future popularity use worth note retrieval propose coordinate ascent learn rank likely also work tweet likely tweet rank unsupervise aggregate previous method number tweet tweet introduce propose also try aggregate performance subsection tweet movie tweet movie base overall extract tweet user movie give opinion tweet movie movie tweet specific tweet contain opinion user movie tweet tweet feature extract extract category use numerical categorical boolean type respectively note normalize analyse exploit elimination retain perform subsection cm cat description user twitter follow tweet user movie rating provide movie tweet user list twitter frequency u day calculate divide membership twitter day frequency day user divide difference movie total movie rate rating tweet user movie people tweet number hash hash tweet tweet tweet age day user twitter tweet average movie hour tweet hour tweet predict tweet predict tweet language method learn build rank algorithm probabilistic basic construct weak rank svm create ranking try ndcg employ inspire use probabilistic rank logarithmic aforementioned fact technique two view totally aggregate increase mentioned aggregation try number final q measure aggregation weight instead number equation rank weight perform randomize cross training consider version dataset challenge contain movie rating automatically user throughout discount cumulative gain top hereafter library hyper randomize search cross learn except exploit source name microsoft report aggregate feature mention retain backward feature category importance user popular twitter boolean none categorical retain categorical difference result ndcg xt extremely randomize tree ridge drop achieve ndcg fs xt learn ndcg emphasize backward elimination demonstrate compare c cm ndcg fs w represent aggregate regression importance consider together aggregate regression far method aggregation aggregate learn rank aggregating method validation achieve statistically cm ndcg rank tweet category user movie show perform user aggregate demonstrate significantly affect method city usa ac ir interaction tweet news post achieve rank medium use recommender paper tweet rating movie focus extract tweet feature movie base tweet category regression learn tweet propose achieve extend dataset provide mining behavioral science social million information social site social recommender let express opinion social give rating movie internet movie website twitter social medium information system recommender study recommender gain comment item comment recommender comment focus movie rating hereafter user add tweet gain contain movie tweet three movie note hidden approach tweet approach globally purpose
view coordinate median contamination multivariate median obtain robust contamination constant determine contamination critical work robustness huber estimator general scatter via estimator various two study robust function give estimator whether possible procedure problem computation provide section dimension relatively great practically difficulty robust location scatter interesting whether problem algorithm discover contaminated write marginally conditioning follow control assume satisfy probability absolute characterize contamination relation real theorem since affine generality two testing elsewhere eq op obvious consider p desire depend thm general consequence following satisfie q constant elliptical distribution combine least least fact theorem argument theorem shorthand union c upper hoeffding finally due relation combine conclusion proposition u u u tx guarantee contradict proof tx u canonical complete characteristic univariate characteristic modify kind depend depend either constant smoothness hoeffding imply therefore sufficiently argument lead probability measure desire conclusion schmidt suggest weak corollary table lemma appendix matrix estimation important accommodate complexity desire procedure outlier arbitrary define concept call depth estimator show huber scatter competitive outlier contamination model covariance last decade rapid development theory covariance seminal covariance guarantee matrix sparse comprehensive work take heavy presence outlier exist outli totally tackle robust estimation high setting arbitrary contamination approximately distribute contamination break huber huber contamination contamination efficiency outlier simultaneously view contamination develop robust optimally concept depth variate semi constant depth parallel depth use location notion verify satisfie multivariate median point robust estimation give use concept covariance may accord one though notion depth definite offer several account structure powerful structured robust matrix estimating matrix sparse principal estimator depth function contamination interestingly minimax unified minimax range matrix classical without contamination quantity modulus continuity work liu contamination given distinguish phenomenon rigorously contamination model besides elliptical specific representation elliptical characteristic scatter elliptical allow naturally elliptical rate claim extra robust besides outlier heavy many work literature elliptical distribution include setting quantify work minimax contamination huber contamination robust small proportion totally robust property affine invariance achieve robustness counterpart contamination suggest huber contamination notion unify discuss depth introduce section structure bind contamination model discuss elliptical matrix estimation elliptical present relate connection proof close introduce notation singular large small singular denote frobenius submatrix cardinality kullback define variant generic constant robust location qp n know average fail sensitivity outlier introduce observation point maxima attain property state absolute high connection valid identity general valid say otherwise long outlier identical median contamination minimax sense contamination usual long achievable optimality perspective characterize outlier natural location inferior via consider obviously upper pn median slow achieve preserve whereas qp depth new matrix covariance observation subspace matrix speak median thus inspire computational reason depth q multiply scalar semi cumulative specify spectra need depth sphere pick cardinality attain state sufficiently computational n square minimax optimal class contamination constant finance order lead notion depth relatively statistical contamination follow state convergence exactly extend robust outlier rate contamination q subset g p q word covariate correlate remain component analysis degree sparsity sparsity depth function define relatively statistical property contamination principal component element row goal robust orthonormal matrix nonzero constant sep absolute sn account case optimal constant minimax contamination model statistical q component analysis whether theory contamination question lie key modulus whose seminal liu modulus contamination quantity measure ability close variation order level interpretation two distinguished minimax modulus state suppose quantity robustness pay dp sparse analysis derivation estimate arbitrary extend set elliptical distribution population scatter achieve depth elliptical property prove gaussian elliptical shape introduce elliptical elliptical distribute sphere simplicity unique secondly random motivated elliptical ec ec always canonical assumption exist elliptical canonical unique object show ec elliptical distribution constant imply constant determined px representation exist special px let estimate scatter contamination require outlier scatter depth induce least estimator small constant absolute constant small uniformly eigen sn ec probability uniformly absolute constant scatter covariance modeling interval subspace elliptical heavy tailed close section elliptical elliptical imply section optimal hard low computation median investigate develop propose adaptation core scatter depth q depth x tu tu tu outline multiple achieve small randomly tie tie back jump back direction back prevent search specify uniform sphere turn present simulation first introduce special correlation th variable define jk correlation scatter scatter ellipsoid find ellipsoid cover ellipsoid estimator covariance determinant estimator find determinant covariance package cover autoregressive autoregressive degree freedom matrix three consider contamination independent package scenario cover error behavior contamination contamination proportion rise though depth three depth behavior contamination compare rise stable estimator show scenario case five contamination table competitive efficiency three perform complete optimal result dependence evidence cancer study great interest co investigation perform pathway tumor exploratory sample contamination covariance cancer since type cancer expression considerably example importance estimator choose mutation disease characterize mutation induce hyper change biological gene involve dna literature raw conclusion create dataset randomly gene pathway outlier dataset matrix dataset difference indicate great
machine use panel support machine cart datum generate learn boolean set rule risk hard overcome use optimisation approximate bad case optimisation find heuristic optimality return training take return empty conjunction select maximize latter favor correctly classify greedy greedy rule conjunction conjunction assign negative force conjunction error consider error cross stop iteration discard utility conjunction negative redundant conjunction effectively example rule conjunction continue conjunction consistent rule lead stop reach conjunction induce stopping reach utility positive differ maximal utility rule small simple genomic number example become example contribute may utility situation likely performance stop mention prevent add rule bad training tb trade early stop stop utility correctly misclassifie u add conjunction represent genome presence overlap least genome training omit discriminate genome boolean rule rely apply boolean phenotype predictor interpret logical fold risk bold cart l propose representation predict grow public concern multi drug start care cost patient world genome combination individually yield dataset often compare regularize cart tree cart pose challenge term runtime make cart necessary filter univariate significance correction fold nested hyperparameter fold include comparison baseline predict tend one svms result svm solution greedy heuristic much difference less cart risk one variant filter preprocessing entire learn high obtaining conduct oppose machine entire require selection result heuristic produce regularizer generalization machine high thank dr loo dr sharing computation universit resource project ad award de pr discovery grants award sup la la sg cover classifier whole genetic exceed three predict machine cart consider feature filter preprocesse biology entire next sequencing lead increase whole
formalize imply exact hardness complete apply whose rank space incoherence passive sample possibly completion q passive subset index corresponding subset input imply passive small column subspace coherent show subset subset know span column error coherent matrix passive passive insufficient notion incoherence base incoherence theorem incoherence passive let column failure hardness column matrix completion limit coherent complexity approximate computing top svd operation take set reverse algorithm remark error selection subset selection target column like enough algorithm fail accurately phase completion excellent illustration phenomenon similar intrinsic probability proportional expected theorem state logarithmic figure plot actual question get bound curve decrease reconstruct sample big practice e figure sampling replacement replacement without replacement column remark iterative estimate project result satisfactory dependency rank conjecture loose believe avoid relative rank make high simulation exponential improve show input low perturb believe sub get input even matrix thank solution group lasso proof direct corollary th assumption apply orthonormal finally complete column represent combination q put bind dominate f f lemma observe r cs ss r span deterministic entry p last small covariance exponentially fraction follow hold put r mean sample randomness r hand projection gaussian rank least x ia lemma note union result prove seminal provide projected term project fix sample replacement th v clear ok desire result prove connect volume simplex permutation ready proof select denote span k k kk inequality lemma q ts f f markov bind lemma cite index bernoulli notation u invertible preserve incoherence incoherence bound subsequently give u r carefully dominate performance consider nonzero position j k n ci j j I ji consequently hold independent eq cs select subset approximated span subset numerous world datum application circuit input propose provably column algorithm matrix propose drawback complexity nice tradeoff employ idea feedback inspire sampling previously task prefer empirical analysis feedback compare input matrix highly aim much specifically compressed norm follow mainly evaluate compare rank large equal form guarantee ideally general error rank perfect remain zero problem example column various population circuit recommendation reader problem excellent class show nearly reconstruct base select column input carefully slight e unify column nearly problem full extensively study hard even genetic variation detection expensive sequence population several include omp explore presence miss pose column subset establish algorithm seem handle elegant identify challenge prevent application theoretical recovery incomplete underlie row matrix weakly selection decomposition effort incoherence obtain column matrix particularly difficult explore possibility gap incoherence need selection set large entry scheme provably scheme ingredient matrix decomposition perturbation fail two infinity norm properly observe complete matrix drastically usually additive strong beyond three entry sequentially feedback drive manner input differ drive science access active incoherent column knowledge column selection coherent theoretical passive error contribution paper provably column via scheme drawback summarize incoherent inferior propose synthetic real sampling error expense expensive addition rank incoherent input however iterative reconstruct matrix distinction entire column norm require entire summary offer column comprehensive accuracy efficiency achieve analysis insight completion comprehensive experimental well modification synthetic world nucleotide image theoretical interesting unknown instance leverage score widely selection regime prefer achieve suggest field spectral unless otherwise specify I generalize definition c type selection select reconstruction output remark always organize knowledge several important proof section complete defer briefly describe miss implementation experimental time background review concept play row three incoherence play decomposition u u always incoherence appear incoherence incoherence assumption subsequently incoherent incoherence assumption row vector norm selection approximate low compressed idea square type algorithm bind pick proportional span volume iterative serve leverage scheme approximation later coherent completion u u right vector relative guarantee approximation column error f matrix employ handle achieve reconstruction input certain structure achieve slow sample table summarize propose observe select active norm sampling column independently norm algorithm construct entry approximation input incoherent algorithm approximation provide show sample dependency tolerance column set x c ti tc u suppose orthonormal ts j ts u ss iterative though input low employ idea select error norm serve scheme relative norm within multiplicative depend exactly eliminate rank resemble completion already column ambient dimension fix span subspace q furthermore step norm satisfied follow bound suppose span column round square probability p mt round accurately estimate norm incoherent defer bind subset pick column corollary fix suppose subset probability volume volume precise cite volume sampling distribution inequality volume error well namely norm volume round strategy eq prove corollary sampling column distribution kk u os c os subsequently apply round e mt take union uniformly deferred theorem note immediately reconstruct coefficient reconstruction recover therefore eq complete note present claim column leverage subset rank right form project project incoherent column consequently row probability subsequently score leverage sampling selection f c previously subset miss theoretical method employ sample matrix mask di index omp use product similar method select manner column project span select nevertheless major difference norm omp product input matrix subspace span subspace exactly decomposition group extension precise propose optimization mask denote standard optimization column consume inexact median report instead leverage quite algorithm p leave plot fair synthetic dataset generate synthetic list first incoherent row space take coherent highlight baseline newly baseline result report number median sampling variant either score replacement result sample exception rate high degradation inaccurate estimation iterative input plus either target high work particularly input column easy gap algorithm block omp lasso considerably observe entry poorly inform underlie highly hand leverage score separation gap sampling bad without replacement coherent column repeat wrong column replacement investigate vary repeat high coherence block omp coherent isometry violate group lasso adapt column coherence decrease theoretical sampling genetic nucleotide human gene select snps capture genetic genome genome capture snp individual apply demonstrate propose entry raw datum selection miss setting omp group
comment discussion acknowledge support research cifar international conference machine true propose recurrent network rnn rnn extend stack control recurrent unit layer recurrent signal adaptively previous propose recurrent short recurrent rnn reveal outperform conventional approach stack rnn improvement adaptively assign different include stack rnn gate recurrent machine reveal rnn promise classification et rnns theoretically long dependency successful promising approach issue rnn e g activation nonlinearity term memory recurrent persistent fast hierarchical pointed help learn term dependency conventional way encode stack multiple recurrent layer recently approach partition hide group predefine feedback partition hierarchy feedforward design rnn call rnn rnn layer stack feedback connection one across fully encourage recurrent layer rnn control strength adapt evaluate conventional stack rnn usual task model experiment conventional approach able arbitrary recursively apply internal state wise transformation sigmoid tangent length let symbol symbol distribution widely model thesis capture term successful fundamental encourage maintain rnn gradient long lstm address learn dependency lstm maintain separate inside update necessary recurrent adaptively input central remainder proposal variant lstm follow consist forget gate gate carry unit control amount exposure memory cell neuron sum previous content forget forget old state current forget unit vector hide lstm memory lstm unit lstm unit gate control similarly output gate previous state memory unit adaptively forget content memory gate carry capture long hand decide forget gate mode happen across lstm unit multiple lstm capture propose lstm content gate gate forget lstm memory content memory content control update gate computed correspond previous candidate content gate memory content content gate compute base previous new memory multiplication traditional transition allow ignore input q long dependency detect feature late gate closed carry content mechanism helps detect necessary capture dependency sequence goal rnn often fast move former dependency ideally rnn capture short rnn partition group implement allow module operate mean module operate precisely connectivity module module b conventional stack rnn propose generalize allow model adjust connectivity consecutive rnn module module evaluation representative train minimize make dataset build english wikipedia character character protocol test vocabulary character token character average bit stack lstm able execute conventional stacking task program include loop logical end character output respectively evaluate sample difficulty sequence target finer grain analysis symbol architecture stack architecture different affine long short memory lstm constrain conduct extra experiment unit unit stack pt stack ex ex lstm stack ex character rnn epoch parameter preliminary result momentum coefficient either case norm gradient update minibatch memory compute update indicate statistically winner different lstm stack feedback feedback encoder mapping output fed encoder rnn state encoder rnn rnn hide rnn encoder feedback connection encoder rnn unit lstm layer unit mix difficulty sample epoch validation prevent evaluate generate length level test contain ht seed stack lstm communication comment increase vi bi pt comment mask material clear feedback architecture try lstm however fail rnn performance fig curve model second rnn train feedback architecture progress number hide rnn optimization propose feedback layer stack rnn lstm validate global layer without lstm conventional stack global confirm importance adaptively qualitatively stack lstm train generate text subsequence character seed read follow probability symbol seed character stack lstm lstm ten seed stack lstm fail trial lstm close tag type stack lstm c stack lstm stack rnn feedback gap rnn recurrent make compare performance architecture rnns multiplicative rnn lstm rnn unit result note vocabulary remove tag
propose encourage science report experience simulate phenomenon often regard ability understand move student complex provide thing assumption check student individual change statistic degree share connection association acknowledgement nsf around calculus thank suggestion comment early cm mm mm em end rgb mathematics college challenge education abstract th look back lead use chance reflect education growth science help ensure datum abundance keyword american history compute education american association mean city besides chapter association ever well exception knowledge ground discuss original agent american education fisher work improve surprisingly since exist somewhat figure american public health survey record record identify health graphical report display survival age four population next primarily neighbor early hence unnecessary cause take year remarkable display display modern methodology algorithmic benefit far familiar rare statistical among many statistical material diverse many formally prediction challenge peak north united population census peak widely mark experience challenging practice turn present stand association internal ensure well encourage development interest analysis lee science progress field ability visualize analyze report hundred thousand encourage growth statistic degree particularly master display student complete master line degree field master level historical program statistic development meet demand distinct add mathematics recent ensure education student complete four year master hundred thousand worker report come small growth decade continue likely insufficient meet demand definition matter raise position formally paper challenge familiar drive broadly big mention page mine big fundamentally traditional remain bag central decision relate familiar development st ensure solid I agree appropriate address suitable analytic interpretation rational need involve connection area visualization datum familiar indicate competitive disadvantage position tend job description cs student tend computation datum easily continue student quantitative topic challenge next generation student think google student course statistic application nan colors ms activity often really mirror world technology hundred thousand school student beyond calculation realistic certainly technology datum address early expand simulation computation I incorporate program issue commonly analyse work understand issue area bring great lack limitation student generally conduct trial make conclusion consider student trial situation student bivariate advanced student statistic program statistic student understand principle statistical student basic principle student mid average test college sometimes figure unconditional association statistically one score ci lower hide behind factor student student teacher teacher tend relatively school student take plan college state act sign average expect ci right student tackle student understand understanding increase principle often another option use student take display group student medium teacher group observe recognize estimate multivariate discrete observational around student tool school without interpretation learn computation develop refine practical theory student analyze aware student need develop mind think work student determine execute implement challenge cycle iterative involve data exploratory interpretation argue load summarize precise visualization report check visualization student technical visualization visualization tool force need beyond manual development decrease difficulty stay preferred student first ensure program excellent place integrate addition help refine student motivate statistic expand master intermediate must new student able life long learner provide answer may provide simulation free aspect allow student answer
context partially enough copy infer fill report highest achieve sampler confident discover start decrease likelihood match accounting find infer infer structure prior complex regular structure recover square express similar starting translate use process summary contribution infer report experimental generate geometric kind generative book group include cut away material product view coordinate specify locate discover understand find meaningful shape structure finite outline definition college google college microsoft model highly symmetric high transform object follow shape sequence compose product compact achieve part process theory shape processes align generative shape propose image term jump chain monte computation feasibility limitation model hierarchy entity notion sciences elementary particle document letter paragraph chapter book music piece etc room may reality exhibit clear mind frequently complex group product shape discover hierarchical representation understanding shape vision graphic long history graphic language create infer history term language account noise ambiguity graphic reality difficult make paradigm focus good graphic geometric tool graphic copy underlie lose full generative consequence preserve structure book generative propose graphic totally idea generative eventually go point far possible shape explain term totally change sub repeatedly complete incomplete shape building super structure introduce stochastic formalism generative give shape appearance process product hence perfectly align hand sketch infer process pixel image jump mcmc approximate view appropriate theory include group concept introduce noisy infer underlie strategy reversible jump framework describe generate geometric characterize shape lead generative object memory set action model algebraic transformation nonempty together property say neutral g gx group transformation act simplify extend third unity affine express matrix multiplication axis rotation origin discretization q horizontal call translate along horizontal direction continuous element copy point transfer location act produce always desirable draw segment translation subset present picture group allow unit discrete often move complex square transfer line side draw square wish transfer group create copy transformation act copy square process start top side imply line segment origin complete history start side cardinality translation transformation map translation generative continue describe element transformation copy orientation construction model initial shape generative history transformation want employ respective operation binary refer act index way let action index implicitly action first correspond history transfer transfer associated description situation shape display integrate theoretical copy application colour part shape like transfer time formation maximize circle square already rotation form origin thus square translate intend want rise characterize trivial origin responsible origin top g origin rotation fold rotation responsible produce abstract mathematic concrete suitable hence mirror fold ng implicit origin structure circle object coordinate multiply deterministic list list concatenation f group model replace add create process account arise generative history intend shape transformation present history account try transfer intend product application generative history exact copy transformation copy receive perturbation noisy thought coordinate subtree act copy repeat transfer transformation big cm cm trivially continuous version translation embed transformation obtain copy I shape bayesian specify uniform follow particular element bound uniformly segment circle vector instance dx noisy sample independently grey infer history shape represent describe generative shape shape belief kind shape leave likelihood trivial domain address use computation abc evaluation follow set simulated live match give reject inference recently specification idea next image define problematic sharp exact match close zero likely discriminate close solution proposal unlikely overcome idea bayesian highlight follow illustrate table generate process image result keep mind although approximated rp pi parameter end assume grey intensity locate interpret purpose reversible refined proposal idea assume upper great impact appearance shape keep level level go level structure u play match look amount keep change scale sample infer might standardize unit implicit interval concern proposal product propose shape keeping change prefer random keep act keep level act nature proposal keep nested structure greatly simplify acceptance
sequence shift parse prediction long principle sequential synthetic practitioner see good structure candidate greedy stanford speech advantage reduction greedy fast label feature map linear high scoring learn index product sufficiently use pair optimize feature model template confident order template score high scoring class aim template template need template pick template group technique compute template scoring label q contain associated label score eq q familiar machine minimum rank ahead notation label rank condition class tb template parameter train initialize test tb ii compute ahead label predict label rank return whole template split subproblem give order template wish learn template want template optimize combine score encourage calibrate stop early encourage single receive equivalent treat simultaneously high template optimize max per indicator define loss add regularization strength speed margin decrease speed large high accuracy three approach template template call norm regularizer encourage template discard strength regularizer learn order group strength technical solution slight abuse terminology induce alternative approach pursuit modification sparse nlp data orthogonal pursuit stagewise stagewise lar linear gradient correlation residual template fit error rapidly ultimately prefer development meta stagewise approach learn order select template development template procedure template hyperparameter empirically find due necessity inspire research cascade score scoring pose whether interact efficiently use requirement incorporate novel estimation separate add approach score add cause overhead cascade use increasingly detection reject sub window early score final avoid incorporation field time budget employ suit e extra pruning offset method confidence nlp stage consideration whereas pruning cascade output context cascade increasingly cascade base parse context nlp template selection test dependency parse technique parse decision graph enforce parse valid employ feature speed prediction dynamically model focus method field general static nlp regularizer template key template test inference problem compare improvement non boost achieve time prediction nlp entire template still require template nlp labeling solution speech parse name entity achieve multiplicative little baseline stagewise x fix base greedy speech employ learn regularization development baseline tag comparable stanford approximately attain divide template template stagewise template start template template add create separately baseline template template pos accuracy provide worth single optimize give speedup nearly could hyperparameter desire learn dynamically template test token template maintain template complicate template hand speedup setting depict template time present template train inference predict template accuracy dynamic template fix demonstrating learn act template group orthogonal pursuit case group correspond experimental setup evaluate group nlp detail good dynamic parse name entity recognition pick template early order pick template parse experiment template pos tag lemma assign head stack token second parse use development stanford pos tag use baseline pos model achieve accuracy lemma automatically label unlabele exclude achieve score speech dynamic train template prediction template dash speech time dynamic parsing label produce list table margin time fast remain time fast maintain template well dash demonstrate predict template successfully fix dynamic greedy right entity template surface feature lemma look token training tuning encoding set tag hardware experiment speedup score test fix fix dynamic learning dynamically select speed structured algorithm fast nlp gain come little art work remove template dynamically score different center fa annotation finding conclusion recommendation express author template discuss setup group nlp template compare pick template pos concern separate prediction rather dynamic successful achieve high template possible dynamic template term method predict template predict baseline speed initially template single instance multiplication template widely different amount time create hash string conjunction take feature behavior interesting template selection help norm naturally bias template include template stage template quickly source arise impact cache weight performance induce template order template run place highly predictive template front template template produce template order unable validation feature order per template offset pursuit pick stagewise order template add generalized template perform attempt find add fit nlp efficiently invert covariance template residual scalable feature problem combine design algebraic group feature matrix template template call template template repeat appear difficult problem template hundred million expensive special nlp feature template hence trivially invertible nlp model pick high template poorly reason apparent notice find subroutine essentially un try template dependent inversion matrix
control slope vary relu learnable relu eqn equivalent relu motivation negligible relu contrary adaptively learn jointly hope activation introduce channel negligible risk variant channel one variant introduce formulation layer represent term gradient deep activation summation channel share sum layer due negligible forward momentum learn tend bias toward relu learn constrain activation conduct deep study e choose sufficient category feasible train relu convolutional conv fc implementation follow imagenet pt c conv pool conv conv conv conv pool conv conv conv conv conv fc fc fc channel wise channel gain relu channel channel share channel introduce compare counterpart parameter critical role gain adaptively shape activation cc top relu channel share channel show coefficient layer two interesting phenomenon table conv conv significantly filter conv texture detector response believe level limited number filter deep conv small coefficient gradually nonlinear early discriminative deep stage network traditional sigmoid activation robust remove obstacle mostly draw distribution deviation difficulty report team experiment pre conv deep local intermediate scaled initialization linear relu follow sound relu initialization extremely deep conv fc converge method forward propagation derivation mainly idea conv response pixel channel filter number connection number filter weight bias pixel activation mutually distribution element product expectation worth relu lead relu put eqn layer initialization design magnitude signal expect proper standard std initialization layer relu layer layer conv layer denote gradient channel vector gradient layer assume symmetric back relu l l gradient eqn eqn relu derivation put eq exponentially zero mean need still overall product sufficient use eqn eqn eqn w w number scale vice versa x axis training epoch center relu activation case initialization lead convergence start main text relu initialization red blue verify gradient converge epoch discussion forward scale factor final signal infinity explain deep network small perhaps appropriate though benefit initialization foundation hope helpful input fc fc conv fc designing table also list modification three conv large feature last roughly unchanged iii spatial pyramid fc pyramid number evidence result augmentation comparable conv feature gpu takes per mini batch k deep extra conv layer wide version substantially complexity b four model improvement degradation depth lead layer speech recognition deep fc similar degradation model extremely layer error run deep degradation depth small dataset suggest conv layer conv severe overfitte attribute training mostly image short side randomly per random color scale fine tuning begin deep initialization help optima decay dropout mini test multi testing dense window pool fc pool slide window average far combine scale multi gpu parallel parallelism conv fc fc perform fc layer fc necessary parallelism besides parallelism introduce overhead fast fc single mini decrease speedup x speedup class imagenet contain rate except rate metric cc relu top top relu wise fair comparison relu epoch also epoch middle relu consistently almost cost comparison next single result publicly comparison relu good believe mainly end well even multi increase width c table improve become factor top relu imagenet team post competition relu imagenet evaluate c team competition net post competition imagenet set multi six train inferior considerable margin top evaluate server publish well winner represent successfully pay attention mini recognize image row col predict label specie display class due intra misclassifie still unchanged human top imagenet train aware existence test class title exceeds report human level human challenge analysis reveal type human come grained class job fine grain recognition specie dataset row grain recognize human recognize human specie algorithm still human require superior particular dataset vision object recognition elementary category believe recognition microsoft microsoft com essential neural network aspect first propose improve computational little derive consider deeply wide net imagenet winner
row index pz represent marginal particular represent pz column pz z way singular vector look projection square sum square standard result algebra third tensor standard path observation draw hmm tuple v calculate eliminate eliminate introduce leaf eliminate continue node eliminate eliminate sum leave marginal structured tuple hmm condition separate form form chain lemma justify recall u I assumption full along root assumption rank imply infinite sample extend path triple tensor full likewise h u u I I u u tree hide hmm generate meta meta equal generate meta meta index meta index I second tree consider index tuple index backward b z matrix contain kronecker second identical note total j px pz pz j pz complete universal following suppose triple input initialization matrix node p h h h h handle inequality meanwhile h triangle handle upon previous assumption follow h u triangle inequality inequality result u triangle h u u h inequality ta u u h u h algebra perturb version tensor iteration input regard decomposition rank completeness universal constant structure perturb q appropriate permutation put canonical appropriate column w w triangle third fact mc tensor permutation conclude provide notational simplicity identity recovery gm I gm I inequality triangle fourth algebra bound z z I triangle fact algebra assumption call obtain condition take get randomness permutation u u p matrix union first recovery accuracy observation final least c item meanwhile u u u inequality inequality equation third inequality follow last inequality equation transition handle already u fact choice u p u u u u u v v first triangle second inequality matrix matrix theorem corollary song chen university efficient spectral motivated mark type markov hmm cell connect main naive spectral cell exploit property hide state spectral current biological sample experimentally biological nine human algorithmic idea variable mark type mark preprocesse bit genome type hmm efficient advantage interpretability extend model relationship type biology relationship type comparative share motivated manner hmm hide state structure tree associate hide tree parent bioinformatics additional node emission biological main hmm genome hmms expectation maximization hmms computationally many hmms exponential property biological achieve key treat leaf improve typically low depth tensor technique full leaf use reveal node key exploit independence emission path version tensor hide model product projection tool implement biological cell result em spectral hmms individually assess spectral spectral rank extend spectral algorithm condition hide model hmms tree model topic mixture involve provide algorithm design hmm parameter hmms modeling sequence probabilistic model hmm position hide comparative jointly structure hide state represent correspond formally represent tuple direct specie represent vector parent denote parent parent structure condition initial iid sequence determine likely typical moderate possible assume node th product whose th symmetric I n j shorthand j j use notation denote co frequency correspond x u u u meta meta meta represent px pz z denote co occurrence meta denote empirical novel tensor learn hmms decompose version third occurrence provide globally solution occurrence co occurrence yield symmetric tensor whitening decompose sequence steady state idea use third co tensor modify learn hmms structured state directly hide implementation design plausible observe generate underlying sample sequence steady observation tree plausible actual instead show achieve exploit idea observation node root maintain correctness naive state naive root occurrence instead node capture root node path project skeleton path occurrence meta appendix even construct take time procedure construction operation take small complexity technique generate individual matrix hmm describe path emission thus store space first therefore construct construct onto dimension could gain biological vs projection co occurrence meta matrix projection would projection differ property efficiently good key observation consecutive hide structure svd denote path root co occurrence h u u recover product technique beyond hmm graphical condition hidden model biology expression sequence successive observation sequence pair triple np assumption generate typically certain parameter parameter hmm tree hide state condition wise ensure root path joint assumption require succeed kind assumption believe future work hold consistent enough learn consistency algorithm iid generate high sample observe run spectral gm follow motivated cell eight mark cell vector poisson background length eight segment combination number hide interpret encode goal discover segment probable cell similar calculate co occurrence successive observation formally use projection round matrix reveal appear tree hmm run memory perform co library application biological compare spectral approximation report structured mean spectral matlab take take iteration suggest spectral variational spectral focus observation matrix row mark condition identify discover weak background state large background mark state interesting biological
nmf smaller add svd identifiable negativity np reference array parallel enhance identifiability represent exist briefly operation unfold th form stack row vector index back example three different appear literature ease number column wise product explicitly q generalize accept two argument help notation exact form rao hadamard eq slight notation loss separable hard incorporated require multi regularize alternate technique cyclic special square iteration discuss especially advance descent consider factorization treat problem become column never calculate cholesky gram compute column forward substitution form respectively cholesky finally substitution take implication least grow grow go least cholesky decomposition nice unconstrained square improve one method qr complexity numerically exploit cholesky decomposition problem cholesky structure tensor efficiently exploit expensive computation j algorithm form rao mode available adopt qr square favorable loss good propose algorithmic factor many possibly st objective monotonically stationary uniqueness exception notice explain coincide improvement update block decrease time commonly propose solving call successive ensure simple put proximal convex strongly strategy formally prove speed unconstrained sub form problem simply sub update everything tensor set ease rao product explicitly problem well want term easy thus omit right appear update introduction universal admm solve optimization follow iterate update equality user review admm reference therein admm split yield update distribute constraint incorporate set outside define indicator admm strongly admm step start least reformulate introduce variable admm important save computation cache dominate forward substitution complexity around become projection proximity operator constraint especially often update constraint reader non negativity wise handle induce ij ij element thresholding want impose structured simplex constrain wise negative projection column smooth add super involve inversion set admm square converge fast approximation obtain naturally take admm reduce alg see cholesky decomposition computation alg dominate update complexity alg essentially calculation plus admm alg order r termination general termination residual admm terminate alg adopt solve introduce notice follow scale dual constraint constraint corresponding lead also interpret least proximal adopt element update list define miss case index fit uniformly corrupted corrupted outlier noise heavy laplacian function resp huber fit huber loss wise leibler adopt loss leibl proximity operation wise k family divergence proximity try fit therefore square loss detailed admm summarize alg termination alg everything least square loss close must minor albeit drawback dual computed admm pay admm scalability consideration become common scalability mind develop big datum usually store regard fortunately commonly become portion huber regard corrupt sparse array close early memory efficient routine propose multi summarize alg alg cycles iteration far away update warm strategy mode operation alg suggest sub iteration precision want actually copy approach adopt store readily without adopt save depend store completely copy early big instrumental proximal fold help loop accelerate convergence admm need stay work initialize use alg previous necessarily alg appeal alternate optimization plug play per admm practice include dominate distribute line preliminary alg fed alg disadvantage alg admm save amount computation initial stage problem derive alg alg admm perform ghz gb discuss world store application treat netflix movie rating movie sparse customer movie customer movie unseen rating recommendation relaxation involve nuclear norm provable nuclear tensor tucker rather tucker value uniquely preference aforementione incorporate netflix movie bias recommender easily equal one movie type tensor completion formulate constrain traditionally handle one problem inefficient even unconstrained case use expectation start zero iteratively fit rank prediction rank recently toolbox use variable relate miss treat loop however admm despite similarity illustrative generate contaminate impose negativity emission loading three chemical use toolbox fig satisfactory cause systematically indicate movie rating consist rating movie include split factorization evaluate correctness absolute error mae attain mae available rating kullback fit fitting compare impose negativity negativity bias seems rate evident negativity reduce rank fitting criterion play role bias prediction mae report admm recommender right believe admm matrix quickly signal represent possibly example fourier compression relaxation procedure recovery resort dl well benchmark clustering dictionary thus svd patch hundred dl formulate q sparsity induce g cardinality norm conceptually scale ambiguity inherent impose well atom th column sharing solve optimal cyclic admm sub routine alg separability handle separately previously cholesky warm ensure sometimes study focus large share alg replace thresholding train least square accelerate
section work fix vx vx help require submatrix z ex kt x write discrete notation similar true need result ex element independent sample theorem discrete I bound binomial note look term assume remark conjecture exercise science pa need norm main gauss main main main norm start review width characterization provide expectation process widely study generic analysis gaussian random one width expectation dual value dual property hull scalar property orthogonal property width connect gaussian net cover every small inequality number norm eq q low converse generality analysis constant choice converse inequality width e lemma move analysis full symmetric net frequently sub random sub exponential random sub value gaussian ex ex ex sub exponential follow center variable center variable center constant gaussian ex gaussian variable norm exponential sub restrict establish relation restrict width start rsc theorem characterize belong optimality q far inequality r z rearrange complete size recall atomic g isotropic construction mm r translation invariant clear noting complete ex recovery compatibility lemma rr rsc triangle inequality sub add q n rsc r add norm compatibility imply plug inequality back section theorem r width square design u sub least length gaussian xx independent isotropic linear mapping equip isotropic convenience need upper expect satisfie lipschitz concentration variance element ex ex ex ex similarly bernstein convenience gaussian gaussian eq combine immediately upper inequality g z apply choice determine ex let assume ex ex quantity norm gaussian show width time gaussian set k ex ex absolute direct complete ex isotropic independent p design isotropic sub sub gaussian ex isotropic gaussian pt bound derivative respectively conversely ex net previously follow result along denote ex result interestingly constant em lipschitz application inequality change inequality instead type design matrix analysis theorem modification sake independent e start center u v chapter form form q theorem proof process eq direct unit condition vector ex design sample literature case sharp gaussian rely constant row note na converse next net spirit lemma lemma subsection matrix conversely ex prove result complete matrix isotropic interestingly continue scenario intuitive p allow problem net cover ng function integrate g require x proof isotropic net cover choose converse convert extend result use proof eq conversely eq follow along gaussian design loss gaussian matrix design sub isotropic random ex ex lemma problem suffice ex ex covering complete pt set ex ex ij x p maintain previous ex apply capture constant norm ij suffice n center sub covering denote ex design isotropic dependent ex show dependent sub correlate na constant ex ex aa entry ex number surely diagonal quantity random minimum maximum
popular key rank recommend list bandit approximate regret recommend bandit rank independently propose whose decrease rank bandit combinatorial learn bandit bandit feedback non parameter bandit recommend observe context study variant whose reward know study optimization whose study action learn finitely many environment unobserved environment bandit view partial outcome hypercube base definition rule b te te together counter happen event optimal fact moreover gap decrease value sum tt tt te optimal item upper confidence complement thank suboptimal argument suboptimal optimal second suboptimal sum lem modular induction claim suppose induction factor decompose expectation product definite integral decrease analytic note minimum proposition observation engine list web page attractive cascade attractive item formulate two prove upper algorithm derive low match upper problem violate web recommend list list click user item user optimal list item maximize user find attractive cascade click cascade agent know time list observe item user click user click agent receive agent attractive bandit feedback feedback rich reward know item attractive attractive five contribution formulate learning cascade sample algorithm semi motivated expect probability web gap fourth bind regret upper finally perform cascade model like regret addition prove relate page engine rank web historical click scan page item web page many search click user differently focus web user attract click item user item attractive maximize attractive item user click practice click item cascade explain several extended explain click cascade cascade attractive reasonably click towards understand online cascade cascade solving simplify write bold problem formally tuple hypercube recommend bandit item time item click user item attractive q particularly eq click click click since click list weight recommend particular note say distribute independently mean item assumption consequence attractive express fa item evaluate item list maximize list simplicity exposition set two solve bandit motivate recommend list large eq discussion probability et te recommend get te te te compute probability number step confidence since compute initialize list specifically algorithm regret suboptimal prove regret discuss result item ground set setting first item item probability suboptimal item item hardness whenever convenient list main difference list choose instead exist attractive define follow item way entry right first event outside bind fourth bind base suboptimal step n four step item apply suboptimal extra finally derive contain item product parameterize suboptimal problem consistent suboptimal without generality suffer polynomial bandit achieve logarithmic instance algorithm regret time condition result observation counter event unable instance thus get conclude attractive agent sufficiently attractive practical close bound item attractive item item bound number recommend item recommend proof position extend upper problem asymptotic low bound p p pn ucb factor eliminate order validate upper item increase even violate rank qualitative value tight accordingly recommend item decrease motivated observe four trend number regret recommend item trend bound outperform surprising payoff arm confidence interval tight bernoulli get close rr order ucb recommend rank item attractive lot feedback could reward depend arbitrarily report decrease future satisfied model datum cascade recommend list recommend last user item
mention instability erm result heuristic stability make incur erm expectation erm exhibit excess dropout problem convex glm precise generalize detail privacy concern machine potentially sensitive medical health record privacy notion effective privacy r change one insight private convex algorithm expectation dropout eigenvalue convex function erm assertion erm training stability random adversarial removal interestingly work complementary dropout adversarial removal adversarial perturbation test feature experiment stability cross regularization uci repository theoretical see performance enjoy desirable stability implication strong dropout preserve privacy privacy setting regularization require aspect setting dropout complexity organization dropout excess lead use empirical provide rigorous class descent actually dropout optima gradient procedure training neural approximate polynomial exact measure w distribution g define neural hide layer underlie architecture assume polynomial point perform f dropout local perturb effectiveness heuristic stress section instability dropout perturbation entail procedure minima help minima reduce reach one link neural represent weight fx constant dropout perturbation degree proof error polynomial polynomial identity notational purpose polynomial along need concentration z provide bind ab problem polynomial complex use polynomial network comparative complex additive oppose multiplicative dropout modulus polynomial ensure oppose either dropout help encounter gradient section model neural descent excess risk x value convex risk excess risk excess descent vector exact dropout algorithm analyze variant capture regularize setup analysis strongly dropout variant enable excess function excess risk tb dropout sgd initial sample normalization ii I excess heuristic dropout draw learn randomness sgd excess ip gb outer randomness provide section excess bind dropout dropout plus use argument dropout square notice even scale risk general strongly hope demonstrate dropping node neural theoretical understanding dropout heuristic behave regularizer underlie problem rate asymptotic taylor dropout behave regularizer rate risk assume come underlie modeling problem polytope parameter provide precise heuristic essence provide fourth open pose aspect rate covariance show differentially private convex private optimization significant attention allow privacy differential privacy privacy information neighbor private measurable think privacy induce output randomize one intuition consequence medical record adversary learn absence privacy output provably good generalization fit dropout detail example linear simplex brevity refer later framework convert privacy guarantee tb set simplex nx ij loss dropout level change possible output slack sample privacy since minimizer refer neighboring differ closeness relate binomial non coordinate exclude ratio bound chernoff happen binomial closeness analogous closeness closeness ensure differential direct implication differ privacy differential laplace differentially test along dropout differentially private function direct tail long p exposition tune optimize linear treatment cccc evidence support dropout result fraction capture belief network dropout perturbation form removal remove removal randomly select fraction absolute refer result remain dropout misclassifie show result value dropout dropout outline exhibit stability version even dropout datum regularization provide appendix adversarial removal complete set removal adversarial removal dropout least study observe dropout decrease rapidly tend similarly regression
million label image object huge convolutional tool cnns significantly medical report store modern picture communication diagnostic mapping hundred thousand create volume remain primary goal extract associate semantic datum mining deep scale database image knowledge report mining scale report document patient history contain imagenet manual google accord prune crowd amazon meet label demand annotation task privacy categorical semantic modeling allocation lda interpretation patient provide patient labeling categorization specificity report image feed forward cnns train work building scale database text image please yet medical interpretation effort learn image object attribute crf annotation computer vision feed cnns recurrent network text cnns medical show benefit domain feature significant conventional sift medical key categorization label describe sentence medical make publicly medical image annotation image convert pixel label local intensity bag block ct study work model vocabulary recurrent image label embedding minimize language train imagenet correspondence imagenet reasonably high dataset dataset predict describe annotate unlabeled dataset present annotation medical semantic document cnn comprehensive semantic use available store national health center year instead patient two dimensional writing notable finding correlate diagnostic semantic unique show occur report leverage exploit make automatically patient mis extract report process nlp sentence list nlp g image basic nlp total retrieve match whereas rest extract modality document k k see mass k k small note propose categorization label text report imagenet often mostly ct show high intrinsic ambiguity define assign semantic sub million report mining correspondence originally propose article method document extract topic among report lda flexible learn coherent study regard special hierarchy common hold unseen document hyper topic document score generally fit score evaluate document although balanced image unbalanced specifically topic primary variety body contain image address result lastly topic mining sentence adjacent sentence score keep count small beyond figure categorization label demonstrate good semantic coherence among list key review validation document topic topic disease primarily disease brain disease mix modality mid concept part diagnosis low level visually image topic imply heavily sub disease mass tumor meanwhile document many sentence semantic image imaging imaging associate semantic addition include image association refer figure supplementary text investigate plausibility via cnns category framework split validation cv test divide cnn learn normally rare imaging protocol topic different document level sub topic map systematic diagram semantic topic image cnn challenge reference slight million consist five follow max pooling layer fc final cnn significantly deeply convolutional imagenet number softmax level sentence tune pre imagenet semantic fine imagenet pre medical modality ct help additionally cnn cnn document level topic less imagenet document topic trace cnns initialization learning cnn imagenet transfer image finding different modality verify imagenet annotate medical date quickly iteration unbalanced among deep already initialization via imagenet closely relate fine less cnn part newly initialize new low rate layer rate set rate key image spatial resolution image level learn classify lda induce different shown skip mapping word train hierarchical softmax slide window frequent word diverse set article learn well keep hyper finding robust learn diverse query close term cosine article show report word cosine list variety mostly disease highly diagnostic disease exploit disease term sentence word disease relate gram description trade medium complexity word show l report reference digit bi extract train cnn vector multiple bi gram per sentence image one bi gram bi ignore annotated sentence bi gram relate representation detecting object configuration employ map bi match disease vocabulary cosine bi gram convert cnn minimize cross line form gram recurrent text vector formulate regression softmax output adopt cnn simple tune cnn predict modify cost text classify category newly layer bi gram bi testing topic document level topic cnn top word map bi lastly topic second half cosine similarity key word high cosine bi disease match actual word k k example figure categorization report score work r word nlp describe sentence truth association association description cnn automate patient generate sometimes generate mining sophisticated nlp parse specific disease section aid interpretation scan nonetheless analysis disease automate added mining disease semantic predict cnns softmax describe disease disease rare exactly disease unified medical language terminology associate resource service sciences create maintain library base comprise concept concept name incorporate control organize define link semantic type consistent categorization concept represent choose structure medical unified retrieve imaging report medical record vast concept word appear find disease mention detect assertion algorithm detect absence clinical determine text scope occurrence find disease detect assertion disease find derive occurrence unchanged occurrence unchanged occurrence find occurrence decide disease occur report disease absence similarly challenge match disease sentence softmax function softmax normalize exponential generalization value softmax among cnn imagenet assign disease image disease absence disease number occurrence show c c mean per mean std std transfer helpful tune topic cnn fine level testing cnn absence top mapping word generation match originally disease specific detect high disease top infection automatic label assignment image sometimes statement possibly would unclear present apparent sentence derive test match originally figure four six contain prediction coherent high top figure label visible characterize support failed detect nonetheless label assign unclear statement due challenge unclear image predict second visible figure second high automate mining enable predict compare image modeling labeling loose couple disease image strongly less find image pair probably mass tumor detect detect image unchanged big loose label rather loose label us word help
rnns rnns formally state previous linear eq non conventional easily use train put aspect trying finally value compute illustration rnns neural time generative define input write distribution state function practice back generative generative recurrent neural generating phase unfortunately al optimization issue train model dependency problem back propagate gradient exponentially zero vanish large hoc restrict go trivial decade tackle rnn speech exploit unit flow state element gate candidate sigmoid matrix show h compare problem rnn capture long dependency vanish problem activation try period attribute time previous input hide unit show two input solid dash clear target signal dataset simple motion mit motion dataset generate position consist information orientation generate row unit toolbox visualize datum frame train recurrent neural unit layer generative fashion feed first feed frame real average generation phase figure solid target dash line use step unit recurrent neural dependency dataset mit recurrent network conventional reference motion rnn recurrent problem machine translation learning model temporal
receive passive remarkable indistinguishable normally implication leverage algorithm like passive bag ever since statistical dominant paradigm learning discover blind partially leverage passive inspire treat visual experience activity learn process visual cast term stream sensor semantic signal correspond sensor video camera feature mapping figure convolutional neural cnn exploit recognition image learn sharp visual feature hand target input seek transformation orientation operation sift cnns rotation powerful representation balance much loss representation capacity impose space exploit observer like signal prior learn learn scene furthermore unlike transformation oppose consider exist whereas application explore application jointly apply three public dataset pure recognition challenge learn accuracy disjoint unlabeled car datum task dataset bag favor special wherein transform invariance valuable descriptor sift aspect cnn like hand design invariance shift rotation image instance preserve operator slow learn vary slowly video gradually adjacent frames temporal cnns dimensionality recognition metric supervision perturbation idea method exploit video couple signal achieve general feature design learn operator design explore descriptor plane rotation observer learns aim sort pose illumination transform auto encoder explicitly object part similarly graphic train autoencoder supervision bottleneck layer variable novel look like method limit sense pose encourage impact unsupervised recognition quantify cnn response specify transformation affine adopt use exist descriptor space train transform infer bilinear multiplicative learn content motion encode autoencoder video neural combine future frame tuple individual transformation whereas abstract make recognition interest computer vision though none signal pixel foreground person solely apparent image exploit robot object reinforcement robot movement exploit learn respect transformation associate pose pose capture may encode observer camera pose position roll subset read sensor pair algorithm pose j j frame multiple video category sensor translate pose pairwise pattern annotation sec define precise seek enhance recognition neural first want pair discrete motion pattern motion pattern might collect control camera prefer approach leverage video end discover pose difference frame typically apart detail simplicity apply though motion pose motion pattern choose speed panel video dataset move angle camera consist center cluster car primary turn forward wish motion motion respect exist feature correspond pixel correspond movement particular direction outcome structure encode fully layer work focus motion pattern observer movement world appearance observer motion motion alone depend depth scene object scene frame depth maps observer motion appear difficult especially newly even target discrete limit preserve every atomic atomic right motion pattern turn head motion diagonal map motion among restrict pattern train design map naive translation would decompose optimization cluster correspond summation deal pair annotate problematic perfectly trivial evaluate learn simultaneously negative motion pattern mode apart mode belong bring close input location highlight temporal hyperparameter encourage nearby motion frame coherence target like passive hold coherence explicitly describe generic representation task addition pose annotate together map denote softmax softmax probability case unsupervise bottom tie stack supervise softmax specification map architecture layer optimize sharing parameter network pair feed initialize identical gradient pass epoch weight tie stack encode wish train array represent stack map motion pattern input stack motion addition softmax replica stack weight label fed stack fed softmax depict implement sec recognition dataset view atomic center accuracy repetition right view outperform baseline dataset selection sec softmax loss adjacent set distance motion combine baseline like popular feature unlike passive video pose recognition additionally access similarly baseline receive pool unsupervise datum dim convolution fully layer bottom two dataset image clean systematically camera image pose vector camera discrete pattern cf sec respectively pattern yield create pair positive negative pair treat neighbor video sensor car drive city road select training validation pose consist position forward velocity output discover pattern frame apart retain turn pair positive equally train apart validate camera frame see know select object split select image split recognition categorization associate category scene scene recognition image unsupervise beneficial label follow convnet recommend cifar architecture architecture relu nonlinearity nesterov accelerate base rate select result repetition motion normalize denote perfect evaluation feature validation atomic training us pattern sec atomic composite invariance absence supervision tend nearly novel transformation optimize easily atomic composite small test transfer first improve classification previous worse obtain unsupervised gain three significantly nearby frame see epoch neighborhood validate effectively space offer exploit frame pattern exploit first require perturbation systematically result challenge noisy pose dynamic traffic object enter scene class road task web image category mostly indicate generality achieve trend show preliminary good view robot move help recognize object neighboring view object uncertainty fact behave identify task detail feature easily baseline qualitatively pair near retrieve pair relate example variety top neighbor pixel difference exhibit transformation turn rotation pixel distance perhaps wrong change box large foreground motion decade method focus almost bag image reflect ease crowdsource though valuable informative physical visual experience approach learning generate show learn beneficial great intel would thank work show substantial domain camera road camera content around belong diverse scene category nothing image color subset text main processing nesterov initialize select identify base fix regularizer retain loss loss uniformly objective include loss set margin motion scalability margin equivalent optima scale
rmse rmse tune good extra tuning search rmse study investigate efficient determination rbf mahalanobis base kernel geometric observation map predict introduction traditional forecasting trend feature normally solve forecast many area uncertain use rbf kernel overview grid search pattern search directly extra pre exist searching overview practice form obtain consider indicator acceptable tolerance natural map build x kx kx j map value map mean space deviation te norm observation describe difference map help te convex reach n n k x jx j mahalanobis kernel gx cx x k use balance performance accuracy deviation function fit see figure value begins map rbf bad x finding solve define reduce number critical big value figure table rd c th th experiment company contain description determination rbf r scale calculation employ rmse error h rmse tune show result solve time area
binary generate build independent input similarly difficult find htb boost boost table boost generate tuning observation term generalization far concerned svms propose toy simulation performance eight diabetes set diabetes ten independent body e create contain instance per business average e concrete create I age etc strength fourth cancer cancer al receive predictor eight clinical cancer etc logarithm specific one measure one response benchmark spam uci repository spam attribute use spam diagnostic breast feature datum identify whether contains attribute instance boost boost diabetes spam select build remainder set evaluating performance randomization time number table iii parameter type simulation easily outperform boost algorithm among real coincide experimentally numerically comparable idea road improve boost lemma completeness proof positive ii contain eq let mean derive contain proof lemma fr differential aid divide two step deduce taylor expansion mf k mf mf mf mf convexity use fact along eq note mf mf mf b follow assumption derive arbitrary mf mf k f mf mf mf mh f f h select later hold get v applying obtain aid ball span origin q radius result proof inequality function q q z x mf confidence assumption q c set direct kk numerical rate variant boost name boost focus alternate scale theoretical outperform theoretically numerical property numerical outperform boost imply reasonable boost boost follow throughout essential reader highlight due theoretical may outperform conclusion partly perform second totally direction improve boost boost guess variant boost work issue report progress future publication accord introduce parameter degree facilitate choice boost recommend zhang yu naturally ask good answer paper practically important slow theoretically think select via strategy leave concrete role reveal support foundation china grant national cb lemma remark boost scheme combine accurate prediction rough one aim develop accelerate rate consequently improve show possess numerical sense tackle problem tight error show boost rate common explore response model loss state activity boost combine produce underlying rule combine boost regard wise fitting additive connect boost problem correspond minimization overfitte problem bit prove rate lie slow hereafter boost accelerate capability include boost via linear truncation specify fix purpose propose accelerate near call boost always one idea greedy essentially yu shrinkage impose composite weak learner help type scale boost classification experimental verification classify performance four aspect deduce numerical result show near secondly capability essentially boost justify build restriction provide flexible experimentally modify outperform boost paper organize compare algorithm study behavior error bind section verify conclusion discussion observe consider eq q expectation learner regressor boost weak update repeat although show slow nonlinear search make show walk exist angle walk comprise strategy control obvious scheme appropriate small numerical aforementioned strategy control rate consequently capability idea iteration regard extent impose operator strategy main idea set shrinkage step compare scale hereafter call shrinkage find greedy relaxation relaxation name relaxed brevity mf mf mf norm exist depend easy verify widely focus rate assumption constant depend therefore least function certain orthogonal logarithmic relaxed optimization slow however well knowledge whether loss exponential logistic cost convergence check give select shrinkage degree definition follow may shrinkage particular al different algorithms check correct finite integer constant select brevity turn consistency boost whether approximate arbitrary depict show consistency certain stop approach bayes impose absolutely derive give stop proof constructive simple stop fairly slow converge speed play role estimator kk consistency remark firstly secondly simple method truncation note computation truncation widely boost however drawback usage entail element truncation indeed estimator help estimator show number deduce study boost regression modify version look vc highlight
operator lasso assume covariate lasso coefficient theoretical view several lasso communication use output inaccurate outlier replaces bound huber mean instance penalty require easier propose novel modeling outlier add corresponding penalty robust linear outlier outlier optimum penalty good robustness result local penalty thus contribution penalty avoid optima estimate optimization recover true coefficient statistical remainder organize theoretical analysis estimate coefficient output performance throughout symbol norm respectively nc cb cb na n linear tn correspond outlier vector correspondingly coefficient match assume zero introduce penalty tuning encourage small outlier type tw use algorithm initialize l converge stop return optimization rewrite express smoothly scad penaltie explicit soft scad penalty recommend pt output thresholding yield penalty q cost solve fast computation first reason relationship state penalty huber penalty scad penalty give characterize property go infinity penalty scad non penalties mcp penalty yield good non suffer able minimum global avoid problem directly computable outlier provide first property solution notation doubly exist row shall provide depend require error identically sub satisfy assume error model gaussian thresholding handle theory negative scad mcp introduce concrete order lasso type simple however preliminary accuracy calculate reason parameter e go analysis matrix detail see condition bound correspond may become exclude threshold jj version hold meanwhile sufficiently simulation examine carlo tuning tune practically tune candidate generate design impact scenario various consider draw true outlier draw independently draw k scad fp fp tp tp error fp tp tp show simulation support coverage preliminary percentage outlier threshold support coverage percentage increase hard fp tp fp fp tp tp tp fp tp square positive positive tp preliminary soft hard never vanish investigate outlier exclude preliminary outlier see moderate quite support interestingly would error extreme preliminary preliminary well opposite true error bad property positive carlo simulation outlier various magnitude positive omit procedure high magnitude come low would hide draw magnitude note magnitude pz g condition correspondingly subscript simplicity I c preliminary go l k k doubly imply q q solving recurrence
leverage score condition matrix equal form eq aim decision minimize make unobserve noisy somewhat approximate sgd computational stochastic monte sa approximation sampling sa follow point idea erm exposition describe regression stochastic recover sgd result basis form randomize suggest apply either sa sa choose assume subproblem compute randomize algebra alternatively sa sample fashion descent sgd solve choose follow subsection sa exist solver simplicity constraint generalize nontrivial choice basis effect erm stochastic problem original require naive choice row equally work toy undesirable first element row sampling require fail large leverage score put row row leverage immediate score recover propose algorithmic leverage regression formally include result result produce sample independent obtain hard show update weight update constant depend affect restrict regression follow undesirable sgd optimal uniform ap ax z ax increase add sgd naive distribution matrix grow condition basis I leverage optimal idea distribution leverage score lead main section main see step implicitly appendix refer find algorithm remove randomization consideration exactly norm alternatively norm assume leverage row accord call sgd phase improve domain instead notice simplified iterate last tb weight sgd determine regression output problem condition range score pick update return objective regression corollary regard result mirror proof result condition computed satisfy computed algorithm return suppose iteration eq x x return dx rf suppose exist traditional algorithm corollary error score factor distortion apply reason expect inverse magnitude whereas empirical important role original choice error consideration subtle simplicity restrict vanish error per sgd dense matrix norm show factor norm section typical algorithm compute condition provide overview condition conditioning score without time provide bound complexity sgd per return approximate solution evaluation real synthetic design diagonal response corrupt gaussian since convergence assessment convergence rate include implement different full diag detail leverage score rate phase diagonal mirror implementation done grid search run upper leverage score apply leverage recover appendix gx rescaled since broad combine sampling assume satisfie conclude connection comment exist sample deterministic stochastic extend problem translate sample size hinge exponential dependency sensitivity approximately result idea need similarly type develop randomize algebra sgd uniform sgd empirically preferable medium precision bound regression point direction work extend would like office advanced project department energy providing condition implicitly norm condition implicitly qr exposition high full embedding well solely way recently short nearly sparsity plus low distortion available compose matrix see ccc type dense cauchy cauchy sparse qr implicit compute form reading row norm norm multiply case lastly additional main corollary theorem f x algorithmic leverage central row state enough result failure independent leverage output approximate well condition pick theoretical text equivalence first observe write follow v equivalent hard verify thus cast stochastic suppose condition range define base estimation run sgd pick generate bregman working apply rule later algorithm perform hold simplicity pick q update weight algorithm satisfy relationship know p solving notice particular unconstrained case actually update let show simplicity use rf see relationship proof rhs rearrange complete equivalently employ rule thing hence complete simplicity hence theorem like hard rf case rhs condition basis zero rhs proof corollary putting recall size choose evaluate satisfie e consist view follow item know intermediate desire old condition basis definition also establish q r binary vc subset lie intersection n less next element fix sensitivity case brief overview mirror sa main sgd exposition convex dual distance generate convex continuous continuously common distance function next define bregman sub r norm stochastic composite mirror descent iterate result analysis appear lemma step q condition side complete output particular fy finally convexity expression minimum rhs fy fy complete stanford edu recent year gradient sgd machine applicable applicable paper regression construct distribution sgd process maintain computation effectiveness consistent theoretical finally also need similar problem descent sgd attention strong performance practical sgd new unconstrained case deviation least unconstraine eigenvector iterative depend unconstrained formulate interior scalability algorithm theoretical thus flexible subproblem solve implementation method precision size two algorithmic approach develop take strength hybrid consist step construct weighted preserve sgd quality sgd convergence low run objective empirically perform medium quickly lp lp sgd sa node online well condition q descent sgd sgd fast q leverage sgd result solver potentially us structure view perspective formally I e answer approximation draw deal sa stochastic approximation mini sample weight useful basis solver weight sa algorithmic improve exploit strong capture construct leverage condition immediate
candidate close possible aspect fit reference need model reference reference model reference calculating project individually approximate explanatory power large relative explanatory explanatory effect predictive general reference discrepancy cross validation outside search search leave find leave indistinguishable precisely denote utility choose estimate utility reference suggest report drawback model time demonstrate may useful investigate predictive ability behave formalism uncertainty specification give list model obtain model speak form average adopt discrete integrate procedure empirically variable context combination review posteriori generate optimal zero utility otherwise type ii propose name median contain marginal include variable sum variable define choice close average squared author admit define prediction mean median problem discuss term utility think fix utility depends observe lead k unbiased biased often beneficial model successful utility selection generalization even black dash due get bias variance utility prototype grey represent model different realization true dataset red due utility choose maxima become far away true optimum overfitte fit demonstrate though optimistic maxima grey selection bias utility increase close true optimum though beneficial model approximately overfitte induced bias important concept little attention discuss validation idea utility depict plot utility considerable become practical selection discuss model show illustrative regression binary involve apply gaussian probit cumulative normal include intercept term analytically integration perform probit markov monte carlo problem convenience binary denote variable specification dimensionality construct bayesian reversible jump adjust belief description section important concept difference correlate rest weight irrelevant get approximately l p cv optimization map maximum posteriori marginal probability median ref discrepancy small explanatory power coefficient regression parameter perform test realization generalization predictive performance negative bad variable respect perform poorly model bad worse comparable dotted line conclusion cover green albeit result high utility method reasonably ref ability ability small empty intercept realization htp replication line denote choose average htp training grey realization black dotted insight closely cv cv row start find high bad word gap two curve empty cv utility average overfitte selection decrease grow error term visible save posterior probability much cv select model especially able variable predictive close result projection selection cv still predictive ability figure selection model function reference model reduce variability substantially close another appealing grey line exceed large main predictive study truly irrelevant versus average ordering seem seem necessarily vary figure model unobserve help explain predictive different dataset summarize deal regression zero log normalize negative target population output value discuss dataset refer list relatively uninformative weight reversible jump mcmc first uninformative last favor posterior plot idea decide include effect prior cancer perform l problem dataset due replace cv importance cv loo cv reduce search model repeat leave measure observation set training cross time sample performance validation time time result mean htp htp dot intercept fold credible dot cv row carry cv loo cv htp select perform point test utility remain grey test split black dotted vertical line variable summarize result select demonstrate cv overfitte overall tend desirable perform marginal division projection performance close tend red choose selection bias black conclude measure five probable three seem indistinguishable small input sample performance projection figure able variability large searching variable kk use outside line utility estimate dotted method also difficulty model choose utility variable size discuss section size estimate validation outside search induce bias estimate independent searching perform projection fold performance fold cv hold small satisfying kk variable merely organize differently final credible apply sense utility worst credible remain suggest substantial case small suggest stage despite effort believe search highly give searching recommend superior review selection performance binary select good predictive real overfitting phase may happen variance utility especially cv relatively reference well complex observe simplify less selection probable variable combination retain complexity despite place automated come good incorporate well correct uncertainty regard prediction projection seem robust search reference selection make problematic predictive minimization discrepancy find solve outside search allow study informative way produce utility estimate emphasize depend input cost control thank helpful manuscript acknowledge science compare widely practical selection highlight recommendation prefer classification perform several optimization relatively utility uncertainty project outperform variable demonstrate benefit cross selection predictive model statistical adopt identify true useful usefulness ability future would concern ability still try assess numerous assessment review review qualitative compare preferred believe study give insight exist article present subset regression discussion selection popular bayesian literature posteriori maximize selection ability loo cv widely unbiased predictive information error none model construct uncertainty simple give nearly prediction selection reference average reference kullback
machine master node parameter solve master carry step classical master local vector multiplication communication master form overall never master vector close small cg eigenvalue base closeness symmetric positive identical suffice implie classical terminate hold terminate choose bind eq terminate norm usually upper bind local iteration discuss complexity master equivalent equivalent end round communication communication precision everything together study efficiency round machine bit machine aggregate back master proposition communication algorithm communication ignore constant round communication notice communication round call corollary call bound extra one compute give good condition adjust inspire describe bound communication self satisfie let large q call q call involve round expression desire call roughly twice follow variant newton algorithm instead simply latter choice direction slightly inexact method two communication round stepsize size round communication work converge still complexity define know eigenvalue inequality imply inequality responsible complexity small result corollary since two round corollary algorithm inexact newton satisfy total communication round q bound quite require slow relatively achieve depend objective upper norm may scale bind global erm scaling quantify local improve minimize assume refined space compact derivative bounded formalize constant w w regularize imply proof analyze erm assumption constant choose respect generate high iw w iw quadratic iw w iw f lemma hessian generate hold assumption apply establishe remark radius analysis concentration instead vector satisfy conjecture especially dependence ready stochastic bind communication optimal take generate self require reach ignore constant suppose terminate iteration denote event respectively law event appendix algorithm return depend eq separately obtain order combine convexity additional remove inequality corollary specifically c put everything replace communication specification formalize regularize loss self minimizer constant communication round require parameter automatically tune least proof ignore desire practice hard replace inexact inexact newton replace distribute bound round communication give combine two consequence local sample help make similarity without need obtain loss rescale self convexity rescale scaling factor grow rely lemma balance scale expect round binary hinge loss q least square round strong result self need initial constant find algorithm input manually choose stop manually tune parameter admm scheme size gradient rule calculate progress communication end follow progress ccc note communication per admm communication l consist round communication another search stepsize communication aggregate solution iteration loop round begin inner interested efficiency plot progress reduce round admm bfgs converge fast bfgs speed comparable grow convergence become slow coincide iteration proportional regularization sensitivity relatively exhibit study standard sketch composite minimization convex simple nonsmooth admit minimize composite use k newton reduce since inexact quantify error follow minimizer set need remain devise inexact minimizer long modify master eq auxiliary condition round locally master solve propose equation accelerate utilize similarity algorithm converge composite inexact newton implementation accelerate proximal state erm cost machine evaluate distribute round often grow due regularization cause sample propose self linear classification inexact method inexact conjugate number communication round grow popular empirical self self function inexact characterize implementation inexact round practice theoretical consequence initial require objective experiment confirm superior communication addition equivalent hold nesterov use function value bind derivative combine inequality relation convexity upper inequality inequality inequality combination assumption sequence fw fw suggest come tolerance need proportional tolerance inequality right eq bound tolerance w arrive part immediately combine combine apply inexact asymptotic occur agree since conservative associate assumption hold whenever eq denote small integer iteration decrease constant side finally suffice terminate terminate v left side side desired prove recall imply inequality population risk sample empirical minimizer population suppose modify replace empirical sum ii fact lipschitz condition q terminology function fact value symmetry also combine result error independent first convexity therefore gradient average zero eq used w variable variance sum equal inequality ii fact v finally inequality last fw di w fw consider loss define assumption iv origin ball radius ball center belong eq point cardinality hessian matrix eq component upper hoeffding triangular union side yield desire number upper bind simplicity assumption similar q give corollary claim microsoft optimization minimization empirical communication overall local loss inexact inexact conjugate gradient self discuss ridge regression smoothed hinge supervise size slowly many learning need access whole process grow happen involve optimization storage machine solve optimization distribute rely inter communication generate whose probability support set distribution deterministic large identically suppose sample machine refer erm linear predictor label loss hinge stability term make loss hinge locally alternate procedure communication simple map operation computation communication speed consumption goal communication round sum iterative two communication per therefore twice constant smooth call quantity characterize ill condition descent master master gradient step iteration accelerate technique alternate direction method multiplier admm assumption strongly admm complexity turn accelerate admm accelerate iteration machine iteration complexity scale admm communication rich distribute high complexity similar distribute method assumption thus obtain look note complexity exploit optimization local zhang et approximate simply communication efficient achieve allow regularization stochastic seem ill motivate propose distribute iteration logarithmic paper propose communication efficient minimize newton convex continuous derivative g point average write gradient master node prohibitive use compute cg specifically master due cg direction newton communication loop outer method loop cg inexact use newton erm linear predictor report cg iteration communication first consider outer newton method analysis reach still depend problem function loss iteration inexact second cg take arrive inexact accelerate admm overcome exploit cg method spectral master depend characterize similarity erm show general effective converge bring overall algorithm self optimization popular loss include logistic erm list communication require several show weakly exclude logarithmic compare round ridge binary hinge accelerate admm deterministic except I improve review self popular empirical loss either self analyze inexact compute inexact use distribute communication linear classification list finally extension distribute theory nesterov interior call derivative derivative third limit convex self q self self book self follow lemma self rescaled self self regularize double parameter binary convex self exist self fw q need appropriately additional since strongly convex bind substituting immediately rely long point freedom broad next take loss smoothed loss concrete example logistic third conclude self self favorable hinge loss smoothed hinge segment smooth lemma derivative second mean accord self scale function initial nonnegative sequence w w satisfied analyze inexact newton minimize loss standard addition exact newton computation newton inexact separate perform approximately budget centralize machine newton system characterize detail strictly
whether spurious necessary selection equation collect verification j covariate avoid identifiable impose statistic regard nan level reject distribution multipli point distribution spurious depend one critical associate q analytic quick validity typically fit residual view nan adjust I nk multipli bootstrap approximation carlo employ analogously eq n ns require intensive large avoid compute note find commonly maximize infeasible quickly hundred thousand trade computational intensity branch pick variable say screen procedure implement rare application computational cost standardized vector covariate dimension take take let upper quantile spurious direct characterize extent bootstrap summarize sd simulate quantile calculate replication simulate bootstrap spurious correlation multipli approximation table good spurious correlation isotropic case sd c identity focus matrix number rescale definite vector covariate simulation table summarize deviation simulation bootstrap fairly heterogeneity covariance estimate sd examine multipli bootstrap serve benchmark discovery spurious spurious rp pr pl post lasso covariate fold select sdp model ii replication compute sdp simulation depict dependency collect highly correlate add difficulty lasso severe reflect sdp fit discovery section life whether technique spurious correlation individual chinese international project introduction think study take response particular ten parameter fit lasso pl pl estimator observe fit quantile approximation bootstrap replication though pl decrease discover discovery direct probe disease correlation value fit response empirical multipli bootstrap approximation sample ccccc pl r observe solid observe correlation dot percentile median bottom percentile blue bootstrap indicate red residual employ nan time multipli bootstrap value evidence depict collect prove proposition line n constant condition fulfil tc ns p b every view term pd concentration upper expectation apply absolute every least I every argument prove take c ns bind semi refer introduction tight note successively lead eq hold previous display k prove anti concentration supremum index anti inequality respectively proposition let random satisfy q j imply take completely deal investigate standardized counterpart ip mean induce ns ns center index variate aforementione consistently estimate supremum prove k ns ss k hold c ns follow ns ss sp p hand side reduce happen divide discretization prove net denote ns notational q display imply c ns obtain right side together suffice apply combine together note p follow display complete theorem n inequality maxima sum play important analysis three argument aforementione discretized finish anti establish approximate net let metric together coefficient ball eq yield carry approximation discrete nd v v g g g j j g corollary random lemma eq three imply turn direct put together c meaningful far put least ns ss follow borel subset p u variant take write n event c together identity deal lead q cardinality observe modification imply least sufficiently side multiple least theorem absolute take proposition theorem section remark section mathematics chinese sciences chinese decade find group covariate mine spurious need validate need derive correlation namely response combination covariate possess process hence approximate unknown residual fit spurious testing covariate mining test model result bootstrap false discovery technology change massive store dimensionality characterize statistical science machine response economic finance statistical method behind datum heuristic example base explain group impossible thousand model consistency restrict homogeneity property despite rarely false scientific set mining spurious fit observe green numerical international ac uk take expression gene alpha expression dimensionality fit tuning cross validation value fit sample post fit response remarkable fit spurious diagnostic covariate residual lasso figure assumption spurious covariate subset correlation random noise sn spurious maximization linear fit maximum block correlation identity top importance spurious recognize distribute random point use demonstrate spurious grow quickly simulation study pp asymptotic analytic depend bootstrap tp e certain correlation approximate center hold ps pn p increment ns ns p establishe size condition ps ps nature establish limit statistic chi freedom integer integral express particular last vanish proof proposition place carlo simulate multipli n normal random observe variate gaussian covariance distribution triplet theorems spurious correlation approximate multipli practically relatively proxy
demand target target storage meaningful content layer separate branch layer multiple resolution section explore dataset thus primitive class detail make branching learn target dataset keep l graphic os hardware hardware layer branch class popular make build corpus represent histogram occurrence word vocabulary disadvantage working vocabulary thousand text lead poor use dense tool skip group similar output entire remove language stop represent word vector post average layer branch create hide bias layer layer unit function vanish relu branch softmax logistic well loss target give figure training cost generate involve situation loss experiment cost minimize also start depict simultaneous training training box plot hide final target hide target testing table box simultaneous branching appropriate level cause meaningful improve training show simultaneous minimized branch enforce information branch flexibility modularity scalability exploit meaningful meaningful representation branch convolutional work computer vision ideal connection desire vision break detail output branch hide exploration neuron directly target branching work output output useful computer practically useful electrical technology neural branch hierarchy final target branch provide enforce help final shared layer modify branching layer flexible inference level situation level result accord paper level target make use branch neural network multiple number make raw effort pre network abstract character layer deep easy problem get optima vanish hyperparameter properly tend proper play important deep layer belief network unsupervise pre component restrict boltzmann rbms tuning model tune supervised training regularizer initialize idea network target level target high learn meaningful layer helpful content activation present deep conclude scope future layer branching target arrange hierarchical fashion learn
mnist decay dropout structure testing weight standard th reference implementation pass dropout mc dropout literature model stress different error none fig fig seem ip cifar convnet dropout suffer passes dropout tendency change randomly subset small evaluate dropout fit mnist network alone ip blue dropout later mc convnet ip dropout fig seem perform dataset fit comparison well well dropout dropout probability mc convnet model publish art convnet fully follow suggest mc two achieve year convolution operation follow interpretation extend function effect encourage name imagenet assess cifar replicate paper evaluate standard mc pass dropout potentially report also section standard average mathematically monte average forward argue dropout mlp dataset follow suggest contrary augment convnet significant achieve might exhibit standard pooling considerably explanation prevent convnet kernels mc optimisation converge thus possess fitting additional explain section lastly worth note long training test average forward pass application hardware allow dropout almost trivially could gpu mini dropout bernoulli unit matrix network generate multiple pass average weight convolutional neural robustness fitting placing convnet intractable approximated bernoulli exist tool require include interpretation convolution might relate exist convolutional furthermore imagenet datum use affected probability would mr comment google european remark identity ex ac convolutional convnet offer place convnet filter approximate require interpretation improvement classification finish art result cifar interpretation neural network extensively literature offer offer small mlp design lead however usually amount kernel also convnet vast commonly perform approximation example approximate try leibler model follow past bayesian fairly computational expensive approximate parameter make increase costly bernoulli computationally surprisingly field interpret gp extend dropout bernoulli variational allow principled convnet dropout layer model layer weight layer implement convnet dropout layer derivation forward mc contribution work numerous dropout connection practical convnet structure approach additional reduce fit small technique lastly dropout convnet literature dropout approximation improve state result insight follow briefly discuss implication convolution review relate inference bernoulli approximate variational denote softmax loss matrix optimisation term add often weight result objective often refer input variable take value layer drop binary binary tool bp psd covariance function layer approximate map gaussian write treat parameter treat variational randomly set binary indicate layer drop encoding value linearity element approximate distribution gps map deep gp explicitly hide unit gp obtain bernoulli precision parameter relate derivation convolution operation equivalence extend beyond operation model interpretation one placing layer matrix mlp eq interested tractable need define approximate variational approximate define layer vector distribute optimisation objective log kl divergence kl divergence encourage explain keep fitting carlo distribution extend development th convnet dimensional extract input input weight convolution matrix k arrange w k pooling nn model variational vector bernoulli set kernel dropout element pooling convolution bernoulli bayesian study insight implement bernoulli considerable improvement attain dropout mnist
equality zero row skew e ratio row row skew smooth analog transpose dataset generality take digit argued long confirm condition fail yes yes yes stock various set last establish sampling find sampling hybrid well plot various clearly digit stock datum via truncation sampling truncation turn truncation however show sampling threshold control noise good threshold reality control threshold hybrid predefine threshold sample produce iterative table list use figure restriction digit various set one pass memory compare sample I incoherence value make list hybrid sampling leverage average respectively leverage align result approximation component us work maintain variance rescale benefit parameterize hybrid superior quality show score distribution suggest optimal distribution align accuracy hybrid leverage digit digit principal component preserve digit category digit rank hybrid digit data c c leverage finally superiority optimal hybrid sampling size hybrid sample support pca digit runtime figure pca digit visualization top digit category take intensity project digit show visual list digit visualization principal component column compute visually close visualization pca project onto approximate similar actual pca h data visualization onto five leave project actual onto projection quality pca produce sparse digit pca digit respective finally matrix element wise leverage score leverage score element wise I score ii hybrid leverage average increase get gradually improve quality large produce variance hybrid bias towards regularizer maintain sample rescale need counter rescale structure show benefit parameterize distribution superior hybrid leverage distribution suggest hybrid align datum require achieve turn digit principal preserve digit category digit datum superiority hybrid sampling leverage sample size superiority hybrid rank h hybrid align table overall present indicate superiority extreme wise also usefulness analysis task fast synthetic real element randomize new sampling recover pca data hybrid ability strictly well performance give recover pca datum entry one user preserve want recover provable matrix order principal subspace sketch top quantity control pca sketch effectively address partially principal additional top iterate benefit ij ij start bold bold g column denote n ij k standard basis ij p ij ij ij sample sublinear product receive parameter identically distribute replacement hybrid return purpose addition noise argue strong linear trend capture top remain top equally low approximation low show respectively svd note problem ideal want structure preserve fundamental time dataset element heuristic properly element probability principal pc project pc pca via provable fast approximation pc synthetic promise along situation element construct estimator svd reconstruction surrogate norm quality pc enough bound matter absolute sampling reason small rescale would huge rescaling result poor keep e entry entry remain simple elegant proof approach could truncation argue would bernstein hybrid wise essentially propertie element bernstein main truncation wise sampling balance flexibility arbitrary parameter desire result generalize sampling know stage sampling pass sampling datum discuss one streaming pass algorithm hybrid sampling fix arrival stream give implement hybrid note element produce indeed sample element f note respectively event I ij j ij ij ij clearly ts pp element event pp note correctness theorem need estimate one case additional parameter twice require sampling requirement obtain triple create proxy need provable computation pca unbiased svd reduce consequently theorem show algorithm sparse sketch pca theorem pca surrogate follow theorem last inherently wise preserve original result derive various element wise let score become rank datum mix various derive sampling accuracy measuring hybrid sampling score respectively pca approximate denote also nr matrix also computation experiment binary noise specifically construct whose
need reduce unique depend increase contain element add signal recover call th without generality optimization problem eq reformulate impose constraint nonconvex subproblem converge alternate subproblem objective function formulate subproblem programming subproblem standard form c eqs definite subproblem problem optimal exist qp solve problem implementation scientific package physical present problem old old old extension line section perform figure signal distribution obtain multiplication mixing generate noisy perform matlab follow three label nmf multiplicative build iii nmf nmf normalize signal respectively perform comparison nmf note signal matrix nmf one three along figure scenario reconstruction capture kind rank however increase one fail notion monotonicity source noisy extend negativity quadratic framework illustrative nmf well exhibit behaviour indicate mm mm definition remark remark thm cccc institute technology nonnegative factorization mixing suffer separation increase mix nonnegative effective suffer order approach nmf alternate assumption relax nmf nmf nmf source nmf source nonnegative factorization blind separation widely area sciences environmental systems biology blind separation text analysis wide applicability fold negativity nmf interpretable seminal lee nmf extract signal give blind bss objective source noisy matrix since nmf suffers order incorporate semi impose negativity constraint nmf multiplicative several algorithm alternate project investigate improvement nmf signal scenario monotonicity constraint nmf semi investigate resolve monotonicity nmf signal semi semi demonstrate datum future incorporation entry matrix factorization nmf monotonicity signal illustrate compare nmf bold capital letter letter quantity denote column permutation column transpose denote indicate nonnegative number dimensional identity vector zero element indicate bss source contain number consider factorization often corrupt noise write contain give signal however bss mix matrix bss
california la ca usa ensemble play unlabele encode identify single suppose binary classifier space prediction accord expectation consider label unlabeled marginal assume bad correspond erm motivate test predictor know prediction apparent classifier case word rule improve rule case distinguish without label give characterization minimax prediction give unlabeled development matrix matrix make minimax example paper organize introduce formalize intuition adversary characterize minimax side slack solution link interpret slack minimax providing give build section computational discussing conclude main paper context hold probabilistic ensemble unlabele denote allow rather change intermediate interpretation extend predictor knowledge development constraint use nn probability simplex column ix h j formulate game predictor first play randomize adversary play predictor equivalently view summarize study follow model label infer apply game linear equilibrium side strategy let weight vector magnitude slack equilibrium strategy strategy slack duality minimax minimax predictor study completely characterize partial purpose specify weight weight convex desire accuracy prediction prediction alone slack test depict function analysis label uniform add h correlation unlabele datum test correlation concentrate specifically classifier e failure thus p subset intuition ensemble make seem type strategy adversary equivalent predict learner predictor increase spirit adversary match adversary hold equal margin qualitatively continue learner perspective subdifferential slack slack function differential weight geometric interpretation equation give weight five taking difference sum negative obtain exactly sum category prediction simplicity prove improvement algorithmic unify automatically see label c unweighted majority vote well vote algorithm consequence closely optima slack definition slack simply guarantee slack directly guarantee adversary label force flip solve game give know error noise weight us noise prediction predict example always statement asymmetric majority vote present step simply optimal weighting minimize slack slack straightforward treat programming storing memory unlabele typically without exploit sgd slack converge convergence suboptimal particularly intersection hyperplane piecewise slack memory convex play theoretically practically slack function limit duality one impose since essentially without multiclass expert weight vote nontrivial receive focus theoretical boost form classic vote purely margin label bad formulation bound slack margin among unlabele bound purely vote hypercube general benefit statistical classifier notably formulation moment thereby handle rich dependence moment long universal optimality linear demonstrate set emphasis set statistical stein new unlabeled individual show characterize function call slack slack computationally tractable streaming gradient descent ensemble support rather combine convergence rate programming requirement sgd bayes limit adversary classifier increase ability systematic core duality argument support describe adversary
appears rule quadratic entropy compose purely expand v naive follow section method value impact sort choose discard variance computation function assume locate close point ignore happen acceptable add value sort discard speedup densely valuable locate approximated work densely previous us interval bx ix preserve width need use bin width variance lead x x q assume small bin projection satisfie inequality similarly sort discard q behave particular change expect actual discard reduction htb go show function add enable vast exist technique adaptive bfgs constraint modification add additional corresponding maximized advanced sphere bfgs complex previously sphere norm stay suffer worth function additional affect state software able evaluate uci repository repository approximation code gradient cg cross validation randomly select across comparable optima acceptable accuracy balanced reduction computation report ratio exp cl bin diabetes easily sort discard pair denote much optimization projection distribute projection obvious theorem increase decrease heat figure original isolated error level introduce approximation significantly higher rare phenomenon accept error color approximate fact many notice consequence rough evaluation act like regularization table significantly simplify number number function conjugate especially line search speed cl bfgs b name bin diabetes heart claim small surface removal element summation technique sphere rapidly cg result order simple approximation fast objective
reason application approximately adjacent reality adjacent syntactic noun may semantic consideration embed view embed reason add option learn unlabeled information complexity simple predictor feature due analyze word triangle regard convolution layer adapt word layer region interpretation cnn illustrate view layer layer unsupervise adjacent region word skip gram train study semi appropriately categorization publicly internet review amazon sentiment label categorization training set unlabele unlabele disjoint review disjoint set make publicly internet article month unlabele unlabeled set semi cnn short type cnn fig base region unlabele data vector cnn learn unlabeled unsupervised minimized z I prediction output objective adjacent though concatenation adjacent unlabeled control purpose side word balance absence theory view concept relevant adjacent region syntactic relation undesirable sentiment meet assumption simple heuristic effective word proposition target vocabulary often lead word use word list provide rest seq seq perform activation convolution multiply top layer character convert meta portion data model r gram gram gram cnn cnn w convolution exclude word neuron view embed fix indicate meta tuned table meaningful comparison model convolution layer exclude seq cnn max cnn pooling report review cnn multi embedding show dimensionality embedding thing w clearly cnn confirm effectiveness framework sentiment relatively outperform w region large cnn use predictive suffer sparsity supervise poorly perform supervise cnn contexts word word help superiority cnn vector cnn word embed cnn embed explore integration embed cnn word embed multi view latter text replacement add turn add chosen replacement except illustrate view really g really view due hard reach training advantage w indeed learn appear combine table region layer triangle concatenation feature receive result nb gram nb lm gram seq cnn k seq seq unlabeled sentence lm ensemble nb lm unlabele unlabele k lm gram seq seq comparison previous knowledge paragraph use produce combine independently non ensemble table good seq convolution neuron layer seq seq cnn l micro macro extra unlabeled micro macro average multi split category comparable entire room disjoint entire test cnn many supervise cnn test compare supervise meta repetition cnn label extremely consume could co performance stop co clearly demonstrate difficulty datum method due focus insight text influential neuron neuron contribution view activate top negative sentiment neuron one poor view though negativity prominent multi show present explain word semi supervise embed cnn experimental decomposition rank matrix denote correspond relationship obtain eq third use x equation ph follow define theorem present feature embed unlabele usefulness explain word embed new learn multi view embed text convolutional sentiment categorization learn apply nlp supervise require train large amount therefore semi supervise semi implicitly svm produce performance via contamination another learn unlabele preserve function additional degradation generate mostly supervise nlp empirically embed learn unlabele additional often supervise nlp alternate structure unlabele auxiliary task improve task name entity often intuitively expectation insight development implicit shifted justification learn limited case paper present analysis useful task allow embedding theoretical supervise framework view availability view come definition view build model internal essence cnn learn region particularly multi cnn exceed art text categorization cnn view unlabele demonstrate categorization first theoretical observe view assume relax conditionally rank sentiment assumption concept concept view reveal informative relation view embedding multi exist predict word everything multi view exist view embed decomposition ph say hold produce view target original arise make task predict paragraph view independence propose supervise cnn learn small text come view build cnn suitable image internal structure adapt cnn convert word view learn categorization use word classification categorization superior categorization explore embed cnn word cnn one cnn cnn forward equip layer pooling layer layer token document region view later associate share unit training region concatenation one representation large region train seq center vocabulary save apply generate convolution embed dim dim max layer feature focus
straightforward note suffice verify eqs begin condition verify throughout lemma hold expand along note claim sufficient eq enough large enough second term eq dominate eq state condition complete proof calculation proposition follow expectation component specific combine result establish proof eq q firstly dominant equation suffice already prove negligible claim third first note negligible term negligible argument n w dominate dominate suffice dominate contribution proof straightforward fourth eq eq claim adjust moment tool control random edge order vertex first couple vertex w bridge vertex set I order face edge way remove face type keep face follow happen subgraph induce tuple show brevity write length pair face star label graph vertex valid exist labeling property label occur abuse write valid matrix constant integer enough probability rescale tr x hand claim one interested suppose exist eq rescale suffice expectation center vanish summation wherein twice vanish term prove rv rv r yield expectation permutation decomposition irreducible permutation overcomplete straightforward check decomposition matrix follow eq verify define eigenvalue prefer derivation span hold large enough span fourth claim claim otherwise spectral understand seminal os proposition could self contain reason note ab tr dm assign summation index cycle suffice dm every obtain vertex label connect unique labeling constant inverting yield claim complete recall expand sum compactly follow term pair product exactly tuple occur instance p g face vertex label class slight abuse vanish intersection useful consequently proof intersect every arrive first one concern eq claim bind r label wherein obvious labeling claim occur connect connect twice recall valid contract couple identify length hence complete inequality I suffice firstly symmetry take conclude proceed multiplication therefore finally proposition j mr mr couple r criterion edge repeat twice show induction every repeat twice label labeling follow happen vertex exist second unique neighboring identify induction range complete proof firstly definition define prove moment claim union graph isolate arbitrary iv v suffice label map unique union isolate isolated contribute label hence suffice prove bridge contain vertex vertex claim contract bridge length map label neighbor identify neighbor bridge length induce labeling induction unique hence unique induction finally term eq since vertex recall star term vanish summation length star note path obtain union path since label vertex deal difference eqs tr mr subset satisfy every couple pair generality vanish satisfy every label argument isolate unconstraine decide vertex consequently triangle yield prove intersection event lemma event bind ba application triangle x prove lemma hence prove ba prove proposition deviation recall define follow let bridge claim claim argument apply minor modification eq bridge length isolate vertex decomposition proposition complete favorable intersection favorable event proposition enough equivalently matrix expand correspond develop explicit yield guarantee eqs claim given consider determine submatrix distribution plant clique plant unbounde suitable despite substantial time succeed fails present improve unless change proof use spectrum os complexity challenge research focus bound study match assume unbounded resource fail clique submatrix category submatrix convention law give random estimation whereby special challenge machine clique plant attract dirac hidden correspond whereby presence absence induce otherwise hide clique section statement graph size exhaustive clique solve hand significant effort polynomial gap performance understand theoretic motivate hard rough clique unfortunately imply computational low instance careful preserve instance specific typically hide limitation statement rely completeness call somewhat change complementary attack prove unconditional low broad chain query formulation prove algorithm close semidefinite remarkably prove hierarchy clique write hierarchy hardness clique strong establish analogous sum hierarchy hierarchy relaxation similar close connection conjecture idea broad class many naturally within hierarchy whose treat propose construction solution bound moment unfortunately contain hierarchy spectral fail unless notice guess hierarchy whenever fall argument present impossible improve except remove logarithmic factor provide apply construction submatrix entry distribution combinatorial certain positive semidefinite matrix os subset depend subgraph degree relaxation consider positive absolute objective claim position derive formal degree maximum clique probability immediately clique far set variable probability large mention introduction generate hypothesis unnecessary technical entry order motivate nearly combinatorial look principal submatrix average straightforward eq state least exist eq fix adjacency matrix ij ij os graph gaussian density choose suitably far hold subgaussian standard small hence hidden hypothesis distribute note scale factor suitable constant high distribute restriction index abuse hence control random I key triangular weak decompose state cf essentially adjacency consist
tweet produce game n n vs vs games european english collect tweet game twitter streaming tweet stream relevant tweet guarantee meaningful involve popular filter game collect least tweet team hour processing yield dataset turn baseline final include produce ability evolve design capture algorithm language opinion mining previous sentiment capture prediction market sentiment analysis library system sentiment framework score collective framework social effectively predict unbalanced analyze twitter relative potential reflect sentiment crowd show around yield achieve negligible strict rigorous odd certainly large great odd grant margin reason part focus unbalanced game unlikely high increase estimate unbalanced game crowd towards enhance odd unlikely offset loss incur likely upon result decrease balanced plan extensively future support award gray media stream forecast outcome political stock fluctuation increase social access crowd forecasting power medium media stream automatically match focus highly argue offer systematic testing medium baseline despite strict baseline system sentiment exploit collect informed prediction behind framework twitter full stream monitoring match major european fa social media twitter source understand phenomena outcome yet happen political office market much failure social predict surprising collective crowd expense hand issue bias opinion directly arguably correlation predict twitter movie generate week g fluctuation market potential leverage medium real event unclear systematically address issue team event offer possible outcome match limit occur continuously lot collect medium million expectation future game social medium systematically implicitly reflect opinion crowd continuously reflect regarded opinion discuss validation occurrence unlikely odd leverage twitter game make popular recent twitter make online platform mention unlikely reason upon importantly arguably odd offer successfully leverage medium game six twitter six game representation discriminate game whose outcome odd unlikely odd translate margin crowd social medium properly odd present procedure specific introduce twitter implementation assess economic yield use collection work summary game la world collect historical repository contain entire monitor twitter dataset refer second live monitoring consider odd odd odd odds hill together odd game define course coincide outcome unlikely upper practical odd experience score correctly game turn generate score exceed arbitrary report finding game likely likely odd likely purpose world outcome game play game definition latter constitute subset former depicted fig game game table immediately see consider note whose odd team game potential final interestingly possibility observe table consist game extract twitter game discuss detail collection turn employ section information predict extra game france usa usa exclude penalty minute seek understand analysis match twitter occurring represent duration minute resolution try event event analyze noticed penalty exception event event decision spike traffic annotate event fluctuation collective sentiment sentiment consistently drop drastically immediate team twitter real high level idea media signal live rare user one user tweet contain team interaction interaction user group dynamic interaction group illustrate one precede useful outcome game exploit sentiment twitter predict potential argue sentiment help collective support social behavioral team express team working game turn sentiment produce precede tweet sentiment score range see retrieve twitter occur hour window minute sentiment tweet separately single sentiment team table live window pass windows minute small suggest significant discriminative turn sentiment minute minute start reader minute game news line weather etc outcome game window total window pass baseline baseline france baseline baseline baseline usa baseline hull city west city city qp city city baseline baseline baseline baseline baseline baseline describe potential discriminate odd collect result game draw consider classification approach explore classifier available library performance well good parameter feasibility advance deep even well performance live monitor train classifier perform twitter sample live monitor twitter streaming stream decide keep separate exhibit tweet twitter stream separately illustrate potential constitute dataset classifier near twitter promise full stream table show performance major european national live improve framework game randomize randomly label game game exhibit presence predictive prediction game solely upon sentiment specifically minute start significant prediction determine potential strategy inform medium precision recall precision recall score determine return systematically odd expert continuously adjust crowd systematic less achievable combine contain potential live monitoring european perform round validation strategy predict game turn team otherwise half half draw dataset realize therefore payoff respectively mean rather average deviation marginal bar surprisingly explanation wish exclude consistently classify correctly single return offset classification therefore analogous odd relative surprisingly high bar odd adopt fold potential game equally regardless systematic advanced strategy amount base proportional odd increase risk fluctuation systematic independently game team
turn satisfied may ergodic mcmc scheme satisfy avoid geometric drift give incorporate scheme last decade despite effort adaptive mcmc behind theory numerous development adaptive routine need valid counterpart establish deep stable mcmc rather modify numerous form instance model intractable equally solution standard simulating dimension grow technique may difficulty mcmc issue hide eventually filter chain algorithm directly doubly term acceptance evaluate exactly algorithmic field ising limit applicability suffer poor acceptance rate resolution metropolis replace intractable idea pseudo evolve iteration estimate proposal analogously metropolis accept otherwise extend enjoy specific construct present expansion offer likelihood approach understand current construct importance sampler thus improvement interest study term gap investigate efficiency derive scaling alternative intractable hasting accept ratio let dropping reject term monte preserve noisy quantify exact development likelihood big computing infeasible approach pseudo marginal filtering indeed filter target simple chain associate mcmc iteration transition since return estimator hasting q general notational demonstrate assess notion gibbs greatly particle follow use complex dynamic dynamical probabilistic approximate distribution base exploit resource seem possibility reach far beyond couple chain construct create adaptive gpu monte even contribution towards massive general explore direction approach ahead simulate next regular hasting back branch high deep path path create move reject evaluation acceptance fix costly computation processor high obviously instance helpful actual chain end approximation sub likelihood see capability find chain consist component seed augmentation auxiliary trick gain metropolis hasting index uniform pick select rejection decrease evaluate proposal ess acceptance rate jump free result processor investigate compare parallel therein namely difficulty handle dataset parallel computer outcome sampler asymptotically sense justification previous result accounting subsampling sampler produce convergence decomposition induce issue restrictive iid subsample contain author suggest final closely product mcmc approximation reason curse curse tail misspecification way mix drawback device avoid operate behaviour estimate likely behaviour true transform help tail finally seem target constant simulate could outside modal transform transform use kernel random related separately compute rejection merge spirit parallel mcmc break run independently parallel chain act converge parallel chain shorter set importantly target mcmc partition sampling regard list author need set chain move thus contribute integral evaluation unclear ergodicity depend e lead region chain stand partition challenge modal area mode indeed explore actually unlikely huge gap go partition hard last comment adaptive wang motivated highly complex method inefficient approximate situation detail variational likelihood version quite may feature come statistical induce balanced budget level evaluate depth wrong computational technique kind intuitive researcher simulate infer forward merge mcmc drawing summary play loss model gate field intractable model first quick elaborate incorporate toolbox nonparametric handle partly therefore reason everything else fail small summary summary sufficient statistic imply information formal rely raw raise wide application strictly likelihood genetic indeed intractable sense get intractable include auto exponential pseudo intractable deviation statistic location core abc see inverse concept conditional value true similar actual far abc actually involve acceptance simulate accept tolerance exactly posterior rarely achievable practice h prior accept acceptance calibration select realistic setting never rarely noise dominate example consider curse candidate leave abc algorithm statistic stress quantile simulate prior order distance draw constitute convergent approximation abc outcomes interpretation decrease precision universal second perspective output connect indirect purely nonparametric yet pair median plus special sample conjugate create compute component properly impact inference discrepancy completely eliminate abc equivalent distribution top sampler reference algorithm local regression central simulation abc summary statistic tolerance rate connect nature already early tolerance precise near ergodicity lack abc replicate involve simulation replicate repeat subsequent gold around mcmc ergodic almost technical zero geometrically ergodic condition prior ball result auxiliary ergodicity efficiency simulating replicate however variance select model rely hypothesis operate demonstrate population choice hypothesis validation primary motivation address impact ik statistic normal fit laplace show factor converge phenomenon naturally summary mean intersect counter expectation summary help achieve show abc prevent model factor consequence simplify approximation beneficial variational bayes decade substitute family exponential sometimes close kullback distance considerable gain term difficult assess would meet past five year approximation operate laplace approximation availability ep start target likelihood group observation term member give current value propagation go select marginal hyperparameter kullback leibler stop stationarity understand propagation practical substitute avoid use simulated actual evidence default implement ep empirical one time tolerance candidate look multimodal posterior program avoid complex drawback amount extend pos te map area receive lot particularly signal processing optimisation computationally integration map estimator space compute calculate major development relate optimisation discuss also useful optimisation concentrate powerful method exploit monotone operator mapping carefully design refer excellent book mathematic optimisation processing time fail element bayesian vision treatment treatment uncertainty optimisation need express decision nature lack capacity fast parameter space focus analytic numerical challenge isolate shape grow bring possibility along code another area use algorithmic tool convex optimisation mcmc computational proximal first decade context posterior density high formulate take proximal subgradient satisfie define subdifferential ng ng proximity q gain analyse opposite operator proximity mapping proximity g g solve sequence mapping long type relaxation relaxation converge backward subgradient point forward construction continuously little place notice convex advanced proximal optimisation either proximal operate splitting proximity mapping find ng possibly algorithm involve linear operator positivity admit compute algorithm converge lipschitz unknown remarkable accelerate class introduce notice several optimisation forward backward project proximity onto convex interpret iteration taylor around case dual ng proximity backward accelerate imply lipschitz computational separable proximity overview implementation proximal viewpoint forward differentiable limit efficient proximity mapping backward many proximal proximal proximal arguably bayesian operate optimisation augment unconstrained saddle admm proximity tailor specific way g exploit decomposition proximal interestingly admm interpret therefore characteristic proximal optimisation parallel architecture invertible express coarse system fine e processor gpu also closely proximal notice lastly optimisation main topic optimisation gradient proximity complexity allow adaptive riemannian speed problem connection proximal development mcmc one connection modern optimisation greatly motivated matrix underlie combinatorial optimisation hard relaxed tractable original development modern optimisation statistical dataset show proximal bayesian recover noisy nh represent spread low power ill pose difficulty bayesian image process improper discrete compute vertical horizontal arguably typically assume associate marginal unimodal concave subproblem optimisation example implement h modern compute mapping use mapping implementation algorithm image enhance estimate algorithm vs compute popular power signal ratio db show solve admm observe remarkably sharp region pixel measure quantile langevin continuously mainly concentrate contour detect presence sharp uncertainty location finally figure produce map second conduct computer matlab certainly fast dimensionality concern apart obvious requirement mistake area statistical community decide become reality science rapidly huge human science part argue four article accuracy easy simply fine change count set spurious big solve old mistake ever mistake think bayes development us carlo exploit environment prevent potential modelling modularity big paradigm entail illustration potential develop bayesian tool answer meta problem reliably apply methodology theory performance limitation start example attract create centre mass away area ti computation time work regard complexity modelling factor often development support technique technique seem justify ask extent cut answer extent cut edge answer answer probably agree entirely something agree fail completely goal explain communication statistic computational people community keep statistic arise good community aware perhaps enough people work interface interface easy research problem write language strongly encourage develop way least without successful somewhat parallel past decade language meta language intend handle solution towards user language successful extent proportion fairly often fail concept bayesian like unable validate gap unclear go modify picture still model cover sound towards locally globally similarly address population program outcome area discuss emphasis progress need past year bayesian technique thousand application application bring constraint inference constraint hyperspectral explore validate approximated reveal therefore boundary algorithm develop simulation optimisation able handle part simple complex handle difficult new way also library technique quick fail practitioner retain inference uncertainty strength might fellowship intra fellowship distinguish france universit paris p decade see evolve move proposal langevin drift theoretical practitioner even dataset address difficulty handle ever dataset likely tool dramatically reduce raw capture aspect next reasonably computational start involve computational something raw incomplete past state future algorithm bayesian turn medium computation obviously challenge long mixture normal dataset algebraic derivation follow hard computer certainly towards computer decade tool need monte hard version back surprisingly much later early bayesian toolbox tool em despite availability computer definite cause partly lag become statistic community surprise significant flexible tool medium integral calculation toy conjugacy provide answer discover mcmc offer chance statistic quadrature develop special analysis precisely quadrature moderately appear paper issue relate sampling year extended generalised focus artificial intelligence drive ode tree aside research methodology branching parallel processor long appropriate formulation long indeed believe incomplete central massive computation abc tool section discuss progress issue approximate highlight modern lack less impact think justify discussion section raise science relevance mcmc could day become tool ergodic researcher rather reality traditional carlo perspective output regular monte carlo attention say development handle process advance accelerate parallel cloud computing within monte carlo take certain reach community compare group sound use output require remove lead close asymptotic meet tool monte drive reproduce metropolis hasting kernel machine return operational compute otherwise flexibility choose curse choice arbitrarily date efficiency complex limit access mala proposal combination metropolis parametric generally transition adaptive markov chain conceptually allow search transition available thus whole hope essentially difficulty practice markovian kernel variation
rd sigmoid rr static rr propose system identification toolbox matlab train remain evaluation quadratic covariance total art maximum take computer model know seem sensible trade load particular likelihood consider energy consumption day daily dynamical approximate converge wise exact gps converge follow converge q expansion follow coincide whole random argument converge proof reduce gram computational load fundamental hold right justified step metropolis within article matrix wishart prior inverse degree pdf sample allow specification bayesian form project set family conduct carefully efficient system identification competitive reliable quantification uncertainty powerful probabilistic behavior gps dynamical topic contribution year state novel formulation superior property variational gp induce attempt instead optimal nystr om underlie induce resort perform linear supplementary gp model define dynamical nonlinear n explicitly parametrization assume tackle amount infer strength gp framework systematically poor figure gp sec world utilize probabilistic nonlinear dynamical include state expansion state literature nonlinear model tool force present dynamical phenomena gps encode dynamical function oppose focus learning replace recent interest relationship propose parametrize pseudo induce expansion second use implicitly base procedure algorithm uncertainty learn expansion posterior weight em probably computational load small magnitude within minute load variational approximate rank favorable property prove outline introduce make representation gps correspond theoretically computational load reduce use synthetic contribution extension gps prior state represent model truncation provide homogeneous represent fourier employ relation pseudo differential operator isotropic operator proof feature hilbert hyperparameter gram matrix interpret parametric function gp function weight also give weight element infer posterior clarity model however extension well infer sampler although involve sequential rely asymptotic thank invariant accord present straightforward iw inverse wishart prefer different quantity brevity rigorously minor four algorithm technique infer smc along state form markov smoothing within trajectory na ia k na I weight condition parametrization available material qp problem inverse covariance function posterior follow analytically sample possible sample q sampling hyperparameter easily utilize hyperparameter metropolis hasting mh hyperparameter predictive result load define approximation basis provide convergence rectangular gp tend mean equivalent follow provide scale oppose furthermore sound property assume mh support formalize expect particle monte prove ergodicity associate example comparison method
human feature common neighboring image tf ic propagate information diagonal matrix incorrectly unobserved operation image code validate strategy participant system make interface build summarize participant shoot choose small intra class dataset expert discriminate leave specie collection capture year dataset annotate extract publicly convnet dataset imagenet challenge tune fully truth produce separate convnet dimensionality convnet perform fine tuned convnet additional supervise produce smoothly space well align student similarity benefit crowd ranking user however convnet balance reduce student code project conduct replicate result setup image ask interactive image ask label interface correct answer phase provide use purpose test image choose exclude set length test feedback student delay encourage drop worth note crowd annotation worker possess concept participant prior participant task knowledge moderate student image always avoid worker reject response testing encourage effort discard participant participant baseline outline baseline centroid student present represent centroid choose intra shot student expect baseline similar batch offline student correctly deterministic student offline directly operate interactive summarize table depict image correctly strategy well performing vary table second offline often outperform random performance dataset acquisition datum oppose imaging control laboratory show participant participant baseline strategy calculate table compare strategy indicate level centroid bad value statistical nan hypothesis method statistically testing indicate bad strategy score calculate student along snapshot trend improve recognition image unlike base strategy relatively outli show challenging give variability student five chinese class due answer previous incorrect answer finally explore understanding unable adapt focus student poor begin reasonable end dataset unique modal leave class look learner assume unimodal would species entire currently attempt past previously similar region incorrectly label still label propagation allow incorrectly present image behavior machine learner human task human human expert automatically adapt perform difficult costly teacher take propose interactive multi informative focus representative image introduce student improve future plan pairwise region part discriminative category investigate concept ensure visual categorization automate information extract student suggest version different explore annotation finally assume student estimate student ability would thank development history assessment project institute token extremely classify image possess hand supervision image learn people first follow e ability discriminate class student vary work propose interactive enable computer human image show student ability progress correct incorrect answer real human varied world manually dataset visual category annotation complete crowd label internet service image begins ask incorrectly specialized acquire training potentially multiple designing challenge group improve collective tend vote weak learn trust expert pose expert improve family offer human computer learn model supervision label rather oracle human help outside automatic domain education image biological crucially needs knowledge focus possible boundary human learn boundary show student classification time unlike computer human limit human generalize unknown majority focus interactive offline feedback interactive student visual learner unlike computer human student student regard model student instead attempt uncertainty knowledge amount take experimentally human participant baseline interface explore strategy encourage development human relate concern task categorization note explore sequential additionally interactive receive feedback feedback adapt free bad predict uncertainty learn teacher truth use assess perform sub optimally seek currently uncertain regard informative interactive visual concept classify linearly separable category label student investigate feature exploration student interactive multi feedback correspond class label goal subset interactive teacher image represent learner teacher student refer teacher teacher ground class class teacher student believe belong student teacher reveal ground truth proceed teacher ability teacher trivially know teacher seek minimize student
make local minima initialization possibility property progress relax note weak constraint weak ensure minimize new satisfy valid two scenario construct scenario g gb tw goodness large scenario valid construction mechanism em first mm instance bias maximize bound progress tend rapidly nearby svms fix concave objective attract condition match explain empirically pick valid sensitive hard performance model mm restrict require progress valid mm exploratory progress g mm simple complex non mrf pairwise enforce bias towards preferred imagine drop mrf unary easy efficient minimizing enforce preferred would simplify procedure mean structural ease demonstrate latent mean belong category worth mm fully capable handle index cluster convex upper fixing minimize quadratic popular repeatedly assign construct center mm exhibit desire issue design bias encourage balanced appropriately generate latent lead valid specifically solution walk configuration configuration change extend structural svm prediction example df control negative variable specifically fix l z bound case configuration lead connect configuration correspond program solve sgd cut plane cut g mm progress bound progress cluster conduct different cloud gmm synthetic cloud reference dataset gmm create place apart dataset three initialization replacement center assign cluster experiment note solution g mm initialization dataset recovers fact gmm dataset initialize well suggest initialization mu mu g mm mu mu cloud em gmm mean em cluster method center assign implement standard objective value trial report use converge progress quality mm line cover deviation average dash indicate solution trial progress allow small g mm sensitive run diversity g coefficient possible b three solid represent area dash line represent good trial color different initialization find near perfect merge incorrectly assign color code truth center code permutation axis white cluster color match permutation axis detail objective detail object l six category level annotation provide class setup gradient report initialization strategy latent reasonable initialization initialization adversarial try example z keep increase fold divide fold fold train fold avoid dimension formal c standard error fold three location center corner random bias inspire fold baseline variant consistently outperform latent initialization location thereby rarely occur top expect validation well test g leave g mm g biased cross fold code differently five fold latent location g mm average perform scene mit scene segment mu regular grid cell part part use describe region pre hybrid convnet region record neuron feature l latent variable assignment multi label reader ls sensitive initialization generative version generative model l svms several part region discover discriminative cut node unary dot score extract assignment encourage specifically neighboring node differently assignment use initialization assign initialization require filter bias latter coherence labeling specifie assignment function system recall cell grid neighbor coherent compare performance mm biased bound bound repeat five initialization seed mean initialization converge pick remarkable boost bias attain attain initialization c c l mit initialization control initial correspond initialization correspond region bound correspond c biased ls mit dataset acc image control mu coherent bound mm bias progress coherent initialization mm random bias generalize g generic framework minimization mechanism sensitivity initialization adopt progress g mm deterministic stochastic way enjoy rich modification significantly counterpart generate sequence converge comment solution converge assume initial must positive must ii optimum stop progress mild get mm let bind objective solution exist eq specific latent svms localization issue fold update fold formal guarantee function g mm repeat structural training order sequence usual latent variable loo latent loo optimize result g behavior equation latent bias multi fold unconstrained define make generate regard support estimate practice fold offer cluster bound mm solution standard mean merge incorrectly origin mm white indicate center code match permutation image result latent object mm mm update object image category annotation bad adversarial initialization section paper discover part space limitation use proxy measure norm likely sample discard apply remain lemma machine mm systematically optimize convex upper function optimizer constraint unnecessary mm
network use application edge perturbation action privacy condition use nearly match produce stability state sufficient match achieve much good approximation guarantee bad hardness stable instance cut perturbation max latter improve perturbation perturbation objective median center objective optimally solve perturbation instance optimally perturbation also perturbation perturbation optimal show algorithm solution nash equilibrium theoretic return give alternative finding approximation structure dense min optimal et optimally protein stability al separation instance low heuristic traversal condition onto center close center center assumption axis beyond notion range cluster privacy point close center minimize formally dp I center instance metric symmetry unless otherwise asymmetric instance introduce change small perturbation perturbation perturbation formally satisfy perturbation unique change partition stay small distance perturb end clustering cluster differently two clustering formally center center strong approximation point satisfie partition close use hard approximation stability cost original function perturbation partition cost stable objective objective mean throughout radius show two third cluster hardness asymmetric show asymmetric center surprising one hand instance absence hard center well come cluster condition center behave together asymmetric hard involve deal asymmetric close center point speak behave symmetric explore define property ai ap ia introduce lemma perturbation lemma perturbation similar result approximation stability stability center satisfy use claim assume construct perturbation contradiction construct follow distance increase formally otherwise optimal center partition equal perturbation similarly define partition say previous must subtract arrive structure contradiction construct replace center increase except formally must center center know therefore center start create graph point apart close structural asymmetric instance candidate dp ep ap ig return instance point distance follow subset structure perturbation stability exception representative asymmetric instance stable asymmetric center asymmetric center return polynomial center even stability perturbation perturbation similarly symmetric center center prove two tight center dominate perfect dominate reduction perfect dominate dominate vertex proximity proximity perturbation perfect dominate yes instance translate least dominating establish instance find perturbation perturbation distance unless introduce center establish property perturbation second call cover ball close outside linkage closure repeatedly create formally closure closure linkage empty subset inside close outside h start singleton internal corresponding perform minimum pruning move require aforementioned crucially property merge readily consider bottleneck indeed arbitrary center median p q dc dp maximum arbitrarily close incorrect form merge partially form challenge us property center next formalize appear al property case perturbation establish proof proximity entail directly perturbation perturbation center contrary dp j dp c next contrary cluster root cluster contradict uniqueness perturbation contrary cluster double exception involve case determine ms dms outside partition lead contradiction dp dp ic contradict unique property structural property lemma us correctness remainder closure center closure ba cluster definition condition closure cluster condition closure closure second hold fully merge partially exploit impose show merge fully form partial center large center effectively create contradict perturbation proof correctness closure suffice show every set current first iteration show merge union include otherwise cluster part merge case lemma optimal contradict perturbation ic lc different imply similarly induce contradiction also closure linkage radius form partially work long show identify near optimal clustering symmetric stability strong necessarily insight behind result cluster otherwise make center leads create point graph contain return cluster define add approximation return cluster edge first direction cluster point b n pa ap ap ap perturbation center find linkage furthermore cluster necessary problem core perturbation large different apart outline distance perturbation keep remove center would score indeed center perturbation consider guarantee far center bad case reaction would challenge create challenge present deal approximation partition challenge construct perturbation case reaction reaction close show point contradiction chain reaction power point center perturbation close perturbation cluster instance linkage return linkage start implication cluster instance perturbation optimal cluster must center perturbation center perturbation often perturbation together use argue perturbation center outline perturbation center center center create cluster close exist assume dc dc dp dc perturbation set center score center must close center similarly leave capture point contradiction distance capture idea close chain reaction capture center exist center center formalize ccc half ccc close unique capture ccc point intermediate argue cluster ccc lemma appear cluster center follow majority exist ccc pair together intuitively reaction become center always center exist careful perturbation ccc cluster cluster close point cluster handle center distance statement statement satisfy center point center ccc say center apply statement statement p x dp I r ds r point close half reach problem polynomial pair close stable perturbation possible center simultaneously theoretic format element rank without rank ahead element optimal hold tie fourth need express rank element rank uniquely prove cluster satisfie point show construct valid perturbation close majority contradiction pick majority let rank tie fourth tie rank high ranking size easily linkage start singleton round merge find correspond return stop center cluster linkage set set last set therefore component merge exactly linkage set need find center perturbation np hard analyze center partition close furthermore give cluster show property point center cluster recognize soon form define example cluster include instance size median cost center weak proximity consequence rely weak analysis perturbation closure linkage center discuss obvious property show enough impose weak linkage outline graph linkage cluster linkage edge continue away prove never point add put instance center proximity polynomial high suffice add induction assume iteration towards contradiction figure merge furthermore proximity imply nonempty subset must merge center contradict add suffice step cluster proceed contain different round edge denote respectively need else connect component proximity already add add nonempty call inductive center contradict add search polynomially many edge condition optimal perturbation constant value hard worst thereby demonstrate power limit approximation stability perturbation work result center clustering require open whether asymmetric show asymmetric solve optimally perturbation interesting handle formation extend asymmetric perturbation achieve contradiction lemma lemma stability asymmetric center show exist eq partition replace center use contradict three give towards center center switch contradict stability assume argument center approximation contradiction contradict property ensure representative proof towards contradict assume asymmetric center radius connect representative connect assume call optimal center center contradict stability algorithm prove center stability call balanced dominate np front matching dm set triple find pair match dm dm dominating give integer balanced dominate set dominating dominate set vertex variant show version another hard establish reduction parsimonious reduction element verify np hard nice crucially point appear mention observation strong david dm every element dm easily parsimonious dm map edge element dominate dominate dm give verify parsimonious reduction center must center size satisfy approximation stability create distance center dominate set
fashion without inversion stage divide group correspond partition product equivalent make approximate regard second diagonal justify careful notably justification experiment possess rest gradient kronecker fisher describe furth inverse efficient inversion describe estimate quantity window process mini batch curvature batch fisher obtain practical robust little manual careful various theoretically optimization local implicitly nd k establish practice type momentum k computational cost way point characterize transformation consider result appendix feed presentation closely network output series bank unit neuron receive output unit via nonlinear sum layer unit output activity precise element matrix additional value bias coordinate think consist stack denote prediction make loss training pair proxy actually predictive objective parameterize minimizing follow backpropagation map forward pass output define network datum input model expectation distribution input perspective geometry gradient direction large gradient natural classical idea equivalent generalize newton case semi definite approximation particular define linearize represent logistic sigmoid cross equivalently network light fisher nd method importantly gradient optimization conversely point bring vast accumulate make book highly method fisher discussion natural gradient challenge natural compute large million impractical initial ingredient computable w q w w I block rao layer deep scale version achieve reach network architecture use exact line block purpose plot linearly kronecker kronecker major limit asymptotic seem successfully capture coarse fisher later section computational gradient arbitrary weight network entry I k g k g interpret approximation interpretation consider approximation error generalization order nd covariance intuitively measure interaction order oppose arise upper high I loose due variable error aware practice network particular error weight roughly whose tie high eqn inverse computation well efficient inverting rao computable follow subsection reasonably approximate make computable restrictive cost product compute give suppose associate inverse say row optimal th inverse usefulness subtle informally equivalent degree equal variable independent linear simply variable setting fisher apply derivative try likely regard reason entry block layer predict forward reasonable indeed undirected graphical potential depict figure reality adjacent accord stand joint reasonably fact approximate graphical basis recent inverse figure average inverse predict inverse exhibit note look block visible inverse factored technique section iteration purpose absolute white level differently due extent inverse fisher block subsection diagonal efficient vector present approximation network approximate block take inverse computing associated approximation block block block block sophisticated deal develop subsection agree block block imply establish efficiently assume assume gaussian depict whose graphical graphical model dag moreover also direct efficiently block low one yield nonetheless distinct label mapping source node whose given simply block generalization cholesky precision give perform follow compute subsection amount multiply correspond product straightforward fortunately invert figure examine approximate exactly diagonal must definition block well likely approximate one arguably interesting picture proportional block approximation meanwhile accounting approximation diagonal neural approximation compare block top right due actually plot note factored approximation network difference compare compare subject factored take purpose plot linearly j fisher would perhaps approximation quantity fisher compatible perform break rise finding detailed discussion poor curvature mini adapting seem efficient entire matrix require batch matrix multiplication multiplication arguably acceptable monte backpropagation target average outer various usual pass target cost additional average quite good gradient backpropagation maintain exponentially decay average scheme particular new old weighted averaging equal depend time proceed estimate kind decay scheme commonly involve diagonal diagonal small ng much process batch notably like deal fisher product scheme implement exact independent seem would amount big practical must process follow riemannian imply fisher matrix view tensor take step space method path objective one follow objective discrete understand traditional theoretic large family experiment matrix negative give pair hessian density view approximation nd taylor expansion whose replace approximation taylor whose kind negative natural update argue natural approximation notably sufficiently optimum traditionally optimizer arguably important practical behave help way apparent strong mathematical think nd sophisticated road arguably reason machine understand take sophisticated technique available crucial role reasonably well adaptive technique adapt adjustment impose spherical region trust model book insufficient update proposal seem rise update comparable method exact unlike equivalently guarantee accurate nd intrinsic model small represent curvature direction fortunately able subsection stage update apply slightly fisher inverse technique add curvature account mf approximation amount add individual modify kronecker use invert try sophisticated diagonal long work block kronecker computation replace expand give work slightly principled expression efficiently b b b negative describe subsection stage scheme exact fisher add one produce use input mini mini batch mini backward predictive forward pass describe adjusting version current modified factored technique describe section compute proposal multiply approximate formula layer efficiency final update product loop gradient invariant way parameterize mean follow small direction natural gradient tend local invariance case fortunately invariance direction respect curvature update negligible affect invariant broad affine transformation network version transform various still main result section invertible updating update immediately characterize transformation path default network assume initialization negligible momentum rate fix relax allow invariance smoothness quickly invariance limit transformation interpret replace nonlinearity sigmoid transformation immediate corollary sigmoid activation function initialization negligible affine transformation normalize activity choice smoothly addition invariant go similarly strong variance diagonal approximation elegant particular end equivalent network transform activity center formally gradient free method linear conjugate cg optimize eqn subject solve main avoid costly matrix secondly curvature lot average oppose fix batch course inexact curvature network optimization block large correspond roughly per therefore despite arguably accurate fisher many approximate diagonal introduce center modify dynamically unit wise activity typically skip connection layer preserve expressive efficiency transformation center network argument use quantity activity notation assume plus center skip connection center interpret transform center intuitively whiten account correlation gradient optimization fisher correspond incoming bias method except approximate discussion difference fisher deterministic spirit deterministic use find basic unbiased stochastic nearly closely fisher feed neural similar factored block approximation approximate accomplish basic factored technique add hand factor adapt combine fisher constitute merely course something crucial observe optimally method neural network dimensional output accurate inverse fisher momentum see additional element kronecker factor numerous include use experience section factor scale momentum maintain matrix kind perform section develop potentially insight diagonal difference determine reweighte effect approximately translate expect specific one basic wise block kronecker fisher term measure quantity change distribution assume condition respective kronecker self intrinsic technique obviously produce investigate deep autoencoder mnist curve face due high difficulty momentum include regularization problem sgd nesterov accelerate calibrate autoencoder schedule approximation curvature experience tend well baseline improvement task engineer hardware batch typical average use technique describe use matlab gpu computer ghz core intel cpu gb memory initialization help use iterate average averaging take average multiply optimizer multiply associated optimizer iterate sometimes mini report paper reconstruction actual function almost perfectly error oppose generalization capability size progress make plot per k tends examine baseline rate progress linear slightly sublinear appear factor momentum optimization baseline extent would seem sgd gpu gpu progress rapidly design exponentially increase schedule mini note neural involve autoencoder schedule stop block diag version indicate version plot row begin plot stage axis last plot vs second experiment autoencoder exponentially schedule momentum good per progress mnist face reflect gpu optimize algebra allow efficiently process parallel mini batch result per mini schedule without partition mini batch computation involve mini batch experiment figure progress order overall progress mostly large mini increase experiment increase although expensive solution noise importance use momentum significantly momentum sgd slow include without appear axis plot recall type momentum allow build define fisher across iteration strong fisher update proposal momentum one might momentum responsible sgd conventional well mnist autoencoder problem mnist face version per progress typically block block cost block overall per moderately diagonal multiplication cost differ increase list perform face unit significant average version sized suggest diagonal great simplicity comparable per progress situation implementation version approximately sgd implementation far synchronization step virtue acknowledgment acknowledge google would like constructive comment early eq lemma scalar independent intermediate lemma end use expectation network say intermediate quantity pass uncorrelated various compute provide instead come valid choice expression accord general relate moment term st order correspond similarly eliminate remain require fact know compute v less likewise rise efficient matrix product stein example numerous recent survey involve simulate multiplication stein equation use stein stein particularly application overhead cost root evaluate multiplication symmetric always application eq inverting side symmetric unitary matrix sized compute multiple application need compute cost future avoid computation considerably simple identity eigen jacobian predictive mini latter operation compute correspond forward pass linearize forward pass rank vector multiplication compute inner additionally compute product similarly obtain reduce cost various
minimize call sparse hard point dimensional position subspace negative independently unless quasi seem recovery basic sense relaxation structural rip compressed matrix allow recovery solve similar checking possesse rip motivated pac seek entry sparse suppose produce upper norm indeed greedy ascent base non unit seek solution find co problem powerful importantly without large additional style property rely approach optimization combinatorial inspire multiplicative cover describe weak suffice formal statement negative learn arise lead paradigm lot success speech mixture model receive attention gaussian gaussian extremely start heuristic maximization em give rigorous mixture albeit separation use method show gaussian component recover time separation exponential show recover surprising mild degeneracy thus mention parameter pac give mixture gaussian find success state class gaussian f weak often improper arbitrary context estimation often hard know proper mixture exponential improve complexity give meanwhile improper know general distribution monotone unimodal run improper mixture unimodal concave extend modal hazard improper component gaussian mixture polynomial usually list one algorithm axis gaussians learn remove gaussians dimension learn dimensional gaussian learn learn something proper improper learning suppose mixture axis gaussian running time ok function try mixture bound careful discretization section propose gaussian time component tradeoff efficient conjecture optimal factor weak obtain approximation polynomial algorithm get unless say necessary cover plant beyond algorithmic technique complexity plant require denote entry integer denote know cover element indicator precisely cover equation additionally connection cover q modification potential multiplicative weight potential precisely vary significantly try increment keep potential sparsity key increment progress intuition ax satisfie next appropriately furthermore scaling maintain may start drop normalization since negative maintain co start potential quantity I easily going check least increase get connect want obtain convenience ax normalize take last inequality index seek q last step eq apply proof know succeed find satisfying conclusion lemma completes add go index check allow suffice variable discretization add align respectively density context interval column solve direct feasible secondly guarantee kind difficult avoid carefully rectangular p denote column potentially infinitely finitely sample generate estimate multiplicative gaussian require coarse partition fine solution continuous partition gaussian apart rectangular grid carefully suffice use partition use rough estimate formalize continuous interval subscript coordinate induce notion give vx algorithm algorithm rectangular group bin coarse much mind run restrict mixture ensure obtain gaussian contain gaussian clarity error tn remain sample ns ns da remark theorem chernoff bs concavity however bs bf hence rw w furthermore bs f w w total close gaussian correspond partition unchanged prove depend interval pi tail since since taylor expansion eq summation add every flat close triangle coordinate inequality tool mixture follow last follow heavy lemma comment section lead complexity lower towards nonnegative sparsity nonnegative ax gives plant problem prove unless plant cover solve inspire hard cover disjoint union outline theorem reduce column equal yes universe hand know set cover word union
statistically direction effect combination loss new large multi separate objective thank financial von electrical engineering university place create overall combination introduce indicator alternative gradient adjust gradient weight require priori enable inner like use high loss multiple new direct mean loss provide self adjust divide describe fits improve error express obviously relate capability depend part current machine create different chance minima rarely attribute use building statistical model frequently although sound problematic involve extensively achievable cost reflect behavior consider choice classify incorrectly classified prefer incorrectly nonetheless place high optimizer choose although toy representative want possible boost incorrectly noise moreover achievable model objective optimization perspective transform gradient high loss like single important highlight although gradient hill conjecture place pressure may remove surface therefore allow minima reach provide provide overview characterize self respectively conclude future multi objective traditional optimization compose decision objective minima objectives minima trade objective optimal impossible objective increase say pareto counterpart pareto multi problem combine linearly become weight although combine mean achievable illustrative linear combination objective go properly non pareto form transform allow standard one objective resort frequently candidate expensive case logarithm call many objective improvement cause point pareto property combination maintain since concave maintain solution expectation previous equally indeed find close would equal solver confident hard weight may prevent straightforward mean technique set merge one take weight incorrectly sample place pressure predict sample may self go currently desire loss logarithm find weighted loss minimization automatically priori moreover high high weight pressure maximization weight loss worst become gradient similar infinity uniform single control objective set mini batch evaluate try version reconstruct digits mnist uniform able use problem already adjustment divide weight gradient order normalize objective gradient epoch slack constant pressure bad number epoch mini batch point increase epoch way define remove requirement absolutely allow arbitrarily experiment perform batch epoch function total generator add equal black corruption hide unit evaluate loss corruption epoch test plot loss maximization loss cause objective provide baseline corruption indicate fit moreover large generalization metric weight impose improve overall align conjecture propose sec corruption influence behavior always achieve baseline difference favorable occur set mean cope noise epoch part part bad median
intermediate square root optimization yield performance mix linearly combine suggest much study frobenius reconstruction reader remain interesting question target decomposition singular eigen psd orthonormal index project rescale therefore n n verify relative good rank difficult present follow ready theorem spectral score remark error successful depend facilitate understanding result small present proof defer supplement remark sampling square root sampling minimize square refer root equal e leverage equal scalar value quantity write flat bind skewed law theorem achieve minimum q skew I always l due quantity flat sampling discussion flat vector come tend uniform exist insight especially flexibility score adjust subsection quick strategy indicate result achieve issue ensure impose minimize easy verify optimization next problem slack search feasibility discussion could variance present generate different allow vary ga ga moderately generate freedom refer synthetic empirical evaluation norm different spectral time average nearly three reconstruction fast iv randomize combination value demonstrate well finally constrain regression synthetic generate bias estimator distribution demonstrate sampling u compare optimization mixing include versus size reconstruction supplement detailed supplement work term chernoff apply bernstein chernoff let finite psd q inequality last inequality utilize verify ki utilize bernstein k prove theorem union clear matrix bernstein derive dependent chernoff randomize frobenius interested norm remain nonetheless include similar phenomenon comment least square structural inequality condition term bind combination yield good sampling bring leverage exhibit bound develop constrained algorithm find lemma thm consider subset novel simple depend sampling probability understand tradeoff probability exhibit insight specific distribution square root uniform leverage score bind demonstrate benefit compare state give compressed select column svd yield maintain close recently problem identify population select minimize reconstruction pseudo norm particularly target randomized select build advanced chernoff bind bernstein inequality novel norm svd quantity dependent sampling scalar quantity leverage inherent knowledge dependent bring several benefit allow tradeoff well uniform well skew iii motivate attain well analysis efficient solve constrain well probability reconstruction establish bind exact closely work section empirical rank approximation spectral column deterministic algorithm randomize select criterion representative category qr variant numerical set fall define sampling representative square known leverage allow bound review achieve qr bind run use make linear target stage select exactly bound sample related reconstruction rank qr work time require show
main analysis response underlying splitting induce purpose partition parent operate repeatedly leaf splitting child require partitioning contain least parent terminal example terminal node observation implement g meanwhile child incorporate parent analysis satisfied induce tree leave kind valid fit infinitely partition name tree forest grow tree general splitting splitting comprise correlated show forest comparison individual forest average forest partition ambiguity proposal splitting depend rely structure paper review tree forest practice recommend original allow evaluated hold bootstrapping effect study concentration show concentration present promise adaptive analysis consistency forest asymptotic still uniformly cube requirement simplicity assumption distribute grow magnitude size principal establish adaptive regression tree apply recursive partitioning adaptive decision tight assumption assumption must polynomially strictly allow ignore sample valid partition expectation practical fit good support valid random forest use analogue directly dimension depth analogue generalization implication guide role cart rule regression cart good intractable problem cart something help bring valid imply compute empirical minimizer valid forest greedy tree may repeatedly question forest datum author true low split concentrate cart split comparison forest forest dimension example actual imbalance partition forest lebesgue rectangle process begin generalize leave hoeffding concentration yield section complement concentration bound guarantee ensemble forest proof give appendix ir r support define finally leave partition bound cube detail constructive construction generalization scan hold volume q approximate set tend leaf lebesgue sense approximate jointly recover thus job volume construct approximate length build form ab integer get guess reason volume become every geometrically cut example immediately exploit generalizing set th observation coordinate specifically establish tail independently whenever coupling tight follow generating lemma follow choose finish generalization forest tree thus forest understand variance us ensemble rate ensemble difficult order analyze ensemble forest far insight us uniform forest surface motivate post approach treat support seek guarantee consistency hand consistency require unknown practice assumption especially learn forest know true prior would forest prefer take shape split reference concentration allow surface conditional theorem apply thank imply condition corollary leave plug sample consequence see meaning decay leave corollary eq everything small immediately remain recall tend meanwhile tend combine desire conclusion complete approximate choice yield desire suffice conclusion directly let denote child parent meanwhile recover choice meanwhile complete show analogous recall guarantee analogously choice define construction proof converge rectangle q event q converge simplify tree construct triangle inequality eq individually last know eq q term similarly theorem mean parameter hoeffding combine bound state chernoff variable binomial desire hoeffding calculus meanwhile q q fact strong leave union multiplicative require apply find must converge set conclusion recursively feature split feature specifically study partition meet partition split index rule node round node event terminal combination split occur differ eq overlap hope comparable within always know partition proceed employ corollary get tail readily eq simplicity standardized variance small detail argument q lemma n together provide write leave determine establish leaf super write approximation reason denominator point fall leave leave verify tend simultaneously valid meanwhile tend arbitrary argument tree define q meanwhile q quantity conditionally variable hoeffding tend bind note prop prop conjecture prop prop prop surface introduce pick split model split formalism forest forest bound perspective predictive whenever forest estimation need consider predictor forest use machine variety field surface especially surprisingly compete network forest stability beyond draw forest believe convergence procedure tree theoretical describe forest pointwise provide tight forest concentration view occur stage stage find split treat tree fit worse fit tree adaptively split tree tree split jumps fit affect position split relate post provide estimate main tree splitting practical give promise asymptotic space training leaf regularity show regression tree forest universal forest split whole tight within modification routine cart original proposal size comprise
outli mle drastically tune present difference full outlier suggest nature equivalently nan outli removal fail reject nan apply et al paper divergence figure nature clear case theorem composite trivial result restriction impose hypothesis illustration composite hypothesis robustness test encounter science restriction perform test tool inferential classical test utilize restriction robust misspecification model outlier well develop robustness test density power divergence paper case minimum discrete continuous focus robustness base test composite provide online supplement introduce density power pd several minimum estimator study restriction estimator parameter restriction measure density parametric family sample space kernel description use x case replace countable f g sa sa supplementary material sa respect restrict divergence equation asymptotically supplementary restriction g independent dominate measure bring curse condition asymptotic help kernel derivation smoothed version density function use divergence subject divergence fitting exist consistent root divergence asymptotically definition supplementary definition independent smoothing restrict sense discrete testing et approximation statistic level give quantile nan p routine help desire composite functional corresponding restrict contaminate contamination contamination statistic statistic simplify influence correspond functional see unbounded consider asymptotic contiguous contiguous tend neighborhood huber contaminate material level let derive general expression asymptotic composite contamination freedom chi degree freedom centrality nan asymptotic distribution f equivalently p supplementary put alternative put asymptotic coincide independently discuss finally power divergence test present clearly whenever nan bound al influence test statistic p illustrate propose mean know population univariate unknown base sample want specify assume know
recommend comparison book instance rank computed test algorithm rank situation pool comprise algorithm pool comprise irrelevant compare algorithm point yet ignore issue ignore equivalent compare whose discuss illustrate example etc comparison post understand regardless specific adopt comparison correction powerful even drawback nan hypothesis test counterpart compare discuss organize denote algorithm performance rank column dataset row depend statistic degree freedom hypothesis establish significant perform perform family wise least comparison control correction also rank valid regardless claim algorithm different mean correct derive probable assume rank configuration probable yet post hoc nan hypothesis present analysis show test test five algorithm accuracy eq rank algorithm compare dataset two difference side sign test rank case favor assume compare nan post standard quantile adjust comparison rank reject result post hoc rank compare reject set simplicity want need comparison sign test numerically power rank statistic significance q claim significance mean test rank compare mean rank experiment finally accuracie seven average naive j locally forest assess validation accuracy alone first rank nan comparison pair quantile rank rank statistic small significant decision mean rank consider run claim five classifier differently different claim significant pool rank classifier clearly alternative classifier assume b z symbol comparison drawback guarantee small equivalent algorithms aspect issue recommend rank test hoc perform compare rank robust assume power sign recommend sign rank make sign sign symmetry datum regardless adjust control discuss adopt sign value always report decision less corrected significance level matlab rank post comparison instead recommend adopt pool bring counterpart overcome drawback nan example htp data machine
panel completely isolate edge isolate risk entity link concentration completely isolate isolated increase concentration subgraph involve american international whether provide augmentation alternative inclusion column variable v eq inclusion indicator involve express tr ij v holding conditional density kind supplementary material matlab implement frequentist use site proposition graph two bayesian literature spike use discuss produce efficiently hundred variable statistical inference among type use represent dependence variable learn refer graph carry follow undirected zero graph problem model induce approach wang model estimation model always bayesian impose sparsity prior positive definite fix prior determination stochastic graphical inherent nature permit theoretical address characterization modular integrate graphical model progress model make year adapt grow publish small day ghz days problem matlab report second edge ghz improvement necessary large call search concentration idea behind use prior characterize normalize update graph continuous shrinkage prior prior exist motivation come successful development shrinkage substantial attractive concentration prior loading handle nevertheless fundamentally distinct estimation estimation continuous shrinkage little know structure positive matrix pose challenge contribution two shrinkage undirected bi graph learn minute dimensional normal let graph model briefly review section concentration model concentration encode undirecte represent pair random except paradigm conjugate wishart prior bernoulli inclusion indicator inclusion degree freedom normalize constant choice directly posterior pp sampling matrix type share framework inefficient large feature manner mean loop iteration feature normalize non decomposable monte unstable complexity slow graph work avoid remain paper would problem computer concentration impose encourage penalize likelihood penalize shrinkage interpretation maximum posteriori exploit efficient hundred bayesian help propose treat consider lead run prior avoid constitute treatment eq maintain scalability insight covariance graph encode dependence bi bi full undirected graph covariance theoretically learn rely hierarchical density specifie structure likelihood b unfortunately quantity normalize decomposable graph carlo importance infeasible beyond later investigate class prior normalize constant decomposable graph portion advance framework early general motivate lasso thresholding likelihood ratio testing approach estimate bayesian derivation excellent report similar gibbs although graph report upon request strength combine approach denote covariance use set small represent normalizing depend component diagonal connect familiar symbol element component variable ij integration constrain proper distribution behind concentrated close appropriately come zero miss graph view indicator control indicator reflect inclusion prior imply inclusion probability chance reflect belief expect edge approximately inference consist whose focus inclusion truly knowledge relation comparison help turn unstable iteration yet concern prior intractable normalize dominate inference concern appear problematic hyperparameter parameter involve depend instead dominate show different reference curve display curve suggest bias introduce introduce fact constraint specify see large impact fix vary panel display imply function different reference plot continue definite force never extremely reflect concern lack challenge incorporation prior example suggest configuration support choice substantially regard practically zero large precisely explicit imply illustrate aspect setting essence aim small density long difference lack calculate infeasible numerical method estimate markov mcmc another perspective choose close mass edge sense mcmc issue standardize mcmc long element usually assign entire plausible experience contain structure insensitive primary scalability one indicator inclusion monte carlo intractable require carlo evaluate generate graph joint generating manner generate depend graph model proposition distribution symmetric zero diagonal full conditional diag last bernoulli conditional column correspond something look indeed imply view information regression coherent fashion interesting hierarchical proof hierarchical symmetric conditional eq kind surprisingly proposition normal speed scalability block sampler evaluate empirically standardized implement hyperparameter computation implement core block gibbs sampler across element column improve solid display minute minute generate approximately minute graph matrix inversion update measure calculate lag sample burn lag suggest efficiency experience usually reliable monte mean far time htbp evaluate scenario real world pattern daily analyze ba exchange website concentration model use true assess positive positive fp evaluate benchmark adaptive graphical wishart prior wishart classical fold validation adaptive seem well model hyperparameter belief iteration graph tp pattern observe compare tp fp positively tp fp fp especially concentration fp partly positively relate treat imply inclusion table benchmark competitive except favor inverse wishart method htbp c htbp wishart tp scenario dependence understand biological relationship cancer gene scenario two graph estimate matrix base panel display correlation nonzero correlation correlation within repeat benchmark propose benchmark graphical thresholding take day evaluation wishart worth experiment little original wishart expensive importance normalize thus slow numerically requires greatly implement website adopt inclusion hour substantially minute require fact fast posterior inverse surprisingly wishart element strong distribution empty follow gamma eq rapidly distribution graph conjecture well estimate support implication bayes factor might truly largely reflect word concentrated allow edge standard fundamental strong perhaps space depend standard wishart allow thorough beyond scope paper safe call hyperparameter relation scenario great benchmark concentration suggest
comprehensive know basis represent control show provide multi resolution representation consider covariance transformation see give three basis scale merely fine resolution capture bt like use conventional conventional form center location basis conventional basis ccc propose function form expression consider ml k practice aic function random tb cc effect estimate give consider six among aic comparison estimate ml cc c resolution performance various predictor krige perform poorly third quantile aic true daily dataset package year weather average daily consider identification q smoothness use obtain estimate apply smoothing select validation know covariance bt divide part consist datum datum underlie sample validation exponential mean surface predict surface b na kf submatrix kf column f na k shall twice trace equality minimize k institute spatial model flexible modeling computationally krige appropriate class thin function degree function small detail lead consequently basis commonly first select total function resolution require variability considerably estimate fourth basis function location effectiveness method fix krige spline process dimensional covariance z tn mutually impose effect w kt uncorrelated modeling spatial depend parameter estimate include commonly radial discrete basis advantageous basis center estimation basis support radius spatial spatial function situation approximate set f matrix non shift control parameter poorly cause significant seven extract thin spline term spatial need model class several advantage commonly select number computation estimation considerably reduce precise estimate fourth location take location location space organize introduce derive simple example daily present develop thin datum observe distinct penalty
ten plot display performance ten ten display agreement add individual sufficiently achieve condition classifier adaboost first since obtain eventually iteration lead robustness environment rather severe overfitting label localized differ rule localize final still average average bagging overfitte increase additional overfitte empirically forest provides far illustrate increase localization result form sample direction force close proximity training rule error interpolation localize proceed quite label comparison continue steady rate even practically yield disagreement illustrate average adaboost prevent performance signal decision small boost ten analogous row show classifier decomposition respect hold represent classify differently bayes rule point incorrectly along incorrectly vary considerably display mistake ten adaboost wise weighted location classifier classify ten still set exceed interesting follow easily theorem non start mass shift recall also I already establish either must completely determine latter probable large proportion b prove adaboost least successful noisy dataset datum continue iterate fit boost adaboost regularization size number contrast adaboost optimization well stage adaboost forest light provide novel intuition example adaboost forest forest classifier evident adaboost ensemble way adaboost actually forest tree average surface nevertheless hope adaboost argue average adaboost behave similarly forest interpolation couple hold belief interpolation argue neighborhood prevent average serve prevent fit random forest desirable interpolation deep tree hope average aspect broad success extend margin rgb adaboost margin however point forest method substantially forest propose classifier predictive procedure rather forest average create adaboost adaboost forest mechanism justification conventional conclude like forest regularization stop boost approach powerful ensemble weighted realization fact conference adaboost world adaboost early success follow effort wise realization lead boost estimation adapt view success adaboost computer science bound margin cast fully understand adaboost implication perfectly wide situation statistical suggest perfectly fitting build decomposed traditionally model smoothly balance extract fit fit hand automatically classical noise irreducible error hard consequently classifier huge g cart create community prediction iteration maintain algorithm claim analogy forest ensemble regard unlike accept boost create tree average canonical example completely exhibit self property generalization forest clear contribution adaboost weight contribution interpolation combine averaging create effective classifier turn interpolation kind extremely locally classifier couple average influence fit become localize adaboost adaboost demonstrate point decrease adaboost demonstrate effect demonstrate adaboost discuss view adaboost main conclusion interpolation correctly provide fit presence simulation discuss forest namely decompose classifier datum perfectly implication run adaboost deep deep allow component classifier bag section theory explain self strength emphasis focus statistical literature development boost briefly adaboost also variant review round version misclassification rate update round final weight datum weight n iy tw iy mf attempt adaboost predict generalization adaboost increasingly overfitte attempt resolve margin think confident label produce margin possible generalization error margin increase observe adaboost iteration decrease margin demonstrate apply maximize suitable separability appeal margin boost margin would hard optimize margin arc lp boost adaboost provably reduce margin adaboost yet error margin loose qualitative crucial large generalization margin view certainly investigation yet provide great view heart familiar program approximate exponential place search combination learner base classifier explanation article recent review boost seminal activity dedicate mathematically adaboost although statistical optimization adaboost surely problem fact minimize exponential classifier introduce variant beta boost except exponential despite beta boost able adaboost present exponential adaboost boost algorithm overfitte avoid learner one overfitte regularization opposite deep deep many recent work tree generalization although one suggest able extract fit maximize however boost fit one excellent job quantile summarize performance cart increase explanation zero continue minimize exponential loss smoothed unlike statistical optimization view perspective iteration adaboost deep tree allow draw adaboost component classifier noisy environment think draw grow reach variable output let rf fit without poor classifier forest serve argue achieve careful perfectly match general semi smoothness quick forest analogy adaboost forest gain popularity often achieve performance respect highly tool many application algorithm review procedure forest cart design bootstrap one average tree far reduce across fit close one bootstrap forest point label vote wish success forest optimize independently index construct fashion adaboost surface analysis hard justify leaf size next interpolation label training fit perfectly generalization come mind near show environment noise fact binary class easy asymptotic error neighbor high generalization problematic perfectly forest claim modify datum else measure insight prevent classifier single ensemble fit smoothed goal create surface small neighborhood everywhere constant fit influence generalization average mostly fits prevent region away average classifier fit continue localize conceptual illustrate idea poorly local point fit wrong relatively neighborhood generalization second sense rapid neighborhood process obvious forest many fit region crucially concept conceptual interpolation help try heavy two blue line locally line influence blue robust spike boost robust extremely red substantial fail little bit strength htp return classification distribute independently far suppose conditional pure general view approximately bayes possible closeness bayes data red red evenly essential htp result fitting learner predictor convention restrict throughout boost sub close vary small classification differ expectation set dimensionality evidence noisy environment large fact rule spike vanish measure obtain consistency stand contrast conclusion necessarily lead would classify rule htp b classifier consistent many display result allow tree bayes near neighbor boost one even rate example illustrate fact additive classifier differ combination flexibility class spike increasingly demonstrate superior performance vary considerably forest ensemble robust forest individually final smooth extremely visualize forest nearest neighbor classifier forest data nn classifier forest less point generalization adaboost subsequent section algorithm tree maximum terminal adaboost take point hypercube choose nn adaboost colored light colored training classify expect adaboost substantially well classifying long boost noise classify visually forest adaboost near sensitive adaboost forest fact noise neighborhood seem degree region forest adaboost visually follow similar dimension iteration adaboost practically still b training interpolation desirable forest previous crucial display forest six decision tree forest visual show majority forest light region region thing reproduce contain bootstrap region localize thin apparent five tend fit decision tree nearby classifier majority vote tree tree poor relatively get small indicate agreement bayes easily imagine wider fitting reduce noise average surface affect point iteration adaboost serve fit localize htp
periodic ar important dimensional application market apply ar type ahead european load conclusion basically possibly covariate stationary contain model autoregressive ar I weak covariate process allow huge class popular error response furthermore situation ny arbitrarily arrange obviously concentrate directly base classical reweighted least square literature g small joint higher impossible non loss many fast motivate resp thus receive common unfortunately never replace perform process practitioner sometimes multiple first computed result part second priori receive n new repeat end sense resp increase use within n v penalty reweighte adaptive special choice w w require usual estimator case worth show different tuning parameter crucial might demand optimal tuning parameter time lasso time framework subsequently optimum almost option information criterion bic discuss generalise criterion amount establish consistent initial option elastic net ridge subsequently ng n n n n l n however process reweighted adaptive estimate compute new l value information criterion reduce computation convenient lasso adaptive optimisation eventually stop plausible resp suggest algorithm n difference asymptotic shown get depend link estimator asymptotic vanishing getting prove achieve basically behaviour complicated point process infinitely n take account concern property notation n n correspond covariate standardize exist nn partial coordinate n n assumption adjust require adaptive make error state restriction grow behaviour weight unweighted property precise sign consistency normality nk furthermore option residual decay laplace replace polynomially decay residual possible variance discuss maximal growth impossible argue possible rate slightly fast linearly polynomial like get relevant growth small last require normality parameter want without choose stick long help several growth observe estimate clear asymptotic help parameter information mention process deal several extension periodic ar threshold multivariate autoregressive index lag process ii jj enumeration everything correspond regressor detail common process recently process restriction absolute moment j I enumeration everything multivariate distribution fix estimation size finite j jj j n parameter I adaptive estimation estimator precisely restriction l provide estimate resp suggest plug common square estimation high positivity parameter residual weakly slightly advanced require non restrict adaptive normality stationary however act shrinkage give computational aforementioned effect mean parameter lt lt j lt lb l periodic weakly stationary weakly periodic factor periodic spline periodic wavelet good mention general nevertheless periodic another application one break ar equation periodic capture periodic take option build triangular lasso particular modelling receive change powerful use inference study past ar switch finance call threshold option threshold lead introduce ar type covariate process threshold option volatility popular regression general interaction weakly full quadratic give eq popular ar ar likely method process lasso approximation idea large residual regressor contain autoregressive regressor matrix automatically iterate receive well principle principle take variance model possibility framework specify every monte model restrict analyse reweighte close consider dimensional ar subsequently aic bic generalise give aic either furthermore freedom replacement monte lag fix uniformly replacement simulate process ignore propose simulation additionally graph sigma range close information bic parameter hence small bic aic propose conditional analog criterion expectation robust tail result distribution satisfy distribute analog clear conduct another bic ahead absolute define h h forecasting reweighte additionally calculate forecast oracle oracle structure autoregressive simulation b dot line correspond sigma additionally estimate namely resp basically relationship well worse one remarkable setting structure usually setting dimensional market application propose ahead price european exchange consider autoregressive lag criterion take lasso iteratively reweighte iteration ni also forecast step ahead forecast conditional influence forecast consistency quite application ar market simulation additionally unknown specification case underlie observation impact series every exhibit finance behaviour analyse penalty parameter high research concern show work sometimes tail situation see absolute
low extension mapping generalizing generalization nystr om propose generalization coordinate symmetric coordinate objective extension function initially generalization many reason general supervised kernel version determine class nystr om apply method generalize supervised function performance likely suboptimal learn application aware make exploit jointly extension embedding measure concentrate around support manifold minimal euclidean interpolation classification class possible error deviation projection manifold practice avoid training datum regularity property magnitude meanwhile likely linearly separable nearly give embed confirm experimentally boundary class ambient especially separable dimension px px correspond direction like interpolation direction coincide derivative magnitude average denote directional induced nn tx normal direction element normalization directional aim boundary relatively derivative different separable directional achieve along meanwhile variation strong directional boundary enhance class interpolation arbitrarily average gradient magnitude along boundary embed formulate interpolation optimization exist explicitly label near euclidean distance let unit direction near neighbor counterpart q expression denote directional x ix average along direction nearest come manifold usually set embed compute supervise embedding sample lie class simplicity embed decompose may preserve learn formulate extension problem supervise learning estimate class label rest focus interpolation interpolation radial basis common rbf property smoothness adopt dx equivalent determination class algorithm construct interpolation alternate construct rbf interpolation kernel center select term gradually n solve compact attain minimum iteration interpolation class close low assign within near neighbor decrease assign near iteration selection center iteration center score compute center high stage score center ff x df k c k ty matrix center embed optimization choice optimize regularization parameter numerous meanwhile experimentally variation regular scale set across simplifie reduce propose solve decomposable scale theoretically meanwhile monotonically underlie ensuring separation directional along separation boundary fast highly interpolation localize around center well sufficiently strong thank underlie separate condition impose training coincide general attain increase overfitte function sufficiently boundary lose strong direction overfitte manifold represent class select concentrated manifold embed sample respectively interpolation fx directions red figures rbf plot display region correspond respectively figure cover manifold accurate interpolation separate two strong derivative show meanwhile overfitte strong directional observable direction overfitte g red yielding find configuration objective rather embed well link objective represent function separate gradient manifold yield scale indicate examine iteration class calculation label estimate iteration manifold rely manifold onto convex near update project onto employ mx mx coincide denote index mx approximate continuity mx ix dy x embed give parameter fitting yield order without iteration propose employ learn extension semi supervise interpolation assign rbf df k high manifold scale subject nn interpolation f I initially method essentially loop determination complexity projection overall step require value x class directional neighbor point k throughout iteration training repeat step throughout complexity section discuss extension supervise rbf interpolation ridge know linear dy modify square adjust weight dual problem product sample product since permit high kernel feature translation invariant family link kernel ridge dimension set I interpolation define write coefficient interpolation give kernel ridge rbf manifold embedding ridge model embed kernel make vector class meanwhile supervise coordinate allow geometric sample concentrate manifold separability preserve performance present extension manifold compute evaluate embedding objective embed remove provide manifold embedding test neighbor dimensional semi interpolation method rbf fit interpolation compute iteration embed map adaptation sample point near add embedding neighbor nystr om nystr om discuss modify nystr om coordinate formula gaussian sum neighbor original field regard classifier semi label first face image individual face database take pose illumination convert subject supervise laplacian embed enhanced separation class separable total directional large due overfitte optimize minimizing avoid final interval deviation meanwhile compute objective embed pair parameter become biased laplacian choice linear reliability may monotonic slightly procedure parameter ratio set embedding display misclassification label experiment early rule apply curve high interpolation add center method well repeat image first image database object category manifold object show figure image normalize learn normalize object embed scale previous experiment obtain object misclassification sample classification database unlabele outperform graph semi regular consideration extension learn supervise building figure differently purely use neighboring vector meanwhile approach function coordinate rely representation experimental confirm attain good performance respectively exploited kernel center learn interpolation classify unlabeled test way iteration assign label nearest classification low generalization image next test highest estimate completely embed computed laplacian well embed compare interpolation extend vary confidence score assign image contain misclassification rate throughout ratio term well propose iteration term strategy may throughout strategy embed influence consequently next iteration embed dramatically even kernel preserve interpolation throughout iteration inaccurate assignment note propose interpolation effect regularization classification accuracy embed construct rbf interpolation function scale parameter rbf fitting sequence interpolation compute misclassification regularization misclassification figure smooth scale resemble regularization objective coincide permit capture parameter learn extension manifold manifold construction rbf interpolation interpolation optimize sample estimate show regularity interpolation control optimize regularization encourage sufficiently strong direction separation ensure effective separation outperform solution application along method classification would fr map sample ambient space low preserve separation supervise manifold available embed known problem learn become especially propose interpolation provide supervise manifold algorithm radial embed embedding manifold smoothness interpolation class interpolation
approximately result parent parent influence target prominent parent avoid parent parent practice indirect improve parent explain distribution decision tree state tree compact learn clarity modification unnecessary complexity key scalable greedy exhaustive well parent subset intuitively influence small influence may detect parent parent detect non parent conditionally strong little influence conditioning parent parent bind cause parent c dynamic states explanation omit possible parent variable contain conditional ensure target strong attractive parent variable long prevent case due large actual parent quantify information parent parent illustrate action implicitly consider matter perfectly determine transition every ensure parent parent add return exist conditioning infer influential parent together useful detect assumption prevent form hardness may implicit dependency separate belong initially true finally assumption beneficial subsequently output hold exist return satisfy divide material derive mdps state mdps function transition realization trajectory inequality subsequently consider evaluate accurately parent policy likely visit trajectory realization infinite capture greedy parent evaluation depend probability trajectory visit constant indicate effect hardness large arbitrarily wrong structure multiplicative parent realization estimation transition must advantage lack multiplicative effective decreased exponentially parent characterize policy target policy policy realization never behavior infinite unlike approach difference parent realization behavior behavior policy visit space domain randomly domain compare normalized furthermore error refer evaluation behavior monte model construct partial free sampling use heuristic ratio flat pair build pair base parent need probability table behavior know wrong parent sample efficient free dramatically evaluation target greedy approach drop select uniform solve discover return problematic trajectory policy policy resolve problem modify return action select useful benchmark true thus rd quantile scale policy exploit structure require evaluation evaluation achieve normalized scale trajectory take structure large trajectory low approach line rl free trajectory similar trajectory adapt high variable parent parent table uniformly ensure distribution sparse return last return horizon randomly policy derive linear episode discount stationary domain modify policy return ensure action state least generate flat flat scale state compare performance trajectory policy fail task artificial slightly well use trajectory behavior probable target evaluation verify factor present challenging bit ram horizon behavior experience linear episode require extract action select trajectory achieve evaluation show evaluation average trial much artificial trajectory complexity sample exploit dramatically large small trajectory analyze three restrict imply weak parent weak significantly relevant parent believe world satisfy learn combinatorial correctly parent structure knowledge say effectiveness structure encourage adaptation l time horizon index number factor notation process mdp previous next variable subset parent parent big high break part derive trajectory need evaluate notice learn everywhere visit likely visit behavior visit never parent number trajectory proposition tell need perform high outcome least proposition outcome q visit event notice infinite never notice l distribute bernoulli least trivially outcome random variable x complement eq completeness use factorize state action v random variable give receive denote target k observe sample realization distribution estimate trajectory apply realization action pair trajectory notice hold directly realization action score realization automatically discard contain parent meet number error hence probability simplify add add parent break distinct parent parent realization realization parent enough hold trivially want sufficient applying probability hold probability use assumption stand realization proof least assumption sure strong alternatively enough realization add parent least combine stage iteration correspond strong strong parent include iteration second weak parent add stop twice hold bind algorithm least union variable add observe parent parent assumption probability inequality since add specific everything trajectory satisfy lemma set less probability induce mdp mdp proposition verify bind equation result corollary problem essential provide superiority policy evaluate policy computationally sample exploit factored dynamic environment sample well high reinforcement rl algorithm rl choose action quickly leave business understand problem customer successively maximize click click ad test company management test unless company exist obtain company exist general generate call generate policy try policy reason construct complete trajectory construct complete little internet technology million transaction world generally million want extremely occur look cf analysis
datum dense noise thresholding let invertible ssc satisfie nc optimality ensures triangle guarantee let triangle use step sufficiently convergence date dense execute thresholding invertible ssc level iteration obtain accurate tail bound chi least q prove satisfied appendix detail give corrupt execute invertible ssc level see obtain vector follow ssc convergence large design discussion give establishes rate least constant rely less unity translate translate make turn challenge get establish general matrix union modify rsc act isometry reader detail along modify give step denote something invertible shall gaussian sub design satisfie get general level corruption improve tolerance result readily vector permutation magnitude e claim accord chi freedom trace concentration inequality purpose perform exercise involve corruption establish exponential chi freedom moment triangle bind norm center second third step markov give repeat give complete ex microsoft com problem robust regression corrupt specifically underlie corruption vector coordinate solely formulation impose strict assumption hard thresholde mild recover exactly sub result generate propose extension sparse fast recovery solver sized corrupted response variant call fast good solver ex error experiment mean give error square least address regression economic computer goal corruption set clean optimization jointly admit efficient solution indeed exist exist provable guarantee observation adopt follow wish corruption value assumption corruption recover penalty result corruption severe restriction either sample incoherent universal less recovery amount importantly extremely unable guarantee recovery result albeit clean error intuitive seem long adopt good knowledge rigorously non setting despite appeal contribution guarantee thresholding mention fc provide select non global dependent convexity ssc smoothness definition rate recover ssc satisfied h allow adversarial value admit universal hold stress rigorously formally completely hence recovery robust fc well large solve address issue design gd geometric rate recover fc hard popular sparse geometrically constant sub experimentally thresholding algorithm significantly fast solver recovery property white corruption solver error organization goal estimate allow represent regressor generate perturbation potentially enforce example corruption clean point unbounded generate clean response sphere compose vector eigenvalue key ssc satisfy strong resp strong uniformity definition ssc sake necessity face adversary precisely ssc bad ex perform hard operator magnitude thresholding operator regressor update regressor fit three try regressor fc regression minimize active progress gd perform update single gradient objective active beneficial noise present along prevent fc expensive execute gd progress adaptively select fc gd active hard thresholding convergence variation also applicability technique setting subsequent sake ease exposition dense analyze fully fc convergence step carry fully regressor set regressor corrupt execute parameter invertible ssc constants algorithm sketch residual set whereas xx c performing give guarantee design actually ssc condition high let least similar prove large ex readily accommodate addition sparse unbounded direct reader setting would like requirement analyse mild assume assume satisfy gd perform gradient rate execute fc make distribution fc hybrid algorithm gd adopt advance pose problem problem fc execute fc gd policy enforce iteration solution reader assumption readily satisfied sub design albeit shall attractive fc approach response shall dense exist objective would recover recovery alone fc fc refer rsc define sparse recovery analyze rsc convexity smoothness shall say constant rsc level convergence satisfy constant see execute thresholding solution particular sample n complexity high constant corruption index fc see solver corruption fc problem solver carry dimensional high experiment offer statistically fast dimensional regressor choose sample select uniformly diagram repeat plot robust augment multiplier implement solve fine grid result fc solver solver ghz extensive comparative study homotopy outperform counterpart recovery extend study solver approximate message pass amp problem solver non phase cpu require compare present indicate run
expansion write linear consist part remain note thank computation v n n n leave side equal follow proof ib b c I ic ic ic I sake completeness herein auxiliary argument conclusion remain condition notice continuity boundedness derivative inequality likewise satisfied therefore corollary multiply additionally gamma euler formula p obtain rewrite j multiply side jt kt jk jt h j jt p jt jt p I apply laplace side equation side let f p f p multiplying multiply get additionally second last calculation denote di apply moment side obtain di divide side equation tt obtain tt divide side tt argument continue side tt change variable similarly compute generating generality minimum index firstly divide side let divide finally divide side argument theorem rewrite c kx calculation iy iy different pairwise multiply side iy therefore let side equation j continue prove appendix integral notice xx x rewrite equation tx dx turn integrable odd turn l c z rewrite indicate element simply argument multivariate find hyperplane tt tt sequel u l uv v uv obtain j multiplying l side fashion uv uv tt paragraph readily formation formation yx ib dx multiply side lx lb lx lf lx dx dm multivariate hyperplane tt rt u jt uk apply equation multiply side multiply ic lb l lie outside l l argument student appropriately symmetric jt jt generalized I multiply equation db put hyperplane tt kt assume distribution minimum side eventually high dt bt tt bt dt tt db k la lt l conclusion assume half circle I follow rx direct calculation additionally notice achieve prove continue care handling variate sequence I j nr p ir ip uv uv fx f identity fx equality convert rewrite proceed equation measure kp n uv j k n kn k k uv du w b extend hellinger contrary argument sequence nk k ij n assumption g g combine choose combination x satisfy p index regard j j likewise divide let system scenario polynomial equation trivial contradiction n divide numerator denote absolute ne combine notation theorem result immediately continue argument without loss p j j easily na p divide numerator case n n n v p I contradiction one eventually imply h p n p exactly p n cc know imply mean h polynomial obtain term positive admit equation admit assumption demonstrate yield h n p h yield polynomial finite happen n thus p case p k nh nh nh n n n n n j k nh np n j nh nh n k n line consequence generality p n n divide numerator denominator scaling argue case divide numerator denominator contradiction argue case divide numerator denominator happen equation admit consequence overall conclusion important get contradiction hold nm n treatment simplicity sign follow equation part b theorem theorem n np I result j notational assume obtain divide numerator denominator obtain result second get two system nz nu ip np n j nx large constant therefore fact assume applicable third order denominator fourth system p admit assertion argue way taylor expansion part contradiction odd number odd without assume divide numerator denominator odd odd already solution generality choose nm j j g n obtain part good scenario low minimax minimax skew calculation appendix divide far infinitely loss dp c np I sum contradiction infinitely hold n dp n np dp n dp contradiction nc dp lemma section support grant nsf nsf dms nsf thank several valuable type study identifiability behavior type fit broad variate show rate applicable include scale shape scale mixture devote demonstrate class structure role determine parameter fit determine rapidly simulation demonstrate identifiable kernel pose mixture density datum understand convergence practical one assess structure work chen metric cumulative line measure use wasserstein major advantage wasserstein measure wasserstein compute effectively popular distance total result well wasserstein distance mix support coefficient practical indicate wasserstein allow people rate paper mix therefore variate popular tool model unobserve associate underlying issue note recent research back continue nonparametric deconvolution beyond mixture contribution chen identifiability mix fit finite mixture oppose fit mixing specification mixing know chen scalar restriction wasserstein provide natural mix convergence mix model study focus per se show condition mention computer science efficient procedure cluster fit mixture carry vary location covariance considerably gaussian scale skew skew gaussians goal variate arise variety mixture euclidean belong know mixture measure location family elliptical family distribution shape skew exponentially location shape matrix rate enable rich heterogeneity among type behavior addition shall setting fit fit later complex consider class function elliptical covariance exponentially shape include modify rate know rate determine mix rr infimum take converge rate atom happen atom atom vanish generally rate mix sharp wasserstein variational distance mix wasserstein distance sharp mixing g extension result dd attempt give power variate space entail amount independence quantitative density several identifiability definition type strong take order involve identifiability criterion worth note tend primarily order identifiability something along additional range actually admit euclidean turn fit mild regularity condition establish sharp method mle induce minimax mixing addition converge rate type develop exhibit kind identifiability identifiability family include student second model type proof characterization theorem insight draw smoothness express characteristic vanish infinity cm n identifiable student exponentially identifiable location exact fit exact generic exact generic generic generic unknown logarithmic fit modify exact fit fit order identifiable covariance I fit gamma generic generic unknown logarithmic skew generic dependent fit dependent fit unknown term point common satisfy either identifiability family identifiable family skew setting location order theory describe exact fit mixture fit gaussian mixture turn separate novel treatment throughout family weakly weak identifiability lead extremely shall able precise non weakly identifiable class algebraic smoothness determine covariance lack identifiability due entail take order minimum value trivial emphasize bind sharp fact convergence find one precisely mix explicit determine algebraic geometry keep use standard addition convergence quickly component shall describe class density one positive order identifiability identity combination true parameter prevent class exclude case measure gamma identifiable class identifiable terminology convergence behavior class estimate fit mixture fit fit happen location bind location logarithmic among generalize skewness skew exhibit family really identifiable somewhat consequence measure generic true measure accord admit exact skew carry behavior subset convergence tie certain polynomial equation turn mixture skew identity second derivative skew manner dependence long adequate brief description general strongly specialized weakly identifiable class sharp choice strongly strong try force taylor vanish inequality sharp resort careful taylor continue key taylor expansion derivative independent back original process equation exponent desired link problematic behavior convergence establish popular gaussian mixture comes assess quality mix mixture mix theory redundant expect redundant complete spectrum logarithmic mix measure deconvolution gamma mixture wide within class useful way identifiability favorable convergence identify avoid organize provide preliminary present strong identifiability address provide devote weakly treat density separately easy consequence maximum many case theoretical bound contain available behavior likelihood mixture use family mixture difficulty traditional deconvolution fail let find bad use le approach wasserstein metric p p n identifiability student identifiability multivariate fit first identifiability general exponential skew exponential fit singular case p strongly weakly identifiable study li g derivative g kn large borel sigma algebra take kp define restrict addition point kp p ij ij tool analyze adopt wasserstein optimal wasserstein matrix relationship wasserstein clearly quantify establish notion probability fp p x via composite regard wasserstein take family combine display arrive wasserstein tx g g multivariate g p g g exponentially modify let independent combine bound I g p g g density multivariate product combine vector g develop accord density wasserstein identifiability essentially need notion identifiability order variate model advantage range allow wasserstein measure hold mix location gamma skew general fundamentally distinct exhibit interested skip variate positive notion fx identifiability fail many instance assume x clearly elliptical anti say r cr first identifiability derive support identifiable uniform depend impose boundedness nonetheless sense close distance varie extend globally measure mild property positive constant g g c verify identifiable thus remarkable result class wasserstein fit move lie interior give strong class identifiability family hold different uniformly r establish fit set family second large element bound suppose fx b fx bind distance sufficiently b elsewhere remark counterpart mixture exact fit setting fit attribute hold set method continue apply part version mild address small parameter away remove impose eigenvalue b estimation hellinger induce iv boundedness boundedness meet bind establish vanish fast subset fit case result wasserstein identifiable fit identifiable uniform k tight proposition private communication mistake correct add note condition valid strong fitted establish simply subset vary subset density family identifiable order admit proposition mix measure set lose identify broad identifiability hold also continue certain transformation vary qp generalize fx fx von distribution modify variate space variate generalized distribution identifiable multivariate odd degree freedom odd freedom modify gamma theorem chen proof however nontrivial conceptually somewhat straightforward demonstrate establish infinity common establish strong interesting next shall meet identifiable class structure theory later class identifiable c fix transform identifiability transform preserve assume order class identifiable modify jacobian conclusion first identifiability strong second sharp bound wasserstein distance useful discrete measure continue class function derivative derivative old boundedness first old relaxed old old identifiability develop class family identifiable class rise gamma skew quite specific algebraic density role determine identifiability mix covariance belong broad gaussian order identifiable multivariate density within broad class location parameter weakly identifiable family multivariate identifiable immediate thank identity whose satisfy x x identifiability mixture eigenvalue obtain sharp equation value equation solution choose check k n n get choose equation show sequel trivial therefore determine case appear difficult method deal basis basis appear equation inconsistent admit rescale precise relationship mix measure gaussian equation p g k sufficiently investigation part together mixture yield interesting convergence estimation mle density fairly moreover entail extra mixing actually see place restriction fx gamma identifiable thank choose identifiable fix vary strong violate neither estimation estimating setting logarithmic obtain lower collect identifiable class consider fit convergence identifiable location covariance gaussian convergence rate fit minimax convergence convergence logarithmic condition logarithmic location logarithmic skew normal fit even convergence rate k hellinger centering set capture entropy fit admit take ingredient rate heart wasserstein low identifiability interest particularly equip variate exact family generalize multiple constant depend give degree fit multivariate generalize gaussian shape part big location gaussian minimax obtain mix skewed density behavior finite mixture skew class skew positive skew distribution support cl g n g give generalize univariate univariate multiple involve achieve identifiability criterion difficulty entropy densitie detail defer iid true mixing likelihood algorithm em algorithm maxima multiple time obtain distance size repeat obtain panel convergence establish confirm finally gamma even difficulty converge maximum question open b g iid distribute accord mixture density density fit mixture exact accord asymptotic mle boundedness regularity smoothness one hellinger distance n pp verify paper sense instance part immediate le cf constant supremum infimum form mle conclude optimal logarithmic mixture gamma skew gaussian fast logarithmic summary obtain number mix collect convergence minimax fit identifiable rate fit fit rate condition generic location exponential rate logarithmic exact convergence satisfie convergence rate condition hellinger center integral denote identifiable non universal fit admit main ingredient lie heart distance weak identifiability able establish well low number variate type exact gaussian positive multivariate sufficiently student family odd multiple depend multivariate shape big part big g cnn cl scale gaussian n class density function like measure skewed class mixture skew assume constant skew fit cl k w mixture univariate depend mixture generalize univariate class sufficiently hard difficulty condition entropy mixture defer illustrate theoretical true obtain possibility obtain wasserstein vary experiment bar panel distance establish panel metric confirm mixture even b rich behavior fit mixture choose class skew accord location distribution figure support scale uniformly bound skew illustrated simulation generic p mix support upper match previous b mixture gaussian exactly mix mixture wasserstein plot sample simulation agreement theory rapidly turn generate iid sample carry generic g remarkable within even finite achieve logarithmic take inequality precise specific characterization present representative theorem fit set identifiable weakly identifiable insight organize gamma skew proof spirit interest proof infimum hold k g n g ng n converge notational replace sequence application p dx g g x replace plan w g j point easy observation sufficiently ij inequality observation follow important taylor nx k nb nx rewrite element argue opposite tend entail p g nx tx td nx g nx nx nx lemma display vanish identifiability criterion establish follow manner find sequence tend nx g g k g nk k fact write ng g ng limit notational replace subsequence sequence ip strictly potentially singular may assume semidefinite use fx shorthand multiple possibly non plan achieve mass couple probability find plan check k sufficiently p n order nx ij lipschitz clear derivative respect order evaluate pair coefficient associate depend shall vanish indeed take summation lx uv therefore p absolute nc nx ix second identifiability coefficient g ni n ni element equal taylor r expansion remainder n dx conclude c w c exposition univariate scalar employ proof p ij point n ij ip fx ip ij enter identity n f f v derivative sum proceed prove trivial construct give k k observe arrive bind hellinger distance w n fx verified fx fx v p g x second form cauchy v g claim turn suffice pass nx combine extracting summation I v n nj j differ one likewise equal divide numerator denominator obtain least element atom trivial entail solution equation contradict vanish hold converge ne differ limit odd linear combination function even coefficient employ entail proof solution I write equation divide b consequence trivial trivial without generality side basis polynomial polynomial equation retain display choose verify basis additionally choose check basis appendix theorem characterization regard strong transformation fit exponential mixture fit skew gaussian fit skew mixture proposition corollary already defer k p I p g p w contrary n g n c w g g g p almost surely identifiability g complete proof wasserstein mean case order identifiability appear guarantee impose condition ii tuple k xx possibility finite hyperplane tx j tu tx first choose finite distinct hence hyperplane tx find hyperplane I j multiply side f e jx jx result tx repeating argument get equivalent rule rewrite follow uv uv uv entail imply identifiable assume contrary modify jacobian matrix possibility complete multivariate x direct check conclusion proof defer assume result ip v
child root th tree path label plain base rademacher class value random think dyadic quantify form norm class small decrease consider parametric sequential covering set cover form nonparametric following rademacher absolute lipschitz however sequential scale yet base contraction give loose regret introduction offset minimax control offset rademacher random value term value notion minimax offset end first sequential rademacher class bind obtain offset rademacher take advantage negative offset rademacher recall conjugate conjugate controlling supremum finite offset rademacher calculation infimum achieve technique beyond lemma cover rather lemma yield bound sequential minimax offset rademacher dimension bound minimax smoothness subset singleton carefully crucially smoothness point fix low offset rademacher match constant long exhibit match constant square loss quantify covering say exist tree depth q value tree lemma entropy rather combinatorial notion closely depend behavior corresponding involve hidden suppose statement statement suppose lemma class exist arm upper ready lemma detail regret assumption growth rate match upper growth combinatorial dimension statement class furthermore factor dependence size devote upper recover absolute loss convex yield match logarithmic properly convexity examine see discussion loss growth cnn n cn pp finite logarithmic factor class bound lower modify growth loss assume check convex third derivative remainder strongly smooth truncated technique universal correct modify minimax function parametric function convex sequential bounded combination function follow pointwise entropy scale scale banach sequential rademacher upper yield forecaster relaxation enjoy specifically ds sp sn problem literature phrase expert possible sublinear regret expert randomize loss picture also define front infimum bound call inexact oracle lead pac yet expert repeat beyond beyond case correct bound discretization bound emphasize obtain equip supremum norm entropy logarithm aggregate procedure net give logarithmic amount aggregation capture indeed conclude slow obtain entropy phenomenon learn concept paper relaxation sequence mapping observe condition one prediction relaxation specifically condition recursive say admissible forecast eq version admissible bound relaxation algorithm relaxation enjoy offset hold rest restrict x tb schema relaxation ty schema proposition closely concrete decrease likewise monotonically decrease happen within admissible base give b outline provide schema set bound regression regret alternatively basis knowing predict thus loss round readily schema design algorithm enjoy bind include limitation partly finding consider space remark distinct exploit aggregate unified technique learn algorithmic one algorithmic aggregate aggregate beyond lie pointwise independent cover covariate difficulty arise bound end notably difficulty recognize cover empirical complexity measure behavior characterize optimal understand covering log partition ball radius aggregate combine regret integral similar statement empirical sequential number phase match logarithmic even rademacher phenomenon notice phenomenon given convert statement sequence thus recover technique many statement relaxation provide improper denote distribution regret third last step observe term outside linearity definition eq view jensen step jensen function function pass upper replace subgradient variable jensen fix jensen bound conjugacy pass far bind step q proceed upper claim statement appear except account bad along path q fix sequential sense e construct cover include soft thresholding tv upper denote let specify later write time bind term term bound choose arbitrarily restrict set possible optima depth choice result observe interval suppose sake contradiction depth least must label path clearly order leave easy leave leave complete subtree size tree tree sign bind choose particular expression definition stay close tree obtain low q inequality since free choose delta function suffice bound growth cover balance change next give turn obtain use second derivative first low bound symmetry binomial give point q grant dms research loss loss curvature affect entropy match online regret forecaster enjoy computationally finite linear predict forecaster abstract encounter observe response literature former fall series form base past set probabilistic instead predict well benchmark strategy latter term
however optimization like get beneficial design architecture well optimization technique identical stream combine bottom meaningful representation roughly resemble vector aid matching perform patch two channel feature width height channel compare patch path patch patch identical multiplication patch combination involve computation result pass intractable maximum map globally produce dimensional combination patch respectively backward pass implement cnn good extract level shrink aggregation image refine pool main ingredient layer consist convolution layer refinement map map preserve pass feature map low twice time small input refinement improve result full image bilinear bilinear approach term start use coarse fine scheme iteration full resolution additionally boundary detect boundary smoothness expensive simple bilinear add field variational refinement see frame truth per frame cm ground unlike neural require task ground overview training ground truth train special motion real stem move observer distant capture optical truth dataset ground truth special realistic version clean effect image ground truth magnitude train cnn provide training image retrieve category city landscape cut image multiple background result view per figure generate sample affine relative interpret camera object move second image optical image number position transformation sample adjust distribution supplementary dataset background arbitrarily strategy neural overfitte augmentation online transformation well quick operation gpu augmentation increase image variety flow accordingly field image sigma sample multiplicative channel image additive change use gaussian sigma result fine refinement keep network nine convolutional form relu nonlinearity connect network size size deep layer layer start fourth deep roughly use error error optical flow euclidean cnn modify show descent momentum fix parameter recommend pixel fairly batch divide tackle problem increase iteration overfitte tuning input although optimal dataset factor dataset term type fine network flow tune clean tune use performance define table fine ft table show public well additionally train realistic real network outperform compete cc cccc train train test cpu gpu ft ft ft one ft even average often smoothed solution interesting qualitative result figure raw optical predict two truth figure show visually error net especially region partially refinement projective transformation encounter network fairly additional fine tuning variational probably outperform aside various net interesting thing bad variational realistic set well cm ccccc cpu layers gpu art time cpu leave aside thank alone enough optical fairly pixel tune fairly augmentation answer augmentation allow draw conclusion strength generalize clean motion suggest though training training heart datum though current setup datum become problem discuss pixel pixel ft px explanation increase computational recent train optical realistic affine synthetic optical flow natural accuracy prove capability cnn perform acknowledgment start grant grant cr ec project visualize field use provide color magnitude color intensity motion illustrate code flow vector pixel magnitude pair main independently normalize pair apply pixel foreground view angle image image set type uniformly size deviation transformation translation coefficient angle translation aim roughly match simply gaussians cc family distribution contain precisely gaussian interval overall bernoulli show flow cut pair pixel histogram translation translation observe sampling gaussian accurate filter filter filter structure converge coarse visible filter correlation layer direction magnitude http supplementary gpu capture pixel show life video http frame video fig false h edu van technical technical university de convolutional networks cnn successful computer especially optical cnns paper appropriate cnns capable optical flow train cnn competitive accuracy many field prediction flow optical precise localization involve image representation optical flow fundamentally differ previous application solve cnn capability train end level scale abstraction help find actual predict surprisingly way optical competitive generic help optical flow dataset art optical material trading
experiment involve effective collapse perform particularly sample count subspace integrate assign time take cluster e j denote collapse lda mixture integrate beta sample produce prediction accuracy comparable dataset interpretability subject experiment involve require show performance compare output visually learn feedback important dataset unsupervise depict digit dataset per document dataset value rescale bin picture lda initialize label incorrect iteration use section learn subspace incorporate sample technique depict assign cluster lda accuracy cluster lda depict capture dependencie full acquire dataset comparable produce digit indicate achieve compute measure prototype prototype generate sensitivity introduce learn sensitive within range reasonable verify interpretability perform incorporate require require name six choose dataset subject require incorporated subject allow effect possible balance half subject participant age answer question question accurately break four question representation I top cluster prototype number number prototype run lda initialization ground truth visually identifiable statistically ground truth label manually code expert author analyst produce ground label statistically spend average per spend style produce statistically difference preference insight experiment demonstrate participant degradation p prototype stick water illustrate learn function later show characterize cluster prototype interestingly highlight make one showing absence loop among cluster initialize tend cluster near refine show third digit tend share sparsity prototype set cluster highlight box subspace prototype cluster ingredient case base prototype come prototype set defining show quantitative interpretability neighbor base reasoning historical back intelligence offer topic balance accuracy interpretability predictive mit edu framework prototype bring framework represent cluster simultaneously play important role prototype interpretability preserve subject statistically participant compare art look people make recommendation make amazon instead amazon customer customer ignore medical large patient favor medical example individual numerous reasoning involve fundamental effective strategy decision decision service usually successful human leverage source decision provide decision make intelligence case reasoning approach rely situation solve world fundamentally limit complex fashion dataset discuss prototype cluster powerful neither situation regardless model model feature model bridge human produces verify human subject subspace meaningful important aspect result participant understand dataset compare output people ai reasoning insight learn result solution provide alone maintain complex challenge backward cognitive load case require manually mixture discover distribution intuitive point interpretability reduce per feature proxy interpretability problematic interpretability model present machine unsupervise learn important subspace preserve interpretable interpretability prototype view three prototype interpretable intend third type explain focus neighbor intuitively generate observation important piece relate movie profile movie subspace prototype observation feature indicate indicator generate bernoulli feature describe wherein row discrete outcome length particular g mostly important consider prototype outcome consider irrelevant look generate next large prototype feature take select agree prototype copy prototype within cluster subspace rest mix wherein important piece prototype dirichlet parameterize hyperparameter index feature observation assign allocation begin mixture though necessarily hyperparameter q extend measure modify square set hyperparameter mean hierarchy plain classifier illustrative subspace compose feature color assume ground two subspace shape define cluster face h cm p cm
contrast often architecture dataset range tf rnn term lstm architecture provide benchmark task human labeling release empirical version briefly exist architecture exhaustive list due space survey resource state use management system interactive formalize agent attempt predict resource allow significant progress field though compare neural architecture contain extract twitter al million extend take long context use triple et however public service human stream message room twitter datum believe room closely micro aggregated resource thus investigation traditional two party dataset interaction seek study c c type word pre topic computer restaurant tracking system human hour track twitter post micro generation twitter twitter triple b micro twitter human extract micro generation focus approach attempt leverage development neural notable et rnn initialize denoise tackle twitter idea superior performance retrieval near approach al exploit structure recurrent decoder achieve poor twitter triple translation al encoder decoder one rnn rnn generate study overall highlight potential neural architecture interactive large research system human neural network ai turn corpus refer internet network protocol participant room channel channel channel technical support issue free channel question potential address avoid confusion day simultaneous channel never occur extract user stop problem continue nature constant stream fairly message room extract intend message clear user old er question perhaps comment corpus extract room party message time tuple tuple easy separate rest intend message trivial sometimes locate user correspond word stop false positive order english user intend message match assume message response frame minute presence name recent identify question say duration along far process standard nlp initial multiple people user user treat dataset issue hour rare researcher filter axis word min turn per avg avg per median property corpus crucial architecture characteristic turn show see turn per approximate aside rest corpus process extract pair triple boolean correctly identify create triple contain actual triple elsewhere test example move response set wrong response l well guess get copy format tag part context stochastically use simple maximum context unnecessary length context select short context often break medium long turn task metric language task agent ask response language perform classification improvement classification lead neural architecture benchmark baseline tf memory pre use library tag word category name location lstm architecture full test triple start rd overlap minor issue rest datum word document retrieval put appear elsewhere word appear context appear classification tf response cosine select response return neural allow time state time previous state diagram rnn tie last rnn context context maximum diagram rnns primary current language rnn encoder rnn encode response upon question answer utilize consist rnn tie embedding embedding embedding respective rnn token tune rnn learn generative measure similarity dot convert label frobenius simplicity training response draw elsewhere layer neuron initialize orthogonal initialize optimizer gradient critical rnns rnn architecture change unit unit sentence embedding dependencie lstm unit determine input old retained error feed lstm overcome rnns otherwise layer configuration number neuron optimize rnn use tf rnn lstm model train evaluate ht tf rnn lstm r various recall lstm tf rnn case likely due ability rnn context overcome correctly classify ht context I response n transfer file rank processing training confirm importance training increase research describe construction availability several possibility research rich preliminary rnn lstm selecting obtain lstm several sophisticated separate trying replicate retrieve support subject interesting difficulty control move false response
close peak mode shift magnitude shift computation current position expensive shift mode perform cluster base merging mode neighborhood increase distance mode neighborhood merge experiment image density use locality lsh find point choose implement lsh lsh available lsh hash per picture lsh present indicator cluster kde result kde areas fine tail capture pick far apart htp different ccccc cd cd image breast heart e e diabetes htp accurately approximation mean recognize radial desire may work kde kde proportion present advantage bandwidth cluster kde less lsh approach perform index hausdorff order magnitude acknowledgment department provide part definition remark mean frequently estimator issue single estimator computation sparse establish incoherence radial construction gain look problem proportion mean quantity form rigorously kernel estimation mean statistic context kde kernel reproduce kernel hilbert motivate kernel review concerned word kernel sparsity seek accurately kernel problem kernel prohibitive efficiently regime scalable argue exist slow primary approximate algorithm radial sparse rest review kde definition formulate approximate sample relate work establish incoherence apply rely demonstrate preliminary appear matlab two review feature address sparse problem although kde ingredient plug anomaly detection detector commonly employ shift kde reach hill test kernel evaluation undesirable prohibitive kernel evaluation shift kde numerous demonstrate kde symmetric definite square semidefinite hilbert reproducing think closed span rkh property reproduce mean rkh mapping mapping derive fact kernel permit distribution hilbert size base n reproduce product embedding mean pairwise inner product evaluation computational approximate sample motivate satisfied density estimation say estimation equivalence view def common radial kernel form q student normalize depend illustrate indeed symmetric definite rkh radial property x x write bandwidth parameter although closed need generality consider abstract h k z form later develop dictionary element hold abstract sparse hard effort time overview matching pursuit originally pursuit atom capture magnitude inner atom iteratively portion note require quantity nz undesirable pursuit cluster point minimize cumulative heavily kde think simpler pairwise computational task speed sum make kernel question yield efficient point effectively reduce evaluation effort rapidly quantity query case construction kde seem concentrate effort computation problem problem calculation discrepancy original consist approximate complete nystr nystr compose detail connect nystr om scheme tailor nystr one algebraic propose coherence base context main atom quantify large absolute atom complete involve coherence however minimization contribution summary contribution sparse error novel radial solve center center approximate running kde important computational subsequent calculation perform address particular automatically select demonstrate kde proportion shift kde separate part find finding value nonzero index pose eq inner optimization unconstraine quadratic z rewrite approximate briefly highlight om ik q nk express nystr nystr nystr om nystr om norm space commonly frobenius solution strategy find orthogonal cauchy confirm existence equality reach basis minimization think incoherence establish q begin inequality due since p z approximate mean radial strictly definition note radial g j minimize translate product pose let ik approximation describe linear algorithm htp choose first choose element output base find determine coefficient burden freedom time possible kernel I example symmetric definite kernel semi conjugate approach advantage evident tolerance value stop record compute overcome indicator let use increase element algorithm max ok max avoid te complexity make simplex k k alternatively account constraint negativity solve complete sparse specific machine task reduction class estimation finally explore result description probability visualization consist create similarity similarity among reduction case embedding notice kde kl divergence obtain start difference distribution symmetric distribution empirical accord construct matrix induce visualize distribution need distance yield computation assume sample inspire flow cancer patient range analysis translation procedure evaluation respect factor small large run htp htp computation total htp range kde kl divergence kde function project indicate construct kl kl two half divergence criterion result f resulting embed also furthermore subsequent negligible htp
degree vertex connect objective separable connect component rest form connect graph sequence precede walk begin circuit edge visit thm decompose algorithm operate penalty consequence representation turn decompose rewrite grouping together slack visit slack vertex admm routine edge along first concrete case crucially fuse efficient combination make efficient dominating reach primarily influence question experiment type strategy approach half first connect walk every description odd degree graph graph decomposition appeal find intensive decomposition phase fix instance little extra time yield open dual quick well fairly maximally leverage solver trivial subgraph add unlikely tractable graph potential instance add set remain generate split merging creating result unlike provably subgraph set ts clear decomposition strategy vs impact build delay present motivate simple node immediate much short conversely grid row dash lr pseudo trial display pseudo impact large speedup substantial algorithm different goal mean high path step require randomly graph preprocesse massive dna algorithm ten evaluate unable adequate slack repeatedly laplacian form benchmark conjugate acceleration hierarchy add subsequent highly efficient available straight spatial example stack proximal split solver benchmark flow idea strategy compare naive treat demonstrate median heuristic heuristic decrease long decomposition update programming package report trial trial create precision vary acceleration exception spatially real world synthetic create square grid adjacency node immediate increase however network often see grid uniformly structure many scientific application computer vision notable exception well understand dna acoustic collection adjacency exhibit dna regularity simulation dna result benchmark example strategy perform perform overall structure significantly iterative preprocessing section light cause performance similar balance aid highly structure grid median balance size example balanced cost short median length dna balance note recursively break half length directly convergence pseudo grid versus random grid approach log random graph little strategy underlie fast note simple row competitive specialize max flow work none solve fuse well empirically method make immediately drop replacement solver currently use regard approach property decompose contain intuitive balanced answering leverage admm convergence work heuristic decompose graph row proximal generic lead adapt lasso lasso nevertheless recover htb tolerance smoothed truncation smooth stepsize determine via backtrack line search practice saddle grid fuse lasso recover substantially smoothed particularly background axiom conclusion theorem exercise author g solve operate predefine technique lasso close flexible set
nk disjoint metric metric cluster representative cluster distance center center minimize distance cluster metric give location optimally problem assign np polynomially approximate kp ip distances location problem calculate new distance ki n dimension become particular proximity query discrimination jj large curse applicability distance space dimension high discrimination plan metric case center continuous previous center power running dimension space prove section result comparison purpose x median necessarily define weighted median median minimizer consider weight rational real rational relax assignment add eq point center square serve seminal paper lagrangian eq zero add k result minimum recurrence together four area inverse fuzzy principle work estimate point function far lie convex center interpolation harmonic harmonic analysis specie harmonic confirm hundred specie importance harmonic establish zhang harmonic distance axiom contour original hard fuzzy simultaneously strength c compute update add center calculate convex combination control assignment assignment classical mean square widely dimensional g choice contrast justify idea classical membership distance decrease well necessary condition probability seem analogy may circuit parallel current circuit optimization well define analog circuit equal instead separable center separate problem cluster serve relax produce membership normalize high exponent close assignment element assignment numerical experience iteration iteratively use exponent update increment assignment w increment arbitrary center mm distance assignment center operation iteration calculate dimensional time assignment center great b iteration close well stop bind iteration c generalization center update center subsequence decrease center minimize center probability exponent increase short distance become probable synthetic cluster datum consist take use rule step record percentage misclassification vice table give misclassification test mean mean mean algorithm table r algorithm mean mean algorithm misclassification algorithm mean datum use mean show r
formulate dual polynomial vanishe vanishe let optimally primal hold z point development inspire formulate dual lp sdp research law space allow g optimal differential atomic solving sdp dual algorithm assertion sdp hierarchy compact translate yield feasible polynomial optimal satisfie place similarly polynomial context er exploit paper er theorem also negative write square polynomial hierarchy priori small speak conjecture theorem standard convex close cone compact cone let matrix semidefinite set kx kx kx jt jj xt precisely moment measure interior differently imply invoke conclude sequence jt coefficient wise prove mm mm mm remark domain semidefinite sdp multi frame algebraic standard focus relaxation gram matrix representation exact primal super square overcome notably super great domain algebraic keyword sign total domain problem pass filter application image theoretical baseline want reconstruct equation enjoy affine study ten super idea property restrict isometry property stability reconstruction lead knowledge super early paper super appear matter invert discrete fourier separation spike inversion apply negative entropy role non negativity regard separation clear understanding role measure subset measure frame point member concentrate deal positivity matrix select member similar quantitative size solution evaluation point family function chebyshev moment negativity one equation frame assumption I unique solution minimize total norm super differ dramatically analysis sensing compressed sense recover super differ formulation see reconstruct e line point paper compress sense exist line continuous prescribed frame super compressed problem go infinity analogy program variation super property primal program polynomial phase issue super resolution frame important discrete satisfy condition fundamental paper huge indeed sharp dual phase support henceforth dual compressed deal resolution frame finitely last point severe practice matter deal formulation extend algebraic domain cope parametrization author infinitely relaxation involve proved solution guarantee recovery inaccurate prediction robustness recover solution often equivalently gram toeplitz representation constraint negativity domain rely programming sdp globally nonnegative er toeplitz toeplitz sdp form relaxed sdp version great sphere expand super resolution implementation frame semi operate tackle norm parametrization decade super contrary approach focus primal truncation infinite minimization univariate sdp standard optimality primal rank extract algebra optimality attain point attain taking polynomial measure er consequence dimension limitation sake knowledge e solve sdp moment measure numerical relative represent polynomial indeed polynomial attain attain red measure super resolution spike deconvolution harmonic investigate spike perturb use real appear naturally formulation sake sphere spherical support purpose consist variate degree relaxation pc solution moment use algebra indeed attain value prescribe prescribe blue appear greater close algebraic whose degree assume compact algebraic sufficiently linearly family index banach sup norm sign banach topological equip supremum disjoint measurable respective nonnegative nonnegative respectively eq decomposition q rewrite lp cone eq mx problem maximization lp gap close compact bound weak banach subsequence weakly element ir dual lp supremum feasibility lp close contain lp attain exist duality condition atomic atomic equality equality counterpart result discrete support converge hierarchy dimensional programming extract key construction paragraph semi algebraic define conversely value identity say represent cone sequel approximate cone degree kp functional acting kp p symmetric gp p construction notation finite semidefinite resp resp index primal lp primal moment
space output message advantage specify distribution characteristic tractable fourier novel distribution approach give fast update quadratic advantage approach query sampler pair prediction update regressor informative forest proceed expectation propagation importance training message operator overview kernel message message embed rkh message describe three artificial regression four demonstrate learner correctly regression incoming change implement distribution j form deal black box paper variable give thus assumption stochastically value expectation propagation ep iterative procedure belief pass message factor see project distribution message message send neighboring variable q numerator evaluate leibler divergence non distribution factor analytic often require approximation expensive integration box fully nonparametric technique alternative integration member qx compute statistic numerator base carlo forward draw support parameter projection could use algorithm suffer sample reliable running variance computational message map tuple incoming fm f ep employ forest message uncertainty function indicate exceed threshold require message importance cf new f via look forest leave mechanism problematic uncertainty uncertainty forest consideration demonstrate heuristic forest inaccurate move initial forest newly initial drift leaf split chain bad storage training finally factor specify tree traversal notably potential incoming expert forward sampling present employ term prediction forest traversal representative tree traversal importance point cost tree forest tree internal involve traversal one cost leaf make tree representative use forest gamma incoming message regressor typically income message around mode incoming characterize thousand leaf propose operator tuple message apply message incoming purpose belief propagation message prediction regression whose inner product directly tuple random feature evaluation incoming message tuple variable tuple tuple reproduce tuple us respective distribution embed product tuple product q kx alternative tuple message embedding dot product bad supplementary incoming message would employ suit grow set even moderately sized prohibitive map yet close kernel transform compute kx yx translation invariant inverse inverse fourier transform expectation approximate monte average frequency follow similar stage fourier expand exponent embedding invariant kx write mapping show kernel feature complex vanish invariance analogous material way store need gaussian input transform translation outer outer x nn incoming node sufficient output message notation feature tuple via estimate message close kernel estimate prior capture importance treat sufficient statistic separate treat incoming arrive sequentially n express covariance cost function uncertain make used evaluate two gamma second capable quality mapping message third fourth ep approximate compound gamma experiment reliably quickly adapt shift message encounter regression problem infer interface default otherwise numerator calculate straightforwardly ht f logistic factor dirac delta fig deterministic incoming p dot sample dataset run iteration message pair ep infer logistic message predict leave cross observe significant improvement feature z q I truth belief numerator prediction histogram kl error message kl evident relationship incoming message ground percentile supplementary ep uncertainty base variance forest kl prediction fix incoming message operator set crucial estimate part message use forward incoming message incoming message plot evaluate pass operator random forest move densely smoothly high set robust importance key forest e uncertainty nearby point check reliable divergence experiment approximate ep loop logistic generation sequentially present generate keep scenario common practice observation share number mini operator initial batch initial batch operator estimation simplicity accord heuristic full heuristic supplementary material output variance uncertainty log predictive setup importance sampler net implementation show predictive predicting upon dash problem shown observe drop stable incoming uncertainty display collect incoming message drop ep infer engine lead repeat zero message show classification error posterior generate infer net logistic factor infer good alternative importance sampler compound tail gaussian follow compound gamma shape infer normally distribute infer two gamma factor specify compound one directly default rely quadrature sequentially parameter fig infer run infer fig plot show agreement net pass factor indicate change income subsequently present learner uci repository learn message point point minibatch parameter early show fig essential rapid first message diverse sharp follow steady indicator message adapt learn incoming message place computationally demand uncertainty efficiently additional train mapping mapping far novel topic current hyperparameter selection adapt anonymous constructive financial pass propagation supplementary kernel inner mean outer embedding depend merely validation likely yield heuristic mini l incoming mini batch tuple let product income tuple tuple message subscript consider tuple kernel tuple message kx lk r x l r define I review relevant random contain invariant correspond infinite feature map approximate feature equality feature cost translation feature generate invariant properly transform complex exponential cosine approximation transform dx suggest extend product draw incoming message message kernel variance kernel unit width message incoming treatment reduce familiar euclidean random straightforwardly mean translation product sr feature consider vanish immediately w rw rw I I rw definition rd c translation draw compute sr equality expectation analytic kx randomly feature trial randomness trial entry wise define tuple one income rkhs kernel income
mod arithmetic acceptable network undirecte recover sign choose one negative node true get consistent reconstruct show reduce start completely indistinguishable equation plausible trivial identical water economic biology overall expensive singular value reconstruction entry acceptable structure compatible network realization need steady state safe steady realize steady causality determine direction every edge plausible acknowledgment dr constructive department chemical institute solve reconstruct network steady column state capture variable comprise incidence estimate fundamental pca circuit time illustrate water flow datum key pca graph identification many broadly effectively predict explain improve capability lot direct identification structure incorporate structural stage open technique reconstruct network steady polynomial structure largely process include energy balance simple write describe accumulation mass term reaction action hence accordance topology connectivity write balance interesting possible topology note connectivity steady evolution insight redundancy centrality understand reconstruct wide infer connectivity flow water relevant include topology measurement involve fit appropriate time model connectivity coefficient however steady rotation ambiguity steady series method ambiguity enable reconstruct topology closely distinct measurement iii realize limitation steady wish matrix steady row full network draw brief algebraic pca fundamental limitation pose transform describe recover realize limitation steady reconstructing involve structure approximate different equivalent relationship network thereby enable look equation cut set tree minimal possible set linear overall polynomial arithmetic graph loop multiple allow incidence structure trivial verify steady essentially structure column entry enter rank working network terminate incidence entire contain represent particularly deal example external assume connect external entry idea illustrate exclude environment sum circuit fundamental tree construct circuit say fundamental since circuit represent fundamental one matrix analogous fundamental circuit cut far span mind fundamental circuit cut matrix characterize ambiguity later mod trivial cut circuit reader book singular widely multivariate statistical provide matrix variable assume suffer rotation ambiguity datum stack corrupt general scale co scale matrix orthonormal singular value orthonormal eigenvector remain entry root eigenvalue principal sort decrease order matrix require computation objective suffice store hyperplane hyperplane hyperplane original recover hyperplane hyperplane due dimensionality pc retain absence criterion percentage idea well interested sub admissible sub find readily retain pc rotation ambiguity multiply invertible row incidence wish recover impose structural case pca orthogonal basis adaptively input indeed develop total square problem shall exploit constraint structural recover denote get denote order option upon e approximate like show pca step flow flow label knowledge network flow equation independent table true add flow computing flow diagonal co sensor apply value close lie constraint assume bad svd specialize remarkably obtain partially clearly true element space adaptively row subspace matrix vector constraint angle angle space span example angle project row sum wise difference projection comparison frobenius calculate experience clearly indicate constraint especially criteria another method experience motivation procedure purpose flow partition completely partitioning set invertible admissible since rotation invertible obtain eq due useful interpretation come correspond act incidence constraint construct discuss forward q fact exactly make thus reconstruct span equivalently old graph circuit discuss former every full pca gauss arithmetic unlike multiply straight forward must permutation column always identity order equivalently get suffer row column know round
methodology context organize health context methodology dedicated engine acquisition equip sensor physical pressure speed measure instance quantity analyze engine anomaly detect diagnostic send engine depend monitor sign engine status methodology introduce engine health experience see survey concrete early sign suitable temperature variation value result variation behavior typical computational remove practice measurement transformation generally record measurement instant record software recognition automatically reproduce segmentation indicator indicator learn actual indicator statistically vector signature identify origin generally specialize mainly diagnostic currently diagnostic incoming information difficulty sign failure discover perfectly decision level furthermore false lead least operator failure issue potential failure long health monitoring automate precise optimally drastically reduce operational event partly current box understand gray partially reasoning help propose integrated decision design indicator present propose methodology engine health focus health event datum temporal engine whose sampling individual record say sensor expert series turn high dimensional outline engine dissimilarity observe quantify anomaly transform anomaly anomaly detect somewhat characterize interpretation feature know major informative interpretation minor failure guarantee numerous variant indicator explain approach field experience feedback coverage monitoring problem indicator decide engine anomaly responsible construction typical univariate anomaly display world identify variance figure case instant roughly window signal fix anomaly detect change illustrate multivariate shift anomaly delay design generally explicitly test aggregation technique calculation indicator variation detector compatible expert expert quantity early normally student test coverage time shift compare summary expert window case expert rough idea inclusion recommendation test possible indicator indicator generally value insufficient training choose indicator rejection hypothesis case point reliable difficulty balance specificity anomaly detector order difficulty indicator test anomaly number nan period reject turn parametric knowledge explore range numerous possible indicator link specific easy temporal apply e classification class possible discriminate anomaly engine explain gray paper choose forest adapt binary indicator high robust performance interpretable cart measure indicator bayes classifier also independence decision naive bayes easy understand thank probability reference finally hundred indicator important anomaly redundancy appear unlike feature projection interpretable expense operator therefore detection technique redundancy maximum excellent dimensional difficult sign sign expert anomaly whose fix look huge amount propose production real world section begin simulate present accord univariate time time e point use time generative purpose actual could potential say follow slightly case noise may seem goal paper evaluate methodology choose distribution test plan associate test observation sample anomalous precisely anomalous length choose anomaly model figure type shift I change parameter change change slope case shift slope sample ij ij procedure correspond anomaly type anomaly shift c explain slide window shift center different position create indicator precisely length series signal window sort position conduct half extract series test notice parametric mean parametric distribution hypothesis indicator significance window window length different level test fix transform window extract level give whether nan hypothesis reject lead binary turn raw indicator produce simple one window obtain way binary classifier indicator process window consecutive window namely binary compute equal window consecutive nevertheless window window experiment derive last indicator indicator value derive indicator length binary indicator addition expert smoothed signal indicator indicator explain several smoothing indicator note used indicator illustrate possibility see summary practice indicator cover anomaly test kolmogorov test average balanced signal proportion full keep classification percentage prediction regardless fitting subset indeed observation constitute prediction observation aggregated decision classification detail confusion insight indicator curse acc acc acc table report forest indicator forest neither curse fitting report bayes performance one confusion perfectly confirm adequate already anomaly variance slope anomaly slope forest satisfactory human operator operate box indicator interpretation review addition bayes significantly low forest one favor forward acceptable performance notice add never remove redundancy issue indicator random forest naive evaluate classification accuracy subset black accuracy bag bootstrap estimate classification indicator give classification accuracy white detail summarize indicator random indicator indicator test tend estimate performance procedure tuning procedure datum confirm moreover reach performance forest set performance naive slice natural classifier observe conditionally estimation practice indicator performance respect accuracy jump indicator leverage efficiency bayes meanwhile detail indicator case difficulty shift decrease classification shift trend indicator maximize indicator maximal indicator many classification naive classifier binary indicators acc acc performance good one indicator show probability get table indicator indicator positive naive variance contrary easily interpretable human htbp ccccc type change variance f f ks paper introduce diagnostic methodology health monitoring expert automatic build expert parametric anomaly plausible hundred cover space aggregation classifier diagnostic binary technique useful indicator allow human operator understand automatic classifier base methodology expert model parametric anomaly methodology
tensor tucker possess unique community tensor hypergraph conditional membership membership identifiable original assume resource community learn approach pure natural expect assumption consist project top rank connectivity resource detect tensor decompose efficiently community pure prove recover applicability dirichlet tight community resource tag resource identify pure construct star tensor use pure pure resource triplet tag tuple tensor require membership carefully moment accurately recover perturbation analyze test inequality connectivity connectivity community impose set strong employ exponential form get regime efficient guarantee community social establish success size scale correlation intuitive make act require constant mix hypergraph graph decay bound body detection popular cluster convex handle membership belong approach community graph world times art star subgraphs star subgraph leave triplet dirichlet membership modify star tensor cp extend beyond distribution star count decomposition tucker learn general distribution beyond membership hypergraph membership learn relationship hyper membership incorporate mixed community densely pure un normalize membership limitation present normalize community membership model node lin towards co cluster co belong community consider community guarantee wang discover mostly modularity provide anchor west communities anchor outer anchor west ml machine outer sep dash corner white label north ml draw corner green fit user south corners north fill fill name anchor west anchor name ml anchor theory guarantee draw thick corner fit north tag green thick west green thick west joint thick thick white thick color blue east color thick node tag occur tag resource tensor denote resource soon become edge hide community community belong community membership similarly provide hyper user tag resource membership community tag resource belong tend resource mainly comprise tag dependent contextual category resource intuition formalize denote community membership tag resource similarly membership resource tag user select tag draw community membership triplet basis u hyper group membership resource resource tag resource tag resource order happen explain eqn resource resource comprise tag resource resource tag select resource context resource convenient model resource tag resource dependent resource resource hypergraph resource edge community membership contain conditionally community membership context resource dependency beyond membership community connectivity give user resource tag unknown community look resource tag topic draw community fraction pure I community stack operator reverse denote hadamard restriction subspace singular top paper realization adjacency clustering usual distance show pure resource community resource tag resource resource community resource look hyper pure resource tag tag membership node issue project xx project classify different pure mixed consider project u whether pure role pure pure resource community learn membership node nod subgraph partition subgraph set pure resource node process membership vector thresholde weak value set membership connectivity community membership resource node threshold vector partition resource part u x rr x u role pure node factor constant membership resource community homogeneous merely remove hence resource resource separation satisfy intuitively imply enough connectivity connectivity across test need intuitive role act tensor alternatively intuitive sparse membership estimate hypergraph recovery membership community tag reconstruction community user tag appendix row membership node guarantee via assume case dirichlet represent guarantee thresholding norm well obtain error guarantee case rao define kronecker vector rao eq rao products kronecker number rao preserve recall user community resource tag tag simple connectivity star membership u r entry vector count particular relationship detail column correspond mixed node big succeed identify node correctness moment full learn community form column test succeed moment perturbation analysis outline define exact upon divide part commonly refer perturbation subspace perturbation begin perturbation analysis define perturbation appendix dominate perturbation subspace perturbation success threshold q node pass pass correctly detect pure eigen pass pure I eigen error control star construct pass novel probabilistic approach modeling propose detect present realistic constraint impose system scalable tractable draw parametric note dirichlet membership limit membership impose fraction resource single expect practice community assumption learn spectral consider note extend award award nf supported award thank detailed discussion extensive visit aa microsoft new membership model acknowledge bernoulli depend contextual variable r chain establish community thus take definition star prop community pure modification connectivity learn membership community resource community connectivity tag resource community pure resource w ab c nk k eigenvector estimate community initialization eigenvalue compute eigenvalue denote entry let denote column respectively absolute proof guarantee without need call combinatorial entry conditionally applicable term dominate product assumption sufficiently lemma decay moment every randomly partition pass test recall line hence q combine pass almost eqn make orthogonal act guarantee perturbation perturbation substitute condition assumption succeed line due hypergraph set w j moment eigenvalue exact subscript whiten impose requirement q initialization dominant error line whiten perturbation whiten whiten perturbation tensor substituting resource dependency reconstruction community vector dominate adjacency random conditionally bernstein zero entry wise indicate hadamard entry wise vector inequality thus u ty enough concentration homogeneous bernstein column apply bernstein p I follow low rao involve
play work paper parameter algorithmic estimation set estimation approach principle apply common panel estimate panel tend infinity purpose implement application author via related consideration reader point call correspondingly adjacent correspond adjacent node correspondingly four connect e path nod r short path accordingly r obviously disjoint quantify later formulate actual spatio temporal rectangular domain e integer noise may interpret field lattice random impose fourth moment non empty assume assume eq set partition noisy statistical non temporal mention time infinity rectangular choose already extended recall partition change quantify change change formally convenient reflect terminology brevity motivation extension think digital array sensor sequence represent light intensity e affect noise article establish cf aim whole sequence record assume across simply rely average obtain recall consistency simulation show moderate assume change scenario observation common change slice set series interpret panel horizon original sub change situation illustrate whenever fit restrict consideration weight q usual attain make sub slice theoretical normalize slice dependent identically cf notice model assumption rely behaviour slice fulfil consistently optimal cf slice overlap slice vertical slice sub slice r dr resemble rd think sub slice vertical counterpart treat series assume slice slice aggregate context w slice restrict admissible require reason step identify induce straightforward critical point identify boundary tend one set critical overlap rule w slice overlap choose assume single asymptotically critical tend exclude theorem sub slice estimate tend consecutive slice th intersect slice condition fulfil line violate contain slice restrictive reliable overlap rectangular spatial sensitivity demonstrate vertical slice perform horizontal vertical pool critical point connect asymptotically contain try identify many carry separately class theorem horizontal procedure pp analogous vertical overlap slice indicate select tend rectangular common connect horizontal intersection horizontal vertical horizontal furthermore ratio slice ensure occur contain sub slice consistency change fact estimation procedure parameter domain noise normally consider usual rectangular shaped round shape shape relation due repetition overlap clear preferable inspection g accordance former latter notice figure base spatio large improve small hand table
combine step give similar note h os rearrange os q calculation strictly increase convexity expansion decompose combine case h give section lemma useful first short consider spherical q dominate mf n pn mp n pn basic find fix column q equality write event pn come appropriately event let equality apply apply symmetry claim definition b consider eq mn b mab calculation least combine give claim moreover second q least j jk z jk pe z pe cn j jk jk cc therefore inequality least proof lemma exist claim write q come np l calculate write q integral infinity moreover result eq dy dy dy pp ct ct pn e dy dy appropriately since ty dy tr pn determine utilize enough dominate theorem datum estimate former rare allow successful pca feature remove whose classical pca aggregation method cluster possibility yield limit colored closely interest test whether model limit insight discover transition threshold partition region pca successful threshold post low interest microarray subject two facilitate class assume dimensional standardize useful signal paper study dimensional strength fundamental separate signal rare impossible correctly label region enough successful cluster yield successful statistical except possibility statistical computationally occurrence three objective discuss important fundamental iid model closely spike especially cancer class label quantitie scientific motivate view normally latent factor great interest recent focus selection e grow scientific increasingly directly validate cluster relatively easy validate result real spike model validate find model extend spike direction help bridge interest spike cancer motivate future application result simply possible feature impossible opposite figure right denote aggregation special sa stand tune determine denote estimate optimize cluster aggregate important aggregation sa mean target case np pca flexible wang particular gene microarray adapt pca select use wise q post consist q skip selection step need effort wang propose screen however pca moderately sparse yes yes invoke weak contrast vector eq mass drive asymptotic fix mostly modern large really extend permutation take performance eq respective fix achievable aggregation cluster tractable tractable eq fix consider model sa q pca successful selection column wise score region phase counterpart boundary separate possibility boundary computationally boundary see limit tight match remark monotonicity monotone le new column column become see hard monotone le comparison colored address sure moment conjecture tight fact mention normal spike tight aforementione theorem challenge fall boundary interesting never flexible clustering wang investigate challenging fall heart screen either trivial range either screen fail signal strong screening calibration slightly calibration screening introduce let transition pca pca singular otherwise theorem random matrix independent entry deal entry conjecture tune version post column screen screen tune design cluster reason unclear white panel impossible screening cluster possible version replace colored matrix generic vary occurrence occurrence color continue theorem prove construct current view add inference hard setting perspective independent row column correlate section bound color replace pca desirable way attack plug estimate deviation say statistic moment discussion far test support selector two fix interior region limit goal global hypothesis alternative model space fix interior region type type ii procedure test limit see figure statistical strength need signal interestingly phase segment statistical limit recovery address recovery limit moderate argument test bottom right panel relatively raise investigate idea section much broad setting error error consist tumor class well study report difficulty apply modification mx mx normalize j post selection leading mean well different feature pca work choice determine fdr control fdr ideal classical skip work suggest associate sample classical apply suggest pca address ht ccccc method error study signal strength separate tractable bind spike scientific perspective scope different model phase phenomenon color dependency among column propose first reduce dimension g idea penalization approach highly reveal interesting preferable simultaneously see recognize cluster recovery testing signal insight case easier rise two cluster recall statistic signal recovery randomness think vector reason ham low problem prove upper signal recovery sa recovery pca study study generic constant normal independent sample q theorem convenient grouping associate together statement theorems theorem follow statement hold aggregation consider sa op sa aggregation sign sa sa pca computationally concerned op bound op claim elementary statistic direct lemma follow restrict restrict claim continue hold adapt therefore item b define sa sa op op negligible hamming fix respective b suffice realize nonzero replace zero b z n l range solve write b op elementary solve gs combine prove ns goal aggregation test aggregation third test recall idea sort hc p sa design unknown complicated newly procedure measure sum curve lower consider see either hypothesis fix testing problem sa sa testing consider hc similarly way test fixing theorem show aggregation sa sa hc omit claim consider elementary sa op op sa n sa versus equal type testing f suffice short drop calculus denote leave three triangle recognize leave side nothing else affinity x hellinger note h p x basic algebra p time x iid entries elementary compare hypothesis joint pearson fundamental follow remain simplicity hx fa equal hx fa df r two connection ii error desire drop let law calculus e impossible signal proof bind write j fs p f p eq second condition theorem q respectively conditional sx pa sx f pa pa schwarz sx copy independence equal recall entry geometric write variable jensen rearrange hand k algebra p p x p small algebra follow exponent negative hand combine claim signal etc p pg p pp lot effort investigate theorem hold leave method fold dimension reduction algorithm dimension hundred screen pca dimension implement motivates array great interest edge singular investigate theory primary problem include related focus hypothesis without careful cluster paper signal paper boundary section also reason theory especially overlap surprising idea interesting problem set paper closely recent computationally low plant recovery model recover recover closely transition closely primary theory bernstein random theory property prove let respective fix notation simplicity omit q least eigenvalue
path connect valid transition exist canonical transition function provide condition motivate follow binary string hamming distance construct thing well end attain kk claim obvious distinct lemma canonical path specify first influential influential hasting flip transition form influential covariate q transition backward flip mh form add influential explain variation transition forward flip update mh form replace influential covariate influential case involve double flip transition valid give rise unique set canonical consist state canonical path state path ensemble specify path connect property two remain valid define understand construction path compose state canonical path notation path lemma important canonical path ensemble pair path canonical path eq take reversible satisfie hasting substitute yield suffice inside make event first guarantee state intersection high side combine lemma bind length inequality complete prove subsection separate part therefore constant matrix corner imply claim block inversion claim return define event least quantity concentration function claim somewhat worth auxiliary characterize select transition second depending case intermediate ensure specify expression integrate determination q posterior give normalization display posteriori penalty parameter conversely pseudo conduct base example profile log correspond aic bayesian criterion prior bic regime high regime interpretation equation focus cp transition randomness moreover imply consistency example need bayesian mcmc mix follow argument reversible markov chain spectral upper analyze mix example notation choice theorem equation equation since projection sufficiently tail display combine q denominator combine yield claim transition influential covariate schwarz fact c combine display complete happen index transition write appendix c last add influential current combine two display obtain combine satisfie use prove q guarantee precede definition step argument construction incur replace influential covariate condition matrix step follow assumption precede apply precede eq claim subset simple linear algebra ii claim jj square last optimality since two express ratio since covariate cauchy schwarz projection ii obtain follow display obtain bound choice consider lemma consequently large step I begin eq display obtain q section assumption ex em ccccc california berkeley department electrical department statistic approach variable relatively mild imply rapid mixing truncate rapid hasting time control spectral markov path ensemble greedy algorithm area science lead exceed size address ill pose address impose covariate optimization incorporate yield nonconvex design placing posterior one report subset posterior oppose single provide theoretical understanding moderate scenario mean influential grow spike variable consistency resemble well dimensional evidence conjecture snp genome wide association confirm theoretically widely fitting design match posterior computational efficiency lag central object markov characterize initial algorithm must bound mix dimensionality interest grow polynomially exponentially hope sample reasonable large positive mixing sampler regular parametric converge normal result reach stationary genomic exponentially length dna goal paper metropolis dimensional selection analysis hasting broad regression marginal past analysis include order move neighborhood motivate search contribution bayesian posterior chain rapidly mix product counter illustrate necessary rapidly assume challenge characterize chain posterior complex object distribution physics markov mix probability property characterize chain condition tendency generate even distribution particular motivated examine procedure greedy use decide covariate overall bayesian concentration property benefit rapid markov draw rapid efficiency somewhat theorem bayesian along simulation proof technical detail conclude markov chain analyze mix response recover absolute weight induce selection think indexing covariate index shorthand active associate subset identification understand similarly use submatrix index analogous notation define specific past work mcmc hasting walk update hasting walk local involve neighborhood uniform move stay choice metropolis analyze randomly select new j empty scheme understand model either opposite switch value metropolis describe reversible e detail describe condition reversible vertex distance give chain difficulty sample polynomially mix exponentially variable give turn consequence sufficient hierarchical vector three maximum number control dispersion model covariate make remark first namely analytical simplify realistic would however difference choice show regime posterior behave popular substantially view indicator mixture motivation theoretical lead likelihood essentially proof et covariate impose vanish many sequel additional rapid mixing response linear rough term formalize fix quantify minimal requirement influential consist index relatively namely coefficient zero magnitude let indicator influential without generality zero regime large involve regression component satisfy simple true trivially hyperparameter consistency necessity rapid letting onto low eigenvalue mild requirement selection projection choose projection matrix neither unit information bayesian consideration motivate establish utility hold set sparsity b exist literature e posterior true characterize consistency truncate sparsity analyze hierarchical reader influential concentration assumption require influential covariate magnitude consistency procedure rest due exactly allow nonzero long noting cover regime exclusive possibility snr regime snr snr hyperparameter completely low snr conversely snr characterization snr regime intermediate regime mcmc exhibit appendix satisfy metropolis grow exponentially seem counter intuitive sharp rapid mixing distinction sufficient condition scheme mixing ensure converge difference assumption sparse involve rapid either snr low constant upper theorem characterize mean initialization state though impractical upper understood number iteration require period state iterate iterate mcmc algorithm match characterize intermediate component inequality base statement conduct simulation example correct succeed mix choosing noise case design signal explore behavior boundary simulation snr size set model figure typical log signal receive stationarity iteration prediction behavior nonzero intermediate fail stationarity within follow design poor intermediate signal regime design h b design grey chain initialize perturbation nan model signal true order performance hasting walk initialization perturbation nan choice empirical nan near markov gr scale chain stationarity gr determination span stationarity effectively gr scale within grow linearly covariate little mixing compare sp h sp n sp successful trial gr difference high n difference quantity simulate dataset h sp sp proportion gr probability model compute gr six see selection posterior high find markov probability model receive correspond weak signal regime regime show ensemble exhibit fast weak regime however markov design among suggest characterize chain slowly interesting see characterize fundamental efficient selection leave question open future reveal oppose relate fairly restrictive condition lasso whereas succeed high necessary theory model condition speak receive negligible covariate
remarkably efficiency retrieval slightly art rotation formulate analytical trade relationship code hashing build analytical nearest ann retrieval database recent accurate vision achieve ann vector quantization study ann high quantization high space small subspace representative product case rotation dimension calculation binary hashing technique efficient retrieval recognition transform feature highly favorable task memory lot propose approach vector base hyperplane method hash eigenfunction derive locality hashing nonlinear hashing propose hash ordinary deriving lower compare uniformly kernel base recently spherical nonlinear hash hyper euclidean calculation hashing get high bilinear hashing treat k high fold unable special orthogonal high paper new binary hashing inspire isotropic hash natural analytically develop efficient isotropic produces yield isotropic main previously state art reduction accurate remarkably fast main cost calculation covariance whereas iteration practically fast although measure hashing quantization bit criterion study analytical naturally mainly applicable distribute expand gaussian discuss low hashing extremely efficient along hash quantization original property binary hash hashing method translation transformation assume center discussion sum error generally center calculate deviation straightforwardly binary purpose group group transformation minimize isotropic hashing quantization measure angle rotation transformation range treat eq angle axis axis symmetry probability plot quantization quantization minimize mean quantization trade relationship compatibility depend heavily trade relationship quantization quantization determinant variance covariance propose algorithm minimize invariant subspace rotation subspace dimensional investigate code balanced interpret minimization entropy maximization possible trade quantization minimization hashing transformation substantially encode yield transformation projection encode reduction treat encode treat reduction work gray permutation variance large apply behavior variance multiplication isotropic graph sort state isotropic rotation variance pair transform basic transformation variance isotropic isotropic isotropic rotation matrix fill two isotropic correspond rotation variance isotropic isotropic step sort dimension take rotation set process denote permutation sort basic isotropic rotation follow variance sequentially apply make isotropic isotropic apply completely isotropic finally factorize highly sparse dimension sparse substantially decrease factorize precede however quantization retrieval balance entropy keep accordingly balance hereafter one simple fill fill pairwise rotation isotropic fig pca angle rotation tuning range pca degree balance isotropic definite basic rotation necessary rotation accuracy application fill matrix sparse rotation isotropic pair rotation pca rotation discuss rotation attain retrieval little introduce factor pairwise rotation hash quantization discretization gaussian property function low expansion viewpoint regard approximation datum consider high analytical considerably toy sift evaluating propose sift evaluate top recall hash gaussian query sift sift protocol k database create original sift data k pick dataset matrix sort rotation rotation set keep nearly counterpart transformation make variance completely isotropic isotropic retrieval differ counterpart isotropic lift algorithm hashing use opposite isotropic trade quantization quantization assignment center thus kind hash nonlinear parameter bilinear like like gaussian sphere sharp sharp gaussian sharp log low row artificial gaussian behavior create covariance eigenvalue log sharp retrieval little notable lower completely
function manifold maxout corruption autoencoder understand like problem broad biology understand related brain learn brain equally inferior aspect effort bridge beneficial away biological neuron salient biological dynamic purpose would know spike nn implement biology energy dynamical system appear naturally suit state machine experimentally bridge gap understanding spike system connectivity neural upon instantaneous fire many statistical framework deep rule activity neuron neuron dynamic method require statistical addition activity connect distant assumption activation function period learn demonstrate similar deep specifically feedforward architecture connectivity utilize deep directly digit collect train classification learn supervise reference operate continuously delay signal model system also network operate discrete architecture without delay encode artificial perform static activity hide neuron connection specify activation general consist external source neuron connectivity pair neuron include delay connection neuron receive need learn may calculate set quantity connect activity self temporal terminology vary delay quantity history event dirac integral allow neuron allow spike rate spike simultaneous strength spike one alternatively strength value simultaneous spike description quantitative unchanged aside spike combine value spike vice fix use sensible instantaneous training output neuron activity integrate period spike operate discretized implementation artificial code interpret spike give activity spike occurrence output learn produce strength spike allow output important focus trajectory use connection delay history activity network different finitely particular range perturbation neuron network neuron four event within neuron spikes ii neuron iii supervise spike supervision neuron spike dynamic activation could also make time periodic showing operation learn neuron spike indicate rule dot vertical dotted line indicate suitable hardware dynamical spike within focus hide pattern indicate neuron spike ii spike conjunction modify occur prediction spike occur occur combination event fig series iterate occurrence relation suppose clearly require meaning cause strict equality nature value occurrence approximately require q current supervision unique way fractional number time single necessarily fix eq require alternatively noise spike poisson supervision target divergence stop converge single mean therefore adjust eqs total require choice restriction unsupervise deep feedforward layer successively autoencoder activity autoencoder neuron direction connectivity causality layer cause hide feedforward function neuron activity vertical line dirac delta height dirac delta learn activity begin connectivity act like encode memory layer input specify period supervise activity layer include train neuron see neuron equation neuron machine typically characterize neuron choose commonly use fig approximate modify calculate need product across delay connection advantage though future wide could sum continuous convolution act convolutional modifying include convolution simple piecewise piece modify delay delay spike model delta multiply spike weight update rule eqs choice update implement implement supervision hyperparameter choose rule within useful period specific could sigmoid change rule predict activity learn future dataset collect dynamic vision type camera camera event light intensity upon pixel change upon event camera output coordinate intensity mnist view demonstrate learn mnist handwritten record camera light intensity camera primarily edge digit camera scene illumination result record across digit contain event relate movement digit relatively number handwritten digit last testing pixel map neuron intensity training selection digit event duration five parameter delay width ms neuron classify current digit layer initialize range initial corresponding connection divide classification neuron connect time decrease half repeat value decay period heavily prediction hide idea training demonstrate layer active neuron threshold learnable cpu operation learn number event
pixel model raw range conventional computer optical straightforward vision visual optical video datum constitute common visual attention background visual optical depicted object category model background one robustness critical situation background intensity component pixel mixture take student outlier method introduce user threshold component drawback dirichlet author aforementione drawback affect illumination object different address aforementioned work author contour extract foreground initially utilize background exploit extract foreground unimodal usually capture background uncertainty mixture gaussians predefine number make asymmetric generalize predefine limitation environment propose exploit foreground make foreground object sufficient dynamically handle aforementioned specific pixel account unknown component advantage change address correspond functional inference incorporate adaptation heuristic accumulate strategy framework avoid issue know utilize derive analytical memory inefficient selection make time analytically derive describe discuss present background task experimental formulate framework reason section behind introduction yield estimation gaussian linear superposition component eq satisfy propose modeling infinity recommend less cardinality I define term marginal distribution component introduce transform exploit observe dataset distribution transform q avoid allow yield analytical solution prior denote set latent estimate maximize logarithm purpose coefficient gamma uninformative prefer dirichlet subsequently model order express preference introduction uninformative prior hyperparameter goal approximate evidence bayes equal minimize exploit show section inference distribution factorize expectation initialize formula initialize associated initialization initial value dirichlet gamma value change em algorithm guarantee convex respect least component describe fit mechanism permit adapt rule sample two successfully successfully close component mahalanobis close new mahalanobi stand denote x upper respectively proportion increase upon arrival estimate around new sufficiently updating call associated close observe model approach model create coefficient standard parameter see mix equal value uniform new create remain unchanged new mix normalize present adaptation use observation deviation initially column new fit generate distribution mechanism overview capture history initialize section training frame classify foreground utilize background initially create observe use train classify pixel foreground represent model threshold classify classified foreground dynamically define threshold define use university international european project camera raw widely illumination weather totally capture frame use capture view narrow perspective video camera height testing background method interest pixel wise figure visually observe satisfactory apply contain high precision low recall pixel classify foreground foreground misclassified seem suffer outperform particular score per frame present good term recall score among frame examine regard load em converge require time requirement method applicable em optimization value use logarithm substituting get logarithm factorize
grow researcher try improve diverse evolutionary reinforcement mention extend examine action fitness anti opposite actually simultaneous entity capable difficult exhibit significance publish clear actually scheme algorithmic begin opposite context year thing exist calculate bring ii opposite suppose meaningful mapping contrast look receive unknown justification intelligence instead sort reward fitness etc enable assess true train learn could contrast propose meaningful accord q priori know continuously evolve manner present relationship available another via inference system evolve depend application whereas evolve review verify correctness section evolve system introduce find initialization guess solution name population genetic policy reinforcement random away exist location search considerably time case become direction opposite chance function random opposite learn continue evaluation measure optimality compare fitness reward report past know guess begin calculate stage evaluate differential seem one successful among early work paper opposite type opposite fuzzy complete machine intelligence volume also notion opposite neural survey overview et al provide compact base publish xu type ii paper independently possibility ii et idea deal look action agent take opposite directional opposite instance environment discount accumulate opposite train episode accuracy approximate interact stochastic environment action value take extract relationship opposite term world mlp due computational expense suggest substitute mlp approximation type calculate focus centroid base landscape boundary unknown look help value interpolation type initially fuzzy idea evolve extend add rule one approximate capture online calculate center recursively data one decision make exist point certain threshold close old cluster evolve instead initial update become however fuzzy inference system way fuzzy preference perform type fuzzy propose online fuzzy rule expand iterative pose approach rule use model comprehensive report advance evolve fuzzy system find compatibility play simple ii fuzzy emphasis find quasi datum adjust fuzzy consist work input expert however world extract available fuzzy operate fuzzy general x x nx iy fuzzy function w nx membership represent logical type evolve find provide assume become increase mining representative generation priori type opposite select continue reliability increase approximate design look calculate straightforward diagram calculate opposite diagram understand diagram find opposite figure produce ii course might validate describe training boundary datum process opposite look domain average opposite sometimes scheme necessary emphasize opposite opposite line may employ fuzzy output cluster rule perform fuzzy ii unseen input report superiority type htb initialization determine opposite iii train fuzzy save conduct demonstrate usefulness evolve fuzzy examine verify correctness algorithm know inverse relation reliable testing matlab implement fuzzy matlab number iteration fuzzy enable formulate exception test mining comparison error error opposite clearly exception function run htb difficult enough demonstrate happen system block one base generate accurately approximate figure standard decrease reach observe evolve long due nature evolve step begin point extract result htb online rule respectively htb commonly without test two low error superiority ii ii conjunction whether guess guess optimization generally simultaneously look continue good result depend continue type exist literature propose question benefit column type third reject simultaneous consideration benefit error run column distribution
clearly location directional derivative independent cauchy univariate rewrite straight dimension marginalization occur distribute provide equation coordinate variable integral clarity bivariate deviation step additional require sum truncate write normal cdf acknowledgement research part nsf provide methodology directional formal gradient previous focus directional follow extend additionally accommodate continuous covariate whose directional also surface assess explanatory include explicit hence theory enable occur post proof methodology set illustration point surface adopt log cox relate point pattern species forest cauchy process directional gaussian cox mat ern location conceptual view realization surface finite covariate explain substantial portion response spatial structure unobserve krige enable interpolation regression coefficient unit provide sensitivity expect may vary locally study local study spatial enable across sensitivity vary covariate weak strong across specie vary spatially within specie approach aim derivative address surface give defines directional derivative enable interpolation region post derivative well researcher consider mean covariate accommodate desire assume interest spatially smooth stochastic behavior response associate process derivative explore contribution spatial accommodate model covariate gradient surface joint distribution significant relationship sensible gradient behavior surface marginally surface response surface covariate surface strength accomplish give derivative another introduce directional sensitivity angular discrepancy inferential tool directional derivative process well mat ern function form mat covariance ij g sd e another directional uncorrelated follow dimensional work ern parameterize n overall conditional ij ij equivalently form implement informative prior prior observe distinguish difficult straight forward result direction correspond environmental response value covariate surface effect exclude unconditional surface directional sensitivity multiplicative overall serve directional mention dx sx directional derivative ratio directional directional describe adopt spatially deviation directional directional ratio call cauchy well simplify clarity involve kn gm cauchy scale normal response surface change normal chi degree freedom process isotropic freedom direction function draw fine surface treat parameterization describe package assume ig ex summary ccc mean truth region contour line produce location indicate location gradient estimate since figure gradient predictive direction gradient gradient denote angle angle section provide region suggest direction small towards surface opposite illustrative forest site exhibit range road site focus regard respective location year tree stem treat give location record specie observation region roughly pattern species cox place point likelihood divide realization gaussian elliptical sampling describe bivariate sx analyse difference response unobserved model posterior sample provide fit table fit specie estimate suggest facilitate identifiability fit credible scale intensity intensity value ccc parameter ht ccc intensity gradient us sd sd independent process center vector consider directional ratio specie result surface majority close suggest recall fairly region region directional derivative ratio coefficient namely occur rapidly large intensity absence tree rapid intensity ht surface discrepancy region close suggest surface opposite surface surface pattern support strong relationship pattern quite intensity nearly opposite everywhere relationship highlight weak region methodology perform sensitivity analysis bivariate associate directional derivative multivariate parent parameter
control account relationship create successful treat twitter behavioral agent receive base valuable service user research oppose learn collect collect social action allow cause seem popular examine contribution formalize contextual encourage behavior execute month agent control live evidence reward signal past aggregate result accurate prediction contribution development define control twitter system agent round reward goal reward formalize multi bandit need answer learn entirely explain express character perform like increase action traditional armed bandit assume content try choose learn valuable unlikely model armed observe tweet status update describe feature describe choose receive reward learn decision reward reward predict reward select correspond enforce enough action user lead time hour round first agent request twitter specialized tweet receive twitter hour request information change agent calculate reward begin immediately exploration try uncertain reward multi armed want agent action learn select action action reward execute machine learn twitter random account control account hour collection tweet twitter tweet status extract one hour signal agent agent twitter experiment tweet tweet coefficient align weight number tweet string flow status language filter make divide three uniform signal extract tweet feature encourage status update update mse utilize introduce algorithm tweet agent hour observe tweet change receive make tweet generate hypothesis signal step hour generate new agent ol instance reward weight mse select one month may generate status account statistically average fold demonstrate strategy raise question experimental gain deviation notice offline ordinary result analyze ridge regression elastic support regression method oppose generalize agent prediction mse test remainder plot day collect indicate comparison user user user examine data reward non probably satisfactory action seem sort small divided instance select yield testing take agent mse train datum result predict train sample sort test datum hypothesis finding prediction oppose examine learn specifically value median
pre unsupervised learn steady assess base replica consequently unsupervised field successively implement learn label pre follow formulate simplified analyze process section nontrivial generalization amount datum summarize label dimensional vector function binary vector margin resemble label joint probability datum number follow p w py consisting label vector meaningful become flat dimensional multi dnn simplify aspect actual perceptron assess nontrivial aspect deep reader sketch conduct label maximize structure utilize hide unit redundancy aspect simplify coarse picture term step precisely classify thing architecture training gradient classify newly often employ e adequate computation reach weight employ weight weight randomness dataset characteristic namely derivative kind quantity compute overlap estimate weight follow spin apply replica energy partition calculate replica replica evaluation introduce simplify calculation energy give solve expectation introduce representation multivariate b ab replica solution impose symmetry replica write auxiliary random unit rs saddle point saddle equation rs variance rs bayes saddle disagreement label datum output newly train perceptron generalization quantity combination unsupervise fig logarithm plot show value tb fine error nontrivial nontrivial solution remarkable nontrivial state several decrease point move difficulty vast attain label however become need nearby hand cause value error high pre architecture achieve origin contrast cause drastically incorporate error exponent characterize decrease generalized integral exponent perceptron supervise saddle solution error label achieve difficulty reach multi neural dnn unsupervise noted study verify numerical message reference modern iterative th component weight numerical gradually label tune independent start water phenomenon label fine tuning tune remarkable performance difficulty require good tuning deep reveal technique degradation cause tb combination unsupervise supervised behaviour classifier hybrid unsupervise remarkable increase sense pre deep essential label datum ordinary iteration optimal deep pre crucial reduce initial condition state specialized nontrivial behaviour involve low role label confirm behaviour water phenomena existence reach semi efficiently technique work simplify essence aspect hope author thank discussion
cubic spline convergence differentiable show grey practical side mean fact thin top constant seem quadrature rule good model probabilistic collecting viewpoint principled procedure procedure might size anneal bayesian quadrature benefit parameter quadrature much smooth associate show thin grey leave smoothness grey quickly converge bar qualitatively comparable convergence arise wrong spline statistical calibrate confident inefficient column identify regressor start calibrate quadrature computation show quadrature calibrate question computational quadrature precisely capture inherent require rule arise strongly restrictive situation precisely check assumption perform computation principle formal form code describe test differentiable future design quadrature know quadrature actually incorporate salient information lead tailor numerical well framework tackle quadrature solve determination node optimally clear upon grid quadrature evaluation stand monte carlo additional uncertainty generator uncertainty worth random generator computational overhead pseudo principal eventually disadvantage evaluation technique improve quadrature common discard advantage course robustness quadrature integration star aim recover produce provide datum analytically intractable modal quadrature smooth secondly beyond achievable classic quadrature strictly uncertainty final contribution permit informative select evaluation carlo metropolis quadrature use quadratic node take probabilistic drastically fast estimate plot computational cover projection arbitrary exact cg convergent improving involve one operation produce cg performance projection numerical measure py x condition policy describe collect work exactly cg element stack dirac width perform strict move x ax choice unit positive assign every computation sense course cg derivation also provide something ph particular observe projection quadrature width scale mean numerical method cg member uncertainty estimate design answer equivalence covariance match cg degree freedom multiplication typical runtime construct scalar number cg computational overhead quadrature valuable formulation calibrate helpful relate however require challenging derivation viewpoint extension regressor universal enable distinct expert progress statistic slightly describe blind deconvolution imaging provide task sequence vary noisy truth spatially vary blind deconvolution estimating problem large I run cg iteration less start mean precede rank low rise computational loop cost increase computational equally method problem art conceptually clear inference method rule increase repeatedly xt xt collect crucially xt xt h sp point computation amount efficiently project tractable perhaps interpretation numerical suggest share strong method probabilistic analysis kind application marginalization prior method capture wide ode filter regression notion recently show connection conceptual make family gauss process match thank property implement give new result identify formulation classic numerical highlight general approximate number yx I require measure tractable structure eq e separate encodes class assign explain tractable number basic role describe classic rule collect precede could independent regular grid markov type previous ode rule associate theoretic example case grid quadrature integration opt ode gradient bfgs post est est aforementioned result classic fundamental numerical problem cast posteriori description often quadrature greedy linear scientific tool viewpoint suggest practical formal stable implementation development various scientific hope reader issue motivation scale computation science uncertainty opinion foundation two complementary goal implicit prior hope either algorithm convergence strong example conversely conservative might cost regard statistic ode severe checking bias increase effort secondly reach uncertainty would challenge interpretation may mass function distribution prominent optimization machine subset problem figure motivate classic shape research gaussian role numerical ideally suited parameter calibrate fix reproduce analytic fit elsewhere algorithm computation perhaps flexibility fundamental insight solve new recent distant motivating field adapt interact conceptual environment action choose encode sequence step marginalization fitting prediction area base solution numerical automate give receive perform computation model accumulation unnecessary convergence input rough machine inference rectangle north north edge edge north quadrature north estimation south prediction north sketch collect numerical inherent uncertainty across target effort allow input output turn along message management chain computer dominate error truncation reach sufficient available algorithm uncertainty outcome explicitly classic abstract analysis estimation problem give rise method task cast establish structured posterior calibrate uncertainty interpretation yet rigorously establish lead remain formulation uncertainty thus active effort hierarchical modular computation rgb draw size width sep black rectangle text black thick rectangle draw minimum sep black corollary institute united university united linear algebra equation return uncertainty uncertainty arise hardware much science decision lead management seminal numerical suggest provide benefit probabilistic scientific open framework calculation e potentially diagnosis source theory scientific evidence validate probability language uncertainty quantity mathematical statement make sense assign remove noise deterministic mathematical rest notion randomness quantify arise solely web page deterministic computation os show factor integer statement factorization integer concept uncertainty quantity early introduce deterministic interpretation gauss formulation add generative might assume useful generative add interpretable capacity become area dynamical krige square loss maximizing add gauss elimination conjugate direction regression language uncertainty computation method call arise solely lack intractable quadrature access finitely answer integration optimization proceed iteratively provide run answer place note approach recent growth automate article connect application show numerical probabilistic inference procedure measure arise interpretation improve performance conceptual clarity draw point computational may discuss overlap statistic pose deterministic problem identify degree computation uncertainty bayesian identify degree pose uncertainty differ stochastic quantify uncertainty computation investigation projection randomize framework already analyse create generally structural probabilistic classic naturally analytical runtime frequently similarly magnitude locally mostly criteria termination internal algorithm thus embed usually algorithm diagnostic tool add hoc argue formal framework uniquely suited construction perspective extension chain directly computable readily available computation even exist numerical inference quantity result show make establish various posteriori class observation picture one classic prior member amount
hellinger obtain response measure reliability ratio ik space entropy let denote probability uncertainty large I k h kk consistency denote discrete proportion answer theoretical answer limit maximal e item corresponding item realize response approximate answer represent occur answer answer early entropy analogue alternative setup reliable replace show reliable number alpha less htbp ex theoretic internal consistency demonstrate capture purpose refine large simulation study measure strong plan power alpha thank ever definition mathematical institute technology statistic university propose alternative popular alpha reliability context strictly function entropy response reliability index track alpha advantage great I represent response level neutral agree neutral agree agree crucial alpha internal let tuple represent initially late thousand practitioner like alpha sum item variance item coefficient item depend variance group sum positively increase correlated px version alpha consistency I refer response item tuple alpha fortunately contribution whereby heart categorical define appropriate possibility type many author develop scalable data concept entropy variation measure kullback leibler hellinger name theoretic concept create several measure consider nj jk jk contain relative question probabilistic essentially response specifically imagine counterpart correspond give value agreement student item
frequently occur segment circle radius hence figure frequency locate right clearly number lie real tailor numerical technique reader result occur search real life derivative point flat gradient differentiable equation measure differentiable alternative parameter smoothness reason equation divide distinguish previously distant margin call devoted differentiable define l formalize pair distance great identical formalize equation sure achieve case frequency smooth version pair depict right plot diversity construct task achieve maximum objective maximize frequency zero range overall learning preliminary converge b plot match upper plot match show plot per smooth score function frequency diversity decomposable derivative derivative derivative derivative step detail start length critical hyper step h input rate smoothness k j c l return start direction dynamically square partial order per segment distance line positive line explain accumulate consequence time illustrative overlap however form conduct maximize learn updating maximized diversity minimize left execution worth note diversity crucial preserve diversity experiment figure experiment diversity title maximize diversity cause similar demonstrate plot concave gaussian demonstrate non concavity use eeg generate value plot axis another frequency maxima dataset term frequent sub sequence could also force sufficient demonstrate superior searching naturally translate superiority approximate force qualitative protocol comparison across learn compute length percentile illustrate percentile segment distance way force search distance way pick value drive neutral manner ensure convergence rate segment slide window threshold percentile frequency keep force execute research code htbp lm vs width width width r pt c width pt width lm lm lm lm lm lm lm lm eeg eeg c threshold percentile high display optimal learn denote indicate lm experiment lm frequency plot assess show dominate lm difference lm deviation zero significant difference frequency well alternative claim learn small minute dataset hour intel processor ghz extract file discuss repeat extract audio file channel coefficient illustration measurement original reading force blue extract length method percentile segment distance value series segment plot find frequency totally improvement investigation reveal k translate concrete segment segment ground indicate specify find word match correspond word important accuracy channel sound series contrast art technique principled optimization maximize objective avoid optima combine ascent optimal form center ball segment experimental qualitative search european acknowledge university california university suggestion improve lar schmidt frequent practitioner interpret phenomenon sequence frequently sub find pattern propose demonstrate frequency contrast search able discover pattern occur sequence life dataset find threshold arguably type domain medical financial video monitoring sensor intensity dataset know practitioner inspection infeasible reason expert understand phenomenon diverse source attract considerable research brief repeat current art discovery segment sub sequence candidate entirely treat variable naturally task formalize principled optimization devise derivative high match theoretically superior case search domain discover pattern real match search exactly threshold discover ambiguity initially define occurring stream paper close segment paper frequently close close series sub variation task force segment effort devoted force length stream rely find line geometry repeat sequential beneficial useful behavioral concept pattern term discover probabilistic distance discover furthermore unsupervised searching involve include learn order exploit find search tree try every segment sequence segment symbolic agglomerative scalable approximately resolution scan utilize symbolic version pair wise discovery large mining versus symbolic discovering another alternative graph implement employ detect multidimensional since previously pattern little length reality discover optimal instance addition inspire attract length classification computing principled optimization lead improve quality real measurement slide pattern point generalize number normalize segment count matching iterate slide check interest segment formalism overall sum concept segment match optimality frequency match optimal achieve important constraint equation
stage unify stage propagate clustering infer stage large tree hierarchical cluster insight exploratory motivate unified series infer index region e census region strategy discover dynamic share stream allow model individual framework share price throughout census model house sale census census may house time sale noisy census accounting house index capture census since focus transaction compute trend datum aggregate focus model census market trend simplify house function compose represent house sale price global market refer autoregressive ar census choice observation census eqs illustrate census sale price intrinsic census house discover group idea cluster stream generate school census increase estimation pooling group stream seek mechanism stream relate series house price price neighborhood price intrinsic cluster dynamic census census treat independently intrinsic price within k census block block jointly census challenge task infer discover order census block add matrix factor membership infer membership particular latent load tp ik ia element matrix diagonal define clustering stream cluster infer dp drive allow related domain specification specification first utilize dynamical cluster intrinsic price dynamic cluster dp distribution infinite probability draw base measure weight break break dp produce discrete multiple identical cluster unique integrating stick break membership join cluster chinese restaurant crp census detailed stream dynamic cluster latent specify nonparametric latent census indicator weight cluster cluster indicator serve dirichlet base autoregressive loading variance place prior parameter hyperparameter supplement emission value supplement house sale price intrinsic census transaction month accounting effect census ar marginally dp flexible prior census latent factor induce price specification bayesian house summarize draw stream census cluster loading I box realization abuse notation strategy let k th stick specifically model house sale transaction weight intrinsic conditional census factor crp exchangeability stream marginalization upon stream covariance eq include ar observational variance upon compute condition integrate kalman include detailed supplement census belief crp prior becomes specify kk backward intrinsic common latent process ar conjugacy exist follow derivation loading loading membership derivation supplement conjugacy normal cluster assignment ar coefficient provide supplement conjugacy census hyperparameter provide supplement sample variance detail find supplement hyperparameter sample provide supplement consider sampler reduce explore marginalization induce crp intensive evaluation census kalman aggregate census serial adopt trick collapse dp similar conventional emission order mcmc processor auxiliary auxiliary assign processor prove remain representation processor indicator price auxiliary framework machine mcmc describe supplement sampler evaluation derive simplify exploit specific stream house sale adjust sale price simplified kalman census change runtime census kalman filter utilize identity simplify sufficient take kalman likelihood evaluation supplement validate simulated section stream sale census census respectively generate price eqs intend observe sale price sale house city house sale price sale cluster census run simulated hamming assignment set label demonstrate cluster converge census blue smooth per census demonstrate posterior latent track house level one metric assess repeat sale include well house price order fairly compare regression effect estimate price sale localize index level available hierarchy examine city computable serve resolution house house price house case city include city city index use case city code census census use index predict house price use denote represent mcmc analytic use house table predictive metric root mean error house sale importantly house sale prediction largely regardless notable improvement compare index test improvement break deviation trend global trend measure latent trend improvement decrease percentile reduction tail well capture neighborhood trend cluster enable improve fine scale important capture deviation trend index census improvement l rmse mean rmse median th city code sale lower improvement th percentile tail might expect city go fine city code suffer result index ill construct lrr improvement city code top sale l point sale examine impact overall comparing treat census one intrinsic census embed em summarize table coincide repeat stage improvement index smooth kalman smoother rmse p central truth home proxy mention section index form house fine scale region bayesian treat bayesian census analysis code dp index construct average census dp component code nonparametric significantly high volatility supplement start dp especially align dynamic census global census towards informed index information shrinkage end extremely repeat sale study middle difference case show tailed method particular look cumulative figure tail case hierarchical however code error code baseline code importance approach region c c index fine transaction significant challenge dynamical utilize approach flexible share region multiple shrinkage individual trend region dynamical avoid repeat sale ability track change local market observation sale house classic sale estimate code city may specificity sale describe imagine longitudinal trend house long process autoregressive side information road school cluster induce heterogeneous acknowledgement wang discussion fellowship work award center grant fa series condition consider observation use sufficient observe house sale simplicity kalman multiple derivation calculate obtain census filter filter belong observation observation distribution multivariate distribution follow multivariate distribution observation sufficient statistic kalman effect remove specific removing variable observational observation operation transaction use recursion sale response include provide derivation kalman filter work forward time implement sampler draw backward jointly autoregressive vector multiply likelihood conjugacy summary write rearrange conjugacy ar parallel condition assignment assignment datum give processor machine processor processor within use hasting assign accept reject current processor cluster machine derivation show material stability multiply choose flat tail hyper prior hyper flat tail distribution variance performance examine figure parallel figure city display intrinsic price plot paper raw space trend clarity additionally estimate p home city level kk estimate price trend without adjust trend comparison figure sale time together market subsequent form dp dynamic variance time value evolve construct coarse region mask price market neighborhood census example census house sale observation aim address challenge leverage multiple census discover bayesian nonparametric build enable infer cluster correlated census scalability computation yield level census code basis dataset market united states service statistic economic analysis change important policy individual nature composition period report price necessarily reflect consequently meaningful price sale sale house house sale serve surrogate house house level sale capture intra sale period largely cause composition body extend original repeat sale modification repeat sale home index core medium sale single sale transaction discard sale area repeat sale transaction house detect repeat sale single property precise propose hybrid combine repeat sale sale autoregressive repeat sale sale without need lead code level sale transaction perform relatively large despite house sale aggregate fine neighborhood census region localize sale census sale per month average transaction make stable repeat sale stability limitation local coarse scale average sale house aggregate house home house appeal weighted repeat sale period addition census nature try house house recent homogeneity particular home history transaction may need problem need adjust house house create index fine method indice valuable house introduce census level index although idea finer inform individual house census include detailed sale price house sale census latent sale census similar dynamic unlike model since spatially neighboring census census adjacent dynamic nearby census adjacent house
overcomplete transform present step sparse operator image processing patch image base enforce coherence overlap patch high field bi level heart unconstrained operator stage row simultaneously update gradient sphere rank penalty except integer serve bind solve operator update stage inspire work decomposition sparse scheme among overview drive proportional multidimensional datum I non drive argument inequality argument rademacher offer discuss co co operator learn find operator avoid additional enforce constraint theoretical result several field self contain f distribution minimizer rely complexity introduce define differ absolute supremum definition coincide closed propose empirical rademacher vanish single definition generally encourage predefine possible bound another bound furthermore random base rademacher surprising us rademacher gaussian rademacher bound noting jensen sample briefly constraint achieve goal require matrix structure refer employ operator learn want operator separable operator subset equation pose form concrete require ability realization task eq ready result draw accord sparsity lipschitz finally hold consider variable bind value change vary within empirical state argument detailed bound last ingredient main center q preliminary take care sample within let g previously separable result consider separately r norm apply lemma cf inequality learning expression right separable previous version signal manifold similar q hold separable remain fact product q row due sgd scale geometric discussion connection excess result derive prediction set optimization provide excess empirical average quantify strategy closely discuss complexity corollary via minimizer f subsequent seminal sgd attract solve iteration computation computation involve accordingly assume independent notation denote batch represent index follow geometric optimization classic introduction optimization manifold manifold euclidean euclidean riemannian projection tangent keep notation simple denote riemannian manifold point search follow straight geodesic topic provide update read sgd base iteration terminate f ki ki ki ki ki ki cost batch total iteration ki stopping terminate fall threshold optimization selection typically continuity lipschitz advance author line minimize cost predefine heuristic disadvantage require propose backtracking algorithm separable filter sparsity compute respect batch come average update problem utilize iterate average fulfilled initial step length successively condition cost calculate function ki respective batch via slide window eq denote fulfil include least step goes stop trial proceed filter ki max ki ki b purpose serve norm property enforce operator rank penalty hence training sample weight impact incoherence handle allow comparison scenario numerical demonstrate filter train learn extract filter separable version enforce thus separable weighting constraint fix initialization visualize value iteration dot framework enforce separable structure separability impose separability cost second terminate update filter offer learn structure separable kernel application analysis operator employ inverse imaging task conduct investigate truth measure recovery permutation ambiguity filter absolute possible denote row column deviation I build confusion account permutation ambiguity method confusion accumulate constraint visit end accumulate sum serve measure low pick twice prevent retrieve match filter recover procedure ground choose separable previous predefine co sparsity truth filter response gaussian standard signal original signal approach framework without constraint separable conduct varied ten I ten generate box correspond separable accordingly right horizontal box trial mid dotted indicate achieve truth indicate see separable fast small optimization framework utilize full learn separable filter conjugate generate ten predefine synthetic stop fulfil summarize result sgd converge execution time sgd reach dropout enforce separability operator trial upper filter avg error avg iteration time sgd sgd separable cg operator structure signal processing denoise rather exist denoise separable filter slightly compare scheme filter impose analysis goal analysis operator extract image include goal couple white operator regularizer formulate utilize denote image represent overlap weighting factor operator weight noise reduce great extend competitive please optimize task couple within ball assume utilize prove co bound depend structure impose furthermore allow incorporate separability aspect design average confirm add present separable operator sense compare respect sample benefit characteristic sgd sparse addition bind common draw accord deduce convexity jensen supremum step complexity rademacher achieve like support foundation european project black thm lemma corollary definition electrical tu mail de co co sparse signal class interest response inverse adapt reliable purpose operator separable operator sort response advantage contribution evidence provide operator
rate compare art introduction paper aim develop status svm learn status statistical svms current status assumption status exceed format common include field mention example infection tumor determine exposure tumor order absence tumor difficult failure importance topic advanced analyze status expectation investigate measure version function censor current order decision current version analyze status datum gamma non cox hazard accelerate survival cox hazard proportional exponent combination failure time assume include cox status cox include status suggest algorithm parameter parametric demand assumption reason estimation decade censor neural splitting mostly censor suggest survival svms survival suggest censor insensitive justification propose censor svm censor optimal suggest survival far svm status theoretic study svm censor simulation tool censor development status development censor prove rate current status discuss quadratic svm correspond dependent theoretical sample contain conclude remark code notation throughout briefly risk I triplet z z take failure censor status indicator contain example testing example exposure presence tumor tumor develop censor definition risk space subset risk df quadratic reproduce kernel rkhs characterize rkh decision current identically function failure censor respectively risk incorporate identity r pf respect function convex loss function loss positivity constant I decision current status censor replace c f denominator status knowledge censoring estimate note censor estimation consistency svm construct censor true censor use additional condition bind svm bayes finite sample svm family learning consistent case censor density novel svm censor learn know quadratic nd censor l pf b sample assume censor r bb h proof definition mean consistency survival identifiability limit consistency prove pf bayes decision theorem great choose converge clearly consistency measure sample censor censor estimated sample utilize bound svm rkhs kernel old function mx mx kernel du mm concentration like minimize include setting difference cox reference censor censor censoring estimation matlab library fit cox status r monotone estimate choose choose knot follow suggest knot knot evenly matlab server r toolbox kx exp choose cross find risk risk iteration time censor generate obtain result cox ph failure ph cox cox though svm ph comparable cox especially sample size however cox produce risk superiority reduce grow coincide failure censor variable generate time variable figure superior datum space svm rbf size multi uniformly failure parameter z risk compare size risk significantly superior converge quickly size grow whereas cox produce svm produce smooth conditional covariate generate truncate present rbf case sample svm compare base svm kernel comparable failure status certain appropriate true comparable approach assume form additionally svm dimension conclude remark svm property believe demonstrate current open remain study estimate expectation quantity status censor failure censor covariate consequence assumption third extend censor censor believe censor great interest please file file nd unique svm decision q h f df eq q error pf thus r pf pf l df p r l df pf p df l df df eq lipschitz thus obtain compact exist first n hz I l hz n nk hz nk hz c hz n k remark hz hz b combine obtain pf bn pr sup pf bn pr sup r ph bn bn l pr last l ph cardinality conclude zero nh nh cauchy schwarz e nh show bandwidth proof theorem thus part hold r n
tree method explain section pair query rank relevance partition datum use compare validation score set apply combination ndcg position select ndcg task optimize gain ndcg check ndcg significant gain position put improvement perspective ndcg fraction tree slice available repository histogram create ct infer dimensional value involve cross fold train set every good rf restrict dropout rate furthermore rf require low loss leave lowest comprise leave fraction per per challenge dataset image face select ensemble prediction get main difference rate significant difference skew label random exhibit accuracy task forest accuracy network additive find accurate task notably web motivated add tree propose ensemble create contribute evenly towards accuracy study several direction adaboost simplicity direction algorithm drop contribution tree target drop exist tree conduct author diverse task practice suffer wherein tree iteration tend impact negligible contribution remain performance unseen certain issue explore address employ recently context network novel regression use dataset outperform margin also issue considerable algorithm shown achieve task well make help sensitivity ensemble adaboost iteratively achieve learn add predictor increase model instance boost ensemble tree negligible towards prediction instance algorithm increase make significant make initially call subsequent prediction regression contribution nevertheless propose neural connection therefore rely example successfully case use feature training phase approach employ wherein different novel employ tree employ call dataset show outperform random margin yahoo learn encourage reason improve address balance boost issue impact instance impact employ example data description similar tree ensemble contribution significant contribution contribution ensemble inherent leave tree tree extreme case algorithm row ensemble color gradient leave stand negative start presentation gradient derivative loss formally loss generating every typically every current denote prediction current loss predict add minimize loss make variety early square logistic define ranking task order evaluation ndcg result summation detail discuss gradient style employ employ address leaf multiplying shrinkage observe describe algorithm place gradient ensemble iteration random create create place new ensemble step close predictor drop try introduce drop tree drop ensemble roughly time magnitude tree drop order drop drop added scaling factor
solution set discrete repetition boundary boundary simple real surface nevertheless datum interest lie low manifold embed ask approximate normalize one begin appropriate closeness eq relate close approximately mass approximately problem normalize actually far ignore essential concerned making normalize one scale tradeoff satisfy self achieve pair iv ax like asymptotically may explicitly construct scale logarithm class guarantee choose likelihood necessarily desire low thus self normalize suitable assumption construction achieve characterization conjunction class good prediction whenever dependent result obtain represent taylor mle likelihood vanish remainder time eigenvalue favorable indicate easy whether nearby normalize analog remain progress develop class natural equivalence say respect denote associated finite arise multi identifiability parameterization input precede summarize predictive low entropy normalization relatively likelihood mixture distribution likelihood gap generally associate experimental initial weight label smoothly high introduce normalize addition tradeoff gap kl self relaxed gap quantity gap gap self procedure basis understanding characterize construct self normalize self normalization address question self normalization community provide open else approximately correspond perfectly parametric relaxed accommodate construction involve relate normalization rise suffer exist low existence variance fall quickly distribution self insight translate inherent theoretical answer question apply provide toward complete understanding sketch lem computer division california berkeley berkeley major computation attract learn analyze introduce unnormalized score extremely theoretical work largely distribution normalize prediction difficulty fit validation prediction general include generalized offer flexible tractable modeling expensive machine translation self normalization cluster unity surrogate particular zero ignore paper aim understand normalization appear applicability spread include prediction expect input finite order million input rich nontrivial seem challenge enough good seem much input choose gap practical experience open normalization seek answer question generalize much distribution believe characterization self characterization self unconstraine provably represent self difficulty conclusion survey feed forward softmax turn log proportional sufficiently vocabulary prohibitive probability must class arrange output product along lead limit learn appear practice output provide slightly measure define formalize normalize normalize example either hyperplane solution upper variance nonconvex eq easy normalize distribution normalize motivated robust improper generalize approximately conditional statement either example make inherently involve input carefully informally self large occur instead concerned
hard analyze use perceptron update application adjacent improve combine perceptron lipschitz perceptron perceptron algorithm dimension sim low learn reversible hard also moderately sized practical empirical index sim consider propose extension use minimize solve follow problem everywhere else perform sort shall invoke routine point dependent g motivated pose underlie goal unseen like provide unseen sim learn fit perform learn loss eq euclidean fit monotonic need learn efficient ignore overcome drawback eq alternate ideally proximal size require derivative perform proximal perceptron spirit perceptron unity q keep keep track estimate hold keep tp square estimate transfer take gradient propose sim try learn let q transfer standard squared logit penalty take sim via sim optimize integral solve minimize update step problem calibrate unlike proximal bring challenge design minimize program efficiently satisfy h qp follow derive calibrate interpolation non alternate procedure initialization achieve initialization demonstrate empirical book step approximate solve line step evaluate calibrate loss loss via fix competitive sometimes glm size superior risk hypothesis excess list technical main excess excess logarithmic notational assume satisfied lipschitz inequality convex see replace expect deviation standard deviation upper bounding equation property maxima collection q excess hold run run initialize first since hold claim hoeffding inequality small get dominate output theoretical guarantee get sign uncorrelated bind prediction result analysis standard multi variate predictor scale high setting dependence need batch sample run competitive number refer test real squared test dimensional sim via mac auto dataset additional result supplementary since dataset dataset consider except well encourage poor sim learning introduce sparse parameter index dimensional excess employ novel superior compare hinge plan sparsity general definition though use concern lemma x following immediately let normal eq similarly obtain equality zero result paper x g universal constant least l large deviation similarly concentration inequality q put follow notational shall follow set know max value get try empirical eq mark monotonicity lipschitz square marked positive lipschitz lemma reasoning upper know g run assumption restrictive practice result large class g shall case rademacher establish w hard bind rademacher consider run run hold validation iterate validation
impact merge dissimilarity partition sequence grid dissimilarity merge merging minimize simplify minimal degradation merge good partition without distinction start agglomerative build keep grid dimension partition partition total number partition negligible optimization choose reach agglomerative hierarchy dimension increase thousand cluster grid point cluster modification intuitively evaluate impact representative typical cluster close cluster visualize heat visualize suggest namely additional valuable frequency dimension mutual mutual partition partition grid cell observe excess cell event conversely cell finally either mi nan quantity expect partition jointly explain w positive highlight characterize set former say excess locate cell highlight interaction visualization cell cluster bre exploit add value grid available validate real experiment follow successful clustering find event underlie discover kind result exploitation tool let event define htbp l e e l pattern pattern e average triplet event value value choose various point event pattern htbp vary adjust rand ari evaluate two discover significant pattern per point per average pattern necessary discover increase hold per increase discovered discover increase datum set increase set per lead value lead similar underlie pattern discover cluster detect w per pattern pattern insight valuable pattern bring cluster pattern relate excess conditionally cluster characterize light blue stand interaction noisy event cell whereas rate dimension relate whole hierarchy event organize view interpretable human figure look correspond science research field communication typical branch label communication recurrent publish event significantly notice also research repository many sub field computer cluster event author visualization see red relate diagonal whole period cell indicate almost singular cluster scale group observation make consider cluster former mainly relate research focus htbp field hierarchy ai db reason field mining intelligence recognize field intuition terminal cluster dm thousand author characterize typical trajectory rest characterize db whose xu yu db highlight db author activity semantic yu w respective birth sub discover cluster big picture discover typical characteristic series base measure exploit categorical series cluster distance fr distance datum tuning classify branch attribute attribute al theoretic grid cluster bottleneck ib stem theoretic paradigm ib aim group ib mutual wang build upon ib extend ib two categorical agglomerative multivariate bottleneck construct interact interaction network exist segmentation free method interpretable summary go progress multi instance mining event forecast future event cluster co evolve mainly dedicated approach efficient know build recent relate suggest effective exploratory categorical sequence group dimension event group whole form grid probable posteriori free grid dissimilarity representative interpret find finding illustrate real plan forecasting direction plan customer customer define communication interactive sale call service period cl exploratory temporal dimensional grid model temporal define three model partition discretized event partition cross multivariate nonparametric event dimension distribution along suggest agglomerative characterize real grid discover categorical application mining temporal discovery nature temporal event mining mining classification datum time g explore medical find behavior time life literature dedicate mining annotate mining receive attention discuss dedicate sequence without annotation biological popular suggest methodology exploratory technique propose summary show along highlight event segment whole result propose valuable trade facilitate analyst computing involve methodology expert knowledge exist cluster methodology suggest co sequence event suggest categorical grid interval partition trade robustness grid method exploit derive criterion information measure contribution discuss conclude toy compose event interval cluster toy event find group event event find two regime event opposite htbp variable variable cell grid interval point event point number value resp categorical length order categorical set define three categorical one object point force co grid generic goal event time say time interval notice partition set however simplify incorrect partition grid posteriori explore minimize bayesian trade accuracy cluster interval result grid event cell belong event analytic datum hierarchical model hierarchy mean minimal partition subset way stand grid term number cluster choice group specification specification fourth line stand multinomial resp locally line stand locally intuition behind priori whereas close prefer priori grid high likelihood grid posteriori length description length principle criterion plus code grid combinatorial partition bn exhaustive optimize use greedy
frame decode scoring time encoder scale parameter vary standard recurrent weight train norm constraint norm low development negative adaptive constraint low fine development batch use attention mechanism direction activation top recurrent layer maxout unit generator treat purely attention eqs network convolutional sec decode stop token start network fail produce show fig wide c baseline conv conv convolutional net model achieve competitive convolutional relative smoothing baseline repeat demonstrate repetition also frame begin slightly visually baseline lead repetition sequence irrelevant apply remain fig help proper help aware network inspection middle long baseline repetition within window happen support sequence use alignment obtain force baseline fail narrow constrain aware network wide window resemble condition attention wide aware mechanism modification decode propose novel end speech hybrid attention mechanism combine order position decode recognize much train recognize speech may language model architecture normalization smooth extract feature apply speech instance attention neural improve generation conduct library acknowledge national center cifar focus h h rl plain keep conv universit universit e de universit cifar generator mechanism show translation image generation mechanism need speech recognition reach competitive error per recognition roughly qualitative explanation failure add new long prevent frame recently variety synthesis translation object relevant extend applicability construct introduce applicable recognize sequence speech perspective synthesis attention compare differ sequence distinguish attention architecture capable noisy input attention speech recognition system hybrid system acoustic hmm dictionary multi hmm recently work leave acoustic benefit aware recognition task mechanism select signal produce extraction potentially speech weighted condition rather mostly second exist translation baseline seem entirely competitive reach per quickly adapt track absolute content short inherently modify attention explicitly add input attention weight convolutional significantly well consider convolutional feature training contribution fold present novel purely speech architecture attention propose attention modification frame example recurrent recurrent network stochastically generate often process output attention work context feature overlap audio frame recurrent step generate focus state recurrent neural attention call call generator memory lstm recurrent recurrent activation content base hybrid attention mechanism g score separately normalize score similar equally regardless sequence convolutional element capacity always location attention alignment generator previous mechanism synthesis speech location attention would limitation hybrid natural candidate informally like use previous alignment short select one confusion content mechanism describe matrix extend mechanism make extract previous alignment potential issue normalization sequence likely focus attention frame input less translation ms acoustic frames coin softmax aggregate frame straightforward inverse temperature frame according normalize issue narrow propose mechanism subsequence predefine median alignment frame indeed sense bring replace unbounded softmax eq bound logistic sigmoid speech rnn follow early signal perform
theorem section start bound arm draw ip iw ix choose proportional explicit name armed bandit side observation performance algorithm due arbitrary regret satisfy eq become arbitrary bind corollary probability least union set armed bandit round tn tn tn corresponding learner sequence propose ix estimate drop weight arm proportional follow establish satisfie eq action similarly switch times learner prove besides fix share know provide guarantee rely enforce exploration update guarantee improves previously track regret become turn environment armed bandit observe arm correspond presence arc imply learner observe show learner variant use start outperform arm evaluate ensure compare actual range theorem algorithm rate varied multipli regret game loss deviation multiplier several type outperform setting perform regime performance identical rate eventually get behind notably respective performance around shift mean suggest control respective fix first elementary inequality define notation jensen definition result w let right jensen inequality put equation last proof build arbitrary sequence straightforward modification z let apply sequence argument g combine last fix hoeffding bind remain difference thus combine lemma surely france address arm problem focus performance prove modification intuitive come guarantee modification learner time essential strong without undesirable estimation ix remarkably clean multi framework robustness technique armed bandit probability armed bandit problem formalize round interact pick environment loss subsequently incur observe solely goal literature measure environment irrespective take learner base past action multi armed bandit call loss adversary choose sequence learner round reward formulation guarantee grow know goal regret need sense concerned bounding learner prove actual regret hold hard serious change make analysis confidence guarantee learner repeatedly easy performance loss even focus force grow steady result high guarantee tend perform sake consider expect regret confidence variant conduct bandit conclusion online inferior version hold well plain still demonstrate high confidence reason online strategy conservative nature arguably elegant previous particular ix strategy propose side loss strategy range expert tracking bandit bound know another work prove reward reveal lead tight survey aware game avoid respective analysis coherent previous advantage game organize regret exploration strategy precise concern concentration estimate conduct simple benefit implicit exploration technique principle online learning exponentially forecaster perturb box appropriate estimate challenge construct base observation follow traditionally reward easy call draw one enjoy pseudo bind around loss bound keep fluctuation appropriately compute help keep enable less standard discussion confidence reward paper short ix exploration reward true allow prove high variant multi bandit bandit bandit
gain score phrase mt decoder outperform ir despite ir response ir capture message rank candidate triple rank mt directly pattern message mt mt improve relative ir penalty even learn ir ir list support formulate previous paragraph since ir solely capture mt gram match gain gram match may select inspection reveal good provide highly mt ir ir outperform respective ir baseline improvement mt albeit low space limit diversity mt gain model hand gain hence ir well retain contextual gram match difference observe architecture gain ci mt mt mt mt ii ir evaluation score mt ir preference ranking investigate mix exact gram overlap information hope hope extremely mind walk month tweet send armed nuclear response corpus mt response system short token response set token overall plausible contain common illustrate book response message nonetheless long content response likely example mechanism maintain component topic especially response incorporation extensive context help yield representation outside likely remain formulate neural generation medium generation contextual extraction metric context sensitive consistently independent baseline mt ir albeit drive self improve system work direction room improvement network word message context interesting potentially promising greatly thorough automate thank ray helpful discussion pt sensitive ji universit microsoft usa ai ca ga usa system train end twitter neural architecture address integrate contextual classic allow dynamic generative consistent gain sensitive baseline open domain vast medium twitter generation system construct statistical translation status twitter translate address broadly speak context linguistic physical virtual linguistic ability key building active contextual mt contextual phrase table increase skew rarely statistical approach share intrinsic semantic occur twitter context sensitive generation phrase compactly encode syntactic argue embed transition dependency word alignment present sensitive utilize language past response relevant typical orient modular easily end without require annotation drive train massive application come introduce automate generalize et paradigm shift work traditional apart management generation task response generation track many typical code particular attribute model completely drive code continuous phrase retrieval ir recommendation translation mt lm successfully embed refine rare phrase translation traditionally affect crucial extend language representation function natural language generate work construct topic generate stop word commonly yet overview recurrent architecture extend sentence sentence parameterize w w initialize dimensional vocabulary token token embedding keep process project vocabulary pass proceed recurrence logistic sigmoid recurrence apply softmax activation recurrence back propagation gradient accumulate distinguish linguistic entity message sequence context generation generation condition straightforwardly give encode useful subsequently response comprise multiple modelling range dependency difficult open present next context encode vector single bag word representation length bias recurrent decoder encoder minimize compact representation leave ii decoder encoder encode bag word representation preserve feed propagation row vocabulary distinct decoder follow eq context force useful help information response address issue mapping bag feed represent prior common strategy representation encoder concatenation token computed efficiency burden restrict sentence cover month triple extract select triple frequent appear corpus twitter triple additionally approximately triple scale yield triple score randomly triple response supplement human pairwise challenge automate response generation reasonable extremely diverse describe status reference ir potential response towards multi ranking align future corpus triple avg min ref tuning triple minimum reference cover triple context first ir ir calibrate triple response original message formally triple bag bm response score formula provide diverse plausible candidate within triple human retain result response token response evaluate parameterize typical mt generation system base phrase mt decoder mt include forward translation phrase distortion twitter response translation phrase million filter long select phrase fisher mt decoder specifically distance phrase exact ir feature triple whose match ir response iff dm mt ir traditionally contextual match match match capture
size sep font node f node b e fill node ab ac node node factor node node factor node ef ac ab ef e circle minimum size pt node factor ab node factor ac node node cd ce ef node ac ab cd ce ef terminology graphical model refer graphical intelligence constraint function graphical describe constraint realization software tool describe seminal complete function constraint finite integer rational hard encode soft realization minimum weighted stochastic translate use exact mode stochastic rely firstly multiplication potential joint potential secondly subset identity basic algebra offer product potential function x elimination function ix ix describe count graphical simplicity insensitive order similarly successive task involve sum include marginal associate define variable compute conditional restrict domain probabilistic linearity use evaluation graphical probable mode require normalize counting task elimination elimination artificial intelligence max see solve exploit generic elimination deduce solve counting model operator problem logical elimination logical combination elimination operator include variant describe elimination recall chain variable elimination formally variable elimination key choice ordering notion graphical elimination elimination variable elimination markov hmc hmc define two realization unknown hmc hide specify initial classical hmc ph ph font circle fill gray sep font hmc realization equivalently q realization exponential viterbi maximization likely variable product operator elimination equation potential create maximizing involve section formalize trick elimination property indeed since elimination one task also hmc elimination counting optimize consist expression elimination first involve operator potential use rewrite follow elimination new potential neighboring except connect together clique edge neighbor instance potential right elimination distribution graphical elimination successively single potential value scale circle fill size scale draw fill gray sep sep b c b circle fill gray inner e e involve create vertex call edge log transformation normalize evaluate graphical obtain elimination change require mode bx I take mode elimination keep successive state space entirely complexity eliminate variable complexity elimination successive elimination elimination decide order elimination illustrate lead two subset efficiency elimination illustrate formalize viterbi hmc shape elimination eliminate unique neighbor graphical first eliminate order elimination order vertex one single generally elimination cardinality model possible elimination neighbor graphical associate length connect adjacent create edge vertex clique elimination fill clique equal clique graph clique elimination scope create elimination require storage quickly exceed depend maximal scope view elimination inner vertex vertex c c g draw inner font node vertex bend leave bend c bend bend bend draw fill font node vertex f bend bend bend bend bend bend bend bend b bend lowest achievable perform call discover graph front equal graph graph follow orient orient follow edge go toward elimination game graph least order call tree graph cluster link edge vertex build elimination tree decomposition tree cluster cluster contain cluster contain connect intersection gray inner font node vertex vertex gray sep vertex node vertex label definition induce graph trivial edge vertex induce tree large width decomposition create order equal hmc tree equal establish equivalent task question drive variable elimination graphical variable elimination apply exact equal elimination elimination solution rely computation sub give graphical counting task eliminate elimination rely tree characterization root elimination toward root eliminate cluster leaf cluster close root start elimination leaf parent assign rewrite expression eq eliminate eliminate cluster x elimination continue elimination scope complexity perform elimination tree decomposition elimination vice elimination elimination eliminate easy decomposition order clique identify define maximum span vertex identify combinatorial elimination relate fourier elimination elimination benefit linearity formula elimination repeatedly serial david boolean elimination backward hmm many elimination impact computation gauss elimination h upper vertex insight vertex quality guarantee time allow guarantee reasonable dominate empirically instance algorithm work well though approach broad greedy algorithm elimination elimination optimize select minimum fill edge vertex add fill fill edge call implementation minimum g graph fill heuristic slow approximation practice elimination graph linear heuristic break tie break random done find four minimum fill randomize iterative fill five mrf disagreement linkage linkage benchmark surface scene decomposition vision characteristic minimum fill second respectively maximum tie iterative version htb problem nb nb nb type mrf mrf randomized elimination randomize iterative exploit first branch good trade memory search effort elimination allocate hour gb ram report able two heavily category fast total gb fill amount second cpu measurement cost search solve instance category version option combinatorial solver option benchmark vision benchmark provide fill fill exploit fill order red blue message pass make external structured message extend elimination efficiently marginal mode elimination compute apply conceptually perform parametrization modification potential external tree describe extension elimination marginal elimination leaf function send parent always subgraph handle message unary variable potential marginal unnormalize elimination define root root subtree turn formally tree message leave first leave leave processed send neighbor message jx x process illustration leave mark marked message link follow c function incoming except message message pass way decomposition previously handle cross approach exact computation expensive intensive exponential typical example algebraic exact alternatively belief another message update termination meet return approximation marginal probability however marginal deep message product max exact approximation logarithm algebraic define compute exact mode compute general graphical message pass pass graphical tree structure joint graphical interest marginal read behind parametrization conceptually simple multiply involve preserve graphical suitable pseudo see possibility potential equal structured say fact pair variable advantage calibrate incremental update cyclic exact decomposition intersection parametrization potential message inside cluster joint calibrate agree exploit jensen incremental calibrate locally set parameterized estimate marginal read rule convergent maximize typical schema reweighte linear normalize seminal publish exact loop potential convexity also use deterministic consistency enforce consistency property define calibration transform desire calibration consistency arc consistency consistency exact maintain exact logical point local closely pass map always convergent calibration may structure submodular problem mainly useful elimination available interaction shape pixel applied start algorithm priori opposed call approximation nevertheless distinction perform sometimes marginalization heuristic principle belief numerous decade understand variational approximation class leibl application continuous variable principle cast variational choose nature variable variable low independent field work class distribution numerous remark remainder key component approximation leibler divergence tractable explain variational constrain accord set distribution marginal normalize constant vertex field fully factorize mf qx ix mf correspond joint respective namely term minimize respect fix relation conditional expect distribution approximation trade opposite guarantee contain focus bethe class enable heuristic liu model factorial approach multivariate state state markov chain apply use gaussian posterior ep minimized field among mechanic cluster variational leibler energy mrf associate qx j qx qx qx coherent tree bethe free draw sep gray draw em width color gray example couple hmm hmm hidden couple hmm assume display tm tm h tp em em tm tm tm h tm leave bend leave tp tp bend tp em em em h tm tm tp involve aim maximize likelihood achieve stand kullback leibler perform observe encode markovian dependency coupling write merge graphical value variable mode compute either forward recursion viterbi
learner solution use solution typical solution construct representative cluster cluster large solution eq likelihood belong auto experimental useful learner denote expression feature solution evaluate first p k also learner make less alternatively benefit common learner help guide also enable automated error early step demonstrate efficacy automatically learner roughly course solution feedback solution dataset response mathematical question set include college level signal question detail statement question pre extract full expression solution randomly sub solution solution similar sub rs sc affinity run gibbs iteration burn ap early mae average absolute estimate mae number sc consistently almost performance likely cluster ap e auto question take compare second computer cpu learner accurately mae moreover eventually enough belong cluster tune achieve balance maximize effort automatically balance around represent require demonstrate efficacy learner incorrect expression occur feedback early carry line become tool learner mathematical question solution learner eq learner enter compute solution expect expect full processing open mathematical potentially solution cluster provide indicate substantially reduce effort large enable visualize help common group learner track step solution enable indicate outcome research currently platform thousand planning extend account ordering expression solution cluster make robust would predictive model multiply answer question discrete time omit calculate discrete fouri simplify answer summation process discussion visit website rgb g f g h thm n z sparfa mean scale aspect education weak kind mathematical technology engineering mathematic language number learner learner language comprise convert response question series develop generic automatically potentially solution track correct enable indicate learner world demonstrate reduce scale capability education provide learner include massive system communication effective mean scale learner stream reading web interact simulation via user online assignment weak link way handle learner substantial restrict language nlp program mathematical verification paper response mathematical science technology education tool education binary correct incorrect knowledge open question shift burden learner need learner obtain develop solution correctness solution mathematical assign partial score likely scope involve mathematical expression include science engineering solution correctness focus evaluate success nlp method analysis short answer comprise main response derive symbolic mathematic canonical solution correct partially incorrect solution two cluster approach numerical affinity propagation ap group learner bayesian solution assign track assignment indicate likely learner develop tackle challenge analyze response mathematical notation mathematical learner refer question admit path yet possibility correctness solution especially question answer learner recognize correctness computer program formulae specify different checking accurately certain kind feature extract language nlp automate computer program response lack structure correctness logic logical operation kind open mathematical calculation involve science engineering cluster question short use visualize program use cluster feedback structure differ answer correctly simplification extract expression feature assume learner extract yield solution expression column numerical representation solution frequency encoding frequency illustrate solution expression unique let four expression observation mathematical share learner true instance suggest limited solution incorrect conclusion particular effectively section solution affinity ap matrix mathematical detail dataset appendix node estimate proportional correctness outline solution define similarity similarity solution entry informally expression two solution figure development future cluster solution similarity cluster algorithm solution sc ap sc specifying cluster ap ap identify figure correspond processing node solution observation answer although able identify correct require able simplify final identify solution connect visualization extremely easily lack adjust accordingly learner solution solution assign automatically construct set break randomly demonstrate auto via outline interpret extension short key cluster assignment denote assignment learner matrix solution implicitly learner analogy learner question assignment cluster consider parameter impose chinese crp cluster assignment crp characterize partition
task require experiment run corpus collect build tf create category include category sub category positive category category error positive writing digit include grey classify digit average overlap result dimensional digit class digit digit positive word rate run size dataset see figure always achieve significant first achieve note regression accurate black power nonetheless similar increase difficulty consume train allow regularization achieve novel transfer purpose classifier ratio test general improve distance linear divergence attractive normalization term decompose relatively later generic law totally verify universal nn universal knn bound absolutely nn probability jensen right side show new apply z n assumption se utilize boundedness se g converge converge via fold validation criterion change iteration gradient tuning mathematic mathematic learn frequently already acquire writing enhance write focus purpose probabilistic transfer sparse parametric model obtain multiply show method achieve promise world text hand digits classifier testing slightly pattern write train recognize user due background large classifier refer label task similar may general purpose task natural idea assign assign contribute target target rise function function risk minimization sample parametric must simultaneously regularization utilize enforce closeness learn task face happen build massive since costly writing come rapid build patch transfer consist separate stage general second light weight translate model intensive involve general purpose dataset sophisticated key difference minor since task handle may handle much simple change especially contain crucial build sophisticated machine software end hope adapt device mobile key idea probabilistic already train multiply classifier composite completely totally still error later efficiently parametric convex conduct motivation algorithm datum yy enough focus scenario relatively desirable boost previous later stage go clearly valid normalize q however approximate normalization introduce ratio log substitute term evaluate pointwise sufficient pair practice sample may pair observe sample may approximate estimation quantity posterior feature therefore together cross notation posterior approximation nn estimate fy reasonable completely make expect transfer method follow absolutely continuous neighbor rely state approximate step model error cause obtain multiplication divergence identifiable q different mean train create kl blue line figure comparison sample plot figure show divergence help improve bar red bar however transfer introduce baseline logistic regression sample modal previous would use ii indicator ratio center show predict transfer dot dataset truth mark
analysis however bad subspace e combination improper although supervise attention drive unsupervised solid foundation development leverage model massive novel product hermitian random streaming generate rectangular identically transform variance empirical surely q eigenvalue inner radius z outer h acquire law spectral show management general standardize past model vector system certain arrange form map far consist n decide length historical big raw area form rectangular hermitian I introduce column z transformation depict single law equal eigenvalue propose transformation raw step raw single conduct visualization unsupervise aim conduct dimensional raw visualize status estimation arbitrary historical time focus drive model topology addition control principal component grid select select pilot orthogonal pilot n train step choose n pilot illustrate scale train historical tag system eigenvector method hard still six package manual white fluctuation sample plus change grid work move event par demand include time area play dominant closely curve decrease event decreasing conduct detect achieve raw datum statistical orient decide node latter high parameter whose decide conduct visualization combination analysis visualization interpolation plot respectively much area change influential conjecture demand system visualization important raw whereas status trend whole trend statistical htbp pdf pdf scale pdf sets set datum realistic node work day node reach size control matrix acquire figure curve data window working detect meanwhile load mathematical visualization conduct status trend make learning method realistic validate effectiveness drive unsupervised highlight even towards power system big unsupervised research radius long goal reach year age grid big pdf pt blue rgb rgb rgb rgb ai xu event become increasingly grid feature vs handle one conduct system hand extract directly propose raw one dimensional statistical visualize reasoning speed traditional spurious bad realistic effectiveness grid unsupervise grid law radius map become resource grid readily various device communication technology measurement unit device acquisition volume velocity vs curse aggregated grid require massive multivariate pattern mean signal grids generator fluctuation error generator execution aim streaming datum dimensionality big resource hardware resource handle tool base extract multivariate multidimensional big establishes inherent system gain rather build assumption already numerous phenomenon quantum system wireless data scope system grid produce label complex power original datum impossible event
even though optima performance likely performance rnn well explore compositional long memory include kb rule rule path kb certain modern tune threshold train feature perform kb extend rule simple multiplication compositional phrase language network successfully phrase neural task language model translation parse recurrent neural like parse classification answer logical semantic rnns attention path connect entity final shot drug shoot neural develop compositional completion recurrent baseline prediction prediction modify ability shoot acknowledgment answer numerous question thank stanford nlp retrieval agreement award google part reproduce notation recommendation express material necessarily cs knowledge kb previous relational relational path atomic z multi rnn path unseen training capacity compositional zero relational triple leverage train embedding construct knowledge reasoning natural triple also binary relation incomplete many fact usefulness infer already kb automate triple leverage triple kb completion focus symbolic learn clause binary relational path threshold triple entity triple entity triple rank greatly efficiency exhaustive walk use relation binary learn imply infer fact later greatly available raw material kb schema relation connect corpus symbolic million treat put aside relation limit applicability modern relation rapidly type relation obviously generalization operate representation relation semantic learn representation relation vector perform prediction kb kb embedding tensor universal schema cast completion likewise embedding kb entity entity evidence propose advantage embedding reasoning generalization network rnns semantics path represent output vector represent extend input step consume relation entity path consume path vector compositional unseen neighborhood atomic allow million path kb separate predict type alternatively composition relation perform shot composition work continue collapse path substitute original relation type path map nearby embedding shoot pre train tuned completion completely additional new large million triple preprocesse kb entity path relation million entity pair k experimental compositional outperform feature statistically compositional substantially strength shot predict unseen background path connect entity employ composition obtain connect entity kb use path connect entity training entity path walk target million path per type improvement neural rnn phrase composition linearity concatenation operation phrase propose kb connect entity relation representation path length kb l binary fact p vector randomly backpropagation thousand without directly describe previous section capable fact type shoot modification composition fix relation unseen relation beyond capability kb composition predict vector relation general composition initialize vector representation update training prediction type never see vector relation sigmoid softmax contain relation unchanged predict unseen training composition use irrespective see compare pre tune hold development iteration use regularizer batch triple avg path avg fact avg instance avg relation run publicly link dataset create node kb object triple sentence entity phrase entity relation length great keep last consume dependency parse relation type triple fact reduce relation relation fact statistic path relation learn rnn quality rnn generalize book book write author person language people language people person book person language location location location bridge location location location contain near people person child people parent people parent target relation example unseen show relation relation mark train create path method replace relation membership path find exactly create pre type simple wise linearity composition rnn initialize rnn use additionally path extension feature compute prediction prediction assign score sort
introduction limit maintain obtain eq variance free fact penalty term final introduce additional cost hdp hmm generative let state transition last state alternatively integrate take logarithm asymptotic expansion hmm hyperparameter cost kl hierarchical bias derivation small asymptotic derivation asymptotic hdp hdp infinite hmm crp crf numerous bayesian model year problem series analysis however mean extremely simplicity large view assignment equivalently dominate asymptotic generalize derivation bayesian dirichlet process obtain straightforward derivation nonparametric subtle aware dp dp hdp probability reference hdp material hdp model hmm serve introduction nonparametric reader also chinese restaurant crp chinese restaurant dp hdp iv dot tensor convention crp observation exchangeability crp parameter restaurant restaurant count repeatedly customer restaurant eq
vector computation track past stochastic gradient refer special case consider requirement set balanced algorithm neighborhood pairwise explicitly purpose sharing gradient neighborhood motivate regard define modify intuitively compute observe generalize exploit neighborhood correction neighborhood practical overhead find well yet plain variant candidate drastically way subsample refined core set g remain sensitive even hash cost affect throughput recurrence imply recurrence utilize e continuity apply parameterized note asymptotic immediately contraction yield contraction factor relative gold sgd thing complicate parametrize share bound pair regard iterate sequence result contraction bring main challenge store e constitute require somewhat complex inspire lyapunov conceptually initialize maintain valid iw quantity expectation lyapunov recurrence without uniformity expectation increment iterate convergence main sharing state sharing provide lot motivation insight see may depend q suitable collect contraction theorem investigate bound optimality factor fw recurrence work get simplify bound identity good maximal result apply claim case case taylor approximate rate note nb q improved yet error vanish control neighborhood neighborhood guarantee share g straightforward get contraction behavior progress towards enough reach ball optimum walk ultimately switch sharing effectively minimizer desired behave sensible neighborhood requirement year sgd paper uniformly superior almost superior albeit sometimes regularizer commonly occur namely logistic million song uci repository logistic obtain website dataset compute exception practically relevant improvement correction everywhere roughly plain sgd schedule axis express epoch really gain significant start plain counting step reference expect actual somewhat gain typically solid speed start difference difference sgd state run asymptotic experiment cross epoch gain epoch streaming present although analysis purely loss also take set proxy generalization somewhat early epoch c novel sgd demonstrate insight role gradient evaluate correction speed remarkably knowledge corollary inf stochastic descent machine know relative sublinear iterate recently overcome vanish step maintain result either full correction disadvantage employ streaming speed neighborhood structure gradient across significant phase epoch investigate family thorough support variance regularizer finding parameter minimize empirical descent straightforward repeat computation become prohibitive new select
problem ease exposition system augmentation develop technique uncertainty drift demonstrate affine dynamical state drift assume lipschitz control control online functional dynamic origin denote eq since invariant xt map accumulate feedback compact controller eq characterized admit continuously solution constitute e differentiable derivative argument control open express solve infeasible actor replace denote replace objective actor controller learn motivated actor maintain substitute bellman eq actor exact approximate system give approximate establish basis function require instead operating domain intractable basis aim obtain aforementioned development cx universal weight compact select cx rx denote function note change system change system change dependent change ideal continuously continuously learn ideal evaluate form actor weight system start initial feedback controller constitute serve indirect close ideal hence gain along system trajectory simulation achieved select nx base law matrix eq constant factor lyapunov improve brevity time hereafter origin facilitate subsequent eq lyapunov select constant strictly regressor regressor regressor hence regressor hence strictly frequency make make function state computationally infeasible high trajectory enough simulation instead establish upper gain law ensure tt q furthermore tt lower positive lyapunov subsequent lyapunov sufficient provide enough kernel kernel small result almost identical approximation select meet assumption gain controller ensure ultimately derivative weight lyapunov express sufficient candidate lyapunov use conclude exact demonstrate simulation perform dynamical system cost select optimal approximate three c tx vertex shrink triangle state shrink point uniform xt ix matrix trajectory generate control estimate unknown ideal ideal trajectory kernel weight system state maintain show control compare signal control show origin ideal figure show decay uncertainty rl uncertainty drift system ideal unknown value xt dt fx approximate select form identification implement origin stack ten record cf computed order initial value dynamic stable simultaneous figure track nn periodic weight converge periodic drift dynamic converge value h nonlinear nonlinear system steady system analytical tracking compare system converge ideal analytical tracking available ideal value unknown drift dynamic dotted error point develop time steady move technique develop tracking simulation run hz simulation run table develop controller resource know quadratic quadratic since basis use inexact generic tracking ten polynomial basis trial since unknown inexact result steady sense use result small trade computational inexact estimate trajectory table develop horizon control solve function aims maintain good value neighborhood efficiency rl allow selection number simulation horizon art horizon problem online entire operate completely value art state aim maintain value kernel sense ideal lose leave unlike memory lemma remark optimization recommendation view horizon online compact stability optimality achieve significantly global popular online system challenge rl action open rl via employ employ challenge controller decay sufficiently determined rate introduce achieve nonlinear system generally basis sufficiently require recorded history rl increase increase rich cause increasingly undesirable experience rl demand rich cause store stack number respectively grow hence drive find novel base rl achieve sufficient undesirable effort like traditional computational technique effort decrease basis key online optimal controller current approximated facilitate summarize reproduce kernel use continuous ideal small would prove facilitate rkh set continuously kernel
language computer double format implementation computer format sign bit exponent bit group integer indicate number exponent plus minus infinity infinite bit bit symbol exp mask exp bit mask bit sign two type available computer implementation arithmetic quantity processor old processor addition slow carry finally order bit field allow exponent bit store mask bit sign computer big little intel past arm architecture summation first summation small preferred method moderate component detail design contrast primarily hardware implementation obvious design sum store bit carry propagate bit loop furthermore sign magnitude handle sign change additional represent carry propagation somewhat save representation high bit low carry defer whose determine sign complement start low order denote eq range code symbol high exp bit bit along represent capacity bit format bit within sum produce advantage arithmetic due sign however canonical produce carry propagation happen whenever correct need propagation start integer high end order whether positive negative carry describe carry negative require modification ie avoid set bit next produce carry absolute value add carry routine remain carry propagation actual subtract necessary per carry justify even procedure mask treat sign sign sign bit point value infinity storing indicator inf highly inf fairly nothing add zero otherwise exponent exponent neither normalize shift mask separate exponent bit bit exponent bit subtracting seem subtract modify bit give exponent order bit operation modify subtract bit modify add determine number quite overall sign operation show array code expand loop function carry routine loop loop check carry define type inf inf inf nan propagate double u value mask exp exp mask exp exp bit else else inf inf exp exp mask exp exp exp mask exp low else add sum carry propagation inf must examine proceeding note round number obtain high point look converted operation operation could replace bit bit highest examine potentially round exponent somewhat round present commonly round round implement round straightforward summation fix zero scan carry final roughly naive operation operation term compare add term modern bit processor probably processor exploit parallelism nevertheless motivate per summation term summing eliminate summation inf checking term decide accomplish bit possible exponent bit still sum initially start view integer shift fill bit bit define exactly least acc holding add remain use use bit bit large count large ix else count count ix exponent entire sign exponent mask add index done keep hold routine check inf check index initialize check index small circumstance routine exponent index pass inf handle associate add process indicating must indicate count two proceed add partial add final add adjustment sign bit time multiply bit remove bit transfer add shift leave bit bit magnitude would small modify consecutive bit conceptually position low bit next however shift instead find left ie exponent implicit beyond top bit appropriate bit top great initialize count small many application typical number sign obvious look count overhead keep array word word skip region scan bit whose bit maintain slightly inner summation operation figure sum array twice fast summing use small evaluation performance sum element array architecture processor speed lead assess summation would run ideal processor parallelism allow depend computed insight assess question perform computer measurement computer assess insight summation characteristic conjunction limited assessment serial many processor core apart implement careful attention obvious attempt branch straightforward might superior decision manually version similarly optimize version order routine add turn double summation use index add allow level supplementary information reasonably implementation improve sometimes good choice appropriate among sum summation seven size try ten cover processor test effect summation divide always fold array fit cache memory cache cache vertical line cache need assessment generator rand avoid term mirror element perform test array performance six colour solid dash line processor year intel bit intel processor high end end intel processor span family x e v processor different pt qualitative picture processor sum term combination superior processor fast array less advantage sum processor nothing array memory dominate advantage method summing array intel core slow slow summation cache memory dominate summation add order summation summing array size summation large intel processor intel distant intel intel ghz intel iii processor processor parallelism case processor summing array large method combination except array overhead cache dominate intel slight sum identically small processor intel intel picture combination almost array small array difference reflect integer processor actually ghz whereas summation perform substantially bit intel ratio time exact summation though cache reflect somewhat specialized processor support operation bit computation modern bit processor perform processor slow summation modern show processor processor bite intel plus optimize overhead look per term processor assume summation model fix model perhaps overhead fairly I array affect stage little effect performance processor one perhaps branch row sign processor branch method much method sum reduce summation produce modern bit fast term I vector square vector sum product method product usual rounding bit implementation intel figure time inner loop method multiplication execute parallel sum dot cache large method processor intel sum computing norm order indistinguishable summation length picture reason expect apparent manner parallelism would however exact great summation supplementary modern processor implementation two summation introduce dominate two result improvement obtain summation less slow order three slow sum vector reduce eliminate method return advantage summation unlike situation summation probably loop small conditional positive eliminate conditionally sign none code improvement summation likely expect processor would allow implementation summation limit discussion exact summation purpose processor way array parallel add together partial sum write routine add straightforward operation comparable produce course integrate might parallelism core eight current core limit bandwidth intel processor memory ns summing array take ns limit impose suggest core sum possible probably maximum may suffice v processor issue cache core evaluation investigate limit regime distinguish achieve wide perform slow summing term ten order summation scan cost dominate term limit summation three two arithmetic overhead keep bit array become bit quickly non produce increase current method fast moderate exact summation improve
follow integer iterate function domain iterate smoothly invertible inverse set verified operator use integer function operation concern continuously since exist inspection equation th integer choose non iterate property hence understand likewise respect logarithm continuous real necessity logarithm argument derive substitution gives examine constant interested restrict z branch fix function infinite plane since contraction mapping arbitrary applying behave affine expansion exponential circle around becomes give repeat application logarithm point circle iterate solution r derivative c z contour application circle radius rotation complex plane branch part iterate application half plane instead thus avoid branch n map half disk series disk thus schwarz hand nn application iterate lead plane give think plane call complex plane plane calculate exponential plane grid cross origin map circle whole integer composition z case range maximum two method net operation adjust straightforward neuron summation value neuron behave like input dependence neuron output neuron compare net real computational necessity output sensible unit avoid neuron accord interpolation transfer tie iterate occur neural net namely computational complexity unchanged conventional neuron function neuron consequently framework replace complex matrix weight neuron value ij task consist binary vector integer output element index efficiently implement architecture fourier dft inverse factor shift definition amount one else neural one output nm encoding pattern shift cell shift multiplicative additive neuron transfer compute dft fouri compute dft shift pattern see automatically neuron net solve continuously addition integrate neural mathematical concept integrate net behave efficiently work technique exponential first sound iterate functional composition monotonic interpolation multiplication landscape ad hoc transition introduce transfer allow back drawback implement area pdf ai combine neuron either inefficient resource procedure present transfer mathematical concept non integer functional neuron smoothly importantly addition multiplication decision integrate backpropagation procedure neuron sum input illustration multidimensional neuron function sigmoid layer context network transfer function arbitrarily sufficient unit though architecture acceptable alternative weighted replace weight use law layer element incoming exponential apply product hybrid summation pose additive neuron solution stack alternate additive skip additive
author interest classical topic world setting paper resolve instead assign fix topic stochastic binomial three author document document mix account multi algorithm model real world capability interest topic mine artificial traditional text mining topic commonly regard mining interest side corpus include author tag label incorporation side lot among topic interest author jointly topic interest variety scenario recommendation recommend interest author surprising author paper rank exist fix number normally dirichlet limit exactly application author topic relax distribution specific gamma three level capture hierarchical vector process simply measure normally gamma process author document introduce gamma closed distribution hide number topic new fixed author gibbs sampling get briefly describe preliminary knowledge propose section direction work second part nonparametric probability originally mine task discover topic good powerful representation music author propose interest document suppose interest detail attract lot work elegant author document incorporate side tag time mixture suppose infer idea multinomial limitation multinomial mp pp infinite property task dirichlet probabilistic model one process gaussian mixture gmm mixture hide extended infinite hide infer process binomial summarize nonparametric apply application model propose gamma circle thick draw black solid pt draw font rectangle rectangle model use author hide paper interest interest graphical interest topic one topic need predefine real scenario process measure product indicator improper get process parameterize process negative also point integer binomial normally count counting variance mean cm black black thick draw fill gray right edge edge normally parameter kind binomial beta gamma base binomial make document base gamma process equivalently augment poisson parameter augmentation tb description word document infinite model author binomial successful fundamentally document three document add process level capture gamma hierarchical author binomial process however thick draw solid black circle gray font rectangle right edge edge edge edge assign topic interest summation however topic assign author author see set multiple author gamma document combination process weight paper gamma interest document frequently around truncation truncation accept potential express good approximation infinite appropriate mixed resolve various distribution far split topic independent assign inference help resolve gamma author gamma author due conjugacy make binomial compound poisson distribution logarithmic logarithmic restaurant distribution theorem finally formulate gibb design list number author ar ni na kp da kl conditional implement procedure summarize n burn stage number output latent infinite topic finite author topic description author paper area database mine artificial description find document training selection two document specialized column requirement requirement sure topic predict author interest widely assess document document well different normalization influence comparison understand big training fig table comparison adjust topic propose hyper rest
node paradigm raise question made round need influence probability learn manner probability assume cascade seed influence probability price seed spread spread perfect minimize spread market structure learn phase exploration knowledge gain phase achieve spread exploitation regret network come mab influence maximization application firstly status seed observe live definition status refer observable seed activate fail observable feedback try face assign neighbor whereas real paper challenge focus ic contribution I influence combinatorial mab motivate regret aim efficiently influence section combinatorial exploitation prove round probability learn online set various minimization conduct extensive dataset effectiveness intend summarize direction graph seed maximize spread cascade threshold variant ic discrete seed get chance inactive neighbor activation succeed probability attempt say neighbor parent inactive multiple activate time parent capable parent ic expect node seed distribution show I nice seed spread gain exploiting et show seed large gain increase spread seed use research around diffusion scalable algorithm multi multi armed bandit mab paradigm reward arm play generate reward continue round reward natural goal bandit goal result play suboptimal regret much hope well learn reward al pure mab bandit paradigm paradigm together arm contain reward reward et al chen et play round consider individual combination al upper ucb algorithm obtain al pure chen et al ad page arm briefly combinatorial armed I consist base arm denote trial support identically unknown play round play arm possibly reward number base mean well play take accommodate attain I arm either live reward arm correspond specifically seed play become live live number diffusion process linear arm maximization oracle constitute approximation output mc serve mean update context notion play role characterize play update world underlie arm reward application bandit I basic assume status live edge possible network arm frequentist formula assume know active key attempt realistic precise status e call compare feedback weak realistic flip estimate challenge node feedback consider parent parent active status edge node level parent responsible overcome adaptation frequentist approach specifically scheme whereby assign assign active assign follow edge feedback inherent uncertainty infer live vice probability feedback bind ultimately achievable verify failure establish effectiveness feedback let denote arm edge probability influence resp use feedback relative learn influence learn situation infer edge live infer live assignment live recall condition fact parent live world hence characterize failure follow pr pr v pr cascade ex pr pr k pr u u pr u j pr u u pr random pr u k theorem use feedback influence edge error status level tx round successful let reward arm play section empirically typical verify feedback cascade input adapt method employ develop cascade cascade action reveal status perfectly align feedback online cascade stream cascade cascade view describe cascade influence probability neighborhood function show maximize propose make need learn improved probability describe propose make offline mle characterize likelihood individual cascade number cascade network l write follow attempt term activation attempt notice concave node separately gradient contain mle maximize likelihood stream online function advance reveal function objective cost advance offline choose minimize online compare offline algorithm depend greedy updating function correspond likelihood cascade round game mle exploration frequentist cascade seed probability within edge correspond round round arm play ic cascade get activate number cascade learn cascade arm follow edge live cascade e game give time probability within number cascade seed choose seed eq follow result cascade learn within relative need combine cascade feedback work round select pure seed spread edge round round combine exploration seed explore intuition active cascade write edge cascade frequently edge define exactly expect spread count activate add seed spread seed intuitively seed large edge call exploration se dynamically se across round effectively result first formally tailor I intuitively give seed seed correspond seed play I spread ic true p hard optimal seed monte mc spread seed know greedy alternatively recent reverse rr use motivated assume notice inherent randomness output randomness rr set true probability know spread seed successive exploring improve seed probability improve choose seed lower quantify randomness output strategy round invoke seed seed size maximize current update budget select seed budget return seed cascade update accord combinatorial logarithmic confidence bind network initially maintain estimate find lead implicit regret feedback initialize exploit cascade arm increment value strategy involve exploration select achieve budget parameter initialize cascade update achieve level proof explore algorithm exploit every knowledge implicit pick seed cascade learn influence oracle influence recently act run briefly diffusion ic obtains spread adaptive near optimal runtime operate generate random reverse rr node reach enough bind rr rr seed node node rr seed seed next cover maximum rr seed al strategy enough idea range probability may method frequentist initialization beta conjugate prior beta update tx act like pseudo formula technique laplace nlp put prior goal various regret regret true generate probability initialize k diffusion sampling world purpose diffusion graph budget influence oracle notion reverse lack refer reader obtain greedy mc iii readily serve runtime value verify run algorithm round find actual spread generate value seed process also learn plot varied pure network edge algorithm feedback exploit mechanism r dataset exploit cm within edge exploration exploration strategy couple feedback base mechanism exploration choose feedback try avoid sample decrease frequentist compare fast node moderate probability typical frequentist quick error depict roc fraction whose probability learn quick round percentage cascade network exploration sample algorithm try learn cascade minimization experiment pure exploitation explore seed regret achieve worse omit completely likelihood learn runtime use minimization greedy set initial set result back feedback cm average increase slow explore edge decrease control amount use see become low end round probability estimate well enough comparable spread know probability almost exploration phase pure exploitation end round reward initially probabilitie lead observe feedback us node feedback failure find number parent big previous cascade vary vary level well plot goes expect choose seed true pure exploitation lead seed initial however bit exploitation explore network seed omit plot observe probability lead spread show spread observe spread always edge feedback irrespective seed study influence cascade adopt armed paradigm formulate interesting regret spread suboptimal set various bandit performance real sampling extend continuous model influence interesting theorem assume paper propose availability diffusion cascade learn offline tackle ii cascade adopt combinatorial armed paradigm formulate spread suboptimal round problem
discuss submatrix localization em theoretical submatrix localization problem noise determine transition boundary hide detection focus submatrix entry relaxation signal noise require submatrix regime wise thresholding well submatrix small snr relaxation improve snr boundary dense scale dimensionality low literature boundary similar message pass snr submatrix submatrix diagonal emphasize localization combine treating achieve set however crucial utilize denote submatrix vector induce precisely latter surrogate norm inner exist universal bn equivalence factor iff sub gaussian random universal random scalar run submatrix answer localization minimal read respectively terminate restriction combine statistical submatrix localization submatrix localization boundary clique hardness phase diagram consider case boundary eq separate region theorem correspond separate snr entry snr small hide clique submatrix furthermore require submatrix run sdp clique mention early localization investigate result provide finish phenomenon figure localization result submatrix intractable region plain beyond localization statistically possible appear distinct principal computational statistical region boundary computational upper introduce extension technical proof defer addition since localization detection boundary submatrix bind upper bind computational hide clique computer science identify hard sense hard problem hard deal seek token difficult clique quasi polynomial describe work precise clique clique clique instance way clique connect connect clique plant tend possibly detailed clique location location nn q uniform clique precisely clique hypothesis recently claim localization solve also localization low localization theorem defer difference construction localization bind ensure submatrix tuple within localization sec pf key idea insight bootstrappe introduce randomness clique field us mixture submatrix submatrix localization need clique support exactly plain reduce clique still exact clique technical please introduce solve localization graph localization submatrix top respectively calculate separate thresholding return several appear secondly require automatically submatrix lemma spectral guarantee spectral succeed exclude exhaustive search boundary submatrix submatrix localization clique hardness proof spectral relaxation theorem analyse multiple grow number single submatrix statistical introduction establish expense begin theoretic accuracy theoretic submatrix search aggregate statistically optimal unfortunately computational algorithm introduce analyze extend gaussian noise localization submatrix sum submatrix report extract large sum greedy fashion submatrix provide achieve submatrix algorithm exist universal return probability boundary theoretic analyze submatrix statistical submatrix high minimax tend boundary section submatrix perturbation combine know thus learn line form sense cut clustering recover summary spectral succeed eq computational localization low detection polynomial algorithmic relate instance proof k n quantitative submatrix time solve localization probability contradiction chernoff bernstein let suppose time stage graph stochastically hide clique easy analysis property clique bernstein least clique long hide clique node clique clique take submatrix set independent subsampling replacement weighted generate matrix q bootstrapping lc stand clique clique n l lm edge condition index clique clique rademacher position matrix submatrix sub estimate submatrix signal side q bernstein inequality position submatrix parametrization slack precisely return submatrix correct go correctly identify clique node correspond clique quantitative lower boundary submatrix amount power correspondingly submatrix introduce clique node tend algorithm rely clique contain clique subset adjacency restrict clique mr lk correct clique recovery complete n induce solve clique problem coincide true go hide hardness time proof testing compose similarly cardinality corresponding measure probability uniform prior space invoke bind leibler q binomial coefficient condition submatrix translate testing pick inequality localization total unique reach need trick row cardinality set use satisfied constant thus pick submatrix sum overlap go theoretical justification submatrix multiple submatrix singular n thus lemma canonical angle observation observation singular use lemma thus invoke basically know learn metric succeed term explicit problem eq feasibility set function although bi relax course need solution relax exact high condition submatrix singular g low relaxed unfortunately estimation simultaneous exploitation relaxation expand original term time nk localization submatrix denote top singular implement multiplier admm disadvantage theoretical hold remark require knowledge submatrix guarantee submatrix universal relaxation mn cm pair w relaxation satisfy find primal q mean feasibility lagrangian associate kkt expand q choose hold thus hence need q see q succeed q achieve boundary upper submatrix tuple hc submatrix going build hide submatrix model q several graph stochastically clique graph property probability clique long hide clique clique node connect otherwise equal take right submatrix index ns ni q case note clique inside variable clique clique n variable thus construct submatrix element clique two side order bernstein computational detection clique school accuracy draw increase computational boundary submatrix localization submatrix contaminate establish threshold term correspond boundary threshold
standardize regularization lasso put imputation fold cross predictor multinomial three death set coefficient vs normal death vs normal penalize likelihood criterion initialization step size backtrack backtrack stop parameter procedure minimize yield discusse exclude consideration outcome individual age outcome age age size associate point nonzero multinomial logit death vs ht vs else person last else far early gender yes white yes heart difficulty arm could substitution task symbol correctly code trust probably else digit substitution symbol code stand gender ever else per day current status never gain lose exercise major status diabetes yes pressure one yes else detail subset validation relative importance longitudinal estimate meaning categorical coefficient display less majority proceed result keep logit outcome record predictor future cognitive status facilitate exposition formal vs odd age lack odd early early range death death variable analogous interpretation low diabetes health status difficulty contrast age digit account odd death odd death show intercept coefficient increase death age predictor broadly risk gender heart disease lack increase provide multinomial fuse demand multinomial fuse cross selection parameter incorporate variable plot typically interesting concern convert categorical selection tuning parameter important yet parameter multinomial fuse illustration consider cross validation aic bic misclassification likelihood cross divide fold usage say pick simplest interpret nonzero aic score fuse tuning number longitudinal degree denote loss multinomial typical aic close cross training bic computationally validation fold set aic log misclassification selection entirely fold fact division evaluation repeat fold rate positive positive rate identify standard selection ht seem favorable balance evaluation model accord majority death yield rate death rate cross true class concern cs simple degree rule absolute g fully multinomial dimensional operate assumption contribute persistent effect fit fuse respectively profile highly proximal demonstrate applicability discuss practically issue selection reach place group categorical variable may encourage sparsity variable complex fuse trend penalty polynomial trend choose application penalty assume variable mostly time trend effect concern fit longitudinal example band select profile interpretation stability unfortunately quite regularization penalty inferential development relate dimensional light work classification regularizer piecewise longitudinal adaptively select gradient descent propose disease health motivate tuning assessment study longitudinal record individual place predict measurement allow employ e extensive regularizer problem lasso encourage active fuse regularizer encourage persistence work cs factor disease ad cognitive people year old matter increase age incidence per age later examine age predict ad age assign longitudinal clear matrix element determine lag generally denote partial indexing multinomial coefficient introduce extension basic number individual index outcome array separate logit time eq eq multinomial fuse notation fact g broad encourage persistence generally generally point piecewise coefficient trajectory kt independence rather longitudinal setup partly role fuse tie across help example normal coefficient remain coefficient trajectory leave relevant outcome multinomial right intercept likelihood dynamic see multinomial fuse estimate pick underlie overall toward prediction multinomial advantage repetition estimate outcome regularize fuse lasso idea extensively community interesting fuse lasso genomic association author setup primary motivation age measure multinomial death individual age mainly complicated factor death death category alternate cox death traditionally cox naturally scheme multinomial hazard depend hazard relate instantaneous failure predictor multinomial death would determined maximize log lasso odd cox describe similarly routine cox minor modification comparison beyond scope current manuscript topic future development next cs discuss estimate coefficient relate discuss numerous approach tune fuse lasso conclude future descent compute lasso regularize multinomial algorithmic implementation direction multiplier proximal simplicity fused regularizer describe number choice size differentiable repeat denote obviously long precise variant descent share proximal descent generalize routine repeat strict convexity criterion define provide computable descent apply map perspective minimize plus quadratic current iterate optimization statistic typically nonsmooth trend somewhat though mapping history community term proximal enjoy convergence rate descent amenable acceleration technique proximal gradient applicable regularizer encounter mapping fast exist compute rely elegant propose negative multinomial lasso fuse lasso formally rewrite convex nonsmooth describe tuning respective computed intercept coefficient penalize proximal consider g evident proximal operator intercept proximal map identity intercept term consider proximal arbitrary predictor fuse specialized problem make compute proximal practical issue arise return rewrite proximal generalize gradient rewrite resemble choice iteration proximal proximal know easily appropriate backtracking line search straightforward algorithm constant shrinkage backtracking routine start initial guess satisfy proximal current lasso map define hand backtrack proximal refined terminate meet common stopping iterate stop meet tolerance outline proximal procedure multinomial backtracking criterion htb predictor c backtrack input ks j k practice individual longitudinal mean outcome predictor individual predictor outcome observe time issue arise penalty another experience sum effective regardless sample effective modification indeed end hundred cover descent interface author website broadly generalize rich thousand cognitive past
orthonormal basis chebyshev power odd I span inequalitie property plug invariance frobenius norm rotation fall span column span orthonormal row expand il eq norm l give necessary return span give proof singular henceforth statement extend guarantee top singular prove align existence ensure perform intermediate span odd contain power fall vector inner inner outer inner outer outer fall span outer remain singular orthogonal outer inner outer inner outer bind first unit vector slightly contain use outer plug span column frobenius norm apply choose polynomial necessary property already inner give ml w w ml argument single outer give place iteration rank probability return satisfy value additive frobenius matlab column explicitly use improve algorithms principal eps algorithms frobenius guarantee tail outperform case error confirm rapidly simultaneous often possible simultaneous long justify convergence gap comparison rapid begin rapid confirm gap comparison frequent singular gap take constant insufficient low approximation special prove svd n draw full rank show k f property note f gaussian distribution row orthonormal gaussians concentration k result high logarithmic chebyshev polynomials actual chebyshev gap exist construct chebyshev recurrence chebyshev polynomial simply degree satisfie place rule suffice eq claim hold derivative chebyshev verified chebyshev recurrence xt xx reduce note prove show chebyshev polynomial decrease also prove equation give thus additive satisfying follow exactly completeness di ji kn technology usa analyze simple value gap iteration approximation within norm give first provable runtime simultaneous give guarantee substantially experimentally despite history subspace method practice furthermore accuracy benchmark issue minor modification simultaneous give nearly far finally fast take advantage improve use singular r r analysis principal want kk vector principal component direction great great principal component denote singular expensive typically time arithmetic inherently iteratively traditional include qr obtain rate environment take hence research randomized seek nearly quickly become practice popular library like learn contrast classical depend gap gap small due identical difficult distinguish inherently depend singular value gap approximation goal randomization avoid need find subspace singular distinguish close fast randomize svd run lower satisfy often insufficient analysis singular noisy multiplicative number suggest remain tail guarantee decade simultaneous subspace achieve become choice practitioner classic simultaneous fast runtime performance section even though discuss improvement simultaneous subspace method highlight decade theory randomize excellent accuracy gap issue error rank output kk strong svd gap note intuitively strong spectral square small singular leaving mind guarantee practice svd amazon co eq principal significantly similar phenomenon popular additionally rank svd address introduce require singular capture I singular classical numerical analysis stress converge singular gap post return guarantee contribution runtime satisfying approximation pca exclude time simultaneous qualitatively weak give gap decay modification large justify start intuition simultaneous power gap progress considerable back full svd achieve rough factor randomization fast paradigm dimension either top leave singular project onto approach refine os recent type reduce multiply frobenius error approximation consider dominate sketch solve efficient regardless pass processor set furthermore small typically norm error seem limitation sketch singular value impossible extract svd singular word inherently norm symmetric simplicity apart accordingly lower much small top singular see specifically effectively method approximate frobenius singular provide xx high value high approximation iteratively repeatedly rough suffice simply column knowledge analyze gap technique gives achieve iteration start develop paradigm mention numerous paper possibility simultaneous experimentally variant none paper bound accelerate simply put well polynomial allow power iteration shift chebyshev nearly run polynomial block subspace polynomial scale lie match find approximation nearly surprisingly efficiently lie subspace span challenge recent frobenius post understand simultaneous block spectral low rely block singular much singular top good unfortunately gap singular lie gap guarantee break separate nearly low fall towards compare proceed full linear use compute note singular value let k spectral norm square j work k r r svd globally simultaneous subspace span svd rank obtain svd k kk norm perform simple norm span column orthonormal orthonormal orthogonal writing k sketch frobenius proof lemma appendix take outline polynomial tail chebyshev appendix chebyshev briefly runtime simultaneous modify way choose column give accuracy achieve per pca however change
approximate eps assume upper left panel obtain upper panel match blue linearization approximate lead predictive distribution approximation desire x control compute expect compute example polynomial gaussians minimize moment control analytically follow describe analytically gradient search j repeat move shorthand tp controller obtain p eq partial derivative I depend representation derivative linearization sec detail chain derive gradient match difference overview evaluation expensive evaluation analytically analytic base bfgs optimize policy prediction parametrization policy evaluation approximate gps detail computation covariance predict gps linearization mean matching moment predictive linearization computationally advantageous iterate integrate input law contain independence start p iterate expectation obtain multiplication diagonal eq th covariance pair iterate represent integral summation integration unnormalize remain integral determine gaussian predictive see obtain desire submatrix approximation exact alternative x gp mean compute gp around moment describe policy control real ideally take account detail implement control deterministic control moment interesting force plan control limit amplitude account limit differentiable function amplitude final policy eq wave normalize analytically moment multiply illustration tb figure convert figure eps convert unconstrained preliminary policy constrain policy although periodic within half wave preliminary initialize produce single period matter practice constrain control signal execute preliminary unconstrained require close control appendix control present representation preliminary computation covariance preliminary policy offset dimension combination row plus offset predictive drawback flexible however controller equilibrium play target center axis function representation deterministic gp rbf gp signal see support variance right additionally regularization contain squared scale gp eq determine preliminary covariance see uncertainty describe u possess parameter control matrix one offset parameter function per target control center summarize compute p gaussian approximation distribution preliminary distribution u x u cross exploit independence integrate lead distribution p fourth analytically gaussian function gp dynamic representation policy see see learn use distance task naturally lead cost quadratic unity large control width reasoning validate typically however cost target predict early predictive describe sec become therefore unnormalized unity analytically matrix unnormalize either immediate partial x state require analytically exploration illustrate fig eps eps convert pdf wide likely substantial tail region early uncertainty largely uncertainty forward automatic state region subsequent state situation tail region region case automatic prediction far target expect lead control hardware applicability outline alg computational burden collect datum assess term speed light approximation state bayesian apply simulated task cart important framework speed bayesian applicability briefly experimental double link see inner measure length zero control controller double eps convert challenge two experience double parametrize deterministic basis position outer choose width immediate cost unity cart cart run cart velocity measure angular controller balance position middle track controller capable nonlinear feedback see sec learn cart length cart prediction horizon constant control order require move horizontal evaluation moment match sec linearization posterior demand sec learn sec dimension evaluate dimensional computed need computational expensive determination uncertainty input scale dimension convert pdf eps convert pdf linearization moment illustrate effort linearization gp exact generate training datum derivative graph train approximate linearization fast eight trajectory controller initialization cart episode correspond double learn learn policy line start controller successful average success cart experience approximate linearization gp fig computationally demand match computationally advantageous linearization average experience reliably cart relate speed rl method bar solve task figure eps convert figure convert pdf moment matching linearization time cart task without use currently convert task linearization learns successfully linearization posterior function success reliably inference match linearization learn reliably reason linearization reliably task get minima largely predictive confident horizon problem focus solely rely evaluation sec linearization match circumstance convert eps convert approximation suboptimal learn fig inner stage gaussian ideal trajectory multimodal deal model controller trajectory unimodal gaussian explain wide require variability however cost uncertainty lead high choose marginally multimodal minimize expressive approach good approximation rl greatly model close look strictly necessary well necessary successfully nonparametric bayesian discard uncertainty long prediction kind policy gp whereas deterministic probabilistic degenerate propagate state tb success track take long term rl uncertainty appropriately learning learn small close target therefore region uncertainty choose prediction iteratively end predict trajectory essentially vanish policy realistic show eps convert eps pdf human longitudinal maintain toward apply describe controller balance use differ conventional learn controller control account separate keep trial quickly trial run policy colored bar position depend angle controller balance desire configuration sometimes due successfully policy challenging task modification besides define tb eps convert convert eps eps convert pdf control interaction control cart cart cart learn sufficiently dynamic confirm cart delay figure eps convert pdf pose feedback learn controller precision arm learn stack block purpose possess six base three open close six duration make radius system additionally joint configuration use camera external visual tracking block robot camera sensor provide structure camera useful object approximately distance continuous value signal comprise learn center object state initial choose camera coordinate robot similarly measurement camera policy e end camera approximately depend robot desire frequency control discretization block split build individual target b bottom top fig robot train share gps deviation comprise robot arm synchronization delay etc movement learn noise level slightly camera signal ten initial generally stage b learning block close sum controller speed learn insight distribution tb eps convert convert eps pdf eps convert pdf learn pay attention cause relatively system coordinate soon collapse video stack robot forward bayesian gps forward expressive inference control application control engine predict state give policy inference e moment exploit gradient approximation long search efficient require compute gradient exploit instance gps straightforwardly exploit high parallelization trajectory sampling limit small discuss exploration result encourage exploration bound type function also get minima approximate gaussian cost deviation key benefit incorporate planning instead treat transition dynamic suffer gp access approach model could transition control location planning robot obstacle function find path ever initially uncertain conservative stay framework learn minimize kullback distribution robot task base gradient advance rl state success principled reduce model long planning rl initialization prior knowledge demonstrate nonparametric hence fundamental role avoid explicit lead receive ec fp grant agreement grant intel process blue rgb decade since datum drive engineering knowledge reinforcement learning system consume approach typically task expert extract learn explicitly incorporate
eigenvalue norm measure chernoff quite powerful numerous application submatrix fix analysis problem closely basic question bernstein inequality mean chernoff bernstein comparison main tail independent explain chernoff spectral random submatrix matrix chernoff result scalar set nonnegative upper result study sequence independent bernoulli trial success chernoff behave drop fast phenomenon semidefinite meet scalar chernoff chernoff hermitian common minimum furthermore proof easy expectation help average term ambient plus reach decay subgaussian second tail decay variable receive chernoff matrix concentration especially large deviation chernoff derive expectation inequality dimensional spectral content next present refinement practice regard observe bind exceed valuable unbounded especially tail chapter sum semidefinite demonstrate precisely identify constant may sharp every side indeed justify display side right obviously suffice always remove term natural q entry elsewhere binomial representation eq logarithm logarithm come mean tail accurately example arise necessary satisfactory expectation appear estimate concavity hand form nevertheless chernoff numerically sharp situation let positive semidefinite chernoff predict determine event occur instance within draw transition verify chernoff numerically sharp occur chernoff sharp chernoff extreme singular deal linear chapter th express refer elsewhere context consider submatrix define include independently remove column obtain expectation submatrix row positive singular random semidefinite eq zero determine vice versa weakly matrix chernoff calculate lk chernoff next row column random submatrix th entry get share total reflect size large ambient common calculation submatrix need extra column refer positive hand calculation analogous omit reach deterministic still control examine eigenvalue first identity positive right apply reach simplify slightly last chernoff direct close spirit treat norm sum random diagonal chernoff eigenvalue therefore submatrix share plus logarithm combine reach expression enyi basic random whether connect vertex address recall element call simplicity vertex include undirected symmetric matrix indicate distinct diagonal equal zero whose degree matrix convention positive semidefinite eigenvector zero modern second small strictly walk connection enyi os enyi vertex mutually os enyi enyi graph nonzero entry possible permutation vertex adjacency diagonal reflect explain os graph adjacency expression translation definition random verify edge entry reflect degree reflect enyi near enyi goal second eigenvalue laplacian strictly chernoff second small eigenvalue random need coincide isometry mutually ensure semidefinite minimum eigenvalue coincide second eigenvector show follow partial isometry direct expectation apply linearity linearity multiplication inside come diagonal note identity displayed arrive smallest bind small unlikely rearrange os seem worth toward semidefinite establish linear chernoff convex lie connect particular q q eigenvalue interval thus imply second applying simply preserve semidefinite eigenvalue bound eigenvalue begin trace monotone substitute master matrix eigenvalue map observation identify infimum admit make variable argument change tail relate minimum eigenvalue state step state trace exponential respect focu reduce fourth line piece minimum eigenvalue introduce master change infimum tail usual continue reference date bound identical proof combine version chernoff inequality case matrix equivalent form substantially bound bound expectation extend chernoff contain inequality information theoretic tail establish proof problem eigenvalue sum character reason closely phenomenon rank one refinement statement work stress result easy elementary matrix closely different principle mention study long history theory provide natural paper functional clean random column submatrix fix literature inequality tool study random study arise submatrix use chernoff sophisticated random graph appear application matrix concentration om develop paper concentration analyze compressed laplacian subspace eigenvalue compression coincide eigenvalue bound report development concentration concern spectral finite satisfy condition sum matrix inequality expectation statistic bind matrix bernstein inequality powerful tool give researcher approximate introduction chapter several technique randomize replace dense proxy approximate multiplication approximate become bernstein effective study approximation nevertheless chernoff happen often matrix explain bernstein matrix bernstein scalar label bernstein inequality apply large focus scalar tail show zero tail subgaussian deviation analogy simple concern sum demonstrate much last paragraph introduce appear moment consequence variance coincide reach law hermitian coincide variance weak ambient feature explain tail scalar bernstein bind appearance inequality informative well helpful moderate tail decay tail whose comparable tail decay fast bernstein result sum mean dimension sum statistic sum immediate ambient example corner everywhere achieve circumstance bind tail behavior inequality relax weak growth moment hermitian set discriminate tail end annotate matrix bernstein strength insight present require appear omit jensen natural also essential suppose symmetric distribution right side comparable always summary match version appear contain matrix random representation ensure remove logarithm justify natural therefore chernoff correctly appearance iterate rely heavily poisson appear bernstein inequality research object structure simple whose target average independent copy number become tradeoff structured example may need construct obtain offer tool assess subsequent explain let identify desirable example decomposition need probability quantifying approximation choose specific insight construct ensure unbiased lot copy linearity small number approximation complexity incur essential obtain sampling write distribute eq relation triangle control statistic identity matrix second expectation semidefinite semidefinite follow definition fact calculation relation positive likewise matrix approximation matrix bernstein suffice bring error examine inequality per reveal aspect proportional achieve phenomenon ultimately central note valuable error involve linear bound use good achieve tradeoff drop construct analogous shrinkage target family abstraction fundamental suboptimal substantially let approximation practical highlight reason suppose singular construct random sampling satisfy j incur q accurate decay quickly bad paper frobenius acceptable interest occur illustrate approximate desire purpose independent spectral error approximation large require relationship approximation satisfie noise datum subject question decomposition approximately freedom approximation capture dense elegant identify proxy randomly select number entry retain several advantage expensive store dense operate recent due analysis immediate end history frobenius easy matrix convention therefore unbiased estimate nonzero copy linearity nonzero challenge quantify perform short replace placing sparsity determine always exceed assume discover replace achieve error norm nonzero entry proceed randomized bound key appropriate one therefore matrix semidefinite q second reach invoke numerical algebra part computer science basic multiplying system focus develop deterministic operation unfortunately become architecture computational heavily communication resource challenge contrast execution useful modern computer randomize fail concentration end task linear multiply compatible complex matrix product form algorithm divide cost approach consider outer dimension set matrix lot row proxy usual column sum method pick frobenius norm use easily cost probability operation product row separately q require unbiased quite large variance combine copy linearity expectation approximate operation determine inner achieve I multiplication beyond sampling express form way represent approximately simplify spectral equal compute spectral reasonable factor rank mean obtain randomized multiplication substantially matrix method error corollary need bind invoke mean since spectral express kind second moment calculation hold one increase matrix semidefinite order reach estimate parameter let technique sophisticated matrix propose david return value angular point write angle vector convention generalization gram euclidean matrix semidefinite positive definite kernel product replace kernel evaluation regression feature advantageous domain sometimes major insight modern big contain entry construct kernel require universe nevertheless datum redundant kernel redundant replace proxy measure approximation write empirical construction sigma assume random pair want matrix set relation form feature definite usual construct value deal modification randomize corollary sampling matrix construct q let furthermore corollary apply multiplication small appeal number norm state challenge refer find start early intrinsic intrinsic dimension relation finally identify stable quantity consideration imply analysis replace dependence ambient stable require proceed concentration inequality intrinsic challenge chapter ambient early transform origin weight resolve transform tail mind adjust real laplace hermitian nonnegative line require indeed tail spectral indicate eigenvalue exceed return discover bind trace ingredient allow semidefinite term intrinsic intrinsic interval positive semidefinite connect eq fall interval intrinsic extend intrinsic begin extract version transform bind state semidefinite identity strong conclusion side reach q intrinsic recall notation line step assumption tangent follow concave semidefinite q relation inequality concave mapping proposition eigenvalue trace semidefinite increase substitute increase argument intrinsic bound usual hermitian set concentration statement hermitian hermitian intrinsic sequence hermitian value intrinsic bind appear sum whose proof hermitian transform imply hold trace right side examine bernstein hermitian matrix intrinsic dimension intrinsic identify depend substitute discover consequence side latter define convex minimal attain tail four develop finally bernstein hermitian point intrinsic induce complete argument obtain integration control integral argument select combine improve concentration concentration random first reduce similar matrix zhang theorem suboptimal essentially somewhat bound zhang easier contain dimension intrinsic chernoff bound intrinsic bernstein bind argument calculation constant marginally well approach sophisticated algebra derivation begin short explain fact relative devoted way core matrix logarithm serve advance trace function exponential chapter matrix hermitian map convex hermitian contain require support result proof chapter symbol positive number convention capital letter symmetric vertical hermitian matrix letter always refer capital letter unless compatible involve include implicit two shorthand formula involve relative entropy relative relative entropy related arise statistical mechanic entropy definite matrix map chapter concave let variable concave fix concavity see supremum take entropy theorem fact entropy formula trace definite relative side formula desire relative proof latter compactly hermitian side concave ensure define concave observation establish deep set extra structure relative relative positive arise measure discrepancy distribution show entropy matrix maps vector entropy ultimately easy measure nonnegative entropy nonnegative function obtain complete rely idea proposition detail establish elegant bivariate univariate perspective perspective define perspective interpretation ray origin perspective point paragraph analytic convex convex interpolation combination combination another interpolation quickly determine identity write identity definition property remarkable extension construct bivariate perspective reach jensen develop idea represent calculation express entropy substantially involve argument investigation establish adequate relative subtle argument convexity relative construct hermitian matrix trace trace function hermitian contain eigenvalue first trace weakly induce function preserve recall relation dominate semidefinite hermitian result range domain convention semidefinite quickly monotone weakly hermitian weakly increase concentration special monotone monotone hermitian continue write argument hermitian eigenvalue list order family orthonormal inner interact complicated way scalar let hermitian eigenvalue contain k matrix linearity identify inner scalar prove state matrix nonnegative real lift formula sometimes nonnegative toward relative entropy logarithm demonstrate convexity semidefinite along monotonicity logarithm decomposition presentation base formula representation logarithm integral logarithm scalar simply integral integral inverse seem thin air motivate monotonicity logarithm introduce abstract monotone hermitian contain fact operator monotone point easily weakly line operator monotone line monotone interval monotone somewhat fortunately monotone inverse monotone define definite rule inequality finally semidefinite relation direction combine logarithm monotone logarithm monotone matrix demonstrate logarithm preserve positive next investigate abstract function real hermitian function monotone cone somewhat family operator convex definite lemma suppose calculate essence elimination bring original block leave positive extract result proof apply top fact finally verify logarithm argument base logarithm line definite invoke integration preserve order statement average automatically average content jensen spirit hermitian matrix whose relation remarkable call combination average rich general average hermitian decomposition form call average hermitian let property convex preserve positivity combination convex though function actually jensen inequality let hermitian identity introduce lie fall apply combination unitary unitary unitary matrix omit label restrict diagonal express convexity work unitary key apply inequality return identity block average look block equivalent block line semidefinite relation interval complete first relation function unitary identity formula preserve block diagonal block entropy perspective go perform perspective property semidefinite scalar perspective function bivariate perspective definite refer denote root remain make perspective perspective matrix hard perspective definite operator jensen convexity perspective interpolation parameter scalar combination eq interpolation observe decompose identity construction express convex give access jensen definition introduce inequality reach relation finally matrix argument entropy matrix simplify restrict attention simple kronecker hermitian hermitian first property fact construction kronecker zero kronecker q kronecker product bilinear usual calculation identity kronecker product importance note kronecker two hermitian matrix hermitian matrix kronecker positivity positivity let definite observe usual unique hermitian must semidefinite discover invertible discuss logarithm role elegant logarithm kronecker valuable logarithm kronecker product exponential equal formula rely apply mixed kronecker complete simply choose identity trace kronecker hermitian pairwise preserve semidefinite valid hermitian column argument course operator evaluate rule arithmetic introduce kronecker calculate kronecker represent relative let tell perspective q preserve conclude relative entropy draw variety range article source book recommend major result paper resolve conjecture concavity certain motivate quantum state uncertainty system control presentation idea corollary difficult concavity complex paper approach author prove concavity trace function implication relatively easy see matrix differ usual quantum mechanic quantum include change proof adapt directly paper nevertheless idea date paragraph relative construct relative divergence entropy divergence say construction suppose bregman divergence square divergence common introduction bregman divergence see ar divergence recognize relative paper recent divergence divergence classical result mechanic monotone characterization operator monotone quantity monotone interval fact scalar derivative monotone function pick develop theory monotone somewhat develop operator monotone convex formula operator write write nonnegative integral closely relate formula monotone logarithm motivate recommend book book monotone jensen establish decade book unable identify idea important direction pair pair prove entropy author kronecker integral construct definite matrix particular perspective year introduce quasi entropy involve influence presentation convexity operator jensen derive relative analysis remove argument kronecker product operator identity interpret consequence definition kronecker right presentation heavily skew chernoff version matrix hoeffding bound inequality value convexity relative still deep convexity chapter contain argument thompson inequality establish roughly martingale bernstein advantage study random show type article tail eigenvalue sum independent matrix chernoff semidefinite type sum establish stein pair inequality matrix sum moment arguably simple markov thesis lead satisfactory logarithmic exponential stein inequality primary estimation roughly bernstein inequalitie unbounded matrix extend hoeffde combine paper combine chernoff chernoff bind random positive semidefinite replacement role establish inequality derive inequality consequence inferior work concentration bound depend ambient develop variant argument control matrix rather ambient contain semidefinite dependence control essentially describe dimension intrinsic variance argument adaptation author present refined obtain intrinsic ambient dimension parameter important development matrix laplace thompson inequality inequality identically concern bernstein completion bernstein concern matrix overview moment concrete literature strong exponential inequality unfortunately typically abstract difficult recently von algebra finite equip power version inequality prove inequality sharp establish random moment growth describe fully argument appendix sum thm thm proposition thm thm thm thm recent mathematic classical expert decade arithmetic aim describe result page matrix discuss herein present coherent exchangeable omit seem broad researcher interested reader article annotate describe influence researcher great friend would like people improve reader inform I manuscript include anonymous suggestion give feedback final acknowledge office air award fa fellowship complete apply mathematic like institute foundation motivate begin connection computational mathematic application examine result assess presentation lie group orthogonal multivariate statistic another appear study behavior algebra solve system linear von arise take estimate procedure typically recent year nuclear early limit energy spectra slow random matrix appropriate quantum reaction random book lead distinct random field os model random theory arise throughout mathematic branch science several distinguish computer serve phenomenon reflect author interest mathematical problem compute develop fast multiply dominant singular input appear explain practice aspect accelerate matrix replace proxy elegant produce proxy entry rescale entry paper contain related idea play fast learn one subsample nystr approximation analysis template dimension many dimension random paper mathematical computer science appear variant idea combinatorial optimization relax constraint solve back round compressed object relatively freedom compare ambient able object refer central multivariate study property typical statistic area typical signal analyze identify column submatrix matrix structure analysis problem orient discriminate stochastic block one describe community assume individual community individual refer quite algorithm extract hold generally random model statistical procedure high theory key wireless matrix recognize may coincide reality allow sense generic intrinsic field phenomenon area random role edge vertice expansion property adjacency argument numerical bad elimination solve numerically however stability arise phenomenon probability matrix condition elimination state banach slice close ball turn slice dimension property quantum information information theory channel property hold capacity result random random theory challenge experience take intensive effort almost everything beyond arithmetic working variety special attention familiar maximum hermitian eigenvalue maximum hermitian eigenvalue probability eigenvalue something spectrum one distribute ask question act geometry three attempt bound application result expect relevant problem branch random fundamental principle describe introduction hope complicate main positive semidefinite star refer transpose operation position zero elsewhere word record covariance statistical imagine eq unbiased estimator know adjustment incorporate fit paradigm type tool theory substantially parameter express variable address substantially question elsewhere establish study extend explain kind return simplify variable center subtract second random sum piece usually together bernstein inequality independent sum proof refer yield substantially truly scalar bernstein inequality independent sum bernstein develop present give detail interpretation bernstein uniformly introduce sum denote proof appear version three salient mean hermitian coincide factor scalar limit include quite challenging expectation bind aspect far contain interpretation mean random introduce study depend assumption bound hypothesis relaxed variant appear random identically apply uniform variance uniform norm triangle hermitian coincide determine fact preserve drop reach square hermitian arrive need estimator invoke bind attain case qualitatively sharp argument corollary building showed essentially overview analysis result little every suppose introduce appear bernstein sometimes significant come factor give describe behavior distinguish tail useful tool practice collect precise book available concentration bernstein familiar exponential many admit focus independent describe random rectangular toeplitz entry chapter rademacher rademacher write fix independent thing random sign sign arise treat chapter chernoff bounds chernoff decompose whose subject uniform submatrix laplacian bernstein concern matrix spectral include multiplication feature paradigm chapter material inequality spectral depend ambient dimension illustrative concrete extend concentration quite useful mention annotate establish concern independent subject inequality extend tail bernstein improve positive semidefinite dimensional interesting obtain concentration martingale hoeffding bind bound norm view martingale bernstein technical explain bound matrix martingale set moment heavy tail reflect matrix simple form polynomial lead inequality annotate include moment inequality another establish argument exchangeable chain concentration moment inequality reproduce stein stein exponential random advantage exchangeable build exchangeable elementary approach take effort inequality material modify unfortunately seem student researcher mathematic random minimal algebra classical elementary beyond good chapter material tail sum matrix significant early example concentration major work concentration inequality practical random small eigenvalue appear literature include optimality necessary phenomenon basic result ease theorem concentration depend annotated contribution hope organization chapter contain background need develop concentration matrix chapter concern chapter introduce chernoff application chapter bernstein intrinsic conclude resource concentration make presentation smooth follow article almost appear chapter elaborate main overview extent concentration inequality careful cross reference able proceed discussion material concern behavior review especially us field write field array complex part depend specify really matrix none require essential consist complex equip complex symbol complex transpose product induce equip inner write complex space entry multiplication space algebra multiply frobenius frobenius induce topology matrix n symbol eq open define norm topology notion open write vector matrix add subscript instance identity basis linear vector write equal standard basis zero elsewhere letter unitary reader prefer prefer orthogonal analogous write hermitian hermitian multiply frobenius bold letter symmetric around represent hermitian hermitian number unitary matrix completely unique permutation eigenvalue denote algebraic eigenvalue extreme map homogeneous careful pass scalar map rarely hermitian aside arise usually eigenvalue weakly confusion prefer set read hermitian term square denote trace existence eigenvalue trace hermitian matrix valuable relation connect frobenius norm calculation semidefinite hermitian equivalently hermitian role nonnegative positive hermitian family form subset combination positively homogeneous geometric describe consequence semidefinite close convex begin consideration definite matrix form cone real semidefinite nonnegative preserve importance hermitian matrix general dimension semidefinite property semidefinite trace extend hermitian eigenvalue let matrix entry matrix confirm sensible fold hermitian whenever power logarithm logarithm definite matrix non power matrix immediate important consequence function theorem hermitian eigenvalue power expansion series real eigenvalue generalization value fail nevertheless inequality extend semidefinite transfer rule hermitian whose decompose immediate allow invoke hermitian equivalently series expansion hermitian always exponential monotonicity dimension establish logarithm logarithm functional q valuable logarithm preserve definite dimension stress monotonicity decomposition admit nonzero value order square decomposition square conversely extract eigenvalue decomposition expression expression property singular unitary hermitian definition hermitian matrix vector coincide q identity application lower express rank stable rank continuous power hermitian matrix hermitian hermitian clear hermitian discover coincide invoke repeatedly justify first column unitary appear calculate norm coincide construction identity inequality eigenvalue derive depend norm matrix singular norm schwarz inequality focus connection prefer abstraction unnecessary frame sufficiently justify expectation limit valid broad circumstance particular position help helpful clarity scalar letter gaussian variable letter take measurable think letter hermitian letter subset simply expectation linear matrix identity far comment markov nonnegative random obeys central tool concentration inequality jensen convexity eq form cone matrix preserve deviation extension concept hermitian value interpret column write value use number eq quantity mean wish rewrite matrix statistic random finite hermitian familiar statistic sum inside statistic relation identity suppose random dimension semidefinite hermitian variance hermitian valuable reduce variance scalar define definition coincide original deeply hermitian direct calculation value second definition matrix maximum diagonal block hermitian interact independent repeat calculation lead summary sum arise everything establish discussion reader treatment matrix analysis excellent book book reference book introduction two process book book comprehensive useful book aspect modern extremely survey work classical comprehensive treatment offer book matrix theory solid chapter core reader concentration may move concentration bounding recommend satisfactory extension allow transform argument concentration elementary purpose chapter application fluctuation hermitian inequality eigenvalue hermitian develop independent challenge arise present idea overcome result allow develop abstract chernoff bound bound heart laplace present hermitian note expectation may varie aim expand formal coefficient refer matrix moment hard set laplace transform start hold tail eigenvalue hermitian extreme classical fix eq hold monotone last identity depend hermitian positive eigenvalue achieve prove monotone identity inequality discussion convexity trace adapt hermitian analog scalar eigenvalue hermitian matrix q positive eigenvalue state relation jensen proposition draw final inequality depend trace positive laplace transform set laplace method sum decompose case subtle indicate thing sequence satisfy multiplication relation sum scalar relation variable imagine perhaps unfortunately hope subject hermitian situation improve result thompson physics analogous eq consider real relation extract logarithm look sum hermitian hermitian q identity fail nevertheless admit satisfactory scalar proof deep consideration discover convexity trace exponential fix hermitian definite analogous result describe complete let consequence remarkable valuable transform section involve hermitian let random hermitian interpretation logarithm functional expectation generalize fundamental approach matrix finite equivalently rule expectation remain random matrix hold iterate invoke formulation follow substitute finally general sum laplace develop apply master matrix sequence hermitian furthermore eq proposition nevertheless sum rectangular hermitian present hermitian include historical establish trick researcher work concern transmission information version substantial number major appear appear generate technical repeat thompson argument hermitian case coincide identically fundamentally weak bind worst unnecessary detail beyond argument appear develop effective thompson establish bernstein analogous bernstein value specialized bernstein constant concentration approach base introduce article recognize content lemma master tail article detailed discussion thompson see get appear seem weak certain advantage however matrix set subsequent research martingale version work report tail independent matrix closely old theorem moment contain extension control fix variable researcher moment scalar random admit overview mention moment play concentration tool quantum mechanic seem distant area study via quantum systems thompson inequality major quantum mechanic book physical book thompson three combination spin first establish important trace main establish quantum system chapter present set fix variable formulation surprisingly range simple precise finite fix independent spectral matrix look express use matrix gaussian represent build attack classical toeplitz idea treat sum recall take new problem spectral sign entry begin overview series bound subsequent describe concentration example substantial part conclude sequence standard routine scalar transform demonstrate turn directly rademacher dimension finite variable appear message subgaussian tail whose decay follow coincide formula bind reduce case nothing subtle scalar see illustration mark dark red vertical coincide tail decrease subgaussian variance answer section claim series argue quite norm available especially begin indeed spectral jensen step use integration tail integral explicitly complete ask exhibit construct series right correct see dimensional gaussian lead logarithm sometimes major computable factor chapter moderate type series remove technology
deviation generate value increase use call pair rank induce comparison improvement select sorting improve noisy small active furthermore seem active keep survey refer aim website image option date million outcome collect purpose result outcome item knowledge generate pairwise despite comparison remain item proceed bt comparison outcome model passive approach experiment passive use realistic figure systematically observe outcome average fitting poisson considerable fraction express opinion difficult probably look preference type large comparison result outcome outcome rare compare active bt necessarily hold bt fit bring quickly passive example reach collect assume order denote observe inconsistent ranking element pl formalize wrong decrease observe bound q di operation item compare misclassifie misclassifie item correctly sort represent precise next operation misclassifie constant misclassifie therefore argument consideration partition proceed main proof partition operation cause chebyshev inequality remark know need comparison convenience different constant value maximum consider operation recall chernoff let call tree randomize step item middle least partition subset condition though middle unlikely small least leaf recursion therefore least item lemma never exceed total item exceed theorem survey pairwise noisy active black sorting enable efficient practice perform systematically centrality recently rank aggregation provably convergent iterative item collection comparison diverse range rank player game preference recommender arguably comparison simple human attempt question aggregate million cope inconsistent noisy outcome bt model bt item noisy item distant item already distant strategy agnostic strategy explore active sort recover basis efficient approximate run sorting induce item rank sort repeatedly budget sorting ignore set pair rank outcome develop likelihood estimator bt start work centrality aggregate finite bt analyze generalization first interpret likelihood ml enjoy essentially provably convergent compute induce bt choose advance investigate strategy sort label bad mistake finally theoretical finding speed outcome sort notation use throughout without loss generality denote preferred preferred define say informally bt observe outcome inconsistent rank decrease intuitive define particular way make parameterization probability prefer multiplicative assume sometimes grow number parameterization model give w parameter parameterization enable make intuitively uncertain outcome concave tractable algorithm centrality aggregate comparison bt square comparison select result mse insight centrality relate among contribution way extend comparison centrality estimator present minimax minimax optimal recover observe minimax instead novel analysis estimate way guarantee mse bt far parameter include impractical dependency ml bt unlike ranking accord ml estimator variation justification comparison noisy model sufficient one noise development provide bt consider datum bt set bt ml pair provably convergent iterative estimate ml centrality towards preferred consider factor ensure stochastic walk diagonal node transition represent unbiased intuitively towards stationary k interpret contract probability reason error interpretable proof bound stay error move relate estimator one centrality satisfie balance compare hand likelihood claim satisfy ml consider approach lead directly expect scale approximation centrality alternatively absence knowledge fall centrality furthermore stationary factor ml w markov essentially let key insight error stationary suffice comparison theorem additional factor recover alternative completeness satisfies w q show pair vanish ml factor newly enable markov via equation error available procedure maximum adapt centrality centrality describe chain irreducible proof mapping soon self irreducible ml estimate define lastly arc similarity maximization solve relaxation however mm whereas gradient comparison random think seek try rank bt consideration change let assumption range grow bt dependency induce easier distant outcome noisy well even actual grow ultimately easy bt arbitrarily make realization bt measure rank one noiseless necessary sort ranking question characterize produce sort operating section sort select uniformly two always rank outcome claim algorithm comparison non ranking read pre traversal notice plug bt theorems average produce bt model ranking sketch proof sort comparison mistake
conjugate pick latter optimization refer shown convert dual high solution allow indeed yield clearly unique verify coordinate point denote point generate minimize coordinate sequence variable start compute hypothesis expectation process govern eigenvalue order normalize eigenvector obtain distance great showing rate generalize denote continuous strongly comprise smooth strongly quadratic minimizing minimize observation since stationary canonical b correspond initialization form far assume precise seem restrict respectively inversion evaluation comment equivalently correspondence soon play polynomial optimization matrix radius sake brevity characteristic inversion lastly minimizer consideration assumption consistency say initialization say broad characterize rule linearly first instead precise descent newton also degenerate vanish ball detail descent algorithm rewrite act repeatedly minimize row popular mainly stationarity requirement fail extension cyclic piecewise future conjugate similarity descent let x clearly form g stationarity imply minimize assume arithmetic computational scale iteration computational inversion time notice coefficient inversion execution inversion level accuracy rigorous fact analogy respect accuracy measure employ theorem provide characterization root long dp root characteristic polynomial evaluate remark constant may consideration minimizer second derivative sake clarity initialization subtle issue regard like point say however employ fortunately although strongly q combining depend distance logarithmic setting equivalent bound imply adequate section presentation case present useful polynomial case finite explain inversion conclude use establish efficient specification see reveal turn extremely characterization condition hold matrix inversion consistent recall consecutive side yield rearrange hand consistency generate convergent limit point formalize system hold precede extensively illustrate significance simplify deterministic free characteristic modulus root ideally like consistency condition maintain seek imply ingredient choose inequality q plug bx apply lower bind e simplify optimization generalization accord derive low root b brevity dependency upper triangular denote order root characteristic polynomial consistency derive modulus root end find section otherwise hold subject obtain design presence see assume consistent root eq reasoning optimization arrive let motivate effectiveness state relate inversion bind compute know super quadratic possible root q consistency regardless strategy balance approximation iteration put differently various restriction inversion turn polynomial rise low quadratic inversion bound optimization scalar meet much wide inversion depend derive low smooth case already hard matrix turn positive meet sake eq maintain scalar inversion split range range low q condition x good scalar matrix overall exist low tight turn suitable reach spectral decomposition minimizer restrict attention coefficient efficient attain see optimization namely inversion disadvantage guarantee convergence satisfy imply since algorithm extension radius coefficient radius us characteristic factor consideration inversion take follow scalar maintain function path natural economic polynomial hope root hold follow equation substitute get linear multiply remarkably enough table equation extension heavy polynomial relate consideration strongly could argue recover specification say necessarily fortunately coefficient reformulate substitute original sequel briefly property canonical extension path optimization nx denote scalar thus rearrange plug q function extension behave answer essentially minimizer unlike guaranteed initialize close enough converge smooth investigation precise principle kind unconstrained sequel employ show theorem tight end lemma modulus root polynomial uniquely attain eq matrix admit state fix need factor characteristic eq accomplish orthogonal positive exist coefficient accordance theorem whose characteristic optimal optimization parameter decomposition use grow yield iteration spectral structural impose polynomial analogy state polynomial spread derivation numerically optimization oppose employ coefficient eigenvector require function vs rate comprise close theoretical gain counter initialize quadratic imply optimization must x eq b somewhat bind optimization algorithm nesterov consider regularization term case shape spectrum preserve demonstrate nesterov low space consequently eigenvalue order derivative relatively distant quadratic fast coefficient shape spectra applicability real simple idea give application analyze allow prove elementary lemma determine recurrence application space converge series note part seem algebra matrix eq modulus exist trivially diagonal index respectively may magnitude row follow equality plug h inequality derive scalar derive bind equation term tend entry zero equal precede constant equation yield u sum power exist claim define direct implication establish update equivalently express new euclidean q mapping convention equivalent regardless initialization improve functional coefficient matrix follow discussion furthermore expression iteration n realization iteration matrix convention multiplication order product multiplication expectation side r h update suppose thus consequently convergent question fast need x previous initialization exist satisfy complexity eq note space sufficiently q govern radius suppose f x f x radius equal radius characteristic combine corollary proof characteristic iteration apply elementary determinant polynomial sake omit precede algorithm implication immediate consequence characteristic polynomial initialization I sake omit dependency moreover verify eq plug equivalently denote degree prove fundamental root root q eq write second yield conclude part reverse triangle remark whereby eq conclude real prove convergence rate inversion optimization whose inversion general dimensional dimensional mx definite note use argument positive satisfy precede rest carry scalar ne develop novel smooth strongly algorithm deterministic focus quadratic recursive turn reveal connection whereby whereas low natural lastly polynomial novel systematic nesterov accelerate rather one solid motivation descent descent mathematical interested solve form eq problem science economic readily express far reach say solve approximate various along convex precisely continuously e wide interesting fast kind problem say answer otherwise address nature computational accept theoretical analyze seminal propose information regard impose resource show employ receive oracle obtain e start view query attain descent seem trivial query exceed moreover class attain e require intensive quite common gradient variant simple indeed quadratic use oppose optimization utilize accelerate heavy ball see sag cyclic piecewise accelerate sdca inspire boundary admit stationary rule canonical algorithm interpret algorithm linear essentially magnitude bound analyze procedure analyze lyapunov mathematical however primarily derive magnitude eigenvalue work low bound merely root maximal modulus absolute root polynomial condition polynomial polynomial correctly rate purely theory modulus root although vast bound root g good bound bound radius adequate consequently new tool argument derive heavy polynomial chebyshev polynomial g formally optimization iteration execute efficiently obtain partially early whose
distribution c generate albeit additional constraint constraint addition normal point associate b broken part overlap obtain alignment synthesis overlap decide point simulate identifiable slight use affine evaluate across alignment accord base outperformed emphasis region component propose variance scale measure propose elastic within measure alignment dataset neighbor bandwidth tune base error clear observe specific dataset outperform scale offset difference trend affine discrimination ratio propose elastic contribute call compare difference elastic distinct generate classification quality elastic difference include subsequence move merge penalty compare elastic overlap measure outperform seem offer perspective offer specialized advantage competitive elastic loss elastic suggest dataset even elastic evaluate dataset dataset good ensemble classifier demonstrate series ever literature propose difference evaluate incorporate ensemble difference measure figure classifier elastic difference dataset measure base nn error method rank dataset rank rank note difference compare elastic describe figure nn absolute exceed might far scale regardless appropriate offset iterative manner update accordingly normalization simulation normalization difference mean evidence well classification want property alignment scaling offset dependent occur motivated demonstrate width crucial sensitivity width sensitive tune ever dataset extract subsequence delay wave medical interestingly tune region width roughly offer handle baseline motivate example thereby nn outperform alignment runtime complexity share recall convergence alignment compare dataset database dataset database alignment calculate randomly select series computation largely follow tune nn exceed almost actual differ multiple dataset require reflect cost large alignment affine emphasis variant set pointwise alignment reflected match variant offset amplitude emphasis combine advantage affine apply globally locally outperform dataset nearest neighbor associate alignment recognition alignment arrange match analysis series finding point match point financial stock market gene activity match time assumption illustrate purely alignment variation connect line visually focus series comprehensive survey demonstrate produce context result context two time amplitude scale offset bias well offset alignment simultaneously fall temperature variation environment alignment temperature subject offset variation figure series peak series normalize extent visually consistent scaling offset rotation mapping impose offset alignment model mapping time transformation arise desirable accommodate scenario substitute pointwise distance interest potential determine characteristic shift degree overlap alignment desirable affine emphasis local one local manner affine offset emphasis subject temporal location place emphasis section heavy amount emphasize short offset temporal variation time scale offset series slight movement objective correctly align rest review method conclude paper easily interest point match assume mention otherwise match time subject search optimal alignment minimize subject monotonicity boundary monotonicity step refer difference programming reduce find alignment constrain problem scaling offset similar way assume run compute time place weight potentially accomplish substitute pointwise measure width distance summation programming manner formula turn observation applicable correspond time width discuss crucial achieve alignment alignment offer target global model scaled offset time emphasis goal minimize subject finding solution em apply convexity set derivative v gs c gs g c well alignment emphasis h scale offset find minimize eq q follow match eq constrain dynamic utilize manner eq backtracking apply appropriate attribute element update element follow observation update thus furthermore complexity introduce bandwidth detailed bandwidth length evaluate reflect reflect stop parameter section unless complete evaluation tune tune
q next rate produce compare coordinate project coordinate even consider asynchronous point old method benefit match assume geometric rate author rate coefficient linear coefficient precise result worse depend broader strong several define project employ project define ready admit relax begin follow structure strongly recently show satisfied convexity hold estimate global property prove fx theorem gap auxiliary eq iteration depend coordinate compute otherwise optimality know q hold remain know x therefore auxiliary imply give hold conclude fx fx x proof optimality use projection plug expectation easy ready choose latter read x w vector expectation observe function q notice minimize q combine gives theorem usa engineering framework descent feasible sdca lasso problem framework linearly duality sdca dual interested convex feasible produce hold every yx include cyclic fit randomize first become question extend randomize give randomized version formulate inexact projection algorithm gradient corrupt fit notation enjoy convexity vector satisfy weak convexity property weak convexity smoothness wise continuous denote continuous lipschitz projection operator coordinate learn literature framework also detail goal label find minimize svm hinge loss smooth j j smooth special strongly double lasso reformulate machine parameter use loss fit square error solution prove feasible enjoy convergence begin locally global property show rate author asynchronous weak recently show smooth global convexity property contribution framework show fit randomize previously expectation duality gap sdca duality gap section compare result briefly review use duality converge linearly sdca apply dual svm brief summary cover gradient cyclic inexact classical method good knowledge framework popular minibatch method fit r randomized descent random property exposition r difference r whereas r weak must iteration deterministic hold framework see later relaxed theorem option r minibatch analyze describe input matrix minibatch option x k option function coordinate strongly problem dual neither assumption hold coordinate descent remark compare cyclic rule coordinate f w nr however
reproduce system capture feature fine picture decompose color distribution equivalently color construct color optimally color color dynamical system statistically sample prescribe distribution consequently equation material lyapunov imply sensitive dependence dynamical system investigate particular convergence mcmc compare sampling hamiltonian mcmc multi modal dimension metropolis hasting require proposal prescribe arise htb supplementary text trajectory generation dynamical system prescribe dynamical trajectory track visit dynamical dirac delta integral rd rd spend coverage domain region distance spherical fourier basis satisfy belong rectangular describe constant easy control describe fouri control maximum equation dynamical evolve trajectory color produce computation potentially multiple different lyapunov signature lyapunov spectrum initial divergence trajectory trajectory rectangular region equivalent picture identical pixel outline equation give eqn note computation lyapunov jacobian analytical jacobian eq q compute wave particular dynamic dynamic jacobian determined along approach compute dynamic lyapunov lyapunov exponent note display dependence initial condition however final statistical invariant hamiltonian investigate extensively rare extensively machine form research hasting slice modal compare compete provide high computation construct fundamentally traditional chain successive pick history trajectory brief description metropolis hasting hamiltonian mcmc comparison hasting popular accept reject hasting tuning propose base distribution much trial error pick distribution modal example hamiltonian hamiltonian hamiltonian resample perform metropolis hamiltonian explicit proposal momentum dynamic typically perform hamiltonian metropolis sample avoid hamiltonian underlie target momentum variable hamiltonian one momentum sampling graph uniform essentially point constitute slice advantage one implementation slice software metropolis hamiltonian pick normalize dimension second gaussian distribution deviation correlation see pick high analytically metropolis hasting proposal eqn mcmc eqn respect simplicity use predict figure explicit dynamical lie reject trajectory pick rejection error modal significantly hamiltonian slice additionally x metropolis hasting hamiltonian mcmc approach require proposal markov monte sampling primarily address additionally development rough integral switch present evolution run euler trajectory trajectory produce additionally correspond evolution capture movie htb vs time lyapunov imply modal use comparison hasting htb convergence sampling hasting slice htb frame superposition trajectory evolve htb sampling first superposition blue frames red green frames compose evolve htb evolve frame superposition frame trajectory evolve frame superposition red green blue red green compose trajectory evolve movie superposition green frames green single trajectory evolve big first frame superposition green compose single evolve superposition blue red green frames color evolve cm united center ct united research center berkeley dynamical da water system aside aspect picture novel reproduce control dynamical property color picture capture refine beyond reproduce expect quantification scalable develop acceleration monte design dynamical availability seem manner modern fundamentally human picture speak pixel determine appropriate level intelligence human objective human capturing randomness prescribe relate theory ergodicity system time average function equal spatial average individual trajectory color desire dynamical exhibit supplementary picture irrespective lyapunov potentially robot applicability challenge lie heart probability mcmc likelihood tight slow complex modal machine big supplementary manuscript present hamiltonian slice low construct euler scheme beyond capture fourier rectangular computed take wave picture force converge great give scale dynamical achieve weight fine one fouri fix maximum higher additionally run burden due fourier underlie mathematical reader supplementary image decompose yield value color obtain color prescribe time color demonstrate
detail completeness implementation simplify equation shorthand quantity start macro remark wolfe community theoretical property investigate application learning focus fw suggest study promise medium benchmark svm classification frank wolfe hereafter fw researcher fw enjoy powerful iteration execution improve rate basic work iteration advantage super complexity result suitable learning statistic bioinformatics classification fw hundred thousand thus provide promising medium focus previously kind benchmark show able accelerate average speedup dual tolerance parameter strict running consider random elaborate advantage drawback general overview fw modification theoretical property examine fw numerical assess fw continuous idea fw exploit iterate direction fw algorithm define k either fw sake primal sublinear optimal algorithm satisfy give computable criterion motivate primal iterate bind well furthermore tolerance clean endow theoretical standard fw exhibit tends become orthogonal date consist algorithmic variation add variant kind frank wolfe method modify fw alternative eq good descent pairwise swap fw iteration value prove enjoy analogous algorithm extensively discus refer literature option improve fw iteration conjugate fw fw hull include paper due variant gap obtain iteration another fw case advantage would possibility explanatory reach approximate observation successfully fw solver motivation mainly fw solution approximation fast large svd prohibitive motivate svm significance allow comparison effort propose traffic fw iteration good scheme investigate basic incorporate average fw iterate via inner cycle south anchor anchor south circle fill fw black dash pos color thick dash fw thick swap current fw direction use obtain look iteration fw tend orthogonal figure consist extra fw case depict directly apparent approach advantageous function scale red dot circle black label pt acc acc pt fw gain dataset cpu count due employ comparable appear issue accuracy possibly capability fw promise solid solver experimental preliminary provide example application fw machine variant
evident hyperparameter automate search community believe benefit greatly project g circuit cancer grant fellowship medical project van cancer via plan bm image rgb frame language title hyperparameter challenge capture element common feature choice hyperparameter complex theoretically sound strategy focus capable capture give coherent within ability predict give discrete range neural parameterized appropriately user learn various result hyperparameter manually hyperparameter predefine approach impractical number briefly inherent combination search current available software key balance learn choose datum poorly unseen overfitte low trade strongly biased hyperparameter control instance via network learn task include mean construct involve parameterize parameterized hyperparameter formalize tuple return give algorithm often split hold obvious train ensemble subtle instance affect training exhibit inherent randomness initialization estimating sometimes typically function usefulness setting optimum fortunately many search probe optimum good post smoothness usually hundred step many case usually type integer hyperparameter architecture ensemble task highly complex space hyperparameter conditional optimize hyperparameter layer induce number particle genetic couple anneal algorithm surprisingly randomly establish relate sequential base use variant
object ds discussion propose crf object characterize sequence connectivity dynamically base inter frame optical flow adjacent facilitate spatial context probabilistic ds crf model facilitate pixel object time target experiment real world capability crf exist track propose ability change inter establish optical connectivity layer deep meaningful temporal change inter frame optical connectivity adjacent maintain prediction factor incorporation object also handle shape ds crf incorporate inter optical via descriptor inter connectivity within video sequence furthermore explore extension crf relationship boundary aim explore ds crf video energy work support science development author z ds crf w work formulation derivation ds crf cm cm department computer mail abstract field crf purpose tracking ds particular adjacent inter layer connectivity dynamically optical flow incorporate context dynamic structured ds allow accurately change greatly well within scene experiment surveillance multiple ds approach tracking tracking introduction predict structure interesting important number object video challenging dynamically change drastically early tracking state kalman filter state observation motion follow behaviour filter address make filter kalman resolve non behaviour motion address behaviour lot use particle filter arbitrary filter difficult especially track appearance change drastically dynamically time recently significant object track generative probability state relax independence assumption generative method study base et al within video must predefine spatio model object object component correspond temporal constraint propose purpose tracking image pre segmentation temporal dependency conditional estimated frame subsequently refine subsequent frame exist segmentation video foreground time different crf similarity spatial continuity motion crf resolution crf approaches limitation crf limited great frame increase limitation object recently concept structure facilitate modeling without complexity incur crf deep crf state improve inter layer et crf linear chain replace sum yu al structure crf compose layer layer track video contribute predict frame efficacy crf model concept deep structure discriminative introduce ds crf state spatially characterize inter dynamically inter optical spatial ds develop efficiently stage situation large change material object classify pixel foreground object goal characterize discriminative ds framework tracking object level frame characterize frame form structure conditional adjacent tracking follow detailed ds crf field amongst use crf crf measurement require sort independence commonly formally undirected vertex crf markov except neighbor respectively crf normalization essentially call partition clique potential maximum crf relationship amongst clique number feature vision segmentation classification undirected crf although early relationship amongst track play role crf inconsistent importance appropriate crf predict base frame without movement two frame see prediction result poor crf object poor tracking tackle appropriate track al incorporation optical crf modeling relationship visual approach promise crf motion frame position motion dynamic change handle motion tracking benefit manner address issue crf crf state model along motion dynamic scenario deep structure random propose detail layer ds crf establish dynamically temporal observation incorporate ds present graph representation ds crf characterize model conditional normalization intra clique feature inter optical target object inter state spatial clique state correspond motion inter connectivity temporal mean inter layer clique inter gray object movement frame feature inter connectivity incorporate crf optical flow target appearance crucial velocity adjacent frame optical optical motion upon sequence specifie move motion optical flow pixel inter frame optical observation utilize unary appearance feature describe appearance appearance unary shifted velocity shift base target scene imply strong target word target frame add target change rough segmentation enforce segmentation frame ise utilize problem incorporate feature function estimate training propose ds maximize log concave global derivative parameter respect exact iteratively ds crf training belief train ds crf graph decode determine state maximum eq ds crf tracking crf across frame annotate velocity two ds crf starts frame crf optical flow optical frame crf rule context object motion dynamic spatial within ds crf model describe base learn ds crf computational complexity crf tracking utilize object field indicate object pixel temporal observation correspond binary issue label suit object appropriate object issue target accomplish determine component field determine detect target matching evaluate ds crf purpose object tracking perform understand analysis different involve motion capability ds handle motion set video human move ds crf handling drastically shape object acceleration ability track object different time simulated acceleration motion object move constant velocity acceleration well frame motion motion track uncertainty motion object appearance life video target ds crf scenario object drastically shape different move used evaluation illustrate capability sequence scene bottom capability sequence scene become person scene illustrate capability top third top scene bottom exist tracking shift tracking maxima achieve previous base measure track tracking detector semi
coincide finally coincide specie sample py us genomic datum express tag est obtain free widely biological consider library cell previously constitute sample c library library count discover step sequence estimator complete specification mention clear estimator exhibit value rely model jointly estimator note exhibit diversity already fix additional sample lie basis library basic py discover gene step decade frequentist consistency major generally accept see study suitably exchangeability frequentist term shall asymptotic kind asymptotic achieve modify among start fix concept first datum assume term type natural neighborhood achieve neighborhood support ensure furthermore desirable study gibbs mixture model gibbs process condition hold suggest type discrete come distribution serious really generate design must w compatible gibbs primarily interested investigate type come strategy show weak say checking identify investigate asymptotic behavior explicit allow guess neighborhood quantity section sample exchangeable choice clearly sure hand discrete henceforth shall stand turn asymptotic discover gibbs type constant eq comment regard order type prior expression regardless mild distribution guarantee trivial exclude discover converge assess consistency limiting think bad concentrate guess learn place visualize case py already predictive distribution imply unless correspond see concern prior focus occurrence phenomena mixture py sufficient state behavior show sufficiently extremely mild assumption mixing require ultimately meet commonly measure one gibbs always discrete tail state lead satisfied gibbs characterize specific mixing focus conclusion hold ensure consistency prior characterize heavy tailed admit bound admit limit second positive integer light since consistent sub prior therefore range increase large limit assign guess identify tail conversely large finally minimal inconsistent behavior even close extension previous refer random object temporal transition stationary least coincide gibbs type important distinguish area random prior outline former drive inferential purpose analytical community concern extremely front besides contribution back generalization dirichlet framework less restrictive dependence covariate date dependence simulation slice factor inferential framework root relate nonparametric underlying construction dimension depend specie allow species rise yield population frequency marginal poisson dirichlet clearly prior author opinion highly possibility interest diversity process constitute concern dynamic hand activity start vast literature concern law use nonparametric prior choice two instance neutral typically since conjugate right conceptual prefer conjugate conjugate convenience prior thing make assumption value depend assumption empirically quantity gibbs counterpart consider exchangeable sequence request invariance nonparametric type exchangeability rule obtain dirichlet turn made nothing implication result concern importantly logarithmic dirichlet spectrum going linearly increase flexible component distributional expression derive appeal retain intuitive relate learn instance concern prediction require frequency sufficient specie frequency accordance key reinforcement mechanism scheme implication sum review provide answer question title confident see well bring foundation obviously prevent gibbs type e concrete drop european fourth associated infinite partition product equivalently depend categorization species model iii specie amount first depend gibbs process case frequency require namely prediction exchangeable hence side coincide side coincide contradict nk iterate step mass characterize dirichlet nk cm remark pr universit di universit di de sigma mx exchangeable induce key address popular surely induce exchangeable elegant dirichlet prior appeal view admit characterization use term precise assumption learn stand iii special besides unified treatment highlight implication nonparametric frequentist concern serve idea intuition inherent class prior dirichlet phrase bayesian exchangeable partition law act several recent review cover use completely measure concept one trade far inference concern analytical represent parameter review also especially terminology introduce adopt py reduce parameter nonetheless distributional py process fundamentally equal class give prior briefly motivate inspection apparent distributional crucially novel prior serve use moreover gibbs analytic issue nonparametric stage highlight allow simplification relevant follow simplicity admit prior notable process inverse generalized gamma prior paper provide survey gibbs type account probabilistic literature point flexibility inferential beyond retrieval survival among bayesian exchangeable framework focus ideally assume distribution use bayesian inferential dimensional typically inferential generally topological desirable ahead distribution prior distribution py say broad gibbs unit concentrated variable henceforth iid general terminology follow sample generate predictive conceptual view observe distinct include namely dirichlet py process whose admissible value integer apparent recover exchangeable another paper essential extremely interpretability induce observe py form py identification lead predictive predictive combination p interpret guess observation prior particular suit address inferential specie sketch framework specification apparent describe structure type proportion model proportion integer correspond nature differ reveal connection terminology adopt name virtue adopt typically distinct specie detect number frequency draw consist specie frequency rare diversity interest biological linguistic play important answer basic building model dependent keep thing estimation define sequence real introduce bayesian date serve cluster distinct group inference great gibbs section provide suitable specie follow overview distributional emphasis discuss type mixture gibbs prior deal frequentist asymptotic discuss extension gibbs prior dynamic context conclude remark title interesting specie generate new induce characterization gibbs type result term probability generating past associate specie specify classify denote distinct value frequency denote k kk model classification generating n px n otherwise prove conceptual view seem generate new suitable specification scalar indeed depend distinct since summarize heterogeneity virtue gibbs specific want increase correspond later iii correspond general information principle operational need hand rise serious case typically quite complicated expression specify observe reflect opinion mechanism opinion finite mathematical type due simplify prediction appear flexibility motivate state predictive provide exchangeable type specie sample negative recursive light reason product accordance mixture dirichlet mix base ii denote factorial depend generally obtain respect still lie gibb clearly specie although preserve conditionally induce predictive distribution guess weight predictive intuitive observed consist value conditionally determine old value past observation weight reinforcement mechanism place see ratio probability assign cluster proportional size cluster cluster frequency represent appeal context reinforcement mechanism mechanism work besides exchangeable element determine size introduce eq limiting term worth dirichlet parameter class nice structure share light aspect important assess property essence nonparametric candidate topological natural nonparametric gibbs requirement full priori positive unbounded possess measure whose prior guess space outline introduction application discrete type prior occur within hierarchical correspond exchangeable density eq particular ingredient model random set index inference numerical mass allocate reinforcement mechanism look induce cluster determination factorial different let eq denote number display suit highlight difference fix component dirichlet control distribution large right imply essentially process role controlling play interesting concern display straight line simplification evident allow distribution yield flexibility py htbp dirichlet type characterize value simple suppose expect number nonparametric fix five process figure clearly informative large variability dirichlet imply specification imply information cluster often furthermore py latter produce k five implication specification end well separate clearly nonparametric specification process distribution hierarchy setup prior opinion wrong whether possess towards namely iteration burn adopt algorithm acceleration depict posterior thing posterior py towards py see value strong mechanism prevent completely wrong prior htbp c py py finally consideration parameter beyond toy represent structural understand g concern considerable heterogeneity inference component well py depict already address prediction specie compose individual biology economic one inferential purpose yield piece information specie label specie last alternatively reformulate specie must typically summarize form induce resort prior characterize novel estimator unobserve sample overall estimating specie correspond frequency thought overall specie hand quantify estimator specie estimate realization unobserve sampling either consist discovery rare specie population compose display specie suit tag serial complementary library rna population goal consist identify compose frequency gene library population due portion whole library overall characteristic framework take example identification status estimate contrast population limited specie type model deal py one display finding carry topic frequentist relevant estimator probability namely proportion specie quantity framework discovery term good estimator coincide sum yield numerical instability arise moderately greater enough appear illustration hand frequentist discovery new distinct
fail able remainder summarize accurate efficiently linear complexity present format format level thing provide evaluation superiority algorithm compute resp submatrix interaction algorithm column basis inner compute non rank storage scheme rank approximation latter vector thus full research scientific aim accelerate partition box interaction potential grow exponentially dimension fail lack scalability transform offer multiplication growth polynomial moderate growth via gold low cost work randomize onto subspace approximation nystr om sampling computationally well matrix uniform score version nystr om column leverage score variant leverage without entire improve random method choose alternatively qr factorization nystr om use cluster real feature euclidean inner problem matrix rank datum cluster block wise approximation require avoid formation method stable deviation trial briefly difference vector block accurate sampling selection rank cluster desire give memory fast product algorithm capture keeping partition denote define th approximation j submatrix submatrix rank memory cost independent store htp linear give detail selection similar fashion point partition shift invariant state shift kernel ft continuous lipschitz partition q proof strategy radius bind reduce appendix differ superior tend evenly cluster already matrix cluster compute basis submatrix restrict randomize svd however form block construct entire impractical massive cost restrict column row u u randomize appendix construct ir achieve satisfy low cost interaction block diagonal achieve reconstruction bound pt therefore rank look low memory proceed memory observe enable minimizer small vector matrix maintain ir inner list include vector benchmark art approximation method ghz memory census house exact computed approximated matrix stand frobenius approximation method deviation behave memory close approximation time addition implementation cost require memory cost carry time demonstrate show value also spread dotted indicate standard speed small increase memory comparable memory exhibit observe deviation approximate low approximate fix get include property get become less kernel go portion memory bad kernel error matrix diagonal increase part entry vary result demonstrate scheme work kernel parameter kernel dark blue cluster roughly describe rank gaussian vary accuracy ptc growth dataset comparison increase enough require behavior performance synthetic dataset behave matlab plot vector total plus case behave differently examine consistent complexity work comparable higher compare method memory randomized appendix center quickly optimum algorithm center centroid centroid assignment cost iteration svd svd approximation briefly describe rank denote apply qr get svd matrix work space seek row apply index denote denote apply qr simplification get side code author column replacement author code matlab interface claim theorem li scientific storage kernel parameter radial mostly case storage idea scientific computing situation construct block approximation block factorization work extend applicability demonstrate deviation superiority dimensional role machine compute
primal reason algorithm avoid optimality slack allow conditional insight operator alternate smooth penalty elegant close fashion dimensional break small subproblem retrieve solution subproblem quick splitting conjugacy simple eq lagrangian lx hold primal point dual idea ascent solve problem ascent therefore iterate appropriate take lagrangian lagrangian add ridge like lagrangian problem q primal minimum convex mild dual ascent iterate dual update ascent upon notice rewrite augmented formula residual shorter augment lagrangian problem bregman iteration compress recovery signal know coordinate observation augment step usual step bregman algorithm appeal divergence idea admm direction augment problem admm similar ascent lagrangian individually jointly pass alternate dual problem correspond nesterov regularity convexity convex bregman divergence induce vertical guess multivariate everything carry bregman family parameterization sometimes repeatedly context envelope recognize dual value value generating parameter induce use variable splitting parameterization expect parameter still canonical bregman split penalty poisson simplification split divide optimisation form add slack divide together use split admm bregman admm divide break hard split global section envelope envelope generate step envelope relationship framework extend possess constant assume proper quadratic envelope possess property pick symmetric definite stationary envelope original satisfy p lx establish property envelope function backward envelope evaluate gradient envelope produce algorithm way iterate class envelope apply norm convex see rate comparison quadratic envelope variable usually understand convex need newton iterative mapping diag lx v b gr bregman objective make envelope divergence attain bregman law establish descent mm insight generate envelope add smooth penalty exponential bregman divergence composite statistical structural making term construction statistical address mapping broadly combination together broad art start summarize function split duality motivation primal lie problem refer formulation primal joint exact exhaustive view formulation dual proper convex specify exact sub form computationally critical related form envelope representation objective slack proximal especially proximal construction proximal like move objective worth efficacy depend proximal connection literature proximal quadratic lagrangian application proximal similar augment squared composite lagrangian admm eq like operator imply appear contain effectively purpose impose approximate produce argument exact see solution backward problem include linearize split inexact context primal demonstrate framework proximal algorithm proper convex lipschitz give proximal namely complete leave sub take minimization next proximal problematic imply iterative two step basic alternate inexact demonstrate proximal q arise second around necessarily split symmetric assume positive like together proximal form equation reflect involve approximation objective solution term point quickly mean per general like attempt use forward objective cholesky decomposition cholesky imply simplified exposition start linearize inspire z proximal base split forward q contraction x note restriction scope illustrate vector non trial number composite envelope commonly use quadratic lipschitz adjust acceleration nature illustrate logit fuse problem inspire quadratic envelope multinomial loss bound envelope perform fuse consist pre decomposition svd thus provide illustrate fused non composite operator since poisson loss function still convex replace accomplish track em likelihood plus convex penalty bridge develop problem proximal form proximal involves find norm norm value result inexact value proximal interestingly kl backward appropriate solution choice map affect property variational convergence half cyclic fashion cyclic derive problem remove similar descent q signal match give plot mean square penalty consist contour plot interesting relationship clinical volume age seminal percent common lasso elastic net exact proximal operator hard figure path major difference jump solution proximal classical descent property arrive implementation mm statistic provide mainly lagrangian originally recently convex broad apply algorithm construct optimization envelope close evaluate numerous demonstrate efficacy advantage proximal fix advance speed approach nesterov acceleration smooth help modify nesterov provide scheme progress couple mirror progress couple convergence help direction statistic explore relationship proximal splitting research combine proximal algorithms r p p p dx x x x p tc strongly accelerate proximal gradient frank wolfe newton l forward semi recall translation operator operator satisfie provide sort quadratic continuity quadratic want fix functional convexity improve step decrease finally compound improve momentum derivative bregman divergence law bound operator w intermediate subscript yield update momentum convex empty global minima non involve range level generate lx lx critical nonempty neighbourhood subgradient trajectory kl alternate typical relaxation function kl one possess kl kl figure proximal useful machine obtain exploit proximal operator envelope half envelope optimisation function smooth objective illustrate logistic bridge fuse provide discussion descent direction future keyword bayes shrinkage splitting fuse regularization divide large solving step function precisely canonical problem together regularization modern statistical curve fit map prior survey alternate multiplier divide dc frank fw split processing tv thresholding maximization mm iteratively fall although machine general work iterative fix banach space useful acceleration smooth gradient reverse descent illustrate proceed section notation operator operator envelope extension rely envelope gradient consider optimisation compute exact proximal general envelope methodology poisson fuse lasso bridge commonly document half list case nesterov acceleration conclude measure impose favorable bridge induce parameter interest vector covariate encode structural trace composite view index function continuous vector pay penalty fused concept useful exploit equivalence constrain slack introduce linear envelope dual quadratic envelope lx supremum dual hold conjugate say function convex satisfie use algebraic take extended line tool semi scalar envelope conditionally least ridge regression normal general proximal proximal operator one provide start advanced differentiable encourage suit iterate take differentiable operator differentiable intermediate lipschitz allow q equality algebra optimum value hand evaluate descent mm convex lipschitz modulus convex ensure optimal obtain meet
supplementary material response vector except carefully ol dimensional need question ol ol view property ridge regardless observation notice hand side consequently ol dimensional dimensional version essentially orthogonal projection projection observe thresholde small exact meaning demonstrating row sample dimension standardize design diagonal visible obtain include variable thresholde far obtain refine term pre selection denote use refinement ty replace tx refer ridge thresholde concrete form need row draw distribution allow various correlation widely illustrate rely restrict condition assume second contrast need bic parameter tune however detail consistency straightforwardly imply exist bic bic state result choose material guarantee rely particular term appropriate preserve magnitude computable alternatively nested form ranking coefficient adopt select well stage ordinary tend assume identify tend ridge condition parameter note constant I algorithm see take form threshold general rank true stage least priori I result condition imply consistent require extensive assess penalize method include elastic biased stage figure respectively lasso noise experiment structure support set component equally correlate factor compare simulate synthetic record iv actual use bic regularize huge computation mc find first predictor know set fold tune ridge finite hard thresholding comprehensive rmse htbp htbp see plot table good well mc time rmse rmse iv htbp ex ex ii rmse iv collect gene week old responsible select sufficient reliable fold variation expression link eight study assess fold offer regularize fair extend record reference report cccc cv average runtime scad error follow select parsimonious interpretability prefer hard performance method exist regularization appendix recall estimator nc c ic follow proposition zero finite variance absolute corollary matrix variable invariant transformation I fact uniformly conditional orthogonal e magnitude know distribute iid notice q put piece accord second use z pz sl match ready prove theorem q great cn obtain x dt define assume singular argument number p dx yu q coordinate submatrix eigenvalue therefore least tx want eq argument proposition dp definition result need quantity much complete singular matrix expansion hold ni know obtain chi bind exist least n proof immediately imply definition applicable sample version ridge propose novel algorithm fitting thresholding intuitively computationally implement consistently compare penalization analysis potential long mild ol widely model unfortunately ol model exceed penalize lasso exploration methodology
specify threshold false discovery show sis pc discover discover large appendix whose unit give expression fix limit limit random variable xy grow rate xy xy note prop remove distribution give role identify transition assumption prop pn undirected fig part th entry denote denote value prop thin match match match dotted dotted dotted dotted dotted vertex connect phase exactly critical decrease increase satisfy xy expression prevent bipartite section screening use I pc recover pc recover prop prop show proposition proposition correlation correlation inactive correlation inactive assumption proposition thm recover thm prop prop function sis recover probability prop sense prove prop accommodate heavy tail sample allocation mse rule online sec w constant increase function stage allocation budget skip prop stage use prop prop stage ol result demonstrate world pc sis pc screening pc compare simulation row dimensional mean sparse exactly entry independent inactive response importance magnitude fast tune minimize screening stage truly sis pc small sis select important inverse variable pc sis evident regularization set cross pc sis propose sample selection compute choose evaluated figure regime sis instead pc stage suffer rmse low sis predictor stage value side pair difference pc sis different value h c c pc sis rmse experiment pc outperform pc predictor sis sample stage coefficient similar reference inactive ar w leave computed pc sis fig leave note estimation sis increase figure independent evident pc sis improve solid rmse pc sis respectively plot pc stage pc sis dash plot marker pc pc support recovery pc coefficient get bernoulli bound sample prop h sample consistent prop predictive health datum set score subject collect subject become ill level clinical record number point include post measure predictor accurately predict measure gene consider scalar response apply predictor consist specified gene previous time take value use logistic ol predictor likelihood logistic coefficient performance evaluate leave cross leave validation trial pc perform predict score l l rmse sis pc validation predictor regression correlation predictor variable stage high throughput selection use dimensional false discovery stage experimental advantage work subsection explain sec proof sis discover discover representation correlation exist column norm show sis entry square calculation define u u u u lie diagonal matrix obtain u u vector discover least discover pc discover notation proposition proof present define p screening proposition represent upper converge dependency define complement neighbor complement play key observation f permutation expression independent von fisher sphere parameter parameter gamma exchangeability feasible write assume f yield n n desire precision intuitive follow prop discovery rate screen know model unit value throughout entry magnitude j vertex n union anti yx yx ni xy conditional joint yx summing conclude second chen method b n xy pn b j b summation give b op combine bound prop weakly proof proof prop prop moreover average dependency satisfy ok xy xy x score correspond dependent large number sequence g converge grow entry hence u thus p conclude p x xy moreover pf conclude partition dependent relation x noting proof prop jointly sample correlation exist cn uniformly cn generality v entry area spherical obtain incomplete beta dt n dt dt obtain cn form f z z z variable em c b independent g detail score without assume score u score z c z r ab e ac ac ad bc ac ac ac w ac ac ad bc ac via let cn increase auxiliary k r constant complete x u x x u u screen prop sec stage stage asymptotic hold outcome expect outcome variable mse constant one wrong ol bias mse select correctly mse rate c depend use ol expect mse minimum since become draw fill black paper propose adaptive budget predictor high dimension screen experimental finance engineering illustrate instance cost linearly stage tradeoff collect small collect pass low regression sample variable online implement false select variable well asymptotic convergence allocation estimation stage much effort objective engineering estimation wireless communication internet gene science predictor difficult normal equation overfitte computational complexity number two method elastic class sis offline screening predictor principal common budget well perform sure screen sis generalize ordinary ol response predictor poisson false specifie phase discovery total establish ol third establish multivariate offline implement try regularize costly large lar interior develop lasso differs regularize objective costly via min norm regularize ols offline correlation also independence wherein thresholded paper transition discovery among sis perform recovery discover absolute large ordinary square ol define min determine min regression inverse matrix coefficient ol pc method pc value
interpret frank refer reader discussion point boost arbitrary limit furthermore lead wide provide small consider scheme excellent boost profile coefficient profile modify approximate herein produce produce predefined grid generate grid warm start statistical sequentially update predefine fix value r k identical structure stagewise step describe rigorous feasibility specifically satisfie boost every hold right theorem recall error boost error specify generalize notion family solution theorem quantify boost approximate part along value respect would ideally guarantee spectrum value guarantee quantity sufficiently produce vanish control learning error correspond summary flexibility control complexity regularization come weak provide value nevertheless error entire regularization parameter array example explore property herein dataset matrix multivariate diagonal entry take choose control signal specify generate take example consider four publicly describe process create package process artificial response take process artificial analyze process dataset form second interaction third form detail refer reader standardized column norm run herein well finding figure discussion appendix explore test iteration regression good optimal know method good performance sparse good play reasonably important role acknowledgement preliminary herein show profile regression c fidelity versus profile synthetic correlation zero run panel profile dataset run run panel panel highlight profile axis normalize horizontal axis scale interval express norm fraction semi assume exist optimal qp hold algebra generality qp write zero column qp equation straightforward qp since establish whereby furthermore root gradient rearrange utilize k j equality seek function yield rearrange invoke I term similarly iii l simplify iv eq q inequality elementary derive q last second note ie complete vi simply derive coordinate status zero present additional coefficient follow hold take root therein iteration iteration complete apply describe indeed link characteristic step value progress rapid progress term progress amount training change residual informally minimize loss towards quantifie decay similar theorem rate dominate point rate begin correlation sample decay slow correlation square long whose variable furthermore eq inequality jensen imply note equivalent normalize follow give scalar elementary hold indeed elementary arithmetic follow rearrange substitution descent size substitute simplify convex value show instance subgradient subgradient descent examine proposition context cm norm covariate standardize run inequality provide iterate residual imply square model iterate index attain inequality use present convex quadratic format eigenvalue substitute minimum attain rearrange prof follow note chain iii simplifying term vi structural completeness show certain fidelity formalize relative prediction pose allow run iteration study guarantee theorem compute fast regard run iii follow shrinkage boost give whereby tolerance need achieve denote bound model produce relative ccccc equation target prediction panel shrinkage relative panel summarize primary goal prediction appropriately determined achieve also shrinkage run reader iteration boost profile profile small small suggest accomplished possible relationship problem square useful function subproblem optimality set result direct scale proposition optimality closely dual weak feasible eq duality q associate solution r let construct least function min drop dual q scale feasible direct yield equality q imply last cast problem convex case duality linearly optimization apply feasible follow tr tr strong ii eq since formula residual recall j tr r j follow rr update finally g g therefore k precisely update write k hold induction exactly descent apply residual translate residual second x follow since fourth follow column norm uniform hold elementary obtaining side prove item quadratic set arithmetic equality whereby inequality take square item essentially boost item iv theorem similarity profile general profile explore research algorithmic lead dataset stagewise incremental backward approximate accuracy produce point model importance sparsity well variable square degree freedom often collection processing proposal apply coefficient machine explore proposal adapt square regression author boost herein study coefficient unchanged coefficient loss holding recover soft thresholding stagewise update propose subgradient frank wolfe analyze perspective subgradient interpret frank update primal wolfe subgradient interpretation algorithms parametric author similarity frank wolfe feasibility parameter induction feasibility j something strong residual give format elementary specifically similar translate iterate iterate namely logic leave right side l third feasibility prove second fourth assumption normalize combine q complete ii feasibility prove consistent last iii coefficient describe perform real dataset four publicly microarray binary covariate approximately process subsample covariate package covariate covariate retain take matrix sample retain generate create create enhanced note example unit run study herein l error bottom panel monotone start reach test panel panel describe c forward error exhibit performance seem marginally predictive sensitive run limit version suggest sensitivity learn array real decrease iteration however sensitive reach furthermore method namely performance well predictive correspond value proper lead superior perform evaluate predictive accuracy know method take run find good obtain predictive excellent solution predictive performance play crucial boost iteration important role obtain quality e model zero well boost dense herein section regularize indeed solution table run value regularization path regression compute reach unconstrained term boost always good automate worth investigate excellent heuristic version achieve good snr achieve snr snr cc ccc example snr method limit boost large sparsity coefficient minimal norm run instance interior well similar term remark corollary author fa mit cat author research mit cat analyze perspective method classic boost incremental stagewise algorithm modification may easily compute may also interpret master maximum loss guarantee several modern computational guarantee statistical boost description fidelity rate method weak learner powerful adaboost boost develop classification particularly form influential boost particular adaboost instance stagewise fundamental tool yield crucial insight underlie boosting provide additive method viewpoint function greedy reader usual py center regression residual regression lead model attractive property popular least fix covariate eq current regression coefficient k k j k know stagewise essentially repeat start iteration find maximal decrease fit residual residual coefficient unchanged evolution root slow strategy describe shrinkage rate qualitatively speak learn compare increase training eventually attain fit reach training shrinkage empirically lead short point tradeoff quantification present freedom non pursuit closely incremental stagewise forward stagewise initialize k j iteration covariate regression coefficient residual factor strategy fidelity shrinkage fashion qualitatively refer reader evolution herein first description quantitie lot contain subtle difference firstly covariate lead choice difference rather plain residual step amount successive differ across square note factor successive loss herein equation difference absolute sign gradient least square expect precise term qualitatively speak update behave shrinkage progress converge globally hold necessarily converge fit operational guarantee square predict albeit sublinear sublinear sublinear difference step size call replace depend version unify view special instance difference versus run draw correlation learn discussion classical tool model forward regression sequentially variable identify absolute residual predictor update quite additional known regularize nature implicit difference implicit explicit regularize regression especially dimensional far exceed parsimonious regularization regularization explicit square solution contrast boost wherein control although boost explore certain profile profile upon profile exactly panel profile albeit fairly strong effort understand angle correlate residual move towards aspect unified version coefficient profile coefficient shrinkage coefficient profile bottom panel cancer profile see coefficient profile probable modification boost path one study algorithm inclusion backward path nice understanding aim view master instance view special algorithm problem regularization term residual residual interpret residual determine assign describe dual subgradient algorithm apply lead new almost incremental stagewise first coefficient coefficient become call path quantify depend learn therein derive herein precise fidelity obtain along compare model contribution boost method exist boost aim residual viewpoint operational characteristic boost computational estimate algorithm towards produce respective demonstrate slow sublinear least square computational amount fidelity boost iteration rely upon distributional generating show subgradient descent regularize rescale every increase evolve towards derive guarantee quantify algorithmic subgradient present naturally boost subgradient descent residual error implication expand computational experiment improve place notation vector vector ball coefficient ax q pp denote subdifferential convex positive matrix small implication generate converge square solution characterize fidelity global linear square convergence coefficient describe shrinkage coefficient change useful gradient equation guarantee shrinkage eigenvalue least linear training l square solution prediction square shrinkage part linear rate write eigenvector large assumption column whereby holds let immediate remark iv error geometric exponential square counterpart least factor part behavior depend pairwise correlation function fix different dataset matrix zero correlation fast rate linear convergence converging confirm behavior boost slowly theoretical justification empirical make stagewise widely least procedure discussion tend correlate whenever updating thereby covariate correlate compete update algorithm progress decrease contrast bring sense line explanation attempt convergence value phenomenon observe reader justification illustrate fidelity present new boost rate show herein algorithmic interpret subgradient residual interestingly result strong section section guarantee theorem herein fidelity consequence guarantee counterpart develop motivate briefly subgradient briefly motivate descent differentiable close function satisfy lie linear intuitive formula kx lie outside feasible onto differentiable virtue subgradient subgradient subgradient generalize state denote point descent generalization differentiable replace subgradient guarantee bound hold objective obtain right side norm subgradient correlation residual predictor residual cm important namely optimization residual absolute predictor therefore problem residual predictor residual least square solution whereby residual vector nonnegative square cm view optimization boost subgradient method cm initialize iteration instance subgradient non iteration I recall k tr choose select k k g projection precisely ii whereby iii interpretation especially traditionally translate light fidelity shrinkage characteristic easily run algorithm show subgradient viewpoint algorithmic new minor solution consider total index hold error I exist square solution hold k versus fidelity theoretical shrinkage regression versus left panel theorem part describe related quantity least counterpart rate shrinkage evolve sublinear decrease dramatically fast iteration theorem difference limit demonstrate theorem nice square unlike towards limit part imply distance solution computational idea figure multivariate gaussian appearing theorem train much value validate three decay carefully explore shrinkage two follow tradeoff tradeoff tradeoff curve level unlike tradeoff range shrinkage error shrinkage shrinkage correspond examine shrinkage alternatively shrinkage shrinkage large enough also shrinkage
metric event diagnosis information measure highlight decomposable metric focus distinct characterize confusion average infinite population class decomposable thresholded practical direct held contrast understand classification aware metric exhibit simple square count sec bridge classification analysis decomposable metric many comprise sign first formalize retrieval family monotonic subset fractional family special case sec population performance study sign thresholding quite evaluating involve computation aside fix test show light complexity computation propose run case full scope manuscript key accuracy optimize could misclassification work surrogate decomposable metric analyze expect chain optimal population equivalence go infinity recently give analysis multi label analyze efficient let label label iid focus decomposable confusion positive negative false negative label confusion simplify sometimes depend utility manuscript instance generally compute utility also pointwise remainder utility population confusion entry utility utility utilize principle identify classifier relate class respect immediate iid sample give decomposable counting sec design difference positive sec n meaningful property consequence metric identify sufficient consider u eq consider tp monotonic tp monotonically increase first tp verify easy tp guarantee satisfie provide tp monotonicity sufficient hold consider performance metric metric metric satisfy sort p v example recover family study metric thresholde contain equivalently result prove tp monotonicity three study easily metric simplify empirical monotonicity property say monotonic monotonically list satisfy metric admit familiar sign thresholded performance tp monotonicity monotonicity satisfy tp monotonicity monotonicity admit optimal familiar monotonic tp monotonicity weak monotonicity consider tp monotonic pf algorithm decomposable performance hard consequence possible light top sort trick evaluate cubic general principle top nk nk compute note evaluate via invoke tb sort nk fractional metric focus attention fractional family decomposable fractional linear efficient solve give generalize rational consider family fractional step I implemented line algorithm v nd j nj k k tb sort j u v ns r nk k k j j k consistent fix consistency estimate depend consistency estimate tp metric show algorithm applicable empirical prediction experimental synthetic serve principle metric second benchmark comparison optimal minimization table namely harmonic pr theorem simulate conditional sigmoid sample standard objective plot probability synthetic verify optimize threshold ii conditional metric pr pr fractional achieve hold aforementioned metric classifier metric optimal training datum select thresholding baseline report news article topic training article article handwritten character letter consist scene consist web page test instance dataset optimize test choose suggest utility baseline baseline refer compute individually average class tp pr tp pr first indicate individually goal gap show metric principle sign literature monotonicity guarantee principle monotonicity large subset
also tensor column row large go proof observe entry th method j column q nn q fashion vector
regularization serial different manner figure parallel eq q new way computation core go multiplication proxy b w b w tune follow u p take expectation get nonconvex yield repeat recursively il ig ix obtain sum get obtain convexity ic smooth give w remark exercise develop new risk enable free sdca moreover define erm allow mini come even convex able mini risk minimization erm successful paradigm learn erm throughout shall assume smooth constant always well effort put realization long state erm idea algorithm belong include sag sdca gd gd cd sdca prox sdca analyze arbitrary alpha accelerate involve computation typically pick uniformly design dependent computation gradient typically equivalent reading work develop regularize erm free sdca mini scheme method example arbitrary flexible scheme useful reason development ii importance aim obtain elsewhere utilize access iv processing mini mean assign reduce setup example lead well dependent primal allow arbitrary characterize update value I example mini sampling define probability example maintain store pick mini scheme gradient follow relation maintain w maintain convex indeed update write sense believe converge circular word describe reason iterative tight formula instance inequality exposure decay b tb tc assume function hold decay moreover equal decay ic tc potential decay moreover erm analyze mini dual cover non illustrate method variant method importance adaptive marked disadvantage mini choose subset random parallel processing processor differ cause
product approximate accuracy alternate parallel transform sample auxiliary core distributional induce address solution constrain make optimization aggregation variational family f descent sgd dimensional sgd variational bayes objective writing optimize derive objective decompose kf segment power jacobian evaluate proof inequality supplement denote jacobian supplement approach give crucially thereby possible without entropy concavity tighter low put everything together relaxed objective paper pose exponential family derive great generality decompose concavity treat emphasize aggregation concavity partial hold arbitrary aggregation supplement family individually concavity entropy density u decompose aggregation variational concave individually assume aggregation decompose concavity concavity relax kf f concave concavity broad setting objective individually family simple capture preserve aggregation meet simple impose semidefinite psd cases psd aggregation suffice general lead therefore sophisticated aggregation preserve orthogonal far crucially da da dd spectral guarantee psd simplex w posterior algorithm mixture gaussian accommodate global center introduce alignment indicate associate partition one worker cluster center estimation baseline gain experiment posterior place moment uniform gaussian average baseline large assess three moment moment mixed moment model spectral aggregation rule restrict diagonal computational point one sgd running optimization normal inverse conjugacy supplement baseline great partition uniform partition run poorly portion bayesian focus variable variance assume sample implementation hamiltonian carlo hmc figure drastically estimation test analyze show boost estimate serial mcmc partition unlike previous indeed method error across method second cost moderate serial batch operate speedup serial marker marker factor schedule estimate gradient factor scope active area assessment probit mixture size sample show negligible sampling second approximately corresponding second error convergence serial sampling number achieve optimization moderate speedup serial bottleneck optimistic increase technique number serial particularly optimization batch marker marker previous parallel also aggregation analyze variable sophisticated lift aggregate recall useful building approximation multimodal posterior overall acknowledgment university support institute california berkeley fellowship nsf fellowship award award award fa amazon web services intel microsoft intractable unfortunately mcmc scale typical recently consensus remove limitation drawing partition combine sample consensus monte variational optimize aggregation function approximate objective relax mild advantage literature demonstrate superior approximation moderate overhead relative reduction measure serial mcmc compare consensus task estimate probit expectation error mixture component gain runtime serial speedup modern inference scalability innovation distribute asynchronous variant achieve similar success estimate chain carlo within former stochastic bayes successfully optimization adaptive apply achieve operating subset advantage architecture motivate parallel asynchronous variant belong communication avoid subset core core sample q centralize combine efficiency procedure algorithm monte proceed intuition aggregate correctly one instance full clear aggregate obtain partition aggregation motivate gaussian raise numerous wide stand first aggregation achieve close covariance weight modify paper consensus monte parallel algorithm bayes possible adaptively aggregation achieve flexibility likewise support aggregation applicable structured aggregation appealing lead approximation assume data partition map global parameter flexibility view possible alternative form within particular posterior place cf scope paper evidence circumstance
explanatory code level factorial keep point factorial typically fraction keep factorial remove carefully refer design factorial composite factorial factorial factorial fractional axis control side origin design factorial may verify way satisfy literature orthogonality eq coordinate vector x hence quality estimation property property diagonal imply vector component make factorial design orthogonal fractional design keep orthogonality design even hard refer factorial fractional factorial design factorial distance origin unchanged rotation orthogonal design order design design determinant justification computer design estimate minimal infinite space experiment multivariate design want design orthonormal n r appropriate appropriate constraint context fourier spline sample drive pca orthogonal norm take orthogonal pls permit interaction procedure pls basis context reference therein functional orthogonality consider see subject order nothing order square directly design verify functional adjoint linear design cause cause circuit incorporate water circuit cause nuclear inner risk pressure heat transfer explore physical develop reproduce behavior temperature heat transfer figure represent evolution depend ccc temperature pressure heat pressure transfer temperature minimize margin failure increase aim factorial pressure heat transfer good consider temperature pressure choose maximal fractional design pressure point around initial heat transfer curve take heat remove point result curve design obtain pressure heat count ccc initial give figure curve direction solid line temperature dot estimate response alternative energy atomic want thank final
observe fx prove martingale counterparts dimension q coordinate bernoulli variant use perturbation algorithm present normalize mse normalize define measurement iteration measurement measurement measurement difference fact use simulation outperform fact iteration measurement result mse comparable c consider direction newton perturbation use aforementione perturbation simultaneous scheme newton know perturbation newton analyze observe asymptotic mean square numerical computationally efficient newton scalar term diagonal term diagonal ij fx expectation rhs rhs simplified dx combine w equality fx observe rest hessian comparison gradient vary bias technique proof consequence fact apply martingale proposition near claim proof asymptotic normality scheme cf segment connect gradient write amenable q fx n recursion estimate gradient establish converge identical imply imply verify order limit c pt height white coordinate thin symmetric perturbation optimization newton perturbation unlike simultaneous perturbation iterate simulation mean achieve compare incorporate perturbation par sometimes newton newton provide latter engineering research network involve system performance concern paper minimize measurement search gradient measurement finite pp require regardless randomly direction unit particularly computationally inherent perturbation observe perturbation cauchy distribute perturbation independently derive convolution see integration convolution regardless parameter perturb parameter direction finite low sf amongst perturbation simultaneous perturbation largely ease observed approach perturbed distribute study square perturbation simulation hessian estimate use iterate hessian objective gradient year considerable aim adaptive newton optimization propose perturbation involve symmetric bernoulli gradient iterate four perturb update project definite perturbation one average objective incorporate approximation certain simulation balanced perturbation hessian estimate hessian inversion procedure effort see similar except computational geometric half parameter certain feedback newton book perturbation first newton contribution summarize benefit perturbation newton simultaneous require perturbation second simulation procedure distribute unlike surface sphere perturbation first two simulation balanced perturb newton also nominal simulation newton scheme gradient sure rate convergence perturbation asymptotic perturbation achieve gaussian propose significant scheme require evaluation rectangle height width em fill white distance coordinate coordinate black cm xshift cm label sim right height yshift width update sim update sim corrupt scheme random estimate illustrate noisy measurement denote I e fx fx denote sequence sigma symmetric uniformly whereas perturbation uniformly distribute algorithm follow continuously bound second assumption ensure ode pose comprise ensure analysis convergent next govern provide refer sketch appendix gradient indicate order perturbation surface computationally perturbation scheme asymptotically differential xt xt compact require hessian th follow project onto positive definite order move hessian happen basic algorithm hessian estimate sa noisy respectively noise direction henceforth second would effort per former require generation perturbation variable measurement require require loss sigma assumption na nf fourth f nx I I si almost surely require system simulation assume c converge ensure recursion assume govern almost surely next additional label assume sketch appendix show asymptotically perturbation define certain setting optimally unknown obtaining average employ employ couple correspond perform adaptive iterate denote obtain scheme dependency
insight measure reliability correspond precisely among avoid combinatorial experiment sec ratio whole fix accuracy measure conditional probability randomly set class fig estimation together white deviation gray plot approximate measure depict vertical ball radius bin expect drop cardinality us insight report remarkable increase suggest effect new confirm trend investigate near extend object class principle actually constant physical explores environment notice cardinality trial high randomly would accuracy perspective expect typical quantify capability report minimum accuracy within level well understand implication relate pass predictor guarantee least accuracy visualization use perspective depend discriminate expect fig human low curve decay fall great could remarkably improve regard change observe object interest train combine frame rule returning occur classify principle beneficial accuracy would evaluate describe stream consecutive classify select frame previous sort label actual since consecutive stream always confidence fig varied frame system clearly benefit probably discriminate misclassification filter boost accuracy achieve accuracy fig confidence gap regard account aspect principle improve day report test separately day line mix observe day predictor train day redundant predictor curve day day day day assess day pt day ex day visual motion around actual compare hand whole perform benefit manually gain strategy consider day system accordingly strategy introduce motion boost suggest presence actually consider large image imagenet often background manual fine approach eventually manual segmentation manual example recognize visual recognition describe interested reference value return recognize recognize visual actually answer ask far ex pt em object pt completeness close brief comparison test architecture visual word implementation convolutional network due reader deep architecture cnn layer remain method carry notice trained cnn clearly outperform gray day day avg ex use test recognition capability robot analysis address currently recognize formulate accurately visual recognize human investigation collect scenario comprise class first would confidence result multiple recognition capability empirically adopt weakly strategy reduce amount extremely principle result hand visual representation architecture cnn visual setting hand extremely challenging far ability visually recognize fundamental indeed task involve interaction accurate scene good visual bottleneck system convolutional remarkable performance visual regard remarkable would generalize possibility take step develop vision platform particular benchmark question recognize experience computer vision approach arguably channel operate human environment visual major agent behavior imply planning progress especially mainly classify image layer architecture recently rapid acquisition train tailor kind ultimately test natural ask system task see offer platform question start collect experience preliminary confirm propose highlight challenge pose lack supervision paper conduct study aim answer object recognize robot interaction acquisition teacher speech label day broad question sec empirically representation application acquisition research many object recognition capability address question divide recognition acquire generalize observed measure expect hold world contextual information offer deal information incorporate recognition observe different robot contextual vision setting unclear employ sec address question artificial able rich expect robot benefit incorporation visual preliminary perform sec life internal supervision ideally robot communication channel speech supervision teacher robot human image manually around object rely strategy motion segmentation eliminate object evaluate fine image vision recognition categorization identify representation datum ideally hand discriminative different separable hand invariant scene rotation class mapping patch optimization loss separately jointly propose local map learn map least square nn availability parallel make cnns accord evidence architecture rich amount use description particularly appeal paper effectively high computational least setting context line research recognition matching often employ suited supervision previously robot supervision hence acquisition protocol collect detail employ employ briefly outline human front object annotation detection track hz localization image pixel fig process representation module information descriptor set comprise distinct object evenly organize category acquire second reduce acquisition frequency image
sensor bs dynamically suggest journal equivalent energy employ incoming intend energy intend algorithm controller amongst compete controller allocate consider comprise system instant controller decide instant amount algorithm controller optimal controller try efficient computationally expensive numerous find allocation follow subsection management policy sensor network multiple sensor discrete buffer sensor obtain energy buffer generally buffer level node buffer slot bit source queue level upon let unit node slot bit data function non decrease shannon channel particular non concave optimal energy sharing form function consider power queue length evolve buffer queue evolves give denote sensor evolve noise spatially correlate arrival evolve depend assumption energy independent x x tuple single controller move prescribed probability state term formulate share mdp set run action tuple comprise buffer buffer note k max k remark tuple simplify state action arrival ks ns energy bit e split refer stationary policy choose set single stage formulate energy mdp generalized jointly arrival arrival easily satisfied history computationally infeasible policy evolution k evolution mdp chapter noise form policy augment sensor transmission long string data string buffer discrete model discretization discretization discretization discrete energy store queue discretization generation energy sdp map energy split run include hence take enable form I policy stage average sdp stage give energy long infer lemma minimize average interested stationary policy deterministic optimally share energy aim minimize cost stage minimize sensor cost delay well deterministic class policy provide optimal small state space large computation problem find optimal tuple tuple update adequate play finding nevertheless buffer tuple amount computation energy increase share energy scenario tackle curse complexity threshold fundamental feature prove source differential easily manner buffer queue buffer queue buffer level differ buffer level monotonicity differential justification nearby aggregated cluster state value policy close property state group arbitrarily aggregation yield policy action go combine guarantee policy representation combine aggregation sa continue aggregation previous formulate aggregate quantization buffer space buffer buffer quantization represent buffer quantization buffer prescribe energy buffer I instance energy buffer partition range x l data bit range bit hold buffer node buffer controller energy two aggregate controller bit node let aggregated action cardinality reduced instance cardinality partition cardinality aggregation explain although information respect counterpart state indicate aggregate action tuple proceed schedule facilitate exploration employ mechanism describe every present buffer level aggregate storing aggregate computational describe dependent aggregate sensor grow energy buffer iteration six buffer compare aggregation must controller I aggregate aggregate aggregate action state buffer level require since aggregate state action optimal indicate level add hold choose energy division sensor system partition suppose energy bit energy bit belong partition aggregate action bit bit remain bit buffer order bit energy node exact buffer advantage aggregation manually scheme result increase partition show sake implement greedy heuristic implement method bit allocate energy available greedy unit share node let action space kt energy need distribute decide upon find learn policy action space action proportion nk bit greedy consider jointly markovian arrival buffer buffer size noise mean keep buffer accord simulation arrival node varied describe markov markovian energy arrival process buffer buffer evolve noise variable component keep buffer buffer buffer cluster partition experiment stepsize greedy ucb I case fig simulation carry fig arrival energy arrival arrival I vary keep energy consider fig data arrival node keep fig policy mean arrival learn design policy algorithm fig policy combine distribute split poor compare mdp energy total max max fig cost policy obtain method since free irrespective arrival fig near plot method average node occur h max variation average aggregation increase well policy single stage cost effect action take tradeoff queue collective observe stage derive take importance queue combined method stage buffer fig indicate rate fig indicate queue value plot queue transmission relate show collective queue average fig occur learn well combine node importance queue cost policy learn energy usage albeit queue length combine q learn aggregation greedy axis aggregation perform method aggregation find greedy available instant storing compare one devise energy requirement naturally incorporate moreover thus naturally early cite work bit require node combine greatly I state node without aggregation learn fast node learn suboptimal fig poorly combine tradeoff policy sharing consider energy sharing sensor scheme problem understand greedy method certain regardless form algorithms sample share action k k buffer buffer share evolve slot must system model observe incur choose require rl computed method exact decide share comparison state
theorem I often give portion time simultaneously runtime th suffice symmetry compute rest find partition size master perform require carry optimization simultaneously apply runtime nonetheless master speed modification give slightly statistic still size master grow approach variable every matrix cd bn time simultaneously appendix proof bind runtime analyze separately compute equal set contribution simultaneously correspond equal size summing give bind analysis corollary undesirable however strategy work typical asymptotically fast heuristic demonstrate difference translate substantial realistic rest paper algorithms simulation data curve square bias noisy bad comparison light present examine simulation I space relationship simulate distribution variance statistic two respect reason usage substantially well instead median perform relationship fact whereas heuristic second result exponent variance expect regime general regime lead regime lead negative regime word detect distinguish among strong value cause detailed recommendation practice extent standard analysis equitability show paper equitability lead analysis sample size conclude respect vast majority examine pearson pdf xy along legend pearson perfect equitability describe th th percentile relationship estimate allow visualize statistic interval plot correspond red high equitability reflect question relationship maximal equitability total coefficient contrast maximal information rather nan supremum estimating supremum characteristic set variable number one procedure undesirable positive population characteristic intuition behind entry characteristic expect small independence power away avoid sum entry property relationship coefficient fit conceptual characteristic prove consistent power define version characteristic whereas denote ordered lead independence test analyze care define quantity grain part grid characteristic sample statistically matrix behave dependent understand bind translate quickly proposition technical information yield jointly integer either give understanding part independent exhibit therefore grid entry without generality argue imply definition element strictly finitely entry column prove claim grid size since allow empty satisfy word last distribute guarantee proposition proposition tell immediately convert write meet tell last grow yield test right test independence jointly sample distribution characteristic suffice monotonic therefore show alternative strategy translate know bind alternative lemma lemma take choice nan independence say alternative hypothesis exhibit proposition grow grow section matrix calculate affect prove turn independence base evaluate pearson correlation choose consider examine simulated estimate right independence analysis compare quite examine three unlike relationship term detailed type relationship size marginal wider increasingly accurately achieve measure dependence independence test latter relationship strength two goal contribution way maximal smoothing supremum infinite consistent computable coefficient statistic arise independence consideration detail show noisy equitability respect quite equitability independence across dependent hypothesis allow quickly enable contribution importance reason characterization limit know converge tell see result smooth mutual uniformly constitute toward role normalization continuity ordinary mutual latter third alternate characterization grid large finite significant improvement approximated evidence basic normalize mutual achievable grid characterize interestingly possible grid excellent theoretically computationally specifically little cost either individually imagine non significant association rank remain strategy leverage reduce burden utilize art equitability effectively course useful nevertheless exploration future sample grid line sophisticated discrete second approach joint infinite seek supremum precision finally characterization representative promise theoretical would advance equitability like acknowledge r constructive useful reference discrete respective whose q entropy rescale analogously proceed proceed proceed bound bind together fact expression fact consequence convexity term complete number function variable result let variable variable eq bind hx hx hx hz hx hx z z hx h fourth line hx hz hx let mass rescale let let normalize magnitude total add go previous lemma remove mass k k h finite independence total result characterize versus jointly distribute speed optimality cd cd entry partition grow therefore supremum partial sum jointly fix grid size quantity state show grid quantity size jointly parameter error rather outside distribution every parameter consistency follow sketch distribute proof th axis restriction contain grow become fine fine fine fine optimal convergence grid pointwise seek abstract lemma index size sequence output sample sequence hold almost converge show exist definition know convergence observe I coordinate analogously straightforward state show ij notation converge sequence sequence begin pair jointly sketch entry zero identical exist take column index finitely large entry together consistent tail algorithm jointly distribute monotonic analyze nan moreover adapt argument apply entry since may follow theorem depend represent number invoke parameter sub partition relationship value compare average finding size introduce size mean grid find grid high sample effect grid regardless exponent use need search grid curve alpha alpha curve alpha alpha david pair retain score statistic systematically assign relationship type avoid assign equally noisy regardless paper characterize population measure way statistic optimal marginal know simulation well bias high set functional nan statistical equitability testing excellent focused depth evaluation dependence show equitability together valuable tool exploratory grow hypothesis science whereby help formulate one among practitioner evaluate pair sort variable scoring low manually examine popular analysis whose value asymptotically trivial relationship utility dependence assess independence relationship dependence exceed explore biological relationship manual study human subject yield powerful direction systematically assign score relationship linear crowd score paper second assess measure equitability informally statistic measure assign equally regardless equitability useful dependence equitability respect strength introduce new bivariate dependence toward goal equitability power begin introduce dependence distribute finite grid cell gives view introduce smoothing sense regularization formal small supremum infinite sequence term dimension time grid greatly simplify associated quantity estimate algorithm call know computable fast hand property guarantee reveal tune toward general study demonstrate equitability relationship though use relationship detection ranking goal equitability independence coefficient later aggregate via rather maximization distinguish signal e distinguish relationship independence arise naturally free computed compare practice addition equitability measure equitability together result maximal excellent one remain computed measuring dependence extensively grid distribution grid analogously distribute mutual information sample case grid row mass column way row column establish matrix norm section maximal information coefficient jointly population way later jointly variable maximal coefficient see regularize grid ensure fall characterization view population matrix denote characteristic relationship additional characteristic useful presence absence rather quantify strength population large limit statistic review result hold hand trivial proof order characteristic define order let define statistic estimating theorem statement recall projection uniformly pointwise since supremum realize yield intuition sample might characteristic turn reason consideration consistent supremum ni suffice characteristic converge particular slow always individual entry characteristic eventually technical heart quantity quickly theorem lemma build grid consider master grid contain bind much grid seek difference seek require entropy contain central idea prove grid impose allow grid cell grid let respective define cell distribution column taylor whose find extend grid grid master grid rely prove later difference let random let cell resp mass cell cell provide away grid replace every horizontal close line resp moving show argument use term bind grid distribution mass sample integer lemma characteristic continuous technical continuity later non normalize mutual without apart mutual pair relate movement prove separately straightforward conditional entropy line magnitude cell column mass leave cell lemma p appendix binary entropy bind result mutual information depend ready continuity characteristic map uniformly step function family consist since consist argument ball remain tell distance normalization respect desired define map therefore continuity family within supremum give continuity map corollary characteristic include fact entry characteristic normalization characteristic continuous function line generality cell uniformly column see give information supremum continuous correlation equal mutual enough sufficiently cause continuity canonical smoothed lack favorable property supremum alternate characterization computing precision second foundation introduce later boundary observation empty know corollary true population characteristic characteristic characteristic show entry fix corollary therefore supremum exceed characterization stem fact express maximization one partition grid equal wish show observe column q q give variable partition bin expression maximization partition rather grid compute characteristic precision utilize dynamic programming brief overview set optimize axis description algorithm optimize paper master return mutual sub algorithm work exploiting condition store subproblem black box mutual theory develop boundary characteristic computing boundary resp computable error numerically mutual claim partition maximize sub master set row get close enough prove restrict partition partition omit since show ok move close
outcome endow validity obviously apply paradigm include produce pseudo helpful comment reference conference conference hold help universit paris paris france fr paris bs calibration france uk paris paris conditions work justify base construction prior analysis arguably central piece since inference define datum lack bayes construction automate extent reject argue non prior via moment maximum even involve consider jointly propose perspective hard long fix infer term truly motivation justify method difficult analyse introduction fisher one author integrate construct within specify produce equally arbitrary transform around freedom happen dirac variable theoretic difficulty analyse somewhat exclusive alternative hence conversely induce joint line define pseudo among define far remove determinant depend reference moment moment condition derive entropy besides produce inconsistent se incoherence specify require possibly valid whether set priori compatible seem highly must transform consequence joint unlikely likelihood likely select fisher example contain borel inference part deduce fine prior hence borel algebra compatible agree regular borel algebra solution abstraction axiom one define derivation reason small density likelihood remain situation consideration notion analytically verify approximation make sufficiently completely distribution regular intractable much alternative bayesian variational suggest suit exist induce distribution item irrelevant wants rely solely observation distance aside author harmonic mean rather poor potential close unclear I stage characteristic available expressed convert adequate available comment aspect true scientific
corollary proof one perhaps way environment gb ga environment list form construct must environment contrary gx gx contradict environment different please list want theorem definition create might think would file discuss environment first title include appropriate form expert name start document file run twice resolve create file source comment alternate file helpful moderately expert read useful please usa format lars rv circle p rv email email document alternate look author page file try include sort title mention produce pdf version software record conference seek conference quality appearance requirement document specify balanced specify size body copy live page specify margin top bottom leave right width news another balanced page class remainder concern rigorous description hierarchical section subsection subsection hierarchy use around name example appear paragraph sentence separate paragraph character already see sample indicate phrase text code change instance handle document take care begin text symbol character english character complete display three text environment symbol structure display display text center display equation environment use symbol available give couple first equation notice environment enter equation handle article conference book list occur automatically citation key item cite proper file key word title item file detail file exhaustive detail specification citation format support split across table table align properly row horizontal vertical material sentence file english comment common wide table live area
like extend discussion formalize argument theoretical claim building layer representation advantage gradient learner hide synthetic allow early computational extended modelling develop modern use transformation mean implication strong promise expand discussion dependence theory probabilistic derive top multi node demonstrate equivalence optimally structured dependence novel relate finally conclusion cast serious consideration provide outperform percentage point dataset aside advanced hundred label publish present variation model dependence underlie rarely implicitly explicitly literature marginal dependence co complete intractable pairwise ensemble incomplete dependence particularly intensive inherently need model measuring entail classifier among scalability much model negligible range time dependence outperform dataset good forest ensemble improvement prediction multi author development year model predictive modelling gap ground truth make risk effort could advance implication label irrelevant word lack one performance classifier illustrate take dependence multi expert human visually classify know contain modelling dependence one label base crucially mean guarantee amount training change change ideal toy guarantee factorial complexity model full limited linear learner perform equally poorly perfectly scenario human conditional base dependence summary perform method dependence condition method due insufficient dependence multi outperform classifier inspire deep remove dependence among top powerful top later base learner separately come result popular lie advanced label considerable success option special consideration neither removal time truly parallel label chain scenario number scalable learner serious division pruning fast cost propagation additionally huge family practitioner go default model basic relevance inter linkage option degree measure also accuracy derive classifier scenario purpose illustration toy imagine represent subject imagine latent event affect composition various event behind add desire case generate independent aside basic illustrate independently independently identify relevant expert piece place write text document label domain expert potentially illustrate impose e linearity recover latent recover knowledge redundant drop structure I interaction unobserved label document bias alternate external combine generic layer output equivalent dependence construct optimally energy improve nothing since even learn build universal approximation approximate label mean linear suffice layer inner radial return though learn restrict rbms study layer find label reverse variable remove overlap novel method rapidly field challenge learn rather unsupervised layer divergence recent adaptive dropout become predict discriminative prediction feature representation multi available scalability reduce subset degradation respect discuss layer inner fashion stack look like middle stack middle combine stack form chain leverage label unit must respect supervise fashion family probabilistic perspective directly approximate probability scalability employ decision tractable create vote get vote st label cast imagine create subset vote relevant label node weight word mention project space generalize function label eq return index use replace representation internal however cascade could could equivalently skip network augment middle flexibility stream machine remove function example stream review deal relate already introduction linearity otherwise expansion technique either suitably arbitrarily high polynomial degree learn input rbms sigmoid cast stochastically stack radial rbf review build linearity unlike basis typical method neural formulate error rbm early label box consistently art pick widely multi text implement dropout competitive compare network even disadvantage multi back learn input back adjust weight adjust stack rbms use tune early propagation layer error idea call hide parameter choose build hide layer obtain huge typical closely fact rbf unlike another difference label feature implement framework library collection statistic type logical audio biology medical multi evaluation empirical predicting carry ten iteration train randomize effect make scalable training disjoint vote propose relu logistic base learner case relate split node obviously trend predictive split average music medical avg music scene medical avg dataset music scene medical avg overall highlight propose explain relatively relatively huge already albeit naturally perform function dataset another proportion label correspondingly proportion synthetic label label incorporate several run parameterization average final slow dataset scalability increase concern drop former time latter extra chain internal split around factorial chain occur second time obtains indeed lead improve include underlie frequently dataset general perhaps ensure label overfitte ten scene classifier ensemble latter simply space thus among avoid ensemble improve mechanism rather give explicitly implicitly use ensemble take ten similar explicitly literature evaluation contribute great depth explain modelling advanced help model dependence inner layer label independence outer show simple base learner create benefit flexibility series novel method exploit space label classification instance cascade find advantage label create combination output ordinary create inner layer concept advanced competitive datum typically beneficial dependence fundamental separate construct hundred effort build discussion deep develop dependence base rather inherent exhaustive analysis inspire remove leverage middle unit base also
experiment active active focus active strategy follow drug interaction measure base round measure batch accuracy stop reach high enough learn entire label dark red interaction represent drug determine truth interaction drug interaction lack denote experiment denote interaction drug target drug approximate drug target kernel prediction product project bayes pairwise provide semi classification use initialization batch greatest sigmoid predict stop active predict point along process predict characterize unique unique uniqueness value interaction measure independence difficult purpose current feature row average prediction fraction drug compute pairwise learn uniqueness parameter kernel distance trajectory learner measure ground truth perform fit validation prediction al drug interaction matrix nuclear evaluate outperform strategy channel reach predict introduction decide stop good active learning therefore uniqueness perform learning simulation red fig datum guarantee true least true stop accuracy simulation rule adopt threshold active great fulfilled occurrence predict accuracy experiment interest begin active amount prediction confident reach peak predict accuracy naturally drop drastically lie range terminate active predict stop evaluate fold set choose reduce training perform make experiment purpose active strategy modify size uncertain prediction target predict acquisition accuracy stop auc result five report stop four datum list rule rule auc channel overall uncertainty adapt base describe predict difference time experiment evaluate define reach pa threshold average time fix threshold maximum uncertainty sa stop well criterion pa pa pa method drug building kernel accuracy furthermore drug limit world uniqueness achieve good matrix dataset basis please limited target produce drug convert replace strategy diversity learn trace simulated feature accuracy accuracy calibrate stop rule achieve drive undesirable presentation abstract conference institute advanced study biology department pa biological sciences engineering usa need drug actually save active crucial evaluate decide criterion actually simulate drug analyze regression stop unseen previously drug effect apply criterion result total highly prediction conduct search affect become find successful require search meet exhaustive selective drive machine refer guide drug method grain exhaustive verify manually limited expert perform time expense exhaustive limit trace drug rule little criterion unlabeled pool consistency instead use stop describe active trajectory use train simulate adopt drug target prediction major
efficiency estimate decide amongst initial stage collect choice repeat feasible low refer choice step typically complicate simple technique suggest behaviour motivate regression secondly regression vary ensure zero regression suggest efficiency choice efficiency use asymptotic see note abc omit return perform subset regression kernel bandwidth manually repeat training relative abc magnitude ess study spatial make ess abc abc include calculation second time core review abc introduce simple tuning provide comparable spatial potential extension include version focus simulation interest costly simulation use appropriately abc process tune approximate bayesian give possible interest define conditional standard sampling version equation w iy l n iw g estimate surely correct inference trade evaluate act since map dimensional statistic vector define typical accept reject normal match produce significant tuning choice consider aspect abc abc split stage refer require simulation considerable simulation stage operation tune introduce function behaviour simulate save division htp algorithm continue simulate relate discussion sketch argument follow assign randomness act weight due theory importance sampling ensure algorithm argument target abc decision converge
result non decomposable recover previous metric natural force suitable respect inefficient alternative provide efficient cg avoid cg feasible form plug consistent confusion knowledge algorithm provably family decomposable also unlike plug method metric novel technique cg solve approximately regret bind smooth metric metric confusion metric mean classification plug apply class metric learn prediction know consistent metric sensitive classifier close optimize multi setting method apply set seek decomposable additive standard performance interest multi express sum loss example loss example potentially decomposable label metric section give multi class metric decomposable non design alternate efficient conditional decomposable metric appendix shall denote simplex yy component closure appropriate space tie break favor set training classification model interested deterministic classifier simplex setting evaluate decomposable metric expectation loss denote class py sd py py metric h fourth column hold confusion linear metric yy yy n concave differentiable us confusion correspond confusion classifier express bound eq macro measure widely text retrieval contain example theoretic look decomposable seek throughout use randomize eq value sample say consistent probability draw classifier loss exist give decomposable method sx regularize empirical understand decomposable metric optimal binary monotonic place certain specific measure geometric median exact also classifier assign empirical suitable statistically decomposable thresholded cost sensitive fractional decomposable metric see little extend sensitive minimization binary metric single generalize class tune class proceed convenient rl r dl start decomposable find classifier decomposable matrix framework characterization classifier decomposable metric satisfie mild maximize decomposable gain give decomposable confusion knowledge general generalize decomposable performance pt begin confusion confusion classifier probability give cast viewpoint useful characterize design multi label random distribute uniformly simplex say absolutely w metric metric n differentiable non classifier maximize decomposable construct h optimal multi metric simple application fractional linear coefficient also metric eq confusion remain show achieve first necessary equation along g give statement decomposable continuity assumption assumption indeed metric unique mean metric classifier worth certain restrict characterization fractional min max decomposable min max sub characterization classifier metric simple plug decomposable search show decomposable also explicit metric like alternate conditional algorithm consistent family decomposable confusion matrix decomposable metric plug suitable class natural approach perform force search gain pick plug classifier see algorithm reduce threshold class probability perform linear number hold exact search maximization grain continuous force method statistically give h n ss g g satisfy using guarantee metric couple lemma fix gain probability entry second give confusion set learn gain sx ij md sd proof theorem optimal let n g g h h fourth matrix one special metric certain like force plug n dl c c c classifier draw constant bind metric non table suitable smoothed metric indicate n performance dl norm learn draw independent theorem metric concave smoothness constant sample draw g mean prescribe table learn concave smooth see form metric fourth find n c nn guarantee plug discuss make strictly high performance randomize classifier high h plug learn fail return algorithm ensemble enable handle general consistency performance derive metric consistency result involved approximate long also point cg objective method hence result general metric show without special concave performance fractional micro linear provide unify decomposable loss metric decomposable performance binary metric learn optimization consistent concave technique novel tool literature particularly thank helpful discussion helpful discussion support fellowship thank technology fellowship c randomize lemma I randomize classifier classifier x f jx x third random f p p integer affine vector take q let entire let denote set hull affine hull dimension denote set ir r g b dr inequality dimensional radius dimensional sphere radius width finally entire space argument small absolutely r base element among component arrange monotonically monotonically obvious scalar column p cc c argument empty constant absolutely continuous singleton maximizer unique maximizer proposition g maximizer equation prove n uniquely lemma complete shall go virtue confusion maximal strictly bad confusion away c f f b f apply equation p f p contradict equation thus two learn g sx x e argue value xy sx implication gx gx sx sx equation g x gx gx py approach thus zero gx r md sd mc observe b bx bx hx g concept intersection correspond vc hold gain I let h h gx em gx h x satisfy assumption dl c let x theorem g p thus g p g lc fourth step assumption last result curvature c observation entry sum approximation h n classifier construct train j due inequality second equation distribution dl norm learn probability distribution simplicity assume exist derivation lipschitz performance constant bounding parameter maximum hessian c n lipschitz norm n n c u n u c c u c n c c u bounding matrix c hold entry assume n table c c c u c
input pooling pool branch branch max fc softmax cifar relu compare activation relu variant baseline observation experiment surprisingly normal relu relu relu lowest relu relu suffer severe overfitte superiority significant cifar cifar dataset cifar big activation test relu relu activation relu relu relu relu relu relu htb htb analyze finding popular activation relu type relu consistently outperform relu reason superior performance justification aspect activation scale type convolutional standard unit relu unit relu randomized unit activation suggest incorporate slope finding belief performance relu overfitte counterpart success various classification object characteristic non relu counterpart g lie aspect vanish second accelerate convergence activation one relu briefly desirable activation pass relu commonly superior relu come sparsity paper want ask broad class activation interest relu contrast relu negative totally drop relu zero slope part predefine author imagenet variant unit relu linear relu illustrate comparison th pass subsection unit variable linear restrict formally activation q mathematically fix set like small parametric relu classification eqn propagation randomize randomize relu dropout suggest competition winner test activation search activation cifar cifar image rgb image pre augmentation ensemble
component despite promise progress pca due whiten tb randomly pick x u u u stochastic summarize reduce per minibatch empirically fast batch version prohibitive fix power show due accuracy fully solve risk unnecessary contrary level efficient accomplished ok concentration try direction simultaneously else whiten present dimensional canonical dataset description mnist image lexical annotate challenge shoot annotated handwritten cca learn leave image extract journal million token vocabulary successfully build embedding cca word sample uci repository million first million ip lexical cca feature lexical feature bank define two sum canonical estimate canonical lead proportion correlation compare well cca relevant capture second cca subspace pose correlation capture truth dataset number amount numerator w algorithms draw normalize hold adaptively regularization add gram matrix people compute practice performance dataset bank capture cca much version big thing notice reveal less denominator numerator mention classical algorithm fail usually prohibitive huge advantage cost generate reveal iteration capture correlation amount correlation description whiten matrix also whiten invert whiten cca compute lead subspace subspace top cca subspace cca back cca dominate heuristic svd randomize correlation increase heuristic incorrect approximately datum correlation paper tackle large cca nonconvex optimization free scalable prohibitive regime concern far incorporate canonical sparse hard cca whiten therefore well corollary remark canonical cca technique structure multi algorithm usually huge matrix computationally storage cca scalable canonical thin matrix usually require store whitening propose generalize especially huge batch prohibitive property suit effectiveness introduce characterize relationship multidimensional fit low unlabeled supplement refine label semi supervise fashion improve prove dimension cca association genetic variation cca algebraic matrix xy np recursively xy u diag identifiability singular vector imply lead dimensional cca computing whitening truncate x xy large common natural corpus computational algorithm whiten huge computational decomposition even whiten dominate truncate svd top svd classical whitening qr perform indicate difficult factorization compute besides whiten dense number bottleneck consider capacity ram system communication ask avoid decomposition huge matrix multiplication online e scalability begin play important role modern attention progress scale technique directly cca whiten several author try devise scalable cca cca thin recently proportion product formulate square exploit consider pca date back compare empirically fast apply streaming setting comes sequentially pass whiten step contribution tackle cca novel advantage state fold multiply width small dimension free eigenvector pair gradient converge proposition proposition leave failure nonconvex project nonconvex domain normalization contrary decrease maintain scheme guarantee alternate multiplication require extra input seem nice behind similarity subtle compare auxiliary unnormalized iterate scale turn fix characterize relation among quantity proof v xy substitute second equality third complete proof connection minimization intuitively close square informally second newton step solve sequence square pair give characterize lead canonical unnormalized counterpart insight intuition estimation actually least truth approximate update therefore enter pair achieve reveal broad contraction randomize capture correlation tb n spirit one compute normalization matrix u multiply thin width
vertex vertex total triplet vertex cluster triplet vertex error perfect clustered uniqueness suppose clustered triplet positive neighbor contradict perfect vertex thus vertex perfect particular perfect cover triplet cluster triplet triplet pairwise triplet force cluster together contradict conversely disjoint index define q perfect vertex namely ground triplet vertex vertex edge triplet error error edge vertex error cluster repeat keep self contain imply total cost total cluster lp tx follow eq pay cluster type contain cluster split reject time cost suffice second accept consider factor time number edge lp definition edge accept low cluster edge proposition definition agnostic label partition cluster try within cluster unlike minimize error enforce partition vertex bipartite problem graph give polynomial algorithm round cluster form object represented label whether go optimization cluster whose cluster cluster error np complete obtain unique approximation theoretical focus special nevertheless approximation graph van approximation approximation complete bipartite van recently cluster outside chen xu classical many science recommender bioinformatic rather cluster np bipartite introduce technical difficulty problem minimax social member community alternatively view constraint enable cluster application recommender quality recommendation organize factor approximation algorithm minimax version counterpart appendix minimax bipartite graph detail proof version minimax cluster integer integer interpretation classical throughout vertex neighborhood positive likewise interpret weight measuring vertex enforce objective classical clustering think minimize scale seek view limit relaxation relaxation round main introduction arbitrarily thus st ts u bound total generate optimal time pay error cost solution thus lp produce otherwise main difference pay error lp cost error complement solution proof follow vertex feasible error depend split safe mark safe x algorithm incur first pay incur total edge singleton edge positive lp must pay edge safe pay mark safe include grow bad pay bad make rigorous singleton inequality rearrange obtain bx uv uv k pay singleton total lp bank pay singleton mark safe bank pay bad safe bad safe bank pay pay uv pay time cost call edge appendix time lp edge outside also time x xx time pay instead cluster cost edge uk positive edge lp possibility inside clustered vertex lp pay bank times lp pay lp leave receive lp also make lp bank lp pay edge singleton k output singleton output singleton lp may pay bank pay lp pay singleton receive time ratio wish plausible optimum incur lp cost edge factor edge still pay bad argument incur lp lp bad triangle yield vertex set hand output q combine rearrange cost independent negative pay singleton cluster total create bank pay bad safe safe bad safe bank describe pay negative edge pay pay edge time still must rest edge must inside factor pay inside vertex inside pay outside lp let vertex xu xx jx u iw lp pay cluster lp cross cluster thus lp cost cross pay lp edge inside cluster cluster cost pay bank pay leaving receive type total lp pay bank times lp pay times pay receive lp times pay bank account lp pay lp pay cluster second lp cluster complete originally state attribute set induce triangle van van triangle np complete np every maximum minimax complete tolerance admit mistake perfect completeness proof regular vertex label graph partition triangle mistake expand optimal cluster vertex essentially augment every clique vertex vertex vertex every vertex clique clique belong clique mistake tolerance exceed tolerance perfect cluster one contain vertex clique cluster mistake exceed cluster four contain cluster triangle partition triangle cluster vertex exactly mistake among cluster cluster neighbor since triangle mistake vertex outside vertex cluster restriction
model event pairwise model distributional group base distance careful determine cluster suffer error robust combine discriminative signal distributional generative correctly mention mention connection gain room think key problem model information distance compatibility repeatedly mention external resolve sampling scale corpora resolution feature discriminative agglomerative generative note easily linguistic exhibit contextual effectively resolve extend operation university pf edu hierarchical incorporation use guide document distance use learnable prior corpus show document resolution identify describe share partition resolution document crucial processing task track extraction answer broadly applicable entity resolution deal noun phrase entity resolution extensively exhibit entity single event associate event decision event example event across document event ht resolution operate extend resolution g pairwise indicate agglomerative merge major decision base decision document propose nonparametric event encode guide toward well limitation dependent resolution rich pairwise bayesian cluster allow automatic determination event rich preference build dependent chinese process dependency extend incorporation learnable cluster encouraging event hierarchy allow group cluster group effectiveness conduct experiment large annotation integrate promise document extensive event less explore address event context extraction specify type group agglomerative cluster entity mention feature extension hierarchical across document consider linguistic likelihood nonparametric linguistic feature learnable similarity iteratively entity regressor quality level merging operation preference dependent chinese restaurant sequential limited work explore model distance topic encodes distance document assume document map segmentation novel employ document inference adopt terminology extend use something situation occur happen involve happen corpus action south location refer mention refer participant noun phrase phrase actual involve particular combination location note text may figure mention contexts document event problem group relation consider document resolution improve within cross extraction event extraction extract text event argument location one action reasonably argument semantic labeling predicate noun phrase capture event extract action participant location event head word matching produce high sentence annotate employ semi markov augment objective boundary rich include pos tag semantic phrase e np hold crf identify participant location head semi event associate mention training event detector argument event heuristic intra action event mention associate customer self affinity observation crp say sequential sequential customer mention refer mention start event relation exist document cluster model employ first second link distance form large cluster process describe restaurant imagine collection collection customers global corpus serve customer configuration structure circle represent customer color customer customer restaurant table restaurant customer customer restaurant table first use calculate path table undirected customer distance compute posterior link configuration carlo chain sampling sampler sampler generative customer calculate choose gibbs iteratively customer customer customer link customer start factorize denote customer belong marginal observation associate customer mention prior compatibility rich set event head pos cosine similarity head word embedding use train dimensional embedding event tf window mention participant tf vector time role logistic regression order document collect document event document tf across document conduct experiment corpus annotation event resolution annotation seminal event divide training train document within chain corpus c cd bl hdp event bl hdp pairwise event extraction annotate participant within cross previous merge document seminal meta seminal metric gold predict merging operation recover b proportion overlap gold gold standard predict bl group event head word pairwise single agglomerative resolution pairwise merging threshold document hdp within hdp feature hdp perform base measure hyperparameter development variant crp cross document reveal incorporate dependency
trust make circle similarity commonly inspire propose domain verify common relational datum structural relational mention oriented level whereas technique factorization trust meanwhile social trust reality extremely hundred friend besides people similar behave trust rank trust problem suitable paper recover matrix np relaxation order solve minimize row svd certain recovery sample rapid matrix completion mention whose recovery world recent utilize notably collaborative nuclear encourage recovery completion formulation handle inspire study completion establish minimax frobenius practical constrained section notation formulate max problem meanwhile accurately expense study dataset conclude paper discuss future capital letter row respectively denote element norm minimum simply explain tight nuclear eq row thus nuclear regularizer row max give approximate eq q expect projection otherwise nuclear surrogate nuclear uniform motivate norm max solver definite element solve obtain solver scalable max combine conduct trust link link dataset link label table link user individual degree form dataset trust baseline tune method choose let metric average error mae square rmse c c observe entry entry cm cm cm mae mae mae rmse mae mae rmse randomly measurement range split run sample deviation mae rmse detailed table result indicate obtain much small mse comparative baseline observe entry mae observed mae comparative term method superior comparative life mse rmse study examine application average entry illustrate accuracy efficiency also mae second second enjoy mae rmse compare baseline order magnitude three trade influence dataset little rank one know actual local minimum optima trust special trust observe uniformly completion solver utilize examine formulation consistently outperform observe achieve accuracy comparable year collaborative filtering empirically theoretically superior popular nuclear norm encouraging promise max practical clustering etc technology mail edu usa edu national university edu sg lemma corollary social address significant problem explore user formulate indicate however challenge trust problem bit sample non handle motivate recent propose since optimize utilize project superiority benchmark popularity network friend recommendation information etc trust negative relationship social user trust friend enjoy naturally social within matrix bit code imply th fraction goal bit pose solve social category first similarity individual people completion
control low rank differentiable strongly constant x deviation e distribution constant resp line frequently uniform sequence zero one resp furthermore notation state assume term depend hand gradient constant take constant large match gaussian completion sampling provide upper error q satisfy bernstein control uniformly logistic bernstein instance exponential noise estimator show coincide oracle derive tool allow prediction risk true belong w integrate bregman divergence leibler optimality provide convex detailed previous inequality imply hold use assume sub state suppose consider uniform satisfy factor depend treat frobenius error appear union combine proof nn u tc exponential bernstein rectangular see start inspire matrix zero integer subset cardinality distinct element difference element leibler xy ny give ia cauchy schwarz imply matrix consequently triangular duality q give sharp event define apply union bind argument let argument give eq ny apply inequality plug eq note duality subdifferential easily subdifferential cone exist divergence subdifferential resp right singular x w r side bound inequality x q holds see first argument therefore bernstein ensure independent n z q bernstein inequality solve q conclude distinguish completion consist reconstruct class nuclear estimator sum distribution family minimizer term penalization upper risk improve completion exponential r leibler translate upper frobenius risk rate completion exponential range collaborative filter much total whole sample index assume conditionally class nuclear estimator nuclear extensively decade prove setting additive sub unknown recover efficiently low rank prediction denote frobenius prove actually factor although discrete first later consider prediction nuclear case belong rich discrete exponential provide upper see match suggest room mild logarithmic penalization provide control noticed exponential additional inspire additive family kullback leibler order inequality minimax factor rest background exponential family distribution give completion inequality scheme finally defer throughout notation integer hilbert schmidt bregman
accomplish standard neural layer embedding feed let ground cross help neural network explanation since embedding sentiment secondary syntactic lose analysis embedding capacity capture word semantic supervise approach task sentiment small space irrelevant use small embedding protocol analyze subsection namely memory consumption setting code reproduce please refer website test sentiment aim sentence categorie sentiment weakly negative dataset testing training sentiment phrase sentiment approach leverage art convolution vary range well may teacher remarkable teacher additional knowledge dominate student dataset teacher express additional nonetheless encode teacher embedding cross complementary encoding embedding fashion straightforwardly long latter compare largely reduce representative recurrent neural rnns convolutional network cnn channel deep short lstm rnn extent indicate notice architecture lstm dimensionality nlp word embedding experiment superiority network combine exist complementary rnn rnn lstm cnn lstm zhang software institute china neural resource address problem embedding nlp task dimensional embedding retain well directly encode knowledge attract past year address probably extract large dataset aspect memory consumption appeal train small aim retain particularly neural application scenario device large much literature feasibility neural feed forward recurrent teacher classification guide argue truth feasible background review despite generic focus embedding nlp specificity bring knowledge word represent multiply table multiplication column retrieve particular embedding supervise apply encode specific original one fold address embedding propose phenomenon embedding notice encoding teacher complementary exist like say training teacher teacher guide student depict typically class variable teacher valuable student class impose training
near neighbor work give nearby stay increase table statistically tight rate perform well datum unlikely assignment several neighbor incorrect label c c randomly draw cube leave half example permutation exactly sample use reduce boundary change role assignment increase stronger increase weak bound distant play ratio neighbor assignment generate characteristic class cube cube cube eight cube improvement rather classify improvement even score likely score improve blind choice produce accurate produce bound example future fast speed test may permutation great challenge avoid assignment working programming likely explicit partition produce partition produce bound challenge friend join testing apply develop interesting concept bound well development new permutation validate bad assignment effective statistic bad assignment especially accurate classifier classifier permutation bioinformatic apply ideal result test bad permutation bad produce set example input work know develop classifier working example validation produce work bad work evaluating cause work example assignment rate assignment statistic assignment evaluate assign work neighboring follow term section review assignment present score work learn label random label input work classifier predict output w associate input eq labels indicator produce probably sequence example among ranking map draw sufficient entry equally likely working assignment likely rank unknown work set compute score score draw draw uniformly eq equally assignment bound highest consistent ease influence design scoring outline develop set permutation random permutation scoring function improve nearest nearby score scoring permutation last nearby among work scoring typically produce strong bound score nearby neighbor contribution emphasize neighbor among set breaking scoring q indicator scoring specify much nearby distant neighbor nearby scoring follow work favor assignment label base solely assignment example neighboring example near bound adjust influence distant affect setting refer scoring figure disagreement much disagreement neighbor example figure increase one neighbor role accept reject baseline scoring near neighbor scoring equivalent equivalent result different value parameter vary amount plot average row table error datum nearest neighbor work subsequent show cell show cell subsequent show mean standard difference mean standard deviation bounding vary quite bit error note cell value estimate trial size uncertainty mean trial deviation
behave factor ensure probability ensure optimal underlie max regret good regret see five estimator trial uniform perform estimator step underlie generate perform poorly furthermore dirichlet prior look figure lemma university california estimate distribution sample symbol learn kl decrease alphabet size min view limit know underlie know symbol equally show competitive reduce uniformly every alphabet essentially advance nearly natural also incur competitive natural demonstrate effectiveness term kl intuitive distribution generate distribution evaluate distance bit encoding estimating achieve estimator namely min min view know quantity collection alphabet simplify alphabet specify redundancy distribution measure consider appear motivated bioinformatic modern fair achieving example english alphabet english vocabulary whose corpus alphabet size constant show alphabet size several modification reflect estimate probability symbol show regret alphabet appeared give time restrict unimodal regret whole consider max competitive regret tight address collection derive drive well reasonable collection consider datum estimator assign symbol symbol exactly every sub person two relaxation come person compete person know restrict particular arise oracle consider know interpret competitive low regret partition part see appear context compression call competitive collection keep know natural question keep know permutation example permutation clearly relation partition let bound partition contain another restriction still exactly force appear time observe assign nx nx since estimator oracle compare drive low eq estimator good estimator knowledge regret estimator q show give permutation partition estimator organize state result section min competitive logarithmic incur estimator nearly however equation fact imply hold every eq big equation consist combine auxiliary denote appear sequence class show natural symbol ny follow negativity kl distribution hence ny ny estimator max
prediction comprehensive comparative feature task computer diverse traditionally technique art quantification tool quantification survey address encode information concern utilize feature texture example type classification several unsupervised style signature encode adapt typically texture statistic purpose highly affected resolution researcher feature base histogram sift sift discriminate comparative evaluated sift sift semantic encodes semantic low conduct al inconsistent texture pattern et al metric influence wide style cover five conduct bar convolution neural categorization lower optimize style preferred indexing large collection explain methodology appropriate combination metric accurate base style image low e object importantly metric learn raw visual meaningful additionally scale collection fine explore find explain type feature represent similarity art publicly knowledge large collection collection fine range abstract etc landscape etc collection close collection target automatic visual feature extract task limitation variation visual specific large intra visual challenging task select ensure testing use subset date restriction least total similarly use subset style follow first depict extract image prediction optimize style induce project optimize feature learn classifier prediction vs svm focus two follow use extract visual feature prediction project separately final training classifier fusion want information vector intractable separately project third project metric optimize obtain vector learn individually take account criterion computer vision unsupervise way learn categorization cnn dimensional characteristic high descriptor mean feature notion object low design scene categorization provide implicitly capture dominant semantic purpose image represent confidence presence image capture basic comprehensive visual apply feature vector follow extract convolutional remarkable categorization cnn layer follow three fully bar et al show output performance style cnn vector find zero function optimization eq trying adjust accuracy avoid overfitte result depend unsupervise mahalanobis distance decompose distance interesting dimension importantly significantly reliable metric supervise differ form objective near start project correctly classify classify member decompose choose rectangular matrix implement happen next solve optimization mahalanobis distance optimum solution involves locally instance target class apply sample stand lead popularity variation introduce call gb experiment assume poor root visual extract fact combination learning find metric combine mahalanobi shown learn merge metric final metric theoretically perform find similarity style rather mahalanobis involve information use measure I via aim preserve indicate start metric set similar euclidean iterative minimum perform sensitive objective leave distance weight minimize leave result variety explain extract visual level feature dimensional base representation descriptor sake fair task analysis eigenvector order type drop eigenvectors cnn feature eigenvectors metric l cnn boost version http www implementation adopt implementation www uci edu smoothly follow implementation author nearest neighbor regard metric fast one computational parameter metric feature category style randomly follow style section aforementioned concept learn metric however metric impractical highly toward list partition find penalty term fold l cnn dim boost percentage metric row metric style quantify similarity metric feature raw boost style great improvement baseline gain boost boost represent finding paragraph big abstract first art verify abstract characterize much active process confusion happen column row effect scale early capture system confusion color agree member color lastly acceptable art note post row th synthetic synthetic later act color perspective get reasonable ten vs feature row different achieve performance boost metric generally classifier less show confusion classification boost investigate landscape nd rd confusion element hand landscape similar daily despite landscape l dim boost vs produce confidence combination boost metric improve except cnn gain good confusion reasonable case row rd interestingly history friend paris interaction one column confusion acceptable american influence look report improve classification independent perform bad boost image perform rooted amount supervision cnn training unlike design bounding box style metric feature individually next aforementioned goal give project project together vector three task table early show fix bar compare report perform style half image achieve variation achieve classification metric project table feature work happen dimensional outperform art image reduce representation gain classification qualitatively feature learn fusion output right fusion close learn base across boost name six row six six investigate applicability metric meaningful metric measure metric supervise put close far learn significantly conduct publicly available aforementione task superior style superior learn work individual information metric across
fairly satisfactory precision estimation enhance efficacy cost flow generalization numerical assume generic pair wise one infer homogeneous precise estimation recent comparable belief empirical future study apply essential perform support grant foundation science massive continue importance rapidly boltzmann physics hamiltonian field interaction learn volume demand big require study boltzmann involve construct even amount substantial amount relevant quantity acquire selection science capture essential generative identify characteristic origin impose wise successful regularization norm pair employ seek number component overcome property parameter wide enable model study resolve lack implement technique often namely minimization reduce interaction problem mechanic exact point optimize organize boltzmann development area resolve fourth summarize ise bias represent sum adjacent component spin configuration gibbs boltzmann ising standard estimation word kl kl divergence evaluation therefore technique approximate partition pseudo function maximum term problem approximated quantity type field pseudo amount implement minimum method inspire impose flip balance q intractable due remove update rule instance metropolis heat probability flow follow flow computation tune expect change divergence combination elementary algebra master lead order monte impose balanced cost minimize estimate likelihood satisfy precision exceed straightforwardly assume assign component interaction bias originally unique equation impose condition estimation approximate description statistical mechanic law mean field validation generic powerful boltzmann pseudo likelihood minimization follow minimization let iteratively technique reach recursive derivative pseudo derivative yield backtrack gradient spin th th configuration backtrack acceleration technique minimization condition conduct several experiment spin configuration generate markov carlo spin interaction bias bias zero interaction restrict neighboring pair know lattice know selection priori estimation likelihood estimation iteration blue top flow fast convergent use convergent estimation correct bias interaction fig interaction confirm estimation interaction characteristic wise bias parameter observe nonzero wise bias small estimation threshold probability panel estimation minimum red slope guide pseudo iteration minimum interaction lattice precision lead estimation wise bias profile interaction show fig comparison emphasize prior sense snapshot indicate generative truncate
unless monotonically decrease happen trend consistent monotone change proportion population hazard big treat patient trial short hazard dominate effect trial baseline hazard combine effect big compare harmonic mean parametric sense confident control group trial harmonic trial effect underlying covariate need survival patient clinical time follow proportional hazard xt h p trial follow hazard baseline hazard variate baseline hazard hazard need hazard parametric subscript stand harmonic hazard aggregate patient realization maximum mle fit pool record follow try true overall treatment treatment concern condition get treatment establish assume proportional hazard true mild regularity prove mle solution converge probability detail asymptotic derive side mix third discussion maximizer minimize kullback mix ft mix mix ft version log parametric kullback distance proportional hazard hazard hazard interval still group patient text formula remain unclear discretized consecutive event history ignore hazard beyond estimate fact patient specifically interval consistent hazard survival true trial patient consider pdf xt pl patients survival law guarantee hence estimate hazard within interval formula investigate various definition patient go quantitative relationship definition hazard follow hazard hazard trial trial always equal follow hazard iii harmonic define true harmonic trivial eq achieve equality le le e equivalently algebraic definition imply comparison rule practical small typically small observe close indicate demonstrate plot three estimator always basis comparison ht pl lc pl lc assume great hazard ratio effect depend hazard ratio various combination l l l liu treatment effect clinical patient characteristic account real practice necessary development hazard cox proportional hazard challenge combine clinical trial formulate treatment effect estimate hazard interpretation analysis involve survival trial trial newly develop variability patient clinical trial efficacy overall efficacy population cox proportional proportional hazard hazard hazard vector log specifically patient interested inference base pool trial analyse comprehensive clinical hazard derive patient collect th trial scheme popular option obvious minimize though researcher recommend ratio covariate value effect effect meta mis baseline method require interested trial far develop meta set hazard unique log hazard trial vary trial hazard ik attribute randomness outcome however effect essentially genome treatment subject true vary patient population heterogeneou inclusion appear align impossible adjust hazard factor detect currently technology treatment pool statistic report patient investigate effect hazard survival method cover li provide another good necessity trial trial advance solely screen reflect patient profile intend capture either difference treat result combination treatment opposite life ready patient overall treatment cover patient concern effect calculation derivation guarantee correct amount significance despite usage completely exact measurement treatment hazard cox hazard obviously regard survival time matter coefficient treatment concern hazard later natural counting event hazard artificial concept cox hazard hazard overall cox population ideal case may performance shape hazard ratio say treatment readily functional address typically paper statistically drug efficacy challenge patient aggregate trial patient clinical survival patient x ik covariate function many accommodate covariate independent censor patient I hazard definition pdf hazard hazard trial collect assume hazard independent censor patient second trial trial matrix setup underlying value indicator arm concern convenient discussion pool line consider pool patient record estimate effect good answer first question obvious require patient originally overall define limit worth effect clinical statistic vice versa different treatment valid discussion provide example impose cox pool hazard hazard care establish true pool hazard control hazard formulation limit develop lin hazard patient pool convenient notation convention lin lin hazard effect censor hazard hand formula hazard eq calculation f xt expand notation proposition nf xt mf nf xt te te te te notation covariate identically order plug definition pdf survival substitute let censor true treatment either trial know denote respectively overall hazard define covariate treat numerically need patient nonetheless aggregate available solution henceforth subscript stand baseline hazard likelihood procedure know regard condition censor definition performance censor noise impose survival trial underlie hazard ratio censor always overall treatment matter censor assume sort randomization analysis use arm hazard hazard calculate pool patient record simplify transformation side strictly overall hazard superior small alternative look overall log hazard combine trial actually censor censor survival censor limit solution mixed population censor length censor hazard trial cox indicator clinical trial censor unity expand definition variable calculate censor simplicity fix equation define pdf integral unity fraction otherwise pdf turn define stochastically cdf make censor maximum indicate accepted treatment recommend report overall log hazard multiple former unbiased censoring provide chance researcher aggregate illustrate censor round patient trial randomization e patient control group survival times standard patient efficacy survival time trial patient censor confidence hazard censor various censoring try report censor trial trial censor pool pl
performance structural broad regime may bits less distribute non previous work good study communication communication machine problem however specify agnostic distribution one one optimize moreover convexity proves distribute split uniformly flip emphasize distribute study setting architecture e machine shared optimize study bound matrix norm denote exist parameter lipschitz g strongly correspond eigenvalue computation communication round computation communication complete computation clearly machine solve measure bit typically constant factor logarithmic domain gradient object model accuracy introduction distinguish relation natural correspond split different statistical often sometimes assign statistically quite similar example context point machine rank early situation typical randomly partition datum essentially communication one function remark relate typical randomly partition datum however construction actually randomly partition dataset partition gradient aware multiple round set initial statistical independence condition quickly seem assumption function impose certain mild operation involve gradient vector involve communication round algorithm sec initially round iteratively compute point every provide span explicit part include machine machine weak lie previous subroutine computing satisfying span incorporate optimization plus previous gradient well assumption satisfied technique aware restrict complexity perform requirement break bound namely origin convenience easily starting accordingly technique optimization construction dynamic roughly necessity communication round progress machine present smooth even machine quadratic sufficiently f round purely convenience decay implie large whether local method gradient round clearly feed result communication round iteration use descent smooth convex yield round constant identical remain utilize round low paper scale total communication round complexity somewhat function open suppose two machine local verify smooth optimum average function diagonal point coordinate progress additional result statement assumption simplicity informally discuss extension smooth therefore adapt namely state even satisfies lipschitz continuous unit round thm convenience together thm convexity smoothness communication round emphasize allow machine arbitrarily many operation assumption matching implementation subgradient actually function smooth aware algorithm performance wide thm idea fix round machine smooth argue result compute non exist satisfy machine without communication round finally result set related multiplying kind structural single communication round still capture realistic interactive distribute fp group feasible point infinite sequence lower lead principal half main q assumption communication recall point round odd machine hold state invertible admit yield absence machine function machine point imply machine progress without subsequently local round consequence communication prove main first smooth realize root choice verify range find machine round bind last must vanish fact strongly smooth bound pick round eq computing minimizer lower depend similar communication sufficiently construct provide type statement explain extract specify later ball q eq convex due lipschitz euclidean subgradient function moment linearity lipschitz linearity smooth case matter choose gain point communication modify differentiable machine point analyze round e execute must machine local machine similar line standard machines eq diagonal therefore mean contradiction assume exist absolute case contradiction remain depend odd valid subgradient satisfy rearrange zero well contradict hence thus generate hold repeating absence round whose assume initially therefore generate hold note contradiction odd term involve case absolute term odd depend subgradient satisfy rearrange term imply q implie eq contradict machine whose function round hold machine repeatedly get corollary round turn namely optimality communication round dimension employ ingredient derive corollary round must triangle bind output flip minimal putting take low well communication round strongly bind apply plug round consider must must construct two machine order provide receive let constant whose symmetric equal establishe definition indeed smooth strongly fact spectral eigenvalue indeed invertible inverse lie upper optimizer strongly show eigenvalue lie eigenvalue smooth hold machine produce machine construction communication smaller formalize follow theoretic lemma bit communication define return satisfie expectation convexity average theorem prove column thought algorithm base return operation get statement expectation uniformly random value eq let show lemma random exist symmetric symmetric whose letting plug random norm principle randomness remains choose e independently speak much information much probability choice recall entry sign bit hold providing recall independent send machine hold conditioning kullback leibler negative recall jensen e root upper root equal information variable compose ii eq second hence corollary institute limit efficient distribute bad room thing objective statistical datum otherwise communication round machine
recall class highly reasonable incorporate cost misclassifie misclassifie misclassifie assignment reflect class penalization boundary cost obvious weight interestingly strong constraint aim discriminant oppose much leave enough generalization performance uci learn usefulness public use input include output median make quality excellent new become combine category process leave randomly remain testing conduct preserve ht nature principal whole illustrate principal plot figure difficult considerable appear scatter plot align weighted misclassified point type misclassification unweighted plot omit omit bar compete ordinal observation three severe article version classify binary simultaneously extra ensure sufficient qp condition integer turn without nk j maximize example dual possible maximize objective ultimately share boundary flexibility usefulness fisher validate ordinal ordinal binary binary interesting include comparison binary denote first product suffice boundary possible consider q write two class positive complete acknowledgement support college foundation thank statistical spend write wu liu theorem condition pt treatment ordinal inherent class due issue ambiguity machine reality area disease diagnosis national security quality tumor iii security five category green blue red order severe quality randomly excellent good ordinal classify ordinal base actual importance ordinal ignore one ordinal one body versus versus ordinal sometimes suboptimal treat equally relative superiority reveal desirable approach utilize available class ordinal sequentially conduct classification combine meta liu simultaneous idea classifier simultaneous well many parallel classify classification boundary maximize pool binary formulation regression optimize parallel separate hyperplane properly ordinal problem ensure fairly either kernel lack framework property example study conclusion remark use ignore ordinal ordinal introduce ordinal lastly principle ordinal classification class prediction py aim classification mc classification rule opt ignore ordinal however suggest wise united state vote party least vote receive north blue much large latter color home classify state red great blue state x seems correctly identify simply break appear underlie root ordinal herein make ordinal appropriate simple lead next randomly state conservative state relatively less conservative conclusion furthermore binary separate combine combine negative label rule observation subproblem aggregate classifier intersection htb bc bc bc table classifiers compare meta observation second prediction reach ordinal prediction pool binary classifier first ordinal top length block class k inside objective kernel svm th minimize objective boundary however rich cover indirect approach call subsection kf slack incorporate wolfe duality come variant multiplier constraint tucker kkt kkt condition item eliminate full whose top primal problem kronecker dual nothing qp equality solve party qp implementation beyond scope k classifier nk lead implementation except lagrangian kkt invertible rest identical sufficient monotonically decrease ultimately condition discriminative need monotonically aim sign regard monotonically constraint I specifically condition logical implication involve number integer seek article exactly impose training vector train datum rich enough sign rather especially norm penalty svm difficult objective aware efficient solve show article simplicity mixed package capable deal available solve qp linear probably statistical integer integer aspect bayes fisher ordinal second normality binary py binary intuitively fisher decision rule fisher classifier function kk
independent build word predict linguistic text however restrict predict semantic benchmark propagate concept never abstract semantic derive corpus apply variety semantic purely symbolic ai linguistic modality lead linguistic often automatically induce collection text visual still drawback generally build linguistic visual concept context visual knowledge modality linguistic coverage apply vision labeling issue upon skip gram construct linguistic context relevant image present corpus human word predict representation jointly linguistic encourage propagation visual representation direct available enhanced achieve remarkably semantic benchmark shoot indirect representation abstract word break cognitive study literature multimodal distributional semantic representative straightforward induction text construct singular decomposition concatenation advance visual representation rely annotate visual attribute multimodal fusion stack autoencoder concatenation visual skip system approach derive multimodal unimodal concept image jointly topic empirically weak propose incremental recurrent focus acquire realistic scale focus affect class less effectively integrate feature word thus linguistic extend model proxy incorporate easily recent address image common induce annotated linguistic retrieve word multimodal follow et al implement subsampling option randomly discard word part softmax context target vocabulary normalization equation considerable actually word fix identity take concept like linguistic often produce visual visual gram multimodal equation ex skip force representation account note visual systematically e generally objective result distinguished multi way force visual representation try directly linguistic representation dimension linguistic induce second linguistic move maximize similarity max use connect margin enhanced word visual advance range visual sample visual act encourage word currently uniform sample control visual linguistic include linguistic representation sketch linguistic onto jointly linguistic straightforwardly substitute modal mapping induce overfitte l regularization corpus wikipedia comprise multimodal visual imagenet occur accord corpus associate imagenet convolutional correspond activation word derive representation facilitate comparison
graph misspecification graph adaptive weight compare final fuse difference correspond elsewhere consider per c penalty select performance clique weight improve clique similarly even clique information c subject group show star star graph star particularly group unbalanced version seem phenomenon behavior group probably theoretical value balance around evaluated computational set variance well penalize bic group bic compare present effect present difference detect parameter enjoy fuse select theoretically fuse nan difference fuse select include effect fuse l real drug stroke reach mainly due anti p improve clinical trial conduct gp incomplete three way interaction pool subject alone alone subject alone subject subject plus patient drug correspond elimination volume individual suppose drug pose small add fix g group penalize algorithm suppose group adaptive use ensure comparable inclusion penalize compose concern certainly low low among de gp experiment concern effect variance estimate fuse allow difference parameter variance iteratively penalize simulation theoretical study future variance volume correlate prediction penalty tackle suppose equal introduce concern bic validate especially dimensional since one subject receive modality could spurious association inter I mixed analyze group usually group group among allow comparison group parameter fuse fix effect penalize couple alternate multiplier solve maximization illustrate compare real drug mix mixed field especially clinical trial clinical modality drug drug clinical trial two patient treat patient population trial assess significant group categorical influence study intractable version em combine criterion group drawback group state reference allow select combination group group consider group difference difference encourage estimate fuse coefficient induce mixed model effect variance effect variance complex difficulty likelihood intractable paper deal semi selection penalize genetic variant penalize maximization square recent apply work use objective fuse jointly several detect variance effect maximization variance sum absolute suggest admm direct penalization covariance admm use group introduce bic criterion section introduce fuse penalize tuning section simulate cross clinical trial study drug drug observation th patient two measurement vector decomposable group patient transformation normally suppose diagonal explain estimate non linearity penalize introduce similarity algorithm maximize reduce criterion close form em divide two simulation individual simulation belong stochastic statistic explicit numerically except joint group separately characteristic similarity penalty within encourage detail penalty penalty encourage objective study potential maximizing calibrate except fuse penalize effect fix tuning parameter h penalize eq random usual update effect fix update expectation least extension alternate problem piece rewrite equality split solve iteratively solve primal lagrangian lagrangian box update z include algorithm tune replace vector expectation complete group depend penalize admm lagrangian primal dual augment lagrangian generally box box variance initialization convergence update solution numerically approximated tuning replace tuning infinity group difference bayesian criterion optimal return minimal q bic define q distinct penalize particular lar ols hybrid relax unbiased select constrain solution nan weight difference high finally correspond behavior simulated subject path depend parameter present impact variance estimation subject benefit weight influence penalty simulate set group joint variance compare variance parameter normally set implement last stochastic step equal evolution
label positive exploit label score negative surrogate bind importantly optimal separate sequel surrogate margin form bipartite select subset position happen item irrelevant scoring assign rank implicitly positive loss surrogate moreover scoring set defer conditionally notion weak condition scoring simply margin iff substantially negative strictly notion binary classification negative seem natural notion weak margin dataset surrogate tight optimal scoring upper bounding imply due surrogate surrogate upper notice rank upper way different surrogate score low rank positive highest rank surrogate well consistent strong margin label margin strong actually much incorporate relaxation tight replace negative consider convex ex q reader proof notable recover original label scoring set margin margin condition scoring margin strictly weak definition fraction assign score assign negative margin weak separate negative positive negative demonstrate three surrogate surrogate formulate mistake condition perceptron maximize performance setting batch gain popularity go away individual instant point top update update negative sake depend perceptron first positive negative score point fail rank score note I extension let receive sort score k perceptron enjoy mistake state loss defer appendix suppose cumulative mistake execute batch mistake also simple score batch mistake become easy hyperplane margin margin binary technique negative mini raise update slightly high dimensional design rank negative score false negative positive negative enjoy let mistake algorithm simplify situation separability definition norm exist condition batch mistake exactly classification bind strong perceptron outperform latter tight mistake suggest fails exploit sgd scale minimization erm pass optimal estimate notice problem mini processing sgd optimize mini batch surrogate make via gradient crucially unfortunately bias perceptron sgd novel theorem online batch guarantee generalization version surrogate score divide convergence predictor population w surrogate appendix well exhibit uniform establish result establish partly term surrogate manner positive labeling nevertheless strong batch compose point I u feed random stream batch generate model bt mistake ensure ensemble return stream c rt b perceptron surrogate well establish execute sake convenience us ks ns ns define thus scoring prove condition label identify positive ns k k case last margin claim mistake suppose let mistake define mistake tell far repeat application start p desire sake brevity step obviously last negative negative positive combine proof update convenience mistake fast mistake mistake prove part however modify prove lemma two sake update since sake convenience note false negative position rank I p crucially utilize help pointwise lipschitz norm establish additive top rank list nature situation analyse universe population arrange without replacement arrange least thm surrogate exhibit uniform convergence four separate subsection replacement sample notation label shall tuple first score let population arrange q application fix uniform fix define large well sample write vc convergence thresholded vc argument establish recall surrogate uniform convergence cover give require convergence identity residual hoeffding inequality tell residual follow order order establishe conclude uniform conclude least sample involve surrogate true label point define true take every two give surrogate definition measure population achievable respectively classifier achievable positive assume step third since fourth application union lemma set element optimality similarly write crucial give far last optimality proceed proof ease simplicity q similarly simplification assume last step since analyze term analyze nature define assign negative label suppose maximize clearly top negative rank positive formalize point arrange negative arrange fact proof exhibit convergence average score thing contain use least q show pointwise lipschitz evident pointwise analyze compose sort negative separately position list score list pointwise uniform lem sup fix analysis lipschitz application lemma tell inequality gives result lem rank universe total item arrange decrease let replacement population arrange decrease assume sake round element bottom sort population sort note bernstein replacement step complete surrogate generalization technique involve prove version thm conv execute stream batch length proof theorem closely theorem confirm precision top find relevance learn severe imbalance popularity significant gap notable lack stochastic optimize heart family bounding surrogate surrogate motivated principled natural margin surrogate novel perceptron provable devise scalable provable bound rely novel convergence structural conclude experimental state cut stochastic maximize relevance several life anomaly rank event rare spam rank accord importance performance average ndcg top rank list informally relevant item widely classification ranking learning remain knowledge reveal general rank aim develop classification agnostic notion distribution give deep framework setting margin condition appropriate recall top notion call relevant item separate irrelevant margin restrictive notably much restrictive item irrelevant notion margin suited perceptron surrogate performance surrogate key firstly secondly surrogate satisfy mention early consistent condition discussion reveal surrogate lie hierarchy gain analysis design perceptron extension perceptron perceptron mistake margin mention early mini style prove bound batch perceptron surrogate novel result require measure surrogate establish sgd algorithm perceptron algorithms organization present formulation surrogate margin condition reveal consistency perceptron mistake
decision fact well match make side conceptually learn interpretability netflix interpret assignment hierarchy movie movie split assignment assignment etc figure induce restriction branch cluster together beyond look branch come root tv range collapse guarantee additive cluster accurate diverse achieve publish domain lead model modularity explicit actor interpretability like valuable support google fellowship national foundation research fellowship grant national science conclusion recommendation material reflect view national foundation thick minimum draw circle draw sep inner black fill blue sep blue minimum google view google com j pa google view usa completion popular tool recommendation take drastically co allow learn surprisingly clustering modeling suggest capture latent preference decision make present classic cluster collapse sampler guarantee excellent efficacy art netflix netflix compare state art rating movie user preference item recommend like refer netflix research propose top item win netflix art large combination amount memory interpret integrate large drastically previous collaborative filter start assumption high completion study competitive conceptually interpretable netflix combination clustering user preference matrix factorization movie weight movie user movie neutral assume partitioning movie might part like correspondingly cluster partition pg rate rate take combination co clustering benefit movie user group instance movie certain age rating shot certain actor take combination attribute motivated order encode combination regardless template row column co nontrivial co partition magnitude small compete require user per number match human decision significant say well aim model real completion finding minimize residual geometry row banach space present devise collapse confirm efficacy completion netflix interpretable hierarchy network believe offer promise direction outline begin discuss related recommendation parametric simple mean co subsequently define bayesian collapse extend bayesian direction probably factorization svd success recommender ensemble relate bayesian ibp assume movie binary cluster somewhat overall similarity ibp simple parameterization co strength within intra improve beyond factorization capable factorization closely membership originally primarily row formulation suit biological wide variety ensemble far co clustering body mainly projection aim recommender parsimonious seek aim combination subset possibly scale per column note interpolation mining find way understanding lead pattern item database discrete rather dataset key template md nm st small simultaneously store sum already indicate general minimize compression trivial codebook store element store bit hold storing error dynamic basis nonetheless accomplished factorization approximation linear combination clustering co solution co cluster subproblem hard inner proceed column replace approximation find essentially new good mahalanobis distance stack case quite assignment exist exist obtain correspondingly entry assignment likewise coordinate count outcome row assignment vector alternate row column cluster objective minimum step approximation round dt ensure minimize loss additive clustering key approximated appropriate sake obtain cover unit ball incur element key relate entropy number singular scaling scale coefficient bound via rapidly decay value one dimensionality harmonic ball cover radius operator multiplicative clustering mean convergence apply guarantee denote singular cluster row singular cluster approximate contain ball unit latter accuracy guarantee residual matrix constructive set submodular bound practice manner penalize regression counterpart gaussian begin bt text height text gamma beta sigma observe thick thick alpha c beta r thick thick sigma gamma sigma fit membership draw chinese restaurant template rating begin co basic template belong draw chinese restaurant movie belong cluster analogously additive draw normal variance via conjugate pick primitive combination flexible formal inverse joint q user movie characterize conjugate likelihood accelerate considerably membership form conjugacy efficiently discuss collapse effectively membership family take additional statistic see movie denote user movie express encourage formation collapse need shall see keep rating integrate rating additive normal distribution vector rating likelihood expression determinant determinant assess beneficial assign user movie operation sum collapse gibbs likelihood assign offset log add fairly rating user movie purpose infer variance check normal parameter note play role classic term stein distribution denote analogously gamma movie equation implement efficient sampler cache per sum movie new matter check operation initialize partition movie n n update assignment analogously statistic beneficial iterate initial disk provide movie compatible possible model column regardless respectively nonzero partition column separate bin entry obviously crp unlikely retain fit rich cover piecewise block correspondingly noise unchanged gaussian inference note though jointly indice tractable overlap intersect expensive tb residual residual instead instead algorithm pass modified capacity modify I follow pass operation setup result real netflix run later practice yield hence implement range avoid simplicity infer assignment proceed burn period assignment many available dataset netflix movie standard three split approximate face black image node create learn matrix treat real value hyperparameter depend result quite factorization model factor svd user movie co cluster assignment column calculate size conservative row contain row since primary motivation filter discuss classic netflix measure rmse avoid divergence number using report publish simple conceptually parameter contextual bit training rmse
unlabeled cardinality introduce density label proof framework probability state extend multi originally object density adopt theoretic multi object suffer compatibility involve product power inner product density object weight normalize geometric fusion rule label object chernoff fusion propose subsequently show particular solution geometric mean derive necessary implement fusion rule result hold weight q l w summarize agent l note quantity eqs thus fusion chernoff fusion pdfs normalize geometric x rule apply theorem find summarize proposition agent agent share birth I dx bernoulli independently eqs overall fusion indeed chernoff pdfs time distribute scalable iterate average subsection agent iterate argument per follow primitive doubly consensus iterate converge global unweighted multi infinity stop consensus counterpart review subsection object represent gm q involve provide gm preserve gm gm depth approximate x b ab jt ab jt ab b ab x jt ab jt ab ab jt ab jt ab j b ab ab jt jt ab ab j b p fusion agent pairwise rule property order pairwise irrelevant notice fusion result fuse b separation component common approach represent single pdf combinations dirac delta kernel square burden resource demand gm recursion paradigm describe scalable multi algorithm along consensus propagate code object tracking order unique index distinguish object object kk clutter finite respectively omit k k density density motion birth death concept convenience label omit object state continue exist step evolve distribute accord birth contain element superposition object k k bi ii x ix bx w il detect generate likelihood multi superposition detect intensity clutter object mapping specify track measurement track track track one instant multi update posterior cardinality update take I p iw kx p ix gm algorithm sequentially carry locally agent operate interval gm produce object outcome description gm gm find consensus receive carry rule proposition perform merge location pdf reduce burden step procedure pdfs extraction gm object forward object predict density posterior k kp x z w k ir reader efficient gm reported sequentially carry gm gm section operate gm produce operation consensus gm gm max r describe object track surveillance wherein sake consensus track e velocity motion model nearly velocity model interval arrival arrival measurement respectively linearity aforementioned sensor order update three different clutter parameter therefore clutter severe mention surveillance scenario surveillance area birth birth summary c birth location state region birth clutter false cutoff average monte object independently measurement realization gm gm hypothesis describe merge gm survival maximum merging threshold truncation birth intensity consensus simulation display mean gm gm distribute algorithm merge lose track correctly algorithm localization gm gm cause gm filter drop track generally able object propagation gm gm similar fig deviation number gm gm gm cardinality gm fail track object become snr fails properly set track factor b clutter lose full cardinality extraction track fail even density show scenario case gm fig gm point gm exhibit term current present tracking sensor use consensus fully scalable way collect multiple admit multi object efficient gaussian implementation test initialization sake density get sum delta delta turn normalization exploit w x holds consider density instead straightforwardly evaluate apply theorem consider density get l prove induction proposition proposition ba edu ba edu address track heterogeneous communication capability theoretic distribute dynamic novel filter namely consensus tracking multi mixture confirm scenario label bayes consensus individual challenge track numerous literature fall major filter wireless lead agent capability net technology picture e benefit call agent operate knowledge flow consideration sensor network central scalable respect size operate operate information combine reconstruct scalability requirement fusion iterate cause presence loop suboptimal fusion ci generalization require multi approach state however uniquely identify hand refer jointly object paradigm explore specifically theoretic together filter filter formulation object track work multi generalizing develop consensus base label trajectory principle moreover conjugate prior development analytic tracking filter filter amenable tractable solution key object suffer like filter rest paper necessary fusion theoretic term leibler multi object bayesian present novel evaluate scenario end conclude notation object h convention generalize delta adopt inclusion generalization indicator shorthand whenever letter letter g throughout consensus unweighted iterate compute satisfy relate property square omit fusion consensus primitive stochastic column primitive doubly consensus collective unweighted primitive path vice versa graph whenever node receive receive information primitive doubly I unweighted carry methodology surveillance application vary association extend consensus methodology general trivial proper dealing object review next
patch patch exchangeability assumption extra beneficial prediction originally study house dataset task interpolation pixel choose random imply heavily initialization run initialize argue decide different see mcmc mf average preserve variable initialization except take gibbs consistently outperform outline study epoch mf dual among reconstruction mf gibb believe appearance reconstruction independence subset consequently much structure test mild extra cm cm cm cm c house mf breaking dependency experiment mf gibbs algorithm let mf need datum mf local optimum optimum gibbs predict mf prediction place little probability neither encourage similar encouraging would otherwise clean possible vast genomic crucial importantly inference cope dataset sparse experiment cancer sample include gene drug focus model around set feature rather interpret biological pathway understand characteristic drug profile randomly capable measure abundance thousand cell second control protein heavy effectively spectrum cell analyse consist heterogeneity result gene converge predictive mf algorithm beta conjugacy natural find preserve significantly intra dependence evident significantly mf mf interpolation model genomic mf suggest maintain dependence global mf variable multi bernoulli type local sensitive gibb also dependencie local variable care need ensure encode variational consider appendix detail variable approximation text paper clear bernoulli mf mf mf optimum optimize parameter maintain descent global analytically conditional gibbs sampler design scale probabilistic assess primarily derive genomic dataset investigate lda specifically demonstrate picture sampling certain dependency effective important lda intra perform decade see development flexible diverse nonparametric powerful enable adapt might adapt interest feature appeal scalability parameter analytically multimodal particularly norm along concern mcmc sample summarize simply give typically sufficient performance computationally inference gradient descent improve mini batch theoretical continuous unbounded influential modeling wang say factor continuous drive availability huge text corpus generate still time thousand high sophisticated analyze simple heuristic pca capture advanced analysis great variational one trade complexity evidence work global vector document vector suggest contrary case maintain dependency ingredient bernoulli increment measurable algebra generate consider simplicity beta may set form beta function measure follow z ik ik process stack dimensional matrix infinite difficult derive chinese restaurant dirichlet process scheme process ibp introduce strong dependency derive variational crucial rather reason dimensional parameterize modelling point belief induce encourage hadamard idea beta bernoulli parameterize beta draw generative model beta gamma gamma separation global crucial sake brevity I n dependency take compute draw simple shall mf spike method maintain spike dependency variable lose analogous mf local use local conditional p I mf mf gibbs latter variational approximation initialize step unbiased idea variational first propose stick ibp promising limit use ep scale ibp parallelization submodular perform limited positive field stochastic scheme interpolation task meanwhile model great develop dirichlet availability text idea initially propose refine million book idea learn parametric though sampling improve negative optimize deal conjugacy change conjugacy improve quality variational approximation exploit conjugacy finding carry stochastic variational inference carry denoise apply genomic transform hyperparameter rate schedule question variational answer
term specific problem poorly behave poorly behave cover imbalance runtime imbalance sensible behave bit term must entirely recursion individually query quantity amount bad rely branch bind technique work prune away success pruning depend show bound sufficient enable runtime hold hence runtime intuition answer query take constant independent large theoretical dependent independent plug get runtime dual tree require analysis constant denote imbalance dependence difficult maximum query reference respective whole follow similarly whole calculate dual bounding already build tight imbalance sublinear closely reflect behavior follow show utility simplify runtime dual tb distance near neighbor simply describe query reference study numerous approach cover near neighbor due algorithm sense respectively compare query point subtree improve neighbor node depend definition eq traversal store current candidate array array represent distance possibly query query candidate correspond notational convenience following take set cover tree prune tree traversal algorithms set c n clearly runtime thing remain reference encounter property r qp rp last point hold true neighbor hold eq r r step center point imply contradiction dp every separate yield behave sublinear represent runtime base kernel assumption adapt tight give kernel accelerate thus attention turn towards absolute dominate theorem search also gaussian bandwidth additionally g note dependence bandwidth demonstrate runtime reasonably choose approximately runtime approximately algebraic give exponential exhibit dependence bandwidth understand dependence bandwidth intuitively consider bandwidth increase thing reference scale allow less pruning level effect opposite gaussian kernel give regardless estimation reduce less relative relative division quickly tb query node kernel node combination q contribute create given prove approximate assume assumption theorem size possible approximate satisfy condition time expansion slightly rule tree relative approximate tree node calculate prove approximate estimation range practically identical size work begin range require understanding sufficiently solve array tb qp tb query reference difficult bounding query define expansion slightly set notion may run reference expansion dual traversal also run runtime pruning query reference reference ignore ball lemma produce necessarily subset assumption conclude take size obtain dependence runtime simplification sufficiently runtime simplify easily exponent get reasonable runtime necessary expansion tree retain parameter bind node framework tree tree traversal imbalance theoretically construction tight bound accelerate play runtime bounding show bound approximate theorem count tree involve maximum reference numerous algorithm core computation input implement approximate bad runtime prove problem runtime dual cover plug derive entire demonstrate plug first guarantee tree tree density estimation search surprising computational iterating near every reference close one require answer size subsequently prohibitive pairwise compute require reference accelerate favorable upon intuition physics dual tree extremely reference query separately query query simultaneously traversal dual algorithm easily understand recent query tree reference prune traversal traversal combination consist reference point hold reference single branch child node tree exist numerous tree kernel span search theoretical neighbor search approximate runtime guarantee span calculation combine generalization develop dual require introduce cover theoretical reader familiar cover tree symbol center tree cover hierarchical originally propose adequate description slightly cover level index tree node level associate parent consequence definition exist node scale child child contain within radius center note cover may easy child node remove child self take node tree construction explicit representation property expansion definition metric dp eq heavily literature dimensionality scenario dataset draw distribution converge see generalization expansion small fx fx close easy however empirical speedup existence dimension small distribute add origin whereas small single add origin encounter much convergence child lastly introduce convenience packing argument expansion subset may trivially sp sp point separate bad pack perfect imbalance lead degradation performance child effectively neighbor cause sort sort imbalance understand imbalance formal imbalance performance measure imbalance another aim imbalance utilize tree already cover index leaf level child need strictly low cover child balance hand child refer cover cover nothing number branch happen practice imbalance far graph dataset outlier away happen node outli point dataset easy outlier structure top illustration chain way motivated imbalance imbalance cover miss parent small root imbalance write calculation cover imbalance easy calculate imbalance tree cover imbalance level perfectly miss imbalance figure entirely like imbalance case imbalance cover reference imbalance near point imbalance leaf parent cover tree specifically aim imbalance perhaps imbalance cover tree actually cover originally intend neighbor approximate neighbor calculation dual algorithm abstraction tree node node pruning pair tree pruning traversal later node reference r r r qr cover traversal traversal cover implementation library traversal originally initially contain depth query reference end recursion maintain node maximum line query tree tree determine aim keep query reference combination reference combination checking line query prune strategy significantly node child node possible combine node pair hold separate reference scale suppose exist implicit child node child implicit representation argument hold set fact runtime notion traversal inter alternate contain recursion reach converse notion scale dependent nonzero pairwise pairwise define top minimum minimum node scale leaf scale reference extra reference recursion recursion situation happen let reference tree cover traversal extra happen query recursion happen recursion reference thus result apply
message contrast message variable incoming amp give behaviour evolution treat appear measure bit decoder analyze exponentially allocation tt remain step q termination amp decoder terminate sufficiently large main follow amp lemma lp block decay allocation error amp decoder measure outer rs one symbol rs guarantee decode rs section modification performance length allocation improvement error rate second hadamard gaussian amp mention consider amp decoder couple hadamard hadamard spatially couple modified power characterize parameter let normalizing ensure power recover section increase increase allocate turn help amp little correctly want amp gets start track large intuition limit correctly must exceed proportional need threshold decode power section decode performance flat allocate power compare decay allocation objective assign ensure final enough analogous limit allocation recover evolution top curve rate predict evolution point different give determined amp exponentially decay curve rate allocation rough guess solid curve describe use constant decoder specify exponential allocation design concentration around flat allocation improve rate bottom fig dash curve prediction power fraction correctly step show step evident yield question good rule allocation allocation section limit section step go proof essentially check good finite length challenge investigation computational decoder multiplication run time remain operation finite decode scale linearly gaussian memory requirement proportional store bottleneck scale amp decoder reduce decode memory generate hadamard design matrix pick uniformly result matrix column norm multiplication denote constitute compute length keep extend equal vector keep hence improvement decode infeasible power performance hadamard ingredient amp system accurately particular show almost ratio limit comparison similar recursively recall vector zero first define sigma recursively distribution product involve ingredient lemma column denote projection projection sigma algebra imply equal almost zero large recall mind determine quantity include function lipschitz statement say statement basis constant define gaussian independent section jointly exist finite lipschitz jointly convergence convergence almost limit strictly ingredient prove act limit convergence zero termination index decay power obtain taylor c know converge surely summarize consequently b orthogonal operator onto fix law triangular array array mutually second pseudo order jointly gaussian invertible jointly strictly positive constant let constant depend stein variable exist ni tc ij c element wise due exclude similarly due exclude exclude ai bi index contain expand taylor series argument z derivative alone ignore keep term side analogously amp give eq e need decay exactly vanish fraction section write write inner expectation jensen inequality get use variable recall rhs least kn eq small positive lm expectation bound recall cdf proof statement simplify geometric formula become simplification obtain use expression induction equal entry cf cauchy schwarz expectation non index jointly marginal side line imply n z together thank amp rv acknowledge support grant leave proposition capacity superposition approximate pass sparse superposition code channel rate codebook combination pass superposition linearly design rigorously achieve appropriate power allocation finite demonstrate matrix paper construct capacity achieve code white channel generate input input channel require goal channel capacity give superposition code algorithm decode decay exponentially despite achieve decoder soft decoder guarantee improve finite approximate message pass amp decoder performance prove decode grow decode proportional design polynomial class belief propagation algorithm dense amp proved particularly reconstruct small commonly compress describe measurement reconstruct though algorithm strong theoretical guarantee fast amp find cost dense infeasible implement pass message complicated value function amp difficulty scalar mean amp approximate approximation equation demonstrate could rigorously evolution hold constant sensing problem comprehensive relate propose amp paper rigorously decode go block tend infinity decoder length demonstrate simulation allocation scheme significantly improve close decode design build follow directly go limit rigorous analysis amp ratio section decoder probability decay probability amp decoder go give compare add allocation role exponentially decay allocation use decay section discuss length design know encoder communication begin length denote specific index scalar size receive generate successive message denote zero equal function variable q component contain brevity understand amp iteratively offline via monte relation follow terminology finite decoder iteratively compute maximum obtain statistic amp follow statistic message property presence reader refer term amp algorithm
birth year birth birth year birth year birth year year birth birth htbp ex description mean intercept gender interaction gender interaction year interaction year year ex abstract challenge survey propose survey relie available several national conduct consist disease mechanism survey generation model rate affect health survey rate way first rate participant non participant participant high death participant participant tend participant economic status education trend rate trend indicator decade trend indicator look mechanism ignore deal non make joint data sensitivity design longitudinal may recently subsample full variable non follow survey linkage naturally health correct illustration datum survey decrease decrease utilize detail survey analysis compare trend approach conclude pt national study project setup health education five risk key disease public health north beginning survey conduct survey area sample systematic people simple draw age sample balanced sampling age extend old north ex design event balanced area balanced area balanced age area gender age gender balanced age group answer daily elsewhere seem investigate period indicator page non htbp care health cause link follow contain date death disease j follow non indicator person background person age area gender variable survey survey people survey sample observe participant participant participant follow consist age diagnosis disease cancer participant participant follow person cause event censor censor age censor death date diagnosis date death person death person disease diagnosis structure concept causal model design represent causal bottom background affect vary depend area gender belonging case age affect people person background q cumulative censor survival baseline north risk describe different baseline stand area differ difference year particular study year variable indicator north area area area study indicator logistic gender study indicator eq gender study area gender year reference participant year birth north non participant lose imputation imputation trend monte regard eight iteration first discard remain store eight realization convergence diagnostic chain model two figure shape mixed autocorrelation cause coefficient good summary appendix posterior predictive population augmentation censor draw obtain participant draw imputation censor censor generate straightforwardly event survey sampling treat imputation full level estimate utilize utilize use rate comparable obtain trend imputation trend consider trend correct trend trend adjustment trend trend decrease difference difference correct correct estimate percentage difference comparison original trend study present ex gender credible north htp approach overcome challenge miss apply population non factor cancer potentially participant level provide cancer event participant model design bayesian utilize knowledge availability follow decade follow cancer unclear extent apply directly
likelihood estimate persistent seed simulation marginal sl sg non persistent seed persistent similar trajectory show walk abc mcmc scatter plot trajectory limitation persistent seed predictive seed extent posterior three sl mcmc row middle bottom right trajectory sl sg sl trajectory sg class persistent random sampler regard stochastic gradient variance abc greatly introduce issue repetition interactive statistical mcmc determine abc one monitoring ensure sampling example noise class work usefulness surface similarly abc surrogate minimize call yet still benefit hamiltonian random mini batch langevin perform start momentum necessary drawback momentum update update current limit hamiltonian dynamic prevent error dynamic hmc avoid directly author naive sampling full avoid b address estimate introduce scalar dynamic act update equation summary practice hamiltonian abc plug gradient implicit simulator gradient keep track random seed allow treat simulation function outside control generator part state synthetic particularly difference gradient high choose hmc abc representation produce close dim problem figure part around sl simple sl input x r x x function deterministic outside emphasize acceptable abc abc approximation free set gradient simultaneous perturbation stochastic work free wish optimize mask name entry estimate side call estimate side two side sided maximum estimation blue circle simulator deterministic smoothly simulator sl limit gradient due gaussian smoothed heavy tailed sum previously work side gradient make exploit step would analogous mini batch side simulation computation seed explore gradient hamiltonian landscape additive dependent stream mcmc use proposal persistent seed seed say seed internal persistent seed chain hasting randomly propose seed time metropolis transition location seed seed propose seed independent uniform ratio leave target distribution qx acceptance simplify could fix still sample keeping noise seed carry step persistent seed persistent seed histogram posterior persistent persistent sg achieve posterior problem let posterior shape simulator generate explicitly simulator seed vary reveal simulation blue circle horizontal indicate suited fixing sl likelihood estimate density gradient sl estimate analytically sl exhibit low sl quickly start remain possible leave persistent seed right persistent seed distance persistent seed walk persistent seed optimal seed persistent gradient consistent result posterior chain sl mcmc version sl marginal sl mcmc give identical space limitation seed gradient step persistent set run experiment table report posterior average sg seed persistent persistent seed sg trace single chain leave trace persistent seed
discovery approach score base constraint test construct skeleton exclude oriented arrive constraint ic tc score assign graph score score np often disadvantage inherent instability structure estimation change outcome discovery conservative develop structure new score causal discovery finite advance subsample search structural model causal scientific exploratory incorporation background constrain produce attention describe base discovery simulate world conclusion graphical section graph node arc e reciprocal relationship cycle direct acyclic four undirected graph edge order triple adjacent causal way state relation drawing equation represent variable cause error assume mutually typically follow hypothesis fit modify typical hypothesis add arc exploratory search literature address search genetic optimization prefer fit data objective propose make optimize objective multi optimal solution dominate bad objective model well dominate front dominate call dominate sketch dominate sort multi several procedure objective well preserve diversity model explain ii develop complexity sort lack ii population mutation form sort dominate sort set front sorting generate form create fast dominate sort new combine forming sort member front next population aspect instability change describe robust subsampling method yield sample wise estimation infer non tackle loss overfitte parameterize estimate objective determine identical end probability randomly subsample element traditional concept cutoff sensible similar model indistinguishable represent derive model particular also belong dag direct dag reversible direct arc undirecte reversible edge relation cause effect undirected members arc method phase phase search combine exploratory search iterative process return pareto front come phase combine ii stability output relevant return compute graph stability model stability complexity divide stability path pair level stability regularization parameter threshold occurrence correspond relationship stability threshold minimal model bic path causal relationship intersect top parsimonious call phase combine node edge edge background visualization interpretation knowledge example denote extend work translate dag specification perform measuring convert outer constraint may violate convert undirected edge preserve constraint return reversible dag produces order edge impose edge reversible return dag edge dag transform fully connect dag transform direct start stability path start path end loop represent loop graph j subset dag make population inner form initially else previous sort use mutation combine sort pareto front outer loop start size contain convert pareto dag stability graph consider edge implement modify handle package handle propose set mixture discrete ii stability ii loop population mutation rate initial represent predict model contain parameter predict minimize objective objective threshold effect equal minimum bic iteration loop subsample continuous compositional difference income light emission emission variable cause filter toolbox set roc vary path approach e region actually show well roc curve explanation stop curve stability value higher tend roc curve approach able find high stability end stability roc curve corner effective causal lie entirely method disease relationship development dedicated causal threshold minimum occur example graph subject originally longitudinal slice focus treatment assess individual strength assess measure patient physical sf implement e range treat continuous variable add eight causal show eight stability line correspond figure causal path path graph accord eight second orient background causal path reliability maximum direct path cause except cause study literature activity sense measure result change control physical sense control self focus whereas consider subject variable exclude instance insufficient remain variable subject assessment use treat knowledge variable
mark mark solid fill black forget crcr mark option solid row sep blue mark mark mark mark option black forget sep crcr mark mark mark option fill forget crcr color mark mark draw forget row sep mark mark mark mark option solid black black forget row sep blue size mark mark option solid fill forget crcr mark mark plot sep crcr mark option solid fill black plot table crcr color blue mark pt mark mark solid black forget sep crcr mark marks mark fill black forget sep crcr color blue mark mark mark solid draw forget sep crcr color mark mark mark mark solid fill black plot sep crcr mark mark option solid fill forget row crcr mark size mark mark solid sep crcr pt marks mark solid forget row sep crcr mark marks mark fill black forget sep color mark mark mark option fill black draw forget row sep crcr blue mark mark mark option forget sep color mark mark black forget table row sep crcr color mark mark mark mark solid fill black forget sep crcr blue mark mark solid forget plot sep crcr mark mark option solid fill draw black forget sep crcr color blue mark mark solid draw forget sep crcr mark mark mark fill black draw forget crcr mark mark mark black draw forget row crcr blue size mark black forget sep crcr mark mark solid black forget crcr color mark size option solid fill draw black forget plot table row sep crcr mark marks mark mark black forget sep mark mark mark mark draw sep crcr color blue mark option forget crcr blue mark fill black plot row sep crcr color mark pt mark option forget crcr color blue mark fill forget crcr color mark mark mark option black forget plot crcr color mark fill black forget plot sep color blue mark mark fill forget table crcr point locate center unit coordinate widely applicable sigma radial explicit utilize scale intersection axis dimension sigma refer building upon integration formula integration polynomial sigma sigma refer order sigma sigma filter sigma sigma filter construct gauss method sigma product dimensional weight disadvantage exact grow exponentially dimension exact formula omit brevity see grow gauss apparent polynomially number evaluation point symmetric formula rgb log sigma legend anchor north fill align table crcr solid row sep crcr color sep crcr color crcr black sep crcr forget crcr forget crcr black forget sep crcr point filter smoothing place low square cholesky covariance k k smooth equation likelihood equivalently three filter smoothing likelihood sigma log algorithm conjugate expectation em lower iterate bind update direct function sigma note maximize unnormalized log unnormalized log low log likelihood side measurement compute filter assume density equation evaluate evaluate recursion sigma equation enable evaluate gradient marginal base equation recursion k obtain filter predict prediction jacobian derivative q jacobian algorithm omit maximization find unobserved log sep crcr solid forget row crcr forget plot crcr forget row crcr forget sep crcr width leave south axis bottom line legend style north column align xshift crcr row sep color solid pt sep crcr solid row sep crcr solid forget plot row crcr solid forget plot sep color forget plot sep crcr color solid forget sep color forget crcr article sensor speed rate noise eq angle gaussian measurement assume separate sensor independent measurement tangent measurement variance sensor covariance know noise ground truth measurement sigma scheme sensor truth value sigma scheme sigma already state implementation toolbox initialize investigate uncertainty coordinate deviation component original initial consistent prior high sigma amongst accurate median mle compare close essentially identical sigma sum sigma negligible addition sigma trajectory simulate smooth simulated trajectory try em practically converge couple sigma rather direct mle estimate xlabel xshift align inside minor legend font xshift rgb rgb xlabel initial location sd ylabel median mle solid option forget crcr axis cs mark plot sep crcr e axis cs solid mark forget crcr e color mark mark option solid forget row sep crcr e axis solid mark option forget sep crcr e cs color mark x forget plot row crcr axis legend font rgb rgb height xlabel initial coordinate ylabel rmse trajectory color mark mark solid forget plot crcr axis mark forget plot sep crcr cs color solid mark option solid forget row sep crcr solid forget crcr solid option solid forget plot crcr color mark mark option forget crcr mark mark option solid plot row crcr axis cs conference various parameter maximization smoothing algorithm point well paper focus sigma transform order gauss particle filter sigma point filter extend kalman particle converge filtering particle assume kalman however high computational satisfactory nonlinearity sigma filter approximation integral cost claim integral beneficial test method case univariate maximum scheme similar gauss order conventional transform suggest utility legend style font xshift rgb height xlabel ylabel forget crcr color solid forget solid forget row sep crcr rgb rgb rgb width height scale xlabel ylabel solid forget sep crcr forget sep crcr forget plot dash row sep crcr dash forget row crcr forget crcr axis cs sensor uncertainty target sigma difference amongst method uncertainty parameter nonlinearity variance since sigma gauss scheme scheme consistent high sigma produce well filter sigma integration derive polynomial degree guarantee high integration rule guarantee filtering well hand approximation gaussian integral accurate approximation tracking mean smooth sigma point experiment increase rapidly initial uncertainty demonstrate local linearization sigma scheme approximate sigma sigma target sigma evaluate considerably suggest dimension produce essentially computation close sigma point tracking vary sigma point use iteration affect scheme sigma fraction sigma measure rule converge obtain reasonable approach sigma accuracy could refined scheme direct support grant acknowledge resource file intend serve file journal produce wish subsection go appendix one go like thank text text publish international conference information fusion sigma filtering consider space optimization em give expression approximation filtering sigma require transform quadrature simulate univariate tracking compare filtering transform accurate article parameter expectation sigma direct filter interest filter kalman filter gauss filter surface form k compute k resort sigma bayesian lie estimate static marginal joint state help equation see directly via linear maximization iterate computed maximization bind parameter require smoothing problem smoothing aim extend show gauss sigma use em model maximization linearization extend kalman computing base although easily sigma derive gaussian enable kalman gaussian equation sigma arise integral sum multi sigma present discuss approximate integral assume density approximately covariance covariance consist prediction result k compute covariance equation expectation take k k iterate smoothing gaussian k k density follow q expectation respect k kp maximization q evaluation smooth need solve gaussian integral integral follow form weight multi generalization refer gaussian sigma weighting sigma unit sigma root choice weight sigma stem trade sigma require quantify high method exact axis blue mark mark mark draw forget row sep crcr mark marks mark option solid black forget row sep crcr mark size pt mark mark solid forget row crcr mark mark sep crcr size mark forget table sep crcr scale x bottom blue mark fill forget sep crcr blue mark mark fill draw black forget row crcr mark mark mark option solid forget plot crcr mark marks mark forget crcr mark mark option black forget table crcr color size mark mark option forget sep crcr blue mark size mark mark option forget crcr mark mark solid black forget row crcr color blue mark mark mark fill black forget crcr height size option fill forget crcr mark pt mark solid fill black forget table row crcr blue mark mark option fill black forget table crcr mark marks mark mark option fill draw table crcr blue mark mark mark draw black forget crcr mark mark fill white forget sep crcr mark mark solid fill draw black forget sep crcr color blue mark mark black forget table row sep crcr color mark mark mark solid draw black forget plot mark marks mark forget plot table sep crcr mark mark mark black forget sep crcr mark marks mark black forget blue pt mark option solid fill black draw black forget crcr blue size mark mark mark forget table crcr blue pt forget row crcr mark option forget plot sep crcr mark marks mark mark option fill forget plot table crcr height axis line color mark size option black draw black forget sep crcr mark mark forget sep crcr pt mark mark option black forget sep crcr blue mark mark solid fill forget plot sep crcr color mark marks mark mark solid black forget plot crcr color mark mark fill draw forget mark solid sep mark solid black forget crcr marks mark fill draw forget row crcr mark marks mark forget plot crcr mark size mark option draw forget crcr blue mark mark fill forget sep crcr color mark option solid fill black forget row sep crcr mark mark option solid fill draw forget crcr blue mark option solid fill forget plot crcr mark marks mark mark solid fill forget plot sep crcr mark mark option forget crcr mark fill black forget crcr mark mark option solid black forget plot row crcr color blue mark option fill forget sep crcr mark mark option black forget sep mark marks mark fill forget crcr mark mark black forget crcr mark solid black forget plot row crcr mark option fill forget sep crcr width line bottom axis line leave blue mark mark forget row sep crcr mark mark mark solid forget crcr color mark mark mark mark option solid fill draw forget plot crcr mark mark option solid fill black black forget table sep crcr mark mark mark option solid forget mark pt mark mark option solid fill draw forget table crcr mark mark solid black forget plot crcr mark option forget sep crcr color mark option solid fill black draw black forget sep crcr blue mark mark mark option solid fill draw forget sep crcr mark mark option solid black draw crcr color mark mark solid fill draw forget crcr color blue solid fill crcr color size mark mark solid fill forget crcr mark mark mark option black forget crcr mark marks mark mark solid fill draw forget sep crcr color mark mark mark option forget sep crcr mark mark fill black
neighbor hence motion locally pca iterate mnist digit smoother anti reduce read vs preserve aspect digit sophisticated denoise average filtering loop remove digit orientation several robust reduction classification completion extreme tangent recover another spectral dimensionality reduction vs extracting cluster degree c shift applicability small converge focused technique subsampling structure numerical segmentation practical combine seem study mean benefit parallelism iteration finite structure know priori segmentation ms without iteration arbitrary neighborhood vary particularly large try accelerate ms ms accurately mode approach mode close datum run typical million pixel attack keep really convergence newton modify hessian newton reason low suited system compare gradient ms sizes effective hessian newton method ms ms ms often suffice move iterate mode newton reduce strategy datum close iterate ignore point introduce update em require near neighbor unless bandwidth portion weight another one datum however run point structure involve grow iteration predict simply ms assigning close differ near neighbor cluster cluster discretization fact trajectory mode keep trajectory soon iterate close mode image code cell iterate converge since cell end reduce shift assign mode discretization run shift r paris accelerate shift spatial classify iterate clustering notion topological persistence geometry enough approximation search neighbor compute truncate become bottleneck near neighbor locality hash lsh acceleration essentially fact point suggest soon replace point happen size iteration original iteration pn section nm nm pn prove replace single component method indistinguishable stop contraction segmentation arise constant bandwidth unlike dominate iteration merging mean speedup ms discuss early mode shift meaningful address categorical take centroid nk assignment kde move centroid mode separate combine cluster assignment suitable bandwidth high bandwidth becomes assign close shift centroid kde define robust homotopy algorithm means gradually optimize objective mean thus iterate colored projection middle plot cluster colored fig towards centroid slowly low linearly shape denoise stop iteration consist cluster keep eventually merge cluster show example location intensity result note ms generally gaussian clustering arise size horizontal although vision sometimes bandwidth patch fig illustrate kde manifold small mode manifold mode estimate reasonable kde handwritten digits image pixel feature ms produce mode uninformative mode outlier panel ms tune nonconvex centroid distant digit look laplacian look valid digit representative c c clustering mode circle right corner mode contour kde red centroid laplacian mode mnist datum assign input classical depend respectively represent shift advantage automatically find mean mode error hessian kde also mixture learn mixture density panel learn mapping robot forward mapping give angle configuration particle learn map speech shape inversion american english sound sound mode square illustration arm mode represent multiple correspond x reach mode conditional whose contour dot black sound shift also video image preserve track surface denoise iterate ms smoothing denoise shift nonparametric shift accelerate popular nonconvex image segmentation mode remain perform neighborhood reduction alternate datum iteration modify eigenvalue future high direct design walk laplacian cluster e distinct mode acknowledgment wang undirected edge subset connect connect depth recursively edge adjacent repeat remain runtime provide vertex threshold define vertex point md nm ball connect component give unless clearly separate reliably step ideally converge stop shift iterate tight mode separate tight corresponding mode connect cost separate tight exist large diameter small connect fig iterate heuristic accelerate dimension soon calculation since typically segmentation pixel pixel cluster boundary computing try average entire pixel compute usually interval segmentation feature pixel need mode must least pixel meaningful produce tight naive apply efficient component assign assign cluster number dataset cm lemma prop example conjecture conjecture remark false n electrical california characterize region contain density nonparametric estimate find practice behind algorithm shift tracking denoise mode laplacian acceleration strategy large segmentation intuitive assume datum whose contour suggest cluster step parametric maximize elliptical optima finding global dependent user initialization selection find shape segmentation optima kde mathematical crucially enable maxima require user bandwidth nonconvex shape need cluster try focus mode maximum kde iteration average one converge review kde convergence discuss extension mean shift denoise respectively disadvantage cost mode mode mode image area categorical radius kde contour kde bandwidth mode indicate rescale jacobian hull follow start cluster shift cluster belong kde contours bandwidth indicate mean shift cluster input multivariate scale call point gaussian weight bandwidth isotropic full covariance presentation scalar bandwidth commonly easy analyze rise simple formula rearrange discuss average simplify elegant bayes pn probability mode mode smooth give rise filter shift ms pt pn connect component mean pn pn stop connect ex ms nm nm stop nm nm stop criterion point assign shift bar give covariance firstly mode happen examine step second stop number would mode numerically merge mode connect convergence lie small graph user mode shift smoothed iterate mean shift iterate filtering processing bandwidth quickly meaningful tight cluster move relatively quite reliably ms component although shift produce specifically bandwidth clustering produce quite fast ms particularly accelerate approximation ms independently practically large iteration asynchronous soon make pick unlikely fundamental bandwidth way bandwidth kde set loss rule point bandwidth estimator give give sense exploratory range ms scale point set log vary clustering adaptive shift probability reweighte inverse bandwidth track object use scale scale operator scale filter space efficient rise shift mean computational efficiency since involve neighboring pair still find finite generate piecewise spurious behavior mode minima bandwidth scale vision image sum pixel equal isotropic gaussian scale give kde components image gray modes kde merging take kernel never create structure reflect mode mode increase single unfortunately gaussian kernel create scale practice rare create point large pixel track location running mode tree agglomerative clustering topological persistence explore hierarchical shift tree tool visualization sensitive cause tree construct indicate pn nm nd nm affinity define nm graph establish regard hence turn use filter scalar one parameter result filter possibly runtime involve involve iterate iteration costly consider find fast runtime sense walk slow ms dataset old new costly kde run ms assigning reflect fact ms whole space point view advantage shift kde nonconvex shape intuitive physical convenient cluster explicitly cluster determined initialization affect kde cluster change successful application pixel color mode depend g medical shift one costly sometimes monotonic meaningful mode create mode mode nonconvex remove mode finally mean computationally discover time derive kde ascent prove convergence stop observe use reduction locally centroid line bandwidth ff therein n early shift attention thank demonstrate success image follow many theoretical shift cluster anneal remarkable regard converge fast character convergence domain surprising area include computer vision attention reference cite sum unimodal mode decrease motivated paper show create mode appear mode need occur create mode see nonetheless two example bandwidth qualitative gm mode even isotropic diagonal far isotropic equal study consist gaussians vertex narrow extremely construction simplex vertex gm component simplex show mode large simplex narrow range mode peak slowly decrease towards perturbation prevent isotropic apart isolated understand location mode gm hull isotropic work shift scale gm identically isolated indeed nonempty repeat etc taylor triangular create ms step make give convergent valid integrate ms devise mode line search ms kernels ms mode minima modes ms gaussian kernel occur iteration sum piece possible subset within piece would ms coincide newton gaussian sublinear convenient relate kernel generalize define dataset probabilistic maximize likelihood gm model point bandwidth translate consist solely locate origin thus maxima occur coincide index origin compute eq ms non update correspond likelihood consequence ms apply em iterate increase unchanged indicate ms jacobian jacobian large along eigenvector associate large eigenvalue always case ms close smoothly principal component hull focus ms covariance initialize far mode ms progress property optimization agreement ms iterate follow property path iterate lie hull datum consecutive shift centroid fact surprising nonconvex ms desirable aspect boundary iterate occur define flow prove shift kernel broad support convergence see set kernel small enough several cluster iteration multiply eigenvalue eigenvalue point coincide broad result cluster converge easy rather integral solve towards axis converge cubic specifically deviation evolve reason extremely fast increase thus component slowly explain show fig iteration bandwidth linearly shape cluster apply generalized update standard axis function depend linear put spectral product pn nm walk posterior normalize commonly spectral eigenvalue structure considerably enhance keep eigenvector second however change iteration collapse cluster slowly remain thus block constant trivially extract piecewise constant eigenvector generalize implicitly without compute rely several give bandwidth computationally spectral solve run perform product connect operate intensity filter diffusion shift basically iterate operate jointly describe range update appear whenever square weighted weight one center another laplacian objective nn nm affinity laplacian optimization mean laplacian objective appear reduction note mean example cluster predict track video histogram simple initialize next current histogram histogram note kde histogram differentiable also kde shift maximize linearize location find pixel region enough track object time robust partial clutter camera et kde tt consider give importance mode mode scale define euclidean space sometimes cluster lie low iterate mode diffusion imaging
describe review art design global statistical redundancy summarize combination applicable obtain audio signature plus minus france email audio name hash powerful audio identification basically concern audio signature usually quick offer wide real art survey audio discuss audio thank increase mobile device internet music recognition song place want tv people want service tv require audio identification match audio signal store database direction signature medium consider monitor audio widely investigate technique recently content synchronization repeat detection live identification audio identification extraction derive relevant audio follow audio collection audio name etc systematically store store audio label match htb world audio signal kind distortion error derive save memory resource property audio requirement short numerous attribute structure centroid continue e gmm though diverse propose paper remain author knowledge comprehensive review eight year survey date extraction audio presentation particularly benefit researcher audio architecture various extensively model conclude depict purpose summarize filter amplitude normalization fast fouri dct wavelet frame input major since directly affect system diversity investigate reduction invariance summary approach first map length frequency coefficient centroid etc derivative feature variation audio signal localize peak top spectral multiscale pursuit feature previous recognition amplitude audio weight triangular spectral algorithm find audio video sequence user implementation music peak amplitude sep argue discrimination sound frequency coordinate peak describe sparse exploit peak automatic audio occurrence together spectral peak widely coefficient audio frame bin range binary difference neighbor wiener relate aspect audio often eq indicate power low concentrate frequency similarly feature also factor promising audio measure audio mass spectrum sc argue addition sc audio experiment world background sc result recognition base form transpose additionally characterize temporal pass post figure adapt statistical reduce redundancy spectral feature characterize ensure discriminative gmm gmm audio signal identification etc audio spectral model multidimensional density conjugate transpose determinant respectively k ml sense global log via em state characterize gmm kk n however gmm explicitly amplitude sound source similar template
leave error incur rather population mmd give synthetic adversarial net illustration generator mmd training noise gaussian generators mmd generator expect mmd bottom dataset distribute distribute datum parameter generator decrease regular space produce cast generative g measure discrepancy optimize adversarial net function net discrepancy incur origin point maximize exist every generator whose close shannon minimize generator mlp minimax propose take alternate along note composition mlp permit gradient paper adversarial net balance optimize suggest maximization step overfitte step bring desire unclear sensitive regardless gb ram potentially tractable adversary sample introduce replace family architecture adversarial mmd net solve reproduce hilbert rkhs carefully rkhs function product reproducing express closure kernel rkh nonempty compact space borel let mmd p fx choose kx underlie rkhs purpose mmd achieve desire detail access mmd n unbiased mmd define proposal input transform let function comprise depend minimization descent carefully rbf depend use propagation net operate mmd find empirical may mmd despite input mmd mmd sup e q fx complexity dimension bl nh ix estimation w q x rp rp c p proof appendix slightly restrictive hypothesis dimension generator take bound translation invariant length training generate appear mnist despite mean test report adversarial net several density second mmd understood optimize subsequent clear connection might explain suffer rbf kernel specific shift invariance image generate train mmd parameter adapt mnist use mlp architecture unit rbf batch generate sample log figure kernel density adversarial net outperform adversarial set digits mmd mnist kde task mmd worth costly acknowledgment discussion rademacher independent independent also fx c pp q take probability p state training least know therefore h p prove eq prove begin fy bl assume fy constant n p independent fy jensen fy fy introduce conditional sum expectation fy n fy random add therefore identically triangle e e fy q p f x assume ny ex proof apart state expand define kx nn x kx n therefore supremum also x unbiased eq split kx kx next define p g w hx h fw fw since bound expectation apply inequality theorem write f difference w hx w rate split q obtain look get finish proceed imply exist eq rewrite rp given depend ball induce infimum training neural maximum university convert theorem lemma generate frame statistic informally speak generator network nan hypothesis statistic unbiased discrepancy nonparametric kernel sample compare al game generator incur optimize empirical mmd draw fix close despite
methodology bayes space successfully statistical due enable proceed density bernstein polynomial natural logical spline density guarantee theoretical property spline reason spline provide mathematic devote spline smoothing interpolation conditional smoothing continue example concern spline spline knot denote spline degree dim b call write q derivative spline write notation upper triangular eq square weight parameter spline smoothing spline denote semidefinite stand spline want sufficient condition minimum linear inverse I matrix class solution minimum solution unique symbol require smoothing spline obtain transform function fulfil spline spline spline knot coefficient g vector simultaneously system perform consideration solution eq reasoning multivariate component introduce spline back transform counterpart visualization real survey national institute lc interval discrete version value obtain proceed approximate spline study option discuss furthermore I cubic knot spline coefficient fulfil spline also function transform nuisance variability effect middle one lc n n h finally present cubic spline proportional knot result smoothed consequence knot negative tail although numerically avoid logarithmic original subsequent spline ignore capture relative high lot respect local policy status behaviour proportion monitoring enable huge amount analysis become statistical spectrum need solve paper handle one smoothing tool purpose spline interpolation example advantage approach clearly cubic spline advance provided mention enable analyze dimensional functional g avoid compare polynomial concept nevertheless similar character paper challenge concern concern knot hand set direction development field science r economic via proper represent challenge task usual fully account character feature space constant constraint reasonable analysis function provide space issue aim spline transform density account reasonable discretized distributional development transformation spline function borel function frequently database individual preserve intrinsic enable meaningful usual seem inherent feature density density contain borel way space pa hilbert density order finite support reference stand integral without density popular transformation compositional carry obviously need
combine process create surrogate separate parameterization uncertainty quantity epidemic actual keyword quantification epidemic polynomial intrinsic parameter mathematical quantify reliability sophisticated quantify interaction parameter force quantification computer carlo carlo comprehensive processes variation create intrinsic reconstruct method simulation characterize range sufficient generate statistical process sample surrogate model parameter uncertainty variation uncertainty model parametrization need prediction range know rate disease typically observe epidemic population input parametric prediction create refer stochastic variation observe epidemic become infected nature community contact input connect specific recover specify input epidemic know uncertainty epidemic epidemic source combine statistical allow intrinsic uncertainty show statistical intrinsic uncertainty gaussian presence intrinsic sample fix account intrinsic variation perform variation parametric analyze add variance increase far situation unimodal distribution satisfy unimodal separately eliminate simulation fall thought study simulation event mode response carlo surrogate model parametric reproduce quickly simulation behave though exact remain introduce epidemic simulation account uncertainty sample throughout surrogate keep mind come epidemic lack epidemic general due use epidemic system sde represent disease epidemic sir differential sir outline derivation individual time infect constant recovered evolution sde eq q wiener infection population represent infect individual recover completely individual recovery rate infection rate run stochastic sir indicate variation sde studying least distribution unimodal transmission recovery uncertainty experimental datum distribution histogram plot recovery account sir quantify uncertainty draw solution record repeating form ht sir infected series effectively h kernel estimation sir detailed paper able reconstruct sir simulation able denote x represent quantity discrete index indicate discrete controlling know absence sir notational translation kl uncorrelated lack aid kl decomposition benefit possibly reduce first separate remove correlation kl find eigenvalue euclidean vector onto basis kl problem independent control eigenvector eigenvalue effectively show compare figure kl coefficient implementation reduce effectiveness kl sir distribution kl reconstruct approximation depict uncorrelated construct decomposition two goal allow sample distribution parameterization although random variable polynomial basis variable show choice polynomial sir basis scheme orthogonal polynomial respect reason basis degree dimension polynomial q standard normal density high tensor multi index polynomial multidimensional order indexing dimensional vector correspond q variable purpose space reconstruction three sir polynomial smooth truncated equation formally formal since live space monte expectation domain standard probability common explicit cumulative jointly onto cumulative uniformly random variable cumulative likewise map expectation numerator note inverse important carlo sufficient equation characterize intrinsic explicit practice fix finite denote conditional cumulative estimate kde univariate tensor product univariate denote denote derivation follow kde equal kde goal build conditional function density easily accomplish increase compute process repeat coefficient decomposition equation approximate simulation pc pc expansion important property uncertainty define probability intrinsic quickly realization intrinsic uncertainty gaussian intrinsic depict sir fine scale preserve well increase sir form kl pc decomposition depict maintain cut due effect truncation kl bandwidth kde pc surrogate parameterization pc arrive gaussian process set review statistical highlight process form coefficient coefficient gaussian fit series way away uncertainty lack realization simulation parameter coefficient k coefficient sample value subtract dividing decompose use truncate construct keep singular build standardize r r independent truncate decomposition nk independent construct come truncate q w length vector p I apply process evaluate point standardize k matrix principle vector relation define normal step sample hasting univariate walk independence sampler one hyperparameter distribution prior ts base I ti model pair respective apply matrix prediction relation coefficient variable
france currently ph degree university technology research interest hyperspectral engineering engineering receive ph security france laboratory university france research interest wireless signal nonlinear system good award signal year review paper france email fr fr nmf powerful extraction apply field technique single propose bi feature account weight bi nmf pareto optimal instead single front study approximated hyperspectral confirm bi nmf art nonnegative factorization pareto hyperspectral factorization nmf provide become oppose principal discriminant nmf decomposition yield tractable interpretation datum recognition blind name approximate rank nonnegative input two consequence hyperspectral cube scene light reflect range available pixel pure material extract recorded estimating abundance pixel interpretation view euclidean product either frobenius generalize kullback leibler intuitive interpretable decomposition temporal smoothness dispersion g bregman divergence oppose research activity nmf work nonlinear extent scope several kernel nmf employ high perform trick map hilbert call know nmf consist product input scheme additive combination map note residual matrix factorization map severe disadvantage basis reverse difficult obstacle difficulty space optimize directly nonlinearity section detail either conventional essence dominate vice datum real fusion nonlinear reveal close ground nmf chen former nonlinearity depend post study bayesian residual share one nonlinear process conventional vertex separation elegant framework estimating combine paper stem define objective exist decomposition deal method literature integrate pareto front multiplicative organize difference optimization propose bi nmf demonstrate hyperspectral work variant nonlinear equivalent consider separately represent scalar frobenius square wise minimize measure coordinate alternate keep straightforward nonlinear variant study suffer pre study consider column norm inner call machine example function apply wise entry residual error analogy descent scheme difference linear sample namely minimize euclidean consider feature two trivial attempt bridge gap estimate mapping pre problem detail literature investigate coefficient implicitly worth framework machine see several nmf extend provide optimize input next investigate combination additive nonlinear nonlinearity outline nonlinearity additive relaxed post nonlinear study residual nonlinearity investigate width nmf approximate nmf solve simultaneously simultaneously two feature see optimize simultaneously function namely ill indeed find dimension bi bring space objective belong beyond optimization multi objective widely literature take advantage bi optimization dominate j j inequality pareto dominate objective improve degradation objective pareto pareto objective successfully evolutionary weight normal name reference respect argument kernel descent scheme differently guarantee easy sensitive lee nmf without generality restrict presentation valid multiplicative procedure derive update rule stepsize additive rule yield become gradient method e nonnegative stepsize page multiplicative page division multiplication element wise dd pt distribution nmf hyperspectral problem initialize column randomly image stop attain iteration reach stop iteration aggregation predefine threshold mention multiplicative unity update imply tucker kkt however kkt concern kkt guarantee multiplicative nmf conventional show provide relevant art propose bi nmf pareto front propose hyperspectral digital sensor top part pixel original image raw channel recommend clean accord build acquire contiguous spectral band removal band band area dominate material employ vary gradually iteration initial objective input determine pareto pareto pareto front strict solver minimum mention nonlinearity nmf problem pareto refer pareto front operate abundance matrix evaluate single point approximate pareto front objective evaluate dominate front outperform objective good pareto pareto maker dm worth pareto solution dm make final dm specifie generate detail regard pareto original multi problem pareto front objective dominate surprising could scheme due solver solution pareto within objective interval obtain solution pareto dominate global pareto point nonconvex global solver generate pareto optimal front pareto solution pareto front nonconvex front attain use case nonconvex front probably result drawback nevertheless pareto pareto dm single objective underlie nonlinearity hyperspectral two reconstruction define denote regardless lead comprise abundance exist estimate connection extraction estimation technique enable estimation variant extraction vertex existence seek large technique abundance constrain use consider sum nonlinear abundance call linear nonlinear bilinear factorization jointly constrain dispersion regularization nmf minimization
w w nx nz partition entirely integer partition addition suppose maximal inside whenever solution g nj allow repeatedly transformation let suppose suppose follow update keep fix clearly partition always ensure terminate step convention proposition k mutually exclusive alternative addition step supplementary algorithm check sort nonnegative project onto algorithm preprocesse appendix supplementary material initialize g nz go previous immediate outer loop z update maintain every iteration pass terminate pass terminate naive bad sort ratio update get implementation relational implementation consist tuple ratio notice assume tuple order view constant boost library remain delay fista computing quite intel cpu matlab b table display generate solve apply level e paper algorithm make available website thank read suggest introduce discuss implementation issue intuitively split partition I I g g jj g jj z I w z w eq expression w constant p z ji gx eq n proposition n eq whether optimality part proof n n w w contradiction argument z g identity imply prove claim g g z group strict contradiction conclusion onto radius weight pass tw z pass tw pass finish example integer nonzero solve unfortunately nonconvex np hard convex instead recently researcher beyond predictor motivated yield nonzero identify take account feature structured predictor select group feature prediction mathematically replace regularizers regularizers include fuse regularizer invariant desire know en grouping development weight see include recently investigate discover atomic norm characterization computation atomic apply frank wolfe cg however nonsmooth fitting include frank wolfe long ball proximal cg get root project onto terminate introduce code project onto norm proximal projection devise currently compute proximal norm compute unlike proximity norm arise evaluate operation introduce partition alternative proposition theorem
par variation I en les si une application optimisation dans cl es analyse energy de abstract evaluation adopt uncertainty optimum measure entropy come negligible propose solution several evaluation integration energy electrical keyword analysis computer energy electrical minimizer construction mu problem end certain adopt classical constructing shannon quantify approach minimize evaluation equivalent mutual reader criterion point approximation carry gauss quadrature need entropy plug range turn sample moderately path noise moderately criterion essence going carry since single limited yield little progress dominate first left idea evaluation evaluation iteration contribution suggest build residual result minimizer minimize suggest large variation carry evaluation iteration evaluation moderately expensive processing possibility update evaluation idea evaluation artificial motivated fact numerical condition k consequence simply respect conditional second path simulation condition available development branch toolbox cb cb ct ct ct ct ct ct ct ct ct ct ct ct cr cr cr cr cr cr represent corresponding sample leave gauss integration strategy energy electrical describe operator connect strict economic requirement value cost fx denote characteristic connection expectation scenario computer program assume evaluation scenario scenario generator identically I evaluation noisy sake simplicity evaluation without initial spaced batch evaluation perform iteration reference iid kriging firstly estimate initial evaluation adjust batch depict entropy fast iid budget evaluation suffice accurately cb cb cb cb cost ct ct ct ct ct ct ct ct ct ct ct ct ct ct ct ct ct ct ct cr cr cr cr cr cr cr cr cr cr cr cr cr ct ct ct ct ct ct ct ct cr cr cr cr cr cr cr cr cr
combinatorial ibp importantly scheme process see miss ibp measurable process indeed exchangeable direct process ordinary due characterize term describe ibp analog chinese restaurant produce bernoulli generalization stable show ibp weak definition theory probability one assume equip algebra lebesgue measure completeness aa note require partition element measure algebra projection value e alternatively iff distinct measurable onto measurable general eliminate agree clear measurable function random say intensity point measure uniquely result characterize poisson process q increment completely random fundamental completely reader ensure space characterization completely note compactly completely random measure purely atomic poisson call q eq uniquely specify component evy latter encode position intensity evy measure intensity let support independent eq evy functional bernoulli beta process mean purely atomic measure evy fix independently measure poisson sx bernoulli ordinary simply evy simple bernoulli process bernoulli process straightforward let ordinary countable variable evy independence increment suffice process point theorem evy bernoulli law characterize mean consider measurable completely purely atomic evy q ordinary implicit refer remainder insight state merely ordinary component infinitely beta process break later due g describe conjugacy censor result observation completely hazard combinatorial absolutely continuous lebesgue rely implie claim omit conditional introduce follow terminology let say condition bernoulli every comment monotonic convergence nonnegative measure direct concentration combinatorial induce chinese table proportional generalization space countable every copy process concentrate measure whose atom atom among atom measurable sequence randomization elegant would work work minimal unique extension measurable extend us law arguably decide scheme characterization equally conclude independent increment rule expectation increment recognize law bernoulli parameter scheme concentration unique moreover parameter condition give agree determine distribution exchangeable bernoulli argument hold countable generate simultaneously parameter ibp space measure sequence verify equivalence denote sequence functional maintain enough every point atom opinion characterization equivalence induce propose follow ibp ibp define later introduce relate exchangeable random characterize generalization exchangeable sequence combinatorial structure partition may empty class complete event nonempty token structure part class among take token integer sequence integer sequence finite exchangeability construction satisfy know completely show scheme hold say de know tail measurable random characterize limit relative token depend sequence way combinatorial arrival token token appear transfer argument scheme bernoulli interested mark bernoulli mark converge scheme every every exchangeable measure independent random version event moreover characterize q pp precede extend define whenever first token extend limit boundedness understand term characterization b measure mean measurable measurable hazard bernoulli rule complete may continuous characterization component let satisfy sum characterize law write binomial distribution variance every particular restriction measure eq claim note poisson probability measurable satisfie let count sum identity complete measurable rule suffice equal identity much focus n k k column let f n eq straightforward verify establishe claim verify law distributional average average continuity suffice nonnegative measurable boundedness continuity measurable dominate complete p fs partial average partial average let q straightforward verify q surely intensity proof note intensity limit average long appear converge limit support singular measure identically support almost surely follow borel characterize countable collection measurable take distributional limit average sure sense development direct bernoulli x existence limit partial version development completely infinite yield completely last equality corollary note following develop stick break like identity calculus measure scheme measure measurable identity argument understand component one take introduce right bernoulli complement support total mass ordinary component upper chain expectation eq final claim immediately see independent study combinatorial structure poisson process lead generalization process special entirely cardinality poisson measurable fact randomness recall appear time indeed distribute trial eq statement multiply identity exchangeability q immediately corollary exchangeable course underlie indeed another note combinatorial exchangeable sense consider exchangeability permutation array exactly equal correspond sharing determine array order develop ibp write order except order adjacent equal view atom label correspond measure atom allocation leave realization order form informally array uniformly random permutation sequence column label array obtain sort distinct column denominator fact indistinguishable copy nonzero symmetric dividing play exchangeable exchangeable induce characterize homogeneous describe generalization ibp exhibit understand similar discount chinese crp exchangeable direct measure characterize law ibp parameter crp deep ibp correspond induce two crp crp define measure vary homogeneous law purely atomic I frequency token chinese absolutely continuous ordinary component recover show stable beta ibp perspective connection parameter crp ordinary atom see define recall mean beta crp multiple dirichlet distribute eq distribute trial even absolutely beta hence absolutely simulate exactly stick break characterization know demand theory make precise particular computable distribution rule work pr provide absolutely continuous characterize line describe simulation produce appearing time ordinary mass know beta structure ibp function special k n token appear st equivalently allocate chinese appear new calculus model make exchangeability n new token st admit additional copy token lead agree appear connect note informally speak condition exchangeability appear summarize parameter ibp discount worth g propose law conjugacy might consider governed binomial make correspond kernel ordinary component beta imply law weakly together imply beta work exchangeable investigate limit purely measure strongly let suffice show convergence complement generality partition countable restriction fix satisfy claim locally map distribution intensity suffice establish complete weakly contrast established follow special counterpart thank author college international fellowship transfer translate distributional claim variable extension underlie measurable space space random measurable may whenever borel space measurable exist measurable space iff measurable proposition var atomic beta scheme describe combinatorial conditionally process share beta measure show beta combinatorial ibp beta scheme parameter measurable scheme ibp exhibit power idea probability rise generalization beta dirichlet chinese restaurant sequence beta process generalization process change introduction ibp characterization relationship extend ibp direction despite beta beta conjugate beta dirichlet subsequently consider measure scheme stable ibp give rise article combinatorial structure de combinatorial structure collection informally allocation component subset element distinct locally countable hausdorff set algebra generate equip generate cardinality recall part simple fix purely atomic atom partition call process informally every block partition independently atom take atom partition permutation carefully equivalence relation exchangeable sequence exist purely atomic give completely mean measure hazard measurable partition induce crp highlight sequence limit necessarily purely characterize chinese appealing argument refer allow vary across outline agree construction measure
information multidimensional decomposition interest decade successfully apply range area vision body tt decomposition similarly tucker decomposition td context contain greatly reduce original core tensor previous mathematical frequently multilinear algebra multidimensional array contain tensor scalar letter letter capital letter tensor tensor I vector third general multilinear sample nk classification supervise category define td single core tensor space tensor far alternatively core tensor tensor core tensor paper common firstly tensor index introduce vector sequence decomposition index arbitrary position core need ensure form rearrange canonical identity position orthogonal extract subsequently core tensor core regard training use classification contain necessary odd order core feature may sound exploit core reduce number core pattern obtain rate image divide two test ratio datum structure average trial high hold several accuracy ratio maximum accuracy htbp database originally order trial high hold ratio hold ratio hold high accuracy ratio respectively htbp database pose illumination image area pose size tensor fourth test core tensor machine v nn nn accuracy plot versus feature obtain high classification accuracy respectively tensor direct affect classification test high necessarily feature large increase ratio seen decrease outperform method color convert image face accuracy multilinear discriminant discriminant multilinear discriminant propose mention classification show use multidimensional supervised require needed classify high tensor recognition problem need efficiency detail multilinear
leave deal nk employ km n nk km n nk km put computation universal constant ii super exponentially case union deduce divergence kn substitution give error nn n universal putting inequality apply reveal c c denote hypothesis recovery output class employ contain let take global offset non trivial l cardinality cut homogeneity recovery ability restrict hypothesis know definition union one k finish establish special without like alternative begin set produce hypothesis denote associated cut fix necessarily mind far cut attain inequality q finally consider alphabet support compose hypothesis guarantee type suggest pick hold put result establish theorem vertex black determine put colored vertex nonempty cut degree repeat argument repeat color scheme eq claim begin expression measure dirac point remain hellinger divergence elementary identity mp mp mp mp indicate mp since recognize apply chernoff yield sn sn union assumption union sn chernoff sn n sn sn employ union sp inequality rely assumption union indicate put bound e p p np complete obeys l represent index quantity cut clearly together rise quantity separately cut ensure empty together q exceed exist secondly total feasible constraint cut size exceed eq put together k inequality hellinger divergence say immediately establish known inequality paper concern jointly measurement imagine take pattern represent measurement channel transition tool decode problem general structure alphabet channel family characterize corruption homogeneous almost irrespective general application outlier lead order recovery case improve random geometric graph various directly pairwise relation pairwise include cluster relative rotation pairwise pair sequencing later substantial consequence joint recovery feasible soon pairwise pass paper explore imagine graph accommodate nature solely channel represent channel uniquely difference ji pose receive field social biology list exhibit community group share feature aim observe similarity member simple represent assignment encode two belong view single angle position rotation several view view pairwise application include vision biology reference image shape physical across input cutting pairwise match aim refine globally numerous graphic people mostly nucleotide position single nucleotide snps associate snps cause various develop sequencing method particularly sequence reconstruct disagreement pair pairwise recent primarily motivated consideration spectral develop provably synchronization choice study manner limit application despite develop instead account similarity motivate graph fed channel information perspective distance success measurement metric graphical insight feasibility exact understand application pairwise recovery turn benchmark evaluation comparison paper towards unified characterization tool measure kullback determine feasibility channel super polynomial coincide broad homogeneous case fix alphabet characterization possibly different rate increase asymptotic tend illustrate effectiveness theory concrete consequence application investigate prior outli problem recover regime side focus characterize theoretic limit determine information theoretic limit genome random graph graph condition general order preliminary recovery structure develop tight characterization measurement aforementione special graphical refer channel whose probability edge previous graphical channel quantify residual output investigate full might interesting notion grid step general let degree two vertex another complete graph denote vertex connect edge introduce widely reference depth way vertex edge eliminate edge effect connect vertex edge upon divergence hellinger unnormalized define particular abuse p divergence hellinger elementary qp vector support mean paper remainder organize describe formal setup develop non special present graph structure emphasis homogeneous graph framework develop general theory specific finding direction proof defer imagine alphabet operation broadly define pairwise operation stand additive partial list modular addition integer multiplication stand represent rotation hence case multiplication capture belong measurement pattern undirected illustrate fig pass conditional py ij illustration abuse observation symmetric mapping say contain corruption oppose code employ across center distinguish shifted version light introduce zero offset factor offset regime vanish proceeding separation development kl hellinger divergence channel minimum reflect channel see various measure constant close hellinger rest see suppose qp qp appendix part quantity specifically self determine number intuitive measurement channel output sufficiently separate quantitative start begin likelihood ml decoder well minimize error prior develop recovery probability characterize tradeoff channel sufficiently universal decoder achieve cn essentially asymptotic limit regime tend infinity exactly degree condition read mn develop setting decode herein concern channel bottleneck minimax present output distinct hypothesis rule code separate say output information distinguish highlight information contain measurement quantify divergence information capture distinct ground truth call bit distinguish exceed interpretation cccc pairwise realization show blue constitute ground reader hellinger kl shoot measurement remark technical unable develop recovery divergence fix divergence grow hellinger divergence stable convenient analyze sufficient hellinger recovery condition account one measurement minimum shot measurement continue replace examine analysis continue hold parametrize py ij I j metric place precede sufficient recovery generalize recovery continue hold probability assess two necessary condition recovery hx p residual concern specifically investigate asymptotically vanish specify hellinger convenient demand exact various datum term alphabet surrogate arise interest complexity tight sequel pay two popular divergence kl hellinger regime case j read multiplicative irrespective alphabet characterize super way total obeys carry see scope exploring rely widely encounter graphical vertex degree vertex degree cut subsection introduce quantity crucial presenting comprise cut particularly cardinality define sequel k factor I cut important homogeneity illustrate ne size little kn interestingly homogeneous interest bound shall constant uv denote vertex uv one highlight concrete graph homogeneous geometric property connect share geometrically close share fraction example worth graph vertex connect away another concern expansion lemma n exceed order aforementioned depth helpful simplify divergence metric channel characteristic begin characterize connect achieve universal hold size cut homogeneity exponent irrespective metric distribution state probability replace fundamental low admit recovery kl divergence characterize alphabet dominant form connect bridge mainly directly hellinger become weak investigate success aforementione emphasize homogeneous turning graph recovery widely adopt homogeneous n n either super arrive fundamental recovery guarantee extra concern distribution truth whose identical vertex determine define logarithmic specifie bit cut hence information theoretic broad include limited various homogeneous cf condition coincide must bottleneck constitute nn apart hypothesis broad oppose rely homogeneous error precede regardless scale discuss full generality distinguish homogeneous graph cut graph edge summarize fundamental geometric graph theorem even gap contrast separate sense differ instead challenging form accommodate variety scenario alphabet decay category testing hypothesis set minimax contrast upon hellinger divergence unify enable characterization minimax limit literature see wise exist block sbm generative partition pair whether fall infer produce interest considerable attention regime structure treat output encode suggest corollary transition exact cluster year precise condition theory factor remark begin accommodate broad regime leave technical interesting observation match fundamental precise imply square distance right recovery recent characterize fundamental cluster determine hellinger depend cluster certain recovery bad situation imagine cluster nn definition theory develop accommodate several include alignment measurement eq rate word act outli consequence present concrete limit model ease restrict consider start comprise evident mp connectivity component apart illustrate precede sequel range alphabet alphabet comparison apply general b bound fall configuration adopt regime g bound graph notably accurate imply hellinger distance quantity recovery give simplify depend substitute respective checking compatibility one immediately say feature interpretation small alphabet increase limit information nm nm increase regime measurement regime fundamental connectivity bottleneck connect single useful measurement hence isolate regime alphabet measurement formulate represent sequence minor allele employ certain sequencing obtain pair stand snps denote assume read realistic sequencing snps geometrically close dna typically median adjacent denote l separation number read fix read nevertheless simplify sufficient capture additionally geometry consequence suppose universal sufficiently condition obtain follow r ii whereas
business term diagnostic deep structure divergence link network company company multi view social network mit student message close evolve snapshot view view citation show whether two keyword title abstract netflix record user record category record review operate ht truncate value kl dataset row matlab corresponding figure describe structure different figure use frobenius norm kl divergence rank seem acceptable large result seem model use number show good component dataset number many kl discover structure two line week minute try however able detect perhaps extremely record pair amazon first product figure reasonably recommendation dataset seek find coherent people recommendation suggestion purpose extract product component remarkably due kl divergence fit similar tend book book book detect anomaly extract complete basis evolve view continue extension cluster forecasting mining bioinformatics text towards number tucker community evolve tensor finally bayesian automatically towards automatic mining algorithm minimize intervention propose mining heuristic provide evaluate method show superiority well real dataset discover meaningful support foundation grant conclusion recommendation material necessarily view valuable sharing observation tool unsupervise mining tensor exploratory extract quality quality interpretation novel automatic mining minimal intervention extensively variety dataset automate tensor mining practitioner decomposition exploratory aspect popularity largely aspect henceforth multi aspect mining growth application citation network name powerful analytical datum tensor decomposition tool challenge attention make tensor decomposition scalable facebook write ever grow decomposition small facebook big category turn e facebook hundred highly scenario exploit scalability al introduce exploit scalability distribute solve problem attention quality baseline enable another assess tensor portion exploratory sort seek extract concept datum crucial extract model datum especially quality variation always tensor seminal exploratory mining component manually link entail measure g validation select generalize label truth hope lose minimum length cost depend heavily application boolean additionally deep operate intuitive decomposition independent require exist influential literature introduce heuristic determine rank decomposition comprehensive aspect contribution mm propose comprehensive methodology mining multi aspect manual trial intervention solution assessment assume divergence effective highly count real explore hide pattern well apply discover meaningful pattern synthetic encourage code publicly available notation subsequent section scalar multiplication kl frobenius efficient ii ii negative decomposition henceforth entry usually represent accordingly decomposition tool admit intuitive latent component see soft use cluster tucker tucker compression super useful motivate diagnostic expressive hard outline introduction heuristic name model imagine fitting tucker tucker tensor restrict tucker super core possible q least f f I ic element core rank high rank model chemical mining application valuable case acceptable reasonable quality e g high mention introduction introduce suitable dense contradict area vast mining application derive behind avoid rewrite problem kronecker product vector potentially big product henceforth refer achieve far extend recently beneficial poisson natural first next exploratory piece usually expert expert provide data process completely ground label impossible tensor ground label provide guarantee human intervention trial attack describe decomposition unified mining user intervention quality frobenius norm least problem kl close iterative apply prominent mm hard minimize minimizer employ use k store use throughout performance exploit break eq compute give decompose expression numerator particular efficiently structure product rewrite respectively kronecker sum property kronecker product conclude put everything end equation iterative kl efficiently tensor diagnostic htp n order minimize human intervention mining automate tool box two offer follow user provide reflect neither require whether count say frobenius norm fortunately equip handle case follow drive let whether capture structure grid measure diagnostic value quality informally problem intuitively objective maximize however get front subset dominate end effective intuitive datum c essentially select max maximize enumeration maximize extract hard example axis intuitively investigation show step point select kl choose discover rank select aim quality expense acceptable select contrary previous component extract acceptable however perform closely depend preferred mining run component maximization output seek good combining
monotonically relu j dirac delta everywhere except poor lack flip relu convex hence encourage suggest relu produce zero leave majority unit close experiment word sigmoid purpose sigmoid corollary sigmoid learn suggest ask show objective term propose aim minimize reconstruct corrupted version usual choice gaussian objective corruption taylor overcome represent gaussian corruption sample though term exact straight gradient monotonically activation practically epoch ignore jacobian aim input auto encoder loss coefficient form hide learn order suggest property corruption dimension derive form iterate corrupted sample loss apart goal hide decoding activation require note value except encoder form enforce intuitive result notice sigmoid activation separability maxout hence guarantee sigmoid satisfy property encourage latter individual drawback relu poor corollary sigmoid hard gain propose activation drawback individual activation j note monotonically increase relu discussion share sigmoid two handwritten digit value train world image cifar size cifar real image objective optimization rate epoch batch size hide unit cifar unless train loss decode zero unit say perform confirm linear sigmoid explain sigmoid sigmoid sigmoid increase record negative percentage activate unit sigmoid activation enforce activation empirically model study dataset bias form attention possibility relu bias term order taylor section corrupt mathematically analytical marginalization practice optimize corruption batch wise manner vanish relu mnist cifar protocol sample activation contribute towards discuss gradient relu generally slow hence conclusion corruption advantage marginalization capture order together lead drawback relu sigmoid encourage evaluation effectiveness learn activation apart mnist randomly choose background digit background pixel image validation dataset train unsupervise hyper extract candidate mnist relu believe trade well relu capable zero opposite relu sigmoid across relu sigmoid produce perform activation consistently relu strong mnist relu relu sigmoid relu relu sigmoid relu sigmoid relu sigmoid relu sigmoid relu sigmoid neuron exhibit sparse establish auto fold encoding encourage pre monotonically increase activation encourage theorem c form learn representation insight activation drawback exist convex section representation absence advantage conclusion combine yield activation whether supplementary es e th j j j monotonically increase activation monotonically f j monotonically convex activation monotonically increase j j increase extend fix j monotonically increase activation chebyshev recall corruption nd nd corruption process approximation yield identity rewrite expand order get square nd h jj cyclic trace operator become upon encoder decode sample decode square j edu mm auto explicitly learn encourage study regularize auto regularization activation play role provide de encoder activation like sigmoid learn together activation insight gain activation sparsity produce par auto sr use heavily distribute representation observe former focus distribute main representation manifold separability power investigate regularize learn distinction distinction encoder decoder aforementione sr follow researcher empirically unsupervise behind convex function efficacy activation since try sr encourage activation hide analyze multiple activation desirable auto auto encoder encourage analyze learn besides comparative tool use exist predict deep understand auto network minimize intermediate encoder part encoder map encoder h back decoder basic motivation repeat though map invariant manifold encoder generally formulate eq fix drive force objective force analyze encourage show activation role achieve j th j proposition go reduce training course gradient practical interpretation term pre activation hide sparsity property analyze gradient activation optimal every iteration monotonically increase negative implicit exist set finite initialization monotonically finite thus long aforementioned set low pre length easily guarantee widely simply constrain lie ball update
computationally fundamentally hardness several g cloud service train produce ideally notion individual differential service output might individual private perform produce beyond address comparison work machine complementary actually correctness guarantee theoretical pac let instance space boolean sequence increase dimension accord target learner select approximate target hypothesis hx cx concept concept choice learner learner sufficient sample polynomial parameter call proper otherwise improper pac learner private set neighboring q privacy call identifiable use hardness private release adapt choose replace succeed identify high showing differentially private formalize follow property I sx cx ni cx ix cx find obeys identify learner efficient trace may relax condition hold learner scheme properly differentially private eq typical satisfied differentially pac differentially private q contradict tuple public key deterministic procedure key output special compare must separate requirement correctness say succeed correct require succeed succeed namely word output correctness notion weakly informally security particular correctly security parameter message strongly comparison informally require length fail security security notion security security key block trivial attack always right challenge message challenge generality message sort message scheme event output experiment message scheme sequence run ib definition security single challenge prove many challenge security hybrid differ message hybrid differ message hybrid single challenge security hybrid first indistinguishable security hybrid security adjacent indistinguishable moreover identical moreover hybrid hybrid challenge security imply indistinguishable actually scheme strongly define class scheme throughout discussion space ideally concept efficiently pac learnable learner comparison public address way example support public use binary string pair produce concept reasonable pair length sequence string let output coin notice concept efficiently description work pac learnable include public example pac work stage determine significant public exactly parameter mas good heavy set learner apply learn pac request ib j observe learner pointwise correctness coin case place place least r f therefore suffice least long receive tf pointwise hardness scheme concept class recall concept example attempt one learner take polynomially close challenge scheme example public need space produce advantage distinguish adjacent distinguish message natural separate completeness e succeed immediately discussion example sample security security reduction sake efficient natural security adversary sequence l nm I nm lb adversary distinguish unfortunately subtle distinguish lose overcome instead security differ message adversary message message agree suggest message guess guess force actually target query receive rx rx place example latter bit search key strongly scheme cc hand conduct search place I rx place answer set next threshold yield good strong correctness strong apply repeating argument correctness strong correctness apply yield contradiction explain satisfy notion correctness protocol multilinear construction noisy term grow correctness multilinear compute multilinear give answer operation compare introduction give generic weakly strongly correct scheme modify add interactive proof result correctness underlie probability protocol correctness protocol multilinear introduce protocol correctness protocol eliminate error gaussian unbounded result center statistically security protocol truncate security protocol un truncate straightforward candidate build multilinear weak correctness scheme map additional perfectly binding randomized perfect bind computational indistinguishable message function protocol string take input reference output requirement security perfect adversary quantity valid satisfy bilinear map perfectly sound assume perfectly bind run rr c cc c b notice component value moreover completeness valid correctness proof valid mean verification c b bm bb c c c simulator namely key game security reduce security security adversary valid distinguish hybrid indistinguishable advantage hybrid negligible break security l lm lb ib bb hybrid probability negligible break strongly advantage lack perfectly proof follow random p public check fail output c k bs result thus security construction rely hardness reason apparent security suppose suppose security show security also prove message adjacent moreover security suffer loss adversary adjacent adversary produce guess adversary advantage adversary security case requirement adjacent challenge receive modify public let skip bm bm bs x compare would particular result program never correctness indistinguishable show hybrid indistinguishable hybrid change security change rely security argue concept separate private representation way learn syntactic place learner want arbitrary circuit follow elegant idea suppose adversary concept concept concept without identify representation actually target differentially infeasible produce argument try properly infeasible hypothesis hypothesis good construct base way signature derive private proper analogous triple private public output indicate signature correctness scheme digital signature scheme adaptively oracle obtain sequence signature game iff super signature function super digital signature scheme fix convenience hypothesis representation pac achieve request ib represent fix k learner place place weight receive place weight get otherwise hardness properly learn properly class polynomially let super digital signature learn class follow mf none find completeness example succeed oracle scheme message produce acknowledge helpful discussion suggestion rgb true theorem fact proposition support fellowship public pac private concept pac learnable fails differentially question al j prove generic enable comparison construction differential privacy learning yield great differential aim enable give strong formal individual note speak differentially learner label presence absence positive concept since require learner work show inherent class sample privacy address complementary differential privacy initial work efficiently learnable concept efficiently learnable limited progress negative question exhibit learnable plausible prove private may interest pac universe draw label accord concept output class approximate different run pac think randomize learner neighboring dataset differential privacy substantial gain gave show properly description size toward private al make powerful observation efficient efficiently simulate differential et concept elimination differential elimination efficient pac fact lead learnable class also efficiently al progress toward pure complexity learner inefficient show generator proper proper substantially proper hypothesis assume learnable learnable approximate privacy detail version learnable concept improper learn powerful differential element privacy consideration unless efficiently learnable plausible resolve al improper learner existence correct efficiently efficiently pac learnable differentially private hold improper learner relax approximate differential remark understanding efficiently learnable different overview construction concept pac class admits learner positive example hypothesis concept minimize error underlying example suffice domain fact totally efficiently learner still example learn nothing modification example efficiently pac learnable learnable distribution place correspond condition correctness message fashion technical contribution weakly scheme strongly able efficiently pac argue differential example security scheme ensure essentially thing give trace back produce build conceptually connection differential adapt learn motivated answer publicly sort order precisely public take input reveal correspond requirement give learn ordering multilinear keep scheme privacy multilinear map construction insufficient purpose issue arise learner distribution include achieve weak message compare probability compare specifie work correctness message valid cause completely learner fail correctness guarantee strong scheme program scheme perform incorrectly comparison wrong multilinear multilinear much argue give generic weakly exist scheme interactive form key comparison procedure check proof comparison
modification state generic define cardinality word functional width fix distortion scale distortion begin rip property combine state theorem distortion successively cover inside generic instead rip mapping near tradeoff vary embed begin state lemma embed belong distortion defer obeys level sign identity q place ready main aim close neighbor fix conclude conclude q conclude complete also identity follow aim hold rip hold together apply every complete imply note prove hence sum identity apply inequality conclude complete proof fa fellowship institute computing nsf award nsf award award amazon web services google blue energy facebook intel microsoft reading manuscript ex ex pt rgb rgb theorem proposition subsection subsection rgb depth reduction structure similarly gaussian matrix multiply provide efficient dimensionality embed optimal obtain via embed certain embedding find engineering perhaps state preserve factor modern form nm ni prove project point dimension later could normal recently multiplication implement efficiently storage please recent detail improve dimensionality arise embed finite preserve precise sphere state measure define mesh unit matrix special point minimal allow matrix albeit constant continue certain ensemble entry dimension factor characterize paper analogue structure efficient heart analysis preserve norm multiply preserve euclidean state transform provide distortion embed also distortion rigorous justification replace scientific sharp connect begin define isometry state rip preserve vector multiplicative distortion isometry isometry sparsity shall restrict lie dependence purpose refine rip simultaneously sparsity distortion level rip distortion isometry distortion rip sparsity inequality require satisfy rip low reduce rip vector look proper satisfied ensemble reduction suppose obeys rip sign obey good embed pattern random ensemble commonly purpose sparsity distortion scale theorem unitary orthonormal uniformly measurement ensemble diagonal focus row choose matrix orthonormal please ensemble isometry matrix resolution rip distortion require matrix hold probability long state complex matrix bound constant logarithmic analogue tradeoff distortion utilize numerous allow sample problem paper theorem result establish analogue hold set mention interesting perhaps restrict isometry set sparsity level et spirit distortion also characterize use ensemble low significant loss distortion establish result suboptimal tradeoff state require sample rademacher one establish logarithm cover relate pseudo incur two requirement
equal measure nonlinearity simulation pearson nonlinearity dependency cloud used generating band eq pdf four type dependence spread different dependency pdfs capture pearson correlation different pdfs pearson characteristic functions pearson depend largely nonlinearity mutual correlation monotonic transformation nonlinearity pearson information remain undesirable incorrect monotonic theoretical generating figure exist show several property pdfs describe efficient algorithm random variable joint variable dependence marginal good distance know hellinger marginal call mutual mutual almost irrespective nonlinearity measure symmetry measure partially quantify random extreme normal first show axiom pdfs maximize band pdfs pdfs addition structure estimator suit avoid numerical integration briefly bl cutoff frequency mx monte pdfs linear quadratic cubic carlo pdfs datum cubic row pdfs work equally square theoretical use carlo fast irrespective nonlinearity generate equally linear normal convergence fast second convergence variance cubic see bottom convergence pdfs showing square different different pdfs cut compute bin contain advantage require pdfs also estimator invariant strictly monotonic transformation correlation achieve mutual showing slow pearson bias show decrease increase time bin fast mutual building paper cut band limited pdf approximate cut band pdf band limited frequency analysis need normalize lie modulus unity correlation case bivariate metric measure distance strictly transformation otherwise computation institute engineering md com edu section satisfy axiom measure marginal date paper parametric band limit pdfs mutual mutual standard pearson know pdfs rate ability nonlinear dependency require converge theoretical capture science several quantify mutual pearson distance mutual thought benchmark quantify pdfs pearson correlation directly estimate correlation directly datum nonlinear slow often reflect dependency correctly enyi axiom strictly transformation axiom table axiom popular pdf product dependence six axiom invariant strictly mutual copula property
corpus experimental parse lstm stack parse tag pos stack lstm parsing model language stack lstm parsing use head stack representation parse lstm classical recurrent rnn exclude symbol comparable test cc lstm pos h cc lstm pos rnn l lstm pos rnn cc lstm pos composition pos rd substantially exception pos chinese parse baseline gold pos tag parse note predict pos tag english add value suggesting parse directly also compose dependency head word implication baseline rnn capable good structure sequentially condition approach first supervise recognize stack shift make top stack decoding finally understand toward large parse exhaustive cube discriminative chart decode lp relaxation parse include feature randomize hill enable feature global discriminative sensitive part stack approach arbitrary stack recurrent art dependency stack possibility learn observable I supervision final far give device learn g observe parse alternative external memory machine supervision stack stack technique reinforcement make early chen office grant support european contract h project edu cs edu dependency innovation stack stack parse element top stack addition maintain stack efficient parse unbounded look ahead buffer income complete iii stack build tree backpropagation parse parse series read sequentially buffer syntactic structure build projective base parse computationally challenge parse action encounter development alternative simplify modeling make recently last line state state history complete partially syntactic global sensitivity parse parse state representation incoming stack although step construct sentence technical variation recurrent neural unit parse three stack represent stack syntactic one history stack syntactic token syntactic tree compute learn chinese english parse section brief stack follow write letter e write letter e scalar letter letter refer input discussion defer cope vanish gradient inherent rnns rnns step apply concatenation pass sigmoid nonlinearity rnns long range difficult repeat nonlinearity result address three control current input memory proportion previous forget update follow sigmoid hadamard product lstm control gate nonlinearity cell improve capacity rnn architecture layer input layer differentiable conventional multidimensional innovation stack always add position location stack lstm new addition sequence stack lstm stack extend stack never add back stack stack stack operation stack must efficiently maintain queue control stack middle box row lstm ever middle cell affine transformation nonlinearity refer vector stack continuous summary stack available refer stack stack influence stack lstm flexibility extract stack knowledge novel recurrent stack stack stack rnn problem preserve structure base buffer process stack construct element stack augment space syntactic additionally introduce third history take stack lstm architecture illustrate stack lstm buffer word stack history take pass relu nonlinearity embed transformation pass softmax layer distribution parse decision representation first word symbol stack time compute stack take update stack symbol contain tree symbol history operation define lstm encoding buffer stack lstm stack lstm encoding pass component relu nonlinearity finally embed stack buffer previous decision valid input arc standard transition transition indicate stack buffer result stack buffer state bold symbol leave stack partially build syntactic buffer keep incoming parsing choose score arc parse construct bottom right head recursively compute construction syntactic structure modify algorithm another head compose strategy token learn type neural representation pos token provide auxiliary pass relu datum lm vocabulary limited parse present lm word ensure parse stochastically singleton parse token iteration create option skip name skip word model skip define window rate epoch recursive network enable phrase stack challenge syntactic arbitrary simplify parameterization combine head expand syntactic syntactic relation satisfied embedding head apply nonlinearity pair triple recursive branching eq construct computation sentence forward computation parse
natural classified embed experimental cr cnn softmax fair comparison cnn softmax embedding cr softmax get convolutional softmax tune validation value cr cnn softmax convolutional cr cr improve et embedding similar softmax word embedding softmax embedding less data c ccc net cr softmax cnn cr cnn softmax cr present classifier fed rich traditional result et neural vector distribute vector method name pos use present cnn softmax employ lexical yu compositional embed derive sentence embedding embedding utilize cr sentence embedding report reach remarkable external resource nlp tool name cr play role informative reverse direction relation towards meaningful various class variety task recognition natural processing successfully different nlp sentiment role deep yu author tackle recursive assign every tree syntactic name recursive sentence embed position lexical extract lexical fed softmax performance yu et compositional embed sentence word utilize dependency name high difference cr cnn wise use softmax top cnn rnn cr effective artificial embedding approach tackle use perform work new state art costly classification use embedding rank deal artificial cr effective extract cr cr cnn relation acknowledgment author suggestion research com ny usa us ny com relation processing system rely tackle relation classification task perform rank cr cnn artificial perform design classify mark sentence outperform art costly additionally softmax representation precision use embedding target nlp task question base decade interest apply availability task classify mark sentence introduction book summary text focus network aim reduce lexical resource nlp dependency entity cnn cr tackle classification propose network learn relation segment convolutional layer produce compare reduce impact artificial extensive cr cnn outperform cr cnn follow representation word embedding remainder neural network detail evaluation previous deep network nlp cr compute embed input step cr transform word value convolutional finally cr compute dot semantic sentence consist word word convert value therefore input word vector vocabulary embed w position classification need determine relation come word al keep role label et instance respectively word embed concatenation embedding use embed word position embed input convolutional w step nn create representation input main challenge sentence variability appear convolutional tackle create size use convolutional vector representation convolutional produce sentence combine use max operation vector sentence convolutional layer apply matrix size successive window concatenation embedding th word order overcome word special begin convolutional convolutional sentence vector convolutional context window hyperparameter choose note vector sentence network compute dot w class c dimension embed size sentence representation network round score generate logistic train cr cnn q difference error term side score class incorrect use minimize loss function like rank task large classifier significant impact learn negative number small experiment sentence choose sgd among incorrect max class backpropagation gradient cr backpropagation group relation relation nine relation group characteristic cr cnn make easy artificial embedding omit benefit prediction step cr term right relation classify actual score otherwise annotate type belong nine main type nine consist predict take consideration cause cause water pressure cause instance macro average nine relation take consideration initialize unsupervised perform pre skip gram tool snapshot english wikipedia corpus removal english substitution character text stanford pos removal less character word substitute digit result clean corpus token tune range cnn configuration show hyperparameter decrease training epoch epoch
alignment right artificial intelligence vertex label xshift fill yshift xshift yshift xshift cm white extend cluster entity relational cluster entity group also entity resolution entity propagate model social community relational discuss apply entity relation assumption triple incomplete entity proceed name scalar letter letter bold letter bold letter stack e n kronecker background formally define graph entity relation knowledge triple entity relation variable existence triple tensor array n whose cf interpret world derive interested triple triple knowledge graph adjacency tensor triple depend close assumption triple exceed type number valid triple actor store actor star movie important issue relationship efficiently ideally graph e linearly linearly relation triple discuss presence absence certain triple correlate certain triple conditionally give relation additional mainly existence triple triple independence write sigmoid form criterion maximize margin triple desire probability via many define discuss proceed general triple triple triple denote parameter strength problem possible loss question contain fact emphasize notation triple generalize way triple negative triple understand valid unknown triple false would generate negative type irrelevant assume type actor triple event reduce encourage focus plausible negative extraction run triple due extraction good plausible negative triple miss close valid incomplete precisely triple triple functional triple triple really general score triple false triple margin first assume example triple likely objective sgd scale well square optimize alternate square pairwise specifically triple triple less likely world close world local world world depend cost use closed often model discuss presence way node latent characterize degree call factor directly latent node latent space factor compute edge distribution sigmoid grouping parameter analogously plus specify detail alternative amongst directly treat mrf variable variable various connectivity markov field case relational mrfs product network triple detail kind whose control loss triple convenience sometimes log occur unfortunately graph observe usually heuristic fitting model specialized loss function get train world indeed false set relational interpret miss open triple exist triple indeed triple assume triple denote occur object triple edge heuristic discard triple set pairwise triple exist triple less use present train world world close world cost graph deterministic rule locate usa infer usa typically pattern true nevertheless power pattern tendency entity characteristic star movie usa relational one kind refer property divide group group might actor star movie science actor consist science movie entity star chain triple involve usa depend city dependency involve entity usa relational able create domain variable conditionally independent latent explain triple via feature instance explanation receive award good explanation entity actor observable award call latent follow latent model award award vector note infer hard behind entity derive interaction possible way derive mean entity number relation number entity l relation positive triple negative triple mean partially triple score triple slice mean vector relation sigmoid relation feature entity size layer layer entity bilinear entity h relational explain triple pairwise triple entry specify interact th bilinear vector model magnitude anti correlation efficiently negative compute triple entity interact property representation entity subject furthermore entity triple subject triple object k entity since share propagate information triple dependencie embedding entity similarity entity representation entity similar latent entity similarity representation act non relational access recommendation tensor factorization compactly illustration tensor efficient factorization adjacency explain entity product triple derive via composite representation information share representation entity compute gradient stochastic assume second eq via efficient update triple non zero iterate update arrive current update tensor parameter runtime update update relational moderate latent scalable product triple capture global relational three provide dataset markov logic relational model cluster factorization prediction aside link task entity cluster instance state predict author publication publication database semantic create clustering entity embed relational factorize adjacency cp tensor web page web predict interaction tensor triple graph dataset recommendation graph number tensor fact apply boolean discrete tensor decompose factor algebra adjacency tensor subject column relation cf unfortunately formulation object lose complexity compute entity require parameter interpret creating triple rewrite equality product feature representation predict existence triple create composite representation please explicitly require layer triple predict let composite alone reason add hide final via difference product approach require disadvantage mlp lot call mlp entity please er mlp global relation project reason show train semantic embed er representation relation compute er mlp put near close parent birth birth parent edu hold job job edu job neural lr lr lr mlp r h h bilinear h bilinear neural precisely slice slice combine bilinear additive mlp bilinear use paper additive interaction latent latent model social derive probability relationship representation entity relational propose social network refer se extend idea relational feature representation entity relationship loss entity relationship se translate offset instead multiplication triple vector note unit euclidean follow h rewrite eq experimental diagonal version prediction er mlp comparison leave future existence predict extracting predict due social parent person could triple existence child reasoning triple via observable directly triple kind neither art relational art superior knowledge strength latent complementary aspect suit modeling computationally triple model suit local graph computationally triple neighborhood entity theoretical factorization inefficient fortunately often via difficult model however easy existence existence edge strength model promise graph training kind model model optimize observable pattern allow result increase solution either logistic square gaussian gaussian efficiently least update relational various joint pattern simultaneously reduction require improvement runtime learn learn relational latent entity representation object composite representation subject object pair efficiently triple triple entity kind include model spirit rate additive information factorization machine allow observable input way stack output er mlp layer stack advantage kind disadvantage need jointly separately bag stack mlp scalar employ mlp flexible kind ensemble interact interaction interaction logic template potential dependency arbitrary relational mrf logical formulae mrfs tool suffer difficulty estimate rule paper general compute hard gibbs soft relaxation system fairly show estimation cast convex quite call pseudo cf dependency flexibility relational mrfs schema predicate content web source annotation second serve compute extract fact extraction train mlp predict combine discuss score fuse derive de page illustration bernoulli label drop mlp mlp employ set mlp system achieve auc roc roc subsequent combine auc neural model observable aspect prediction achieve combined score slightly classifier achieve triple integrate google probability name logistic fit million triple approximately triple reason triple low perform triple predict million triple substantially big structured repository give extract triple triple include extraction triple multiple indirect fall accept cause former belief final fused combining method extraction triple calibrate probability plus perhaps date handle unary unary relation statement property entity person row column tensor approach unary modify say see high relation graph relation via express two actor star movie loss auxiliary actor movie character entity auxiliary triple format relationship without transform related truth change google page schmidt fact correct fact annotate begin construct represent auxiliary however duration fact necessarily usage auxiliary easily order high neural impose hard useful powerful language language formulate computationally demand fortunately machine face evidence deterministic triple relation dependency usa north triple add knowledge triple constraint knowledge graph apply entity domain limited modelling manual constraint consider scale relation greatly reduce relation type although range triple induce city etc mutual deal entity mention knowledge consideration current newly latent representation entity calculate approximately explain relationship relative current calculate relation already probabilistic triple might quantify expensive involve handle review relational conjunction machine read build show massive machine memory many application represent kind human possess notably miss representation fact water thing knowledge email etc represent reasoning ai expert relation type triple label nsf award technology grant mt rgb rgb rgb rgb center base xshift yshift name north east arc cycle lr lr study method relational structured paper predict world discuss relational massive dataset latent second graph observable decrease finally discuss information automatically google project characterize categorical main relational object datum form node label relationship goal node pattern arise analysis social biological pathway far de logical article review community relationship entity knowledge google discuss cause application relational grow automatically
helpful mathematical california technology plant flat introduce strongly difficulty plant flat flat rapid field increasingly naturally notion complexity motivated algorithmic aspect consider jointly pose hardness understanding algorithmic attract lot successful link arise abstract extensively theoretical computer hardness random hardness approximation primitive hardness improper hypothesis plant primitive detection subsequently high understand come randomness computationally manner investigate treat detection show flat flat formula clause exclude introduce problem instance testing plant unknown make flat rate minimax base various able inspire successful sample discuss plant significantly affect detection plant solution flat focus flat detect plant phase instance transition compute successful plant describe dimensional flat determining whether alternatively whether take independent j flat random yield constrain coordinate clause flat flat satisfy flat exist element asymptotic underlie plant uniform denote independent identically distribute uniform dimension linearly independently plant uniformly identically denote generate linearly contain satisfying assignment consist transformation fix contain particular contain uniform particular procedure bit result description description tuple require allow invariant confusion representation actual oracle flat base flat purely make membership oracle list basis flat formally uniform uniform contain q write v flat study flat result collection doubly compute suffice element derivation together flat flat exponentially small sharp phase regime probability transition equivalent choose independently among interpretation independently correspond term flat flat equally behind jensen variation consider lemma yield approach variation plant sided observation problem distance converge powerful flat regime since view check cover detection flat constraint multivariate flat equation hard lift system equation obtain quadratic general technique take embed equivalent instance flat solution intersection constraint intractable order relax solely constraint flat associate equation flat solution equation consequence always recall kn multivariate multilinear exist element distribute uniform eq aside tight obtain take test remark go time linearization analogue plant sample linear benchmark plant vertex clique greater primitive hardness plant assignment study bound type come statistic let consider sum tuple show behave differently great typical deviation powerful typical sign sum triplet sample version light show nature suggest modification successful plant flat hypothesis alternative happen dimension uniform define therefore v tackle statistic maximum flat hypothesis n variable hoeffding union q hoeffding result prove powerful direct consequence second point bind divergence similarly optimal still
yes yes c average fan adaboost wang adaboost diversity deal majority work different combine final learning cite regression keep online wang testing procedure last accurately interval record classifier adaboost select record run scheme scheme comparison deal feature dimension set context context belong list reference mis classification set importantly among technique accurate accurate poorly accurate datum expert case prediction presence drift concept slide scheme w slightly able adapt quickly change context fig decide classifier send fig refer decision exploit context time instant select equally context relevant select expert datum learner learn instant drift irrelevant automatically decrease time exploit obtain context help reward window quickly bottom decision make affected drift w concept scenario c c set achieve information label low scheme adaboost adaboost two time label adaboost horizon good happen q ta ta variation ta p ta regret exploitation reduce order bind regret contain happen level level happen level level must eq maximum level hypercube happen pair interval context consider counter select upper bound original scenario scenario scenario type multiply maximum possible exploration bad interval great interval bound achieve exploitation level exploitation hence level event analysis ta ip ip tuple inaccurate action tuple td ta tuple contain relevant action candidate ad sp chernoff tuple imply happen happen tuple variation ta ta ta ta conclude exploitation small regret bad context active interval contain arrive happen guarantee contain maximize since maximum hypercube level hence hence need contain consider update counter select interval tuple regret due tuple type tuple let configuration level different type different interval regret tuple interval tr subsection equal corollary recommender medical diagnosis security require go example difficulty present big available valuable integrate efficient learning curse formalize maker dimension advance dimension different action relevant contextual armed add exploit bind number relevance absence good contextual exploring observe outcome suboptimal action arbitrarily breast diagnosis news article contextual recommender drive diverse source diverse include document transaction file big surveillance health monitor stock market etc stream continuously dynamically evolve way decision stream tackle online big challenge exploit application embed know advance relevant action decision build bandit formalize stream perhaps process characterize process act e receive vector take generate context context action reward meaning depend security context attack context gender reward indicator item reward reward context context application reward relation relevance relation advance decision arise naturally practical treat disease patient image medical often treatment drug close indicate patient care characteristic past strongly home relevance avoid curse bound relevant dimension dimension summarize phase general bind growth gd incur phase observe exploration phase perform control even observe reward costly arbitrarily confidence select exploitation provide medical organize formalize learn relevance action type numerical summarize c c contextual similarity continuity bad always contextual bandit relevance contextual bandit problem paper lipschitz contextual similarity reward action come stochastic good action context regret achieve cover compare work regret dimensional contextual bandit problem consider linear context learn corresponding assumption generate context arm reward assume context process regret space work arbitrary reward lipschitz lipschitz take reward decompose reward function directly reduce bandit reduce graphical bandit action ai imply bandit problem reduction find contain approximately instance project onto low adaptively representation base work action different relevance relation work observation efficient consider function bind online prediction consider lie predictor derive similar work desire never however assumption form expect continuity special datum stream special relate costly assess label stream provide unlabele instance active deal sublinear exist base combine reward combine expert update goal take ensemble analytically expert reward hence expert contrast benchmark context bandit action choose denote give dependent variable subscript vector unknown relevance di kkt ta choose maximize cost consist ti ii element infinite notational value lie correspond ta generate process know priori l learner know need algorithm result show aa due compare oracle choose denote learner learner denote choose learner observe learner benchmark definition relate work cost call active learner reward balance cost incur able two observe reward regret achieve sublinear growth rate depend section well simultaneously relevant reward context way estimate control operate active learning parameter type analytic perform know enough operation summarize adaptively compose type vector estimate tuple tuple estimate explore observe reward cost reward current action tuple action variation hypercube interval similarity dt I explore ta ta ta ta ta ip p ip I ia exploit slowly sublinear reward action good relevant tuple type big type relevant action adaptively space disjoint interval denote let arrive small tuple type past observation vector lie form mean reward create balance sample mean reward due past calculate form keep counter counter exceed duration level create example otherwise remain duration next describe keep keep tuple interval determine exploit tuple tuple let element tuple tuple value type let tuple assign depend cardinality active reward reward action guarantee action context close type expect action expect contain ensure within hypercube exploration guarantee enough sublinear compute explore empty select explore observe learn cost reward eq ii estimate form action tuple type reward fail tuple select nonempty compute variation ta ta relevant mean calculate tuple find tuple type mean select reward tuple interval know hence compute reward tuple interval different learn action high reward tuple however learn high form tuple greedy type sample reward set tp show arrival process independently explore sufficiently many exploitation contain context problem pair form reward relevant action subsection derive sublinear bind prove online regret exposition simple numerical section randomness action incur incur flexibility learner objective minimize label cost exploitation learner trade exploitation run control number instantaneous reward select relevance total exploitation tr context arrival dependent exploitation choose zero come reduction step exploit I require exploitation take stay nonempty regret value state theorem relation proof give appendix duration get give sublinear regret increase run state exploration exploitation balance order exploitation order exploration contextual focused balancing exploitation context vector reduce proved contextual bandit relevance result say reward pair require relevant however comparison action prove work case action great never impose explore begin reward relevance duration control level relevance probability tr sublinear duration parameter control eq gd gd match relation independently sample reward reward equal average know select action average reward estimate type type action relevance relation develop relevance give figure general regret keep mean action tuple tuple tuple time similar explore newly reward estimate tuple similar maximum mean tuple relevance relation action sublinear regret know bandit algorithms breast cancer diagnosis iii simulation learn accurate prediction classifier
cognitive business web specific user click three ad query describe entity relate precise view learn view wherein complementary naturally view learn basically usefulness correlation approach however fail explore paper advantage feature view contribution high combination order allow different error logit list basic use view click user ad query description aspect click click represent view click predict array second index index definition tensor mode product tensor index value mode product k multi view view wherein complementary interaction extra w mi view view kf overfitte assume interaction rank w I basically factorization element wise transform order multiple view tensor factorize th vi denote flexible order interaction interest learn sometimes intuitively redundant scenario overlap view construct full interaction order outside investigate view complexity largely equation time interaction reformulate model risk overfitte importantly choice conduct popular choice loss number differentiable least etc logit model otherwise possess property gradient independent iteration loss eqs moreover eq initialize deviation accord eqs learn search hold improve much hard memory training mi convergence discuss extension multi include vector svms machines factorization svms margin hyperplane essentially svms integrate hinge loss view concatenation shown obviously explore restrict remove svms svms implicitly interaction nonlinear svms interaction exist nonlinear enough reliably instance either estimate interaction svms svms factorization eq instance allow interaction effectively train investigate interaction view machine high interaction decomposition estimate g reliably higher critical low interaction high rank latent achieve machine advantage svms order fm j v interaction include multi redundant correlation within view thereby group achieve pairwise interaction I p qx robust svms instance main order completely
capacity free beneficial appear triple regularization capacity encode section kind validity different datum introduce previous control way way capture approach training embedding way different embedding space capture embedding pre combine stage work benchmark also systematically scheme add embed add thorough strategy besides benchmark prediction embedding insight behavior organize follow work scheme benchmark discuss art modeling embed method one simple hold tail close label hierarchical asymmetric relationship knowledge basis modification entity hyperplane translation idea except shall additional performance kb dramatically hard perform high context link modeling assumption one dimensional however represent bilinear correspond tensor criterion parameterization criterion neural tensor combination way way share entity entity embedding combination embedding improvement difference interaction parameterization argue maximum degree parameterization embedding term linearity nonlinearity embed overview difference recently purely embedding entity relationship framework explicitly extension blockmodel entity entity similar relationship entity way share discuss symbolic way path go multi relationship weight represent relationship project conjunction symbolic also embed fix entity relation triple head index label relationship type learn score set triple receive triple unlikely learn low triple embedding head entity triple canonical dot appropriate term embedding entity even dimension embed dimension constraint relaxation translation special entity unit exactly basically entity besides parameterization embedding interaction constraint way image leave relation score embed space function result magnitude strategy indicate train strategy depends jointly denote sum fine tune training accommodate train directly without separately strategy combine fine tuning parameter unchanged hence follow combination rank later bfgs additional version discuss parameterization classification good netflix item bias contain embed embedding depend bias see mode factorization parameterization play collaborative play bias analogue collective factorization matrix b exactly little argue factorization bias space way type interaction motivate choice score embedding add hyperparameter add hyperparameter reasonably remain collaborative filtering critical feature bias collaborative rank square kind leave aside idea singular value rank pattern bias exist weakly strong capture allow offer control capacity translate useful end capacity closely add entity quadratic form gain ensure useful control capacity regularization parameter well turn effective admissible embedding lead conclusion really embedding absence clear concrete capacity believe embed space less expressive useful prediction hand motivation term relational basis relationship like capital head country tail huge entity identify type entity person filter prediction term correspond bias head embedding entity feature predict head predict natural share embedding first two b last h intend entity capital city connection france country city link position respective type embed feature use type object diagonal change keep reverse rotation preserve regularization come intuition direction choice triples paris capital france rather france capital paris invariant direction relationship inversion direction task invariance replace letter h w e ex stochastic design triple fact express kb fact suppose triple provide kb positive triple corrupt one carry discriminative approach create replace triple may create wrong negative triple rank set triple application h define gap stochastic minibatch set disjoint triple triple keep whole pattern lead initialize model disjoint pre tuning stop validation weight stop convergence margin rate initialization entity embedding normalize subset metric deviation connect subject relation extract wikipedia act subject object rank metric proportion rank raw triple random triple example epoch validate epoch validate validation criterion validate radius determining fix radius validate among apply parameter validate among tuning rate select learn margin regularization training carry way alternate experiment version impact shared share soft hyperparameter grid hyperparameter performing method extract main hyperparameter dimension compare model head label occurrence head experimental configuration model c soft hard soft soft soft top variant bottom perform bold filter raw l mean rank c hard hard hard soft soft soft recall ft lc combination provide model alone significant combination bring basically somewhat one potential impact constrain head tail entity regard come triple kb entity head totally uninformative highly rely interaction well irrelevant interaction may poor turn r kb side reach completely conclusion kb automatically regularization setting display outperform simplicity advantage something lead improvement kb information encode complementary except wide roughly counterpart hard soft performance perform share confirm pre different embedding essential properly collect embedding constrain pre encode complementary performance performance soft model type classify relationship cardinality tail argument variety head vice versa pair classify respectively result constructive predict tail remarkably relationship filter model cccc c predict head predict soft ex previous regularization similar soft term bad confirm embedding actually different via e good relationship seem happen bad model around twice h insight relationship behavior detail notice simplicity predict triple use counterpart present triple triple triple subset contain triple learn one triple rank decompose overall adequate particular original paper counterpart triple use expect well counterpart train soft counterpart instead account prediction entity relation test triple slot answer triple prediction display row want among find make team topic answer type country movie may operate relationship could actually attribute relationship entity head website website entity little among relationship leave hand nearly impossible cm project embedding project use usa cluster separate except correspond thompson cluster appear triple heterogeneous category illustrate however look neighbor embedding entity entity like work movie tail together triple form act object triple predict fit expect answer triple top enter make take enter leave move join lead join conduct carry convert release produce include base become establish dominate name enter ex like good third instance show heterogeneous relationship explain good fit express channel lead rank target list pair similar rank high triple unbalanced frequency rare rank much bad appearance tend rank sometimes match due influence frobenius norm relation impose norm norm impact importance frequency tuning enforce example answer triple provide carry use save visit enter come know include join bring reach become join say help leave make involve support take carry carry move leave release produce become include take include leave run name take call move form establish dominate break relationship translate entity embedding explain argument unbalanced factorization combine good pattern embed phase pre benchmark different strength actually conclusion hyperparameter soft soft soft hard hard configuration soft c hard soft soft c hard hard hard configuration soft com facebook research de paris france universit universit cs france problem basis entity previous attempt complex connectivity pattern overfitte rare relationship capacity frequent capacity simple train variant kind regularization combination strategy show result benchmark tool rise retrieve digital kind area domain biological purpose kb google knowledge kind provide capability knowledge engine language internal kb answer language processing task translation formalize relational entity encode various kind
accumulate big annotation enable outli contrary majority voting outli detection vote green correct annotation red outlier number vote receive majority voting remove bad save receive one vote voting detect examine outli example comparison label validate closely huber robust contrast rank statistical ranking train datum huber ranking rank score store rank instance huber otherwise regard outli loss huber robust outli outlier ranking design huber huber huber difference robust cost low instance main objective rank thing common ability detect framework could remove introduce low huber differ consider low critical huber instance denote decompose svd complement svd project column space compute outlier solve sparse approximation dimension huber lasso huber lasso cyclic ranking perform operation rewrite eq e analysis huber effective detect outlier exploit low projection able well identify outlier especially pairwise annotation training huber always pca sure huber outlier validate dimensionality expect ridge level pair dim net age original model pca outlier five benchmark dataset fall category visual video estimating attribute recognition human age face set human however model comparative deviation plot qualitative box annotate interesting success box detect box failure box agree annotation later annotate image consist scene comparison annotation human annotation reasonable pairwise rank ranking mean order predict method voting outli pruning remove outlier vote score regression learn image dataset enough robust outli conventional huber follow estimate use outli annotation clearly significantly outperform global local outli superior joint outlier enable result outli order reliable hundred comparison per met suggest weak global majority interestingly comparable annotation discrimination majority affected see improve pruning rate outlier big showing stop see box bottom ground indicate nice predict odd hold camera visually unlikely capture camera aspect video digital product video give complete interesting video annotation noisy invariant feature sift coefficient incorrectly annotate attribute answer failure case cause unique building colour consider age key evaluate truth person enable perform depth significance alternative factor annotation outli accuracy measure directly age individual label truth range compose generate error accord pilot pairwise collect age fitting error age human age error introduce bad worker provide label human crowdsource mixture thus setting error error result add around unless ground truth give compare method experiment four show similar correlation comparison show global robust rate feature dimension chance identify outlier peak importantly stay annotation compare majority voting compare majority voting prediction accuracy pruning rate pass aggregate pair outlier aggregate voting effect ratio roc measure relationship pruning age vary comparison amount error train prune fix show true deal non effectiveness employ outli examine decrease rank list comparison accord outli relationship pruning ground outlier large age tend first conservative pruning obvious reliably framework visual advantage voting detect outli detection ranking prediction formulation conventional outli comparison effectiveness alternative validate effectiveness outli also human going include extend application denoise iterative field economics fu degree university degree china currently post video understanding degree currently university research interest model machine vision mining interactive major international member degree computer national reader associate school engineering computer science interest include vision machine international co behaviour semantic currently degree china research interest topological high datum electrical receive college university research interest include computer vision wang wang china vice institute digital medium medium video technology receive electrical engineering ph california work ph dr wang research interest computational vision digital visual institute technology mathematics ph mathematics california berkeley stanford sciences china interest topological geometric vision member american mathematical statistic apply area mathematics neural ac school sciences university china email edu cn correspond wang school china email wang cn estimating attract increase visual image intermediate visual recognition challenging make recent crowdsourcing tool video interesting give separately introduce outlier rely majority voting annotation require amount pairwise collect detection cause principled way annotation visual property problem outli jointly pairwise label together rank lead well annotation outlier benchmark alternative property detection ranking path computer image video scene indicate scene category image object object interest e represent bounding box face one age gender person property little ambiguity estimate less variety example estimating improves automatically predict people video start prediction world increasingly rely retrieve increase video application useful visual recognition people face like meaningful refer visual prediction challenge primarily difficulty obtain annotated score range cast problem low level annotate value annotation people example especially note human pair visual easier exist comparative predict pairwise amount annotation instead compare study thus resort crowdsource amazon economic conventional laboratory bring crowd affected worker provide wrong annotation cause nature regardless worker rank familiar figure face number comparison big instance crowdsource tool annotation remain e pair compare deal outli majority voting annotate allocate annotation pair vote limit infeasible cause error effectively majority voting base inconsistent pairwise ranking pair ranking eliminate ranking ranking cause locally consistent vote outlier focus outli method property collected crowdsource first outlier majority follow formulate unified robust framework jointly voting operate integrate local together correspond receive vote thus comparison operate sparsity optimisation formulation statistical make suitable unseen video formulation video dataset relative attribute demonstrate method state art effort aspect include relate correlation people al systematic contribute preference refer certain type natural input video receive less attention perhaps hard understand mean liu frames essentially treat work benchmark video video early cast problem absolute social collect pairwise comparison crowdsource majority voting remove outlier comparison employ learn rank unseen video compare experiment unified robust majority vote formulation broad attribute base gain popularity recently intermediate attribute use include shot shoot previous binary attribute relative predict semantic focus due intra interactive address annotation outlier majority voting necessity heuristic primarily annotation sparsity majority voting respect global voting theory study computer aggregate huber lasso potential robust local unseen learn addition order address critical problem theoretically experimentally solve outli ranking prediction novel visual pairwise comparison rank first detect outlier theoretically experimentally superior exist majority voting ranking early version work focus image video model noisy pairwise video training instance represent instance comparison tool direct comparison label give comparison strong save save aggregate cast vote ij indicate similarly e edge word ij indicate instance vote carry node consist task remove outli problem prediction consider feature predict coefficient level formulation introduce vector edge notation convenience vertex ideal vote outlier vote majority outli crowd propose globally jointly end variable outli coefficient unified edge model magnitude edge outli expect discrepancy annotation nonzero whole ec e e annotation keep sparse note vote many discrepancy need vote represent edge sparsity outli unify robust identify outlier globally integrate ideally sparsity
adopt representation reflect group aim representation tool classical invariant reader invariance haar kernel haar integration binary classification highlight conceptual advantage perspective explicit subset radial act haar unitary invariant integration see z haar group action framework since turn classification give label class minimize empirical lipschitz belong hypothesis kernel call space optimal nx ix f gx nx group invariant translate assume follow x x x identity function rkh posterior since unchanged endowed group eq misclassification dx preserve core imagine cardinality cardinality review haar setup reduce core kernel haar integration test time computationally virtual transform make usefulness contribution fold feature derive theory introduce haar integration around kernel large unseen approximate rkh assess empirical minimization mnist random theory haar integration start cumulative variable draw accord haar latter define truncate cdf gaussian vector rejection template behind control concentration theorem unitary concentrate rejection hold sphere sx sampling control dot pt xt pt uniformly group cdf template independently gaussian ready section study geometric state explicitly advantage map outline store invariant category plausible feature invariant proof supplementary material haar compute distance expectation invariant hold sx gx gx hold restrict sampling group asymptotically constraint relaxed r invariant result dot product around invariant large bin template element n n put corollary capture distance distance x c universal constant assume template unitary draw equivalent indeed gaussian rejection template proportional template assess template kernel interesting template achieve minimum point future set unseen architecture aim risk cdf sx combination x dense constant approximate via empirical restrict sampling restrict infinity relaxed practice n template feature arbitrarily function core unseen hence integration achieve invariant random space rkh govern template summarize datum perform f f cdf onto complete toolbox explore sequence element alphabet character assign character regardless invariant permutation version template sequence translation rotation template translate pixel degree child digit speak template play speed detail template template outperform bag invariant invariant good template match bar remove clarity supplement z g unitary virtue unitary compact particular template eq g gaussian variable rotation invariance note write gaussian analytically chi freedom bound upper tail chi chi freedom upper together equation put equation z k sx e noting unitary gx z eq product sx z x symmetric proof two template ss sx turn cdf cdf cdf sx z n nx nz z sx j nz cardinality note dense sx dt pt ff prove need preliminary assess approximation certain function function f jx jx tf fm lemma x j f j equation follow function translate exist proof dx dx risk sample fix ready f f let approximate know template element optimality union template cumulative empirical let lipschitz respect rademacher iid symmetric bernoulli take create dataset exploit permutation invariance provide group access take alphabet give total character choose target sequence position character likewise character binary positive sequence preserve permutation sequence belong version another character vector every position form represent character invariant representation standard pool cdf baseline dimensional split positive remain template encode datum template fix number template improve accuracy bin
domain guarantee existence local minimizer conversely constraint restrict consequence descent local never elastic mkl lagrangian identical lagrange correspond elastic optimizer include minimum show necessarily manuscript consist step alternate problem composite kx k kx efficiently solver attack sum elastic net constraint manuscript novel simple solution sub remain section solution optimization abstract original mkl please vector like problem change implicitly account elastic scaling find practice minima corollary c gx sx sx gx sg cone show simple calculus give efficiently iterate iterated stop meet later provide behind ease hereafter next interpret gradient offset constant p sx substituting follow p gx hyperplane level find tangent guarantee decrease monotonically fact point sx make start descent algorithm give ref ref outside limit convergent subsequence continuously function mapping continuity condition simple easy sx sequence exception condition satisfy provide name n follow q first unfortunately origin inequality violate consider readily q importantly equality start show imply convex equality define point condition sequence converge new iterate lie positive cx x hx hx g suboptimal know exact condition predefine q terminate iterate satisfied hx hx focus bind elastic algorithm detail kernel coordinate classifier optimization compare exist external library wide applicability readily open source library report assertion follow state name series show open suffice remain evaluate x sx gx sx sx cone hz sx x convexity sx sx sx substituting rearrange gx gx gx condition gx x sx gx gx statement proof sx gx hx sx gx function global unique reasoning strict convexity sx gx sx statement side r sx sx substitute rearrange n z concavity square
order tight leverage recently svms helps sample risk ridge regression feature design future direction feature provable nsf problem square analogue power selection leverage score randomize spectral leverage sample risk perform synthetic world indicate popular machine penalize simple square long space require technique deterministic provable numerous empirically randomize like provable empirically failure accurate feature algorithm feature irrespective feature class label provably provable non bad feature fix design subset deterministic spectral provably feature unsupervised setting score selection error failure feature complexity pick training provide approximation guarantee guarantee training score unsupervise ridge design comparable risk full feature guarantee feature selection set single set information gain qr report run time offline experimental indicate spectral perform dataset observe small number require deterministic achieve good whose dimensionality identity singular orthogonal diagonal singular value singular q nr indicator one row respectively sample matrix rescale dimension hilbert rkhs square class consist throughout eqn eq function generalize classification goal perform algorithm train proportional rank transform dimension giving subsequent portion similar second lie outside span consider real ridge norm coefficient towards fix ridge rr dual tn n dataset study svd full closely circumstance selection base sampling dominate svd single set randomize setting however focused reduction combination hadamard lower subsequently solve regression dimensional spectral technique one vector symmetric definite low respectively potential measure low lt dominate satisfy q construct lemma rescale combine sampling score sampling vector train choose probability norm leave select trial scale row parameter describe bound ridge bss invertible definite let theory show guarantee bind bss depend accuracy feature bss leverage technique definite bss ram provide I suggest pick break tie previous column never inner pick column use single matlab vector compare bss numerical algebra community rank qr slightly abuse matrix thin polynomial preserve rank uniformly replacement serve get sample five score randomness pick presence absence strategy whereas unsupervised bss ig bss ig bss feature bss experiment synthetic relevant working randomly probability choose relevant among power follow run ht value bss repeat ten compare bss ig leverage sampling table across set top pick bss good pick ten frequently select relevant able bss document matrix project comprehensive web maintain contain pair category binary collect bag document systematic use bss remove group category whose appear perform fold cross validation repeat ten time dataset regularization parameter offline music us uk product us bss chemical laboratory school music business south north leverage score analysis analytical library service service south bss score regularization show bss ig bss average document fig bss well leverage score achieve bss bss bss outperform sample bss due worse supervise metric list word bss seven validation experiment name document matrix experiment
come improve query j return strategy query query group rotation iteration suffer acquisition easy regret factor regret section surprise move coordinate update k perform bayesian update axis maximum produce run pt optimisation dimensional expensive scale tackle challenge additive expressive additive function dimension scientific naive additive application optimisation evaluate example tune machine strategy scientific interact bandit reinforcement either optimum exploitation bayesian refer tackle challenge unknown pair optimisation successfully tune hyperparameter high field design knowledge date identify challenge scale dimension lower often exponential sample reflect regret optimisation heuristic attempt high effectively concern work challenge treat mutually exclusive acquisition high regret dimension additive experiment simulator detection matlab implementation online next set acquisition include improvement thompson upper confidence interest literature study variant acquisition batch vary work carefully restrictive experimental method meet expressive contain along entire kernel dd ex additive additive additive assume additive even statistic optimisation force regime function develop additive additive naive query dominate application precision monte exponential bottleneck paradigm evaluate expensive result complexity believe restrictive cost online learn act second real time smoothness tractable bayesian paradigm sample kernel exponential respectively write convenience gp analytically keep sample gp independent covariance imply gp formally mean need define call act act variable kernel look natural run kernel true still alternatively approach fraction query group easy approach budget group proceed approach place hyperplane entire suffer high elaborate sequential additive first gp pair time gp next since dimension z bayesian condition aspect specify tend require tradeoff exploitation note reality rarely gp base tuning recommendation point always treat marginal likelihood decomposition infeasible randomly select decomposition choose exhaustive decomposition random part rich risk kernel bias low fairly budget hope recommendation practical observe specification original uniformly random hyper
difference parameter generate obtain assess trace check meaning spread beta noise uniformly hyperparameter simulate dataset sign noise simulate dataset error simulation display th exploratory patient model covariate trajectory cross patient cancer collect treatment month period survey send respective treatment response cancer index answer treatment age cancer condition filter author training filter goal opt constitute remove curve patient filter fail criterion remove patient secondly whose representative constrain treatment datum patient patient report row trust report patient patient note fail criterion sum criterion filter patient tb identify potential recovery curve fitting post shape scatter plot correlate identify treatment patient correlate scatter relationship likely decide categorical whether patient age categorical level author clinical experience pre variance patient covariate bin age thus patient belong depend four interval pre treatment lie add visualize patient patient category general model patient pre value justify show superior analogue consider paper first possess value patient average average scale value patient average curve figure analogous separate patient feature word patient scale regression prediction scale treatment inverse assume see predict scale superior sample tb difference time population parametric shape curve unimodal cluster important visually extract likely give along cluster say clear result certainly make closed predictive boundary peak tb shape treatment patient series respective recovery level dependence show fit categorical see recovery separately figure examine patient curve treatment control patient year level see patient age large old value however level pre asymptotic patient range distinct small proportional drop initial pre treatment appear depend age age patient drop function treatment scale patient treatment value high patient year age level treatment level monotonic unlike time entire post trajectory regardless still compare finding model function treatment low age link binary satisfactory age past analysis pre measure link post function agreement finding unlike previous dependence longitudinal function treatment patient depend emphasize prediction shape however strength visually recovery goal facilitate flow statistically interpretability believe medical practitioner easily particular predict recovery curve domain furthermore visualization encourage recovery curve clear used analyze patient post patient age produce agree supplement past finding medical quantified model produce believe produce benefit context acknowledgement support national foundation grant discussion beta show beta gamma vary tb figure value parameter hyperparameter tie remain tie hyperparameter aforementione style gray sep pt style rectangle draw dash inner sep fit author technology curve curve interest cancer patient utility aid produce interpretable relationship supplement medical event disease level extent perturb recovery many follow stroke exhibit recovery curve initial instantaneous drop towards function predict patient available aid treatment would particular function patient post prediction decision close interval treatment medical widely adopt aid merely patient readily particularly curve restrict curve thus event trajectory curve level drop smoothly lie treatment furthermore encourage predictive trajectory curve plot visually interpret posteriori apply express measure index evaluate convert answer cancer affect stage usually treatment effect localized affect able crucially past study illustrate average patient function time select patient post treatment patient method post prediction score obvious naturally cutoff serious logistic longitudinal outcome whereas post treatment trajectory longitudinal relate treatment growth rich past exist possesse curve monotonically pre recovery growth growth trying predict trajectory include place enforce regardless applicability contexts shape medical growth technique sum score basis incorporation patient specialized stroke time specific contribution expect possess ensure predict trajectory actually shape clinical curve function value satisfy q parameterization thus post event trajectory post event trajectory event parameterization refer scale post denote patient normalize treatment shape scale value parameter asymptotic drop drop scale function drop recovery mean recovery curve block parameterization pre patient appropriate throughout introduction tb observe patient post treatment towards adopt hierarchical bayesian observation patient accord function shrinkage accomplish let covariate model generalize analogous section post treatment curve patient support recovery profile support patient pre patient outcome pre constraint model beta respectively distribution unimodal example constrain analogous constraint ensure beta gamma interval come elaborate patient shape happen shape curve model center section curve parameterization section generalize vector specialize beta spread property pt sep latent pt ib ic iy east xshift yshift patient draw sep pt fit white sep south east xshift yshift b f curve support beta gamma ex specialized parameterization beta detailed mode
superiority propose visual dimensional propose achieve new art wikipedia basic program china cb fundamental project address institute university china advanced technology china email com edu cn com key laboratory vision china university china department electrical computer sciences berkeley electrical engineering university technology datum audio widely internet example text web page conduct recent decade retrieval across modality attract modal span different space heterogeneous characteristic challenge media retrieval address media semantic propose modality observe focus couple mapping project modality modality maximize subspace closeness sufficient media retrieval since modal semantic unite common subspace although modal may text streaming wikipedia http paper cross media retrieval different method retrieval method semantic modal semantic unite common latent fig wise closeness modal learn projection couple projection reason accurate image semantic query hard retrieve image label correlation term optimize main contribution dependent media retrieval datum different modality media retrieval validate effectiveness compare state powerful feature far evaluation feature publicly available remainder organize briefly work media retrieval cross media retrieval experimental past numerous medium retrieval try subspace project representation modality directly modal popular cca pls find couple mapping variable cross media retrieval investigate media retrieval abstraction hypothesis obtain generate medium media retrieval correlation generic multi analysis work view introduce separation modal address nearest hashing approach large search medium propose cross hash hash hash code maximize sparse multi modal obtain code modality modal cross modal development medium deep semantic identify visual label obtain document mapping mechanism modal inter relationship source stack auto beyond problem bi media bi directional ranking embed medium inter media heterogeneous cross medium visual medium retrieval retrieval dependent retrieval text e space cluster instance I original assume ij th media problem map dataset consist pair wise closeness modal semantic mapping subsection framework address media retrieval image retrieve respectively denote frobenius norm matrix paper define media retrieval retrieve regression semantic framework present optimization unconstraine convex problem local solution design stationary fix fix specifically minimization q partial updating summarize procedure I easily semantic matrix n cv size convergence v w evaluate systematically wikipedia totally text utilize available sift lda besides media cnn latent feature base token use description firstly base token stop utilize lda text annotation remove pair belong category treat utilize firstly feature token compute topic semantic text experiment euclidean retrieval precision map retrieval specifically precision ap query item rank pls cca sm cca mainly cca sm semantic three cca cca discriminant wikipedia firstly publicly I sift lda experimentally optimization map score method see effective compare learn validate necessity medium term eq could compare dependent htbp wikipedia unified scheme wikipedia cross media retrieval division average image text map
network exploit recall variable moment draw approximation bind approximation distance scale nonlinear line target x frequency magnitude spectrum linearly overcomplete linear independence frequency lift suppose learn lift satisfy theorem arbitrary neural magnitude fourier formal bind frequency make intuitive fluctuation second lift factor sample method network term neuron generalization impose vanish idea complete proof appendix lift part fourier technique estimate label neural architecture label score tensor decomposition network specify proved function yield operator complete magnitude phase lem manifold fourier exploit argue bound impose set say hilbert satisfying term multilinear multilinear form bind matrix technique vanish assumption sample lift know see tensor exact prove sample propose follow magnitude relate sample lem fourier fx fourier transform entry noise equality equality zero final equality manifold compute uniform ss lf integral simplify I proof property delta q use equality introduce change q sake implicitly second delta property repeat finally eq argument impose l thus integral limit magnitude weight concentration desire nn lift satisfy prove concentration use label denote final use apply inequality concentration h v proof phase actually generalize norm rgb derivation claim observation example time bold bold ff non optimization backpropagation descent novel neural efficient input tensor provably set mild degeneracy standard descent sgd neural moment tensor decomposition network vision recognition understand neural paper training guarantee error ability unseen analyze training overfitte poor new classical bound extensive guarantee neuron np local per optima example analysis guarantee lead relevance network wide guarantee hardness refer bad condition input tractable tensor formulate tensor finding sum fit achieve computationally technique mild degeneracy model address trivial question work use tensor linear activation adapt set behave perturbation establish address question efficient term tensor lift sgd training scale train layer feedforward layer start neural high lift decomposition use estimate bias lift operation fourier dataset complexity comparable lift transformation transformation depend learn manner without many theoretical distribution fitting error nn lift polynomially etc guarantee lift well approximated architecture natural expect guarantee meet continuous input discrete reduce function collapse single precisely characterize redundancy neuron network weight distribution target thus achieve generalization matrix etc moment correlation lift tensor yield third identifiable tensor realistic assume orthogonal require vector tensor network exceed input overcomplete exceed incorporate tensor recent perturbation despite convexity decomposition provable lift nn lift layer recursively layer principle analysis control estimation establish formulation column refer operator th rao member product euclidean space second refer rd order outer unit say cp transform call frequency neural know continuous feedforward neural nonlinear fit target neural estimation network finite ability hide label weight overall estimation establish combine generalization detailed minimum pt sep draw fill cm mm circle cm dash mm f name name name name observe width line width width red width red line line red mm line width mm width mm red blue mm blue width network name lift use component denote second estimating bias unknown compare part explain third pdf mild regularity probability derivative differential operator discuss next see various score score addition learn auto find encoding decode reconstruction denoise unsupervise unlabeled argue approximately first estimate recursive estimate high score score representation extract use training neural ia yield show eq th refer notion reason behind score recover tensor tensor appendix power power rd tensor ti l multilinear guarantee iteration orthogonal tensor develop literature whitening apply tensor iteration perform start mini small initialization overall complexity nn lift parallel iteration first auto approximate dominant computational complexity tensor comparable computational backpropagation processor estimate weight bias decomposition fourier frequency fouri entry dimensional manifold intersection sphere actually spherical draw spherical coordinate angle directly cone angle draw pseudo spherical estimate density function l v need function network function second case high cross network overcomplete product full overcomplete similarly extend degeneracy vector full singular target overcomplete product smooth
matrix g projection useful important reader understand embed intuitively follow vector x speedup solve problem frobenius error hold strong speak rank approximation almost column solve svd svd expensive line matlab reader try near randomize hadamard sketch projection combine know seminal gaussian implement line code c guarantee advantage implement line matlab quality sketch even sparse hadamard transform hadamard fashion uniform nine line matlab notice product perform low complexity n subspace embed count sketch property sketch apply uniformly reader notice sketch fact entry line code true j sketch memory keep pass theoretical rank disadvantage sketch attain accuracy improvement sketch matrix cs product subspace hold nearly efficiently small prove sketch count projection cs satisfy q property property subsection sampling leverage random projection column selection visit entry preserve negativity sketch whole avoid every entry matrix leverage define equivalently coherence great leverage score small good study define column leverage column score roughly leverage score k theoretical leverage sampling satisfy sampling accord score leverage expensive svd leverage score practical way score little uniform leverage effective heuristic finding representative zhang centroids class centroid sketch heuristic little make simply solve centroid unnecessary centroid run centroid suffice local dataset supervise associate centroid sketch row correspond datum contain label regression computer science economics inverse svd solve cg machine cg cg attain x roughly time cg heavily slow ill condition long subspace embed solve gaussian count sketch inexact implement matlab matlab sketch sketch thing inexact sketch logarithm thus high subspace indicate z sketch score find standard discuss cg efficient matrix cg let triangular matrix qr compute one easily thus factor probability qr r cg initial subsection extension svd efficient extension describe solve section particularly cs complicated solve implement matlab function section uniform nearly sampling complexitie logarithm iteration weakly gap block implement u c end attain machine pass infeasible memory trade cost go pass store volume disk ram keep sketch iteration svd nk kk kn describe frobenius norm target matrix qr n low gaussian matrix sketch property hold minimization problem matrix find orthonormal matrix randomize depend implement line code empirically discard line algorithm keep prototype solving randomize even fast reader inexact present trick cs embed solve frobenius randomize svd approximate decompose sm cs cs p r p svd k sketch sketch sketch sketch rd clear thing go pass cost k line remove l program fortunately pass ram store disk block consider kernel social matrix fisher find low symmetric show sketch row rbf kernel compute matlab sigma minus form rbf matrix computed function sigma presence million kernel fortunately sketch efficiently matlab code sigma sigma require solve exact solution time approximately need identity expand gradient naive low rank inversion possible spectral fix approximation besides diagonal even approximately discard way low approach svd show chosen error apply follow prototype count line code go pass fit store disk enough despite several drawback cost visit entry serious apply kernel point time avoid every entry reader notice column approximately tb integer column qr try leverage proportional norm row high probability sampling overall matrix sketch contains empirically enforce improve empirically large sufficient line k true z kp give give row rbf sigma sigma true unique sigma computing propose w become approximation call nystr om nystr om use literature figure illustration nystr method thing nystr nystr om rough moderately accuracy thus inverse numerically inverse drop bottom many discussion nystr om implement matlab code nystr om w w sigma sigma rbf tune enhance notice small affected approximation nystr om kernel efficient nystr om inexact mean top efficiently use thing speedup square svm speedup eigenvalue significantly unstable reader well implementation extension nystr om selection c much nystr reason speedup subsection rectangular matrix matrix form multiply large generalization kernel fortunately matrix merely give cr prototype prototype every compute let solve kernel avoid whole leverage column quality approximation speak uniform sampling well suffice quality pc c c pr pr pc pr unique u pr x rbf procedure c sigma r pr c sc sr sigma pc pr pr sr pr sc sigma pr generalization spectral transpose kernel rbf goal extract feature step datum th kernel th entry feature eigenvalue empirically accurate nystr om cost thus fast matlab lambda sigma k sigma clear u lambda lambda lambda k u lambda sigma end row output perform use test user decomposition normalize uk fourth scalability spectral al nystr om make spectral cluster scalable form accurate nystr om line matlab sigma n sigma times replicate fast instead efficient replace sigma gaussian bayesian hyper parameter transpose rbf form tune label compute inversion empirically apply speedup discuss similar nystr nystr matlab sigma alpha l sigma alpha end intensity train matrix entry compute predictive four sigma sigma apply generalization straightforwardly speedup sigma max r sigma c orthonormal column b b solution qr decomposition full row
broad art model special significantly remarkably neural performance comparable vocabulary problem language application speech parse task generality translation paper neural translation fall category encoder decoder sentence automatic represent sentence bi directional alignment network target sentence rnn alignment decoder achieve less instance superiority efficiency mainly mechanism avoid vector dynamic mechanism external inspire novel architecture name carry task series input different eventually intermediate stacked layer stacking generalize network introduce tailor sequence special case architecture importantly deep capability superior cc start discuss read operation memory generalize form illustrate transformation memory read head controller controller operate read memory modify value location memory write basic machine architecture implementation infinite size determine memory implementation always instance determine controller controller operate read write head discussion put simplicity read layer high another reading convention get controller influence controller read address head controller location core controller long memory rnn lstm state controller read write return read turn update reading allow one example suppose omit notational simplicity read memory unit main respectively operator read writing relatively effective reading write address address read address run determine time important suggest go forward read parameterize differently address address f implement dnn un memory mechanism advantage address way therefore introduce flexibility hybrid address address read read address controller worth note address address read head address allow read different base address address writing simple keep unchanged write determined network tn normalize weight dnn clearly transform memory embed shape specify parameter argue offer complicated introduce flexibility address read get unit rnn stack right read invoke dependency spatial order recover deep layer read operation address read writing offer major address add layer design especially couple next layer deep later learn base representation need read transformation induce read address strategy list since combine address read read amount specify omit due design different way address writing stack together stack architecture analogous representation suited layer transformation stack greatly efficiency language chinese english figure leave stack apply transformation low layer base layer stack manner entire diagram right start layer reach layer read operation output rely lstm generating memory read follow symbol guess state generate lstm read flow layer target pure address reading stop generate token different read learn different homogeneous mostly transform activation sensible transformation greatly performance little future read write reading allow read specifically follow cc memory require strict alignment together create structure section read layer potentially inner read right address read address read head write reading memory accordingly flow correction scale signal start output layer signal back control machine lstm read write location address optimization practice descent sgd control discuss four representative show proposal c address intermediate layer diagram right address address read read layer put together layer address sequence read different memory use memory layer differ address address form together predict target deep architecture address strategy generate layer address write address address read head bundle layer layer put intermediate among special efficacy address write form address read memory write address read address address layer transformation read address cc interestingly seminal translation automatic right employ address read address write address target intermediate nontrivial write operation empirical english art machine training sentence corpora million chinese million choose mt mt mt mt number sentence mt mt mt mt insensitive evaluation significance segmentation chinese stanford nlp english frequent chinese map token translation corpora direction grow diag adopt model gram
additional latent transformation volume transformation operator mcmc langevin elegant approach iteration true disadvantage langevin hamiltonian one gradient throughout effect flow inference training monte estimate version parameter sample version result schedule go variable deterministic maxout linearity windows maxout window take mini batch collect time average score estimate insight normalizing flow set unnormalized list e w w characteristic normalizing length transformation diagonal volume preserve flow nice achieve performance grow flow flow far parameter flow nice initialize matrix nice figure height fig posterior posterior kl divergence digit contain image ten handwritten digit train latent flow nf approximation volume preserve nice nice different summarize systematically kl approximate approach normalize nice wide result specification nf nf nice k nice nice c consist rgb size extract patch convert logit x summarize increase length systematically improve log posterior ccccc develop density transformation complex normalize flow inference clear improvement view normalize flow unified perspective closely flexible point conclusion flow rich approximation normalize flow convergence class see able competitive default make rigorous research normalizing flow simply flow base alternative transform design g lie transformation allow thank radial flow always invertible linearity choose condition solve parallel expand take dot product yield invertible h enforce modify produce compactly write always invertible splitting uniquely scalar take q suffice impose constraint impose choice inference application employ family inference approximation impact specify flexible scalable construct normalize whereby invertible transformation view develop category flow view theoretical true combine variational interest increasingly complex problem increasingly large core large model default chemical despite advance limit power wide default limitation choice address know approximation approximation imply solution method inferential regime unable recover posterior rich sigmoid belief field posterior auto clear evidence limit posterior provide exposition widely choose posterior result map rich typically mean incorporate dependency within powerful evaluation present approach specify variational base inference carlo section propose flow distribution transform probability invertible mapping sect normalize sect show normalizing flow admit us regime variational present unified view normalize flow sect normalize flow systematically compete sufficient marginalization variable integration marginal likelihood latent variable principle jensen prior latent often refer approximate prior act parameter variational use mini batch descent scale address variational log approximate tool way log expectation approximation analytically mini carlo compute center approximation affine backpropagation involve latent know location transformation backpropagation carlo monte carlo variate exist alternative backpropagation variable continuous backpropagation model variable among compete use map compute variational time cost generalizing deep deep deep latent gaussian deep direct hierarchy latent variable transform composition form successive distribution transformation law know expectation write jacobian depend invertible flow reduce point towards interior reduce outside formalism normalizing flow give variational appropriate transformation factorize length increasingly modal normalize term transformation partial differential evolve time langevin sde wiener process vector diffusion transform langevin flow transformation kolmogorov transform evolve often langevin importantly density evolve sample langevin sde accord density term result machine make hamiltonian allow scalable inference normalizing flow jacobian straightforward equation g invertible approach jacobian determinant dimension furthermore gradient jacobian determinant several involve numerically unstable normalize flow allow jacobian determinant transform define transformation series expansion hence refer map modify density around parameter radial density flow visualization show spherical successive form invertible discuss satisfy appendix posterior flow free write normalize flow free variational generalize variational construct
security additive interactive database original statistical query statistical decrease perturbation interactive differential result interactive correlate perturb get get survey present berkeley edu present private aim adversary sensitive subject intend possible perfect privacy regardless preserve sufficient health sharing grow volume phenomenon storage social law public patient lack etc even whole use area interest collect health technology health patient monitor status intervention overall scenario collect privacy preserve patient privacy life cycle processing finding focus phase sensitive private thus aim prevent adversary intend linkage maintain privacy private subject might consider become argue private private circumstance reveal little adversary private piece privacy converse also complementary follow survey relate motivation discuss experimental discuss conclusion future research technique security database intersection randomization survey analysis privacy field conduct brevity brief study survey area recently attention health spread health record medical concern regard health research medical therefore domain reader privacy database statistical database review lattice limiting risk lattice control security privacy security rigorous survey datum privacy statistical database privacy grow privacy differential privacy quasi subject anonymous every record anonymous individual individual extension include closeness differential single record differentially private sd show attribute differentially diverse review privacy preserve piece private exchange use shorthand capital realization respectively variable marginal function instead write mass primarily monitor health share nature infer piece like health patient weight generally want piece infer private information guarantee information potentially use inference passive circumstance ability infer auto thick circle fill draw font edge edge leave node concrete follow health index already know status category consider however category infer want ensure privacy create encode public status group encode privacy preserving sense status category security objective adversarial infer private text perform know complement private ability adversary infer towards consider send send information passive information exploit know decode send e ability inference send minimize private similarly define piece encode send call encoding output since use message describe treat continuous reader substituting function cover case nature use appropriate function need adversary different subject model recall information intend know intend send message requirement finally like minimize carry sake infer adversary present reader correspond concern discrete mixture discrete continuous yy intuitively bit mutual independence conditionally objective class information hold privacy communication transaction piece information belong transaction piece information send belong apply function address question formulate condition space pc pz pz dx pc privacy adversary conditionally prior adversary already possess need ask privacy ever privacy message extra adversary surprising utility usually utility kullback define know get pc pz mapping fact pz r valuable privacy mapping observation case membership function adversary regardless auxiliary serve privacy verify origin zero mean similar omit every distribution shape cc describe publicly toolbox question privacy category hand order maintain weight status base scenario monitoring fit assumption motivation early matlab toolbox affine extra degree problem c yield affine class ground depict class note calculate know train classifier procedure encode datum confusion information drop highly indistinguishable since weight category class trivial predict datum bias category category class histogram category preserve piece without look category decode weight decoding point also infer derive preserve private theoretical perfect privacy preserve utility show achievable provide close perfect privacy adversary knowledge show perfect adversary auxiliary knowledge information subsequently discuss demonstrate control weight status set drop bind guarantee achieve approach private information infer propose serve alternative say adversary get access message suffer curse dimensionality number framework estimating appeal option model mixture computationally mutual extend scenario clearly general wide applicability present indeed message sense well belief adversary private research include mapping necessary condition change approach solution analogous minimal oppose privacy acknowledgment discussion significantly improve team research science foundation toolbox toolbox toolbox similar code keyword subsection reader familiar helpful toolbox organize short manual toolbox describe engine privacy structure parametrize definition dimension hand two keyword begin code subsection worth note nested block allow skeleton skeleton begin variable atom user search parameter fx variable implement toolbox parameterization convenience var shorthand var variable either expression convex constraint var expression expression object mathematical involve repeatedly equal code engine create constraint constraint later engine feasible problem treat general object pass separately learn subject mark begin constraint nothing line keyword toolbox provide keyword toolbox current implementation datum histogram allow bin example weight another private class class name string call one try variable useful example class provide store variable come convention list keyword toolbox ht variable cn expression parametrize end end definition block program rule satisfied program start end maintain consistency assume toolbox therefore toolbox assume dimension toolbox computation follow assume variable constraint latter create instance map inequality v symbolic constraint implication later relaxed fix correspond entry exist
relation relation yes counting list set simple knowledge conjunction compound reasoning induction reasoning task language present learner learner ai hope loop solve believe language solve ai fail task go ai beyond variant also highlight extension hope task fail motivate loop develop task believe language understand training task ai learner ai present memory interesting beyond highlight propose extension still several supervision support typically require task human couple additional supervision signal hope develop solve lead research produce applicable reasoning language building agent goal argue usefulness set proxy evaluate read answer simple many design aim many set researcher identify recently introduce able motivate motivate semi supervised equation uci continue component dataset latter relevant work synthetic amount datum researcher elaborate base datum example gram compete far researcher synthetic try break work develop automatically ai response task answer question think cast propose capability common build unify classic whereby actor object interact kind hope task current help loop break task sec benchmark result analyse failure development propose memory show unable solve open project recently unlike task like especially scenario appeal research question human child require ai organize collection question intend machine reader aspect remain complicated indeed able task answer etc capability improvement modification project come scale feature acquire corpus argue actually understand extraction representation remain highly rely lot choose collection failure success feedback capability schema challenge thank help receive help result challenge mostly center around system background diverse self train need setup amount test relate one lambda compositional semantic provide task software computer ideally test simple aspect subsequent build publicly successful perform supervision answer statement answer may task correct else cut noiseless human potentially try choose human reader formal semantic logic representation simulation character object move around interact location generate many giving include task support provide person thus office already simplest hard answer support question office pick support difficulty letter alphabet dataset ip I ip vb ip language english produce task task le support fact pick office drop office statement answer answer recognize subject consider extreme sentence bag office north north office word different answer separate argument task give receive give question two actor mention test support ability answer question pick yes perform count operation hold design answer ability produce answer list entity pick pick hold database operation union apart one type support fact imply office office office office yes yes statement describe possibility office test simplest detect office typically label study sophisticated phenomena address multiple subject task refer actor office far implicitly understand expression school school go world evaluate via property induction color white induction scope produce induction test spatial component red sphere right sphere square yes red triangle yes task yes question inspire reasoning schema box fit yes three yes task north east west north ask certain address actor behave game generating location object hope well model within environment real complement version task class apply memory index learnable execute feature convert internal representation memory I output component use parse simple incoming example leave responsible reading g calculate produce actual module produce feature support scoring support memory support square output module produce recurrent rnn limit response ranking dictionary function feature role map text every representation depend support model gram ng multilinear ml indicate task extension last analysis column amount achievable training c c strong supervision c fact support fact relation yes question fail set fail conjunction compound time reasoning induction fail reasoning extension model former order need know sentence directly sentence old triple pair question carry gradient answer support try task fail bag sec max involve support fact unless rnn provide set required finding improvement variable number support fact dependent ask support x support fact stop embedding learn hard loop computation stop module word iteration conditional I I rx I way model bag variant gram bag disadvantage dictionary rapidly neural multilinear map position position employ whereby word position follow nonlinearity mapping tag rather consider nonlinear embed network side bag gram follow nonlinearity compare I long recurrent network describe gram baseline produce bag gram share least answer use gram use filter method similar answer support fact disadvantage testing hyperparameter give result experimental separately outperform consistent still task build failure expect fact answer bag failure yes linear scoring query yes answer interaction response sec approach gram ng multilinear ml plus combination give straight forward support multi output remain difficult gram modeling clear improvement task gram seem substitute embed outperform gram especially yes fail g combine ng improved task multilinear useful gram ng example perform example quickly task require require latter solve pick subtract drop task finding solve even advanced build
order random tuple bundle ii kn label enforce pseudo label pseudo random tuple pseudo say q exist efficient input variable pseudo random q conclude return almost term kk arbitrarily case choose c kn google fellowship research author thank discussion many I work terminology counter hypothesis theorem conjecture learner access use prove complexity formula show guarantee even favorable strong version logarithmic arbitrarily low case proper access hx learn parameter learner run emphasize general improper good bound efficient poor assumption exact algorithm theory establish hardness gap problem unclear belong learn recently bound possible certain hardness certain indicate low recent hardness learn hardness average problem yet natural rather hardness recognize direction overcome proved hardness spirit work hardness formula framework strong allow hardness e proper hardness currently low base concern study assumption hardness approximation public tuple coordinate collection tuple denote collection tuple distinguish formula hard gets gets solve seem light problem evidence addition performance hardness instance approximately interpret hard semi formula relaxation relaxation sum problem relaxation hierarchy low bound another analyze bound statistical solve formula extensively formula efficiently make cn algorithm trivial distribution output underlie belong family time imply constant section outline implication hardness upper efficiently linear much hard currently guarantee achieve marginal hardness assumption security hardness derive formula show assume marginal nothing hardness concrete algorithm bound hardness approximation restrict algorithms classifier computational already soon theorem later consider author assume efficient distinguish instance instance even rather methodology hardness basic certain hard learn restrict boolean cube sample fair denote distinguish easy output efficient contain toward return distribution maximal random example lie bit describe bit efficient use l generate choose random example l use description oracle n weak end explain course reduce problem conceptual problem correspond constraint namely problem formulate minimal problem distinguish n distinguish point sense yet measure next address easy reduction strongly hard error map reduction address replace independent second show produce property independently uniformly new indeed w either produce note together reduction reduce specify w fail reduction reduction n reduction randomness h test give next deal ks r describe pseudo property convenient whose un concatenation indicator product remain nk class consist therefore show ks ds ks ds j indeed ds j c strong pc strong theorem unfortunately explain pseudo set close check fraction tuple polynomial h nk c cp k eq polynomial z kp prove completeness assumption arbitrarily close conclusion aspect theorem restrictive give majority completeness toward realize define eq nh w w n show efficient distribution distribution analyze formula assignment variable every mapping partial output different remain size leave remain unchanged partial tuple hoeffding tuple pseudo tuple formula every z fix z j note u j
error assess finish small scheme associate may fail probability inequality sample probability imply finish task interval inequality depict hold short upper furthermore present transition behavior confidence critical exponentially quantity performance reveal provide extreme possess generalization regression kernel localization domain control essentially linear thus automatically square expectation differently present exponential purpose term ill employ guarantee number bind deduce attain minimal guarantee suitable generalization capability study capability kernel whose lead specific regularizer smoothly toward exactly vary form choice application purpose show utilize impact capability range real may bring suitable notice assertion conclusion heavily behave square distinguish regularization range regularization attain highlighted knowledge among encourage reader lasso regularization see accord lasso generalization capability choice capability determine take non generalization consideration smoothness explain domain near construct arbitrary sample estimate section respectively deduce x rewrite sample subsection type describe rkh norm define addition formula since v fx u k p dt u vx dx dt nn u dt imply k n proof complete lemma deduce subsection standard typical one bernstein inequality probability space deduce q q imply eq q desire difficult empirical bind main norm denote cover bound properly inequality deal class bernstein single almost everywhere everywhere everywhere everywhere apply eq hold provide hold rate ridge regression proposition ef ef confidence cm r actually chapter constant complete simple traditionally space error regularization regard nature data dependence attribute essential characteristic scheme excellent localization frequency reproduce guarantee present formula spherical polynomial help deduce approximation deduce probabilistic inequality spherical polynomial definition space u u generate operator interior help deduce real without small convention set space q follow lemma note polynomial taylor formula ep qx p np cn since polynomial e deduce eq confidence hold np np prove functional operator restriction hold old introduce decomposition ef ef e ef e ef ef z q ef ef ef since inequality ef ef qp ep present error deduce old implie represent deduce regularization cn ef confidence set tackle spherical due perfect localization frequency paper suggest usage spherical contribution firstly selection kernel totally require computation set excellent localization truncation add sense optimality mean discrete parameter reduce computation secondly bridge bring utilize kernel capability ridge excellent property domain recognize tool tackle spherical datum domain nonparametric due localization regularization ridge arbitrarily consequence excellent localization property far associate include bridge possess almost reveal utilize choice strong capability modeling arbitrarily smoothness computational complexity nonparametric spherical scientific explore predictor variable phenomenon euclidean neural support machine appropriate exclusive useful recent focus considerable spherical spherical poor spherical localize spread sphere formulate theoretical alternate sphere reproduce hilbert utilize spherical popular polynomial similar drawback method remain exclusive spherical sphere localization requirement develop technique cope nonparametric regression exclusive localization paper organize follow introduce generalization capability ridge capability scheme regression main result useful sphere integer homogeneous harmonic sphere spherical class spherical spherical degree course comprise restriction polynomial
forward fairly smooth parameter mse linearization approximation enter approximation fix smooth linearization quality estimate wide comparison map ml kalman filter kalman kf kalman filter state lag pf backward maximization link e mail liu se division mail electrical engineering mail newton gradient hessian compute gradient identity explore linearization linearization computationally model cost ml control overview focus denote time function dynamic denote solve hessian quantity smoother explore recursively sensitivity derivative analytically intractable result solve end linearization likelihood extend smoothing gauss particle filtering smoothing method asymptotically estimate computational beneficial linearization approximation small linearization typically vary evaluation focus approximate likelihood subsequently newton approximate finite difference approach ascent particle challenge skewed lastly method gaussian optimization method advantage estimate product intractable distribution section use linearization return solve ml hessian schedule newton gradient hessian pt obtain label use use gauss newton similar particle height pt run label sample multinomial iw first give section locate hessian equal counterpart consist compare propose initialize estimate set present parameter positive plane low alg far alg alg alg term bias argue accurate alg r alg alg alg scale dynamic apply necessity would expect compare
cm fix design device implementation dct architecture version expansion correspond expansion design design list input dct array device via interface hardware processing toolbox define within bit typical ai architecture throughout ai encode ensure offer accuracy architecture resource device bring term slice look table resource though design expansion possess accuracy compare design hardware resource architecture require considerable amount ai propose real bivariate ai encode ai encode dct hardware completely ai domain completely quantization free final ai tb dct operation entirely ai intermediate ai dct quantization noise final location quantization channel dct noise level remain hardware test ai encode bit relevance bit realization operational frequency ghz propose dct embed resolution word dct publish architecture arithmetic row column dct transform precision affect dct acknowledgment proposition definition cr tag cr dct architecture integer exact mail algebraic integer base architecture propose ai encode discrete cosine dct free dct without dct ai user dct multiplier expansion architecture high digital video computation propose validate hardware implementation realize bit size simulate among bit input design imply ghz embed image device dct digital video demand video image automatic surveillance traffic security wireless video system operate associate hardware throughput complexity efficient circuit capable operation numerical need video resolution minimal noise consume possible two discrete cosine transform dct system circuit dct relate noise circuit consumption video device dct successive call apply dft area point dct require multiplication computational rational dct implementation employ operation introduce address employ algebraic integer ai ai encode possibly integer exact dct multiplier computation ai back usual besides quantization dct dct coefficient correlation noise video signal noise concern dct ai dct architecture form ai complexity multiplication eight thus naturally low foundation propose optimize architecture require reconstructed format column mean reconstruction code enter dct ideal propagation intermediate propose digital dct throughput quantization ai concept reconstruction truly occur structure prevent computation result correlation final coefficient totally doubly ai dct precision dct speed section dct base fully architecture fundamental difference intermediate doubly ai scheme architecture characteristic dct absence operation quantization architecture aim bi encode ai basis optimize multiplier realize gate array review exist ai bring description hardware architecture detail propose test measurement report conclude ai digital dct bivariate encode bivariate dct dct area processing circuit dct core conventional arithmetic architecture architecture dct buffer implementation dct block size suitable application available linear array dct hardware realization array architecture forward dct report dct core cyclic performance transform employ scheduling dct array algorithm base dct realization dct block cycle ip core dct synthesis arithmetic dct processor dct architecture unique describe ai dct computation wise application dct core ai architecture realization also application dct dct refer encoding ai dct quantization report implementation synthesis ai encoding make realization prevent adopt ai mapping link integer array major classic exposition widely clarity depth bring explanation emphasis integer follow focus practical ai useful number call algebraic root coefficient algebraic mathematical form multiplication ai encode follow format integer array integer always integer arbitrary decode operation array constitute ai basis hardware ai basis require ai ai ta integer represent decode principle limit employ integer encode exact multiplier dct encode dct algebraic ai sequence represent ai interpret modular multiplication polynomial ai particular illustrative multiplication fast multiplication consideration ai possess constant represent error ii integer element representation small facilitate encoding decode ai constant yield transform dct cosine arcs adopt dct element particular dct encode adopt array encoding encode scale encoding specific ai cosine possess free utilize integer arithmetic moreover employ hardware shift ai encode dct arbitrary real usage essentially exhaustive search however unnecessary encode express term usually hereafter identify bivariate ai coefficient representation indicate encode integer element emphasize ai encode algebraic multiplication modular hold tailor technique handle ai dct mathematically express usual dct notice correspond column application dct dct result dct use ai encode decode section place column dct operation intermediate introduction quantization noise component contrast employ bivariate ai encode maintain computation ai arithmetic avoid arithmetic error ai dct ai dct column tb ai dct block wise architecture five input circuit ai encode dct block fig column ai buffer connection obtain iv ai dct computation eight circuit ai encode complement format via connection input format already block block stack pixel dct modular refer modular input bit serial fed rate aside stream optical transmission throughput drive sequence row mean eight operate eight consist ai code computation ai dct wise dct core architecture employ arithmetic entirely ai transformation ai encode channel index modular hardware ai parallel channel ai encode integer cf four connection channel ai transpose buffer ai show pre wise dct partially transform wise dct represent encode ai tb fig operation eight cycle ai ai column wise eight therefore output buffer require transpose dct connection cross brevity ai tb period master subsequently ai element ai dct core operate parallel continuously partially dct component bit channel row ai dct provide input rate dct input dct every eight cycle signal perform row dct order complete dct doubly ai representation output channel ai number complement format architecture architecture differ circuit compute indeed circuit employ dct prevent dct channel quantization uncorrelate doubly encode element final dct summation return term rational number two remain binary close sign list number consequently respectively cl input fix
smooth distribution achieve study learner decision expert learner set follow loss also multiplicative weight bind generate base entropy think loss return learner generate assumption eq q f te sf nd nx q fact lemma contradiction mm find center minimum denote entity w entity remove large remove small element mm word correctness see overall reduce half stop round except need two round overall communication institute technology agnostic setting noise general boost concept space noise computationally communication prohibitive demonstrate scalability increase amount attention datum common fit want process utilize one entity evenly inherently entitie example scenario scientific customer deal datum partition traditional algorithm care bottleneck communication baseline entity center vc communication advanced communication complexity recent work distribute boost logarithmic standard set noiseless impossible communication inefficient boost agnostic much enjoy communication baseline efficient concept finite dimension insight example learn weak hypothesis boost result challenge design agnostic set agnostic boost weak list open work rate algorithm communication distribute identify boost agnostic adapt agnostic require call learner previous learner set class agnostic learn flexible weak learner learner centralize distribute make easy confirm theoretical empirically synthetic promise introduce agnostic problem access low often within denote common function agnostic entity entity act learn much convenient count problem paper bind communication complexity boost vc boost hypothesis assumption agnostic setting even set poorly set access learner agnostic rate discussion existence centralize reason boost agnostic tend weight end put noisy overcome smoothed boost technique enjoy nice shown originally analyze hard first weight additional current weight bregman projection technique find distance bregman boost always generate verify convex hypothesis center call weak entity sum center across sum vc rate underlie sufficient find error good weak entity summation center communication index find sort coordinate inefficient fortunately advanced find median median potential large em subset mm mm em mm mm mm mm entity entity w w project set complexity direct adaptation centralize proof correctness search run find respectively must remove median center half stop except update round find easy base find candidate communication agnostic algorithm theoretical rate use round involve per boost draw ks centralized theorem thus achieve communication round communication weak bound note vc generalization weak learner tn iv center agnostic weak learner union return update normalize project algorithm mm empirical boost algorithm synthetic dataset adaboost logistic implementation three amazon ec trial still run synthetic dataset interesting potential boost choose odd sample coordinate equal set example machine approach poorly achieve adaboost experiment real million datum repository yahoo yahoo positive example
incorrect right sentence action formulate problem action world encoder decoder infer action lstm encode alignment lstm decoder sequence neural encode encoder state encode version rnn use arrive decoder hide maximize determine sequence approach decoder employ function term dependencie vanish gradient alignment salient abstraction detail illustrate encoder take natural sentence treat word vocabulary rnn summarize relationship word annotation annotation sequence architecture affine sigmoid forget lstm cell activation cell summarize forget lstm gate affect hidden encoder encode backward direction recognition machine hide annotation encoder annotation include word improve decoder level context also word permit match salient sentence g weight extent position around match model perceptron architecture lstm decoder take current previous matrix learn encoder decoder action give draw corpus demonstrate loss train sequence find posteriori action learn search sequence search initialize sentence list step action world agent line domain include define bag word world publicly contain six virtual pair sequence deviation ht sentence train fold retain latter tune repeat fold weight fold later refer training procedure adopt whereby train decide empirically effective decay decay find increase epoch converge within epoch regularization use early metric end strict evaluation metric position exactly orientation goal still challenge error overall sentence benchmark present study first investigate ability intend language report overall accuracy sentence directly use linguistic resource sentence really amount paragraph art employ specialized linguistic semantic multi strategy sentence additional enhance reinforcement path correction chen consequence accuracy vary different aforementione average five previous art stable randomly use evaluation ensemble improve ensemble single multi sentence table ht sentence reach exact model reach fraction reach reach encoder encoder understand encoder experiment directly randomly initialize embedding decoder rely alignment table present demonstrate encode sentence information sequentially help resolve turn encoder utilize focus salient effect training vector unweighted eqn encoder rnn evaluate single sentence execution competitive multi despite work train linguistic resource study currently extension embedding reinforcement add acknowledgment david chen prop pt edu sequence language natural recurrent rnn sentence environment propose salient region alignment help resource e seed benchmark sentence dataset give competitive sentence able understand execute people environment ambiguity amount detail require specialize annotation goal end linguistic knowledge I compositional semantic raw pair able language propose neural long memory encode action suit task temporal machine action use base approach representation achieve sentence prior seed use exhibit stable paragraph perform specialized series study primary conclude direction form calculus weak supervision learn multi inference generative structure include model spatial discriminative compositional factor express correspondence linguistic object location action alternative formulation treat end neural
map subsequently compactly ready next contain advantage apply arbitrary nonempty define expansion sample expansion cf assume coefficient sample sufficient converge coefficient likewise make ensure expansion assertion converge proposition mutually constant assertion converge conclude prove consistent sum turn expansion necessarily formal qualitative end approximation unknown along construct kernel close point appear lead effectively enough work definite uniquely substitute arbitrarily rkh consistent least since state estimator technical assumption e dominate sufficient dependency issue consistency asymptotically arithmetic virtue order estimate parametric gaussians z point variable jointly need value characteristic general expansion think propose applicable dependent interesting field consider jointly causal direct acyclic dag statistic causal direction suppose eq simplicity eq independent copy basic expression multiplication division scalar rkhs independent proxy lead sufficiently estimator range approximation kernel reduce size optimize coefficient rkh analogously I evaluate kernel gaussian rbf kernel bandwidth choose distance distinct depict operation three see sample increase bivariate inference interested identifying cause cause observational datum cause benchmark x square fit degree illustrate next direction score specifically decide rbf heuristic speed adopt map see value rate force range scatter benchmark depict pair benchmark method achieve force developed base approach value propose rkh cause encouraging material remain unify describe hope future le song comment eps stroke acknowledgement mail statistical mathematic mail ac mail reproduce applicable respective distribution probabilistic programming structural derive structure crucial computation carry string determine well operation permit typically composite simple one operation applicable propose build applicable approach nonparametric categorical general pay either input require characteristic mapping associate represent distribution hilbert space generate kernel reduce weighted operation expansion remainder article organize describe analysis conclude limited represent reproduce hilbert attract rkhs briefly start live nonempty experiment kernel equality strictly kernel pca hilbert map positive allow canonical whenever write satisfying reproduce space g string possibility helpful map order one generalize point eq density provide nonnegative map apply kernel take negative normalize moreover would kernel map map symbol borel hold support distribution sometimes distinguish map special sometimes instead retain represent object map summarize result table moment see conclude characteristic universal include moment rv equal equality homogeneity independence latter estimate aside stein construct cf probability requirement use rkh sufficiently bound approximate kernel substitute alternatively method match way road linear algebra question interest far rather conditional update connect lead realization rv induce probability instead distribution value random define operation density belong elementary form spectrum resort operation real world system uncertainty arithmetic measurement connected serial establish arithmetic independent suggest fourier transform people propose cumulative become loose repeat use computation consider numerical arithmetic represent piecewise chebyshev long behave approach scientific g goal another generalize propagation help inference conditional variety probabilistic programming propose map expression operation apply expression show desire property operation rv resort exist complexity result expansion limit benefit three fold first data domain kernel finally density intermediate dimension function take since speak leave I measurable consistent moreover convergence
value positive let cm relaxation supplementary lp relaxation tight df satisfy condition solution lp rr energy bind follow lp responsible marginalization w paragraph relaxation constraint redundant potential order potential equal first potential labeling two mrfs potential reduce marginalization pairwise lp extra lagrangian potential method discuss sec approach dd perform mrf create mrfs semantic mrfs mrfs connectivity group grid value lower bind bundle segmentation dataset reasoning bfgs aggregated bundle remain bfgs author implementation implement dynamic max implementation programming iterate normalize energy energy expansion unary potential code dd subgradient bundle material detailed plot outperform optimization outperform solution energie energy expansion robust semantic segmentation segmentation construct energy unary unary potential piecewise train unary potential unary potential boost grid potential shift segmentation produce segmentation bandwidth colour domain pair colour set model choose optimizer decomposition split horizontal diagonal order potential form subproblem unary potential evenly horizontal within variable potential individual variable bfgs step routine apply image call bound energy bound energy label subproblem aggregate converge obtain sophisticated could potentially intersect intersect equivalent report expansion energy optimality instance find energy model exclude energy global equal energy table report energy median gap analogous running second robust potential cc exp indicator interactive segmentation segmentation task synthetic neighborhood grid unary potential pairwise potential significantly different unconstrained energy linear constraints loop loop solve lp outer lagrangian low lb lp unary potential solve lb dd solve simplex energy constraint conclude converge fast primal comparison energie energy global percentage pixel consistent curve propose mrf comparison exist hypergraph tree require art method potentially high theoretical provide equivalence relaxation acknowledgment kolmogorov valuable discussion anonymous great support microsoft technology additional plot paper segmentation main iteration version minimize pairwise main potential positive elaborate true sec pairwise affect potential retain long submodular minimize cube solve standard lp lagrangian constraint describe follow pseudo apply equal standard lp concave piecewise relaxation minimization specifically assign unary equal set explore experimentally sec low pairwise equal lp minimize set unary correspondingly lp relaxation look lp lp target set feasibility negativity feasibility complete standard express lagrangian consistency constraint add corollary together trick lp versa define set comparison gap dd set sec specifically energie label unary potential potential weight note report table tight trick cccc percentile pairwise energy pairwise potential maximum prove min energy unary potential eq z ff z equality finish fact case potential function sum pd hold sum new dual function contrary point exist label l converge consequently mean number belong segment restrict generality analogously show opposite equal contradict opposite possible contradict feasible contradict form specify statement ph work bayesian supervision currently team sup research interest receive sc ph degrees state university head science school economic method research include computer researcher sup paris mail fr science school economics university e mail address mrf motivate numerous approximate propose submodular unlike energy minimize mrfs take account global property experimentally field combinatorial cut field mrf many apply computer paper one important inference posteriori refer mrf energy inference type combinatorial mrf minimization potential two unary potential set energy np exactly polynomial define energy submodular one pairwise energy potential image representation potential preferred energy potential lagrangian consistency constraint pseudo boolean binary submodular efficiently max min relaxation expression experimentally applicability world rest paper follow sec well present sec analysis sec sec way analogously call cut lagrangian relaxation encoding run expansion generalization minimize nevertheless pairwise w submodular label submodular possible applicable cut rely cut multiple time oppose minimize energy popular bind graph approach problem try agreement solution provide give obtain energy ensure drawback method two type subproblem aware way enforce agreement message pass message pass energy flexibility relaxation lagrangian relaxation define acyclic graph tree low point option submodular problem variable submodular subproblem flow submodular generalize energy potential put subproblem several apply task without create take subproblem relaxation reach agreement advantage depend result take joint denote mapping ci ai number cardinality correspondingly hypergraph accord hypergraph mapping potential convenient notational ci ic unary potential energy notation take indicator minimize number indicate popular approach perform solve formally get eq continuous strict convex relaxation paper dual primal energy r unary polytope number intractable polytope marginalization constraint constraint local polytope call g recent paper mrfs sec potential global version indicator rewrite energy constrain constraint label assign describe way approach potential simultaneously either compact representation linear indicator important special class intensity image pixel separately number node take g lagrangian inequality submodular indicator thus dual piecewise concave thus convex maximize construct approach ascent maximize bundle smooth proximal directly applicable oracle evaluate function compute subgradient flow request computation proximal prox log solve sum explicit conjugate lp min cut aware possibility prove tradeoff optimize face subgradient short bundle bfgs compute value subgradient subgradient optimum solution overall tp ip primal stepsize mrf energy bundle collection compute parameter intend keep current step bundle perform bundle replace suggest choose adaptively serious bundle another size choose step work try default size typically line search bfgs implement hessian maximum mm l select potential min old lagrangian min label old rule potential intuition mm coordinate fig order analytical variable numerically give dual practical issue relaxation framework primal solution subgradient maximize dual aspect heuristic
number simulator deterministic simulation deterministic case worth improve dual abc carlo close small possible tell result possible optimization apart execute communication equally solid blue grey jacobian describe weight stem change proportional length line segment indicate contour statistic help illustrate one stand idea delta peak replace average evaluating arbitrarily inside ball derive posterior small enough jacobian end optimization since assume define remain restrict inside next expansion volume last compute normalization accurate optimization solution distance observation assume reject reject reject difficult mix situation depict trace form surface may intersect manifold intersect q pseudo around ball define manifold volume I crucially optimization sort particle smc use sequential round smc simulator newton smooth random compute sided expense place max simulation round note time lack error bar deviation break notational convention exactly translate pseudo pseudo statistic explain unknown smc use ss ss ss value ess normal ss ess decrease ess remain result ss experiment ss ess dropping drop smc ess ess significantly remain whereas allows effectively switch run expensive fine grain still effective v ie bottom queue plot std sort plot ess ess discrepancy ess work explore control simulator transform procedure simulator parallel quality optimization apply problem allow jacobian computation note high expensive infeasible include ad library incorporate incorporate add expensive method start view pseudo simulator number outcome know simulator prior jacobian ensemble monte parallel handling resource sample procedure validate demand spectrum whether biology tumor cancer research galaxy weather forecast science movement economic physics aim likelihood simulator base model target correct distribution simple rejection process synchronization e inefficient benefit title considerable aim sequential smc particle cascade introduce stream minimal management processor independently trick generation simulator simulator piece simulator inverse jacobian result get value optimization core mcmc free model note simulator reach use alternative randomness relate estimation indirect abc make create development indirect independently manuscript jacobian dividing restrict introduce abc novel evidence correctness primary simulator simulator pseudo treat auxiliary
boundary discuss recall order q indicate ucb average budget rank context time rank event e lemma error detail proof supplementary give ucb agent heterogeneous contextual bandit horizon identify algorithm near optimal multi method ucb ucb regret boundary achieve regret insight design contextual future study system general contextual bandit constraint context exploitation bandit context identical approximation combine ucb expect unknown ucb ucb certain system contextual armed mab exploitation tradeoff contextual bandit observe receive function agent historical potentially motivate example crowdsource sublinear regret contextual bandit context recent computationally efficient however traditional bandit capture characteristic constraint resource action horizon crowdsource pay worker consider work mab arm budget constrain mab observable contextual bandit time agent horizon incur context bandit process total reward case setting possibly shown achieve regret benchmark computationally inefficient resource bandit focus static action unclear extend arrival context address challenge practical cost constraint agent achieve consider contextual bandit still bandit system current context remain theoretically scenario computationally curse regret implement manner total third start unit cost context normalize oracle approximation context expect context capture reward well bad context less agent unless remain budget remain context incur make decision medium context balance expect reward resort specifically bind lp budget constraint static lp suboptimal propose adaptive replace performance use budget near within boundary insight reward note algorithm order reward estimation method expect short systems ucb systems two system set ucb relax assumption system heterogeneous agent statistic coupling achieve unit ucb identical summary regret contextual bandit round accord identical generate condition cost incur action context insight constrain contextual bandit cost cost take observable round reward begin round observe agent round round otherwise agent action neither reward paper horizon end agent run contextual bandit observation reward context reward budget greater compare know include let interested infinity point unit system capture expect reward expect good action context reward suboptimal agent oracle context present recall cost capture reward know k j oracle action context skip depend budget unless verify context arrival general computationally horizon resort approximation constrain linear lp propose bind denote time budget lp threshold verify value view reward consider entire horizon reward hard oracle hard budget horizon budget upper reward oracle later upper reward time propose programming randomization probability instantaneous remain budget replace follow round remain therefore reward take total verify remain remain nothing budget consider replacement draw take action draw white ball budget follow budget symmetry evolution budget property distribution ready investigate bind regret budget unchanged thus stay static change budget stay average budget budget boundary critical threshold achieve good certain cumulative optimality possible decay please refer consider similarly supplementary contextual bandit agent reward still focus know order reward probability short combine ucb ucb contextual bandit take reward reward pair traditional ucb well ucb states action reward long execute property widely armed lemma minor well constrain input horizon remain remain j kt j jj j tb kt kt kt maintain ucb implement define next seem regret
overview run server challenge recent key processing analyze meanwhile problem framework example thousand problem interest become ever organization wikipedia example automate document document classify rare remain despite available available class add imbalance level hierarchy statistical pose challenge new specific major challenge number web repository million account complex relationship wikipedia category different scientific event include semantic indexing answer international challenge series imagenet challenge http www net challenge extreme research microsoft com en people classify web series challenge aim assess hundred thousand hundred thousand multi setting track main corpora wikipedia www www datasets http gr may run dataset get performance rank source http know http www consist indexing indexing removal create indexing description page manually format file example instance sparse format category comma category correspond feature feature internal indexing ignore token map unique year track track mapping use track instance validation train datum file participant free create use file belong category meaning keep participant track file file child file challenge file hierarchy parent depth deep depth parent visit root parent omit allow leaf artificial child track label challenge split track vector type less hierarchy track cycle track medium number stem category track third addition medium text instance process track b track flat prediction include gold wrong gold hand account predict gold way wrong various measure gold flat first negative tn positive tp count many label truly gold label gold fp prediction accuracy precision divide category macro version multi flat implement version micro calculate micro micro follow false positive false negative significance evaluation test p two flat challenge account thorough evaluation challenge run attract world challenge subsequent challenge conference briefly present regard track participant track track flat description polynomial svms online approach centroid description svms online centroid knn bm similarity meta feature prune multinomial naive centroid track track win couple post score one knn top system close participant first track track hierarchical competitive system multi class meta usual hierarchical construct meta extract tree meta thresholding classify top learn hierarchy pruning improve multi class flat competitive multinomial optimization strategy
query produce text wikipedia training feature text author text lda apply assign view feature text inference overcome unbalanced dimensionality image feature text representation quadratic variational text placing without account learn text optimize search variational distance latent produce ranking evaluate curve l query avg sm cm art algorithm match sm correlation fisher paper query drop compare report image query lack input capability introduce make latent scaling introduce switch explicitly provide integrate spike acceleration dimension model spike state modal retrieval structural become important spike ibp prior enable composite process fig process choice provide principled dimension way scale effectiveness real view process share latent manifold determination spike view art apply cross modal task reduce dimensionality dimensional gp nature use enable compact gp various domain gp cell gene gp verification human dimension large overfitte choose dimensionality dimension large covariance negligible therefore dimension length relate automatic determination ard ard limitation threshold hand non kernel like whether slowly decide really drive spike latent allow discard use principled however intractable form marginal close form likelihood efficient switching spike problematic variational switch closely datum simultaneously determine active literature spike monte switch enable representation regression input select nature spike unsupervise code truncate covariance extend learn explicit view determination space amongst different view particular formulation inter dimension assigning decide parameter must ignore space switch unnecessary principled spike introduce counterpart spike gp extension effectiveness dimension new aim latent simplicity th maximize fitting therefore introduce derive variational gp formulation rely induce becomes induce log x marginal jensen w integrated lead lower marginal latent rely call scale e covariance typical scale principle latent usage variable determine latent switch variable control usage dimension wise binary distribution prior marginal eq tractable inference introduce approximation spike variable define variational posterior dimension view posterior representation view consistent operation posterior view view latent posterior fall integrate latent lower correspondingly exactly divide evaluate distribute way demonstrate signal multi dimensional switch latent dataset correspond curve colored variance posterior learn different color reconstruct artificial source evenly interval signal recover st rd transform generate way combine nd spike offer signal latent st nd variance two dimension first explain big difference use match gp perfectly match apply spike generate share recover assignment dimension signal latent recover variance dimension private recover nd signal infer signal significantly true signal infer view parameter st nd switch nd th scale answer quantify digit take mnist digit take image training spike dimensional optimization purely label use latent dimension scale
different reformulate positive semi compressive sense demonstrate choose relaxation provably recover semidefinite quite notably consider choose probability assume random subsequently initialization provably vector number measurement logarithmic modify remove consider restrict demonstrate sub vector recover follow provably show appear ball center explicitly compute minimizer fall global minimizer method global strong actually result sub measurement one semidefinite empirically reformulate manifold exact present broad problem rank generalize matrix relaxation technique derive remain restrict noiseless sub satisfy eigenvalue th moment calculation random vector fu positive aim sign measurement high minimizer finding condition normalize eigenvector eigenvalue iteratively via output rule around global minimizer convexity convexity result sub require namely moment include first main convexity hold sub gaussian refer reader broad measurement thereby establish quantitative strong guarantee state I gaussian measurement fix sample vector covariance probability initialization lie around satisfie remark one entirely tolerance fix accurate want unstable saddle point rest paper convexity result concentration initialization prove produce region finally numerical robustness equivalently semi neighborhood draw initialization assume moment convexity coherence proof quadratic polynomial insight bind region loose provide enough information guarantee stochastic via concentration uniform ensure convexity section lemma refer reader make geometric exist depend norm bernstein improve state indicate unique minimum region however overview begin know consequently would normalize norm produce regime conditioning hold broad sub measurement thereby quantitative convexity parameter fourth parameter x eigenvalue quantify strong guarantee incoherent eigenvalue remain initialize result initialization sake completeness state gaussian eigenvector main extend wide study initialize noise numerical present follow measurement ensemble experiment unit noiseless meta matlab build quasi newton measurement ensemble problem global numerically meta noisy mean zero noise ratio meta relative reconstruction signal random ensemble additive rgb
version domain suppose ideal verify distribution firstly domain marginal secondly prove kind result theorem domain marginal hypothesis defer theorem open pac adaptation one domain learn imply trade direction domain bind improve distribution proved rely triangle inequality detail therefore seem good distribution improvement rely contrary domain imply bind close equation stand disagreement consistency except instead abstract pair obtain finish equation probability negative mr apply inequality find bind leave measure exact r th td proof follow guarantee abstract distribution every proof process theorem appendix rescale result bound theorem apply j sg nh abuse notation classifier j correspond r define mm kl j least choice substitution et de universit st fr universit universit st universit france contribution well hand propose previous average tight adaptation bind classifier show sentiment annotation task generalize adaptation allow domain adaptation make adaptation pac human think education student course make acquire previous learning learning draw strong real task adapt spam filter system poorly another another need tackle framework arise generate differ generate situation learn come refer domain approach weight covariate shift direct exploit labeling transfer source unlabele common unlabele execute source g present source target hypothesis consider behind look preserve good measure easier much related issue loss deduce bind distance divergence value loss generalization kernel situation domain adaptation view multiple trade adaptation divergence marginal exploit different propose take prior majority prefer construct model contribution explore domain adaptation situation sometimes domain family pac focus distribution evaluate divergence disagreement many last estimate disagreement derive domain adaptation averaging provide improve easy thank independent three contrast majority tailor multiple imply quantity pac divergence risk correspond structural domain deal seminal work pac completeness time deviation tailor adaptation pac adaptation section section derive comparison provide review seminal measure domain consider adaptation space sp tx sd tackle challenge target identically b ss objective learn target lead low expect agree respective empirical source disagreement target depend source I view disagreement assign label respect adaptation target impossible solve derive domain adaptation source domain easier differ marginal different function happen take account labeling hypothesis situation target representation marginal close work adaptation divergence hypothesis perform source h distance marginal bind depend disagreement hypothesis quantify marginal last domain act quality adapt vc bind trade complexity symmetric td divergence bind differ disagreement tight equation trade disagreement rademacher hypothesis detect difference perform divergence disagreement essence pac pac introduce succeed tight vote rely set various derive new machine pac orient create adaptive traditionally pac vote hypothesis distribution learner aim find distribution vote classifier risk classifier gibbs draw error disagreement suggest fix numerator denominator great joint pac one sg kl main bayes kl divergence bernoulli handle least every low sg sg interpret risk give close task pac bayes theorem first propose straightforwardly least suggest minimize perform trade minimization complexity nature control distribution theorem become grow describe closely relate isotropic us trade classifier dot restrict gaussian specialized pac classifier vector property prediction gauss finally base theory bayesian completeness conference algorithm supervise work hyperparameter explore choose thank descent vector similarly express classifier regularizer trick substitute recover kernel many extensive achieve accuracy time replace derivative note figure contribution domain theoretically bayesian domain pac adaptation adaptation present classifier disagreement domain adaptation pac derivation relaxed generalization guarantee behind bind equation h disagreement triangle jointly error disagreement easy need disagreement maximize minimize instance family modification contrary pac quantity disagreement divergence posterior simplicity type disagreement prior choice defer straightforwardly set pac bind derive adaptation deferred theorem domain disagreement put indeed grow choice differ term logarithm pac domain disagreement notice marginal p mm g pg therefore sg sg e sg general trade term risk domain marginal expect joint target source accord good adaptation possible deviation target labeling come discussion provide next pac pac source disagreement bayesian domain hypothesis pac domain marginal prior number upper theorem respectively q choice theorem agreement distribution imply negligible pac theorem notice adaptation bind risk divergence justification section design inspire adopt pac theory linear spherical classifier sample negligible minimize minimize recover disagreement sg section equal minimize function descent even task empirical convex name algorithm source hyperparameter algorithm gauss function derivative function evaluate trick allow dual augment space kernel term vector toy sentiment minimize function implement source domain adaptation light library source domain adaptation algorithm try iteratively self library co adaptation look train show fold cross via fold reverse circular source target point logarithm crucial rely validation approach tune similar circular fold fold target parametrize tuple follow firstly label example set unlabele secondly use algorithm reverse finally reverse summarize repeat reverse cross across fold reverse circular domain classical inter accord seven angle angle difficult evaluate algorithm make kernel problem repeat ten table co nice adaptation probably seem situation appear illustrate source maintain source focus behavior domain c svm mm rotation angle green target grey plot correspond source tune highlight adaptation black eq q q popular amazon review review amazon book review follow set equal star keep appear ten time task remain process standard tf weighting correspond task book label evaluate competitive costly increase run clear advantage jointly step tackle adaptation ask whether combine stack denoise autoencoder new denoise autoencoder representation reconstruct reconstruct execute source input value execute hyper select representation representation using execute amazon slightly tf source execute representation sentiment cccc db kb select hyperparameter validate svm result hyperparameter achievable advantageous mix label cross strategy exploratory perform reverse interestingly section analysis consider source
learn mapping space word vocabulary learn share contiguous sentence book try reconstruct sentence I home color indicate share token book use near sentence run inside still copy I sure say I party say turn although start become pressure vision could ram behind far chance stroke head towards house probably answer sharp I say come place piece break reach framework encoder decoder gain lot encoder english translation english encoder decoder pair convnet lstm lstm dynamically attention translation activation decoder identical encoder translation show well lstm conceptually simple rnns model decoder tuple embed part encoder encoder word produce state sentence encode iterate drop subscript gate gate decoder language condition computation update gate gate hide decoder second decoder separate decoder exception vocabulary weight connect computing decoder next analogous computation hide decoder iterate drop subscript denote tuple optimize probability forward backward sentence condition encoder tuple c amplitude neuron encoder vocabulary word rnn vocabulary large w space un regularize matrix map query vocabulary initialize rnn word softmax vocabulary decoder hundred avoid train character capability training learn encoder sentence involve computing compute describe detail train linear extract fine backpropagation skip restrict gain throughout experiment non scope strength representation become induce skip subsequently skip sentence sentence dimensional skip training recurrent initialization recurrent weight initialize batch use roughly also report concatenation vector skip result vector since extent gain trivially skip skip vocabulary rnn encoder purpose skip thought vocabulary vocabulary though skip train skip pre sentence nlp mean rnn lstm skip bi skip combine acc dp skip bi skip skip combine skip semantic metric pearson second microsoft corpus metric autoencoder sentence score related sentence average relate dataset come predefine derive image metric pearson difficulty employ engineering heavily lstm task take completely sentence skip component wise together score setup regression compute derive train obtain exist table result previous remarkable simplicity approach lack highlight skip think suited semantic task dependency lstm dependency expensive collect embed perform lstm table challenging drastically sentence person little look little little look little look drive car car drive drive car stream stream stream person person remove task microsoft sentence predict training consist positive pair sentence component whether sentence semantic two sentence result baseline well publish right skip alone dynamic pooling use recursive pooling skip combine basic competitive incorporate much promising pair fine grain detail signal cccc search ranking gmm skip bi skip skip rank sentence publicly available description annotate consider task annotation search annotation sentence rank reverse retrieve good query development split set image k rank within retrieve vice versa close ground result rank well rnn encoding sentence sentence jointly represent strong rnn baseline compare experiment use skip pairwise input loss skip sentence incorrect image sentence similarity margin model performance development using skip thought sentence get image skip thought representative combine also perform well high image available quantitative commonly evaluate sentence dataset movie mr customer cr opinion dataset skip train classifier top pre define train split tune l mr cr svm paragraph skip bi skip combine skip combine skip nb group properly bag nb skip thought give alternative easy bayes nb improve performance present skip bag baseline sentence sentiment learn unsupervised big skip nb particularly mr new baseline text skip bag train skip remarkably sentence skip property language also generate generation sentence generate previous book read albeit
computational algorithm risk erm overcome acceptable achieve robustness acceptable computation unlabele poor short robust solution construct minimize inside good never maintain parsimonious specify cover practical label analysis present demonstrate effectively maintain label tight label disagreement algorithm substantially superior tractable show substantially well theory key aspect technical analysis appendix address empirically reveal degree effectively erm extensive simulate active streaming call superior array summary figure show fraction dataset test sub different query rate detail regime classifier simplicity relax use respect label example w hx h regret take empirical receive label hide unless query goal label decision pick whenever inverse weight specifically unbiased h importance radius schedule ss unlabele I add increment update epoch epoch long technical always query label epoch compute maintain note consist label radius level consist accord notion measure empirical epoch determine constant state epoch schedule difference come solved consideration epoch query erm time obtain compute essence problem encode generalization maintain specific optimization objective encourage might odd objective encourage query barrier algorithmic constraint importance key ensure later bernstein style constraint estimate measure weight apply example region label importance weight rhs ensure feasibility always regret make satisfy crucial complexity benefit see might force disagreement region one hypothesis include slack implementation alone adequate ensure concentration impose region predict bound albeit biased meaning describe efficiently indeed feasible see consequently make suffer choice compute provide next counterpart crucially rely disagreement erm capture inherent problem recall overall describe section epoch maintain identical amongst critical proof exploit fact label introduce bias favor thereby drop classifier since erm always classifier additionally set h actual imply however corollary start set suppose unlabele set good control disagreement region constant bad disagreement classifier follow epoch defer appendix worth note rate value epoch drop leave reader algorithm query thereby guarantee passive generalization guarantee label complexity favorable begin agnostic quantify extent passive disagreement define intuitively learn good label term set aspect attain difference label close comparison dependence corollary also epoch defer label indeed example match theoretic corollary query recall corollary highlight improvement attain result completely method entirely query disagreement result refined disagreement however completely query oppose use region query rarely even regard illustrate quantify complexity virtue disagreement region classifier predict differently gain single classifier query disagreement everywhere imply finite distinguish let uniform hx rx h h h hx h h problem ideally uninformative however determine query uninformative different see uninformative consider focus query region fix constraint mx region pick rhs consequently satisfie find label sum thing check query baseline algorithm erm operation passive testing schedule still follow efficiently apply adequate solve hx big challenge every constraint bind infinitely primitive available true expectation access true expectation challenge variable difficulty lagrange call erm become clear classifier violate level level barrier notice objective barrier parameter x solve large variable dual ascent access erm lagrange multipli hx algorithm approximately present degree rescaling solve violate reduce call erm constraint q hand risk sample may scale last approximate violate constraint erm appropriate detail primal approximately execute appropriately erm primal epoch p constraint solution iteration vary solve unlabele substitute solve sample original follow expectation replace expectation slack every solve draw streaming collect sample size additive slack intuitively solution since large concentration argument guarantee proof statement finite boundedness replace example query solution initialize importance classifier stream estimate I point p ix iy iy w erm store query may need demand discuss test setup epoch schedule assign new epoch explicit dependence current entry correspond elsewhere explain connection start erm threshold instead erm erm stream without store oracle logistic computing maintain intend weighted update coordinate ascent derive use stream numerator denominator enforce negativity point via online detailed appendix algorithm slight modification maintain estimate query threshold quantity decrease variant current disagreement maintain label batch erm style predict label sub draw exist dataset feature characteristic dataset appendix run evaluate testing goal trade query algorithm select available look set thereby individual detail parameter appendix query performance label pass achievable query rate dominate agnostic except strong reveal difference hardness dataset scale error dataset hyper minimum minimum error quantile dataset par query query high specific dataset relatively example possess level advantage ap right horizontal reveal well vertical reveal typical h setting select optimize query hyperparameter dataset figure optimize cumulative achieve mean rate test competitive extreme much superior rate algorithm vary marker vertical bar th quantile relative error achieve across error comparable query w ap w independently much improvement hyper examine different hyper possibility hyper setting may dominate achieve generalization baseline diverse present broken regret appropriately control constraint term epochs corollary set appropriate inductive claim intuition precise notation introduce prove technical notation z importance define sequence population restrict region expectation importance epoch center around error concern example also term entire biased define regret simplify sometimes shorthand play bias early notation h h adopt convention zero epoch notation biased introduce favor evaluate favorable h key ingredient appropriately control term hold lemma intuitively importance disagreement disagreement well behave highlight natural keep handle lemma hold handle event proposition concentration regret analogous erm epoch concentration epoch epoch proposition prove general version corollary prove give proof theorem corollary follow clearly statement establish case hypothesis establish epoch conclusion inductive event hold epoch lemma intuitively h appendix lemma algebra finally epoch directly second observe empirical furthermore yield manner indeed hold epoch invoke lemma substitute lemma eq substitute yield substitute obtain complete inductive claim almost establish proposition condition complete simply use pick epoch rearrange desire trivially h c last rearrange agnostic active streaming guarantee good generalization label favorable show disagreement complexity special condition additionally interesting highlight structural complexity improvement entire limitation achieve careful defining probability refine datum estimate also extensive online well baseline indeed comprehensive diverse agnostic active knowledge believe reveal characterization disagreement likely fine grain easy active disagreement development number example need ideally solve epoch perhaps important future attractive implementation impractical obtaining close would close acknowledgement author thank helpful initial adaptation martingale adapt exist quantity depend define direct present several threshold define satisfy inequality empirical h imply finally prove schedule summation second inequality second inequality provide proposition prove lemma pick h lemma properly difference epoch bind desire clutter round pair instantaneous ix ix h I associate measurable form martingale adapt accord identify inequality use independent past consider concentration empirical random martingale difference furthermore event choose desire proof proposition inequality control q note q rewrite disagreement simply mh substituting obtain far substitute back cauchy schwarz inequality use assumption eq rewrite bound application cauchy inequality complete proposition lemma
singular value svd singular leave orthonormal leave freedom unit norm freedom constraint unit total degree need completion space unknown completion uniquely pick space minimize affine propose nuclear singular via semi thus become nuclear affine strict minima probabilistic natural element construct sample want entry matrix subject research year application collaborative dimensionality reconstruct correspond low matrix heuristic solve replace nuclear objective observe see th entry entry freedom via uniform sampling contain entry unless almost observe impossible zero space extract value product vector therefore row restrict recover observe eliminate exactly requirement rank incoherent inner recover probability sum leverage observe element recover require incorporate column leverage reconstruct probability entry sum observe exactly nuclear norm also obtain rank incoherent row arbitrarily coherent provable leverage score relax leverage leverage improvement recover finally achieve additive improvement size even sample incoherent briefly notation natural bold scalar entry th component clear transpose respectively trace square product act letter singular op I unique sampling independently define leverage singular denote negative column space orthonormal state main n towards score reconstruct leverage observation exact completion notice subtract relaxation element recover matrix regardless accord match relaxed leverage score observe incoherence two completion well know bind achieve sample dependence show observe recover exactly case consequently entry discuss completion incoherent domain datum adapt incoherent column high observe step nuclear minimization problem recover correctness leverage score algorithm total sample couple exist theorem recover exactly via exactly comparable failure theorem arbitrary recovered leverage reality phase completion knowledge budget entry uniformly score second estimate I heuristic synthetic zero poorly relaxed leverage score replacement probability nuclear relaxed leverage similar rest organize size proof condition intermediate lemma experimental matrix software write sample observe constant entry leverage optimization recover solution say success successful behave close suggest seed collaborative web edu rating movie user rate movie low perform truncation create choose repository text categorization th entry th appear choose remove unit observe spectrum rank figure score close incoherent nature reasonably coherent dataset high power law relax leverage sampling singular value leverage leverage score constant c use relaxed score leverage overall result use relaxed leverage rank via outline unique optimal give closely need notation th span complement span operator similarly onto indicator sum ij road map optimal solution hold op subspace lemma optimality universal prove proposition discuss scheme satisfie proposition recall I e eqn bernoulli I lemma sample hold claim hold follow hold control norm set hold least eq hold construct op apply fail failure write apply hold derive fail recursively lemma failure sum total failure exceed failure lemma call tight depend related failure leverage relax set sampling need leverage score entry independently
remain tree small different reduce poorly poor subset matter trial try simply try forest also instead well reduce performance adaboost adaboost adaboost want create grow forest find adjust tw happen adaboost fail otherwise constant grow update repeat use intuitively weight grow next tree force concentrate case random tree make aggregate unlike aggregate prediction weight median algorithm read document score word score score relative document score multiply word score document divide square statistically different scale test score text frequent text score scale dispersion metric document scale raw transformation raw score deviation scaling show sample total batch score cover country I country corpus remain corpus extraction lda extraction asymmetric tree forest correlation table correlation com ccccc forest rand topics lda poorly correlation high ratio every lda specification coherent influential drive drop score weakly good call automate ad available com ad ad ht range limit english world low middle east ad follow ad summary ht mean std ad house etc generating behind score ad bias raise conservative bias investigate possibility mean test country country dataset political orientation ad two ad difference country std n ad country seem tend check biased toward economic country economic median score ad std error mean reward free market perhaps right positively ad less capture economic state resource country sure whether ad inefficient ad economic policy bias favor market policy circular partly merely ad country ad ad less country get case unite indistinguishable ad united states statistically indistinguishable country country indistinguishable bad ad reason ad one million article word total word go denominator word ad something believe model categorical like analysis extract latent behind house index categorical conclusion house etc fine grain regime surprising grained difference subtle ad ad address ad narrow ad effective already country expert collect create training replicate exist ad ad could incorporate regression variable daily index country text help news day would enough produce pick month period precede date want score nlp de n proceed n allocation state comment g processing code score ad ad article period indice ad enough ad produce semantic analysis combination regression one economic political economic affect country go political concern question researcher assign score house index mixture competition yet none adequate uncertainty rely directly country expert checking box box check score odd increasingly country boost adopt policy rule much minor political operate code must moderate political challenge regime always observe moderate bias test association regard measure house house instead proper house give prevent know say statistically indistinguishable regressor whenever almost conclusion create al ordinal latent measure among house come confidence interval quantile big improvement indistinguishable year year moment write pair diverse regime new actually use create idea article say north contain news article article relate create ad I try work try method regression outperform I I news repository content list american york usa post daily france english internal identify select contain regime particular choose article one tag international security system news database period actual coverage vary source ad cover provide reliably retrieve news union east country think regime news million article total country spurious association proper help prevent high news country occurrence proper noun remove country year news article document matrix try news proper frequent language corpus corpus million adopt case learn word score frequent knowledge period country unified sample sample select extract text tell b appear frequency column entry article country instance france transformation entry multiply number word appear transformation word appear appear whole corpus increase appear lot detail step document rarely news united appear news unique large normalize column transform ready decompose weight step extract principled choose large something topic svd decompose follow singular vector value whose vector conjugate transpose create decompose keeping row call truncate truncate matrix map topic run collection medical result look topic topic mutation heart heart large absolute text diabetes contain weight corpus word also etc topic topic cancer real usually large common top across two topic real corpus extract topic three turn onto document collection article product look ht document document diabetes disease diabetes diabete importantly extract order first column variation third topic topic decompose topic weight expect generate create topic row topic influential improve clear topic change corpus try memory limitation word core truncate work onto topic interpretability get know represent document say motivation allocation lda ng whereas free text gain interpretability word weight topic clear mean unclear tend result topic extract try lda transform text lda value assume generate draw heavily document draw poisson topic continue topic diabetes heart disease cancer ready draw word may diabetes document specific word variable everything else randomness datum generating observe inference algorithm create suitable also though topic document draw topic corpus motivate explain topic score principle ols explain year score respective score predict sample ol run topic may need recursively allocate leave split leaf follow branch branch parameter try point homogeneous use gr non need specify interact like large
nuclear incoherence probability exact sufficiently observe although problem formulate completion rank want upper bind enough challenge practitioner prefer nuclear regularize efficient problem due hand rate linear certain unclear correspondence unknown happen compressive solid theoretical efficient bridge gap theory investigate full like kind compressive classical completion relative summarize result general study good error tighter develop compare small notice never vanish denote nuclear absolute respectively brief mathematical prove obey incoherence convex great highly elegant analyzing give bound bind require assumption simplification completion alternative study matrix also however mention completion completion side completion coherent universal completion name amongst many study completion characterize behavior investigation rank completion follow upper restrictive view least theoretical generic generalization additive ignore investigate contain propose condition relative work high completion subspace subspace multi step algorithm recover matrix unclear formulation follow norm constraint thus restrict convexity incoherence derive upper contrast contain study define two operator eq replacement simplify guarantee nm n eq simplicity factor comparable rely incoherence lemma simplify result bound relative tight tight additive derive interpret describe arbitrarily make necessary utilize theoretical bind relative bind please although require easily completion correspond bind theorem bind become work analysis upon guarantee rank q q optimality base convex analysis main bit two intermediate nm go lemma throughout care except last provide construction partition least subgradient subgradient convexity get form verify complete continue proof three utilize conclusion put get basic mn constant instead ensure later due ta plug substitute eq plug inequality
policy horizon main penalize uniformly respect policy framework penalize completely case detail markov process two armed sharp devoted regret analysis weak limit rescale arm proof go n multi armed arm first independent success occur value sequel generality real define sequence probability ni na distribution otherwise behaviour arm fail keep case success represents spread arm weight decrease case study recall fast success exist wrong ns bad remark competitive game prediction step forecaster choose receive use next arm step seminal reward assess compute action paragraph refer minimax strategy supremum possible general overview minimax several kind replace statistical analysis pseudo quantity bernoulli reward uniformly integer refer first uniform order focus pseudo order lead bind good accord lead conclusion growth absolutely competitive constant take last ns convenient drive ns well ns rely failure use probability arm modify opposite penalize ns carry failure note procedure probability select ns decrease failure decrease whereas probability possess become theoretically uniform pseudo capacity control uniformity define random q introduce play precisely modification increment weak mention role exploration efficient completion penalize procedure ni probability arm fairly arm alternative draw arm penalize state main understand ns regret arm ns page armed stationary generator act stress constant may minimize seem potential minimize r h make comment armed order competitive bind replace theorem regret competitive viewpoint sequel penalize penalize algorithm completely recursive horizon trick section dependent horizon bandit simple handle numerical view arm state result n armed penalize lack generate mean sequence carefully explain penalization illustrate let armed furthermore choice make term obtain figure remark upper sharp penalize satisfy uniform red leading converge result boundedness numerical bandit ucb therein well kl worth phenomenon evolution p penalize color dash colored penalize algorithm multi armed situation describe pointwise penalize armed bandit provide sharp weak establishe toward unique act compactly limit markov normalization study argument spirit existence thesis case may write one resp exponentially jump equal positivity start well jump could easily represent h exact trajectory drive right pointed depend relation jump unique invariant existence uniqueness wasserstein uniqueness show strongly ergodic left limit ergodicity positivity distance wasserstein variation state wasserstein distance initial law bandit drive p recursion upper exponent open appear drive distribution almost generator start one dimensional deduce build wasserstein couple bring path sufficiently wasserstein stick really intrinsic explicit exponent particular property different integer binomial lead hx formulation exhibit contraction x h contraction useful soon sequel remark recursion r previous n aim apply lemma defer section argument deduce key need section bind application careful inspection increment decomposition satisfy thus competitive regret choice careful inspection lead deduce remark multiplication expectation sum x p n also r r r controls eq since reach maximal function ip z sharp reader keep mind penalize need third polynomial way careful coefficient lead fulfil remark soon possess consequently unique simplify negative computation fulfil idea use sharp tt p c integral conclude consider power increment z plugging series yield eq lead regret sketch variant remain rr adaptation want argument first satisfied remark q soon rough estimation yield previous fulfil exist resp deduce result proposition penalization bandit multi permit ix main increment v n n x j c ode method ode possesse number equilibrium identify equation equation straightforward recursion equilibrium discriminate decide stability lyapunov close vx j x soon unstable arm increment consequence true toward start martingale increment event toeplitz deduce put together last conclusion boundedness ii argument line prove generator martingale invariant detail rest let differentiable generator sake clarity iy p rewrite p iy I fy fy n fy iy f iy f iy iy n f approximation iy iy fy ix ix ix us decompose equation part f iy iy iy behaviour iy iy iy ig c iy g c n g iy I iy iy end trajectory bandit armed ergodicity generator particular us convention relation control exploit obtain suitable possesse position jump follow jump jump procedure generator symmetric invariant drive exchange act immediate check use imply integration consequence inequality result argument base path close wasserstein try jump path couple establish couple bx xt yx xx aim build sharp triple bt x st naturally law independent pt deduce consequence coupling denote te bt deduce use decrease conclude plug ergodicity w distance start idea wasserstein couple stick alternative b x wasserstein preserve every b bx try deduce moment small optimization e check author numerous motivating n introduce bandit competitive point result competitive bound precisely penalization modification penalize make explicit exist penalize convergence process suitable finally multi process ns bandit type algorithms recursive markovian seminal bandit necessary study slot arm play arm none aware want design strategy linear define represent select
remainder section mlp write r b rl l n k activation similarity form lin fix particular radial rbf amount associate r mlp unit space similarity fix space select equivalent elaborate suffice mlp mlp constant r z x z conv sim implement layer process processing color mlp describe extension locality sharing focus layer enhance context convolution field extension processing accordingly refer across incoming successively stack coherent bank summarize follow principle mlp incoming map summarize via fig location similarity template rl classify rule ij z patch sec relate underlie base patch word patch extension maintain patch kernel addition operator density sec capture desirable coordinate seek linearly transform independent refer literature independent would input multiply measure similarity weight template rise dimensionality reduction matrix low component produce dimension whitening follow conv sim figure dimensional filter match template produce similarity output one similarity conv sim structure similarity sec conv particular filter perform whiten intend construct sec whiten conv sim arbitrarily deep start account sim depth network general sim similarity income weighted template similarity map max use classify final final local classification class conv sim template conv conv sim follow briefly describe pre layer conv sim filter template property scheme unsupervise ii rise channel conv sim layer forward define suffice consider conv recall conv sim linear transformation reduce input measurement accordingly filter turn initialization weight input template weight define define gaussian prior coordinate nonetheless regularization art recognition challenge hand enable evaluate architecture color partition category hold learn implementation toolbox code near future softmax nesterov acceleration momentum weight rate least choice consistently momentum epoch mostly initialize naturally estimation experiment similarity conv convnet choose design convnet accordance layer convolutional relu max pooling align spatial size relatively whiten compare vary run convnet validation accuracies convnet similarity plot operation classify computational budget comparable convnet fall behind publicly compact convnet recent deal none choose cifar convnet network follow relu pooling dense relu layer score ccccc convnet layer convnet cifar accuracy classify learn compare convnet general outline maximize alignment convnet channel pooling summarize convnet cut see accurate play contrast network specify much problem architecture expressive burden explore reach ccccc maxout c supervise net art cifar augmentation exclude comparison channel layer field average case pool window augmentation nature input conv sim datum augmentation rescaling improve orthogonal convnet distinction simple architecture comparison reach augmentation art check extremely compact three dropout multiplicative leave parameter classify inherent outperform compare call generalize architecture drive operator product non capability generalization interesting architecture operator neuron incorporate locality sharing realize type simple feature exponential gaussian include special dynamically generalize multiple equip argue abstraction trait mobile application cifar validate concern thus abstraction advantage architecture endow scheme unlabele besides aid determine channel hide pattern determine unlabeled capability probabilistic unsupervised also plan evaluate intel grant definition lemma proposition university university university deep architecture generalize convolutional architecture call drive similarity inner mean exp neuron space realize simple setting space powerful include case even dynamically learn learn contain abstraction convnet enhance impose mobile empirical resource also concern vision speech domain train end rely manually feature introduce preserve effectiveness inner lies convnet inner control conventional argue design abstraction mobile approximation high level abstraction generalize neural role activation capability beyond conduct cifar accuracy perform complexity limit feed connect comprise operator layer neuron activation mlp forward parameter bias neuron two function template r map linear similarity note unlike mlp
penalize neighbor voting proof need essentially continue job establish follow consider end event op cd lp op b op op op op op theorem term right side lemma appropriate exercise corollary proposition false chapter claim corollary lemma exercise proposition section analysis guarantee procedure theoretic limit space computationally proportion model regularity two stage penalize consistent apply assignment achieve competitive numerical cluster analysis spectral clustering become topic computer science observe among subject computer people instance underlying process belong algorithmic great advance make threshold among effort state art solution yet reach comparable problem know limit computationally major present network propose stochastic provable statistical optimality describe sbm adjacency matrix sbm community zeros bernoulli label node respectively connect community refer proportion wrong permutation shall break regime proportion justified physics recent possibly size established ensure misclassification strong equal later arguably intermediate vanishing grow usually call consistency literature weak strong consistency network go among various way study vanish refinement guarantee important devote investigation later tackle em another likelihood mle relaxation indeed achieve definite recently zhang establish misclassification sbm weak form size minimum enyi order precise statement achieve computationally none tractable likelihood base error match exponent lie computationally feasible provably misclassification proportion establish adaptively weak cover size regime addition compute even node bind match misclassification exist boundary weakly condition strongly necessary sufficient could even polynomial word enjoy statistical core refinement detection estimation initial certain refinement able improve high separately optimization step completely drive hence penalty play ensure community probability cluster normalize variant satisfy need subsequent refinement scheme consider fashion essence shall local stage desire also localization idea play lead problem example phase retrieval high dimensional closely relate linear instance therein refinement method literature provably achieve optimal wide configuration organize present method demonstrate simulate section discussion investigation defer notation frobenius usual probability determine context independent notice may change line problem two refinement shall node wise neighbor voting discuss several initialization cluster tailor current stochastic completely symmetric label sbm community nearly equal size assume paper grow assume parameter community connection bound connection proposition throughout rest induce community structure label permutation symbol therefore error misclassification define stand permutation main refinement community penalize combinatorial intractable wise know reduce quantity neighbor first node connection advance label apply exclude submatrix column remove htb nk neighbor ij j define consensus submatrix row able cluster node category present description refinement step first basis obtain give assignment assign node except penalty add connectivity equal way connectivity assignment voting rhs count penalty vector basic step community assignment determine aim consensus look consensus possibly assignment truth algorithms note apply cluster speak spectral clustering call unnormalized spectral normalize introduce study bound sparse adjacency node replace argue remove particular denote unnormalized clustering normalize spectral cluster another important popular choice mean establish spectral community close look small proportional setting lead inferior address inspire center population kt u last algorithm spectral consistency state property govern critical quantity enyi bernoulli probability throughout p hellinger distance distribution space tend parameter essentially community community stage misclassification proportion essence lie long refinement constant suppose sequence addition conclusion continue achieved reduce simply consistent rate misclassification converge give wrong community misclassification proportion least space extra estimating connectivity adaptive estimation without directly check initialization step either conclusion improve require different refinement state suppose compare condition condition sufficient weak consistency follow characterize spectral sufficiently sufficiently q conclusion assume regularization dense dense regime moreover due conjecture bind theorem quantity step full three different dense community dense community cluster lead report set base draw achieve simplified algorithm obtain different refine simplified initialization thus simulation similar consider precise refer reader generate node community consist four spectral achieve refinement regardless reduce misclassifie around use remove discussion sensitive recall average mis standard respectively hence community great misclassifie different initialization reduce ht version much stochastic block simulate around misclassifie initialization either initialization reduce misclassifie simulation four initialization consider agreement theoretical property political connect contain large component pre conservative naturally panel figure likely node group panel node group political summarize simplify dataset average removal node initialization misclassifie misclassifie except initialization refinement na na misclassifie stand application initialization method application simplify refinement multiple misclassifie node keep misclassification misclassifie refinement misclassifie depend initialization three misclassifie node converge within include due inferior iteration art method score achieve comparable worth note correct fit sbm presence spectral design semi definite method lead political competitive well important issue relate theory misclassification bound recall suppose addition hold vanish exponent replace drive arbitrarily define neighbor voting need truncate theorem obtain initialize hold parameter replace achieve mild misclassification weak consistency long regardless behavior ensure regardless behavior strong comparable consistency algorithm case fix without regime misclassification paper multipli exponent comparison result much broad last provably strong consistency grow algorithm initialize slightly use initialization initialize sufficiently hold parameter space point key large eq space replace instead large replace condition refinement version description similar simulate iterative simplify keep drive misclassifie great interest establish simplify iterative certain think interesting knowledge date drive research propose block validation information ratio paper
mutual fix become variable update conditional small making average actually marginal average similar backward note solve empirically convexity calculate triplet look bind mutual observable pair apply selection mnist noisy mnist empirically backward improve convert thresholding pick digit datum conduct initialization em close drop poor component mixed digit mix proportion digit training set digit refine prevent able regard regularization improvement active digit upper part look pixel pixel contrary pixel bottom strong two firstly digit style bottom style want hard get stagewise marginal datum model marginal calculate mutual add active component mix proportion component eq one step usual diagonal whose th active decide add need stagewise em stop criterion q mutual q mention stagewise big forward decrease empirically find empirical mutual start split e active local convexity empirically converge local digit result stagewise job poor job stagewise local experiment dataset stagewise perfectly mutual drop zero sufficient stagewise em mnist mnist bottom mnist stagewise em learn digit lot digit original em experimentally converge local maximum maximum mask small stagewise em dramatically converge quickly stagewise em open component experiment split encourage science new develop bernoulli information theoretic propose backward active guide em stagewise em analogous stagewise irrelevant mnist approach diverse biology tool generalized categorical theoretically identifiable condition learn mixture develop em continuous gmm well identify parameter information problem backward strong interaction show robustness importantly tn behave tend weak much pattern contribute eliminate question order eliminate selection forward
belief sample belief guide reward convergence estimation observable probabilistic infinite first chapter represent mix problem simulate anneal schedule metropolis comparison figure evaluation annealing despite ability tune anneal schedule finite infinite continuous case may boundary randomize span beyond estimation base powerful general u fa david frank ac uk search posteriori program ascent probabilistic mutually dependent countable random map search compare map artificial intelligence planning reasoning utility recommendation map problem wide artificial intelligence representation graphical typically represent powerful model plan program represent expressive probabilistic separate allow inference restriction sampling finding map would scheme posterior single joint multi contribute optimize say optimize simple setting bayesian monte countable dependency program program construct draw value define probability program probabilistic property initially argument return argument return without upon return terminate repeatedly implicitly denote deterministic trace proportional discover implementation programming usually algorithm valid program probability finding maximize inference paper map propose extend marginal exactly advanced probabilistic anneal simulated annealing sa constitute approach anneal gradually change analogy physical anneal acceptance course sa fail annealing sa program bayesian monte monte unlike information assignment know planning game playing planning root certain number must determine simultaneously program often finite countable variable variant mix type open use introduce way independently search estimate previous go quality solution algorithm compute weight trace trace previous line belief log log ig k weight discover improve select domain random high randomize thompson many context scheme maintain belief reward select belief belief sample extend randomized domain maximize know type choice unified maintain reward guess reward belief choice choice add choice base
relate form control issue method assessment parsimonious logistic introduce hasting compare discuss limitation report database individual logistic article binary specifically suffer explanatory indicate presence absence drug drug consumption define influence drug coefficient zero coefficient group induce belong observation write q obviously log event profile appendix solve likelihood mle also mle eq absence presence drug consumption suggest account compete set model high uniformity hold distribution laplace logarithm integrate compete model huge exhaustive compute bic therefore hasting perform unique stationary propose move neighbourhood copy iteration current element sample mix candidate accept pr return maximize different algorithm visit r present compare comment method evaluate performance contain four event study nine case control across interest support clinical report ccccc positive extract consumption mention drug among control negative status four study htp choose method report odd ratio reporting statistic drug pair cccc ref mid result apply regression obtain ten fold misclassification permit signal intercept zero difficulty roughly weak event event bic compete event dimension ccccc negative detect compete ht ns cv good control find lasso explain misclassification constrain obtain detect control worse negative probably relate detect could cpu number times nb signal control control minute require realization profile ccccc signal control strongly reduce appendix allow list detect obtain poor penalty determine misclassification event moreover deduce control black relate criterion indicate dot hard well present penalty result optimize figure reality nature seem calibration practice individual reporting database avoid drawback co effect lead throughout parsimonious regression lasso challenge calibration reference signal detect method event show present evolution event exhaustive compute compete selection whole database several day propose investigate drug reducing grateful il support drug obviously coordinate belong many profile profile eq drug ccccc event ab positive bb positive kx l bc ac positive ac ac aa aa bb aa positive aa ad positive ab c ab ba ax ec bc ac ac j ac aa positive ab unknown ba unknown ba aa c db bf n ba aa ab aa unknown em phone event detect drug use onto contingency association method regression penalty approach limit drawback influence logistic regression selection metropolis hasting bic penalty threshold database approach reference drug association propose
neighbor f offline help offline fast process occur batch ms completion time adapt rate cause benefit particularly evolve mention initial gibbs converge allow impose computational load propose online sequential hmm capable batch incremental main unsupervised adaptation batch accomplish dynamically balance batch memory far sequential adaptation hdp hmm include evolutionary thereby test segmentation accuracy improvement thank hdp solution attention literature intervention candidate streaming application stationary load significantly balance effect posterior inference parameter observation accumulate summary online parameter q accumulate however control versus impact accumulate conjugacy conjugate prior derive posterior investigate gamma conjugate iw rate inverse wishart derive hyper canonical parameter simplify try thank remove ideally affect hyper initial dependent conjugate scale parameter conjugate derive posterior general conjugate conjugacy expand derive hyper parameter presence single extend case observation proportional appendix explore mean inverse wishart unchanged scale draw scale whereas former tending nee sequential context daily surveillance stock flow address focus pre principled capable class delay streaming context far enhance responsible balancing extent streaming observation evolve remarkable evolutionary sequence unseen segmentation attract domain segment classify finance understanding annotation human computer interaction date main slide window hmm structural covering spectrum discriminative margin increasingly dataset challenge adaptation dynamic remain address limitation accommodate model hierarchical prior hide hmm exploit adaptation adaptive joint incremental set hdp hmm sequential ii buffer tune class unseen continue entire iii bootstrap supervised manner operation process stream obviously life problem learn rate biased adapt pattern evolve rest present hdp expand compare benchmark amongst hierarchical hdp principled nonparametric typically inference variety hmm data state decode dynamically find domain varied problem approach entire learning system obviously stream response demand dedicate processing mini batch inspire recursive study online refer repeatedly online optimisation formal seminal work monitor stream class comprehensive time appear adaptation adaptation drift knowledge periodic ad hoc costly absence expert balance problem assign likelihood complex choice dependent dynamic exponential decay recently step adaptation introduce novel statistic supervision slightly evolve significantly life continuous follow time tackle thereby dynamically dirichlet think distribution infinite control base measurable location repeatedly establish name hdp process similar top level dirichlet process various element application continuous take parameter wishart yet hierarchical hdp property diverse collection book genetic marker population hdp switch markov hdp interpretation step current property explain hdp observation sequential hdp pz group coincide add hdp worth hdp tendency segment unbounded add towards change yet brevity hdp extension estimate distribution hyper derive extensive mainly gibbs sample variational simple significant slowly remain local minima mixing variational usually fast derivation analytical suffer low approximation initial rapid accurate indicator meaningful correspondence ground truth label classification obvious hdp correspondence hamming ground truth frame adaptive online inference extent annotation comprehensive costly annotation brief variable truth reach conclusion process batch batch alternative batch pass stream fy figure hdp emission transition mean posterior next adaptation imply accumulate non nature carry buffer unbounded latent dirichlet unbounded buffer extend memory requirement process respective batch propose system learn note responsible set prior parameter current accumulated adapting likelihood worth note batch compare play relative accordingly pseudo respective distribution belong exponential bayes property canonical accordingly rate bold font notation convert canonical canonical prior ultimately please proportional purpose thank coefficient merged exponent scale canonical parameter sample hmm hdp dirichlet member unify form section learn text mention early exponent merge hyper convert impact ultimately convert transformation learn conjugate derive prior sufficient gamma however wishart gamma parameter derive posteriori hyper parameter video datum importantly category hdp context introduce segmentation recall detect boundary segment regard correctly detect interval frame ground truth percent segment additional boundary positive detect action ideal report table colour instance estimate plot label provide plot view colour univariate around dirichlet matrix generative hdp hmm replicate absence please configuration run sequence train leave batch size time unit batch propose online hdp segment percent probe far add increase considerable overlap despite noise significantly accurate percent level repeat datum percentage undesirable row accuracy figure term precision infer l cardinality ex stationary noisy evolutionary evolutionary evolutionary combine evolutionary bottom half adapt truth class learn colour combine challenge slight decrease extra nevertheless considerable performance visible synthetic evolve distribution involve shift unseen class deviation examine appearing shift generation demonstrate distributional comparison drop undesirable class shift yet around class colour learn consistently batch class percent mostly thank adjust adaptation rate highly percentage combination new need distinguished fold mode misclassifie ii merge shifted experiment close challenging distribution give hdp prove highly percent cardinality thank mechanism hdp perturb observe evolutionary increase keep prevent drift evolve around zero absence hdp overall undesirable drift exist considerable allow evolve ultimately merge neighbor single section video action video actor action sequence segmentation way action frame feature centroid centroid actor contour c cardinality hmm online offline hdp hmm ex offline avg c hdp remarkable qualitative segmentation offline variant represent batch run comparison offline study yet operate class fix trend due stationary test addition accuracy offline consist activity contain annotate separately actor leave relevant reach something reach subtle even action boundary even annotation segmentation sequence provide leave validation test run typical one order run actor frequency occurrence compare show frame similar sequence vs minor frame difference infer visual colour segment distinction leave subtle back blue figure freedom number explain sequence emission change transition adapt due remarkable mainly one phenomena object show sequence hdp consistent future model inherent
problem work direction become compute processing consensus consensus admm additionally fitting machine domain variation admm variant decentralize inexact subproblem update solve ax enable begin lagrange multiplier generate ax k ax ax stepsize ix global solves update f share global finally lagrange multiplier force iteration regularize problem jx dx b scalar lasso consensus solver ix tn consensus form compute admm single entire equivalent td tb solve server store single aggregate server compute cloud use admm forward split latter transpose exploit complex dataset consensus subset aim solve remove yield fy dx step solution represent proximal decomposable minimization line simple solve square exploit transpose reduction decompose iy server place datum gram even far converge central iy ix ix ix classifier mapping kx admm proximal optimization solution compute form solution solve hinge give proximal simply note svm form support solver require variable fitting reduced setting compute happen column row fortunately handle dual f ball otherwise formation server rather admm general result guarantee rate rate iterate admm multiplier thus iterate primal dual feasibility decrease formulation iterate satisfy optimal feasibility large ask optimality go lipschitz constant global constant spectral begin write optimality ty optimality equivalently fy dx fy k know result note satisfie rate convex accelerate convergence possible reduction consensus synthetic transpose consensus optimization resource center range extremely implement transpose consensus consensus routine author use stepsize parameter substantially tune tune scale stepsize make solver regression svm warm start subproblem limit bfgs method warm accelerate transpose reduce require square accomplish backward splitting solve svm solver well warm solve solver use feature core b datum transpose admm minute band geometric every star feature interaction require space run decrease show transpose converge consensus experiment experiment store different core node little transpose method far efficient load report core advantage transpose confirm transpose powerful realistic oppose used core computing core core classify consensus admm terminate core transpose reduction consensus transpose consensus nearly amount problem transpose grow consensus transpose whereas require transpose gram send consensus local particularly overall solve short note however total computation time shorter transpose subproblem entire corpus consensus portion distribute apparent heterogeneous across datum different homogeneous consensus method datum across problem differ tendency solution consensus consensus take heterogeneous transpose transpose solve entire insensitive heterogeneity transpose reduction figure datum admm tradeoff node solve problem consensus admm require despite transpose highly dramatically server send consensus consensus admm solve node regression svm consensus contrast admm close transpose reduction stay consensus admm communication consensus terminate time especially across node inner overhead admm cause call algorithm allow stay naturally transpose model problem global square distribute consensus distribute across transpose particularly advantageous consensus solve consensus apparent original put form svm solve approach attack dual advantageous act coordinate efficiently approach popular powerful consensus solve method inspire implementation iteration solver bias consensus central server treat differently warm accelerate outperform problem core processor solve feature exclude oppose sub svm dimension processor average second solver core per core total corpus truncate transpose second computation transpose second second total consensus core day day day day day day day day day day day day day day day day day day day day day day f day day
connect represent connect distribute engine computation iterative adapt factorize force graph oppose automate partitioning factorize improve balance assign also minimize communication edge partition undesirable cut apply vertex span multiple partition minimize copy vertex definition directly cause inter detailed method uniformly onto compute vertex add node master vertex add vertex induce computing vertex respect master node master vertex receive vertex update master master integrate implement computation graph model edge cut node master vertex overhead incur message master replica reduce reduce communication overhead total hold since replica provide efficient balanced complicated evaluate diverse include light field hyper face illumination study behavior consist select atom light field array collect light patch ii consist atom light light consist produce band video dictionary patch image produce image illumination addition decompose level decomposition light cpu processor ram light field large node machine amazon ec core two intel ram per synthetic evaluate computing core processor ram per level abstraction enable implement architecture map efficiently accelerate use framework matrix library compressed format use system distribute factorize core cluster dotted scale behavior almost linearly minimization utility denoise employ vector approximation signal zero select make signal lie certain classification show fista full gram fista square vary fista remove decomposition obtain full gram zero zero light light field ii capture slightly viewpoint combine enable view represent observer position device trade capture light result image resolution resolution complete light collect dictionary field different fista decompose tailor dataset regular dense peak ratio ratio maximum recommend db fista norm fista fista runtime achieve order magnitude fast compare decompose reach comparison achieve run fista decompose db eigenvalue relative runtime decompose normalize evaluate power light various power error expect accumulate decompose versus significant improvement power finally propose synthetic decompose advantage regular format efficiency sparsity model zero would overhead base vary density decrease degradation worse overhead represent number scale processor processor single processor inter purpose scale processor gap large processor baseline depend specification select systematic future b processor introduce distribute apply iterative framework dataset dependency component underlie platform load balance significantly communication method demonstrate significant offline decompose subsequent computation overhead example light section core complete less overhead justify consider patch use light matrix ask operate zero store advantage communication approximately diagonal reduce communication communication proportional difference large general specification objective complexity massive dataset decomposition scalable subset svd require costly create scalable decomposition introduce successfully implement create decomposition execution densely lemma conjecture platform aware framework dataset introduce aware end execution massive iterative platform scalable mapping enable arithmetic platform message pass incur update available resource subspace flow iterative trade level base facilitate automate perform optimize power dataset amazon ec usage execution compare prior many modern prominent algorithm application belief propagation achieve matrix multiplication involve gram single update become highly challenging communication number distribute iterative adopt parallel run parallel gain communication partition effective movement parallelism accelerate machine readily exhibit non format store addition infeasible exist datum dense dependency wide field medical image problem find densely dependent execution broad apparent low lie union property overhead impractical dataset transformation analysis critical system execution memory usage overhead accelerate matrix multiplication require dense structured rewrite far data automate method partition decompose factor within bind computational depending decompose iterative model base pass vertex write programming develop computation decompose write amazon service utilize available explicit contribution domain specific knowledge extraction dependency structure hardware resource scalable onto contain zero result dependency systematic way tune desire application efficient model partition decompose datum aware demonstrate magnitude domain specific use provide well rank extracting improve learn setting span powerful approximation kk kk singular truncate analysis seek subspace approximate least leave provide find good rank sparse dictionary dl apply large column independently batch omp zero independently store small column write property dataset predict accuracy factor decomposition impact predict decomposition exhibit lie exactly linearly low characterize decrease factorization select column difference good rank decrease exponentially another gain low illumination structure signal low subspace union effectively bound sparsity sparsity column independent number zero increasing control algorithm guarantee select span ambient exactly subspace approximately low introduce number far increase discuss associate introduce naturally control also achieve question specific application connection error gram accuracy include exploit aim generic approach specify introduce iteratively compact decomposition already establish decomposition framework learn tune map result decomposition decomposition resource large value resource
distribution response regression see random cdf without z diag diag case special design subset due another kind rank sampling perfect use used error subset design matrix stochastic f likelihood give r n rx follow sampling design condition rx ne g n r r rx rx rx content counterpart analytical numerical tables rx rx negative proof theorem effect member counterpart value report table comprise replication table effect unknown content size content content compare counterpart first calculate content sample fix size perfect different ranking scenario value order match scale provide value element size efficiency simulate replication apparent key parameter observe design moderately design parameter increase rank effect model propose design design clutter covariate compute probability estimate compute design carlo comprising table explore error content sample mixture exponential handle calculate counterpart ex ex logistic uncertainty structure aspect include shannon nevertheless worth play role inferential design likelihood ml cs ex ex ex exponential ex ex ex ex ex content ex ex ex ex exponential ex ex ex content ex ex information propose aspect concept engineer shannon entropy quantify quantify uncertainty inherent technology censor datum testing therein shannon entropy design ranking perfect associate integral shannon quantitative information technology computer practice shannon sx dx complete enyi data r enyi entropy enyi entropy include enyi engineering etc enyi investigation enyi eq pdf sx sx dx u sx complete kullback leibler measure quantify kl measure kf dt quantify instead rank design underlie design denote compare sampling design one interpret hypothesis within dy fy dy l sample content perfect informative observation population simple set sampling extra information measure enyi perfect interest suggest test seem goodness investigation criterion appeal university fellowship unbalance unbalanced unbalanced size sub group cycle th cycle say unit end observation unbalance unbalanced cycle subset second cycle respectively unit subset ccccc present unbalanced continuous pdf q pdf th statistics latent iv iv sum I iy unbalanced design u subset fy obtain unbalanced counterpart ex ex ex let distribution unbalanced n ie I unbalanced I unbalance counterpart size follow example field research university department statistics mb abstract different environmental study superior fisher compare rank counterpart uncertainty enyi kullback leibler kl discrimination several subject phrase shannon kullback leibler rank sample underlie sampling fairly accurately without actual measurement little measurement costly unit rank environmental estimating stream area management association exposure cancer stock abundance previous rank initial sample take unit call perfect small unit rank small ask rank difficult unit particularly rank consequently partially rank order aim burden require flexibility unit subset rank subset observation collect mean propose statistical selection measurement basic post information formal perfect give joint l u vector rr furthermore marginal easily give first analytic size give modelling involve play theory behaviour maximum likelihood rao regularity calculate derivative function matrix size indicate negative
measurement contain allow compare classification bic compare body diabetes follow name component clearly perform compare mixture family body fit heavy eight gender heavy tail component mixture diabetes model classification select close also imply shape mixture four freedom four select model freedom despite differ greatly respectively ask comment parameter fit restrict three ari ari model overall real note bic pick datum set commonly bank fit ari mixture expand model improve procedure propose previously tail tail suffer datum often show gaussian mixture student handle heavy light tailed component allow thin flat well purpose range eight eigen outlier restrict exponential become enable fit estimation mixture skew future suit asymmetric lastly mixture power may high outlier work grant engineering award research density scale valid see appendix different present comparison estimation estimate parameter hold initialize log heavily determination convergence point axis procedure plot dimensional conduct extensive equivalent scale concavity exponential log q derivative fix update get jacobian p j analogous ig context ig ig log write calculate iteration log update form newton update g ig alternatively implement function obtain closed update utilize increase accelerate search orthogonal manifold construct surrogate employ maximized step list initialize depend check go back constrain equal group alternatively implement involve update base ig refer construct use support hyperplane maximize lead g construct simplify ig ig ig ig ig ig ig ig take eq q ig ig ig ig ig accelerate orthogonal orthonormal tangent objective reasonably decrease ig q unconstrained descent tangent hence smooth curve move qr decomposition herein denote ig ig ig g ig g surrogate q maximize obtain update identity g g ig ig two depend use ig ig ig q ig ig ig derivative obtain q g ig ig ig ig ig accelerate orthogonal manifold function minimize unconstrained gradient show space refer isotropic derivative hence recall scale refer yield detail group ig ig ig update decomposition ig ig ig weight less follow proceed similar ig ig ig ig ig ig minimize matrix eigenvalue idea ig ideas ig ig ig ig schwarz commonly development extensively bic maximize likelihood size put observation otherwise iterative initial need em poor anneal pick run multiple start initialize random mean cluster constrain degree freedom mixture sometimes constrain acceleration commonly progress criterion stop result acceleration iteration herein adjust rand determine classification group label ari rand chance calculate ari perfect agreement ari random thorough diabetes class name mm mm multivariate fit skewness receive vary tail parsimonious eigen maximization lastly model benchmark popular investigate heterogeneity classification refer advance mixture popular presence become tackle weight deal skewness lin mode herein utilize base generalized parameter kind characterize peak heavy tail peak thin tail quite flexible furthermore difficulty estimate shape yet distribution parameter none previously geodesic unconstrained newton focus special impose model term decomposition previously family five see log negative infinite herein guarantee monotonicity make accelerate line manifold estimation wide family weight mixture toy suggest random definite identifiability issue give determine special distribution covariance multidimensional denote distribution scale th mixture previously identifiable practice component geometrically diagonal matrix proportional orthogonal eigenvector order eigenvalue equal variable parsimonious eight parsimonious option constrain structure natural extension structure structure group denote eigen spherical align axis align equal pp g p g algorithm complete likelihood maximize conditional replace em maximization computationally expensive step rather maximize expect give numerical schwarz criterion acceleration adjust rand assessment detail appendix compare simulate modify package use program utilize work dimension metropolis rule easily refer bic family bic select ari family median similarly ari even select ari select range value scatter mean give ht deviation parameter family success overall distribution dimensional common scale generate dimensional generating follow report frobenius norm bias clearly generate simulation investigate parameter estimation perfect ht lrr family component compare family mixture sample binomial zero component generate scale perform run five select component low size seem tailed component cluster unique group similarly time generate family heavy due fact clearly separate mixture model mean ari similarly
track display improvement hold network hold task similarly fix consistently suggest amount task add hold apply number type white area bar smoothed represent useful hold datum growth curve learn growth curve hold fold validation note many bad baseline initialize dataset never positive negative transfer stronger train large well average auc absence include training curve experimental across hold run randomly result section benefit explain consider question active biological target context imply contain inactive inactive inactive reason active physical mechanism hence similarity plot call occurrence compound dataset compound coordinate odds auc eq dataset single odd reduce auc baseline discuss exclude moderate portion effect determine likely benefit many collection exclude correlation improvement improvement exclude qualitatively gain suggest framework datum overlap odd improvement vs sided unique improvement sign target give confidence suggest unlikely affected target collection investigate virtual screening collection significant explore aspect performance still task introduce additional contain large amount observe effect strong investigate possible active moderately correlate improvement biological accurately model target efficacy availability amount critical possess private measurement argument increase sharing benefit maximize achievable architecture algorithmic publish deep virtual aware comparable metric field direction use realize target another improve unsupervised explore chemical deep offer drug discovery process remain deep couple field field optimistic acknowledgment support foundation fellowship acknowledge support nsf gm nsf american recovery act architecture drug discovery architecture public source dataset measurement across target aspect study network accuracy significantly method improve task add contribute significantly improvement sharing innovation drug disease challenge traditionally drug year move start rate suitable identify first target million drug like attractive automate virtual screening attempt replace throughput screen virtual machine learning apply virtual supervise target virtual screening active special handling care must inactive artificial virtual screening impact learn drug great predictive learning combine multiple flexible predictive facilitate share limit particular aspect virtual screen collection million train achieve improvement machine learning method add task yield collection significant feature extract contain presence moderately class rich discovery predict drug activity relevance voting combine belief networks retrieval virtual relate deep recursive neural predict extract feature small discovery notably competition molecular experimentally predict activity hold team model set virtual choice hyperparameter forest drug major concern size well occur gain justify virtual work target train far million nearly million highlight gain network focused improvement collaborative virtual screening network network outperform great language unify recognition pattern introduce go winner publicly divide four e database wang virtual maximum group interaction protein target drug test construct contain dataset detail fold classifier train fold recommendation literature comparison network perform transformation respectively layer nonlinearity feed softmax predict learn backpropagation softmax dataset number capability provide boost architecture network chemical benefit subsection series table highlight architecture outperform include whose performance lr extremely dataset fold average comparison uninformative exclude subsequent affect performance dataset show basic include single lr cross give lr rf task net hide net neural consistent network overfitte dataset positive hundred overfitte issue strong motivation wide narrow layer implementation lower expressive narrow specific train task good understand design work datum indicate substitute alternate present combination layer size architecture shift show sensitive datum demonstrate section understand increase growth curve visually performance initially improve fall initially performance improve construct dataset ten hold target hold precede
ft proof equation ft ft norm ft ft term term minimum eigenvalue large result ft side respect know exist q figure title show title first show line real dash residual display plot variance realize ba realize realize covariance theorem forecasting asset crucial finance availability frequency suffer curse model factor realize extensive parameter significantly maintain autoregressive dimension forecasting covariance volatility asset crucial financial field portfolio asset pricing asset lead realize covariance major arise realize covariance transaction different frequency secondly observe frequency price price think noisy version several way tackle overlap moreover realize realize covariance fit covariance natural value automatically generate impose wishart put wishart autoregressive wishart centrality fix autoregressive distribution dependent wishart wishart instance issue realize dimensionality covariance entry realize matrix grows need quite challenging model practice probably reason study literature realize limited say realize counterpart covariance build improve realize average volatility method technique construct volatility matrix inspire realize volatility estimator realize length day infinity parametric realize vector autoregressive var covariance significantly need covariance var factor matrix fit need factor article approach realize matrix overcome var fit extract several advantage matrix impose additional secondly excellent approach indeed less combination extraction modeling study also propose thorough theoretical setup extract section middle asset conclusion proof theory price process continuous diffusion standard motion matrix integrate volatility th define inherent trading trade asset day allow several arise asynchronous number adopt threshold volatility denote attractive dimensional integrated use construct raw realize covariance kind estimator thresholde dimension definite normalize hand let next obtain corresponding eigenvalue finally calculate fitting ft ft ft ft ft latter fs tt central wishart freedom order coefficient still grow factor practically efficiency coefficient support tend covariance history variance covariance study find achieve performance parsimonious require however b q carry likelihood bfgs optimization positivity maximization empirical analysis root large eigenvalue frobenius theory use follow model satisfy slowly constant factor xt xt ergodic matrix condition consistency paper use nd te addition go go infinity assume value eigenvector large go maximized datum dimensional finite fourth least model adequate series solid fit residual plot residual day ahead var day realize day ahead plug forecast model get frobenius first day day parameter take average period error except far good need need imagine become need prediction day ahead forecast performance check matrix actually positive covariance stock trade research start end totally day firstly outside pm exchange open pm eliminate open transaction price multiple transaction use price entry price outlier treat price sample mean around price enough price entry realize covariance realize variance realize variance covariance skew skewness realize variance covariance big tail variance year graph subsection show diagonal day treat evaluate choose factor drop less let corresponding factor volatility matrix fit diagonal series order namely every log parameter fit var r criterion ft ft dimensional square white process moment absolute ensure stationarity var check almost forecast realize ahead factor expectation forecast covariance norm day estimate upon inverse contain follow frobenius except parameterization order parameter var need day ahead addition day forecast find deal covariance become dimension model perform require model frobenius spectral parameter latter realize covariance mean entry stock sd skewness realize variance ba cat realize ba cat ba cat cat ba cat fit var aic sc final prediction report model show frobenius sn spectral sn sn inverse var sn sn var realize variance covariance dataset skewness
stick optimize choice strong baseline trial experiment accuracy good accuracy generally search resource bayesian hyperparameter easy realistic environment certainly impractical increasingly machine trial separate learn e course trial evaluate network take sophisticated simple classification logistic choice example include character gram word different language representation document serve initialization nonconvex work approach text representation categorization sequential identifies sophisticated sentiment relatively link linguistic big effect see black nlp raw manual tuning acknowledgement support project fa computing amazon em institute school computer science pa usa edu apply nlp choice make text big researcher module sequential sophisticated sentiment towards nlp system manual tune nlp amount compare machine learn differ way text bag weighting lead big little consistency across task language learner suppose decision automate way hyperparameter selection strength lasso space choice interact decision hyperparameter argue high gram need training iteration decision human al work hyperparameter popular little nlp range task consistently perform baseline linear train network overall goal likelihood etc hold proceed map representation transform n c composition learn concatenation datum simplicity rest clear context perhaps focus select wish carry select optimization family select et iteratively make evaluate trial search option choice algorithm th select function surrogate assess hold probabilistic nonparametric trial initialize ty describe acquisition surrogate use acquisition return either predict uncertainty high balance classic tradeoff choice exceed perform discover acquisition combination ei widely acquisition show et estimate expensive exactly density previous trial less use quantile prefer draw note give explicit need joint depend compute trial hyperparameter th range reweighte count occurrence place set great distance neighbor multiply probability every version multiply probability relevant path exclude gaussian like preliminary configuration advantageous exploit allow research implementation publicly available library function treat box newton hyperparameter experiment consider choice categorization representation type gram scheme removal gram length minimum term include side least predict vote speech segment topic task first classify topic classify vs realistic remove information article often stanford amazon review science graphic benchmark result eight four table dataset l r dataset acc weight strength conv sentiment tf amazon review n graphic classification accuracy acc accuracy regression max correspond gram gram removal regularization strength conv tolerance strength round baseline case publish svm expert method overall always evaluate testing split case development set datum set summarize hyperparameter baseline use weight baseline art recursive vector show logistic comparable recursive acc na I paragraph stanford sentiment base convolutional neural feed zhang use varied log frequency vector weight finding outperform achieve weighting consider ht acc svm gram nn gram lr sequential amazon score amazon come feed restrict acc gram gram rbm gram nn gram lr grams lr bag words cnn sequential cnn comparison rbm nn gram outperform base weak baseline well acc svm link u svm svm method include apply learn author elaborate logistic baseline use normalization net binary weighting
parameterize significant redundancy deep remove integer activation exploit linear appropriate rank et compress quantization storage one memory reference codebook quantization orthogonal pruning attempt replace global network art benchmark adopt learn new problem et al motivate enable pruning complexity fit optimal brain brain hessian loss pruning prune weight decay scale activation testing activation thus layer remain layer randomly hash connection hash value pruning point et sparsity minimize hash hashing pruning give pruning employ via unlike conventional final connection remain connection significantly choose performance prune non result pruning dropout prevent adjust account dropout regard drop drop regarded hard dropout pruning chance informative fitting pruning capacity original work dropout pruning follow retain pruning cnn co layer layer network layer retain less back propagate entire suffer vanish deeply make pruning error hard recover prevent co un connection pruning follow find accuracy method boost pruning compare single pruning iteration connection prune pruning connection connection pruning connection connection zero contribution loss lead connection neuron automatically prune modify add mask tensor network pruning choose quality deviation carry prune l parameter l l ref top parameter l ad ad naive cut parameter inferior work cut layer much reduce single svd neurons achieve rate mnist convolutional convolutional layer mnist prune pruning achieve number activation prune percentage operation prune sparse act weight fc fc weight k conv k fc total pruning region pattern layer colored correspond digit network center performance prune dataset example million parameter top accuracy take train prune whole rate require layer original size reduce pruning connect act weight conv conv conv conv conv fc fc fc iterative pruning give show green pruning still much drop much connection reduce accuracy regularization accuracy pruning dot close green towards extension regularization pruning phase mode adapt prune solid red solid line circle dot curve green pruning drop achieve pruning believe reduce layer convolution pruning layer prune memory operation appropriate hardware activation column multiply ratio layer layer convolutional intensive convolutional figure pruning layer sensitive pruning convolutional sensitive pruning due redundancy layer small layer etc histogram fully panel weight tail drop quickly almost prune center remove parameter adjust spread also convolutional pruning pre prune pruning conv fc fc fc x promise whole size yet chance layer gain prune connect reduction layer image process reduce requirement prohibitive right part brain pruning highlight imagenet show convolutional reducing without accuracy network capacity
sir state sis eq sir constant mild assumption cf ess adaptive particle particle ess technical rigorous justification speak allow fall parameter behave regularity condition ensure effective particle sir measure effective ess bound whereas ess measure ess ess ess condition pg geometrically specifically regularity pg geometrically soon gibbs sampler geometrically condition particle exist pg exist furthermore condition exist maintain ingredient smc pg provide concern smc primary expect smc measurable knowledge investigation propagation kl sense kl investigate encode information lose beyond justification study appear importance estimate particle informally region low probability small measure produce sample
statistic use procedure reduce support assertion attain approach readily gain interest graphical multivariate aid multiple connection vertex lack connection vertex edge vertex connect distinct edge undirected series encode direction distinct degree moderate practically challenge assume model value time direct undirected subsequently direct absence partial frequency involve hold partial coherence jk terminology assessment indirect partial exactly test every suggest partial frequency nan approximation see considerable kullback leibler stationary determine know graphical interaction autoregressive var ic select appropriate exhaustive exhaustive search topology selection penalize term reflect pair partial coherence determine everywhere nan determine missing edge constrain order ic case ic select possibility misspecification arise address world identification leibler kl divergence test simple nan allow test particular subgraph determine edge pose real constraint iterative less computationally costly especially decomposable statistic employ impose select much identifying model instead iterative exactly miss hypothesis hypothesis method error test nan exceed specify obviously decrease fitting simulation original review time series model correct graph summarize work computationally employ statistic two approach empirically statistical power section conclude provide refer otherwise state take edge connection uncorrelated precise remove residual u prediction residual partial jk partial correlation graph gaussian nan partial correlation independence conditional spectral jk jk f assume exist denote element coherence jk jx partially uncorrelated corresponding concept use graphical correct impose way fig cover cover impose edge true correct brief relevant var process jointly stress applicable autoregressive vector value white process vector covariance pp fu I uncorrelated purpose name series therefore satisfy constraint correct result determine correct calculate observe difference suggest correct uncorrelated uncorrelated spectral pf pf ft nj average derivation sufficient e estimator expect window cross recursion purpose along estimate leibler divergence fully miss graphical construct form hypothesis thus concern exist correct incorrect miss correct true incorrect hence obvious v v e h v procedure z accept accept z l l h c test iterative consequently easily family number conservative common statistic distribution function critical formula level z c c h mean estimate edge compare difference miss classify miss reject hypothesis reject find calculate proof edge total number statistic sparsity asymptotically regardless length statistic miss edge exclude increase remove time time test make direct comparison calculate reasonable compare second fig plot expect rapid algorithm generate large gradient fig generate completion different specify two ii var case base replication outside enough simply comparison purpose ignore take infinity miss true true false fig construct multiple varied replication step record proportion replication recorded replication essentially miss hypothesis create varied form concentrate carry replication fig effective hypothesis state procedure fig turn miss boundary c replication effective state miss perform relatively dimension dimension think moderately fig algorithm take day type encounter averaging model repeat column percentage ratio accept type ii ratio graph number present value varied derive rise connection satisfactory behave expect contrast dependency statistic cpu upon calculation assign due compute simply core calculate scale high processor inverse processor eeg control rare clinical patient discuss detail interest detect brain
nmf thresholded plus standard entry entry quickly apparent structured introduce significant whereas compression heavily seem mixed character book pt active link character book exhibit characteristic real life network character appear jointly book column th zero coefficient identify character column nmf correctly identify thing human character ff nmf correctly character recover use sc compression compression summarize bring nmf introduce compression seem cost reconstruction consistently compression library perform loading main regular resort first publicly regular use column algorithm structure compression adjust r produce matrix vary fast explain qr always investigate fast fix varie qr approach propose reflect observe compression explain compressed curve towards generate error conclusion explain analyze qr approach shape compress order qr decay reconstruction column increase row extraction extremely efficient compression nmf last representative frame video examine frame movie comparison least magnitude video truly rank matrix yield qr compression impose comparison extract element fact reflect example compare matrix compression greatly enforce help find b qr qr core core pt core control comparable x second sample frame per channel compression representative normalize coefficient take compute respectively test complete source movie movie minute long process two matrix file gb fitting compressed extract extreme frame second process gb structure nmf formulation technique namely nonnegative algorithm compress random projection compute compute extremely matrix useful decomposition currently investigate replace norm fast cauchy suitable structured rescale sparse author thank scientific help library direct simple direct stack factorize component qr stacking factor compute multiply matrix eq orthonormal matrix multiplication orthonormal matrix qr matrix decompose matrix block extract store need indexing hold augment ij use method multiplier successively respect fix closed set nr r nk k ij k preliminary propose follow closely tucker kkt hadamard accumulation kkt plug clearly limit point leave non combined get ij ij identical argument apply prove accumulation kkt kkt kkt ideally kkt provide fig rgb edu nmf establish numerous usage situation challenge year increasingly grow science exploit nmf separable nmf subset formulation representative show result technique shape limit presentation numerous structured projection analysis rapidly economic collect speed database transaction social rapidly raw insight guide management whereas usefulness theoretical challenge information science aspect become algorithm cope present tool power rich create increase big communication algorithm broad include secondary pass even substantially technique lastly massive parallelism model numerical adapt environment benefit boost recent nonnegative nmf frequently since good way model recommender system audio nmf seek I matrix popularity combination factorization interpretable nonnegative formally nonnegative entry appeal advantage present np pose matrix exhibit efficiently exist stack separable present nmf denote choice frobenius easy improve year popularity partial decomposition partial decomposition key observation identify project matrix desire low robustness thorough propose algorithmic computing compute decomposition beyond focus compression loading memory need use projection increase boost multiplicative active nonnegative admm structured projection reach nmf case rank random arbitrarily available organized provide propose diverse technique medium scale finally conclude remark describe projection guarantee limit deal way overcome limitation desire factorization follow become define whose entry realization independent identically follow step compute approximate range orthonormal factor factor exploit try technique compression let h nr r l decomposition rank execute iteration note hypothesis assume failure beyond prove theorems justification grant freedom grow execute r extra factor note analogous open gaussian fourier significantly use give automatic speedup running times structure leave multiply preserve reference therein however structured compression achieve compression agnostic whereas structured theoretical research justify gap reduce necessary computation arise happen number even store secondary I line qr completeness design parallel computation cost perfectly use implement algorithm involve matrix compression introduce scalable many matrix decomposition e nmf work nuclear norm already decomposition commonly norm become increasingly popular recent year widely audio frobenius matrix contaminate noise adapt right investigate fast alternative goal large aim new nmf make try decomposition lose might already conceptually compute structured compression difference qr let r r storing magnitude become huge gain speed fast easy understand compute qr decomposition ratio decrease fast trivially detailed qr order nmf similar model negativity row similarity problem replace pseudo norm possibly solve contain reason present alternative present numerous example support structured random projection nmf speed compression allow core algorithm disk fair structured compression slow approximately small matrix overhead matrix per block evident well impact algorithm performance summary great compression respect core compression present slow exhibit computation come slow compression representative use multiplicative update variant admm structure matlab far show nmf variant although structured end reflect variant converge observation nmf variant counterpart admm gain structured compression lastly compression come indicate case pt sparsity generate uniformly zero stand structured mean sc generally original match sc nmf hyperspectral emission visually nmf compression error seem several statistic reflect visual computing time statistic south east north thick thick thick multiplicative admm sc south thick rectangle rectangle south east north west thick thick image south north west thick thick image north thick thick thick rectangle east image north thick rectangle thick thick sc image west thick thick south rectangle thick thick south east north west thick
signature expert algorithm component couple describe combine resample svms report svm bag ad hc boost hc discriminative svm code patch quantify deep extract either filter convolutional layer study pre stack autoencoder tune convolution operation contrary p ad hc vs hc ad patch ad hc vs hc ad ad hc hc way ad hc ad third hc vs hc patient baseline hc longitudinal subject baseline vs hc hc hc vs vs set vs hc hc data ad hc hc hc ad hc hc hc ad hc vs convolutional validation evaluate performance unseen give architecture superior comparison well hc comparison superiority interpretation deep architecture class fourth autoencoder l ad hc vs design pattern autoencoder network primarily assess relatively patient experiment local boost classification albeit small future hyper system layer pre autoencoder improve complexity stage would share pattern recent autoencoder convolutional predict disease status scan historical network outperform report result refer disease unite million mild cognitive mild change without field create try ad human machine predict ad great develop image able discriminate hc artificial autoencoder convolutional image yield performance slice experiment report obtain hc ad hc ad vs hc describe approach offer brief review literature discussion part disease clinical genetic early dataset originally ad hc international brain template template weight normalise divide deviation result voxel slice scan whereby initially autoencoder filter neural whose use autoencoder interested compare present convolutional detailed autoencoder neural network extract feature autoencoder layer several input map input decoder reconstruct bias function analogously estimate identity decoder real sigmoid would tie intensity error autoencoder hide decide try autoencoder overcomplete autoencoder hide autoencoder overcomplete layer autoencoder minimize reconstruction potentially experiment autoencoder investigate autoencoder autoencoder enforce operation advantageous context encourage may underlie controlling variability image mean activation hide average try unit hide unit try extract spatially digit artificial neural convolutional connectivity hide sharing describe hide spatially beneficial number one number architecture modelling detect part unit hide whole reduce number within position input array let filter connect let bias feature add filter array convolution map give output obtain convolutional layer basis sparse autoencoder previously learn convolutional applying basis convolutional patch basis term apply sigmoid activation likely discover topology convolutional follow pooling feature adjacent every neighbourhood retain pooling approach apply max operation pool feature ignore output pool stack map input e output hide unit sigmoid unit softmax activation represent class ad hc network number summing function label decay early network train batch randomly layer momentum
frequently mobile device read book user use old mobile device percentage medium video making indicate rather technology dynamic seem finance across ad category finance continue finance video respectively video seem group finance ad completion figure mobile device percentage finance fitness interact ad product appeal user aware user mobile device profile correspond music video social medium preference probably phone playing game rule day profile target profile ad video rule video lift video leave support lift ad rule example class play confidence health aware video lift video lift mobile user ad simple suitable numerous rule lift rule small result rule average lift could accomplished algorithm future work describe cluster analyse medium find profile ad future expand study investigate ad investigate association sufficient ac uk mobile rapidly success user interact application mining mobile whole interaction ten consider base percentage validate investigate difference way ten interact various association perform find user certain interact difference interact interact rapidly expand mobile ad mobile uk digital ad growth international overall operate focus mobile ad network mobile package operate side side work medium globally customer ad month interest receive interaction click identify association age interact ad display refer attribute cluster mobile user determine interact mobile ad researcher medium mobile ad phone email web history good knowledge investigate cluster cluster ten investigate difference interact finance ad profile association profile day ad interaction demonstrate type user time follow section analyse profile ad million user period interact ad ad ad user click ad video either finish ad play ad view video video general record stage load video video video video video however point ad play video unique user anonymous use multiple ad list mobile device date ad play etc name site site ad user nd nd file consist approximately million million people ten type ad ad ad ad finance ad device list date site finance finance pm site finance finance pm site u pm site finance finance load finance finance pm finance finance pm finance finance pm site finance finance site example collect site record cause whether video looking association interaction unlikely user datum mean preliminary investigate distance sum distance performance increase however undesirable perspective represent vector represent user binary likely issue curse computationally expensive consider map user category normalise user correspond feature space category category category dot category vector user divide ij table finance finance category finance rather one interaction feature map category f normalise treat squared point cluster ten interact ad profile calculate mean k specify first cluster cluster denote receive user user play video l specify interact investigate association profile day total rule mining relationship consequence commonly restrict support consequence occur item database rule consequence support confidence extremely efficiently issue overcome left support instead contain lift value cluster ten classified group many classification total total day occur find create profile time u cluster assume profile record low user minimum support interest filter rule lift rule rule consequence play video video store access r library mine ten present score profile percentage category profile low total consist category interaction ten profile finance ad ad ad identify user percentage game less average percentage user numerous centre profile mobile device work user index interaction low video consider finance due financial playing finance video getting decrease profile finance ad ad investigate suitable health aware medical weather making slightly health fitness average health aware tend mobile device guide health weather may often aware profile finance ad respectively user video finance ad aware user average continue suggest design ad modify encourage ad profile suggest finance user tend mobile fact video likely people user alone index finance ad finance ad value ad drop play video interested ad ad index dynamic ad increase video complete click ad interested majority search percentage business consider business looking high finance suggest life indicate mobile device profile mobile device interaction category video present play finance index finance decrease index completion less ad value interact video expect finance explain interest finance ad lack ad
widely availability include parametric find build work also consistency nonparametric density build consistency condition set completeness organize model place bayesian slice nonparametric combination weak true provide study weather predictive daily return daily wind speed paper statistical expert tuple cdf let pool combination cdf beta interpret calibration act calibration function linear pool pool admit lebesgue probability serve achieve seek combination weight assign importance pool flexible certain calibration mixture beta k interval beta mixture illustrate flexibility mixture raise general mixture bm flexible transform flexibility unknown treat unbounde mixture bm calibration aggregate alternative interpretation combination major whereas imply continuous model express beta pdf symbol denote correspond cdf horizon predictive time respective realization calibration map cdf use calibration combination aggregate predictive subsequent leave ease burden aggregate cdf eq bm w positive parameter approach beta density proportional gamma proportional uninformative prior parameter respectively adopt datum augmentation introduce allocation likelihood bm draw ii number desire ahead distribution beta transform propose gibbs account namely approximate advantage credible interval calibrate easily output mixture number beta procedure choose study series instability thus pool dramatically converge select subset properly instability scheme select give finite beta increase one benefit infinite answer finite mixture propose estimating include uncertainty infinite calibration assume bm dirichlet standard result dirichlet eq breaking stick base base stick break calibration ty gd ty ty first sign driving introduce new dispersion depend crucially infinite usually assume provide allow dirichlet method dirichlet truncation mixture propose use observation complete finite conditioning slice introduction auxiliary variable introduce allocation finite observation complete k dimensional breaking dispersion complete assumption distribution ty joint adapting describe collapse gibbs sequentially full ii ahead cumulative weak mixture density speak put near general cover simulation forecast non heavily ergodic markov spectral posterior calibration density space say converge weak neighbourhood prior assign satisfied formally kullback leibler neighbourhood size kullback leibler hold short kullback via parameter density see joint calibration kernel dirichlet sake pooling parameter case dp state recall f proof assume strictly turn check similarly assumption check satisfied gaussian student consider cumulative need check interior g end mixture variance c yy use ii check analogous consideration easy satisfied calibration base concentration turn analogous corollary give necessarily spirit w dp p kl combine normal location equal hyper prior posterior affect calibration less secondly improper distribution model possible still improper lp pool estimate recursive score equal bm beta pool bm component bm define bc mixture bm mcmc iteration burn purpose arbitrarily bc combination respectively arbitrarily bc cc bm bm bm cc bm bm bm bm table empirical transform standard black dataset bc cdf green uniformity nc difficulty combination part ccc flexible cdf linear calibrate cdf blue line close show decompose mixture consider solid left figure contribution mainly part component result weight bc calibration assign weight solid line component component pool calibration mainly positive part thank assign weight model quantity choice p cc graph nc gray black blue random figure infinite beta burn calibrate line belong interval cdf calibrate account uncertainty component chart dispersion model see always row infinite particularly accurate also wide gray line concentrated suggest informative second distribution degree freedom predictive combination nc bc result calibrate calibrate cdf evidence bc tails bc calibrate tail assume unknown infinite mixture lead calibration calibrate middle nc gray prior black posterior number component panel panel bc contrary calibrate tail see distribution mixture change dispersion parameter substantial hyper investigate period kullback leibler utilize forecast density compute variable density though compete sufficient average rank cdf pdf consider daily version ml trading year ahead first ahead form combine density score non property split sample period calibration investigate sample therefore period relate period times window day day ahead forecast confirm cc individual measure period degree ideal confidence nc calibrate tail nc differ substantially simulation number beta concentrated attention average forecasting provide score perform approach model combination normal period nc begin version apply improve predictive consider ten ground prediction european centre weather forecast restrict prediction speed ensemble bilinear interpolation initialize
long period temperature rapidly increase dimensionality constant add per month science advantageous computational large first vs consist pixel use pre convolutional extract train remain size gb non normalize mse also attain varie need case reflected training process difference error difference minor slightly datum worker error speedup speedup achieve correspond speedup average run obtain duality gap vary small iteration iteration achieve speedup attain decrease run suffer split number observation remain dimensionality decrease together shorter plausible illustrate splitting row present generalize wide variety worker result bound notably worker task achieve convenient smooth loss however demonstrate function communication deterministic ahead beneficial additional block feature physical location clearly theoretical improve exponentially research amount communication minimax supplementary lemma random value column raw random worker loss problem raw proceed result relate quantity ease notation subscript follow definition define cauchy schwarz write combine definition contain contain row sum simply obtain sampling orthonormal balanced partition quantity subset row coordinate without replacement denote row express q uniformly replacement chernoff b subsampling fact eq take block define plug low upper chernoff ease paper chernoff semidefinite dimension replacement low different bound compare optimization variety estimation form depend covariate furthermore convex lipschitz ridge long even deal distribute datum many core cluster common portion incremental update number point sparse setting slowly fundamentally approach worker access portion preferable reason well scale challenge separable across high encountered bioinformatics science furthermore beneficial vector deep privacy block feature medical record include solve ridge locally number wider intuitive tight regression dependence rather yet good guarantee experimental dimensional world vision art ascent show optimal implementation environment asynchronous propose solve memory environment large possible asynchronous datum dense slow distribute impractical cost al coordinate resp worker make update allow trading communication speed notably show worker thus consider assumption hand relatively assume recently ridge distribute across round optimize sub portion master structure able setting indeed split able entire extensively use sample domain factorization square context randomize hadamard consist subsample independently normalize hadamard key sum operation impractical applicable distribute sum random dimensionality worker random combine aspect single worker worker compute solution sdca worker way final primal discard feature consequently especially consider matrix subsampling return bind final solution obtain worker union mainly determine relie rank often fulfil high arrive application recovery estimate primary distribute implementation detail implement increasingly community benefit many easily library diverse task greatly facilitate application worker worker require feature locally sum scheme illustrate increase number worker simply layer load benefit increase relatively lead demonstrate practice datum well compete hinge therefore suggest
acyclic panel node neighbor node arbitrarily panel edge restriction absence imply showing may constant normal may edge db use show edge differ negative candidate fit select direction include graphical model appear literature divergence structure stand information complement row second similarly column equal obtain inequality second inequality mutual result immediate consequence schwarz rearrange yield distribution let information respect nonnegative term note expression accord integral nonnegative conclude conditional kl equal uniquely determine l ex ex theorem kl set corresponding model bound sample order increase popularity scientific network vision molecular biology year substantial advance broadly estimate edge structure encode independence relationship goal treat separately reduce dimensionality subsequent advantageous graph observation scenario propagate misspecification may g estimation procedure edge albeit fairly need consistency explore incorrectly close fit respect graphical restrict constant true close single conjunction true accuracy order relate kl gaussian mutual discrepancy mutual procedure propose al focus kl recent ise unclear whether kl ise paper organize relevant statement kl term parameter accord kl extension kl example separation grow manuscript write similarly write constant use determinant mean covariance p undirected miss cx j exposition cite therein distribution quantify distance via divergence fix distribution wish infimum range distribution subgraph infimum hence least impose impose frobenius diagonal analyze ise heuristic identifiability follow signal covariance furthermore fact semidefinite quantify conditional equality impose side possible express conditional mutual information next accordingly quantity submatrix index definite quantity stem relationship necessary validity equal graphical message true graph constant result al upper hamming distance model estimate whenever hamming divergence close already lie whereas inside conclusion may interpret divergence kl divergence inverse matrix make edge introduce corollary intrinsic equality explore separation example distance affect purely appear account suppose set graph model analyze distribution let let respect minimum discuss frobenius remark corollary objective agnostic minimizer draw multivariate pm require size graphical size somewhat impose regression incoherence take min assume previous discuss parameter frobenius inverse matrix class somewhat undesirable frobenius norm g jointly create even requirement impose
skewness gm furthermore gm tail near five four suffice model location spread skewness filter pf cope skewed computational increase state dimension smoother retain computational kf introduce flexibility skew heavy model skew bayes vb simulation compare pf approximation skew et series unimodal introduction skew univariate skew spread shape density pdf gamma student freedom pdf six shape skew skew introduction multivariate version skew univariate cdf skew multivariate factor letter general assume pdfs covariance index read conditionally skew parameter whose shape mutually entry letter derive filter bayesian smooth vb hierarchical diagonal denote squared denote gamma pdf analytically tractable approximation vb kullback factorize expect side constant recursion convergent integrate hand discard normal filter posterior derivation expectation kk cp xy ax kp k ax kp kp q xx cx kk u k kk c cp c xy cx xx cx k k u r ii ax kp ap qx kk simulation skew skew bayes smoother compare particle pf kf kf kf component innovation quantile kf covariance times covariance computation walk vb change process low square error rmse kf kf kf slowly discount kf large error ht rmse skewness kf kf th position bias vary linearize linearization negligible process international service average carlo replication computed study vb speed slow increase fast outperform reduction negligible slow iteration iteration pf numerical acc acc delta show rmse rmse levels box quantile replication measure correlate vb posterior work dominate variance snr rmse rmse histogram distribution set histogram rmse replication indicate robust real compare algorithm skew percentile difference small filter smooth smooth bayes simulation outperform symmetric skewness present burden depend simulation cost lemma
similar computational complex summation ensure suffice proof theorem define chebyshev converse fix arbitrary present theorem measurement require vanish concentration inequality proceed step separately converse start former observe condition mean ni si chebyshev q finding constant function define precede remain unchanged leave implicit throughout suffice generally within set vanish overall case logarithm therein vanish impose condition dominant heavily concentration use theorem combine probability conditioning low recall function ensure achieve restrict similarly always put everything replace pair weak lead significantly nearly proof key difference step theorem obtain sufficient condition sequence sx characterize use probability make dependent vanish ensure negligible vanish contribution negligible remainder vanish cf remark construct deduce step deduce experience procedure come less concentration chebyshev converse obtain converse however provide tight concentration bernstein analysis generalize rather repeat attention q analog theorem often powerful small one theoretic limit recovery use end combine refined argument analyze section incorporate argument discuss advantage precise mutual require turn require straightforward discuss main difficulty concentration add converse add difficulty derive converse exist technique powerful comparable recover observation focus throughout example use mention vanish inequality chebyshev moreover lie follow hold chebyshev fact moment general bernstein observation cs section group general converse explicitly concentration inequality consider form generality consider fix vector noise information procedure trivially contain support concentration accordance kk yet fashion readily kk hold term behave rearrange suffice behave suffice imply condition imply converse analogous left hand combinatorial reveal combine precede apply precede setup provide conversely converse hold equality decoder additional bad therein behave numerator behave final numerator arise permutation low term behave immediately dominate objective behave iii condition dominate behave kb numerator scale b readily verify maximizer numerator factor two identical threshold coincide advantage handling strong converse converse part notable however maximize easily evaluate corollary dominate term law would allow coincide ideally factor former characterize avoid brevity analogous law comparison discuss constant snr logarithmic iii corollary state regardless achievable corollary reveal enough coefficient yield constant provably suboptimal optimize constant strong converse contain simplicity accept suffice intuitively enough handle large thus sharp converse setup two change distribution recovery recovery eq auxiliary see proceed follow characterize magnitude define variable increase magnitude one empirical k surely immediately small converge integral write verify typical set distribution entry consequence proposition behave ensure contain focus also focus realization simplifie give vanishe imply converse weak combine previous setup whenever obtain numerator coincide remainder factor k op behave factor vanishing without generality readily obtain turn proposition ok dominate hold numerator follow continuity handle focus dd k desire converse law maxima achieve hence coincide numerical counterpart uniformly absolute one quantization multiplicative mutual precede mutual mutual quantity iv variance correspond give simultaneously present step setting trivial chebyshev proposition choose therein behave second vanishe provide set describe part handle theorem thus substitute combine apply asymptotic corollary precede conversely whenever part identity remain term factor equality numerator hold equality evaluate achieve set identity precisely case increase factor describe counterpart necessary behave bit require super ambient dimension counterpart seek setup appendix along obtain counterpart focus limitation handling avoid bound function set gs minimize amount fix converge converge accordance make concentration proposition set ensure term control typical see small mutual information observation behave hence provide imply converse part analogously weak suffice previous obtain conversely whenever usual moreover identical vanishing remainder factor right simplify whenever similarly converse behave tend bit unbounded corollary bit set snr high similarly present represent db asymptotic corollary replace arbitrarily normalize number divide linear setting necessary interesting bound converse measurement multiplicative great generality behavior grow discussion moreover model kk bernoulli measurement depend one depends readily grow well understand noiseless testing q proceed trivial singleton inequalitie accordance write precise small sequence appendix converse immediately effort summarize finding choice kk idea dominate provide well behavior corollary combine previous precede setup noiseless conversely consider part maximum substitute slowly q identity p objective growth eq whenever bound behave thus maximum remainder condition maximum arbitrarily value behave achieve arbitrarily one converse part maximize satisfied equality readily verify coincide yield threshold test limit counterpart denote fall see attempt provide follow accordingly proposition proof conceptual analog let optimization follow precede setup noisy conversely numerical matching converse noiseless precisely characterize tail term dominant sufficiently term tend imply assume achieve logarithmic hold limit oppose see partial recovery cf fact testing follow proof corollary concentration converse multiplicative corollary e recovery reduction multiplicative exist switch see bound bound combinatorial matching pursuit dd algorithm assumption c asymptotic converse adaptive key implication asymptotic measurement important decode scheme unclear dd noisy e narrow converse adaptive adaptive yield asymptotic optimal limit density output alphabet notable example group generally strong variant probability vanish argument replace eq side right vanish coincide recover recent combinatorial additive behave whereas dominant wide section provide proof theorem change result mention proof mix channel due part discussion code density search exist setup decoder choose exhibit symmetry say subsequently remain bound intersection event one mu mu ss mu mu mu indicator substitute count simple I fix apply writing term upper bound union term equal ss write eq logarithm appear density bound recall theoretic converse bound reveal important entry prove symmetry pair permutation index claim fact among function maximize low realization appear respect permutation vector generate uniformly subset reveal vector estimate occur modified setup converse converse first condition indicator follow shorthand recover write joint distribution namely ss section coincide realization permutation immediately already amount support search oppose converse section immediate recall without generality cardinality almost surely decoder term remove restrict change disjoint count fall set dp channel appear develop bit test exact threshold number converse direction focus interest consider converse constraint move model structure sparsity probabilistic guarantee minimax poisson interesting similar analysis argument triangle subsequently give form proceed value variable distribute direct substituting apply hand side definition statement define probability follow moment combine lemma obtain result identify e substitute identity calculation conclude fact p substitution random freedom definite identity b relevant value x similarly partition write minus amount multiply substitute derivative part eq use difference part one upper substituting note fourth argument therein decay term part iii assumed may also loss imply factor simplify assumption easily taylor convenience obtain taylor expand middle follow qx qx qx e qx qx follow substitute complete show behave follow follow integral half verify handle similarly briefly integral thus decay part assume x iw function vector imply derivative writing logarithm difference lie hence yield effort accord w respectively combination upper f w case universal case two account I combine recall throughout oppose bayes independent write mutual information negative p write upper therein follow take expand hold mention split expand integration bit quantization mutual latter recall without loss generality precede use continuity binary analogous limit use begin brevity sp ok consider k identity assumption expansion measurement respectively imply one combine statement right use fact chernoff tail binomial write first case logarithm summation note follow summation provide substitute analogous along law readily suffice fact reduce group attention differ noiseless appendix begin later substitute proceeding follow precede probability information behave example follow expand logarithm analog noisy testing sequence level index obtain recall average apply expansion x h section index depend hold identity bernstein base expression common numerator denominator let right identical proposition keep rather towards choice remark consider slowly maximize yield upon write kk numerator form keep clearly set write translate verify decrease obtain set compare hand write verify occur symmetric remark support arise setting compressive sense unify sparse channel converse characterize error measurement variant law necessary threshold matching advantage broad parameter converse fail vanish tend support recovery limit compressive sense compressive sense recovery determining arise sense cs vary significantly considerable interest unify sparse observation
initial fitness greedy fitness action bt composition explain bt degenerate bt change change bt composition encode decrease fitness bt perform continue success agent run fitness fitness stop bt learn bt bt goal anti possibly inefficient node due window due act tt simulated environment fitness value fitness fitness accordance complete fitness point acquire game character agent spend game bt tree reach genetic operator size fair fair selection gp generate bt gp restriction bt reduce size bt property bt unnecessary enumeration run bt fitness latter tree create bt new procedure stop find present code procedure tree experimentally ai super game initially ai control character namely level view walk state act jump one onto collect score collect presence life currently small big benchmark walk walk field choose pass left end around agent move end show phase illustrate bt learn move phase show bt action fig illustrate bt version benchmark learn bt www bt node bt conventional scalability free ap involve greedy goal whereas relevant base framework detail address subject ap learn experience exploitation knowledge simulate benchmark work illustration available result encourage art work examine ai extensive dynamic accordingly inspire work plan possibility supervise bt regard develop model bt strength lie http supervise implement example example play remark em height se definition automate ap impractical fail drawback conventional system finite machine describe plan ap represent valid alternative present term modularity framework programming gp bt observable environment illustrate source character include evolutionary genetic planning branch artificial ai concern realization action typically unlike must multidimensional concern automate ap example pattern finally ap word environment evaluate case environment online extend toy planning present come despite suffer planning description environment goal exact planning ap recent tackle control planning well plan fitness execution composition computer game namely bt modular robot task artificial intelligence meet player flexibility ease human popular grow attention compare transition switch leave actually many language transition govern call return pass flexible call modern language exhibit many gain call add remove reveal connect state avoid redundant transition bt relation define child relation gp entire yield optimize plan base generate bt agent achieve goal lie modular dividing goal successfully evolve behavior methodology mobile apply generate simulation environment benchmark evolutionary outperform reinforcement learn possess ambiguity player work bt evolutionary create ai controller agent game use genetic learning micro air application evolve bt conduct control robot demonstrate bt compare bt take bt go free work framework robust observable make work behavior graphical modeling execution become popular intelligence computer hierarchical network bt direct tree check return accordingly node running generate label biological bt use gp optimize randomly bt determine bt ability tool genetic mutation part fitness function satisfy reach final bt use exist bt fitness generating bt necessary control optimize bt gp child child mutation three parent mutation replace bt function perform bt sub bt cross use avoid unary mutation mutation replace node versa gp call mutation several population bt gradually start avoid fitness diversity goal population population select proportional sub goal fitness naive fitness divided fitness individual ensure rank high sort population fitness fitness define follow fitness individual survival lie fully environment initial final take fitness derive fitness bt fitness value move second bt execute continuously fitness fitness assess determine course use gp provide increase significantly pure achieving bt satisfy goal bt consist execute find execute second fitness keep action add bt find bt assignment consist action
fix size name encode size rely constant factor word mechanism use word provide unique code word long factor properly select language fed enable dependency corpus compression outperform lstm implementation assume vocabulary adopt word vocabulary representation independent representation use history encoding symbol namely symbol represent representation first history eq control influence position symbol vocabulary obviously discrete recursive context power property language gradually nearby role code vocabulary symbol always perfectly recover simple value without ambiguity since many far context close countable choice theorem almost countable chance choose isolate extremely almost quantization verify run element wise code less figure case show chance allow appear time obviously normally run corpora nine show except feed lm scale base figure encode neural encoding direct platform computation multiplication particularly mini run code code compute multiplication code word triangular code position mini consist mini compose n sequence code mini batch eq code show compute activation project look matrix embed vector calculate gradient propagation bp vocabulary limit k preprocesse validation compression benchmark article part validation testing set vocabulary replace vocabulary token reader reproduce c evaluate word gram neural projection hide layer per net initialize initialization sgd mini keep fix set continue six epoch training epoch net architecture mini mini compose several word truncate sentence investigate factor order nd experimental figure factor lm rest paper summarize test outperform lstm indicate long without feedback nd knowledge kn gram st order gram lm rnn lm lm far examine much large text corpus article wikipedia gram ii traditional rnn lm sentence speed examine base input two st output initial batch sentence rnn experimental outperform popular lm yield paper almost uniquely work apply neural network nlp sentence machine translation part technology development china fundamental central china discovery microsoft constructive suggestion everywhere countable choice code ambiguity decode multiple happen happen equation either word ahead root eq total total root fraction root except choice
goal easy satisfied arbitrarily either induction satisfied focus choice eq complementary satisfie set restrict formalize separate mass deduce couple useful side go induction arbitrary result outcome couple marginally couple measure simply shorthand couple exactly interest couple miss distinguish couple confusion couple force identical since depend probability sample differ symbol miss claim probability positive absolute explicitly fact coverage couple identical allow event divide localization range probability coupling equation family see probability coupling compute meanwhile conditionally simply complement least value appear k k choice verify event always away zero step combine complete establishing begin proof validity couple equation coupling recall sake similarly hold event distribution violated formalize q hand recall combine choice occur occur occur satisfied event contradict establish closely relate simple task tail exceed miss problem concern probability completely positive subsequence pac tail sketch theorem force tail unchanged adjacent concern sample coarse instead concrete justification tail theory concern even clean characterization sufficient learnable family cover let family distribution pac miss mass cite technical rely chebyshev readily albeit much demand surprising dirichlet process produce tailed distribution mass paper unseen symbol discrete contribution learn completely well failure light tail place discussion continuous tail estimation show light learnable assume nothing familiar notion kind failure study landscape reveal open concern establish mass paper family heavy tail learnable smooth parametric learnable plug question concern establish rate fact established slow convergence lack uniformity lastly learn make fail accounting critical geometric plug symbol mass estimator miss geometric coverage parameter eq let chebyshev inequality indeed eq q bound convergence care symbol true portion convenient segment formalize segment segment contiguous coverage coverage segment localization segment give integer appear induction turn geometric give nothing thus integer event complement event n claim union analyze hold neither hold put show pac mass equivalent fix great satisfy large intersection give see equality numerator denominator sx claim proceed step yet prove basic positive number base fraction z induction continue previous large z z effectively bound converge roughly instead regardless probability satisfy desire school california berkeley california rare structural obtain outcome coverage event miss distribution learnable relative proof constructive rely couple rare one heavy event light tail give datum traditionally event rare rare symbol answer mass arbitrary distribution precisely sequence miss say consist mass relative empirical trivial answer mass mass good interpretation derivation cross contribute fundamentally derive form framework later refinement also focus shift relative pac property good power geometric tight concentration interpret learn light tail include addition power leave exist pac learn miss fashion insight justify assumption event failure good light tailed distribution barrier success law
margin demonstrate jointly video number truth performance step illustrate car lift car one three frame language appearance video despite large variability text manual supervision please project website assess cluster vision order cluster allow relation align even similarity car car however want able refine group direct car experiment vision improve similarity recover recall alignment step represent object relation obtain truth manually give similarity precision curve similarity average nlp nlp vision object relation average varie list nlp use step previous remove object relation signal case overlap direct consider evaluate perfect stream encode alignment video text constraint order video encode assign video type iii let feasible still optimize relaxed frank give detail step frank solve separately linear quadratic video follow frank wolfe oracle q program entry equal recalling mean exactly type ii th column go jump time subsequent possibility continuous path column cost value every end size illustrate transpose red gray cost gray entry page interested additional constraint alignment region wolfe stack definition look among initial relax several option turn use z stack automatically change car video contribution develop natural take two internet video contain car experimentally automatically within video demonstrate joint people change flat machine video cognitive ability would virtual work video sequence video demonstrate car consecutive car remove addition learn visual linguistic video discover step video task expression variability video little car start appearance vary video people viewpoint perform vary finally variability step slightly challenge joint appearance benefit modality assume sequence step call nlp input video individual learn modeling video advance video task link joint output discover step temporal video validate compose video supervise relate like natural description discover scale purpose scenario differently video internet event description help description lift car car video work rely scenario differently aim video consider latent structured perceptron align video laboratory protocol whereas form speech video computer recognition video weakly action video address explore training annotated discover event video description discover structure video action category unlike video exploit visual modality approach exploit improve nlp linguistic expression appearance people video visual video recent video model cluster fold address automatically discover video unsupervise improve video frame stream task raw text process token main step give input input link achieve stream video stream sequence latent variable indicate step interval token problem video stream cluster raw text video convert direct illustration video sequence align direct align together sense g right main description step roughly preserve overcome challenge complete involve interaction representation specifically direct pair compose complement dependency object extract correspond object video signal sequence token length key convert raw easy direct formulate relation together occur frequently keep maximize pairwise similarity key match together direct object measure object particular sense find constrained change sequence threshold step common align object step exceed case less output extract formulate video stream vector extra feature regularize video stack design classifier among video video stack matrix similarly concatenation function measure target class assignment square simplifie respect additional optimize matrix encode incorporate obtain constraint link section cluster video section use cluster task people perform approximately strongly together cluster second constraint encode sequence order assignment link single clustering detail constraint encode mention video alignment action alignment action video issue first global action happen interval encode direct assign video note encode video interval object match constraint identity video strong text encode alignment maintain order constraint similar particular video step see appendix previously summarize without constraint fix text cluster subject frank wolfe give satisfactory video far similarity text cluster ground top head lift lift continue continue annotate video feature sec localization video visual description task search keyword video speech manually correct obtain video length video frame second video manually annotate video ground step task video truth represent sequence relation dependency object object represent give object respectively frame represent appearance histogram optical descriptor aggregate frame vocabulary appearance obtain descriptor video output descriptor aggregate aggregation descriptor dimension descriptor normalize single represent discover report result alignment step step step mostly recover sequence repetition car fine lift include quantify precision proportion step recover step bottom partly cause coarse
meet forward pass scoring configuration normalize score soft computation probability iii back gradient iv state complex severe output prevent line amount configuration score efficiently representation normalize chen discuss extend scoring scoring score small restriction function local require marginal restriction pass width approximation reweighte alternative iterative method summarize forward pass compute local marginal approximate backward pass utilize repeat propagation message pass backward pass parameter fail decomposition assume computationally require restriction densely pixel possibly pixel densely yield segmentation densely consider context log set aim fully convolutional discuss within probabilistic model get approach mixture detailed simplicity assume function high two scoring generalization begin computationally efficient approximation n leibler assume field require valid due update iterate convergence update marginal point neighbor densely bottleneck arise involve densely complexity marginal marginal require importantly dimensional marginal achievable mixture formally label compatibility feature ensure convergence cost restriction pairwise formally compatibility semi definite readily convex term convex proceed iteratively result linearize program linearization solve filter linear system entropy relate filter marginal cost observe address restriction attention finding parameter exact expensive field marginal surrogate loss truth labeling perform parameter loss w marginal perform w carefully investigate field update recursive derivative x depend early iteration hence desired successively track fortunately back substitution gradient require back refer reader regard assume give logistic crf generalizing gradient arbitrary gradient marginal truth predict unary scoring connect repeat via backward pass update compatibility kernel summarize functional network subsequently via efficient tracking evaluate approach summarize challenge segmentation background training addition report intersection metric validation training fine convert aim large probability mask skip take employ fact two adapt object assume dimension size spatial however unary crf parameter size intermediate probability image bi interpolation scoring variable image perform update original loss track propagate r unary term bi interpolation network shape compatibility crf detailed update many successively due stage connection present tune imagenet dataset due gb memory gpu mini data car cat c tv valid convolutional compatibility shape crf arise pairwise employ use contain position contain dimensional pixel channel nine compatibility parameter shape ccc mention part neither tune unary approach unary second training crf compatibility shape fig indicate unary visualize performance number peak largely accuracy peak report chen visual approach segment image variation pose challenge present also boundary jointly train convolutional fully generalize make publicly dense fully long employ modify architecture present joint incorporate unary potential tractable even li investigate objective linear remain regressor computationally pose heuristic normalization convex obtain ccc ccc convolutional neural neural approximation
way correct sentence describe artificial model model explicit et find tree generalization decaying gradually find decay generalize structure find recognize syntactic natural syntactic task simple encode sentence linearize lstm syntactic also develop encode structure massive token leave learn unseen object show learn understand obstacle kind structure define logical consequence hold draw suited sentence model develop sentence task force allow representation effectively mutually exclusive logical distinguish equivalence semantic independence tp p p brevity show language et highlight complexity straightforwardly interpret statement logic label interpret logic crucially conjunction new sentence allow arbitrary conjunction arbitrary sentence datum come use model word tree sequence model embedding structure sentence corpus use multiplication rank third tensor find help rather sentence encoding transform jointly embedding input word eqn activation activation rnn sequence include parameter minibatch descent negative regularization tune test train epoch largely converge peak figure reach accuracy slightly generalize structure familiar tree example size reach test sentence smoothly quickly lstm fall lstm considerably well set baseline frequent short bin unlikely level lstm lag far behind four exploit sentence complex unseen bias interpret architecture role sentence recursive syntactic explicit make small set architecture constrain suggest linguistic natural language sequence exploit structure acknowledgment acknowledge google advanced project filter air force laboratory contract fa national grant department office research grant finding conclusion author reflect google nsf ex citation stanford stanford nlp stanford computer science network geometry sentence network outperform model neural like fact discover compositional structure possibility recursive compositional learn tree well able informative sentence real value vector array nlp include parse analysis base build representation base linguistic phrase model principle align linguistic
desire subsample variability benchmark except state benchmark sample candidate normal draw replacement early methodology replicate detection detect anomaly logistic discriminate anomaly normal al parameter choose validation point normal response anomaly provide quantitative level difficulty score score difficulty score difficulty score create benchmark bin dataset pd normal class decide impact practical consuming partition equal apply separately difficulty anomaly influence comes create compose entirely anomalous control contaminate benchmark draw candidate anomaly benchmark anomaly draw candidate anomaly benchmark candidate anomaly benchmark draw anomaly candidate anomaly configuration level yield anomaly reach rf normal target normal achieve candidate normal anomaly benchmark normal state impact anomaly detection relative anomaly normal desired dataset anomalous normalize sample variance normal point anomaly normalize anomaly exhibit semantic candidate anomaly algorithm set cluster measure euclidean distance location point cluster choose anomaly difficulty candidate anomaly cl level cl benchmark anomaly cl cl cl normalize normalize cl normalize benchmark anomaly score mathematically purpose score densely anomaly anomaly pose anomaly unlike many benchmark create instead run anomaly datum set ran choose clustered anomaly measure bin benchmark contain discuss early detect outlier assume offer set purpose irrelevant introduce add pairwise increase level average ratio ratio point feature sampling replacement original point marginal status point preserve real determine irrelevant feature expect distance vector want need extent set set anomaly four pd pd pd pd difficulty pd summarize anomaly c pd pd pd note anomalie pd even benchmark wish ignore point entirely anomaly exhibit undesirable multiclass majority candidate anomaly candidate pd pd many pd suggest offer great flexibility control difficulty benchmark benchmark entirely point difficulty vary anomaly point difficulty induce neighbor induce success normalize nc success nc great summarize near binary neighbor near benchmark anomaly especially binary multiclass anomaly usually create anomaly draw binary less anomaly requirement anomaly create benchmark anomaly great flexibility difficulty level truly flexible methodology unclear setting note cluster anomaly create match application domain anomalie binary difficulty effort seek create target level pairwise ratio level report add irrelevant assess effectiveness conduct describe tune employ cross validation maximize make effort maximize conservative implementation one shift datum away search boundary separate employ available radial basis execute reliably distance determine point support vector find available radial outside surface distance determine outli algorithm nearest compute average near neighbor roughly point anomalous significantly neighbor package choose small reliably detection probabilistic point anomaly model em numerical analysis retain single gmm robust select combine generate member ensemble varied value bootstrap replicate randomly replicate bag discard less log point compute assign gmm work fit know complicated robust et employ radial basis optimize forest liu anomalous isolate parallel forest tree subsample data select split uniformly observe datum totally isolated point leaf contain depth anomaly score depth intuition outlier easily isolate anomaly subsample anomalous anomaly address liu et criterion forest employ axis internal forest implementation subsample entire benchmark benchmark micro auc roc achieve micro probability anomalous analyze four anomaly bernoulli random begin robust package pd relative frequency anomalie pd rf model believe portion benchmark optimize apply auc expect particular benchmark logit prevent auc assume uniform beta much correction parameter place prior anomalous rank fit model level impact predict intercept remain fit tell change odd benchmark baseline baseline configuration pd pd rf rf cl perform factor factor produces yield large examine value auc figure quite anomaly benchmark construct learn show assessment classic excellent surprising research recommend design improvement worse recommend although control variation disadvantage make hard interpret intuitive display median average away similar drop third contain show relative pd notion difficulty intuition difficulty result easy pd pd hard pd anomaly confirm auc show intuition supervise anomaly easy intuition anomaly become imagine case additional anomaly become distant away anomalous span tend general become match plot advantage analysis steady plot performance mix capable produce cl cl impact relative easily methodology create benchmark show serious maintain find surprising euclidean poorly seek uncorrelated hence begin micro discuss sensor drift produce consequently factor present analysis point impact estimate good tune hyper maximize still tune write assessment propose publicly significantly bad matter concern argue discuss employ liu contrast publicly implementation liu examine code liu differ suggestion make forest sample algorithms parameter notably prove impractical level projection outperform easy recommend implementation real rarely know anomaly encourage perform irrelevant irrelevant understand add performance anomaly implication confirms extremely solve relative suggest practitioner g anomaly use domain obtain anomaly anomaly improve reduce anomaly eliminate e pose important challenge research anomaly handle clustered well suggest practitioner reason believe supervise method anomaly feature well recommend use setting may aspect produce big try forest recommend use start methodology detection control important benchmark offer thousand demonstrate four strongly influence anomaly benchmark accurate robust kernel estimation problem dimension great influence systematic anomaly benchmark anomaly suffer lack realistic construction limit ability understand determine anomaly anomaly thousand supervise benchmark difficulty anomaly anomaly superior anomaly variation four dimension choice advanced project contract nf finding recommendation material view research office anomaly task application security novel phenomena broken failure cancer cell field anomaly detection evaluate study hoc dataset publicly dataset consequence first compare progress understand dimension anomaly problem influence anomaly guide development standardized methodology statistical detection anomaly create realistic dataset systematically demonstrate experimentally exist assess anomaly detection method set requirement procedure construct realistic benchmark validate evaluate lead systematic anomaly detection clearly inferior third anomaly significantly understand lie choose anomaly anomaly detection value normal anomalous detection cite anomalous point anomalous natural alarm recall fraction anomalous composite area roc curve focus auc point understand sufficient training anomaly understand anomalous computer attack supervise study base discover machine failure law fail cancer disease rather whole understand mechanism anomaly anomaly rely target benchmark anomaly detection heavy combine assess anomaly kind anomaly synthetic dataset dataset construct supervise treat anomaly dataset help publicly security dataset anomaly anomaly decade experience complex validity finally retain e treat dataset without exception anomaly another et anomalie dataset treat anomaly idea systematically property exist anomaly benchmark systematically feature power irrelevant methodology achieve list first transform dataset repository classification anomalous distinct process rather distribution requirement anomalous criterion feature define dataset ensure construction benchmark work uci repository uci begin match follow multi time categorical ignore value exception anomaly cover detection focus common identically iid explore nominal ordinal network yield collection segmentation array segmentation letter optical recognition handwritten digits page concrete compressive year
analyze way interest principal dimensional work differ classical encourage factor author exhibit favorable array exceed make practitioner exist decomposition encourage outperform pca pattern interpretable penalty encourage gap propose tensor margin array present trivial intersection detail method recover factor exhibit interesting sparse temporal design smooth filter structure recover accurately exist array statistical ease array three observation hide however vary smoothly locally indice situation array spatial main generalized array penalize decomposition main challenge face result however unlike sparse unconstraine contribution exploit find penalize research case penalize work successful protein mass measure microarray use interpretable fused encourage neighboring solution filtering trend area need comparative genomic genetic close along different different column reference component array dimensional interpretability feasibility propose successfully apply area tensor decomposition describe sparse point sparse outperform decomposition directly penalize generalization tensor incorporate solution include relate follow notation definition throughout mathematical formulation brief state use highlight literature component discuss orthogonal decomposition address tucker experiment extension connect datum introduce material capital bar thus th formally likewise follow outer outer nj nx case three yield third mode tensor j matrix successively nn nn n wise scalar frobenius generalize scalar product tensor frobenius eq conclude scatter tensor scatter u ny ng nu u nj j nj n j ng j nj ng j ng j q lagrange eq q lagrangian observe separate special eq focus q dual eq path therefore
hessian adjust apply determine length matrix covariance kalman calculate sensitivity derivative likelihood use obtain evaluation derivative direction nontrivial posterior difficulty factor properly normalise analytically integration amount intractable intractable distribution monte method use approximately particularly integrate latent stochastic process tackle shall simulate markov construct call interest mean spatial average coincide average chain sure note commonly systematic way construct metropolis reject iteration distribution design newly probability newly add establish nice algorithm mh random walk proposal analogue present result run discard burn think way link amount treat together use augmentation terminology sequence naturally simple contrary via suggest make address bayesian identification identification procedure relate complete log likelihood iteratively step monotonic increase likelihood intermediate require quantity q directly intermediate tp tp covariance kalman reader explicit implement blue em conditional augmentation complete firstly provide secondly many sample trajectory follow generation variable form mutual represent method call admit stationary ergodic markov use expectation interested store discard worth pointing method instance sample exactly valid replace mh similarly nonlinear available trajectory gibbs apply implement trajectory simulation rely filter denote model density sample posterior gamma part sampler mh note augmentation implement approximate sequential associate non gaussian particle smooth approximate smooth marginal intuitively kalman nonlinear approximation distribution often spread particle represent system possible probable strategy arise identify derivative order find search likelihood come augmentation strategy quantity order particle principled filtering specific result kalman filter general particle application use expression generate approximation importance particle proposal account discrepancy particle assign proposal normalise q approximate furthermore result inspire proposal mixture choice part proposal mixture step use randomly component particle component particle time component ix refer particle conditionally also refer particle propagate selection replacement among weight account discrepancy proposal weight particle empirical complete particle freedom jx particle particle simulate time bring complexity sample work particle early influential arguably simple nevertheless propagation distribution particle appeal due choice unfortunately suboptimal account simulate particle proposal well strategy reduce computational weight computation instead explicitly introduce importance sampler mixture computational refer particle simply refer random simulating advance around refer particle generate generate variable depend identify interest x pf compute investigate convergence bound exist book give limit theorem weak regularity pf state reveal rate carlo ask variance recall approximation interpretation exponentially fortunately scenario ensure pf analogy pf identification pf approximation compute importance predictive proposal sharp first tn expectation randomness pf sequel estimator convergent normalise hence albeit however selecting derive approximate q approximate result index trace particle get full serious limitation smoothing know due resample particle weight resample degeneracy propagate degeneracy degeneracy important direction make simulate complete approximately smoothing particle promise smoothing introduction particle filter distribution position outline standard optimisation method possibly via identity intermediate log approximation smoother discuss hessian example identity note gradient way step method gradient asymptotic identification via algorithm intuitive idea simply replace intractable sketch auxiliary quantity auxiliary marginal distribution use unbiased dependence operate despite employ likelihood eq fact q recover extended target despite explain interpretation likelihood obtain current sample sample practice track likelihood estimate store development generate particle constitute one member seminal extend sampler use standard target proposal parameter together target discuss posterior pilot upper present result indicate dotted augmentation strategy treat integrate algorithm introduce closely link discuss correspond smooth natural particle smoother discuss solve subproblem idea monte approximate new simulated approximation current discard entirely result work inefficient particle construction serve subsequent member sampling intractable situation exact introduce systematic particle aforementione state retain particle interested connect compute x jx ix input trajectory return trajectory view trajectory refer usefulness come mcmc ergodic generate particle intermediate quantity variable employ intermediate quantity later mcmc nonlinear run accord base outline particle particle contribute factor efficient iteration fact even particle sample use parameter kernel complete state block invariance ergodicity property sampler arbitrarily conditionally set draw make parameter need available generate rejection instrumental dotted point direction challenge believe start provide inherent much applicable continue possibly entirely new concrete also call imagine weather example spatio promising tackle high dimensional know duality learning problem couple various fundamentally learn policy control work concrete trend couple various powerful continue evolve believe play role contract contract system mail se united mail
interaction video human recognize activity paper hierarchical video cut hmms semi graphical model divide category generative discriminative assumption concern reflect attribute probability regardless environment combination correlate temporal sensor scenario recognition crf popular linear chain inference limit capture within extra therefore names crf crf hide spatial interaction type action object object node within segment fully connect modeling interaction outperform solve less exact linear convergence study latent latent layer et al solve inference inference solve latent variable approach utilize explicitly independence transform wherein efficient human feature learn directly drive latent outperform graphical fig temporal video kk source pose accelerate correlated nature model implicitly learn consider semantic clarity imagine type action greatly video however capture formulate total activity recognize variable five potential score observation feature concatenation connectivity avoid conditional conditionally case consider aforementioned example dependent potential score coupling either represent potential potential characterize compare potential model rich contextual contain also latent fourth potential model compatibility among consecutive activity scalar compatibility activity compatibility activity global interpret potential sequence potential potential evaluate equal joint space objective therefore explicitly rather latent learn however initialization therefore great graphical loop general make semi maximize generally np hard applied efficiently acyclic loop transform chain tractable become result chain crf cardinality efficiently perform programming chain compute maximal evaluate record contribute max segment optimal segment know assignment track good assignment solve cost activity show max margin graphical example nn na activity unobserve automatically train goal objective avoid normalization provide balance fitting make incorrect activity compute loss return zero truth indicator loss view form previous recognize leave graphical unchanged framework track incorrectly predict action regardless directly involve substitute surrogate factor add exact inference compute solve concave tangent hyperplane serve term transform constraint add slack infer datum constraint solve cut learn em variable state learn decrease objective global avoid local minimum presented extensively transform action activity activity model insight show outperform contain sequence collect sensor skeleton person quite dataset label action subject room office include contrast video activity include show activity detect object illustrate challenge perform actor actor differently term view front b large label video partial also dataset video actor completely difficult body directly compare ensure feature consider action three baseline detail introduction evaluate dataset contain label environment label unchanged environment activity predict video sequence level zero focus action label structure svm margin rescale slack adopt initialization initialization state apply set well minimal object upon label categorical encoding transform therefore drive single activity learn structured upon baseline infer action classify activity multi augment activity refer successively hierarchical joint activity video manually annotate segmentation motion together segmentation enable fair two generalize performance evaluate fold fold choose hyper segmentation subject training testing validation training video perform generalization across data result average fold recall enable dataset consider remain imbalance report report score precision single al model latent precision recall f precision layer et dataset score report performance different action ground testing result pose wherein video activity label achieve average segmentation standard percentage point use truth term activitie information transition aspect activity table similar prediction percentage hierarchical approach exhibit improvement percentage reduce importance variable recognition activity notably recall percentage ground truth base performance percentage recall significant latent case illustrate contrast table start good vary level complexity adjust successively recognize show segmentation notably activity increase point percentage recall gain recall layer label prediction provide environment learn experiment outperform art precision percentage achieve precision score percentage group base location train test different term room al et std compares dataset al activity estimate action activity activity notably ground truth activity improve percentage row show confusion task overall recognition stacking value simultaneously activity exploit joint activity traditional focus use result jointly benchmark ph institute research interest robot vision human activity sc intelligence software china research ph computer university focus automate university computer track human across camera machine network sensor research interaction human sc china currently ph research interest include computer ambient university digital life research focus interactive device apply service ensure health security user artificial intelligence system paper scientific book issue conference surveillance association ai activity recognition contrast approach successively recognize activity action unify embed capture rich contextual although loop overall chain tractable therefore learn learn structured structured drive therefore manual result two model human activity people daily currently numerous provide physical fundamental recognize activity decide physical robot recognize person people continue need recognize activity robot interaction people propose hierarchical activity care water detect water rgb recognize activity build recognition object skeleton skeleton human activity type sensor task adopt
line number vary pdfs discuss experiment skewed density multimodal decrease htbp ccc integrate htbp pdf mse pdf type require kde multivariate multivariate multivariate bandwidth kde article notation report section multivariate require multivariate multivariate dimensional spread multidimensional kernel vector bandwidth determinant matrix symbol bound symmetric matrix give simplification diagonal precisely direction definition product derivative taylor series notation expand origin let dimensional vector differentiable df natural expanded taylor expand taylor relationship permutation put element keep unchanged eq derivation multivariate find derivation generalize gram series characteristic multivariate vector follow fourier take inverse transform bring convolution derive equality q pdf identify series satisfy multivariate reason notation use kronecker product differential variance estimation equation q kde assume near pdf required bandwidth multivariate use kernel product whiten direction derivative equation q need kernel q kernel bandwidth derivative visualization article address bandwidth kde multivariate series assumption estimate gaussian density unimodal case skew well multimodal density skewed asymmetric multivariate kde estimation generalize rule various encouraging realization parameter definite bound mostly quantify measure kullback divergence smoothing bandwidth obtain base criteria square expand taylor expand pdf sample obtain equation small whereas bias depend preliminary kronecker product repeat operator product dimension size general np nm f f jacobian match calculus operator also apply article f q matrix stack column one operator derivative differential obtain jacobian list scaling simplification simplification constant estimation article derive novel extended kde exist bandwidth minimization mean error pdf integration pdf well rule series expansion verification derive article extend gaussian gaussian univariate kde kde multivariate elementary calculus calculus derivation differential notation derivative kde gram derivative estimation widely processing upon select decide bandwidth limited pdf avoid either variety rule vary drive bandwidth asymptotic mean estimate actual criterion require square function estimate satisfy selector reasonably good estimator integrate square functional computation use fast order kde accuracy article pdf restrictive infinite selection extend gram empirical selection unimodal include skewed outlier univariate similar kde multivariate derivative estimation kde multivariate taylor expansion derive gradient vector calculus derivative involve polynomial use elementary calculus repeat product differential expression also elementary comprehensive counterpart use derive notation overall multivariate taylor series polynomial multivariate rule derive univariate exist datum bandwidth derive kronecker taylor series others derivation multivariate derive vector kronecker product series derive kde derive realizations pdf estimate bandwidth mostly follow property accuracy pdf quantify measure pdfs norm integrate error divergence optimal smoothing bandwidth measure integrate appendix identify I slow increase large optimal minimize total give derivative vary criterion vary criterion various rule bandwidth differ way name rough estimate gaussian accordingly deviation parametric family rule base computation plug rule derivative functional actual require order pilot density pilot bandwidth rule assume optimize put performance rule selection cross bandwidth criterion list rule low bandwidth selector restrictive estimation derive pdfs expansion verification concept pdf gx gx order approximate quantity obtain gram series bandwidth give separate selector performance test density
sufficiently additionally reconstruct word far reconstruct describe representation tf tf weight document document task word train language along regularization encourage align parallel corpus look representation phrase mention linear mapping separately train skip gram learn language align also useful representation phrase extract neural architecture training discuss tree autoencoder technique enable capture language language closely publicly language classify document achieve leverage corpora corpora language language directly language english en de english en en es embedding roughly million language pre processing remove remove label experiment document top category pre corpus mini batch speed merge adjacent pair pair hyperparameter tune performance portion available language compare use language regularization encourage aligned word embedding document describe mt document translate training phrase mt default embedding document report embedding mt baseline summarize report vice versa good outperform last table network document autoencoder aim make align close align sentence train embedding publicly en sentence accuracy respectively en one train de plus document cost en de en improve model still en en good en en report last de en en es mt majority supervise brevity observe importantly remarkably learn meaningful en cc en de en pr say pr office shall microsoft en de en microsoft microsoft markets competition competitive business also capture across language en english close word distance word language excellent merging bag word still sentence level essential merge bag merge several sentence word embedding exact essential reach good h l l en de en en fr fr en en label example application setup addition study consider determine wherein whether cca english language pair equivalence common approximately pair character represent english serve test standard news title original task give english word title pair obtain dimensional correlation representation threshold tune news seen clearly share learn cca mae common representation view propose capability one ensures learn two correlate scalable benefit world learn language representation learn believe view application amount example parallel english language common english act provide code valuable correspond author deep neural network autoencoder common abstract common representation wherein embed attention analysis approach cca joint representation maximize subspace learn representation reconstruct approach cca outperform task scalable approach neural explicitly mention approach correlate representation far employ cross language representation learn state datum view modality video movie audio video may available lot common representation view view application motivate importance view transfer item view learn reconstruct autoencoder reconstruct view reconstruct audio video available detector movie common representation view detector compute view common representation fed finally name write one language language view subspace item view correlate common formally view concatenation respectively highly correlate possible reconstruct vice versa canonical cca correlate representation suffer drawback scalable course try make cca scalable come capability reconstruct view cca put severe disadvantage view one view multimodal autoencoder common modality idea kind view notice mae explicit encouraging capacity view word develop mae view subspace verify report cca mae variant produce representation lack capability mae aim self guarantee representation representation combine approach main propose method allow cross reconstruction unlike cca mae capability useful view reconstruct view unlike mae ensure particularly item view descent mini batch modify world view three use mnist digit reconstruct ability common representation usefulness setup digit view language project parallel two language subspace employ representation task perform finally aim view correlate representation well cca mae organize describe architecture representation present characteristic cca mae language conclude remark highlight describe early common two reconstruct cca two view correlate autoencoder together autoencoder two propose variant autoencoder view goal section describe training propose layer single input layer consider discussion denote compute projection activation function layer try output activation architecture parameter sub outline goal formally view would self reconstruct minimize reconstruct view w b objective add common take create step deep layer common make hard describe specifically training minimize error training cross error contrast stacking outline objective pre align describe neural explain differ three feedforward neural network view single maximally train maximize learn covariate reconstruction clear neural cca single cca objective correlation maximization self self multimodal autoencoder mae neural though mae mae three error reconstruct ii reconstructing reconstruct fourth mae force network correlate secondly manner accordingly try minimize procedure three separately cca employ maximally maximize correlation experiment cca cca mae ability reconstruct learn representation mae handwritten digits dataset represent image use image set tuning list mae view half suggest reconstruction mae mae correlation view obviously learn view well correlate self reconstruction reconstruction half equally reconstruction emphasize show follow capture dimension four mae check plot correlation tune hyper dimension embedding aim transfer task digits image learn representation dimensional representation consider use provide classifier fold accuracy two leave right right cca mae decrease image less perform mae term analyze function term consider term capture reconstruct learn repeat function performance term row immediately wise row immediately
normalize statistic appear throughout regime illustrate efficacy real bi occur point work mix yield improve estimate property learn extensively symmetric whose entropy distribution distance algorithmic result establish thorough development area specific closeness identity versus main know summarize previous thought relatively essentially intuition element et optimal matching low bind distribution pp rd small mass remove tight sufficient introduce community refer version sample low factor al closeness slightly closeness analysis two set sized asymmetric closeness determine require versus distinguish large et al imply distinguish two distribution bind gap result testing mix chain go back expansion pick testing graph independently provably rapid test computation relate symmetric nonzero task achieve use test markov often comment testing hope additive strictly difficult distinguish result test distance estimate sample unknown theoretically wants distinguish test distinguish exact bad sample distinguishing understand though case logarithmic linearly exponent go closeness constant factor bad sample theoretically distinguish lower uniformity step propose two versus regime algorithm incorporate appear appendix robustness hypothese complexity application extreme mixing factor oracle mix finite chain query versus time obtain mix improved theorem testing chain closeness run begin state throughout portion assume sample distribution number occurrence expectation complexity set obtain complete calculation employ defer testing testing finally section empirical suggest statistic illustration gram contain word testing give extreme regime extreme modification basic wish test partition sample denote respectively occur draw I check accept reject intuition probability capture empirical frequency frequency second set element use modification size similar instead numerator statistic possibly note seem poor performance additionally unweighte easier heavy light ease empirical suggest want certainly perform heavy light independent copy tractable need extreme component qp n I I appropriately choose accept asymmetric assumption algorithm appendix variance draw variance threshold tailor design distinguish hypothesis unbalanced modification closeness case follow algorithm work whenever suppose pt b I reject otherwise reject summarize probability worth robustness extreme involve modify present take probability versus proposition closeness size markov matrix stationary start step step product zero everywhere chain xy markov al closeness test closeness improvement markov mix test mixing involve every running start state runtime markov chain sample average state whether accept characterize markov sample versus least stationary apply geometrically repeat one obtain formal involve likely provide statistic core small primitive natural language distinguish surprisingly small select occurrence random occurrence books corpus google books dataset henceforth refer set bi gram involve convenience whose illustrate statistic range rather essentially identical pair grey differ variation preference bi contain first word second vary provide reference value correspond e depict red variation empirical distribution word versus different size subtle
sub exponential loss long sense characterize thresholded descent size match retrieval explain unchanged change banach simplicity detail thresholded apply thresholded numerical illustrate relative estimation size sparsity signal thresholded proof give defer thresholded solution due avoid away initialization crucial provide justify assume give magnitude statistically heavy deviation square recent progress modern preferable due square empirical update step reach appropriate condition gradient flow global complex direct ideal phase retrieval utilize signal contamination noise incorporate priori end guess update thresholde tn compute update recover avoid greatly result reality knowledge additional descent intend behavior restrict update thresholding thresholded flow noiseless retrieval thresholde flow drive motivated independent act combination thresholding literature although choice justified notice validate later ht j select crucially first minimizer thresholded propose yield diagonal thresholding collection estimator focus marginal coordinate select coordinate least focus treat constant choice later thresholded flow j independent sub maximum fundamental sub absolute efficacy lemma explain thresholded defer interpret follow noiseless probability signal noisy mt upper intractable contribution fast ignore thresholde guarantee condition sensing previously principal hardness plant paper size computable discretized establish bound gaussian design future word thresholding idea interesting converge rate thresholde iterative project optimality satisfie property convexity smoothness thresholded aim dimensional apply risk satisfy rsc matter sample show optimal precision recovery rsc thresholded regularize widely dimensional thresholding e alternative sparse risk non interesting guarantee precision local optima long condition possibly satisfy strong convexity appear empirical eigenvalue se however phase rsc optima consistent penalize version strongly large global lie sample minimizer optimal question matrix respectively cl ki si consequence moreover eq let exist satisfy c l k j notice first know sub bernstein see absolute chebyshev bind eq q l x l eq next probability k mp ss moreover probability least eq eigenvector unit e c support upper separately lemma inequality provided least provide inequality summary guarantee absolute describe lemma argument thresholded iteratively n se ne soft eq q know support moreover support assertion establish show q eq p theorem imply condition straight constant take satisfy proposition due lipschitz function whereas imply satisfie proof independent exponential constant probability least sub eq chebyshev inequality let absolute sphere exponential bernstein probability probability rgb chapter remark exercise conjecture subsection height consider phase recover measurement goal computationally optimal rate retrieval thresholded show adaptively achieve minimax sparsity provide retrieval recovery thresholde researcher recover signal contaminate rise terminology extensive theory ray array quantum information impossible observe one treat multi refer interested scientific engineering background relate sparse deterministic linear transformation without rest paper case setting may
would streaming scenario unique near neighbor find offline forest need sub complicated test answer decision tree question r tree end search portion see slow metric informative online scaling learn bf projection speedup real tracking pass raw information boundary learn dna come dna protein dna letter mnist protein number recommend dataset repository data dna letter mnist letter dna letter protein dna forest bf supervise forest store see datum online bf crucial application need input flexible learn speed setting art learn often practical situation seek learn ii fast iii iv satisfy particularly importance problem stream forest bf boundary bt store bt structure quickly modify relate near boundary different class arbitrarily shape fix bf retrieval include geometric neighbor nearest build structure need access entire bf tree ball costly require volume dimensionality bad case calculate computation example tree bf latter decision forest find method preferable decision make bf projection speedup maintain create tree fact tree well tree online latter library naive near neighbor extensive try reduce store near neighbor add compression enough build forest root consist represent tree start show example independently trivially associate retrieval associate specify output real need child tree move compare child query unless close child case none child locally close child could potentially get child add show finite low negligible label compare vector position boundary node consist child tree child smaller add close break position call take set vector training call min bt min x position label vector iy process leave locally close happen task retrieval close close approximate give position locally vector weight answer use would bf locally close tree locally query example label vector tree whether new position label edge decision classification label need add bf follow root bf practice root training emphasize training strictly find power assess training example accord ask fall say good fraction fig within law time bf example note bt bf node bt comparison bottleneck bf algorithm observe time sublinear switch happen bt appear child understand consider artificial situation point remove go node child recursively stop connect node stop traversal child stop probability eventually logarithmic coincide root scale bt train dimensional hypercube bt behave like bt plus correction node child child grow metric root child child argument set tree behave root child query balanced metric power child must root query root increase inner bias bt train hypercube increase phenomenon cover ct handle neighbor train online use ct approximate nearest ann ct study previously ann ct point ct little speed important base original suggest publicly default change increase scaling amount bf rescale bf thing ct ct triangle obvious bf ct draw hypercube example bf see bf neighbor computational maintaining since traversal bf informative bf perform implementation multiple give well good tree visit number lead give repository use metric intensity even though give generalization bf ghz intel cpu
ii iii fixing vi moderate size correlation function increment simulate set material ridge probability size increase importance tend sample size rough recognize quite make concern screening screen use computation efficiency sis fix record fix sub vary sis compute inefficient comparison provide sis efficient sis forward sis incur approximately active time preferred computational complexity step refine evaluate screening selection stage method six choose scad use extend determine minimize model stage sub use extend model compare scad scad sis fr fr scad lasso scad apply sis measurement negative wrong wrong true select true false negative second simulate sis sis calculate discuss cpu responsible probe logarithm normalize link gene high nine study examine fold cross validation prediction report error final choose nan ccc final scad sis scad fr scad sis scad cross might interesting gene full detailed supplementary material choose report scad marginal efficiently compute successful computation extensive among good screening circumstance resource dimensional ridge go zero concern close degeneracy matrix dominant magnitude may dominate illustrate phenomenon ii screen propose issue theorem ridge xx middle employ combine ensure subset divide close great deal regression paper study nonlinearity compress sensing sense satisfie fourth graphical elsewhere thank constructive comment wang es national institute environmental health sciences definition lemma far exceed independence screen sis reduce sure screening rarely reality overcome simple screening technique ordinary possesse screen consistent correlation ridge simulation correlation disease illustrate inverse ordinary square sure rapid advance technology complex exceed ordinary square estimate ol long lack sufficient recent develop handling set assumption affect response loss sparsity lasso group selector elastic accurate estimation discrete although dimensional case computation concern desirable reduce refined analysis concern dimensionality quickly sure preserve property sure play role sis screening hazard response extension correlation deal marginal correlation aspect screen design operator consideration computational screening estimator possess sure assumption otherwise sis operate estimator efficiently scale sure important away refer hereafter set often correlate highly hand important correlate marginally reason iterative sis repeatedly apply sis finite classical name projection motivate sis efficient sure screening restrictive correlation discuss version theoretically possess sure interestingly ridge retain true tend extensive elaborate motivate compare analysis confirm conclude proof material familiar error alternatively realization distribution assume invertible true nonzero cardinality look map sis emphasis screen important maintain nonzero relatively q combination preserve satisfy part would part argument least square degenerate identity motivate kind material particular long term dominate quickly iii iv propose diagonal scenario pattern singular decomposition orthogonal diagonal belong see supplementary material intuitively impact random prove supplementary dominate dimensional least square ordinary square follow select retained screening projection ol project onto use capture forward regression another screening give dominant likely contrast forward goodness whenever mark analysis motivate ridge sis ridge inverse formula supplementary material gives see extreme opposed let real often ridge degenerate implement sis computational sis scale sis invariant affected predictor response denote respectively standard easy identity tail behavior different family tail zero independent follow classical admit characterization analog show include random exponential moment integer various place denote constant symmetric large small assume error tail sis need easily violate correlate weak concentration directly screen fact noiseless sparsity establish present see interest strong consistency far specify simply thresholde state choose guarantee select seem surprising however consistency condition parameter pre consistency screening assume standardized depend become recall control similar property satisfie hold recommend suggest screening procedure practical preserve choose extended simplicity mainly provide screen robust fr numerically assess various successful screening screen procedure screen sis forward evaluate report motivate sis investigate iterative variable entry iterative slightly much consideration decide simulation adjust theoretical high covariate six simulation report include true selecting due cost predictor coefficient structure use distribution coefficient factor reduction independent distribution choose specify correlation pattern independent control comprehensive generate motivate example response predictor predictor predictor extremely important predictor l sis ii iii autoregressive group vi symmetry iii autoregressive extreme supplementary material summarize noise
elimination state indicate gain assumption first result large difference result difference consider set r n reduction difficult recommend obtain use effect compare activate closely successive elimination benchmark two world begin multiply loose recommend theorem ever value suffice size preference simulate respective moreover theorem state log web set microsoft rank approximately query contains label microsoft data label ranking feature whenever ranking feature rank relevance document document result flip find winner alternate hypothesis differ index score select arm arm since assume pa j b j iterate arm prove technical bernoulli ji bernstein bound repeat triangle ti j jt ready probability hoeffding begin prescribe arm definition never side greater equal n tt rt arm sparse discard arbitrary I theorem analogous hand equal inequality step recall remark electrical computer engineering armed bandit pair arm new arm gap suboptimal arm particular new sparsity winner experimentally synthetic result sparsity improvement classic armed action rather arm represent bit response people ask paper exploit different notion primarily concerned winner arm probably every drawback winner unless underlying comparison show winner winner exist make impractical drawback prefer choose winner always assume paper winner winner armed score order oppose quadratic winner winner consider motivated existence multi armed bandit exploit structure explore capture behavior candidate compete winner mostly irrelevant predict winner concern successive arm well experimental standard approach dependent low winner show multi armed essentially logarithmic structural structural assumption top arm distinguished arm bandit show complexity improvement naive multi armed experimental demonstrate modification arm bit indicate well prefer constant th entry exist focus exploit furthermore make existence winner winner arm winner arm say high winner attention winner winner reduce bandit arm simulated arm arm however far winner winner web winner matrix allow comparison violate practice secondly winner winner winner would prefer case arm winner winner arm marginally arm prefer marginally preferred winner design robust estimation matrix vote winner winner finally define win choose uniformly unknown become bound depend preference preference score hard another course apply algorithm winner collect matrix exhibit kind difference motivate preference matrix index number arm gap like approximately argue certain unknown permutation winner recall ignore argue permutation winner winner probability arm uniformly random hoeffde reduce arm identify arm consider index winner easier reduce top two guarantee winner scale argue gap winner ask know e index learn exploit answer argue dependent complexity arm general algorithm tight reduction bandit pac select winner inspire bind j pac bandit algorithm winner choose imply preference extreme real dataset aspect winner distinguished subset find winner sort structure look two dataset consider microsoft web list suffice arm arm cardinality contain great indicate reach plot winner order small away message unnecessary estimate score arm gap
user request receive uniform width know give sample guess sequentially randomness query return return passive strategy budget avoid outside let use passive estimate noise margin characterize around threshold give constant boundary half boundary together threshold getting seems substantially easy threshold capture brevity denote passive analyse minimax equivalently mean mean follow label risk nk n mx threshold interpret quantity exponentially e rate rate paper prediction noise true change noise noiseless passive learner notice remain noise noiseless active learner rate passive vary smoothly coincide coincide well furthermore might inequality intuition help noiseless active learner add however subtle claim rate get small large even behave quite information idea quite flat convolution noise less behave linearly region nearly regardless drop assumption threshold shift quite trivial ensure behaviour flat linear shift intuitively help contradiction control passive describe handle making cross see lead leave first technique bound passive active eq lower follow passive intuition generalize passive passive approach bound feature suffice hypothesis let measure kullback leibler hold minimax loss metric threshold two threshold least use noise distribution risk learning determine model follow past passive iterate need passive reveal convolution noise regression grow region differ iii region vary active learning choose passive apply since differ interval rise passive similar passive minimax calculate notational convenience shift q explicitly understand e implicitly threshold respectively point proposition verify bind length difference point na n verify level large point equally stay epoch end phase induction always start epoch epoch phase nk setting detect noiseless argue work noiseless suffice unchanged proof bin except bin leave claim I r get point two phase phase large second verify must design intuitively query geometrically query successive epoch noiseless error shrink claim w final propose threshold setup analyse learn already feature passive level minimax noiseless continue large level achieve passive passive seem powerful carry beyond get emphasize possibly due denominator class constant observation function convolution uniform seem figure without quite task base tight acknowledgement thank support nsf big x cx cx calculation satisfy whenever c prove detailed calculation check prove cx cx cx w w cx dx w cx boundary cx dx cx k f w cx k w k k k cx dx f f w k k k proposition allow w dx cx dx boundary w w cx cx k dx w cx cx cx w w k w cx w cx cx k k f give k cx dx cx w cx w w k respect k cx cx hence k f verify complete ease presentation assume expansion look like particular let break form claim immediately see outside b small enough make set epoch feature noise epoch induction high trivially epoch epoch epoch threshold point stay epoch epoch imply final n active notice choose er ec r ne epoch since become theorem theorem definition department department university user sequentially transmission oracle returns label feature noise error variable study extensively study additive uniform set one
tackle jointly exhibit profile survival particular offer direct natural extension risk order survival survival event condition common fitting lasso prevent enforcing efficiently descent report breast cancer experiment outperform cox logistic predict task consider survival analysis make associate censor tuple censor whenever class patient expect survival patient risk model consider supervision viewpoint logistic specific patient interpret score correspond either respect look survival event compute patient patient still since patient risk see patient low also express aggregated partial event event cox model survival except censor computation restrict define event time conditionally compute naturally formulate cox enforce net solve gradually decrease model include zero extent feature covariate normalize partition group first group feature predictive survival group represent combination parametrize feature run model cox hazard select column illustrate feature survival report term correct harmonic outperform approach task predictive cox breast study repository sample distant common patient feature large variance objective tumor low versus equal survival report replacement various feature predictive run harmonic mean aggregate representative performance overall benefit model original mm generalize predict survival jointly classify induce offer embed discover informative specific generalization multi continuous
arm fix eq identification sample population solve confidence draw reward obviously computation induce solve ergodicity readily available finite regular reward one regardless gap complexity bound reward index arm correspond sample reward th variance estimate arm I drop notational simplicity context optimal identification adjust finite set algorithm maintain confidence law iterate logarithm high update algorithm terminate reader detail ucb achieve complexity reward arm could ucb terminate criterion arm sec ucb terminate reward replacement convergence take maintain mini sub stop remain equal schedule leave whenever alg stop within iteration computational former price specific alg arm adapt draw tight positive typical inference size mini uncertainty index find good eliminate sub gd I I gd c inequality replacement improve later subsample mh miss valid uncertain empirical bernstein bound concentration make assumption provide prop let replacement match mean covariance p central joint assumption uncertainty intuitively cdf standard choose normal equation union account notice runtime mini batch provide arm reward nd depend gap signal technique reduce compute efficiently reward adopt expansion reference point gradient analytically moment typical choice mostly depend reward conservative trick exploit close trick extend rejection optimal fix tailor discrete problem ucb latter assume distribution reward hold algorithm subsampling mh respectively evenly free take schedule replace original binary sampling also alg family subsampling sampler discuss sampling adapt ucb tb interval proportion desirable adapt ucb reward close heavy tight pairwise estimate draw loose ucb sharp large direct uncertainty joint author mention author name parameterize take empty case name sample subsample important mini union label label f use varie vary score almost identical negligible relative evaluate initial dependency algorithms framework guarantee improve original also evaluation speed inferior arm future relax assumption efficiently distribution condition prop almost I reward chernoff population without replacement adapt second valid tight theorem adapt never big prove output adapt ucb update change arm eventually original arm return mean alg iteration arm become require g sub eliminate last correct prove arm alg marginal union none happen eliminate use x high estimate alg marginal estimate pe define alg rhs plug stop upper bound alg pe follow proposition vary iteration visualization release code numerically e e e perform plot pairwise also variance perform empirical never exceed adaptive exceed heuristic theoretical ucb perform significantly ucb thm perform significantly bad ucb term scale log scale confidence show sample plot log scale sample overlap easy log scale estimate sample normal stochastic volatility let logarithm return asset remove problem auto treat inference mcmc introduce complete model assign regular without reversible mcmc ideally model conditional except mh normal control likelihood could order magnitude sure sample often wang approximately fix compare separately stochastic langevin bins quantile notice subsampling size approximately thousand already mini batch abuse use empirical error mis independent exploit modification pairwise strictly score communication discrete algorithm sampling suffer high burden typical scale efficient approximate solution connection arm bandit finite population provide empirical robustness efficiency algorithm synthetic real conditional core operation necessary component filtering apply challenge burden type large large random describe chain monte
model trajectory system besides fail long term mark separately trajectory cost c e next control system line start angle balance reconstruction face difficulty angular velocity image discretization error pixel exact markov property stack inner sep draw west center cm edge west showing position topology model trajectory space almost perfect reconstruction learn meaningful position remarkable table well stable dynamic balance explain non magnitude unstable real globally high six six input reconstruction visual classical history raw robot pixel image latent dimensional cart control arm dimension control use previous space depict image execute slightly obtain operate real start material experiment representation deep autoencoder ignore transition g stream employ learn raw non autoencoder forward train neither prediction learn dynamic control approach recent variant deep observe desire transformation transform learn receive considerable year refer recent overview implement multiplicative interaction find model filter recent discussion system stochastic stream state benchmark reveal embedding ease system acknowledgment thank provide link would discussion partly foundation grant program parameter propagate start approximate identity without network truly sample method main simulate dynamic simulator produce input robot arm link link compare always report use real cost goal state execution achievable control model cost dynamic create training one subset prediction autoencoder extract combine dynamic method fundamentally achieve reconstruction hyperparameter minimum make phase unfold trivial minimization architecture relu conv dimensionality encoder relu relu relu relu relu sigmoid relu relu c dimension dimensionality relu decoder relu relu relu relu except c dimension encoder relu relu relu relu relu relu relu relu conv relu relu relu input action relu relu max pooling conv relu relu relu relu relu relu relu conv relu relu relu qualitatively predictive accuracy latent latent position multi start angle velocity predict force apply angle force prediction cart long cart unstable fix dynamic cart move change angular velocity move right c prediction bias consistently cart attribute angle make accurately image input left depict case trajectory omit image figure system one ahead prediction combine model predictive control supplementary deal space cart robot arm compare locally model always velocity always robot angle slight advantage h rgb de uk non system raw deep belong autoencoder learn locally linear latent support prediction exhibit variety dynamical system broad reinforcement prominent aim algorithms linearization combine problem relatively ultimately need capable complex difficult apply usually thousand content typically advanced algorithm raw turning locally high identifying locally learn latent class autoencoder derive formulation probabilistic trajectory plan train fully compare aside briefly review dynamical introduce locally finish derivation dynamical step system smooth use notation restriction require control perform learn map dimensional solve account noise equivalently approximated transition tf system analogue assume control instantaneous cost tf minimize control optimal locally time reference trajectory linearize offset efficient control weight c formulation control result trajectory optimization locally trajectory formulation appropriate pz property information enable reconstruction prediction latent next observation require highly non crucially hard subsequently transition linearize directly impose desire prediction next property representation formalize pz z state generate access follow bayes resort outer sep draw corner height outer sep line pt thick angle black fill solid north mlp south leave leave sigma sigma gauss east edge gauss east gauss center box z z north east south north tr west box right tr z north east edge tr north edge west tr west z east fill white xshift white north north yshift fill white leave close sigma sigma gauss inter gauss sigma close gauss center north gauss mlp west east south east north shift north z east west north west south mm cm none sep encode width east draw none width mm cm draw inner shift th diagonal z x n compute network activation hide layer learnable encode weight bias variance natural expressive come gradient enforce yield generative generative operate aim dynamic reconstruction image decode network mean white include make learn linearization offset covariance transformation base network offset tn z prediction require distribution planning enforce latent lead possible valid action since come prediction fail model markov chain form model datum tuple obtain interaction dynamical true log loss per inference generative isotropic mean unit additional contraction weight agreement chain transition kl expectation via take give sample minimize descent highlight layer evaluate visual visual balance cart control link arm detail type connect moderately encoder accordance adequate throughout state action cart list parameter hyperparameter choice open
decision treatment decision sensitive receive consider use target ordinary learner aim find minimize give learning viewpoint individual aim little viewpoint company want decision job include include gender viewpoint wish base job supervise say aware attempt avoid minimizing minimization misclassification therefore need trade misclassification dependency resolve accomplished term add minimization lead give dependency predictor empirical low sensitive generalization job predictor exist might fair decision job empirical decision dependency except theoretical provide dependency unify aware universal divergence exist variational divergence aware erm aware predictor upper bound generalization dependency extra restrict low upper bound estimating divergence aware state constrain divergence guarantee tight divergence achieve generalization divergence provide expand divergence second bind loose purpose tight divergence introduce discrepancy formulate erm aware employ generalization generalization rademacher complexity regular erm thank compare estimation divergence divergence constant aware measure represent readily solve solver setup describe elimination viewpoint insufficient achieving viewpoint indirect job train exclude hypothesis may address may correlate indirect exist work construct naive employ discrimination difference aware erm total preserve propose ff learn design couple dependency measure unseen dependency derive error term dependency iid extensively distance divergence loss divergence moment match property conjugate derive estimator divergence analysis derive upper dependency exist obtain measurable addition seek measurable rate viewpoint misclassification generalization learner directly empirical erm find risk relatively number theoretically dependency viewpoint general measure dependency difference divergence suppose compact absolutely continuous divergence divergence measure generality subdifferential confirm change include divergence kl attempt rf f trade parameterize eq generalization learner empirical evaluate underlie procedure divergence upper empirical objective aware maximum mmd estimating estimation estimate evaluate estimate expect mmd mmd fr function universal discrepancy dependent choice u statistic give regularizer mmd ensure empirically addition regularizer estimation surprisingly rf rv addition true state mmd contain subdifferential almost q divergence divergence mmd divergence divergence minimize mmd order divergence large lead rate empirical cover divergence formulate procedure aware define indicate nf mmd us effect estimation depend small hellinger divergence various convexity respect assumption convex simple hypothesis form rkh give appear appear problem rewrite relax prove claim convex problem error bind algorithm theorem x n obtain error aware learn hypothesis appear generalization fx learn divergence upper divergence compare erm accordingly hypothese divergence reduction cause restrict divergence rademacher x fx appear aware misclassification viewpoint contribution estimate aware estimation introduce erm aware bind generalization mmd considerable product include
head associate intrinsic intrinsic set consider always appear appear simply include conversely satisfy condition head tail h ji ta vertex edge remove cause subgraph standard run nest markov parameterize parameter index see detail consequence algebraic define closure irreducible strictly simplex semi irreducible closure dimension prove section fairly technical particular cone marginal nest space prove py ax py vx keeping span space uniform totally cone around collection empty subset direction vector contain tangent cone idea nested graph constraint lie show model perturb rv thick minimum mm sep control control control u u achieve random function formalize parent parent assume value variable parent label vertex value fact fixing initially independently sample lead completely uniform observe clearly generate control distribution us direction tangent know argument direction x x direction give additional since parameterization direction cone tangent locally restrict loss concern marginal satisfy run intersection order edge share contain order intersection call choose rather idea terminology dag edge may decomposable simple cycle run intersection edge share different call addition vertex order q ordering since take remain latent attack result paper contain exactly variable residual behave visible parent vx vx value take say decomposable fix dag replace structure b give variable dag contain parent rv minimum inner sep distance u control rv thick sep mm parent take vertex parent set construction direct remainder induce conditional variable represent dag satisfy parent three component additionally parent parent lastly formulation tell rather parent must remainder set lemma vertex exist proving within remainder I exist vertice root degenerate q lie span connect I change nothing connect rv draw thick mm mm node nest non empty consider cc apply around uniform nest marginal associate induce inequality joint nested marginal identical theorem far fix space without tangent compose subgraph circle mm sep mm xshift xshift subgraph suitable tc span tangent subgraph variable choose mechanism share parent result distribution satisfy markov distribution uniform perturbation generate follow choice tc proof cycle nest graph subgraph however ii graph q obey nest use subgraph apply see span contain tangent nest model tangent together everywhere manifold boundary describe algebraic nest closure irreducible variety furthermore property set interior marginal irreducible variety closure subset exponential nice statistical asymptotic boundary active distribution generally complicated effort computational generalize graphical criterion obtaining np algebraic elimination latent variable discrete without generality marginal define inequality probability conjecture however graph marginal state space potentially cause latent model nest maximum ml latent mle claim loss denote px ax q lastly count dimension give whole build degenerate sum degenerate degenerate degenerate matrix clearly degenerate degenerate replace degenerate degenerate degenerate sum degenerate disjoint particular degenerate degenerate j db particular degenerate last general method proof though observable f vx v c I w w aa w suppose degenerate degenerate letting combination follow linearity part summation degenerate sum identical summation first construct definition impose maximal claim begin inductive imply root subgraph lem lem proposition lem lem conjecture lem remark lem mit imply latent nest nested well possible variable avoid inequality extremely complicated asymptotic identifiable represent exponential easily fit parameterization acyclic dag widely machine causal discrete jointly greatly flexibility unobserved cost create consider latent identifiable regular asymptotic specify parametric latent variable additional assumption generally difficult implicitly avoid characterization rv thick sep node mm rv color five show graph represent treat write form margin dag deduce word satisfie four distribution constraint subset immediately model might restriction constraint sufficient describe model inequality equivalent markov property consider conditional presence refined account model know large marginal always sense model yshift bend bend black black circle n sep node xshift xshift size mm tc dag situation lie within nested sharing tangent full algebraic characterization model represent sensible complicated nested work easily fit addition regular suitable propertie discovery currently causal structure without make assumption hyper associate dag model margin carefully nest outline main show marginal manifold begin elementary collection distinct parent similarly denote vertex dag visually dag vertex edge generalization dag introduce dag vertex restriction vertex disjoint rv circle inner rv yshift reduce dag denote square one round identification associate probability determine density obey dag density reduce familiar extra represent also factorization dag exist random noise variable criterion equivalent always property model structural obeys satisfy obeys dag possibility integrate latent marginal random vertex margin principle however existence dags dag instead additional figure latent rv circle mm treat random hyper element inclusion fix aspect change theory necessary nest dag node circle rv thick inner sep node u u yshift control represent dag replace edge child vertex canonical dag dag spirit let marginal model distribution construct margin distributional marginal define dag represent appear restrict variable parent vertex edge edge b rv draw inner xshift cm together parent direct ignore subgraph
recurrence relation appear mainly additional epoch define I recurrence n simultaneously difficulty infinitely overcome bound net infinitely still need probability enough compare already bound epoch large small period epoch precisely epoch bound epoch phase end epoch denote note end epoch bind finally event key appendix mention small enough lemma failure probability complete determine certainly check consider recall qr decomposition r h qr I turn block power refer block sample allow different precisely step block block later basically grow call space easy result focus dependence simplify analysis bind n recall would like suffice start fact next I I q r ii rely suppose eq q fact estimate begin save sample keep lemma appendix enough sample algorithm remain bind achieve rely put complexity represent bound omit experiment thousand match normalize generally report optimistic algorithm fix follow block l propose block insufficient ratio run dataset evaluation value lead visualize clearly figure check algorithm pt choice competitive figure small block achieve error sometimes block slow keep block ensure update result less block begin far reduce error error match reduction choice easy tune favorable rather figure similar affect update less slow result tune deep proper show support converge however immediate block stream family fairly theoretical empirical side sgd principal family dynamic enjoy fast original study demonstrate dynamically justify family block new research competitive result substantial hide conjecture hand suggest hence worth study could improve use omit proof well appendix remark therefore assume induction accord recurrence appendix instead look net happen eq therefore assumption I lemma satisfy recurrence relation q finally choose n give u u matrix mean chernoff bind eq first second fact lin national chi study recover stream family previously set exist easy analyze representative family moreover set dynamically empirical family real advantage datum goal component principal component covariance store moderate algorithm run batch streaming recent streaming store feed solver streaming huge arise modern pca main restrict usage amount measurement goodness streaming project point low spectral span solution span component th eigenvalue th reach somewhat hard spectral algorithm meet consider pca along regret guarantee reconstruction order convergence choose proper size real contribution perspective block easy guarantee well conduct experiment real provide concrete recommendation notation enough let integer spectral streaming point assumption sample
simultaneously contribute result conclusion plausible consider product product force inference jointly plausible individually noise configuration conclusion efficient inference variable useful sum variable want conclusion max specialized transition probability hmm state forward backward viterbi product hmms step process convolution viterbi currently perform max convolution valid sum utilize max convolution speed convolution yet max convolution x n case prefer sort guarantee useful decrease amount preserve unitary sorting explore event adapt convolution replace pair max product require fast try derive equivalent max challenging convolution use operation value number convolution apply probability operation convolution convolution regardless convolution addition two l operator prevent exploitation lagrange convolution exclude highly specialized achieve contain value convolution applicable runtime bad certain expect proceed sorting list head update yet still index overall runtime become despite pose suggest due overhead sophisticated algorithm neither sort value suggest result theoretical suggest subsequently extend min alignment wherein collection circular align sophisticated highly complicated short path max runtime improve pair short solve previously mention solve practically moderately sized max easily library transform input output turn max chebyshev shift rewrite consist rewrite chebyshev ignore expand back original place appear r yield strategy introduce result library nonnegative use numerical max convolution r v r v often loss precision input late way unnecessary recognize since normalize estimate computation final divide dominant parameter max l r l r l convolution implement automatically naive numerical naive overhead roughly expect running reduce briefly numerical compare implement package convolution implement fast k pair vector uniform element draw result ok convolution fast demonstrate substantial speedup curve speedup axis dominate empirical demonstrate max computed value respectively demonstrate replicate numerical naive different value low high value scale manner large satisfy compete large significant maximal term contribute although sophisticated yield exploit relative fairly intuitive formula relative consideration normalize maximum convergent substantially zero high numerical yield improvement call note piecewise extend increase runtime decrease routine terminate index adequate estimate convolution nonnegative return max l optimize specific application p f enough issue avoid lastly way piecewise convolution code code solve simulated people person price person knowledge cost bin fuzzy amount try amount spend gaussian likelihood variance plus problem instance likelihood appear convolution several case second distribution naive fast plotted demonstrate product sum close ultimately setting mention sum far computationally occur numerical many problem practical high quality probabilistic subset problem solve convolution mode indicate bar large sum operator less discriminative curve connection max pairs short path fast method science numerical theoretical solution depth would decrease one transform convolution argument series perform suited transform represent locally choose value uniform max close zero thus high chance high index indice maximum toward take element current dividing may number large could accurate property solve max bound subproblems subproblem numerical initial subsequent call qr way use furthermore obtain always interest improvement ever method traditionally empirically max inference perspective rather perform product inference view hyperparameter position event end product value sense convolution already moderate max convolution piecewise available acknowledgement edu perform transform convolution min convolution estimate max nonnegative two mass length numerical demonstrate fast fast inference arbitrary contiguous convolution specialized runtime viterbi path problem mass similar mass occur small nuclear mass read rna location genome sum many copy read protein source responsible discover protein knowledge individual information inference particularly increase field sum biology effectively utilize sum ability meet collective turn convert several back address peak multiple read discard limit direction might substantial effort distinguish result peak binary discover arbitrary discrete decompose information
machine aspect share preliminary compose expect insight boltzmann brief boltzmann machine truly boltzmann element note english slight article originally recently boltzmann machines boltzmann without number pseudo boltzmann machine use successfully boltzmann investigate system boltzmann truly boltzmann machine compose element although machine two give dynamic large boltzmann since variable boltzmann machines spin physics boltzmann probabilistic addition internal state boltzmann internal namely initial r eqs boltzmann machine variable map boltzmann machine map rectangular boltzmann follow state space differential rectangular move space change rotation rotation ergodic uniform distribution move velocity brief note boltzmann quasi periodic probabilistic rotation number rational lebesgue carlo zero
length tree help sequence true would expect great long sentence fig relationship measure specific bar omit clarity observe significantly sequential short encode structural information applicability variety nlp task substantial learning long build rnn avoid confusion neural tree rnn associate child composition numerous variant basic tree rnn phrase representation word classify sentiment sentence generalization topology arbitrary branching factor lstm two task control dimensionality lstm outperform sequential suggest role sentence thank anonymous valuable stanford university natural project exploration filter program air force laboratory contract finding conclusion recommendation express view recurrent modeling lstm far exhibit syntactic combine lstm structure lstm baseline sentiment stanford representation phrase sentence model value represent mean fall model bag word representation example representation contrast token lastly phrase sentence syntactic insensitive insufficient mean syntactic tree model relation syntactic interpretation sentence natural extent tree oppose address art nlp structured generalization x x chain structure lstm structure branching due arbitrary length recurrent rnn modeling recently rnn architecture effectiveness capture successfully variety prediction notably speech recognition program execution standard lstm topology superiority represent sentence lstm current lstm hidden arbitrarily unit special case lstm internal child evaluation demonstrate sentence evaluate architecture pair sentiment sentence movie review experiment tree outperform available rnn input receive previous token commonly transition affine nonlinearity problem rnn decay exponentially learn address introduce able preserve numerous lstm version define lstm state lstm lstm multiplication intuitively forget gate control extent previous gate control update gate hide lstm unit therefore internal memory vary element basic consist step lstm concatenation forward backward setup allow capture future hide lstm lstm long dependencie input sequence propagation extension lstm variant rich topology able incorporate tree lstm unit index input dependent unit additionally single forget gate lstm contain forget gate lstm unit incorporate child tree preserve rich lstm lstm lstm head take word node right mm child node sum tree follow component unit hide unit child dependency application gate open input word since child lstm unit well suit tree branch child head highly variable child dependency lstm use branching factor order index node child follow eqs eqs reduce transition eqs introduction grain regularization hyperparameter value integer sequence ordinal scale indicate ground human sentence pair tree lstm model sentence representation consider distance empirically outperform multiplicative interpret sign expect rating predict rating function kl divergence pair th sentence lstm sentiment sample movie review sequential dimensionality state summarize predict sentiment review stanford fine grain five neutral classification grain classification neutral exclude annotated sequential baseline predict sentiment phrase representation lstm lstm train tree sec tree sentence sentiment span match sentence semantic involve compositional sentence score completely rating assign eqs layer produce dependency lstm lstm grain rnn paragraph static lstm lstm tree initialize sentiment nlp rnn lstm lstm layer lstm lstm lstm baseline structure development initialize available sentiment update task hold improvement tune train minibatch regularize minibatch sentiment classifier regularize dropout gain dropout semantic system fine grain state tree least dependency dependency corresponding match span find fine word boost fine minor gain gain originally train capture sentiment summarize pearson square metric correlation evaluation compare baseline rnn vector dependency sum transform child follow nonlinearity dt transformation baseline lstm perform system semantic share mean nlp generally use combination lexical lstm outperform without achieve dependency lstm model receive supervision contrast sentiment supervision
bound le easier relate performance risk define detailed treatment variational trade measure proximity hard distinguish proximity distinguish action must corrupt wish decrease experiment advantageous work variational invoke generalized common theoretical hellinger alpha divergence divergence follow collection ii function lemma define yield bound fashion many function divergence often apply sake conceptual proceed corrupted experiment processing divergence le rank corruption invertible meaningful allow processing divergence variational divergence seek prove occur kernel tp possible matter require kernel k mx markov apply tx ki provide processing theorem generic compositional kernel hence occur call ergodicity stationary loss simple define ever le repetition clean experiment repetition matter suggest measure amount summarize e ft one clean many machine corrupt problem label le kt suggest cost total problem admit greedy choose high pick high previous upper bound r hence r greatly occur increase particular interest get compositional reconstruction furthermore statement first use follow norm implication optimistic bad arrive bound apart know consider present rank proxy theorem thing theorem case loss theorem combine worst insensitive choosing corrupt clean slow rate reader preliminary corrupt fast minimizer surrogate link ultimately work v convex notice predict expect learn proceed proper attempt careful leave corrupt corruption rank corrupt make informed decision introduce corrupted bound tight facilitate ranking corrupt bad future refine proportion corrupt problem corruption directly losse particular classification case original even loss rescale shifted loss ie take margin develop result noisy directly symmetric l cc tend marginally less easy semi r confirm unbiased one simply leave behind label average l cc omit rational present three class variant noisy ccc r follow partial label spurious label add ccc case give complicate available closed good bound pac assess appear pac prior furthermore combine generating lead appendix corrupt first generating yield max risk reduce consider supremum inequality relate find quantity mean act simultaneously classifier infimum take firstly definition complete follow convexity finally fourth forward reverse implication must positive summing e k proceed follow inequality definition definition definition focus bernstein loss ex finite theorem pac bernstein least draw choose position bernstein erm fast erm erm bernstein eq define relative utilize yield erm minimize side meaning generalize secondly bernstein chose always high version question ask bernstein condition rule erm learn quickly slowly converse true compatible bernstein compatibility condition final corrupted one symmetry right condition need q easy confirm take useful pair bernstein separable classifier class noisy pt identify pattern joint label pair provide low learner many real corrupt many type corruption yet mean ease corruption develop introduce risk corruption inform economic corruption process sense coefficient ergodicity calculate proceed appear early goal relationship function learner comprise iid empirical risk minimization problem label learner observe label variant multiple usefulness type theory inform economic decision place abstract decision develop risk corruption problem abstract main unbiased early theorem corrupt learn mean bound corrupt theorem answer corrupt contribution progress toward final goal inform regard main deal start actually action decision maker rather observe corrupt different corruption triple convenience ideally compare le form l le year label term term tr transformation form learn include label
average encode observation fitting training unbiased cross evaluate mle construct penalty promise adopt convention understand true powerful compare difference parameterization ic approximations compute first originally rise criterion aic great distribution unknown freedom criterion aic encoding parameterize mle aic selection many problem fail context reason failure aic mle normally especially precision determine eigenvalue poorly failure laplace information criterion analogy aic aic approximation consider true complexity write predictive call analogy evaluate mle mle value large analytic elsewhere difference nest represent general increment specify level direct aic expression exploit approximation let information implicitly dependent eq harmonic procedure chi square random freedom harmonic freedom parameter discuss derive rigorously information criterion bic mathematical statistic argument statistic bic approximation bic result bic large bic mathematical prior ignore principle aim I aic ii dependence encode measurement super sense simplify particular bin bin instead smoothly periodic equal bin clear mathematical necessity list know intensity daily parameterization therefore experimental discrete represent finite datum bin simulate fit expand discuss mle encode sequential underlie coefficient vector select respective mle identically execute add fourier integer except cutoff index determine aic aic perspective complexity count complexity ambiguity mle predict invoke argument q clearly low towards fourier greedy fourier follow fouri row correspond fouri initialize encode parameter execute sequential procedure choose magnitude already cutoff complexity aic parameter one add expect expect initialize large coefficient mean complexity identifiable ambiguity recover aic fouri coefficient chi correspond piecewise respect panel argument aic contribution fouri coefficient integer sensible uninformative bin assume source visualization true simulate sequential greedy qualitatively red green fourier frequency model dot cutoff correspond coefficient red correctly identify red function optimum cross entropy happen cutoff cutoff index sequential encode fouri coefficient per greedy slope dash complexity cutoff sequential prefer compare algorithm distinction bin panel index since poor true slope independent observation greedy complexity like proportional note number observation greedy like regime see index predict eqn complexity capture vs aic bic fail correct scale aic predict correct bic cutoff large bic extremely strict reverse greedy aic weak lead whereas correct scaling even coefficient incorrect accurately complexity scenario determine encode complexity sequential representation unlike formalism depend encode oppose formalism key presence greedy mle small consequence equivalent encoding need resolve ambiguity arise consequence zero fisher result non fisher lead identifiable complexity general system give sequential clearly scale scaling like counter example model therefore scale like elsewhere conclusion information tractable parameter generally must unlike applicable approximate hoc prior need implicitly understand frequentist specify confidence typically inference therefore generally applicable plot simulated encode fit panel coefficient magnitude plot index cutoff dot qualitative agreement fit selection identify illustrate true encoding information information information blue dash index solid represent equivalent criterion match simulate sequential dot algorithm dot slope slope cutoff case true predict solid application analyze degree intensity gaussian variance intensity derivation intensity model relevant discuss encoding
stream rate wherein hence uniform cdf distribution cdf clarity presentation discovery choice goal namely ed denote cdf marginal apply eq arbitrary exponent rate denote apply surely exist sequence q finding condition corollary discovery define condition discovery mixture gaussian get alternative side follow classical performance alpha power consider setup concern normal statistic random lead power compute significance choose way apply decision rule alpha denote discovery propose context incorporate substantial hypothesis reject truly appear exploit scenario scenario mean stream explain power underlie domain research hypothese reject stream early stress relative alpha show procedure scenario total figure several almost identical procedure reject hypothesis via note alpha drop substantially acceptance henceforth metric five procedure scenario drop procedure support discussion compute ii several figure discovery rate scenario relative scenario relative evaluate adjustment dependency trial statistic diagonal uniformly cope dependency control interestingly adjust show rule presence dependency microarray wherein gene arrive stream would one horizon cancer expression control cancer patient compare tumor tumor cancer patient test hypothesis gene cumulative degree since expression adjusted describe one subset return fdr online manner recover state fact deterministic get first proceed proof clear sum result next discovery false union fact nan moreover due rule adopt q true false nan note complete apply I e occur clearly eq rearrange recall rule arrive false occur write rhs false time equivalently write use th discovery mixture c I decrease q straightforward concave arrive bind relax eq eq apply inequality plug suppose induction clearly claim use prove rhs equivalently use rhs eq inequality successive inter discovery elementary pt minus pt minus conjecture consequence replica test core inference proportion false nan control pre procedure multiple testing possibly hypothesis must whether reject access well first control whose alpha rule manner develop lower truly nan independent accord nan hypothesis adjustment address arbitrary procedure synthetic datum compare alpha scientific discovery typically hypothese significance family fdr microarray expression level thousand gene cancer patient gene association cancer hypothesis say nan expect procedure large false false particular truly get finding time stress challenge increase especially case line numerous hypothesis time researcher central carry account generate cancer environmental stream false association previous could obtain research run cancer raw issue e carry instance hypothesis control decentralized nature prevent decide basis evidence information previous remark motivate formal hypothesis test need one implement discovery step unlikely seminal widely serve acceptable reduce spurious false strong interest hypothesis instead predictor sequential hereafter fdr testing briefly significance level define reject every test significance address describe control hypothesis arrive value aim ensure remain pre assign rule function one testing allow flexible use collection hypothesis reject hypothesis increase reject gain power stein control false alpha spend hypothesis occur alpha toward refer alpha fdr online fashion work online discovery adjustment cope choose hence test truly fix arbitrary independently accord non nan hypothesis validate management concern desire limitation currently database assume quality underlie motivated concern practical insight query paper fdr building upon alpha procedure computationally search pool alpha avoid leverage control incorporate procedure feature selection provably year hypothesis conjunction coordinate selective control testing context regressor regressor fall short address nan past discovery current score accept alone achieve nan indicate asymptotically throughout online level stand number second discovery occur stand level choose give equation less control simple condition alpha exploit control rule adapt discovery discovery sequence next discovery let recent eq show control every control control follow control rule procedure introduction control capture among test relaxed assumption subset nan control conduct prove control basic significance budget spend rule increment hypothesis alpha rule test outcome alpha stay alpha proceed sequentially might
applicable symmetric approach crf validate apply previous future speed method quasi eigen vector product arbitrary property l l write u equivalently reformulate binary problem correspondence k x sdp relaxation drop solve u denote variable w u u n l use requirement far bf pt bf claim conjecture token conditional crf widely conventional neighboring pixel long contextual approximate sensitive initialization make develop yet fully sdp rank algorithm tailor quasi specialized sdp dual apply fully connect solve pixel level co image segmentation vision category clear satisfactory contextual image successful semantic pixel solve posteriori contain unary potential typically feature texture potential consist term disagreement pixel contextual relationship crf encouraging contextual fully challenge stem crf pixel million usually infeasible case approximation inference fully base accelerate filter crf relationship depend relative spatially inference kernel semidefinite programming sdp relaxation accurate solution relaxation interior sdp constraint method map alternate multiplier admm estimation wang present still work sdp map large significant improvement present integrated sdp accelerate expensive part term mixture filter method field much applicable achieve superior knowledge first level co super make tractable sdp relaxation generally project quasi newton semidefinite notation list rl bold letter bold low case letter cone semidefinite identity scalar wise diagonal whose indicator otherwise product hadamard two kronecker product derivative order factorial crf energy conditioning drop notational assume unary map inference follow I I unary term represent compatibility l l l pixel compatibility base accelerate two briefly respective limitation approximation marginals crf suppose kullback leibler kl divergence recall complete factorization kl divergence view keep marginal fix form solution equation iteratively monotonically decrease limitation converge one problem optimize convex consequence non sensitive define bottleneck update equation express matrix naive need time next speed pairwise gaussian convolution r viewpoint signal processing convolution proportional standard filter filter convolution complexity filter approach limitation general euclidean feature dimension filter complexity time lattice work accumulate semidefinite programming convex minimize semidefinite q b integer denote number relaxation develop optimize typically solve follow lift relaxed round original solve method scale poorly associate requirement solve approximately solve solution respectively accurate close advantage much firstly intersection contain column rr prove several strategy sample representative mean method column entire nystr positive semidefinite approximated th summation sdp paper compatibility arbitrary label compatibility function discuss define j f quadratic x encode constraint drop relaxation constraint follow major improvement scalable key fast interior issue address show drop several efficiently several spend eigen decomposition variable spend prohibitive bfgs convergence condition continuously necessarily differentiable algorithm map nr k h next improvement scalability initialization n traditionally round round carry quasi converge round quasi dual value increase dramatically also drop quasi newton adopt round semidefinite round scheme express step standard variance discretization bottleneck eigen require iteratively accelerate utilize lr specific structure w constraint l make u j accordingly lr descent eigen decomposition computational crf approximation superiority segmentation image co experiment iteration lr set work pairwise image color pixel respectively similarly appearance adjacent compatibility function product c k f p k k operation bring memory image resolution need perform position nystr om low representative original image c unary mf mf c l c unary mf lr times na truth provide respectively field cpu memory vector product nystr om evaluate refer pixel around qualitative result segmentation mean field quantitative demonstrate complexity optimize filter limited gaussian achieve significantly energy viewpoint unfortunately superiority performance actually evaluate similar lr mf scale level perform sometimes converge undesirable see detail lose c mf co
requirement qualitative level statement concrete formulation course practical instance satisfy non trivial requirement however consideration serve filter restrictive generative formalize requirement instance satisfy give requirement significance run instance input guarantee satisfie terminate however problem far bad desirable ability namely advantage requirement allow direct checking one collection representative practical cluster input extent requirement somewhat relate namely requirement explanation success cluster lead main question follow rather metric euclidean euclidean explicitly instance find instance set cost depend course objective simplify cluster dx k kx give cluster denote format objective ig objective sum distance objective hard np hard approximate datum furthermore otherwise discuss algorithm difference approximate probably objective clustering require arise different type specific clustering define center clustering say c solution require measure implication notion particular mean c roughly discuss imply cluster year line clustering appropriately exhaustive major notion perturbation general sake similarity notion namely formulate center optimal however scale add diameter stability multiply multiplicative perturbation robustness optimal every namely cluster remain small multiplicative relaxation optimal introduce discuss objective definition initially respect cost factor easily hold relate optimum significantly optimal respect objective cluster every instance satisfy condition q optimal cluster center obtain center instance list almost except perturbation yield imply vast separate provide version characteristic notion showing carry efficiently sound plausible concrete support plausibility paper present quantitative assumption essential evaluate plausibility currently desire expect satisfy keep focused relatively level major notion determine concrete need evaluate gap optimistic thesis distinction context concern mean hardness cluster determine clearly input exhaustive search cluster feasible refer task space cluster g take runtime dependence feasible requirement demand relevant target cluster find instance diameter input recall np consider get runtime allow depend runtime respectively obtain similar optimal perturbation max cut consider cluster show existence cut focus note cluster task small find clustering mean propose variant solution arbitrary cluster clustering clustering distance optimal clustering r objective output cluster cluster pruning examine list value suffice efficiency corresponding requirement think average point input setup imply hardness obtain probably somewhat relevant exceed center somewhat trivial cluster allow find cluster viewpoint relatively gap np almost stability become claim extremely strong formula average optimal exponent minimum distance runtime claim every grow pick one cluster point follow implication condition bound cluster center distance cluster bound bind average distance read every outside consider aim overcome vast distant outside cluster comes test key clustering np hard whether conjecture np property clustering say clustering imply evenly spread singleton cluster datum show algorithm follow stable every efficiently find pruning concern cost pruning define datum weakly median clear I arbitrarily stable come notion test guarantee aware currently variant popular come show application yield quality clustering relax recent view ask convex recover address generative balanced notion range requirement range rather trivially suffice currently rather requirement currently thesis thesis well case many yield clustering way know clustering sense explanation result may proof come notion list notion close match hardness imply np vs notion almost variant rely implication prove efficiency condition believe yet light notion way separation demonstrate assumption restrictive yield finally paragraph remark argue really wish well current practice varied detect record basis patient advance aim extent input like transmission objective usefulness optimize compression distortion furthermore restriction clustering cluster make cluster hard imagine realistic situation number focus analyze case meaningful currently satisfactory conclusion intend intuition stem cluster obvious open status thesis challenge section technical whose answer meaningful complexity condition stability notion arbitrary euclidean though relatively significance optimize objective stable become hard significant open question carry contract whose leave singleton pick dissimilarity node already create agglomerative tree stable input datum pruning notion linkage cluster output tree properly nice pruning notion cluster mean nice median cluster notion nice exist nice programming relaxation come naturally approximate approximation guarantee hardness grateful david discussion concern input instance attention lemma theorem example theorem cluster hard optimize practice cluster provide discrepancy notion distinguish hope provably hope matter believe extent line critical conclusion thesis formally requirement meet validity thesis list imply requirement examine exist requirement outline open two fold I biased overview research concern assumption work quite community arise motivation technical aim attention gap encourage theory quantification resource worst understand approach hardness hard exist infinitely many compare experience np solve instance actual hardness give approach notion well expect come behave behaved area paper
quite complicated derive implement common ep traditional mcmc poorly dataset gradient langevin draw estimate include monte carlo improve fisher langevin dynamics infer simplex etc mcmc whether sample ep vb thing plug need memory big since store experiment dnn store paper parametric carlo teacher student student online give detail past parametric student teacher mixture student online large dataset use deep neural also train student teacher extend approximated single training student fit net teacher level teacher generate improve classification reliable prediction datum dark knowledge represent hidden teacher student bayesian dark knowledge first combine mcmc network kind prediction lead score compare recently ep vb train minimize kl teacher student form teacher monte approximation copy network single furthermore architecture student deep net fewer wide problem uncertainty capture softmax neural multimodal unimodal py fx gx fx gx variance fix independent kind multimodal dimensional dnn output dimension train student predictive teacher teacher network expressive effect nn I test want prediction teacher network integrate train student denote point ground label teacher choice control make accurate prediction uniformly eq quite integral tractable however sample put together take show low hyper control teacher student spherical precision strength datum teacher two reason teacher single whereas student predict argue second teacher pass input minibatch iteration schedule teacher teacher minibatch j teacher softmax approximate use also softmax output eqn function cross loss output compute propagate py py fx w train student network twice teacher predict avoid deal positivity train back propagate section approximate sgd ep vb hamiltonian carlo hmc nets sgd library hmc perform ideally apply enable open source code ep support vb number third small toy compare dataset vb hmc mnist start toy illustrate performance dimension per class fit perceptron mlp layer relu softmax output result decision boundary high bottom corner figure hmc true wish discard keep every th see monte student mlp train random encourage student predict accurately location include student teacher capture student get well kl hmc pointwise grid qualitative cccc c k consider mnist digit problem example preprocesse tune strictly comparable whole relu activation minibatch final hyper minibatch rate dropout believe perform averaging use unweighted teacher mc generate student gaussian sophisticated datum two teacher use rate reduce every network furthermore make small teacher make sgd sgd sgd report run unit sgd report sgd start toy regression order visually illustrate mlp unit activation student see density vb ep finally incur lot computationally avg ep sgd regression dataset training repeat hidden relu remain hyper minibatch noise table fit iteration interval well teacher initial rate reduce every student use precision every log bad ep method vb show kind seem
amenable normalize bethe amp bethe likelihood normalization evolution behavior via two parameter express variate bethe appear definition evolution simplify physics intuition behind bayes signal describe chosen overlap randomly propagation amp evolution statistical physics widely weak notably mmse compute state evolution log likelihood amp state fix trivial mse sign completely noise whenever evolution mse sense problem hard study fix expand equation trivial matter linearization away linearization contraction mean transition behavior translate iterative amp small quite remarkable universal detail agree phase remarkably spectral transition canonical transition mean variate gauss dirac criterion amp matrix fix point give informative preserved identity play assume note invariant always rank line result evolution instance mse reach uninformative se amp reach red middle mmse several investigation amp evolution likelihood different compare amp algorithm excellent agreement amp able find transition mmse green nd informative transition gauss trivial stable case mean single mmse previous limit r depict result e zero trivial translate fact amp theoretically blue amp amp mmse red mmse amp suboptimal region blue red nd informative mean phase transition remarkably density depict density amp transition mmse support size analyze probabilistic evolution rely statistical physics state fix amp large important topic future signal regime region sub picture signal asymptotic mmse mmse false negative mmse amp unless hard region stay barrier algorithm part european union analysis zero employ pass algorithm state theoretically minimal amp size prove phase transition suggest amp fail matrix consist gaussian noise stem constrained underlie analyze minimize large theoretically minimal square mmse achieve approximate amp estimate marginal se rely exactly amp experience reach principal technique describe small component pca search facilitate describe variability constraint pca simplicity report model straightforward comparable abundance pca algorithmic development theoretical study g many concerned recovery possible number zero zero probabilistic amp bernoulli state evolution describe remarkably achieve amp theoretically transition mmse rank derive asymptotic amp least reason question possible
cumulative usual learner q decision environment depend decision deterministic achieve sublinear assumption decision learner set round choose accord learner suffer observe framework accommodate plan minimum span cut set accordingly considerable attention differ make round regardless mean information scheme learner observe associate ii seminal learner minimize fix choose concern exploitation mix result distribution modification concerned variant exponentially average forecaster match normalize forecaster refine popular scheme sp information come combinatorial optimization overview consider paper problem name full set online mirror method use prove expect semi coincide completeness forecaster know attain regret semi case outline work full scheme picture weighting efficiently hull describe constraint work prohibitive list efficiently efficient semi bandit perturb later offer efficient idea draw perturbation minimize perturb loss conceptual simplicity efficiency due reason good scale bandit efficient regret straightforward obstacle expression importance issue efficient guarantee online contribute wave concern besides mention show implement form hull decision perturbation regard recently show intuitive excellent information scheme call recurrence efficiently principle besides full concern variant regret information access increase result close gap performance semi section main recurrence weight geometric change name concept broadly use statistic specific armed bandit access basis decision decision vector sigma round notation estimate prediction estimate otherwise I operate utilize importance operate many algorithm fall arguably online operate recurrence weight even computable variable repeat execute round geometrically distribute random construct estimate whenever notice surely case time might offer problem combinatorial offline combinatorial optimization feedback critical recurrence combinatorial action mapping let algebra history algorithm pick action problem importance close recurrence weighting manner follow learner draw well everything ready follow perturb recurrence define st draw component exponential implicitly cumulative equivalent perturbation emphasize additional construct bandit compute static overhead loss computation sample take sample control high observe I recurrence weight statement notice prove martingale follow suggest running probability proof statement concern present recurrence present tool analyze put theorem idea recurrence weight replace importance amount distribution yield unbiased want sampling termination introduce bias concern matter first expression estimate generate estimate fix satisfy simplicity write combine important recurrence estimate ensure rely sense second property ensure learner loss estimate statement q hold control calculation take last concern loss estimate well current satisfy infeasible somewhat surprisingly fix simplify copy geometric law analyze component synthesis style idea nevertheless combinatorial know tool develop gap semi bandit statement know work perturbation step let virtual pick eq crucially conditionally exploit use numerous first virtual sequence proof completeness also slightly improve replace usual fix refer side follow md prove next relate actual rely loss trick relate rule fix arbitrary kp lemma notice proof sum everything ready expect fix put regret prove central done remain arise begin use consistently simple inequality key trivially fix multiply side sum relate start increment mb obtain q holds imply together increment least prove lemma order arise fix hold martingale increment statement theorem also enable particular exponential full define satisfie become combine online semi tune properly study weighted forecaster whether remain gap
uniformly draw validate follow introduce benchmark dataset handwritten digits interesting classifier generally level label facilitate new classification create employ artificial building image principal contribute flat pyramid great challenge significant occur capture second cover various kind air water partially cast tree great dataset extreme pyramid unbalanced classification edge employ extract image prototype characterize primary ridge type synthetic fig synthetic validate handwritten digits uci repository totally handwritten people set generate dataset interpolation configuration rx process channel experimental benefit utilize sec scenario encode input rbf employ classification image reconstruct fed convolutional cnn create digit recognition summarize comparison cc visualization dot synthetic synthetic distribution synthetic bridge synthetic autoencoder autoencoder reconstruct encode layer comparison handwritten digit get c cnn corresponding autoencoder c thing synthetic achieve real synthetic result synthetic reconstruct encode encode much appearance reconstruct correlation reconstruct synthetic shown intuitively help datum ht identify synthetic solving problem could learn synthetic novel multiple standard synthetic gap jointly robust facilitate image introduce method validate supplementary material branch target balance balance branch optimize turn cause two branch avoid branch minimization quasi regularization add difference oppose parameter exact gradient vary control source one red accurately control roughly later difference synthetic real note directly autoencoder autoencoder purely place synthetic output gap reconstruction contrary pattern validate extend bridge gap reconstruct divergence cc sf opt sf edu fu research fu propose synthetic classifier normally bridge jointly propose show possible learn experiment type two validate efficiency model methodology large normally crucial world adequate even crowdsource amazon necessary classifier per object mean object extensive cloud label classifier label consume expert labeling effort practically point solve problem sample transfer hold instance nevertheless attribute nontrivial learn ability angle utilize synthetic g learn well b synthetic synthetic development cognitive artificial intelligence vision learn parent example svm work create sense train extremely challenge firstly generate shift illustrate obstacle synthetic potential useful knowledge address literature practically label available automatically synthetic novel sparse autoencoder synthetic real datum try enforce transfer synthetic generate applied facilitate image dataset need challenge instance demonstrate handwritten uci learning repository datum generate basic result approach highlight contribution knowledge synthetic gap propose gap vision community annotation several image classifier synthetic create mesh simplification visual quality recently build point cloud indicate semantic cloud normalization employ one space degradation help line handwritten apply moderate training boost enhance document degradation degradation degradation degradation texture degradation handwritten recognition success limit methodology handwritten digits aim applie find helpful sentiment page zero shot image video unsupervise transfer fall transfer nonetheless previous domain task synthetic gap cause shift feature real idea autoencoder vector input autoencoder follow pre train deep autoencoder different activation layer sparsity layer sparse autoencoder train autoencoder purely place synthetic bridge synthetic problem reconstruction complement synthetic real real synthetic synthetic identical datum leverage rx new autoencoder channel encode enforce reconstruct common task divide autoencoder channel task two channel decode flexible autoencoder knowledge reconstruction together balance channel speed optimize minimize require cause fast situation channel channel propagation newton balance task compute please material autoencoder learn pattern simultaneously capture another input sx r rx autoencoder autoencoder autoencoder learn configuration unbalanced optimization biased autoencoder topic similarity difference augment datum aim classifier nevertheless bring preserving highlight stage synthetic well could generate stage interpolation set respectively propose synthetic represent model simulate real appearance point prototype location iteratively minimize associate connect get match synthetic htb real point position prototype image initial converge generate generate prototype manually learn section generation prototype control propose zhang pre knowledge prototype design
ng default induce may recommend next covariance value mix try trajectory age ng vary summarize number bayesian criterion bic describe posterior class class include subject subject classify classify classified posterior goodness fit assess l age false panel residual prediction weight mean observation accord age option prediction contain soon covariate specify include associate standard tool seq length age age col age year normalize mmse r type c add na predict trajectory model implement intercept random age link beta age link data spline default spline knot mixed default difference parameterization intercept residual rescaling ordinal might probit mix rarely complexity integration log likelihood assume estimation recommend satisfy inclusion show data threshold age data threshold take hour depend object mix fit age subject link node criteria aic bic discrete log aic aic per per per subject likelihood longitudinal se intercept age covariance intercept intercept link se spline spline spline spline spline intercept constrain involve link give involve approximated spline knot provide probit guide evaluate nonlinearity relationship longitudinal marker normal estimate link provide confidence band col col add col col col col add col legend linear spline quantile col q legend na col n latent process option beta knot quantile confidence band spline knot quantile plot observation provide break outcome compute accord covariate default draw line code computing seq age var draw trajectory display col age legend n col gender confidence band class class use section mixed multivariate implement trajectory cognitive since entry entry far investigate effect gender marker correlate marker process mix beta cdf function summary center age time link beta latent maximum mmse time subject link beta observation function cdf mmse cdf criterion iteration derivative goodness maximum aic longitudinal se intercept age center mmse coefficient sum effect intercept intercept se bm mmse error beta mmse beta mmse beta mmse beta beta beta beta beta fix summary object common marker global test covariance marker specific along standard finally marker provide link plot band col c col mmse mmse mmse mmse mmse mmse draw asymptotic distribution vary depend seed percentage compute percentage measurement error call explain datum process input apply object joint mix implement study trajectory risk indeed cognitive closely study natural cognitive dynamic risk cognitive simplicity illustration account compete risk death incidence trajectory model assume delay give latent one latent class default survival ng age mixture age survival hazard ng age survival hazard ng function table bic proportion latent compare select latent specific effect intercept bic proportion automatic choice maxima criterion reach class solution default systematically try different example illustrate use initial similarly class mixture age survival hazard ng mixture random survival hazard ng class value beyond example table b g latent provide note latent bic avoid computation latent outcome risk maximum age age ng model event event baseline risk criterion iteration e derivative fit ci p maximum reference class se intercept class event se intercept intercept intercept age class age class effect intercept residual similar summary depend whether conditional longitudinal survival although provide longitudinal option predict longitudinal option age decade weight subject model predict longitudinal use covariate profile plot risk baseline option predict baseline survival class seq r b c decade year year ht marginal survival class summarize longitudinal time class longitudinal classification provide classification objective dynamic function use plot visualize var age age r age main validate estimate col col true col main c add col legend col c c observe model surprisingly change incidence give predictive low simple class although joint perform age finally indicate computation individual include illustration purpose class good candidate highlight year c horizon draw col age age year main old subject would thank implementation ne implemented share subsample de grant extend latent mixed theory include mix longitudinal ordinal longitudinal multivariate longitudinal outcome latent gaussian outcome censor compete setting modify strict criterion base second derivative likelihood provide various fit goodness trajectory predictive constitute introduce give analyze longitudinal outcome assess longitudinal study enter outcome gaussian ordinal e absence presence longitudinal especially biological measure life longitudinal process disease observe may exist unknown disease gene cognitive complexity longitudinal directly one gaussian variable asymmetric distribution death jointly finally population subject group toward estimation function heterogeneity latent trajectory model model theory powerful iterative goodness compute v organized implement analysis function conclude package subsection type longitudinal subsection dedicate longitudinal marker subject subject vector asset linear mix measurement subject individual visit greatly follow mixed vector respective vector shape trajectory design fit spline measurement brownian stationary w parameter involve model random effect cholesky longitudinal marker measurement covariate effect entire marker longitudinal outcome scale extend longitudinal outcome longitudinal marker define mixed latent process model separate structural interest link observation continuous longitudinal flexible observe measurement parameterize latent quantitative monotonic linear mixed rescale cumulative canonical reason follow ij basis spline knot splines ordinal marker probit cumulative ij latent mixed intercept process parameter separation longitudinal observe longitudinal marker unique marker multivariate mixed covariate measurement extend set take specific longitudinal marker measurement differ subject marker flexibility account aspect intermediate marker effect marker induce take intercept capture would capture marker model marker cdf univariate constraint require identify set latent constrain intercept location intercept intercept allow structural model tb assume population mixed consist population heterogeneous subject profile subject membership equal latent describe multinomial intercept covariate identifiability scalar covariate predict class probability profile covariate latent mixed model standard fix effect gaussian outcome previously still call distribution proportional identifiability error apply latent mixed replace structural constraint intercept constraint remain remain assume heterogeneity affect underlie interest include marker longitudinal process simultaneously longitudinal survival death disease capture correlation family ensure positivity piecewise tn knot spline specify tn lt cubic spline three family baseline parameter restrict square transformation exponential paragraph event multiple cause censor nature occur censor hazard covariate cause baseline cause baseline function proportional class model p g mixed vector involve individual likelihood contribution likelihood matrix link process function jacobian transformation rescale ordinal link function level define conditionally moment q ij variable presence effect integral random gauss quadrature unique gauss quadrature currently continuous function process individual determinant link covariance definition matrix block ik nn ni identity contribution linear variance row mix longitudinal marker individual specific cause censor q cause longitudinal outcome class delay entry contribution maximize type model choose speed find extend update default knot time first knot knot regular event time knot manually risk function transformation square imply specification hazard note unconstrained range event specification suited number baseline function cumulative output iterative initialize generate package one default yy respectively couple survival risk z n presence least crucial program put model log multiple maxima might converge maxima algorithm ensure convergence recommend initial manually aware begin grid work discovery program automatic ensure automatic deriving initial assumption value g assumption automatic initial actually estimate estimation analysis symbol subsection apply likelihood directly give matrix maximum likelihood latter triangular effect directly summarie cholesky estimation error variance compute function test calculation class goodness model class collect longitudinal latent complete class membership subject membership ig provide correspond probability base longitudinal fit selection discrimination derive two classified subject table compute belong class perfect would elsewhere indicate belong n ig n g ig g ig g ig mix empirical longitudinal four linear empirical li equation univariate assume predict transform marker kk ki involve ig z ig g mix z ig process mix involve empirical bayes g ig residual estimation residual ss ij mr specific ij ij ig marginal prediction ig ss g ig ij mr ij ss ij ss prediction residual ij mr ij ss mr ss transformed provide marginal specific graph membership link compute marker compute object multivariate latent threshold longitudinal marker standard cumulative function predict trajectory marker profile compute computation longitudinal computed refer marker class posterior approximate monte large marker value maximum link function option compute inverse carlo use estimate value ordinal link constitute cumulative mixed effect acceptable aic function risk option survival
list seven intervention absolute gram diagonal violate mechanism fit well mechanism intervention activity abundance activity change mechanism abundance abundance global seven intervention approximately mechanism experimental intervention case violate concern relation connectivity correspond activity target point success occur american intervention variance peak american value contradict cycle exist row write generality fix definition j l equation furthermore vector analogously replace reasoning invertible furthermore diagonal element path cycle hence cycle show cast complexity let write analogously row diag eq p define loss minimize exist theorem cyclic cyclic cyclic record strength necessary fulfil almost three distinct pure observational demonstrate simulated series discover causal effect fundamentally challenge various public study economic application life acyclic include observational alone assumption interest characterize relation self loop semi matrix existence equilibrium term govern equilibrium invertible also converge iterate e condition large eigenvalue assumption strength feedback cycle eq product clearly graph cycle strictly identifiability see strictly small identical cycle intersect cycle cycle intersect eigenvalue strictly situation cycle cycle solution iteration stable either still theory arguably little interest summary interesting observational alone cyclic show parent contribution effectively intervention occur income location node neither limit exact strength often different environment see financial time series call uncertain bayesian intervention simply give variable determine intervention assume demand variable discuss section detect location leverage environment matrix sufficient identifiability simulate flow let two external input member flow assume external explicitly consecutive observation section furthermore except except environment j let mean center version n state matrix invertible thus strictly latent connectivity identical intervention uncorrelated c matrix imply characterize aim reconstruct observation environment unknown intervention strength environment additionally detect assumption reconstruct main transform setting change matrix hand stem shift define setting imply intervention shift side let restricted space one product assumption l j alternative replace throughout counterpart subsequently enforce important step detail compute minimize counterpart constraint invertible scaling row diagonal element result challenge product follow lead cycle seem variant last cycle product one satisfy return met op problem compute cycle product problem exploit difference observation environment shift intervention specifically equal variable shift environment another gram read strong proceed wider adapt replace gram practice unclear approximately weak exploit location intervention namely environment difference intervention convention minimal intervention alternatively observational serve intervention variance intervention environment identifiability provide appendix solution intervention intervention variance variable must environment variance intervention shift across identifiability environment identifiability intervention environment absolutely lebesgue relaxed achieve identifiability generic environment present synthetic set various property besides assess stability retrieve code compare cyclic case observational data cyclic generate specifically environment intervention draw intervention act observe strength intervention sample present sample intervention tp minimum inner mm font circle circle draw blue blue circle blue dash dash dash dash blue w metric precision coincide point exclude close return achieve absolute value illustrate relative magnitude hamming show hidden hide intensity illustrate estimate width edge coefficient absolute retain setting assume cope present pool interpret come variable follow pose cover satisfied five obtain recall increase adjacency improve causal require worse somewhat well positive identical converge value precision accurate setting set increase intervention strength false positive return stability selection
size profile contain give profile assign assign class standard inner train model label nb distribution popular include discriminative aim class otherwise label helpful prevent overfitte svm hinge loss meet world compositional square lead technology numerous computational challenge implementation large read model approximately base profile distinct genome reach million train thousand reference may choice redundancy still useful properly intra inter specie explore large interesting dimensional real life actual hundred massive multiclass scenario reach efficiently dedicated exploit approximate sgd require fast scalable exploit train lead storage refer interested relevance sgd long disk count map hash impact reference database refer database genome cover list different generate one specie one database represent situation reference validation complete filter sequence accord keep less filter genome short specie represent specie pick sample genomic sequence remain sequence reference database adding describe represent genome refer therefore involve solely database database use base database represent gradually increase batch cover nucleotide coverage maximal value length complete train performance computing specie proportion correctly median multiclass bias performance axis color start respectively sufficient still increase beyond systematically increase steady length increase coverage marginally dataset involve size drawing length consider increase bring improvement vector hashing feature hash multiclass hash divide consider store hash observe increase actually decrease great middle mean specie micro compare comparative compositional set profile abundance use affect genome return maximal pick specie correspond repetition discriminative never outperform alignment nb performance shorter obtain outperform nb bp show level cover genome gold performance report dataset right bp species predictor train cover coverage equal solid compositional naive green dotted alignment grey dot experiment relevance feasibility performance establish learn database specie large database learn configuration database allow respectively species hash database database compatible evaluate read base genome around sequence approach previous concept nb reference specie median number little performance vs compositional nb dropping ability reference grey compositional performance report reference database median accuracy bayes nb reference compositional approach performance performance perform error sequence challenging sequencing read sequence read simulation sequence error commonly g able read evaluation systematically length evaluate impact error kind drop small reference impact compositional drop less case nb drop use database severe impact drop around nb consider reference almost nb respectively compare alignment approach around profile read mutation mean half read show impact see relatively severe compositional mutation mutation error implement empirically mutation current probably calibrate short median mutation read agreement publication impact length modify model increase mutation default configuration type alignment mutation median mutation performance drop large hand obtain compositional mutation database drop mutation drop even severe remain around mutation great nb keeps reach high mutation outperform nb configuration current experiment significant drop compositional moderate mutation rate especially mutation performance nb respectively impact alignment realistic alignment show high performance median reach database figure grey approach evaluate accuracy obtain nb grey solid line rate turn comparative compositional aspect indeed large volume generation sequence constitute motivation base measure time take base involve experiment read read mutation database allow investigate involved reference sequencing read compositional computing specie dot classify obtain dot product efficiently procedure procedure nucleotide encode convert memory lie define cpu gb memory summarize show variation across reach classify around read read increase sequence impact need database compositional systematically offer prediction time read top bottom horizontal line require represent ratio take modern scale datum extensively performance scale regard iii specie involve reference detail robustness simulate read baseline comparative compositional generative demonstrate impact estimate model highlight configuration reach svm compositional offer high nb classifiers competitive alignment tool involve sequence error result however compositional still limit hundred specie compositional exhibit alignment approach sequence error confirm compositional systematically list compositional approach species level emphasize provide memory scale linearly database could compositional alignment fast memory sequence improve learn simulate allow tune sequence technology producing read provide model properly characterize reduce memory could straightforwardly learn store prediction multiclass suggest address issue art compositional broad spectrum emphasize remain amount sequence error specie
censor enyi node noiseless span graph impossible happen recover turn limit average isolate average grow average remain fix ask question infer assignment plant positively quantity strictly overlap vanish guess unity recovery task positively assignment task knowledge belief bp conjecture part prove practical bp describe spectral show rigorously sense additionally without knowledge gap method large fast trivial interpretation connect try community membership censor observation relationship graph cluster discuss contribution development detect recovery spectral operator traditional adjacency interestingly statistical spin plant spin line notation know spin backtrack sec threshold non backtrack bethe property backtrack operator relevant bethe non backtrack bethe backtracking act graph motivation uninformative sec entry neighbor favor graph edge ensure positively plant backtrack call bethe bethe otherwise relation second lead stability bethe hessian arbitrary turn algorithm belief locally optimal overlap achieve bp strictly small avoid propagation optimal detecting observe overlap bethe always superior backtrack bethe h noise vary instance positively correlate soon overlap transition require concern assignment positively plant notice unweighted backtrack spectrum uninformative contain disk plane informative disk follow theorem generalize main enyi graph average vertex uniformly random backtracking decrease magnitude tend positively plant illustrate fig straightforward assignment positively correlate plant sketch proof proof orient transpose start ef ef eigenvector easier ef p problem case know contiguous trace allow graph neighborhood radius cycle moment e symmetry symmetry prove small eigenvector adapt bound h allow eigenvalue compute quantity explain ball large neighborhood branching process natural branching generation path yu natural martingale reasoning couple branch martingale translate backtrack operator eigenvalue spectrum circle eigenvalue plant relate spectra generalize define ki v site convenience define note bethe value thus must zero turn eigenvector eigenvector eigenvalue need limitation
v nm nm sn sn mn contradiction clearly x rr sample implicit possibly dependent construct sequence min tm tm limit constant recognize convergent since follow xt contradiction form accumulation statement development stochastic involve mild function set requirement map conclusion upper semi pointwise analyze stochastic generalization asynchronous couple field general classical approach I algorithm differential system approach stochastic track inclusion follow map reader step exposition book heuristic reinforcement execute couple iterate size satisfy martingale lipschitz function couple iterate could project onto compact ensure euclidean tackle problem generalize word follow couple recursion h create single asynchronous notation paper I al present reference di km dx mm xx da dx compact invariant call subset follow convex map radius close represent couple recursion eq h k scalar bn bn square sequence k generality k upper closure contain globally lyapunov standard essentially wise map course clear key requirement link slow iterate mild show enable describe start analyze exist say convenience claim f proceed gx sequence convergent sub n nk w nk gx yx remain bx requirement next martingale proof sake similarly prove enough q follow prove technical exist convergent sequence nk n gx n nk gx contradiction statement gx nk n convergent nm nm proceed trajectory let construct trajectory bin sn sn sn ns sn sn corresponding ty write I surely recursion asymptotic reader refer trajectory evolution iterate surely satisfy assumption I trajectory track follow trivially
insight many theoretical recent paper stochastic mechanism exploit motivate recent optimization present threshold minimax rate adapt unknown noisy sign along gradient line solve provably achieve rate gradient part stochastic convex noisy repeatedly perform optimal adaptive smoothness convex active seem inherent nature field role feedback action condition bound technique however unclear idea common field aforementione new inspire adaptive uniform design parameter simple pool active access learner subroutine randomize procedure uniformly simple return noisy sign coordinate full value gradient result adaptive two preliminary insight describe minimize function query diameter convex convexity arise dual strongly equivalently deal parameter estimate internal randomness algorithm query return optimum alternatively error oracle sign internal randomness paper motivated application computing gradient huge amount compute coordinate compute expensive multiply however require expense vector keep track coordinate proportional expense sign weak actually obtain noise zero next return sign derivative easier small circumstance calculate value could much easy require expect spirit power crucially sign round error precision get round precision flip assume length draw side half hence allow sequentially dependent label guess close formal cf common minimax exponent version condition classifier strategy measure excess threshold minimax notation clearly idea subroutine optimally unknown prove active subroutine convergence rate adaptive argue access noisy switch sign sign exponentially deterministic fx fx fx fx uniformly dimension uniformly jx mathematically use bounding bound error setting exponent exponent mention require condition growth tight similar around directional minima growth strong smoothness strongly strongly function strong relate unbiased around uniform reproduce clarity sketch sign convex jx jx boundary switch erm vc argument ignore passive learning procedure ball know contain threshold close threshold constant kk k within argue risk point stay sized region bind high ht diameter budget passive choose r generalize epoch repeatedly passive epoch en epoch radius epoch epoch otherwise threshold passive least treat clarity exposition factor diameter least limitation algebra appropriately epoch analysis sufficiently eq lie range equation issue completion though start might far away radius round round imply secondly round may geometrically epoch far like epoch close summing word hold epoch radius actually mathematically assume epoch notice previous completion would er e something strong epoch e epoch least epoch upper lemma justify result conclude dimension stochastic subroutine gradient sign optimally knowledge simply coordinate approximate search subroutine active algorithm call descent coordinate vector choose due approximate accomplish optimal fix time search set diameter budget stochastic sign oracle return let number epoch approximate use subroutine allow q subtract denote take cauchy k c lr epoch subroutine subroutine also adaptive appropriately calculate summary give information unknown convexity smoothness minimize concern store limit affect gradient remain unbiased might first surprising reveal rounding flip sign drop possible return
equation information sparsity specify fix specify spatio kronecker van covariance ik qr contextual kronecker kronecker rank temporal kronecker matrix widely physical output passive sense non move array target completely contextual target case specify kronecker kronecker unknown kronecker theory recently discuss estimator structure incorporate entry mean expectation mse relative decrease piece information complexity covariance form map log mse directly table map uninformative contextual information specify glasso add l penalty glasso precision determine algorithm contextual kronecker kronecker kronecker kronecker sparse kronecker factor laplacian type factor add contextual assess study sample complexity asymptotic dimension maintain go spatio mse take rd row contextual information contextual delta c kronecker sparse represent prior contextual nd rd information specify spatial coordinate norm type contextual row column correspond gauss markov field sufficiently large kronecker corresponding row rank kronecker factor kronecker complexity regime th row various quantifie value quantify change mse contextual kronecker kronecker additional determine show e fix plot set constant mse variable knowledge kronecker valuable illustrate right structure kronecker information integer curve contour row plane equal curve reduction require contextual inverse alone kronecker alone curve case inverse label dominate covariance maximal free value per one primary support correlation correlation measurement model consider estimating model year learn sparse inverse entry model broadly method bayesian space sparse mining gaussian penalize method pseudo base base entail sparse develop maximize penalize likelihood inverse quantify line problem complexity estimation tend infinity grow dimension via complexity recently graphical seek maximize eq element matrix denote th iterative along cover property state detail covariance vary accurate exist hold large denote diagonal entry n jj furthermore ii iii selection establish iii hold minimizer ij establishe precise spirit literature remark consistency guarantee size consistency tail heavy size grow polynomial correlation seek discover topological characteristic precision treat screen presence connect node high correlation topology easy covariance table high specifie structure inverse block screen edge perform apply estimate partial correlation place exceed plug inverse correlation develop study discovery local node variety include gaussian regression testing sparsity pattern screen illustrate complexity determine also vector bound block go go one relation pp ne ga nn problem screen occur false constrain rate type I control remarkably true attain give number rate zero correlation decrease critical direct thm great positive following quantify intrinsic variable large size require reliably detect great needed ten screen higher quantify value curve surface similar positive detect curve panel phase differently reveal detect reliably increase small correlation desire often mine possibly existence highly value specification confidence reliable quantification require critical task inference science recall require go infinity summarize regime rd regime increase screening correlation detect mean correlation partial false correlation screen false specifie increase infinity rate converge satisfy support covariance support include priori apply union subset cardinality bind sum pp p infinity thm regime regime critical report table recovery derive particular tend detail variable impose coincide convergence relax determine frobenius norm square limit mse set estimation specify example critical region optimally anomaly estimate nan outlier square function empirical density minimax risk asymptotic critical screening pe pn n covariance st specify rd sample row increasingly size require limit detection existence asymptotic positive one give existence magnitude mixed limit false critical mixed asymptotic misclassification correlation asymptotic square frobenius error covariance finally performance bind high asymptotic mse borel constant conclude unlike screen scalability glasso reduce computational contrast screen non due building thresholded ball ann ann datasets million hundred issue appropriate inferential classification decision inference lack account credible inference completely dataset population variable focus mining infer population reliability inference limit mathematically associate specify ensure complexity regime infinity purely dimensional go comparative sampling correlation mine different regime screen govern purely rate quantification require screen require acquire adapt inference estimation uncertainty quantification acquire strategy prediction regression acknowledgement partially air office scientific grant award office grant nf w nf foundation award national health grant technology us energy national nuclear security award support air office scientific award fa national foundation dms dms dms research project smc corollary conjecture ann usa stanford usa reliable draw context answer implication scale like dataset rich acquire replicate far neuron recent focus understanding especially dimension grow gap unified quantifie sample various inferential task divide category size go comparable purely go regime scale problem correlation regime mine dimensional covariance task keyword big mining correlation correlation screen graphical increase availability drive science big phrase business scientific media concentrate issue research community issue statistical largely recognize stand success scientific consequence insufficient especially inference big big column row index statistical theory develop big correlation discover correlation limit mathematical reconstruct population covariance sample underlie mining significant challenge term use specify requirement latter challenge covariance include finance communication sense science differently depend entire covariance far explore thresholding correlation covariance relate error support zero vector interest context special reliably mining presence correlate might accurately estimate entry matrix densely population structure emphasis set correlation network annotate human thousand human subject correlation level sometimes significant spatio temporal clutter full spatio clutter hundred thousand bin discovery profile thousand ip address point profile recommender preference category music fmri brain activation brain currently researcher hundred pattern brain practitioner face spurious correlation essential understand intrinsic requirement study requirement fall control asymptotic theory inference small regime cover go regime lebesgue plug covariance sparse say say function additive natural addition correspond specify pairwise global graphical also node zero support zero indicate correspond correspond covariance reason scale version give predictor inverse covariance depend inverse covariance many classical discriminant analysis variance fourth entry th coefficient prediction residual physical sparse last inverse example physics poisson integrable laplacian differential operator solution poisson heat transfer extract graphical discretized convert smooth discretized equation diagonal spatio drive gauss structure diagonal full parsimonious visualization realization simulation discretize support partial user adjacency
hx hx hx il contribution lp il h sum give b r r h thus ij ij nk nk n nk nk nk set indeed proof observe nk novel constructive kolmogorov importantly subsection state basic intuition support dimensional metric turn intuition formalize I distribution cdf let namely map invertible distinct uniquely distinct cdf make non exist neighborhood continuously jacobian point define distinct statement continuously argument jacobian entry lemma parameter matrix matrix calculate jx ix ii quantity respect equal multiply take particular q ij p jk p contain complete ready interior inverse mapping specifically define n k completes take neighborhood ii lemma argument sd ns bs intersection hand disjoint intersection jacobian open open proof structural constructive cover kolmogorov metric must identical packing subset denote cdf immediately suppose contradiction lie point inequality cdf kolmogorov cover union volume volume ns proof complexity prove prove theoretic argument subsection tv main theoretic absolute complete define I apply root coefficient ix ix ix qx assume proceed far simple give two z z z c z x another application triangle x complete characterize statement kk x variation equal k must mx ok ok enough lemma c bc x x contribution small discrete x prove x I must mx ok eps expectation large enough lemma x achievable obtain cover easy need deal single cover indeed straightforward discrete appear produce approximation fortunately require negligible large element mean couple minimum c increase expectation assume component leave unchanged k cx nk nc nk pair symmetry may integer constant long c let complete approximate element efficiently give appropriately variance construct x theorem universal support deviation exist integer mode satisfy generality mode know eq similarly term recall basic fact concern variation start processing domain function draw next total use variance tv pt claim observation theorem observation ac uk california edu university com sum independent near sample variation use nearly cover admit ok cover transform structural transform namely analytic argument concern integer variable order distinction distinction sum independent arise special poisson trivial binomial know survey fundamental form special chernoff hoeffding long research random decade near efficient tight upper constructive explain cover elaborate context motivation work variable definition learn natural analogue well pac boolean unsupervise set topic rich extensive literature context year body study perspective computationally gold setting theoretically ideally near main run output description hypothesis learn variation require sample sample provably logarithmic case previously near run polynomial high sample algorithm understand computational consideration theorem give tight problem would conjecture complexity case distance distinguish fair biased coin perhaps surprisingly learn arbitrary distance separation conjecture learn theoretic rely understand space elaborate cover say cover cover cover exploit variety cover kolmogorov role theory book statistic book upper bound distance upper constructive cover prove construct polynomial size least comparison non cover quasi cover theorem consequence learn game theory combine algorithm imply output run sort enumeration cover equilibrium show constructive upper size standard along additive nash equilibria correspond hence constructive cover approach lead implication hardness computing equilibrium anonymous support denote cumulative cdf variation distance variation kolmogorov f give overview ingredient cover moment matching agree tight proposition explicit agree variation unfortunately moment moment periodic structure distinguish odd integer proposition explicit agree work limit regularity variation support integer close force proceed hypothesis bottleneck arbitrary exploit beyond aforementioned upper transform tool give fail type fourier fouri random product fourier transform similar starting essential new small assume effective know extremely simple point transform everywhere exploit sparsity fourier transform complicated transform precisely cover show transform necessarily transform logarithm approximate interval actual root fourier circle therefore provide coefficient logarithm description geometric defining probability mass distribution function fact interior understanding allow expectation roughly change region distinct effect word effectively size remark structured family polynomial approximation provably apply sense lead piecewise necessarily incur structure bind sample additionally exploit dependence idea I detail completeness tv discrete transform dft function integer dft dft dft onto give intuitive explanation fourier transform apply fouri good likely error bind believe may interest fourier effective standard fourier big effective effective support could dft idea hard compute b otherwise proceed sm depend fourier transform appropriately small effective proof efficiently ii constructive beyond remark upper analysis since copy note consider dft write ij take relate claim integer exist integer application claim proof ii follow rhs integer uniformly note claim q ij integer ready theorem run dft calculate ok ok correctness e henceforth give tv size learn total henceforth assume indeed application kt come consider high automatically absolute mean chernoff nr union nr bind get nr ok contribution case error ii eq note lemma compute expect q like outside use show I x real complete upper size proceed construction minimum size upper proceed start size case desire polynomially cover base point order close discretize random constructing cover belong theorem translation subsection moreover ii cover claim sub variation cover support discretized variance large variance want discrete interval discretization geometric grid proposition cover size last inequality reduction note proof cover proof notation identity pdf view function circle plane entire complex agree e conceptually logarithm additive taylor polynomial assume desire true reason logarithm arc base arc lemma magnitude divide root arc aforementione cover transform logarithm appropriate nearby relate variation transform equality analyze polynomial defer fix root qx qx suppose root list qx x jj triangle lemma claim root standard proceed prove difference first jx nk nk eq multiplicative next replace q assume seek hx h j mx rhs satisfie assumption f hx hence complete induction require proposition size z fact tv nk ok associate follow lemma root consist part real I show cover claim ok dd first relatively arcs number take possible arc note strong arc I z z z part w taylor give near cover time integer exist run ok kn k ok kk build establish subsection cover case close exploited cover spurious point large efficiently construct proper cover spurious careful argument defer possibility constant possibility observation coordinate variation suffice find probability henceforth main possible taylor fourier transform additive dynamic program problem let sufficiently divide unit circle arcs associate root mc ic ic b c ic ii concatenation near exception less algorithm
cnn latent crf recover object condition compute feed forward energy yield microsoft capability mit recover coherent category appear room together measure improve classifier take category entire gain object gain combine pre classification performance unsupervised node image neighborhood latent activation node potential instance scene image variable scene diverse traffic another represent various kind framework capturing unsupervise scene mit scene probability match use misclassification baseline neural capture scene engineering combine strength learn expensive dataset employ advance technique instead small finally pass category detect contextual co occur tree graphical dependency incorporate category detector probabilistic use simple detector degradation contrast pre train incorporate context many contextual addition thus framework object classification imagenet vision task popular number train cnn object framework plan future independent incorporate probabilistic localization object bayesian optimization grain believe setting learn cnns scene classification scene label available training training automatically label scene localization segmentation task perform multiple coherent account spatial location interest body work contextual future plan scene expect recent attempt probabilistic combine crf joint joint framework deep learning feature account dependency variable train network mrf network train latent lead finally work multi technique text rest overview fc train model dataset compute likelihood eqn image train imagenet consider extracted effectively multiple image predict achieve dependency structure relate dependency object label condition input allow tractable structure leverage extract moment rkhs distance recovery kernel conditional conditional give setting modal component transformation x rkh empirical distribution reproduce hilbert rkhs yx xx iy given employ rbf estimate tree gram parameter l l work among available provable guarantee statistical k tn employ cl learn many structured probabilistic graphical structured absence design machine view special energy energy particular output compatible potential graph use configuration eqn define find perform parameterized net gradient loss classifying multiple category b ms training image label object independent classifier recall tree network correspond avoid potential covariate use along map compute back use dropout viterbi message latent l l l l l l l l l l ht measure ex classifier classifier layer learn recover structure relate appendix tree role divide scene node object room cluster around car instance object precision layer conditional train layer feed network classifier label decision category neural gain significant object like percent percent percent percent percent b contain test different marginal precision comparison investigate activation potential image result high activation effectively capture contain result image appear belong scene relevant use scene scene capability mit room optimally misclassification never scene use marginal probability hide neural table show input hide result misclassification ex place co appearance train set gain neural network capture semantic distinguish level information image information manner co knowledge apply like california art classification imagenet unify strength multiple deep microsoft image incorporate contextual latent co condition extract fc train imagenet pairwise object occurrence take fc object learn conditional significant gain measure ms especially object capture scene infer alone scene mit use present scene deep performance computer task scene parse pose focus train imagenet consist object far challenge currently framework use simple approach predict category mutually exclusive classification decision however natural label mutually exclusive ignore label share knowledge prediction explore expensive
case significantly hope classifier accurate perturbation theorem vertical horizontal independently risk similarly two robustness adversarial straightforward calculation unlike use perturbation switch illustrate result practical classification confirm identify linear quadratic large adversarial robustness suggest linear svm svm rbf width classifier validation close perform find satisfie procedure obtained define I point robustness follow perturbation opposite baseline adversarial say uniform line find large condition estimate svm svm adversarial perturb f switch cubic rbf classifier g original perturb g ht vs first mnist handwritten digits digit task train small translation image unit euclidean report adversarial perturbation despite perform fairly well small adversarial perturbation visually translate perturbation instability adversarial perturbation surprising table addition improve classifier important implication establish limit interest hence classifier though random robustness classifier hope adversarial perturbation get close classifier design specifically account robustness identify limit perturbation theorem would identify towards understand deep net human successively event observe moreover cauchy schwarz inequality fx fx rf schwarz together adversarial conclude sphere spherical show follow sample sphere prove bound note armed concentration n deduce result n f fc decrease pd negativity take side obtain result prove conclude inequality first exist generality lemma successively inequality get negative z similar fx follow use get norm rf p last use note hold get conclude solve perturbation label thank point reference arbitrary small possibly robustness perturbation adversarial perturbation express result task involve adversarial robustness random perturbation perturbation knowledge theoretical address phenomenon instability recently limited give explanation adversarial instability proportion misclassifie evaluate robustness perturbation highly desirable particular change paper robustness classifier classifier perturbation differently average lack robustness perturbation data adversarial vs car task small car plane therefore seek understand perturbation formally study robustness perturbation set robustness linear fundamental robustness adversarial perturbation express specifically classifier robustness classifier implication involve small classifier misclassification compare notion noise robustness much former showing notion classification task value illustrate newly concept theoretical run practical task surprisingly unstable adversarial receive instability adversarial perturbation raise challenge generalize unable correctly paper show perturbation flip classifier theoretically adversarial instability network several attempt network adversarial perturbation relate explore argue nature high dimension contrary network go general trend problem involve flexible adversarial even low risk security adversarial attack work e decision counter attack robustness adversarial extensively differ classifier paper sec introduce problem introduce throughout sec quadratic classifier conclusion adversarial conclusion leave future associate simply take misclassification focus classifier perturbation ambient perturbation noise flip nature perturbation perturb point outside support robustness adversarial perturbation minimal perturbation flip estimate label note independent adversarial perturbation perturbation robustness definition assume observe region sample label robustness radius center sample classified illustration give fig perturbation outside quantity quantity risk robustness adversarial perturbation uniform introduce run robustness risk consider binary vertical resp horizontal constant class image background resp illustrate permit separate line vs vertical valid task separate despite detect visually image orientation classifier exploit achieve indeed resp risk achieve capture fail orientation separable adversarial minor computation robustness satisfy maximize adversarial perturbation unlike orientation direction robustness classifier fig great extent classifier image difference robust adversarial noise example fact perturbation orientation unlike classifier f say capture essence equal evaluate similarly perturbation world partial enough concept essence robustness class adversarial perturbation classifier adversarial random perturbation equal distance hyperplane classifier assume q intercept eq robustness adversarial constant represent vs diagram diagram region attain importantly interesting quantity intra intra class geometric transformation vision even task adversarial average class robustness perturbation linear illustrative diagram achievable robustness classifier adversarial perturbation bound behave
true separate consider confirm converge htbp simulate inversion procedure determine run linear cost stop iteration multiplication multiplication power multiplication first risk reduce cost nr fast nr hundreds great low agnostic exception smoothly low explain iteration slight improvement eigenvector guarantee interpretable tradeoff risk computational cost estimating setting suggest single update allow risk computational illustrate iterative simplifying various allow subsequent choice estimator tool measure aspect together employ whenever study throughput read development methodology spectral numerical preserve edu foundation mathematical research department university partially science foundation grant dms office nf office grant research institute advance grateful constructive greatly research recent need trade tractable practitioner computational theoretical give problem tradeoff computation analytic risk computationally constrained estimator conclude risk termination iterative computation family massive field curse improvement cost largely field gradient descent relaxation conversely introduce describe procedure bag little implementation bootstrap massive classical problem constraint chain carlo consistent mix order detection detection compute algorithmic complexity likely never bad aspect understand fine specifically framework address cost risk understand gain assess degradation risk exponential outline basic section illustrate normal idea general robust extend idea problem estimate value random value denote estimate seek r optimality principle maintain add formally compute statistical estimate equip algorithm compute hence together runtime straightforward example manuscript property compute outside set collection storage keep much processing exploit put among feasible estimator must know estimator plot illustrate achieve risk balance examine investigate computation identically variable explore index generalize allow allow linear compute operation look compute store datum perform operation sufficient paradigm look perform mle omit algorithms q consider streaming near zero extremely select intuitively say computing streaming setting possibly use collection estimate index index impact computational cost assign indicate b unique change blue cost illustrate row row risk fisher overall risk signal present note optimal great regime figure time family density q parameter model ie n n p maximum convenient index subset estimate fisher pairwise intersection kk kl analog parameter covariance sample verify well frequently compute runtime compute sum risk relative importance component able estimation general inverse nontrivial task generally especially graphical finding compute np parameter nonetheless runtime tradeoff less investigation possible consider estimate define arise contaminate central degree scale percent contaminated proportion datum decompose time pairwise sum comparison data simulation computation sample approximately describe compare replacement pair replacement median estimate three estimate cost simulate contamination
far pf exhibit centralize pf ii require communication neighboring sensor iii consensus average implementation sensor communication distribute achieve account decentralize pf multi network involve motivate broadly mean parsimonious representation term low dimensional facilitate interpretability enhance predictive deal decentralize algorithm rank argue internet traffic anomaly measurement moreover rf subsequently development layer wireless cr decentralize linear algorithm outline aforementioned ip abstraction node operational origin traffic flow denote traffic link interval count across single adopt mean traffic link source accordingly horizon count flow traffic relate term carry flow traffic matrix temporal traffic periodic traffic typically intuitive validate term failure attack attack service let traffic flow explicitly flow traffic carry error anomaly traffic flow effect link anomaly anomalous spike interference flow stem miss link measurement operational reality rely indirect measurement traffic link measurement tuple available introduce l traffic compact operator entry keep unchanged flow traffic matrix link traffic rate short relative flow suppose anomalous instant row flow put plus decomposition albeit natural criterion np optimize surrogate accordingly sparsity control optimization appeal accelerate complexity develop network interestingly link subsequently exploit spatio temporal link anomaly shoot estimation turn outperform latter value leverage sparsity jointly red anomaly continuously traffic monitoring aggregation anomaly operational adopt network associate minimize reduce translate miss raise central represent isolated point traffic anomaly anomaly carry locally rely count cardinality block likewise edge terminate term oriented incidence incidence orient denote small nonzero eigenvalue algebraic theory establish define np lp notational convenience use rewrite form lp lagrange constraint lp collect multiplier associate multiplier augment back dual make local cost local cost gradient constant hold function point I f b aggregate aggregate gradient guarantee constant hence aggregate cost differentiable sufficient convergence decentralize admm far sequel well respective consensus attained form stack comprise stack copy optimal may multiple strongly exist lagrange converge one establish dual lie space convergence prove analysis define convergence several contraction distance euclidean context ergodic rate establish prove refer speed successive primal dual vanish optima convergence next four monotonic namely hold show monotonically ergodic primal solution prove straightforward kkt condition main establish decentralize multipli primal solution initialization admm local cost close specify dual solution indeed primal solution ultimately ergodic iterate difference prove recent decentralized contraction inequality linearly convergent convergent meaning convergence decentralize admm algorithm initial multipli c iterate lie convex lipschitz multipli column guarantee converge lie converge contraction indeed lipschitz continuity eigenvalue orient eigenvalue laplacian admm penalty arbitrary insight influence find value right aggregate condition note graph condition dominate imply contraction cost dominate acknowledgement friend author extract j wu support grant gm wireless communication science technology china china ny tx put framework decentralize acquire importance communication central cost privacy reason term network paradigm decentralize maintain broad decentralized comprise wireless medium internet use refine local hierarchy local estimator fully exploit maximize suffice accurate task decentralize favorable structure direction multiplier admm iterative method back process decentralize price wide encourage single inter communication th single neighborhood inter symmetric undirected edge represent communication clear domain wireless wireless electrical cognitive example across scheme decentralize local processor refer fc internet collaborative agent perform centralized raise concern processor represent isolated failure objective develop decentralized setup exhibit coincide correspond keep overhead communication neighborhood argue admm wireless decentralized solver classify operating handling constrain domain subgradient variant incremental gradient average subgradient inexact achieve price stepsize order primal form local applicable depend local iterate without admm numerical convergence chapter solve subproblem demand fortunately subproblem solve run iteration burden remainder describe admm heart algorithms chapter network section focus estimation unsupervised inference deal estimation collect sequentially internet spectrum wireless cr network motivate fundamental admm state straightforward decompose overcome idea local represent local per network formulate equality constraint coincide neighborhood extend turn amenable decentralized leverage favorable alternate direction method multiplier g employ minimize fashion whereby converge centralized facilitate application one variable eventually eliminate multiplier j lagrangian coefficient entail comprise decomposable separability come primal turn lead step admm decentralize algorithm jk admm decentralize redundant eliminate say track redundant end store node iteration local cost attain consensus likelihood mle square posteriori formulate minimization centralize fall short capability spatially sense outline decentralize far outline depend technique accordingly centralize mle capturing pdf weighted vector map yield decentralize decentralize example decentralize recent advance nonlinear least monitor grid arise ac global interestingly estimation decentralized programming leveraging base centralized sdp sensor employ yield wish blue fashion local I unconstraine admit close decentralize case fitting offer decentralize wide rule local suitably average originally allow decentralized exhibit inter quadratic centralized mle problem tackle task reformulate outer linear know structure formulate ensure matrix outer drop non semidefinite solve decentralize decentralize complex node system magnitude newton tool iterative linearization therein suffer variability grid capability decentralize multiple control attract grow three area centralize htb decentralize neighbor error converge decentralized sdp successfully address solver overview along decentralized estimation variety task rely sense resource constrain message wireless ap limited capability desirable decentralize sensor attain another environmental monitoring local sense decentralize framework sensor network available everywhere fc diverse challenge inter node exchange allow overhead decentralized detection framework decode task wireless communication scenario know codebook belong assume symbol symmetric channel conditionally sensor know characterize pdf noise unable reliably message information sensor global centralized centralized ml decoder likely multiple propagation approach centralize even cardinality exponentially introduce burden sensor centralize decode become il py likelihood ignore decode objective give n clearly statistic equivalently bit interestingly discuss admm decentralized framework section allow attain sufficient length complexity decentralize decoding since posteriori rely average sufficient extension alphabet consider tb bit versus snr demonstrate performance decode code numerical test involve ap scheme sensor curve mark initialize decentralized iteration correspond iii decoder consensus iv admm decentralize decoder exhibit convergence consensus average counterpart iteration suffice bring decentralized relate common ap message map entry finite per sensor admit form sensor channel additive assume uncorrelated ap neighbor ml covariance suffice wide constitute argument decentralize decode lead attain locally constitute decentralized order centralize demonstrate task develop decentralize minimal centralized significant reduction communication svms centralize setting task surveillance often limited acquire central processing costly scalability communication overhead environmental structural monitoring diagnosis medical condition record available seek follow slack scalar allow discriminant mapping possibly centralized decentralized reach decentralized svm take pair slack decomposable structure decentralize identify b I decentralized iteration algorithmic decentralize admm svm incorporate consider node draw gaussian matrix respectively optimal depict global training set centralized iteration admm rule centralize counterpart local unsupervise exploratory infer structure collect set design decentralize capable joint processing various centralized centroid denote prototype element amount specify centroid error minimize convex coefficient program consequently admit solution rise term cluster optimal membership suboptimal proceed r fix least nonetheless require availability information per challenge reason topology offer address yet decentralize leverage neighbor albeit decentralized extension decentralize method environmental typical monitoring option less motivate decentralized processing decentralized scheme sensor identify temperature measurement group connectivity average centralize test include note converge iteration whereby datum exchange reach available description interest varying motivate decentralized scheme node collect datum recursively refine develop tracking approach kalman particle scope facilitate processing network decentralize adaptive possibly nod collaborative fashion communication linear criterion per instant node without generality estimator interest jointly well wiener tt develop decentralized building form amenable via stochastic handle variation statistical step instant apparent obtain root equation cost solve available statistical acquire algorithm find expected size local constitute counterpart section decentralize first stochastic approximation step parallel admm constant tracking capability operate presence node see expression mean relate slowly vary vector specifically link variance depict local evolution mean noisy ideal link closely follow theoretical trajectory steady accurate penalty link fig adaptation affect adaptation respective adaptation closely track variation fail square well online estimation signal track especially attractive rate offer valuable admm decentralized scheme distribute spectrum claim set minimize form history past discard enable tracking decompose global estimate utilize follow decentralized iteration q involve node recommend recursion estimate converge invariant bound mse along comparison diffusion regression arise spectrum monitoring suppose sensor comprise interest spectral peak reveal heat source channel contaminate additive sensor source frequency band operate lead decentralized estimation evolution signal
completion mc preference incomplete feedback ill fortunately structure texture motion lie subspace dimensionality recent convex remarkable fact complete rank select well suffer robustness even minor sensor failure environment mc far ground resolve issue effort devote mc solid theoretical chen al truth subspace detect column corrupt expansion please advance physics quantum apply exist try resolve show extend robust mc coefficient basis column intrinsic corrupt filtering able solve numerically polynomial expansion however perturbation might fitting far original reduce sometimes expansion coefficient possibility fouri due failure signal carry outlier success mc quantum r experimental overcome basis severe commonly justify exact robust dataset corrupt try segmentation face observation miss resolve issue relate subspace model call robust low lrr thus mc remove outlier suppose whose range probably mc formulate recent sense objective nuclear envelope ball eq worth note standard propose general mc ground cardinality traditional issue sensitive minor occur failure environment mc mr range space probably principal component analysis pca outlier even corruption resolve successful pca via convex xu work truth sample outli pursuit apply subspace alignment texture etc unfortunately value work worth note mention mc outlier mutually limitation recent mc complete detect simultaneously correspondingly relaxed chen range sufficient column input exactly corrupted sample report mc recommendation research basis limit challenging task extend robust mc general demonstrate extend robust mc succeed traditional basis ambiguity fraction comparison significantly robust regularization parameter choose reduce algorithm incoherence relate extend lrr algorithm immediately subspace fine structure follow describe setup present detailed proof establish theoretical application cluster validity theory section conclude column general consider exact corrupted recover element hope recover column mostly rank cover situation problem mc low original normal unfortunately replace relaxed brevity rewrite project onto rank total word approximate successfully low several example sparse identify expect space low svd problem condition adopt incoherence condition please table explanation incoherence incoherence suffice column sample bernoulli entry non select corrupt incoherent identify clean guarantee analogously issue column sparse isotropic ambiguity assumption ambiguity number comparable ambiguity condition geometrically zero scatter matter main entry measure assumption position specifically position determine event guarantee exact noiseless l severe work summarize use notation support matrix grow product capital zero column whose sum column truth optimal solution h xx result surprisingly column close measure w fraction column exact robust mc recover probability subject model parameter automatically imply distribute traditional low rank mc seek rank bind consideration arbitrary partially recover rank matrix fraction rank miss recover corrupt robust al incoherence ambiguity extend recover extended result extend mc elimination extend mc recover succeed solution mc ks km succeed lt lt argument bernoulli high c probability nc fix small proof I j assume provide eq nc p extend mc accord feasible note h prove brevity construct inductive eq obeys lemma q three inequality first eq dual exact h norm l hence f first inequality mp remainder construct q triangle dual suffice assumption obey dual condition check net unit cardinality show adjoint accord automatically bernoulli variable hoeffding inequality eq accord stand I ie unitary second inequality hold fact proof complete mc efficiently alternate multiplier admm nuclear minimization problem fast term algorithm speak recover ground truth select norm least speed scale subproblem recover suppose I theorem form submatrix brevity submatrix solve scale mapping restrict section bernoulli column rl ks r regression admm nearly fortunately separability norm decompose subproblem equivalently matrix problem admit l l r filter subspace randomly sample recover seed solve conduct line remain recover low dimensional subspace filter range column end step probabilitie recovery seed column line span range range recover justify recover ground truth subspace seed though row check range examine column outlier informative select line bind parameter intuitively property element element zero high suffice guarantee two measurement connect incoherence define eqn space see condition definition corollary incoherence ready follow illustrate result suppose bernoulli sample well condition hold large happen coherent enough exact rank low select roughly typically probability suppose provide appropriate chernoff obtain theorem lemma ground subspace seed fulfil incoherence succeed numerical constant filtering justify outli outli matrix e l condition incoherence identify even complexity filter case bad algorithm require recover seed factorization multiplication r mn r subspace svd multiplication converge significantly discuss missing demonstrate validity application subspace clustering aim lie e face motion etc probably effective lrr clean representation low mathematically solution cluster cut lie robust lrr widely handle situation commonly failure extend robust modify lrr np incur great difficulty efficient show mutually form solution mc conversely lrr find approximate original obtain extend lrr far conduct validity algorithm I matrix column sample I optimal compare range hamming run illustrate regularization enable recover range show independent magnitude record truth vary succeed matter magnitude range simulation plot region succeed comparable working ht speed admm list cpu hamming distance significantly comparable c admm filter admm filter admm filter fraction
universe physics feasible thank development extract everything feasible range growth universe analysis invoke transformation estimate act prominent example appear instance value quantity routine invoke fouri combination non computable calculation give remainder formalism implicit provide perspective science summarize sec implicitly deal set storage computer computer routine implement mapping science technique determinant covariance require constrain denote adjoint exploiting estimate wise vector analogously operator trace method spirit require purely stochastic cost phrase trace subsequently linear split case implicit operator physics approximation violate introduce q time sufficiently together represent determinant operator evaluate time integration order taylor expand determinant dropping pseudo case deal dominant determinant coarse numerical correction computational cost partition ref community signal stochastic novel implicitly previously impossible see address bayesian determinant homogeneity physical field parametrize power assumption detail become space respective spectrum position kk x determinant form q set diagonal dominant whereas diagonal fig ht matrix refer explicit implicit explicit well determinant regard separation apply well perform integration realize furthermore study value dependency interval fig precise matrix eqs applicability discretization interval chosen see particular illustrate numerical determinant fully minimizer step calculate selection vast topic henceforth present example find describe measurement signal independent gaussian e represent operator variable relate covariance q signal respectively denote phase bayesian evidence might deal implicit perform integral last method nest wants infer switching exchange affect determinant calculation field reconstruct ip explicit dependency follow calibration integration perform produce general contain instead routine implicit matrix probe variety scientific affected extraction background calibration realistic unknown calibration amplitude parametrize specific assume gaussian gets affect mask cut still uncorrelated measurement equation device calibration calibration regard external calibration sufficiently strong infer amplitude simultaneously approach prior sec calibration noise generate realization part numerically calibration give data efficiency use eight trace peak eq determine determinant determinant integral involve discretization integration operator obey fine discretization necessary fact might keep cost universal inference comparison deal instance calculation determinant acknowledge realize determinant operator expand method expansion approximate theoretically single pseudo step enable derivative integrate pseudo integral representation determinant integral representation ref
wavelet wavelet efficient compactly wavelet useful wavelet introduce soft family wavelet look sort haar wavelet beta wavelet fine tune keep loose orthogonality property haar address wavelet remain wavelet approximate wavelet advantage I cycle iv central signal nature compose cycle signal successive probably wavelet cope another detect analyze power system investigation dr lot regard limit theorem continuous wavelet beta distribution limit theory compactly wavelet recently insight wavelet present reading wave wavelet link wavelet infinitely wavelet wavelet transform wavelet unbounded support wavelet wavelet order interpretation average smooth derivative derivative derivative wavelet transform low pass version kind central limit unbounde compactly variable chi square play central wavelet unbounded wavelet concept impose beta efficient concept entropy recently reveal wavelet one link beta valuable practical role unbounded support ip random ji lattice dirac accord kolmogorov density q wavelet well know application discover wavelet compactly couple factor generalize factorial function beta function easily variable transform unity extreme ia guarantee wavelet cycle wavelet smooth wavelet spectrum function q wavelet prove spectrum carry wavelet sense spectrum symmetric haar wavelet support check reliability computation wavelet etc occur expected wavelet half cycle first spectral due unimodal feature wavelet henceforth refer wavelet algebraic handling expression order play beta relate h couple beta wavelet aim investigation potential wavelet write happen compact beta wavelet ability provide wavelet efficiently concern focused location main drawback haar wavelet contrast
risk suggest genomic promise spurious association identify variant disease high throughput popular important genomic level achieve due restriction share level summary datum score frequently share significant shared commonly value normal transformation generalize weight combine issue dominate setting irrelevant high cause wrong inference correlation lose disease associate genetic identify study conduct summary variant simultaneously genetic comprehensive review perform genomic genomic genetic novel drawback genomic genetic complex besides genetic disease trait genetic new genetic genomic possesse work admit single genetic study second two share genetic signal solid support third produce unique minimizer solution method conduct comparison experiment method formulation selection discussion mathematically multiple relate express entry transform score study goal detect genetic variant sparsity indicate please example study identify type genetic recover genetic genomic variant irrelevant noise rank correspond causal snps disease trait correspond causal disease trait component measurement zero count intractable effectively nuclear singular surrogate rank low prove powerful prove solve alternatively theoretical reduce regularize least problem follow value singular singular value rewrite close ij refer optimize summarize global variable thresholde soft input two control probability recover state value adjust specific matrix number snps share snps rarely use step expect snps sparse absolute deviation four method method search association distinguish result meet predefine cutoff default setting use decomposition try result method precision suitable irrelevant method trait relate trait material study snps disease trait disease trait convert pc investigate work apply share snps present rank component individual snps sparse recover disease trait edu college pressure pressure density tc disease trait causal snps finding disease trait cluster together relate detect snp snp snp map gene supplementary material besides snps value moderate snp whose college detect snp method snp report value map gene severe include fold pressure pressure publish additional low snp respectively gene pressure identify causal trait snps match confirm disease trait supplementary material clearly show recognize detect snps moderate disease task disease trait discover year hundred carry systematically investigate comprehensive genetic complex disease trait disease snps divide share trait snps individual recover noise formulate optimization demonstrate method conduct several different setting method outperform many study successively datum discover share propose easily well understand disease mainly development technology annotation structural acknowledgment support grant university grant explore genome data table table simulation table four four simulation description disease trait body index height ratio adjust five spectrum tag total density high diabetes pressure pressure social college trait disease disease diabetes body disease http www pressure pressure cm attention major diabetes type diabetes htbp c c snr snr low recall htbp c snr precision f cm true genome thousand individual widely identify disease increment explain genetic variation disease miss disease disease common genetic variant explore correlation promise remove spurious identify genetic complex disease identify genetic scale genomic dataset formulate aim multiple disease trait trait convex solve dataset experimental show reconstruct wide scenario matlab code human disease diabetes cancer influence environmental health great interest find advance insight complex substantial support
example reaction increase course depend incorrect branching splitting step strongly brownian motion start brownian motion euler method time choose inverse order magnitude run highlight proper splitting version happen yield consider exactly namely maximum small reaction precisely version biased modify section work newly replica explain remark remain may pick work maximum level equal end probability eq consistent updating formula exactly resample step work small initial current level equal accordingly strict version enter increase modification increase iteration explain result estimation give stable value check standard direct carlo table e estimation version negligible probability version incorrect implementation order easily extract modification size go variant maximum see go recall variant present htbp c e e c e choice reaction efficiency dimensional langevin dynamic wiener euler denote simulation numerical scheme g condition give plot figure potential connect channel minimum saddle open around trajectory temperature lot interested time reaction associate perform run empirical confidence solid line blue circle represent low bound line evolution htp small bias phenomenon sufficiently large agreement fact unbiased reaction coordinate fluctuation lot reaction seem computed consider empirical dramatically phenomenon apparent large get reaction computation plot equivalent kind evolution confidence jump already use large reaction relate pathway relate paper one potential reaction value situation reaction pathway estimation reaction coordinate carlo sizes dynamic langevin euler depend double minima saddle whereas latter saddle decrease x reaction coordinate figure reaction green red circle reaction criterion q always average realization take test behavior go saddle evolution realization overlap reaction comparison standard monte able reliable direct reasonable realization saddle figure evolution interval realization realization realization temperature fluctuation reaction realization interval section pathway rather pathway symmetric role reaction apparent split htp summarize finding always simulation accordance result lot reaction coordinate poor average limit remark reaction thus tail interval reaction coordinate trajectory go unlikely reaction coordinate particular reaction multiple reach certain level relative channel reach maximum example reaction adaptive reaction update get close algorithm direction reliable branching resample implementation build section property check minimum reaction coordinate particular recommend simulation reaction minimal realization regime scale instance parallelization trivial namely thus acknowledgement grateful position grant grateful conduct european european agreement grateful project would many proposition heuristic method example discrete dynamic path chain estimator rare event choice practical illustrate experiment efficient reliability molecular us molecular let discretization langevin dynamic q giving position position energy configuration variance remain long call locate reality molecular event denote disjoint reach molecular path path assumed simplicity deterministic trajectory start chain reach smaller naive carlo reliable estimate example molecular technique quantity sample splitting splitting ingredient q use advance towards reaction molecular call useful requirement existence path markov call system stop remove path keep discuss generalization sup soon fit path remove replace fit remove path sample go computation determine remove remain fit path iii resample path iterate one obtains successively stop feature level remove maximum threshold iteratively fix priori deterministic adaptive splitting generally standard sequential carlo chain stochastic reason mainly discretize numerical discrete monte carlo markov context raise question context resample whether time remove exactly implementation section main appropriate implementation splitting yield estimator rare event splitting enter call framework various classical remove reduce sort procedure moreover chain numerically toy example implementation splitting numerical experiment recommendation reliable see property possible reaction quality concern number minimum reaction coordinate interpretation reaction spirit mutation resample analogy precise see reaction know resampling accord condition reach remove case practical interest condition meet monte statistical error crucial show concern choice discussion reaction large relate apparent stress resample definition suit trajectory another terminology mind static consider describe actually splitting detail write variant yield highlight property produce unbiased quantity theoretical illustrate efficiency rare event go denote recall standard disjoint borel give trajectory probability distribution endow test probability measurable associate probability transition test notation dx respectively value rigorously variable measurable element endow x n endow denote write ensemble subset replica label n replace space goal define markov unbiased treat time treat many continuous chain transition generality deterministic random condition introduce sup corresponding path endow measurable disjoint borel mainly occur region neighborhood resp resp close stop probability see interest rewrite generally allow stop time stop endowed ingredient importance reaction coordinate molecular value q aim choice unbiased estimator impose define stop reaction coordinate q stop emphasize strict time equivalent stop average memory splitting advance contribute copy fit accord resample kernel denote define law q stop identically generate probability stop reach branching resample resample modify dirac follow specific trajectory unbiased generalized splitting reaction two minimum iteration replica decrease iteration order work maximum equal keep resample procedure replica use resample kernel complete trajectory time label end th iteration construction level denote subscript retain terminology refer stop generate iteratively resample estimator observable consider set namely maximum equal union refer working label estimator position detail algorithm define initial n th order remark remark satisfied case step work equal label denote branch notice I criterion fulfil parent replica replica procedure replica branching replica old one construction observe weight replica soon replica branching replica q q eq set th statistic x time loop consist none first replace step imply stop three maximum level split resample occur representation replica small replica beyond replica reach maximum procedure replica refer replica maximum level replica square replica action replica replica iteration level work iterate one obtain even replica create level next especially discretization time process observe carefully definition test phenomenon describe illustrate splitting resample unbiased estimator bound observable realization highlight choice bx obtains contribute one retain specific observable bx give namely work investigate introduce contain refer framework splitting framework fit highlight essential mathematical produce prove framework estimator propose variant organize main framework introduce path markov analogy branching resample mutation precise variant illustrate flexibility order introduce section consistent context space assume section path chain q aim estimate main introduce two section framework structure index therefore application measurable construct consider subset z back e z convention consequence introduce stop level stop level level stop field characterize particular interest application z x ingredient field consistent resample resample probability index resample use resample continuity mapping right introduce dy consistency introduce assume consistency relation distribute eq assumption law replica necessary finally mention assumption view implicitly accord measure ii initialization step resample framework aim introduce general refer adaptive sequel section introduction successive step splitting satisfied random iteration n x become precise iii level correspond item many variant section framework adaptive splitting index measure initialize z stop perform branching satisfy replica replica old new child replica thus q q q label update child parent replica label parent replica resample procedure replica child parent branching replica set q field field sample next level assume level increment go follow iterative procedure labeling way framework field index z endow field necessary three random property branching assumed sample conditionally estimator claim emphasize requirement level instrumental optimal easily get property emphasize measurable stop section moreover replace algorithm section martingale algorithm thus go describe prove section endow topology explain define pz assumption consequence continuity property crucially open implie kernel precisely resample piecewise x ax xx x explain practical splitting criterion branching indeed branch splitting iv check satisfy requirement notice branching positive particular strict weight formula ng n nn consistent g p framework n let requirement satisfy computation actually result highlight variable subset measurable consequence fact last result assumption convention q partition I thank sure mass indeed weight induction satisfy estimator define notice actually one obtain explicit algorithm sequential monte smc sampler familiar smc method reaction value understand sequential importance introduction algorithm highlight reaction coordinate label successive rather iteration iteration system index level check standard sequential iterate step reach set accord split total level path stop define smc crucial smc resample parent section precisely resample one unchanged check discussion method comprehensive mathematical particular dynamical interpret discrete path stop iii hard obstacle reach differ presentation use change picture construction unbiased form standard smc language normalize ratio average rely smc reaction extend reaction consider discrete section variant generalize improve section particular three illustrate setting enter dynamical setting path markov design possibility level lead exactly require resample simulation variant resample define modify follow sample kernel chain random variable distribute use stop reach order markov chain stop time z additional markov chain euler langevin dynamic see initialization since algorithm sort entire idea level th flexibility parallelization notice modify branching branching number affect important spirit sequential importance bi enforce probability branching visit channel sufficiently implement branching present stop reaction enter path dependent reaction coordinate duration continuous process jump branch homogeneous stochastic etc bridge brownian interpret dx x mx distribution lemma fact natural deterministic build gaussian empirical prove content state section test function
instead arise anomaly unknown proportion mixture mixture proportion mixture set address development sec jointly optimize scale many discussion datum contamination let empirical population consider membership property convex q simultaneously observe contaminate parameter search category line distribution category dash show divided point mass numerical conduct deterministic input run show mixture conduct solve average total second implement broadly classify fit testing recent bernoulli quantify contamination address probability fisher exact solution contamination pearson limitation approximation category employ optimization category ingredient readily thus eq complete combine confirm particular assume contaminate dx b three step first write second property regard separation property close solution valid lastly separation equivalently form unique large factor let order show kkt eq simply suffice proof confirm complementary verify primal feasible lastly check primal trivially kkt satisfied range strictly approach infinity approach examine behavior constraint allow minimize increase monotone existence uniqueness fix decrease statement require arbitrarily objective realize bind close divergence ready difference must q result thm remark thm prop university identify anomaly variety estimate contamination contribution contamination control appeal contamination series program contamination goodness testing detect wide environmental science motivate anomaly contamination communication computer system application management internet broadly distinguish include dimensionality anomaly base g distributional threshold identify anomaly establish norm threshold anomaly false alarm see section base estimate anomaly level consider contamination free specify comprise distributional standard method compare contamination testing base base answer question consist empirical member distribution subset attribute inequality problem whose geometric application model number model finite lastly show category denote index empirical occurrence leibler jointly lastly probability simplex distribution category entropy mixture distribution correspond concerned specify significance sample random model contaminate quantify define statistical significance contribution herein order sample q xx iff order definition typical empirical interpret become require increase sequence typical original insight continuous create discard contamination contaminate contaminate must attribute contamination agnostic limitation case contaminate full empty significance level time consistent report zero contamination reporting proportion minimum know important grow result grow theorem exception involve category directly check contaminate alternatively deviation particular contaminate kl provide way contaminate level numerical efficiently check contaminate empirical size single check contaminate contaminate follows exclude long answer question discard kl still remove empirical create discard possibility provide twice discard violate exclude interpret number time appear discard kl empirical sample remove check contaminate note meet convention imply empty contaminate
mind expectation randomness learner literature find slowly regret environment feedback achieve regret bound least logarithmic factor bandit notable exception achieve set bandit guarantee extension set also propose version match though substantially realization sequence improvement specific arguably improvement replace round loss action standard bound take exist superior cognitive channel service answer question full aware bound show prove previous armed discussion scale size simple implement combinatorial computationally efficient similar combinatorial bandit approach perturb show appropriately tune largely minimax whenever become notice inferior bound know tune adaptive sense way bad guarantee armed bandit action e tend large e online good ad satisfactory bound worse worse increase problem consider much type provide sequence stay condition variation obviously easy construct linearly summary conclude order depend quadratic variation capture loss discussion full reference comment bandit set vector every combinatorial bandit name bandit give bit confusion combinatorial bandit learner observe tt line prove highlight algorithm distribution dependent note compare regret bound expect regret rather argument regret actually optimal refined assume expect regret argument variation explain key underlie many know bandit algorithm entropy regularization tuning depend loss prove may easy give reasoning replace keep close bind course bias challenge rate schedule information perturbation perturbation know achieve guarantee bandit satisfied action perturbation specifically perturbation variant exponential truncate tuning ix algorithm hold truncated perturbation implicit exploration parameter draw technical going proceed comment probability computable close efficient equivalent expectation resort presentation otherwise implement access efficient introduce answer deeply reader might reader answer without price however relate truncated perturbation select use exponentially perturbation vector play round particular establish total relate quantity ease upper similar important highlight generate entail hold algorithm follow integrate side md md concern state hold proof defer armed ready prove thus dl statement expectation substitute achieve bound requires trick overcome difficulty issue modify tune solely observation tune corollary notation notation ensure hold tune random known analysis adaptive deterministic largely simplify see treat performance guarantee result regret simultaneously nonnegative together hand ready theorem plug equation expectation jensen prove bit care rule bound eq solving auxiliary equation truncate perturbation allow step forecaster lemma result replace provide bind term assume prove final lemma quantify proof observe discuss implication extension really hold truncated perturbation perturbation bound along line arise perturbation additional become note induce paper essential result high suggest would handle corollary proving confidence variant leave future acknowledgment high education research author thank france optimization repeatedly combinatorial loss associate learner action propose improve scale loss action feedback combinatorial combinatorial feedback
intercept constant f trial count choice iterative initialize reconstruction implement range grid reconstruction fig reconstruction realization regularization optimize reconstruction visually several weak may attribute element average realization poisson total count minimize total increase fig reach continue improve thank sparsity constraint employ improve reconstruction objective fast time reconstruction measurement identity adopt link intercept solve criterion performance integer reconstruct substitution toeplitz element constant select element gate average cpu show normalize measurement yield reconstruction unknown small compete almost second well similarly large performance unknown time slow known omit density simulate orthonormal haar level circular mask sense transpose fan platform circular mask ray rotation center platform image pixel choose space vary correspond collect detector ray detector maximum element number projection c reconstruction projection take show reconstruction projection reconstruction visually reconstruction list reconstruction signal truncation truncate thank projection well bring take minute converge reconstruct nonnegative transform proximal scheme nesterov acceleration propose iteration decrease computationally state reconstruction signal discover crucial size handle fidelity constant avoid convergence develop sparsity poisson remarkably focus incorporate sparse ray maintain publicly package unknown substituting q upon ignore maximize ignore establish convexity remark concentrate hessian prove positive apply cauchy schwarz eq q r label step invertible except subgradient minimize solve proximal objective construct ordinary optimum optimum dual minimize algorithm j proximal close use get direction since make large step take scale relax orthonormal linearize linearization regardless indicator quadratic denote instead linearize ordinary condition english english proposition remark edu accelerate proximal reconstruct nonnegative motivated signal nonnegative represent material hyperspectral band adopt data fidelity indicator accelerate account vary provide numerical accelerate construct sense reconstruction gaussian generalize wavelet achieve method reconstruction compressed compressive paradigm transform domain number much appropriate signal vector negligible magnitude idea compress value p noiseless measurement compress focus nonnegative encounter hyperspectral dna monitor hide see transform nonnegative practical application ray ray correspond material map activity pixel concentration region interest nonnegative transform domain recently consider linearly toolbox adopt difference onto computationally impose hessian norm poisson link advantage paper also adopt unconstrained minimize scalar constant quantifying term impose orthonormal c transpose zero identity logarithm soft I follow poisson often adopt optical hyperspectral count particle detector poisson c mean identity ignore term linear summarize function hessian identity identity count optical emission deep light physics identity scan adopt identity link simplify intercept link function account imaging refer treat disease mapping combine norm intercept unknown replace ignore concentrate appendix regularization constant define certain intercept thought nonnegative constrain problem integrate toolbox substituting linear propose nesterov numerical conclude whose u proximal discuss nesterov acceleration achieve iteration second momentum accelerate iterative denoise perform impose size satisfy acceleration iterate monotonically place monotonic carry accelerate discuss j j usually set obtain sum exact orthonormal appendix linearize see onto nonnegative yield objective wish outer consecutive return index respectively within criterion I I j sufficiently improvement loop achieve compare loop return sense outer size backtrack inner splitting part threshold identity interest identity second derivative decrease global constant indicate coefficient minimize large step towards noticed analytical size design seek conservative iteration size consecutive iteration otherwise multiply keep subject signal reach type without increase step would fail constant algorithm illustrate advantage adaptive backtrack signal sparsity employ equivalent approach employ scheme modify monotonicity condition accelerate scheme regularization desire return initialize especially parameter usually unlike decrease convergence initial final indeed range parameter keep close reasonable scenario identity repeat u map intermediate threshold c guarantee ensure decrease together reduce intermediate general inspire adapt intermediate example follow explain regularization hold ref regularization constraint converse hold minimize appendix discuss implication motivate serve scheme thank use focus gradually decrease convergence threshold among realization fig reconstruction sense optimal average imposing improve greatly metric impose sparsity final minor reconstruction reduce fig omit report time partly matlab measurement realization sense matrix measurement group traditional signal sparsity separate achieve thus incorporate via active signal possible implement line truncation group bring method regard fail reach reconstruction much employ explain valuable small converge achieve convergence start fail black color solve fast within competitive term attribute step explain similar unstable behavior objective function benefit convergence scheme thank adaptive run run convergence occur complete reach phenomenon trial completion explain threshold section consider ray conventional image generate matlab orthonormal construct haar decomposition full circular mask transpose operator platform circular mask detector array set initialize reconstruction store allow storage method implementation accept vary achieve fig show average compete fig two method group ii reach upon fig projection fix choose equally vary coincide projection projection fig time coincide benefit signal fix projection achieve projection achieve accounting bring significant benefit time projection time perform poorly projection consider converge gap employ show reach low reconstruction similarly reach reconstruction time fast reconstruction group consume gain reconstruction increase fig converge slowly perfect fail achieve objective compare realization projection beyond point proximity minima convergence threshold reduce far result decrease convergence signal separate method reconstruction reconstruction space realization show gray start reconstruction true subsequent plot suffer inferior nonnegative group signal sparsity reconstruction impose sparsity achieve quality pt mm adopt sense stability generalize divergence small cause nesterov acceleration make infeasible would address logarithm q observe continuous step size parameter minimize regularize model applicable package option fitting
fdr aggregation conditional uniformly symmetry interpret shall expect likely original odd enter early also hypothese hypothesis likely early improve sequential decentralize center normal variance randomness filter pilot order value aggregate hypothesis let summary statistic measurable evidence correspond particularly interested function decrease condition include x mx x iw one size order place aggregated simply number behind feature nonzero binomial translate refined far summarize filter get shoot communication aggregated statistic fdr call false rate fdr special accept reject rule introduce randomized assign hypothesis close fdr motivation fdr close whereas randomization undesirable practical fdr potential find internet randomize decision occur frequently randomization testing g mention refined statistic motivate nan matter away false nan nan jointly binomial stochastically generalization incorporate obey interested reader nominal eq convention reject hypothesis present control set summary function q lemma parallel proof lemma let martingale run backward know variable proof theorem idea rank argument prove without get eq stop time stop theorem stochastically would control interpretation favorable reader refer attractive translate control information improve still control aggregation communication decentralize send sign piece encode bit require logarithmic respectively original median low control fdr maintain big instead decentralized simple achieve vanishing start column far extremely augment design orthogonal take cost otherwise uniformly cube denote reject last tend aggregation slowly nominal obey hence aggregation capable distinguish move send model center hypothese piece message non interactive bit budget interactive hold tend exponent constant coin hamming hence signal design strength row draw length randomly level save hold take scenario universal detection access decentralize allow summary simulation ip achieve level vary potential case meanwhile aggregation least square ol procedure estimator obey pm ignore value hereafter choose send center majority vote share signal illustration show sd ols lasso validate e fdr despite validation still term satisfactory aggregation ol nominal though aggregation around nominal ol spread sometimes information contrast ol undesirable aggregate run decentralize aggregation enjoy fdr provide exhibit address framework cover broad link room investigation fashion last interesting incorporate lead much strong manuscript false hypothesis similarly hypothesis conditional concatenation proof respectively binomial extensively use know hold without hypothesis agree exchangeability proceed jensen reveal obey right rhs attain obeys give jensen give rhs shall decrease consequence remain hence tend permutation rejection rule take almost vanish generality replace make extensive eqn sufficiently eqn give shannon consequence move since exponent obey next proceed finish nothing v pt theorem proposition theorem definition lemma remark section stanford usa control false spurious target possibly manner start filter shoot fdr asymptotically signal setting scientific significance summarize explore big number difficulty hypothesis simultaneously sophisticated tuning arise chance care address community decade fdr elegant fdr
furthermore provide hill simple problem asymmetric insight experimentally asymmetric likelihood markov stock index flexibility introduce also insight entropy organize constrain asymmetric normal condition compare asymmetric normal one real world advantage finding characterize existence underlie split segment must base provide segment continuous weight describe simplify associate create laplace constrain x pdf argument partition describe continuity pdf mixture guarantee preserve build force pdf place side map constraint underlie pdf redundant use volume unitary place part describe non pdf split base exponential parameter behave therefore produce stack sufficient produce stacking hope without fix partitioned separate parameter mixture asymmetric introduce laplace normal version optimize appendix separation place mode follow avoid mixture laplace prove generalize asymmetric laplace constraint give respectively arrive laplace define family equation match laplace get close negative less exhibit describe asymmetric let q satisfie appendix arrive normal asymmetric define exponential show asymmetric case adjust model involve indicator belong current markov asymmetric identify parameter specifie give make simultaneously optimize either formulation likelihood optima closed optimize numerically describe alternatively define exponential show sound new intractable approximate conjugate therefore optimize important highlight particular likelihood give equation entropy divergence therefore third likelihood depend move implicit come define increase fitting loss become asymmetric asymmetric optimality pdf optimum weighted median optimal look check partition optima median create value outside require alternatively consider show conjugate give equation equation asymmetric conjugate density give gamma gamma distribution laplace format write distribution parameter bernoulli beta gamma parameter hyperparameter link way partition optimization asymmetric pdf normal give optimum similarly asymmetric look partition valid similarly laplace instead asymmetric define equation asymmetric conjugate asymmetric give equation prior eq beta distribution agreement prior variance prior section show maximize likelihood hill initial tolerance adjustment hill algorithm go algorithm keep compare move direction maximize likelihood repeat asymmetric example choose fix behave hill allow change able avoid first solve minima flexibility asymmetric new application version understand explore deeply normal frequently standard therefore asymmetric able adapt gamma regression may user may people familiar asymmetric normal since use emission flexibility linear parameter compute offset asymmetric asymmetric weighting c figure asymmetric straight relationship dot since standard fit point concentrate simulation value uniformly set symmetric fit dash line since describe symmetric symmetric case likelihood asymmetric asymmetric higher perform however symmetric asymmetric likelihood asymmetric fitting decrease asymmetric explain distant incorrect noise model equation prevent error weight equation prevent inherent equation impose much line confidence clearly far parameter sample like instance draw estimate specify noise outperform motivate asymmetric able interpretability insight use state hmm asymmetric improve initial estimate hmm emission iteration every emission asymmetric hmm method describe last price consider distribution economic compose logarithm consecutive day sample th day happen sample miss day symmetry normal may reflect reality market return market expect introduce emission asymmetric state asymmetric flexibility state increase subsequent symmetric asymmetric b emission subtle component large preserve considerably certain none symmetric indicate likelihood appear b evaluate become fourth transition entropy clear transition c miss entropy reduce entropy occur state histogram normalize entropy divide hmms considerably state version version less spike emphasize figure quantile represent identity high first reach latter almost always equivalent quantile low additionally curve consider indicate asymmetric lack introduce use normal create new asymmetric underlie keeping prior moreover distribution inherent term regularization directly likelihood impose avoid symmetric underlie compare asymmetric understanding operate symmetric version asymmetric must asymmetric normal hmm stock flexibility allow distribution
encourage powerful et al et purpose logic logic al coding kind knowledge subsection arrange representation spread representation representation min percentage representation et b et et hyper sphere representation et hull logic fuzzy fuzzy fuzzy et et al al representation stack b genetic programming x fuzzy representation group category present rule schema format correspond representation partition space result paper apply value part define interval handle range interval represent tuple value match classifier argue many vary interval phenotype phenotype mapping perform mutation operator truncation must keep bias frequency appear condition representation name value match dimension truncation comparison frequency interval issue order operator might produce infeasible order previous name phenotype one mutation provide pressure unlike mutation pressure raise inconsistent block architecture ga run al present name maintain encode et problem equivalent unlikely change htb difficulty boundary long successful extension classifier hyper condition axis advantageous new condition base sphere shape sphere common condition deviation axis parallel condition hyper axis name transformation indicate fully ellipsoid represent ellipsoid center ellipsoid transformation ellipsoid diagonal initialize zero hyper angular represent ellipsoid encode mutation operator act al decrease successful evolutionary investigate condition rotation hyper ellipsoid three angle hyper ellipsoid highlight mention htb activation center current activation use match parallel ellipsoid hyper respectively promise approach report al continue general condition dependency representation note suited boundary parallel effectively another represent value concept hull depict lie inside form representation convex hull fine complex region asymmetric htb represent hull hull present angle hull hull variable sized condition hull base representation fast condition classifier consist rule base system interpretability must considerable besides fuzzy logic mechanism fuzzy goal behind effort capability fine fuzzy online briefly comprehensive notable approach fuzzy logic integrate fuzzy fuzzy message list researcher fuzzy fuzzy fuzzy task al many reinforcement reinforcement et reinforcement relation membership especially control et al address classic competition versus fuzzy style name divided produce classifier base apply reinforcement agent et learning framework call analyze model classify literature al al fuzzy logic rule format issue system successful ability proposal et name fuzzy mining comprehensive description fuzzy fuzzy fuzzy represent label fuzzy evolve consistent fuzzy rule mapping real nominal common rule consist fuzzy input linguistic operator linguistic meanwhile represent linguistic classifier code schema integer code schema action propose bit linguistic dimension dimension linguistic part classifier bit linguistic bit appearance linguistic utilize linguistic show portion input cover linguistic rectangle htb example fuzzy visualization area fuzzy classifier white part fuzzy feature offer miss fuzzy support absence consist promising robot simulation online learning increase adapt fuzzy well issue modify predefine fuzzy linguistic employ produce fuzzy modify fuzzy manner original fuzzy linguistic knowledge well linguistic fuzzy technique modify inclusion cause improvement expression environmental well understand condition classifier value exceed add feature translation ga representation phenomenon happen occur important evolution ability illustrate like partition expression tree form accord different namely encode phenotype classifier way classifier form seven encode gene initialize terminal arbitrary visualization cover area phenotype advantage fine environmental linear tree also verify validation operator attractive make gp like representation report show environment slow compact dynamical representation term dynamical graph condition wherein boolean present node input node correspond input connection common benchmark computational rl scheme name classifier like perceptron mlp condition ga mechanism besides number node rule see extra whether member member correspond high test single step name show addition examine root fuzzy radial component evolve scheme namely ability problem continuous action promise include also applicable single hybrid propose ga self adaptation explore system well adaptive improvement optimally complex one al replacement make advantage ability issue name drive mining maintain predictive part encode predicate et well structure space htb space visualization highlight et define produce cover hyper rectangle extend code code common successful tackle learn space map overlap partition input hyper prediction resolution ga replace match specify input partitioning suggest extend test take car evolve reach converge technique strength effectively system suited identify problem area proper whole complexity effectively problem setting relative handle problem category worth category boundary aim kind solve environment word researcher propertie knowledge string interval good boundary htb positive instance white negative instance boundary representation technique kind simple popular technique might classifier space ability parallel boundary continuous action handle round interval independent method use produce recent htb region white problem properly consider boundary contrary hull representation decision due shape flexibility cover model obviously ability cover unknown dimensional p applicable environment implement well contain boundary limit value integer value mining al medical mining al data set et al suit wherein dependency dimensional shape applicable real value environment length complex generalization interval convex ga fuzzy fuzzy applicable capability maximal set interpretability fuzzy support action handle mixed attribute miss produce limited freedom flexible adapt mechanism et reinforcement al present among ga operator relational well suit nominal able offer ignore input add relevant applicable value offer great ignore add step datum free able produce interpretability every accept necessary multiple et hard number problem problem string one attribute real subproblem value value attribute subproblem solve mixed attribute representation attribute gp like fuzzy technique mention representation remarkable provide table representation main advantageous domain test technique exist far improvement knowledge seek prominent capacity efficacy decade interest propose representation grouping incorporate category representation schema representation technique partition extensive knowledge illumination domain comparative analysis hope stream research usage since representation like properly survey interest researcher practitioner choose knowledge stream research key rule include insight partition turn prominent generalization whole attention mining community efficiency find comprehensive yet elaborate knowledge group incorporate schema format support precise technique partition extensive experimental provide view technique comparative interest researcher practitioner research topic framework cognitive year originally general principle evolution cognitive framework ensure reward evolutionary population production genetic reinforcement technique production system population methodology address classifier system inspire university de classifiers system usually university member ga complete population form mechanism reward tool lose rise reinforcement agent simplify use reinforcement accuracy system classifier system wherein fitness rule use effectiveness paradigm extend wide computational economic mining light control al al diverse ever valuable domain author try research indeed aim development et give past try look ahead system various domain focus sequential introduce historical major algorithmic difference principle interested develop specific good exist try common consider provide research area might efficient enough progress regard handle dedicate survey individually focus modify give consequently survey attempt description knowledge attention mining term efficacy identify environmental support system attempt elaborate exist knowledge incorporate schema precise additionally comparative exist conventional current providing choose problem address none trivial issue hope survey understand stream research choose representation interest researcher practitioner stream provide description describe incorporate subsection schema format explanation comparative conventional general include conclusion remark system belong combine genetic ga paradigm solve specific representation component provide environment reinforcement responsible incoming discovery create mechanism evolve ga idea keep notable solve compatibility develop style environmental detector environment eventually reward select action efficacy reinforcement try receive produce current reinforcement subproblem subproblem usually use evolve review mark appearance successful popular recent et show online mechanism continuous classifier call empty beginning randomly different condition action integer payoff apply fitness etc receive environmental match set environmental classifier match operator predefine current
design small advantage increase design reduce optimize advantageous large fill dimension situation preferable remain computationally expensive variability specifically application random compute eq nd measure variability n take lebesgue functional distance distance variability variance cause uncertain quantification simulation fine grid estimate method generate fashion quality grid thus possible two select possible precision simulation simulation simulation grid distance distance transform transform variability equation benchmark realization realization simulation conditional field simulate time design use obtain three grid field krige algebraic mark benchmark grid obtain interpolation simulation obtain simulation approximate enhance example show substantial cost optimize b well point optimize benchmark behave measure reconstruct point optimize criteria b time optimize interpolation total approximation benchmark could point results optimize show value reconstructed obtain simulation point blue dotted level denote represent variable kolmogorov kolmogorov statistics test distribution optimize point rejection distribution optimize point estimate volume reconstruct computational reduce cpu simulate design total second intel ghz gb ram simulating realization however random fine impractical fine approximation realization conditional literature define distance variability variability appeal grid show cost uncertainty quantification could improve volume monte carlo simulation fine design attain volume six indistinguishable attention regularity predict design predict realization correct center issue need bias field approach extension optimization generic box analytical dx dominate prove pick almost everywhere conversely zero word everywhere almost density standard thank helpful insight remark lemma quantify uncertainty evaluation budget adopt objective function realization rise approach analytically tractable expectation variability carlo rely random choose realization fine design predictor computational cost simulation enable prediction dimension fine grid uncertainty volume six process number application behavior practitioner get input forward prescribe inverse problem scalar quantifying equivalently pay reliability engineering describe configuration lead nuclear science phenomenon costly evaluate typically consequence systematic space fine reach reconstruction interest evaluation rely net mainly focus modeling become evaluate rely drastically reference therein approximation enable quantify uncertainty conditional idea appeal context estimate like reference therein realization simulate field location question arise notion scatter variable context quantification present base field carlo often obtain simulate fine design need especially choose approximate predictor obtain set unbiased predictor point introduce way specific reconstruct set possible divide section introduce formula present explain introduce limit advantage present optimize simulation application show g approach allow transform uncertainty quantification transform heavily quantification dimensional volume monte method unknown reduce computational time definition come theory close main continuous objective df paths kernel range borel tt simplicity generalization interval pre image concept theory notion role approach algebra cs one expect distance borel close particular notion measurable review another expectation notion variability empty set define appropriate space close respect variability let infimum say addition define notion heart discrete simulate e moderate dr become impractical involve rapidly consist fine simulation rely basis quantify uncertainty actually end trace fine simulate propose replace remain construct way expect evaluate essence affine predictor deterministic krige predictor case review simulation krige context notably uncertainty complex address purpose criterion measure borel eq mean bivariate krige ne k ne ne eq random conditionally prove conditional moment tm ne ne e ne conclusion field suffice nk ms simulation consistent necessary proposition established select ordinary predictor algorithm find assume simulation fix package characterization field several technique already design simulate approach lead rely analytical gradient slow follow heuristic optimize previous current add reach bivariate time optimizer rely krige krige model sequential krige formula new approximate bivariate bivariate package fast standard bivariate cdf
possibility one employ hardware device restriction small sized baseline ann describe neural wireless able prediction come layer output activation ann output hide step wise step activation available weight bias equation vector previous consequently validate database real accuracy demand simple architecture hardware wireless network promising advance wireless physical wireless capability provide application monitor health physical environmental ambient network sensor home environment house develop home paper temperature home resource study european primary energy demand consumption home half consumption air conditioning thus major overall necessary develop home demand efficiently consider plausible artificial intelligence soft compute artificial learn apply wide devote system problem normally historical traditional back propagation could minute hour number stopping criterion receive datum consume lot mention nevertheless could trained totally learn successfully regard bp application line arrive soon observation discard without necessity store historical imply necessity storage prior many generalization learn computing resource idea integrate technology cost concern idea resource variable consumption sequential objective hardware device intelligence forecasting environment cost far monitor ann ann require storage whether feasible ann resolution estimation short period innovation bp time income feasible device hardware resource hardware describe work wireless topology slightly forecast back bp neural depict implement resource experimental conclusion explain draw future power environment devote monitor current activity control actually embed capability distance wireless medium thousand sensor typically part internal circuit embed device sensor node also sensor sensor correspond memory communication advance mesh propagation technique network besides unique characteristic constraint state dense sensor sensor impossible change severe constraint resource focus sensor node application sensor physical failure topology change topology node channel identification address overhead identification traffic pattern multiple certain requirement different objective capability typically influential design sensor node cost low consumption self reliability tolerance security wireless acquire usually devote collect come computer store persistent device want acquire pc network node wireless network add figure display pc validation purpose four sensor capture room allow power extend four sensor technology tx cc ghz system wireless combine excellent rf enhanced memory kb ram powerful limitation cc suit consumption several advanced operate systems wireless configuration also power within temperature security transmission exchangeable four sensor room temperature ambient calibrate digital point temperature two general output expansion sensor addition reach depend connect hardware sensor wireless sensor sl monitor control security alarm sensor implement learn advanced architecture nevertheless highly accept lot area market control moreover excellent application device competitive really device implementation consumption record forecast future energy resource record repeatedly record interesting formalize scalar process appropriate model internal consider forecasting method smooth fit autoregressive move restrict storage much possible forecasting linear perceptron perceptron mlp purpose comparison perceptron estimate result standard storage complex computational expect however comparison assess base device propose summary difference vector measure mae mae several mae sample paradigm frame deal frame modify time failure raw incoming frame hardware resource buffer incoming time variant back propagation probably noisy computation high model compare batch bp random traversal well series statement implementation device algorithm skip allow ignore incoming depend skip additionally powerful device buffer sample also skip rule behavior take consideration least training scale tackle adopt belong input unique capture solve incorporate regression response predictor concern predictor non complete one condition variable could additional column consequently coefficient write represent identity predict receive value series must element simple forecast consequence build need start represent assumption auto correlate use incorporate feedback violate introduce lead biased consider skip primary building estimation prediction acceptable storage perform scientific absence available assume bayesian could incorporate parameter context estimation informative prediction predictor first demonstrate solve inverse employ term storage remain expensive must future way introduce estimation context informative datum utilize treat additional last assessment previous follow read represent thus predictive assessment high process storage cost high hardware concern standard model simple computational resource requirement model ann consider device resource train ann state propose sequential bp compare baseline perceptron mlp matrix bias input ann describe bp ann layer moreover influence bp kind need algebra operation dot product logistic activation implement library requirement implement ann element bp real perceptron real mlp hidden output correspondingly bp number resource bp mlp implement ann memory data temperature experimental node room house continuously hour pc mainly configuration place room home central control energy purpose competition loop show start initialization minimal rf interface issue wireless protocol rf aim rf aspect correspond ann establish random ann system start decide core main receive sensor sensor place room temperature among message come wireless subroutine receive temperature average min ann average one ready main loop negligible requirement use pass consumption communication depend frame implementation goal wireless h subroutine display subroutine receive iteration responsible second happen call compute temperature nature consider implement aggregation temperature aggregation consecutive integrate datum pair second use store call time miss bt store return case consecutive consecutive different non lose case temperature slope compute intercept begin line segment second increment temperature complete mean value subroutine frame system negligible use aggregate lose temperature subroutine compute consecutive auxiliary buffer length buffer control forecast buffer counter equal buffer counter use static call execute compute increase unit success circular equation condition check buffer forecast follow cumulative add input call static circular buffer number memory consumption whole bp control add current counter need buffer buffer size assume position buffer least matrix initialize start use different model iteration forecast add algorithm forecast state train aggregate second system plot mid window receive delayed value forecast condition air temperature relate receive pass directly study complete first generate large baseline real application uci repository dataset house world competition energy structure ready explore simulate consecutive randomly modify
mt mt sentence mt sentence limit corpora direction use diag final balance gram portion million chinese stanford trees vocabulary frequent chinese english approximately map special token descent train minibatch word gram lm cnn convolution layer dnn multi perceptron softmax insensitive gram baseline decode string configuration cnn target target side hereafter test hierarchical phrase base dependency integrate table proceed give report clearly cnn cnn baseline average mt indicate decode worth informative fact avoid propagation alignment cnn cnn win table signal complementary cnn cnn cnn head cnn pooling cnn pooling cnn strategy max extent max pooling max pooling replace local pooling layer pooling layer guarantee clearly see pool pool well conjecture relevant translation mainly source sentence pool seminal text recently context word model clearly relate work instead ad hoc window cover model effectively leverage form sentence cnn rnn decoder representation source therefore inferior directly integrate apply decode signal weighted sum proceed importantly nonlinear retrieve summarize source propose devise consider linguistic enhance cm lemma proposition conjecture conjecture thm remark institute technology chinese com adapt centre school city recently neural gram systematic treatment convolutional decode design architecture part target source form representation language word fed network dnn strong experiment english task achieve improvement space source language attract much statistical translation model neural sentence encoder decoder encode part decode process notably model gram achieve architecture dynamically cover entire sentence effectively part information language decode convolution architecture sentence unified representation together fed dnn purely decoder cnn signal decode joint art chinese english translation show able improvement outperform cnn b cnn start overview convolutional key decode experiment report cnn cnn predict language probability target source p stand cnn index source stand cnn figure translate target proceeding word gram lm source word source cnn generate cnn g generic architecture encoder basic generic cnn encoder six form length shorter put begin convolution layer simply sum feature window size global final convolution operate slide window carry high sentence ff ff eq previous cnns nlp take convolution fusion select value map soft template keep convolution separate release score convolution composition overlap window map convolutional layer segment strategy merge embed logistic model windows layer parameterized assign weight representation train along embed proceeding target dnn soft word procedure corpus seek maximize one parallel optimization perform back propagation batch cnn describe layer cnn extra embedding indicate word treat regular embed tag parameterize supervised predict propagation learn put make stand adjust cnn predict learn alignment unlike tell location cnn proceeding encoder retrieve essentially attention alignment attention generative network rnn decoder basically proceed window specifically window index convolution dnn sigmoid retrieve sentence transform retrieved segment representation word target language cnn proceeding provide complementary verify decode improvement purely integrate decoder adopt integrate gram hierarchical extend include index word align align source word integrate joint dependency translation decoder efficacy describe dependency mt art dependency string dependency string employ rule head string head tree top
empirical max convolution nearly identical max method even fairly rough fast second second naive approach speedup increase dramatically term essentially viterbi claim state track train vertical viterbi right track viterbi fast affine nearly exact substantial particularly multipli particle observable universe slow speedup basic approximate chebyshev accurately suggest possible compute contour value real real magnitude numerically approximate convolution history long would interesting rational rather highly increase precision arithmetic operation small increase optimize error runtime base contour search contour base optimize trade runtime max convolution affine modification matrix vector tensor likewise element multidimensional norm norm enough piecewise multidimensional convolution tensor likewise via row transform demonstrate fast numerical convolution method run tensor without speedup tensor convolution considerably tensor reason speedup become even increase dimension width cost convolution mean graph adjacency use distance node concrete example max convolution naive compute tensor replace argument size convolution likewise additive transition multidimensional allow dimensional american deconvolution computing already stable denominator large piecewise method absolute piecewise piecewise bad contour correction negligible must nearly vertical scatter plot exist index create scatter correct conversely entirely contour zero achieve requirement simultaneously point absolute individual scatter bad case affine absolute absolute code convolution numerical would thank suggestion max closely convolution convolution occur field approximate chebyshev derive error propose bound viterbi markov approximate viterbi image calculus equilibrium generate wherein probable identifying protein convolution operate semi mean identically convolution operation convolution min convolution operate semi convolution effort quadratic max vector function analogous nonnegative max convolution find l max convolution equivalent find probability event version subsequently numerous small drawback category accurate method worst conversely type bind category solve convolution either complicated sorting create numerical solve convolution equivalence process index vector chebyshev norm step choose yield fourier algorithm date demonstrate good task particular convolution efficiently probable discrete probabilistic generalization sum hard evenly spaced possible value analysis formalize closely note value multiply back conversely suffer less perform numerical bound worst demonstrate begin compare analyzing method improve variant use max summarize introduction max give numerical convolution two parameter scale maximal standard still numerical error maximal chebyshev exact note converse numerical achieving performance significant convolution high method cutoff previously method increase empirical numerical stability boundary formalize convolution argument fast convolution approximation stable boundary replicate max perform source see depict left mode mode large index regardless occur go match tolerance piecewise implementation make accuracy result dominant convolution employ large choice characterize well approximate unstable unstable pose improve give number runtime respect term maximum introduce compare high piecewise method convolution vector value estimate r I result l derive scale mention refer scale demonstrate problem stable fast convolution easily element chebyshev norm p u p denote one element simplify therefore reason bad contour bind piecewise step achieve desire practically particle observable universe full approximation stable many method stability vs index tight elliptical contour slope contour lower depict scatter plot exact piecewise every exact correlate contour slope approximate contour slope contour bound constrain scatter plot point inside envelope figure previous ideal affine contour exploit correct bias small within contour f u max single index result via naive quadratic long index already cost small contour affine compute correct contour specific trend approximate exact numerical estimate error use possible return convolution slope I I contour contour I I I r combination absolute affine piecewise also qualitatively point affine choose manner value value propagate affine thereby avoid error affine absolute affine affine transformation dramatically error fast negative viterbi additive transition arbitrary property compute variable viterbi algorithm exploit self small specialize hmm leave max vector multiplication backtrack thus enable left pass note modification perform complex pass vector pmf max matrix use describe return viterbi valid j additive transition economic country price predict figure
communication allow error begin input additive implie perform subset matrix argue construct define associated row let construct verify eigenvalue canonical consequently u I suppose randomize introduce shorthand thus compute solve study show bit scaling suggest establish scale complexity achieve rank tight question special compute easy algorithm communication rate prove tight low special many precisely rank compute large give rough connection observe rank matrix determine sum computing reduction series binary search whether turn goal testing psd testing reduction machine inclusion coin complexity moderate amount definite vector norm decide note minimize solve particular ix strictly solver allow importance rank estimation characterize communication acknowledgement partially grant office partially laboratory research office contract grant nf fellowship monotonically refine eq summary chebyshev guarantee yield th row subspace orthogonal th great suffice satisfy sample dimension loss dimension expand projection expand reach projection note project sphere subspace q define since combine claim completeness degree variable replace compare conclusion parameter exponential apply last exponential notice occur since plug proposition definition ccccc zhang berkeley electrical university california berkeley hold separate machine deterministic solving must randomized bit match demonstrate semidefinite eigenvalue generalize large analysis expensive order determine determine robust pca collaborative filter algebra scale divide approximate number locate determine motivate set decompose matrix store estimation formulation suppose want determine aggregated root determine rank paper computing exploit law decomposition entry exact limitation delay bottleneck efficiency reduce communication moderate eigenvalue chebyshev polynomial pass cost degree chebyshev polynomial algorithm establish derive efficient result deterministic communication deterministic approximate algorithm able corresponding algorithm randomize approximate bit relative bit establish communication low randomized eigenvalue randomize easier long history seminal characterize communication algebraic question low task non arbitrarily requirement practical applicable allow inexact practice li test inverse solving linear well know whether related streaming distinguish model generalize rank well semidefinite denote give generalize usual rank motivation terminology assume machine sum store distribute protocol machine exchange arrive close sense bind strictly eigenvalue close distinguish basic communication complexity see book detail standard party communication etc input string string communication scheme player common read protocol consists construct early player information communication protocol bit deterministic deterministic broad class protocol allow public randomness access string message randomize protocol correctly least probability framework minor define correctness public set master communication notion correctness model protocol error rank protocol satisfie definition deterministic study quantity allow contrast communication input substantially hard round discuss sequel discretization little communication complexity devote consequence trivial essentially assume bind deterministic surprisingly large scale matrix analyze encode receive order bind party holding suppose substantially slow composite function stage recurrence prove numerically stage substitute evaluate overall qx ia chebyshev expansion polynomial generate compute fa repeat overall combination generalize q logarithmic pre factor experiment algorithm practically suppose receive point uniformly random datum ccc eigenvalue b version estimate matrix nr ji sample sum generate choose choice motivate degree repetition let output evaluate square run experiment distribution eigenvalue eigenvalue generate show centralized set panel achieve communication efficient distribute case approximation chebyshev composite replace pass x chebyshev expansion method substantially communication bind spectral interval rank satisfy section true communication match choose bind communication achieve length answer code length open interesting natural machine deep investigation party communication say obvious currently upper bound lower defer low state hold sum operator rank addition mutually exclusive alternative hold use particular reduce two machine hold achieve rank otherwise conjunction test bound say orthogonal orthogonal appendix shorthand n conjunction lemma side inequality orthogonal use stre binary string communication perform reduction string encode communication complete proof psd find ib bit entry define final inequality yield recall q
replace represent entity abstraction abstraction abstraction word belong abstraction name type abstraction ability next abuse x abstraction part contribute sub describe match weather paragraph meaningful pattern mining graph discriminate one roughly evidence efficient success fortunately abstraction uniquely minimal direct mining reduce jointly side mine avoid find match remainder however mining abstraction grow graph discriminative ability start simple tweet recursively grow size side remove pair threshold grow efficient pattern give algorithm tweet extend nm nm nn nm mine abstraction abstraction name entity resolution grow counting replace entity appear side group pattern x stand merge give match abstraction stand similar abstraction x note tree correspondence sentence occurrence explore superiority tweet fig assign high spurious york occurrence mining pattern relationship text match text contrast text translation translation dependency deep incorporate determine text diagram text obtain look dependency convert binary text match learn neural network raw layer suit dense continuous building sparse demand take refer connect approximately underlie much go representation architecture sigmoid hide significant triple objective e margin control margin measure match basically tweet calculate candidate select score one original good chance average tweet translation tweet variant model calculate text representation since tend tweet text word text perceptron mlp concatenation input topic network layer neural logistic pattern input view roughly pattern embed base make trained descent insensitive mini present report large remove architecture performance setting detail performance architecture match regularization dropout prevent influence salient hidden generally architecture deep bring improvement slow margin one pattern l baseline test vast gap pattern embed fairly one one performance drop dramatically versus nine maintain drop contrast suggest space certain match case reliable fold hyper greatly improve suitable pool accuracy represent show vs observation partially deep cc response determine role abstraction improve p vs real cc want response candidate abstraction stand name entity role assign specific filter mining processing use matching extend notion match subgraphs domain common subgraph capture hierarchical relation pattern simply type string generate pattern learn difference consider tree discover vast match dnn task tweet margins work china national cb liu partially support ce adapt centre city university institute computing centre next city university translation answer matching sentence text propose approach consist mining discover matching text define product dependency text build matching tweet social medium hard match outperform margin central importance problem language processing formalize match two text match information retrieval matching modeling translation sentence meaning need appropriate response neural suited processing embed building embed answer short text match text represent relation sophisticated text correspondence hard capture embed study short text matching name discover subtle corpus pair short text network dnn decision text pattern task tweet chinese
north anchor dense north south white softmax softmax north south white north anchor south dense mm north anchor softmax softmax mm north south cnn north anchor south softmax north south font sep height text fill draw inner sep minimum height center fill outer pt sep minimum cm cm text outer sep inner width height text center bend right cnn cnn cnn xshift cnn right cnn xshift cm cnn east pool pool cnn north west south pool mm pool anchor dense dense north anchor south softmax softmax north anchor south font rgb rectangle inner sep center text height height cm sep fill rectangle thick bend bend cnn cnn cnn xshift cnn cnn cnn cnn xshift cm xshift cm yshift anchor yshift height minimum anchor south outer xshift yshift cm xshift yshift rgb pt width text center outer sep height text angle sep true height inner black inner width cm height cm text black fill inner sep black rectangle inner sep text center fill thick bend bend conv right xshift west west minimum conv north west anchor west conv right xshift xshift west east north south width conv north anchor west dm dm xshift right xshift cm mp dm north south xshift densely thick fit mp cm south north softmax north anchor south south anchor anchor right east inner sep west anchor cm base frame architecture network span video recurrence pool mp temporal briefly recognition overview depict figure pay fully connect number cell base individually mention optimize architecture model cause difference hyper preprocesse single architecture work video nevertheless much image contribute convolution layer stack overlap shorthand architecture denote map layer layer show promise second exploit strategy suggest ng et connect across video network window event lose across frame core rnns create memory temporal conventional recurrent build frame wise recurrence direction formally microsoft depth sequence user record front camera performing stand pose stream microsoft sign mean contain several performance pose vocabulary dataset kind video minute sample hz include varying position imbalance frame annotate contain skeleton show achieve good depth end architecture mini work mini exponential rate initialization describe file consist recurrent produce summarize channel shorthand optimize cell model base rnn generally locate frame correctly forward backward feed frame frame cnn frame recurrent optimize architecture frame score test frame well outcome cnn fine temporal max pooling give slightly observe temporal dimensional pooling show consider frame target frame video slide single many deep train pixel vertical direction horizontal rotation factor temporal factor interval value video online furthermore conventional spatial temporal overfitte recall pooling conv rnn lstm conv lstm follow challenge score dataset competition score category binary rate among category sequence architecture prediction baseline pooling vs max pooling last network surprisingly act cnn feature cell small temporal long combine temporal convolution architecture rnn cell improve score network multi table work outperform et al rgb remove depth preprocesse rescaling achieve need depth pose vs b l c depth knn et multi dnn conv lstm video information usage stream image prediction architecture sequence frame classify accurate difficulty boundary prediction recurrence temporal user feature map inactive without move activation movement suggest motion feature tb inner pt outer thick bend right bend outer anchor west bend bend layer map architecture without extract strong activation move learn paper recurrence rnn pooling architecture need account aspect add architecture notable impact able motion cell equally rnn recurrent able beginning frame great model future build subtle part write language annotation translation simultaneously channel sign translate audio acknowledgment gpu lead innovation science reference van van recent machine video however question temporal aspect architecture neural incorporate temporal recurrence crucial approach dataset art core human interaction become increasingly enable device towards subtle due vary performance camera hand motion target cnns de computer vision cnns abstraction impact like pose video classify video frame aggregate cnn pooling apart collection frames rnns either short memory lstm cells temporal dependency allow researcher achieve recurrent extraction motivate cnns rnns cnns spatial add capability recurrence video recognition almost play motion particular category beneficial spatial explore end apply wise challenge
leave invert thresholding class separate distance think moment anomaly technique information compare window simplification detect anomaly regular approach identify anomalous state estimation class classification color etc disadvantage encounter know available representative normal behavior probability moment normal compute observation anomalous threshold challenge infinite sequel optimization find financial economic wide application hold back intractable advance problem formally borel sign borel integrable stand sdp solve efficiently thereby tool attempt solve medium state solver relaxation optimality optimistic moment aforementione demonstrate anomalous real mean semidefinite column polynomial dimension denote polynomial coefficient variable th give understand column contain element illustration eq check positive reader detail refer polynomial matrix describe subsection discuss variable express indicator sdp relaxation moment tell optimality reach include completeness generally semidefinite program attain optimal bound approach work small density incoming anomalous classify compare threshold receiver operate roc curve metric standard way assess obtain matlab kde toolbox matlab automatic tune standardized option reader simple decide anomalous anomaly select incoming neighborhood sphere box form select fact set like account measurement equation moment present power scale quickly direction fewer available sdp solver whiten technique discuss unit may possible moment size resource whiten subtract univariate process store transformation obtain density anomaly detector moment contain two especially point increase three recover contour binary pose tool window mode estimate test consist outlier experiment moment c c complexity pdfs greatly multi distribution portion svm point outlier table moment include another trend continue h roll equation create consist h familiar
etc non mathematically respect dominate I possibly share refer set independent prominent stochastic covariate parameter life application covariate researcher rather fix situation include clinical pre treatment level etc I nh residual residual variance covariate robustness general nh case nh minimize average originally density excellent property approach also model alternative tail suitable property nh boundedness applicability test generalize glm covariate present supplement comparative remark end proof supplement refer condition ensure normality nh supplement make section nh take correctly specify density measure correspond datum density f satisfie coincide distribution I observation different nh statistic remark present define pm satisfie approximation testing significance nan significance test consistent pre solution require least robustness nh observation nh multipli nh functional sample size depend size contaminate contamination contamination direction contamination derive f l eq r p w derive condition respect supplement true parameter define supplement case always independent normal regression covariate invertible limit test glm directly set glm nan z ti base power direct develop report brevity robustness hypothesis denote first reduce significance statistic z contamination restrict fix order form nuisance glm like need row column result asymptotic q similarly element along perform brevity present online supplement popular originally discuss test hypothesis distinct robust robust unknown specify consider value parameter unlike case incorrect estimate robust h theoretical contiguous pure decrease robustness term power increase increase choose suitably robustness property test mostly extent compatible proper nh set proposal desirable yield inference hypothesis contiguous alternative hold present supplement
connection biological target feedback unit feedback put burden propagate message separate channel allow layer target direct connection output backpropagation obviously case implement feedback feedback slow distinct recurrent rapid dynamic tune rapid g combine generative recognition circuit feedback connection serve note weight typically associated neuron problem transmission neuron income output reach neuron principle close substantial degree organization neuron spatially feedforward neuron require transmission spatially close nature depend notion target channel geometry regardless channel weight target feed ultimately change taylor remainder system full hessian interest expansion unitary approximation interpret term approximate precision level bit bit turn bit real direction correspond component gradient thus main question bit budget per gradient box around cone unitary expectation approximated good channel include instance curvature case like approximation significant practically useful improvement learn compare target weight transmission backward channel interested estimating limit precise primarily interested computed example epoch want scale term elementary operation elementary include computing transfer function transfer backward essence implementation computer assumption computation forward pass backpropagation network scale send back target information send back hide derive double represent instance backpropagation bit bit backward divide information correspond least operation weight consider ultimately improvement dot descent g perturbation perturbation stochastically produce notion large magnitude various direction produce g u unnecessary generation uniformly sphere approximate calculation equivalently norm tend normally simple tend normally calculation normally tends implement descent descent associate identify w perturbation target weight either binary perturbation indicate improvement presence repeat brevity local global offer additional insight perturb amount perturbation since decrease however detail stochastic algorithm amount real represent change computation weight different deep target binary thus essence sign component locate random unitary direction target unit perturb provide derivative use compute perturbation produce case binary feedback perturbation lead except perturbation real feed dot small whether deep perturbation propagation lead w first small forward measure e ij back computational scale propagation step bit final thus bit unit well information improvement perturbation hyperplane correspond good refined version worth use perturbation unitary normally distribute mean direction scale deviation multiplicative constant provide feedback per direction produce dot high direction orthogonal select unitary take backpropagation furthermore unlikely backpropagation maximal improvement conclusion backpropagation algorithm maximum improvement h concept play role network learn six concept systematic beneficial mathematical expression adjustment describe physical study neural capable associate stack feedforward input even available learn feedback capable information deep deep implementation channel reverse connection carry feedback interpret bit capacity channel gradient divide require per capacity calculated backpropagation capacity remarkable optimality necessity biological neural must biological relevance carry simplified match biological biological question extend reinforcement analysis carry feedforward recurrent carry question whether capture biological obviously biological neuron machine seem favor descent biological neural obtain biological deep must locality principle provide uniqueness network view network connect connection asynchronous minima vector function store minima energy induce acyclic orientation dimensional hypercube isometry polynomial hypercube need case construct force rule derive spin spin appendix rule logistic goal factor actually convergent rule simple supervise supervised decay decay version eq q decay depend alternative bound range weight version gradient appendix deep target target activity follow activity forward may consideration isolated target instance may minimize respect procedure generalization use autoencoder schedule inner loop well provide include layer alternate pass along architecture variation backpropagation convolutional architecture momentum dropout adjustment phase backpropagation target rather layer apply focus derive leverage contain monotonic figure jump beneficial avoid poor hidden activity correspond entire guarantee stay hence case perfect exhaustive convergent convergent stochastic deep target target target hide successive refine target part award physical neural processing rule adjust available post systematic framework studying must nature functional tie capability network discovery rule stack deep feedforward deep output target layer input backward nature target information provide divide capacity backpropagation outperform theory concept learnable explain sparsity learn discover network unsupervise backpropagation problem view away try important backpropagation backpropagation vision energy physics attempt biological unsupervise within general precise capability limitation backpropagation core idea could require state case adjust activity appropriately input suffice deep think plausible book organization rather cell b repeatedly cell fire b neuron book appear thousand cite lack become obvious soon raise simple rule molecular biology opinion progress address basic happen feedforward network capability partly observation far rule backpropagation familiar perceptron delta variation create situation newton raise broader particular discover delta origin distinct idea associate neuron learn depend spectrum possibility narrow end spectrum proportional activity pre neuron activity pre sense ultimately govern environment possibility organize study systematic implementation adjust clarity must give backpropagation error variable backpropagation local g fire concept decide local model learn type rule also transfer function g threshold topology autoencoder feedforward line batch identity bias input value unit assume corresponding formalism recurrent primarily feedforward issue important within formalism activity activity formalism also instance consider component perceptron assume issue section share possible local look coordinate treatment beyond scope change bring narrow apply sense formalism problematic always transformation apply use transformation transform rule quadratic homogeneous system multiplicative sensitive behavior sensitive generally change slowly average epoch however neuron connection different different fourth consider consist connect well characterize quadratic quadratic orientation hypercube edge neighbor orient store set acyclic orientation hypercube hamming see kind elementary sign happen simple lead acyclic orientation hypercube show ultimately acyclic orientation hypercube dynamic towards isometry hypercube yield new vector hence rule new acyclic orientation see cubic although form rational rule although system invariant neuron apparent effective depend degree effective shall classify expect word recommendation local assume ij local rule could denote quadratic would include correlation require average form possible also rather cubic complexity concept degree input line stochastic fluctuation presentation term behavior average epoch assume rapidly datum compare assume epoch analysis weight constant change training epoch instantaneous govern write epoch analysis must recurrence restrict unsupervise et I supervise epoch architecture primarily feedforward architecture feedforward close feedforward layer input essence problem expectation case case input output consider case linear purely limitation backward channel section time move away learn replace definition goal systematically study local rule feedforward network reduce behavior local expectation let correspond training polynomial solve standard great general precisely less covariance require compute systematically transpose vector diagonal square component component represent elsewhere order moment vector moment ei ii notation compute thus expectation list expectation cubic moment datum term diag n diag diag eliminate recurrence expectation precisely compute epoch iterate furthermore invertible write symmetric matrix power cd c diag equation become kb c I iw diag w iw w www diag w diag iw w diag w w diag w iw h var diag diag w n diag ei tn vector epoch independent write give second example version get magnitude linearly epoch remain case quick approximation grow direction center rule lead notable learn anti rule bias include vector properly descent converge linear rule solve covariance effective rule recurrence relation must e reason let drop equation sign origin sign either asymptotically decrease try boolean fan architecture train learn boolean initialization learnable learn least one single train layer adaptive adaptive target learn decay linearly boolean rule learn converse deep rule learn demonstrate complex function layer top see boolean total learnable learnable boolean function total circuit comprise recursive method boolean function instance boolean function input learnable single fashion learnable layer network combination question learnable hope threshold gate perceptron solved perceptron gate implement perceptron state linearly separable local converge separate gradient separable perceptron behave sense relatively small compact supervise target independently update training linearly separable without separate simplify throughout separable learnable separate hyperplane every condition put ensure target pair learnable every learnable learnable first square cosine uv cv supervise canonical bias necessary hyperplane learnable start vector equivalently orthogonal learnable sum row column since canonical epoch learn gate epoch sufficient effect alternatively decrease rate condition thus cause term epoch sum allow simply take epoch preserve sign rotation special note fair coin learnable length binary equation become obvious true vector bias necessarily start component finally canonical previous check rule vector lead w I ei ei learnable separable go sufficient learnable angle lie equivalently learnable start length learnable sum cosine multiply target weight initialize update gate logarithm variable logarithm boolean polynomially therein monotone boolean learn single polynomial bound argument short bound function learn fraction boolean learn network significant possibility iterate deep architecture simulation learnable learnable year attempt seek plausible alternative local simple autoencoder local first broadly try learn purely local rate hyperparameter attempt fail feedforward layer layer layer layer activity process fairly arbitrary differentiable input extend take differentiable supervise consider local feedforward deep critical locally globally weight deep show target weight likewise depend layer layer point input target strictly learn scheme deep weight input thus reason stack autoencoder backpropagation course local stack technique globally would completely phrase exclude difficult capture precision entirely datum input simple feedforward architecture physical reach locally architecture physical target back deep raise question regard nature channel channel target see implementation target remain target weight deep previous local target become local incorporated rule I local deep deep target algorithm depend thus deep q see target available adapt work practice deep solve good deep deep target target unit backpropagation view
centroid independence analogous neither depend operational scheme generalize cope simplicity exposition counterpart alg alg although line alg notable difference validation estimate augment dimension vector n confusion pdf zero alg overall application phase r around r core validate involve test full ii rp scheme rp score iv algorithm association randomly datum attractive attribute draw initialization usage initialization keep finally association result distance centroid cluster percentage point assign cluster cluster pc core gb ram test run server eight bridge processor ghz gb memory matlab inherent capability core server plot curve average carlo realizations model per integer centroid cluster cluster randomly select demonstrate alg approach execution separable map prescribe linearly separable kernel end handwritten digit b accordance sigmoid store accuracy time second accuracy require scalability draw execute parallel moreover exploit capability multiplication rp fig parallelization beneficial compete idea novel algorithmic propose member tailor streaming mode cluster third member family trick separable fourth intermediate complexity synthetic art projection research rigorous implementation appendix vector product limited look r span notational convenience express moreover evaluation let th entry n follow linearity r term show solve task r r r alg efficiently term evaluation follow edu response tune introduce efficient huge possibly build consensus context robust operate batch fashion streaming operation separable family offer member user select trade family minimal subset mean iteration extensive algorithm competitive huge datum collect image mobile device medical big big volume impossible traditional stand alone processor e examine face comprise refer group numerous prominent thank point via hyperplane term probabilistic tool cope key question contain huge informative efficient computation retain albeit distinct introduce task angle combinatorial scheme well big require solve latent sparse efficient dimensional subspace sample randomized scheme non term score unfortunately leverage computation svd impossible computationally rely rp leave multiply agnostic rp reduce employ flexible attribute rank datum offer trading develop include step efficiency streaming mode operation member big datum even fourth extensive numerical highlight massive population art rp alternative letter indicate matrix vector letter stand respectively denote massive centroid accordingly model comprise centroid say association euclidean centroid per iteratively association assignment initialize iterative solve success denote square hard outlier k metric centroid term need carry qualitative long become otherwise g various distance centroid generalize potentially dimensional transform cf association probabilistic iii incorporate regularizer generalization unify replace per map linearly whose term knowledge association confirm whereas canonical k readily yield centroid recognize also pdf parameterized mean k multiple mixture pdf kk iteration maximizer likelihood unknown although module sec algorithms novel scheme unified remain follow trial mean dimension upon validate phase repeat final starting per realization dimension uniformly nk move phase procedure assess draw select draw extra dimension cf eq n centroid measuring association cluster space per draw trial validation phase prescribe realization last r k alg phase draw row obtain run centroid k close identify validation cf straightforward vs f unbiased large separable cluster light calculated validation burden numerator fdr choice avoid concentrated incur plus alg available resource computation alg dimensional probabilistic argument determine practice draw along denote realization meaning mean association mean characteristic draw repetition probability quantify carry leverage informative di I capture locate confidence centroid pdf per draw informative independence cluster interesting percentage validation ranking realization r randomly centroid initialize obtain sample k alg associate identify validation drawing validation phase may computationally especially prohibitive motivate feature per augmentation add flexibility meet consideration development alg phase alg remain alg phase one cf likewise small current memory draw test reject possibly bad clustering perform satisfactory augmentation difference across augment drop alg smaller equivalently detailed next cluster full prominent separable define distance induce pre kernel simplicity novel big hard extension similar norm form mean store step implicit association substitute eq distance list alg comprise realization trial line phase initialize comprise column distance centroid involve step evaluation cf operate dimension line major tailor deal across huge phase specify alg centroid centroid map cluster group distance generate summarize I randomly centroid draw versa common change validation q realization trial identify alg alg store alg incur data centroid randomly associate close centroid centroid cf associate centroid nonetheless practical term select run investigate single limitation require random draw sample section introduce number assess complexity perform examine draw center assess draw end follow pdf stand define parameterize pdf kernel link well population translate actual select clearly representative estimate fig pdf whole population
along theorem case regularization base start support employ example gradient evident pointwise distance euclidean readily minimax theorem condition vector exist last scalar projection reward vector mix compute build generate vector next satisfie observe consider apply discussion vector satisfied repeat satisfied choose w rx tr rx eq actions eq restrict require reduce restricted set handle concrete apply gradient specify evaluate since set modify q one set examine relation coincide use equivalence convergence convex algorithm recall choose action via need show give substitute turn attain scaling accord support positive induce sublinear summarize compact regret consequently proof recall tw sf fr eq bound integral emphasize simple lead rate recursively rather view rely logarithmic acknowledgement author helpful preliminary foundation follow outline proof lemma accommodate prove induction therefore eq strongly maximizer generally observe follow proposition hold trivially proceed logarithmic lemma point coincide outer unit normal due shrink property projection tr eq unit hold particular obtain sum yield regret bind small affect sum need establish proceeding bind obtain conjecture remark notion repeat game payoff introduce along geometric condition rely average payoff set regret set embed high original convergence regret learn decision presence adversary address feasibility repeat payoff payoff nature geometric extensive implication play game agent obtain action nature action strategy regret sub regret adopt last decade community online extend offer overview online recent survey may know already explicit show optimization online target convex carry original high present direct mentioned support along relation euclidean algorithms may require sequence vector concern present recover via propose bit logarithmic general observe algorithm still meta algorithm rely generic discuss demonstrate obtain outline conclude remark standard product norm euclidean diameter set maximal programming focus player extend mixed action bilinear stage stage action vector pure history action pure nature strategy restrict attention agent independent across stage furthermore smoothed reward average rather reward exist strategy exist exist strategy arbitrarily strategy recent dual avoid propose strategy elaborate smoothed reward useful benefit probabilistic nature online may meaningful martingale smooth reward agent nature mean reward extra pure action affect reward mix particular restrict assign action past mix
low formulation short moreover cost prove cost take full polytope subset denote index component index basis cardinality optimal solution low spirit tool cost find failure shape obtain linear program necessary basis use follow linear basis optimal restrict preserve satisfactory basis expression easily summarize independence component formula turn short path represent linear instance shortest equivalent whose component incidence orient degradation start end incidence row column index edge extra column edge incidence program encode path correspond nonzero totally every submatrix drawback incidence rank recall rank introduce extended incidence incidence graph path solution tucker equation cost mc nice result contribution vector feasible surprisingly bad possible remarkable show happen replace bind trivial cost assume independent exponentially bind minimize one hand index thus intuitive prescribe program replace cost far safe providing bound address inspection explain main random application failure complex degradation state residual incomplete gamma function define distribute moreover omit integrate obtain readily see positive crucial commonly encounter satisfy direct explicit distribution consider component program let satisfying theorem variable thus subsequent eq easier involve obtain trick triangle function belong simple moreover e desire path optimal path programming extend mc nature application subsequent property section inspection time reasonable shortest direct degradation sometimes class linear program cost upper programming short may describe failure degradation scheme study path problem degradation identify system degradation node system suppose evolve degradation neighbor assume distribution expert number merge start system start policy reach degradation
model single pf improve informed way consider subset assign close fig row single dataset htb initialization efficient mixture ise explain beneficial method field well ise overhead future selection centre coin introduce problem physics ise learn propose ise synthetic physic challenge intractable normalizing inference forward parameter form markov chain approach widely another successfully apply e go recover high amount currently paper parameter learn model overhead ise model ise superposition ise mix ise external field ise intractable ise learn bn n k q maximize side sample represent course due constant mcmc accomplish build iterative estimate mention field propose e instead efficient optimization ir ising show good single mixture coupling surface row curve agree generate obtain curve dataset bottom neither contrary surface bottom see coincide peak htb dataset ise leave mixture basically ise accomplish collection pl
framework impose extra formally background use solve difference provide notation denote mapping empty denote absolutely almost unique every inclusion induce ordinary differential equation format pose every real number great ii chain aa invariant neighbourhood topology dynamical apply result application trajectory neighbourhood neighbourhood call lyapunov converge consider e lipschitz continuous globally stable trajectory converge lyapunov continuity initial fundamental neighbourhood reader lemma couple iteration jointly latter q lipschitz uniformly latter condition depend martingale increase field scalar individual specify respectively state kernel need space countable ball equivalent uniformly continuous iff recall relatively tight definition assume follow eq invariant prescribed prove map close close let I f z w w compact follow close compact upper w ng nz ng z I pointwise uniform w g w w I nz nz w w similar continuity require analysis inclusion singleton inclusion h exact correspond first single restriction space use reference single scale eq time piecewise linear e dirac solution family lemma sn sn sn sn sn sn un un un n tn km limit point meaningful measurable note member fix e ease understanding point limit problematic far explore auxiliary tracking lemma lemma one trajectory track hence iteration control drive difference require modify version precisely tn tn tn rest tracking lemma topology surely limit countable zero martingale martingale bound quadratic martingale converge eq fact eventually claim fact compact use inside fix one c ns z sg sg sn sn sn sn sn sn due uniform continuity un compact union continuous argument lebesgue distribution control noise iterate see unchanged solution suppose every absolutely continuous moreover hence w z bn converge almost differential w converge track solution let construction tn tn tn te k note correspond continuous jointly integral pointwise convergence fact eq show every satisfie integrable martingale process convergent theorem choice eventually increase latter property follow due z continuous uniformly compact w proof ns f un z z sg un help ga compact union lemma thereby lebesgue e absolutely satisfy inclusion main almost set differential state specify transition behaviour sub approach transition keep discard triplet next policy increase weighting trajectory use allow unlike introduce currently target policy generate sampling sample policy td policy introduce importance weight policy gain reason policy trajectory policy analyze previous behavior represent action reward find value discount factor x px ps rs temporal reward nx hence project bellman mean bellman error descent contain expectation model iterate expectation correction importance weight iteration weight irreducible behavior px n ne tx td e w iii due homogeneity section value finite transition mdp iii q state argument third uniformly w third iv vi martingale increase field w condition iii fast clearly globally tx slow nx xx assumption equality assumption statement assume iterate condition stability markov subsequently put stability satisfied theorem boundedness iterate operator projection trace call irreducible e depend iterate
detailed e multiplicative namely four go purely appear replace transform arithmetic multiplication diagram depict transform matrix hadamard transform addition hadamard hadamard matrix govern arcs ba aa consequently therefore absolute combine hadamard arise matrix point algorithm aside element new step represent agree next make observe post complete multiplication depict derive transform extend derive figure diagram fast transform general proposition general decomposition achieve multiplicative value additive digital processor acknowledgment partially lemma conjecture cr tag cr tag email de mail r de discrete transform discrete fourier dft transform transform fast derive low dft achieve approach transform field role application area integral year dft existence ft computing connect promise transform field transform paper minimal dft signal pair cc dft vice besides real self exploit ft minimal point multiplication transform correspond give follow matrix value
increment keep walk price define drift suggest return term one write student generic explicitly integrable become moderate resort increment computation still impossible financial market recursive part nk k computation nn correction student bias walk keep expansion sum ns become become expansion contribute negligible accordingly relevant fact student distribution converge record first term walk find q formula confirm limit reason number record first increment see correction heavy tail important record mean drift make possible derive insight derive state distribution tail large discuss come different student tail keep law intuitive argument student part law tail start e student approximate derivative sharp transition line student z find numerically approximate record increase linearly ce record increment walk increment one show surprisingly increment record previous price record ensemble stock daily price record day result global negligible may back stock likely back compute record record probability uncorrelated case ratio directly expansion equivalence price warm statistic stock price least coincide unbiased exponent really package implement daily yield confidence standard determine allow keep mind valid record upper record average I measure discrepancy approximation reasonably accurate determination record problematic record number stock asset log price path log return permutation record interestingly two version version uniformly powerful exponent focus straightforward permutation distribute infer er record ratio infer dr spline straightforward deviation various completeness record price increment estimator efficiency interestingly close single sum estimator statistic upper price record estimate ratio return regard outlier contribute lack estimator record ratio numerically hour computer conceptual limitation record statistic uncorrelated variable main computation ar alternatively permutation estimator fa suggestion code available com gray rgb much use finance yet record interval increment variance record uncorrelated attractive new expect record increment tail remarkably expect numerically asymptotic record analysis nothing else hence finance live world arise bias correction serial correlation block actual price return price return fundamental walk depend precise walk asset price approximation connection price context study distribution low walk quite remarkably universal depend increment latter occurrence record record robustness r non parametric ratio trend price passing reason unbiased generic review classify treatment record number time importantly increment focus student upper asset walk increment first time increment notably zero inverting estimate
plug resources university liu national speedup platform processor ghz study core gb speedup memory mix lda sampler compare fully sequential collapse standard include start seed iteration collapse sampler gold topic indicator collapse pc lda run collapse sampler collapse mcmc topic spectral package table gave report conclude efficiency collapse sampler pc sampler collapse gold two sampler quite situation extreme dominate propose sampling contribute systematic token investigate effect time seed sampling threshold corpus small runtime run clear figure short sampler get seem independent popular use ad topic indicator effect initial core partition effect mode converge joint study core sure due unlikely randomly study effect seed seed remove h exception tendency converge core insight convergence ad lda display sparsity decrease increase core interpret posterior towards end collapse collapse end see end partially collapse sampler run example sparse mode core indicator detail decrease progress gibbs topic indicator ad lda token core number mode partition different probably find word result especially situation large token prior influence speed sparse lda different sampler clear benchmark actually pc relative switch alternative see total medium sized sample corpus determine runtime parallel scaling characteristics setup measurement aspect runtime want fast burn convergence occur sparse likelihood definition collapse sampler recognize gold previously assess aspect time sampler burn ideally initialize seed sampler core prior sparse ad speedup good program take lda fast sampler configuration good program pc lda real speedup eight speedup time eight core roughly core dataset offset gain parallelism compare core matter core relatively get corpora sampler go topic offset indicator thus gain parallelization speedup characteristic sampler ad eight probably cache ad describe never reach collapse configuration leave figure core total speedup penalize get speedup core core topic dataset roughly five eight characteristic sampler increase increase core corpus core real compare pc sampler relative execution core sampler seed topic topic table figure cm speedup core pc sparse ad lda hour ad hour pc slow pc lda make large conclusion speedup characteristic sparse ad term speedup parallelization overhead notable relatively several characteristic large topic affect therefore heavily sampler choice speedup generally lda scale eight core dataset handle weak scaling core ad small medium lda number pc lda already variable force strong become much quite introduce situation quite towards reduce spike prior small perform count draw model speed lda pc lda enhanced partially collapse lda art important indicate parallelization efficient contrary commonly collapse sampler nearly mcmc gold sequential collapse sampler enjoy parallelization moderately corpora core corpus important conjugate regularize topic pc spike give increase thereby solution interpretable collapse lda algorithmic improvement pc fast model collapse ad acknowledgment part systems united ns university david allocation model text collapse indicator sample draw popularity sampler stem balanced combination efficiency inherently implementation growing size complexity lda model computationally infeasible sampling therefore indicator basic exploit far collapse contrary parallel implementation collapse sampler well know corpus partial speed parallelization corpora keyword topic parallelism latent dirichlet probability distribution word indicator th denote document lda develop trend supervision inferential markov carlo collapse block advanced topic suffer sequential nature practically impossible way still generate sample serious number computational since approximate bayes solution use distribute collapse algorithm approximate bound error check inference sampler topic document remain sample iterate topic conditionally document row regard regard collapse sampler partially collapse note quickly mcmc collapse generally efficiency increase must benefit parallelization show collapse lda compare collapse actually small setting theoretically complexity token nonzero document basic pc version table search scan less frequently partial elaborate fully collapse use gibbs model study together collapse indicator collapse gibbs topic indicator word corpus scalar hyperparameter type sequential nature conditionally dependent indicator whole corpus efficient collapse approximate lda lda idea processor work parallel count processor collapse processor sample processor guarantee converge find ad sampling indicator number processor total parallelization suggest improve speed collapse sampler lda attempt sampler introduce document evenly document heuristic load document collapse sampling since fully collapse sampler job allocate core load sample job performance overhead large job introduce synchronization job decrease topic way topic indicator typically store topic indicator cache sampler collapse sampler add cost number token basic collapse sampler token complexity reduce collapse sampler grow determine overall pc language quite law relationship token often topic dirichlet pc lda sampler nk di nk eq exist extreme pc topic moderate section topic start costly scan type corpus typically rare scan frequently common type reduce compute much scan gibbs probability iteration start sampling theorem scan ergodic
construct equivalent bn subsection sampler sampler ce markov exact model reference therein build probabilistic constraint surprising equilibria game continuous known presentation technique book ai well specific book subject linearity simplify ce need clique consistent full joint following adapt tell sufficient condition pairwise clique let decomposable clique locally exist joint distribution unique possible value random index large sx sx state bn direct decomposable well appendix alternative brief process page decomposable necessary force marginal clique clique hypergraph decomposable systematically extra edge become decomposable compute decomposable graph whose linear large therein alternative throughout remainder decomposable construction feasibility ce make clique every constraint ce pairwise exist need discussion clique clique variables system constraint w constraint l lemma paragraph pairwise marginal clique decomposable decomposable game game equilibria marginal clique decomposable system feasibility linear clique decomposable submodular discussion game hypergraph game equilibrium polynomial game well bn ce corollary correlated equilibria action clique large take thus mostly grow true yet algorithm still practice open guarantee adapt present online ce ellipsoid attempt reasonably sized fail consistency marginal unfortunately guarantee problematic unclear test consistency whether convergence sampler reasonably keep add increasingly consistent np estimation mrfs instance ce idea leave future process compute joint tree primal linear system join joint assign join constraint local make sure truly consistent clique ce normalization join join player intersection edge local ce player express marginal clique ii ci must join neighborhood originally hypergraph clearly ni ni ci ni ce pair ia ni ia ni new extended player player node appear easily local player ci ni ia pair ci ia ci ia ci join tree time join thus local number ce variate marginals ce clique join uniquely decomposable graphical size variable represent see model believe process ph page page pdf thus simple game assume game provide mrfs lot work equilibrium researcher economic bring view current go algorithmic advance mrfs immediately advance graphical model ce games result yet width find practically e intuitive perspective property ce ce etc unclear I kind property ce circle motivation concern could result original apply avoid truly idea field simple computing equilibria game graphical equilibria bayesian implementation equilibrium distribution bn respect direct acyclic finite factor table bn hypergraph primal undirected parent arc implicit variable conditionally parent px bn arbitrary mrfs permit stochastic bn simple cumulative px I tf apply parent node value parent uniquely determine space connection undirecte graphical direct still graphical bn mrf bn assume course mrf x likely mrf cx normalizing function concern outcome evidence index evidence variable connect remove turn mrf clique c c posteriori posteriori map likely mrf computing normalizing equivalence presentation problem belief inference mrfs general np hard although reference therein usually characterize graph deterministic exist result equilibrium game summary book introduction one equivalent open problem ne ne games pls common statement ne think bad computing game essentially mirror mrfs constraint polynomial intractable probabilistic ai therefore share characteristic heuristic graph ce games feasibility problem include graphical game joint strategy individual player action I ix notation clique except player clique strategy respect play mixed call equilibrium ne payoff accord nash equilibrium ne ne ip ix p ix possible mixed strategy player differ formally I ix ip corollary remark game originally early economic existence equilibria graphical potential game applicability area engineering beyond economic intelligence study resource g public segmentation graphical potential back several game consider literature local game party game game leverage know probabilistic model particularly work establish playing imply game originally economic important class game equilibrium pure originally inference equilibrium game note version game potential inherently nash strategy game fundamental broad introduce potential back special potential engineering economic area artificial intelligence machine network social dynamical resource allocation public image segmentation selection spread social please describe paragraph implicitly special graphical potential specifically relaxation labeling ai vision see connection recognize dynamical ne mrf introduce goal propose metropolis minima annealing base exploration also work explore ne new class interaction game lattice game party instance graphical game graphical game payoff player neighbor game player potential game game leverage literature major contribution present establish playing rule imply player graphical delay economic recently science end large game sum clique neighbor node graph game symmetric previous important impose imply must game playing games mrf structure game run consistently converge graphical game play play rule steady distribution neighborhood game game special trees cycle grid terminology theory finally certain playing establish via connection monte preliminary terminology concept graphical game graphical role establish connection mrfs equilibrium game note characterize game certain kind establish mcmc gibbs sampler random mrfs graphical shift reduction belief mrfs equilibria potential call game introduce basic notation model ia undirected set include clique mutually connected useful concept generalization hypergraph think clique graph acyclic e direct length denote node e direct start elegant graph theory impact modern effective language system correspond miss mrf respect I variable clique joint mrf function mrfs familiar mrf px class suit fine structural graph extremely fine graphical wide variety probabilistic game theoretic offer respective hypergraph structure gibbs hypergraph px gibbs mrf primal conditional mrf hypergraph maximal behavior outcome rational individual paper individual maximize utility act want good central equilibrium equilibrium set player let denote action pure play joint action action player compact representation game graphical model ai inspire mrfs payoff node player game payoff player hypergraph payoff cx ci actions player ni unclear game polynomial contrast player clique fact several player exactly game j player player graphical addition game graphical pairwise multi define direct player arc hypergraph clique payoff player ix ni cx ix ni multi player clique clique implicitly cx ni respectively clique singleton obtain game clique define neighborhood undirected version game game game property unclear I game exactly polynomial correlated equilibrium contrast game player clique local appear summation game game set clique pair involve every player subset game game player particular game game dominate representation payoff emphasis representation remainder considerably small game neighborhood symmetric size dominated clique matrix exponential clique size game size game matrix equilibria game equilibria equilibrium x player improve payoff prescribe stick like problematic sensible come strategy try way randomization yes player distribution capture joint play component player play player except joint mixed play ne strategy play every ne approximation version equilibrium except gain ne one interesting contrast ne ce players conceptual equilibrium external player implement draw thing infer player mechanism encode conditional qx qx qx I action receive switch payoff would player switch formally equilibrium ce qx qx x marginal play qx qx qx ix ne q ne relaxed deviation gain approximate ce replace qx qx ce conceptual ne payoff achievable guarantee consistent something ne response player player play role mrfs game game game game player payoff matrix satisfy q weight last replace weight game call introduce probabilistic model facilitate derivation graph function strictly transform context play transform preference transform transform payoff ix transform transform games player potential ordinal graphical neighbor satisfy ix potential exact potential weighted potential characterize transform potential local potential function clique clique game potential graphical transform game graphical transform potential potential derivation mrf game gibbs graph clique local potential normalize respectively define strong transform potential potential ordinal potential potential ordinal graphical corollary dimensional game scale ix ix iw simply say payoff totally open neighborhood node subgraph connect totally neighborhood tree cycle grids potential scale payoff maximal corresponding addition neighborhood symmetric respective neighborhood potential potential implication reason expect potential say difference payoff would tell payoff simple dimensional matrix generalization dimension game local use potential hypergraph undirected graph node clique player play stochastic adjustment literature learning game sequel let sequence let playing plan select play formally player player observe action graphical game say neighbor graphical type conditional give every compose consecutive joint outcome joint play empirical round round undirected graph vertex set clique potential define play transform mrf consistent conditional scheme regardless condition play always weight player scheme correspond leverage game theory graphical model establish strong connection answer question computing equilibrium last discover develop early computational game equilibrium may relate area plan strong belief artificial intelligence equilibrium largely development back advance understanding belief single mrfs computing ce ne graphical game follow section mrf graph gibbs potential node clique neighborhood player hypergraph game order mrf individual payoff player game graphical game mrf game exact ix x ix remark converge finite regardless play refer player observe maximize dynamic implement assignment mrf guarantee maxima mrf mrf game characterize maxima ordinal potential game game induce mrf potential whose maxima solve local mrfs pls mrf ising model symmetric similarly network game party game arbitrary reference support notion strong player payoff map game induce mrf short heuristic mrfs simplify expression definition mrf maximum one whether mrf induce game reduction implication reduce likely perform mrf would theory characterize normal game therein consideration game additional insight mrfs graph expansion property limit poisson expect game enyi sufficiently average high mrf games low games something maxima suggest max maxima critical mrf state induce rich game go begin establish strong connection equilibria player joint condition equilibria mrf express simplify equivalent simplification highlight need induce ce thus maintain size ce linear programs ce alternative mrf algebraic get remark useful denote marginal play player condition ce mrf game kind mrf entropy summarize mrf equilibria game player ix ix px leibler qx qx player player equivalent hence ce mrf game kind optimum critical kind summarize remark mrf equilibrium induce satisfie local ce ne ne strategy marginal joint action play player except probability player entropy variable imply hold turn express ne locally field except ne mrf nash game induce note property imply field view compute mix formally player ix I I call game player strategy nash assumption mrfs extra game continuous set strategy equilibria reasonable action nonempty euclidean quasi concave infinite want mixed strategy mixed strategy game infinite gibbs potential product mix derive individual mixed pure equilibrium game local payoff also connection game learn repeat game existence equilibria difference involve mixed regularization factor strict player consistency infinite payoff overall play behave depend indeed good player mrf induce variational recursive condition equivalent minimize function parallel strategy monotonically optimum equilibrium nash equilibria game surprising property broad game interest modify ne go opposite also suggest treat mrf play game converge ne explore connection learning propose play estimation game mrf maxima potential game everywhere maximize support measure pure ne game might equilibria optimum critical point connection nash equilibria support limit ce mrf heuristic high fact argue desirable lead well capture aspect would approximations multi modal approximation mrf induce game polynomial ce polynomially sized distribution case ce action play correlate product follow concept correlate equilibria ce potentially ce iff player qx derivation seem novel improved mixture polynomial ce algorithm correlate equilibria mixture regardless one ce mixture ellipsoid ellipsoid practical algorithm interior simple arbitrary polynomial guarantee guarantee ce describe section concept introduce way spirit see g reference therein connection game equilibria probabilistic inference future end recent work equilibria comprehensive evaluation relaxation labeling ai vision connection game yet recognize connection game reduction pure concentrate
ba double double double double cp p space double cp double string true string scenario describe generic scalar intercept double lambda cp cp space cp rand double cp block wise cp cp cp cp true mr job mr double double double mr double double mr matrix mr double double output label cp matrix double double cp cp double double double double cost scalar scalar example plan runtime plan remove remove directly relate execution code variable specific describe program dag small translation see first operator transpose self matrix multiply exploit unary computation transform prevent transpose construction runtime available meta scenario memory execution mr runtime accordingly job make generate hybrid runtime plan mr mr aggregation aggregation select call multiplication small cache rewrite execute transpose mr transpose exceed local budget mr job fourth mr mr single mr job share prevent decide cp read partition w task input demand prevent repeat read optimizer runtime column prevent optimizer select map operator mr small parallelism transpose prevent intermediate scenario already given generate similar mr configuration budget block constraint third combine operator mr summarize major plan characteristic decision runtime bottom runtime computation important runtime dag rather size runtime reflect give plan use box compute cost model job entire program runtime aware resource see memory constraint explicitly parallelism estimator allow size disk parallelism available virtual core resource execution runtime plan skeleton cost track memory pass runtime program compute estimate per aggregate accordingly program track fundamental individual dense format weight read bandwidth compute maximum multiplier operation introduce early requirement convert execution assume example correction ghz processor cost white mr job consist task time reduce read compute computation degree parallelism reduce mr job account without aggregation job job read cost result write cost effective degree parallelism available parallelism block take mr degree parallelism put runtime discuss main program cp lambda cp cp generic cp cp x cp cp rand cp cp cp e cp cp cp cp cp cp write plan simplify annotate program total plan plain cost show couple cost e operation dominate execution cost generic c cp cp cp line cp cp rand cp cp c mr job input map mr mr mr mr ix cp c plan scenario cost mr job pure simplified scenario annotate comparison adapt increase operator read second generate mr job factor contribute include job reduce parallelism reflect read degree intensive job include cache partition time time actual read third despite remain memory hybrid exchange account accuracy example estimate cost within actual simplify fundamental limitation give general reasonable complex ml program run however conservative scalable ensure plan validity however mr job infer case lead commonly make optimizer pruning partially buffer cost address box buffer live sake buffer box acceptable buffer pool usually small fraction total many ml flow loop branch recursive especially number heuristic predefine reflect loop body execute repeatedly allow code motion iteration future summarize allow runtime program reflect decision importantly analytical without relevant aware runtime execution learn true false red black pt ml aim specification ml algorithm level language automatic range memory computation mr framework exhibit advanced cost technique model decision share runtime generating runtime automatically successive phase runtime loop branch cost time advanced resource global optimization optimization state system aim ml language ml construct ml algorithm underlie runtime cluster program automatically memory runtime level full physical independence representation efficiency scalability multiplication decision distribute operator quantify several optimize potentially program characteristic runtime potentially rely run orthogonal cost cost intermediate program available parallelism aware cluster resource ml program loop branch call complex program simple robust runtime several world example algorithm feasible rarely w ordinary problem read intercept lambda intercept one tx beta ask compute intercept program construct write discuss input cluster characteristic generate select optimizer task leverage runtime create x ram intel ghz ram disk storage gb map reduce used default memory budget ratio max size overview scenario size c size ds ds scenario input plan generation give range use case detail show dense follow discuss select dag program runtime scenario well memory estimate selection multiple transform prevent unnecessary intermediate memory intermediate accordingly challenge
contradict part shorthand choose prove n v reward prove follow case less classifier know guarantee since maximization two apply following make maximization invoke note step valid measure benefit value value lemma follow conv measure reward point hoeffde online batch guarantee length stage challenge procedure respectively es union ds require accurate frequently mild severe label imbalance requirement classification measure non decomposable measure pose challenge application family possible implement family pseudo linear measure core contribution adaptive scheme technique truly base update concave dual pseudo alternate method demonstrate significant similar datum severe label imbalance requirement negative include spam classification anomaly medical imbalance classification suited situation trivial optimize predict class instead measure case entire decomposable include consistent effort optimize year result broad approach convex indirect include cost approach solve plug approach rely fairly scale large memory style approach cut plane well plug solve class approach prevent moreover take large streaming take preferable however decomposable recently surrogate optimization maintain prohibitive state develop novel two broad family decomposable truly wise buffer pass intuitive level linearization amenable sgd feed see write include mean etc exploit variable parallel linearization maximize stochastic mirror fractional need outline structure develop optimize combination via alternate strategy converge optima strategy batch validation experiment class fast plug style measure surrogate base indirect applicable measure learn compute exist dedicated optimize pseudo linearity maximization cross validation considerably improve multi label decomposable plug style challenging role designing solver non decomposable define challenge generic therein style buffer maintain guarantee special application care denote instance denote positive sake calculate r averages reward concave lipschitz value range publicly available benchmark dataset mnist severe imbalance min also compare specialized sake unweighted compare stochastic method implement rest testing plug cross validate split hinge execute level actual solver implement since rapidly pass datum epoch every allow runtime solver allow q accuracy greatly accelerate fairly find accelerate fail slow least find classification due imbalance report similar competitive accuracy f accuracy similar else slow reason confirm buffer method cause rate secondly style poor accuracy compare plug work measure plug explain acknowledgement fellowship lem write iff region us primal dual exclude prove direction conclude vector great dual inside prove direction region radius eq define f result stream execute feasible shall notice update write interpret respect involve constant note monotone conjugacy far tr tr functions individually p monotonicity second stability inequality concavity jensen write eq projection concave involve show ascent observe linearity step concavity third conjugacy measure reward satisfy probability eq
proceed previous thank notation ratio hull result optimize intersection radius complement ball radius x converge zero j proposition use obtain eq dot previously propose straightforwardly modification adapt elastic net allow early discard derivative propose version duality consideration create safe region zero screen wide convergence support cope solver descent significant safe mid attract attention explanatory context square refer lasso pursuit processing enjoy theoretical name many issue accelerate dimension indeed method rely hundred non scad mcp lasso often mention homotopy lar homotopy choice particularly method problem seminal exploit discard guarantee allow reduce burden introduction safe potentially variable post operate choose solver safe safe static orient towards commonly well drive manner consecutive know call strategy road think warm start screen warm sequential safe rule keep improve screening efficient proceed argument leverage safe safe safe contribution introduction safe look associated safe concept screen rule converge rule rule equivalently inactive also rule build dual converge safe safe region gap safe sequential descent solver propose standard report safe denote observation approximate norm norm control fidelity estimator solution primal eq denote feasible formulation read see refer onto rule center radius dynamic dynamic safe lasso primal link eq tucker kkt kkt primal screening consider safe rule exploit equation soon challenge unknown safe construct set contain safe helpful benefit construct region denote define cast differently safe primal explanatory thank close convex hull restrict safe safe explanatory safe safe safe safe region possible safe safe set contain support lasso solution safe whose rely safe sphere center radius simplicity safe commonly safe review safe strategy consist safe region however evaluate priori also experiment gap safe safe sphere convex dark blue provide safe build converge let duality provide light dual feasible thank see tangent let read late insight one recover safe primal dual objective resp gap solver next establish oracle radius quantity pick dynamically available safe ensure converge eq converge primal sequence define safe safe safe safe sphere safe include gap radius safe rule respectively property satisfied radius warm start safe make safe inherently sequential approximately handle approximate produce safe screening safe sequence next equivalently safe evolve follow sequentially safe gap safe replace definition one access primal solution safe still approximation screening choose well suited especially coordinate require process commonly operator wavelet thank x implement rule coordinate code level necessary dense array sparse pseudo present pass stop gap involve know safe norm evaluation gap stop strong rule safe need processing also sequential require exact solution lasso prevent converge phenomenon present safe proportion safe screening depend much gap safe especially tuning scale safe rule especially test bring improvement cost gain hence prescribe safe dataset present feature graphic
rich assignment heuristic cope make outcome parse dependency parsing may similarly complex combination simplicity removal concern employ cost multiclass algorithm ensure error policy oracle competition advance overhead neural remove essentially write decoder oracle english nine language competitive recent publish label strong parse complex broad learning approach include perceptron production search policy take action search word n word n output build framework write decoder parse loss state consider middle deviation complete loss write speech annotation aside library computation reference trivial hamming prediction tag previous machine arise answer try one yield efficiently epoch execute trajectory loss alternative execute act loss context learn policy able initial step deviation trajectory vary policy manner regression instance mixture epoch subsequent epoch reference reference l stack buffer arc root root root root root cc thick node style base b bend leave bend bend bend leave bend cm every anchor base bend leave bend bend leave derive gold framework implement stack maintain buffer keep arc triple ease buffer stack dependency arc direct word terminate arc parent derive parent word parse assign tag arc dependency head unlabeled extension label top arc hybrid transition system arc buffer contain stack arcs root parsing take buffer stack e terminate take one move word add arc arc valid show execution parse parse dependency parsing associate derive gold transition action move root ref mention framework decoder reference oracle policy optimal derive annotate pseudo decoder dependency discuss take stack buffer speech tag list template generate configuration change parse oracle lead minimal library learn system policy automatically therefore action implement arc predict action configuration predict annotation wrong loss effect decoder implement unlabele label loss tree gold assign arc feature feature library fold decode system second framework library ease quadratic cubic mechanism provide unified allow base learner argument modify mention framework reduce sensitive reduce framework employ study base analyze learner baseline recent baseline greedy stanford wide different language show achieves language transition perceptron dynamic avg assumption language hence obtain substantially bad language root exclude conduct language chinese convert head split testing pos tag evaluation stanford accuracy split last datum development need gold pos tag learn policy policy decrease round round experiment reference preferable roll reference roll dynamic tag embedding regularize particularly neural hyperparameter transition stanford network setting arc hybrid system exploration thus set development seed test external resource embedding randomly suggest setting fair exclude run unlabeled exclude evaluation compare tune cc leverage well learner update state base learner stochastic gradient improve metric importance single hide neural hide nn regularize multiclass multiclass nn rule gold label oracle learn detail l bi gram template comprehensive transition average parse algorithm combine learn explore principled way search sensitive classification evaluate end treat bad approach labeling resolution graph unified first interface parse broadly structure probabilistic programming language however rely language describe implement furthermore provide wide advanced optimization language extend room line stanford support code parse stanford implementation usually code comment gd sensitive h offset initialize finish label valid valid action gold stack arc right arcs arc arc arc ec variable task action option add option value ensure sentence label value arc label index ex index get bc bb dd ff ef db df triple vector triple triple condition feature return l cost task task label gold tag stack v cs ex child ns size mask f ex ex sum ns ex begin ns index ns ex arc hybrid datum task array stack stack gold gold tag gold tag tag tag v array child child stack return last stack child stack stack child stack stack child stack stack size stack child tag stack stack stack gold tag stack stack return stack child child stack child tag stack last stack stack stack return extract search ec task mask mask multipli shift stack stack tag datum ec ec ex ex mask multipli ec ec stack ec ec ec ec ec buffer ec ec ec ec ec feature stack sl ec sl ec ec sr stack empty last stack ec child stack ec bl ec bl ec child ec ec stack stack child end ec stack I fs ec begin fs ec fs ignore continue offset ti offset ec k ec fs k fs quadratic offset mask multipli ec fs ec multipli ex offset mask multipli stack empty stack stack stack stack child stack last min stack empty stack tag child additional offset offset offset mask index datum ex data ns string begin end count ex vector
require solution datum distance except privacy differentially q private naturally prevent attack power differential interpret theory reader several privacy firstly dp dp privacy automatically allow dp dp advance make simple boundedness single posterior denote preserve free classic consistent differentially preserve alternatively domain g preserve privacy result thing boundedness l lr familiar notice mechanism preserve output exactly posterior exponential simply notation specify thing effort posterior privacy b b x boundedness usually small decrease super exponentially paper convenience practice predefine threshold release perform great generality consistent briefly consistency bayesian sense great consistency apply consistency frequentist sense prior posterior consistency hard consistency prior promise bayesian find distribution either equivalence weakly suitably bernstein von posterior distribution von theorem hold obey normality independent interesting class near similar class leave work proposition asymptotic relative function key idea scale different fitting include mild converge mass mle remain bayesian whenever hold bernstein von asymptotic near optimality mle rescale posterior obeys likelihood correct equality likelihood close generate since difference scale minimum regularity invoke modified bernstein theorem say converge nj nj note interesting remark proposition log sharp previous intrinsic eigenvalue depend implication essentially generalize classic result confidence interval test generalize ratio private use trade powerful easy extend handle agnostic leave claim privacy sampler rare complicated often option sampler never something privacy sampling preserve differential privacy sampling procedure preserve dp commonly proposition clean interface need privacy bad news g approximation easy sampling lda suggest arbitrarily near log concavity distribution imply confirm differential privacy constraint nice thing modify go dp whenever tractable provide insight bind seem barrier privacy rather hold achieve differentially private erm objective perturbation work function prior differentiable strongly threshold restriction hinge huber privacy hard work view stem intrinsic privacy require implementation application convexity need add additional may give sample privacy many section answer look technique year show differentially free simply release differentially private sgd advanced composition allow privacy composition advanced mechanism dp fold adaptive eq constant simplify expression see apply taylor addition dp subsampling randomly evenly random sampling sensitivity gaussian differentially minibatch regularizer empirical gradient tool would avoid iteration update parameter mini ordinary stepsize strongly minimax optimal convergent later prove choose iterate stepsize stochastic minibatch converge gaussian stepsize slow discretization approximation obeys due translate assume mean estimator study minor modification burn minibatch number pass initial minibatch coordinate define burn period lipschitz collect carlo differentially private minibatch also smooth preserve differential privacy every iteration access l technique use tt advanced failure accordingly proof choose level big reduce failure converge alternatively suggest pass variance langevin use iteration collect stepsize need overcome estimator calibrate initial already posterior long stepsize valid claim different ensure privacy internal bad however run strong privacy modify balance get privacy practical drawback mixing describe attempt resolve use auxiliary variable counter use stepsize gradient therefore briefly langevin hamiltonian ignore hmc proposal distant enable rapid hmc author restrict arbitrarily long simulate correct noise dominate get quickly become discuss still exactly true correct trivially variable serve similar appropriately describe interpret momentum sgd get flexibility range gradient chain posterior parametric von idea use inverse fisher key stepsize speed stepsize far differentially learn near release bar true true go noise benefit principle many collect stepsize collect able adaptively adjust temperature unbiased sense equation unstable train pass sufficiently large involve fine direct way privacy constant matter use decide perturbation degree often conservative bad stochastic differentially private sampler logarithmic thousand well illustration sampler linear illustrate converge like become able produce unbiased level evaluate page uci repository logistic hybrid risk minimization privacy result figure improve classification privacy use laplace mechanism perturbation solve bfgs numerical long confident minibatch pass choose plain initialization perform equally slightly especially curse constant early first aware develop focus mostly conjugate point boundedness differential computational use provide efficiency aware scheme normal strong result require unbounded semantic point develop tool perform privacy completely different post processing procedure aim denoise integrate far boost investigate effect beyond scope differentially stochastic private party modification gaussian match logarithmic confident contribution extension preserve disjoint require setting applicable pass well replicate find objective perturbation originally version compare sample solution differentially private sample intermediate iteration conceptually inherently differentially get exponential estimator parametric algorithmic langevin dynamic variant preliminary practice theoretically practically meaningful provide intermediate think case exploit randomize hash dropout thing hope differentially private movie recommendation goal
challenge near prediction quickly curse overfitte limited mostly univariate strategy ideally power grow combination prohibitive limitation drastically reduce possible predictive causal make tractable criterion demonstrate nonlinear delay even forward selection suggest fit improve prediction index ni predict goal traditionally since neighbor neural ahead nearby mostly state reconstruct take nature multivariate information dimensionality nearest impossible predictor redundant information perspective variable mutual curse predictor mutual searching subset lag exponentially search strategy prohibitive due therefore propose demonstrate approach predictor theoretically recently serie much allow globally search strategy case additionally criterion selecting subset predictor compare even cross much run forward suggest framework also problem series underlying mechanism understand firstly also drive fit free relevant efficiently improve understand mechanism index ni causal sect information selection prediction sect explain sect causal criterion discuss computational sect sect sect analyze prediction apply sect evolution function drive possibly time lag represent drive quantify shannon entropy latter conditional level predict past entropy uncertainty maximally perfectly truncate lag dimensionality many actually carry merely poorly goal thus carry still new possible avoid combinatorial search predictor include selection iteratively conditional lead computational sect globally strategy might drive fail sect key use satisfying state remain term theoretically parent add variable increase parent parent describe sect drive drive series drive selection parent causal sect p start iterate combination node hx hx h stop combination test cardinality else one iterate initial previously need underlie independence order guarantee graph entail conditional relation violate certain assumption fulfil algorithm imply predictor inference yield analyze scheme far backward step algorithm ix propose ix n mix value sort numerical positive come surrogate drawback adapt predictor fix threshold level low causal predictor complexity optimization illustrate mi fail prediction select causal parameter predictive mutual subset analyze mi certain combination large causal maximal mi
omit take w matrix identity diagonal normalization element rotation rotation rotation step k compute h estimate source comment insight account rotation numerical bss moreover show bss whiten numerical cost increase linearly life environment vary adaptive estimation utilize slide sample show matrix encounter parameter complex rotation follow resp resp avoid non one typically need sample convergence method algorithm present mm criterion show comparison bss performance signal interference ratio filter matrix channel source system symbol pass generate mean specify signal snr channel simulation htb db htb different examine number figure compare draw unchanged perform compare db figure propose snr expect significant compare notice perform pre whiten g low obtain algorithm suitable notice db number nearly db well snr respectively figure sample notice nearly however high figure iterative batch name use pre whiten operation reduce recursive unitary unitary rotation modulus instead maintain mainly deconvolution signal favorable bss notice number case criteria together alphabet eq double angle identity write bottom write element subsection order generalized write order scale scale al laboratory france mail fr also department electrical engineering technology mail address multiple deconvolution bss algorithm modulus criterion show design quite real maintain modulus whitening rotation improve size rotation whitening occur bss interference symbol bss blind modulus rotation source bss implement pilot symbol utilize reduce bandwidth efficiency g pilot symbol valuable tool meet demand rate wireless system pilot contamination output system bss signal estimate source unknown bss source signals bss find criterion multi criterion attract interest cm criterion mm utilize separately several present mm numerous cm algebraic name analytical constant modulus capable separate mode overcome drawback numerical batch bss similarly mm modulus outperform mm minimize name firstly bss implement manner two batch bss communication unitary unitary rotation utilize present name algorithm pass whiten operation unitary unitary filter filter signal convert rotation iteratively find algorithm slow convergence fast developed show propose case manner provide comparison perform much bss organize bss brief bss rotation real design algorithm rotation detail propose section present notation along th transpose complex conjugate transpose pre filter modulus part matrix real element symbol pass channel model symbol instant independent source white noise signal utilize prior source inherent bss bss channel mean value signal add r n h bss receive matrix w receiver vector global system receiver rotation large whiten efficient unitary decompose rotation lead size whiten inefficient rotation unitary brief review unitary rotation identity two diagonal angle unitary rotation except h pp qp like algorithm decompose rotation decompose product rotation denote order rotation compute desire unitary transformation accord write seven involve simplification thus complicated rotation motivate come deal mention challenge work previously difficultie version use receive convert contain maintain rotation rotation necessary preserve sequence rotation show rotation apply successively parameter rotation rotation shift diagonal criterion explain iterative transformation accord q rotation unchanged modify rotation angle assume express identity express similarly replace last I irrelevant determining solution minimize eigenvector correspond eigenvalue rotation similarly apply successively function norm eigenvector initialize summarize table whiten construct use rotation separation matrix small whiten effective channel unitary unitary real rotation rotation overcome limitation product elementary rotation rotation refer transformation rotation
derive mutual group computer science biology algorithm ref different evaluate find important perform model plant partition political annotate expert partition community reference partition similarity easily label maximize range permutation see overlap partition roughly partition label refine normalize overlap however problem group modularity ill another accept well study randomly select mutual information leibler kl detect nothing detect ground practice joint approximate group shannon joint gain know gain knowledge nothing obviously similar evaluate one way bound identical become popular consistent fig partition plant sbm call plant plant generate independently commonly size distinct un phase network plant modularity three give plant configuration come modularity report guess group bb line bar top bottom use evaluate similarity partition give systematic statistical value compare configuration detect could configuration large significance compare nan use science structure modularity compares expect edge graph configuration compute average usually already less plot algorithm find plant sbm plant detect generate stochastic block benchmark happen plant partition plant overlap consistent un perfectly phase maximize permutation un similarity partition fig algorithm modularity benchmark measure work benchmark right tell use community detection mutual network fig average realization network modularity bp benchmark different size exponent distribution exponent size show numerically
hardware round operate quantization precision asynchronous nonzero entry high sgd algorithm precision increase result corollary negligible unfortunately rate sgd expect particular martingale arise track principle unit eigenvector base simplicity focus outline entry condition update randomly index uniformly origin equivalently recover show require incoherent unit ease run bound times martingale save initial value expression problem horizon parameter cb determine initial appropriately run failure bound analyze convex illustrate analyze asynchronous precision sgd complex include validate matrix implementation precision like run update bit input limit decrease modern row bite k gb gb conjunction terminal explanation color package graphic terminal graphic ltb lt lt lt ltb lt lt lt bp r logistic regression bit ltb ltb conjunction explanation color package graphic explanation graphic macro ltb lt lt lt ltb lt lt r sequential ltb version claim discuss application show precision change ran analyze report forest music music glm report logistic display speedup sgd axis six core ram low arithmetic algorithm sgd combine data table update compare convergence ten eigenvalue differ somewhat randomness asynchronous version behave qualitatively dataset take run take second speedup unified produce rate asynchronous precision random stochastic martingale base sequential easily give asynchronous modern hardware resource algorithm acknowledgment thank helpful author acknowledge contract air heterogeneous graph stream mid da library language high dna sequencing specific national energy system stanford parallelism fa program simplex national science foundation nsf award office research national image big http www american view conclusion herein policy either express imply nsf detail body long tx w r x k next g x g indexing apply h k apply continuity v gx h r side produce h h r k h r tx apply update distance r h r tx x r x e use k entry substituting conclude algorithm success occur actual take apply hardware law expectation state convex sgd horizon section result except quantization set prove lemma purpose piecewise logarithm lemma order concave armed prove lemma optimum tx tx x f bound fx assign fx x side jensen rate occur x success occur occur negativity rate verify lemma statement occur lemma tx definition tx tx clearly expression maximize tx x lipschitz mean value theorem next lipschitz index fy apply round apply lemma lemma appear state first version specialized update include use another combination lemma proof x define stop stop stop b tx x occur stop yet stop negativity tx rate next bind time first give x therefore x x n assumption give bind x j u apply incoherence j agree assignment produce proof substitute c f produce desire result simplify martingale consider elementary tx x noise due delay update x I e e k c x entry value substitute therefore prove secondary literature stanford electrical engineering computer stanford stanford machine researcher technique runtime asynchronous execution capture rich specifically use new way relaxed sparsity asynchronous sgd completion design analyze asynchronous low arithmetic experimentally algorithm efficiently variety modern problem eq sgd wide range application machine widely learn poorly success practitioner asynchronous asynchronous include deep recommender system practitioner way also asynchronous version stochastic stochastic proximal producing propose asynchronous unfortunately sgd approach entirely precision ideally could martingale enable extension unify technique relax assumption asynchronous sgd matrix asynchronous sgd quantization fix point validate experimentally algorithm theoretically describe analyze asynchronous challenging copy asynchronous core separate copy cache core write handle possible central store atomic solely dependent function write denote write think independent reasonable though since delay delay occur equip convergence asynchronous continuity collect
efficiently reconstruct input usually sparse classifier risk framework drive truth mapping space kernel carry representation solve neighboring code reconstruct neighboring pixel sparse code pixel encourage neighboring pixel sparse pattern extend drive joint pixel extend general drive dictionary use section generality input consist pixel label parameter jointly minimizer choose quadratic joint input difficulty function define row active perturbation locally active gradient bit involved omit limitation optimal kernel theory observe yield satisfactory nonconvex properly university spread pixel range band water remove processing randomly available training range spread pixel split training datum rest pixel choose per set parameter outline drive propose purpose name enforce neighboring set neighboring jointly evaluate propose kernel svm l latter construct prior accordingly university hyperspectral show table formulation achieve enforce among pixel propose performance competitive compare dictionary construct dictionary propose compact dictionary translate computationally formulation enjoy among pixel show equip university hyperspectral image formulation dictionary readily task research topic sparsity test corollary example laboratory md successfully discriminative dictionary constrain prior advantage domain hyperspectral classification formulation dictionary jointly optimal performance supervise hyperspectral suit prior enforce neighbor illustrate hyperspectral hyperspectral increasingly become target shown achieve construct collect sample pixel lie form generate unstable enforce code neighbor pixel joint neighboring lie stable improve base learn generally dictionary aim find yield task supervise drive art task jointly machine high dimensional counterpart rational class space class sample typically subspace discriminative code
dft number input compute multiplication complex coefficient implement compute polynomial polynomial real number polynomial division division multiplication two multiplication necessary multiplication compute dft polynomial division autoregressive fed circuit dft component dft fast attractive procedure dft need compute introduce root inversion factorization euler dft component polynomial combine produce multiplicative single dft iv section dft consider zero write autoregressive q polynomial division multiplication component multiplication due multiplicative multiplication assume polynomial multiplication indicate hardware q compute hardware implementation symmetry numerator polynomial therefore form algorithm multiplication multiplication attractive fed shift circuit dft element namely lead dft obtain output correspond hardware multiplication multiplication require j algorithms component transform well complexity far compute respect fix
combination specifically assumption dispersion around spherical contain compactly ps ps ps laplacian identifiable structure carry necessary ica natural seem ica however properly progress work unconstrained model source visible size difficulty deal limit attention image reveal basic block multi architecture propagation handle transform reference therein compose block block handle diagram amenable implementation rapid source row matrix q kronecker identity probability contribute space represent distribution flow network branch forward message usually numerical stability version backward combine propagation direction multiplication loop branch compute normalize element reader rigorous translation marginalization flexibility bi directional message generation delta three propagation distribution variable version display result follow delta backward propagation distribution factorial code decoding pattern subset available bottom delta backward miss step propagation collect product observation forward reduce elsewhere omit reason therein report mnist binary pixel architecture figure delta maximum block prior generation figure increase forward message picture number increasingly accurate pattern character shape build representation definition factorial code learn marginal less kronecker present set delta posterior source act soft configuration factorial code present note code sharp column decode graph decoder sharp figure network backward pixel forward posterior miss ht ht question mnist ica matlab retain density confirm ica source try look patch natural ica sources generative preserve structured composition set figure mnist image unsupervise addition information backward block learn bar represent encoding row naturally experiment architecture backward show could consider bar simultaneous ht forward posterior encoding propagation unified framework image datum code correct flexible alphabet greater report elsewhere currently universit pe te le belief bipartite factor graph inference full image mnist dataset show factorial code implement source contribute build generative ica information propagation become aim capture visible popular map source source visible variable signal ica filter seem converge pattern visual explore possibility constrain factorial code feed difficulty naturally product limit attention perhaps
ram hour train show imbalance varied address remain operational search fine grid could operational baseline choose satisfied drop operational ii instance mean free violate operational meet cv test cv final produce constraint favor mixed ensure operational constraint htbp ptc instance free cart c ridge elastic choose lin svm interpretability train operational constraint satisfy lr ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc elastic ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc lin ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc none ptc ptc ptc ptc ptc ptc difficulty operational among elastic net produce operational constraint cart produce regularization ridge lin rbf unable require level operational include emphasize point pt operational crucial implementation adjust model sparsity incorporate suitable operational correctly handle operational high extensive never operational r choose parameter predictive several fold free maximize predictive operational free unfortunately operational htbp lr lasso elastic lin svm rbf cart acceptable produce acceptable scoring produce significantly expect minimize elastic surrogate operational sparsity max sign net need least sensitivity plot sign score net roc evident sensitivity specificity lin sign coefficient vs real interpretability acceptable head head operational lasso elastic htbp point align domain knowledge sign large coefficient model screen tool poor sensitivity elastic high sensitivity ability provide relationship response score suited provide kind qualitative understanding quick computer help user work example human cognitive entity association may also help influence input sparsity require score help follow simple rule interpretation sparsity system popular train sized minute ran learn summarize choose explore varied nature process categorical value process resource htbp breast breast cancer high risk heart disease detect breast cancer predict mail spam baseline publicly available package subset accuracy validation sparsity interpretability run baseline time grid free training run per allocate minute ip ghz machine gb ram hour train scoring method dataset function htbp cart cart default rule penalty lasso value figure report represent coefficient model lasso ridge elastic lin leave cart rule base c box svm dataset bar cv addition regularization path error varie level ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc range ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc error error ptc ptc ptc ptc ptc ptc ptc ptc scoring unable show figure sparsity evidence minute sparsity trade directly optimize necessarily accuracy discrete lasso baseline restriction arguably mainly suggest accuracy dataset find restriction relax ridge lin interpretability interpretability focus nice comparison predictive figure model omit attain perfect cart far lin integer express line mutually exclusive prediction hand help find use hand hierarchical make assess input interpretability benefit interpretability notion depend light benefit practitioner directly encode interpretability requirement operational point htbp mml htbp text center height font draw gray near fill text center font green font none em yes leave safe root root end yes narrow right safe leave narrow safe block right yes leave safe yes yes line yes yes node end yes node yes cv none narrow none narrow mean c lr lr lr lr specialized problem interpretability set interpretability interpretable heavily exclusive interpretable set require scoring system train accuracy train scoring ip l p j x j r j identical binary ensure interpretability interpretability ensure interpretability specialize base real convert convert feature value threshold use discretization form unchanged categorical yield binary jt threshold tt benefit compute table originally model net stand fully form coefficient n gain require c tumor rule cv solve r j j ip thresholded version suit value intensive use threshold feature optimize real exhaustive value classifier threshold interpretability small rule per include constraint agree ensure maintain monotonically system ip p j max variable p big parameter identical interpretability penalty interpretability least one rule count value variable limit binary rule constraint constraint encode create drive scoring show scoring system optimize operational constraint come computation approximation surrogate practitioner need integer programming software real integer practitioner way allow choose acknowledgment thank comment addition dr dr general acknowledge version vector loss using ensure example define I statement fact cauchy show ensure classify three margin margin lie margin I I definition minimum expression whenever put yield nz c minimizer theorem set q remove remove sign mean statement sufficient assume satisfie use would contradiction iii iv look rhs inequalities incorrect management institute add multiply assess risk serious medical quick extensive computer american medical currently whereby system create scoring difficult create integer explicit operational number approximation round surrogate round address operational impose hard false calculate impossible mean loss operational constraint free parameter model operational alone score integer optimize current classification produce scoring optimize sparsity wide complicated operational without contribution pt approach learn score system operational advantage derive particular present discretization bound sufficiently addition relate novel reduction portion reduction laboratory tailor significant million people state alone eight classification publicly accurate sparse matter paper rest special accommodate operational constraint medical system discrete present reduction laboratory create tailor screening report experimental specialized stream medical scoring integer classification medical medical systems pt iii ii iii patient death detect criterion score sparse model small scoring system use medical scoring case heuristic ii construct multiply integer approach odd fact solution programming system hand suggest history point factor increase rr subsequent year dm stroke easily create medical scoring cognitive poor trade loss attain good theoretic accuracy similarly restrictive rarely recover lin linear balance rely greedy part stream create linear discrete minimize small coefficient formulation optimize hinge loss discretization novel model discrete reproduce create converse necessarily method oppose sparsity difference may eliminate produce operational train integer ip goal restrict integer tackle albeit formulation minimize refined formulation integer comparison since goal simultaneous tuning encode important operational constraint restrict whose lp significantly tighter design norm minimize discriminant overview mix lee early misclassification feasible dataset mathematical body focus improve scalability procedure cut specialized branch remove redundant dataset x ty label represent coefficient intercept value datum may great encode absolutely control balance soft set interpretability separable directly optimize restrict finite operational purpose restrict objective coefficient add dropping adjust trade additional adjustment factor choose drop entirely remark feature penalize training scoring finite coefficient e bad real minimum large magnitude training classifier less round resolution set discrete coefficient attain classifier baseline attain directly margin resolution margin resolution denote coefficient train coefficient set parameter discretize construct discretize discretize py follow provide uniform important bind q classifier obeys hypothesis space motivation indicate include increase exclude provably suboptimal scoring linear every classifier obeys appendix parameter model theorem exploit system coefficient express discrete classifier linear classifier integer obey counting use plot improvement show reduce classifier improvement associate model discard redundant technique classification carry well suited optimization reduction general computationally represent feasible represent function aim discard change solution require specify easy initial surrogate identify redundant training solve provide reduction work problem q objective surrogate denote level original
compare stein shrinkage en estimator penalty scad iii stein estimator restrict preliminary shrinkage ridge rr en en en represent estimate en en elastic net rather elastic estimator ridge estimation retain notation ridge lasso lasso en en portion define see elastic easy among estimator rr en facilitate indicate superiority estimator efficiency finding summarize ridge efficiency estimator dominate figure shrinkage positive always well find zero estimator term relative stein dominate simulation scad estimator result table begin various configuration increase observe relative may order stein dominate neither scad stein dominate outperform covariate correlate elastic ridge en decrease value en dominate type stein estimator en dominate significant simultaneous whereas stein methodology focus en en en inf rr en inf inf en en inf inf al scad al scad l scad l scad l c scad scad scad scad scad al scad al scad scad scad scad scad scad al inf inf en en en inf inf l en en en inf inf l al en en c inf inf en en cm lemma section development predictive key formulate search popular adaptive scad elastic net analytically characteristic ridge rr elastic net en versus restrict preliminary stein space rr dominate scad en dominate neither stein dominate lasso scad en significant uniformly dominate stein dimension efficiency en penalty depend en dominate stein rr estimator analysis error keyword phrase stein shrinkage uncertain classical normal admissible quadratic give birth partial document preliminary stein stein type estimator expand include stein paper appear estimator shrinkage rr minimization thus least criterion penalty bridge generalize criterion ridge become smoothly net devoted characteristic like elastic net stein ridge stein type simultaneous exceed dimension ridge estimator limitation conclusion base efficiency organization follow discuss penalty preliminary test estimator table graph provide unbiased penalty simple example namely coefficient estimator ls criterion sphere minimize yield ridge parameter regression problem linear minimize q pn bridge reduce ridge estimator variable popularity later glm course estimator good penalty unbiased unnecessary modeling coefficient estimator avoid instability well possess property estimation call smoothly absolute deviation continuous define q tuning expression scad adaptive consistent vector equation minimizer computational methodology propose elastic net ridge component ridge estimate mix cross validation elastic advantageous highly predictor elastic net lasso shrinkage elastic oracle property elastic property group variable traditional group inefficient inconsistent overcome extended group tool oracle regression vs chi freedom df may optimality assess lose continuous stein type inherent problem due happen interpretation become another namely stein five group improve preliminary estimator asymptotic uncertain hypothesis consistent function unity p distributional cdf chi square give difference give perform whenever hence consider optimum
addressing I example denote behaviour noise variance quantify selection eq rv provide magnitude show well define combine pure ranking dependency examine tend prove important relate behaviour variance since become noise infinity free consistently prefer consistently outperform rs whereas consistently prefer bad consistently select example alone impossible directly term possibility leave e rewrite give magnitude proof namely unbiased quantify combination selection determine entirely behaviour random exceed likely argument pool comparison argument serve illustrate rs guarantee receive existence unbiased estimator algorithm capture ideal unbiased ignore central estimation expect new include component estimate label multiple quantity dataset raise statistical choice term bootstrapping I testing three three I bootstrappe replacement give two clarity variation estimate estimation classifier estimate estimate classifier close bayes loss increase small precisely estimate near optimisation reasoning suggest practical application algorithm simple component broadly reduce component estimate second classifier base compute loss immediately estimate classifier also loss produce suffer use lead argument require component two estimate na I motivate development term computational evaluate pool popular random seek way provide reasonably estimator bootstrapping class classifier third together estimate estimate sampling pool j k j e median seek show define c statistical generating independent component component unbiased ideal dataset unbiased completely unbiased application neither classifier large test available open research suffer development algorithm ideal estimation estimator asymptotic rate result suggest reasonably class raise whether estimation merely reasonably bias selection size prediction cost estimation differ explore describe varied al substantially classifier capability nn na I appendix explore group experimental use another conclusion experimental explore variation evaluate al compare classification fall natural benchmark se third define method logistic forest classifier diverse open research weighting sometimes recommend literature however weight theoretically weight primary competitive method weighting defer exploration performance continue entire label label final loss much metric describe classification monte replicate use experimental experimental al evaluate performance iterate produce profile loss label label class learn experiment nn al metric assess improvement al rs quantity method literature performance evaluate function label metric create single curve iterate rank employ al total overall rank overall avoid metric experimental assess evaluate seven method tie brevity overall rank table reasonably insensitive al address variability label pool test draw source namely initially label c relative al discover perform whole end loss average al imply ranking finally classifier c se rs classifier replicate al al method mean ranking average ranking clarity method classifier ranking se se rs se rs calculations ten monte replicate loss ranking group average overall ranking three average appendix e overall rs classifier ranking rs twice replicate problem table central study ranking method rank rank rank rank se al conclusion different suggests consistently outperform se outperform rs experimental argument algorithm consistently outperform section examine method literature consistently entropy leibler somewhat way pool classifier rs clear benefit exception rs individual classifier estimate experimental despite difficulty performance achieve statistical behaviour via classifier practical insight al competitive statistical begin target al individual batch iterate definition examine know exactly suboptimal unbiased motivate construction algorithm comprehensive experimental study several result competitive recommend choice estimation algorithm various bootstrap e resample sophisticated estimator research motivate superior subject work al heuristic experimental effectiveness univariate gaussian rv sized cdf dt rv b rv giving chebyshev hence apply rv strictly since six six discriminant na I bayes logistic analysis lda classifier discuss na I independence logistic svm popular classifier r classifier detail lda use implementation r implementation covariate covariate na I r implementation e assume regression svm svm radial score mle computing optimisation diverse al fall set problem datum split group another large problem error uci provide term property covariate b dim class dim illustrate probability balanced uniform prior problem multipli multipli mixture boundary appendix nn I logistic show table six problem monte rank mean tie se rs se se rs se na I rs se b c c rs rs se rs se h se se rs se logistic se se rs se rs section ranking rs quantify rs regret naturally difference give actual benefit rs behaviour treat reduction estimation novel provide framework framework central motivate allow al behaviour heuristic make abstract al outperform reveal issue turn motivate new al experimental al competitive effort bias improvement central certain case learn al seek select base classifier example include medical diagnosis approach review method performance assess experimental al consist classifier improve systematically formulation raise classification loss suggest optimality suggest reduction quantity basis improvement novel statistical framework strong advantage theoretical practical describe framework formally define al behaviour abstraction behaviour heuristic optimal context ideal unbiased crucially motivate algorithm strongly compare estimation type different explore source variation classifier abstract result perform background classification define illustrate abstract estimation scale experimental conclude background context al brief method later model covariate prior denote produce probable allocate objective somewhat example index indexing dataset may division subset discriminant regression regard fix give fitting notation extend become contain store near classifier role produces use denote prediction assess performance assume assess example error quantify focus allocate denote empirical performance reason generalise loss log denote hereafter refer example al abundance label expensive al select obtain oracle expert provide label classifier improvement pool usually small label denote consider scenario al repeat al time generate define al exploration amount label grow al contrast iterate selection occur iterate step critical iterate label covariate create example pool turn rs preference receive label example thus provide reasonable benchmark al classifier even benchmark receive training explore assessment rs hence address relative rather loss label blue horizontal uci nn shannon al reduction number classifier form fundamental al seek metric comparison common significant comparison goal label need single pool random expect define label form denote denote loss one give denote loss j kt future enhanced difference define loss pool example novel abstract problem indeed iterate generate covariate extend al smoothed classifier batch pool batch consist example expect improvement examine label actual reduction example select pool expect example denote batch analog incur major huge candidate consider pool batch candidate size candidate jump batch al generate selection candidate present major calculation selection require require calculation greatly severe make estimation extremely challenging batch al option recommend estimate individual al target iterate rest foundation framework abstract classification illustrate character calculation reason pool explore al examine denote make shannon imagine binary balanced c boundary rate split equally pure subset sampling hold prior classifier classifier denote calculate
follow resp provide resp range accordance integer fix say corollary quantify probability sparsity relate independence suffice comment length corollary equal q complete statement thing remain many way deal satisfy decrease eq integer side give sign edge correspond strategy probable growth result positivity part assumption separate kl divergence continue assume still appear straightforward counterpart satisfy begin recall notation collection definition least possibly define pa ps correct fix satisfie argument long throughout expression maximize subset cardinality pg pg version appropriate hand differ second belong account account place criterion complete simultaneously factor appear proposition proposition claim general three reason number third parameter choice condition equivalent interval nonempty take eq final task accomplish way namely sum condition thereby parameter quantity least derivation explanation second fourth obtain final eq change relation part also take bounded condition quantity form state throughout probability emission necessarily marginal list elementary distribution q q lemma taylor write taylor test taylor expand base base eq use hypothesis define cm monotonicity prove compute zero critical point inequality uniform expansion number contingency least upper bind bound bind difference define together find amount quadratic analytically fix let assume formula purpose proof substitute accord quadratic interested nonnegative must positivity proposition return inside numerator numerator q carry outlined beginning eq minimum purpose mutual information set say follow give claim probability completion depend make statement yet expression responsible asymptotic maximum one involve know exponent denominator make ignore asymptotic asymptotic see case use b asymptotic theorem refine throughout proposition ultimately chernoff ultimately draw lead sparsity boost specify growth sequence consist draw great statement apply statement integer least consequently let least observation apply union multiplying form complexity hypothesis precede proposition replace second statement estimate dominate large usual statement state involve asymptotic explicit expression make arbitrary adequate term first order proposition achieve leave subtract decrease coefficient result precede term obtain third accordance first quantity theorem large aid elementary asymptotic behavior log sided eq apply inequality right recursive identity yield claim asymptotic recursive familiar single say apply finite complexity behavior non positive integer well let follow immediately fix difficult formulate q obtain last divide n appropriate union multiplicative second q multiplicative second explain part computer describe concrete implement code package package follow theory type alphabet whose integer outcome element sum terminology frequency equivalence belong class associate define namely characteristic indicator aa ap expand emission probability reduce notational clutter reader marginal carry compute offline reasonably pair file table produce interpolation raise offline computation main issue interpolation main issue precisely carry statistic derive answer question computation carry store slow desire consideration balance another step obtain want table ram possible find explore first computation method exact computation second combinatorial integral approximate replacement integration scheme determine iteration concern construct selection point accord pick weight scheme heavily second covering weighting develop method exactly feasible cdf random mutual step finite store structure keep track point cdf hash structure part practical type length type possible advantage type amount account separate various list arrange rooted tree root locate parent leaf root level pass tree iterate give method tree traversal reason traversal recursive dft def datum process def boolean generator return else return generator def child else process leave tree element proper processing node input must emission type calculate constant finitely function implementation depend used list time certainly pr emission order compute make call n parallelization e plot dependence calculation algorithm experiment possible improvement far serial simple parallelization computation break branch subtree branch separate return object accounting type branch return merge accounting although parallelization agree closely paradigm carry parallelization auxiliary accounting list cdf generalization internal object carry parallelization modulus collection cdf library one available core processor entire eight processor hour processor call constitute color branch language marginal marginal p follow explain family define finitely increase ratio finitely many nf pdf gaussian variable unit standard term paragraph finitely approximate shape path explain pick maximum path still consume though marginal path statistic namely scale calculation marginal define manner segment contain center ram reality sort speed linear thought suffice fortunately theory analysis discover follow dependence likewise relationship emphasis statement care moderately small moderately large explain demonstrate claim affect store statistic linear marginal adopt statistic ultimately illustrate give intuitive make relationship conjecture great projection onto form relationship htb plot statistic present case small present demonstrate fix care strong thousand figure linear already except consider boost least safe interpolation range relationship trend difficult estimate become range end unnecessary algorithm whose intermediate procedure close nearby record table enough reflect edge left reason come last decision table inform consideration section cover statistic precise meaning meaningful determine illustrate reliable grow practice seem experience adopt table experiment dependence recall denote positive real distribution statistic note order direction obtain parameterization good think unit care getting become concentrated towards produce scheme object invoke generate list store member invoke signature stepsize rational separated unit rule bt def return bt interpolation actually separate associate contain soon encounter function convert kl send already explain choose crucial sparsity boost function simply evenly side statistic replace practice boost reason example could rise sample possible boost difference matter find reasonable return belong small path marginal path pass reason factor drop location length whether contribution arithmetic portion boost huge prevent sparsity show put threshold empirical mutual information computation clear power observe make entirely alternative interpolation language def gamma small near gamma gamma gamma interpolation linearly simplex interpolation neighboring point rigorous manner close reader exponent exponent believe exponent exponent result boost finite boost term boost function rule skeleton boost structure finally polynomially restrict consider approach attempt usually independence lead greedy relaxation max running constraint orientation produce initial relaxation relaxed contrast identification undirecte skeleton final network edge greedy scoring hybrid still separate constraint approach distinct step rather independence term directly score remove need heuristic one believe could valuable outside implicit experiment research compare publish result bic score quickly every boost boost evidence test preliminary experiment maximization suggest local able scoring sample issue find prune parent parent parent exclude consideration variable significantly section criterion parent whose contribution like cut regard parent view parent set independence line investigation choice overall ii error currently implement marginal restrictive alternatively fix way insensitive marginal somewhat adopt strong marginal must marginal every approach approximate without concrete future explore incorporate order boost effective certain lead point discrete structure throughout field paradigm hypothesis learn researcher paragraph characterize precisely onto least onto projection onto illustrate need text understand projection considerably simplify interest close marginal entirely constant serve justify interest uniform component conjecture decrease conjecture consider product symmetry distribution distribution identical far product write let fix correspond partial derivative say variable derivative q principal real unique branch map interval fashion namely decrease interval graph critical increase value follow order lie uniquely solution one symmetry point claim come function trivial interval principal branch htb conclude give conjecture marginal marginal point divergence nearly case p quickly move away enough lead long conjecture small conjecture empirical show heuristic chance appearing state minima exist state recall notion relate likelihood field probably exception characterization kl recall notation conditional joint use entropy conditional parent dag use term nb arbitrary relationship express dag collection bn know appropriately count extend naturally relationship kl divergence since involve entropy characterize accord side factor marginalization minimize kl simultaneously set matter resp mostly probability underlie bayesian scoring assign among closely though content empirical elementary state directly prove eq estimate eliminate absolute sign use condition small inside cite union theorem conjecture incorporate score mm com david cs edu scoring bayesian traditional dependent work property become prove polynomial distribution generating whenever exist perfect generating distribution although new score together explanation relation hide conceptual automated reasoning prediction concern system concern discrete write factorization number basic estimation perhaps fundamentally give structural factor world artificial intelligence speech biological medical even factor factor system sparse preferable representation task wish tractable proportion appear formally pair acyclic dag follow condition node edge correlation influence relation give explanation simple connect dag term every independent see follow equivalent represent rewrite conditioning except parent parent dag imply call independence one bayesian framework overview attempt comment literature simple case bn call bn notion bn change statement connect devise observation distribution distribution avoid possibility error seek identical study classic discuss evident different situation differently relationship imply terminology learn strictly consistent network distribution true clear consideration probability unobserved perfect map thus generate network framework count costly map two avoid complexity bn bayesian entirely obtain dag undirecte long share skeleton cause situation cause common effect cause dependent cause oracle conditional distinguish break problem follow structure reason appearance v responsible orientation learn structure vertex sift thing many recurrence relation bn complete even restrict bn independence constrain bn naive iterate structure take advantage achieve much relaxed study construct scoring assign score fit penalty bias structure fewer simple principle bic score score know justification make maximize np hard hill program seek conditional observe one approach commonly refer constraint key parent run approach drawback propagation difficult together knowledge combine main approach statistical testing view true keep function incorporate hypothesis add bic skeleton bic experimental log score act prevent boost close sensitive dag parameterized depend observe first score weighting penalty contribution perhaps go parent node paper parent bound iterate possibility second potentially produce sparsity boost exist hand conditional fix minimum sparsity intuitively boost calculate two really dependence truly exist want ensure independent zero mutual go empirical reference dependence independent inside logarithm event strength remain remain test pearson nan independence observe cdf justification namely pearson mutual information really reason threshold decision pearson small denote type relationship powerful explanation pearson classify evidence minus complement sparsity ii pearson test instead produce associate powerful henceforth theorem refer case equivalent elementary correct correct correct sense sense second learn network skeleton dependent ultimately learn network mention depend contingency represent conditional occur set express contingency stay contingency less integer contingency restrict set size subset contain pair substantially tolerance parameter appear variable role function sake formula main readily deduce n n would relatively remove however network basis access table finitely software look accurate chapter approximation publicly point generating contrast divergence generating benefit familiar relate representative put achieve polynomial number may exponential compete rather recover skeleton make discussion hybrid approach author acknowledge useful discussion definition need sample course follow invoke detailed classic theory deviation introduction topic refer notion concern chapter section chapter composite problem structure examine contingency table conditional variable denote convention cardinality denote summing quantity eq often restrict various proof random product xx l simplex identify contingency table sum element standard parameterization contingency table number consider denote identifiable equal sum row column say contingency contingency table denote atomic event atomic contingency sum second distribution notation represent certain product first distribution marginal case denote special distribution fundamental hypothesis mutual kullback leibler divergence minimize kl constraint onto share marginal complementary edge network denote term set consist one family distribution share parameterization distribution binary contingency range range share say binary definition course interval fix kl unique positive sharing marginal reference clear tool quantify strength quantitative strength parameter important quantity sake idea probability consider frequently bn bayesian acyclic dag without tuple obtain joint assignment variable write distribution write level sequence n p g bayesian network distribution competing obtain normalize produce conditioning order objective minus place element convenient shorthand understand informally mapping subset family bn collection wish learn consist vertex bs ga gb b ga behind separate collection think distinguish dropping edge another reciprocal structure strength notion separate make polynomial line bic boost mainly place state order finite sample complexity elementary need make involve define thing understand statement need lemma complexity allow learn return quantify bn state divergence divergence arbitrary notion familiar accord onto factor solution map namely distance simply word map denote bayesian satisfying map cm namely cm network whose distance quantity though content independently deviation specific need quantitative concern probable estimate various entropy multinomial mutual entropy mutual combine standard theory deviation probability estimate concern continuous function primarily case effort lead variable let frequency exceed fall theorem easy side chernoff achieve large deviation chernoff lemma theorem need complicated appendix simple one side inequality multiplicative elementary analytic sequel draw bernoulli second breaking argument derivative interval term approach infinity derivative value entire unit occur absolute value divide thus find denominator namely elementary monotonicity zero differential calculus increase interval clear slope possible interval value attain value bind denominator turn attention chernoff exponent use claim interval attain decrease keep letting interval actually suppose contradiction side consequently assume proof claim possibility mutually exclusive possibility exhaustive proposition able derive concern involve note state quantity apply know entropy minus entropy far side observation define principal variable follow sign desire estimate factor outside bound term mutual proposition collect fact function simplify formula eq monotonic minimum reason decrease two though still decrease monotonic decrease q decrease fact three decrease eq q drop differentiable say eq define point derive decrease similarly eq one inside less quantity multiply side decrease function impose eq statement relate statement prove accord unlike application probability generate learner control marginal go naturally large must condition hold calculate inequality proposition implicit circumstance preferable condition conjunction q imply follow form let upper eq express finite error local distribution theorem parameter relevant state come first complexity concentrate case refer initial boost consider straightforward complexity sparsity boost weight complete analogous formula technical future choice hyper affect asymptotic growth choose quantity q probability examine result may arise free particular obvious close optimal appear denominator comment hyperparameter carry however merely nod eq asymptotic dependence appear determine dependence replace say element cause dependence namely instead chernoff cm great define sample probability function combine chernoff bound asymptotic quantity finite complexity node mutual minimum edge strength perfect separate define let free pa pa ss quantities follow let q reader theorem edge graph refinement asymptotic strength free large q asymptotic nm part theorem asymptotic theorems theorem begin collect factor asymptotic asymptotic actually assume elementary list last logarithm suitably difficult go define dominate completes verify dominate asymptotic function define n asymptotic dominate asymptotic statement properly comment fewer technical essential corollary reason boost boost objective extra edge false quickly term network second presence boost false miss false proposition key quickly network skeleton distinct structure possible distribution fully boost mapping objective reflect composition first test generate marginal derivation case point quantify converge enough moderately multipli time proposition penalty grow moderately proposition finite multipli order growth start choose conclusion upper readily probable bound boost let satisfying condition assumption q reason large new convenient corollary first derive
generate monte actual cost hierarchical substantial sample consider procedure previously moreover certain generate augmentation emphasize location apparent assume parametric parameter choose scale location write proposal exactly independent pdf possible prior simplicity draw draw proposal pdfs drawing distribute accord closely resemble target interpret estimation x clearly infeasible main sampling obtain layer instance base combine mis role devote level tune mechanism know scheme walk mh monte method implicitly subsection notable case independent walk mh collapse technique layer although measure previously target proposal chain matrix mh summarize pdf x strategy well target parameter properly many mode mean proposal choice tuning provide depict proposal proposal walk mh burn iteration already reach stationary burn proposal walk method burn period imply walk generate follow hierarchical interpretation implication draw target walk well independent proposal roughly tune non generating procedure certain denote write pdf density q estimation clearly walk generation build figure importance sampler population step initial draw x accord tt generating cast nn advantage play approximate mc technique adapt cloud mis population proposal alternative literature static mis mis sampler highlight challenge consist population pdfs refer mis target assume exactly exactly ensure robustness scenario mis mis mis mis pdfs sample distribute mis yield statistically dm mis pdfs evaluation proposal evaluation target weight accord load instance mixture dm mis p mis divide proposal disjoint form set case definition capture proposal mis weight jj weight whereas mis htb mis c mis mis dm mis sample set integral measure proposal pdfs monte past scenario pdfs subscript proposal framework adaptive multiple include eq characterization show adaptation many procedure instance build mis discuss figure representation show spatial proposal pdfs htb xt proposal dm mis case partial mis appear weight single iteration form use scheme sample employ htb mis mis dm mis suggest mis dm mis summarize different grouping proposal pdfs divide proposal index pr cost increase total grow indeed proposal evaluate hence number perform scheme build use remark pdfs generate advance usual adaptation algorithm convert mis normalize simple iterative adapt proposal mode iteration version th previously need show c express indeed final estimation eqs express recursively estimation state start n ms see appendix estimator simply proposal pdfs weight ratio finally consistency version choose proposal pdfs specific generation draw weighting given normalize return pair tn adapt pdfs scheme markov adaptation location mis proposals mis section adaptation procedure estimation argument observe adaptation underlie markov depend update parameter technique type adaptation variant like procedure walk interact pi doubly interact importance detail algorithm cost simple specifically mcmc technique coincide proposal pdf motivation one hierarchical mh underlie approximation interact markov base markov markov markov location mh draw nm normalize easily pdfs number iteration step specify mcmc adaptation first one parallel chain interact adaptive pi pi version doubly interact case initialization choose pdfs mcmc technique step draw mn normalize eq return simple applying mcmc consider mh pdf n n scenario location pi layer partial dm mis manner explain build pdfs incorporate adjust langevin mala sharing introduce hence pi detailed mh mh working location sequential different adaptation rest discuss extend coincides pdf subsection interact pdf fig simple technique type transition form draw pi accept differ describe probability dramatically suitable purpose q idea potentially burn period one iteration consist candidate choose correspond accept difference one iteration need already compute step mh sequentially mh draw pdf total employ pdfs mis final population change diversity preserve unlike resample update section alternative describe update involved total big specifically target pi mh gibbs generation acceptance choose pi mh uniform extended multinomial evaluation require recall base evaluation general monte bad easily rest observe jointly different n robustness result suggest approach sake covariance covariance strategy consider proposal suggest performance benchmark tackle issue carlo consider parameter wireless propose multimodal specifically multimodal gaussians I eq matrix approximate monte normalize adequate approximation ability square mse three describe partial dm mis parallel independent pi method moreover static mis mis dm final fair implement total evaluation pdfs specifically order proposal approach importance level pdfs different matrix initialization denote initialization region mode thus improve specifically select show initialization initialization mode eq result experiment respectively table highlight face computation per sake good among simulation pi choices initializations pdfs initially localize pi improve adaptation pi depict circle configuration proposal density pi pi htb literature nature mathematically carlo approximations true exhaustive deterministic thin order mse pi proposal randomly n case mixture algorithm keep number mixture component different st sn order pi adaptation pi pdfs pi see simulation small highlight outperform example also display circle proposal run represent approximately mass output triangle diversity pi ensure well cover htb adapt covariance mass localization wireless sensor plane realization range sensor locate identical pdfs also p prior receive observation sensor fixing compute expect order deviation pi three scheme description proposal initialize randomly pdf consider proposal also isotropic diagonal fair pi choose average pi outperform robustness pi algorithm walk proposal mh furthermore use adaptive importance scheme class employ reduce dm strategy include different differ extent term scheme consider partial dm computational confirm benefit sampling project foundation de grant european network grant mathematics university es circuit de represent approximate complicated multidimensional target simple proposal draw
abstraction principle bellman mdp regardless ai reward element ai aggregate mdp aggregate mdp algorithm vi mdp us option consist termination tell primitive expect reward primitive analogous discount give option execute state pi state pi matrix matrix option format introduce format function value option evaluate brief survey hierarchical stress reason macro run date except option vi generalizations bellman require mdp macro operator largely focus hierarchy discovery option discuss hierarchy also controller temporal abstraction differently abstraction option construct hierarchy policy hierarchy mdp hierarchy describe mdp termination start plain vi proceed notation state eq select action rewrite correspond come execute note multiplication vi next reward define associate pick improve update execute policy policy stop value reach possibility termination stage termination therefore terminate terminate conceptually think specification termination diagonal however summarize termination behave terminate identity actual update iterate tend go state state exceed induce policy introduction action give formally boolean action allow follow q benefit irrelevant whole become macro give rise state solve immediate availability use macro operator hierarchy contrast convert mdp aggregate state stress valid describe reach macro repeat fast aggregate example vi eqs state use help mdp model convert new transformation aggregate compute option state model option termination termination policy eq state follow state q aggregate terminate towards evaluate row option valid primitive accord follow contain row option terminate row option primitive add action action original time g mdp macro action macro original supplement observation algorithm mdp bad thing happen macro iteration take look eqn may domain run four computing vi complexity take version aggregation get iteration complexity algorithm need jump due converge version see aggregate state map original position gets map aggregate proceed stage vi obtain aggregate value time less original combine stage compressed action get iteration complexity fast add action original action move sensible movement converge run plain fig vi make comparison fair speed vi version summarize construct stochastic move qualitatively combine option stress deterministic follow iteration many follow cm aggregation option cm aggregation option try different aggregation speed compressed extract use aggregation happen leave apply never intend aspect system eliminate vi tuple disk take denote action disk disk vi iteration speed abstraction solve e abstraction ignore place sub ignore proceed solve move linear mean speed vi plain vi whole state times vi plain computing options vi plain vi plan denote allow place map onto mark belong use alone trial able reach time plain intuition behind already time amount speed figure bellman equation sound solution medium sized mdps combine option abstraction notable problem finally experimentally option realize apply appendix background information concern aggregation mdps adapt entirely due concern ai I vector reward ai aggregate define architecture value conversely define state aggregate matrix represent aggregate state state bellman mdp call exactly q aggregate equation operate short course since present contain operate expand follow aggregate lead row state column operate state introduce namely exact single aggregate equation aggregate state transition mdp solve equation mdp action
direction constraint w partial derivative lagrangian tangent strict saddle regularity smoothness problem strict saddle project descent algorithm output defer apply stochastic give strict saddle ica decomposition goal find symmetry valid symmetry tensor saddle property solve problem warm saddle apply prove stochastic descent iteratively careful try single component misspecification straight view satisfy strict saddle stochastic gradient unstable expand form scalar rewrite tu l li tu permutation sign global program strict saddle minima defer appendix strict minima permutation base objective oracle application tensor multilinear operation tensor multilinear gx u j u u orthonormal transformation transformation orthonormal observe use technique simplicity follow multilinear z ta ia lemma verify closely relate u function therefore construct order gradient one sample straight share inner take apply result predict reconstruction formulation orthogonal carefully measure f way sample compute gradient sample set easy ica introduce compute stochastic variance use mini size simple generating reconstruction converge exhibit cause saddle significant negative new ica stay decrease rate htb bc bc htb bc bc paper saddle descent converge decomposition step gradient saddle property handle symmetry give detailed strict randomness step assumption add simplicity gradient extend di descent saddle bound smooth exist minimum factor dependent proof analyze behavior three assumption iteration choose equation initial inequality w specify correctness never always make saddle know locally neighborhood smoothness w know imply initialization fw tw tt max relax max max sketch sequence update next lemma approximation around sgd simplicity analytically follow simultaneously q substitute q hoeffding summing union directly eq finish prove generate sgd substitute sgd w carefully event enough fourth know contribution come product hoeffding equivalent finish ready proof denote fw fw fw fw carefully bind choice max finish finally fw know definition locally close minimum bind probability know p I sum since enter hold repeat lemma discuss equality constraint mild point w project argument unconstraine could slightly convert standard unconstraine interested satisfy easily di introduce constrain optimization material technical deal modify constrained want condition quantification common problem say independence constraint gradient linearly constrain unconstrained introduce properly regularity well define easy check curvature everywhere quantitative bound case exponentially close partial lagrangian tangent iw dm know interpretation tangent normal complement vector tangent space assume w w fw give know q necessary due fact optimality please suppose continuously multipli tucker equality lagrange multiplier sufficient equality constraint lagrange multipli kkt know multiplier satisfy therefore strong implication thing unconstrained constraint effectively consider feasible point technical relate equality space much constraint w w w q give give see serve constraint serve curvature next tangent nearby point eq calculation conclude constraint c projection iw since close hand know proof use lemma add project back feasible smooth p first inside ball w w constrain problem continuously tucker know eq ready saddle constrain follow next smooth run direct manifold bound curvature local dynamic locally similarity unconstraine point saddle function smooth lipschitz lipschitz proof condition exist lipschitz bound lipschitz thus eventually need smooth feasible follow without ambiguity calculation linear derivative inverse every transpose rd finally lipschitz bounded lipschitz list essential require modification eq small theorem notation follow calculus know saddle neighborhood everything else proof notation tangent previous subsection define characterize couple sequence lemma notation lemma around tangent project e tw satisfy hold q notation w update gradient remainder project space denote p tw immediately tw later prevent ambiguity event carefully know w recursive easy case hand also choose combine w w fact us saddle point combine prove proof lemma optimization problem strict saddle trying first specify dynamic equivalent section compute gradient lagrangian multipli second partial function constraint therefore strict dependency far strict saddle respectively intuition choice parameter I eq symmetry pick pick line pass neighborhood diagonal mean local finally proof immediately thing local minima transformed problem investigate lemma satisfy know close local must lemma minimum symmetry change coordinate perform change effect I fu support constraint give express coordinate coordinate q derivative lagrangian index indice satisfy strict try dependency exactly strict saddle follow version around saddle relative choice parameter I ik eq suffice j diagonal swap coordinate argument would j know quadratic swap either suppose hand since therefore know negative entry clearly coordinate u du u u concatenation know unit tangent finally c eq proof finally ready theorem saddle lemma thing local permutation sign argument proof chi optimize convex reasonable concern update saddle point strict saddle convex use polynomial knowledge give stochastic gradient descent convex function saddle decomposition rich class optimization formulation tensor saddle property tensor basic solve problem pair convergence gradient descent understand convex stochastic backpropagation success transfer optimize np hard come non convex many minima among even minimum saddle minima point local discrete analog pls deep network main bottleneck minima saddle gradient particular saddle saddle rely hessian usually computation time gradient empirically saddle answer give saddle efficiently call saddle intuitively local progress first efficiently give framework tensor core latent see saddle issue different permutation valid symmetry creates exponentially optimization analysis give online scalable twice call stationary stationary saddle identify convex strict hessian saddle negative seem counter intuitive stochastic descent show stochastic help saddle strict output step saddle may application orthogonal discussion strict saddle give orthogonal correspond valid analyze stochastic setting get first decomposition guarantee economic work property twice function call point stationary could minima maxima saddle hessian minimum criterion positive semidefinite equal point saddle degenerate say strict saddle twice local minima stationary intuitively saddle point taylor
histogram next form protein final element experiment outlier small remove outlier protein cluster row product structure element cluster shape transformation one see inside similar detect protein get rand evident good shape protein class class present assume flexible configuration dirichlet realize chinese restaurant distribution reasonable automate partitioning visualization cluster curve shape broad scientific technique shape assume cluster elastic metric joint wishart assign carefully automatic markov carlo chinese cluster protein shape chinese restaurant wishart automate object area large choose homogeneity minimize homogeneity cluster address researcher g component clustering assume e maximize useful quantification probability population population population introduce comes assume upper assumed convenience parametrize multivariate alternative appeal infer cluster almost focused primarily availability center cluster g shape unlike cluster center quantitie quantification homogeneity obvious important use cluster object preserve scale parametrization cluster however choose exist shape shape project preserve cluster euclidean avoid mean cluster base cluster datum summary encode cluster salient elastic preserve transformation inner product wishart configuration induce chinese restaurant avoid idea configuration model organize follow start study mathematical metric product specification present cell shape protein conclusion study involve protein biology discover structure state protein term protein structure evolutionary origin protein manual classification protein snapshot protein trace evolutionary structural automate shape diagnosis genetic research extract contour shape extract contour medical diagnosis database describe b cell contour paper article shape shape cell visually denote appearance method object broadly base riemannian pairwise develop wishart cluster instead method computationally product specific square transformation drawback nonparametric elastic introduce inner square velocity let parameterized curve domain attention absolutely open curve close curve p purpose study shape square integrable translation elastic curve nice interpretation simplification term care variability protein biological camera distance rescale curve unit c follow rotation element preserve composition parameterization join accord riemannian rotation analyze equivalent translation scale product shape give curve inner well perform optimal optimal rotation whether product u n article distinction specify irrespective illustrated pose curve inner wishart generalize product non sum easily parameter shape inner respectively denote partition represent membership come sub population shape population place inner product panel three block define enable identity encode cluster information introduce conjugate prior intuitively control similarity inner indicate association goal develop infer membership prior let place constitute prior induce partition chinese restaurant crp induce bi n ji th calculating calculate euclidean membership matrix onto threshold cluster thresholding calculate mean euclidean posterior observation cluster find threshold thresholded rare posterior b wishart crp w crp section euclidean discard number trace plot main mode euclidean model px gx gx w control within crp produce keep fig three cluster class contain histogram upper panel right appear compare lead tight intuition right upper right cluster small big experiment present investigate relation fig upper correspond observation infer exceed certain infer increase reasonable crp prior induce typically situation strong sensitivity relation specify fig heat range fix tend tend role probability small recommend choose preliminary association strong tend tight class cc dataset gaussians lead inconsistent finite mixture prior mixture assume assign panel panel cc comparison gaussians unlike case mean euclidean result panel show gaussians gaussian crp confirm convergence slow although theoretically cc c gaussians data shape shape shape shape shape observation definition cluster final result number cluster shape class I histogram number freedom wishart euclidean cluster robust compare shape method class shape result shape assign recognize misclassifie dominant shape class number shape sensitive rand quality similarity ground ground truth quantity belong different assign class rand compare rand index method fourier descriptor vector markov weight elastic shape pairwise model wishart crp elastic inner crp rand I include cost generating mcmc calculate elastic inner since know hmm neighbor result c hmm classification rand index section shape crucial datum hard rough estimate visually grind available provide
importance sec reduction learn sec call adaptive sgd factorization modify lead support generalization first useful thing view one learn use sampling training minimize reduction addition aforementione like estimate conjugate potential straightforwardly properly second method slow however estimate quasi limited memory bfgs sgd tool motivate sgd keep family absolutely include via omit clarity estimator estimator use variance denote variance exist almost nan reduction minimize respect step average equation f sgd lead machine loss function reinforcement scenario follow sequential change variance speedup sgd sample uniformly importance evolve goal expectation accord base iteration instead gradient properly I would depend reasonable rate decrease schedule build function solve sub gradient gradient variance equation e approximately stochastic involve sample direction simplify standard since remain remain valid strongly convex upper weighted standard sgd benefit define minimize moment sgd moment simplicity differentiable strictly minimum assume convex exact appendix strongly smooth step exist inspire reference adaptation rate practice constant optimization imbalance hyper parameter standard mining positive negative importance positive important negative cross good imbalance expensive instead sgd biased sampling mm acceleration convergence benchmark rest feature pre image classification provide strong consider image negative image odd sgd initialize sgd regard parameter sgd sampling sgd fast acceleration sgd parameter gradually learn hard also depend decomposition observe loss softmax square sgd currently one integral policy algorithm expect reward reward predefine trajectory action p grid environment four grid change distribution trajectory canonical squared consider discount terminal state locate successful terminal start optimize correspond sample variance tune learn grid close improvement success sgd sgd benefit sgd problem three connection reinforcement application reduction strong inference integral sgd integral also intractable machine potential importance without technique around gradient compute h minimal strongly convexity minimize inequality tt argument tt smoothness assume objective know proposition www w w line large reason property prove suppose respect condition satisfy definition evolution class negative negative correspond object varied class exception positive intra note dynamic stochastic process value self tune exploit extra cost memory access ns inspire know multiply factor access time weight divide sample achieve sgd plot algorithm time time account sgd speedup compute epoch divide slow access big overhead read memory local seek sgd sgd overhead choice epoch sgd speedup slow actually correctly convergence try use notice sgd improvement factorization column parametrization embedding non slightly fast behave index index prove reduction choose integral efficiently minimize derivative derivative equation main decrease zero unbounded ax qx ax qx minimize equation non sampling optimizing need store implement store denominator axiom conclusion theorem criterion theorem theorem problem theorem summary sgd online machine accelerate adaptively example first estimator sgd optimize show convergence
symmetry get desire sx differentiable l q use mean hoeffding union bernstein maximize exponent though low long original final sx sx dr ff contain q assume hoeffding bernstein bernstein exponent middle exponent final statement solve suppose lipschitz tool slight special collection although result thus pick mean via hoeffding generate entropy except distribute except additive condition event error shift however q obvious achieve convenient mean cast q absolute value l q agree define hold argument get guarantee sign extremely error shift invariant phase thing university approach lead scalability dataset handle approximation error bind give use learn surprisingly two elegant traditional contain attractive approximate shift exploit embedding scalable kernel date embedding fourier transform eq continuous transform let eq b b obvious twice add non formulation use remainder embed count publication practically aware implementation learn library popular use kernel previous analysis increase nystr om fouri embedding problem study complementary view embedding preferable popular prove bad constant expectation concentration discuss effect evaluate two analysis kx sx bt green let approximation error claim diameter embed bind definite diameter fx achieve probability long place net centers net hoeffding replace inequality originally low exist moment finite first moment finite moment material bt gaussian place increase constant though many fx kx full note unfortunately integrate minimum use kl appear unbounded bind truncate relate gaussian kx increase otherwise note somewhat let kernel bernstein style f f claim lower let f tight concentration possible concentrate high stochastically less finite justified kernel likewise p x thus exact result decision use training value eq k kx ii say sx k z hx hx require loose achieve induce p unbiased strong sample serve compare microarray different situation merge database approximate embed bias unbiased estimator simplify noticed subsampling pair reduce favor approximate solver approximation test slow far block quality fx f advantage bind apply uniformly set single two tight consider x evaluate change tell exponential unbiased easily combine unbounded extend uniform evenly spaced matrix embedding curve predict z expected numerically integrate loose first depend tight constant loose old loose predict empirically mm survival mean former low slope black color mean mean squared uniform natural substantially constraint turn distinguish I
learn principle optima many binary hash nonconvex nonsmooth much issue image hash optimize assume continuous code learn code affinity way round optimally continuous code thresholding introduce bit result classifier hash various technique several approach produce function retrieval precision need code binary affinity since million optimize one competitive precision art slow improve provide option three code learn hash join learn code learn hash code element problem code hash incorporate optimize lower learn hash function iterate correct version code iterate emphasis optimize mac cause stage review function mac base propose evaluate use hash focus hash perform emphasis attempt preserve attempt preserve similarity achieve affinity affinity subset optimize hash embedding hash supervise hash sequentially element affinity although approach find affinity usually approximately code alternate hash number embedding free parameter encode base unsupervise representative l nm try project close projection laplacian locally nonlinear embedding objective exist separate point optimize embedding code use two hashing long embedding though focus difficult thresholded hash recently meta construct nest proceed stage transform recognize constrain deterministic hash second augment lagrangian former simplicity unconstraine dependent penalty third apply hash iterating later optimize code projection close try modification optimize target except latter would nothing correspond optimize practice start initialize mac result slowly optimize nn hence unlike method affect optimization optima give overall mac hash optimize affinity ex pca approximate minimizer cycle stop appear term simply hash minimize hence hash classification problem even enforce eventually increase svm optimize slack penalty simplify code complete fortunately alternate would correspond hash surrogate start alternate bit remain fix next start describe modification regularize binary base hamming distance binary consider well mi replace help rewrite define n optimization q tn quadratic qp definite qp qp qp binary use spectral relaxation constraint relax eigenvector small truncate eigenvector qp solve bfgs np complete expect skip update avoid minimum bit bit np bit form variable propose proceed possibly group function code binary code etc alternate bit ignore rewrite rewrite equation block submodular proof use alternate block submodular objective combination hash dataset art hashing focus elastic linear svm hash train radial center svm use gram use svm give use feature sift image image sift image cifar extract feature image test digit feature original near unsupervised training label supervise dataset retrieve nearest hamming hamming inside indicate loss schedule mac hash initialize mac increase code stop stop find consistently low objective unsupervise state art algorithm method preferable mac optima introduce framework effectiveness retrieve neighbor use bit hash compare way optimize step hash mac use subset hash mac svms mac linear hash top iteration mac show mac operate augment trading decrease enforce step typically bottom show reduce sift sift sift retrieved hamming hashing supervise hash sift bit nearest retrieve near image query search mac quadratic step thresholde two hash iterative quantization locality sensitive hashing hash spherical hashing mac create affinity neighbor mac subset affinity less achieve mac take wide training point method mac mac truth near retrieve neighbor practice one would increase retrieve use sift mac training neighbor sift retrieve near set plot mac outperform precision bit wide truth retrieve hamming hamming ex sift retrieve ground truth ground nearest training near hamming precision bit precision retrieve neighbor ham hamming hamming digit retrieve hamming hamming ex cifar neighbor hamming distance ex retrieve cifar ground retrieve image search binary code digits precision bit use bit neighbor search binary code mac use step hash hash iterative quantization supervise mac create mac point show achieve mac overall winner consider retrieve neighbor hamming distance bit change retrieve almost mac achieve well show result cifar bit hash distribution equal code extent hash discussion precision depend user ground compare single mac mac indicate mac achieve well well note retrieve drop large mac ham find number expect precision sift k cifar algorithm bit sift cifar plot code bottom row hashing code dash line horizontal dotted algorithm set mac mac cifar dataset hash hashing proceed greedy find code fit hash final randomly cifar dataset create label iteration mac training bit row test hash first iteration mac function almost case precision bad happen many point neighbor ham precision retrieve indicate avoid available last systematically try number retrieve precision cifar linear svms cifar runtime case mac well precision subset noted work quadratic try nonzero limit hash subset relation well enough amount entry use problem subset precision bit retrieve neighbor ham hamming hamming retrieve cifar mac top mac precision hash bit cifar image ground retrieve image search retrieve bit kernel hash entire originally propose method mac hash mac hashing hamming hamming b precision retrieve neighbor cifar retrieve hamming hamming c bit truth point query image retrieve near image hamming search binary code bit range neighbor code learn ignore interaction hash nonlinear learn tradeoff gradually term find make hash gain function know situation arise selection classifier classifier accord filter hash act produce map input neighborhood role hash pattern pattern preserve relation regardless easy mapping code function hash act objective nature attempt hash optimize fit strict suboptimal separation hash mac still objective involve code hash iterate penalty code close hash e achieve hash quality retrieve code dimensional reason hash retrieval ease hardware software library etc optimally hash simply involve change runtime mac approach iterate besides iteration except warm
important bayes shannon sensitive loss produce function information bottleneck bregman attempt extract solve e regularize mutual information derivation follow concave bregman firstly secondly insensitive seek assume column two surrogate learning assess function page distortion curve firstly loss plot one see mutual greatly mistake classify class error way fp recover mutual information processing inequality eq case mutual algorithm optimization department science theorem proposition definition conjecture feature aim automate fashion drive behind current trend method characterize generalization lead learn linear code independence test unsupervise course old fashion learn familiar flow chart method exist measure performance sake seek overall end feature scheme present transfer result distortion understand well unsupervised possible framework characterization lose map understand surrogate distortion throughout denote instance action space arbitrary include amongst denote norm denote denote divergence shorthand measurable markov give x space learn draw observe learner incur part place distribution act act loss p suboptimal distribution action r xy xy function ie markov xy purpose risk randomization help bayes respectively standard p p apply rule normally restrict sample focus information computation curse wish possibly randomize learn protocol draw observe incur feature move map loss gap feature versus raw non closely notion independent stein theorem label zero contain predict average x posterior sufficient prior assign zero well lead class pick iff lp xy surrogate surrogate log lead bottleneck bregman divergence losse another surrogate minimization include alternate rate distortion theory first something involve third restrict practice know care surrogate greatly contain difference loss varied study field know comparison experiment focus eq sense noise notion appear stein theorem randomization lp suggest calculate material fast alternate behave toy drawback interest relative predict many compact might seem one could use many case much well structure model automate search structure behind deep restriction relationship marginal learn xy l p minimize lost need reconstruct deep belief network highlight reconstruct find structure interest regret choose nearby equip metric wish map equivalent ex dx x xy material require high good feature surrogate entropy prior small reconstruct map view principle reconstruction many surrogate reconstruct divergence restrict normal standard autoencoder autoencoder justify autoencoder terminology involve end performance complete capture distortion rank map loss mutual map obtain form surrogate ideally calculate distortion question surrogate provide least answer information distortion fp xy tight information algorithmic implication ultimately restrict lie know optimize general rather deep hierarchical rather map learn final map n composition map chain final scheme layer fashion analyse system invoke union reconstruction material belief surrogate semi learn classifier comprise normally via map label allow analyse generalization joint something know sample complexity combine complexity work square allow supervised give scheme operate give example particular sufficient greatly contain particularly hence bayes show classifying however jump say gap sensitive generic interest insensitive weighted bottleneck toward class map feature map ht loss determine conditional bottleneck separate function versus experiment fashion example sort concentrate one cost map row stochastic prior plot
hyperplane dt candidate generalize nonlinear probabilistic use probably bind ensure learn safe inductive invariant program abstract hyperplane form generation benchmark literature benchmark tool dt simple ml fundamental verification loop construct program hard approach come boolean inequality reason abstraction learn label task learn unseen ml formalize probably pac one sample diverse bioinformatics finance vision artificial intelligence program consist loop verify value reach program record execution loop head program execution terminate fail loop maintain loop begin maintain existence prove nothing binary classification two good bad loop ml algorithm condition worth note classifier counter point classifier good linear arithmetic program must invariant must ml describe set simple elegant method approach discuss illustrative end inductive invariant restrict program control program begin execution state reach start consistent good start loop head loop assertion fail directly go assertion fail assertion side state safe inductive indicate hyperplane bad automatically assume else em overview overview bad domain default hyperplane translation domain learn dt correctness invariant pick satisfying collect state point good reach bad state good choose invariant benchmark domain inequality hyperplane domain transform bad sample dt process dt rule binary point path equal threshold child leave specify list easily split last bad split state right bad state state child good child pattern pick bad fall computed dt dt follow path root lead good conjunction path simplified annotate invariant pass verify invariant prove program think transition transition safe state want program remain respect program least similarly complement show imply separate good invariant generation space label hypothesis approximate often term think common instance view compute safe set program label characteristic safe safe inductive program assume partition consider represent domain represented dt binary inner node child inner leaf evaluate trace tree inner node hyperplane many dt np separate well follow leave separation normally entropy commonly low reach homogeneous co formally look reach define splitting point reach condition split perfectly separate bad heuristic pick entropy measure use dt dt get abstract program return matrix program hyperplane describe procedure surprisingly get sample function transform dt learning transform tree correctly formula simple procedure program annotate verify box necessary safe inductive encoding formula tt generation dt convert formula reach predicate thus classify leave path leave conjunction predicate formula recursively learn sample output boolean combination inequality return moreover assume generation sound terminate return inductive hard generation dt learner could refinement loop make role less incorrect could potentially refinement large justify probably note key sample justify include pac empirically successful formally guarantee et hypothesis output true vc might class bind finite boolean hyperplane large label unfortunately point arbitrarily point leave end leaf labeling learn arbitrarily nod basic dimension polynomial probabilistic invariant time measure however run routine hyperplane point set sample compare cover hyperplane consider take candidate ease generalize nonlinear add column final occurrence easy correctly require nonlinear benchmark require see algorithm library dt cart learn greedy simple consider variable state run loop state reach state state margin run program fail assertion state program program benchmark domain program class appear learn work state lower dimensional would reduce run algorithm good pca respect basis point form program allow verify static mainly focused consider tool abstract ice ice version use pick similar run version well benchmark base learn meaningful sampler implementation default abstraction interpolation software interpolation inference tool linear combine solve testing benchmark various benchmark choose benchmark among solve reflect tool instead valuable technique dt second sc total follow tool second safe invariant find program arithmetic tool safe repetition case ghz benchmark minute table experiment depth detail ice program solver run ice stop limit ice solve predicate constant ice similarly difficulty sc higher also run easily handle benchmark mainly specialized dealing boolean weak slow handle spend dt learn benefit approximation guide ad hoc benchmark implement verify modulus certain finally one specifically guide overhead algorithms generation discuss sc learn tree benchmark well learner hyperplane hand time hyperplane hyperplane candidate slope sample point yield generation view binary loop learn inductive unclear good ice implication refinement handle however able infer nevertheless consider ice become complex paper infer ice learning refer
theory chapter first real connected follow riemannian th point connect send send point independent rank connect row connect send manifold nearly identical operator depend particular bandwidth ad else unknown graph connect bandwidth compactly support directly connect choose article choice use use principal ideally l denote construct final modify elimination eigenvector evaluate define q elimination denote sample column divide make note accord close complete generate eigenvector create map x eq ball center radius lebesgue ambient small amount embed horizontal circle radius horizontal noise show show eigenfunction pass algorithm color green blue h thm thm example thm lem prop major computer science geometric topological geometry survey space topological begin support lack topological concentrated approximate statistical laplacian manifold related use eigenvector laplacian datum orient persistent homology use directly heat topological community well drive exploit cluster required also non find outside usual proof long apply rigorous justification validation study automatically choose kernel
second equality chebyshev equivalently dt z e protocol give compute normal probability otherwise x bit protocol x stability normal cumulative var var
cg dual erm variant note dual dual runtime sdca could dual repeatedly dual produce run un minimize erm randomized way desire running yield run achieve fast dual primal mapping map subsequent mapping viewpoint warm begin bound center change error dual center sub measurement strongly convex invoke obtain q strongly convex eq recall lemma establish primal let iteration notational convenience fx f g dual oracle q fact combine induction case invariant since hold combine result subsection evaluation rate grain match terminology spend stage pass gradient analysis iterative meanwhile stage dual current coordinate negligible overhead valid time primal dual xy sx nice property solve error rapidly switch empirically size appear inner take amenable mid tuning dual sdca convenient derive update single sdca square locally efficiently decrease algorithmic require tuning end optimizer sdca sgd demonstrate tradeoff approximate proximal advantage shrinkage yet improve stability stage desire bias mnist cifar protein mnist classify vs rest classify vs mnist cifar transformation significantly normalize row take k meanwhile protein pre train randomly hold sdca decay sdca minimizer notion inner period exact simplicity advantage count stage report erm introduce vanish bias towards point investigate minimize erm regularize erm run optimizer extent center help sdca regularization erm heavily meanwhile convexity place cifar recall convexity add sdca least sdca low final line erm sdca legend l erm lastly demonstrate extent statistically desirable cifar take sdca achieve protein effectively erm sdca plot incur sharp meanwhile sdca converge smoothly exhibit degradation stage take research institute berkeley nsf research fellowship several stand alone lemma regard smooth convex combination quadratic strongly smoothness convexity quadratic function convex consequently erm objective error erm notation convex duality optimization turn saddle lagrangian mapping imply duality equality attain primal primal substitute recall primal dual error furthermore f recalling yield international machine france email accelerate fast minimization erm linear wide setting classical provide strongly accelerate box exhibit stability advantageous strongly term correspond erm predictor minimize sample focus linear logistic regressor z capture problem five year increase mild despite solve dependence impact running application erm notion arise correlation solve erm solve erm know scale solve erm option bridge black box erm approximate erm problem regularize erm accelerate erm inner precise approximation operate ascent run erm summarize improvement accelerate proximal convex suppose possibly convergent minimizing order multiplicative previous art well multiclass structured variety erm instance space smoothness exposition clear capture argument illustrate several machine proximal accelerate proximal enable acceleration impose believe presentation simplify broad loop employ accelerate proximal sdca lead erm regularity smooth refer case erm subject comparison algorithm consider interaction primitive refer denote context dual convex denote certain present hold make erm operate namely decrease dual primal risk gd gd sag sdca ap sdca l naive r acc mark regularize mark algorithm system mark minimize square denote run least square briefly explain run context erm gd refer canonical gd nesterov accelerate reduce sag stochastic gradient ap sdca accelerate sdca accelerate latter restrictive solve proportional time general guarantee time minimizer number use run ignore problem variety three slight simplest apply strongly complicated apply prove require erm natural choice conjunction minimizer operate popular desirable analyze proximal concern involve general use derivation technical lemma simple introduce sequel abstraction minimizer finally design quantify minimizer accounting ensure abstraction inner give oracle property typical erm erm accelerate oracle primal complexity erm state one time duality gap error primal induce apply perform dual mapping yield proximal section geometric contraction possibly immediately time un give erm within immediately oracle take yield aside effect namely prove relate minimum convexity immediately contraction every multiplicative prove oracle accelerate minimization factor compare yield accelerate run run part oracle require fix constant primal py x central regard accelerate erm p primal erm accelerate section erm remark similar result
hide update parallel ps visible accord visible hidden versa step update full gibbs long step collect calculate average square gs bethe gs indicate accurate instance grow strength reasonable operating cavity remove node function gibbs reliable replace mean gradient likelihood numerical inference locally visible reach expect bring insight understand rbm role work science institute study brain education technology restrict boltzmann function restrict boltzmann advanced bethe theory evaluate free gradient compare expensive boltzmann rbm deep belief able learn internal representation structured rbm biology activity rbm hard log every accomplish tradeoff require careful way evaluate log likelihood cavity bethe efficient distribute rbm remarkable efficiency confirm likelihood visible layer layer layer visible external connect visible hide visible assume matrix identically layer distribution mean denote ratio hide visible node fig complexity become advanced certain simulation propose bethe original fig boltzmann cavity visible contribution follow consistent equation normalization neighbor factor denote neighbor except visible auxiliary quantity represent product correlation bethe stability due summation iw nearly imply iw b I recursive message eq dependency drop cavity understand node cavity pass pass approximation factorize near neighbor bethe approximation bethe free express e I pass rbm basically densely typical precisely initialize cavity factor iterate converge prescribe time amenable cm free apply confirm result gradient log analysis also equation rbms strength vary display size theoretical enumeration external step single energy bethe fast reach also apart cavity evolution iw instability spin bethe inconsistent single fig step maximal become
computation poisson intensity hyperparameter iteration coverage correct compare interval credible hierarchical model uninformative interval percentile bias correction correction zero bootstrap quantile bootstrap bias obtain resample iteratively correct medium replication correct close nominal coverage minor coverage near behavior bad interval vary bootstrap coverage basic correct percentile fail near coverage compare correct correction iteration credible percentile basic interval number coverage horizontal gain correction simply percentile interval bias iteration improve increase figure prefer produce interval correct suffer bootstrap without resample play role uncertainty coverage pattern situation effectively attain nominal average interval peak difficult correct close nominal coverage part spectrum estimate bias interval somewhat online supplement small number illustrate unfold real large invariant publish decay show histogram count unfold mass intensity tail bias cover peak whole spectrum particularly somewhat physics unfold involve quantification correction technique way hyperparameter performance appeal choose strength cross discrepancy well require specification method beyond unfold frequentist quantification correction correction crucial frequentist base resample credible raise interesting correct govern iteration appear able coverage expense increase interval iteration might optimize within nominal supplement evident interval near boundary space improve quality confidence usually systematic fact parametric unknown generally however may need uncertainty incorporate loop confidence finally point find situation unfold reconstruction smoothness appropriate instance true intensity contain sharp rapid biased correction sufficiently penalty vary second consideration highlight inference family interpret mind acknowledgement thank discussion ed physics unfold estimate elementary resolution consist observe unfold proceed one intensity frequentist solution propose form regularization strength marginal observe credible interval bootstrap achieve frequentist problem inherent introduce iteratively correct confidence enable achieve frequentist coverage methodology apply experiment regularization em call unfold arise european organization research world powerful property produce energy detector vast produce order law physics pose challenge unfold detector detector production angle induce detector version observe produce physics unfold study momentum production challenge unfold ill sense space trivial exhibit solution two denote unfold intensity relate via detector hand inference intensity technique valid unfold account rarely close challenge unfold use maximization smoothed consecutive em smoothed stop analysis variant regularization terminology somewhat svd call account effect incorrectly multiplicative correction recently main iteration physical interpretation regularization stop svd unfold nature enforce positivity solution strength quantification handle standard heuristic worst simply quantify uncertainty confidence propagation little know coverage aim satisfactory regularization frequentist quantification correction bootstrap interval unfold properly take positivity spectrum impose curvature physical helpful unfold separate subproblem point quantification construct frequentist pose discrepancy technique context alternative bayes advantage literature statistical inference process empirical follow use form point like variability energy frequentist statement preferred interval good frequentist coverage challenge otherwise ill pose simulation credible propose employ variability correct estimate interval remarkably length explain unfold unfold process explain detail simulation study world analysis consist unfold invariant close conclude reader supplement technical quantity detector corrupt detector section supplement observe energy follow always additive furthermore sophisticated one usually close analysis detector simulation measurement detector point unfold measurement analysis analyse know phenomenon particle obtain unfold discovery analysis unfold play indirect attempt discover need arise purpose direct compare measurement distribution monte generators spectra spectra published directly compare might measurement sometimes since alternatively detector prediction could reveal exist document server four paper make unfold unfold use physics bin bin em form penalization expect similar mechanism model e compact interval random process intensity fs mb nb poisson set detector spectrum compact negative intensity two relate bounded kernel reality associate unfold process problem pose case operator na I unstable fluctuation understand unfold denote dirac identically sx detector thing happen limit efficiency mathematically indicator z pz intensity point constitute poisson whose identify notice unfold deconvolution unfold formalize methodology discretize histogram intensity model spline b function sampler carlo quantification percentile correct pointwise enable principled unfold uncertainty quantification detail argue technique natural inverse observable discretize first observable histogram application discrete detector reason analyse million observe treating would partition denote see histogram employ follow function basis denote unfold reduce spectra high energy physics function spline attractive whose restriction interval degree interior interior knot freedom order cubic spline consist polynomial continuously order give spline spline conceptual simplicity unfold literature go toolbox spline recursive order detail negativity spline negative negativity note condition positivity spline impose restrict subset spline spline bayesian regression scale decide bayesian reason mass plausible way empirical bayes explain truncate smoothness I interpretation curvature hyperparameter smoother enforce boundary result improper depend orientation posterior undesirable bayes require furthermore boundary near condition smoothness penalty introduce matrix hyperparameter plug obtain bayes posterior point j available intractable hence resort markov sample computed mean monte unfortunately elementary sampler unfold full conditional belong family proposal different scale able efficiently hasting denote sampler posterior conditional sampler tractable density full truncation negative negative replace tail detail supplement provide mix inverse attractive admit strength marginal introduction bayes appear bayes hyperparameter hyperparameter maximizer respect trivial close use integration monte one question rough issue use maximization poisson inverse originally image reconstruction later receive little unfold read complete log step unknown spline conditional hyperparameter depend hyperparameter step guarantee coincide em enable hyperparameter integral monte expectation monte hasting sampler em call monte monotonicity property iteration reach maximizer clear iteration summarize find hyperparameter iterate compute intuitive summarize understand tune vary match sample become closed take normalization hence q constant plug iteration size mcmc hasting result summarize start chain step devise rule mcmc large extent fully bayes unknown allow joint mean compare hierarchical bayes gamma parameter convenient conjugate full single component metropolis loop metropolis correct acceptance attractive unfold base analyst belief typical physics consensus unlikely especially quantity regularization uninformative specification uninformative unfortunately vary limited amount highlight major issue pose frequentist quantification nonparametric inference generally problem see chapter approach bootstrappe various construction bayesian interval argue coverage provide guarantee coverage interval significant demonstrate building intensity estimator problem major bootstrap overcome issue directly variability construct confidence first correct interval similarity quantification de problem methodology seem reduce bias confidence ill pose iterative interval improve procedure indicate stop correction attain nominal coverage interval probability close nominal residual length phenomenon bias bias bootstrap point context model previously employ ill pose nonparametric bootstrap g replace word bias version obtain use estimate correct estimator ignore reason replace less bias naturally I replace version positivity constraint reason value resample hyperparameter correction bias single metropolis compute element correct spline coefficient correct intensity variability basic bootstrap probe band fs property fs fs e pointwise inference point multiple issue elsewhere generate maximum observation r reason obtain bias correct intensity form interval deviation form standard suffer due skewness induce positivity intensity interval formal justification approximately sampling section first demonstrate unfold use intensity intensity fs refer sample interval noise unit discard far deconvolution error discretize bin uniformly place ill pose boundary computer setup core ghz intel processor outer core exception iteration parameter sample sampler start negative least square
logic interpret logic infinite logic truth extend truth enable concept completely false proposition sensor high protein capture represent naturally scale actual sensor level continuous valuable many entirely extension require conjunction operator correspond boolean logic operators max eq mrf variable logical map identical relaxed logic subsection relaxation convex program reason generalize mrfs kind probabilistic mrfs preserve scalable modeling additionally rich dependency mrfs variable semantic variable generalize interpretation explore round represent naturally degree ranking describe mrfs unify objective several semantic eq examine least occur also observe explicitly unweighted far relax easily logical clause linear distance define clause discuss domain satisfy higher penalize relaxed relaxed constraint satisfied tool enable exclusive possibility background restrict feasible aggregate far include value equality introduce mrfs useful treat relaxed constraint mrfs component region linearly hyperplane loss useful section thing piecewise make winner preferable highly reduce weight example follow optimization optimizer term ambiguity objective potential probabilistic preferable reflect smoothly objective term hinge loss optimizer influence function intuitively presence require exclusive strength two exclusive possibility could many prediction specialize similarity optimizer include evidence support square relative optimizer informative complete inference either hinge loss mrfs choice potential map subsection state convenience definition fully empty domain vector constraint index denote constraint give nonnegative free loss energy mrfs place input constrain energy condition probability explore mrfs wide range structure problem mrf rich purpose programming language soft mrfs easily define datum repeat include strength tie people network closure predict exactly define across repeat dependency template template define abstract constraint single dependency template model template random template introduce model template program mrfs parameterize provide interface hinge template logical rule arithmetic mapping clause hinge provide additional wide hinge potential constraint essential support many mrfs setting development kind mrfs convenient definition potential many name letter zero digit program universe must string universe program element universe constant denote person person person program string double character encode within constant constant represent constant logical predicate refer either predicate name every name example friend relate string entity attribute subject predicate combine create atom atom call ground atom reasoning unknown interest take atom whether friend ground atom stand substitute type state template hinge potential induce mrf predicate predicate atom observe predicate unobserve atom atom maps either unobserve valid map redundant sense ultimately specific different scenario agnostic define specify base universe universe eq universe six four type predicate include type close predicate relationship open predicate final observation value atom could value default value remain text formal annotate tag atom type example atom list map map specific rule mrfs induce mrf atom include variable define hinge mrf rest rule logical atom atom use refers minus atom value logical rule logical unweighted logical annotated express logical boolean logic operator de law implication rewrite body head therefore implication body head valid kind logical weighted rule template potential satisfy logical rule end logical induce unweighted logical template logical induce enforce note emphasize weighted rule atom agree atom map produce contain rule interpret induce mrf without loss generality contain replace ground unify objective ground set ground likewise map logical rule annotate potential mrf annotate add mrf unweighted constraint hard consider q atom interpret hinge logical efficiently mrfs inference objective logical basis objective mrfs potential rule arithmetic template logical unweighted arithmetic arithmetic logical ground arithmetic define atom example substitution define arithmetic flexible atom sum atom augment term augment substituting constant atom augment dependency arithmetic possible variable select logical arithmetic predicate constant sum statement restrict corresponding statement evaluate statement affect precede clause ground treat imagine restrict summation arithmetic constant satisfy property select constant arithmetic coefficient piece count without arithmetic atom enable rule depend piece coefficient build coefficient implementation include distinguish take scalar rule maximum far define template loss arithmetic template weight arithmetic instead hinge arithmetic define potential completeness definition equality relate combination augment arithmetic annotated arithmetic rule unweighted atom rule replace agree atom consistently map appropriate possibly statement coefficient arithmetic input arithmetic unweighted ground add instead arithmetic rule unweighted add set index add arithmetic include rule weight equality potential include arithmetic rule annotate induce flexible language usage come many predict among predicate background entity exactly arithmetic express say functional alternatively imagine task predict relationship student one write finally imagine aligned person person align rule say predicate additional argument range incorporate predicate predicate multiplier copy potential mrf potential convex hinge could solve interior expensive admm update find quickly optimizer region region optimize optimizer correct region check replace optimizer optimize case optimizer gradient vector optimizer solution trivially solve inspection definite via cholesky potential often share structure perhaps cholesky among potential optimizer modify fact modify modified objective problem reduce projection operation subproblem easy equality inequality feasible define constraint optimizer optimizer local copy lagrange multiplier local correspond lagrange q likewise mrf em initialize copy appear initialize converge em lagrange multipli copy iteratively update lagrange multiplier local copy subproblem local depend iteration update back subproblem parallelization fast scalable one interesting mrfs map subset must potential map state amount time identify map perform potential repeat section method mrfs maximize find discriminate truth shared potential template mrfs mrf template template associate weight partition template potential template shorthand potential template mrf th hinge loss weight template energy learn mrfs maximize derivative log perceptron direction average point outside region project back smooth ascent divide th template intractable current variant likelihood intractable condition I occur derivative integral admit expectation interval accurate linear group e set block sample uniformly represent mutual label drop interpretation view mrf prediction large shift produce accurate model produce accurate map task structure describe large margin cutting intuition behind truth alternate continuous margin disagreement expect separable relax slack large user rescale infinite follow subject grow iteratively add update subject plane full objective infinite inactive bad separation oracle augment inference potential mrf truth loss simply augment mrf violate standard augment mrf ground interior base loss oracle non interior ground local optimum since concave portion loss objective ground current great round flip variable state correspond svm objective iteratively invoke separation oracle find violate violate solution add repeat one margin slack mrfs square potential potential always distance ground case slack quadratic margin intelligence interest predict inductive programming structural logic several many dependency correlation implication compactly specify proposition adapt domain inference proposition consistent limited uncertainty approach hold dependency rare broad research probabilistic model distribution relationship explicitly allow compactly parameterize represent assignment random describe compact benefit probabilistic operate conditionally piece wide mrfs bayesian construct design usually researcher approach first relational relational schema schema dependency template use server query schema logic boolean mrfs logical template set proposition whether potential ground satisfied proposition value energy probable clause potential unweighted way similar boolean mrfs discrete information relationship mrfs mrfs basis field structure generalize multiclass task appropriate score true structure structure energy model mrfs connect structure show mrfs objective train train structure incorporate regularize broad class structure admit cut plane learn terminate size large margin violate accomplished function often equally challenging structure distance view mrf search structure np focus approximation tractable technique inference view optimization potential indicator certain polytope marginal polytope local consistency relaxation relaxed marginal potential state sense sum quickly predictor research highly message approach dd solve descent dd solve pass solve optimization iteratively nearby relaxed solution allow fast another primal objective optimize binary mrfs support example analog ad use admm optimize objective approximate programming relaxation enforce individual order consistency technique relaxation mrf configuration particular explore relaxation previous relaxation relax graphical bipartite potential condition nevertheless case useful criterion admit researcher relaxation local code also convex relaxation programming therein search discrete dd formulate recent programming discrete mixed programming relaxation inspire relaxation guarantee solution mrfs already effectively domain include vision drug natural language traffic model user prediction easily mrfs discover medium student massive open communication researcher mrfs develop implementation mrfs graphical model boolean logic fuzzy logic capture relax boolean fuzzy model mrfs allow programming mrfs mrfs refine algorithm set tool enable practitioner large accurate model relational acknowledgement would people suggestion development mrfs grant contract pc reproduce purpose annotation view conclusion contain herein necessarily policy imply appendix objective equivalent unique begin hierarchical polytope inner program mrf fix program objective mrfs clause potential constraint redundant simplify analysis make give deriving reasoning maximizer guarantee parameter fully sense potential simplex show fully constraint summing component bound imply complete parameterize fully via tucker kkt maximization kkt necessary optimum write kkt reason variable relevant kkt value constraint simplex also kkt constraint resp fully lemma reason prove lemma imply since constraint exclude convenience fully value trivially yield case definition multiply complete equivalence logical clause nonnegative max vice versa optimization max relaxation definition ex mm hinge markov mrfs suited prediction derive mrfs generalize three scalable consistency field reason fuzzy logic mrfs logic language mrfs pass exact mrfs well algorithm domain well model jointly example biological web vision relational inductive logic seek ever grow capture rich rely combinatorial mean approach scale graphical scalable rich call mrfs mrfs mrfs proportional functions discrete mrfs mrfs continuous variable lose mrfs useful discrete class trade complex connectivity dependency computationally challenging learn mrfs address crucial hinge admit highly scalable without restriction connectivity wide range useful relationship expressive technique structure three approach scalable inference randomize markov field reason continuous mrfs generalize unified inference hinge feature mrfs generalize reasoning relational logical retain language mrfs relational explore class markov discrete mrfs dependency dependency model build hinge logical clause probabilistic enable tool aggregate mrfs applicable probable assignment unobserved variable mrfs leverage map mrfs show decompose method multiplier subproblem loss potential mrfs easily million potential novel run mrfs overlap constraint show mrfs margin margin rely inference estimation mrfs excellent mrfs core collective mrfs offer accuracy comparable dramatically organize follow structured logical clause approach mrfs precise extremely scalable pass inference discuss wide range relate useful tool dependencie domain usually noisy address logical incorporate task one logic probabilistic mrfs popular class graphical rich structured informally mrf logical clause potential mrfs assign configuration mrf behaves assign configuration potential prediction logic excellent formalism define relational logic clause variable index logical clause form expressive equivalently structured clause express dependency condition imply
runtime modify stop probability still maintain true finding contain number operation event tree algorithm different precisely runtime implement seem transform signal advance translate estimation sparse covariance typically whereas nonetheless r sub sparse corresponding sparse albeit say gaussian every sufficient condition assume ct entry significant vice versa probabilistic strong large replace tending find operation leave number sparse entry nn direct runtime lsh library use select diagonal definite increase chose space complexity algorithm store exceed memory simultaneously process coarse modification theoretical particularly reduce space probability dimension increase use small large entry value lead discovery large lsh hash pick number whereas table large runtime various surprisingly opposite sign runtime factor tree slow runtime search number lsh design cope lsh set small thus increase low dimension direct clearly preferable slope direct clearly large slope application artificial decide previous examine ignore effort spend process use average query runtime false choose way inner runtime simulation success single lsh lsh projections lsh reduce unchanged query lsh lsh nn direct scenario near query unlike tune lsh preferable direct dimension importantly runtime query operation sub calculation lsh operation constant logarithmic aspect detect give either requirement underlie demonstrate one raise question future oracle give precise low raise theoretical consider g could lead algorithm require efficiently parameter datum definition q output satisfy union bind conclude proof detect q thresholde entry fail contradict step calculate operation algorithm step exceed query multiply number conclude alternative eq respectively concentration derive alarm divide node disjoint set say visit execution algorithm tree visit fast avoid process tree thus suffice w visit th since sparse therefore suffice occur enter skip denote complementary readily obtain prove absolute consider variable non positive absolute reduce center sum follow proof coefficient sub gaussian let matrix eq since entry absolute probability show sparsity sr ts every either condition gap union bind argument instead imply acknowledgment discussion ram gray reference intel intelligence ci computer science matrix fundamental analysis estimate computation slow population approximately raise question assume approximate detect much theoretically large entry sample operation sufficient approximate real value random covariance denote operation computation storage modern computer several however approximately whereby small even precisely array interested possibly detect location compute randomize sub quadratic reduction fast fourier task multiple call recently simpler prove sparsity furthermore p operation suitable normalization datum sample whose section assumption absolute value sample thresholded statistic study statistical main motivation estimate assume threshold slowly various concerned effort thresholde mainly approximately reality may population approximately provide p approximate empirically entry addition include near potential corpus represent fast multiplication currently fast multiplication square complexity hence expand matrix good knowledge nonetheless work relate problem goal inner product retrieve generalize provide runtime dataset recently simple reduction exact search study decade reference example exact algorithm lsh suitable dimension lsh algorithm approximate approximate covariance perhaps matrix slow runtime guarantee empirically sub tree relate goal recover form partial closely task exact suitable mainly hash algorithm rapidly pair similarity uniformly distribute apart correlation detect correlate vector method sub exponent fast offer op tree weak correlation value entry entry read coordinate priori demand entry simultaneously succeed suffice invoke output row union index run row corollary run suffice inequality omit multiple ij potentially compactly prove appendix runtime guarantee accuracy row compact entry p li r runtime operation entry method detect entry assume dimensional level tree coarse fine subset simultaneously whether contain query way resolve query compute calculation require sample q estimate variance calculation operation construction subset form describe pre process construction suffice lead large row apply start check divide disjoint continue reduce lead efficiently construct full binary height start th vector tree require future denote node level consider
tree recursively bottom work use matrix child parent bottom compute reach root node training minimize propagation structure rnn employ parse parse dependency parse semantic role network direct field neural extension tackle relate translation right role history activation condition back propagation time vanish quickly propagation addition dependency input long enhance lstm idea maintain back vanish receive preserved compute output cell memory gate forget gate gate sigmoid output activation correspond element bias sigmoid activation see intuitively network gate decide output gate access forget gate rnn figure combine child parent child option store information later simplicity lstm two forget child lstm activation gate parent decide gate moreover forget gate child gate forget gate child store compute composition instance beneficial information pass higher gate forget gate cover main interestingly child contribute correspond gate forget gate happen rnns sentiment lstm rnn rnn composition lstm cover word neutral softmax see equation assign embedding leaf similarly leaf leaf node inner let dimension word matrix leaf inner size inner regularization sentiment representation training batch thank propagation method analyse rnn forward phase classification complexity backward gradient multiplication carry sentiment multiplication inner sentence assume unary branch node node inner approximately eq lstm rnn approximately lstm rnn times high difference take core modern sentence second evaluate sentence stanford sentiment sentiment label negative neutral phrase sentence standard splitting also sentence development testing support sentiment remove neutral lead sentence development give initialize train word symmetric n unit lstm test function softmax choose representation epoch network achieve high accuracy accuracy final grain binary task lstm rnn outperform rnn activation function activation seem rnn lstm rnn fine classification rnn fine grained often show work experiment lstm work sigmoid agree common research lstm rnn tensor convolutional neural deep neural rnn keep rnn multiplication composition purpose make deep rnns cnn cnn convolutional pooling handle length hierarchical convolutional lstm rnn achieve performance lstm rnn clearly bad task bad grain cnn lstm rnn focus rnn four model cnn word whereas word fact grain task binary task perform conclude embedding use dropout neuron dropout strong regularization co adapt share neural thank fine grain experiment boost inspire try embedding dropout work lstm might dropout corrupted training difficult pay embedding train hyper parameter experiment select get high development dash lstm perform par grain task taking outperform propose extend long memory lstm recurrent lstm rnn rnn lstm rnns lstm recurrent explain lstm behave pass filter focus lstm keep information region region composition could compression auto boost lstm work compare rnn give lstm embedding gain significant thank dropout lstm boost topic future acknowledgment thank comment propose neural memory allow tree store memory much vanishing allow
possibly estimate suffer high guarantee estimate exactly obtain nonlinearity approximation aforementioned end structure parameter call context address e maximize ml square property oppose decompose nonlinearity coefficient impulse make throughout connection propose give identification toolbox identification formulate modeling give experiment conclusion end output time relation static nonlinear measurable signal strictly stable describe impulse response output corrupt combination input convenience impulse large static know scale fact input output equally suggest introduce impulse gain e symbol indicate product bilinear property dimension denote give introduce result throughout paper lemma extend toeplitz construct dynamic mean vector contain constitute parameter kronecker formalize kronecker let uniquely vector column complete constitute regularize detail frobenius assume drawback square possibly despite suffer large see square error estimate equivalently gaussian vector property subsection design system incorporate first zero smooth typical spline multiply impulse response hyperparameter redundant working case kernel kronecker kernel rank gaussian highlight estimate eq determine step solve establish decompose estimate next become scheme vector review system base measurement relation spline description parameter estimate via maximization solve square estimate next strong produce kernel decompose kronecker product dimensional thus vector equivalence recall bilinear kronecker prove measurement marginal lemma ml prove ml equivalently eq recall bilinear product recall q since statement follow estimate detailed impulse response nonlinearity procedure ht identification solve decompose consist carlo run accord follow pick uniform nonlinearity coefficient depend different snr c cccc snr impulse follow paper problem initialize sample variance implement least review working condition square linear detail well know accuracy index impulse generate experiment static nonlinear op ratio snr estimator identify snr estimate version rank l op need estimate method nonparametric order system particular start spline kernel novel regularize impulse
inform support expensive base incoming message consider message ep infer variable px mf I parametric g family approximate ep f ix x xx I know cavity projection satisfy factor message typically factor complicated integral defining become intractable quadrature numerical integration technique learn message signature v f numerator inference cast message parameter moment importance send offline inference need firstly assume factor conditional sample incoming message rough incoming message ep need sufficiently cover relevant incoming message distributional input assumption message characterize dimensional hence simplify regression contain characterize input treat vector regression allow parametrization operator message analytically opt simplicity online learning inference rich support consider guarantee computing expectation output uv seek regression square loss I define enter straightforwardly message appeal set choose make expensive unseen point eliminate fourier v back nothing generate incoming message study distribution rkh incoming message lr ls random expectation cr dl define distribution lr expect product preliminary experiment logistic use regression incoming message fx operator choose truth message operator output expectation moment kl kl number choose see improvement ridge incoming message convert compute multiplying virtue derive active inference variance similar incoming message importance otherwise message efficiently
entropy fit author average entropy plot see slight benchmark evaluation identical solely ph well link lda cluster remain lda tie mention marginally representation turn surprising cluster cluster cluster leverage leverage learn author show truly representative community understand examine recall reader set agnostic intersection hold user benchmark select another document great us document ph co similarity select another document possess tuple summarize lda low ph co th extreme high high optimizing within cluster explain cluster document rather document belong author prevent community present augment blockmodel item community community membership content fit real accuracy distinguish world community give nsf nsf fa provide helpful david liu community infer community latent item ignore item item tendency cluster share content augment datum enhance community item state arise interact article highly respect metric representative pattern motivate scientific document generative document associate scalar indicate membership stochastic blockmodel interaction document application motivate develop fine grain categorization paper currently fine grained daily newly paper within preference visit website date physics researcher discover people entity interact interact document video form bipartite kind interaction document ignore community tend differently community tend interact document community interact frequently bipartite alone add document cluster observable occur argue community argue hold paper interact tend article preferred community community community interact model co provide document distinguish interaction observable model distinguished comment co advantage access match community distinguish co cluster document refine co advantage cluster address grow interest item content recommender community discovery user detail deal user potentially encode feedback define relevant item belong community user belong membership cluster membership paper completely find explicitly model assume encode sample community follow thing assume blockmodel without notion content item represent trait occur force item vector item choose place thick black gamma gamma right beta w edge connect connect edge connect connect w edge connect c expectation j xy xy describe stochastic blockmodel follow variational dependency grow cluster community document link consider author jointly text allocation mixed stochastic relatively response link draw content article powerful attribute study node node attribute vertex closeness distance introduce initially popularity assume describe node community membership depend popularity membership belong specific node technique variable associate parameterized kl topic choose co category high energy physics user visit website date category vast representative frequent primary validate benchmark lda model allocation lda mixed blockmodel learns satisfying structure lda benchmark treat user cluster al generative preference learn preference vector document recommendation rating document draw eq offset recommendation fit go benchmark et propose link formation depend extend incorporate content require assign content model benchmark modify without another article train vector text pdfs article character appear vocabulary cluster article train ph th article choose benchmark proper increase qualitative evaluation conference proceeding conference conference practitioner research paper present paper author link dataset study conference represent field cluster evaluate form cloud frequently occur form scalar proportion paper belong cloud scheme popular across cluster frequently occur th limit appear display clearly mathematical community separation relax real website complete well purely retrieve research community evaluation ph ga create category go ph ga ph ga paper discuss galaxy paper discuss galaxy ph enforce interpretation galaxy ph ph ga communities interested differ paper separate old paper ph ph ph ga item content paper ph co compare clustering truth ph co ga truth manually naive
essentially require monte carlo rapidly range technical review appear comprehensive survey currently journal area sufficiently finance integrated practitioner reason carlo option sde section motivation performance build conclude brief sde drift motion euler compute approximation stepsize increment sde roughly convergence mean result impose sde standard satisfy constant eq euclidean appropriate euler order absolute euler understood borel small argue relevant sde important make convergence remark sde complete sde financial growth infinity also unbounded occur interest positive constant break nonlinear sde euler requirement fix concrete wish position independently answer would euler path monte average overall error independent depend accurately method approximate sde path discretization bias arise sde remain solution decrease reduce stepsize describe weak behave width arrive like word scale random generator either per proportional hence argue euler like replace argument scale like establish extra place achieve euler carlo computational suppose exact expression sde apply numerical interval evaluation overall quickly sde curve infinite truncate affect later fine detail build resolution fine impact picture wiener six show monte require box sde approximately box euler comes turn sample long exploit obtain quickly large path low frequency path might idea next focus wish mind represent asset risk neutral european style option european call exercise simplicity argument payoff satisfie lipschitz cover different discretization level level stepsize precise simplicity think limit index large stepsize covering interval one refined euler apply sde variable linearity operator carlo expansion hand side indirect hand think widely usual path q illustrate general brownian suitably construct variable knowledge nearby tractable nature sde knowledge explain author sde similarity geometrically refined grid grid resolve important mind distinct notion pass refinement cycle early monte sample via discretization type analysis summarize constant p estimator bind estimator replace euler european style option make achievable formalize confirm euler fail weak asymptotic closely relate direct whether combination euler carlo sure euler event euler rare extremely unlikely gaussian sde euler convergence analysis indicate improvement confirm potential practice matlab code http people ac uk asset geometric european digital price payoff digital option event code monte carlo request picture path level decrease added right indicate term precisely dash line cost predict picture equivalent computation large grow give digital version see monte carlo htb output carlo code site show number target scale call option digital summarize advance relevant option comprehensive maintain http people uk source date coarse refined payoff logic behind euler path close refined payoff lipschitz european put option must refine european option option value depend option problematic sde path payoff close asset change barrier option sensitive barrier logical must able path exception case european call put accept slight digital option consider barrier digital option far option option option barrier option american options monte common variance integration conditional produce estimator quasi carlo low replace outperform finally methodology extend asset brownian motion aim explain manner idea behind base financial monte widely heart rely tight sufficiently implement straightforwardly gain circumstance approach specific developing scenario carlo
review publish gaussian noise represent error summarize satisfy function follow spectral eq j modify expansion formulae k integer z logarithm derivative eq q euler therefore g g l polynomial constitute orthogonal hilbert either admits decrease open condition unknown belong sense say nd satisfied parametric eq weakly limit dependent error asymptotic obtain certain range define rewrite include consideration condition formulate spectral measure gaussian singular spectrum asymptotic vector weakly multidimensional b ccc kb kb b assumption limit obtain class q nonlinear regression q g behavior play crucial role distribution properly coincide b capacity polynomial case random stationary process seem weakly consistent sense seem concern asymptotic normality observation normality simulate value generation quantile chi degree number square mahalanobis plot graph distribution multivariate apply simulated set compose replication note rate need increase verify suggest estimator discretization step numerically matlab function base graph contour form equation ie constant contour study inside plot represent red dot ellipsoid tt right bottom case tt bottom h right discretization b leave top bottom h case h subsection aim normality perform section test plot combination contour plot subsection figure consider b h h tt discretization case b top case bottom tt discretization bottom tt discretization size case h bottom right gaussian harmonic constitute parameter check limit result prove confirm simulation validity spectra regression convergence differs include partially european ga research grant dp corollary theorem theorem european ga partially grant dp harmonic address property numerical prove consistency asymptotic parameter prove two long science harmonic technique formal treatment matrix formulate assume zero count continuous nonlinear regression independent weakly therein nonlinear slowly dependence obtain volume present asymptotic distribution class estimate nonlinear estimation study consistency zero transformation process process complete follow
reconstruction application denoise noisy diffusion recover smooth relate diffusion relate retain embed noisy clean coordinate mapping encoder decoder recover variation recover low new autoencoder clean version point provide denoise assumption addition add clean clean smooth representation implementation net autoencoder procedure neural train unsupervised input previous initialize yield poor deep quantity little employ gain large beneficial encourage unit hide corrupted set autoencoder average divergence mass impose unit specific limited bfgs bfgs line matlab package deep bias layer normal zero mean regularization experimental examine new constraint mini typically dataset show gradient entail multiply problematic mini point local influence eigenvector extract subset mini batch output walk manifold constraint unnecessary mini batch differ deep imagenet cifar etc perform offline typically nystr om nm retain embed train matrix necessary thus train nystr om geometric save embedding training matrix additional application memory approximate laplacian derivation equip bi mean smooth compact equip locally laplacian extension combination eigenfunction exist real demonstrate autoencoder constraint especially noisy finally outli demonstrate verify agree training examine add eigenvector encoder toy effect network layer point diffusion point th great train datum average encoder mse single calculate mse encoder hide row output add know compactly represent layer number unit hide reconstruction embed add hidden improved serve noise decrease encoder trivial minimizing impose constraint yield twice high noisy run vary train layer unit average mse realization several consistently noise include proper estimating diffusion result perform dependence unit give clean I autoencoder denoise capability decrease order fit encoder autoencoder previously diffusion train autoencoder encoder layer decoder fig denoise capability diffusion std example fit scenario autoencoder outlier state necessarily apply distinguish display image acquire sized center image separately patch image sized training dimensionality training set autoencoder image easily capture main structure datum differ diffusion autoencoder reconstruct mapping autoencoder embed properly b point extension point embed due true embed decompose geometric eigenvector similar rotation experiment sec heat equation thus solution pde eigenvector eigenfunction eigenvector eigenvalue take curve diffusion embed rotation embedding share calculate rotation calculate error realization value embed encoder solid encoder much new deep manifold propose designing encoder vice propose training encoder preserve locality embed demonstrate encoder enable efficient linear embedding decoder enable stack together deep autoencoder enable denoise autoencoder properly represents present noisy scenario demonstrate focus perform training processing manifold embed dictionary lead good datum evolve develop instead regular add layer tune datum autoencoder remain recover embed work affect harmonic constraint enable minimal surface encoder decoder decoder decoder provide theoretical decoder expand net average texture synthesis examine affect decoder research examine determine number maximal need system addition explore deep neural implicitly incorporate manifold embed support foundation support award dms author thank suggestion rely bound n let suffice moment fourier rely proposition let limited band bx dx need intermediate address claim fx dx n let transform clearly band mean bx l ball show kx bx bx dx f notation manifold learning enable sample datum embed decoder stacking encoder autoencoder net extension detection net constraint preserve prove encoder also storage space world yet embed ambient reveal year learn develop geometry within neighborhood include kernel laplacian maps affinity capture representation noise outlier capture structure unlike reduction world lie hyperplane preserve apart mining processing compute entire complexity eigen embed extend new representative common approach processing application deep learn gain popularity achieving handle deep net increasingly abstract perturbation representation globally without incorporate geometry laplacian autoencoder locality preserve affinity pre autoencoder formulation reconstruction regularizer embed ensure representation paper approach apply incorporate manifold embed address embed image datum outli representation insufficient three goal encoder layer approximate map perform train inverse decoder recover point stack two term diffusion diffusion outli encoder due outli diffusion denoise reconstruct clean memory unnecessary retain efficiency enable quantity organize provide manifold learn deep neural propose learning enable pre proof encoder present discuss map technique various signal review diffusion see graph point connect measure similarity negative radial use neighbor point neighbor point create normalize set view transition eigen eigenvector eigenvalue calculate diffusion two stationary depend spectrum approximated equation set set general diffusion propose extend new point example create harmonic eigenfunction analytically nystr give extend upon nystr om treat instability due extend eigenvector significant scale dependent eigenvector adapt complexity eigenvector separately implicitly cross validation avoid minimize distance embed embed respect covariance incorporate property datum determined perform necessary memory affinity euclidean complexity add enable network typically layer layer feed cycle loop layer net layer densely computed mapping layer term linear element denote net successfully etc achieving result task minimize supervised predict consist weight prevent layer weight regression multi square q penalty net train variant stochastic minimize loss weight compute backpropagation start output learn use manner autoencoder autoencoder encoder decoder stacking encoder autoencoder train minimize reconstruction try unit dimension tune output autoencoder autoencoder autoencoder classification denoise image lie smooth compact embed calculate euclidean structure address problem detection purpose extension extension pre calculate back calculation datum aim new newly good nystr om scale kernel x near prediction affinity evaluate zero work however uninformative lead mistake indicate outli net encoder decoder deep autoencoder instead autoencoder middle see datum embed output layer layer map manifold output impose constraint preserve decoder inverse back pre stack decoder obtain compute diffusion autoencoder denoise present ht diffusion initialize encoder pre stack parameter tune new initialize decoder stack optimize network tuning cost calculate image point decoder ht stack decoder alg encoder alg autoencoder reconstruction autoencoder outli score learn encoder
fmri particularly create contrast pearson correlation adapt brain theory mutual information pairwise information measure benefit stable test cluster stability cluster area prior prior covariance choice one rule first priori sort covariance allow consistency covariance hierarchy wishart derive prior integrate result turn inverse wishart last simplicity family prior advantage marginalization wishart freedom strategy freedom still correlation identity since correspond uniform correlation admit content inverse prior simultaneously seem variance ideally separate variance likelihood could issue contrast algorithm perspective influence behavior outperform exist method variable mutually independent block one solve difference stem inverse wishart wishart difference relate covariance expect base confirm behave automate stopping yield arguably intuitively structure normal exist start consider rigorously normal grid support de support study case normal mutual usual use matrix e suffer systematic bias additive chi freedom kullback besides I eigenvalue mean determinant mutual lead mutual see mutual information value mutual favor merge marginal correlation systematically give likelihood optimization contain equation equivalent e fmri liu publicly project fmri tr analysis motion estimate time dataset inter median fmri dl transform template body fmri functional nuisance parameter voxel basis hz pass white well principal six body parameter fmri volume spatially mm isotropic condition time spatially brain include cluster apply subject exclude enough volume subject exclude fmri visual thus actually degree scale equation manuscript translate wishart z lead derive wishart matrix j proportional freedom paris universit paris france paris france paris department center op centre de universit mail fr measure agglomerative hierarchical clustering raise correction merging multivariate procedure naturally empirical priori g automated scale measure dimensionality log asymptotically additive dimensionality criterion mutual encouraging derive outperform well toy lead advantage automate fmri dataset identify establish mutual systematically hierarchical keyword cluster model mutual normalize mutual analysis vast variety agglomerative hierarchical sequentially cluster measure shape critical use distance popularity notably information shannon kullback cover tm feature interest univariate apply arbitrary dimensionality suffer dependent mutual normalize version mutual dimension normalize however correction paper normal measure compare covariance element admissible log bayes log marginal sum restriction express matrix automate cluster remain term local global asymptotically I sample enough explicit variable mutual dataset aim asymptotic dataset toy imaging detailed introduce framework namely likelihood assumption respectively section examine compare union decide whether compete model marginal restriction block respectively identically log ratio integral j dependence quantity parameter q calculation j tw lead normalization scale b yield n j wishart degree scale j hypothesis block block submatrix introduction j j I far sake I turn marginalization I inverse incorporate equation wishart degree incorporate equation equation yield quantify amount support multivariate distribution start define covariance right hand expand k k term sum use fact k n taylor expansion give plug multivariate see independence development aim e step read quantify marginal likelihood quantity similarity submatrix previously l expand turn degree diagonal compatible incorporate equation calculation correspond cluster unchanged consequence th obtain merge bayes c quantity nothing use successive degree low optimize cluster yield covariance freedom matrix yield distribution note equation involve advantage bic automatic fact belong consequence therefore likely pair mean probably stop correspond apply cluster automated stop behavior examine synthetic bayes method automatic comparison implement pearson correlation linkage method hierarchical mutual algorithms purpose block precision maximum penalization penalization sm version criterion transformation unconstraine base von either original partial correlation approach repetition mean cluster measure define namely sign absolute time perform code implementation mean simulation perform equal occurrence take normal sample wishart freedom rescale correlation matrix correlation j student equation length vary increment assess various thresholded quantify use proportion classification adjust rand fraction cluster correct rand pool number length multivariate bad index adjust rand percentile adjust rand percentile adjust rand smallest rand proportion classification definite consequence run algorithm operational simulation htbp c rand classifications htbp computational time adjust rand right summarize figure globally affect classify well never outperform method already publish automatic compare particular outperform variant prove simulation analyze conditional study investigate early diagnosis child various component g count statistic give spectral repetition result discard htbp summary correlation partial correlation clustering give table hierarchical started confirm behavior classify accordingly g cluster create variable partition variable cluster partitioning compose g yield cluster represent note identical bayes merge successive strongly bayes analysis lead follow favor independent belong g cluster step bottom row cluster part grey cluster favor hierarchical tool organization brain state fmri identifying brain region highly good simulation see brain aim establish cluster algorithm yield solution examine several literature similarity cluster subject rand rand index subject result capture similarity visually identify cluster rand large include generate rand index close large rand cluster split cluster correlation mutual similar method know plausible sp analogy variant mutual test generate solution decide cluster rand subject panel associate consensus brain panel weight visual brain order hierarchical examine cluster variant yield solution consensus ensemble individual consensus accumulation represent adjacency area across subject method element code brain cluster use criterion one consensus brain region cluster probability subject figure stability method output
early geometry processing shape albeit heuristic interpretation insight many improve compose map cycle attempt method patch manifold likely inaccurate due diffusion visualization score compute global shape bundle mean correspondence manifold analyze discretized bundle point goal global make analogy terminology flat induce flat bundle geometry bundle triplet datum union shall denote value product neighbor edge neighbor moreover also terminology pair block edge symmetric matrix since triplet graph also construction eq set distance decomposition eigen decomposition since eigenvector define length h eigenvector graph theory vector spectral block frobenius norm eq inner produce base data base affinity wise non closely map adjacency laplacian connection g negativity negativity eigenvalue allow power power theorem view addition base capable entry th segment could euclidean preserve automatically similarity simplicity call component bring common template close embed reconstruct interpolation map build correspondence map implicit sometimes version unit sphere laplacian define multiplicative map goal relate differential importance tangent riemannian extend build adopt notation sample bundle review support manifold accord shall equip riemannian tensor inner tangent q unless adopt summation product shall denote q connect curve compact positive geodesic define orientation preserve isometry tangent parameter w symmetric isometry definition standard rotation borel riemannian call define respect normalize asymptotic relative tangent riemannian tm tm tm constant bundle unit tangent proof sufficiently whereas theorem volume see modify geodesic induce metric still symmetric notation shall tangent contexts notation eq unit manifold constant depend appendix laplace project well define proposition tangent completely counterpart practice much bundle base make finite unit tangent sampling tangent tangent map acquire proof appendix strategy technical assumption dimensional riemannian two compactly derivative therefore automatically compactly derivative inverse polynomial demonstrate compactly datum satisfy step point projection respect recall difference euclidean use define probability practice show procedure good basis tangent suffice repeatedly point truly space manifold tangent characterize coordinate basis express approximate map pca coordinate change choice describe detail simultaneously throughout summarize near eq carry svd tangent plane singular singular arrange q decay take decomposition neighboring point schmidt norm minimization efficient namely eq basis coordinate explain parallel compose basis expansion summarize definition collection uniformly unit sd tangent unit column space q plane isometry uniformly projection isometry j therefore parallel q assumption suppose b realize sphere compare near cloud sample unit length unit circle collecting denote neighbor among neighbor choice explain sphere explicitly rotation finally normalize entry diagonal choose various observe experiment investigate influence ratio eigenvalue size laplacian approximate laplacian approximate manifold limit moreover coincide eigenvalue laplacian spectrum similar sampling sphere cloud discretization sampling tangent space tangent k bn construction noiseless non generalize obtain sampling b embedding diffusion similar specify pairwise point template euclidean embed take place illustrate unit bundle stand unit black choose unit tangent give rise discretization tangent bundle interpolation canonical connection vector close discrete extend interpolation neighbor geodesic segment construct close possible topology truly manifold interpret generate connection connectivity store mesh use fundamentally cloud manifold oppose structured triangular mesh formulae provide interpretation experiment follow coordinate eq index consequently definition depend differential operator coordinate jacobian invariance reason define volume respect volume element geodesic flow bundle tangent bundle manifold base manifold see tangent induce volume tangent shall work compact manifold constraint prefer compact non compact motivate unit tangent bundle compact notation riemannian manifold inner metric induce volume know geodesic flow arise horizontal extension extension flow start start write coordinate eq equation define parallel word isometry similarly construct convention everywhere otherwise compactly compact manifold radius parameter sufficiently operator eq moment operator scalar curvature put geodesic geodesic coordinate orthonormal basis eq recall normal tensor meanwhile read q thus symmetry domain kernel vanish argue explicitly characterize coordinate conclusion follow study riemannian geodesic coordinate parallel denote geodesic connect normal coordinate chart center field equivalently geodesic iteratively equality geodesic normal coordinate derivative expression q jt lemma high expansion geodesic hand meanwhile geodesic parametrization combine crucial computation expansion homogeneous coordinate drop armed ready start investigation incorporate parallel deal soon denote horizontal define consider geodesic around sufficiently geodesic radius center contain neighborhood geodesic support geodesic put th lead q geodesic taylor expand coordinate rest simply substitute taylor expansion integrate symmetry computation drop thank equality use geodesic coordinate bundle laplacian define note compatibility isometry naturally isometry carry prove assume dimensional horizontal spherical operator scalar curvature integral ball since orthonormal direct give eq expand numerator expansion denominator numerator conclude compose computation bundle version instead proof proof recall drop higher argue prove path lemma bridge manifold ambient close yx reason diffusion geodesic euclidean ambient diffusion quantity replace euclidean since construct euclidean exact parallel estimate parallel establish asymptotic version lemma compactly assume close manifold embed radius kernel euclidean integral operator associate constant moment laplace curvature fundamental form expand put geodesic coordinate neighborhood geodesic support accord high order geodesic normal coordinate taylor expand around eq note symmetry proof adopt volume denote fundamental sphere integrated place go affect conclusion specifically numerator still replace fact apply expansion pick I key establish last piece large step strategy point I generally directly tangent bundle number stand component explicitly next note manifold variable observation iterate limit leave generally iterate expectation expectation respectively density real value density respect probability denote purpose due bernstein individual summation stem independent come fp bn bn therefore last replace fix trivial notation like manner easily bound reduce compute various uniformly explicitly recall remark interested small note suffice moment moment create notation coordinate lemma side apply determined obtain remain plug expansion back scenario thus direct computation yield short bind constant interestingly term sense control bandwidth find positive q lead error determine notation shall merely dependency bound root q later rewrite q second noise result order accordance reflect accumulate grow linearly increase make unless one appropriately term equivalently q prevent instance pointwise case law shall estimate case replace would since like union interested long asymptotically heat laplacian eq constant small bound high specifically eq eq depend bound ensure probability establishe provide estimate adapt notation compact dimensional subspace minimizer orthonormal frobenius geodesic eq taylor notation large I fact f assumption theorem theorem introduce generalize map massive affinity diffusion vector geometry likewise consider scenario possess structure interest investigate tool study augment goal obtain nearby analyze tangent connection sub riemannian geometry family operator massive science challenging analyze understand datum wide inference mention interest direction laplacian base hessian alignment map diffusion image text etc abstract vertex similarity connect pair build dm interpretation appropriately smooth eigenvector precisely eigenvector preserve appropriate lead euclidean manifold estimate original opposed diffusion wide task precision robustness early instance construct random manifold translate reveal orientation associate algorithm eigen name analogously finite although embed much benefit incorporate tangent sign successful geometry dimensionality tangent utilize bundle notice origin eq isotropic embed blue beyond contrast incorporate tangent yield see sense algebraic geometry curve x dissimilarity score intersection belong distinct parametrization use methodology broad context geometric indeed structural typically encode circumstance detail vertex face mesh text collection shape desirable variation across collection shape score simplify similarity always clear similarity practical heuristic addition situation shape surface set persistent diagram direction laplacian analyzing carry huge degree freedom contain feature oppose characteristic g triangular persistent case admissible surface correspondence surface substantial miss mining generalize dm take different path scenario individual consideration manifold dm denote dimension augmentation around augment look like universal template manifold intuitively parametrization template sense compatible appropriate restriction compatibility datum picture family play role development geometry shall underlie adopt terminology geometry universal template manifold step transition occur either adjacent within bundle refer look application augment object also incorporate bundle formulation transition distinct certain directional impose analogy counterpart manifold lift walk base mild manifold name eigenvector differential new turn couple parameter informative partial bundle geometry experiment eigenvector though focus tangent study tangent bundle differ terminology describe characterize graph tangent shown conclude propose include geometry tangent implicit freedom reasonable ambient semi supervise supervised build label training high reduce simplify data manifold differential bundle manifold manifold parametrize manifold bundle consist class distinction bundle coordinate use act open neighborhood component element correspondence neighborhood base manifold state bundle approximately bundle object single assumption reduce manifold bundle especially bundle manifold piece intersect manifold bundle get trivial bundle restriction understand equally understanding become insufficient datum interest collection surface biological shape govern freedom learn manifold apply diffusion shape distance yet infer variation geometry individual shape away point add interpretability coordinate keep similar coordinate belong
normalize resort rejection sampling ram proposal arise sample proposal perhaps type reasonably behave target example unimodal chain long logarithm dependence proposal manifold extension cost provide nontrivial proposal rw proposal algorithm explicit discretization certain diffusion variance take scale translate inverse number require almost impractical limit target dimensional effective target design problem discretized refinement mesh towards independent introduce weighting proposal discretization sde go use adaptively posterior covariance space adaptively discretization langevin sde general amount elaborate world adjoint possibility avoid code valuable motivation algorithm present attempt combine well without arise discretization arise euler discretization diffusion infinity versus viewpoint former even herein preserve empirical past yield adaptive behave gaussian nonetheless gain quantity limited dimension much large become factor operation traditionally multiplication operation prevent high dimension algorithmic advance outline cholesky inversion immediately cost assume evaluate logarithm unnormalize feasible use low update acceleration algebra operation fundamental level benefit hardware stress significant hierarchy attain hardware operation limit fast negligible memory accelerate hardware gpu memory device thin express magnitude bandwidth gpu illustrated across gpu reduce slow operation increase memory bandwidth gpu processor logic nearly physical due frequency forward intel etc future value force parallelization traditional inherently nature nonetheless justify merge parallel chain within potential scale reduction diagnostic parallelization work work explore partition forward herein parallelization efficiency chain periodic slight strong black principle apply note elaborate parallelization recently tackle consume core available gpu operation provide optimize e hardware thank advanced sampler multi way operation propose great impact black rest precisely finally present diagnostic justification experiment acceleration extend multiple highlight probability indicates give number assumption correlation assume readily evaluate present notation algebra cause confusion know observation observation vary inverse problem formulate discretization limit measure mind big scenario statistic need imply explain dimensional space increase potential increase inverse observation parameter intrinsic property full informative space effective bayesian problem albeit connect expect markov transition dx qx follow notation qx dx analogy quantitie stochastic measure ergodicity tv x rate approximately function calculation identity time limit effective metropolis refine version perhaps amongst essentially arbitrarily compose accept follow satisfy give define many behavior kernel one subsequent turn large size metropolis hasting right computation mention subsection basic hasting indeed advanced black box mention sec finally present work begin increment motion euler discretization give walk rw n identity although bayesian maximizer turn symmetric distribution use start point posterior furthermore multiplication q notice preserve imply acceptance nothing discretization one extend extend introduce proposal reversible general proposal name derive incorporation inform proposal allow effective play role inform direction gradient present work upon proposal plug identifie propose step acceptance consider presentation substitute account eq I else may approximation case kullback independence target acceptance point non proposal coincide turn proposal trace proposal par target necessarily target additive performance although dimension exist investigation parallel serial proof rely nonetheless diagnostic chain covariance diagnostic justify chain chain batch interval follow within merged moment mention potential scale diagnostic chain merging start quantitie chain chain variance indicator systematic begin normal generate eigenvalue standard log fix eigenvalue dense matrix multiplication pde forward solver realistic simple shaped target exactly computable case consider diag I order correspond jacobian determinant change maximizer furthermore course two distribution spread target practice reduce highly covariance big target reduce clear target problem posterior example pde forward decay spectrum parameter big hard sample gaussian acceptance initially four autocorrelation measure rw panel autocorrelation function projection eigenvector eigenvalue subtle competitive although suffer target whose condition increase show algorithm perform eigen bottom perform worse expect give multiplication involve fast fourier argument indicate roughly eigenvalue inverse eigenvalue prior pre least turn outperform outperform outcome simulation aside specific length burn parameter affect proposal case run separately various varied reach convergence fig necessary show due operation case increase curve due effect average simulation converge multiple value target suffice experiment total effect cholesky update large operation consequence large memory discuss section illustrate convergence reduction factor describe require right hand stop chain relative covariance fall convergence latter euclidean gpu accelerate library hardware landscape x call cpu gpu thousand core computing capability magnitude peak compare standard speed gpu memory cpu bandwidth cpu specification challenge gpu order magnitude memory application technology e parallelism move communication reduce communication software stack hardware kernel available particular dense linear operation thank regularity three basic algebra e multiplication multiplication mostly bandwidth kernel thank kernel bottom software chain critical parallel architecture library library mkl library implement last cpu high inversion flow target x x ref x ref x x accept accept n compute batch update lag target evaluation weight every cholesky matrix inversion bottleneck increase operation therefore basically compose generation triangular operation perform general perform general matrix operation perform mostly compute symmetric factorization mostly compose gpu library implementation determine kernel need platform operation well symmetric dense multiplication highlight cholesky inversion compute intensive overall parallel frequently hand memory exhibit arithmetic lag investigation exist implementation operation library operation occur intel mkl library data movement cpu gpu slow try operate persistent gpu memory motion asynchronous overhead bridge library ensure parallelism thank parallel rely join parallel programming parallelism batch batch processing share facilitate safe synchronization load imbalance sophisticated parallelism another cpu intel library chain implement mkl mkl running chain core therefore critical various define cpu specification bandwidth stream benchmark
error statistic look small minimize realistic parameter skew high estimate distinct fold mid however optimistic frequency compare versus stream observe limited agree stream estimate platform counting frequency distribution cv frequency rather rely large body toolbox deterministic work quantile vector linearity imply processing frequency statistic design distribution support query effective skewed overhead relevance monotone statistic encoding update outperform particular sampling use sketch sample roughly obtain replacement sample replacement skewed repetition overhead sample query weight linear presence conclusion statistic analysis accurately frequency estimate compute bring distinct frequency look ahead framework extend boundarie attention use start partial function weight start get particular function claim sampling randomization use randomization assign element adjust observe randomization density threshold correctness change remain value determined value depend randomization preserve claim claim respect initial case density u du maintain count condition notion dominate distribution threshold dominate dominate fy dy ready upper cv build function distribution seed dominate th small exponential exponentially distribute condition small difference order seed exponential combination equal order parameter sample weight segment cv extend take seed whereas since consider weight population key perspective small seed dominate suffice upper bound expectation different square variance inverse estimate minimize unbiased estimator sum per surprisingly cv segment cv condition contribution dy inequality relation seed take zero analysis work seed exponential condition seed density seed key xy dy e decrease hold xy dy dy eq consider since substitute dominate small seed draw dominate latter ready inclusion key eq inequality relation substituting remain treat establish maximize size e recall already assumption denominator minimize substitute establish tw last ready conclude dominate obtain key showing segment proportion cv variance outline pass conditional tw tw coefficient bind tw eq theorem w inequality use dx tt subject w dx w e increase e te w maximize useful work set think cache drop threshold back queue depend rand return rand kx xx x threshold x design compute segment estimate stream construction store need key counter stream cv logarithmic present logarithmic apply stream string return score counter stream key stream stream key string string key thus counter would rough inherent counter weighted question extend objective sample feed program optimize inverse eq relation obtain substitute last var tw py exp title exp exp title x title thm thm example thm thm google ca usa pt pt hash seed diverse source web service ip traffic utilize express segment apply frequency key segment parameter statistic computation instead would include easily aggregate costly state active ideally without aggregation present pass stream two pass design classic provide decrease gold tight single stream utilize segment special define frequency active practice sample sketch approximate exist design stream general frequency estimate monotone benefit unified single service interaction service ip traffic key universe stream distribute storage aggregate consist active occur total uniform frequency key proportional frequency frequency query nonnegative segment population typically carry contribution less prominent moment frequency eq special moment sum element segment mid typically frequent mid limit typically target segment say place pose facilitate interactive exact frequency aggregate representation aggregated produce distinct resource process stream need element maintain translate communication pass discard ip statistic live address retain sampling include replacement poisson weight approximate segment frequency understand tradeoff segment value cv normalize root square segment fraction interval actual segment cv gold design estimator aggregate pass maintain query hold family another sum query suit distinct query distinct support unbiased frequency meet cv bind contribution sampling specify scoring element tailor want cast stream cv gold frequency offer arbitrary spectrum algorithm parametrize exceed maximum derive admissible statistic specify integer continuous parameter derive continuous everywhere statistic differentiable surprisingly perhaps spectrum elegant simple estimate statistic upper cv estimate unbiased monotone decrease nonnegative make transform frequency invert transform inverse transform minimum variance unbiased nonnegative meaning optimally inversion estimate flow sample spectrum estimator address application estimate multiple close propose tradeoff spectrum aggregated notion simple technical resemble application include demonstrate accuracy suit universe set appear key key pass maintain size execute sequentially location quick scheme dataset specify frequency interpret key estimator key key guarantee estimate segment normalize well bad scheme statistical fix size quality correlation scheme draw seed scheme treatment successively key seed threshold take seed take weight sampling inverse depend available seed interpret condition randomization covariance query sample pre turn statistic cv respect aggregate generally moment cast distinct sampling cache sample scheme distribution seed key minimum sample include small seed value seed apply exact datum detail pure streaming setting platform scheme fully pass stream flexible execute process provide process first pass identify seed score fix summary small summary summary summary merge union seed summary seed summary exceed compute summarie weight merge two summary union pass stream sampling distribute parallel rand hash kx w fix threshold sample initialize discrete initialization continuous rand hash return xx cache stream element key xx algorithm discrete algorithm maintain keep count seed processing element increment result key seed fully reflect seed currently repeat key seed set iterate count either become latter rand xx supremum range work stream per maintain queue active cache distinct final count place low value count element queue decrease get queue probability number key seed attempt key seed way seed roughly whereas roughly regardless illustrate key select terminal font line set left title lc red title lc blue title lc green title lc title form express parameter th contribution sample express f non element express element otherwise grow suffice entry use write note suffice entry pass inverse invert streaming estimator random express relation triangular iy q computing sampling entry substitute total unique admissible need computed sketch show monotone claim positive induction show rearrange nonnegative follow hypothesis key nonnegative nonnegative nonnegative monotonicity sum present continuous scheme update continuous offer fix explicitly maintain value implicitly derivative spectrum multi stream base hash draw return minimum key detail minimum exponentially variable happen seed satisfie qualitatively exponentially result property roughly pass scoring need key weight sample statistic proof appendix cv smoothly cv inclusion gap aggregate relative gold large ratio conjunction terminal option explanation load color graphic graphic macro ltb lt lt lt lt lt lt ltb lt lt lt r p ltb ltb r ltb maximum normalize streaming pass perform compute break otherwise break initialize assign mass return xx cache stream size algorithm maintain threshold key working begin randomization key randomization compute randomization threshold maximum remain elaborate key computed cache score randomization necessary process element key enter simply entry rule key enter cache enter cache rand hash xx initialize cache w z xy h adjust b verify key depend cache statement condition randomization provide key comment cache expect however reduce modification cache full th present next rand hash x initialize cache least xx size transform state obtain relation coefficient seek unbiased nonnegative continuous differentiable unbiased treat first coefficient count count consider expect fw w fw fw fw unbiased dx dx continuous monotone requirement cv appendix cv bound cv cv establish estimate statistical query statistic suffice improve basic instead set improvement ensure sample proportional coordinate randomization randomization constitute variable element score express seed include value fewer surprisingly perhaps fix na
polynomial like arise newton like method walk adjacency diagonal weighted degree walk induction undirected graph walk consequently walk polynomial size dense laplacian algorithmic spectral recent technique degree paper nearly spectral walk observation design utilize mathematical critical walk weight negative construct spectral zero walk constant even construct eq approximate handle degree directly invoke even degree degree om build overcome matrix appear would expensive due multiplication power rely clique specialized polynomials newton like find equation cubic polynomial root factor one challenge slow complex new series careful adaptation error spectral condition approximate simple mathematical algorithmic advance middle turn matrix induce walk offer enough loose similarity algorithm together preprocesse query regard effective logarithmic due connection widely nearly useful variety undirected vertex transition walk analysis make spectral semidefinite positive semi definite usual approximation positive scalar situation lower relate electrical flow view recall potential current else equal obeys add graph suffice guarantee weakly critical upper bind apply standard number upper efficiently two support integer fraction draw draw fix u preprocesse draw edge two random walk length time combine algorithm efficiently low g run even effective edge first path accord integer subsection put give spectral eq lemma r om complement vertice regard complement semi definite rearrange add eq extend routine walk walk side composition use expand since q us write support edge laplacian om nonzero support first q u describe walk replace uniform proportional additional occur chain approximation however remain spectral combine degree invoke lemma reach approximation invoke approximation final dd rd g dd term effect sampling splitting split diagonal laplacian upper difference need extra lemma analyze error matrix definite nonnegative nonzero nonnegative r n om decompose sum column incidence odd correspond weight u upper walk exact lemma exist q via multiplication nearly motivated equation high degree routine call range mathematical future nearly routine widely well explain effectiveness symmetric prove laplacian sum follow induction ii unitary scalar claim graph na entry everywhere theorem algorithmic question theory random undirected matrix recall walk graph polynomial become hence enjoy nearly challenge path precise calculation would expensive multiplication power nearly edge laplacian polynomial well numerical equation motivated problem classic sample transition reversible application random fast challenge polynomial slow efficient structure multi reversible markov task field mathematic encode various physical taylor fast numerical responsible engineering range weather forecasting galaxy polynomial arise mathematical dynamical matrix equation adjacency undirected graph use transition matrix walk power step walk graph prohibitive memory walk motivate walk close representation classical newton reduce radius sampling gaussian graphical model random field specify gaussian distribution representation convert vector
article informative feature yield give block theorem embed human md usa center md usa mathematics statistics md university fidelity manifold dissimilarity multiple dissimilarity optimize fidelity modality stress transform special inherent efficiently compute transform greatly measured modality yield object dissimilarity fidelity object modality common dissimilarity one optimize fidelity e preserve within dissimilarity across modality preserve raw multidimensional exploit employ dramatically procedure see synthetic matching common dimensional wherein joint investigate application computer machine example survey manifold match broad entire correspondence proceed raw multidimensional represent inter treat missing represent procedure embed miss embed scaling dissimilarity potentially embed see cross dissimilarity routine attempt minimize raw criterion row euclidean fidelity versus embed optimal preserve dissimilarity expense correspondence cross correspondence expense dissimilarity optimal subject see detail raw stress optimize fidelity regard cca optimize embed regard fidelity see pairwise distance fidelity pair distance point modality parallel combinatorial define scaling write iterative generate transform derive hence algorithm closely metric multidimensional stress multidimensional scale weight embed initialization I value expensive fortunately application sometimes simplification familiar unit permit much simplified calculation algorithm first notation mn hermitian follow block dissimilarity weight dimension configuration orthogonal onto x I remark output iteration diagonal block denote qr additionally iteration algorithmic give algorithm algorithmic parallel iteration algorithmic complexity serial parallel potential dramatically increase achievable nm nm plot replicate increase versus variety ghz processor gb let set represent measure modality nm nm identical iteration average monte replicate average average mc replicate relatively increase dramatically figure serial ratio time suggest contrast fix vary average mc replicate ratio versus nearly factor utility multimodal wikipedia article english wikipedia geometry exist article link consider undirected correspondence article wikipedia associate short path graph cosine semantic article across modality serial factor embed embed wikipedia article article preserve pairwise result dendrogram two cluster dissimilarity preserve two call dendrogram merge histogram article article article less dissimilarity preserve across label dendrogram height rand ari clustering ground
sum regular radius provide minimax estimator prediction appendix base standard pack critical fundamental quantity appropriate benchmark estimate consequence notion sketch achieve achievable corollary type randomize choose illustrate experimental simulation tu u column correspond eigenvector lead whereas remain sketch matrix whereas formalize sketch follow noise upper sample sketch great matrix sketch apart design randomness randomness proof source project noiseless define whereas fit space elementary sketch matrix induce range estimation error lemma approximation prediction sketch however computational conjecture dimension compute nonetheless various randomized construction sketch various form randomize proportional dimension previously throughout require sketch equation constant purpose state high function guarantee gaussian I problem satisfy universal three overhead sample statistical sketch scale statistical scales cube x illustrate perform begin u unknown spaced theory kernel ridge hadamard note predict prediction confirm plot prediction error three curve sketch sketch tend prediction versus decay simulation sketch dimension panel rescale remain x versus rate complexity nystr approximation om sketch row error nystr column match number substantially dimension performance nystr om poor nystr om sketch empirical diagonal k kk sketch take degenerate depend lead substantial approximation ccc approximation panel panel rescale top covariate arrange unit om compare nystr sketch return kernel h interval type nystr om show regular function namely perturb case original closely contrast nystr om approximation behave intuition precede nystr om recent work sampling involve extra leverage obtain kernel matrix minimax regularization scale put piece sampling must requirement large prohibitive statistical contrast whether approximate leverage score statistically optimal theorem definition remainder technical lemma theorem matrix lemma conjunction accordingly prove two simplify proof rescale statement program must feasibility applying obtain basic least appendix auxiliary analysis basic inequality imply sketch randomness imply inequality rearrange universal exhibit recall u u constructive transform block assumption recall triangle j put piece sketch value turn remaining previously state combine two randomize specialized broadly analyze save computation hope result support office grant dms air grant microsoft fellowship show end nystr om kernel constrain program qp estimate solution via I turn write constrain nystr om formulation base om apply formulation begin g j consider x version subject draw index condition I remains pack mutual ball packing norm q take packing kl regular yield prove w standard jj variate tail union underlie noise vector consequently note union combine last display z violate bind k standard consequently concentration gaussians eq moreover inequality ii radius recall rescale set tail completes prove schwarz inequality suffice show choice side family since increase setting eq equip bind violate ba b violate vector second lower bind early claim theorem state projection guarantee formal split straightforward discretization argument j consequently union part v measure gaussian turning inequality follow piece mt taking complete argument analogous letting sketch c eq euclidean thus result lipschitz eq row c claim follow section assumption definition em university california berkeley electrical computer science department kernel ridge reproduce respectively prohibitive dimension preserve optimality prove sketch proportional regression goal parametric regression make prediction covariate vector say covariate covariate standard assume function regularity enforce reproduce kernel rkhs short assumption estimate lead estimator process appropriately yield minimax attractive property complexity take different machine base partition combine average divide approach zhang give split guarantee forming rank nystr complexity exclude estimator section classical widely algorithmic context project choose subspace involve complexity approximate processing complexity suitably projection tt cluster pose connect definition projection constrain project dimension minimax optimality result estimator contribution several projection organize devote background regression reproduce dimension statement sketch consequence class confirm nystr om devote proof matrix advantage orthonormal dft hadamard say detail focus orthonormal bound meaning entry dft identity p identity understand sketch identity orthonormal use randomization nystr om characterize definite u correspond rescale sum truncate function g uniqueness radius radius bound
dependent word embedding consistent model produce discriminative counterpart word embedding candidate select candidate mean contextual word across language similarity principle phrase recurrent rnn recursive autoencoder long convolutional semantic sentence accurately architecture task largely synthesis sentence one hypothesis distinguish candidate contribute procedure turn reason set work context phrase context improve performance obtain score integrate deep architecture dependent translation way exploit contextual information target partial novel translation statistical translation convolutional language specifically design encode semantic translation pair context phrase similarity translation pair adopt classify medium gradually ability represent phrase context use example difficult experimental significantly outperform conventional translation translation construct word word align corpus phrase however utilize translation surface form capture translation pair utilize phrase distance phrase incorporate method prove decode pair treat context accordingly fail local context context capture propose neural semantic similarity phrase pair language phrase pair sentence phrase source phrase layer perceptron classify triple phrase negative distinguish phrase negative train similarity phrase dependent semantic convolutional matching phrase system outperform proving incorporate local context context similarity phrase section learn strategy section research build phrase employ word phrase employ sentence context word guide treat distinct exploit contextual part tag word pair similarity sum matching score phrase directly similarity convolutional strength network focus capture context instance match occur sentence different precise sentence sentence moreover phrase derive build train large phrase match representation phrase similar language translation semantic mean continuous project source continuous language phrase exploit two semantic capture similarity pair context context useful match dependent sentence convolutional sentence capture dependent translation phrase compare perceptron symbol indicate turn model consist convolutional sentence mean phrase compare perceptron sentence phrase project feature compute matching take train elsewhere summarize mean layer reach convolution take slide window involve composition slide convolution gate function whether activation relu convolution unit I word vector slide term distinguish pair additional embedding phrase transform embedding convolutional take composition slide slide phrase exploit contextual source phrase zero dependent phrase vary embedding language embedding capture across language capture level could embedding language similar language representation utilize mt encourage similar word embedding embedding contextual inspire study word embed exploit word align nearby window word word representation return embedding matching score accord word train eq either machine refer sequence strategy learn easy gradually increase learn benefit give rise randomly present organize gradually negative difficulty distinguish phrase target phrase sentence target phrase want semantic target phrase medium semantic phrase vary context difficult mix mix reach local minima random mix example alg different training consist example easy increase line use sgd meanwhile capture language reach minima reach terminate propose baseline outperform context counterpart embedding counterpart translation chinese english score local translation initialization conventional embedding embedding counterpart indicate embedding well experiment chinese english contain come dataset portion language portion corpus use mt mt minimum insensitive translation neural pooling layer map slide window development produce high phrase decode collect phrase obtain phrase pair phrase context phrase least corresponding phrase phrase remove undesirable mt mt two baseline system system translation phrase convolutional previous degree phrase contextual information serve phrase phrase baseline matter significantly translation gain
hold add equation acknowledgment discussion lemma let increase calculus increase j eq use k k substitution hellinger simplicity depend j kp pp c constant depend decrease substitution ratio multiply therefore adapt constant square substitution remain let k x substitution I c constant sum get hellinger distance bind q combine k kp lemma part constant expression regime chernoff depend appropriately substitution compute hellinger get sum hellinger distance use let whereas computation previous similarly let chebyshev enough tight relie relax collection gene section split equal sized disjoint sub convenience proceed infer compare quantile bin algorithm succeed probability whenever sum repeat lie monotonicity p c come q vice versa follow together consider three specie triplet molecular rgb rgb berkeley reconstruction establish seek connection trade reconstruction gene branch corrupt typically application establish error explicitly boundary paper establish signal reconstruction multiple gene population genetic evolutionary subject study survey particular grow body understand research history theoretical connection derive boundary trade need accurately reconstruct signal extract gene method e evolutionary evolutionary relationship species leaf problem common gene seek reconstruct principle time difference formally estimation root leaf leave root one simple markovian model molecular evolution mutation mutation derive continuous markov mutation associate root u u two adjacent root combine leave also identifiable long evolutionary biology survey learning problem theoretical sequence theoretic upper general molecular evolution context g access gene abundance new survey evolutionary gene deep horizontal loss sorting phenomenon paper accounting leaf label binary specie past divergence share consideration associate gene mutation pick specie mutation parameter inter time gene conditionally specie sort give combine genetic explicit perfect base method distance distance sufficient accuracy reconstruct detection lower label leave early distance estimate differently require leaf limit distribution respectively notation gene sparse positive total variation distinguish bound spectrum imply need bind recently spectrum regime regime gene tree gene reconstruction decrease analysis whenever test distinguish reconstruction molecular trial equation recent formula boundary bind instead hellinger q prove proof regime sample test test gene specie gene event different exponential depend effective theory get specie success property give leaf specie specie gene conditioning leave look branch exponentially independently branch merge population proceed
integer sum partition subset number instance consider inequality evenly partition verify lemma assign sum equal lemma continuity decreasing must satisfy meanwhile minimizer minimizer interval discussion interval overlap since overlap element thus must global lie proof due lemma case strictly need equality become fan comment definition thm wang keyword nonconvex concave penalty penalize statistic past decade high review recent advance refer reader two smoothly scad concave mcp concave penalization enjoy computational concave penalize due intrinsic structure many local quadratic linear however regularize remain author penalty np hard assume accordance assume main monotonicity decrease concavity concave hard literature fall category global strongly arise naturally select generalization penalization penalization correspond bridge obtain thresholding penalty satisfy widely scad mcp penalty study function also prove result follow decrease follow concave lipschitz np strictly np hardness among example work concave penalty addition proof considerably proof end property penalty furthermore
autoencoder look quantitative aware imagenet generative convolutional imagenet lead finding include layer preserve color show color present retrieve object convolutional pixel layer precisely layer information non zero activation precise value object small supporting allow generate somewhat look feature representation principle visual representation could modality acknowledgement acknowledge start grateful sharing thank comment supplementary material figure similar one main reconstruction reconstruction reconstruction reconstruction autoencoder figure contain reconstruction autoencoder reconstruction vector two autoencoder example generate feature standard shift leave percentile normalize feature multiply average generate significantly complicate th main feature autoencoder less realistic image image way every say neuron look class reconstruction somewhat interpretable much image cm cm autoencoder cm cm cm cm cm cm cm cccc cm rgb particular good learning representation extract new inverting convolutional imagenet numerous insight representation color rough contour reconstruct activation even class convolutional network cnns large impact technique vision nonetheless typically towards understanding cnns allow image perturb vector insight task invert trivial usually pose train noise illumination mapping many indistinguishable interested inverting n inversion impose manually implicitly natural convolutional allow supervise squared reconstruction natural cnn imagenet insight probability sufficient reconstruct input image activation detail propagate activation identify responsible activation activation make extra maxima information crucial inverting also feature seek minimize loss regularizer enforce optimize solve feature produce feature approach map try hard distinguish care moreover feature representation relatively gpu costly inversion forward per image gpu representation example reconstruct base invert traditional computer invert restrict invert neural work make backpropagation advance architecture allow modern convolutional architecture generative study pre website layer process please h c c conv conv conv conv conv fc fc drop fc relu norm relu relu relu relu relu relu channel cm c layer conv conv conv channel c fc fc channel train set convolutional minimize image speed computation image slightly improve convolutional map value original corner comparable varied architecture generative depend layer architecture reconstruct convolutional layer reconstruct connected architecture reconstruct fully connect three five layer convolutional layer depend layer reconstruct slope imagenet use optimizer mini gradually learn towards design low square interpretable favor suppose visually even reconstruction cm cm net decoder representation encode autoencoder stay fix decoder autoencoder reconstruction net autoencoder invert high layer neither much error reconstruct qualitative reconstruction autoencoder even reconstruct autoencoder training estimate lose autoencoder reconstruction reconstruction actually convolution max pooling much beneficial slightly reconstruct deep deep reconstruction even deep layer error reconstruction autoencoder low cccc cm cm cm check color preserve high network green image layer softmax give large pass inversion zero lead classification color precisely reconstruct probability depend maximal top small probability probability predict carry test high preserve reconstruction preserve motion interestingly high image indicate horizontal image cm bin level preserve rich represent gain insight representation perturb change reconstruction carry reconstruction dropout sign vector unchanged try except feature negative set non dropout normalize unchanged normalization qualitative perturbation feature layer quantitative surprisingly dropout reconstruction quality although know convnet well reconstruct important code activation binary training autoencoder sensitive perturbation code dropping dropping
rapidly network efficient way analyse modern genomic network analyse genomic association genomic year dna breast cancer platform cancer genome pre process mapping snps remove removal gene annotation remove probe replace knn imputation annotation information platform package check effect component phenotype batch significant would individual patient survival age individual potential dna clinical validate expression node gene rna protein domain alpha I domain contain fusion protein activate ii protein box sort protein activate protein alpha gamma beta family member neural box family signal protein relate early cell contain cd cd adjacent ap bind protein family member family interactive rich rbm rna bind active reading member nucleotide interact protein beta protein protein neutral neutral heat bind protein contain contain node box alpha protein member homology c open member contain member breast associate domain contain repeat protein like bind dominant negative protein alpha activate death nr member alpha neutral ab cdr protein cell sec h alpha gap like bind node l protein alpha four domain alpha b nk bind sub member sort six six open reading repeat protein code rna protein protein di alpha protein open read rna cd b body protein activate member protein bb bb dna domain member alpha similarity member g ph domain transfer protein growth b bp ph domain protein l open read activate protein domain like alpha sec member associate dna sorting b member reading frame dependent member protein st alpha alpha channel repeat factor group box repeat domain long non protein code rna g domain domain contain member activation half bind protein gamma alpha interact protein family protein activate alpha like iii domain contain contain open reading protein h couple interact loop translation gamma cd cd protein induce protein bind protein pt process dna increasingly disease cancer change reflect environmental amongst pre change dna promise develop dna base measure pattern assess genomic analyse group interaction quantity happen genome dna measure genomic infer genomic community science system model know include social science network much shift gene pathway gene cell examine gene successful principle group gene statistical possible gene mean expression particular gene amongst store modification chemical dna di information acquire phenotype interface genome also environmental dna change human offer identify develop early dna think promise development wide play major reflect change dna highly change dna individual dna whereas lead datum noisy dna gene analyse show activity genomic region associate way represent analyse large quantity produce balance fidelity efficiency widely study blockmodel observe interaction block modularity quantify observe particular community present certain fitting blockmodel modularity spectral network genomic decompose module functional biological community detect genomic sbm think module recently reasonable sbm mechanism optimum blockmodel representation mean blockmodel biological genomic law mean visible use module identify isolated biological fitting cancer network arrange phenotype advantageous cancer patient outcome breast could gene interaction gene dna dna measure infer genomic node indicate module within network gene interactive term dna behaviour link relevance network patient dna represent new methodology dna dna numerous form dna analysis cca novel interaction gene single patient dna dna quantify extent dna profile gene dna profile gene act surrogate extent pair may include co type interaction gene presence product amongst cca cca discover combination variable explain way combine location particularly particular deviation profile probably gene along correlate cca across cca seek dimensional space xx profile patient measurement gene gene patient dna interaction profile analogy equation eq cancer gene reflect well correlate behaviour patient measure dna typical explain profile c gene vary typical equivalent varie vary interaction number measurement ordering identify interaction gene represent adjacency node accord measure problematic carry level edge pair network cox proportional model quantify dna patient cox hence adjust clinical covariate dna normally variance mixture gaussian infer lead mixture equation interpret function take likelihood observe standard mixture gaussian prior tail use practical implementation slight mis specification gene find pairwise comparison mostly detect correspond generally detect high gene adaptation degree obtain interact social functional biological detection allow constructed interact differently cancer predictive advanced term detect interact relative serious disease interact serious community fit correct blockmodel community divide network histogram community module identify way represent dna network community summary accord gene whereas gene correspond increasingly positive care network community magnitude network interaction pair gene pair community multiply cox cox fit predictor hazard factor event g death occur hazard coefficient issue fit dna interaction measure generate follow network infer cox dna disease green increase score dna interaction decrease poor interact examine dna amongst genomic interaction calculate gene expression gene patient intuition non zero measure interactive statistic tool breast cancer cancer genome initial batch dna train together clinical disease disease dna dna profile potential community datum set clinical validate training gene infer adjacency presence dna interaction associate edge extract connect community range size relate ht adjacency blue detect outline table indicate label survival divide median score initial hazard value calculate cox community survival patient group community divide training display value cox validate note identify potential network carry infer cox calculate community unseen test association survival unseen test sample five potential validate way outline cox network community datum clinical covariate test l ci age disease network r l ci network ci disease ci
provide human fm translation visual help ambiguity sentence modify component multimodal show failure case mistake reason incorrect method say actually try focus small look similar fourth answering output sign e try issue incorporate visual linguistic information question university california present question short lstm convolutional network extract lstm store linguistic answer component question evaluate chinese english answer human mix answer provide human human score indicate quality human rapid progress deep deep cnn recurrent memory lstm annotation play discuss description image interaction preference paper answering need answer content address four figure lstm encode sentence dense second convolutional extract representation train imagenet fix third component lstm encode previous dense fourth predict next jointly train fourth answer weight sharing also allow layer fully layer answer fm detail ms chinese english allow content annotation contain ai position object e base visual column encourage study area variability answer accurately evaluate visual human mix answer generate label human determine human addition also ask e pass treat human average failure ask answer tb recent significant progress neural natural computer vision cnn classification rnn lstm widely speech recognition inspire task deep cnn vision rnn extend ask question image use question pre template contain kind language translation set requirement complementary lead interesting topic visual answer lstm question answer word prefer single answer feed lstm answer fm dataset annotation dataset four pre color location dataset weight share manner different different component please see short extract semantic convolutional cnn extract lstm current answer linguistic incorporate part generate next jointly four word answer e non zero index sign begin answer use generate answer stage stage input image start start search keep probability accord softmax repeat question extract embed memory cell function layer map semantic feed dense neural design vanish layer serve bridge e lstm gate calculate stand activation word denote matrix multimodal word softmax layer share word softmax detail similar use sigmoid activation relu non cell non activation intermediate answer question mean answer embed third component weight softmax layer word embed function pseudo word reduce concept task cnn imagenet adopt word answer function answer fourth decrease epoch stop epoch chinese answer treat equivalently english train question answering describe annotation publication start image newly ms image annotation annotation amazon type long get question beneficial question annotation monitoring set quality labeling monitoring question answer per satisfactory correct interesting question require reasoning give rest set good bad example answer reference annotation image far refine portion question answer tb answer annotation length question answer chinese answer question ai capability contain simple understanding question g object person among computer object e color contain question vision language answer hold tool problem answer part answer phrase e give answer automatic metric word similarity wu answer dataset complete sentence metric image metric answer critical tend give keyword evaluation suffer severe answer conduct generate well grain evaluation ccc rate rate score visual answer determine base answer pass
side sup sup f contraction last sup sup sup sup h old jensen previous chain combine give part inequality hoeffding n sup I sup use combine bind well bind complicated simple clarity occur definition f nm f iid value hoeffding remain thm bind among task representation task highly beneficial study statistical general rigorous justification benefit illustrate linear specification iteratively linear transformation sigmoid representation learn analysis case specialized consist function different choice note feature function hilbert method apply large representation would hour height ne ne pt skip ne compatibility abstract chapter head part head head head head head head compatibility conjecture proposition compatibility compatibility true true false mu false size ex h mu mu th h mu mu align align end end end end end end align style style use style array type allow true tag tag true hour depth ne f ne pt abstract head head head head compatibility corollary proposition ex em compatibility compatibility compatibility true em em mu mu mu mu mu false mu mu h mu mu align align end end end end environment environment style allow use allow false false tag true uk science college uk ac uk uk method justification setting method advantage task focus derive regime beneficial independent sample number intrinsic feature learn multiple jointly become increasingly range preference object mention similarity study mid connection neural structured perform task arguably intelligence experience task influential research transfer mean jointly task ai hierarchical representation multiple empirical remarkable increase interest component work remain discuss advantage domain derive representation enough reliably model precise half propose vast paper incomplete considerable representation learn inductive learn analysis perform paper main goes consider base cover reproducing avoid factor dependent subspace good main specificity beneficial worth effort half noiseless isolate task method match performance know advantage good agreement theory organize problem learn far half rigorously error suggest interpret member interpret learn model pair output predict constant predictor handle simple scale task simplify task composition map define specialized sequel class refer representation specialized require multi sequel exponent iid representation solve concerned rather propertie hand important assume probabilistic law keep parametrization learn guarantee correspond decrease typically contribution first play observe average independent tn many interest include kernel machines gaussian rbf discussion learning vanish term govern equivalent map specific special phenomenon considerable sample subspace learn show operator size understand support task like dependent eq q play basis quantity assume subset hope obtain suffice hypothesis albeit unknown subspace isometry bound radius specifically k k appear dictionary call certain allow nonlinear activation atom drop allow trade class specialized identity express simply theorem let g theorem least excess eq compete bind bad whenever approach slow course break distribution obtain excess risk occurrence norm covariance imply large number term trace covariance easily marginal concentrated eigenvalue c term appropriate available highlight potential order computational task superior really dimension regularizer large proportional empirical burden dimension vanish section explain case noiseless classification half performance superior marginal unit sphere classify loss function let define without partial isometry mapping onto unit sup da f excess get probability expect use estimation vanish limit roughly share bind class transform orthogonal algorithm produce correspondingly algorithms rotation misclassification unit least every u bind bind high advantage learn purpose setting beneficial binary task namely generate ground vector orthonormal haar select sampling build hinge optimize weight assess consider report repeat report average input task average suggest despite find axis instance experiment experiment help diagram accord experiment scheme previously error similar average test trial diagram generate dark compute bottom column reader parameter difference parameter partly accumulation loose derivation another apply agnostic nature bottom agreement dictionary truth regime could permutation change overcome similarity tr permutation ds tr vertical represent horizontal task theorem establish right excess
flexible whose tucker decomposition depend denote column usually need advance however due apply significantly less stability seek elegant multilinear avoid employ sparsity prior latent straightforwardly place independent prior interaction inaccurate multilinear account n precision place propose student let core gaussian integrate term tucker l laplace prior simplicity thus estimate logarithm however map aim infer distribution treatment perform inference scalable sampling inference technique conjugate address inference provide appendix vb aim seek approximate posterior p factorize optimize expectation distribution variable follow variational posterior q linearly relate evaluate show expectation computational polynomially complexity scale introduce theorem scalar eq detailed scalar spectral factorize product lead operation individual eigenvalue decomposition matrix operation diagonal matrix significantly memory significantly reduce format product posterior multilinear operation evaluate memory multilinear kronecker avoid explicitly kronecker note factorize operation cost gamma rigorous approximation play automatic determination multilinear rank distribution conjugate accord expectation square account automatic model f n factor unnecessary slice machine precision expectation laplace prior lead difficulty solve problem inverse distribution inverse gamma inverse hyperparameter l mode hyper variational represented posterior straightforwardly mode avoid computational modify essential difference student one n lie student laplace manually hyper straightforwardly derive mode modify function variational hyperparameter update expectation residual alternatively multilinear hence computational reduce denote iteration bayesian tucker tensor completion position tuple index entry tucker multilinear incomplete tucker consider new tucker represent model tucker incomplete ill role successful determination multilinear significantly affect strategy cross multilinear varie dramatically ratio multilinear occur core minimum multilinear automatically efficient elegant repeat many memory overall complexity n nr memory scale polynomially suitable multilinear automatic reduce rapidly iteration computational decrease automatic determination multilinear enable low tucker secondly uncertainty problem uncertainty deterministic scalable convenient related ard tucker carefully performance ard tucker automatically tucker fit vary show discrepancy baseline three automatically complete vary noise level range db range f theoretically rapid surprisingly improvement snr ard tucker automatically infer however db fig exactly except snr ard tucker infer accurately runtime second ard tucker ratio range snr mr rapid increase able optimize base tensor rapidly infer tensor condition snr mr runtime summary mr completion evaluate recover miss test different water pe b sample represent ideally three original corrupted snr db tucker need predefine tensor infer tensor rank underlie capture stable estimation datum performance case ard tucker always well noise computation ard tucker also randomly datum repeat present tensor infer sensitive five htb noisy original noisy noisy original ard r n runtime runtime flow global generally prediction become challenge globally rank small block tensor overlap tensor completion tensor handle performance adapt vary noise condition outperform require medical test evaluation rank specify infer ratio performance rank global yield completion visual quality method c tucker structural introduce group prior hierarchical inference especially multilinear observe significant advantage completion propose theorem multilinear operation scalability empirical validate superiority ph degree china laboratory advanced signal interest vision brain computer interface paper international zhang ph degree china department engineering china interests cover theory visual publish paper receive ph dr electrical technology team laboratory advanced processing scientific paper translate proposition zhang bayesian tucker completion modern attract extraction compressive sense challenging determination multilinear especially probabilistic tucker decomposition structural multilinear fully inference inference model efficient multilinear automatically complexity multilinear principle comparison synthetic remarkable recover ground multilinear entry tensor tucker multilinear structural completion aim seek multilinear array technique modern mining factorization structural model multilinear lead dimension compressive tucker decomposition attract tucker multilinear operation factor core basis core tensor multilinear whole tensor basis dimensional coefficient profile datum tucker decomposition contrast strict limited become low compact representation one fundamental problem determination tensor matrix numerous study either tensor always manually general need tucker possibility exponentially tensor automatic determination exploit fail true tensor case highly decomposition value decomposition focus tensor focus semi aim predict element tuple index important issue completion relate miss underlie tensor key appropriate obtain whereas tensor specifically applicable correlation observe decomposition develop tensor rank determination cp contrast multilinear degree framework assumption minimize relaxation minimization avoid specify manually selection problem nuclear tensor completion attractive year impose exploited type obtain carefully implicit issue nuclear correspond weighted rank denote dimension core model previously another nuclear tucker tensor multilinear rank solely datum automatic student facilitate enforce group sparsity framework non miss parameter addition multilinear rest multilinear modeling induce present bayesian tucker tensor tucker completion issue discuss experimental tensor matrix ni n standard tucker nn
order detect signal pressure patient might event offer attractive unknown base particle mix slowly base maximization size nonparametric furthermore counterpart degenerate maximum directly observe case inferential challenge jump predict patient show degeneracy take develop trajectory parametric variance asymptotic proven infer extend connection gaussian variance gaussian approach idea obtain motivated objective scalable challenging dirichlet hmms approach parametric function suffer solution degeneracy lead robust inference case construct exponential nonparametric jump parametric jump attractive improvement case reduction mean error prior provide degenerate solution probable comparable outperform reconstruction finite space probability state exponentially leave transition kkt ts kt k states trajectory state state time inference expectation maximization state suitable finding carry efficiently degenerate spent visit far thus slow often carlo propose mcmc address issue probable model contain pz pd pz approach limit map variance simplify likely time time simplicity interval let place gamma prior detailed rely asymptotic stable maximize limit variance exponential two shape hence mean scale ft gamma scale writing drop modify long markov stability maximum ht kt kt grow penalize expect penalize manner however remain long interpretation trajectorie usually trivial system almost mle mode superior valid difference case observation case consider observe thus objective trivial know jump combinatorial sequence continuous complexity resort alternate optimize spirit modify viterbi jump optimally distinct jump jump point modify viterbi keep dependent upon state infer eq restrictive jump jump eliminate find optimize jump mean similar place indirect viterbi step initialize converge poor bl jump nonparametric ms pressure run em number hold state synthetic synthetic ms model exponential process construct gamma generate state parametric replace scale hierarchical gamma h component mt error jump uniform scan hyperparameter respectively mcmc jump em par error jump em run obtain significant state generate probability em likelihood use uniform initialization amount jump jump mean hide state jump perform slow trajectory slow likelihood disease cast patient state represent disease trajectory aid disease care real world phase clinical trial drug dataset track different randomly evaluate value record initialization tb infer jump mean patient ms dataset jump significantly achieve reconstruction fig provide trajectory jump maximum include reflect realistic patient amount trajectory produce take account stage picture pressure collect hour period observation length patient keep test initialize uniform matrix hyperparameter particle state inference particle category pressure run em report good em bt evaluation category jump significantly reduction error iteration cpu need example infer color assign pressure degenerate trajectory tune show histogram run fraction run setting lead large hence study robustness jump choice hyperparameter runtime jump increasingly handle scale linearly decrease increase bt jump inference asymptotic algorithm experiment obtain degenerate offer parametric art problem acknowledgment comment air office scientific research national fellowship begin scale natural uniquely integral ft shape let ft incomplete gamma state give bregman divergence multinomial multinomial writing trajectory family apply expansion fact process base partition define gamma key ep conjugacy conjugate family proof analogous proposition key additional insight denote b suffice xt exchangeable times exchangeable integrate generative manner restaurant obtain dirichlet omit transition reach likelihood k logarithm expansion term mt retain asymptotic asymptotic rewrite integrated ignore yield
error plot cost probability observable n exist cover nonetheless implement numerical enkf probability near slightly slow achieve approximate rmse sde analytically transform q noisy introduce eq upon define noisy sde artificial ensemble numerically integrate ensemble mean ensemble provide return integration solved numerically validate exist integration euler scheme mesh parameter set gold rmse slightly enkf measure rmse mean plot measure attempt monte equivalently optimality kalman however limit view linear filter convergence limiting model sequel publication support rt member quantification sampling carlo filter enkf yield kalman provably superior asymptotic carlo filtering kalman sequential parameter system sequential incorporation complete estimation probability conditional solution formulae kalman filter close form resort probabilistic leverage kalman enkf approach converge kalman filter solution incorporate evolution give may computable integral approximate grid dimensional even solve enkf intractable close less accelerate idea iterative solution pde early iterative pde pre become random context sde pde beyond discretization proportional cost computation sample monte context markov chain knowledge author yet extension methodology explore enkf implementation enkf limit distribution bayesian something else rest organize review enkf first sub favorable field equation direction filter gaussian state section implementation kalman filter ensemble enkf time sigma generate positive definite track aim variable variable seek admits evaluate exactly rather associate confusion concern herein follow sde w satisfy follow condition fit framework wiener principle come unknown must lead hierarchy approximation notational time refer non easily extend merely notational convenience approximation solution follow depend solver verify cf arise coefficient dependence map simplify uniquely iterative computing covariance give derive use one may derive verify formula derive verify fix side quadratic obtain shorthand summarize classical kalman formula describe equation nonetheless responsible enkf appear particle enkf propagation kalman filter n consist nonetheless step one interval realization sample random variable realization realization confusion latter become apparent suffice assume map numerical satisfactory complete path mean ad hoc indeed implementation formulation filter know perturb nonetheless functional converge lipschitz limit particle increase discretization let denote simulation numerical increment consist update pairwise couple realization level level unlike return similarly enkf ensemble I enkf predict level compute level convenience convention introduction second argument sde initial hierarchy section use constant contain implication scheme possible sufficient regularity sde euler norm assimilation vs monte cf repetition notice realization give boundedness tolerance large step grow polynomial exist scalar locally polynomial growth infinity cf sample update formulae enkf distribution sequence line high practice introduce limit l evolve covariance gain limit formulae intra level member identically I independent solve limit system approximate replace last sample perturb limit independent come correlation crucial proximity two particle require great mean lemma bound boundedness predict ensemble ultimately ensemble covariance virtue convergence consider omit avoid unnecessary gain follow micro control let notice normalize notice eq continuity depend notice ml q cost denote predict final system forward let triangle turn lemma term respectively discretization boundedness contain second inequality comes triangle complete observing imply difference hold define bind surely expectation quantity triangle avoid similarly aa ba ba cauchy arise sure depend cf use hand boundedness eq next covariance see continuity covariance similar term inequality term old proof show predict ensemble error carry make rigorous ensemble begin use old plug hand summing recall induction remains induction actually able definition denote treat separately relate
quadratic perform allow stochastic gpu implementation show converge correct low structure goal structure input explain want input avoid serious utilize hide cost spurious spurious signal must subsequent supervise additional expert generative pp median regularization approximate constrain normalize model posterior maximize leibler achieve extract desire structure impose generative pt factor loading component unit correlate fig center consist stack analysis posterior maximize give expectation em minimize first constrain I component wise hide solve project feasible start newton try reduce fail ensure alternatively method project fast projected newton require define gradient projection step iteration program benchmark mnist speedup project solver ratio mnist cifar speedup update essential computer restriction quadratic reconstruction eq step gradient allow gpu dropout gradient using require euclidean posterior euclidean euclidean projection positive non positive otherwise supplementary project project scale active column posterior alternate dropout zero predefine dropout long hold cm project gradient project n cm kk complexity step project project step gradient alg alg converge maximize sketch alg ensure minimum alg decrease gradient projection requirement fulfil generalize thus update view gpu implementation covariance capture good ridge sample noise strength randomly noise deviation large evaluate method code generative percentage value small sum reconstruction supplement detail hyperparameter average overcomplete unit instance dataset supplement confirm level yield since variational low yield ex performance factor network stack obtain pass architecture layer machine stack denoise autoencoder stack autoencoder iv restrict machine rbm stack boltzmann machine report bold overlap validation select bad nine experiment perform significantly p ex mnist basic rand ex ex ex benchmark dataset mnist iii random iv mnist discrimination discrimination rectangular image discrimination generic category validation center selection set default fine fine stop rate learn bold significantly good nine perform project lead company aim pde project project expression identify event expression gene event stem formation panel rare small negative feedback pathway event detect relevant project example design previous code row gene red green inactive gene formation confirm analysis green panel feedback pathway cancer factor construct code normalize alternate minimization prove correct code low yield sp improve deep network drug detect rare module unsupervise relevant study large code factor network efficiently sparse dimensional rare interference unit explain covariance generalize derive posterior unsupervised autoencoder rbms ica sparse code reconstruction test deep vision superior rbms autoencoder expression drug discovery detect module highly insight learn advanced representation relu advantage representation code importantly much use representation break bioinformatic dna human representation code event vast majority suppose would fluctuation
answer answer set output hide call map feature map follow output hide sentence word slide windows input convolution traditional combine sentence joint pair formalize input sentence perform sequence key cell time cell gate gate gate lstm study discuss time memory gate forget gate equation lstm keep context discard forget gate add compute update cell unit matrix accord conditional answer answer sequence softmax training test conduct dataset challenge training contain answer answer good bad accounting answer question bag model dirichlet crf crf approach word apply belief predict answer multimodal learn cnns representation question answer answer score development hyperparameter joint modeling question sentence pattern question notable powerful svm crf answer potential svm crf answer tendency good answer reason distribute deep architecture capture semantic crf suffer noisy question answer f crf cnn cnn cnn superiority pair sentence cnn convolution operators r cnn tensor sure representation semantic feature bag improve answer sequence answer complement learn context previous answer modify valuable pass rnn main improvement cnn potential answer much cnn potential answer answer intermediate r cnn score multi ok please response indicate question easy performance cnn correlation answer bad cnn sequence model integrate lstm unit cnn successive relevance answer explore improve overall label world support part national china foundation china anonymous comments intelligence institute technology chen cn answer question answer regard sequence task novel approach apply cnns representation question pair firstly use joint long short lstm learn answer question answer conduct answer valuable knowledge information retrieval matching answer typical question explore study exploit syntactic measure semantic answer require external resource directly disadvantage semantic question answer selection figure answer intuitively answer answer recently especially short memory superiority long term short work use convolutional cnns sentence sentiment labeling identify answer cnn answer pair use lstm bad answer study answer generally treat problem explore pair structural represent machine classify train lr classifier answers fields crf answer match additionally language translation suffer perform applicability symbolic belief net semantic learn
triplet visualization generalization independent triplet gain joint task measure difference triplet triplet consistency average triplet learn embedding large triplet triplet consistency synthetic pose significant gain approach triplet high consistency gain significant tb cm cluster triplet consistency jointly measure triplet view specify specific mahalanobi demonstrate conventional although hinge easily generalize trade view real application triplet expensive jointly metric preferable empirically triplet consistency view view great gain show multiple error future study similarity classification label annotate top leave illustrate five triplet translation square match work categorie attribute p attribute shape cs similarity multi similarity specific view perspective jointly exist view achieve triplet generalization grouping learn independently improvement large triplet role application content recommendation speech similarity object abstract represent explicit parameterization inner product matrix gram implicit representation operation complex demonstrate embedding object triplet supervision embedding comparison proximity use word similar back head human embedding reflect similarity comparison easy absolute scale ambiguity head head back ambiguity annotation result poor desire perspective measure view enable human loop interactive fine grain thereby main drawback view comparison undesirable triplet jointly exploit view training car take angle model onto play view view iterative dataset realistic domain namely pose dataset crowd similarity collect datum per view low triplet naive independent approach propose learn well cluster leave joint embedding base triplet wise ordering dissimilarity set kernel crowd alone triplet relation van triplet similarity embed cluster study four focused supervision aim learn object cluster extend multiple task label different fold triplet comparison learn instead multiple aim embedding instead triplet capture complementary user effort collect triplet triplet collection embedding triplet dd distance agree j set study van propose loss hinge non crowd stochastic triplet distance jj minimization q trace gram convex embed embed however optimization true similarity case object multiple measure obtain aspect comparison triplet corresponding notion embedding view end hybrid approach combine global gram base view correspond global object gram define mahalanobi ti formulate learning generalize literature regularization term add trace produce ambiguity scale scaling hinge respect since descent via iteratively take project result cone summarize term scale carefully trace geometric reach value depend product hyperparameter view argument tell view dimension much enable well leave whose throughout classifier van triplet lead error triplet necessarily low metric aspect conduct triplet task adopt package author cross sample hypercube center randomly hypercube six project datum five triplet possible triplet pose construct image base pose translation associate range vary view additionally leave point evaluate produce unbalanced belong class similarity color public figure public face create consist attribute real value appearance person aspect ten attribute attribute group image label specie collect show user various contain triplet collect crowd manner ask nine image interface specie display partition nine set triplet specie l sim triplet constraint acquire region various fig procedure triplet cast similarity whole cast localize region breast triplet test balanced manually super class challenge sense situation triplet relation nature crowdsource triplet incorporate human feedback recognition embedding data conduct learn triplet generalization leave error plot achieve triplet generalization cluster dataset triplet learn triplet error error pose embed triplet object lie leave orientation triplet embedding show triplet generalization learn triplet become indistinguishable see joint independent pose public embed triplet set triplet generalization error triplet error reduce triplet increase dimension understand bias triple learn understand term leave continue well triplet embed dimensional visualization bottom jointly show triplet embedding dimensional
impose draw vector selection existence meaningful approach learn bayesian class sample base factorization single vector aforementioned control class discriminative infer use desire dictionary restrictive overcome c weight standard precision distinguish parameter r arrive representation conjugate place mention latent dictionary atom datum dictionary notation assume place hyper normal distribution f gamma representation model variational et effective easier relate sampling process conventional sparse svd expression conjugacy analytically start analytical expression posterior isotropic simplification significantly approach atom dictionary atom atom dictionary codes dictionary atom drop update contribution atom write aforementioned eq expression probability concern therefore ik eq sample normalize bernoulli simplify arrive expression light expression sample conjugacy sampling eq sampling give weight assume express isotropic conjugacy distribution must sample write arrive set vector k discussion desire draw estimating dictionary mean size present regard closely take discrimination appear model simplify c dictionary require vector non later happen ok atom simultaneously accord drop atom bring remove redundant arrive probability probability c atom commonly represent class word arrange large learn appear location infer clustering discriminative dictionary character six probability different training extend scene respectively plot represent vector plot distinct query follow methodology encodes contain query assign component technique joint optimize separately appear class computing denote modeling matrix write h framework gibbs use instead infer new infer code query learn discriminative dictionary underlie learn classify regard exist coupling probabilistic exact hence jointly coupling keep term match omp query greedy pursuit efficiently search omp code infer select omp support initial initial equally effective atom initially getting select finally serve similarly vector compute vector initialize computation complete drop category categorization scene category evaluation representation consistent svd learn separate datum dl comparison unsupervise use dictionary acquire public code implement toolbox public perform intel cpu ghz ram mention carefully difference illumination ar illustration follow subject test pixel image projection ten lc lc whereas result expense computation time reasonably lc lc original distinguished section result lc dl residual tolerance small give classify denote work l accuracy time lc lc svd lc lc term accuracie fairly I reduction rate require propose exist l c c zhang al et wang dl lc comprise object class tree sign number vary sift descriptor extract patch densely pixel extract spatial pyramid extract grid codebook pyramid train pyramid protocol select experiment repeat accuracy experiment number lc lc result sparsity et suggest give result select well lc dl clear consistently compete approach case lc propose increase favor sample result precise posterior distribution setting inherently technique inference lc verify testing batch class contribute efficiency table include propose sec lc scene category category kind country etc pyramid descriptor vector feature consider propose value set suggest al lc dl original database lc dl approach propose lc pyramid lc lc l lc dl comprise channel video include dataset evaluation protocol performed fold one five summarize lc lc parameter along accuracy action also dl take literature optimize report outperform validation database propose well art claim insensitive large precision clean noisy give less training class increase availability clean therefore precision large dataset among similar without easily verify mention desire atom plot training represent complete training correct mention convergence initialization work code learn dictionary sample svd lc fair parametric employ infer bernoulli atom say atom specific datum also correct hierarchical exploit classifier code instance use evaluate scene comparison art discriminative representation outperform prove advantage exist base us dictionary secondly principled dictionary atom manner make inherently online specific prior knowledge principled signature hyperspectral spectral signature adapt hyperspectral image classification future arc grant dp lemma edu discriminative dictionaries dictionary atoms association dictionary atom class parametric infer dictionary exploit separately classification instance encode learn fed face scene public state discriminative representation experiment propose consistently discriminative redundant root human technique digit well instance dictionary redundant atom effective g wavelet domain decade favor unsupervised learn signal supervise dictionary discriminative representation discriminative dictionary use training dictionary atom query assign associate maximally representation query result achieve become considerable dictionary allow computational learn force dictionary specific associate constrain learn exclusive class separate atom exist strategy assign atom adjust accuracy principle perspective representation beta adaptively build association atom character fig bayesian discriminative learn atom bernoulli code later learn atom wise code datum classify infer classify code exploit bernoulli distribution test database action database scene art approach efficiency exist paper follow review explain propose propose experimental setting conclusion main approach learn dictionary al recognition atom dictionary et term texture segmentation sparse code compute specific dictionary action atom coefficient negativity training apply detection recognition use encouraging incoherence among specific dictionary allowed represent incoherence mention mainly associate atom directly single minimum query stage representation dictionary approach single force encourage discriminative objective already author coefficient common learn classifier joint dictionary learn zhang li enhance along coefficient minimize task classify sparse code learn dictionary classification stage dictionary also fall category take hybrid discriminative representation dictionary et
equivalent kernel quadrature rule definite measurable lead eigenvalue integral logarithmic particular quadrature general beyond match special preserve cm topological equip borel integrable family matrix reproducing also reproduce element respect weak rkhs kx dy kx adjoint semi definite trace df h extension sequence eigenvalue eigenvalue dense element orthonormal ne covariance rkhs dx operator eigenvalue cm generic make equipped probability measure consider additional give term show x equal always attain cm usual decomposition kernel fouri kernel form periodic definite negative kx expectation usual uniform traditional space kx geometric decay eigenvalue kx tx tx fourier split decay study decay decay decay dd extension tensor decay integrable derivative multi integrable derivative last decay avoid linear x strong expectation x nx surely kernel uniform subset function rkh key difficulty general include characterize approximation measure look possible possible definition n e n scaling choose well respect n v cm square integrable integral combination allow depend fashion cauchy equal quadrature formulate quantity possible mean standard weight note correspond respect require fix well integrated accommodate respect respect robust cm mx many always x gx quadrature expansion note form approximation approximation qx gx many constant set compact line interpolation decay uniformly fourth quadrature one integral basis orthogonal quadrature polynomial derivative good quadrature orthogonal quadrature rule gauss quadrature lebesgue measure sequence point univariate adaptation smooth generalize interval typically quadrature essentially quadrature paper quadrature weight positivity property integral preserve constant exactly constraint require kernel novel conditional gradient improve several setting comprehensive example space good quadrature integrable property eigenvalue integral thus allow extension manifold quadrature rule sequentially improve quadrature error characterize space go recover partially perspective optimize adaptively interest outside scope minimum noiseless problem guarantee bandit bound outline section quadrature section eigenvalue operator quadrature expansion proposition I rely f properly bernstein concentration column sample namely small explain may quantity refer open order relate directly state eigenvalue also allow degree freedom decay max q prop thus need logarithmic sample decay eigenvalue get geometric decay cm cm em surprisingly tool constrain norm interpretation tolerance quadrature prop end quadrature namely equivalently computation density quadrature approximation norm strict qx e converge tend decay recover upper quadrature gx dx gx dx several kernel space cm output fx minimizer sampling feature lead prop base require worst n however regardless cm build evaluation amount noise decrease happen smooth rkh h l op rkhs number quadrature max rkhs compute quadrature bit get note estimation quadrature regularize consider characterize norm I op decay notice eigenfunction constant periodic simplicity dr prop degradation plus show quadrature rule expansion positive approximation quadrature applicable improve within work variety quadrature framework kernel parametric improve consequence stochastic support centre european author write paper three eigenvalue eigenvalue easily eigenvalue regular number multi easily time beta uniform optimize report space function integrate parameterized quadrature parameterize average convergence matching integrate less quadrature potentially bad compare quadrature gauss quadrature take uniformly spread compute kx large computed point happen dl norm eq introduce equal minimize define heuristic equal ax ax v dx adjoint overall f f f f performance goal v x eigenvalue less
display display fig reality collect generate cm generate period retain obtain histogram marginal long histogram together design observe posterior display appear decentralize towards posterior retain shape seem gain correspond layer gain characteristic design posterior posteriori show agreement slight overall red black incorporate bayes discretized x j kernel variance reference kx top fig posterior burn period retain obtain distribution display significant divergence shape seem information gain present location collect low informative something assumption reality prior present maximize information additional address issue well computational optimization apply problem identification contaminate system medium flow location measure set adopt code simulator derivative optimization process address accelerate surrogate crucial otherwise validate inference show location informative posterior true limited resource performance acquisition highlight monitoring environmental risk economic I I precisely proportional clearly control control bayesian two phase crucial restriction informative bayesian translate update expect design task alternative design criterion addition burden address concern contaminate area validity simulator approximation evaluation methodology demonstrate set field increasingly need area accelerate expansion range health consequence monitor never key credible procedure characterization across ultimately physics assessment flow functional assessment issue resource vast challenge year decision work worth result uncertainty reduction next address analogous recently formalism experiment characterize flow media work worth phase phase make utility quantify worth collect evaluate general design latter usually add economic concern role uncertainty design attention present arise objective explore criterion candidate worth without concerned economic providing decision criterion functional fisher counterpart design criterion nonlinear choose utility work maximize gain high model intensive surrogate monte involve evaluate criterion optimal design rational basis decision mention fig paper actual site collect site locate figure site dot site specific investigate step storage result initial flow medium correspond specific observable distribution observable concentration kullback kl fix optimal location determine distance location provide update improve prediction concentration criterion bayesian elsewhere scope address algorithmic implementation monitoring strategy organize optimal inference stochastic approximation toy apply site subsection accelerate computationally intensive experimental validate generate scenario conclusion summarize fix resource reduce uncertainty sense design fix thus instance smoothing kernel alternatively present vector coordinate plane also consist nonlinear functional denote prior attain restrictive attain reduction uncertainty numerically conceptual difficulty infer parameter posterior posterior update rule integral pd shannon kullback leibler kl divergence quantify gain spirit define expect output set gain experimental trivial unknown evaluate approximated replace estimate evidence prior carlo identify exhaustive grid search limitation computational expense report easily become infeasible design involve expensive forward incorporate adapt monte carlo instead adapt gain maximize arise application direct entail appear proportional sample require also furthermore variance loop sample application include hundred prohibitive contrary bind see unbiased loop something achieve accuracy much low number sample denote measurement normally relate typically incorporate purpose rather assumption among measurement finally use jensen namely q low equation eventually need term datum new linear regression problem minimize expand eq see carlo derive difficult design solver evaluation become prohibitive maximized turn root noisy root optimize iterative constant respect explicit attractive versus update step appropriate random zero correspond success probability goal explore bind substitute gain numerical two respect uncertain quantity prior initially explore carry second location observe case direct monte posterior sparse quadrature infer reproduce gain distribute sake illustration display perform carry present real show use finite simulator module simulate volume energy consist estimate increment step several density etc various r module intend water change decay phase model law decay decay interpret decay explain dependent characterize r water air parent phase simulate locate approximately deep discretized cell dimension paragraph investigate uncertainty site take domain although mainly effect assign half molecular describing detected name molecular gram molecular weight gram simulation c formation heat heat j factor initial pressure pa initial assign pressure close state ground initial mass zero pressure initial simulation assign initial condition location approximately area storage see purpose inactive volume cell assign value per randomly simulation minute finish implement thousand impractical necessary create would evaluation run unknown material choose make cover cm si semi material find solely material independent uniform admit q modulus variable independent gaussian gamma basis multidimensional polynomial paragraph polynomial version write number expansion coefficient fashion need general done implement convenient simulation see coefficient calculate multidimensional method calculate evaluate root least linear regression take tn form select square give exist demand simulator impractical processor one week allow n use hypercube expansion include coefficient close choice concern convergence polynomial fall scope purpose coefficient sample first know statistic test truncation estimate statistic expansion leave regard particularly expansion thorough include fit along east boundary truncation error expansion median upper quantile good bottom paragraph expansion model output bottom perform bayesian enhance potentially gain analysis observe depth generality good location moderate appear satisfactory validate beyond x x subject additional indicate involved information gain bind derivation solver sophisticated deviation proportional quantity factor contradict derivation red location source plot maximize performance case monte vary variance unbiased maintain three evaluate result run objective evaluation iterate much maintain low fig idea
perform convolutional datum independently first see fully convolutional hand pathway demonstrate depth intensity significant quality alone describe modal pre share one correlated taking suboptimal resolve architecture require test blind fusion fundamentally correlation see capture layer hyper modal initially separate intensity video channel effective stage nature hand modality correlate rarely beneficial channel hand fuse cross complementary skeleton motion audio initialization specific pre fusion pre relate network train network effective fusion quick degradation share fusion strategy gradually powerful strategy among fusion strategy mean classifier complex gradient descent implement early non straightforward dropout activation geometric output arithmetic well quality consistency geometric fusion output layer initialize architecture diagram indicate matrix conventional visualize interpret structure clarity vertical size modality hide specific hide target share layer size output weight think matrix unit column share specific block meaning initialization phase force procedure modality train cross modality capture impose evolve eq initially relate notation stand share output channel relate number first contain weight responsible inter correlation force comprises unit modality layer block softmax activate else initialization force output fusion mean modality initial force stage relax later fusion multimodal concept number weakly separate modality shared avoid modality handle channel key share would meaningful modality expect signal formally consider model represent q output ground regularization weight initialize diagonal relaxed objective formulate indicate modality formulate fine path modal drop certain accordingly non zero minimize correspond advance follow network activation input denote come output weight unit l ns come unit minimize cross target drop eq pt consider corresponding modality modality drop preserve formulate activation output relate involve bernoulli selector activate channel activation concern output unit gradient correspond q drop bernoulli e get expectation expression approximate exception selector derivative calculate derivative sum weight minus modality need stress correlation involve multiplication channel cross product analyse case channel unit come modality uncorrelated network expectation eq q pt network lyapunov lyapunov central product input tend result centralized magnitude input assume vanish number regularization prevent interesting belong modality positively product grow logic apply input correlate enforce correlation accordingly correlate modality act cross regularizer discover signal dropout multiplier proportional sigmoid activation magnitude weight mid range play less role weight introduce adaptive regularization input unit vote strategy fusion single introduce weight meta classifier quickly per output prediction obtain rate increase wider slide window stroke post stroke overlap pose class stage appearance vast temporal employ simple address additional period activity precisely point fully pose descriptor frame example frame right consider negative thus motion module frame output end typically noisy boundary close switching point detect boundary people rgb stream vocabulary category recognize rest participant explore dynamic modal version annotation distance phrase due alone surprisingly challenge dataset augment take neural table architecture identical temporal unit module tangent optimize early prevent overfitte additional fusion temporal scale section deep implement library operate frame gpu c filter unit share pt evaluation challenge adopt quantify sequence prediction frame mark rest addition ensemble iterative descriptor purpose explore relative beneficial combine depth intensity extract hand pose plane histogram pose third dimension third comprise depth reflect temporal hand extremely randomized fusion iterative architecture baseline recognition pre order word phrase period activity class list treat c team team chen et wu author modality cm cm pose video video audio localization challenge win hybrid combination deep baseline second note multi achieve one work percentage optimize architecture video skeleton path employ advanced fusion procedure challenge neural architecture modal per test useful capacity typically temporal pose correspond temporal prediction refine localization module stream contain also insensitive spatio temporal nevertheless duration roughly length cover participant channel propose alternative pose video subset entry al competition wu validation test test competition video index localization virtual visual modality hybrid visual modality isolate provide architecture mostly alone gain obtain alone localization couple experiment localization module contribute significantly c precision recognition comparison recognition audio extend introduce speech dataset actor result gain performance modality audio alone next quality audio temporal performance dynamic pose duration overall speech alone perform partly result audio localization annotate phrase moreover style either delay ahead alone predict poor compare representation baseline involve recognition report accurate localization audio possible recall case detect temporal truth context employ drastically improve recognition different start audio fusion multi modal classic deep consist handwritten digits augmentation hide convolutional digit formulation obtain dynamic modal optimize architecture modality channel aspect c fully training dropout dropout visible segment segment clean segment corrupt corrupted segment segment currently fully activation mnist exploit strategy redundancy switch structured network separate layer connect capacity case uniformly modal optimize unit channel turn due drop drop separate capacity restriction place row error table mnist sensitive dropout pt modal optimize balance operate hard constraint real capacity experiment insufficient modelling specific lead degradation whole typically thorough hyper fusion initialize fine layer share layer block section speed observe bias critical mnist optimize validation drop degradation architecture comparative analysis report provide per indicator path layer pre block case block effectiveness observe positive interestingly dropout network noisy channel regularization result respect dropout signal audio channel hand method modality information spatial body whole operate temporal extend augment channel depend sensor pathway without structure video scale integrate explore aspect multi modal term complete modality drop channel wise obtain stable input corrupt partly g student france work action recognition modal aspect taylor university google interested computer vision motion science technology national institute science university machine inspire vision emphasis university thesis co lead different project dedicate human robot de france work computer team human rgb multi spatial modal base scale multi modal capture motion body operate temporal strategy initialization modality fusion dropping channel cross preserve representation recognition track modality allow classifier well noise channel ensure robustness miss channel produce available modality demonstrate applicability fusion modality nature augment audio neural learning deep rapidly grow human interaction effective variable take typical scenario infinitely kind motion real constraint computer demonstrate previously object localization recognition galaxy claim reach face extend explore partially explain orient version competition core aspect approach employ network call dynamic scale visual modality integrate intensity depth pose make decompose multiple spatially grain pay special develop label challenge strategy network robust corrupted channel scheme augment channel arbitrary audio classification major present develop modal detection localization augment channel arbitrary nature inclusion fusion multiple target co ensure missing audio enhance immediate recognition address raw multimodal fusion action distant recognition video extraction spatio descriptor follow classification near accurate reconstruction dedicate infer multi resolution spatial pyramid frame path pose output per score modality depth video
rate far warm next initialize latent price entail w yield department engineering david computer university price develop probabilistic learn strategy set historical fit price new modeling decision estimation solve variety mechanism variable network scalability space price company minimal high transaction company bid bid price maximize company want price future bid imagine company advantageous exactly pay company price run million item learn historical word item advantage paper predict might potential average predictor price maximize second price bid dash smoothed approximate actual typical value function asymmetric high bid bid bid formally fig put bid predict bid bad price regressor fail reflect price advance yield probabilistic difficult seek specifically formulate price study predictor historical maximize problem turn posteriori new objective variable price parameterize draw objective note objective model imagine parameter prefer parameter find spirit technique decision decision find help decision generalize neural price price ref demonstrate optimize price quantify yahoo previous optimizing build idea research demonstrate mechanism nonlinear sec relate recent idea reinforcement markov process amount maximize binary reward likelihood solve similar addition learn simple policy set high bid bid various characteristic date time day date average price open market execute price determine receive price illustrate historical price feature feature regularize regularization regularizer optimization price set price mapping bid consider highest high high second b much predict price much account directly high bid difficult convex address iteratively dc solve result dc expectation price center around become probit principle parameter however latent em update e replace nonlinear predictor parameter specifically high bid next bid interpret relate pz b ir around observed outcome maximize center linear equal smoothed plus involve thus smoothed tp line distinguish set mm distance mm thick connect connect edge connect latent price red proportional time historical attribute imagine variable regularizer compute given take respect previously model price ascent bind optimize price standard appendix integrate real posterior expectation function complete model predictor step amount ridge initialize price update step integrate price terminate change threshold least square advantage change parameterized prediction technique nonlinear predictor outperform linear algorithm nonlinear unchanged e change feature price become kernel gram product degree gram without evaluate operate space technical demonstrate replace lead work computational nonlinearity network layer h analytic instead neural dc term compute oracle know bid advance report ten train split exist
criterion program propose tracking technique ucb armed propose preserve reasonable compare original probabilistic computation argument value value return draw argument call upon return argument upon return program run termination produce sequence induce pair distribution program trace choice simulate invariant mh drawing sample distribution reject next mh offline adjust mcmc criterion proposal target dimensional either systematic step modification sample metropolis hasting select subset proposal point vary probability provide target scheme program course via joint trace differ algorithm random select execution resample start initialization provide lie support random precede choice sample choice equal form trace let trace define modification accept reject accepted output description algorithm indeed essential aspect one influence output quantify influence must translate trace part computation extensive literature probabilistic user output program variable mcmc objective accept indistinguishable reject acceptance parameter proposal propose variable new accepted change bernoulli modification change choice change consideration program probabilistic produce type choose identical output quantify introduce quantify fraction output choice define total hamming adjust compute reward generative modify program remain trace update value accept early reflect program section probability variable variable due update scheme modify delay variable variable maintain component variable show line maintain history ensure cause get modification ensure ergodicity condition degenerate equilibrium selection weight accept change trace accept analyse sequence shall subsequent arrival change occurrence reward count begin sequence add history sequence reward count end probability geometric unit match proportional shall analyse summation substituting appendix note program dimensional distribution program unit reward variable zero ensure unit reward family bandit ucb compute factor ucb lower preferable different bandit arm adaptive expect equilibrium proportional arbitrarily history mcmc ergodic fundamental algorithm adaptation informally must decrease zero technical program broad specification density many adaptive invoke concept versus choice crucially preserve restrict admit program choice reduce adaptive mh algorithm ergodic suitable necessary satisfied ensure across positive language restriction induce regularity leave precise program adaptive scheme adaptation ergodic next demonstrate program evaluate many observe verify number adaptive program sample effort engine kullback kl time number difference scale plot run reward hmm transition trace predict divergence bar plot reward choice adaptive exhibit fast whole many approximation median quantile bar provide insight bar bar reward bar right bar plot height bar unit exploration low unit reward final immediately select often unit converge study define form mx ax bx kx x program value hyperparameter observation predict infer distribution run take bar reward sample count adaptive range bar choice choice predict require low acceptance bar green bar unit reward converge dynamic choice selection involve large amount classify specie dataset observation indicator variable fit split leave dataset belong specie run exhibit fast half many classification kalman previously describe dimensional impose additional assume simple velocity predict prior posterior condition simulated matrix qualitative consecutive chain
would aid start knowledge rate another trade payment help otherwise pay truth scheme payment tackle challenge digital scheme truth identify probably side summary complete prototype impact prototype remainder organize prototype experimental work section adversary payment exhibit behave digital independent e group view omit view due besides act indeed want server specification different assumption different scheme simplify assumption purely digital zero weight happen weight scenario view truth view suppose view separately adopt weight weight prototype server start want specify payment server server apply payment finally trade confirm payment define view introduce three prototype three stage detail st payment confidence pay whereas pay specification weight server server nd mode view server total view assignment server evaluate calculate rd truth payment else function payment e payment trading rely prototype assign view calculate introduce prototype assign view implement server start several truth reliability derive mean error view view start server eq adjust view divide group adversary average equation calculate view restrict suppose maximize great value maximize small combination gaussian unfortunately verify consequence unable derive likely namely weight assignment normalize view probability prototype usually statistic source say influence external limited china statistic approach relationship provide indicator ground deviation view ne bb statistic growth year weight prototype view assign median value calculate result source implement k vote spirit relate category vote let assign view view weight equal difference growth voting source initially year statistic full name error capital formation production worker fix balance secondary ti cccc source payment function different payment sort view v mc v unit ccccc ccccc method vote voting source source method vote voting source ccccc ccccc voting source payment th payment use ground trading growth rate say confidence level confirm payment list payment voting see among weight improve three factor prevent find improvement per payment truth payment fast payment follow view see payment grow increase payment consider variance factor factor randomly see grow rate change payment decrease receive payment least level small similar design scheme crowd relate focus issue firstly blind digital signature trading consideration signature construct signature publicly fair recently add truth suffer quality view find heuristic knowledge fact sensitivity specificity use reliability source calculate crowd
take related gradient even represent however helpful naive implementation hmc update iii term order result sec dynamic desire system framework consider correction distribution outline momentum r scenario explore complex dirichlet large wikipedia find rapid trait sampler kl divergence right bar aim correctness assess high full choose sampler naive implementation converge correct addition efficiently show pre hamiltonian help element explore contour plot versus number wikipedia include fisher membership corpus wikipedia three run lowest report expand parametrization sampling distribution discuss incorporate riemannian riemannian sampler benefit gain hmc pt present sampler markov construct sde two matrix skew devise continuous process prove cast particularly stochastic sampler propose scalability method streaming wikipedia fa mr wu complete write far decompose term q equality compact constructive theorem existence notice matrix hand fourier multiplying arrive write side substitute nice eq n variable clear skew arrive inverse fourier sl process turn new skew ik convolution real dimensional q differential ingredient motion position surface r discretized hmc practice continuous careful show naive hmc interestingly author prove correct eq stochastic noise r interestingly physical interpretation interpretation term langevin dynamic framework hmc rely additionally detailed momentum langevin correspond take variance finite stepsize simulation accurate lead score information metric sampler dynamic q fall correction term take correction lebesgue determined framework provide correct method incorporate idea far auxiliary algorithm take r ccc dynamic framework university edu markov adaptation define transition explore mcmc via subsampling gradient require physical modify account gradient general sampler include gradient trivially previously stochastic propose adaptive streaming sampler become mix model scale poorly decade rise provide efficient hamiltonian monte carlo define explore landscape enable proposal gain burden large quite langevin minibatch mcmc notion langevin dynamic adding amount iterate posterior hamiltonian monte build incorporate provide hmc momentum term naive efficiency mcmc leverage hmc show complex desire stationary novel dynamic ensure challenging require physical natural gradient minibatch method target quite importantly jump hmc variant development stochastic positive diffusion matrix skew symmetric matrix stochastic dynamic explicitly vary explore mcmc maintain stationary completeness although provide take avoid significant specific modification leave question define explore direction choice framework new building sampler synthetic streaming mcmc drawing distribution like hmc auxiliary desired represent augment state discard perform desire marginalization hmc translate simulate sde discuss characterize stochastic simulate stochastic sample mh straightforwardly meet step costly entire mh correction short period stochastic dynamic correction sampling write sde deterministic relate dimensional wiener clearly devise red continuous represent continuous define choice theorem stationary stationary blue corresponding method discretization sde lead rule calculate computationally intensive eq potential form unbiased full gradient key gradient distribution analyze impact make result hamiltonian update variance satisfy rule term stochastic gradient get bias distribution design meet maintain target avoid need however small practice bias tradeoff sampler addition pt mcmc choice sampler mistake implementation ingredient hamiltonian simulate motion object position simulation special u stochastic discretize hmc update arise careful hmc interestingly author naive indeed compare see physical interpretation term langevin fit hmc sampler correct mistake rely intuition readily sampler sampler propose momentum
instead difficulty know polynomial even norm tensor approximate nevertheless approximate performance theoretic completion get good connect appeal atomic observation agree hierarchy universal broad polynomial end natural upper sort tradeoff present previous inverse considerable recent interest understanding area possess inherent run efficiently problem information widely however assumption prove problem assumption preserve input sense step powerful think explore machine algorithm design something provably well reach sum hierarchy sharp phase impose phase transition section decomposition numerous learn hmms modeling detection perhaps random contrast tensor observe tensor tensor highly tensor factor constrain orthogonal enable decomposition tensor tensor tensor whose work elaborate tensor useful keep view offer satisfie make precise think standard transformation clause observation informally maximal agreement readily upper clause refer clause hierarchy bind fraction clause improve work formula clause go clause algorithmic machine also connection make powerful tool complexity need round straightforward hierarchy originally natural hierarchy upper extend tensor follow unfold author complete picture work view pseudo see eq observe entry j triangle inequality chernoff feasible good true type optimization design atomic balanced atomic even approximately well computationally hard balanced require dual hard third instead induce square atomic operator pseudo polynomial suppose say every behind dd converse true nonetheless upper pseudo satisfy kx throughout definition size set exist call sum atomic bind resolution norm completeness see tensor independent bad give j j concavity rademacher random k k j complexity generalization bound replacement set move model invariant think triple index choose tensor tensor convention section sum order triple triple inner remark straightforward rademacher atomic discretization considerably let convenient suppose arbitrary expand moreover chernoff bind q set soon nearly good hope algorithm bound norm albeit dependent resolution strongly formula six repeatedly degree expectation pd bound bound respectively eq think map case multiply maximum care remark index index come clause decompose matrix separately matrix pseudo hence hand matrix separately part claim expectation proceed pseudo nonnegative claim easy section follow eq row still index clause instead triple use otherwise vice versa eq event triple contribute contribute high probability moreover contribute ensemble ensemble q sign independent expand trace k u v indicator cover early ignore variable odd time encode encoding appear distinct distinct give encode convenient encoding question appearance appear appear step value visit visit visit already answer easy answer arise work remove encode compactly term bind encoding uv occur exactly identical encoding step new visit current similarly expectation exactly distinct distinct set positively mutually variable return task triple q conclude equality recall arbitrary element rademacher q readily make plug square follow convex sized plug concentration feasible return error rademacher theorem translate imply refer satisfie fraction clause satisfied whenever recall constraint balanced fraction clause desire hold moreover rademacher immediately low absolute typical value think invoke prove os rr fraction let k k invoke solve satisfie noiseless noise fr definition atomic show rademacher relaxation resolution norm follow introduce hierarchy boolean f v f f kf f f satisfy feasible program solution relaxation think convenient correspond character round hierarchy u ks switch feasible clause constraint feasible solution function complement moreover feasible consider identical since thus complete particular multilinear define multilinear replace multilinear z construction repeat multilinear need c verify accordance complete random formula permit immediate rademacher tensor resolution norm even relaxation round square optimize trivial hold tensor atomic th norm immediate general formula completion ignore entirely almost conclusion predict truly possess measurement need inefficient conjecture formula clause condition fraction clause imply clause thank note give show upper fraction clause direction implication like term computational semidefinite interesting explore several notable possible speed semidefinite programming e round even hope speed application find speed prediction provable minimization work orthonormal guarantee like many helpful discussion copy rgb lemma theorem restriction corollary conjecture definition rgb rgb support mit google award accurately tensor sum hierarchy work attempt hierarchy moderately theoretically suffice broad square hierarchy linear natural characterize rademacher connection formula advance broad solving problem recover unknown object possess special structure perhaps compressed sensing show incoherent observe general inverse nonzero low challenge turn obtain relaxation q solve efficiently interest succeed compressed px nuclear result computationally theoretic significantly prediction define many phase retrieval principal resolution
overcomplete accuracy compete recovery vector removal impulse handwritten remark future direction selector incorporate overcomplete dictionary signal reliably suppose sensor noise overcomplete spike concatenation transform concatenation orthonormal basis component admit selector sense dictionary express use overcomplete two representation bernoulli element fix largely incoherent employ sample bernoulli overcomplete basis frame give isometry successful compressive sense selector fix equation involve proximity equation finding let convex indicator exist therefore also proximity selector incorporate overcomplete proximity proximity guess generate criterion meet construct signal iteration ideally terminate reach meet b selector tend support let define whose element selector solve least square begin equation stage compute multiplication stopping criterion meet contribute length vector iteration complete separation composite signal demonstrate code implement intel ram observation sense sample unit noise respectively algorithm cpu run recovery fourth world united service handwritten digits composite signal overcomplete individual composite composite accuracy however significantly complex dictionary problem table follow second recover value separate wavelet cosine transform composite wavelet length signal select signal observe overcomplete form level haar discrete cosine transform cpu component simulation c std std std std std mean std std std noise composite one dirac spike signal location coefficient experiment vector support entry illustrate numerically value simulation standard simulation std mean std std level std std std level composite signal select set overcomplete concatenation fouri specify signal observe accord plot signal htbp separate std std signal handwritten handwritten digit class nine collection vector form form abuse whose vector select test random overcomplete set principal two digit composition recover digits integer column overcomplete dictionary apply component yield small dictionary coefficient vector explain digit composite appropriate overcomplete decomposition leave vector recover composite space principal identify reduce determine residual generate match accuracy separation image algorithm overcomplete training experiment read composite recovered strength noisy composite signal experiment table composite separate use demonstrate distinguish range figure separate smooth impulse demonstrate separate overcomplete component fairly increase system practice demonstrate noisy underlie overcomplete element yet fast moreover readily involve dictionary use real introduce selector dictionary separate composite signal additionally iterative experiment support cpu applicable wide problem foundation separation signal include image moreover advance compression communication composition reliably component another add compressive composite selector incorporate overcomplete collection composite selector noisy minimize
distance dx kf f dx dx essentially quantify close disjoint band present km introduce km replace define set step psd bt accord window identify entry step construct z normalize km observation carry psd bt x compare distance meaningful comparable average henceforth adjacency determine form component assign come generative hence eigenvalue laplacian bt psd compute contiguous fw white variance across autocorrelation function f moment l correspond length result false connection connection although property alone guarantee sensible appear difficult finite length accord describe km guarantee thank km easy proof theorem km overlap observation contaminate provide sufficiently window bt psd quantify tradeoff overlap k rhs vanish particular increase large km observation length unknown quantity large like come close spirit show come sense via psd distance positive measure finally process cluster cast vector demonstrate clearly thank exploit available synthetic performance solid index table mark dot x index plot anchor west xlabel xlabel xlabel height entry km km font style font legend font none x mark none dash black index mark none dot black mark none black index human activity experiment motion contain sequence activity marker body record optical sequence respectively cluster marker subject difference length length normalize bt psd power ce confusion matrix value ce report subject perform subject outperform algorithm km c c c ce ce ce human proof prove condition theorem say condition observation model close measure follow property imply km select observation contain generative noting suppose km l g generative underlie run guarantee iterate contain model f triangle lead similarly rhs imply upper bounding u g I e sequence e r r last remain bind accomplished tail obey u r toeplitz covariance consecutive element tail concentration namely toeplitz f b red green rgb definition cm cluster finite ergodic nonparametric knowledge generative algorithm dissimilarity term near knowledge rely via simply km initialization consider literature albeit dissimilarity km provide length tradeoff noise synthetic stationary want knowledge number generative divided meaningful processing example audio video sequence production dissimilarity euclidean divergence g divergence distributional fed cluster infer posteriori effective analytical result mostly
put subsection theorem sequence f induction lemma generate f dx estimate boundedness learn q exist desire firstly yield increase must hence part eq complete know rademacher average class function rademacher random rademacher average define g independent state principle useful gaussian gaussian process countable average let j index parameter let variable process follow denote lemma x k hold q jj gx desire f critical part let jk combine complete analogy easily counterpart guarantee give tx jk gradient z partly derive sufficient state hold e x induction theorem notice apply put third certainly verify eq recall eq b bt r uniformly recursive q analogy argument fourth classification learn guarantee establish explicit rate decay size contrast mainly focus online novel refine property average direction firstly instance least loss achieve rate remain secondly kf remove loss clearly future remove loss function popular hinge loss remain algorithm hinge lastly would interesting convergence last iterate acknowledgement thank comment suggestion grateful lemma paper grant proposition corollary explicit extend refine continuous establish guarantee iterate establish first polynomially decay tool refine reproduce hilbert bipartite ranking complete consider learn identically classification univariate predict label predictor generalization classification notable bipartite auc aim pairwise true pairwise online pairwise hilbert rkhs semi rkh linear satisfy reproduce learn throughout specific define follow concept introduce loss typical loss loss logistic purpose online usually draw tx square vary deriving refine square loss gradient contrast size form step iterate proof soon new simple powerful handle learn rkhs eq pairwise involve g turn rate explicit uniformly furth f maximal loss gradient maximal theorem immediately since bound f derive role h old l l definition desire role part eq b generalize general case complete end comment main algorithm inequality old part part take
subsection utilize min band threshold data vary size available lie outside explain optimal day slot detail supervise unsupervised algorithm size accuracy svm attain htbp comparison vector machine naive tree hide member paper technique technique naive vector unsupervise study highest detailed perform classified status utilize secondary spectrum share numerical svm unsupervise propose new support vector cognitive compose type secondary core behind cr access band interference understand spectrum towards realistic spectrum usage various spectrum cover wide study lead researcher spectrum characteristic depth exploit spectrum many statistic spectral frequency autoregressive availability achieve datum secondary utilize transmission purpose less switch control cognitive series evaluate tool limited assumption require derive whether tool tool machine assumption conventional ml spectrum cr aim comprehensive investigation analyze motivation often good ml list propose study use advantageous capable environment region space sense ml spectrum cr management sensing discuss analyze collect walk aggregate cell band dt svm unsupervise hidden classify status status far utilize evaluate status supervise modify hmm lr investigate spectrum approach outperform well technique outperform unsupervise organize ii detailed explanation contain eight band eight band bin band example band bin band bin arrange row frequency represent four constitute minute column vary number bin band user band interference let slot frequency bin total frequency bin energy time slot bin q represent zero computation explain status bin three minute frequency bin decide bin minute quantify chance slot decide rule minimum consecutive bin represents consider vary band frequency day evaluate order guarantee transmission lie apply apply evaluate slot free bin slot vice versa b ml construct feature train train n fed classifier successfully ready test tp I n sequence assume divide reference status slot class therefore correctly give alarm occur evaluate slot allow utilize slot free transmission occur length consecutive present start index evaluated utilize predict status supervise dt lr unsupervised motivation five characteristic naive bayesian call dependency feature account slot represent status response status evaluate bayes classify classified find rule explain iv b decision build tree leaf dt divide subset node label case tree regression tree label label represent record belong iv fraction affect classification unlike dt prevent svm separable datum separable classifier vector define division represent divide divide give q separation margin box I try light intensity evaluate find another intensity determined represent position flexibility motivation slot criterion select square criterion aic square increase unsupervised need sequence recover observation state state value define array produce array main emission probability utilize sequence produce viterbi likely generate viterbi match accuracy forward matrix emission maximum estimate evaluate viterbi eight statistic
edge edge edge edge edge edge edge edge edge nonparametric hand document q equal convenient slice beta constrain property propose gibbs distribution truncation big technique difficult truncation form distribution conditional proposal eq finally w w summarize implement parallel initial k update eq truncate commonly accept maintain consume elegant call resolve adaptively version slice decrease construction gamma posterior sampling distribution update update update eq whole slice summarize effectiveness number topic usefulness generate explore hide choose set ground generate global dirichlet distribution parameterized interest topic parameterize interest follow number word firstly draw document word finally row word relation document inner product relationship retain one adjust dataset topic count bar rough gibbs mix citation link publication dataset consist unique citation publication absence unique cross performance split five stage four scalable toolbox document link evaluation design link link link document interest topic document word link vector word evaluate topic topic interest probability link influence model e topic link document word accord interest fold show fig clarity denote slice algorithm slice version slice keep initial guess normally mixed left setting fit word outperform every notice less accurate see link come topic tend reach link possible number within topic around knowledge topic argue absence accurate domain well discover topic document world predefine relax distribution relational necessity time introduce difficulty therefore present truncated slice experiment dataset real world dataset ability future interested scalable network mrfs mrf constraint support research arc grant dp china foundation china receive china currently technology research interest machine network web associate engineering technology interest support publish book five research discovery grant grant excellent serve international intelligence special issue international six international zhang engineering technology mathematics university mathematics university china associate interest decision fuzzy fuzzy four book conference four grant xu receive engineering south ph sciences university technology school compute communication computer vision computer china currently receive master technology china grid technology chinese science main interest cognitive co appear science computation experience computing etc transaction system technology also number include program co member zhang da xu traditional way discover hide document prediction benefit reveal hide advance impractical relax relational topic probability generative elegant bring spatially document tend document resolve assign global subsampling design discover simultaneously capability importantly nonparametric markov corpus instance paper person understand corpus discover corpus topic service paper organization resolve concern help understand interest person provide accurate service citation example link citation link document nature apparent discovery study network successfully develop mining topic make link consider topic drawback dimensional multinomial gamma require hidden advance normally domain difficult fail document drawback topic remove necessity fix simply dimensional express infinite gamma process two tendency feature find requirement formally relational network document database document retain design superior performance hide topic note use example contribution nonparametric relax assumption use truncate version summarize model derivation synthetic conclude study discussion briefly review aim network link generation link trivial call mixture suppose infer traditional dirichlet normal avoid process replace former distribution limit satisfie dirichlet dirichlet good alternative dirichlet three method schema chinese restaurant dirichlet schema cluster chinese restaurant process constructive process infinite mixture infinite finite gaussian dirichlet dirichlet topic allocation chinese restaurant number dirichlet infinite properly successful mcmc inference successfully many many relational topic gamma extend finite relational propose nonparametric detail handle need gamma document topic process
frame tight frame ds invertible frame shift invariant frame leave discrete index label shift instance illustrate base discrete directional wavelet atom label collection invariant countable q cm circle cm circle tensor discrete discrete right fix translation thank key let frame upper l minor thank ingredient argument band function yield operator l da integral kx du l dl k kx end xx kx inequality complete q remark convolutional specifically call identical frame prove translation invariance purpose importantly frame frames wavelet layer class transform invariant result wavelet base feature particular algebraic frame may want detect handwritten digit temporal location handwritten digits robust linear success practical theory modulus wavelet translation stable linear moreover effective deal signal dominate natural transformation audio transformation contribution goal theory cope transformation wavelet translation major contribution stability wide structural transform simplify short proof invariance wavelet behind theory continuous notation material dx j rx df f fx operator dl dl derivative rapidly decay df dm fx operator denote gradient jacobian jacobian vx vx filtering technique modulus extract pass filter function label wavelet filter satisfy reader short discrete frame wavelet apply element wise prove prove state q distance mm child node f fill child child fill none child fill none child parent none child fill parent draw child child child atom reader might want frame frame applicable g frame course wavelet introduce network generalize allow addition layer require modulus convolution index frame I l f operator note f inequality gene network frame put piece nn atom semi define mm mm circle child parent none none parent child none parent none parent child node pt child circle fill circle fill circle child pt fill child child fill none parent child fill parent none child circle child circle atom discrete frame associate result feature translation time wide one pass band stable depend appendix result retain feature derive condition easily normalize frame accordingly neither translation see technique algebraic frame accomplish state employ argument
quantification order assertion due apply corollary duality assertion follow apply calculated conjugate programming duality safe empty assertion substitute expression open variable value belong uncertainty optimal set e ks ib ns coincide sample outside analogous reduce inside polytope major challenge implicitly define program linearly uncertain problem observable loss generality dependence stage two stage programming uncertainty polytope corollaries feasible non empty compact give parametric empty compact vertex eq assertion infimum indeed follow classical applie assertion follow well assertion rely equality hold due strong apply set exploit elementary observation express pointwise linear assertion ii expect ambiguity free variable case reduce computational wasserstein reduce program piece underlie except uncertainty program scale polynomially description tractable number vertex polytope program hx kx reformulate program hx q hx hx contrast examine wasserstein rich first problem view separable study uncertain maxima concave assumption bad uncertain stochastic assume jt appear instance open loop induced wasserstein reduce norm summation express combination result would solution may overcome wasserstein define process small satisfy every bad case summation separability auxiliary hold provide inf note tractable case wasserstein ball satisfy irrespective sequence decision whose supremum wasserstein attain convex program omit brevity exposition emphasize theorem discretization unless affine pointwise concave function may loose upper expectation wasserstein loss equal bad expectation coincide exactly radius small ball contain effective conjugate term arithmetic interpret bi domain conjugate conjunction minimax maximization obtain denote seem simple arbitrary uncertainty maximization lead corollary employ function conservative substituting trivially coincide extend sample loss adjust proposition highlight theorem mm lipschitz e directly conjugacy equality explicitly consequently ball imply portfolio problem simulation provide optimization capital market return capture short range simplex mx iii aim single q portfolio quantify average high portfolio replace definition solve counterpart wasserstein set proceed ambiguity optimal portfolio wasserstein ambiguity set portfolio portfolio ambiguity analytical consequence computational non x x n nx dominate equality substitution hold minimizer portfolio cone readily recognize cone portfolio wasserstein uncertainty k n x uncertainty nonnegative shift asset definition unique portfolio assertion observation positive price far return decomposable systematic asset constitute high exponent say distance whereby reduce portfolio asset bottom dark asset dark red solve figure average independent run numerical confirm insight weighted wasserstein portfolio constitute average run line critical wasserstein fact across provide adopt robust figure portfolio solid represent event jx likely wasserstein improve consistently unable validate theoretically visual inspection suggest wasserstein respectively indicate empirical consistent indeed exponent wasserstein radius wish portfolio outperform various benchmark quantification section quantify portfolio distribution unknown dataset bind compute moreover replace interior another weak linear line line dataset visualize green line solid wasserstein respectively figure empirical emphasize basis dataset constitute rare bound coincide contain drop wasserstein reach imply almost fall proportional agreement radius magnitude small first large wasserstein curse whereby portfolio portfolio keep optimizer curse grateful valuable grant cm thm thm corollary thm definition thm robust wasserstein guarantee program uncertain finite training wasserstein ball seek view wasserstein state quickly mild wasserstein ball program leverage concentration solution risk portfolio uncertainty quantification powerful paradigm uncertainty generic stochastic find uncertain problem encounter increasingly world never must infer dataset two dataset term optimizer overfitte integral constitute affine distribute hypercube robust paradigm expect ambiguity characterize seminal gain modern optimization decade robust benefit adopt optimizer curse characteristic tractable even though stochastic surprisingly may offer decision counterpart easy ambiguity set ingredient ambiguity set rich confidence ambiguity exclude would conservative ambiguity easy ideally tractable structured program solve robust ambiguity therein metric leibler divergence wasserstein metric etc set distribution close nominal prescribed metric adjust radius ambiguity underlie radius drop ambiguity singleton nominal ambiguity stochastic wasserstein ambiguity identically sample wasserstein ambiguity empirical wasserstein modern measure concentration generate wasserstein ambiguity around large robust confidence achievable wasserstein offer guarantee maker control display property tractable wasserstein ambiguity counterpart art wasserstein ambiguity fix atom bad program via effort optimum paper bad wasserstein ambiguity numerous efficient constructing attain otherwise attain asymptotically wasserstein ambiguity set polynomial performance result theoretically main contribution wasserstein ambiguity coincide pointwise finitely generalization maxima concave bad compute modern finite convex approximate dimensional bad expectation reduce wasserstein metric affine function indicator polytope indicator complement open polytope parametric side linearly wasserstein ambiguity bind optimal validate numerical uncertain fix subset ambiguity substantially reformulate tractable elegant regression list tractable robust practically realization discretization technique ambiguity set one closed bad various measure solution come expense believe evaluate bad function subject ambiguity kullback attract attention portfolio distributional asset nominal focus kullback leibler ambiguity offer model flexibility chance constraint involve respective classical chance nominal rescale probability moreover robust ambiguity set leibl ambiguity fail offer paper generating ambiguity center indeed kullback ambiguity assign continuous kullback leibler ambiguity irrespective fail contrast wasserstein center distribution far elaborate fall scope drive optimization spirit robust seek pass prescribe hypothesis wasserstein ambiguity view fit training rest proceed drive introduce ambiguity establish performance bad expectation wasserstein ambiguity reduce program develop linear programming quantification problem derive extend scope broader whereby two denote conjugacy preserve indicator define define dirac probability distribution represent view stage interpret spirit solve partially observable past realization comprise view govern support drive throughout dependence constitute object govern evaluate hope tight feasibility seek performance constitute bind depend respect datum amount program loss problem argument whereas leibler replace wasserstein metric divergence modify ambiguity conceptual fail continuous variation distance conclude generate continuous rule meaningful choice kullback ambiguity variation positive contain irrespective kullback leibler nd leibl ball inner wasserstein ambiguity ease notation throughout function pointwise elementary j sign real arithmetic whereby dominate focus pointwise maxima remainder convexity close concave mild much generalization subsection case distribution constitute dimensional intractable demonstrate program leverage wasserstein modern distribution wasserstein metric constitute dual pair linear another describe wasserstein represent plan ingredient subsequent bad equal conjugacy represent conjugate support wasserstein bad case expectation eq law construct result generalized moment argument inequality operator dirac introduce auxiliary allow exploit maximization restriction result express conjugacy substitution program reduce virtue extended result continue hold reduce singleton constitute maximum elementary infinity minimax apply ij imply proper identically coincide closure map assumption infinite generalize nonlinear constraint necessarily concave constraint admit theorem immediately clear bad expectation evaluate whereby formalize approximate reduction program stress experiment instrumental decision stress test wasserstein systematically program case bad expectation eq irrespective decision wasserstein attain supremum highlight evaluate extend imply contrast evaluate regardless low mf z q conjugacy imply proper low perspective apply coincide theorem lagrangian evaluate convention duality appear show dual minimax fact feasible simplify objective reduce vanish whenever evaluate least reduce prof coincide term supremum statement marginal dual wasserstein mass upper bind feasibility conclude nk trivial equality construction bad notable program radius wasserstein distribution force objective thus simplify emphasize attain general attain supremum discuss admit bad distribution existence problem amount within wasserstein radius attain
tx sx lemma weak x x assumption strongly monotone establish three lemma scalar satisfy negative integer next hold eq inconsistent read read plug strongly generate q inequality take expectation side desire ready derive rate quasi monotone monotone q assumption straightforward problem goal implementation parallel machine ram code eigen library operation code publicly author website parallel operation affect cache addition parallel load finish every core assign feature either iteration number figure core reduce imbalance load explain core give ratio imbalance grow nearly number core load delay small work well core parallel epoch htbp news speedup time parallel implementation regularize speedup imbalance core conclusion coordinate sure strong assumption preliminary illustrate traditional parallel parallel acknowledgement organization paper would thank grateful helpful discussion l asynchronous agent machine processor core randomly asynchronous special novel decentralize converge strong performance present numerical linear sparse advance storage rapid diverse area internet involve modern grow analyze fashion asynchronous processor core agent execute parallel next determine speed parallel request parallel continuously memory access asynchronous failure asynchronous propose novel parallel coordinate note find hereafter convenience widely equation machine convex km nonempty k strongly km linear differential equation include iteration proximal splitting splitting operator alternate method multiplier admm algorithmic solve random asynchronous mp k update counter whenever agent update iteration apply coordinate whose normalize cache use memory apply parallel due parallel establish appear state include generate properly weakly addition fix point quasi monotone fix strongly fix make assumption selection impose c essential tail leave employ coordinate discuss advantage disadvantage prevent agent general store global pass network secondary disadvantage generation require assignment event advantage user every agent power amount therefore assignment load tolerance numerically coordinate selection convergence furthermore coordinate nonconvex fashion extend continue operator example generate scheme compute sensor area point mention simplicity solve nonzero diagonal equation note give multiplying adding follow equation nk agent continuously kx lipschitz monotone assume monotone hence reduce complete addition generate converge solve nonlinear nonlinear solve ordinary equation ode consider differentiable tx ss q easier evaluate rather give program variable guarantee jk modulus converge convergence term comparison recent bound require decay similar assumption solve decentralized consider agent connect differentiable hold decentralized gradient mx iw I doubly consensus express di see compute iteration eq l poisson otherwise activation agent equality constraint unconstraine iteration q convergence agent decentralize select assign update agent consistent follow summarize view asynchronous decentralize nonsmooth following processing well regularize proximal operator forward backward average apply separable th huber indicate box jk guarantee assume strongly quasi operator monotone function differentiable modulus quasi operator contraction operator convex know think nd paper case projection splitting proximal component block evaluate huber indicate separable evaluating avoid backward backward splitting compute small update evaluate splitting constraint operator indicate evaluate speedup asynchronous nonsmooth consider close proper convex relaxed operator operator solve problem solution finer naive intermediate next special subsection discuss feasibility nonempty intersection intersection otherwise feasibility formulate minimization z implement update efficiently hold maintain random read compute maintain memory agent accord implementation computing reading parallel admm operator lagrange subproblem involve plug structure efficiently block matrix correspond separable need decentralize admm subsection parallel admm consensus optimization consensus eq q nonsmooth admm dual update select therefore arrive parallel h k I k algorithm survey al decentralize agent agent decentralize consensus introduce auxiliary reformulate write proper follow whenever activate present ie ki li update associate asynchronous associated iteration side communication edge derive consider adjacent activate period compute delay inconsistent correspond stepsize lasso follow eq logistic hinge penalty specify reduce fuse asynchronous number assign worker subproblem solve inexact asynchronous proximal admm consider eq differentiable proper multipli condition monotone forward backward splitting problem monotone asynchronous iteration eq admm method purpose definite choose eq update calculate substitute asynchronous kk ki kx x ki mp k kk solve programming problem solve pursuit svm classifier eq kernel function previously list assumption bound delay delay assume order actual delay independent update assumption quasi subsection
fix linear response quickly calculate demonstrate scalability learn gaussian simulate increasingly interested datum past paradigm practitioner complex practitioner quantify approximate bayesian popularity fast runtime scale major approximate multivariate information uncertainty interact family individual elaborate exponential define posterior perturbation previously derive particularly point parameter multivariate gaussian employ mnist handwritten produce monte mcmc even variance dramatically accurate magnitude mcmc wide theoretically gaussian number point fast influence mention unobserved dimension give model belief bayes factorize kullback divergence factorization obey rest denote exponential natural write scalar entry otherwise notation guarantee contain eqs across mapping factorize distribution denote I solution technique improve estimate log perturbation probability interior feasible ball fact generate far perturb p vector scalar perturb perturb success often derive interpretation individual approximation derivation q substituting writing arise improper eq q normal length variational factorize component case posterior correct variance estimate equality location appendix asymptotically transform fisher amount datum go detail unknown include nuisance nuisance mixture assignment treat nuisance directly compute inverse impractical covariance able sub divided grow variable field factorization factor eq sized identity respectively finite mixture gaussian model investigate analogously much bayesian posterior perturbation convenient formula covariance add notational sensitivity dx nm refer derivative connection covariance version parameter posterior belief assumption sufficient account correlate cause proportional q detail value use influence score note covariance require covariance draw covariance influence divide three main nuisance parameter also distribution use perturbation nearly result perturb inverse take appendix mixture constitute application may illustrate efficacy multivariate gaussian normal multivariate dimensional mean component employ trick pz nk nk augmentation component multivariate univariate nuisance distribution influence speed gibbs augment function implementation heavily algebra datum world mnist handwritten digits principle center intensity keep evaluation project onto subspace training separate handwritten result keeping calculate expectation posteriori expectation count classification majority label test measure test stress feasibility practical covariance estimate sampler treat shape real interpretation generally interested marginal uncertainty pose standard restrict regime switch sampler avoid prevent mode simulation component dimension uncertain point cause mis perform sample calculate mh produce truth derivation deviation particular alternative deviation covariance closely mnist take second whereas explore detail eq linearly polynomially also experimentally sample fast though preferable grow length vector product involve sufficient redundant parameter simplify though perturbation argument require interior nuisance normal replace sized sized directly cubic scaling simulation scale slightly vertical examined demonstrate mnist derivative score score find new numerically calculate influence point mnist point range concentrate one score perturb dimension mnist covariance calculate second process score mnist score two approach practically indistinguishable simulation fig see pattern datum depict moderately graph assign sign change effect rgb seen overlap distant nearly ht upper influence mnist label random particular one dimension recall mean wish score value respect influence point figure correspond axis logit component component different mode mostly handwritten influence amongst component variational demonstrate estimate scale point quickly calculate score influence multivariate traditionally hope work model allocation prove factorization abuse write whole vector exclude exclude aid sometimes explicitly component let denote th product define way intend make multiply lemma additional always via eq suppose collecting way k allow exchange expectation expectation proposition proposition exponential family interior feasible ball radius within origin define cf approximate track true mean vary equality eqs depend write part factor dimension dimension zero analogous lemma know exactly follow x posterior multivariate normal let index final constant invertible invertible equation stack assumption follow interior exactly log interior derivative e covariance exact draw connection correction expectation asymptotically transform fisher matrix formally argue variational block unlike covariance
marginalization significantly rely rigorous suited alternative bayes computational load posterior parametric unknown form encode hyperparameter assume accord gp distribution conjugacy gp tool trend encode gp greatly choice choose likelihood point multimodal happen point integrate could beneficial obtain marginalization average range robustness marginalization marginalization treat integrate predictive computed unfortunately analytically intractable weighted approximated independent law hyperparameter new hyperparameter marginalization multimodal hyperparameter posterior far propose simplify tune load hyperparameter ability handle monte carlo smc sampler closely popular smc particle smc extensively understand apply parameter bayesian audio smc sampler respect internal smc application compare demonstrate marginalization computationally demand dataset target marginalization marginalization previously literature implement depend tuning scale slice propose sample solving approach possibly adopt hamiltonian monte carlo chain carlo tune curse problem propose approach smc connection pl marginalization make transition marginalization require particle particle smc idea construct density easy reader familiar section construct sequentially py n give evolve alternative prior geometric path guide sample smooth weight usefulness eventually particle particle residual resample decrease resample invariant sample proposal adjust otherwise repeat update increase mix smc online another easily point transition decrease design choice proposal concern proposal brevity adapt maximize markov explore part center rao give covariance py p accord sample sample interval size use choice piece summarize compatible toolbox computational evaluation number smc number mcmc transition evolve high instance experience often could beneficial particle decrease represent probably probably couple dot stack introduce spread lot close opposite cluster probably carry smc monte importance solid line red dot grid computation axis equally thresholding adapt vice versa absolute bound adapt theorem justify vary particle integrable far let nn sampler adaptive particle eq neither induction lemma benefit compare common method world different well affine function square noise measurement good repeat run seed give similar apply adaptive importance marginalization fast comparable might competitive curse sensible however note load substantially decrease hyperparameter utilize conceptually figure deterministic clearly initialization cause computational alternative although initialization illustrate marginalization competitive hyperparameter base online application marginalization efficient hyperparameter sampler problem fit inverse seven degree robot arm map joint total end use separate involve hyperparameter point subset select ignore propose hyperparameter point report prior variance method regressor standardize divide well otherwise
define rbf suitable learn dimension distance wang label active criterion reduction exploration exploitation initially explore annotation acquire exploit perform refinement density uncertainty hyperparameter complex automatically measure surrogate seek make overall unlabeled discriminative operation example minimize stem unlabeled pool error feasible demonstrate superior criterion field serve baseline cope must consider subsample selection assumption explore evaluate unable perform refinement statistic label information within cluster also representation differ advantage ability refine boundary review detail pool base active initially know pool weight space node supervise learn harmonic energy minimization harmonic laplacian diagonal entry matrix part matrix except label conditional distribution learner learner quantify predict true multi class case base marginal sample integral produce error true expect greedy error combination unlabele q add calculate label provide select expect risk remainder risk criterion seek framework example size address approach expense exhaustive desirable hierarchical criterion note previously similarity correspond affinity similarity radial depend histogram set choice unlikely dataset want region dimensionality reduction non similarity conditional probability interpret pick variance center value neighbor choice enforce criterion give calculate send oracle want specify sufficient node adaptive unlabele every represent bold use evaluate contain evaluate bottom set expand advantage cluster walk transition summary cluster differ strategy seek reduction first unlabeled child start proceed expect expand active child add expect oracle hierarchy top bottom cccc ccc dataset rand entropy pca mean despite evaluation ht around represent vary run toy illustration advantage refine two need ask oracle low hierarchy toward occur tree reach depth make unlikely alone open make hierarchy potential respective graph near initial query hierarchy code produce inferior graph graph constant node elsewhere knn bandwidth segment mean outperform baseline compare algorithm seven baseline entropy approach strategy empirically perform summarize area interestingly outperform computation greedy necessarily optimal encourage set hierarchy observe offer performance high variability iterative propagation prevent expense marginal illustrate subset ccc segment depict present scale linearly soon impractical query perform accurate al human generalize slightly accurately well par complexity matlab default combine pick oracle big keep interactive validation variety situation supplementary far across dataset lead area across run plot apart hierarchical construction give graph compatible graph perform graph choice part file effective propagation future work either exist learn inductive would also account effort never future information online ep ep token good semi expert inconsistent dataset new evaluation variability release circumstance active costly dataset practical active vary sparsity dimensionality run human expert match tune supervise specify classification image massive expert suggest unlabele choose give oracle unlabele pick query quickly train repeat query unlabele therefore popular image vertex encode label feature select inference benefit influence criterion decide image naive put individual receive label method construction good crucially exploit building construction hierarchical allow ask unlabeled image heuristic overall curve significant hierarchical cope dimensionality balance exploration good refine decision boundary specify benchmark construction establish across dataset cover thorough active learning tracking categorization object semantic segmentation video segmentation human automatic body relatively facilitate interactive perform far active learn propose voting feedback user however unlabele pool interactive co informative current wang annotation supervise labeling
sensitivity confirm case asymptotic behavior default diagram focus diagram score forecast tuple forecaster respective outcome compete issue set prediction convenient interpretation empirical probability average empirical expect compare difference namely diagram expect intersect unless support analytic score fortunately establish compare well argument quantile empirical forecast dominate dominate n n x omit functional case score right vanish I jump similarly case slope change slope median give example issue forecast artificial estimation forecast diagram panel forecaster world infer dominate remain give rise relation scoring reflect consider joint low empirical superior economic dark unique share attains attain share score diagram economic setting diagram empirical score visual appearance stem score depend forecast supplement diagram difference autocorrelation west view underlie band pointwise nominal forecasting united states survey bank datum source use motivated mean forecast another survey choice except fourth figure forecast diagram show curve score survey intersect suggest survey whereas survey explain well match survey band fairly relate rich forecast cover forecast forecast probit short interest period economic choice find original probit probability forecast gray bar indicate tend probit forecast period diagram row attain band score difference small confirm superiority probit forecast partly probit demonstrate probit improve information play forecast switch introduce et autoregressive hour ahead forecast wind wind united states refer specification terminology datum period forecast period forecast bottom quantile forecast exceed outcome nominal forecast calibration forecast ar wind nonnegative quantile identify suggest superiority forecast benchmark forecast diagrams elementary confirm mean expectation functional forecast event aspect interpretation problem differential perspective argue forecast score specify quantile scoring target rather merely specify functional relevant class consistent survey specify specific function whenever way another scoring participant forecast platform inform scoring start competition receive available utilize community situation international weather loss apply center behavior routine diagnostic evaluation rank forecast center decomposition reliability extension diagram connect huber traditionally estimation quantile asymmetric piecewise choice respectively contrast logarithmic infinite lebesgue half generally raise focus interpretation paper criterion consistent extend generalized quantile mixture functional study whether representation score rule assign predictive proper variable forecast apply answer extension probability forecast general feasible member lie negative encourage widely rank equal forecast event threshold simplicity assume quantile invoke relationship integration quantile evaluating weighting scheme family scoring motivate justify european grant reference mm mm generalize quantile economic measure quantitative finance journal r probabilistic weather forecasting road journal american verification interpretation forecast weather journal convex journal mathematics binary structure work predictive accuracy journal business economic b square probability outcome misspecification working department economics california wang risk prediction journal economic j evaluate forecast journal american association american forecasts quantile scoring business economic journal forecast journal e calibrate forecasting american association hand classification american role forecast management p fan global forecasting competition international journal forecast huber j estimation location quantile quantile convexity quantile order probability forecast verification tool probabilistic forecast management relate specie economic management science proper forecast ratio situation weather review forecast verification weather decision forecast value asymmetric west positive semi autocorrelation consistent matrix volatility comparison volatility journal forecast paper economic relative economic journal verification guide nd j forecasting power yield business journal american association method statistics quantile david eps forecasting application journal research thompson weather forecast weather production possibility inform journal economic theory partially inform forecast bayesian journal american economic forecast market economic e coherence permit subsequent immediate quantile h hx straightforward consequence every eq increment unique turn bregman variable representation hx consequence x finally increment mix proposition elementary scoring associate coincide argument right vanish latter expectation handle analogously elementary strictly proof complete forecast c cdf density analytic score forecaster scoring quantile scoring elementary scoring xy accounting scoring event mm scoring ranking receive functional evaluate forecast scoring purpose hand score score functional parameterize probability forecast constitute important quantile along function admit economic interpretation threshold decision give forecast dominate sense preferable consistent empirical average element score call diagram comparison compete forecast key word phrase economic sensitivity forecast forecast past broad develop forecast nature predictive quantity forecast reason make report situation predictive distribution potentially set key compete forecast score consistent class score prominent expectation alternative classical show condition score functional subgradient arise forecast basis forecast broad forecast ever ideal compete forecast scoring observe among special correspond predictive success obvious reason prefer raise question many alternative motivate regularity x write important forecast score preferable elementary see represent oracle future empirical forecast let consider triplet compete forecast forecast respective versus display show forecast wind wind versus along pointwise band forecast preferable section generally quantile apparent readily interpretable sense represent expectation forecast special correspond level remainder organize mixture representation relate economic functional differential aforementioned diagram forecast comparison proof defer quantile review forecast emphasis consistent first introduce denote measure borel lebesgue follow right continuous rectangle yx cost forecast regular jointly measurable rarely evident implicitly notion set functional mapping functional moment generally correspond quantile value specifically functional map one quantile scoring consistent relative eq forecast admits score optimization attention functional quantile mean probit logit class consistent score quantile mild scoring form decrease prominent arise piecewise similarly level convex example arise estimation regression representation quantile parameterize decrease function function subgradient general neither uniquely version finite point leave decrease let hand everywhere function purpose family regular scoring function quantile apparent every admit member admit form satisfy h g furthermore member admit mix unique hx hand respective pointwise qx ex measure assign finite interval quantile asymmetric lebesgue e measure asymmetric square lebesgue special case exponential homogeneous line lebesgue remarkably case distinction elementary measure consider sense member mixture representation elementary extreme true admit average build therefore family need similarly denote class situation fully toward deal level cdf exceed threshold economic meaning functional far house suppose put score consistent respective strong scoring order relative continue inequality strict borel finite quantile score sensitive quantile every score functional relative representation sensitivity scoring scoring sensitivity apply strictly relative suitable closely et al characterize regularity compactly score rank representation notion follow seminal prediction consider distribution forecast probabilistic cumulative outcome respective tuple utilize measure language sigma cdf value measurable quantity forecaster mm z mm denote cdf specify joint tuple perfect sigma variable ideal forecaster unconditional outcome ideal uninformative sigma forecast extract focus quantile exceed forecast forecaster forecast case identify tuple forecast information x x extension might case quantile straightforward forecast sometimes prediction specify tuple cdf represent forecast define notion forecast forecast turn forecast set suitably penalty score proper rule therefore careful scoring induce proper score predictive probabilistic forecast dominate relative rule turn quantile regular functional quantile forecast outcome prediction forecast definition forecast outcome forecast x notion respectively essentially forecast dominate another inferior type case quantile dominate preferable least inferior predictive consider functional quantile straightforward dominate recently show rich dominate result carry include limited quantile perfect sigma generate
plug prove pp remain next l fact number conclude complete section lemma nonlinear dimensionality factor forecast forecasting index high predictor extract forecasting deep explicitly state sufficient forecasting correctly presence nonparametric forecasting extend asymptotic direction method multiple forecast component index forecast rich economic finance include forecast production forecast use asset outcome high microarray vast effective cross think vast dimensionality predictive turning curse dimensionality vast predictive reliably response portfolio management testing predictor leverage development forecasting problem forecast focus similar forecast predictor first principal pca underlying factor follow many point relevance forecasting lead improve forecast predictor similar fashion predictor take forecast generalize partial fundamentally forecast extract forecasting point model especially thorough often additional gain paper forecasting factor volatility estimate factor significant return effective effective class nonlinear cognitive formation pose challenge estimate work completely forecast introduce favorable call forecasting introduce seminal arbitrary unknown forecast independent unobserved put way forecast relate unobserved goal forecasting go dimension effectively reveal index enhance greatly forecast study improve especially forecast one advance method incorporate deal identify present technique well reduction aid factor high dimensional regime size organize forecasting fall demonstrate forecast simulation conclude remark appendix framework forecasting forecast unobserved index forecast function factor predictor loading drive response ease lk orthonormal model mention forecasting forecasting illustrate advantage scalable explicit kt input via model nonparametric target predictive index word reduction direction span identifiable subspace span throughout dimension direction index canonical matter principal linear forecast forecasting explore predictive embed common factor create index unobserve nonlinear forecasting forecasting situation effort tailor factor fundamentally limited forecasting traditional focus covariance forecast optimal forecast nonlinear forecasting utilize consider inverse salient feature impose ks thus estimate direction investigate practice inverse obtain predictive conventional follow conditional information underlie divide slice direction observe orthogonal eigenvector positive orthogonal factor natural estimate predictor since sequel interestingly exactly loading constrain estimator equivalent estimate eigenvector converge central yield consistent predictive index unobserve factor construct large eigenvector index make forecast loading constrain extract factor eigenvector sample counterpart yy influence introduce double refer slice observation slice form analogously show alternative way estimate loading example procedure characteristic model lead equivalence forecasting include determination slice factor estimate direction point smoothing may slower show rate convergence matter sufficient forecasting long estimation propose determine eigenvalue distribute forecasting model many latent determine literature estimator eigenvalue enjoys grow consistent conditioning unless vector represent norm large value large eigenvalue first detail forecasting function loading loading linearity impact correspond center contain reduction impose strong generating mix dependence consistently loading residual every establish estimate inverse curve norm far imply convergence index assumption enable obtain fast direction large eigenvalue eigenvector eigenvalue give direction linear factor regressor forecast affect forecasting extend predictive use forecasting coincide forecast consistently condition predictor let view forecasting find estimate effectively onto dimension fix cross dimension regime methodology applicability common factor vast run forecast nonlinear directly forecast link misspecification central subspace weight response normality specification link fall central specifically converge sufficient direction model normally forecast asymptotically direction multiplicative check fit suffice employ nonparametric factor forecast predictive belong subspace reveal dimension subspace power contrast entirely contain estimating try effective direction capture drive comparable significantly conduct assess forecast target linear plus prominent example asset could book ratio forecast aggregate market return specify I let predictor combination predict loading draw standard ar draw simulation standard take infeasible principal regression forecasting typically regression sample suggest ccc ccc sf sf pc simulation sf forecast principal pc result summarize table sf denote forecast predictive denote component principal pc perform poorly table sf five regressor sf consistent whereas due poor target economic influential examine financial generating e factor response plane algorithm forecasting involve strict subset irrelevant factor three suffice factor evident correlation estimate factor variance median correlation factor replication next principal section examine sample direction span ensure factor loading meet calculate invertible still forecast coefficient evaluate measure use reduction calculate forecasting direction cc forecasting square replication deviation ccc ccc median percentage replication use predictive extend matrix predictor sequel replication observe series increase term pick sufficient forecasting pick sufficient direction consistent evaluate index build forecast interaction sample report purpose principal regression extract first principal top denote principal direction relatively low interaction two pc predictor incorrectly picks exhibit large investigation sufficient forecasting dataset series take make forecast target involve recursive factor estimation datum begin forecasting category may target second category eigenvalue time small pick representative simulation sf sufficient single predictive index linear forecasting sf
error empirical risk dp good practice denote become risk proxy minimize limitation hence e approximation family express opt e est opt analysis sgd effort translate chance sgd reduce expect large limiting rather perform convexity last observation base neuron assign consideration consider stream sample risk equation datum descent gd update represent gain gd parameter matrix gradient calculation effort consider gradient available converge large importantly engine algorithm imbalance modify modify express imbalance run count datum instant derive analysis oppose section ease express represent far unbounded lyapunov increase w sgd output persistence amplitude boundedness prediction long bound true application apply streaming learning engine engine specification auto stroke mix air form cycle retain reaction rate air engine suitable c engine stroke stroke mm ratio lift peak lift degree angle negative engine control fm angle angle center include temperature manifold pressure flow pressure temperature air engine indicate angle net mean high pressure reading please refer identify variable dynamic operate boundary appropriate require model engine operating envelope variable capture steady conduct naturally condition vary fm every engine make amplitude pseudo use engine frequency suitable consider acquisition pressure angle per collect combination input engine drop bar phenomenon lead undesirable engine operating range conceptually constrain require without complete occur ratio result increase lead variation feedback operation temperature high ratio heat release rise rate engine phenomena input engine engine nonlinear dynamic use output represent past order respectively augment measurement convert eq sample classification definition ahead predict time parallel architecture explain exist use become convert model long prediction control requirement base engine variable cycle ahead action impact engine variable engine fundamental represent engine consequence engine input predictive control orient modeling exist identification slow eliminate hand purpose control engine condition approximate extreme learning different state os sg baseline baseline purpose baseline offline behavior completely offline offline produce purpose consist unit fix parameter cycle engine update sequential cycle ahead prediction compare step ahead represent recommend os layer initialization fair sg determine trial prediction robustness outside measure normalize square performing os sg initial sg growth keep os govern add leading consequence value os os sg implication theory result reflect summarize rmse incomplete training convergence aim engine may rmse os sg sg computation dimension datum one os win marginally accuracy marginally model ultimately feed predictive control framework step ahead generalization parameter well demonstrate sg linear baseline adopt nonlinear identification engine summarize prediction experimental instant allow several step see outperform offline online behavior operate note limitation sg model sg os converge stability consume system os tune properly initialize predictive operating engine develop prevent engine towards dynamic operating envelope engine model offline paper operating envelope sg operating envelope common unstable operating envelope manually unstable depend engine heuristic engine number stable unstable imbalance imbalance imbalance cost heavily sample os sg compare classification offline previous include justify adopt nonlinear offline train effectiveness online capture operating envelope control engine temperature pressure envelope measurement history dynamic sign indicate future q section learn experimental engine define label consider sensor fm fa engine cycle process engine update cycle step ahead ratio nonlinear approximate initialize extreme learning covariance consist unit study portion use initialize sg weight handle imbalance imbalance datum ratio number majority instant imbalance conventional misclassification bias classifier majority label predict label undesirable metric skew correctly geometric ta classifier accuracy equally class result imbalance table performance model perform well identification accuracy problem achieve counterpart os perform offline completeness sg slight train os prediction accuracy sg slightly os indicate sgd online subtle os sg accuracy indicate predict well sg tune fail tuning class predict htbp gm os sg model os sg unseen operating envelope within operate manner envelope engine fm instant record datum point predict engine operation unstable mark predict plot dot instability dot indicate misclassifie understand fm also understand fm plot whole os sg engine fairly amplitude inherent experimental fm bad choose engine fall limit exhibit dot work predict unstable available accuracy os clear predict dotted plot true sg sg inferior predict unstable evident set sg compare os stability lyapunov sg involve tuning suffer sg develop operate envelope engine suggest generalization os sg sg advantage design sg appear comprehensive perspective hardware explore operate range development upon department project direction pi account work unite usefulness use herein specific constitute united author herein reflect united stochastic article gradient extreme develop sg stability stability estimate identification nonlinear reduce os square order demonstrate algorithm advanced engine system online imbalance operating envelope engine indicate art add reduction effort stochastic extreme online online imbalance lyapunov homogeneous operating envelope prediction engine engine homogeneous reduce consumption compare traditional highly operation advance nonlinear control challenge factor contribute include absence direct narrow issue engine physics model significant associate cost engine calibration key engine engine several operating effect far engine limit engine operate engine engine operating envelope engine state engine state engine require pressure respect engine state represent engine temperature pressure mixture production engine engine narrow operation operating envelope operate engine operate engine significance operating envelope introduce operating envelope crucial insight engine load condition use enforce envelope could enable engine diagnostic event lack engine normal monitoring emission control engine detection essential meet requirement envelope alarm engine article detail art os stochastic gradient derive along background engine section discussion sg conclusion thank setup output feature include capability get minima iterative well capability label problem
step bit set locally constant implicitly rewrite set drop ease partial side element column transpose column column eqs respectively simplify chain l respective ns complete derivation similarly active row modality non active optimality condition section example ray laboratory md mail dictionary use input advantage feature fusion sparse input sparsity enforce multimodal dictionary simultaneously multimodal dictionary feature code multiclass flexible fusion modality multimodal three application multimodal recognition face recognition counterpart computationally equipped achieve dictionary multimodal sparse individual fusion fusion fusion aggregate different classifier decision individual train classifier fusion study fusion mainly due stack suffer curse concatenation among noisy redundant classifier limitation improve classification attract researcher successfully acoustic structured dictionary usually stack class expand feature joint show assumption simultaneously multimodal dictionary modality sparse construct suffer become demand neither discriminative face recognition dictionary compact dictionary svd aim adapt achieve dictionary adapt call utilize rather reconstruction minimize incoherent train dictionary minimize atom small unsupervise dictionary reformulate scale factorization solve recently tackle supervise solve compressive sensing majority drive dictionary applicable single view dictionary common learn application recognition view dictionary exploit correspondence dictionary pattern share formulation atom error fusion heterogeneous multimodal learn extract typical template multimodal feature template represent modality localization multimodal multimodal dictionary jointly modalitie encouraging utilize dictionary atom independently utilize among modality focus dictionary present overview propose framework different enforce modal classifier major contribution multimodal task drive algorithm fuse heterogeneous train classifier enforce modality code optimize binary multiclass classification unsupervise product version bi task multimodal solve framework fusion modality feature modality pattern modality performance multimodal multi multimodal counterpart efficient sense still rest organize supervise source information review joint sparse multimodal review propose drive multimodal comparative several benchmark conclude bold bold letter symbol vector column column represent index row index index form index element x ij widely compressive principal learn orthogonality condition statistically convex set atom dictionary exploit unsupervise often draw train reconstruct input robust feature code ground truth label associate input measure task formulation classifier parameter emphasize task drive dictionary show set set optimize classifier representation level fusion source multimodal dictionary different modality multimodal regularize correspond sparse alternate admm encourage encourage modality enforce modality present reconstruct add extend within result source fusion multimodal dictionary potentially represent supervise dictionary adapt unsupervise multimodal dictionary unsupervise multimodal derive characterize modality sparse code frobenius joint sparse case optimization assume draw use project orthogonal algorithm typical problem minimum practical minimize reconstruction modality level optimization parametrize observe take discriminative convex twice binary binary label belong multimodal classified sign monotonicity intercept easily add use bilinear learn multimodal regularization replace frobenius norm bilinear rich classification need careful fitting multiclass vs vs handle vs set softmax loss softmax regression define vs multiclass turn change coordinate rest optimal test classify heterogeneity modality impose sparsity reconstruction relaxation sparsity multimodal replace intuitively group dominant modality small encourage reconstruction modality design modify multimodal problem active row define update non ds rest unchanged multimodal dictionary ar face multimodal choose quadratic multiclass multiclass vs select use atom compare value use except use positive select heuristic constant anneal iteration whole try retain empirically selection considerably also note competitive cross comparison mkl multimodal classification modality include right face ccccc modality svm svm lr faces pose illumination expression seven sample portion use optimize five modality face modality ar modality show pixel zero dictionary dictionary atom mkl lr classification dictionary mkl rbf kernel equip classifier result multimodal classification way utilize modal algorithm namely multimodal fuse zero enforce row sparse denote multimodal norm unsupervise multimodal denote algorithm prior algorithms modality fusion well fusion right modality agree modality highly multimodal dictionary improve recognition performance fusion art evaluate propose supervise multimodal dictionary mix level fusion aggregate score vote independent decision obtain modality fusion compare include classifier mkl algorithms ar performance moreover achieve performance classification joint prior number atom ar cccc test sparsity equip dictionary dictionary confirm reconstruction achieve dictionary perform discriminative dictionary keep per dictionary moreover available meaningful unsupervised multimodal sub dictionary approximate stack final indicate indeed formulation enjoy especially number relatively recognition task real application hand fit expense discuss dictionary size typical multimodal fig increase advantage state compact dictionary dataset contain face use training classification different table modality couple modality reflect lr svm lr propose dataset consist expression record span head height camera subject present manually protocol view angle scenario pose handle multi modal divide view test way dictionary size class individual modality performance multi face recognition supervise learn outperform fusion study application multimodal dictionary learn one tune dictionary individual mkl modalities subject gender corrupt modality four modality modality subject overall remain transform preprocesse modality input atom dictionary modality split table achieve dictionary multimodal fusion fusion fusion modality fusion modality different cumulative roc originally recognition competitive modality recognition perform multimodal performance extract among modality comparison joint sparsity
total intermediate probabilistic latent seek dimensional latent equation white component iteratively prove model pca principal datum projection vector intuition behind first make guess principal state find give log parameter expect repeat method converge reconstruction analysis present base principal per show need iteration complexity communication state complexity work show pca without pca store intermediate redundant computation redundant tp library eigen matrix svd probabilistic analysis complexity eigen covariance cubic complexity prevent dataset time dataset dimension addition communication complexitie two high communication cost still method pca efficient assume small typically two evaluation however reveal employ communication use prevent promise pca large analyze pca show computational communication scale dataset indicate two eigen intensive complexity cubic multiply quite hand show svd pca performing suffer communication promising pca recent design scalable outperform often algorithm assume computing comment limitation support dataset analyze complexity consider software library report help researcher pca library characteristic design pca day sensor web site value business information machine volume machine learn new challenge distribute execution machine may intermediate carefully bottleneck regardless compute complexity communication consider metric complexity terminate intermediate analysis library library linear algorithms parallel machine help software library knowledge environment use report matlab name include compose letter indicate star transpose respectively element rest organize distribute execution analyze eigenvalue base call svd analyze pca algorithm summary conclude analyze two metric needed terminate well state pca node exchange among total intermediate complexity phase end phase must produce execution start amount delay increase execution bottleneck delay speed platform use code exchange intermediate storage virtual communication total abstract detail hardware architecture give target dimensionality summarize follow center computed subtract matrix compute multiply transpose multiplying none fit memory eigenvalue eigenvector put formula column eigenvalue cholesky factorization sort vector principal result intensive complexity eigenvalue decomposition size complexity incur substantial make describe section programming language language extensively use include scalable implementation include pca mean give several dense library framework svd algorithm optimize offer speedup memory usage matrix explain use compute provide converge apply singular compute qr qr matrix singular assume second value singular accord singular formula principal point operation either cubic dimension disadvantage routine intermediate overhead bottleneck qr computation result matrix intermediate matrix intermediate element store sparse application however implementation library take employ svd library obtain principal many mean subtract sparsity svd prohibitive provide svd slightly variant work quality rapidly iteration exceed value solution avoid support enhance quality eigenvector fix initial choose enhance describe recently randomized compute approximate matrix compute svd stage compute approximate via svd stage al describe computation compare randomized accuracy rest stochastic require orthonormal matrix contain importantly target compute matrix approximate vector extra add great computational set without change discussion term symbol step compute describe approximate comprise regard one requirement output compute slowly decay I vary accurately motivation behind singular value high power
influence assess sign color vi modeling assess sign color reflect class ask question influence pseudo unsupervise vi training fig rwm similarity influence vi determine hyper control direct impact result rwm critical never result almost together parameter much center attract rwm hyper variance shape shape consequence rwm similarity set consist five generate uniquely sign density large component small rwm different impact component rwm question vi three learn result rwm similarity training span principal project visualization normally first rwm rwm svm set normally rwm standard rbf two j rwm capture gmm base approach advantage rwm implementation question rwm always lead exploit property realize library cope psd parameter rbf penalty lin two dimensional parameter contain shape set sigmoid kernel htbp lead heuristic combination small generalization kernel straight minus cut good combination gray circle color along line svm rbf cf red line gray circle fig search span set uci machine repository correspond exhaustive circle parameter combination heuristic work rwm rwm kernel parametrize rbf define rwm unsupervised rwm partially rwm kernel gmm see rwm shall rwm underlie component process varied vi svm rwm colored fig show classifier accuracy slightly component htbp vi contain vi rwm depict rwm information model rbf consequently rwm yield namely rwm state experiment rwm parameterized rbf htbp value often handle categorical sample categorical one scheme extend rwm kernel eq weighting value respective encode categorical checking dimension equality also categorical part component identity matrix rwm behave encode categorical compare kernel gmm kernel svm call matlab adapt cope multi visualize kernel set benchmark mention detail summarize htbp visualize behavior rwm take five artificial suggest uci machine learn repository mixture email normalization five conduct fold apply exhaustive density underlie rwm gmm data rwm gmm first da fold color correspond algorithm whereas remain use kernel rwm black line decision boundary gray colored rwm gmm correspond locate indicate large svm worst surprising label usage derive unlabele significantly furth kernel relatively even generate produce shape data gmm rwm mixture rwm svm overlap clearly interestingly rwm well set set gmm kernel straight rwm responsible two assess new numerically conduct publicly conclusion conduct heart page uci low rwm gmm rbf comparison function parametrization average fold significance significant least state kernel higher high however correlation vector classify table limit accuracy svm combine small accord highlight set degree value nan hypothesis critical difference significance rwm kernel gmm rbf kernels classifier rwm rwm average significantly rwm combine vector regard need cm rwm rwm heart page block seed rank experiment increase also reject hypothesis rwm combine kernel confirm cd show svm perform svm kernel significant show svm rwm kernel perform good average advantage rwm rwm slightly gmm kernel column table htbp cm rwm gmm rbf rwm heart seed two experiment label svm completely supervise supervise classification rwm gmm datum reject rwm belong top perform classifier svm cf close table high small rank comparison rwm kernel state rwm gmm rbf experiment bring column combine structure information presence label rwm may kernel rbf rwm perfectly rwm gmm kernel c rwm kernel gmm c heart page seed rank evaluate run intel processor fold cross rwm gmm rbf solve unlabeled rwm train test density rwm size small function building comparison rwm gmm time estimation time much estimate offline train svm small rwm take comparable building matrix htbp r time rwm gmm rbf train heart page seed rwm rbf rwm whose estimate vi step give rwm mahalanobis distance sample rbf allow learn experiment optimally especially label reason combination yield validation experiment additional regard solution user advantageous rely good exist density shall technique parametric parametric generate clearly rwm cluster vi offline unsupervised manner besides gaussian anomaly third gmm vi influence also precisely gmm appropriate sampling adopt cope restrict isotropic rwm set article avoid discussion psd definite clearly formal always semi use experimental rwm g another numerically stable weighted mahalanobis rwm data key advantage follow kernel label suit semi due rwm way easy fact svm implementation provide heuristic svm easily adopt encourage investigate future approach cf e svm build step cf variant vi cope set theoretical rwm kernel density expect model rwm kernel machine need besides radial basis tailor assessment structure space inherently contain mahalanobi derive mahalanobis rwm basically two responsible capture structure kernel many advantage rwm easily algorithm sequential optimization know kernel rwm sample machine mahalanobis supervise pattern sigmoid polynomial support weight application time specific task attempt modify matrix mean essentially identify hide generation describe embed sample article datum training mixture case mahalanobis distance similarity measure space classifier dissimilarity similar maximum call rwm similarity influence determine matrix term respective rbf rwm similarity define rwm svm build rwm black solid result illustrate rwm produce input sign green circle recognize sample gmm component gmm overlap information gmm model show sake euclidean rwm rwm similarity consider information gmm case label build vector rwm fig regard fig curve rwm gmm distance rely fig illustrate classifier accuracy test essentially minimization boundary regard use thus line sample rwm decision become nearly shape curve svm rwm rbf kernel rwm advantage supervise outperform kernel capture structure laplacian svm regard training implementation rwm kernel extension rwm parametrize rely strategy give overview relate define rwm propose rwm respective property datum set finding rwm particularly thus focus aspect set typically amount unlabeled instance sample output conjunction aim consider capture unlabele improve many algorithm implicitly unlabeled call claim major idea idea move close similarity membership purpose descent soft classifier svm differ step mahalanobis metric svm train solve quadratic boundary conclusion underlie try place decision boundary svm implementation much lot major mean classify unlabele available without classifier regularization empirically label shown yield supervise generative model view unsupervised cluster adapt known algorithm algorithm well consider information unlabele parametric mahalanobis rwm base rwm define investigate integrate explore extend input contain model start input training model sum rule densitie conditional motivated application eq
concept formal neighbor cross fold context form cross correctly object fold object fold run base classifier whole prediction keep classifier building formal concept formal minimal classifier count upper object process near select metric form yield concept mostly ignore select concept classifier rule base building neighbor fold validation array train neighbor kx x extent concept machine gb uci digits p cm rbf c logit knn bag decision sec sec sec sec sec sec sec sec sec sec cm rbf logit knn p bag adaboost iteration sec sec sec sec sec sec sec k classification classifier bagging svm outperform case turn bag dataset digit paper underlie classifier bagging stack turn base implementation bag multi class continue direction explore impact metric base attribute importance type classifier investigate condition preferable bagging improve thank high school economic influence study gray briefly multiple classifier system describe improve accuracy label correctly correctly classify assign object formal idea multiple classifier well machine appear name mixture expert fusion base early formulate decision vote add probability majority vote tend similarly great boost compare bag boost stack paper present type recommender base classifier classifier label correctly correctly classify organized follow bagging stacking provide toy synthetic describe well many one application learn impossible bootstrap multiple source replacement may overlap result usually accurate single classify prediction prediction aggregation bagging change result unstable classifier misclassifie boost combine boost start base relative instance evenly account also receive multiply meta chance less hard appear overfitte test rarely classifier formal concept construct one address context concept chapter concrete formal lattice concept serve rule frequent mining obtain formal let toy synthetic comprise test set attribute class train try attribute classifier run leave classifier far fill object correctly object train label let classifier refer classification context lattice formal context toy draw lattice diagram build lattice keep top top classify object find neighbor distance use ham near object set neighbor maximal intersection set classifier concept object cm p
period ensure slowly newly specify boundedness usual gaussian mutually assume eigenvalue decomposition j algorithm assumption define define eq ensures refer frame subspace slow projection direction frame change great rely actual along change particular great direction drop time also nonzero therefore necessarily explicit direction direction part say special whereas old mutually mean let background around mean autoregressive thus q q quantity slow change example every frame mutually set video foreground static move always reach scene frame go top bottom scene appear fig denote distinct value ni nk object since static one move mean move support disjoint condition move one direction become vector notation vector correctness give explain algorithm subspace new estimate get use hold corrupted bound video translate subspace change enough necessarily frame along direction value fast change tighter require choose many pair background fast need foreground moving imply diagonal proof let become nonzero allow r variance matrix knowledge explained enter foreground way enough direction subspace direction assumption assumption use triangle show matrix theorem basis corollary need nonzero entry outlier seem counter since magnitude whereas expect low nonzero try simpler simple explain really small affect low outli assume assume mutually everything else set conclusion hold proof follow exactly fashion treat extra follow three fact place denote condition far relate online completion mc require maximum eigenvalue entry parameter turn delay explain require frame follow retain vector initial short video mc another initialization technique adaptive mc model place slow eigenvalue gradually explain slow eigenvalue increase foreground discuss constrain long entry g r accurate early often video zero easily current appropriately none moreover exact recovery analyze fast need storage complexity need importantly highly entry easy allow ti pattern every support constant strong reason result j instead correctness generalization secondly correctness estimate subspace automatically know algorithm bound advantage develop lemma detection false correctness analyze algorithm knowledge prove approach recursive assume something use subspace piecewise sparsity weaker weak detailed simulation demonstrate thing available recent parallel recursive track underlie convergence optimize second explain subspace discount projection observe reconstruct via update recursively retain historical observed converge optimize initial knowledge subspace use idea introduce automatic work provide go thing subsection insight simplification subspace lie slow change onto orthogonal complement rewrite support recover nonzero ls e q argue small recovery far pca modification projection need p sum estimate see standard uncorrelated argue close condition become argue compare large perturbation subspace apply bind address issue considerably pca change form threshold change frame project change p phase pca reason pca see pca however even argue good perturbation small explain subspace estimate change correctly detect detect direction detect within short occur direction recovery exact present recovery eigenvalue occur interval interval detect show use input u whose eigenvalue recover method estimate rest remain formalize subspace detect occur within frame subspace recovery exponentially pca follow hoeffding lemma subspace bind hoeffding need use matrix j definition summarize definition pca j detect kt kb kb lemma bound appropriate conditioning see give dominant term expression thus recall define event j notation r sometimes j complement precise u j j kk k condition u previous eq whose eigenvalue check detect whose define later bound bound conditioning model use imply slow along bind decay definition theorem hold tt condition satisfie furthermore condition satisfie eq perform direction add decaying exponentially successfully false change imply frame definition respectively definition pca accurate correctly successful detect get bind notice imply accurate subspace chain assume prove lemma precede lemma proof theorem j lemma k j u j kx case proof fact j u j j j inequality fact event imply use u x x lemma p thing estimate number correct j k u j proceed observe combine u u appropriate fact conditioning lemma k define k j j rest remove etc everything similarly p j e first observe form independent lemma need condition corollary hoeffde hermitian random hermitian ii b hoeffding condition u apply use assumption recall r apply q probability x k combine lemma remark apply recall lemma proof give definition noting side satisfie let various eq cauchy schwarz find term apply term proceeding much tight need summation fact q apply apply corollary use use get expression pca prove corrupted entry time change reach bottom start vector start column schmidt last column subspace occur draw uniformly v show noise orthonormal optimization first perform prove support exponentially step program fail average simulation code matlab run processor black zero simulation plot section subsection result prove proof indice small integer respect process exceed new appearing old object consist index index start assume let index eventually walk object may come nothing general index contiguous moving move object long support minor scene reflect back start move direction hence bernoulli prove time ensure tail move move claim ensure static move least index notice object k third every frame figure behind first generality top u fact index number leave simple state motion start I I si exception set exactly correctness result modification algorithm remove estimate subspace newly subspace accurately prove one assumption cluster frame key weak able currently valid tool autoregressive allow obtained apply algebraic sum modification explain conditioning past value tool also analyze background approximately model apply modification video tool recovery noisy analysis due assume simple pca expect simple partial obtain correctness online accurate initial knowledge slow corrupt one quantify need give assumption matrix parameter mention section besides various problem correlate interval construct mutually subset I model assumption recall index word plus number l u mutually disjoint model u u u l line plus proof proof special case kb rhs increase assume theorem j thus proof difference use assumption use bound cs begin bound cs j follow j j j j u j jt definition follow assumption theorem k k sum vector schwarz schwarz line side edu paper partly nsf grant theorem example theorem pca mc define separate low true use important video try slowly analyze mc open develop modification online online mc main contribution obtain mild obtain short video surveillance change least every often pca tool frequently reduction compute small orthogonal call principal variability relatively accomplish via outlier pose pursuit batch solution assumption solve norm appropriately scalar program later et amount batch performance need recursive online assume outli separate video foreground background layer automatic surveillance streaming foreground move sparse image usually gradually move move forest subspace change dense valid initial background initial foreground recursive structured impose change part vector entirely complete column track foreground object intensity know significantly different support simple nuclear norm subject amount mc problem miss entry possibility infinity quantify set problem complete surveillance application short background miss mc include convergence optimize second old recent another result differently neither estimate pca
algorithm build ensemble repeat time classifier result rf pool combining approach vote vote class final weighted vote exist classifier training subset divide vote datum use test frequent c train benchmark multiply weight rf precision label failure decide weight rf rf sum high score datum result class failure ensemble classification compute apply ghz intel processor gb take benchmark control continuous equation roc threshold classified failure decrease true grow positive different pr display sensitivity curve great correspond average precision well imbalance figure obtain benchmark appear first two benchmark due aggregated day incomplete begin approach display roc pr curves benchmark individual roc categorical predicting failure failure actually identify appear expense precision plot clear dependence value classifier become less failure imbalance plot clearly critical obtain good roc pr describe ensemble line roc pr maximize two figure display failure failure identify label failure failure misclassification depend position latter situation failure appear time machine implication relation classifier study assign point hour would whether different impact vary failure display next decrease misclassifie misclassification failure actually negative close verify assign incorrect label class irrespective real next failure next true positive tp false tn fp benchmark look misclassified positive failure classified failure misclassifie negative classify negative failure panel divide tp failure correctly failure classifier distribution plot many positive misclassifie failure moment suggest recognize failure classification failure moment approach fact hour large especially vs benchmark event hour divide failure next machine fraction entire day end still failure fp failure tn negative failure fp compare tn many give failure hour publication google trace include goal characterization statistic node identify high level heterogeneity system especially compare grid user profile usage shape characterize trace usage various management example trace perform model study use google trace far validation early simulator parameter simulate job status load prediction cpu ram history load future divide load level failure prediction year application file email etc monitoring error report introduce fall element tracking error reporting failure job event concentrate scale failure track resource failure failure distribution inter job cloud naive failure job name trace amazon ec run scientific application reach performance setting point total point similar benchmark never job failure reliability availability svms network nearest algorithm outperform reach precision analysis perform neighbor forest anomaly cloud select relevant principal correlate component identify outside range study cpu disk relate relate control house high production trace failure trace failure failure mention concentrate result study result possibility controller center model happen various technical require change build collect streaming compute aggregated feature correlation window previous window store dedicate basic negligible event aggregated parallelization time average window second newly datum second window time stage time feature feature speedup cost storage cost currently mb original raw gb day translate gb day day time last day need store analysis day require gb day new use google engine train eliminate need ensemble entire ensemble take since independent minute provide study classifier take negligible expect take minute parallelization negligible amount cloud scenario however storage resource require monitoring cluster trace extraction cloud platform ensemble day tested overlap day length trace repeat produce benchmark platform useful obtaining feature sometimes process month worth daily cost varied area precision curve false word identify failure classified look failure failure achieve datum extract ensemble training parallelization also explore machine obtain machine failure present suitable day scenario simulate benchmark would train overlapping day benchmark day failure day training account time ensure test live reason voting secondly predict cloud google grateful google regard trace center ever reach ultimately computational intervention limit setting goal rather management control build update path operator center study public collected build predictive failure google amount characterize many classifier hour false component live analysis publicly website google trace failure modern center internet cloud services device power utilize million user service many aspect daily availability center availability manner user current automated management tool limit allocation monitoring capability operator stream display optimistic per situation center ever compute technique center complex characteristic building control loop compute try system state undesirable capability undesirable action place center vast corresponding course numerous internal external include power management server removal network modification modify electrical change physical server storage device benefit modern new generation capture centre political environment towards failure study google event management month period employ google platform massive exploratory suitable allow contain reasonable combine ensemble rf unbalanced mainly occur rf initial limited tree rf combination day result day false day comparable failure study field contribution modern center adopt generation drive towards extensively develop potential drive secondly tailor subsampling bagging weight provide quantitative evaluation run website public describe section issue drive controller argue conclude publish contain several table monitor status machine task day gb evolution task usage gb interval feature change third correlation window start cpu memory disk table consume require aggregation step amount process gb hour window per feature hour tb similar platform google trace report machine cause failure machine software update event cause distinguish discussion google investigate way distinction suggest interest failure perform software ensure event failure failure use relatively threshold hour require typical base threshold total event failure predictive sure precede hour window completely remove completely fact feature add two feature entire last gb data negative positive certain less assign failure
calculate updating occurrence initialization incoming event I n joint n calculate adapt operation operation cluster merge sum merge cluster eq gram track symbol must substitute count new gram include boltzmann pattern formally boltzmann consist binary update rule connect represent initially boltzmann implementation consist consecutive symbol binary node initially nodes symbol softmax rule temperature converge boltzmann machine boltzmann zero symbol connect state reach rule rate yield aim threshold concatenation symbol strong cf iteratively unit create represent length length variant machine symbol boltzmann retrieve implement call act operate model machine change removal merging update adjust atom newly pattern node represent annotated classifier predict analysis annotate annotated evaluating agreement annotation suggest pearson chi square rand index evaluation since natural set partition generate annotation derive x xx pair annotated rand since rand baseline partition gets maximal ari partition ari annotation clustering draw number ari establish alternative one entropy measure system ari evaluation feature entire symbol segmentation input sound wave transform symbol occurrence include contain index equal symbol partition ground segment yield annotate generate symbol cluster table repeat pattern length g annotation audio low duration unsupervised category automatically annotate record simple slow country complex audio annotate audio website consist entire chain symbol expectation audio processing system give repetition length reach cb noise robustness cb receive determine next symbol symbol annotate symbol generate explain annotation bm ari constant assess scale average ari partition length gram reach perfect ari event repetition cb seem converge reach ari high event increase slowly involve window detect frame vector range grid tb ari length ari ccccc table ari mean audio event window test annotate result expectation evaluate module predict tb cb cb cb work lot cb result sequel assess evaluate extraction entire symbol stage use symbol annotation use evaluation table detect ari conduct tb cluster versus analysis window length column cluster window measure c ccccc task event symbol annotate tolerance ms configuration yield ari ari tb row audio website prediction performance adopt optimally annotate symbol find calculate annotation iteratively entry thereby establish connection row annotation column entry maximal entry htb horizontal map event indicate black line match due tb ta ta fig fig annotation cluster map ta pattern capture annotate match wrong initial line red square symbol symbol pattern occur annotate match regular process prediction middle event event wrong pattern gram perform correct prediction tree sound website sequence gradually amount versa mixed yield sound begin website sound event analyze sound third sound gradually system predict next sound event operate audio start perform sound event pattern currently limited sensitive context prediction alignment combine incremental cb limitation inspired imagine human machine may novel idea machine finally stop machine suggestion currently associate speech research university unite european ph music study engineering institute des sciences france school write scientific several international project van research speech music analysis ph institute des study mathematic pure mathematic brain interface paris stanford scientific paper grant music play music create present segment cluster predict audio unsupervised adjust dynamically flow segmentation detection discretization incremental g drive sound event symbol extraction symbol sequence gram newly introduce conceptual boltzmann machine sound respect rand ari entire ari music retrieval adaptive music novel music music automatic either symbolic audio base pre train classifier cope new label every new appear severe flexibility system new human mind novel unsupervised paradigm concept base e tree input deal decrease sound similar use unsupervised n gram couple split symbol count system prototype learn begin reasonable predict symbolic predict distance pattern cluster root build operate audio segment stream initial unsupervised clustering build maintain set sound result sequence gram phase adaptive learn music sequence document online cover gram previously method conceptual advanced overview system component segmentation introduce rand condition module give audio example support website fig four extraction incremental giving sequence incremental drive symbol discretization cognitive node distance align font execute em execute center inner draw corner segmentation xshift input block feature block right feature incremental xshift em anchor north yshift anchor north yshift anchor yshift line bend anchor shrink bend anchor south sequence explain generally applicable employ domain difference soft induce transform complex th build distance actual bin complex spectrum amplitude phase two precede map bin quantify stationarity summing bin consecutive frame yield detection median ahead window length control order eliminate occurrence smoothing window define length subsequent analyze coefficient cosine frame dct represent sound clustering receive symbol symbol important state system cluster arrival symbolic create continuously build hierarchical object assign create object leave node represent model heuristic use utility version extend cluster previous contain deviation feature vector cluster specificity dimension utility quantify specificity instance dimension standard deviation dimension specificity ii specificity utility upper maximal minimal control discrimination incorporation process count operator create
alm around mac around simulated mean general mac evaluate multiple effect construct square alm example alm mac exhibit behaviour mac tend well alm example apply alm mac reservoir case appear alm exhibit due nonlinearity reservoir chen total anonymous valuable help comment suggestion acknowledge integrate realistic thank matlab toolbox mac maximization iterative algorithm iteration construct prior solve maximization update update pass knowledge correspond function mac th e mac assimilation update k k unknown deterministic determine subsequent criterion determine functional may choose view conditional view enkf reference therein optimal dirac assume addition q denote iteration compositional define bayes functional construct q equivalently suggest pdf also note impose compositional functional term subject condition system observation error make rule nonlinearity non construction beyond current mac operation histogram alm mac repetition visualization horizontal axis figure calculate difference initial day alm mac right alm box alm mac model mac alm alm converge visualization vertical axis logarithmic step iteration along horizontal box horizontal median bottom range determine individually plus panel ensemble alm mac bottom alm mac difference dot indicate model member column alm nd column mac rd estimation average score ensemble alm c mac plot different step alm mac production top bottom estimation ensemble column ensemble alm nd column mac rd iteration red dot measurement forecast ensemble column matching rd comparison production history water production plot estimation leave mac bottom panel alm mac box difference step layer reservoir ensemble reservoir alm reservoir dot indicate location field alm leave field use production production rate top middle p column ensemble alm nd mac iteration red dot datum forecast respect ensemble st matching profile nd column water plot difference iteration step field ensemble alm middle mac field production year cross validate history reservoir alm production year validation production p case initial alm nd column mac rd column final step dot historical forecast st ensembles nd rd vertical figures nd rd separate decade figure water example eps mm international institute alternative iterative smooth formulae derive adopt approximately solve minimum mac alternative understanding analyze behaviour aforementione insight illustration compare mac system reservoir even assimilation way assimilation algorithm ensemble kalman enkf sequentially assimilation collect observation simultaneously ensemble smooth example variant enkf widely reservoir assimilation history matching problem engineering application es reservoir assimilation investigate recently es include enkf reservoir benefit avoid reduction circumstance latter computer es certain method formulae follow formulae involved adjoint enkf error call es lm discard lm alm es mac apply solve cost involve obtain kalman enkf behaviour aforementione thus iterative es example implementation base illustrate performance alm es alm reservoir reservoir simulator certain assume mean covariance ensemble suppose j en ei iteration formula focus define follow root product es make enkf error appendix end formula I I I total number iteration perturbation enkf satisfy call hereafter covariance simulate randomized likelihood lm sense ensemble update iteration I positive derive line accordance expensive j formula algebra suitable iteration consistent technical enkf approximations final compare eqs formulae distinction call alm alm start average reduce value stop consecutive regularize regularize assimilation similarity es alm implication also ensemble study approximate gradient minimization concrete prove solution cost square e use enkf also alternative interpret reader deterministic reservoir match square minimize weight convenience discussion model ill pose undesirable e uniqueness large might solution become weight assign datum term solution tend extreme approach tend coincide apart relative choose covariance correlation model space choose comparable instance threshold reader choice straightforward often iterative algorithm construct linearize I I positive scalar convenient want far simplify square eq simplify weight later intuitively provide control taylor valid I previous gradually presence bind prevent use analytically sequence locally noise level adopt technical prevent fitting wish let stop comparable often course iteration stop iterate pre scalar discussion b change consecutive iteration implementation number step reach early alm control alm experiment extension issue take account firstly algorithm typically assume stay sufficiently reality addition discussion order derivation exact must certain efficiency put practice likely iteration lead high iterate order increase extent execute iteration essentially track optimization theory study ensemble smoothing alm examine study iteration square namely distinction call mac mac point cost jacobian matrix may member jacobian algebra see formula j c one I j I formulae alm iteration mac general differ construct possible member close ensemble work coincide result mac mainly differ alm different centering circumstance model mac alm identical case distinction may alm mac thus alm stress although superiority one broad theorem intuitively alm mac optima superiority case make use around order theoretically sound computationally expensive around neighbourhood strategy ensemble around mac incorporate term onto ensemble beyond kalman formula aspect carry numerical discuss similarity es alm focus behaviour root matrix step decide iteration start parameter constraint alm mac total number alm mac mac explanation mac cost mac member taylor alm degree nonlinearity nonlinearity iteration ensemble member alm accordance member initial condition alm panel member ensemble spread ensemble member alm slight alm mac low alm spread alm monotonically decrease iteration mac reach minimum around final ensemble may certain local distance truth small apart sampling member contribute depict ensemble member please box indicate vector I whose observation ensemble compute mac ensemble alm difference step initial figure nature tend mac toward decrease result tend reservoir simple reservoir water reservoir vary spatially md channel md background water location mark rate day bottom bar matching water simulation day plus certain noise day production bar enhance channel eight statistical measurement include sample define neighboring see area maximal body extension body fraction volume total evaluate ensemble assimilation measurement variance statistic also run alm mac include assimilation close tend follow level need enkf sign convert matlab toolbox version ensemble alm mac addition parameter adopt maximum coefficient alm mac third f respectively see despite update alm mac final alm panel mac matching reference score final mean correct figure seem alm mac channel match certain seem alm alm mac alm stop early relative mac stop iteration step despite alm mac convergence speed final alm deviation std mac final alm follow performance water production rate ensemble alm mac compare obtain mac improve alm well mac history initial production water rate reference highlight ensemble alm mac water production tend visible alm seem final mac plot iteration alm mac one ensemble situation box observe rmse value enter tend alm alm decrease slight mac alm comparison figure normalize I step ensemble mac normalize difference flow non tend difference history match mac field dataset contain year production decade decade production decade history production production decade history match matching control production historical production water cut historical pressure production net ratio consequently alm use model provide mac drop member alm mac reservoir indicate layer realization reservoir one mac although final alm mac see seem result difference way function hence subsequent previously box plot
generally speak use system smc maximum learning recently develop importance smc strongly choice proposal match size monte variance resample mcmc complementary research distributional inference particle filter particle construction limit propose box automatically flexible proposal assess kullback smc leverage literature outperform proposal include community performance translate complicated handle parametrize challenge filtering smc approximate model briefly smc importance sis resample sir comprise space hmms markovian notation sampler simple distribution distribution convenient take supplementary normalise weight recursion sis elegant compute pass I suffer severe distribution become many sequential sir multinomial particle weight replace particle importance sir require trajectory importance particle need let represent index particle collect employ resample previous smc supplementary proposal critical employ produce trajectory quickly posterior proposal care optimal choice proposal time posterior intractable filter fail incorporate current house employ distributional extended kalman filter inaccurate poorly behave outside apply neither criterion introduce new adapt distribution limitation proposal parameter explicit adapt objective four main sample global lie advantageous exclusive fourth derivative approximated negative take final step smc unbiased intermediate filtering bring advantage derivative time trivial update update proposal proposal full perform maximum update usage similar proposal j p contrast employ analytic distributional loop global particle special previously rich proposal go work base smc literature optimisation available option train however example proposal proposal generate smc place filter variant use apply briefly flexibility neural network generally technique unsupervised setting wider hide multi modal due uncertainty sign latent experiment interested modal diagonal three md recurrent help recurrent improve spirit variational proposal proposal dynamic state adaptation proposal generative supplementary sequence bootstrap nn rnn implementations guide implementation acceleration significantly bootstrap term ess rmse estimate simple md indicate multi modal proposal outperform recurrent transition dynamic rnn interestingly cut converge correct box plot estimate ess ht rmse std mean std rnn rnn f md rnn md ess marginal rmse system consider drive white ode consider infer cart orientation noisy location supplementary material significantly directly admit md model successfully proposal ess mean angle md learn higher ess infer often loop prominent example particle chain latent context proposal metropolis mh accept form e walk smc sample use smc full detail model section follow random model md md setting number particle small enable significantly reason quickly model adapt costly adapt particle slow burn mix propose dataset different hour simultaneous lstm layer output fed dynamics supplementary lstm generative optimizer bootstrap tune standard comparison approximate smc particle bootstrap three marginally well state state approximate marginalization smc c c rnn bootstrap similarity free employ refine exclusive variational integral approximate posterior spirit way except entail approximation proposal accurate therefore proposal lead advantage energy severe kl avoid compute entropy prove problematic approximate tailed variational employ refine context extend adapt proposal smc long contextual descent outperform
evaluate carlo estimate trace draw equal component carlo trace satisfy q multiplication appeal expensive absolute interval knowledge loose low first section section algorithm definite subroutine estimating give matrix initialize chebyshev eigenvalue g remark multiplication matrix input theorem ready present determinant generalize singular interval initialize tc equality singular easy problem obtain counting span determinant general power small compute require multiplication matrix bind input value quite straightforward fact become condition number chebyshev require degree I sign multiplicative determinant zero state corollary input singular follow eq give limitation count span undirected graph span one classical counting problem application model denote degree let laplacian spanning correspond give follow algorithm q supplementary material theorem choose error chebyshev intuitively approximate chebyshev polynomial bind minor length chebyshev q chebyshev notational convenience plane analytic hence rate z hence observation version eigenvalue eq estimator give sampling know interval proof ready definite imply combine theorem follow experiment first generate random row five uniformly distribute matrix symmetric position entry sum add run scale roughly determinant error comparison relative c complement taylor mean determinant run compute cholesky decomposition complement use inverse million author algorithms accuracy taylor expansion fair number report chebyshev expansion superior accuracy propose field graph capture property extensively application computer spatial definite sparse graph grid million matrix node four generate use sample j likelihood report likelihood provide track spatial million log presence let j z solver linear thin two interpolation fit value tool g solver eigenvalue computation matrix decomposition important computation admit often infeasible set linear logarithm exact computation cubic furthermore easy multiplication numerous thank chen chebyshev complement hold corollary c mn provide vertex vertex tree complete proof claim positive machine determinant involve cholesky cubic number prohibitive approximate matrix call chebyshev efficient multiplication multiplicative error propose cholesky compute million variable scalability learn extremely model increasingly attention prominent optimization randomize one variety machine compute determinant precision also variety machine include variational addition learn form adapt involve bregman become determinant probabilistic recommend determinant cholesky cholesky cubic thousand aim accurate definite sparse million literature compute diagonal band filter count approximation score stochastic gaussian taylor expansion
importance weight active learn unbiased hypothesis particularly go terminology relevant explanation exposition equip dual space dual shall entropy proceed stream one step calculate probability notice point step calculus component exponential proper loss formulae estimate instance estimate bx tp bx labeling initialize tp bx ty technical receive evolution risk aggregate different collection excess eq jensen subgradient build put inequality lemma stream eq approximate bx mention approximate get choose allow flip report coin turning good hull combine majority decide query order boost bag assumption assume hypothesis hull version base possible combination datum weak learn boost admit somewhat algorithm build fashion run next generate random look margin label point budget decision along weak characterize dimension threshold expense far label dataset query table report datum stream report sample see stream note large convex aggregation see stream suffer loss large seem imply comparable unlabeled believe excess c c dataset budget rate mnist comparison query four scale expect stream mnist scale sublinear stream query clear eq excess upper scale discussion know expect large demonstrate trade make small additional unlabeled round choose fair budget report rate budget inferior possibly budget experiment stochastic mirror use unbiased excess aggregation experimentally direction tradeoff error query help tune excess guarantee establish need key state result normalize mirror simplex dual differentiable strongly dual pair definition equation q last old increase sequence sum sum definition mirror fact replace put together equation get assumption conjecture ex learn aggregation good aggregation make want query see mirror descent risk aggregate return risk demonstrate uci passive query accuracy passive machine physics biology web finance availability great machine discover problem supervise require unseen label domain easy unlabele hard speech recognition require supervision passive access label domain sample distribution sort empirical return whose collection much rich model ensemble view perform aggregation gradient via sequential boost base model learner aggregate boost capability final aggregation outline aggregation hull whose excess risk hull small remainder assume access underlie define give label unlabele query label study convex good aggregation access stream sample I introduce mirror shall show one mirror iterate aggregate excess risk aggregated number essentially mirror descent stochastic simplex simplex regularizer mirror follow step excess bound consider essentially pass mirror base algorithm stochastic bx stochastic mirror construct unbiased iterate length stream return excess risk convex decay algorithmic mild dependence desirable mirror descent introduce algorithm
version appendix power wikipedia collapse initialization dataset hybrid hybrid min wikipedia dataset topic implement lda tensor whiten fast baseline collapse gibbs wikipedia detail word reach set hold dataset document pick testing mix solve dd report hold spectral different hash length accurate table lda collapse gibbs iterations spectral lda different length collapse comparable hold run collapse initialization collapse topic already perform sampler x also report collapse build page hold build randomly pick topic get topic obtain table collapse initialize spectral lda much shorter category corpus usual wikipedia dataset email v norm wrong min specifically explain symmetric upper triangular note u u u u u subsequently exactly nonzero inner product accelerate alternate square experimental toolbox widely consider alternate least popular tensor decomposition maintain iteratively low rank good nevertheless index hash km bt ki v c ij similarly eigenvalue show cp different roll asymmetric kk ambient tensor bottleneck one c hold computed operation list wrong min exact compare various synthetic use matlab toolbox fair plain accelerate algorithm sketch length powerful review lda decomposition lda propose detail propose fast list ik parameter lda dictionary particular topic topic sample parameterized lda rd robust third know quantity algebra q extract vector procedure specifically find whiten orthonormal complete tensor decomposition perform moment v w w word moment co exact moment accelerate mention section help lda computation whiten bottleneck bottleneck come sketch tensor decomposition computation w pose challenge w sketch technique remainder efficient sketch mention efficiently dd w j km number word per document decompose slow application decompose efficiently sketch efficiently word decompose whiten sketch share sum document number speed sketch operation efficient computation w decompose consequently trick compute definition true variance I consequently chebyshev obtain prove I notational simplicity omit consider permutation everything demonstrate preserve product lemma state inner variance hash right prove hash tensor hold assumption real tuple pp b l expression r case definition l l wise l r r automatically generality tuple l l concatenation finally position combine get immediately add everything term equation easily refined section demonstrate noisy overall part v initialization randomly sphere eigenvector lemma approach exact ground noisy come tensor present noisy separate magnitude noise eq separate u u u tt examine numerator use assumption I v h yield eq prove term first vector since u v u subsequently noisy satisfy chain triangle eq principal matrix u level imply lemma plugging section upper principal orthonormal follow v ib bound upper manner everything section present appropriate hash detailed eigenvalue robust tensor tensor randomness independent follow condition I satisfy check tensor inequality addition hash assume induction lemma lemma hash yield index tensor hadamard conjugate n inner product matrix case tensor rao product n b reference theorem lemma cp decomposition wide learn randomized decomposition introduce tensor novel tensor explicitly tensor encounter alternate also design save combine idea whiten tensor power iterative technique fast quality method uniformity cp decomposition sketch domain modal relational important tensor decomposition cp decompose numerous variable modeling estimate mild importance interest world tensor fall tensor sketch tensor power tensor update decomposition dense tensor framework modern training attractive guarantee propose construction tensor input tensor factor empirical tensor sketch operation operation sketch force computation sample take express involve multilinear combination power alternate whiten directly compute sketch tensor tensor force th contraction addition directly empirical learn huge tensor contraction sketch symmetric tensor application hash avoid operation though hash computation depend property hand previous magnitude practice dense tensor processing propose randomized degradation accuracy competitive result tensor model implementation small collapse time gibbs within sampler much iteration propose efficient optima accelerate burn numerous work implement execute one implementation procedure whiten result significant high many bad moment decompose alternative carry decomposition batch extremely suffer paper handle batch therefore variance randomize tensor expensive sensitive particular uniform handle adopt require sketch pass inner take tuple simply nk I I em c n propose batch reduce noisy tm stand v sketch tensor eigenvector present bottleneck robust power u speed sketch approximately u nb hash go detail key idea behind tensor computation tensor inner approximated tensor whose build consequently approximately compute operation sketch u completely satisfactory tensor us I u f prove inner eliminate right side contain entry detail defer space run time exclude significantly improve method method analysis order plain method ccccc tensor factor tensor contraction tensor design style sketch build idea hash symmetric entry etc build hash random permutation address issue rademacher complex domain rademacher divide entry symmetric otherwise characterize
bind focus solely risk highlight whereas highlight exceed always define weight make equivalent form relate q margin notion place majority chebyshev reach correlate sake focus ignore distribution moment second moment second majority vote random first follow straightforward connection margin gibbs disagreement majority vote line proposition show condition consequence prove proposition twice risk vote small risk show average always risk pairwise concept mm note random qx inequality consequence moreover whenever large independent even large arbitrarily close increase proposition consider distribution eq apply motivate relate risk clearly individual predict majority vote great criterion boost algorithm quantity appear moment qr learner uci round study majority per dataset see however figure generally strong almost suited characterize quantity contain insufficient black r l heart ab letter letter test validation vs round sign well criterion every sign nan criterion evaluate tool adaboost dataset however split example contain remain example adaboost empirical majority classifier low bayes risk stop bayes round select early perform validation number round validation finally vote low bayes adaboost pay set compare risk majority well criterion test binomial suggest task validation perform suffer cross conclude surprisingly criterion boost pac theory estimate base recall classical pac bayesian majority vote classifier third pac contain act bayesian bayesian chosen observe pac theorem gibbs classifier rely quantity kullback leibl distribution note obtain present general pac convert bind risk majority proposition define pair loss allow pac theory pac entropy measurable equation jensen concave equality share present bayesian loss pac theorem specialize expect risk similar pac loss kullback point different relative weighting especially situation good note negative logarithm measure leave side f f easy many variant pac among kullback leibler bernoulli probability shorthand notation order loss let binomial fm fm f however case verify pac still apply obtain next risk derive theorem independent swap present classical pac pac majority vote find interpret corollary indeed gibbs corollary well pac replace corollary slight bind gibbs obtain inequality omit great classifier kullback leibler corollary gibbs risk eq multiply factor methodology bind trivially valid otherwise consequence close follow paragraph compute turn since hand convex constant show application pac package explanation load graphic explanation terminal need macro ltb lt lt lt lt lt lt ltb lt lt lt lt ltb observe intersection limit pac pac bayesian section expect disagreement definition binary disagreement define closely relate notion two usual equation define x q h either correctly rewrite margin risk rewrite bind prove define kind pair tuple pair loss pair pair new loss define recover directly equation explain classical pac theorems corollary multiply present rely third risk another disagreement corollary pair new bound majority supervise another semi q joint vote see definition q mm equation f ij pf pf mm mm finally e hence give pac bind risk vote distribution q always strictly two bind bound bound therefore suitably apply corollary replace major tight however achieve label affect huge tight tighter supervise large unlabele corollary obtain label follow pac obtain solve mm pac unlabele last section require approximation another extension pac enable new instead pac valid result tight bound simultaneously loss pair inspire real value q ij f ij notation ij f ij appendix give exploit increase definition p convexity ij p q q ij ij ij ij proof straightforwardly subsection notable former distribution variable kullback shorthand notation corollary apply inspired contrast base list distribution dm multinomial three outcome ij outcome totally notation mm ij ij ij convex ij ij ij f f ij k ij f last prove pair directly q swap ij kl ij since new tight notation indeed reduce achievable property bind correspond consider pair later valid idea equation pac supremum assume empty numerator denominator positive supremum mm attain subset pair closure constraint supremum closure imply trivially supremum attain case valid mm supremum equation probability valid q q slightly pac bound via remove define replace obtain theorem pac bind concave equation decompose nested maximization optimization twice risk need maximize kl black correspond marker vertical upper star marker tight bit pac replace optimize bound show bind present far obtain use adaboost uci contain dataset split training testing boost gibbs majority variant pac conjunction terminal option use load package graphic terminal need graphic macro ltb lt lt lt lt lt ltb lt lt lt lt bp bound ltb ltb ltb ltb ltb train ltb round boost tight pac obtain substantial improvement almost pac see pac bind testing round boost continue decrease drawback due fact denominator margin form moment slack unfortunately majority vote pac structural leibler distribution priori theorems surprising attempt minimize surprisingly kl poor regularizer attempt localize dependent prior inequality term provide notion contain kl term posterior regularization action inspire pac divergence hyperparameter restrict surprisingly latter use good assume possibly infinite set introduce suitable name introduce uniform prior align say finite set uniform quasi pair equal low upper rise kl necessarily small pac quasi theorem corollaries pac bound getting achieve loss gibb align need change term one fact result expectation f posterior distribution pm case distinction pac theorems section step linear posterior note version result theorem classical align posterior pm equation proof rely pac bayesian straightforwardly proof reason align term follow deal pair instead loss result case theorems shorthand define recall pf lemma pair distribution align j cf change expectation ij qp change appendix f f ij pac pair give df require property result obtain rest derive use inequality inequality finally therefore posterior give rise pac disagreement pac mm q corollaries direct application theory set overcome difficulty theorem section classifier depend problematic view suppose surrogate linear classifier induce turn similar support equation curse restrict posterior isotropic classifier base consist pac minimization sample pac theory directly deal involve notion conversely approach measure deal access refer information I I ks imply give sequence message compression priori observe either compression strictly compressed since message simplify framework message string bit sequence sc weight output sc really general depend compression therefore sequence gibbs risk mm pac vote sc follow issue draw instance belong sequence bias replace factor compression version average compression compression simplification allow character key reconstruction output independence simplify set describe complicated rewrite sc compression sample replace mm kl theorem present pac compression reconstruction output sc result calculation present precede generalize exception minimize bound compression generalize self output sc note proof reconstruction disagreement function pair obtain align let reconstruction output ij calculation p eq majority compression reconstruction self reconstruction rise sc always output vote margin vote gibbs risk disagreement express follow vote proof identical pac compress pac theorems direct pac express bound disagreement form rely justify vote design minimize express quadratic program case vote vote majority risk situation margin remain maximally uncorrelated tend use quasi see moment margin force become justify pac dedicated quasi posterior dedicate posterior always lead uniform minimize pac pac justify vote shall restrict margin quasi uniform vote value minimize reduce margin control random forest idea margin justification type directly consider recall form moment majority vote first attempt minimize without give instability estimate ss way restrict section upper bayes hence accord bayesian effect prevent compression necessary kernel next restriction vote let self quasi uniform distribution give vote nf f q ix nm nm nm definition point rise vote pac together restrict uniform overfitte consequence posterior margin uniform exist majority vote let quasi eq q quasi quasi distribution vc know capacity vote capacity relate degradation produce instability explain instability uniform interestingly equivalent margin constraint equivalent pac always represent margin call self quasi distribution empirical finding translate program qp remain turn qp self column qp represent definition quasi minimize rewrite qp qp weight recover sure solution qp uniformity property n stay give program attribute calculate normalize rbf svm empirically even significant among experiment implementation scale black test vs context interest learning algorithm handwritten digits split task intersection set example avoid binary result binary dataset table show handwritten context binomial context outperform uci help answer well dataset remain risk algorithm heart letter ab letter segment test l vs l binomial explanation statistical classical perform represent product task amazon user language review convert dimension consider least chen show r name book comparison test l explanation context test sign draw majority vote maximally uncorrelated sound adaboost classical binary sentiment highly gain handwritten digits context observe pac pac majority vote moment fix moment validation close basically severe degradation pac former small value value individual decision implementation maximum depth high datum pac corresponding pac multiple uci dataset construct remain example testing figure tight risk majority vote value tight nevertheless pac guide rely validation expect pac bind pac justify hyperparameter majority vote moment side chebyshev mild moment empirically vote theorem estimate risk pac allow function pac property kullback leibler pac together give end nice solid perform compare svm regard ever nevertheless machine supervise datum enough sophisticated framework several adaptation structure co armed bandit reinforcement national engineering discovery grant perform grid universit e innovation du comment vote anonymous jensen side markov obtain note straightforwardly kullback kullback leibler bernoulli kullback distribution straightforward expectation vector independent consider value probability success mm step one generalize lemma countable possible generalization give martingale sequence expectation value correspond nh h h give independent give take integer n combination extreme prove induction induction hypothesis denote vector element last term couple eq hessian semi eq section inequality pac bayesian generalization require provide countable moreover pf let tuple vote x probability binary number next pac bayesian self align note change expectation align obtain obtain expectation jensen inequality f f self f side lemma jensen f f therefore theorem convention universit mm analysis vote binary particular introduce risk vote average extensive observation training contain learning end allow sample analysis basically minimize reduce quadratic aside theoretically achieve vector vote pac theory mathematical approach empirical experiment bound algorithm art majority vote firstly bag known majority
make tend dimension fix use stability dynamical system chapter section number converge interior need close approach capture eq et adapt px px px result step expand around py follow part apply px px except exponentially small function critical mode density risk exclude boundary spherical cluster mode show despite variation distance perform high dimension difference component randomly replication achieve component separation far draw weight mixture unit measure mean replication vision much nonparametric mode dimension develop mode risk density region even several bandwidth regard believe possible low fail part boundary merge separate cluster chen mode nonparametric mode risk cluster cluster risk even beyond core idea assign mode find risk cluster cluster core portion moreover hold cluster core condition literature worth expand dimension might get estimate function type hierarchical superior rather simply find mode clustering cover case outline use estimator outside core noise consider risk mode introduce related density region several cluster tree density local eigenvalue denote usual eigenvalue denote ball give brief detail need terminology good theory point regular non hessian critical point maximum maxima minima starting satisfy ascent manifold manifold mode figure useful exclude point disjoint flow critical flow density perturb particular result lemma finitely derivative vanish follow function cluster core boundary core compact finitely critical another define cluster mode mode point cx x projection exact c let gd c mx finitely remark lemma exponentially apply theorem exponentially small beyond core theorem core fail risk separate assumption spirit noise assumption use high classification mode part capture cluster say boundary tail multivariate normal tail continuous derivative risk
thank triangle side know literature positive minimax therefore proposition whose section suitable regularity negligible principal affect rate consider infinity uniformly presence distinct label advantage factorization regard probability measure mode associate intensity intensity assign proximity view sake notation generalize probability one concern consider real asymptotic condition pc volume thank conditional pc combine asymptotic behaviour dx word see argument versa concern fact term unimodal particular unimodal sense unimodal assume large mixture illustrate bivariate gaussian joint factorize specified belong approach mixture view algorithm maxima form estimate pc part attention notice theorem fine different providing attention tuning enough avoid curse care latter choose fraction large criterion criterion decay provide curse dimensionality view approach nonparametric estimation full parametric see gauss moreover finite procedure strategy g graphic library contour criterion fulfil might sufficiently fast applicability theorems decay greater interested reader slightly modify adapt cluster procedure one linkage procedure estimation exploit diagram implicitly suffer bandwidth selection pc bandwidth identify axis parallel estimate depend smooth bandwidth identify axis orthogonal oriented pc fast decay well unless provide estimation choice bandwidth phenomenon nonparametric remark concern pc study difficulty neighbourhood spurious mode explain help detect look great grid within fix recognize alternatively multivariate namely pc connect thought display cluster procedure classify algorithm sample curve label nearest mode curve equivalently mode pc shift shift loo ray lin yu reference therein pc label curve illustrate berkeley growth phenomenon bring kind dataset year curve smoothing fit discretize detail ram curve curve instance regression classification cluster aim exercise retrieve gender subject posteriori assess performance remove phase curve step recommend merge amplitude variation removal performance agree second principal eigenvalue estimate concentrate pc explain total remark limit may decay provide justify eigenvalue type remark conclude analysis first useful besides eigenfunction mean minus suitable multiple take weight display j appear monotonic describe fan effect integral second eigenfunction appear eigenfunction mean curve perturb subtract illustrate main matrix comment behaviour section smoothing structure apply automatically group connect perfectly group group mainly cluster correct rate retrieve subject apply micro sub assign cluster illustrate factorial prototype automatic multiply modal modal figure relatively segmentation detail recognize whereas separate reduce composition variable correct recognize outperform competitive algorithm curve analysis appear homogeneous suggest reduce repeat pt retrieve gender subject algorithm homogeneous interpretable group interested whenever tend zero cumulative process span boundedness derivative depend fx linear center latter fix first eq fix concern asymptotic proposition proof proposition converge sure monotone convergence order eventually away algebra thank thesis notation drop hilbert since clear argument follow denote denote assumption norm thank iii np finally behaviour similar hold member get thank choose choose computation allow negligible author grateful hold university thank discussion visit towards grateful density cluster la le di proposition em hilbert value rigorously exploit fact asymptotically proportional principal pc depend pc core define parametric joint pc application hilbert exploratory tool technique whose reveal structural difference collection sense heuristic multivariate context orient primitive back wishart region connect joint along explore author cluster easy depend difference contrast absolutely drawback connect reference therein worth local density identify region look maxima family research reference apply go framework curve surface contribution multivariate always due see survey worth propose inspire seek propose new inspired illustrate density orient dominant oriented issue surrogate derive briefly see reference tend intensity theoretical tail deviation focus asymptotic estimation non evaluate convergence see well effort widely discuss reference framework study instance break choice center behave worth candidate play density finite consequently knowledge characterization direction factorization hilbert process determine decomposition component particular besides concern operator smooth show term lead dimensionality univariate turn quite restrictive intensity independence point principal independence independent implementation drawback independently drop independence expansion simply optimal way coefficient explain candidate orient motivating implement obviously density multivariate procedure involve instead one convergence obtain constitute intensity assign proximity real apply well functional connect go introduce concern asymptotic theorem application collect clarity four part asymptotic tend two effect behaviour asymptotic negligible intuitively close rate zero effort provide go basis consider sufficiently latter condition since purpose hilbert speak tend convergent assume infinity tend eigenvalue worth moreover convergence contrary behaviour drop extra theorem claim proposition arise result suitable choice term hence plug condition highlight suitable operator whenever algebra worth note hand side converge asymptotic hand eq exist instance guarantee vanish choose exactly approximation dimensional formula moreover hilbert author metric orthonormal hilbert space worth explain strictly play resolution fine despite fact intensity intend besides extract unless additional remark reduce hypothesis latter argument one density exponential normal pp whenever
align pool balance preserve texture face date subject contribution traditionally align similarity feed cnns however alignment rotation overcome limitation propose camera cnn base identity unlike cnn fusion patch feature form powerful grey extract grey length network convolutional fully layer use verification train inter intra variation far propose add supervision improve big discover selective robust work propose another face recognition learn use cnn contain around subject publicly motivated train deep cnn table train convolutional filter avoid texture along architecture powerful layer typical cnns clear architecture choice rather motivate evaluation benchmark cnns trained cnns cnn architecture since extensive training improve discrimination detailed subsection design good architecture layer avoid design cnn architecture adapt architecture size cnn medium cnn cnn convolutional fully connect filter cnn convolutional activation linear relu softmax layer exclusive rate batch fix table conv st pool st x conv pool st conv st pool c st conv c softmax softmax aim make class separable often face verification extraction hand fed model method verification task intra extra hypothesis base map respectively discuss computing first explain model six face recognition cosine correlation achieve recognition cosine among euclidean city cosine grey vs colour cnns grey colour grey colour impact type comparative face grey colour grey colour colour contain significant augmentation flip augmentation technique face recognition use evaluation little analyse impact test pair fusion fusion learn score score compare flip score improve performance fusion statistically fusion preprocesse step pixel usually normalise space original normalise motivated feature normalise cosine recognition rate dimensionality reduction learn feature dimensionality storage crucial mobile device scale separately feed face capture spatial train separately fusion fusion improve face implement fusion corner different patch original evaluate performance network face learn fusion network face face fusion actually fusion idea hand pattern multi dimensional local network well fed compare accuracy database improve face show compare method art method feature dim face convnet convolutional neural attract lot attention field work rigorous empirical evaluate architecture cnns face fusion recognition powerful greatly fusion metric factor cnn subject support union horizon innovation agreement contract ep audio home contract ep l union project acknowledge use ac uk uk ia ac cn deep cnn achieve recognition cnn focus cnn architecture rather reason conduct evaluation recognition easily cnn unlike cnns train architecture architecture compare architecture evaluate identify useful property dimensionality traditional exploit crucial good performance fusion make source publicly consist face alignment face important extraction phase face dramatically unconstrained environment face cover complex intra variation pose illumination ideal unconstraine environment last cnn deep unlike intra notably top three face report database face achieve cnn cnns recognition stem fact gpu greatly cnns generation effective despite promising cnn unclear good experimental cnn make task object recognition face align similarity transformation pose correction conduct alignment object recognition cnn choice publish cnns database available cnn contribution component cnn face system
per extract descriptor feature implement descriptor bag word use square normalization baseline classifier achieve also one kernel baseline classification rate literature reporting feature sophisticated geometric volume solve multiclass flow probability h py p k il r note geometry l f gr f ij g j dx gr f tr r ii ie gr w r p f summation convention basis orthonormal invertible derivative let directional derivative direction ii ie b r k r j r j smoothing penalty check g assume class notation subsequence leave q compact x k contradiction summary claim hold aa n h speak pointwise length maps convergence follow I k b follow jensen inequality k proof hard plot evolve decision boundary geometric representation framework amenable rbf initialize boundary leave column vertical department mathematics wu computer science university geometry multiclass propose geometric find measure volume intuition overfitte fast hence gradient flow move initial towards minimizer function establish mild multiclass compare probabilistic training label new label multiclass py plug get h f balance datum erm empirical balance erm complexity enough pixel human image change face regard measure fitting geometric method finds iteratively curvature priori assumption field point move show initialization algorithm bayes experiment multiclass uci repository green dots simplex value unlabele grey dot show position simplex flow htb example evolve inside simplex method towards deviation training boundary show additional contribution multiclass fitting term method consistent produce numerical result introduce knowledge current literature formula erm term closely plug regularize smoothness decision term paper geometric support contrast support follow regularize erm scheme scheme allow treat multiclass case simultaneously seek flat difference goal reduction flat minimizing towards flat volume dimension curvature associate geometric gr f l approach map remain flat impose penalty consist ideally vanish e distortion distance penalty ii md field necessarily exist map open neighborhood vector theory g curvature intrinsic curvature curvature r curvature correspond inefficient volume tr calculate geometric explicit formula volume penalty gradient field convention repeat penalty l f e ph pp flow approach functional speak hope estimator field recall neighborhood flow line tend bayes multiclass follow regularize gr gr k mt nn appendix flow sequence independent critical g p close treat practical choose radial la g ij ia plug rbf total evaluate center rbf reasonable field function role specify rbf extra summary learn summary input training every accord actually proper automatically distance vector way enforce geometric take center project tangent
thresholding pair experiment although admm run short time show converge admm test reach stop amount different three row experimental left much perform however class thresholde fine mean potentially speedup compare three term go supplementary solve group demonstrate effectiveness although focus group idea also extend admm graphical quadratic lasso thresholding solve graphical extend identify estimation precision shall fine partition institute precision gaussian model propose novel algorithm identify graphical lasso subproblem superior individual scheme split much fast thresholding validate feature extensively distribution estimating world formulate solve take advantage sparsity also screen develop detect entire precision matrix study relate distinct class underlie statistical power aggregate class formulate joint graphical non among fuse solve node base class number similar graphical lasso couple thresholde joint subproblem algorithm uniform decompose precision distinct split effect uniform precision require partition exactly graphical problem subproblem admm multiplier graphical partition scheme group efficiently fuse supplementary material bold letter k denote covariance precision matrix lasso sparsity pattern satisfy union element equal element partition denote strictly fine strict refinement denote describe element two submatrix ji concentration graph exist partition obviously base merging correspond divide kk k class feasible fine follow partition feasible partition first edge mix union one least contradict feasible paragraph contain contradict component pattern include typical graphical formulate represent penalty encourage structural pattern lasso entry precision accelerate graphical screening variable precision correspond computational efficiency divide group shall meanwhile method thresholding e employ partition method fuse node employ thresholding partition concentration split b connect divide disjoint without increase even separately another partition non uniform except white color identify feasible uniform partition kf ki ki theorems non supplementary material proof partition pair must hold feasible partition j jk condition non uniform covariance thresholding uniform screening utilize j uniform partition hybrid algorithm feasible partition generate good condition global toy three class three divide hybrid thresholding prove file hybrid screening algorithm feasible satisfy section admm admm minimize augment k admm iteratively variable insensitive require eigen k solve require eigen shall eigen decomposition upon update updating plain non thresholding detect
certain negative backpropagation training proceed sample ensure indistinguishable domain straightforward sophisticated architecture appropriate state art learn discriminative image let notation network parameter label prediction note preserve theorem class loss training layer saddle previously saddle estimate equation stochastic sgd comprise fed predictor domain gradient would convenient procedure sgd accomplish introduce associate act transform backpropagation gradient level sign pass precede layer use package propagation backpropagation multiplying update define domain classifier backpropagation multiplied effectively implement pseudo forward define run implement sgd sample well source section note result present subsection obtain stochastic consist gradient parameter crucially usual direction maximize respect minimize classification mm represent dot study variant problem one source contain example remove unlabeled toy share architecture hide layer neuron procedure keep regressor hyper execute algorithm source risk regularizer domain regressor toy boundary regressor nn relate look four detail show boundary predict label nn two decision boundary perfectly affect pca set map dimensional project cluster lie tag four crucial original column b nn happen conversely opposite corner resp locate pca well suit classification problem regressor precisely source classify otherwise regressor discriminate hidden representation prevent explain domain regressor nn learn domain generalize topology hand nn regressor capability seems roughly capture rotation angle neurons neuron regressor line observe nn group straight boundary classification however able roughly rotation observe regularizer prevent kind produce plane vanish one way domain datum hyper variant call tuple parameter unlabele example classifier reverse sample classifier multiple value select reverse architecture validation use early stop validation accuracy initialize reverse network nn svm l book book book book book c svm network machine svm amazon review pre process compose book encode star rank star unlabele separate target procedure logarithmic rate hyper training nn hyper among logarithmic present hyper criterion datum part algorithm poisson nn svm domain representation target improve representation art brief unsupervised source sample learn space reconstruct original svm reach propose source target different objective describe representation corruption number layer execute procedure concatenation layer search binomial performance report foundation claim toy confirm proxy various representation run nn section estimating source representation construct equation split subset equal large range low error firstly compare lead raw nn layer tends increase representation adaptation clearly low experiment one hand notice raw clearly help theory use parameter value improvement provide pt ccc ccc cm mnist source mnist number sign mnist mnist l difference non background adaptation perform row label da da considerably cover black amazon target sa l sa outperform state art extensive evaluation alignment sa adaptation synthetic number address testing view house synthetic digit latter image window vary include digit number orientation background stroke color degree choose structured clutter background backpropagation technique target label contrast sa accuracy probably information reduction case mnist mnist test mnist significantly appearance stay avoid minimum use obviously direction mnist equally diverse reasonably mnist appearance separation domain feed cnn solely network difference explain improve mnist scenario opposite sa perform either mnist da synthetic sign similar number due obtain image simulate target sensible datum unlabele domain additionally provide reveal label per add predictor change method beneficial set thorough verification semi supervise left work office datum office three distinct amazon discuss office spread large crucial successful fine cnn pre imagenet make comparable mean classifier previous work task commonly protocol adopt label source unlabele abundance unlabeled method previously state adaptation especially amazon domain slight moreover switch classifier apparent serve regularizer task identification people camera person person probe person disjoint illumination pose low make problem difficult human descriptor descriptor match commonly identification recall probability descriptor probe descriptor image several paper however drop descriptor handle evaluation camera constitute significantly identification report cross scenario training adaptation improve identification descriptor set camera camera appear contain capture every image view person two refer include extensive serve source descriptor probe probe correspondence p domain experiment serve view camera exclude image camera datum domain domain cccc cccc cccc mirror calculate similarity score correspond mirror image camera person average experiment architecture incorporate relu max pool dimensional descriptor within cnn middle share convolution training similarity feature batch batch domain adversarial architecture include convolutional relu include fully layer include intermediate representation verification descriptor predictor parameter pair train momentum schedule dropout concatenation output sized cm cm cm p eight hardness annotation adversarial ensure match source distribution iteration adversarial consistently performance pair far demonstrate adaptation descriptor p experiment propose approach feed annotate amount da domain backpropagation training behind predictive uninformative target implement new feed introduction simple gradient layer flexible benchmark sentiment convenient adaptation almost architecture backpropagation towards demonstrate experimentally feed architecture descriptor refer consist position product optimize em simple gradient update crucially opposite adversarial parameter hide adaptation propagation backpropagation f regularizer I f j network construction lead update use minimization discrimination minimize distinct update constitute special domain indicate index adversarial label pair potential strong gradient domain quite however observe rgb rgb different figure small domain mnist involve adopt set baseline package office domain adaptation identical layer bottleneck branch somewhat adaptation anneal progress follow restriction train et de universit universit universit institute science region new representation adaptation suggest achieve prediction make discriminate implement ii shift domain adaptation behaviour feed result backpropagation stochastic implement effort deep package sentiment analysis art standard achieve descriptor task context person application domain adaptation learn synthetic analysis person identification machine obstacle progress neural advance state across variety task datum training deep suffer shift datum come fully sentiment review movie classify review product book shift mapping domain target compose approach domain situation domain unlabele unsupervise label supervised focus hard generalize semi rather straightforwardly unlike work focus process decision make way feed network applicable target domain motivated adaptation representation domain identify origin observation focus combine invariance jointly discriminative operating predict minimize feature loss label classifier classifier update work classifier encourage course crucially three appropriately compose feed layer algorithm gradient modification momentum generic create exist architecture backpropagation rather layer multiply scalar backpropagation adversarial architecture architecture part label predictor success adversarial well sentiment image task traditional mnist office benchmark obtain finally person good retrieval apply domain adversarial label train loss demonstrate considerably often obstacle learn develop exploiting generalize focus situation similar example sentiment want distinguish review one type movie want book domain try generalize unlabeled review book approach classifier large body exist classifier linear recent denoise stack denoise autoencoder representation corruption transfer control domain suggest include work towards neural descent learn objective adversarial network confirm toy datum performance regular sentiment reach take rely explore many large literature focus mainly recently increasingly study notably exploit principle robust domain approach sample seek map source one aspect match dissimilarity space axis attempt distribution accomplish modify measure separability perform autoencoder gradually replace source improve separate representation learn autoencoder jointly considerably office adaptation deep feed tune easily available related building network way minimize discrepancy architecture minimize mention arise early dissimilarity use suitable applicable focus feed network minimize datum potentially rkh indistinguishable compare office adaptation arguably seminal optimize note representation word posterior regularizer task argue learn objective optimize reason part publish conference extend incorporate bring terminology depth justification extensive sentiment version descriptor person identification relate unsupervised adaptation feature approach source target aspect approach mean kernel reproduce axis distribution approach accomplish modify feature representation geometric use base separability target change among way autoencoder gradually domain improve simply train autoencoder actual learn autoencoder perform domain jointly single backpropagation simple conceptually achieve office benchmark unsupervise adaptation adaptation label target domain feed label target building network sample way discrepancy discrepancy focus feed network regard seek alignment achieve domain adaptation datum explore however recently non become increasingly neural notably denoise paradigm contribution domain another complementary robustness cross domain literature hmm also inspire addition sentiment classification optimize rely representation propose fair set result directly linear classifier distribution assume representation classifier note adversarial adversarial instead model possible source domain domain label draw I build information tackle challenge target error notion method justify
ordinary wind sensitivity capability reference initial observation three deviation height magnitude wind magnitude wind background background error covariance model follow error average magnitude reference background error wind magnitude reference wind reference ensemble mean covariance initial kalman cycle hour uncertainty wind synthetic multiply wind covariance calculate average covariance last per hour version long create several var background ensemble smoother investigate future assimilation three assimilation window interval assimilation window window regard spin assimilation synthetic observation available observation two window window accord experiment scalar day force background covariance lead covariance error solution rmse rmse trajectory assimilation window carry routine toolbox optimization process stop process iteration converge experiment use ensemble member assimilation window empirically tune setting keep fix hamiltonian keep number constant scale burn smooth approximately approximately cm p fix evaluation require window iteration optimizer evaluation hmc depend configuration burn number drop stationarity adjoint evaluate desire step hmc smooth adjoint rejection criterion run make cost equal assimilation window ensemble reject hmc case state collect run assimilation var hmc smooth assimilation summarize total forward run var scheme window hmc smoother roughly hmc smoother handle calculation computational hmc smooth replace burn suboptimal var course decrease impact assess technique increase smooth could acceptable information sample immediate consequence background error assimilation smooth monte uncertainty smoother build together calculate mean analysis moreover analysis ensemble begin assimilation window popular covariance analysis hmc smoother require adjoint var var must hmc smooth practical investigate strategy enhance sample smooth nonlinear gaussian acknowledgement laboratory reference construct smooth assimilation hybrid hamiltonian smooth time methodology unlike meet mode smooth variance conjunction assimilation window numerical compared ensemble assimilation da combine observation consistent da large great da gain acceptance first control variational three strategy variational find posteriori successful assimilation ensemble enkf root filter ensemble kalman filter efficient implementation kalman enkf provide variational identical error var enkf instance advantageous smoothing var assimilation time var update assimilation window ensemble hand posterior time assimilation window tangent adjoint challenge inherently var quantify matrix inconsistent scheme enkf approach var analysis statistically require additional observation unlikely ensemble ensemble uncertainty offer practical yield mcmc provide generating chain whose stationary distribution distribution mcmc gold limitation considerable state accelerate continuously hmc accelerate sampling hamiltonian generate hmc high good knowledge hmc da ill pose posterior approximated nearby anneal process hmc non solve four datum assimilation monte name smooth extend distribute hmc smooth ensemble distribution time smoothing carry sequentially assimilation distribution begin window assimilation background distribution assimilation present attempt reduce chain ensure independence hmc hamiltonian several important var framework system uncertainty water equation sphere moderately multidimensional test part assimilation hmc discuss direction review assimilation var initial assimilation background initial produce window analysis assimilation window background operator generally observation usually dimension determine initial discretize partial differential realistic numerical methodology propose small tangent jacobian dynamic initial solve adjoint k adjoint tangent linear adjoint challenge dimensional complex practical improve prior prior probability sampling give perfectly reality new contain observation act observation error background observation point perfect bayes operator functional compute observation mode var characterization uncertain dynamical system calculate infeasible approach describe directly act smooth system totally var enkf lead inconsistent uncertain dynamical system kalman represent represent matrix column forecast member ensemble q matrix forecast adjustment kalman calculate uncertain dynamical past solve smoothing smooth ensemble kalman employ state kalman enkf repeatedly write interval smoothing ensemble specify time condition available later version interval interval smooth computational fix interval single beginning observation highly violate show hmc smooth herein capable handling produce hybrid method ensemble trust mcmc distribution require distribution often consider impractical drawback take burn discard sampling target independence sample usually drop intermediate another drawback carlo member rapidly require control survey fast multi modal fail mode carlo hmc limitation mcmc hamiltonian consist momentum ordinary equation term hamiltonian advance solution exactly reversible common position advance hamiltonian time satisfy careful good accuracy practical consideration split evolve small sub flow exact flow hmc variable variable variable choose sampling joint distribution hmc variable position hamiltonian system serve markov chain kernel probability hamiltonian distribution position mean independence momentum momentum gaussian know view auxiliary acceptance rejection summarize hmc probability markov target auxiliary discard accept reject continue distinct propose q stage three stage energy length hamiltonian step tune covariance momentum choice converge sampling take identity ideally know approximated stationarity hmc sampling state fast generate parallel algorithm mode smoother simultaneously account assimilation initial assimilation window analysis distribution seek distribution potential energy var functional coincide var sequentially forecast analysis forecast assimilation window begin forecast forecast window propagate current background begin current include error considerably enhance analysis assimilation herein forecast hmc initial current assimilation give background state distribution follow calculate base forecast fix forecast ensemble generate ensemble initialize good suboptimal drop number helps propagate assimilation begin assimilation window initial forecast assimilation window use final provide assimilation test hmc smoother nonlinear sampling compare conventional var move similar use two local act initial choose
periodic arrange base particle diameter magnitude non scale denote box box formulae ratio volume particle height homogeneous mixed initial two particle carry contact contact coefficient detail contact mm mm profile flow depth depth red circle denote coarse scale track illustrate log block label denote scale order field essential proper coarse irrespective question scale make valid coarse length individual particle strongly thus lead spatial coarse suitable scale ratio steady simulation flow reach steady whose steady flow fig steady profile average time averaging thereby volume fraction flow fig fig ratio see plot depth profile coarse ideal scale scenario simulation scale average similar fig volume coarse width depth denote fig strong statistical temporal ensemble average probably particle particle scale similarly slightly compare fig label range particle latter usage illustrate construct coarse volume fraction fig fig average denote near base particle effect particle nearly decay base surface pressure pressure thereby large alone momentum velocity contact show sub particle length denote observe near base smoothed particle denote understanding mixture scope paper publication nevertheless spatial coarse scale particle cc steady mm illustrate centre steady spatially profile plot temporal averaging thereby dash spatial average steady range towards investigate issue spatially coarse grain field thus previous spatially average coarse carry depth concern temporal particle sec rather sec approximate correspond steady see fig fix interval min total define interval begin gradually result spatial averaging spatially average alone profile density increase increase fluctuation gradually fluctuation quantify error spatially observe proportional steady flow average average coarse smooth boundary solid dotted line steady denote temporal window profile large alone temporal contrary effect denote effect similarly coarse scale produce field block section apply expression datum steady flow particle underlie phenomenon predict thereby step application cg state system channel describe particle segregation happen see fig vertical centre particle coarse flow particle segregation dynamic particle alone consider investigation simulation approach spatial coarse spatial coarse result spatially profile result dimension average time average sec fix interval average temporal spatially average time focus size illustrate particle profile fig effect different averaging scale fix scale effect two old spatial scale invariant rectangular invariance region purpose something pick tracking flow one fig track fig average average take exist temporal block coarse scale invariant flow coarse scale average field cg thereby temporal scale average construct track flow depth analyse display plot exist irrespective average rectangular dominate fluctuation exist spatial coarse scale nevertheless invariance different coarse expression sec flow needs specify temporal ii similar steady flow range temporal flow technique average coarse flow applicable system investigation flow discover spatial scale coarse grain fig region fig imply exist coarse spatially average field coarse grain unknown validate shall develop formulation use approach micro macro coarse available processing reduce store etc couple pressure release like support project rgb rgb validate model micro momentum discrete position micro restrict micro macro flow average method advantageous interaction force construction boundary stress force determine component mixture segregation require average efficiently investigate simulation molecular dynamic etc illustration generate simulation mixture rough channel coarse flow source coarse part solver predict validate micro macro method momentum stress particle force include approach interested reader therein micro macro call average coarse advantage satisfy equation mechanic boundary spherical coarse contact shape ii contact contact instantaneous often micro macro situation therefore generate equation coarse deviation concern coarse see flexible use discrete molecular particle datum coarse successfully allow flow boundary analyse flow convex shaped flow material nature material crucial diverse designing extensive carry material past static material failure static spherical nature material particle shape external etc phenomena particle segregation tool arbitrary solve law interaction break particle contact computationally nevertheless million particle mm represent flow magnitude life flow environmental environmental flow simplify material closure relation law etc assumption parameter determine give flow call calibration year focus multi simulation application include mixture channel particle segregation application particle segregation often tend distinct alone flow channel eventually particle end free particle appear near need property material motivate need average coarse recently steady alone besides extend multi particle density channel focus upon topic steady flow technique ensemble nevertheless coarse average alone flow necessity temporal coarse field extract average field coarse cg derive case channel sec flow steady length coarse scale scale determine averaging flow illustrate spatial average spatial averaging field invariant main finding circle ex extend spherical system extend coarse formulae classical law mass momentum thereby expression stress particle mixture multi point construct distinct coarse formulae formulate media flow flow water mixture ice concrete deal partial variable mechanic dirac delta particle density field sec function integrable real function benefit later fraction define eq thereby expression spatially coarse grain unclear concern coarse expression thereby briefly reflect characteristic possible need possess certain ensure positive momentum delta w result differentiable efficiently manuscript polynomial three cut radius range coarse polynomial polynomial allow average gradient integrable analytically direct different make delta coarse choose coarse mass density momentum velocity field derive section coarse derive equation chain derivative coarse grain mass grain momentum follow momentum notation arrive mass balance hold particle mixture velocity ratio momentum field grain velocity mixture continuity exclude momentum velocity satisfy balance law balance momentum state expression stress momentum derivative force particle expand term represent index q second substitute force density force stress rewrite branch illustrate substituting expression allow force along vector identity rewrite contact length branch within sized receive big moreover contact force independent velocity particle I substitute stress field thereby total stress field contact stress field stress type expression consider fig coarse fig magnitude arise contact far derive mass velocity force
descent solver svms however systematically investigate incorporate posterior monte subgradient non max margin discuss detailed balance several hmc subsampling efficiency subgradient langevin method svms devise effectively integrate variable far hmc mcmc converge target posterior fairly previous attempt carlo langevin carlo stand also extensive max organize hamiltonian stochastic subsampling propose hmc version mixture svms experimental bayesian svms svms bayesian become increasingly big application overfitte account adaptively infer via growth fortunately scalable inference carlo readers review particular gradient prove dimensional space optimization technique representative example include stochastic langevin manifold high posterior differentiable little systematically doubly log arise vector svms max margin smooth differentiable metropolis hasting walk efficiency space sampler datum efficient inverting benefit view extra stochastic subgradient extensively differentiable objective many svms lasso regressor extension none systematically investigate idea subgradient markov chain systematically mcmc posterior generalize hamiltonian dynamic subgradient able detailed cost subgradient replace unbiased anneal subgradient empirically previous attempt subgradient hamiltonian carlo langevin little distinction distinguish hmc like hmc technique subgradient theoretical scalable paper organize review hamiltonian subgradient monte carlo sparse conclude hamiltonian dynamic mechanic describe momentum hamiltonian energy definite decompose hamiltonian eq carlo hybrid hmc classic combine besides conventional bayes posterior problem represent u conventional typically write give convention interest position hamiltonian sampler differentiable potential hmc simulate euler conventional stepsize discard augment sample one langevin momentum discretization retain invariance distribution deal u subsampling langevin extend hmc hmc stochastic posterior subset size noisy turn algorithm lot review hmc euler introduce fluctuation follow stepsize polynomially decay mh hamiltonian dynamics central gradient gradient hmc method hmc formally hamiltonian subgradient hamiltonian ordinary energy posterior non differentiable accumulation laplace hamiltonian assume hamiltonian hamiltonian dynamic hamiltonian stochastic subgradient mapping another turn back primary dynamic hamiltonian hamiltonian equation indicate subgradient potential energy differentiable hamiltonian keep property hamiltonian mean flow energy flow transformation separable hamiltonian regardless property metropolis subgradient hmc hmc subgradient subgradient energy ordinary subgradient initialize discretization stepsize analyze hamiltonian detailed balance subgradient potential mainly case zero implication hmc dynamic may result energy construction save space sample subgradient hmc draw approximation detail go sense converge hamiltonian subgradient follow subgradient hmc either euler leave balance subgradient leave readily subgradient langevin replace log formally simulate noisy subgradient exist recommend decay save mh correction langevin proposal stepsize properly subtle thus discretization stepsize scheme properly make relatively adaptive subgradient paper stepsize experiment scheme beneficial fast tb stepsize dynamic likewise version hmc generate via iteration mh correction adopt anneal properly generate sample subgradient inference svms dataset binary naturally svms interested classifier per unnormalized py ic regularization hinge differentiable log adopt subgradient sampler eqn eqn tb stepsize initialize tp latent mixture capturing consequently learn instead one description extension infinite svm mixture associate assignment work distribution gibbs classifier discriminative function hinge adopt build variance sampling wishart conjugate augmentation infinite usually fix simplify reader detail input subgradient partial probability introduce hmc step gaussian parameter subgradient hmc develop mcmc crp use randomly draw calculate sampling subsample inspire doubly doubly classifier doubly assignment subsampling formally take subgradient q gradient save assignment improve efficiency alg stochastic parametric svms extension max margin svms bayesian achieve still sample high dimensional baseline synthetic consider stochastic dimensional observation distribution input sec augmentation sampler obtain accurate sampler stochastic rather stepsize omit mh correction besides diffusion number sample burn give mark visual mcmc accurate almost indistinguishable method namely uci dataset binary space training use test stochastic metropolis good justified successfully decay stepsize diffusion carry validation discuss detail choose various respect time mcmc reach moreover subgradient augmentation method draw mini walk metropolis subgradient observe method mix fast dimensional rest testing walk metropolis sampler three sampling walk metropolis set choose stepsize adaptive respect running scale see ten fast might particularly curse decay well flexible stepsize decaying give analysis sensitivity subgradient reflect tradeoff general often computation efficiency consideration right subgraph e curve dot mild green serious typical right bit accuracy mild benefit result mixture svms doubly subgradient hmc f contain test use stepsize final efficiency doubly hmc greatly improve svm use stepsize hmc significantly also stochastic hmc currently svms still remain sample gibbs systematically subgradient apply linear svms mixture svms doubly subgradient posterior prior extensive empirical study subgradient mcmc improve systematically subgradient mcmc large laplace svms piecewise hinge g volume ordinary hamiltonian subgradient deal effectiveness mcmc efficiency draw subgradient differentiable posterior class see hmc subgradient method bayesian posterior investigate detailed balance subgradient hmc generalize dynamic efficiency correctness test well future subgradient stepsize acknowledgment acknowledgment go acknowledgment final reference supplementary differentiable potential ordinary hamiltonian dynamic construct smoothness countable infinite smoothness energy satisfie polynomial interpolation existence ordinary would hamiltonian dynamic see g solution generalize hamiltonian eqn correspond equation flow differentiable finish show determinant jacobian determinant finite except eqn write hamiltonian detailed balance hmc similarly hamiltonian subgradient hmc volume euler volume subgradient
internal split split expand fashion visit terminate empty present sequential generative process denote iteration empty tree assign split split location child iteration leaf internal child node iteration assign iteration terminate applicable expansion encode node expand expand expand fashion input initialization j pg j briefly propose limitation present pg sampler distribution bayesian sample gibbs loop tree associate condition remain j j j propose sample condition j summarize choose move split right child child leaf thereby leaf randomly internal node parent child parent child issue tree compute hyper encourage affect tree computationally involve deep likelihood subtree affect computational propose sampler illustrate section sampler next address concern propose ask complete tree indeed see particle markov carlo dimensional pg sampler instead sampler smc smc particle current pg nn contain residual integrating likelihood posterior top particle approximate complete substitute filter however leave version smc leave refer detail build filter decision stationary reduce clutter old tree write ccc end sequence partial particle old particle e tc particle generative process model decision potentially deep old hence whenever particle associate partial likelihood nod leaf node step smc typical smc resample old particle proportional resample resample none contain node stop loss return sampler summarize pg per iteration sampler explore likely reject decision internal since internal leave subtree empty pg training particle old pr w ct tc tc tc tc tc e tc ce ce w tc stop pr split normalize tc w tc tc tc tc tc tc tc tc tc tc tc experimental pg sampler contribution work efficiency algorithm popular box interested sampler run particle early sampler multiple condition equivalent inference residual label sampler mix consist lead tree true heart mix create depth control hypercube vertex offset generate generate vertex hypercube explain increase default leave mcmc illustrate material trace plot pg converge mse tend leave train indicate sampler posterior behavior sampler dataset compare computing ess mcmc ess discard iteration burn ess generate differ additionally ess ess likelihood ess ess pg ess ess pg pg sampler c pg pg hypercube consistent prior dimensionality dataset training point single characteristic c sampler ess ess three sampler achieve tree similar ess sampler increase observe pg exist pg sampler pg pg real pg unlike move pg complete confirm dramatically true consist pg well pg prior tree backward sampling show pg pg future direction bl fellowship college international fellowship receive ep agreement popular linear bioinformatic demand prediction size metropolis hasting chain local change slow mix fit deep present gibbs pg particle filter tree change individual tree pg propose complete fit pg sampler international conference intelligence cp volume ensemble heart broadly prediction reduce wherein explain remainder extremely additive serial like tree additive tree additive extremely popular variety include protein dna automatic spam detection drug additive avoid overfitte fitting structure prediction time inferential credible well measure variable time predictive comparable forest network mcmc particular introduce local tree expensive large consider subset however move poor produce inaccurate poorly chain use scenario user focus pg use purpose decision partitioning input align tree rooted finite collection exactly except distinguish node child child without child child parent denote split denote location tuple decision leaf give leaf
could good estimator outperform sample target propose covariance concern riemannian compose offline asynchronous perform subject classifier distance riemannian possible consider setup stimulus stimulus modify signal frequency result henceforth filter trial belong trial center input center compute center predict algorithm find offline brain nonetheless asynchronous one trivial introduce period eeg record observe record eeg overlap epoch n consecutive record denote epoch hold slide process continue reach epoch identify enforce negative occurrence criterion trial ensure detect user present good flexible offline online record eeg channel reference right hz hz hz hz setup stimulus minute trial record stimulus record varie record minute trial second detailed trial channel time stimulus remain vary effectiveness evaluate term classification matrix investigate bootstrapping replication assess compare trial length second affect accuracy estimator increase attribute trial producing appear shrinkage affect epoch direct independent shrinkage estimator epoch maintain correct large small condition matrix condition ill condition second trial singular b eigenvalue increase eeg length integrated vary indicate baseline integrate improvement ability complex apparent difference favorable long trial length compatible shrinkage offer well inferior hence significant covariance quality computation consider training center figure show matrix tangent star visualization principal figure riemannian riemannian riemannian remove highly allow remove outli center filter offline center lowest illustrate center estimate affect filter bad poor cluster riemannian could reject close far mean central b line black star riemannian riemannian subject riemannian part first evaluation delay online subject display signal filter stimulus blue hz hz visible synchronization second axis previous trial high trial power decrease capability low trial stimulus frequency axis line signal high low eeg plot subject low representation rectangular row trial hz hz hz band ht accuracy almost trial second crucial synchronization stimulus confident second asynchronous delay eeg epoch second ht hz hz hz show visualization tangent matrix classify stimulus attribute classification offline art offline opt show take response take online contain online show improvement acc take trial classification delay offline end trial second rejection epoch delay long trial classification accuracy optimize online moreover asynchronous strongly interaction offline offline opt acc acc acc delay acc delay c delay work efficiency riemannian classification feature rely covariance eeg signal apply investigate robustness verify estimator yield high accuracy online stability speed epoch perturbation curve misclassification eeg past riemannian determination offer powerful include adaptation riemannian dynamic successfully covariance mean report paper attribute eeg riemannian time bring like thank geometry brain computer interface asynchronous brain computer steady visually potential true mm geometry brain computer electrical engineering des de france la paris universit france paris brain interface signal yield study eeg allow common source variability electrical biological construct invariant matrix might advantage definite comprehensive review tool could matrix thorough conduct main contribution propose classifier riemannian space subsequent assessment steady visually computer visually potential human interaction rely capability computer scientific eeg dataset inter individual lead hand intra adaptive subject several signal vision work try maximize one minimize principal canonical technique covariance eeg cca aim also classifier largely handle euclidean space space reduce riemannian manifold inherently account approximate euclidean condition feature riemannian partitioning build emphasize brain outcome art performance version steady visually potential subject concentrate frequency focus brain stimulus subject propose performance brain response acquire external implementation phase apply dynamic criterion speed system allows dynamically determine trial riemannian estimate point similarly adapt manifold laplacian adaptation pdfs matrix interpolation filter datum riemannian invariant complete manifold point reach finite last kind instead adapt riemannian apply eeg eeg assign class close classification filter component back riemannian another unsupervised filter removal reject lie beyond computed window robustness affect riemannian divergence two different class correspond cognitive matrix algorithm eeg metric eeg selection spatial filter application riemannian geometry mention mi paradigm mi subject
component variance gaussian quadrature express use integration define reason introduce filter next quadrature approximation fix sigma usually enable relationship quadrature however often beneficial square consider sigma point location one square quadrature approximation sigma analogously sigma formulate sigma quadrature respectively prediction sigma propagate sigma dynamic sigma propagate sigma point measurement filter measurement think formulate sigma smooth sigma weight quadrature smoother start filter sigma k n propagate sigma k n k q cope additive augment filter lag analogously reference smoothing affect weight equation discuss choice choice sigma point sigma transform gauss machine choice make gaussian quadrature computing close cf turn computed sigma immediately possible depend observation one sigma variance variance respect close moderate optimize bfgs covariance choose mat ern covariance least chance quadrature sigma could similarly linear polynomial regressor form gaussian method evaluation classical integration see special detailed polynomial use find covariance positive definite covariance multivariate polynomial select evaluation weight become furthermore exactly classical method weight quadrature set theorem e se fig nature transform interpretation polynomially move diagonal datum transform shape clearly polynomial fit flexibility quadratic flexibility certainly see outperform integral show use rd sigma point integration rd spherical sigma process quadrature se covariance rd spherical integration use rule sigma place intersection coordinate origin bit divergence plain vary bit state slightly bad growth often follow extended gauss filter quadrature smooth point quadratic variance bar figure see quadrature filter low smooth close evaluate target track dynamic iii linear dimensional noise angle measurement sensor write q sensor two rmse error sigma outperform sigma quadrature gauss practically sigma point htb quadrature smoothing sigma criterion polynomial order well gauss kalman process well outperform previously sigma point filter e orthogonal polynomial inherently generalization connection quadrature multivariate function form orthogonal hilbert polynomial orthogonality denote function expand polynomial product deterministic gaussian covariance eq theorem algorithm section proposition remark corollary gaussian sigma use advanced kalman sigma method interpret quadrature suitably also multivariate gauss integration spherical discuss sigma polynomial method numerical gaussian integral regressor approximated regressor sigma integral predefine call point multivariate certain exact particularly multivariate context turn regressor quadratic closely relate polynomial gaussian process regression approximation regressor perform much selection weight function particularly useful smoothing kalman filter integral form sigma approximate example multidimensional type quadrature base integration transform sigma interpret belong numerical integration conversely quadrature interpret case sigma point taylor interpolation gaussian integral polynomial numerical present process quadrature connection sigma multivariate integration sigma kalman filter filter gauss kalman filter see case suitably generally quadrature gauss rule symmetric integration special criterion sigma error conference article analyze process linear article connection extend symmetric rule sigma selection well provide kalman filter k kp non state filter page smoothing see filter filter linear kalman q initial respectively update sequence smooth non non smooth smooth smooth moment matching comes transform additive transform moment variable point smoothing method generally approximate filter sigma point call refer weight sigma lead form sigma correspond unit sigma transform recall dimensionality algorithm unit th axis sigma sometimes computation concentrate direct method special g form process gp evaluation integrate gp regression predict test j contain model set q gaussian prior parameter equation want approximation joint covariance point posterior everything describe typical dimension conceptually practical typical independently gaussian quadrature idea point integral scalar simplicity interpretation quadrature interpret polynomial function aim still link due integration integral
others framework forecast decide time tracking produce use period assume historical activity week level material metric mean mae estimation produce ar current table summarize estimation estimate outperform period column table regular since cm whole c week week al ar naive ar naive ar naive naive blue ar grey background reflect suggest suggest panel period regular week regular paper week week year htbp train panel estimate close show post uniformly square absolute correlation correlation partial r avoid root absolute potentially alternative well publication internal lead available historical necessarily inaccurate week report website period period outperform accuracy metric essentially change suggest table challenge us article website counterpart change update affect estimate search website day month estimate despite change stable et result superiority term accuracy compare track google search methodology lead accurate access input google use incorporation key enhanced week activity year show reflect correlation integration series prevent spike series remain significant result increase query week average week smoothness substantial understand google historical complement period slow response activity current effect correlation activity google search detect public reaction investigate movement towards activity increment increment naturally smoother uniformly behind competitive evident globally select google autoregressive close figure tendency level evident wave h result self correct week series google search domain peak across term week important note algorithm update gold activity immediately activity display superiority method rely behavior google search user methodology change place future easily generalize spatial disease social amenable search moreover framework strategy track internet service twitter normalize volume google independent raw volume divide volume volume comparable available website www google trends date standardize normalize query zero year google estimate impose invoke kind penalty penalty dynamically week window week find hyper hyper however since window dynamically datum week cross validation need hyper combine validation autoregressive google tend unnecessary section still remain hyper considerable propose detailed specification hyper square rmse mae target increment two co movement hyper parameter explain aim test methodology respect come improve variable level transformation google time formal x py normal eq year window date specification hyper restrict propose google search autoregressive lag restrict validate google autoregressive lag term autoregressive restrict validate penalty google lag l l penalty google lag htbp partial sep lag specification autoregressive lag specification sep term autoregressive lag elastic autoregressive lag period prior third fourth rmse mae relative naive summarize specification apparent perform penalty exactly redundant penalize counterpart besides specification penalty error absolute similarly penalty outperform square net penalty elastic show restrict global validation restrict zero drop use want remain specification penalty google term lag table give flexibility improvement fixing process separate deviation well impose decide restrict report new incorporate publish report subsequent week available recent inaccurate presence model schedule report week typically delay week week activity week value eventually week train activity sense future beyond week incorporate training week metric activity aforementioned schedule outperform absolute train main indicate l whole week week website week year year study article example activity http www activity week available http www available week activity study robustness website http www google com trend variability input common cover version despite considerable suggest incorporation extreme focus entirely series formulate input insensitive datum good portion period google raw volume disease low quality uniformly datum mean mean absolute htbp mae correlation increment different common period partial cm cm accurate public health decision save propose track google search publicly online statistical previously google trend even quality publicly available capture people flexible scalable powerful temporal big activity pattern million internet service show great big set epidemic predict price movie google digital disease google activity traditional collection center considerable regard digital lead theoretically sound estimation still outperform tracking cause united ability availability activity duration remain clinical track lag http www lag make data forecast activity recent year internet google yahoo
admit form evaluation section fix modify whereas place mass neighbourhood optimize rate view integral vary depend expert general first conjugate introduction correspond special compute numerically conjugate max regret b become overhead rate know impossible reason achieve bind put neighbourhood issue instead desire regret bind depend put amount mass cost rate price slightly bad prior put constant mass density proportional run finite section adjust integrate quantile constant motivation suggest obtain suboptimal factor careful happen cv cv integrate improper highly integral e process would remain improper yet turn improper define still desirable equally numerical stability improper prior improper expert combinatorial combinatorial concept combinatorial reflect loss edge play expect loss p simplify algorithm concept expert upon proper notion combinatorial come strategy fundamentally suboptimal range mirror improvement exploit exhibit matching domain expand range efficient combinatorial obtain quantile range expert suboptimal find r k tv whole variance inside expectation thus identify mix turn mix exponential module weight combinatorial mix new variant component prove regret component module aggregate module learn predictor summarize proof find linear defer regret express quantile cumulative coordinate v discrete discuss discretization mis instance point grow number imply miss affect factor exponentially uniform grid arbitrary vector regret factor loss quantile switching share aggregate space yet another loss bound order see multiply ambient intuitively try learn separate rate example dependency combinatorial component case reflect integral potential function whole way optimal raise perspective explain rate issue future work find low available issue aware bound hold range round adaptively tune take characterize pareto optimal expert regime assume loss normalize normalization adapt question incorporate proof section otherwise let two plugging implie maximize non hence interval consider suppose x numerator everything q satisfied consider take left side choice complete go improper careful jensen bad expert arbitrary imply expert follow implie put everything together together use everything give minimize component instantaneous mix regret generator negative satisfie bregman mix regret hence equality design weight tell ensure exponentially spaced rate v side care weight expression improper look involve contribution form outside use range argument around fall directly c w design decision make adjust difficulty general combinatorial task satisfy minimax rate datum popular formalize bound notion sophisticated exploit one paper new construct show prove incorporate quantile core instance prediction play assign incur dot product good without specifically expert round respect every ensure tight loss yet ask case indeed line improve scenario line obtain stand kind like typically line independently parallel reduce dependence expert whenever expert perform occur construct expert call reciprocal prior mass bad expert guarantee obtain sufficiently consequently quantile imply closely bound prior study two technique develop prove bound denote reference excess loss specify common type variance factor small weight happen grow linearly obvious like imply expert losse unique good vary rate prior call quantile improve combine quantile improper efficient operation round applicable sophisticated instead expert action theoretically online subset schedule tree communication fix etc reflect sum coordinate loss natural example component family analog combinatorial quantile method combinatorial combinatorial quantile keep combinatorial regret average entropy expert expert follow concept instead straight case call collapse coordinate separately bound easy expert achieve scheme learning build monotonically decrease prove multiple rate diversity aggregate reproduce certain expert motivate second extend
online h continue define next hypothesis regret translate aforementioned online statistical prediction essence ask mechanism ability mechanism may points limited budget though observe portion parameterized budget assume post mechanism pricing price transaction transaction mechanism signal note difference step set suffer regardless guarantee randomness give algorithm arrival strategy price induce obtain arrival price expect brief sketch price state regret constant strategy general prove advance knowledge regret focus slightly agent pay cost variant tractable bound low variant apply problem obtain pricing minimize importance first important strategy hypothesis randomness lipschitz set meet regret mechanism improve factor pricing factor several rely quantity measure financial difficulty understand answer four question also explain aspect examine regret aspect descent correlation value converge expect side alone cost compare regret pricing obtaining point decrease vice versa arrival equivalently burn unit simulation full paper success along burn rate turn guarantee continue true main setting follow pricing choose give knowledge algorithm implication cost contain run case small enjoy regret improvement reflect hope vc capture instance one depend capture candidate case scenario expect approximate example z guarantee run weak knowledge deep investigation present easier apply unlike pricing intuitively pay decision private propose simply draw price strategy pricing exist price cdf pricing give h illustrate price arrival benefit obtain f cost regret depend expected budget regret equality cost cost cost fix bind datum immediate pay meaning obtain less datum long meanwhile strategy price give mechanism mechanism simply run aggregate final mechanism run respectively specify online statement state regret price main show variant formally transaction rather price minimize subject budget upper theorem give regret show hold main begin easier suppose pay take bind minimize lemma subject form normalization sequence key price mechanism optimal drawing distribution exceed remain price actually induce want draw price state mechanism mechanism pay arrival pay difference main bind cost give theorem open classic mechanism separation complexity cost know advance every provide zero flip otherwise biased coin usually tail tail coin distinguish require gives extend heterogeneous idea begin benefit expensive expect regret tt mechanism h datum handwritten digit feature digit depict mnist handwritten digit http mnist ask digit digits digits task drawing otherwise therefore misclassifie give implementation descent include train randomly dataset baseline every naive offer every budget knowledge far comparison adjust instead initial future reason use base pick price cost reason na I implementation compatible strictly explore implementation third party mention direction price mechanism agent reveal explore propose contribution pricing hold pricing mechanism interface box depend rely guarantee interest good mechanism one hope direction know marginal correlation unknown broadly motivate set crowdsource consist label mechanism offer price obtain label build importance paradigm acknowledgment discussion participant week theory science foundation recommendation express alone regularizer algorithm randomness choice h eq could quite consider however importance importance let probability clear expectation condition proof outcome algorithm follow regularize convex regret guarantee reference q take side separate q weighting hold implement assume underlie regularizer expect depend expectation choice let th wish choice previous expectation condition equal outcome algorithm thus every pick sequence equal zero otherwise let respect regularize bound upper reproduce regret h come fact since last inequality get complete q probability arrival tune later summation budget constraint take point pay completely goal summation step optimize sequence pick budget choice expectation randomness lagrangian eq imply complementary optimum case simply complementary c budget tight let call valuable complete lipschitz take expectation statement pointwise include advance budget third note extreme specific outcome suffice furthermore budget upper guarantee expect improve beyond pricing knowledge appear upper obtain subtle simplify proof budget decrease constant choice parameter valuable point summation regret let achieve subtle argument derive summation constant plug set may even guarantee case prove summation bound summation give lipschitz eq meanwhile k easily approximate jensen line q line algorithm online every regret coin suffice prove biased tail require coin algorithm coin output tail tail biased coin set budget round coin budget coin coin behavior coin except guess bias correct procedure coin bias everything tail biased coin round coin loss meanwhile expect regret tail inequality hypothesis large small coin actually coin need budget exceed online regret well regret follow still hypothesis coin fix coin point either tail coin irrelevant regret proof order loss either thus theorem mechanism expect constraint eq proceed budget plug term bind constant cost consider draw induced probability amount spend density amount spend arrival arrival budget simplification hand side satisfy difficulty quickly statement split correspond regret use function obtain expect still modification give argument analogously eq pick knowledge jensen plug observation rgb chen mechanism datum hold task model cost challenge past future mechanism classic resource budget robust many guarantee significantly due active analysis cost constraint couple regret order behavioral artificial address edu edu edu edu interest drive generate algorithmic tool accuracy make extent value leverage apparent million competition netflix foundation google ml thorough precise reduce beneficial must marginal paradigm imagine interface sequentially greatly label learn learn cost indeed world task obtain hold interested agent seek hold agent heterogeneous correlate mechanism compatible optimize efficiency inherent question scenario consideration company store patient medical public yet company offer contribute private record heterogeneity patient correlate content medical disease know website order target customer offer social facebook profile customer statistical hypothesis unseen hypothesis member point consist encode predict theory attempt characterize task inherent usually quantity classification difficulty capture vc achieve budget round cost cost offer randomize price long learn select offer minimize limited budget capture randomness depend quantity rough rough similarly resource constraint quantity also simple bound continues require rough stay post predict interact pricing agent price agent transaction agent transaction mechanism output completely design problem mechanism price predictive goal mechanism minimize final input arrival agent actually mechanism agent price reject obtain agent transaction emphasize focused pricing intend develop note implementation straightforward accept transaction could imagine learn disease act patient proceed
htb track subsequent object density tractable filter filter filter recursion close solution filter proposition target time birth next multi fig respectively grow filtering generate component new density component component I predict k h discard small review recursively filter sequentially proposition since direct sum density respectively k j jk k I stage disjoint label birth track factorize depend mutually exclusive predict separate short track run shortest augment set birth track track track generate update l compute rank intuitive drawback truncation survival birth portion component hence computation number rank cubic approximation truncate density involve track birth track track subsection joint update separate prediction procedure involve truncation preserve filter performance instead aim filter one current filtering specifically derive I essence extend include association identical except survival track denote base association follow establish simplicity target multi multi next k filter proposition track birth track measurement j extend k desirable association simple determine without update via eq depict j h h rank optimal use enumeration track valid association stochastic distribution extend proportional corresponding allocate extended association assign difficult solution sampler marginal computed prove generally contribution state element p give q simply state ensure track z conditional pseudo sampler directly rank htb k kk k h z h tt h h algorithm gibbs initialization length roughly distance true one good initialization gibbs either start track sample term sampling length therefore complexity present alg pm fast target gibbs generally assignment update counterpart fair rank tracking probability clutter rate numerical scenario adapt straight duration target origin pair target track velocity target k h process survival b x b p clutter intensity average false fair comparison maximum figure cardinality distance localization htb cardinality equally similarly second sampler except clutter scan performance number hypothesis trial e probability target gibbs expect pick target rank approach fig average time order new conventional construct track require truncation due elimination inefficient intermediate strategy provide superior counterpart method theorem axiom theorem condition exercise paper propose multi bernoulli combine prediction update perform rank assignment truncation drastically superior propose extensive study set filter refer estimating number trajectory drive application lie heart diverse text popular track bp systematic systems foundation development novel filter density filter design computer pr cell sensor scheduling rv zhang label multi lead development object attractive conjugacy family propagate forward filter update operation result shortest component implementation intuitive highly inefficient specifically truncation rank assignment truncation perform separately predict component negligible weight rank assignment inefficient truncation procedure innovation predict original implementation filtering filtering rank admit chain innovation generate component generating advantage first cubic exploiting characteristic background label present prediction gibbs conclude remark section rest g unlabeled one etc space represent use convention kronecker delta argument etc inclusion indicator single dynamical label space define integral incorporation identity object bayes multi suppose target take space finite evolve
computing sift cpu times imagenet descriptor gpu state standard patch gaussians point subset multi correct negative incorrect evaluate roc thresholding feature space six combination test combination denote combination take place train stream deep stream nd nd part follow iii branch branch vi reduce branch branch decision layer consist branch shorthand use follow filter pool layer apply report architecture also briefly conclusion architecture something indicate network network closely importance matching help achieve score sift art interesting try learn pooling utilize pooling stream well come image patch pseudo also experiment tested replace branch descriptor test branch extract descriptor compute distance network descriptor never image patch benchmark number comparison sift bad network filter subset filter fig display filter convolutional part layer filter worth part basically mean learn two patch match patch train choose contain sequence produce pair baseline choose image matching compute produce art descriptor choose cost channel extract patch estimation batch patch result pair point descriptor match distance case branch hold worth computed lot similar mean treat connected layer branch network branch location forward cost would require channel image pair network compute subsequently unary pairwise mrf set grid qualitative mrf visually verify cost network well channel architecture show estimated depth map fine detail exhibit sparse network eliminate quantitative comparison focus plot ground across threshold depth six threshold pixel plot pixel across threshold curve winner take mrf optimization local descriptor consist change gradually technique patch input size factor context descriptor branch extract patch test cnn architecture boost suggest great architecture compare patch conclusion already simply good distance outperform imagenet raw form cnn adapt extremely problem show precision average transformation precision curve material lack among architecture superior far accelerate architecture stream resolution turn significant boost importance compare consistently quality result far actually consider paris est fr manually compare fundamental opt appearance study architecture specifically adapt outperform ec project fp across fundamental vision subroutine variety range structure motion matching building image characteristic two factor final appearance include viewpoint variation illumination camera fact need rise many descriptor sift huge vision manually design descriptor unable aforementioned appearance patch gain access generate software question automatically aim generate patch manually design instead directly annotate inspire advance deep fig interested address explore architecture exhibit advantage train raw patch match non database readily contribution manually implicitly wide baseline illumination variety neural model network benchmark dataset show significantly art descriptor manually design learn efficient descriptor sift recently method descriptor descriptor pooling dimensionality recent involve various performance convolutional descriptor imagenet dataset show convolutional descriptor sift derive sift imagenet consist narrow broad appearance include baseline already mention neural image impose limitation colour patch increase however state exist choose patch exception describe patch give size may dimension several way architecture branch adjusted training flexibility much network next hand maintain notion descriptor simply two patch input image directly feed convolutional layer convolutional relu pooling layer give module consist fully provide compare two patch test patch break convolutional layer x relu suppose increase inside decision discriminative might observe technique architecture turn consist layer layer relu activation shall also contribute accordance architecture enable central resolution stream receive resolution patch low resolution receive original stream process use architecture describe use make stream improve match central patch twice high resolution stream implicitly put help
ap schedule throughput discount indicator define true inform state mdp practice state ap energy accordingly appropriate decision node active mdp extend state characterize action node active node belief whole system state ts otherwise unbounded space mdp section schedule certain simultaneously active ts ts state ts equipped transmission simultaneously rf transmission state node correspondence node normalise rewrite normalise current scheduling ts since active ts evolve belief monotonically increase proof belief ts belief state sake clarity throughput ts ap interested scheduling accord throughput optimization bellman function sum respectively expect instantaneous mp belief mp rr ts send bottom list ts monotonicity denote permutation contain node position denote permutation vector vector concatenation operator solely ts node accord mp ts belief mp mp permutation position position position position symmetric monotonically j ff boundedness throughput bound regular pseudo symmetric swap use decrease order mp monotonically map apply study structure ts mp ts lemma show difference different difference value differ certain mp optimal ts ts ts swap need scheduling mp mp value contain node order former kn optimality pseudo particular belief position value element monotonically un lemma function correspond ts swap q map p induction already mp assume mp optimal ts mp ts ts lemma hold prove complete study throughput state optimality study reward probability lost immediately energy available transmission ts expect energy available transmission node belief probability monotonically affine node schedule discount throughput pseudo denote node process node function order change function value pseudo swap j mp process node send queue queue monotonicity mp continue scheduling non note argument prove mp swap w backward provide mp attempt channel mp prove perfect channel error I channel detection optimality horizon discount horizon discount upper horizon problem horizon reward study show bound scheduling ts scheduling exactly ts relax impose ts relax belief steady probability ts throughput hence bound maximum node impose therefore solve polynomial section numerically schedule mp case rr cyclic policy regardless history schedule average throughput repetition unless state channel ts node period horizon case figure throughput channel throughput increase number increase throughput value mp curve effect throughput clearly occur throughput channel reduce throughput quickly capacity high performance scheduling policy h figure throughput different amount energy notably mp policy ts throughput high throughput fall due state low reliably action mp scenario mp scheduling access system model optimality node energy simultaneously process state suggest scheduling require scheduling schedule wireless sensor device storage node decrease p p p belief n nb monotonically nb np z nb mb z nb first increase induction respectively first hold regularity note distinguish rs k rs k boundedness belief km ks w j rs first similar node state must ts distinguish correspond case kn respectively use induction summation equal boundedness p r j n j I either w j un follow immediate reward ts belief ts belief belief km element belief value belief pseudo function pseudo value function equal ks value ns ns ns ns un j un j w p un j p permutation hold combination belief belief ts boundedness lemma monotonically increase series induction trivially realization four since case remain denote hand definition conclude n n p l l p k p w case I pseudo side linearity need ns ns remain swap induction receive universit ph college centre wireless college thesis european center carry universit cover device cognitive making receive ph electrical school research university stanford university electrical department associate university transaction communication student college center co conference co european school interest theory theory emphasis joint energy communication nd uk wireless equip device study ts probability availability channel arrival model ts ts node ap scheduling policy throughput study armed bandit scheduling mp compare numerically multi access scheduling observable armed wireless network machine sensor energy technology extend constrain energy availability energy scheduling policy wireless wireless network offline availability online assumed realization communication decision mdp mdp dynamic numerically many state mdps prohibitive much scheduling order avoid important characterize behaviour scheduling case knowledge statistical scheduling learn scheduling wireless ap equip device slot ts probability availability model ts
optimum expect insensitive limiting value note optimum panel maximize different value essentially threshold concern negative appendix value panel well alg nevertheless believe general alg interestingly measurement suppose estimate estimating sign provide experimental validate recovery sign measurement alg small coordinate assign alg sign result report scan predict theoretical alg top coordinate even measurement experimental setting time report medium error bit bit scheme multi threshold parameter bit reader variance actually variance example I value much estimator bit far reduce sensitive threshold alg estimate harmonic label measurement well compressed hardware sign reduction transmission retrieval current satisfactory require measurement decode scan sense use tailed design bit provably fast current focus nice relax recent direction n let l convenience ij I ij term q computer science nj usa stable projection compressed sign measurement stable projection become measurement reduction propose decode conservative sign experiment relate projection accurate recovery technical bit stable distribution cs topic research mathematic engineering design sense sense e stable distribution g signal pursuit omp develop compressed sense study literature compress many hardware sign transmission appear bit sense accomplish goal show length decode desirable scan coordinate projection scan effective conservative well stable norm stream cs signal stream fashion signal advantage I scan extremely heavy tailed nature recovery procedure significant recover transition vanish major disadvantage heavy tailed require storage substantial follow first reader want excellent book cauchy summarize scan sign signal generate formula measurement coordinate algorithm reasonably study assume reliably measurement sec bit alg paper simplify analysis practitioner burden simple report coordinate increase measurement sign intuition reader interested performance please eq probability
within domain dimensionality transform represent dimension zero represent augment particular correlation cross classical correlation cross method computing section error cross cv error explanatory regression inference matrix sample second matching success weight web small source validation generalize unknown observe appropriate regularization denote true fitting true cv rescale properly value compute cv numerical show error accurately diagonal column sum vector example similarly match matching rewrite respect weight say correlation although explicitly throughout generally well error find minimize symbol distinction thus k strong extra uncorrelated avoid laplacian spectral introduce regularization stability regularize properly nonnegative write matrix optimization denote cholesky eigenvector characterize part eigenvalue correlation part k correlation type correlation explain hold factor b k kb unweighted rescaling define consider section consider theory section match arbitrary weight omit notation omit although matching actually depend fit let proportional comparable adjust split repeat several matching cross error formal error weight weight trial success symmetry match cv fit cv compute several example time replace rescale k elaborate account structural convert grid add randomly element nonzero triangle mean domain diagonal proportional adjust grid scatter recover look match fig matching become observation appropriate choice choose value look choice agree circle plot rescale constant rescaling right hand change term fine evaluation column total nonzero assume x ik n ik ik x op op assumption describe eigenvalue zero negative however observe hold accurately look cause expansion p element greater express p define respect ij bias eq eq also fit cross change two equation p part pi ii ii p ignore extract part equation replace substitute substitute p substitute simplify look also write g jj ki jj ji ji ji jj jj ji ji ii get let k k ij jk b k kk ik k kk kk c kk kk rearrange formula last ij ik jk jk jk first lm lm w lm lm w lm summation follow lm lm f lm lm lm lm lm lm order role play give pe rewrite ik g ik ik ik kk kk kk note note derive expansion like omit define describe express lm w lm lm lm lm lm ij lm lm lm lm lm f lm lm f lm lm lm lm particular expression give use eq ij finally take w lm g lm kk h lm kk lm kk lm jk h lm jk lm h lm jk compare acknowledgment discussion grid mm pair nonnegative matching define transform matching matching matching weight
closeness input file likewise class name super noun use string string al information word specific mention string string appear versus string vector name insensitive mistake name instead word type mapping class uniquely opposite multiple string distance two string class corpus distributional evaluate occurrence semantic context define class usage argument detailed table observe repository distributional similarity previously describe corpus similarity occurrence contexts repository distributional similarity use entropy p pass occurrence definition would method count occurrence method occurrence statement size would count occurrence mean one purpose relation way define language normally common package organize infer belong structure hierarchy extend implement type package relation class within since extend implement website question user extract nlp package study text exclude stanford noun conjunction dependency total librarie code repository source datum source file interface package repository abstract feature sim string sim code sim refer software name additional aspect commonly noun attempt select mutual pair appear rarely corpus hard noun score noun highlight bold font see software frequent variable name method name class repository class contrast term sim type display report cross coordinate corpus distributional similarity corpus code note significantly corpu baseline accuracy code report feature code feature distributional similarity extract conjunction additional manual label similar coordinate term classifier corpus predict classifier predict pair full classifier whereas code text table top ten coordinate successful top hierarchy class indeed top label match pair class package belong code common exact visualize aggregate entity determine edge color entity determine centrality degree community indicate note prediction highlight connection group package one highlight within package basic represent simple entity map implementation library code extract usage class location corpus distributional achieving prediction add entity software f label predict build code connection common usage department coordinate term entity refer examine technical domain entity suggest couple lead improve statistic context appear validation dataset dramatically predict pair discover semantic text critical system variety nlp application temporal parsing examine semantic thing share et normally compare corpus statistic associate entity entity appear context domain world object name plausible nlp statistic couple world lead discovery discovery entity potentially corpus website user answer question development label coordinate attempt pair distributional entity library code distributional similarity additionally base repository able calculate organization demonstrate cross accuracy accord human high predict towards software entity software method application language domain capability application software domain token visualization similar work semantic relation discovery discovery approach certain lexical relationship pattern analogy relation second approach contrast approach normally recall extract language mapping sentence include physical world entire align supervision relative rich complex software code constrain discovery software adapt software enhance task
use first fitting practice typically thousand comparison less experimentally favorable balance analysis machine favorable function tensor nonconvex empirically sufficiently structure tm robust efficient require favorable tradeoff polynomial network introduce polynomial function draw unknown formulate reproduce polynomial specify numerical precision couple approximate underlie kernel nystr om approximations incomplete cholesky fall inherent namely knowledge target exploit target significantly small remain another supervised rely modeling consist recall nonzero exponentially al attempt relevant attempt learn sparse online select form next fitting polynomial computational cost use neural network guarantee learn layer reach predictor careful al cubic manner cubic class machine relationship factorization coordinate success recommendation application linear size length homogeneous polynomial represent polynomial involve contain power high drawback machine evaluate fm operation computational claim order order use approach polynomial exploit concentration directly dimensional random approach polynomial homogeneous polynomial sign map polynomial hashing transform feature outperform require large combine either approach motivate machine section polynomial degree polynomial machine correspond decomposition recall array inner tensor treat rr comprise polynomial equivalence homogeneous us attack tensor factorization rank feature coefficient predictor correspond tensor span tensor factorization predictor two comprise define vector coefficient implicitly search low target polynomial machine attempt low rank drawback try ensure represent machine impose polynomial machine propose instead fit machine low tensor since obtain minimizer proxy kernel machine objective machine couple machine empirical observe machine expect risk indicate efficient locally optimal avoid variant tensor machine equivalent formulation fit tensor machine measure noise rademacher take theorem train datum observe risk risk rademacher main rademacher tensor grow converge optimal class rank constrain lie accord surely rademacher rademacher spectral tensor proof world dataset demonstrate solver bfgs provide mark schmidt investigate use quasi newton solver use provide fit tensor tm tm respectively influence tm accordingly initial well set batch tm contain choose time reasonable original matlab conduct intel processor ram nonlinear algorithm publicly characterization basic feature euclidean ground slice binary forest govern interact iteration tm epoch tm layer width major record running mean performance tm batch dataset evaluate machine fitting fm available unclear plot algorithm relative median third detail tm across reliable tm converge tm significantly machine might expect nature fast algorithm tm batch tm low h ccccc tm batch tm learner year census forest rna framework solver tm model require fit solver long parameter error tensor machine tm batch tm census fm well small slice relatively factor census interestingly census lead census error assess vs slice determine vary tm batch tm considerably pattern mini update dataset explore tm comprise pair item website subsequently visit classify point remainder also fix epoch table classification grow figure tm give tm second evolve target tm decay theorem constrain rademacher satisfie proof straightforward calculation establish complexity specify structural theorem definition drop subscript rademacher euclidean slice replace sum
gaussian divergence flow optimization covariance derivative rewrite q simplify solution consider vector unit direction write solve must term three fortunately analytically quadratic formula yield put optimal four solution root choose situation choose identity root ensuring full belief next conclude optimal obtain component spherical begin note form scalar univariate align translate axis paper probabilistic dynamically incorporate stochastic arbitrary probabilistic maintain belief weight unlike incorporate small evaluation thompson computationally tractable transform flow optimally update theoretic principle intelligence cope even datum stream bioinformatic problem rich computational minima overfitte work formulate online cast enjoy guarantee filtering belief optimal overfitte prohibitive belief prescribe calculate take specify velocity field track answer mid conservative measurement prediction integrate parameter uncertainty update prohibitive well unknown possibly exploitation trade precisely state art method illustrate modelling flow field close discard belief input round belief parameter round vector consider regression network specify weight different layer supervise provide minimize predict rate vary perceptron need offset advantage flow flow flow belief shifted scale utilize leibl mid entropy principle show close form let transform vector mean respectively express span unitary equal eq small choose select find act subspace flow diagonal spherical diagonal unitary match posterior follow case independently constraint pick hyperparameter deviation preserve form scalar unitary align update factor previously produce landscape desirable flow flow transformation small typical spherical unconstrained covariance unconstraine described numerically maintain positive way predefine minimum training calculate calculate flow correction vertical transformation rotation subspace sample contrast translate basis independently spherical act radial rotation full unchanged diagonal flow force shift converge update intuition briefly trivial update show sample weight center move decrease occur first finally move correspond diagonal third flow regressor importantly sgd effect regularization sgd regime consider application flow binary linearly dataset model parameter sigmoid binary kl become sgd diagonal flow spherical compare draw isotropic assign true train label online test round predict give noisy update report mistake differential entropy online sgd outperform single optimum flow entropy equilibria invariant gradient model flow several classification dataset sgd bayesian langevin dynamic exception combine margin regressor input model sigmoid generalization mistake pass average remain additionally test inverting set dataset instance characteristic classify uci contain categorical expand forest instance classify take winner balanced feature eeg discriminate census aim exceed eeg sgd give good regularization avoid heavy scheduling draw keep choose eeg experimental classification within fall exploratory beginning outperform generalization compare come regime monte feedforward belief flow sgd aim choose avoid architecture test well digit basic plus background patch black image digit split dataset error plain sgd dropout sgd dropout pass choose unit update sigmoid gradient kullback output example either sgd dropout throughout online discarding average plain build minima attain flow difficult online
assume moreover heterogeneous address ca partition context match define set corner locate origin hypercube estimate matching relevance contexts hypercube hypercube tradeoff hypercube hypercube locate assign tn tn p ia I I ia ia ir ia ib ir I jx jt j htb x ix n kt kx k htb pt j j p pt j I j operate user phase another content relevance select score exploitation choose relevance score minus another request score user number minus score user time confidence content user user training come contrast help build accurate relevance score exploit current keep function describe time content training phase let context hypercube request content keep estimate phase counter update number content exploration exploitation identify choose give exploration content content second high training high exploration exploitation minimize time describe increase call value affect accuracy hence phase randomly content explore identify candidate similar hence balance possible reward report pt j empty select empty imply hence action set empty exploration randomly select request explore exploitation select matching relevance score matching collect exploration exploitation relevance score time take time context union request relevance match set account cost net relevance minus maximizer randomly select feedback I operate let content explored explore user exploit content user I regret l constant exponent horizon hypercube context phase content hypercube except phase phase contexts exploitation phase train train explore ca integer sublinear averaged increase context similar give final time call divide round length instance instance begin round modify overall user user almost parameter appendix technical user particularly provide service user quality recommendation low type user analyze feedback content feedback unknown give feedback large case feedback available algorithm reveal eq p proof report miss however user real online distribute mining spatially pattern uniform intuitively loss choosing minimize large carefully distribute begin differently adaptively estimate arm single hypercube context fine region explore focus space ball distribute ball context structured address basically learner learner space exploration exploitation learner principle hypercube hypercube denote contain mutually exclusive active depend time hypercube learner activate hypercube divide context explore exploit arm exploration control arm exploration exploit arm reward arm different function inaccurate reward learner expect reward reward learner learner train classify keep learner learner send activate hypercube learner train learner enough always therefore reward explore update select arm k tt reward exploitation step activation different scenario learner case train learner much high hypercube active let high hypercube high active hypercube hypercube arrival regret due hypercube hypercube q lemma whenever hypercube suboptimal arm give suboptimal set suboptimal hypercube hypercube regret suboptimal exploitation e omit outcome exploit arm choose event suboptimal suboptimal th equal chernoff hoeffding active hypercube combine hypercube choose hypercube hypercube hypercube exploitation hypercube suboptimal exploitation eq remain optimal action determine much hypercube lemma bind level hypercube lemma level tight desire regret bind form arrival worst bad minimum case locate inside hypercube learner learner correlation learner learner characterize extreme time intuitive q order intuitive arrival form context adaptively time order start split case arrival start begin regret order regret large case parameter problem regret logarithmic difference c arm active hypercube level hypercube inactive hypercube active small activate activate arrival memory estimate currently active activate reality memory hypercube arrival single hypercube requirement require trick multiply order content score context context exist speed change content call capture drift learner learn dynamically change observation relevance group round window round keep separate relevance context window round round w round passive call sub sub round overlap fig round round passive sub round round operate begin modify base passive learn active round form horizon stability assumption use action round sub round matching accord content matching mean relevance passive matching take fig result relevance round sub round start already depend passive exploiting whereas spend start sub past observation context current relevance assume past current length impossible sublinear regret theorem run denote large stability time proof appendix online decay round relevance action suboptimal matching decrease round propose cost choosing affect news article yahoo page compose recommend content iii click relevance equal recommended content assign content also access network content divide hence process different number click click match contain contextual rating student randomly arrive content music rating music reveal dataset scenario content randomly click next news article day news article stay popular several content implement centralized run select content user run control divide md divided provide service exploit performance user algorithm compute matching assume select high history observation explore good matching action probability adaptively create ball action index ball action ball high context context simulate set value find range well see differential service context achieve learn algorithm percentage exploitation phase different simulation expect increase action explore accuracy exploitation explore hybrid table avg length equal single parameter value click value good average click em hybrid service characterize learn match relevance characteristic content suboptimal content dynamic depend approximately learn match content another user increase immediate chance user switch denote logarithm hypercube p center symmetry hypercube ip matching action hypercube relate suboptimal time hypercube sum regret regret matching lemma slot exploration slot content match ca sum realize since I first process sample reward good sample bind use artificial event suboptimal content th time exploitation phase select suboptimal exploitation otherwise lemma bound match suboptimal select yield outcome need run parameter pt pt last upper running similarly inequality ca time choose suboptimal probability result near upper next come sum bound lemma q difference expect hypercube imply slot one suboptimal near action miss feedback matching lemma hold estimate accurate exploit exploration content match feedback feedback denote number training user binomial feedback therefore basic due relevance regret due relevance round balanced omit time set suboptimal eq net slot incur show regret show due suboptimal matching characteristic due matching regret sum r r electrical engineering university california receive engineer middle east degree electrical ph university ann research interest arm problem mine receive university electrical fellowship award van van electrical engineering california interest network theory online communication distinguish communication transaction journal topic signal paper award transaction circuit system technology award cite award communication journal circuit system award compression streaming international hold lemma van diversity content source news medium etc diverse content content numerous key challenge accurately content evolve preference propose aggregation demand content content characteristic preference contexts content aggregation priori online match bound speed importantly operate efficiently feedback preference evolve illustrative highlight system distribute online armed bandit web tv video news aggregation generate source interest often responsible mining numerous source find evolve source request request content user characterize value preference assume arrive sequentially requests either another connect gender query content etc device phone match suitable content source content change match learn matching explore content match help learn characteristic content source call content application business news aggregation business collect source recommendation specific need music music content instance music facilitate share music collection user g model framework test dataset aggregation music type direct indirect visit website indirect user request content receive indirect achieve match current I obtain request trivial collect way vast user type dynamically user content content content jointly content build bandit notion content matching give characteristic user content characteristic sublinear regret preference characteristic slowly change achieve change characteristic remainder organize highlight difference describe decentralized content aggregation content matching scheme regret optimal scheme unknown static characteristic distribute analysis user numerical online conclude recommender armed bandit recommender characteristic user maximize recommender learn preference online base true relevance user propose relevance score moreover due nature online phase phase account link long run characteristic online learning centralize recommender preference characteristic item exploit past recommend content collaborative similarity user examine user user high relevance relevant content match content prior know drift distribution concept drift concept drift take account speed drift window concept drift deal ad manner dynamic period popularity may type popularity due event certain type popular certain gender shift case content recommend ask another another payment make display incur website payment token assume payment return content mostly ask justify whenever user obtain benefit content recommend ca popularity increase pool content strategy
mu claim mu mu mu mu therefore mu mu mu mu mu mu mu mu z gradient regularity satisfy function lipschitz probability curvature convexity analogous cauchy schwarz nd line term cauchy line third term put inequality q curvature combine obtain q q f freedom bind expand q calculation show proof prove let respectively minimize regularization minimizer affine constraint q auxiliary equality augment lagrangian spectral norm ball otherwise rewrite transformation I ia variable form decomposition fact update iterate singular soft multiplier converge I corollary definition descent semidefinite semidefinite method mathematics signal arise derive relaxation combinatorial promise runtime current handle problem thus considerable scalable semidefinite programming relate family nonconvex program surprising effectiveness classical explore recent area signal process classification gradient stochastic optimization lead remarkably efficient attack large problem build closely program linear optimum rank find matrix interest generalization sense optimization np hard alternate isometry rip least semidefinite ii n orthogonal solve addition applicability affine connect semidefinite decompose squared residual take phase retrieval develop descent optimize initialization contribution concern descent recover probability bind potentially carry alternative review detail analytical result proof experimental relate present semidefinite rank descent invertible minimizer minimizer hence nuclear norm next sdp coincide nk isometry small furthermore attain word find satisfy affine semidefinite constraint minimum rank solution positive semidefinite coincide ignore comparison nuclear nontrivial affine rank replace norm already nontrivial minimization perform singular thresholding expense prevent problem recently propose project eq bx rx rip heuristic suffer subsequent propose alternate square algorithm avoid svd factor rx uv uv b uv iterates linearly converge considerable semidefinite scalable provably convergent exploit goal vector magnitude author global phase value see illustration method generalization flow turn initialization course present initialization spectral start unbiased fact z sm z v yield rate converge gradient hold nevertheless local achieve subsequent constant replace tb minimizer gradient tb I kk main sketch matrix note define distance recovery gradually denote ratio exist probability proof supplementary material step give regularity enough property mu mu show close iterate condition event small specify expectation regularity next suppose z z v eq finally valid sufficiently nu constant r n I number scale conjecture could experiment relaxation justify observation use admm approach simplicity scheme could compute partial ghz intel core per three affine nuclear svd update compute multiply vector partial require gradient via gradient dense affine transformation dominate remove overhead cause small dominate enjoy low conduct slow generate take frobenius define f approach regularization fast three three scenario general entry nuclear step value choose figure nuclear significantly tb ix recover refer successful trial transition rank confirm linearly connect case program constraint recently retrieval develop descent procedure conjecture sufficient condition significantly broadly technique semidefinite
good variable produce parsimonious absolute much attention year xu lasso least regularize solve become initial lasso yield level lasso bold letter face capital face letter denote transpose represent absolute design vector component vector result minimum notational indicate variate mean centrality follow identity stand respect indicator distribution restrict shrink study analyze application error risk level information depend assume effect non sample priori restriction test vector imply restriction linearly restriction known theoretical consideration tested eliminate redundancy description see analogy ol estimator small risk may bias inconsistent plausible follow preliminary take acceptance rejection critical value distribution proposal upon use test base computation everything easier highly level nature simplify making shrink toward major shrink double shrinking consequently combine stein shrinkage stein stein shrinkage fix equivalent investigate respective performance cc restriction rather many situation e q bias mse weight distributional k rl nz c hz rl c hz hz th h value chi f hz th hz class local rl qx q c c h ols lasso estimator setup non chi h n rl accord estimator proof follow w procedure generate assertion dominate large relative increase present decrease nan c ccc ccc htbp ccc propose estimator al study examine specific clinical measure receive description variable description remark specific response age seminal percent center second restrict lasso preliminary stein shrinkage sl shrinkage variable table min estimator average fold validation fold divide datum subset predict version specification repeat show error sd estimator construct decrease confirm visually demonstrate variability estimator paper impose restriction preliminary shrinkage preliminary lasso stein type sub size propose improve dimensional situation estimator analytically numerically configuration error classical centrality degree misspecification vary dominate perform get efficiency near present analyze validation deviation shrinkage stein dominate error sense conclude lasso estimator fu know th q vi iii obvious obvious fu making prove iv follow immediate proving eq implement fashion fact algebra bias significance ann l instability selection fan j penalize zhang zhang generalized statistics ann ridge estimator j study united j prediction york pt orthogonal fu asymptotic ridge subset ed usa test stein united statistics introduction integrate pt text york j diagnosis ii journal xu iterative ann regularization variable elastic axiom claim conclusion exercise theorem remark summary school school mathematic statistics university abstract suppose interest relatively respect absolute operator suggest restrict lasso class selection restrict performance propose estimator risk show double shrink lasso word double preliminary stein shrinkage primary draw regressor response vector goal response ordinary square ols tx standard estimate ol regression unbiased large coefficient reduce subset component regression scad net fan li variable objective intuitive large exist cope selection backward elimination method derive
use former repeat pointwise optimal limit eq ahead bellman prove number leave side repetition contradict complete control convergence ahead establish engineering south school technology rapid city email theorem corollary optimal dynamic horizon cost investigate control problem optimality follow uniqueness solution bellman furthermore iteration result ahead study pi scheme sometimes dynamic program value vi pi load per iteration vi pi remains adapt control convergence analyse pi control viewpoint two establish pi compare implementation variation pi know denote positive integer dimension number positive denote select I minimize minimize control policy word hx within initial select domain compact policy asymptotically define boundedness meet select continuity bind compact value admissible within conclusion value policy admissible trajectory convergence give two control admissible bellman nonlinear policy form approximate origin start respective eqs answer two lemma admissible eq evaluate bellman equation bellman contradiction contradict step policy optimal solution show pi equation former lead eq pointwise decrease value converge limit policy pi hence bellman per theorem everywhere admissible give requirement asymptotically respective pass vi establish former monotonicity state trajectory utility order need vi pi conduct guess former merge vi find reference guess subject evolution vi admissible policy guess vi pi seem similar load vi significantly pi vi pi eq pi vi aim analytical admissible converge q inequality monotonically therefore eq assume monotonicity claim prove
report respect one cg yield iteration infer start concrete possible proposal hasting mh algorithm scalability gp apply concrete allow initial acceptance ten run mean deviation line mh cg threshold step size start gradient batch computation decide fix advantage solution initialize system propose finally hardware parallelism even capable quantification gp without impose quantification primary interest gps calibration computer model report direction langevin scaling ideally scale gradient fisher compute unable due gradient despite demonstrate distribution covariance relevance determination covariance possibility extend gp g gp gps spatio temporal aspect calculation algorithm improve present acknowledgment mf quantification uncertainty accurately langevin distribution negligible need marginal advantage gradient solve system unbiased unbiased enable scalable sense quantification impose number gps offer possibility datum function underlie behavior quantification primary interest necessary accurately argue carlo involve store complexity set gp covariance matrix e space compute common several machine approximation usually subset application gps name extent approximation quantification uncertainty application regression quantification primary interest avoid propose langevin draw require computation solve iterative conjugate cg product idea put optimize despite complexity complexity solve slow practical compare speed fast yield gain large unbiased algorithm stop highlight batch alternative contribution scale batch dependent complementary aim rather iii complementary approach noisy likelihood gp regression cg obtain determinant remain calculation log likelihood far transform unbiased unbiased likelihood employ carry inference parameter gps computation day parameter hardware knowledge attempt enable full uncertainty reduce vector impose gps motivate infer gp variant present gps demonstrate methodology gps marginal conclusion employ function determine covariance whereas marginal comprise noise bayesian forward label require q posterior analytically necessary resort approximation tackle approximate sample draw mechanism accept reject log cholesky cost require inverse multiply computational complexity scalability gps unbiased log space practical determinant obtain negligible bias briefly describe adapt gps idea stochastic gaussian way transition langevin parameter except stochastic optimization local transition langevin proposal require reach langevin enough acceptance therefore possible avoid introduce negligible langevin phase monitor gradient define gradient eq langevin produce motivation log marginal require q give yield unbiased version consideration methodology involve solve much easy solve conjugate cg popular carry without store complexity variant speed computation section regression apply concrete datum uci repository compressive strength concrete describe tb threshold number iteration system cg algorithm idea calculate solution minimizer employing initialize iteration refine cg gradient characterize orthogonality conjugacy namely mutually orthogonal iteration remarkably cg implement trade accuracy theoretically cg lose condition cg take sample gamma rate distribution reasonably number encounter inference gp covariance slow numerical quantifying extent apply gps impact solve need calculation sake brevity efficient accuracy show cg versus versus obtain algorithm back cg double cg implement precision iii relative absolute tolerance parameter calculation compare large double calculation offer low lead achieve draw conclusion whether implementation hardware order would cg converge would reduce solution would cg convergence cg yield possibility approximate expensive propose introduce solve cg well make convergence speed inner considerably necessity cg need fig accuracy version single version reduce standard cg iteration count inner inner cg gain iteration different experimental might
rate fully local focus global sample turn provide guarantee alternate iteratively keep optimize require rip incoherence establish geometric formulation convergence version gradient batch full heavy burden stream random initialization fast computationally update lastly manifold include optima conjugate trust name instead rate analyze output understand reliable empirically operate manifold giving empirically initial slow local angle metric principal close evaluation offer angle angle u discrepancy discrepancy column zero one angle cosine zero frobenius discrepancy measure discrepancy span subspace angle angle discrepancy entire sum square start principal angle accomplish iteration come represent state rely novel initial random initialization theorem improve behavior reach reach isotropic iteration enough without figure though identically admit tight leave second require though experimentally behind version rate dependent regard improvement per state theorem phase uncorrelated imply tu tu u determinant strictly analyze prove determinant increase toward non attract stationary global problem point strictly great determinant monotonically stay away convergence norm discrepancy proof theorem standard gaussian use theorem may guarantee tu bind discuss support rank subspace sample uncorrelated identically distribute recover incremental constrained take manifold justified section scenario norm slowly minimum frobenius must create orthonormal vector choose initialize subspace characteristic initialization iid bernoulli quickly low determinant fig htbp fig provide tight fig expect rate optimizer expectation situation tight average trial run initialization iid reach compute corresponding take entry noiseless empirically small noise illustrate convergence generate low datum orthonormal coefficient bernoulli fig finally factor recover fm w subspace unitary matrix unitary simplify quantity reference lemma entry zero identical discrepancy share take result strong monotonic frobenius construct orthogonal matrix span give orthogonal complement unchanged multiplying allow follow thus replace first leave frobenius orthogonal determined term complete equation difference maximize take span prescribe incremental globally improvement minimum sum perfect justification analysis motivate seek future expect theorem rewrite let follow ready finish main simplified expectation side consider random determinant discrepancy discrepancy prove increase determinant novel initialization recall define orthonormal matrix gram exist pick jensen lemma complete reasoning give theorem expectation nonnegative incremental zero mean uncorrelated distribute happen phase loose number iteration require local region minimizer many reach trial monotonicity tight require accuracy hope deep understand proposition context success rank factorization instance scenario seek natural incremental initialization also span neighborhood match tool process numerical method impose orthogonality constraint computationally algorithm attempt speed actually therefore guarantee svd convex several algorithm solve expand publish kind gradient convex factorization algorithm gradient suited problem regularizer svd contribution incremental dimensional subspace special factorization orthogonality span algorithm incremental rate part local minimizer match convergence application stream subspace track medical communication environmental science extensive discussion variety
limit version sequence convergent point van convergent recursive nonzero digit expansion interior recursive convergent recursive base condition hold cell condition convergent without generality convergent recursive representation digital via fold potentially space even copy index complement denote consecutive rule enough context make split integer volume recursive volume geometric volume cell net nest recursive split volume set weak geometric half interval enough point weak net eq q maps convergent recursive preserve section lebesgue measure end lebesgue convergent recursive split nb c aa thus equality n ia proof extend multidimensional uniformity let convergent basis transformation base uniform preserve element go geometric net require digital convergent transformation let nf adapt wavelet recursively represent explain depend f sf sf fold integral leave unchanged version haar wavelet adapt base split cell turn product f kf lebesgue theorem converge convergent everywhere function unlike nonetheless know multidimensional recursive basis narrow entire study average net digital map split base proposition simplify property elementary orthogonality property define must correspond plain improvement plain carlo net put coefficient particular otherwise give smoothness rectangular take leave unchanged extension series rectangular bounding point shorthand later use interior anchor set set figure integrable anchor mf jj convention version calculus circular disk center anchor rectangular origin exclusive center diagonal region extension anchor restrict account appear make extension appropriate extension mixed interior form ignore useful fundamental calculus though partial continuous everywhere move arbitrary region let might eq convenience replace subscript keep substitute introduce rewrite q vanish assume empty fail non interior region like include domain great multi continuous extension function non interior use domain one require boundary measure fundamental apply induction suppose hold shall fix therefore induction continuous apply integral integral induction complete hand side certain smoothness require extension variance product section recursive ss u jj u convergent recursive diameter define extension close boundary measure regardless whether place alternatively supremum essential supremum argument smoothness either know integral smooth index depend make put write equation diameter box finally put large ratio rectangular similarly cell parallel angle triangle condition therefore algebra j consideration point randomize base convergent gain net zero regardless whether know may suppose substitution test see get plug desire result integration fold root square rmse plain might attain advantage net net attain really size effective compose well use smooth component move triangle however call induce infinite uniform result notably demand limit average nest uniform stochastically square discrepancy split could relax deterministic construction fail weak net net proved extend nothing split never dimension become extent nothing retain diameter split assumption develop integration accuracy plain triangle product space may numerical product space special interest point transformation net recursive partition integral unbiased smoothness fold quasi quasi unit cube region cube space plain unfortunately composition smoothness integration product triangle triangle triangle reach incorporate shape relative lattice like construction attain discrepancy van generalize construction unit replace van sequence digital net digital net order quadrature compare survey randomize graphic transformation map throughout whenever integrate plain monte finite net uniformly digital net variance via net primary behave integration allow dimension background material net present geometric split van construction product present product domain domain rectangular domain rectangular domain latter result compare plain net dimension conclude tensor simplex constructive ordinary case plain number root variance improve usually uniformity via discrepancy q dimensional sense account numerous construction interest construction net definition integer precisely net discrepancy box approximated box digital net base subsequence base integer digital net handle half track triangle valid prefer zero point split basis digit split panel label new new right look algebraic description case describe describe digit rule obtain multiply operate plane recursive fold split collection exactly member call say level member recursive split split need cell split cell level split figure arbitrarily result version van appearance level latter similar transformation triangle yield get shape triangular set split disk disk limit angle split angle disk
map partition principle mechanic cell product division imply reverse evolution sensor property conjugate transpose related arbitrary index existence derive equation possibility single illustrate possibility partition orthonormal formalism follow row integer moderate number make become consequently partition obtain eq example present music estimator sufficiently besides resolve closely operator pp e j life formalism quasi rao available extended operator geometry long consecutive half sensor spectra rao ideal spectrum detect ideal spectrum side fourier transform superposition magnitude present result eigen briefly describe eigenvalue rotation partition element array characterize invariance input create partition array subspace ne u computer verify performance consider source frequency sensor spaced total array angular limit identically process possibility due possibility take operator snr peak sharp operator easy lack peak explore matrix coefficient square spectra vary last spectrum music music last fluctuation paper source coherent drawback know hence eigen paper coherent high generate distinct version resolution operator angular position source formalism mechanic channel diversity suitable array elementary order prove power computer implement simulation matlab operator r r rp p r n rp p thm element set compute angle arrival moderate give possibility dependent elementary keyword array angular spectrum resolution possibility next third section elaborate description use concentrated negligible acquisition locate region direction permit linearly noise medium model ergodic process receive instant angle operator half signal additive model matrix array lot characteristic focused angle arrival decompose four channel second subspace equation presence variation due presence power diagonal block give situation suppose array comprise hundred question operator
denote transform model integrated average underlie wind window smoother characterize coherence illustrate link process coherence weakly continuous accord unity linear average theoretical use process introduce move case convolution square integrable separable convolution convolution suggest attain nontrivial coherence framework notion coherence examine random spectral define possibly define g interpretation gain phase process function asymmetric covariance shift angle exploratory suggest exhibit visually assess amount multivariate value matrix real cross spectral gain testing spectral give develop ern marginally describe mat ern functions mat ern class ern impose ij valid mat ern popular continuously path interpretation index act control mat ern interpretation analogous interpretation link behavior interpretation function mat ern correlation simplify covariance ern note constant imply linear band great seem suggest interpretation smoothness amount serve illustrate function common perhaps suggest parameter induce coherence parameter particular examine distinct small high behavior non coherence complementary share illustrate coherence concern cross yield flexibility parsimonious ern imposing produce inferior fit version mat ern class coherence bivariate parsimonious mat ern constant close empirical illustration random field realization high bivariate mat ern spaced grid low pass pass filter low pass filter bivariate bivariate mat ern range smoothness low b pass panel suggest complementary cross coherence low high exhibit positively coefficient panel pairwise filter contour e mat ern linear compete build combination univariate matrix strength dependency uncorrelated uncorrelated process useful follow unity exactly coherent multiplier simply case yield gain relative contribution mention spectral variate observe grid fourier tf natural asymptotic uncorrelated spatial framework increase asymptotic ever direction series complementary asymptotic sometimes call domain asymptotic point ever fine anomaly anomaly forecast hour location region day day day empirical yield pass compare various forecast calculate day validation filter marginal denote smoothed smoothed coherence lead substantial increase longitudinal band coherence appear relatively sensible great south pressure horizon begin build mid pressure long forecast frequency substantial forecast scale horizon bivariate mat ern forecast physical spectra statistical behavior spectral density kk suggest additionally across frequency appear empirical plot follow ern spectral density day forecast band minimize coherence estimate day tp forecast forecast decay decay almost exactly h horizon fitting parameter horizon cross covariance day whereas marginal day idea hypothesis substantial strongly word table grid estimate guarantee mat ern sufficiently flexible field area spatial height pressure science regime anomaly unite low surface core stream anomaly temperature interest science height anomaly represent pressure anomaly height vary day qualitatively bandwidth pass calculate square day pairwise coherence coherence pressure level moderate coherence frequency behavior pressure apparent play crucial role formation high coherence frequency capture one explanation frequency assimilation anomaly level pressure constrain observational weather anomaly frequency band pressure approximately km typical substantial shift pair model utilize value shift spectra notion gain literature multivariate spatial yield physical relative amplitude process analogous process extend coherence phase smoothed exploratory tool function useful detect readily capture coherence interpretation multivariate mat ern insight future research may coherence process multivariate perhaps manuscript process covariance spectral continuous symmetric fourier df additionally cross involve calculation involve include vector bb ib ib ib detail admit dm ij representation complex fourier f dm eq z fourier transform immediately lemma lemma constant frequency acknowledgement helpful development national science foundation grant dms ex plus minus ex corollary coherence increasingly rely alternative viewpoint model develop coherence multidimensional see band coherence fundamental construction suggest interpretation parameter mat ern class smoothness index frequency imply dependence smoothed illustrate interpretation forecast pressure examine insight difficult detect formulation keyword square coherence spectral density stochastic nearly development spatial recent review relatively explore pose construction flexible explore empirically dataset compare cross know krige fundamental open extent construction theoretically follow quantify spatial dependence gain multidimensional question previously lack sufficient suggest insight mat ern direct interpretation manuscript correspond develop autoregressive move rational know squared interpret quantification relationship sciences numerical weather forecast version generate forecast consider state surface level pressure daily forecast hour show coherence diagnostic forecast band involve pressure phase highlight pressure level difficult construction construction pp stationary j obstacle process develop flexible value nonnegative say nonnegative nonnegative
naturally specify procedure traditionally think setting parent generate body reinforcement powerful sequential decision paper encourage transfer tool extend drive advance model sequential change view lstm connection direct reinforcement develop training sequential imputation approach imputation horizon direct cover show successfully model sequential imputation loop generative implement mdp train model policy use motivated examine qualitative quantitative difficulty mechanism imputation baseline significantly baseline develop benchmark imputation imputation model special case direct gain popularity relative undirected counter part reason rapid available computing view precede decision investigate form indicate may factor eqn arbitrary variable conditional restrict exchange eqn permit approach interpretation take search available indicate compute control stem guide use policy direct interpret finite process terminal encode mdp place I initial draw train autoregressive variational low dirac delta training describe previous paragraph guide trajectory generates guide rewrite distribution z prefer define x px term eqn become variational training direct generative interpret search eqn enough model precede g author tx px non horizon tx x abuse sum trajectory integral target reverse stochastic transform write xx generate tx trajectories tx guide log eqn tractable construction basic reverse process eqn trick guide trajectory terminal start material subsection derive bind trajectory distribution capable learn primary trajectory recursively represent hide visible lstm lstm input connection author indicate affine lstm govern guide trajectory guide policy x z diagonal give affine lstm guide govern affect q read primary px primary policy p recursively eqn diagonal variance state change conditioning add feedforward alternate indicate affine lstm turn construct train care variance feedforward layer examine good upper benchmark alternate fine tuned fig show provide code pc imputation concern density expand cover standard generative shrink regression imputation policy mask complete step initial state reward initial policy reward trajectory px pz pz pz z definition imputation maximize log producing discuss train introduce guide produce trajectory x qx px I define imputation trajectory partial imputation sec lstm base c feedforward network relu primary generate trajectory feedforward relu represent step primary construct guide similarly policy imputation information incorporate guide give likelihood guide policy primary guide feedforward relu indicate train monte roll appendix provide full implementation code primary add second lstm include read policy read policy execution state r w tc imputation update add tag lstm govern subset parameter test read affine govern transform trajectory policy paragraph update ts tc distribution read guide observe compute backpropagation imputation three convert miss test completely removal pixel source unconditional test four type imputation baseline sample use hold test add lstm lstm model step lstm add jump autoencoder imputation template template matching imputation run multiple reconstruction input match template significantly outperform lstm outperform direct use naturally objective template imputation imputation valid likelihood fig high quality modal behavior swap non imputation imputation provide policy train lstm raw tune lstm jump close loop overfitte sec mm mm present view direct model reinforcement guide grow sort imputation unconditional imputation unconditional modelling show train comprise million search outperform appear qualitatively g improvement describe px x p qx tx x tx bind trajectory produce start
histogram panel aggregate green area window code red three digit rather feature pixel nearby train small traditional topic use highly contiguous grid generate grid document window topic usage therefore window cg count considerably histogram individual dataset embed room window window overlap grid window magnitude capacity thousand traditional require advantage count classification visualization question remain train small overlap evidence indicate direct imagine counting imagine contiguous create grid learning may suboptimal may coherence minima arrange coherent answer bring contribution family hierarchical collapse mathematically maxima important especially variational learn count consistently traditionally standard deviation contribution quality outperform evaluation study participant name name tb name name name pi dot name name name name n name pi dot solid w name pi dot w name pi dot name pi dot l pi w pi l pi k counting count grid counting stack count dot circle represent parameter represent grid grid dimensional index extent index grid grid generate bag list word value grid count window dimension bag pick grid location inside place collapse sum variable write bag count single window bag number capture occurrence explain single counting grid multiple thus large highly window prior location grid turn inference fast sophisticated train topic model join force neighbor explain cg tend exhibit topic move away mean topic go gradually shift grid attractive minima grouping certain break suboptimal fig digits window contain even however nearby add digit rich one component variation three peak location combination create contiguous stroke distant likely illustration build location map derive feature count add top location grid particularly another grid layer linear mix inherent nearby feature around peak also slight shift leave layer model mixture cg terminate uncorrelated arbitrary stack sake brevity generalize count grid utilize sum training cg show allow omit index expect vocabulary grid location word formula act higher firstly introduce factorize posterior true grid write entropy algorithm iterate status update grid reader directly collapse second variational presence summation window big resolution peak hierarchical stage stage smoothed deep incremental document contain word illustrate acquire document tuple retrieve last depend total sample ambiguity without relevance tuple individual document tuple uniform document employ sample stable complexity grid document overlap arrange grain statistic greatly improve despite strongly outperform bit around third despite allow correlate topic enable lda process index tuple perform task originally coherence topic coherence six order subject outli topic five high word six subject target often fail correctly detect lda small instance question perform sense actually topic meaningful human artificial lda would coherent one grid instead pick sample pick choose start select location respectively procedure wikipedia article amazon http www result show euclidean user able interestingly even pick show worth outperform cg model benefit dataset www com mc compose class previous subset vary similarity compose similar complexity classify document use show l cg lda possibly cg grid sparse intersection document clarity capture intersection topic intersection rather rbms often stack deep though change intersection modeling optimize advantage visualize end application expert uniform dropout grid first section main mnist digits cg bag represent location virtual appear time image learn portion cg assume window nature rather nearby relate overlap window learn mnist digit window rather nearby window digit index relatively rich posterior general hierarchical grid paper build stack cg stack deep course derive architecture grid previous specific illustrate stack counting grid put grid grid total top grid general network place would conditional factorization link link layer formula evident location act specify joint joint intractable inference resort firstly factorize posterior assume factorize multinomial grid location free energy h variational last yet last third add variational convergence distribution status reduce top place cg window token update posterior variable therefore employ place inference utilize cumulative slow individual detail section qualitative measuring micro originally topic coherence original task six user find model word select word six coherence micro slightly micro grid select average start started select cg wikipedia amazon http www word lda cg e cg lda cg cg lda
well impose bt weight use orthogonal joint derive easy bellman bellman equation uncertainty bellman analogous robust guarantee project risk bellman contraction w unique solution projection space orthogonality give point project law large order implement iterative algorithm repeatedly solve inner problem intractable section robust trajectory probability nx x aa empirical kkt section solution optimization problem obtain empirical inner material section effectively formula coherent thus shall derive saddle risk measure mean formula kkt multiplier saddle solution analyze gradient case envelope risk sensitive bellman equation report neutral material generate stage function q coherent neutral wise cost function probability saddle policy become impossible sampling unfortunately risk neutral bellman stationary policy action mdp exact value intractable summation compute approximate project address trajectory x np calculate use nx indeed structure case may replace transition np nx phase measure transition induce probability x policy j j wise approximation bind reader supplementary imply policy follow decrease increase sample p probability function close thus want infimum attain compact infimum attain strong markov empirical statistical limit show q tx nx tv quantity bound repeat claim strong imply bellman unique point whenever let corresponding multiplier x kkt x skew objective magnitude get recall function since affine equip kkt kkt x easily x second sufficient condition hold analysis implicit sensitivity kkt optimization know x sample x p exist probability complete identity q follow write notice maximum note every element p fact argument one x stanford electrical engineering department technology remark theorem sensitive method cost coherent risk accept finance research spirit dynamic value reinforcement generalize extend previous consider involve maker various application finance operation research objective gain popularity variability discount reward markov process mdp objective apply rl var risk actor percentile optimize view take preference another rare highly influential coherent desirable measure satisfy measure satisfy term financial coherent sequential mdps another desirable programming style property end optimal recently markov coherent measure work present rl coherent risk generalize focus coherent total discount return markov coherent formula coherent convenient gradient coherent risk programming policy coherent relate actor algorithm generality sensitive sensitivity variance optimize studied study dynamic risk coherent measure static coherent planning robust approximation suitable g rl style mdps robust part investigate stochastic system trajectory dynamic dimensionality motivation risk actor outcome sample event parameterize ease restrict finite without omit brevity space cost realization order cx state distribution discount parameterize denote draw policy real risk z intuitively risk ensure asset also intuition asset risk refer reader coherent risk show exist state worst suitable coherent risk envelope sequel risk measure envelope paper canonical convex programming formulation satisfy envelope parameter envelope coherent affine equality twice differentiable envelope know form risk hold risk semi account temporal structure dynamic measure take stochastic primary measure issue consider world less tight consistency evaluation risk illustrate multi optimize inconsistent reader risk insight markov particularly mdps length markov coherent measure coherent policy note coherent risk transition px aa depend sensitive random risk correspond discount cost cx cx cx trajectory induce mdp parameterized mdp coherent risk define hope neither tractable risk complex try calculate interested trivial analytically static correspond trivial case case devise suitable descent sgd learn structure dynamic think estimate static static coherent policy parametrization define assume requirement satisfied application management financial engineering survey fu fu value lagrangian denote write presents subsequently formula particularly hold point eq application dynamic risk expectation gradient return coherent material assumption wise develop sampling compose calculate sensitive update actor analysis follow highlight refer reader version supplementary material challenge space large dynamic curse exploit mdp modify algorithm robust actor main thm sample involve estimate trajectory mdp trajectory next estimate analysis actor incur supplementary illustration importance designing criterion risk trading agent return return third asset pareto widely financial train e train policy asset vs training return lower expect policy counter intuitive rational case stochastically dominate static risk gradient style combine convex thereby sensitive improve especially rare event conceptual explore maker preference flexibility cost variability dynamic importantly coherent theorem naturally relate make mdps markov maker able take sense principled trivial scope potential risk misspecification note assumption optimization duality assumption family absolutely saddle envelope result write trick equality assumption define f contact expectation gradient eq envelope theorem expectation trick back naturally base estimation let denote function randomness lagrangian recall e saddle lagrangian lagrangian set empty bound value continuous enough l chain definition lagrangian constraint l point page sequence p finite condition wise iv vi show derivation theorem n furthermore objective constraint interior follow interior l contradiction n n saddle l n must must saddle term guarantee
mutual information able automatically balance population ratio class mathematical interpretation mechanism mutual similarity truth baseline relation hold fig relation joint cross modify include divergence fig goal mathematically machine human identify machine novel learn linguistic argue exist unified behind purpose extend conjecture description study computational function drive law various ia ac cn position evaluate adjust selection briefly study theorem unify one come thing nature mathematical machine construction increasingly develop goal description principle loose reflect fundamental universe suggest interpretation mechanism relate subject mathematical principle mechanism mechanism mathematical principle belief principle critical brain purpose briefly review empirically measure base conjecture information processing novel distinct complementary great necessity describe three level learn engineering application study show address decomposition basic methodology novel fig level problem level give four level learn identify problem linguistic reflect description language expect cognitive science information notation include process design implementation utility subject concern complexity include realize evaluation selection adjust behavior machine adjust intelligence flow problem four neither exclusive exhaustive fig illustrate context learn scalability cost provide loop intelligence critical benefit utilize show example adjust level intrinsic methodology offer power four novel perspective learn dataset classification compatible error classification whenever target wrong unable goal another character describe un similarity linearly separate circle htb example learn linguistic similarity original semantic link namely direct describing linguistic inverse connection opposite direct distinguishing reflect difficulty direct way linguistic inverse way call ill target selection compare study learn systematic generative target selection advantage application machine provide drive law rule cost wu translation chinese classify accord rule dataset english describe chinese refer search consideration derive two principle support bayesian machine need shannon introduce concept mass variable variable call fully asymmetric list criterion discussion h joint symmetry information symmetry kl z dissimilarity machine mathematical principle margin
cnn deep feature mean five layer fully feed hash generate code please local normalization classification hash add reduce invariance capture subtle distinction connect layer diverse toward appearance hash hash represent composition function bias term omit binary computed single label label label hamming however multiple preserve essential point keep neighbor hamming distance semantic semantic database calculate share accordingly last share none label obtain sort similarity level truth ranking ndcg measure ensure ndcg ranking construct surrogate directly try practice rank triplet hash length h distance disagreement incorrectly rank svm rank definition ndcg top score well reflect rank retrieval system pay wish predict treat inspire modify level database eq ndcg normalization constant ndcg suffer assign wish hash give encourage sure decay sign optimization relax eq logistic function facilitate rewrite hamming distance inner objective observe function actually summation triplet loss triplet hash code mini batch mini batch image randomly retrieval six type word base sift histogram histogram wavelet texture block dimensional discount ndcg map use retrieve evaluate calculate similarity within position rank actually mean average big weighted number effectiveness propose influence performance ns ns weight adaptive weight quality ndcg expense performance less relevant layer hash toward appearance utilize semantic similarity b b hash illustrate ndcg use performance hash capability exploit usually compare activation pre train imagenet activation feature recognition show activation boost margin construct deep feature representation hash code utilize supervision fit hash advance superiority fine retrieval I evaluate well entropy achieve semantic rank supervision well preserve semantic structure label cca worse unsupervised consider similarity relationship fine supervision unsupervised layer hash attempt activation cnn even bad hash cnn paper employ hash preserve rank supervision jointly mapping binary code problem nonsmooth triplet stochastic descent effectively experiment demonstrate outperform hashing method term rank quality acknowledgment work program china cb china ia ac cn rapid hashing receive interest retrieval effort compact preserve however hashing semantic yet deep rank hash preserve semantic multi convolutional incorporate jointly learn hash avoid limitation power meanwhile guide surrogate loss solve intractable nonsmooth superiority hash large content base retrieval attract due storage hash aim code maintain similarity hash locality explore hash hash mainly metric structure datum spectral hashing euclidean semantic similarity preserve semantic structure label hash optimization base learn hash image multiple similarity measure normally require handle well hashing extract like representation representation code deal relatively structure semantic novel base ranking image view framework term hash use neural cnn hash rich feature learn hash supervision ranking list derive query database stage surrogate result stochastic propose couple compare activation experimental semantic significantly outperform rank semantic multi convolutional ranking supervision apply ranking image organize discuss semantic hashing formulate optimize conclude roughly divided category independent datum dependent preserve focus iterative quantization cca utilize cca minimize quantization pointwise guide learn preserve similarity assign classifier hash code uncorrelated motivated structural hashing propose pairwise upper binary approach learn feature semantic preserve hashing alignment similarity euclidean hamming use basis triplet ranking preserve relative triplet capture use triplet supervision hashing minimize hamming space discover deep scale column hashing combine boost
document car publication retrieve output new publish car car query wikipedia lda well dimensionality dimensionality space document output almost equivalent dimensionality similar representation fast traditional collaborative challenging relative content informative scale content digital recommender allocation lda belief low document carry public benchmark digital medium provide online platform article come toolbox tailor evaluation concern net latent lda bag conceptual query dimensional net toolbox implement comparison ability due architecture representation retrieval visible hide topic must dirichlet distribute comprise lda package conduct platform reading acyclic bipartite layer ability deep autoencoder da reconstruction consist visible layer layer rbm rbm input partly pre document count count rbm layer rbms execute rbms apply gibbs update give visible hide logistic sigmoid unit visible visible unit visible binary bias visible unit visible softmax softmax unit value hide document bias learn gibbs joint p document query possible proximity neighbor momentum weight initialize variance bias initialize epoch fine batch line search define deterministic perform comparison compare output output consider output internal input output train dataset corpus article category use category connectivity category wikipedia corpus wikipedia business consist document document wikipedia business provide category wikipedia lda model topic measurement bad topic evaluate evaluation wikipedia business higher dimensional measurement fig evident document number outperform twice size two identical show limitation visualize ht pca
start gradually beyond evident increase monotonically capture attribute include go popularity tb recommend article red use abstract string document text search manuscript figure list recommend article top proportion form count year publication recommendation construct five article red recommend cite recommend pairwise cite proportion test recommend recommend topic likely allow article exhibit proportion pairwise able topic quite pairwise link article topic article recommend exhibit mix small proportion look citation count article count capture degree popularity rank citation alone compatibility mix article sense article accumulate huge recent short period highlight offer article relevant row average document right year estimate one comparable publication interestingly per year rapidly publish later showing tendency bar proportion arrange color represent average figure proportion proportion increase transition citation count rate red article topic citation blue phenomenon demonstrate citation topic assign low citation rate citation count mathematics molecular biology mathematic molecular biology raw citation mathematic assign high citation average article issue tackle field citation activity share scientific impact publish citation publish special citation area citation network arguably mix citation connectivity content connectivity citation detection community citation content probability article membership topic citation within domain publication article citation topic variable adjust indicate likely cite locate domain raw merely account activity improvement link helpful scientific field propose enable predictive bayes acknowledgement national fellowship university impact scientific article bias citation properly field evaluate article derive joint probabilistic amongst lda mixed membership blockmodel individual article citation citation behavior recommendation account pattern fitting method control measure article compare researcher award consideration recognition indicator valuable various consider author reader unify journal journal enable article usage activity recommendation web twitter facebook count raw impact article index attempt author publish bias use impact scientific article accounting factor citation publication journal know relevant factor variation certain social typically cite molecular biology compare raw citation address procedure normalize count respect standard recently receive belong scientific model assume citation scientific influence article level citation account potentially useful scientific article name derive joint text whereby article citation citation belong article position citation compatibility research article relevant unobserved quantify profile journal since publication account citation network relational content information combine establish relational mixed membership blockmodel scientific detect citation intra community detect citation integrate community introduce variable article citation due compatibility topic act understand citation citation model article article citation field search paper technique keyword method relevant article citation count may yield citation scenario article recommend citation metric adjust citation citation rate intra citation relevant identify adjust citation count article external provide article citation monotonically fully besides framework fit develop efficient posterior real citation often massive analyze interaction scale square develop variant subsampling network iteration optimize variational objective reduce storage detect community pair informative set define mcmc space adapt link assume close suitably adapt model requirement scientific science research benchmark high physics paper model study join article date absence publication significantly organize introduce alternative recommendation conclude combine lda act adjust model generative probabilistic detect assume vocabulary represent vocabulary document topic vector probability topic document word draw assignment word slight abuse hand mix within data node community group node membership blockmodel group membership indicator receiver element refer generate text draw topic link em mixed pair indicator dd dd citation relational pairwise combine identify community topic suitably incorporate would improve em draw proportion position topic j ij topic receiver dd dd document circle variable indicate corner link cite violate world citation research affect citation cite high cite author attribute cite variation topic generation link latent variable draw citation blockmodel element one document topic citation factor cite citation probability topic characteristic citation topic proportion attribute area take link lda identically issue whereby datum content connectivity citation multi assume latent rise tractable distribution simplification topic pairwise probability weight bayesian logistic minimize structure laplace place word parameter log text lda annotate entity realize entity blockmodel link ball document instead random offset proportion cite document connectivity due pt sample document corpus pair perform success denote set successful update ij update half update optimize gradient ascent parameter denote take pass write stochastic ascent replace unbiased step satisfy lie column example stage estimator variational outline algorithm lda baseline lda follow regression link covariate take hadamard th text separately serve baseline method modeling text account link structure assume consider explicitly imbalance lda lda link variational lda implementation suggest author link initialize link text application article researcher search look paragraph keyword text link mean recommendation intra citation citation nature come citation scientific article reader reader preference amongst article document training fit perform document proportion iterate convergence update assume knowledge link document true variational posterior hold ability predict give rank model fit proportion document rank hold document actually predictive rank fit evaluate also able blockmodel perform lda lda computation topic proportion topic document mean citation cite element interpret citation visualization vocabulary inspire retrieval examine subsample add subsampling publication time take add package vocabulary us cpu divide fold fold training topic minibatch tb predictive rank runtime predictive rank cpu cpu hyperparameter predictive attain improvement close subsampling time maintain around topic concentrate fit fold blockmodel estimate close dotted agreement repeat blockmodel run quantity correspond agreement greater slightly right count trend increase citation capture citation citation high capture mix characteristic account citation among topic popularity document set illustrate incorporation performance example document segmentation classification cite organization rank give table rank low take account lda index high improve rank improve performance lda article recommendation represent total cite display font activity topic width arc total arc come infer colour origin visualization blockmodel word citation activity figure tends dominate topic citation hand citation individual article citation topic figure tends cite nearly besides high tendency helpful vary area even physics competition rank first cpu times rank cpu provide
learn begin size time thereby da consider easy expectation expectation big da sequentially achieve visible least one update let ideally practice sample da relate corruption set visible belong da da replacement require equal autoencoder distribute accord f distribute com class pixel instance comprise voxel water intensity scalar grey template parametric http www ac uk image accord variance even number imagenet comprise category collect apart hierarchy comprise million broadly http www image five category imagenet database amount million image center theorem lemma corollary corollary corollary unsupervised learn denoise depth interesting guide gap mechanism backpropagation objective incorporate two mechanism denoise autoencoder deep building upon level support empirical evaluate unsupervised success last year practical treatment theoretic concern generalization formulation relate ensemble analyze quality network choice fine proxy categorization question ask analyze importantly line like establish rate consistency output recall optimization specific analyze objective start nonconvex describe gradient analyze deep denoise autoencoder dropout net analyze recurrent activation subsequently interact seem influence concept type corruption importantly structure influence dropout estimate idea depth size corruption certain convergence certain choice last year dropout follow notation describe objective backpropagation dropout denoise autoencoder dropout reconstruct corrupted correspond learn dropping unit corrupt version unit bernoulli focus loss layer dropout sigmoid activation bias handle use sigmoid linearity activation minimization via gradient noisy k iteration randomize allow either priori train offer theoretical exist implementation analysis serve result da corruption compute gradient kk rp proceed bound architecture correspond convexity function estimating size discuss repeatedly da include supplement number let layer nn serve give dependence requirement refer proof backpropagation adaptive convergence impose restriction may lack study gradient present address general nonconvex oracle believe context serve da potentially widely convolutional net loose supplement choice ensure expectation construct presents nn run randomization fold layer layer require fold surprisingly fold need small minimum idea base nn work autoencoder closely section present da insight da correspond da autoencoder rate maximum da step size eq knowledge remark result relate da corruption autoencoder denoise also interest observation follow da ideal stronger visible layer weak support minimum show necessary provide evaluate bind interaction estimate denoise autoencoder instance da depend structural statistical characteristic correlation across exploit size unlabele need chapter consistency representation ensure gradient fall sigmoid estimate via refer predict decide exceed imply suggest present trend choice extreme right lipschitz scalar choice main b vs da vs impractical sample fewer empirical several suggest recently show deep pre corruption da since da module bipartite minimization da equivalent include carefully whenever setup little room da unit rarely thereby requirement seem explicitly disjoint share visible ensure solution summarize autoencoder remark small refer lemma ex choice decrease improvement increase infeasible combination dark blue exist improvement da da feasibility observe increase fix increase infeasible combination leave half denote dark general case drop iteration layer layer dropout iteration optimal size dropout instance dropout layer net address via comment hide bound reduce observe correspond averaging may little backpropagation therefore layer distant layer phenomenon partially obtain layer da dropout broad regularizers da learning fraction available rate dependency compare gradient denoise da trivial early analyze architecture present sample root interpret consider hidden layer length dropout rate reduce retained unit derive setting hide expect retain layer dropout layer ensure easily length predict depend check accuracy many satisfy generate bad bound present convergence backpropagation ensemble maximum dropping half mnist imagenet refer imagenet present interested plot htb gradient vs multiple color vs layer third expect sub asynchronous inherent supplement top mnist imagenet normalize respective h gradient call four trend show stepsize decrease stepsize gradient optima computation gain imagenet test bar error dropout gradient black blue gradient vs layer imagenet show trend trend three figure correspond decay layer black vs fractional da gradient strong black green red figure gradient decrease green reach increase blue report refer deep gradient help distribute vs least refer twice master via pass initialize run whole iteration expect decay strong attribute factor trend rate show imagenet increase speed increase rapidly fall vs distribute da give denoise dropout influence denoise b gradient corruption da denoising agreement depend da dropout efficacy use choose corruption rate convergence exist strong denoise context framework construct backpropagation net analyze interaction scale convolutional recurrent also boltzmann nn iteration let lipschitz recall eq sigmoid f u aa bb b aa ba bb ab aa bb significant start noisy gradient w k step denote rearrange sum estimate inequality randomization construct noisy stop criterion random however point iteration update markov process take expectation last expectation last eq q finally constant monotonic resp term decrease balance substitute stepsize stepsize q change nn instance require markov q sense
use problem dimension strongly regularizer induce exact gs efficient gs connection section generate regularize dimension difference performance strategy gs cyclic selection substantial lipschitz sampling narrow gap remain gs gs rule non randomize zero away compute gs rule coordinate seem improvement gs rule despite cyclic gs plot number gs advantageous perform label dataset dataset connect high base supervise implement efficiently optimization normally cyclic coordinate cyclic randomized coordinate gs gs constant gs rule randomize applicable approximate gs randomize similar justification exact optimization justify exact coordinate expect block parallel dual ascent successive boost algorithm strong like thank anonymous gs idea rely bound constant column column max give gs compute structure exist unchanged element change update expensive update modify total q differentiable denote strong rule variety notable least non process gs maintain contain product vector contain value store max allow gs cost reasonable cost cost structure update depend update gradient cost relationship convex imply similarly strongly imply strongly two relationship norm along equivalent derive strongly convex gives conversely strongly hessian h line scale constrain quadratic stationary combine give obtain fast convergence gs occur value exact gs one tight bad gs coordinate combinatorial graph maximize much particular case mode alternate alternate mode must eventually cycle one node cycle weight burn period repeatedly go mode final step finish several consecutive maximizer constant burn burn period implication away fast descent gauss eq dual strong put logic appendix establish relationship different convexity square norm use establish q expression nonnegative vector use less section additive gauss rule choose assume generality progress lipschitz continuous norm prove dependency subsequently give less rely show follow show lipschitz q subsequently continuous gradient substitute apply inequality progress hold eq imply inequality recursively although small usual notation need gs rule state turn gs rule analyze case lipschitz imply notice define contain notation gs use gs optimality eq notation unique relationship gs reduce q lie show gs first min gs progress gs min bind progress subtract give gs apply gs progress consider gs derive add subtracting note select gs upper would choose gs use continuity gs strong gx make substitution q side rule gs method eigenvalue correspond norm proximal indicator gs eq gs thus choose zero obtain even gs rule either hand gs rule progress clearly turn show gs possible rule proximal value function gs rule progress ratio gs choose obtain progress ratio bound substantial margin gs find counter able produce counter gs rule detail implementation gs offer runtime hardware rgb significant recent application randomize begin achieve gauss suggest computational selection rule comparable gauss give rule showing coordinate exact sparse problem propose gauss rule fast rate analyze gauss proximal substantial optimization seminal give coordinate minimize random coordinate well gs nesterov randomize later paper randomize contexts expensive suggest use context gs practice suggest class descent discuss context gs rule gs strong standard smoothness assumption randomize restrict fast show usual gs optimization provably fast certain sparsity result show benefit exact update optimization variant nesterov randomize lipschitz strategy variant optimize separable non show case perform update mean coordinate descent minimize express element family include core machine lasso svms solve form include quadratic propagation assignment continuous graphical general gs expensive however often gradient max implement gs randomize two slightly base facebook detect disease example friend friend gs degree maximum thus gs optimize gs inefficient star instance problem implement time maintain efficient gs rule solve near although factor general gs function product solve wise lipschitz twice differentiable base uniformly alternatively choose coordinate large directional bind make positive side uniform sampling subtract side notation progress imply gs definition obtain gs rule descent fast rate cyclic selection rather rate gs lose avoid e side q make use fx fx gs square norm one gs rule gs obtain extreme guarantee future iteration choose variable graph gs alternate large show gs structured maximizer consecutive maximizer consecutive implication edge conjecture value form correspond non fast review gs gs scenario relax backtracking proceeding lipschitz special fast gs extreme differ sample fast context section sampling factor nearly large benefit rule selection extreme gs lipschitz rate gs fast lipschitz gs lipschitz gs obtain fast lead call similar argument place obtain convexity constant appendix thus always fast rule lipschitz close minimum gs quadratic use rule optimal coordinate size iteration boost rate apply mi improve strategy require interesting gs near residual denote solve refer return justify approximation gs gs rule special power interesting rule lipschitz constant formula thus towards gs problem rule compute gs inefficient practical gs
topic topic disjoint ranking permutation general item center permutation permutation partial within probability permutation permutation row lemma conclude rank approximately separable combine paper approximate prior full consistently eq form select roughly apply k note sum give center ranking complexity except proposition novel comparison inconsistent user probabilistic share key insight connection comparison insight advance separable discovery separability appear restrictive outcome latent world extreme ranking provably new empirically competitive diverse application system pairwise comparison item inconsistent record web transaction click predict pairwise comparison new mixed membership comparison ref ref permutation pmf center heterogeneous inconsistent noisy mixture capture heterogeneous multiple furthermore capture factor consistently generalize perspective fit observation provable polynomially component mix pairwise comparison model receive attention yet theoretical guarantee learn extensively unclear view user comparison latent topic topic discovery provably separability geometrically inspire work ref ref generalize separability require pair item prefer probability restrictive formally separability arise set preference provably generalize angle establish reference computational allow user htb component pairwise provable pairwise provable available provable combinatorial vertex available top provable pairwise vertex provable full extensively study setting decade ref rank ranking component cluster heterogeneous preference type attention pairwise comparison full ranking provably correct base handle impractical within user view alternative pl study model relate validate adopt membership perspective permutation permutation inconsistent guarantee motivate another mixed ranking topic pl summarize relate separable discovery consistent discovery topic separable separability many topic perturbation date establish provable guarantee perturbation go augmentation improve ratio provable degree approximate explicitly derive separability provable similar separability strong user dominant full satisfied considerable preference rating influence share population ref come different mixed membership perspective organize introduce approximate separability summarize step demonstrate synthetic sec htb describe process universe item population assume un independently distribution outcome comparison denote order dispersion parameter permutation normalization comparison ex ex rank token z ex mix characterize order time item share reduction formally pairwise compare ex item prefer sample behavior ex topic enable let ranking define ki I ex prop infer prop infer item entry therefore pairwise ranking correctly hence ranking prop note topic document compose word topic weight sample ex column mixed membership topic observation pairwise condition ex model thus prop section family mixed membership rank membership ex dimension model ex key separable come consistency favorable enforce total ranking precise geometric occurrence matrix pairwise estimate splitting user k topic co ex separable discovery ref rank exactly separability order definition ranking separability recall high item arbitrarily close prop propose rank approximately negative separable e order separability order refer novel pair separability uniquely reference ranking seem share next model scale sufficiently fast negligible fraction satisfy approximate separability draw reference ranking permutation ranking sample dispersion ranking would small property prop supplementary loose separable separability approximately geometry row pair circle region angle exactly novel row circle separable row perturbation ideal ideal hull point hand non close form novel detect approximate novel normalize solid extreme probability row strictly approximate ideal become extreme separability solid angle close hull form pair solid angle angle correspond angle consistently approximate isotropic asymptotically estimate pair identify prop specific weight prediction inference ex main detail alg alg estimate regression scale alg prop alg ranking equivalent rank define component tolerance reference ranking I p j jk novel ranking precision I j k k k j k pairwise sort computation run proof loose parameter order moment rank consistently ranking propose algorithm fail normalize form detailed supplementary provide prop note complexity spread hardness smaller require achievable identifiable validate assumption demonstrate preference experiment suggest specifically projection ex measure ranking since align ranking distance ex ground reference ranking movie rating parameter normalize reference ranking depict estimation vary dispersion ground ranking reference ranking carlo comparison dataset rating due public partial viewpoint suggested frequently rate split convert rating rating rating ignore prior dirichlet evaluate performance log approximation likelihood phase compare topic tm setting summarize tm htb ex c pmf tm train rating aggregate properly test purpose optimize rich rating rating rate convert training movie rate tie train rating movie
email concentration invariant parameter vary isometry singular digital processing compressive identification compressive pursuit recovery match orthogonal pursuit compressive sampling pursuit tucker impulse response transform discrete cosine operating curve pearson external basis least absolute shrinkage selection negative output compressive spatio wind forecast autoregressive autoregressive portfolio prediction direction switch artificial neural root square square operator spatio ann least terminal routine wind power short forecast wind present incorporate datum inspire compressive sparse recovery dimensional structure collection exploit cast forecasting recovery signal propose east compressive spatio wind speed improve short forecast widely wind energy grow world global another total wind power year reach capacity wind make balance integration service load forecast one directly forecast wind convert wind wind generation wind forecast wind forecasting method group vs ii probabilistic forecasting paper short point forecast temporal wind forecast neighboring forecast spatio forecasting method later incorporate wind forecasting introduce probabilistic et al speed advantage markov model prediction wind aggregate graph spatio regime switch wind direction study various forecast error methodology density field wind power comprehensive review compressive cs usually exist collection weather forecast end cast linear propose algorithm forecast measure weather east york result considerable advanced spatio temporal forecasting wind benchmark conclude remark model variable present autoregressive generalize condition suitably interested reader signal block concatenation eq additional design due flexibility recover block computation recently topology uniform assume generalized target relate high target sparse concatenation zero generalization block block correlation adjust prediction wind speed east wind speed higher state speed datum weather report east york fig depict study red locate subject wind profile low area correlation wind mainly time simulation period wind throughout year compare spatio temporal wind forecasting forecasting simply forecast improve persistence advanced capability capture nonlinearity wind speed series wind behave sub band subsequently sub carry mode speed reconstruct high band wind perform recursive period hour ahead forecast compare ht spatio temporal forecasting method spatio spatio depict incorporation improve forecast ht ht new obtain every hour equivalently step hour recursive prediction speed prediction predict wind speed recursive continue effectiveness forecast wind forecast list speed measure moreover calculate spatio good example reduction persistence reduction spatio temporal ann st st tb illustrate calculated horizon confirm spatio temporal
conceptually requirement evidence meet derive example present address causality eqn work concern notion drive confusion cause measure direction influence assign rewrite eqn pe yield alternative pe expression gives require probability parametric relationship investigate use counting notion causality infer might drive causality zero conditional eqn former two cause appear influence two bayes use series mean series occur assume cause causality determination intervention absence cause perform action rather lack absence assume cause identify causal causality attempt remove assume cause eqn issue account time see dynamic assume assume cause causality argument causality address article term causal causality relationship time cause effect high assume address positive cause assume cause cause assume yield cause assume cause difference causal inference effective probability cause assignment series use calculation cause effect consistency causality cause must natural assignment assignment px assignment unobserved two effect probability g appear appear accounting must shift shorter counterpart single calculate library assumption cause cause observed mean observed cause cause cause agree intuitive causality unobserved cause cause incorporate averaging cause cause mean make cause calculation conclusion would useful cause show cause effect cause assignment series initially x effect length use cause effect assignment weight naturally conceptually account causal influence cause effect causal follow time noise could five see discrete calculation address estimation calculation eqn instead relevant py noisy simple noiseless weight tolerance large enough probability cause become tolerance motivate noise little calculate size become find tolerance sign compare know level time causal require become zero series signal observe calculated agree intuition e expect result depend library specific pair assignment calculation eventually causal inference agree intuition thus three value calculation way observe cause pair causal series cause rare effect impulse cardinality would tolerance imply weight median course agree library reason believe inference increase basic observed cause effect I calculated cause algorithm complicated cause complicated conceptual cause assignment set usefulness tool inference test directly intuitive understanding drive system dynamical specifically consider drive periodic impulse amplitude response drive ht cc large show tolerance increment intuitively show meet short increase example always inference agree intuition know deviation mean bin deviation bin show method ccc deviation bin yield expect part series independently exploratory assignment useful causal assignment exploratory causal analysis example different assignment near expect lag appear dynamical create synthetic dynamical eq instance eqn observe noise level use tolerance domain spread reasonably cause assume period counter peak assume drive poor value insufficient reliably cause eqn lead intuition relationship drive response signal tool conceptual restrict linearity datum dynamic example agree intuition tolerance causal inference generate nonlinear similarly intuitive ht agree causality tool point limitation causality exhibit behavior system eq pair couple model introduction convergent causality vice difficult justify instance seen see strong even strong present paragraph support calculation ht cc along domain use calculation standard tolerance intuition figure x enough implication irrelevant domain effect determine causal system calculation standard provide intuitively ignore causal serious usually seek causal two strong bivariate causality causal try drive relationship series see explore calculation eq directly intuitive despite dependence intuitive causal intuitive domain mean cause effect inference agree though consider calculation implies also seem imply case imply counter imply unable identify drive situation drive driving occur interaction imply autoregressive result consider cause insufficient previously analysis effect test cause include assignment weight assignment calculation case calculation part exploratory must assignment try understand assignment expand autoregressive definition extension article understand exploratory causal dynamic pair repository repository datum assume relationship temperature series daily expect temperature tell small difficulty exploratory analysis determination cause effect tolerance thorough calculation tolerance bin histogram close library bin deviation cause set purpose article series detail comment regard confidence exploratory however convenience take simply causal cause effect pair show point tolerance domain imply inference agree causal test pair assignment highlight determine decide inference depend max truth weather collect include collect national center b cause effect assignment could domain set l sample assignment plotted weighted cause assignment show inference line algebraic set algebraic mean aforementioned set weight linear system periodic impulse example eqn apply causal relationship eqn observe inference symmetric increase towards tolerance library length causal eqn library spurious use assignment library length spurious example imply relationship calculation care causality involve investigation experiment article explore exploratory proof dynamical system observational alone field physics analysis involve technique entropy te te tolerance attempt make causality token causality technique method include acyclic temporal logic causal connection broad causality leave general logic interpretation framework regard interpret cause assignment assignment stronger interpret
value predict uncertain prediction leave weight assess uncertainty relu squared covariance show mc dropout lastly fig blue colour represent half deviation confidence plot none capture mlp predict mark dash clearly sensible increase uncertainty effect predictive uncertain behaviour capture mc figure increase uncertainty increase relu stay relu covariance appendix whereas relu different different dropout dropout initially optimisation uncertainty interpolation repeat experiment relu network layer setup segment minus various interpolation show interpolation miss gaussian squared red green blue mean relu dropout increase uncertainty miss uncertainty capture deviation error bar uncertainty number forward draw number mean neural mnist dropout relu operation usual dropout train iteration reference scatter scatter uncertainty digit axis scatter forward pass softmax fully softmax predict softmax softmax output digits input axis image rest fig predict look envelope envelope class input softmax softmax softmax reasonable return middle high would ask fairly moment confident reinforcement learning receive various aim time try low reward pick instead great agent decide advance rl make network action agent state lead game agent human greedy explore uncertainty estimate use converge code world point angle ahead depict one different angle different reward reach look white approach batch purpose experience network step initial relu rate weight decay descent momentum batch original implementation change q burn additional dropout linearity dropout uncertainty greedy perform single propagate sample average blue batch plot batch move thompson reward within batch burn batch bad batch move greedy interpretation model uncertainty demonstrate deep reinforcement development exist new additional burden uncertainty estimate assess corrupt incorrectly high pixel change considerably space corrupt lie increase compare uncertainty variety gp approximation like thank dr chen mr dr mr van mr wu comment european fellowship remark ex ex ac uk tool gain attention bayesian offer come prohibitive extract away model computational accuracy exploratory dropout uncertainty task mnist biology name tool ever mlp know dropout convolutional network however many field towards classification confidence uncertain prediction softmax softmax confident point classify pass softmax reflect take appropriate result high uncertainty classify happen post office sort responsible uncertainty reinforcement value quality action often agent estimation explore thompson dash line solid area mark ignore offer reason uncertainty come prohibitive perhaps surprising often change optimisation dropout interpret approximation know avoid tool exist dropout mlp extract away often exploratory different dropout represent extensive assessment task different architecture important task mnist concrete lastly uncertainty set reinforcement similar deep reinforcement learn know converge place weight study extensively computational variational inference inference inference neural dropout ik similar well variational indicate unit layer drop linearity extension approximate place distribution map deep appendix variational divergence full kl sample bernoulli identical scale obtain dropout approximate moment predictive empirically l monte pass network average derivation uncertainty estimate multiply uncertainty obtain variance equal forward pass mlp precision embed ratio set modal approximate place matrix result layer modal mlp predictive pass exist mlp result dropout often
ssc b ssc b sc b normalize undirected node compute use convenient algebraic assignment recover subspace cluster ssc noiseless datum step optimization matrix ij e desire self property similarity make connection early sep obtain similarity could sep various regime albeit ssc subspace complete sep cluster list remark corrupt extension noisy present present concern require degenerate make algebraic generally mild example surely generate fix linearly position assignment truth permutation cluster identical property connect position point require reconstruct linear hand self exist connect otherwise contradict eq contradict identifiability position could drop relaxed notion identifiability union union point start repeatedly increment assign new subspace sep minimal minimal truth truth cluster subset certain regularization achieve structure intersection intersection otherwise address advantageous compressive likely yes evidence iterative hand shrinkage would reduce formal treatment idea suggest regularize degree freedom good generalize well everywhere ssc answer minimal union subspace span point order point sort subspace previously fix ball body hull give cluster restrict ready consistent condition design matrix noise furthermore self satisfy every highlight consequence feasible general reduce noiseless show nonempty noise level addition differ maximum differ cluster component pick apply pca point connect robust potential procedure subspace merge robust restrict problem nice complete proposition later notational x proposition equal follow low sign u plug first long hold construct range contain point belong subspace theorem yield eigenvalue nonzero connect component natural point noiseless input assumption regularization satisfy self c contain necessity imply c imply result contradiction order product note dc ij x dc argument substitute upper get bind inequality due imply result desire contradiction rv cluster separation belong would r merge never mistake subspace ssc year show noiseless processing step discovery additional data point condition provably ssc deterministic lastly ssc address often advantage ssc research improve ssc empirical evaluation singular u u q rgb corollary affine abstraction statistic recently line recent guarantee seminal subspace ssc extent justify motion face get condition ssc obey self property ensure subspace cluster together sufficient correct thank issue post mild general position robust bound margin subspace application physical law subspace human body illumination model cluster membership reveal source datum much wide application fall category image compression identification identification modeling study privacy movie recommendation algorithmic subspace back maximization method plane factorization early theoretically justify past decade spectral recently cluster ssc arguably due elegant strong provable guarantee condition connect use assume union subspace handle separation edge subspace self sep drawback within connect segment subtle originally partially address reach segmentation general position counter position long graph sdp condition previously sufficient condition exact sense subspace sep exact hope drop notion subspace subspace overlap completely identify ssc clustering lose ssc regularize reweighte robust ssc deal within intuitive cluster number might interest subroutine vector noiseless dimensional subspace point union intrinsic ground assignment permutation deterministic additional underlie subspace noisy consider white provable subspace progress theoretical regime beyond original definition may contribution list assumption weak model additional instance semi place sphere polytope non value subspace algorithms assumption subspace capital refer applicable ssc highlight table understand result optimal sep subspace substantially overlap subspace cluster
create dimension test effect transfer letter pre imagenet list extract region image first pixel body extract state art bag document previous cluster feature pool pyramid combination horizontal recursively split recursively bag original split vertical bag result dimension h dimension classification descriptor ensemble descriptor act performance representation ensemble represent document create cnns basic benefit distance descriptor sort list pca enable keep memory table accuracy cnn ensemble cnns perform achieve well worse pool cnn perform ensemble region cnns margin cnn compute first outperform every imagenet cnn improve imagenet descriptor suggest category gain spatial pyramid pool perform well interestingly descriptor representative retrieval seven different signature image similarly document content may lead author pca remarkably loss reduce dimension compression cnn perform art document image feature cnns extract cnn alternative document show training enforce unnecessary train approximately cnns trained show representation exceed acknowledgement discovery grant hold helpful discussion acknowledge use retrieval use cnn scene net capable abstraction explore confirm superior cnns compression cnns transfer well analysis enforce unnecessary training label collection contain document image across category cnns visual letter write motivated structure document image digital library document store process optical character tool indexing pre analysis graphic index image stage analysis arise fact correspondence document spatial header body spatial template often similaritie article form perspective circumstance classify retrieve intra variability inter challenge object classification current state inspired cnns presents extensive cnns deep cnns retrieval learning cnns object recognition surprisingly net significantly focused suggest capable region add past base image power structure business assume document distinct visually component business letter typically date extent document document letter fit configuration template transformation drawback manual template document document definition flexible structure herein treat document document bag histogram vocabulary document potentially feature position result geometric successful classify document template less domain template recently attempt bridge gap feature pool several stage whole proceed small small global pyramid categorization mind retrieval type represent researcher representation learn research domain concern structure geometric configuration toward goal base document building decision convolutional report well report spatial pyramid matching yet apply retrieval learn area computer recognition cnns currently performance margin cnn domain traditionally ill suited detection grain recognition cnns grain object relevant analysis field challenge distinguish ii powerful therefore grain object challenge major cnns fine cnn recommend train problem challenge regularization technique cnn effectively potentially information train train unnecessary entirely seek whether insight document cnn retrieval form abstraction low therefore extract near cnn vector light previous paper follow first evaluate deep toward present design compression cnns cnns non document transfer explore strategy embed ensemble cnns interestingly retrieval basic perhaps available new label structured document graphic element share cnn cnns additionally explore different initialization first cnns entirely feature weight train fine implementation computer input process stack layer convolutional vast hierarchical organization responsible feature classifier convolutional neural network activation geometrically invariance cnns datum add image architecture scale specificity cnn beneficial make treat region differently align capable region specific automatically cnn train activation near cnn dimensionality reduce involve distance query descriptor every descriptor sort sort document account cnn image aid grain discrimination category letter illustrate figure consistently short letter full address learn automatically classify document learn idea cnn cnns region dependent extract total four region header cnn train entire base build descriptor concatenation compress cnn extract region illustrate full vector use network classify goal transfer take share structure facilitate cnns initialization initialization strategy cnn alternative pre complementary training put challenge object category extract imagenet challenge fine tuning target question transfer imagenet feature document feature address whether initialization challenge result initialization document cnns seek usefulness transfer feature unseen version collection resolution public record american seven document label tag tag image tag version list image collection label work relate ten image letter category present full dataset category collection
prox encounter far use projection bregman unable use bregman divergence exactly problem project divergence maintain simplex optimize structure factorize marginal polytope bethe tree strong negative bethe entropy strongly interior marginal polytope consequence definition bethe project simplicity intuitively appeal oracle average objective gradient modify nesterov acceleration technique convergence solve stable belief entropy inexact marginal define mean field inference non well unlike propagation composition first order loss appendix energy find also work problem experimentally good tb example oracle distribution c crf max tb input example clique structure max seek crf respectively mrf parameter highlight index possible datum family convex prevent conjugate duality relationship family iterative procedure present crf structure variational parameter surrogate crf give surrogate gradient break show crf I clearly parametrization measurement amount equal gradient marginal truth marginal loop doubly simplicity implement learn derivative notation case sometimes overall doubly solely converge yield recall experiment update parametrization ht f energy dd variant ht nlp citation string author etc closely segmentation soft labeling constrain predict name name number last name measurement would enforce hinge constraint dual style crucially rely hundred soft impose hinge inference tune development ignore measure baseline high optimization difference local energy match dual dd programming whereas plug energy experimental configuration relax expectation preference map demonstrate algorithm use underlie analyze gram class mm next recognition setup equally fold train validation result fold achieve extremely clique state excellent clique unique people easier solve mind convex local energy intend lie standard image marginal sum marginal give expect unique word intuition global non eq variant differently approximate fouri simply multiply pointwise non linearity baseline input order letter vocabulary cascade motivate marginal length th distinct train energy add chain version note structured cascade give structure cascade course dataset different arguably much create logistic allow choose length map variant tune fourier much expressive indicate local global important method cardinality model dramatically aggregate represent via graphical node successfully time variable correspond count location poisson rate proportional infer patterns map perform likelihood observe count hard provide additional alternate surrogate procedure expensive solve inner loop applicable synthetic code solve learning framework tractable model marginal possibility gradient substantial work problem domain agreement grant reproduce recommendation author reflect learn reasoning response distribution bind structure show minimizer mrf clique structure section configuration constraint implicitly since bethe inference crf distribution behavior citation inference local disagreement modification along rough proof case energy heavily significant self contain work associate bregman composite euclidean update distance barrier visit build regularize average different minimize associated bregman mirror descent original algorithm conjugate duality actually slight tb energy function prox first similar energy convex convexity average online convexity mirror order converge stationary large tell norm notion order differentiable surrogate surrogate composite euclidean surrogate admit establish composite mirror descent strongly bregman surrogate gradient strongly convex smoothness proposition directly bethe entropy bregman unbounde corner polytope domain neighborhood lipschitz practice mirror barrier iterative never polytope effectively purpose minimization intuitively plausible iterate corner polytope constraint learn satisfie asymptotic follow note asymptotic choose lipschitz bound set smoothness constant rough convergence energy heuristic effective future believe examine parameter contribute barrier prox g l large accelerated procedure fast polytope measure bregman us cs david edu solve structure soft lagrangian semi broad objective capture statistic maintain tractable provably inference non project generating bethe structure achieve task novel inference procedure providing highly collective graphical apply show dependency graphical relationship often dependency word phrase cyclic dependencies nlp constraint sentence likelihood token predict marginal pose clique marginal tradeoff inference score clique us enforce way work objective optimize parametric entire linear enforce property whenever euclidean project bethe generating passing maintain iteration convex convergence abuse terminology objective generalize algorithm convex marginal framework utility preference motivate repeat call see success impose similarly constraint expectation distribution supervise expressive domain rather algorithm solve parametrize use implement black box experiment demonstrate power generality achieve discriminative art learn word algorithm improvement inference structure apply let define conditional ns capture joint clique go model field expect sufficient eq specifically go often dependence combine clique surprisingly approximate structure upper parametrize compactly mrf analysis technique convex benchmark undirected graphical particular convex joint marginal tractable factorize induce partition clique marginal product full involve simple base energy base augment clique include tractable potential clique repeat outer tensor product node non generalize allow
proof angle combine come order use prop face set interested analyze eigenvalue matrix index r distinct zero add laplacian graph graph definition side complement small eigenvalue laplacian square angle use consider model mrfs capture technique belief propagation minimum submodular insight powerful specific divergence variational produce furthermore message confirm scalability quality benefit potential probabilistic central provide foundation make uncertain general problem one amount attract community notably propagation size involve optima inference process assign like equivalently bernoulli set indicate concrete show task want foreground pixel traditionally indicate foreground define quantity pixel foreground employ view set make e function submodular implication approximate emphasis submodular special dpp modeling diversity tractable even ise provide optimize partition function submodular lead problem interaction slow impractical computer show problem problem algorithm handle indeed inference image hundred thousand insight agree mode secondly connection namely specific light type log image segmentation message lastly image segmentation demonstrate exist technique provides formally say submodular set add without arise measure typical ise task introduction edge connect neighbor pixel neighbor preference assign place neighboring pixel segment penalize weight attractive behavior model go far would modify potential concave concrete example assign segment assign otherwise segment function modular see analogue function say modular arise factorize q evident modular vector modular interested polytope q modular add restriction empty though many inequality optimize question configuration log minimizing result fast know combinatorial evaluate expensive ground perform well wolfe solution minimizer barrier perform computation normalize literature compute technique common optimization variational quantity optimize technique modular analytical function submodular lower modular idea parametrize modular optimize minimize inequality separable polytope divide solve error marginal frank even map require submodular costly convergence frank wolfe method minimum contribution surprising result crucially objective consequence submodular minimization point substantial performance gain seek optimal equivalence extract map marginal exact marginal point algorithm become demonstrate extremely parallel attack partition employ factorize parametrize upper marginal distribution end turn measure enable prefer quantify dissimilarity argument pick interesting minimize factorize probability set completely prominent example kl divergence interest enyi factor minimize factorize factor minimize infinite factor current sequentially I one alternative guarantee factor generalize setting change passing describe follow norm arise structure sum belief propagation vector base base problem exhaustive propagation size send receive message projection parametrize separable polytope divide solve factor follow differently every store extract factorize incoming message variable step coordinate incoming message formally see descent discuss describe possess message parallel important maximal connectivity new coordinate depend extend analysis extension variable message pass linearly specifically optimal initial graph h h h b specify whereas offer marginal dynamic range segmentation marginal ground truth segmentation compute exact ideally wish proxy quality area roc curve ground truth classify pixel roc pairwise interaction unary potential potential shift pixel unary potential belief fractional fast relatively converge minute pairwise variation less test leave validation generate grow boundary foreground accuracy boundary std avg std column curve auc fourth column deviation precede htbp compare accuracy aggregate roc approach auc whole image challenge boundary poor alternative attribute confidence verify optimize non iteration lastly order around qualitative characteristic result marginal method minimization low prefer side propagation confident four exactly concentrated around around strong prior low mainly unary procedure last two preserve boundary benefit variational inference log interpretation reduce minimum making tool optimization available approximate inference show factorize return type immediately useful model pass exploit connection strong natural approach rate lastly challenging demonstrate marginal produce variant move high potential become intractable inference variable bf hence
langevin target mala hmc proposal preserve distribution us proposal mh general proposal acceleration evaluate ar include acceleration technique algebra use first lemma give function theorem need check similarly jump transformation obtain first hold show easy v eq eigenvalue matrix eigenvalue entry factorize equivalent spectral decomposition l ht apply show hence section proposal proposal discretize langevin diffusion mala discretize hamiltonian dynamic hmc splitting analyse efficiency ar gaussian mcmc metropolis hasting target give current draw target kernel state approximation matrix give vector draw ar vice versa radius metropolis langevin mala ar discretize langevin hybrid discretize hamiltonian far analyse although identify useful designing choose gaussian mn proposal ar proposal may ar mh accept idea quadratic approximation objective iteration concern chain monte ar target proposal target ar target scan accept reject redundant algorithm showed accelerate ar efficiency ar mh accept reject step normal ar proposal gaussian proposal hard analyse question efficient accelerate gaussian require equilibrium interested sample mainly arbitrary burn integrated autocorrelation length reduce proxy independence per autocorrelation article case feature mh correct simplification analysis keep transition change analyse accelerate analyse accelerate moreover every local distribution approximately normal simultaneously ar restrictive see several mh analysis test behaviour quadratic function gaussian case accelerate accelerate ar proposal mh accelerate normal use idea solver observe example fourth mh case mh discretize discretized hamiltonian proposal ar replicate exist absolutely continuous inverse unknown prior hyperparameter conditionally hyperparameter involve brownian purpose splitting similar make solver algorithms operation product infeasible directly section analysis jump mh ar process proposal section apply langevin dynamic see proposal splitting assess conclude remark provide ar converge spectral process q radius satisfied substitute spectral radius matrix symmetric formulae see since efficiency average acceptance show walk metropolis rwm mala hmc require expect nm lemma algebra symmetric mh define orthogonal stop spectral theory algebra coordinate algorithm transform lemma analyse need lyapunov central limit theorem expect variance theorem x cumulative distribution equilibrium eigenvalue define normal cumulative matrix e dm n g ix iy I eventually lyapunov c mcmc usually integrate thought sample markov give see unable directly splitting depend concern proxy expect successive chain eigenvector precision eigenvalue mala depend study finite jump superior mala rwm mala burn like gap eigenvalue kernel determine rate aware analyse langevin mala accept reject mh correct converge usually mala wrong target wrong depend radius slow per mala require multiplication mala langevin diffusion discretization euler positive langevin differential eq motion time current proposal target ar mala table several mala langevin mala langevin call identify langevin correspond avg v symmetric eq q symmetric yield proposal target redundant analyse accelerate performance mh proposal would theorems fix algebra mh mh moreover proposal mh splitting expect size constant equivalently satisfy mh algorithm normalise eigenvector although langevin diffusion particular gaussian also efficiency consider expected jump require independent sample mala compute multiply assume another fit proposal hamiltonian see e treat state particle momentum accord particle proposal solve hamiltonian particle modify hamiltonian proposal hamiltonian l block vector time mala immediately hmc process still process split hmc splitting hmc imply hmc target proposal hmc iteration eigenvalue system alternatively may momentum attention try theorem matrix precision simple coordinate hamiltonian hmc correspond coordinate mechanic split precision matrix hmc reveal algorithm avoid restrict extension hmc matrix balance optimize convergence rather extend independent mala suggest alternative numerical suggest variant infeasible designing proposal mh challenge distribution job make hard mh algorithms focus ar proposal high new criterion evaluate ar process proposal guide construct proposal ar process
assumption limit small pixel refer parent shift invariance pixel sharing constraint mixture conditional take covariance variance dimensionality neighborhood far introduce factorize additional sharing neighborhood derivation multivariate supplementary describe memory spatial lstm cm pointwise product depend memory lstm memory precede state forget sequentially read produce hide vector pixel hide fed factorize predict pixel px ij recurrent much large region pixel far stack image c boltzmann rbm try weight sharing rbm et al autoregressive unit et al sequential manner draw bernoulli treat make difficult mean one video optimize al try step pixel contrast try pixel intensity heavy well modal momentum bfgs except early stop indicate recurrent augment conditionally whiten let pixel causal conditional whitening replace dependent evaluate change variance jacobian neighborhood ensemble pixel improve simple trick produce ensemble without need transformation leave invariant ensemble k simply mixture image yield ensemble argue leave natural invariant nevertheless boost dim gmm deep layer em dim layer layer recent image patch sample berkeley dataset strength capture correlation follow rgb turn account discretization split image contain dc live bottom pixel discard validation patch evaluation factorize pixel pixel fix find outperform single deep model table outperform ensemble knowledge currently density dataset try compute pixel pixel lead explanation intensity zero indicator neighborhood pixel treat infinitely image bfgs pixel causal neighborhood epoch range backpropagation log analogously average likelihood directly ensemble approximated evaluate lc layer able scale axis xlabel neighborhood ylabel log legend align legend anchor south east font xlabel near ylabel major font style axis mark densely width color coordinate mark solid line mark option color red green two comparable transform rate dc jacobian transformation two comparable sense patch would independently highlight result achieve gray large benefit patch rate apply large capture patch factorize approximately gmm frequently dataset van contain dataset pixel linearize intensity evaluation account discover section several model large patch patch model simple procedure correlation leave stationary statistic model pixel recently multiscale image greatly improve hand yield add lead layer recurrent par ensemble previously result dataset improve simply causal increase instead likely cause cm figure figure cm cm figure cm figure cm figure cm cm cm cm figure cm cm cm cm figure sample train texture illustrate capture recurrent kind capture capture try pixel pixel select purpose train epoch range pixel model correlation marginal periodic although reproduce periodic well suit failure likelihood indistinguishable region model correlation region miss sample resort miss initialize candidate one large sequentially overlap pixel metropolis proposal via accept patch joint density costly gibb introduce recurrent insight generative superior performance quantitative important abstraction collection factorize version large neighborhood parameter perform block video long network apply natural prove generative conceptual model abstract level author foundation challenge partly extend hundred pixel range dependency number problem recently generative short memory modeling image arbitrary tractable art texture synthesis see progress lee drive improvement supervise potential
hoeffding product lemma distribution noise easy reader paper integer linearly draw simplify notation generalize parameter recover definition resp distribution kullback return resp failure cost reduce solve negligible bound sample product transform remain return small bias advantage average large algorithm sample bias part except hoeffde mb kt main prove dimension natural distinguish yield uniform variable sample hence distinguish thus crucial gaussian elimination sample produce sample block iterate consecutive eventually sample obtain sample ultimately consecutive later adapt improved modulus switching vector merely may perform round produce essentially reduce balance decrease complexity round costly add round contrast make modulus hand maintain decrease allow balance modulus switching point view entirely idea result later combine sample repeatedly attain repeat sample quantization produce center modulus simple proven depend index point center without improve modulus might code partition k presentation setting specific exist true later coordinate define fix terminate distinguish equivalent terminate run accord noise independence uniformity add get noise indeed hand optimal amount baseline list empty final bias balance convenient auxiliary decide dependent parameter choose failure induction lemma choice part superior hence solve time choose parameter fulfil amount get kb kb bx ik finally already quantization else amount error find bound previous discover work correct except return negligible lattice problem factor goes use author incorrectly round p q reach actually assume part solve possible work big final need propose heuristic exponent sum sample independent lose aspect negligible practice associate opposite value coordinate second step sample center whose center simple pick thus short ever gain somewhat add cumulative vector gamma amount suppose fixed proportion formula sum within ball radius able much technique keep proportion instance twice fall sample notable newly coordinate previous completely ability expect norm minimal count within complexity factor bernoulli practice transform high match another significant improvement linear quantization quantization center basis decrease besides available help entry transform sample try amount available impact practice continuous modulus multiplying reduction assume variance correspond keep vector test complexity bit operation use multipli optimistic complexity c reasonable optimistic error optimistic distance thus solve repeat solve close access free lattice follow lattice last stem assume modulus eq lattice oracle lattice call n since bind remove solve apply large solve reduction solve polynomial zero lattice vector shorter impossible q b need prove intersection ball radius divided volume ball solve infinity negligible failure use call oracle lattice oracle law uniformly center origin return complexity k probability let summation bb definition n polynomially previous chebyshev previous polynomial prove sufficiently easy operation integer order circular generating let axis align cube length radius sufficiently solve solve solve independently trivially sum possible cube radius include ball subset problem q failure failure therefore break lattice slow nm nk generalization gr hard lattice theorem justification claim prove assume efficient use solver distribution either sample reduction distribution integer uniformly whose sample else distinguish sample distinguish follow uniform uniform counterpart rank result modification uniform accord return remark dimension take coordinate switch feed output uniform therefore statistical efficient distribution take error resp solver let uniform poisson property hash bias negligible failure hardness apply introduce failure error apply particular round direct solve lattice nn recover significant bit use principle coordinate radius ball radius contain sample ball add output add element coordinate clearly coordinate maximize fouri odd bias distribute small fourier introduce time lower universal introduce bias stage determine approximation use fast fourier transform recover continue get final repeat whole distortion first iteration input large small inequality reduction except bias lattice lattice short inferior lattice stop call basis proportional v dc algorithm lattice basis vector ba cb binary since count clearly difficult lattice subset sum density lattice complexity find remain precisely form column lattice coordinate inferior select nd n b binary apply require exponential number disadvantage vector preserve decision give deduce horizontal axis represent modulus classical gr hard colored contour attack choice vary increment q distribution failure choose n negligible apply switch distinguish output vector failure new except union n np fr paper interval introduce variant rely quantization generalize fine modulus significant gain front exponent dimension introduce variant require solve analysis require break security solve subset independent hardness decision n nb ne q concentrate given come average approximate bad show reduction modulus finally
component scatter plot expert fit propose quasi identical line correct tune cc right right predictor actual training experiment skew ghz processor gb fit bottom right except figure expert differ see one htbp c c estimate component bic give criterion poorly bic correct correct number component correct suggest ccc ccc ccc k aic aic bic aic laplace mixture expert htbp e value anomaly choose criterion bic aic except provide evidence htbp l ccc ccc ccc c bic aic anomaly normal skew suggest tail suggest heavy tailed noisy datum infer model successfully confirm evidence normal alternative propose successfully include tend aic poorly analyze obtain version future mixture natural future extension regression rather universit france universit france model heterogeneity cluster expert component group group observation asymmetric behavior heavy use expert fit introduce normal expert issue possibly skewed skew skew respectively develop dedicated em monotonically maximize log present experiment carry propose term show usefulness change keyword expert skew skew em expert study statistic fully proportion density expert analyse different context cluster usually expert expert well outlier may affect paper attempt overcome propose adapt deal possibly recently regression maximization mm normal skew normal skew beneficial asymmetric namely asymmetric univariate skew natural robust integrated develop propose recently robust univariate mixture sufficient asymmetric mixture skewness tail skew mixture skew expectation extension bayesian multivariate skew robust univariate regression laplace regression mixture expert mixture regression mix proportion mixture robustness deal asymmetric attempt overcome limitation deal asymmetric tailed contain use skew normal commonly skew normal accommodate asymmetric tail regard outlier skew tail unconditional normal expert mean covariate maximization model maximize model expert show maximize case stable log monotonically step maximization parametric non hierarchical expert expert model organize maximum estimation em derive estimation technique show perform non linear model perform conclusion mixture expert context including consider regression aim covariate via explore unconditional distribution modeling took distinguish regression regression generate hide categorical random component observation mixture proportion sum k although aspect difference consist conditional model model softmax expert analysis modeling proportion covariate proportion logistic covariate vector mix proportion densitie expert conditional proportion regression normal expert follow semi parametric case log likelihood observe respective log step em calculation follow posterior estimation update complete expert expert vector coefficient consist analytically linear proportion update reweighte heavy sensitive skewness propose normal expert fit propose expert tail skew expert skew expert component describe integrate follow skew skewness density pdf cdf skew normal skew denote skew hierarchical introduce skew skew mixing assume skew mixture extend skew framework conditional distribution skew skew expert skew parameter skewness expert component skew normal obvious see skewness stochastic representation derive representation skew normal representation skew covariate follow multinomial I I hide label th observation stochastic representation lead inference introduce iff th respective vector maximization log however framework maximization maximization algorithm dedicate variant mainly several maximization divide sub space sequentially one coordinate block algorithm complete z I I propose perform start cm step function expectation eq ik label correspond membership observe correspond hierarchical show mixture bayes expectation calculate analytically calculate parameter function adopt extension consist maximization decomposition step calculate skew skew maximization close reweighted square respect iteration newton update gradient hessian take maximization expert solved analytically calculate analytic calculate maximize k consist equation update skewness skewness update correspond standard characterize handle skew tailor skewness contain heavy tailed affect sensitivity outlier tail mixture propose robust normal distribution describe representation handle accommodate tailed univariate location mahalanobis distance gamma give gamma bx distribution give hierarchical express expert extend component mixture form univariate mean linear proportion expert expert location degree vector k approach inference procedure let hide iy follow categorical variable multinomial hierarchical present mixture distribution hierarchical unknown maximize perform maximization describe maximize vector e expect function latent complete complete observed estimation n ik ik membership probability easily step maximize km update iteratively mixture calculate kt mean analytically update note ml component multivariate algorithm modify may calculate degree freedom equation scalar root mention constitute tail derive describe single model give consist take maximization cm degree expectation tail affected skew mixture expert attempt accommodate heavy tailed expert component skew normal respectively skew hierarchical representation cumulative cdf freedom introduce random variable normal univariate skew freedom skew hierarchical stochastic skew give skew expert first introduce framework skew component skew proportion mean mixture see robustness approach flexible accommodate tail skew categorical label component generate skew say skew mixture expert skew hierarchical maximum model perform dedicate consist variable label hierarchical representation log k ik start cm step complete log current see conditional ik ik ik follow expectation namely case skew propose ik ik note adopt approach expression rather carlo mention exact expectation place maximize provide maximization carry update iteration parameter calculate tm tt expert provide update skewness update maximize show degree fix update calculate equation find component update hand degree update predictor expert therefore model training via prediction predictive q compute expert give k variance model describe normal variance expect propose case expert calculate respectively eq follow mean model mean easily thus expert variance perspective assume component skew skew expert interpret mixture dedicate algorithm provide represent fuzzy partition membership respectively membership apply maximize posterior base problem form one aic bayesian integrate observe criterion maximize express observe complete convergence correspond free proportion transformation covariate univariate covariate work case expert linear regressor covariate correspond univariate covariate dedicated evaluation simulate evaluate clustering implement matlab mix initialize randomly equal initialize partition fitting initial membership replace robustness initialize randomly skewness randomly stop log process likelihood illustrative value output I expert partition mix cc fit normal toy analyze experiment impact generation accord model generate mse component estimate one error average trial l table obtain three decrease confirm property mle mixture decrease pt mse parameter mse component estimate c mse one sample addition previously figure quantity counterpart plot show minus twice true middle plot true function middle show counterpart bottom em show mix probability clearly estimation correspond additional propose generalization expert
simplify positive q fairly verify complete controller sampling finite population specify time assume variable unknown controller policy expect sum outcome equivalently regret lack asymptotically horizon keyword sample armed bandit family support population population controller f discrete indicate controller bandit dependence controller grow equivalently restrict bandit ensure bandit bandit would bandit bandit bandit surely identify focus unknown kullback generalization part policy bind specific eq give policy policy achieve maximum optimal first show asymptotically policy construct bandit number index index current estimator eq choice n therein automatically satisfied condition see c condition essentially k f optimality policy often take space potentially bandit herein bandit sample early include asymptotically consider derive policy achieve include poisson distribution belong exponential far bernoulli thompson optimal sample likelihood respectively maximum policy indicate would lead n equal eq q break tie arbitrarily ta satisfied fail optimality technique insufficient verify may negligible tie arbitrarily remainder optimality additionally horizon bound remainder refine somewhat work convenient bandit take bandit recall follow sub optimal delay briefly present result infimum complete remainder eq q inequality apply interval take I proceed bound three observe hence observe indicator term account possibility second q inequality ti minimal span follow finite joint simply convenience observing result point convenience constant nice refine strong remainder complicate particular suffice take building take bandit value
also connection contraction property hilbert kalman contraction map metric thompson indeed fundamental property endow kalman filter prediction dynamic incoming order internal semidefinite discrete algebraic kalman invariant converge step show covariance monotone limit line formula contraction convergence kalman iteration flow contraction hilbert contraction understand assumption strict convergence kalman iteration combine iteration indeed ii underlie metric contraction riemannian hilbert metric drawing link contraction property hilbert metric metric definite metric pointwise filtering end cone matrix pointwise introduce strict positivity contraction base hilbert banach space cone empty iii k mx x hilbert cone positive say relevant triangle metric discrete expand definite strict contraction hypothesis contraction thompson discrete expand hilbert positive kalman q far mutually kalman observable component hide countable common use setting countable interested parallel transition countable state space measurable qx yx qx stochastic process value common space transition namely formally space hide determine equivalently x wiener process work filter measure initialize induce normalization pointwise map split ahead projective forward filter let composition three map iii expand hilbert see amount isometry thesis composition operator kalman span variable kalman iteration see gaussian literature see combine contraction briefly kalman kalman model emission constant connect kalman filtering recursion kalman recursion emission virtue priori give virtue
expression parameter partition free energy complexity familiar degree freedom regular aic suggest frequentist penalty singular contain fisher finite aic propose frequentist singular pearson hypothesis hope asymptotic well likelihood g detailed description objective frequentist weight posterior distribution parameterization understand model name weight prior perspective important significance resolution objective machine consequence weighting increase also complexity dataset information weight rather interval I true prior maximize optimize analogy predictive unified discuss directly maximize far prior also understand although strict improper rigorous long history successful improper jeffreys despite considerable approach propose improper prior perspective uninformative since scenario describe state interpretation uninformative parameter maximally uninformative sense maximize content interpret demonstrate regular desirable short come weighting interpretation believe desirable mathematical necessity three principle statistical objective bayes probability interpretation maximally uninformative information aic regular asymptotic limit free ad hoc complexity complexity applicable bayes methodology formally bayesian unknown second formulation begin gauss inference although arrive conclusion summary use principle implicit indistinguishable parameterization model understand maximally uninformative establish prior connection demonstrate regular bayes criterion introduce observation parameterize parameterization probability distribution bayesian realization parameterization introduce marginal partition nee normalize marginal parameterization posterior bayes update meaning initially prior free q closely performance discuss ability bayes model predict convenient formulate model information laplace assign exclusive equal prior lead unclear mutually exclusive exhaustive uncertain origin parameterization cutoff volume specificity gaussian mutually exclusive intuitively know difference sufficiently distinguish small observation value always resolve exclusive divergence assume indistinguishable exclusive sum exclusive write volume drawing volume fig simply time information learn machine drop dependence keep purpose retain factored sum make definition density model depend describe divergence fact need propose precise density first bayes q indistinguishable discussion indistinguishable unity improper name unbiased improper long give eqn eqn eqn predictive last probability connection meaning analytically continue temperature identify free angle parameterize physical average gibbs follow understand model indistinguishable indistinguishable illustration take key avoid validation true interpretation wish interpret code parameter careful validation posterior normalize integrate cutoff determine total strictly convergent interpretation respect uninformative wish code informative localize intuitively expect uninformative prior result content uninformative since gain maximally uninformative argue uninformative analogous use code understand maximally uninformative locally possess maximally uninformative regular special
certain predictor fold calculate use likelihood compare choose baseline global trade considerably recent decade international increasingly global air network price customer air trade express forecast decade attention surprisingly little air poor business air consequence significant service level reliability deviation arrival customer cause delay production service incur storage handle cost risk customer risk routine deviation refer survey name management strategy empirical study air chain literature interesting phenomenon observe datum include etc show figure figure risk observe datum clearly distribution positive well around concentrate day peak largely fail later usually international hour gap thus transfer gap peak detail empirical study primarily focus arrival delay review delay positive concern assume delay unimodal adopt ordinary least multimodal observation ols air predictor unimodal normal distribution mixture delay need develop new model contribution accommodate multimodal risk introduce art bayesian tool rapid development decade accelerate ever several year diverse finance reference therein estimate independent identically observation arise give parametric bold assign discrete finite process adopt specific formally importantly computational conditional create genetic develop variable risk characteristic range cover risk risk within allow relationship risk predictor include etc explore way reliability demonstrate risk estimation ol dramatically ols fail level importantly ols risk insufficient management drive powerful general assessment receive risk management attention practitioner co business company assessment risk assessment risk must execute update experience involve record long experience business detail carry quantitative quantification analytical management step develop negative event assessment assessment consist estimation hazard occur term notice distinguished impact part chemical use environmental conceptual risk chain work focus call advanced computation correctly implication alternative management develop tailor service customer service resemble capacity risk component risk assume particular bernoulli risk introduce organize introduction air challenge question exploratory lead introduce posterior gibbs propose several model operational conclude future detailed checking air public operate necessity brief explain motivation standardize examine air four short chain air provide company origin date detail piece service pick share connect map term refer customer air service use service upon request request several economic certain percentage include combine leading enable participant reliable entire air operating plan develop monitor air allow control major ground etc see appendix member management implement different control confirm create share describe along customer agree plan agreement essentially combination profile duration completion define system alarm correction take responsible party back meanwhile exception system illustration end keep party customer directly compare service customer direct include predict risk risk reliability help address customer volume demand variable month decision duration description customer provide customer next elaborate demand differ dramatically across air service level factor month demand weather e trend air service finish month approximated piece fail capacity large usually valuable high analysis help reveal dominant substantially across variety factor connect strong predictor variable duration number greatly reflect send early onto early weather available allow distribution company datum update five north sa south figure see depict service available around column figure direct service variety impact service cc l risk predictor table facilitate cause delay failure company datum helpful delay code delay day code appear denote use exception provide motivation mixture model decompose two part mixture model discuss selection provide ols aggregate histogram empirical accurate inference first datum usual multimodal mixture rely investigate risk demand decision categorical one stream double kernel popular mixture model joint dirichlet normal response normal express prior b specification involve carlo limited capability high require algorithmic proposal need carefully adequate gibbs update conditional superior tractable strategy augmentation specifically observe correspond replicate replicate drop replicate observation help indicator gibbs classify category represent gamma conjugate q n xy x indicator multinomial probability multinomial conditional l x variable n beginning equal augmentation scheme simplify following augment prior due similarity updating scheme coefficient j posterior easily generate slice discuss finite value conservative many utilize explore need label move mode set mode negligible probability framework switching greatly improve appendix explanation move consider parameter location mixture case global observation rough argument scenario prior weakly lead improve improper discuss prior transform convenient parameter simplify assumption component mixture dependent constraint baseline build hierarchy heterogeneity small lead set especially sure yield size describe characterize limit risk fitting inference range quality assess gibbs sampler iteration iteration burn matlab ghz intel dramatically improve efficiency rely recent development prefer diagnostic adequate lack fitting validation check visual cross check sample capability overfitte log strictly forecast log follow appendix model ols delay replicate ol separately check specification expect something column resemble test ol mean histogram represent posterior solid vertical dotted ol largely hour delay recurrent hour ols datum solid dot deviation predictive true check fitting level histogram draw real interval narrow predict capture location weight predict cm p p p mean interval thing range deviation day normal narrow flexible parameter range variation certain present distribution narrow estimation name great heterogeneity measure among confirm aspect close inspection reveal whose coefficient impact base ol significantly huge estimating independent mean playing peak ol detect impact considerable summary table hyper heterogeneity huge risk aid decision use provide demand variable help preferable service help price customer demand requirement choose service service initial start customer expect customer customer interesting several generic aid pick arise risk neutral delay early proper delay certain example deviation function expect loss analytical short figure present expect choice service service play dominant normal service estimate offer price different increase price sensitive price unlike business solve integrate set integrate practical certain decision service reliability critical interest replace capacity pricing generate comparison baseline certain effect thus type impossible baseline effect plug level sample distribution differ location peak allow comparison offer much rich comparison metric meanwhile rich comparison single truncate risk plus minus initial delay applicable zero otherwise author argue short incur incur author wise international air analogous schedule calculate ratio effect exclude thus calculate intrinsic service baseline
lf ef lf lf ef lf ef lf ef lf ef lf ef lf ef lf lf ef lf ef lf ef ef lf ef lf ef lf ef lf ef lf ef lf ef lf ef lf ef lf ef lf ef lf ef lf vary learn inductive self clarity ep lr omit ccc l co train ef lf ef lf ef lf ef lf ef lf ef lf ef lf ef lf ef lf ef lf ef lf ef lf ef ef lf ef lf ef lf ef lf ef lf ef ef ef lf ef ef ef lf ef lf ef ef ef lf co train round ef lf ef lf ef lf ef lf ef lf ef lf ef lf ef lf ef lf lf ef lf ef lf ef lf ef lf ef lf lf ef lf ef lf ef lf ef lf ef lf ef lf ef lf ef lf ef lf ef ef ef lf round ef lf ef ef lf ef ef ef lf ef lf ef lf ef lf ef lf ef lf ef lf ef lf ef lf ef lf ef lf ef lf ef ef lf ef lf ef lf ef ef lf ef lf ef lf ef lf add round decrease amount observe omit sake brevity suit image lr fuse round baseline ensemble strategy powerful result descriptor extract train convolutional neural network consistently superior highly tune art visual feature implementation cnn image annotation convolutional extract last leverage three discriminative cnn alone see apply four scene four significantly discriminative cnn baseline mid level feature want substitute unsupervised kernel pass exploit unsupervise lf multiple used view experimental scene scenario plot confirm suit label image prefer label htbp cc qualitative class instance add round correspond part fig add round belong current green box red belong image add round pattern find occur see moreover classified increase confidence move leave class decrease right namely different training randomly contain number test experimental ten label similar plot show suited prefer available remarkable improve cnn htbp ht ef lf round ef lf round ef lf round ef lf round ef lf round ef lf co round train report height test image decrease classification confidence classification unlabele classifier scheme early fusion test scene three different experimental inductive test co outperform art class training round result image add label mid cnn art effectively leverage discriminative feature boost I e co co iteratively build two view view learn combine conduct experiment recognition feature combination extract ensemble unsupervise couple result semi supervise analysis supervise supervise account machine particularly instance many semi generative co difference classifier datum independence view success view technique work exploit way first representation unsupervise label train build sub separately feature strategy co unsupervise component schema fig view single feature recognize concept classifier exploit accurately recognize argue scheme together unsupervised make effective co build unsupervised component assess conduct set datum set scenario unlabele test coming set verify efficacy also unsupervise couple logistic semi result show outperform learn sake brevity discuss supervise book co verify web page basic idea train classifier confident concept co annotation recognition traffic analysis speech recognition retrieval image object individually conditionally label show co error support view detail year success framework observe automatically effective training boltzmann auto encoder network notable method conceptually occur unlabeled mean widely purpose vision lead variant representation briefly use vocabulary extract similar visual counting occurrence visual vocabulary feature unlabele learn noisy multiple accord representation implicitly embed kernel combine information issue recognition fusion bring complementary improve retrieval modality fuse fusion early position whole early usually refer single refer combination stage obtain comparison universal strategy prefer give task fusion semantic indexing get indexing fusion mkl support svms single computed svms mkl learn determine different different former choose correspond multiple different fusion consists label l view early fusion early ef use ef fusion lf unsupervised learn representation lf ef l ef ef ef lf lf lf lf ef lf view co train classifier regression co train view e ef ef l lf lf iteratively confident confident pseudo classifier ef ef lf lf ef lf round training choose vice us call candidate unlabele belong unlabele ef confident confidence extract confident q choose view pseudo label complete outline em ef ef lf lf ef lf ef ef ef lf ef ef ef ef ef ef ef ef ef lf lf lf lf lf v v l v ef lf lf l ef lf ef ef u lf lf parametric projection unsupervise train logistic latter another scene scene divide scene category category image category imagenet datum self current version imagenet come pyramid histogram orient gradient rescale scale pyramid neighbor use learn ensemble train plain projection learn prototype pseudo label indicate belong image sample prototype prototype large keep classifier prototype vector feature project follow logistic classifier unsupervised ef fusion lf vector make single ef lf kind compete classification co three scenario label semi training classifier unlabele take imagenet unlabeled test set evaluation average precision recall test scene e random label split strategy built label concatenation available ep ep specifically test train classifier operate fuse employ ep lr lr lr differ pseudo add round maximum add two co classifier number per scene five svm result report show ht test baseline variant available htbp ccc scene ep svm ep lr lr svm ep lr svm st svm
four predict display capture corresponding provide technique employ c mesh bs bs ls region analyse level certain hour around united air environmental bs simple spatial intercept spatial version bivariate spline stationary represent b location basis table basis direction representation employ cross score table bs log score bs g generally bs bs g representation yield small stationary bs display fact score b high stationarity selection aic beyond stationary predict united states bs display prediction present south corner north east corner north area stationarity htbp great triple directional coordinate directional directional derivative directional integral inner product calculate precisely area define note triangle basis construct easily spline coefficient spline triangle contain bivariate function test leave identity fact q th entry diag diag diag representation spline follow lx lx lx product integral e f bivariate spline suitable easy j associate cc eq piecewise result prove general note recall bernstein approximation arbitrary df let x fact cauchy schwarz next eq constant similarly put estimate length edge proposition approach bivariate spline field computational bottleneck gain partial use element new polynomial representation degree infer bivariate numerical also keyword spline non field especially structure additive name flexible extensively statistical computation typically order computational approximate bayesian assume integrated nest fast inference mcmc mat ern modify nominal correlation marginal integer determine underlie noticed mat ern fractional innovation white unit q g require suitable construct element gaussian piecewise carry dramatically closely finite recent propose field basis function rate finite refinement underlie alternative high degree polynomial element polynomial order multivariate employ conventional finite easy piecewise polynomial various spline show efficient conventional element solve spatial thin place concentration forecasting spline advantage piecewise adapt efficient spline review link establish approximation discuss extension stationary several numerical conclusion discussion proof space eq polynomial degree continuous possible continuous spline triangle form spline construction locally calculation appendix triangle coordinate polynomial call bernstein polynomial triangle spline basis spline eq find sensible set least square gaussian operator bivariate mean least square diag diag b recursive solution sparse make mass diagonal next application mention element previously bivariate theoretical section integrable finite bivariate spline span finite sequence edge th appendix spline converge obtain derive bivariate spline field triangle clear able spline high magnitude matrix approximation bivariate projection bivariate basis base solution covariance spline result show g norm edge spline decrease fact illustrate bivariate spline show extend stationary parameter location vary slowly domain representation basis guarantee domain associate see easily minor modification bivariate representation square solution locally interpret mat ern field local mat ern automatically conduct simulation spline finite term parameter ern model integrate nest laplace approximation brevity bivariate denote bs bs bs element study bs l fitting surface surface space surface mat covariance whole fine equally spaced surface nf four different include element spline triangle associate cpu denote recorded weight complexity operation stop function b function mesh present second bs bs l level different surface efficient reach low bs l bs bs bs right precision dimension surface reach mse high level bs bs l basis function compute bs generally gain bs bs mse surface previous bs degree level around reach level bs low level bs reach require quite neither b l bs reach require basis l efficient level function low b note shape surface comparable gain bs cm study bs prediction national center de choose cover relatively near three bs rmse location aim surface suggest estimate surface close validation employ embed predictive log six figure extend
fourth b rule inner eq jensen denominator plug mean identical update inequality algorithm k coordinate inside bind q term since divide side inequality prove lemma trading ingredient decomposition exploitation first reward explore decomposition schwarz deviation w regret exploitation regression matrix constant response feature true vector thus write subgaussian independent subgaussian reveal universal lead exploration round give exploitation regret failure union union event since exploitation challenge part phase exploration small probability follow lemma analysis algorithm characterize consider context denote feasible action round eq vector action action apply hoeffding realization round random randomization leave dependence randomness easy since hoeffding reveal round regret bound last randomness action regret round q us characterize regret action cauchy appear give account randomness algorithm notice bound deviation regret insight regret stop round two round regret round associate round bound q threshold set tn w probability bound hold proof know reference rather first bernstein type hoeffding assume value hoeffding due subgaussian fix surely probability hoeffding adjoint self almost rgb rgb pt learner receive action crowd search domain analyze enjoy linear transformation explore learn use achieve unknown algorithm explicit enumeration consequently computationally fully set partial great recent motivating observe prescribe patient work internet user reaction click article content framework learner observe take capture aspect learner maximize many round application decision internet application recommend information rich learner tuple receive composite simple play additional feedback unless relate refer rather feedback derive contextual semi bandit goal build contextual enjoy hardness composite interaction compete mapping composite tree neural net access warm idea identical efficient run logarithmic contextual bandit linear composite encoding valid move still composite feedback feature however still stage algorithm choose empirically scale call generalize contextual rich setting form grow body work bandit refer bandit majority contextual work generalize bandit contextual bandit require explicit enumeration instead impose linearly linearly work bandit assume sum generalize reward attempt strategy cope space linear bandit reveal include action unknown playing assume relationship reward composite action crucial action context let mapping weight policy one length bandit weight learner play round context learner class instantaneous success regret eq avoid enumeration exponentially access henceforth comprise contexts weight oracle composite structured mechanism composite action define marginal vector contain round aggregation composite avoid unbiased define counterpart nonetheless empirical context typical weighting scale typically desire set mix action could achieve uniform composite smooth uniform action play efficiently action indicator well composite action probability maintain non place default policy smoothed project always capital letter composite action letter action contextual semi bandit weight exploit failure zero kp tt tx ta ta l ta p structure bandit policy use smoothed play action distribution interaction op feasibility look thereby place ensure variance bind policy ensure exploration amongst exploitation tradeoff encourage place good encourage main op modification constraint action obviou simple action action constraint lead tight weighted improve regret modification reward estimate without equation really modification modify reward influence universal algorithm scale feedback additional relate result fall exploration direction restrict affect efficiently optimization oracle bad discrepancy contextual setting style contextual aware involve qp ks computational bottleneck solve focus subroutine classical contextual update violate equation add policy constraint long violate shrink weight involve calculate easy together decrease potential update shrink execute construction bad alternate update prove theorem horizon failure tx ta tr ta ty pa ir ia remain round context first explores explore thing happen action feature accurately expect policy horizon subset simple display round reliable weight vector well policy take inner policy remain play analog setting would action round estimate reward exploit exploitation thing round least square line use explore condition behave policy exploitation ensure phase difficulty condition round proceed also upper exploration regret accumulate intuition reveal least regret guarantee sublinear simple feedback epoch greedy greedy action hard exploration relate minimum enable learn weight regret exploration reveal ahead feature action classical bandit ignore algorithm shorter leave sketch proof detail intermediate stop exploitation term true stem deviation provide provide first term straightforward grow exploration suffer may eigenvalue exploration round ta follow cauchy schwarz imply cumulative equation explore uniformly subset round hoeffding sample concentrate feature round immediately translate accumulate equation come argument come deviation composite observe tuple simple regret provide regret algorithm leverage feedback scale action arise regret feasible contextual combinatorial bandit bind obtain avoid composite action partial feedback many application call beyond transformation hope address question careful inductive op first deviation variance estimate use suitable variance deviation spirit theorem argue version union tp schwarz inequality prove lemma lt bound one game empirical round martingale moreover use schwarz claim equip main note since hold policy set straightforward prove round achieve trivially choice deviation kt p deviation constraint policy induction definition lemma
form vote monotonic function much answer vote majority worker odd weight rule assume among class odd confident grow linearize theoretical desirable weighting monotonic exclude label property focus worker label aggregate refer worker problem analysis aggregation select worker require unfortunately weight majority voting majority vote unbiased label probability depend worker term gap vote I weight majority function worker give assumption coin worker coin model bound score coin natural unfortunately true unknown unbiased I tend informative answer side worker balance encourage worker ii simple estimation directly term worker favorable confident concrete question number answer worker question estimator interval defer modular modular worker pool question constraint worker sort select rank evaluate worker find worker achieve set objective maximize actually pareto exist feasible improve result intuition small rank perform good worker naive top worker algorithm tends select top rank specific practical experiment worker complete question platform ii task subset worker distribute answer iv final label though worker derive em worker ratio worker top worker worker algorithm aggregation perform bad omit plot clarity trial algorithm item randomly collect estimate item trial store average base com question know truth worker finish question typical question year internet create widely include use worker select worker htb task identify wikipedia option highlight sentence refer truth available complete typical example follow file run framework page runtime option http wikipedia http en wikipedia http en wikipedia http en run ask contain b worker actually supplementary worker achieve worker show worker true objective optimum different unbiased confidence optimize margin confirm select mainly worker reliability close truncation prevent go worker improve pool increasingly less bad estimation match decrease reliable worker select crowdsource labeling demonstrate simultaneously worker future advanced model supplementary material guarantee weight weight vote aggregated potential introduce ambiguity label event relation provide voting weight apply hoeffding hand straightforwardly depend thus valid say e desire thus move term construct lead follow lemma straightforwardly interval base form note collection variable hoeffde bind meanwhile cover least optimality give treat maximum worker select worker worker globally configuration value global optimal worker optimum maximum yield exactly therefore globally worker mention multi score worker actually multiple pareto theorem improve objective htb liu department computer science california crowdsource worker pool maximize accuracy natural worker allow analyze typical crowdsource worker simulate world able quality worker perform sometimes budget crowdsource possible collect human intervention large cost micro amazon crowd human task short payment unfortunately pay worker label low expert redundancy crowd answer answer much individual worker sometimes crowd diverse weight properly aggregate answer large body deal diversity method often majority voting answer majority weighting account use answer von liu liu necessarily well idea bayesian able procedure inference however worker never perfectly add worker extreme completely random answer attempt label able perfectly zero worker dominate quality worker recent empirical aggregated number accurate assume pool test gold question label aggregation want maximize budget worker I worker high
interaction environmental location assign label office achieve spatial understanding rather precise efficiently facilitate carry mostly modality informative place exploit two map knowledge new environment map insight place problem raw encode environment feature raw opinion hold fully exploit achieve high encode spatial consistency spatial proximity field corner contain enough office room robot mixed class input paper edge relationship information recursive input input contain field view represent increasingly classified raw automatically impose adjacency similarity map map feed learn form semi step predict label make tree carry remainder paper organize introduce construction tree make supervise validate section environment demonstrate effectiveness camera classifying place al al extract feature et place base range easily sensor vision nonnegative nearby contain include clutter fed label environment al classifier class furth address solution place specify unsupervise deep include object natural recognition discover extract success unsupervise end feature locate research characteristic consideration consistency feature process invariance work implement graph model utilize base effectiveness embed reduction paper view rise classification field boundary propose construct multi classification follow represent generalized topological successfully general meet meet adopt resolution level number layer denote layer denote sense local represent assign connection describe adopt angular l e r le le l implement demonstrate build low layer explanation contain without child connect obviously carry neighbor end detailed fusion otherwise eliminate high layer layer apply times th process illustrate figure move node decrease elimination illustration layer map node distribute high structure abstract consideration compose compose compose red eliminate layer recursively generate describe construction generate state end respective integrate give mapping layer carry achieve transform frame assume knowledge robot pose virtual generate ray angular range dimension different interpolation fix fact completeness proportion measure uniformity however interpolation range interpolation apply pre range keep sequence input follow v fr end eliminate layer black preserve reveal connection illustrate red position blue environment ray sequence one input obtain predict tree figure correspond maximize intend tree structure parent range reason layer number leave root factor compute classifier input sequence give lc c l ic j ic j lc ic I tree denote label optimize label tree structure decision consecutive layer layer always optimize confidence optimize optimize label tell change child confidence advantage optimize obtain optimize label leaf evaluate leaf clarity classifier separately construct classifier predict node assign label layer show child decision firstly initialization upper tree respective node label finally compare optimize figure datum testing capability automatically learn indicate raw label thus omit note map semi rich spatial classification convention denote way denote label give illustrated figure firstly feed red difference scan consecutive difference rich extract practical sort feed stack auto encoder build deep decoder learn encoder stack sigmoid decoder reconstruction weighted encoder impose classifier network parameter random discard pre preserve auto encoder regard work softmax learn stack auto belong whole preserve softmax consistency regularization fine tuning cost follow last layer respect input two cost cost cause error classifier impose hide learn weight regularization term detail construction build step firstly weight employ adjacency force close regularization input euclidean weight closeness input connect belong office although force close however keep validate end multi layer conduct datum international environment include centre university technology research centre artificial intel state robot collect maximum horizontal view place information class therefore target number office room room laboratory among six leave many utilize testing
k mle therefore limit easy approximate small ease illustration strictly abc whenever refer dc dynamic dc figure exact mle corresponding course hide simply gamma shape schedule set dc decrease iteration observe rate satisfactory keep last draw take corresponding error return good variability high dc accuracy find variability growth model model result cite reference explicit solution w wish preserve positivity conduct b determine since deviation observational normalize stability dc usual keep decrease small threshold increase schedule last acceptance comment dc relevant beyond acceptance stochastic markovian issue exact abc dc obtain system determine estimate vice versa let subscript maximization taking transition time likelihood z pz z obtain error asymptotic use observational time value estimate error also identify difficulty space abc dc thresholds acceptance rate determine satisfactory notice concentrate posterior maxima though strategy integrate approximate initially sampler beyond mle enhance however simultaneously criterion highly become increasingly unlikely accept increase abc dc produce reasonable inference even work value keep fix vary furthermore start mcmc mcmc start abc mcmc stage use possibility metropolis propose stage methodology rely within partly value value increase free differential equation present methodology estimation applicable main question since abc produce generating compare accord well inference unable rough locate mode conduct approximate basically mode switch likelihood propose draw use sampler realization datum might dependence go work assume model method stochastic unknown think measurement assume unknown measurement ease notation inference analytically distribution implement draw posterior carry chain monte carlo mcmc embed mcmc procedure see bayesian method strategy lead term consider enter enter state write independence nature latent integer copy stack time say generic individual latent imagine series result px k gx depend index integral write result mean choice sampling call generalization general proposal distribution convenience vx px initialization fix generate calculate acceptance large enough produce stationary distribution discard obtain mle returns covariance mle choose degenerate mle simplification take call expression simplification solve ready density deal posterior posterior increasingly deep mode existence fix start value enable rapid help analogous issue occur proposal difficult model process example trajectory result quite distant value posterior rarely simulate blind exploit many tune smc proximity forward smc smc large enough mle abc later abc phase enable surface acceptance approximate ii explore reach typical peak point value start iv complete maximum course posterior locate hence increasingly explore surface schedule rate reduce drastically iii value stochastic volatility model adequate equation unable go dc choose kernel generalise student freedom scalar weight ease read produce dc free dc give use start integer independent value denote conditionally correspond corresponding calculate step execution version abc dc cite propose keep fix remove anneal walk explore high enough metropolis face difficulty subsequent discovery identify mode possibly abc threshold etc mcmc accomplish implement current kernel perform interested generate calculate go check jj accept abc generate jj previous schedule otherwise abc parameter proposal random function exploration surface small abc
drastically I unseen input convolutional reduce albeit pool layer input side figure desirable convolutional high network input large convolutional constitute vision problem effectiveness vision research restrict neural network drastically require image exploit symmetry galaxy exploited version sharing exploit symmetry explicitly additionally image interpolation whose align row pixel galaxy classification raw vote excess galaxy galaxy raw interpret preference brain apparent universe bias brain galaxy probability contain rotation variant bias bias exist reduce refer map layer aggregate extract viewpoint reduce time centre galaxy informative viewpoint extraction modify width conv width width height minimum rectangle minimum conv dense width cm height width height height conv width anchor font base font anchor font anchor base align anchor anchor north east west east south edge west edge conv edge conv conv conv north west east edge west conv east edge south west dense develop setup overfitte behind successive set five viewpoint convolutional implementation draw black width cm input cm preprocessing viewpoint extraction averaging average augmentation augmentation convnet convnet section preprocesse augmentation convnet augmentation west convnet east west describe provide dataset answer evaluation image competition competition reveal image score image platform split real model network learnable million high capacity strategy unchanged dropout reducing parameter exploit prediction first reduce input middle background fit square approximately image rescale speed pixel subset operation object either angular measure object allow centre image far processing effect though rmse make competition provide format galaxy website colour colour considerably despite artificial intend nevertheless model correlation train instrumental example randomly rotation angle symmetry shift pixel relative direction limited centre random rescale colour adjust describe eigenvector factor adjustment first four transformation collapse together mean augmentation computational augmentation randomly perturb image never augmentation extract corner pixel centre right corner patch constitute shape image red outline outline total overlap corner patch extract galaxy centre corner patch patch view colour allow affine avoid interpolation indexing also image fidelity augmentation viewpoint minimal array rgb architecture process stack four convolutional layer position separate max pooling fourth convolutional stack maxout output maxout instead relu reduce maxout convolutional prove architecture manual competition million convolutional relu convolutional relu convolutional relu relu maxout dense maxout dense last describe initialization bias see incorporation constraint produces convert pass linearity normalize categorical obtain follow normalization however decrease predict rescale question ask ask question ask question unconditional probability answer decision see incorporate consist constraint must high level question result addition network purpose averaging network differ make slightly include layer three filter dense layer filter layer train minibatch nesterov momentum gradient decrease neural performance perform million improve convergence learning decrease decrease million million first output network ensure layer manually proper bias positive getting region although strategy layer improve uncorrelated computed affine image rotation horizontal unweighted average train see average fashion increase train result average total aspect library allow gpu acceleration effort able automatic simplifie train perform cpu image package training take reproduce win galaxy report perform variant error table metric score galaxy challenge average across transformation average important fast practical prediction generate million combine large impractical public transformation network competition fundamentally capabilitie interpretable fashion classification high participant predict count classification match probability classification fashion cause discard easier interpret agreement galaxy participant affect entropy option entropy minimal entropy answer equally likely select entropy range agreement maximal disagreement maximal condition use probability predict relate evaluation bin answer prediction bin average perform network average graph circle show accuracy agreement classification accuracy confidence show horizontal horizontal option overall indicate level agreement galaxy participant decrease low accuracy achieve perfect agreement arm low agreement near confident useful determine able trust expert could small annotated manually expert greatly expert confident accurate majority would allow largely assessment smoothness question classification accuracy practical q would manual input level annotation dataset analyse evaluation thick horizontal dotted chance number image included indicate able various precision recall score individually score classification list strategy classification least galaxy participant question number occur frequently category exclude galaxy effect attribute generally precision rare rare answer example rare construct smoothness disk yes bar yes yes shaped arc medium arm traditionally neural often treat box informative sometimes try convolutional image layer filter interpret visually layer filter individually three channel separately weight channel filter sensitive sensitive pattern phenomenon train neural look radial image b viewpoint unit convolutional note image still apparent activation except third reason third layer input layer pooling map layer map map map pooling map map map viewpoint upper visualize neuron learn activation type sensitive maxout type visualization clearly discriminate galaxy scale invariance observe direction seem multimodal galaxy activation value image across centre pixel depict turn try replicate participant tend classify image galaxy answer though answer answer galaxy web interface seem feature b loose arm tight look evaluation get idea strength rmse value center rescale varied fairly various way figure motivation additional center galaxy classify round b b b b fine grain architecture exploit galaxy project reliably predict aspect galaxy extraction enable quantitative galaxy scale exploit symmetry art win galaxy challenge win average model collection galaxy source code publicly hardware prediction highly reliable confident grain scale survey perform analysis research future vision scale galaxy paper provide modern though train model effectively improve dataset galaxy annotation recent galaxy take ensure generalize slice dataset number without survey annotation crowdsource possibility raw inspection include structural automate radial symmetry occur interesting architecture deeply small acknowledgement thank van anonymous valuable feedback acknowledge david help galaxy thank capital financial competition classification effort individually grant aid circle thick fill rectangle corner gray draw black corner font department school university st mn usa measure galaxy key requirement formation survey digital survey result availability wide galaxy traditionally mostly inspection train time consume attempt automate able level galaxy project successfully crowdsource strategy answer question unfortunately increase availability galaxy neural galaxy symmetry international annotate galaxy galaxy participant able reproduce consensus confident highly accurate make collection image expert manual greatly reduce large result survey processing galaxy galaxy shape property age formation galaxy course galaxy formation evolution probe physical survey galaxy complicate relationship environment deep survey start take surveys survey result availability million manually impractical individual build automate galaxy reliability scientific galaxy project accelerate crowdsource classification galaxy member public contribute classification web platform entire annotated follow two development automate feasible large primarily although network decade recently research available technique unit regularization possible build network section description reliably annotate galaxy available success galaxy classification size augmentation sharing average become day galaxy classify image deep coverage crowdsource expect expert classification logical necessary convolutional galaxy tailor efficiently exploit image several image classification model international competition base annotated galaxy project first place information enable study galaxy galaxy galaxy challenge discuss convolutional rotation invariance report analyse finally galaxy online crowdsource ask colour galaxy project colour classification project participant ask question determining ask question tune answer total galaxy sign disk smooth disk star disk yes bar centre galaxy prominent central galaxy obvious dominant q odd yes q end completely end arc end galaxy centre tight loose subset question many classify directly pixel convolutional task complex statistic detector hard recognize complex necessary feature selection representation participant challenge literature apart whose galaxy classify galaxy predict classification make galaxy grain task network question data galaxy bar strength star merging etc development crucial galaxy exploit galaxy however besides neural define convolution operation generalize pixel locally invariant affine learn representation convolutional feature invariance encode subsequently learn symmetry ability symmetry input image similar spirit work major effective computational idea discover consist hierarchy subsequent layer abstract build transformation optimize neuron compute combination follow
dropout entropy test epoch prove sgd unbalanced verify despite train sgd balanced unbalanced initialization curve exactly unbalanced considerably sgd case error range plot display observation often fast sgd cifar sgd also implicit regularizer generalize similar role deep dropout figure poor unbalanced except fast generalize path outperform sgd achieve fast implicit analyze look plot epoch provide stepsize momentum term could compare geometry relu suggest geometry geometry beneficial concept deep momentum heuristic enhance believe method plug hope also regularizer perhaps rescale relu path sgd certainly rescale might simplicity choice mirror appropriate seem non convexity acknowledgment nsf award intel discussion cifar cifar cifar cifar mnist cross test title claim choice training rescale sgd descent wise regularizer max regularization easy lead gain deep question generalization issue deep heuristic train training architecture slow open many initialization stepsize momentum inherently tie geometry tie descent norm exp least tie corresponding potential example gradient divergence view regularizer therefore regularization implicit performance align inductive driving geometry geometry geometry desirable enable fast also regularization link appropriate deep focus relu activation incoming edge factor yield invariant seek inspire max norm regularization maximum incoming seem inductive decay max max discuss measure express regularizer sgd regularization rescale classification feedforward network compute direct acyclic input output apply internal unit compute activation depth direct length unit define network hide relu homogeneity rescale change rescaling give node rescale edge weight rescale easy rescale compute f rescale goal minimize step update form descent stochastic sgd mini set rescaling affect rescale call rescale rescale rescale start rescale weight vector separately remain rescale unfortunately invariant opposite invariant update gradient poorly unbalanced network might regularizer relu activation homogeneity effective main relu norm unit rescale regularizer among rescale surprisingly feed forward rescale network efficiently compute forward see vector total path equal weight along regularizer establish path involve nest rescale approximate path rescale interested derive exactly instead update partial derivative follow call normalize direction regularizer let e know path sgd approximate respect regularizer whether path rescale next prove rescale rescaling neither incoming however incoming moreover get ec therefore similar argument path rescale however calculate forward backward step mini setting mini time batch moderate runtime typical mini batch hundred thousand show sgd balanced unbalanced commonly conduct benchmark handwritten digit cifar image
observation provide web survival probability state two state mechanism possibility covariate individual absence individual record individual observe covariate always mass alternatively covariate random correspond covariate presence model generalise covariate observe reporting incorporated omit far fit optimal aic statistic pt shift binomial initially four lead bad omit covariate stationary try two estimating capture assume parameter obtain probability state ii time distribution specify shift negative binomial time geometric markov display remove dependence probability retain though constant survival house absence distribution aic aic htb survival difference survival markov semi predict high contract covariate suggests recently recover duration explanation misclassification model distribution population assume stationary htb inclusion memory specification within incorporate specify attractive increase level increase number previous model state lead notably regard interestingly fit incorrect order markov refer disease status possible biological regard absence follow structure accurately markov crucial efficient order become literature usage almost certainly development modification handle probability computational argue distribution cover capture scenario series capture house word infect immediately individual suffer model markov recovery immediately form include event partially recovery directly event semi state transition observation process need assignment advantage specify capture separate able via penalize extension site status conceptually straightforward curse remain notably semi probability specify probability however non trivial move covariate modeling within development acknowledgement like house thank anonymous comment gray consider traditionally schwarz model first chain though fit biology specify order stay one specify expense significant number schwarz specify semi generally shift distribution expansion apply order result semi tractable markov schwarz increase selection procedure use semi important state spend state feasibility house state keyword markov multi often population uniquely mark record subsequent individual record marked individual lead study particularly individual covariate dynamic individual covariate status disease status life covariate refer state also observe discrete history may history represent observe time state recover individual time schwarz review homogeneity involve spend time individual already process stay one realistic particular state alternatively correspond infected time individual non geometric biological process follow mode distinct exponentially two infeasible memory semi model yet parsimonious specification model spent apply semi markovian shift negative shifted simply distribution process markovian state probability covariate fitting propose general semi markov specification flexible memory markov memory manuscript describe formulation analyse capture house correspond absence section individual covariate let potentially observe initial observe relaxed covariate underlie recently long typically interval occur drop subscript clear live extend hmm capture recovery present conditional essentially decompose state conditional survival modify deal scenario recover recovery decrease individual capture initially calculate sum possible initial mathematically fs calculate hmm define time shift shift flexible specification mixture estimate equally easy implement extend hierarchical survival state formulate semi state essentially mean markovian markovian represent arbitrary semi model expand state advantage hmm hmm applicable markov integer first chain state semi expand k observation role responsible time state semi kk geometric x jk j k zeros different interpret follow survival transition aggregate govern diagonal determine probability specification give length spend assume initial capture dependent state restrict equilibrium initial capture equation assume stationarity state alternatively stability stationarity aggregate q specify analogously definition diagonal entry diagonal aggregate representation approximate semi time spend tail definition make arbitrarily small sufficiently finite geometric ensure appropriately summary approximation arbitrarily accurate choose semi individual simply capture history routine know optimisation confidence inverse bootstrap underlie criterion aic investigate simple markovian semi semi three binomial poisson mean covariate state specify specify simulated individual capture event correctly
obtain theorem detail defer sketch iterate random eq core combine event q sketch sketch q bounding size recursion remain recursion sketch constrain algebra lead hand du subtract term triangle eq vector smoothness lipschitz hessian obtain inequality substitute rearrange find claim convex function book define update whereas proof unconstraine self function classical newton sketch involve fx prove high randomness newton tangent cone cone gaussian sketch sketch newton newton sketch phases magnitude constitute fx fx complete dividing phase phase c next gradient necessarily convexity acknowledgement office national science foundation grant dms mp microsoft fellowship independent optimality feasibility define algebra basic optimality feasibility subtract sketch consequently definition piece finally guarantee e imply construction satisfie g yield fx perform observe fx subtracting factor newton final repeating conclude claim proof lemma accordingly nesterov combine c hessian relation proof newton v lead basic optimal feasible consequently add subtract gx remainder proof broad give twice collection constrain cone tangent cone follow gaussians consequently apply bound inequality cm theorem theorem ex ex berkeley california berkeley department electrical computer science department second newton sketch perform newton self super convergence probability number dependent substantially newton extension equip illustrate program regression generalize program modification newton slow see another tuning size optimal smoothness optimize g twice method convexity conditioning whenever self suitably provably newton reason system pose challenge million common issue approximation quasi newton form hessian gradient computationally example bfgs scheme book disadvantage weak method restriction super paper propose sketch projection projection hadamard sufficient sketch regime low sketch moreover show nd convexity unlike hand strategy consider sub partial prove quadratic condition point arbitrary constraint sketch barrier iteration pre specify begin background classical measure include version constrain illustrative convergence theory devote convergence unconstraine setting additional aspect defer begin section see background convex uniquely eigenvalue f assume modulus initial newton modify globally convergent choosing size backtrack search procedure apply central development book number typical constant give restrict normalize base generally base entry distribute terminology matrix theoretical perspective well sub perspective disadvantage multiplication multiplication consider multiplication perform class hadamard fourier hadamard fourier respectively random vector diagonal multiplication another sketch independent take canonical leverage sub background width randomize gaussian width banach space theory cone sphere width substantially cone calculation background sketch number guarantee constraint motivate sketch current iterate perform take simple root square efficiently instance suitable hessian root newton update precisely sketch isotropic generate iterate recursion realization simple eq intuition isotropic sketch original analyze form additive sketch lead analysis setting dimension newton update dimension lead intuition newton apply lp polytope barrier denote inspection twice version hessian sketch require barrier refer central compare central ordinary newton polytope trial dimension middle sketch sketch sketch black interior step point vertex represent optimum optimum sketch central newton covariate case model count collection covariate glm convex user enforce case least square well set objective function guarantee explore return set sketch must order good behavior sketch geometry term tangent cone fx recall gaussian width sketch tolerance constant square root dimension width case achieved unconstrained substantially problem illustration constrain measure smoothness twice differentiable objective define lipschitz convergence sketch newton initialization bind probability linear convergence specifically rate step total guarantee notable depend sketch illustrative example portfolio linearly program section newton solve different size calculation appendix suffice sketch dimension quantity require theory convergence newton different consistent suffice case newton iterate super sketch logarithmic curvature constant practice theory local take iterate origin seek classical appropriate self appropriate backtracking begin unconstrained discuss sketch update equip exponentially high strong unconstrained optimization convex bound impose include logarithm affine transformation result accurate impose sort analysis scale depend backtrack self fx approximate via convergence newton parameter step nd case close barrier constraint solve many hessian highly instance constraint usual simplex hessian barrier problem structure frequently arise regularizer ill inverse regularizer regularization e norm strategy sketch retain previously provide step include choice line sketch dimension start function parameter matrix f f dimension depend iterate sketch reader recall let barrier implement sketch suffice see self barrier section barrier sketch newton barrier interior particular provide rigorous bad precisely problem form exist unique trace g optimality barrier successively update also newton method lie heart barrier fast newton algorithm provide different dealing sketch apply tolerance sketch backtracking update lead arise particular application barrier sketch g newton sketch sketch upper iteration instead interior solver discuss sketch particular various newton contain barrier sketch flexibility partial strategy sketch primal generalize enforce include enforce sparsity nuclear enforce differentiable norm variation enforce smoothness suppose sketch current iterate computing iterate solve cardinality effective apply homotopy lar optimality start focus choose suitable sketch function suffice sketch size constant us example u r u typical sketch sub newton sketch project intrinsic cone sketch semidefinite program illustration ia semidefinite semi norm establish definite sdp pre regularization encourage relatively rank solution standard self barrier psd barrier sequence hessian barrier hessian sketch first ij exact classical sdp interior solver consider portfolio problem find
constraint measurement identical maximize residual repeat variance small unity small unity estimate constraint residual area indirect element indirect diagonal error covariance analytical solution robust estimating variance difference indirect early dr constraint whereas describe simultaneous covariance flow assume need constraint element identifiable standard provide variance six obtain clearly indicate constraint estimate eqs subspace angle estimate flow difference estimate angle accurate knowledge rmse known show reliable drive obtaining estimate deviation knowledge sd section dr far demonstrate possible identify constraint require accurate estimate measurement identify derive connection pca dr measurement measurement relate pca constraint error covariance apply simultaneously constraint apply dr system note apply pca matrix identify relate measure relate measure reduce project dr measure relate variable balance argument derive eigenvector small rotation reduce constraint estimate identify bring process determined singular describe section hold difficult order great force conversely investigate datum partition small choose subset linear combination go estimate constraint indicate unbiased even constraint great estimate give orthogonal value subspace estimate show overfitte make recommendation conservative overfitte heuristic demonstrate effect deviation value example consider demonstrate concept apply different assume great model five clearly assume incorrect unity equal systematic identifiable value variance unity singular value obtain assume increment unity unity true model observe theory noise assumption nonlinearity consider ambiguity select clearly order choice lead introduce estimate marginally increase order overfitte lead overfitte primarily regard statistical useful denoise pca method steady state entirely measurement iterative pca covariance simultaneously necessary extract exploit constraint perspective dr integration fig historical process dr technology fellowship cccc control institute technology dr pca high preprocessing technique develop primary relationship lead unified dr collaborative datum extend partially incorporate partial principal derive consistent estimate develop technique software package plus software package chemical benefit apply dr estimate material typically dr constraint derive first principle material correlation specify additionally covariance derive historical multivariate processing popular principal method primarily develop regressor variable chemical engineering monitor diagnosis regard multivariate statistical technique pca author pca relationship simultaneously process interpretation apply purely measure difficult measurement technique matrix require rigorous diagnosis pca diagnosis modification incorporate partial impact incorrectly estimate model actual constraint application pca tool organize section introduce dr identification section partially matrix discuss criterion model conclude process section dr steady discuss measure know define exploit linearly constrain process operate water stream measure flow operate sample draw steady state steady relationship q label constraint subspace error instant error denote expectation minimize regard error estimate using normally distribute measure steady operate dr steady apply independently pca operating apply arrange value give follow illustrate flow flow consist six balance write order flow six stream measure noisy simulate noise rate variable base variable simulate steady normally fluctuation add flow flow eq steady normally random value steady steady base sde dr apply rmse computed table rmse report rmse indicate variable rmse dr rmse feasibility measure refer system partially dr provide value estimate dr variable uniquely estimate observable redundant observable dr describe variable label label matrix construct get constraint variable define q unique estimate obtain redundant observable algebraic completely book redundant dr measure redundancy visualization ease flow flow stream measure add connect stream flow contain whose flow flow balance reduce flow subject reduced technique measurement stream eliminate non flow original cycle flow rmse redundant accurate improvement flow compare obtained reduce dr component view describe steady pca linear uncorrelated principal variance pc variable hence uncorrelated variance pc pc eigenvector obtain datum scale eigenvalue orthonormal eigenvector remain diagonal whose diagonal element root eigenvector order magnitude pc give variance pc pc heuristic pc retain look sharp eigenvalue heuristic book alternatively denoise technique retain pc viewpoint importance article discover underlying variable identify author explore analyze measured covariance matrix generalization matrix simultaneously section observation draw steady datum lie subspace relate apply retain pc equal give orthogonality pc eigenvalue process constraint focus eigenvector retain identification concern small main constraint variable lie correspond orthogonal square identify subspace lie square identify matrix prove n sample matrix steady due measurement n make imply systematic true value true assumption steady linearly steady eigenvalue recommend operating eigenvalue orthogonal constraint corresponding n row matrix furthermore knowledge examine small know normally use result conclude error normally distribute identical variance mutually derive asymptotically unbiased note differ singular substituting estimate give orthonormal eq orthonormal eigenvector q identify obtain identify purely error different also purely estimate satisfy estimate closely pca dr may rotation pca differ desire constraint precede identical constraint pca furthermore identically distribute may note dr identification transform appropriate apply triangular transform let transform n apply transform estimate corresponding estimate constraint singular value part relate constraint datum equal provide systematic size numerically check small unity error constraint constraint assume error variance transform order singular four nearly original rmse rmse obtain pca
operate would main strong use lstm unit gradient descent initialization gradient training pair observe overfitte e understand purely intersection hull coverage self intersection fail simply report present area net mistake come aligned mistake hull input affect true sequence processing step update convex overcome problem describe whole modification focus point side create net lstm attention key inherently variable length half length uniformly effective single able degradation even satisfactory learn simple lstm attention give lstm fail attention lstm net lstm net net hull fact triangle coverage triangle case permutation net accuracy middle symmetric hard finding implement unclear would capacity solely feasible small importance good provide reasonable show unlike hull decoder unconstraine net sometimes decide produce row show optimal feasible net try beyond seem able could train paper describe architecture token correspond net learn work sized yield sized something sequence attention previously memory base output location neural assumption try sort combinatorial acknowledgment le thank help final google berkeley google brain introduce architecture conditional position address sequence number target input sort sized belong mechanism attention attention decoder use select member call net alone net us dictionary learn generalize beyond hope encourage broad discrete recurrent neural network rnn function three decade limit available introduce paradigm remove rnn decoder contextual information possible rnn domain art core language processing parse execute output limitation input output satisfactory model approximate purely fashion input rnn code generate rnn output feed generate content attention equal contribution architecture net effective deal fundamental length dictionary softmax distribution net distinct trivial algorithmic involve geometry generalize net learn competitive small drive approach approximate baseline section sequence compute I pn learn maximize example short p ii end input reach symbol termination independence assumption rnns symbol rnn typically lastly state feed step recurrent note become perform significantly well sequence hull output input nevertheless simple reduction describe modification us combinatorial dictionary softmax size attention follow u w j softmax propagate copy correspond attention mechanism note target problem output could rnn map back prediction long video follow protocol coordinate convex hull case sequence associate set hull ht hull c hull token understand difficulty combinatorial drive approach solution number consider position special represent plane every empty exact solution example triple token
wide ill probably various method five close typical class exploratory contour depth remove depth report assume define boundary neighbor outli locally tend typical outer relative extend near many influence several subsequent work come neighborhood isolate early detection outli score mahalanobis instance locate fall unified linearization method randomize pruning resolution suffer curse model increase suitable great handle extreme typical invariant outli detection datum low subspace outli reduction process help explore vast solve unconditional across attribute increasingly recent year contextual outlier song et generative response outlier although share similarity song al representation mapping parameter exploit decomposition make large require expectation step limit outli testing utilize piecewise probability individual improve significant sensitive outlier outlier joint infeasible section briefly learn assume free outlier unseen include outlier phase probabilistic multivariate instance unlikely detail outli detection outli accurate probabilistic relate define different datum study extensively multivariate able automatically assign tag keyword document posteriori purpose outli assess classification second equally assignment output assume response separately suffice world among important build among define via univariate product parent output variable directly depend model relation among response variable representation generalize component empty decomposition output circular view status work many like conditional vector probabilistic naive logistic building use estimation py regression use section present apply testing identify conditional phase unseen phase like score metric towards pseudo likelihood decomposable structure estimate likely unlikely dimensional response multivariate along set scoring metric transform test previous outlier exist outli method sensitive datum pattern utilize identify model propose piecewise individual useful evaluation outlier unsupervise nature outlier dataset assumption data process assign probable observation small portion outlier comparison fraction influence building model create conduct consist part realistic context eight outli six competitive adjust number wrong outli three show multi include video image label text categorization clinical patient consist context table label configuration c medical clinical biology plausible context find everywhere example label irrelevant tag clinical diagnosis patient inaccurate diagnosis gene sequence simulate outlier follow sized c medical multivariate outli rd run evaluate datum refer use use norm svm outli fair comparison radial fix penalize logistic cross lastly metric individual response input train auc curve true positive auc high table mean equivalent paired level bold follow level statistically superior mark produce score outperform four outperform produce well rest local understand condition information handle unconditional attribute efficacy due analyze benefit x different group score gray show bar significant good work even method testing actually improve dimension e sparse instance method three video annotation categorization labeling table characteristic control adjust clinical response space outli auc pr auc auc pr pr curve conservative auc sensitivity outli detection pr method axis indicate auc pr axis indicate use color dot simply dot pr superior general intuitively dimension hard outlier figure start bottom plot gradually increase usually auc dimension increase obvious auc rapidly baseline invariant dimension exploit posterior help outli special variable review exist outli detection multi method outli multivariate transform unconditional solve effectively accordingly five score usa outlier expect particularly define combination paper outcome space transform detection outli unconditional outlier rely classification probability probability output score use detect outlier multiple outli show artificial expert outli datum statistic anomaly useful annotation preprocessing help remove noisy irrelevant utilize beneficial surveillance disease clinical monitoring despite huge exist detect unconditional outlier ordinary response label unconditional method easily positive negative want detect suppose unconditional outli detection annotation due patient rare correct incorrectly image modern assign image image label outlier become one moderately respect patient include become consider child unconditional become apparent detection seek response outcome unconditional outli detection seek instance detection dimensional output pattern multivariate output correspond fall keyword incorrect detection particularly challenge pattern detect build classifier use discriminative briefly observe space express treat account output output method define separate due procedure together outli instance dimension score statistic outlier output image process label label randomly chance dimension network attack keep decompose covering help
preference threshold literature grouping divide classification sorting depend maker dm sorting assigning predefine formally predefine order category set criterion criterion relation category profile limit category characterize assignment alternative category criterion profile one four partial criterion criterion partial index profile threshold case weight coefficient computation partial index know profile preference threshold cut lambda permit assignment alternative relation two optimistic assignment procedure profile category main parameter preference follow phase formulate input application take american census approximate record synthetic census ds report research cause difficulty analysis facilitate record miss ability generalize play important role alternative importance lambda lambda increase consider substantially lambda parameter case namely link situation dm interesting preference machine propose record linkage matching criterion solve record application start initial demonstrate application good show performance experiment confirm record linkage preprocesse matching say measure well scheme search lc machine ordinal classification propose linkage preliminary experimental show correctly identify machine scientific concerned allow computer learn intend recognize make closely relate field mining recognition artificial intelligence theoretical computer machine learn type supervise generate map recognize supervise unsupervised combined impact environment translate guide principal statistical linkage matching network short propose record time linkage answer challenge machine record criterion linkage provide classification performance application light development context pair record heterogeneous maintain distance linkage describe record linkage use record section preliminary simulate remark conclusion generally speak record unique record source linkage methodology record two file file record linkage linkage solution record file identical match linkage file information record linkage big answer well understand suppose want age suppose follow cccc name age road read furthermore contain address age h st b contain unit probably match modern record linkage begin et odd ratio rule match year yield computer idea computer sciences operational research mathematical linkage theory demonstrate optimality crucial probability file formally file probability agreement comparison set match classify pair analyst decision three
specialized regret go bold face let finite space begin concave domain observe nominal convenience call definition agent action ta tf ts make identically unknown allow decomposable aggregate objective constraint handle define policy randomize policy policy context appear context nonlinear strong compete policy equivalent similarly define p rp vx rp expect reward enumeration impractical purpose access employ contextual bandit previous max policy max since optimize frank wolfe repeatedly ft consistent terminology average amount regard special refer constraint aim average nontrivial problem study call contextual bandit break component vector maximize vector budget form never allow budget nothing get take round need certain lipschitz detail bind call bound term optimum need scale regret feasibility problem reward ta ts measure algorithm share bandit change necessary constraint completeness allow ta select op algorithm policy class proceed length begin compute right policy ideally concentrate order probability large enable computable define bandit technical dealing constraint describe problem denote action take reward observe take select action way mixed choose complete every v straightforward unbiased reward empirically sp define regret obtain q op empirically get finding return smoothed assign minimum op regret small constraint one compete compete constraint derive implementation op call average due achieve hold trivially come mixed sketch give true second op high give op every empirical regret mixed section maximize ensure feasibility error alternate tradeoff however appear regret suffice regret well budget sufficient give rest intuitively optimization lemma appendix constructive small variable f rp instead construct idea algorithm section new combine op recall equation factor norm policy good policy estimate achieve sketch step use op lemma additional parameter new actual constraint op regret regret amount bounding term rp rp rp rp rp rp aim maximize ensure return however budget stop budget resource fully violate ensure happen early compete ta ta add regret would objective capture quantity suffice easy precise property lp similar general discuss property applying require great equal suffice outcome play uniformly general problem early lipschitz sure budget violate enough constant set aside entire follow round pure exploration aside amount run time full actual component budget essentially accounting budget appendix real v e version chernoff support least b feasibility require solve op op nontrivial sampling input mix op current minimum probability proper default pick schedule allow ta op op op describe main easy combination policy eq coefficient op tp convexity op version feasible feasible early side q jensen inequality trivially feasible op op problem solve descent assign weight shorthand op describe instead h e k p use loop start call loop initially compute ix remain solving sequence concave elsewhere denote elsewhere p maximize number time apply section constant since prove represent specify one suffice show mix need another choose compactly schedule quick place increase p record history round v statement hold epoch round epoch choice lemma get first epoch ix furthermore definition reward let mi sum union bind choice note e mi v mi tp tp definition rp p j side tp rp tp rp tp assume event round definition v v get substitute get round choice event epoch tp k universal constant fix immediately equation event observation op constant assume event tp tp induction epoch epoch triangle distribution tp rp tp rp tp tp rp tp rp tp rp tp follow fix epoch round epoch distribution tp tp inductive always inductive use fact therefore whether part epoch tp k inductive hypothesis simplify matter whether combine gives ensure inductive ready hold epoch follow mt r jensen trivially epoch round epoch tp q tp k c substitute next show th component recall easy see hoeffding martingale sequence apply triangle far get regret dt trivial dt convexity follow trivially always suppose rp satisfy exist rp lagrangian program duality get l l f optimal variable program statement concave gradient small hold epoch epoch policy tp tp induction case rp rp tp rp z rp rp tp tp rp rp rp substitute side z dp fp z dp rp z tp p tp rp rp tp tp inductive epoch round fix epoch policy event step ready theorem hold suffice whenever projection appendix linear mt mt jensen rp ft r rp rp inequality jensen dt rp use condition p tp apply bind detailed bind equation bound condition rp ft ta l otherwise budget p bp c contradiction regret algorithm loss regret regret order consumption round hold arm estimate tp da pick policy support observe eq component solve relaxed policy achieve maximum
behind entity compositional worth domain name contain average always cc entity randomly average word vector initialize work everything bilinear entity relation ideally simple horizontal translation traversal red expect parent square dotted red large compositional entity think intuition incorrect reveal query error along edge training encourage incorrect however close discrepancy path phenomenon empirically well path traversal operation path entity rank confirm relation compositional training precision divide path length box show compositional decrease precision decrease little irrelevant knowledge completion embedding reasoning knowledge compositional technique improvement help compositional compositional knowledge completion answer reduce incorporate compositional incorporate path approach use regularization walk social introduce answer query incomplete technique vector show compositional answer completion key path represent perform believe great stanford university stanford stanford cs stanford edu compositional question however fact query suffer error compositional training improve ability query compositional training act novel regularization reliably base basis reasoning question answer know suffer incomplete coverage distant entity elegant space control force fact reason compositional hope parent ability compositional query ask generalize propose path entity traversal drive vector transformation present basis broad high implement edge traversal interpretation encourage modeling include bilinear three two finding first compositional answer substantially base somewhat surprisingly answer query compositional regularization exist formal task answer entity relation knowledge graph triple triple answer query entity reach eq evaluation detail candidate answer incorrect answer completion candidate answer query motivate present technique compositional describe illustrate experiment entity relation bilinear likely use motivate technique adjacency matrix entry entity easy count relation positive q interpret recursively begin entity apply traversal new reach point traversal apply traversal much learn bilinear compositional scoring q score model every perfectly present optimize naturally suggest compositional training membership traversal explicitly encourage notion vector query different vector sequence another traversal transformation preserve traversal objective completion query path length compositional substantially answer base completion insight one completion scoring vector membership operator traversal handle visualize compositional bilinear diag bilinear relation view naturally matrix neural reasoning optimize initialize size validate query length form relation bilinear initialize bilinear diag inverse initialize gaussians entry bilinear yield performance code consist section reasoning subset exhibit subset bipartite person source entity contain relation perfectly correlate inverse relation edge easy inverse triple exclude query extremely amount training overfitte base follow sample entity mm relation current next via practice repeat except plus dataset remove appear training statistic train show compositional training substantial edge demonstrate inferior outperform state art numerous use query include answer rank normalize rank account candidate quantile answer quantile range average quantile normalization important predict gender query receive rank mean quantile several query match trivial answer query exclude query cc ccc ccc red red h vs compositional quantile percentage cc c single vs compositional query exclude compositional improve model show surprisingly compositional improve completion across bilinear term bilinear deep look query divide query subset source path never see training query explicit traversal subset
good web multiplier kkt theorem axiom rgb liu rapidly crowdsource crowdsource crowdsource worker minimax conditional truth unique labeling worker item difficulty measurement principle principle variety multiclass ordinal crowdsource minimax world costly considerable research semi recent year crowdsource service amazon associate collect domain drop dramatically enable large low cost provide worker lack worker overcome worker manner voting assumption majority worker good obviously assumption reflect imagine worker confusion worker correspond maximize worker stay item many item difficult worker may minimax conditional crowdsource worker ability account item ignore reduce measurement measurement conditional aggregate collect crowd derive regularize minimax overfitting generate label propose extend label present ascent empirical crowdsourcing report present principle aggregate multiclass primal show minimize kullback leibler divergence index assign item denote belong label true true worker approach build upon tensor tensor refer confusion tensor observe item tensor refer class worker worker worker item worker worker row match row assume observe unknown attack simple worker item enforce count worker enforce confusion item counterpart illustration c item item worker worker item item entry label class unknown intuitively understand entropy worker connect lagrange multiplier tucker kkt combine constraint yield q although mathematical understand intuitively worker worker worker worker confusion confusion substitute labeling equation dual minimax obvious extend define stay class true equivalent sketch moreover deterministic crowdsourcing collect collect limited count fluctuation probabilistic label item uniform either ask item overfitte formulate replace match approximate fluctuation generate label formally minimax true label subject relaxed worker slack fluctuation slack positive fluctuation normally distribute central motivate entropy objective generate regard deviation labeling turn log jensen inequality maximize conditional restrict slack say answer worker law correct answer note percentage worker additional problem express somewhat natural labeling equation consequence objective involve also worker independent involve comparison worker mathematically objective principle event formally item item formally obvious worker worker choose nonnegative probabilistic labeling equation immediately instead requirement result requirement multiclass ordinal ordinal web product since ordinal special multiclass label previous section worker multiclass labeling summarize adjacency assumption formulate introduce adjacent observation ray breast cancer cancer screening rate worker formulation parameterize worker relation ordinal consecutive integer eq subject constraint exclude hold draw shape center true observe give ordinal us equation ordinal label label section trivial check ordinal label indirect value ordinal label set choose label region partition example table define constraint sum equation similarly define set constraint item discussion constraint restrictive result disjoint degenerate thus explain ordinal write count item belong worker outcome label reference observe outcome enforce kind dimension match counterpart equation enforce count kind dimension empirical counterpart partition four write multiplier multiclass equation worker item confusion structure ordinal class expect ordinal write item constraint choose problem q worker define first event principle worker independent independent arbitrary worker reach ordinal labeling describe simple coordinate ascent minimax model regularize either ordinal ascent expectation maximization aggregate vote iteration give current estimate confusion estimate label close experiment multiclass label gradient ordinal worth unnecessary exact intermediate ascent suffice reach initialize repeat regularization true subset refer validation choose choose fold partition crowd label finite confusion item plug leave average likelihood go choice large average parameter simplify gain motivate square magnitude dramatically number super linearly scale relate generative ability confusion represent worker label worker item probabilistic item q generalize jointly estimate model task write worker usually coin coin assume propose coin achieves rate fix essentially belief propagation update assume worker equal coin achieve achieve spectral impose beta confusion true generate logistic full include belief test illustrate response model item location trait person ability variable response incorrect item mathematically difficulty response item trait correct special theory measurement model adapt integer scale let person location refer rating rating latent minor label item labeling difficulty easy worker confusion generalize multiclass worker cumulative standardized normal worker z unobserved parameter think crowdsource reader make crowdsource entropy propose apply texture field multiplicative mechanism crowdsource worker answer question skip error datum crowdsource publicly detail contain image crowdsource worker worker label image error worker amazon student price bin create student decide bin student systematically bias tend present two sentence ask check hypothesis sentence infer sentence pair annotation ask
wolfe probabilistic variation capture spirit wolfe derivative accept I direction strong wolfe condition write exactly bound strong wolfe replace limit overhead black box inner loop precede introduce six far decision eliminate search noise level runtime objective form one problematic learning rate wolfe threshold problematic annealing threshold new probabilistic motivate value free upon back envelope computation achieve compete empirically rarely observe either clearly condition scale eliminate scale ensure range digit line search take division cause seem notable deviation exchangeable variance batch overhead approach already statistic square beginning search amount finally empirical expensive summation simple overhead batch run average batch necessarily converge fast estimate separately capture project line search along weight demonstrate step size mis scale fit line propagate iteration initial initial search next search line search search put extremely loose control neural net nonlinearity mnist use cifar dimensional search deal univariate subproblem task empirical evaluation aside computation independent practice optimal effectively remove exploratory rate tuning mean std repetition central nuisance sgd potentially schedule decrease theoretically decay rate empirically decay often exploratory schedule find cifar respectively first sgd sgd decay schedule probabilistic fig epoch learn error base start quickly reaching report kind architecture without decay outperform search optimal one exploratory come lead start single let overhead objective ms mnist instance closely search search first thus optimally additional plot raw vs error dataset rich picture show choose tune progress use elaborate analytical convergence search widely accept noisy design combine principle idea user complementary combine evaluation reasonable quickly optimal matlab implementation available publication article additional network architecture cifar mnist plot evaluate batch sgd constant decay search enhance sgd keep line smoothed unstable black instability control regardless reach development step vanish size costly remove dash decay dash dotted line search initialize vary search accept course optimization nontrivial level gradient sgd instance accepted gradient noise level line smoothed magnitude search converge indeed mnist cifar dash green horizontal decrease search start slowly appear simple nontrivial objective pick cause jump minima already line performance drastically figure show relation encounter course symbol color little cifar instance circle behaviour sgd suggest line sgd typically prevent truly beneficial sgd search ensure stability efficiency formulate construct probabilistic search combine structure notion surrogate belief wolfe condition effectively remove rate highly multivariate gradient batch neural network regression arise exchangeable loss I limit distribute despite popularity inefficient main even noise sgd well individual step former adapting address meta newton direction auxiliary et adapt none size grow individual ht optimization search direction finds follow acceptable insufficient decrease point gradient exclude curvature wolfe strong free subroutine conjugate bfgs free search deterministic optimization easily small become space efficiency construct objective cite search change direction adapt cite present search essence explore reach operate univariate along scalar gradient xt search search likelihood eq regard three surrogate formulation wolfe termination allow discrete point evaluation subroutine requirement fulfil wiener process zero gaussian semi irrelevant regard give whose cubic generalize spline ill case observation crucial typically generic inference gram measure
result good semantic hashing net code autoencoder preserve distance encourage quantization autoencoder fast note binary autoencoder stack restrict boltzmann machine rbm binary output decoder autoencoder rbms differentiable involve normalization different nonsmooth binary autoencoder combinatorial code optimization rbms approach ignore approximated relaxation truncation possibly hash code auxiliary coordinate able break optimize parallelism armed optimization efficient particularly encouraging give autoencoder objective intuitive code nonlinear hash mapping straightforward much towards hash function mac framework nsf award helpful simply involve dual variable global optimizer optimizer small vector additional optimality optimizer possible tight condition complicate compute minimizer relax global minimizer solution relaxed optimizer relax global optimizer binary particularly interested computationally comparable objective besides arithmetic evaluate sufficient fast global stop relaxed minimizer training code cm thm thm thm thm hash binary autoencoder electrical computer science university california false attractive map low hash constraint autoencoder reconstruct code auxiliary optimization easy decoder optimize precision recall code hashing problem hash hash fast database space use factor hardware operation false positive negative retrieve verify ground still year code try capture notion thing reduction note binary real optimizing learn hash run reduction procedure filter pca thresholded minimize thresholded projection jointly mapping threshold optimize code binary show joint binary actually carry reasonably general solve complexity focus mac describe derive mac carefully binary carry function several reconstruction entropy show optimize autoencoder mac nonlinear sophisticated function hash basic locality sensitive hashing lsh lsh outperform dependent specific give unsupervised define either achieve space example essentially binary relax thresholded code variation eigenfunction obtain hash classifier spectral label optimize instead embed parametric thresholded threshold relax try nature code threshold subset number binary code couple learn hash code close binary autoencoder quantization fast hashing obtain code apply pca seek rotation make code latter base find continuous laplacian spectral continuous local minimum optimization function thresholded pca code optimize binary autoencoder relax code semantic hashing use autoencoder consist stack rbms threshold encoder round encoder backpropagation forward ignore round broad composition encoder effort apply mostly encoder pca code vector bit write tt act bit n code layer nonsmooth difficult exist nearly everywhere call later optimize pattern hash hash fit code bit filter ba optimize take separable break nest functional equality penalize coordinate minimization introduce I equality note binary augment increase eventually function optimally reconstruct individual reasonably still parallelism describe result step encoder classification decoder one operator iteration decoder decoder reduce optimize step stop convergent differentiable objective binary problem instead vice valid choice solve mac autoencoder stop change minimize n let prove even set function exact set ba independently ba end r ba lb minimizer limit set small ba ba g iterate stop parameter regression necessary equivalently simplicity multiplication binary hamming objective number misclassifie separate label classifier perceptron solve closely misclassifie optimize margin plus slack code surrogate make local optima generalize well maximum penalty constraint linear optimum warm initialize previous note decoder train independent np practical intensive computation parallel spend make good cholesky square precision error triangular l minima since depend speedup triangular e henceforth enumeration iterate optimize form early initialization far binary hash population intuitively hash code use equally preferable bit distinct hash ideally bit bit vector spc integer code normalize work large code usage measure real hence code use number available number distribution code entropy large available code entropy induce dataset hash preserve neighbor crucial necessarily precision recall code easy code necessarily pick hash half half half generate decision cut impractical hyperplane per internal distribution axis thresholded principal long hyperplane contain half hyperplane orthogonal thresholded generally see generally projection code projection approximately gaussian mind useful hash important ground size code indeed binary reconstruction retrieval cifar image ignore wide contain extract subset contain image sift image sift feature use retrieve neighbor either ham hamming evaluate minimize ba hash runtime code reconstruction b precision hamming purely ba mac approach pca find wide wide result ba dominate reconstruction precision expect worst hash ba competitive method mac inexact use warm relaxed qp fig cifar criterion objective surprisingly warm start ba fig warm solid line relax early warm good relaxed warm relaxed initialization result optima almost binary size eventually converge almost warm likewise fig alternate enumeration vary course runtime middle inexact step model learn remain unless enumeration reconstruction relax bl bl bl bl bl bl bl bl bl minutes runtime mac optimization ba step optimization warm vs initialization cifar matlab loop iteration processor observe scale particular parallelization rough ba cifar bit speedup nonconvex result code mac guarantee improve leave unchanged validation tends generally schedule double skip past occur schedule well seem thresholded locality sensitive hashing hash spherical hash use sophisticated near ba hash minimize use ba depend neighbor report small ground query retrieve near hamming neighbor neighbor curve recall result cifar retrieve image size depend
modality maximally space wise co vector space dnn ordinary follow learn representation produce fuse predict leave parent contextual leave parent category share bilinear across leave account audio visual fine grain contextual think share pool operation formally overlap joint propagation rule bilinear dnn share leave leave parent set bilinear softmax sharing keep error bilinear softmax layer bilinear softmax audio propagate term pass network bilinear influence give completeness diag diag equation keep control frobenius norm u j bilinear softmax sharing rule factor architecture initialize vocabulary architecture bilinear architecture audio visual fuse architecture dnn architecture improve average posterior three gain posterior bilinear deep multimodal audio modality demonstrate clean acoustic speech bilinear dnn audio modality technology di mit edu com com present multimodal speech modality automatic deep train separately fuse space deep network audio alone achieve phone clean vocabulary audio model visual channel phone second present deep network architecture use bilinear softmax modality bilinear network significant phone yielding per speech pose often carry information effect multiple party human read order enhance recognition clean speech help automatic audio video train build visual audio information enhance work visual show visual indeed scenario multimodal multimodal find modality interaction framework dnn audio restrict framework deep learning modality validate section training audio consider joint representation phone bilinear modality network bilinear posterior well clean speech organize vocabulary extraction visual fusion video video add central frame visual audio dependent cluster refer phone multimodal second feature note classification canonical modality would model audio feature objective stochastic joint visual hide network keep deep fused final layer fuse achieve alone achieve visual carry build fuse substantial audio visual audio visual task interestingly per h audio alone alone dnn softmax separately modality bilinear dnn dnn linearity sigmoid unit shown consider simplicity exposure assume
particularly rnn cause composition property dark transfer know include hard weight balance relative soft target respectively hard sample conventional force teacher transfer study dnn rnn target dnn risk fit fitting largely model soft hard target reasonable target refine target transfer conventional fine target easy however information soft refinement informally firstly discriminative dark conventional training approach restrict boltzmann rbm auto simple stack dark function discriminative though structure possess totally orient train layer pre train complex clear focus pre view discuss sample learn one dark regard train fine tuning interestingly regularization view pre closely training essentially place reach discuss train acoustic experiment noisy profile standard largely gpu dnn start construct plus train provide frame window lda dnn dnn architecture involve layer layer equal gmm entropy sgd dark transfer dnn teacher rnn rnn lstm structure dnn empirically fa speech train tr fa fa target train dark transfer dark transfer target employ soft soft target rnn target role hard soft target role pre regularization empirically htb fa dnn hard rnn rnn soft rnn soft rnn hard rnn dnn baseline much devoted momentum rnn inferior fa rnn additionally interpret suitable report rnns rnn well largely solve dark rnn system obtain dnn dnn rnn arrange rnn fa learning accuracy close set compare dnn baseline indicate soft fa well cv improve sense combine target pre fa improve confirm hypothesis role rnn confirm two dark knowledge rnn bad cv confirm high generalization dark model knowledge pre involve complex train deep rnns investigation probabilistic edu cn recurrent rnn acoustic automatic successful rnn highly research dark model teacher idea model use transfer target simplify combine rnn without scheme hessian dark gain success powerful rnn rnns rnns long speech signal back inefficient difficulty dependency cause nonlinearity vanish address architecture memory successfully architecture odd recently variant hessian successfully rnn problem computation demand recent momentum rnn reach performance address difficulty rnn e optimal e g simple powerful rnns work logit dark involve rich target training research focus term complex ensemble model employ dnn large employ transfer train research try teacher treat teacher smoothed step fact extend rnn task database improve organize dnn dark basically train dnn play teacher target posterior probability identity deterministic hard target target temperature formulate introduction target training rank information class reflect additionally apply class additional teacher additionally information class soft informative knowledge hence need appropriately task dark soft boosting also complex soft lead smooth objective function compare intuitively soft arbitrary hard
simplicity chernoff conditioning give eq take probability event remove submatrix line exponentially entry part split diagonal concentrate subgaussian coordinate hoeffde subgaussian equality complete estimate subgaussian random identically early easily kernel upper close low bound generalization formalize sensitive row concatenation estimate hamming attribute private estimate attribute achieve privacy operate mechanism every ridge regressor attribute achieve attribute operate time subgaussian individual vector subgaussian weak subgaussian part dominate computation distortion privacy becomes imply comes draw match proposition comparable thereby hold high natural minimization also lp program context handle complex z minimization acknowledgement grateful helpful discussion theorem pt author method extremely popular technique use many analysis practical performance develop implicit function return feature kernel evaluate kernel matrix entry dimensional main tight commonly application bound need privacy privacy definition regression perform outperform domain privacy distortion technique assume realistic scenario bound release restrictive recent year development kernel practical range multiclass ranking outperform heart notion value power define could nonlinear kernel evaluate introduction nonlinearity optimization ingredient build formally kernel ni jk j asymptotic kernel form establish matrix kernel require matrix property recent mostly focused bound input draw distribution satisfy subgaussian spectral construct roughly spectral high assumption subgaussian kernel correlate exist directly overcome argument combine subgaussian norm matrix role rademacher analyze conjugate gradient value establish application arise database want attribute code age literature partly medical operate linear attribute approximation attack privacy loose goal privacy record assume know public mechanism attribute private consistently reconstruct attribute privacy mechanism add privacy mechanism attribute private setting ridge privacy magnitude imply attack attribute attack release unlike attack analyse goal property matrix focus asymptotic infinity let symmetric random kernel f df limit matrix asymptotically kernel limit recently investigate development traditional area geometric compressed sensing understanding utilize release global property database privacy information database privacy notion privacy tailor differential roughly outcome lot differentially various application objective release follow complementary seek distortion release sensitive attack attack reconstruct accurate attack privacy differential privacy translate distortion reconstruction attack first consider context mechanism random database direction close privacy bound marginal linear parameter table per attribute non notation bound attack fraction arbitrarily attribute private show extend release problem point look singular however attack analyse subgaussian analysis dimensional subgaussian thereby applicable subgaussian analysis hamming letter transpose euclidean denote frobenius identity sphere center origin z independent brief introduction theory kernel book let empty set hilbert space reproduce hilbert space df refer allow computation know explicitly trick allow solve note infinite mining rank kernel pa ab polynomial dimension frequently radial x control locality indicate vice versa extension formally subgaussian subgaussian subgaussian say subgaussian dimensional marginal subgaussian subgaussian random variable arise naturally analysis spherical variable fix dimension subgaussian convex isotropic isotropic isotropic subgaussian subgaussian subgaussian random subgaussian random constant use net subset net cardinality follow standard x large kernel triplet entry additionally subgaussian need bind vector random center subgaussian dc subgaussian take net claim c establish spectral random subgaussian denote obtain kernel diagonal independent entry deal dependence entry provide follow center subgaussian p pn dc split diagonal let represent part norm bound substituting assume variable subgaussian fix j j whether easily simplify lemma c take union
win heavily study optimal rely social agent choose choose well whether player question learn order exploration range agent bandit characterize payoff strategy bandit agent plane intelligence payoff laboratory multiple parameter choose social intelligence interactive game game player aim maximize payoff round agent square integer write payoff bandit round store piece time obtain obtain information piece bandit move exploitation bandit provide bandit exploit agent previous round bandit randomly agent information obtain old bandit exploit bandit bandit necessarily receive among bandit intuitively upper hold agent observe payoff agent learn agent multiply represent player game agent game advance sequential round agent denote player store three piece bandit information player choose round start player observe agent environment website agent round store piece bandit interface show observe bandit would payoff bandit information large bandit obtain good bandit good report room student mainly school science subject room brief experiment sign experiment document game start round min subject subject among subject reward top subject environment could environment approximately addition relate subject ask three game environment randomly experimental experiment subject game environment optimal mean could player know know round denote exploit payoff choice estimate player game value remain assume piece information player information exploit expect bandit quantity expect bandit summing divide expect round payoff obtain assume player continue new payoff round front vanish payoff one zero otherwise obtain payoff payoff exploit large might optimal expect payoff per obtain bandit payoff round change account age bandit compare determine action maximum payoff bandit choose round conversely hold expect payoff choice exploit player highest choose choose learn condition player maximum payoff round payoff expect payoff per pi agent bandit exploit four simultaneously round strategy simplicity agent term tb thick solid boundary dot beyond plane thick boundary lower leave I great delay bandit might dotted boundary comparable payoff exploit good bandit exploitation trade social dotted noise social trade bandit previously small learn instead try good bandit exploit exceed intelligence intelligence intelligence subject choice observe intelligence human subject comment intelligence choose good observe exceed intelligence advantage estimate perform experiment reasoning player everything relate game experimental human subject analysis subject calculate environment divide round average payoff subject represent subject pi agent cc subject great expect intelligence fact h pi increase pi value pi pi obtain bandit however near intelligence subject could see agent find change provide bandit payoff payoff pi increase pi depend bandit great bandit hand obtain bandit payoff exceed agent bandit big performance pi depend large great intelligence payoff payoff low region pi good succeed obtain bandit observe intelligence variation examine make strategy linear multiple performance predictor kind proportion move round number round predictor agent program subject notice frequency former predictor include predictor regression average payoff round linear htb intercept n negative suggest effect observe rather two trade develop interactive player option bandit exploit environment scope exploration making observe payoff optimal restriction knowledge exploit observe plane strategy intelligence intelligence intelligence plane optimal I experiment intelligence effect intelligence subject proportion factor proportion agent make believe human factor mind elaborate make basis agent option round maximize
entry note I merge discretize well gaussian gaussian quantify amount distribution suppose parallel axis induction handle proposition general th lemma induction recall structural integer let variance independent positive take confidence use k guarantee return round describe original preserve minimum shift shift determine try integer express onto consider hypothesis select require take access om output stage distribution differently use lemma sample kolmogorov distance discretize tell result lemma rescaling get within true tv tv z triangle subroutine select remark fact definition pt pt mit mit mit sum independent multinomial discretized multinomial applying requirement factor minimum eigenvalue distance significantly cover term dependence particular result multinomial nonparametric multidimensional family ball perhaps characteristic bias towards bin mathematically distribution support basis specify understanding question via gaussian behave discretize dimensional exhibit variation hard multi limiting quantify finite dimensional assign general multi matrix typically tend provide bind distance discretize establish old use stein multinomial summary distribution discretized random covariance desire interestingly direction arbitrarily sparse add direction approximate provide intuition mean nash equilibria player share say player depend action player utility otherwise show total f approximation equilibria whose cover intuitively nash profile player affect payoff anonymous cover interest interesting feature discretization cover exponential discretization polynomial provide asymptotically space equilibrium anonymous game polynomial cover consequence theorem improve cover sample motivated application cover cover make cover contain form see least count count probability namely polynomial obtain namely polynomial view directly learn ok kn generalize poisson sample run vector binomial correspond multinomial random recent establish already work complexity exploit connection optimal pose projection onto vector poisson binomial indicator discretize support light one like aggregate heavy discretized gaussian light small correlate even polynomial unclear pay dependence projection onto vector behave log unimodal exhibit mod modal bernoulli give thus respect mod projection permutation vector identify show multinomial discretize independent multinomial roughly explain heavy explain light light heart proof approximate poisson issue application arbitrary decrease structural technical lie avoid latter cost version procedure shift equal sufficiently far round combine argue round result original variance partition eigenvalue span logarithmic repeatedly partition sort vector logarithm central fall bin vector poisson structural detail approximation comprise several sparse gaussian sum gaussians equal discretize quantify induce detail cover advantageous discretized characterization achieve exponential reduce dependence size naive one moment first profile leverage result size cover remove dependence exploit cover characterization multinomial sum discretize independent dependence cover discretize dimensional dependence challenge candidate discretize gaussian independent suffice purpose unknown easy variation intuitively answer yes multi feasible sample access case moment discretize dimensional movement mass aware broken arbitrarily vector round matrix picking round near minus notion discretize care gaussian live disjoint gaussians direction non simply non consecutive zero diagonal add matrix result multidimensional multi form ignore block difference sample structural state gaussian describe multinomial close preserve round multinomial minimum least main replace sufficiently far original motivate operation note central minimum covariance matrix perform round guarantee summarize round matrix efficiently start fix consider coordinate move light analysis relate careful coupling binomial single long approximately round eventually far analysis round lie relate multinomial gaussian preserve round plus start bound total discretize careful partition merge discretized gaussian assign cardinality discretize leave obtain original sum discretize preserve rounding preserve round overlap dimension merge repeat merge leave structure preserve rounding would cost care discretize gaussian two bind must previously merge discretized clear indeed swap merge describe pair cover structural poisson discretize multinomial grid vector cover overall size cover moment technique component write sum derivative multinomial drop total parameter evaluate point derivative multinomial two matching moment profile roughly derivative close count moment whose lemma section dynamic two lemma mention use style take remove cover try possible guess partition sample mean vector convert search spectral cover semidefinite identify take consistent sufficiently variation learn guess diagonal acceptable formalize component within fill coordinate block gaussian variation cover distribution sum mean following guess accurately enough order discretize suffice match section look fix prove real range affect direction multiplicative second additive direction error direction component additive error sparse due show covariance challenge significantly matrix close cover read ok contain close good pdf
acquisition result trees acquisition acquisition forest grow end incorporate acquisition cost forest grow greedy minimax split optimally cost intractable greedy approach output respect optimal cost classification low feature acquisition subject cost risk constraint acquire also low generalization theoretically characterize random forest superior curve full training extensively cascade belong acquire generalize multi propose however reinforcement learning capable wide supervised study risk framework cascade complex leaf sensor node various forest tree collection constraint tree low cost tree operate evaluation notion despite problem prediction pair learn budget user budget acquisition cost use example cost make problem iid subject budget forest learn cost write follow rhs rhs feature trees forest motivate tree feature budget forest strong feature acquisition cost vector acquisition label monotone helpful measuring mostly iy pt rt fs fs tt rt fs node build tree subroutine tree return subroutine greedy return leaf searching classifier minimize outcome intuitively feature reduce cost choose partition recursively apply allow return predict predict majority acquisition cost tree subroutine hope reduce maintain main broad class max classification pair internal leaf direct along path example reach acquisition denote cost incur path feature contribute subsequent acquire incur aim decision give cost criterion bound cost loose max number real leaf feature max function decision tree enough optimal strategy subroutine function negativity example set always scale outcome classifier say path contribute fs fs tree tree associate subtree root min fs fs fs maximize inequality choice fs fs sr fs fs fr fr leave subtree root leaf child max construct achieving example admissible feature optimal inductive induction verify base induction subset choose cost reduce choose child choose feature choose algorithm pick show fall admissible call paper object proof neither monotonic entropy entire therefore traditional max admissible power small offer advantage pair please detail conclude discuss implication subroutine build leaf meet random forest tree high acquisition forest constraint meet conversely yield due add meet illustrate toy figure circle triangle upper figure rest classifier draw leave evenly equal either useful reflect plot reduce choosing reduce set choose towards feature classifier contrast appendix subroutine call opposed child split lead cost split cost setting emphasize adjust threshold cost figure synthetic achieve compose belong fix feature integer range range label respectively carry unit cost correctly classify every max early stop comparison one show high prediction massive acquisition explicit achieve use fraction meaning example feature plot understand cost art configuration example exceed among run standard deviation bar world clear method error less yahoo consist document document relevance document algorithm take query acquisition extraction provide yahoo query precision predict sort rank predict irrelevant document reveal appear increase precision run tree leaf aggregate leaf node class sum show fast cost build thus require user search task goal distinguish signal validation forest number meet budget choose achieve every point budget feature achieve contain choose high whereas decrease acquire believe partly distinct categorical highly cifar datum combine initially test budget outperform curve budget comment observe achieve whereas low terminate early budget budget fast setting powerful incorporate forest algorithm issue incorporate strategy limit acquisition cost matlab rf default setting forest replacement grow select set rf show tree number feature example test achieve cifar quite rf yahoo even high c tree forest rf cifar rf j accord polynomial admissible singleton integer exist show summation index involve leave sum dominate another power various study building compare uci repository assume feature cost unique single instance label
content indicator reference corpora scope content corpora york article tweet twitter author article project public text extent wikipedia mixtures many language majority english attempt entry phrase length inclusion corpora process cross execute splitting list piece experiment likelihood distinct union accept list code experiment truly phrases code positive truly phrase code positive discover upon perform average spaced plot operate characteristic roc fig area auc expand list twitter wikipedia live present miss discover gram phrase length corpora material prediction accord consider gram likelihood frequency date live correctly discover reference add phrase produce filter corpus analyze phrase twitter far away reference corpus wikipedia filter filter gray likelihood reflect discovered horizontal dotted number discover filter filter short list take red dotted vertical gram effective miss list filter observe output perform cross day http rate think http video I I well change twitter background video http facebook video http http think video http go channel http http could live stream daily back thank rt http filter corpus frequency filter automate phrase definition manuscript pilot program lexical table phrase highlight filter summarize look lexical table closely corpora phrase likely name corpus consistently family know corpora phrase hand general filter also corpora pure twitter corpora extra english expression highlight rely beyond extent phrase tu ne syntactic english ne straightforward construct language indicator language fair phrase predict form define dictionary rise list well corpus twitter likelihood integrate auto correct possibility construct syntactic indicator part whose possible precisely present sec scalable understand language text aim lexical organization fall family partition importance universal applicability interest appropriately applicable research order categorical g phrase primary lexical unit object employ word phrase online collaborative dictionary develop miss phrase entry short list lexical extraction expand knowledge english shannon appearance symbol shannon assign word place occurrence probability find production english frequency still early modern language guess phrase roughly equally measure spend arise make people actually say size availability text shannon generally information association extract extreme many aside information theoretic syntactic scalability make shannon ic associations length shannon gram gram predict word length relationship sentiment ic word however gram special representation corpus level plot line frequency gram consideration concern shannon context appearance property window gram recall gram define page appear gram seem text scale parse practice sentence boundary significant new format five apply gram frequency express also advance gram corpus text grouping occurrence lexical unit informative word produce informative text quantify text frequency pmf phenomenon relate informative power applicability frequently human capable infer previous scalable partitioning since balanced underlie word frequency length norm generalization phrase length word removal set collection removal phrase joint phrase form produce page phrase external relation semantic external context interest actually development phrase internal context pattern removal phrase phrase phrase analogous removal pattern phrase length clear semi formalize template formulation difference restriction contiguous word I phrase mechanic secondary partition weighting context phrase accomplish secondary partition process follow relate phrase phrase observer sub phrase phrase retain sub phrase probability proportional preserve phrase conditional q utilize work eq define preserve context beyond point document normalize convenience derivation expectation section back line densely draw white lc lc lc lc cm gray lc distance fu gray mc gray leave right cm contrary gray contrary node cm contrary fu fu eps eps fu contrary deriving mean definition phrase
notice grow clustering grow infinity seem uninformative illustrated fig ari dataset improve c rand hc km ap b notice automatically detect span art topological maximally filter practice compute filter thank cluster build retrieve co distinguish return interested extent apply display market take lead meaningful practically odd trading day day partition partition obtain price return stable conclude highlight leverage lead fine novel could machine series describe relevance walk scale website lead area finance aggregation give article cl design rgb capital management paris capital management paris pre distance learn algorithm work identically distribute process split dependency metric synthetic benefit series market website field metric advance art many claim fair mention combined behaviour difficult conclusion fast develop restrict scope series identically mathematically subsection present similar copula namely classical pre dedicate synthetic correlate different also financial series market whose dynamic model market stock stock market cf website available future direction methodology usually step pre article study distance exist measure process classify two quantify divergence distance copula ignore property discriminate motivated perfectly correlate return normally risk hence illustrate benefit primarily finance obtain variable propagation mean cluster grouping object object cluster different cluster cluster dependence differ dependency stable stability desirable perturbation obtain resample spirit preserve researcher financial notice poor representation thus capture variable perform distance yet suit consider value discriminate obtain expect equal variable perfectly discriminate distribution actually half appropriate comparing grows take blue distance dynamic wavelet pattern distance take distributional random absolutely cdf split marginal distributional mapping transform represent follow element replicate seminal theory apart yet random let particular case hellinger quantify dissimilarity two measuring express implicitly hellinger trivial verify separation axiom yy u addition monotonic desirable unit device model bivariate dependence copula perfectly want discriminate two gaussian dirac function follow proxy tailed capture apply parametric realization continuous distribution statistic yield coordinate multivariate copula underlie approximated realization permutation function ix hx xx I distance use parametric use estimate suggest mix reflect information cross
show aggregation follow finite quantifie method nice stage prior knowledge beyond bounded enjoy rademach routine argument empirical happen minimizer mean otherwise model specify deterministic theorem focus present clean abuse dimensional sphere radius star algorithm statement outside optimality interior inside strictly contact location three range maximize argue extreme geometry cone rearrange prove extend claim h h bind immediately deterministic star multiplier empirical term start discrepancy multiplier rademacher inequality bound unbounded former statement function require tail excess surely contraction argument remove multiplier appendix offset rademacher offset rademacher localization phenomenon beyond contraction ball assumption condition analysis quadratic expect function multiplicative ball somewhat phrase low isometry isometry say depend mild behavior ball heavy tail assume ball condition plus comparison small satisfie ball arm isometry control via offset rademacher isometry exist absolute constant stochastically dominate rademacher requirement mild offset proof appendix extend classical offset investigation summarize show excess star high appropriate property estimator offset rate cover contrast radius way offset end offset empirical theory offset eq unbounded armed upper offset complexity class probability union cover objective process star shape isometry critical statement offset fluctuation offset process control second offset large obtain term radius concerned offset rademacher complexity gram conditionally loss offset interpret transform expectation order symmetric aggregation offset rademacher class define offset bound eq observe due fact finite pass offset rademacher star hull class case offset star hull offset rademacher complexity rate initially one offset define online regime parametric also estimate vc easily plug upper bounding offset offset minimax define offset rademacher complexity matching take star hull lie eq invoke support state prove stand respect term jensen rademacher observe unchanged precede contraction upper bind combine uniformly hold q jensen bound operator respect copy expression q expectation drop back sign excess later probabilistic exist term rademacher isometry bounding last term chebyshev whenever regime eq write claim q move unbounded n use proceed exist write argue term element triangle positivity bound keep diameter indexing proceed high ball let sphere choose compare supremum rademacher ball inside apply isometry conclude probability offset bound restrict within radius denote minimax excess uniform cn proceed exactly replacement conditionally upper hold offset lemma equation probability valuable regression offset excess inequality risk recover boundedness determine regression arguably substantial generality class verify bernstein condition relate variance increment localization phenomenon optimum analysis large part heavy obtain tight excess especially unbounded sided control tail control mild analogue localization offset online supervise supremum offset lower establish nature offset supremum high convex star estimator assumption even offset provide intuitive rademacher let rademacher index stochastic offset rademacher capture magnitude quadratic act
kk later term time begin decay continue number coefficient depend introduce compute store call consideration return purpose detail neutral bridge eq facilitate put fisher proposition recognize simulate bridge neutral mutation step complicated appearance address follow subsection x alternate combine monotonically converge take pointwise evaluation hard actually employ approach strategy triangular coefficient k property analogous drop convenience coefficient multiply alternate simply provide member decay use lemma analogue property respectively combine proposition order amenable k alternate q odd monotonically converge bound j sufficiently explicitly summarize convenience also simulate v z algorithm fisher diffusion candidate employ reject impose continuously differentiable z detail express recognize simulate give xt algorithm skeleton e find numerically minimize bound easy since table selection candidate result length parameter efficiency great extent length prohibitive nonetheless still feasible simulate collection short string together long diffusion mutation except path run mutation total number per path poisson simulate number l r l l c improvement underlie algorithm vary inspection permutation start may improve refine must compute recall dependence summarize evident coefficient algorithm value various exception observation relevant grow quickly know instability expansion unable separation implementation small fortunately much simulation apply distribution bound point product possibility simulate neutral fisher diffusion even dimension fisher dimension neutral equipped sigma algebra weak topology mutation mutation reversible stationary dx evolution dirichlet product simulate simulate exact simulation fisher diffusion exact neutral extension interesting currently wide perspective believe develop exactly process brownian function treat show subsequently decrease monotonically follow finite hand drop routine soon subsequent right hold exceed express mode separate two continue hold substitute j rearrange maximize note last inequality compare note pair head hence md md subtle increment use get proof immediately proposition helpful discussion department mail uk cv mail primary secondary keyword phrase simulation process diffusion bridge population span department fisher evolutionary widely population finance simulate fisher diffusion difficult know formula function drift simulation key approach yield exact simulation fisher leaf tree application evolutionary mutation perspective model simulation great chi inference serve evolution genetic variant randomly diffusion dimensional sde drift coefficient evolutionary recurrent govern natural fitness individual number copy allele trajectory become available dna evolution fisher finance evolve discount price signal filter evolve simulation transition exact transition kolmogorov quantify empirically however another recent diffusion process use algorithm motion construct path recover admit process brownian start process goal less require necessary occurring realize obtain path infinite determine simulate simulate simulate xt j tb u j b skeleton point finite skeleton restrictive use certain brownian motion
equal vector easy return factor perturb order zero convenience show rough roughly sum get work offset corollary conjecture assumption symmetric symmetric plus eq operator equal eq vector sphere vector nd determine single matrix recall noisy side maximum calculus skew matrix zeros th section noise orthogonal coherence intuitively map satisfy natural simplify analysis derive eq q observe turn form first technical follow vector randomly union establish desire line cauchy assumption q incoherence stage ik jk component another basis symmetry sensitivity lie union center radius permutation combine find term whiten useful whitening simple decomposition
greatly different symmetry allow integral analytically weight numerically completely integral increase accuracy mcmc perfect expect convergence markov chain mix intersection curvature plane ignore isotropic take account curvature volume reduce eigenvalue blue curve integral geometry weight paper red top figure thing worth curve rejection curve compare integral geometry smooth achieve benefit part variance weight geometry variance high intersection occur rare reason search small subspace dimension huge intersection volume difference angle curvature aware eliminate validity curvature motivate curvature mathematic concentration geometry weight intersection volume dimensional blue depict red volume close intersection volume exponentially logarithmic curse volume traditional avoid curvature volume variance volume informally cauchy rotation measure formal discuss situation point unit inside cube choose plane isotropic center cube idea choose isotropic orientation use origin orientation technical issue relate effect cube lack spherical natural conditioning cause subspace greatly much completeness spherical euclidean theoretical application fix non usually haar special generalizing euclidean geometry bit careful case issue poisson subspace g ds haar measurable drop plane element wish life infinite measure intersect volume proportional subspace introduction formula volume spherical gauss curvature manifold euler gauss theorem gauss curvature form connection intrinsic curvature jacobian determinant gauss determinant hessian manifold express tangent space gauss theorem usually relate curvature euler characteristic gauss way relate volume curvature come curvature sufficiently allow many weight measurable unweighted histogram go infinity unweighted slow term greatly variance hence greatly convergence cauchy obtain reduce intersection figure probability subset finite volume constraint accord point unbiased jacobian tm observe integrate x formula measure layer apply illustrate measure cauchy formula exchange hold theorem ds dot say dot differential manifold green dot instead weight greatly cauchy individually manifold apply first algorithm isotropic mcmc oracle dimension generate isotropic dimensional spherical qr decomposition solver unnormalized sphere radius full metropolis reweighte sphere unweighted conditional I whose gaussian h tx standard normal jacobian oracle search iteration random isotropic origin heavily nonlinear unnormalized xx x unweighted correctly accord find compare geometry greatly mcmc geometry differential symmetry haar depend choice moreover constant regardless generality uniformly intersection w w convergence identically slow give wishart determinant iid normal nonlinear solver intersection introduce great traditional ideally correct chain randomization pair great corrected require great randomization nonlinear weight slowly traditional weight great dimension manifold event high intersection however intersection assign point factor simple jacobian depend sphere constraint intersect much intersection intersect sphere curvature level manifold intersect plane pass near slice slice small density gauss curvature slice exactly reweighte gauss relate curvature volume would intersection heavily orientation respect mean converge absolute gauss curvature intersection gauss curvature general unbiased distribute guarantee unbiased distribute determinant inside originally intersect orientation projection complement denominator radius instance compute analytically gauss fact total always always volume sphere converge need uniformly without introduce curvature beyond curvature cauchy curvature uniformly manner function keep vary first haar orthogonal tangent plane suppose operator gauss satisfy boundedness curvature weight algorithm condition arbitrarily intersection occur cutoff fraction average intersection curvature manifold uniformly cut likewise volume curvature cutoff convergence manifold gauss corollary form sake completeness far beyond prove suffice plane ss measure euclidean restrict increasingly neighborhood geodesic cube search treat assume orientation subspace curvature location small remainder proof consist part place extend entire k ball radius tangent ball independent short line contain denote orientation write surely equation fact conditioning independent remain event symmetry formula coordinate origin rotation subspace span multiply rearrange expectation side q expectation respect condition intersect rhs exactly orientation tangent write place equation observe eq poisson wish think subset uniformly form symmetry volume assumption numerator denominator lipschitz denominator cut countable compact jj almost every must q side q nonnegative term expectation subset follow use indeed boundedness everywhere pre gauss together improvement step jacobian connection possibly derivative heavily solver support center step metropolis reweighte restricted sphere curvature original applying row also determinant haar expectation determinant definite polynomial matrix algebra form curvature random determinant numerically perhaps carlo also easy volume algebraic theorem always rare alternatively one might density density certain region search cause variation unless briefly new gauss interesting introduce serve introduction motivation curvature manifold chain algorithm gauss gauss hand may quantity pre generate unbiased intersection volume argue gauss case order point section estimate curvature equation ks implement represent information ability divide euler characteristic high estimate possible euler characteristic well euler characteristic respect statistically ss property gauss curvature manifold general say nothing assume statistically nothing quantity ss like know locally second pre attempt local make guess characteristic intersection would order probably hard implement reason nonlinear solver newton may topological riemannian manifold gauss curvature curvature another differential partial pde define say integral equal product invariant pde idea pde attempt curvature form curvature pde differential form vice versa whether elliptical implicitly argue exponential volume corollary gauss curvature involve sample imagine random subspace imagine subspace intersection volume speed intersection different volume volume theorem metropolis need many converge intersection volume cause reweighted volume avoid variance intersection volume volume volume vary greatly depend sphere fact intersection sphere curse volume wish sample sampler gauss intersection exactly deal variance intersection volume increase exponentially close although subspace radial direction accord haar spherical represent small spherical let affine subspace accord result deal spherical spherical concentration say variance volume exponentially dimension probability volume use generate show yet geometry make soon show haar dimensional great spherical radius exponentially dimension subspace convergence traditional within intersection volume gauss curvature regardless exponential volume geometry generalize algebraic manifold formula theorem volume algebraic manifold long analytical argument curvature convenient subspace degree bx algebraic intersection arbitrary integral absolute gaussian derive formula intersection corollary additional vary much direction manifold corollary beyond scope paper know large perform principal random matrix include unitary ensemble point process large eigenvalue limit limit model converge limit condition eigenvector statistic algorithm case discretize iid stochastic discretized matrix knn normal cutoff cutoff due decay eigenvalue decay like discretized operator already iid constraint simplify nonlinear solver subroutine approximate instance metropolis temperature randomness subroutine randomness search step isotropic random radius deterministic nonlinear solver start intersection weight independent approximately accord fx iid normal I weight solver subroutine introduce solver probably intersection solver would normally compare weight scheme use simplify numerical implementation beyond rather purely deterministic solver metropolis contrary section traditional scheme together plan perform numerical metropolis solver briefly explain fast rejection eigenvalue deviation since condition rare probability event big oppose equal value since equal event dimensional manifold remain situation eigenvalue dependent reasoning come week majority involve neighbor hence test general situation week conditional dependency even probability would geometry histogram blue agree probability rejection black weight traditional histogram skew either blue greatly bias intersection point h histogram use rejection sampling six would cause rejection metropolis subroutine geometry agree approximated rejection weight extremely skewed histogram probably theoretically traditional weight nonlinear solver intersection skewness especially simulation event chance find unless search indeed tell vs sec algorithm hope rejection would day make reasonable amount subspace subspace rejection integral fairly close obtain blue weighted traditional much skewed implying reduce skew solve restrict plot error histogram case skewed right obtain weight
regret discretization supremum norm attempt start contrast complexity analogue discretization expert time suppose tx ft ft depend invoke suppose order indeed satisfie conclude view hold term absolute ignore constant minimax respect balance choose euclidean loss additive logarithmic exhibit imply tight turn field gradient restrict still gradient descent constructive mention function exp infinite barrier self barrier protocol local bound independent together bind take boundary appropriately importantly sequential cover hilbert class ability hessian barrier give simple yet due appendix regret hence crucially self regularization ensure gradient linearly surprising lead ball loss offset sequential complexity problem offset rademacher complexity via scale sensitive cover supremum take entropy expectation deal drop indeed outcome p tv eq suggest key tight keep uniform give concentrate first addition tree notation optimal sequence reasoning specifically subsequently probability surely choose take argument side lemma random variable probability proceed fashion become become shorthand sense course scale zero element eq supremum choice n h lemma term conditionally cauchy far q indeed clearly q w eq q statement proof closely refer define recursion proceed induction induction proof induction fix tree accord sake contradiction aa construct root rest gradient barrier without loss inverse consider reproduce completeness ellipsoid statement acknowledge grant dms assignment loss expert upper term sequential complexity factor bound loss logarithmic employ bernstein intrinsic assignment expert parametrize hilbert ball discretization barrier interest study observe loss allow history forecaster expert specifically consider singleton set contain far may view forecaster optimal extensively alternatively forecaster slightly adversarial fashion forecaster markov expert time invariant case act formulation interesting affect minimax regret eq shorthand forecaster nature upper attain last year root analyze bring assignment study rich class unbounded choose later let thresholded class truncate function check minimax modify via sequential dependence bernstein sub tail behavior expert well static obtain set discretization style expert unit supremum set approach contrast employ idea dependent notion technique attain optimal section loss square matching open attain finish introduction sequential assignment theory code mostly exact case class connection compression interesting algorithmic compression method thresholde state lemma abstract view sequence notational definition appropriately say respect set mapping purpose constraint reflect minimax regret stochastic key difference bias coin logarithmic dual approach worse bernoulli range tree supremum random index mean supremum indexing eq possible act display bernstein crucially consist q collection infinite recall sequential number sequential respect value depth cover denote become tree eq readily identify immediate q section theorem sublinear sequential finite define respect equivalence thank summary complexity sequential covering number match balance day upper quantify soon control sequential covering number could directly many calculation say value
eq policy decision bandit give asymptotically interesting different uniform prior time indicate left table produce regret random bandit additionally interesting small variance indicate exhibit tail policy superior largely science foundation nsf grant du joint classic z observe induction q complete room improvement bind arbitrary power influence result utilize instance similarly event normally simply proposition equivalently eq rhs simplify limit complete convexity suffice relevant sufficient c c dc h consider problem sequentially population specify sample assume normal population expect outcome total equivalently lack simple index additionally controller sample unknown controller convenient define maximum bandit additionally paper largely non policy controller bandit n pseudo take due controller controller would expect follow eqs introduce context per nr constructing modify along two play winner derivation strong say however slow well controller make trade turn strongly policy existence present pt width additionally therein give policy exist fast primary motivation population considerable bind imply guarantee population sample logarithmic therein population I policy achieve motivate definition convergent within convergent n blue arc pt nn establish sufficient therein express conjecture open conjecture appendix fail I asymptotically optimal technique establish insufficient lose establish demonstrate thompson sampling achieve horizon provide remainder thompson paper conjecture optimality depend probability tighter possible paper chi square bound demonstrate second giving worth improve use version eliminate term simplify take infimum complete define dominate linear follow prove make good achievable choice bound yield tight still optimality growth convenience value regret basic define quantity q follow express indicator account term let third recall chernoff bandit u I ti maximal optimal bandit hence aside would play optimality essentially successful conjecture observe dependence integral extend
empirical truncate spectral b range employ backtrack implementation detail alternative main work appropriate perform descent q size conventional progress objective backtrack line repeat constant definition mainly calculate product tm product step extra incur backtracking compare set algorithmic parameter determine truncation threshold backtrack fix employ backtracking adopt extra herein fall range table proceed understanding start uncertainty presentation let direction helpful intuition identity average direction unbounded consequence non negligible influence issue one separate individual directly whole absolutely sufficient ensure truncate truncate obeys account bias truncation look direction sufficiently align reasonably angle away towards step size appropriately regularity fundamental rapid procedure descent specialized sometimes rule plant nonzero reasonably contraction z z otherwise finally connect former guarantee graph distinction stem pair fix simultaneously stationary point truncate objective neighborhood suffice scheme section report practical applicability numerical conduct current concrete parameter unless employ initialization iterate iterative refinement series concern free design independently draw claim return success rate trial iteration ambient dimension find experiment code describe depict success indicate value sake empirical default success rate suggest fast exhibit behavior experiment demonstrate stability varie vary snr cf I generate accord show scale function match stability scale slope predict mse vs snr section prove theoretical absence noiseless mainly truncate similar argument return obeys exceed constant suffice state local contraction consider noiseless case monotonicity reasonably attract geometric rate hypothesis everything constant proceed event begin truncation two fact universal bound prove eigenvalue ft illustrate immediate consequence cauchy together two demonstrate eq satisfying follow event resp statistically independent close inspection reveal quadratic facilitate interest follow inclusion prove well imply refined provide tell establish suffice uniform form formally derive measurement condition constant expect second decrease convenient inclusion reveal leave right explain influence truncation discard reasonably please recognize term right side I share nonzero rare upon rigorous first subsequent non move second rise influence satisfy fraction put quadratic rate constant recognize necessarily q come give amplitude inequality pick appropriately get simplify precede restrict poisson carry broad nonconvex objective result continue hold within neighborhood time sharp concentration truncation truncation might randomness leave future power rank imagine wish known problem computational hope develop modify maintain operate truncate spectral initialization successively measurement low evident precede add scheme concrete application imagine instance image align pair denote cyclic one shift efficient make q quadratic moment respectively make outer I I proceed conclude homogeneity suffice indicator function proceed orthonormal resp resp identity arise since I gaussian inequality deduce obtain net n discretize unit eq lipschitz guarantee arise place unit arbitrary putting complete make convenient lipschitz definition apply probability unit cardinality arise demonstrates claim difficulty handle indicator work auxiliary function purpose tail I inequality sub indicate sufficiently proceed uniform control sphere follow lemma put deal observation basically tell none consequently fix recall eq soon large everything come make inclusion control one I substitute collect useful observe sufficiently besides hoeffding yield sequel first vector pair distance obey packing argument inequality take union r derive convenient numerator denominator stochastically simplify presentation affect I enforce group consequently existence equation fortunately put satisfie k later light collection precede notational move many vector notably remain argument proceed apply bind markov together vector set consist form lemma eq probability thus consequence come lower define bind proceed identity computation truncate state proposition truncate obeys homogeneity case imply non isotropic sub gaussian deduce besides repeat argument omit justify condition tt let obtain addition indicate consequence far union conclusion claim apply effectiveness backtrack search contraction keep noiseless difficulty optimize constant boundary size backtrack notational throughout I truncate evident start scalar get simplify observe plug two identity yield I term consequence combine gm secondly follow mi gm mi I mi put together yield backtrack seek satisfying criterion taking argument one criterion omit acknowledgement support nsf grant award foundation nsf thank long manuscript grateful many flow rgb problem system equation start compute nonconvex approach distinguish feature notably operate fashion drop careful quadratic time soon exceeds extend nearly example random quadratic hence title square imagine set form know priori nothing phase sign product nonlinear nature alternatively pose recover magnitude boolean example equal letting indicate one formulate system check np complete physical sciences technique ray arise record intensity notably upon object form however intensity measurement leading magnitude magnitude spatial depth motivate line think record intensity always noise noise shall pay poisson reason variation optical imaging noise impulse seek maximum denote outcome poisson unfortunately log surrogate propose particularly vector choose quadratic trace relaxation scheme performance guarantee many aspect achieve near optimal exceed applicability another high paradigm iterate nonconvex promise suitable successively rule namely iterate exact mn presence formalize advantage hope achievable convex relaxation enjoy spirit propose novel adopt subtle informally proceed stage guess observation remark firstly data varying correspond result well recommend either take determined backtracking instance appropriate take stage vector product desire truncated gradient detailed specification defer reader practical illustrative impossible sign evaluate solution represent sign signal real shall throughout straight numerical concern square b solve arguably popular least square cg go condition equal ideal cg fig show cg iteration cg design observation applicability image digital code set discrete diagonal entry delay mask illumination quadratic code generate band green separately carry equip ghz intel core gb truncate gradient total cost color band recover display iteration take extremely concern noiseless numerical extend draw poisson model independently snr snr var mse e solution phase reveal addition give away mle cast program illustrate plot incur extra db loss ideal mle reveal phenomenon please precede promise exponentially recovery noiseless complexity nearly minimal mean square offer finding assume tractable shall use absence noiseless size backtrack universal estimate specify explain take make precise truncation threshold take appealing equation optimal since one measurement I cost outperform provable enhanced refinement stage proceed mean operate upon contribution control take away compare movement broadly must guarantee estimate represent claim backtrack eq q least estimate specify constant poisson exist event satisfy reader material universal simultaneously noise stronger noiseless prove use inform complexity rapidly logarithmic put way arrive guarantee snr emphasize e even approach formalize derive fundamental minimax error minimax obeys eq q infimum numerical measurement proportional energy plant theorem achieve vanish match optimality careful reader naturally normalize importance optical employ detect sensor receiver typically practical black apart nonconvex procedure phase iterate alternate favorable fall theoretical support except call attain exceed
correctness step must atomic processor put write cf algorithm execute master worker set worker master return average master fourth execute master cf memory master worker share processor read processor memory snapshot algorithm share master worker worker master processor access global master perform update possibly date gradient pass back independently processor share memory master master processor evaluation mini finish update apply asynchronous overall optimization time involve read clear reading processor use execution depend processor cyclic delay mini batch processor update decision receive termination satisfy establish property optimizer eq iterate average converge residual slow step correspond inequality tell side negligible mini batch processor algorithm therefore processor parallelization furthermore update roughly quickly processor mean speedup depend advance easy optimal minimize second inequality q reduce obtain serial mirror descent size master worker algorithm interface library although argue section atomic flexibility environment text categorization document span decide related classifier regularize token assign document document document scalability document tolerance meet figure speedup accuracy speedup average number asynchronous mini regularize smooth iteration algorithm run vary closed rate penalty negligible speedup experience confirm instrumental argument recursion suppose iterate number gradient problem subgradient bregman plug equality know rewrite left side equality result generate rest convexity recall obtain rewrite error seek quantity turn convexity norm substituting simplifying complete relation assumption iterate subtract leave sum precede inequality drop left conclude strongly convex zero norm see history last expectation side imply inequality prove theorem since increase completely integral verify substitute guarantee assume describe clearly ready multiplying use sum dropping yield term hand substitute definition tt simple type indicator se mini batch optimization powerful paradigm art mini batch cyclic order worker capability delay cyclic leave slow complete asynchronous loss suitably strongly iteration negligible near speedup worker expect confirm implementation arise signal processing expectation loss possibly nonsmooth term elastic stochastic descent nonsmooth approximation develop mirror composite explicitly account stochastic cite inherently place processor access whole happen unable handle amount cause develop able split processor therein one simple point recently processor compute gradient span processor drawback rest run processor paper propose mini regularize overhead synchronization processor perform update gradient similar asynchronous mirror stochastic asynchronous mini interestingly show delay compact set extend value contribution regularization running iterate around algorithm size function compact set prove average iterate time vary residual rate improves previously know delay mirror vary long establish iterate processor rate asymptotically strongly optimization serial review essential formulate mini report natural number include endowed definition refer distance modulus generating bregman another strongly respect simplex bregman function q motivation usual convexity throughout generality indeed scale stochastic support nonsmooth extend eq differentiable denote unknown situation occur application machine learning application time identically impose optimal q continuous effective possible include set
adapt intrinsic validate help even represent euclidean space correspond interesting practitioner include variety measurement uninformative distance powerful technique learn notion distance emphasize spurious measurement decade leverage domain notable mahalanobis quality explicitly prediction task popularity study attribute dataset theoretically practically vary uninformative measurement change formally modality develop two pac popular empirical two framework use objective small optimize base objective mahalanobis cluster comparison proxy optimize prediction incorporate hypothesis learn interesting example regime metric quality learn metric help structure framework lemma absence assumption generalize previous light early uninformative weakly expect metric formalize term intrinsic metric way intrinsic refine framework variation minimize erm jointly observe bias expect intra balance erm algorithm regularization metric efficacy criterion benchmark indeed metric adapt learn weighting remove arbitrary literature minimize notion underlie want metric metric label base reasonable way early explore regime popular quantifying amongst point class opposite weight yield short distance pair distance constraint rise error denote want become generic loss compute weight mx upper limit optimize computable criterion look keep amongst total amongst opposite class variant low limit distance rather hard opposite triplet focus relative distance triplet draw discuss variant triplet triplets metric maintain gap opposite comparison neighborhood affect performance make distance comparison act optimize explicitly incorporate insight retrieval incorporate learn principle constraint formalize framework consider hypothesis shall hypothesis real weight space good study ideal minimizing size definition discuss sample sample grow sequence pair sample mm bound convergence unknown lemma q note I mm ms conclude sufficient never dependency necessary distance error bound weight make distribution make bad classification metric explore effectively complexity vc dimension real complexity error excellent b pick hypothesis class line key achieve sample note find hypothesis hypothesis class complexity lemma absence specific datum pick function vary degree content must solid concept generalization emphasize contribution spurious thus reflect quality individual feature measurement learn performance intrinsic norm feature refine canonical metric framework start follow weighting metric sample loss frobenius quadratic weighting complexity consider class weighting help yield discuss automatically account induce data distribution still base feed recall feed hide arbitrarily enough incorporate hypothesis feed feed forward specify metric also criterion compare uci benchmark dim dim dim unknown synthetic simulating regime large uninformative uci synthetic covariance matrix entry draw set drawing ambient dimension uci split validation setting pick rank coordinate average near uci notice dash noisy introduce uninformative quickly unweighted poor interestingly consistently high whereas yield regularize solid improve performance remarkably degradation classification noise robustness show regularization encourage complexity noise metric generalization optimization framework instance pairwise distance complexity sublinear characterize specific likewise ability triplet partition training representation perhaps work similar erm criterion erm metric bound generalize lipschitz loss dependence alternate weighting distance result lemma loss result lipschitz importance structure formalize intrinsic metric rate tune typical focus complementary representation arbitrary hypothesis class partly success base regularization regularization design high measure interested bound I second uniform width depth let h satisfie x hx hx rademacher complexity class choose expectation exceed level failure well regular simplex later assign show consideration minimize metric restrict weighting simplification note solve binary classification vc bound detail weight pick belong f x mx mx x rate optimal equality ii occurrence observe note let quantity return follow note satisfied imply moreover would suffice height vector q vector definition define unit moreover non empty bi mean centroids knn n set point map p j ip ip x constitute random uniformly independently suppose sequence value function note inequality maximize select well distribute eq h class matrix value minimize cover call note observe I combine namely form fx bound failure probability pt spectral cover volume know v construction universal suppose value bound distribution find see shall pack generic value class domain value let cover resp packing minimize cover resp maximize packing follow maximal x hx determined net case distinct b apart packing
identity set easy get combine monotonically lagrangian function second equality fact convergent integer x k x k let k hence boundedness convergent relation q q k k let continuity optimality moreover x admm implie hold optimality condition get inequality hold monotonicity apply side identity k x k bound convergent far show fact I suffice solution replace k sequence complete block admm x imply immediately remark define apply globally range cover conclude free satisfying range motivated fact block parameter global convergence admm counter show block impose look sufficient however usually admm block also iteration yield add inequality increase sequence inequality k yield kf convexity k monotonically sequence yield boundedness give boundedness sequence k reduce k k apply three hand imply furthermore hence moreover x third k k sequence corollary zhang method multiplier admm apply structured optimization superior admm extensively literature prove admm implement ensure chen al study admm usually require small difficult compute small admm still parameter solve commonly cover keyword minimization square structure arise processing computer survey particularly separable minimization usually admm certain place block solve eq lagrange multipli dual convergence admm study extensively literature nice block parameter restriction prove block admm admm particularly attractive solve admm solve block variable q lagrangian convergence chen et show far impose block stable pursuit robust alignment semidefinite great convergence restrict relaxed chen lin zhang restrict bind sublinear convergence admm admm study variant block require strong boundedness constraint lin zhang admm lin zhang far propose approach convergence strongly trade penalty affect block admm suffer alternatively opt modify classified category class step add admm jacobian gauss manner restrictive also affect arise effort acknowledge admm admm probably restrict value convergence parameter free give great globally convergent term regularize square next decomposition seek decompose certain decompose fitting admm solve et advantage subproblem easy especially subproblem zhang admm regularize statistic lasso zhang reformulate numerical conduct note lasso vanish interesting example share literature interested component pursuit aim wise formulate corrupted respectively form admm surveillance aim extract surveillance frame surveillance find move foreground pixel restrict add physical et molecular pattern discovery identification reader component pursuit rank sparse observe small set measurement q note unconstrained interesting compressive measurement similar paper globally
fluctuation observation km row red solid red blue dotted versus subject subject accuracy repeat procedure relevant last trend probable mode notice appear first mode maxima compute choose standard second change mean deviation trend characterize essence difference people patient develop analysis time contain part series use decomposition low component apply heart interval ability extremely slowly strong ability conjecture activity partially support nsf fa cm new iterative decompose series extract mechanism underlie show many measure time key word outlier heart interval heart many heart attack health exercise complicate apply classification application commonly focus aspect series dimensionality randomness etc tool use technique include restrict deviation empirical mode series transform characteristic nonlinear representation great success biological medical sciences engineering texture chinese main purpose time two modern come belief contain reflect basic system variability mathematically represent low frequency frequency motivate frequency quantitative wavelet second intuition come perspective lot decrease statistical perspective set represent motivate time careful practical examine pure structure motivation series analysis learn redundancy support classification heart heart disease heart failure heart decrease heart failure literature propose analyze heart name incorporate allow purpose fold enable diagnosis kind heart health mainly decompose secondly outlier pure informative interestingly filter denote limit operator iteratively roughly speak noise cubic spline connect low envelope cubic spline connect use lack foundation new pass generate mask convolution iterative convolution rigorous mathematical foundation mask finitely crucial method wavelet trend characterize profile signal detail application need extract priori without priori get proper statistic dependent heart interval illustrate construct application record interval decompose function previous heart heart motivate statistic large deviation statistic term outside statistic characterize maxima compose amplitude series hour heart period activity think people motivate idea split whole suppose series correspondingly mode denote th quantile total component fundamental mechanism individual may trend trend represent three description list deviation st st deviations rd rd standard subscript subscript statistic series less omit compute maxima notation part diagnosis find irrelevant firstly almost diagnosis eliminate eliminate svm feature size small might feature lead inaccurate refine eliminate least feature repeat iteratively conclude diagnosis essence make heart conclusion datum heart series hour activitie activity period classify slight iv severe use method heart people patient people patient
restrict context acknowledgment grateful thank anonymous conference whose remark answer air office uk ac uk theorem output prediction allow q interested computable list computable besides make infinitely derivative side infinitely differentiable side satisfy list computable loss prediction corner eq loss later computable check smoothness obviously computable strictly measure probabilistic intuitively regard get function essence attain loss smoothness intuitively corner new typical repeatedly simplicity set number suffer say place precise proof consider trivial spherical function fix small call intuition behind prediction whose sequence long finite stand define section never arise replace random prediction identify measure universal say respect probability applicable computable whereas former computable randomness element ignore uninformative object equal within randomness respect prediction continue ignore informative randomness respect function special use log theorem follow quantitative computable role ignore coincide randomness previous randomness prediction e proof prediction notice pass translation moving map l computable proper loss function function result q computation eq spherical sign explicit simplify loss fundamental expression condition eq suffice compare criterion see back lemma criterion give partly check kf taylor convergent hand randomness log grow simplify spherical function fundamental cutting end correspond easy check case restrict restrict coincide title least typical ask question say log lead randomness respect loss parameterization replace impose parameterization requirement ensure randomness existence computable necessary yes intuition behind behind notion set suppose curve straight segment point set canonical namely tangent line cf stay transform correspond say selective preferred spherical
hdp infer double manner estimate double embed also use synthetic continuous speech represent sequence categorization task speech recognition whose acoustic manner acquisition child continuous segmentation boundary speech give isolate direct knowledge word problem solve list process direct acquisition access speech recognition current automatic speech modern language knowledge distributional well acoustic represent knowledge linguistic corpus however access acoustic raw acoustic speech signal human discover continuous speech et distributional co rely relationship speech child detect co entity consider distributional speech accomplish month old solely distributional seem age imply fundamental word segment distributional fundamental distributional help segmentation viewpoint acquisition consider language finding distributional explore discover signal distributional unsupervised learn directly double organize word feasible acquisition acoustic develop newly probabilistic generative hierarchical section section present hdp extend hide acoustic sequential computational kind method decade program recover boundary source segment maximize text sequence improve include process sophisticated calculate gram word context treat infinite segmentation method account nest language letter gram embed word gram backward boundary mention recognition error learn recognize knowledge knowledge acoustic recognition become overcome word method occurrence enable robot word multimodal show cognitive raw sensor human et al interactive interaction unsupervised et enable robot linguistic communication speech behavioral viewpoint basis online word concept build category acquire name multimodal dirichlet increase co occurrence multimodal categorization show category update categorization co occurrence name pair mobile robot acoustic word selection criterion carlo localization localization robot result ill error solely speech signal al unsupervised et et outperformed experiment lattice text discover language iterative report improve propose jointly word learn sound recognize acoustic ill recognition distributional acoustic train manner insufficient constructive acquisition raw hence unsupervise acoustic acquisition acoustic categorization transform continuous include hmms acquisition use category learn acquisition categorization overlap sound al word effect lee al discover proper sub word acoustic unsupervised manner language lee discovering letter sound rule automatically determine acoustic several study simultaneous acoustic language small method simultaneously integrate acoustic propose enable acoustic find descriptor parallel technique viewpoint point segmentation acquisition mutually theoretically integrate acquisition acoustic integrate theoretical author double analysis view unsupervise discovery raw regard double represent double structure structure period discovery become double et al double hdp hdp nonparametric extract motion sequentially model generative recognition categorization letter hdp hmm unsupervised terminology letter latent basically conventional newly conventional apply drive double conventional purpose mining topic respect drive drive compare raw driving letter conventional raw speech background mention paper double acoustic assume infer variable double unsupervised novel double hdp double generative hdp series potentially extend hdp name hdp contain language basis hdp word hdp next latent word basis illustrative overview hdp sampler briefly hdp hdp extension unlike hdp conventional markov hdp explicitly model duration hdp breaking super distribution emission hide next semi super state hide determined duration super categorical super super time tt assume emission efficient base construct gibbs hmm super pass reduce cardinality duration super order backward filter constant double structure extend hdp super state fundamental th generative hdp generate letter furthermore latent letter letter output word language lm respectively latent sequentially latent letter word hmms duration explicitly latent time letter th letter latent draw duration duration duration latent hdp latent letter draw emission distribution map letter word generate assume datum double viewpoint language acquisition review compose machine hdp sequence letter transition probability correspond letter regard conventional hdp consist inference generative inference acoustic simultaneously sampler hdp letter language acoustic structure continuous propose unsupervised machine overall hdp sampling adopt instead naive sampler sampler backward sequence backward sampling procedure make message super hdp follow super state super transition super represent obtain st emission condition duration state easily procedure calculate backward message pass hdp word st message occurrence become partition duration substitute message hdp look complicated efficiently latent latent letter recursively formula message calculation backward procedure employ letter hdp super iteratively use backward message please refer original hdp concrete letter word sample accord latent word word generative model regard super state letter sub hdp letter sequence sample ordinal hdp word letter sequence latent sample latent sampling resample sir define word kp represent hdp way hdp propose sir procedure employ proportional sample model letter letter I update update letter sequence parameter acoustic update state sir sampling accelerate result sir acoustic sample hdp hdp overall sampling initialize initialize message initialize super state word pz super state ss model sample letter sequence super word sir basis hdp time variable analyze word manner validate propose infer latent double apply hdp synthetic series comparative generate five word w word sequentially th letter letter poisson parameter emission index emission compare word pair represent follow six word letter seven emission comparative hdp hmm average fig trial result work appropriately gradually probable increase contrast speech acquisition double viewpoint precision adjust rand quantify ari estimate letter ari letter conventional decrease conventional ari ill ari ari gradually latent variable show generate sequence top show letter latent word infer word estimate show procedure work estimate tb eps bt c effective double embed series evaluate method applicability datum ask sentence use ie five five five sentence ie ie ie ie ie encode data size shift datum frame hz language set number seven hyperparameter maximum letter seven duration emission dimension conventional hyperparameter hmm similar possible hyperparameter conventional gibbs iterate seed trial open speech engine dimensional feature speech acoustic speech conditions dictionary encoding unsupervise conduct discover model unsupervise propose software fourth word contain ie use acoustic contain acoustic manner label dictionary base acoustic letter speech letter indice bt c lm conventional check
train recent work adopt deep supervised encoder decoder pair entire information convolutional encoder relate unsupervise semi loss add loss fully encoder train loss jointly nature depicted q intermediate constraint loss construct convnet phase encode decode convnet forward pathway part pathway feed opposite encoding pass convolution relu pooling pooling layer complementary preserve add pooling within pool pathway operation pathway basically place right position region encoding pooling figure loss indicator sample input encoding reconstruct decode successful architecture stack attempt useful widely sub manifold one manifold carry identical perspective experimental amount discard decode pathway fall convnet go purely unsupervise deep auto architecture deal miss joint flexibility ease switch loss collapse indeed common auto identity cause case avoid code must jointly introduce supervised scheme tend task label pre take account recognize generalize pre bound henceforth restrict training drawback due supervise improper epoch latter offer control interpretability play reason add list prevent otherwise work properly secondly avoid situation reconstruct middle upper regularize intermediate intermediate reconstruction shape light statement digit cause change equivalent nearby meanwhile major component sub direction introduce invariance sensitive basically sub explain digit exhibit rotation range supervise middle b right architecture layer mnist supervise semi mnist label unlabeled size sure uniformly several round new dataset form report along fix identical well computed choose hyper train union training validation set configuration digit kernel denote pool layer pool region jointly regularizer regularization architecture include dropout connect dp fc train without basically dropout model report aside follow softmax encoder denote tool fine entire encode softmax write c fc convnet pl present publish besides supervise label basically use regularizer use improve dp dp fc knn na knn na na dp fc task bayesian zero construct unlabeled class effect regularization show well publish get supervise architecture lastly drop add mnist become highlight address separate whereas merge step wider likewise tune validation induced performance reconstruction help display classification couple convnet semi dataset label remain unsupervised unlabeled video edu novel architecture stack generative pathway unified essentially net convnet couple objective include convnet produce feed position feed decoder desirable mode learn desirable property generative train wise stack feed pathway manner fail mechanism unsupervise another boltzmann boltzmann rbm kind encoder deep rbms procedure sampling tend inefficient main stack mapping implement feed forward pathway conversely mapping implemented feed back generative pathway e reconstruction deal rbms category sampling tend complicate inefficient feed reconstruct good invariance desirable mapping layer convnet invariance max subsampling idea approach layer complementary pool reconstruction model consist feed convnet couple feed back stage auto encoder convolutional layer relu follow layer max next complementary switch incomplete information feed
svd orthogonal matrix negative eigenvector principal quantum feature quantum picture mapping quantum pca way component quantum representation quantum eq representation compose eq representation representation imply inner product factor representation principal linearly classical respect linearly perform measurement return yes answer answer probability return output q positively classical likelihood new quantum digital image analysis classification copy digital support national centre st national institute mathematics technology quantum von quantum image paradigm computation obvious quantum computer construct field quantum rapidly many create analysis measurement behind training pca divide signal variability mostly noise lead quantum classify encode quantum measure quantum image processing draw quantum elementary system basic choose hilbert vector represent state combination operation join state system big eq also quantum system column hilbert orthonormal represent product iff outer quantum measurement outcome assign corresponding measurement request measurement operator first state execute propose recent year lattice representation intensity encode real quantum serve position pixel encode quantum inspired pixel image computer responsible encoding encode vector kronecker product responsible enhanced digital state form sub deal discuss publish recent classical already quantum cosine wavelet technique quantum state example author circuit representation processing number quantum author projective store operating processing basic algorithmic
expand detail rough google compare several image variant google aspects factorization approach scene factorize add region add scene factorize greedy dataset greedy add attention scene metric moreover benefit region base attain previously show except art also qualitatively two dynamic abstract mean visual influence generation pixel first pixel distinction foreground focus background small ie focus region highlight region highlight foreground please refer visual scene category significantly predict scene scene hold scene lda topic topic drastically correspond regard hold slice topic scene impact generation scene exploit structure contribution process generate attention visual introduce scene lstm scene system popular attention scene context combine model intelligence advanced project via department laboratory contract nf c additionally nsf google award fellowship award nf reproduce purpose view conclusion herein represent express imply partially cb image dataset provide patch example patch classifier output patch adapt describe vocabulary less discard token begin take denote start sentence well size token begin sentence adaptive model advantageous effective especially scene factorize multiplicative optimize jointly regularization dataset ensemble observe minibatch give minibatch take day gpu validate k table metric table rough google google images sentence proportional assume occur less probability model system top retrieve sentence length quality cccc several compete need typically compare image evaluation metric rate return sentence image sentence low perform less par good group fig recognize object image interaction localize patch scale softmax output focus function softmax feature word mean visual illustrate determine sum foreground attention focus ie also region well sequence fig show sentence go allocate word occur patch experiment contain go slot match patch rule match match patch choose match learn learn match inside patch semantic significance modal red cat em em em em theorem proposition university china edu california equally em comment send progress image salient meaningful paper propose exploit parallel experience impose alignment characterize novel introduce context language generation specific benchmark contrast several improve furthermore attain recent generation image greatly vast visually information language nonetheless image shown describe salient meaningful attain leverage crucial vision learn represent rich visual localization capable generate sentence progress remain challenge task sentence far probability decision represent language model information encode visual fed predict sequence govern select text cf arguably start understand object reason object focus salient generate relationship linguistic determine sequentially order word sentence keep salient secondary information generate follow exploit parallel structure sentence conceptual diagram correspondence detect like word generate align experience region impose alignment characterize mean share visual description recurrent neural next also another novel contribution scene specific context encode place activity people model word scene instance unlikely scene rather context affect word recurrent neural differ detailed defer localize scale visually salient object represent ground concept unable grain collection stage patch word correspondence region word parallel contrast publish either specific context combine attain art compete follow detailed sec sec image generation long traditional pre define template detect retrieval image retrieve sentence generate sentence recent language learn log bilinear propose multimodal recurrent architecture visual feature extract rnn image detector word patch generate
deep argue approximately way benefit clearly illustrate benefit value classification without upper bind incur n careful benefit preferable result cost analysis assume label probability explicitly obstacle strategy nontrivial literature major style calculation argument mistake correspond spirit ensemble decision vote stochastically mistake oppose et cause order example n motivate near primal max min advantage order j eq strategy unable choice useful suppose optimal minimax outline example effectively present analysis aggregate formulate pac analysis learn manuscript enable appeal nontrivial strategy without error allocate aim argument future proof suffice game payoff maximize predictor predictor performance raise nature progress satisfying extract every advantage leave slack first example v payoff check kkt example select n procedure constraint I desire proof play regardless n suffice predictor force play primal play play v minimize maximized nature I value irrelevant set call force therefore constraint prop yield variation nature trivial without think nature adjust raise budget budget payoff proof remain budget satisfy little n w game duality use reasoning throughout substitute game keep mind order regardless neither identically example prove p contradict strategy purpose lemma bad simplification predictor predict optimal depend play chance incorrectly incorrect give lemma thm thm conjecture thm definition em california california consider advance derive minimax rate prediction set pac rule distributional readily predictor allow application binary classification drive rate concern classifier encoding approach averaged choose predict vote vote extreme fairly slight variation rough argument true suggest gibbs traditional pac spirit focus average weighted paper well aggregate consider predictor nature sum game example nature choose predictor maximize correlation unlabele constraint correlation predictor therefore clearly combine ensemble question motivate contribution predictor game minimax pac minimax predictor average quantify enjoy extend predictor early scenario formalize rule predictor unlabele example tt argument encode predictor predictor jx correlation choose true example prediction average predictor rated bad predict prediction immediate make gibbs average identify outperform make true ensemble effectively impossible without outside minimax prop minimax game without game defer label blue minimax simple training statistical compose hx h kl kl observe label choose use pac set distribution convert p classifier scenario well game label token incorrect probability uniformly training randomness benefit voting exp h low due benefit disagreement many nontrivial significant fraction g extend class rate part source robustness pac bayes work part admit would achieve notable generic extension online replacement classification game predictor treat relative early predict output rest I randomize cost suffer nature gain incorporate
cc cc pt sketch sketch recover also reasonable datum pixel white cccc digit stock sketch sketch sketch identical mix center run h f sketch sketch f r observe sketch uniform extensive comparison recover speed digits text important bias component topic pca complete south frame transaction principal discover pca h c order w parameter word appear pca algorithms sketch match closely pca mixing sampling well small show r uniform sketch comparison factor sketch also bias toward large look stock complete table show stock appear pca sample discover validate corresponding gene multiple principal since c gene symbol top occurrence gene respective cancer co eight eight gene list characterize gene incomplete disease identify additional report suboptimal construct suboptimal sketch reveal popular toward large work svd leverage follow score square norm row row matrix rank project low project space span preserve digit onto top datum top respective separation three component leverage sampling digit stock rank sketch leverage score range optimal sketch possible sketch incomplete mention good feature identify pca column research remain pca getting point optimize natural look datum minimize objective maximize theorem corollary one matrix sketch data one particular principal sparse text biological financial sparse incomplete drop projection preserve equivalently reference principal classic challenge interpretation combination original variable significance biological financial desirable factor interpretable small feature incomplete recommendation privacy preserve get carefully pca provable work demonstrate mn dimension effective rank often kk kn component solution principal optimization non entry sparsity input pca np hard heuristic provable typically address get top principal component address know incomplete datum sampling else pca sketch instead datum perform try optimize measure result solve closely approximate capture much sketch completion sample high element sampling sampling solve solution datum quantity give speak ignore multiply stable price far incomplete sparse thresholding denoise observed matrix perturb principled small quality sparse principal component algorithm input benefit fold summary show optimal sparse find component np approximate take heuristic heuristic practice may able sample rather establish recover outcome reasonable likely really really negative bold bold denote element k xx give fluctuation sophisticated element pca reasonable necessarily variable lipschitz setting maximize bind surrogate set tf singular von trace fact simple incomplete small zero happen large noisy setting treat ij fraction energy lose zero datum create truncate satisfie eq show appropriate signal element recover near keep particular spectral wise element way bias high rhs upper gx fx fx f f follow step set instead keep toward proportional element element trial sketch wise sampling index entry note unlike deterministic small intuition expect outcome probability define element appropriate deviation suggest choice summarize create sketch select simplified version least stable away suffice sketch tolerance use two depend performance pca hard heuristic heuristic next six
quantitative quantitative feature qualitative mapping quantitative example qualitative store qualitative along compare experiment consider qualitative quantitative method backward elimination lc lc perform lc ap complexity pruning standard number five fold pruning fold node run respective deviation experiment induce tree example remain approximately example belong aim ccccc example heart breast cancer bc bs bs ap lc ap ap lc bc ap ap lc lc lc ap lc ap perform quantitative ap design split qualitative qualitative feature purpose ten five accuracy ten report tree tree ccccc feature qualitative set misclassification cost split discriminant minimum node splitting bank deviation produce accuracy dimension produce comparable matrix reflect split matrix whereas dominant empirical obtain method perform tree furthermore capable classify quantitative node quantitative node one example complete class eigenvector since parallel one reflect space complexity finding reflect node decision region homogeneous particular tree cart algorithm space split boundary boundary potentially simplify limitation induction new tree utilize series consider axis split reflect datum appealing classifying iteratively partition disjoint region dt tree terminal non terminal call consider hyperplane sub hyperplane obtain recursively reach homogeneous region terminal node misclassification play classification aim accurate depend node dt split include split split partition axis parallel desirable align feature hyperplane split feature appeal boundary align axis split split decision boundary split arbitrary shape easily noise induce differ study tree therefore become increasingly dt dt non node decision tree specifically attribute reflect eigenvector reflect mechanism reflects reflect reflect axis split axis reflect original search enhance classification problem class eigenvector explain propose version class dominant eigenvector terminal available find covariance whereas dominant eigenvector split coordinate axis reflect parallel separate already axis parallel split hyperplane find algorithm satisfy misclassification child user equal algorithm mp pi ji ji h construct reflect good parallel hyperplane ji ti grow raise question search axis size split twice second
abundance report reconstruction spatial coherence improve generate mixed figure abundance fourth include detect part face part return localize sparsity comparable sparsity abundance element numerical generate trade sparsity remark reduction powerful important nonnegative factorization nmf extract localize technique identify sequentially particularly suited incorporate localized look approach comparable state hyperspectral matrix sparsity imaging dimensionality technique tool well take interpret nonnegative nmf classification air emission rank nmf look via combination nmf impose basis g pixel intensity interpretable imaging hyperspectral image cube provide scene hyperspectral sensor vary material energy signature material certain hyperspectral use material pixel hyperspectral cube convert dimensional vector stack row signature pixel mix signature pixel combination nonnegative signature contain signature hyperspectral cube dimensional nmf matrix signature pixel signature row represent th th pixel figure hyperspectral cube road kind row abundance map abundance pixel unfortunately difficult np non rank reason nmf refer recently allow compute sequentially nmf rank time try localize feature factor residual compute nmf advantage pca mild pca sequentially factorization priori part enhance decomposition part successfully hyperspectral imaging modification nmf reference therein precisely propose add abundance localize art particular hyperspectral prior localize image describe introduction nmf sequentially dual fix lagrangian relaxation uv scale vice versa note trivial satisfy contradiction generality base optimize variable update lagrangian scheme write vice versa u uv tm convergence share similarity rank incorporate neighboring coherent desirable contain isolated pixel tv adjacent pixel evaluate spatial pixel column correspond pixel indicate neighbor account preserve oppose smooth spatial information incorporate feature contain imaging translate fact pixel abundance author incorporate decomposition entry approach incorporate sparse nmf information process add compose relate classical least residual sparsity abundance improve coherence balance relax formally ht small uv locality matrix w nx bp z z bx lipschitz fx w nx original give close subproblem iterative perform require operation cost lagrangian multiplier nmf sensitivity surprisingly critical role mention many classical completely prior supervision tune quantitative conduct nmf accelerate suggest author nmf sparse add norm abundance sparsity abundance hence nonnegative nonnegative use quantify percent amount abundance normalize normalize column term trade three measure factorization lead coherent fair sequentially generate sparsity optimize zero handle refer processing method detect hyperspectral abundance hyperspectral show correctly detect widely mit matlab code equip intel cpu core ram gb code assess hyperspectral technique consist precisely see effectiveness test variant conduct experiment show wide abundance extract abundance figure indicate seven abundance penalty penalty spatial vary lose contour sub figure b constraint sparsity abundance seven basis vary observe spatial parameter detect figure locality high pixel abundance element improve take zero spatial coherence influence supervision necessary localize introduction extract well reconstruction coherence display abundance quantitative algorithm sparsity spatial follow nmf generate localize image qualitative confirm quantitative nmf low since focus coherence bad spatial despite give solution unable figure abundance image surprisingly coherence return sparse different abundance map material abundance sub generate coherent surprisingly provide material rather noisy dense add increase spatial coherence achieve trade sparsity low show image set right top
comparison preliminary preliminary stein stein shrinkage lasso characteristic simultaneous exceed conclusion make efficiency carry test estimator paper shrinkage classical error analytical asymptotic risk carlo behavior propose application life organization contain stein improve lasso section setup discuss detail application estimator demonstrate life conclusion multiple regression far know preliminary shrinkage depend lose vary want estimator classical least shrinkage belong restrict minimize l write tune explicitly yield n pn solution estimator selection computational later angle efficient glm estimator good nearly unnecessary threshold set zero datum multiple restrict vs condition n value estimator optimality assess mse stein give note replace inherent change sign value define estimator positive stein eq define improved estimator six penalty shrinkage stein shrinkage lasso estimator eq stein rule estimator alternative center estimator pn q alternative equal coefficient remain take ns thus se chi eq next compare take whenever le difference preliminary test lasso relative estimator study mse relative le conduct sub test partially linear regression study full shrinkage setup simulation distribution consider simulation indicate nan set hypothesis realization variance parameter subsequently least square preliminary read secondly generation setup accommodate function translate response generate least square consistent neither stein dominate follow life pre set center predict interest use regressor le improve preliminary stein regression validation fold validation validation randomly aside term remain model predict observe predict varie run average deviation standard deviation analyze seven represent covariate species km km km area km covariate convenience specie figure h min max r corrected display validate deviation summarize error well le notably le estimate population life percent mean capital city square response summary table correlation predictor display present study observe predictor rr correct sd le table give average average follow large error visually yet highly error widely seven predictor price national people armed force old people employ data matrix highly average variability small plot prediction error demonstrate max sd l correct le propose estimator stein rule preliminary lasso stein shrinkage stein study compare configuration size variance relative estimator vary degree misspecification table configuration dominate mse among uniformly neither one efficiency estimator near decrease preliminary stein average le outperform picture outperform life try correlation predictor moderately estimator equally case estimator would set among predictor h c c h c h c h
base prototype calculation dp approximation likelihood work entirely considerably work due asymptotic interpretable recover outperform benchmark dataset behind formalism agree cluster closely discrete categorical select draw point maximum mle denote belong p uncertainty across cluster force close extend insight enforce common whether categorical log modeling make separately intensive elegant k repeat center choose point assign create select feature consist binary dirichlet avoid priori hyper underlie exchangeable probability tune start explicit categorical respectively whether bernoulli parameter assume prior categorical cluster index categorical beta value formalize small variance select around select draw independent log detail pair elegant categorical differently term control would turn cluster provide supplementary categorical feature feature take select categorical draw initialize categorical assignment compute generate cluster choose feature low categorical nd along feature outline center need center follow asymptotic cost assigning exceeds would information feature guide feature select specify constant later modify recover exact implication dp mean supplementary denote feature aspect estimate often need application minimal tuning statistical turn via prior readily informative overlap away introduce covariance inverse vanish asymptotic hyperparameter absence asymptotic thereby computational available resource selection trade enhance feature benchmark feature per cluster experiment facilitate specific subspace result experiment synthetic subspace categorical disjoint include comprise evenly split subspace independently bernoulli cluster overlap second third accurately dataset comprise evenly gaussians unit respectively cluster disjoint add isotropic distribution standard modify cluster contiguous additionally completely subspace contain may subspace allow modified dp cluster method indicate select real dataset two compute frequent normalized labeling divide cluster assignment lie close henceforth fair bank spam c c bank comparison dp dp determination categorical extend retain dp retain comparison outperform extend importance entropy selection art benefit accomplish art unsupervise besides dp bank highlight global selection finally besides time spam execution second attribute benefit mean style oppose require intensive spectral feature datum asymptotic set vary code website various show derive retained b binary categorical particular dp mean assign distribution find shape ensures assume uninformative conjugate I gaussian categorical contribution contribution categorical categorical nd assume draw independent global categorical likelihood beta provide posterior equivalently simplify quantifie select since uninformative contribute simplify specifie equivalently maximize minimize simplify cat k quantify change cluster constant must enable feature thereby bernoulli discrepancy control mean joint cluster log eq first k nd nd nd analogous asymptotic eq data mean cluster initialize mean indicator compute pt nk generate k distance within point assign otherwise cluster start assignment successive objective equivalently write mean characterize uncertainty try thus force point come together ensure point absence regularizer singleton cluster thereby lead trivial case uninformative conjugate contribute negative retain letting obtain compute dp objective recover dp imply get derivation automate binary draw bernoulli feature proceeding section dp c reproduce contribution proceed contribution different also avoid value underlie noting set q put everything objective
expectation update equation mean maximization log lead precision equation unlike monotonic improve performance become large instability kronecker prevent iteration posteriori contribution rescale often algorithm side hypothesis side nuclear minimization variational completion normalize square keeping row linearly draw form choose compete algorithm repeat measurement repeat simulation vector realization independent compete algorithm use follow toolbox estimator compare factorize eq block induce vb reconstruction vb nuclear require knowledge compare rao rank setup mention valid lower technical fulfilled estimator absence verification hypothesis experiment figure plot well sn respective side second good sn experiment consider nuclear norm robustness study improvement nuclear region vary confirm noise deal completion measurement consider inferior measurement use show sn typically find vb investigate vb find improvement sn experiment varied result attribute vb arise away relate rank type side laplace conjugate model maximization equation name relevance name simulation precision side outperform nuclear estimator though nuclear outperform completion second order around since expand minima integral em help derivation regularize result occur equation formula find occur remove penalty update lemma learn low determined system rank rank relation justify kronecker structure matrix parameter numerical inherently popular sparse reconstruction setup reconstruction regularization regularization bring type rank penalty literature eq matrix mention nuclear penalty literature optimization exist compressed sparse solve nuclear reweighte solve algebraic approximation convex priori signal noise absence priori preferred capable measurement prior posteriori estimate type information hyper sparse reconstruction form machine bayesian learn gain popularity ii bayesian pursuit monte iterative via technique low help characterize hyper treat follow precision determinant sense estimator derive evidence compare numerically learn convex aware evidence hence unable organized learn sided matrix derive side compare machine reconstruct measurement measurement eq main learning lead vector latent assume laplace lead couple solve repeat maximization type concave convex solve conjugate appropriate two example rank penalty follow penalty c nuclear base penalty wishart scalar instead prior easily right precision question stem estimate side otherwise develop sided sided precision enhance model random matrix relation l notice evaluate bring suitable low function nuclear establish direct connection indirect space respectively interpret skewed comprise correlate skewed correlate column relation presence highly r ij strong auto mention qualitative sided model side base unable capture side precision estimator amenable
appendix building propagation snapshot projection model snapshot architecture highly architecture literature heart decision rich enough relation fairly broad example consequence take action provide exposure attempt target failure mode motivation expansion allow agent state agent endow depend agent capable seek instant operating pick act randomly domain guarantee knowledge snapshot restrict endowed realization sensor sensor action agent name precise x accept system various available differ nothing control mind invoke set purpose control outcome outcome action reflect viewpoint impose restriction enter moment must precise principle restrict example action intersection set outcome action force interpretation action generalize every outcome equal move contradict opposite aside generalized set admissible define matter action tt regard simple example endow mutually exclusive atomic action leave correspondence pure observation oppose evolve structure represent two restrict interested interaction set coherent fact duality median embedding provide consider unit along integer length formally environment agent action enable exist sensor realize hold exist relation indicate reach relation product relation equip snapshot derive mind notational agent task predict tt jointly decide action agent invoke record complete complete represent observation tb define see assign reach possibly position serve recall snapshot tb direct path path implement update produce propagation explain snapshot update propagation use graph tool turn rest variant expand record vertex visit sensor snapshot snapshot observation first u v fashion corollary ts reverse vertex zero implement prohibitive process time network plan kind ability action ability behind form lead direct point snapshot order allow sensor necessary formalism carry sensor aside future contain abundance sensor environment action consequence consequence apply propagation mechanism predict outcome provide snapshot among sensor fidelity perform planning snapshot sensor rewrite thus sensor q whose far expand figure large geometry geometry geometry propagation ability immediate theory greedy decide target characterize region represent desire may considered action guarantee possible may select tie break completion lemma directly motion planning euclidean plane absence approximate path point next kind arise presence occur determine selection weak capable matter motivate follow model complex form implication record class incoherent sufficiently review homotopy requirement place consider example kind example realize equip collection position identical I vertex simulation equip label vertex adjacent belong remain snapshot sufficiently statistical nature learn agent weak set complete could space vertex adjacent short prescribe attempt sufficiently agent implication responsible overall planning example b circular model integer action operation subtract relate simplicity complete structure without system let specify target origin accommodate intersect separate circle satisfy geodesic set pass yet constraint action enable demonstrate strength target agent signal example sense information regard transition absolutely topological provide reference planning snapshot stage threshold seem plausible however cause threshold relation principled exist sensor function become must failure adjustment mechanism evaluate agent system paragraph introduction close loop control suggest range multivariate human mechanism absence current motion simple seem possess decrease fix point environment single simplify sense stay put specification first failure action behavior different stress result precede agent suffice structure associate requirement close control exposure relation necessary offer vary representation precisely map patch integrate map record annotated learn known topological presence topological valuable information loop closure otherwise observation extensive effort notion topological map hierarchy allow plan vary scale leverage topological efficient structure motion planning geometry sensor family algorithm know employ neural pose cell engine underlie cell field dense configuration make also represent system observation cell connect place evidence recently introduce encoding spatio context drawback cover intersection guarantee combinatorial duality turn idea leverage relation among geometric recognize well snapshot entire close snapshot agent capable fact snapshot introduce sense quantify reward g reinforcement innovation planning capability variety enable even contribute topological representation encode facilitate drive sensor inconsistent model architecture result improve connectivity snapshot capability immediate day neural network demonstrate ability perform include symbolic sensitive hierarchical structural feature physical even feed forward network show capable control token environment merging architecture capable match human play raw output internal representation maintain encode symbolic term solve simplification realization controller direct neural code hope snapshot architecture provable symbolic reasoning understand purpose whose property model mechanism expand stable threshold line characterization organization code perturb structural topological constraint analogy obtain coherent snapshot take nature analogy investigate collection threshold strict geometry snapshot could snapshot architecture may expand hand symbolic architecture discrete operation completely nature maintain evolve snapshot size quadratic sensor plan propagation architecture order weak set structure characterize half account equivalence provide agent duality symbolic space interaction dynamic formalism rigorously symbolic planning efficiency snapshot architecture certain find exist architecture represent predicate predicate course lies propose attractive principled analytical computational compound symbolic abstraction relate duality weak appendix predicate symbolic abstraction clear snapshot extension acknowledgement air office fa foundation duality go successful envelope positively term provide review element support memory overview provide intend current duality necessary formal actually job mainly result elegant exposition duality weak necessity structure weak endow call say negligible negligible element negligible weak two preserve denote relation formally construct symbol set fix intersection addition equivalence order notation stand relation derive generator relation compact set partial satisfy empty identify symmetric operator translate pointwise respectively realization weak empty point relation endow obvious denote realization structure identify duality base construction selection denote metric fixing explicit isometry hamming thought skeleton cube combinatorial complex diameter vertex say incoherent pair dual skeleton skeleton illustrate whose diagram form set realize augment complementary question implication implication record observer endow cube proper correspond face redundant use question coherent subset planning sense put vertex weak characterized interval define vertex fundamental quick finite coincide well connect graph triple modern generalization presentation another state precede dual finite median formula coherent determined majority vote value strong recall say deduce subset p aa say median intersection family pairwise convex convex subset induce graph subset vertex subset median preserve map underlie space odd via finite dual practical offer understanding geometry aside view categorical duality category possible element satisfy us realization tell weak nest pairwise space restriction cube relation face incoherent vertex finds remove improper element exercise tree nest robot capable move suppose sensor position say turn turn turn form set path whose please note choice point order matter correctness discretize sensor imagine description appropriate indicate exclusive case ignore underlie join sum external union endow iff iff abuse notation identify natural representative easy proper element cube product path appropriate value dual precede coherent vertex agent incoherent agent vertice spread circle agent capable question arc position question agent sense symbol agent agent result v observe representation relation difference clearly advantage deduce must note vertex white vertex form incoherent family nest example give none fashion symbolic category quick review notion refer reader category one introduce major unnecessary connect every assignment rather let easy median preserve map appropriate yield composition notion category construction together duality fp correspondence ff composition correspondence duality order statement theoretic speak aspect cover interpret term geometry conclusion translate boolean deal survey contribution category theoretic application recall fact proper restrictive business verify duality weak map coherent negligible weak denote median making indistinguishable concerned obtain weak set flexible structure easier evolve dynamically possibly since sensor binary pair sensor equip free aa nothing special never observe transition write implication believe imply correct vertex realization call realization motivate consistent rise maintain record discard incoherent view organization state introduction contribution connect non positively detailed non positively metric space suffice hadamard generality collective effort graph skeleton median skeleton develop paragraph topological space realization collection cc vertex result homotopy circle hand back example explain qualitative threshold possess observer model realistic extend graph situation define absence solution include cube cube embed convexity canonical piecewise thus although graph describe dual geometry dynamically real set motivated analogy idea kind update capture identity map observer yet regard nature pair maintain set represent observer identical belief dual underlie dual g strong complicated symmetric definition satisfy identity loop consistency constraint rewrite form q verify identity anti pair end turn trivial case exactly pair none proper vertex path particular contradiction therefore cycle evolution trivial snapshot empirical snapshot indicator hold satisfied compare conversely presence suffice snapshot write form trivial weighted unit pair satisfy iff snapshot proper fact selection coincide counterpart generality one choice conclude consistency decrease must selection finally snapshot triangle name dissimilarity observe undirected edge induce connect nothing equivalent chain mind whenever connect direct metric inequality conclude iff triangle actually structure need convention observe assertion observe lemma imply turn proof summarize progress snapshot triangle operation point iff edge acyclic let propagation eq map preserve suggest weak yet contain therein prove insufficient support planning study close point projection inequality v lk know gate pair non empty subset gate apply proof consequence gate exist equivalently kind reasoning empty lk lm uniqueness force study technical necessity explain recall notation order identity coherent coherent aa aa show coherent iff disjoint ba aa coherent complete also coherence vertex define coherent observation weak adjacent coincide range characterize follow weak correspondence point projection coherent short element replace step iff algorithm projection projection kt leave suppose precede proposition disjoint corollary converse case already coherent hence intersect henceforth lm u lemma mean propagation recall j jt jt tt since precede second precede self capable degree planning explore problem beyond produce viewpoint space formal notion ai notion space draw category fundamental able formalize early come equip engine notion extend define automatic category allow notion snapshot capable handle observation boolean despite obvious current maintain control evolve control learn sensor immediately mind duality problem motivate possibility together theory brain discuss introduce approach gps base human processing yield advance understanding role visual space dominate become numerous include attempt effect reason comprehensive perhaps evolve advance become fully base machine use operational sub goal reasoning effectively problem solve search flip serious memory absence form symbolic common follow concern uniformity formal capability unbounded resource reflect management difficulty discuss deal regard search leave ability intelligence return system disadvantage cognitive phenomenon limit modern course imagine provable property become setting human ideally bridge enable extraction problem conversely like formal connection cost storage cost planning implementation life emphasis symbol entity ability set well dealing mean player eventually self subsequently produce intelligence character organization collection word category quick introduction cognitive architecture category whose finite category object whose median preserve map specify space state suitably equip sufficient map agent universe realization capable operator present present sense simple basic cognitive space equip snapshot specify coherent collection atomic input loop produce heuristic heuristic combinatorial distance take automatically model serve solution agent work exposure snapshot architecture avoid symbolic symbolic abstraction way symbolic abstraction snapshot addition agent maintain bank already term engine present rise goal pre total abstraction encode aside future cognitive form symbol base agglomerative reconstruction architecture use reach search pattern formulate production happen substitute g snapshot architecture memory nevertheless implement snapshot architecture weak snapshot analog employ say agent observe resolve unless boolean exercise care topology result advantage snapshot drive another snapshot drive agent duality explicit formal extra example duality pose preserve snapshot ready learn one argue reasoning place agree seem wide periodic sensor expand space admissible include sensor statement importantly algebra quantify come necessity convenience quantify natural meaningful class operate algebra value think real value uncertain propagation would construct snapshot way recurrent code code cell stable al word proposition code analogy code stable pattern raw geometry perhaps possibly general consideration reader might characterize convexity theory evidence viewpoint section thm thm thm thm thm electrical systems school engineering rd pa usa school engineering university rd architecture capable support learn absence information agent enough ensure sensor requirement quadratic execute complex agent internal minimal every class subject agent state capable homotopy state provable property memory structure symbolic discrete positively rich convexity cycle obstacle memory human memory seem functional hierarchy system vs split scale address science action task explore map intelligence architecture one stand formal notion space memory system comprise history format argue architecture notion domain vast discuss agent universal learner optimize gain suggest result insufficient broadly formal advance provably intuitive property environment encode generic minimal obtain develop arbitrary encode observation whereby atomic provably correspond near projection generic sense equip sensor action behavior give instance interact environment natural power generally accept must support enough account exact abstract planning require representation eventually account transition review obtain description absence strong sensor obstacle rather impose precise characterize exactly small effective object call snapshot keep track state collection quadratic history implication atomic cycle crucial architecture formally cat see skeleton snapshot update encode transform contribution informally provable architecture absence encode impossible distinguish skeleton chapter rich topology chapter quadratic sensor storage quadratic time pick action learn result walk limit planning action search process height chain implication provable appear contribution briefly review topic arise distinct intelligence present explore relation implication trivial ambient avoid collection geometrically represent fundamental planning topological literature planning membership reduce storage plan planning model play role traditionally euclidean generalize strong convexity enable cost greedy demonstrate oriented topological use encode causal symbolic generalize formulate self evolve pairwise intersection record necessity planning idea encode fairly specialized additional principle available signal interact result control mechanism may realize simplify simulate analogy intersection activation sensor sensor topological mapping competitive necessity pose general formalism ability plan essentially flow construct maintain allow curse guarantee sufficiently rich account class approach come largely ii result topological shape basic drive agent efficiency mechanism agent necessarily state possible control early stage feasibility mechanism agent gain choose formal weak geometry space repository elsewhere discuss observation numerical implication claim iv contribution control validate extend discussion result literature appendix environment sec integer snapshot snapshot new snapshot update direct satisfied follow snapshot weak iff direct path snapshot construction snapshot maintain frequency trajectory agent try trivial snapshot snapshot say trivial constraint snapshot snapshot eq choice vanish justified evolution snapshot observation snapshot snapshot snapshot obtain snapshot snapshot k characterize snapshot snapshot trivial return define snapshot accordingly imply acyclic acyclic henceforth utilize paper restrict attention endow agent trivial snapshot
form provide influence simplify influence bound function consideration eq er rao bind fisher mle point design robust behavior hard set come include local regularize coincide property high order advantage nonconvex loss consistent loss suboptimal viewpoint high scale oracle unless exponential proof paper existence local proof eq q rsc inequality old eq scaling imply element imply inequality give finally existence define interior program argument adaptation theorems primal define c subgradient condition local point rs apply interior program satisfy imply equation zero subgradient condition fundamental mean condition restrict region sum obtain q oracle implying imply u u fu w u w covering argument let cover triple furthermore analogously imply hand side arithmetic mean eq finally invoke concentration result average imply take union least plug inequality far note argument establish inequality inversion relation return assume desire selection imply plug proof eq bounding cover hold complete construction finally minimum program condition minimum norm regularize apply program rsc apply inequalities eq recall interior follow identical left derive remainder rsc provide local note combine q feasibility hand lower bound optimum inequality inductive rsc q together q give combine inequality conclude side hypothesis scale conclude complete suppose iteration simplification rsc denote hence hence index inequality eq inequality eq complete provide technical proposition establish statistical section conditional condition bound variable I desire sum sub main supporting provide subsection general proposition everywhere definition lie triangle theorem truncation well particular lipschitz truncation eq expectations function truncation provide hold eq note quadratic unit exponential proportional guarantee w inequality inequality extend domain replace accomplish follow fix inequality finally proper rsc cauchy eq gaussian finally q proof gaussian condition denote homogeneity inequality property calculate define process gaussian calculation expectation lie inequality lemma function inequality nm inequality apply arithmetic mean define define event inequality define eq eq analog lemma cauchy schwarz side sub I average exponential parameter proportional hence version put piece arrive notation define event I proposition modification replace every follow arrive familiar inequality remainder identical proof fairly consequence eq condition satisfy u p nn careful inspection reveal restrict attention exactly restrict rsc al imply satisfied hold estimator appear body paper rsc imply conclusion stable eq may old second inequality consequently exhibit exponential ordinary note hand hence eq finally solution eq lasso cm ex ex em department school pa applicable contaminate tailed covariate fairly loss curvature within radius minimax lasso nonconvex place fact equal correct support case immediately nonconvex local loss useful consequence optimization regression regularizer nonconvex possess outside descent initialize linear point region optimum convex regularize regression obtain increase efficiency result finding ever robustness statistical scene box toward quantify procedure notably huber huber estimator property class theory construct high mostly g paper light estimator paper globally arise possess curvature new curvature linear type ordinary view normally ordinary sub least square converge whereas usual assumption show covariate weak covariate normally distribute estimator inconsistent observation contaminate response exist wish extend dimensional estimator estimator version dimensional estimator set high robust deviation contribution provide condition optima statistically presence condition strong convexity true conclusion strong convexity previously traditionally condition loss robust function interest restrict convexity main provide curvature covariate agree least sub study estimator distributional consistency estimator question contribution estimator advantage nonconvex huber dimensional reason nonconvex convex justification viewpoint function rise nonconvex cauchy prove regularizer scale sparsity normality number use correspond dimensional sense regard nonconvex strong provide technique construction extend optimize propose solution devise region inconsistent even dimensional stationary within consist separate situation nonconvex optimize obtain sufficiently initialize nonconvex rigorously second curvature successive iterate lie stationary statistically suffice huber covariate optima consistent optimize huber optimize possibly nonconvex estimator literature note involve huber step optimize loss step paper resemble technical regression estimator paper category addition notion optima optimization regularize go beyond composite gradient another relate fan al develop robust huber strictly estimator analysis huber still relevant provide gradient apply step differently tune accord distributional additive noise reveal choice function albeit factor analysis convex cover nonconvex suggest consider primary alone remainder organize basic concern regularizer concern robust distributional proposition concern estimator conclusion oracle conclude variety brief proof proposition contain supplementary universal constant write simultaneously write restrict gradient subgradient provide background generalize cover theory eq I function program general setting regularizer may nonconvex include feasible scenario convex imply stationary certain statistically wish function outlier misspecification classical regularizer appear program encourage appeal review estimator treatment basic concept book cite define observation estimator q always choice error function exist first check exist everywhere degree freedom heavy distribution nonconvex desirable view explore cauchy maximize check although third always turn result intuitively equivalently q outli contamination literature exists completely eliminate give estimator nonconvex expense estimator measure article et al whereas estimator describe outlier covariate concept intuitively behave motivate large estimating follow define sequel allow distributional covariate e form weighting estimator consider choice take indeed effectively elliptical likewise close g influence hill estimator around effect leverage term variance function note take equation reasonable see remark estimator form finally regularizer analysis composite objective function satisfy property amenable regularizer scalar satisfies vi say amenable everywhere define amenable penalty fan li amenable due mcp amenable amenable amenable regularizer oracle point discussion normality concern general statistical stationary restrict regularizer next interpret consequence generalize proposition covariate hold high lastly provide establish equal nonconvex amenable feasible slight minima interior local maxima require satisfy rsc rsc note impose outside radius rsc use region nonconvex cut behave main condition stationary local region function rsc amenable suppose eq stationary contain distributional covariate error come play rsc prescribe scale one local rsc alone fact case robust stationary actually oracle truly actual neighborhood around global lie ball omit local program suppose rsc condition guarantee section regularizer rsc within min state unweighted similar assumption well give twice differentiable ball amenable addition proof build upon develop simple radius completeness modification necessary obtain form previous concern optima oracle careful local optima essentially optimization previous simultaneously huber upon nonconvex concern estimator wu extend allow grow prove normality convex program oracle nonetheless convex standard estimator number apply normality sample unweighted case type normality amenable program v provide derive slightly modify useful estimator composite guarantee region rsc hold denote rewrite q composite iterate stepsize soft thresholding q iterate take descent near close enough denote remainder strong convexity satisfie slightly relate taylor rsc exactly rsc repeat restrict smoothness condition fairly mild simplicity scad mcp regularizers appendix composite iterate linearly rsc amenable successive composite descent obvious radius iterate remain expect hold rsc result composite outside rsc cause stationary point outside region proximity ensure trajectory nonconvex estimator derivative estimator bound view efficiency long guarantee composite inconsistent optima nonetheless theorem stationary radius optimize robust even converge statistically function convex output initialize dimensional consistency produce optimum appropriately composite gradient converge stationary final consistent agree oracle use amenable penalty optima single initial asymptotic efficiency property finer grain al optimize order possibly method mostly justify composite estimator step theoretical efficacy importance step result throughout generate model simulation consistency robust various failure minimax n standard stable suppose scale problem lasso establish proposition ordinary yield estimator n rate cauchy scale run level equal huber regularizer huber loss two huber initialize penalize yield consistent curve align error plot rescale b p huber cauchy dot loss size tailed huber cauchy robust yield consistent predict proposition normal also statistically rate losse significant normal represent contaminate constant value otherwise rise statistically also ordinary robust upon run relaxed distributional mean cauchy scale run trial huber initialization theorem huber difficult see nx exponential large huber yield statistically relatively slow simulation large huber run cauchy path huber panel huber choose distribution optimum preliminary log initialization plot error red roughly huber sublinear convergence initially convergence locally within radius indeed plot outside rsc converge unique global tolerance implementation cc show path huber loss iterate enter initialization huber slight perturbation local restrict green initialization converge predict iterate point need proper initialization statistically huber random initialization green blue initial iterate satisfie
ignore validity linguistic law linguistic law k kind law unit kind type law link law measure length term linguistic write text law law corpus large letter refer frequency text representative law law see name word database recurrence range autocorrelation lag entropy size length law lexical network law good linguistic ref historical state accord th type frequent word modern analogy tail motivate frequency formulation map attention part intend describe example idea variety see ref therein word word database word book increase token end draw english article separate dot trivial database strongly law summarize capture text linguistic list quantitative observation motivate corpora question address law determine around allow law discuss linguistic law law ref notion scientific law quantitative obtain special validity science probably work scientific ref rule violate law straight forward identification linguistic law statistic theory linguistic law notice law affect production meaningful short persistent text strict text sufficient law linguistic law syntactic law role statistic quantitative distinction language law conventional language law nature universe modern physics discuss law refer interpretation statement law frequency decay interpretation collection text vocabulary size text mention unlikely law modern physics point predict corpus determine magnitude law include possible statistical subject linguistic law detail linguistic law language degree corpus ii relevance law quantitative rich principle entropy sec argue linguistic law fig availability linguistic law linguistic translate precise address describe law compatible representative fundamental discussion importance three list linguistic law visual inspection analysis linguistic law widely fitting often combination transformation law straight axis logarithm law visually valuable fitting scale fitting uncertainty distribute justify quantify goodness insufficient evaluate unable assign suit rigorous central fitting assume validity search account correspond multidimensional parameter law kind log ml power distribution law review article ref g cut law third list regard gaussian fluctuation assume j maximize fitting comparison form compare likelihood function criterion calculate average validity linguistic law value compatible linguistic low strong violate computed fraction realization assume linguistic law first kolmogorov linguistic law second third kind fit scale plot correspond english wikipedia representation formulation linguistic law conclusion fit law scale linguistic formulation likelihood compute case frequency frequent count contribute observational quantity occurrence count count point large dominate fit ref fit straight log across statistical either frequent fitting frequent asymptotically formulation law assume datum fit law describe fit reflect different weight high case surprisingly vary database large computed bt word draw ref failure surprising nan statistically previously alternative description publication ref generalization assume assumption fluctuation assess validity unclear negative validity violate text letter obviously show word fluctuation usage write likelihood violate affect analysis book thought approach account correlation small approach come method law straightforward exist show position book agreement generally asymptotic value generation linguistic law sampling affect model unclear extent bt article solid line dependence reveal scale ref law text natural relationship generative process ref range text skew recurrence law fluctuation underlie nan need nan word typical consider every probability global frequency usually text formulation frequency lead nan implicitly explicitly derivation figure connection law usage reproduce fluctuation observe particular fluctuation vocabulary scale linearly central limit ref taylor different structure book claim linguistic valid close inspection claim chapter critical linguistic law evidence support argue linguistic law sense selection compatibility datum compute statistical test plausibility choice original matter picture straight application linguistic law good description strict law linguistic law capture see unable text existence additional process ignore describe violate write p value linguistic limitation necessity able linguistic law incomplete meaningful linguistic allow fluctuation generative relevant ultimately model text interpret explanation linguistic law despite attention fluctuation scientific linguistic law fully long range variation fluctuation estimation text quantitie linguistic consequence retrieval generation law use independence applicability law artificial text fluctuation generation text impose constraint apply law linguistic rejection emphasize law rigorous test reference assumption observation treat case nevertheless law fitting scaling law acknowledgment appendix list project book filter supplementary remove symbol letter string symbol consecutive space english wikipedia filtering keep symbol separate letter law appear fig unique word word type word count dictionary b book word word wikipedia success wikipedia size large rare word strongly database universal language fit availability accuracy fluctuation fluctuation much simplify
architecture linear relationship cell fundamentally pathway trace plot large scheme also effective mode efficiency effectiveness attribute hamming scheme change across mutation hamming sample large exhaustive enumeration impractical block require significantly effort simulation fail lead scientific model discrete object regression analytically material would observed response covariate contribute explanation observation model problem quantitative trait concern nucleotide phenotype observation response covariate explain redundant perfect consequence set challenge truth range mcmc sampling massive block sampler conditionally block size hamming sampler ham block gibbs sampler block hamming bottom cpu times integrate autocorrelation effective ess estimate compare plot hamming block indicate able identify relevant frequently inclusion strong dimensional simple hamming ball sampler integrate time ball sampling block require exhaustive enumeration latent probability sampler particularly effective example utility involve covariate application hamming factorial hmm chain represent discrete whose correspond hide length challenging comprises rely approximation sample conditional sampling scheme easily become condition three different ball give balance radius block strategy big application hamming sampler performance standard sampling provide implementation actual toolbox currently computation trivially advantage hardware graphic processing unit hamming sampling update also scheme evolutionary finally believe many explore field conduct develop description tumor deconvolution example pair read variant allele site distribution allele p ki attribute tumor prior hierarchical equivalent automatic selection tumor population specify probability ki ki simulation si tumor deconvolution sparse response total small represent follow variance assign conjugate gamma hyperparameter scalar distribution obtain density c g hyperparameter hyperparameter prior inference si typical sequence interpretation presence absence different different row ki x k bernoulli parametrize additive k w whole determine px ball step separate gibbs p x I ff time normalize forward pass ff bs time inference model si factorial hide markov support uk research new grant ref mr trust z university introduce hamming markov involve discrete iterative polynomial generalize conventional big datum control statistical illustrate generic algorithm statistical across include modelling typically rely mcmc posterior object proposal explore effort distribution state receive example classic wang unobserve value discrete matrix conditionally py intractable exhaustive entire metropolis gibbs sampling subset q exclude sub vector allow resort hasting major difficult possibly incremental lead local mode exhibit address mcmc high name hamming employ auxiliary slice slice slice significant spectrum block strategy express novel scheme computational ball enumeration realistic approach panel ball vector hamming joint factorize auxiliary indicator normalize ham I maximal hamming column hamming ball consist whose behind hamming ball sampler augment admit marginalization recover target hamming step update alternatively hasting accept reject q q generalization scheme latter radius become enough si hamming sampler crucially ham conditional summation admissible matrix inside hamming cardinality enumeration inside hamming would slice element necessary ensure scheme ergodic differ e step draw observation necessary ergodic figure illustrate ball hamming require subset specify address deconvolution mixture efficient block could factorize p factorial pool dependent divide may use operation sequentially precisely split sequential scheme hamming incorporate iteration special equal purely ball scheme algorithmic illustrated detail si find time hamming block block hamming scale accord applicable hamming ball sampling flexible control ideal hamming hamming update shall outcome actual beyond simulation circumstance advantageous flexibility balance hamming ball conditional denote maximal distance allow si b discussion alternate
auxiliary particle define empirical dirac conceptually step step put emphasis particle step target distribution simulate correspond well suffer dimensional setting dominate importance proposal degeneracy distribution inference fairly forward adapt limited space proposal graphical coupling simulator instance couple sampler backward construct efficient adapt proposal arbitrary latent space derivation class simple key presentation orient class degree w iw qx x replace obtain consistent motivate auxiliary explicit relationship particle specifically make justify properly weight properly pf take px interestingly construct target possible density point wise access class approximated procedure typically correspond internal carlo unbiased member function place proposal distribution validity interpret sampler qx x qx u properly furthermore standard example refer appear generate correct implement modularity fact nest mm execute categorical return properly properly repeat precede return inference let denote wise sequentially resample weight often kx x kx particle notational give kx probability z kx draw approximate form loop resample particle step equal z analogue initial accordingly access k z multinomial probability z proposal weight replace procedure despite condition denote point converge accuracy procedure leave work relate ideal procedure modular sample use generate properly uniformly definition path degeneracy sampler time improve procedure use backward simulator smoothing algorithm particle pass backward approximately uniformly w categorical probability x assume unweighted particle conjunction procedure particle appendix direct use chain standard special imply proper weight interest recent couple problem variable sampler component internal sampler construct way require estimate nested algorithm relate develop implementation spatio temporal dimension distinction sampler comprise simply correspond variable regardless sampler easily particle effort construction call particle furthermore match ease essentially cubic markov implement target distribution dimensional particle smoother review however inconsistent approximation systematic mention validity mention increase three bootstrap distribute knowledge result tb ess resample filter filter latent measurement dd state sampler fully adapt proposal constitute target level properly sample operate bootstrap proposal actual sampling conditional datum exact filter standard evaluate kalman independent involve reflect mean intuitively correspond resample eq effective particle resample step computational bootstrap present display conduct experiment agreement trade possible maintain large probability improvement block imply running give around perform satisfactory particle final study measure location look north year decade spatio define compare region model location year rectangular essence figure consider configuration relaxation c north c north north number estimate site rectangular method level target distribution operate proposal structure problem agreement receive uncertainty illustrate three level coincide visible year particle keep low proof concept particle attain thousand close challenging project contract contract proper weighting theorem manuscript turn square relationship serve article section consider manuscript htb international conference france methodology require
solution tensor mode condition projection mode multilinear conditional subproblem project vector line correspond differently unit eigenvalue identity size include lagrange multiplier indicate orthogonality vanish non substitute get maximize eigenvector n multilinear respectively eigenvector calculate large eigenvalue limited rs rs fixing without maximization idea motivated model fix freedom impose orthogonality reduce variance rs model differ start dependency strategy generally subspace also summarize rs remove set c order extract capture variance effectiveness relax tensor subset face subject challenge subject probe binary face subject rest random repetition rate follow set split repetition rank five exist full select sort classification recognition recognition much worse include variance study save ccccc cc pca face std repetition highlight top bold easy rs compare well rs five least rs size overfitte top highlight consistently pca outperform rs rs outperform still face extract rs face capture rs face clarity sort variance semi orthogonality discuss sec moreover though capture less consistently rs less surprising maximize rs rs iteration convergence evaluate method get experiment relaxed help rs improvement achieve rate rs rs outperform average face performance rs improvement rs improve recognition rs improve rs rs control experiment relax multilinear pca explanation rs investigation multilinear pca setting name multilinear relaxed rs learn tensor capture orthogonality impose semi orthogonality capture feature achieve generalization fix start projection recognition show rs good overall compete addition effective semi future learn rs feature mode separately acknowledgment grant special grant remark china edu component pca learn multilinear extend multidimensional tensor tensor multilinear orthogonality paper propose novel tensor impose capture orthogonality generalization rs fix vector increase variance datum rs compete whole relaxed effective classical input pattern tensor video tensor datum monitor tensor break address multilinear tensor directly main base dimensional low rank generalize generalize side sided projection reconstruction multilinear extend high base tensor minimize greedy uncorrelated multilinear maximize successive derivation low mode usage orthogonality tensor build n consist tensor projection consist present rs derive successive conditional introduce relaxed start orthogonality follow available sample kronecker product consider pm project projection
minimal inequality tell give enyi degree concentration variable minimal value copy imply weakly collect reasoning precise give heuristic give value graph replacement imply two algebraic connectivity note sampling main estimator figure value various observe observe fit well estimate case greedy sampling sample chernoff hoeffding follow inequality kullback leibler divergence mean kk expectation proof lemma os adjacency I need u directly inequality divide bernstein inequality heavy degree discrepancy variation bind exponentially space approximate depend surely part light heavy u iv u iv u u n v bernstein e nn union next property every degree least edge discrepancy property probability property ab b da da bm ba eq right side side b replacement qualitatively simulate independently plot pair contaminate sampling op database video assessment pair live attractive dataset pair live include video distortion compress simulated ip transmission wireless video video comparison therefore comparison ground obtain pair video show experimental video live database interesting collection increase without replacement ranking assessment total publicly live live distortion totally number internet pair occur pair comparable three score pair live database stability exhibit replacement dominate practical initial transition point graph random replacement random sampling performance sample large gap vanish performance rely comparison make easy adapt situation adopt greedy may gap scheme vanish enable reliable rating helpful tool exploit pair comparison xu national china china national program china grant cb cb grant project crowdsource extensively pairwise pairwise via sample estimator estimator stability graph limit vertex finding compare greedy initially replacement item replacement computationally trivially world analysis crowdsource os enyi internet growth crowdsource crowdsource employ community crowdsource researcher participant economic cost laboratory researcher internet conduct computer approach test result control control experimental propose randomized conduct accommodate combinatorial aggregation incomplete inspire statistical rank theory computer mechanic rank norm triangular flow harmonic flow perspective provide general model pair possibly incomplete crowdsource sampling replacement sampling replacement pick whole regardless replacement pair chance possible simple sampling replacement pair os stochastic start edge uniformly dependence experience could design crowdsource exploit topology clique os r enyi least necessary rank edge collect maximize equivalent maximize small nonzero algebraic unnormalized cost greedy prohibitive large effectively algebraic connectivity os enyi collect internet crowd passive benefit collection trivially weakly viewpoint simplicity generality situation online rating crowdsource interest paper greedy attractive crowdsourcing trying scheme theory experiment paper scheme measure value enyi replacement estimate value value associate random approximation increase sampling replacement random replacement graph consider analytic conclusion support base compare recommend stage initially replacement recommend computationally trivially remark future term crowdsource crowd distinguish public crowdsource benefit include crowdsource amazon probably popular provide seek internet crowd task request website besides diverse rapidly idea internet crowd provide break platform million community bring crowdsource ranking ranking avoid member item website create survey either help image document recognition mining game worker complete micro crowdsource researcher find expert aggregate non expert could crowdsource reasonable ranking rating pair comparison widely variety science machine rank centrality draw algorithm able aggregate global provide pair e uniform angular ranking map pair may receive comparison partial comparison apply combinatorial flow flow rating harmonic flow base active subsequently categorization video co knowledge gain effort active problem rank rating reducing must collect scoring reflect euclidean sampling approximate np minimum arc active complexity rank vector rank apply crowdsource scenario rank small number sampling maximize nonzero arise subject analyze sampling pairwise discussion voting choice rapid growth spread internet crowdsource technique scenario typically pairwise choice use scale purpose preference look lx pair item lead feedback arc set problem complexity benefit square global ranking extended clique triangular subgraph admit decomposition satisfie flow satisfie locally globally local characterize triangular cycle involves long arise cause sampling without unnormalized algebra laplacian characterize subsection sensitivity score perturbation give parameterized system
weakly lemma positive ml concavity likelihood concavity model identify imply asymptotically normally establish convergence estimate procedure likelihood use solution present theory option wish correspond maker ten option softmax temperature observation parameter explanatory estimator decision simulate make figure explanatory draw gaussian response quasi newton converge convergence observe estimate represent solid represent horizontal theorem plot distribution compute great closely distribution converge importantly statistical amount biased insufficient bias parameter correspond choice choice option treat vector interval imply ensemble parameter repeatedly simulate hold explanatory black mean parameter option figure apply parameter estimation figure value total explanatory draw accord response scalar behavior mean confidence represent true dash around estimate clarity omit figure show estimate width interval scale true dash line value repeatedly hold explanatory confidence address problem objective nonlinear nominal value linearize stochastic make multi armed bandit problem option call analogy slot option probability solve decision pick receive reward option maximize value reward decision agent reward must arm reward reward arm information tradeoff machine bandit subject machine show excellent armed bandit algorithm know case attribute subject structure design stochastic human human armed bandit would belief facilitate design system design solve reward ir part maintain depend belief decision introduce assume belief reward parameter belief reward spatially embed arm spatially reward interpret absolute belief complete confidence posterior select compose agent option respectively belief identity choose heuristic function value arm temperature schedule assume inverse distribution decrease softmax quantity linearize recover near nominal relative prior fix value include deviation simplicity exposition inverse denote nominal root follow get element diagonal element deviation must imply upper lower small bound value linearization valid variable linearize heuristic q explanatory linearize define form provide parameter describe simulate algorithm various figure parameter linearization linearize true converge linearization linearize objective correspond true true horizon confidence implication robust linearization realistic empirical study horizon statistically guarantee amount get depend linearization true value algorithm sufficiently reward gain regret initial effectively make useful except uncertain noise confidence estimate width order magnitude display exhibit away confidence interval precise value simulate true estimator converge observation value repeatedly parameter imply linearization point grow dash value form repeatedly simulate confidence imply simulate weakly informative linearization algorithm line linearize grow ensemble repeatedly compute line panel normal confidence interval much estimate omit linearization local effectiveness sensitive nominal linearization linear linearization fortunately aspect unique objective estimate know linearization estimation linearization point result estimate linearization intuition choice linearization show broadly linearization relatively insensitive linearization behavioral intuition parameter base datum subject experiment section fit experimental select nominal linearization linearize model fitting behavior review run bandit tasks nj usa protocols participant amazon web task platform participant play could point goal obtain part arrange grid reward choice report game dynamic reward structure participant task stochastic participant value option arrange thought landscape reward landscape landscape landscape landscape flat dimension follow along landscape landscape sophisticated strategy landscape task task rate cumulative approximately achieve remainder low subject landscape landscape high subject outperform frequentist subject quantify wish stochastic make high subject landscape task level four landscape combine identically iid apply estimator subject subject subject value performance fitting matrix iid population four category individual subject table population four column deviation parameter deviation nominal compare standard deviation high likelihood original comparable subject clearly differ level noise uncertainty value decision represent great placing factor encourage value explore help reward subject region allow answer question subject task performance category separately deviation difference landscape confirm subject precise side word distinguish subject match human linearization objective difference cc cc e regret growth reward landscape high significantly surface cc power cc law motivated decision make make function derive use formulate important softmax make objective nominal point parameter perform linearize use could true value depend parameter represent prior variance easier readily hold extend generalized logistic softmax objective procedure nonlinear linearization fit develop human subject science technology develop provide quantify multi armed bandit facilitate principled development machine corollary example department university nj towards human contribute systematic infer make behavioral softmax figure human make softmax making derive likelihood asymptotic distribution show nominal fit credible limit human significant difference relate variety decision option scenario agent receive maximize example air traffic controller select reward option task challenge especially reward air human enable much research decide option condition lead make empirical define option model option operation differentiable operation softmax operation plausible operation goal decision objective explain observe q form constant decision softmax make relevant several behavioral decision making process identification seek design step determine determine equivalent call step rigorous softmax fitting infer intuition represent develop present estimator algorithmic credible armed setting qualitatively reproduce experiment infer belief class since commonly motivate pick two option option rate increase discriminate slope explain decision option scalar parameter picking option softmax study particular explanatory belong multinomial regression class softmax logistic value vector multinomial generalize explanatory objective appear restrictive locally human decision work fast parameter ensure converge instead likelihood convex imply estimation optimization contribution condition maximum parameter convex condition matrix operation derive rao another contraction operator block product analogous hadamard develop composition estimation procedure general nonlinear nominal linearization datum remainder define softmax define softmax review iv softmax converge vi model parameter softmax model apply linearization fit predict expect know explanatory explanatory height posteriori estimate unlikely ml solve problem frequentist true framework standard summarize answer depend concavity statistical q identified yes concavity observe fail vector mi unable value follow weak ensure design estimating answer mild regularity expect hessian limit hold use n permit tool interval estimate test obey er study likelihood condition reduce optimization problem condition ml concavity operation product product prove operation hadamard matrix value real block block size denote block hadamard whose define e thought analogy hadamard product rao compose sized hadamard product two semidefinite liu analogous rao let partition square compose block preserve hadamard special rao
tx f ta next particular consider gradient demonstrating aforementione unchanged iteration iterate unlike similar however rather biased update iteration end gradient iteration highlight result motivate schedule update store cost however since storage finally store low expense slow optimal update epoch storage frequency optimal I epoch iteration denote straightforward new combine specifically schedule show schedule exhibit storage depend likely incur computational per iteration hand cardinality I ss concluding framework incremental second platform analyzing help asynchronous designing sophisticated special setup assume gradient estimate iteration ease exposition epoch case epoch replace randomly brevity quantity epoch quantity q epoch size choose iterate schedule immediately obtain linear specify value theorem corollary setting satisfied sufficiently epoch similar ready asynchronous version capture make manner key algorithm read iterate read schedule schedule iterate update incremental algorithm schedule update processor hence change iterate correspond maintain counter track denote iterate delay integer capture parallelism typical read time section key asynchronous sparsity convergence norm depend small frequency warm asynchronous analysis asynchronous epoch asynchronous schedule consider rate epoch asynchronous variant calculation give fast since run sparse linear speedup asynchronous case complicate unlike epoch iteration positive epoch modern processor constant asynchronous variant schedule give free asynchronous compare decay version dataset convergence implement algebra operation eigen website normalize lead lipschitz constant choose recommend speedup asynchronous speedup define runtime speedup achieve surprisingly speedup higher furthermore low speedup report experiment compare stochastic particular variant describe performance variance empirically verify asynchronous variant figure complexity epoch versus runtime core outperform qualitatively similar version see outperform observe b sim news right right dataset core descent develop primary provable obtain asynchronous variant like asynchronous obtain speedup typically encounter explore analyze variant bregman depend index clear expand epoch define follow follow schedule substitute fashion third substitute particular gx apply choose strongly inequality eq following use recall algorithm term expand manner equality simple algebraic calculation way directly nature second repeat triangle gm fact lipschitz finally last definition equation gm inequality third inequality get gradient add substitute equation definition get index manner define use follow follow manner insight differ change follow fashion lemma since epoch turn identical since combine theorem detail fact mention manner q recurrence notation ease last delay particular substituting bind get q use substitute bind get q bregman use follow inequality constant eq follow fact linearity negative bregman remark sum exploit suitable stochastic reduction thereby sgd although advance scale still process sgd key new asynchronous parallel method provably inspire influential two core framework discussion ii asynchronous parallel formal special key broad understanding attain linearly processor concrete illustration asynchronous reduce
study de cat response lead ability cat pl knowledge support generalize nominal response select fisher cat response category independent belong prove mle selection consistent go infinity mle ability efficient significance nominal full capacity choice item comparison treat false second cat nominal major address cat allow answer cat efficiency none operational program allow traditional paper become reason program decide cat mode clear provide environment reliable ability error mistake long feature cat propose cat argue impact response cat carlo simulation foundation cat rigorous assume multiple item response algorithm give previously wrong setup allow previous item however experimental cat select information accumulate scale cat response conditionally response pmf give previous answer ability maximizer conditional give observed decision cat incorporate need regular cat nominal probability switch correct wrong item pool probably organize property section cat cat establish property finding study illustrate conclude response cat quantify denote response item category nominal number satisfy follow identifiability determine c simplify datum nominal recover pl difficulty particular imply likelihood take depend positive upper first consequently item draw bank whenever bank useful result gx illustration jointly universal cat item category response govern nominal response define scalar I select sense however parameter cat response measurable response structure cat able fisher information obtain mle asymptotically knowledge try adaptive nature maximize level belong course number restriction exposure benchmark performance expect asymptotic sense call estimation ability process conditional log likelihood response nominal response item unfortunately root acquire q cat item item resp value strategy resp acquire response focus asymptotic root item normality maximize adopt heavily martingale score nj item moreover therefore measurable vector follow martingale n complete establish consistency selection selection let item strategy proposition martingale martingale law around consistency guarantee fraction last remain suffice j continuous infinity recall rewrite q subsequence follow establishes complete second jointly establish information maximize strategy maximize strategy denote start show show need nj therefore martingale increment variation g obtain eq cat response allow choice go e previous impose result item unlike include though item correspond formulate correspond item complete previous algebra response attempt item algebra response contain cat assume govern every I c pmf nominal govern nominal conditionally independent assume observe item specify completely left probability determine nominal response whether ability beginning case cat possibility random denote indeed formally reveal stop select information say vector characterize value measurable cat accuracy final maximized maximize fisher nominal current provide response maximizer conditional likelihood item strategy conditional probability follow define response cat every property go martingale score selection increment finally complete response proceed choose previous measurable respect measurable martingale respect square q consistency without item martingale moreover strictly martingale follow taylor vector response event j prove consistency claim need go ratio strong consistency continuity x go strong condition normality selection indeed regular cat need eq item current regular cat nevertheless q theorem particular information number case probability difference indeed martingale stop increment martingale limit take lie follow ratio go application theorem cat distinct since review item study illustrate result cat category follow interval pool analysis respect item recursion possibility item rmse summarize ability square exception circle response dash square line achieve standard cat interval cat result validity illustrate actually c cat dash square response dash information ci cat plot I allow first design cat nominal response belong consistency mle fisher binary nominal reduce pl one indeed assume unbounde rather item bank moreover mle heavily proof general nominal response cat design response propose estimator strongly
recover eeg contrary measurement produce analysis nearly zero theoretically compressed isometry adapt rip property equivalent signal sparse put zero draw zero analysis eeg signal recovery recovery eeg signal sparse representation incoherent way coherence super eeg third eeg signal approximately signal eeg system channel process jointly eeg slightly dictionary channel way channel support generalize single straightforwardly code correlate signal preprocesse eeg analog analog eeg signal sample sampling channel eeg signal measurement coding exploit tensor channel eeg less correlated low compression motivate channel eeg signal eeg find newly eeg singular nd try piecewise exploit structure enhance signal recovery exploit channel signal simultaneously encourage method criterion optimization transform multiplier analyze eeg eeg recovery recovery simultaneous achieve error mse mean cross eeg paper organize section exploit structure system simultaneously multi eq put sequentially reconstruct compressed exploit low structure formulate singular variety method eeg signal structure channel eeg measurement formulate simultaneous low nonconvex norm sum e sum newly programming besides step process core processor computational decrease experience acceptable eeg group detail material recovery eeg signal kind gap candidate measurement proper recovery simultaneous matching simultaneous greedy pursuit argue nd eeg recovery cs eeg dictionary wavelet subsample quantify value quantity mean value imply formulate eeg channel vector eeg signal reconstruction variant form percent mean index similarity eeg mit eeg database http www bin intractable without anti intervention channel international eeg position eeg recovery segment channel segment eeg eeg channel eeg eeg frobenius take segment eeg compress reconstruct reconstruct omp fig omp different gap gap slightly care accuracy analysis computational complexity b b section interior admm mse cpu interior optimization value interior admm outperform one speed admm fast rest worse acceptable recommend admm candidate channel eeg nd dictionary eeg exploit recovery compress eeg recovery norm encourage use low solve nuclear optimization criterion exist eeg cs channel eeg show optimization van deal single channel compress eeg enforce rank channel eeg alternate eeg reconstruction computational candidate compressed sense method enable successful compressed eeg sparse compressed consumption wireless eeg multiplier admm recovery rank eeg take signal wireless central eeg frequently
algorithm demonstrate text network lead speedup importance continue grow big enable wide amount increasingly machine significantly fundamental couple important machine spread machine big fit storage one require storing keep machine likewise distribute inference efficiently furthermore server face communication iterative nature algorithm produce huge amount traffic document specifically bandwidth memory bandwidth potentially bottleneck achieve large amount process constrain fraction need communication key pattern scale
three validity nonzero element goal separation nonzero two number sample correctly reject normalize arise nontrivial regard support support nonzero insight whether adequate reflect total correct value classification large measure correct decision reject incorrectly classify correctly classify measure correct block give value induce give insight approach classification correctly incorrect classification nonzero
frequency proportion share across let atomic measure conditionally identically base draw precisely atom hdp share place location detail describe come population population segment boundary segment pick origin distribution prior proportion correspond population stick impose order atom atom efficient describe monte mcmc update turn probability slice allow measure update make forward filter backward hasting proposal update model metropolis detail matlab implement scheme measure hierarchical dirichlet process model sequence index dna sequence assign chinese restaurant hdp auxiliary form hierarchy infinitely simulate address sampling
provide obtain positive th q exist configuration besides leave investigation parameter configuration dual stepsize constant strength convergence decay use propose uniformly uniform sampling well adaptive consider erm handling extend wider allow several regularize conduct competitive sdca sampling provide fair dual
channel datum manually identifiable pattern formalism wavelet transform via fast wavelet pyramid decomposition forward inverse analog wavelet cascade consist filter pass uniquely bank end decomposition detail scale fine detail coefficient vector
compact well map situation nonlinear comparable space computation complexity suitable natural term structure matrix past randomize embed locality sensitive come suitably matrix efficient achieve complexity matrix entry variable define require reduction matrix space transform denote
formulation algorithm calculate let q use remain unchanged however converge figure viewpoint soon totally influence intermediate estimation relationship monotonically decrease decrease first figure smoothed pattern
basic property kronecker apply define problem apply cca recursive correlation combination uncorrelated determination tensor constraint decomposition tensor canonical x pp pp th suggest classification collaborative linear instance linear end kernel extend case projection project high feature map dimension infinite accord follow pn proof appendix trivial follow least positive definite cholesky decomposition similar rank recursively maximize obtain pp mr instance complexity straightforwardly offline complexity dominate accord complexity determine size respectively small effectiveness image annotation follow label used percent datum specify accuracy structure least
need mapping unstable accuracy physical function unless efficient inexact exist closed inversion target I term memory become propose span reduce rank alignment impose plug product matrix reduce set training benefit requirement practice raise accurate empirical solution differ depend adapt eigenvalue generalize project span eigenvalue notation projection eigenvector complement projection residual k ns n squared
compare vary conditional satisfied relevant undirected separation dag separation supplement structural emphasize apply much wide furthermore fact undirected cycle connect direct acyclic connect structural criterion simplify detail partial property directly translate consider implication random describe determine polynomial result describe inequality come gaussian graphical monotonic necessary condition structure occur analysis various translate miss choose select designing survey science building search may application design markov procedure see etc component vertex underlie supplement detail
label baseline quantitative first dnn recognition denoise recognition evaluate mnist add clean accounting clean testing correspond noisy testing use denoise baseline autoencoder quantitative method baseline competitive baseline cc b lrr dnn cccc lrr wiener dnn task output prediction boost classification total belong unbalanced purpose answer camera helpful fold validation fold testing fan deep leverage svm
vector directional difference estimate throughout reasonable eventually speed epoch take approximately hour ghz intel figure observation efficacy outlier part formation perfect image manually mistake difference symmetry efficiently mind momentum synthetic dataset might predict sgd extra progress low qualitative say momentum free cg progress progress exist determination
riemannian inner exploit account symmetry tucker framework riemannian manifold nonlinear algorithm end representation comparison outperform across address tensor estimate generality order tensor tensor operator I I f r r mode unfold n nuclear term unfold generalization lead applicability especially necessity exploit tensor algorithm tucker unconstraine study uniqueness tucker build upon suggest manifold matrix completion connect tensor symmetry novel optimization tucker manifold develop list art instance implement toolbox development
true true section corollary edu cs private answer statistical high database fail first smoothly privacy privacy answer connection lower purely answer arbitrary standard laplace gaussian mechanism achieve bad case guarantee factor privacy preserving enable rich database individual guarantee privacy significant influence control privacy upper
simplify lagrangian q constraint marginal cross
network anomaly choose newly sample weight vector define anomalous across satisfie xy w ty xy ty anomalous vector eq follow direction far ty neural anomaly detector activation follow ty xy ty anomalous
interest discriminative take input depict confident th hierarchy representation input categorical make back partial respect objective matrix composition correspond bias apply dag structure recursively consider left formulation diagonal span play recurrent allow prevent vanish recursive neural net composition gate lstm allow propagation benchmark sentence phrase visualize project pca qualitatively paper detail review sentence review overall sentiment customer product classify customer
bootstrapping algorithm estimator within subject lagrange f px xx df worth involve minimization subject idea behind et average suitable obtain lagrange multiplier rt r rr th order respect bootstrappe df treat size k rt tw rw rw multiplier bootstrappe balance unbalanced estimator lemma entire second use et study bootstrappe well
guarantee paper interest understand remark enjoys kernel question possible norm foundation supplement metric measurable family z rademacher sequence ss proof section proof prove family definition expectation term rademacher rademacher metric making entropy get smoothly function separable q difference cm g choice bounding existence word tr db
curve red curve sd sd sd sd sd sd curve dash correspond replicate dash color black nk component curve replicate plot intercept plot gray dash correspond black solid dash correspond proposition section height support research business usage business replicate usage usage give usage condition time thus nonparametric estimate state error frequentist world technology operational
consequently sign importantly transition spectral spectral almost vector opposite sign sign community generate trial std std std std fraction fraction network noisy edge randomly critical value eq n surely fact
show tradeoff term frequency come term depend recovery graph signal choose w step w ki ik signal frequency come size graph contribution column similar leverage evaluate column random component bandwidth algorithm bandwidth tradeoff expectation discriminate
lexical semantic future take account development embed try thank anonymous comment le thanks helpful self parse supervise parse iterate ir start rich parse tree achieve et al parse supervise common sentence percentage token system parse produce model dataset unsupervise parse generally use attempt tackle parse mention nevertheless aspect would use third state parse cost overfitting make disadvantage linguistic upper performance annotate performance
global find close sort I eigenvector summarize positive lp j feasible modify follow point generate necessity proposition matrix although function constraint become product two scale solve similar reveal solution assumption global minimum assign limit generate robust gauss arbitrary matrix update block convex term diag diag unbounde immediate implication unique point stationary application loop surrogate easily impose structure update problem constraint update demonstrate impose covariance
deal vocabulary size serious nlp point normalizing neural solve tree use representation approach instance metropolis softmax although mh unnormalize input unnormalize great efficiency efficient name relate length enable improve use deep undirected machine extend
fit evaluation sample exercise bt variance reliability purely ranking sample unknown bt self assessment student ranking perform ordinal pairwise per exercise consequence time insufficient accurate estimation inaccurate decrease error approach outperform point us reason amount self collect student around study enough lead assumption artificial whose agree assumption show work reasonably model major
call aspect make surrogate classifier build theoretic density approach easily translate language additive show family distribution attain zero also related entropy aim schwarz divergence sphere density simply big
eigenvalue decrease point decomposition flexible situation parsimonious estimate parsimonious propose formulation parsimonious formulation analysis parsimonious gmm propose algorithm mixture case type issue parametric possibly type penalize criterion bic likelihood use compare namely mixture model describe well adapt realistic recently parametric process chinese restaurant crp principled cluster offer principle jointly cluster derive restrictive chinese restaurant base assume generative dp first dp ng representation variable underlie occur atom dirac probability atom atom independently base hence dp property among process dp add give generate dp add dp example density multivariate compose matrix inverse wishart dirichlet property make crp dp connect crp share partition distribution label ii crp provide infinite predictive
hoc member privacy security economic security network content member run laboratory group researcher recognize nsf award student award receive bs electrical engineering university college institute technology early department electrical computer engineering interest detection education unite engineering contribution fusion distribute open claim proposition mit edu technology edu continuously person device collect analyze behavioral mobile device period novel due duration modality absence restriction large environment organization user device likely come modality soft visit location device gps
sf tf f z mentioned naturally simply recover extra log prior knowledge play zero sum define matrix like strategy player player prescribe payoff player player action drastically use kl bit distribution produce dynamic game letter refer player use prescribe irrespective I prescribe q priori w sequence action
true reporting probability expert outcome observe expert proper rule report expert score logarithmic rule divergence scoring euclidean derive
competition participant offer daily side effect new use produce forecast use summarize environment evaluation scheme present conclusion regressor successful result use series implementation library benchmark methodology series auto moving originally describe ar support
use thompson innovation true allow theory quantify certain thompson elegant fundamental martingale bayesian armed bandit current state reward next current reward distribution property thompson essential markov martingale property proof shorthand expectation notation shorthand underlie markov martingale underlie time furthermore assume frequentist regret smoothness almost surely regardless consequence see avoid posterior much small value tend thompson
insufficient variation terminology al view database experiment use recently receive literature drive gene expression profile gene module reference therein representative profile activity profile relevance retrieve require feasibility profile model dataset learn relevance infer slightly retrieve likelihood store experiment database introduce instead query learn measure suitably dataset beneficial extract characteristic query way dataset importance
theorem square definition role pair require provide dataset cancer gene task come dataset merge dataset datum point five retain ct dataset relative ct slice validation time obtain dataset prediction six preference united price target price date date day date date price price etc normalize measure detail dataset convert day dimensional vector via answer proposition conjecture axiom underlie dimensional dimensional regressor sparse coefficient deviation light dependent
admit equal cluster sufficiently study iid vector rotation support take summarize lp small center recovery disjoint two different ex yes exercise theorem ex lp theorem conjecture summary recovery separation give recovery recovery dimension
percent community mixture forest cover type landscape close open system forest cover forest exceed forest exceed cover less water present water cover follow multiple system classify build cover forest component comprise landscape ice ice throughout never year water either water activity temperature problem framework additionally approach involve specify look detect type shift form bayesian collection sense put emphasis explore change index series temporal detection break ii method demonstrate feasibility process free spurious due change method identify step metric use cover
map cluster final unique state respectively decompose temporal light cone cluster system follow serve weight quantity use reconstruct predictive unique set light light nonparametric replace final weighted mixture reconstruction attempt spatio temporal forecasting real world material state experiment frame hold experiment effectively consist slice frame
find sensitive initialization fine little progress tune behave training architecture later initialize local appear fortunately good fine sufficiently mini batch fine tune note unless layer deep fc second conv fc layer place nd evaluate single view region whose short imagenet number layer speedup conv conv unless decomposition single evaluate layer layer unchanged involve solution decrease nonlinear consistently activation relu relu indicate relu substantial portion activation conv rate conv conv conv degradation speedup pca little degradation quickly small need drastically speedup ratio whole experiment conv speedup conv conv conv conv conv conv conv
likely outlier affect collection rarely cluster exist set qualitatively five normal normal simulate three give shape convex separate cluster half separate split convex examine technique ct hereafter seven half normal intend repetition simulate visually true cluster calculate ai datum assign calculated software describe randomness repeat mean ai repetition ct first familiar microarray quality really red one white drop hard implement dropped ct rarely perform assume cluster describe eight example choose du implication straightforward choose eight six technique convex three generate normal apply seven set normal leave roughly merge htp seven clustering
boundary none far ii entry ever replace iterate satisfie ever error replacement original theorem give claim verify inductive iteration auxiliary score eq recall moreover obey precede c rise long replace valid continue repeat increase turn establish low limit minimax proceed construct base notational simplicity index respectively permutation abuse value satisfy rank informed impose alternative generate probability verify error bound bound accommodate introduce l generalize p come assumption arise version arise start iw w yield hypothesis location location bound rely bernstein simplify bernstein probability know explore rank aggregation consistent rank reveal preference popular pair item top quantify gap rank rank return item
whereas diagonal generate block argument extend kronecker measure state present subtracting recall kn u kn tn algebraic evolution regressor rest recall stability radius argument eigenvalue triangular focus step size satisfy guarantee estimator kn mn rely necessary combination negative negativity apply iteration ensure stability although derivation sufficient mention simulation become show instability justification constraint result instability framework norm element stand trace entry q expectation side denote tractable replace steady kn tn sense
iteratively add fashion train mnist black digit overlap intensity add great digit suggest piece visual image view house preprocesse house image two visually rectangle indicate attention patch digit writing size h consistently extract image centre house highly realistic fig reveal lstm read mnist cifar challenging draw cifar natural cifar diverse
supervise task regression task share task learn minimize neural purpose labeling firstly stack auto allow build feature trick output order hierarchical pre output first relate architecture vanish gradient trick layer backward fashion start original pre layer learn output structure discover allow incorporate apply fashion auto encoder mlp reconstruct
I minimum achieve note naturally compute matrix system use efficient system unconstraine theorem contrary property allow linear gradient subsection condition depend value contrast h chi small figure write use illustrate function orthogonal scalar basis system condition uniformly linear obtain finite illustrate drop fine mesh representation form illustrate small coefficient without theorem localize accord theorem localize iterative illustrate diagram present pyramid fine soon apply scale space write interior node similarly element ib kn n require operation approximation compare present complexity robust lack regularity pde rough involve uniformly support air force office scientific award fa computation department office office advance scientific material extreme contract de theorem method rough method discover decision theory identify operator incomplete pde hierarchy elementary gamble orthogonal pde enable compression
model enhance recurrent unlike feed forward neural rnn history summarize hide rnn decay less frequently lstm replace recurrent equip principle discussion digit translation art benchmark encode human automatic translation translation digit capability context lstm train global cell lstm track distant indicate train music rnn fail chain inherently input carry merely structure lstm may capture
estimate variational table show trick yield dropout substantially high dropout early additional estimator stable compare top layer stochastic epoch epoch regular separate efficiency modern gpu optimization I take per efficient speedup test error choice hide unit variational equal adaptive counterpart difference especially variational dropout dropout network come part beneficial dropout kl divergence seem prevent dropout efficiency global translate noise locally instead globally obtain trivially low extension dropout infer
knowledge augmentation broadly posterior assume exponential field specify family copula parameter condition separate dependency red fit blue alternate third field right fourth set free variational aim free energy term reward variational joint mass crucial posterior dependency use copula particular dependency dependency factorization cdf say I densitie information transform among bivariate copula define
question convolution significantly require secondly multimodal section image treat semantic component match compose question far natural cnn employ illustrate field share utilize cnn capture rich composition word sentence layer convolution pooling perform sentence unit map layer segment convolution convolution unit parameter unit share window slide begin cnn embedding word high composition representation question pooling process follow convolution representation quickly make pooling select
option input termination history reward begin execution option hierarchical option bt whole e g termination child return return call primitive perform interaction core depend aspect selection execution learn see action termination action termination learn result option guarantee condition division converge node behavior speed convergence validate action node version could execute trial iteration experiment divide room possible action room chance agent save room chance type chance must leave room
immediately inner loop main mini component kt f exactly sag work nevertheless impractical result comment optimally satisfied px mb b gd special obtain special case recover equal focus post choice translate analyze modulus moreover fact mini inner reasonable target guarantee epoch focus fix epoch sense minimize minimize recover mini mini target evaluation follow present formula attain less mini batch quantity stepsize work gradient
value procedure patient selector parameter display indicator vector patient well patient higher order figure illustrate tendency diagnosis accuracy thresholde true value spread easier accurately distinct iterative selector propose find selector upon compare alternate result loop alternate method approximate yet significantly less use acknowledgement author
feature vector sample symmetric outcome sample distribution xx constant learn xx stability linear nx randomly generate symmetric sample n define square achieve stability rate benefit become mostly par decay aim classify document belong hinge classification accuracy regularization log exclude rate prescribe determine original inner
question important variation reveal via distribution via suggest testing specific document cause patient infer ask generate newly sample decide generate observed closeness try entity different example generate author suffer distribution like whether one researcher tend infinity reference recent two independent sample unknown either high namely constitute graph show recently low take high maximized suffice closeness every error closeness application lead scenario suffice monotone concave element require support optimality outli collection test poisson collection consider direction consider support
fig test constraint surrogate induce zero tumor eps constraint risk extensive cancer algorithm algorithm loss onto therefore however use activate subgradient information elsewhere ms I ds I dd classifier classifier property linear bound satisfie conclusion cm cm theorem conjecture pearson
recover rank noiseless solve feasibility noiseless primarily study argue compressed study constitute choice present recovery rank eq q devote herein limited covariance obtain attention convex therein modification enforce obtain eq value receive attention recently treat nuclear regularization enforce work refine state impose without result prove concern complex semidefinite unit trace matrix consideration notable contact von low constitute enforce establish least square geometric upper indicate competitive world
distribution py fy pl expectation give uncertain censor observe otherwise censor pseudo expectation eq generality p f uncertain update point condition lagrange propose show terminal condition data life
label competitive incorporate kl tune newly htp c neutral entropy kl space mac financial top feature pool movie use positive unbalanced document expect preference case select label label feature without label significantly outperform incorporate neutral feature kl lda label among control unlabele apply constrain
one step system unlikely generate energy two dimensional eqs working limitation approximation molecular show boundary configuration visualization experience sequential main ensemble ensemble anneal proceed comprise follow initialization initialize energy eqs approximation initial new monte carlo energies pool visit new energy histogram annealing desire entropy namely determine compression small overlap successive ensemble anneal slowly fast risk anneal harmonic ground canonical ensemble inverse q kl ratio successive relative result geometric
frequency intersection bin know coefficient number per use cluster instead suggest singleton identify proposition code sufficiently high singleton identify dft singleton bin least operation please claim synthetic theoretical dft used phase instead arbitrary dft coefficient perfectly reconstruct dft corrupt successful support zero dft coefficient recover perfectly observe error recovery successful evaluating signal r rather feasibility promising demonstrate random support dominant dft practice cluster mr image reconstruct dft fix snr well empirically validate scaling sample sparsity time successfully ambient signal simulation setup dft coefficient random obtain varied corrupted noise db period bin front varied least plot average shown cluster decode singleton bin recover dft overall coincide reliable singleton bin point proposition discrepancy weak total front recover dft
nd order ensure match bl outperform convergence bl seem knowledge error implement build matlab use compute nd th approach remarkably positive band limit one b normal bl pdf kde kde estimator keep cutoff bl create time calculate test potential potential potential spike carry characterize glm cell train open radius period minute area spike
availability include require fit question km area rt artificial network krige method rely ability prediction approach residual simple content national france add spatial component give might transform apply mapping stock stock national preserve median map essentially model whether local krige area although neighbourhood issue regression could forest model residual opinion consequence spatial likely compare efficient state rely sophisticated rather datum modelling state scale solely include france country short certainly model new extent might occur demonstrate study add increase analysis uncertainty map solely quality complex many could improvement possibility improvement obviously candidate drop study
variety outcome describe routine predict miss entry four baseline art variant baseline sample rmse group validation running performance law exponent power predictor exponent residual predictor per exponent exponent sum square scheme use score velocity distance adjust velocity predict art model entry column percent nuclear completion nuclear propose rank completion follow local paradigm extend size circuit minimization circuit outline modelling circuit co circuit event distance close reader way completion predictor law bag power weight one line predictor rank triple weight radial ps aggregation bagging predictor approach aggregate approach completion circuit variance use notation matlab usual subscript stand whole also boundary affect original ht performance estimate denoise entry event close restrict index row except row ia I variant repeatedly rank event choice obtain component component correspond consider scalar sub sample four top validation completion singular compute rd singular nd rd st column precisely residual summary manuscript third summary obtain coefficient computation significance equation error prediction bootstrapping performance per consider error region
discovery proportion simplicity assume wish coordinate statistic false note proportion define structure simplicity nonzero stochastically normal th loading provide estimate square estimate follow estimator spike relax much weak proportional obtain assumption convergence rate attain second relax allow estimate conduct simulation demonstrate behavior eigen proportion constant generate histogram standardize eigenvalue low b c jj histogram asymptotic diagonal position stochastically observe report correlation element three eigenvector base repetition correlation theory b b j normalize uniformly result sphere normalize q normal distribution pairwise angle realize ccc
author threshold observe distribution pair distance formula triangle geodesic yes cholesky yes yes yes jensen bregman yes yes yes cm cm cm invariance rotation frobenius cholesky yes yes yes jensen bregman yes yes yes yes yes yes log infinite euclidean geodesic euclidean euclidean distance
price vector good assume contradiction minimax primal price gp price must induce bundle price price approximately lagrangian bundle price induce bundle satisfie assumption note v p detail reduce price bundle subgradient access bundle subgradient lagrange price bundle gp easily bundle perform project lagrange price return contain center subgradient x difference guarantee project descent know obtain induce project descent lagrange subgradient work concrete preference abstract unknown utility game choose action response reveal space bundle utility minus cost producing utility approximately maximize formally player action unknown u l function associate game action action induce tie break rewrite objective note optimize approximately algorithm utility function player need game concave lipschitz space follow space diameter first target action want learn q observation polynomially action action
multiplicative function cosine value hadamard employ discrete multiplicative observe algorithm multiplicative dft denote calculate present ccccc transform
optimizer sigma variable repeatedly p pp integral partition unnormalized distribution define follow direct measure eq hold scalar hold insight bind old bind possible integral right several upper quantity provably convergent polynomial finally bind partition inequality imply
aa n aa aa aa possess therefore max notice conditioning index bind obtain part eq use combine union two obtain q dominant require certainly notice precise fast dimension reduction feature method moderate variable via variable reduce discard substantially screen crucial fast selection algorithm fail method establish consistency particular dominant concrete screening subject design limitation challenge facilitate improve decade field focus recover
affect space observe quickly propose chance accept problem fast region original problem classify datum
state experimental table achieve lc well lc improvement respectively classify dictionary fast lc randomly acc acc sr lc ht compare lc art sr shown compare lc perform among method confusion misclassification error store high setting category give per used evaluate pyramid lc lc improvement c lc ht examine performance
addition turn technique virtual also introduce virtual symmetric anti case anti pi simultaneously pi pi unitary eigenfunction operator five operator consideration eigenvalue arrange eigenvalue operator eigenfunction belong since reference analogously adjoint orthogonal q operator accord share pi preserve common pi pi either anti symmetric
worker worker time worker read worker obtains share worker guarantee precisely one modification component atomic parameter share share memory early miss analyze asynchronous parallel share select update analysis assume early memory practice physical consume step reduce trick ask multiple cycle update update write q coordinate compute index express index iteration allow jk might practice probably asynchronous sg asynchronous basically serve abuse mini age inconsistent read result hold
year next execute initialize extra iterate pass reverse label year title nan format title max sort order sort begin execute begin execute call execute end pt pt xt thank title title width width ex ex sect mark pt conference plus minus pt sp reference page em theorem condition principle outline diffusion achieve bridge develop mathematical simulating path important widely phenomena broad economic finance life computational class markov jump diffusion markov sde denote instantaneous diffusion brownian motion compound process jump jump coefficient typically ensure weak naturally simulate path infinite random avoid simulate broad jump simulate tackle simulate jump path represented sde jump diffusion diffusion purpose restrict impose coefficient simulate induce construct simulate condition simulate jump construction appropriate measure
imagine ml process relate entire loop relate solution enable rapid solve problem material context formalism general cubic neighbor hamiltonian unit density center section seek machine output solution vanish gap level self chemical advantageous entire remain
monotonic reason explain heterogeneity low whereas link nod link consider probit include dyadic probit except fit rgb psd status age row status col col col age col characteristic appear indicate fit model regressor multiplicative effect bin goodness fit discrepancy proceed examine regression effect interpretation multiplicative proceed identification association multiplicative compute effect ordinal characteristic association effect via plot u status status age formation although multiplicative additive model associate office binary extend accommodate latent provide treat nuisance ordinal approach simplify specification general ordinal computation use level model dyadic record dominate dyadic nature heterogeneity lead dominate lead scenario dominate unlikely able dominate available plot age particularly dominated ordinal probit
transformation usually easy arise abc evaluate proposal accept define alg implement hasting mh rw mh smc proposal view stationary algorithm far characterize behavior induce assess pseudo mh function pseudo find objective paper could proposition expression statistic jacobian note es need r suggest parametrization make change employ lead broad class abc constant closed proposal restrictive extend assume exist intractable reason density metropolis construct tail abc simulate enough simulate sample standard draw complete importance weight necessary update argument genetic analyze role except
dataset unbalanced structure require choice despite tend uniformly comparable growth comparable general produce growth curve process glm np thank provide dataset provide le di theorem proposition study curve rest probability mixture factorization small ball approach principal datum computational attention propose base kernel density estimate introduction supervise unsupervised role base call require development deal belong space introduction refer book consequence orient classical multivariate implement suitable put tackle follow put coefficient underlie process instance technique another aim refer principle ball associate depend term reflect underlie
would response overall impulse impulse open proof criterion expectation define parameter eq iterate estimate two problem respect collect obtain unconstrained solution consider derivative expression spline factorization find reasoning consider ml argument theorem definition conjecture se advanced learn contract identification regularize
discover rule mining consequence generally event high proportion proportion contain input constraint rule greater great medical still interested occur ignore rare support medical health outcome unlikely support unless low left support constraint detect lift association chi square significance thin gender code parent read code code level item medical thin total gender read mine medical support minimum propose refinement unsupervise signal drug interest health
margin multiclass way extend theory complex output develop multiclass pac de france universit universit al tight vote output provide generalization multiclass label output majority
contradiction constraint lie edge polytope rgb straight ex ex ex minus plus minus paragraph em claim corollary edu study predict linear partial constraint generalize predict rational agent unknown reveal mistake learn algorithm learner program thing rational objective change control program may day partial learner systematic dimensional information single fix study capture follow utility decision day day observe price bundle learner bundle price face round constraint optimize broadly objective change constraint predict agent constraint unknown
lebesgue density satisfy consider piecewise polynomial partition orthonormal hold lebesgue form orthonormal polynomial sup suitably control localize piecewise sup orthonormal localize histogram existence interval polynomial localize sup localize convenient set integer l give explicit strongly localize assume q jj admit localize
develop leave algorithmic closely study past recurrent symbolic symbolic system deal link symbolic symbolic network design simple recurrent generalize promise truly count mostly pattern data gr tackle work architecture internal symbol generalize network gradient able learn simple context context unit choose linear count constant store amount recurrent investigate mechanism context memory oppose store new al lstm build roughly resemble
show learn solve example tree learn determinant nuclear dictionary involve fits k n empirically degenerate solution shape gaussian difference variance phenomenon impose shape model hyper incorporate joint learning still wishart gaussian pdf derive term please wishart learn dictionary address parameter properly initialization rough consistent ml much rough contribute still please k learn require manual annotation particular corner similarity initialize orientation determine although initialization transformation difference identical statistical overcome learning shape compose linearize problem actual fig effect matrix fig determinant well tree shape obtain connect localization liu find shape shape assume independence liu learn reduce summation node take span root optimal configuration tree cope variation parameterize pose change index part correspondingly dictionarie z use dictionary belong overall dictionary component similarly dictionary ideally learn first face image pose subject learn dictionary separate set recognize select fully automatic pose experiment face pose neutral expression
sparse statistical establish connection view dictionary coefficient center draw moment center lead kk th reduce mean dictionary pm original e characterize compressive k specify important heavy tailed tradeoff rate signal increase
matrix reason store secondary storage pass call would pass deterministic single cost performance trade conditioning invoke round original state one pass conditioning run round separation non along matrix compute size svd n p size ram increase block conditioning ram round replace embed low rank running result substantially condition practice trade real application subsection embed method relative basically meta os embed present four category discuss discuss method first structure distortion embedding p follow distortion embed ps nm solve subproblem terminology low distortion distortion qr decomposition condition low equivalence similar result qr rounding condition c ccccc name pass er er er er er ct qr n mn qr qr compare trade conditioning qr take round well conditioning trade theoretical certainly affect distortion embedding distortion preserving embed ps nm embed closely j l point ss global mapping construct subspace scale simplify improve moreover scale maintain latter storage although os geometry vector os arbitrary dimensional subspace transform n ff eq apply transform include well transform able refined spectral orthonormal approach whose normal random g dense use multiplication embed compute transform start algorithm give orthonormal call product matrix sparse approximately hadamard projection simplify subsequently refine analyze name preserve chernoff sign hadamard scale dimensional uniformly essentially combination dimensional eq distribute although might matrix store like issue embed run order polynomially exactly stream extremely write matrix column independently subspace heavy base norm orthonormal base algebraic subsequently
imply move generally outside temporal clutter phase calibration slowly across calibration ccc clutter clutter filter whiten traditionally however pca component number sample sample unstable problematic reliably estimate filter inverse clutter span clutter clutter filter project orthogonal clutter effective parallel acquisition clutter since require relatively free enough target partially target especially problematic online implementation take inherent space kronecker covariance kronecker plus noise clutter iterative excellent applicable essential matrix p minimization derive principal identifiable advantageous kronecker appendix hermitian lr initialize
depend three merged applied affect desirable receive current trend distribution appropriate homogeneous among sake final protocol storage several stream homogeneous database use storage formal framework mining system refer slide window section sequence characterize briefly reduction merge let monitor incidence city city whether incidence disease city one report merge report incidence upon composite picture able handle requirement algebra framework monitor impact composite map approximate map map design like language language operation area combination language grid offer image processing combine mathematically prove control combination wireless application forest temperature temperature change also strength collection processing sensor life infeasible sensor capability summarie average sensor construct base observation summary rich goal cost use may wireless sensor simplify pruning send
subsequently correspond optimization demand make direct impractical many problem pixel perform pca apply feature deep auto standard rl xlabel frames ylabel success legend style axis cs anchor south bar cd plot plot bar display average success rate error deep together tailor success around solve quickly graph learn far behind truth rl solution achieve rate trial frame auto red fail explain auto
correlation graph sequentially matrix matrix row identically dispersion vector elliptical decrease matrix spherical assume assume common dispersion dispersion pre change post change respectively take value different realization random maker either sampling change occur
perhaps nuclear near rank constrain selector analyze propose weighted depends sampling empirically study minimization rank noisy constrain least estimator loss approximate matrix genomic matrix smc subset goal reconstruct whole observe genomic integration introduce model study association expression ensure adequate one goal marker power power phenotype practical feasibility genomic genome sequencing provide information genome wide certain wide genomic extensive genomic include analysis extent genomic observation suffer miss propose year include decomposition many often observation unclear statistically extend genomic integrate study extent genomic arrange structured submatrix miss construct rule datum significantly improve row whole value column
sample system output cast linear cast framework follow depend context kernel identification advantage kernel like parameterized decay generate impulse response namely impulse square mmse common kernel hyperparameter maximize likelihood follow square estimate output
agent bind marginal connectivity follow update agent truth exponentially trade communication private adequate agent highlight communication unnecessary agent interaction neighbor recover signal digit threshold consensus almost versus randomly observe involve load analyze group try world rely private signal sufficient information private adequate communication show
side figure highlight method mean test guarantee permutation approach f f among level permutation note lead conclusion focus description multiple detect match behavioral classical window potentially cover whole interval trial record implement enable permutation parallel window delay count return b set count multiple non period count significantly considered window train necessarily discovery fdr table precise package code set simulation assume process homogeneous trial delay correct permutation always whereas fail comparable result fdr basic value trial robust much permutation shown figure share correspond see formula self permutation perform overlap window run trial resp
big small root recent common pairwise distance grow balanced ensure small fall short path generation generation generation necessity tree grow produce vanish come generation upper bind decay tree prevent unbalanced grow grow unbounded second balanced quasi tree satisfy assumption growth node rate differ upper bounding uniformly determined contain parameterize iterate iid several certain gain tree tree allow node sub within condition survival grow conceptual match fourth balanced distribution typically bound certainly interesting node potentially result various subsection investigate social publish simulated tree two network
coordinate requirement distribution real dataset lead parametric case facilitate least scale approximate situation collection frequent fraction small perturbation coordinate perturb assume small upper approximate via perturbation coordinate adopt mixed analysis let incremental mixed coordinate q decompose direction fisher compare fisher n difficult analytically fisher bound turn direction fisher proportional equation calculation maximum incremental fisher inverse fisher constant scale square information analysis parameter information hierarchy high confident confident additionally confident neutral indicate hence implement replace confident neutral reconstructing tailor l verify preserve tailor coordinate begin draw result show ratio preserve ratio maximally preserve fisher distance surface denote half square parameterization ellipsoid center general maximally upon coordinate determine maximally preserve rao metric general
firstly algorithm sift descriptor algorithm cluster norm rewrite membership indicator cardinality element belong convex phase call code codebook coarse overcome yu al relaxed instead put regularization nonzero formulation turn another
give identifiability tight measurable identifiable mm identifiable identifiable identifiable identifiability monotonic measure identifiable identifiable contradiction l identifiable contradiction identifiable follow exist mixture b j view small value minimal heavily geometry tensor product space section tensor product hilbert space tensor hilbert basic intuition hilbert space tensor tensor completion
costly mt formulation kernel arrive problem describe one implementation mkl mkl solvers tailor hinge prove integrate module toolbox occur fast convergence overview mkl survey kernel kernel string kernel space tailor large considerably previous top core alternate weight improve variable precision initialize initialize optimality satisfied descent compute decrease pt compute weight module module illustrate table mt approximately later context multiple carry line maximization infimum attain boundary step carry solely fact descent objective presentation choice optimize optimize I purely involve support infeasible carry hold procedure solely vector one track coordinate way course explain rely inner product adequate computing row substantial gain mt argue aim express analytical solely thus dy mx x
image image send svm result summarize generative convolutional develop propose enjoy
let currently associate k max contexts belong create word np differ sense discrimination probabilistic observe context pair remain train vocabulary snapshot wikipedia corpus approximately million article token occurrence context occurrence unless hyperparameter amount manual skip np noisy word train initial regularization describe word quantitative et np skip gram skip show implementation size corpus
hypothesis contain algorithm linearly lp solver factor slow sort sample bring overhead factor close iii learning achieve piecewise piecewise constant hypothesis show piecewise hypothesis approximate gmm note slope curve log term perfectly ii factor close roughly obtain htb format n avg minus low high gmm avg piecewise beta table avg minus plus gamma txt avg time error high piecewise gmm x avg low plus piecewise beta txt avg minus low plus txt htb format error avg piecewise txt avg piecewise txt std piecewise gmm avg std piecewise linear avg error txt requirement roughly also robust essentially acknowledgement thank early thank lee discussion help dedicate recall statement run partition return j condition split three partition interval jumps create follow vc suffice equation triangle sign change therefore eq interval interval contain jump singleton interval jump cover assign simply follow negativity finally interval create merging jump triangle since triangle along start bound prove complete summing use pair merge suppose interval recall iteration apply empirical interval jj sign li j contain jump jump interval combine obtain complete require elegant tight polynomial unit change function analytic modulus principle since fix symmetry also choose thus therefore pz w claim polynomial uniformly bound interested polynomial integrate constant relate bound use bernstein let degree ready py
use pass whole determine value index operation separately partially sort turn sort index sort turn around also compute bad operation implement practice conditional test layer binary solution fix reinforcement reduce transition capacity bp algorithm generalization transition occur transition teacher around infer teacher find improve theoretical case point solution teacher batch classification fig maximum sample started fail eventually result decrease depend max
simple homogeneity combine possibly end sample kernel associate tolerance approximate simple schwarz control accuracy figure visualization discriminative dimensional red negative representative pick boundary define loss misclassification tight generalization misclassification loss margin svm amount allow margin q simplicity e must similar spirit perceptron produce
free energy free motivated entropy reweighte free energy temperature specify polytope hypergraph counting number henceforth refer energy restrict marginal polytope reweighte recover typical bethe choose reweighted algorithm choose correspond appearance probability span reweighte energy form point investigate likelihood mrfs demonstrate bethe energy guarantee marginal moment match marginal bethe maximize approximate mrfs reweighte investigate via bp approximation theoretically necessarily double loop bethe piecewise whereby divide small subgraph combine subproblem inaccurate piece bipartite unbiased utilize rank vertice mle estimate converge result improve method free energy learn variable minimize compact invoke minimax yield minimize ellipsoid slow sequel wolfe free energy bethe energy bethe work mle dual follow mle specifically unfortunately convex
notice carry throughout require yahoo likely page top derive consumption customer unable news display collect yahoo piece news indeed intend reach yahoo ucb news cluster quite performing probably inductive bias cluster actual meet yahoo reporting time get payoff retain world world dataset dataset yahoo theoretical omit relate regret simplicity result hold vector incorporate provable need advantage size translate practical universe tends determine major behavior one frequent behavior user profile uniformly positive suffice arbitrarily item I condition payoff
statistic explore large grid define also however generally bad notable method explore grid use mutual possible grid include dynamic sample test measure dependence variable either variant estimator estimator convenience define include analogue pearson different covariance point schmidt general reproduce hilbert space intuitive pearson coefficient however perhaps give search measurable maximized finding widely include randomized search linear ideal assess instance know present reality bad equitability measure aim good equitability relationship type assess small set general context noisy functional broad class strength coefficient determination ensure test along many include add distribution analysis equitability across sampling equitability characterize broad trivial equitability analysis include functional additive regime combination set correspond evenly along curve add dependent definition size examine evenly spaced generate realization relationship regard parametrize assess equitability setting significantly equitability present default parameter mutual equitability value test supplement section generally equitability quantify interpretable section equitability denote interval contain bad equitability analysis red plot case interpretable short mutual range along case equitability offer salient equitability normal strength quantify interval indicate red equitability interpretable reflect generate relatively weak question equitability respect represented parametrize whose parameter affect equitability parameter equitability marginal test natural ask direct mutual achieve equitability appear vs mi equitability plot worst interpretable indicate red bad list statistic sample equitability marginal mutual information correlation equitability independent variable marginal distribution version table equitability mutual sample equitability
de du fr une est e en pr l sup l analyse es pr trait es de la k jj pr des dans dimension identification un le de la figure des observation le par des et des dans ce de
singular eigenvalue reduce avoid undesirable appropriately specifically huber remark surprising despite outlier purpose indicate forward distribution isolate outlier spectral distribution finitely outlier corollary reasoning normalize line display scenario outli value n eigenvalue isolate isolated reduced font style densely yshift anchor yshift west fill north font xlabel ylabel near n bar major scale coordinate axis cs u st diag outlier insight bt font densely dash yshift yshift pt west xlabel ylabel width coordinate font style densely anchor west anchor north east font xlabel bar width pt false scale mark repeat plot
complementary four space extremely specifically high internal orthogonal tend well symmetric link design performance complementary log model future provide well corresponding consider fitting logit give taylor expansion around eq first proposition edu tr ny edu justification long claim probit despite similarity researcher dedicate carry study aim characterize similarity predictive equivalence model explore various way probit link
example consider graphical factor generate copy normal f ft f nan replicate choose estimate significance increment power clear indistinguishable indicate report type clear tb b graphical precision precision compare scad likelihood report
split constraint know assume gaussian show unconstrained point split two ccc middle unconstraine right approach spectral sl flexible constrain clustering via sl modify weight matrix edge connect link weight edge zero spectral satisfy user solve full generalize eigenvector correspond incorporate class sdp aim adapt code link encode
call importance analysis empirical replacement element sample replacement uniformly immediately use cardinality call replacement nf indeed expectation randomly complexity sign term appear denote I risk slack replacement subset uniformly inequality inside find sect compare new bind show
second need avoid summary create human still perform music classified summary remarkably whole segment duration counterpart summary sometimes discriminative also state summarize dataset size well song individually acoustic vocabulary besides classification also fast transform peak information retrieval kullback leibler semantic coefficient music evaluation exchange music information retrieval relevance point value transaction audio speech constraint lead performance segment leverage text summarize diverse appropriate segment make good music binary multiclass music task obtain full centrality performance summarize difference make state summarize music decade address music algorithm human orient summary people song summary entail extra besides non people generic algorithm however focus diverse
spaced graph interpretable statistical reader build case interval indeed tailed test distinguish demonstrates examine contrast mutual respectively extensive varie noise marginal compare curve summary curve indicate interpretable bad interval indicate analysis via nan hypothesis plot legend describe functional informally equitability dependence us relationship across broad relationship give conceptual motivate equitability different way motivate equitability begin though asymptotically detect deviation independence datum relationship absence strength relationship estimator robust detect relationship outside perhaps value estimator relationship computing parametric know relationship whether difficulty consistently propertie measure dependence approximate version consistent lead equitability equitability allow dependence approximate relationship formalize equitability interval show equitability equivalently state correspond different strength power distinguish trivial statistical independence equitability property fix detect relationship pass certain threshold across show threshold straightforward equitability converse low
bayesian combination kernel present assume hypercube original consider hyperparameter accordingly adapt towards area g pos intuition behind surrogate model could search region near many problem difficult near high variability important note initialize local kernel small capture could local scale smooth result less total
channel keep depend respect zero arrive closed output covariance weight feedforward finally formulate fully rewrite recursive scalar perform neural minimize online streaming phase update activity neuron activity local rule feedforward anti sign connection neuron except free cumulative neuron square end argue single neuron derive representation input activity recover feedforward weight present neural take zero arrive linear equation solve output component interestingly right feedforward fix descent single linear stimulus presentation repeat coordinate descent algorithm neuron activity neuron update algorithm update convergence always spectral whereas prove
stack queue input top read input form rnn controller output controller roughly rnn controller stack queue integer encode present start symbol end symbol symbol separate source integer encode symbol convert embedding embed separate mapping encode input embedding target symbol vocabulary symbol uniformly replacement source vocabulary ignore deterministic apply source sequence entirely generation sequence follow test symbol sequence symbol form source odd index th symbol form examine sequence context translation interesting
add local arithmetic multiplying add sum add bn add reader thorough discussion add introduce notion return simplify notation polynomial unnormalized boolean multilinear indicator summation boolean polynomial boolean expansion unnormalize rooted dag leave whose sum negative value child child root set indicator terminal indicator follow sum scope consistent appear another decomposable iff scope clearly unnormalize distribution sufficient necessary focus probabilistic semantic complete node define root compute weighted root root child product induce sf root htb x paper discussion keep boolean state straightforward discrete random variable plus bn size graph decomposable add time represent thm immediately exist boolean bn add bn theorem corollary simple bipartite terminal boolean sum node
exact quite balance moreover well standard relaxation recursive inspire tackle directly cut asymmetric cut propose information cluster contrast balanced cut competitive respect balanced cut recently factorization also amount learn encode similarity instance cluster partition partition cut graph vertex cut furthermore x b order sf aa set submodular extension f j submodular balanced correspond balance towards attain perfect
assume take deviation summarize demonstrate sampler comparable preferred factor model likewise compare model specify detect variance value suggest point probability demonstrate work well normal true sd c vs variance variance mean sd sd vs c number still correct replicate entire previous change replication iterate gibbs sample obtain collection number variance show
degenerate realize common format likelihood covariance bind eigenvalue invertible treat derivation ridge numerical flat sample algorithm mle entry adjustment find missing proportion square four grid square point miss exp exp points square miss generate estimate latter encounter sensitive point optimization appear convergence approach reason favorable approach exclude approach exhibit range large lattice case effectively grid preserved accuracy provide inversion last gradually space reduce miss zero obvious bias severe quick among high take entry slow efficient take second take achieve fast take huge functional compare fast estimation occur suffer convergence reasonable extend functional demonstrate facilitate propose
mean addition gaussian proof path almost follow simply almost surely rkh specifically closure expansion ft norm inner give functional regression range operator wherein dimension reduction fall consider rao measure binary control separation easily covariance covariance following maximize schwarz solve eigenvalue link study linear rao xt c
number obey family subset nr mi contradiction pseudo triangle result
radius also fact specify net n nf n u obvious gamma stick construction gamma time technical random back measure dr stick break representation surely yield variable laplace transform get arise identically whole stand q prove theorem sequence regression straightforward satisfy wavelet n nz nz get proof together quite abstract conclusion validity consistency kernel coherent type wavelet type imply belong surely verify coherent wavelet admissible ensure coherent straightforward yield generality metric easily check constant enough choose exhibit correct trivial assumption wavelet recall situation
mean rank tweet select case bottom bank mistake gold ratio narrow arise optimisation relate unseen investigate conclusion method spectral rbf lin lin mr mr move domain adaptation first half significant difficult training like previous loo train good tweet report process rbf hand rbf report spectral together classifier spectral examine cluster automatic determination ard whereby
orthonormal basis see detail call sub solve tn en ij theory property tt compute project compute treat nevertheless quantity correlate play characterize substitute obtaining tx subsection obtain easily eigenvector call reflect discuss aim relation particular since understand analyze square situation trivially moreover eq square estimator panel pointwise dot asymptotic latter see panel pointwise
element sequence unbounde claim shorthand quantity index initialization subset eq lemma vector cc condition event equality I step lemma coordinate separable minimax applie scad remain discuss base relaxation et reformulate optimization function boolean many standard programming hierarchy relaxation possible pairwise interaction incorporate constraint polynomial conjecture contain achieve prove conjecture penalty strongly result imply compute bad local minima descent interesting algorithmic broad rely give broad initialization acknowledgement support grant dms nsf
preliminary use calculation occur curse cut scheme frequent number architecture hide calculate average minibatch epoch current evaluate performance utilize calculate translation regularizer dropout technique neuron performance later different report configuration translation system model train different vocabulary table en fr vary vocabulary size help dramatically neural affect translation word english well achieve notable
group g aim disease modern datum center total represent expression brain cancer package package apparent dataset focus state right apparent comparison apparent misclassification rough diabetes brain na na na na na na seen work diabetes exception winner pp indicate technique explore performance variant pp robust rf discrimination replication summarize figure come provide detail later another aspect computation perform good operate infer predictive inherently line somewhat rely existence unstable pp huge spike prediction depict cause hand figure last right pp variant
take standard wise effectiveness spatial refer dimensionality cnns impose capacity bottleneck approach pooling reduction project onto frequency issue approach sharp dimensionality reduction encourage invariance capacity approximation loss window often represent well exploit uniformity natural power concentrate frequency high frequency minimal addition pooling permit map manner since truncation frequency exactly correspond resolution supplement pooling pool additional convolutional network convolution truncation batch well fourier basis
favorable insufficient regime bind q tight fairly bound simple analysis bit allocation allocation two weight need result weakly geometric geometric decompose remainder correspond remainder term bind ease write remainder discard given consider eq therefore condition codebook decompose term sphere fix observe k k complement choice k k summing lemma vector quantity suppose lemma define side schwarz lemma therein sum sequence sequence notice take expectation apply follow turning allocation assignment allocation go sum follow q give thus matter conclude let lemma specify hold
amplitude infinitely way cosine format amplitude frequency infinitely smooth general favor take amplitude quantify cc parameter unique locally theorem describe different form tt ms ms represent lt constant right hand universal focus representation follow generalize amplitude phase instantaneous amplitude take ft instantaneous call always positive constant harmonic behave harmonic error encounter ingredient processing concentrate dirac feature visualization
equal denote output eq function prove necessary lemma correctness main write feasible solution otherwise say
usually solve repeatedly adapt cost multiply synthesis algorithm computationally practice problem popular dictionary svd learn follow keep alternate enhance replace penalty importantly globally transform rest briefly derive cost demonstrating also show brief denoise conclude transform discuss w degenerate simplify positivity value help remove admit exactly penalty learn singular value typically little image denoise hand denoise function proposition hence encourage transform transform become condition tend scale specifically learn number close depend application condition invariance trivial minimization sparsity learn learn low minimum value equal pair sparsity exist underlie transform transform minimizer therefore sense model interesting admit solution minimizer pair row permutation certain setting learn poorly exclude crucial help overcome condition replace version recently transform weight penalty p extended previously solve keep step transform gradient low synthesis transform fact
bias fall network dnn acoustic speech recognition single work hmm specifically dnn produce phone hmm produce probable although desirable dnn train dnn force parameter acoustic acoustic correct force alignment word architecture unit softmax hmm target ms advance frame predict version acoustic strong train dnn acoustic hour example system achieve frame word predict procedure initialize find create diversity significantly outperform explore diversity model
construction marginal distribution variance precision integral inverse want marginal new joint transform advantage mean reduce transform joint q original eq compare add addition correction conceptually effect might seem
index u number activation activation bandit time sufficiently n n n n kn eq bounds aid take additionally convenient finite q similarly sufficiently observe aside finite bandit activate infinitely iterate time since apply n n proposition let eq proof proceed analogously iterate logarithm define hence relationship eq proof three come term inequality iterate bind far finitely time surely almost surely maximal u I last iterate simplification event constant combine fix recall index notational convenience
cumulative linearity gain randomization consider q k instance sake gain gain algorithm equivalently definite gd maintain projection old new set predict choose sampling belong cumulative randomly allow l g one easily deterministic include obtain force strategy act add hence adapt find adaptation
east west east west east computation posterior intensive activation distribution posterior greatly consist activation bottom weight top locally unit use softmax hoc however identify assume label unlabele identify activity represent bottom middle neural derivation activity interpretable crucially however gradient space rate systematic change point elementary minimal rate constant refer neural apply implementation benchmark handwritten mnist investigate weakly divide part proportion label label report independent unlabeled refer feed forward simply contain feed ff ff layer approach deep weakly train dimensionality unlabele feed layer formulation incorporate top regime availability take tuning subset tune free normalization randomly e
multidimensional beyond high frequency channel wavelet transform kind nonlinear iterate filter bank desirable transform subsampling transform subsample avoid drive discuss degree freedom appropriately strictly summation window partitioning subsample wider
dm uniformly exist finite integer would dm dm weakly acyclic strict dm policy asynchronous denote baseline eq integer generalize multiple dms policy strict possibly dm strict discount acyclic policy deterministic lead large game weakly acyclic stage cost figure dm dm dm choose dm dm game dm single I dms dm strictly well game probability dms dms patient discount factor error dm perfect equilibria equilibrium dm equilibrium equilibrium dms joint strict equilibrium
pdf subscript integrate power may several pdf sample structure consideration quantity term ip ip pdfs cross cross potential obvious force define potential done bring mass infinity field contains apply force ip sample derive novel blind separation bss volumes region bound hyper surface equality derivative pdfs pdfs result independence indirect bss direct target potential information potential derive field rip place point closed expression field square multiplicative pair place pair paired kernel affect choice bandwidth estimation computation rule help achieve two blind bss potential method function simply blind separation unobserve available mixture interpretation derive bss focused kullback base independence interpretation approximation statistic interpretation independence towards new bss bss mathematical solution always spurious optima existence spurious optima large bss balancing derive subgaussian source bss base parametric measure kernel ica independence may estimation actual source
test send widely metric click number email customer click place order ccc date avg avg lift significant order extremely hypothesis give online confirm add capture temporal customer click order suggest recommendation mining stream temporal recommendation attention recently et factorization movie music rating propose class moreover aim differentiable rank et al formulate temporal filter one wang recommend meanwhile diversity recommendation another problem make attract extend memory rank rather item factorization extend optimize orient ranking relate approximate extend factorization optimize like ndcg instead non couple convex optimize low reciprocal differ optimize rank bias rank propose filter partly class label classification still widely construct class bias class accuracy recommendation item matter
rnn initialize rnn many epoch day core analog nlp induce class induction pos operate token algorithms obtain embedding factorization ignore transition token embedding require stochastic token context token spectral method similarly offer mle moment efficiency rnns nlp include translation language parse replace term interaction however rnn careful stochastic scalability favorable preliminary use initialize nonlinear encourage work latent multinomial develop sophisticated practitioner start use nonlinear recommendation differ rnn good nlp consider observation dimensionality latent completely choose maintain either fix stable fit center eigenvalue less lag
separate slack margin modify convenience e elsewhere incorporate denote estimation become unstable x mn induce tr tr fit error explicitly degree fitting avoid constraint enforce validity adopt dag advantage dag employ topological order bilinear moment skip dag concentrate solve optimize hyperplane strong duality svm multipli j dag package eqn matlab many difficult learning algorithm let compute nj minimize eqn eqn eqn optimize feature optimize svms imply discrimination discrimination far instead method binary minimum maximize minimum generality false label identify sample whose assignment maximize target maximal separation class naturally class replace false irrelevant eqn maximize likelihood sample maximize hard formulation enforce false slack indicate margin denote eqn iy iy use kl define constraint dag constraint enforce order constraint please refer eqn solve summarize solve matlab step programming input eqn fix fix optimize eqn enforce dag
algorithm select whereby play reward k player cell reach total least improve total fair player propose mm mab player load accord learn accord probability distribution kt action select change receive fulfil fulfil receive reward update whereby non action k simulation configuration scenario version system simulator three bss macro randomly macro mobile
variable previous yield shrinkage onto nonnegative propose multiplier ascent term update per hold verify n output abundance datum technique sparse lagrangian recently nonnegative method admm involved metric execute different whose detail finally abundance reveal algorithm hyperspectral reason sparsity lagrange multipli parameter admm lagrange multipli regularization influence extend efficiency correspond value admm algorithm compete experiment root abundance stand reflect ratio estimation error window abundance abundance dictionary spectral band interval signature subject
also really xt recursion assumption eigenvalue part origin recall hence convergent approximation control well sufficient approximation control markov study algorithm weak requirement analyse future direction extend approximation actor theorem claim stability sa iterate iterate track solution define measure control
evident round mode adopt precision il precision representation matter consideration perform fix define scheme round probability round eq rounding round possess desirable round il irrespective round outside operation point format inner product also represent split produce number think enough precision sum product width worst rare low convert convert limit round fractional rounding mode error fix round mode round fractional adopt hardware product hardware hardware unit implement addition accept accumulation hardware overhead implement stochastic rounding simulate library fix hardware optimize apply
noiseless unitary even maximizer defer da zero formally column permutation non recover ica rise simplify suppose free ica scale e zero entry consistent notion optimality minimum signal latent signal recover kb pearson k fact desire convention let non minima set restrict investigation gx gx mm x km yy maximizer minimization first equivalence note change demonstrate recover extend ica source provide variant discussion ica appendix ica work construction
use word relation simultaneously subsection solver leverage pair relation like reflect example optimize answer cosine similarity index word analogy list apply optimization select answer form word pair question property distribute belong co information training word word accord possible candidate index choose sense close solver word locate word solver translation entity knowledge specifically offset embedding candidate candidate word offset close well question explore solver distance solver offset relation might solver co occurrence vector relation lie embed solver skip first conduct examine result word embedding publicly corpus text snapshot wikipedia process meta english word token unique vocabulary accord specifically
pc severe restriction pc reliability hundred restrict search true approximate explain work phase add variable remain variable shrink irrelevant variable prune irrelevant parent child datum children xt datum variable parent child parent child xy xx hill discuss ss draw strongly phase hybrid time appear idea employ sound identify hybrid identify hill bn begin empty continue recursively difference search add discover phase list explore change change maxima ever algorithm terminate score clearly false positive allow enter burden impose ss label synthetic eight benchmark size benchmark bn term various investigate well dependence match true benchmark implement code code pc publicly carry ghz ram running bit size c c output bn benchmark table repository experiment repeat skeleton pc cb measure false positive I divide edge divide number edge combination precision euclidean distance assess quality phase
achieve g r take negative positive derivative vice versa increase vice versa lemma proof loose curve loose low treat curve vary estimate unlabele given roc pr explain section figure translate pr optimistic loose yield loose applicability approach roc curve set estimate truth ranking simulation test independent fully label set enable ranking set negative positive negative produce estimation discard discuss remain supplementary
detail able preserve variable method margin gain lose independence admit upper dependence preserve topic copula margin know appropriate parametric margin marginal exhibit skewness marginal tractable inverse cdf mixture highly flexible ideally proposal recover jacobian term correction analytical monotonic flexible via kernel
initial stochastic mode search chain rapidly density partition state overcome pdfs development twice differentiable concave application pdf definite case pdf identify hessian property slice hmc embed state partitioning replace discuss context state handle convergence general case carefully twice violate unconstrained deal constrain mix sampler within slice capable deal therefore assign subspace handle relax twice concavity
number accuracy bfgs bfgs use gradient competitive intersection multiple iteration minimize figure sd iteration fast size bfgs immediately
training divide unlabeled vocabulary rank batch unlabele reveal select sample report ap annotate concept crowd build extensively annotation annotation histogram effective technique text content feature color wavelet frame randomly select video concept set size virtual character platform library point
store unnecessary resource stream observer find occurrence streaming four characteristic na framework benefit choose closed form sequentially test nan use surrogate unknown use average performance contribute contribute concept consistent present assume stable consistent change indicator eq thus mse synthetic streaming stream confusion drift scenario accuracy classifier drop choose cc algorithm rate algorithm report detection sensitivity rigorously compare color horizontal line grey characteristic
reduce potentially example realize code implement approach filter dt generalization character roughly speak irrelevant present environment introduce projective ps ps physical intelligence processing experience weight represent sequence observe activate walk walk walk randomize toolbox analyze scheme physical thereby relate artificial agent quantum walk walk potentially agent quadratic ps realize internal interaction dynamically increase well step see detailed learning trial make rl ps perform standard task grid car ps evolve experience exploit similarity network mechanism notion abstraction importantly knowledge generalization ps rest characteristic list learning ps generalizations agent scheme require curse bellman agent consider environment irrespective resource ability learn agent follow ps enhance mechanism detail analytical ps
select precisely question worker question like worker possible worker compatible compatible mechanism mechanism turn unfortunately compatible guarantee worker establish processing storage human average distinguish seven state human verify subsequent establish fine human verify incorporate establish option worker belief one situation totally follow worker belief lie wish full worker assign low mechanism compatible coarse mechanism coarse belief mechanism evaluation worker answer gold question payment worker gold question payment answer otherwise
image goal primarily image large image search color shape descriptor typical system database store query similarity feature fix low necessarily semantic call context intervention form rank ask identify irrelevant system retrieval rank process continue discriminate image retrieval consider image region attribute texture adequate satisfactory image several researcher precision cluster segmentation cluster uniformly effective database propose modify rf firstly novel feedback cluster relevance weight utilize content color shape occurrence
derive depict formalize sphere divide behave euclidean exhibit local either negative riemannian capable reduce least number spend away minimizer objective r independent q since embed riemannian riemannian hessian taylor approximation unconstraine semidefinite hence f q act choice tangent n translation translate tangent tangent manner restriction derivation restriction establish x generate adequate write sphere negative consist union sign vector consist center around vector cover step take similarly corollary always minimizer subproblem constrain boundary lie hence constraint force case condition various context page next riemannian trust region reduce descent trust trust subproblem point page descent lemma goal convenience state statement section suppose trust section trust l page np h region decrease assume region trust numerical carry current iterate w g c proposition h p trust decrease objective similarly region proposition w n decrease obey near minimizer convex unlikely lower bound decrease movement nevertheless trust take iterate drop next concern constrain trust give q trust subproblem page canonical section canonical exist cn w see page section np w definition constrain trust value eq numerical constant section claim carry next iterate decrease
good performance worse sag order magnitude reasonable top quantify require dataset naive sag storing store unary marginal mixed pos due sag application involve mini batch reduce mini batch need use twice pass requirement sag sag sampling crf extension examine large regularizer proximal variant see structure adopt sag might suit implementation acknowledgment like thank well helpful comment engineering research company conference provide institute cognitive proof strongly section subsequently second
feedback user parameter pure explore new standard regression overfitte insufficient provide dataset extract user click user characteristic click dataset span day ground test day day one query contain default search user example click information click day pass list click contain pass irrelevant relevant labeling document click click strictly relevant click unit highly document click short unit document associate click action pass click interested click

map reconstruct encoder reconstruct possible denoise auto reconstruct corrupted auto make deep stack show deep architecture decode construct auto dnn stack report propagation deep auto issue transpose weight transpose training auto encoder deep minimize locally procedure construct auto encoder
start initialization termination stop fall exceed simulated real find well depend component assess addition define ratio large matrix spherical component test medium number gray lr rr rr algorithm time time equal table apparent em computation notably experiment table condition predict converge slowly confirm cg cg less cg gray rr em performance mixture
frequently ten five classification assess superiority remain challenge objective relevance redundancy complementary dispersion pairwise need study direction include multi programming additionally causal inter concern dispersion always effective evaluation thus approximation study correspond thank fellowship foundation science research innovation science technology china section section gray gray gray attract datum mining past decade feature eliminate redundancy inter correlation whereas correlation ignore item feature evaluation criterion additionally interference false positive redundancy pairwise correlation classifier ten superiority representative redundancy dispersion fast fast field
proposal abc mix tune pilot paper adapt abc early demonstrate bfgs require pilot computationally hessian encounter demonstrate proposal different state proposal stable model prohibitive posterior use statistical method operate construct chain construct iterative candidate use proposal reject
even set set imply unique equation guarantee include unlabele result though expect likelihood supervised model improvement simply test contrast get compare look case well percentage supervise first odd concern number expect outperform statistically difference relatively small basically regard relative improvement provide column sometimes none semi supervise close optimal improvement explain supervision supervise semi supervise latter increase unlike improvement case turn report optical semi supervise supervise low likelihood rate still phenomena artificial display check regular increase rate classifier decrease increase go semi supervise supervised classifier less set hoc approach look obtain regular reason approach often probably explain approach likelihood large
operate insufficient pt soft ji second operate abc parameter use measure abc readily euclidean arguably abc genetic past year address kernel genetic genetic association string graph realization approximate become mmd give mmd testing mmd consider proportion true summary empirical e dimensional soft algorithm mean summary statistic
particle weight perturb draw distribution particle current simulate pass accept reject repeat maintain pool particle probability favor particle high reject region choice originally abc weight algorithm view abc package contain make publicly available detail appendix dataset set create empirical particle algorithm commonly trivially advanced particle particle pool could assign cpu
improve classification number asymptotic concluding summarize multi within assign close centroid fan procedure feature curse selection term model interaction feature identifiability impact classification depend different characterize interaction interaction strong measure contribution feature may effect vary hence sparse dimensionality
typically problem bilinear approach relax factorization factorize column aside solve hard relax norm efficiently simple rank via decomposition factorization nuclear norm generality desire negativity etc mean address consider convex optimization factorize relaxation include much broadly convex natural wide decompose multidimensional possibly multilinear term hold closely encourage factor requirement factorization convex regardless due multilinear tensor factorization mapping factorize multilinear mapping factorize mapping multilinear capture factorize linear follow desire parameter solve appropriately linearity max although multiplication easily operator network contain allow variety problem
point case harmonic analysis use analysis depend operator map denote array scalar array integer function relation formulate eq chapter chapter line constructive bound use idea approximation eq eq large simple lemma next specify
reach curve method curve train also attain learn minimum maximum setting rate numerous value way surprising allow beneficial though dataset imagenet architectures book neural source discuss range hyper give practical suggestion layer gradient involve practical setting rate learning rate early adaptive estimate learn gradient discuss divide magnitude gradient fundamental adaptive build upon hessian gradient automatic decrease learn hand demonstrate schedule limit gradient appear
component apply variety nlp task process sentence string feature capture attention mechanism useful neural problem list sort apply nlp state task different basis open use neural dependency lot approach right answer unclear reason phenomenon answer reasoning solve manual us mode understand necessary capability analyze attempt everything like parse logical rule classification stanford sentiment become neural representation journal bank semantic memory
closed solution account penalty thresholding introduce dominant parameter penalty adaptive pls product fit sparse pls select non response residual select estimate non estimation logistic pseudo p non predict logit pls pseudo accordance pls complete approach inspire hyper sub prediction reduce parameter decomposition train learn pls compression regression try response pls glm solve problem pls sequence define previously pls pseudo weight solve least step pls prevent modification generalize pls principle present
protein performance evaluate svm svm yet substantial gain svm original perform bad fewer worse imply make nmf vote neighborhood cosine string annotation observe dimension topological ask conventional reduction similar improvement end majority cosine pca gap topological constrained conventional aim model distribution dimension imply surprisingly performance comparable use dimension svms pca nmf poorly evidence find good summary current micro score full combine novel network notable upon
lp ellipsoid search subset avoid characterization proper give proper polynomial theoretical partial implication redundant characterization algorithmic could detect produce work computer theorem pair semantic sort implication define datum confidence partial rule redundancy implication far two exploit note duality characterize arbitrary identify confidence threshold conclusion useful present research decade context incomplete whereby process result great particularly context machine logic probability mechanism already expressive usefulness price feasibility reflect feasibility within polynomially tractable imply serious hundred explore difficult balance limit machine mention reference cite book learn particularly
allocation ideally eigenvector measurement use estimate error true covariance entropy outcome determinant accommodate ellipsoid true signal track progress covariance measurement update update covariance signal analyze measurement covariance small small update
cv select voxel vary different fold voxel stable disease label evaluate gain denote introduce stability coefficient suggest stable voxel disease stability positive voxel one drop around instability cause largely undesirable mid lasso large explore nonnegative fuse selection greatly improve
concave fact make strong currently formulation dominate pn compete relaxed enforce via alternate disadvantage difficult adjust user default pca true although outli run runtime name otherwise use reconstruction true eq close feasible ccc dimension span noise outlier half sample
label example draw convergence rate hand choose indicate md example regression likelihood xx constant order I regularity choose u logistic xy information matrix matrix suppose bound covariance py satisfied tell provably optimal convergence round apply expensive efficient greedy rate open
note statement obvious converse invariance argument illustrate constant refer complex complex real descent narrow function impose example loss loss huber loss complex define influence degree huber loss huber huber residual early huber maximum solving lead depend replace
f paris message pass amp inference compress sense amp framework modularity choice hierarchical utilize boltzmann rbm prior well rbm analyze rbm factorization signal handwritten experimentally rbm decade research occur inverse encounter compressed via deep recent year amp propagation problem description amp signal work application complex prior prior leverage hybrid amp promise present amp attempt correlation
base orthonormal estimate phase rotation attractive phase modulus instead sdp application phase sdp many orthonormal equivalent fall scope rank semidefinite global sdp relaxation extensively notably particular interest develop solver solver solver review complexity much characterize especially large allow grow determine desire specific tight closely synchronization proof stochastic block transition program also typically rely deterministic proof synchronization availability close form expression easy semidefinite relaxation form appear particular estimation rotation explicit constrain hull group orthogonal synchronization match doubly powerful class synchronization generalize framework mention scope indeed involve orthonormal different parameter throughout thought block block subscript indexing refer column refer stack column norm symmetric matrix positive dd coordinate interior extreme black dot matrix smooth matrix isolate matrix product circle degree freedom redundant factorization unique remain degree notice redundancy smooth nonsmooth code yy search geometry numerical tool optimization former riemannian manifold two metric u view riemannian
european fp agreement numerous discussion var style height department sup paris france international centre physics sup paris france universit paris paris france restrict boltzmann undirecte many include initialization multi main reason success unsupervise alternative iterative field physics provide sometimes persistent evaluate easily approximation systematic improvement machine unit restrict rbm undirected surprisingly dimensionality collaborative filtering rbms stack network form net architecture representation rbms
properly although section forget gate rnn current number advance cell begin therefore forget great keep concern important remove forget gate reduce learn sequentially embed semantic lstm vector feed lstm activations gate gate cell show respectively index horizontal axis code activation fig cell valuable gradually information rich semantic output evolve fig input gate word green color appendix bar figure range reason clearly appendix interestingly semantic representation sentence query lstm keyword focus active cell final representation whenever cell assume detect cell activation top query also cell word observe word mean number cell ht c cells lstm model keyword belong
usual deviation proportion left display sample mixture change component change c contain fourth mse correction equal strong despite slightly lose error error test differ region energy size detect contrary mse partition summary signature estimation associate represent mode minima fundamental mode list prove limit still limit resample bootstrap treat derive estimator optimality conjecture smoothness support finitely critical differentiable vanish boundary index basically lemma sufficiently lemma notation flow start mode assume mode gradient field boundary condition project project point away boundary angle whenever line point boundary line always move away boundary distance boundaries dx idea project nearby boundary flow boundary lemma every boundary flow region come flow within early start
normally one dimensional change statistic connection frequentist intuitive locally convenient partition slowly partition complexity binary state dimension aic accordingly introduce change point length segment length new computationally possible assume roughly varying significant point binary complexity aic complexity acceptable circumstance sample penalization state wide expression compute dependence penalty converge monte determine penalty
item infinity item modification gradient carlo rearrange mse follow power p bias I k eq break v v v similar j k j term vanish lm term v p estimate k p kt k p kt corresponding calculation go need collect theorem growth poisson equation solution equation eq q multi priori theorem obtain formulate bounded equation apply high first calculate assume front ff kf kf km km ij noise f kf kf hand e fact situation euler term q obtain introduce appropriate noise
approach miss publicly available good extensive study study feature information set entity kb single entity aim infer entity kb give entity infer kb wikipedia describe construct representation type entity observe infer entity type instance example snapshot type entity entity article give entity external entity entity text entity type entity type kb easily extraction snapshot strategy motivate importance evaluating globally
learn cavity nf n old choose cavity f q ff global motivate vi vi optimisation version descent sep optimisation alpha understand loop power
co occurrence subtle demonstrate simultaneously collection third modification gene dna mutual observe et able multiple module character pathways relationship exclusive gene exhibit co occur occurrence dot gene dot line ran remove back treat marginal mutually exclusive module module include gene pathway pathway publication module member pathway indeed know pathway genomic surprisingly occur co gene emphasize overlap module exclusive set module tumor enhance degradation think play role report cell identify important place module pathway publication module pi pathway pi pathway module mutually exclusive weight individual mutually exclusive gene specific association cancer previously role manually publication include pathway identify gene style character name cancer merge run run classification classify integration molecular examine relationship introduce sample mark mutual identify exclusive module module gene module specific include gene study contain suggest might pi strong signal surprising appear dominate module output six module unclear include gs exclusive study interact cell formation interaction perfect mutual pair subtle need
naive achieve increase validation optimization literature actual patient dramatically solution effective scheduling account comprise consider identify patient trajectory integration prove optimal dramatically optimal solution simple theoretical contribution novel entire general patient account literature applicable movement temporal user website movement cell phone user among network schedule patient week adopt operate patient automate algorithmic appeal ad manual employ validate simulate datum estimate cluster generate confidence level validate efficacy patient trajectory schedule achieve perfect dynamic exist naive significantly bad show access increase report average spatio temporal flow potential cost gray email school institute ga school business ability forecast census thereby management literature focus schedule largely patient schedule inaccurate paper scheduling patient
estimator dag dag step step dominate overall moderate thus reduce dag u pa z pa convenience run time dag p pa sampling discrete z z dag two multinomial trial cell actually relation cell provide store memory creating avoid one save running time run sample worst create take bad run save value concentrate accordingly j n z j log z log j log correspondingly likely multinomial concentrate dominant probability candidate usually become effective policy create strategy computer order store create especially store represent pre memory reach dag sampling use store currently accord usage serve memory use pre interval memory get order sample close equal share every posterior sort likely order share component accordingly sort increase experimental result dag please dag sampling dag sampling modular effectively order modular due modular essentially modular different modular eq common equal set relation hold order appendix accordingly eq note dp use importance sample unfortunately dp much efficiently estimator respect directly correct order method follow order draw order draw unique dag pre large result treat importance strategy please detail term
hereafter method life maintain memory consumption linearize mathematic sparse consumption replicate life software implement article image collect subject stanford center cognitive head collection procedure stanford diffusion coverage acquire scan spin water diffusion direction acquire isotropic acquire diffusion acquire scan acquire isotropic resolution collect slice trajectory volume acquire scan respectively volume estimate motion gradient rotation apply motion correction spin correction long rf acquisition software available scan segmentation manually track perform toolbox matter matter voxel use seed harmonic step orientation amplitude cutoff create candidate individual method brain brain matlab comprise cell neuron measure signal within
trajectory current mini trajectory trajectory context trajectory label alg model trajectory eq two mdp define two approximated model let trajectory approximate assumption trajectory approximate error batch approximated applying identify stop correctly exploitation next obtain batch achieve mini batch set must observe additional utilize trajectory soon algorithm computationally expensive practice line cluster essence mml goal simultaneously empirical matrix lose ignore despite effect variance sample get trajectory subsequently one question infinitely many bound
rv ne I n find expression less unless r v v cv r h I I p edge bad approximation run return constant nr r imply n r v e w mn bounds factor mn e h fall term multiply way eq case result sign sufficiently great eigenvalue assign work need run sometimes fail multiple vertex edge edge assign run I output run take execution basic eigenvalue run assume condition majority vertex basic give output median value desire one attempt assume let tuple already run bad approximation bad bad later eq probability goodness vertice community vertex community strict classify r assume already compute run product approximation community otherwise average minimum nonzero algorithm vertex approximation time execution succeed product assume computed vertex product minimize value generate properly execution vertex succeed less I vertex run must seek iff return start classify
algorithmic remain important matrix tensor increase polynomially address challenge inverse unlike relaxation neither scalable guarantee conjunction unfold result third unfold achieve also scalable apply measurement raise interesting regard relevance tensor section section definition tensor central modern machine target rank tensor measurement mechanism tensor suitable tensor completion constitute efficient rigorous tensor recovery adaptation tensor experimentally provide perspective problem domain video processing collaborative processing analysis recovery measurement ill simple adopt much rank example video model order tensor low scene spectral linear tensor specify refer reliably pose structured pose even ambient vector result straightforward mechanism separable tensor represent product merely four slice slice prove complexity mention relevance practical multi low rank gaussian sense compressive relevant machine collection inter consider predict assign naturally completion completion task jointly item consist item tensor framework setup rating user
system post sufficient cost constraint transition represent use programming follow stochastic vector policy state decision employ correspond practitioner scenario approximate policy optimal readily due stage reduction construct dynamic programming applicability limit growth challenge stochastic dual combinatorial exploit key stagewise ts decision resource partially arise partitioning resource outer collection affine cut hyperplane resource visit forward pass program ignore backward iteration construct cut hyperplane subproblem please hyperplane necessarily tangent strictly emphasize post state notation assume construct problem aggregate pass update approximate grow solution period feasibility clarity presentation statistical simulating realization interval gap practice often criterion version exist utilize consider separate history solution update new problem scenario realization update feasibility
active progress advantage offer flexible appear come performance gap count first practical factor numerous pathway biological problem appear support benefit paper simple examine parametrization labeling adaptively adaptive vertex selection helpful concerned adversarial deterministic query graph observe query collect statistic passive semi exist good paper label mention predict rest component correspond labeling perspective improve able close quantifie learn labeling quantify achieve hamming error sound ham valid tree guarantee nonetheless set induced labeling guarantee vertex call noisy fixing vertex return equal oracle design label accurately label design nothing label equip adversarial labeling towards end label I
prediction gp reference polynomial approximation surrogate amount gp stochastic gp representation hyper parametrization alternative incorporate problem construction pc compare direct expansion hyper issue basis true prediction likelihood appear hyper turn mcmc greatly accelerate posterior introduce stage acceleration hyper parameter determination require dominant subspace covariance function constitute severe limitation simplify cpu roughly future coordinate pc expansion surrogate prediction line sampler truncate mode average optimal choice density variability reference process pc improve surrogate pc introduction problem infer profile gain hyper parameter infer improve pc surrogate seem quite suggest possibility moderate pc particularly pc error finding plan posterior involve transform sampler adapt structure regard pc surrogate construction reduce accuracy accelerate sampler coordinate appear key element handle pursuit currently consider publication science technology ok acknowledge
ergodic admit rely generator spectral identification method study crucial determined operator change diffusion change order estimator stochastic practice value could omit view determine surprising rate coincide frequency clearly apply low randomly use time step introduce reflect main rate section proof stability eigenvalue loss generality measurable volatility process brownian satisfie part non increase schmidt drift volatility continuous strictly topology endow borel give point write observation independent law process weak restriction
rank infer factor coefficient coefficient e spike high anomalous subset actor search page website national uk actor report insufficient component perform search actor along top wikipedia page provide far anomalous infer year date range search interpret ht three shape grow mode grow slowly property apparent vertical half arithmetic expectation show two presence rare event yield prediction factorization equation auxiliary expectation equation factorization equation relationship lee tensor factorization pmf make connection implication perform generalize kl lee equation sometimes converge value factor set due update correct euclidean small prevent
unknown variable message uniform distribution message terminal try content address bottom smooth step message propagation message correct uncertainty distribution forward backward propagation message correspond layer block shift basic patch dimension layer need layer learn vector generic level pixel
design new carefully project update usual many pursuit program penalty code simple multiplication set keep coordinate coordinate recover sign correctly column wise sign correctly progress explicit rule initialize decode initialization pair work high section interest solve bring properly proper primary trivial initialization analyze plausible sparse architecture implement algorithm light code accomplish nature closeness sign hope simplify coordinate wise sign fix sample geometrically strategy correlate never get elsewhere probability ingredient update rule prove near formula amenable decode p x ix negligible constant calculation column step algorithm ok np close invoke probability function event x moreover happen notational plug equation b b happen support nonzero use ir ib I I matrix lemma decode complete near prove theorem simplify expression need assumption model distinct variant step geometrically currently need subsection design converge geometrically
precision least rescale must claim admissible j j ji consecutive copy go line exposition suppose mean particular imply notice happen claim apply symmetric line instead straightforward claim rescale parametrization replace piecewise rest parametrization often describe distinguish denote mixture r rescale transformation know simply iterate interval interval see parametrization fix allocation allocate decompose rescale allocate interval empty x omit necessary fix imply inequality component guarantee component admissible let possibly let mean algorithm learn gaussians informally first distance rescale gmm analog omit proof identical approximately minimize binary technique feasibility form let ki almost exactly encode range perform constant polynomial rescale shifted support break tie arbitrarily iw let k also weight return must program k theorem contribution take density
use mask corruption experiment add input noise layer unit layer noise approach describe network dropout unit achieve mnist universit universit de universit cifar project dropout network thereby yield explanation show augmentation dropout result significant normally undesirable mathematical aim mean decade play brain deterministic incorporation noise strong developing place beneficial
space semantic space statistical get sequence example minimize expect difficult joint perform divergence widely output measure like marginal power call gm histogram ground distance eq predict distribution increase mass distance accord ground wasserstein duality mean lp optimum subgradient costly entail prohibitive propose efficient subgradient loss importantly solve dual identifying constraint sum well efficient algorithm know lie simplex directly optimal ambiguity pair correspond vice versa
cv estimator fact correctly contain quantity asymptotically pseudo estimator root residual distribution sphere carry goodness residual another natural diagnostic might individual disjoint include measurable therefore nan compare statistic reference potential p intersect p e ty es ty ty j ty truncate gaussian truncation interval constraint involve e right hand approximately restrict interested testing propose assume sufficient validate scenario gaussian
approximation dct spectrum closely examine require interpolation aim expression allow efficient act elementary identity establish dirichlet kernel dirichlet value instance imply translate x expression connect interpolation function efficiently delay offer form draw let near function matlab fractional interpolation essentially govern indeed accord case say plot intuitively use linear effect remain negligible also since overlap act function conservative employ propose heuristic calculate
algebra chain irreducible stopping depend finite f fx derive easily arm satisfy min j jt ts jt nx jt nx jt jt expectation min j n jt x jt armed markovian computation slowly arbitrarily incur exploration incur exploitation event also observe event l j jt x e l max b jt jt inequality fact max q present easily precision suppose precision assume monotonically bandit markovian reward precise computation give choose regret arbitrarily omit decentralize mab arm markov player get arm model irreducible reversible x
fx consist state markov chain behavior boundary f tend third purpose infinite dimensional generalization matrix construct transform apply deal operator hilbert calculation discrete fourier transform matrix unitary f kf k matrix unitary transformation function h nf h unbounded operator assume simplicity periodic condition fact dimensional notation subsequent theorem proof characteristic characteristic spectral unitary operator approximate discrete hessian converge generator counterpart exploit relationship corresponding function sequence sequence spectral characteristic distribution converge characteristic calculation straightforward shift fs fx k fs fs
combine last statement provide low subspace nontrivial let deviation since w w refined deviation convergence almost difference careful convex thank inherently optimum strong without establish hypothesis pair b statement start apply taylor every since whereby jensen due start lipschitz satisfy mean strongly optimize eq term lipschitz satisfy eq r class q property rademacher choice consequently bind grant combine precede nonnegative finish case yield back ii iii support result establish relate primal optimum hoeffding range secondly final discard low consequence measure eq purely particular remainder choice prove contribute secondly choose satisfie w place restrict part fix simplification finish display purely briefly draw let draw treat treat sample multiplicative chernoff piece condition failure draw set invoke control via failure note crucially e exist specialized agree particular apply suffice lastly term per handle manually appear bind discard fact desire lastly consequence q
detailed flow decoder pressure information concentrate denoise autoencoder encoder discard typical irrelevant pyramid top true learning autoencoder prefer build connection
sparse hadamard randomness yet main intuition loop line improve approximation pick since success pick designing enough begin pick random choice try estimate good improvement turn option increase subsequently sample running nearly consider choice instead minimize perform respect submatrix random choice randomize choose determine obtain stack use simplify j n n uniformly kx minimize x yy parameter randomize analogue since time randomize receive access vector analogue absolute parameter suppose rr bipartite row least execution bad sequel hadamard application specifically incur figure provide start input choice random moreover let complement proof query general later attempt optimize incur logarithmic elimination time need computation basis arithmetic hash linear family informally hash output remark computable mild asymptotic appeal
evolve base start production induce evaluation evolve responsible historical bias discard user profile guarantee stationary item record probability offline offline evaluation
strong combine transform residual use distribution hypercube multivariate distribution cdf tail student financial return scale denote univariate student degree log denote correlation margin correlation r copula newton compare margin use smc conclude margin degree freedom variation return equally weight three copula use asset exceed adopt monte simulating copula simulate filter residual repeat equally weighted portfolio advanced computing case present student copula margin quite especially leave reader decide illustration section abc compare require robust log provide extract costly proposal pilot require include adopt gp tailor idea interesting tailor gp would fall part efficient abc area discuss acknowledgement work model
mi give predictive proper gaussian predictive test show uncertainty would expect covariance key encode wish example exponential among informative point principled follow part covariance real important value matrix proper skew symmetric conclude reproduce framework
htb substitute filter software consist optimize compact run hardware collective context conceptually software maximize cost improve reduce time market cm big public framework collect preserve incoming save believe approach eventually enable software hoc need software hardware piece win either help gradually power consumption community try reduce consumption cost numerous rapidly evolve system several decade practical inspire science wikipedia may help software connect domain often coherent top public repository mind engineering community gradually frequently specie input specie execute piece software mobile center collective mind public notably behavior community gradually environment manually popular big result integrated specialized version cost across many hardware software tuning continuously adapt run hardware continue improve performance minimize usage current hardware engineering gradually create diverse benchmark public continuously improve help improve hardware simplify convert generic library cm avoid many project vanish publication public development already share specie collect feature meta semantic code os hardware cpu public repository cm allow validate approach major demonstrate enhance house heuristic production detect validation hardware finally
risk vote call classifier true risk return usual predict risk h pac deterministic risk
rank rank toeplitz use exponential matrix corollary recover measurement random toeplitz toeplitz satisfy anti anti operator maps toeplitz unitary adapt toeplitz follow state rank recover exactly gaussian rank toeplitz let toeplitz satisfy n universal unique section numerical experiment improvement numerical signal application signal complex uniformly randomly generate
sequentially development plan explore improvement estimate estimation quality column appear grateful state university help create rich motivating award mixture inference economic partially award national tool mutation sciences university partially state stage article pass away like around composite problem email statistic usa edu biology institute school public health usa edu state usa arrays allows simultaneously produce along g observational unit contiguous segment order markov structure likelihood subset simulation application validation composite array grouped take serial arise economic record pool one group contiguous period nuclear
rate constant exhibit phase illustrate htbp optimal smoothness fall determine entirely specifically function eq right dominate become write bound away happen vector come ball zhang yu coincide variate dimensionality early exact yu phenomenon universal approximate pointing vanishe sufficiently smooth phenomenon situation
feature upper body player entirely involve action skeleton static pair feature extract window well across actor dimensionality dynamic feature reduce pca extract deep rbms joint normalize dimensionality evaluate different size work movement baseline different classify evaluation task predict level strength set consist instance fold validation accuracy feature baseline combination six annotation demonstrate effectiveness detect furthermore demonstrating start skeleton feature superior class classifier person frame grind sequence length metric partial
close resample enkf resample improve figure maximum enkf resample enkf weight ensemble particle drop describe filter unless resample strategy panel resample enkf panel filter distribute particle filter idea particle understand optimal derive enkf target approximate approximation marginal become weighted particle equation plot weight unless implement leave panel resample weight rigorous covariance matrix ensemble enkf enkf would impractical nonetheless enkf perform application contradiction assimilation enkf estimate often mse model enkf member frobenius norm line wish dimension enkf size study scale linear enkf localize draw central limit assume state combine sample expression mse central limit mse quickly mean find mse test enkf may insensitive measure covariance idea covariance wishart dimension go infinity order agree find forecast ensemble enkf ensemble huge moderate
mention take concentrate situation evidence copula mainly final among language algorithm th row simulation uniformly sample draw approximation posterior critical implementation base moment strictly provide course pose issue ease unnecessary
actual achieve program program hierarchical continue hierarchical realistic atom weight vote distinct factor exist satisfy upper k construct graph achieve variation semantic special factor variable experimentally different semantic converge voting illustrate vary sampling semantic logical seek program set semantic voting logical ratio xx rule evaluation hierarchical query boolean rule overlap logical semantic trivial simple non program exponential rather tuple asymptotic grow logical example unbounde contribute proof construct another attain define space coupling define time coupling couple sampler run sampler choose assign assign prove vote logical voting weight independently logical ratio projection semantic vote semantic world remove ratio semantics fm fm p argument apply next bound logical semantic running sampler least variable parameter violate sampler argument event event couple variable couple know meanwhile since run coupling occur cn prove lb logical logical ratio minimum variation must require vote linear voting semantic exist choose flip sample sample
loss infimum training converge almost empirical n consistency well chapter erm uniform law number cover via dim think general
explore enforce number arbitrary appendix tune propagation configuration message pass confident discard datum assume cluster limitation overcome inclusion intra address first metric property extend higher modify enable constrain enforce equality case datum four possible state pair possible participant discussion
whether state curve generate filter criterion aic autoregressive capture autoregressive ar state model use improve ar filter coefficient approach time series modeling forecasting behavior stock market variation eeg cause brain refer state denote assume belong
behavior requirement regard logic logical conjunction implication temporal eventually nest combination always variable finite atomic proposition formula atomic formula formula operator formula evaluate truth atomic proposition execution atomic proposition appear iff iff iff iff iff imply hold formula execution position formula execution execution formula something bad never happen formula thing happen proposition state atomic proposition say complete win formula run express qualitative specification requirement environment protocol map set finite memory strategy strategy regard case singleton
descent inspire mirror particle approximate posterior density competitive scalable latent method capture posterior nx posterior hence intractable pose challenge one variational besides challenge arise large pose challenge scan dataset practical address issue approximate descent space point filtering maintain correctness convergence stochastic mirror optimize objective functional prox mapping subproblem long solve control connect optimization possess number apply even line code value different
embed embedding recent compositional structure take account cnn recursive rnn several designing feature engineer design annotate chain rnns linguistic annotation dependency name entity relation difference word appear role task nlp extraction entity sentence tackle compositional rnn yet order achieve assign word assign treat way compositional approach significantly pure compositional give entity head embedding accord entity linguistic rnn enhance rnn type tag feature embedding engineering task parse role relation
run second include thus fast speedup synthesis slice rich acquisition image employ normalize fig far sparse db db b preliminary transform blind sense investigation elsewhere patch directional extend overcomplete transform boost performance present transform blind formulation exploit transform voxel formulation nonconvex block problem update guarantee objective define formulation guarantee minimizer usefulness promise mr reconstruction usefulness blind inverse study minimizer follow problem iterate proof prove iterate sequence accumulation accumulation accumulation iterate critical difference successive iterate accumulation local minimizer establish input initial sequence fig fix transform step code step alternate code step similar algorithm g furthermore regularizer cf sequence b convergent subsequence hence accumulation standard boundedness trivially square barrier negative function singular bound immediately conclude boundedness h h previous argument constant sequence optimality accumulation sequence accumulation equivalent index iterate simple due obviously constraint also continuity singular limit property convergent limit product convergent limit arrive every accumulation accumulation subsequence accumulation optimality
similarity matrix form singular vector separable trick map lie kernel encode pair short path along surface manifold map singular svd require review dimensionality geodesic distance nystr om method improve speed base classification approximate psd psd psd matrix choose collect partition tn contain row column index note generality column column nystr om svd tn since computation much fast svd complexity enable singular large desirable calculate store selection method type nystr om approximation nystr om random sampling theoretical number norm appeal computationally accuracy matrix redundant large draw sampling rank regardless must form store approximated exhaustive
design minimize measure uncertainty variety product monitor example uncertainty adjust surrogate square realization space divide hand square integrate indicate right side independent sample term hand ability directly evaluate design assume draw loss adaptive hyperparameter evaluation integrate therefore simultaneously posterior additional interpretation covariance operator understand exactly similar optimization find simultaneously update design close loop alternate batch numerically quadrature monte mc general quadrature low moderate carlo generally carlo offer flexibility domain replace variance set operation sample optimization simply replace quadrature minimize readily analytical directly derivative derivative quadrature form eigenfunction situation desired form unknown eigenfunction maximize right eigenfunction integration eigenfunction homogeneous explore work become sequence suppose find procedure popular include location experiment space design design tailor regression comparison design seek uncertainty measure entropy candidate location seek represent output simulation input np
lda focus topic topic treatment absolutely distinguished obviously well grouping lda embed word tend mix word representative return eventually integrate lda successfully
statistical expert intend knowledge provide service cluster request handle since request organize manually service program ask service request result tool sample pick pattern demonstrate paradigm machine restrict cluster searching agree cluster set partitioning distance potential potential distance formal paradigm cluster center center value point cluster arguably center lack incorporate domain hoc translate
reasonable assumption gap upper bound direction cubic moderately continuous time reason future chain specific parametric chain chain factor kernel future result hope insight area empirical markov sample inequality simultaneously follow bernstein chain marginal chain case combine obtain tail ni generality assume ni follow gm second bind prove deduce probability least bound devote range bind non randomness immediate combine application tail analyze sum
unit generalize maximize eigenvalue reduction np hard np hard recognize factor encourage multi scale dimensionality notation
unit sensor compression datum acquire central indirect indirect coding pass channel sequence rate compressed receiver produce distortion another characterize trade quality distortion amount maintain notice follow observable fidelity reconstruct symbol realization appearance intuitive
nominal disagreement indicator adopt report auc performance apply problem goal impact parameter quantization level estimate rank indicate asymptotic thm thm level underlie htbp auc bayesian thresholde generative density detection insensitive quantization simple k bayesian htbp c anomaly http attack conduct experiment use set http forest uci repository set c c forest cover http cover http randomly nominal rest datum hold memory use test point auc report fast comparable density bp comparison class due however svm training single percentile different
suppose equilibrium summarize global agent collective relie reader refer somewhat together second inequality inequality give goal compute independent notice inequality upper inequality obtain inequality nan give important aspect equilibrium arise play filtering algorithm equilibrium diffusion strategy undirected follow attract equilibria part action agent equilibrium reveal preference probe associate action probe minimize type detection subject world detect agent social multi reveal preference equilibrium network traditionally economics sciences rational pattern agent comprise limited capability agent interact network theoretic notion equilibrium describe content reach game theoretic arise long adapt em address paper relevant broad equilibrium paper social possess capability reach fashion formation characteristic facilitate network scheme collective behavior converge equilibrium sec non social follow illustrative social jump possible reason answer yes friend different behavior inspire jump due restriction behavior tendency social
linear cubic test sized corpus principal component capacity readily corpora scalability deep rich scale elegant load method generate approximate conjunction method randomize scalability linear introduce shift kernel kx
empty expectation iterate j iterate expectation take definition use calculus use th harmonic term use derivation elementary finally compare I variance auto reconstruction maximize informative normalize explain decoder primarily historical auto optimize symmetric explain hand objective algorithm report reconstruction error objective auto encoder discuss psd matrix eigenvector large algebra ccc eps eps eps eps observe good predict sparse explain surprisingly respect variance argue optimize implement variant theorem section approximation simple call sparse correspondingly version principal iterative auto batch principal auto encoder art sparse pca generalized method operate
schwarz p n op c relation basis basis relation theorem n e I np claim assertion result combine elementary statistic since prove proposition four lemma establish n pl element v n constant depend exist lemma relation u enough probability tend imply lemma consider variable satisfied follow asymptotic nh kk kn jj e pp n statistic limited relation neighbourhood order th
unitary amplitude hz hz scale wavelet may better wavelet scale difference level odd wavelet wavelet retrieve regardless symmetry symmetry analyze priori wavelet may st wavelet later improvement odd wavelet odd hilbert wavelet signal
coincide label high correspond widely estimator distribution usually stage supervise label training phase usually case label challenge behind quite label binary label two example co interpret high capture kind application domain label summarize notation toy label classification label implicitly circle triangle element active description binary datum multi work method later discuss ct build posterior complexity bottleneck set ct offer performance cc classifiers conditional dependency multi direct graphical naive multi know l graphical implicitly target output dataset rule b increase extra propagation incorrect estimate affect always serious ensemble exist avoid exhaustive opt chain propose individual
long place layer graph lemma inductive know node otherwise get path long since path inductive argument note one inductive prove therefore v prove lemma path f consider path weight exist return wu wu activation therefore eq conclude sum turn add together complete subject hypothesis w hardness short pac intersection realize network bind intersection half margin unit incoming input layer sure title title institute characterization feed network capacity feed neural understand hard activation sample feed network logarithmic parameter vc understood depth activation vc depth train capacity class bound
careful summarize start ij mt mt repeat gaussian principal gaussian partial w exponential binary smoothness mat ern smoothness therefore exponential leave find exponential performance evaluation logistic pc parameter calibration observational natural observational natural observational intercept discrepancy dimensional vector j supplementary uniform cover plausible density scale prior n independent infer via discrepancy important inspire discrepancy location sign output observational iy r cv discrepancy persistent setting procedure design plausible hold translate discrepancy pattern logit natural choose large cause issue simulate nice capture figure supplement heuristic appear well variety remainder positive infinite sign great output represent different dimensional vector logistic reduce dimensional detail covariance construct discrepancy variance estimate calibration ice calibration parameter reduction ice observational description calibrate scientific interpretation approach simulated result implementation I
solve g achieve approximate algebraic computational feedforward machine n n ii matrix weight fix bias neural small result equation hamiltonian efficiency exploit regularity function using usually discard early set train algebraic approach see compute neural potential move cc activation target single feedforward unit training bias set function shape hyperplane approximate use property run henceforth monte hmc phase exploitation initialize prior distribution
price stock share shape link market increase geometric brownian capture fact vector move significantly rapidly adjust volatility stock share shape closely link stock row three high volatility financial affect volatility across market price top row market market brownian motion volatility volatility however major spike confirm respective stock change stock life half due volatility year collaborative kalman dynamic filtering object location brownian allow preference present drift geometric motion predict result time since player team performance automate ask collaborative environment would estimate question
common practice crowdsource amazon become powerful collect collection preference rating response online engine datum training machine understand sequentially finally massive another pairwise choose pairwise comparison product crowdsource ask identify well search involve sequence model think estimate item player search engine comparison noisy variety pose subject arise competition randomness important latent comparison relate compared design fundamental aggregation broad special namely variety theoretical paper similarly closely case case complement base gap achievable contrast show constant rate tight tight comparison aggregate ordinal different approach parameterization partially rank aggregation setup setup ahead fashion literature sort assume noise pairwise rank actual rank assume embed outcome comparison distance auxiliary variable item compare instance individual make comparison objective consider crowdsource present spirit collaborative measure individual ranking item probability rank case belong broader analyze paper concave norm dimension upper bound
x eq hold binomial gaussian norm observe available thus entropy eq combine low forward trajectory x design constant entropy final reverse trajectory design x rewrite recognize transform probability note entropy divergence analytically compute perturb set body normalization write original original distribution substitute substitute identical integral achieve show equation behave b x x x f x x x
experimentally initialization generation besides novel estimate experimental small em good initialization model candidate necessity conduct necessary discrete context project cluster multinomial novel evaluate identify appropriate different parameter statistical automatically necessary generate generate maximization however
number evaluation thank force considerably require subsampling subsampling review advance mcmc divide distribution manner individual chain grow small approach face keep likelihood evaluation per original show strong ergodicity assumption mh satisfied practice experiment extend general scenario methodology even iteration however gain context excellent improve subsampling approach negative bernstein von model achieve couple pass observe far demonstrate applicability difficult bernstein von good importantly von acknowledge discussion convergence ns moment convention note come proposition write com france uk markov monte often computationally intensive practical big also approach recently learn group divide aim first comprehensive guarantee leverage understanding limitation posterior evaluation able far propose display good bernstein von excellent scenario bernstein von individual aspect statistical bayesian demand yet mcmc often intensive mcmc mh bayes approach prefer fully justify scenario function differentiable application quantification uncertainty preferable
j remove acoustic remainder per derive ht r r source remove complex interpret avoid representation relative frequency channel add perturbation add transform similarly per express leave identical delay frequency angle regression average additive magnitude response unlike everywhere magnitude concatenation original magnitude convolution constant across specify per division predict scalar target variable variable bayesian assume realize f linear draw normal index input generality gram characterize pairwise semi establish kernel omit space derive multivariate normal random kx wise evaluation input represent latter respective measurement coordinate model coordinate
n tw method primal exactly standard sdca dual tool ultimately development good iteration prove set play primal dual contraction dual positive function need sure serial sampling ascent sampling sdca unnecessary scale apart
state centralized obtain coefficient optimal insight numerical result average gain perform rest parameter also parameter first determine notice hypothesis comes pose achieve pdf learn unknown parameter mn network observe come neighbor respective hypothesis explain unknown neighbor ty I likelihood mle write expression write give learn iteration parameter alternate maximization algorithm come th jj z pz expectation current set eq compute lagrangian derivative eq summing zero jk initial achieve follow decision
denote limitation far parametrize equivalent specify depth since specify minimum splitting h nj nj nj nj x x nj nj n nj nj nj j nj predictive nj like forest want distribution distribution nearby smoothing approach associate prior via marginalization common tree label block independent label take variance label label leaf follow convention discuss inference
need choose subspace encode encode pair represent number number classify belong lack decide solely state assign arbitrarily quantum encode consist basic scheme encode separable hybrid partially exploit quantum state use quantum use representation alternatively representation fig wise translate
present world important discovery high sequence oppose sequence conclusion sequential pattern raise complex scope address pattern body sequential mining introduction major derive compact pattern discovery grow body several variety score interest pattern frequent mining database world numerous include web biological mining frequent sequential database seminal paper researcher extraction sequential high sequence note support extract pattern subsequence occur sequence sequence subsequence application domain decade interesting sequential pattern score explain score pattern author score heuristic construct sequential pattern direct consequence use derive nan independence hypothesis derive expect study extract pattern study significance study take
condition result hx parametrization conclude proof assertion prove distribution differentiable easy derivation conclude complete frequently lemma k q part part first establish constant whenever reach projection onto projection exist difference onto proof span remain asymptotic bias du
real adjacency physical previous example exchange health care patient room temperature sequence continuously weather derive low discrete manifold describe computationally tractable datum weight possibly represent literature matrix graph test processing series big generate store finance medium running example protein protein patient record care customer power water natural utility phone wireless service financial
computational bottleneck cholesky storage belong induce view approximation subset regressor induce exact covariance correction prefer specify induce point likelihood rise storage efficiency gain often severe expressive kronecker toeplitz kronecker introduction chapter multidimensional x pm pm pn per efficiently separately kronecker product scalable exact gps eigenvalue inversion trivial eigenvector product storage popular kernel rbf already structure require multidimensional input severe extend kronecker dataset grid structure g image miss grid miss due virtual observation virtual augmentation virtual efficiently kronecker
relate execute cpu multiplication batch small maintain take memory show convolutional cpu device figure core core observe vary speedup small batch speedup size compare underlie unable optimize example severe execute batch device memory permit entire partition partition partitioning equivalent coarse grain employ show full end ec physical core horizontal axis
condition nx hold remark appendix optimal optimal norm note derive nx loss sparse x remark second eigen reveal sparsity dimensionality approximately contrast analysis implication randomize reduction implication randomize separately lemma possibly assumption individual version assumption four randomized randomize hadamard transform hash corresponding implication recovery employ reduction projection sub variance e ii rademacher
discovery human loop g representation crowdsource hierarchical propose kernel assume etc unknown present abuse write recover entire name effort feature discuss batch consideration design want possible worker suffice return return query triple query simulate triple example distinguish known outcome triple though also advanced feature form tree internal single leaf path set feature triple triple none b query query triple terminate root internal aside example feature root leaf otherwise reconstruct standard internal child proper binary proper tree triple find query never
trade oppose label available control trade domain pac bind justify empirical classifier adaptation dataset set deal adaptation conclude latter pac introduce stand bayesian study first tackle objective adaptation distribution provide belief observe aim learn lead nice
yield integral analytic target majority tracking place sensor contribution I represent sensor mapping sensor function know clear take measurement target sa filter measurement gaussian measurement filter sensor analytic bayes multi identically iid brief review sa filter subsection propagate cardinality filter mixture particle sa filter multi estimate challenge perform space subset distinct inner delta inclusion write letter multi represent g unlabeled one etc bold important distribution filter standard distribution
ability software simulation form available page supplementary material additional result paper none foundation carry program national penalty lagrangian vanish complementary equivalent permutation rely simulation numerous hypothesis replication rather quantile replication involve select decomposition adaptive ridge q diagonal adaptive diagonal entry calculation th convention design estimate concatenation original ba complement j perform removal secondly permutation thus eventually cl ex optimize ex remove average intermediate calibrate control snr rr rr rr rr ex design em ar h ridge ar compare ridge calibrate
give arbitrary reduction filter thm red fast attribute ignore simplicity constant confidence parameter red application thm run e k ok applying since time label rgb em mistake sequence question question tradeoff mistake improve et factor presence
estimator prior criterion assume prior sequence minimize contain hyperparameter compact optimal cv case parameter distribution practical application question hyperparameter cv minimize theoretical hyperparameter asymptotically neither cv want measure cv hyperparameter hyperparameter hyperparameter small use useful complicated mathematical numerically approximate heavy paper hyperparameter average onto important future study support education aid scientific keyword keyword important cv widely
case know sample krige eq r experimental give krige apart meta krige technique develop year development krige aspect estimation hyper parameter adaptive additive may adaptive structural reliability analysis global meta model expansion vanish function loo krige model exact analytical loo krige spirit pc expansion denote polynomial krige experimental version pc interpret krige uncorrelated dirac show leave derivation block one error combine inverse see leave krige expansion approximate polynomial behavior orthogonal polynomial krige call krige combine modeling technique expansion cast orthonormal polynomial stationary autocorrelation parametrize hyper building krige meta part truncation hyper set sparse polynomial evaluate universal framework krige combine various way approach pc spc pc krige krige experimental
noise produce prediction module assume reference therein identically result independence across sequence yet th component realizations independent stochastic denote signal namely impose condition variance satisfy complement let reconstruction quantity direct right variance note variance know small pursuit state reasoning iteration u signal estimate sequence perfectly guarantee bind variance estimate way insight large negligible tell mostly give select namely perfect reconstruction frame reconstruct visualization gray cs gray reconstruction frame
apart simple alternative univariate know paper argument sequential kolmogorov test power argument univariate parametric classic test nan simple test seminal clear emphasize tackle hypothesis secondly one also test computational lastly sequential provable argument stop hoeffde even context line context union arbitrarily inferior v hoeffde far outside scope
comparative largely govern minus drop see change concept gain concept derivation thresholding computation function reveal outperform high environment stream environment would memory repository exploit accuracy care spectra combine simply similarly perform spectra maintain single spectra show outperform include memory speed believe effort involve keep low coefficient residual coefficient spectrum income another parallel research spectra compact version decision obtain apply capture concept highly concept long research recurrence reveal term classification world pattern machine recognize concept occurrence efficiency advantage system time use make become auto pilot avoid smoothly environmental action take interest occurrence pressure couple
gaussian free issue per quantity adjust hyperparameter pick parameter update hyperparameter change datum easy change try thus learn poor less move poor yield local gaussian density th mixture tail prior component resemble spike prior amenable optimisation optimisation upon amenable minibatch often epoch optimisation data partition equally subset fully gradient propose minibatch cost minibatch cost way
sphere chart treat spherical constrain inverse adjust change volume implicitly red vertical boundary green vertical boundary horizontal boundary map sphere result automatically back ball section norm constraint common domain quite address transform ball sphere automatically fall domain adjust follow obtain unit hypercube map illustrate jacobian determinant rd detail discuss quadratic multivariate since exactly analytically expensive type spherical augmentation range last l invertible follow change variable formula spherical augmentation handle impose hmc applicable wave hmc handle wide quadratic write spectrum decomposition need type map method original domain operation need comment general constraint unconstraine constraint deal map unconstraine sided constraint change
model consume limited power capacity exploit structured training structure alternate direction coordinate descent task formulate structured problem assign constitute example image scene parse co document fully exploit representation structure essential train current impose disk capacity structure large volume limit linear develop distribute little except develop distribute structured notice
covariance ideally mse prefer estimate distance analytical percentage kl less present divergence mml estimate map ratio determine preference nan alternate negative logarithm give freedom rejection nan conversely exceed reject estimate compare mml equivalent hypothesis alternate reject percentile degree evaluate value rejection critical value p estimate control fix present generate distribution increase magnitude start behaviour estimate analyze estimate use illustrate mse mml base version compare mml mml low map see frequency mml estimate suggest transform parameter estimate ml agreement mse ml f show p variation mml however value across suggest modelling estimate mml comparison result present map mml estimate mml bias mse mml divergence mml percentage mml c mml estimate number mml estimate majority figure observation observe mml hypothesis model moment map mml significance follow map mml mse proportion time discuss parameterization map affect parameterization amongst mml respect describe ht behaviour range low moderate clearly mse datum figure illustrate prominent mml version especially mse mml map mml divergence highest accept
determine adopt aim positive neutral user tweet tweet precede hour period publication tweet user activity create tweet number tweet end neutral sentiment publication tweet exposure possibility sentiment neutral four prior bottom positive bottom stimulus response negative tweet three neutral sentiment generate observe produce fig stack identify sentiment neutral tweet prior tweet tweet neutral
desirable distance single graph short answer point graph hardness formalize median therefore unlikely near pair pair require distance exact computation quality normalize root expect randomization square difference estimate actual unbiased average ratio chebyshev mean cv imply mean roughly probability decrease size size get relative polynomially node review median metric natural centrality determine distance result weak detail show average uniform use centrality albeit identify approximate heavy true average dominate recently obtain computation metric space distance distance obtain small error sample distance estimate relative improved bound sample suffice argue uniform al also show project onto
aggregation procedure probability aggregation dictionary reference follow assume dictionary consist bound lead subgaussian every corollary present attain dictionary tend cm introduce star star moreover star erm erm infinite dictionary suggest erm benchmark direction bound target unbounded procedure function bound assume moreover subtle subgaussian different diameter note slower close introduction constant mention abuse write function specify integration perform ball unit sphere specify present essential proof former heavy
make overfitte image although share need embed multimodal layer datum yield rich multimodal description concept concept difficulty firstly concept may concept solve fix secondly concept example intuitively roughly proportional word baseline new address fix three involve make activity dataset dataset annotation construct concept occur standard give performance entire start sign sentence color recently progress neural language recurrent rnn long lstm achieve nlp task computer
reduction pac access run example thm formal monotone together lower embed bind pac bound main section equivalence rademacher complexity proof approximation section version version boolean hypercube fundamental boolean monotonicity literature structure spectrum monotone boolean uniform start investigate hypercube closely relate submodular monotone addition monotone share formula inspire monotonicity hypercube build technique develop aware technique submodular review submodular submodular multiplicative reconstruct submodular factor match briefly detail find application submodular random example come subsequently algorithm essentially multiplicative release submodular constant lead order build give random low approximate submodular submodular approximated approximation pac submodular submodular imply within norm work bound function improve bind
parameter observation varied estimate moment mle approach benchmark estimator weight error estimator figure function estimation superior estimator function package incorporate source package refer reproduce document cell constitute preprocesse large center scale study use file cdf package base microarray datum laboratory yield study size range cross normalization quantile normalize common cumulative lastly dataset gene study sis integrate scatter top pool study investigate em yield subsequently scale gene gene obvious contribution within
recursion decide density four normalization explain try replicate hmm path viterbi last evolve move probable ergodic mixture probable state quite transition probable state jump right possibility conclusion forecast totally uninformative applying calculate predict order comparable cm cc analyzing mean cover wide possible loose
random approximation complicated likelihood mention spectrum transfer overview literature markov perturbation might restrictive perturbation geometrically chain perturbation iterate geometrically chain perturbation lyapunov stability one estimate geometrically ergodic related focus constant main qualitative ergodicity perturbation early induce finally recent contribution present wasserstein chain approximate uniformly important whole supremum norm restrict thus probability rely lyapunov type approximate section wasserstein distances highlight functional analytic formulate interpret ergodicity result present perturbation bind wasserstein distance show geometrically chain perturbation model langevin algebra probability define wasserstein probability measure measurable
hoc specific fashion claim function optimize structural stress correctness drive empirically hyperparameter adaptively budget bayesian hyperparameter adaptively exception bayesian attempt optimize promise empirical result view complementary hyperparameter extend principled fashion mini setting interesting ht propose strategy learn hyperparameter optimization setup outline amazon ec memory base partition dataset different algorithm divide amongst different budget time interpretability budget warm start dataset aside collaborative normalize dimension train descent trial
network investigate vertex direct pair vertex loop twice application absence generally arbitrary graph adjacency edge pair equal direct vocabulary theory graph call graph relax remarkable characteristic remain second vertex connection vertex single set adjacent pair nod small highlight degree property property lot
daily return rate variance cause big effect match event detect date stock contain collect axis accord behaviour use evaluation number receiver curve evaluate performance different positive show outperform speech first hour news min acoustic extract segmentation two variety great challenge segmentation bic reduce false experiment show outperform bic slightly bad c
part correction consistent crucially subroutine potential neighboring difference use sampling budget subroutine correction open powerful cdf query monotone solve available class refer miss error interval sensor network sensor spam filter lose distribution monotonicity fall monotone stage detect testing give knowledge interval rejection limited amount available use possibly crucial situation bit expensive therein physical rely device undesirable want sake parallelization uniformity grant complexity convolution improvement von trick optimal closeness deal noisy incomplete science variant paradigm consist one likelihood make distribution resemble use similar g distribution datum yield theoretical science perspective local receive program code filter knowledge first correction et close pac style noisy sampler primitive whether problem total variation modal n author logarithmic size use e monotone particular compare distribution essential difference discussion write work concern totally order respective cumulative possibly increase variation processing inequality domain randomize independently q recall fundamental informally say cumulative taking let define take deal consider monotone histogram partition k k k show distribution state monotone approximate
performance currently one specific include therefore small pixel challenge refine refine layer observe tendency detection cnn equip score refinement gap still improvement precision curve early precision decrease precision decrease truncate low cnn positive bootstrapping work novel classification present towards exact box object top approach suffer initial object work firstly art object bottom approach approach scalable class extension object include thing low hard state boost thresholde positive mining bootstrapping promise candidate also
particular width scale axis line axis line style font outer black every style mark mark mark solid forget plot sep crcr color mark mark forget crcr line red variable let form orthonormal basis column let whose column satisfy factor determine svd turn rely trace c k choice orthonormal assume combine last conclude decay hold theorem theoretical conditioning adapt
precision combine get step estimate regression past decade much attention goal present diagonal precision perform comprehensive empirical evaluation residual relaxed likelihood estimate error estimator realistic estimate relatively residual precision partly tight precisely individual interesting association especially association different partial measure particularly generally partial entry precision way population practice important compare suitably important therefore relevant simple inverting covariance pseudo inversion poor precision impose computationally efficient procedure convenient setting meet concrete fashion random individual row object size row challenge attract attention past decade encounter comparable even commonly namely maximal
sentence ambiguity salient close heuristic allow stanford library type build distinguished string capture stanford type person respectively entity distinguished entity feature type entity partition coarse speech category jj nn pp vb everything six pp vb feature pos entity pos seq make scalable corpora variational approximate classical approximate dirichlet multinomial respectively usage requirement
interpretation cost convergence reason verification thing sbm dense graph variation k direction initialization work certain emphasis paper note seem since implementation organize discuss relation provide motivation derive cost establish empirical proof material material last decade generate huge literature community stay within leave assign partition perhaps introduce spectral see graph simple overlap partition sbm set edge belong otherwise well serve benchmark lack power law detail reconstruct graph sbm possibly component include
feature ordinal assign many application field finance gold reference optimal bipartite subproblem transform ratio kf k write way degree eq illustrate example many find certain order form erm learning suggest estimation study minimizer naturally fluctuation length problem probabilistic hoeffding adequate g generalization follow extend major expect pool point satisfy consider scoring fulfil soon supremum learn essential uniformly relaxed tail subsequent classical truncation however number compute generally prohibitive usual asymptotic refer much
distribution normalizing effect marginal standard accurate unlikely case scalar improved extent reduce approximation inclusion derivative expansion idea context high use computation order strategy design effect laplace improvement scalar preliminary approximate although proposal slightly finally differ integrate nested approximation approximation require laplace univariate approximation density provide estimate explored likelihood sampling general solid
employ confirm accelerate aspect mathematical asymmetric numerical confirm langevin dynamic study reveal reduction regard aspect explain mathematical result regard hope mathematical tool algebra acceleration steady development mechanic explain beneficial acceleration reduction frequency energy existence bottleneck toward minima help particle particle typical transition equilibrium case path force recently mechanic force elsewhere viewpoint consider viewpoint convergence steady aspect mechanic concrete equilibrium acknowledgment grateful
run heavy every occur interference v expect prove implie channel size heavy channel without let heavy theorem run probability channel least possibly channel frequency estimate item eq keep whereas item implicitly assume great frequency less complete transform protocol distribute histogram private protocol report bit expense overall public randomness user modification distribute protocol protocol iv public randomness report let server integer server report report public give protocol algorithm htb input parameter string ni iv ib ip server server collect report protocol note also cost computing efficiently preserve computational protocol protocol output valid hand side differential public give easy e feature string iy iv protocol set take protocol first user sample original essentially affected formalize metric two user construct sampling point estimate negative q randomness characterization respect negative bit efficient protocol private protocol histogram protocol probability parallel channel describe user moreover hash report execute basic string algorithm pair j compute item string otherwise hence pp protocol privacy seed I k iv encode oracle k protocol privacy protocol histogram histogram private theorem bit protocol histogram protocol proof protocol sampling user subset item item pick extra approach distribute necessarily frequency private protocol user single expense add original public randomness protocol introduction modification compression user server iv protocol output binary length server server report report randomness estimate bit htb bit protocol public string ni iv ib server server report estimate preserve protocol protocol step valid public two easy iv iv feature construction public exactly server view view actual report randomness thus original respect error affect generic transformation essentially transformation private histogram key differentially private protocol argue probability compute efficiently algorithm item
set score simple score belong design engine monitoring aim classify short series series length normal domain statistical instance reject population select population point apply population
mean level variability across condition type magnitude source phenotype across quantify variability variability across variability dominate highlight determine level protein across type measure start concept common display magnitude absolute type estimate level protein variability protein gene constant predict accurately specific protein level indeed level explain quantify protein level variability gene measure variability fit capture variability regression aggregation
validity note since score always demand martingale notion satisfy stationary write geometric underlie orthonormal reveal proposition general obtain corollary grant universal establish omit detail validity expansion serial context long operator key entirely present essentially serial often financial considerable instance stock often display martingale financial discrete equally absolute display even memory cf us iid satisfy structural martingale behave differently desire relevant estimator still mild estimator contrast employ dependence memory convenient kernel frequently analogue cf eigenvalue eigenfunction due j calculation reveal upper schwarz claim proof lemma preliminary provide sequel also proceed derive lemma orthogonality dominate yield cauchy schwarz lemma proceeding e supplement likewise lemma absolute eq observe hence triangle proof note backward inductive
graph source cascade except edge except sparse graph paragraph adjust margin report number cascade depth great benchmark maximum experiment fast easy approximate valid validate discrete model whereas graph cascade realistic rarely patient algorithm edge g recover return number experiment achieve high precision interestingly previous graph draw follow graph work recovery thresholded adaptive
remove training add sensitivity difference low analysis see percentage bound sign depict tight computational thick curve actual incremental table depict sufficiently tight bound label operation large variance incremental algorithm computational option op second op use op speak instance upper greater small zero incorrectly classified sign unknown op run op stop merely add classifier low obtain logistic
hellinger depend complete construction entry joint sum row sum therefore p practice section construction power randomization statistic demonstrate another commonly statistic powerful expression power randomization statistic difficult carlo alternative randomization test randomization small construct alternative marginal construct marginal use carlo procedure power allow hellinger share marginals hellinger result draw
large publicly student exercise tag three dataset outperform lstm neural lead auc notable improvement auc marginal auc previous auc triple date synthetic lstm predict knowledge variable model incorporate concept difficulty exercise transformation exercise hide concept mechanism select subset concept span notable mix par exercise deep look next student mdp accuracy synthetic suggest graph influence perfect
class dataset sentence sentiment regard sub sentence individual test whole root label stanford preprocesse baseline svm na cnns cnn rnns matrix lstm variant tree lstm variant recurrent lstm bi lstm avg paragraph detail tune change website convolution hide layer dimensional english slot slot pool back mini batch add penalty dropout layer drop embedding model task sentiment prediction short worse achieve art include
variable multiple location elimination polynomial root include uniquely determined ready argument whether every write row uniquely conclusion finitely proceed define finitely must terminate finitely desire polynomial hence dependent infinitely finitely missing obtain order
regressor test different value second course economic economic ol sense combine forecast testing know date break date option break variable break read perform break f statistic break unknown large statistic also consider processes u q l process u u combine autoregressive move average define u lag also remark relation univariate simultaneous equation traditional characterize
integral function increase consist series investigate dimension generate mat ern c correlation select quadratic evident area correspond infinity vanish prediction scatter plot prediction versus respective correlation time set associate bar area low fluctuation excellent initial series mat ern process ern correlation logarithm precision dark area online correspond red correspond value prediction versus correlation star validation mark dot online deviation cube function select cube set term validation optima give I mae rmse initial determine scatter algorithm space
computation define arm play note bernoulli time use ks bernstein inequality rt rt low usual would yield probability imply get jt reward round j jt step combine jt regret round arm round recall jt equivalently eq event rhs least emphasize play moreover time least fact hold would apply jt jt contradiction jt u jt jt jt expectation jt jt
even far pattern experiment dynamic search switch long behavioral great variety spatio global eqs pure without e confirm plain point motion start unit normalization use everywhere copy experiment sec eigenvalue spectrum linearize sensor turn sensor learning video heavy perturbation control internal produce motion sensitive perturbation fall matrix sec initialization dash perturbation lead behavior video normalization sec artificial provide insight system environment work physical reality simulation largely particular motion physical although large integrate sensor trick choose scale start choose central maximum break innovation introduce modify action law generator entirely external curse restrict coupling
rapidly sequence close design would result correct one I il iteration chain l l target distribution possible update form jj posterior conditional accurately stage mixture kernel construct estimate n aim particle fitness threshold time mutation take give n ensure particle smc detail mutation kernel smc genetic class mutation mutation kernel optimisation stochastic limit activity generation skew symmetry constraint preserve evolutionary operator new form th q mutation kernel type move element use operator particle element boundary q mutation move element update mutation use denote inverse wishart small happen uninformative effect region fit matching sample successfully solution solution generation factor constitute day trade european chi period consideration note chi secondary exchange list national exchange six amongst maintain trading market case complete primarily stock american select
address importance motivate operation keep track pick minimizer valid potential additive give degenerate appropriately derive estimator unbounded arbitrarily risk problem importance hyper trade small value induce large enumeration inverse variance consider prediction explore confidence agnostic importance include empirical need capacity hypothesis class class deterministic convergence refer small cardinality contain ball center covering condition capacity sample size n vector
convolutional input video separately single propose convolutional network demonstrate recognition use network language empirically confirm context generation generation open application generation short term lstm unit lstm maintain usual hide gate element multiplication gate context word word memory memory content update forget update encoder eq new distribution concatenation softmax allow interpret lstm decoder q generate sentence lstm instance symbol high probability al decoder description generation work e effectively temporal video temporal
category especially box outperform feature naturally infer box show local generalize quite unseen accuracy unseen cnn although employ augmentation dense feature outperform densely feature combine layer cnn fusion fusion execute map note trade fusion improve cnn train deep cnn base cnn part cnn fusion report state method pre additional deep framework outperform significant aid method additional category demonstrate category map degenerate aid box achieve ap ground propose accuracy result deep achieve deep fuse image
get gradient em mnist mnist b mnist sample c benchmark mnist test remain validation rescale pixel intensity preprocesse dataset network auto encoder hide relu logistic sigmoid unit layer encoder decoder auto decoder auto encoder sigmoid output mnist separately cross loss layer fine layer jointly layer auto pass encoder code train code code code
probit dt multivariate bf probit approximation divide restrict even independently baye hypothesis logit kp difference logistic jacobian probability modify exercise logit little exercise logistic asymptotic bank logit exercise exercise except parameter estimate logit full walk restrict restrict log bf logit bf logit probit except large factor contingency categorical variable build pick treat exclude contingency count dirichlet deduce probability eq imply multinomial table comes restrict uniform associate improper deduce median series distribution normalise moreover therefore closed value imply expectation section marginal describe tag year total capture far note irrelevant code posterior posterior conditional distribution q conditional q track repeat day day give expectation prior derivation mean equal deduce define proportional increase likelihood give extension capture observation proportional n increase likelihood episode prior converge prior informative conditional reproduce switch exercise modify code book conditional direct prefer metropolis conditional simulate nc nc nc prop I prop log prop p nc nc extension capture capture give extend capture probability episode capture another recover lose mark extension lose mark observe
reconstruction fouri slice unbounded activation universal transform filter constructive consistency also imply network activation denote correspond substitute regularity property construct considerably relu truncate unbounded polynomial relu radial rbf relu say vanish empirically verify analytic noting show unbounded activation polynomial functional seem later strong detail harmonic constructive network learn backpropagation
dataset annotate video result yield fine convnet initialize yield marginal fine yield convolutional layer weight feature layer contrast despite surprisingly average scaling extract ia large first learn yield softmax class match use table
table grid use indicate previously always use indeed average note mean standard format across fold accuracy high support parsimonious translate figure hardware c sample acc acc band planning relax circuit exploit parallelism
edge outside bp sbm introduce effect external current estimate h rt rt rs belief propagation put field need complexity edge bp marginal node identical incoming account obtain partition assign well know marginal optimal sparse asymptotically succeed
pp q straightforward choose k since give l however separate variance source l n yield learner sparse regret tight conjecture match low apply desire early one realize hierarchical use great robustness challenge provide theoretically notion well utilize guide hyperparameter prior batch learn regret certain good show convert bound may risk bound student formalize share investigate suggest theoretic hierarchical uncertainty place great misspecification
might fortunately per pseudo operation pair document topic thing first ensure topic must contain topic exclude intersect word outside inclusion document topic document chernoff union word well document intersection support identify handle document intersect yes intersect topic ensure set time contain list hold pair indicator denote probability dominate dominate topic dominate inclusion pairwise chebyshev document plug bound topic pair imply eliminate construct discriminative configuration generator fact configuration intersect topic appear intersect indicator variable fact topic inclusion variable intersect chebyshev putting size appear correspond generate topic option limit support intersection support lemma probability yes set configuration support else configuration inside existence word two set either support intersection topic remove non easily intersection existence topic ji dr every score topic instant j add give one document ji add topic topic add initialization property update topic word identify dominate anchor progress sense word word anchor lower dominate topic anchor properly identify topic word drop reach identify dominating topic large weight
fold feature without additionally crowd generative model recently progress variational learning direct graphical model major graphical component supervise use crowd weak supervision similarity generative specificity crowd constraint otherwise early usage context nature constraint connection framework present terminology sparse crowd improve process crowd figure model treat triplet difference unobserve approximated parametric tackle crowd provide weak supervision informative triplet implicit semantic triplet
autocorrelation develop additional constraint spectral design sequence spectral band autocorrelation pm n show complete thm low division access integrate widely share monotonic algorithm fourier thus adapt design flexible outperform sequence autocorrelation integrate digital communication sequence whose low measure sequence synchronization code division target synchronization purpose additionally generation amplitude analog digital usually modulus
highlight temporal sentiment pattern positive sentiment highly mainly characterize understand express short affect online strategy introduction focus system understand communication diffusion characterize medium understand ability enhance political influence online facebook twitter individual day sentiment quantify sentiment diffusion recent study affect language relate devoted extent sentiment medium affect feedback behavior twitter sentiment explore diffusion popularity class temporal highlight different
l mf nj estimation see compute good atom separable next negativity follow coefficient post step atom guarantee exact covariate block constrain bc omp orthogonal matching pursuit greedy method solve select provide belong update select atom availability prevent two atom detail covariate residual atom weight p j become square eq basis transfer function multiplying coefficient vector shown iterate spline termination meet terminate negativity weight initialize entry bc omp variable use spline analyze bc omp update respectively dominate omp operation spline complexity operation spline equivalently assume simplicity problem drive complexity fix iteration complexity transfer per additive independent omp bc omp building
x x x htbp convolution convnet approach convolution could operation convolutional neural efficiency implementation represent gpu research project describe convolution deep gpu reaches typical deep
unobserved score account feature approximation exponential availability anomaly detector inspire approach nature order explicitly concept key analyst anomaly computing intend specify jointly responsible anomalous great anomaly employ try detector anomalous explanation develop specialized explanation anomaly et directly search discriminative explanation contrast density estimation fraction detector approach explanation set large methodology contribution anomaly detection data account fraction application anomaly anomaly point usage anomaly manual anomaly point detector address identify anomaly outlier outlier anomaly analyst outlier decide anomaly say analyst anomaly able enough
tf compare long type publication record case computation build specific author subsection architecture name could run expert modification incorporate compute representative component research string compute feature name dnn probability determine name pair belong system dnn aggregate bag train retrieve fold distinct take dnn
second replication aim reliable attempt discover independence generating cause hand worth independence facilitate understanding understand help scenario inspire domain characterize transfer thank zhang grant nf research grant research definition zhang development structural modeling produce several usually distinguish cause impose substantial constraint functional point view causal direction determination condition cause involve
dd j modify cholesky root triangular conventional cholesky diagonal ensure e diagonal diagonal definite minimize take factor commonly close tuning approach implement square cholesky factorization column course iteration algorithm add dense cholesky derive carry usage definite tensor simplify manifold adjust langevin mala adaptive step metric curvature length typical mala tensor constitute step develop automatically model information admit factorization moderate methodology perform alternative carlo kernel analytically unnormalized integral researcher tackle challenge highly rely time process ergodic ensure integral
potential function base learn final contrast optimize value pass base apply crf recursively calculate potential propose cnns pass potential function directly learn accommodate message pass belief bp calculate encode label variable compute message message reason operation derive message pass unnormalized variable connected factor exclude factor message exclude message message substitute definition factor graph variable exclude pairwise connect
quadratic name description mnist patient test list subsection solver matlab toolbox symmetric implementation solution vector svm logistic centroid l run measured object template intensity vector base group seven water type seven class aggregate five mention degree class member bad case verify regularize centroid template expect curve r place centroid note template successful boost bad compare centroid template decrease centroid avoid various present change r centroid template regularize template visualize effect
treat remove remove save treat remain memory zero precisely line require operation compute inner line need total computed gram schmidt operation require smc server server make row row non reference row bs streaming memory limited problem observe streaming produce estimate vanish square ambient entry exploit technique address since remain bernstein concentration independent set adjoint surely inequality independently lemma constant compute uv therefore exist ix uv
leibler divergence competitive benchmark great significance divergence enyi nan quantify empirical small community well method modularity suggest strategy future community slight significance thing community thus within total number total within community implicitly communitie significance address explicitly actual issue community separately distribution derive fraction blockmodel er nan focus community type rather compare partition
lie representation label softmax softmax back major element derivation mathematically perform encode generative message node send fine level channel instead abstract e factor graph commonly hoc derive precise entirely determine role perspective marginalization nuisance intractable exponentially many affine transformation convert otherwise intractable marginalization abstraction levels eqs relu max variable contrary graphical model training maximization develop dataset old step parameter probability complete probability class variable template noise isotropic covariance statistic template would separate introduce assign cluster e wherein likely likely equal eq true nuisance em intractable require exponentially l form template result enable infer probable truly instead slow g deep realization extend previous training input weight activation input grain output essence abstraction convenient form mathematically em iteration g switch early independently batch scale bias batch activation batch deviation activation activation costly matrix bias normalization derivation normalization eq whose dependent google consist unit drop output corruption encourage data dropout brevity refer reader dropout dropout correspondence exact bias distributional misspecification relax allow seem ad approach distinction classifier former know distinction distributional assumption significant distributional risk practically generative discriminative achieve distinction type model transform discriminative gaussian modification procedure generative classify rule pick classifier
response hold stimulus examine representation outside incorporate software share publicly fmri public grant national science information nsf science agreement additionally nsf fellowship thank discussion share analysis figure analysis cca initialize hyperparameter cca mapping cca dataset variance explain fit subject response stimulus subject cca surface colored experiment accurately visual describe histogram maximum statistically bar average across voxel interest
fixed use train ideal set classifier alternate since one additional practice capacity tradeoff generalize focus training discriminate ratio term problem line trivial useful classification relate calibrate denote discriminative likelihood ratio one parametrize capacity focus usage systematic uncertainty search term nuisance inference search parameter measure particle mass easily include nuisance formalism always static class event parametrize physical
chemical protein property word cluster mean smoothly quantitative chemical property make artificial gram table contain protein versus space different normally map contraction protein physical chemical structure suggest train space gram h l protein van volume strength classification obtain family exist primary alone template show sensitivity specificity accuracy family structural protein ht specificity sensitivity surface ph associated protein beta
reservoir enough proof independent sum random hold lemma index eq q eq arm definition know constant depend u f arm upper confidence arm note arm instead happen eq large implie imply otherwise bound enough eq arbitrarily index constant enough together arm lemma previous three eq case probability large assumption associated variance arm reservoir q arms reservoir bernstein constant arm learner arm arm set easy oracle low
conjugacy smc ed cox technique log sequentially could template rest follow formulate scientific problem quantity notation throughout model inferential ed technique discuss calculation graphical problem energy mean intensity template log collect tn total count bandwidth unknown exist template frequentist determine template truth quantify uncertainty away observational setting ignore true template region address naive collect available fit might availability hour multiple due inefficient adequate address choose template template mixture template summarize
turn finish demonstrate need singular matrix density label calculation k v arbitrarily density e thus obvious implication taylor expansion limit follow lemma probably gaussian direct prove let definite matrix induce n denote measure generate sampling prove scalar infinity vector measurable event n simple calculation stand show turn second pick suitably necessarily obtain part supremum terminate matrix zero program function value kkt equation solve equation condition exponential spectral easy verify proposition exponential continuity dominate convergence algebra verification exponential covariance compact interior n confirm continuity demonstrate algebraic difference appropriately positive scalar dominate notice continuity covariance interval degenerate center need investigate toeplitz polynomially decay entry play crucial analysis simplify symmetric periodic nf ns shorthand n n toeplitz covariance generator indicate sized depend polynomially decay entry correspond see simple counting argument inequality rearrange eq admits terminate substitute skip algebraic lack cn n follow analogy cf
playing allow demonstrate focus action separately level task limit previously believe turn impossible human instance replicate one generalizing never success deep domain vision language deep previously input classify deep modal structure build extend neural handle modality completely datum type cloud trajectory label crowd expert crowd build platform crowd expert platform component standard incorporate various unlike point around differently shape object object share part problem handle modality crowd crowd platform public web cloud part three euclidean color g set vary object part obtain together
multiplier simple say column correspond eigenvector maintain algorithm instance schmidt characterize apply start time constant unnecessary acknowledgment grateful science foundation grant equation v n follow expression relate quantity orthogonal eq explicitly measurable lemma determine expand lie recall insensitive skip normalization update rule final intermediate
order previously bandit suggest past future dimension mathematically somewhat surprisingly derivation show reason derive finite normality otherwise amount htp typical infinite highly infinite hmms ergodic ref leave possibility divergence divergence ergodic level ergodic process highly organize divergence separate architecture sec ref distinguished low process e g commonly process level process general recurrent statistical nonetheless generative process infinite
compute remarkably original proof section functional delta every hadamard conditionally w thus accord fp equation theorems bb randomly replacement size assign point datum accord bootstrap draw subsample break ls estimator bag far bag bootstrap show still draw bounded mm estimator high finite estimate broken conclude draw bootstrap perform statistical inference big process store compatible
follow otherwise infimum polynomial fall th concavity jensen inequality use expert q infimum follow summing fall jensen net entail bind omit optimize rigorously take substitute level j na thus separately thing together whole storage round fall position bound france paris universit france problem nonparametric arbitrary sequence constructive fashion regret term metric sup optimal order magnitude optimal old adapt sequence deterministic choose forecaster instant reveal forecaster observation incur standard possible algorithm
deduce lemma l ep j triangle inequality term first centralize lasso n sn convergence straightforward consequence show dominant simplified nk nk nk centralize gain theoretical simulation estimation average linearly estimation centralize centralized simulation study thresholding centralize average compare centralized average versus study machine average vary machine estimation
combinatorial class know solve query problem threshold interior totally differentially private release differentially solve interior point private solve interior formally formally threshold database database database converse e reduce threshold universe handle every universe query equivalence threshold combination much small universe idea reduction partition block roughly solve interior block answer threshold base answer describe reduction factor interior sample complexity actually removal row solve whenever database less subsample answering threshold database database sort set nd r rd r arbitrary interpolation database loss noise noise partition accord partition index may partition differ partition removal density execution answer threshold every succeed interior point ensure interpolation eq item probability hence execution succeed union bind complete release view query release fix equivalent differential collection differentially private differentially first direction database enough low require item restrict applicability proof differentially programming qx succeed long answer distribution feasible post processing argue follow close consist union totally domain correspond kolmogorov differentially totally accuracy accurate learner direction equivalence differentially r concept differentially equivalence learner answer learner run
approximate many hide complex generative fitting mass major layer argue deep generative potential thus generalize machine concept introduce powerful intractable generative approximate model perform idea enhance many autoencoder backpropagation inference reweighte approach rely obtain generative machine spirit variable deep generative approach model
infer eigenvectors bethe rmse initializations large systematically remarkably infer essentially achieve oracle contrast row rmse bottom rmse estimate size limit bfgs maximum compare oracle infer svd tr svd regime rank tr ir look ratio minimize completion ability reliably fewer give
error propagate affect address dynamical dynamical sequence method incorporate type flexibility design estimator implement new become rearrange view instrumental regression technique coefficient estimate linear instrumental note g two ordinary formulate dynamical supervise analysis behave instrumental variable stage instrumental variable generalize ordinary linear counterpart quickly converge describe instrumental learn system guarantee instrumental linearity enhance method model sec show replace performance correctness explain connect perform dynamical belief observation inference task observation range
require operation use free optimization maximize number hierarchical typically approximate likelihood review particularly procedure scale counterpart hierarchical likelihood maximize likelihood approach treat inconsistent random comparable observation objective optimization descent take identity value tuning approximately employ validation disadvantage inconsistent effect sample employ ten estimate estimate weight similar approach amount hierarchical develop hierarchical method specific effect use moment computationally unfortunately require restriction seem become prohibitive recommender us method spirit arbitrary effect roughly initial specific combine sense across estimation effect estimate combine effect matching remove restriction fix variety gain intuition matrix size share last response zero square denote restriction notably coefficient despite effect row unconditional expectation covariance effect relation estimate estimator response predictor specific mean specific conditional mr positive definite section combine estimate invertible moment base
pass evaluating may elementary conventional trajectory memory store million ram gb thousand mini batch time epoch store history imagine could trace training start work back trajectory reverse storing descent momentum store precision arithmetic gradient physical force exact reversible computation affect loss tt cm exactly reverse td dd td td hessian reverse forward decay velocity point sufficient deep fix problem information
computer imputation compound approach highly result nine approach rescale da image compound use analysis via cnns explain deep network deep greatly speech vision cnns extract sharing become art unsupervise diverse complex unsupervised unsupervise belief auto integrate restrict machine rbms inspire train imputation task auto encoder extend apply inspire learn consider visually recognize classify science physic
assume allow increase payment player deterministic uniquely maximize equal player payment expression bound noise equal inequality r ridge differ datum strategy player player use simplicity expectation differ player lemma difference expectation I r except sufficiently addition increase decrease privacy player nothing privacy symmetric approximate nash output require concentration regression parameter compute long expand r remainder long q use hold let report strategy within add bounding term recall add database differ definition player database player expectation also bound term probability pair database take third plug final union two q budget characterize total analyst run budget private mechanism player privacy accept player
instead see wide probability uniform ergodicity noisy acceptance probability desired iterate noisy approximately simulate langevin dynamics series sum require exact exchange approach previous exchange auxiliary internal exactly regularity condition tend use mcmc abc sl sections exact comment computationally approximate exchange refer method intractable summary denote jacobian determinant arise concentrate case sufficient estimating become close use abc calculation monte exact carlo insufficient statistic might resort abc success simulation summary might sum imply appropriate sl proceed make sl mcmc summary distribution approximation unlike choose additional sl unbiased exact mcmc expect additional introduce effect internal section sl simulation indirect method
video assessment video frame compression qp degradation relate display encode frame sequence compress frame visually indistinguishable compare hardware resource consumption exact test matlab realize field gate validate hardware
scalar equality consider possible identity natural option whereas option adjustment w iy iw affect direction composite approximation expect phenomenon particularly important level relate independence block move likelihood substitution keep geometry assume triangular correlation component asymptotic infinity covariance matrix map cholesky
iterate however project require prescribe radius question tune appropriately directly unconstraine parameter section show last iterate converge almost surely secondly discuss online amount univariate general call back stochastic simple stochastic gradient operator dimensional rkh share similar univariate strongly hypothesis space close online follow descent heavily randomized estimator randomize unbiased true
cover dominate collect fix know know concentrate number fact fix event ball cover probability average dominate fix covering easily concentrate dominate union ball se se constant skeleton outlier note outlier come outlier skeleton point ball base fix assign conclude ne c se ne w take core kn rhs already notice suffice find already se se least point cluster hold get cluster core error regard make inequality union come
functional q eq turn input weight thought perceptron terminology network exploit convexity also learn implement algorithm state virtue duality easily bound classical consist operator therefore remain corollary hypothesis conv whole adopt two concept attain rank quantum either clearly quantum effect achieve similarly consider quantum achieve quantum measurement consist projection mathematical conv scenario problem sense quantum access code code code mutually unbiased attain upper equal hilbert exist quantum hull effect tr quantum state contradict operator four effect quantum quantum demonstrate rich projection sphere sphere provide geometric picture concrete extreme play region convenient metric sphere correspond schmidt e conv schmidt efficient sphere representative states unit ball class scale hilbert since size functional measurement g perceptron sphere operator operator representation measurement simply learn worth quantum quantum state q good ccc quantum
consequently converge quadratic rigorously full step still quadratic proximal r r algorithm iteration tolerance exceed see therefore though case metric employ hessian newton bfgs update backtrack automatically apply bfgs proximal newton omit framework advance comparison accelerate quasi newton sophisticated backtrack profile profile build
order pair neuron correlate signal neuron compute signal range robust noise maximal minimal value different obtain follow q performance simulator include typical real technology limited neuron
kt stationarity meanwhile solve stein stein sort uniform stein c program I optimize boundary point indeed introduce slack constrain quadratic neighboring problem amenable stein identical show program bound optimum every feasible optimum c x combine lipschitz calculus give evaluate hand side bind yield finally satisfy generality integrate side inequality yield function feasible lipschitz gx gx I tm extension lipschitz magnitude bound satisfy ensure satisfy I bm know gx I bm root continuous root exactly combination combination suffice conclude x rw x portion portion hence root hence unique gx bm I imply construction gx extension lipschitz moreover reasoning establish notation
easily show loss attain py classifier imply small act diagnosis learn classifier recommend minimizer appealing predict conditional detector role hierarchical classification material main default without reference see classifier depend consistent estimator probability class vs consistent surrogate conditional much problem piecewise surrogate minimize expect successful
word train align summarize mean layer length final convolutional unit rich slide deep layer convolution among layer ff relu denote convolution slide window sentence max pooling two convolution fold representation quickly filter undesirable composition see sentence fairly readily eliminate caused add convolutional gate zero filter gate pooling sigmoid keep layer actually create hierarchy net contribute forward
introduction purpose namely excellent rapid training speed relative state efficacy task ever mnist performance literature previous imagenet give digit nature imagenet filter result increase error increase point less note cifar gap test enhance dropout convolutional datum cifar nonlinearity response reflect high convolutional dataset improve cifar iterative g front batch aware newly publish
cluster multi network optimize differently hand allow calculate correlation ideal really biological activity almost task fail unbalanced community atom group center define c drug chemical compound determine effectiveness investigate implicitly encode train multi task deep chemical compound activation absence indeed correlation demonstrate unit neural visual inspection layer tend learn often focus group group see cluster involve match attract crowd
mostly step example problem uniform hard increase dependency long problem signal ignore noisy signal example rnn hide notice rnn start h gradient take long method computation result lstm expensive expensive advantageous
module solely topology reveal pathway co expression fundamental phenotype rapid accumulation datum lack bottleneck process especially human subject moreover researcher phenotype report trait survival inference poorly trait furthermore qualitative categorical quantitative reason boundary category often arbitrary distinguish category lose develop quantitative phenotype miss genomic intensity specific phenotype trait quantitative individual tumor quantitative response patient drug genomic incomplete trait record focus vast accumulate microarray human microarray systematically phenotype diverse disease disease method training profile phenotype strongly phenotype microarray phenotype phenotype aim estimate pp profile value eps gene correspond gene color green red gene coefficient degree gene coefficient signature new relative intensity phenotype derive intensity profile phenotype depict grey color estimate association profile profile gene phenotype anti association phenotype directly train thus datum platform microarray microarray stage human covering microarray sample phenotype profile consistent phenotype description show phenotype dataset discovery factor comprehensive generate value publish illustrate microarray description phenotype gene association profile phenotype phenotype description compare phenotype simply phenotype description gene term phenotype predict phenotype new determined find argument close weighted expression value assess profile trend signature calculate score pearson assess statistical compare permutation illustrative two microarray stationary growth microarray phase phenotype transition serve prediction correlation temporal predict phenotype profile recover order highly visible profile accurately logarithmic phase width order eps stop hour phase hand measurement remarkable phenotype signature accurately sort demonstrate occur growth phase microarray process disjoint sample group baseline least group phenotype value categorical statistic set threshold association associate validate need dataset description exactly phenotype identify predict phenotype profile phenotype sum predict phenotype
active although multiple time thus peak active detection keyword flat incorporate keyword unknown well active know passive active keyword know priori initialization keyword prior knowledge remain unknown phrase answer outside home make ten keyword
ccccc g sn sp acc lr life death predictive resource patient contact medical arguably important specificity basic regression meet care service line responsible effort local scale operate svm almost improvement provide merge clinical scale miss skewness challenge recognition technique support sensitive deal classifier svm method tackle combine feature
imply decrease centre specific critical confirm analytical size exponential practice good analytical effect seem significant form measure probability configuration remarkably confirm role bias choose choice uncorrelated bias performance lift criterion uncorrelate report unfortunately analytical calculation substantial relevance depend smoothness whether uncorrelated exist extend study ensemble direction would concern direction extend present analysis replica soon grant aid program matter partly institute volume saddle saddle write integration volume get saddle meaningful otherwise come lead scale volume dominate replica generate trace write around saddle parameter point component easily column transpose matrix candidate replica equal tend span hereafter order well identity upper q equation vanish replica solution give eq impose orthogonality choice replica eigenvalue third obtain
organize basic try reconstruct weighted histogram coefficient code histogram sample organize code reconstruction also code objective code histogram traditional apply norm introduction histogram histogram bin bin ground th demand define fill variable denote constrain prevent demand encourage
conceptually backward pass perform backpropagation derivative propagation input description inverse dy dy dy know us function argument compare true double next case function pass derivative net use weight act activation net propagate activation multiplication weight derivative allow implement ibp layer standard bp functions function jacobian vector immediately thus correspond ibp operation almost pass derivative pass derivative approximately ibp transform multiplication
assume function metric insensitive canonical issue arise instability pixel sift representation computer vision robust natural formulate insensitive leading vanish basic intuition mahalanobis hessian mahalanobis transformation easily minimizer objective psd see basis notice split orthogonal since
far quantify seed sparse clustering normalize vary cut bi define point dataset experiment cost cut column ssc nn omp exhibit complexity make impractical ssc base approach method cut rand observe ssc omp good random scheme approximation cut ratio visualization embed union overlap intersection embed via embed aid visualization cluster display blue fig display cut ratio six decay seed normalize ssc omp ssc omp seed sample performance leverage appear ssc dataset produce cut ratio ssc omp fact grow dimension generate ssc weakly cut representation produce seed expressive basis contain incoherent thus seed graph produce small cut seed compute column sparsity column version spirit denoise near denoise
rating agreement number rate rating user item weight agreement total number average profile knowledge attack calculate subset profile rest profile max f p target mean set item profile profile rate profile rating feature total item rate recommender denote total item user rate otherwise rate recommender boundary point size popular rate entire rate size ratio rate item recommender ratio rate user item rated propose specific score minimum item attack select item attack item attack high attack item rate maximum score reverse attack table item attack random attack rate score
experimental real weight backpropagation fully connect recent neural area signal dnn speech event often massive resource thousand hide advantage dedicate hardware enable per attractive effective train bp gradient straightforward expectation backpropagation training experiment text promise extension
run simulated annealing approximately log thank stage region nx r optimization convexity recursion formula radius far critical pr critical multiplicative overhead optimize sketch domain possess covariate analyze compute w rw would central minimize happen current current pass randomly index result central computation outside world check repeatedly reduce yet latter mean repeat allow learn interestingly noise present procedure change discuss formulation take stage problem
cycle marker xlabel ylabel performance google l google correspond training default whether produce compute near word cat dimensionality seen figure capable capture semantic cat syntactic project onto default com theorem property proposition conjecture claim embedding via rank amazon university california recent efficient via unclear naturally view insight framework efficiently measure word analogy benchmark art produce meaningful accuracy
formulation classification function rule furthermore however cs sense lead imply bit motivate loss side could establish follow besides loss system different measurement measurement loss half margin give mean class helpful side robust cs introduce algorithm subgradient side norm summarize htbp l subgradient c parallel user need give number good noiseless investigate different drawing average run fig average side marked generally conclude improve performance coincide htbp observe significantly
average gradually increase value increase impact infer component identical infer great evidence correctly infer average b infer component sign slope beyond curve number relative estimating coincide imagine coincide datum accurately depend dimensionality appear comprise value direction available high ability search infer appropriate situation text investigate compute representation metric central analysis argument transform text length motivate model mixture propose search mml devise ideally far improvement merge mixture employ dataset intermediate mixture mml equation mutual mi assignment message length use mi one compute actual frequently word generate tf word bm unit directional infer greedy document category document component show category one category split specialized category distribute category overlap category infer fine segregation mixture algorithm confusion assignment component htb c comprise document belong cluster apart song measure mi mml mi message message song score mml component mi obtain mml song mml avg message mutual information dataset natural report component good cluster may combine appropriately however method unsupervised news result evaluation mml distinguish news mml mml htb c mml length avg f information apply mixture number strong component mml message length mml mixture mixture mi metric mixture mixture tradeoff explain observe normalize mixture model kind nature alternate strategy modelling set model split true merge would close true prefer operation split ignore split may merge would length mixture perturb use merge operation convergence directional arise orientation protein protein adopt largely
atom suitably whereas learn tensor information find one tensor np fortunately mp make cost accordance pursuit economic mp relaxed mp pursuit nonconvex weight along solve mention provable analyzing sect besides advantage low storage sect convergence analyze tensor specifically least converge nonconvex include establish contribution tensor convex present provable ratio analyze convergence function tensor tensor formulate sect specify completion select tensor detailed numerical sect sect draw conclusion tensor inner product x x r n dd result th order unfold mode merge unfold remain mode merge unfold specifie th unfold
mean mcmc evaluation produce first imply comparison mcmc analyse likelihood evaluation necessary number need compete hardware toy flat tail method mini wrong almost happen despite converge rapidly mini batch reveal chain quickly raise question way way exploit expectation demonstrate geometric truncation expectation assume hmc burn partial taking mcmc run iteration evaluation iteration burn remarkably maximum cost median replication mcmc estimate sum mcmc iteration complete burn log bar trial per usage posterior experiment situation involve inference apply methodology logit positively true significantly magnitude mcmc contrast geometric
strength team match weak team lie match rank graph node noise detail extensive world impose significant player associate simplicity think value underlie truth player truth offset pair player comparison noisy version truth entry measurement pairwise rank intensity setup commonly encounter theory rank perhaps noisy pairwise ordinal player consistent rank summarize measurement vary comparison robustness player independently experiment complete node rank test robustness detailed remark add result measurement skew mean offset offset distribute whenever position offset available enyi experiment enyi outlier available practical extent correct compare sdp summarize sec rank centrality name rank sec glm serial glm centrality sup rank superiority score synchronization sdp solution popular distance one recover ranking level miss equal compare figure ordinal similar ordinal plot recover rank favorable noise sdp enforce phenomenon explain al investigate amount multiplicative bottom enyi average outlier enyi year english home game pair home away pre several way game outcome build comparison matrix raw report scenario pair aggregate total play winner aggregate take game play winner user interpret win consideration game rank finding minimize denote player compute prefer possibly incomplete matrix rely count order contribute whenever order contradict order rank induce eq cm team glm sup city united west west nr final obtain measurement final denote show method plot similar across type l sdp sdp score alternate place procedure b across different beta period head head remove degree discard maximum deviation histogram degree obtain score across input l low ranking score achieve good college match regular pair team play early significantly therefore play explain recent year game year
similarity seem suppose influence cite proportional semantic content cite semantic cite likely cite influence semantic similarity access cite cite age benefit efficiently even reference include full cite cite surrogate five title cite pt label sim sim sim sim similarity title introduction conclusion feature able abstract summary section feature specifically piece first type token vector appearing keep stop remove improve reader semantic body near mention cite citation cite citation surrogate paper title citation pt label sim sim sim sim similarity citation title abstract conclusion average context title similarity feature cosine similarity different window range citation sentence sentence citation give indexing purpose citation influence way full text cite citation inspire likely influential window word citation paper score relation citation indicate explicitly mention citation et citation indicate citation mention together citation three feature may bias citation format various supervise whether useful particularly experiment feature mean pt start pt pt pt start label manually relatively whether citation cite especially citation cite citation cite extreme citation cite name feature kind list table give full
rd e choose appropriately claim follow existence succeed desire prof suppose notice ex ex ex f ex tv r concentration e tv v union leave define construction entry prove upper enough random use sphere ij get subset cardinality technique parameter conditional j ip j j j u hessian optimization key technical lemma ce c interested regime problem attention convexity alternative set outcome comparison lemma divide prove hold ex ex definition minimization ex prove ex independent matrix entry wise ex last get desire tail dd quadratic bound remark k ex ex happen ex upper bind
attribute describe hard visual force water attribute address aforementione attribute attribute problem relation hypergraph multiple since common hypergraph vertex share hypergraph cut hypergraph cut attribute minimize attribute cut hypergraph cut hypergraph embedding try align encode attribute predictor mapping space space hypergraph attribute hypergraph cut illustrate information information consider encode cut class formulate class ability predictor encode produce version hypergraph incorporate nonlinearity summarize contribution far attribute supervise hypergraph approach predictor classifier cut attribute
fashion unseen node proof enjoy adversarial advance present flexible arrival forecaster day rest provably make prediction idea relaxation generic deriving moment forecaster ahead forecaster predict furthermore regret relaxation condition term learn make draw regret expectation turn sequentially lift assume introduction method style forecaster draw otherwise term expect class forecaster integrate randomized enjoy performance refer detail come relaxation previous prediction generate constraint specific forecaster solve problem per randomized let set assign relaxation prediction rademacher stand coordinate relaxation drawing vector provide randomized round generate
complexity sag incorporate batch acc prox acc prox incorporate nesterov acceleration whereas acc prox incorporate acc prox applicable strongly overall complexity constant logarithmic sag moreover acc prox quickly
traditional training process extremely fast concept applicable exhibit response line beyond digital demonstrate variety water optical device optical device circuit digital physical offer speed massive parallelism great power learn scalability optical device find task optical header optical recovery fast loop paradigm suffer drawback inherently nature inefficient expansion reservoir approximate output however rely increasingly difficult become massive descent important shape nonlinear automatically neural analog dynamical extensively paradigm delay feedback reservoir input encode dimensional incorporate perform computer effort encode high
intersection figure mostly uci repository web site book site kernel svms etc http possible conventional effort report merely sake please contact possible source agree transform favor similarity try site already scale train min intersection letter letter rand protein segment k spam svms min deep net simply max kernel close
tensor th order unfold kronecker simple impose norm us control mode wise nuclear achieve ideal tensor wide range become latent variable consider generalization mathematical property tensor corrupt time algorithm convert unfold recent show nontrivial decomposition order standard show signal noise even order tensor also np analyze tensor completion intractable maximum would wide ideal achieve
g stable heavy unless always rarely manually heavy tail project utilize behave dimensional sign sign although nonlinear expand pay learn focus vision datum histogram work essentially bit hashing relate valid stable tune miss mentioned
ic ic ic ic example analyze data split elastic net penalty candidate five cross mis classification comparison logistic tune numerical ghz processor percentage split observe elastic elastic net elastic get message standard competitive also notice sparse fast almost explanation
characteristic dct counter series dct possess mse dct nevertheless exhibit compression ii dct transform standard order prescribe select reference dct employ dct overall rd quantization point implementation software encode perform color fig depict rd frame dct chen f rd curves reveal dct absolute db frame show stream dct qp frame db confirm approximate dct transformation introduce architecture implementation nm application synthesis section explore hardware discuss algorithm dct offer digital realization measure metric hardware resource implementation digital computer architecture real propose architecture employ
powerful moment generate arbitrary stopping time convert uniform stop u side stop generate hoeffding analogously tool prove theorem kl reader technique generalization pac first stop
arise observe x random column interest additive use signal column statistical property estimate additive convergence study unify programming estimator slightly choose penalization regularize basis selector selector adapt subgaussian independent cf statistical compose subgaussian row auto correspond pursuit directly knowledge compose need e functional priori
differ identity need far require specify advance validate propose state art model illustrate evolution brain cancer representation reveal underlie phenomenon empirical make practitioner numerous past include variety method increasingly often access since measure pairwise ex exist linkage linkage clustering potentially underlie however research frequently measure point order security record behavior vision stream sequence scenario dynamic address evolutionary dynamic multiple however evolve exist evolve still bridge gap novel evolve model directly tailor direct access vector able detect popularity get rich phenomenon rich rich seem plausible many stay variability size arrive capacity automatically result thereby share neighboring related vary markovian carlo applicable
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indicator categorical continuous black white proceed mixture nb homogeneous within ga contribute sorting ga use numerous poisson nb ga apply ga summarize see frequently fit aic interest
chinese split chinese convert tool head finding follow gold tag top chart optimize train finally mixture embedding wikipedia english chinese parameter evaluate different vary achieve base increase still baseline overfitte negative limited base learn achieve search advantage xlabel ylabel legend grid style txt index xlabel ylabel legend legend pos south index txt index txt oracle accuracies pos improvement
brownian motion drift probability process b immediately simplify preferable follow convenient martingale drift process construction maximum estimator transformation account integral elementary
spirit key use architecture encoder decoder efficient b handle static pair convolutional inverse graphic dc decoder learn conditional distribution approximation contain factored important graphic engine help apart generalization capability respect dc encoder parametrization gradient obtain trick statistical train face connect express interpretable main consist interpretable variable subset variable target use graphic pose light trivially
encourage neighbor separate margin minimize term introduce visualization differentiable operate denote write
equality ai uniformly assumption last last concluding td suffice equality addition element schwarz precede td precede therefore last establish q appendix precede establishing establish equality last equality thus establish conclude n td td cauchy jensen inequality invoke conclude precede td td equality prove trivially last distribute appendix lemma show martingale difference array denominator bound ii numerator ii already uniformly I theorem iii theorem since denominator bound away bound establish equality last equality equality due therefore establish asymptotically prove denominator suffice eq row row note kkt condition equality due recall positive constant make precede arbitrarily small sufficiently reasoning lead validity probabilitie event eq due uniformly last sufficiently continuity n second inequality inequality make normality continuity conclude fact uniformity take supremum let infinity next turn equality due equality let positive difference precede take satisfying
appendix analogous often rigorously ambiguity consistent well singular perspective important note similar mathematical interpretation propose piecewise complexity complexity implicitly piecewise show reasoning accept support difference series computation calculation use expression brevity application elsewhere diverse key information frequentist first principle pearson alternative nan hypothese subset direct analogy extra always proceed limit test statistic reject approach assume write distribution give cumulative large acceptance usually ad hoc cutoff compute inverting
policy relate maximum reward shift achieve success large bayes evaluation parameter solve reinforcement less address agent reward trajectory policy fix k kn global kk discount accord trajectory empirical approximate hence offer learn decentralized algorithm infer policy explicitly expert knowledge accomplish measure proportional marginal simulation straight mcmc costly storage minimize kullback approximate able vb method reward since kl equivalent maximize bind optimization decentralize field optimize decentralize
case filter computable filter pf exploit tx follows plug conditionally analogously procedure approximation keep complete history consequently employ rao use numerical guarantee approximately step exhaustive mean assumption idea subproblem solve guarantee convergence carry optimal modification argument fw step mmd search sample fw exhaustive search interpret subproblem frank approximate though interior polytope appendix inequality triangular array guarantee even motivated problem bottleneck whereas continuous appendix comparison clear mixture high fw column higher significantly high dash line linear axis gaussian randomly normalize uniform additional mixture perform pair difference increase fw fw ls fw clarity decrease fw generate use use
stability theorem present discuss refer prove another connect recursive brief outline one result summarize discuss assumption describe definition al easy n follow compact differential di guarantee reader refer detail say km lx dx mt chain exist sequence let dx dx neighborhood invariant tr dy k interpret accumulation sup martingale n x
threshold code section overcomplete find nonzero nonzero synthesis possible np hard approximate place solve minimize decomposition efficiently dictionary form superposition representation representation type coding dft dictionary dictionary learn invertible dictionary even dictionary signal white transform apply operation zero threshold operation shrink toward corrupted
efficient computational lda singular decomposition operation hand operation large datum sensible quickly detail localization attribute span embed subspace two combine metric distance boundary describe classification individual class group generates assign discrete kx r dx k ki misclassification
fuzzy concept fuzzy applying set assign fuzzy membership fuzzy accordance fuzzy membership get membership hyperplane distinguish form negative hyperplane membership coefficient eq eq hyperplane respectively distance hyperplane construct hyperplane
high organize review section illustrate application open problem compare suboptimal control discussion comprehensive review stream geometric structure belief stream solution divide policy asymptotically hypothesis testing examine acceptance structure integrate hypothesis belief optimal implementation method heuristic base parallel normal two heuristic extend hypothesis region representative appeal asymptotically obtain substantial foundation decentralize control dynamic involve almost policy none claim optimality mention extension simultaneously rule
computer grant l alarm child link x x size alarm definition bayesian call hybrid perform hill subroutine combine idea incremental method parent child conduct experimental hill art benchmark pc term code pc test bn probabilistic form structure acyclic dag distribution graph bn independence infer encode attract great dependence global one call terminology basically cb method systematically conditional independence orient representative bn search evaluate graphical structure hybrid attempt skeleton cb approach
alternate fix two updating stack alternate step decomposition multiplication f full column require reasonable assumption redundant ensure condition block break present update solution give update reformulate effort go compute fast transform simple multiplication form concrete form update processor take focus estimating inverting
tune algorithm probit right variance mcmc mostly simulate gibbs metropolis equivalently step hmc supplement plot support statement hmc well walk already hmc random dataset seem phenomenon hmc type algorithms mention outperform pass per explain bad practice require much sampler probit able well dataset gaussian probit section supplement scheme probit similar except outperform attention dataset strong dna large uci repository covariate include intercept perform well laplace dna covariate accuracy laplace importance effective section set algorithm hasting metropolis provide ep fix expectation figure report cpu panel posterior estimate run outperform consistently across second offer follow insight despite significantly probit show strongly surprisingly despite calibration error ep amenable architecture outperform implication finding selection binary resp one exclude simplicity cauchy prior discuss distribution discrete small enumeration value importance sampling next section I close smc
calculate distance average birth instead lie space persistence diagram bottleneck distance give landscape landscape calculate average make procedure hope practitioner persistence average birth death pair user persistence also pairwise respective persistence provide use persistence landscape degree degree output main algorithm calculate complexity persistence landscape envelope numerical demonstrate implementation describe implementation output main pair birth death persistent homology pair persistence calculate birth death achieve remove infinite implementation ask define maximal truncate persistent homology persistent homology filter growth sensible interval element evenly space birth death often rescale assume persistence combination persistence k piecewise input death represent sort
economic activity locate north west north business traffic excess correlation working call traffic belong cluster locate neighborhood east city fine city neighborhood locate north call office hour week correlated presence people area note call start around pm pm business economic lag pm spread country area except excess whole h trajectory use week day hour study simultaneously week connected period filter mobile mobile characterize frequent user cluster week day cluster post introduce simplify enable number week day hour week divided day hour occur around pm interval pm
still feed forward pass test pass although practice standard deep model cost autoencoder consist simulate model require sum configuration recommend efficiency autoencoder minor cost rbms polynomial make require autoencoder require efficiently lc rbm cd architecture without randomization work go explore agnostic interesting interpretation mask structure autoencoder mask test different uci mnist
hide sigmoid universal state output unit unit well hide universal stochastic feedforward capability feedforward network study condition function domain study minimal universal approximation limit hidden feedforward network commonly refer deterministic address less attention approximation function markov minimal feedforward output
incremental likelihood applicable sampler posterior arrival piece artificial individual adaptive produce approximate posterior draw k proposal n kk k move leave invariant set w size ess fall threshold ess degeneracy define adaptive resample apply particle prove sampler reason resample step call invariant proposal posterior converge step step ess threshold constitute extend various algorithmic article property advantage posterior multimodal effect criterion respect smc yield resample step step justify particle eq yield estimator inclusion sampler study obtain empirically advantage estimate model evidence original mh could incremental likelihood reasoning simplicity smc algorithm article application produce smc particle smc avoid particle particle number particle algorithm initialization draw k x dx kk n c particle
random prove yes randomness far unfold take clearly opt least return hope randomness take hand consistent value tensor enable schwarz expand quantity equal clearly regime defer hard deal rhs naive polynomial replace key insight much well I j tm direct intuition write b tight cause treat psd weight bound happen idea variable ok ta different scalar
segmentation noise confirm use centroid unknown deal provide refinement segmentation detail propose call represent figure feed forward stack layer artificial neuron new neuron act detector recursively deep neuron detect high detector decompose edge let input architecture simply weight bias feature pathway later specific apart centroid convolutional pool high layer representation merge representation capture complex representation learn connected layer process position neuron convolutional input neuron contiguous intensity
share force reach optimum low long path suppose start pick well know place force visit similarly constraint check attain optimality move move neighbor exclude parent path key jj consequently length hypercube neighbor vertice code monotonically maximal cost traversal appear capable policy currently update mini force step policy make action first large cost sensitive minimize cost sensitive roll accumulate imply force local optimality acknowledgement work carry microsoft predictor reduction sensitive multiclass sensitive one word class train regressor predict natural approach sensitive predictor simply zero elsewhere common predictor separate one
connection graph loop rnns component intra parallelism intra parallelism inter stream parallelism stream acceleration exploit gpu rnns processing suffer parallel parallelization rnn challenge recurrent dependency different generalize rnn structure cover long short term memory parallelization explore rnns great single stream multiple parallel stream rnn term parallelization graphics gpu deep quite pattern
understand mechanic base co proof regard relationship broad work inter performance topic summarize sp sp sp representative unlabele call stop agreement prediction example stop stop statistic consecutive round consecutive background regard agreement measurement agreement human receive drawback recognize agreement agreement agreement agreement differ compute metric category consecutive chance assign particular formally compute
day week tweet tweet tweet partition day tweet illustrate tweet language feature tell tweet english predict globally tweet tweet tweet tweet tweet predict exploit rank user individually try supervise aggregate tweet tweet regression exploit extremely bayesian ridge regression regression task extra build splitting randomly choose split combine generalized try
decompose conditioning half p ty u exist immediate consequence least inequality probability proof proposition conclusion sufficient minimax lower consider case n j event conclusion let ij event inequality establish e j p argument conclusion us lemma assumption moreover ij hoeffding inequality small hence estimator sufficiently facilitate ty ty ty ty union get inequality property right bound probability spectra least complete since take op u ij ij imply op proof proof bias u op argument repeat theorem use triangle eq op op ss op op sp ssc sm cc op eq pick least pair parameter dp p dp imply
variation biological state implication clinical diagnostic guide drug provide insight biological phenotype genome nucleotide snp snp exist occur within computationally affect favor inspire bag heavily text genome contain genome discovery feature accurately predict phenotype lead
never unable difficult incoherence project column noise exponentially satisfactory factor nearly go fix column zero distribute column subset expect k unnormalized leverage j update column subset present base score introduce partially input attempt leverage directly score technique score column construct c approximate theorem incoherent reason hold compute provable observe reduce generalize selection input reveal drawback approximate leverage sampling need column level column relative suboptimal multiplicative matrix incoherent column index output probability kk provide technical detail defer divide step column yield additive similar second kk carefully constant lemma c defer appendix give separately input low plus incoherent low incoherent projection span estimation sampling incoherent perturbation rigorous statement incoherent fix subset index cs span noise incoherent randomness independent ensure typically
correspond different stack layer explicit module module operate stack stack pattern rnn stack module current hide gate wise gate single see global gate concatenation associate weight input word control scalar connect transition feedback fig stack rnn flow recurrent layer rnn however recurrent flow recurrent fine describe lstm unit stack rnn state layer hide th module layer control global eqs content lstm case similarly evaluate rnn character program
develop capacity without expand science lot rigorous make happen place facilitate effort member play home odd association statistic use challenge big video big conference understanding well statistic evolve new bring refined address time public association united associations united states association country association take association kind heart develop upon soon united opinion combine serve public organization help
translate perform h group sided arise exact recovery preference reasonable assumption deterministic choice unit rotation translation affect slight group cccc try regular grid predefine group prefer shape score maximum high posterior prefer fourth pixel pixel inference copy application trade fidelity input ccccc lastly representative architecture architecture ccccc apply previously particular generalize haar fourier inverse generative history propose hierarchical usually employ bottom generative couple validation use image set hierarchy automatically sketch employ recognition vocabulary vocabulary generalization
pick regime cs edu significantly computation speed dot heart nlp accomplish partitioning reach fraction feature parameter arrange maximize simpler well suited nlp right speech name entity recognition base dependency parse typical preserve parse reduction run increase speed task parse name entity recognition solely object production run inference hardware center paper describe pair computation many nlp heart prediction dot sparse vector bottleneck combination feature operation feature expensive dot product involve graph string hashing however case necessary speech word many string operation accurately feature g confident noun simple novel
take model team conv filter st nd rd layers std std std conv conv gradient roughly case signal range softmax normalize input impact factor weight among fc number initialization easy initialize relu eqn comparison adopt average adopt forward consider forward std std std completely layer compare make start investigate relu initialization clear superiority compare extremely layer conv fc add conv layer initialization make extremely contrary investigate observe deep aforementioned top degradation
emphasize tensor permutation state directly state permutation easy node size concentration recall large raw moment h h h h u h u next svd give range recall orthonormal column onto algorithm node angle equation result l u u v u p equation item column imply suppose invertible second moment consider truth third order invertible invertible u ht ht likewise h h u u u next result concentrate assumption define condition event hold q show h h h first inequality happen follow first triangle bound individually f h
show interpretability constraint loss carefully adopt admm computation unconstraine warm naturally reduce iteration thank proximal handle efficiently proximity operator include constraint impose least criterion monotone decrease loss property recent advance generalization traditional guarantee point matrix extensive main claim plug play tensor co new hybrid alternate alternate direction multiplier update admm naturally accommodate great constraint almost loss fitting computation warm outer coordinate descent help fast special non factorization constrain tensor simulation real effectiveness broad applicability framework widely cluster machine blind separation application diverse square rank tensor principal component singular svd tensor alternate yield
almost h z supremum bound space e chapter product subsequent discussion constant discussion eq ex u b set since width rf rf pf pf analysis hold proof correlate first obtain simple show arguably heavy applicable style px em imply lipschitz constant variable let take weak converse q next extend lemma
solve computationally inefficient problem reason pair work monitor reward dependency determine reward algorithm prove logarithmic algorithm inefficient bandit feedback index item observe combinatorial bandit cascade search prove address limit assumption violate several optimal indicate learnable bandit work bandit address issue line generalize network fail probability view want refine explain reverse order recommend around te interval confidence hoeffde
operate stability hold mnist mnist set unit layer stability remove dropout exhibit dropout accurate pt theorem definition belief require extremely exist gd arbitrarily poor local paper rigorously avoid technique heuristic randomly drop layer certain decrease multiplicative flip erm act gd assertion dropout glm moreover stability dropout differentially prediction validate surprisingly benchmark dataset network system success prediction
semantic slice variation ct modality high concept demonstrate variation appearance disease occur different accuracy accuracy document distribute false discover correlate confusion matrix find deep well level sub body part visual distinguish image complex deeply require resource consumption train level topic seem amount task imagenet dataset imagenet top rate moderately higher versus error comparable encourage also uncertainty unsupervised algorithm multi cnn light parse image database top level sentence view layer hour topic section automate interpretation expensive consuming examine keyword image key topic semantic image description language cnn text mr expression image modality image tumor address label ambiguity transform word article ni meaning project close example visualization principal
rnn rnns feedforward share consequently theoretically rnn capable capture arbitrary unfortunately difficulty rnn past decade recurrent et al perform much
pairwise distance annotation measure receiver operate characteristic train value begins reach datum temporal since response offer strong prior show near various per block difference pair motion pair indicate block give task help recognize object exploit behave identify next limit simplicity k nn pair form pair output probability return class histogram feature per cf sec transform wise information select would predict qualitatively impact near neighbor retrieve motion space pair relate strictly wise kind approximately practice answer obtain close query example cs edu image behave crucial aspect proper development yet method regularization convolutional learn exhibit systematically distinct outperform visual task test show capture drive platform scene recognition
j x x x gx gx gx j gx j I gx x gx x gx gx jx gx
approximation regard detailed projection point paper relax greedy boost utilize need tune parameter time successfully avoid theoretical behavior address issue regard convergence generalization main assumption dictionary boundedness certainly introduce purpose derive fast rate concrete loss relaxed localization indeed state arbitrary small number weak learner widely weak neural spline note concern dimension rademacher learner already adopt point concern boundedness mild r mf actually convexity smoothness condition strict certain step step smoothness arbitrary smoothness
function half lead eq sparse non bind reduce shall technical require exclude go infinity although derive follow show constant go regard error vanish increase probability go shall condition exclude advance output exist constant sequence k immediately k define previous preliminary
replacement five test unconstrained regression figure term converge diag theory dataset diag around proportional diag large number accurate accuracy completeness algorithmic leverage sgd record method range pair efficiency solver randomize display much method become favorable feasibility size sgd fast sgd equip rate quickly competition medium solution diag slightly e medium fairly large might solve sample become advantage figure cc error axis sgd optimization use main question notion amount negative problem work author problem form set case algorithm pick return central sensitivity mf present two algorithmic leveraging first result
reference onto model although loo cv obtain predictive ability stochastic whose dataset large overfitte induce bias select overfitte become candidate highly select far probable tend however reference typically well due despite reference variable demonstrate validation searching model assess organize section discusse illustrate induce experiment paper discuss comprehensive review discuss ability review method table mean assume candidate one section complete view form model construct notation assume predict input scalar vector utility open cross predictive criterion approach view predictive predictive view posteriori median model use leave simplify logarithmic score
directly communication complexity strong scale choose minimize probability note also algorithm parameter hold well distribute accelerate high iteration two round enjoy communication convex function binary far bound give usually scale become scale factor scale discuss smoothed hinge loss result table satisfie rescale function self standard assumption scale plug constant communication eq ignore communication round slowly minimize hinge show consequence verify apply smoothed bound smoothed hinge loss enjoy numerical experiment art admm accelerate method bfgs quasi newton algorithms bfgs well rich admm distribute straightforward bfgs implementation gradient master master iteration complexity stay centralized involve describe distribute propose al iteration round first communication local q communication quadratic loss n ordinary name number sample c ccc news gap vertical communication algorithm three news theoretical analysis suggest
branch simulate n correlation identity simulate depict simulation result though seem cover several derive statistic combinatorial far technical spurious fix technique bootstrap limit statistical correlation organize section concept spurious introduce spurious extend give two proof defer material let random vector n independent I n spurious sparsity pearson coefficient anti sign eq express invariance diag x q resp exponential follow impose moreover process maximal covariance ps role notation two write sufficiently enough write
hide cost regularization hide layer perform descent explain momentum aim achieve accuracy use library simultaneous update epoch update ignore epoch fine fine bias network respect layer account variation loss repetition final
test optimisation stop deviation statistically mc dropout augment show previous convnet mc significant augment lowest augment none repetition give deviation mc dropout consistent lowest augment dropout mc imagenet use offer well imagenet much perhaps label collect imagenet obtain imagenet strong suggest work question give suggest sample dataset mc experiment within converge deviation test deviation analyse explain improvement result
model moreover provably recover hybrid experimental suggest use strictly achieve significant pca maintain approximation sampling achieve tight sampling experience datum data example gaussian follow noisy behavior index accord separately produce sparse sample rescale hybrid towards control mostly noise lot produce rescale regularizer datum preserve balance sample reproduce small element tt element accord element n assume eq optimization come bernstein flexibility compute reproduce produce tight plot plot axis
old return scale proceed subproblem multiplicative use framework decrease produce point slow demonstrate idea constraint mix nonnegative nmf nonnegative entry reformulate negativity analytical update rule obtain algorithm old new old old return
definition slack optimal ensure slack simplify simplex minimize achieve optimal strategy empty wider make good v n always never ideal obtain classifier column equal choose true classifier error lead particular case must heuristic belief combine want maximal erm recover unweighted vote
mail pl kde entropy closely kde simplify complexity make dataset discard similarity point impact process phase wide optimization require bound core operator estimation project
problem reality datum view without predictive help embed word map vector view similar embed become predict sense proximity usefulness supervise relation reflect relevant unsupervised embed datum task embed provide method embed mention study low dense focus gram within correspondence view token task pos feature mapping skip gram word word word word vector embed w context task well relation factor hand task produce view role word context vector decomposition train necessarily individually might reason skip sized omit
shall matrix positive semidefinite act irreducible permutation moment characterize submatrix together positivity useful problem illustration analysis cover os sublinear namely degree square lower find clique technical concern spectrum imply clique submatrix problem present brief technical unless subsection introduce association slight develop imply semidefinite state proposition clique os vertex edge vertex think adjacency graph size subset head respectively tail denote indicator convenience let e cg imply give proposition check derive hide clique clique independently relaxation degree clique semidefinite obviously size maximum clique introduction replace clique clique soon
tweet manually annotate neutral tweet sentiment report sentiment tool movie review adopt naive benchmark show achieve ever report label neutral comprise hereafter use configuration neutral tweet neutral tweet neutral neutral social medium stream various movie success stock political predictive power box movie signal seem popularity framework tweet mention political house general model political sentiment social medium combine sentiment base highly twitter daily zhang collect twitter six percentage tweet fluctuation display correlation work call predict keep machine algorithm black box reason design simple dynamic entirely observable rely single sentiment hypothesis root recent
convergence schedule use monte carlo may limited birth death explore metropolis jump generic metropolis hasting recently accept overcome require expensive intractable effort devote design appropriate estimator take direction monte seminal way key insight diffusion brownian approach sampling euler approximate apply literature chain evolve choose tradeoff improve measure improve soon follow proposal let h langevin equilibrium comment centre community systematic adjust langevin mala evidence gradient mala type tailed grow cause precisely contour mala geometrically ergodic tail decay metropolis geometrically ergodic lack ergodicity quantify operator context general equation drift invariant f proposal improve ergodicity nontrivial recently mathematically riemannian probability mala dependent mala differ precise version specification replace absolute value hessian robust metropolis ergodic target tail metric term strictly behaviour sequel signature start hamiltonian hybrid monte hmc stem physics like mcmc also differential efficient augment hmc add lead hamiltonian speed statistical create auxiliary moving preserve marginal exactly solve approximately correction g dynamic induce reversible volume preserve need jacobian update rely commonly nd level updating via euler arbitrary consider drive dynamic govern make augmentation scheme metropolis likely accept modify proceed avoid mcmc monte avoid walk metropolis simulation calibration quite influential mean approximation choice metropolis crucially smc metropolis hasting appear due must
three forecast inside particle filter ahead check account performance evaluate provide full gp theoretically utilize structure example load capability competitive variational model par establish natural knowledge present extend parametrization mh dynamical time approach counterpart gp time acknowledgment project contract reference material material domain basis
task human visual candidate acquisition human direct reader internal process treat human stochastic learner convenience divide batch fix interactive online general machine teacher goal example offline early focused like work make simplify memory assumption world theoretically motivate interesting subject teacher student student know teacher maintain unlike computer capability human motivate limited capacity improve visual learner offline try classification attempt encode unable student order fix interactive adaptive student noisy stage learn
relaxation freedom design well training problem membership lead tend mm structural svms latent difference way progress respect requirement give rise progress leverage objective directly conceptually computationally avoid bind valid select convex machine understand miss em step progress sharp minima attempt mm framework mm generalize bound concave instance function successful learn particular initialization expensive latent ability application information modification drawback relax constraint objective closely work require may intersect objective use requirement framework binary energy mrfs surrogate
statement induction list say high respectively triple rank high must rank case rank prove inductive unique ranking property rank respectively list remove would rank rank highest contradict symmetric np hard instance diameter perturbation center add additional point define finally let radius put leave optimal maximum cluster clearly center max perturbation original achieve keep cluster must partition corollary condition definition conjecture theorem note cs edu center canonical study many application form version tight bad case symmetric version go take result symmetric perturbation perturbation distance state partition perturbation asymmetric center problem optimally approximation center center illustrate surprising asymmetric stability unlike solve asymmetric center optimally small constant perturbation long place throughout city city distance center satisfy inequality give distance symmetry want center center image classification symmetric find simple asymmetric center problem center find ratio et build paper establish hypergraph interest though
long history scalar quantity method perform vector scalar reduce one versus matrix product trick second effectively approximate initial interestingly view approximate traditional cg suggest cg obtain cg reasonably well suffer much strong mini exact demonstrate versus simple raw update poor unless factor compute factor block x factor describe develop relie scale whose curvature add equivalent modification constraint spherical region depend theoretically adopt current mini intuitively try small possible implicit trust region maintain property sense accurately predict value get convergence exact sufficiently minimum enough convergence apply every iteration set could efficiently usual pass remaining need reasonably avoid truly situation factor technique maintain independently section separate constant adjust end reasoning modification theory trust meanwhile compute batch exact scale perform add multiple exact good multiple making help conservative ultimately useful proposal quality negative adjust greedy every iteration current metric must multiple well practice add cost quantity compute find obtain obvious momentum helpful stochastic version momentum arguably even version work final update previous effectively optimization iteration similarly momentum initialize value mini
take coordinate call roughly long event ax upon expansion hard rearrange precisely sake trace invariant cyclic rotation compare trace uv xx yy negative coefficient proof define last average real index exist sum contradiction first rhs multiply theorem efficient solution negativity update learn align output gaussians roughly
try balanced maximization model try objective contrary weight note notation learn objective loss area machine objective weight regularization characterize objective limit number differ importance balanced loss parameter become sec maintain information iterative step pressure value loss learn hardness significantly problematic providing
algorithm explicit large rest list entry satisfie set correspond clearly lie particular lemma result comment constraint feasible discuss approximation construct low time spectral select accord score develop interpretable application least give row rescale matrix pointing compute derive opt opt previous case analysis reason become clear
encode confirm approximate give rectangle select characterization cardinality previous show approximate tree lebesgue lebesgue uniform result show high induced training chernoff hoeffding style poisson hold choose exist hold possible leaf interest choose enough value theorem corollary corollary present series lead poisson process classical chernoff variable consequence relation course want issue valid corollary understanding leave satisfy result prove reduce algebra work convergence forest convergence bound weak main suppose intersection unity detailed approximation stochastically low
nan notation proper hypothesis construct indeed estimator asymptotically replace respectively statistic substitution estimator smooth asymptotic distributional hold refer condition inference testing matrix seen start nan satisfie value asymptotic statistic note nan et
conclude presence difference rank well never dataset ten real probably suit ten suited large difference rank classifier correspond happen compare exist rank normally distribute pool comprise also follow q
study level panel subgraph subgraph structure shift period form major life company meet cb service group risk four among connected financial capital american express company subgraph display panel shift differently dependence component connection severe network able adjust treat dependence market htbp propose concentration limitation structure point simple suggest bayesian run simple oppose remarkable facilitate modular since matrix contain mass prior concern posterior concentrate covariance concentration limit experiment distribution normal indeed concentration average correspond zero depend hyperparameter prior normal mass problematic weak shrinkage refinement density regression heavy tail offer comparable point prior g maintain
poor function approximation another figure although large clearly depend instead basis involve center order degree basis significant improvement nine function six six method obtain basis vary surprisingly example regard
classify differently bayes rule number improve occur neighbor rule repetition classifier classifier equal adaboost throughout consider terminal adaboost large sample substantially performance statistical theory predict large instance preferable rule seminal book tree interaction nearly large tree likely rough fit fail enough fit smooth outperform adaboost rate forest correlation return five simulation additive display point differently bayes sample adaboost hold differently tree fact seem suffer overfitte increase htp qualitative serve share adaboost idea well noisy give zero training worse attribute self enough adaboost average forest explain sum view adaboost practice rule explain adaboost let weight classification expect reasonable uncorrelated error follow constraint positive odd integer I assign integer comment result reduce misclassification justification increase proceed degenerate trivially formalize mathematically inequality coordinate second leaving create
x receive h k asymptotic normality consistency q term estimate nk nk ta n w w w n get last x n w triangle nk n receive equation nk proof normality proof explain unconstrained lasso complex involve assumption ng l mm minus pt rgb shrinkage almost well autoregressive residual behaviour counterpart suitable fast fashion several extension like periodic finally simulation load parameter increase many type grow especially asymptotic stationary usually stationary standard shrinkage attractive autoregressive detail unfortunately literature like deal rarely
linearly high rarely reduction flexibility nonlinear typically difficulty nonlinear method compute embed pointwise low initially available training test need manifold generalization work focus learn eigenvector show nystr om eigenfunction coincide nystr formula regularization impose kernel extension rely construction interpolation domain interpolation sparse hilbert space sparse extension low similarity method manifold learning propose multidimensional interpretation square unfold meanwhile face problem image concentrate extension unsupervise applicable supervise popular kernel order meanwhile manifold depend pair pair nystr om manifold generalization manifold order embed supervised novel compute radial basis interpolation domain class interpolation interpolation extension account regularity interpolation interpolation optimize minimize regularization interpolation sharp directional boundary attain
receive rather policy one fall policy receive behavior determine probability policy unfortunately suffer variance target able exploit environment environment drastically improve policy dimension factor process model infer generally world rarely ideally like apply generally efficient computationally relate contribution paper novel describe notation main present space action action state process map state policy start cumulative policy finite horizon batch trajectory initial policy target policy aim minimize eq discrete variable factor factored mdp mdp compose domain lie domain variable call parent small
corruption tune corruption matrix corrupt ssc gd set tc ensure since condition give expression separately give design enough theorem lower bind use identity kk elementary fact similar true execute gd may enforce shall fc gd execute denote function use analysis fc gd fc gd gd fc execute fc fc gd result two type gd guarantee corrupt execute invertible constant obtain n recovery rsc level constant model sparse resolve notice proof use
gamma derivative combination outside order observation lead mixture fix true k generic I sufficiently case consider b construct gradually conclusion turn gamma mixture set give gamma j necessity restriction gamma strongly call hellinger obtain consequence bind fit mixture word conclusion crucial guarantee restriction polynomial wasserstein far point zero lie low wasserstein distance inclusion location wasserstein special restrict positive differentiable condition definition behavior slow actually special fixing location direct calculation algebraic identity location except non constant identity k location exponential unlike gamma location slow fit minimax parameter logarithmic f fx scale parameter fix asymmetric skewness sign rich skew gaussian note identifiable reveal combination prevent skew family identifiability condition skew family case rich varied seen skew density skew component scale say underlie least additionally mixture skew behavior strong component skew gaussian ii allow presence iii fit sufficiently hold small link system polynomial admit polynomial equation satisfy odd number odd polynomial arise give value describe role gaussian fit assume ig g sufficiently bind establish nonetheless estimation entail assumption fail polynomial infer b mixture fit gaussian fundamental identity density identity skew exception identifiable second see analysis eq depend make skew gaussians complex gaussian mx identity go hard give type skew gaussian assume condition subset odd exploit entail fully nonlinear produce theorem iii iv contain second shall behaviors mle measure hand strongly identifiable introduce despite extremely failure converge rigorous remainder shall implication strong theorem identifiability order g w strong identifiability fail identifiable low behave hellinger approach calculate simulation integral restrict rectangular distance p ij kp kp wasserstein distance programming yield freedom multivariate g plot panel prove panel multivariate identical plot present line panel bottom respectively bind distance density wasserstein p estimator fit mixture set replace n boundedness sufficient regularity hellinger pp applicable precise give le minimax low set determine infimum upper mle location exponential skew bound entail
prediction rather duality capture phenomenon like lead yet practice analogue commonly growth notion accomplish present loss give reader state measure familiar number analogue scale behave grow square loss rate sometimes factor part establish curvature reader us conclusion rate excess theory datum empirical cover complexity introduce later deep phenomenon concern square informally affect convergence geometric investigation set prediction ensure consistent history line omit extra overhead introduce notation overview sequential complexity low bound established calculate minimax rate question develop
arbitrary cnns realistic fine leverage gpu cnns optical flow per among variational dominate optical improvement combinatorial term relate information aggregate fine coarse sparse max manually term put even emphasis match merely boundary flow convolutional optical optical learn regularizer statistic optical flow mixture predict optical flow flow motion video task factor boltzmann special autoencoder autoencoder control activity video competitive realistic video backpropagation show perform scale give apply cnn vision estimating
mm draw n lambda alpha connect connect p connect lambda connect structure characterize indicator human interpretability depict represent random advance parametric mixture denote come cluster denote value hyperparameter explanatory standard assume parametric characterize intuitively important identify prototype hence intuitively define prototype cluster maximize element prototype prototype prototype good represent characterize select indicator variable vector generative generate
option machine short twitter give reasonable response score score response actual embed standardized use effective automatically expense highly model slot unable standardize meaningful motivate time channel box extraction address address ex de introduce million turn million million word resource research base property tracking service twitter describe task response converse objective intelligence ai build ability diverse topic target logical system slot ai recognition break year worth successful
use indicate either lsh abuse notation clustering index first measure call discrepancy shift away full counterpart set bandwidth shift accord hausdorff performance hausdorff clustering denote hausdorff pixel subset equivalent notice define ba hausdorff clustering distance element clustering let indicate indicate algorithm lsh lsh computational half lsh hausdorff htp shift sparse paper density argument improve lsh density list kl divergence kernel mean choose heuristic performance divergence choose perform test quantity plotted approximation uniformly want stop complete random pick element dense
loss ordinary fuse lasso grid graph refer denoise exist efficient programming routine graph correspond grid gaussian rapidly desire grid arbitrary idea exploit basic theory decompose form proximal closed form primal set subproblem result flexible algorithm fuse criterion relevant also derive different investigate offer quickly minimal
high finding line performance observe grow need optimality scalable valuable insight cost high scientific medical database pose challenge advantageous less curse new compute center know interpolation membership approximate decompose problem solve separately straightforward complexity
formulation sdp formulation support set sign two nonnegative measure sdp sdp must finitely nonnegative measure sdp multiplier distinguish problem size sdp polynomially sdp moment rank atomic sdp algebra retrieve example super resolution phenomenon matter measurement moment numerical carry interface design relaxation lp multivariate notational ix x n notational polynomial chebyshev preferable numerically sdp relaxation solve primal matlab code public interface sdp solver fr software want lp
mm college com propagation produce message operator replace integral classical ep analytic train incoming message approach two fast feature principle mean request training substantially operator modelling language wish complexity approximation conjugacy reality simplicity intend way user widely prior expressive choose expensive quadrature make challenging impractical run propagation wherein parametric incoming factor potential project achievable close thus ep update numerically detail due estimate integral procedure instance input sampler neural network income disadvantage training type g assess event uncertain network forest uncertainty prediction predictor uncertainty empirically become away update unbalanced prediction training size rather message regression input measure represent embedding
stack alternate manner undirected cut would incidence q correspond exactly originally e incidence process scalable primarily graph fundamental circuit input work however direction need reconstruct network graph circuit set since indirect estimating network biology model particular would provide alternate chemical reaction steady state recover circuit matrix traversal front sequence problem reconstruction involve fig major component pca svd obtain edge linear find topology
study confirm notice classification way evaluate performance engine health monitoring engine monitor imbalance secondly health area impact important asymmetric misclassification proper evaluation plan methodology evaluate complexity complex indicator universit e paris paris france author detect sign failure system goal failure operation optimize monitor representative field anomaly collect engine article introduce allow early build upon human remain human make idea generate binary anomaly selection naive give interpretable classifier interpretable methodology design reproduce anomaly engine detection anomaly major numerous generate scientific application engine health monitoring aim detect failure application make reliable operational event engine jointly rate
popular year consist user tag resource music file review collaborative annotation user keyword enable retrieval social resource tag scalable operation explore resource community understand formation human human interaction hypergraph consist tag scalable challenge limited membership belong interest tag resource community heuristic novel model guarantee naive scalable realistic scalable hypergraph popular mixed stochastic blockmodel stochastic generate informative intuitively hyper multiple paper exploit practically relevant membership hypergraph generation hyper edge resource tag impose natural membership independent tag resource resource theoretical apply machine allow user tag resource depend user various tag application latent independence user look tag category example
account regard depend estimate spurious closed expression clear switching behaviour notice property stick single scenario vertical overlap estimation local single result summarize procedure outline perform horizontal horizontal vertical proceed vertical proceed manner modification clearly previous series dimensional vertical slice th
latter part statistical limit match former sparsity moderately sparse limit case closely wang low section phase transition pca prove discuss recovery theorem theorem study fundamental hypothesis introduce conclude discussion detail generic vary occurrence see section x two hellinger distance small characterization post tight spectra feature unclear exist directly need post use cover show event event negligible realization set feature introduce fixing notational play elementary ratio assumption recall conditioning realization cp number turn insufficient constant condition singular fall q c singular short hand expectation perturbation matrix end note suppose hold condition realization z cn combine lemma definition condition lemma r fix goal denote short h h h prove realization claim case claim limit recovery prove together compare
satisfie intermediate canonical influential flip mh influential covariate flip consequently precede ratio include influential influential canonical involve update mh inequality satisfy obtain contain influential influential well moreover ensure inequality q treat upper valid divide assign receive receive determined lemma event event control equip simple counting yield express stand numerator numerator indicator follow numerator fact function fu last combine finally proof let iterate satisfie claim study bayesian regression sparsity bayesian computational example insight markov model good contraction statistical estimation rapid selection behavior simulated understand result direction nonparametric regression investigate acknowledgement yy office yy additionally national science foundation grant appendix solution problem equivalent penalization make posterior covariate large noise large
think improper number sift ts product encoding since dense evaluation speed fig ratio naive theoretical operation cut per almost sum would explanation theoretical improvement learn pairwise rotation encoding trade relationship entropy phase loop hash high dimension dimension still room exclude reduction minor key issue appropriate even demonstrate performance pair selective substantially scheme near binary hashing visual
weight average converge toward toward input specific stop field tune toward pattern predictive tend patch encode neuron without dropout denoise connectivity form mapping neuron connect previous layer eight three weight field summary introduce base scheme form focus may train artificial activation apply inspire broad applicability provide field derive learn neural network instantaneous fire conditional part neuron activation simple online local rule occurrence spike idea
variable model present solution factorize multinomial form quantity optimize dirichlet proportion th gaussian parameter centroid th component distribution nx optimal initialize prior introduce uninformative parameter distribution justify affect zero describe mean uninformative em utilize k priori create component equal reason small number centroid parameter parameter stand cluster
train desire evolve fuzzy refine incorporation future test test evolve aspect investigate benchmark correctness usefulness approach employ axis rule however literature arbitrarily able one improvement future work investigate certainly author investigate benefit optimization problem hope result future anonymous us point thank form two discovery grant notion evolutionary algorithm agent become fast notion opposite opposite seem alternative receive
surface angle inversion care function maximum bivariate project univariate process marginally direction define refer angular discrepancy gradient flat direction magnitude large direction angular discrepancy meaningful area magnitude plot describe imagine scenario zero case utilize gradient surface response differentiable transformation surface derivative sd uv sg analyze cox gradient surface sd linear covariate probit regression simulate gaussian ern
entirely oppose formalize contextual bandit content choose tweet design reward month aggregate experience agent prediction finding deep social people short tweet status update call tweet tweet mark large twitter provide machine world evaluate deep insight alone without
foundation science deep machine performance pattern particular extensively deep extract structural precision essentially consist iterative coarse grain procedure level redundancy feature extraction part pre kind organization organization dnn structure neural procedure update weight I take greedy
plot plot progress problem dominant linear spread understand linear axis xlabel log ylabel ylabel solid forget plot crcr e e e e width height ylabel ylabel black solid forget row sep e e e e e e e e e e e e e e e e sequence utilize probabilistic interpretation also top image model convolution vary spread encodes convolution deconvolution instance solver plot residual jump residual deconvolution imply interpretation solver progress increasingly note decrease step deconvolution magnitude span spread example highlight quadrature algebra numerical posteriori establish experiment picture numerical across boundary method bfgs rule bfgs cg bfgs specific generalization conjugate thing model gradient currently area probabilistic ordinary equation tx
variation straightforward adaptation alpha reliability simply take transpose number reliability poor type alpha coefficient rigorously speak average ordinal difficult bad year researcher work
non concave minimization problem effective maxima optimization multiple run achieve formalize recognize frequency counting match implementation figure illustrate left frequency demonstrate normally normal yield portion within algorithm determine term distance respect computing gradient translate lk force quadratic segment worth learn force primary strength addition protocol compare computed length threshold represent consist among continuous sensor measure difference contain national consist collect front human many frequent pattern segment broad scope
census comparison apply kalman smooth census fail census refer supplement dp benefit hierarchical enable census indicator trend observation census substantial expect observation simulation factor process factor large trend compare scenario importantly price dynamic bayesian without nonparametric component improvement rmse rmse rmse rmse rmse turn city form census separately estimate trend capture city wide dynamic would regression remove h h sale roughly per month decompose trend trend component noise discard term trend attribute change list transaction circumstance sale set training chain discard half burn remain sample factor illustration census posteriori I probability intrinsic census demand cluster expensive slow difference cluster occur period follow intuitively region affect highly supplement price u census assign large rich crp examine collect time share cluster neighbor lack code enable heterogeneous effect dependence discover price pattern adjust kk color index describe
determine non opposed row response filter subspace hyperplane filter vector information encode zero response analytic fuse difference relative frequently concatenation algorithm find optimal minimize average overview provide co denoise super analyze different modality analyze propose blind compressive cf learn adaptively finally medical representation eeg signal sparsity success learn performance rapidly author cf offer filter reader separable benefit phase learn reliable co sparsity co theorem section number impose consequence operator play role investigation co size condition novel evaluate confirm sense separable
zero lemma assume proof corollary corollary consistency learn pr pf pc cn proof failure compute quadratic loss close l nd c z unique addition nf hence intercept input rkhs j unique method lagrange multiplier lagrangian minimize yield substitute strong duality theorem equivalent j c yield n c v c quadratic programming constraint intercept term interesting attain differentiable f l nc ic nk performance art mechanism
make ensemble tree see tree negligible datum point indicate leave briefly shrinkage reduce impact tree entire bias slow case order shrinkage drop ensemble contribution notable algorithm specialize significantly rate indicate slow expect contribution much tree drastically unlike continue tree tree ensemble control forest
million decode train output weight origin train simultaneously raw score success parameter simple parameterization believe capture self normalize related begin related training classifier distinguish objective log estimate treat eventually converge fix case evidence suggest exhibit self sum quadrature space softmax replace large
sim provide generalize glm sim method model response monotonic transfer typical example probit function regression generalize nonlinear single sim semi parametric glm introduce efficient algorithm provide setting ambient dimension modern problem biology number exceed sample simple call iterative vector monotonic lipschitz iterative use calibrate sim generate
red blue contrast contrast despite level characteristic event white show nan cell generator cell excess interaction c result computer million mainly conference science author year journal publish year publish year still incomplete describe million event skew marginal less come half event appear time htbp b confirm robustness picture apply knowledge round end day relate evolution first grid day stop generally allow analyst author publish appear thousand computational grid hour round offer day experiment remove event computation datum year potential whole grid make
reliably repetition phone check force alignment long generator correct alignment inside ground truth properly correctly align w consider sequence model attention convolutional align sequence fail align behave examine fail understand mode failure material cycle behavior track capability jump end contrast aware fail stop frame phone issue sec irrelevant frame network slightly concatenation
ix loss estimate define I draw exponentially weight variant simplicity round fix observation satisfie particular kt kt eq proof fundamentally particular thus bind logarithmic high aware weak maximal acyclic subgraph side conduct demonstrate superior bandit independent mean arm loss arm change round round well half arm
encode bring ii improvement align improvement mt ir list match interact semantic relationship positively co estimation gram match away might human use ask output pairwise string pair summarize pairwise confidence generate identical string incorporate assign pattern result clear consistent preference versus ir versus mt sensitive system context mt automate ranking raw I I thank thank much book really nothing thank way trust trust
elimination concept variable pass main systematically graphical question present algorithm focus graphical decade discussion tackle challenge offer learn undirected graphical approximated tractable enable usual variational branch recursively conditioning prevent exhaustive use pass prune simple rule involve compute reasonable convergent message global stochastic global continuous counting family analytic pass recent passing rely orthogonal update reference deal message passing exclude sample sampling need biased estimation hashing enforce fair potential suitable review sort statistic deterministic plus minus et es france paris paris account structure constant marginalization review exploit yet standard condition principle elimination eliminate graph characterize algorithmic efficiently algorithmic elimination problem review technique belief link elimination illustrate parameter couple computational message complex object locally interact graphical form relationship enable heterogeneous capture wide speech bioinformatic application consequence calculation manner estimate derive appeal memory consuming procedure store perform increase probably widely monte particle filtering tool starting become practice essence
next develop approach convert solution numerical process small reduce human discussion research collection later section numerical well informative feedback explanatory since correctness ignore text deriving feature potentially generate nlp treat text numerical classification cluster response mathematical mathematical together text mathematical include identify expression contain learner extend bag word model mathematical expression coin phrase novel mathematical extract library symbolic mathematic powerful capability simplify expression simplify way equivalent expression result might simplification perform simplification verify
classifier available interest step case principle allow lie posterior posterior formulation methodology ratio multiply class ratio minimize kullback kl kl divergence domain proceed composite q eq minimize obtain task upper transfer easy task tight mle likelihood technique maximize ratio feasible unique thing kl divergence sure
grow pilot one rather one quantitative normal improper pilot bad core event area impossible event must something scale even sometimes hard simultaneous reverse streaming method drive unsupervised one foundation without moreover raw label unsupervised traditional challenge well spurious bad g inaccurate case standard
compute representation node representation output type make path value composition learn representation relation predict contain relation figure element composition concatenation vector similarly representation accumulate neural prediction path vector sigmoid r exist fact treat unobserve rnns challenge path connect predictive select close fast training rnn triple entity pair observed fact connect variable whose predicted assign unobserved fact rnn predict predict relation parameter
approach hmm combinatorial integrate integrate crf development exactly free transition use hmm add cost
impossible case stochastic gradient alternative version provide find additive variant geometric rate guarantee sgd require vanish gradient ensure finite size mention contribution analysis sgd insight correction previous present correction stochastic fourth experimentally algorithm regime streaming give investigate stochastic descent update q
controller entire system feedback small neighborhood current implement controller approximation small would fewer opposed domain operation reduction require refer kernel accurate use technical system lyapunov section incorporate ideal continuously differentiable exploration without rl load vary pe regressor vary input dynamic include effectiveness maintain approximate control approximated scheme system gradient approximate current gradient state application interest function evolves maintain value gradient value state
without therefore compute introduction sum add round occur positive lose large problem many application improve summation somewhat little computational try subtract core compute adjust result expression three digit digits many sum round close preferable non computation advantage exact change inexact summation depend contrast add sum depend perhaps run availability processor serial modern exploit parallelism dependency allow parallelism sum focus summation fall standard point arithmetic hardware processor summation arithmetic hybrid see small large resemble enough binary digit exponent range twice exponent add sum bit format high product extend high sum arithmetic number slow dot single bit summation easily small paper incorporate largely fix termination term cost use carefully write inexact summation sum allow increase
z composition iterate composition satisfied composition give continuously differentiable plane meaning iterate exp function highlight operator continuously elementary refer analogous n employ operator parameter use shorthand q argument solution want argument argument produce monotonic interpolation maxima existence pose exponential calculate use
likelihood likelihood topic topic document take reach reach stability document keyword interest topic corpora process adopt fix dimensional probability author measure document document design gibbs learn infinite author potential include label partly research arc discovery grant china jointly national science foundation china da xu incorporating corpus author tag
pick result cascade mle offline lie cascade front last last seed minimize round probability correspond possible distance minimize evaluate offline learn offline let parameter offline cascade batch parameter round round node round l greedy decrease result gradient round influence value various lie range loss batch batch cascade batch significance online mle algorithm nearly learn offline cascade round true base make assumption cascade result hold random criterion overview come next problem round seed may edge spread network explore appropriately exploitation feedback node feedback adopt approach exploration regardless round thus pure exploration influence exploration strategy choose random round pick probability feedback feedback mechanism observe cascade random frequentist frequentist update mainly focus important round cascade influence probability feedback random say diffusion seed sum incoming v straightforward node bound node budget seed
extend assumption extension spectral localization localization top vector u rv follow spectral assume eq succeed non overlap latter submatrix singular perturbation canonical angle denote define far invariant us perturbation derive particularly useful tell q submatrix hold unitary see mild moment require explicit sub let gaussian isotropic one eq universal zero satisfy isotropic probability reach eq addition q mean definition canonical angle vector independent follow pure subsample computationally recall operator projection assume sub operator concentration theorem case q sub lemma invoke eq j bounding approach split term second
natural factor age risk clinical independent death associate risk highlight risk substitution death physical arise exercise predictor death analysis link risk death exception gender take add grow importance account especially among chose year individual age year old wish window present argue short window age survival increasingly less old model perturbation set assess application stability gain popularity regression measure variable tree additive expansion fuse lasso subsample individual absolute magnitude across stability across datum compute adjust typically real meaning role rr reflect coefficient simple appropriate ahead employ like discussion selection final coefficient take care sample perform cross
norm translate additive spectral norm specifically eq combine frobenius norm satisfy singular value adjust guarantee algorithm return satisfy note I I apply modification run accelerate method decay focus block usual simple simultaneous avoid potentially gap block ensure singular value large actually sufficient separate top significantly value specifically know follow property exact dependent satisfy long architecture multiply even additionally post size finally classical gap take advantage fact look approximation precisely dependence gap experimental paper mention justify randomize simultaneous sketch focus demonstrate problem offer significantly
convert multiple deterministic probabilistic plausible small transition severe often become place plausible uncertainty early express uncertainty planning policy propose probabilistic process deterministic evaluation improvement learn action directly applicable system article detail key assess two hardware article discuss related work idea dynamic detail particular sec property conclude sec controlling decade robust treat uncertainty stochastic control uncertainty closely rl adaptive often system nonlinear parametric control knowledge rough uncertain sufficient locally idea evaluation real life trial manually suited model range promise dynamic weight regression deal parameter account temporal correlation treat approach dynamic programming discretized space build treat require space infinitely plausible dynamic nonparametric gps dynamic training function policy require effect function point necessary statistically impractical model error usually rl search policy currently value x cost
kernel many accurate continue random concentration valuable feature map angular define construct map classical sphere relation diagram figure light formula reproduce immediately angular rectangle node arc arc arc cycle right node node arc arc let nonzero red map representation translation positive yield complex value term short sum formula cosine map radial kernel reflect regard theorem radial use q short introduce kernel assumption similarity measure justify see far intrinsic view empirical satisfie randomize kernel small apply second uniform upper come map feature calculate semidefinite introduce unbiased eq invoke bn arrive sum hermitian begin hermitian result explain bernstein sum hermitian matrices hermitian hermitian introduce q furthermore theorem theorem information hermitian apply emphasize eigenvalue less zero mean lead directly state hermitian matrix exponential positive therefore exceed allow introduce relation bind argument series taylor fraction view function series identity preserve semidefinite hold extract logarithm monotone bernstein hermitian hermitian begin invoke find invoke monotone semidefinite identify rest line follow mapping third compute argument computer confirm next master tail infimum proceed elegant bind finally explain immediately hermitian matrix two hermitian real linear theorem next hermitian coincide finally invoke wide bernstein application inequality brief david approach concentration inequality mathematical statistic involve variance significantly large coincide version inequality statistic accomplish elegant thompson constant sum unbounded moment paper theorem interior bernstein extension use recognize even spirit version appear matrix subsequent inspire develop context banach consider hull radius cover express depend procedure banach type error approximate banach concrete banach matrix sample empirical probabilistic follow hull idea reference cover behavior row fouri transform empirical appear wide although paper recognize mention construct difficult identify mathematic corollary require bernstein indeed weak accelerate spectral appear propose use accelerate programming mechanism initial researcher randomize pointed inequality paper accelerate idea full treatment significant improvement analysis adapt early book scale kernel feature translation attention year product random feature draw et presentation approximation recommend date ambient dimension ambient improvement nevertheless benefit nontrivial version chernoff intrinsic describe bernstein involve intrinsic intrinsic bernstein theorem argument idea reader may wish intrinsic chernoff study random submatrix multiplication bernstein intrinsic attractive interpretation development intrinsic intrinsic bind consequence bernstein require transform beyond describe powerful dependence intrinsic bernstein little content far discriminate among example intrinsic semidefinite quantity intrinsic significant spectral make term eigenvalue verify attained attain identity intrinsic homogeneous insensitive intrinsic monotone semidefinite chernoff inequality control eigenvalue intrinsic chernoff intrinsic random hermitian define appear theorem chernoff concern let attention key dimension matrix instead decay improvement extra two limitation frame exactly conclusion bind result minimum value develop refined column variable study norm random invoke intrinsic intrinsic dimension term may logarithm extension matrix tail bound depend intrinsic variance intrinsic bernstein sequence upper variance quite may challenge intrinsic monotone simple intrinsic intrinsic exceed side length tail always pay restrict attention neither estimate integrate similar intrinsic ambient bernstein expectation bind close look quantity intrinsic block reflect phenomenon intrinsic dimension comparable intrinsic matrix become hermitian
comparison reduce comparison factor work recover comparison systematic gain select surprisingly agnostic able approximately rank noise understand novel formal mse future ranking could investigate sparse comparison sort acknowledgment grateful rich wish careful comment give bound prove centrality arc corollary fix triangle furthermore proof convergence rank centrality inspire interior simplex onto use notation matrix note zero entry irreducible show contraction therefore banach furthermore q kt therefore eq line refer consider bt outcome probabilistic inconsistent ranking
initialize enough see indeed far slightly bad function prevent converge globally minimizer general impose matching scalar whose light scale may seem surprising define approach situation focus admit cost exist scalar believe precisely diagonal inversion prove low field learn sag often allow necessarily interval turn indeed optimization establish hessian admit densely stress detailed exhibit spectrum admit project algorithm thereby question question root opposite say correspond optimization admit iteration surprisingly systematic allow sdca coefficient argument set specify extended scheme deterministic coefficient either constant scalar else motivation coefficient matrix efficiency seek characteristic compatible parameter linearity characteristic simplify characteristic denoting expres radius characteristic whereby characteristic polynomial translate whose scalar consideration inversion take scalar
subproblem formulate alignment constraint first table reach formula construct backtrack start additional constraint eliminate reduce band recommend table path backtrack backtrack alignment increment allow arbitrary amplitude scale offset offset constraint brevity subject aim apply finding suboptimal hard p v optimum condition fulfil v c e apply equation set manner similar constrain general version subset
coefficient cover cyclic randomize cover sdca achieve duality let simplicity ax sdca fx rearrange want sufficient choose hence gap sdca sdca comment identity correlate choose particular impose method strong rate cyclic coordinate theory tell stochastic descent show sdca
statistical inference design application quick achieve increasingly application imagine trivial part desire average achieve precisely head use system desired average system biological collection time statistical build micro analyze develop property model dynamical system thus dynamical gene expression desire property problem design system construct dynamical picture picture green blue note spatial corresponding picture color green respectively like construct already visit trajectory dirac delta color converge coverage
analogous improve stop linear possible predict stop issue prominent fw field train definite matrix simplex whose principle apply rise convex feasible svm convenience geometry yield formula word run constitute substantial fw nonlinear suffer curse unable
characteristic equation domain discuss consequence model resource may considerable day time train multiple employ reliably estimate generalization hyperparameter amount evaluation hyperparameter
denote clique intra impose smoothness target inter adjacent node object motion inter layer connectivity carry unary dynamically adaptively derive ds crf three frame modeling frame provide reasonably acceleration thus handle situation short step incorporate motion type clique connectivity inter clique inter adjacent motion connectivity intra inter incorporate crf inter create inter clique spatial location temporal lattice tracking temporal shape structure little motivated inter connectivity crf manner inter dynamically inter clique layer base inter frame optical node clique direction e manner illustrative inter layer show inter layer establish
measure diversity degenerate mass almost surely point correspond new value drive look case py close distribution sample assign mass component predictive correspondingly probability assign mass add generating phenomenon new effect imply new factor reinforcement generate intuitively obtain growth thing intuition correspond mixture everything proportional value gibbs normalize gamma later analogous though difference gibbs prior exchangeable random integer impose function cardinality reasonable choice random reduce gibbs type measure induce coincide type prior use prior verify satisfie admissible parameter reduce dirichlet coincide family py normalize py process discuss special gibb worth look py compose dirichlet stationary py distinguish actually basic obtain specify coincide specie crucially reverse hold type process measure gibb arise specification obtains introduce completely use heavy specify distribution another process n stand availability expression name attribute motivate define normalizing therefore class independent increment interestingly belong prior type specific far case start generate obtain transformation completely random background need go beyond scope interest challenging satisfactory far
linear basis restrict entire submatrix carefully sub column ideally preserve randomize appendix row size submatrix svd avoid machine norm choose equivalent minimizer pseudo inverse avoid form restrict write full subsample denote sub way extra cost choose computing uniformly skip block sufficiently memory rank memory cluster large come inner empirically apply provide produce index find
proximal operator envelope approximate proximal operator specify envelope control trade list throughout view algorithm suitably envelope different optimisation first envelope interpretation operator intuition proximal highlight relate envelope perspective behave descent motivate observe proximal step proximal generalize ordinary projection onto suggest proximal think generalize constrain equivalent unconstrained proximal term hessian term involve optimization envelope highlight familiar informally start proximal operator convergence reach envelope thus finally property proximal identity allow algorithm dual also different algorithm describe region suitably arise therefore imagine operator relate many intermediate compactly proximal operator likelihood simple operator value special proper show simple envelope black dotted envelope whose minimum envelope operator close circle point envelope operator must eq
solution issue ol sparse achieve model estimation despite success issue usually mild condition expense concern practical motivate alternative fitting linear model ols answer question ordinary square algorithm provide fairly ol consistently recover type non nature compare performance estimator selection ols ridge methodology ridge ol strong easily violate hard spirit hard nevertheless
variable response variable supervise unsupervise compare various ranking sampling advance compressive sense ds thresholding sense recover large seek strongly predictive response furthermore unlike select variable paper follow biology stage theory allocation section gene expression motivate stage biology moreover motivation relevant increase microarray throughput full costly test gene situation sensible stage throughput motivated stage procedure stage perform selection stage refer construct sample screen marginal sis thresholde ordinary square correlation screen pc ol variable asymptotic total optimal stage unit square mse cost sample notation response predictor cross block covariance matrix diagonal vector sis pick
vi fidelity specification boost quantity appear without run substitute solve unconstrained square since standardized coefficient index formula express iteration intuitively two hand residual amount decrease ingredient exact amount optimality gap convergence property eq training error multiplicative global odd understanding view norm ball least square extend define well surprising herein per se exhibit derive least interior boost profile typically tradeoff merely statement rapidly iteration least square fit moderately pair describe tradeoff compute boost simplicity combine follow profile bound suggest theorem profiles fidelity eps profile obtain panel synthetic profile extract axis horizontal axis regularization trace problem serve bind correspond upper q note shrinkage profile ccccc ten profile profile document comprise sample unit panel show exhibit rapid convergence show monotone progress slow square uniquely concern square solution boost close least square solution least quality group iteration datum coefficient linear confirm intuition family correspond slow model ratio close play determine present hold measure coincide linearly least thus hence positive restrict interestingly adaptive automatically versus boost iteration sort index three value update vertical axis update axis value update number large algorithm reach square reflect figure coefficient insight gain substantially thereby part theorem towards suppose standardize every column appendix discussion insight
identical ns iy ns iy net proposition denote vector j tp indicate j py equation tp tp tp contradiction p p q fix denote use high probability finitely put output provide expect decomposable metric f utility metric provably equivalent sign probability gap light recent decomposable utility maximization style principle cubic dataset decision
definition incoherent
stepsize pick single whereby obtain come guarantee even convex convex toward understanding enjoy rate finally experiment load balance technique utility mini scheme ti w ia family encode mini
fix obtain variational aggregation optimization switch weight greedy fix minimize evaluate vector experiment availability parallel study linear fix average empirical comparison goal usually serial test body report function supplement show partial axis scale assess function moment brevity pure second supplement assess group pure pure supplement error probit function augment augmentation implement rapidly sampler dimension supplement decrease
optimality design nuclear keyword response introduce minimize cost maximize factor pressure proportion explanatory use suppose region observe consist often take account surface
newton require newton future work extend scenario parameterize multiscale horizon context actor appendix proof lemma recall taylor expansion obtain hence last fact dx verify govern mention value sequence bound random almost verify b establish bias see verify argument martingale inequality attribute martingale inequality claim pp observe expansion estimate observe obtain inside even multiplication analyse
capability namely incremental example per indeed suited novel provide update quantify motivate fig confidence classifier train number per notice impact overall effect learning process recognition capability collect day training discuss say impact day consider three classifying object accuracy day day day hand acquire incremental predictor remarkably day contain similar information observe seem limitation switch report seem day beneficial experimental compare acquire day accuracy classifier train create take set
state basis adaptation improve threshold deterministic gradient approximate fix run parameterization policy function also optimize random slow well policy shall develop prototype implementation thank three comment significantly manuscript project development science proposition pdf remark consider energy policy comprise sensor node source energy ambient source generate buffer sensor transmission store energy minimize delay transmission infinite horizon mdps efficient namely ucb incorporate action tackle cross incorporate policy parameterization policy outperform heuristic greedy keyword energy sharing sensor sensor environment network weather monitor sensor node environment fusion fusion obtain several sensor carry fusion node equip sense large stop thus
show remainder proof proceed follow x write union imply cn cn cn line concave bind come mean also sum across index get horizontal line desire lemma yield entire characteristic jointly th entry characteristic characteristic cn inequality every lemma hold ensure pair number probability uniformly pointwise let write nd fm f nd fm f fm vanish first establish via say random continuous replace continuity supremum neighborhood map uniformly latter option impossible grow define establish normalization minimal mutual interpretation normalization estimator appear second characteristic simple object efficient computing portion characteristic matrix function practice base form start characterization jointly supremum later continuous ab equal sort space equip uniformly uniform continuity strong characteristic begin reason function increase uniformly establish need choose way statistical distance try continuity entry characteristic uniformly however obtain infinite statement fix resolution normalization characteristic problem grid away statistical distance change extent non uniformity show away mutual change quantity
offer se convergent lead consistent validation remain production occur exclude occurrence statistic example work asset pricing start assertion difficulty I perspective namely problematic big coherent current question address grateful
create new source code primarily intend camera convert least paragraph body document brief sample follow still make exception book article nothing use acknowledge grant author
exist view learner see composition meta deterministic composition space ensemble researcher probably deep learner order multiple label treat multi stacking layer stack attribute ambiguity subset section feature search early toy label begin bad propagate problem identify dependency difficulty decide label effort create undirected directed rely gibbs level label add synthetic beginning build classifier synthetic label namely create choose unit synthetic build slightly expand dropout dropout mask dropping note standardize augment label label beginning improve real eq extract toy regardless original label dimension chain augment synthetic per chain responsible one show create deep
capability exploration alternatively use iteratively build drug instead rely thus spend improve verification account chemical drug process large drug prediction guide decide stop goal reach calculate reliable predict current drug drug remain drug factorization drug combine drug target drug target similarity benefit report predict
leave abc application short extreme estimate abc hard expensive calculate statistic triple stage simulate distance statistic calculate remain triple
constant micro convexity decomposable gain certain force consider obtain confusion case fractional micro confusion parametrize scalar make method previous section consistent decomposable certain parameter grows require section alternate cg concave force plug cg form optimal underlie specifically pose decomposable optimization problem confusion solve optimization confusion general gradient solver instead cg instead sequence decomposable g u j jx jx base access underlie maintain classifier approximately maximize decomposable derive extend shall access classifier confusion solution plug estimation shall important linear show base conditional gradient decomposable performance metric ix jx j jx dl draw algorithm showing maximization use along smoothness theorem classifier j j
negative randomized range national science competition reduce overfitte randomized nature empirically kind small dataset relu variant consistently relu convolutional favorable reduce investigation kind
benefit storage complexity compare store online variant moderate suited stream knowledge cca partly answer rest scheme section extensive real conclude future work motivate scenario later begin nonconvex constraint show alternate converge canonical square second current natural valid solve yx rise project summarize algorithm single gradient project constrain domain avoid invert huge unfortunately demonstrate fail output incorrect x either
rigorous intuitively optimality hold minimize appear analytical achieve roughly minimax correlation bipartite label minimize bad vertex instead seek error user object quality service need appendix one side np give optimize minimax criterion analogous impose clustering let denote round round algorithm plausible attempt complex require correspondingly point nu uv u st st st feasible input cluster error constant show bit bind safe initially pay cluster lp pay cost two pay pay positive negative
event nominal associate participant mention suggest play role predicate extension chinese event document form process toward capture log syntactic feature feature gold chinese restaurant chinese process partition assignment customer sequentially customer proportional customer already table customer draw distribution determine parameter associate exchangeability customer change exchangeability dependency allow incorporation dependency infinite encourage group process sample customer assignment customer customer uniquely customer link determine customer reach link link assignment
factorization simplicity inequality step check projection rule converge summarize initial iteration compute rule compute iterate max fold project non pick optimum actually iteration stop algorithm bit e I ij satisfactory evaluate completion introduce setting comparative encourage result two matrix
also obtain low logarithmic state introduce ratio bind bound treat right vector resp denote projection definition q use lead case hence q directly control l obtain inequality mu mn md function yield hand dominate therefore statement yield
convolution embedding process architecture embedding one comparison neural layer word corpus match mild teacher model mini decay apply validate validation run time smoothing match c compare difference usually
share introduce boundary sample beyond near work boundary bound produce work near neighbor among result strong c show diabetes available repository maintain california class example eight input input scale normalize dimension mean zero standard example example trial permutation nearest work likely
partial equivalent namely know incur bad competitive partition competitive namely know partition incur part competitive regret least part wise know oracle part contain know competitive estimator max middle equality one estimator q partition
classification verification retrieval work past fine collection rapidly collection come need develop retrieve measure essential level extract learn comprehensive feature optimize measure predict style art interpretation aforementione year collection publicly grow collection come develop system retrieve pool collection modern one annotation date automate recommendation retrieve like metric similarity computer make digital recognize image person look sophisticated inference individual art historical landscape style create obviously level exposure goal semantic domain art historical question arise include visual feature encode visual achieve
stable smoothness gradient identify absolute incorrect estimation utilize point consider optimization derivative quadratic solution property gradually technique quadratic substitution minimization broadly reconstruct output regularization original signal utilize parameter short optimization problem change independent type mean field
hazard case combine modelling closely idea indicator observe survival censor follow proportional hazard hazard ratio hazard equation survival harmonic hazard modelling substitute solve equation denote solution efficient estimate principle functional derive consider estimate assume mild regularity dominate system similarly formula delta method bernoulli covariate specified variance log hazard delta var var var overall use test statistic various treatment approximate patient highly positively biased treat treatment mix clinical perspective patient converge overall log hazard time censor bias censor real asymptotically semi parametric estimator statistic increase size despite censor data hazard procedure survival type semi
produce machine result round communication get arbitrarily infeasible establish strong convexity similarity measure compare communication return optimum machine dimension exist quadratic smooth return randomness machine j unless communication quadratic optimize trivial round communication early result round optimum strong merely average focus responsible provide output essentially choose moreover equal th machine column hardness need enough however small know carefully use theoretic tool communication learn local local power polynomially bad function straightforward descent mild employ provide quadratic sometimes question design open bound round communication focus complexity complexity algorithm research institute foundation grant
top bottom correspond great ordinal desire would class class vertical cut block principle mode ordinal ordinal via binary easy support statistical discrimination learner unified boundary separately annotate iii boundary properly particular classify argue ambiguity star htb boundaries cross boundary mathematically equivalent monotonically decrease fairly difficult implement subsection article add function radial kernel k kk k j
suit purely cover manually annotate generalize purpose bring significant purely linguistic multimodal space retrieval limited collection multimodal automatically induce purpose representation embedding process research start word mathematically maximize around determine word
difference depend group specific similarity similarity link together related group interest course structure put forward naturally appeal clique make group parameter penalize star graph penalize covariate order graph edge parameter fuse fuse penalty encourage group connect random component variance penalty demand alternative efficiently issue parameter penalty rapidly work normally weight far prevent introduce take penalty write gp penalize consist iteratively
least several class publicly benchmark severe imbalance positive well sgd cutting solver perceptron sgd pass batch method implement insight maximization surrogate indeed beneficial solver suboptimal batch perceptron method offer dataset cut plane expensive perceptron make frequent identify accurate tight surrogate converge solution whereas loose surrogate show large across task bound surrogate suboptimal work tight surrogate work novel surrogate outperform work allow stability run c thing span experimental acknowledgment thank google fellowship tight claim score prove dataset q scoring satisfie margin rank positive step third false negative fourth bound well integer last number whose form hand recall observation
mid notice friend branch friend fall old fall learn concept break way hierarchy demonstrate understand use movie tv cluster movie movie ham see combination movie tv obvious succeed city return six city notice take subtle similarity movie beyond follow another david return movie movie search child show mid subtle albeit tb aside success across approximation valuable method working large secondary experimentally drop rmse visually approximation room grain bayesian ultimately movie distribute distribute spread across later notice high assignment movie user capture preference rating infer useful clustering co bayesian
g pdfs define weight operator property p pdfs scalar heterogeneous depict mathematical viewpoint direct arc receive jj receive measurement shift operate knowledge scalable operate without node single network central node knowledge topology formalize markovian dynamical measurement transition conditional pdf kx kx k z p kp kp simplicity dependence measurement access measurement z recursion k x p agent density appropriately ingredient estimation object provide mathematically manner cp way pdfs relative weight eq coincide pdfs fusion fusion e ci fusion notion pdfs scalable consensus exploit tool computation entire iteratively repeat operation
make complicated goal inference member kl family form distribution f conjugacy global variational step global parameter variational ascent riemannian fast stochastically full global natural conjugacy ii th sum notation general inference shall entirely dependent form structure describe dependence simplest commonly make q ik q function mean approximation baseline field mf identical maintain structure generalize suggest local parameter I mcmc
relative flexible requirement tolerance solve regardless geometric accelerate build subsection consider absolute relative decrease bandwidth imply restrictive kernel fall tree density absolute kernel algorithm tree partial density estimate begin traversal list traversal combination difference rd combinations update evaluation traversal actual query calculate fp visit calculation accomplish simple gray version runtime implementation tend tb output tb query reference list estimate node f r q subsection query reference constant give algorithm traversal define lemma clear time bound thus bound split empty obey thus interested possible value assumption center r bound rp statement rp r h r traversal take extract traversal node tree traversal
qualitatively amp decoder observe fit amp decoder despite way amp structure corrupt gaussian amp implement decoder large length approximate message pass superposition amp propose replica recently report improve couple mention bit code lattice code design good grow length organize code describe amp decoder intuition decoder min message amp length contains denote denote lie base measure code integer whose term fig think column formally exactly one constant satisfy bit choice pair example shannon codebook number choose split stream input segment segment compute simply
vector element actual event censor censor denote censor cancer cause deal kind informative censor compete participant reality participant differ outcome censor alternative discuss htbp mechanism overcome mechanism equation far mean observation understanding rest restrict relation indicator participant sub model indicator form model time shape person age begin eq
persistent trajectory digit mnist despite simplicity represent run use sg starting explore compare sg gradient mini batch randomly onto evenly exhibit behavior positive gradient hamiltonian abc propose likelihood free build connection preliminary show feasibility problem large innovation persistent seed simulator smooth simulation landscape local case benefit hamiltonian
direct cause decade causal see theoretical development discovery algorithm divide disadvantage currently discovery inherent instability small optimal score base discovery subsampling selection exploratory search incorporation background experimental reliable structure yet far properly dependency cause act subject slice rich h line b c line edge acknowledgment receive european grant agreement attractive researcher recent decade divide two score disadvantage currently exist constraint inherent instability structure causal robust advance stability subsample
likelihood formulation em density state hold fix respect hold furthermore maximum respect eq q lower bind conditional expectation step low eq replace computing follow em decompose employ measurement smoothing distribution consecutive p sigma approximation expectation smoothing sigma smooth covariance sigma approximation depend sigma model function coefficient covariance contain model optimize sigma smooth value q get log alternative direct approximated sigma gradient place later call gradient demonstrate sigma estimation two dimensional growth illustrate compare simulated problem track measurement focus estimate sensor variance variance actual tracking univariate change typically quadratic model sigma hold parameter curve obtain sigma toward curve em sigma curve first exceed sigma evaluation sigma approximations sigma parameter grid close proximity figure rule height unbounde xlabel axis bottom legend draw align leave dotted width pt line round inf inf inf inf inf inf inf inf inf inf inf inf inf inf inf inf inf inf inf crcr color crcr plot table crcr color forget sep crcr axis solid forget table row sep crcr cs dotted width forget table crcr nan rgb width scale axis xlabel name axis bottom leave solid line forget crcr color solid forget plot crcr solid forget solid line forget sep crcr color line forget table row sep crcr solid forget plot
fast shift obtain mode kde mode split homotopy tend major avoid dimension unlike disadvantage mode cell minimize hard assignment mode assignment add encourage neighboring affinity nm program variety solver cluster obtain kde centroids soft assignment unlike spectral equal simplex two dependent centroid assignment posterior probability belong hence laplacian see spectral obtain probability cluster maximize parametric laplacian assignment optimize follow consequently unlike cluster assignment helpful uncertainty use use smooth conditional field centroid kde mode centroid valid pattern representative typical pattern disadvantage valid may manifold digit somewhat denoise typical yet area representative kde mode indeed implie equal weight mode individual kde local average remove thus achieve form denoise compare cluster centroid model nonconvex whether density whether assignment mode em centroid valid valid valid valid depend yes yes yes extent density yes yes yes assignment hard soft illustrate point effective ms code dimensional application point location intensity color texture pixel feature scale approximately spatial feature example intensity correspond intensity white affect carefully spatial feature introduce spatial coherence nearby although sometimes one euclidean gaussian bandwidth result color mode mark pixel ms
often value range depend signal euclidean match normalization recommend post vector robustness importantly many block feature time fit audio characterize include gmm negative nmf decrease redundancy processing post step straightforward present detail extraction audio far scope select present speech
illustrative mnist digit face mmd perform density hold generator apply number illustrative generate seed digit digit produce approximately hour kernel newly digit furth iteration mmd mmd right mmd iteration axis later train mnist choose activation radial rbf evaluate rational laplacian kernel find rbf parameter use rbf neuron suggest digit perform minibatch resample
datum set data survey bank record region value exclude process whole quantile omit consequently single class know range correspond proportion single region stand count character occur make transformation possible imputation model result collect interpretation purpose
parametric use source know sample run define parameterized source control construct yield source uncertainty output accounting lack sampling allow infection rate infect percent population peak epidemic percent application quantification scale utility ability require human supervision especially case parameter depend surrogate like future type numerical support energy contract ac grant model author national laboratory university provide valuable comment release la pt px quantify prediction make mathematical model broad quantify determine identify affect
therein method objective among easily implement objective problem objective objective belong convex pareto front thus change objective appropriately front drawback pareto frequently spread pareto point pareto point weight pose np bi sum tackle transform bi aggregated convex objective call optimal general spread weight approximation pareto front obvious nmf extreme closed form solution drawback moreover nonconvex make propose iterative optimal vector approximate pareto front aggregate become objective form expand expression term nmf nonconvex subproblem alternate keep element fix
modify merge tuple terminate throughout w I merge likewise see modification I total demonstrate utility file adopt slightly modify generate vector da standardized measurement generate see finally solve splitting plot pdf backward backward fista optimal
th thick obtain global presence noisy several going point propose optimization outperform plain original national project ref acknowledge plus var approach mm evaluation application integration strategy et des france em et universit paris france
identity rewrite side back complete section scheme exchangeable family generalize sequence locally countable every arrival token sequence concentrate equivalently law clearly exist also say induced say scheme exchangeable sequence characterize parameter process one fix measurable exchangeable computer one generate measurable bernoulli produce rich unique token event token let concentrated process imply array process therefore begin characterize j define straightforward infer characterize array highlight distinct role play atom bernoulli nonnegative measurable family nonnegative characterize law variable component let variable exchangeability condition zero finally distribute variance sum follow corollary measure beta completely ordinary informally sum give complementary identity representation chinese restaurant equality second equality second class representation representation perhaps type recover stick later calculus chinese restaurant stick break construction representation follow identity stick break
step concatenation tensor concatenation common test datum tensor common assume index core n vector svm nearest nn training core tensor university library extend use initially pixel
substitution theorem hellinger divergence let ground truth divergence obey hellinger divergence put way notably divergence measure identical long necessarily exceed inequality utilize follow besides claim sufficiently large precede b vertex type whose whose edge ease color white feasible contain white vertex blue cut develop intuitive understanding notion consist vertex area blue cut lie boundary cut lie boundary singleton vertice white keep read start examine combination color exist vertex argument thus combination take imply n distinct way select vertex assign color cut piece type vertex color cut uniquely determined whole regard remain cut vertex generality reveal black whose exceed black suppose instead connectivity uv inequality contradiction vertex reveal degree color remain vertex already suggest follow fact uv white black vertex color color unless color reveal remain situation black type hence connect neighbor within vertex color color black vertex result color say long white
subsequently model choose representative component map particular temporal area business city difference peak equal height peak business peak week capture highly center convention week fig high activity outside international customer daily peak activity collective activity drop people city business analyze help traffic pattern demand times city cluster product self complete guide job book book visual book great book rise transform community life technical book begin database business manual software book history world american vote perfect plan dataset extract use dataset
activation ultimately recall case j encode form trend constraint cifar figure clearly sigmoid trend activation sensitive suggest explanation relu regularization relu figure gradient every relu gradient result lead poor relu propose regularization constraint interested objective activation equivalent objective section sparsity plot cifar sparsity trend plausible explanation coefficient bias surprising part trend relu relu generate
strictly well fail overall security give formal theorem n space case follow sample chernoff completeness produce verify respect scheme suffice build adversary static challenge security convert challenge many explicitly adversary message space sort permutation challenge nm I j j jt ls b r choice message adversary fix notational simplicity probability advantage least term assumption contradict natural pac able example concept access query error oracle value cx tolerance polynomial query issue statistical run investigate pac learnable learnable polynomially learnable concept class inner concept class formula assignment efficient base elimination pac learnable learnable either good learner theoretically unless concept share pac learnable learnable learnable complexity interest hardness hardness learning concept learnable polynomially two target place nearly good place learner use comparison parameter bit public learner key
make dependence explicit bounding apply triangular note proceed get section derive shall detail arrive eq combine latter add bounding imply definition l repeat triangular equation use second term analog bound arrive difference study second bounding derivation l l finish
carlo ess mark whereas assume monte measure use cut identically distribute mutual give cut respectively also describe respectively approximately consistent pdfs estimator distance mutual finally estimator evaluate relative implement carlo quadratic cubic pdfs band respective row
impact score report decode likewise generate recurrent network although substantial phrase adaptation dependency parse leave behave differently syntactic project learn embedding plot vector allow briefly relation modify furthermore relation quite capture intuitive ambiguity divide line tie kind stack stack neural stack top stack stack feature although stack latent author predict action neural dependency manually global recursively head network chart parse demonstrate phrase read obtain generating
word closest cr l pos dependency google gram embedding pos cnn embedding word position word embeddings cr cnn embedding word embedding publish c e cause cause cause cause cause cause part comprise contain entity put entity origin derive member collection political numerous product representative cr class trace back responsible score summing position create result sentence appear sentence value sentence increase score
article relationship triple relation section existence triple fact e entity relationship play character science movie via set triple actor play combine triple entity subject object type represent sometimes example thick vertex picture vertex star right sf star sf thick fill white font fact actor thing constraint another person thing triple encode relationship interpretation exist triple triple example edge star movie associate either example miss star star approach actor movie complete cast place birth attribute might think would typically semantic web world discuss local baseline bend vertex label usa vertex intelligence pos font pos font usefulness influence special classify kb group triple manually triple create manually group triple extract semi text wikipedia regular triple automatically text language l auto auto elementary auto construction basis lead accurate wikipedia place birth attribute people though schema recent find wikipedia consequently automatic base construction method attention divide wikipedia high project include however cover try read extract fact language page reduce automatically combine knowledge extract entity kb entity
similarly statistic sum tuple modification example plant light plant plant therefore plant plant change plant statistic sign plant affect significantly manner solve informally need light statistically lemma tv square chi divergence write yield uniform contain flat equation rhs v rhs ex theorem axiom partially
learn weight label ensemble logistic ni low two prediction priori priori training require label observation lipschitz numerical relate find perform highlight finding regard suboptimal exploitation click recommend feedback costly availability action feedback include weight dataset arm since reward hybrid click average recommend item high identify relevant action high reward play table percentage type exploit many relevant high bc relevance ni assign assign em action high rd type rate recommend formalize prediction recommendation etc take stream big relevance context action sublinear regret time iii select suboptimal via numerical simulation show high identifies type variety include application medical diagnosis recommender interesting research learning context action bandit action let type inaccurate ta ta event pairwise mean accurate ta candidate ta type action candidate relevant type type exploitation theorem give ad tt reward chernoff ta happen lemma
factorize multi design compose view interaction partial interaction order theorem corollary theorem yu learn laboratory medical available disease diagnosis reflect person view view provide complementary expect learn predictor name machine interaction view factorization
minibatch initialize sample batch record update parameter former hard project entity embedding minibatch regularization penalization entity embedding embedding capacity relation regularization relation frobenius scheme loss e r big flexibility soft hyperparameter scheme implicit entity entity h case various competitive respect several benchmark version art database use r example protocol experiment experimental evaluation metric result disease metric split fold cross last available triple triple fold match fix validate epoch criterion randomly choose keep validate every epoch subset generic fact triple entity rank triple replace score procedure repeat mean entity rank raw setting triple rank rank remove triple set target call filter triple epoch generate positive unit triple entity head corruption implement knowledge unobserved triple use previous triple filter filter criterion work lead test split
video category video video report sift employ facilitate computation approximate kernel make comparable replace rank rank evaluation compare annotation figure show experiment gap densely annotate performance much advantage stronger close experiment bottom video clearly interesting note video character background video outlier annotate one green annotation detect indicate positive video rate chance scene attribute etc consist category attribute pairwise annotation collect label majority average label e compare mean image annotation extremely sparse colour feature alone belong scene dataset design attribute except rather class amount annotation annotation visual inspection annotation outlier perhaps relative attribute simulate situation random comparison outlier lead extra outlier original dataset attribute dataset experiment vote well scene probably familiar face scene scene vote wrong many value image vote outli qualitative result dataset
perspective work kernel approximate linear conjunction nystr om random fourier translation kernel sample unlike projection induce representation relation feature rate focus kernel integral represent transformation unlike invariance kernel integral induce closely descriptor build introduce histogram correspond cumulative distribution neighborhood image consider composition haar convolution specifically show capture learn linear generalize unseen point validate near simple algorithm correspond
inequality apply prove algorithm mkl please multiple extension particular kernel elegant constraint algorithm elastic net algorithm implement extensive piece depend library g elastic net code depend library wider include source library
follow hold opt opt opt q sample prove opt opt eqn invertible bind substituting follow matrix invertible spectral need represent eqn fact value expand conclude proof get value get next guarantee new component bss q lie entirely constant bss ridge matrix svd hand inequality z set ridge eqn feature method ridge
manuscript proof defer summarize region nonconvex information obtain choose bandit r set interested bandit desirable optimisation progress dimension additive refer e treat element dimensionality index g time notation theoretical assume handle unknown decomposition non
enough reliably determine treat hyperparameter predictive via grid hyperparameter sensitive particular setting use chain burn assessed parameter assess likely understand illustrate curve section perform obtain prediction examine test time fold patient time particular series patient fold predict fold part plot baseline merely average patient shape compare performance common separate generalized linear scale patient scale section use link high model label make patient look patient median function patient scale patient figure roughly expect time prediction solely smoothly increase scale guarantee realistic practice figure patient scale interpretable patient encourage pointwise
category please fig comparison precision category h dataset cca sm cca wikipedia average image text cross media challenge focus effective media retrieval model image text use propose projection medium optimize linear mapping gain retrieval extensive sentence superiority state dependent media media retrieval image text exist medium usually one couple text content use projection different
statement argue closure additional prof chain closure denote result novel base decomposition layer computationally guarantee sample input neuron tensor parallel dimension comparable parallel backpropagation future extend great explore discriminative model thank point bound support nsf award award microsoft nsf award nsf award award award additional require material form tensor consider mu u multilinear three main piece detail recover analyze fourier vector recovery finally follow expand use lipschitz short third operator wise function abuse order derivative take rd linearity layer
section parallel random approximation section offer elegant example kind optimization demonstrate high theorem definition conjecture frame mm operation decomposition world memory precision one randomized make computation write review randomize differently derivation implementation people algebra computation algorithms manuscript user implement reader follow understand randomized selection square eigenvalue approximation nystr key modern inversion etc memory expensive limit computation apply past decade impossible randomized comprehensive rigorous theoretical linear algebra difficulty implement differently manuscript implementation reader familiar basic algebra little manuscript describe provide code understand easy translate language provide matlab code manuscript cover topic review algebra familiar generating problem arbitrary section introduce
role powerful representation translation conjecture far verify address intermediate layer analysis translation short possibility address write hard need initialization thing layer read always help unit unchanged observe architecture design notably address write stage address address prevent transformation temporal propose architecture sequence build architecture stack improve verify machine translation task thm lemma prop thm definition conjecture remark architecture learn tu li liu institute chinese sciences ac cn tu com neural novel
linear specify deep joint latent observation appropriate condition deep network model factor suit backpropagation view encoder decoder present combination inference auto encoder plus low rank true open variational examining allow use limitation methodology due even posterior family distribution highly flexible contain true path normalize flow describe sequence invertible mapping change flow invertible mapping distribution basic mapping inverse transform variable distribution result q equality invertible arbitrarily complex successively apply successively
label simplify model information implication imply adversary view membership belief discuss implication limit degradation incur class simplify tractable allow performed consider eq parametrize privacy mapping parameter model require histogram approach implement suffer curse dimensionality exponentially allow computation order optimize genetic fitness fitness proportional evolution genetic operator reader motivate disease control health survey body measure portion mass individual weight status category consider weight status age subject category age gender since age inference status different depict percentile svm sense category rest treat category classifier treat category datum point svms training pick good boundary classifier vote class agree confusion
text coherent artificial comment firstly understand want learn move well task real world secondly release simulate world compose location etc operate entity entity internal state object box actor size color nearby place lie east encode actor pre actor execute randomly actor execute simulation consist object put object actor drop examine impose action something already place location already actor constraint define actor act need g drop go office question question world look text lexical employ automate assign get pick take drop replace drop discard put object actor crucial language example far entity vocabulary typically
work note random instance random event pseudo function threshold distinguish strongly ds randomize label c c x nk section clearly degree assignment tuple degree k kp pz kp step define distinguish consist problem efficiently reduce decompose deal replace define coordinate ds clear form connect dot r n dd specify check choice hold enough
manifold recall manifold frequency sphere kernel control frequency need excellent localization property come spherical frame perfect furthermore isotropic spherical domain prove possess approximation aim present article theoretical method spherical excellent localization fast totally localization attribute besides generalization world estimate possess us bridge capability behave least distinguish regularization aid probabilistic excellent property kernel attain provide accuracy capability regularization depend
newton sequentially stochastic notation respectively likelihood amount marginalization gradient state marginalization carry form compute eq logarithm brevity two hessian approximate estimator
cm hz frame rate achieve video throughput operate bit dct case yield dct embed processor pixel eight dct flow example ghz consumption total power consumption circuit consist dynamic tool device fig fig estimate area metric circuit speed circuit speed concern offer area architecture purpose multiplier directly bring comparison brevity aim algorithm ai circuit table provide cm cm cm al et cm propose architecture yes yes dct single dct single dct dct single dct multiplier operating n technology quantization yes yes yes buffer pixel cm et cm propose architecture design yes yes structure dct dct ram dct ai yes rate technology yes stage transpose buffer pixel hardware
guarantee weak learner hypothese initialization call weak ti make call weak learner learner agnostic access realistic agnostic learner achievable idea boost smooth agnostic weak rate give inequality utilize smoothness h error alternatively adapt technique adapt boost claim base boost turn boost communication complexity iteration original boost result projection distribute boost desire
addition extra great deal attention allow virtual agent refer mapping linguistic external manually mapping environment statistical language learn symbol observe natural language define linguistic semantic model language corpus correspond acquisition treat map language prior linguistic employ semantic alternatively specification carry agent language sequence alternatively frame language context learn production upon weakly meanwhile convert
structural noise substitute parent express know test image bivariate case special inference case fit fit estimator analogously backward direction function well residual noise likewise strictly speak use converge additive comparison mutually far additive noise model kernel characteristic proof appear interpret imply contradict identifiability additive sample estimate function speak estimate estimate show converge additive reason causal identifiable thus
constraint target variable boolean ignore edge label potential constraint index consistency lagrangian computable lagrangian unary potential submodular lagrangian relaxation problem piecewise smooth computed well low family maximization eq gap effect happen exist label subproblem contain index ip ip solve minimize consist energy label equivalent minimization consistency constraint us order potential set select potential perform reduction special propose review eq minimize cut add dual lagrangian l dd low energy analogously low technique generalize order formulate indicator min transform via add variable submodular variable nonnegative introduce lagrange multiplier absolutely high order potential potential cardinality correspond gain variable take stand allow select truncated operation outer scheme deviation substitution context c
mistake typical manifold hyperplane procedure less interesting one summary statistic experiment mc smc correctness histogram along three jacobian simulation ss threshold ess ess fraction effective ess way detect particle particle discrepancy option optimize reach variation
know ucb thus budget stay probability ucb ucb go divide regret error regret error difference context supplementary theorem ucb boundary regret may dominate action pair insight unit system heterogeneous limitation detail find statistic bandit roughly good j jk expected decide action context solve constraint one context show boundary boundary regret additional reward ucb ordering ratio case I ucb reward open design
intersection challenge vector track test track facilitate development mapping token difference run allow leaf node l content train test basic full per track dataset category add non decide keep decide move multi track first previously track extend track track dataset less
input dimension gp prior spike switch q change cross indicate variational spike switch variational part related field keep adapt kernel switch zero determination view simultaneously assume share aspect retain view view share dimension private share ard soft view share fig wish variable latent view
output require tolerance parameter runtime algorithm moreover choose descent rely minimization gradient numerical suggest ambient consequently refine recent hessian form minimizer minimum fall analysis guarantee broad nonconvex machine completion etc possibly achieve provable guarantee
sample help pac adaptation competitive imply generalize analysis pac theory give rise new deal priori paper informative consist center origin domain adaptation direction exploit available point learn objective address would kind reverse validation technique take particular link adaptation besides derive domain interest indeed consider take unlabeled distribution disagreement seem importance part grant european european grant agreement perform discussion part let mm jensen function convex q measure measurable set counterpart abstract abstract quantity kl apply logarithm side eq side last jensen inequality last equality obtained obtain empirical state four kl kl theorem empirical lie process negative choice follow binomial bm give choice inequality equality
dataset generic representation robust develop algorithm compositional open language year several developed operator word recursive recurrent method representation pass order composition sentence paragraph vector language model abstract alternative loss apply operator task sentence propose model sentence mind skip instead use context around sentence modify call skip skip thought training corpus contiguous book book book contain wide character furthermore towards learn semantic newly skip generic feature image benchmark extract skip representation
decrease ascent prove leave x round least increase total objective rescale execute maximize denote modify argument expectation hx argument average complete pick denote sample h x argument simultaneously slack constraint absolute follow constraint prove statement boundedness technique focus statement hoeffding event henceforth condition well objective value satisfie follow turn relate first optimal slack apply establishe feasible combine provide eeg news letter bank activity census dataset number average number hyper go quantity number input expression actual input hyper reduction regression active try rate active table c c algorithms lie interior different fewer include three fraction minimum strongly predict agnostic active may undesirable highly minimum dataset variant outperform algorithm em bb ccc cc microsoft ny ny develop stream severe erm newly optimization analyze conduct first experimental across wide complexity learn active yield improvement
recovery low matrix freedom recover sample additional observe accord proportional relaxation oppose give problem experiment far incoherent restrict leverage column incoherent without know leverage achieve completion leverage randomize accurately appear many domain incomplete rating product make prediction user product matrix sensor pixel image tracking failure surveillance mathematically tractable impossible unobserved element limit say freedom
reflect increase house political freedom house year country correspond relationship linear kb mb ht low ad house political ht vary ad year otherwise see correlation weak ad score partly house ad run year run year one run freedom house sample training instance scenario start scenario country wise notable ad show large positive difference ad ad large positive ad high mostly small little news hard country level ad repeatedly country true ad course possibility exclusive ad unbiased extent filter article political regime bias ad imagine biased favor news article focus ad ad ideally score hand rely assumption noisy et bias
theoretical nuclear formulation examine provide incoherent recover matrix relative bound top perfect recovery build condition regularize first proved completion least low rank recover low theoretical advance applicability include collaborative sensor learn unknown sample
path jump trivial coupling jump time spirit drive proof corollary coordinate probability couple detail study regret penalize bandit explicit notation summarize strongly sharp study important sequel division handle remark conversely derive upper strongly difficulty penalize armed careful term usual consequence increment easily n drift act sequence understand behaviour one generally drift deal q become bandit algorithm always good consequence remark contain second dynamic check q fulfil triple false mean guarantee weak uniform since finally note neighborhood opposite h difficulty idea introduce area properly element eq
draw ij gaussian template via word channel probabilistic heat patch shape patch similarity pass bias make heat exact scheme induce automatic convolution channel conv channel patch whiten e similarity mixture estimate patch via mixture effectiveness experiment run single layer convnet convolution similarity network second experiment layer perform publicly convnet aware third three layer design demonstrate compact load theoretical sec exhibit expressive machine simple become expressive elaborate burden overfitte
eq note small constant yx last display hold first inequality boundedness last due combine q k note sequence argument involve argument gives desire u tv inequality proof misclassification proportion bound proof sufficiently assume replace combine adjacency undirected statement es v x prove sufficiently us nod subgraph go universal lemma imply eq apply union node es last least give least argument uv bernstein inequality uv v c c nx uv op bind uv vx graph degree complete obtain column op jj jj op op p lemma control satisfy exist lemma eq q q imply constant op r op op r
dimension much illustrate straightforwardly refinement algorithm work show modeling activity bernoulli may circuit interact record cf active neuron address simplicity illustrate mixture component satisfie component belong optimize
produce algorithm anneal schedule sa keep correspond empirically list consistently outperform algorithm high map estimate blue blue smoothed median go gradually assignment fast map differ number way model combination randomize straightforward empirical
event post drug surveillance report induce event huge database describe drug reporting suffer association event association propose projection onto popular proportional reporting report odd detect event contingency involve co association none comparison contingency spirit conditionally sparsity drug relate drug strictly occur consumption influence rigorous computationally demanding validation
inverse gamma ig wishart ig univariate ig posterior see parameter wishart issue scalar use large eigenvalue determinant determinant thorough become dimensionality costly preferable iw single consider infinite merge iw class observation degree iw impact multiply learn ultimately back format conjugate definition detailed proof conjugacy weight statistic class infer adaptation emission main hdp jointly represent occurrence illustrate impact dirichlet knowledge intercept learn metropolis hasting mh jump use several study valid move sample accept acceptance p new accept current batch accept identical notational clutter parameter adapt differently datum respective distribution show appendix case drive infer follow closely extensively aim effectiveness hdp variety examine adaptive basic demonstrate challenging sequence datum challenge
core vary experiment size corpus core linearly horizontal axis horizontal core vertical illustrate transpose corpus multiple apply transpose solver synthetic record time amount perform communication diagnostic necessary optimization include use synthetic study consensus suggest penalty zero remain perfectly linearly rule case heterogeneous create heterogeneity test consensus essentially arrive quickly node source else consensus logistic trial core per homogeneous seen behavior transpose guide star million star measurement spectral
costly big generalize idea component iteration result sparse strategy factorization upon column solution factorize approach decomposition setting employ subspace set base successful example include become application parallelism propose datum graph format notably computation execute parallel efficient densely furthermore tool mostly vertice densely result dramatically slow design sparse call decomposition behind ii sparse factorization find sparse sparse greedy omp alg must employ adaptive column method upon batch equal subsampling column input tolerance column initialize normalize column batch omp select normalize write q pursuit omp solver enforce fix fix approximation
description sample information counterpart include location shannon entropy remark size partition mutually exclusive first unit assign actual inspection select another assign unit draw quantify sample construction sample includes assign two order subset partial unit small rank assign unit within likely quantify unit h cccc subset throughout otherwise specify use denote suppose probability pdf cumulative cdf
construct illustrate family scenario light tail heavy tailed higher low size simulate distribution recovery mixture package package implement mixture facilitate mixture package implement compare additionally identity start initial ten algorithm success scale simulate three run bic select component time component respectively similarly component clearly family deal light tail time ari ari yield ari family value range contour select model example figure close true initialization mean initialization time select bic run hence explain observation multinomial mix proportion simulate dimensional lastly
categorization preprocesse remove experimental may whose cause interference experimental allow interaction target notable exception process maximum connectivity extend atom extend atom assign unique bit application similarity chemical fail evaluation validation imbalance present vary train fold inactive remainder dataset skew result affect fold however hyperparameter serious issue receiver operate roc evaluate roc curve plot discrimination varied area roc curve auc auc mean
condition focus var minute stock market datum obtain stock york stock exchange american express ba cat gs home pg unite former index american international group daily realize xt jt except half hour period start totally trading day exclude generate realize covariance provide show space property realize skewed rather realize variance covariance big except realize subsection estimate point factor factor show diagonal use comparison namely numerical time var aic schwarz final ft
tf choose remove construct furthermore regularizer criterion gram slow regularizer hyperparameter experiment option combination huge actually strength tolerance value exhaustive search trial preprocesse report unseen hyperparameter weight tf tf remove stop hyperparameter bottom five categorization stanford sentiment sentiment movie review website task goal neutral review review amazon review amazon text summary section review obtain movie sentiment movie review vote
requirement stanford edu com intensive intensive embed training start architecture limitation order magnitude accuracy connection redundant connection three connection tune connection imagenet dataset million find network reduce network vision recognition language vision handwritten digits imagenet competition face et scale b considerable memory bandwidth mobile resource become prohibitive
iterate smc algorithm tb line indicate either sis sir respect lebesgue
f stationary definite twice continuously k assumption denote dominant test statistic statistic rate powerful standard test fact might necessity question statistic good v iv stop statistic assumption correct I z n hypothesis critical conservative would remove test step unclear related overall choose evaluation run significance level test interpret edge test stop missing lowest likewise low likewise miss statistic stop final
algorithm projection enhance big price reconstruction algorithm iterate general formulation multiplicative variant alternate nonnegative particular minimize end eq find propose representative respectively set multiplicative update nr n r nk multiplicative update nonnegative least square achieve reduce fast solve framework compression gpu equivalently define propose q significantly employ whereas former simultaneous sided may great however study behavior light get challenge compression easily multiplicative framework version greatly reduce communication cost implement large volume nature execution million row matrix separable art technique computing approach extract index literature refer column literature focus separately trivially basis key find extreme column trivially notice qr column nmf low
impose constraint typically constraint term function choose penalty kullback divergence quantify mean hide hidden penalty move effect previously hyper add third cost weight decay reduce overfitte hyper control decay autoencoder randomly extract filter spatially patch training patch train overcomplete autoencoder unit patch patch training patch mini batch divide mini mini batch minimize batch weight
appear interested ad finance range ad finance ad b interesting show ad simple user category number finance interact finance video percentage comprise average validate ten interact three ad study therefore advanced mobile use user investigate specific ad user future work record ad static user user might interact prefer ad historical ad ad user ad could style user day profile number ad interaction finance ad video example rule play finance confidence lift class play video support lift aware play finance video lift lift index rule lift mean user
predictive predict accurately high low gain substantial multimodal low mode concentrate wind speed second lower skew thus nc shifts probability mass predictive tail predictive accounting framework mixture beta rely flexibility beta achieve combination mixture use calibration allow component property methodology show adequate multimodal heavy stock return wind infinite beta provide calibrate forecast predictive calibration combination functional work mixture dirichlet parameter bayesian nonparametric calibration suitable multimodal density density forecast daily wind secondary forecast forecast beta bayesian forecast statistical source address one estimate forecast point forecast
benefit never moderately regression solution vector compress interpretability furthermore original guarantee optimum projection map attain solution clearly define project define importantly original assumption new back obtain dimensional worker similar guarantee overlap across maintain dependency feature overhead rewrite explicit let sub raw block block preserve feature worker concatenation raw feature worker
underlie kl graphical model although ise would lead necessary picture estimation since structure large research technology grant anonymous feedback mutual claim nonnegative finally integral kl integral equality satisfied conditional kl equal begin prove computation provide completeness normal distribution respect begin covariance
filter time kalman skew skewness kalman filter smooth kalman kf linear filter optimal normally tail real kf occur smoothing outli measurement many involve tailed distribution cause error show histogram distance fit distribution bayesian
condition available scale law theoretic study square compare largely early system adopt work significantly analysis similarly entry exact latter fail fraction criterion compare understand limit provide base contribution advantage support recovery apply specific focus law result necessary sufficient measurement multiplicative thus number measurement thresholds general snr work converse converse necessary distinction converse refined argument ex model sufficient necessary partial discrete eq model snr q bernoulli noisy recovery bernoulli eq specific overview state asymptotically term remainder discuss contribution case threshold handle broad see detail converse near necessary snr scale law bad behavior partial corollary various sufficient law sublinear scale form moreover sufficiently threshold noiseless necessary proving measurement prove claim observation slightly remainder generalization formally converse apply model proof bound draw letter case bold character collection scalar index submatrix contain index symbol variance usual mutual counterpart differential continuous distinction clear q asymptotic notation ambient dimension subset cardinality observation number support entry entry entry accord specific common theoretic support recovery realization decoder deterministic successfully recover follow fact remove clarity list subset measurement
node incoming parent control flow parent child one parent child flow place execute left execution bt begin execution parent status execution yet selector execution execution selector selector node child return success return failure child return return selector child selector box fig return return failure success child return child run box h child success success failure run node return user status child perform return complete failure action complete return run
isolated point mail cn cn fix almost uniquely encode representation word sequence position work feedforward neural recurrent outperform recurrent role traditionally back gram language model art model feedforward recurrent neural rnn layer word smoother generalize lm usually limit fix language lm
verify fail sense capture maximal know learnable sparse failure learn miss seem hold estimator family empirical define plug learn respect mass gram prominent transition never corpus likelihood likelihood truth essential ingredient propose gram tail language often heavy tailed rare hard tail family lead parameter dirichlet worth part contribution beyond scope soon inference technique closely
manual adjust split step change car step evaluation text majority localization text nlp vision conclude localization improvement hc loose car nlp nlp nlp discover main video cluster benefit plan text automatic recognition could research google grant appendix explicit video sequence text video stream analog stream encode recall contain video explicit yield bias consider output label note convex imply q convex minimizer case loss equation simplify expression strictly quadratic interact use frank wolfe localization solve assignment video assignment object alignment optimize constraint concept text stream video
pose estimation generation mainly due achieve recently net prove segmentation perhaps invariance field mrf refinement boundary seminal inference mrfs potential performance demonstrate segmentation later fashion unary weight parameter enforce able train potential dependency effectiveness describe deep take account interest lie domain sample model configuration maximize maximizer monotone normalization concern training
bin logical long train evaluate training bin training us fall cutoff reasonable decay gradually build model tree leave right feature predict seven relation sentence model force meaning need neural network layer softmax layer plain recurrent
anomaly degree anomaly versus describe anomaly detection search outlier distinction requirement benchmark tail generator develop large expand ensure prevent overfitte gain insight strength ideally identify anomaly might control dimension difficulty measure anomalous outlier become hard happen target extreme extreme anomaly normal aspect anomaly adversarial detection point experience difficulty refer process anomalous knowledge oracle value anomaly discover anomalous semantic anomaly detection anomaly kind attack cancer cell benchmark highly process correspond generate benchmark construct anomaly create phenomenon relative frequency incoming anomaly issue detection contaminate anomaly anomaly long reliably easily frequency contamination anomaly relative anomaly rare may anomaly anomalie high establish irrelevant performance irrelevant believe irrelevant great anomaly perspective exponentially dimensionality surface volume contain increase geometric lie normal fall application tendency increase irrelevant feature label user
iterate converge set tensor vector entry w I I equal standard lemma plus minus em width depth smoothly vary generalize exist decomposition penalize
quantity formulation prediction formulation metropolis hasting approximate treat solve sampler strategy model kalman associate primary decade start rapid constitute compute unobserved gaussian solve arise find cover aim key principle complementary bayesian maximum section describe augmentation strategy arise self implement several essentially way systematically key deal identification problem treat paper among first amount thorough see secondly amount theorem q discuss identification slight abuse formulation early available use convention result fact sequence obtain via marginalization close expression expression whereas likelihood state sequence express q tight state development work identification couple
microsoft activity capture structure light researcher extract feature sequence extract location show person relate reach placing complete relatively short sequence activity usually sequential divide segment segment approximately manual annotation automate temporal extract world desirable whereby activity usually duration activity address separate task need infer contrast paper activity unify activity simultaneously demonstrate beneficial recognition case activity label action versa refer activity gray refer use sub consecutive enable rich global graphical representation recognize augment hidden training contrast able variation imagine represent difference temporal preserve latent avoid make graph whereby consider structure apply inference margin discriminative thus provide
assume order moment component write variance estimation equation asymptotic large small whereas bias also dimension indicator simplification derivative equation compare assume direction derivative compare minimize bandwidth kernel derivative actual pdf multivariate bandwidth multivariate gaussian bandwidth standard deviation infinite notation take twice equation simplification vector solve integration use equation rewrite dimensional multivariate equation rf simplification operation equation gram series
fourth mean representation view hide representation dimension representation view autoencoder help view representation third view alone fourth sure correlate hide unit representation view sgd parameter mini minibatch approximate three unit hide iii sgd hyperparameter hand tune using exactly compete algorithm hyperparameter objective function error hyperparameter rate view potentially view instance observe often view text amount datum parallel available abundance single consider suitably modify match conventional autoencoder type set mini batch feed batch random order perform objective obvious allow network view train step modify connect connect view common view
polynomial markov factor query oracle improve relatively language distinguish whether unknown test square regime size nevertheless modern customer web quantity size fact phenomenon dataset empirical genetic find separate discrepancy mutation cite explain difference importance statistical test asymptotic available go infinity significantly small size surprisingly despite statistic science hypothesis question remain extremely consider largely
prove p mutually ep notice variable brevity demonstrate thresholde independently take error relation choice plot show decrease thresholding increase increase continue relation curve plot instance relative average effect fix generate independently depend error case thresholded lead poor sparsity average relative increase sparse gaussian exponential thresholded achieve high parameter vector constraint sparse approximation low recovery
amount time time add bf child change time training linearly retrieval sublinear misclassifie store tree requirement store bf property show zero pass data set pseudo boundary tree study bf retrieval within hypercube qualitative mixture orientation mnist picture hypercube unlabele interpret intensity vector preprocesse present line law half separately scale try fit bf tree maximum point bf return train close
scad scad sis scad fr scad lasso scad scad sis scad fr fr scad scad scad sis fr scad sis scad fr fr lasso scad lasso scad scad sis scad fr fr scad scad sis scad scad fr scad lasso scad nine show supplementary material method set relatively autoregressive correlation likewise complicated factor scad small scad negative follow closely fr scad positive scad bic reduce although gain note speed scad scad sis scad efficient term speed lasso improve attractive efficient stream expression week gene interest code
coin monte simulation set respective entry use simulate comparison time good score behave unlike respect different algorithm return winner single correctness insensitive complexity winner half become lastly data support argument winner winner winner winner arm rank second state technical lemma completeness stop divergence arm bandit free number upper alternate cumulative upper kl position prove winner bandit pac
spaced divide bin cover bin bin average bins bin classify region leave operate query domain query linearly let bin e suffer bin except classify correctly probability indeed appendix hoeffding bind bin nh cn ne epoch round optimal passive round restrict hence inspire similar let epoch radius budget label return recent sample radius repeat run stage depend initially passive behave noiseless become behave since
annotation scenario relaxed consider semi acknowledgment jointly fitting classify set feature survival condition naturally label order survival vector breast cancer approach cox predict
much conservative justify surprisingly perform regardless mini batch size fail applie exponentially evaluate normal problem auto stock price include discrete adopt augment subsampling gradient langevin dynamics correct tune adjust posterior real control apply exchange composite index consist reduce magnitude consistent runtime subsample fix sub sampler empirical adjust posterior run auto obtain twice bias scale graphical database scientific cluster author random
process proceed reduce compute divergence give gaussian simplicity compute convolutional cart help overcome pixel link still decoder high generation thus decide deconvolution network powerful mirror encoder step decoder architecture convolution use simple enforce previously representation video representation enforce similarity divergence eq seem coherence similar latent depict main paper term suffice space linear prediction control feasible compare variant baseline reconstruct basic necessity reconstruction planning must correct transition encode coincide planning action achieve goal locally trajectory necessity successful latent criterion reflect reality reflect reconstruction current cross entropy function
zero yield give claim decompose subdifferential regard mean decompose statistic f exponential addition unbiased x jj solution inequality rf q u technique proof bound school school title aware task erm classifier gender method low guarantee unseen sample exist learning design diversity theoretically analyze propose general
order explain fact slight marginal graph value equivalent margin network algebraic represent model clinical clinical circle mm xshift b therefore trivially lead latent result dag treatment arm determine function causal purpose mathematical question outcome herein function draw apply precisely evaluate ultimately sum combination evaluate return represent induced brevity upon think clear choosing ignore uniform contain set tangent cone cone upon uniform vector tangent cone uniform follow degenerate formulation result cone delay end dimension degenerate valid tangent cone contain tangent around ce q result show appropriate hence e subspace tell model tangent latent adjacent connected multiple
memory guarantee meet family tackle spectral block family solve optimization sgd streaming algorithm interestingly standard sgd minimize reconstruction guarantee general turn trivial contribution family namely extend batch block key block matrix serve batch mini sgd sgd block pca converge spike gaussian generalize broad spike block streaming dependence make difficult
peak favor advanced separation mass discard share peak even may large percent likewise discard datum instance two specie discard lose strong distinguishing closely fast dramatically improve product discrete pmf unity convolution store convolution step exploit convolution polynomial multiply product enable represent represent unique turn permit elegant storing pmf runtime convolution nature partly implementation optimize heavily retrieve information exponentially possible outcome inefficient either become fortunately propose decompose sum pair discrete
evolve differential constant define internal follow internal move inside square pseudo distribution
conditioning state unit child child lstm application correspond noun phrase suppose case advantageous emphasize phrase forget preserve parameterization child forget gate parameterization flexible propagation child allow hide state binary forget gate child large impractical tie apply lstm child distinguish receive leaf paper dependency tree key tree child model lstm architecture wish predict tree tree correspond phrase span predict input node subtree root negative log label node
act maker perform observe space ultimately maker act incur abstract development wide corruption henceforth work markov eq markov parallel kernel restrict finite matrix composition co n e action action size supremum l normally loss experiment problem normally observation know replication experiment grow quantity le subsection readily main distinction one focus focus sort explain supervise special letting supervise instance label map
throughout analysis expand intensity basis expression eq limit eigen function normalization form eqn information frequentist criterion frequentist paper provide simple application simplify analytically mechanism give phenomenon selection motivate justify narrow predict observation model absolutely complexity dimension brevity simply quantity amount character give character value neither mathematical scaling change result offset convenience base cross
alpha observe often fdr regard total introduction definition fdr procedure online alpha control procedure mixture evaluate alpha section defer order nan concern generality assume denote let value hypothesis indicator hence discovery incorrectly criteria relevant hypothesis eq false reject false rate control realization list line increase effort reduce risk research finding publication bias lack multiple comparison paper share database preserve share amenable testing maintaining pay price form sample add depend correspond control
stand eigenvector dual u see detail find eliminate prove simplified problem nice yield continuously necessarily twice adopt dual quasi bottleneck calculation equivalent sdp discuss contain equality key contribution semidefinite low large crf propose low memory requirement compute semidefinite restriction feature dimension quality approximation depend eigenvalue kernel rank approximation norm norm inefficient computational achieve linear nystr om incomplete fourier feature homogeneous please adopt nystr rank approximation matrix nystr method
belong argue rather point efficiency cluster polynomial furthermore run robustness metric datum actually stability property hardness center hardness cluster far restrictive requirement datum ball cluster ball shape become space consider cluster connect center fail strict requirement exceed cluster upper shrinking cite actual constant parameter significant result word benefit parameter demand help severe requirement mean input cluster point center readily subset satisfie let finally satisfy condition separate surprising aim pick stability degree comparison discuss target approximation target trivially meaningful vast strong probably condition result initially stable
langevin unfortunately store prediction describe monte carlo approximation form deep expectation perform well simple implement time deep network art prediction confident bandit fusion principle tackle nx th output predictive py estimate nn plug unfortunately ignore py x unfortunately confident reinforcement rely accurately approximate bayes variational vb product ep approach inference predictive ep
minima possible generic rank eq surface sphere dimension estimator nonzero gauss signal likelihood trivial mmse behave exactly zero e amp reach prove concavity evolution stable becomes log develop tell enough point mmse
reader simple implement zero z est ct est est est k subroutine access optimization list recurrence think possible way define problem recurrence weight proxy thank obtain hope terminate reasonable reliable require depend path motivated work use appropriately replace rule variant provide algorithm ready guarantee satisfie eq expect regret main contribution regret arbitrary least satisfie bound attention note efficiency crucially availability problem variant straightforward
share domain thus benefit bridge minimize preserve emphasize useful essentially real mapping autoencoder vice versa flexible learn channel channel share layer denoise capable application bridge channel datum channel real input meaningful keep channel attempt discrepancy cc autoencoder autoencoder network decompose process instance transform affine function kk decode parameter autoencoder attempt reconstruct impose mapping follow nonlinearity q view normally impose input short autoencoder synthetic decay add generalization autoencoder importance mapping autoencoder
hessian ii gradient likelihood derivative compute convergence criterion log tn seem relatively large three criterion simultaneously criterion ensure drawback converge stability setting latent mixed latent relatively flat systematically matrix package include currently call estimate univariate model possibly latent multivariate possibly include latent rely write systematically function mixture ng argument define linear regression covariate side define covariate variable latent membership rare argument whether indicate proportional stochastic default include argument name longitudinal format management omit default provide threshold indicate maximum iteration algorithm structure ng false nan action detailed argument specify nature rescaled cdf probit spline knot knot default knot knot internal argument use rescale beta indicate consider spline beta finally output call ng datum na action name hand include covariate marker link link name vector link function knot manually transformation spline cdf marker consider measurement call ng survival hazard specific nan nan argument call define class specific survival formula define survival package survival specific cause cause cause family baseline family cause risk presence compete program hazard
intervention variance financial series clearly stock block connectivity intervention strength intervention market technology correctly dot com american proxy p european proxy cyclic assignment computational number bad identifiable strength furthermore j j p j latter repeat one entry dm invertible us row want show observe mm cycle contradiction conclude cycle product analogously sequence
sequence however compositional limit involve sensitive sequencing confirm compositional fast respect read level sequencing access genomic within hour reasonable analyse genomic sequencing give medium characterize resolution raw obtain throughput sequence dna analyze different goal estimate abundance purpose affect read unsupervised rely read operational individually arguably challenge necessary application notably aim micro assign read purpose alignment read sequence alignment tool li fast compositional learn bayes nb classifier label read compositional offer similarity computationally compositional typically free nb explicit train abundance
free spectral problem bethe plan write bethe energy inference minimum censor describe algorithm provably recovery remarkably require knowledge previous provide transition european fp grant france partially censor edge problem application synchronization base backtrack bethe
couple give inclusion lyapunov compact contain fundamental prove follow lyapunov stable exist ax tx hence lyapunov get ax x k max slow aforementioned solution substituting point wise need
identify threshold true discuss early fx obtain excess risk function learner similarly sign sec fx binary call efficiently set one need theorem query approximately dimensional convexity point yield dimensional line perform active threshold mean minimax quite regarded procedure design inspire dual algorithm subroutine epoch adaptive passive subroutine pool set access unlabele passive bound
high setting regime slow availability sample purely screen application discuss sec short regime sequel asymptotic c fix rao pearson chernoff yu wang fan address suitable adapt dimensional regime extremely big datum pose practitioner include computational challenge control regime surprising benefit law scalable complexity advantage regime mining neighbor correlation purely rate regime double accommodate order without comparative different various apply inference procedure supervise maintain level statistical similar notion description statistical tradeoff specific specificity form outline review definition partial treat screening inference task conclude remark distribution exist vector e define expectation symmetric semidefinite element assume generally component variable I predict without precision dependency sparse marginally component scale important correlation respective invariant element non along diagonal retain distribution p
tailor resp exist distribution universal integer defer ideally whose unfortunately direct difficulty disjoint use indirect essential lemma start pdf appropriately hypercube distribution guarantee perturbation consecutive integer ready hypercube follow similarly I inequality complete give explicit example agree similarly show elementary polynomial value root st moment agree variation differ imply variational suffice support first agree exist support support odd show polynomial root argument z nz z negative purely root moment complete da equal equal within additive random independent modify replace terminate component inequality use remain bc dc dd da error introduce time observation tv tv db complete lemma fix root list polynomial qx qx prove sum follow root prove root absolute coefficient write similarly magnitude induction sum absolute sum j claim
probability fc observe discriminative tree crf belong exponential family restrict generalize potential represent learn output neural potential node belief covariate configuration covariate framework parametric kernel evaluate randomize estimate cl serial version neural potential message multi indicate set edge tree condition use partition potential instead restrict potential generalize represent learn object occurrence strong wise object training section occurrence fc occurrence sufficient coherent
robust adversarial perturbation third classifier slightly one small control flexibility degree interestingly relatively easy classification task rbf svm good adversarial perturbation result gap adversarial classifier theorem maintain fig rbf svm train adversarial cifar digit natural computation cifar database contain image restrict report robust accuracy around robust illustrate digits measure predict low task classifier robustness instability classifier perturbation essence task correctly capture classifier use latter seem possible limit room existence limit adversarial
limited allow access mean look consider estimator sum sum sequential provide estimate statistic simply assign abstraction reality compute depend implementation language believe reasonable analysis generalizing allow define unbiased streaming mean estimate variance streaming moment independent explore great detail discuss versus quadratic streaming alternatively get tend essentially eq verify asymptotic
submatrix link connect local goal global anomaly least close centralized counterpart amenable function available estimate obtain space effectively reduce view belong span column projection subspace next consider follow alternative bilinear leverage follow towards obtain decentralized anomaly identification partition adopt separable regularization optimality less optimal stationary globally inside couple proceed copy represent local decentralized q consensus neighborhood carry agree separable admm network iteration unconstraine quadratic program refine exchange directly overhead remain employ offer however evidence non convex structure convex potentially extensive test demonstrate open attain global desirable centralized performance domain sense cr rf amount fashion psd capturing across cg medium identification available band activity psd interested reader cg approach basis psd spatially collect receive sampling determine introduce virtual spatial grid introduce narrow band broad locate active operational nonzero correspond band active estimate psd problem decentralize path variant cope due inaccurate channel spline psd estimator also capture psd span sub assess rate convergence decentralize outline batch local function static introduce agent arc represent indicate per adjacent set
lemma use therein complete prove operator know bernoulli want invoke mean z numerical bernoulli ij bernstein variance furthermore q n bernstein obtain proceed lemma operator plan c q unitary identity eq identity separable since zhang master china join department ph optimization lin ph currently key laboratory machine school computer science university university institute chinese science interest vision recognition area associate transaction intelligence journal zhang ph degree china post laboratory laboratory school engineering computer science current research interest process visual cn crucial completion mostly concentrate recover coefficient result special mc recover suggest theoretical model
discretized lrr ht explicit representation pseudo determinant implicit matrix remarkable relatively average size avoid safe numerical trace convergence respective representation fig matrix especially part determinant part interval add correction result instance determinant practice investigate influence huge advantage calculation single
normalize zero unity wavelet distribution derivative unimodal wavelet unimodal one cycle beta easily wavelet beta wavelet refer cyclic balance length causal causal piece instant transition second wavelet wavelet
overlap sometimes report contain select state threshold positive negative adopt pattern study generate share several signal distribute value generate tu pattern extend add signal given generate simulate item ratio snr divide low detail material fourth study suffer miss identify trait fourth almost reason parameter fit adjust parameter observe perform equally three often favor precision recall
reliable estimate observe realization variance explain observed illustrate aspect case event sufficiently accurate quantitative observation temperature reaction coordinate realization importance large value small realization n realization contribute reaction characteristic tailed distribution differ lot agreement statistical indeed fluctuation htbp c e average realization mention tail reaction exponential namely probability random particular heavy reaction coordinate since behavior explain possible go upper realization one distinguish go channel one go path trajectory channel resp precisely replica channel go lower need explicitly th let introduce realization realization divide associate reach channel reach channel reach obviously realization go realization realization contrary estimator reaction coordinate see approximately channel estimator realization go channel realization contribute average comparison htbp e e reaction realization reach reach channel observe realization experiment table reaction coordinate fluctuation admit close notice minimum since go reaction coordinate reaction coordinate channel around upper resample back replica may replica satisfie contribution leave reach even large see degeneracy branching tree small typically realization plot bp fact proportion contradiction quantity form dynamic estimator bx ax well realization interval agreement value compute reaction coordinate estimate bx ax reaction close upper take channel
aforementione question result quadratic analytic solution point pearson normal approach strict contamination bound contamination multiclass focus proportion extensive relate topic anomaly outli entropy contamination estimate lastly security development early identify anomalous behavior spike associate attack traffic component tuning provide significance present proof first theorem set empirical sample
derive first armed bandit base principle prevent pick maintain compute arm compute remove intuitive effect operation poorly perform eliminate loss property providing give little intuition obtain generalize complicated setting able come conceptual implementation variant place probabilistic seem either tool convex analysis current sampling suboptimal property much perturb equip enable bound choose vector cumulative perturbation study
sequence regularization gap need decrease signal cause need case decrease four constant value performance sensitive default generally example initialization bb evaluate threshold ensure unstable fail converge regularization challenge scenario remainder outside simplify terminology nonnegative step intermediate mapping goal close subscript impose numerical adopt signal limit sec limit sense iterative example ray projection operate matlab propose two category linearly toolbox herein aim solve parameter solve initial size backtrack without adaptation place u signal transform signal base synthesis square imply synthesis solve provide integer option keep default mean noisy vector noiseless design signal overlap shift triangle fig wavelet large magnitude achieve use initialize simplify show center objective wish bring
observation scientific inherently decentralize hypothesis group work access dataset aggregate maintaining identify imagine intend cause research dataset hold aggregate institute hold behavioral million individual common part communication formalize problem design response interest nonzero contribute return discovery fdr access protocol aim achieve fdr control make receive center remarkably exact fdr single validity variance sequence correlation apart list general asymptotic
distribution similar generalize asymmetric laplace normal know perform fit version distribution fit diverse however require probabilistic like mixture markov latent complex flexibility motivate possible introduce skew skewness normal expression additionally modify lost keep interpretability fit shape maintain happen introduce mixture distribution differ whole overlap partition parameter guarantee keep segment analyze separately like mixture laplace case partition design expression show asymmetric distribution maximum likelihood
accuracy system name rule logic improve relationship among task modify clause part action symbol place classifier action atom background logic relational relate variable place condition figure inductive logic rational reinforcement restriction logic logic result near traditional near accuracy standard firstly allele sized gene gene test allele value fix bit string specify eight sensor obstacle alternatively obstacle south position match show cover bit htb example classifier visualization match state environment mark star reach optimal analyze population classifier order cope specification classifier develop probability extend mutation match enhance expression similar structure gp firstly name part classifier namely figure condition classifier matching represent string attribute terminal symbol genetic operator genetic programming b many influence besides phenomenon et al common gp almost develop overcome extract final compact overlap instead extract population promise suggest symbolic base extract stack condition linear consist token correspond attribute stack genetic operator account generate redundant explore
easily point university intel ghz cpu gb ram approach conditional simulating predict fine naturally define set regarded volume compute distribution volume monte volume set moderate domain fine achieve volume prohibitive help six analytical consider test function six moderately mat ern field consequently obtain volume proportion come simulation reconstruct fact predict smooth linear nature predictor introduce realization observe volume modification classic volume step firstly estimate secondly center mean simulation
sensor code learn line historical obvious nevertheless paradigm generate prediction feasible temporal study place low system totally dynamic wireless term accuracy state descent sampling real device improve architecture issue operate air affect forecasting past delay enough extend information system determine room de wants acknowledge valuable suggestion provide valuable team besides development besides discussion idea write discussion section describe informative posterior could estimation symbol denote carry estimation consider predict step linear estimation estimate way prediction column parameter vector linear prediction predictor product furthermore last new treat additional new datum estimation freedom freedom calculus estimation resource mathematically bp widely help understanding requirement state ann output input need function number layer
cnn cnn mt dependency string baseline top generic cnn cnn cnn number stand surprising achieve gain intuitively cnn encode entire representation cnn thank sophisticated source signal cnn cnn crucial cnn encoder indeed learn cnn generic counterpart cnn cnn far benefit rich source language use improve section bit layer tag whether incorporate dependency information extend add embed part dependency word I help dependency
great scale bound derive piecewise complicate case contour second contour contour next contour must stable property absolute fact contour k contour stable index factor small leave room likewise small qualitatively contour mode appropriate contour middle contour contour choose guarantee contour bad contour quite nonempty maximum index approximation error produce mode choose top contour minimum middle therefore specify contour root contour error use length solve trivial contour roughly formula contour
hold bound decide agent show reduce matrix substantially randomize low match various regime scalar piecewise step semidefinite exploit rank find cone psd spectral ensure let gaussian useful polynomial degree iterate equip natural chebyshev approximation first chebyshev expansion consequently chebyshev kind eq threshold chebyshev expansion error interval composite degree converge degree chebyshev polynomial reduce expansion level degree necessary strategy chebyshev guarantees approximation composite precisely consider respectively chebyshev polynomial fair chebyshev composite composite polynomial chebyshev sub
train stable range generic back adapt sparse active prop learn instance least positive instance consider stop turn parameter report consist million tweet triple give tweet select response tweet training nine pick tweet around tweet matching enhance performance model tweet hard response tweet individually rank response tweet precision
transform frame compute hide sequence forward hide hide recurrent type memory cell modern lstm connection cell type prediction softmax sum forward input cnn temporal enable extraction capture high layer approach lead dimensional temporal nonlinearity opt nonlinearity step maintain compute frame position feature channel every spatial architecture model frame cnn dimensional spatial temporal dimension architecture slide window approach wise classification computationally intensive figure network max pooling stack rnn lstm temporal dependency slide implement frame wise look track interaction dataset throughout modal capture
process operation contour ability recent development polynomial elegant computationally anomaly use provide observe sample without salient datum enter finite update since affect detector reach peak performance svm make decision anomalous available
case covariate covariate estimation extensively expect much consider comparison conjecture corollary ph author life statistical modelling highly non misspecification identically independently develop statistic base influence robust contamination contiguous illustrate
converge solution formally equation simple important rule eq neuron rule decay decay obtain expand constraint vector converge rule discuss broadly theory linear transfer tangent threshold expectation linear linear furthermore deal power essence non expectation approximate could center approximate although quite origin normalize prove valid slope generally transfer differentiable expand taylor mean fs es es es sf g deal expectation possible obtain function taylor taylor need assume term dependent q dropout center reduce begin summary rule solve moment align continue input pair perceptron backpropagation rule output depend general grow linearly unless target independent input ei epoch weight tend depend epoch last convergent transfer rule convergent properly tt tt remarkably form discover rule recursively rule rational property provide issue major concern simple positive remain remain positive happen cubic neuron target van gradient occurrence descent version list convergent yet introducing range thus keep range eq local force supervise become independence acceptable associate differential linear converge q rule demonstrate effectively weight another alternative mechanism schedule learnable feedforward architecture successively follow would understand output function learn alternative propagation begin motivate conduct experiment various rule
hypercube accommodate tuning separate mean throughout test compete mean rp fig accuracy dimension low since latter moreover rp achieve rp utilize module reduction rp change demonstrate whenever massive multiplying rp result real spectra patient spectra individual cluster grouping employ performance rp mean require alg depict database grey image group number exhibit much alg fast comparable test winner university run randomly report svd computation number augment exhibit performance however alg alg compete note memory algorithms memory requirement linearly alg full mean although kernel accommodate cluster nonlinear linear fig show
sublinear additional assumption counter reformulate follow obtain lead regret convex suppose curvature bound outer euclidean smoothness formulate interior sphere hold logarithm consequently see establishe algorithm assume observe reward due mixed action pure logarithmic algorithm strong finally requirement smoothness imply generally consequently observation maximize optimize
row lagrange precisely third impose select optimality hand positive always readily imply solve path ax corner index last constraint know optimal deduce apply random shortest expect
use ise datum trial e belong ise obtain full distribution make follow eq variable ise
situation approximation consist update considerably area signal non noise mention need framework stochastic approximation markov drive iterate reduce scale control take space use embed metric lie require countable sufficient hard verify take asymptotic relate define control random process irreducible policy problem difference policy reinforcement rl solution available least square temporal trace well td feasible dimension td solve make
digital system efficient complexity object interest fourier minimal multiplicative sequence dft dft give table discrete transform cosine
compatible number record asset stock belong whole problematic financial index must perform past need historical stock price asset plot record ten overlapping period day record symmetric walk number record likely explanation since excess almost small average fluctuation overlap negligible time influence number window
make zero topic speedup table sparse draw traditional collapse metropolis table metropolis proposal document reduce complexity metropolis approach problem cumulative topic solve indicator reduce still use sampler integrate thereby make dependent indicator corpus joint follow sampling step posterior due conjugacy advantage document conditionally follow subsection lda ok straightforward extended pc sampler j b token topic count calculate iterate reduce approach lda calculate document indicator search complexity conditioning give couple advantage method collapse metropolis conditioning type table table constant store every th create collapse see variable speed far hence topic advantage
play conditionally converge play surely distribution play infinite round round tx let playing scheme play play globally convergent convergent joint playing preserve ix I ix x ip ix p f ix play ix x ix game case markov play corollary procedure gibbs transform gibbs play procedure game transform game globally convergent game convergent graphical game game convergent furthermore potential transformation potential shift sampler gibbs sampler regular probability gibbs ix normalizing conditional mrf right last monotonically eq statement f ix ix game complete characterization game play procedure gibbs potential playing use playing game gibbs gibbs game play player theorem positive mrf scheme sampler gibbs potential thus consistent gibbs player transform ordinal game open follow conclude remark work graphical game accord os enyi local unlikely unlikely full distribution full long chain large etc payoff random generate ix x ix ix ix interpret conditional run gibbs conditional converge version playing would pure game variant connection variety field anneal search test global hard polynomial test general apply unclear whether even belong polynomial solution question problem consistent marginal problem connection potential game mrfs economics cs community connection roughly play logit call nature steady playing game establish steady state work steady graphical call mrf graph specific type game play characterization case study rate play perspective convergence simplify considerably appear direction proof leave statement maximal neighborhood hence difference potential definition clique payoff define gibbs potential local hypergraph
empty symbol runtime plan live first create handle remove modify live size maintain variable mr job memory input allows correctly reflect persistent pay reading allow reason runtime cp finally optimizer program predicates block sequential branch weight branch scale aggregate iteration e loop reflect body execute correction read loop persistent input maintain prevent recursive skeleton complex runtime flow execution memory input output statistic cost contrast white box bandwidth multiplier operation remove cp consist size format state via specific input
class analyze fall situation severe imbalance sensitive classification require performance utilize measure find measure define concavity notation denote link well performance popular representative p monotonicity acceptable reward reward range range tp fp nm tp fp tn pseudo fractional coefficient popular usually entrie confusion sake shall find useful negative shall proportion denote skew pt shall art
safe soon popular safe ball half kind safe region center radius orient hyperplane hyperplane illustration trivial get length defer commonly denote iterative safe aim discover safe region increase need see define dual proportional close primal safe unnecessary distance dual ball safe insight proceed close similarly occur soon proceed safe q yet stop thank
tag sensitive ec cs cost head cost cost root tag tag exception gold gold tag tag datum ec task array stack gold valid valid action gold tag tag valid ec stack stack child ec valid action stack stack stack gold next shift leave reduce right set set action history set learner gold gold tag last search learner last tag stack last stack stack tag rgb college md edu microsoft york microsoft com demonstrate dependency build use assignment remove level simple performance date parse avoid various randomization feature dependency long
preserve also variance sequence information result rely ellipsoid transforming imply direction state spherical minibatch number pass stepsize public update fisher let x f post release composition follow sensitivity difference adjacent upper rgb bayesian surprising bayesian approach individual utility specifically get differentially near interest optimal recent hmc preserve minor modification algorithmic demonstrate perform art differential private method real successful class tool stand conceptually model uncertainty decade bayesian image brain activity gold inherently privacy design differential privacy appropriately randomize release noise upon perturbation datum sample connect assumption intrinsic randomization produce approximate differentially private
left length discovery interest task low significance causal pre algorithm hand significance high result predictor heuristic minimal causal scheme prediction series length short model reach true error mm institute impact research p department
symbol filter operation e source signal receive source relate output process whiten operation follow whitening transform unitary time reduce denote whitening matrix consider noise free case whiten unitary whitening unitary literature express source I permutation non literature like cost review utilize dispersion cost square modulus output phase ambiguity usually exact symbol propose criterion dispersion ii undesirable iv implementation bss upon suitable convergence separation product
stochastic deep sbm label propagation belief propagation plant average partition lead finite entropy finite effect entropy correction system number positive group two partition finite
copy condition gradient difficulty saddle broadly algorithm repeatedly step I update failure main asynchronous even non elegant multiple stochastic initialization asynchronous delay within standard analyze sometimes core contribution martingale modify base must construct informally true must second hold run many without success use process recursively q event tw rearrange statement show construct sgd algorithm try iteration cache protocol typically ram provide consistent produce write produce atomic read add write
poor initialize initialize respect regularization learn training randomly label close pixel code row direct update project l size lr lr aa mm svm l lr evaluate visible imaging
multiplication paper field typical decode version derive essentially decode corollary construction da cp pe mail
soft knowledge variable synthesis come error present delta bottom propagation distribution collect correction backward bottom backward message code branch train set available system algorithm inspire likelihood iteration block backward message
classifier small clear penalty set enough never score attain pareto minimize loss accuracy attain vice versa produce attain prevent additional due formulate operational sparsity see free parameter remove scoring coefficient attain attain use formulation accuracy control value score j variable big constraint mixed represent score example misclassifie set lower define big absolute practical ip loss norm formulation norm integer restrict allow big approximate sufficiently large bound mixed tight gap improve ip discuss accommodate range operational predictive tailor free parameter tune remark encode loss formulation sparsity produce heart attack predict heart specifically curve specify specificity encode ip solve parameter example ip control positively assuming score correctly expense constraint prevent label attain high model error label encode arbitrarily complicate logical either practical create structured indicator input add constraint eq constraint scoring include ensure also preference practitioner specify distinct coefficient ensure use gain problem miss instead
penalty scad notice scad perform small correlation compare stein type net en consider mixture ridge elastic net allow eliminate ridge somewhat compare e compare estimator range equal elastic net type estimator positive elastic en portion space adaptive lasso estimator outperform shrinkage elastic estimator preliminary stein parameter space base relative ridge dominate scad
ratio performance tractable heuristic great great membership probability decision method shannon entropy uncertainty entropy justification al space al efficiently select disagreement focus predict vote entropy predict probability leave al exploitation pool selection motivating could suffice optimistic potential theoretical contribution introduce loss select two error author replace average unobserved define calculate approximate use pool entire pool pool intend capture uncertainty example propose eq label training use classifier estimate uncertainty posterior potentially problematic case uncertainty example question uncertainty assumption motivation train estimate reduction selection valuable however current examine define individual estimation component raise choice term describe explore omit choice use behaviour section use section address section improvement theoretically application statistical pc classifier vector sd target statistical batch loss j j form dataset sample intend classifier label label label label actual label great q turning
illustrate parallelization indicate type fall take severe imbalance branch branch contain modulus core essentially whenever assign branch finish assign standard quadrature sum integration iteratively criterion obtain provably integral sum integration integral basic estimate information assign upper cardinality possible see equivalently obtain appear reciprocal product set q q p nh q j interior line set attempt emission directly sort interpolation simplex complicated procedure numerical quantity base derive considerable approximation instead upper tight order bind tight approximation cm extend hold denote lebesgue sum put previous numerical integral monte stop parameterize remain fix software take return value integration integration measure small percentage say confidence percent choose probability integration induce monte part explain particular contribution high input parameter relevant application probability absolutely continuous frequency integer integer precision percent integer typically output percent true integral statement rather empirically framework language def standard k iterate reach f iteration iteration invoke criterion break return variable easily application example heuristic monte carlo achieve quantify carlo integral would condition center see quantity calculate simple language f satisfy assumption priori outline simple justification criterion achieve error margin support per percent time value improvement accuracy estimate sample nothing besides extremely uniform contribute much less proportion speaking jump even frequently choose uniform test close point account proportion even jump iteration affect interest fortunately call explain part calculation potentially open contingency embed corner contingency mapping scheme coordinate coordinate provide coordinate hand convenient think coordinate cube think coordinate deviation explain section validate jacobian mapping constant compute claim ff lemma lebesgue coordinate pdf integral location adopt simple justify really fortunately pdf integral function list principle integration wish approximate emission numerical approximation variation primarily inside oppose simple q central locate remain scale inequality heuristic say specify precisely true hand certain convenient constant convenient value proportion mass lying say implement calculation define
normalize evidence several normalizing several investigate instability furthermore easier appealing depend movement reason mcmc target intrinsic nature seem monte rough evident wide success common walk order behavior several parallel jointly strategy space expense several share among chain improve hierarchical procedure monte carlo scheme conditionally proposal level draw bag density metropolis carlo apply mis alternative call pdfs mis use technique standard adaptive sampling finally framework algorithm chain drive underlie mis adaptation employ mis novel combine strength estimate normalize call walk importance moreover population choice mis trade cost location accord chain interact case adapt pdfs one algorithm parallel interact sampling pi parallel global exchange rest devote hierarchical introduce importance use pdfs adaptation
action expressive come specifically alternative converge operator matrix state mdp name minor p mm reward separately treat semantic column make next pick next reward dynamical markov chain aggregation compress architecture q show pair uniform produce outside circle compose vi transition spectrum circle convergent however action lose base behaviour option vi several include aggregation vi familiar plain vi vi figure domain detail subsection stress produce
convexity work consider admit procedure rsc strong minimum saddle near minimum case give example desire polynomial try orthogonal simple application behavior decomposition permutation symmetry know generate saddle perform reasonably case none many include coding permutation iterative base different symmetry invertible connect saddle throughout norm spectral hessian matrix aim evaluate random stochastic gradient oracle strongly twice assume bound strongly hessian strongly require hessian hessian two th mostly tensor th th construct tensor nd generalize tensor say write vector satisfy sign tensor define multilinear bilinear tensor matrix define multilinear another multilinear orthogonal decomposition know decomposition central include gaussians orthogonal problem estimation approach successfully ica topic function discuss saddle saddle behave minimum
dirichlet specify assume choose assign constant distribution recall control strength within tight control prior cluster choice degree represent rank explain total bayes eigenvalue trace determinant analytically observe element substitute integrate analytically ps dd yield get fig suggestion specify hyper table monte detail suggest tight degree wishart posterior prior product matrix index discrete step normalize update complexity complete repeat sample discard burn configuration however membership huge posterior explore moderate instead devise configuration accurately set membership matrix restriction
eq index component experiment mini tuned yield average run initialize column decrease keep unit multiply experimentally benefit see half show rate axis epoch value converge sgd require converge mm successfully factorize handwritten algorithm sgd less reach speedup showing zero handwritten grids substitution nd line recognize digit show change supplementary experiment simulate distribute sgd
exponential bind invariant positive function somewhat denominator similarly appendix fx kx cost simple denominator give reproduce hilbert close study situation hypothesis instead ridge construct gram prediction component primal weight still proposition test feature analyze fourier feature regression label thus suppose per sx want rate machine offset embed find bind
learn optimize binary hash ensuring auxiliary code hash gradually enforce design choice loss hash simply exist software much slow suboptimal iterate hash consume part code np binary objective function result reliably optima future research another sophisticated hash go hash take hashing feature sift learn code nsf cm thm example computer university california want learn code application nonsmooth relax optimization posteriori suboptimal optimization achieve hash affinity optimize binary code see iterate optimize code hash guarantee well hash demonstrate experimentally unsupervise addition retrieval web example interested image essentially relevant close sift distance dataset hundred slow image hash map bit binary fast crucially compute fast disk disadvantage inexact
vector fix infimum program fix take alternate mean insensitive q last x ex dx rx dx xy xy composition let return easy confirm hence chain reconstruction chain bound true reconstruction joint q n proceed distortion channel capacity experiment define distortion q mutual information distortion distortion obtain distortion form distortion q key bind processing inequality chain distortion slack function iterative direct derivation bind strength distortion theory justification
return number infer inductive program formulate machine sample bad ask generalize separate state bad property classifier propose decision candidate boolean inequality program algorithm safe challenging benchmark invariant sample inductive verification static automatically false abstract scalability precision analysis fine tune property careful engineering manually analysis across many program refinement adapt give hand automatic adaptation static ml likely test learn boolean ml label unseen instance partition learn separate classification purpose invariant good safe state two safe static analysis fail spurious refinement loop refinement candidate learner dt label call value encode
norm uniform measure manifold variance triangle introduction actually obvious discuss return future increase produce diameter component
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matrix multiplication inversion fast counterpart also list randomize variant parameter close independent improve tolerance typical plain accelerate study however mainly iterate produce minimizer assume begin runtime runtime present body literature approach survey independent develop iteratively obtain algorithm solve time stem generally iteratively proper update proximal algorithm inner way erm proximal essentially relaxed multiplicative accelerate erm solver accelerate accelerate minimizer erm erm dual algorithm operate warm minimizer accelerate erm solver yield accelerate run erm problem
correlate analysis put constraint correlation magnitude imply estimation message scatter bethe gibbs far confirm propose range result ground practically particle bethe contrast bethe yield one cavity conclusion theory rbm theory correlation
elementary decay excellent unfold know decay detect energy measure stop particle supplement brief description detector information write unit reconstruct invariant two track preserve decay invariant transformation frame reference rest equal invariant measurement enable rest mass cauchy know mode often call maximum contribution background channel true proportional dominant source invariant mass working mass interval ignore resolution replace decaying order account loss constant choose density function location transition control mapping kernel cb kt center energy discretize bin width narrow central part event reference therein function divide draw bin trial bin bin histogram marginally mutually poisson event belong unfold cb cb mass range assume true part fit cb cross check fit function intensity carry unfold mass interval result histogram event side spline knot spline coefficient choice unfold correction take minute bias correct approximately figure intensity pointwise percentile bias correct intensity shape intensity
require explicitly reasoning entity similarity explore similarity mrfs modeling similarity predicate make entity resolution degree name dependency implement possibly specialized string function potential equally probable probable attribute sparsity thing infer goal prior absence consist prediction imagine preference act task quickly grow amount example link predict among handle scale difficult handle entity partition entity alternatively finer grained pruning entity entity domain atom block logical ensure dependency entity magnitude consider predict interest correlate along social network interest face rule relative challenge answering template rule person friend person scale reflect type aggregate relate person relate friend interest aggregate person interest scale person complex whether two reference friend social rule pair suffer express friend eq mean log concavity linear inverse aggregate set lower define mrfs language posteriori probable assignment mrfs exponential maximize fundamental prediction weight map discuss mrfs distinct general minimize interior polynomial complexity variable potential bad big structured problem algorithm design mrfs leverage connectivity world consensus subproblems mrf first equivalent potential map copy concatenation satisfy infinity likewise drop use enforce domain easier decompose finally operator define wise consensus optimization reformulate inspection equality easy solve multiplier admm form concatenation vector multiplier lagrangian parameter find global exist feasible assignment measure update convergence describe implement lagrange
norm product fourier complex I th say every th radius give consist level quite give location require operation key whether present analyze instance calculation fast fourier vector op sparse know past problem algorithm follow reduction sparsity end discrete fourier dft fast dft operation sparse possible evaluate particular randomized compute fourier transform time formally vector pair sparse end w dft dft thus significant index sublinear k correspond original computing entry require total matrix applicable output
linear base function semantics multi neural linear activation sigmoid approximate already attractive try question base recursive rnn convolutional network advantage procedure success rnn parse enhance net range like parse sentiment rnn motivation recurrent rnn tree vanish error back addition send root lead capture recurrent tackle idea allow store cell use tree structure vanish
kernel merge nonlinearity stable theoretically result would remark european research advanced contract contract dynamic consist nonlinearity invariant cascade coefficient sample impulse estimate adopt tailor uniquely impulse coefficient nonlinearity compare compose block invariant engineering reason year
determine acknowledgment acknowledge college ep incoming message produce involve estimating allow computation incoming message message automate kind approach automate inference broadly category inform case full
assume variational represent notational convenience write involve condition evidence posterior optimize ascent ascent equal conditional corresponding conditional equation summarize coordinate correspond directly cluster dataset visit ph new time read least article paper read link paper visit read day visit analogously select physic read least article manner link standard represent representation truncate abstract remove vocabulary ph co evaluation paper force size text allocation treat document count document let appear dirichlet
path euler stepsize combine increment path early information new htb straight clarity euler euler stepsize bias achieve appeal behaviour euler globally follow conclude variance deduce evenly level level estimator scale remark course purpose final count requirement guarantee tight coupling path true strong rate rely moment upper appear conclusion drive method long optimal beneficial analyse mention control simple control wish compute
generate simulation dimension lower triangular cholesky firstly secondly realization different moreover term function concern normality simulate check study tw w bt tw bt bt tt chi apply random belong recommend several overall alternative chi quantile plot mahalanobis
show bound empirically achieve ht dash line corollary use layer unless encoder try value affect decoder solves extend embed space enable visualization manifold example clean give diffusion map calculate first cover space circle use predict top left bottom left display embed colored radius display view predict point color color within origin handle smoothly space diffusion locate circle boundary range extension limit next periodic extract patch sized pixel obtain display example patch diffusion circle layer diffusion patch dimension decoder visualization patch point diffusion position patch circle decoder radius represent amplitude periodic origin diffusion smooth opposed patch figure diffusion map decoder reconstruct assign amplitude decrease autoencoder encoder stack train noisy calculate autoencoder phase decoder mse autoencoder decoder autoencoder decoder reconstruction separately
mean matrix keep hyperparameter deal formula interpret measure equation restrict require simple behavior synthetic derive demand operation computation element possible every initialization update formula independently increase variant substantially accelerate deterministic rule select merge probabilistic pair result last extension could implement generalization method potentially different structure characterize variable straightforward modification present perform assume element independent common underlie similarity measure framework would account wide variety situation generalization possibility marginal posterior criterion aic variant schwarz appear manuscript hierarchical connection deal datum improve scope dirichlet bayesian decide merge agglomerative procedure result closely correction dimensionality alternative turn beneficial simulation implementation generalization random present hierarchical cluster acknowledgment grateful member functional de universit de al provide resource
understand apply diffusion equip canonical tangent bundle see vertical metric map imply consideration total manifold though still insight value duality generator change relative size bandwidth depend riemannian choose mention link bandwidth kernel diffusion bandwidth relation explore detail space essential duality differential geometry broad purpose index theory parametrize horizontal metric send thus geometry topological carry extract similarity proved base manifold flexibility us diffusion dm focus analyze tangent riemannian tangent foundation motivate concern wide deep algorithmic set shape persistent diagram tangent space horizontal possible show establish foundation knowledge address laplacian spectral eigenvector small graph way possible upper recently connection central interesting practice eigenvector laplacian globally manner multiscale massive bundle structure meaningful possible simultaneously diffusion high store applicability thus expect develop performance real bundle summarize geometry unit tangent metric jump start unit tangent collect horizontal base manifold diffusion operator riemannian bundle coordinate chart basis denote projection usage distinguish connection suffice tangent call vertical vector immediately eq symbol split sum horizontal choose impose orthogonality riemannian symbol verify horizontal differential operator lift instance tm
b contrast scale shrink require chain still consider form sense reduction number converge likely chain translate update experiment perform multiple memory node beyond ten core equip memory environment use message pass interface communication node collective communication require generate move however communication running box herein name illustrated integrated autocorrelation efficiency underlie tend infinity algorithm metropolis gpu accelerate operation serial tend conjunction allow target scale core parallelization acknowledgement publication technology member compute research member quantification symbol david law consider space adaptive herein parameter dimension gaussian result refer metropolis accelerate library chain justify posteriori gpu improvement competitive intel mkl alone strong long fewer necessary example dimension excess markov carlo big hastings gpu acceleration activity area quantification
place number distinct correspond place scoring estimate bind achieve streaming greatly simplify independent weight key depend random estimator review individual seed depend inclusion apply estimation streaming pass include partial count depend derivation correspond invert inverse mean unbiased sample size st small key sample randomization th key scheme moreover property us segment sum estimate cast sampling scheme distinct sample hash fix retain distinct reservoir stream element hash distinct equal weight key cache key scheme enter sample therefore sampling equivalent cv grow rapidly sample contribute specify key cast element therefore seed transform sample actually obtain estimator final unbiased estimator x monotone perhaps pass cv bound exceed skew dominate mostly would htbp package terminal option package graphic explanation graphics macro ltb lt lt lt lt lt ltb lt lt lt lt lt r ltb ltb ltb spectrum parametrize distinct classic estimate element score hash key element
showing walk union u necessarily good path subgraph laplacian walk effective odd eq triangular conclude odd lemma vertex subgraph length middle path triangular hold sampling walk effective follow crucial graph rr u eq expression path extend end iteratively summation step recall path sample integer random perform step one end perform walk keep
cross embed close matching capability versus enhance indeed combine sparse dissimilarity representation large capability simultaneously dissimilarity lead computed globally minimizer convergence transform exhibit suboptimal embed analyze accelerate step transform newton optimize essential next multiply block set yield yield
technical conclude hilbert regression quality way difficulty control reproduce reader book endow integrable respect abstract functional concrete rkhs psd v moreover n w closure rkh provide example follow natural base nature place hilbert let ij scale matrix solve exactly via qr large prohibitive addition require problematic approximation original generate precisely via kernel across column analyze say many choice sketch row rescale rescale row I orthonormal include fourier dft sketch matrix I diagonal matrix I sample
moreover gate eliminate zero set output zero input non window convolution layer convolution pooling source target similarity use multi similarity score state score mlp ideally positive source phrase correct translation negative source phrase bad translation context max triple matching consist parameter mlp train encourage example low score aim capture contextual distinguish good translation candidate bad embedding start contextual equivalence dependent semantic semantic phrase difficult
defer needed level overview reader familiar particular result leaf label generate accord define leave leaf label specie sort active property identifiability incomplete picture requirement base simplification apply way infer distance base molecular hypothesis linkage e special root close expectation ab ab
concavity verify continuous inequality show exist function contradict focus exist strictly increase minimizer follow
duration epoch epoch deep one qualitative apply layer several additional understand quantitative image imagenet qualitative show material middle autoencoder bottom reconstruction various use reconstruct look progress drop reconstruction quality reconstruction convolutional preserve location figure interestingly reconstruction reconstruction look position reconstruction layer match representation quantitative inversion coefficient reconstruction plot support conclusion reconstruct roughly twice reconstruct fairly object high visually reason color match happen
anchor contain alpha protein protein domain protein contain repeat domain family member alpha alpha nr box repeat protein contain cat candidate bind protein protein complement component patch protein protein sort family alpha read frame alpha contain cell line transform contain scan contain contain h j cluster homology contain family node family member box protein contain anti b family member box auxiliary gamma gamma domain member protein light dna link release activate derive ab ab rx x channel family rich repeat protein dna protein rna reduce cycle member I family interact contain protein associate protein b bind protein read frame contain repeat rich family member family disease protein st st alpha p like
gate gate gate gate read lstm cell forget gate memory date feed believe question input image supervision memory treat sentence component convolutional network cnn generate representation image paper cnn remove softmax deep cnn connect remain top embed lstm structure activation memory cell word feed answer separate answer lstm share word answer share word embed first third fourth specifically
make conceptually specialized want class give fundamental hope practical purpose hope belief assumption statement reason agnostic difference task average true solution c average equation give appearance essential denominator average sup sup ti data specific consider retain unknown course law task common environment environment induce q mixture interpretation also induce h learning algorithm n problem interpret namely select x replace expectation well still excess follow associate eq solution gaussian make previous control simultaneously term equivalent order
select therefore prior variable tucker completion enable infer multilinear well noise level solely partially treatment differ likelihood indicate core parameter derivation provide core multilinear due sum explicitly denote memory prevent dataset scalability achieved employ multilinear operation apply explicitly result memory scale observe index interact memory posterior tradeoff residual hence update force expectation expression noise update entry computational multilinear operation essentially solution model addition also predictive entry uncertainty prediction important bayesian prior employ represent hierarchical prior model e ensure solely firstly manually denote slice associate evaluate recover tensor infer
retain detail counterpart via penalty state transition state non penalty formation introduce arise similar optimize depend major accept reject addition optimize label state time use function analogously step every pair transition overall new back old synthetic ms mean vs runtime find supplementary multinomial observation parameter mean assign quantitative jump compare evaluation hold performance jump maximum learn simple baseline ignore run generate rate observation
correlated mean moment level moment appear definite guarantee algorithm devise preserve positivity covariance min kalman gain correct measurement perturb sequence I pair become situation next ensemble subsequently next via eigenvalue truncation interest ensemble e shorthand limit non e nonlinear define mean enkf easy show enkf kalman show nonlinear fully process long model strictly small enkf counterpart error denote approximation assumption slight
equation ridge solution trace multiply trace q quadratic fan norm norm trace accord error reconstruction depend ensure em code sp er co sp er co ex ex rbm ex ica fa ex assess factor autoencoder restrict boltzmann machine hide laplace hidden unit component analysis fix nine dataset result like pathway activate
window input gate size forget gate gate memory cell stochastic gradient propagation prevent serious overfitte dropout dynamically activation relu word embedding language summarize score class cnn macro average expect main take advantage correlation answer lstm relationship learn cnns rich
especially triplet explicit vs implicit parametrization employ parametrization implicit gram eigen require scale gram fast gram parametrization scalable eigen problem I proper counter eq counter choose proper counter dataset consist different pose two pose rigorously triplet compute attribute image specie triplet crowd experiment experimental embedding size triplet view check quality embed addition view comparison draw triplet large number embedding split triplet generalization test triplet whose triplet relation correctly model triplet choose embed target label
use work difference favor approach experimental protocol follow approach whereas select whereas except scene category use experiment extend people acquire illumination different expression descriptor project onto database choose sample perform randomly experiment accuracy report locality coding take accuracy lc lc set lc lc lc keep optimize result atom atom result lc value use error tolerance also result accuracy optimize setting setting ar database original refer reader list report show local classifier lc dl report result lc c ms svd dl lc rate well require classify test instance dl comparable lc come multi color ar database variation term expression
qx quadrature become qx h g put integer exist distinct element ff ie associate indeed j nj less pack sphere lemma concentration hold plus plus minus minus pc pt minus pt minus quadrature integral kernel decomposition give logarithmic bound distribution result quadrature recover bound moreover extend general full norm result result improvement need preserve guarantee integral area machine signal generally mathematic bayesian
use sample prior variance maxima graph agreement lower bind design partition design space existence maxima former first estimate carlo smooth analysis prior observe although low fall also negligible sample see analytic calculate qualitative observe combination maxima maxima scenario surface line result b demonstrate formalism actual bayesian inference uncertain contaminate aspect limit resource great enhance concern locate california along cross reach depth accordingly field domain use regard achieve capability software paragraph limited assigning horizon cross section horizon matching merge exist point calculate point counterpart opposite automate marker line horizon grid form section interpolation inverse weight scatter distance scatter cross section increase final fig separately
pose pixel hand segmentation position body modal integrate way extract box contain fuse extract propose covariance represent spatio temporal augment audio architecture supervise unsupervised subspace autoencoder unsupervise learn invariant spatio temporal high video sequence convolutional rbms network explore preprocesse input employ convolutional mid explore sign language video wu hmms video stream propose allow classify restrict temporal pose fusion fusion score early fusion representation investigation early object mkl additive multiplicative individual et strategy feature model recently modal employ rbms correlation audio visual speech isolate letter digit al modal boltzmann fashion tackle integrate annotation al challenge video scene region wu deep video author modal convolutional explore multimodal pay various modality training image describe first challenge strategy address deep neural network path parallel perform video audio path aggregate additive fusion scale notion frames spatio early allow prediction
simulate datum use dc linear dc programming grid surrogate art price algorithm evaluate ref give efficient sort percentage oracle bid high bid well dc mapping exhibit scale nonlinear neural world working become neural real world maximize historical develop optimal method set outperform simulate normalizing computed line equal
adaptive counterpart outperform probabilistic easy implement due negligible present adaptation approach believe programming benefit additional potential acquisition dependency exploitation sensitive bring clear explore model agreement fa definition output sensitive express extend light adjust probability output correct equilibrium distribution adaptation convergence adaptation ensure sample equilibrium metropolis within decompose stochastic schedule select next component adapt schedule modification probabilistic
mean verify without exist digital exist digital help paper digital trade stage prototype decide quality performance prototype impact digital transaction processing business community digital various scenario instantaneous transaction transfer precise possibly meanwhile data digital statistic g source influence limited exist digital mechanism help precision datum
construct stationary propose writing term target skew effect hmc process diffusion attain detailed influence provide importantly stationary stochastic eq integral lead positive semidefinite skew symmetric variable unique distribution restrict skew posterior h describe evolution density compact form verify equivalence detailed supplementary material completeness pt portion continuous iterate stationary sde stationary matrix define possible sampler sde stationary density ij integrable skew constructive proof cc
minimum often atomic thought operate set information theoretic set atomic powerful approximation theoretic limit show tensor incoherent algorithmic attempt hierarchy moderately constraint bounding rademacher sequence prediction third close predict denote denote tensor appendix thought spirit incoherence location entry typical assume uniformly goal recover observation completion completion thus mathematical right type data clinical observation three observation try predict assumption low miss first weak strong recover arguably well suited remark product within theory output hypothesis eq achieve error appeal property almost typical tensor algorithm sum relaxation atomic norm find analyze
removal impulse smooth separation handwritten digit separate linear admit sparse consistent infinitely candidate selector sparse discussion selector include comparison selector aim whereas try candidate residual tune properly lasso selector minimization although corresponding selector selector gene cancer compressive sense guarantee projection encountered signal perfectly
km ce observation model bf bn normalize process overlap significantly bt psd window deviation km ce average generative insensitive result ce consistently outperform km performance km note window experiment cluster satisfied h xlabel xlabel xlabel yshift height legend style font font legend mark none
introduce difficulty univariate research direction conduct convergence establish work least square analysis loss last iterate pairwise learn polynomially decay size concrete example illustrate tool refine inequality average relate convergence satisfy derive explicit theorem maximal rate achieve choose since
day amount day day day discussion compare month band dt day observe attain lr computation iteration lr hmm second compare hmm dt fig day take observed train hmm attain dt train dt thus short time performance affect illustrate depict conventional attain compare expect evident predict svm hmm imply fail consecutive
explicit commonly area model network social category role social network task investigate social room utilize address properly social analysis extract blockmodel topic node content node citation topic network link unlike block adopt consider citation relation introduce indicate cite keep field mrf communities citation relational traditional topic topic fix world situation advance try resolve nonparametric review thick cm draw line circle thick black fill gray rectangle right edge relational graphical word
circle child fill child fill circle child child child fill draw fill none parent none fill none child pt fill circle child fill notion semi correspond particular layer generalize one wavelet pass filter output terminology atom note require atom write
converge surely even assumption continuity acquire poor situation additional costly address account true offer expensive specifically design confidence whose requirement ambiguity singleton set tend infinity irrespective problem less conservative polynomially large independently wasserstein metric ambiguity ambiguity set condition represent space use wasserstein define wasserstein constant wasserstein metric wasserstein integral examine ambiguity wasserstein common light distribution spirit light tailed exist exponent distribution modern concentration establish guarantee concentration positive depend priori outside wasserstein ball small wasserstein probability represent prescribe yield radius confidence radius tend increase rise wasserstein metric assertion wasserstein triangle inequality construction sample virtue borel surely conclude note respect wasserstein recall assertion corollary ensure wasserstein ambiguity ball tend wasserstein contain scale quantify behavior potential far wasserstein ambiguity favorable wasserstein significantly solve correspond two probability popularity kullback total distribution leibler respect assign event virtue jensen whenever possibly highlight symmetric
imbalance variation linear present synthetic bandwidth whenever dataset dominant entry independently follow name bandwidth ii share core usually lead large delay serve core component evenly partition coordinate component varie core update one assign coordinate epoch epoch depict size parallel speedup different finish plot residual running core nearly hence core nearly closely gauss enjoy gauss phenomenon work logistic news table cccc news logistic since coordinate uniformly nearly memory store product gets update cost store entire parallel implementation achieve scale explain implementation core
read calculate z h nk h h b derivative h nk bx ax x nonzero calculate influence nk x nk hx hx nk involve zero conditional nm x give n multiplying taking eq assume perturb partition interested matrix result perturbation nearly necessarily variational multidimensional eliminate cc matrix subsequent h z complicate simplify allow eliminate order taylor expansion corner x v perturbation belief next right corner q v department institute mean field runtime set major
sensor interpretation point note drive explicit tuning parameter take geometrically hyperparameter scale hour etc hyperparameter segment fix marginalization hyperparameter suit online problem several ability show competitive commonly handle multimodal posterior development smc sampler mention improvement possibly improve challenge contract research thank dr se regression parametric encounter
greedy include child show repeat query lowest great harmonic proximity create redundancy node produce hierarchical cluster figure exploit local neighboring reduce candidate latter coarse represent cluster search use active trade achieve labeling region boundary exploitation exploration refinement leave learner exploratory mode effectively propose allow exploration trade still dramatically perform hierarchy provide illustrative hierarchical shift lee feature space technique node calculation explore steady state hierarchy
family elementary member proposition quantile score scoring scoring quantile success forecast respective nonnegative interpret cost decision regard rule forecast event rule twice close section note scoring quantile probability relation repeatedly economic interpretation score along functional either interpretation decision quantile relate exceed payoff realize market independently act enter actual zero enter strictly expression determine motivated format bayes table payoff enter end positively orient payoff orient payoff payoff payoff irrelevant multiplicative correspond classical cost loss distinction payoff regret choose threshold quantile payoff cc mm regret mm mm cc payoff positively relative multiplicative factor value consider amount company exchange future company losse payoff payoff independently payoff represent act payoff scheme enter expect vanish payoff strictly cdf analogy quantile strategy relate score relative deal see determine score forecast binary obtain ratio threshold fix loss
tx constrain result easy schwarz schwarz two p tu bind point tu schwarz inequality ep p tu due follow show begin note note pp pp completes find model regression ready slice version h finite moment therefore pt schwarz h pp pp pp third hence second direction eigenvalue eigenvalue eigenvector eigenvalue orthogonal basis eigenvector constitute direction note similarly tend theorem write bt bt ty use show triangular bt ty bt b bt term right respectively term q number term hyperplane orthogonal decomposition independent conditionally contraction thus
operate dynamic operating envelope however research offline collect engine take offline however requirement capability task offline ambient temperature pressure valid condition model implement expectation condition require offline develop engine velocity streaming pressure produce day infeasible store development learning process datum advanced engine like engine insufficient outperform step ahead online exist survey sequential extreme learning os survey popular context efficient least os achieve accuracy quick know parameterized ill conditioning recursive sometimes regularization unbounded growth prediction estimation boundedness base descent lyapunov base notable lyapunov radial basis map calculation linear estimating basis function aim retain simplicity os stability control purpose gradient engine follow extreme machine use lyapunov engine well online operate boundary remainder
zero choose multimodal dictionary multiclass vs equal note multiclass large generally require performance variation dictionary multiclass linearly share consider parallel paper multiclass allow differentiable optimization respect optimality zero loss assume set belong real couple mild require generalization modal multimodal admit continuous twice differentiable reasonable deal acquire sensor state main paper active assumption jj sn ds sl appendix dictionary classifier convergence batch strategy sample factorization admm code case single drive guarantee unique word practice representation set regression initialize properly poor unsupervised multimodal learn assignment row sl project code sparse rely modality group impose example scenario modality
scalable computation implement popular learn computing triangular perform summarize decomposition triangular matrix reduce mention step perform triangular computing rotation describe iteration multiply rotation element series rotation continue convert output rotation total rotation convert qr iteration svd avoid different qr apply iteration compute describe qr apply work necessarily qr computed orthonormal triangular previous similar hence matrix moreover
world machine addition data set suggest suggest meaningful new rwm requirement set life number unbalanced em datum categorical heart block vi capture consider measure dissimilarity density window measure kullback hellinger distance cross outer validation fold keep parametrization fold presence label size experiment experiment fold box precisely lie experiment ten parametrization result set three fold capture structure information class assignment whole laplacian case rwm gmm kernel rate consider randomly fold build penalty vary account kernel rwm weight continuous dimension factor varied heart attribute cf assess sense good classifier get low classifier average rank classifier compare classifier average claim classifier rank statistic distribute accord degree freedom hypothesis nan reject hoc significantly different rank statistic divide test plot difference result rank paradigm
call send level accurate voting probability level predict triple set attribute object possess denote call formal common attribute describe cluster attribute concept case computing reduce formal concept choose concept index concept relationship order relationship might represented building
p pca denote expectation accurate far example block matrix bound band matrix exploit bind let estimate outlier support bind use completion low section case next support row column begin next follow lemma recovering change subspace change add successfully dimension pca proof section lemma entry support set prevent give change enough unable enough accurately fill follow quantifie purpose index move change would represent remain give area frequently change product th interval mutually disjoint subset mutually disjoint interval equal define take choice trivially good appendix case semidefinite generalize correspondingly choice attain rest remove subscript ease notation row I correspondingly ki similarity necessary call therefore row band side everything analogously proceed band central band summation sub band interval term away summation time easy define describe algorithm detection appropriate write change detect final change kt j
feature period aggregate trace datum cloud entire online web event several minute total feature count number task minute usage additional interval load level term result window correspond machine minute obtain total feature platform table run status require feature feature machine load window require join single row gb analysis experience extremely regular aggregation sd cv correlation sd cv windows sd cv compute aggregation minute deviation deviation hour various statistic event gap hour hour example failure grey probably need extensive back aggregated table point point hour average contain show evolution past hour table consume require second quite range hour tb hour aggregate interest significant platform even feature handle table although criterion
high cb seem prediction ari cb sequence give sequence symbol alternative symbol ari run cb ari failure extraction prediction repetition basic cb guess ari appear robust cb cb switching graph cb bm reach random guess ari switch ari increase switching symbol cb guess ari relatively small appear pattern length serve concept dynamically vary comprehensive evaluation extraction size evaluation manually annotate subject manually annotate consider evaluation threshold ms assume permit estimate annotate use detection sensitivity ahead threshold measure datum notice length improve length reduce precision extraction event locate attack order assess cluster process annotate assess sensible
threshold addition comparison alm mac implementation truncate singular formulae pp dimensional identity order energy reservoir estimation field estimation nonlinear nonlinear potentially behaviour alm mac equation cyclic test assimilation weather assimilation discretize fourth hour initialize draw state avoid effect discard assimilation reference initial subsequent observation observation assimilation window hour instant odd variable add normal total assimilation draw whose mean alm mac simulate apart ensemble member size mac alm mac number iteration roughly valid iteration otherwise subsequent study way apply tune adaptively stress expect repeat background random choose figure alm mac repetition member rmse truth give tr alm range interval mac interval interval suggest alm alm modal single peak mac peak close interval alm intervals low mac overall suggest mac alm box alm leave different data mac alm maximum alm alm follow tendency box phenomenon
current computationally short term deterministic vanish recurrent gaussian mixture network mix proportion covariance another neural lstm neural experiment adaptive inference benchmark smc inner learn use truth smc normally compare state assess hard metric truth evaluate root square approximate variable variance common effective sample ess ess equivalently ess alone sufficient metric absolute effectiveness often benchmark
chebyshev expansion chebyshev polynomials determinant input multiplication efficiently example time propose grow non general multiplicative approximate determinant analytic chebyshev ratio imply additive obtain scheme counting number span certain class graph find likelihood random size million infeasible cholesky experiment order fast solution accuracy million minute single
symmetric add mention experiment uci error stream every get query alternatively test w see hypothesis consecutive plot rather randomized report average passive query column dataset report query represent training return l mnist comparison return dataset test see stream slightly well stream performance passive mnist
provably perturbation tensor base initial u keep subsequent obtain via implement storage improve avoid ambient apart tensor cp tensor factor run sketch permutation propose novel hash build space limit main tensor necessarily dimension whitening could intrinsic count sketch hash bernoulli variable pi ps ph need wise hash error tensor proof appendix pt u v mt f stand inverse moment scale explicitly take cubic main decompose moment u r follow wise u I reduce element fouri therefore know rank u sec b bn moment rd computation factor list alg reduce approximation failure help mini
risk gibbs occur vote risk limit analysis commonly unable evaluate whether framework help produce individually tackle risk classifier disagreement vote show vote moment bind call consider together chebyshev present make pac guarantee majority vote base present recover bound way risk disagreement fundamental expect disagreement rely well supervised method improve bound supervise bring new pac derive kullback case problematic make define apparent call section basically way originally build vote finding minimize respective build even quadratic program confirm adaboost section conclude point recent pac tackle classification convex hull paper use convention uniformly properly normalize majority vote vote classifier sometimes majority vote case choose simplify accord md counterpart uniform training simplicity replace pac theory traditionally vote hx otherwise one fx exactly simplify majority fx output space imply classifier value package color terminal option load graphic explanation terminal graphic ltb lt lt lt ltb lt lt lt lt lt lt lt bp ltb risk distribution loss vote vote vote output majority closely classifier classify choose gibbs classifier later order follow classifier hence pac l risk bound twice risk extend general definition gibbs either depend prove accord pac bound risk majority vote usually risk even circumstance distribution give case expect linear perfect majority vote inaccurate gibb indeed bind fact consider population
monotonic non regular great flow mode closure finitely many mode iid estimate pairwise minus rand index involve derivative state simple generalization state lemma also q exist necessary figure approximate find mode shift risk preliminary concept
balance devoted probability square integrable endow inner denote introduce expansion associate covariance pc uncorrelate necessarily project eigenfunction provide orthonormal function dimensional span pcs dx admit strictly assume choose uniformly fact add pc decay process strictly x standardized pc equivalent boundedness second probability component fact w finally worth hilbert zero univariate variable variance pc j aim trade fix like extra behaviour turn technical try order define suppose equivalently eq dimensional density pc volume task behaviour term expect interpret correction truncate version process whenever pc behaviour strictly related depend radius pc hand whenever fix
variation eq follow unknown regard face face face depend contain training divide predefine split nine define protocol evaluate recognition subject identity recognition sample augmentation capacity open implementation train cnns cnn base recognition system architecture architecture extremely small overfitting converge rgb colour fed cnns feature cosine performance table recognition indicate architecture cnn offer little discussion cnn feature good face compare
free volume minimize relative estimate scaling difficult moment manifold mt initial dense line approach local find well allow treat gradient boundary ignore implement dimensional manifold topology fr topology fr open tangent induce riemannian speak gradient flow unlike case existence line flow local neutral l assign equal class gr map
eigen consume subroutine partition complexity much axis logarithm file propose method real cancer number gene cancer control treat compound joint inference task gene absolute divide entry running iteration compound show screen efficient fast fast moderate much network reasonable topology significantly
number domain adaptation vision model domain reveal serve assume domain compare propose da method base boost principal domain maximize sa activation last layer final mapping domain adapt label predictor svm four domain approximately office directly compare use publish result general pick configuration different architecture experiment first show office adaptation architecture three mnist choice adaptation attain architecture binomial restriction architecture select small costly training schedule momentum adaptation parameter gradually change schedule schedule update ensure latter train batch image know comprise projection visualize feature network domain figure version adaptation classification accuracy domain overlap good da momentum architecture way without domain one assess performance system effect hyper success error successful error high layer pick computing suggest cnn activation procedure incorporate training red adaptation make discuss train domain shift domain experiment deal mnist obtain digit extract color define image coordinate channel patch inverting position digit task hard compare digit still mnist background perform feature distribution adaptation
state discard ensemble guarantee independence ensemble rmse panel rmse update drop select error hmc smooth assimilation windows report case background keep assimilation analyse similar second case slight notice trajectory rmse assimilation window error begin assimilation line hmc smooth sample show smooth noisy var hmc keep fix assimilation next show figure hmc var analysis keep background smooth var analysis close reality accurate well smooth update assimilation scheme forecast crucial uniformly grid overhead forward forecast assimilation hybrid error window update hmc smooth adjoint compute gradient var calculation var hmc smoother require forward backward adjoint water adjoint
associate measure experimentally basic mixture state overlap fraction percentage law momentum angular momentum momentum law momentum fundamental law principle momentum temporal spatial velocity denote account relative account body force generally act appear sum make calculate simply sum convention denote variable stress represent pressure common mixture volume fraction pressure interaction type boundary therefore system category boundary comprise type denote depict dynamic particle act computed sum force body force force subscript denote contact max pointing define contact ij ij irrespective contact centre contact form overlap particle centre contact contact point ij category theory sec define density sum species boundary momentum balance exclude boundary velocity respectively additionally newton law equivalent define section systematically arrive point mass
mcmc method walk stochastic walk metropolis stepsize adaptive mix sampling feature coefficient histogram various metropolis mcmc reveal subgradient accuracy bayesian converge subgradient get ten convergence set stochastic sigma variance walk metropolis parameter stepsize stepsize figure metropolis space local minima auxiliary good select stochastic mcmc show standardized frequency table subgradient walk dimensional magnitude margin superiority various challenge setting analogy augmentation technique tackle easy computationally efficient svm furthermore hmc within hmc svms experimental wide effectiveness bayesian continuous log posterior sparse bayesian max margin act analogy subgradient subgradient inference deal experimental problem demonstrate effectiveness prove stand popularity
across setting describe gibbs pg framework sequential speak pg proposal monte sampler mcmc key pg instead proposal sampler tree tree fit residual pg explore efficiently sampler one could move scheme propose tree move reject slow high setting pg succeed non move non pg sampler require one sample tree acceptance ratio computationally efficient impossible easier organize review pg pg briefly review decision tree closely
point correctly heavily noise rely obtain eeg comparative estimator contribution comprehensive riemannian geometry eeg asynchronous thorough information paper divide follow review application riemannian geometry relevant propose online introduce experimental tool machine feature matrix product differential riemannian geometry algorithm consideration three kind rely onto tangent successfully log provide rich representation trick provide inner product lie reproduce hilbert allow extension svm kernel apart kernel map onto vector mapping datum euclidean tangent rkhs adapt riemannian adapt
covariance equivalent function clearly evaluation unique one property cover ct matching kalman smoother high order analogously rd weights sigma function gauss covariance eq matrix multivariate polynomial product root polynomial multivariate integral exactly polynomial together uniqueness specific set classical good choice polynomial sigma quadrature exponential strong quadrature method informally polynomial family c squared covariance argue converge indeed happen way sigma monte carlo would could sigma sometimes monte carlo normal quasi
dataset search frequency manually search query http www google com trend date provide volume query integer scale analysis date indicate background analysis weight date website record week week subsequent activity report week http www activity week www google trends current google time google date motivate transform intrinsic interest lag vector intuition online search markovian formal logit obtain google volume obtain google search frequency growth divide add transformation observation lead detailed transform google thought choose capture
detail grid vector min usage l cp u u section describe combinatorial subroutine space interpret put expert environment mix loss goal name distinction protocol component usage alternate binary relative scheme entropy projection fortunately compact polytope permutation optimization like mix sub construct quantile regret perfect scenario clearly concept fully still coordinate predict usage define fix bayes b discussion vs action combination potential expert loss guarantee close loss desire
good thorough overview convert follow regret introduce convert mechanism statistical box rely utilize output hypothesis price interact problem budget subset tool entire aggregate convert develop pricing minimization enough budget later detailed pricing present pricing give main learner budget follow deep understanding pricing analytically variant mechanism pay pricing minimize algorithm budget proving mechanism easier guarantee appear price pricing scheme appear mechanism statistic draw marginal value progress budget pricing focus noisy sample budget broad agent datum literature may present advance active somewhat point feature height age formally body derive mechanism provable finally appear datum object parameterize broadly endow convenience add scaled loss scenario space generic canonical parameterize number budget accord arbitrarily consider bad cost instance datum correlate case operation
k extend assign assignment unnormalized weight generate filtering algorithm sequence equivalently assignment htb h kk h normalize operate thereby target original latter show generally important association component generate iteration component exceed filter performance iteration nonetheless appeal rank stochastic carlo gibbs directly mention conceptually association proportional component association association low weight association diverse obtain
pair process case reason test variety section describe three network architecture channel pseudo patch testing patch essentially architecture attempt similarity compare compute descriptor descriptor skip descriptor proceed estimation addition variation concern architecture variation mutually exclusive patch resemble idea descriptor branch network share exactly weight branch take relu pooling branch output fully relu test network connect unit separate relu activation layer branch network descriptor module two patch descriptor independently branch match
either unit scheduling ap learn state ap interested expect sum throughput within horizon numerically expense prove optimality policy special schedule ts obtain scheduling mp time affect policy optimality energy previous ts upper ts show numerically mp bind wireless multi section throughput grow communication scheduling computationally prohibitive scheduling author assume arrive intrinsic also variable policy threshold dp lp reference channel state mp rr throughput consider capacity find static optimality exploit infinite optimal paper model classic model chain arm play reveal channel cognitive mp mp
sign large analysis compress least easy task signal expect measurement hope develop bit prefer technical otherwise reader bit extend family stable entire fact briefly reliably measurement essentially essentially additional number accurate follow full
rescale unweighted constant rescaling replace hand side unweighted similar rescale value match along fig expect version fig fig overall tendency error large expansion sample weight nx ik asymptotic parameter represent distinct assumption unbiased ignore high comparing follow mention lemma subsection discuss lemma lemma appendix put p attempt leave
low code corpus sim string sim corpus sim code directly relate classifier original second weighted code probability noun keep pair noun must map string format name character noun pair distributional entity associate corpus frequency evidence interest extract software factor bias software focus corpus affect simply fundamental software introduce user motivation highlight
establish eq rank international institute berkeley demonstrate decrease testing agnostic thousand achieve acceptable accuracy explicit feature name tensor connection polynomial machine tensor behave real compare technique significantly parsimonious adopt data success fact succeed capture inherent problem form dimensional explicitly often exploit form store prohibitive consequence fit draw attention year random satisfie k matrix considerably reduce model form
hyperparameter unchanged vertical lead posterior panel sample reflect mean belief update posterior flow update stay relate precisely belief verify prior likelihood necessarily semidefinite flow eigenvalue imply encounter away figure show pseudo sequence spherical belief temporal dynamic belief think black center mass red correspond add circular eigenvalue round belief accumulation belief
type arrive central decentralize heterogeneous connection neighbor connected ca user stream one twitter topic relate stream content tweet recommendation video search content device receive content popularity predict video popular social trend social media domain twitter knowledge may predict high popularity user facebook context provide advantage approach work recommender systematic recommendation recommendation priori knowledge preference learn high characteristic change iv work recommender summarize c distribute old confidence yes yes yes yes recommendation problem decentralize contextual bandit news article payoff agent contextual bandit exploitation phase instead phase contextual form way utilize iii content learner need request content set event happen sequentially context match content iii content action nd age gender map establish normalize age gender set business music formulation operate european news content manuscript content source locate region context gender locate country access content local country request locate content discover business content recommend construction news price distribute music classical allow music recommendation instance user music track recommend addition music two type characteristic static change content content correspond scenario dynamic especially social medium content hence
hence sum z similarly q norm multiplicative f equality hold last assumption rely let technical let e rewrite tail inequality follow deviation since identically hold combine statement obtain orthonormal unitary identically suffice apply n conclude partial derivative certain suffice unitary two vector proof need prove easy
le n c h l hc p r mean r r distributional risk cc cc estimator lasso shrinkage part estimator estimator evident hypothesis preliminary find conduct simulation respect different generate multivariate range scheme way vector setup generate hypothesis realization obtain square risk relative whose compare estimator setup slightly indicate indicate translate mention
repeat time letting drop two policy lemma simply apply admissible hold lead repeat reason contradict term hx complete proof simple less apply apply least
inverse negative log mode subset way rough posterior black solid gray discard approach demonstrate valid reliably quantify gps check ten parallel chain mean solution vector ensure fig median percentile ten reveal computation conventional census use regression house region composition market experimental condition cope datum compute five chain run machine eight ghz processor graphic graphic carry ten day independent hour novel infer carry gp comprise day machine
real would schmidt datum directly subspace seek onto current iterate get see explain rank combination towards discrepancy monotonic size choice q decrease uncorrelated angle wish several comment bind hold monotonically loose compare close empirically determinant decrease discrepancy choice determinant norm datum iterate make progress determinant discrepancy distribute tight phase two approach allow phase initial determinant frobenius
attain discrepancy improve digital net expansion base expansion apply digit yield way permutation present probable permutation independent notice digit nest apply nest nest apply digits proposition base sequence quadrature uniform digital use replication obtain directly surprisingly mt remove carlo nest net reasonable informally speak power net bad plain nested base e van construct begin inverse van van low base digit determine place digit place triangular sequence measure van discrepancy notion splitting split except intersection fold borel interest overlap unit partition
angle order sample matrix conjugate transpose operator sensor orthogonality span signal contain eigenvalue diag element subspace form decomposition angular subspace use divide block linearly exist subspace p inverse use
property asymptotic raw multivariate low frequency overcome simulate asymptotic series asymptotically uncorrelated fourier frequency estimator smooth gain analogous series assumption nonparametric generate uncorrelated distinct multivariate true k value asymptotically consistent density smooth iii function naturally call directly fraction coherence ki kk estimate dataset science pressure datum produce weather generation forecast forecast hour increment hour h control dataset region region approximately hour increment day increment band define region quality forecast coherence forecast example forecast well decay horizon expect short forecast long forecast begin forecast subtract
computation graph management manually lstm lstm underlie relu test increase unit latent mnist available computation somewhat architecture localize read autoencoder proceed step previous combine mutual marginal equal part perform inference experience perturb guarantee remain present perturb basic imputation imputation trajectory imputation trajectory largely policy primary policy uninformative initially policy toward gradually improve convert connect broad generative sequential improve idea imputation perhaps investigate unconditional
practically oppose single cg easier visualize broader stack collapse mathematically grid define single describe grid merge new dataset refine training initialization consistently likelihood cg cg basic convergence clarity collapse grid regular propose procedure relate indexing usually even learn compare human science year one step remove stop show count window sufficiently model times average word token inferior report diversity curve show tuple diversity clarity content evaluate straightforward sample grid location obtain tuple repetition tuple tuple allow check clarity content quantify term
envelope trick demonstrate utility theorem example generalize also enable formula level coherent cost envelope saddle formula prove optimize continuous several hold regardless considerably simple capture variability coherent follow devise replace expectation average proof proposition saddle plug lagrangian saddle point analytically may define replace define efficiently interior let multiplier eq involve condition supplementary set saddle empty ii function continuous assume nz assumption proposition mild note iii mean summarize exploit coherent measure envelope theorem likelihood style coherent risk sub routine treatment dynamic coherent dynamic risk approach static formula theorem sensitive abuse notation markov coherent
classification fig describe two important aspect reach information reach exact connection interpretation relation classification information conditional divergence although fig table problem measure extraction machine recognition datum sense dissimilarity important describe relation correspondence empirically cf therein achieve define
feed hashing label benchmark quantitative evaluation unsupervise method iterative quantization learn mini batch overfitte respectively ground create item sharing query item without common item one report imagenet dataset million convolutional connect hash pre train hashing method compare carry use fine social annotate object moreover label concept salient query database training image pyramid descriptor wide relatively concept challenging diverse semantic website image learn
reconstruction optimize weight decay momentum unlabele provide way generate output pre deterministic probability deterministic way sigmoid output train decide much fast dataset label label human
unobserved factor incorporate corpus text relationship text affect composition date political rating overcome incorporate data sentiment introduce multinomial inverse use inverse conditional sentiment inverse multinomial mixed distribute dimensional modeling document structural topic linear incorporate level paper exploration document interesting model close approximate tractable beta implication jensen inequality leibl divergence give analytically expand expectation obtained denote discussion coordinate algorithm update respect gradient prior variational message pass method natural form however guarantee resolve issue variational step reduce positive increase otherwise former update nest loop cycle update pt update converge q eq cycle update convergence ij ij ij ta algorithm scale approach classify common node iteration minibatch optimize ascent converge objective document trial trial trial document illustration sampling initialize
sub da operator drop subscript update size update nature estimate inequality take variable also definition finally q condition recall true ij give ij derive refer scalar derivation combine dropout nn q heavily layer backpropagation wise layer represent note abuse notation subscript run distribute correspond layer k f layer sum get follow eq finally theorem layer dropout sample dropout iteration would parameter share iteration share q inequality datum operation proof exactly dropout eq exceed eq solving going dominate three correspond instance mnist digit recognition image part mnist http
prove correctness formula gs rule change reasonable select way expect gauss would eq case intuitive selection separable case harmonic hessian divide harmonic notable furthermore interpretation working process finish correspond fast worker interpretation gs provide benefit mean fast task alone intuitive scenario gs benefit worker worker together worker slow work benefit q typically avoid shift convexity gs selection scenario logic choose worst distinct coordinate coordinate remain discuss incorporate whenever optimization able numerically even search form coordinate optimization equivalent practice coordinate optimization well gs distinct exact coordinate gs row guarantee exact coordinate improvement simply alternate consider minimize
consistently ranking share order pair effect solid separability row novel solid angle define angle novel row replace jk ki close c union therefore jj topic combine solid angle novel separate angle non define maximum j require neighbor require two k distinct error accumulate distinct novel denote ideal row row constrain access separability establish j kk ex denote kb circle minimum first k scale j k k order learn kp prop jj prop denote k result eq combine row pair I constrain ranking
approach multivariate measurement wind weather write coefficient associate th time lag noise let wind speed format find explain appear one weather contribute distinct coefficient model entry cluster call call recovery signal nm nm nk na infinitely candidate certain
agree intuition always drive argue technique tool q expectation quantity sign e cause eqn presentation mean approximately figure causal agree lag cyclic near small lag eqn lag appear nonlinear dynamic dynamic neither discuss exploratory dynamic physical system source discrete use value physical expect weight unlike noise dynamic numerical use ode series create interpolation time require ode cc ode analytical start agree intuition report calculate agree consider physical source ode solver setting would agree peak peak peak peak peak peak peak peak peak peak would lead agree
model come prohibitive computational cost uncertainty parameter require reasonable estimate unnecessary mlp mathematically approximation stress simplify applicable use constraint reader review variational dropout softmax loss dimension output mlp corresponding mlp optimisation result objective often input layer unit drop value pass derivative deep allow b example function section appendix deep parametric model dimensional layer map I
dimensionality exactly go partition converse ssc union contrast ssc require say ssc regime optimization subspace application input norm solve pick angular subspace r adopt propose variant subspace cluster resemble intrinsic difference identify subspace due subspace angular distance subspace canonical angle angular dimensional key eigenvalue impose singular assumption constant th thought version noiseless also position margin slightly version present adopt restricted datum
cnns suggest region interestingly small cnn size cnn initialize weight appear imagenet substantially cnn spatial pool imagenet perform well cnns suggest tune eliminate cnn perform margin small approach drop category perhaps ii training help cnn show cnn pyramid cnn header cnn cnn body retrieval mean compute average formally version retrieve equal document higher relevant retrieve divided number retrieve determine first dataset summarize tuned cnns follow tune interestingly imagenet descriptor perform descriptor approach pyramid
crf finding marginal equivalent computing though general tree correspond pose find equivalent structured marginal map programming tractable local polytope local term arbitrary parametrize allow broad score global extraction structure function constrain negative maintain linear marginal provide complementary interpretation tractable yield precisely variational motivate second help characterize output augment objective clique form lagrangian stationarity collect relate polytope proposition stationarity characterize joint distribution mrf even namely dual ultimately configuration infer proposition parametrize mrf parameter avoid predict locally maximize yield feasible introduce act yield learn parametrization global mrf perform characterize
enyi divergence terminology approximation conservative attempt mass mass turns factorize say minimize find imply central consider problem follow immediate log find close factorize r specific divergence polynomial time whether like necessarily exist log counter claim structure exploit write sum simple function q submodular set either potential ground set bipartite connect node function enjoys write interested discuss exploit descent look clear variant propagation approach specialized idea approximate factor completely factorize procedure factor replace
jump autocorrelation approach rwm mala hmc square component need appendix suppose value generalization markov definition exist becomes consider squared case proof n define e condition meet cauchy schwarz simple algebra yield situation acceptance rate derive eigenvalue small proposal mala euler langevin process satisfy equation desire distribution expect discretization process preserve desire mala identify prove follow mala matrix splitting mala apply mala theorems recover omit eigenvalue mala mala tune result practice
combine discriminative generative image reconstruction generative b li try bring flexibility challenge contribute line model recurrent neural mixture specifically form variant particularly successful modeling work generative ability range condition hidden neural mixture image instance image h connection north connection connection connection h south draw draw connection east west connection north south connection h south west east west xshift draw connection v
reasonable optimistic practical improvement modulus parameter modulus claim time sample sample hour single equip core intel mostly compute naive different problem enumeration condition remove always rejection part gaussians though q constant q truncate parameter hand unconditional acceptance sufficiently add product coordinate correct line hoeffding prove I assumption execute time hoeffde unconditional solve modulus apply sample independent therefore kullback leibl error reduce enough easy subsection reduction simplify decode factor really lie lemma assume gaussian care dimension modulus distortion negligible probability oracle summation formula q poisson summation formula
choose expert original criterion bic criterion model set adopt identical leverage apply way plot normal normal outlier case compare laplace result fit affect may attribute proportion affect partition provide previous fit quasi datum outlier differ fit situation outlier show mixture fails affect propose leverage impact fit proportion htbp cc data outlier actual show likelihood show ten add left bottom outlier coefficient component heavy component c world surface temperature surface temperature resolution reasonably establish datum present update primary department anomaly compute recently two expert component provide period slight temperature anomaly expert minus twice estimate pointwise log four model also laplace expert anomaly set bottom right year response anomaly anomaly leave anomaly twice cc temperature upper anomaly model quasi identical see skewness zero skewness
arbitrary know cyclic construct form prior lower asymptotically optimal dynamic programming density interval fall seem gap treatment care take subset infinite notational convenience take
recursion step step expression contraction projective contraction property operator aim invertible conjugacy ii thesis map introduce underlie contraction model refer word markov
therefore advantage bayesian author frequentist hoc due may principle par cross validation bootstrappe generalization mathematically interpret optimize validation formally sufficient predictive central motivated work repeatedly instance always training lead model one approach predict consistent nest rise unnecessary predictive hoc device hyper result analysis true
although provably practice conditional without nuisance poor arise convolution kernel deviation literature reference li dependent keep fix poor include air atom allow location variable international peak daily peak demand height peak observation fall peak peak delay day peak conditional arrival height induce difficult lead limit applicability overcome adopt regularity recall stick breaking ratio finite predictor construct cumulative cdf reciprocal probability pdf transformation build transformation allow researcher normally model simplify latent rather heterogeneity control dependence possible express degree knot knot knot knot weight piece skew ensure identification ar retain conjugacy normal prior spline predictor build hierarchy parameter fx bx enforce specification x moderate normal accurate conjugacy calculation recall take replace
ef lf ef lf stable respect map self outperform method include consider among small lr result label classifier fuse representation generally uniformly well original ep lr svms poorly outperform representation fusion opinion happen projection concatenation representation concatenation e fusion group fusion induce among row blue light green place right fig e explain effectiveness difference performance lr classifier produce co obtain denote ef ef lr lr detail performance strategy increase color line ccc scene scene self inductive lr training inductive five map improvement respect ep recall round always round co add wrong label result concept drift represent label tend homogeneous class label image lr conservative round
linear application accuracy representation structure solution model however current mapping tool use map appropriate promise spatial room bivariate suggest basis approach global fact framework bivariate spline degree polynomial triangle automatic triangle method able effectively manifold spherical apply mat ern manifold smooth mat ern however difficult implement high representation bivariate smoothness impose constraint potential smoother smoother difficult still implement bivariate spline smoothness bivariate spline provide whose coordinate uniquely
guarantee coordinate algorithm op algorithm iteration one prove proof equip see total call execution due style maintain mention pass partition vector round warm coordinate computational detail argument detail start simultaneously version inequality v thing composite reward estimate tight control deviation deviation leverage lead reward inductive probability empirical inequality actual playing show apply regret associate regret smooth algorithm regret satisfy satisfied history efficiently iteration store burden check contextual bandit create come rescaled regret context essentially everything check end sized large constraint bind base potential unnormalized fact potential violate shrink positive
terminate generate simulated drawing reliability dominate worker select relatively small worker even entire available line find tend bad correct estimator worker reliability tb b variance reliability influence worker show vary budget increase accurate worker make decision worker select worker addition decrease show fix worker high increase improvement worker performance improve worker affect worker dataset collect crowdsource platform amazon worker ask
note office room room especially cause narrow massive corner room office room office room office room work compose vector field crf slightly well ccc fr average ccc multi regularize implement fed guarantee experimental lead effectiveness layer perform semi show help classification keep tree achieved achieve preserve future type representative discriminative institute china laboratory mail edu classification ability possess nontrivial classification attract artificial inspire propose end deep contribute accuracy compose
ease sampling evaluation discussion ease notion consider error proposal threshold closeness abc draw augment approximated intractable intractable posterior region therefore like ii assume constant marginal closeness indicator alternative article sum abc algorithm statistical feasibility avoid usage abc summary problem experimental fact result rejection propose slightly typically couple dynamically decrease reach moderately surface produce acceptance rate burn start constant burn around maxima
answer aggregate set vote decision tree vote classify million people classification galaxy classify people galaxy project already galaxy formation galaxy classification level accuracy annotation context galaxy online international organize galaxy capital platform hold competition one galaxy image galaxy competition galaxy full goal survey total number limited imaging elimination uncertain category colour elliptical galaxy proxy would purely colour galaxy decision competition datum vote transform probability high colour position explicitly competition probability oppose determined rmse prediction set rmse put emphasis question tree probability classification certain build bias galaxy correspond discrepancy participant platform automatically score public evaluation compute public score immediately reveal competition score competition final participant image could private new technique artificial decade neural initially star galaxy discrimination galaxy recently estimation galaxy extract limited surface log type radial profile svms typically dataset handle least parameter fit availability survey trend image feature galaxy feature originally galaxy attempt form extraction neural network svms raw work work g require hour engineering
network balance norm incoming roughly example randomly generate balanced mnist unbalanced weight rescale unbalanced figure weight unbalanced change compare balanced rescale w measure network group go type simple group correspond regularization effective relu regularization income similar bind norm income output unit feed correspond decay
covariate record time submatrix compose covariate element state diagonal contribution individual initially column state initial initial initial capture distribution estimate covariate entry entry process restrictive time geometric process semi markov survival define however semi survival distribution shift shift state survival cumulative conditional transition probability probability transition determine long markovian state comprise visit e transition homogeneous semi model geometric shift denote shift represent failure occur correspond easily include
iteration may contraction repeatedly round therefore consequently iteration backtrack stepsize move iterate outline define backtrack search stepsize x monotone auxiliary lemma provide fx probability shorthand yield add obtain fx lemma low part auxiliary proof prove two inequality hold bind yield final complete prove remain sub paper analyze sketch randomize independent sketch twice differentiable various sketch include use barrier constrain combination sketch interior fast body optimization newton sketch always complexity moreover either denote barrier much large newton hadamard suit parallel environment processor decrease central computation sketch significant specifically complexity scale would lower threshold access
pca relate variable matrix relate variable estimate measure observable redundant however perform drive examine estimate correspond measure constraint redundant illustrate flow process covariance pca measurement three order obtain indicate variable constraint estimate constraint principle coefficient flow estimate report table pca cc sd pca simultaneous identification knowledge relationship relate subsection extend pca partial constraint describe unknown specify identify component estimate project note satisfie correspond project combination singular almost project matrix correspond transpose estimate constraint constraint matrix example demonstrate covariance variance singular value constraint rmse estimate report infer utilize constraint obtain angle space degree estimate indicate incorporation method make regard
encoder rnn inference learn sequence probability impractical combinatorial size symbol thus prevent solution computational deal costly hull attention mechanism add computational capacity entire recognition rnn generative attention rnns neural use attention entire notation purpose rnns use multiply activation attention vector softmax length mask input decoder
outli score conditional detection subsection describe five scoring metric outli outli n notational us quantity nm datum dimensional define outli score metric first scoring interpretation complementary widely outli technique variant mahalanobis maintain process score eq covariance outli norm score along response outli l complementary outli score em score outli outli factor extend neighbor essence summarize density score find margin datum slack define instance boundary estimate location decision metric convert percentile evenly range score outli validate demonstrate response improve detection
cut lambda implication whose procedure phase dedicate estimation weight cut estimation profile estimation lp category h problem classification require dm linkage structure ordinal sorting problem whose strict preference relation criterion method dm want find classification assigning category
either increase optimization multiply optimum corollary definition research microsoft research contextual armed global context concave dependent important optimal generalizing version budget match answer classic exploration exploitation cumulative environment armed bandit allow action observe take action observe take advance successful application major lack world action arm level consumption certain number price certain price sequence sale limitation resource budget constraint application one price crowdsource capture resource pre agent resource consumption reward consumption budget bandit reward resource consumption round confidence ucb technique guarantee non contextual bandit context dependent remarkably access linear enumeration previous computationally achieve
incorrect significantly degradation increase correspondingly green accurate set reveal perform single edge base somewhat whereas play predict parent location birth body infer generally consider relation focus clause entity implicitly meet criterion entity relate keep capture imply meet cause put incorrect entity empirically also job condition fix define angle corresponding measure inner product entity capture compositional
search perform poorly age ordinal web estimate large behavior multiclass method regularization somewhat entropy aggregate crowdsource worker probabilistic distribution label infer entropy condition item generate empirical crowdsourcing validate aggregate minimax entropy structure different protein speech worker structure confusion acknowledgement thank contribution chen discussion derive minimax substitute equation obtain definition depend put piece together dual regularize sum write lagrangian l kkt maximize q maximize respect substitute lagrangian verify dual objective solve problem group item instead update variable value optimization subject constrain
gradient cubic spline write fig noisy gp solid thin standard sample path bivariate validity weak wolfe middle three I acceptable joint gray wolfe dash search suffice close accept acceptable constant slope demand decrease condition wolfe replace figure conceptual line condition exposition model section add global search functional process popular expensive ill suited line search efficiency optimizer need follow add overhead access noisy gradient gaussian
runtime cost constant practice perfectly without optimization group enumeration use experiment find near optima intuitively make code code lot initialization early relaxed constrain quadratic program qp speed special objective sum warm continuous qp previous step continuous minimizer instead greedy efficient suboptimal optimally already pick relaxed solution warm run alternate decrease monotonically enumeration involve cost roughly multiplication enumeration incremental evaluate code solution minimizer bind minimum scan code thus code keep code second evaluate stop soon exceed run easy appendix recognize reach keep relaxed qp qp initialization schedule use schedule fortunately simplify reason occur stop unseen
dnn correlation two modality propagation bilinear vocabulary extraction vocabulary hour clean along video frame vocabulary audio video audio extract nine consecutive lda audio neural frame audio visual start detect utilize scale extract level coefficient region discriminant replicate
experiment dark aspect use study use complex teacher know provide rough guide model exist teacher use related rnn however employ rnn still employ dnn teacher guide basic teacher rich boost often simple teacher knowledge logit encourage activation norm dark encourage teacher transfer learning dnn
privacy attribute adversary asymptotically mechanism say consist dependency overfitte add regularization shrinkage ridge try account reproduce ridge regression estimate loss dataset weight q theorem form regression coefficient plug solve result get achieve predict regression make prediction construct define attack privacy noisy unknown try set attack semidefinite public less eq hold great h privacy attribute private probability establish privacy ridge regressor
intelligence interactive game condition intelligence many agent observe many agent environment exploit agent instead environment bandit agent provide player showing number think show provide make intelligence intelligence grant aid challenge exploratory research financial service com intelligence interactive armed bandit payoff one bandit good choose bandit agent specify threshold exploit uniformly learn intelligence conduct laboratory subject intelligence social armed bandit intelligence interactive exploitation good know bandit mab typical environment payoff
dimensional apparent come theorem discretize x xx suffice bound big generalize structural poisson poisson multinomial latter sample covariance relate x xx xx small loose bound approximate x structural argue unfortunately eigenvector ratio large produce approximate detail cover small cover discretize multi select throughout repeatedly set class start multinomial along give necessarily parameterized whose row define probability draw return column histogram recover binomial distribution frequency may multinomial dimensional zero vector law identical row sum column vector whenever column make total tie
neighborhood compose categorical disease compose feature diabetes cancer feature property cell identify contain classify ex house vote dramatically pair power cost power function pair f accord subroutine rt rt chosen threshold rt rt set strong preference balanced theorem proposition reduction acquire additional user specify acquisition budget forest tree cost forest grow acquisition cost strength base establish near optimal guarantee benchmark demonstrate art surveillance retrieval expensive complementary namely acquire often acquisition maximize learn
fold frequency alone miss fig discover little see fig short list prediction corpus length present frequency c york walk piece mind right take I go go never note long know check http person check lost keeping pick every self dr order pick post take live http http video video
series swap observation daily price million point default counter fix price arbitrarily pm stock trading fix price avoid spurious arise price intra market correlation inter since trading hour series assume price follow walk increment aggregate look information cluster precisely deviation fig
complexity enter cover simply size cover discuss unless integral finite upper cover pay rate aggregation finite plus pay convergence erm ball compute quantifie arise loss may balance q present balance function provide analogue class cover contain universal constant offset abstract class class class original precisely significantly large sense critical offset
computer filter markov condition perhaps brownian motion regard diffusion boundary easily solve transformation boundary boundary diffusion regular develop handle single boundary difficult extend boundary boundary simulate approach neutral diffusion solution drift rejection target process candidate expect diffusion drift interest discuss extend many dimension give brief paper detail simulate diffusion assume
form one bind error easily simplicity simplify first generate therefore chernoff constant high turn
certain rejection event condition rare region exponentially whose grow chain monte manifold unfortunately still suffer fix imagine dark blue imagine volume region generally consider dot search subspace helpful sphere jump take sphere mathematically exploit sphere become blue dot high great circle ambient space subspace sphere great intersection green dot curve blue intersection blue worth distinction angle intersection intersection angle weight distinction traditional become inefficient accordingly algorithm focus circle knowledge isotropic red circle angle work go weight angle specifically become arbitrarily therefore fraction green contain portion probability cause intersection slowly situation intersection independent approach mathematics integral geometry cauchy formula intersection work choice subspace nearly orientation north orientation almost sample sample favor east west orient orientation favor north south east suggest situation without search use geometry numerically thereby dark metropolis subspace light center sphere make convergence dark perform eigenvalue rare event histogram weight weight figure
proceed thresholding difference take turn development lead linearity bind expand around remainder consider derivative dominate continue interval establish remaining check direct analogous obtain threshold deal hard thresholding example gradient smooth author thresholde interesting investigation minimax make constructive relaxation computationally sequential shorthand bound take henceforth range understand infimum sign quantity infimum indeed minimizer
policy provably bandit bandit remark slight replace effect provable significant modification bandit know connect way however follow problem therein work reference therein reference knowledge outside develop finite cyclic implement exist herein variance recently
subsection reveal claim tight truncate least eq plausible together take arrive definition proposition constant justify contraction fall establish guarantee concentrate stage stage collect shall inequality truncation rule arises discuss estimation error regularity hold shall regime prove regime contraction noiseless long fortunately move possibly regime guarantee jump summarize obey geometric either stay regime jump error exceed namely justify truncation cauchy sufficiently scale since analyze separate inclusion e I since c one proposition obey arise last consequence well weight take bound yield since establish theorem substitution complete goal establish kullback result useful hypothesis collection q word exponentially around yet see hypothesis distance hypothesis center say logarithmic quantity vector obey q inequality occur hypothesis rescale new hardness select turn connect keep generalization extension point general
policy upper usually addition close convex addition q satisfy vary ensure consist error delay theorem case becomes evaluate asymptotically nonsmooth problem optimization show convergence delay vary instead asymptotic delay obtain rate subsection restrict composite strongly strongly include convergence rate bregman next read strongly iterate q regard composite strongly asynchronous mini match achievable serial
net leave weight metric layer loss lipschitz norm quadratic weighting lemma bind metric select unknown fortunately uniform intrinsic complexity unknown follow size bound negative measure sequence classifier base error particular f encodes absence prior belief weight underlie complexity weight metric highlight weight return proportional cf intrinsic metric lower intrinsic well partly explain empirical success criterion metric design optimization minimize error regularization proportional explore efficacy analysis regularization adapt want effect select popular near classification quality algorithm implicitly margin explicit regularization let practitioner
convergent admm imply hold monotonicity note proof theorem conclude bound cluster whole converge extend admm make strongly ease presentation apply value admm q subproblem rewrite eq present subsequent admm assumption block subsequence use k iteration x let continuity f k x
mode deviation people patient observe statistic heart conjecture extremely heart might characterize item plus deviation minus deviation natural question account answer outlier firstly notice fluctuation balance patient item large plus people stand outlier due balanced noise percentage standard close much patient involve probably conclusion plus time
additionally loss function equivalently q stand usual see later sequence random respect straightforward loss loss fundamental cf proof go computable call constant perhaps multiplying constant loss regard
recognition unsupervise pointed table estimate ie ie ie ie ie ie ie ie top show latent letter vertical axis represent sampling inference work datum adequate tb simultaneous acoustic continuous speech unsupervise manner purpose generative hdp extend hdp generative extend hdp enable robot language acoustic speech experiment synthetic shown infer embed experiment sequence result conventional stage sequential baseline challenge language language natural signal child speech future extraction extraction gain integrate deep problem intel ghz particular gibbs latent duration improve accuracy acquisition paper language acquisition obviously language acquisition suggest make occurrence accuracy acquisition hdp therefore heuristic advantage plausible constructive acquisition direction beta college science em ci ac school engineering ci word directly novel purpose model
encoder convolutional hence think train unit successively differ train single auto whole decode train generate experiment train exploiting generator reconstruct digit explore role feed encoder feed reconstruct exchange display reconstruction motivation experiment one digit aforementione change shape display result reconstruction role store analysis pca encoder move along major component integer use inside
important algorithm algorithm component analysis characteristic subspace take principal component crucial quantum previous principal quantum base schema output subspace create component classical
lstm represent output forget state properly transformation sigmoid tangent stand wise lstm bottom corpus assign membership scene scene vector purely infer train perceptron predict image target output mlp layer size last train cnn cnn cnn optimize predict location feature connect scene vector lda new scene predict image english treat evaluation randomly early monitoring construct training lead evaluation follow protocol training
verify optimal strategy suppose order minimax strategy game thm I fig elegant function similar case monotonic nature approximate increase predictor act relative ensemble indeed example quantify benefit oppose classifier prediction uniquely relationship ensemble determine relationship easy insight rate notable exception ensemble thereby quantify benefit vote predictor unnecessary predictor tighter incorporate accord
match closely use complete match f r pt c stocks cl stock ten stock top principal stock pca stock order stock r sparsity mix sample run significant performance recover biased toward outperform sketch actual sparse principal pca observe sample optimal respective sparse follow hybrid irrespective choice well computation top
lc accuracie five classification list clear increase average significantly bc bs tree size consistently average respect tree bs bc eight relatively lc propose work relatively bs tree respect size b high show accuracy dataset dimensional problem experiment
reweighte project finally descent apply scheme optimize keep line wu paragraph uv original algorithm order take feasible suggest least approximate w diagonal entry diagonal w nu line objective differentiable term approximate gradient singular since large several constant priori know coherence localize difficult scaling multiply propose respectively range parameter high would give coherence generate see course local coherence
hypothesis act increase estimator nan hypothesis relative superiority lasso indicate carry package study datum generate estimator present visually various estimator separately various figure horizontal facilitate among indicate superiority lasso relative efficiency finding summarize indicate dominate estimator function figure around however dominate
categorical overlap importantly fig evenly feature cluster varied increment procedure tuning find range search fig recover fig categorical increment dataset categorical dataset pdf pdf setup k advance heuristic cluster find initialize iteratively round distance small point round distance respective heuristic evaluate objective objective entropy mean randomly input initialize find cluster center tend separate highlight dp mean importantly conduct
gamma student hx function student relation function right plot characterize dominant vector value rank definite random interpret relation motivation design laplace establishe transform integral real contour region convergence characterize trivial analytically mode approximation find mode laplace minima derivation laplace appendix denote term normalization find
read eqn realization snapshot derive sec record sec observation snapshot def snapshot def prop derive snapshot derive prop derive structure without index def skeleton def realization def dual begin formal statement weak learning capability throughout produce agent system turn mathematical hence toward much situation calculus set basic trajectory define abstract transition transition fold refer trajectory map refer transition transition possibly implicit fashion refer run endowed sensor finite assign sensor system macro sensor trajectory trajectory time avoid value super come endow satisfy virtual sensor trajectory subset use range database maintain record sensor record encode sensor requirement translate treat planning inclusion positive implication order force replace requirement weak time interpret eq hold sensor sensor identity terminate state mean transition kind statement essential planning informally implication interaction boolean implication maintain set consist partial compare experiment characterize encode subset incomplete selection remark space however redundant implication contain observation coherent pair topological space skeleton skeleton successively cell choose vertex coincide join vertex condition graph appendix topology space regard capability unnecessary return possibly give incoherent represent raw resolve current keep requirement appendix complete technical agent contradiction price record knowledge correct current agent basis architecture loose notion snapshot database structure require vertex abuse edge snapshot snapshot vertex assign denote learning carry subgraph denote original motivation snapshot representation assign coherent implication suffice quantify relevance e frequency snapshot illustrate ab graphical snapshot automatically orientation ba abuse symbol direct point weak closure orientation cycle mainly deal acyclic snapshot trajectory trajectory agent indicator evolve represent cumulative identity indicator eq identity motivate probabilistic snapshot snapshot coherent probabilistic confusion fundamental snapshot satisfy orientation eq direct appendix put snapshot graph snapshot setting acyclic iff imply orientation every apply element part update snapshot set
fall shrink ball radius stationary local usual penalize illustrate derivative ordinary square conclusion still follow tail see stationary ultimately convergence interpretable sufficient condition section proposition suppose bound highlight ordinary least hold differentiable nonconvex careful huber proposition assumption simplicity establish fairly assumption application unweighted hold gaussian nonetheless hold bound furthermore odd far highlight aside possible symmetry heavy tailed distributional setting contaminate sub sub leverage define lead decrease bias rsc proposition fashion statistical strong require rsc condition treat curvature exist assume choose take usual equation eq probability tail via may sub outlier contaminate decrease small agree intuition contamination deterministic qualitative describe behavior rsc select estimator rsc condition hill rsc weak suppose draw sub exponential bx suppose bound suppose hill bx satisfy presence distributional requirement requirement impose proposition weak bx sub proposition rsc distributional covariate bind radius requirement explore proposition heavy tailed outli sub precede two subsection stationary whenever suitable assumption distinguish aspect lie concern function er condition minimize asymptotic mle influence reveal section oracle penalize robust estimator stationary estimator agree oracle attractive stationary
well bold shift exp book wikipedia application sec critical assumption intend describe per compute give deviation word fitting number parameter two function shift law database well next consider distinction fitting division formal point view procedure correspond free parameterization fitting test validity linguistic linguistic law however rank interpret drawback process ranking introduce estimator rank ranking rank different one affect bias one large rank contribute law negligible bt instead unnormalized frequency e occurrence word database ml simplex fit straight fit obtain calculate eqs english books frequency linear
structural propose filter ff bs exhaustive enumeration simulate ff hmms practical x w generate power instant sum power device sequence detail si model presence together parameter block gibbs sampler three tumor deconvolution hamming auxiliary bs posterior figure scheme sampler particularly sample high configuration sample close latter block signal variation finally time cpu advanced sampler sample efficiency si sampling density noise hamming ball statistical involve space generalize
furthermore consequently dimensional proposal distribution another arbitrary high motivate problem particle inference bioinformatics particular inference integral kx x state sequential manner expectation distribution x interested mm consider call spatio representation spatio monte review inference dimension limit strategy drastically dimension rarely proposal address require proper providing within proposal arbitrary three nested sampler
popular orthogonal projection orthogonal impose full orthogonality pca new multilinear rs sec orthogonality call accord semi orthogonality full orthogonality tensor orthogonal multilinear fixing small increase reduce experimental compete whole new strategy tensor background notation vector letter tensor g index index letter range letter
bb valid let get least desire discrepancy follow establish os enyi concentration independent deal chebyshev lemma asymptotic use last taylor least prove reverse chebyshev inequality imply eq go give nf statement real scheme simulate real numerical code com simulate illustrate difference rank score ground truth uniformly complete preference sampling without scheme repeat estimator record truth mean associate world edge probability performance random without replacement random case small grow scheme greedy without graph dense
response predict child height kind inaccurate generalization unify glm glm relate term explanatory variable option binomial success expect q invert link option response variable option avoid explanatory spirit observation explanatory variable response observation assignment correspond motivated statistical model weight class introduce multinomial constraint concrete functional appear softmax decision reinforcement option rule select option stochastically select limit standard option high limit option equally probable option form unknown identical literature refer allow softmax schedule slightly softmax temperature example simulate good functional represent form softmax learning assign pick agent decision function option similar fmri initial value know nonlinear form may transformation put softmax solve provable guarantee convergence softmax model provide converge softmax observe estimation likelihood maximum logarithm interpret likely model adopt
quantity q iterate asynchronous schedule specify size normalize like highlight emphasis case qualitative difference guarantee accumulate epoch asynchronous modify one empirical experiment logistic correspond interested sparse ix update similar recall relationship separately update term scalar maintain need aggregate b sim center news evaluate algorithm asynchronous schedule read atomic
cat design item reality pool call modification scheme robust moreover require response ability strategy modeling address implement relevant open model corollary lemma corollary conjecture test cat question response accurately sequential design binary multiple full available cat heavily allow novel cat allow item infinity item asymptotically ability item finding support asymptotic assessment accurate kind latent trait conventional paper adaptive testing cat item response efficiently end cat technology kind report life
nuclear inequality equivalent programming solve interior besides use individual rewrite consensus global new constraint admm dual eq scale I update update scale update algorithm optimization gap method value straightforwardly easy htbp f lot generally guarantee q nonempty saddle point optimization
worker might traffic worker parallel submodular neighbor max delay subgraph ask ask server start request containing receive hyper load neighbor set contain choose live journal click internet company vertice bipartite bipartite bipartite undirected bipartite bipartite fortunately many efficient partition overhead formulate partitioning algorithm theoretical highly implementation dataset distinct friend social importance dataset partitioning np poor partitioning parallel
constant compare classifier different rejection reject fraction rejection classification quality reject option accuracy include reject marginally expense define operate require equal operate guarantee region base reject easily note equation classifier maximization accuracy loss concept quality instead maximize number maximization correctly classify reject incorrectly classify reject denote assign rejection rejection minimization loss become
genetic use software implementation structure cluster occurrence proportion correlation european however fourth explore another verification person additive count minor allele two snp count convert genetic pc axis reflect positively european american capture european pc european pcs supervise european pc pc consequently correlation cluster table third genetic method c cluster investigate genetic allele snp take new panel snps primarily contribute snp snps rs rare allele recognize allele population contain minor snps rare fourth cluster major population mean four major colour linkage infer genetic assignment random criterion
add focus last firstly configuration obtain expectation apply configuration negative verify combine inequality inequality q eq combine complete ac uk saddle structure cover framework saddle incorporate stepsize theoretically stepsize achieve compare since amenable scale apply regularize minimization state art set
matching minima coordinate plane typical surface human value good parameterization generate normal surface wavelet parameterization representative human surface depict circle correspond haar show axis ccccc optimal compression wavelet depict worth wavelet
alternate adapt change discussion fact optimize sgd near perfect obviously overfitte regularizer lead test marginally alternate provide reasonable ignore fouri one interesting certain kernel neural back sgd sigmoid relu uci dataset batch step optimize satisfactory
clean test test image consist come clean image respectively ground obviously dataset increasingly noisy feature test set convert feature classification pose image patch patch dimension reduce actually classification clean mathematical
mode proof mn outer element additionally vector accord compare remark xu edu sg rao edu edu cn proven foundation success practical view limit capability world hoc simultaneously explore numerically deal order discover cca straightforwardly naturally generalize handle number analyze aim correlation view crucially approximation tensor solve efficiently different view explore reliable addition extension present various challenge task web annotation effectiveness mining task dimension extract multiple view page usually web classification sift descriptor image dimension seek low compactly heterogeneous
amazon per da train label adapt dataset subject neutral available pixel classification repetition neutral neutral neutral subject align expression three class give work dual effectively three domain space classify additional high require subject align reflect classification subject little trait subject cccc subject introduce exercise address learn call align pair
accord whenever graph path connect separate connect separate path separate suggestion component association conditional subset measure many beneficial partial specific covariance counter unless certain article subset conditional independence markov read thus various graphical article relationship qualitatively kind comparison condition keep fix partial submatrix mutual information information correspond qualitative apply well condition square compare change information graphical markov separate lie dependence model path meet ie separation see ensure ac criterion however
neural predict handle multi task visual classification view multi object deep fan optimize mention task visual label class recognition former peak signal ratio addition denoise improve digit testing test denoise fig noise structural digit structural fig heavily image handwritten digit capital multi calibrate camera place high two high rest annotate consecutive frames frames situation view frame accord
select limit correspond quadrature day modern cpu instead formulate problem drop factor present neighbourhood explain particle orientation shift equation become log regularize least mixture orientation slice typical mixture component frequency share many high suggest behave long coefficient motivation consider restrict behave higher fouri gradually explore acquire provide cell resolution particle second
size os figure behavior figure hyperspectral image hyperspectral mean os identical error reconstruct os especially low five superior propose htbp os os scale htbp os os os asymmetric comparison r r medium comparison htbp os os square htbp b f hard hyperspectral image os mean square theorem section theorem department electrical school system communications li riemannian numerical show section depend properly structure end focus
requirement first satisfy code assume condition independent bound apply prove technical proposition meaning become recall lemma help correlation define function gp pp part lemma proof gp prove something look lemma linearity suffice column randomness difference firstly former suffice independence lemma rgb bind
provide derivation paper optimize cnn introduce
vector transpose imply hx l h dx hx hx contradiction anomalous leave class consist anomalous dx dx exist pointwise thus generality lie x n cd dy h n n dy suppose
capture sequential nature capture sentence sentence incorporate neural recurrent network convolution level interaction learn output fix retain depend hand self maintain inspired recursive convolutional neural pyramid direct gradually compose intermediate representation phrase recurrent recursive nature flow unlike pyramid representation pyramid adaptively depend illustrate architecture compare recurrent neural network network summarize novel short explore new multiscale
approximated nominal bit nominal level follow closely exponential distribution htp c n exponential table display alternative percentile confidence power proportion time interval power conduct present similar size apparent get c ccc pt ccccc ccccc ccccc pt pt pt bootstrapping technique rank order clutter denote order significance choice
characteristic fourier transform x kx characteristic study detail literature hermitian also logarithmic match absolutely lebesgue om sure consider retain characteristic vanish consequence would nature hold aware use norm supplement compact r show grow fast show prove supplement r improve factor provide well diameter om dm corollary appropriate latter even fouri
also maximizer j replace lemma obtain diagonal leave replicate k maximizer plus short formula maximizer file supplementary material manuscript fit curve condition red nk c red c nk correspond fit tool build user consumption improve energy curve consumption expect consumption weather intensive consist smoothing reduce exploit simplify flexible fit limit shape consumption show usage
successive discover community edges practice incorrect background analyze algorithm transition community detection type stochastic external connection community generalize community community aggregate multiple validate transition empirical threshold apply estimate
score leverage strategy unbiased frequency experimentally enyi star result recovery signal signal generalize concept process bridge connect function theory perfectly recover recovery year variation sampling theory perfect recovery experimentally design sampling graph recovery sampling experimentally design
high benefit expect new iterate ir train manually annotate unsupervised corpus moreover show unsupervised parse lexical semantic lexical semantic syntactic walk walk may recognize similarly decide parse parse train annotated ambiguity cost increase difficulty ir manually annotate employ outperform corpus paper lexical unsupervised parse bridge connect research area unsupervise parse supervise counterpart avoid confusion name worth note supervise first outperform
prove group symmetry numerical decrease cost constraint alternative solve author impose estimator semidefinite relaxation problem propose consistent nevertheless demand contrary formulation derive enjoy toeplitz impose covariance arrival entry diagonal since convex construct surrogate convex surrogate taylor equality concavity function bind surrogate consider possesse function robust structure positive definite matrix since function invariant constraint equivalent conclusion follow subsection lemma show update eqn sdp
represent document rbms variant softmax hide vocabulary equal dictionary rbm assigning document energy share rbms refer count document free analytically integrate normalization hide equation softmax multinomial efficient word probability activate ml intractable divergence cd
result individual little proof book solve necessary requirement student exercise every group platform together assessment criterion online template try place way student read student individually self ii student double blind result tb right skewed distribution draw random box plot solution make reasonable pass moreover assessment exercise anonymous reveal half student student ta accurate recover adversarial
optimization machine general require require result perform perform hyperplane namely information potential risk quadratic negative logarithm q arbitrary learn linear define project ij j v infinitely angle vector
assume likelihood model call integrated difficult approximation approximation metropolis approximation mode free hessian covariance computed sample sample number component hessian mode estimation suggest ht c bf negative select bad factor simulate cluster simultaneously select rand rate estimate equal consider situation model cluster stability regard hyperparameter consider perform several factor likelihood rand actual partition compare ten initialization run generate burn remove high factor experiment mixture experimental protocol estimation framework non know consider extensive extensive monte paper experiment simulate six generate two spherical diagonal respective c mixture variation volume relate orientation mixture poorly separate separate achieve mixture result structure situation separation component
strong indicator identity therefore contribution high window parallel architecture classifier modality usage fusion address characterize mobile device period day achieve device minute fuse fire rate system associate technology bs ms university degree university computer engineering university currently associate interest around mathematical network implement test fusion architecture system quantify overall multimodal decision fusion active location application people american store mobile device email map location service yet take inaccurate discuss several percentage phone monitoring device phone entry fail recent
dual natural shorthand respectively argument regularity statement regularity sequence involve tune learning environment establish regret notion tu u tx tf f tu tu bound set budget bind modify learner advance observe establish setting well gradually horizon comparable assume minimizer scenario expert even round
much provide generally proper verify primarily wide evaluation bias coin flip user click natural product ask make
suggest gradient learner use tree purpose average generally result observation training previously misclassifie observation compute successive tree precede tree maximally negative gradient whole subsequent stage hard learner utilize beneficial intrinsic selection issue
straightforward computation recall tp dynamic programming define policy start thompson sampling induction decrease three devoted showing decrease decrease decreasing also decrease show satisfy claim every equation induction induction monotonicity thus thompson thompson prior regret thompson obtained true note past draw distribution depend first second follow true reward first argue always upper proven
relevant annotation nuisance relevant retrieval expression interested exact influence gene expression partition together similar pattern integrate level cluster co expression reveal retain clustering associate involve characterize retrieval characterize minimal central expression essentially expression gene case normalization carry house european bioinformatics institute seek show gene gene formulation explicit partition exchangeable furthermore
slice outperform study propose model parameter regressor compete way subspace extend norm shall definition take rademacher bad complexity rademacher nf prove without enable miss sake convenience sake bound rademacher guarantee upper g equation fact rd plug obtain substitute appear equation inequality dropping term gradient eq substituting since quantity parameter regressor simulation synthetic dataset compete context tool dataset constraint one dataset
ie I ie j program feasible bound optimal value optimal notational index forward indicator th also term entry q column accord cluster block equivalent sdp sdp satisfy sdp ab complementary equality b ax
tell conjugate advantageous flexibility exploit cover occur particularly suited change accommodate benefit incorporate location consist discuss occur multiple occur value select representative posterior q markov adopt routine dataset often million arbitrary update cyclic updating update change start arbitrary convergence proceed systematically span possible exclude value objective significant threshold stop detail derivation inferring imply variability arise miss capture pixel plot k modal pixel belong region code implement available material involve amazon routine need parameter
spatio spatio temporal video however conceptually compact ease simple spatio temporal modeling share aim scalable power experiment markovian transfer spatio temporal originally aim natural qualitative well formalism interest sufficient suggest new limit light cone speed poorly use end future fast nonparametric efficiency present property future regard randomness
account develop solution result focus layer enable asymmetric reconstruction rapidly accumulate multiple approximate widely use achieve whole speedup merely also accuracy degradation object detector convolutional cnns continuously cost increase great success wide task substantially early may suffer cloud thousand new request second device may like object thus importance accelerate cnns deep cnns one promise speedup ratio whole imagenet accelerate remain imagenet acceleration decomposition optimization response descent work character imagenet sgd based optima moreover solver accelerate error approximate multiple rapidly may exhibit great beneficial uniformly accelerate propose account nonlinear nonlinear nonlinearity importantly enable account approximate accumulate method determine acceleration reconstruction whole acceleration control imagenet furthermore
suggest convex well separation clear entirely dissimilarity write percentile cut search extra ahead step find behind dissimilarity represent basis add length dissimilarity axis dendrogram precisely value axis horizontal dendrogram correspond vertical remove step form dendrogram sl cut dendrogram remove give sub cut cluster number cluster motivation let cluster disjoint symmetric diameter k nk support consist finitely set step sequence begin go dominate term bound say absolute factor pre assign arbitrarily consequently complete item remain show contradiction suppose open indicate center radius sequence item rate establish theorem stem mostly choose mean ensure union approximate regardless support disjoint straightforward cluster euclidean distance similarity
np w inequality give arise whenever combine basically reveal iw iw w exceed precede l j j control magnitude inequality rely take union reveal least imply eq equivalently finally ready put presentation np lt nc give implication c lc indicate spectral output remain error satisfy universal product result upper w w satisfy notably divide stage phase constant definition hypothesis tw tw w spectral replace place necessarily replace immediate consequence establish constant sufficiently definition cf np np np small put inductive hold reveal q auxiliary candidate however omit accordance score mle initial minimal obtain spectral successively spectral mle identification minimal mle furth numerical mle encounter
similarly steady although achieve performance convergence steady analytical extended analysis expression evaluate scheme finally numerical diffusion rule network gain one asynchronous network scheme resolution partly project support partly support theory communication diffusion call keep fully estimate strategy convenient adapt combine theoretically implement case heterogeneous useful diffusion simulate stationary estimation problem become competitive scenario estimation attractive interest g diffusion present advantage failure definition cyclic run nod incremental consensus tracking ability reason network application localization
classify ignore clutter outside sequence classify translate mnist size attention lstm recurrent receive selective operation number softmax classify similar ram except refer differentiable ram error ram network patch step error ram ram ram ram generative study compare draw generative art ht except column close column network
size architecture train train configuration configuration momentum momentum control overfitte auto regularization code decode decay termination criterion validation use except epochs momentum coefficient everywhere tangent adequate every pre training square distance ground product shape inter ground shape shape specific database test equal curve plot vary usual comparison area auc configuration dnn
h h jj constant aa lead square root I computation localize energy satisfied observe particular use cauchy l lemma eigenvalue imply aa aa easy numerical implementation fully discrete discretized fine mesh write piecewise constant mesh square resolution mesh piecewise q finite discretization span square numerical selection subset triangle constant subset work index precisely write mesh generate hierarchy subset connectivity numerical example identify index interior resolution illustrate matrix form note particular note affect localization property non integral support core positivity element proof integral illustrate identify ce h gap necessary discrete constraint determine diagram method pyramid virtue exponential pde decompose localization interpretation form martingale hierarchy measurement scientific
forget independently allow child vector output gate block pass child parent merge cell reflect multiple indirect cell structure capture block right forget gate matrix hadamard product weight matrix indicate output gate backpropagation unlike lstm discriminate child discriminate child obtain formula list facilitate discus error pass gate forget gate right forget gate input gate derivative logistic computed abuse activate child
drawing separate trick respect minibatch trick term basically easier much gradient b variable hand effect lose appendix argument rigorous regularization neural fully dropout minibatch current layer nonlinearity hadamard product noise draw later interpret variational develop dropout justification dropout implicit interpretation useful extension dropout principled way normally dropout rate activation make propose activation draw report good result type bernoulli argue central note arise multiply
pair copula determine bivariate copula node share figure copula copula represent correspond copula g correspond bivariate far last nest tree pair specify complete copula formally tree generalization construction canonical tree tree tree tree tree specify product edge factorization copula copula pair copula variable respectively edge bivariate copula conditioning cdf conditioning calculation require pair flexibility
cnn datasets answer cnn outperform model except word threshold accurate demonstrate superiority propose secondly treat image answer word treat class generation adaptive question answer compare answer base necessity dataset word compare introduce guess output answer treat equally answer via approach language lstm model lstm lstm lstm two lstm encode question direction propose cnn outperform specifically propose cnn achieve improvement well include cnn classify
child node represent marker work child return node propagate node marker child return return double notice exception parallel propagate sequentially create behavior tree behavior node child vertical view branch root branch tree low controller avoid fall environment please address capacity agent act supervision e model arrival piece formulate reinforcement task mdp time step observe admissible action system accord expect immediate state action maximize discount correspond unique unfortunately equation pair take therefore receive observe
coordinate estimate gradient due adapt update respective application epoch coordinate gradient ms gd operator regularizers explain cover stochastic epoch start mini mini batch iy ks dy problem efficient popular regularizer regularizers htbp l ms perform update predefine regularizer regularizer ms gd efficiently update distinct three accord let operator define let sm ms gd batch accelerated benefit parallelism theory mini
curve represent iteration simulation vertical away accuracy much require iteration figure standard deviation algorithm note selector approximate figure display standard cpu mean iteration loop simulation indicate whether study test patient test patient indicate diagnosis training use large submatrix q selector phase patient test value near
continuity hessian assumption weak assumption almost subgradient scalar gradient one implication enable probabilistic furthermore reduce solve family narrow act normalize fact non literature convergence hold full supplementary material term lemma whereby measure rest analysis decompose full theorem supplementary implication er rao average prove sequence
equation lemma chi equation bind find tuple find prune small success tuple probability tuple error first calculate number call hence use tuple complexity closeness claim definition exercise intuition theorem property theorem etc pt fact california whose run sublinear domain test response equal specific give differ complexity thereby match information theoretic closeness datum equal complexity provide well open sublinear original several testing depend query sample spirit paradigm additional show testing significantly let require identity closeness dependence q closeness testing label property property pose ask closeness sampling partly show
class consider I training iteration perfectly close inequality statistical estimate ii type ii performance constraint surrogate classical red cm cm acc eps genome unbalanced since sample tumor dataset whether tumor use eq close mean deviation focus type upper convex employ
basic estimation computationally much require solve ols prediction unbounded past year consequently successfully additional notably rank roughly regularize estimation problem computationally give rise estimator constraint impose advantage square turn estimator performance norm regularize tune case remarkable property constrain improve unconstrained top etc square block drop follow statement orthonormal improve constant orthogonal matrix eq measurement compress sense study serious limitation associate use measurement implication
pdf cumulative fix experiment failure failure censor unit stop failure time failure remain remove unit remove failure type ii scheme observation c j frame mass basic plausibility eq
ip ip regularization involve reference justify label class evaluate dataset unless expectation range sentiment web page science bag word stage classify movie unbalanced unbalanced unbalanced randomly document label feature label mutual pool information gain simulate knowledge second select probable topic sort give topic class probability
estimate sampling state contribute gray leave ten state gray line produce ensemble annealing error panel state anneal line parallel line panel temperature black heat capacity match peak diagram anneal readily protein model degree length angle energy comprise generic non energy ref distance contact serve ensemble anneal hybrid structure ensemble run anneal optimize exchange replica transition simulate estimate histogram small protein domain code compare agreement high energy
front singleton decoder successfully variable reliably bin author address front chinese graph singleton completeness provide description front interested reader assumption design address frequency singleton estimating frequency amplitude corrupt location even implementation author computationally attain rao moderately briefly snr approximation noise periodic amplitude henceforth notation snr approximation complex noise snr consider complex fig zero snr validated snr db write mean snr colored noise author mmse unknown optimal whiten mmse author express measurement contrast interested frequency get probabilistic high snr please uniformly carefully chinese remainder delay sample delay carefully incoherence rip delay chain big dft small r decoder generic front architecture show fig delay stage shift sub output identical shift chain stage fundamental convenience sufficient
normality extend rao bind plot fisher parametric counterpart prove acknowledge dr manuscript dr dr valuable discussion national nsf nsf chen nsf award q md university school band bl pdfs propose estimator binary trivial pdf bl trivial exploit band kde high data bl infinite band pdfs remarkably algorithm intensity point record cell state quick density parametrization optimality estimator
stock entirely measure alternatively concentration improve calibration complex validation entirely study f model yield cross scheme value produce inverse distance ordinary yield produce study among rigorous km r magnitude area km comprehensive remarkably uncertainty map measure quantify study variable present measure previous mapping variable validation study assessment difference difficult might different study issue distribution stock error stock residual normal model robust krige back currently ready solution l model comparison compare show mean bias logical method transform stock back prediction stock examine index prediction back mean prediction prediction prediction exponential preserve introduce mind partly error perhaps measure
zero rmse mae performance remove validation percentile outlier event predict measure repeat year year completeness treat additional detail mae benchmark outperform sign prediction error outperform year percentile outperform power rmse power reach fast paragraph rank accuracy different rank event display restrict top performance order predict improvement middle e middle imply assume matrix singular svd singular square log coordinate log time prediction high power law apply short resemble speed individual display break behaviour world record deviation line second explain broken trend fit record iv three coefficient entry iv correlation exponent distance positively display non association middle distance coefficient middle association three coefficient notable individual exponent correlation iv display data base appear qualitative middle summary world uk data compute table highlight top exponent hold come record exponent iv bottom positively iv cross individual exponent decrease year subsequently c l exponent score score interpret see individual exponent panel scatter phase transition two exhibit exhibit transition shift second first exhibit
correspond factor imply reveal interesting eigenvalue surely focus eigen relaxed natural general set addition take technique flip treat degenerate switching role allow eigen specific independent simply distribute still extremely understand illustrate entail behavior eigenvalue eigenvector sample matrix rest section theoretical conclusion asymptotic regime result estimate risk control discovery proportion finite proof non one factor consider mm identity p remain three spike assume matrix though allow covariance growing mean grow see orthonormal empirical eigenvalue invariant translate j z sub norm z pm eigenvalue j entry
propose distance frobenius frobenius bregman treat euclidean triangular cholesky fold training parameter cm cm frobenius cholesky divergence bregman cm cm matrix cholesky decomposition gain frobenius summarize conclusion riemannian geodesic euclidean geodesic distance cholesky decomposition perform poorly experiment object category dataset category different different category image convert pixel image
suffer play general minimax lagrangian price descent contribution guarantee agent eq realize utility principal utility maximize price induce contribution contribution homogeneous concave run interior price price loss restrict vector eq follow evaluation realize gives obtain realize constant price induce vector b gaussian variable least ready utility paper optimally feedback reveal preference function highlight application natural function utility bundle find maximize draw efficiently main challenge bundle behave even concave thank discussion condition concave discussion flow reveal behavior general give game edge demand specify agent infinitely agent flow select aggregate decision flow polytope ff game feasible path crucial lemma equilibrium whenever decrease convex network social equilibrium flow impose rise potential flow vector equilibrium find approximately minimize social cost efficient compute
transform define st eq consequently since multiplication multiplication post dft let transform pair st addition order pre rd addition define
diagonal bind integral univariate integral integral compute integration quadrature application factor considerable designing integral dedicate interpolation fast run quadrature time one expression term dimension factor log integral product correlate vb symbol truncate problem linear put truncation I integrate truncate
orthogonal row te begin define degenerate degenerate form write vector write orthogonality orthogonal dimensional orthogonal projection orientation eq relationship transform condition magnitude independent equation imply numerator optimize regularize exceed regularize fail guarantee variable convex regularize might expense dramatically concern propose concept screen dimensionality size comparable coefficient preserve screen screen preserve violate extend marginal gain different approach attack screen variant forward type eliminate
chance illustrate discrepancy measure validity challenge synthetic well real epidemic day care center main difficult discrepancy
subject subject normalize medium datum several total per category range category room database face image dimensional vector distribution extend database feature image database sift descriptor patch dense stepsize pyramid feature sift grids codebook pyramid database pyramid pyramid sift descriptor codebook spatial pyramid pca initialize mean regularization choose epoch layer experiment denote indicate
large map linear follow g inversion order prove combine due second linear pi last low analogously limit eigenvalue claim theorem corollary problem pairwise learn anti symmetric anti transformation reduce result term world phenomena relationship entity relation anti symmetry prior application protein protein conversely anti preference
quite analysis stochastic unbiased say gradient say assumption constant positive gradient asynchronous asynchronous example parallel convergence gradient also algorithm please study sg prove ergodic nonconvex optimization mini multiple stochastic gradient modify convergence follow variant long processor remain variant partially asynchronous node receive broad attention start rapid hardware resource asynchronous parallelism coordinate asynchronous stochastic inconsistent achievable smooth nonsmooth study asynchronous coordinate symmetric show consistent asynchronous version coordinate
term term division calculate bb bb find bb p p p name cs ps bb search p p bb p sp sp bb bb bb bb angle hundred hundred height p size bb angle hundred height hundred empty ps version def ps sp author chapter journal key month organization page title volume label year sort mid mid string mid sentence write block output write output mid write empty month empty month output sentence sentence output empty empty skip sentence empty sentence skip empty swap integer format name name jj format format skip al label name format skip et name name jj skip format annotation field nan name format name ed
energy machine term ii material ed parameter know n interpolation abstract multidimensional descriptor space label dataset heavily weighted kernel material choose validation particular also find choose really formalism type first test scalar particle weight
w w psd much vary possibly dyadic negative make effect decrease long dyadic observation simplification design understand asymmetric I former justify matrix mean th express asymmetric represent low pattern regressor pattern eigenvalue state u probit specify illustrate dataset dyadic count simplicity analyze average dim distance response ordinal fit symmetric burn symmetric regressor must specify mix mcmc low slow reason psd psd strong association country specific attribute dyadic positively interpretation might refine could evaluate term describe heterogeneity explain dyadic relation model mean point direction direction side origin usa link generally opposite origin end software dyadic datum relational exhibit dependency order
approach proposal scheme see variate jacobian multi calculation jacobian method linear derivative proposal component force diagonal matrix order guarantee definite parameter mcmc illustrate along jacobian pm mh distance distance way common characterize quantile consider derive one separately estimate f determinant jacobian speed worth jacobian respective calculated pilot run interpolation inverse grid total computationally demand remark flexibility mainly interested r jacobian non summary calculated package equation point newton acknowledge approach propose line recommendation statistic illustrate four scalar model
paris aim detect relationship second analyze quality medium quality curve plot membership patient highlight classification whereas misclassification error allow deviation misclassification three represent candidate play finite factorization link component relaxed idea mixture full possible nn paper pseudo density principal functional approach coefficient paper propose tackle hilbert oriented mode bayes rule introduce theoretical focus computational tuning different bandwidth prevent spurious mode complete special attention discriminant present spherical devote illustrate case define hilbert endow usual condition latter start latent deal focus
green kb ic input improve estimate kb monte realization improvement impulse really initial prior indicate output carry close estimate spectrum design mixed map jointly hyperparameter unconstraine scalar standard truncation present length approximating since information assume order adopt stable spline information attractive output may
refine association rule measure adjust risk drug methodology database four outcome within thin database refinement refine health outcome table discover rule three death cause bias patient take drug ab result drug death temporal read record day total instance association rule minimum confidence patient record r association lift average instance chi medical record record contain association whereas side effect
algebra argument derive set multi every bayesian multiclass version generalization vote classifier bag forest learn majority vote vote see set majority classifier machine view vote
constraint also hyperplane another generalization objective may arbitrary vary across specifically specify unknown feasible goal change feasible simply vary price preference model different decision member organization learner day bundle study al may rich constraint represent thing beyond price view predict behavior rational decision maker choose objective unknown goal observe behavior reliably predict future variant learner example learner incorrect ever mistake strong pac study specify mistake polytope also precision constraint allow learner exponential
ks ks ks ks n see quantity good penalty justify sake clarity def half assumption constant opt asymptotic hold penalization remarkable classical lead suboptimal procedure penalization allow difficulty fold hold bring unique price increase however seem difficult complicated resample
interesting may capacity learn add external pattern doubly link action structure stack network operate stack persistent operation remove add stack first step softmax action stack size capacity grow top top stack structure equal top stack store depth stack rule stack stack update q recurrent stack stack clarity stack minor replace single stack serious
extract illumination recognition early classic template availability pose expression practical view neutral recognize probe pose expression implicit representation probe face invariant appearance virtual face recognition still align form cope illumination across pose appearance densely solution undesirable minima consequently localize unseen localization shape node base shape constraint simplified ignore recently object particular adopt structured encode node dependency use consequently develop illumination carefully align probe face recognition choose probe indeed human geometry appearance change vary change face piece correspondingly piece improve easy opposite consider paper partition part characterize face alignment piece shape use similarity tree part model appearance probe face model fit appearance structured shape objective likelihood solve compose step part former solution face recognition take appearance evidence probe align alignment face readily recognition aggregate robustness choose recognition rich appearance need appearance constraint criterion batch alignment globally alignment hand shape graph likelihood couple
mathematic email process compressive measurement receive attention compressive showing include one precisely compressive give theoretical projection memory design analyze volume allow memory access know pca capable world task well compressive combination element atom dictionary
specify follow nb ccc name condition score stack unchanged divide divide stack nb ng denote replication projection directly leverage score sampling consider aware evaluation use input sparsity projection sampling uniform randomize transform speed throughout projection size worth embed experiment quality solution elaborate sized share intel core ghz gb sized perform master core ghz gb ram evaluate kind method describe dimension capacity embed compute three relative compute matrix score include fast score even transform behave although run yield meanwhile leverage score give dimension particular relative throughout embed see embed dimension minor relative error amount lie denote across nb fast leverage investigate quality score embedding hadamard projection method quality nb leverage hard score implementation projection way embed kind embed quantity leverage cccc e e enough approximation typically large reliable score projection general approximate leverage equal leverage crucial sufficient underlie invoke sampling solution evaluate leverage leverage score figure ccc quality score parameter poorly quality would matter less use leverage much explore scalability solver embedding nb stacking nb dimension result coherence leverage score coherence get projection base behave relative remain dimension approximate increase embed quantity perform median report solver evaluate nb fix value nb coherence fix embed objective projection behave go meanwhile seem strong low dimension ccc c error trial one invoke iterative use conditioning quality nb computing ng try small embed fail see detail rate phase depend quality imply projection tend yield similar among need embed dimension reliable clearly tend translate
roc curve poor slow lr superiority spatio temporal spatial quickly slowly converge superior filter plot know statistic move target auc kronecker remain whereas lr confirm impose range pixel range location bin spatial high interpretability sophisticated prior reasonable leave work frame image spatio temporal kronecker amount image pixel similar kronecker standard kronecker target amplitude red spatial filtering achieve image still remain inferior improve kronecker exploit simulation random object covariance correspond filter use image result rapid convergence kronecker method sample require
receive toward base merging consumption merging accuracy average exchange consumption predictive attribute predict target attribute value without target apply value attribute example label classified goal technique svm hyperplane maximum obtain linear popular metric split represent yield tree percentage rule conjunction attribute distribution tree twice meta learning build collect deduce exchange merge another speedup homogeneous partition either provide subset subset partitioning unit homogeneous create researcher merge kind employ induce merged transform merge rule rule argue handle unlikely set distribution train limited homogeneous basis could prevent use find approach meta technique design homogeneous must focus examine one quantify difference update model require ability merge potentially correspond attribute contain element requirement kernel study investigate learn generalize classifier svms informally shape
high b controller correspond frame per trial learn encoder angle consider consecutive frame z feedforward neural architecture nz assess take control vertical play central role obtain step ahead sequential divide grid point image display validation feature separate illustrate ground row long future ahead ahead encode image corner corner dynamic auto learn training auto structure section result move link robot weight start
e generalize ratio technique base unknown test paper solution set scale establish statistic denote derive random purely regime thresholded statistic family purely apply summary rule asymptotically term sequence purely dimensional regime
singular observe minimization pn pn pn km h th nuclear data p finally parameter nuclear rgb rgb corollary attract many statistic mathematic completion entry application genomic propose smc matrix smc establish certain class matrix study sample variety configuration apply integrate several cancer study genomic enable rule survival genomic completion array attract electrical completion system localization vision among write block block display way recover genomic rank observation column exist constrain paper rank see operation recovery complement complement estimate block use svd remove thresholde robust small perturbation bind estimator require gap true rank perturbation accurately whenever value
gibbs briefly rough sampler similarly compute need complicated estimate maximize cost non good binary satisfactory performance exploit carry different posteriori mean square bayes run pick zero randomly pair phase estimate impulse realization unit q use
converge I investigate signal simplify tv investigate I I regard agent surprising however belief become certain surprising precede agent characterization informative signal switch epoch receive uninformative neighbor set neighboring request neighbor particular neighbor uninformative particular whenever private informative accordingly appear symmetric
count procedure analysis delay center notice precisely consist standard whereas asymptotically prescribe process distribution assumption value train formula unknown firing rate test fire develop assumption quite far spike train agnostic spike train step without estimate reasonable statistic usual prove mild present denote sample level plain black cumulative poisson firing rate hz tail tail look reasonably look informally reader roughly way q illustrate h two line f mean deviation marginal line actually value practice mean center variance second need n c fluctuation perfectly c line randomness account u conclusion purely consist may large
markov seed create decay exponentially branch branch apart much study tree structure rank exactly within homogeneous result make require minimal social markov expand distance chain identify abstract identify function relate markov distance probability function second subscript grow thompson asymptotically tree degree show rate decay two regime tree sense characterize ht obtain estimator converge slow widely spectral measure west seed belong east west concept formalize bottleneck create core separate piece constitute throughout social provide formula bound bound tree satisfy random
sample size cv rand cv divergence cv rand cv tend select full difference cv rand marginal powerful describe rand worse consume cross cv indicate balance learn nontrivial connection could rand way select respect work density change ratio preserve news partition cd manually infeasible dimensionality hamming fit let denote document evaluate randomly hamming calculate kind compare model rand complexity evaluate achieve significantly well performance wide range insufficient good rand bm hide particularly unsupervised characterize discovery bm model learn one connection visible investigate selection visible unit affect connection visible unit self preserve maintain artificial number selection rand standard rbm baseline note actually add rbm kl evaluate kl divergence
excellent fairly compare sift descriptor patch codebook train parameter sparsity parameter set threshold empirically pyramid histogram chi negative scene category image scene country select per use data
combination affine bernoulli measure similarly lie subspace mass distribution iid bernoulli identifiability random binomial variable identifiability identifiable tight construct identifiable sample contain finite l measure follow proceed induction induction exist unitary via unitary l l f unitary transform n unitary continuous u nh nh h n
duality hermitian numerous primal convergence implement style solver mkl aspect reconstruct full experiment biology purpose gene solver include discuss available part toolbox computational effective statistical learn analyze ultimately understand biological regardless signal site protein modeling frequently optimally combine multi transfer enjoy grow machine learn year mid thing easier learn human single insight human build learn idea multiple task early first assume later non couple task potentially convex identify convex relaxation approach equivalence cluster relaxation assign learn constrained covariance directly show basic use inspire identify relevant challenge remain find similarity parameter ignore background base approach candidate measure ground assume task mt mkl thorough analysis use duality solver combine framework advance
apply layer summary result feature refine send support svm classifier tune
word focus work context token label token embedding pass embed obtain use jointly sense include computational expense take week learn token multiple advantage approach sense learn assignment token contexts representation vary type make parametric counterpart np build skip maintain word online token word token close create proportional demonstrating benefit approach sense skip gram neighbor non method previous google
amongst case property sign inductive sign value still alternate atomic least sign alternate three interval sign right share figure say sign I case condition interval partition existence condition impossible analogous contradiction right interval know maximal kind reasoning contain lemma maximal interval conclude lemma case conclude use fact produce sequence suffice construct lie turn efficiently evaluate solve discretized version process correctness suffice compute os moreover run run claim section near normal mode space multi modal complexity time logarithmic know entire line mixture distribution probability assign cost finite hx gx function log concave class concave broad gaussians gamma receive economic piecewise give nearly follow combination remove factor piecewise agnostic real line variance identifiability univariate gaussians proper output gaussians time e piecewise degree structural nearly agnostic nearly factor agnostic gaussian ok density space constitute piecewise polynomial considerable attention theory amenable multiscale piecewise scaling coefficient agnostic approximated piece degree factor algorithm monotone monotone non monotone context mle implicit monotone approximated piecewise result guarantee algorithm run kt discrete set lead mixture unimodal pmf modal conditional pmf unimodal modal n kt main mixture similarly learn mixture distribution approximate kn mixture distribution binomial constant piece show binomial poisson use piecewise polynomial agnostic mixture binomial addition guarantee section also good
otherwise misclassifie I unit device compatible teacher perceptron problem bipartite single node error ms involve message indicate direct opposite relationship quantity scalar tw tw rhs concave assignment ms iterating eqs computing either message converge limit reach speak graph furthermore even tree overcome dependent reinforcement eqs analogously reinforcement speak low step scales break compete configuration breaking field reinforcement purpose straightforward
want subject theorem lemma classifier working infinite exist approach focus machine post hoc procedure adopt latter begin algorithm classifier range let abstract product loss p px px x output classifier loss assume classifier misclassification risk classifier inner possibly infinite
fortunately maintain ability interact map bethe marginal minimize suitable fw use lp map provide bipartite demonstrate favorable speed fw versus bp speed fw bethe purely fw preferable bp regime except precise mle combinatorial structure include image bipartite vision university assignment intractable efficiently solve lp local polytope matching flow polytope case globally consistent extraction setup graph observe perfect wise approximation impractical learn learn estimating combination map learn suppose edge matching reweighted polytope prove polytope early perfect convexity bethe quality matching solver suited derivation specific form technique make particularly certain log evaluate parameter maximize rw bethe well probable exact mle perform value rw
argument practice edge number remain actual bring substantial updating connect involve experiment test bandit baseline use com social website ad user limit week drop click ad ad frequent ad turn retain user offline evaluation payoff policy discard coincide recommendation order offline estimator g choice available follow retain create along item draw item occur selection website history user collection represent fm song original dataset tuple song create bandit original dataset list song user payoff past payoff
independence plausible equivalence prove equitability strength achieve power nan hypothese consideration imply establish parametrize examine equitability independence range bad equitability pareto front beyond existence boundary support trade perform analysis equitability equitability tradeoff axis plot worst equitability every parametrize plot strictly preferable front front non trivial indeed exhibit trade parameter control maximal resolution show low regime mid high grain high resolution distinguish different compare along dimension size figure show offer equitability front maximal give maximal resolution control speed versus trade consideration choose care power distinguish expect consideration test likely require resolution grid explore estimator grow include increase independence equitability analysis optimize regime analyze alternative complex alternative alternative material equitability equitability likely grow resolution statistic hypothesis equitability test equitability always runtime extreme consistency give decrease indeed equitability appear balance equitability equitability runtime suggest value equitability sample size examine compare statistic determine fine discretization characterize bias future seem good performance appear moderate dependence assess set since depends present recommend maximize equitability independence equitability limited budget compute search fast equitability affect achieve equitability parameter maximize heavily different power necessary certain performing quantify online statistic use appendix size c use pre computed respective function package standard package write fast population computable analysis thing default theoretical search runtime comparable fast quite fast large sample note feature since estimate involve substantially independent bad
des matrix de de via une des et des la pour de est dans composition en la des lin des les des des class en des des la resp uv pour les de semi positive es dans un dans la les
projection subspace etc weight tailor consideration leave difficulty dedicate list corollary divide fix equation matrix interference interference positivity interference equation prove uniqueness clearly increase imply eq x h ni remain eq h imply introduce turn focus boundedness immediately zero follow monotonicity simple assume essentially elliptical ii controlling quantity asymptotically part mirror mainly significantly study n solve generality ib n solution q core lemma thus proceed bind therefore subsequence f finite subsequence become alternatively opposite restrict
sign htbp probit logit probit spam encounter spam datum set total observation logit median skewness cv clearly depict four function explore conceptually computationally similarity commonly theoretically light structural reason univariate function logit provide complementary throughout life demonstrate theoretically variable binary seek use vector cdf link link along cdf logit extensively field engineer economic education logit commonly probably
graph external represent adjust component graph construct residual application conditional relationship z k coefficient absence coincide span denote motivate z kt section statistic eq edge scenario interest example motivate modify first loading pls follow
cut preserved minimization unconstraine ratio positively subdifferential negative positively algorithm sequence terminate cluster critical function ks sf sf f want easy check form equivalent note w inner rewrite lie empty minimizer euclidean give arbitrary element onto result smooth solve efficiently gradient guarantee rather loose rescale well tight
sign proof base couple lemma cube plus minus follow close hamming vector contain sign tv coupling note conclude consequence core eq e e minus sign sign term case plus sign remark mapping thus lead conclude zero
experiment choose highly reason summary varied likely produce representative enough influence experiment structural lack element remarkably performance characteristic type positively influence improvement sometimes classification summarize even outperform perform idea music share music purpose drawback summary successfully apply application music consumption orient diverse information definition relevance people reflect synchronization result recognize oppose final people ignore requirement music effort try piece knowledge relate music beyond task summarie sufficient relevant feature account human relevant redundant thus improve portion processing portion signal fast disk usage music set music automatic consumption evaluate binary class music take middle use show rest
intensity color tail distinguish confidence high equitability give equitability relationship strength concept fundamentally signal weak able different make heterogeneous relationship exhibit ignore relationship equitability translate differ dependence analyze functional property would independence right statistic evaluate exponential relationship vary result yield right test distinguish conventional three reason composite relationship noisy hypothesis composite non also composite whereas conventional independence consider alternative time simultaneously set alternative understand news bad news concrete equitability dependence clear motivation behind equitability equitability correspond type consideration power independence several give equitability expression early several dependence thousand significant relationship detect five scenario reliably relationship focus significant equitability datum interested deviation independence rather set consideration relationship equitability even size setting simply detect strength situation still cause concern result easy imagine despite
optimization suggest space partition homogeneous example variable discrete evaluation hierarchy imply hierarchy principle evaluation input separate low number considerably reduce hyperparameter discrete component hierarchy suitable discrete gaussian might evaluation suggest allow proposal model etc convention ern relevant determination hamming method test etc burden every
filter filter take outside filter next calculate evolve rule perturbation state filter span eigenvector high sketch proof simplify eigenvalue span I ready right give change one equation equation stable identical eigenvector correspond high eigenvalue analyze stability filter inherently function rotation vector input affect perturbation decay converge filter synthetic colored arbitrarily covariance channel component neuron magnitude asynchronous figure anti triangular feedforward learn connection unlike sake network argue metric first offline matrix rapidly drop derive drop quantify subspace subspace row principal deviation rapidly neural quickly drop
simple constitute neural toolbox linguistic parse tractable helpful stack derivative equation indicate equation q rule assume terminal generating rule terminal l st st si vi b w f n ne ng result well perform describe lstm lstm lstm stack lstm lstm layer lstm queue lstm layer lstm stack lstm lstm layer lstm lstm lstm lstm stack google google google deep
gray line add alg alg node create increase bound combine outer iteration increase bound decomposable def induction height base boolean polynomial immediately step height consider root restrict follow induction node addition hypothesis decomposable def decomposable sum alg construct w v return condition modify line also straightforward height height node child alg return hypothesis law induction hypothesis v complete decomposable last def complete decomposable normalize alg normalize weight check sum bottom network polynomial distribution terminal child last remove thm htb bn normal show useful view example naturally suggest sum therefore define implicitly conceptual understanding help child root select th branch restrict child root go root decomposable admit factorization
community balance know np thus practice relaxation approach find spectral vertex laplacian relaxation loose one approach frequently spectral strongly ease handle weight e c n relaxation crucial constraint continuous yield round work exact basically ratio cut problem sf l sf extension cm simplify denote vertex desire question get typically assign attain tie break row round unique solution weakly partition
unknown measure space constructive breaking present transition infinite illustrate right restriction stay upper triangular equal different band regime model ii p ik I draw profile prior integrate denote sample limit tend take popular place hence potentially state infinite transition therefore multiple model evolution impose process transition restriction transition state state
surprisingly seem likely lie analysis method autocorrelation specifically inversion usually increase matrix sum approximation transform discrete increase future facilitate also cluster selection provide nice spatial utilize spatial reduction small knot alternatively focus resolution except component regardless compatible integrated potential extension knot lattice process remove lattice large knot multiscale eigenvalue involve dimension work many spectral close correlated facilitate model material fourier integration evaluate ht fit component ns stick iteratively us ny ls ls l l l lm functional identifiability l simulation dimension multivariate size square ern eigenvalue run setup efficiency approximately second test mat ern table correctly next move dimension lattice isotropic dimension well flexibility high second point
condition exist bound bound operator g affect procedure slice function rkhs observe give slice form reproduce involve q force unique chosen occur span whole freedom vanish happen entire long induce uniqueness utility penalty slice subspace eq representation modify bivariate risk eigenfunction f
side equal q inequality obey accord iii
seem check define almost expression finite surely coherent wavelet surely expression satisfying banach x k almost almost surely immediate consequence denote sequence sum q q surely surely absolutely surely unitary irreducible integrable integrable z condition belong almost surely establish fx k easy arise derive relie henceforth include first remark hence belong choose family compact k borel consequence prove df weakly uniform finish notice restrictive assumption topology weak topology assign borel set superposition eq exist ff x open v I prove prove exist eq surely almost eq f dominate I I decrease
good validation evaluate hyperparameter model tune validation preprocesse preprocesse replace extended character remove specificity user tag simple apart method demonstrate twitter regression partitioning word similarity normalise pointwise mutual cluster centrality word cluster document old run apparent central word assign make stand useful transfer cluster tweet comprise frequency weight centrality component tweet kernel hyperparameter rbf reflect
decomposition limit sequence sequence appropriately ahead remark sufficient bound remark see even odd last zero among write odd conclude show discuss introduce square two integer dd mean worth restrictive always limit generate eigenfunction since term consider coefficient I remark increase obtain fix
penalty illustrative example penalty whereas mcp regularizer reach global step additional condition stepsize reasonable stepsize bound small step complexity multiplicative regularizer integer scalar column condition update terminate satisfie local minima difficult regression regression compare refer base mcp implement adopt implementation path regularization scad fan mcp maximum concavity mcp penalty scad mcp recall choice factor past convex regularizer lead error manually error choice use run range plot plotted scale scad mcp
dimension increase adapt vocabulary sentence review describe neural replace architecture lexical sentence would learn lexical translation represent source sentence sentence index target sentence binary vocabulary figure depict lexical translation forward connect h encode contexts v h lexical translation sentence sentence q investigate impact configuration build target sentence comprise sentence appearance target
establish perform quite intuitive determinant simply final indicator location parameter estimator scatter parameter eq practice matrix multiplicative unbiased yield maximal determine scenario function obvious little extension regularize whereby interest even essentially intensive subset many suggest shall later computation drastically performance one limitation lie class group treatment potentially inefficient presence group drawback drawback turn deal less matter combine situation arbitrarily pursuit pp unfortunately fail later pp relatively number intrinsic space inherently robust class ability adaptation thank supervise discriminant
dft dft linear transform conjugate intuitively dft coefficient think basis various length see dft basis dft attribute development standard library dft powerful convolution spatial multiplication fourier loss loss frequency domain statement dft allow quickly assess input affect make representation follow propagate gradient dft dft layer frequency field apart achieve freedom reconstruct since conjugate symmetry dft necessarily meet symmetry observe optimize embed need close
upper discard class class monotone complete estimation community mention introduction shannon sharp rate view correspondence study continuously quantization precise asymptotic regression rate realize scheme radius ellipsoid achieve automatically regime specify analysis phenomenon filter attain regime nonparametric estimation distortion reverse water source play show case ellipsoid level quantization euclidean ball distortion communication closely ball analyze distortion minimax rate distortion quantization estimate basis alternative view say differently statistical compressed appear community analyze linear transformation sparse zhang problem constraint problem consider across aggregate pool central parametric paper certain rate introduce distortion degradation location finally
intrinsic te ft ft te th functional penalty front clear functional next hermitian property symmetry claim assumption note q consequently convex minimizer hence ambiguity rewrite ambiguity blind source optimization scaling consider implementation follows numerically consider discretization axis restrict dim period suggest partial straight finite difference matrix approximate discretization instability use ft finite fourier transform sake simplicity discretization operator denote discretization take discretization evaluate result
correspondence ik iy kt feasible lemma rewrite calculate value calculate prove optimization stable
learn employ attention focus recently formulation condition signal whose code denote deviation transform also f alternate sparse fix code transform synthesis model formulation hard synthesis remain highly alternate project non project low p solve follow thresholding operator subscript index vector mention proof condition occur row column value definition equally solution similar theorem corollary although step minimum update prove instead write singular decomposition transpose code matrix minimizer write singular invariant choice tr w yy yy positive definite square simplify determinant eq entry inner use tr maximum non cost transform solution invariant yy brief latter root x yx yx l obvious using uniqueness corresponding matrix follow matrix obvious optimal unique uniqueness aforementioned uniqueness value distinct svd unique scaling still zero singular map repeat extend repeat value say
non softmax class prevent early stop train class effective tendency special solve helpful transfer speech suggest initialize retain nearly special soft class hard target currently explore approach expert network choose expert assign example assignment cluster expert cluster make training expert keep second network need expert rarely task huge training multiple confusion matrix train subset entirely prediction model ensemble model remarkably transfer
clinical trial compare either outcome severe infection code severe infection code record seven always baseline visit treatment ij ij I model individual log histogram posterior correction correction accounting correction hyperparameter precision
activation become true justification force activation asymptotically sample infinitely almost finite may q mean sufficiently past mean bandit inferior activate force number sub optimal activate winner period increase unbounded observe exceed activate activate take section assumption finitely surely strong surely positive u ij jk define take relationship via pointwise come indicator condition sum hold one surely finite structure prior observation finitely almost surely prop yield sum equal activation bandit total bandit time optimal activate
replace principal component datum point project onto maximize total principal e belong projection pca per trial choose goal obtain gain trial cumulative projection maximum difference cumulative cumulative however gain predict projection definite matrix trace gain gd trading bregman divergence bregman onto convex
find strong random fluctuation compute center stop drop move move average stop good error occur iteration peak report mnist benchmark r feed forward ff art approach competitive version tuning protocol comparison exceed available label point test classification training unlabele set datum decision decision boundary fine semi blind blind approach deep approach usually adjust unlabeled blind I fair test datum give group label train training tuning validation run label set additional label label model training setting respective tuning overfitte free parameter use overfitting per set test ff ff fully label mnist set range generative label
integer range satisfie range absolute usually instead consideration please distribution absolute integer real distribution ex remainder focus absolute robust spectrum absolute deviation spectrum fundamental sense absolute preferable absolute
dms ahead term average dms dm dm track frequency past decision opponent dm response base dm frequency decision dms since know convergence team problem dm games team weakly game game establish dm acyclic games dms involve game two dms game dm payoff I factor mm size iterate cl I ix ix ix cl closely visit hand discount strict dm update hold exist exist integer discuss constant throughout phase dm indeed
property absolute invariance number integrable integrable power absolute equivalently combination real include limit potential particle mass proportional usually decay bring analogy information particle pdf potential field particle interact potential individual ip assume integral incorporating generalize bss bss overall give low blind without demand bss direction conventional entropy interpretation pdf independence interpretation research start focus alternative derive bss interpretation incorporate nonparametric new bss direction quadratic characteristic pdfs independence measure cross information euclidean cauchy schwarz base quadrature propose ica inspire trend derive independence interpretation gradient pdfs bss nonparametric computation direct estimation bandwidth contrast estimation pdfs equality joint pdf product pdfs imply hessian marginal pdfs independence zero independence contrast function bss restrict computer work definite may valid case newly stage method advantage quadratic nature concept reference potential rip rip ip basis close expression verify
top period cover raw immediately coverage transaction transaction prediction frequency unbounded value regular transaction bias level recommendation performance observable big difference recommendation map lift baseline lift imply learn bias nevertheless final initially lead high map test three optimize ndcg acc biases acc ndcg acc zero ndcg item sort bias acc notice positive suggest item bias try discount popularity optimize ndcg ndcg larger optimize acc balance recommendation update bias daily accomplish via warm bias purpose optimize acc subsection base bias kind orthogonal wide mf category recommendation mf momentum netflix competition collaborative filter standard factorization approximate item product gradient involve overhead implement order
quickly time tx history steady satisfy quickly steady stationary unconditional steady smoothing substantially steady unconditional sequence steady filter asymptotically short sequence see exposition learn steady require expectation kalman average since least obtain recover em second compute training fortunately avoid employ steady relationship recursive relationship system average switch operation average covariance horizon covariance unlike lag never average scale multiply initialize identification moment combination statistically perform perform hill local marginal yield empirical gain relate two likelihood surface minimax normality
identify include miss false list indistinguishable bn obtained see six alarm identify water method inferior experiment dag constraint bn structure edge favorable exploratory mb gs tc tc alarm th c mb gs tc alarm water mb gs tc tc alarm chain water dag constraint discriminative framework consider classifier vector induce whether power kind classifier robust assign class svm fisher capability allow maximal squared initial average randomly partition training group varied optimize process plot line individually h plot bottom kl improve discriminative individually well row setting classification individually learn drop possibly insufficient kl still maintain classification classifier fig number second row optimize discriminative help build kl induce upon feature optimize cross third row notice worse learn cause serious improve fitting base lead computing fisher classifier fourth fisher vi discrimination square error explicitly application tolerance fit potentially bring discrimination ii gain insight population visualize brain compare
macro drop around macro refer ik km angle channel operate bandwidth resource every instant bs expand coverage db macro bss decide bs ik allocate bs standard mechanism measurement filter measured margin mechanism execute biased plus margin condition bs define margin bs bs aim total long
contain admm seek sparse utilize lrr low abundance lrr assume spectral signature abundance linearly dependent abundance nuclear properly adapt cost alternate minimization abundance take utilize slide window odd contain signature adjacent lie usual paradigm column matrix rank low naturally abundance due dependence respective abundance reasonable independently individual vector impose structure deal learn sparsity rank rise least fitting weighted abundance minimize incremental proximal alternate imply name operator state demonstrate extensive letter vector denote moreover respective one denote large
result also lyapunov stable subset n n lyapunov ax tx I I hx h map compact xt vx z note assumption iterate invariant assumption invariant associate recursion satisfy temporal condition build work consider form eq
stochastic round number integer indicate epoch bit also show comparison mnist cnn comprise filter relu activation second convolutional subsampling pooling pool overlap pooling pooling layer connect consist relu neurons way softmax exponentially epoch momentum train network computation value create jump allocate expense reduce format represent output figure near adopt stochastic rounding use achieve correspond slight degradation round consider commonly image cifar consist rgb cnn subsampling layer subsample layer pooling connect way exception normalization epoch
accurate fourth fourth ica design clutter ica ica achieve highlight medium regime ica reasonably well recovery something turn share rely whiten preprocesse linearly noise free whiten mix rotation orthogonal approximate true mix identity estimate ica bias violate actually whitening show gradient positive inner ica clarity ica section throughout property fourth though construction even order version fourth capture zero mean variable definition different scheme construct order value algebraic ica homogeneity
algebra al propose arithmetic automatically logical besides whether machine progress ai question simple however think generalize build distribute attract base dimensional neural maximize deep technique quality word representation al neural unified suited nlp simultaneously representation similar maximize context slide work occurrence embedding context amount bias give quality effort learn word address knowledge base completion investigate side coin leveraging enhance early attempt al leverage yu incorporate syntactic semantic word particularly additional auxiliary supervision several matter whether bring confusion solve et context cluster et perspective enhance word aforementione method knowledge effective test intelligence scale full typical name cognitive question e induction
base boundary pc hour pc interpretability time exponentially mainly computational overhead parent optimize gain pt pc c c lp max scene label min scene medical mb mb pc image medical c c medical conclusion h local around specific thousand recall keep low practical bn resource bn guide structural heuristic capable thousand maintain pc potentially day time far skeleton thousand shall next prove biology identify behavior turn lead diagnosis characterize protein trait throughput dna mutation bn huge difficult author exclude overcome focus search utility hybrid bn learn multi many dependency intuitively experiment dependency identify induction motivated approach serve toward annotation categorization protein drug categorization hybrid hybrid extensive experiment h art significant edge super pc source online application label challenge many world domain theoretical condition minimal irreducible factor markov investigation principle result characterize decomposition lp
boundary treat mapping rank estimate bind cdf denote bound formalize function rank rank compute standard bootstrap confidence cdf satisfactory result narrow distributional demand become band empirical true pr criteria contingency rank instead single roc curve correspond positive construct curve methodology outline great contingency great roc corresponding least correspond important understand estimate correspond roc computing equation greatest low curve shift conversely roc shift
modal gaussians employ logarithm mixture address limitation variational broadly skewness copula copula margin form advance automate variational bayesian minimal write copula describe structure variable conversely copula joint distribution term marginally partial derivative qx f pf jx jx jt cdf copula modeling parametric
function exponential predictor beta diag line code x burn summary prediction sd ess prediction nominal mean sd ess infinite prediction deviation uncertainty around discrete quantile continuous standard metropolis variant center local seek pdf uncorrelate extreme perfectly uncorrelated high poisson regression hold explicitly use mode pdf r poisson mh diag sampler burn end iteration
line power individually mark schmidt see bottleneck calculation carefully mention calculation search compare problem minimize coefficient smoothed percentile
engine decision explore measure however ambiguity label dim relevance relevant assign denominator irrelevant choose diversity start cluster sample first perform pick score value pick optimal label share sample share create unlabeled original old extent amongst sample mapping
negative confusion matrix change cc cc thus error limitation drop class product classify rate I would go imbalance address limitation due concept drift task underlie recall unfortunately without drift score remain unchanged type unlikely detect unless true statistic concept specify reason frequently drift detection stream slow two component confusion matrix coefficient batch drop user threshold drift error monitor length interval possible great many algorithm
try nearby guide rather traffic shape category alone square whereas adapt environmental old shape generalization flexible rl learn free underlie equation extensively paper begin short description present give rise study absolute ps cope analytically scenario category beneficial summarize basic principle detailed description reader reference ps agent call represent node transition fig ps fig action random carry c thick style sep thick font bend edge node cm bend bend bend tuple specification physical category dimension pick restrict vary edge dependent value ensure take place dynamical internal reward come update agent forget past environment generalization usually compose translate precisely
feedback amazon indicate indeed practical many learn denoise crowdsource worker aggregation aggregate obtain propose end aggregation collect interface mechanism aggregation voting label crowdsource voting interface belief worker complementary single answer mode belief often mechanism principled appeal uniqueness mechanism preliminary experiment amazon practical expert crowdsource platform worker traditional line aggregate redundant worker open specific response interface vote social theory vote removal labeling task crowdsource theorem corollary ccc berkeley microsoft research microsoft
paradigm present couple measure query image component system extraction process database th store extraction image subsequently image database fix user relevance feedback briefly image base distance measure throughout mention early semantic commonly aim bridge intervention query rank similarity image irrelevant system use present improvement user provide feedback implement relevance assign weight discriminate enhance retrieval retrieve image retrieve denote deviation set non rf obvious iteration
compression wavelet basis evenly divide patch stack datum solve plot repetition relative difference empirical answer specifically orthogonal basis patch design dictionary cosine dct wavelet seem compare discussion also divide image convert stack column derive concrete method classic development respect fix form denote thresholde acting seem regardless heuristic therein include equally phenomenon observe dl take nonconvex heuristic x r diagonal sign scale reasonably make follow complete class dictionary overcomplete tend powerful representation nevertheless dictionary orthogonal competitive certain admit encoding necessity dictionary apply structured dictionary frame matrix bernoulli model ij mutually write compactly high whenever cn identification pose model imply probability may imply control intuitively recovery hard large x sparse observe scale version dense possible easily find surprising e dl give provably recover per comparison overcomplete main x first recover subsequently linear constraint remove homogeneity scale problem reduce program image differentiable yield qualitatively preliminary nonsmooth huber handle technical modification spherical nonconvex priori unclear admit algorithms optima surprisingly descent exhibit x phenomenon negative height function exhibit region finally region direction curvature moment suppose e spurious minima minimizer row illustrate point take project equivalently plot minimizer minimizer apparent nonconvex landscape move successively strongly curvature
graphical structure memory store marginal product computational recently randomize optimization descent stochastic quickly less maintain training bias various context dependence substantially usual might sag scheme sag key ingredient algorithm often find sag see fast scheme degree uniformity grow subsequent backtracking choose obtain take step size non sag recently dependence pass simplifie require full sag analyze appendix show scheme also fast able iteration line satisfied backtrack line example
current calculate article receive normally gaussian distribution period normally span within last hide trade gs trade gs trade nice friend restaurant query express semi markov generate observation mean I duration duration stay also emission q algorithm sequence forward backward backward state end duration sequence duration expectation rather rank offline predict predict next ranking implement optimistic contain solve

sound speech deep denoise encoder distortion difference denoise encoder speech synthesis semi contain static delta delta streams band limit build dnn synthesis five dnn linguistic auto encoder acoustic hide preference auto synthesis speech system conventional synthesis dnn require
likelihood maxima agree component log huge empirical unconstrained variable manifold unconstraine two direction line sufficient convergence manifold descent tangent curve introduction scale direction generate current point adapt point manifold tangent directional direction tangent space curve geodesic step point riemannian map generic parallel descent direction memory store summarize quantity manifold riemannian manifold riemannian
classifier select range nb low number range superior range error rate nb ht ht redundancy many eliminate redundancy measure inter correlation among former item feature make far high effectiveness redundancy dispersion take adjust pairwise inter correlation conduct effectively reduce play discovery bioinformatic recognize acquisition storage etc feature attract attention researcher thus performance method affect computationally test candidate rely irrelevant discrimination study focus also selection searching evaluation selection discrimination trying discover unlike method individually evaluation discrimination
include include propose system ergodic behaved function ergodic geometric mixing obey central variance proportional markov proposal initial algorithm compute specify proposal influence mix different version make accept essentially scale semi psd dependent ideally proposal
relation likelihood error monotonic high necessarily optimize check heuristic never improve fourth point technique equation ad put broad self learn competitive logistic especially name simulation prove array drift letter letter particle optical recognition optical handwritten handwritten digits segmentation heart uci name contain name table set generate unlabele fair computational burden high conditional leave supervise ill set variance remove numerically apply variance retain remove note reduce transformation attain provide indicate purely column give size label unlabeled set gain employ keep small class instance training size ht rr dim large test yes letter yes yes yes description rr
similarity embed infinite rkh suitably kernel embedding order moment kernel nonparametric testing construction mcmc computable notion measure discrepancy characteristic commonly include inverse mmd mmd select role mmd apply gaussian operate correspond e abc experimental approach robust overview rejection soft abc indirect abc experimental discuss start
star galaxy summary iterative thus step choose particle pool create represent parameter parameterize propose new object post remove property abc gradually reach quantify technique determine pdf prominent kolmogorov ks individually insufficient problematic object non plot create py diverse exist quantify divergence variant jensen near kernel measure mahalanobi
infinity number yield regime centroid class condition classification let observation slow evaluated infimum take l immediately natural successful grow growth average per zero condition minimal significant feature respectively insight effect significant contribute increase significant equal q rhs decrease fine fine end
allow formulation factorize large global purely descent algorithm provide theoretical increasingly neural offer regularization facilitate involve across technical example relevant machine form technique factorization dictionary typical factorization might approximate require property negativity naturally lead function desire unfortunately case pca vast majority disadvantage associate able convex factorization sufficient always possible reach global purely strategy two factorize constrain regularization allow require factorize deep increasingly unit relu satisfy homogeneity empirically speed network increase relu use partial phenomenon performance neural success range discuss vast majority disadvantage optimization convex challenge certainly gradient quasi guarantee local minimum norm low
formulate let upper achieve constructive greedy construct result approximation formulate upper constructive type embed corollary eq constructive however prove constructive greedy type begin bound read repeat complete consider function estimate additional frequency prove
show provide bias unbiased ba gradient advantage propose multi fundamentally policy rate although beyond scope generally cifar convolutional neural provide cifar along hyper file train file figure quickly considerably iteration stop lost drop time alternatively learn drop order schedule quickly quick rate drop show fully manually disadvantage run cifar infeasible resource essence term long term beneficial idea learn rate value adopt common rate boundary rate numerous
wikipedia semantic concepts fact initialization semantic nlp complex store basis capture triplet entity answer module compute memory iterative attention part specific module iterate provide newly relevant retrieve fact several pass summarize answer module produce module module generate predict module relevant fact retrieve later module think compute final representation module memory module question module memory process hide state sequence embedding
pls pls versus pls sparse pls pls resp vs discriminant pls da vs da compression simulation table compression stable mean compression prediction converge suitable qualitative response da log converging indicate check sufficient assess relevance nonetheless combine ensure prediction error rate da pls da log nevertheless predictor pool select predictor perform compression relevant construct evaluate compression determine specificity proportion regard illustrate phenomenon false sensitivity one especially grow select log select positive tend rate
edge presence calculation transition take single network use string approach integrate edge build protein report database undirecte easily direct pair edge merge redundant weight merge test method annotation protein database organize consist protein repeat test difficult candidate label repeatedly protein validation remain metric calculate assign protein micro assign top prediction contingency treat protein string network without representation different number tag protein address diffusion similar
linear programming give polynomial lp exponential second really bit vector check come set check however np seem alone hard comment efficient exponentially study knowledge relaxed implication natural abstract concept concrete relax semantic allow form implication precise often focus relaxed endowed semantic conditional implication implication semantic whereby implication either want hence semantic dataset count see equivalently mean implication aware work partial implication name partial much association rule survey partial implication confidence implication clearly start preferable variation confidence redundancy turn logical allow reach assign least contribution
tail probability wishart form draw initialize draw signal random computing average repetition linear original following specify condition initialization ns absolute constant matrix signal greedy sensing albeit update greedy robust dimensional low dimensional lie subspace gaussian
adjacent otherwise name patient nc ad nc include datum phase include datum preprocesse commonly mm voxel great matter serve task svm cv applied summarize experiment lr variant unconstraine analysis test accuracy outperform voxel use good several spatially connect region lasso
one look assume uniformly row corrupt theoretical trivial choose choice slow induce pc formulation maximal interpretation propose new pc center motivation completely require additional observation row assume outlier able outlier manifold ik dimensional subspace dimensional affine minimal focus describe subspace empirical outlier outcome
likelihood ml restrict maximum minor minor kind possible remark high sample happen satisfy label point remove future long convergence lower low term restrictive affect log likelihood respect never essentially characterize empirical minimizer erm estimate
model model signal signal zero row identify location equivalent identify know circular source equal power arrive array half inter element compound cg tail accurately clutter covariance snapshot music peak music uniform thus true localization large peak music
message amp modification adapt amp rbms visible prior aid oracle support interesting projection use move amp modification algorithm successful reconstruction amp adapt produce even well stack rbms perhaps work allow rbm lead result receive european research fp height em style coordinate department sup paris france france es utilize boltzmann machine rbm train class rbm amp interesting
formula simply smoothness latter exist dual local optimizer kkt kkt point kkt condition feasible definite satisfie optimizer point generalization optimality condition unconstraine riemannian formalism adjoint lemma dual j unique theorem complex kkt semidefinite statement construction semidefinite fx xx block fx notice riemannian make read analytical smooth geometry uniqueness rise former study information local first neighborhood claim x sx make regard uniqueness particular x optimizer optimizer another segment hence dual hence strict basis ensure x general remove globally strict optimizer likewise globally strict cost conversely uniqueness optimizer strict optimizer enough rank example coincide summary optimality whenever tight optimality globally global optimizer extreme global optimizer unique optimizer return kkt useful order critical critical since expect hold extra two theorem critical order critical point matrix vector simplify semidefinite kkt kkt theorem rank invertible optimality imply hence proof per point either thus rank hope minimize critical kkt
theoretically rbms receive european write maximization intractable fast specifically cd persistent rbms entirely clear propose deterministic rbms mechanic network early network boltzmann representation unlike visible rbm unit rbms cd little apparent deterministic rbm training go beyond na I approximation extend commonly physics rbm improve na I along nature systematic bipartite wherein visible fully connect let
active query assign cell cell c c c cell c c number cell microsoft com address sentence research recurrent rnn cell propose lstm rnn rnn rich go sentence word whole lstm rnn click web visualization understand salient keyword furthermore keyword lstm belong cell semantic vector application automatic allocation lstm lstm search lstm embed exist state deep learn input task train vector encode mean sentence word learn sentence salient sentence thus similarity text unify different language sentiment information retrieval rnn lstm english meaning sentence another lstm rnn sentence paragraph sentiment sentence retrieval properly
denote set main mode level unstable practice level define define cluster assume small allow every every combine consistency distance soft mode assignment induce soft denote confidence normalize instance indicate transformation connectivity assign occur highly overlap summary structure mode greater discover geometric cluster assignment permutation element consistency connectivity cluster sufficiently show convergence estimate rate pick turn yield estimate distance connectivity way difference blue dot green dot signature norm location mode minima nonempty note measure cell mode thus cell call bipartite summarize summary graph summary visualize signature use package implement statistic capture encode summarize cell idea piecewise visualize local mode plot mode green dot
detail os asymptotic cumulative distribution call expression well cdf allow euler drop low line low os expression cumulative distribution lead expression gamma drop step location visualization walk brownian bridge red law iterate visualization limit flexible tool transition unify new transition state free wide state transition discrete canonical page mid point great field change
expression derivative expression obtain analytical expression mse increase use step computational budget divide notice proportional budget comparison optimally optimisation advantageous degenerate euler method expectation bottom go algebraic estimate go behaviour obtain file artificial expression mse depend case grow depict euler method outperform surprising already case one choose average mse analytic denominator mse moment artificial consider bias small grow variance estimate choice effort even analytic solution form solution euler euler depict become small gain euler method term computational effort reason much minimal confirm minimal effort
impractical due large add least negative entity negative much instance train snapshot award snapshot snapshot snapshot contain fact examine snapshot snapshot manually snapshot award close world kb snapshot incomplete human kb mean type treat account frequently large entity frequent type thousand miss map ability prediction measure gap micro evaluation multi classification entity entity otherwise entity reciprocal rank metric reciprocal biased ordering
point stochastic ep expectation sep start analogy ep ask theorem importantly also limit paper proper ep partition dataset piece true assign factor dataset kf
table occur table frequency b notation occur sample number occur contingency fix observe margin equal multivariate contingency table statistic contingency table one occur probability observe margin contingency margin note side test contingency table many freedom test statistic mid average observe extreme mid tail due since pathway collection nan require table statistic count contingency table margin contingency fisher contingency table contingency table branch heuristic use specialized approach still table problem table seem enumeration exhaustive margin demonstrate example million table randomize contingency table e therein provide guarantee efficiently table statistic maximum possible degenerate perfect table evaluate table highly accurate enumeration value tail enumeration contingency table least exclusive occur exclusive cell strategy generate exclusive iterate contingency uniquely cell first co occurrence less exclusive constrain set contingency exclusive allow distribution know nan occur independently exclusive give exclusive sample tail find provide exact test value occur consequently
naive demonstrate patient patient type infer probability simultaneously conventional rather redundant prevent suboptimal cluster study estimate benefit application world trajectory patient claim attribute trajectory patient heart disease trajectory transition center begin stay transition heart operate get although patient heart disease trajectory closely patient might severe heart relatively heart require employ see patient trajectory patient identify enter either heart operate leave patient center similar heart cause possibly full base patient relate base trajectory cluster chance patient cluster estimation rs increase maintain service percentage respectively schedule flow service similarity cluster scheme oppose admit type validate real patient method far achieve approach traditional technique
cost much iteration partial much short time five mean use significance small use four accept method moderate infeasible dp account portion run dag dag randomness variability actually small case range much ratio remark reduce dag run dag set child value decrease effectiveness clearly finally choose try run sample direct run totally discard iteration burn sample show bar previously correspondingly sample mean run time run second see small return variance total reach short mcmc second datum letter perform please material supplementary material iw mcmc art estimate posterior modular mcmc available moderate use dp modular use performance fair score since rr rr rr dp iw iw tumor letter e e child e iw iw c letter child dp mcmc iw make fair tool dp phase change criterion computation implementation store usage mcmc issue original matlab code store hash table new perform run window intel cpu gb memory proposal discovery iteration perform run totally sample dp sign table mcmc run iw result note letter dag six case letter tumor experiment tumor good due memory
kn column array convert entry unfold ni nx slice e span array array multiply matrix array array match specify multiplication multiplication give array n nn note operation product mode b rd array horizontal slice array mode unfold array table information tensor scalar mode mode fix slice fix ni ni I na ji ni pt life evaluate contribution predict measure diffusion signal contain focus signal signal isotropic right signal voxel predict direction orientation specific signal side difference model extend voxel white matter voxel white voxel n solve constrain formulation life model
reward measure top right behave trajectory available perform well well decrease optimal identification interesting result portion model result plot exploration mdp markovian exist offer flexibility model observable maintain certain overcome problem line within contexts contextual mdps modular improved technique inefficient consider trajectory orient improved incorporate exploitation phase scheme solve approach accordingly another context learn uncertainty mdps directly rectangular result investigate rl infinitely many practical importance present rough setup precise issue important despite availability
use rv eigenvalue select median q sufficiently determine pair community solely compare vertex every fairly fact vertex inequality side strict large small focus vertex plan vertice community pick attempt vertex actually vertex community approximation particularly randomly bad repeat previous good classification classification half classification fairly element furthermore classification together sphere comparison detailed use obtain classification improvement neighborhood key testing error posteriori unknown largely unchanged add step base neighbor require handle necessary prove graph approximation modify determine value classification neighbor conclusion estimate sbm bad preliminary classification agnostic set group community subset call agnostic comparison edge determine belong profile computed section call new estimate likely profile compute simplify political classified conservative link use modification standard basic tool use leave ball edge leave measure prevent dependent would result increase vertex make somewhat less reliable fairly secondly fact thing affine also
trial introduce recovery low know pf sample restrict slice show framework build tensor decomposition sample complexity guarantee setting interesting follow naturally consequence context measurement variation measurement completion rely fundamentally tensor problem randomly framework fraction rating assume rating assuming require e rating context activity set naturally user item context model sample make slice third provide rating slice restrict slice solve one entire concern name tensor provably recover complexity logarithmic tensor completion specifically achieve sample respectively order recover rank factorization addition unknown application involve tensor factorization weak small rank assumption incoherence assumption computationally operation convex norm compete tensor unfold true underlying decomposition conceptually simple analyze algebraic tensor decomposition recovery especially hardness focused algorithm tensor approach insight work know tractable extension nuclear tensor important sketch contraction seem appear expand upon form slice slice
lead requirement distinct approximation possible post maintain cut period suitable problem sample autoregressive decision resource preserve cut increase avoid pass path piecewise post backward cut hyperplane please construct decision description tr kk x h x x r discrete equation k k specify present policy one need consider extend neutral treatment beyond scope presentation markovian adapt turn mathematical numerical need tune present reliability computational performance method construction sequence numerical concern subproblem regularization sequence attempt value sequence case insight high e optimality gap prefer bound reliable result sequence various success experiment anchor north legend legend name name title stagewise independence xlabel ylabel coordinate coordinate width major xlabel coordinate stochastically generation
object represent goal label predict subset costly expert therefore reveal especially interested sequentially highlight learn essence stand operate pick mid path least connect demonstrate complexity novel focused connect refine cluster cluster problem boundary e fx fy cl vertex least design noisy suppose noiseless query behind build repeat many majority vote proposition straightforward chernoff bound keep sequel oracle noiseless work note extend vs thorough investigation name fact connect automatically cluster cut figure represent class explanation work sequentially adaptively vertice budget need well wide problem budget specification merely completely agnostic nature subsequently section short short x edge lx sub iv
marginal figure respective plot kl coordinate coordinate assign posterior conclusion case show brevity ccc kullback divergence plot compare difference scale around essentially variance cc improvement result first hyper assign profile case figure depict true see true profile smooth profile consider suited hyper parameter ccc run quantile plot respective profile plot figure pre previous case hyper affect quantile nearly significant improvement partly median also variability covariance hyper robust run inference quantile repeat spatial figure profile three leave right bottom improvement median significant smooth yield true accelerate rate gain observation consider ccc pre covariance row bottom finally median profile figure irrespective smoothness true profile quickly increase demonstrate pc surrogate coordinate report profile come pc surrogate ccc present bayesian order unique spatial mode
describe traffic time interval rate concentrate zero type cover dirac measure govern distribute intensity denote subset ensure imply ergodic admit invariant measure focus normalize essential analysis formula note condition generator diffusion densely operator f furthermore see operator hilbert adjoint compact see spectrum eigenfunction large eigenvalue order strictly construction spectral invariant nontrivial invariant measure together cf ergodicity easy invariant measure construct matrix estimator share eigenfunction generator generalize transition crucial laplace calculus sense spectral laplace situation
compare method factorization involve likelihood cp decomposition minimize researcher generalize standard involve multiplicative originally lee nature negativity constraint interpretability give slice sort country overall slice toward upper slice e high count slice finding equal generalize kl divergence validate factorization kl generalize datum vary degree sort country actor overall activity receiver slice tensor toward corner property divide observe test randomly slice index time define test slice model performance intend test handle upper portion complement portion set experimental analogous collaborative infer slice leave portion parameter direct point reconstruct slice geometric expectation
binary implementation totally carry confident similarly paradigm may interesting challenging task alphabet message f discrete comprehensive contain obtain vector normalize keep message normalize poorly condition product message branch forward leave block incoming jk di k jk alphabet py
j ok lemma directly give proof algorithm access share top close corresponding element suppose sample support unique element contribution close proof support unique two support dictionary know dictionary dictionary spectral element part entry incoherence know small large singular sa large singular combine know share unique dictionary every element contain dictionary hand dictionary use intersect distance least incoherence correctly identify analyze rule algorithm take extend generalization framework correlate desire solution random desired relax correlated expectation desire solution strong expect random q theorem expectation side proceed ok long guarantee next lemma work preserve iteration formalize ok mn high satisfie bound various euclidean near I ok correlate computing reweighte average matrix algorithm defer notice sample use know initialization ready conclusion could proof perturbation singular true actually choose follow infinite invoke counterpart give lemma
polynomial inequality develop encode closeness behave density estimate optimize subsection shape polynomial algebraic predicate shape encode map know inequality constraint encode ok develop far behave htb rescale shifted correctness introduce assumption see prove lemma first quantify robustness standard pdf distribution ever center know claim notice guarantee quantity suffice claim follow restriction density rescale k w I triangle bi give prove behave pdf distribution behave claim step time restrict polynomial bind system inequality subsection time complexity solve system propose succeed occur rescale behave triangle inequality almost step recall lie moreover become feasible note return via relate unknown scale back line fact good density step algorithm prove piecewise polynomial fact degree unbounded length assume interval intuitively different scale formalize intuition
multiplication zero approximate distribution unit element square clear interestingly proportional set mask replace hyperparameter call proportional gaussian autoencoder successful suggest play fundamental view huge parameter apply bag adapt generalization come report population express ambiguity think basic augmentation multiplicative formulate dropout deterministic significant autoencoder style corruption bias discriminative sample along also augmentation
connection uniform appendix wasserstein encourage artificial handwritten digit dataset digit encourage digit predict digits b approach treat digit true digit digit become evenly digit converge apply wasserstein yahoo two tag tag word tag unit tag redundant prefer tag tag find combination wasserstein train set control relative weight loss additionally second redundant difficult tag tag amongst decrease hard loss baseline effect
selective selective selective concrete example pair q eq selective formulate build know set question choice procedure choose term lasso perhaps interested away family sufficient nuisance key regard carry generate law question sided law law law test law know truncate surprising distribution selective hold second often describe data lasso power hold approximation unbiased selective clarity though selective communication follow selective estimator estimator claim obvious claim justification splitting
solid mathematical arithmetic transform theoretical advance arithmetic transform precise interpolation scheme spectrum length interpolation heuristic follow examine tool dct section arise introduce act discuss arithmetic conclusion remark dct regard eq hereafter examine act tool inversion tailor identity present inversion nan exist inversion formula unitary inverse sequence integer simply refer inversion development usual theoretic behavior function
include decentralized plot growth shown figure outperform performance curve single bandit logarithmic deterministic phase well suited use performance policy prior policy deterministic logarithmic truly remain subscript omit refer known expect exploration phase phase policy exploitation thus eq computation play exploitation jt lemma event policy one event q substitute b l expression chernoff hoeffding l without line illustrate relax without event let th play term expression last inequality come chernoff inequality note line come fact cdf inequality standard binomial hoeffding set define monotonically
generator generator exploit conditional facilitate characterization multidimensional generator equivalently partition introduce multivariate fix partitioning approximate generator partition multivariate characterize locally component dimensionality dimension aggregate global give proposition calculate fact express x generator p z analogously operator cn prove express cross marginal cn x approximation correlate decomposition mt exploit orthogonality operator follow stage introduce variation generator introduce let zero compute variety dependence achieve purpose specification copulas product copula distribution marginal decomposition transform multivariate copula joint new copula copula unit unit follow every
necessarily immediately thus nontrivial mean nonempty c attain integral indeed everywhere must else nonzero proof part yield lemma optimum difficult whenever result note satisfy property primal dual measure primal optimize primal minimum attain remainder subsection towards loss theoretic essential suppose entail dual optimum associated loss measure primal optimum e q obtain dual positive course q meanwhile show automatically thus grant development difficult tie show state despite existence arbitrarily good finite canonical consider hand complete whereby grant let set whereby grant choose whereby grant u definition sum b combine derivation desire thank precede similar difficult every loss portion portion claim follow consequently obtain every proceed similarly derivation merely issue surface first direct evidence place bound hypothesis vc combine grant probability display finish grant failure plugging lemma let hypothesis z n j kn remainder proof discard failure desire first univariate map rademacher deviation
thin connection normalize reconstruction projection dimension encoder autoencoder act perceptron supervise task encoder depict expressive minor modification layer convergence input
hadamard transform algorithm basis v element vector v r index element similarly row element entry resp recurrence q subroutine operation recursive tool ready finish row choice assume zero submatrix would whose I fact transformation generator observe correspond inner go claim modular reduce algorithm run design recovery sublinear interesting application reconstruction matrix guide search obstacle sparse sketch crucially use box specifically design augmentation compatible restrictive hadamard transform query restrict augmentation bit augmentation implement hadamard column entry bit row index wise product nn nm bipartite associated access bx b adjacency stacking contain query access deterministic tool section integer exist rr adjacency bipartite selection entry long compute time prove conjunction main absolute positive rr first argument
biased offline correct efficiency permit impact offline stationarity filter collaborative correlation recommendation various factor influence historical offline recommendation production tend item inspire
smc histogram show alternative continue abc right grey indicate tolerance bad curve small tends grow wide trade effect tolerance parameter compare alternative good operate approximation direction tune achieve performance smc possible map quantification uncertainty quickly three smc however leave upper return future period jump financial model way stable jump return stochastic volatility denote stable stability behaviour intractable specific recover three general abc rely abc posterior obtain abc solid abc histogram mix quite log posterior overlap alternative enjoy computational make abc forward filter simulator black seem log price present north produce adopt commonly marginal model dependency structure outline repeat abc estimate
physical system isotropic well construction ensure quite freedom design correlation delay value splitting composite sake composite value problem proper scenario intuition measure benefit inversion availability estimate hyperparameter face derivation cope derivative performance include algorithm
piece extract cm repository example help automatically extract share piece several project rt develop ad favor consist share exploration module share diverse computer source device share software experiment way public repository mind continue grow support source transformation fine transformation mind optimization remain hardware decade due intel carefully hardware enough demonstrate concept follow gradually move fine physics gradually quantum mechanic p max auto loop loop array frame block loop array heuristic optimize propagate loop optimize loop loop share software piece select share build generate format run select system characteristic cost process pareto record update win software piece optimization versus hardware cm repository table win combination production environment meta optimization gradually influential one good solution unified service far fast efficient system internet thing device balanced mobile device though speed preserve encounter serious number begin node immediately repository science year effectively mine classify physical biological specie leave thus reduce required engineering collective mind repository software piece specie hardware behave differently depend hardware include system hardware interaction specie execution
disagreement et small david h notion joint pair h h h therefore improvement theorem
well index class index I give bind c I I km ki bs pc ks pc k begin find index class must vertex vertex entry eq j define easily bound know available j possible cardinality result real need position choice position remain yield let estimate term
also importantly methodology likely happen one cross compute straightforward genomic preliminary use methodology genome array genomic identify mb begin genomic tend strongly contain proxy activity h structure strongly broad feature many genomic apply instance series economic indicator record item theory specie capture molecular genome different molecular cluster column array assume identically entity nature genomic refer along nuclear dna markov row observable independent law parameter specific outcome way comprise know special effect discrete model serial article feasible extend longitudinal moderate array even employ
rkhs mean smoothness decay ball axis know rkh fundamentally capacity ok space order purpose estimate constant depend interest complexity depend follow argument van shall assume see eigenvalue fix rest also non addition treatment model e constant ensure nontrivial bound
interactive participant typically working block stack game participant simplicity block involve action interact physical concrete object ensure stay dyadic interaction player kind building build build desire complicated design player player require build restrict build enforce color place player know rule player book certain objective need rule process try rule player play piece block sensor record video video track pair fig audio capture order ensure skeleton tracking slightly
diagonal thus infeasible feasibility error error increasingly feasibility label kind series converge uncorrelated component average use feasibility frobenius variance must decay correlation uncorrelate consideration connect assimilation scenario grid mesh frobenius suggest covariance illustrate posterior interpret weather blue average weather weather however little weather frobenius norm leave weather current weather small gaussian exhibit variation cover various weather lead forecast informative various sample exhibit domain boundary frobenius one forecasts accurate sense assimilation infeasible frobenius large point function decay asymptotic may enough feasibility scale state enkf enkf distribute enkf particle enkf enkf enkf idea enkf draw enkf enkf read summary enkf enkf whereas enkf approximate enkf ensemble posterior assimilation enkf ensemble joint sample sequentially logarithm
partially multivariate distribution analysis area application quantum mechanic finance copula probabilistic tool joint step distribution estimate however dependence fundamental feature economic huge important copula broadly satisfactory joint marginal pseudo parameter copula perform account uncertainty yet remarkable
tuple bundle require bit storage approach require space proposal estimate date sampling rejection metropolis investigate metropolis approach acceptance original still answer fraction accept key another evaluation generate factor regularization sample draw use variable let kk v nz ij intuition behind store original factor approximate small simple correlation determinant penalty want understand strength implement input new binary potential intuition come factor first gibbs sampling covariance solve create correspond entry new update approximated c execution approach dominate need variable expensive operation approximate factor time tuning understand news vary parameter quality change observe safe region five even minimal impact figure support large tradeoff empirical performance different differ factor number report tradeoff axis pt amount affect select weight result tradeoff exponential size observe variable slow sampling variational contain focus pt acceptance rate one order high require gibbs acceptance g fast approach low operation happen development show fast variational approach slow sampling graph discussion strategy method sampling sample true variational subtle
dim notice compatible density assume surely show erm classical empirical mean guarantee sure follow density learn theoretical application
pass graph cycle belief explain bethe message two tuple figure corresponding message update function node receive receive message sequence transmission iteration end configuration iterate unchanged message rule first message eliminate need lastly auxiliary reader show call ij perform include
circle calculate outperform gap gap gap condition past survey autoregressive use stationary usually regime switch normally n lm ar ar state analyze general ar world capable basic ar sequentially reasonable stochastic behavior cycle financial stock describe
strategy win strategy win maximize stay time win move game win strategy guarantee win strategy special even able winning strategy transition reinforcement priori strategy separate correctness specification priori reward compose strategy reinforcement strategy unknown concern combine strategy strategy recall set run induce strategy strategy say furthermore induce run win win strategie inclusion relation game non win win include win game win strategy add tag combine win win game win result representation maximally
run report illustrate significantly improve performance begin treat update refine pass propose scalable provable mirror root maintain tractable keep strong model true importantly particle mirror descent direction connect carlo promising acknowledge gm nsf edu cc school engineering institute technology bayesian scalability tackle challenge scalable yet typically variational model approximate kernel flexibility modal scalability kernel compute functional sample scale dataset
w extraction portion corpus unfortunately validation treat domain news domain union another half remainder remainder gold entity recall relation extraction focus entity log set consist carefully feature style exclude country log integration accomplish model p integration easy embedding compositional highlight advantage relation sentence select early stop fine report publish c bc f st st baseline domain table entity unable embedding baseline feature combination baseline baseline fine yield
per outer fig method call measurement outer scale sparsity per iteration scale per iteration cost much imaging practice therefore advantage synthesis typically translate net present solve highly constraint involve previously establish noiseless underlie interested minimizer degree involve g metric backward iterate point objective show violate jx j objective formulation certain one derive convergent fig mainly code transform unique approach step machine list definition domain fr e critical thought subsequence problem p penalty objective violate barrier unitary violate formulation replace barrier problem formulation write objective respectively denote cf unconstrained formulation minimizer formulation result also accept complex value input argument derivative part proper iterate magnitude magnitude e initial constraint iterate generate sequence decrease say iterate achieve finally accumulation follow optimality follow depend specific region column half b iterate accumulation every accumulation exact value vary initialization importantly accumulation critical well minimizer accumulation global iterate point minimizer hold sparse accumulation arbitrarily perturbation
find column sample deterministic sampling see detail naive implementation sis inefficient step require calculate fortunately update use block formula new column invertible column block inversion obtain invertible non terminate column wise already form need next iteration equation initialize starting formed form next entry update formula iteration symmetric store column choose index ki r matrix expensive however store require memory memory among processor submatrix copy column select column node column size communication essential distribute setting large million tractable call parallel detailed single central receive compute determine point sample dataset arrange
act eigenfunction integrate consistent less informative change integrated minimize large quality design arise mi base domain variety compare design design mi isotropic square alm choose select randomly location amenable hypercube monte experimental procedure find report f construct circular fix measure design greedy full spaced spread point domain place boundary desirable radial see mi behavior attribute poor mi yet size set many domain eps eps eps eps eps eps eps eps effectiveness perform regression describe design include greedy clearly outperform optimality bayes greedy strategy well design eps eps unbounded endow design maximum entropy procedure choose among brevity show generate difference unweighted maximum design spread region parameter weight entropy error figure illustrate domain via full minimization design isotropic correlation length objective describe point randomly design start cover evenly distribute design point well interesting performance superior mi approach eps eps eps
vocabulary run topic skip topic representation evaluate topic via select word topic topic lda select top high topic topic low vector cosine topic select similarity
intend semantic pick seem systematic incorporation selection way address elaborate hand user perform replace metric performing center suppose base size suffice generalize intuitively mapping algorithm notion vc method supervision method sometimes supervise supervision objective constraint close keep search objective function metric mark consider two objective usually hoc clear
entry denote euclidean entry simplex dd stationary denote associate follow start distribution account strongly symmetry system identical order ergodicity chain construct fully quantity mention introduction chain bound moreover also ultimately term make achievable rate gap necessary achieve multiplicative even additive estimating multiplicative estimating chain sequence length necessary estimator input length chain state two ergodic next length accuracy visit uniform pick chain
z corresponding matrix v k z subgraph node clique np complete provide evidence possibility opt
symmetric binary hamming distance source hamming fidelity give fidelity realization special source channel treat indirect version consider root whose additionally upper indirect coding introduce conclude cm distance source indirect indirect depicted encoder observe channel produce
somewhat meaningful rank technique intermediate knn outlier score estimate score purpose lsh db direction complementary employ lsh speed propose statistical exist superior score base learn unseen algorithm order detail asymptotic analysis real sample test look functional nominal define correspond acceptance value prescribe significance sometimes say nominal fall false lebesgue acceptance follow volume seek set capture briefly graph test volume anomaly simply check hence connect score measure define anomalous motivate multivariate value attractive viewpoint time complexity nn prohibitive thus
cardinality distribution brevity interested reader refer proof step next global comprise neighborhood abuse notation still q globally asymptotically stable usual proof use lyapunov stability omit brevity look nevertheless instead study estimation sec introduce concept correlate equilibrium generalization nash equilibrium scenario past naturally decision diffusion implement among information pattern agent real change underlie good first diffusion strategy play game agent collective behavior polytope correlate equilibria external action reveal construct parametric agent fundamentally model widely process convex construct reveal preference interaction introduce detect action agent concave external sec utility maximization agent sec play sec concave game statistical simultaneous external influence provide test develop sec games detection nash equilibrium consistent nash potential restriction compare still utility aspect nature take parse ordinal human make ordinal convert attribute scale make decision matter ordinal symbol aa humans symbolic ordinal theoretic social graphical game restrict
algorithm supervise example precision point pca variable non approximate draw recently map square performance dense speech corpus x method certain call hilbert rkhs setup lead rich algorithm rkh norm optimum
top principal component bind bind therefore asymptotically converse encoder column prove state encoder offer elementary construct encoder encoder combine decoder give bad ratio every encoder loss encoder achieve explain give result encoder loading dimension least bad sense batch construct refer row distinguish iteratively batch factor construct iteratively adaptively among might maintain combined sparsity iterative prove construct factor residual previously construct advantage recall algorithm target encoder residual iterative step encoder reconstruction satisfie batch encoder decoder rhs k g note iterative produce loss encoder orthonormal observe also
accord relation p hand pn n k eq replace obtain decomposition n l n l obtain n n n n relation together vector notation relation n n expansion lemma side n relation assumption eq equation relation n n eq furthermore I b bn proposition tend combine obtain martingale construct martingale condition apply step step construct martingale also consider I martingale
wavelet analytic wavelet coefficient nan similar proposition wavelet multiply factor fouri analytic asymmetric hilbert complex analyze signal wavelet kernel recall tt simple motivate wavelet take sum difference real hilbert us
state pc sparse reduce infer label order deep structure problem control evaluate factorization undirected eq subset undirected drop simplify graphical natural domain relational suited g unlike classifier classifier probabilistic undirected graph belong connected undirected py lf x tailor undirected final family label adapt fairly see suffer chain similarly attempt yet tractable graph resemble undirected randomness properly find structure modelling dependency dependency performance return get heavy random method lx label structure ensemble marginal order classification cv choose two small could cv dataset real music machine fit default provide implementation justify try overcome suggest factorial fitting fact relatively dependence surprisingly
convex convexity triangular inequality p wu fw maximum neuron norm norm neuron neuron set edge change neuron let divide original incoming input propagate ratio layer always scale layer new weight neuron input neuron equal w equality weight give complete weight p function due norm input internal last generality dag incoming edge discard direct say length since internal incoming vertex vertex otherwise norm understand capacity alone feed forward rely network regularizer admit capacity capacity behave central question analyze result unlike control potential class limit control norm convenient incoming per unit
handle address spatial need simulate avoid special type count incorporate flexible discrepancy example approach input reasonably considerable observation ice observational acknowledgment grateful liu year thank logistic web com national nsf nsf management agreement nsf statistical science partially support pa support dp solely author hc dim pd factor dim pd slide modern pd multiplicative pd relaxation pd heat pd ice non dim pd line dim spaced parameter multiply exist equation number call pd pd show ice capable reproduce observational follow interested ice slide description density peak exception modern ice coverage red line calibration green vertical vertical bar slight predictive change mode important sub ice slide section constrain calibration dash prediction bar rapid ice west rise significant risk lie region west ice rise computer calibration unable calibration datum inferential challenge utility
dirichlet allocation pp hamiltonian monte statistic pp chen stochastic fundamental hamiltonian monte subsampling computer pp simulate hamiltonian dynamic process speed bayesian statistic hamiltonian carlo regularity parameter space model report surrogate manuscript feedforward universal pp r parallel book pp pp b neural inf pp markov statistical science asymptotically intensive manuscript success bank systems carlo journal american pp htp cc hmc htp hmc hmc pde
rating movie user movie substantial online arguably rmse disadvantage notice figure individual predict movie preference plot example begin begin toward end interest clear evolution movie reasonable people fundamentally change able find individual movie movie valuable percentage dynamically stock return observe treat stock section learn stock drift brownian find within filter converging enforce closely note stock figure see stock increase assess ability stock price indicate capture histogram log tracking distinguish visually signal mention tracking performance stock price space capture degree freedom stock
term diagonal matrix trace algebra claim result pseudo inverse single zero restrict cauchy schwarz orthogonal throughout impose restriction score shift invariance necessity condition identifiability ordinal verify bound sake completeness absence error devote comparison high score indistinguishable error unbounde bound away expect unbounded name empty empty name display tag tag tag display tag make tag name macro cr cr cr cr name l topology crowdsource pairwise arise domain include among include widely use work minimax tight comparison induce compare may principled rate rate ordinal non expert human preference product directly despite literature characterization moreover guarantee assess pairwise comparison specify accuracy derive pair analysis reveal spectral gap certain scale play pairwise versus often ask human subject would adopt estimate superior whereas show ordinal identical pre comparison ordinal term measurement measurement obvious measurement denote begin evaluate
corresponding trajectory sequence normalize intermediate demonstrate multiply markov new instead derive appendix new equation modify fashion smooth perturbation reverse kernel perturb perturb flip kernel form multiply closed coordinate course trajectory choose convenient x reverse trajectory reverse trajectory straightforward forward process learn reverse transition p bound reverse eq low depend trajectory x analytically dataset roll bit seq seq pixel leave bit
real text evaluate criterion moreover comparison increase fail success successful conduct issue reasonable rate comparative study multinomial inefficient propose em hierarchy merge naturally component compare method multinomial mixture mm explicitly
number evaluation mh drop proxy necessarily unlike store evaluation likelihood next per second proxy read choose perform likelihood additional computing proxy leave implementation summarize run average evaluation proxy proxy manually assess proxy less average show thank force considerably imply delay quickly converge fast gain gamma q gamma distribution shape parameter parameter assume additive dataset nonnegative run chain iteration drop explain section pdf line correspond budget mh correspond average per proxy proxy increase science big much effort scalable broadly classified divide divide separately individual limitation mcmc introduce divide literature rest devote marginal section focus illustrate improve control approximation sampler break barrier limitation expansion research scaling material inference mh also detail conditionally likelihood parameter bayesian unknown unnormalized application focus method mcmc hasting mh approximate algorithm illustrate mh mh weak assumption suitable k accept reject generic
kx recurrence indice union gram rank give w te identity allow require update variance determinant update prior function input posterior gp assumption prior mat covariance evaluate update update matrix aa ab k b bb n product ern unbounded q ab bb dx contain improper integral valid converge limit improper integral q appendix posterior mean test gp evaluate gps specify full input frequentist f fx gps mean angular separation show error localization fraction randomized sample become return select generalize well pd visual column
coordinate calculate need purpose algorithm epoch sdca sdca sdca type function smoothed hinge loss smoothed hinge eq problem smoothed machine loss note smoothed hinge see dataset option sdca number iteration quadratic smoothed loss option
assign update rather fix primitive nonnegative reverse weight eigenvalue primitive achieve design requirement assign multiplication satisfie assign transformation distribute consensus ik iw jx jk property modify satisfy frobenius later utilize consensus let connect modify property nonnegative eigenvector eigenvalue circle primitive tt tw j jj strongly connect irreducible matrix primitive modulus author I undirecte topology employ consensus ik ik iw jx limit frobenius theorem primitive right leave primitive nonnegative respectively kx ix gain insight
draw treat ensemble variance ensemble process broadly variational computation operate gaussian gps building variational robust combine operate subset forest popular investigate use rf optimization probabilistic forest uncertainty implementation regressor fit leaf unlike forest node associate attribute highly forest predict message uncertainty kullback moment applicable regression even application estimate good produce categorical label online prediction comment estimate mf popular
construct example formalism summary provide conclude remark introduce quantum information classification mechanic vector represent quantum accordingly q express combination us store measurement represent product space tensor write formalism scalar element scalar product represent dual deal label denote class discriminate pattern
specifically much likely one would score pattern generators pattern paradigm particularly potential disadvantage sequential show interesting conference name word frequent word extract closure interpret generator focus traditional principled partition token frequency well finally introduce efficient search interesting sequential database sequence exist head element remain sequential set often call dataset record pattern record subsequence notational convenience pattern derive framework pattern tackle introduction unlikely would pattern core explain frequency frequency example interesting sub sequence follow measure assess deviation develop formula length intuition definition sequential pattern modify pattern iff support aim sequential pattern partition subsequence
square prove particular smoothed bound consistency consistency split three panel parameter risk apply splitting bandwidth shift algorithm result dataset row bottom character splitting panel smooth remove noise optimal digital survey huge galaxy galaxy galaxy universe uniquely slice universe
sensor adjacency graph close among thing source time structured nature challenge first principle alternative great significance develop network graph become describe relationship low compute make inference task topology influence network problem first infer problem adjacency estimating assume graph markov effect draw provide brief overview notation section infer information brief
computation storage computation toeplitz computation storage storage proceed gradient product machine evaluation role virtual observation alternatively product eigenvalue determinant accurate small possible take fast vector complexity potential kronecker toeplitz accelerate approach unlike induce might exist formalism uniquely substantial efficiency induce trivially correction approach easily still understand predictive gp regression kernel find induce mean kernel could interpolation case incur aside ideal perform gp regression popular neither accurately conventional induce induce replace gp kernel interpolation inverse
library study gain transform hope work arise compute et convolution parameterize optimize analyze tradeoff characterize recently optimizer release optimizer input idea optimizer tradeoff rely learn topic google microsoft project core across design focus cnn plan study believe effort case lead gain support project fa national foundation nsf award national
set binary website datum achieve performance directly training perform much computational overhead name space hashing square classification present dual recovery mild reduction recovery benefit good acknowledgement like anonymous helpful part r strong strong modulus combine eq eq proof inequality conclusion q modulus optimality q hold thus result conv conv extend
distinguish small index gender salient difference stand also probability crowd triple probability distinguish therefore crowd salient distinguish triple sample triple unique distinguish identifiable triple eventually argue identifiability draw iid features moreover identifiable triple query totally distinguish feature feature could multiple people discover distinguish reality crowd completely natural help light help turn make triple query grow triple triple independent frequency infinitely query adaptive triple query triple non triple interpret triple query feature query query seek
bayesian adaptation disagreement equation pac bayesian analysis stochastic classifier result state e good domain reflect usual adaptation deviation respect sg provide pac analysis domain number sg empirical kullback result source minimize however optimization argue negligible achievable major specialized linear classifier dot building
notice modify filter place demand require history particle backward smoothing design approximate estimate full filter density k avoid evaluation transition suffer particle sensor due arise snr efficiency exponentially target particle transition evaluate combinatorial fortunately evaluate multi continue evolve addition new target distribute birth define birth cover bernoulli superposition target independently transition combinatorial sum simply term target weight particle drastically reduce recall approximate measurement approximate measurement assume noise
human shown focus single nucleotide explanatory categorical encode aa bb correspond snp level marker snp genomic ce ar ce net ex ex rs rs screening select snps fdr ridge rna ols select snps identify rs note cluster intra correlation negative correlation study context ridge ols conservative magnitude stage connection ridge screening stage magnitude screening bring improvement improve improvement clean uncertainty whereas stage operate valuable situation post beneficial regression respect penalization regard error clean discovery extend sensitivity stage often ordinary penalize second employ
confidence run solve algorithm binary random n hx bound example sample trivial chernoff x chernoff chernoff return challenge show run inherently require rate polynomially sufficiently break barrier e
central limit eq central arbitrary random equal hence j e w w w immediately derive show lastly validation loss subsection minimize inverse px eq l p k f p hence chapter several mathematical summarize model theoretically cv variance little cv cv approximate cv cv rigorously true cv general appropriate predictive subsection cv applicable apply find property
high krige character calibration krige compute krige increase krige consist krige calibration krige spc krige approximately intend technique realistic response apparent krige resource lie run meta study large experimental scheme generalization size error cosine polynomial expansion dominate krige sample size ordinary krige outperform approach krige work well sample sample interpolation auto krige model subset neighbor meta approach krige magnitude computation though reality meta model krige optimally loo function present design loo dash loo predict polynomial loo solely contain whereas relative generalization although polynomial krige loo procedure pc krige meta decrease loo polynomial also leave loo polynomial regression part context assessment design repeat run
reconstruction I good henceforth cardinality suppose reconstruct require function side quality contribute theorem measurement perfectly bind signal dimension sign magnitude reasonably pursuit q indeed negative depend section scheme counterpart informally section noisy reconstruction noiseless positive sequence sparsity kn generate gaussian measurement kn scheme reconstruct measurement scenario straightforward discuss later essential e camera initialization signal reconstruct initialization measurement accord reconstruct pursuit henceforth initialize side subscript part loop input measurement prescribe reconstruct compute match bind measurement acquisition
maximum bound last analysis confirm precisely suffice right hand side check clearly iterate logarithm suppose give eq university california access distribution constrain basically guarantee false offline constraint notably distinct computational many unknown analyze empirical iterate nonparametric homogeneity independence make broad applicability decision pose alternate dataset control positive false negative control
robust next superiority please drift end pool namely four situation memory plus pt pt ep gain pool outcome environment ep well implication accuracy conduct strength pool counterpart high accommodate spectra pool tailor detector incorrectly partial develop store introduce fluctuation change fluctuation fail fashion poor situation drift detector delay property capture would couple simulator classifier store repository live live use store classifier occur live simulator also repository future exhibit sale make improved capture store compressed repository concept future number challenge firstly compression storage dimensional grow nature stream ever
dropout use interestingly range bayes great replace weight still perform examine network noise top plot plot see modality ratio cdf separate peak peak drop suggest related removal require compressed error r begin ensemble twice parameter store run pruning ratio time proportion still maintain encourage spread successfully show example network region ordinary choose noisy range
section lagrangian involve update velocity term exponentially grow velocity update extremely avoid vector update geodesic flow sphere obtain apply x evolve xt tt go back velocity value u v l calculate summarize theory hyper type constraint map jacobian rather complicated large q consider topic view identify positive component sample sphere transform simplex root simplex natural define sphere belong category I kx ik tn kn posterior simplex component calculate simplex k metric intensive document hence refer recall omit adjustment dd regard go therefore illustrate method toy riemannian langevin method use expand mean might figure set run k compare far generate low right panel real datum compare
admm optimize primal perceptron perform task admm confirm rate slow inferior investigate solver consume dp portion training spend achieve pos machine already multiple time less use bandwidth multiple address challenge training propose algorithm outperform structured svm capacity structure learning volume software public use edu training difficulty structure
estimation result mode shift per transformation mode show emphasize central parameter satisfy invariance ht aforementione straightforward remain four leave unchanged five parameter appendix derive mml explain derivation mml message distribution prior take two sphere density main mml fisher later computation first moment proceed mml follow normalization constant adopt represent distribution expression whose mean minor axis align coordinate setup provide mutually axis orient fashion axis align axis rotation axis axis distribution deduce second partial negative log parameter angular fisher scale derivative comprise expectation identity expression provide fisher symmetric element thus length formulate fisher message length mml message length mml map library routine information calculate computation derivative computation form message presence derivative provide formula section explain quantity logarithm logarithm consecutive sum term convergent express term calculate previous logarithm normalization implicitly derivative equation eq equation equation note formulate relate explain convergent series derivative eq substituting hence give series overview model directional parameter w estimate mixture likelihood involve em
devise content user tweet post exposure baseline exposure less negative adopt general adopt technical increasingly play twitter million impact discussion response medium offline central issue content media affect behavior year longitudinal suggest pass social long effect pass also recent facebook suggest even interaction perform correlation word post possibility suit question existence raise
integer inverse sum unbiased provide detailed probability concentration use size relative least roughly proportional sample use coefficient approximate satisfy however universal independent node universal interested dominant component single pairwise query universal pass coefficient represent collection sampling node require time estimate single source distance probability relative polynomially arbitrary member query computation preprocesse query distance computation weight desire point cv exceed polynomially small primitive computation metric sum cv single exceed polynomially distance computation exceed polynomially small provide claim use computation provide later concentrate computation c compactly outer representation draw pair use dependency unbiased approximate storage approach would oracle store
q perform block eq event follow ball copy small chebyshev selector copy concentration least lemma recall integer name later eq eq least nj hence assertion sl use may next respect class indicator probability observe depend probability r star sl mi jx subset homogeneous star shape event recall depend good part least every thus least third independent begin infimum star shape
base recurrent network multimodal description relate image task specify word shoot learn associate attribute amount co idea show closely shoot category learn object target task paper base adapt base develop main modification side improve performance original significantly secondly recurrent lstm layer recurrent neural vanish briefly detail weight sentence word image represent indicate component vision layer lstm convolutional remove final softmax deep cnn top fully connect layer layer
contribution therefore equivalently sum switch sum conclude function polynomial therefore apply approximation answer submodular logarithmic build technique seem know expression happen handle influence imagine bound derivative coordinate side submodular gx easy sketch argument lead totally submodular totally symmetric sense note concave function influence case large totally symmetric ni fx opposite function totally fx n j fx assume fact modify region adjust function show simplify prove note assumption partial derivative method bound tail certain level choose trivially trivial totally influence need handle separately suitable prove influence decay influence purpose approximate coordinate wise define terminology boost depend denote indicator
draw four consist choose elliptical highly elliptical scenario orientation orthonormal basis differ jj training misclassification repeat time rgb c misclassification equal spherical spherical able covariance invertible stable inversion similar setup low equal spherical difference prominent space hold spherical see always scenario consistently superior large lda potentially toolbox modeling across straight forward pool alternative article advantageous inter variability virtue quantification inter estimate exhibit largely versa attempt aid basic study heterogeneity demand improve derive nan yet intractable
information evaluate reasonable high state number hmms work calculate viterbi recognize remaining go deep hmm correspond price depend viterbi histogram mixture approximate histogram time base euclidean distance tend price order try initialize even state level transition stay shape big quite outlier
condition choice increase theorem difference assumption stationary stationarity let lyapunov condition lead artificial hold finite eq trivial perturbation number bind wasserstein distance distinguish ergodic chain hold geometrically ergodic helpful wasserstein estimate wasserstein corollary lyapunov kernel sufficiently quantitative perturbation control variation perturbation geometrically markov call ergodic irreducible markov ergodicity uniform ergodicity geometrically ergodic respect constant establish connection wasserstein due also define q wasserstein similar argument suitable upper satisfied elementary obtain assertion lemma theorem geometrically ergodic lyapunov vx perturbation
arm number become repeat explain sum stochastic tight consider identify good predict term quadratic optimization efficiently gradient ta quadratic identify problem fortunately agnostic function adaptive arm discover ever algorithm behavior difficult uniform allocation let end budget arm equal return unclear method actually next say tight bad real naive allocation necessity budget exist sequence loss budget multiply quantity
modularity position series community detection survey iterative detect community removal community path vertex community edge go edge account short run moreover walk random would path iterative produce dendrogram situation belong modularity select division network edge vertex community modularity hierarchy rather build dendrogram focus computational method proceed greedy merging
change treat candidate via diversity decomposition quality candidate change define around well change item close parameter represent take kernel rich dissimilarity metric could tailor use kullback leibl segment compare follow give ratio numerator follow plug segment homogeneous occur candidate cross green line method five world firstly classic segmentation characterize hard
algorithm allow particular must subsection difficult improve property class need assume run one close correct distribution exist efficient hypothesis hypothesis produces belong furthermore introduction agnostic efficient class point inefficient generate illustrate corollary first monotone hazard risk class risk property demonstrate approach monotone piece indeed learn simulate per simulation class approximation learner say agnostic combine learn obtain agnostic rough namely efficient constant learner distribute hereafter happen probability correctness sampling imply mean output class agnostic guarantee get efficient class proper learn binomial stress sake illustration binomial instance aforementione require agnostic binomial design sampling exist knowledge although agnostic suggest sample strategy inspire try guess good agnostic learner good good agnostic learner total bind hypothesis satisfy condition remain guarantee variant procedure failure exist make n use well come existence efficient convert testing testing connection test correct estimator non corollary sake clarity reader may pass sampling access accept hereafter happen estimate hand case exceed step procedure know estimation sample instance guarantee modal observation know property quite modal useful derive bound monotonicity
refine none base single class detection though intrinsic handle diverse pose successfully converge window human body robust diverse human pose carefully combined target bounding proposal clearly strength promise cnn demonstrate gap cnn cnn error strength cnn yield multiple primary plot curve curve cnn tendency corner take decision corner box weak confident box score bounding box achieve pt l extra refine cnn refine cnn evaluation method demonstrate human detection performance cnn boost
draw color sep x mark sep crcr acknowledgment support collaborative institute intelligence ci lipschitz progress towards w tw algebraic summing obtain value divide case would every write optimizer let minimum applying first feasible attain claim feasible fan symmetric decrease increase equality proceed prove symmetric respectively since orthonormal doubly claim thus differential matrix clearly
correlation future square could mle penalize mle precision paper pt rectangle rectangle rgb rectangle circle circle rgb rectangle rectangle circle circle cycle cycle cycle rectangle circle rgb rectangle rgb cycle cycle cycle acknowledgment author grant du pour lemma write cc ii use triangular matrix eigenvalue eigenvector definite conclude also thus matrix eigenvector definition corollary accept date estimator call vector precision underlie estimate form problem computationally estimate diagonal degree precision matrix popularity year rely diagonal element recent contribution aspect find view nonzero element convexity
vb pos seq corpus sampling optimize algorithm include guarantee fit corpus carry search choice substantial nonetheless evaluation term substantial effect vary ht relation european per un counterpart nn european trade nn jj trade nn un special nn cd market jj nn market jj nn dt jj jj maker bid market jj nn
repeat independently suppose single coordinate accord single grow walk together summarize additional replace simplify improve empirical present detect community disjoint subset usually random vertex th walk start measure probability act empirical observe sample degree walk tb walk component initialize algorithm essentially k non measure unchanged strictly defer finite terminate cost rewrite somewhat distribute started let iff shannon entropies interpret maximize step algorithm another algorithm minimize graph relative need take account cluster resolve issue explicitly second
statistic naturally writing variance plus detail therein empirical easy approximately bootstrap investigate article angle collection learn subsection show approach numerical assumption vc reveal collection extend incomplete version maximal symmetric straightforwardly deduce bind incomplete empirical proposition eq subgaussian deviation incomplete counterpart class previously tend maximal minimizer require empirical require slow hence remarkable yield preserve upper summarize empirical nb statistic automatic crucial cv situation adequate level vc incomplete statistic penalization split extend kernel
core gradient hessian start numerical nonlinear response covariate regression independent term df aim bf normal integrate whereas integrate concern independent bivariate scale df recommendation prior jeffreys take consider logarithmic compare laplace correct laplace approximation comparison correct result posterior multivariate distribution df inverse modal also obtain c laplace
case nontrivial vector straightforwardly case large relaxation state second eigenvalue anti hermitian hermitian utilize know anti hermitian apply characteristic ai since large picture root characteristic ensure slope asymmetric near root point hermitian root prove hermitian anti hermitian satisfy unitary transformation unitary change large positive definite find large eigenvalue symmetric acceleration nontrivial accelerate desire langevin force potential second rotation field
come outside could public use go construct private idea originally inefficient protocol private estimation private lemma moreover entry enjoy note finally review code private histogram code length subset rather mapping satisfy constraint know relative distance fraction error word code decode several construction literature property generate differentially basic bit hypercube symbol pick string bit scale bit become construction bit serve notational describe user item basic z pair bit choice randomness input however represent require index output fact privacy hold independent randomness help ensure come public situation server receive describe construction private projection construction three construction provide oracle estimating frequency heavy randomization copy give guarantee difference carry oppose private oracle item theorem affect confidence guarantee public construction generate wise independent namely protocol protocol privacy confidence di server compute length report run basic efficient run easily verify fix item oracle item privacy privacy frequency private randomness output proof good product formalize let copy sequence take basic put hoeffding claim linearity inner second least union show oracle construction three identify frequency randomization user copy give pure opposed protocol protocol output oracle object measurement report frequency use simply inner encoding user protocol encoding length user item utility construct follow construction oracle user construct protocol projection input upper asymptotically tight rely concentration inner aggregate encoding item encoding histogram use private frequency subsection call fraction server hold item private histogram universal protocol item server server learn item rate relative say quantity asymptotic construction thesis example fix code encoder decoder code obtain server report code vertex instead round near hypercube running randomization round sufficiently precisely describe protocol code
low score build classifier knowledge high time break size score acceptable propose selection turn score redundant feature finding threshold score naive approach finite set
informative variability protein explain scale gene correlation informative exactly across type biological protein implication simple variability protein efficiency reflect degradation refer ratio variability poor statistical change protein across investigate scaling figure scale instead protein scale figure gene vary extend pair c smaller observe scale despite similar protein c substantially unlike specie individual protein level vary much support
I j simplify notation often dependence relevance rich include general intuition let discuss translate cf general fashion eigenfunction nonetheless eigenfunction deal subtle issue sequel happen possible actual impact simplify sequel universal universal hold j I assumption depend deal sequel due universal preliminary working though particular inner result focus setup translate eigenfunction eigenvalue universal j h degenerate impose score discuss dependence hold sharp dependence condition condition long however show carry pose sense memory long show necessary hilbert detail finally necessary key appearing variance eigenfunction decay instance reflect discuss turn much weak shall nearly implication j note mild include encounter result moment like growth expansion weak formulation state quantity
model cascade expect convergence rate estimator state theorem cs solution convex condition convergence must smallest possible lf n crucially independent recover support set recover false assume rarely network particular realistic parent recover cost small recover result limit formalize consequence solve recovery guarantee interpret restrict degeneracy apply hessian essentially reduce strong hessian strictly
q quite quick sensitivity tell bound original score appendix modify easy confirm cost bound difference upper bound sensitivity would parameter zero bound become loose third e depend amount measure norm old coefficient bound coefficient bound rather next sensitivity upper score let arbitrary dimensional interested label leave whether correctly classify step
index fill equal diagonal row correspond row sum r j p need p r entry label label numerator denominator already entry hence compute uniquely construction method imputation construct condition kk kl j iv follow prove imply row substitute last kl way diagonal remain
equal entry represents predict student answer correspond network get negative log response encode exercise time binary student q minimize stochastic overfitting apply prevent gradient hide knowledge student future past continuous power good sequence give estimate hide rnn calculate particular exercise exercise knowledge choice intercept next markov classic rule education literature mix topic answer particle particle
mix similar deep cnn include flat recursive rnn utilize sentence parse rnn build tree represent dimensional sentence code supervised parse categorical exploit rnn enhance improvement vector capture logical sentence rnn propagation leaf rnns neural long gradient vanish make difficult term memory lstm data rnns rnns tree structure lose cnn sentence dependency variant c subtree feature detector call tree
every span span span uniform aid dimensional previous associate denote measure dense dense vice statement true also r open dense build generative column draw dense open subset lebesgue column shorthand play
introduction key ingredient study characterize toward series various method financial market yet attention moreover momentum medium small strictly great effort read economic piece technique dependency market give specify copula technique new situation account important close span characteristic highly interesting people market put apply already cite find economic characterize effective forecast real market completeness split project part aim give contain introduction univariate attention relate time delay resp
size statement improvement processing capability art experimental method powerful framework processing use question combine idea develop visualization set whereas practice gap fill mean imputation interpolation various krige learn neighbor artificial network similarity process suit high drawback scalability datum algorithmic complexity memory degree achieve technique parallel approach employ approximation reduce local approximation involve method composite pseudo involve dimensionality precision field mrfs also locality mrfs datum initially structure grid field mrfs via mrfs propose interaction model
appear click ucb explore use continue rather one strategy inspire article would strategy case exploration tradeoff though relevant click rate explore click click setting dynamic advantage success multi armed algorithm past key element exploitation reward period possible extension advance multi besides connection acknowledgment member ed ta exploitation inspire thank helpful project mit time arm play mean rule decide rewrite z round set separate bound playing inclusion
neural combine generate fitness argument evolution addition combine mechanism rule trait mutation mutation system new trait alone fitness evolve enhance develop achieve later encode trait robust also loss capability mechanism come concern act evolutionary advantage attempt explain ignore quantum randomness explanation controller feature break system trend apparent deterministic network study neural unit interaction physical self internal brain brain rest coarse understand brain robot normalization p normalization entire neuron individually normalization frobenius norm use row regularization near keep normalization scale slow activity converge mode appropriate lead exploratory impact result normalization behavior neuron contrary normalization activity example robot initially body learn
model presence intractable grow either specification costly evaluate wise resort estimation inference sampler computation carlo sampler typically posterior ef numerical chain give asset ss n j tt feature market trading hour trading producing vector value record build observation set grow interest computationally intractable evaluate result modification task sampler develop employ essence reduce observe low summary replace new target computationally intractable embed joint observe simulate obtain c parameter weight auxiliary observe dataset via smoothing argument case point elsewhere marginal impractical choice smoothing free abc incorporate summary conditionally ft specify kernel ht ht posterior extend general proceed augment sample posteriori discard sampler quantity directly approximate integration evaluation eq q approximation reduce otherwise enter variance likelihood free sampler poor propagate particle direct target abc sampler allow place great
optimize bfgs approach regularization report crf access aside select hyper calibration training method begin w adapt stochastic approach entire randomly choose subset create different collect report average test across run statistically pair together tailed difference experiment optimization variant learn substantially improve design learn indeed beneficial ht significance c crf second run perform parameter search result skew regularization able recover good optimization
oppose deep widely action recognition dataset cnn formulation consist feature art image fully volume convolutional network spatio cnn video video feedforward increasingly recently stream instance term extract video rnn patch video explore couple domain description video description corpora suit evaluate automatic generation description video total dataset video description vocabulary unique dataset open domain topic music validation large video description video description corpora video along video wide situation suggest toolbox
fine network pooling proposal handle problem window extract scale window across densely across multi feature different aggregated stacking sum present integrate external svm iteratively boost object classification ambiguity mining difficult complete fair comparison augmentation substantially fairly cnn train separately classifier intel cpu employ generate proposal combine extract object scale selective still code proposal extract hardware tight application recall less overlap performance much average maximum proposal parameter pca feature preserve
trick vector universal mmd kernels taylor get cover ability salient appeal capable latent density learn effectively density mode area capable mean responsible complex transformation original involve deep array approach outline class learn boltzmann normalize typically intractable usually expensive carlo mcmc fully sigmoid network autoregressive admit image take parallel method sequential nature relate work devote recover
pa outside exercise list incorrect reciprocal polynomial root show produce inside deduce inverse root simulation design around repeat code virtue inside last root outside disk extend code sd sd k call code consistently module show convention eq q distribution conditional recursive value compute distribution conditional proportional proportional costly realistic value get derive coefficient term construct argument variance useful horizon model obviously ability horizon complete deduce model f independent full chain also observe hide figure book marginal deduce identifiability mixture since distribution stationary although markov switching indicate exercise chain simulate comparison variate simulate variate repeat simulation use book gain back formula involve summation summation later multiply order confirm irrelevant method case doubly framework hide part case particular posterior distribution dirichlet book label switch introduction hide posterior gibbs symmetry likely biased implementation involve pick high averaging conditional counterpart prediction switch recursive formula exercise obvious development distribution ji update fx r stationary state obvious question conditional joint agreement conditional infinite full distribution
extend show belong neural unbounded theoretical find non admissible work pass interesting work activation deep essential output scalar cascade multi play key role relation reconstruction cascade transform coincide filtering point view structure gray gray thm thm thm remark property function relu de learning relu respect neural integral transform neural analysis note old theory regard neural transform constructive al
fine convnet nearly match spatio temporal deep convnet give describe transform spatial learn convnet imagenet convolutional training convnet imagenet goal convolutional weight convnet convolutional layer remain originally order time e kind initialization consider consecutive image
plot point draw axis represent fig plot show fig derivative normal matlab bit window operate intel I processor gb ram show linear benchmark uci learn accuracy dynamical accuracy htbp dynamical svm voting
bp denote community long imply work leave plane backtracking approach eigenvalue disk several eigenvalue fall disk example eigenvector eigenvalue yield community position correlate structure group obtain infer transition conduct numerical generative dynamic choice community maximally
bound inequality combine everything l odd nn bayesian weight write place prior one hierarchical predict modify simultaneous hold hierarchical long note magnitude magnitude hierarchical even must magnitude regret derive eq datum hierarchical share big special journal comprise political public health robustness statistical limited commonly cite practitioner hierarchical contain obtain vision illustrative benefit employ category car label example visually label object category group tailed category example label
structure short seem issue ground truth level expansion size topic belong discriminative topic weight per document topic might document already view local anchor document pair document dominate topic variational relaxation closely description number topic document fractional count word way pz I factorize optimal distribution say variational update show one q document word close expression optimize via gradient assign clear would like ideally optimum work vanish contribution simplify remain focus e update update try approximate large value e become q convergence modify equation slightly modify use negative factorization author update preserve f iterate update way version minimization perform make modification step natural go add fractional average reason behind study kl put weight document term actually variational inference thresholde em min step start previous initialization look focus inspire use way treat pure fractional find topic first long topic word ensure topic overlap support assumption topic dependence topic conditioning analogue distribution roughly appear document small large
wish distance space theoretic advantage add triplet principled shape normalize metric triplet location associate variable one belief triplet whether draw parametrize distribution accurately serve triplet relative maximized acquisition triplet ask distance statement similarity ranking encode relaxation similar alternative probability pz pz pz amenable thresholded formulation smoothly constraint flexibility relative introduce instead rely prefer use model data triplet unsupervise oracle belief family mlp transform
lag subsection design length autocorrelation band sequence spectral algorithm name minimization guarantee acceleration scheme numerical generate sequence design autocorrelation complex without generality assume modulus sequence design periodic define goodness autocorrelation periodic note important goodness mf central practical good effort devoted study focus sequence early extend later frank exhaustive evolutionary heuristic suggest generally capable design
information production consumption million log facebook news read become interaction although affect still effect sentiment work quantify sentiment whether broad vice versa social medium type sentiment message suggest call exhibit sentiment evolution diffusion material sentiment effective stream predictive purpose sentiment analysis date sentiment design short text sentiment adopt promising tool provide advantage tweet employ linguistic etc application datum effective
albeit covariate learn rbf latter training moderately suitable besides interpret attempt interpretability store easy numerical number task rbf function value store result store weight coefficient find fig show predictive performance value bad regardless average candidate ht transfer correlation hour st day week fig display transfer obtain correlation atom estimate hour day note transfer depend condition bc omp omp clearly satisfied interpretability learn study customer relate activation hour day type vs experiment choose negativity trick customer function activate model look intuitively consumption customer peak day activation week business available transfer function customer percentage week
indexing calculation goal efficiency variety convolution shape report respect minibatch practice lot layer parameter input conv conv x conv conv l network architectures convnet code propagation believe effective
detector employ point outlier consider different detector explanation density detector operate treat usual anomaly normal joint point mixture gaussians approximate inference g mcmc noting consider method anomaly detector term anomaly situation anomaly point reason relevant anomaly help analyst anomaly since critical outli make refer model analyst classifier assume anomaly uniform reasonable absence analyst would likelihood threshold since compare marginal analyst anomaly choose particularly method method add feature compute set compute inherent minimizing approach focus quickly manner explanation sorting feature increase marginal computation offer alternative feature select value serve understanding
publication similarity among assign author try create attribute name keyword etc heuristic manually predefine well specific originally poorly solve recent internal dnn relatively help deal build author publication record new ambiguity additionally author combination name present author learn section experiment work survey
indicate complex dependence relationship direction recently approach post causal give cause assess nonparametric draw generating fortunately bootstrap way successfully validity demand necessarily enjoy generate structural estimate indicate variable method validate real system artificial understanding relation predict usually causal discovery attract much find property causal make use conditional
exploitation sparsity decomposition currently investigate hmc member target member iteration mass integration modify cholesky hmc remain member hmc hamiltonian work step section numerical apply equation associate proposal particular euler euler specific proposal modify cholesky small l j l tt construct process current markov probability mh propose restriction satisfy minimal strategy choose liu strategy proposal distribution hand low acceptance thereby distance still lead fix variance extreme note target magnitude many aspect additional distribution joint latent different scaling support consequently
augment evaluate message pass cnns unary potential capture crf potential message construct variable node pairwise unary message pairwise relation formulate estimator output network initialize available learns capture contextual follow crf prediction potential potential final network slightly achieve compare potential function one negligible enable perform simple augmentation training extra scale denote segmentation supplementary method result method
oppose instance bad complexity grow matrix additional help robust exist formulation regularization soft vector develop lagrangian kernel trick nonlinear present confirm effectiveness improvement enable regularization overfitte discover structure scalable variety alternative centroid brief column center involve bad center vector construct correlation linearly independent template centroid complexity iterative geometric formally template direction angle outlier template reasonably existence fig template return toward recognize
qr require inner product operation qr compute time operation qr decomposition operation qr algorithm column module next code aim recover bernstein theorem power pm eigenvector accurately pm perturbation negligible pm find accurately b make row column ba b ta input initial every randomly b uv smc present smc main step reference denote principle explain detail show singular explain contain reference line code column
insight obtain proportional variance community community small discriminative significance value close behave relatively similar nonetheless fewer well confirm validate er modularity benchmark compare formulation binomial root child create add keep tree prevent numerical issue partition good node graph usually large expect benchmark distribution plant community size small large plant exponent play crucial plant community link
individual decision tree overfitte historical introducing prevent overfitte average train known introduce completeness bag complete mapping derive bag explicit dependence ci ci typically bag entail previously unseen computing switching point include full approximate marginalization subset tree parameter intractable map e separately draw entire average mutual label mutual play role softmax regression layer predict give classifier entropy layer calculation across level second max dynamic programming third theoretic related diagonal noise equivalent employ result collapse version serve prevent overfitte despite similarity difference naturally max pooling second affine act multi channel channel connectivity rise locally fully switch critical architecture base representation inspire idea accord perspective feedforward aspect visual processing representation irrelevant pose etc sense perspective qualitative explanatory architecture serious success deep architecture precise type nuisance transformation group relax build share goal theory explicitly notion however differ impose nuisance wider include naturally pool probabilistic nuisance arise direct consequence comprise theory complementary approach deal energy focus template discrimination question future notion nuisance invariance approach invariance series wavelet nonlinear modulus pool wavelet nuisance rotation transform modulus moment st image model determined wavelet template maximally strong learn st well consistent bias dataset successful therefore bias st world nuisance learn vast search
detail publicly available correction scan low frequency impulse voxel subject consist repeat averaged signal noise volume cover voxel identify motion selective area v inferior temporal voxel interest retain cca subject result voxel voxel voxel compute cca bold response three subject ten appropriate size start later recommend validation cca subject response bold response voxel surface interpret high low weight interpret discover
make interested compose monotonic part per event contour monotonic contour challenge monotonic notational totally hand function high sx equal sufficient base q integral ratio think discriminative ratio far likelihood univariate density generalize case free density classifier observe never parametrize pre compute evaluation use hold composite generalize ratio test presence nuisance nuisance break component nuisance fix obvious work particular work
classify versus order level respectively construct distribute sequence training word nlp corpus fed number corpus need train representation biological sequence rich protein database manually annotate review representation sequence sub simple common bioinformatics length overlap gram gram extraction utilize model embed model train adopt extraction n overlap window windows list shift validation window vector overlap versus gram show embed splitting h eq represent
reservoir assumption precisely optimize infinitely setting first optimize regret ucb design extension ucb design arm identification purpose comparison par ucb constant reservoir take right ucb perform worse empirically confirm remark ucb equip arm reservoir compare experiment time infinitely bandit potential regime efficient acknowledgment education national research project extra ce write reservoir reservoir reservoir express infinitely bandit assumption modify regularity version assumption equivalent generality consist arm reservoir crucial layer sample arm arm true object decompose arm gap
discuss inference ed derivation equation supplementary material convenient covariance conduct use inferential choose simulation strong reality perform maximum likelihood flexible exist one observe object unnecessary framework allow select template compatible frequentist assign prior clarity presentation another moment exist encourage term hierarchical interest process covariance mixture cox process main seminal shannon information precisely ed maximize kullback leibler denote historical want design decision extract study gain template wish signal
direct j j identity nj universal depending fact q conclude proof apply ti n follow appropriately vanish inequality explicit lastly aim obtain right occurrence trivially matrix nu necessary control alarm detection existence belong get q lemma lemma show line hold obvious suffice right compact side continuity sequel tend notice k k vanish probability proof analogy proof conditional occurrence go notation variance inequality employ q guarantee continuity universal u path apply one proof almost skip proof similarity develop conclude threshold proceed note hypothesis n jk jk identity imply dominate lemma derivative assume turn algebra trivial almost surely thus infer sufficiently quick derivation simplicity vanish sequence depend omit analogy straightforward hellinger henceforth tight spectral parametrize moreover cn analogous show generality get assumption taylor expansion chapter zero density upon similar p ns first
actual environment rather movement rare configuration affect represent orientation robot arm trajectory enough make coordinate regardless orientation shape negative align principal axis object orientation change trajectory similarly part shape direction take modality trajectory match bad last modality give cloud language deep modal handle completely modality cloud trajectory solve problem structured convert binary f language output match language trajectory goal learn separate layer learn relation modality cloud modality linear activation eq learn node predict label trajectory crowd datum ground crowd equally sec input
deviation integer generate pick put together successively drop goal n space realization meet portion technical need realization crucially measurable restricted expectation schedule eq towards prove suppose sample space handle observation repeat application hold moment intermediate martingale assume goal final epoch period lemma
kolmogorov grow equal offset memory time intrinsic measure randomness organization aforementioned complexity explicitly depend see algorithmic complexity almost intrinsic intrinsic process learn intrinsic class choice remain problem class always necessity ever something place practically rarely intrinsic develop phenomena physics mention exhibit arbitrarily especially relationship construction follow coin coin choose user distribution bi infinite coin
avoid estimate equation w estimating finally song uci repository n ni pp audio song feature song song conduct regression I h subsample subsample nan accept compute provide result number reduce exclude mm computational frame outlier comparison robust introduce aim find statistical scalable compatible system bootstrap number distinct point bag little
ti n eq infimum level fall thus time therefore jensen entail follow expert tuning parameter get page round fall apply jensen obtain entail grow small regret roughly rigorously closely split incur cumulative inside second regret position part whose multi split proof start supremum norm gradient achieve cf right side expectation center subgaussian increment statistical empirical minimizer show integral appear constructive algorithmic turn online regression class notion instead sequential fortunately notion example leave modify algorithm entropy bound ease
covariate response predictor thing distribute name averaging study satisfy n k mc nk term ep put piece particular decay match enough average converge match convergence centralize occur subgaussian subgaussian subgaussian norm conclusion independent subgaussian subgaussian norm absolute bind imply pe subgaussian subgaussian q state satisfy lasso drawback dense
consider price three computation threshold release kolmogorov pac task privacy impossible universe infinite fact must grow universe previous improve problem allow grow pac differential think universe individual privacy individual significant output randomize differentially differ pure differential provide gain pure differential dependency body query query release problem accurate differentially private error interest query predicate individual extend database average query release count widely privacy query release private mechanism constant remarkably complexity much release family family significantly point iff totally order query release family produce histogram cumulative respectively function dependence thus open private algorithm threshold universe grow size universe resolve sample complexity universe privacy differentially threshold infinite universe present simplification roughly pure privacy pack match standard laplace threshold build construction pack tight pure privacy approximate privacy family estimate pure release threshold closely distribution cdf closeness kolmogorov distance weak closeness function closeness work total g show privacy task threshold equivalent complexity prove kolmogorov release amount approximate approximate without query release distribution q al learning privacy pac sample
opposite direction efficiently gmm inference bottom inference datum geometric log bind pressure target geometric mean rise name mean stay recently jointly majority da mit latent top distribution stochastic contrast explicitly generative gmm discuss theoretical property distribution demonstrate section vector
equivalently interesting matrix entry bayes set notice completion completion phenomena section bethe hessian graph spectral density eigenvalue bethe cavity delta peak linearly recursion turn graph remarkably show asymptotically spectral enyi random numerically result spectrum demonstrate open around begin small ij
dynamical observable representation replace track discrete latent predictive state observable window formulate belief prediction correspond inferring observe noisy due overlap window correlate noise na I employ instrumental instrumental use part correlate instrumental overlap future therefore instrumental detail moment correlated maintain dynamical instrumental pick extend tf estimate h tt possibly train average realization start state give depict modeling reflect stage supplementary material choice framework extend manner filter infinite
moment specific degenerate impose let orthonormal spanning span condition maximum issue specific outcome modify particular modify generalization effective predictor exist reduce combination group dispersion estimate pearson base dispersion approximately chi freedom scale generalize specific singular value decomposition decompose full modify score group specific dispersion unknown pool dispersion otherwise set positive semidefinite replace onto semidefinite use choose set require normal full fitting order total choice discuss report operation cost take operation follow compute require bind conservative scenario notably specific dispersion procedure reduce procedure trivially balance demonstrate regime analyse procedure precise facilitate analysis sequence index dependence statement text fix effect distribute vector moment ir follow hold mn invertible satisfy let th exist specific satisfie moment relation dispersion finite constant vector mutually dispersion parameter identifiable combine column ensure identifiable hold assumption linear
reversible must converge equilibrium discard try parsimonious extra section address section store momentum operation fall store fine grain store information lose multiply less give exactly velocity parameter integer rational divide integer division reverse store buffer integer would single bit store bit add analogue multiply eventually detect store else support division multiply store integer division bring stand integer reverse process get integer buffer divide use
likely quasi without condition help imputation angle da actually find training different actually gaussian filter create novel imputation demonstrate competitive good imputation use raw da benefit dependency preserve prototype correlate raw future recurrent neural stream architecture image generative apply edu deep computer vision image angular summation enable speech typically frequency coefficient coefficient researcher trying build
rational mechanism require analyst claim make space defer mechanism estimator differential affect appendix differential privacy build show equilibrium agent threshold threshold probability least player player threshold marginal large threshold threshold report allow symmetric privacy strategy nash biased technique add preserve allow control source convexity payment show player payment reporting accuracy confidence output error preserve difference induce reporting model short report predict report payment closely report precisely induce sensitive use score payment hold random scoring event payment report extension payment event agent uniquely player report occur payment parametrize rescale transformation strictly proper remain strictly rescale scoring rule criterion scoring report payment rule set hold generate generate report bit analyst analyst report payment player
intractable exchange point exchange high inference exchange suffer limitation note limited describe readily space extension smc empirical examine present mcmc section outperform investigate consider sampler evidence bf simply weight expression mean sampler directly importance likelihood sl look toy consider inexact auxiliary method mcmc q evidence yy yu obtain variable view estimator although unbiased weight large extension algorithm use give auxiliary common terminology estimate marginal weight evidence estimate abc abc estimate bf sufficient outside parameter true bf comparison sl describe reasonable summary abc sl understand property investigate take bf ny n rewrite
transform hereafter factorization forward approximation signal flow direct depict obtain summarize arithmetic demand fast ignore quantization step transform shift order image public bank image transformation subsequently coefficient retain
level graph indeed graph ks configuration neighbourhood vertical horizontal interior lattice strength differ direction considerable interest composite likelihood index lattice composite write special term term composite likelihood singleton contiguous square block exhaustive would collection inference lead approximate surprisingly little composite
grant example conjecture usually refer involve notable metric auc setting reproduce hilbert rkhs refer pairwise contrast exist iterate restrict strongly objective target unconstraine establish guarantee without underlie polynomially decay kernel methodology mainly depend operator inequality compact grow important problem contrast classical involve function express formulate
probably work fine l mostly method partly center separable object work cc fast run varied reason offline bit expensive serious cluster newly come snapshot offline special evolve layer difficulty b outlier produce even outlier skeleton replace eventually fig skeleton two skeleton use point gradually big skeleton present truly shape presence massive stream skeleton continuously dynamically adapt change data space produce experiment nonconvex produce hybrid combine offline investigate framework crucial maximal execution bind provide conclude find bind merge wrong skeleton point
lipschitz training easily satisfy denote risk resp occur probability last introduce induce eq triangle inequality bound definition sample sample risk fx depend hold follow present upper relate discrimination goal learn unknown effect training note outcome due correspondence effect functional functional form present tackle measure hypothesis set effect banach unit ball conv denote hull hypothesis consist functional element eq schmidt inner product apply dimension measurement integer q outline dimension check unbounded pseudo theorem relate since function obtain uniform also rademacher random lemma probabilistic quantity n cn result find upper rademacher series independent rademacher variable hold absolute denote prove gaussian preserve reduce convex optimisation attain realization I prove tight effect pure every di bf paradigm calculate formula rademacher quantum hypothesis complexity rademacher duality formula last follow cover
exhibit strategy backtrack world synthetic strength composite like multinomial gradient specific theoretically match author consider propose algorithm optimally structural emphasis different introduce basic condition derive variable propose describe variant real synthetic adopt subdifferential continuously gradient hessian vector
neuron still task organization goal competition reconstruct structure activity neuron reconstruction criterion cause algorithm require lead receive lot simplicity successful result quick reconstruct network correlation coefficient quantify variable
langevin couple drive share wiener fix derivative remainder establishing lipschitz lipschitz hessian equation x dt py dy dt hz x dt hz dt relation relation coupling use fact next hx hx h dt hz dt hz kt dt hz h weighted introduce shorthand inequality continuously schwarz second hx hx hx dt hz hz hz kt dt kt hz dt x hz hz fix lemma difference demonstrate existence derivative directional u hx integer eq hx hx hx hz cauchy derivative directional derivative fix bind hx hz hz hz directional continuous lipschitz relation u begin establish eqn u u hx hx v hx v hx hx hz x hz x op
find surrogate multiclass hinge note surrogate multiclass interestingly construct surrogate yield loss algorithm due get class reject also way give surrogate design surrogate surrogate yield consistent restrictive fundamentally difficult evaluate class great partition classifier figure excess relate excess multiclass excess frame surrogate co solve generalization vs
slide window slide convolutional differ network rnn path phrase external natural instead multiple pooling every adjacent window type would rich convolutional supervise tune model architecture composition task limitation largely take synthesis sentence simple convolution layer max sentence feature soft template detect local sentence structure architecture layer pool window discussion section propose convolutional matching sentence different nonlinear similarity enjoy flexibility
option filter marginally datum well good mnist small cifar done indicate scale comprise pixel corner dimension rotation pixel like increase log axis rate good filter mnist repeatedly well show increase pc core gb ram mnist time generate second large filter use generation benefit multiplication large batch simultaneously image individually carry generate employ matlab
predict keep environmental activate gene overall task stress indicator stress heat response element heat cell activate dna sr also predict p activation pathway cancer pathway activate stress pathway high probability general sr task several c subset allow task measure pathway er er correlated pathway stress response stress split know use rank participant public private label initially
question book separate hundred page yet powerful input come long dependency fail information forward backward formally vanish hold dependency use several overcome long lstm address problem enable irrelevant property dependency
microarray follow scenario predict pp vs able separate additional subject turn almost among compare signature gender goal cancer separate construction surprisingly separate four correlation suggest systematic difference cancer suppose sampling bias often limited availability phenotype phenotype profile http edu provide comprehensive description database identification molecular share disease phenotype phenotype difficult throughput lack comprehensive phenotype phenotype describe approach phenotype method numerous outline phenotype profile help disease treatment categorical phenotype reality phenotype constitute spectrum datum study direct different thus available microarray platform directly microarray aspect variety phenotype latter benefit derive correlate dataset principle become focus phenotype depend example construct take discriminant discriminant thousand gene per fan fan high similar often multivariate phenotype characterize reduction phenotype microarray derive gene sample aim pp profile magnitude gene phenotype association term form deviation expand minimization lagrangian lagrangian optimum problem pp profile signature pearson correlation derive provide association expression profile profile high consistency gene association derive microarray gene generate less gene yield two disjoint baseline discard gene common dataset annotate short description paragraph annotate word phrase annotate systematically phenotype microarray unified system map description sample description description program disease concept mesh part cell concept hierarchy concept order concept phenotype test dataset title member description inverse document tf tf calculate dot essentially identify good dataset testing dataset take possibility could reverse group include concept title description human microarray concept annotation share concept annotation sample concept tf group get tf present formula notice concept ideally annotation group reflect phenotype mask concept annotation tool identify discriminate phenotype monitoring level thousand experiment
keyword correctly detect accuracy recognition accuracy keyword overall memory lstm keyword suitable scope importantly network scenario identity keyword identity oracle active recognition accuracy similarly keyword keyword informative attribute mixture model keyword framework detect
uci measure specificity rna clean popular classify comparative ratio implement miss imputation level accumulation result well general miss value moreover compare nb lr letter rna ccccc letter rna life patient order low motivation original study much medical clustering base alone precision use accuracy classifier per class lr correct
drastically fast simulation seem around I shrink sample varied value step average distribution however analytical finite effect small transition I behaviour show finding numerically accurate configuration show fig symmetry target probability configuration check symmetry recover histogram histogram two configuration characteristic shape gaussian behaviour resemble shape two possible upon probability learn former latter sample histogram configuration examine target accurately htbp histogram dash line fluctuation dash vertical fig regard center excellent agreement reach sample histogram show constraint fluctuation location peak height histogram fig spin correlation subset htbp line fluctuation variable configuration reproduce admissible interest addition target admissible tolerance bias entropy measure grow exponentially act back unbiased concentrate understand effective principle distribution compute analytically property generally configuration compute typical
change care lp original contain fix limitation additional variable totally training dimension sample cause serious overcome reduce reduce usually amount summarize iterative basic histogram learn coding sequentially basic novel histogram solve obtain code histogram initialize histogram update fix update basic histogram histogram
digits black image total contain layer fc length variance cifar cifar benchmark consist cifar cifar convolutional filter fc epoch mnist house use digits house number cifar mean perform normalization employ cifar fc reason rate accord sum bp invariant good cifar cifar statistically significant sum standard invariant well final full dataset error version without ibp regularizers times dropout improvement ibp dropout see connected lead additional layer employ
major learn know simple example rotation incorporate metric manner experiment mahalanobis transformation learn per consider lead learn comprehensive non local margin mahalanobis find mahalanobi global weakly single local invariant least sift adding
terminate need na I update coefficient sake completeness brief description see kernel span correspond diagonal maximal region contain inner representation column normalize unit normalize generality diag si accelerate omp efficiently representation batch pursuit omp recovery norm constrain either total constrain sparsity introduce seed outli contain norm sample alg seed column column step storage seed omp thus compute roughly total contrast seed complexity neighbor omp develop sufficient exact recovery span give back exactly prove thm begin select lem gram provide rank exact linearly independent describe linearly gram linearly provide entry invertible linearly select form inner newly previously index invertible provide complement complement
although effectively attack observe failure certain area nr attack attack size attack especially detect nr attack area adaboost impose enhance negligible alarm comparative experimental conventional supervise svm knn attack surface attack attack alarm low attack attack performance optimistic attack publish attention effectively attack mainly material indicate attack nr attack attack serious filtering recommender system small conventional improved extract profile base attack make classified gradually emphasis predictive difficult attack effective feature profile feature base description discriminate profile diverse addition neighbor profile concern attack profile size axiom claim criterion definition exercise theorem theorem
university school pa real binary network efficiently dedicate hardware task backpropagation capability use multiclass task performance unit examine filter study besides investigate different network explore backpropagation introduce implement introduce notation letter matrix non capital letter denote indicator
q dy measure induce define minimum definition base let building concave follow unbounded walk log concave level ball induce constant although author effective diameter distribution would close truncation implicitly technique require handle set concave random walk induce contraction mix concave associate approximately concave dc theorem anneal warm carefully pick closeness describe mix denote scheme proportional distribution run step maintain precision sampling main paper exhibit
predefine amount pseudo worker narrow observation surprising converge stochastic take scope partition r w c w c predefine converge legend legend pos south east font legend align leave benchmark analogy perform especially well corpora embed preserved language natural community introduce neural call state word task retrieve word analogy answer word vocabulary answer query concrete performance lead try empirically
color stand ep evaluate experiment introduce bit two simple help improve recovery evaluate highlight loss suitably minimization assume recover solve algorithm though bit sign flip snr measurement value repeat experiment matlab core ghz gb author select display performance improve confirm fix flip performance noise error advance yet still next performance estimation fig accurate estimation bad sparsity true approximately reduce estimation svm plan passive error efficient ep passive
amount reach true rapidly drawback weight happen free weight eliminate effectively demonstrate behaviour generate average apart search infer analyze message give demonstrate base mixture scoring observe mixture length demonstrate mixture search unable infer infer part term mode increasingly separation hence incorrect kl increase plot shift correspond infer kl mixture search widely mixture mml mml formulation advantage neutral closeness mixture distribution world merge perturbation determine convergence require component initial number routine examine respect mixture htb great value require number bivariate average great iteration propose well discuss cost result section require close however infer well fig result stop accommodate component membership correct significant overhead regard example univariate data search compare infer search message mixture mml approximated mml htb mml mixture gain bit see base mml mixture mixture evaluate mml scoring bit scoring component index index use mml mml scoring score mml mml popular datum specie comprise representative component membership length mml mml scoring formulation evaluate complete mml compression make twice compete mml scoring h data c species specie mml scoring infer mml mml mml current section test mml concentration mml newton traditionally however hence result mml estimate follow demonstrate aid inference mixture experiment search study dimensionality concentration sample previously mention approximation respective report calculate absolute average simulation percentage song mml song mml e e e e e e e e e e e e hold across accurate shown reflect mean drop error average drop drop decrease appear clear change mml base
factorize rank one tucker order tensor tuple rank value define equivalent spectral hard tensor seek minimize work denote continuously differentiable nonconvex cp encourage learn tensor low rank specify infer low partial tensor tensor sum let k positive integer cp learn index example rating contain aspect jointly yield tensor restrict multilinear employ task finite train ii w w share w completion quite present section introduce matching solve w k w k lipschitz guess divide update tensor important update compute case square mp l mp economic orthogonal mp relaxed
tune generality range chain tool pose serious methodology infeasible recent dataset draw asymptotically reduce amount expense introduce bias preserve asymptotically correct invariant expense requirement might applicability construct monte consensus theoretical guarantee contribution approximate naturally increase increase particular big arrive useful methodology unbiased ii framework underlying free tune example show posterior expectation fast method addition state world aim inference perspective core consider unbiased functional value mcmc focus rather expectation example regression variance employ solely propose mcmc systematic subsequently careful estimator sake assume sense time estimate address situation close prohibitive amount build differential
surely find clique long perhaps various entry w unweighted hope clique detect approach another subgraph hence plant subgraph similarity adjacent pair extract eigenvector produce weighted matrix random laplacian large sort consider sort top corresponding final sort decrease tie break vertex figure ground perfectly recover ground value ensemble across term distance accurately would run expect yield especially numerical meaning uniformly n truth synchronization preserve rank phenomenon recover plant totally average method ensemble average solve propose al matrix serial order I spectral lie chain adapt polynomial provable robustness guarantee serial recover underlie fraction comparison corrupt completely dense high pairwise serial rank similarity ordinal ordinal similarity pairwise follow sign count match third reference contribute summation similarity rank compact form similarity summarize main step graph laplacian vector eigenvector correspond small induce ordering choose minimize linear set pair comparison independent preferred item rank glm model et al propose play match compare comparison ranking player global ranking provide serial glm glm glm consistently noise refer figure glm newly additional english set table b cm cm team l glm sup sdp hull city united w cm cm team l glm sup united west west nr score c cm team l glm sup sdp city united west west score w synchronization group element anchor terminology sensor give node element compose possibly noisy group synchronization shall refer cast eigenvector information synchronization constrain relie sdp briefly summarize approach refer reader pass synchronization motivate anchor eigenvector incorporate combine contribution sensor sensor contribution sensor sensor synchronization write denote sensor anchor anchor correct sensor interested quadratic tu np quadratic program
pt correlate individual influence count convenient feature cite effective modify citation count increment citation cite body cite reference cite citation count count consider author seem unlikely reference field well first title title conclusion pt similarity feature sim sim sim sim sim show feature right feature feature g count predict citation title cite pt sim sim sim sim feature similarity citation abstract context follow context conclusion title context citation citation pt indicate name citation context word citation pt type aspect citation context sentiment differential category word citation kind sentiment citation citation influential even sentiment neither correlation gold split negative split eight figure among eight high correlation great predict citation pt position reference might influential position location body base pt feature count correspond position variance benefit pt show citation previous cite paper cite however correlation influence influence cite influence cite author take self citation gold influence final influential range pearson coefficient cite seven old paper recent paper year old year poorly consistent define influence paper supervise chose function
q bernstein inequality tail probability equality hessian follow j l prove conditional alternative user eq q equality hold much violate I ex ex argument upper ex ex ex ex ex ex ex ex ex ex ex exist union follow provide h ex ex k hold choice inequality recall e prove desire concentration utilize alternative outcome utility wise distribution draw unobserved item standard cumulative cdf independent follow random appear j case inequality upper contraction randomness alternative outcome three partition summation apply argument generalization match involve three triple round round triple scheduling match lemma round random ready first inequality supremum supremum contraction ci ie ex ex
relation class prediction hypergraph construct attribute remove class actually contribution occurrence attribute incremental attribute space hypergraph employ attribute attribute structure group provide preliminary visual attribute information approach additional exploitation part consider task multi hypergraph cut enable hypergraph shot recognition category use attribute linguistic visual abstraction core attribute prediction integrate attribute readily attribute shot relation problem hypergraph hypergraph classification specifically hypergraph hypergraph reformulate predictor chen hypergraph capture introduce hypergraph perform embed new hypergraph utilize hypergraph derive label hypergraph embed start
integer program prove gap specifically lp problem hierarchy strong separation assignment cost exact state combine important exact invoke cost incorporate cost per since theorem rademacher thus far extend side polynomially concrete example happen encode term side constraint term side consider problem imagine flexibility individual remark labeling among problem function constraint cut ise furthermore soon well say gap metric labeling section show algorithm develop term binary expect regret predictor bind lower time rademacher summary gap polynomially indicate process predictor david discussion acknowledge nsf grant dms decentralize
strongly respect define bregman mirror take appropriate fast nesterov accelerate moreover nesterov entire procedure sgd reduce stochastic gradient set stage perform sgd randomly unbiased
optimize significant compare broadly applicable analog computer significant role generation signal hardware explain convert update use compare simulated physical encoded demonstrate special setup see special explain feedback physical setup source optical act delay circuit optical intensity filter drive measure signal differential factor filter circuit loop change power delay system input offset control bias identify find stable always fall absence start delay couple infinite depend cover delay property motivation couple paradigm suppose input denote total
difference bit experiment full keep bit scheme completely discard simulation reliably mse bias right column empirical together binomial curve bit scheme full bit unbiased scheme avoid small typical start nevertheless curve essentially overlap somewhat figure two order small bias th practical serve bit plot even
vector obtained decompose second minimizer k moreover term hand suppose divide hand fold follow inequality hold inequality hand side derivation lead side follow sufficiently choose plugging parameter rao product easy matrix full full verify rao k recursively expand fold calculate tensor third line affect norm spectral k k definition recover subtle
sign value min max kernels max original report wide projection regularize bold accuracy well projection linear min basic rand protein spam figure detailed sign dataset regularize experiment experiment time result projection g projection
defer wise define f j f j cycle unless must stop answer show kkt condition conclusion objective change cycle algorithm correct kkt support competitive sparse elastic penalize good prediction elastic penalize dominate situation consist validation test class validation observation select path prediction
subsection employ power low approximate dct mathematical matrix indeed identity well present fast orthogonal work refer method transform follow minimize analysis hardware right transform cyclic notation notation case component index accord unchanged represent round dct dct obtain possess arithmetic fast propose describe permutation approximation replace zero dct approximation tailor rf accordance transformation consist follow factorization permutation aim derive dct candidate matrix possess define operation require computation candidate operation complexity constraint nan matrix orthogonality constraint preserve dct like point dct intractable exhaustive eight candidate find
bernstein prefer basic differently correctly martingale result non anti optimality martingale suffice bayes additive uniformity also pac traditionally theoretic introduce classical measure technique examine idea
bring sense parsimonious fitting dependency column wise dependency row dependency subgaussian definition pressure effect patient top measurement entry independent assume development continue eigenvalue condition independent isotropic subgaussian definition condition low condition tolerance lower easy relationship condition suppose low hold k hold low
patient balance prototype represent cluster prototype sentence evidence indicate patient ex word sentence observe chain checking ability see indicate patient stable insight automate medical patient state patient patient infer patient classify patient suggest clinical patient step technical cancer treatment work evolve datum probabilistic able handle summarize fold cluster problematic embed operate pairwise evolve ii enable give iii cluster advance validate compare hierarchical use brain cancer patient patient cancer patient inference optimize exist patient thank david helpful discussion suggestion partly science foundation acknowledge national cancer patient cover cluster point utilizes give adjacent evolution object pairwise similarity likelihood kernel structure particularly cover
let q
aic prevent overfitte ic bic criterion form aic respective derivation formulae fm previously deal whose poisson observe fm effect equation recall come
parse decision rnn different parse dependency widely syntactic reflect word two child binary network ease unit word dependency terminal node pos tag interaction head word value word embedding stack retrieve embedding update back word head relative map randomly initialize embed neural experimental traditional encode subtree fed convolution model interaction link terminal two representation phrase subtree
change follow substitute order statement lemma rewrite take operator compact self adjoint eigenfunction omit since compact self adjoint mention countable
visual function office span degree decoder learn engine encoder etc keep decoder generate several vary light dc three network reasonably predict static seen separately profile pose light linear demonstrate complex encoder straight training make novel view network representation versus representation baseline network representation identical dc train procedure network input decoder angle simply
slow temporal spatial mainly appropriate operator successful supervise architecture inspire stage encoder comprise
hence definition kt I I consider follow positivity et estimate merge lemma stochastic valid corollary become arbitrarily n ap large conclusion let say integer index arrive oracle pr note inequality valid j nz c hold jensen inequality norm norm assumption invoke arbitrarily eq provide j lk lk op imply precede arbitrarily manner invoke nm precede arbitrarily lastly q thus q inequality follow definition n invoke probability become inequality th inequality imply definite due fourth get sum equality dominate equality precede inequality q away z j j hand side due next first definition equality due c j q equality theorem imply imply b imply assumption assumption choose latter simplify equal na n ta h ta nh ta p end note q well uniformly away show
index important model refer exist generate shannon information measurement precision amount character code specify parameter complexity precision unit information change result change offset mathematical unit measure information central importance understand write expectation shannon cross since mle denote subscript call parameterization infinite information identify parameterized parameter three interpretation ii minimize loss approximate ii divergence select model clear interpretation identical approach parameterize mle code call predictive predictive
ki pz z pa update appendix determine controller check reward node assign great visit sum hence decentralized policy obtain calculate see var n n time summarize episode magnitude magnitude refer episode dense nonzero reward terminal step algorithm linearly episode agent number scalable separate appropriate behavior expert guide agent random long want learn process keep proper suboptimal policy learn quickly execution efficient confidence bind heuristic controller strategy greedy might inefficient
fw mmd fw method radius lie within fast fw practice theory detail frank quadrature mixture gaussian optimize difference non optimization approximately exhaustive search random density state px px py initial inference compute filtering px filtering summation get particle pf provide marginal smoothing bootstrap predictive randomly step come particle sum un sampling accord propagate bootstrap particle filter obtain maintain uniform predictive unbiased obtain mechanism resample practice replace low distribution normalization particle quasi propose monte contribution frank wolfe set dynamical model practice assume emphasis mixture history still fairly fw compute quantity subtle I define depend though history past
start observe wise boundedness property conclude compact prove aforementioned let wise follow xx yy cx xx n associated become h cx xx cx tu tv cx tv nx start arbitrary tu c xu n uv tu tv tu c tu tv nh xy yy h nz cx n nz cx nc c cx n h u yu c prove divide approximately interval length
function lagrangian lagrangian expression modify gd range dictionary try simulation combination combination range resolution classifier set train combination training set good create bit finally traditional dataset safe whole high good classification conservative proportion chance use six paper could dataset
studying explore subspace future estimation fitting error purpose helpful model discriminate separation fact advance base analyze way good risk define combine acknowledgement thank motivate author problem valuable discussion comment pt department mathematics nc usa edu classifier find
fuzzy fuzzy fuzzy membership hyperplane fuzzy fuzzy two hyperplane carry svm obtain accuracy future work concentrate support extremely fast svm algorithm hyperplane solve fast unable cope problem first impossible assign single importance world
stop hypothesis admissible suppose stop accept termination incur penalty incur period belief true prior hypothesis true bayes belief total horizon minimum clearly bayes rule ny iy dynamic variable dimensional belief space equation interpret minimum cost immediately period collect period decision suffer grow take optimality choose accept soon implement compare belief illustrate panel procedure two change independent policy statistic py bb p py
one considerable suited acknowledgment author thank package report constrain dag excellent ss advantage cb quick unstable sense early cause ss knowledge probability capable ss deal slow converge prevent optimal bn currently computational therefore intractable large burden mind restrict cb construct node whole bn balance several hybrid min hill conduct show specifically outperform efficiency reconstruction dependency greedy greedy hill search variety rather optimum function bn capable thousand enhance small hybrid
processor scalable huge optimize computation cost algebra form matrix storage require pass order huge run expensive avoid filter average map estimate pass superiority compare alternate order alternate filter blind deconvolution many produce signal activation technique program hierarchical bayesian map incorporate reduce completely eliminate deconvolution blind separation ica incorporate shift constraint
start loop position course exercise preserve time construct velocity position start pair leaf backward iteratively leave either backward stop grow turn randomly leaf preserve move position refer reader implement seem effort cover implementation efficient programming automatically numerical sequential iteratively use importance resample step allow produce approximate progress sampling multiply n get indicate current get time particle particle get time diversity apply leave interpolation convenient approximation ep second smc temperature describe base efficiency section equal pre default value involve stop numerically normalise supplement resample update leave smc amenable walk metropolis calibrate automatically end obtain box dataset smc move sampler importance previous standard representative numerical study big quantity posterior coefficient latter define lie since marginal criterion probit logit logit boost library standard computer except explicitly version uci except page book super set dataset I predictor plus dataset
dd kb kb kb persistence landscape correspond birth death birth death critical loop graph graph list sort list pass exactly construct last iteration outer loop list outer length decrease one increase length terminal list terminate outer loop take variation want add birth pair persistence landscape list list take finally copy I landscape size initialize size list decrease first may look find segment section geometry persistence landscape also bad persistence landscape optimality envelope line segment part visible equivalently segment envelope importance efficient available practically persistence segment start envelope family however
call month st record value call hour precision date call make number day first day datum span day passed consider week call week day hour trace week period million value categorical time precision date identify inside consider hour consider set spatio dimension represent variable week day hour grid several categorical main variable multidimensional whose define partition variable work slightly incorrect choose bayesian posteriori minimize implement robustness define follow probable maximum give limitation appendix hereafter tool exploit large keep mind free need cluster interval
represent path demonstrate autoregressive definition equal construct assignment imagine unit sample autoregressive neural advantageous input context reconstruction mask one direct framework describe previous generalize architecture indeed deep autoregressive pass fully layer layer autoencoder autoencoder input binary order necessary principle maximum autoregressive property probabilistic example light correspond dark depend hidden use unit hide index connect
equal universal hence knowledge universal markov feedforward network preferable either undirected network kernel especially would architecture detail adapt cover architecture feedforward universal kernel unit verification leave relation analyze stochastic feedforward topic attract mathematics sciences upper distribution output output
correspond prior odd correspond simple I dimensional parameter model enough believe generating case specification probability interpret criterion likely model logarithmic scoring prior mis hide ideally integral general evaluate measurement variance matrix linearity various instance filter distribution gaussian filter product likelihood allow standard name identification linearity markov chain thus integral approximated technique filtering tractable argue scientific practice generative linearization kalman filter filter ensemble particularly linearization systematic bias let look pz variation successive index growth hidden stochastic growth day species eq transition jointly differential equation q time equation could piecewise continuous process accord however discrete next ordinary approximated precision pz lie randomness integer measure easy measure water simplicity unknown free generating datum distribution inference
prove pseudo satisfie must exist integer randomness kt schwarz averaging argument lemma follow section randomness unfold suggest vector break proof claim step probability easy sub concentration know pa clearly valid schwarz pseudo pseudo expectation side use induction inner tc tc tc c tc multilinear define c c I claim noisy handle
coefficient misclassification former final size window report result use rate tune code base expression consistency patch coordinate patch network notice manually use use centroid testing never segmentation return manual contrary part illustrate patch manual misclassifie voxel tend lie deep network automatically challenge competitive contrary base rely verify method query region volume propose region ensure current memory gpu scale system intensity
corpus test unlabele pick reference roll learn roll mixture include notable difference draw learner policy suffice theoretical support online mixture sensitive work otherwise learner table result multiclass pos dependency parse qualitatively agree reference roll reference reference bad idea mixture perform well let policy consequently role application eq immediate policy roll round complete action take recall simplify expectation combine dividing complete round regret exploitation exploration exploration vector exploration round let policy exploration round exploitation invoke round exploitation round far however yield valid chernoff round complete
close rnn architecture ratio sequential recurrent gpu get stream intra parallelism small cover lstm forget graph among feed forward input length recurrent rnn employ feedback suitable whose dimension fix automatic rnn language rnn contain feed loop affect rnn train history considerable especially short lstm solve long lstm rnn
spam macro dataset variance contingency sp stop fold display contain macro variance estimate success method generally augment potential exceed stop reduce get user sp expect sp publish stop test demonstrate operation remainder describe agreement subsequently threshold bound table count count numerator numerator learn agreement classification stop classification truth classification
tweet predict tweet sort user rank model tweet approach rank tweet extensively exploit retrieval recommender tweet contrary try consider tweet rank maximize accord improve challenge report aggregation news social several address several provide important role probabilistic model predict previous popularity video observe popularity period al
apply use original covariance evaluate performance deviation operator clearly performance bad three among rd figure sample dataset correlation support estimate matrix depth estimator robust outli example application failure heterogeneity influential wrong hypothesis recommend apply study heterogeneous disease problem expression outlier rate still preserve number outlier minimax contamination condition reduce long outlier consistent contamination notion quantify give discuss huber consist outlier robust influence outlier proportion point point measure totally appear usually position apply modification contamination qx n n contamination ratio counterpart allow bound influence contamination low contamination rigorously loss discuss implication thus minimax optimal estimator word automatically huber contamination study robustness develop estimation
genome sequence throughput individual serve genomic diagnostic aspect fraction feature likely highly belong family able retrieve extremely feature avoid overfitte propose whole predicting phenotype rely greedy learn dna microarray boolean highlight specific dna covering
handle coherent datum well use gene east consist chinese letter describe gene individual convert raw matrix snp fix gene otherwise multiple window reconstruction w reader step window length different preprocesse selection perform snp average select range range iterative leverage margin truncate close low one score sampling replacement snps capture merely norm simply verify replacement replacement discuss image compression observe subset column entire depict pixel column scale mean selection iterative leverage small refer reader select middle relatively white bar value sample column show column approximation though relative much leverage result compare baseline truncate svd percentage column cm accord say sampling entry request mainly passive uniform priori passive sampling know coherent bind subset passive passive achieve unless observe incoherent column
wikipedia directly comparable rnn test sequence difficulty rnn regardless unit gap stack rnn rnns subtract b color large concentrate right sequence easily stack rnn especially grow increase stack rnn feedback experiment challenge character language consistent feedback helpful train also stack rnn amount capacity able previously character rnns scalable rnns outperform stack previous record notice use sophisticated thorough investigation feedback connection role activation acknowledgment author thank
exist train mathematical repeatedly future continue recommend exercise highly illustrative uniform question root quadratic begin straightforward note root rgb surprisingly get slightly appear analytic define py fy dy dy world student able tackle solution far otherwise book illustrative incorrect answer adopt answer implication motivate exercise I get sort wrong approximation particularly involve answer answer develop analytical assessment student program make argument approach political
intermediate f cm discover noisy part firstly define regard exact sample perform type accounting similar initial experiment assess impact importance track instance draw image rest experiment carry infer model primary measure quantify structure successfully infer infer level group account assess instance last example measure lot control group recover quantify problematic equivalence possible structure construct explore capability structure complexity lead dense copy high properly information usually look see quantitative cm c exact c shape visit higher
pick template dynamically determine per prediction token previous tag template calculate dot template order prediction determine compute reach cascade increasingly cascade perform detection dependency maintain tractable cascade increasingly high recall stagewise classifier template change improve structure dynamically well suited feature task speech parse overhead method already fast part speech achieve run five time parse baseline name entity recognition speed score feature template frequently nlp solve use dot meet scoring acceleration dependency parse reduce pos sentence part speech
result visual convolutional cnns accuracy traffic sign face digit year due advance direction building overfitte become capable training increase activation sophisticated design well generalization augmentation scale advance neuron relu success conventional sigmoid rarely focus aspect drive new relu learn accuracy difficulty deep explicitly nonlinearity relu theoretically sound method help deep directly explore powerful imagenet multi error improvement winner good knowledge first recognition challenge derive initialization lastly architecture sec activation activation activation function task definition activation
segment mark low model state genome background make estimate significantly type k ac type state mark cell distinguish suggest branch parameter yield biological probable decoding test define compare spectral algorithm hmms without spectral assess spectral six nine type harmonic recall find accurate six cell low specificity spectral assess spectral spectral predict hmm type except hamming tree gm experimentally hmm hmms hmms em bioinformatic analyze currently thank spectral berkeley estimate observation theorem input sample estimate compute r u u node sometimes denote index index z j j px r x un px
inversion one cache replace show one plain lasso sub initialization interesting regularization dl negativity maintain simplicity constraint proximity alg negativity non negativity illustrative mnist handwritten digits collection gray digits digit image form negativity setting show whole approximately exact expensive unconstrained tensor constrain desire writing easily naturally incorporate handle specialized constraint need negativity sparsity induce simplex constraint need latent establish processing differ classical ignore case without constraint latent formulate constraint impose tensor formulation unified handle would algorithmic incorporate alternate method framework provide constraint efficiently iteration sub multiplicative factorization
pt bounding argument realization variable express isotropic scenario sub denote probability respect substitute tr tr proof pt condition glm glm log r maximum negative assume regularization recovery analogously rsc glm apply twice rsc need consider always rsc rely non suitably consider compact assumption set derivative assumption characterization rsc isotropic matrix assume x norm e suitably design
item probability attract click probability user examine user list observe regret accordingly click click item item objective maximized weight order also reason click satisfactory simple click multiple item satisfactory could click recommend experiment item experiment setting click result experiment learn explanation theory ucb final compare bandit base likely statistical efficiency reason bandit encourage
use example b marginal dropout bad regularization well mean belief stability hold model also make extremely challenging since get might optimum local randomly drop several setting general heuristic understand especially seek fairly general layer neural minima stationary constant dropout objective multiplicative recently show rigorously descent polynomial perturbation descent apply setting see additionally easy
irrelevant mining disease could disease total utilize enable make labeling challenge big become seem probably automate mining big application advance nlp comprehensive summary address relate image unclear extend success vision medical imaging define huge medical deep extent scale image semantic diagnostic medical topic associate document neural language learn assign disease addition match disease demonstrate promising scale communication database large ever representative huge diagnostic semantic decade explore way scale hope encourage clinical establish resource research le ari le ari despite vision database deep extract semantic interaction national research picture communication processing image description automate manner sentence scan topic level key word frequent disease present scan scale record modern deep topic processing imaging imagenet recognition challenge
dependency hmm hmms condition state hmms well introduce train mit motion demonstrate generate point neural feedforward rnns idea enable
margin fact apply transform bring thorough training usually run roughly epoch stack major difference find batch compute despite gpu machine task take minute minute sec propose normalize denote map follow evaluation hold validation cf sec paper compute map explicitly use subset unnecessary atomic composite table trend among big atomic present sec lower motion right close establish large cf sec generalize task atomic composite u r atomic support principle temporal independent unlike understand neighbor non pairwise parameter distance domain shape motion examine question movement eight birth keep dark environment except hour day one could passive along movement force move
small area fit figure whole rmse tune three small tune produce well tune result solution rmse h
dictionary endowed omit dictionary consequence arbitrary rate logistic eq encourage reader rademacher exponential fast note truncation operator impose adaboost conduct series toy real promising boost I regression boost cart build week learner task toy split week learner task vary two univariate multivariate validation boost choose set element localize shrinkage spaced performance root squared rmse htb piecewise continuously document standard error report observation draw capability variant essentially secondly simulation capability preferable choice toy
end proof positive dominate never contradiction evaluate middle lemma k meanwhile ij write term dominate organization system minus pc pc cm minus abstract study sparse response add estimation outlier impose coefficient outlier introduce algorithm go phrase algorithmic statistical minus large fundamental overcome
develop idea descent update contribution deterministic solver section develop regression non use rate show return relative objective section regression perform obtain compete solver regression complexity extend idea numerous error size time depend implement state art solver ram environment moderate sized input regression construct preserve slow distortion embed leverage small respectively problem via random sparsity plus complexity algorithm implement scalable extend constraint regularize accelerate convergence sgd favorable denote column column full usual condition element subsection definition pp notion well condition crucial condition give define minimum basis degree polynomial notion score important dataset score score
leibler measure candidate use another accept information however specific instead datum expression refer kullback divergence generalization proxy true generate utility datum divide fold utility subset conditioning utility variance prefer leave one loo cv would fit analytical approximation loo offer appeal estimate fully widely applicable calculate expectation loo generalization loo solid justification sense still method criterion theoretically since fit parameter utility principle practical especially criterion unbiased selection open view assess predictive property complete lee million believe prior desirable empirically properly shrinkage somewhat reference approach posterior candidate
need q newton algorithm appendix hold fw mention exact newton fw newton error inequality conclusion cut half linear small slow quadratic exact method see would require effort round inter denote integer equal strictly imply one within use side whenever k k k fw eq stop choice practice another stop criterion self conclude serve regularize self standard minimize scaling compute eq strong choosing choose stepsize adjustment intuitive newton type local smoothness become stepsize target erm factor example regularization large relevant counter grow gap inexact newton distribute involve question inexact newton answer question communication complexity accordance specifically eq speak sample make comment local become objective effectively sag method pick sag predictor stochastic ascent recent accelerate stochastic theory compute aggregate set u tr tw need approximately hessian give conjugate define
spurious address correct nn view differ factor theory remain valid replace coefficient sparsity typically mcp scad mcp strong focus scad pp scad tn tn tn residual spurious except replace spurious center gaussian realization sub e pd j k hold triplet maximum spurious correlation variate center vector concave problem local initial problem initialize via step scad mcp penaltie lasso spurious replace let prediction minimal triplet variate question technique well chance statistical spurious fit predictor discovery spurious multipli bootstrap quantile critical
final layer learn final second hide branch layer bias final target use cost use update final hide meaningful layer govern help provide layer overall target branch arrange
setting fit training mnist various architecture mc dropout empirically evaluate need improvement finish state exist configuration file online experimental analyse dropout weight convnet convnet considerable mnist convnet dropout throughout traditional fully layer alone ip originally every dropout model cifar set convolution instead last every convnet dropout connect alone ip dot red mc show dropout perform blue evaluate run
element value average perturb I respectively dataset document describe topic word remove divide frobenius norm table c dimension handwritten six nine treat pixel first column transpose row stock market price stock collect temporal stock price order compare review equality hold skew ratio zeros smooth analog assume loss transpose without digit l digit stock five observe variance perform noiseless element hybrid capture structure mixing maintain right regularization accuracy summarize various review metric
extract knowledge negativity constraint meaningful physical insight negativity popular non negative nmf nmf nmf aim nonnegative nmf solve impose efficiently suffer ambiguity solution local optimum initialization order pi order hence formulation partial
primal equivalent maximize lagrange every q rewrite slightly eq become lp program since lp complementary thm examine rewrite I first rewrite I I q meanwhile example inspection describe body example note lagrangian theorem nc assertion lp duality lp duality assertion cd assertion prop instead prop thm assumption university
order constant bind specific adaptive give close equation acceptable error show view extensive valid call classifier act might mathematic computer science maximize core bottleneck
detail two convolution task region cnn unlabele supervise learn convolution serve view final label intend next regard convolution layer adjacent region illustrate convolution give convolution indicate layer use adjacent learned convolution correspondence help assign positive negative region g sensible make predict train predict e say half embed nothing integrate convolution multi embed train option replacement replace convolution layer neural option add convolution replace result view layer minimize option update assumption make add option empirically case obtain plausible
prove need decompose along assertion order clique crucially space admit orthogonal proposition entry index intersect conjecture outline deal portion term number accordance decomposition ignore prove view devote random norm conclusion care typical significantly correspond product turn consideration impose condition turn related clique hierarchy detect hidden clique establish satisfy sum let yield consider similar condition index row equivalently demonstrate threshold semidefinite estimate throughout one complement indicator write understand argument belong context proposition control matrix theorem check suitable two lemma simplify expression
win team win marginal marginal deviation game possibly interestingly strategy impose loss tie benchmark produce marginal systematic odd employ twitter attract event ever note systematically collect store sample entire stream focus occur th isolated tweet keyword recommend team name team combine team identify yield corpus tweet game occur competition isolate produce hour begin game analyze result five representative fig note keyword game character save team game decide characterized involve match team name manual validation dataset tweet three manually tweet tweet collect game precision consistently game contain
spectral gap chain limit trajectory often underlie secondly tool validity significantly fail converge intend target indicate care validity inferential heuristic adaptive consider markovian function average walk proposal increment sensible rescale converge hence diffusion infinity surprisingly correspondence acceptance example increment verify iid target medium even combined simplicity easy scale theoretical mcmc application motivation scaling extend address discrete target mala acceptance establish confirm setting stepsize hybrid spirit conclude order metropolis algorithm need explore mala hmc study reach scaling try mcmc delay mcmc optimal numerous result give rise mcmc successful one consider increment proposal covariance sample apply dimension dependent scale recursive version application motivated contribution rejection adaptive I shape distribution rather hence suitable heavy tailed target analogous mala hamiltonian carlo adaptive scaling result substantial area adaptive reversible limit implementation fully variable hoc address variation trivially effort strong algorithm sampler elegant theoretically markovian appeal practitioner complex dimension metropolis validate control dependency ergodicity apply adaptive refine martingale theorem suitable version stochastic approximation contribute develop interact address adaptation algorithm interact simplified coupling establish successfully weak adaptive sampler adaptive asymptotically distribution two distribution every ergodicity satisfy adaptive start probability adaptation subject discussion hand restrictive ergodicity weak c
separate training evaluation performance rmse log concept square prior case eigenfunction variance increase regime benchmark synthetic equation mat ern result gp particle gp fair realization implementation intel ghz involve performance regard load turn outperform rmse test comment report rr model wiener rd order
appropriate classification teacher student probabilistic classifier expect loss strategy student unable adapt lack adaptation present concept directly student feedback strategy relate sampling computer strategy image whose ground disadvantage propose image current local optimal sampling method find active unlike choose great update estimate distribution label correctly advantageous boundary context learn exploration versus exploitation relate concept approximate student propagate student unobserved benefit directly similarity flexibility allow use extract image
flexible bias change latent outperform counterpart optimize objective less initialization detector weakly optimization behave iteratively sequence optimize bind two successful drawback practice sensitivity initialization function inspire process mm allow large valid bound maintain algorithmic max require several height height pt height pt objective wish minimize generate minimize upper iteration valid member function rest mm progress construct measure progress true mm mm measure respect value original g pick set optimize see figure progress progress mm iteration constraint objective requirement
vertex vertex dominate center center miss ccc dc dc dc r score set score cluster else contradiction must exist center center majority center close center else close original center would therefore must majority case majority center center majority center majority distance majority ccc center center center center center point undesirable line call design algorithm satisfy property rise well work bad view strong asymmetric center instance perturbation natural perturbation notion notion stability cost center approximation optimal solution stability optimal size stability hard approximate yet optimally value give find improve et al perturbation moreover show perturbation give perturbation cluster call weak proximity close center recognize form surprising unlike ratio asymmetric center symmetric asymmetric optimally asymmetric additional location stability uncertainty fluctuation city traffic time drastically affect optimal view instance satisfie
scalar roughly network transformation accord invertible use place simple induction induction simply original one view original similarly adapt invertible objective invertible satisfy update h lemma suffice j therefore kronecker block fisher transform invert j first property respect definition take default center ia indeed standard choice particular g observation efficient descent network factor curvature invertible neither completely approximate matrix plain objective momentum natural quality hessian work highly stochastic regime store inverting associate important despite work wise scheme sophisticated approximate newton update gradient augment momentum computed local progress per gradient fewer practical firstly hundred current batch cg iterate go less amount suit cg potential much apply locally cg potentially much fast overall function cg spread cg tune descent momentum cg character tune sgd momentum unclear preliminary evidence cg fall analysis motivate rely cg primary study inverting diag fisher within provide compare sgd momentum practitioner favor sgd network highly properly curvature plain gradient objective require method diagonal direct quality diagonal curvature rely cg could method whose method kronecker factor curvature much tune sgd momentum benchmark main sophisticated approximation neither block
therefore become possible distinguish yes unless cover hope obtain cover plant cover yes none algorithmic distinguish situation planted plant solution significantly formally find approximate system cover system let whose suffice convenience template ax x consider quantity row otherwise define follow simply variable inequality z claim imply z w row way coordinate negativity setting wish set sparse zero system equation fundamental simple
happen increase noiseless happen loss noiseless difference less pair paper datum combination propose indicator trade loss align metric require determine priori instead determined possess parameter experiment mnist denoise autoencoder achieve validation increase occurs optimize cause converge may minima change loss always validation large improvement
select deterministic selection nk k bind bind worth understand comparable derive could lead computed approximately could yield deterministic selection criterion example fourier construct space establish fourier transform approximation nystr besides successfully apply lead interpretable least square statistical least leverage leverage suffer uniform less tradeoff sampling square root suggest
random deeply become empirically propertie random adapt dimensional bound whereas instability well small e forest grow view show grow growth ever introduction property forest explain prove forest tree use without use guarantee discuss forest work post post tree analyze asymptotic multidimensional process similar complexity decision generalization extend forest analysis notable extension original survival forest alternative forest far understand work classifier bag adaptation validation estimate random adaptive concentration hierarchy forest predictor split begin formal
show testing limit nx nx nan simplify ratio statistic consideration generalization statistic composite give asymptotic nan hypothesis composite exactly behind interesting real contain order difference control match model mean first
typically comparison multiple comparison rank hoc machine well include lead situation pool comprise pool comprise suggest depend sign post recommend set nan post hoc carry adjust correction post hoc
except prior classical also bayesian drop thus generally perform hyperparameter favor g e sensible bayesian speed routine c pt wishart run c apply default swap contract know exchange spread depend view aim structure risk entity risk important example risk risk financial fail service daily spread american entity five year widely spread analysis assess variation year move particular month daily month month choice intend accommodate time period begin continue total estimation period correspond run change number edge range possible indicate reflect temporal variation steady trend mid mid event include series market integrate period tend provide graph detail
function imply f give property basis degree nf kf enable spatial spatial show matrix eigen orthogonal expensive since eigen eigenvalue efficiently obtain qr package available
light cart recover bayes classifier still prevent fitting point uniformly hypercube inside square circle figure bayes outside give error cart examine classify much region classified forest cross cart adaboost overall cart example allow circular split allow random adaboost pruning rather rectangular nn suffer interpolation localize classified incorrectly forest adaboost noise affected b b error datum display rule proportion point classify proportion peak fact iteration overfitte smoothing agree fact completely iteration thus exactly adaboost interpolation training differ bayes forest agree point example random adaboost yield respect argued initially exhibit self signal keep localize noise smoothing forest average obvious adaboost boost beyond classification occur noise lead overfitte good knowledge perspective key pure produce early take successful rewrite define every
square type proof normality frobenius vector hold argument illustrate n tucker r analogue receive pa pa n far schwarz n inequality receive taylor receive n n different yield n argument receive pa pn b receive argument j cover far consistent weak normality residual classical residual go conditional lasso additionally formulate problem come solution future recently set estimator significantly well apply construct reweighte lasso mention suitable underlie series third motivate subsequently asymptotic several extension
estimate jointly potential extension classification manifold low useful treatment problem among benefit unsupervised sample recent seek preserve learn meanwhile learn embed data sample work high datum embed datum manifold algorithm low preserve drive coordinate nonlinear dimensionality generalization embed already effective image show extension briefly overview supervise extension manifold discuss aspect analyze ny dx come dimension search significantly reduce preserve objective compute geodesic preserve construct nearest typically edge entry kernel solution eigenvector eigenvalue give row seek neighboring nearby version solve projection recently separation coordinate vary slowly neighboring sample neighbor different class class denoting seek embed employ alternative formulation order obtain embed variation work initially include embed whole set high ambient
experiment expect denote realization parent consecutive realization policy omit brevity policy early know framework dependency fall meet algorithm complete degree parent active work focus provide simulation approach try reduce bias trace evaluate artificial trajectory concatenation cost substantial trajectory infer naive greedy small j c mdps evaluation acceptable parent inner set far consider old additional parent add output enough adequate variable unlikely parent high parent likely prominent find stop parent available simulate find provide probably
presence superiority fc fc achieve figure appendix variation vary trend low recovery error comparison bring interesting magnitude corruption recovery become corrupted unable exploit relative solver error magnitude corruption corruption fc solver recovery ill condition perform plot spend considerable allow run unable desire residual converge fc solve gd slow convergence algorithm condition combine fast variation varied recover offer perfect recovery set infeasible setting key slow identifying clean around guarantee fc noise present extension gd
v repeat ip contradiction without n nf contradiction let v divide numerator respectively imply p n lead contradiction divide denominator trivial happen already know mixture equation rewrite ib multiply equivalently repeat argument without b k b g n taylor remainder op n nj nb p jx j n jx r g imply boundedness r nn x np xx np hx achieve observation begin suffice hold repeat aforementioned invoke I b I I multiply side contradiction conclude proof part choose p n b np p np I g n r j interest b nb r rr r computation demonstrate proof choose I n n last identity expansion constant exist sequence p n j mx direct calculation lemma kx scenario fails rewrite first j identifiability violate rewrite rewrite multiply side x proof denote large l part show np v n n taylor expansion summation remainder expansion fx I I fx I fx l n n I ji n na linear n j j I dp n argument step argue appendix infinitely hold n k k divide numerator contradiction however formation n contradiction infinitely nk numerator denominator real denote organize nd therefore n show combining since dp n I I coefficient go contradiction happen argument n part come identity three equal least n nx w nx p apply combination I coefficient go fact coefficient vanish absolute contradiction complete condition construct p n conclusion mean expansion conclusion also satisfy mix optimal give sketch contrary odd np ij ij define ij invoke mean derivative involve linear fm coefficient derivative ip ip g imply contradiction due proof therefore consequence combination coefficient differ collection conclude proof address remark regard removal ip v k na taylor
focus paper outcome formally loss round forecaster quality forecaster cumulative forecaster keep encode belief predictor forecast error fact class obtain generative show admissible constructive manner relaxation forecaster outcome restriction remove truncation recall protocol online round subsequently reveal subdifferential support minimizer belong calculation study extract upper bound minimax value introduce minimax minimax range guarantee guarantee perform technique introduce rademacher supremum conditionally introduction complexity number cover finite discretization compare discretization subtle notion small capture nature depth root labeling label right
dataset world flow cnns research matching representation cnns train unsupervised match patch spatial post processing flow include segmentation depth optical flow per architecture recent progress review conventional slide fashion class many drawback implementation usage intermediate map per patch nature account property stack per interest eigen refine depth iteratively refine feature coarse coarse high level part network good take approach flow dataset ground flow train network directly simple stack input feed network decide enough optical
prototype subspace color shape important differently lda different depict representation cluster lda middle column particularly feature lda identical learn differently record prototype green depict lda regard order special constrain version lda weight distribution value carry prototype within lda topic control depend independence many single perhaps unlikely would prototype subspace indicator cluster label encourage inference well also
recall consider similar measure cosine perform grain semantic meaning pick tf focus see step slot candidate identify engineering typically response ever reason algorithm yet good preferable metric progress performance case reasonable progress major availability substantial power development new architecture leverage progress lack aim barrier multi consist almost million person receive order tracking challenge recent answer service twitter turn long furthermore target namely development ai application
htp range set label training proportion represent proportion derivative parallel kde instead simulation necessity project setup handwritten digits mnist image evenly five comparison compare mean location simplex reader probable mean probable entry probable half bandwidth bandwidth minimize matlab full case computation expensive sparsity available depend iteratively construction half result also estimation uniformly pick computation notice main bottleneck computation optimal ten process ten task form segment pixel dimension describe position feature kde denote image shift lie surface
offer preprocesse associated undirected graph goal paper propose operate tend optimization problem error expense long convergence versus careful rapid match real purpose exposition square loss trivially losse poisson binomial derive discover simple illustrative benchmark present conclude decompose begin definition vertex edge every even odd
winner summary outperform fuzzy mean distance probabilistic principle exponential guide number direction future extension incorporation technique cluster problem possibly principle global optimality establish face distance propose linear outperform dimension need indicate denote vector p
lp linear read r dimensional programming vary moment growing monotonically converge low solve relaxation let resp atomic support resp support resp feasible value optimal moment constraint matlab toolbox primal denote degree infinite polynomial cone module define g mx px belong amount function lp q read maximization turn duality sdp e deduce subsequence set kp kp converge optimal sequence compact see subsequence since cone close optimal problem sdp
message lr l l lr fouri expectation map dl q compute true gram frobenius feature average suggest improve kernel may suitable compare collection incoming e collect time iteration pass convergence message subsampling collect message feature leave incomplete cholesky widely cross randomly validation ridge incomplete cholesky factor ground truth belief estimate learn kl expect embedding feature sum product kernel product sum embedding joint forest extremely randomize forest income significantly compare prediction randomize forest learn toolbox tree random forest tree empirically observe kl gain ep coincide accurate confident kl make confident fact training operator reference kernel
flow estimate element correspondence evidence process remain reconstruct share circuit equivalently mean correspondence form cut set circuit graph incidence identical possible share permutation share space circuit correspondence converse also graph column case incidence fundamental argument uniqueness steady understand incidence write distinguish alone realization fundamental circuit realization graph determine edge structure stage round sign incidence zero column hence perform operation perform
forest indicator cover selection technique base indicator one performance pair classifier indicator naive collect health monitoring need multivariate numerous system anomaly performance display change forward computationally demand namely per engine hour per via operation health external evaluation monitoring message include engine status overview specific engine send anomaly early degradation anomaly detect send company operating engine sign degradation despite avoid minimize customer inspection prevent availability avoid possible general methodology human operator final decision leverage expert build existence sign anomaly engine health select standard forward automatic classifier complex interpretability requirement focus
matrix denote follow eq assume e drop positive standard bernstein thm pt w n n require exist constant strong enough bound extension random variable random exponential zero say exponential equivalent notion sub employ version inequality vector entry employ proposition top leave directly q less rank rank I rgb rgb conjecture claim fact time bold bold ff community community tensor learn class hypergraph resource membership community separation tensor social system detect movie political mostly movie name person illustrate movie community tag make hold importantly distribution mixed community membership work dirichlet membership early
usage horizontal procedure plain change shape consistent cf figure overlap h ks ks ks ks km ks k loose least gain setting section proposition definition assumption l p title em date spatio temporal change framework multivariate series fix increase receive denoise
entry fall eq combine fall q next definition suffice sphere call symmetric subset q p short prove realization submatrix probability q xx equation suffice independent verify probability combine elementary algebra moreover ratio claim apply elementary note due spherical see em lemma generalize satisfy fix occurrence occurrence see b show item combine inequality q symmetric short z correspond eigenvectors elementary elementary least exceed p op give least elementary random combine large time theory q right side show item union show q claim easily vector correlate overcome difficulty independent additionally combine see first second similar noting prove let cdf inequality claim eq short q q calculation generating assumption elementary taylor eq taylor expansion third derivative
removal goodness weak show interesting recovery intractable search know necessity condition ensemble verify threshold consequently h lasso cross matrix row generate variate replicate ratio select median bayesian performance expect logarithm select true begin rapid mixing certain purpose transition eigenvalue consequence eigenvalue know gap markov suffice universal probability transition intermediate claim thereby posterior low relation exist constant accordingly remainder devoted gap associate markov weight direct edge order pair edge distinct unique ensemble show reversible markov choice gap ensemble quantity e construct operation path intersection overlap edge canonical path inspire variable path variable procedure construction prove helpful respect intuitively intermediate towards property connect central clearly canonical ensemble function rise canonical path consist
obviously inferior although isotropic reasonable discuss section infinite isotropic lift hashing tend favorable fact inferior angle pairwise rotation lead pairwise almost rotation hash c bit reduction bit reduction across sift next common isotropic attain good achieve remarkably reduction exist pca dimension reduction sift propose high method b method contribution examine dimensional state art attain contrary inferior novel hashing rotation encoding
occur memory whereas smoothed identifiable prediction often digit correctly input digit feedforward describe mnist neuron vertical red divide neuron frame activity input correspond connection activity input hide activity network present activity square error inference available movie error vs training connection layer train low vertical indicate point end test field initially feedforward self supervise move image dynamic sensor network deep level object detection speech rapidly develop theoretical framework learn understanding neuron
q depend logarithm dirichlet logarithm depend q bayesian automatically number gaussian simultaneously analytically inefficient highly accurate mechanism permit system dynamically comparison method precision keep frame correspond reflect scene fact equally scenario time visual optical affected illumination texture privacy people scene persistent surveillance optical video unique challenge pixel temperature issue
benefit contrast ii simultaneous consideration guess opposite guess plausible benefit ht c c run function global search domain third reject simultaneous low ht run run simultaneous consideration guess already always show ii fundamental assumption simultaneous consideration shorter purpose paper fuzzy introduce evolve core mining quasi level accuracy increase report gain accuracy reduction employ introduce type evolve mining multiple year
analysis occur directional derivative derive process order directional sensitivity local variation insight spatial angular discrepancy process surface surface two process spatial relationship tree forest log cox local sensitivity specie extension application observe multiple relationship involve incorporation temporal leaf trait relate well highlighted gradient analysis spatial gradient chain allow model location ratio two depend integral consistency permutation
value single space contrary increase agent generalize single use move window datum mse generalize user size increase dramatically big set agent per history generalize multiple history datum electrical twitter research focus turn relate may vary
financial dnn auto make minima deep tuning supervise elaborate dnn precise architecture focus learn highlight implement deep theoretical relationship recursive confirm indeed without degradation direct layer early deep learn boltzmann simplify dnn basic simplification light fundamental origin
fit task call prove uncertainty offer new approach point process quadrature generalise note work intuitive challenge assign computation symbolic form deterministic real analytic sense quadrature rule optimize come strict interpolation line quadrature rule challenge uncertainty two thick line thin density posterior line integral rapid estimate thin line match real estimate calibrate grey two repeat spline calibrate confident banach hilbert space quadrature may vanish spline wiener finitely kx show grey hypothesis differentiable gaussian close projection univariate measure condition collect another x definite optimize lead place grid piecewise linear weight posterior spline equation bayes bayesian quadrature spline prior uncertainty process piecewise function red elaborate quadrature rule high spline interpolation chebyshev change rule rule value increase quadrature
suggest careful relationship theory provide kullback leibl closely relate mutual extensively define vector far extensively learn distance eq coefficient overlap q use
overlap state consecutive segment slide window subsequent segment share close increment slide offset trivial pass risk potentially miss experiment focus search primarily concerned try segment propose scope scalability etc upper length segment tr p j describe wise naive qualitatively implementation compute series frequency segment force output discovered occur explicitly unfortunately infinitely candidate show locate real segment comparison restrict restrict segment thus dimensional h frequency interpret threshold distance pose hyper method
sale certain census vary dramatically neighboring census census price spatially university trend behave compare neighboring census census heavily student rate instead rely census solely house price accounting census assume evolve leverage build drive discovering dynamic cluster c neighboring census color offer advantage exist nonparametric feature attain shrinkage improving estimate multiple section likewise bayesian consider uncertainty together parameter price sample census examine census similar dynamic house transaction census individually correlation couple explain component overview outline step challenge implement parallel simulation house sale transaction census city sale include house house census code month sale house covariate variable sale sale attribute association home house price region examine region joint modeling underlie evolution desire index house sale infer kalman smoother embed em census independent jointly relate independently kalman smoothed cut tree certain vector observation kalman smoother share latent dynamic smooth figure exploratory analysis consider hoc
separable rule experiment algorithm co separable ability decade commonly synthesis extensively investigate prove validity many change regard validity co publish assume certain synthesis read signal combination operator improve case letter bold letter bold face capital letter letter consistently entry entry mode define apply result kk rewrite vector product notational comment provide section point come author svd dictionary co phase consist stage operator row orthogonal signal meet operator uniformly normalize training sample output operator achieve stage operator project subgradient signal alternate multiplier concept operator call
n n theorem difference denominator recall empirical true censor r l b r n h net pr pf bn pr sup pf df bn pr bn v v lipschitz continuous q derivative nb p cc minimize eq conclusion behave pn pm depend depend lemma thm remark skip author nsf grant format failure monitoring status obtain support decision version sample novel oracle inequality true conditional
drop reduce problem drop amount drop extreme tree drop forest drop set vary drop report employ technique drop drop drop round well mode define tree ensemble nd mt xt tm task scale publicly compare rf consider whenever leave per c leaf c yahoo
distinction penalization bad unbounded direction application odd ratio expect zero close natural quantity normalize say approximately depict self normalize set geometry paper statistic parameter regularize kind every hypercube general whether train feasible exist self distribution already exactly consider ax every characterize
small validation precisely upper apply high iterative dataset figure incomplete remark conjecture axiom observation classification model combination weight regime learn dimension computationally statistically establish experimentally validate advantage hypothesis return experimentally sim compare commonly rest wish solve method perform thorough evaluation several appendix algorithmic therein learn
bottom principle grain event adjacent interval perform merge improvement merge improve grid solution evaluation grid give grid cell step evaluation grain time show allow sophisticated algorithmic exploit start grain set cell cell perform advanced stem dependent hierarchy component cluster property interval perform grain concern grid dedicate preprocesse locally improve final solution optimize partition move optimize concern round initialization optimization technique world study make hundred event million interpret issue simplification together choose rank insight meaningful simplification iteratively merge interval merge least degradation
recently able speech without rnn output rnn alignment posteriori alignment search deterministic mechanism keep marginalization monotonic speech recognition hybrid neural step compute base address deep describe hybrid attention embedding disadvantage see recognize width perform corpus train split sa stopping scale bank together temporal total feature rescale train phone extend extra token similarly
observation number maximal acyclic ix performance bandit general least particularly proof notably involve important bandit mention clean proof algorithm elementary advanced inequality loss game tight bound advantage demonstrate variant problem arm expert track good arm bandit level confidence level tune despite property latter g notable previously know
contextual triple denote baseline contribute candidate message lrr mt reference random mt mt ir ir sensitive result mt ir list subscript indicate relative improvement frequent order avoid mini neural recurrent initialize orthogonal scaled performance hold training size bottleneck response vector candidate response phrase decoder ir mt create human less issue comprehensive list weight task suggest human baseline response extract choose amongst reference status make three evaluation broad pattern mt ir help improvement outperform baseline
calculation intractable exceed ht em em ht tm em tp em em tm htp tp ht htp restrict tractable maximization log critical efficiency choose break dependency couple hmm factorize replace influence tm tp tm em tp em em em tm tm h tp heterogeneous markov chain eq tm tm tp em em em h tm tm tm em tp em em tm tm maximization maximum involve provide approximate negative bethe definition depend involve compare variational joint conditional step approximate whereas approximate optimization computer standard purpose characterization definition enable exact marginal operational answer enable acceptable complexity obtain elimination adopt unified algebraic presentation inference marginalization solve elimination elimination task ingredient algebraic instance write evaluating require multiplication one enable operation algebra inference artificial intelligence name elimination elimination rely elimination either marginalization correspond order apply topology calculation elementary operation elimination correspond degree start leave elimination reason elimination inference propose parallel mathematics minor circle
analogy chinese restaurant model crp prior customer condition solution belong exclude assignment flexibility start enable infer satisfie scale number impose multinomial impose symmetric k conjugacy multinomial goal cluster assignment gibbs inference cluster apply initialize step solution remove assignment bayes non remove correspond parameter new sample otherwise solution cluster describe gibbs series approximate interest inference important estimate take mode number iteration mixture model switch cause cluster arbitrarily overcome first terminate well assign index mode
see true fit poorly posterior due insufficient sharp e indicate logistic negative observe thank method successfully recover posterior transfer learn axis happen region explain superiority region posterior differ insufficient performance since center corresponding region learn hold likelihood figure draw vary average run datum provide result show representative strategy
event unlike detect tag method conduct solid mathematical drive unsupervised handle challenge solid matrix system parameter measure visualize unsupervise supervised pca auxiliary visualization visualization study validate effectiveness high data resource management grid discussion wide area monitor wide wide data analytic xu mathematical architecture early algorithm
prediction result describe publicly implementation instead path rnn well rnn surprising extension cluster conjecture strength important unseen statistically significant p perform well table shoot order fully train rnns evaluate shot relation set train shot two rnn supervise rnn shoot supervision path map rnn shoot shoot explicitly result perform affect local optima rnns train apart rnns individually stop improve use
customer customer permutation object disjoint hdp stick break prior concentration group hdp crf
correction define asymptotically imply correction maintain correction select stochastic gradient update correction use get adjust version property correction additional cost variant practically convenient investigate advantage correction note correction variance randomization simpler suggest sample variable otherwise expectation per iteration parameter option save store
let lx tw space evolve approximate essential smoothly radius restriction hilbert rx b z rx rx motivate correspond universal asymptotically find degree achieve function small kernel small approximate function facilitate lyapunov loop let function time associated ideal eq function time continuously respect weight continuously differentiable use rl implementation gradient base law learn vary ideal time base online exact via
overhead mean dominate fast essential sum close produce correct round small integer computation implementation science engineering research hold research statistic efficiency page http fr summation page parallel core architecture page far reduce truncation error digital computer exact accumulation product pp p scientific fr quick http team project statistical http stream page summation science two summing round high rounding application guarantee parallel serial use seven bit next allow carry propagation alone one small carefully modern exactly array take twice inexact serial processor summation array limit impose bandwidth attempt accuracy parallelism exact test intel processor show modern processor sum thousand inexact exact method sum fast large always small except sometimes two slow bit processor old exception bit processor architecture discuss implication improvement term processor core implementation
additive layer drawback result ideal learn function unit multiplication operation obvious operation neural backpropagation iterative could initialization operation neuron train determine allocation optimization particle genetic satisfactory computational allocation whole must hour moderately size distinction neuron additive multiplicative standard approach organize
different document importantly make widely applicable middle fig active topic document topic drop burn learn amongst number group group effectiveness propose performance document document share topic interest document latent show bar except burn predefine standard get deviation reason topic change walk sampler result predefine middle traditional natural machine learn branch work incorporate
finally effectiveness usefulness via dataset last propagation motivate feed name model aid diffusion behind propagation certain node node diffusion activate literature direct budget seed activation seed achieve I cascade ic generalization solve paper heuristic body cover assume input influence assign set incoming set however assignment learn influence kind cascade past network al maximization approach probability cascade ic frequentist approach influence quality seed present likelihood use ic diffusion transfer rate develop infer cascade time irrespective approach propose diffusion work depend cascade dataset cascade unfortunately probability raise network influence manner generally I cascade influence contribution ic multi armed mab problem maximization intuitively seed amount play seed budget play seed playing attempt select seed reward knowledge knowledge choose well seed round seed tradeoff detailed effect seed cascade seed activate
identify time strength capture exhaustive submatrix phenomenon essential statistical submatrix submatrix signal problem variety assumption sparse principal look statistical detecting concern pose computational challenge computationally efficient draw emphasis decomposition computational trading accuracy focus computational statistical problem regression investigate accuracy submatrix localization noisy matrix formalize eq zero formally form submatrix submatrix simplicity focus submatrix extend fundamental associate whether submatrix consider goal exactly set clear least exploitation ratio statistically quantify phenomenon boundary possibly go minimax sense exhaustive successfully find submatrix adaptive find later work submatrix
old complex age variable cognitive involve cs show risk predict clinical allele education medical risk presence diabetes cognitive consider include exploratory summary hypothesis survival regression none accommodate predictor small often requirement particular thousand introduce section accommodate predictor use define avoid risk cognitive status predictor measurement age coefficient multinomial logit lasso force identify important thousand cs fuse piecewise coefficient j kt justification predictor record cognitive range outcome outcome cognitive status one instance cognitive propose predict presence death adjust death factor factor death predictor compose record cs study variable miss value rule cause past measurement exception either categorical outcome convert
span arithmetic improve simultaneous iteration multiply svd multiply time achieving improve number traditional block previous block recurrence sized block large compute recurrence issue qr furthermore typically dominate cost block compute avoid poorly algorithm analogous costly subspace time finally multiply takes claim next return basis frobenius start give intuition proof return first column main singular l l intuition norm low polynomial align intuition outside distinguish small rather cost column outside cost accumulation capture intuition separate value achieve frobenius trivially statement case explain polynomial assume
planning controller effect problem art achieve robot control control reinforcement e environment rl state space fairly dynamic thousand trial control scenario rl knowledge extraction expert knowledge realistic explicit use flexible promising extract datum learn td learn main rl suffer inherently resembles issue task illustrate affect tb figure convert ep find build upon evaluation analytic gradient sec subsequently employ obtain approximate long alg controller record probabilistic gp eq sec get j cg bfgs record implement tuple u leave remain specify positive depend characteristic transition scale prediction distribute gp k target conditionally gps gps input uncertain long marginal ahead p one uncertain gp notational convenience omit conditioning episode tp u approximate gaussian provide assume distribution integrate gp compute exact predictive analytically intractable
gaussian deviation control provide together two display exponent tail sometimes big application less matrix chernoff bernstein study concentration value random many provide reasonable tail decay recommend apply scalar try two type extensively literature precise benchmark sharp comparison specialize similar conclusion begin independent represent compactly gaussian eq precise independent theorem hermitian square fact technique long calculation argument next standard independent standard express elegant infinity yield matrix variance satisfie conclude term factor belong comparison produce general principle matrix less independent rademacher entry satisfie lead row expect sign maximum case admit find coincide establish match logarithmic obtain quite sharp main like involve combinatorial inequality offer toeplitz application toeplitz row matrix variable take value variable toeplitz matrix act column vector place introduce bottom shift place first square instance term line switch order rewrite conclude correct constant know nevertheless eigenvalue toeplitz standard conclude toeplitz index scaling lie final substantial combinatorial one certain optimization rademacher far indicate difficult approach solving become relaxation relaxed round procedure back round change value substantially class quadratic subject convex quadratic refer desire solve family specification relaxed family scaling variance satisfy q important scale therefore massive ultimately solve objective factor maximum value chapter theorem gaussian series explore application development random setting therefore inequality hermitian hermitian hermitian dimension standard introduce matrix series variance statistic sum q eq independent rademacher proof result proceed series formula hermitian next eigenvalue maximum eigenvalue q second identity relationship consideration tail minimum eigenvalue result hermitian indeed theorem concern produce side instead two sided tail bound improvement really hermitian matrix maximum valuable two see chapter continue exhibit behavior describe rademacher master sum identify matrix hermitian let may standard normal satisfy normal establish formula therefore vanish series representation expectation compute extract recall logarithm exponential quickly hermitian consider hermitian finite normal line introduce reach third eigenvalue fourth spectral use formula infimum attain proof tail invoke master step calculation infimum achieved involve argument proof piece reasoning simple hermitian first compare inequality observe apply rule probability hermitian rademacher hermitian sequence rademacher rademacher series follow identical justification obtain semidefinite leave increase substitution monotone semidefinite series hermitian rectangular bound norm hermitian formal device hermitian hermitian theorem recall hermitian two hermitian hermitian use conclusion term random employ preserve spectral calculation coincide invoke reference analyze appear argument similar depend factor discuss give chapter rademacher gaussian originally follow simple inequality concern two elementary exponential moment analog strong nevertheless useful practice long contain concentration act use inequality covariance lead activity researcher require parallel researcher probability difficult lead concerned quantum researcher mathematical advance lead optimal researcher reasonably analyse variety effort literature nuclear physics book overview book complete maximum eigenvalue analysis present rectangular limit name almost sure limit value rectangular ultimately process due use sign van elegant independent toeplitz surprisingly paper obtain limit toeplitz iid mark establish toeplitz iid toeplitz entrie identical reference toeplitz whose entry variance simple modification semidefinite moment reach pointed moment imply concentration inequality robust presentation achieve chapter present concentration analogous chernoff bound set extreme eigenvalue semidefinite consider independent hermitian sum
item mistake exponentially partition consider incorrect comparison cause operation partition comparison bt available high combine follow simple enable recover leave exponentially average realization test bound value accord loose indicate maximum appear tight seem contraction effect far expand sort noisy non trivial sorting fall side natural question induce sort section pair sort constant comparison estimation sort comparison sort repeatedly last truncate retain outcome discard sort procedure comparison first preference collect passive standard
point computed scale condition smooth convex weak execute efficiently technique meet low case last concern nesterov gradient method publish gd obtain order gap bind researcher lead nesterov intuition heavy primarily sophisticated algebraic e non trivial time admit complete satisfactory surprisingly design optimization polynomial constrain ball coin summarize appearance mention unlike dimensionality exist matching attain state calculation inefficient execute yield systematic gradient heavy root scheme offer exploit obtain spectral gap scalar letter letter equip diagonal symbol ia spectral eigenvalue root characteristic polynomial abuse root modulus root quadratic matrix frequent quadratic sequel analyze strongly convex motivate show present generalize various sdca case formulation apart inspection give subtle lastly stochastic ascent sdca solving regularize minimization great sdca recursive simplicity exploit assume sdca work
evaluate compare evaluation variation series impose impose simulate series alignment simulation series monotonic maintain manner true sa accordingly ease impose respectively white deviation slight alignment identity simulation temperature across aforementioned subject variation series series standard deviation noise average tuned noise low account variation simultaneously emphasis behind start outperform emphasis across comprised superposition base length component series location scale window denote rectangular triangular window uniform
whereas cyclic descent update whereas update extreme one fit framework problem allow follow require subset coordinate randomize let satisfied element iteration theorem hold option ii hold option option ii assumption convergence convergence hold produce eq
da figure green blue show green single color continuous eventually picture individual three trajectory evolve eul movie display remarkable evolution look human fig material bayesian mcmc various statistical approach form active methodology metropolis hamiltonian slice popular modal high fast computation sampling chain fundamentally unlike markov point pick entire supplementary exposition two dimension sample wave effort focus address undesirable methodology conclusion work dynamical approach system
accelerate fw advantage fw especially apparent tolerance arguably due theoretical randomization employ result tradeoff complexity kind different application investigate acknowledgment european european union framework reflect view project grant medical science policy office dynamical author project
expect attention efficiency package various dedicated search package use library provide package though package towards automate hyperparameter automate still way
frame region context tracking identify within particle track dirac particle importance function particle suit tracking visual method field visual motion energy function shift tracking method implementation design tracking object tracking ds ds crf situation object significantly shape track tracking tracking target completely track ds track method capability ds crf next crf object object time object sequence track particle ds able person result capability propose ds crf object examine ds crf track figure object object tracking method filter track target path object particle filtering track bound hand ds crf track
provide partition factorial implication predict serve estimator good py process measure specie explicit size additional basic genomic one library consist million apply context useful credible derive central factorial resort credible asymptotic fix stand function u eq obtain diversity recover turn use determination interval still nonetheless avoid allow quantile derive rare specie recently distinct less threshold abundance sample display loss decomposition distinct frequency detect old specie appear frequency arise quantity new distinct additional old frequency frequency sample interpret overall explicit expression specie old specie rare variety statistic specie equal sufficient predicting derive focus py considerably simplify worth note determination specie pose estimate possible additional species special py establish rare species spirit context specie x k random predict distinct specie rare term proposal nonparametric good discovery species note suggest discovery latter minor km yield analog specie frequentist counterpart integer variety yield process analog reduce counterpart
high kernel nk limitation lie lack scalability require multiplication even write full prohibitive try avoid low matrix nystr approximate scientific computing way deal approximation know fast adopt compute rank form wise generalize low learn informally field kernel interaction datum far design kernel particularly light result example decay score uniformity width sparsity traditional approximation
proximal converge rate find fix forward operator subdifferential calculus connection characterize follow operator composition operator subdifferential repeatedly operator precisely proximal operator subdifferential subdifferential respect necessary satisfy even value also apply proximal apply gradient shrinkage primary multiplying iterate multiply proximal simple penalty contribute negligible complexity require improve notice imply implement linear approximation naturally high expansion calculate instead directly approximation employ newton bind way interpret quadratic newton proximal information proximal accelerate within intermediate momentum slack evaluate advanced technique common variation proximal describe conjugacy relate algorithms describe primal redundant parameter slack encode consensus requirement affine redundant certainly family generate generalize slack lead arise consider variational connection explicit mixture mean scale envelope section detail variable splitting fit couple objective primal original
penalization small nonzero minimize square additional coefficient norm penalization motivation different support require generalize dimensional consistently sparse three highly design consistency oppose strong require recovery remarkable f rely additional third communication large hard parallel parallelization amount machine communication ordinary consist loose provide consistency evaluate implication
pc recovery discover ol assumption sis pc admit support sis instead pc performance predictor suffer inverse subsequently ol screen sample covariance convergence obtain allocation proportion second introduce asymptotic notation proposition surface often score specifically vector define formula spherical stage refer stage asymptotic generate I response remain inactive comparable make study realization mean probability density g differentiable positive u score weak unlike commonly heavy tail analysis support impose also introduce impose magnitude relate regularity concentration moreover incoherence score similar satisfied set population matrix weakly b sparse j op assumption limit screen yield accurate poisson
unbounded right compare fidelity theory achieve fidelity furthermore algorithm denote incremental stagewise regression rescale coefficient update introduce version refer adaptation descent may interpret natural popular boost like coefficient amount residual factor select solution least guarantee spirit fidelity shrinkage boost iteration notion implicit shrinkage literature notion herein new unify subgradient regularize cm structural similarity forward stagewise connect role regularization later e amount coefficient update coefficient index coefficient description residual coefficient rate initialize k j share previously additional rescaling rescaling control demonstrate play connect modify modification lead index correlation note reduce minimize residual predictor residual far via duality equivalence insight coefficient whereas equivalent correlation characterization least square show subgradient boost extend subgradient regularize computational boost equivalence subgradient descent state regression exist l I n cancer coefficient scale boost appear panel panel panel evolution panel middle take norm interpretation imply theorem fidelity boost demonstrate computational describe trace profile reflect item function rescale shrinkage series iii iteration characterize complexity applie iterate hand characterize item training prescribe appropriately shrinkage trajectory interior profile lead visit well predictive trading bias desirable suitable minimum figure show profile similarity present impose shrinkage sufficiently profile may draw analogy e profile profile regression run maximum top cancer dataset coefficient bottom constrain profile respective
recover sec estimator principle reduce cubic quadratic case suggest connection indeed show asymptotically tend classifier detail extend future extend design efficient classification algorithm fix distribution denote sample decomposable binary indicate set sum apply decomposable evaluate decompose sum accuracy desirable tradeoff confusion positive tp positive true negative tn negative
extension
degree choose instead coordinate assign load computing hence minimize processor return coordinate make take pass difference preprocesse w w uniform serial sampling sag gd gd sdca sgd different
demonstrate substantial outline remainder relate parallel variational variational concave low constrain parameter aggregation one enable aggregation sample serial replicate carry challenging experiment summarize several work datum bayesian strategy serial differ employ communication spectrum parallel core sampling aggregated lead design aggregation procedure construct approximate posterior sample motivate denote core average na I heuristic motivated consequently covariance suggest covariance treat aggregate sample aggregation
helpful careful cm cm propose covariate instance key combine dimension reduction technique multivariate design nonparametric calculate appropriate still therein recent multivariate design describe fit linear
moreover use hold choice affect compare bernoulli report ratio achieve bernoulli result huge optimally require objective vector take well denominator depend ratio one know objective fx fx use per use order denote minor denominator quantity problem require balance result turn expansion equality reader appendix taylor expansion expectation hessian observe calculation expectation simplify follow reader refer detailed proof
second low cost collect assess system acquisition protocol day available organize category representation acquire originally train imagenet library propose module visual method machine library indeed comparable even support machine set provide system incremental implement sec question provide reader capability recognize visual indeed realistic system investigate possible benchmark generalize run application sec identify sec briefly comparison recognition system reference answer motivate predictor vary test problem particularly limited object offer
form system require illustrate greedy form irrespective good require energy energy present mdp determine amount every instant queue length energy split profile curse reduce experiment show involve apply enhance focus sensor deal usage energy sensor wind body etc electrical energy network performance metric reduce delay transmission though potentially yet require therefore energy consumption amount energy performance sensor additional node share available node fig arrange pressure sensor sensor could energy energy sensor efficiently nee dynamically data sensor queue length transmission delay minimize paper develop allocation comprise multiple base bs maintain separate queue
horizontal line contain contain perform sufficiently proposition eq desire numerical integration associate integration numerical make small give finite boundary characteristic mx precise conservative approximate advantage show equitability advantage bias compare property introduce computable new computable alternate prove previous boundary characteristic boundary compute individual characteristic involve find optimal grid entry require employ section formalize idea object characteristic matrix entry maximal achievable mutual grid achievable despite question get fine fine become indistinguishable formalize boundary case consistent supremum indeed define row note presence ccc pdf column size large result mutual instead column whose grid axis partition analogously define convention quantity presence jointly variable denote characteristic define partition row equal mutual fine g chapter partition equality follow entry entry consequence jointly analogously characteristic let consistency estimating use quantity quantity sample whose sample sufficiently sample abstract continuity state formally equip projection zero uniformly pointwise obtain efficiently via entry characteristic subroutine present axis master partition subroutine find axis mutual induce grid way use
model replace substitute available intractable density close versus statistical prior parameter seem matter simulate massive scientific suggest primary truly abc abc one assumption stand overall
location desirable immediately file document l enumeration table table wide split top page nearest proper environment document table want table environment forget note use construct logical construct axiom
scalability label meta stacking supervise class dependency method label model general classification task e speaking seek class label infer indicator label test classification individually h l diagram practically multi label suboptimal reason fix meta stack meta skip correct error subset mapping employ frequent frequent vector consider find label exist typically small hamming family like label rather classification family attempt trivially label method classifier classifier relate extra label original year e reason improvement label ensemble recent performing search analyse hundred chain configuration purpose order method model focus paper discuss dependence respect chain point argue leverage attribute
goal want target experiment experiment confident systematically stopping design previously kernel show drug interaction interaction factorize project project project target multiply prediction drug entry use truncate drug factor kernel specific kernel kernel similarity drug knowledge e main powerful estimate prediction iii evaluation drug target superiority
ess typically equal sum core abc give asymptotic weight standard abc importance expected give control time natural practice remainder detail simulate
c n n c c n c smoothness bound confusion bind entry hessian assume w v c bound nn n n nn constant c take n u u u n n c smoothness c thm thm remark lem corollary computer institute algorithm confusion express sum example performance micro class understand consistency property know decomposable unify decomposable metric problem confusion achievable distribution continuous metric generalize decomposable metric cost see cg feasible confusion provably family multi decomposable real task multi decomposable classifier include micro mean class understand minimization decomposable metric decomposable metric tuned scale general decomposable metric metric confusion confusion generalize
input x pooling x avg avg softmax national science competition classify award image private divide image competition multi augmentation team competition experiment scale
dominate cost reduce datum sparse spectral perform svd covariance whiten whole want direction procedure whiten instead try simultaneously normalization potential cca therefore deal parallel lemma k k kk k top canonical identifiable cca linearity offer project work let exist mapping f f jk object feature canonical n k logic prove replace grow stochastic dominate theoretically give effort spend develop
triangle vertex clique vertex least mistake exceed hypothesis perfect one sided cluster bipartite np np ground element tolerance edge side bipartite graph also suffice perfect np one sided cluster view special reduction side nontrivial instance construct pair triplet vertex iy ks iy jx iy ks positive negative edge call correspond triplet lie triplet perfect contain since follow contradict clustered property cluster vertex cluster call immediate vertex cluster vertex
hdp effect incorporate mention describe except sample belong concentration customer conditional create table cluster assignment customer assignment crp report iteration find mention similarity well development set event resolution hdp suggest improve hdp cd thresholding mention clear hdp prior cluster gain indicate far comparable cd help precision explanation merging tend cluster list top sort weight mainly discriminate event head context word argument head sim sim sim sim sim correct hdp
rank binary utilize issue trust entry analogous binary handling fortunately margin binary problem idea utilize location entry gap lot completion algorithm tackle nuclear fold well trust utilize max sdp utilize show bit benchmark investigate decade social indicate shown figure social rely
unknown I distribution assume follow entry ease matrix bregman divergence exponential bregman leibler divergence resp resp kullback conditionally kullback leibler divergence introduction continuous information commonly use enough successive twice statistical guarantee norm penalization log observation nuclear
probability ground truth teacher student aware introduce word embedding index map capture certain aspect semantic maximize scoring feed speech sentiment formalize notation element look matrix
show difference mean score statistically figure trial line trial example size form training form working error compute scoring difference statistic figure ht p c c set maintain california two type binary class example training working indicate scoring permutation make except figure improve substantially go
big min n lemma let appear time total symbol appear time divergence distribution estimator eq sequence assign appear appearing give formalize eq thus competitive natural mass convert slight modification lemma paper
feature margin find outperform measure historical optimize address provide answer benefit researcher computer art dataset describe texture shape line movement unity contrast add physical analysis investigate encode encode color texture variation effect investigate depth need carefully design visual visual encode aforementioned advance vision show advantage feature however would impractical feature encode concept annotation concept image large obtaining annotation art typical alternative investigate different range semantic use metric specific ultimately goal art retrieve direction high semantic concept annotation task widely test metric visual feature aforementione
aid interaction reproduce emphasize minimization substitution seek interaction bias demonstrate technique priori square hyperparameter investigate amount red green tb boltzmann probability relevant task prune irrelevant beneficial solving cost function method technique express interact effective body independent test method adjacent
gray line overall hazard ii censor censor report account log hazard serious clinical trial breast censor dataset display unbiased wide datum censor note require baseline hazard hazard trial maximization unknown discretize baseline hazard hazard shape hazard return equivalently estimate hazard undesirable effect event treatment trial event group trial without loss generality censor provide patient effect clinical trial pool overall treatment effect nx b overall hazard trial harmonic define case calculate hazard version overall attribute baseline hazard hazard asymptotically patient patient treatment numerator always censor pool lead arbitrary parametric baseline hazard
convex prominent minimize average dataset subset compose average challenge involve little communication size become single grid optimization recent year example communication require paper opposite limitation assumption case room possible major study desire problem feasible large several sec precise merely study optimization important type loss relate random machine value gradient local differ sample e study study mild assumption satisfy reasonable aware round local low smooth function match accelerate descent many round may get machine smooth strongly straightforward distribute local quantify smooth strongly match logarithmic getting alone strongly study
fisher e x x x discussion classifier fisher ordinal rule ordinal ordinal classification generalize minimizer tf k subproblem generalize discriminant small risk aggregated subproblem one risk distance summarize simulation well costly reason probably perturbation add help generate radius within centered label add range realization two perturbation boundary classic class outside thin constraint boundary class class versa report show appear time
former linguistic dimensionality dimension softmax encoding set subsampling option representation thus maximum interaction modality include extra layer act modal result vision input generation abstract concept incremental suited cognitive acquisition plan acknowledgment semantic support
plot structure attain effect effect get difference value red green description variance numerically increase variance value scenario cccc variance discuss favor parameter weight group simulate hope fuse term consequence span regularization adaptive span blue adaptive without green curve blue correspond return selection difference penalty select distribute parameter fuse lasso grid structure star graph clique structure group theoretically clique graph suggest overcome
mini surrogate perceptron section decade rank early problem rank measure performance portion optimize one practice aware directly structural svm cut stochastic implementation use suited note bipartite rank emphasize ndcg recent problem tailor ndcg adversarial limit nan label rank goal rank subset one label scoring permutation function rank shorthand positive top scoring otherwise score surrogate act proxy surrogate regularity surrogate well requirement family surrogate consistent consistent surrogate notable seminal surrogate refer crucial design broad surrogate surrogate output exponentially space label data surrogate point large score negative however candidate labeling
mb report c mb mb mb mb mb mb mb mb mb achieve completion netflix par anchor size list tb netflix prediction netflix approximate classic co valuable analyze well fit netflix image face netflix use svd svd handle use classic set domain compactly netflix miss face quality observe approximate classic ultimately design effective tb city city city go city dr city particle city city I bring pilot c spin city stein nine star episode law star iv vi order lose list away world seven star material color great fellowship part care file file decade er preference
object vary time due appearance etc environment clutter know estimate state observation history formulation directly describe indeed system space essence finite value density random characterize unnormalized trajectory object uniquely identify unobserved discrete countable I integer set object identity essence mark label label bold distinguish measure follow discrete object consist special l I represent object time history association function track label measurement track measurement represent efficient association preserve association sensor filter approximation thereby drastically reduce form cardinality iw iw result case propose tractable multi element avoid match
question hold experiment gibbs mf gibbs mean average training finding high per versus start gibbs whereas stochastically require local notice predictive close gibbs sampler correct variance issue scheme converge unbiased sample converge biased burn chain burn combination discuss burn fix burn severe gibbs length burn experiment chain worth fix burn apply research metric quantify reconstruct peak signal ratio root
definition expression logarithmic identity loose later purpose case use tree traversal reference query recursion goal reference final query recursion traversal whole query observe part observe node happen query show cause large pairwise hand reference tree extra cover traversal reference query cover tree tree algorithm eq reference set recursion maximum runtime part recursion reference runtime query recursion large query line full reference visit query lastly total root query type although size show consider situation child imbalance exactly recursion parent possible reference recursion recursion lastly may runtime query trivially dual cover traversal take recursion maximum runtime consider arise near arise node visit reference
eq use independent q together distributional equal square non entry role quantity assumption quantity consequently denote vector q assume section interpreted expectation emphasize lemma decay power consequence strictly nice section mass correctly allocate additional cc correctly correctly amp statistic decrease monotonically effectively amp decoder green curve red indistinguishable evolution curve amp terminate termination size dictionary precisely fraction vs specify theoretical show amp decoder aim highlight similarity amp derivation remainder index node update obtained iteratively update pass traditional amp cf message eq
applicable result analysis censor treat estimate cause death censor censor proportion could correct proportion would compete risk disease inclusion dependent process realistic effect year disease diagnosis age five reality may start stop survey arise survey give recommendation situation change allow estimate whole population survey participant grant related year birth year
show right seed persistent seed reduce walk behavior observe variational sl interaction hyperparameter persistent seed persistent minimal persistent persistent sl abc l compare abc problem apply bayesian versus dimensionality gradient simulator statistic population population e broad prior problem produce degenerate nature average quantile population peak difference abc sl
directly seven causal path leave stability causal path seven path line appendix stability graph connect relevant edge orient background add fact cause direct edge orient relevant since relevant two path loose stability infer annotate reliability score edge relation gender cause attention attention structural approach exploratory incorporation background constrain produce real world causal topic decade especially since advance variety discovery algorithm
draw black forget sep crcr color blue mark mark mark mark option black draw forget plot table crcr color pt mark forget row crcr color mark option solid crcr blue mark mark mark option forget plot table crcr mark mark black forget plot sep mark solid forget plot crcr mark size mark mark mark option fill forget table crcr color blue size option solid fill black forget table sep crcr color blue mark mark option forget sep crcr color mark mark black forget plot sep crcr color mark mark mark solid fill forget table crcr color blue pt mark solid plot sep crcr mark mark solid fill black forget row crcr blue mark option solid black forget crcr marks mark fill black forget sep crcr color option black forget plot row sep crcr color blue solid fill black forget plot sep crcr blue mark mark option solid forget table crcr blue mark option solid fill black forget plot crcr color mark mark option solid forget plot sep crcr color mark mark solid black forget plot row sep crcr color blue mark mark solid forget sep crcr color mark mark option black black forget sep crcr color mark mark mark option fill black draw forget sep color blue mark mark draw forget row crcr pt mark
computer vision rotation positive matrix riemannian manifold square manifold although proper kde manifold extended graphic shift surface mesh mode later lie nonconvex manifold shift operate application relation pair represent allow live euclidean distance along approximate short graph kde parent root mode neighboring cluster kde rather maximize nk algorithm improve merge topological persistence directly relate use criterion riemannian riemannian center updating shift square unlike shift maxima kde rather local give well clustering update see iteration accelerate variation use type shift estimate gradient step et derive shift obtain matching least square true shift update iteration fast function data original kde give shift criterion ranking mean permutation stop number step functional surface live shift ascent surrogate shift essentially provide nearby specific describe work lie add iteration replace point eventually structure usually denoise ability note graphic literature element represent surface record eliminate laplacian replace cloud usually typically kde matrix keep laplacian smoothing lie boundary manifold away boundary shrink manifold shape handwritten digit along e rescale object volume graphic extension local manifold correct eliminate motion shift project manifold estimate use
audio apply audio give audio learn feature high exploit gmm hmm capture side match feature nmf technique help consider music especially audio vector approximate negative cost column multiplicative hadamard division compactly apply energy shown identify audio researcher decade
generator deep classify randomness direct although connection autoencoder generator understand generator learn approximate great deal progress approach base boltzmann deep boltzmann beyond build proposal take indirect network train recognize difference generator player cast minimax differentiable iteratively perform greedy give careful gradient clear balance network adversarial replace form adopt mmd
account statistical analysis compositional extend currently exist function occur rarely aggregation individual discretize approximated functional number devote cope functional spline turn appropriate tool inherent obtain even spline function get quite logarithmic simplify spline without deep background
state realization form pc coefficient build intrinsic concentrate equation explicitly variation variation control kl represent arise control dependence lastly gp relate introduce originally input lack quantify give way evaluation simulation permit simulation discrepancy determine realization sir eq three pc approximate overall maintain importantly uncertainty separately variable similarly contribution lack simulation realization kde reconstruct problem uncertainty uncertainty quantification generate accurate unable model require lot resource address stochastic
context restrict span view convex nmf nmf active method experiment kernel adopt kernel nmf kernel gaussian embed basis nmf feature aforementione require computing input space pareto c aggregated increment aforementione observe solution outperform exist method map different road recognize pareto optimal solution gaussian three region pareto nmf abundance nmf compare pareto also notice nmf even zero poorly bi nonnegative matrix decomposition simultaneously input derive analyze nmf feature several also investigation china receive b mathematic economics degree security
finally regression n I norm analysis form consist follow transformation algorithm convenience vector operator convention reduction w project project reduction simple suppose otherwise problem z exercise kkt equation z record solution special projection onto simplex simplex
cr cr cr cr cr cr cr cr cr cr cr cr cr cr nh box central mark th th de ep france I une un les r position de par et es par
homogeneous characterize notion exchangeable develop exchangeable partition block nonempty integer block among element block e almost frequency singleton distribute cf random eq independent poisson independent measure convention appear measure describe atom ordinary hazard limit q process collection crp concentration mean combine crp concentration discount bias dirichlet collection ordinary thus merely recover stick beta refer though opinion arise process process ordinary finitely atom appear prove yield give combinatorial random restrict event event token frequency appearance token remainder ordinary measure collection intensity stage finite truncation right ordinary equivalently latent combinatorial primary j j invariant combinatorial identity relate exchangeable exchangeability identity simple worth highlight ibp crp f ibp concentration recover crp recover parameter ibp parameter scheme process beta
component td well proposition difficult analyze work tensor extraction due paper large tensor contain reduce feature extraction increase multidimensional analyze process modern computer curse dimensionality information beyond essential
resp establish sufficient condition replace model move new substitution compare result prior fundamental coverage total perfect resp capture total read resp minimal sample obeys match characterize proportional notably sample asymptotic fix infinity characterize read tight capturing read simultaneous recovery measurement formulation numerous community computational develop unified understanding kind pairwise represent channel transition minimum divergence cut moreover various homogeneous rely metric spectral benchmark algorithm attention pairwise framework broad measurement denote example operator x ax mb analysis concern full left minimax configuration family component spread entire alphabet addition establish namely situation recovery remain pre recovery nontrivial gap away acknowledgment chen thank discussion channel xu discussion theory helpful chen part science grant fa partly support program recall hypothesis parametrize comprise hypothesis conditional error hypothesis simplicity presentation vertex denote edge mind follow hellinger divergence definition arise l w lemma value depend offset input pairwise without constraint maximize solution small index strictly n contradiction strictly
furthermore assessment extreme tensor toolbox provide storage make available evaluation ram true setting create synthetic tensor toolbox matlab standardize tensor sparse zero dense factor value range noisy baseline tensor automatically determine simple record consecutive small previous baseline accordingly expect effective baseline well baseline case noiseless estimate randomize run fig observe rank outperform boost due absence side outperform baseline tractable encourage baseline baseline establish set real table result mining month person communication day movie user user
activation monotonically function negative corollary encourage auto zero activation sparse monotonically encourage expect corollary activation unit thus become immediately negative low average value mention sparsity entail majority unit de activate usually discussion bind h monotonically usage activate couple property hide keep activation straightforward convexity increase turn encourage low pre proposition hence monotonically activation imply reason sparsity monotonically iteration unlikely activation activation relu maxout sigmoid maxout applicable satisfy property
valid security meanwhile procedure must weak correctness comparison comparison consistent correctness proof perfect build outline apply scheme basic study answer fraction record count family release release answer preserve differential privacy improvement show privacy even capable answer dimensionality moreover work rely hardness private query release et al scheme help digital content conceptually obtain hardness private connection certain bilinear recently scheme base record specialize private algorithm private produce synthetic row query synthetic dataset answer way private produce synthetic refine extremely nevertheless synthetic rule possibility preserve structure family restriction g answer certain family place syntactic expense utility theoretically efficient polynomially query unless efficiently learnable class efficiently pac polynomially et show simulate differentially separate polynomially query learner existence learnable polynomial query complexity learnable strong hardness barrier detail even though result computational hardness rely crucially theoretically
state rip rip originally state version remark analysis continue denote q define row level immediately obey rip distortion critical tool due rip random sign rip set suppose satisfying distortion diagonal entry equal obeys least differ two way state side author verify
dependence mutual widely density pdfs yield inaccurate estimator dependence directly maximize function set band limited pdfs pdf parametric density estimator see compute directly perform integration inaccurate inefficient converge various type maintain always
single next hold optimization perform learn epoch momentum initialize sample version parse lstm hide stack embed chinese word embedding speech embedding dimension parse pos tag relatively development future carefully optimize report balance expense apply parse two parse report english chinese parse likewise predict action english stanford sd closest publish split tag stanford negligible non projective arcs zhang tag rule speech tag ten projective embedding english portion english
use word pos window hyperparameter discuss approach idea relation convolutional target close impact distant appear cr configuration indicate full whether use essential jump report text impact small strategy art dataset also suggest text sentence ccc yes yes comparison cr cnn assess embed class noisy bring noise account f cr avoid artificial class yes yes impact last line remain mean class recall
combine fact extract read g adopt google knowledge project explain structure discuss knowledge main technique variable capture describe combine good discuss learn automate project present logic artificial intelligence web represent machine enable agent operate vision semantic web realize particular concept gain relational description relation represent globally people person entity extraction refer name system triple birth etc clear triple refer triple thing birth disadvantage l current base project classify data schema schema machine property regard ingredient knowledge million google engine use entity entity microsoft kb integrate semantic graph search answer service prominent demonstrate graph answer able human graph instance knowledge graph answer decision concerned predicting correctness exist knowledge graph often incorrect relationship significantly ml see describe prediction base construction plausibility triple fact suppose extraction return true place birth store model relate fact infer include linkage object identification matching object refer underlie entity object assume propagate matching decision object pair schema automate entity name store tb thick ai design child vertex
regime method section testing rely fact flat linear reason improve would distinguish sample polynomial show learn know recovery n dimensional vanish form uniformly element take uniformly flat hold instance distribute probability distribution distribution unit weight placing let remark detect plant flat planted distinguish grant dms thank
ensemble dataset recommendation click recommend include multi show without need ensemble accurate vector click recommend extend include information consist extract breast give information uniformity finite arrive learner online fashion slot operate subset belong else create randomly connection record connection attack take recommendation yahoo front page internet news website recommend item iii click recommend click otherwise map user include gender age history give select consider briefly multi armed assign select availability algorithm keep action action randomly take form context reward exploitation high reward type select fig except action version useful reward cost except observe select refer adaptively create ball ball step ball high pair contextual reward action consider group context compute action action linear combination type action high average adaboost goal base classifier action vector perform active adaboost whose label use learn adaboost receive end slot window require
contrary bias obtain combine factor number fm many embed gradient descent learn multi svms tensor factorization mining volume multiple subset generally view task facilitate wide accurate diagnosis laboratory medical imaging
multi relational fact kb triple relational recommender user relationship thousand million entity fact portion since question answering capable generalize acquire missing fact limit question query internal kb correct correctly consequently completion via manual automatic mix divide task extraction new entity kb prediction add formalize triple like head label tail argument triple yet kb contain fact influence entity relationship fact ex entity connect entity rarely connect diverse characteristic present look subset entity indicate triple roughly connection head vice versa diverse big relationship location precise everything link prediction pseudo symbolic prediction logic walk representation recently prove efficient vector act embedding scoring learn triple score unobserve capture allow predict triple relationship share across share concern relationship entity share task score head tail head label label entity generally capacity
term penalty cost set subproblem mathematically tx tx tx avoid instability assume introduction eq solver respect iterative algebraic grid outli plug formulation outlier identify substitute becomes let outli outlier unknown number none heuristic rank estimator knowledge value outlier may validation applicable edge variable also unknown leaving hold set root generate outlier moreover alternative aic information bic unstable rp gradually parameter piecewise efficiently package outli outli greatly increase lasso accounting deviation deviation assign nonzero accord word whose outli outlier complementary estimating outlier element whether outli path instead substitute estimate clean pseudo consist annotation outli pruning detect solve outlier outli vector unify robust prediction identify outlier globally advantageous majority voting image compare correct ranking term among pair majority four however globally clear outlier deduce detect outli specifically estimate vote voting
add template improve dependence accuracy bin template depend transformation curve flat degradation bar template bin template average template bin random map raw increase template close matching template split remark brain machine mit theory invariance signature transform analysis defines haar integration kernel invariant uniformly define equivalent sample learning algorithm encoding similarity unsupervise image state performance task label example well augment reflect virtual transform identity
write exploit statement label mkl classifier elastic constraints rkh hinge net involve go minimization jointly elastic net attain minimum elastic tight exploit disadvantage convex admit phenomenon occur example require
word category leverage sampling belong category music category bss select word select leverage score document pair pair top closely select relevant word library relate fix datum synthetic know unknown ran bss leverage ridge risk full bss sampling sample observe risk full bss score almost bss leverage provably accurate prior bss
likelihood step condition modification construct kernel acquisition attain cumulative regret acquisition z could approach explain recommend reader proceed sequentially outline desirable place additive note additive hope additive decomposition explain mean query allocate group solution suffer instantaneous
figure parameterization distribution detail unimodal example satisfy restriction property beta meet parameterization parameterization unimodal sm sm spread tb purpose unimodal parameterization unimodal necessity specialize necessity parameterization necessary unimodal spread ax ia b would unimodal property recall property come specifically complicated proportion covariate limit become flat spread undesirable regression parameter patient thus encourage spread towards let parameter shape patient patient unconditional light equation prior examine simulated examine fit model first choose value simulate observed cancer dataset calculate median mean plot absolute simulate
query text query auto encoder medium validate powerful I text experimentally achieve outperform utilize reach wikipedia please image nd although image failure text description query text text experimentally result compare even pair media retrieval good performance work cross medium projection different cross medium couple jointly optimize modal text
remark generalization elaborate require whitening iteration component overcomplete additional case tensor set impose vanish affect translate guarantee overall require activation also weight sign identifiability vector ambiguity note integer add phase avoid ambiguity bias mild simplify presentation suppose assume density mild overcomplete nn lift number sample lift satisfie neural network complexity state estimate overcomplete layer idea product possible additional order approximation approximate combine train neural generalization neural fix
leverage great demonstrate noise singular cost cost seminal capture bind feature symmetric spectral cluster newton section define decomposition singular manuscript let denote ambiguity either entry diagonal zero linear let frobenius qr let matrix column triangular let svd eq principal semantic spectral cluster interested singular truncate svd singular vector analogously close eigenvalue decomposition semidefinite eigenvalue svd exist full rectangular matrix rank pseudo comprehensive inverse pseudo b complexity operation list multiply sparse qr decomposition assume gap inversion eigenvalue decomposition pass efficiently entry visit comparison spectral compute go memory pass place volume ram constant ram otherwise ram disk highly expensive manuscript visit
target train automatic alignment denote default scaling set use state translation vocabulary parameter architecture implement variant architecture design default less comparable baseline default architecture phrase indicate indicate good model behind consistent translation remarkably sensible design clearly architecture baseline set baseline less suggest deep architecture capture representation essential address reading sequence chinese english inspire stack read nonlinear design parameter layer transformation complicated machine translation distant language propagation text
modification summarize inference able graph gradient gradient update matrix adaptively minimization algorithmic jointly sampling compute jacobian deterministic map flow average flow length latent overall quadratic making overall competitive large distinguish two flow mechanism differ jacobian linear computation jacobian design jacobian determinant posterior flow finite component nice easy compute form jacobian triangular result determinant transformation capable flow alternate forward nice general partitioning separate disjoint nice nice factorization hamiltonian variational
vector although limit toolbox way thing state class add entity minimize engine parametrize parametrize affine ib stack corresponding matrix reference pass pass reference external I design reference step procedure histogram membership engine parameterize reference definition object type variable toolbox reason perform translate representation matlab pass optimizer order use histogram calculate engine matlab engine genetic ga initialize optimizer variable optimal differential constraint engine value optimizer matrix type non high dimensional histogram computed bin reference suffer curse dimensionality approach shannon differential variable define absolutely probability abuse w source privacy example netflix security call knowledge inference care criterion
column ng task rather show aspect understand reasoning proof beyond model text highlight hope fail motivate publication camera ready review material build nlp semantic preprocesse costly stanford system mention role label run argument rank support fact supervision support fact less scoring function exhaustive unlike greedy scalability fact match construct simplicity look sentence gx gx word one indicator sentence pair indicator indicator argument support fact similar structure stage tune pair indicator support fact pair external resource perform hand build worse support fact many mistake greedy important external resource l c weakly c supervision support resource lstm single support support
u formula pseudo mapping realize degree polynomial tuple tuple recall whose except coordinate also vector th chernoff support z exist fully determined namely polynomial coincide coincide whenever hence eq lemma support requirement let pseudo pseudo distribution degree polynomial tc gap majority odd kn exist efficient use assume first randomly tuple equally size
spherical normalized symbol establish connection spherical admissible spherical formula description easy possess possess perfect localization reproduce hilbert rkhs space spherical observe without surely generalization q minimize integrable setting scheme let measure enter competition nontrivial impose potential embed theorem employ easy truncation employ truncation name capability associate exist depend eq several remark real world application
quality alg alg comparable plot right consider parameter determine log likelihood smoothed nonlinear hessian approximate simulated linear linearization log
representation fully hardware circuit integer multipli circuit constant multipli shift multiplication power circuit bit minimize input yield accordingly multiplication power two architecture parallel delay flip align constant quantity identity expression notice summation grouping eq require simple block fig employ integer facilitate usage integer find quantity z real satisfie minimization rounding introduce difficulty solution non resort limited space thus could integer require hardware optimal relatively error write imply expansion integer multiplication efficiently implement hardware consider elimination multiplication shift require amount eight five calculation represent coefficient multiplication one accord bring encode cc multiplication stage dct typically employ several architecture depict eight test measurement cm cm cm percentage
information randomly error table bold run fast however finish within example dimension datum run adaboost rna yahoo rna yahoo agnostic setting give agnostic boost error adaptation enjoy flexible variety weak improve communication efficient prohibitive promising result world dataset acknowledgment nsf grant fa thank amazon grant amazon service call
without prior linguistic structure manner synthesis decoder synthesis whereby image embed decoder decode identify sequence give world decoder alignment technique translation machine action learn language correspond action observable learn produce previously unseen arise fact numerous aware agent action within virtual consist blue intersection explore ask another overview virtual world overview path
probability aside value two fx unbiased omit copy fx fx keep cost limited use operation outcome quickly become computationally prohibitive fall point vector expansion instance solve program simple sequentially rkh demand suffer lead expansion way approximation usually kernel map almost orthogonal expansion carlo e effectively choose lie retain vary information thus approximation think rkhs perform conditional mean
solution analogous heuristic dual allow bound analytical possible associate construct within equal equal agreement labeling consequently define statement ip labeling agreement assignment primal minimize supplementary agreement condition maximum agreement labeling reconstruct gap trivial still set consist minimum lagrangian non zero situation binary labeling consistency lagrangian label horizontal subgradient define cm horizontal dotted typically start agreement coordinate dual exist satisfie summarize material pairwise potential case mrfs lagrangian dual standard relaxation one tight bad mrfs relaxation uniform label relevant later order mrfs mrfs maximum equal relaxation energy lp relaxation term elaborate lp solve mrf start two lemma program eq lp supplementary material problem contain empty interior problem equal direct finish
continue q choice discrepancy account statistic critical computationally demand large induce sampler rejection carlo rejection simulation discrepancy accept otherwise particle accept rejection instead weight extend adapt particle round posterior smc generate differ completely parallel inherent assumption call generator produce random
ucb action execute enough time contextual bandit ucb context fortunately context pair ucb action ucb round kt follow find supplementary material ucb logarithmic cost context ucb rank contexts error action lemma ucb ucb ucb algorithm expect reward context ucb may decrease depend actual difference horizon k kt j tt bc kt next ucb ucb
integrate rank centroid extend scale hundred thousand challenge site one various challenge goal boost classification east name base xshift east arc cycle name xshift west challenge aim assess performance classification scale class hundred thousand along detail track quick
switch make optimize test latent classifier compare point predict training choice compare choice posterior switch comparison number axis latent accord dimension significant th similar according relate different relate potentially look object lot illumination etc purely datum might difficult representation resolve people name text collect wikipedia retrieval need
thm thm reconstruct quadratic case isotropic recover computationally convex sub gaussian general radius converge measurement initialization radius believe initialization global prove broad acquire quadratic hope recover complex type optical record understand independent phase perform map
unlabele I marginal possibly similarly issue domain distribution mixture source domain denote true gibbs domain pac learn learn domain try vote situation life long treat prior one generalize disagreement definition disagreement source j easily extend theorem prefer clearly tight disagreement source notice various obtain empirical particular sake result present set share denote hypothesis hand together final stand equation hypothesis real deferred appendix bind disagreement generalize theorem joint domain equation indeed build pac adaptation hypothesis define theorem respectively theorem generalize important theorem prove distribution possible another kind detail discuss section optimize adaptation domain source uniform probable possibility create source deviation algorithm secondly difference pac
sentence skip sentence classifier compare task skip task highlight representation skip thought objective yet include encode likely case exploration quality representation acknowledgment suggest skip hill xu valuable comment work cifar google grant unsupervise distribute continuity text book train encoder try sentence encode syntactic thus allow expand million extract sentence sentiment tune skip generic consider difficulty word vocabulary encode sentence sentence wikipedia unlikely book word train learn bag word
use inductive lemma use simplify inequality third cauchy schwarz invoke whose assumption obtain inequality schwarz give term consequently bind simplify equality convention index use obtain consider hx inequality x use disagreement coefficient number query upper w similar h h n disagreement region corollary immediate condition hold epoch make statement since clearly claim epoch condition far disagreement one recall disagreement inductive yield rhs induction observe clearly exponent corollary satisfie plug statement yield disagreement begin violate oracle follow framework solve x derivation equality hence third equality use hx expression suggest find cx cx cx drop subscript cause dual except instance increase plug dual p primal rescaling never cause much streaming agnostic active particularly noise I overview literature assumption place source common whether never decision active concern set agnostic zhang require explicit enumeration classifier imply
total observe require om fail probability lemma fail sum exceed bind calculate finally hoeffde expectation nm proof closely row pick mc np follow lemma inequality frequently note adjoint xy xy xy xy xy xy lemma simplicity notation section exactly via relaxation leverage small experimental analysis logarithmic discussion edu many
ignore particular I try prominent compare subsection explain turn random forest heavily tree say dependent split independent want put subset want arbitrarily want make come keep recursively subset fewer well bad generalize rigorous tree look like extremely concrete look like split observation independent splitting point immediately grow tree subset contain subset number way criterion split stick leaf year year eight group people eight year year conventional non interactive order freedom decision however course life big hundred thousand suggest forest tree population decision simply conventional perturbation drastically random prediction outperform conventional rigorous try less forest would produce
q thus denote line cauchy schwarz verify formulation top incoherent extensive tight certain world scenario far noise immediately result rest almost current nuclear investigate future thm thm seminal assume pose
expect attempt tend limit behaviour trivially ode result presence several noiseless instance advance necessary bandit range application finance big introduce optimization use forecaster gain appear course strategy realistic probability estimate convergence practical number framework clinical trial instance arm step see definition literature optimize allocation contexts book therein know several policy rely tradeoff point sophisticated confidence sense develop paradigm exploitation recursive reinforcement learning rely penalization penalization omit ns possess imply difficulty ns convergence two ns background ns knowledge context question ns competitive viewpoint lack ns viewpoint section ns section one modification robustness uniform
smooth generalize softmax ie generalize operator consist take coordinate b additive inner bias omit bias design drop implement soft maximum analogy single sec one result hide unit template denote output fed rl h rl l maxout maxout generalization suggest rl l p pl p generalizes coincide maxout negative ii try maxout whereas mlp implement predict maximal mlp machine machine exponential replace linear similarity abstraction level mlp multiple engine
range popularity limit future research direction achieve statistically correct block theorem prove remain appendix establish result ok tv uniformly svd nonzero entry svd bernstein deduce lower ok applying definition hence u nr bt ball center intersect proportion permutation center note assumption speak negligible proportion center center away define intersect jt ct c deduce j deduce mis ct prove conclusion finally go establish mathematical contradict therefore definition exclusive l bound contradict therefore l contradict argument adjacency independently svd n u first np show la q satisfie supplement achieve block university university nk method I ji
empirically improve prevent entropy backward series parametrize active moment nothing keep perform easily mutually component nature distribution minor artificial seek satisfy principle structure distribution bethe graphical studying entropy
belief good reward belief good reward choice log reward ascent map assignment randomize line perform randomized match efficiently choice unbounded represent vary wide initial hard guess uninformative equivalent maintain variance belief attribute reward correspondingly uninformative
parallel author go drug detect international collecting penalty notably empirically subset detection avoids threshold calibration penalty logistic good maximize bic unfortunately huge exhaustive compute bic compete perform use find discrete maximize bic develop efficient advantage paper roughly drug procedure observational medical outcome
irrespective close crf hdp online create offline hdp crf principal remarkable improvement transition mainly learn encourage hdp translate l l cardinality ex c seq seq seq ex seq ex seq ex seq seq ex seq seq ex seq seq seq seq seq seq ex seq seq sequence avg avg crf avg avg ex avg avg ex whereas one human order contain action blue model colour adaptive cause improvement performance infer actor actor ex actor one comparison show accuracy decrease gibbs sampler mix execution mix emission log shown sample contribute trend rate run value prevent immediate tendency fit change evolve similarly value ensure towards transition hdp adapt change stream cardinality
day day day day core day core core day day day approach heavily consensus entire transpose strategy node solve massive without put simple avoid inner loop consensus demonstrate efficiency classifier tb distribute optimization function store distribute across stack store problem admm build solve optimization rather solve sub involve transpose availability enable entire application way large optimization current art support result smooth decade poorly sublinear stochastic
essence predefine half guarantee decrease provide polynomial hold result decomposition learn accuracy numerous figure iterative update execution execution base introduce decompose memory operation computational bf thresholding approximation normalization carry base vertex base model update decompose base store array compress column format dense sparsity eigen column uniformly node partition number column divide allocate locally column send central next central node usage zero input computing usage multiplication communication edge matrix non value store store format node memory number correspond denote bottom layer layer edge
definite si cdf design fy gx subject equation get hence content regularity size non complete matrix hence complete worth case complete complete denote matrix theorem superiority sample within family linear location scale simple notice influence observe considerable gain investigate fix effect number counterparts family pdf q cdf member value content complete
family ari scatter show assess family implement package five obtain package measurement front width length middle depth blue mixture equal freedom outlier add variate point replicate model fit perform minimum show probably unable increase decrease small mixture gaussian approach suffer extreme gaussian value outli third lastly benchmark diabetes commonly illustration literature bioinformatics body age weight individual package measurement observation observation age specie observation package lastly microarray round package sample correspond four include gene www methodology herein high line datum suitable known top ten rank
collection randomly series hold collection ten time plot region growth black dot average result hold dataset show subsequent one achieve baseline limit collection agree mean increase alternate previous add relative namely task amount associate previous randomly cumulative contain ten hold dataset contain k task network time datum point hold perform present appear decide across entire combination point realize enough repeat experiment random seed experiment axis
stock skewness c ba realize ba ba fit factor package var namely schwarz choose l j c report norm sn spectral order sn sn inverse sn sn inverse inverse h convenience part proof theorem easily order eigenvalue q prove entry go prove first result prove part hand equation ft equation need ft ft f p b mn mn ft denote ft pa calculation ft ft pa b ft
well result acc rbm gram output lr classify document topic rbm result compressive lr gram vs I achieve achieve achieve slightly amazon sentiment vote dataset green dotted trial reasonably comparable sophisticated achieving automate text number etc optimizer additional feature option like grams nlp choice need train hyperparameter choice need whereas amazon except tf nlp trial researcher nlp choice
bandwidth cost energy access coefficient network costly connection neural network hz access envelope typical mobile device prune large network run mobile device operation bit fp fp connection manner preserve original phase connection threshold dense phase learn connection remove phase prune reduce network much create pruning little connection network typically
increase kl quantify kl divergence divergence tb b concern exist comparable typical
record value process edge connection assign purpose patient patient determine patient fig connection group connection control suggest patient tendency graphical kullback leibler improve exhaustive heavy burden notably long require allow derive rate control intuitive order test conservative offset detail procedure methodology investigation additional scale calculation hardware appendix specify diagonal modulus unity replace reconstruct modify control support multiple use kullback divergence statistic
image east north west rectangle thick east west rectangle thick thick rectangle south north west thick high image compose slice slice highlight compression sc identifiable numerical see detailed explanation high low sc impact use sc decrease original highlighted green lr multiplicative sc sc active sc nmf believe low rank technical datum explain enough accuracy seem periodic compression counterpart compression introduce visible compression fast three interestingly row north south south north visually sc introduce center duality bottom noisy compare sc daily arrange grid since form nmf active time center structure reach visual inspection classical nmf popular network contain particular type character book book appear
weighted average gradient remove optimisation keep lead low validation comparative boost performance compare capture patch slice patch slice scan size autoencoder architecture sigmoid reduce pool stack slice output comparable architecture possible input connect parameter try structural modality ad support area relate sample size study dataset protocol table kernel grey matter
group user profile cluster centre news social medium seem finance music reference rely mobile activity purpose phone finance finance seem generally ad average mobile video social medium category particular user less profile interact finance ad video complete show ad ad finance ad finance suggest finance ad aim complete interaction display profile likely news user mobile device education mobile device dynamic ad video less finance ad video video likely video user able ad finance ad interact end percentage user tend
hour horizontal km resolution hour daily wind speed daily member day wind minute hour daily maximum speed hour frame forecast verification period forecast case period equal day purpose period compete tn extreme tn maximum estimation sample exercise period report tn reports ideal nc perform poorly quantile number exercise focus exercise bayesian linearly combine forecast field macro economic routine central finance asset strategy base moment operational weather research shift combination prominent calibrate moreover traditional scheme nonparametric calibration mixture transform pool develop know dirichlet extension
loss logistic use modify accordingly sdca loss sgd extensive validation step mini batch observe omit comparison variant mini data global modeling ensemble forecast air pressure result pressure point two normalize prediction error reference cf pressure differential north top temperature pattern project projection introduce recover coefficient prediction estimate return solution attain accurate
purpose graphical et class covariance consideration certainly necessary furthermore compare frobenius priori diagonal requirement diagonal know missing divergence study low case corollary reasonable scaling hamming graph meaningful explicit constant tight generalize consider illustration analogous via express convenient generalize statement theorem structure
liu se liu se receive project time tail posterior skew measurement filter smooth conventional low alternative criterion skew skew two component mixture gm histogram despite tail asymmetric miss robust heavy tailed propose
measurement measurement goal find condition support either study limit study determine proceed capture cover take important role non linear several practical variety pay attention bit equal negative interest limitation capture recently small item within subset biology indicate contain denote think work information theoretic limit focus model necessary sufficient condition vanish however vs support recovery class vs combination recovery non entry decoder know goal derive converse strong introduce terminology hold compressive sense literature generally write place emphasize eq counterpart fold appear distribute remain notational convenience part analysis average state function pdfs integral necessary apply information theoretic definition brief technique code introduce recall context directly logarithm threshold expression use yield capacity refine analysis mixed channel conditionally figure proof mu partition condition empty left still allow introduce precede definition letter work discrete clarity exposition ratio average respect condition mutual q play make subsequent exact counterpart former throughout low code dominant involve tail probability information density mutual thus subsequent sufficient showing deviation specific start proof
node bt fitness bt execute fitness change within compose randomly subtree fig add bt greedy search find action subtree subtree whole bt execute action increase fitness value find subtree bt process continue action gp subtree fitness iterate goal unnecessary node bt apply anti address ap represent bt bt satisfied environment use agent robot obstacle similarly game character front reach point collect etc algorithm present pseudo perform aim learn framework
hold adopt delay architecture dependency language widely report outperform rnns rnns need propagation due recurrent feedback complexity many vanish architecture solve lstm enhance implement recurrent use learnable promising sequence model recurrent handle gradient vanishing add make learn along language simple however
rare tail organize detailed give comment context motivate lastly conclude miss dependence dedicate every impossible pac missing effort parameter divide parametrize choice proof tail miss methodology outline subsequence depend choice proceed induction exist least induction infinite readily mostly set simultaneous sufficient induction satisfied therefore case fail choice
annotation bring annotation multiple split mm discover time video event often time parse miss constrain output goal evaluate localization car validation adjust keep entire dataset annotation thus evaluation average localization manually map table ground truth evaluation evaluate whether every truth correctly detect evaluation video predict exactly interval correct fall truth incorrect recall across video correct possible truth truth irrespective recover recall every video happen detect positive demonstrate uniform entire second video discriminative class strong third video produce sequence score discover present illustrate difficulty baseline
dataset additionally label prop corollary neural recently excellent high classification semantic segmentation segmentation convolutional provide wise feature traditionally semantic show encourage convolutional achieve variety vision human input assume independent identically place score generally repeat forward pass pass function rule exactly configuration typically refer sum possibility convexity efficiency mini summarize criterion
structure compositional derivation improvement lead expect head improvement sentence mean representation thereby tree design syntactic natural language believe paper lstm use guide interpretation case syntactic handling length limited recursive structure generalize
majority remove central goal methodology normal anomalous requirement normal inform semantic employ distinct choose candidate select anomaly class instance final benchmark subsample candidate anomaly anomaly constitute majority define candidate anomaly class benchmark anomaly variance anomaly great along low semantic variation single difficulty create exhibit point difficulty transform problem treat transformation compute regression thresholding extent generative distinction anomaly point near anomaly benchmark derive regression control point near median cluster near varied dataset maximize many impractical class employ approximation begin forest multi point compute estimate confusion whose unnormalized vice compute tree color color maximize confusion normal anomaly tend semantic anomaly many benchmark allow flexibility choose anomaly despite difficulty nature original partitioning prove difficult distinguish benchmark four pd relative frequency rf set measure bin correspond choose level iterate level benchmark limitation
recall strictly solution unique kkt therefore solution update partition satisfie variant constrain q find unconstrained proceed iterate decomposition parallel intersection study coordinate present characterize real penalize tensor method
division automatic mail liu se identify possibly system monte carlo decades numerical solution arise identification solid system strategy create implement strategy identification discrete time consider unknown distribute notational loss drop know consider model unobserved random place identification wish difficulty state state illustrate algorithm concrete formulate illustrative gamma involve real consider modelling ice year change ice location year bc description appear state quantity interest case one density computable investigate possibility illustrate strategy integral solve direct optimisation rewrite integral typically optimisation typically denote search tell search direction positive
set pressure contact recognize daily activity people home recognize recognize base rgb data fusion evaluate benchmark compare art perform well approach efficient summary contribution paper crf activity level outperform open paper address research question add activity add layer activity state generalize organize describe work formalize previous single layer nature method activity recognition particularly depend complexity duration activity category hierarchical approach recognize human activity hierarchy simple short require activity category activity approach level activity infer et video node interesting activity object people interact node inter object across enable
vary pdfs trial table table rule correspond pdf pdf time second normal mixture density mixture pure gaussian density cost second complexity unimodal density case skew outperform multimodal density wrong skewness outperform asymmetric multimodal option unimodal estimation feature ghz gb ram windows matlab ccc pdf estimator number bandwidth interpretation log plot dash indicate dot
function representation transfer evaluate already compare mae previous image mnist use linear provide validation image view view right testing view hide unit hide contain notice perform well correlation representation representation document classification language art performance word representation language contain vocabulary word vocabulary bag achieve datum encoding word correspond word follow linearity view column act vector embedding word word source target pair like language align translate representation column sentence ensure bag reconstruct binary slow million sentence individual bag propose trick assume list bag adjacent sentence simply merge mini batch bag result per epoch divide mini batch size experimental trick good representation
use go size size distance small indistinguishable depict phenomenon synthetic plot empirical large sample represent range depict trial bar repetition vary observation science stanford california usa stanford closeness testing sized sample target draw distinguish case size result resolve question practically informally element successfully probability size tradeoff smoothly sample necessary natural
count efficient method community seminal effectiveness rely prior successive provide insight observation recently inspire multiple illumination retrieval priori signal semidefinite gradient alternate importantly remarkable guarantee noiseless prove phase retrieval regard retrieval result establish literature establish noisy retrieval establish however optimal imply establish minimax optimal noisy sparse exponential novel thresholde vector value idea naturally light
ram run mac os find bf accuracy computational bf tree ct fast train table study much offline bf tree version test error bf ct dna mnist protein bf bf nn r ct rf bf bf core visit online nn rf feature recommend repository p r data bf rf ct mnist classification table bf similar error naive emphasize arbitrarily number scale
sis fr iv sis performance seem screen extreme vi poor iii cause residual weak predictor select impose strong achieve satisfactory screening sis one eliminate poor sis iv vi improvement remarkably remain good simple superior structure screen strategy achieve yet fail extent structure feature adjust sub size unlike limitation sub choose estimation select probability improve ten forward lack degree marginal require sis sis important predictor jointly correlate marginally uncorrelated specification verify marginally set true plot well sis htbp counterpart separate guarantee claim convenience
appropriately score arm k h randomly choose visualization purpose leave right gate propose exploit kind complexity inspire elimination armed bandit implement criterion exploit sparsity algorithm successive elimination sparsity maintain active arm winner choose bernoulli denote define I quantity p quantity p idea definition winner guarantee eliminate become start remove exploit distinguish set arm terminate input solution one least return constant would
deal function condition upper demonstrate passive noiseless noise effect threshold together behaviour beneficial even label provide feedback correspond query feature feature value help analyze happen situation include sensor corrupt source storage oracle well study literature cause minimax become instead see deconvolution estimator uniform represent observe difference start feature determine intuitively request return generate address literature conjecture qualitatively model rule classification rule classifier regression
look easy would extension softmax suppose clinical survival informative survival aim age environmental value death cox proportional hazard patient tumor collection sample supervise naive
efficiently finite denote computation frequently machine unnormalized node potential unnormalize potential examine scalable metropolis mh slice markov runtime could monte proper mixing scale propose novel sampling efficiency mab finite trick attempt contribute library sec unify subsampling sampling discuss variable discuss work sec model conclude domain normalize cdf solve method complexity save computation avoid consider unlikely discuss armed mab bandit slot
implementation together dynamic variant remove dynamic network estimating planning variant latent thorough complicated nature autoencoder autoencoder train image autoencoder refer detailed architecture e latent iterative plane operate except experiment control perform predictive horizon give pass encoder state trajectory optimize trajectory state cost transform cost offset direction circular move bottom white cost additional closeness inner sep sep pt line corners east control latent show figure also clearly fundamental advantage autoencoder fail underlie cost say space test accuracy trajectory whether reflect reality start action evaluate reconstruction reconstruction difference accumulate superiority globally
reproduce rkh empirical calculate universal rkhs canonical induced suppose mmd equivalent mmd achieve simplicity represent v obtain base hypothesis learn suppose surely decrease subdifferential zero claim subdifferential aware divergence dependency measure estimation convergence divergence aware reveal mild erm importantly aware dependency measure divergence recently increasingly technique
subgraph density marginal factorization follow child variable tell characterization dag use restriction rv mm node mm xshift perspective property restriction size property variable figure represent dimension pairwise parent inequality model b strictly constraint property obeys obeys respect subgraph property respect apply criterion markov figures second criterion vertex give marginal close without child nest markov sound marginal hold latter result ccccc lastly eq intrinsic child obtain intrinsic consist random vertex connect intrinsic intrinsic reach operation strictly small say strictly small reach use operator eq collection index eq precisely appear odd difference fundamental paper exist connect
covariance column eigenvector know know equal singular initial qr classic call note computation first closely one overcome case n attempt work potential epoch constant start epoch would establish epoch know constant epoch I rank crucially function nice
u k k kn could specie abundance specie discretize fast convolution make algorithm taking grow big form fairly likely significantly inherent optimize implementation exist utilize individual variable nn multidimensional discrete third parameter convolution information sum nn integer kronecker chance reach tree without change sum n ni j qualitatively max product consider joint event hide forward event model define path complementary advantage weight high however also high likewise product mutually exclusive
boltzmann become rational case adjust make rotation rotation boltzmann rotation discrepancy well machine
conjecture tree lstm tree lstm shorter aggregate dependency lstm cc word similarity cutting child gate child gate play play lc dependency tree cut play ball play front crowd set example tail window list neighbor retrieve rank dependency lstm cosine vector sentence lstm desirable query sentence root retrieve related word preserve emphasize distant great robustness play tree lstm phrase play phrase front crowd overlap phrase sentence
work inverse inverse property expectation corruption correct reader corrupt let pac one markov kernel ap probability erm finite uniform convergence optimum versus clean ratio versus clean final inform economic acquisition wish quantify comprise corrupt example clean generally make ik I ir ap hold appear follow low pick budget far erm find corruption occur constant well answer develop le powerful vc present differentially private set evaluation convex generic statistical tool
energy treat limit technique mention computation information information encoding define understand true expectation entropy determination entropy subscript parameterization collect mle minimum cross equivalent interpretation approximate probability discuss validation validate independent identical predict parameterized failure loss degradation model cross validate estimator entropy parameter encode dataset whenever discuss estimator evaluate mle bias equal bias
stop testing reject note value one reject rule become general guarantee fdr discuss special stop deterministic control p adjustment aware analogous introduce broad call significance level make therefore positive effect large level hence particularly hypothesis truly nan number step whose likewise leverage knowledge truly nan arrive namely yield increase power study simulation ensure exploit choose sequence decrease arrival truly pattern batch procedure fdr fdr never fdr discovery use outcome previous
foreground image unary potential pairwise potential image otherwise regularization parameter diagonal see submodular entry perform base experiment segmentation field sdp mean pixel default setting limit prevent converge undesirable local optima gb union segmentation illustrate achieve accurate segmentation field bad demonstrate variable time method field quantitative method energy mean field field sensitive energy initialization field improve show c kk drop several bind dual sdp stable field sdp rank product make
apply type key behave np instance partition formalize criterion notion behave instance type focus notion complexity cluster useful paradigm term vary discrete np whether hardness meaningful argue wish extent matter thesis median start think requirement notion body section notion meet requirement strict condition imply point center distance center analyze notion list distinct cluster least stability cluster imply cluster point center center vast center least conclusion currently thesis non parameter rise dimension publish allow unless imply open opinion proceed begin state requirement notion satisfy apply support thesis
work recently propose base thing utility end useful contextual bandit sophisticated sg distribute sg way algorithm minibatch parameter explore investigate confident example discuss acknowledgement sharing code grateful comment help corollary theorem pt pt com university neural important density application online monte several problem posterior multi skew predictive integrate run double parameter limited mobile
moreover link plant clique scale analogous tractable identify transition mmse region amp wise present generalization evolution mmse distribution pass evolution reduce amp simplification correspond q derivation later graphical propagation expansion mean simplification corresponding depend weakly exploit lead lead amp
refined open understanding trade efficiency online generally recurrence weight immediate weight readily think efficiently weight difficult efficient optimization observe product pseudo inverse readily importance recurrence observe copy construct go ahead paper main technique nevertheless recurrence crucial constructing truly project es thank ari thought unit permutation step use minimize perturbation q define since prove martingale var b xx ok minimize
adjust prototype various kind similar share characteristic ridge intersection manually characterize pattern synthetic prototype intersection segment connect edge extract please search control minimize accelerate summarize case prototype digit prototype digit prototype digit project prototype image image evenly boundary detect prototype image map digit generate synthetic image intermediate image prototype generate intermediate transform synthetic prototype step image close boundary pixel intermediate situation algorithm image prototype synthetic transform update status converge interpolation
category plot unique cause cumulative computed profile dynamic computed event longitudinal collect n x ij z ij si si si longitudinal membership survival define section finally cumulative incidence incidence eq cause specific instantaneous hazard cumulative cause incidence numerical integral gauss quadrature perform vector compute distribution approximate confidence interval base accuracy compute risk history cross validate estimator accuracy also compare compute track compute individual random subject identify aim investigate functional collect along visit diagnosis visit diagnosis education gender case mmse mmse package analyze mmse replace minus divided center division model effect effect create display mmse age age age implement next age ng mix fit age age subject ng process goodness fit maximum likelihood longitudinal se value intercept age age intercept se residual subject default number convergence number criterion converge correctly criterion log table estimate effect display residual along standard change age formally use multivariate pos age intercept line estimation model latent value age subject ng mixture ng age age
close strong hide strength intervention increase setting obtain unable cope also bad absence variable union network reflect edge reconstruction stability detail thresholded intensity reflect magnitude comparison panel edge illustrate procedure publish external differ environment several nine different contain roughly measurement agent single setting observational establish figure thresholded retrieve edge find study edge three discover stability one notably feedback loop validate extent mostly think agent rather mechanism whether check plug
hence useful encounter may diagnostic bioinformatics l france centre paris france paris france contact com characterize environmental challenge read volume operate base compositional approach assign fast potential generation sequencing read profile base sample genome increase reach implementation scale competitive well establish alignment involve moderate species genome nb simple implementation svm need limitation competitive svm investigate however also raise involve million represent compositional modern learning carry toy demonstrate necessity consider method extensive realistic investigate compositional sequence compositional profile count occurrence letter profile dimensional although
spectra definite decrease gain new negative apply eigenvalue positively plant future worth also propagation bp bethe bethe alternative call uninformative assignment local bethe operator partial detect soon available represent edge infer latent recover
note equal iterate prove assumption pointwise function compact proof chapter proof sake completeness almost point view uniformly relatively follow follow z xt nk nm nm tn xt surely form
still need logarithmic epoch epoch exponentially noiseless sign use high deterministic sign round drop place precision etc minimize exponentially rate relax exponent strong consistency bad rate lipschitz assumption hard rate coordinate smoothness budget adaptive know recently stochastic adapt adapt uniform convexity strong convexity special also idea learn threshold paper recently explicit field
thresholded specifie estimator six fig recover critical value fig h driving spatio matrix six recover sharp threshold threshold appear theory predict critical sec empirically common norm frobenius error covariance arise finance array adaptive spatio likelihood ml maximize ml ml ml commonly use yield positive regularization suitable penalty penalize prior posteriori p penalize ml towards penalty sparse toeplitz kronecker call un penalty encourage induce penalize match effect interest practitioner scale comparative penalty model match decrease effective number spatio temporal gaussian model child dimensional snapshot spatio g random snapshot qr qr symmetric definite unknown contextual precision value quantify several specify problem covariance spatio sensor network information physical environment vary law lagrangian mechanic flow wave field
follow claim ok n iy tv first moreover remain statement store exception true reason proof proposition show large list root first nearby root range unless basic dynamic come representative collection let h h end prove induction element achieve clearly true imply claim iii return appropriate runtime element generate final bind cover section desire explicit bind packing follow useful hold root exactly tv follow n triangle suffice note prove therefore elementary inequality find en explicit packing cover must assume generality smaller appropriately pack cover cardinality pack empty tv ij tv tv n empty imply cover fix cover ji section prove explicit fix follow take pdf convenience packing theorem lk j lk lk lk lk tv triangle claim ij statement exploit close ignore coordinate contribution separately firstly namely contribution n lp p iy
dataset incorporate contextual category co occurrence scene performance eliminate false coherent scene unify variable combine pre deep art know learn differ task neural pre train enhanced capturing contextual contextual input allow incorporate effect pre feature capture various category label object work impose moreover make access learn category manner capture scene framework scene combine strength learn improvement art deep learn challenge conditional tree small potential via large
linear noise perturbation tight perspective empirical choose classifier adversarial perturbation achieve upper exact curve zero upper tight adversarial give focus noise robustness noise robustness robustness equal dimension adversarial perturbation besides use linear adversarial similarly exclude trivial label satisfie q impose eq impose follow adversarial moment risk eqs denote upper adversarial task small perturbation case task distribution risk robust quadratic priori possible suggest adversarial
choose contamination show statistical trade operation bottleneck gauss elimination practical naturally computational iterative method section consider square standard attribute explore inversion nr clear build eigenvector symmetric square norm second compute step second stop second purpose exposition compound diagonal matrix large approximate rank likely full inversion newton practice need approach fast throughout
ar ar denote drive source channel contaminate observation since autoregressive process white variance term purpose determine peak observation know sensor variance source sensor end acquire decentralized power hoc sensor realization source sensor nan ar leave depict actual spectral psd source bad form filter feed involve perform inter sensor exploit diversity spectral l problematic fig tt ideal communication apparent improve performance steady sensor cost incur decentralize suitable signal statistical model derive employ exploit physics problem availability kalman see kalman filter filtering scheme network briefly outline initial centralized kf see average correct correct error inherent delay operation scheme vary state communication need consensus acquisition consecutive measurement issue instability decentralize kf approach detailed incur inner motivated consideration decentralize smoothing matching acquisition lag sensor local smoothed acquire measurement decentralize related consensus noise exchange cost decentralize decentralize ks communication reduce strategy exploit redundancy provide individual sensor collect sensor collaborative gain wireless link filter widely statistic adopt decentralized leverage admm track term yet consensus see wireless cognitive agent equip sensor measure available sensor employ
conduct consist object motion trajectory body single cluster challenging coefficient list accuracy first obvious obtain sequence paper investigate extended coefficient cover incoherence able allow unobserved necessity tuning regularization extend work mc moreover model application relate value immediately segmentation synthetic national research china cb cb national science foundation nsf china lin china grant national nsf china microsoft research collaborative program chernoff hermitian assume large expectation ready small probability x ie md definite obviously invoke chernoff chernoff prove latter term strictly well proposition property pseudo problem robust mutually immediately verify unfortunately storage obstacle entry information carry partial great interest datum
prove diagonal dominant hermitian positive integral yield max universit role volume act routine implement multiplication meanwhile solely multiplication frequently determinant logarithm determinant operator introduce log involve expression determination enable keyword current physical huge analyze particle physics scientific field structure carry remarkable resolution order extract
beta wavelet frequency resolution narrow spectrum corresponding haar software especially interface wavelet toolbox kind wavelet wavelet infinitely wavelets compactly support compactly support wavelet wavelet implementation matlab file beta www
disease gene full identification genetic complex disease human genome international project focus goal identify single nucleotide snps disease diabetes trait snps report associated trait wide significance finding genetic disease snps identify diabetes large disease expect study effect genetic variant however find variant identify number multiple investigate tend randomness requirement consuming indicate disease relate mean disease genetic
intermediate empirical weight x n stopping stop stage branching number denote label I q measurable system resample n end sequence relate main sequence weighted framework assumption branching number computation suppose surely finite almost mass deterministic thank framework stop next section conditionally random distribute resp conditionally independent explicit integer qx n proposition average proof proposition proof prove notice proposition martingale assumption q chapter corollary q intermediate conditionally notice continuity property explicitly assumption hold z I first consist system sufficient range function measurable nx get q measurable resample family distribute n iy identity conclude main claim namely z back stop strictly level let stochastic process measurable uniformly nx thank step right stop q equality separate conclude proceed measurable function exactly recall initially independent distribute distribution qp qx notice resample identity hold sufficient define equation finally equality branching assertion direct branching rule indeed branch replica branch iv behavior algorithm various involve langevin formula reaction coordinate know refer realization estimator obtain algorithm denote investigate numerically dimensional situation independent organize illustrate branching splitting section dimension algorithm study parameter recommendation
integer program solve translate specifically distribution fraction total discard remove joint empirical optimization answer conduct check follow let hold theorem detail optimization series distance capture appendix implicit reason produce precise statement extract proof width hypercube increase eventually outer close kl divergence intersect outer practice precise
improvement losse multi allocation cognitive maximize overall quality service user preference dispersion channel channel external quality match channel make round goal sequentially select round learn instantaneous allocate channel loss never problem interest framework formalize learner pick combinatorial loss suffer observe feedback simple observe vector situation learner observe precise arise cognitive measure total time paper allow argument
allow exist adopt form infinity large scheme scenario nonnegative discussion distribute zero use section orthogonality use initialize simplify lemma ta interesting approximate become exact immediately scenario lipschitz make proximal slowly run number iteration getting impose optimum return immediately optimum going time large fig sufficiently function realization parameter number gap previous indirect small slowly small poorly reconstruction may fig mind low reconstruction reconstruction center objective reconstruction constant clearly minor performance reconstruction plot back center objective function realization noise sense constraint use case convergence reconstruction fig simulate activity construct matrix satisfying haar circular mask need collect equally spaced radial implementation lead sense variation known detect lead generally nonzero
statistic fdr fraction claim claim motivation enhance linear apart aggregation aggregate summary fdr randomize rule shot factor theoretic exposition filter term entry number identifiability exist combination sum normalize column construct first equality force correlation construct p augment reference suggest lasso augment design pilot recommend statistic notation ease exposition furthermore pilot lasso square angle procedure
datum use well transition hmm probability considerable reduction miss transition transition transition version analyze know prior close expression normal mode effect distribution family besides theoretical trivially satisfied since side satisfy trivially since side converge satisfy z pdf give I independently value minimize maximum associated likelihood laplace median eq value provide optima write let convex fx increase convex prove derivative convex optimize minimize likelihood strictly mixture provide
population operator cause population predefine classifier eliminate regard fitness experience classifier compute fitness equation cl cl cl bid current choose subset receive environment environment previous action hereafter notable regime environmental discount current rate follow term begin calculate fitness discovery predefine ga ga mutation selection fitness parent rate allele result population predefine threshold classifier fitness improve generalization capability utilize population ga find accurate cover newly cover eliminate population add ga process action sufficiently eliminate parameter cover increase interaction environment one solve must regard responsible modeling represent consequently able set also rule decision classifier achieve accuracy provide proper representation find able regard set responsible rule represent make consequently solve efficiently cover problem properly rule therefore e essential role classifier objective task provide rule representation essential whose main part generalization decision without modify commonly term part simply alphabet easy analyze understand categorical eventually change computer system contain mixed attribute researcher
quasi monte function genetic optimum evaluate population spread sequential fitness evaluation computationally cdf estimate consideration closely integrate error maximize equation sum probability underlie integral moreover choose lead maximize maximization I evaluation require bivariate maximization implemented decide analytic point sequential maximization p locate uncertainty set location point present multiply normalize function mat ern kernel package assume evaluation evaluation hypercube krige realization fine simulation indicator reconstruct realization distance design hypercube show simulation integer space optimize
range original moreover bp figure show mae mae error evolve ahead calculation period second show smoothed mae window length randomness table depict mae dataset ann structure choose output moment eight predict eight step comparison bayesian baseline mlp available information operate ann fast mlp mlp achieve learn lin mlp school competition integrate system consumption database exploit develop ann research project cc temperature min equally spaced temperature utilize show datum mae mae behavior illustrate evolve smoothed value randomness time mae dataset ann receive eight value next short observation period simulate baseline evolve able ann experimental seem promise simple hardware device publish journal kind algorithms able prediction low period make
head act translation rule variant easy implement comparable basically decode decoder decode cube pruning derivation derivation decoder translation probability lexical gram joint baseline decoder contain eight pseudo rule ensure rule decode via rate decoder set rule stack threshold rule design language improve cnn compare generic helpful improve max pool extract pair part long word training sentence million english
usual release responsible advantage short economic indicator price hmm past history often short current stock price discretize random retrieve yahoo historical adjust retrieve u claim adjust retrieve department week week investigate two effectively viterbi datum estimation discretize stock discretize model prior stock price build prior claim counting bin transition function count occur interestingly roughly resemble great form quickly distinct lastly stock use display depict index claim share stock index code viterbi naive max convolution step second use
hand inequality yield combine claim part correspond eigenvalue eigenvector algebra random I freedom analyze concentration behavior notion appendix characterize property sub negative weighted sub sub parameter variable bound f sum sum piece sub exponential exponential find randomize complexity subtle round message bit consequently number degree inequality suggest choose quantity order round integer evaluate rounding error chebyshev expansion construction iy ia universal binomial piece overall bound establishe prove set player element goal additive randomize
service complicated text exist contribution algorithm mining matching learn use match efficacy scale first represent text treat subgraphs dependency tree skeleton distance dependency tree represent necessarily product graph edge v v v direct tree left direct product tree right panel fig l interaction lexical syntactic sentence next abstraction abstraction vertex entity
capture temporal enable recurrence model essential recognition cnn provide baseline baseline ng scheme dataset reason frame pooling architecture another motion convert dense optical flow motion spatially core stream human pose descriptor optical complexity win recognition modal operating feature develop training part architecture learn convolution layer ji temporal lstm combination cnn use font style thin rgb rgb minimum cm text text inner sep fill outer sep cm cm fill thick bend minimum cm north cnn xshift cnn right xshift cm cnn cnn
trend excellent training illustrate recommend automatic use c include potential contour attempt contour direction neighborhood pixel image image since many contour pixel lower occurrence window pixel contour pixel reader
choose mean small loss property test brief exist literature propose general covariate work pearson decade many work develop procedure type statistical develop arguably general ratio classical maximum unstable attempt testing receive attempt
proof target layer start would form transfer function accommodate transfer accordingly solve activity backpropagation view target provide target exactly target depend search alternative backpropagation investigate exist deep availability optimize hold rest availability provide target place measure subset unit case architecture obvious boolean perceptron algorithm exact linearly layer delta rule descent perform deep loop outer inner cycle deep feedforward architecture cycle unit successively layer top hold key whether backpropagation available target use sample online layer produce target sample activity vector different way train proximity perturbation g case logistic transfer probability activation short finally produce vector ideal selecting error minimize ensure target current layer activity algorithmic varied additional remark describe boolean autoencoder differentiable propagation apply directly develop four layer autoencoder gate hamming layer unit layer connect weight initialize zero cluster cluster start centroid additional example perceptron exhaustive since layer comprises plus make schedule cycle cycle trajectory demonstrate target train reasonably armed understanding implement learn capable reach minima deep nature semantic nature hardware possibility either channel essence digital computer use transpose matrix backpropagation thing channel forward electrical backward propagation chemical evidence existence molecular cascade dna conversely chemical electrical short evidence support
avoid contrary modal thick fig rate potential candidate toward assess draw quantify pdf among distance pdfs schwarz reason divergence leibler ease obtain numerator cf side simplified choose gaussian hard verify become th simply last become phase respectively realization center randomly center r term divergence alg discover differ adaptively take recorded check realization draw stay since reliable remain change augmentation whole notice record divergence iteration moreover case alg store calculation draw memory incur complexity close find close centroid capture confidence center centroid high assume independently probability locate
action sequence randomization definition equivalent requirement action extend function minimization let sequential observe choose define supremum function effective regret linearly introduce online gradient q element subdifferential gain denote state regret online sequence regret upper class regularize algorithm additional convergence recall call last bind establish
complex pose acceptable inspection simplify analysis degradation proceed rule among acceptable inspection weight length short task general achieve practice low expect length short path organize program address problem random study
couple problem pl family pf pf align length contact model tp rate pair try pl initialization rate
rigorous temporal difference trace function behaviour single value map rl application chain irreducible lie thus extend field value map yet section thm thm proof first asymptotic fast slow additive markov martingale differential time scale measure solution algorithm parametric function set trajectory weighting develop analysis trace sufficiently stochastic additive markov analyze handle consider recursion control cast stochastic sequence thus
variable substitution recognize multiplication matrix post express compactly nontrivial diagram let pair correspond column addition
ensemble record statistic asset upper record average log return asymptotic distribute return time increment identically independently run record number define number jump run demonstrate interest fully characterize persistence price characteristic z provide constructive way consequence unbiased pr rr symmetry bias random walks knowledge elementary
systematic ergodic convergence way selection row select need randomly select word type proportional token systematic array count type token give word token probability pc complex model derive topic assign cut interpretable topic selection associate increase interesting sampling use variational priori exactly lda define complement I token th dirichlet k prior restriction conditional dirichlet follow except integrate straightforward appear follow pc sampler implementation call ad implementation reduce collapse sampler core three evaluate pc pc word remove rare together whole occurrence remove evaluate sampler respect depend initialization sampler initialize sampler exact state implement version sampler also aim come
potential game inference configuration style minima simulated annealing information game response exploration modern connection game property convergence seem recognize none make inference approximation notion equilibria game mrfs induce game game converge distinction ok game correspond equilibria equilibria considerably case surface probabilistic store equilibrium equilibrium equilibria ce games unlikely leave open possible design interior technique equilibrium idea effective practical model extension ce equilibria structural achieve sense translate connection structure implement ce feasibility system compute ce advantage structure game apply produce algorithm linear useful discussion possibly correlate mixed game hypergraph marginal player joint strategy every hypergraph strategy hard payoff former opposite clique payoff simplify presentation notation refer hypergraph game primal follow representation every equilibrium possibly correlate hypergraph need represent main correlate multi local hypergraph except observation differ omit ce instead neighborhood observation joint marginal issue address zero probability access marginal need set assume correspond solution variable uniqueness problem feasible entropy concave result appropriate derivative must value multiplier consistency clique marginal indicator gibbs fact clique equivalence equivalence ce depend payoff mix lead within payoff equivalence ce induce behavioral determine particular ce mrf infer large variety conditional property ce without look specific ce definition mrf conditioning neighbor player well ce player implementation ce change ce player behavior call immediately neighborhood extend statement disjoint player separate path pass player conditionally player compactly mrf infer conditional conditional summary make behavioral structure equilibrium ce game efficiently
accord type cp pure read multiplication system dags dag runtime program budget mb mb mb parallelism local program e mb e mb cp mb mb intercept mb cp mb cp generic false mb cp rt cp ba mb cp mb cp mb cp mb b mb cp e mb cp ba mb mb mb scenario row column non operation cp main program generic cp false scalar intercept scalar true cp cp false true double double cp cp rand double cp true cp cp double cp cp
u tr tr tr tt batch get p application eq get thm conv execute offer value excess condition yield proof e great achieve linearity v c establish show prove notice rate calculate performance look look measure reward give range give approximate sign happen pseudo linearity pseudo pseudo measure contradict q
vs tf stop occur focus require regularization solution default learn package safe gain speedup interest high gap safe one safe screening much speed see screen active wide slow compute formula correspond net introduce key concept second create rule benefit dense extend group acknowledge big datum program appendix detail safe test
string triple begin triple count size ex array stack depth valid shift depth valid action valid datum stack stack gold valid valid action stack stack last return stack last action stack gold stack gold stack size stack I stack action gold stack gold stack stack action stack action return good action action return search ec task gold gold tag gold tag tag tag ec tag gold gold gold history aspect study predictor train building get previous automatically translate specification parse complex prediction something wrong third prediction exist may wrong ignore train lead
differential privacy extent solve life fisher scoring newton solve estimation solve highly non iterative update linearize score expand rule score covariance mild hessian avoid stochastic gradient quasi advanced guarantee algorithm review section whenever efficiency large scale langevin dynamic differentially change stepsize correlate yet finding privacy explicitly valid empirically well minimization erm solver previous e knowledge comparison family hilbert parameter observe parameter update condition datum mode entire treat rich ignore close expression often monte carlo sample scale combine method equation langevin monte show tool differential
blue grey box prior predictor mi selection select lead decrease grey problem already prediction avoid black box fig causal red causal predictor scheme correctly identify large variable physics institute systems biology united forecasting predictor
rotation another opposite preserve dispersion parameter brief minimize row represent end dispersion rotation double identity q expression expression I lagrange multipli taking equivalent eq lagrangian polynomial equation equation root value cosine us rotation remain rotation conduct explain modify rotation note rotation opposite angle rotation us eq set derivative r explain rotation apply rotation rotation dispersion transform receive transformation identity
analytically normalize detection systematic group fix normalize mutual consider statistically significance value partition entropy put future work refine implementation support institute grateful helpful size wrong expression
asynchronous hardware rate bad hardware compare statement sequential delay additional many interpret need distance noise delay update large q occur straightforward logic use bound asynchronous first hard prove third theorem asynchronous sgd asynchronous really analyze sequential analog turn attention application couple construct rate case exist first case asynchronous sgd moment gradient rate rate sgd asynchronous result difference sparsity structure require absolute corollary precision introduce system
unsupervised know svd schmidt independence dependency minimize misclassification task drive generalize way use second generalize bi smooth achieve task compressive sense band formulation usually set unsupervised code probability stationary
dft dft compute different multiplication dft component dft introduce implementation transform engineering dft sequence number
build architecture discrete component therein block belief mnist dataset network learn accurately version transform belief mode unsupervise mnist report conclusion section hide variable must condition whole ps ps ps ps hide single group variable note marginally component
specify set surrogate relate remove remove objective classifier I stage variant surrogate surrogate include force classify lie outside surrogate level contain know label figure reduce surrogate level htbp f f classifier fit variant additional initial ni n reduce training classifier surrogate f f reduce problem minimize z pt require data reduction whenever obey easily apply lp ip use solver feasible determine surrogate ip relaxation ip width surrogate level discard scoring remove ip compute ip computation coefficient eq discard solution discard high quality feasible solution laboratory use flexibility approach world problem tailor binary feature health patient imbalance pr specify simple operational cm pt pt maintain rate many feature understand establish relationship incidence patient scoring coefficient address operational without tune add loss yield high feature would sparsity add ensure model predict train subset fold final ip parallel ghz
ridge give therefore dominate dominate behave conduct carlo simulation experiment performance stein compare squared mse study test estimator setup testing setup discuss generate depend comparative setup setup nan hypothesis bias stein estimator rr estimator simulated use formula calculate efficiency estimator relative compare estimator accommodate zero indicate translate previously would generate carry discuss simulation
classifier wrong classifier straightforward cdf choose classifier denote boundary reflect calculate c p cx explore examine show case great correction close intuitively reasonable figure great two hence close true threshold classifier sign mean offer great classifier improvement together even improvement estimate mean estimate dotted great correction move use two se behaviour boundary rs pool rs selection marginal contrast rs stochastic selection maxima show blue dotted indicate se figure select central thereby se select improve third bad four se never improve classifier great improvement specific suboptimal abstract turn nature rs quantity four rs suboptimal notable rs usually rs select se exceed rs performance formal motivate since rs generally make rs heuristic lack kind argument section scenario examine pool consist target label population somewhat pool unbiased cx cx estimator unbiased relationship quantity
follow condition consequently estimate least n close case multipli condition satisfy condition together form complexity result variable nonempty proposition account combination come proposition since probability complexity stand reason nonempty secondly asymptotic number explicitly expression side since express concern nonempty follow nonempty choice hold give asymptotic four less zero inside arrive follow four quantity enough us condition condition apply give probability arbitrary imply define nonempty choose notation difference network probable positivity symmetric difference symmetric g g b g corollary contrast somewhat node difference term time fortunately function adopt follow appear degree parent maximum space exponentially application bound follow union simultaneous lead bayesian term entropy apply individually estimate log achieve complexity result stress idea prove effective version fix expression contingency expansion entropy distribution subset relative entropy express ordinary entropy example ig number entropy entropy independent bernoulli fix likelihood binomial entropy expansion arbitrary empirical context analogous concern appropriately condition one able prove represent empirical corresponding underlying expression entropy entropy rewrite entropy ordinary entropy form event respectively estimate probability sum resp suffice accuracy us bn dag network range consist motivation setup unknown perfect assumption conceptually imagine dag log learn make look maximizer distance positive return close precisely score learning achieve order instead statement think add say score recall g arbitrary network np p take take statement relation expression us margin structure certain margin fix function eq network empirical likelihood regard quantity opposite paragraph precede aspect quantification proceed node network belong able high eventually linear sparsity boost sum explain node constant logarithmic positivity likelihood positivity assumption perfect meaning imply line sum reason proposition stress map degree latter cardinality edge skeleton parameter bn number sparsity boost contribute quantify much overcome state keep track let consist point number subsequence sometimes comment notation matter set eq straightforward
performance easily employ carlo ensure equivalent automatically expense moderate hierarchical know metropolis mh carlo provide unify employ scheme population proposal adaptation drive interact adaptive sampling parallel mcmc currently represent widely throughout carlo mc integral involve success monte represent research area certain intrinsic attempt successfully development briefly strength benefit update location parameter pdfs numerical nonlinearity contain brief present target handle simulation us variable pdf function unnormalized pdf goal computing pdf depend observation precise since observation remove simplify address method impossible general candidate weight proper several pdfs mc discrepancy namely difficult statistical target issue focus pdf proposal procedure employ carlo parameter use technique play pdf location sample equivalent proposal hierarchical
give vector aggregate two approximation replace small aggregate vi emphasize vi stress solve bellman utilize david university present combine combine combine full benefit approximate mdp solution discount mdps important problem discount obtain mdp maximize instead vi iterate bellman equation conceptually bellman matrix problems vi aggregation bellman equation primitive long path vi rl option model primitive state unlike demonstrate improvement runtime reduction vi style temporal abstraction
expansion strict property negative small local however clear improvement robust saddle property true intuitively local eigenvalue purpose gradient traditional main noise every direction allow algorithm saddle oracle add noise saddle optimize strict strict exist least algorithm descent polynomially focus dependency give dependency presentation use rate converge local strongly decrease part sketch defer function polynomial number number future close stochastic gradient descent convex except local convexity appear point close saddle gradient depend fw fw tt max hope update couple local second dynamic descent analyze calculate smoothness long two update sequence remain martingale detailed easy theorem long always decrease know stay appear many briefly adapt constraint future optimization manifold constraint every project manifold constrain compute
rate shape cell introduce include cell shape cell cell cell cell shape shape cluster cluster cluster class shape corner three corner cluster corner shape four corner shape corner small ccc big cluster pool shape shape cell cell pool two one shape cell expect shape automatically cluster class shape blue thin shape separate shape star instead rate rand index high rand index experiment choose protein protein protein raw n protein allow degree remove rotation inner apply show rate ground cc
deep factorization uniformly iii reinforcement optimize intractable integral average combinatorial approximation intractable evaluation stochastic type learning task thank good maps function method many application technique unknown step sgd view two ii gradient link improvement parameter reduce estimator mini sample time variance technique use conjunction aforementione use focus amount minimize step objective simulation slow experiment sgd
xy kernel bandwidth absolute error perform predict absolute decay interval decay investigation popular fourier exist exact change approximation verify aspect bound embed half phase gaussian kernel part fa dr centers il z subset measure satisfies function integrable g e z assume thus sx sx
start average cost approach auxiliary context hash autoencoder ba encoder decoder mac hash code binary autoencoder optimization code advantage step fast easy ba objective pair neighbor scale linearly compute affinity find neighbor costly reason ba affinity ba disadvantage less goal desirable retrieval view general framework hash affinity suboptimal code control bit objective hash ideally minimal space often recall optimize hash objective many preserve original distance binary g loss function code image ham nm affinity space neighborhood within objective describe model dimensionality spectral laplacian locally elastic embed supervise hashing supervise hash input simply apply nonlinear descent mild example wolfe search would otherwise optimally
distortion mutual hellinger channel horizontal vertical distortion hellinger th see automated method remarkable empirical current insensitive method finally usefulness utilize generalization learn material greatly proper highlight connection loss bregman divergence let arbitrarily proper action risk good action play reconstruct hence calculate risk achieve homogeneous super three usefulness super c concave eq element verify make
compare well ice iterate template boolean template integer arithmetic check solver linear quadratic template boolean structure threshold dt automatically parameter ice algorithm search advance boolean pick abstract give randomly walk hill find invariant satisfie search template boolean threshold hyperplane algorithm dt try simple one benchmark simple procedure invariant generation considerable technical learn fall abstract boolean combination set predicate decision believe ability infer boolean technique infer abstraction abstract many analysis infer loss refinement future explore complement acc return ensure var old assume inequality threshold dt sample numerical yield
laplace operator see equivalently eigenfunction heat describe rx r compact sample embed topological number belong parameter laplace fact
first equality independence
specialize primal inner series bound main accelerate comprise lemma requirement meet oracle decrease lemma lemma convexity asymptotic convergence strongly cauchy therefore x equation quadratic look respect obtain v regardless know q last expand consequently equal assumption know eq fact set function factor know show bound amount eq whose point primal primal continue x f fx lemma operate explore abstraction dual minimizer step runtime primary quantify objective progress rate similar section formally define requirement
effect increase right point decay tell unstable iteration rbm converge step instability rbm training follow h visible although quantity require phase message theoretical fix algorithm alternate gradient
denote obtain low medium bias percentile percentile bias correct bias stable sample hyperparameter medium size supplement case obtain point minute take hour medium hour time cut substantially modern cloud computing platform figure intensity correct intensity three size band consist pointwise percentile interval bootstrappe intensity capture shape apparent peak cost covering intensity without realization interval cover sample medium everywhere interval property supplement empirical alternative section answer compare bayes uninformative yet proper former flat replication generate bayes hierarchical difference empirical three statistically significance side sign comparison sign test square bayes although attribute slow mixing sampler highlight another major difficult construct contrary reliably test firstly difference strongly bad seem hierarchical limit well hierarchical medium hierarchical slightly large empirical hierarchical perform comparable achieve depend situation even coverage section supplement effectively regression e regularization poisson enable coverage interval similar supplement detailed coverage similar worth
imply consequence logical describe define potential subsequent mrf preserve structure flexible clause long express weight express logical basis mrfs appeal one formalism induce mrfs clause constant clause weight compactly specify entire mrfs task define structured challenge probable assignment unless tractable approximately mrf integer intractable admit programming relaxation programming tractable mrfs discuss section view map program obtain disadvantage general large graphical consistency complementary highly message quality guarantee identical solution unify inference logic fuzzy interpretation objective extremely equivalence algorithm accurately generalize unified inference derive hinge mrfs program goal objective weight satisfied clause compose clause annotate max rounding boolean expect optimize respect round randomize yet show tractable optimize boolean optimum objective variable relax show approximation function way method conditional assignment achieve score round maximize condition previously assign greedy maximization clause quick specifically need clause small tractable purpose map subsection graphical another approximate random relaxation start map variational formulation optimization distribution inference true mrfs far appeal suit mrfs guarantee tractable fractional mrfs define relaxation equivalent relaxation potential logical clause consistency max specifically optimum vice appendix max round max relaxation scalable consistency apply logic subsection logic model whether perspective use reason naturally fuzzy value clause boolean
zero store entire distance child node draw child draw child node child node tree mention sample tree perfectly distinguish give consider continue correspond subtree suffice correctly w row processing starting contain entry query continue require describe separately row algorithm algorithm tree I compact large tree x mi k large approximately appendix gaussian inequality zero constant sparse w succeed large importantly runtime operation p return sparsity alarm necessarily w p entry runtime operation bad one
show outperform composition stanford move lexical compositional semantic answer nature function lambda variable bind long function question include simple addition define layer multiplication sentiment analysis experimental composition traditional composition brief neural network neural recursive neural lstm sentiment analysis show e xx multi neuron neuron activation weight layer bias assign matrix minimize objective back algorithm efficiently descent neural rnn
coefficient static nonlinearity retrieve suitably method system popular identification base propose identification approximation focus study er static nonlinearity model linear impulse system model combination nonlinearity coefficient call square decompose nonlinearity impulse response square
parameter income high come information engine often technique inform build construct require manual relevant focus ep ep capture incoming message gap inform flexibility message
classifier classify papers ga ph require training classification access training determination auxiliary poisson factorization lda bag word galaxy classified paper scheme bag representation closely paper consider link presence incorporation zero content hard exploit across lda misclassifie lot content document point ground community able exploit discover evaluate community belong dataset take write paper form proportion document determine word concentrate scientific majority paper community entropy author cluster author
evolve finance author award digital fellowship grateful place code use flexible finance quantity variable define propose remarkable numerical require computation run quickly adopt introduction carlo option euler financial formulate product option model differential sde simulate sde numerically path payoff path carlo notably monte flexible enough cope wide range sde model carlo typically seminal together stochastic order efficiency sde carlo approach require accuracy
value regression dependent linearization normality could matrix converge atomic spectral atom non process spectral diagram formulae phenomenon include asymptotic prove limit function confirm addition spectra section outline result concern asymptotic consistency prove illustrate broad conclusion
mlp minimize output net facilitate training walk matrix add encoder eigenvector affinity impose smooth locality maintain geometry encoder contrast output locality locality kernel define maintain value force origin solution diffusion method impose decompose approach affinity spectral unnormalized laplacian add encoder cost bias encoder incorporate propagation extension learn efficient affine make efficient memory architecture encoder new decoder bias decoder output enforce tie autoencoder decoder solve pre decoder learn enable diffusion visualization cover diffusion enable increase application benefit decoder perform embed space calculate centroid cluster formulation constraint decoder pde regularizer function surface surface train encoder network stack autoencoder decoder stack autoencoder autoencoder denote output q new autoencoder training perform outli framework mahalanobis embed anomaly detection
highly matrix area subject present cluster closely match cluster part van clusters apparent stability consensus decomposition fmri lead spectral bayes result plausible absence ground truth comment relevance interpret behave cluster difference reflect quality fast htbp ms ms ms ms ms ms ms ms ms ms require dataset method agglomerative datum ground partition hierarchical procedure adjust rand lead automated fmri method behavior approach dimensionality mutual extensive measure motivated introduction estimation mutual bias describe mutual information tend large fmri heuristic general solution dimensionality contrast dimensionality principled way provide quantitative normalization rather information pair bayes j merge equation access fit automated stopping behave could fmri introduce paper
shape intuition bundle view bundle theorem interesting contain bundle global globally manner progress relate homology human input manually place shape collection task perform correctly recently automate surface compare surface merely large consistency visual interpretation feature trivially use comparison pairwise remarkably quality fig surface direct bundle framework inherent pairwise comparison model short sequence sequence tangent bundle total manifold meaning carry structure tangent exact canonical horizontal tangent bundle connection specify vertical tangent smoothly shall call vector keep mind concept build connection bundle x mh follows immediately imply exact uniquely eventually enable lift base define ode uniquely determined curve connect sufficiently lift start obviously horizontal tangent parallel unique connect super index shall interpretation even implicitly base continuous smooth though trivially achieve ode return geometric similar map manifold assume surface distance equal moreover parallel along obtain map piecewise map shown cause closely characterize connection say trivial sphere along connect bundle structure orthogonal tangent bundle uniquely group consist tangent carry analyze note tangent bundle aim goal even tangent act equivalently operate bundle euclidean interested solve
ghz cache size mb express total three gpu memory gb provide intel mkl gpu iteration complexity size intel prefer library prefer tuning gpu cpu indicate becomes scale high limited bandwidth memory accelerate hardware algorithm show scalability use mkl sequential mkl mkl mkl hybrid performance library combine mkl curve mkl gpu library run mkl sequential technique mkl mkl fitting three sequential make read begin gain mkl mkl serial chain essentially core memory need limit spend synchronization see big largely progress ever powerful computer force algorithmic advance may spatially problem context lead analogue motivate present big former measurement posterior quite big datum distribution differ significantly via high something sufficient probability carlo arise introduce another evaluate distribution
na order th exactly position contiguous interval form amongst small among interval amongst obtain tight randomization since inclusion individual single elaborate computed otherwise contiguous st value piece sampling maintain proportional distinct list maintain small amongst flow budget multiple objective force grow statistic frequency statistic overhead need pass pass r r pass pass r pass pass r pass r r continuous r r r r pass r pass r r pass r r pass continuous pass r pass r r r r r r r pass r pass r continuous r r r pass pass r r r evaluation aim understand provide gold standard aggregated pass assume case frequency replacement skew error library air mac mini computer attempt run counting scale stream range work fewer distinct per element use use stream computation estimate fix use coefficient obtain pass element discrete scheme outline computed relative use hash
vector numerically approximate precision dominant matrix representation inverse need cubic power dense make exact instead directly algorithmic find parameter similarity operator start application particularly markov conference email receive company need transition transition poor year reversible chain first approximation extension newton formula similar equation find root obtaining algorithm polynomial start
figure cluster article ari hierarchical truth dissimilarity identical inference figure branch dendrogram height although article highlight similar across highlight english significantly article modality study hope understand versus would correct topology unable identify
kernel kernel follow empirical process theory e illustrate reproduce space work determine underlie population level q equivalence follow standard population empirical rademacher integer function u generate scalar correspond generally radius rank eigenvalue consequently long consequently intuitive free lebesgue fact critical parametric small space consider generate function think everywhere differentiable generalize space function additional book hilbert operator respect give relation calculation critical familiar convenient achievable first define kernel index show control bias variance critical scaling relation whenever
improvement mt grow capture counterpart show contribute addition intuitive phrase list interesting improve phrase score phrase translation translation pair rather semantic occurrence corpus complementary grow fully train model frequent semantic similarity adapt context translation candidate information context c ccc mt mt mt mt tb embedding initialize word embedding word result word consistently report relationship language embedding machine translation performance finding list case
specifically convenience parent topology occur incomplete sort failure prove generic computation consider correspond density previous admit independent density support away let eq abstract hellinger control dominant turn come event detail describe behavior consider write hellinger consider divide constant combine constant combine
arbitrary property value follow lemma theorem function consider transformation prove hard strongly np partition follow give
much drop layer activation drop activation decrease exact much manifold correspond reconstruction natural reconstruction vector smoothly indicate learn together autoencoder method feature randomly train reconstruct representation feature hence principle wrong actual lead multiply vector increase contrast layer perform abstract sample fully layer much typical model cifar learn natural look vector find fit qualitatively image supplementary material
stage cox derive community score age residual ci age disease r ci age residual community stage l ci score residual cox score community community gene become indicate interaction become disease network cancer community functional module network appear mixture represent module cancer death gene examine dna interaction measure co genomic effect interactive behaviour gene level equation appear indicate community histogram indicate genomic interactive term assess might influence code ht network measure pairwise gene genomic identify network detection methodology network amongst interactive community network behaviour functional finding likely genomic interactive level score community patient patient phenomenon relate genomic dark genome new insight discovery change insight evolution non code field science
machine question fm implement baseline answer baseline similar feed blind answer blind together answer human conduct visual show table answer treat human blind perform pass question pure linguistic rough answer conduct fine wrong perfectly answer partially general right category rate answer look mistake randomly task people visual perfectly correct score paper sentence phrase validate question answer fm answer answer
risk tn tt theorem ti ti tn contraction ti tn define theorem f ti ti ti ti sup sup ti ti ti sup f ti ti sup ti ti ti ti arrive sup substitution ti ti ti ti ti ti ti ti x variable ti tn middle minimizer union definition parametrize also two begin let ii rhs generalization function ts inequality I sup f n lipschitz property member ti term rhs
nr nn represent term tensor I matrix eq symbol denote tucker represent element wise q multilinear operation firstly multilinear explicitly computational paper clarity multilinear induce considerably powerful spike however conjugate likelihood difficulty automatic relevance ard widely powerful analysis ard conjugacy result ard essentially marginal eq q student induce student marginal whereas laplace employ bx ig sparsity example manually although gamma induce hence ap prior random contain enforce random property sparsity group sparse therefore derive ab ap laplace student multivariate r multi yield latent tucker tensor generate tucker form infer solely prior place appropriate
objective minimize synthetic dataset jump comparable performance versus accuracy jump mean implement plot boost experience compute adjustment note optimize per show reasonable dataset hide sequence hold bin phenomena disease rna path degenerate inferential performance small derive asymptotic case obtain degenerate gamma jump widely use approach state state matrix I lx observation trajectory directly observe discrete accord model system rna path rate important signal patient
relate bias triangle bind expression inequality old inequality boundedness write virtue ii bias relationship imply recall final together enkf numerical cost vs consider motion consider example indeed analytically solid error illustrate sde sde realization observation generate eq noisy hierarchy integration take solution gold become kalman update covariance enkf solve mesh mean gold kalman filtering term error rmse denote enkf standard figure measure rmse vs respective expect magnitude fast expect enkf
construct hierarchical representation suppose rare side effect drug rare customer affect pattern representation paragraph current unsupervised deep autoencoder boltzmann machine rbms explain code negative generative depend negativity generative model like value therefore mean sparse posterior separate dependent representation massive computational code prior solve iteratively prior input code constrain see solve
representation cnns achieve nlp sentence identify via mean convolutional structure rich matching pattern language object label matching answer approach link answer relevance summarize recurrent motivation model encode sentence use joint lstm apply model answer pair step convolutional cnns representation
plane visualization triplet middle independently right learn jointly embed illustration embed independent embed dataset triplet first view view triplet triplet triplet triplet triplet show error leave error vertical show add triplet first compare dimension dimension high triplet bias triplet triplet increase interestingly coincide view complexity asymptotic obtain view except view triplet nd triplets getting embed view extremely learn embedding evident purely triplet different mixed embed group class member
face face fan employ wang dictionary dictionaries incoherent dictionary hierarchical categorization hybrid approach reduce comparison fall category decide balance part hybrid x c data frobenius non fix iteration complementary namely code label information exploit contain loss label code jointly classifier intend label get induce dictionary utilize computing dictionary correspondence solve class appear label binary appear dictionary atom instance thus label bring improvement note label pre association discriminative successful achievable mention discriminative result optimization illustrate face keep value ar testing instance half half testing develop parametric bayesian automatically training valid vector draw draw beta number component column sparsity
use improve imply similar spirit projection imply allow prop obtain kx eq control approximation useful prop quadrature appear equivalently sample gx gx g moreover inference study system paper aim approximate integral potentially knowledge remain measurable structured structure technique situation plain quadrature problem space factor evaluation kernel representation point compute sufficiently replace ridge go quadratic contribution cm problem
maximize formation certain pattern obvious place point also characteristic widely spread area around around accumulate provide conclusion majority converge location close form clear quantitative argument exact run choose one gain diagnostic reason sample point value significant vary axis away might design second optimal concentrated completeness nothing logarithm shift cover mean histogram interval case unimodal histogram eventually converge experimental analysis previous two specifically approximate previous achieve good design optimal design use low update wide sample target period stationary avoid autocorrelation among powerful metropolis hasting mh new last accept mh develop step account prove autocorrelation sample explore realization field gp site design
audio contribute appearance capture descriptor video stream class perform body dimensionality concatenation skeleton signal individual isolate follow task discriminative follow joint fine meta detail share employ nature gray video stream joint pose body skeleton modern depth purpose exploit correspond head central formulate descriptor logical calculate position descriptor angle pairwise skeleton playing coordinate position body size proportion shape start normalize skeleton segment average normalize temporal skeleton position form triple virtual angle pose coordinate angle orientation angle position descriptor normalize descriptor descriptor sampling partially redundant occurrence stack descriptor theoretically unnecessary stream serve information pose bounding box around eliminate camera keep hand approximately size normalize hand frame form dynamic frame square sum frames spatio temporal converted scale intensity variance leave video vertical training training hand introduce additional noise switching detect adjust hand respective skeleton sum axis either assign channel
dc nonlinear sim simulate large collection hyperparameter hold objective variable outperform exist method world contain discard draw contain feature intercept generate introduce world rescale deviation large validation decide fit predictor make replicate set describe implement compare regression predictor price parameter grid use predict study kernel neural fits price ascent
sample burn exhibit strongly correlate summarize convergence output assess adaptive dynamically adapt represent probabilistic variable program program contribution choice output mh proposal scheduling application adjust language facilitate expressive programming language goal execution constrain program expression hasting propose change single value reject program simplicity make arbitrary language programming model program program manual dynamically schedule select modification discuss
value crowd trading connect server complete prototype assignment level characterize prototype impact thm thm problem university chinese china chinese china ac cn current digital scheme provide instantaneous exchange precise take trade back proceed normally give process scheme digital find challenge trading
langevin noise stepsize accurate fisher score generalize diffusion fall ij theory provide incorporate u ccc dynamic discuss relationship provide intuition matrix sampler remain design diffusion adaptive account hand hmc focus combine constant sampler potentially convergence distribution mass might region quickly adaptive level set facilitate sampler sec adaptive diffusion theory sampler try guide sampler term hmc easy distribution consider
compress grid thought hull normalization problem following true rademacher base domain random probability framework algorithmic variety utilize common arise square hierarchy immediately rate multiplicative algorithmic hypothesis achieve distribute tensor satisfie fact namely typical asymptotically random interesting factor use go constant emphasize arbitrary product sign entry weakly presence observe fraction anti random result particular norm nuclear nevertheless norm sized turn informally goal polynomial satisfy advantage random clause relaxation weakly clause clause even relaxation long corollary discuss previous tensor rademach complexity could analogue much particular
norm candidate scope selector recovery separation proximity noisy simulation alternate produce additionally experiment application basis signal overcomplete frame selector inspire overcomplete compressive develop finding scalar element norm row concatenation concatenation column conjugate transpose selector incorporate overcomplete dictionary propose presents numerical demonstrating matrix
concerned length regime attract contribution estimate power term near neighbor simply km consist km segmentation tend gaussian process km analytically contaminate analytical result true generative letter letter entry column xy give I real random corresponding process algorithm psd
side note q complete step boundedness q g therefore eq tx know directly yield imply consequently part g convexity j put back turn induction inequality inequality choose know go c
band explain iv parameter well cdf power band fig eight band main group group band band deviation group discuss selection slot minimum power band seven separately use monotonically threshold increase classify relationship band find band band band cause band usage pattern band band mobile base determine periodic mobile activity affect
base concentration complete measure realization gamma function improper infinite denote assign assignment two two document component topic topic eq share global hope component fall gamma know point independent process bernoulli realization lead gamma furthermore document indicator distribute denote document link document gamma subsampling subsampling field subsample gamma network energy mrf part also subsample subsample therefore minimum draw black gray font scale rectangle right edge
wavelet cover construction wavelet frame trivially impose call define good stability bind apply finite difficult analytically wavelet tend characterize apply space wavelet besides wide instead atom namely isotropic directional wavelet scale give short frame index countable translate call discrete semi
tends fact attain bad sake observation second exist existence worst guarantee case compact loss accumulation achieve function supremum case guarantee imply atom neighborhood cumulative atom datum amount wasserstein ball worst exist outside distinguish ambiguity induce metric leibl see case expectation ball decision wish perturbation induced weak coupling highlight program amenable parallelization decision variable sample couple result offer substantial solution efficient could convex concave pricing management generic piecewise frequently approximation smooth piecewise affine uncertainty appropriate dimension affine evaluate expectation evaluate row assertion immediate applie hold conjugacy operator strong duality set empty assertion assertion ii also concave follow linear duality hold maximization assertion substitute free reduce variable penalty belong optimality analogous distribution penalty optimality sample great economic engineering system safe goal quantify probability bad system system safe uncertainty quantification suppose polytope polytope nonempty intersection
proof upper denote hold q schwarz inequality plug gives result maximal sequence convergence variable measurable let eq linear mx fundamental hold er monotonicity er consequence next c hold k x apply ii mp x tu triangle note proof worth statement depend ii convergent subsequence k ji jx triangle imply sequence strongly I x iii converge weakly weak iv unique weak complete weak weakly convergent subsequence convergent subsequence use iv value
correction central apply prior view may view latent formally analogy paper avoid distribution interested correspond keep fix mean mean calculate normal denote denote th generating px nz c pz nk nk pz x nk tx improper trivial notational convenience drop formula parameter distribution z n save denote complete stack without subscript keep redundant sufficient c express take account redundant keep b ax update covariance standard correction uncertainty sometimes methodology exponential family across posterior
regressor run hyperparameter result numerically matrix report probably posterior fairly close routine package job consider sensor identification event relevant treatment gp online hyperparameter change detection detect algorithmic recursive although hyperparameter use therefore key consecutive need fit particle run two gaussian often unknown great
exploit update interactive annotation unlike work learn access unlabele front concern active unlabeled graph node encode near operation propagate graph harmonic gaussian random unlike cut formulation produce broad balancing receive neighbor expense method regularization imbalance quickly become perform matrix inversion scalability effective parametric anchor eigenvector base highly edge remain distance away nearest neighbor distance thresholding suffer graph guarantee edge computationally costly quality regular graph step
empty set forecaster dominate predictive certain dominate measurable induce forecast table perfect forecaster ideal relative sigma sigma dominate forecaster ever set score forecast empirical validity function respectively fortunately forecast quantile dominate forecast qx qx forecast ex diagram order suppose value random sigma quantile forecast scenario argument put median forecast specifically forecast mean generate argument dominate sign forecaster median note corollary comparison forecast special diagnostic point forecast forecast forecast elementary dominate forecast equal asymmetric piecewise forecast plot graph elementary forecast dominate forecast expect asymmetric event corollary probability dominate forecast weather type thompson example et distinguish diagram decision orient diagram forecast positively orient take unconditional forecast plot expense forecast forecast diagram utility forecast utility diagram diagram default orient quantile appearance connect limit curve quantile mm mm binary vertical dash figure diagram sign table forecast exceed expression perfect dominate sign intersect diagram suggest dominate forecaster functional
employ additional predictive sf fit regression explain cc sf sf pc ip price exchange expectation ahead sf build factor make forecast sf fit predictive regressor pc first principal sf sf exhibit due fact factor account nonlinear create find sf estimate date effect macro target consistently htbp low display run introduce high forecast forecast multiple index provide nonlinear explicitly point forecasting dimension reduction high regime demonstrate efficacy improvement beyond conventional two subsequently proof order correspondingly suffice b easy proof proposition matrix identifiability assumption continue assumption meet ii constant addition verify eq complete show normalization
value refer real becomes solve state distinction make availability offline online learn sequential offline addition available could enable offline typically model demand high velocity feasible quick simultaneously incoming preferred process effort storage record online stable online offline online batch adopt regularization target radial n respectively layer output represent output assign element map space assign remain process reduce noted eliminate become least l eq training obtain optimum design handle skewed data modification datum weighting ratio majority step os process update store step initialize h base generalize recursive updating learn decade severe poor method indicate sgd powerful sgd perceptron mean develop extreme machine show potential velocity streaming justification sgd briefly follow encountered approximation
equip dictionary mostly similar compare algorithm per keep fig scenario see dictionary choose table indicate propose equip compact formulation study recognition cccc lr mkl multimodal drive jointly dictionary result bi level gradient general scenario sparsity study sparse multimodal task drive discriminative improved performance achieve utilize framework algorithm experiment heuristic develop tool fast develop algorithm multimodal fusion tree structure future adapt multimodal multimodal view action image super resolution subgradient norm proceed next transition hold ss elastic bound compact impose element statement let n sd sd rewrite convert sd sd column rest proposition everywhere fact twice differentiable everywhere expect measure
weight decay summary formula orthonormal decompose upper triangular step requirement rotation multiply rotation rotation rotation continue convert triangular rotation multiply orthogonal need convert triangular rotation rotation r define qr ab form multiply transpose store use b eigenvector matrix whose entry multiply decomposition eq svd dominate cost bottleneck multiplication idea simple structure sample several intermediate matrix
determinant know I mahalanobis distance special htb gmm dimensional level curve correspond surface covariance observe call assignment consider well example cf mixture determine perform estimation call whose advantage robust avoid vanishe start vi see mahalanobis measure di respect model gaussian mean covariance component assume gmm mixture di j define measure zero positive triangle euclidean j respective coefficient mix relate determined rwm sample want mahalanobis weight dissimilarity sample negativity symmetry inequality drop former property investigate rwm want distance gmm rwm synthetic mixture consist gray background gaussian sample locate center indicate curve mahalanobis respective rwm gmm curve reason rwm consider sample consideration fig scale factor influence coefficient component two input illustrate rwm distance isotropic scaling rwm influence mix mix coefficient show
near neighbor neighbor recommend classifier object correspondingly lastly would ignore classified recommend occur turn good classifying correspond comprise one argument another would vote among consider input dataset idea applicable classification problem recommender base classifier intend accuracy classifier attribute hamming building
contaminate outlier sparse far datum discuss relate detail modification compressive sensing solve correctness generally online application video surveillance need batch fast e conference correctness restrictive moreover exploit temporal change allow correlate corrupt sec overall provide insight need result new need almost analyze batch explain procedure result applicable algorithm online foreground background extraction batch slow use transpose induced norm integer complement contain entry column refer hermitian matrix denote eigenvalue decomposition column size eigenvalue hermitian integer similarly etc isometry number orthonormal column quantify range mc discuss discussion explain go within describe insight need proof form miss key section lemma experiment discuss extension conclusion initial algorithm eigenvalue newly change version subspace change line get subspace model simple lie identically iid zero impractical change albeit perfectly zero let subspace datum estimation change subspace accurately moreover low resolve enough
subsampling subsampling deal subsample imbalance procedure negative positive point plus explore use feature account top principal component original mechanism next failure correlation absolute nan attempt well rf train perform decision rf mechanism evaluate approach employ cross validation point correspond minute random selection may realistic train datum training day test day last day omit order apply predict future failure similar class far perform way varied testing always base point also performance well satisfactory build classifier select combine enhance power low perform diverse answer create bag match well subsampling overcome rare event effective class imbalance classifier classifier training dataset build positive subset negative early parameter value create
node child prune object primitive transform hill thereby sequence symbol meta symbol alphabet expectation achieve persistent partitioning reduce perform new incoming event partition event operation remove child n conceptual boltzmann machine little long shorter next thereby sound symbol gram exhaustive gram symbol occur gram forward online gram exhaustive gram compositional compositional gram count keep separate pattern whose statistic hand exhaustive reflect appearance symbol gram gram actually occur pattern length occur order th length occurred iteratively see frequency calculate pattern pattern length pattern confidence occurrence appear rarely stream appearance pattern integrated pattern
alm mac iteration alm stop due stop stop iteration alm close final alm mac depict history profile year reservoir column match reservoir model alm production counterpart reservoir rd red dots represent historical curve forecast column history matching rd consistent alm mac close result also difference calculate model mac difference small correspond also reservoir tend plot reservoir ensemble leave alm production year mac right value final ensemble alm mac reservoir history match lower initial alm mac mac depict profile middle year reservoir alm nd ensemble mac rd dots historical blue forecast ensemble nd rd contain separate production period decade alm mac second decade alm predict mac alm decade mac alm instead predict mac figure work formula use smooth multiple assimilation approximate ensemble maximum alm mac specifically mac simulator order taylor around common mean current study result jacobian kalman enkf jacobian root matrix formula similar alm square
mechanism burn mix deep smc adaptive application particle thank source code helpful university thin sequential smc method intractable distribution importance dependent proposal bad adapt kullback leibler divergence flexible support online powerful rich neural adaptive carlo indicate adaptive filter indicate translate parameter learn subroutine hasting able generative smc scalable sequential carlo smc simple construct proposal suit filtering
cauchy appear chebyshev expansion use taylor expansions solver chebyshev log degree chebyshev trace rigorous result design estimator determinant chebyshev trace chebyshev analytic polynomial chebyshev chebyshev q k nt chebyshev chebyshev determinant last equality matrix polynomial approximate chebyshev rigorous main calculating
return review md sub md solve bregman power md stem adapt fx mirror analysis mirror subgradient provide function try mirror would obtain subgradient iteration current stochastic subgradient round decide subgradient decide subgradient subgradient vector long subgradient problematic descent access subgradient objective counter weight take take subgradient bx ty allow
asymmetric hash need strong function order wise due fix define map wise independent sketch speed process tensor nonzero build one build compute need evaluation construct decompose provide theoretical power mainly extension setting due place tensor u proof defer u r u I hold analyze method tensor sketch approximation detail find appendix k eigenvector obtain lk randomness sketch product provably approximately decompose rd together comparison complexity drawback contraction evaluation conjecture development v norm exact effectiveness sketch tensor synthetic tensor world experimental intel ghz gb single fast tensor input generate basis tensor constraint input reasonable minute
vote secondly bayesian vote produce machine vote q pair represent interpret alternative weight majority vote confidence vote multiply vote output opposite similarly last artificial majority practice classifier algorithm individually vote dramatically classifier phenomenon boost aim bound vote theoretically justify learn combination provably majority vote improve understand present idea pac bayesian suited majority aim probably approximately guarantee guarantee consider nevertheless use datum account training pac risk associate gibbs pac consider gibbs classifier well vote twice gibbs unfortunately weak indirect vote tight pac give tight majority vote happen community error overcome compare take individually consider disagreement h h h mm mm marginal notion define notice value example expect section majority definition definition study margin vote margin bind suggest extend moment finally section vote vote variable draw majority vote example nice vote complicate handle statement margin majority vote gibbs eq rewrite gibbs disagreement margin therefore second disagreement risk negative follow desire distribution equation b transform gibbs justified consider vote margin zero apply provide qx qx qx qx directly highlight solely moment subsection chebyshev inequality present form highlight property illustrate behavior interesting point proof chebyshev appendix eq mm side chebyshev lem qx qx present form risk vote perform trade disagreement bayes usual
control outside noise part assumption nx show covariance separate fix k suppose claim definition course note hence eq last quantity density bandwidth suppose tend dominate imply boundary risk
right side eigenvalue trade provide behaviour eigenvalue decay relationship decay rate equivalently easy consider theorem hold hypothesis infinity case thesis combine gamma super decay decay argument always asymptotic fact slow exponential decay concern rate view adopt employ much lower attractive less straightforward point introduction component point present still basis hilbert space exponential construction factorization introduce di abuse hilbert clear estimate matrix pseudo operator ji asymptotic widely study plug estimate component eigen abuse notation replacement may convergence estimator effect study sake simplicity case nonparametric occur windows lipschitz integrable e observe pseudo
systematic conduct make train publicly contribution include filter layer impact network introduce augmentation pixel colour evaluated analyse metric boost cnn learn finally source code cnn available already public extremely competitive baseline face cnn considerable recent year cnn briefly review cnn patch fusion c l l image k subject c researcher facebook layer layer locally connect transformation local texture pooling recognition
iterate omit next component correspond htb I net bn weight knn tree na I radial bold dataset knn c test classification four multiclass dataset setup ten fold dataset select dimension input estimate step geometric attain top five eight report study classification multiclass dataset six method world conduct database category
entry admm test synthetic control admm specific e ghz dataset generate sure precision dataset matrix diagonal structure generate respectively converge plain e admm h correctness objective admm iteration table admm slightly pass show supplementary confirm hybrid
good similar domain adaptation use early work classifier however hypothesis permutation label use architecture distribution distinguish even learn example example source label target approximate equation subset combine bind sample tell exist tell classifier vc risk divergence representation indistinguishable possible source original aspect neural classifier generalize well ensure contain origin preserve label develop idea possible describe generalize architecture us nn architecture layer map representation parametrize r l l neural represent source classification log optimization problem domain shorthand notation prediction th heart divergence end output layer unlabele correspond representation hypothesis hyperplane inspire proxy distance scalar r thus hence loss domain come source source example add domain regularizer optimization problem implement hyper use tune rewrite complete saddle backpropagation regularizer network parameter propose tackle make opposite maximize stochastic estimate make sample compute average complete training parametrize compete adversarial domain adversarial network attempt either ability regressor whether green deep predictor together forward architecture add domain connect
background condition error assume normally obtain equally spaced initial use normalize density peak deviation peak left occur assimilation nature posterior center peak gaussian peak standard capturing peak hmc smooth representative posterior analysis ensemble conclusion obtain traditional hmc collect test hamiltonian empirically length step choose forecast lie support burn omit test number burn collect drop consecutive stationarity histogram ensemble obtain hmc show hmc smooth analysis ensemble match generate analysis ensemble likely locate give multi modal var minimum close observation insensitive sign behavior opposite confirm water extensively simplify essential wave propagation mechanism water angular speed wind longitudinal radius acceleration discretization longitudinal lead
expression field discrete average appropriately coarse expression house open package illustrate work coarse sec briefly package coarse package house particle solver operate describe one build execute list type pass later specify coarse domain scale define statistic define type direction define window window file field average particle file name particle velocity angular store assign suitable value efficiently interest follow type package file compatible although average improve quality field coarse grain average store file contain window momentum momentum momentum normal tensor heat local angular momentum angular momentum contact couple stress density sec besides static coarse expression consider channel depict setup fully three dimensional type refer type mean diameter fraction particle
generate stepsize use decay might decay absolute adaptive decaying ht detailed gradient svms gradient psd constrain take decomposition assignment application hmc non bayesian still consider classification set input sigmoid prior vector laplace non conjugacy sigmoid jump mcmc test mh death subgradient mcmc log subgradient ever subgradient justification give stochastic subgradient hmc rest recent year flexibility structure learn possible way bit discrimination regularize successful svms perform posterior inference generally challenge mcmc sampling hinge non hasting walk suffer fairly non unfortunately newly discover augmentation people gibbs sample normally force sample efficiency descent optimize non smooth objective subgradient
function label additive variance training training tree capture describe comprise root leaf internal decision child event split independently force split child indicator split keep informally deeply high location draw independently uniform denote contain valid block mean hyperparameter shift adjustment node towards prior hyperparameter control shape square unconditional shape locate specify use sequential tree prior start partial use review stage stage stochastically assign
imagine movement usually synchronization area riemannian mi link directly embed obtain rely event apply capture define riemannian would distance dissimilarity weighted power density classify eeg neighbor definition geometry literature hausdorff point tangent differential short smooth space span pass endow inner tangent vary smoothly point definite tangent tangent manifold mapping eq computation operator straightforward u eigenvalue vector geodesic eigenvalue square distance q iteratively eeg trial record vector choice estimator crucial verify condition computational usual estimator technique
regressor input approximate scalar article integral appear easy square multivariate integral distribution available show sigma transform analogously show sigma quadrature sigma accord predefine weight determine j sigma covariance vector cross principle unit sigma good let enable unit integration integrate element dimensional kk regressor select determine point g give result actually stochastically reason mean integration beneficial affine transformation also variability correspond hand argue stochastically
article query crowd self report mobile near activity subsequent interestingly google never impossible reproduce improve exact behind limitation identify evolve drift lead inaccurate aggregate people appropriately behind ignore intrinsic activity produce level address aforementioned multiple justification methodology contain new dynamically automatically google estimation improve long record variable move desire date period capture recent search though low acquire method significant statistically speak autoregressive google employ potentially autoregressive capture historical activity exploit google penalty achieve automatic generalize system aim track health
combinatorial conclude open quantile expert protocol play loss expert k central sequence instantaneous length schema distribution expert rate parameter ensures investigate positivity quick let variance k imagine simplicity prior put implie immediately quantile raise question potential always role apply bind increase last identity hold choose sketch rigorous prior mass admit computation close integral potential cumulative essential addition mean latter delay might introduce alternative weight potential
pricing determine learner pay learner bit begin data ingredient price regret minimization assumption hypothesis nature adversary etc choosing observe algorithm adversarial discount incur choose often randomize loss say utilize broad class algorithm include specify usually strongly convex norm multiplicative regularizer strongly regularizer respect case close rule compute tf indeed assumption guarantee respect suppose design arrival arrival algorithm section abstract randomly section external mean step observe goal setting obtain notice crucially unchanged still regardless observe key technique idea get unbiased sum take check event expectation machine independently expect outline depend notation give explore coin bernoulli implement regularizer sequence depend expectation randomness recover classic note far regret small tool batch may leverage online batch technique far feed hypothesis predict mean hypothesis average suffice take hypothesis batch
target multi object multi either detect observation detect intensity track track assign track generate measurement notation iid multi system capture information target individual target object recursively commonly bayes special label provide kolmogorov conjugate object discrete discrete space satisfy pair track association track association map corresponding density probability label unique cardinality characterize enumeration track h h p
improve precision matching image reduce also practical advantage network require need therefore need arbitrary dimension sift instance way patch adjust pool proportional patch maintain resolution idea recently net pyramid pooling network layer convolutional size adapt architecture achieve consider model train supervise square regularization objective q network network matching momentum decay batch overfitte training patch overfitte train allow store memory efficiently retrieve pair augment convolution library descriptor gpu slow
prove condition transition function include arbitrary state optimality mp arm non chain case channel mp consider wireless depict divided duration ap schedule available channel ts otherwise buffer channel chain assume positively likely process ts ts transmission assume fundamental energy make transition equip energy ts state ts b system node function single allocate channel ap ap ts ap node active ts rate linear ap throughout energy simplify notation energy transmission throughput ts ap schedule node ts know information ap receive ts note ap active transmission use energy scheduling observation
eq cumulative function similarly log mention early advantage basically harmonic therefore f lx explain alg note divide coordinate must complete lemma use tail understand concern threshold eq q minimize lemma optimum
whereas treat investigate resample matching cross rescale unbiased respect weight validation resample weight domain directly applicable domain search domain matching matrix transformation n minimize location association spectral embed dimensionality reduction transform cca relation matching transform cca call domain respectively vector word hundred thousand million would retrieve image alternatively retrieve query match across
token analysis variable name language structured prediction discover semantic entity coordinate classification depict step detail software corpus code repository library goal relation potentially pair corpus distributional closely class closeness similarity noun name class define similarity usage type classifier extract extract corpus assumption context al empirical noun noun similarity due many class
entry jensen inequality readily success close take concentration phenomenon must must enough select good function associate massive necessity generate uninformative implicit expressive yet simple enough sample question relevant efficient parametrize family select parsimonious effort sample radial basis answer random introduce radial basis arguably arbitrarily accurately principle high use connection introduce tensor paradigm dimensional input cost degree tm
robustness concern conjunction locally function train descent velocity field field family model ensemble density velocity field kullback single must exploitation error try new avoid minima thompson sampling dynamic complex scenario possibility gaussian conjunction technique linear suitable flow field outline work theoretical bound investigation relation model anonymous suggestion improve manuscript national science foundation office research department
shift goal content total model contextual cost translate reward formally define solution compute knowledge user characteristic due content benchmark always context correspond select reward matching action benchmark kx relevance content normalize function long learner choose choose benchmark relevance score cost network correspond matching hence correspond centralized act hand ca subsection relevance cost sublinear user time subsection regret incur due system regret select action respect regret sublinear next sublinear static ca user action action content content static prior learner mechanism recommend without source content learn analytically characteristic static content relevance context formalize assumption indicate instance user similar gender call similarity exponent depend characteristic content question become experience match current answer propose relevance type algorithm partition past observation use relevance include mechanism algorithm content user task implement task content numerous large content provide content user user match hence able relevance payment mechanism incorporate type directly connect type source connect exchange source horizon interest arrive create website day different regret hold since
nonconvex may effective wide acknowledgement part grant grant ni h mu hz second derivative ingredient respect I yx entry eq therefore yx z x direct substitute back equation matrix two least bound line tool u u z e replace f q goal bind mu expand first sum bind cross eq mu mu mu h mu mu mu
number linear original explanatory may purpose difficult interpret regularization selection continue improve trade decrease biased discuss regression zero penalize regression minimize x index frank case subset regression family family fan li act among ridge disadvantage include penalty ridge quantitative gap practically lasso toward irrelevant shrinkage lasso predictive method parameter part statistical use
induction former probably conservative convergence pi vi policy variation pi equation repeat ahead regular pi surprising fast pi calculate section convergence ahead optimal theorem show
analysis speedup calculation convergence substantial gain employ traditional seem beyond novel accelerate idea stop cg correction ensure variant early linear incremental define early reach iteration rewrite series coefficient focus add w carry convergence reach early give rough report number different cg stop cg stress show calculation stochastic gps instead estimate implement run cg track early stop cg involve one propose way weight
factor low dimensional visualization capture give storage massive precise non convex optimization firstly optimization convex variety specifically desire singular factorization streaming orthonormal wish put identical update analyze orthonormal use constrain update problem decade deal reader guarantee focus gradient streaming maintain manifold establish convergence directly via scheme prove global rank stage martingale global
bad aspect ratio define aspect ratio cell ratio circular radial centroid show decomposition disk cell aspect ratio decomposition recursive figure eight disk cell centroid arc binary area splitting disk aspect ratio unity triangle great circle convention consider spherical triangle internal angle spherical triangle great circle original triangle four equal area triangle arc general circle want splitting circle great circle generalize great circle abc split two find bc area step write construction uniquely decomposition bad aspect ratio base line segment tail triangle
give elaborate array origin coordinate channel diag dependent present source fail arbitrary information whether angle incidence eq formalism pi pi follow denote frobenius decompose decomposition equivalent ta scalar base
nonnegative definite density lebesgue multivariate nonnegative density fourier covariance theorem primarily build multivariate specify matrix density definite series frequency coherence assess series notion entry density coherence coherence unity indicate band relate prediction smoothed univariate krige kernel function weakly stationary bivariate value everywhere integrable predictor optimal great high coherence corollary coherence attractive amount variability process development popular covariance construction coherence compare flexibility bivariate specify definite approach contribute cc ji jj structure integrable square c convolution follow restrictive coherence necessarily square integrable function covariance
though trajectory thus add define xx training generative visible state run step sec variational unit sample convert initial hide state primary feedforward network single use minibatch trajectory guide policy divergence step step analytically provide include momentum order like rescale learn imputation add lstm guide lstm step value thing try work could probably block lstm model model formulate capable represent policy network train test policy imputation difficulty direct
e ht cg cg count grid individually overlap rectangular use group build count reasoning help minima produce extraction number small diversity clarity index word discuss deep architecture model grid word think massive document intersection among little word sum document piece grid great label base corpus grid evident visualization though train arbitrary dimensionality discrete overlap group define rectangular window sum window e aggregate much w portion mnist digits cg model average cg bag location virtual intensity bag histogram show image
consistency bound equality see n strong duality wise function unfold expression continuously obtain x result section first formula develop risk use formula actor style sampling function value encodes term calculate programming become curse risk neutral problem bt td popular purpose td approximate sensitive discuss affect estimate actor closely discussion sequence value variable random cost along mdp parameterized parameter z cx interested mdp parameterize variable risk envelope markovian measure use dynamic bellman style markov coherent risk denote aa cost induce bellman risk state enumeration bellman curse dimensionality iterative risk sensitive risk vector belong low space find order
systematic necessary answer basic process deal relation criterion limitation criterion entropy behind entropy transform order pattern seem properly light learning target study yet optimization information generic learn machine mechanism take cost theoretically unable support rule classifier
hash deep convolutional neural cnns annotation object capability cnns explore wang use rank triplet cnns al incorporate cnns image hash deep hash similarly et deep rbms hash pairwise hash cnns hash code explicitly impose problem hash treat mapping project code hash code semantic label hash obtain desirable hash early conventional extract learn hash limit semantic jointly raw pixel mapping hash code non hash capability advance learn fig incorporate
lda belong category document apply latent evident generate dimensional oppose st nd principal component evident spread output category category dimensional pca output exploratory map category
well predictive pairwise link publication time prediction accurate publish cite document publish attain blockmodel citation blockmodel probability citation much interest citation among corpus occur visualization citation patterns proportional citation strength estimate blockmodel element topic font topic next focus topic citation interesting trend landscape tendency topic however citation deal aspect vast link topic worth tendency topic body aware experimental article focus constitute topic topic string claim object matter string band particle emphasis dimensional concept stre dimensional important concept string relate topic citation relationship two topic respective tendency popular topic early successful energy particle relate fractional spin force matter stre force expense mathematically concern cite black mini topic investigation narrow document citation general however citation vary wide
integer go use follow limit observe whenever upper need rearrange last without satisfied prove limit recall whenever denoise autoencoder denoise da variance recall inside end continuity proceed kk nf expectation term proceed replace gradient denoise autoencoder run number datum q constant operation denote average time step use distribute da da denoise learn denoise objective show size without correspond visible corrupted unit refer e unknown visible hide input un reconstruction simplify term summation corruption correspond objective da summation extra sigmoid minimize complete corruption pre train size
rule approximate rate selection approximate multiplicative error regime gs choose satisfy basic progress incorporate rate gs must time error gs choose satisfy rule substantially maximum likely unless regime close condition error gs hand repeatedly update switch randomize detect key method q smooth operator convergence possibility gs rule choose negative directional arbitrarily gs eq effective constrained maximize progress intuitive seem theoretical far gs gs bound conjecture actually appendix counter example gs gs rate random gs lead compare efficacy coordinate rule instance set entry multiply ten induce lipschitz keep gauss rule coordinate row set
method movie rate comparison rate movie train rating movie tm rating pmf pmf latent latent predict integer integer table tm restrict hand rmse match outperform pmf come benchmark approach accommodate world rank main recall distribution j ki proposition item th consider indicate prefer ex ex repeat ranking sequentially place th length item procedure irrelevant without item show hold ex item partial assumption therefore r true conclude j separability reference ranking set rank separable xx prefer
worth shorter compare forecast instance simulation prediction small spatio temporal spatio wind call exist case study short forecast direction pressure etc wind forecast another path yet idea probabilistic forecasting method useful economic technical risk convert result remark electrical california berkeley usa berkeley edu electrical
cause calculated assignment histogram consider step toward article exploratory series exactly te calculate way two technique opposite causal causality exploratory far quantity bivariate series straightforward causality depend may exist causality tool structure probability scientific rely system implement control experiment current study datum collect control observational difficult correlation different tool drive series exact drive primary driving causality relate straightforward use classical mechanic development fall broad entropy causality reconstruction find field include economic introduce series causality directly causality causal relationship discuss strength causal inference system implement define causal straightforward indicator relate two however fundamentally
evaluate exploratory study different assessment dropout architecture classification mnist lastly give quantitative performance use previous begin qualitatively dropout mlp uncertainty come across co dataset air reconstruct assess dataset centre evaluate convolutional softmax uncertainty assess realistic plot assess unit either relu non run batch optimisation fairly co decay scale red dash blue line standard mark dash line point away predict
result amount arbitrary particular substantially nevertheless subspace instance completely obey reveal hard lasso ssc set resolve small mild however subspace theory reason fit gap noiseless post step resolve identifiable ssc assumption paper provide provable lrr provably dense model facilitate generalize ssc ssc considerably much lastly mining refer completely coordinate instead nonetheless also theoretical result applicable independent subspace adversarial false lrr ssc ssc noisy
category letter email budget news presentation publication scientific report specification selection categorization category well large restrict category represent category perfectly distinct label tag potentially several eventually select dataset relate split proportion split retrieval median retrieval split proportion imagenet validation cnn letter cnns implement softmax top cnns network layer network extract cnns output first network architecture list cnn hyperparameter document layer pool relu take architecture imagenet extract take case cnn extract cnn perform length large vector compress pca dimension
interaction citation global local ascent marginal perform structured leverage tractable family decomposition link must alternate direction multiplier marginal tractable linear provide practitioner enforce pr closely relate perform conditional maximization come directly inference onto expectation property pr projection lagrangian regularity extend assume learn goal different depend fully data approach section differ finally framework employ convex consequence project gradient pose algorithm framework wide algorithm material present negative entropy convex simplex restrict strongly asymmetric bregman entropy allow compute solve call requirement proximal project since project onto euclidean perform fact square tb
submodular projection expand enyi modular modular ignore expand log partition capturing arrive submodular function optimum minimize connection follow bernoulli function submodular extension w submodular inequality follow k indice submodular reach optimum q minimum entropy lagrange note feasible objective work lagrange v r bf ii dual duality optimize lagrange multiplier inner conjugate equal project subtract clearly project define definition primal simple close change please reference therein terminology
formalism identify show convergent ar write splitting proposal mala langevin discretize ar correspond scan gibbs sampler conclude hmc fix albeit normal split matrix feature mh splitting function target high design balance act integrate jump proposal distribution efficient ar small action target desire target sense mean small small quantify discretize generalise langevin choice matrix efficiency square choose balance induce hand condition matrix require independent infeasible high spectral result hmc langevin hamiltonian proposal
much source big scene understand multimodal often unlabele unsupervised unsupervised natural optimally predict image scene semi generative connection cm xshift cm yshift cm cm cm prediction black green graphical representation limit small region visualization recurrent two represent feedforward depend recurrent red dimensional demonstrate validity compare building pixel suggest distribution extend work al far make simply note share apply parameter drastically increase
prove solve polynomial modulus approach technique algorithm originally view special case exponential control evaluate problem build variant public cyclic hardness relate work present bottleneck first improve stage unchanged approximate factor show lattice magnitude free polynomial solve gr present regularity contribution improvement introduce generalize switching require rely assumption tackle lattice close ask coordinate basis matrix infinity reduction density time interest technique density lattice recover polynomial security assumption heuristic come rotation hidden attack recommend identify x obtain concatenation logarithm lattice dual definition usual preserve distinguished probability
precise addition actual therefore set model fit accord cc true estimate observation cc estimate simulated experiment robustness versus exactly observation generate outlier outlier uniformly apply consider sample size impact mean estimate mse calculate I square see outlier four model slightly outperform situation one contain generate outlier generate two outperform majority also situation much outlier compare expert skew outlier comparable see model even support c c mse function true varying indicate highlight robustness figure generate outlier rough model clearly expert outlier freedom degree freedom heavy tail fit datum fit generate accord outlier fit set generate outlier real world anomaly scatter temperature go back study use robust model laplace pure fundamental play harmonic usually ask tune adjust variable consider predictor response
specific nice define follow regret remainder taking complete would acknowledge support national foundation nsf appendix additional proposition take result eq proposition rhs convention partial suffice convenience inequality simplify suffice
posteriori recover inversion discussion kalman filter imply contraction address consequence filter recursion filter endow metric see hmms kalman contraction space kalman explain contraction endow riemannian show
sufficiently simple interest number freedom gibbs equal determinant fisher jeffreys jeffreys propose invariance compare principle correctly jeffreys factor scale precise correctly eqn play critical role clearly eqn imply significant prior increase except marginalization eqn prior complexity compute recursive analogous total parameterization expression unknown still eqn
distribution news fact calculate ratio probability report convert report ratio circle circle estimation obvious calculated rate averaging exclude weight thus comparison baseline calculate metric extreme international air investigate assess forecast risk arrival arrival special multimodal probit stick mixture demand probability change predictor flexibility weak ol inference demonstrate help baseline compare report rank broad show distribution serve deep air datum report focus availability top mind get support research project confirm say data know critical shape rather future hope obtain differently root service service drop extremely delay cause error retain treat distinct avoid noise cause contain contain less observe filter selecting table typical explanation table ccccc exception element probability probability since symmetric uniformly stick break density drop burn indicate equation dropping
lr ep svm svm lr svm label scenario middle self clarity omit ccc ef lf ef lf ef ef lf ef lf ef lf ef lf ef lf ef ef ef ef lf ef lf lf ef lf ef lf ef ef lf ef lf ef lf ef ef lf ef lf ef lf ef lf ef ef lf ef lf train round ef lf ef ef lf ef lf ef lf ef lf ef lf ef ef lf ef lf ef ef lf ef ef lf lf ef ef lf ef lf ef lf ef ef lf ef ef lf ef lf ef lf ef lf ef lf ef lf l co ef lf ef lf ef
triangle avoid basis mesh method bs htbp htbp mesh bs bs show estimation prediction bs score bs generally bs case log b mesh component mesh bs l bs g bs bs ls mesh polynomial example mesh decrease mesh bs associate number change may element spline well good rmse location meanwhile confident prediction mark choose bs ls mesh ls mesh mesh mesh ls mesh rmse combination cost select efficient
give fact follow use definition claim lemma trivially inductive similarly term combine bound q recall c k give similarly inductive twice give inductive twice round round round fact increase bound multiplier ingredient constraint actually ensure eq q second place remain regret optimization constraint must add relate explore differ straightforward adaptation ready add tx ta sequence clearly probability application use round play round exploration round use order collect descent use decrease five exactly shrink update violate potential kl kl p first fairly straightforward later marginal convexity fact unnormalize marginal regret
aggregation majority voting select give resource base majority select rank worker na I combinatorial globally comprehensive experiment world dataset worker predefine eliminate instead focus highest rank discard threshold distinguished assignment formulate worker problem combinatorial optimization present section give far conclude assume crowd worker item label item question item assign worker possibly worker assume reliability worker control reliability label I write
respectively input show configuration consecutive respectively dimension output represent thus dimension addition perform interpolation choose conduct evaluate input classification rich cccc fr average graph regularization adjacency fr result trend rise accuracy comparison table perform regularization improve keep fr cccc map accuracy regularize fusion table compare layer table fusion accuracy fusion fr reach train part firstly layer environment level secondly feed impose deep architecture adjacent layer fuse tree overall validate graph furthermore automatically raw robot
dc stage use draw walk sampler propose next large help avoid mode posterior increase reduce support current independence sampler end step balance numerator denominator go instead numerator back employ numerator chain effect remove short already notice transition draw variance reject maximum parametric construct draw rejection maximizer parametric uniform obtain confidence procedure principle large covariance first mle behind abc
linearity bias layer choice activation rise possibility rise output feed use output layer jointly optimize make quantify use hide learn predict height hide n nm nm nm hide l si si si si si fit text east fit nm nm east rectangle background east f si si anchor anchor anchor anchor anchor center right determine dataset use parameter repeatedly opposite hyperparameter traditionally model non processing commonly use information vanish layer make layer significant available facilitate deep network stack use initialization avoid gradient vanish unnecessary introduction dropout regularization remove zero scale layer different remove evaluation remove rescaling prevent unit force utility remove neural constrain connectivity layer exhibit kind pixel structure neural connectivity convolutional stack colour learnable produce stack output implement feature represent convolution operation represent map obtain sum feature previous spatial bias vary image south east north west pool txt txt reduce subset unit across figure orient edge enable consequence convolutional traditional
house color house number collect c dimensionality test cifar mnist r cifar cifar mnist test experiment mini batch consider validation point pick reach fast set training retain balanced unbalanced initialization incoming initialize gaussian distribution unbalanced setting pick randomly replacement multiply incoming edge edge randomly cifar cifar error
order latter consequence accounting display time run either representative parameter formulation deviation ht mean deviation model incorrect geometric geometric distribution flexibility negative state life future population approximately unbiased display mean bias decrease increase essentially nuisance parameter primary order embed subsequence duration markovian clearly inaccurate estimation problematic lie switch interestingly survival robust regard mis specification distribution study capture house collect study new york study capture covariate correspond state individual unknown
return let return semidefinite stock datum via window method canonical row one barrier penalize consequently sketch part rank keep result diagonal plus apply sketch hinge loss comparison sketch popular large scale per newton besides newton method backtracking accelerate gradient descent acc adapt manually tune stochastic gradient sgd step choice hessian newton sgd trials stepsize plot expect newton fastest log plot bottom panel see newton sketch fast lasso take regularization strategy problem program barrier dual need sequence formulation first last two via partially calculate newton sketch strategy solver per run duality versus barrier blue barrier although iteration iteration reach duality gap barrier barrier
report dr process constraint order good true variable rmse example small estimate element constraint matrix constraint matrix derive principle element estimate principle eq space matrix compare criterion make angle estimate estimate subspace row hence angle constraint obtain pca experience angle criterion quality estimate especially practically estimate matrix process set independent base dependent rewrite respectively matrix relate dependent manner derive matrix constraint technique steady practice variance covariance change replicate measurement steady state operating period clearly steady operate challenge simultaneously constraint without replicate steady surprisingly description estimate combine
permutation represent triangle coordinate order choose without order find arise science dna short city return opposite np allow test capability model pair similar describe represent represent consistency city generate hold algorithm produce solution costly solution find factor extensive search though gain
detect effectively output context output experiment dataset outli process error space outlier signal formally detection investigate section describe outli detection conclude value response goal approach precise outli approach decomposable phase phase dimensional learning apply model unseen ode technique pattern classify organize define multivariate outli review research outli experimental evaluation lastly conclude paper multivariate outli response n x nd goal identify fundamental challenge model exponential notational also name py data community accordingly approach conduct
constitute partition record intersection set linkage criterion priori go introduction criterion decision aid object performance real alternative criterion value dm type criterion pseudo criterion pre criterion imply preference method type affect error uncertainty imply big imply binary relation
achieve maximum round pure proof therefore problem regret resource consider version reward contextual bandit budget contextual budget obtain contextual contextual regime pose maintain optimal study generalization agent observe agent ensure inside policy problem bandit concave reward henceforth arbitrary constraint reward budget interesting contextual need modify minimal fashion substantially achieve optimal many regret provide bind furthermore need special precise statement bound first feasibility extend sketch efficient achieve action work share type early paper general contextual policy nonlinear combination challenge bandit add bandit regime ask open achieve efficiency maintain
ability training existence subset least edge source robustness subset dataset answer compositional dramatically reduce compositional training hard along able dataset dataset interpretable illustrative cc bilinear bilinear interpretable query bilinear bilinear x parent people country x reasoning accuracy negative make directly result previously inferior compositional training entity entity table report conjunction compositional outperform
occur worker annotation error worker amazon ask age integer put dataset worker worker require rating excellent good fair bad average different label around pair worker label one ground expert worker microsoft team page worker require web page spam around worker web spam worker item worker worker price spam c ds mf search spam dataset ds mf age error evaluate follow baseline jointly worker maximize label variational mf worker confusion hyperparameter maximize calculate mf c multiclass source source regularization regularization validation show method crowdsource multiclass minimax conditional entropy show ordinal entropy perform
wiener else evaluation piecewise spline three vanish always cubic crucial purpose value respectively minima analytically denote search cubic one scalar arise approximately cubic high minimizer node large evaluation decide amount show variability bottom wolfe mark extension ii positivity projection variable gp thus bivariate wolfe bivariate normal coefficient readily line call compute evaluation node wolfe accept return requirement motivate fix
report trend ba significant close ba cifar quite competitive consistently ba ba large image neighbor hash bit retrieve initialization code affect optimum mac find optimum effective wide horizontal line one method lie close reliable indicator good precision reasonably notably especially precision neighbor within clear precision small retrieve small code achievable make use code hamming neighbor particular explain ba note early continue autoencoder consistent suggest well mac dataset code increase bit fig select wide histogram vector map code use unnormalized middle plot uniformity contribution reveal effective hash binary fast ba filter code iteratively hash ba correct suited autoencoder encourage hamming neighbor autoencoder
standard separate fusion capture intermediate modality produce posterior v v bilinear decrease complexity factorization bilinear motivated cca fw fuse small entropy bilinear eq projection maximize alignment modality cca hand posterior wise hence fix hyperplane fuse cca like
target exist classification speech difficult phone frame due effect uncertainty soft soft target say function less smooth objective certainly much easy optimize go objective flat formulate two fig depending sample entropy represent target smooth easy model toy deep practice highly expect target smoother attribute involve soft less model htb objective desirable
entry frobenius plug probabilistic bind complete chain start exponentially improve omit extended expectation tight bind establish norm random center subgaussian upper matrix trivially tight fact obtain subgaussian distribution vector uniformly sphere isotropic subgaussian drawn distribution situation change additional assumption subgaussian draw univariate perturbation part frobenius upper bound experiment da jk g ij n satisfy additional center straightforward event eq independent
total basis agent make human impossible understand decision make experiment collective intelligence effect estimate value neither mixed strategy condition effect performance great latter intelligence except dominant study experimentally game human player budget variety circumstance strategy recently mab social interaction individual social might trade multi armed bandit bandit payoff independently distribution payoff round agent obtain payoff bandit exploit round obtain old exploit choose explore extremely agent outcome
distribution close conclude expressive demonstrate applicability particular say either low limited enjoy structural exploiting close search generic probability distribution total variation kolmogorov algebra state otherwise mean total q kolmogorov variation distance total variation lower bound kolmogorov hoeffding hoeffding independent give kolmogorov sample x I variation proof say two random statement processing fix possibly let independently hypothesis roughly set give access pdf algorithm make h nh expect algorithm give eq generality done note direction variation e x n ic recall symmetric dominant symmetric eigenvalue
object datum perfectly classify additionally transform discrete spaced cost outperform power early power threshold pair threshold pair power house vote compose record member house record party member contain level different identify response goal identifying represent dna sequence predict attribute level unique neighborhood feature ground truth propose minimize acquisition budget construct wherein tree easy yield although suited supervise forest present forest account acquisition cost term power example cost cost obviously undesirable element amongst tree connection diversity amongst construct
phrase access sec ranking derive phrase definition community live experiment define phrase dictionary tell phrase reference replace norm forward likelihood ic present phrase randomly phrase phrase phrase still indicator never see dictionary write dictionary prove primary language learner spirit phrase need phrase likelihood frequency occurrence upon frequency phrase list value double sort list phrase frequent sort definition reference despite lack automate generation investigation utilize
stability irrelevant besides overfitte toward underlie ground truth sample asymptotically test cluster mod otherwise partition variable play mix notation mean discriminate see add noise matrix mix distance describe test case information compose hierarchical linkage hc km affinity table
fluctuation involve supremum term expect small average offset canonical offset mild condition familiar ordinary square statement well contribution offset excess extend behavior offset number recover aggregation latter present boundedness excess offset indicate intrinsic bound require statement isometry hold subgaussian class offset complexity complexity ordinary square jointly q
fisher fact implement computation assume goal fisher diffusion drift density transition expansion pmf death surely transition representation natural term fisher diffusion infinite leaf mutation purpose appear pmf subsection simulate distribution pmf inversion q accord despite evaluate pmf q km km compute modification say require exception first term check condition km km
want start simplicity matrix orthogonal singular degeneracy technical yet take
obtain accurate rejection move right explanation second large large eigenvalue reduce effectively large eigenvalue move large allow second move right reduce subspace rare plot close integral blue histogram obtain traditional red smooth take long either solver skewed imply greatly help towards bottom large eigenvalue right move large eigenvalue remark perfectly hasting volume intersection would concentrate unless gauss curvature would slowly reweighte much bad geometry even national engineering program mit mathematics department nsf dms extremely grateful chen discussion definition lemma curvature formula condition exploit search subspace intersection orientation manifold exponentially dimension search variance subspace dimension otherwise rare event unlikely intersect rare reason support prove theoretical volume algebraic manifold apply manifold arise many mechanic molecular biology interest geometry greatly reduce geometry normalize median time small bottom simulation converge accurate histogram section traditional probability greatly take weight smooth histogram speed convergence geometry place weight rx ix intersection point manifold isotropic random gibbs intersection intersection bottom
hold combinatorial relation prior result class value exist large depth order stand class extend case step obtain analogue lemma order sequential extension section class discretization supremum finite expert cover much covering consider one define case requirement consistent read contrast pointwise metric uniformity gap probability element last agree sequential take f minimax
observe regret figure study set six population variance provide implement high implement horizon activation average regret time activation remark limit show bound regret regret rhs additionally nb thompson sampling asymptotically arc pt initially
none algorithm come provable realistic reader convex discussion alternative iterative efficient involve theoretical em code scheme truncation herein might improve truncate reverse encounter motivate principle design sample obey eq represent direction update unfortunately along precede come meaningful solution helpful vector often arise direction assign average monte showing varie figure component point form descent value fig return poisson truth appropriate truncation gradient give truncation give truncation light fall outside remove numerator denominator enforce recognize denominator obeys lead law hence numerator remark extra continue modification constant truncation summarize algorithm general fashion apply presence extra term q default eigenvector order rapidly amount whose lead eigenvector unfortunately initial due heavy tailed quantity moment generating much tell vector lead phenomenon prevent method return discard truncation merely theoretical concern substantial issue show truncation fig compare
application satisfy one give unknown want minimization role produce mirror computing mirror guarantee execute parallel processor exploit multiple processor iteration rate compare art processor decision processor capability processing access datum update need synchronization conceptually mirror load storage location vector q shared implement way processor execute independently store processor read twice round gradient last
therefore pack combine pt unit packing exist v see v v generality hypothesis function real algorithm margin ff f rademacher particular follow lipschitz argument height pt depth composition note eq second ii iii note complexity rademacher complexity lipschitz base depth every rademacher complexity conclusion immediate divide failure individually finally deviation define inequality metric failure inequality note minimizer per lemma iv equality theorem definition theorem corollary height width author seek pac style complexity give show leverage notion intrinsic reveal metric regularization
admm free hold condition hold full rank objective function denote indicator eq bound f f restrictive norm norm assumption whose solving limit constraint well denote sum separable notation presentation ambiguity euclidean prove theorem theorem respectively block immediate prove convergent generate admm inequality obtain
roughly separate average heart people heart slow consistent people severe might essence underlie separability iterative convolution step specify mask experiment choose calculate really essence stable heart hour around record show heart motivate find heart correlation analysis mode sort compute correlation disease plot figure statistic fluctuation
place use popular extensive important loss often lead loss art believe bound coefficient merge quantitative theorem growth degree function replace randomness function offer answer experimental alternative answer raise really
lm use language map outperform conventional average contrast model hdp acoustic contrast generative result row adequate outperform conventional discovery system acoustic manner acoustic model speech contrast train speech signal word acquire contrast acoustic adaptation acoustic must performance naive recognize acquisition letter word sequence conventional ari ari dramatically improve acquisition contrast improve word ari letter ari dictionary keep recognition recognition field widely language could ari letter ari adequate ari become bad letter ari error acquisition procedure describe directly latent letter sequence achieve language inference typical conventional boundary sample word ie divide index letter word ie although ie single ie conventional acoustic hierarchical hdp hdp hierarchical semi hdp derive gibbs originally simultaneous acoustic hdp procedure
feed back reconstruct auto penalty feed pathway state feed forward pathway system manner feed feed pathway desire output minimize top cost reconstruction output feed stack auto encoder bm change supervise mode thing particularly suitable amount unlabele way know transform auto fit input pre adjust among decoder relevant train stack pair oppose
develop quantum translation map picture new colour quantum quantum test answer exhibit create classifier principal pca successfully classical paper face compose stack sample sample
sentence patch image visual context model close spirit compute fix size investigate information exist system localize encode mean sentence system component representation image network attention focus generating word visual describe comparison evaluate several task image retrieval evaluate qualitatively attention scene discuss qualitative protocol validation detail follow server essence ground public server google system publish publicly sound call computed keep
regardless minimize stochastic nature nature maximize duality calculate value strategy n value bound j enough among intuitively appeal appear interested approach simple near favorable facilitate rest paper game order strategy binary classification game incur induce bad minimax ideal theorem nearly suffice disagreement among latter bayes almost minimax notably
heuristic principal magnitude toolbox principal component sketch sparse sketch c sparse component center value pca benchmark highlight good algorithm confirm pca nearly complete sparsity briefly handwritten bit gray repository text categorization dataset document bag remove letter stock price stock stock price form row expression cancer gene expression database tumor control platform annotation primarily qualitatively parameter performance small include
availability eigenvalue zero eigen solve modify eigenvalue simply omit suffer problem perform mixture qualitative information may qualitative qualitative empty level possible axis qualitative qualitative feature split explore capable qualitative transform qualitative order qualitative feature exact induce qualitative
bottom abundance obtain hyperspectral compose image htbp material green figure display abundance report quantitative material clearly material abundance abundance variant eight material remain abundance material basis sub coherence abundance make line abundance element th drawback fact clear except approximation interesting influence constraint sparsity coherence could improve improve preserve coherence datum reconstruction experiment identify material image mit face face center biological mit face display
h r example remark note estimation technique stein preliminary shrinkage carry configuration coefficient variance several usefulness stein keyword shrinkage preliminary lasso classical front admissible give birth class various setup document stein estimator stein reformulate include asymptotic nonparametric stein appear cover application popular devote preliminary test stein
computer science intelligence laboratory mit computer artificial intelligence laboratory institute technology mit present feature framework derive log formulation base intuitively scale minimal parameter tuning need principled selection selection global cluster news article regardless unsupervise challenging unsupervised overlap subspace infer balance handling feature vast majority however categorical contain categorical web cluster binary categorical treating relationship despite handle derivation asymptotic
direct handling lack practical type estimator dedicate design new ii use expectation standard square update rule improve rule benefit establish property improve finally deal log determinant equation rule log derivation update equation precision fix precision singular norm log determinant prior name sn respectively maximization l update maximize q
motivate well structure initial stage action formulate available consider set vertex label action endow draw available agent loop markov nothing restrict abuse let introduce prop direct intend contain learn fix threshold iff contain total matrix view appropriate depend action fix reversible action converge strong broad left topology guarantee simulation setting along path forward learn gps sensor agent walk consist sensor describe agent right gps sensor along agent perform path sensor along separate sensor cubic sensor random position empty spread one replace dark run though subtle four similarity difference weak mean abundance complete definition performance lag completely nest set matter mode value set random relation record agent small false record recall represent ground truth copy axis sensor axis counterpart graph project two stay put force view environment something qualitative behavior investigation notable datum space need maintain integer value sense entire history motivate mechanism whose snapshot snapshot advantage discount applicability arbitrary snapshot probabilistic clear preserve discount snapshot implication compare implication decay relation put record consecutive occur relation sufficiently small value hard false maintain qualitative requirement period place learn threshold snapshot might vary value threshold individually aim flexibility square employ analogous mean simulation emphasize kind show discount change geometry topology sensor performance discount snapshot walk immediately discount monotone term optimize deviation observe learn environment run observation make topology subtle implication record precede reason place become logical equivalence square agent equip adjustment require agent reason extension serve probabilistic snapshot define add direct adequate requirement snapshot say inequality indicator example atomic measure sum discount fall formal interaction however probabilistic snapshot triangle agent snapshot arrive action choice distinguish well identical simplify exposition introduce basic snapshot begin introduce formalism treat discrete sub structure formalism snapshot mechanism section require classical cover underlie greedy cover
rsc green region however descent iterate actually stationary point exist regression function converge undesirable run verify section side comparison huber cauchy run normal select prescribe regularization versus penalize statistically trial recover agree support curve stack horizontal rescale furthermore support transition happen sharp drop error equal dimensional oracle plot empirical first component cauchy huber indeed correspond mle furthermore align corollary directly huber huber huber oracle regularizers theorem function prediction threshold see agree empirical roughly rsc stationary curvature tail outli contamination estimator convergence penalize gaussian nonconvex amenable regularizers rsc agree asymptotic regularity conclude asymptotically procedure nonconvex first sufficiently initialization composite provide constraint program ensure stationary cone condition unclear condition necessary constraint redundant tune property solution asymptotic function robust parameter potentially hard robustness regularize hold penalize estimator lastly asymptotic normality draw conclusion asymptotic variance valuable trade variance one quantify point sample replace point estimator type another function hard nonconvex decay wang al nonconvex precise require suitable take concept robustness population estimator mass influence twice
lack statistical test fluctuation linguistic law meaningful fluctuation e generative linguistic text tight expect paris study year linguistic law law frequency frequent linguistic quantitative linguistic law text language production linguistic law increasingly modern estimation vocabulary text law discuss next automatic generation language know linguistic law usual text linguistic law consider bt besides linguistic law sec sec sec possibilities law sec availability database improve law careful typically confirm motivate law inspection increasingly law test design evaluate validity situation law allow description present discuss interpretation linguistic law correlation fluctuation account often
form uniform uniform auxiliary ball hamming ball space motivate tumor sample heterogeneous cell population dna sequencing population insight genetic architecture model identify mutation profile framework set unobserved mutation mutation mutation tumor attribute sequence like simulate explore configuration compatible inference conduct deterministic massive initialization overcome interest full characterization three sampling approach gibbs strategy proceed one column weight sample hamming matrix correspond exhaustive summation hamming sufficiently derivation simulate explain
proposal furthermore produce properly weight construct efficient sampler complex high motivate efficacy filtering dimension high spatio university link ac uk united se sequences sample correct volume concentrate concept monte body section well relationship development present section dimensional fact low cost profile method extensively art compete datum modularity spatio temporal constant intractable resort constitute treatment
orthogonality denote scatter eq p orthogonality mode normalization impose corollary extract mode upper high mode impose constraint choose dimension feature constraint primarily heavy quite capture sec extract extract successive follow derivation determine constraint follow alternate obtain locally optimal
crowdsource regard induced crowdsource obtain laplacian span case norm square small graph provide insensitive noise pairwise error matrix laplacian algebraic experimental information edge sample edge replacement weighted graph edge replacement os enyi random graph motivate estimate characterize behavior associate use degree boost random scheme maximize connectivity connectivity np follow base laplacian vertex graph iteratively maximize iterate repeat sized obtain key evaluate graph due whose dominate os enyi random graph least degree minimal establish os adjacency constant aid
product obey positive square size block definite semidefinite theorem denote contraction semidefinite similar relationship relate hadamard block follow square matrix size ij nonnegative ij c hold inequality fact upper bound hessian explanatory recall option k ki show kronecker operation rewrite hessian sum let convergence concave iterative solving estimation derive state mm theorem k obeys bounds symmetric random variable draw however variance prove hessian kk suffices b semidefinite theorem k semidefinite therefore k k concavity help convergence estimator identification concavity property concavity condition identification define transform explanatory variable observation ensure identification explanatory let transformation moment definite choose suffice exist q give practice replace estimate sample must rank add give trivial case experimental summarize condition definite approximately per estimate
asynchronous empirical close primal precise thesis algorithmic trace randomization help proximal accelerate leave variant finite within asynchronous sgd algorithms parallel know coordinate variant share assumption work gd mini batch parallel allow mini batch convex variance describe framework function maintain additional parameter denote general iterative updating specify subroutine subroutine crucial mechanism thereby rise approach responsible reduction
asymptotically asymptotic cat item select fisher relative total result view message cat capture response aspect incorporate cat nominal response provide rigorous management armed service heavily response modeling response probability specific parameter ability scalar parameter pl probability correct answer difficulty pl case pl add parameter step logistic suggest select parameter suggest wu base parametric originally inefficient data estimator normal wu pl pl design cat assume operational cat unable incorrect answer efficiency avoid
signal continuous eeg lead wireless eeg device operational wireless analog discard digital compress low signal compress measurement cs signal compress discrete compressed signal represent transform zero measurement sampling computing norm count current programming pursuit omp thresholding etc eeg signal sparse transform exploit sparse
